a ee See eee eer mere LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-‘CHAMPAIGN The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN NOV 3 49071, NOV 15 KE MAR 2 3 198 NOV 12 {088 oct 18 wet are ~ Cy Tian +) R 13) y | es Y 3 3 ri ‘eee. : NOV 16 eCD LIBRARY OF USEFUL, KNOWLEDGE. THE DIFFERENTIAL AND INTEGRAT CALCULUS, CONTAINING DIFFERENTIATION, INTEGRATION, DEVELOPMENT, SERIES, DIFFERENTIAL EQUATIONS, DIFFERENCES, SUMMATION, EQUATIONS OF DIFFERENCES, CALCULUS OF VARIATIONS, DEFINITE INTEGRALS,— wItH APPLICATIONS TO ALGEBRA, PLANE GEOMETRY, SOLID GEOMETRY, AND MECHANICS, ALSO, ELEMENTARY . ILLUSTRATIONS oF THE DIFFEREN TIAL AND INTEGRAL CALC ULUS. BY AUGUSTUS DE MORGAN, F.R.A.S. AND C.P\S,, OF TRINITY COLLEGE, CAMBRIDGE, PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON. Tavra 8€ trois uév modXois Ka} #1) Kekowornkdrecon roy Parjoey tmoapBdve rots dé HetaheaBn Kal Tap HeyeOéwv, Tas ré yas, Koopov, Tehpovrixdreo st, ToT \ 4 > > , ” y+ Kat Tivas ovk avapyootoy ety er: Habnudrov otk etmora KOTETOL, Kal TEeplt Tey dmroornudrey ~ , —— 7 Kai Tov dXiov, Kab ras TeAnvas, Kat rod édov A 4A ‘ > , 3 om ‘ 2S a dia ray arddeE éoceiobus, Avorep anOnv eneapnora Tav’Ta.— ARCHIMEDEs. PUBLISHED UNDER THE SUPERINTENDEN CE OF THE SOCIETY FOR THE DIFFUSION oF USE FUL KNOWLEDGE. ag tae ee eens eee LONDON:—ROBERT BALDWIN, 47, PATERNOSTER ROW. LONDON : GEORGE WOODFALL AND SON, ANGER COURT, SKINNER STREET. Peer TS PREFACE. Tue work now before the reader is the most extensive which our lan- guage contains on the subject, being (exclusive of the Elementary Illustrations, at the end) more than double in matter of the Cam- bridge translation of Lacroix, and full half as much as the great work of the same author in three volumes quarto. I state this because students are sometimes apt to be discouraged by the apparent slowness of their progress, which they measure by the pages read, without any other con- sideration. This extent of matter is not due to fullness of explanation or abundance of examples, but to a variety of subjects exceeding that which is usually introduced into elementary writings. There are many works which enter more largely into the simpler parts, and elucidate them by more copious instances, both of which my specific object has prevented me from doing. That object has been to contain, within the , rescribed limits, the whole of the student’s course, from the confines of elementary algebra and trigonometry, to the entrance of the highest works on mathe- matical physics. A learner who has a good knowledge of the subjects just named, and who can master the present treatise, taking up elementary works on conic sections, application of algebra to geometry, and the theory of equations, as he wants them, will, I am perfectly sure, find himself able to conquer the difficulties of anything he may meet with; and need not close any book of Laplace, Lagrange, Legendre, Poisson, Fourier, Cauchy, Gauss, Abel, Hindenburgh and his followers, or of any one of our English mathematicians, under the idea that it is too hard for him. It may be admitted to be desirable that some one writer should endeavour to attain such a result as that of placing before the student all that is requisite to put him in communication with the highest investi- gators ; and it will readily be seen, that unless a very large work indeed were written, no such result could be obtained without condensation, par- ticularly in the higher parts. If much difficulty should be experienced in the elementary chapters, I know of no work which I can so confi- dently recommend to be used with the present one, as that of M. Du- ‘amel, cited in the note to page 681. PREFACE. The method of publication in numbers has afforded time to consult a large amount of writing on the different branches of the subject; the | issue of the parts* has extended over six years, during two of which cir- cumstances with which I had nothing to do stopped all progress. The first number was preceded by a short advertisement, which I should desire to be retained as part of the work; for I have no opinion there | expressed to alter or modify, nor have I found occasion to depart from | the plan then contemplated. The principal feature of that plan was the rejection of the whole doc- trine of series in the establishment of the fundamental parts both of the Differential and Integral Calculus. The method of Lagrange,+ founded on a very defective demonstration of the possibility of expanding ~(x+h) in whole powers of h, had taken deep root in elementary works ; it was the sacrifice of the clear and indubitable principle of limits to a phantom, the idea that an algebra without limits was purer than one in which that notion was introduced. But, independently of the idea of * It may be convenient for reference to state the dates of publication of the \ different Numbers, and also the corresponding Numbers of the Library of Useful Knowledge, each Number containing 32 pages :— | Differential Calculus, No. 1, Lib. Useful Know., No. 219, July 15, 1836. \ | ie No. 2, es No, 221, Aug. 15, ,, of No. 3, ¥ No. 224, Oct. 15, ,, 1 | 5 No. 4, re No. 227, Deex By 5; | | 6 No. -4, AE No. 229, Jan. 2, 1837. es No. 6, ye No. 236, July 1, ,, a No.7, AA No. 253, Jan. 1, 1839. | nt No. 8, Ae No. 259, April 1, ,, | rh No. 9, if No. 260, May 1, ,, } rae No. 10, Pr No. 266, Aug. 1, ,, j Ny No. 1], m No. 273, Dee. 1, ,, | Bs No. 12, ws No. 276, Feb. 1, 1840. | | 4a No. 13, $5 No. 282; April's, ey, \ ay. No. 14, » 141. SA cos (aé+a) must have aroot. 142 and 143. Other ex. amples. 144 to 152, Geometrical applications. Development in general, Calculus of Operations (pages 303—320). 152, Burmann’s theorem.* 153, Expansion of wz in powers of gx. 154. Deduction of Lagrange’s theorem. 155, Expansion of g-y in powers of x 156. Reversion of series, carried farther than page 158. 157. Altered form of the series in 153. 158 and 159, Simple examples of the calculus of operations. 160. Herschel’s theorem. ‘Expansion of fe” in powers of x, 161. The last reduced to gv =@(1+A).a. 162, Comparison of Burmann’s and Herschel’s coefficient. 163 and 164, Bernoulli’s numbers in terms of the differences of nothing. 165. Expansion of cos (a:*). 166 to 168, Consequences of Herschel’s theorem. 169. Expression of a linear function of differences as a function of diff, co. 170. Expression of a linear function of succes~ Sive values, as a function of differences or of diff. co. 171, Expression of gz+O(7+1).a+... in terms of 9x, Px, &e. 172. Finite summation with terms of alternate sign by means of diff.co. 173. Example of the last. 174, Another form of G2+¢(¢+1). a+... 175, Another form of 2y,. 176. 3a" and sx~!, 177. Slog x and more correct approximation to 1.2.3...¢. 178. aAu, —dq A*u,+ &e. and agx-taig/x.h+... in terms of gx, Px, and ¢(x+h), ¢/(x+h), &e. 179. Ary, in terms of receding differences.+ 180 and 18}. J yzdx in terms of receding differences : method of quadratures and example. 184. Development of x: {(1+a)"— 1h in powers of z. 185 and 186. Summation of a series with interposedterms. 187. Connexion of Sand A-!, 188. Deduction of the method of quadratures from the preceding. 189. Summation of series with interposed terms and receding differences. 192 to 199. Properties of the roots of unity and multisection of series by means of them. Theory of Dimensions (pages 320~ 328). 200, 201. Explanations relative to Chapter X. 202 and 203. Critical value of e in gu : (Yax)*, defined and determined. 204. Results of the “last collected. 205. Algebraical dimensions of a function, the dimetients being 2, log 2, log log a, &c. 206 and 207. Mode of determining the dimensions. 208 to 210. Application to the determination of the convergency or divergency of a Series. 211]. Preceding errors corrected. 212 and 213. Development in positive and negative powers. Arbogast’s Derivations, Combinatorial Analysis, Calculus of Generating Functions (pages 328—340). 214. Theory of derivation. 215. Tasxxt of derivatives. 216, Example. 217. Generalized derivation. 218. Deduction of differentiation. 219, Example of differentiation. 220, Arbogast’s own mode of obtaining derivatives 221. Reversion of series. 222. Reciprocal of 1+-4a-+er+... and (altblat,, 5 ie (I-+be-+...), 223. Combinatorial analysis. 224. Development of (Gotae+ &c.)*. * Appendix, p. 774, + Ibid., p. 774, | Lbid., p.774, CONTENTS. .). 226. How to deduce Arbogast’s deriva- 298—233. Generating functions, xvi 925. Development of ¢(a +bafcx-f es tives. 227. Most simple form of (aba ore)" principles, and various examples. Cuapter XIV. APPLICATION TO GEOMETRY OF TWO DIMENSIONS. Differential equations of curves (344—347). General theory of contact (349, 350). Tangent and normal (350—353). Enveloping,* orthogonal, &c. curves (354—357). Polar ion of tangent and normal (358). Involute, evolute, and radius of curvature ff. co. (368). Second diff. co. (369—372). Theory of dimen- sions applied to these diff. co. (372—374). Transformation of equations, intersec- tions of curves with the axes, properties of the tangent '(375, 376). Asymptotes (376). Flexure and curvature (377). Point d’arrét and cusp (378, 379). Multiple point (379—382). Conjugate point (382). Pointed branch (382—384). Area and length (385, 386). Problem illustrative of the singular solution (387). Theory of signs (341—343). Branches of a curve (347, 348). equat (358—368). First di Coaprer XV. APPLICATION TO GEOMETRY OF THREE DIMENSIONS. Notation for higher diff. co. (388). Double integration, earliest process (388, 389). Geometrical illustration (390, 391). Extension of the method to the case in which the limits of each variable are functions of the other (391—395). Examples (395— 397). Surface of a spheroid (397, 398). Surface of revolution generally (398, 399). Classes of surfaces (399—401). Characteristic of a surface (401). Connecting and connected surfaces (enveloppes and enveloppées) (402). Connecting curve of the characteristics (aréle de rebroussement) (403). Developable surfaces (403). Surfaces on which a straight line can be drawn (403, 404). Surface passing through any number of curves (404). Remarks on the coordinate planes (405). Tangent plane, normal, line of greatest declivity (406, 407). Curve, its tangent, normal plane, and osculating plane (407—409). Osculating and polar surfaces, curvatures, flexures, and evolutes (410—415).+ The screw (415—417). Expressions derived from different forms of the equation of a surface (417, 418). Species of contact with the tangent plane (419). Surfaces generated by the straight line (420—426). Normal sur- faces, and curvature of surfaces (426—436). Method of drawing figures (430). Lines of curvature (434—440). Various problems proposed (441). Shortest line on a surface (442, 443). Expressions for arc, volume, &c., and axioms required (443—446). Cuapter XVI. ON THE CALCULUS OF VARIATIONS. Illustration of the object in view (446, 447). Fundamental definitions (447, 448). Variation of diff. co. (449). Variation of fg¢dx (449, 450). The same when @ contains another integral (450, 451). The same when faa is given by a differential equation (451). Variation of /¢dady (451—454). Illustration of the use of this calculus in mechanics (455—458). Maxima and minima, full considera- * Elem. Illust., pp.» 22—29. + Appendix, p. 776. CONTENTS, XVii tion of the instance of the shortest line between two points, and cases not usually considered (458—461). Generalization of the problem, and instances (461—465), Relative maxima and minima, and instances (465—468). Collection of the different cases into one form, and application of the form to the brachystochron in a resisting medium (468—471), Instance, with two independent variables (471—473), Solution of a partial diff. equ. (473). The surface made by the revolution of a catenary about its directrix is one of equal and opposite curvatures (474, 475). Cuarprer XVII. APPLICATION TO MECHANICS.* Object of mechanics, laws of motion, connexion of pressure and velocity (475— 477). Principle of virtual velocities (477479), Perpendicular on a surface from a point infinitely near (479). Coordinate motions expressed in terms of rotation (479), Every motion compounded of rotation and translation (480—482). Trans. formation of the expression of rotations (482, 483). Geometrical consideration of rotation (484—488), Every change of place, one point being fixed, reducible to one rotation (488,489). Other matters connected with rotation (490—493). Integrals depending on the arrangement of the parts of a system (493—496), Properties of the ellipsoid necessary in rotation (496—498). Momental ellipsoid, moments of rotation, principal axes (498—500). Statical application of the principle of virtual Velocities (501—503). Dynamics, motion of a point, tangential and curving forces deduced from the motion (5083—505), Inverse problem, fundamental equations (505, 506). Astronomical form of these equations (507—509). Motion of a system, D’Alembert’s principle (509, 510). The six equations of motion (511, 512). Transformation of the equations of rotation (512, 513). General principles, centre of gravity, conservation of areas, Sir W. Hamilton’s method (514—518), Lagrange’s general method (518—522). Variation of parameters (523—530). Lagrange’s general forms (530—535). The fundamental equations connected with attractions (535—541), % Cuaprer XVIII. ON INTERPOLATION AND SUMMATION. General modes of interpolation (542), Ordinary restricted interpolation, with examples (543, 544), Interpolation by preceding and succeeding values (544—548), The same, without direct use of differences (548—550). Interpolation when the values of the variable are not in arithmetical progression (550—552). Relation between sums of series of powers (552). Summation of an infinite series by some of its terms and an integral (553). Tass of the sums of powers (554). Summa- tion when the terms are alternately positive and negative (554—556). Hutton’s method (557—560).+ CuarteR XIX, ON THE TRANSFORMATION OF DIVERGENT DEVELOPMENTS. General theorem by which this transformation may be frequently effected, and consequences (560—565). Particular modes of effecting the same (565, 566), * Elem. Ilust., pp. 25—31. + Appendix, p. 776. CONTENTS. CHAPTER XX. ON DEFINITE INTEGRALS. General considerations (566—569). Mode of rascertaining whether a definite integral is finite (569, 570). Remarks on the case in which the subject of integra- tion becomes infinite between the limits (571). Periodic integrals, how connected with convergent ones (571. 572). Certainty of the appearance of agit Menges! in this subject (572). /3* tan—"ddé and fo a™dax : (1+2") Se camp dies wcejbuclcbor' series not to be depended on after integration (576). fo cos bada (12%) and Se sin bx. awda : (1+) (576, 577). Definition of Tx (577). Series for log r(2+1), new constant y, how obtained (577—579). Exercises from the preceding (579). fiw —2)tde= (579, 580). .Tx.T1—2). fos mde, (580). More convergent forms for log T(a-+1), series for Bernoulli’s numbers, their convergency (580, 581). General property of Tz, reduction of I(m:n) to the smallest number of distinct transcendentals (581—588). Correction of some of the preceding reasoning, and more rigorous deduction of log T(a+1) (583—585). Resolution of <*e—*, sin x and cos a, into factors (585,"586). Series for log cos # and log (#:sin x) (587). Tasie of Tr. (587—589). Mode of reconstructing Legendre’s larger table (589). Tap.e of Tv for the twelfths of a unit (590). Determination of Tw by continued fractions (591, 592). Interpolation of form (593—597). Differentiation with frac- tional indices (597—600). Laplace’s method for functions of high numbers (601— 605). Periodic series, cases and difficulties of (605—608). Geometrical illustra- tion of their discontinuity (608). Expansion of functions into periodic series (609— 614). Poisson’s view of the same subject, and digression on symbols of discon- tinuity (614—618). Fourier’s theorem, aud remarks on proposed verifications of it (618—621)* Examples, with particular reference to discontinuity (621, 622). Application of Poisson’s theorem to summation (622—625). Remarks on the con- vergency and discontinuity of periodic series, and examples of Fourier’s theorem (625—629). /e-@ cos br.a" "dx and fe—**sin ba. a"—'dx (630, 631). Considera- tion of the case in which’ the subject of integration becomes infinite between the limits ; Cauchy’s method of deducing definite integrals, and examples, with a slight description of his term residval (633—640). Remarks on indeterminate‘values, with particular reference to sin o and cos 0 (640—642). Results deduced from S75 OC ah) Fo(aphe™ avdo (642, 643). Parseval’s and Murphy’s theorems 644—646). Calculus of operations applied to fgue~**dv (646, 647). Modes of approximating to Fresnel’s integrals (647—649). On the celebrated property of series with terms alternately positive and negative, with application to some hyper- geometrical and factorial series (649—652). Remuant of Taylor’s series expressed by a definite integral (652). /e-vordy from v=a to v= (652—654). Other inte- erals connected with the last (654—656). Mention of elliptic functions (656, 657). Slight rapiu of fe—dé (657). Connexion of faculties or factorials and Y-functions (658). Spence’s theorems and extensions (658—660). Soldner’s integral or f dz : log x (660—662). Tastx of Soldner’s integral (662, 663). Integrals depending on the last (663, 664). Theorem connected with Laplace’s coefficients (664, 665). Method of using A” /<-*Jvdv (665, 666). Miscellaneous integrations, from Le- gendre (666—668). Expansions of sin aw: sin ba, &c., in series deduced from continued products, and resulting integrations (668—670). Abel’s’ expression of sox, &e. by definite integrals (671—673). Abel’s general formula, with extensions (673—676). Miscellaneous integrations from Poisson (676—678). Extension of the Eulerian form /12™(1—)"dw to a form of several variables (678—681). * Appendix, p, 776. CONTENTS.’ , xix Cuapter XXJ. ON DIFFERENTIAL EQUATIONS AND EQUATIONS OF DIFFERENCES, (This Chapter is divided into Articles, to which the references are made.) Equations of one independent variable (pages 681—709). 1, Solution of y= px. 2. Solutions in which a transcendental is evaded. 3. r=fy!. 4, y=fyl. 5, y=xoy! ty’. 6. y=¢ (2, y’) made to depend on the linear form. 7, Reduction to homo- geneity. 8. y' +Py=Qy". 9; 10) 11, $213) On the integrating factor. 14, So- lution of $2 -+py=¢ (2-+y). 15. Reduction of diff. equ. of higher orders to simul- taneous equations of the first order, and the converse. 16,17, 18, 19, Integration of simultaneous equations. 20, 21. Case of coincident solutions generally. 22, 23, 24, 25, 26, 27, 28, Ny! +Py2+ Qy+R=0. 29. The same by a continued fraction. 30. Method of series, 31, Reduction of y/ == Py? + Qy+R. 32. Riceati’s equation. 33, 34, 35. Linear equations generally, 36. One diff Co. given in terms of another. 37, g (x, y,y'=0. 38. Deduction of a linear equation from ¢ (a, Y, y's.yY¥™)=0. 39, ? (a, y), yFt),..)=0. 40, Py"+-Qy?=R, 42, 43, 44, 45, 46. y!!4-Py!+ Qy+R=0. 47, 48. Consequences of different kinds of homogeneity, 49. Change of the independent variable. 50. Solution of y" +ma-y! — ary —0 by a definite integral. 51, Uses of the preceding. 52. Reduction of the same by generating functions. 53. Verification. 54, Reduction which applies when m js negative. 55, Riccati’s equation reduced to the preceding. 56, 57. Poisson’s method. 58. On our means of expression. 59, 60. View of the origin of the sin- gular solution, 61. Extension of Clairaut’s method. 62, Way of deciding between an ordinary and singular solution, 63, 64. Singular solutions of higher equations, Equations of two or more independent variables (pages 709—736). 65, 66, 67. Con- ditions of integrability. 68, 69, General integration of @2— dax*-L-dy?, 70), ray 72, 73. General methods for partial differential equations, 74, 75, 76. Primary, general, and singular solutions, 77,78. Examples. 79, 80,81. Detached artifices, 82. Integrability of rdx*+-2sdxdy+tdy?.. 83, Formation of equations of the second 2 2 order. 84, 85. Solution of such equations. 86. Solution of oe 87, 88, dx® dy? Other instances, 89,90, Linear Equations. 91, Arbitrary functions. 92,93, 94, 95, Discontinuity, 96. Limitation of the number of arbitrary functions, 97, 98, 99, 100, 101, 102, 103. Solution of partial differentia] equations by definite integrals, Equations of differences, by ordinary methods (pages 736—746). 104, Comparison of functional €quations and equations of differences. 105, Formation of an equa- tion of differences: two distinct solutions. 106, 107, 108. Linear equations of the first order. 109, Continued fractions, 110, 111. Equation of the second order, 112. General method when particular solutions coincide. 113. General linear equation. 114, General method of reduction. 115, Particular case of reduction. 116. Simultaneous equations, Application of generating functions (pages 746—750). 117. Equations of one variable. 118, 119, 120, 12), 122. Equations of more than one variable. Application of the calculus of operations (pages 751—758). 123, 124, General formule. 125. Linear equations of one variable, 126. Partial differential equations of the first order. 127, Of higher orders. 128. Reduction to a definite integral. 129. Linear equation of differences of One variable. 130. Successive summation. 131. Case of coincident solutions, 132. Linear equation of differences of two variables. 133, Equations~ of mixed differences, 134, 135. Instances of partial differences, 136, Intermediate equations, xx CONTENTS. Equations with indefinite increments of the variables (pages 759—761). 137. In- stances, distinguishing the cases which admit of general solution from those which do not. 138. Discontinuity of such solutions: proof of every differential equation having a solution. Circulating equations (pages 761—766). 139. Definition and instance. 140, 141, 142, General method and instances. 143, 144. Isolated equations which admit of solution: their use. 140. Remark on the suffix of differentiation. APPENDIX. Abbreviation of proof in page 68 (767). Addition to page 103 (767). On the calculus of operations (768). On 0: 0 when it is 0 or o (770). Addition to page, 190 (771). On singular solutions (771). , Shorter solution of partial differential equations (771). Correction of error in page 206 (772). General mode of solving equations of higher degree than the first (7 72). Fourier’s theorem (772). Onthe dif- gorences of nothing (773). Abbreviation of page 305 (774). Alteration of Arbo- 4 gast’s derivatives to meet the case of 9 x ag x < g (otietes +e 2-3 + ee . ) Aybar Ay gts 7-3t o0ce and Tanz of derivatives (774). Addition to page 410 (776). Alleged improvement of Hutton’s method of series (776). Expansion of functions in series both of sines and cosines (776). TasiLe or Errata, page 778. Exempnrary Inuusrrations, pages 1—64, These are referred to in their proper places in the preceding table of contents. INDEX TO TABLES AND OTHER MATTERS OF REFERENCE. Pages Change of variable. . A ; : : 153, 156 Reversion of series™ . : ‘ : ° 158, 306 Lagrange’s theorem .« . ° ° : 170 Unusual singular forms ° : : 175 Bernoulli’s numbers} : : : 247 Differences of 0 “ : 253 Common summation . . 4 : 256 Coefficients of quadrature ; ; - 262 Differences by differentiation . 5 A 263 Differentiation by differences. : : : 264 Ultimate integral forms 4 : ‘i 285 Burmann’s expansion : : : : 305 Transformation{ of summation . . . ’ 311, 552, &c. Hutton’s summation . : z ° . 557,776 Quadratures 7 F i : . é S13. Summation with interposition . : ; : 318 Arbogast’s§ coefficients. . ; : : ; 331 Ditto for divided powers : . ° 5 774 Curve formule A : g ‘ a ‘ 345 Legendre’s double integration - 2 : s 395 Table of Sx” . 4 3 4 é P . 554 Table of log r (1+ 2) : : ‘ : A 587, 590 Table of fe-?dt ‘ 2 i , 657 > 662 Table of fdw: log a. . * Also Penny Cyclopedia, Reversion. + Ibid., Numbers of Bernoulli. t Ibid., Semmation. & Ibid., Taylor’s theorem, ADVERTISEMENT. THE following Treatise will differ from most others, for better or worse, in several points. In the first place, it has been endeavoured to make the theory of /imets, or ultimate ratios, by whichever name it may be called, the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expansions. I am not aware that any work exists in which this has been avowedly attempted, and I have been the more encouraged to make the trial from observing that the objections to the theory of limits have usually been founded either upon the difficulty of the notion itself, or its unalgebraical character, and seldom or never upon anything not to be defined or not to be received in the conception of a limit, or not to be admitted in the usual conse- quences, when drawn independently of expansions, that is, of develop- ments under assumed forms. The objection to the difficulty I have endeavoured to lessen in the introductory chapter ; that to the name by which a science founded on limits should be called, I cannot fee] the force of, or see what is to be answered, I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclu- sions logical. Secondly, I have introduced applications to mechanics as well as geometry, in cases where the preliminary notions are not of too difficult a character, and I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. The parts of the former science which can be understood by a learner at any stage of the latter, are, I suppose it will be allowed, necessary to a proper view even of so much of the latter as precedes the point supposed. Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If 80, why are not multiplica- B 2 4 ADVERTISEMENT. tion and involution in arithmetic made to follow addition and precede subtraction? ‘The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold—but I do not feel it necessary further,to defend placing the question of finding the area of a parabola at an earlier period of the work than that of finding the lines of curvature of a surface. Experience has convinced me that the proper way of teaching is to bring together that which is simple from all quarters, and, if I may use such a phrase, to draw upon the surface of the subject a proper mean between the line of closest connexion and the ine of easiest deduction. ‘This was the method followed by Euclid, who, fortunately for us, never dreamed of a geometry of triangles, as distinguished from a geometry of circles, or a separate application of the arithmetics of addition and subtraction ; but made one help out the other as he best could. At the same time I am far from saying that this Treatise will be easy ; the subject is a difficult one, as all know who have tried it. The absolute requisites for the study of this work, as of most others on the same subject, are a knowledge of algebra to the binomial theorem at least (according to the usual arrangement), plane and solid geometry, plane trigonometry, and the most simple part of the usual applications of algebra to geometry. The Treatise entitled ‘ Elementary Illustrations of the Differential and Integral Calculus,’ will be bound up with this Volume, and referred to in the proper places. A. Dr Morgan. London, July 1, 1836. DIFFERENTIAL CALCULUS. INTRODUCTORY CHAPTER. Ir the mathematical sciences were cultivated wholly for their practical utility, as it is called, meaning their application to the formation and management of all the mechanism by which the arts of life are advanced, it would not be necessary to consider any magnitude as having existence at all, unless it were sufficiently great to be either useful or noxious to some object connected with some given application in question, And the human senses would fix what we might in that case call the limits of quantity ; namely, the greatest of the great and the smallest of the small, among those quantities which actually are measured and consi- dered in astronomy or navigation or manufactures, &c. The longest line would be that drawn from the spectator to the farthest heavenly body whose distance he had measured 3 the shortest would be the smallest line his eye could perceive when aided by the microscope, or by any machines which multiply small motions. There would consequently be as many systems of mathematics, or sciences of calculation, as there are practical applications differing materially in the nicety of operations which they require; from that of the joiner, to whom the length of the hundredth of an inch may be considered as non-existing, and who com- pares one length with another by means of a rule warped by the sun, worn by time, and divided into parts by deep and broad furrows, to that of the astronomer, who lays one rod by the side of another by the aid of a powerful microscope, haying first levelled them by the most accurate instruments, and then consults the thermometer to know what length it will be proper to consider the rods in question as having to-day, compared with what they had yesterday. The first considerations connected with number and magnitude always enter the mind in connexion with some application to the rough pur- poses of life, more or less approaching to exactness* in different circum- Stances,—and as many different systems of rules are formed as there are different modes of dealing with material objects, each by itself relatively more perfect than the rest, that is, better adapted to its parti- cular end,—the consequence is, that the various terms which imply relation, that is, which are used in speaking of one quantity or magni- tude as to how it stands with respect to another, are really used in * The child of an artisan exercising any of the more ingenious manual arts, or of a savage in the state of life in which arts have made the progress which is possible without division of labour, might perhaps be considered as being most advanta- geously situated in this respect: but we think it beyond question that the children of the middle and upper classes in England, it may be throughout Europe, are in as unfavourable a position as any of their species, 6 DIFFERENTIAL CALCULUS. many different senses ; or, which is much the same thing in the diffi- culty which it creates, in many different degrees of the same sense. It is hardly necessary to insist upon this as to words which imply pure relation, such as small or great, when it may be known by those who have tried that the same variety of degree enters into the notions which have been formed of positive terms. If a class of boys beginning geo- metry at school (that is of course geometry, not saying Euclid) were thus put to the question: “ You all know what a straight line is?” there would be but one answer, and that in the affirmative: one would call to mind a stroke on a slate, another the side of it, a third perhaps the length of a street, and so on. To the question, “ Can two straight lines enclose a space?” there would be a majority for the negative, con- sisting principally of those whose primitive straight line had not been part of a bounded figure. But still the proposition is not a “‘ common notion,” because its terms have not a common meaning. When the question, ‘‘ Can two straight lines be made to enclose a space by length- ening them ??’ was proposed, all would answer in the negative, not as to the notion they had previously had of a straight line, but as to the new one they would form out of the terms of the question. And by further asking, “Can two straight limes im any direction whatsoever enclose a space 2”? it would in some way or other appear that all the straight lines had been horizontal straight limes, and most of them parallel to the sides of the ceiling. The student of the Differential Calculus may by such an illustration be brought to think it possible that the terms and ideas which that science requires may exist in his own mind in the same rude form as that of a straight line in the conceptions of a begimner in geometry. Remembering the acknowledged difficulty of the subject, he must be prepared to stop his course until he can form exact notions, acquire precise ideas, both of resemblance between those things which have appeared most distinct, and of distinction between those which have appeared most alike. .To do this sufficiently, even for the outset, formal definitions would be useless ; for he cannot be supposed to have one single notion in that precise form which would make it worth while to attach it toa word. One reason of the great difficulty which is found in treatises on this subject has always appeared to us to be the tacit assumption that nothing is necessary previously to actually embodying Cc the terms and rules of the science, as if mere statement of definitions could give instantaneous power of using terms rightly. We shall here attempt at least a wider degree of verbal explanation than is usual, with the view of enabling the student to come.to the definitions in some state of previous preparation. Very little progress, even in arithmetic, makes the student aware of the existence of problems, which, being absolutely impossible, are yet of this character, that numbers or fractions may be given, which shall, as nearly as we please, satisfy the conditions of the problem. For instance, we wish to find a fraction which, multiplied by itself, shall give 6, or to find the square root of 6. This can be shown to be an impossible problem ; for it can be shown that no fraction whatsoever multiplied by itself, can give a whole number, unless it be itself a whole number dis- guised in a fractional form, such as $ or *. To this problem, theu, there is but one answer, that it is self-contradictory. But if we propose the following problem,—to find a fraction which, multiplied by itself, shall give a product lying between 6 and 6+ 4; we find that this problem admits of solution in every case. It therefore admits of solution how- INTRODUCTORY CHAPTER. 7 ever small a may be: for instance, we can find a fraction which, multi- plied by itself, lies between 6 and 6°00001, or between 6 and 6:0000001. We have here introduced a word which by itself has no meaning, namely, “‘small”?; but it must be observed that we have not introduced it by itself, as if we laid down a distinction between small and great, but in connexion with the word “ however,” meaning that whatever a may be, and whether, being what it is, it may be called small or not, we can find x so that w x shall lie between 6 and 6+a. This use of the word small runs so completely through the whole of the science which we propose to treat, that it demands the most complete elucidation. We must observe that, though in all grammars “ small’? is called positive, and ‘‘ smaller ”’ comparative, yet in fact the latter is the only absolute term of the two, while the former is purely relative. Assign two numbers, and the smaller of the two can be pointed out; but assign a number or fraction, and it cannot be said to be either small or great, because these words depend for their meaning upon the circumstances under which they may be used. The number ¢en stands equally for a large family of children, a small school of boys, a very small number of men to be lost in a battle, an enormous number of candidates at an election. But nine is always smaller than ten, whatever may be the objects of reckoning in question, When we say then, that x may be so found that vx shall lie between 6 and 6+, however small a may be, we merely imply that if a be named at pleasure, any number whatsoever, or any fraction whatsoever, then x can be so found that xa should exceed 6 by a smaller quantity than a. We can conceive ourselves engaged in two different kinds of metaphysical disputes on this subject, as follows: Firstly, A denies that the word small ought to be used, on account of its indefinite cha- racter. We answer that we can, with more expense of words, dispense with it entirely ; and that all we mean is this, that if he will assign the value he chooses to give to a, we will take a sma/ler value (a term about which there is no dispute) and find x so that xx shall lie between 6 and 6+ less than a: and that the use of the word small is merely to remind the reader of this, that whatever he may assign to be the value of a, it would not interfere with our power of solving the problem; he might, with equal. certainty of receiying an answer, have made a smaller than he actually did. But B, on the other hand, thinks he has a notion of a fraction which is actually small, but differs from us as to its value. We have said it may be, “let a be a small quantity, for instance, “0000001,” whereas he is not inclined to call any quantity small, which is greater than ‘0000000001. We answer, that the matter is perfectly indifferent; it is as easy, In every thing but mere labour of calculation, to assign as the unit of smallness, any fraction which he may please to name. What we mean to say is this, that we never use the word small, unless where it implies, as smadl as you please. Similarly we neyer use the word near, unless in the sense of as near as you please ; or great, unless in that of as great as you please. And the same with all other terms which are purely relative. We reject them in their relative sense because the relation is indefinite ; we adopt them again as a mode of signifying a relation which we may make what we please in the extent to which we carry the idea of the relation in question. In the questions which occur in arithmetic and algebra, relating to problems the conditions of which can be satisfied only as nearly as we please but not exactly, it is usual to create a solution by hypothesis, and to say that. we continually approach to that solution, the more 8 DIFFERENTIAL CALCULUS. nearly we solve the problem. Thus it is never said that there is no such thing as 2, which makes 2@ actually equal to 6; but it is said that there is such a thing as the square root of 6, and it is denoted by / 6. But we do not say we actually find this, but that we approximate to it. If we take the following series of numbers or fractions— : Perens: 7. 2°449490 Des ar 8. 2°4494898 3.8 2°45 9. 2°44948975 4, 2°450 10. 2°449489743 5. 2°4495 ll. 2°4494897428 6." 2744949 12. 2°44948974279 and multiply each by itself, we shall find the product to approach nearer and nearer to 6, and always exceeding it, so that while the first multi- plied by itself exceeds six by 3 units, the last multiplied by itself does not exceed 6 by so much as the thousand-millionth part ofaunit. We thence get the idea of a continual approach to the fraction which satisfies the problem, though in truth there is no such fraction ; but all that we can say is that we have found a fraction which has a square lying between 6 and 6 + one thousand-millionth part of a unit. And also, which is the essential part of the problem, that we might have made the last- mentioned fraction still smaller, to any extent, and have found a corre- sponding solution. This non-existing limit, if we may so call it, actually has a more defi- nite existence in geometry than in arithmetic, but only when we take a sort of supposition which is practically as impossible as the extraction of the square root of 6 in arithmetic. Let there be such things as geome- trical lines, namely, lengths which have no breadths or thickness, and let it be competent to us to mark off points which divide one part of a line from another, without themselves filling any portion of space; then it is shown in Euclid that the side of a square which contains six square units is a line, which, when we come to apply arithmetic to geometry, “must be called ,/6 whenever our arbitrary linear unit is called 1. And the lines represented by the preceding ‘twelve fractions will, in such case, be a set of lines which, being always greater than the line in ques- tion, yet are severally nearer and nearer to it. This line can no more be expressed by means of an arithmetical fraction than v 6. We have then got an idea of a limit towards which we may approach as near as we please, but which we can never reach, We shall take another instance of a similar kind, in which the limit, though equally unattainable under the conditions prescribed, is yet a defimite number or fraction. Take a unit, halve it, halve the result, and so on conti- nually. This gives— 1 1 a + e|- r eo g Add these together, beginning from the first, namely, add the first two, the first three, the first four, &c. The first is Li 08 02, yell Apa) The first two give 3. “Or Bil eis ae ys oe tree i SOF grat |) 1 . four tee ee ee 4. bah «Si five ~ fl OF, 2 lens See eile rear ae Oia mS $3. OF sod. gas ay INTRODUCTORY CHAPTER. 9 We see then a continual approach to 2, which is not reached, nor ever will be, for the deficit from 2. js always equal to the last term added, And the reason is simple. Let AB represent 2 units | | nn a A C Da Halve AB by the point C, CB by the point D, DB by the point E, and soon. Now, whatever degree of approximation may be made to the point B by passing from A to C, from C to D, from D to E, &e., it is clear that as much remains to be passed over as was passed over at the last step, nor can the length which remains ever be passed over by passing over its half. We have then here a case in which there is a limit unattainable, by the process described, but capable of being attained within any degree of nearness, however great. : The following phraseology is in continual use. We say that — ak eC ee ee ee ee kee is a series of quantities which continually approximate to the limit 2. N ow, the truth is, these several quantities are fixed, and do not approximate to2. The first is 1, the second is $, and so on; it is we ourselves who approximate to 2, by passing from one to another. Similarly when we say, “let x bea quantity which continually approximates to the limit 2,” we mean, let us assign different values to x, each nearer to 2 than the preceding, and following such a law that we shall, by continuing our steps sufficiently far, actually find a value for « which’ shall be as near to 2 as we please. In the second place, 2 is not the limit of the preced- ing sets merely because each is nearer to 2 than the preceding : for by the same rule, each is nearer to 1000 than the preceding. But we cannot assign one of the set which shall be as near to 1000 as. we please ; though we can assign one which is as near to 2 as we please. The following is exactly what we mean by a Limit. Let there be a symbol a which has different values depending on different successive suppositions of such a kind that any one of the suppositions being made, we can thence deduce the corresponding value of x: let the several values of x resulting from the different suppositions be —_—_—. i Og Te tie gs, OCs then if by passing from a, to da, from a, to a, &c., we continually approach tova certain quantity /, so that each of the set differs from “by less than its predecessors ; and if, in addition to this, the approach to 7 is of such a kind, that name any quantity we may, however small, namely z, we shall at last come to a serles beginning, say with a,, and continuing ad infinitum, An Qnty An+9 . o e e &e. all the terms of which severally differ from 7 by less than =: then /is called the limit of x with respect to the supposition in question. When, either in the way of hypothesis or consequence, we have a series of values of a quantity which continually diminish, and in such a way, that name any quantity we may, however small, all the values, after a certain value, are severally less than that quantity, then the symbol by which the values are denoted is said to diminish without limit. And it the series of values increase in succession, so that name any quantity Wwe may, however great, all after a certain point will be greater, then the DIFFERENTIAL CALCULUS. » without limit. It is also frequently said, when thout limit, that it has nothing, zero or 0, for its t increases without limit, it has infinity or & 10 series is said to zncreas a quantity diminishes wi limit: and that when 1 or 4 for its limit. For instance, we may ask what is the limit of Sal ; That is, supposing we give to x a set when a increases without limit. der and without limit, what will the of successive values, increasing in or , which correspond to the values of 2, have for a set of values of = limit, or will they also increase without limit, or dimi Let us choose for the set of values of wv in question, 1, 10, 100, 1000, 10,000, &c. nish without limit. 2 When x= 1 we i en x Pal 5 2 When x = 10 Pal = der and b+ b! 4-4...» that de Pian eae 148 b + b'+..6- Sh t lie between the greatest and least of a, a. ..., and there- alled small, if all the set a, a’.... are severally small. : f this mode of defining nearly equal will suffi- ciently appear in the rest of this work, and we therefore pass to its 7 most important application. It appears that two quantities, however : small they may be, are not to be considered as approximating on account — | of their smallness ; for, in fact, they may be possibly receding from | each other, even while they are absolutely diminishing, or approaching = to 0. The following instances will show this to happen in certain cases. where (2 mus fore must be c But the convenience 0 SS “a et a circle be drawn of which any diameter AB is taken. Let any point P be taken, as near to B as may be chosen, and draw P M per- pendicular to the diameter AB. From O draw OT perpendicular to the jj same diameter, and produce B P to meet OT in T. We have then a J rectilinear triangle MBP, the sides of which become smaller and smaller as P is placed nearer and nearer to B, in such a manner that, by making P sufficiently near to b, we may render either of the sides as small as we please. If P absolutely coincide with B there is no such triangle at all. The question is, what relations do P M, M B, and BP, as they diminish, assume or tend to assume, not with respect to any fixed, or given, or constant magnitude, such as O A, but with respect to each other? As P approaches towards B, it is evident that the angle OBP increases. For the angle POB diminishes, and Two right angles — 2 POB Z POB Z OBP = ——— = A right angle— 2 Piet” 2 As P approaches without limit to B, the angle POB diminishes without | limit, or the limit of the angle OBPisa right angle: that is, the | line B PT continually approaches to a state of parallelism with OT, ot | INTRODUCTORY CHAPTER. 17 the point T recedes from O farther and farther without limit. Place the point T ever so far from O, and TB will cut the circle somewhere. If O B were one foot; and if O T were a hundred thousand feet, still P would be a distinct point from B. It is true that the arc PB would hardly be the thousandth part of an inch, but that has nothing to do with the comparative dimensions of the triangle PMB. It is perfectly within the limit of geometrical conception to imagine all the diagrams of the six books of Euclid drawn within the compass of a square, having for its side the thousandth part of an inch: perhaps many of our readers have seen the Lord’s Prayer, the Creed, and the Decalogue written within the compass of a sixpenny piece. In the first case, every figure would have the same proportions existing between its parts as in the largest diagram ever displayed in a lecture-room: in the second, the length of two letters would preserve the same proportion as in the largest handwriting. Hence all we know of the sides P M, MB, and BP, being that they become small together, smaller together, and finally, as the phrase is, vanish together, we cannot from this alone affirm any thing as to whether or no they approach to or recede from equality according to our definition of such approach or recession: for this depends, not upon the absolute mag- nitudes of the quantities in question, but upon how many times, or parts of times, each is contained in the other. Two quantities may both be small, but one may be a thousand times the other : two quantities may both be great, but one may contain the other only one time and a thou- sandth part of a time. Hence we must examine the figure itself, and from its particular properties, as distinguished from all others, we must ascertain the manner in which the daw of relation changes (if it do change) while the triangle is diminished. Since the triangle PMB must be similar to the triangle T O B, we See that, whatever may be the absolute magnitude of the former, TO bears to OB the same proportion as PM to MB. Consequently, as often as OB isrepeated in TO so often js M B repeatedin M P. Butas P approaches towards B, the point T recedes without limit from O, that is, there is no point so distant from O but T must reach it before P reaches B. Therefore, there is no number so great, but M P will con- tain M B more times than that number before P reaches B. This is the most difficult of all the fundamental points of the Differential Calculus - two quantities both diminish without linut, yet as they diminish more and more, one contains the other more and more times without limit, so that 2f we wish to destgnate any number, however great, we can do it by assigning some position of P near to B, and saying vt ts the num- ber of times which PM contains MB; and the greater the number we wish to designate, the nearer must P be placed to B. This result as announced must appear surprising at first: but it is sufficiently evident by considering that, as to proportion of its dimensions, the triangle TOB is only a magnified representation of the triangle PMB. The difficulty of the proposition lies, firstly, in our not being used to consider that the proportions of figures do not depend upon their size, but upon what Euclid terms the ratio (Aoyoc) which he says* is (if we * The translators and commentators of Euclid have first cut this definition to pieces that they might quarrel about putting the parts together again. To English readers every word of Euclid is curious, and we shall therefore show how they have Managed. Simson, and all the recognised editions in our language, express them- Selves to this effect :—* Ratio is a mutual relation of two magnitudes with respect to Cc * 18 DIFFERENTIAL CALCULUS. may coin such an English word) the number-of-times-ness, Or quantu- plicaty, of one quantity, considered with respect to another. Because sve seldom have to consider small quantities except as parts of larger ones; we carry with us our notion of smallness to the comparison of two small quantities, where, in propriety, the notion of smallness ought not to enter. The second cause of difficulty lies in our being apt to run to the limit at which our suppositions cease to exist, and to say that if PM contain MB more and more times without limit before P can reach B, then’ when P actually reaches B, PM must contain M B an infinite number of times, or one nothing contains another nothing an infinite number of times, To this we must say, in the first place, that the result is not absurd, but only vague and indefinite, for nothing may be supposed, without palpable contradiction, to contain nothing just what number of times we like. In the second place, we have seen that 0 must be con- sidered with reference to the way in which it was obtained, before we can attempt to say what are its properties. And in the third place, that whether the two preceding arguments be good or bad, we have nothing to do with them, but content ourselves with asserting what we can prove, in circumstances which we can understand, namely, that P may be placed so near to B, as that PM shall contain MB any given number of times however great. If you* name a million, we can calculate to any degree of exactness you please, the angle POB which will give P M a million times MB : if you name a higher number, we can do the same ; name any number you please, which can be named, and we can do the same. What have we here to do with either nothing or infinity ¢ We say, that as P approaches towards B, the ratio of PM to MB increases without limit, which is our way of stating the theorem just explained more at length. If you say that you cannot conceive P con- tinually approaching to B, and its consequences, without forming some notion about what will become of these consequences when P actually reaches B, we answer that you are at liberty to form your notion, and it may be anything you please, or that you cannot help; all we say is, quantity.’ The old Latin versions simply call it a “ certa alterius ad alteram habi- tudo’? Billingsley, the oldest of the English editors, calls it a ‘‘ habitude of one to the other according to quantity.” Williamson, in the last century, who prided himself upon his staunch adherence to Euclid, gives it correctly in a note, but not in the text; Cotes saw the propriety of an alteration, but did not go back to the Greek to make it, but says it is a mutual relation “secundum communem mensuram,’’ while much discussion has ensued upon the meaning of the mangled definition. We cannot say what they would have done in France, for their editor, Peyrard, has omitted the fifth book altogether, but quotes it in the sixth. The words of Euclid are Adyos eee dbo weryebay ouoyevar n nord ANAIKOTNTA pds BAANAG Wore cicéous, the seventh and eighth words of which were rendered by Wallis and Gregory secundum quantuplici- tatem. In fact, magnitude itself (weyetos) is Euclid’s term for quantity in the usual English sense. ‘The definition seems to hint at the very distinction drawn in the text. It is, when we talk of ratio, we do not talk of one quantity or magnitude, for it is a mutual relation between ¢wo quantities or magnitudes ; nor do we speak of their quantity, or of how much they are, but of their mutual quantuplicity, or how many times one contains the other: so that two magnitudes, however small, may have the same ratio as two others however great. or may give the same answer to the question, how many times does the first contain the second? It is true that the word used by Euclid does, according to lexicographers, mean quantity as well as quan- tuplicity ; but as Euclid had already a word for quantity or magnitude, we think the sense in which he employed it is sufficiently clear. * We have taken a locutory style as the most easy to write, and, we believe, the most easy to understand. INTRODUCTORY CHAPTER, 3 19 that your case és not included in our theorem (whether it ought to be or not, we neither know nor care) ; all we have said (and it has been proved) is, that as P approaches to B, the ratio of PM to MB continually increases, and without limit, If a supposition of your own, superadded to ours, raises a difficulty, you, who made the supposition, must remove it as you may. But we can show that the difficulty comes too late ; and that, upon your own plan of adding suppositions to the expressed statement of theorems, you ought to be in the middle of the first book of Euclid, without any hope of reaching the second. For when it is shown of all triangles whatsoever, that the sum of two sides is greater than the third ; and when it is added that this remains true, however small the sides of the triangle may be (which is a necessary conse- quence of its being asserted of any triangle whatsoever), there comes the difficulty implied in asking what the theorem means when the triangle is diminished to a point, and all its sides are severally nothing. Are two nothings added together greater than a third nothing ? But are we necessarily obliged to suppose, that, because P continually and for ever approaches to B, therefore it will at last come to B? By no means, as the following reasoning will show. Suppose a circular Z M B X arc B Y (whose centre is Z) falling perpendicularly upon one of two parallels X Z and Y W, Along Y a point V travels at the rate, say of a mile an hour, and at every point of its course the line ZV is drawn, meeting the circlein P. ‘It is clear first, that as V proceeds from Y along Y W, the point P will move towards B, for V cannot progress in any degree whatsoever to the right without requiring a line ZV which shall place P somewhat (be it ever so little) nearer to B. But P cannot teach B, for to suppose that, would be to suppose that Z B produced meets Y W, which, by previous supposition, it does not, be it ever so far produced. We can then actually suppose P to move for ever without reaching B, and as we have shown, during the whole of that motion, the ratio of PM to MB increases continually, and without limit. The third cause of difficulty lies in unlimited diminution removing figures out of the province of our senses, which are a very great assist- ance in understanding the elementary propositions of geometry. In algebra, the difficulty is not so apparent, because the senses do not give the same assistance in any formula which has the least complication. Compare for a moment. the degree of evidence, independent of reason- ing, which attaches to the two following propositions. Algebra. + Geometry. eo ee Z Any two sides of a triangle are to- SE Tata gether greater than the third. me ——(y This difficulty arises from the student depending Somewhat too much On ocular demonstration, and not entirely on reasoning, in his preceding ~ Course, and can only be overcome by close attention to the reasoning. c 2 20 DIFFERENTIAL CALCULUS. Wé have the result of all that precedes in the following proposition. If two quantities diminish together without limit, their ratzo may etther is Merits! ic 4's A : increase without limit, or diminish without limit. UB is an instance of the first, and a of the second. For to say that PM may be as many times M B as we please, is to say that MB may be as small a raction of P M as we please. But we also have the following proposition. Lf two quantities dima- nish without limit, their ratio may ezther increase or decrease, but not without limit, that is, may have a finite limit. Let us suppose the suc- . cession of quantities diminishing without limit, ee eee the ratio which each bears to its predecessor will be an increasing ratio ; for, dividing the second by the first, the third by the second, and so on, we have $4444 % & which is aseries of quantities increasing for ever, that is, it never ends, and each term is greater than the preceding. But the increase ts not without limit ; for sce every numerator is less than its denominator, every one of the fractions is less than unity. And unity,as the limit for the preceding series of fractions, may be thus represented,— 1 Fp 1 a a ee Z 73 1 which, being generally 1 ——, may be brought as near to one as we n U please, by making 7 sufficiently great. We now return to the figure in page 16, and ask, what limit will the ratio of PM to P B assume, as P approaches without limit to B. The only thing we know immediately from the nature of the figure is that P B, the hypothenuse of a right angled triangle, must always be greater than PM theside. But as P approaches to B, does the inequality increase or decrease? Can we, in the manner proved of P M and M 5, place P so near to B, that PB shall be a thousand times PM? Since P Mis contained in P B in the same manner as IO in TB, we must examine the change of propor- tions of the two latter, while T recedes without limit from O. And since the two sides of a triangle differ from each other by less than the third side, it follows that T B can never exceed TO by so much as OB. And since, by sufficiently removing T, we can make OB less than any given fraction (say one millionth) of TO, it follows that (since removing T brings P nearer to B) that by sufficiently approaching P to B, we can make P M differ from PB by less than its millionth part. — Consequently, the limit of the ratio of P B to P M is unity; for, as we can take P so near to B that the equation oe 1 RB oe 1 PB=PM+ zs PMosy—it = shall be satisfied where n may be as great as we please, it follows that “i second side of the equation shall be brought as near to unity as we please. We may make it appear by the following method that it by no means | follows that the mere diminution of two quantities gives the right to infer anything as to the alteration of relative magnitude, A and B INTRODUCTORY CHAPTE R. 2] diminish together, but it may be that, while A loses one half of its first magnitude, B loses three-tenths of itself. This is one method of diminu- tion; and if we call a and & the magnitudes of A and B at the first stage, then 4a and 57,5 are their magnitudes at the second stage alluded to. At first, then, - is > 3 but o is afterwards 4a—~-7,6 or an less than before. But if, while A lost its half, B did the same, the ratio would be the same in both cases. And if A lost only one-tenth of itself, while B lost nine-tenths of itself, the ratio of the two would be increased by their diminution, Consequently, nothing can be inferred of a ratio from the diminution of its terms, unless the simultaneous pro- portions of themselves which the terms lose be given. The next difficulty is one which should be of a more serious nature, because it does not arise from the preceding views of the student being too limited, but from his not having had the necessary considerations presented to him in any manner or degree. Let us Suppose it made perfectly clear that two quantities may have limits, to which they approach together under the same circumstances; and, moreover, as in preceding instances, that though we may approach the limits as near as we please, yet we must not consider the supposition pushed to the extent of their being actually reached, either because we have then to deal with nothings, or with infinites, as in p. 20, where we cannot, in any finite number of terms, reach the limit in question. The difficulty is, how are we to reason upon cases which we are not allowed to Suppose? The actual state of the problem in which a quantity has reached its limit is expressly forbidden to be considered. If the limit itself be known, this may seem to be immaterial; but it may be that the limit itself is to be found, by means of other limits which depend upon the same circumstances. In this case, we can only determine the unknown limit by means of an equation which combines it with the _ known limits. But such an equation we are not allowed to form. The question is, by what method are we to proceed ? There are two gencral ways of proving any assertion: the first, in which it is expressly proved that the assertion is true, in all the cases which it includes ; this is called direct reasoning : the second, in which it is proved that every proposition which contradicts the assertion is false ; this is called indirect reasoning. It seems customary to look upon indirect reasoning as being of a less conclusive character than direct reasoning, and therefore to be avoided if possible. Perhaps this may depend upon the mental constitution of the individual to whom the reasoning is supposed to be addressed ; to us it seems equally conclu- sive whether we prove that every equiangular triangle is equilateral, or that he who asserts that any one equiangular triangle is not equilateral, asserts at the same time that the whole is less than its part. Let us suppose that there are two quantities, P and Q, of which it is the property that P is always double of Q; and let any supposition whatsoever make P and Q approximate at the same to the limits p and 7, So that it is allowable to suppose P and Q respectively brought to differ from p and q by quantities less than any we may assign, however small. Here P and Q are what are called variables, namely, symbols which have different values upon different suppositions, but which at the Same time are always connected by the equation P =2Q; and p and q are fixed limits. What we have to prove is, that p=24q: but weare not 22 | DIFFERENTIAL CALCULUS. at libarty to say that P ever can be actually = p or Q to q, but only that P and Q may simultancously approach within any degree of nearness to p and q short of absolute equality. That is, if we say let P=p +a, and Q=¢ +B, were al liberty to suppose and 6 smaller than any quan- tity we may name, put not absolutely nothing. We shall not prove this proposition p= 2q to be true; but we shall prove everything which contradicts it to be false, Now, what are the propositions which con- tradict p is equal to 2 q? evidently only those éontained in the following— p is greater than 2 q, or Pp is less than 2 q. If, then, p be greater than 2q, let it be 2q + ™, therefore we have Pa=pta=2qt+mr+e Q=q+t pand2Q=2qg4+28=P or; m+-a=26 ma2hP—-a; now since p and q, are given limits, not changing when P and Q change (being in fact the fixed quantities to which P and Q in their changes continually approach), it follows that m, the difference between p and 2q, must also be a fixed quantity throughout the changes of P and Q. Therefore 28 — « is always the same: but it is allowable to suppose o and f as small as we please, and therefore « —2 6 may be as small as we please. That is, a quantity both has a fixed value, and may be as small as we please, which is absurd. Thence p= 2q + mis false 5 a similar train of reasoning will show that p= 2q—m is false, what- ever may be in either case, provided it actually have some value. But either p=2q + m or p= Qqorp=2q—M; the first and last are false, therefore the second must be true. This will give an idea of the method by which it is possible to prove propositions with respect to limits, without actually supposing the quan- tities in question to have attained their limits. We shall now proceed to a rough-and practical kind of Differential and Integral Calculus, preparatory to more exact methods. Draw a circle with a fine pencil, and nearly cover it with a straight- edged piece of paper, and more and more nearly until none of the inte- rior is visible, but only a small part of the circumference. That this can be the case at all arises from the roughness of the edge, and the thickness of the circumferent line: for it 1s impossible that a geome- trical line should coincide with the boundary of a circle for any length whatsoever. Draw two straight lines meeting each other, and cover them in the same way, and a similar effect will not be produced, at least not nearly to the same extent. And even if a geometrical circle could be drawn, and a geometrical straight line apphed to it, provided only we could conceive these lines without breadth to reflect light, and be visible, the same effect would be produced. Let A B be the imaginary edge of the paper (supposed perfectly straight), and AD B a part, either D Nill ORME RS of the circle, or of the intersecting straight lines, according to the figure INTRODUCTORY CHAPTER. 23 chosen, while CD is in both cases a perpendicular dropped from the highest point upon AB. Let us now conceive the edge of the paper moved up parallel to itself very near to D. As our eyes cannot per- ceive lengths of more than a certain degree of smallness, let the minj- mum visibile (least visible portion) of length be named; it matters little what it may be, say it is one millionth of an inch. Then let the edge of the paper be moved up until C D is in both cases less than one millionth of an inch. The consequences will be very different in the two cases. In the straight lines, CD B will always change so as to remain similar to its first form, that is, the proportion of CD to DB will not alter. If we suppose DB and D A together to be five times C D, then so soon as C D is less than the five-millionth part of an inch, there will be no visible length in the triangle A DB, and nothing will be seen but a point. But in the circle, if we suppose the radius to be one foot, it will follow that when CD is the five-millionth part of an inch, AB will be more than fourteen-thousand times as great as CD, that is, nearly three times the thousandth part of an inch, and will there- fore be a visible length. This depends upon what has been already proved, that the smaller CD is taken or the nearer B approaches to C, the more times will C B contain C D, and this without limit. In practice, then, a small are of a curve may be considered as a Straight line, the words, in practice, always implying that there are lengths so small that they may be absolutely rejected as inconsiderable, and without sensible error for the object in view. Suppose now we were to divide a circle into a thousand equal arcs : measure each are vey accurately as if it were a straight line, that is from end to end along A C B, instead of round A DB, and put the whole results together : would the total sums of these measurements be a tolerably correct value of the circumference of the circle ? By no means, would be the first answer which suggests itself: for, however small the error may be in taking each individual are to be a straight line, there is an accumulation of a thousand errors in the summation, and we do not gain anything by measuring twelve separate inches, each one-tenth too small, to avoid measuring a foot upwards of a whole inch too small. But the preced- ing answer is not correct ; for it happens that, by diminishing the arcs, we not only diminish the absolute error made by reckoning an are to be a straight line, but we also diminish the proportion which each error is of wits whole arc*. If CD be the five-millionth part of an inch, then ACB will not fall short of A DB by its fourteen-thousandth part ; but if the are AD B were one-sixth of the whole circle, AC B would fall short of AD B by more than its twenty-fifth part. If we estimate an error, not by its actual magnitude, but by the proportion it bears to the thing measured, then the error of the first measurement is less than that of the second in the proportion of 25 to 14,000. To illustrate this, try the following experiment: Draw a fine circle of three inches in radius, the circumference of which is therefore extremely near to 18°85 inches or eighteen inches and seventeen: twentieths of an inch. If we take an opening of the compasses of three inches and carry it round the circle, we shall find it contained exactly six times: or taking chords instead of ares, we then find eighteen inches as a first approxi- mation. Now, take an opening of one inch, which we shall find to go round the whole circumference eighteen times, with an are over, having * The student must particularly attend to this. If any one sentence in the whole book ought to be called the ‘ Differential Calculus, this is it. 24 DIFFERENTIAL CALCULUS. a chord of about thirteen-twentieths of an inch. Subject then to the errors of taking chords for arcs in this second measurement, we con~ clude the circle to be 183% inches, considerably nearer the truth than the first. Now, though in the second measurement we have accumulated nineteen errors, while in the first there were only six, yet each error of the first measurement amounts to this, that the chord falls short of the arc by about its twenty-fifth part, while in the second measurement the chord falls short of the arc by only about its two-hundredth part. Consequently, the total error of the second will be less than that of the first in about the proportion of 200 to 25 or 8 to 1, which, in the actual rough measurement we have given, is not far from the truth. In this way we may see, what will afterwards be more strictly proved, that the following assertion, Any arc of a curve ts equal to the sum of the chords of its parts, is of this kind :— 1. Itis never true: for every chord is shorter than its arc. 2. If the whole arc be divided into a moderately great number of parts, it is sufficiently near the truth for practical purposes. 3. It can be brought as near to absolute truth as we please (that is, the error involved in it can be made as small as we please) if we are at liberty to divide the whole arc of the curve into as many parts as we please. When we speak of one false proposition as being more near the truth than anothér, we mean that the numerical error made by acting upon the first is less than that made by acting upon the second. And by saying that an assertion can be brought as near the truth as we please, we mean that, by some particular disposition of the circumstances which it leaves at our disposal, we can make the numerical error which it involves as small as we please. For instance, the preceding proposition is an assertion about an arc divided into a number of parts which it does not fix. It is never true; but the greater the number of parts of which it is supposed to speak, the less will be the error it asserts, and that with- out limit. The consequence is, that if we imagine the arc first divided into ten parts, afterwards into 100 parts, afterwards into a 1000 parts, and so on, and if we add together the ten chords in the first, giving A, the hundred in the second, giving B, the thousand in the third, giving C, and so on, we shall have a series of terms A, B, C, &c. which approach continually towards a certain limit, which, however, they never actually reach. With reference to the problem of finding an arc of a known curve, the Differential Calculus ascertains what is the form and value of the parts which are to be added; the Integral Calculus adds them together and gives the result. At least this is the first rough defim- tion of these terms which can be given to a beginner. In the following form the preceding assertion is strictly true. The arc of a curve is the limit of the sum of the chords of allits parts. No addition of chords will be sufficient ; we must observe the sum of the chords of 10 parts, of 100 parts, of 1000 parts, and so on, and find from the proper- ties of the series of terms so obtained the value of their limit. It might be said that the proposition, ‘* The are of a curve is equal to the sum of the chords of all its parts,” is actually true if all the possible parts be really taken. But the determination of all the possible parts into which a whole can be divided, is the same thing as the determination of an infinite number, which is impracticable even in imagination. Every part of a magnitude is itself a whole so far as subdivision is concerned : that is, it admits of as many subdivisions as the whole from which it INTRODUCTORY CHAPTER, 25 was obtained. And it is therefore impossible to subdivide the magni- tude until there is no such thing as further subdivision. But the theorems which we have been considering, led to the notion of infinitely small quantities, the most convenient of all simplifications, when proposed in a proper manner. Seeing that every magnitude can be subdivided into parts which shall severally be as small as we please, it was imagined that all quantities could be said to be made up of an infinite number of infinitely small parts, each of those parts being in magnitude less than any assigned fraction of the whole, and yet not abso- lutely equal to nothing. On the glaring untruth of this conception, positively considered, it is unnecessary to say a word; but it is never- theless one of those assertions which can be made as near as we please to truth. For a quantity can be made up of as many parts as we please, each of which shall be as small as we please. And all the con- sequences of this assumption, properly deduced, will be true; so that it may be considered as an abbreviated way of representing the necessity of dividing quantity into parts, which are to be supposed to be as many as we please. The only danger is, that the student should fall into the error of treating the assumption itself as an absolute truth ; but from this he will perhaps be saved by observing that though the doctrine of infinitely small quantities appears simple and natural, owing to the mind being always accustomed in practice to reject quantities on account of smallness, yet that its immediate consequences present unna~ tural absurdities. Allow, for a moment, the notion of infinitely small quantities, and in the figure of page 16, suppose PB to be infinitely small. Then P M and MB willbe infinitely small, but the latter will be now an absolutely incomprehensibility. For since it has been shown that the smaller P M is, the more times does it contain M B, it follows that when P M is infinitely small, it contains M B an infinite number of times ; so that M B is only an infinitely small part of an infinitely small quan- tity. This beats all our power of imagining subdivisions, and therefore (which may appear strange) we may be justified in retaining the terms of the infinitesimal Calculus as a method of abbreviating stricter pro- positions, when properly understood. For, if the student should ever for a moment imagine that he sees reason in the use of infinitely small quantities, absolutely considered, he has only to recall to mind the idea of an infinitely small part of an infinitely small quantity, and he will surely remember that the modes of speech employed are only abbrevi- ations of assertions which are to be reasoned on in their strict form, though expressed for shortness in one which is not absolutely correct. In algebra, the use of the term “ infinitely great’? is universal, though the notion attached is not that derived from the etymology of the word. To use the words infinitely great in any sense, and to reject the correspond- ing method of using the words infinitely small, is to accustom our- selves to false distinctions. If it be proper, 1m any manner whatsoever, ee Loe to say that x is infinitely great, it is equally proper to say that = 3s infinitely small. It is usual to say that when wx is infinite, 73s nothing ; and the meaning is simply this, that there is no limit to the smallness ge haa : of —, if there be no limit to the greatness of x, or that by making x x . I sufficiently great, we may make 7s small as we please. When we 26 DIFFERENTIAL CALCULUS. have to compare os with a fixed quantity, for instance, in the expres- av sion @& + -, we may indifferently use the phrases nothing or infinitely small, because, in every sense in which it has ever been proposed to use them, they here mean the same thing, The notion of infinitely small quantities is in fact that of comparing different nothings springing from different suppositions, as if they had relative magnitudes depending upon the suppositions which produced them: a method of reasoning which never can be admitted in any manner or to any extent whatso- ever, What we here mean to illustrate is this; that the forms of speak- ing, which such an hypothesis would require, may be made to give use- ful abbreviations of propositions deduced from stricter methods. It must be remembered that in mathematics, as in everything else, no definition of single words is always sufficient to define the meaning of words put together in a sentence, and the following explanations are to be considered as the meaning which we intend to affix to the sentences in italics. 1. Two infinitely small quantities may have a fintte rato. Two quantities may diminish without limit, and may still preserve a finite ratio, which is either a given ratio, or which becomes nearer and nearer without limit to a given ratio, as the two quantities diminish, The ratio may or may not alter as the quantities diminish. And when we say that two infinitely small quantities have an infinitely great ratio, we mean that the first divided by the second increases without limit when the quantities themselves diminish without limit. 2. When x is infinitely small, B is equal to C. By this we mean that, by making wx sufficiently small, we may make EG 3 nearly equal to unity as we please. 3. When x isinfinitely small, B is infinttely near to C. This is the last in a different form, and will illustrate what we have said, that the theory of infinitely small quantities, in the absolute meaning of the terms, is equivalent to giving relative magnitudes to nothings. If we have to consider C without reference to the difference between B and ©, and if the diminution of «, without limit, give the limit 1 to B “EG we simply say that the limit of Cis B. But, if we have to con- sider the diminishing difference of C and B, and to compare it with a or any other simultaneously diminishing magnitude, in order to see whether the ratio of the two remain finite or not, we then simply say that, instead of considering B and C as equal, they are infinitely near to each other, or their difference is infinitely small. ‘ 4. Of two infinitely small quantities, one may be infinitely greater than the other. By this we mean to abbreviate the following :—T wo quan- tities may diminish without limit, so that the more they are diminished, the more times does one of them contain the other ; and this without any limit to the number of times just mentioned. The term infinitely great is used as an abbreviation of corresponding propositions relative to magnitudes which increase without limit. Thus, when we speak of two infinitely great magnitudes, one of which is infi- nitely greater than the other, we speak of two quantities which simul- taneously increase without limit, but one of which increases so much INTRODUCTORY CHAPTER. 27 faster than the other, that it may be made to contain. the other as many times as we please, by making both sufficiently great. And here we shall observe, once for all, that I. When we speak of a magnitude increasing without. limit, we do not mean that it actually increases so as to be above every limit which could be named, for that is impossible; but that we can make it greater than any quantity which we actually do name. 2. That when we speak of a quantity changing its value, we do not mean, or at least we need not be supposed to mean, that the quantity itself grows, or flows, in the language of fluxions; but that we have a symbol of magnitude to which we attribute different values in succes- sion. Bunt whether we take, for example, straight lines of different lengths, and compare them together, or whether we take a straight line, Suppose it to acquire different lengths by the motion of one of its ex- treme points, and compare together its length at one time, and its length at another time, is perfectly indifferent. In future we shall use the theory of limits in all reasonings ; but when we abbreviate the results into the language of the infinitesimal calculus, we shall inclose the paragraphs so introduced in brackets [_ ]. We shall now proceed with our rough sketch of the principles on which the Differential Calculus is founded. © Our object is to show that there is no great refinement or abstruseness in the nature of the fundamental ideas of the science; but that they do, in fact, sugeest themselves in various cases which occur in common life, wherever a dis- tinct notion is to be formed of the actual state of a variable magnitude at any given epoch of its variation, It is observed that when a stone falls to the ground from a height (the resistance of the air being first allowed for) its motion is of this kind. Let ¢ be the number of seconds or fractions of seconds elapsed from the beginning of the motion, then the height fallen through is very nearly 16, x ¢tin feet. We ask, at what rate, or with what velocity, will the stone be falling at the end of three seconds, when it will alto- gether have fallen through 1634 x 9 or 144% feet. By velocity, we mean the space actually described in one second when the body moves uniformly ; but here there is no uniform motion, or the lengths described im successive equal times continually increase. Still, if we examine the lengths described in successive very small times, we shall find them nearly equal, and more nearly so, the smaller the intervals of time in question, and so on without limit. To show this, let us call 16, feet a measure ; then the number of measures fallen through in ¢ seconds is tt. Let us now suppose a very small portion of time k, and let the position of the stone be A at the end of ¢ seconds, B at the end of t+ k seconds, © at the end of t+ 2k seconds, &c. Let Q be the point from which the stone fell. Then by hypothesis, the values of the lines expressed in measures are as follows :— Q QA= QB= (t+ ky QC= (t+ 2k)%, &. : —£ AB=QB—QA=2tkh+ R= (2t+ ky i BC=QC—QB=2th4+3K=(2t43h)4 =o CD=QD—QC= 2th + 5h = (t+ 52)h, &e. em or the relative proportions of the successive spaces described in equal intervals, each being the part k of a second, are those of 28 DIFFERENTIAL CALCULUS. Qtith, Qt4+3hk, 2t+5k, 2t+ 7h, &e. to each other. Now it is clear, 1. That the spaces described in successive equal times are never equal, for no two of the preceding can be equal, however small may be. 2. That if ¢ have any value whatever, that is, if we commence the comparison after any given period has elapsed, during which the stone has fallen, we can take the interval k so small, that the lengths described in successive equal intervals shall be as nearly equal as we please. For k 2t43hk 2h cas pee a eee ees Le 2t+ k 2t+k BC AB o4 k t which can be brought as near to unity as we please, if k be made a suf- ficiently small fraction of ¢. Therefore the notion of equal lengths in equal times, or uniform velocity, 1s one which approaches without limit to the truth. What then is the velocity, or rate per second, to the effects of which the preceding motion more and more nearly assimilates? It is 2£ measures per second: not that any thing near this rate is conti- nued through a whole second, but that the rate of uniform motion which would carry the point through 2¢ --A* measures in a second, approaches without limit to the rate of 2¢ measures per second, as k is diminished without limit. For length described in a con i fhé teattioh k fe length de- — motion during the fraction k of scribed in a of a second whole second; and if we suppose v measures per second to be the necessary rate at which 2 tk + k? measures will be described in the fraction k of a second, we have QtkRek = kv or QE- REV} the smaller k is supposed to be, the more nearly will v = 2¢ be true, which is the proposition asserted. The notion of velocity is one which it is always customary to define by means of uniform motion, and, this mode of comparison being taken for granted, the preceding is the only way in which a body moving through unequal lengths in equal intervals can be said to have a defi- nite velocity. At the end, then, of one second, the velocity is 2 mea- sures per second, at the end of ten seconds it is 20 measures, the mea- sure being merely a term of abbreviation for 16 feet 1 inch. There is one remarkable case of exception, which will ilustrate the manner in which, throughout the Differential Calculus, particular cases may require rules of their own. If we count the small intervals 2 from the very beginning of the stone’s motion, that is, if we make f= 0, we find the total lengths described in k, 2k, 3k, &c. of time to be kh’, 4k” 9 }?, &c. or the lengths described in the successive intervals to be k®, 3 he, 5 k®, &c. which cannot be made as nearly equal as we please, for the second is three times the first for every value of k, however small. But here we find the velocity, as obtained from the preceding process, to be 0: that is, the rate per second with which k® would be dpacmen in the fraction k of a second, diminishes without limit at.the same time as R. This follows from #®? =kv orv=k. INTRODUCTORY CHAPTER, 29 In the preceding manner, let the student deduce the following propo- sition. Ifa point move along a straight line in such a manner, that at the end of é seconds from the beginning of the motion, the length described shall always be ¢* + ¢? + ¢ units of length, then the Velocit which that point must have at the end of ¢ seconds, is always 30? + 2¢4] units of length per second. [If a body move as just described, and if to the time ¢ already elapsed, an infinitely small time k be added, the infinitely small space described in the time & will be uniformly described with’a velocity at the rate of 3¢*°+2¢+ 1 units of length per second.] Prostem.—The curve OPM _ is of this VW A nature, that the area included between any ; abscissa O M, the corresponding ordinate P M, LA PY and the curve, is the third part of the square described on OM. Required the algebraical expression for the ordinate PM in terms of the seigeatdad abscissa O M ? ie cad We «es Let OM contain a units of length, and PM y units: take MN h units, and let N Q, the ordinate to O N, exceed P M by ZQ containing k units. Then, by the law of the curve, the area OQN is one-third of the square on ON, and contains 3 @ + h)? square units, while the area OPM is one-third of the Square on OM, and contains 5a square units. Hence the area M P QN contains ad 3(v@+h)?—isx* or Beh +1} square units. But this area is less than the rectangle MWQN, containing h (y-+-k) square units, and greater than MP ZN, containing hy square units, Therefore, whatever may be the values of A and ky $ch-+'th* must lie between h (y +k) andhy or. $ PER Lee & - y th and y. Now h and are so related, that by diminishing the first without limit, we diminish the second also without limit, and xv and y are, with respect to h and , fixed quantities. Consequently, y must be 22 ; for, if not, let 3 OM exceed PM by any quantity, however small. This excess of 2m above y doesnot change when / and h are diminished. Butas the pre- ceding relation must be true for all values of }; and k, take k less than the excess of 2.x above y. Then y + k must be less than = and there- fore less than 22 -+4+1h, or 2x + th cannot lie between y+ and y, which it has been proved to do. Therefore, $2 cannot exceed is neither can it be less than y, for in that case take h so small that $2 + yh shall not be so great as y, in which case it cannot lie between y and y + k,as required. Therefore, ¥y = 3 x, or the curve (as we sup- posed it) must be a straight line passing through O, and inclined to OM at an angle whose tangent is 2. In this case, since the relation so obtained holds for all points of the curve, we have YtkR=3(x7 +h) ork=2h, and we see that 22 +14h lies between y +k or2x2+ 2h and y or 2a, [If MN be infinitely small, QPZ is an infinitely small part of QPMN, and QPMN of the whole QON.] The preceding is a problem of the Differential Calculus ; we shall now take a corresponding problem of the Integral Calculus, the 30 DIFFERENTIAL CALCULUS. algebraical difficulty of which lies entirely in a proposition which we shall here take for granted, namely, that the sum of all whole square numbers, 1, 4, 9, 16, &c. up to n 1s 1) (2 1 Pee PG ss ceed)? Hamers wena 6 this may be easily verified in individual cases; thus, hg! BPS 2.2.9 3.4.7 -_—- ile ee ee i =a & e Pe ate 1+ 6. 144-49 err c Prosiem.—lIn the curve OM P, the ordi- aves nate M P (y) is always a times the number | of square units contained in the square of the [7 | | abscissa OM (x) ; or y= axe: required the Pe number of square units in the area OMP? a | ; f=} Divide O M into n equal parts, being any Sais = whole number: that is, we mean to trace the o M consequences of dividing O M into a number of equal parts as great as we may find necessary to choose. We represent this in the figure by dividing OM into such a number of equal parts as the dimensions of the figure makes convenient. By drawing the ordinates at every point of section, and completing such a construction as is seen in the figure, we have to notice 1. A curvilinear triangle, together with n — 1 rectangles, all falling inside the curve, and making up an area less than that of the curve required. 2. A number n of other rectangles having severally the same bases as the preceding, but each exceeding its portion of the curvilinear area by a small curvilinear triangle, and altogether, therefore, making up an area greater than that of the curve. 3. A series of small rectangles diagonally cut by the curve, the first of which is a rectangle mentioned in (2.), but all the rest of which are the differences between the rectangles in (1.) and (2.) The sum of all these smaller rectangles is equal to the last rectangle in (2.), or that which has the side P M, for all the bases are the same, and the sum of the altitudes of the rectangles which are diagonally cut by the curve is equal to the altitude of the rectangle on P M just mentioned. Hence it follows that, by making the number 2 of subdivisions greater and greater, we continually make the sum of the rectangles in either (1.) or (2.) approach to the area of the curve required; for the area of the curve must lie, as to magnitude, between the sum of the curvilinear triangle and the rectangles in (1.) and the sum of the rectangles in (2.) But. these only differ from each other by the difference between the rectangle adjacent to P M and the curvilinear triangle at the commence- ment, which may both be made as small as we please by increasing the number of subdivisions. Therefore, by increasing the number of sub- divisions without limit, we shall find the required area of the curve in the limit towards which the sum of the rectangles in (2.) continually approaches. Let OM be a, then the several intervals between the points of section are equal to and the distances of the points of section from O are severally, INTRODUCTORY CHAPTER. 31 v 2b) 3a @ Seite; kn Laks Seedine ATID UNL GF n n n n the corresponding ordinates to which are wt 643% 92” ears : a Or -++. up to an’ — or aa, n n and the areas (in square units) of the several rectangles are @ co a 4 x? x >it rt oy, ss, pe tte Up to xX an? — n n n n n n the sum of which is, a Ge Cl 4 Ob ib gi + (n+ 1)? + n?2) n' pe he n(n+1) (2n+1) or 2 n. (n+ 1) (2n +1) n 6 6 Mss n 3 3 7 ae n+) eect Ae jets ated 45 n n 6 n n this expresses, for every value of n, the sum of the rectangles in (2.), and as 7 increases without limit, the term — diminishes without limit, n so that the limit of the preceding summation is, ax ax® —. 1X2 oye ee, pe 3 But that same limit is the area of the curve in question, whence we have ax x#xeadr LY rea O r 5 5 namely, the third of the rectangle described on OM and MP. {tis obvious that the success of this method depends on our being able to sub- stitute the definite formula & 2 (n+ 1) (27-4 1) instead of the indefinite formula 1+4494 severe + (m—1)24 m7? and that a similar substitution, if we are able to make it, will enable us to find the area of any other curve. We have examined cases in which the limit of a ratio has difficulties arising from the unlimited diminution of the terms ; we shall now show a case in which the limit is to be singled out from an infinite number of results, all of which appear at first sight equally possessed of. that cha- racter : for instance, when two straight lines intersect each other in a point, and then continually approach to coincidence, shifting their point of intersection with their changes of position. When they are actually brought to coincide, they have all their points in common, or every point Is a point of coincidence. The question is, which among all these points of coincidence is the point towards which the point of intersec- 32 DIFFERENTIAL CALCULUS. tion always tended while there was intersection. Let QR be a straight line which always moves perpendicular to the tangent of the curve PQ, while Q moves towards P; and let PR be perpendicular to the tangent at P. As the point Q approaches to P, will the point R recede from P? If so, will it recede without limit, that is, may any point in PR, however distant, become the intersection, by bringing Q sufficiently near to P? Or will it recede with a limit, that is, though always receding while Q approaches P, will there be any point in PR beyond which it never can be found? Or will it ap- R proach to P, and if so, will the approach be without limit as to near- ness; or can a point be assigned in P R, within which and P, the inter- section will never be found? The answer to these questions depends upon the nature of the curve PQ; we ask them here that the student may be able to see whether he still retains notions of limits derived from anything but demonstration. In the ‘ Elementary Illustrations, &c.’ *, page 22, a case will be found, in which the limit of an intersection 18 deduced. All. works which treat of the Differential Calculus, for the most part make more or less reference to the discovery of the method, and the celebrated dispute upon the right to the honour of it. We shall here state in few words as much as we think necessary upon that subject. Unquestionably, the first whom we know to have solved any problem of the Differential Calculus was Archimedes, in whose treatises on spirals on the quadrature of the parabola, and on the cone and sphere, are £5 be found processes which depend upon the comparison of curvilinear figures or curved surfaces, with the inscribed rectilinear figures or plane solids. A method of limits is really introduced, the basis of which is the proposition, that by successively taking away more than half from any quantity and the remainders obtained, the last remainder may be made less than a given quantity, and a process somewhat like that in page 22, is made to furnish rigid demonstration of the results. Taking all the curves and surfaces which were considered in his time, Archi- medes has produced most of the results which even the modern Differen- tial Calculus can express in finite terms ; and he was stopped, not by the inadequacy of his method considered with reference to the distinction between the Differential Calculus and other. branches of mathematics but simply by the want of a more powerful instrument of expression; such as is algebra when compared with geometry. He could overcome the difficulty which answers to writing in(n +1) (Qn + 1) forl+44+9+4+ ..6.0+ (n ~ 1)? 4+ 7? “but he could not obtain the approximate expression (314159... =)? een for 1 +44 db veers RGus the language and ideas of his time hardly admitted an adequate concep- tion of the preceding, or of anything equivalent to it, and the methods of operation would have been utterly unable to discover it. * Nos, 135 and 140 of the ‘ Library of Useful Knowledge’ INTRODUCTORY CHAPTER. 33 Between the time of Archimedes and the end of the sixteenth cen- tury, there is nothing to arrest our attention. The discovery of a very few new propositions having just this affinity with the Differential Cal- culus that they are €asy Cases of it, is all that can be adverted to, Vieta, the first user of general symbols in algebra, that is, of letters designat- ing any quantity whatsoever, and Ves Cartes in applymg the algebra go obtained to geometry, by what is now called the method of co-ordinates, were the original creators of the power of algebra, and they were fol- lowed by a multitude of partial discoverers, who added isolated theorems on series and developements to the general stock. At the same time the general theory of curve lines was receiving similar accessions, and the multitude of analogies suggested to several the idea of combining them under one general form. In the first half of the seventeenth cen- tury, Cavalieri proposed his notion of indivisibles *, and Roberval his nolion of fluxions. We Say notions instead of methods, because, in fact, no methods could spring out of them, unless by the application of a more powerful algebra than was then possessed. It is difficult to imagine that either idea had not occurred to Archimedes, and been used by him as a method of discovery, though rejected as one of demonstration, Roberval considers curves as formed by the motion of a point ; and by assigning the law of description of the curve, and the consequent velocities of the point in any convenient direction, he obtains the direction of the tangent of the curve by the composition of these velocities. He also lays down the connexion between the method of indivisibles and of infinitely solved, and which contained more hints for future discovery than any Newton and Leibnitz had independently come to the consideration of quantity, and each made the new step of connecting his ideas with a Specific notation. If one line depend upon another, and both increase, Newton supposed the first line w to increase or flow with a velocity 2, in consequence of which the second increases with a velocity 7. Leib- nitz supposed an infinitely small increase dx to be given to 2, in conse- quence of. which y receives the infinitely small increase dy. These almost amount to the same thing: if we Suppose an infinitely small | time d ¢ to elapse, during which the motion supposed by Newton causes the increase supposed by Leibnitz, we haye * See § Elementary Illustrations,’ &e., p. 61. t “ Pour tirer des conclusions par le moyen des indivisibles, i] faut supposer que foute ligne, soit droite ou Courbe, se peut diviser en une infinité de parties ou petites lignes toutes égales entr’elles, ou qui suivent entrelles telle progression que ]’on Voudra, comme de quarré 4 quarré, de cube a cube, de quarré-quarré a quarré. quarré, ou selon quelqu’ autre puissance, “Or d'autant que toute ligne se termine par de points, au lieu de lignes on se Servira de points; et puis au lieu de dire que toutes les petites lignes sont & telle chose en certaine raison, on dira que tous ces points sont a telle chose en ladite raison.”—Roberval, Traité des Indvisibles, Roberval’s Fluxicns are to be found in his * Observations sur la Composition deg Mouvemens,’ the work of a pupil from his instructions, with his remarks, Both treatises are in ‘ Divers Ouyrages de Mathé- matique,’ &e. Paris, 1693, folio. n ae. 34 DIFFERENTIAL CALCULUS. , d drnadt dy=ydt = The merit of this step being granted to belong equally to both, it only k 3 ee y : remains to ask which did most towards assigning the value of kis its equal oH in every possible case. And here there can be no question ip that the binomial theorem of Newton is a much larger constituent of the difference of power between Archimedes and the immediate successors of the former, than anything else whatsoever, unless it be the step made by Vieta, already mentioned*. It is perfectly true that Leibnitz advanced the Differential Calculus, in conjunction with the Bernoullis, to a much: greater pitch of perfection than Newton or his English contemporaries. Our preceding remarks are only intended to draw the attention of the student to the distinction between the metaphysics and notation of the subject, and the algebra which makes them serviceable. The notation of Newton, which prevailed in England till after the commencement of the present century, has been discarded by all writers ‘n the universities, and by most out of them. ‘There are those who object to the change, and who consider the fluxional notation as at least equal, if not superior, to that of Leibnitz. Without discussing this point, we are inclined to consider the universality of the notation of Leibnitz throughout the whole of the civilized world, and the fact of most of the discoveries made since the time of Newton, both in pure mathematics and physics, being expressed by means of it, as itself a sufficient reason for adopting it. But we shall in the proper place give both notations, and explain the method of converting one into the other. We shall also endeavour to teach the Integral Calculus at the same time as the Differential. The separation of the two which takes place in most works, though convenient in some respects, and those not unim- portant, yet deprives the student of the means of learning, at the same time, subjects between which the analogy is as strong as between addition and subtraction. * Leibnitz complained that when he spoke of the Differential Calculus, his oppo- nents answered him by reference to the method of series. M. Montucla remarks on | this, that “‘a geometer might have been in possession of the method of series, and : have been able to square a multitude of curves, and yet not have been in possession — of the calcul des fluxions et fluentes.” But what those words mean when abstracted from the method of series he does not state; but goes on to add, “ the expression for the ordinate of a curve being reduced into a series, if the, case required, the methods of Wallis, Mercator, Cavalerius, or Fermat, would have sufficed to find the area.’ Considering that Leibmitz himself admitted the priority of Newton in the method of series, and that there is no question at all of the labours of Leibnitz in this sespect being in no degree to be compared with those of Newton, this is some- | thing like conceding the point in question. It is difficult to see what Montucla means we should infer in favour of Leibnitz, from his admission that, with Newion’s method of serzes, there were four integral calculi in existence before Leibnitz. DIFFERENTIAL AND INTEGRAL CALCULUS. Cuapter I. ON THE PROCESSES OF DIRECT DIFFERENTIATION, Tue rules by which quantities are differentiated must be studied until they are perfectly known, and easy to practise. Without demonstrating them, therefore, or even defining them, we prefer to place them by themselves, and to recommend the student to practise them while read- to any letter which may be named. If the expression do not contain that letter, the differentia] coefficient is 0; but if the expression contain the tndependent variable. The expression differentiated should be called the dependent vertable, but the phrase is not found necessary. Every expression which in any way contains x, or depends for its value upon the value of z, is called a function of a. In what follows, the independent variable will always be 2. 1. The differential coefficient of mx ism. Thus @ gives 1, 22 gives 2,42 gives 1, — x gives —1, —Q2,y gives —2, 2. The differential coefficient of 7” isma”"—, Thus x gives 1 2! oy = or 1. as before; a? gives 22; 2° gives 3a°; grt gives (p + 9) grHHl, yf gives a xt 3 © gives —3a* oy — 3 y+ The following are instances; over the columns of functions in question is written ifr, meaning the function of x ; over the column of differential coefficients is written f’x, which stands for the differential coefticient of Pe fa fix fa fia r | 5 xt x? or — | —Q2x-3 or _ —, x 100 99 =3 1 4 1 x 100 x* Yi) OF | 3a or Sa ‘ x 32 5 ; 1 ‘ 1 -e 2 I c*, £ are ye, BNO) ys wre ae or — a v D x2 i | ss ] b: 1 3 hy 7 x La or = ety -S 3 x*® 4 ! 1 ~ I 1 1 6 Be Or yf a) bes Sr — rT La-F 2/2 ery! 1 ; tor -- | ~];z- or —— pt OR eee T 36 DIFFERENTIAL AND INTEGRAL CALCULUS. 3. If log 2 to the base abe the function, the differential coefficient is Bi where M is the modulus of the system of logarithms having the base a, or the logarithm of et 2:7182818) in that system, which, when @ is 10, is ‘4342945. But in this subject, and, indeed, in all branches of pure analysis, the only system of logarithms employed is the one which has ¢ or 2°71828..--> for its base, the modulus of which is unity. Consequently, in this case, the differential coefficient of log x is —. 4. The differential coefficient of a” is a*loga (here logarithm of @ +s taken to the base €, which is always meant when no other base 1s specified). The differential coefficient of «7 is e” itself. The differential coefficient of (a + 6)’ is (a + b)” log (@ + b), &c. 5. The diff. co. of sin a is cos & Fo Ca aie ee COSL.- — sin 2 _...tane.. 1 + tan*z or ; cos *w 6. By sina, we mean the angle which has the sine 7; by cos‘, the angle which has 2 for its cosine, &c. Thus, if a= sinb b == sina, &e. ; , 1 The diff. co. of sin l,j J1— 2 ; 1 . of cosa 18 — J 1—a* , Ha) 1 OED EOL LAT ae IS ———.. 1+2 All angles are measured in the manner described in the ‘ Study of Ma- thematics*,’ namely, by the number of times which any arc subtend- ing the angle contains its radius, and an angle so expressed may be turned into seconds at the rate of 206264'8 seconds to a unit, and thence into degrees, minutes, and seconds. 4. To differentiate the sum or difference of any number of functions, differentiate each separately, and put the same signs between these diff. co. as are between the functions they spring from. Thus, The diff. co. of a” + a° + log a — sing — cose 1 oe a x ; 1 18 nix*-! + a? log a+ ee cos x — (— sin 2) + (- at 1 ; 1 or na"! + a log a + — — cose + snr — —|.. x r u Diff. co. of a* +c is nat ?+ Oorna™' 1 Rae 1 1 Oo BS ee OOS ak 1B Pe eg Ga AREER r A tia wf 1 — 2 s . oe }—zZ is o—1 or —l. ™ Library of Useful Knowledge, No. 90, pp. $4, $2, 116. ON THE PROCESSES OF DIRECT DIFFERENTIATION, 8. The diff. co. of a function o differentiating the function, an 37 f x multiplied by a constant * is formed b d then multiplying by the constant. Thus the diff. co. of ¢ log xis c x Ee or~— r e Diff. co. of ca" — cat + ac logx — (@ +c) tan—z ; ac at+e is new"! —¢2 gs lope opal so Cee X x 1 +22 : : } e 3c Diff. co. of p sin7'2 — gsing — —_ — — x ; Pp 3¢ 2c 18 it Ls Ce arth oe: Diff. co. of l+2e+ 38a2 +4 4 ho bat 6s is 2 +62 + 122° + 202° + 30 24 9. To differentiate the product of two fun the diff. co. of the other, and add the results ( ctions, multiply each by thus, with their proper signs) ; 1 the diff. co. of 2” log vis na" logx +a x Or na log x at, 2zsnmzis 1 x sing + © X cosvor sing + cosa. ; 1 1 1 TOTO eae > tan@ is —— , tang + — Serer sot ax = « cos’s ole crete Ai, (1—2’) (w+ 2°) is (O— 22) (x + aw) + (1-2) (1 +3 2°) or 1—5 24, 10. To differentiate a fraction, form the following fra ction— Den’ x (diff. co, num") — num!" x (diff. co, den") (Denominator)? ete BPE E AS Sos on 0 : xv Diff. co. of ao is Toe 24 : sd ee : = sing , cos X cos t— sin x (—sin x) ] php COS & se COs 7 hente Cos rv l+e i (l—z) (041) — (1+2) CeO near *e@esereacs 1, REO OB yc, Fas ETT (l—2z)2" 1 Ths x O~1 x (log a +2— ) foo aaa Rises 6 0 6 0 “vlog a 1s re (og 2)? or — x* (log a)*" Sa o ivsing . (1+ sinz) (—cos )—(1 — sin) (cos x) | 1+sin x (1 + sin x)? 2cosxr (1+sin 2)?" | 11. To differentiate the product of any number of funct _ ™ A constant, with respect to x, In a* , Gis a constant, if change in ions, multiply is a function which does not depend on x; thus, © produce no change in a, 38 DIFFERENTIAL AND INTEGRAL CALCULUS. the diff. co. of each function by the product of all the other functions, and add the results. Thus, the diff. co. of x sinz X cosax* e” is (re- member that ¢ does not change by differentiation ) rsin acosz €” + 7cos*@ &* — xsin’re® + sing cosre’. Some examples of these processes will be given at the end of this chapter ; but the best examples are those which the student forms for himself in the followmg manner. Take any function which can be differentiated by one rule, and throw it into another form, in which it requires another rule. Differentiate each form by its own rule, and see whether the results can be made to agree. For instance, all the following forms are the same function, 2°. 6 —4 3 x x 3 2 2 x — ae ed t«z)—- x grt ( ) ‘ and their diff. co. are, a. 6x — 2°. 32° gt(—4a4°)—a (a! ae ———__——— + —$<— ie? ; a a Del +ry+e(04+1)—24 show that the latter three of these forms are severally equal to the first, ae We have now differentiated—1. the fundamental forms a, a®, loga, sina, Cos@, tana, sin7'x, COS” wv, tan 2: 9 All functions of them made by the fundamental rules of addition, subtraction, multiplication, and division. It remains to point out how to differentiate more complicated functions of functions. Rule.—To differentiate with respect to x a function of v, where v is a function of 2, differentiate with respect to v, then differentiate v with respect to x, then multiply the two results together. This rule needs some elucidation, but, when understood, will be found the best help to the memory. If we have, for instance, the double function log sin x, the logarithm (not of x, but) of sina. We see that 1 ——? sin @ Yes; when differentiated with respect to (not x, but) sn », We have here made sin # stand in the place of x. To differentiate with respect to x, differentiate sin v with respect to 2, giving COs @, and multiply the / ; ; ; 1 ; . in the preceding rules logw gives —- Does log sing give preceding result by cos 2, giving cos 2, or cot x, the result re- sin © quired. fay fis fa fit Spee e i 1 ey age g sin £ : . COS & log’ a™ | — XX na or” — sin @ = a” Ls! log cos & ani x —sinz log a” 1 atl * log a “aie g api. og a or™ log } A - i ogtanz | -—, (1 + tan *r) sin 2° RES a X22 * Account for the simplicity of these results. ON THE PROCESSES OF DIRECT DIFFERENTIATION, 39 Par f'z ae S'x cosz* |/—sin2® x 2a sing! I -# : 1 ke Le sin. log z| cos. log x x — es : sin- | x sin) x I ee eo eX &* ‘ i 2 VJ ae e* E X cos wv (1 + 2°) 6 (1 + 27)? (04-27) oat eX nat" Qi — 2)" rl eaen (=p) (220°) 112 (a®+2°)" (20432°) n (sin & + cos r)"7! X (cos 7—sin x) ettee ettir x (0 ze b) Ske e-* X (— 1) Diff. co. of (a+ bu+c2°) is n(atbat ex*)"* (b + 2cz) l — 72 7 a ee Nal ia 1s he (sinz+cosx)” 5 x (0O— 22x) or —-—_—_— V1 —2 cos (cos x + sin v) is —sin(cos# + sin) x (—sin«-+cos v). We can now differentiate functions of functions of functions of v, &C. Suppose we have log sin a*"*, By the last rule we have, 1 orc: Met. co..of (sin a”) s1n a” Diff. co. of (log sin a*™*) is (sm @*""*) is cos a®"* x Diff. co. of (a‘"*) ( an?) is a®™* . log ax Diff. co. of (sin x) (sin x) is cos z 1 4 =a & cosa*™* X a™"* log-a'X cosa see Cldziein g 9") is — . S ) sin a*""* Diff. co. of (x + J 2x? = 1): is2(a+A 2-1) x Diff. co.of (2+ 2?—1) x: ES +721) = Diff. co. of e+ Diff. co. of Vx?—1 snes x Diff. co. of (2? — 1) =1+ a V22—1 Shy 22—O _ ce _&+nva*—1 oV7—] AV ?—1 Vv x? — 1 ceca Po1) EOD coof (2 -+ Y=)’ 2 ENR H DY) x? —] In the following symbolical recapitulation, every case of which the student must refer to its rule preceding, x, Wax, yx, mean different functions of x, and ¢’a, W'v, x’x, their differential coefficients with respect tor; also (baw)! means the differential coefficient of the product of @2 and wx; and so on. (2+ wa -— xz) = Pat wa— xe (cPr+epr—hys!=cHp'r+ewr—hy'a 40 DIFFERENTIAL AND INTEGRAL CALCULUS. pay woegp's—Ppxu pr rwery = rwzt+t Payer |= ee (or yay =924 “ —s (px wr xx)’ = prpry'r+ pe weyr + Pryryxe gat yayry =m (prety Get yD {(p2)"} =m (pxry"™ Pla fe Ga er pe U {log da}! = = {sin@ al! = cospx. dla {cosh ajl= —singa. Pz gx anes oa tan @ die ve= areca gin) Ox}! ce ee ee t cos? x { V1— (2) : ; o'r ni aia ox os oft SS tan oct oo ee, By ¢yar we mean the same function of yr, which ¢@ 1s of x: thus, - if pa be log x, PY x means log ya. By d' we always mean that func- tion of 2 which arises simply from differentiating Pw; thus. in p/w a, we mean that after Px has been differentiated, we substitute y instead of zr. We have then, foway ays. we (opxay Sox. WXe. xe The differential coefficient of the differential coefficient 1s called the second differential coefficient ; the differential coefficient of the second differential coefficient is called the third differential coefficient, and so on. The several differential coefficients of ¢ # are denoted by $’, ri gi"2, Pr, &C.5 and it is customary to use Roman numerals to express a number of accents, when they are too many to be conveniently written. Thus, the tenth differential coefficient is written gr. But when a letter represents a number of accents, itis customary to place st in brackets: thus, the nth differential coefficient of Pz is written pe. This process 1s called successive differentiation, and its easiest cases are as follows :— 1. Let fax be x"; then Pris n a, plz isn (n—1) 2", gx is n (n—1) (n—2) 2", dir is n(n—1) (n — 2) (r— 3) a"*, and so on. Inthe following, the function differentiated is the first of the line, and it is followed by its successive differential coefficients. eo eat, 4,327, 4.3.2.2, ci Tiss ba ee 0,0,0, &e. ae, Sat, 5.4a°, 5.4.3 a, 5.4.3.22, 5 4°3.2.1,0,0, &. 1 heehee hie 2.3 24 9.8 459 a emt on ee, ea Se Le = 5 ty Aaa , &C x fap 2 a ae a 1 n n(n+1) n(w+1) (+2) n(n+1) (n+2) (+3) & a” 2 grt ? grt ye yrs ol a" +3 ? G U rat — 5 Z 1 8 a ig ar, ¥v ’ 3 x0 ’ yea: ae ee —3 ee =x Bi &e. 3 ely fers 12 1 3 ped. 3 5 soe: 7: a ee eg ae} a ee F &e. m m-7 m—-2n m—3n on, se mm—n = mm—n M2 > =—-2@"> ——2 > ———— —-— # » &e. rm n n n n n ON THE PROCESSES OF DIRECT DIFFERENTIATION, 4} aa" gt log a, a? (log a)*, g? (log a)®, a? (log a)*, &c. Oi ser, &*, €*, &*, nk &c. 1 1 4. log x pe ge = (for the rest see last page.) 5. sin ar cos?, —sinz, —cosyx sin x cos zr, &c. > 3 3 >] 6. cosz, —sin 2, —cosa, Si.) Gos 2, ae vt, &e. Memorandum.—Observe that in every case a function of x + ¢ does not require any second Process in differentiation, for instance, Diff. co. of sin (© +c) = cos (x + ©) X Diff co. of (@ + c). But the differential coefficient of x + ¢ jg ] + 0or 1. We shall now give some examples * for practice, Let ou ed or (l1— x) 3 p'r = me At? : 7 ee (1 — x*)? v uy V1 — 22 dit CO. of V1 — 23 I a pxr= one A ia l—vz (1—.2?)® ee Wd —2 glo—v!} + @ diff. coof 4/1 — 2— JI —2 diff. co. of Jl+a et +2 I+e@ ] Vita ——== X (0—] SUN Tig 7 aoe ides gion sual 2Vi+ez l+ux bes GO +2) += 2) 1 ~ 2042) geht iba aL (1 + 2) Vi—@? Reductions, such as are here to be made, and Success in which depends on the expertness of the student in common algebra, form the greater part of the difficulty of the succeeding examples, Vee a? = feap atic , oo a adie @ a Ne b6+2cr =— AV, gh GY ci cea eames a Sao Dex Net bx oa 2Jatbrtes fi Py 5 JE seb AE la ae cm jhe 8 a+ bla (a’ + b/x)? ie ost 1 ] vise 5S I Tae RGD r= be (+a) * There are two works in English, which are express collections of examples for the learner, 1], « Collection of Examples of the Differential and Integral Calculus? By George Peacock, A.M., &e., Cambridge, 1820, This work is now out of print and Scarce, and we have been frequently indebted to it. 2. ‘A Digested Series of Examples,’ &c, By John Hind, M.A., &c. Deighton, Cambridge, and Fellowes, London, 1832, This work would be very useful to the student who wishes for more examples than one work can give, 42 DIFFERENTIAL AND INTEGRAL CALCULUS. Rure.— When two functions differ only in the sign of x, the diff. co. of one may be found from that of the other by changing the sign of 2, and then changing the sign of the whole. The last is an example. se nia} eg oa eae ey eee ba) se eer ie CES Bhs Abs 5 1h 9 eee | ot a r+l (is aes ) ev+oetl (a + #2 +1) b x The process may sometimes be rendered less embarrassing by the use of logarithms, as follows. Suppose we wish to differentiate Badly aah ~r=FRs mT Ch); where all the capital letters are functions of x, and P’ Q’, &c. are their differential coefficients. Take the logarithms of both sides (and let » stand for log) 1 YPa=rNP +2XQ—XR—AS ER Pe en +r~W —AZ); differentiate both sides (it being true, as hereafter noticed, that the _ differential coefficients of equal functions are equal), and we have beat es ay Sra RN ZI! os ox P Quek S n YW T]he. ie whence we get 9’ by multiplying together (1) and (2.) The student should first try the following example by himself, and when he has completed his result, may consult the following process. ~~ b+a ak x Process. Apaer(a+2)—ACO + 2) +ir@’ + v)-LrA(?+2) we A ose gx atv b+2 ed ies acne 2 @+2° om eae oot a Cae —~ (a+ a) (b+ 2) (b? + a*) (a? + 2") b—a (a? — b*) x = Gin Ota) Gt CF”) -11(—a) ou + e%)— Gt b) (a 4%) dah “1 Gta Ota @tr) C+”) (a? + 2°) (b? ++ x") — ag? b? + (a? + 6°) 2 + at (a+b) (at 2) (b+ 2) 2 = (at byabart (a+ ba +(a+b) When the numerator of the preceding fraction is ON THE PROCESSES OF DIRECT DIFFERENTIATION, 43 a’ b? + (a? + B?—at b?) xe + xt— (a4 b) (aba + 2°) or’ a*b*—2aba*+ xt—(q + 6) (aba + 2°), or (ab—2*)? — (a+b) (abe +x’), —_——__. He = b— a) 242 VUE Gd —A— (a4 5) abate a +e Veta (+9 (b+2) (+a (Ft2°) nt \ SEY ‘ 8 ne ayere x) a ee), (9 + x)’ (a? + a*)® (62-4 2?)® In the following list, each function is followed by its differential coef- ficient. G +z log —_—— j ee Sy — — log (log x) , e” (2? — Qe 2), sin @ i te x” (log x)", va (aloga + n) a" (log x)" (m log w + 2) e* &* @ (1 + 2)? “ig sin 2 x sin(nm—1)a& L—SiNne cosa, Zsin? ae =a SN sin” x sin”t x sin (sinz), cos sinz. cos 3 V1 a> UE CATE ih cell a+ bcosx .a+b cosa’ cos' (42°— 3 x), — cos~! _ Having thus laid down the mere rules of differentiation, we proceed | to investigate and apply these rules, 44 Cuapter II. ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS AND DIFFERENTIATION. Wuen any function of xis given, we can determine by common algebra the value which the function receives when x receives any given value, say a, and also the change of value which takes place when x becomes a + h, by which we merely mean, when we pass from the consideration of the function of a to that of the function of a + h. Thus, “let 2 = 4,9 followed in the same problem by “let 2 =a + h,? does not mean that we make these suppositions both at once, but that we consider 2 as changing its value, or ourselves as changing the value we attribute to 2. Of course, the consequences of the two suppositions may exhibit any sort of difference. ' When we consider # as having some assigned and specific value a, the function @ 2 may exhibit two distinct species of phenomena. 1. It may have a finite and calculable value, positive or negative. Thus, 2-+ a” is beyond all question 6 when the value of vis 2; and — 1 when @ is— 4- 2. It may exhibit one of the varieties of form which arises out of our supposition being followed by an absence of all magnitude, or 0, in @ place where the general form of the function would lead us to suppose there is some number or fraction to be operated on or with. Such forms are, 1 1 $y 10's Go Mba) ae For instance, in the function (1 — x) °-, we see that the supposition of « — 2 offers no difficulty, for the function then becomes ( —1)~* or —1; but when 2 = 1 we have no means of operation left, except such as are implied in the symbol 0°, which offers no ideas of numerical value. With regard to such cases, it may or may not be proper to say the function has existence and value: but we do not enter into that ques- tion, We examine, in such a case, not what (1 — x)'-* becomes when a= 1, but we ask to what does it approach without limit when @ approaches without limit to 1. If we can prove, as we may hereafte1 do, that the preceding function also approaches without limit to 1 wher x approaches without limit to 1, we may then abbreviate the preceding proposition into these words “ when x is 1, (1—)' is also 1 :”” but we use the preceding sentence in no other signification. Therefore we havi the following definition. Derinition.—The function is said to have the value A when @ ha the value a, either when the common arithmetical sense of these phrase applies, or when by making x sufficiently near to a, we can make th function as near as we please to A. In the first case A is simply calle a value, or an ordinary value, of the function: in the second case Al called a singular value. Postulate 1.—If $a be an ordinary value of @z, then A can alway be taken so small that no singular value shall le between pa an ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS, 45 @ (a+ h), that is, no singular value shal] correspond to any value of x between 2 = a andx—q +h. . The truth of this postulate is matter of observation. We always find singular values separated by an infinite number of ordinary values. If we lay down all the possible values of 2 on a straight line, measuring then lay down the values of the function upon lines perpendicular to the values of x, placing each value of the function on the fine drawn through the variable extremity of the linear value of z, and measuring it above or below the axis of z, according as it is positive or negative, we have the well-known method of representing a function by means of a curve, which is the foundation of the application of algebra to geometry, as given by Des Cartes. We have drawn the representation of a function below, so as to exhibit every variety of singular value, and more than the skill of the most practised algebraist would at present be able to find a function for. The stars mark the singular values, or rather the places at which-there may possibly be a singular value ; all other values are ordinary, however near the singular values they may approach in posi- ‘tion. And we see that, however nearly a, the value of 2, may approach to b the value of x at one of the singular points, it must be possible to take a+ h lying between a and 8. Postulate 2.—If ga be any finite value of @ a, it is always possible 0 take h so small, that d (a + h) shall. be as near to Qa as we please, ind that @ x shall remain finite from r=ator=q + h, and always ie between @ a and (a + h) in magnitude. | _ This again is a part of our experience of algebraical functions. It 18 fenerally assumed under the name of the daw of continuity. The latter art of the postulate may be true of the whole extent of some functions: hus, however great h may be, a? perpetually increases between a? and a+ h)?*. _ It is possible to imagine a function which does not observe this law, of singular values, find the instance, in the following EF is discontinuous at B Q /R 46 DIFFERENTIAL AND INTEGRAL CALCULUS. and D. But we have no means of expressing such a function in common algebra. We may call the law expressed in this postulate the law of continuity of value, to distinguish it from that of the next postulate ; and we may say that functions, which do not obey this law, if any, are discontinuous in value. Postulate 3.—lf any function follow one law for every value of z between v=aandr= 4 + h, however small h may be, it follows the same law throughout : that is, the curves of no two algebraical func- tions can entirely coincide with each other, for any arc, however small. If pax be a” for every value of a between & and a + h, however small h may be, it is a” for every other value of z This we may call the law of continuity of form, or permanence of form. Exceptions to this law may be represented, but cannot yet be alge- praically formed. As in MN P QR, we may conceive a function which is represented by an arc of a circle joined to one of a parabola, which is itself joined to a part of a straight line, and so on. Such a function would be called discontinuous in form, and though not now exhibited algebraically, may actually occur in practice. Suppose, for instance, @ spring of the form MNPQR fxed at the end M, and disturbed at the other end. The number of ‘ts vibrations per second might become a subject of inquiry. . Let ox be a function, continuous in form and value, which we always mean unless when the contrary is expressed. Let us take two consecutive values of 2, namely a anda+h; but instead of supposing # to be a, and then to become 4 -r h at once, let it. pass through m steps altogether, becoming successively, a, a+9, a+ DOM OR Ss fh (n — 1) 9; a+née: that is, let 29 be h, so that by increasing the number of swbaltern incre- ments by which a becomes @ + h, we may diminish each increment ) without limit. The corresponding values of the function are 4, o(a+ 9), PD (biG). a ser Ue to @ (a+ 78) or (a+ hn). The several increments * of the values of the function are then— $ (a +0) — 99, 9 (a420)—9 (a + 6). .¢ (atn0)—$9 (a+n—18), Let ¢a be called P,, let ¢ (a + 0) be called P,, &c. up to ¢ (@+ ne which is called P,. Consequently the increments of the function art P,— Po P,4Pin Pact toa 30: P= Pann number) the sun of which is P, — Py or ¢ (4 + h)—ga. We have then, (P,— Po) + (P.— P,) se + (P,,— Pa) = (a+ h) —pa ) Cer) A aCe aie ago ok 0 —- 260 > =a (P, — Py) + (PsP) + OF Sie REF 4+ (P,—-P,-) _? @+h)—94 7 Gisiede WL Yee 0 ‘a — so that (h and a being given) the fraction made by summing the numé rators of P, Pra Py P, ES Pi ry Sr r) 6 ee ew Ne Soh) oie for the numerator, and the denominators for a denominator, 1s equal the same quantity whatever may be the value of 7. * If the function decrease instead of increasing, we must either use the wo decrement, OY apply the term increment to both positive and negative quantities, negative increment being a decrement. We take the latter alternative. ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS, 47 If n increase without limit, @ diminishes without limit, and so do al] the numerators of the fractions in question, which last therefore all approach the singular form ¢ and we have now to ascertain whether the limits of all or any must be finite, or whether they may severally increase without limit or diminish without limit. Yow (we refer the student to the lemma fol- lowing this) they cannot all increase without limit or all diminish hae! as oe ean Cs, Os, ent without limit: for it is shown that among the fractions — eee &c., there must always be some which are algebraically greater, and some which GO - gf ee at are algebraically less (some means one at least) than —__.. bd _—— . the only possible case then, unless there be finite limits among them, 1s that some increase without limit, and all the rest either diminish without limit, or increase negatively without limit. P,—Pp P,— Pp ee 0 ) 1s said QE 28 a oid seis ks Now, whatever these quantities Q,,Q,.... may be, a law of con- tinuity must exist among them, for they may all be made from the first, by changing a into a + @ time after time. Thus, 6) — 1 6) — Q. or ‘eaenetd imi, a is made from Q, or BEL a by changing a into a +0. And we have reduced the question to this alternative : either there are finite limits, or some Increase without limit and the rest diminish without limit: if the latter, we shall have two contiguous fractions, one of Which is as small as we please, and the other as great as we please : or we shall find, fora sufficiently great value of n, somewhere or other in the series Q, Qy. .-. a phenomenon of this sort, Q, smaller, say than ‘00001 or anything else we may name, and «+i greater than a million, or any other number we may name. Or Q,. will be positive, and Quit: negative, both numerically as great as we please. This cannot be true of ordinary and calculable values of the function, and can only be true when Q,. is the fraction which js near to some singular value of the function, or when q + k®@ is near to a +] Corresponding to a singular value P(a+l),atl lying between a and a@+h. Butas h may be at the outset as small as we please, let us avoid this by taking a new value of Ah, namely h’, so that a + h’ is less than a +- ]. Repeat the whole process and argument witha and q +h’, by the same reasoning it will appear that if we refuse to admit finite imits to some of the set Q, Q..... where n6 is now h’, we are driven to suppose another singular value of the function corresponding to a+ k’, lying between a and a+h’. Avoid this again by reasoning in the Same way on a and a+h" where is less than k” ; we shall be obliged to admit another singular value, and so on. Hither, then, there are finite limits to some of the set contained in the general expression P(a+ k0) — P(a+ k—18) TRS tess, or the function admits of an infinite number of singular points between @=aandx—a-+h: that 18, is not according to the postulate. There- fore, we have the following theorem. ? x being any function of v,andaandath any consecutive values 48 DIFFERENTIAL AND INTEGRAL CALCULUS. of x, where h may be given as small as we please, there must be finite p(r+9)— Pr. A. Rox AR ad: Fa ee in which 6 diminishes Wl! out limit, for some values of x between + = @ and math. The limit of isis —* is called the differential coefficient of limits to the fraction ¢ x with respect to x, and the theorem just proved is as follows :—Every function either has a finite differential coefficient when x has the specific value a, or when it has a value a + & where k may be as small as we please. There are points in the preceding demonstration which lie open to certain objections, depending upon the way in which the terms of the postulates are understood. The student may, if he pleases, consider it only as giving a very high degree of probability to the fact stated, since we shall presently demonstrate of all classes of functions separately, that the preceding fraction has a finite limit for all values of x, with the exception of a limited and assignable number of values for each func- tion. ; : aa! Lemma referred to in the preceding demonstration. it Pw? Br be aseries of fractions the numerators of which are of either sign, avd the ! denominators‘all of the same sign, then SE a must lie alge- braically between the greatest positive, and the numerically greatest negative, of the preceding fractions. To take a case, suppose the fractions to be ie Wii at ren (3 2 4 3 adhe which are arranged in algebraical order, the algebraical greatest being first, and the least * of the same kind last. Then we have BS or 3= 5 2 | Sa ges ae i <50 1<>.4 | — or = 2 a 3 3 or 1>=z.4 3 chor —5<5-? 3S 3>—5 3 Hence, by addition 3 —5 (3+1—2—5)<52+44342), (3+1-2-—5)> 32 +443+2) or 3+1—2—5 3 3 i527 Pies 9444312 2 2144342 v.46 * See ‘Study of Mathematics,’ p. 49. To avoid confusion, it would be desirable to talk of the smadlest of quantities, when we speak of arithmetical magnitude, am of the /east when we speak of algebraical order ; but the necessity for the distine tion seldom occurs, ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS. 49 and any other case may be treated in the same way. We have adopted an instance, to keep the ideas of the student fixed upon the algebraical relation of greater and less, which is necessary to the proposition. If the denominators were all negative, the same thing might be deduced : thus, if the set were 8 ] —2 wens Farlahy jay | ae mal since if p lies between q and 7, it follows that — p lies between — g and — r, then, since 3+1—2—5., 3 =i 2444945 lies between — the greater and — the les Ba ae 3 i Be ch hen 3+1—2—5 3 —5 " Sey pe) FS tet ty eT ORE eg the Jess and =o the greater. The object then of our first investigations must be to determine the : 0) — Se : Shed Rou himit of ECR De ke when @ diminishes without limit, in every possible case; which we shall sce amounts to substantiating the rules given in Chapter I. But first we must acquire some more precise idea of the meaning of the preceding. We see that x is first supposed to have some specific value a, which is changed intoa + 6. It is usual to write x itself for its first value, and to call 6 the increment of tae Tet A x be the abbreviation of the words difference of x, or increment of x, we see then that 6 is an arbitrarily assigned value of Az. And ~ (a+ 9) — dais the increment or difference of ¢ x, for it represents the alteration of ¢ x made by changing x from aintoa +6. But it is not arbitrarily assigned ; for dx being a given function, and a and a +0 given values to be used, (a + 6) — dais given with a and 6. Hence Ag¢ex represents (a + 6)— a, or if u=dax, we have Au = P (a+ 6) — $a, or the differential coefficient is the limit of the fraction = which we cannot ascertain from this form, because when A x= 0 Ls that is, when the value of the independent variable is not altered, Au = 0, or the value of the function is not altered. For instance, let GNeats 2 1 the function in question be iii We have then sig Nd OD ve 2 e ae ashe wii x(x + 0) Au ] OM aries yey we use A x on one side, and @ on the other, which must appear a super- fluity of notation, because we thereby, on the left, preserve a better repre- sentation to the eye of the process which is going forward, while we have a more convenient working symbol on the other side. The limit of the preceding fraction is easily ascertained from the ‘ : l l : second side of the equation to be — —— or — 7 For when no sin- LD ; gular form is produced by making @ = 0, the latter gives the way to ascertain the limit towards which we approach by diminishing 6 with. K 50 DIFFERENTIAL AND INTEGRAL CALCULUS. out limit. But this supposition, namely 6 = 0, is merely a step of the work, and not a necessary part of the reasoning. TurorEmM—lIf p, q, &c. be the limits of P, Q, &c. to which they approach when @ diminishes without limit, and if none of the set P,Q; &c. exhibit singular forms when 6 = 0, then the limit of any function is found by substituting instead of P, Q, &c. their limits p,q, &c. pro- vided no singular form be thereby obtained. Let us take as an instance - the limit of which we assert to be Ee To prove this, observe that Pop. Pq Gp a ~ th Lond let P = 0 + oF Q Sos Q 4g Qq a whence it follows that: and « diminish without limit at the same time as 0, This gives Pp ct Oa es Croan Q 4 (q+")¢ ee att the last fraction has a numerator which diminishes without limit with 6, and a denominator which continually approaches to the finite quan- tity q®» This fraction, therefore, diminishes without limit, that is, Q approaches without limit to p. or the latter is the limit of the former. hes Au, du ; It is usual to represent the limit of ie by aon which the student should now read the remarks in pp. 13—15, of the ‘ Elementary Illustra- tions.’ This latter fraction does not mean a quantity du divided by a quantity da, nor are its parts to be separately considered in the theory of limits. [But in that of Leibnitz, pp. 21, 29, it is said that if dx be an infinitely small increment given to v, du is the corresponding infi- nitely small increment thereby given to the value of w, and the diffe- rential coefficient is the ratio of these infinitely small increments. Thus it would be allowable to say, that if 1 du 1 1 us Esai 7a oF dus a3 dx. | When «# becomes x + Ax, we suppose that uw becomes u + Au, P be- comes P + AP, &. Let us now suppose that P, Q, and R being functions of x, and Ca constant. Let P,Q, and R have finite and determinable differential coefficients. This relation, being required to remain true for all values of «, exists when z is changed into x + Az, and gives u+Au=(P+4P)+(Q+4Q)—(R+AR) +0C, the constant not being affected by a change in the value of x. Subtract the preceding, which gives Au_AP AQ AR An Aw Az Ax’ the A « diminish without limit, in which case the fractions in the last Au=AP+AQ—AR, ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS, 51 : ns du equation severally approach without limit to what we represent by Th’ v ad a and a which give dx ; di dz ; a du dP dQ dR We see that the constant C does not appear in the result. If it had been a function of x, we should have found re added to the preceding. But at present, if we suppose any other term in the last equation, it can only be +0. It may be said then, that when C is a constant, i is 0. The proposition to which this may be considered as a sort of 2 limiting theorem is the following. If a function increase slowly, its differential coefficient is small: the less it increases, for a given increase of a, the smaller is the differential coefficient. Finally, if it do not increase at all when « increases, the differential coefficient is nothing. cee ak =o EG) u+Au=(P+ AP) (Q+ AQ)=PQ+ PAQ+QAP+AP.AQ or as before, Au = PAQ + QAP +APAQ Au AQ AP AP Barr ein Cet ee Ph Rs the last term of the preceding consists of one factor which approaches a finite limit, and another, AQ, which diminishes without limit. All the increments Au, AP, &c. diminish without limit with Aa, though their AP 7 Pa ratios do not. Consequently, the term ak A Q itself diminishes with- out limit with Ax, and we have du dQ a dat dp tb Qa. 37, Rule 9.) Letu= PQR=(PQ)R. . : du _ dR d (PQ) Then, as just AEG prac t NE8 segs +R ze dR Ee are ey dR dQ, dP PQS +R(Pe Qe = PQ +Pr@s rae. And by carrying on this process, we may obtain the following general tule : to differentiate the product of n quantities, differentiate each and multiply by all the rest. If wu be the product of m functions PQR... then the product of all but P is Z , and so on; whence we have ’ du BBP Qoy: wwidR dz P dz Q dz R dz Metts Satie Rarics ks EQ 1d, uw de ~ Pedy "9 Q.'dx™ R’ dx eb +. U. OF ILL. LIB. §2 DIFFERENTIAL AND INTEGRAL CALCULUS. This remarkable relation is intimately connected with the theory of logarithms. If \ P mean the logarithm of P, &c., and if u = PQOReS: it follows that d(au) d(AP) _ dQQ) Au =AP+AQHTAR+.-- + 7G 75 Sg ee and it will afterwards be shown that d(ku) _ 1 du dQP)_ 1dP ie de u dx dg. Pda eee Pp gre Sie cal y ara Sle i@ak MOnedQ gAP_ pag Ay QAP = PAQ at 6 yn Aiocraien Aik Pe ae nO RI Ax QV+QAQ taking the limit, and remembering that Q A Q diminishes without limit, we have | dP dQ By apie lal et Maino STEN ] : oe: a (p. 37, Rule 10.) We shall now proceed to find the differential coefficients of the fun- damental forms. But first we must premise the following consideration. If w be a given function of 2, then w is also a given function of w, though not always an assignable function. 1 For instance, if w= 2°, then « = vu sifu = a then.c =u", ft bs 2 Te ait: bes eee ee 2a : we see then that a function may have more values than one for the same value of the variable, and we know from algebra that such functions will arise from the inversion of any direct operation, except only addition Thus, if we consider the equation u= 2*+ x, and if the question be, given 2 to find u, we have but one value of u to every value of x: but if it be, given w to find z, we have to solve an equation of the second degree, with two values. In the differential calculus we must always distinguish these two values as if they arose from different func- tions; thus, there are two differential coefficients, one to each value. With this restriction we apply the rules separately to every different value of an inverse fnnction. Thus, when we say if ua, then let x= Ww, we mean, let % xu be one or other of the values of z obtained from the first equation; but whichever it may be, do not use one in one part of the question, and another in another. It is usual (or rather it is becoming usual) to let ¢~' w stand for the value of # obtained from u== Px; or to say that, in such a case, cr = Po! wu. If, when v7 = a, u=b, we are at liberty to say that when w= b, x == a is one of the values of x corresponding to that value of uw. If therefore, w= gx makes r= Yu a necessary consequence, and if b= ¢a be true, then a = Wb must be true, not must be the only true consequence. If then the value of w corresponding to a+ Aa be ( + Ab, orifb + Ab=o (a+ Aa), andifu= de makessr= bu ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS, 58 a necessary consequence, it then follows that a+ Aa= (6+ Ab) is atruth. That is, we may consider Aa and Ad as simultaneous incre- ments of wu and 2, without asking by which of the two equations either is derived from the other. And the same of Au and A «, when we drop the reference to specific values of w and x, which we have used for dis- tinctness. If we use the first equation uw = ¢ 2, we obtain cake = Bee Sayre ie and the limit is a function of a. a Aw If we use the second equation # = wu, we obtain ae —_ Hahah AU) salt 4, and the limit is a function of yu. Au Au Calling these limits @’z and Ww/u, as in the first chapter, and remember- ing, that for all values of Au and Az we have a eae 8 4 eater ped =o Av Au as in p.22. That is, d’x x ¥'u = 1, which will be reduced to an identical equation 1 = 1 by the substitution of a instead of U, as in the following example. a Let wor Pv=— + 6, then # or %¥u = Ds A ae A = 1, we see that limit of —” x limit of —" = 1G Ar Au a u—b Aw 1 a @ \ - a taco t? (E+5) oe, v(v+Az)’ the limit of which is — or d/z = — %, av av i { a be aN t a Au AulutAu—b u— bd. . (u—b) (u + Au —by’ the limit of which is we pit Wu = ne ; a a , then will — ar: x © a ye not universally, but only when the (throughout this process) permanent : a : f relation «= — + 6b is also satisfied. And we see that the latter rela- v ae a tion gives u—b = —- and therefore @ a a ai a y a song ‘, ke , ae (u— 6b)? — gq? (aar)> ©), ; Ria » du ?’e obtained from u= 2 has been signified by ie dx yu obtained from «= yw will be signified by — du dx dx 1 therefore a x Fai 1 or Vester dx We have illustrated this at length in order that the student may not 54 DIFFERENTIAL AND INTEGRAL CALCULUS. think he sees it too soon, which he will always do, because there is a b between wi and ee. a resemblance to — and — of common algebra, dx du b a which leads him to think that the preceding equation must be as true as EES 42 1, and for the same reason. This is the dvsadvantage of the a b ie: , du. notation, but it ceases to be such when it 1s understood that — is not dx a symbol in which we can separately speak of du and dz, but an inde- composible symbol, the parts of which, though they serve to remind us of the manner in which its value is obtained, have no separate meaning in connexion with that value. du derived from u=@2, arbitrarily stands himit or ? («+ Az) pe Eads dz for Aw dx implies a consequence of the preceding init of v(utAu)—du du namely « = yu, and stands for Au ; Cover the left side of the preceding with the hand, and see in what degree it is evident from algebra that the product of the two limits spe- cified at length is 1; for that degree of evidence, and no more, should . dx attach itself in the mind of the learner to the equation 7 x 5 dx independently of the demonstration. In the same manner ay which seems most evidently = 1, must not be received as such without the following. If wu = a for all values of x, and if increasing x by Ax makes u increase by Aw, we have u+Au=a2-+ Az, and, subtracting the Au ; former eqiation, Aw = Aor er 1, which being true, however small ma A x is taken, has the limit 1, and now * we may say that a (which is de) dz Let us now suppose that u~ is a function of y (Py) where y is a func- tion of # (a). We have then u = py from which we can find limit of = or 7 y = Ww 2 from which we can find limit of Ay or gy. : Au wid? + . ad but we have no equation from whence to find = though we can make x * In the beginning of every science comes the difficulty of understanding why some apparently self-evident things are proved, and others not. We cannot here enter into this question, but we recommend the student to inquire, if he has never thought of it, why Euclid shows how to cut off a part equal to the less from the greater of two straight lines, when he does not prove that a straight line can be drawn. We have hardly thought it necessary to prove that if two functions be always equal, their differential coefficients are equal. It is evident their increments must be the same, the ratio of these increments to that of the independent variable the same; and variable ratios which are always equal must have the same limit. ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS, 55 one by substituting the value of y in wu, giving u= (we). Yet, if x become # + Aa, y will receive a certain increment A Y, in consequence of which w will receive an increment A yu. And, from common algebra, Sle AY h 56 Bo Ay Ay ence, p. 50, limit 2 = tim. A” x tim AY op MY = du, ay Ax Ay Ase) dices Open ead which also seems evident from algebra, and the preceding remarks apply. In fact, retaining the notation of Chapter I., and supposing that % (¥ x) is xa, this equation might have heen deduced in this form xe = $'y X W'x, which does not appear self-evident, and is only true under two implied equations, namely, x v7 = $(% «) and y = we. Thus, ifu= y° y = 2° giving u = 2°, it will be proved that i = 37° “ = 22, and also that 2 = 62’, each equation in the lower line following from one in the upper, in- dependently of the others. But from the connexion of those in the first line follows this connexion between those in the second, namely, 6° = 3y° x 22, which is evidently true ieee a In the same way we might prove, if of the variables u, v, W, Y, Xx, each is a function of the following, that du du dv du du dv dw du _ du dv dw dy eee eee — dw” dv dw dy dv dw dy da dv dw dy dx d where —, ——, —, 4 are directly obtained from the supposition: ae but Tp its ies that u has been made a function of w, which can only Ww be by substituting in w = dv, the value of v from v = YW w, and so on. Let us suppose u = x" (n being a whole number ; observe that by n and m we always mean whole numbers, unless otherwise specified) that is, let w be the product of x functions 2, ®,@,...... (m). Then by the formula in page 51, we have ab ee dry dt — = ih. in all a pe re eat, (n terms in all) x Len — nx"~', (p. 35, part of Rule 2.) Now, let w= x" or u” = 2", Let p= u", where wu is a function of x. dp dp du , du ; — = = | — the last, but also a”, herefore Pag Seereali i Ur by the last, but p is also d whence ee = mz", Therefore dz ee ae ean. Pee: a _m ™_, /p. 35, Rule nu age es ieee) ae. Vs im park m mn—m wm—1 m == — —i a x Aw Aw Arce ave ; tg é—I and the question is now reduced to finding what limit has i when 0 ——= CE 0 6 diminishes without limit, the singular form being (6 = 0) I-11 0 6 cannot appear in a function which (when a proper form is given to it) 0 : oo ane ; , or 5° as in other cases. This limit must be some function of a, for : : oh a*—1 , is found by making @=0. For the same reason, the limit of ——— is K the same function of a, if k diminish without limit. We obtain, there- fore, the same limit if « be a function of 9, provided both diminish with- out limit together. Let « = 00, 6 being a constant. Then we have 6] ie é— ] limit @ iis limit — Ci ee b\4 But © 1a = ; : (7) I which second factor only differs from ag—] in having a substituted for a, and therefore its limit is the same —l. el hs is of a. Let the limit of this latter ; aé function of a’, which that of be fa, then we have 1 (a*)¢— 1 (a’)s — Ime SV are pc Se = 5 lim eee pf (@); consequently (1) the function fa is such that if (at) =f (a) orf (a) = bf and a and b are independent of each other. If a’ be q, we have log gq = b loga, whatever a base of the logarithms may be. This gives ha) Sd tia FE 14a) = a0 " Tog q qd — Jog « a and q and a may have any different values we please, for though g=a, yet since 6 may be what we please, it may be so taken (exactly or with any degree of approximation we please) as to give q any other value. limit me a ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS, 57 . ‘ toy as ; Therefore fa is such a function as to give x this property, that it remains the same if any other quantity q be substituted for g, That is, or ; oo 7 1S & constant independent of a, which call C. log a “. fa =C log a; but the equation Au dag x limit a**?— ] ives —— = a” x limi Ax Ar 8 dx Ar du ; Re Xeane, = C loga x a*, where all that is known of C is, that it is inde- L pendent of a It must clearly depend on the base of the logarithms chosen, and it will afterwards be shown that when the logarithms are Naperian, then C =1. But this point must be reserved till the next chapter. Remember, that for the present, all differentiations which contain a” are not finally demonstrated until it shall have been shown ‘ du : that if w= a", [Pn Nap. loga x a’; all we know is that, taking these logarithms, it must be of the form C Nap. loga x a? where C is not determined, but assumed, for the present, to be =]. From this it will] follow that if @—e— 2°7182818....the base of ‘ du Napier’s logarithms, or if loge= 1, and if uw = é*, PP teat a 8 2 (p. 36, Rule 4.) Let u = log x to the basea or x — a" h dx $ t Ode X loga=cloga where M is the modulus * of the system of logarithms having a for its base. Hence, since loge = 1, (p. 36, Rule 3.) u 1 if w=] Sere nee Nt lf u og & . - (Read here the proof that the limit of the ratio of —s is 1 when @ di- | Minishes without limit, given in the ‘Elementary Illustrations,’ &c, p- «4, : : ‘ é 0 tet inte Ai, ain (t+0)—sinz eos (2 + - }sm- 9. SS a 2 2 7 = 9 Az A sikacraat b Reet ] eG sin — =cos (4+ 4 * —5—,» whose limit is cos xX 1. e * By a well-known relation, log x (to base y) X logy (to basex) = 1. Hence = logs (basea) = Modulus of system whose base is a. ] log a (base «) 58 DIFFERENTIAL AND INTEGRAL CALCULUS. du : Hence wu = sinw gives ae =cosz, (p. 36, Rule 5, n part.) 6 6 —2sin (cx+— } sin = Au cos (v7+9)— cos 2 ( 5) 2 Let u = cosz op ns Penn Oe Eis Ea Ax 0 ae sin— whose limit is —sinz x 1. ; 6 ——sin (2 +5) 7 2 d ; Hence w= cosw gives — = -—sina, (p. 36, Rule 5, in part.) Au _ tan (x4+6)—tanv sin 0 “ua tant. > —_—— = _—________— tet Ax 0 6 cos (4 + 9) cos. sing sinb sin (a—)d) (Remember tan a— tan. ate oe oe cosa. cosb cosacosb 1 sin 0 ey x — whose limit 1s x 1, cos @.COs x ~ cos (+9) cos z ; u 1 ; or us tanw gives 7 = oo a 1 + tan 2x. (p. 36, Rule 5, in part.) Let wu = sin“ or t= sin wu du 1 1 1 ] ; ae =< dx meee Biehl aa fare OL, a ay aie’ (p. 36, Rule 6, in part.) du Let usx= cos") x or # = cosu du 1 l 1 t oT i» = = this » (p. 36, Rule 6, in part.) Fa by —WP du Let u=tan'a2 or x= tanu du WA 1 1 ; aa z= Tans ina? (p. 36, Rule 6, in part.) du _ We have now differentiated the component parts of the common func- tions of algebra, including trigonometry. It only remains to show how to differentiate the compounds of these elements. | Let u = (px)”: if then we denote Oz by y, we have u= y™, y=O, du dy —_— = m—1 a f 77 my ip Q'x, du _ dudy mat dy 1 a | (p. 55) aT bide ie 5 = m(poar)""' Px. d Let wu = (cosr—2z)” =m (cos 2—x)"-'(—sina—1), du dy ik eae ah? ap a’ log a Pos a® loga o'a, ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS, 59 =] a TCS er ae Pa Le GGT de y dx o¢ , ‘is du tp dy | } u=sny y=der te cosy = cosP2r. P/zx, &. u—=sin“y yea ee Vet a os nde ees &e. BET i. 29a dx VI—(o2y The following cases deserve special attention :- du da u=y -y=¢e py a 202. $’zr urNy Y= Pr —_— = —- = gal! si du ldy x Se ed ea du 1 a BON, ae = Teme % ~ 28 Va a du y dy = SE 25 — —_ = — u=Va— 7 y=da, d V@ayi de ‘d ] a-—wZ MOND reemty ote ee yy hee (2a—22) = — dx 2/oa7-3 2ar— x? Phe following equations are the fundamental relations of trigonome- try im another form :— sin-' z, or the angle which has x for its sine, is J 1 — x? 1 i ae z atl cos! /] — 72 tan =——, Cot » sec™' 7=—=——=_,, cosec™! —; i 3” J 1—2 x l—z cos-'.z, or the angle which has x for its cosine, is V1—2 : I ae | Gf ood 4 1—2z v NA 1-2? tan™ x, or the angle which has x for its tangent, is . oe ee eae = tare ' x 1 1 yh ot )sm~ ————. egg! jas. CON geet? Af OR oat VIF at Vj +2 + x2 a ‘ x cot~' x, or the angle which has 2 for its cotangent, is ] v ] V1 2 Fett. in—! = —1 -1 +27 sin in = cos — » tan iat sec : cosec7! V1 +2?; +a +a sec™' x, or the angle which has for its secant, is 3 a Nee —] ] i & YH » cosec-? =, Vz?—] Vx? =] Cosec™*, or the angle which has 2 for its cosecant, is 1 V9] —— sin~* —, cog-! ——-, tan 5 fot 2 Vat, sec! — : x v r—1] v— ] v 60 DIFFERENTIAL AND INTEGRAL CALCULUS. Beginners usually fnd some difficulty in comprehending these rela- tions, owing to there not being distinct names for sin” a, &. We shall call sin7 x the enverse sine of x, meaning, not that x 1s an anzle and we are speaking of its sine, but that x is a sine, and we speak of its angle: an inverse sine is the angle which belongs to a szne. The following are the most common formule of trigonometry trans- lated into this language. 1 w 1 T 2 6 2 sin (sin"'#) = # COS (cos"'x) = 2 tan (tan x) = 2, &c. tans le an — ——— 4 7 T = oe is cos? a + sin 2 = ot cot—12 + tan 2 = 5 sec 2+ cosec 2 = 3 sina + sin“ y = sin™ (@ Ji-yt yn 1-a*) cos7! x + cos y = cos™ (#y -F Ji—a? Nis —y’) tan + tan y = tan” (724). 1 ay In sin (sin™ x) we see something analogous to (Ja), x-+a—a, and other cases, in which two operations are successively performed on 2, one of which by definition destroys the other. The question, “ What is the sine of the angle whose sine is « ?”” is not readily answered at first ; but the difficulty vanishes when we use more familiar objects— What is the form of the letter whose form is A ?”?—‘ What is the name of the man whose name is B ?” An angle has but one sine, one cosine, &c. Therefore, sin p, sin (sin7'q), &c. have but one value. But a given sine has an infinite number of angles, as is shown in trigonometry. Thus, 6, O+2n, 0+47, &e. a—0, 37-9, 5r—-8, &c. all have the same sine. If, then, sin® = 2, 0 is only one of the values of sin~?a, the others consisting in the several terms of the series just written ; and the same for the cosine, tangent, &c. We shall return to this subject. Since the expressions in the six lines above cited are equivalents, their differential coefficients are also equivalents. By equivalents we mean formule which express the same value in different forms. The verification of this assertion will furnish thirty useful instances of diffe- rentiation. We shall take one of the most complicated at full length. x Let u = sec" = sec 'y where y= J/x?—1 Vat—1 d du d ‘ qu _@U CF which two are to be separately found. dx dy dz a Oy i. 1 d.cosu _ sinu Ae Bane ae, cler aac CORN ig Oe as Me OB Nee : seal 1 = cos*s my 1 \/ 1 of ee = ee ipetctem Pu Dios 7a cos? u secu Seeks i y y=yVvy : du . dy 1 V(@-l)-+2 a—] ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTs, 61 (Observe that when P jg a complicated expression, it is typographi- te cally more convenient to write — P than — . dx d L dx d v Noes ee, fag, Afat Eo] Sie dy dx dx J?—] — hear ne nD ariel a ——______, dx xv— ] 6 gna | oe ] pese2 LE ees at oo —_ 7 (z*—1)? (v*— 1)? dus du d A ies) | 1 ] Therefore — oy ee res Se TRS cara in cage Teme eee Xv Xv x aus Y (v’?— 1)? ve Vx l “ae oF 1 Again, let » — Cosec” @ or 2 = cosec 4: == = sin u dz 1 d.sinu Cos u ] 3 ——_ nee SOR eg tee ea tt Ee 1 ———___ cogec? y du sin? % du sin? yz cosec?x du BEE ] 1 ae Sele bee Mi Nariemere — dz du 2 1 a V x2 — 1 ies that is, cosec— x and Bec eee have the same differentia] coéflicients, xv — J] as they should have, being equivalents, e have hitherto considered only the first diff coeff. and a function of only one variable. But successive differentiation is only a repetition of the same sort of operation, and it merely remains to find a proper notation to express the diff. coeff. of the diff. coeff. or the 2nd diff. coeff., du “dx d du ; vif of du cB or aye a O express Qu. Co, pre du fy ta dz oe ae a. d. dy ‘ d du Leet iN — a es I y dx y dry y y du agit ig eee x a a zr CT ee —s—sin — % — > 7 A dy y dy y y eae du 4 du 4 oy mest) ng =, oe dy : u=o(aty) => @) wherever ty du du dv ? dus du.-—s dv Pavichd tk Pate Bee } Pea Sed sto! l. Se io eae dy dv dy aye Theref jee an wmportant It herefore u = > (w+ y) gives ape portant result. The student may think, and perhaps ought to think, that, in applying , the reasonings hitherto given to functions of more than one variable, we are extending our conclusions, without further proof, to cases which the preceding proofs did not embrace. If so, now is the time to make him reflect, that from the beginning we have meant by a function of a, @ function of 2, and a constant. These constants, wpon, other suppost- tions, might change their value, that is, they are constants only with respect tox; a change in 2 does not change them. We are therefore justified in applying our conclusions to the variation of any single varia- able, with attention to the proper rules: we must only take care mn practice not to apply to consequences of the variation of one variable, the supposition that they were produced by that of another, except where we can prove the variation of both to give the same result, a8 in the case of ¢ (a + Y)- | To familiarise the student with these considerations, we shall take this opportunity of pointing out that relations may exist among differen? tial coefficients which are not derivable from one or two particular func tions, but from an infinite number, that is, are equally characteristic of * all. And, firstly, as to one variable only Let w= #- ¢, where ¢ is ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS, 63 du any constant. Then a 1, whatever c might have been : thus, aly u=uta, us Gt b, &. all give — = & (3 dx du di Let u= cr + x? = Sele", C= Lon dx dx oru= (SF + 20) a + 4, dx a relation which exists whatever e may be, provided only it is constant, This is the distinction between an arbitrary constant and a variable : the former may be what we please, but must keep one value throughout the process : the latter may be differentiated, which infers variation of value, as one of the steps of the process. ‘Thus, the answer to the : : du question—“ What function of x must w be, in order that | PPR "r is unanswerable in definite terms. It is y — © +c, (at least this is one Case ; we are not to infer now that because u = x + cis an answer that it is the only answer) where ¢ is any constant whatever, Prove the following ; : du du \3 fu .* t= — , — li w ce +c u oF: o4(= Cc ] du du. iu fy bet 2—(0- i i Ths ee bic if 2 aa pt OG: if u = FP * du du f — cead (0 fe; —_- = pn aed é uu = cr —loger ne a5 log x Whence we have the following theorem :-—if u, a function of 2, also contain a constant, that constant can be eliminated between the values du : : : of w and Fp? 2nd an equation produced which does not contain the con- Stant, and is true for every value of it. In considering a function of 7 and y, such as SF (2% y) it is important to observe that there are two sorts of indeterminateness in its form. Under this general symbol are contained I. All the functions of x FY, (C+Y)" log (x+ y), &e. 2. All the functions of ry, (ry)” log (ay), &e. 3. All the functions of ary, (a+y)" log (@ +), &e. &e. &e. &e. ad infinitum. in the first, let x and y be said to enter through « + y, in the second through xy, in the third through a? + y, &. And we shall now con- sider, not the general form f (z, y); but some restricted forms in which vand y enter through given functions of wand y. We have alread aad one result in the case of % («+ y), where x and y enter through D+ y. Let u= 9 (2° +y’) e+y— y u= dv du du dv du du dv j —_—-=— — —7’ 2 2y. dx dv dr Rath Ags dy dv dy ED x Be du du Eliminate ¢’v, and aa “th =), 64 DIFFERENTIAL AND INTEGRAL CALCULUS. Here is a relation which must exist for all functions whatsoever of 2+ y": thus pa Tah ss du 2a du 2y du_ du _o u = log (* +y")s dc aty” dy «x+y Vie dy ao Cle ae. du ate ‘ u(r ty), HA ty") .2% FT —=2(a°+y*)2y | in both cases. dx dy du du du du Let u=P(e-Yy), CRE Yay u=p(ry), ere imo du du ; he du du — 5 Laer pam (se Mea aly — — 22. u=adb(mrrny), 7 ar eae dy 5 ees 6(<) a 4 vay 0 Let u=29(%), fy aan 1 en Ou, x Xu | d d = — nx pv+a” ah = nu" ov+a" Pv . ip du Bo Oh ll aia dy dy dy’? dx a dy « du du from all these deduce that « Th +y Te what particular case has Y been already found ? We have chosen such instances as we knew to give simple results: let us now take u=a¢(a—y loge), du $ y Pa b(x#—y logs) +29! @— y log x) a-4) d a7 xo! (x —y loga) x (—log 2), from which deduce a a- v) es log =. 0 bee dy ép dx We thus see that, however # and y may enter through a function of x and y, we can by means of the two diff. coeff. of w and the given equation, eliminate the arbitrary function altogether, and produce ail equation which is true for any form that may be assigned to it. When any specific value is to be given to an arbitrary constant, which remains such throughout the process, it is immaterial whether the specific value be assigned at the beginning or the end of the process. For the rules of differentiation are the same whatever the specific value of the constant may be. The simplest case of this is as follows :-—If du A ae ; = cr, = 6. Now, if all this time c be = 5, we may either diffe-_ ; ites, OM du rentiate w= 52, giving 7 = 5, or u = cx giving 7 = ¢ in which we’ 2 then make == 5. This remark, however slight it may appear, is of great importance. With regard to the results of differentiation, observe 1. that all rational and integral functions (aa* + bx + ¢ for example) are lowered one degree by it. 2. That when <9 is a factor of w, it is also a factor of the diff. coeff. Thus, if wu = 2?” x Ya, | ON THE GENERAL THEORY OF FUNCTIONAL INCREMENTS, 65 moe’ x yr te | dr. wae? LWatd¢'e ya, of which €*” is also a factor. 3. That no factor is ever made to disappear from a denominator; but on the contrary, is introduced with a higher exponent, mile Sasi 4: Bs t Thus u = © gives 2. 2 He —y 1 Se Oe fy hae Wa dx (Yr a)? we (Ye av)? We are now to proceed to the application of this calculus to algebra. We must call the attention of the student to the fact that we have not assumed any algebraical development into an infinite series, directly or indirectly. He may therefore dismiss from his mind entirely (until further proof shall be offered) all such developments and their conse- quences. The assumption which is usually made in algebraical works for the establishment of such developments, is that certain functions of L, m (a+ 2x)™ for example, can be expanded in a series of whole powers of x of the form A+tBeo+Cx2+E2 + &. where A, B, C, &c.-are not functions of z. Of this no legitimate proof Was ever given depending entirely on algebra. Nor is the assumption universally true. That we may make use of infinite series, we shall find ; but it should be matter of proof, not of assumption. By rejecting infinite series we are unable as yet to complete the differentiation of a’. We have only found it to be ca*loga, and have assumed that ¢ is 1 when loga@ is the Naperian logarithm. This assumption, which is excusable while we are only inquiring into what will be its consequences if it be true, must be abandoned in all applications until we can pro- duce a proof of it. Cuarter III, ON ALGEBRAICAL DEVELOPMENT, AssuMING w= Oz, we have shown how to find another function px, h(x A _ which has this property, that oes) of may be made as near as we please to ¢'x, by taking Ax sufficiently small. Let the first of these differ from the second by P, which is therefore a function of x and 4x, having this property, that whatever 2 may be, it diminishes with- out limit with A z, There may be special exceptions in each particular function. For instance, if w—=log (7—a), “ = —, which is finite for every value of # except only =a. These cases, observe, we except for the present ; that they must be finite in number, or, if infinite in number, belonging only to a particular class of values, separated by intervals in which no Such. thing takes place, appears as follows. The only cases in which we can conceive them to happen, are those in which such a value is first assigned to x as makes a numerator or a denominator, or an expo- F 66 DIFFERENTIAL AND INTEGRAL CALCULUS. nent, one or any of them, nothing or infinite. Now, in all known func- tions, the values of x which satisfy such a condition are separated by in- tervals of finitude, and there is no function which is nothing or infinite for every value of x between @ and a@ + b (for any value of 6 however small) in all the functions of algebra. If there be such, we have notified in the postulates at the head of Chapter II. that they do not form a part of what we have called the Differential and Integral Calculus, but their consideration forms a science by itself, This condition is €x- pressed or implied in every treatise on the subject. Let there be two limits a and a + fh, such that neither for them nor. between them, are there any singular values of p2. Thus, for logz from ¢ — 2 to v= 3, there is no singular value, nor is log 2 or log 3 either of them singular. We have now P, a comminuent * with Aa, whatever the value of w may be, between a and a + h. Consequently, P and Az will still remain comminuent, even though, while 4 x dimi- nishes, « should vary in any manner between @ and @ + h. Thus, for instance, Aw and vA are comminuents, even though, while 4 # dimi- nishes without limit, v increase from atoa+h. Let us suppose Aa to be the nth part of h, so that Aw diminishes without limit as 7 increases without limit.’ Let P, which is a function of x and Aa, be denoted by f (#, A x), and we then have o(a+Axr)—o2 _ pe blot fm A0)5 now substitute successively «+A. for w until we come to have o(@+n42) oro + h) in the numerator, which will give the fol- lowing set of equations (z in number) :— ¢(@+h42)-o2 Ag v BN 0) = ha = 4) (w+ Aa) + f(a+ Ax, h2) e+3Ax)—$(a+24 $F 8A2) 9 OHI) = gf (w42h20) + fw + 2ha, Aa) = ¢gat+f (@, Az) a ———— — e e ) e e e (o+n—lAt — o(at+n—2Ax el — asda seca? Foe go! (c-+n—2 Ax) +f (@+n-2A2;A2). o (@+nA2)—¢ (a+n—1 42) Pic meme — ee Se (etn- Ants (v+n—1Aa,A2). Form the fraction which has the sum of the numerators of ‘the pre- — ceding for its numerator, and the sum of the denominators for its deno- minator, 1t being clear that all the denominators have the same sign. This gives * To avoid the tedious repetition of “a quantity which diminishes without limit when Av diminishes without limit,” I have coined this word. If ever the constant re- currence of along phrase justified a new word, here is acase, ‘There are sufficient ana- logies for the derivation, or at any rate we must not want words because Cicero did not know the Differential Calculus. Hence we add to our dictionary as follows :——-To° comminute two quantities, 1s to suppose them to diminish without limittogether: com- minution,the corresponding substantive; comminuents, quantities which diminish with- out limit together. > To comminute has been used. in the sense of to pulverize, and is therefore recognised English. . ON ALGEBRAICAL DEVELOPMENT. 67 p(e+Ar)-dr+¢ (@+2Ar)—4(04 Ar)+..+6(«+nAz)- ¢(@+-n=1Ax) — on Aa b(@ +n Ax)—oex o(@+h)— ox ee ee or ee nh ov h which must therefore lie between the greatest and least of the preceding fractions, or of their equivalents, all contained under the formula P(@tkAa)+f(e+kAz, Az). Now let the first value of x be a, and let C and c be the values of x which give #/x the greatest and least possible values it can have between @=aandr=a+th. (We have supposed that $’x does not become infinite between these limits.) And let C’ and K’ be the values of and k which, give J (@+kAa, Az) the greatest value it can have between the limits, and ¢’ and &! those which give it the least. Then still more do we know that ' aah aad lies between @C + f(C/+K! Ax, A x) and gc + Si +h Az, A 2), in which the two functions marked J are, as we have shown, comminu- ents with Ax. Now, if a quantity always lie between two others, it must lie between their limits: for if not, let it be ever so little greater than the greater limit, then we can bring the greater quantity nearer to that limit than the one we have supposed to be always intermediate, Or, in illustration, suppose P and Q to be P A B = Q ere moving points which perpetually approach the limits A and B-: if X (a fixed point) must always lie between the two, P and Q, it must lie between A and B; for if not, let it be at X, then by the notion of a limit, Q may be brought nearer to B than X, or X does not always lie between A and B ; which is a contradiction, The limits of the preceding, when 7 increases or Aw diminishes, are ¢C and ¢¢c: whence we have the following THEOREM :— If ¢x be a function which is finite and without singular values from ?—d tov=a-+h inclusive, and if the differential coefficient be the Same, and if C and c be the values of x which make $/x createst and least between these limits, then it follows that o(a+h)—¢ hie f CoroLiary.—Since, by the law of continuity of value, a function does not pass from its greatest to its least without passing through every ; ath)y—pa, My : Intermediate value, and since eto 1s an intermediate value of ¢ pw between PC and @c, and since g + Oh where 6 lies between Oand 1, Is, by properly assuming 6, a representative of any value which falls etween a and a+h, and consequently between C and ¢, it follows that ath)—doa A § true for some positive value of @ less than unity | F 2 hin lies between ¢C and pe. 68 DIFFERENTIAL AND INTEGRAL CALCULUS. As an instance, it must be true that gh Ae Gh a" = 3 (a+60h)* gives 9< 1 for one value, U To verify this, expand both sides, which gives ——— ee —_—— 8a@+3aht+h’ + Je+ ah +1h?—a at+oh= ne g fae - eaeateas main SGM So which, taking the positive sign, gives 9< 1; for @ . ah +4? is not so great as a + 2ah 4+ h?, whence the square root in question 1s less than a+h, the numerator less than A the denominator, and the fraction less than 1. Let there now be two functions ¢2 and y 2, the second of which has the property of always increasing or always decreasing, from x= @ to 2=a-+h, in other respects fulfilling the conditions of contmuity 1 the same manner as $ 7. xr+Axr)—wWwe Let AOS A) ae Wa +f, (@, 4), Ax whence f, (2, A) is comminuent with Az. We have then, as before, a series of equations of the form b (eth Ax)—6 (w+ k—-1 At) Ax meena marae =a y (nth Av) —¥ (ath —142) L(a-k—lAa)+fi(et k—-1 As, Az) or b(r+k Av)—o (e+ h—1 At) _ oat Rel An)+f (@+hk=1 Ag, Ac) Ww (a+k Ar)— wv (a+k—1 Ar) w(a+k-1 Ae) tfht@tk-1 Av, Ar) from which, by summing the numerators and denominators of the first sides, which gives iam ee if the first value of v be a, and if u(ath)—wya nAwzh; by observing that the denominators are all of one sign by the supposition either of continual increase or decrease in wx from r=a to a=a+h; we find the preceding fraction to lie between the greatest and least values of the fractions on the second side of the set, and therefore (using the preceding reasoning) between ! ! Wives be and ° the greatest and least values of cid w'C wie 5 uw! from azatow=a+h. And this must as before correspond to some \/ o value of = for a value of x lying between ea and a=ath. Let it be «a+ 0h as before, and we have the following THEOREM :— If dx and Wa be continuous in value from 2=a to x=a+h, and if in addition @’x and Wx be the same, and if also y w always increases or always decreases from «=a to x=ath, then p(ath)—ba _ bath) ye (a+h)-ya Y'(ateh) 0<1 ON ALGEBRAICAL DEVELOPMENT. 69 Corontary.—If the two functions be such that Pa==0 and wad without any discontinuity or singularity of value, we then have P(a+h)_ $'(a+ oh) Wath) ¥"(a-p Oh) Boy below: (hy Let us now consider @’x and y/a as new functions of x having for diff. co. 6x and yx, and take the limits v=a and x=a+06h (0 being determined by the last equation) and Suppose that in addition’ to the preceding conditions y/x continually increases or decreases between v=adandx=a-+ Oh, and also that ¢/a=0 w’a = O without discon- tinuity or singularity, and that dx and yx have no singular values from x=ator—=a-+ 0h. The same theorem then gives P(a+0h) __ p'(a+6, 6h) o< W'(a-+0h)~ -w(a+o, Gry he Now consider $x and wx as new functions of x having diff. co. pe and wx, which give 6a = 0 Wa = 0, without discontinuity or singularity from v=a to x =a+% 0h, &c. from which the same theorem gives P"(a+0,0h)__ $!"(a+0, 6, Oh) 6, <1 (3) Y"(at0,0h) ~ wl"(a4+0,0, Oh)? Pra and so on. Now remembering that we know nothing of 0, 6, &c. except that they are severally less than 1, in which case all their products are severally less than 1, we may include all the terms a + Oh, a+ 6, Oh, &c., under the general symbol a+60h (9 <1), and if we collect the several sets of conditions under which this theorem will apply to all functions up to the mth diff. co, inclusive, and observe that the first side of (1) has @ succession of values found for it in the second sides of (3); (2), (3), . . . we have the following TuorrEM * :— If there be two functions ¢ x and Ww wv, having the series of diff. co, pail sy le eee Ion 4 1),,) all continuous and without ea? ih oie a ee pra ete ie singularity from «=a to ° / / n n+} S 5 hb SG Tat i B, Us Ue @ ° ° & LS Mb Us 2 ¥ b) P| H a Ys ¥. E Soccal 7 Aa DY) : and if as a second set of conditions, ga=0, fiax=0, db/a=0.. . up to AMax=0 ya=0, y'a=0, w"a=0. . . up to Ya=0 and if, as a third set of conditions, Yor, Yia, lr, . « . up to wr be functions which either continually increase, or continually decrease from v= ato xr=a+h: then there is a value of 0 less than unity, which will satisfy the equation Plath) Ot (e+ Oh) Uw (a+ h) tA Uw") (¢ + Oh)’ If we were at once to proceed with the consequences of this theorem, the student would not be well able to see why so apparently cumbrous an apparatus of proof is necessary to obtain what is called Taylor’s * Remember that whatever is assumed to be true from x — a to t= a+ hy is true from «= ato r—a + @h, from r=ator—a + 0: 6/, &e., if @, @)° &e. be severally less than 1. 70 DIFFERENTIAL AND INTEGRAL CALCULUS. Theorem: we shall therefore make what is often given as a proof pre- cede what we consider as really a proof. TuroreM. If it be allowable to suppose that ¢(a + h) can be ex- panded in a series of whole powers of h, of the form another a third : a fourth ; prota + (eae xa (0 of x xh +( G0 of x x hi &e. then that series must be the following, and no other : 4 h2 h® h ogetde hte. > + 6!" t= + dx - 54g em , duh db: & We have shown that u=6 (w + h) has the property = a : if pos- sible, let d (ath) =u=A+ Bh+ Ch? + Eh’ + Fht + &e. ad infin. and let us assume (which we consider as rather a questionable assump- tion) that the property which is true of ¢ (w + h) is also true of its ex- pansion. ‘Then we have (A, B, C.... being functions of a, which & isnot, and A, B,C.:.. being not functions of f#: all this is m the original supposition, ) du Sager dB Re AEs di ee Se os | ae dx dx which we will write as follows :— wWoA'+ Bhat CR+ HR+ FR + &e. d But ae +2Ch+ 3ER+ 4Fh? + 5Gh + &e. h? + OL oa aie. dx du du and ah — w’ or - for all values of x and h, whence by the common H Lv theory of algebra, called by the name of that of ¢ndeterminate coeffici- ents, we have TA’ it jo prem sf 20 = Bl = — which call A” Pos dx L 2 dUpol aA J 1 ae (ae PR ab dar, sancti fj 3E=C cpg tre 5A o E=>,A Gera OA ea eae 1 40 = SS = he Ay Ge E30 dal at aD and so on; whence substitution gives —Peo@+h)a=A+ An + Ar 4 ne ap ees 2 0.3) eee ee It only remains to determine A, to do which another doubtful assumption* is usually made, namely, that when 2 = 0, the series just * Observe that we do not say these assumptions are wnitrwe, but not self-evident, and therefore not to be assumed without proof. We may readily see that the sup- position P=Q when h=0 is very suspicious, unless we can show that, by making & as small (near to nothing) as we. please, we can make P as near to Q as we please. Now, in the series in question, though by making & as small as we please, we cat render all terms after the first individually as small as we please, yet it is to be ON ALGEBRAICAL DEVELOPMENT. 71 found is reduced to its first term. If so, then by making h = 0 p (x + h) becomes 2, and the equivalent series becomes A : therefore p2= A, and A’ A,’ &c., are the successive diff. co. of A with respect to x, whence the theorem will follow. We shall treat the preceding process as nothing more than rendering 2 it highly probable that ¢ (@ + h)anddat+ pa.h+q" - + ave, have relations which are worth Inquiring into. But as we are deter- mined to know nothing of infinite series without proof, we shall take a finite number of terms, h? Beas Bee Pat+Pa.h+ha—+....... -up to + Aq ——___ 2 2.3...7 which we proceed to compare with d (a+h), as to its excess or defect. Or rather, as we have used $x in a particular theorem, we shall use Ta here, and proceed to consider h2 hh f(ath)—{fatfla. h-+fla a Fee $f Oa = Let a be a fixed quantity, but let +h be variable, and let it be called x. ‘Then substituting x~ a for h, we have the following function of #:— («—a) (v—a)" — = tes ~~ Ui ST > eee kee ae Tyece (m) ities aan fa—fa—f'a (a~a)—flla 9 f S 2uaa, Let us suppose 1. that fv is continuous and ordinary from 2 = a to w=a+th. 2, That the values of its diff.-co. when, asia, namely, ‘a....f/a are none of them infinite. Let this function be called dx and let it be differentiated m times in succession with respect to a. (v—a)? (w—«a)" — a ae ok my nf ee peo (n) Case Oa a) pu = fe-fa—fla (a—a) fra 2 t QA y (z—a)? (t—a)""? lnm fl yn fly I Nt Fa ce see aC ne De) ne”) gle ak Oe coh al eg OT a | ae uw msg (v—a)"~? ROSE al Gc Fla ma) I 2.3..(n—3) POV =fePa—fe%a—fMa (x—a) O™ x = fz — fa PEt) x = f Say: The student must ascertain that in the series x—a)? (t—a)? (xr—a)* Gs. ake ; OI PE Re peer o each one is the diff. co. of its successor, or to differentiate any one, that he must pass to its predecessor. The general process is, remembered that the number of them is infinite, and we have no evidence whatever that here will be an unlimited number of small quantities, whose swm must be small too. For a sufficient number of parts as small as we please will compose any quan- tity, great or small. It is true that we shall hereafter prove certam cases in which We are justified in the assumption to which this note is written, but we never saw a proo which embraced every case, 72 DIFFERENTIAL AND INTEGRAL CALCULUS. d (x-a)" 1 d(x—a ee ) Sef BE ae ay =o dx 2.8...n 2.3... dx n (x—a)""' sn ee NN ee eee 2.34..2—1.n ( ) 2.3...(n—1) He must also observe that a constant fa in the first, f’@ in the second, &c., vanishes at each step, and a new constant appears, resulting from the differentiation of the current term of the form p («—a@) which gives p. But the best way will be to try several particular cases, such as the following (n=4) :— Q (x—a)? (x —a)* _ (a-a)* Pees hay ace ehh pies eed Fg) 2 oe Fy SS Oe F QU ————_—— p c=fe-fa—f'e (a0) —fla—g—— fa I te a ans ee paxfle—fla—fla (w-a)—fl"a Stil —f' a v2 : Se 2 pla=flv—fla—f"'a (4-4) oo PE teams LER —f"a—fra (x —@) Wyo fie—f"a Gen] ae On looking either at the general or specific case, we see that fa, f’a, flla......up to fa being all finite or zero, this function can present no singular values for any finite value of cz. And moreover, when 7=a@ each expression presents a finite number of evanescent terms, and we therefore have da=0 fpax0 ga=0.... Oreo: consequently this function completely satisfies the conditions of the theorem in p.69. We have now to look for a form of yaewith which to compare it, this function being determined by the conditions to be such that va, wa. . . up to ya are severally =0, that wore does not give singular values, and that ya, wie.... are all severally increasing or décreasing throughout the extent of the function from a=zatov=a+h. It will be found that («—a)*t' complies with all these conditions, and the general and specific cases will be as follows :— General. Specific (7 = 4.) wo=(e—a)” Wwe = (a—a)’ ye (n+1) (e—2)" Wi e==5 (x — a)? ap"a=(n+1)n («@—a)" ye 5.4.(4—a)P ya (n+1) 2 (n—1) (w—a)"-* y!n=5.4.3.(a—a@)? ’ ; : ; : wi'e=5.4.3.2 (4@—-@) wn =5.4,3,2. WOr=(n-+1)n....3.2 @—-a@) yee =(n+t1)n...-3.2 In which it is clear that all the diff. co. up to the mth inclusive, are in- creasing from =a or z—a=0 to r=a-+h or a—az=h, and also that they all vanish when z=a. It is moreover evident that the (m+1)th diff. co., being a constant, presents no singularity of form. We have ‘then, writing a+h for x (p. 69.) :-— ON ALGEBRAICAL DEVELOPMENT, 73 P(ath) po (a+eh) % (ath) we) (a+ 6h) h > B)—fi— fa — OC, SP Ogee f (a+h)—fa-f'a.h Mies 3..n ft (a+h) ee. oe 9< 1 or ice where @ is less than 1; or we have : E | hh f (ath) = fa + flaht+ fa ats + ai Bat Aa as yeh and eee 2; Tae 0 subject only to the condition that no one of the set fa, fla.... up to fa is infinite. We may carry this series (if no diff. co. become infi- nite) as far as we please: it will afterwards remain to be pointed out what are the cases in which we may legitimately suppose it carried ad infinitum. Whatever these cases may be, in them we haye 2 hh? S(ath)=fa+fla.h+f'a. Spa F = + &c. ad infin. which is TayLor’s TuzorrM*; and we sce that we may stop at any term, and give an expression for the value of the rest, beginning at that term, by writing «-+ 9h instead of a in the term we stop at, and expunging all that come after, the value of this accession lying in its having been proved that 6 is less than 1. This is Lacrancr’s THEOREM ON THE Limits oF TayLor’s series +. If we call C and ¢ the greatest and least values of #"F” (a+6h) from @==0 to 0==1, we know that by stopping at + fo (a+6h) 2 ; al 4 n+l fa h we commit an error _ Ch pee ch 2.3..n which lies between 2.3...7 2 Son, We can now demonstrate the binomial theorem : for if 6a = 2” we have ¢'c = nu”~', p’2 =n (n—1) and therefore Pa =a", P/a = na™, &c. This gives 3 h2 h a--+h)" =a*+na**h+n (n--1) qn +2 (2 —1) (rn — 2) a? — + . 2 ys het OSE RTD EEA ai Seal ee toe ay AS +n(n—1).... (n—-p—1) (a+6h)"?-? MET TR y or (a+h)"=a" +n (a+ Oh)’h n—1 =a" +nah+n— 5 (a+ 0h)" he _ bie Wi Gai 2 | =a"+na" thn —_ a? h? bn — —— (a+0h)”"*h’, &e., * Dr. Brook Taylor (born 1685 at Edmonton, died 1731) first gave this theorem in his ‘ Methodus Incrementorum,’ published in 1715, in the same year with his ex- cellent treatise on Perspective ; the latter being as much the foundation of most of what has been done since in perspective, as the former of the Differential Calculus. + D’Alembert first gave a proof of Taylor’s Theorem which involved a method of determining the limits, but this was only incidental, Lagrange first formally took up the subject in his‘ Lecons surle Calcul des Fonctions,’ first published in 1801. 74 DIFFERENTIAL AND INTEGRAL CALCULUS. where, however, it must be observed, that though 4 is less than unity in every one of these cases, it is not the same in all. sin (a+h)=sin a+ cos (a+ 6h) ich 2 = sina+cosa.h—sin (4+ 6h) a h2 3. = sina+cosa.h—sina Ck (a+96h) 73 &e. We shall ascertain the truth of the first line by an instance, which will also serve to illustrate the way in which angles are measured in analysis (a point on which the notions of most students are remarkably confused : see Penny Cyciopzpt1a, article ANGLE, ‘ Study of Mathema- tics,’ p. 89.) Let a be (in common degrees and minutes) 35°, and let hbe 10°. When these enter under a sine or cosine, it is most conye- nient to express them in degrees, minutes, &c., because the sines, &c. are given to those denominations in the tables, and are the same for the same angles in whatever way we may measure the angles. But when an angle enters as an angle, the truth of all theorems yet obtained depends upon measuring that angle by the fraction which its arc is of the radius*. The angle of 10° must be expressed by ‘1745329. The assertion then which we wish to verify amounts to this: that if we find 6 from the equation sin (35° + 10°) = sin 35° + cos (35° + 8 x 10°) x °1745329 we shall find it less than unity. : sin 45° ='7071068 log *1335304 °1255801 1 sin 35°='5735764 log 1745329 124187713 1335304 log cos. 40°5/ —1°8837028 } Bie 35° + 8 X 10° = 40° +1, 6= sii = 501 = + nearly. We ngw come to a modification of the preceding, which is usually called Maclaurin’s Theorem, but which should be called Stirling’s Theo- rem}. If we suppose @ = 0 to satisfy the conditions under which Taylor’s Theorem exists, that is, if we suppose f0, f/0, f/0.... tobe all finite up to f"0 we have, by Taylor’s Theorem, , he 8 SOFM HPO FOREST + f0 H+... 4/70 + f° (0-402) —— Qi ai selnnly and remembering that h being anything whatever, we may write w for h, we have Nha 2.3.37 * It may be worth while to revert to the fundamental step on which this rests. It is a theorem derivable from ‘Elementary Illustrations,’ p. 5., that the limiting ratio of a comminuent sine and angle is 1. Now this theorem is not true of the number a Oa tn an angle; but only of the fraction which the arc of the angle is of its radalus. + Maclaurin, in our view of the subject, was the first who wrote a logical treatise on Fluxions, The reader who would verify the assertion implied in the text for him- self must compare Stirling’s ‘ Methodus Differentialis,’ London, 1730, p.102, “ Hine si ordinata Curve, &c.” with Maclaurin’s Fluxions, Edinburgh, 1742, p. 610, “ The following theorem, &c.’’ The fact, we doubt not, would be, that both Maclaurin and Stirling would have been astonished to know that a particular case of Taylor’s theorem would be called by either of their names, . ON ALGEBRAICAL DEVELOPMENT, 75 v= f0+f0.a4 fo 24 +f —= a a Soe 4 SOREN, ite a 7s AT (n-4-1) ant Paro 5 ye? TS ae ee ee a of which the following is an instance :— fe=sina, f'ex=c0s 2, fle =—sin a, f"c= — cosa, f*x=sin a, &c. fo=0 f0=1 f%0=0 FLO Hr so uf Dee, Ser i ; Fi sin v= 0+ cos@x.x2=041 x &— sin Ox a. 3 Octet Av gu cos Ox — ears t— —_-— 8s ov — NS 2 2.3 2 x a ; =0+1x2#-0x — —-1x — + sin Ox a , &e. 2 2.3 2.3.4 : ; ave ae Or sin = cos Of . x = e—sIin Or — — ® — cos 0a —~— 2 Buz a ee g x* x 4 4 v = LQSI{LQSI[LQSI(LQSI¢z)]}. Now suppose we change the order in which these operations are made to the following LLLL QQQQSISISISI dz ; the question is, can we get a clear idea of what we are doing, and can we advantageously make that idea serve for the further elucidation of higher differential coefficients than the first. This we proceed to discuss in the next chapter. 77 Cuarrrer LY, ON THE CALCULUS OF FINITE DIFFERENCES. By the word finite we here mean that the theorems of this subject sup- pose quantities to have given augmentations or increments which do not decrease without limit. Not that we debar ourselves from using all legitimate consequences of any theorems which may arise from supposed diminution without limit, but that we thereby change the name under which we view the subject, and pass from the Calculus of Finite Differ- ences to the Calculus of Differences diminishing without limit, or to the Differential Calculus. Observe first the consequence of forming a set of series, each of which is made by subtracting every term of the preceding series from its suc- cessor ; a b—a c—2b+a e—3c+3b—a f —4e+6c—4b+a, &e. 6 c—b e—2c+b f—3e+3c—b g—4f+6e—4c+b ec e-c f—2e+e g—3f+3e—e &, e f-e g-2ft+e &c. ed Ke. mc. &e. ~ Observe, secondly, that when an operation is performed two or more times in succession upon a function, it will be convenient to make a symbol for the result by writing the symbol of the single operation, with the number of times it is repeated in the manner of an exponent. Thus, if Ay denote an operation performed upon y, and if the operation be repeated upon the result, it will be convenient to denote A (Ay) by A’y, and A (A*y) by A’y. Here A is not a symbol of ‘quantity, but of operation ; A" is not a symbol of m quantities multiplied together, but of 7 ' Operations successively performed. Let wu be a function of w, and let Aw be the increment received by u when Az is added to z. This gives Au = $ (x + Ar) — dz; without proceeding further in the Differential Calculus, repeat this operation again. Let x become x+Aa, and find the increment of Au. This. gives A (Au) = (6 (a+ 2A2)—¢ («+ Ar) — (dh Sire aie this is what Aw becomes this is Aw itself. when x becomes r+ Av. or A’u = $(«#+2Ar)—-2¢ («+ Ax)t+¢e. Repeat the operation again: when x becomes x + Aa, A’u becomes $ (1+3 Ar) —2¢ (w + 2 Ax) +¢ (w+ Ar) Aru is P(a+2Ar)—2h(@+ Ax)+¢z; and (A*u as changed) — (A?u as it was) or Yu = $ (vx+3Azr)—3 ¢(2+2Ar)4+3¢6(at+Ar)—ha2 Proceeding in this way, and supposing u=P ev m= («+ Az) Ug=P (74 2Ar). 6 ..tr=P (@4+n Ar), 78 DIFFERENTIAL AND INTEGRAL CALCULUS. and writing w, w,, &c. instead of a, b, &c. in the preceding page, and also putting for each subtraction the symbol by which we have agreed to represent its result, we have the following table (only altered. by writing each quantity between those of which it is the difference made by subtracting the higher from the lower) :— Valuesof First Second Third Fourth &e. the F° Diff, Diff. Diff, Diff. Ub Au Uy Aru Au, APu, Us Leu, Au, Aus Aus &e Us A*U, Atl, : Aus Pus : Us us ; : Au, é R Us . ; : Z and the actual performance of the operations indicated gives Au =u,—-u A?y =u,— 2u,+u Aeu =u, — 3ulgt 3u,—U Au,=Us.— Uy, A2uU;=Ug— 2Ugt+ Uy Aeu,;=u,— 3u3+ 3u.—U, Au,=Us — Us A?ug=U,— 2st Us Lous us — 3ty+ 3g — Ue &c. , &c. &e. &c. &c. &c. The general law is evidently that of the coefficients of the binomial theorem combined with the successive values of the function in the fol- lowing formula (7 a whole number) :— n—1 n—1 A"u=Uu,—NUjz1~ +n TR ee cin +n él Us - Niy+u e n—Il — n—l AM = Una, — NU, tn my Wee ee and so on; the upper sign being true when 7 is even; the lower when’ 7 is odd. This may readily be proved; for if we assume the preceding to be true for the present value of x, we then have for A’w,—A"u, which is. the same as A”™ x Uz + NUg + Wy n—l 2 —7 SUn—M+1l)u+ atl 5 Un — &e. which follows the same law. But this law, being proved by inspection as to the second: difference, is therefore true of the third, and therefore of the fourth, and so on. Now let us suppose w and all its differences to be given, from which we are to recover the original succession of values 2, % v3, &e. uu +Au Au,=Au + A2u A?uj—A*u +A®*u- &e. Us=u,+Au, = Aus=Au,+A2u, A?u,= A, + A®u, -&c. Ug =U, + Aug Aug Aua+ A?u, A®u,= A*u,+A®u, &e. &c, &c. &e. SC. &c. &e. ON THE CALCULUS OF FINITE DIFFERENCES, 79 as is evident from the table preceding, the method of its formation being recollected. We have then uu +Au UWy= + Au=u +Au +Au +A%y, —y +2 Au + Au Us=Us+ Aug=u, + Au, + Au, + A’, =u, +2 Au, + Av, =u + Au+2 (Au +A’u) + Aut Au=u+3 Au+3 A’ + Aru, Similarly Aus=Au+3 A’u +3 Asx + Atu U, OY Us-+ AUs=u+4 Au+6 A’u+4 A%u+tAty and the coefficients of the binomial theorem (when 7 is a whole number) again appear as follows :— n—1 n—1] U=uU +-ndu +n zo. Aut eee tn bn Ay + Au ed nam—] 2 from which as before it follows that Un+Au,, or Au,=Au + n A®u +n —] Au + een = Anu tnd" -- Ay, Unt = w+ (a+ 1) Aw +(2-+1) 5 AM eee tb (+1) Au FAHY, or the truth of this theorem for any one value of 7 enables us to infer its truth for the next higher. We know that Aw, A®w &c., are comminuent with Ax, as also are Au, A’u,, Au., &c. In the same manner Ad (v+p) is comminuent with Az, and the same remains true if p itself be comminuent with Ax. And the following equations are easily proved. If w= y tv Aw = Au + Av, ifu=cv Au=cAv. And Az, remaining the same In all the processes, is a constant, as are all its powers. If, then, u=v X (Av)", Au = Av x (Ar). And we have proved that Ar)? @(e+Az)= data. Aa+ "(e+ 6Azr) sty if then we write w (for convenience) on the second side instead of Ax, we have for ¢ (vx-+-Av)—¢z, or for U;—uU, or for Au Au=u' . w--d! (1+0w) — a alt FF By the same rule we have (making w’ or dx itself the original func- tion, and therefore "a and #"z its first two diff. co.) Ww? ? (w+Azr)=P'r+¢"2 . wp!” (e+ Ow) > 6<] or Au =u", wd!” (246) ~ v2 Similarly Au ula 4 hb (n+ 0,0) — Di I - w? Au™ =U Veg hot (v+6,w) © 0, is Where by w’ wv’... ...u™ we mean the functions obtained by successive differentiation of u, in’ the manner already described, and which it is 80 DIFFERENTIAL AND INTEGRAL CALCULUS. our object to compare with the results of finite differences. From the first of these equations find Aw, by equating the differences of the two equivalent forms (remembering, what we need not express, that in db! (x+Ow), @ itselfis a function of 2 and w, but always less than unity in value) and using these equations ; If w= w+v Aw=Au+v: Lf. vescz Av=chz If w = u+ecz Aw=Au+cAz. We have then A?u = w Au! + = Ad" (+0) 2 2 = w (w'o+ we need know nothing except this, that it remains finite when diminishes without limit, the first term having the limit d!x, and the second term having for its limit a differential coefficient, as is evident from the form of the frac- tion, Let us call k, the term in question: we have then 3 W 2p, ——azll 7.2 ; Au =u" o + ky >: Repeat the process, which gives 3 : w Lu = w* Au! + — Ak, 2 J D) 2 ee w? w® — wy? (wr 4- (xv + 9.) =) +. = Ak, = ul o® 4 (a (70,0) + — alt! w? . ks w*, Ade} a ei 2 where k, remains finite when w diminishes without limit, as before. Proceeding in this way we come toa gencral equation of this form between A"w the nth difference of w, w or Ax the difference of x, and au” the nth diff. co. of w: k, being a function of z and w, of which all we know, or need to know, is that it is finite. Mu a u™ w" + k, wo, If we divide both sides of this by w" or (Av)", we have A"u (Az)" the second term of which is comminuent with w, and by diminishing @ without limit, we have =u” +k, w ; A"u Limit of (Ar)" <= uw the ath diff. co. of wv. ON THE CALCULUS OF FINITE DIFFERENCKEs. 8} As an instance, we shall find the second diff. co. of x, without find- ing the first. SSR Me Le Pi, ates (2 + 2w)? A’u = ug — 2u,+u=— (x + Qu)? —2(¢ + wf + 2 = 6 rw? 4+- 6 w? Avu sh *5: yee), Aye = 6+ 6y, the limit of which is 6 x, Ax® Now, if dx = 2° P'r = 32° Dn 6 7.4 From this, a notation may be obtained for the successive diff. co. of w . : or Au du with respect to 7, For since the limit of Al has been denoted by ae x ‘ d du. eS. and since we haye now found PD Pia the same thing as the limit of 2 dx 2, du : Te to which as Qi a total symbol, the remarks in pp. 59 and 54 apply. ‘The diff. co. of the : du. OE ay diff. co, of being found to be the limit of coats we may denote it by ax i (Az) (dey 3 and soon. Hence, to connect the notations we have used, we Ox ae iP am (Aa)? it will be consistent to signify the latter by have the following equations (it is usual to leave ont the brackets in What would be denominators, if the preceding were algebraical fractions) du du Mu anes 4 penyOL , ree i adaoe I, , i her ora p'x ge a hea Dx, &e. The usual way of reading these is “du by dx,” “d two u by dz square,” “d three uw by da cube,” and soon. Thus Taylor’s theorem becomes the following : when x becomes x th @uh? dy fe deat Gag t ke. du wz becomes wz + ar h v When we wish to express a diff. co, as it becomes when the variable ‘ eae du receives a specific value a, we shall sometimes write jt thus (3 : but v/a in this case it is more convenient to write dx for u, since ’x then ex- presses the general diff. co., and a the particular value. Thus we have when wx changes from a to a +h du» du he u changes from (wv). to (uw), +(3), A+ Ga). ot We shall now proceed with such results of the Calculus of finite ifferences as will be useful in future parts of this work. Let us sup- pose a series of terms connected according to such a law that a certain difference (say the fourth) is always = 0. Then we have, wu being any term whatsoever, wu, the next, wv, the next, and so on, | G 82 DIFFERENTIAL AND INTEGRAL CALCULUS. Aty = Uy — 4s $6 -— 4H = 0; hence we can express any term by means of the four nearest to it, either on one side or the other, or both, For instance, Loe et Us) — (wu + Us) ii 6 If the fourth difference, instead of being absolutely zero, should be a smaller quantity than 1s requisite to be taken into account, these theorems will be sufficiently near the truth for the purpose. It is plain, by the method of constructing differences, that the (m + n)th difference of wis the same as the mth difference of the nth difference of a, or that Us = 4Ug—6 Ue + 4Au,—u, &c¢. u, A™ +, — A” (A"w) ; and if we attempt to give meaning to such symbols as A°v, Av'w, A~*u, &e. it will be convenient to assign such meanings as will satisfy the preced- ing equation. Accordingly, A°u must be the same as U, in order that we may have A" T° = A” Aw or A™u = A”™ Aw. . We now ask what is the proper meaning of A-u, Since we are to have AA ‘wu or At Au the same as A'~'w or A’u or w; that is since A Au is to be* u, then Aw is the quantity whose difference is u. If, then, we take the series of terms UU, U...-. and ask, not what are their differences, but what are they the differences of, we find that, taking any quantity we please, C, to begin with, the following frst column has the second column for its differences, the third column for its second differences, and so on. Dr of the function. 1st Diff. 2d Diff. 3rd Diff. &e. u C+u Au Uy Leu C+ut+u, Au, &c. Us Aru, Ctutumtue Aus Cut uy -ust Us &c. &e. &c. &c. Hence Az is an arbitrary constant C; Atu,isC +4 Amu, is C + w+ u,, and generally Anu, is Cu 4 Uy + Ue fee FE Un-2 1 Un From this being a summation it is customary to signify Av'u, by 2t, : thus, Ce te Be ny ow) eae is denoted by 2x O44 oes s.deo i De... Be Ge meaning by 2 dx the sum of all the values of oa, for every whole value of x from any given number up to 7— 1, increased by an arbitrary con- * Some students may, from their previous reading, have an idea of this sort of . process, but most will not. Observe that what we are here doing is not tracing the properties of defined symbols, but finding out how to define a symbo},so that it may have a certain property, re ON THE CALCULUS OF FINITE DIFFERENCES, 83 stant. But unless the contrary be mentioned, let it be presumed that the arbitrary constant is 0, and that the series begins from the first term of which there is question in the problem, Thus, in treating of the suc- cession of terms x Uy Be » by Xu, we mean the sum beginning with u, and ending with Mt It may be that we have a number of terms given, but not their gene- ral law, and we wish to ascertain what law they do follow. This is always to be found from the equation n—I1 Biba ae ete Nee AMR shalt for we thus have a function of m which expresses the 7 + 1th term. Suppose, for instance, we ask, what is the general law of 1, 4, 9, 16, 25, &c., shutting our eyes for a moment to the evidence of the terms them- selves, in order that we may deduce the law by a method which is not simple observation. Taking the differences of this set of terms 1 3 Uu=1l Au=3 A%y—2 A’u=0 A*tu=0,&e, 4 2 5 0 n—] 9 2 0) ON TIA Sori DE Oe Op ae 7 0 16 2 0 ' 9 0 =1-- 8n + n? —n = (n +1)? 25 2 1] the (7+-1)th term is (7+-1)? and the nth term is . Let the student take some simple formula, such as x (x+1), Give VL a number of whole values beginning from 1, and then reconstruct the formula by the preceding method. Thus x (v+1) gives 2, 6, 12, 20, 30, 42, &e. Per Au= 4 pg ihe fi), A®’u = 0, &e. n—] ge a A eS apn Hane tn ea) (2 + 2) this is the (n + 1)th; to find the nth term write n for + 1orn—1 for n, which gives n (m + 1). The utility of the preceding method is most obvious in a case in which all orders of differences vanish after a certain number. And we shall prove that this is always the case in a rational algebraical expres- sion. Take for instance, US ax” + ba”! 4 egm-2 SS a pe+dq; and let x become x +- , giving u,. Expansion will immediately make it obvious that the highest term of each disappears when w is taken from %, and that we have a result of the form Au=ama™ 4 Azo, -+ Pr+Q A, &e. being functions of w. The same reasoning applied to this pro- *€S8 gives a result of the form qa 2 ) 84 DIFFERENTIAL AND INTEGRAL CALCULUS. Ata = am (m—1) 2" + Oe Ce eaie and continuing in this manner, we come to Any = am (m—1)....3.20@ + E A" uz am(m—1)....3.2.1 a constant AtHy = 0 Anu = 0, &. &e. In this manner we can always arrive at a finite algebraical expression for the sum of n values of a function, provided that function bea rational and integral function of the variable. For let Uae. Ure S44 U.=Ct+tutu Us, C tut t us U, 2 OC Fut uy t ty tb eeee tb Un By the general truth already proved, we know that (U,2 U Fn AU + n™ = Btu + teoe tn AU + AU; but U is C, AU is u, A®U is Au, and generally A"'U is A" uw: while U, is C + ut .-ee tb Uni: Substituting, and taking the common term C from both sides, we find that n—1 Ut Ub oe eb Una = NU MS Au-i.see tnd ur APs a°very convenient formula, if all the differences vanish after a certain number. Let us apply it to the finding of 1 + 449+... + 2%, which we may denote by 2 (7 + 1)%. It appears that w= 1 mr4.. tan Au=3, A’ux2, A°u=0, &c., whence n—1l n—-ln—-1l 1 4. eese > ee ——3 ce eras En es ar we. i aL +n ntn 5 + 2 5 3 Biante 7 __ On 4 Qn? —9n 4 Qn?—6n?+4n 2 (n+1) (2n+1) a 6 6 oF 6 : which is the formula assumed in p. 30. If we now consider 1? + 2? + 37+ .... +47’, we have proved that the differences of ? vanish from and after the (p + 1)th and that the (p)th difference is p(p—1).-- 3.2.1. We have then (calling ¢% Cy ++ Cpy the first p differences of 1?.) Peiacdy alti git n—l, n(n—1)...€m—p) , + 2? + + 2 n+ n —>— teh cathe ra aee ie Ona but c,=p.p—1...1, whence the preceding sum is (we shall soon see why the last term is particularly attended to) pe ve! os Be Pee, ; nbn 1 mee n—-1ln—2 Bute iv (n—1) (n—2).« (n—p) 2 Pa ptl This, it is evident, might be expanded term by term, and afterwards — arranged in powers of n. And since in each factor there is only the — first power of 7, it is obvious that the highest power of 2 comes out of that term in which there are most factors, namely out of the last. In this last term, there are p-+1 factors, 2 the first, »—1 the second, n— two the third, &c. up to n—pthe (p-++ 1) th. Its highest term 18 ON IMPLICIT DIFFERENTIATION, 85 therefore n’+" : and no power so high can otherwise appear in this factor, because no other term js compounded of all the n’s; nor in any other part of the expression, because in no other term whatsoever are (p+1) ns multiplied together. And from this, remembering that the last term has the divisor (p+1), we find geri BA beep ate B + An? + Br?" + ...,Pn4.Q where A, B, &c. are functions of p, not of m, Which might be ‘found by expansion, but with which our present object gives us nothing to do, except to remark that, being functions of Pp only, they are not changed by Supposing n to change. This gives Me eon ae eee pg net PHL a tae tee bot; ' and now we sce that the greater n is Supposed, the smaller will al] the terms of the second side be, except the first which does not depend on n, This first term is the limit when » js increased without limit, and we thus have the following theorem. If the sum of the pth powers of all the natural numbers, up to m inclusive, be divided by the (p+1)th power of the last, the greater 7 is supposed to be, the nearer is the result to ] ae we ET and this without limit, (Elementary Lilustrations, p. 33.) We shall now leave the Calculus of Differences for the present, and proceed with the methods of differentiation, ie I gene ee eee a Cuarter VY. ON IMPLICIT DIFFERENTIATION, In ‘all that precedes, w was given, as it is called, explicitly as a function of x, that is, the function which x is of x was expressly stated, and in ho degree left to be deduced or inferred. Such a case we see In u= cz, But we may imagine wu to be given, for example, as in the equation “ = C@ + eu, in which w is a function of «and uw; and though it be true that u must be a function of x, yet it must be found from the equation what function it is, And though in this case it is easily found that Cx ; ; u =F Yet there may be cases in which this step, at present abso- =e lutely necessary before differentiation can be performed, may not be possible with existing algebraical forms and methods. Such, for Instance, oe = © — asin wu, in which y can only be expressed in terms of x by an infinite series. But still w zs a function of w, that is, a given value of x will allow only a certain number of values of w, an increase of x gives an increase or decrease to u, those increments haye a ratio, are comiminuent, and their ratio has a limit. The question is, how are we to extend our power of differentiation to such cases, 86 DIFFERENTIAL AND INTEGRAL CALCULUS. We must first consider functions of several independent variables, in which all the variables increase together independently of each other. If w be a function of # and y, it is indifferent as to the result, whether we first change x into z -FA, and afterwards y into y + &, or whether we allow these changes to be simultancous. If the changes be made successively, 2*y becomes successively (« + h)*y and (@ + h)? (y +), the same as if both had been made at once. Here h and k are supposed to be independent of each other. When w is differentiated time ‘after time with respect to a, the results are du du du du du du 15 de da” &e. : and u dy dy? dy®? U when w is successively differentiated with respect to y. But we may differentiate times 1m succession, sometimes with respect to ' du one, sometimes to another. For mstance, we may have Ki au © Oe d, du ; ' mh. the first of which directs to differentiate w with respect toy, dy dx and the result with respect to a. The method of notation is thus ex- tended (a reason for which will be afterwards given) : one is written ee algae is written ok: dx dy dx dy dy dx dy dx da is written cas i be tai is written ahs dadxdy dx2dy dy dy da dy? dx d d du ee Pu d d du 7 du — —— ; ——_-—- — = — is wr —____—_— dy da dy dydxdy da dy du he easanirle: dy dx where the apparent numerator (p. 54) shows how many differentiations have taken place, and the apparent denominator, looking from right to left, shows the variables employed and the order of the operations. We now proceed. When a is changed into x + A, w is changed into du u+ — .h + Vk? by Taylor’s theorem, where all that we need remember of V is that it must be a function of x and y and A, and does not increase without limit when / is diminished without limit. If in this we substitute y+ instead of y, a similar process shows that wu becomes w+ ee w+ WR dy du. du d du ae aR nod . ° e it ee i= boa ae 9U 2) dx’ dx dy dz caidaak ; 2 ON IMPLICIT DIFFERENTIATION, 87 d u + = h + Vh? becomes y +. hi h-. ai ‘y dx da’ dy 9 + Vas jE hk+-Wie dx dy dV + = Wk-+- TAH. Li*h?, dy where W, T, L, are certain functions of w and y, &., which might, were it necessary, be expressed. When we have a set of terms of which it is only necessary to remember that they do exist with finite coefficients, we may merely put the parts of which we desire to be reminded, by them- selves in brackets; thus we write the preceding result du du Ub Tht ab + AR, hie IP, Bh, Bh, BIA} which is to be considered as equivalent to stating that there are certain additional terms of the form Ph?, Qhk, &e. The preceding is what the function becomes when x + handy +k are simultaneously substituted for « and y; and the Increment of w is therefore du dx Observe that if x only had varied, the increment would have been du ITER h+ oy k +- {h?, hk, &c. } du du dh 2. erie a ae ee re A+ {he}: and =k + {#}, if y only had varied. When and y vary together, the increment, as far as the first powers of h and k are concerned, is made by an addi- tion of the terms just written, but there is an intermixture of results in the remaining parts. Thus, a variation of gives to wu the increment du : « only oe h+ {h?} du ] —~}; -2 y only Pea AL u du wire both x and y ~h-+—k + {h®, hk, k*, h’k, k*h, h?k?\, dex: dy If we now suppose a quantity z, which has hitherto lain constant in w, to become z + /, we find by a repetition of the process that the total increment of u is now a + i + ae L+ th’, *, P, hk, &. &e. } dx dy dz and so on: whence if we denote by A.w (as distinguished from Au) the increment which x receives from several variables 2, © 3, &., we have this result. d d du Ee ae Av, + as At, + a6 Av; + &ec. dx, az, dx; + {(Az)?, (Az, Az,), &. &e. } 88 DIFFERENTIAL AND INTEGRAL CALCULUS. Now this being true for any values of Ar,, &c. remains true even if those-values should be so taken as to satisfy given conditions, and even though 2 7», &c. themselves enter into those conditions. But as this is a difficult point, we prefer to take a more simple case in illustration. ° Return to the equation 2 bt hy = ort or ht oY @ +m) S all that is requisite being that neither da, p'x nor $"a should be infi- nite. This being true for all values of 2, remains true, even if for 2 we substitute a function of x; but it would not be convenient to deduce it on this supposition, because we should need to remember that x becomes x + wa, and contains an x which varied, and an & which entered with the variation. But having proved this equation for all values of h, we have proved it among the rest for all values of /, which are also values of any given function of x; that is, we may substitute ya, or Uw (a, y) or anything else, for h. Indeed, we ought rather to say, that having proved the equation for all values of h, a fortiort we have proved it for those of any given function of x. Let us then take the following case: wis a function of 2, y, and 2, of which z is a function of 2, y; and ft, and y of v and ¢, and « itself of ¢, or ua oh (a, ¥; 2) 22% (a, y; £) y = x (2, 1) rot db, xX, and @ being functional symbols. We might evidently make a a function of ¢ only by substitution, for we have y=x (ott z=%¥ (at, x @t, O), t) u = > fat, x (at, £),¥ (at, x (at, t); t) } where ¢ only enters. For instance; let usacyz, z=ry, y= t-+-2,.e=snt g=ttsnt, 2= sint (¢-+ sin?) t ux sin’t (t + sin f)’.¢ : u it ae from which last formula we might find rk But the question is, how al... (o) ey: Btls shall we find er without this intermediate process of substitution ? First, let us consider u as @ function of a, y and 2 only, and take the universal equation du Aa dx du du Ara+ a, Ay + a Az+{(Az)’, (Az Ay), &c.}e ees (1.) du d This is true for all the values of Aa, &c.; but the diff. co. oe ii, at P dx dy’ dz are partial, cach supposes -ts variable to be the only variable, our theo- rem showing how to form the total increment out of the partial incre ments. This theorem being always true, +s true when Az has such @ value as would be given to it by assuming the second equation 2 wu (2, y, 2) which gives dz dz dz sofas eee Rye tas aga f Sivas Az= re ao + Ay4 a At + {(Ax)?, &¢.5 Rene ba ON IMPLICIT DIFFERENTIATION. 89 These two equations are true together for all values of Az, Ay and At, but not of Az, for that must have the value just assigned. Suppose, then, that we assume the third equation y = x, (2, t) which gives dy di ay a Ar ++ At + { (Av) Gay we. (3.) The three are true for all values of Av and Aft, but if we assume the fourth equation c= at, we have dx . Arm G Att 1 Ad ey and the four together are true for all values of Ad, but Ad being given, they determine Ax, Ay, Az, and Au. Before proceeding further, we shall observe by the following table in how many different ways ¢ enters into z. St ey F aa err ak 2 agen t as Rivet Y asc t t Hence it appears that « contains ¢, after all substitutions are made, in seven different ways, as follows :— 1. « contains 2, which contains ¢. 2. u contains y, which contains x, which contains ¢, * 3. u contains ¥, which contains t +. wu contains z, which contains x, which contains é. 5. u contains z, which contains y, which contains 2, which contains f, 6. w contains z, which contains y, which contains ¢. 7. u contains z, which contains ¢. du dt have in our result the effects of every one of the methods in which ¢ enters. With what we know of the rules of differentiation, it is incredible that two functions should contain ¢ in different numbers of ways, and not exhibit some sort of difference in their diff. co. We proceed to find Now, before proceeding to find —, we may presume that we must ‘4 the actual value of aah dt In the third equation above deduced, substitute the value of Av from the fourth, in the term which has the first power only. This gives dy (dx da a= “~ Gy ft + {at} ) + = At + {(Ar)’, &e.} dydx | dy dex di* ae In the value of Az, substitute the values of Av and Ay. dz dz dz (dydx | dy dz —s — — —— —— At ek ear a —s At “anit At Ar eee ar * = da di * dy \ dx dt i dt = dt ae 3 ; or by = ) ae + {(Ar)’, &e., (Ad"} 90 DIFFERENTIAL AND INTEGRAL CALCULUS. Then substitute in Au the values of Ax, Ay, and Az. du dx du ¢: daz 2) ip du dz dx PEE Pata ae Ky sehen er ip di dy\da dt‘ dt dz dx dt du dz (2 dx . dy Sa, dz 7 dy \dudt Uh ae ++ (terms containing powers or products of Av, Ay, Az, At.) We now come to the reason why the specification of the higher terms would be useless. When we take such a term as PAvAy, and divide it A ; eat ; by At, we have PAx ee which, since y has a finite diff. co. with respect to t, is itself comminuent with Az, that is, with Ad: for P and ad remain finite, while Av diminishes without limit. If, then, we divide the preceding equation by At, and take the limit of ey all the terms included in the brackets disappear, and we have d.u du dx | du dy dx du dy | du dz dx dt da dt ' dy daw di dy dt dz dxdt dudzdydx . dudzdy | du dz eee — «—— ssa dz dy dx dt ' dzdydt ° dz dt iin k's MAE Te du : ; : We write Tr instead of oe to remind us that we have a differential coefficient which implies several different entrances of the variable : this is called a total differential coefficient, when it is necessary to dis- tinguish it from the separate terms belonging to the several ways In which ¢ enters, which are partial diff. co. Looking at the result which we have obtained, we see seven terms, very closely connected with the seven ways in which ¢ has been shown to enter vu. For instance, ju contains @, which sam | Hence ot du dx \ Seer | tains 7. al g ju contains ¥; which con- re ea Se du dy dx tains #, which contains 7. dy dx dt wu contains y, which con- j du dy ' tains ft. pee the term 7m me } and soon. Hence we see the following general theorem. If u be a function of ¢ in different ways, find out each way in which i enters, and if one of those ways be thus ascertaimed, w contains A, du dA dB dA dB dts having found all these terms, add them together, and the result will be the total diff. co. of w with respect to 7. We see also that, in taking the increments, we may express all except the terms containing the first powers of the variables by a simple &c., since they disappear when the final limits are taken. If we forgot them altogether, the error would not affect the result ; we could not be said which contains B, which contains ¢, take the term ON IMPLICIT DIFFERENTIATION, 91 to have reasoned correctly, but such an error of reasoning has been shown to produce no erroneous result. To make the principle of the preceding more clear, we.shall now take a more simple instance. Let u= $ (2, y), where y = wu: that is, let w contain 2 in a two- fold manner—1. because it actually and explicitly contains z—2. be- cause it contains y, which is a function of x. Give x and y any incre- ments Av and Ay; whatever they may be, the following equation (when the meaning of &c. is properly remembered) follows from Taylor’s theorem. _ du du a.uU= —Ar+ — uC. § A.u as Ax ay Az -+ &c but if we require that the second equation shall exist, it gives l Ay = —/ Ax + &c. ~~ dx d du d or py apa sa jcc Aaa z+ «c., dae dy dx divide both sides by Az, take the limit, and we have d.u du | du dy de dx " dy dx’ which, by the preceding rule, would follow from u contains x directly, and wu contains y, which contains x. d.u du ee It appears that aan and Tn ote totally distinct, as might be expected. x “3 The second merely supposes that in the equation u= ¢ (a, y), v receives an imcrement, and y remains constant; but 7 this case dx implies that another equation exists which makes y a function of 2, so that x cannot be changed without y changing also. If we suppose Uu= ty’,y = wv’, we have du du dy d.u ame 9/2 eo a 2. Let x be the independent variable. We have then Lane nis aay oth eae 68: oe 1—b 1—b dz” Ib ‘dat. 1p 3. Let y be the independent variable. We have then a—b 1—b d.u a@—b dia lb CR Ny eet eer al Ged Te ce dU) poral But this’ previous reduction may be mconvenient or impossible. If we now take the general case u = (a, y) u = ¥ (a, y), we see that : Sie: du we shall have two diff. co. to signify by qa? one from the first equation, x one from the second. To distinguish between these (which are not the same) write the functional symbol of the equation which is used, instead re and the second pa Both are diff. co. of wv, but dz dx under different circumstances; the first a consequence of u = ¢ (a, y), the second of w= % (vw, y). The co-existence of these equations may lead to relations between the two, but is no reason for confounding them. ‘This co-existence requires the co-existence of _ dp dp Aap o ze Aa a dy Ay + &e. of uw; call the first dus dfs Au = — Ar+ —A &e. dx : dy ge in which Au, &c. are to mean the same in both; for though each equa- tion is satisfied by values of Au, &c. which do not satisfy the other, it is ‘ON IMPLICIT DIFFERENTIATION, 93 not of those values that we enquire, but of values u, #, and y, which satisfy both, of the changes of value under which they continue to satisfy both, and consequently of the increments which satisfy both the : Sods ges, Lite equations of increments. Now, to find the limit of the ratio Az? We ,r must express Av in terms of Az, or eliminate Ay from the preceding, which will give d dp d lus ip ap at = (a ane Ar-+ &e dy dy dx dy dx dy dpdw db do due dg dou dx dy dx dy Qt dy dy Pn ae ih du db de dws dp? dy dy de dy dx dy ; Ul d.x : i ; we might write these —— and =, and this notation might be conve- s d.x d.u % nient in some cases, but where one dot is sufficient, the other may be dis- pensed with : it being always remembered that the diff. co., with the point, distinguishes a diff. co. derived from more than one consideration, whether the additional considerations be expressed in equations, or implied in Suppositions. The preceding method is one by which these questions may always be reduced to first principles, but the rule already laid down (p. 90) will be sufficient, when understood. To repeat the case just solved, let us suppose Uu=* (a, y) u=¥% (@, y), from which it follows that x and y may be considered as functions of 1, Taking this additional supposition, differentiate both sides of these equations with respect to u, observing to write the dotted diff. co, Wherever the supposition is used ; and, also, remember that x is sup posed * a function of u, and y a function of u. We have then i1_% d.v , db d.y dx du dy du } _ aida dbd.y ad dx du dy du’ 4) eg d.4 from which two equations re and a can be found by common alge- | du dit bra. These, as found, may be made to coincide with the result of the particular case in the last page, namely, P(Qyv=a+y % (2, y) sax + by For we see that * uf . EE ei Ta a b dx dy dx dy Let us now suppose that w is a function of x, Y, and u, or Uu=d(x, Y,u), from which it follows that there are two independent variables : for x and y being taken at pleasure, the equation may be satisfied by finding * Observe that these suppositions are always implied in, and may be deduced from, the equations, 94 DIFFERENTIAL AND INTEGRAL CALCULUS. the proper value of wu. This equation implies that w is a function of « and y only: thus from é ot y ua=ex+y—wu can be obtained w = 3 s d.u GG aa using this supposition, we want to find el and ag which are partial diff. co., but not the same as and i The dot denotes the intro- duction of a supposition more than is dérectly shown in the equation, namely, that w is to be considered as the function of xv and y, to which it might be brought by solving the equation. Taking x as constant, and considering # (2, y, wu) as containing y two ways 1. directly ; 2. as containing wu, which is a function of y; and differentiating the equation u =? (a, y; #) on this supposition, we have dp d.u_ db . dp d.u dew dy dy dy du dy. dy — , a du Again, if we regard y as a constant, dp d.u db dp d.u du dx dx dx du dv age dp * [ss du : do, dow a Goes For instance, if w= x —yu, we have 5 tag Uy a aaa Th 1 d.u —' i therefore iho ce Ee aay GO de il+y dy his Se the supposition which gave these, in an explicit form, we have 2 d.u 1 dw © ry —U Thy Caro. bh edy Mig hg) lee which agrees with the preceding. In most treatises on the Differential Calculus, there are but two terms of distinction between diff. co., total and partial. The reason is, that the additional distinction we have made is left till particular cases re- quire it, and is not usually formally proposed. We now introduce the following additional distinction of explicit and implicit diff. co. and the following definitions (the two first of which agree sufficiently well with the senses * in which they are commonly used) will enable the student to apply to each of the processes in this chapter its proper name. Partial.—The function differentiated may be considered as of more variables than one, nothing expressed or implied in the equations given being to the contrary, and one only is supposed to vary. Total.—T he independent variable enters in different ways expressed or implied, or both: and is considered as varying in all. * They cannot altogether agree ; for the distinction of partial and total diff. co. is frequently used in more senses than one. If, therefore, the student, at any future’ time, find himself puzzled by the use of these words in any treatise on the applica- tion of this Calculus, let him ask himself whether the distinction of explicit and implictt be not intended. Now, if we actually produce U ON IMPLICIT DIFFERENTIATION, 95 Explicit.—No variation considered except as it affects one given equation. All common differentiations, as in Chapter IT., are explicit : no supposition (except assigning a given quantity as variable) drawn from other source than the equation itself, affects the result. Implicit.—Any other than explicit; affected by the co-existence of any other equation or supposition. Total diff. co. are implicit, but distin- guished on account of their frequent occurrence. The terms partial and total are not contradictory, as might be sup- posed from their etymology (consistently with common usage, we can- not avoid this inconvenience). A diff. co. may be partial, inasmuch as it supposes only x to vary, and not y or z; but totad with respect to a, inasmuch as the function differentiated may contain x directly, as well as through p, g, &c. For instance, let u= ¢ (t, Y, 2; D, 4, 7) where P, 7, and r, are themselves each a function of a, y, and 2. The explicit partral diff. co. of wu with respect to x, is simply ei but the partéal x diff. co. considered with reference to every way in which 2 can enter (which we should think might be called the complete partial diff. co. to avoid the objectional phrase total partial) is iu | lp d lip d Id dr ‘ Bhs tds, deep dp dq © as in p, 90. dx ~~ dz dp dix dq dx It would be impossible to specify all the various methods and combi- nations of equations which present results of differentiation worthy of a distinct name. We shall proceed to take some of the most important cases, Let u= (a, y) = 0, required the implicit diff. co. - The sup- wv position is, that, by solving this equation, we may make y a function of 2, If « = 0, that is, if the values of x and y are always to be so taken simultaneously that «= 0, we have A.u = 0 for all changes of value : ae Athy of x and y which the supposition will allow, Consequently, as x tat be ulways 0, and its limit is 0, or — = dp udp. dbdia d. de Now 2 — & ere or eee pee dx dx dy dx Ax do ai For instance, let 7 — (ogy)? = 0 = 6 (a, Y)s dd dp oe 1 a I — (logy)? . log log y an = — x (logy)*"'! x 8 ey _ y—y (ogy) log log y aah x (log y)* To verify this, observe that 7 — (log y)* gives loga =a log. log y, or og v log x log x ere 2) e @ S 1 —log x >=. x & x ; ii Gar dx tes 96 DIFFERENTIAL AND INTEGRAL CALCULUS. : let the student try to make these results agree, remembering that . definition €°8" = a. Let ¢ (a, ¥; a= = 0, whence it follows that z must be a function of x | ‘nine the icaplndly pirean te wale ane ae | and y. To deter mine the implicitly partial duit. Os ge =m an a : . anes d.u d As before, u==0 gives the complete partial diff. co. ar and ey | severally = 0. This gives ao dp d.z db db d.z._ ae ie a =e, dy ide dy ap ap d.z_ dv d.z “dy dg. dh dy. a dz Jawa DOV tad, wa ae et tee mC )* avin 1 dp wa 2 oe] Wig) ee, SE Show hae ee Se ge Play ee ae Let u=z > (y+ au), from which it may be inferred that wu is a function of 2 and y. Required, on this supposition, ae an nd = : “. Let y + vyeu = V, which gives wu = $V. x d.V lyu d. cam ut ee SE yu tay se ae it betes yankee * Saaa ais “du ee ae | du dVd.V_,, , du | dee ae de 208 (yet eS) d.u _ IPV d.V ; saa en d.w P'V yu do pV dx Tad wu dy 1—a2'V ww’ which gives this simple relation sS = Wu For instance, let w == ¢’+*°s" (show that this amounts to supposing | an | Pee gs eae i Oe L (v-1)? dy a-1 OV sl ON eee’ ue log a : ul d.u _ us’logu du _ ue Mro uae! dy wae show that these agree with the preceding. MEANING OF AND PROCESSES IN INTEGRATION, It must be observed that if x be a function of wand y, and if du , du re ae ; n> I re where P is a function of x and y, this same relation is true for any function of x. For, let ; fu be any function of u, and mul- tiply both sides of the preceding by / “uy which gives dfu du _ Pp dfudu — dfu eve: Efe du dx ds dy” ae dy’ Show that if ube a function of z, which js itself a function of x d.u dz d.u dz dx dy dy de > °2 where the dot reminds us of the implicit supposition, and y, tenet Cuaprer VI, MEANING OF AND PROCESSES IN INTEGRATION, Tue Integral Calculus is the inverse of the Differential Calculus. Thus one question of the latter being “given a function to find its diff. co,?? the corresponding question of the former is « given a diff. co. to find the function from which it came.”? The original function is called, with re- Spect to its diff. co., the primitive function : thus, 2.x being the diff. co, of 2°, x” is the primitive function of 22, Thus we may easily see, that 2 . ” ee ° av . & ° with respect to zx, the primitive function of — js ame, but with respect a y to y, the primitive function of = is x log y. But a primitive function, merely considered as the inverse of a diff, co., would not be of much use. The following theorem will show the point of view in which the necessity of finding primitive functions actually presents itself in practice. Let dz be a function of x, and let a4 anda +h be two limiting values ofr. Let h, as before, be divided into » equal parts, each of which is @ or Az, and let xv pass from a to a + hk through the steps a,a+u, Mer 20,. . .a.+ (n—1l)w,a+nw or a +h. Let every one of these values be substituted in the function, and let all be added to- gether, giving $at+d(atv)+b(at2Qw)+. , -+¢(at+n—le) +46 (a+n); each of these lying between given limits, the sum of them all may he made as great as we please, by taking a sufficient number, that is, by taking n sufficiently great. Multiply this sum by wo, giving iPatd(atw)+o(at+2w4.., - + h(a +n) } 0, Which we do not now affirm can be made as great as we please, for the greater the number of terms in the first factor, the greater is 7, or (since n® =h) the less isw. And we can even conceive it. to happen that the taking a greater value of n should diminish the preceding product, or i 98 DIFFERENTIAL AND INTEGRAL CALCULUS. factor should be more than counterbalanced f the second. We can immediately show, however, that the preceding product can neither imcrease nor decrease without limit, provided a be always finite between 2 =a andaw—a+th. Let C andc be the greatest values it can have between these limits; then the preceding product must always lie between _ (C+C+C+.. - -+€) and (¢+e+tep. . + c)w nm +1 terms n+ 1 terms, that the increase of the first by the corresponding decrease 0 or must lie between (2 + 1) Cw and (n-+1) cw, or between C (nw + &) and c (nw+w), or between C (i+w) and c(h+~). That is, there must be a finite limit, lying between the limits of the preceding, which (when n increases or » diminishes without limit) are Ch and ch. This summation, of which we wish to find the limit, we shall proceed to illustrate by a few cases, as follows:— Let dv = 2, then the summation required 1s ee et ee eae +] or (n4+ljawt+e'U+2t3sr. « : +n) or (n+1) aw +o'n nt vw + nw? he+h or (nw 4a) ace = or (ht o)a+———, putting h for nw. We have thus eliminated » (which is to increase without limit) by means of a relation which is always to exist between n and w (which diminishes without limit), and in the form to which we have now reduced the product, its limit 1s evident, when w diminishes Poe ale h? without limit: that limit is ha + ot and we may observe that as diminishes the preceding diminishes towards its limit, thus verifying the surmise above thrown out, that the increase of the first factor might in certain cases be more than compensated by the diminution of the second. + ne) Next, suppose Or = 2". {P+ (atw)?+ (at 20+. - - +(a + nw)*}o which may be easily reduced to (ntl)ao +1+24+3+ . . - +n) 2aer%+(P+2'+. +» +-n*) W's We want then to find the limit of ie h for w write its value -, and the preceding becomes n 1 cooet i 7 hier ht MCRD i (1 oe = ease B + "+2 ag a we hh? n nN in which if we suppose 7 to increase without limit, and write for the two latter fractions their limits obtained in p. 85, we have for the limit of the preceding summation h? ha? 4+- a+ >. 1a? +- Wat 3 Let u = log: we wish then to find Slog a + log (a+ w) +...+ log(at+ nw)} ow, and here we are stopped, for there is no process of common algebra for representing in a finite form the sum of m+ 1 terms of a series of MEANING OF AND PROCESSES IN INTEGRATION. 99 logarithms, such as here appears. We must therefore look for other methods ; but first we shall lay down names and symbols for summations of the preceding kind. The limit of the sum of a series of terms, such as igat+d(atwyt.., +9 (a+ nw) }o or $aXwt¢ (ato) X w+ oh + (4+ nw) x w, is called a definite integral: an integral, because it arises from putting together the parts of which a whole is composed (or rather from the mit of such a process) : a definite integral, because the first and last values of the variable, a and @ ++ Nw, or aand a +h, are definite, de- Jined or given. And since each term is-a value of the function inter- mediate between ga and $(a + h), multiplied by the interval between the values of x corresponding, we may make dz x Ax the representa- tive of any one term, and 2 (dx. Ax) the representative of the sum. l And, agreeably to the analogy by which we made a (a total symbol, Ms en, Ai See p. 50) represent the limit of re an algebraical fraction, we shall cause /oxdzx to stand for the limit of the summation 2 ox Ax, when Av diminishes without limit. The symbol /” is, or was, an italic #7 We must have some symbols to denote the limits of the integral which were used, and the method of doing this has not been well settled. by custom. Sore would express the result by Si bade, others by I dedz, from x = a tov=ath. For ourselves, we prefer the first of these two; but should incline to write the limits above and below the last z, thus SJ pr daet* All, however, have their conveniences, and we shall adopt the first, simply because it is used jn many works of high reputation, particularly on the continent. When we say that a-+eh h?2 f ade =ha+ —, @ a 2 we mean that the definite integral of xdz (why we use this instead of ‘x will .be afterwards explained) or the limit of the summation, the extreme am : : h? values being the lower limit, and a+-h the higher, is ha -- me Now the value of [2+*dedx, when deduced, may be applied to any value of a+h, or of h, provided no infinite value of px occur between da and o) (a+h). And since ath is a value of x, let a itself (the general symbol) stand for its superior limit in 4 a*'phedx, which gives in the particular instance first cited, . ded (w--aP—s_ x? —g? Me Ok == (2—a) & - ——__* me Cu oe 2 This is generally denoted by S dxdzx, meaning the limit of the summa- tion in question, from a to v, or the indefinite integral beginning at t=a (sometimes it is said ending at z=, which is an awkward way of saying that the last value of # is indefinite). And in this expression, when « only varies, its initial value a may be what we please, or an 9 a. ee, a arbitrary constant. Whence — 5 isan arbitrary constant (only in this ; a” Particular case, it must be negative). Let 5 be called C, whence H 2 100 DIFFERENTIAL AND INTEGRAL CALCULUS. 2 we find > + C for the above indefinite integral, where v may be what ends upon the arbitrary value of v, at which we we please, and C dep choose the summation to begin. We have thus two new expressions primitive function, or the function w it. 2. The indefinite integral of dxdx, meaning the limit of the sum- mation above described, beginning at any given value of a. Now we observe that the primitive function of mx must contain an arbitrary constant: for by the rules, if yr differentiated yield dr, Y2 + C does ihe same, and is therefore a primitive function. And we also see that the integral of drdx contains an arbitrary constant depending on the ‘nitial value of. We have given these two new things different names, because they are derived in different ways: but we now proceed to show that they are the same: or that the primitive function is no other than’ the indefinite integral. This will easily be seen in the instance of 2, 2 and its indefinite integral the same. connected with pr, namely, 1. Its hich must be differentiated to give whose primitive function 1s ey + C, Let us now return to the equation 2 W $ (atu)—¢a = ba. w +b" (a+ Ov) 5» +20, &c. él Gio and supposing 2w=h, substitute successively a+, @ side of which, as a+nw or a+h, adding together the results, the first before, gives 6 (4+h) — da, and we have b (ath)—a = {Patd! (ato)t. +: +9! (a+n—lw)}.o + (iat Gu) +0" (at 6.) + . )\ > tm Ge in which we know that @, 0, &c. are severally less than unity, and in the highest of which we see @ + (2—1 + 9-1) 5 which is less than atnuw orath. Let C be the greatest value of @”x between 7 =a an 3 ¢] _— : ; W W a—a-+h, then the second series must be less than 2C ey Cnw 5 or W Ch a One term added to, and afterwards subtracted from, the first series, with the preceding consideration, gives p(at+h)—pa=(P'atP'(ato)t--. +¢(a+n—lo)+¢'(a+ne)§ 0 —¢! (a+nw).w + less than Ch = : the last two terms of which are comminuent with w. Now the primitive function of p'x is oxr+C, C being any constant : while the term con- taining the series has for its limit the definite integral of $'2.dx from n—atd@naoath. Let drmer + C,, the primitive function ; we have then Py (a + h)—,a = p (a > h) — ga, and finally diminishing w or increasing 7 without limit, we have $b. (a +h) —da = fetdle. de, OF making a+h= «as before, that is, letting « represent its superior limit, we have : MEANING OF AND PROCESSES IN INTEGRATION, 101 dx—ga= {thle dr, and a being an arbitrary constant, so is — Pid, giving at last So'v.dx =¢02+C,= dx +C+C,; 80 that the two apparent arbitrary constants are only equivalent to one. For the condition that C and C, may both be what we please, merely tells us that C + €, may be what we please. The indefinite integral and the primitive function being the same, we shall use the former term, where distinction is not necessary, to denote both. The following will now be easily intelligible, du If ——=e utC= /zdr dx + i dv “dr dr ee lor > + C — =logxr—loga — = log b—loga of x ay ie soa r Yr dix dx 2edx —=loge — =logxr — ] —) aloe 1 av é ; & av We thus see ourselves in possession of a method for finding the limits of the sums of series, in cases where the sums themselves can- not be reduced to any more simple expression. Thus, in the last example, we have found the limit of ] sai ftetagte toh is acinar when w diminishes without limit. [The language of the infinitesimal calculus is very well adapted to illustrate the relation between a diff. co. and an integral. If x increase by an infinitely small quantity, 2 is increased by the infinitely small quantity 2rdz: so that the transition from a? to (a +h)? is conceived to be made by the successive addition of an infinite number of infinitely small quantities, namely, 2adx, 2 (a + dr) dx, 2 (@ + 2dzr) dx, and so on. But the total of these being that by which a? is increased so as to become (a+ h)?2, is (a+ h)’—h?. The whole difference of two values of a function is conceived to be made of an infinite number of infinitely small parts (as in p. 26); but for each of these infinitely small parts is substituted another, infinitely near to it, so that the Sum of all the errors committed js itself infinitely small. Com- pare this with the reasoning by which the second series in (A) is shown to diminish without limit. The real differential of 2? is (z+ dx)?—2 oy 2rdxr + (dr)?; but if dx be infinitely small, (dr)? is an infinitely small part of dx, so that 2 (dr)? when n is infinite, being ndv x dx or hdr is infinitely small. For it is the condition of this process that 2 and dx shall be connected by the equation ndr=h, We have here (as we shall always do in the remarks in [ ]) used the language of Leibnitz in its broadest form: the student can omit it entirely without breaking the chain of investigation ; but we should recommend him always to consider the language here used, in reference to every problem he meets, for when the method of rationalizing the Single false assumption in whith the whole error of the system of Leibnitz consists, is once understood, he may depend on it that there is no other like it for giving power of application. ] It is not necessary that in the transition from a to a +h, the incre« 102 _' DIFFERENTIAL AND INTEGRAL CALCULUS. - ments of the value of x should all be equal. They may follow any law which makes them all comminuent, In the process of page 100, let us suppose this alteration, that a first becomes a + w,, next a + 0, + a, and soon up to @-+ wo, + 2 T+: . +o, orath. Then we have, as before, P(atot. - to) — (a+, + Rag cs Re Opes) +h(atu,+-. - ~toa,n)—Pa@tor: - . & w,—2) +h (a+ 0, +o) — > (a +o) + (a + w,) — Pa = o(ath)— Pa: e 8 9 and also: $(a + w,) — $4 = dla .o, +b! (a+ % 1) = ”) (abort W2)—P (ato)=P (a + ) + 9" (a+ or, +0, we) = ke. or, d(ath)— da = Pa. + d' (ato) or - ee Cat ep ct ley sis) oll w,2 w + fp’ (a + 9 w) my +o" (ato, + O, 2) 9 Tue 2 + Pl (a+ wb Wy toe. + On On) a Now, since o, + @, +..+. FOn = hycand.0, 942+ 4. are severally less than 1, there is no value of x here employed, but what lies between aanda +h, both inclusive: let C, as before, be the greatest value of | ox, and let © be a quantity greater than any one of w, @y ++» but fad comminuent with them, so that 7@ is a finite quantity, and * we have o(ath)—ga = the first series above written, 2 0 + less than nC see Che ; so that taking the limits of both sides, it appears that (4 + h) — da is the definite integral with unequal but comminuent increments, But it is also the definite integral with equal (and therefore of course comminuent) increments: these two methods of integration thereiore give the same result. A yery common case of this process is where it is required to integrate dx . e ° . ° 7 fu X a? where a isa function of ¢, and the integration 1s to be with respect to ¢, from ¢ = btot=— b+ &. dx he de Nees vo ; If we suppose 2 = Vt, ra y't, this is the same as requiring to find * The completion of the first series, as in page 100, is not absolutely necessary, for the additional term 1s comminuent with o,. MEANING OF AND PROCESSES IN INTEGRATION. 103 ” Sie St. Wt. dt. Let fx be the primitive function of fx; we have then* - dfiz dfx daz . ax (*, dx dfx “a fae rea 2 a | fea a= ee dt. c Now since, dr + C = fid'e . dz, and x is —, we see that *ddxv af oe dx is $x + C, and therefore 4 b+Lk : de b+k Af vst : i fe F dt of 7 dt = fie (b + k) — Tied =fi(a +h) met Tate supposing @ = Wh ath= &’(b +k), aandath being the values of & or wt, corresponding to band b + & for values of t. But this last result (fiz + C being the same as Jc fe dv) is the same as Ji ** fe dz ; whence we have b-+k 1 ath x fe i Gf fe dx, b provided only that 6 and b + k are those values of £ which give a and a-+hfor x. If ¢ and x themselves stand for their superior limits, we have - t ax ec . J eZee [i feae We shall now proceed to some methods of integration ; but first we shall remark, that though we can differentiate every function, we cannot integrate every function. Integration is an énverse operation to dif- ferentiation, and though we found many functions appear as diff. co. yet it would be easy to name functions which neither appear, nor, in our present state of knowledge, could have appeared. Imagine, for ex- ample, a given ellipse, and let a starting point be taken on its circum- ference, from which measure the variable arc $ on one given side of the starting point, and jet A be the variable area included between the arc and its chord. Then A is evidently a function of s, at our present point wholly undetermined. We do not know whether our means of’ expression are sufficient to express it or not. We can take powers, logarithms, sines, logarithms of sines, sines of logarithms, &c. of s or functions of s, and combine them by addition, subtraction, &¢., but we Cannot say whether any finite number of such processes can compuse a formula which shall represent the value of the area required. Suppose, which may happen, that it is inexpressible, it does not therefore follow that its diff. coeff. is inexpressible ; consequently, we may have an éx- pressible diff. coeff. with an inexpressible integral. To illustrate this, let us Suppose we had commenced this: subject with common algebra only, and without geometry. By common algebra, we mean to include the operations of addition, subtraction, multiplication, division, the * We shall not stop to prove that functions which are always equal have the same primitive functions or integrals. We take as an axiom, that the same operations performed on equal quantities give the same results, 104 DIFFERENTIAL AND INTEGRAL CALCULUS. raising of powers, and the extraction of roots, together with all com- binations of them in finite numbers, that is, entirely excluding all infinite serics. We should immediately observe that our differential J ‘ : ; calculus never caused — to appear as a differential coefficient. We © should find ourselves able to give the integral of 2” generally in the form va +l ; ‘ 7 + C, but if we attempted to apply this to the case of #7 ', or 17 ] a we should find a +1 1 ; it aera + C, OF 5 + C, an DO eg form. If we took the following expression, b prt qrt} p+} as qnt} fone ee ee : n+l ab Sk n+l 1-1 - 0, for —-, or= fo —1+1’ 0 the preceding expression, and should conclude that the integral required is the limit of the preceding expression, on the supposition that 2 ap- proaches without limit to — 1.. It would not be very difficult to find this limit in any particular case. Say that a=2 b = 3, and, to get an approximation to the limit, make 72 very nearly equal to — 13 say n==z —1°0001 orn+ l= ‘0001. We should find the limit in question near enough for most practical purposes by calculating we should see that the supposition n = —1 gives 000 060 3 Le tee 9 01 ‘0001 of arithmetic, since a tedious process would enable us to extract the ten-thousandth roots of 2 and 3 to any degree of exactness. And by calculating for a number of values of a and b, we might thus get a table of values of wf ba—tdx sufficiently numerous in instances, and exact in each instance, for practical purposes. But these tabulated values would give no information on the properties of the function of @ and 6 in question. Now it so happens, that this process has been already forestalled in algebra in another shape. In looking at the equation y = a’, it appeared that to find y when @ is given, is an operation of common algebra; thus, , which is (with difficulty) within the compass of the rules 2S 3 st is not difficult to assign 2” (12)° &c., with any degree of nearness. But to find w when y is given is a perfectly new question ; for instance, to find what value of « satisfies 3= 2”. It is true that certain pro- cesses may be found by which the value of x may be approximated to, and that these processes contain nothing but common algebra; yet whether we consider the question as one of common algebra or not, it is obvious that we have a new process, not contemplated when we laid down the most simple relations of magnitude. By giving # a name to designate its relation to y, by calling it the logarithm of y to the base a, and by investigating the nature of logarithms, we come to simple rules of computing them, and to methods of making tables of them. Hence, } when we begin the Differential Calculus, we naturally ask for the diff. co, of a logarithm among the rest, and having found that (to the base | MEANING OF AND PROCESSES IN INTEGRATION. 105 . a ° . e . 1 Which is ascertained to be the most convenient base) it is—, we are vi ; : dx : . prepared to assign the integral of —. But let it be remarked, ‘that x this is entirely owing to our having been led to pick out from an in- finite number of equally possible suppositions, the relation y =a", and to investigate the nature of the connexion of v and y. And this truns- scendental (as it is called) log x, has an algebraical diff. co. But it may happen that there is an infinite number of other relations which require new names to express them, and yet undiscovered properties of expressions to compute them, having all the while either algebraical or known transcendental diff. coeff. Tf this case ever arise, we are in ; ee dx’ #, precisely the same situation as we should have heen with | — if we x had not previously considered the theory of logarithms. Our first methods of integration must be the observation of differen- tial coefficients, and the reconversion of each into an indefinite integral, Understanding always by Sox dx the integral with an arbitrary, but given, lower limit, and 2 itself for the higher limit, we see that if Px differentiated gives @r, then Sibxde isPrx+cC. It is usual to omit the constant, as an attendant of the integral sign so well known that it is unnecessary except where we are actually applying the integral cal- culus, and may be dispensed with when we are merely ascertaming integral forms. We can thus find. the following theorems : l. S tu +v—w)dr= S udzx 4 Svdx — fwd. To prove that these are the same, observe that differentiated they give d the same result. For LP nin mir, consequently, ve d ; af ut v0) ae = (u +v—w) 3 : . l 7 OS hi a (Jas + fvde —/ wie) = udgy + real vag = i uwdyx =u+t+v—w, But this is not true for all values of the constants appended to each integral, but only for such asmake the total constant on the second side equal to the constant on the first side. 2. fbudx = b fudx, b being independent of x. For differen- tiation gives bw for both, d dv du : ; 3. Since — (uw) = u—+y ——, the integration of both sides dx da dx gives i eer) dx =f. PE Fa [oc ad aos 3 dx dx 4 da or (page 103.) wo = Sudv + Jvdu Judy = ww — frdu, 106 DIFFERENTIAL AND INTEGRAL CALCULUS. We have thus the following theorem Jfudv can be found whenever fvdu can be found. The process is called integrating by parts, and is of fundamental importance, as we shall find. The following are evident from differentiation : fade = “ fe Ot. ce = at fu *da — Qn? / 13 - 2 | [eae == ifar doe ae ‘fdas we dx the single exception being fi a—'dx or ff an log « 2 faa+ b) dx = facde + fodx = afade + bfdz= — + be ial Se a cx? ys 4 3 5 Be, a b C a b [CS aritee A oe ees z a As (ax? + bx + cx + e) dx [I fat log ade = a* = loga fords .. fade = et eae & &, foosa dx = sin a fsine dx = — cosa dx da Sioa — dx dy = tan a, —————— —— Sin wt, = cos wv V1 — 2 V1 — at ee tan @ aay as ‘ It must always be observed, that the arbitrary constant must never be neglected, except in finding forms, and must be applied whenever we wish to compare forms; otherwise, an integral obtained by two dif- ferent methods may give two different results, apparently, but which, in reality, differ only by a constant. For instance, we have found by ob- serving differentiation, dx 4 dx ; Ste Ol av —_ + ee cos |! & J Vi — # Jia But A dx dx dx ’ 5 aie Ue ee ———— =— sin 2; V1 — 2? V1 — 2? Vv 1-2 apparently then cos~ tw = —-sin7 ‘x, which is not true. But for the first take cos~!x -+ C, and for the second — sin~‘w -+- C’, and equate these, which gives cos a -- sin“'y =C’—C, But cos~'r + sina T ' sai constant (p. 60); hence this comparison produces nothing eX- cept the condition that the two constants of integration here introduced — must differ by 5 MEANING OF AND PROCESSES IN INTEGRATION, dx ] W to find —-—— € now propose to finc fj ee ve r 7, ae. dy Let 1 + «= v, whence dy = 1s and. we may write the pre- 107 ! 1 d ceding f; dx; but by p. 103, we have * 1 dv eid eats fe Mae =lgv+Cc= log (1 +2) +C; the difference of the inferior limits may make stants of the two, but at present we of the result. Let vy — 1 — a, then dx Hs dy 1 dv % ] a difference in the con- are only inquiring about the form 1 = ee Le = ¢ erent ; f{ dy Required aafa? —xadx. Let a— x =», 5 22, L I Res oe = 1 dv We Fics 1 —dv ve # ade = [45 ( 4, ig) = nears 9 [seas ] pe 1a: (a? — x)% —-—-~ Vm eel p? See =e 5 | ede Te 3 The preceding example belongs to a large class of integrable cases, contained under the general form fpaxr.a’x.dx, where a'r is the diff. co. of ar, and dx dz is easily integrable. Let y= ax, and the pre- ceding becomes ay : J py — dr, which, p. 103, can be found from Sey dy, by using, dx as the limits of y, the values corresponding to the limits of x. It is not our present intention to enter largely into the mass of methods by which detached integrals are found; we shall only give Some examples-of the method of integrating by parts, and shall then /proceed to some simple cases for which no rule can be given. The student may, without absolutely breaking the chain of demonstration, omit the rest of this chapter. ada It is required to find ——— (na whole number.) e/ J ar — x? The theorem to be applied is Judo = w — Jvdu, and the object is, w and v being so taken that udyv is the function to be integrated above, vdu shall be more easy of integration than udv. For in the , : ~ 8 wr ee ape equation last written, / udv is made to depend upon jvdu. Now the diff. C0. of a? — x? being — 2rdzr, if ‘we resolve the numerator of the pre 7 do, i ~ a" and —2ade, we ceding, namely a" dz, into the two factors, — 5? and — 22dar, we nave 108 DIFFERENTIAL AND INTEGRAL CALCULUS. adr fl pose pecs == [(-5 oe d.(a? 1. (a? —2*) —_ = sete ae i ‘ ; V w— x? 2 Veo 2 Vaio dV = iG zen) © -— “anpiere Veo a ae 2 NV ea : where, perhaps, for dV we should write ca dx, seeing that we have w not yet used dV alone, where V is not the independent variable, but a function of it. But here we mustrecal the theorem in p. 103, in w hich IV it is proved that f°; eae dx and fUdV are the same, provided we take such limits for V in dhe second as are values of V corresponding to the limiting values of z. By /Udv we mean the limit of 2(UAV), obtained in the same manner as in p- 102, where the values of AV in the several terms are different, but comminuent. Again, since diff. co. V + * V is the diff. co. of 27V, or 2 diff. co. 7V, the last form of the integral is reduced to {(-3 >) 2, dVV or SC-— av) d. VV or —NV a" — INV de (— 2"), which is rs ie . VV ath Ba) LAY on 1 aes or — VV a" +n—l} INV ar? dx ; because fi Cyd acm cf yd wep. 105. Therefore, x” dx Va — 2 We have therefore found that the given integral depends upon that oo fae + on i [ve — « a" "dx, of Va — a2 x*-%dv. But whenever a square root occurs in the nume- rator of an integral, such as VV, it will generally be found convenient to remove it into the denominator by substituting V + VV. In the present instance, Sig — ox Ct pee win Sd ee (oe Th _ : Va? — 2 atom 2 Vata az x" dv uv" dx Sian» dx xv dr — — 02 J va 2 Vea Hy Jao @ ted Var— 22 Substitute this value in the preceding, which gives x" dx +f RS n—2 >] — = Vee + (2 — 1) a? tia Vat~ at Naa MEANING OF AND PROCESSES IN INTEGRATION, 109 Let us now signify the integral to be found by U,, and any other similar integral into which 2” enters, mstead of 2”, by U,,. We have then from the preceding, Uo = 2 py cad + (n- Po U,-2 —(m- 1) U,, —] Whence U, = —— = alge a a ee rene Un il n and we have thus made the integral U,, depend upon an integral of the same form, but with a lower power of 2. Apply precisely the § same pro- cess to wv. which gives 1 eee TAM poaey = —— "3/5 2 + “eu = n—2 ao 9 a v n—2 n—4 9 which, substituted in the preceding, gives (V = a? — zt) Ly fy na— 1] = (2-1) @—8) , — eel ee ONT hes Bee 2 ~n—3 oes mis rIalVv Gl ara VV + GS a’ U apply the process to U,_, and substitute ; continuing thus itis evident that the series U,, ‘U,-2: U,_,, &c., ends with U, when 7 is even, and with U, when 2 is odd. But v dx dx f v Uo.= — = | —— ced fs ge Va? — 7 az — a which is thus deduced, We have, from what is known of differentia- tion, and from p. 106, —=—== = sin“z, in'Which let > — A tees dx ] —— ly = ite Das Vi-2 ie aay “ay yt Je—yi Gi d or sin='y — Bey 1.e. sin 2 = ia Va — x? & Va — y? edu 1 (‘dV Again, U,= Veta? Ye A ie © Hence, by carrying on the preceding series, in the case where x is eyen, which we indicate by w riting 2m for n, we find 1 = 2m — J » jo 1 ad = 5, Flea — ; — a pons JV 2m 2m (2m — 2) Im — In — Am = 1) Oma 38) oy base 2m ( (2m - ~ i (2m — 4) _ Qn - =), (2m - 3).. ~ 2m ¢ (Qn — 40) Rep ae aie 2 “ (2m — mth} ) (2m.—- 3).. cf ea lee I om sin —5 Qn Qm — a ae Bei or 2 NV 110 DIFFERENTIAL AND INTEGRAL CALCULUS. Of which the following are stances : 1 ——— if Pee Ue ag” VV Re a Use poy say aaNV + ae : a aa y 5 5 ous 5.3 Pike NV .a 6 Sires U=(~ 53 RA here dine 6 AD ae and so on. . When 7 is odd, write 2m + 1 for n, and tr te Im é, —e eR NI 5 Sec e e SE VV Dams ne v (m+ 1Qm-1)° * Pignnniees AE: ia a (Qm+1) (Q2m—1) (Q2m—3) Im (2Qn—2Z) eves ~4.2 a aes ( ree ) a™ VV : ~ (Qm+1) (Qm—1)......5.3 of which the following are instances : U, = —¥V_ (which is also in the process) ] ae ( ‘ara [OP — i @NV ty al any , 3 3.1 1 — 4 = "Sy — §= —laV —paWV-— ‘NV U: ae te a aa . 1 — 6 —— 6.4 — 6422 = | U es 6 af We 2 vt bea ae: Fie 6 7 nw? Vv rare VV serge Nae ey a VV, and so on. In this way we may see that it will sometimes be prac- ticable to make an integral which contains an operation repeated times depend upon another which contains the same n—1 or n—2 times, in which case, by continued reduction, the whole difficulty is at last contained in finding what we may call the ultimate form, which either does not contain the operation in question at all, or else only once. The general principle of this reduction is as follows: let A, and B,, be given functions of m, and U,, a function, whether involving mte- eration or not, of which we know only this, that for all values of 1, | U, = A, + B, U,_;. Then it is evident that U, = Av+ 5, U..4= A, + By AL it 5. Ue — A, +B, Ani + Br By. (Ans + B,-2 Uns), — A +B, A.) +5, B,-, Ano +B, Bi1"B, 2 Ans + Dae and proceeding in this way, we get U, = A, +B, Ana +B, Br Ano + &. +B,..-BeA, + B,...B,Us whence, U, being found, U, is found. MEANING OF AND PROCESSES IN INTEGRATION. 11] But if we have U, = A, + B, U,_s, this gives Uren A+ BAS BB, U,.: | =A, + B, Aig 4eB, Big An ek BY Baw ies iw, and so on, which gives, according as 7 is even or odd, Us = Aan + Bom Agnes + &. ng oP os Re >, eam 3 As+Bom+ +++ By Us, i Asmti+ Bins Aden. -L &ee Dy cotahes oc B,A;+ La Mabe RR 2 hd 9 As an example of the first, take fe” xv" dx, which is also fa" de”, In- tegrate by parts, which gives fea'de = are — nf ex" da, Let ai e*a"dx = U, then U, = vf "dx == s* naa 62", AL, = e'st-|. &e. B= —n, B,.1 = — (n— 1), &e. and the negative sign of B, gives the signs in the series alternately positive and negative, so that we have of Sada = ex" — nex" + (n— 1) e*x"-? — &e. Patni —Ty. 2a n(m—-1). . . 2.12. As an instance of the second case, we take Ssin"@ do — Jfsin’0 d(—cos@) = —cos@ sin"—9 —f(— cos @) d.sin®— 6 (write C and §S for cos @ and sin 6, when not under the integral sign) = —CS*"'+ @m— 1) fcos*6 sin”-*9 = —CS""! +.(1-1) J (sino —sin"@) dé; or U, = — CS"? 4+ a— 1) U,-2— (n— 1) U, ] n— I ] U, =~ = OS"! + —~—— Us 35 A, foes fees Ost, 7 Tt Nn ] e n— J na—3 —= —"— Cs", &. BL = eee Anas n—2 bee oe ed Po igs ae = f sin’e (ide 6. 1, = fsin@ dO? = — cos@, 1 (2m—1) ; ae oC ils jp a Von Qm : 2m (2m —2) (2m—1)...3 (Q2m—1)...3.1 &e. CS? Ses —— CS aoe 2m... 4.2 + P41 OPT Pe 2m ; — 2m i: OS*=? & : Vana Q2n+1 (2m+1) (2m—1) 4 2m (2m—2) ...4.2 (2m+1) (2m—1)...5.3 We have already had this integral in another form, as follows. Let «=a sind, then J a? — 2? = acos 0, and* dx=acosé@. dé, which gives C. dy dx! in the form dy=pda. The justification of this process is contained in the theorems in p. 54, in which it appears that diff. co. have the same properties as if they had ordinary numerators and denominators, * It is much more convenient in many instances to write such equations as 112 DIFFERENTIAL AND INTEGRAL CALCULUS. l. 1” sin"@.acos 0 dd : ee 3 Oe oe oe Va 3 _— 72 a cos 8 Verify from p. 109, and the last process, the equation x de ; it = a"f sin" 6 dé when 0 = sin? -. Var— x? N a The method of integration by parts is almost the only systematic rule sn the direct Integral Calculus. In most questions unconnected arti- fices must be used, of which we proceed to give some examples. { ae The denominator is the product of a+2 and a—2; and a wea MY he ; eh ; 4, aa 1 it is obvious that, ae ? whence “A — X° sa dx dx d (a+2) %4\ = SH 1 ——_——=log(a+e C—x 4 TF alk (= a+a ate (a4 dx d(a@—2) a ses en see eh — 2). {= ea OPE i og (a — 2) d 1 —— ] w@—a 2a dx dx 1 a—wv She aT dees eo ae log r—a “— 2 2a a+e2 cos9.d0 __ d.sind, 1 1+sin @ <= ~ €0s?0 | = Soa 2 °5 1—sin 0 1 a( 6 dx ] ‘ ; eh if = po =~ tan7? = (from differentiation) xv+a x il a pf if de Wn Wie be eee PSE Ce Ss ee BOS 1 o 2) (t= ae Ee {an ek pale ; aR]! apie b fa, aoe an” ae oh p+ The following reductions should be practised till oe are easy. 9 b* at br +e ma- anu cb ink ae ay, Gr p b? b mn Ne oka Saale : Cc dx : To find if. Me ae Assume = peg > fs Ve dx = da’, ; MEANING OF AND PROCESSES IN INTEGRATION, 113 f. dx ] i ‘da’: 2 ree We a! ) oe ah SS Peto —— a+ botox Vc fae Oi V4 ac—b V dac— b 4c 2 =, 2cn 4.6 =— tan : 4ac—b? J4ac— bh? If 4ac—b? be impossible, that is, if 4 ac be less than 6°, this in- tegral appears to be impossible. But, p- 97, if all the elements ofethe form yAx be finite and possible, the limit of =(yAr) must be the same. ‘There can be no real impossibility therefore in. this integration, and we must look to some anomaly in the method for the reason of this peculiarity of form. In algebra we find that the alteration of a constant from positive to negative sometimes does, sometimes docs not, produce results possible in appearance, and impossible in reality, or vice versd: but frequently, owing to the comparatively simple character of the results, and the closeness of their connexion with the fundamental definitions, we are able to tell at once what effect a change of sign will have. In our present subject we are dealing with more remote con- siderations: and whether we consider Sydx as the primitive function of y, p. 100, or as the limit of the summation expressed by Z2yAv, we cannot in either case pretend to carry with us from y to Jydx any such perception of connexion as will guide us either to the form or magnitude of the latter. We have already found the two following results, dx ] Ve = dx 1 z SS == log > Ce ie are ie “Tee Wee tanh) ee which are only general forms, p. 106, and must, before we begin to Compare them, be taken between the same limits. But both forms vanish when x = 0, and are both therefore taken to the higher limit L, from the lower limit 2 = 0, The first form becomes Impossible when x is greater than lo, for in that case the integral becomes the logarithm of a negative quantity ; but at the same time we see that in this case a value of x (namely, /c) which makes the function to be integrated be- come infinite, lies between the limiting values of the integration, T his than vc, reserving all other cases for future discussion. We now pro- ceed to another point; the first of the preceding integrals is changed into the second, if we change the sign of ¢, or change —c into + e. But the second sides of both become impossible under such a change ; dx 1] V—c— 2 shane RES ai ings = log = 2 = >— tan — >; +c Q/—¢ —c+o2 dane en Ve and we thus obtain f. dx ] ies 1 (= —) === fane! —— shor ee Io = a MN NG RAT id ee I 114 DIFFERENTIAL AND INTEGRAL CALCULUS. do 1 tg (AEA) oe Rite w—e ove Vote vc ae 2NVc giving a possible and an impossible form for each : the latter sub- ject of course to all difficulties of the passage from possible to impossible expressions. The only question for us now is this: are the preceding possible and impossible forms the same in algebra, such as it is to the student who commences the Differential Calculus, or shall we be obliged to make any extensions in the meaning of algebraical terms, before we can consider them as the same? Let us equate the two expressions for the first integral, and consider them as identical, that we may see whether the consequences of such a supposition will or will not be consistent with those already known. 1 v ] Z —_c— wv Assume —— tan’? —=> -—= log bass : J—c—2« Vo Je 2s—c Assume «= Vc tan 0, and substitute, which gives Ao Unis 1b = <=, 2 Sail J—1 + tané eV ETS V—1—tand — —|— rat tan 0 v—1 + tine med +- oa tan 0 whence tand. PE} wa ( gov pe (<2V1 x 1) a result well known to those who have studied the higher part of trigo- nometrical analysis, and on the method of finding and interpreting which we shall enter in the next chapter. We shall now return to the subject, with this result, that so far as we have yet seen, the possible apd impossible forms of integrals are identical, and lead to the well- known relations in which trigonometrical functions are expressed by algebraical functions involving the symbol J/—1. The student will observe, that we do not in this place profess to remove a difficulty, but only to show that, whatever it may be, it is only such as is found m algebra. In the integral last found, p. 113, we have the form l dx! , b? 1 . i) Oe x where C =a — 7 = 7- (4ac—bt): if c be negative, we have already impossibility of form in the con- stant factor, a case we shall presently mention. Let ¢ be positive, then C is positive or negative according as 4ac is greater than or less than 0*. The first of these two cases has been integrated in a possible form ; im the second case, where b?is greater that 4ac, let C be —C’, and the integral then becomes (co sie ¢ (0° —4ae) ) = MEANING OF AND PROCESSES IN INTEGRATION. 115 = [ dx! a pra na Vo'— =) Ved) t?—C' ~ o/cq ® VC + a! i ‘ log V 40 2V ea! ) Pm 4 ac V4 C0! + 2 Sea! but b+ 2cr +2 Je x’, which substituted, gives i) dx 1 ( /b?—4ac—~ b — =) Sc oe ee SP EMME. Case ‘al > p12 5 see pee ; aQtbr+ezxz 6°~4ac V6?+-4ac + b + 2exr, which is the possible form when 4.ac—8? is negative. And in this, it must be observed that the case where ¢ is negative is included; for in that case b2?—4ac must be positive, unless @ be also negative, and B<4ac. But the case where both a and ¢ are negative is treated by the following reduction da dx dx 1 eS ee a I a A Brey nrg a — bx+ cx? It makes no difference as to form, whether 5 he positive or negative. The most important integrals in practice are those which involve Square roots, and which we now proceed to consider, using various methods of reduction. We shall frequently, vithout formal notice, substitute throughout for one variable, such a function of another as is convenient. Thus, in the first example which follows, we do in effect say let x= ay, and we thereby find the integral in terms of y, and thence by restitution in terms of wz. dx d.ay dy Yirr Lr fe ee ee gi y = sin“! —, Va? — 7? Ve—a% 2 EP y” a dx . { =-. Let@’?+e?= y’, whence xdx = ydy, and ydx + ada = ydx + ydy, whence Of fe, [ast dy a ey log (w4+-Va?-+ 22) Mi as) 9 OEY yh) Urey ; This is a specimen of an artifice of integration for which no rule can be given, We might have used the preceding integral as a method of dis- covery, thus: i d.(a 4 —1) ERS of —1 ” da Le, Weel = sin7? —_—_ ——$<$<$ — __________ — or Va*—(2/—1)? * Ve+e Ja] - But, as will be seen in the next chapter, cosO — sind J/—] = «-*V=i oy _9 J—1= log (cos 9 — sin 94/41). 2 ef ii re See > Let sing=-/—1, COs e\ 1+—, tA 1 Ee 116 DIFFERENTIAL AND INTEGRAL CALCULUS, ‘ piel iP A Pa g—sin-'{—_—— ],. or j= sin : a v=] a == log O/1 si 2: (Shea ) = log ( Ja?+ 2° +a) — loga, a eee (43 a result which differs from the last by a constant quantity. It must be remembered that since ox and px + const. have the same diff. co., we are liable, in using artifices of integration, to produce results which appear different, but which in fact only differ by a constant. This dis- crepancy does not appear when the integrals are taken between definite limits, since da — Ob and ga + C — ($b + C) are the same. dx Assume 22 — a? = y%, and proceed as before, which will x’—a* give as the result log (@ + V o?—a?), pea ae dbs)" 7) sol ana Jane SN (in) | VE) = ee ba 1 “ —_——, pec 2yae aC Sor) — log (Vbx + Va + ba*) Jatbe Vb) Vas (you) dx ae 2 Veda _ ea d (b+2cx Vatbatca N4ac—b?+(b+2cxr)? Vo J 4ac—b?-+(b-+ 207)" dx ] —_________— Je 8 log (Qex+-b+ N40 (a+ bx +cx°)) Va +bx-+cx NE { dx 1 il d (2cx—b) } Se eens Gy semoemcsranrer. samme: LSC maser a eee ee sim. [ee ee Va-+ bax—cx" Ve Vv 4ac+b? —(2cxr—b)? Ve V4ac =) dx Sake fae ——— = log (ota+ VJ 2ax+a2) + log 2. (Omit the constant.) 24x +2? dz L— a . v ————— = sn! which may be written vers~’ —. \2an—a° a a We do not say the two last are equal, for they differ by a constant, as follows :— . T Seater yy a xv av —-+sin (G —1)=cos7?{ 1 ——)})= vers’ = a a a a —— dx xdz | vor di = a et Vu? +2 Ve +a ad: —_——. a Pin Me (|= = | rd (J a+ x?) =—aNae +27 — fy etx de) a+ a? = 1a log (w+ Ja’ + 2°) +12 N@4@ fv Br — xx = |W @ — a sin’? d (asin) = wr | cos 0 d (sin 9) MEANING OF AND PROCESSES IN INTEGRATION. 117 J cos 0 d (sin 6) or /cos’é d@ = cos@ sin@ — fsin 0 d (cos 0) = cos @ sin d@ + fsin’@ d? = cos @ sin 0 + [do — {cos’6 de {vas dt = 1taVQ—z +a sin“! a _ We shall close this chapter with some examples of the preceding integrals taken between limits. We state again the theorem proved 1n p. 100, which establishes the connexion between a primitive func- fzon and the limit of a summation. If Wwe be the diff. co. of gx, and if @ and b be two limits of which 6 is the greater, and if we pass from @to 6b by n steps,a+0,a+ 20,.°. . upto a+nd=b: then the limit of (Wa + (a + 6) + .... + Wb) 0, on the supposition that 7 creases without limit, is 6b — da. b htt n+ a a — a dx i a das ; == log'a, edu = s* — ] - n+] fae 0 girth n+] . +5 z a cos tdzr = 1, ii cos xdr = 0, ff ‘cosadrz= 2, [ sing dr=1 0 i) ee 0 an ep 7 Bg le ys T Jes via” Jie; txaf dP (n—1) (n—3)...3.1 7a” $ Aaa “Sohn (n—2)...4.2 2 _ (n—1) (n—3)...4.2 ap or “i DS 3 2 (1 odd.) ‘a ex" dz = Fn (n—1)...3.2.1 according as m is odd or even. — 1 +™ dz atm Se Oe i at+-m ear ‘ 3 — 8) ; <=, 109 es “ ioe Bot) Mage aa 1e—a og °° a—m } ,a@—a? When a definite integral is infinite, the product in the theorem in- creases without limit. (2 an integer) SJ it ade = 0 when n is odd, = when 7 is even. ws] 4 (2 even) ; 118 Cnaprer VII. TRIGONOMETRICAL ANALYSIS*. Ir we apply Maclaurin’s Theorem, as in p. 795, to the determination of sin 2 and cos 2, we find that they may be expressed by any number of terms of the following series, the error never being greater than the next succeeding term, (being in fact that term multiplied by the sine or cosine of 07, 8 < 1,) a i at ers, ty aik Sao Ts Meee x2 xt x d cosv = 1 —~-5 +534 “5 Fa erG a. & cue) ee ea If these series be sufficiently continued they can be made as nearly equal as we please to the sine and cosine. For the following relations will easily be seen: In the first, (2 + 1)th term = (mth term) xX ace x (2n — 1) Qe’ in which, whatever « may be, » can be taken so great that the (n + 1)th term shall be as small a fraction as we please of the nth, and still more the (7 + 2)nd of the (2 +1)st; and soon. ‘The terms, consequently, must at some point begin to diminish, and from thence must diminish without limit. But the error caused by stopping at any term is less than the first term rejected: that is, diminishes without limit. ‘These series therefore, carried on ad infinitum, have sin # and cos x for their limits, and are said to be convergent}. The same may be shown, as is done in p. 75, of the equation In the second, (7 + 1)th term = (nth term) x a 9) e 4. x f=-lt+a+—+--+2>357 + &..... 3). 2 2.3 2.3.4 ( ) The development of ¢* consists then of the terms which appear in the developments of sin # and cos a, and of no others. If all the terms in (1) and (2) were positive, we should have sina + cosv= &"; but as it is, no simple algebraical relation appears to exist among the three. But compare cos x + k sin x with e**, writing (a”) for 2” 1.2.3... and we have cosa+ksina = 1 + kx — (a*) — kk (e*) + (#5) +2 0’) — &e. ef 1 + hat h(a") + RB (a*) + ht (at) + (we) + Now these series can be made identical, if we can make Bo sie Peek md, Poh, See * This chapter may be considered as a continuation of the Treatise on Trigo- nometry. It may be omitted by the student who does not wish to go into the more difficult parts of the subject. + See the “ Elementary Llustrations, &c.,” p. 9, for the usual definition and cri teria of convergency. TRIGONOMETRICAL ANALYSIS, 119 of which we may easily see that the first is impossible; but that if the first. were possible, all the rest would follow from it. For if k2 = — J, then & = — kh, k= —h* = 1, &e. If then-we assumé the identity of these two series, whatever may be said of the fundamental assumption k* = — ], it involves the whole of the question, the identity of the re- maining parts following from it by the common rules of algebra. Let us first investigate the algebraical consequences of this assumption, con- sidered without reference to the truth or falsehood of the assumption itself. If we take #2 = — 1 or k = /—1, the preceding series become identical, that is 2 v= pie a ae Nea cox + ¥ —] sing = €& —! andcosa—V—] SI eo ET ee The second of which may either be deduced in the same manner as the first, or may be obtained from the first by observing, that the series from which it is obtained being true for all values of x, we may write — z instead of z, observing that cos (— xv) = cos 2, and sin (— rv) = ~sinz. By the addition and subtraction of these equations we obtain —s ] |/— f= 4 1 / een ee cos # = £( ev + SP ee) sin a = | Sere 4d), \ pas a These expressions will be found to have all the properties of the sine and cosine, but it must not be forgotten that they involve the expression 4 —1, which has no algebraical existence, either as a positive or nega- tive quantity. They must be considered as abbreviations for the series, which expressions treated algebraically may be made to give the series, but which cannot be considered * as algebraical quantities. It must be remembered, however, that all algebraical expressions are combined and reduced by rules, which, though derived from notions of quantity, will produce the same results, if we alter the form of the primitive ex- pressions in any manner, consistently with the rules, even though the new forms should no longer admit of being considered as quantities. Suppose that we have a set of symbols, a, b, c, &c., representing quantities, and that we are going to perform an algebraical process. Let us, instead of a, 6, c, &c., perform the process on a+tVvm— Wn, b+V im! — Vn’, CHENG Hill Vnl! &e. As long as m, n, &c. are positive, the process and result will both be intelligible ; and if, after the process is finished, we suppose m= n, IP anata m =n’, m!=n", &., the result will reduce itself to that which it would have been if we had commenced with a, b,c, &c., in the manner first contemplated. Now so far as results are concerned, the applica- tion of rules will have the same effect whether alm , Vn , &c., repre- Sent quantities or not, provided only that they be used as if they were * Of late years these expressions have been considered in a manner which places them on the same footing as negative quantities with regard to their definition and use. For an explanation of this method, which is not yet made a part of elementary reading, the student may consult Mr. Peacock’s “ Algebra,” Mr. Warren’s Treatise “On the Square Roots of Negative Quantities,’ Mr. Peacock s “ Report on the State of Analysis” (British Association, Third Report, 1834), a review of the algebra of the last mentioned author in the ninth volume of the “ Journal of Education,’’ or a “ Treatise on Trigonometry” now in the press, by the author of this Treatise. 120 DIFFERENTIAL AND INTEGRAL CALCULUS. quantities. If, then, instead of m,n, &c., we write —1 at the end of the process, we shall produce the same results as if we had commenced with a + ot ae eat J—1, &c., thatis, with a, &c. (because since oes is to be used as a quantity, ¥ —l — VJ—1l= 0). The preceding is exactly a case of this sort: cos x, which has no real algebraical equi- valent, is connected with the expression 4 Ge 4 aoey A.) by a re- lation of this kind, that if m the expression, J —1 be treated by rules of quantity, the series for the cosine is the result of developing the exponentials e*V-7, and e~*’-', and of taking half their sum. The student who has duly considered the theory of negative quan- tities knows that every problem, the result of which is negative, is connected with another which has a positive result. To complete the analogy, we shall show that the sine and cosine, as deduced from the circle, and which have no possible algebraical equivalents, are connected with a sine and cosine which may be deduced from the hyperbola, in such manner that the properties of the two kinds are very analogous, with this exception, that all the relations which involve impossible quan- tities in the former, have no impossible quantities in the latter. B OA>=a OM=2 BM=y OAS a’ : i QO’ M’'= al Beni 2" We have here a circle and an equilateral hyperbola, the equations of which are as written under them. The sector AOB is 3 a°@ in the circle, where 6 is the angle AOB, (are BA + rad OA,) and if A be this sector, we have, according to definition, for the circle, 2A a 2A a8! 6=-> = <= cos (=F ore ) ¥ xin (= ore). a a ae a a” Now let us, by definition, create an hyperbolic sine and cosine in this manner: let the sector O’A’B’ be called A’, and let 2A’ a” have its sine and cosine, namely, let us lay down, for the hyperbola, (remember, however, that 6’ is not the angle B’/OA’ as in the circle,) | ; TRIGONOMETRICAL ANALYSIS, 121 SAY x 2 Al y! ‘OA? 6’ = — 22) COS rape ea tL = sin ( =~ ord! }. a"? a al? al Og It will hereafter be shown that the value of the sector O’A’B’ is as follows - a’ ee iy a! y! ee i Ar = ae log (Z + r) ore’ = ; ae vag off e y! a! y! But (3 mipsel wench eT!) ee al ta q! a’ ; al aig whence, by addition and subtraction, EF > 1 a, ete | Si 6 ; U7 ahs Meee | fara: —6 cos 6 = 5 (5 +e¢ ) sin ot = 5 (« soni » corresponding to the equations obtained for the circle, namely, ] | epee —— “ 1 % |j-— aan cos 0 = — Ge “i ee) sin @ = =(" — g-0V=i , 2 2V—] We shall now proceed to show that these latter expressions have the properties of the sine and cosine, on the supposition that we use / — ] as a quantity the powers of which are aA eee ck OL Pom aang Rel name Teas) Hs Let us first construct sin 6 cos ¢, 1 = “> = we sin@ cos¢ = (ev a ene (© VT gp VEi ), 4f/—] 1 aly eO+PV= eo +e)Vii eg O-OVa1 _ g-O-9)V7i 1 shal. Pere = Wane V=1 sin (6 +6) +2V—1 sin (0) ) _ 5 («in (P-+ 6) + sin (¢ — )), a well known theorem. Let the student take various relations which exist in trigonometry, and make them identical by substituting on both sides the exponential values (as they are termed) of the sine and cosine. We shall now take a couple of instances in which results of more com. plexity are obtained. ProsLem. To expand cos "0 in terms of cos or sin 6, cos or sin 20, &c., n being a whole number : = — 1 1 IN Let® 1 — e,.thne4 = = » CoS O=—-( r+ -}, i % ee x ee * Observe that we do not escape the impossibility by substituting a for e?¥~}, l l The equation cos ¢ = 3 ( x 9) is impossible, fur x ce can never be less x than 2, (which prove,) and 2 cos ¢ can never be greater than 2, 122 DIFFERENTIAL AND INTEGRAL CALCULUS. Pe ka, vaso Rnb yee gnoN=i =x", then e "9° = a2 cos nO = 5 a” + =p 1 1x" 1 1 n—- 1 1 m= — ~) =—(2a"+4+na"'-+n a? — COS 0 Sl v ne yn : + + 9 2? + Collect together the first and last, the second and last but one, &c., which, gives 1 ] 1 n-1/ ] cos "== — ba” - —+ ape"? 4+ — |) + 2 | 2 tt [tee Ze usd eB a 2 # \ cos (n—4) 0+... } n— 1 = eat (co nO +n cos (n ~— 2)0+n If n be an even number = 2m, there will be 2m-+1 terms in the development, which will give m cosines, namely, those of 2m8, 2(m-1)@.... down to 20, and an additional term corresponding to the middle term of the development, which is 9m (2m —1T). ..(m+1) , 1 Qn (Qm—1). . . (m+ 1) a Se OEE Legh ingle aaame |" ike kee. eat eure This term, which has no corresponding term, does not follow the law of 1 ; the series, for though we write 2 cos 29 for «° + —, we cannot write mM a 2cos 09 or 2 for 2°, which is 1. But if m be odd, and = 2m + 1, there are 2m + 2 terms giving m+ 1 cosines, namely, those of (2m +1)9, (2m—1)@.... down to 4, and there is no middle term.” Consequently, we have the following theorems : Q2"-) cos?” @ = cos 2mO-+ 2m cos (2m—2)O + . am (2m 1) bow . (tip 2) cos 20-4 Qn(Qm-1)... . (m+ 1) Te 92... nue (Ne) VE Pe ee m 2" cost! 6 = cos (2m +1)9+ (2m+1) cos (2m—1)? +.... (Qm +1)2m.. .(m-+ 2) saa ——— co eR Be a eae ~ bo] = s 0. An instance of an odd and even power is as follows : Pe ad of i 1 1 1 1 abel cos °O= 58 (0 +62 = 4+ 15.2% 4 +. 20x" -= +152 or + 6x 3 + oT i £2 a4 v+a* vet eg =—{— 6 — 0 5 ( 9 + 3 +15 5 + 10 ) 2°. cos°@ = cos 60 + 6 cos 49 + 15 cos 20 + 10. By proceeding in the same way, 2* cos°@ = cos 59 + 5 cos 30 + 10 cos@. TRIGONOMETRICAL ANALYSIS. 123 These results may be verified by the common method: that is, by means of 2 cos 9 cos @ = cos (0 -+ d) + cos (9 — d) 2cos*@ = cos 20+ 1, 4cos*O = 2 cos 6 cos 20 + 2 cos 0 = cos 30 + cos 9 + 2 cos 0 = cos 36 + 3 cos 8, S cos*0 = 2cos 0 cos30 + 6 cos °6 = cos 49 + 4 cos 20 + 3, &c. Prosiem. To expand sin ”@ in terms of cos or sin 0, cos or sin 20, &c. We have, a 1 ] ( =) IR ge agree 5 ermal ip ae ar (7-1) ey which gives four different cases, corresponding to the four forms of (Vv 1)", namely, PPD) = (YIN eT, Gaye Ff ( Ee os eee F a 1 ; When n is even, the first and last terms, the second and last but one, &c. are of the same signs, consequently the expansion presents cosines only; but when 72 is evenly even, (of the form 4m,) the sign of the whole is contrary to that which exists when 7 is oddly even (of the form 4m +- 2). Proceeding as in the last problem, we have, making P, sig- nify the coefficient of x* in the development of (1 + 2)’: 2" sin*’@ = cos 4mO—P, cos (4m —2)0 +P, cos (4m—4)0—..., =~ Pin €09 20-4 5 Py, 2” sin 6= sin (4m + 1)0 — P, sin (4m — 1)0 + P, sin(4m—3)0— . . . +P, sin 0, aime* sin*™*?9 = — cos (4m + 2)0 + P, cos (4m)é ae Ps Cos.(437— D8 4 i Pon COS 20:4 oy Panes aimr* sin "+9 == — gin (4m, + 3)0 + P, sin (4m + 156 — Pysin(4m—1)0+ . 2. + Peau ain 0°; a complete set, for the student to consider first, 1s as follows: 8 sin*? = cos 40 — 4cos 20 +- 3, 16 sin°@= sin 50 — 5 sin 36 -+- 10 sin@, 32 sino = — cos 66 + 6 cos 40 — 15 cos 20 + 10, 64 sin’6 = — sin 70 + 7 sin56 — 21 sin 30 + 35 sin 0. These may be obtained from the following theorems : 2sin@ cos@ = sin (0+ o) +sin(@ — o) =sin (6+ 0) —sin (P-9), 2 cos Ocosp = cos (0 + ¢) + cos (0— 9) , 2sind sind = — cos(¢ +6) + cos (p — 6). Thus, 2 sin?6 = — cos 26 + 1, 4sin°@ = — 2sin@ cos 20 + 2 sin = — (sin 30— sin0) + 2sin@ = — sin 36 + 3sin0, Ssin “O = — 2sin@ sin36 + 6 sin 20 = cos 40 — 4c0s 20 + 3, &e. These results are frequently convenient in integration ; for by them, Jsin"6 dé, and Joos "0 d0 may be reduced to the addition or subtrac- tion of integrals of the form Ja cos m0 dd, or fasinmé dd; but-we have 124 DIFFERENTIAL AND INTEGRAL CALCULUS. fa cos m0 d? = Ree feos mo d(m0) = © sin md, m m a farsin mo dé = sa [sin moO d(md) = — a cos mé. m m Prostem. The equation tan =k tanO existing between ¢ and 9, required a series for @ in terms of 0. We have “= sin db 1 gov Aa. ae l e2oN=1 esa | ie far eg ee oa ee the last result being obtained by multiplying the numerator and deno- minator of the preceding by VA Let &°V = F, and Y= T. Then, using a similar formula for tan 0, and recurring to the equation of condition, we have ton @ = ny fei ip F-1_,T-1 pe LektCtHT_ pT Peep 1 we “al ee (hh Be, Lees ns Xu phone ae whence log F = log T+ log (1 ++) — log I+AER 1+ | 4 Now from the theory of logarithms (or from Maclaurin’s Theorem, which the student may here apply, if he be not acquainted with this series) x a vt ] —7— — — log (Ep eta oe a: BE tc Weare ko 1 log F Slog - (T-) + 5 (TP -~( ° — a) + de But log F=log <%=! = 9gV—1; log T = log e*Y-? = 99/—1, 1 ee. pos ae lj go 207 on — 1 sin 270; whence 4 ih 2gN—1 = 20V=1 — 2WZT=i sin 20 De Ore cord Pieatemgeyiece et OUT Gt Big ee Stina sin 69 + &c, é De 44 M8 @ = 0—Asin 20 + e sin 40 — 3 sin 66+ &c., a series of considerable use in astronomy. When k is near to unity, » is small, and the series is very convergent. In order, as much as pos- sible, to verify results obtained by the use of impossible quantities, we shall proceed to show the truth of this series without them. Differen- tiate both sides with respect to 0, and we have sin m0 = m cos nd) d dé TRIGONOMETRICAL ANALYSIS. 125 if a = 1 — 20 cos 20+ 22 cos 40 — 208 cos 60-4 . But, dp k(1-+tan’@ tan ¢=k tan 0, (1+ tan’) =k (1+tan26), or — aa 1—cos 20 1—k 1—A tan?@ = — and A = —— 9; = an Tbeos99 22 Tap Bives k ae? ‘ . I an? =e which gives AS re iS Ree eh 1+ hk? tan°9 ~ 1+ 2d cos20 +r? We should have then, if the preceding be correct, 1 — 2? Our object is then, to ascertain, without the use of impossible quantities, the value of the series \ cos 20 — 2 cos 40 +&c. This we may do, in this particular case, as follows: take the general equation 2 cos 20 cos 2n 0 = cos (2n + 2) 6+ cos (2n — 2)6, multiply by d’, and write the series of equations for all values of n from n=1 upwards, giving a negative sign to the alternate equations. This gives 2 cos 20.cos20 = 2 cos 4A +X, — 2d’ cos 20 cos 40 = —)2cos60 — r2 cos 20, 2d* cos 20 cos 68 = ~—° cos 89 + A? cos 46, —2X* cos 29 cos 88 = —2*4 cos 100 — ‘cos 60. &e. &c. &c. Let the expression for the series required be called S; if then we sum these equations ad infinitum, the sum of the first column is 2S cos 20; that of the second is — S + A cos 26 divided by \; that of the third A—AS: so that —S+dAcos20 - A* + » cos 20 9S ‘roe eee —i/ S g ets SERGE a ye ee 3 cos 20 i -+A—AS or S LDN come 1—2S — pe. I otha which verifies the preceding : ~ T+ 2nrcos 20-2 VN I 2 Now, as an exercise, let the student substitute ‘> Caer ian*ye 3 (v%*+v~“) , &c., for cos 20, cos 40, &c., v meaning eoV=r, the series will then be reduced to two geometrical series of the form 4 cae 1+ AP?’ by adding the two fractions thus obtained, the same result will be found for the series as is given above. The fundamental expressions et#V-1 — agg 9 + /—1 sin 6, lead to the following relations : aa (Y= )" or cosn6+-V—Tsin nO=(cos 0+V—1 sin 6)", AP* — P+ -+ A°P® — &., the value of which is "lala ~ (eas ) orcos n0—V = 1 sin nO=(cos 0O—V—=1 sin 0)"; 126 DIFFERENTIAL AND INTEGRAL CALCULUS. : ] 1 and also to the following: if 2 cos 0 =a + = then 2 cos nO=a"-+ ee and it also follows that hs Mtg 1 oe es 9/—1 sindO=ar— - and 2V— 1 sin nO = a? — —. x L These, which are the same in different forms, are called *De Movvre’s Theorem. The preceding considerations have led to an extension of the theory of logarithms. By definition, the logarithm of x (the only one used in analysis) is the value of y, which satisfies &” =, where ¢=1 + ] ] ‘| pad, : + 2 +L > 3 4. 2007182818 . <>.) and ws given. There is only one arithmetical value of y, which is accordingly the only real logarithm. But one of the consequences of admitting ¥ — 1 among the objects of algebra is this, that every quantity has an infinite number of logarithms, one of which is the arithmetical logarithm, and the re- mainder of which are of the form a+6V/—1. If in the equation eV) — cog 9 +V—1 sin), we suppose 6 = 2mm, m being a whole number, positive or negative, and (here, as in every other place) the ratio of the circumference of a circle to its diameter, or 3°14159..., ; = ; we have then cos2m7—=1 sinQm7=0, or @™ = 1. This result, which, considered by itself, is one of the most singular in ana- lysis, draws upon no other principle except the one on which impossible quantities are used throughout this chapter, namely, that V—1 is to be used as if it were a quantity, so far as rules are concerned. Let this be done, and we have Ps eins Am? 3? Sm? nr? p-— gam s/-1 oe 1 a oN aad — 9 asad aoa A fie & ] ae &c. ° 4m? 16m* 1+ — 8m? = pai are ee ———— — &e. ony (2 a eoee | ee 61s aA Or ce nt If the “student, taking any value for m, say m = 1, and making 7 = 3°14159... were to calculate the value of each of the series, he would find the result to be 1 -++ /—1 x 0, true to as many places of decimals as he took into account. If then y be the arithmetical loga- rithm of x, or if : NA = v=, wehavealsoe’ xe’ '=ax x1, or gybimeN-1 a ps that is, y + Oma — 1 is also a logarithm, where m is any whole num- ber, positive or negative. If then we take log «, as usual, to represent the arithmetical logarithm of x, and Log x (with the capital letter) for the more general logarithm, we have Log « = log @ + 2mm —} Log z = log z-++ ona —1 &e. tT ooime=lor re (ms ery. i sog rze=log rz +2 (m+n) -1, Log — = log—-+2(m—n) ay 1, &C. * Having been first given by De Moivre. They are in his “ Miscellanea Ana- lytica,’” 1730, but not in their present form. TRIGONOMETRICAL ANALYSIS, 127 Whence we see that if we add one of the Logarithms of x to one of the Logarithms of z, we have one of the Logarithms of «rz, &c. A negative number has no arithmetical logarithm: but it has a Logarithm of the kind just found. If for 6 we take (2m +1), we find eQmti)r V1 cos (2m + 1)47 + J—1 sin (2m + 1) r =—1+0xV¥—1=— £1. Hence Log (—1) = (2m + 1) « J—1, where m isa positive or nega- tive whole number. We have then : Log (—.r) = Log a + log ( ek) ee loga + QnxV—1 + (Qm-+. ] )rV¥—] or Log (—2) = log x + (2m +1) rV—1; for 2n + 2m + ] may be written 2m + 1, since m and m-+te 7 are equally indefinite, meaning merely any whole number. The value of Log ( — 1) gives 2m +. 1) EF v=] This result is usually deduced on the supposition that m= 0; and it is said that Wop 23) V1 314159... « a result which Must appear surprising, if it be not remembered that in using Ning by the rules of quantity, the sign = also undergoes an extension of meaning. We must remember that the result (A) can only be thus interpreted in the algebra here used: if ever, by the use of a negative quantity, intentionally or unintentionally treated as a positive quantity, we obtain Log (—1) + #—1, then the real process, if the funda- mental correction had been made, would have given some odd number of times 7. 1 Taking the general equation Log 2" = — Log 2, we find n 1 i 2m Nf nr Log a — 1 (Qmr peas or ] —eE n A) 274 — | Bmx = cos - TN ARS BAS peated n n 1 1 Q2m+1_.4/— Bowl npeite ia ae seg Log (—1) =~ (2m+1) x J—1 or (—1l)" =€ n 2m+l)r - ~—~— | (Qn+l1)¢ tae Sc + V¥—1 sin ( ” ; n and thus we have expressions for all the roots of the equations 2”"=1], 2 =—1,ora"*—] = 0,2" +1=0. It might appear at first as if an Infinite number of roots were thus obtained, since any value may be faken for m. But if we begin, say with the first, and make m — 0, m=1, &c, in succession, we haye the following :— 128 DIFFERENTIAL AND INTEGRAL CALCULUS. ue Ist m=0} Ist value of (1)» =1 aw . 2nd m=1)} 2nd et ees = cos—- +/—1 sin 7 Ar aver 3 3rd m=2 1 3rd RET oo. = cos—+4—1 sin — ath |\m==n—1| nth... .. =cos Qn — . QInr ro ‘ ] } ee ee ee aoa Sa A oes i ee (n+1)th| mn | (w+1)th cos—— + sin — (n+2)th \m=n+1} (n+2)th...=cos —— n &e. &e. &c. &e. &e. on et el aes a nN y Qn6 ; ’ But since —— = 27, and cos 27 = cos 0, sin 27 = sin0,the (n+1)th n . 2n+2) 7 27 value is the same as the first; and since Gad tas = 27 oranda n 3 ; cos (2= + =) = cos =; &c., the (x+2)th value is the same as the n second; andsoon. The first 7 values therefore recur in periods, 7 in each; and the m roots in each period are all that can be obtained. The same may be proved for the roots of —1. Suppose, for instance, that we would have the four fourth roots of — 1. The first four values of 2m + 1 are 1, 3, 5, and 7, and the corresponding angles are 11, $a, Sa, and 4, which, expressed in degrees, are 45°, 185°, 225°, 315°: and we have cos45°=1V2 cosl35°= —iV/2 cos 225°= — 42 cos SIbe =a 19, sin45°= 142 sin135°= 12 sin225°=-4V2 sin315°= -3V2, whence the four roots are, firstly, 12 (1+ —1); secondly, aJ2 (—1+4 J —1); thirdly, + V2 sd ee fourthly, LQ (— VP): Hither of these raised to the fourth power will give —1, Square of Ist root is 4. 2 /—1, the square of which is —1, Square of 2nd root is 1x -2V —1, the square of which is —1: The roots of -+-1 are of great use in analysis, and possess many remarkable properties. ‘The method by which they are obtained rests entirely on this: that a” undergoes the extraction of the nth root by TRIGONOMETRICAL ANALYSIS. 129 substitution of = stead of x; that every whole value of m gives cos 2mm +V—1 sin 2ma equal to 1; that this latter expression is of the form a’, being e%""¥-1. ang consequently that one of the mth roots ‘ hy BOE g . of 1 is made by writing —— for 2mz in that expression, n Every whole power of an nth root of unity is also an xth root. For, if « be an nth root of unity, that is, if a"=1, then (a@”) = (a")” =(1)"=1 or «#" is an nth root of 1. This is also evident from De Moivre’s Theorem (p. 125); for if 6 be 2r—-n, one nth root of 1 is cos mO-+- /—1 sin mé@, the pth power of which is cos mpo +- f—] Sin mp9, another root. Consequently, « being one POE, "a", anya) Laney (a” or 1) are all roots, but it does not follow that all the roots are among them, for the same root may be repeated twice or more. To explain this, observe that if n be a composite number, say 12, which is 6 x2 and 4X3, among the 12th roots of 1 will be found all the 6th, 4th and Square roots. Let 6 be a sixth root of unity; then 6*= 1 and ($*)2 = (1)*=1, or $"=1, therefore $ is also a 12th root ; and so of the rest. If, then, we take a 12th root of unity from among those which are also 6th roots, the series of powers of such a root will never give the complete series of 12th roots; but only a continual recurrence of the Toots which are both 6th and 12th roots. For in such a case the series of powers will be 9, $2, 3°, OO p= 1, SF, MSY Se Ee. But there are 12th roots among the powers of which are found ad/ the 12th roots: to prove which we premise the following TuHEOREM.—It is impossible that sin z = sin y, and also cos x=cos Ys unless x and y differ by a whole multiple of 27, or a whole number of revolutions. For the solutions of the first are all contained in =x t2mmr and y = (2n + 1)a— 2, and those of the second in = 27 +2m’r, and y = Qn't — x; m, m,n, vn’, being whole numbers, positive or negative. But no whole values of m and »’ will make Qxu+1)r7-«#= 2n'r — 2, or 2n+1 = 2n!, consequently, the solu- tions common to the two equations are all contained in y= 2 + Qmr : which was to be proved. Now, to apply this theorem, suppose 6 = 27 — n, and let « = cos@ + /—1sin@, the powers of which are a* = cos 20 + “—] sin 90 ’ 2mxr +e. a"=Cos md -+- /—1 sin md, and mé or —— cannot exceed @ or n 2a : ; : = by a whole circumference, till m= + 1, that is, the first 2 roots must be different, and therefore give all the nth roots (which are but 2 in number). Consequently, cos4 + VW —1 sin 6 is what is sometimes called a primitive nth root. Again, let s be a whole number which is prime to » (or let m and s have no Common measure greater than unity) : I say that a’ or cos s@ + /—I sin s@ is another primitive nth root. For let its pth power be taken (all its powers are also mth roots) : then ps®@ can never differ from s@ by a whole number of revolutions until p=(n-+1). Forif psd — s0= + 2vr (v being a whole num- ber) and if for 2x we write its value 0, and then divide by 0, we have (PS —s = +n, all being whole numbers ; which gives K 130 DIFFERENTIAL AND INTEGRAL CALCULUS. S v 63 ; oe ot yen is reduced to lower terms if p—1 be less than m, n — p— n or p less than m+ 1. Hence s and have a common measure, which is against the supposition. Consequently, by the same reasoning as before, o* is a primitive mth root. If « be a primitive 12th root of unity, then 2°, a, a, 0°, x, and a’ or 1, are sixth roots ; o', o® and a” are. cube roots; ewe be . and a are fourth roots; «° and @ (—1 and +1) are square roots ; and a, @, a, and a’, are primitive 12th roots. If we take p+q=n, or pe+ qd = nd = 27, we have p@ = 2% — q9, cos pO = cos 0, sin po = ~ sin 9, that is, if cos pO + ,/—1 sin pO, be A+B v—1, cos g@+V—1 sin qf is A—BA —1, or the first and last, the second and last but one, &c. of the roots derived from the lowest primitive root cos 6+ /—1 sin @ are pairs of the form A+B v1, A—BV—1. If n be even = 2n’, there is a root which is not in such a couple, namely, when p = n', g =, which case does not give two different roots. But this single root is always = — 1, for n'§ =ind = 7, and cos 7 =—1, sinz =0. A similar theorem may be proved for the roots of —1. One great use of this theory is the resolution of the expression a2” + a" into factors, for the purposes of integration. It is known from the theory of equations that if an expression beginning with a" have a a ...+@n for its 7 roots, that ex- pression must be identical with the product (w—o) (w—a%)..+ (v—a,). First take 2"—1 from whénce (m%, a ++++ &n being the 2 nth roots of 1) a" —1 = (a@—ay) (@— a) (1 — Hs) « « » Oo es (1.) Now assume Fes a ae ee ge +H A. » 2 hae a—l @r—-a@ @—ay UL = Oy Differentiate both sides of the first, which gives eae Lied of sie oe of all| Pat aise of ae a but r—a, but r—a, J but r—@, in which when z=, all the terms vanish except only that which 18 free of x —m, and so on, whence Rot oe (a, o ay) (a,- os) ee (a,- Ons Rag = (a— ) (o9- 03) es (a_- On) &e. n But ¢,"=1, &c., whence ne,"= —, &e. ay Multiply together (1) and (2), which give | as the first side, and as the second the sum of A,, A,, &c. severally multiplied by the products in (8) ; make w successively = @, @, &c. and we have, 7 a 1 a A, xX (a; — a2) @e (a,—o,)=A, ——_ OF Ay =e shy Ag =e ry &e. a, n . n ay Hg a, ome + e e e of a—l @—a &r—a% L— On If we proceed exactly in the same way with a"+1, the only differ- ence is that o," = —1 (@,—o) . » (%—%) = — 2+, and we have ‘y n oy Oe Ay 0, Hoe. being — ee Th Glib at Lk css 9 oe. ol eee ais ( of (—1)" x” + z Cm Ay = Xe E = 'On MEANING OF DIFFERENTIAL COEFFICIENTS, 131 A real form may be given ‘as follows: Let A +BV—T be a couple of corresponding roots, as proved to exist in p. 130; then in the first case, A+BV=1 " A-BYV=1 _. 2A («—A) —2 B? x—A—B , Ay z—-A+BV-1 _ (v—A)?+ B® ~ So that each couple gives a real fraction. We shall resume this sub- ject in the sequel. Previously to closing this chapter, we must observe that, when we take the logarithms of both sides of an expression, we must, if impossible quantities be in question, take the general loga- rithms as in p. 126; so that in p. 124, Imx yan 2m'r./ — 1, &e. Should have been annexed, the effect of which upon the result would have been to make Px (wh. no)w =O9—XAsinge+ ..., but this agrees with the original equation tang =k tan @; for @ and + (wh . no) x, have the Same tangent. If the nearest values of @ and @ be sought, then nothing must be annexed to ¢. Cuarrer VIII. ON THE MEANING OF DIFFERENTIAL COEFFICIENTS, AND ON THE FIRST PRINCIPLES OF THE APPLICATION OF THE SCIENCE TO GEOMETRY AND MECHANICS. On a perfect understanding of the reasoning contained in this Chapter, it must depend whether the student will hereafter apply the Differential Calculus to geometry, mechanics, &c., or only its symbols and mechanism, The derivation of differential coefficients has been sufficiently ex- plained ; we understand what they are in relation to their primitive functions, which are algebraical expressions. But when we come to apply the primitives, and make them representatives of concrete magni- tudes, such as spaces, times, forces, &c. &c., we do not carry with us _ any relations between the diff. co. and the magnitudes in question. | Our first question is this: ga being a given function of x, and @’z its diff. co., we know that for any value of x, x if a possible quantity, is either positive or negative ; it may for particular values of x, be 0 or x. What do these several states denote ? If we suppose the variable x to pass through all stages of magnitude from — « to + «, that is, through all values, positive and negative, the function $2 will pass through all its stages of magnitude; and we Shall now proye the following TuzoreM.—So long as ¢’z is positive, x and ¢z increase together, or decrease together ; or, let us say, take similar changes: but so long as Px is negative, if x increase, gx diminishes, and if x diminish, dx increases; or x and gx take dissimilar changes. We shall first give an example; let dx = 2°, ¢'x = 2x, which is positive or negative with «. That is, when a is positive, 7 and 2? ‘icrease together or diminish together, as is evident, But when a is K 2 132 DIFFERENTIAL AND INTEGRAL CALCULUS. negative, an increase of » diminishes 22; for instance, let x imcrease from —7 to —6, and 2 diminishes from 49 to 36. Increase and dimi- — nution are to be taken in their algebraical sense. Let # increase to x+Ar (that is, let Ar be positive) ; then, if the: diff. co. be positive {@ (r+ Az) —fx\—Ar is either positive, or becomes so when Az is diminished. For it approaches without limit to $’z, a positive quantity, and therefore must become positive before it attains that limit. But Av being positive, 6 (e+Azr)—¢z also is or becomes positive, that is, ¢(e+Ar) is greater than @x for finite values of Av. So that x and @x increase together. But, if Av be negative, or 7+Az less than 2, then @’x being positive, and {¢ («@-+Ax) — 92} Av becom- ing so before Ar = 0, it follows that (w + Ar) — gx must become negative, or (x -+- Ar) becomes less than @zr, or 2 and ¢x diminish together. Considerations. precisely similar show that when g'x is negative (x+Ar)—r must become negative before Av =0, when Az is posi- tive, or positive when Az is negative. If dr = tan a, 'x = 1 + tan*a, which is always positive : the angle and its tangent are always increasing together. Let the student verify this theorem round the four right angles. In the first right angle the theorem is obvious: but when z = 42, tan 7 = «, and here, we might at first suppose, increase must stop; but the following extension is a necessary consequence of the algebraical definition .of increase and decrease. When a quantity becomes 0 or «, it may change its sign, but it may not. The only restriction is, that it cannot change its sign for any other values. Now, 0 and & are themselves of dubious sign ; where they are accompanied by a change of sign, they themselves belong to neither sign more than to the other. In the case of ¢v=—tan a, we have a change of sign when 2 = 47; consequently, tan 5 is+ a, considered as the final state of tan « in the first right angle, and — dy y or $x its ordinate, qe tangent line, or line of direction of the curve, makes with the axis of op at the point whose abscissa is «. Examrte. In the curve in which the ordinate is the Naperian loga~ rithm of the abscissa, what is the angle made by the tangent line, or line of direction of the curve, with the axis of x, at the point whose abscissa is x= 10, and whose ordinate is therefore 2°30258.... . Here d 1 : fee : y = log 2, i ars ‘1 at the particular point in question. But °1 or g/x is the tangent of the angle which the is the tangent of 5° 43’, the angle required. Since the tangent line passes through the point * (a, b) or (a, pa), and makes with the axis of 2 an angle whose tangent is d’a, the equa- tion of the line, « and y now meaning the co-ordinates of any point in it, is (Algebraic Geometry, p. 23) y — ba = P'a(«—a), or y = pa + 'a (x — a); see page 135. 2. Curvaturet. We shall consider the curvature of a curve as a quantity to be estimated as follows: take three points on the curve, the first being the fixed point in question, the second and third being points near to it, which we shall afterwards suppose to approach without limit to the first. Three points determine a circle; and the nearer the two latter points Q and R approach to the fixed point P, the more nearly may the arc of the curve PQR be considered as identica] with the arc of the circle which passes through those three points. Let (x, y) be * This always means the point whose co-ordinates are a and 6, + The beginner may omit this article. 138 DIFFERENTIAL AND INTEGRAL CALCULUS. the fixed point in question («’,y’) and (o!,y") the contiguous points. If there be a circle having its centre at the point (m,n), and its radius p, and if X and Y be co-ordinates of any point in that circle, then (Alge- braic Geometry, p. 36) the equation of that circle is (X —m)? + (Y—n)? =p. But (2,y), (7,y'), (2",y") are to be points in the circle; whence the equations in the first column below: those in the second are obtained by subtraction of the first from the second, and of the second from the third— (a—m)"+ (Y—2) =P) (a! —2)(a! + 2—2m)-+ (y'—y) (y! +y-2n)= PNM apt rene y —Yy) (y’+y-2n)=0 Gade ("= aaa! 2m) + yy Vly! + y'-2n)=0 Subtract the firstin the second column from the second, which gives al2— Qa! + g2— (a!’—Qa' +x) 2m+ y'?—2y? + y2— (y’—2y' +y)2n=0. But if P be any function, which on two successive suppositions be- comes P’ and P”, then (Chapter IV.) AP=P/— P, AP’= P”— P,, A?P = AP! — AP=P”—2P’+P. Apply this to the functions 2°, a, y%, y, and the preceding becomes A*(x*)—A?’x ,.2m-+ A*(y’) — A’y.2m=0. Now, if we consider y as a function of 2, and suppose a to be- come successively 2’ = a+ Az, x” = 2+ 2Ax, which is the suppo- sition of ordinary differentiation, we have then A*x=0. But let us take a wider supposition. Let y not be given in terms of 2, but let x and y both be given in terms of another variable ¢, namely, by the equations x =yt y= wt, from which, by elimination of t, y = @r may be found. For instance, in the curve called the cycloid, instead of giving an equa- tion between z and y, it isfound more convenient to express both » and y in this way, y=a (t— sin 4), w=a (1—cost). Suppose that x be- comes a! and wv”, and y becomes y’ and y”’, when ¢ becomes ¢+Aé, and ¢+2At. Divide both sides of the preceding equation by (Ad)®, and then, to find the relation between m and », which is perpetually approximated to by supposing Q and R to approach P, let At diminish without limit. Then, (page 81) we have "2 2 2 2 2 2 d? (2”) Peo ay’) d de dé det ae Se Si an Bt POD o( HV 5 nn PU 9 (HY 4 ay di dt? dt dt. "de dé dt dt’ leg a d?y doo dy\? or («—m) Tr + (y—n) is + aa) + te em th _ Another relation is obtained from the first of the equations in the second column above, by writing Az for 2'—a, Ay for y’—y, dividing by Ad, and taking the limit, remembering that 2! and y have the limits wand y. This gives d. (v— m) E + (y — n) 2 cS 10, From which last two equations we easily obtain MEANING OF DIFFERENTIAL COEFFICIENTS. ’ 139 P 2 2 A a 2 = ee cf oe fay ae gee cel a dt \\ dt dt/\~* \dt d&~ ae aes’ ra at dx\" “al . {¢y Px dx ey y n= AG+(Z 7 de de ~ ae ae! 3 och ee (Gi) + (3 vB Bete. ee ax dz uel Eater dt di)S = \Vaede ~ at aes? the third equation being formed by, adding together the squares of the first two, and extracting the square root. It might at first appear as if we might obtain as many different circles as we can make different suppositions with respect to ¢: but it will be shown hereafter that there is. only one such circle; and this circle (by an extension of the same kind as that under which the curve is said to have a definite direction determined by the tangent) is said to have the same curvature as the curve has at the point (a, y), and its radius is called the radius of curvature of the curve at that point. Let us make the supposition that t=, in which case we have y= Xz, x= ¥x, the second of which must be made identical, that is, the function ¥%x mast be @ itself, and xx is the same as dv. We have also, dx dx dy _dy dy dy eee —=e BOT. page Ske) as OT Ne da?’ 3 3 dy be oy {1 + (p'2)* }? = + BBS BTR SSA Ih ae ER lla Me Re Splie: {} = (+) Bia’ 3) OM : neglecting the sign, which we shall consider elsewhere. Let us suppose it required to find the radius of curvature at any point of a parabola whose equation is y¥2 = 4ex, We have then gr = Alo Vx ba =r/%, 1+ ay = 2=*, 8 8 i, aw w+ c\3 tg: a-c)e gle = — We x? p=( a ) ost (~4Ne x +) 29 @toF xv , neglecting the sign. Hence, since the curvature of a circle is evidently the less, the greater the radius, it follows that the curvature of a para- bola diminishes as we go from the vertex, where it is greatest, the radius of curvature being there least, and equal to 2c. We may easily give a sufficient proof that the circle thus obtained is closer to the curve at the point P, than any other which can be drawn. For if possible, let a circle (A) fall between the circle of curvature (K) and the curve (C), immediately after leaving P. Then the circle drawn through P, Q,R, which approaches without limit See to coincide with (K), cannot approach it nearer Q aN «than (A), which is absurd. Give a similar proof Teen 7: that no straight line can le between the tan- gent and the curve. More formal proofs: of both propositions will be hereafter given. 140 DIFFERENTIAL AND INTEGRAL CALCULUS. 3. Length. (Read again the remarks in page 23, and also the pro- cess in page 30.) We now proceed to find the length of any portion of a curve whose ordinate is x. Let it be the arc contained between the points which have a and a’ for abscissze. Divide the portion of the axis of 2 which lies under the given arc, a’—a in length, into n equal parts, each of which is Aa. Let MN (figure, page 136,) be one of these portions; and let OM= «, MP=y. We assume as an axiom, that the arc PQ is greater than the chord PQ, but less than PT + TQ. And we have AEC RCO ans ae (dy? PQ=V(Ax)?+ (Ay) PT=V(Ax) +(Ar)’ tan*TPZ= scn/1 +(2) GY nisit! jae fa a ne TQ= Ay~ sae ng eae =a AP, where* « and Av are comminuent. Hence we find that The arc PQ ey Ay? a/ (avy lies ey Ax 1+ (=) and Ax 1+ ae + a (ON, bo (AS ton Writing $'x for o, and making V1+(/x+-a) == V1+ (Pa)? + B, we see that 6 and ew are comminuent, as are therefore 8 and Ax. Re- peating this process for every one of the parts into which the whole arc is divided, we see that the whole arc in question must le between S| Av W14+(@x)* + 6) § and Ej Ar Wi+(@2)* + «)}, or = ( ArV 1+(@x)2) +26Ax and = (Ax J 14+(@'x)) + SaAc. Now, when 7 is increased without limit, or Ax diminished without limit, (2 Aw = a’ — a) a and £ are in every portion of the arc dimi- nished without limit. Consequently, A and B may be always greater than the greatest of the values of « and 8, and yet be comminuent with Ar. In that case nA and 2B must be greater than 2« and 2A, and nAAx and nB Ax greater than 2(a¢Azx) and >(BAx). Remember that Ax is the sameim all. But nA Aw = A(a’—a) and nBAw=B(a'-a), which last are comminuent with A and B, and therefore with Az. Consequently the limits of the two preceding functions, when Az is diminished without limit, are both the same as that of 2 (AxV1. +(@'z)"); which (page 100) is 7, aN1+('x)? dx. Hence the arc of the curve, which always lies between these sums, is itself the limit just found; that is, the arc of the curve whose ordinate is éx, contained between the points whose abscissee are a and a’, (and called s) is _*amay be reckoned positive, though the expression it represents may be nega- tive. We have nothing to do but with the fact that its numerical value (indepen- dent of sign) is comminuent with Aa, MEANING OF DIFFERENTIAL COEFFICIENTS. M1 oa —~; af dy 2 d f s= 1+ (2)? de = 1+ (4 wi ea a ak Exampte 1. Required the length of the arc of a parabola whose equation is y* = 4cr, which begins when x = 0, and ends when z=a. gr= V4er “glx = ee VJ1+ Gay = Ri gt x f af abe pi 2 foe oe, fietcet Petey (ueitacke Gy a te dx v 12 Ve + cr 2 V2? + ox _ {d@+cer) ec dx Vertex 2 V2? + cx the last being obtained as in page 116.. Hence we have, s/s dz = Va? + ca + = log (a+ S84- Va $ca) —£ log ° x ae 2° 2 2 — “ated ave ten) =u x ox +< log (a+ 5 4Ve-+er), Cc aN TF ca + 5 log ( ExampLe 2.—What is that curve the arc of which, beginning from v©=0, is always = J/2ax? The diff. co. of {v1 + (wa)? de is NA 1-++ (¢’x)®; and therefore since {vi + (o'x)? dx = WV 2ar we haveV1 a (o'r)? = - 0 av a —_ bathe d: ef 2k eer 2h. d (Qhx + mk | de 2 J 2ha— x? 2 Qhe — 2? V Qhe — x? = V Qhr — x? + Qk vers~ i+ constant, (page 116). Any value of this constant may be used. In fact, if the constant be made =p, then the curve which has the two first terms for its ordinate is raised or lowered from or to the axis of w by increasing or decreasing p : but the arc intercepted between any two ordinate lines is not changed. 4, Avea.—The number of square units in a rectangle is the product of the numbers of linear units in its sides. Let it now be required to find in square units, the value of the portion of space contained between the points of the curve y = ¢x which have a and a’ for abscissa, bounded by the arc of the curve, the ordinates of its extreme points, and the 142 DIFFERENTIAL AND INTEGRAL CALCULUS. axis of a. Let the portion a’— a of the axis of x be divided into ” equal parts, each =Az, as before. Then (figure, p. 136) let MN be one of these parts, and draw ordinates (as in figure, p. 30). Hence the por- tion of the curvilinear area MPQN is composed of the rectangle PMNZ having the area yAv, and the curvilinear triangle PQZ, which is less than the rectangle contained by PZ and ZQ, or less than Av Ay square units (neglecting the sign of Ay, if it be negative). Hence the area MPQN lies between yAwv and yAv + Ay Ax, and the whole area of the curve lies between SyAa and Yy Ax + ZAy Az. But, Ay being com- minuent with Az, it follows by the same reasoning as in p. 140, that Ay Az is comminuent with Ax ; and thence, that the two preceding sums have the same limit ee “ yd«, which is therefore the area in ques- tion. That is, the area bounded by the ordinates whose abscissz are a and a’, and the arc and axis of x contained between them, is S. a yaa or fn’ px dx. Exampie 1.—The area of a parabola, whose equation is y° = 4ea, contained between the vertex, the axis of x, and the ordinate whose —_—— 1 3 , : , abscissa is @, 18 I¢ OV cr dx = 4c? a® =4 abscissaa xX its ordinate. In this is condensed the whole of the process in pages 30, 31. ExampLe 2.—What is the curve, whose area contained between the ordinates to the abscisse @ and 2, is always (in square units) clog=? ce x . ut eee c We have here | ydx=c log = and differentiating both sides y = ss or wy = c, the equation of an hyperbola. Observe, that this area being an integral between certain limits a and x, must be of the form ya —wa, and we have accordingly assumed it so, in clogr—cloga. The are is also an integral, and a similar assumption is required. It was made in the second example of the last article, for the limits are there, 0 and 2, and a Vx is ava — a0. 5. Solidity or Volume.—The method of finding the solidity under a given surface must be deferred until we have more developments of the Integral Calculus. 6. Density——When any solid (or fluid) contains equal quantities of matter in equal bulks, from what part soever they may be drawn, the uniform density which is then said to prevail, may be measured, for the purposes of comparing one density with another, by the different quanti- ties of matter (or weights) contained in any one given bulk. If the same vessel filled with fluid B, weigh twice as much (independent of the weight of the vessel) as when it is filled with fluid A, then without knowing the content of the vessel, we pronounce fluid B twice as dense as fluid A. But as it is generally more conyenient to employ absolute than relative terms, we obtain the necessary language in the samt manner as in the case of length, by choosing an arbitrary magnitude, and calling it wntty or 1. Let pure water be said to have the density 1; then any substance twice as heavy as water, bulk for bulk, has the density 2, and so on. An accidental relation in our metrical system makes the descent from the mathematical notion of density to the terms of common life immediate and easy. A cubic foot of water weighs (very nearly) 1000 ounces avoirdupois ; so that if we say the density of gold is 19'362, we infer that a cubic foot of gold weighs 19362 ounces avoir dupois nearly, Let us now suppose a thin rod of matter whose uni- MEANING OF DIFFERENTIAL COEFFICIENTS. 143 form density is 1, or a cubic foot of which weighs as much as the same of water. And let there be another such rod, not of uniform density, evidenced;by our finding that any two equal lengths of it have different weights. Let the law of the weights of different portions be this, that winches taken from one of the two ends, which is specified, always weighs 2 ounces ; that is, the first 2 Inch weighs 1 0z., the first inch 1 oz., the first two inches 4 ounces. In the case of a uniform rod we might always find k by dividing the weight of any portion by that of an equal bulk of water: but in the second case we have no definite measure of density, though it is clear that the weight of equal portions goes on increasing. t I Le ye Let AB be a part of the rod in question = a, and let BC=BD=Azg. Then the weight of DB is a?—(«— Ax), and that of BC is (v + Ar)?—a*. These are 2vAxr — (Ax)’ and 22 Ar + (Ax)? ounces. Let the weight of a bulk of water, such as that of DB or BC (which must, ceterts paribus, be proportional to Ax) be eAr, then the density ; fing t ae . 22 — Ax of BD, if the matter in it be uniformly distributed, is ———— and that e .» . 2 Ax of BC, on the same supposition,is —-— These two suppositions are not correct ; nor according to the definition of density, can we say what the density of the rod should be at B. But we may see that the Weights of the successive’ equal portions DB, BC, approach without limit to equality when Az is diminished without limit, and that ne ; Bh 2x the presumed densities approach without limit to es. Let us say .. 22 ' : that the density at B is —; we have here an assertion which will be e nearly verified by a small portion of the rod taken on either side B; more nearly on a smaller portion, &c., and in this sense we may admit the assertion. Similarly, if the weight of the length x inches be gz oz., it will follow in the same manner that the density at the point whose distance is x will be ¢'x divided by e, the weight in ounces of one inch of water. And hence it follows that the density being given at the distance a and called y, the weight in ounces of a! — @ inches taken between the points which are @ and a/ inches distant from the end is ef. ydz. 1. Velocity.—When a point moves uniformly, that is to say, describes equal portions of length in any equal portions of time during the motion, it is said to move with a velocity which is measured by the number of units of length described in a unit of time. Thus taking feet and seconds, with reference to these units the velocity 10 is that of a point moving over 10 feet in one second of time, 20 feet in two seconds, 5 feet in half a second, and in the same proportion for every other time. Hence it is evident that v being the velocity (length in one second) and ¢ the number of seconds (called the teme) vt must be the length de- Scribed, which call s; hence s — vt. Hence, knowing the length described in any time, or knowing s for any value of ¢, we find » the Velocity by dividing s by ¢. It may help the student to make him remember that as ¢ seconds is to one second, so is s the length described 144 DIFFERENTIAL AND INTEGRAL CALCULUS. H sxl in ¢ seconds to or the length described in one second (the velo- city). When speaking of length moved over by a point, it is usual (but incorrectly) to call the length space. Thus it is said that one point moves over more space than another. Let there now be a point which does not move over equal lengths in eqyal times ; but suppose it to move in such a way that at the end of £ seconds, it has‘always moved over ¢ feet. Suppose, that in the last figure, D, B, and C are its positions at the end of ¢— At, and t + At seconds. Then the lengths described in the At seconds* (or of a second) immediately preceding and succeeding ¢ seconds elapsed are 2 — (¢ — At)? and (¢ + Ad)’ — P, or 2¢ At — (At)? and 2¢ At + (At). For by hypothesis AD = (¢— At)’, AB= PAC = +. A) then, DB and BC were uniformly described, the velocities (length per second answering to those lengths per At) would be the preceding lengths divided by At; or 24 —Aé and 2¢+ At. But this supposition is incorrect. Nevertheless, if we speak at all of the point having a velocity at B, we must assert that velocity to be 2; and this assertion becomes more and more nearly true on one side and the other of B, as we take At less and less. Let us then say that the velocity at the end of t seconds, of a point which has then moved through ? seconds, 1s 2¢: not that the point will continue to move uniformly at the rate of 2¢ feet per second for any portion of time however small; but that the length moved through in the ensuing At, is nearly as it would be at that» rate if At be small, more nearly if At be smaller, and so on without limit. In the same way it may be shown that, gf being the feet moved over m t seconds, the velocity at the end of ¢ seconds is g't; and if v (a given function of t) be the velocity, the length described between the end of @ seconds and a’ seconds is Fe ‘’vdt. Moreover, the time of describing a’ from a feet to a’ feet from the origin of the measurement is f —, if v v a be a given function of s: or, a oF om fot me 5 dt v a lan? what function is this same velocity of the length described, the length being measured from the beginning of the motion, so that when t= 0 s =0. Here we have ExampLe 1.—The velocity at the end of ¢ seconds being ds a aa ee = alog (1+¢) + const. But when t= 0, s=0, or O=alog (1) + const. or const, = 0: whence s == alog(1+¢). Hence we have, dt 1 reeds : t— or 3@ At—3t (Ad)* + (At)* feet per second : and in the interval from ¢ to t+ At seconds, an accession amounting to (¢+ At —# or 3@ At-+ 3h(At)? + (Ad) feet per second. | Now, if an accession of Av be made to velocity uniformly throughout the time At, then the force (corresponding accession in one second) is found thus. As Af is to one second, so is the acceleration made in the time Af Av x1 Av pny (namely Av) to ory? the acceleration in one second. If, then, the preceding accelerations had been uniformly made throughout their several times, it is obvious that the forces producing them would be 3t — Bt At + (At)? and 3@ + 3¢ At + (AZ)’. But this supposition is imcorrect ; nevertheless, in saying that the force at the end of the time t is 3/2, we make an assertion which is the more nearly true the smaller At is supposed to be. And in a similar way, if dt be the velocity at the end of the time t, @’t is the accelerating force at the end of that time. Similarly, if f be the force at the end of the time ft, the velocity at the end of a’ seconds, communicated in the in- terval from that of a seconds, is f’ fdt: so that Vel. at end of a’ sec. = Vel. at end of a@ sec. + Lk jee The following are then the equations connected with the motion of a point which has described the length s (or s feet from the origin of measurement) and has a velocity v, and is acted on by an accelerating force f. vo aT f= a eh 2 aE pe = gs or Ae dt dt ds The last equation finds the velocity directly when f is expressed as @ function of s: for by it we find v* = 2 if ‘fds + C; and if we know the baci of the velocity when s is a, and want to find that when s is a’, we ave (vel.)? at distance a! = (vel.)? at dist. a +2 f° fds. _ Let the known velocity at the distance a be A; and let the superior limit a’ be indeterminate. We have then, ds ) ds en eA +2 f 2 fds Soe B —-—— + const. a VAP + Ofc fds where the constant must be determined by the circumstances of each particular case. . MEANING OF DIFFERENTIAL COEFFICIENTS. 147 We shall end the chapter with some examples of this method: but we have occasion first to consider the preceding cases in their connexion with each other, as well as in reference to the distinction between posi- tive and negative. 1. It has doubtless appeared that terms which seem as independent of the conventions of our science as direction, density, velocity and force, have been treated rather as if they were mere definitions springing out of and were well known to the student, as he may think, before beginning this Calculus. We have proceeded with common ideas, and common phraseology, so long as uniform density, untform direction, &c., were in question ; but when we come to consider a point which has a varying motion, &¢c., we no longer deduce a function, and say, this ts the velo- city, &c.; but we say, let the term velocity, &c. be applied to such and such results of the Differential Calculus. Has, then, a point in vary- ing motion no title to be considered as having a velocity, &c.? Such will be the difficulty that must at first occur. But it may easily be shown that the preceding process is only such a refinement of the rough Differential Calculus which all people who deal with material objects are obliged to use, as is rendered necessary by its inexactness. If we assign a definite direction to the motion of a point over a curve at ever we assert a stone falling freely to have a definite velocity at every point, but one which continually increases, it is because when motion changes gradually, we think we may take a time so small, that the motion may be actually uniform during that time ; which is not correct. All these Suppositions spring from one common falsehood (in mathematics) or truth sufficiently near for practical purposes (in common life): namely, that every whole has parts which are such small fractions of it that they may be rejected without causing any error. To this, the answer is that there is no such part of a whole; but that since for the last four words may be substituted “ without causing any error greater than one which is form the ground-work of this science. And it is an evident corollary, that since the common notion is an approach to the more exact one, ithe results of the former will always nearly coincide with those of the latter. The student must avoid the notion that he is dealing with densities, velocities, forces, &c. as real things, and must remember that his symbols stand for nothing but numbers or fractions which are the mea- sures of the sensible phenomena in question, upon purely arbitrary sup- 20sitions. For just as owing to the resemblance of certain algebraical md geometrical terms, nine students out of ten have a mysterious lotion that a straight line multiplied by a straight line is a rectangle *, Vhich is nothing less than supposing that the addition of numbers ogether places two straight lines at right angles to each other, and * Which ought to mean that if the nwnber of times which one of the sides con- ans a foot (au arbitrary length) be taken as many times as the other side containg foot, the resulting number will be the number of times which the rectangle con- ains the square whose side is a foot, | Lu 2 148 DIFFERENTIAL AND INTEGRAL CALCULUS. , their extremities ; just in this manner, we say, | many students are perplexed for a long time with such notions as that the force multiplied by the time gives the velocity, using the words in a sense as concrete as occurs when we say that force, if allowed to act, will | in time produce velocity. To avoid this, we recommend our reader perpe- tually to recur to the definitions of all numerical measures ; for instance, frequently in using the preceding proportion, force X time = velocity, to remove the mystery by remembering that it means nothing more than this; if that which gives a feet of velocity in every second be allowed to act for b seconds, then ab feet of velocity must result. Finally, he should recur to the notion of matter having velocity as implying merely the being in such a state of motion as would, if continued unaltered, cause it to describe a certain number of feet per second. 2. Several of the preceding cases may be considered as belonging to one general proposition. In the last, treating of force, we have directly the notion of cause and effect; and in treating of the diff. co. abstractedly (page 135) we are easily led to a mode of speaking which looks somewhat like the supposition that the diff. co. is the cause of the snerease of the function. To avoid the possible misconception of the words cause and effect, let us speak simply of a precedent and a conse- quent, the former of which has a numerical value a, which, allowed to remain the same, makes the consequent =az, or gives « for every unit mn 2 If, then, the consequent, instead of ax, were px, the precedent, if consi- dered as existing at all, could not be @x, unless $ (v + Av) — or were equal to @ Az for all values of x, which is not true except for Pr=aa. But d(a-+Ar) — bv= pla. Av + $$" (w+ OAx). (Ar) O<1; and the first term on the second side is to the second as $a to 1 "(w + 0Ax) . Av, that is, can be made as nearly the whole as we please. Hence the supposition @ (x + Ar) —ou = g'e . Aa, (which would result if the precedent were ¢/r,) may be made as neat the truth as we please; and if we should say there is a precedent, no longer uniform, but variable, that precedent cannot be considered as having any other value than ’z. 3. We have to consider what are the negative suppositions which correspond to the positive ones we have made. In the case of direction we need say nothing more; in that of curvature, a purely arbitrary distinction, if any, must be made; but to this we shall return. The occurrence of a square root in the consequences more than was in the premises, is generally the index of a power of selection as to sign. This applies to the question of finding the arc of a curve; though here we lay it down as convenient that the arc should be measured positively in the same direction as the abscissa. But with regard to area, we must take care to distinguish the alge: braical amount of all the rectangles from the arithmetical one, in al cases where the ordinates are negative. It is evident that (a’ > a) if 4 be negative from r =atoxr=d, S ydx between the same limits 1 negative also: both from the summation of which this is the limit (p 100), and from this also ; that if fi ydx generally be ¢x + const., W! have a “yda = oa — ga; draws parallels throug] « : MEANING OF DIFFERENTIAL COEFFICIENTS. 149 and $/x or y is negative from x =a the less to v=a' the greater, whence (p. 131) da’ is less than da. If, then, y be negative for any interval between the limits of integration, all the area obtained from that interval will be negative, and will be subtracted in the result, For instance, let the ordinate in feet be the sine of the angle made at the centre by forming the abscissa into a circle (repeating the folds if necessary) whose radius is one foot ; or let y=sina. Then the area from the origin till the whole circle is completed on the abscissa, is ae if snadx: but Wet SJ sin edz = — cos x So sina dx=(—1)-(-)D=0, or the whole area OMANB = 0, which is not true unless we consi- der ANB is negative, which it is lig AR in the integration. To find the o0 B arithmetical amount of OMANB, ee a we first integrate from 2 =0 to 0 #= % giving OMA= 1 — (— 1) or 2 (square feet) : then integrat- ing from « = 7 to x= Qn we find (—1) — (1) or —2, which, arith- _ Metically considered, is 2. Therefore the whole area, in the arithmetical sense, is 4 square feet. But if we remove the axis of 2 to O'B’ _ (00'= a) giving for the equation y = a + sinz. we find 2ra for the area, namely, that O'OMANBB/ = rectangle OO’BB’, as is sufficiently evident. In this case the arithmetical consideration of ANB would _ lead us wrong. _ With regard to density, we have no idea corresponding to that of __hegative density, except when we consider it as immediately connected F with weight. If the weight considered be in air, and if part of: the rod _ were lighter than air, then the tendency of that part would be to rise, and the density of the corresponding part must be considered as negative. Velocity is negative when a 8 negative, that is, (p. 131) when in- crease of time decreases s, or (s being positive) when the point is moving towards the origin of measurement. Hence, if we would solve the question of the motion of a point which moves towards the origin with a velocity, which, absolutely considered, is dt, we must form the equa- tion a = — $t, and integrate. ; LD yh : ie aaa, Force is negative when ra negative, or when the velocity diminishes as the time increases; that is, when the force lessens (algebraically speaking) the velocity. This amounts to saying that the force must be directed towards the origin of measurement, which lessens both kinds of Velocity, for the negative velocity is thus made arithmetically more and Negative, the positive velocity arithmetically less and positive. Se (ee at A Pp B _ Examrre.—A body at rest at B (AB = a feet) begins to be driven or attracted (according as the cause of motion comes from behind or ore) towards the point A, with a force depending upon its distance, 150 DIFFERENTIAL AND INTEGRAL CALCULUS. so that, when at.P (AP=s) the force is ms; that is, if, being such asit is at P, it were allowed to act uniformly for one second, it would add ms to the velocity in the direction PA, In how many seconds will the body move from B to A ? dv ds dv The equations of motion are — = — ms, —~ =v V_ = — Ms. J : di dt ds Integrate the latter, which gives v° = const. — ms*, But »v = 0, when s = @ = const. — ma’. 0 8 ds, ds v= m (a*—S*) ese | oy | ee io —V m (a—s (We use the negative sign because the velocity is towards A.) ] ds i bi C ) + pee — —— | ———_ > 7 — COs aon const. =. Vn’ Vea—s vim a) 1 But i= 0 when s=a —— cos! (1) + const. = 0 const.=( NM d ] ls Time from B to P = —— cos ( ‘). lm & T : 1 Place P at A, or makes =O and whole time = —— =<: dm 2 This result is independent of a, that is, wherever the point wé placed at first, it will fall to @ in the same time. This result will m appear strange when it is considered that the farther the body is place from A, the greater the force which begins to act on it. If, therefore, number of points were placed at different distances, the farthest wou! immediately begin to gain on the nearer ones, and all might con together (as has been shown they would) at the point A. The who, velocity acquired is Ym « a. 2: n Exameie 2.—Other things remaining the same, let the force be 3 , 2m 2m =--—-— oe ———+ const, 0 = — + const eee 2 S a Lido [= lees fom Wn SS i=- — —d : on (; :) \ 2m a— § : if / sy { sds 1 — Jsds ana: 5 ES Sitciee pean See ee ee “\ a—s - J as—s? 2. Vas — §? LM a s- 28) 08>) Gln aoa ; ——_—_-——— 2 Nas — ) as—s* ———, , a ys const. * as — s+ — vers)} —° p a Ih * Observe that though this term is immediately multiplied, we simply write Col: as before, because it is as before nothing but an undetermined constant, a” But ¢ = 0 when s = a, or 0 = const. 4+- 0 — vers 7) 2 ; 22m B 2 =f Ta and + = vers~! 2, whence const. — —. 22m 8 —— 3 ‘ ra? a Qe 2s Time (from B to P) = 72 — yo Vas—s-+—— vers* —, } 2V 2m 2m 2am a 2 iets a0) (s=0). Time from B to A = He 272m The velocity increases without limit (numerically) as P approaches a ; the reason is that the accelerating force increases without limit. € now pass on to some extensions, which are necessary in the further application of the methods contained in this chapter. or % CuHapter III. doe c ON THE CONNEXION OF DIFFERENTIATIONS OF DIFFERENT ey KINDS. : 4 , _ Wuen we propose an equation between two or more variables, it may be differentiated in as many different Ways as it allows of expressing one Variable in terms of others. If we wish to consider one variable as _ actually expressed by means of the rest, the equation is written in. the 4 U= (2, y,2z...); but if it be merely required to signify that a relation does exist between such variables, we write @ (u, a, Ye oO. In the first case w is explicitly, in the second case implicitly, a function Mey,z. . | Certain values of all the variables being taken, which satisfy the equation, and increments given to each, the permanent existence of the telation ¢ (u,...) = 0 gives an equation between the increments, from Which any one may be determined in terms of all the rest. Thus taking %, 2, Y,...- 80 as to satisfy the equation, Az, Ay, ... may be assumed at pleasure; but Aw must then be taken so as to satisfy @ (u + Au, @+ Ar, y + Ay,...) = 0. But there evidently exists this mutual Coexistence of the same values of the increments ; namely, that if Ar =a, Ay= b,.... will permit Au =m to satisfy the equation, then Au = m, 4y = b.... will permit Ar = a to satisfy the equation. _ For this condition being fulfilled, it is indifferent which of the incre- Mhents is supposed to be determined by the rest. Hence, one equation _ only existing, and any admissible supposition being made as to the man- ~ 7 : 152 DIFFERENTIAL AND INTEGRAL CALCULUS. 2] ner in which Au, Av.... shall diminish without limit, the diff. co. | du dx — and ay ote reciprocals. For whether we suppose the equation to lu dx assign win terms of x, &c., or x in terms of u, &c., any values of Aw and Az, which are simultaneously admissible on the one supposition, are the same on the other; so that Aw + Az, obtained on the first sup- position, is the reciprocal of Ar Au obtained on the second; and their limits are, therefore, reciprocals. But it is far. otherwise with au et du ax. ; Are —- antl, and ——, &c. The first requires successive increments dx du® dx* du® of 2 and a relation between them, namely, that of equality ; x becomes a+ Ax, then « + 2Aa,&c. The successive increments of w are then determined ; and will not, generally speaking, satisfy that relation of equality which, by a similar convention, is the foundation of the process 2 by which aa 8 determined; namely, the supposition that w becomes U u + Au, u + 2Au, &c., from which successive increments of 2 are de- termined, which, in their turn, are no longer equal. Observe, that we are considering the 2nd diff. co., not as the diff. co. of the first diff. co., but as the limit of the second difference of one variable divided by the square of the difference of a uniformly increasing variable (p. 80). Though the two results are the same in form and value, they are obtained by different processes, and the second process 1s frequently the more convenient origin to suppose in reasoning. | The only relation in which successive equal increments to w give equal increments to u, is any one of which au — br = O is a necessary conse- pS ea du dx quence, and in this case both Ay and —— are = 0. 2 du n oT) . d"u Lig Let O(x, v) =O and abbreviate —. and —~ ito u (eaccents) atm ; dx” du” | v (n accents) ; Let wu = ¢2,0 = Wu, follow from y (a, uv) = 0, so that w/ may be found as a function of 2, or w asa function of w, namely, u! = pe vw =w'u. And w' and a! are reciprocals (p. 53), whence wu! Oo ¢'z . Wu= 1, which will be found to be a necessary consequence 0 & (a, u) = 0. We can now solve the case in which w' is given as % function of u (not of w as in common integration). Let du eas U, then dz du Ante U Then the solution of «= fu gives wu in terms of «. which suppose = fu. du ‘ : EXAMPLE. 5, sin: required wu in terms of «. x dx 1 page du _ (‘snudu “d.cosu du sinu sinu,) 1—cos’u 1 —cos’u 1 1+cos u eg? C7] sm ees oe we dT F ts ae mie] “ay eer = > log +C, whence w= cos ae] ) 2 1—cos u where © may be any constant whatever. - DIFFERENTIATIONS OF DIFFERENT KINDs. To find the relation between w” and x” proceed as follows :— 153 du ; 1 i ie eae 1 dw'u oo == ~ t= a ere IAT 7 = matt aoe - ax wu dx dx wu (Y’u)? dx a 1 dw'u du _ du? du _ =) d?x ; e. (euy de dr dx}: due. dx) du d a : er Ori ax To remember this, write it az dx® + Tat dura (, xt & u ____ Differential equations are frequently written as if the diff. co. had pf - distinct numerators and denominators > thus; = =P is written dy =Pdz. Je ‘ ' Remember that the second implies only the first; and that as far as first diff. co. are concerned, we see in p. 53, that they have the ordi- _ hary properties of fractions; but it would not be safe for a beginner to _ proceed in the same way with higher diff.co. For instance, we should ~ not recommend him to write the preceding thus, du dx + dx du — 0, though it is certainly true that upon the implied suppositions with _ tegard to the successive increments, A2~ . Ar + A2v. Au diminishes _ without limit as compared with (Ax)*. As far as the mechanism of the operations is concerned, this process is safe enough ; the risk is that the student should forget, when there are several variables, which of them received successive uniform increments in order to form the several 2 second differences. 2 EXAMPLE. u = sina, what is aut p <2 du? | Mu ax _ dx® an. = BIND a U dae (ay ~ (cosa)? GQ—w)? dx ah # Ce.” ae leer ” Verification. 2» = sin u, = Ga niass Gres OS ee ‘ : au . ° L s Let = be a given function of u, =U. Required wu in terms of 2. av d*u ‘ cad | ri Oe: Or a? V being / Udu au dV du du i 5 dV du ad dV dx? du? dx? dr dudx. dx * ae dudu d /du dV d ray 2h ye el —s ° = at. ~ pct 9 : ~~ But 2 Tne dn’ (=) which therefore =2 a a (2V) du ee or (z) = C+2V=C + 2/U du, = + VC+2f U du. ___ The sign is to be ascertained by the conditions of the problem, as also _ OC, the arbitrary constant. 4 +. 154 DIFFERENTIAL AND: INTEGRAL CALCULUS, daz l du — SSS eee LC — AES. PSE, du VC+2f Udu VC+2 [Udu and wu being found from the last equation, the problem is solved. C and C’ are specific constants when the problem implies any conditions for determining them ; but when the quéstion merely is, what function of u has a second diff. co. equal to a given function of u, they are perfectly general, and may be any whatever. Verification ; Se a ec Piha (C + 2fU duy? du J C+ 2fUdu +C'= fu; 1 Lord l fduy —=— — — 2 oO _ ) Z ) . (C+ 2/f U du) ha (C+2/U du)= — (3) x 2U du? du du? da du® ( 1 fees ae ee ef SP ee ef ol “x2U }o4. dx? ) du? ar Nn ae =) ) a EXAMPLE. _ oe US ee gee if Vayu w v : rN du ile Bot: ee CS SGaw =log (u+ VC + u3) +C! eC = y + NO+U us ie-%—1Ce @-™. This result contains a complication of constants, which is reducible to simplicity, as very frequently happens in the results of integration. The preceding may be thus written : wate Ce —1CeCe™®. But 4.¢~° may be made anything we please by giving the proper value to C!, and then—C x 1 ¢~°' may be anything else we please, by giving | the proper value to C. Hence these two coefficients simply amount to arbitrary constants, and we may simply say that w= Ke*+K’e’. Examp.e I].—Instance of the transformation of an equation into another of a totally different form of solution, by the use of impossible quantities. In the preceding equation, let « = 9 V—1. Then wu may be made a function of 9. And we have du du 4 ee 1 aa 2? dp d0 dx 2 a0’ dtu; od ( 1 du do 1 du 1 Rael dp 0 dO Jes eo Pde ORY al d? du fee A Therefore — api O15 +u=0 givesu = Kev tit Kier? (p. 119) = (K+K’) cos 0+ /—1 (K—K’) sin @ = C cos 0+C’ sin 8, : on similar reasoning to that immediately preceding this article. We shall now produce the same result directly. Pu du =—u, Uz—u, 2/Udu=—v 0= | Fi u wu, 2f Udu u ire = > a , DIFFERENTIATIONS OF DIFFERENT KINDS. 155 or 6=sin™! (Je )e: u=NCsin(6- C)=V GC cosC'sind- V6 sinC’ cosé, Assume = / VC cos 0! = K, —¥C sinC’/=K’, or tan! = —, C=K? + K®, and u = Ksin 6 + K’ cos 0, in form as before. Exampte III.—Instance of a more complicated integration, attained by preserving the less complicated form, but generalizing the constants ; Pu mto variable functions. Let a6 +u=T, a given function of @. Whatever the solution may be, it can be represented in an infinite number of ways by Ksin@ + K’cos 6, if K and K! be functions of 0. If it were w= 6%, and if we chose to assume K + K’=96, we can satisfy the conditions K sin 0 + K’ cos@ = 6 and K + K’'=96 by the simple method of algebraic solution, which gives K=(@—6 cos 0) (sin@ —cos@) K/= (6 sin 6—6°)—(sin 6—cos 6). Therefore, not only may we assume w= K sin@ + K/ cos 6, but even then we are at liberty to assign any relation we please between K and K’, which does not contradict their being functions of 6. Let us make the assumption, which gives du ; Fie der dk! 6 cos 0 — K’ sin 8 + 7 sn 6+ a) cos 0, dK. dk! Let our assumed relation be —— sin @ + — cos0= 0. a0 do du ae, Then — = K cos@ —K’ sin@ dé du : : dik aK?! ©. dk dKar —— — ~ |. —— = —-U+— cosd -—— 0, rc K sind-K cosd-+— cos0 76 sin8 ut oF cos Wo sin dK aks dk — O. -— 7) —_ = 5% ) rel 10 cos 70 sin Pt 70 T cos whence dK dK’ dK! : = — gj ek ae Aa = era | a) and O aT: sin@ + a5 cos 0 70 T sin by the ordinary solution of algebraical equations. Hence J K=/Tcs@d0+C, K’=—/Tsine do +c u=C sind + C’cos@ +sin@ fT cos0 d0—cos0 fT sind dé. _ The above solution makes use of the following notion. When T=, we have found a solution which contains two constants. It is not un- likely, then, that a similar form, but with more complicated coefficients for sin 6 and cos@ than simple constants, will be the solution of the more complicated equation. This of course is no argument, but only reason enough to make it worth while to try u = K sin@ + K’ cos @, in the manner preceding. Our suspicion turns out to be correct in this case, . 156 DIFFERENTIAL AND INTEGRAL CALCULUS. 76 , u=Csind+C’ cos 6+sind 0 cos do—cosd [0 sin8 dé J 8 cos 6 dé = Osin 0—f-sin 0 d6 = 6 sin @+ cos 0 fe sin 0 d@ =—6cos0+/f cos 6 dé = —6cos 0+ sin# u== Csin§.+C’cos 0 + 6 which may easily be verified. d*u 2. Rack u=cos 0, u=C sind + C’ cos+ sin 0 fcos*6 dO —% cos 0 fsin 20 do [cos*0 do=f (443 cos 20) dd = 46+4sin 20, Josin 20 dO= —- 4cos 20 w= Csin0+C’cos04+40sin 6 + 4 sin 26 sin 6 + 1cos 6 cos 20 — Csin9+C’cos0+40sin 6 + 4 cos 0 = Csin 0+C'cos04+46sin6. (Explain this step ?) 2 3k 2 tue", w=C sind+C’ cos6+sin 0 [e’cosd do—cos0 fe'sin® do fé cosd dd=e’ sind— fé sin 6 dé, fe sin@ dé= — é’ cos O+ fe cos 6 d0 fe cos 0 dO= 1 ¢° (sin@ + cos 8) fé sin@ dd = 1° (sin @ — cos @) u=Csnd4+C’ cos6+ te. We shall afterwards have to return to this equation. | aac au Show, in a similar manner, that OE I he X (a function of xr) gives w= Ce* + Cle* + he fe*X dx —4 ery e* X dx. We have placed the first two differential coefficients hy themselves, not only because it is comparatively uncommon to see third, &c. diff. co. in applications, but also because we are, as has been seen, in possession of a general method of solving the inverse cases, or those ‘of the Integral Calculus. That is, we can reduce the solution of wu’ = U, or of uw/=U, to the finding of a common integral. But we are not in possession of any such method with regard to ul" =U, u' =U, &c., and these equa- tions can only be reduced to explicit integration (with our present knowledge) in a very few particular cases. Bx dir Mu du r) = pa yr Pall Sa aE 1 woe ESE Fe ES ProspLem.—To express qa” dak? &c. in terms of Ta? dat? &C. 3 tt tt u dex dx d u nage (153), a = — —, = => =| OS pas ; u'? ~du® du du u'® " / : ul? du me EE du! ata of BN pay he da dx uw?) \du ow nae u!® a! ul! — 3y!2 oe ul N B. In differentiating a fraction of which the denominator is a power of a function, such. as P + (Q)’, abbreviate the rule deduced in page 52, as follows :— Differentiate as if the function were PQ with these alterations, } | atall, DIFFERENTIATIONS OF DIFFERENT KINDS. I, After differentiating Q, multiply the term by 2. 2 in the denominator, write Q"+?, 157 2. instead of Q2 gaP a) dP dQ d pe dp PQ ie Qa a — dz Q” Sir ane aaa Qn déx = dr!!! d (3ull?— wy! yl" d (3u'l? —y! y!" dx 1 a Ao 5 see ee ee —} or — dut du du u!® dx u! dus u! Lu! (Gel a! a! yl" — yy! uw) —5 ul! (3y!2 — uy! wl!) u u!® * —" — wu — 10 uu! uw! + 15 yl 7 We leave the following to the student: oy wu — 15 ul? uw!" ui® — 10 y/2 a! 4-105 (ul ull — ll 2) alts —_— DRE aT ar ng epee 7r 96 errata LN aa Die aN Cae Lea uy! ? The problem which we have solved amounts to this: given “= x, and therefore the power of differentiating « with respect to a, required the diff. co. of « with respect to w, without the necessity of actually inverting the equation w = 2, and making itr = Wu. Hence, whene ever Maclaurin’s Theorem applies, we can from u = $x, not only expand u in powers of x, but also x in powers of uw. For we know that in every case where an infinite series is admissible, we have (p. 74.) dz da u? Ber u® ade d where by (2), ae &c. are meant their values when wu = 0. Now, Uu ~ when wu = 0, let x= k; or let dk = 0: then (2') (x), &c. can be found by making x = k on the second sides of the preceding relations, in Which w/, w”, &c. are all functions of x. Let A, A, Az, &c. be the values of x! x" x!, &c. when «= k; then we have Bey Alu AL ge a. et eal ST baie om rah eat Exampie.—Given u = ax-+ba® + cx®+ ex! + fr’ + &c. required a in terms of u. e e e Here, when u=0, one * value of w is x=0, and we will therefore sup- pose uv and x to be beginning together from being simultaneously = 0, by which we shall produce a series for z, which will be true until we Come to another value 2 =k, which makes w =.0, after which we * There may be (often will be, we say, but perhaps the student may not have come to the point at which this is proved) an infinite number of other values of x which will make x =0. The practice of assuming that a=0 is the value (meaning the only value) of « which makes x—0 infests elementary works, both English and French, to a great degree, The consequence is, that when the student has finished his elementary course, he learns that several of his general theorems are not general J 158 DIFFERENTIAL AND INTEGRAL CALCULUS. must take another series beginning from the simultaneous values u = 0, aah, &c. Consequently, we find wo =—at2be+3 cx’ +4 er +5 ft'+.. A= a PY ha eae 9% +2.3c~r +3.4 exv®+4.5 fP+ eee, Agr=2b y= 2.3c +2.3.4e7+3.4.5 fe+..., As=2.3e wt ms 2.3.4¢ +2.3.4.5f¢ +.+.,Ag=2.3.40 w= 2.3.4.5f +...5As=2.3.4.5-f &c. &c. &c. (oy= oe nt ()\=- bis — ad sf) = Baa a Bee te As ar’ A! a 1v A,A, ai 10A,A,A, + 15 A? 24 ae —— 120 abc oe 120 6° @") = — SO («") = A®A,—15A,2A,A,—10 A2A2 + 105 (A\As — Ay”) As Bia Sate cikioke Sale AM ONE “2 atthe AG & 420 off — Tee a’be — 360 ac? + 420 (6ac — 46°) b* a? Substitute in (1) and write the terms in a form alternately positive and negative, which gives 1 b 2b? — ac ae —5abe + 56° — us — ———_— 1 Baer intima 42 camer : ag Fe ch a al ‘ Gavbe + Sar? — & f +71 (20° — 3.ac).b° ee eee ee Thus w=er+22°+3a°+... givesw=u—2u+5u?— 14ué+ 42u°?+ Ke. | We recommend the student to try various cases, and shall proceed to | observe of this reversion, as itis called, of the series aw + bu +..... that n terms of the series determine 7 terms of the reverse series, so that two terms of the latter are given when a@ and 6 are given, three terms when a, b, and care given, and so on. We now proceed to another case of our main subject. Instead of supposing wu to be an explicit function of x, let us now sup- pose.u = yt, r= Wt, so that w and @ are not connected together by a given equation, but by one implied in the coexistence of these equations, and which may be obtained by eliminating t. Let accents now denote differentiations with respect to ¢, and let the question be to find the | diff. co. of u with respect to x, in terms of those of wu and x with respect to ¢. } deed dt vw tu. d (=) dt — v'ul’ — ula" dx. dt-dx” z' dx dt dx a * du a! (a'ul— ae") — 32" (aul! — we") _ $ az er, dz Phil a’?® dtu a! *(alu® — ue )—Fala! (a!!! — ula!) -3(a'a' —5a!*) (au! — ula") das" ee Exercise.—If u=at+b2+cl+... and v=a,t+b,0+¢,F+... find the three first terms of « expanded in a series of powers of a. DIFFERENTIATIONS OF DIFFERENT KINDS, 159 The equation u==dx ¢ an be made to result from two others of the form u=yt ome wt in an infinite number of ways; for assuming y¢ at pleasure, w¢ can be found by determining w from gr = xt. But -_whatever xt and Wt may be, consistently with u=x¢ and ~=Wet giving u=x, the function (a2! u!! —u! x!) x’? will always be the same func- Qa, 25 tion of wv, being always — Thus w= x follows from any case of the following, ie u=xi r= (xé)%, giving w= yt, wu! = xt, . U=3 (xt) x/t, 0" = 3 (yt)? Xt + 6xt (x’1)%, or Roy du _ 3 (xt)®.x't.x"t — yt {3 (xt)”.x"t + 6 yé (y't)?} dix? 1 3(xé)? x’¢ } 6 1 Le .. ie ~ 5H Gi> =— 3°3 x3, the same as from wu = Wo, _ Exercise. If w be a function of t, t of v, and v of x, show that ! au _. Pu di? dv* d*t du dv? du di dy dx?” df ‘dy? dx? dv’ di dz® ~ dt dv‘ dx _ and verify this in the case of u=@, t= v, v=a*. To avoid the . 2 * . ad * ° ° . du 3 mconvenience of parentheses, it is usual to write Ie 3 instead of (dus? We now resume the supposition (page 151) of there being several _ Variables independent of each other. To take the simplest case, let us | Suppose w= ¢ (x, y). We have established all that is necessary re- _ Specting successive differentiations made on the supposition that x becomes e+ Ax, 2 +2Aa, &., in succession while y remains constant, or that y becomes y + Ay, &c., while # remains constant. But we “have as yet said nothing of differentiations in which first one and then [- the other is supposed to vary. | ; ae du , | The diff. co. with respect to x is written Tn and that with respect to ; ex : % a” 2. du ay But we cannot too emphatically remind the student not to “extend the analogies which (page 54) have been shown to exist Peiween diff: co. and algebraic fractions when all the variables are _ Connected, to the case where there are variables independent of each _ other. In the present case y may vary independently of x, and x of Y3 the variation of w takes different forms according to the different sup- positions. Hence Au springing from a change of # into Az is altogether ferent function from Aw which comes from changing y into Ay. If - ve occasion to use them together, we must invent a symbol of dis- m: but since we want nothing but diff. co. or limits of ratios, the ent denominator is sufficient distinction. n: 160 DIFFERENTIAL AND INTEGRAL CALCULUS. du ; ro ; | When we see a? we know that it was the variation of 2 which made x the variation of w by which this fraction was obtained. Similarly, as to second differences, Au may either represent the dif ference (x varying) of the difference (a also varying); or the difference (y varying) of the difference (w varying) ; or the difference (wv varying) | of the difference (y varying); or lastly, the difference (y varying) of the difference (y varying). In all, A®w is the difference of the differ-. ence, but to each repetition of the word difference a supposition is im- plied as to the manner in which the difference was obtained. The two cases in which the variable is the same in both have been already treated, the only difference being in the notation. For whereas hitherto there has been only one quantity which does or can vary, we must now introduce another quantity as a possible variable, but which, so long as it does not vary, has all the properties of a constant. Thus hitherto we have included, for instance, 2ca — 2° under the general symbol Pr: whereas, in future, if we mean to imply that we are at liberty to make c variable, we shall write it @ (a,c). Thus Ad (x, c) = o (« +4a, ¢) — (a,c) is an equation of the same force and meaning as Ag@r= db (a + Av) — px, with this addition only, that we remind the reader of the quantity c, which might have varied, had we thought fit, but which, in the preceding equation, does not vary. We shall take A’w where w= (2, y) on the four possible suppo- sitions when w only varies Au = $(«-+42,y) — @ (a, y) when y only varies Au= ¢ (a, ytAy) — O(a, y) x varies twice, Yu=o(e+ 242, y) — 20(r+4z, y) + (a, y) x varies, then y, | Atu=o(a+ Ar, y+Ay)— Pa, y+Ay) — P(@t+he, y)+9(% YY y varies, then 2, : Au=p(atAz, y+Ay)—o(a+Aa, y) — O(a, y FAY) + pcr, y) y varies twice | Muxo(a, y+2Ay)—26(@, y+ Ay) + OG, y); the second and third of these are the same: that is, in a second dil ference, formed from one variation of x and one variation of y, it i indifferent which is supposed to vary first. From this it may be show that the order of the suppositions as to variations when these variation are altogether independent of each other, is itself immaterial. For moment let D and A refer to # andy. Then A (Du) = D (Aw),i which it is usual to omit the brackets. Then AADu = ADAu ¢ AA. Du = AD (Au) = DAAu, that is A’. Du = D. Au, &c. &e. Ge nerally A”. D"u = D", Aw. : Let us now expand each term of the differences by Taylor’s theoren applying the theorem of Lagrange (page 73) at the second differenti: tion. | Let Av = h, Ay = k, and let differentiation with respect to 2 onl to y only, be denoted by an accent above or below: while, when the x ON ALGEBRAICAL DEVELOPMENT. 161 are two differentiations with different variables, the one which is made first has its accent in parentheses. Thus d/d é d/fa ¢ Gt, dhs Bi Fo p (, y) ) and %) (a, 9 7. (9 (a, ») P(t + Ax, y) = (x,y) +B (a, yh+ P" (w+ Oh, a: ey | $6944) = 06 y) + &, (2, y) rite Nie? 1 In the first write (#< 1). .. y+ 4y)= (2, y)+9, (x,y) k+-¢,, (2, yd) “ y+Ay for y, and develope the two first terms x 2 ‘ ath] d’ 1s y)+¢, bd tal k+ h(a, ytpk) — att +9''(x+6h, ytk) = ) Uhm the last increased by # (2, y) subtract the sum y y of the two pre- ceding, which gives A?~ (where both x and y eb once )3 or A’ u=o,/ (2, y) hk +$,(a, ean” fo a a ai (v+0h, y+kh) — b" (x + Oh, 08 fu An b,” (a, y) +34, ee a idisces ytvk) .h, in which if we suppose / and to diminish without limit, we haye a= 8" N= 7 (FdGy ). pid we had Sst m the same way, with the exception only of ituting 7+ Ar in (2, y+ Ay) instead of y+ Ay in ¢ (w+ Az, y) Should have found limit of 5 th limit of 5 a = Poy) = 5p ee )). * : Vhere i in the first we ae A’u (x varies, then y); in the second Aty (yvaries, then z). But these two are always the same, and therefore | the first sides are identical, being limits of the same function. Hence 2 second sides are the same; or when two differentiations are per- med with respect to two variables independent of each other, the der is immaterial. : ris du du | : ae ee 2 . 22 | te ; instance wu = e* gsiny — — &* . 27. sin =n ee COM : ; J dx 4 dy Y dd (du du a = , Rt. = €* 2r. cos c ifs ay (ae iz) = e 22 cos ¥ alg ha y. M 162 DIFFERENTIAL AND INTEGRAL CALCULUS, tp = a Roe dx? dy i al ONE TUN sc. seh | yt df d0\) -y haat oe dey di = "4 ya log aay) = log # + me ay. ie : Now, as 74 18 60 denoted, because though originally obtained thus wv in like manner lei d /d A? (=) , it is shown to be the limit of as ; dr \ dx a e 4 a, . . (a) be denoted by as bs , because it is shown to be the limit 0 dy \dx dy dx A?u iy . 7 Ay Rus And if we place on the right hand the increment of the va riable with respect to which differentiation first takes place, we ma) express that the order of the differentiations is indifferent by the follow ing equation, hi I dy dx ~ dx dy In a similar way it may be shown, 1. That A”*"w, where x varies 1 times, and y varies 7 times, is the same in whatever order the variation may be made; and also that m differentiations with respect to x, followe by n differentiations with respect to y, in whatever order they may b made, will give the same result, namely, the limit of San —. which limit we represent b iad Az™ Ay” : P y da™ dy" But it will materially facilitate the transition from ¢(x) to ¢(@,y where x and y are independent, and both vary, if we pass through th case where y is a function of a. In that case we have the partial diff. ¢ (page 91) just considered and the total diff. co, connected together the equation d.u _ du du dy in aie eg ae d Repeat this process, (remember that ~ does not contain y) © @.u d/dw , dudy d (du | dudy dy dx. dx Bas >. i'd + ay kala hag aa du du dy , du dy du dy _ dudy? ris dat Bs dxdy dx dy da? dx dy dx dy? da®’ a du du 5 du dy dudy? dudy de tn de tes .dxdy ae dy? dx® dy dx?’ If we take the simple relation y = ax + 6 we have the following :— : dui dtu ig) eee d*u : dtu’ lo Clans eet dat” cat dy * * aay 8" + Tay 48 + ts : , the law of which, and its connexion with the binomial theorem, is - obvious. ' Now apply Taylor’s Theorem with the theorem as to its limits to the | expansion of 9 (# + Ax, y + Ay), y + Ay being a(v+-Ax) +4, and let | Us, be the nth total diff. co. just obtained. This gives (Ar = h). | . . du d”.u h? hn | P(r+ Ar, y+Ay)= ut a A+ gre Hae tf UM, nak 53 (B). _ To expand the last, observe that if Pir (x, y) represent the partial ~ (m+ n)th diff. co. in which x varies m times, and y n times, we have © Ua, pean = G" (w+ 9h, y+ Oah) + $," (2+6h, y+0ah) na + | J Substitute in (B) from the set (A), which gives, making ah or Ay=k, ¥ .. : : du du L/(aa Pu Py LN: | Oe: ie Te h du The 5 ae | paar Vaatdy tt a aya SHE + aa ) ies P a Y ‘e 1 u ‘ 1 me es re Wr erie +. Geen eat +.) e i | + the result of writing #+06h for 2, y¥+ek for y, in .. “i 1 du i due oY du at | a 2.3...n! dz" dxu""'dy oe ho aaa c | which equation contains 2x, y, h, and k, and not a or b. But it is true : or all values of @ and 8, that is, true for all values of y and k. Con- ' Sequently this equation is always true whether y bea function of z or not. A theorem of the same sort may be found for a function of a, y, and 2, by making y=au-+-b, z=0en+ f, and proceeding in the manner above. But the following consideration will tend to fix the method in the Memory, as well as to introduce a remarkable view of the subject. x _ If there be a number of operations successively performed. upon 1, - denoted by A,, A,, &c., and if they be all’of what is called the con- | vertible kind, namely, if A, performed upon A,w gives the same as A, Performed upon Aw ; and also of the distributive kind, by which we Mean that A, (ad + 6 — c)is the same as A,a + A,b — A, c, &.; we Sin M 2 & = ‘164 DIFFERENTIAL AND INTEGRAL CALCULUS. may for every such set of operations invent a new algebra, or show that the old one has been more than necessarily limited, as follows. If we exa- mine the processes of algebra, we find that, so far as the juxta-position of letters is concerned, whether by multiplication or division (which is a case of multiplication) it is the convertibility and distributiveness ot the operation denoted by ab which gives the form of all processes after addition and subtraction. Let us suppose we know that a, }, &ec. are magnitudes, and that we assume addition, subtraction, and the rule of signs. But let us not be supposed to know anything of the meaning 0} ab except this, that whatever it be, it is the same as ba, and also that a(b-+e) and ab + ac must mean the same things. That is, let us assume ab to be 1., some magnitude in its result 2. obtained by a double operation of a convertible and distributive character. Again, of - a b a a | —b or b y means a. Then all the resi b | of algebra follows in the same forms of expression as when ab meant multiplication. For instance, | (a +b)(c + d) means (a+ b)e + (@+ b) d, of which (a + b)c means c(a + d) or ca+cb, and (a+b) d means ad + bd, and soon. Now among the operations which are convertible and dis tributive we have 1. successive diiferentiations with respect to the sam variable, an ( d™u Or ae at a™ d™u d™v ) =5(Ga) an ek 8) Sloe us know nothing, except that dr™\ dz") da" *" da™ da™ dx” * 2. Independent differentiations ; for instance, ddu _ d du d (du Av. 6 d’v dxdy dy dx dzu\dy ° dx/ dxdy dx 3. Differentiation and multiplication by a constant, d dv d dv. dw | a7 (hv) =h Ane (v + w) = hv+hu, ay (v+w) ee + Ta and the same for finite differences. Now consider the theorem du 7 es (MF uphu=ut 7 At oes hate 1. mutt ee We make a step, the details of which the student cannot follow further than to show the coincidence of some of its results with thos, already obtained. Weassume all the formulz of common algebra in th case of convertible and distributive operations. The last equation (lookin merely at the operations performed on w, and considering differentiatio | Beet | with respect to x as an operation whose symbol 1s ae’ which for a m«¢ ment we call D) is + weer wae ae h* D.k anes | > L$A=14D.4+D'S +0 ON ALGEBRAICAL DEVELOPMENT. 165 + the last symbol must be to the student at present a symbol of abbre- _ Yiation, derived from looking at the expansion, and remembering what it would be if D were a quantity. That is, if we treat A and D as quantities, in the equation (1 + A)u=s?*.u, until the expansion of both sides is made, and u replaced after A and D, D*, &c., and if we then restore to A and D their meaning as symbols of operation, we have atrue result. Now let A’and D’ imply a difference and diff, co. with respect toy, and we have accordingly (Ay = £) utAutdA’(utAu=u 7 Au+D! (u+ Au) k+ sees or collecting operations as before, Q+A)1+Au— err heP*) a, ee EL DE oy ; the last result being that which would exist if D and D’, &c. were quantities. Let us try the same mode, namely, treat D, A, &e. as quan- , tities until the development is completed, and then restore the original meaning. We thus have (1+4) (14-4) u= {1+ (Da + D/k) +3 (DA4+ Dk)? +... bu = ut+hDu+tkD/u+dk (A?D%u + 2hk DD’u+kD"u) +.. ‘which evidently agrees with the expansion in page 163. y 4 | We shall now follow the preceding method freely, in order to show : 1 : that its results are true. Firstly, what should A> mean, or = , consi- A 3 dered as a symbol of operation ? By definition A (A~' xv) means w, or ee aw = 2, w= A-'z. ) But Agr = 6 (w+ Ar) — ¢(2), and if Adr= wa, we find that the following is one solution, if not the only solution, of Agr = wa. gx =C+ (x — Ax) + H(x@ — 2Ar)+.... ad enfinitum, which, by changing z into 2 + Az, and subtracting, gives Ag z= wx; C being any constant whatsoever. This we have introduced merely to show that the relation in question is capable of being satisfied ; what- ever the general solution of the equation Adz = wx may be, let it be denoted by dx =A'ywx. We proceed by assuming that the form in which the binomial theorem enters remains true when we make the exponent negative and = — 1, and we obtain the following, in which _ the first side of the final result is a symbol to be explained, the second _ Side (if the peculiar assumptions we are considering lead to no error) i of explanation. | A= eDh 1 A‘ = (

If our process be correct, the expansion of the second side, in powers or) as a quantity, and the subsequent restoration of the meaning of | Dru, should give an explicable result. That it will do so, we shall show —™ a subsequent part of the work ; at present, we shall take an instance _ We can more easily verify. Since 1+A= &, we have Di=log(1+A) hDu = log (1 + A)u. 166 DIFFERENTIAL AND INTEGRAL CALCULUS. On expanding the second side, and restoring the meaning of A*u, we have d h— = Au—t}t Au +} Au—tbtut - We may easily verify this result on particular cases. Thus when ux, Au=3eh+3ch +h, Mus 3hQch+h’)+3h, XU = 6h", Au = Osc, | il 3 2 . . du Au —} Au +1 A®’u = 3h x, which is also h ae We may now consider ourselves as having advanced by the route of analogy to a theorem which we should never otherwise have suspected, but of which we have not yet got demonstration. But having the’ theorem, it is easy to furnish a demonstration. Firstly, we shall show that * dx The accents denoting differential co-efficients, we have h? 2.3 Take the difference A of both sides, which gives A’u = h Au! + 4 h?Au! +. . . and in place of each term write its corresponding series derived from using the theorem just given with w’, u', &c. Thi) gives for A’ u a series of the form Au aul he + Mu" + Nuvht+. . Repeat the process, writing for A w'’, &c. their values, and we have Buu! P+ Mu ht +N wh +. - | and so on; where M,N, M’, &c. are specific fractions determined in th process. Substitute every one of these in the series A Aw + B Au 4 C A’u-+-...., and we have may be expanded in a series of the form of A Au + B A’u +. 2 Au = wh 4 ul = + uy!!! oie h? h® t Ligan mnt A(wh + 5 +u nar + Ou +...) + &e. ) + B(w'h? + Mull? + &c.) : ; ; d which can be made identical with h — by. A ely eo B Sa 3 A+MB+C=0, &. It is not necessary to determine an term, for as soon as we know that any form of hu! can be expande into AAu + BA?u+.... where A, B, &c. are independent of tl function chosen, and of A, we can immediately find a function whic shall point out what these co-efficients must be. Let u=(1+a)*, aw let h= 1; then we must have Au = (1 + aya Au=(1+a)y. &e. (1+a)’ log(ita) =AQ+a)’. a+B(1+a)’a+C0 +a)’.a°+. or log (1+a) = Aat+Ba’+ Ca’+....=4 —40 +4048 whence A=1, B= —4, CH}, &. ON ALGEBRAIGAL DEVELOPMENT. 167 . : . Let the student now interpret the following, and verify the second. d d d P(x+Ar, ythy, z+ Az) = HN TE™ By y, 2) d LU “h d gq” 9 ie , d\" POA + Typ, (Pe) ae ey, P. ay cc : By D™ or (=) » We are to mean, by definition, a function such r Wr? BR aes r dig See | that D (Du) =u, wheres ni =U whence (=) uw is Jude. Let us apply this to the last, and see whether we can derive a verifiable theorem from ed d\ dd ) ) ae ‘ P Tv) a ot a — <* Cae) —— — (=) Che jig (1 4 =) Pica (1 ay + (= &e, P, oie 2 dP d?P ei ee (Pex mre) This may be immediately verified by parts; thus | th dP ith ines ou PPyiy: OF oat H frie=pe— [Rede = ve res + [ee dz, &c. __ We shall conclude, for the present, with another remarkable instance, _ Taking Taylor’s Theorem, and changing h into — h, we have h? h® : ht | Ae = La Nf tt MES ae Ss NF iv pear 3 at h) = dx —P'eh+ ox : d faint 2 aes &e. As h may be anything whatever, let it =, and a simple transpo- _ Sition gives . 2 3 px = (0) + P'x t— "x 2 +.o"x re — &e. Or du au x dui x? vant : u= (u) + an Gos v “ da +a a37 where (uw) is the value of wwhen v=0. As this is true for all functions, . Substitute the nth diff, co. of « instead of u, and we have . ba d"u Y d"u d*t wu +2 uU pd ‘ dat \aa) * dg! * ~ Ga 3 Nets : _-ty this when n= — 1, on the suppositions hitherto employed. Phen | dees? du du x* au at = Sudx = (fudz) + edie, ry &e. &c., da.2.") dai 23; * . du Here we must ask what 7,0 Means? Since in the method by hh. x . ¢ which these extensions are made the symbol D is used as a quantity until the end of the process, D® will not occur except where it is unity, A . 168 DIFFERENTIAL AND INTEGRAL CALCULUS. when considered as a quantity. Hence wu itself is D°ws but we will make the theorem just obtained the test of the correctness of this, so far as one instance can be a test. If we assume the point, we have on one side fudz — ( fudz) or fudz — its value when x = 0, that is feudz. And thus du x® du x dz 2 dz 2:3 — &e. feudz'= ur — which is called Jonn BeRNoOvILLI’s THEOREM. It is verified thus: fudz Suz — fadu = ux — — ardz, du x iy sd du q — uw—- — - Lh coe, «See dz 2 9dr dz : du x du =a wu =a —=ur—— — + — —— dz, &c. dz 2 ix 235 7 ee es | We have not yet applied the great principle of the convertibility o independent differentiations in any problem of primary importance: bu we shall now proceed to establish what are called Lacrance’s and La pLACE’s THEOREMS. They are contained in the following :—Given Fa gx, and wx, and the condition that w must be such a function of x ant z as is implied in the equation u= F (2+ 2x gu) Required the development of yw in powers of a. Since wu is to be developed in powers of 2, and since it must bi (with w) a function both of x and z, the co-efficients of the developmen) will be functions of =, and considering x alone as variable, we haw (page 74) (dhbu ‘(Pwu xv Yu = cu) + (SE) 2+ ( = > +e ae ly \ where the brackets indicate the values when r= 0. The determina tion of these is the point on which the solution now depends; and th consideration by means of which we succeed is the followmg. Whe: a function is to be differentiated with respect to 2, and 2 is then to b made = 0, we have a result which can only be indicated, unless th function be explicitly given. For if we made x = 0 before differen tiation, we should only have a particular, value of the function, or a con stant. But when we have to differentiate, and then to make a constaD = 0, these operations are convertible, and either may be done first Thus to differentiate dx + cyzr, and then to make c= 0, is to tak d'x + cy’x, and then g’r. If we invert the order, we first reduc gx + cy x to oz, and then take ¢'x. Accordingly, if we can express th diff. co. of w with respect to x in terms of those with respect to z, then a in the latter case 7 is a constant, if.may be made = 0 before the dit ferentiations. We proceed with the problem as thus reduced, which i simply this :—Given ux F (z + rw), du dU du dU du ee If Pras 4 re aS i oi or d é ; ‘ bu ra is a function of w; and is therefure the diff. co. of some function ; then a 4s , wu du dV du dV dee wdeides dude d2; ' d : . to express dy im terms of diff. co. of U with respect to being any function of uw (remember that z and = Let du z+rpu =v; then u= Fo, Pps dv dou dou du MME SLi veto Listy otek ‘cle? dv dou du dx = bute aah ei. “du da du ; , du du as = Fv d+ vouz): aac du dx Bi : du whence it is plain that esi pu a result independent of the Res Fr. See pages 63, 64; where it is Be shown that an equation between partial diff. co. may be true for a whole class of functions. @U_ ddv d dV Wr dridestsdede Be Bir, (ci Sarhle a) du dx ae ad Fv dv du "dy de. da \iv Ip 1 —2w¢'u F’v du F’y 1 — rputk’v e ON ALGEBRAICAL DEVELOPMENT, - Ps ’ d dV de ‘ed ~ ~ only, U 2 are independent). ad¥v dv dv aV du dU du =i (o du dz z) a = ( @u sah du’ dz) ~ =; (@uy i oh d _ for the equation a = du oh is true of al/ functions of w. nV GU ©) ode, @U dV du + ae eee hell => iL. me 1et (ou) du du’ Ben da? du a)=@ a @U_ av aU ee) > at? d@” dz** d® ~ dxdz*~ dz 5 ¥ fi, i Oa d2* ~ ae ~ dzt du dz “= dU — 2 hs Puy >) #* (uy 2 dU du du dz dU dz dx’ ie) "dz 170 DIFFERENTIAL AND INTEGRAL CALCULUS. au xd aU To give the general law, let ge oe (ou a) dU dV d"U dV d bi Assume ( w)" ee a Miners da” = “ie an) A d*'U 44.0" ay uae d” ( as) uu Crested dat tet ieMadaie det d” dV du na AU du yt = 7 (ou 5) = dz" (GW a = (0 : = or if i Bt dU aes, Ms hos — nl “de de aos ( (oe H a then ee de oe (on ) “ay But this law has been proved to hold true as far as the third diff. co. therefore it is true for the fourth, &c. We have now expressed diff. co. with respect to x in terms of those with respect to z; and making # = 0 on both sides, and taking yw as the function represented by U, we have (=e) = (fou Fh ) but on the second side we may make w = 0 before differentiation. Now when v=0, z+ «fu becomes z, and wu or F (2 + xou) becomes Fz. Consequently, (yu) is ¥Fz and (= ae) or (ym a) is ore AE i and generally ds n du ‘ qr} ‘ dsFz Gs te) as dz a (bu y" “iy ) dz"— ($F z) ats. Hence by substitution in (1) we have d ayek x dusk ote a GES ‘ eat a i ool Os Bek ee st ay which is Lapuacr’s Turorem. If we take the particular case of Fer = 2, or u= z + xu, we have yusyFe+gF2 mee d i ry 3 puwat be. ye at ((éey v2 )5 oy cal 22) ) sgt ke which is LAGRANGE’s THEOREM. The most simple case of this 1s where Yuz=u, ¥/u=1; in which case uxz+axrpu gives es i aie) Na 2" u=2z+hz.0+ rp (pz) ao | +52) ‘33 + &c. ON ALGEBRAICAL DEVELOPMENT. 171 Taylor’s Theorem is a particular case of that of Laplace, as follows: let du be a constant =a, and let wu=u; then u=F (z+az), and IF 27 i hie 3 2 3 a sea SMa aot ca az 2 dz 25 and making ar = h we have the well-known development of F (z+h). u=Fz+a + &c. dz ExamrLe 1, w=sin (z+ 6"); required log uw? Fe=sna, ¢r=e, wa=loga gFze=s%"* dl Fz d (log sin z) Pioerars = cot z, dz dz log w= log sin z+¢°"*.cot z.7+ d (2° "= Got fs ai GM tt x Lome ‘ ae ae 0.2 sin % 2 8 3 ap a ae (é€ cot Ee qit ado &e., and the differentiations only remain to be performed. Example 2. u=2z+4 sin w; required 2 tan7! (a tan u). This is a case of Lagrange’s Theorem, or Fu=u 3; ou=sin u, 2 a 2a 2=—2 tan" (a tan z ‘2m eee Je eats RS Fo ¥ ( ) ¥ l+a*tan*z cos*z 14(a?—1)sin®z 2a sin 2 1+(a?— 1)sin?z" * d 2a sin 2z x + Fae TsrP Taw + eee dz\1+(a@—1) sin *z/ 2 EXAMPLE 3. u=z+2sin u; required w. d x” a? a =z+ sin 2.x7 + — (sin 2z) — — (sin z)*. 3 as Mae Taees | ) ae The student must not believe that theorems have been invented or perfected by the methods in which it is afterwards most convenient to deduce them. The march of the discoverer is generally anything but on the line on which it is afterwards convenient to cut the road. Wallis made a near approach to the binomial theorem in trying a problem which we should now express by the question of finding J i (a?—x*)"dz. Newton, following his steps, did what amounted to expanding the preceding in powers of x, and afterwards found that the expansion of (a+ 2)" was involved in his result. In the case of Lagrange’s theorem, Lambert (of Alsace, died 1777), in endeavouring to express the roots of some algebraic equations in series, found (for his particular case) a law resembling that which we have just developed. He published his results in 1758, and Lagrange generalised them into the theorem which bears his own name. Finally, in the Mécanique Céleste Laplace made a still further extension. We now proceed to the consideration of singular values. 2 tan~'(a tan u)=2 tan-'(atan z)+ sees CHAPTER X. ON SINGULAR VALUES. By a singular value we mean generally, that which corresponds to any form of the function which cannot be directly calculated ; and the only way in which we shall say the function has a value at all in such a case is this: if = a give a singular form to the function, then the limit of the values of the function when a approaches without limit to a, is the value of the function. That it cannot have any other value, is readily proved by the process in pages 21, 22, and perhaps a proper method of considering the symbols 0 and oc, as bearing a tacit reference to the manner in which they are obtained, might render it easy to say in absolute terms, that the singular forms of functions have values*. But with this question we have here nothing to do; our object being to find the limit towards which a function approaches, as we approach the singular form. The language used will, for abbreviation, be that which calls the limits so obtained values of the singular forms. The most obvious singular forms are, a + eh ee ae 0", Dts Co» OG iy, OG ee, arg 1 » &C. Thus with reference to forms merely, 7 = a gives e—a 0 cosec (4 —a oc oe ia (a — a) cot(rx—a)=0 Xa, one Oe , cot (v7 —a) a x a® 9 2 2 a © 1 ) when P diminishes and Q increases is included jn the first : 0 case. But of the rest we can say nothing. For o> see Introductory Chapter; and the rest not hitherto mentioned can all be reduced to this form. Let dx and %z both become nothing when a=a, then, page 69, if the diff. co. ¢’a and wa be finite, we know that ¢@+h) _ SP (@+ 4h) wath) ~~ wi(a@+h) Now, h diminishing without limit, the first side approaches the singular 0<1. 0 , : er form = 5: but its equivalent continually approaches the limit , a sued iF : ; wa’ which is the limit required. If ¢’a only be = 0, then the function in question diminishes without limit: if ¥'a only = 0, it then increases without limit: but if both ‘a and ya = 0, then by the theorem already cited we have Md vt OED BOHM ag 5 ue tim subject to similar remarks. But if 6”’@ =0, w’a=0, then da and ¥’"a must be used, and so on. Hence the rule is, to find the value of Nhs ty x apa afunction in the case where its form js mY substitute for the nume- rator and denominator the first diff. co. which do not, for the value of | &, assume the same form. 1 — cosz sin r . : EXampPte 1. where 7 = 0 = iat =O or is commi- nuent with c. x 1 Exampte 2, ———, (whenz =0) = — = 1. e— J.’ s* 2% sin xr—r T 2sinz+2rcosar__ Parr ae |) Ee COS @ 2 —sinv r— a)” Rive see ’ ° EXamrve 4, come = ) when « = a iseither 0, €~°, or «© according Ee" — as n is greater than, equal to, or less than, 1, 174 DIFFERENTIAL AND INTEGRAL CALCULUS. : ; 0 Remember that in the relation which produces the form 5 any letter may be treated as the variable. For instance, — Qy 1 4 af ; 4x® 5 ee etic when y = 2" has either 5— or for its limit, Ct Al, 2x which are the same when y= 2"; that is, we may either suppose 2 to approach towards y or y towards «, and the relation which produces 0 : othe : ; 5 makes the results of both differentiations agree. But if, as in the case of (8a°—2ay— 2) + (3x? — y) we observe that without assigning any general relation between w and y, the form 5 occurs when « = 1 y =3, we are not to expect the same result by substituting 1 for 2, and making y variable, as we should have if we substituted 3 for y, and then made x variable. The two processes give 6 — Qy 82° — 6r — 2 2 salways: £208)" end: =a y é 3a — 3 ASG) Let dx and wa be functions of x which severally become O and @ haying limit = 3. ae when =a; then x X Ya 18 2-7, In which fraction both terms are = 0 when c=a. This case is then treated by the last as follows : Gas hi Fe —— i Thi pe —_ It must be observed that any finite value of 2 which makes ya in- finite, makes all the diff. co. infinite: for « can only arise, in such a case, from the denominator becoming = 0; and page 65, no deno- minator is ever got rid of by differentiation. There 1s then the form Pa +n the denominator of the preceding. ‘To this form we proceed. Let @x and wx both become infinite when c=a; their reciprocals then become nothing, and we have 1 wile eae to. GPeNs ak (SEN eo” Oe alee Mev a i OM un] Pa wo Wn” pr (px)” the rule for this case is then the same as in the first, but pie w'e oo 4 also has the form a It will however frequently happen that a factor disappears from the numerator and denominator, or that some other reduction may be. made, by which the value of the original ratio may be more easily found. Some instances will show the mode of pro- ceeding. ON SINGULAR VALUES. EXAMPLE 1. 1 — log a 1 . 1 i oy fe. x ——7 — (when c= 0)= — so (# x P+ &c., or powers of 2 begin to appear in the denominators: and generally, if «= V"P, we find yimV" PV EVP’, ul==m(m—1)V"-? PV? + 2mV"™ PIV! + &e. ua" PV" 4 a, VIO PVE + eb ae. where a), @, &c., are functions of m. If m be negative at the outset, V is in denominators from the beginning : if m be positive and m- teger, V never comes into a denominator, since the differentiated term previously disappears by introduction of the factor 0: if m be positive * The beginner may omit the rest of this chapter, and it is perhaps necessary to give the more advanced student some reason why this subject is treated at such length. Until very lately, all analysts considered functions which vanish when r—a as necessarily divisible by some positive power of a — a. This is only one of ‘ a great many too general assumptions which are disappearing one by one from the science. It appeared to be true from observation of functions, and is so in fact forall the ordinary forms of algebra. But observation at last detected a function for which st could not be true, as was shown by Professor Hamilton, in the Transactions of 1 the Royal Irish Academy, some years ago. The function in question was ee or tog-"( — ape which vanishes when « is nothing, but is not divisible by any oe . positive power of a, as can. be independently proved. From this hint I have been — led to the classification of functions which is here deduced, and of which I will not © undertake the unlimited defence. But I feel disposed to maintain that the con- clusions of this chapter are more rigorous than any demonstration which has been given of Taylor’s Theorem, except only the one in Chapter IIL, which is founded on that given by M. Cauchy. ON SINGULAR VALUES. 177 and fractional, then the series of exponents m,m—l,. . . has no term = 0, but in time negative fractions appear. If then a par- ticular value of « make V — 0, say v= a, then the diff. co. may be either infinite from the beginning, or become infinite, according as we have the first or third cases. THeorem. If a certain value of m give P= gr (x — a)" a finite limit when x = a, then every greater value makes P infinite, and every less value makes P vanish; and if two values both make P either infinite or nothing, then every intermediate value does the same; and if any value of m make P infinite, so does every one greater ; while if any value of m make P vanish, so does every less value. And there is ai most but one value of m which will make P finite (m is supposed positive throughout). All this will immediately appear by looking at the following equations, and remembering that when x — a is small, division by any positive power of it increases, and multiplication diminishes, any expression. =e femoral X (e--a)" ea eae (7—a)", We must now consider the various singular forms of a diff. co, ; and we shall confine ourselves to singularities which are created by differen- tiation, and did not exist in the original function. If « = @ make any ¢ 0 diff, co. assume the form > then we must presume that the factor which the numerator and denominator contain in common, existed in the original function ; for differentiation introduces no new factors into both. And the same applies to ° x «&, and to all the other forms. Moreover, an exponential never appears in a diff. co., unless it were in the original function. All this is to be taken as very insecure reason- ing, for the purpose of pointing out the cases which, as our knowledge of functions stands, require or do not require a particular consideration. It has been of great disadvantage to analysis in general that there has existed a strong disposition readily to take for granted theorems which appeared to be generally true, only because they were true of the most ordinary functions. For instance, it is only very lately that the follow- ing proposition has been doubted : “If dx become nothing when w=a, then ¢x can be expanded in a series of positive powers of zx, such as ax" + be’? + ....;? and the reasoning was as follows, sanctioned by the name of Lagrange*: let x be expanded in a series of powers of x (the possibility of which is assumption the first) ; then if there be Negative powers, there are terms which will become infinite, and the series will become infinite (demonstrable when the number of negative * Perhaps the object of the Théorie des Fonctions has not always been fully com- prehended. Did not Lagrange simply say to his contemporaries, “ You found your Differential Calculus upon a mixture of the theory of limits and expansions ; I will show you that your algebra, such as it is, is sufficient to establish your Differential Calculus without the theory of limits.” This appears to us sufficiently apparent, when he says “ it is clear” that radical quantities in a development must spring from the same in the function. What makes this clear ? Certainly not native evidence in the assertion. It must be then the ordinary algebra to which he appeals. And those who are acquainted with the controversy upon this subject know that the ©pponenis of Lagrange (Woodhouse, for example) are at the same moment those of that part of Algebra to which he appeals under the name of the Théorie des Suites. N 178 DIFFERENTIAL AND INTEGRAL CALCULUS. terms is finite, but the truth of this when the number is infinite consti- tutes assumption the second): this is against hypothesis, therefore all the exponents must be positive, in which case the series is evidently = (0 when «= 0, because all its terms are nothing (this is assumpion the third), The third assumption is demonstrably true when the co- efficients, a, b, &c., are such as to render the series convergent for émall valiies of a. But in the case, for instance, of 1 + 2¢ + 2. 3x? + 2.3.42 + &., it is easily shown that the summation of terms gives an infinite result when x has any value, however small. It must be proved then, and not assumed, that the equivalent expression for this series becomes 0 when # = 0. We shall point out some instances in which distinct sin cularity of form appears, without denying the existence of others. Taylor’s theorem readily applies, as has been proved, to all cases except those in which a diff. co. becomes infinite. But there isa possible case in which all the diff, co, vanish; in which case the following theorem (page 73) must be true :— . h @ (a+ h) = (a+ 6h) = O9<1; ee eft in which there is nothing like expansion in powers of &. Weshall now give the instances. 1. sina, r+ (144°), tana. All the even diff. co. vanish when + 0, 2. cost, a-+(1+ 2°), & . All the odd diff. co. vanish when Ge 0, 3. a+(1 +a"). All the diff. co. vanish when xv = 0, except the Ist; the (2 + 1)th, the (2n + 1)th, &c. In these three cases there . is no singularity ;:certain powers of w disappear from the development called Maclaurin’s theorem. vi 4. («—a)*+ («—a)2. The first and second diff. co. vanish when x =a; the third is then =6, and all the succeeding ones are infinite. 10 | 5. («—a)s. The first three diff. co. vanish and all the rest be- | come infinite, when z = a. 4 i 17° ° re . q | 6. € =. his, and all its diff. co. vanish whenzv =O, For the» mth diff. co. will be found to have the form, the several terms of which are of the form 1 as” “e a2? — ,.0r—, wherez=@% whenw= 0; or & which may easily be shown (page 175) to be = 0, when « = 0. Turorem. If x = a make ¢v infinite, it also makes ¢’x infinite. This was matter of observation in preceding chapters ; we now prove it for all functions. . ON SINGULAR VALUES. 179 For if possible, suppose ov n of to increase without limit as x ap- proaches a. Say then, that however near a shall be to a, ¢/a, shall not be greater than A, while by the hypothesis da, may be made as | great as we please. Divide h, the interval between a,—h and a, into n ~ equal parts, each = Aa, and take Axd'(a,—h) + Ax $'(a,—h+Ar)-+.. up to Ar¢/(a,—Azr). This sum is therefore (as in page 100) always less than Av. A xn, since each term is less than Av.A; oritis less than AA. Its limit consequently does not exceed AA; but this limit is ¢’e dx from «= a,—h to T=, OF > (a) —(a—h).. Now that gx icreases without limit as x approaches a indicates that whatever ¢(a—h) may be, da, may be made as much greater as we please, or $(a,) — d(a,—h) may be made as great as we please, which is absurd, it being less than hA. Consequently #‘z is not always less than a given . | quantity A as approaches to a in value, or g’x increases without limit in such case. And this is our primary signification of the phrase “62 = ow when a = a.” , Corollary. Hence, if gr=e when x=a, every diff. co. is infinite, For $a being infinite, its diff. co. oa is infinite, and so on. A function which has some diff. co. finite, pr becomes infinite, can have all the difficulty of i to that of another in which all the diff. co. pre function itself, vanish when x m—1 diff. co. be A, A, function aE eceding the nth which ts development reduced ceding the nth, and the =a. Let the function itself, and its first - A, , all 0 or finite. Then ‘the px — Ay— (a — a) A, — (w —a)? BNW age ts a Anat 2 DESitigs, vanishes with its first — 1 diff. co. when x = a, while its nth diff. co. 1s "x, and becomes infinite whén x — a. Turorrem. If @z be 0 or finite when z = a, and increasing from #=atoz= x, but if d/a be infinite, then { (¢z — ba) dz must be greater than 5 ($x — ga) (x — a), or at least must become so if x be taken sufficiently near to a. For by definition of a diff. co, (pz—a) > (2 — a) increases without limit as + approaches a; let then x be so hear to a, that from z=a@ to x =a the preceding function shall be always increasing ; that is, dze—ha . ae - or (a) (fz-Ga) > (~x—a) (z-a), z—a 0 a consider these two last as diff. co, with respect toz. Then, since they remain finite from =a to z = 2, and since, from the process in page 100, it follows that P being always greater than Q within certain limits, I Pdz is greater than J Qd:z, both being taken within these limits: it ‘ollows also that ae—a) S (bz — a) dz >(d#— a) /[@- a)dz from za tuz=2, ir 1 (c— a) { (bz — pa) dz> (¢r—a) X 9 (7—a)*, n 2 180 DIFFERENTIAL AND INTEGRAL CALCULUS. 1 and S (pz — ga) dz > 5 (ox — da) (2 — 4). (px — pa) (w— a) . aru 7 Sis less than 2 for lue of 2, how- Hence Tbe — Ga) de is less than 2 for any % ue ever small. y As x approaches to a, the last fraction approaches the form 0°? for the denominator being /7Pdz is of the form fe — fa. But being always less than 2, so must be its limit (at least it cannot exceed 2) ; and this limit being determined as in page 173, by the ratio of the diff. co. (the denominator being considered as a function of x) we see that the limit of p'x (x—a) + Pr—$a eee ere" does not exceed 2, (or pxr—pa ib : Plas emeseo < oY = le pxr—ha does not exceed 1. Hence, if a =0, the limit of oo If a be finite, then ¢'w (vx—a) + $x decreases without limit : for ¢'n(a—a) > o'r(x—a) pr— pa pr gx—ha AbD. (1) hind | the first factor of which remains ‘finite, the second diminishes without © limit. i Also since ¢z-¢a.< gx—dga, f (¢2—$a) dz <(ox- ga) faz, or less than (¢r—¢a) (a—a). Hence —a) (¢x—a) (x—a2) Ws olan Dufi(ge—payds i > 1 and limit of ie Sen Turorem. If everything remain as above except that 9’a = 0, then the limit of ¢’v (a — a) + (¢z—¢a) must be greater than unity. | For everything is as before, except that (dz -—$a) + (z-4) dim , nishes without limit; that which was the less of the two integrals is | now the greater, and the final result is that > 0 or positive. as ‘ng (a—a) . limit of Sita oer 4) is greater than, or =1, which was to be shown. xr—a If da = « and therefore ¢/a= «, let dx X Par = 1. Then Pilg as, wa o'r wa a) Ue Ne bv AEE Un G2), and the limits of these are the same with different signs. But ya= 0, and therefore one of the preceding cases applies to it. And the limit of » wa (x—a) + wax being always positive when finite, that of b!a (a—@) | — px is always negative when finite ; and can never be = 0, because the only case in which this limit = 0 for yx, is when ya is finite, which cannot be if da= a. Turorem. If da, d/a,- . .. up to $a be-severally = 0, but if $"q and all the rest be infinite, then the limit of $’a (x—a) + $2 ON SINGULAR VALUES. , 181 ! lies between 7 and n + ] » or at least is either x, or n + 1, or some | fraction between. For by differentiating the numerator and deno- 3 ? : F 0 | mmator of this fraction, which takes the form 5 when x =a, we find pe at fo Yy ae eae ) limit ee ie, of 2~ D+ $n _ 1 + limit of L2e-D px zx '. | (repeat the process) , “2 (r— a mn (op — a = 2 + lim. of se Od) eoees = 2 + him: of ARS 2. a a | but because $"x =0, and Ry ae _ whence the theorem. Hence it appears that the more remote the diff. co. which first be- ' comes infinite the greater the limit in question ; or if the diff. co. ad infintum be = 0, this limit is infinite, or 6/a («—a) + px increases | withont limit. When all the diff. co. are — 0, then by the usual pro- cess br — (x — a)" is = 0 when r—a@ (0 —1.2.3...m) for every whole value of m, and therefore for every fractional value (page 177). And it will immediately be proved independently, that if $’x («—a)— bx had any finite limit, this could not be the case. Turorem. If $x be nothing or infinite when w=a, and if its diff. co. be all infinite (as must be when pr=«) or all nothing up to a certain point, and then all infinite, it will follow that, p being the limit of /x (x —a) — dx, the function px itself, divided by (7—a)?, will be a function which does not vanish when z—=a. oc, this last limit does not “exceed 1; pu RO) akae In this case ——— ., which call Wwe, 1s ~ or —, for when dao, (t—a)? OE ie p is negative, as was shown. We have, when c= a, — log pu —log Ya ae Oa: ioe lim P'x (x —a) i lim We (7—a) Malontin a). oy pu v observe that, the first fraction being always p, a finite quantity, and its denominator increasing without limit, so must its numerator, therefore even if log ya= , the numerator oo —cc , must increase without limit. Without this remark, there would be a tacit assumption of the question ; namely, that wa is finite. But by hypothesis, the preceding equa- tion is ‘ut (a—a ; ‘ew (a—a p= p—lim. mere or lim. gh = 0. Therefore wa must be finite: for of all suppositions, this is the only one on which the preceding limit = 0. Sins Consequently, when the (n+ 1)th diff. co. becomes infinite, make the preceding diff. co. vanish by the method in page 179, and suppose the function then becomes of the form pr—ga—(x—a) $'a— &e. This then may be written («—a)? xx, where p is the limit obtained from gr —da—. . . and yz does not vanish for c= a. We have then (p lying between » and n+1) pa px = pa+(x—a) p'a+.... + (v-a)" ————. -+- (r—a)” yx. ian he 182 DIFFERENTIAL AND INTEGRAL CALCULUS. The development of (a + h) becomes (exa+h) n | ‘ia ox = ba+ dah + "a ae che «2 oh o's he + hey (a+h), or the (7 +1)th diff. co. becoming infinite when w=a, isa sign that the development of # (a + A) contains fractional powers all higher than n. The process must be repeated with xx, if any diff. co. become infinite. But if ¢a= «, then at once determine d'e (x — a) — 2, and its limit, and we have then dr = (vw — a)? x2, where p is negative, and ya finite. Hence (a +h)=h’x(a+h), and negative powers occur in every term of the development. Proceed in the same way with x (a + Ah). 1 But if all the diff. co. become nothing, the development of ¢ (@ + A) cannot be made in the form hitherto specified, which contains ascending | powers, and nothing but ascending powers, whether whole or fractional, — whether beginning from 0 or from a negative power. The only re-_ maining case is that in which the development is in descending powers, — that is in ascending powers of 1 + /, in which way therefore all func- tions can be developed in the case in which all diff. co. are = 0, or in no series of simple powers whatsoever. The formal application of the preceding theory will not be necessary, © since the instances to which it might apply are generally such as are easily reducible by common methods. But its use is to complete the theory of development, and to prevent the student from imbibing the © notion of the universality of the common form of Taylor’s Theorem. — In the case of 1'x° —a’, for example, which is to be developed when oazath, we see that da=0 Pasa: and the function may be L : a Sy i Be L written (c—a)® (x + a)’; when e=a+A this becomes h? (Qa+h)? | the second factor of which can be developed in the common way, and the whole development will then be in powers of h of the form 7 5s : where 7 is a whole number. dy . avy When on is expressed as a function of , it can only take the form a 0. : ait consequence of factors being both in the numerator and denominator » of the original function. But if this diff. co. be expressed as a function e e 0 . . at . ° both of y and 2, its appearance in the form ig a sien of its having several values, as follows : Let dy _ $(@y) dn) ab (a. wy. de or Pp eee el dx and letra =a yb, make p= 0, Y= 0, it being understood that the’ arbitrary constant of integration must be so assumed that in the original function 2 =a, when y = 6. Differentiate both sides with respect to a, of which y is a function: then dp dp dy Oe R pipe ed pe ay —O... (A). de ° dy dx dx - dy dx / dv "dx a ON DIFFERENTIAL EQUATIONS. 133 o Let p be the value of © song Let p be the value o dy SCught: then making x= @, yb, w=0, in the last, we have (‘dp ' f dd ‘ds db if SP) {(#)- (Dlo> Beco mY, Law dp where dai &c. are the values of 7 &c., when w= @ pon. Tf dx v these be finite, there is an equation of the second degree giving two values for p. But if pas determined from this equation be - 0 0 9 dif- ferentiate (A) again, and it will be found that the terms containing y’ and y'”’ disappear, Jeaving an equation of the third degree to determine p, which has therefore three values: and so on. There will be further illustration of this point in the sequel. We now pass to the considera- tion of differential equations. A Cuarter XI, ON DIFFERENTIAL EQUATIONS. Aut that we have yet done has been in one sense or other on dif. ferential equations; but this term is more particularly applied to rela- tions between diff. co. and functions, where we wish to find the primitive relation between the functions. We have already (p. 154) in the course of investigation come so near to some very important diff. eq., that it was worth while to stop and solve them. A differential equation is considered as solved, when it is reduced to explicit integration, as in m. 155. Firstly, how does a differential equation arise? By differentiating a function, no doubt. But our present question is, how does that dif- ferential equation arise which belongs to one stipulated function, and to no other whatsoever? Not always by simple differentiation; as in the ML fags Oey ou, : case of y = ax, which gives Sin certainly a differential equation, dx and certainly true of y= ar, but not in the sense of being true of nothing else ; for it springs equally from y =axv-+ b. And it is clear that since integration always introduces a constant, there must always be at Jeast as many more in the primitive equation as we need inte- grations to pass to it. If then we would have a diff. eq. which belongs toy = ar only, we must so differentiate that a shall disappear in the process; or if not, we must eliminate a between the primitive and the differentiated equation. Hither 1, Write y = ax thus 184 DIFFERENTIAL AND INTEGRAL CALCULUS, pie yp We BD 2 ey. he Oy a > Y aes Te Bollcive the raul fae ep f ag): oth give the result y — 7 Seti a O8:Y =5 (24 Examrie 1, y=e™ @= — ese ae log y). DY ives pele a ae ge ae ies Or Pi ir aa both give 2 Fal y log y = 0. EXAMPLe 2. .y mcr ey eS a's Vx? — Ay, dy Rigge: d x y ——-— 0O=-—+)] + —, r—-22=4+2—4 Sa Be a iy Wes os square both sides, to avoid ambiguity, and we have dy" dy PE ad ore +y =0. eae Pi dy dy? Or, yer — Cc, Bee a 52 — at as before. We see thus how it happens that we introduce one constant at least in every integration; but may not an integration introduce more than one constant? We are not to conclude that because differentiation destroys only one constant, and explicit integration introduces only one, Be Ril d that therefore elimination of one constant between U = O and ee = 0 dx will never eliminate more than one. There are cases enough in algebra in which two quantities so enter two equations, that one cannot be eli- minated without the other. Where is the evidence that no such thing can take place in the two equations just mentioned ? Assume y= ¢@ (a, ¢, c’), c and c’ being two constants, and let the common diff. co. be denoted by ¢’ (2, c, c’). Let y= 9 (a, ¢, ¢') give c= ¥& (a, y, c’), consequently direct differen- tiation makes c disappear ; if possible, let it also make c’ disappear. Now since % contains 2, directly, and also through y, direct differentia- tion gives dis ade adie das) p sdy voli pede Oz re ay sae oC ae Ae at (A), ay) isiad age ; which answers to the way in which = is obtained without c, above. Compare it with Example 1., thus : _ logy dy _ logy dy ot logy , ldy _'9 eee feared — ——_.. oe ae en a ody xy a ° ay dx ON DIFFERENTIAL EQUATIONS. 185 Now the fraction (A) can only be independent of c’ in two ways, 1. by | neither numerator nor denominator containing ¢’; 2. by their both con- . . / s 2 . . . . taining the factor C’, a function of c’, and not containing c’ in any other 3 dus ; way. In the first case, since ee does not contain c’, then % must i ax j have the forma (a, y) + p (y, c’) (« and 6 being functional symbols) for c’ can only occur combined with terms which disappear in differ- —_—_— entiating with respect to x, that is, with functions of y. And since Be ay }does not contain c’, % must be of the form ¥(@, y) +3 (2, c’), for isimilar reason. Hence a(2,y) +B (y, c’) = y(a, y) + 8(a, c’) jor 6 (y, ce’) —3 (a,c) = x(a, y) — « (2, y), a function of wand y only ; }consequently 6 (y, c’) and (a, c’) can only contain the same function of c’, disengaged of all functions of y and z respectively, for if c’ could senter combined with a function of y in the first, it could not disappear iby subtraction of the second, which must not contain y. That is, ithe preceding forms must be B(y) +C and 6(7) +C, C being a function of c’. Ory has the form f(z, y)+C. But Ur (x,y, c!), or Ys, is =, or the equation between a and y may be reduced to the form f (x,y) = ¢ — C, in which the two constants are in reality only one. But if * and have a common factor, a function of ¢ only, which call C’, then % must have the form O’a (x, y) + 6 (y,¢’) and Cy (x,y) + 2 (a, ce’) for reasons as before. Hence 1 1 Q / AN +BY =YOY + Fae); he second terms of which are only other forms of f (y, c’), and (a, c'). The same reasoning applies, the two sides can only have the orm f (2, y)+C”, and y can therefore only have the form O! f(2, y) + C” C’, which being c, we have Oe OH OR ancien 2,4) = a which is equivalent to but one independent con- itant. But may not both numerator and denominator in (A) contain a factor, vhich is a function of c’, x, and y; c’ not being contained in the other ‘arts ? If possible, let ~ =o BLY Hy = MW, M containing c’, but and W not containing a Then es have ay dM ._ dL dM = — \ = Ww dade dc’. "Pdydse dc’ ’ ‘om which we find, putting for V and W their values, addy 1 dMdy _ 7 ddyw 1dMd _ dxdce’ Mdc' dx” ” dydc! Mdc’ dy dd dw ddw dw _ 0 dy dd’ dx dxdc’’ dy” 186 DIFFERENTIAL AND INTEGRAL CALCULUS. the last of which (by the lemma in the next page, proved independently bet d | of this) shows that et must contain x and y only through y, or 79 He de! f0#)5 giving di die i, At d dis dis LE PS AEN aitt same () CE ae ee — = Q, dx de’ J Sh hau Gal ae PY) dy 1 dM 1 dfs Fy eg Ea ca UP tae Pa oy 1 oO ou Cee bc a Lot eee . which, with the preceding, gives M rate Ws, Aue Me ls dlogM fed _ dlog fy —. or log f(%)=log M+Z, | whence —— it ot = de! fis de’ A RN where Z% is a function of a and y (or may be, since # and y are the con- stants of the last integration). | Hence M is of the form fy.Z,, where Z, does not contain c’. And thus we have . and neither VZ, nor WZ, contains c’. Let f (fw) "dys==xy, then VZ, and WZ, are its diff. co. with respect to vandy. But neither contains c’, hence yw itself can only have the form F (a, y)+C. But since the original condition gave c = w%, we have therefore yo=F (a, y)+C or xc-C=F(2, y), so that the two constants are equivalent only to one. Before we proceed further, we must require the student to remember that there will be between the diff. co. employed a distinction analogous to that of known and unknown quantities in algebra, If we actually | assign a function of » and y, say zy®, we shall never need anything to remind us that its diff. co. are given, for we absolutely write them, | namely, y®, and 2xy. But when we reason upon a given function which | is not specifically given, but merely assigned or Jaid down as given (like the known letters of an equation in algebra), we are in danger of coufounding the diff. co, of a given function a(v, y), which are given without an equation, and which we can specify as soon as we specify the function—we say, we are in danger of confounding these with such hs aida ro} 7 diff. co. as ann which have no existence except under an implied equation. What are the diff. co. of zy*? Answer, y? with respect to xr, Qry with respect to y: this question is answered without an equation expressed or implied. What are the diff.co. of w? Answer, with respect to z and y both equal to nothing, for u is not a function either of « or y. But what are the diff. co. of w when it is meant that u is an du du always =zy?? Answer, — = y°, — = 2ry. Hence then we see Ax dy 4 du ; that such an assertion as w= P, therefore 5 eee &c. is not use- less tautology ; for it implies that we have u, a given function of x and y, with diff. co. which can be found, and the second equation of the last ON DIFFERENTIAL EQUATIONS. 187 pair is'a symbolic imitation of the process of finding the wknown on the first side by means of the known on the second side ; an imitation which cannot be rendered real till we specify P, in which case an alge- braical result takes the place of the symbol of differentiation on the second side, but not on the first. Lagrange, in his attempt to reduce the Diff. Calc. entirely to the principles of common algebra (in the Théorie des Fonctions), adopted the following notation: f(x, y) being a function of x and y, its diff. co. with respect to 2 and y were denoted by f’ (w, 7) and f, (7,4). As this notation will be frequently convenient in functions of two variables, we notice it here. In ike manner w’ and u, may be the diff. co. of w. We shall adopt the following notation. Let ¥(2, y,c)=0, give y== A 1 ?(x, c) when solved with respect to y ; and let = = x (2, y) be the resulting diff. eq. : Lemma. If p=a(a, y) and if at st a ch — 0, then » dx dy dy dx cannot be any function of 2 and y other than some function of p (the converse appears in page 97). For if possible, let u = f(x, y), such that finding y in terms of p and z from p=a (2, y) we obtain u=F (p,2), where x as well as p appears. Then w contains x directly, and through p; but u contains y only through p. Hence du _ dF dF dp du _ di’ dp dx" dx ° dp dx dy ~ dp dy’ u dj lu d dF, d and ae nee = — fF 9 (by hyp.) dz dy dy dx dx dy dp . ; .. ai But — is not =0 if p be a function of y; therefore — = 0, that is, dy dx F is not (as was supposed) a function of x directly, or F (p, x) is only of the form of some function of p. dy : d ; Turorem. ‘The equation Ter nO (x,y) cannot result from two lx different primitives y = (a,c), y=a (2, ’) of different forms, with _ an arbitrary constant in each. For, let both the second and third be | primitives of the first; and let y= (2x, c) give c= ©(2, y), and let Y=a(x, k) give k=11(2x, y); then the diff. eq. of these primitives are db do dy Ul | dildy _ —— an se ee beeps at SoA), dx dy dx ee sey: he ; We da dy . jst oh are Which are both satisfied by = sar, 7), oF A is the same in both. ze ; Eliminate this, which gives d@ dil dod _ ——— —~—,—- =), hence ®(2, y)==some f° of IT (a, y) dz dy dy’ de when (2, y) Or, c= f{il(z,y)}, k= (a,y); letcofe givez= fe. 188 DIFFERENTIAL AND INTEGRAL CALCULUS. Then fic = TI (a, y) which gives y= w (4, fc) one primitive, ca I (ay). she oy Se(2, c) , the other, or the two primitives only differ in the form of the constant. Consequently, a differential equation of the first degree cannot have a primitive with more additional constants than one, nor two different primitives with the additional constants entering in different manners. It only remains to ask, may not someone particular case of another primitive, made by giving its constant some one particular value (and thus making it cease to be an arbitrary constant) solve that diff. eq. whose primitive, with the constant, is y= @ (a, c)? The preceding case includes this as well as any other, for whether & be supposed to have a particular or a general, but constant, value, the investigation is the same. (The student must always remember the difference between “ let k be 10, or 11, or any other asstgned constant,” and “ let k be anything whatever, but let it not vary,” which is the character of an arbitrary constant.) It should seem then that the question is answered; but here we are obliged to remember the con- dition which runs through all our reasonings, unless the contrary be specially mentioned, namely, that diff. co, must not be infinite. And it is essential before we proceed to show why we did not find it neces- sary to allude to the possible case of ®! or ®, being infinite in the last theorem. When we differentiate a simple function of ., specific values of x may make y’ (y= 2) infinite, as already discussed. But when we come to functions of « and y, not only specific values of 2, but specific forms with unlimited numbers of values of x and y, will produce the same effect. Instance, du a ——— dx Ve+y—1 u=Vet+ yl =. if y=V1—2 du y dy A734 pI This was immaterial in the preceding theorem, for since ® (a, y) was ! eee 8 eas if y=V1— 2 ! Betas . aD without an arbitrary constant, so were its diff. co., and if -— had had a v denominator « (x, y), then « (2, y)=0 could not have given a value of y in terms of 2, with an arbitrary constant, which was necessary to every case then under trial. But now, when we are considering the possibility of some specific case of another primitive satisfying our equation dy Pl! (2, y), we are bound to consider those relations between rand db A age which make an ie infinite, for they may now (that we are con- sidering relations without arbitrary constants) be the cases in question : and no others can be such, since the preceding theorem is conclusive as to all the cases which it includes. Returning then to the preceding theorem, it appears that we must devote our attention to the cases in which the diff, co. of ®, or any of them, are nothing or infinite, and to But y =$(a, c) (caf°ofx) gives ON DIFFERENTIAL EQUATIONS. 189 the relations between y and x which produce that result. But having thus defined the question, we have a more easy method of proceeding than direct investigation of its several cases, as follows : The equation y= (2, c) may be changed into any equation what- ever y= wx, by making c, not a constant, but such a function of x as will be obtained by finding ¢ from ¢(2, c) = ax. Let us then suppose ea function of 2, and let y = wx thence obtained be the particular case (if there be any) of another primitive which satisfies d : ede d: e = x (2, y), obtained by eliminating c from y= (a, c), ar=ol(ay 0), ly dp | db de ‘ dx dx de diet 0 ); mre dp where, since i supposes c constant, Ae =" 6" (2, c)} aX di d é dd de and since ae x (2, y) satisfies (1), v (2, y) = P'(a,c)+ Reeth: But x (2, y) = 9'(2, c) is satisfied independently of c by y= (a, c), because y= (a,c), y’=¢' (a, c) together give y= x(a, y) by elimination: so that x(2, y) = ¢/(a, c) is made identically true if y= (2,c). From hence it is immaterial whether in y = ¢ (a,c) : dp dc we suppose c constant, or any functionof z, Consequently de dp must dc : = 0 in the case supposed.. Either then er Q (or ¢ is constant, v Saree do ‘ which reduces ¢ (x, c) tothe usual primitive), or i = Ohmiuat. is.88 lc certain function of x and c is =0, from which c may be determined in terms of a. For instance, in y=2-+(c—<)’, we have, to form the diff, eq., ses d eee ey = 1—2(c—x) : eliminate c, and a ie ov y Sik, cmunken ys dx dx di dc If c were a function of x, then* 5 = 1 — 2(c-2r) + 2(c—z) a Now required ¢ so that (2) shall still be true, or that (y being 2+ (c — x)?), ici. de 1—2Vy=7=— 1—2(c—a)+ 2(c—2) = : aaa dc Observe that 1 — 2V/y— x is1—2 (c-x), therefore 2(c-2) om 0, and either c is constant, or else c=, in which case y=v+0=2. And * Though the following caution appear rather trivial, yet some difficulty to the student may be avoided by it: the sign = includes all the moods and tenses of the phrase “ is equal to.’ In the present case read it, would be equal to. 190 DIFFERENTIAL AND INTEGRAL CALCULUS. be pie eh peomaclpanty heaps Teal y= a satishes FF te 1 avy L 5 but is no case of the primitive y=a-+ (c—2)?, ¢ being constant. Itis then the only particular case of any other priimitive which satisfies (2); the primitive of (2), which has a constant, being y= @+-(¢ = 2)’. This new kind of solution has been called a singular solution, a par- ticular solution, and a particular integral. We shall adopt the first title. . The point of view under which the singular solution takes its most remarkable form in geometry answers to that of a species of connecting function between the different cases of the primitive, such as arise from giving different values to the constant. Thus y’= y (#, y) is true for y==o(a, c), whatever the (constant) value of c may be, It is equally true therefore of y= (2, c) and of y= («,c+Ac).’ Now (a, c) and (x, c-+Ac) are generally of different values; but there may be specific values of x for which they are equal. Let us consider then the case : do d?p (Ac)? ; (a, c)=O(a, c+Ac) = O(a, c) + ae Ac + rue a dp , dz ag or Wait deg Bites alalyitecet Us If Ac be very small, then the resulting value of z is very nearly that ob= tained from —0; if still smaller, still more nearly ; and so on without limit.. But if Ac=0 absolutely, then @(2, c)=9$(@, e+ Ac) for all values of x, and of course among the rest for those obtained by = =(0. Stee C the solutions of the last equation have this property, that the values of # for which the two functions have the same value when Ac is small, ap- proach nearer and nearer to them without limit, as Ac diminishes. For — example, in the equation y=a~+ (c—<)° already discussed, if we inquire | for those values of 2 which make | t+(ce—a)? = «+ (c+Ac—z)? or 2(c—z) Ac + (Ac)? = 0, we find that 2(c — 7)+ Ac=0, ‘or = c+ side, which. approaches _ nearer and nearer to = c (the supposition from which the singular solution was derived) as Ac is diminished. We return to page 186, in which it is shown that no case of any other than one primitive will satisfy a diff. eq. so long as the suppo- | sitions implied in the demonstration exist ; that is, so long as none of the diff. co. employed have singular values. Whence it follows that the singular solution really obtained must belong to a case in which diff. co. have singular values. d dy db addy And since —— © = 6/46, = — + --— nee” da | ae de. (do, we cannot have, by one supposition, both ©’ and @, = 0; for that sup- position (say it is y = fr) would show that ® (a, y) is by y= fa re ON DIFFERENTIAL EQUATIONS, 191 duced identically to a constant, and this case is therefore included in the primitive y = (2, c) or c= ®(a, y). We cannot have 6’=0 and , infinite, for if we suppose c=auw to be the value of ¢ which gives the singular solution above, we have then wr=0 (2, y) and ar=0'+o, x(a, wx). But ,is= %, and wx not being generally infinite for all values di ; ’ ‘ of x, we can only have y (x, av) = 0 or a 0, which is not uni- > py x ad b>] v versally true; for the singular solution, as well as the ordinary primitive, gives _ a function of candy. Neither can we have ®’= « and ®, =0, for then @’x =, which cannot be generally true. There only remains then the case where © and ©, are both infinite, so that (remem- bering that algebraical quantities, upon finite suppositions, only become infinite when a denominator is made = 0) we have the following theorem. If y= (a,c) give y’=y (a, y), and c= @(z, y),; then the sin- gular solutions of y/=y (2, y) will all be found among such equations J(@, y)=0 as make ©’ and ©, infinite, or a common factor in their de- nominators nothing. Observe, we have not proved the converse. There may be values which make 6’ and ®, infinite, but which are not singular solutions. ExampLte 1. y=a+(c — x)’, gives c= a+ /y Soi » Which dif- ferentiated with respect to x and y, has only 2V y — «x in the denomi- nator. ‘Therefore, if there be a singular solution, it is y = a. Verification. ‘This is the singular solution we found. Exampir 2. Let y=c?—2ca, caatvy +2, As before, if there be a singular solution, it must be y=—a*. Treat this by the other method, and we have P(x, c)=C2—2cx, ee 2ce—2r2=0, c=, or yr? —2e' = — 2°. € As yet, we have only found the singular solution from the primitive itself. We now proceed to show how it may be connected with the diff. eq. From y=¢(e, c) giving c= (a, y), we obtain dy dy @! 0= 8 +4+0— or “= = ~ — = y (xz, y) by reduction. tik dx dt 6, * (7; y) by But if we prefer the direct elimination of c, we take y = @(c,¢), dy _ dp ind = a 2 function of x andc. Let this last equation give x i, : P( a, 7) , then the diff. eq. is fi d ; di y=D\x; F (« -) equivalent to sn = ¥ @@,4/)'} 192 DIFFERENTIAL AND INTEGRAL CALCULUS. d ; | so that the substitution of x for a in @, as last written, would make | y= identically true, independently of # andy. Orwe have y= (a, c) d nl | is made identical by c= F («. aL), if “7 be made = x(a, y). dx , dx This equation, then, on these suppositions, may be differentiated par- tially with respect either to x ory, and thus we have dp do dc dp dpdc . dpdc dx wi dy dv dc dx 0 Soh Se ee oe it SSIVINE, ae dx de dx dc dy dx dz dp dc de dx dp decd 1 ] dp ae eX giving, dx oe . dc dy dy dy dp de dc dy As an instance of this process, we take y=x+(c—z)’=((a, ¢) dy dp 1 1 dy dy —~ =1—-2(c—av) = — SF 5 ea LS ee — di (Cap tia. ade” Gee Se a Leewlady ys ( dy y=et(5-3 2) = 6(a F(a p) een.) dy —- which is rendered identical by Fs J ON, y—x = x(2, y), dp dp dc Gz, de dp) lie howe) + 2(on 2) Min 1 dy dp dc | ¢-2 "ae iret aeh Bary Pen eerie ee ye lay 2(c—2) X 5 ge 1 1 dpdc tr pie Bo Ny-x y dy ; ae ae Wc—a) ¥ —5 Now, returning to the general expressions, we know that the sin- ' ; ‘ dp gular solution requires c to be such a function of ras will make = = 0; eo d Oy te and therefore —* and = infinite, unless it happen at the same time dix dy WE) Ae sake d dy. do. that — is infinite, or else nothing. But = is es, in both cases; the ie 1x dx de. ; last therefore cannot be: and to suppose oe infinite would be to sup- x , : d a pose that F (w, x) = ¢, re-imverted into y = —, gives <5 = 0, or thai dx c x does not contain c, or that y= (2, c) must be of the form y=fare ON DIFFERENTIAL EQUATIONS, 193 4 case we presently consider. There remains then only this case; that a ly l ™ being = x (2, y), all the singular values of y make x and —X ; dx both infinite. : In the preceding, we have supposed x (2, y) to be really a function of both x and y; but if it happen that the diff. equ. be of the form y'= x(x), we may sce at once that the primitive is y= fyrdx +c; ae dy while if y’=yy, we have r= |—— +c. The singular solutions of these are only such as can be derived from yx = x and yy= x; as we shall now show. TuEoreM. If ever we imagine a letter to be a variable, and differen- tiate with respect to it, while under our implied conditions it is a constant, then the diff. co. which we expected to find finite, will be found infinite. Suppose, for instance, a=a-+hyt, which we imagine to vary with ¢, but which does not, because, as we afterwards find, k=0. If we then differentiate y with respect to x, we have (y being really variable with ¢) dy dy . CE ay =f dy _ —~—- — = —~—_—~=c« ifk=0. dz dt “dt kwtdt ~~ ' dy dy ed ‘Vg ee and if r=a make yr= «x , then a 3 and #== d, Ww Q wv Z or a constant value given to a, satisfies the differential equation. But this is an extreme case of singular solutions, and will be satisfactorily illustrated when we come to apply the subject to geometry. di ae : . “2 Examp.e 1. = — Ve—y?. The singular solutions, if any, are dz =-+2, ory=—vz: but neither of these is a singular solution, for > to) : v* eee dh ; neither satisfies the diff. eq. : they give eA = +1 or—1, while L oy a2 shi L: —— v— y°=0 gives oo —="0. But = —1+/22—y? has y= +2 for its v singular solution: it is usual to add, unless it happen to be a particular | Case of the primitive ; and certainly the not being a case of the primitive which contains the arbitrary constant, is the fundamental definition of a singular solution. But as it may happen that a particular case of Y= ¢(2,c) may have, with the single exception of being such a par- ticular case, all the characters of a singular solution, and particularly all the geometrical characters, we shall not attend to this distinction. di —— = melee IN EXAMPLE 2. y — =V2+4+y—a?—2. The singular solution, if dx any, is y= tVe—2@ , and this does satisfy the diff, equ. e bd E 7 ‘ ’ € ‘ Lt fer We are now in possession of all the possible forms which can satisfy an equation of the first degree y/= x(a, y). We shall now take several Oo 194 DIFFERENTIAL AND INTEGRAL CALCULUS. leading forms which admit of complete solution, reserving those which require particular artifices for a future chapter, or specific application, dy t os : Nt Fi f(x). This evidently gives y Sei Ar ag +o. 2 di BY oe ppencly g xe ee eth fai . (4 2. phigats bie oe {a +c or y= W(x —c) where, {% being «2 xv is such that ay = © —c gives y= W(“—Cc). 8 4Y) i k di } 1 Exampe. = = Uf dl Ol 0 Y Bae ree dx Yy : Cc—2 dy d ; 3. = em fe iy Se. en f fr dx +e. dx fy dy l 1 x +e owl a pee 7) Bre oe EXAMPLE. 5° = ay; log y= 5% +e, yous da pi ; re. oF a" iy) (“) . Under this general symbol is included every sy homogeneous function of and y, meaning either rational and integral functions, all terms of which are of the same degree, or any functions of them made as follows. The number or fraction 7, positive or negative, is the degree of the function. Aas aN r+ ay + y? or a ( 1+=-+ = } is of the degree 2, r X a : \ a+y > ityre ———— or v ——_-——_ . . e . . ° 0, 4 at) l-y>et 7 5 Seer eT sy 7 ] ve ty or x (4 + d i Ra ce Naa eee ag x 2 ome -2 |l—y+2 3 Sees ew Bie? Or & A PARE ere ° ° e e 6.) Soe re Ni + of? v1+ (y—- x) 2 Assume y= 2u, Then we have ; CAL ih u+e—=2" fu, dx ‘ which is immediately reducible to integration only when 2 = 0. Sup pose this, and fa + au ER yf 1 i se = — a ROO EL Cee 108 Cr fi—a dz ya” fu-u e hee for instead of c, which is perfectly arbitrary, we may write log c. Let du | fue Here by W~!u we mean the function inverse to yu, so that YO =u = yu, and let wusv give u=y lv, then y=ay" (log cw). ON DIFFERENTIAL EQUATIONS. 195 l ‘ff , d We have thus integrated cy ye to which P + Q ~/ =) may dx xy dx be reduced, if P and Q be homogeneous functions of x and y of the same degree. dy Oy a te Mie 7 i. 2 ry = 42 J me aye Lhe, ea EXAMPLE, 2*-+ xy dp 4 Bives = £5 eae 1 du ak asa suey — fudu = me Ue OS fa u or 2 = —Qlog ca > oor yo V9, 2r/ tog(+ ; 2 cx Verification. De ee ha] AY A a —-— V2,/-logecr Mea — {00 Bn ES a eee oe J J -logcx +N2 x at log ca) ( 5) a ae d 5, + Py =Q;_ where P and Q are functions of x. Let y = uv, which may be satisfied in an infinite number of ways, and we are at liberty to assume one equation between u and v, or to assign a value to either, the other remaining to be determined by the diff. equ. We have then 5 d id ’ u ee v a + Puv=Q or u ie + Pe) +y - == (). Let dv dv im 44° Pn=0 or nh —{Pda4-c or vy see Pete eo, ert Per, for which we write ce“"™, since ¢’ is merely an arbitrary constant. du du We also have »v —=Q, or — dx dx = M Qe , c 1 : Hence u = — | Qe?" dz +c', c! being another constant, c yee Sere. [Qe?* dx + ce, (writing c! for c! xc) in which one constant has disappeared, and only one distinct constant temains. We may verify this result as follows: — e—/Pdz {— P) : [Qere dx =a gvtes Qelt% + cle Pas (—P) ax = — P(e PP (Qe dx + de P*) + Q= —Py+Q , dy : A Be [2% dx + ce EXaMPLeE 1. i +ay=Q_ givesy=e e* db : EXAMPLE 2. oe + Py=P_ gives yo i + ce’, - 196 DIFFERENTIAL AND INTEGRAL CALCULUS. EXAMPLE 3. Let Y= oty, P= -1 (edn = om ctl iN eres fre * dx = —ye* — &%, yr — au —l4+ ce’. Qs nape dy c+ f dy f being any function whatsoever. The eae Es dx)’ 6 ; integral evidently is y = cx + fc, which gives = ='¢.’"''Ehis’ primitive a is remarkable for its singular solution, found from #+ Flies 0. fae give c= Wa, then y= xpa + f(x) Is the singular solution. dy dy EXAMPLE 1. — —~y7 + sin —’ gives y= cu + sinc. Its sin- J dx i dx 8 J s gular solution found from xzi— | mnt f) or co = , 1s ¥ =—Vz*— 1 + sec™z. z+ : Vic? 2 d dy - EXAMPLE 2. y = = + (3) gives y= —3 (=) , the sin- gular solution. We are now in possession of the means of integrating equations enough to illustrate their theory; and particular instances can only acquire an interest in connexion with problems which produce them. : dy The most general attempt to integrate P + Q Ss <= 0, where P and Q dx are any functions whatsoever of x and y, is one which fails by requiring the previous solution of another species of equation ; but its principle is highly instructive. We return to y = (2a, c) giving c = @(2, y) and 0 = oe Hs Fy ds? which latter is in fact the differential equation, since it does not involve c. But if ©’ and ®, have a common factor M, so that 6’ = MP3 a he ee da : ©, = MQ, substitution and division show us that P+Q st = 0, which dx may be the diff. equ. in the form in which it is first presented to us by a problem. Now, how are we to know whether a factor has or has not disappeared? By the following simple process. If dy db d&dy me 9) - "4 ] fame pa RE P+Q a presented to a be really ma + snp 0, to which direct derivation from the primitive would bring us, then, be- cause d d® d d® dP dQ ae ee, OER ee fee oO oy? y 5 he (CO =) —— ae ae ae ibe ag (page 162) we must have qe awe di ae ee ae Thus, in a? + 4? = = (0, we.see that a = ee = 0 (partial diff. co.) ON DIFFERENTIAL EQUATIONS. 197 dy d(«r#+y’) ab But in e+ y24 a? aE EF — Dy, _ i y dv’ dy hing dx an In the first, therefore, we have no factor to look for, in the second a factor has been lost. This equation o. aoe dQ is called the condition dy dx of integrability, and we shall see that integration can really be per- formed without further preparation when it exists. Let as — ue : dy dx posing y constant. Integrate on this supposition, then SJ Pde + const. is the primitive. But since y was a constant in the integration, the Jatter term (const.) may have been a function of y; for such a function may have disappeared by differentiation with respect tov. Let there- fore f Pdx + fy be the primitive : then, because Q is the diff. co, of the primitive with respect to y, we must have d. d a (fPde+ fy)=Q or ay / Pax) + fy =Q d then in P+ Q » Pisa diff. co. obtained by sup- @ Let [Pdr = V, then P = aN Bi Pf er Le Ns du’ dy dydx dxdy dP d dV dV d —dre= |—.— .dr= — = — or { iG dx SJ eseay dz 7 a (f Pdz) ; so that, in like manner as the order of independent differentiations is indifferent, so is that of a differentiation and an integration with respect to independent variables. dfy dP ty dP Hence Tate, Q ay. dx fy ={(@ 4 ay ir) dy. The latter integration is made on the supposition that y only is variable. This might appear to require that a function of x should be added ; which, however, must not be, because by such an addition the condition already satisfied, namely, that the diff. co. with respect toz is P, would be undone again. Hence, the function whose diff. co. with re- spect to x and y are P and Q (which call U) is U=fPae+ {(Q— far) dy. Differentiate for verification, remembering the theorem just proved, and iP d Bey. P+ {(2 — ge. dy =P, because pee! te a. dv dy dy dz’ iP dP Be g dx +Q— |—dr= dy dy dy Exampete. From what function springs d d(x+2ay) . d(a? +?) w+ Qry+ (a* + y”) 3 ( et ae dx 198 DIFFERENTIAL AND INTEGRAL CALCULUS. : 2 ) fPde=f@t ay) de = > + ay, dP Gilony ieee yw ape f( ok a Pees ] i and the function is : a? 4+ a%y + x y?; from which we infer that the solution of dy l 1 v-t- 2x ety) 2-0. isc aa wy t gy. ary H+) F gots hae In the preceding operations, observe that none of the signs mh imply the addition of constants, those having been considered in the pro- cess. And also that the term annexed to y, though it appear to contain aw, is really a function of y only, since d ‘uP nota Wide \a ae da) (Of ay) =| a )a=e InP+Q “= we have hitherto supposed that y is some function of @, it is not known what. If we make the preceding = 0, then y is the function of # defined by c= U. We have reserved the notion of differentials (which we may abbre- viate into diff’’.) as distinguished from diff. co., till we have come | to a point at which the occasional rejection of the latter in favour | of the former will produce an advantage more than compensating the liability to inaccuracy which the former are said to involve*. (Read | here pages 14, 15, 38—41 of the Elementary Illustrations.) If | uz h(a, y)- give Au =! Ar+ $,Ay + &. (page 871) we write | du = o' dx + $, dy as an equation 1, which approximates without limit | to truth, as dz and dy are diminished; 2. as one which gives the limits, ; s0 soon as ratios are formed by division, upon all suppositions. The * The author takes this opportunity, once for all, to dissent from notions which — have been lately promulgated in English works, relative to the rejection of differen- tials. To such a point has this been carried, that the very striking and instructive analogy between 2yAxr and /yda, as compared with that which exists between A 1 ; : — and a has been lost to the eye by the introduction of /,y to stand for c ax a) fydc. But has this great sensibility to notation been accompanied by a similar feeling with regard to the assumption of principles or theorems? Have those who di, ’ imagined they were more accurate when they wrote = = p instead of dy = pdz, dx rejected the assumption that f(#-+h) can always generally be expanded in whole powers of A, or the attempts at @ priori proof, after the manner of Lagrange, that fractional and negative powers cannot enter? And have they been equally attentive to phraseology? Have they rejected the expressions about the failure of Taylor’s Theorem, which would imply that the said expansion, not having the process by which it was declared universal before its eyes, but being moved and instigated by the vanishing of a factor, did wilfully and of malice aforethought, refuse to be true in Chapter IL., the same being against the proof in Chapter I., its truth and) generality ? Until these questions can be answered in the affirmative, we are reminded of differentials by the relative sizes of a gnat anda camel. ON DIFFERENTIAL EQUATIONS. 199 | only warning necessary is, never to separate a partial differential from its denominator without making a proper distinction, since the removal of the denominator removes the existing distinction. Thus du d 1 dus — de + d ; igh eny + ay y cannot be written du = du + du, though we have du (when both vary) = du (when « only varies) + du (when y only varies). aN : d lu Which might be written du= d,u+d,u, but oe oe a dy will be found more convenient. We shall now suppose that in Pda + Qdy, the condition of inte- grability is not satisfied. Let M be the factor which has been lost, so that MPdx + MQdy is a complete differential. l AY L (IV IM J} i Then @MP) FQ pIM_ gM _ yy (4Q_ oP) | dy dx dy ak dx dy Thus, if we wish to render ydx — «dy complete, we have dM dM “ Beh) Ca a y— +2 = — 2M ; dy dx or we have to solve a partial diff. equ., namely, to find M, a function of x and y, between which and its partial diff. co. the preceding relation shall exist. This we cannot do generally, but thereupon, seeing that this proposition is true: “ given the solution of all partial diff. equ. that of allcommon diff. equ. follows, both being of the first degree,” we may suspect the converse, namely, that we shall be able to solve all partial equations, so soon as we can solve all common ones. And this we shall find true, with just enough of variation to remind us that the assumption of converses is dangerous. Tueorem. If N be a function of x and y, giving dN = pdx + qdy, then the equation du= VdN is incongruous and self-contradictory, except upon the assumption that w is, as to x and y, a function of N ; or only contains # and y through N. Let N = (a2, y) givey = x(N, x), and suppose, if possible, that the substitution of this value of y in wu gives u= 8 (N, x), x not disap- pearing with y. Then, xz and y varying, dB dN dB dN dG du = — — dr + ——— dy + —- da, adN dz dN dy da 12 RQ ap d = or du = — dN + in dx = VdN, dN dx which equation being universal, is true on the supposition that x does i, djs Ne mot vary, or that dv =0. This gives — =}; , aN rT IAT GB rT INT or du = VdN + —-dx = VaN, Ae ip) because and V being independent of the variations (as are all aL 200 DIFFERENTIAL AND INTEGRAL CALCULUS. diff. co.) whatever relation exists upon one supposition exists upon all y dp ; ; others. Hence ae = 0, or 6 does not contain 2 directly, but only as dx it contains N. We have purposely introduced this demonstration here, because it gives the opportunity of dwelling on the point most likely to | confuse a beginner in his first use of differentials. In the equation | dN = pdx + qdy, which is true of dN, dz, dy, not in the ratios which” they ever can have, but only in those to which they continually approach | as they diminish, we can no more suppose dx = 0 absolutely, than dy or dN, except only on the supposition that # does not vary at all. The | smallness of dy, if it be supposed small, is no reason for the rejection of gdy as compared with pdx. Or let dé be a comminuent with dN, | dx, and dy, and let | dN dB dN | dp dr Vv dN —— &c. be limiting ratios as usual, whence —- — — = VY — dt ss ‘ dN dt dx dt dt is absolutely true, upon all suppositions. If then x do not vary, we | have ans dor dN” dGdr —_. dN doy MAN ag Mer Md" car deme which being true independently of ai? must give rE 0, as before. Again, : dN dN d dN d /_.dN = VdN = V— d — dy gives—(V— )=—( V— du=V In dz + V a y gives ty (v 7) AG (v zt dV dN , @dN __ dV dN adN .dVdN _dVdN _| dy * dv dydx dx dy dx dy’ E dy dx dx dy ™@ whence (page 187), V, as to xz and y, must be a function of N. Let it) be fN, then du = fN.dN, u= [fN.dN + const., a function of N, | Hence, du = VdN requires both V and z to be functions of N. TueoremM. du = Pdx + Qdy, u being a function of x and y, cannot be true, « and y being independent, unless F = = du du du du nce CO Daun) | less apie tt Ae erp i a and unles ae: dy Q Slving dy An we may easily show that no given function of x and y can be = 4, unless upon a supposition which connects x and y. ‘Thus, in the case of du= dx + xdy, we cannot, for instance, have w= a®-+ 7’, unless we have 2rdx + Qydy=dxr+ady or ay = Ait Ses : dx e— 2y | ON DIFFERENTIAL EQUATIONS. 301 which is only true where ¥ is one particular function of 2. Similarly, | wecan only have w=2y-+y, where y is another function of 2, and so on | for every function of w and y which w can be. But in du = xdy+ydz, ————e we have w=ay, whatever y may be. This latter sort of connexion between w and a function of x and y is therefore impossible in the pre- ceding case: which was to be proved. Where one equation only exists between two variables, as in y= oz, or % (x, y) =0, there is one independent variable. But there is one only when there are two equations between three variables, three between four, &c. To take the former case, let us suppose $(z, y, u, c) = 0, | Y& (a, y, u, c’) = 0, each equation containing an arbitrary constant. If we differentiate these, we have dd dp | oO) dy dw dis d a tt a, yy shy izp aU ee pt cia 7, tu 0, dx C dy ° d from which four equations we may eliminate c and c’, leaving two equa- tions between , y, u, and their differentials, or when more convenient, the diff. co. of any two with respect to the third. We may also in the same way obtain singular solutions, satisfying the diff. equ. by substi- tuting in the equations the values of c and c’ in terms of a, y, and u, it el) ee = 0. dec dc! equations contain both c and c’, except with regard to the singular solutions, which we shall have to consider hereafter. And the diff. equ. may be obtained directly (as in page 184), by explicitly obtaining c and ce’ from @=0 %=0. Let these give c= (2, y, u,), cl = V(2,y, u,) from which we obtain diff. equ. of the form Mdz + Ndy + Pdu =0 M'dx + N’dy + P’du =0, derived from All this will be also true when both where M, N, P, do not contain c or c’, and are either partial diff. co. of ®, or diff. co. stripped of a common factor. And the same of M’, N’, P’, and’. But we are not to conclude that these will always be the diff. equ. presented by a problem of which the result is that¢ =O w=0. For if we multiply the second by V and W successively, and add the results to the first, we have (M + M'V) dz +(N+N’V) dy+ (P+P/V) du = 0, (M +M’W)da+(N+N’W)dy+(P+P/W)du = 0, | the truth of which implies, and is implied in, the truth of the first pair. ' And these, with some particular form of V or W, may be the conditions at which we arrive. But now suppose we require, not that the preceding equations should be both true, but that w, v, and y, should be connected in such a way, that either of them will be true when the other is true; that 18, either is to be a necessary consequence of the other. Supposing the equations to be so combined, if necessary, that the restoration of a factor shall make the first side of each a complete differential (say the first of ® and the second of ¥), then our requisite condition is this, that db shall = 0, when d¥ = 0. This will be true if such an equation as dé —Vd¥ exist, that is, if @ be made a function of Y. Hence, we have this 202 DIFFERENTIAL AND INTEGRAL CALCULUS. TuroreM. If the diff. equ. of @ (u, 2, y, c, c!) = 0, and w(u, a, y, c, c’) = 0 may be so connected that either shall follow from the other, | then ® and © being the values of c and c’ deduced from = 0, % = 0, we must have ®=/f(‘¥): and conversely, (it may be shown from | do=f'¥ d¥) that © = f/(¥) makes either of the diff. equ. deduced from = 0 | = 0, follow from the other. Though we have shown that Mdr + Ndy + Pdu =0 is incongruous, — except only in the case where N M du = — Dp dx — P dy is a complete differential ; yet two such equations existing together, have meaning and rational | results. For by eliminating du we obtain a relation between dz and dy, which implies that y is a particular function of x; as also appears by eliminating w between the primitives 6=0 y% = 0. This isa sufficient | sketch of the theory of stmultaneous diff. equ. for our present purpose. What function of x and y is u, so as to fulfil the condition ar A Peete ek ee .Q) where each of X, Y, and U, is a given function of the three variables x, y,and uw, all or either. To begin with a particular case, let us take du du r : ee +y aio Now w being a function of w and y, gives @ ly l du . dit it du = Bre dz + a dy (for all functions) = Big Bh ae “u—wv—— | dy dx dy dx y (for the case in question). du That is, ydu — udy = = (yde — rdy). ... » (2). This immediately shows us that w must be of such a kind that ydu = udy = 0 follows from yda — xdy =0: of which the first gives «= cy, the second y = c’x. Hence,in the theorem preceding, ¢ or ®, or u4— U] ] tor) > e Y; must be a function of c’ or Y, or y +a, and therefore ¢ du du dew: + Y dy wa = uw have a solution, its form is u= y r(%) : The next question is, will any form of f be a solution, or does this | require any particular forms, and what? ‘To try this: observe that (2) may be immediately reduced to u a? du (*) AG y u y \ d ° rik = aie ane © ah 7 oe a d = 3 f nO ke ie —_ — Cr a J = Fae): Fat eD= re then should — a = = ed (“) or du ee y" re) : dx x which, w being y f(y +2), is true for every form of f. We now pro- ceed to the general case. ON DIFFERENTIAL EQUATIONS. : Ii In the value of du substitute ta from (1), which gives ay 1 Ydu — Udy = = (Yda — Xdy). vonsequently, w must be such, that Xdu—Udy — 0, and Ydx— Xdy=0 shall follow one from the other. If then at primitives can be found, ind the two constants deduced in terms of a, y, and w, the value of z must be among those derived from making thé "expression for one of the sonstants a function of that for the other. It only remains to show that che one may be any function of the other. Let c= and c’= ¥W be the values of the constants above mentioned ; whence @= /¥ is the form to be tried. We know that (by the manner in which ® and ¥ are obtained) do d® d® dy ay ay — du lr y=0, —du+ — dx + —dy=90, du aa da dy dy ae odo a8 dy Wie may be transformed into, and imply and are implied in Ydu— Udy=0 Ydx — Xdy = 0. If then we use the two last, and eliminate dy and dx from the two first, we produce (eliminating a quantity from equations which are the same in different forms), zdentica/ equations. These are a Oe wey 0, ee Ke VS SO du dx dy du da dy These results are necessary consequences of the manner in which ® and ¥ were obtained. Now I say, that the supposition of @=/f¥, makes these render the equation (1) true, whatever j may be. For, differentiating the last with respect to w and y separately, we find do du . d® : pods ditt av [I Ie here or aiaeals du dx | dx ate \du dx ax J Sola al O38 du dy — dy du dy * dy / Multiply the first by X, and the second by Y and add, remembering the preceding equations. We then have dd /., du du db H. (ae du du adv — Meer yar, fot Vow | eee du Ce ae 7) du Lie Ya ( dix dy du > fu) x xt +Y a oy \= i. d® du dd fly (= du av du du Axe dy Consequently, whatever f may be, we have either dD 4 AE _ du A Rat aa ae -aath 1 or X=—-+ Y=-— ja GQ; du * du dv dy of which we shall show that the first not only re quires a relation to exist identically b yetween ® and WY, but is even then only true of one form of f. Assume the first, then from equations (A), we have the following additional equations : 204 DIFFERENTIAL AND INTEGRAL CALCULUS. d® Weg do — =f’ —, . and Le ee dx dy ~— f'v gh : dy which three relations imply that © and f¥ are zdentically the same, or at least only differ in constants, or in quantities not containing either u, av, ory. Now ® and ¥ contain nothing arbitrary, being entirely determined when X, Y and U are given: the one therefore cannot be) made identically a function of the other; and even supposing that we had obtained a case, in which ® was a certain function of Y, the first could only be one definite function of the second; that is, f could not be, as was supposed, of any form whatever. Generally, therefore, b= f¥ gives equation (1). And we have thus obtained the most, general solution ; for if not, let the more general one be a (a, y, wu) =0,) which is such, that when we substitute values for 2 and y in terms of u, ©, and ¥, from ® = O(a, y,u), Y= WV (a, y, wu), we do not find u disappear also, but suppose we find x (wu, &, ¥)=0 giving O=f(¥, u)| instead of the former solution. ‘The equations (A) then require the ad-; dition of terms to the second sides arising from f containing x and y through u, which enters directly, as well as in YW: that is, terms of kaye! df d is the forn aya and i Gly The multiplication and addition then du dx du dy makes the final equation become (f’¥ meaning now the partial diff, af co. av db ay du du df [du du Lie ee 4 tess grils Pari Pee Pe pea al G EES. 7 ae (G a ia) (xe rat dy v) iaaacle Se 4¥5)s and this does not satisfy the equation (1); for the admission of that equation gives 0= —U Now, if U be finite, this gives! dus df : | EAE 0, the very equation which denotes that w does not enter where a it was supposed to enter: but if U=0 the preceding equation is then reduced to | he Ce = vo) (xq +S = 0. du du — du dy ) The first factor does not vanish, by reasoning similar to that already given. The second factor therefore vanishes, or the equation (1) is satisfied ; but our new supposition ® = f(%, w) still exists, as a solu- tion; has the equation really a more general solution when U= 0 than in other cases? If we return to the diff. equ. we find that U=0 (Y being finite) gives du = 0, Ydx — Xdy = 0, and one of the pmi- mitives must be w= c; that is, w itself is either ® or Y: be it either; still d= f (¥, ¥) or b= f(¥, ©) show, either directly or by deduction, that ® is a function of ¥. | Thus an arbitrary function is in partial diff. equ. what an arbitrary constant is in those which have only one independent variable, a neces- sary part of the most general solution of any one, however simple. We now give some examples :— 5 } if 7 —U. HereX =1 Y=0 andthe diff. equ. become Udy=0, ON DIFFERENTIAL EQUATIONS. 205 Kdy = 0, or y=c satisfies both. In fact, owing to only one variable cing differentiated, this is a common diff. equ., in which the other pos- ible variable is constant. The arbitrary function is one of y. du du 5 } 5—-=—. X=1 Y= -—1, U=0, and the equations are dx dy iu =0, dx + dy=0, the primitives of which arew=e x-+ y=, nd w= f(« + y) is the solution. (For the converse, see page 62.) du du , has da ost fone gives u=f (x — y) Aeon j ives u= f(ay — bx) + —. du dui... du du ——_. (ant — o — 2 2 Sigs sae —_— (4, LF re a gives w= f'(2® + 7?) ete a i 0 ives w= @ (2? — y”), 5. Let X, Y, and U, be severally a function of x only, of y only, and fwonly. Then the solution is the value of wu derived from ites dy dy dx rivet o+i($- =). du d 6. tig! gives u=as(2), rt 7. Explain the following assertion :—If f may be any function, then (P —Q) + P, and f(P —Q) + Q are the same zn form ; and so are s(e) mars) We have thus completed what it is necessary the student should now on equations of the first order (of differentiation), and of the first egree (as to powers or products of diff. co.) both for two variables (one dependent) and three variables (two independent). With regard to 1ose of the second order, we have already integrated (in page 154, &c.) yfar the most important of those which occur in practice. Those of a igher degree than the first are not of primary utility. Without making ther application than is necessary for elucidation, we shall content arselves in this chapter with pointing out the most important general Insiderations connected with them. Let there be an equation y= (x, C,Co,. . » ) Containing 7 ar- itrary constants; three will be sufficient for our purpose. We may 1en form different diff. equ. of the first order, according as we eli- imate one or another constant. From any one of these we may elimi- ate a second constant, and thus we shall have equations of the second ‘der with only 7 — 2 constants in each. Proceeding in this way, we lay by means of the primitive equation, and the m equations imme- lately deduced by n differentiations, eliminate all the constants, and we 1all thus have an equation of the mth order contaiming no arbitrary mstants. For instance, suppose y= ¢,2* + 2° +c¢,x° (A) whose ifferentiated equation is y’ = 4c, 2° + 3c,a* + 2csx, from which 206 DIFFERENTIAL AND INTEGRAL CALCULUS. . . . . , abe } Eliminate c, giving 4y— ay = CaP ir Btu es cp es Cotten BY re wy Se om Crete Grete ec B, o «6 e C3 eve? Qy — xy’ = — 2c, 2° — Go kiss ci taee The differentiated equation of the first is By! — xy! = 3c, 2* + 4c, 2, | from which, and from either B,, Bz, or B;, another constant may b eliminated. Proceed in this way, and show that the first equation in which al the constants are eliminated, is gy! — 62° y" +- 182xy! — 24y= 0, which equation has (A) for its complete primitive. It might be sup posed that there are 12 equations of the second order, namely (denotin; by BJ’ the differentiated equation of B, , &c.), two from each of the fol lowing pairs, according as one or the other constant is eliminated B, BY) B, B’,, B, B’;, and one from each of the six other pairs B, B’,, B,.B’,, & But four of these twelve contain c, only, and are identical, and th same of c,andc;. However an equation containing c, only may arisé it must be, with one order of processes or another, the result of elimi nating c, and c, between A and its differentiated equations A’ and Al Hence there are m ways (supposing 7 constants) in which one constan can be omitted, or m diff. equ. of the first order; 5(2 —1) waysi which two can be omitted, giving as many of the second order; an finally, one only in which all can be omitted, or one of the mth ordey Thus, in one equation with 4 constants, there are 4 equ. of the fir order, 6 of the second, 4 of the third, and 1 of the fourth. Hence, 2 is the least number of constants which an equation of th mth order can have in its complete primitive, and also the greatest. Thi last point is one of which a complete and final proof cannot easily b given ; we shall therefore (here at least) content ourselves with remark ing, that as our only method of reducing an equation to the next lowe order is common integration, which introduces one constant only é' each step, we know that a primitive with » constants, independent (| each other, is the most general which we have the means of finding We shall now proceed to consider the general properties of the expres sion dl” d"~*4 da Pa ni Praga te tio + Py+Q= V, where P,, P,-,, &c. are any given functions of 2 and y, and V=0° the general diff. equ. of the nth order and first degree. If for y w substitute any given function of wv, then V becomes a given function ‘ x, and is integrable, or supposed to be so: we shall hereafter show thé approximate integration, at least, is always possible. But there may | cases in which this function is what is called integrable per se, that i whatever function y may be of x; that for example, in which Q + P; is such, has been already investigated. But what we have at present show is this, that excepting only in the case last instanced, or in that ‘ Q+ Py +Piy’, the preceding function cannot have arisen fror direct differentiation. Nothing more is necessary to show this tha actual differentiation of a function of g andy. Let the function be t and let U', U,, U”, U’,, U,, &c. be its partial diff. co, with resper i ON DIFFERENTIAL EQUATIONS. 207 jto wand y. We have then, 7’, y", &c., being the diff. co. of y, the following results for the diff. co. of U, considered as a function of Ee both directly and through y. Ist diff. co., =U'4U,7/ 2nd diff. co. = U"42U! y'+U,y2+U,y", '3rd diff. co., =U" 80 "y+ 3U,/y? + Uy, y42U" y+ 2U y'y!'+ (U/+ Uy ys Uy". It appears then that the mth diff. co. of wu, thus obtained, contains not only y’ y”, &., but powers and products of them: so that V cannot be such an nth diff. co. when P,, &c. are simple functions of x and Yy. ‘The only exception is the first diff. co., since Q + P,y + P,y’ may be identical with U’ + U,y’. But if we are at liberty to suppose P, , &C., functions of a, y, y', y'’, &c., then V may, in particular cases, be an exact diff. co. independent of any specific connexion between y and a. We shall proceed to ascertain when this is possible. _ By integrating Oh Vdzx by parts, we can now attain the condition (for there is only one, as will be found) under which this operation ean be performed independently of specific connexion between y and 2. Let us take the general term d”y rte Re Nitec’ d™*y fi eae m P,,— da which is | P,,d.———* or P,, ——? — u | { er tase, va it mG per Ot En i] dart Um - r | Any Acn"'y) Amy For, Av being constant, Age ot Siena pa =) . ] Write dP,,, in the form "dn, the diff, co. being total (throughout C this process, y is an implied function of #) and continue the process, which gives im m—1 sca m—1 [e.g a=? e f(a da da” Wi hee de dz" gery _ dP Caieted 7 {< Pde ty ts Pacem dz da"-" dx dx™-? it. p ay " dP» d™— Fay dP, any i dP. defy azn dx dz” dx® dx*=4 da* dz™* : Bt d™'y ar. ” sea &P., dy + ad"? udz me de dr dx"? FT eo RO Os AE ene " "p dy q p Gils Oks Gif 0 Fs : (*d? Ps ute oy | dg 3, dxt dx da’ da* J | deo Substitute these several terms, up to m=, in | [ Vdzx, and we have : “dP dy dP, d®?P [Vdx= [Qdx+ i Poydx = 5 By | 7g yet P, ae Beant Tal 208 DIFFERENTIAL AND INTEGRAL CALCULUS. P ey ED sp iy ie bac eine ‘y— | = ydx + &e. * dx? dx dx dx dx" ( LP Oe ates eee 2 a ass Sek Sy \ ee eee ee a eeee d 5 ik ‘te v u(P dx | dr dx® ai = ar ; dP dP aby Meodge wh dy dP, | d"-7ya 4 (2 Sie ae Eat) L(Po oe a dx ac” dx d? dP dP, Pog i a(R eooee + aE ) + seve a (Ey. Agate rade But the integral in the first line is not attainable without specific connexion between zw and y, unless we suppose that Q, y, Po, &c., ane so connected that the multiplier of dx is a function of 2 only: let it be | xx, whence the following theorem, obtained by equating that multiplier to x2, and substituting the value of Q thus obtained (we leave out x2, because yx dx alone is evidently integrable; and if the whole be inte- grable, and one of its parts, so is the remainder). The expression n, n—1 2 nD peed epee cee sed tay Ps os tt Sl dx dx dx dx? da TG is integrable per se; and its integral is +...+P, a’? dP, \d"~*y d'- dP, | P u +(Pa- : + (Pr) Lt aty (Peo 7 to) ! “da de fda dae dy dP, dy dy dP, “PR ixamples: P,— — , and P, = — ——— MEADDPISE later ee ae anges Pecos Bi in 9 \ i are integrable ; the first we know well already ; the integral of the second | is pw + (2, -- i) y, Which may easily be verified. dx dx y f These are the conditions upon which one integration is possible; we might apply the same method to ascertain those upon which a second integration is possible; and so on up to m integrations; but as this would not be useful, we shall merely give the results of one case as an exercise for the student. What are the conditions which make d*y d’y dy : eek, a8 + P, i’ +P, rn +P.y ... (A) completely integrable ? iP d?P, he That first integ". may be possible P) = put Na Sse | P, dx dx? dz . ‘% \ di ee First integral is .... P piled i (2. cai dP dy - (p, by aR ay d =) * dx? i dy ras dx dx Condition of 2nd integration, dP d?P dP dP 1*P, d?P [2 icrang aes eek) Taka a SE alc oP, =o dx dz? dx dr? dx d d.x* ON DIFFERENTIAL EQUATIONS, : d Second integral P, rc eps adP, dP; dx dx Cond". of 3rd integ. =P, ~ obs GBs — 9 2P as, ¢ Third and last integral P.y. | Show, from the conditions, that dy dP, d?y dP, dy ee P ee eS a —- et a As * dx? dx dx? dx2 dx Bi dy? J The student should attend importance in the Calculus of Suppose now that V, instead particularly to this process, as it is of Variations, to which we shall come, of being integrable one step per se, is not so because it has lost a factor, as might have happened if V = 0 be an equation given. We shall confine ourselves to the second order of diff. equ. Let M be the factor 3 consequently, d? l : MP,—* + MP, “ +MP,y is integrable, and MP, = eo A From this last, if M can be found, we can integrate V=O one step. But this is itself a dif. equ. of the same degree as V0, and we there- fore appear to have only reproduced the difficulty in another form. Nor have we done more relatively to the order of the diff. equ. ; but at the Same time observe that all that is necessary to M being a factor fit for our purpose is that the last equation shall be satisfied. We do not want its general solution, or even a solution with an arbitrary constant ; any solution will do. For the preceding process makes it evident that the mere existence of the condition, arise how it may, is sufficient to destroy, or to render a function of x only, the indeterminate integral part of f Vader. We have then made a particular solution of one diff. equ. the only condition necessary for a step towards the general solution ofanother. For instance, I propose the equation d i : F hi = _ 2a + 2y =— PSs P,=— Qn, P, = 2. d(—2x«M (2M) , Let M be the factor; then 2M = go 8a) aE ee. pa ee dx dx vhj F 2M. dM We aia vhich may be reduced to Oo + 6x qe + 6M Now su hat M Chen Ne et a, epee Dee eso a a integrable ; it gives eee y ppose by trial, or other means, we arrive at the knowledge = 12° will satisfy the last, which it will be found to do. Consequently, page 195, 210 DIFFERENTIAL AND INTEGRAL CALCULUS. 2 os ee y= file fe i dx + | =—cr4+c2’, which is the complete integral of the given equation. This method can only be applied with success to cases in which Py, P1, &e, are all functions of v. Let the student apply it to es Py Q, and show that the factor which makes the first side v integrable, is log~ ‘ Pdzx), whence let him deduce the solution which was obtained by a particular artifice in page 195. When P,, &c., are all constants, the equation d” qr da as teal Og ot be Hy Et ay = XP. of x), dx” dx av admits of complete integration. We shall take the third degree as a case. Let M be the factor which makes the first side integrable; then, taking the equation of the third degree, the condition for determining M is ; dM a’M A aM — a, Tp +. Che SFE = sz — dx* dx* A particular solution is readily found. Assume M = «7; then we have ee(q, + dk + a, h® + as k*) = 0, which: is satisfied if & be either of the roots of a tak + &. =O, Tock. Bycvite >. Res abe..these three roots; use them one after the other, and we determine the three primitives of the second order belonging to the given equation, as follows (multiplying both sides by ¢-™, inte erating by the formula, and then dividing both sides by €~“*) ; ad di ; eo ils a +. (da+ eh) oak (ay + dah + ask") y=! Sf Xe dx , Py dy Pr yee Og a? + (dat Asko) eo (a, + ekg + asks?) y= ee” JS Xe~“e* dv, da di As ae + (d+ Asks) =A 1 (a, + dks + dgks?) y= e"s" af Xer*s" gee It is unnecessary to integrate further ; for the elimination of y/ and y" between these three equations will give y in terms of the three explicit integrals, each of which contains an arbitrary constant. To perform this elimination, determine ), p, and v, from M+p+tv=0, Rrk+thptkhyv = 9, which are satisfied by N= hy—hg, p=hs—hy, v= h— kes Multiply by A, p, v, and add, make Ak, pk? + vies = K ; then Ne*i* ‘ : : ekot ckgt 3 a3y = K [xetar -+- is K [xehart al e—*s* dz. ON DIFFERENTIAL EQUATIONS. 211 If X=0 the integrals are arbitrary constants, and we have, writing C1, C2, C3, for the complicated coefficients, which are in reality arbi- trary and constant, y= cet” cies" 4 C6887), If two of the roots be equal, say k, = k., then »y = 0, and one of the preceding terms disappears, whence the solution not having three arbi- trary constants, is not complete. In this case two of the three primi- tives of the second order are identical, so that having only two distinct equations, we can only eliminate y”; do this from the second and third, giving di a(t, — Is) + (ky—ks) { dy + as(hy + he) Lysate ve Xem"s"d x — es” Sf Xe-"s*dx But a, + a, (ko +h,) = —azk,, in all cases, by the theory of equations ; : d. or the first side of the preceding becomes a@, (k,— kg) (4 —hy) ; the factor which renders this integrable is e~4:; multiply by this, and antegrate, which gives (since k, = k,), as(ky—h,)ye—*s* = fdz | /Xe*do} —f| dre"s-')* [ Xemkee de} ’ which, involving four integrations, may seem to introduce four arbitrary constants; but this is only in appearance. For the second side of the preceding differentiated twice successively, gives [Xe-* dx — e%s-')* [Xe dy and (hy—ky) e%s- 2" [XerKs* dz, whence aye = f ' du f (dx.e%s-* [Kerk dx )} ; in which there are three integrations only. (lt is always possible to make a single integration appear two or more; thus SPQdr = P/Qdx — f{Z few har ). When X=0, the first integration gives a-constant, say c; the second gives | Cc | (kg—kg)x / d finallv a kar — g(ka—ko)x +cr+tel esTl2* +. ¢, and finally a,yé 5) Th ne (hs — ha) or y = Cerse 4+. (C! x + C”) ee? When all three roots are equal, the three primitives of the second order become identical ; and we should then integrate the primitive of the second order twice successively. But the form to which we have reduced the case of two equal roots does not lose a constant when k,=h,, and gives (with three integrations), / being the root, ayy .e-* = f{dx f (da f[Xe“dz)}, ff 212 DIFFERENTIAL AND INTEGRAL CALCULUS. when X = 0 y == (C2? + Cr + CC") &*. The most important case is that of the second order, or d?y di 3 ay - +ay=X, Ag dx? and proceeding as before, we find that the factor is either e~* , or €72%, h, and k, being the roots of dy hk? + a,k 4+ ad) == 0: the two primitives of the first order are da ‘ Pe ae oe ay = 4+- (a, + aki) y= ett [Xabi dr, d as ~ + (di + aah.) y = est [Xe da, iving.. Ge(Ri—Re) ye et f KeT's* dx — eh” Kevttdas AAI g 8 y If both roots be = k, the integration of either of the first pair gives (remembering that ak + a= — a,k, and that the first side becomes L ay @ -ho) , of which the factor is ¢~*’) aye = fdalfXe"dr}. -. «+ @B). when X = 0, yocen* + cee, or (Cc, + Gr), according as the roots are unequal or equal. But let us suppose in (A), that k, is a variable which approaches to k, as a limit, in which : 0 . case the value of y in (A) approaches the form 5: Differentiate both numerator and denominator with respect to k,, remembering that (2 a. l POP . and k, being independent) —— /Pdz ee dx, and the value of | € dk, ty : 1 a,y will be («ince + (k, — kz) =) - dk, de*i ke dei : ds y= Th [xe dxa+eh* fs apres | x of Xe" dxr- Yi Xne*day Ri dk, To which (B) is immediately reduced by parts. If the two roots be impossible, we have (ki = a+ pV—1 hy a — BN—1), ett [Xe drs" (cos Bx+ 4-1 sin Bx) { {cos Ba-NW —Isin Sr }e~ Xe | a" (Xe edna e**(cos Be-WV —1 sin Bx) f {cos bat -1 sin Brie" Xdu 26V lay = Qe*,/ -1sin buf Xen cos Brdx — 22° —1 cos Bx of Xe- sin Prdx aye ** =sin Px fXe~* cos fv dx —cos pu f Xe-* sin Bx dv. If’ = 0, we have the case already considered in page 155. ON DIFFERENTIAL EQUATIONS. 213 | The following theorem is the synthetical construction of the solution _ of such equations: If y be multiplied by <4”, and the product dif- | ferentiated ; the result multiplied by e@2—")# ang the product differen- _ tiated; the result multiplied by e%3—')* ang differentiated, and so on | up to multiplication by e¢e-"»-»* and differentiation : and if the result | be then divided by en—'n-)?s the final result will be d"y a y da dgt tft eit OS Be + ay + aay, Where d,_,—h, +k, +... Ons = Rhy he +h, hy +.... &, , Qh leatak ne® We now come to equations of higher degrees than the first. be sufficient here to consider dy (WN (AYN ee (34) +P 7) tf) 4R=0. i Oy where P, Q, It will and R are functions of x and y. This equation gives three | da : distinct forms for , answering to its roots, considering it as of the third degree: let them be d d S ae ce = As BE a As (Aj, As, Aj, f* of x and y). If we can find the primitive of either of these three, we have a solution of the equation. Let the primitives of these be V,=0, V,=0, and V;=0; either of these then satisfies (1); but no others satisfy ™ V.V;—-0: consequently, let V,, Vz, and V;, be combined by mul- liplication, and let y be deduced from the product. This value of y will contain three arbitrary constants, contrary to what is proved in page 184. But it must be remembered that in what we have just said we have tacitly extended our meaning of the term differential equation beyond what was allowed in the page just cited. The equation (1) gives a ; di choice of three forms for e » and may be written dy dy ; dy vg 9 (Gia) (r= A) (haa. =0': lal, See (2). {nd V, V, V,=0 gives a choice of three primitives. If we choose V,=0, ef di ; ¥€ satisfy (2) by means of the factor = — A,;=0, which follows from dx Vi=0. But y as obtained from V, V, V; = 0 being differentiated, md ¢, (one constant) being eliminated, will the result be the equation Pe To try this, suppose the three primitives to be written ¢ = W,, 2=W,, c,= W,, when (ec, — W,), (co — We) (G,— W;) =0 1s he complete primitive, as far as we have yet gone. Differentiate his, and we have 214 DIFFERENTIAL AND INTEGRAL CALCULUS. d.W. dx d.W, (co— W2) (6, = Wire 9 (c,— Ws) (¢,— W,) d.W, + (c,—W,) (¢s— We) Ge Se 0. Eliminate c, from the original, which can only be done by making c, = W,, and the preceding is reduced to d.W, dx which is not the diff. equ. (1) or (2), but has a factor in common with it, so that both are satisfied together by c, = Wi. For by supposition WE oes W;,) (cs —W;) mort BF . di c, = W, and V, = 0 are simultaneous, and the latter gives = -A,=0. But if we make c, = ¢,== ¢ 80 as to have only one arbitrary constant, the elimination of c, will lead to the equation (2). Suppose (to give & more simple example) we take the form (1) but of the second degree, everything remaining as before, except the suppression of A3, Vs, &e. Then (c — W,) (c —W.) = 0 gives d.W, . d.W, d.W. d.W. c?—(W,+ W,)c Ws Wats ( dz “ie da ye W, is +W, i ton d.W, d.W; Eliminate c; then (W, — Ws2)? bar ae = 0 . (3) But d.W, dW, dW, dy dy d.W iy — Pay: “dy de and Pi s Ai=e follows from det = 0, whence ise ined Weg dW, d.W,_ dW,/dy i at Se _ . ..dW,dw./dy dy | w—w.y? — —|— - — — er Suh 2) dy dy es af VG .) Me which is the primitive diff. equ. affected only by factors not containing d Hi Rey ie ; — Hence the real primitive, in the sense used in page 184, is the: product of all the primitives with the same arbitrary constant in all, dy? d For example, let ie —(a+ ae 4.ac—=0, which is satisfied either d da eee by Sas or ~ =, the primitives of which are y —av— c=0, and y—40°—c=0, and y? — (axt 4a°+ 2c)y+ (ar+c) (42?-+¢c)=0 ) is the complete primitive. The student must here remark a distinction which has ‘no specific name, but is of considerable importance. The ambiguity which exist ON DIFFERENTIAL EQUATIONS. 215 9) | in algebraic expressions arising from the occurrence of the radical sign, has two characters, 1, when the root in question can be extracted in a more simple algebraic form ; 2, when the root cannot be so extracted. An example drawn from geometry will do better than anything else to illustrate the difference. Let y=V and y=W be the equations of two curves, V and W being functions of x, Let it be required to find an equation to both curves in one; or (a, y) =0 is to be satisfied when z and y are co-ordinates of a point in either curve. This may be repre- sented by means. of the ambiguity of P+ Q?; let P+V7Q =V, and. pP-VQ=w, and we haye - 1 P=3(V+W) - Q=i(V—W)?_ y=3(V + W)4+4(V?-2VW4+W?, which is either V or W, according as we take one sign or the other for the square root. Thus, under the appearance of an ambiguous Single form, y may have either of two pertectly distinct forms. But if may 1 re we now consider y=a+wx*, we have two varieties y=atNz, and y=a—Nz, belonging not to two different curves, but to two different branches of the same curve ; where by the same curve we mean the same to common perceptions. We can geta circle and a parabola into one equation of the first kind, but y=a+ Je and y=a —/«x belong to two different branches of the same parabola. Thus the equation 1 pies eres y=(2?-+c)® exhibits an hyperbola, or +7z?-+c and —V224+¢ are ordinates of different branches. But let ¢ become =0, and we have 1 y == (a°)* ; that is, y= +2 or y= —z2, and these two branches together form two straight lines. It is true that this system of two straight lines is an hyperbola, according to every definition that can be given of that curve: but it is equally true that this is an extreme case of the hyperbola, which presents a peculiarity of its own ; namely, that for this single case, the hyperbola degenerates, as is sometimes said, into two other lines which, both together possessing the properties of an hyperbola, are yet each complete in itself. The last diff. equ. we took was one which belongs either to a straight line or a parabola; but let us now consider one which cannot rationally : : dy? be resolved into factors, say ( =) =y. We have then either d a ie ee — | — vy or —==— Vy and vy =tr+c or — Vy=h4e+<, the complete primitive is (Qe+c—Wy) (dat c+Vy)=0 or y=(Bet+c)’, the equation of one parabola, each factor being that of one branch. We shall now proceed to applications of the differential calculus Which are valuable in themselves, as well as for illustration of prin- Ciples. We have before us the fields of aleebra, geometry, and me- chanics, which we shall take in the order in which they are mentioned, placing a chapter of examples on the subjects of all the preceding chap- ters between those on algebra afid mechanics. 216 CHAPTER XII. FURTHER APPLICATION TO ALGEBRA. A runcrtion of two variables may have a maximum or a minimum ; that is, it may be possible to assign a=a, y=6, so that p(ath, b+k) shall be always greater or always less than f(a, b), or become perma- nently so from certain values of h and k to anything short of h=0 k= ~The law by which these values are to be determined is obtained as fol- lows: such an absolute maximum or minimum remains if we suppose ¥ any function of x, subject to the single condition of that function being — when v=a. For if all species of values of h and & satisfy any condition, so do those which arise from supposing k=a(a+h)—aa; and conversely, k may be made = any given quantity, @ and 4 being given, by choosing a proper form for «. ‘Thence O(2, ax) is to be made a maximum or minimum, whatever may be the form of « ; that is d.¢(2, y) ee dp yi? Y Facecky 5 CUTE aS (OF et oe changes sign, whatever o/ may be (page 132), in passing from xn=a-h toa+h; and this, however small # may be. That there may bea maximum this change must be from + to —, or the last function must be decreasing ; for a minimum, it must be increasing ; or, for a a al rp , 1 ae dp _,, (must be = aa dady” i dy? iain dy 2 mnust be + We shall confine ourselves here to those maxima or minima which arise when ¢/+6,.0c/r=0, (it must be either O or « ), and since this must be true independently of a'r, we must have ¢'=0 $,=0. Making ¢)==0 in the last, which is thereby reduced to bp! +2) a+, (ox), we know that this cannot be always of one sign whatever ax may be, unless the values it would give to e/«, when equated to nothing, are impossible or equal; that is, unless #’,, be not less than (¢/)*. In this case @” and ¢, must have the same sign, and this sign determines that of the expression. Consequently, for a minimum determine all the values of 2 and y which give idee TU te dx ay aN ; ; : dp dd dd 2 then f h — —_ —- (| —— itive si en for any pair which give da? dy? ule a positive sign, : : : P ; (a, y) is amax. or amin. according re and 7 are — or +. We also exclude the possible case in which #”, d/,, and ¢, vanish with ¢! and ¢,. Example. (a, y=e?+y'—ay —32, P=2x—-y—3, PH 2y—-% oo, Pi, =2, pee et bb, >(¢/)’; ¢'=0 and¢,—0 give v=2, y=l. Consequently ¢ is a minimum (=—3) when ¢=2, y=1. FURTHER APPLICATION TO ALGEBRA. 217 We have introduced this method here as subservient to the demonstra- ition of an important theorem in algebra ; namely, that every function of z, whose diff. co. cannot become infinite for any finite value of z, can ee aes ibe made =0 by giving x a value of the form a+b As 1, where a and 3 lare possible quantities, positive, nothing, or negative, finite or in- finite. The assumption made with regard to impossible quantities is, ols the processes of differentiation may be applied to functions con- taining them, and all general conclusions applied to them. This being premised, expand f(v+y/—1) and S (a—y f~1) by Taylor’s theorem, which gives ak ri) : i : f(@ety¥—1)=P4+QV=1 P= fax—f"'x a tits Z q7 ke. 6(~—-yV—1)=P—QV/-] ans mln eee, ev (ay ) Q Q=/"r.y a ae are &¢ | a dP_dQ dP dQ ; Whence we find that dr dy’ dy ae CA)3 dP dQ. dP as. dP en; TQ. dx’ dedy — dy®” dx® dz dy* dy? whence P’’P,, —(P’)? and Q’Q,— (Q’)? are necessarily negative; that 3, Pand Q are of a class of functions which cannot have absolute axima or minima. Turorem. If P andQ be real functions of x and y of the form just iven, and if f’z can never be infinite for any finite value of z, then *+Q* cannot have any minimum value unless there be simultaneous alues of w and y, which make-P=0, Q=0. Firstly, since /’z can never be infinite, and since ‘ —~, dP dQ ;— : rare OP aQ ;—. ety —1)=— + V—1, f(@ yN T= aI either can P’ or Q’ become infinite; for such @ supposition would rake Oe tyN—1)+f" (e—yN =I) or 4g (w+yN—1)—/! (w—yN—}) ine or both infinite, which cannot be. Next, if P?+ Q*® be a maximum ¢minimum, it must be when 2 and y are such that (for their particular alues) d a qQ_6 z gB 0 a a dx ot Wy f dy Sa Os gates aL Je Now, if Pand Q be neither of them =0, these equations will give ae dQ — i qQ ==0, the brackets denoting that we do not dx dy dy dx ssert this of all values, but only of those in which for x and y haya 218 DIFFERENTIAL AND INTEGRAL CALCULUS. been substituted the particular values which satisfy (B). But equa- tions (A), true for all values, show that the last is equivalent to dP? . (dP? ane": Th (=) +(&) ==0, which requires artes ih dQ dQ _ and also from (A), do ==, cH — iP dP l IG If Q be =0 and P be finite, we have wo =; —=0 aQ 9 (0 a dx dY..a\ «Ab yict on from (A) and (B), and, similarly, if P—O and Q be finite. Finally, if P=0 and Q=0, the equations are thereby satisfied. Let P?4+Q’=u; form wv", uw, and u,, we have ce? d@ _@P..@&Q = eee Hua does Bh Biel ae Siaee Me? (Gat ax* a ae? dx? dP dP , dQ dQ dP PQ L2G) ypotae pantalones ast (S a dy dx nm dx dy ° as dy ee Hence in all the preceding cases, except where P=0, Q=0 (since P’=0, &c.), the condition of the minimum requires that d’P PQ arp d’Q ak P?QY (P ae Q ap) (P ape Q i) -(P dx ae @ dx dy should be positive or nothing, for the values of x and y in question. But, using P”, &c., for abbreviation, this is Pp? (P'P,, —P*) + Q? (Q’ Q,—Q®) + PQ (PQ, + P,Q" —2P Q) the first two terms of which are necessarily negative, and the last vanishes, for, from (A), P'Q, +P, Q’= P,Q a P/ Q/. Therefore there cannot be a minimum, unless there be one when’ P20, QU: If we suppose P=0, Q=0, and if P’, &c., be finite, then ull, — w= 4(PP+-Q”) (PP+Q')—4(PP, + Q)'=4(P’ Q,— P/Q; and is necessarily finite and positive, being 4(P”+Q”)* Now, since P?-+ Q’ is always positive, there must be some one value: which is less than any other whatsoever, or a number of equal values: which are each less than any other whatsoever. And with regard to these equal values, they must either be separated by finite intervals, in which case each is a real minimum, or there must be such a relation possible between 4 and k& in b(x+h,y+k), where $ (2, y)=P?+Q%, as will by taking h and & accordingly give $ (%-+ h, y,+k)=const.; where a, and y, are values which give $ (a, yi)= the same constant. That is, writing « and av for 2, +h and y+ which is determined by it, there is some function which gives $ (a, ax) =const. In this case d 1 1 = age PE +QF ale=0, ; db. , dP ra Calas Osa dy : FURTHER APPLICATION TO ALGEBRA. 219 | But since the values included under @ (wv, ax) are less than any others, ‘it follows that every value of @(x,y) in which y=ax has the pro- | perties of a minimum for every change in a and y, except only that which imakes Ay=a (w+ Ar) — az. But if ¢’-+, 6’x must change sign for every form of x, except only | Px=ax, we must have d’+¢,6/7=0 independently of 6x, or ¢’=0, ¢,=0 'for these values; and the other conditions of a minimum must hold. | Hence by the same reasoning as before, P=0, Q=0, are the necessary ‘conditions of this case also. Buta minimum or a collection of consecu- tive minima there must be, which there can only be when P=0, Q=0; | consequently P and Q can be made equal to nothing for some possible values of w and y. Hence P4+QV—1 or f(et+tyv =1), and B_—Q/—1 or f(a—y 1) can both be made =O by the same possible values of x and y. From hence it follows that every algebraical equation of the form Ay2"*+A,2""'+.... +A,12+A,=0, (1 a whole number,) has 7 roots, either possible, of the form z=a, or impossible of the form 2=a+bV—1. The common proof of this, granting that every equa- tion has one root, we presume to be familiar to the student. Supposing 1, T:....7, to be the roots of the preceding, it is then the same as A, (2—7,) (2—72)....(2—17,). If two of these roots be equal, say ™=rz, then 7, is also a root of the diff. co. of the preceding with respect to 2, for that diff. co. has either z—7, or z—r, in every term. If dx be an integral and rational function of w, of the form A,a"+- A, x"~'+ &c., and if its diff. co. 6/x be made a divisor, and the common process be followed for finding the highest rational divisor, we have a Series equations of the following form: remembering that the remainder Is always one degree at least lower than the divisor, so that we must at last come to a remainder which is not a function of x, but of Ay, A,, &c., only, if the expression have no equal roots. Let the quotients be Q,, Q., &c., and let the rth remainder be that which is constant. We have then a set of equations as follows: or=F'x.Qit+R, ¢7=R,Q,+R,, R= Rh, Q,+R,...-. 66. yg Q,+R, Now suppose the same process to be thus modified; let V, be the first remainder with its sign changed, with which proceed to the next ‘equation, and let V, be the next remainder with its sign changed, and so on. That is, suppose or= 9/2 .Q,—-Vi, Pe=V,Q,—V,, Viz Vale —V3°. eeeevee Veugs Wao Q, 12g Ve where Q:, Qs, &c., are the same as before, or differ only in sign. We Shall give the result of both processes, in the case of #3 —2®—47-+3= $2, o2°-27-4=¢'x. Observe that,in the same manner as in the common rule of algebra, we may multiply any dividend or divisor by any number or fraction, without affecting the sign of any subsequent quotient or remainder, or the conditions under which it is nothing. We omit the quotients as immaterial, DIFFERENTIAL AND INTEGRAL CALCULUS. Common Process. BSa’—2a—4) a®— a? -— 4x43 (x3) 34° — 32°—1244+9 32° —2a*?— 4x ——— U8} — 32°— 2474+ 27 —3277+ Qr+ 4 ben 28)32°—22—4 (x 26) "822—522— 104 "Sx? —69r ( x 26) l7zr— 104 4427-204 4427— 391 ep aoeeras! Signs of remainders changed. First remainder —262-+- 23 Sign changed 262 — 23 267-23) 32°—2r—4 Second remainder —2313 Sign changed 2313 dr= a— a—4r—3 b/a=3r°—Qr —4 Vi 267 —23 V.= 2313 V, and V., as written, are not the expressions which would satisfy the equations above, but multiples of them: this is of no consequence, as : our only concern is with the sign. Now the theorem* we. are going to prove is this; that in all cases, the number of real roots, if any, which lie between w=a and e=6 (greater + than a) can be determined as follows. Note the series of signs which | xa gives to the series px, $’x, Vj, Vo, &c., and compare it with the | series of signs which «=b gives to the same. Then the number of variations (from + to —.or — to +) which is found in the last falls short of the number of variations which is found in the first by the | number of real roots which lie between a and 6. But if no real roots are contained in those limits, the variations of sign are the same in number in hoth series. For instance, in the preceding, x=:2 gives to bx, b’a, V,, and V., the signs — + + + (one variation), and #=3 gives + +--+ + (mo variation). Consequently, there is one real root * This theorem was presented a few years ago to the Institute of Paris by M. Sturm, and is published in the Mem. des Savans Eirangers. It is the complete | theoretical solution of a difficulty upon which energies of every order have been em- ployed since the time of Des Cartes. A translation has been published by Mr. W. 4H. Spiller. John Souter, St. Paul’s Churchyard, 1835. ec FURTHER APPLICATION TO ALGEBRA, between 2 and 3. If we wish to know the tot we substitute for 7, ~@ and +a, both s | three first of the same signs as their first jthan @ shall have the same effect (the possibility of which is 2 common theorem of algebra). The signs will then be — rt for vo | and + + + + for t—=-+a. There are then three real roots. This theorem is demonstrated by showing that if we suppose 2 to increase from — o to +, through all magnitude negative and posi- tive, the series of signs of dz, d’r, V,, &c., will always lose a variation iwhen v passes through a, a root of px, and will never either lose or gain |a variation in any other case. We suppose there to be no equal roots of ip, so that dx and o’x cannot vanish together. (If there be equal roots, ithe equation may be cleared of the factors belonging to them by com- mon methods, and the remaining expression treated by this method.) lAnd no two consecutive ones of the set px, o'r, &c., can vanish together, for then the equations show that all which succeed would vanish, and ‘there would be equal roots, since the vay ushing of the last remainder ‘Gwhich is no function of x) shows a common factor in ox and g’r. | Firstly, let ¢a=0, and let ViV,...». be all finite, ‘Then however near a may be to a root of zx or V,, &c., atu may be taken so near to @ that all shall remain finite, and with the same sign. And (page 132) Pp(a+u)—d¢a has the sign of ¢/a, while ¢ (a —u)—ga has that of i-—¢’a. And da=0; whence ~(a+u) and ¢ (a@—u) have different Signs; that is, (the other signs all remaining the same, since w is taken 80 small that no root of d/z, Vi, &c., lies between a+-w and q— w,) the prder of signs for ¢ (a —u), &., is either —-++, &c. or +—, &., and shat for d (a+v) is ++, &. or — —, &c.: whence a variation zs lost when x, in its increase, passes through a root of px. _ Secondly, no change of sign can take place in any other part of the series except only where either @'2, or V,, or V,, &c., becomes nothing. Let V,=0 when t=h; then, as before observed, both V,_, and Viet ire finite. More than this, they have different signs; for V,_— Vi, Qi — Vis, from the hypothesis of formation, in which V,=0 requires eS —Vi4yi. Take w so small tbat no root of either of the last shall le between A-+-u and h—u; then whether V,, change from + to — or rom — to +, we see that the part of the series of signs arising from mn, Vz, V;_, is changed, when az passes through h, either from +—— to + + —, or from ++— to +—-~-, or from — —-t to —-+-+, or from —++4 to ~~ +; m all of which we see a variation md a permanence, so that no variation is then lost. Consequently the tumber of variations in the series of signs is neither increased nor liminished by any of the changes of sign of d’r, V,, &c., but all the flect produced is, to remove a variation from one part of the series to nother. Hence the theorem follows immediately ; for if da give n more ‘ariations than ¢ (a-- 6), there must have been n epochs between r=G nd z=a-+6, at which g~x=0. The number of impossible roots is termined by finding the number of possible roots, and subtracting that umber from the dimension of the highest power in ¢z. The following instances are from the Memoir cited (remember that ‘» &c., here given are multiples of their values in the system of equa- ons) : Pr= 1° —27~ 5, P'x=32°—2, V,=42+15, Vi-=—643, 221 al number of real roots, O great that they shall render the terms, and that anything greater a 222 DIFFERENTIAL AND INTEGRAL CALCULUS. There is one real and positive root. Pare att ila 10277 te All the roots real; two positive pla=S2* + 220 — 102 both between 3 and 4 Vi 8542 2751, Ve==441 The method of approximation to the roots of equations called after Newton is based upon, the theorem d(ath)=ha+h' (atih).h. Tf we have found m, which is nearly a root of an equation, and if the real root be a, let m=a+h, and we have d(m)=G' (m—(1—8) h).h. Tf h be small, we have 6m=’'m.h nearly; in which it must be observed that @/m must be considerable when compared. with gm; for if not, dm—-d'm, or h will not be small. We shall now proceed to the theory of series, and to the consideration of the conditions under which we may speak of an infinite series as | the subject of algebraical operations. The subject of their arithmetical consideration has been discussed in the Elementary Illustrations, (pages 8—10), in which will be found the development of the following asser- | tions. Derinition. The series @,+@.-+ 43+ &c. ad. inf. is said to be con- vergent (and by an arithmetical series we mean only a convergent series) when there is a limit L to which we continually approach by the addition of terms of the series; and this limit is called the sum of the series. TurorrM. The preceding series must be convergent if @,4:—-G approaches to a limit less than unity, when 7 is increased without limit: may be either convergent or divergent (that is, one series may be cons vergent and another divergent) when unity is the limit of the pre ceding; but must be divergent if that limit be greater than unity. THEOREM. ‘The series @ +4 0+ Ge B24-&C.. 0s 05 VE Onan have any finite limit A when 7 is increased without limit, must be convet- gent for all values of x lying between — (1--A) and +(1+A) ; may | be either convergent or divergent (in one series or another) when @: has either of these values; and must be divergent if x be numerically greater than (iA). And if @,41:-@n diminish without limit, the series must be convergent for every value of @, however great, while if On4i-& increase without limit the series cannot be convergent for- any value of 2, however small. In convergent series, we include those which begin divergently, but afterwards become convergent. Such, for instance, as the development of «*. Here the direction to form the (n+ 1)th term from the mth is: multiply the nth term by #, and divide it by n. Ifv==1000 the terms continually increase until x= 1000, and the 1001st term is the same as the 1000th @ but the term after the millionth is only the thou- sandth part of the millionth term, or at that part of the series the convergency is rapid. And since we are not now speaking of methods of summing series in practice, but only of the way in which we can satisfy ourselves as to the fact of there being or not beimg a finite limit, great or small, we do not weaken our reasoning by the supposition of a million of million of terms being divergent. For a million of million of finite quantities is a finite quantity ; and if all the remaining terms have @ limit to their sum; so has the whole series. When the terms of a series are alternately positive and negative there is certain convergency if they diminish without limit. For any even “> Ae FURTHER APPLICATION TO ALGEBRA. 223 number of terms of a= b-+c—e+.... must be less than double the number of terms of a—b+-b—c+c~ é+....+, which is either g—d or a@—c, or a@—e, &c., that is, less than a. Consequently, in the series @,—Q,+a,—a,+..., the remnant* Anti—An--.+. is less than d,,4,3 that is, diminishes without limit. But in the case where a,, Grit, &. approach a finite limit, and diminish, we cannot, by pure arithmetic, assign a finite limit. For instance, in 3—254+235—214 &c., the limit of the individual terms is 2, and counting from the first term we see that no subtraction is ever compensated by the next addition ; consequently, if there be a limit, it must not exceed 3. But counting from the second term we see that no addition is ever compensated by the next sub- traction ; so that, if there be a limit, it must be greater than 3—2H, Then between 4 and 3 lies the limit, if there be any, which is all we can now say. We cannot show by the preceding process that the remnants diminish without limit. By considering a series algebraically, we mean that we do not inquire for any arithmetical limit of the sum of the terms, but only treat the series as the result of applying rules of algebra to algebraical expres- sions, or formule. And though the algebraical consideration includes the arithmetical, yet the converse does not apply. All arithmetic is algebra, but all algebra is not arithmetic. For instance, suppose an algebraical problem gave as a result r=] +ax, an equatton which has its arithmetical cases, and its cases which are not arithmetical, the latter when a is >1. We proceed to solve this by the method of sue- cessive substitution, the principle of which is to suppose the required whole made up of parts, and to endeavour to find these parts suc- cessively by any steps which given relations point out. This notion of the-whole made up of parts is at first purely arithmetical ; and we proceed accordingly. If our process be such as if its own nature cannot have an end, we cannot thereby completely attain z And one of these two things will take place: either our method will give us Con- tinually smaller and smaller parts, whose sum converges towards a limit Which we can ascertain, and in this case we have arithmetically found the unknown quantity ; or we shall at last come upon a part (a supposed part) which more than completes the whole required, in which case the next process is not arithmetical. Our first notion would be that the next part should turn out to be negative, a result we should immediately comprehend. But it may happen that we choose a process which gives us continually greater and greater parts without end; are we then to conclude that the quantity sought is infinite? We shall immediately show that, sometimes at least, it indicates that the quantity sought is negative, and that we have proceeded to determine it as if it were posttive. Let e=1+ a2, and, presuming x positive, it must be >1; for it is I+ar. Takel as the first part; then 1+a 1 is still too small; for since wv is 1-+axz, then 1+a less than 2 is less than x For a similar reason 1+a(1l+qa) is too small, or 1+a+a?. Se, therefore, is I+a(1+a+a2) or 1+a+a’?+a*; that is to say, l+a+a?+.... is * The jerm remainder being constantly used in connexion with subtraction, and the word « rest,’ answering to the French reste, being of too general signification nm our language, I have borrowed this phrase to signify: what is left of a series, when a *ertain number of its leading terms is removed. Thus, in a—b--c—e-+-..., —~e+/—..,.&c, is the remnant after c - 224 DIFFERENTIAL AND INTEGRAL CALCULUS. always too small, however far we may go. Now if a4 this is intelligi- | ble; 1, 14, 14, 1%, 144, &c. &c., are all too small. The reason is evident; | the answer is r=2 (21-444 2), and our method is of a character which cannot terminate. But if a=2, then proceeding as before, 1, 1+2, 14244, 1+2+4+4+8, &c. &c., are all too small, or x is infinite. This result is wrong; the fact is, that z=—1, (—1=1+2x—]), and the fundamental supposition a>1 is incorrect. When, therefore, we write —1=14+24448+4164 &. ad infinitum, the student must not think we intend to assert any arithmetical equality, or other arithmetical resemblance or analogy of any sort or kind what- soever, between —1 and 1+2+&c. Every attempt to establish any idea of such connexion must end in utter confusion. But we mean this: we assert that 1+24+44+8+&c. is the result of an attempt to procure an arithmetical result, upon an arithmetical process, to represent a quan- tity which is net arithmetical; and = means, as in every other similar case, that the two sides of the equation are thus connected: the first side is the quantity which was attempted to be found by the process ending in the second side. And this result being obtained in strict | conformity with algebraical rules, the first side and the second will be found to have every property in common, if we consider the infinite series as an infinite series, dropping every notion of its numerical character, and considering it as a whole. It has no connexion, for instance, with 1+2+4-+8, though the latter expression contains some of its terms; nor are we to be considered as making any approximation to its value by stopping anywhere; such idea being reserved entirely for arithmetical series. And in a similar manner, we consider the equation 1 AP tn ae lk red oe ltat+a’+a?-+-+-&c. ad inf., arising from c= 1+ a2. We shall now apply the ideas here laid down to methods, by which we’ shall in various instances return to the finite algebraical expression from which divergent series are produced. And, firstly, we shall apply the series just obtained. Let | U=Uy +a, e+, 2? +a, +a, 0*+...-, where d;, a, &¢. are not functions of «. Multiply both sides by (1—.), which gives u(l—a2)=a,+ Aa, rtAa,v?+Aa,v+.... Let uu (1—zx)—a,; multiply by 1—.«, which gives uw, (l—a)=2 (Aa+A* a, v+A%a,2’?+A° aa? t+....)5 let wu, (1—x)—Aa,.x; multiply by (1—2), which gives us (l—x) =a" (Aa tA%ar+A®a, v+....)5 let %3=1. (1—x) —A’? a #2; and soon. We have then a set of series, the first of which, w, is the one in question, and 2, te, ws, &c. are con- nected with w (or uw) by the general equation Unti=Un (1—2) — A” ay ee, or U,= sae Tyenieg We now invert the process, and apply successive substitution to the last equation to determine wu. We have, then, making 1~(1—2)=X, “> FURTHER APPLICATION TO ALGEBRA, u=a)X+u,Xma,X+ (Aq.xX+2, X)X =a, X+Aa,.x2 X*+u, X°=a, X-+ Ady x X?+ A? ay. a? X2+ uu, X? = ay X + Ady x X°-+ A® ay v® X84. An Gy x" XH ay XH, nN to ws » 1 v a OY a a U+a a+, ———— a +Aa ar ta A’a at ae ae Par er eae A ot Q, 2 ve ral 0 te er, "T-zy ) ( ) If ao, a,, &c. be such that all the differences vanish after the nth, that is, if a, be a rational and integral function of » of the nth degree, we _ then see from the method of formation that V,4,=0, and w is expressed by a finite number of terms. We thus obtain 1 Rh l—vx l—z (1—wx)? Ne fi 38x D7? \ 1 v M28 p39? 22-b j e (4+ h+ : = ee l—z 1-2 (—2)?)" (—a)" 1427r4+-3274+....—- If we change the sign of x, we have pie 1 A x LA? a (F Ay~ , & a3 0—..6.= Ady— Aa oe Bs eeF b Oo: a eee in Ee eee iB) e jt a ie ei i (1+z7)° ) 2 3 x a Let us now take w= A+, +d,—+a,——+.... 2 263 22 3 Multiply both si ry € =] — 24+ ——— ultiply both sides by l= 2 9 9.3 3 a 7 which gives we~"=a,+ Aa, x tA g 0 0 5 0 we 6 ® 2.3 Piaeees 2 ( Pes : v : v a+ a, r+ as CRY oboe eS (dota e+ a8 a <4 ceo 6 ) ene (C), = 9 2 Mp 04 2 + a, —~ ere =e *(aj— Aa atAtay eee ) a) ~e(D). By integrating the expressions A and C_with respect to 2, we obtain, provided we may suppose the right-hand side to vanish when x=0, (see p. 157, note,) 2 3 tz £ x v da rar M X+a,—+a,—+....=a, ( —— -++ Aa, ent aii a Te 2 ay J ol—zx o(1—x)? , x ae vr > : : eS Zl e@ R et Be in Met a Qt ezotazas + aie, £ i du+Ady fi s*adx+. sh): We have thus obtained a large number of cases in which equivalent series may be found, and which become finite expressions if all the differences of a), a,, &c. vanish from and after any given difference. To these we may add all the cases which can be expressed by the deye- ‘opment of f(«@+ 2x) by Taylor’s theorem. We shall now consider Uu=pr+ Pu ht Plrhe+P"n M+. .., Which is the evident result of successive substitution applied to the ; du : . p quation u=pr-+h Te which gives (p. 195) Reh FING O24 uC g4— — am ¢ *hedx, I © 226 DIFFERENTIAL AND INTEGRAL CALCULUS. Let U, be the value of the series when 7=4@, Fhe (Pe oat drt Pla ht Plu. W+.s-s cont) NO —ze[s ‘bar da ‘af 7] = pes or—Pia h+o"x Ree. ee +E 7s a ekdrdx; V, being the value of $a—9’ah+.++. Though all these reductions may occasionally be useful, yet our prin- cipal object in making them is to show that there is an abundance of series, including every variety of form, which are by the common pro- cesses of algebra, or otherwise, reducible either to convergent series or finite expressions, or definite integrals ; or, at least, can be shown to be precisely what would arise from the process of successive substitution applied to an equation. Wherever there is anything like successive operation following a known law in the coefficients a, a, &c., then a,+a,e+é&c. can be materially altered in form. With regard to series, all whose terms are positive, we can only make arithmetical use of them when they are convergent; and the limits of the value of x within which they are so must be determined as in p. 222. But when the terms of a series are alternately positive and negative, it has this remarkable property ; that if it converge for any number of terms, and afterwards diverge, the convergent part makes a perpetual approximation to the arithmetical value of the original function. For example, let us take the series ; a re log (1 +2)=r-Zt+a— 7 tke ad. inf., of which the individual terms sooner or later increase without limit when ais anything greater than 1. Let us suppose x==1°3, in which case the series becomes 1°3— 3845+ °7323....—°7140....+° 7426—(increasing terms. ) Now so long as the terms are convergent, the error committed by taking convergent terms only will not be so great as the first term thrown away ; for instance, 1‘3—‘845+ °7323 will be too great, but not too great by -"140. The sum of the first is 1°1873; and the logarithm of 1+1°3 or 2°3 is *$329, and 1°1873 exceeds *8329 by less than *7140. The general proof of the proposition is as follows. Assuming Ay — A, & +a, x —&e. to have a definite algebraical equivalent, pw, we know that ¢ (0) =a, $'(0)= —a, &e.; for by p. 75, the only series of | whole powers of x which can be algebraically identical with a is b(0)+/(0)a+.... And since a, a, &c. are all finite, we have antl — t av / n a" n+} f ae} (0)+4'O) at... +9 2 jrermmera Oro 3; jai a Now since ¢"“« begins (when r=0) with a contrary sign from énttz, as long as it preserves that sign, gtx must be im a state of decrease if o"+2x be positive, or of increase if negative, when considered algebraically ; that is, in a state of numerical decrease in both cases. Consequently, if v lie within the limits in ‘which ¢"*°(a) retains its first sign, @"*'(@z) must be numerically less than ¢"t'(0), and FURTHER APPLICATION TO ALGEBRA. 227 $"t"(Ox) ana mt numerically less than ¢"*! () os a RF ne] Bevel or £4,120" that is, at any point whatsoever of such a series the arithmetical value of the remnant is numerically less than that of its first term. The student must always remember that the above can only be applied to cases in which no diff. co. of $2 up to ¢"*'x becomes infinite between 0 and x, and where ¢"*+x preserves one sign within the same limits. This will be the case in most of the necessary applications. And the theorem is not untrue in the divergent part of the series, but only useless, since the convergent part alone gives a surer approximation. It is also true when the series is altogether divergent. Nor need the terms be alter- nately + and —. If the series have only one negative term, the theorem is true, within the proper limits, if we stop immediately before that term. ? TuHrorEM. Whenever the series A++a,¢?+a,x°+&c. is the deve- lopment of a continuous function, the value of that function, when t=0, is a, even when the series never becomes convergent for any value of x, however small. For if, a, and a, being positive, we suppose @ to be negative, then the diff. co. being all finite for r=0, the value of the invelopment* will lie between @ and a,+a,x, if x be taken of sufficient numerical smallness. And its limit, when 2 diminishes with- out limit, is therefore a. And whatever may be the signs of a,, &c., the theorem’ may be proved by taking 2 such that two consecutive terms may have different signs. The theory of series is both difficult and incomplete; but the difficulty is not of the kind which a student perceives, and the deficiency is also unseen, because, in fact, the imperfect theory which is first pre- sented to him is more than sufficient for all the series of which he has any experience. He grows, therefore, in the conviction, that whatever series may be proposed, or may occur, the theory may always be made satisfactory. Now it is my present object to prevent the growth of such a conviction, by showing the difficulties of the subject. A complete theory of series would be contained in the answer to the following question: Given a series Ag+ A,+As+As+ A, +A,+ &c. ad infinitum, im which the terms are connected together by known laws, so that any _ one of them, A,, can be assigned, required the finite algebraical expres- ' sion which may in all cases be substituted for the series, and from which the series may be obtained by development. But if there be no such expression, or if different expressions be necessary for different sets of values of any variables contained in Ay, Ag, &c., required a eriterion of determination of these several cases. The preceding question is one of almost as great a width as the follow- ing: “ Required a mode of solving all algebraical problems whatsoever.” This is the first point on which most students will find they have a wrong notion. Instead of being an isolated branch of algebra, the theory of PTs; : * The inverse term to development: thus nas the invelopment of 1l—x+2?— Ke, Q 2 228 DIFFERENTIAL AND INTEGRAL CALCULUS. series is an infinite subject, in which, as in geometry, every question answered will point out questions to ask. We shall first consider such series as arise from successive substitution. Let pr, vx, px, be functions of 2, and let (jr) be abbreviated ito pea, (p (pr)) into pre, and so on. ' Let ox be a function of x, which is ascertained by the following equation, hxr=wxt+ vx. Gar ; or bx is that function of x, which is such that a similar function of ax is reconverted into the simple function of v, by multiplying by vz, and add- ing a. We have then the following series of equations : prez px+vr.dax, parwat-+ var. parr, parr —uaetyar.dadr, baie=paeo+ yoru pata, &C.$ which give by substitution Pr= x + VL. Lar +yr.var. para = E+ ve. [Lat ye yan. pare yr.var. yar x pare, &C. : so that the function @x is composed of, 1. the infinite series mabye patpyx. var. Lore ye.var. vex, ware + &e. 2. the limit of the set of products vv. par, vx.vax. gar, &c. &e., which we may denote by 2 Cc les) Y PSV OL « VOL cic wha ete le eva “.ga 2 Let the limit of the series av, a’r, a®x, &c., or a 2, be denoted by L; and let Yr be any function which satisfies Pr=yx War, Then by a similar process of successive substitution, we shall find yar=yx.vax. pore S72. Var . va 0. Wo VE VEE Vs cg sls va tw, or the limit above mentioned is Wer WL pu — Yr ry eee fyxr.pat-+y2.yar, wo + &C., where L is as yet wholly undetermined. Now it is not uncommon, in the theory of series, when such a case occurs as pxr+yx.par+ &c., to observe that it satisfies the condition da=px+vx ax, and having ascertained what appears to be the solu- tion of this equation, to equate such solution at once to the given series. For instance, suppose busx— eta (a’—a*) + 2° (2! — 2?) +27 (@8— 2") + &., which appears at once to be equal to 2; being —a?--a*—a? 4+ a7— ke, But it also satisfies the equation ¢v=xr—a.a?+a¢ (a?), and oraxr-+a! is a solution of this equation as well as @x=z. Though, therefore, the series satisfies the condition ¢r—=pu2r-+-rxrdar, yet when this equation has more than one solution, nothing but attention to the preceding pro- cess can preserve us from error. _ With respect to the equation @Gr=pe+yex gar, it can be shown that its most complete solution is as follows. Let wa be one solution, and let .~L; so that we have FURTHER APPLICATION TO ALGEBRA, 229 xx be one solution of the equation yo=vxv Wax, and let we--xx. Ex be the most complete solution. Then we have Deb xL. Fe wat yx (war +xarv. Far) ; but by hypothesis DU=wUrLrve.wav, and xr==vx.xar, therefore far=fx, or with the particular solutions above mentioned, nothing more is necessary than to find the most general function which remains un- changed when wv is changed into az. In the same manner it may be shown that xr.£x is the most general solution of ¢r=ya.dar. “We have then ~* . px wxtxx. ex parr _ waretna'r. ex We oe ee \ 3 a call Ee a E« being the function which is absolutely unchanged by changing vinto ar. If n be increased without limit, we have then gl _aL+xL.ée eee el wi ° “xen a ice Ah oar i so that the equivalent obtained for the series is the same, whatever solution of the equation was takes, We have thus obtained the absolute arithmetical sum of the infinite series ; for the process was equivalent to finding the sum of n terms, and. then increasing » without limit. Whenever the series is divergent, the term xt.a@L—+xL will become infinite. Thus if we apply the process to v+ar+a’x+&c., which is obtained from px=2X+O (ar), where PU=X, ve=1, av=ax, a"x=a"r, we shall find as the sum of the series « (1—a* )—(1—a) which is finite only when a<1, and infinite in all other cases. The preceding is literally nothing but a modification of the method of taking 7 terms of the series, and then increasing nm without limit; but it will lead us to the following conclusion ; namely, that the algebraical expression for a convergent series may be discontinuous, or not always the same function of x. This we shall show if we prove that L may have different values for different values of x; or that ar, when mn is increased without limit, is not always the same for all values of x. For instance, let ar==2°, then vexr=2, a®tv=2", &c., as to which it isobvious that they increase without limit if x >1, remain always the same if £=1, and diminish without limit if e< 1. As it is here my object to prevent the formation of an opinion, and not | to establish any general method, one example of every difficulty will be sufficient. Let us now consider the following series : a? x as x? at—zxt g@—gs qs—yie "tt" Looking at this series, we should suppose it to be one which we might safely use as a common algebraical quantity, for it is always con- vergent, except only in the single case of w=a, when every term evidently becomes infinite. To prove this, form the ratio of each term to the preceding (p. 222), and we have a? x at x* a x a ee IS perenne &C., ' ate a+ 32” ae ae 230 DIFFERENTIAL AND INTEGRAL CALCULUS, which must diminish without limit; for every term may be written in either of the following forms: (xa)? (a+r)? and either v--@ or aa is less than unity (with the exception above cited) ; so that one or the other form explicitly shows the diminution without limit, when p increases without limit. Now observing the terms of the series, we may readily see that it is derived by successive substifution from : xv r ae ae +¢ (=). of which a particular solution will be found to be dvr=1—+-(a®—2”). Applying the result of the preceding pages, we have ya=1, a particular solution of wer (2a); prema (at—a*); ve=1; and We Sd 1 or—vxe —= —-— — <, ¥ ¥L @—a2*.at—TH a 2\2 4 8 rE l vir a Now ar=—, &r=— | — } —=—, air=—, &c.; so that L must be a aa, as a the limit of #’--a?~’, when p increases without limit. According as # is less than, equal to, or greater than, @, this limit is 0, a, or &; so that See 1 x when, a, useaieicl ary teg Oy (Mor a —iz v—a? The terms of an infinite series must be connected by some law, other- wise the series is not given and distinguishable from others. A finite number of terms may be written down, and each is then given; but an infinite number of terms cannot be written down, and can only be said to be given when a law is pointed out, by which, when 7 is assigned, the rth term can be found. Let us now consider the ordinary algebraical development, namely, a series which proceeds by whole powers of a variable quantity. Let the (r+ 1)th term of such a series be F (a+7r/) a"; so that the series is Fe+F (#+l).a+F (7421) @+F (e#+3la+....3 which is derived by successive substitution from r=Fa+ag (x+l). We have now this question to consider:—1. Can the equation dx=Fr+ad (r+l) i always be solved by a continuous function az, when F is a continuous function ? This question will, as we shall see, bring us to the following: Can a continuous curve be drawn through an infinite number of points sepa- rated by finite intervals? We know that through any finite number of points, however great, an infinite number of continuous curves can be FURTHER APPLICATION TO ALGEBRA. 231 drawn : it is quite certain, for instance (as will appear in a subsequent chapter), that if we had ten million of given points, nothing but opera- tions of impracticable length would lie between us and the power of obtaining as many continuous curves as we please, each passing through all the given points. As an instance, suppose the equation of a curve is required which, when r=a, gives y equal to either A, A’, or A”; which, when w=, gives y either B or B’, and when x—c gives y=C. Let xy be any function of y which does not become infinite when x js a, b, or c,and find y from the following equation : (y—A) (y—A’) (y—A") (@—b) (20) + (y—B) (y—B) (w—a) (0) +(y—C) (a—a) (x— 6) +-xy (a—a) (—b) (v—c) =0. Here, when x=a, the equation becomes (y—A) (y—A!) (y—A") (a—b) a—c) =0, which has three roots, y=A, y= A’, and y= A", and so on. Seeing, then, that through any number of points, however great, we may draw a continuous curve, it may appear that we can do the same through an absolutely unlimited number of points. On this postulate* the following considerations rest: let it be granted, that whatever is true of any finite number of points, however great, is true of an infinite number of points. We now return to the equation ox=Fu+ag(v+/). Observe, that we do not want a solution of this equation for all values of x, but only for c=k, w=h+l, x=k+21, &c., ad. inf., where k is some value assigned to x. Multiply the equation by a**’, and let a’ ‘ox be called xz. Then we have yarmat Fa+yx (a+/), or ya—y (a@+l)=a? Fr=fr. Draw the curve whose equation is y==a"*’Fx, and on the line of abscissee cut off k, k+/, k+2/, &c. | Let A, B, C, &c., be the points of the curve y=a**! Fx, whose abscissee are k, R+1, &c., and let MP be taken for xk... lake Na=AP:; then Na=MP—MA=yk—fk=y (k+l). Similarly, take Qb=Ba, Re=Cb, Sd=De, Te=Ed, &c.; we thus obtain an. infinite number of points, and the curve drawn through them, if it be y= xz, satisfies the equation xt—fr= x («+/). * Several other methods which I have tried of obtaining the same conclusions end in the necessity of the same postulate, 232 DIFFERENTIAL AND INTEGRAL CALCULUS. Assuming then the existence of wr, a continuous function which satisfies ¢r=Fa+a¢ (r+), we find yr=a * to be a particular solu- tion of yr=ay (v+l): whence the arithmetical sum of the given serics 18 ae (z+!) paren ae or wr—a"a (x-+nl) ; We— A mig in which n is to be made infinite: it being always remembered that aw is a function of @ as well as of x. If we assume v+7/=z, the pre- ceding becomes z a] a wr—-a 'Xlimitof(a'wz) {z=e}; and the limit in question may be nothing, infinite, or a function of a, which, for anything yet appearing to the contrary, may be continuous or discontinuous. And upon this limit depends the convergency or divergency, continuity or discontinuity, of the series. It is my object now to show that discontinuity cannot take place without the series becoming divergent at the epoch of discontinuity. Let us suppose the series to be convergent for every value of a, from axa’ to a=a”, both inclusive. The continuity of law of a function is not to be presumed from the simple continuity of its values (page 45.) ‘To return to the geometrical illustration: two different curves may join in such a way that the value of y increases continuously in passing from one to the other through the point of junction. If they have a common tangent at the junction, a1 ; i ? oF may also varv continuously in value; if they have there a common : 5) Cia may do the same. And two curves may be dis- tinct, though the value of y and of any finite number of diff. co. increase or decrease continuously in passing through the point of junction. But if all the diff. co. increase or decrease continuously, then the second curve is only the continuation of the first. Now if wz satisfy dr=Fa+e¢ (t+), it follows that wx satisfies dlezz F’x+ap'(«+/), and so on; and whether we differentiate the result radius of curvature ar a x Lim.(@!' wz) =Fet+F (#4+2).a+.... n times, or whether we treat the equation P”r=F™a+ap(x+1) by the method of this chapter (and by pages 172—175) we find the following: wed Loe aie: fe a™r—a (- = ) x Lim.(@? wz) = FOP 2 + FM (atl) a+ ..0e5 * l so that the convergency, &c., of every differentiated series depends upon the same function as that of the original series; namely, Lim. (@*'@2). If, then, the first be convergent from a=a' to a=a", so are all the rest. Name any number of them, m, which may be as great as you please. We have then m-+1 convergent series. Let ¢ be a number of terms so great that for no value of a between a’ and a” can ¢ terms of any one of the m-+1 series differ from its arithmetical sum by so much as @, where @ is 7 -— FURTHER APPLICATION TO ALGEBRA. 233 a definite quantity, as small as you please. This is evidently possible, though to bring some series a little within the limit it may be necessary to take é so great that others shall be very much within it. Let the sums of the ¢ terms of the several series be represented by 2, 2’, 2”, &c. It is clear that X, &’, &., are a set of continuous algebraical functions, finite, rational, and integral with respect to a., And the values of wr— (the limit in question), and of its m diff, co., do not differ by so much as 0 from those of 3, >’, &c., for any value of a between a’ and a’. But if there were any discontinuity of value in any one of these expressions, this could not be the case ; for the discontinuity must take place at some definite point, and be of some definite amount. If possible, let @ be the abscissa of a curve, and let @wv—&c. be discontinuous in value between a=a’ and a=a”’. Let AB be the arc of the curve y= >, contained between those abscissa, and let PQ, RS, represent the dis- continuity of value of y=awxr—&c. Take 6 less than the half of the discontinuity QR; and let the dotted curves be those whose ordinates are always greater by 0, and less by 9, than those of AB. Then PQ, RS, by what has been shown, lie entirely within the dotted curves, which is impossible, since QR is greater than 26. The supposition, therefore, of discontinuity of value in any one of the diff. co. of nF oe ed z . wt—a !Lim.(a'awz) is inadmissible as long as the series which it represents remains con- vergent; whence we have the following Turorrm. If A, B, C, &c. be coefficients independent of @ and following any law, the series A+Ba+Ca?+&c. ad. inf. can never change the function of @ which it represents, in passing from one value of a to another, without becoming divergent in the interval between those values of a. Hence we have no further occasion to consider the possible discon- tinuity of such a series ; for if it become divergent for any one value of a, it is divergent for every greater value; and the discontinuity, if any, takes place in a function, of which all the values are infinite. But in periodic series (see next Chapter) we shall have occasion to use this test. We now see a reason for the appearance of discontinuity in series of other forms, which does not exist in those we have just considered. Looking back to the general expression Wa" x DE-XKL Lim.(5 =purtyr. wartrya.vax.oerx+ &e. ad. inf,’ ‘ i { Xa" Xx we have seen that aw may have different limits for different values of 234 DIFFERENTIAL AND INTEGRAL CALCULUS. v. Butin the case before us,* eral, kw=a+2l,.. oe xvaet+al, and « is the only limit. In the example of page 230, the discontinuity arose from (x+-a)” being 0 or &, according as ris a. I have now carried this subject far enough for the purposes of this work; but the same conclusions might be extended further. It is always true that a series cannot change its equivalent function without passing through divergency, or some other singularity of form. I now come to the question of convergency or divergency, considered apart from the connexion between a series and its algebraical equi valent. Turorem. If P,+P,+.... andQ.+Q4+...-- be series, of which | the terms continually approximate to a finite ratio, so that by making 7 sufficiently great, P,+-Q, may be made as near as we please to the finite quantity c; I say that these series are either both convergent or both divergent. Begin from the terms P,, and Q,, and let P,-Q,=c,; then P,+Pry Hee Cp Qn t Cnt Vapi tess: And since 2 may be so great that Cny Cn &C., Shall be as near to c as we please, they may all be con- tained within c+6, where @ is as small as we please. Certainly, then, Cr Qu Cnii Quart... lies between (c+) (Q,+Qrit...-- ) and (c—9)(Qa-FQnryit-+.)3 Or Prt... lies between (c+0)(Q, +...) | and (c—0)(Q,+....). If, then, either of the two, P,t+.... and Q,+...., imcrease or diminish without limit, or approach a finite limit, so does the other ; which was to be proved. Let two series, in which the limit of P,—Q, has a finite ratio, be. called comparable ; those in which the same limit is nothing or infinite, incomparable, Turorem. If dn be a function of m which increases without limit with n, then @n—n° may have a finite limit, but only for one value of e; every higher value giving diminution without limit, and every lower value increase without limit. The first part of the theorem is well known; the second is thus proved. Let ¢@n—n* have a finite limit L; then if f be positive, on=n' is (pn—_n’) xn, and its limit is L x0 or 0; but (dn~—n'*) = (dn—~n’) X 7’, and its limit is Lx o, or infinite. The value of ¢ is easily found; for since n’—gn takes the form oc +-0, when n=«, we know that its limit is the same as ‘that of en’+'n, so that the limit of nf’n—edn is unity, or e is the limit of no'n—on. Jf this be infinite, then n’+-dn has the limit 0 for every finite value of e; but if it be nothing, then n’+@n increases without limit for all finite values of e. The properties of the limit of n*<@n,, when n=, may be readily deduced from those of dx+(ar—a)” m Chapter X. Derinition. If P,Q, have the limit c,let P,+... be called higher than Q,+...., when c is greater than unity, and lower when c is less than unity. But when the ratio increases without limit, let the first be: called incomparably higher than the second ; and when it decreases with- out limit, incomparably lower. If, then, a series be divergent, all com- parable series are divergent, and all incomparably higher series ; but ifa series be convergent, so are all which are comparable, and also those which are incomparably lower. And any divergent series is incom- * Also ya=a, eax=Fa, xr=a-*"', FURTHER APPLICATION TO ALGEBRA. 235 parably higher than any and every convergent series, All this readily follows from the last theorem but one. ; 1 ] : Turorem. The series MAS aoe gerat +++ 18 convergent when e is (no matter how little) greater than unity; and divergent when e is equal to or less than unity. Firstly, let e be less than 1; then the sum of » terms of the series being greater than n times: the least of them, is greater than nuxn~ or-than n'~*. But this increases without limit with n; con- sequently, the sum of n terms of the series increases without limit, or the series is divergent. Secondly, let e be equal to unity; the series then consists of eae ed AY i Bae ig oT, ie haar i (+++ +( —~4+—4+—4— eight terms ending with en Bind Bit loGsine oh BLgare 8 1 Liat ee ie 7a) + (sixteen terms ending with a5 HBC. 5 which is evidently greater x x 1 ] — +16% —+é&c. is last is Tt 6xs5+6 c. But this last is ee tie eee the divergent series as: + a le ae +...., which, being ex- ceeded by the series in question, the latter is therefore divergent. Thirdly, let e be greater than unity ; make parcels as before, and the series is 1 Bi wh kaha] 1) 1+s+($ +E +(4 terms beginning x t(s do. do. *)+ &C., which is less than 1 1 1 vt ieee er Eee than Mig TeX + X= FBX 1 pu 4 8 Ite ta te tg the. and still more less than 1 2 4 8 as epider ial e l 1 i ] Q-e 2 or game a areas arr + &c., or 1+2-°4+v+4 v0? + &c., where 2 _ Tueorem. If gn be a function of n which increases without limit with n, the series l l ] l! l ee ed. : iy Gia) SG) in) Ga may be convergent. To ascertain whether it is so or not, find e, so that n'—on is finite when n is infinite. If, then, e be greater than unity, the series is convergent; if unity, or less than unity, divergent. But if n’dn be infinite for all values of e, the series must be divergent; if nothing for all values of e, convergent. To find e, ascertain the limit of n@’n—@n, when n increases without limit: but n’—on increases without limit when 2¢'n+n diminishes without limit; and diminishes without limit when no'n—on increases — i cet? o=(> - In this case, therefore, the series is convergent. 236 DIFFERENTIAL AND INTEGRAL CALCULUS. without limit. So that the complete test of convergency or divergency may be stated as follows:—the series whose terms are reciprocals of dn is convergent when the limit of nd’n—dn is greater than unity (infinity included), and divergent when the same limit is unity, or less than unity (nothing, negative quantity, and —« being included. ) — The proof of the preceding is obvious. If n’+-n have a finite limit, - 1 1 the two series 2 — and 2 Bi are comparable, and are therefore con- n n vergent or divergent together ; that is, convergent when ¢>1, divergent when e= or <1. Butif the limit of n°+@n be always infinite, or that ore UAL. of -—-+—, then, taking e<1, the given series is incomparably abovea dn n divergent series, and is therefore divergent; and in this case the limit of nd/n——n is nothing. But if the limit of n’+d@n be always nothing, 1 1 aes or that a then taking e>1, the given series is incomparably . ant below a convergent series, and is therefore convergent; and in this case the limit of nd’n—dn 1s infinite. If wn, the term of the series, be used instead of pn, its reciprocal, we have 1 1 1 Examece Il. (r—1)+(2?—1)+(@* —1)4+-(r71—1)4.... 14 - wn Re th Here Meno" — 1, AN gent A Se a wn n(a~—l) The limit of the denominator is log x, whence that of the fraction is I, | and the series is divergent. : | / n Examr_e Il. 1+a7+4a®+a°+....¥n=a"", —n eae —n log x. n If x be <1, the limit is +c, and the series is convergent; if v=1 or | ee 4 ; ae a be >1, the limit is O or —c, and in both cases the series is divergent. A negative limit denotes that sort of divergency which is shown in the - series 1+2°+3°+...., where e¢ is positive. Ps he ea ce ears Examete III. x+—+—-+.... : on= Lee ? e's It will hereafter be shown, that when 2 increases without limit, Late. wos dP OAT 21 iesk0i ay oo (2n—1) approach without Jimit to 1 1 3 Vom cure baa" Nn e-n : . a Onn #e-", and 2 2 2" e~", from which the application of the rules shows the series to be always convergent. Examp.e IV. F(@+/).a+F (@4+2l) @+....Yn=F («+nl)a" Win | I’Cr+nl) ] ren! pais tye pa + 084 f=—wlog Ge F (@+nl) se, —nN Ww 7 2 ‘s F’(x7-+-nl+86l) t ] & } d l l —| o HK (7-+-7)/ Mey ARS 3 * But log F (r+nl+l) ne} (24 nl) +55 G@anleen (0<1); FURTHER APPLICATION TO ALGEBRA, 237 whence the limit of e+)‘ js that of rth it aa F (x+nl) finite, the series is convergent when the limit of aF (w7+(n+1)1)~ F (v+nl) is less than unity, and divergent when it is greater. But when that limit is unity, the convergency or divergency of the series depends, agreeably to the rule, on the limit of ee eons F (v+nl) l 1 Bi ot EXAMPLE sya 2" (log 2)? + 3° (log 3) *t a JS always divergent when If this be —n log ia. @ is unity, or less, whatever may be the value of bd, / n ht mete The expression ae the limit of which is greater than unity rn whenever Yn is convergent, may be written as —n x diff. co. log wn. Phe limit of this, when n increases without limit, is not altered by Writing «" for 2; in which case / n d —72-— becomes — — (log we"), wn dn °8¥ ) Phe result may be stated as follows. To ascertain whether the series %yn is convergent or divergent, take the function Yn, or any more imple one the ratio of which to wn neither increases nor diminishes vithout limit when 2 is increased without limit, and find the most onvenient of the following expressions: nm din mn d(wn) dlogwa d 1 d(nyen) ~- — = ——, ~ —(logye"), } = — Bead! Cen) dat 6 Tan dy Bye") wn dn f, then, the limit of the result be greater than unity, the series is con- ergent; if unity or less than unity, divergent. But first examine b(n+1)+¥n, since this test can only be necessary when the limit this is unity. As to series of the form P,—P.+P,—.... we have seen that hey are necessarily convergent when the terms diminish without limit. ‘onsequently, the series is convergent, all whose terms are positive, rovided they can be represented by P, —P,, P,;—P,, P,—P,, &c., where 1>P.>P3, &c. But this last is not altered by adding the same Uantity to both of every pair; that is to say, the series P,+A—(P,+A)+(P;+B)—(P,+B)+(P,+C)—(P,+C)+4.... ems Convergent whenever P,, P., &c. diminish without limit. Thus a * The reasoning here given is correct only on the supposition that F!(a+nl+ o/) (Peay Gea F’(a--nl) ¥F (a+n/-+-é6L) ¥F (a-+nZ) proach the same limit when x is increased without limit. For accounts of the tests of convergency up to the proposal of the present one, see rofessor Peacock’s Report to the British Association, in paye 267, &e, of the second lume of their Reports ; or Grunert’s Supplement to Klugel’s Wirterbuche, vol. 1. ige 416, TI have another proof of the correctness of the test, founded on entirely flerent principles, which will appear either in the sequel of this work, or elsewhere. 238 DIFFERENTIAL AND INTEGRAL CALCULUS. series of alternately positive and negative terms may apparently be con- vergent, even when the terms increase without limit; andif A=B=€ &c., we have then a series, of which the nth term (independent of sign) is P, +A; and because P,, diminishes without limit, this has the limit A, And we might certainly suppose that the preceding series can mean nothing but P,—P,+P,—.... in a different form. Is it possible that there can be an error in the following reasoning ? If P,—P,+P,—.... be a series, which may by summing its terms be brought as near to M as we please, then certainly the sum of P,—P., P,—P,, P,—Ps, &c. can be brought as near to M as we please, But P,—P, is the same as P,\+A+~(P.+A), and P,—P, as Ps + A— (P,+A), and so on. It follows, then, that P,+A—(P.+A)+(Ps+A) | — &c. can be brought as near to M as we ‘please; or that if sucha series as the last should occur as the answer to a problem, we may con- clude that M is the answer required. . I say we have no right to draw any such conclusion; and the” reason of this is contained in a principle which cannot be too often remembered by the student of thissubject. Whenever a deduc- tion is made from purely arithmetical principles, by means of purely arithmetical premises, it must not be extended, without proof, to cases in which the premises, or any of them, cease to be the objects of arith- metic. In the preceding series, P,— P,+P;—.... approaches without limit to a fixed arithmetical quantity, and an accession to the number of terms taken always brings us nearer to a certain limit. The same is true of (P,—P.)+ (P:—P,) +. ..., each term of which is compounded of two of the terms of the preceding series. The same is also true of the series whose several terms are _ first, P,+A—(P,+A), second, P,+A—(P,+A), &C. 5 which is, term for term, identical with the preceding. But the same is not true of the series, whose terms are first P, +A, second P,+-A, third P,+A, &c., alternately positive and negative. For since Py, Ps, &e. diminish without limit, the series may, by proceeding to a sufficiently distant term, be represented as nearly as we please, from and after that term, by A—~A+A—A+...., which has no arithmetical signification. 1 Tet Lehn. Thus, if we take Leinny Hager a -+&c., which has the limit 3 and add Gees Pers! one to each of its terms, we find Aleks se gph eRe tee Let the terms of the first series be collected, and we find the set of ties vewaes flay ES results 1, 2 a? 8? 16’ 30 &c., alternately greater and less than 2 ge AS Miia 5? but perpetually approximating to it. Treat the second series in the he US ae same way, and we find 2, 3° 4? 8? 16° 39 , &c.; of which the even 2 5 terms only approximate to > while the odd terms approximate to | If, then, we were asked which is the arithmetical limit of the preceding | a ¢. 2 5 series, we should have no mode of deciding between gs and cy. ~ FURTHER APPLICATION TO ALGEBRA. 239 In the preceding theory is contained all that the'student needs, to enable him to apply the theory of series to questions of geometry and physics ; and I shall now retapitulate the principal results, desiring the reader’s attention to the summary, as distinctly marking the point at which we have arrived. 1. An infinite series, even when arithmetically convergent, may be the arithmetical development of different functions, of one for one value of 2, or set of values, and of another for another. Or, the con- tinuity of any series must be proved, and not assumed, (page 230.) 2. If the series be of the form a+ br-+cx+...., or developed in whole powers of x, it must represent one function of x, and one only, throughout the whole range of values of x, for which it is convergent, (page 233.) 3. When a series is given, and nothing is known of its invelopment, it cannot yet be used m any case in which it is divergent. But when the series is produced from a given function, the necessity of absolutely considering a divergent series may be avoided, as in page 226, by using the theorem of Lagrange on the limits of Taylor’s series. I shall, in a future chapter, consider this subject further, and shall conclude the present one by giving some theorems which may be con- sidered as instruments of operation merely, not giving any proof to their results, except in cases to which all the preceding reasonings will apply. Turorem. Let dr=a+a,r+a,a+....ad anf. Wwe=ab+a,b,r+a,b,v?+... ad. nf. x 93 where Ab, A*b, are the successive differences of b, obtained from 8, hy by, &e. N. B. We have already had cases of this theorem in page 225. ‘rom page 79 we have b=b+Abd, b=b+2Ab+A%, b,=5 +346+ 3A°b+A%), &e. substitute these in the second series, which then becomes b(a+a,¢+a,a°+....)+Ab (@,+2a,7+3a,2°+....) 2 FA°d (d.4+3a,2+6a,¢2+... ) 2+ A$ b (as+4a,7+10a;a2+... .) Now Pr=at+a,r+a,c* +a,2° +a, +a,0° +.... ED BS eee +2a,% +3a,0°+4a,7°+ Baat+.... 2 Then Yor= Dpt+Ab.p'0.2+A%. px. +A). gla sate or dv ; , i dg +3a,0 +6a,0°410a,0°+.... pr a ag +4a,2°4+10a,2°+.. ., &.; nd the results of this set, substituted-in the preceding development, ‘ill obviously give the theorem in question. J Exampte I, aay =l+ar+a2+a°+at+.... (page 225.) —L < 240 DIFFERENTIAL AND INTEGRAL CALCULUS. Let we =b+h, +b, P+b,0 +b, a+... b Ab.&® 4 MO sss; Then Wr eat ¥. au 5 a nage > sta Bee 3 ax hae 4 7 * o a)y—1r— — a va wie Exampe II.* log (l+e)=2« at ed 4+ oak idl AR ye Sbe-— +5 pura wa enage _ Abx 1 Nb. at 0) eA pada (b=0) WX Tolepeeay a 3 asa anime —l 2-2 Exampue III. b+70, vpn by: e+; rite bh, ae+ ete —b(1+r)"+nAb (1+2)"" a+n LAD (+a)? a+ pane) th met 2 a =(1+7) pirat tes (aes) +. 43 4 ExampeLE IV. b+40),0+6,— Shs ran bine 4 +. =e {b-+Ab, r+A% aS ye ea %: 2 Rb 3 AS EXAMPLE V. ba—b ai sae +6 3 g = COs ee rar ae ny (s—a0 5+ wt .) and a b 19 3.4 3 eee x a? F os aa “ a ore — 8} u b.t— . ee eee cos |b LA? b me “i sino ab. v Ab + ) where the differences must be taken from the complete series }, d,, bs, dg, b,, &c. We shall see more of this when we come to treat of interpola- tion. This theorem enables us to give a finite expression for wx, whenever ox can be expressed in a finite form, and 8, is a rational and integral function of n, (page 83,) in which case A”d is nothing; for all values of m which exceed the degree of 5,,. The theorem itself will afford an instance of the truth of results obtained by separating the symbols of operation and of quantity, as im page 164. The symbol for 5, is (1+A)"), and the whole train of operations performed on 0, to produce ab+ a,b PEE ey is fata,(1+A) r+a, (144)? a?+....}, or ete / * Let the student apply this example to the case of Ab=1, A%—=1, A%+—1, &c., and explain the result. FURTHER APPLICATION TO ALGEBRA, 2 {or+$r.cd+ pln A®-+e oe. } b, 2 |bb2+Ad.G/2.0-+ A2b pa 5+ ste ef I now apply the preceding theorem to the transformation of 6+, cos 0.2-+ by cos 20,27-++ b, cos 39.2°+.... id b, sin 0, w+, sin 20. x?-+ 5, sin 30.2°+.., he first series, writing z for goV-1 » (as in Chapter VII.) may be thus ritten : Q 5 {bet by zu+-b,2?a°+.,..$4+= plete th te. by the use of the theorem, a Ab — Stet be aor als 2\l—z2 (1l—ez)? 2 2 cE 0 i—— (| ]1—— 4 id the two, collected, give a series of terms, each having the form an 1 “m ym 2m — Am aed Ss ye c (l—z2)"" i: ioe mafresse (A). omaet ut 1—zx= 1—27 cosO0—z sin o.V—1, L : eae 1——=1—weos 0+2sind. V1. ssume 1—xcosO=rcosf, xsind=r sing, hich ¢i r?=1—2x cos0+n%, tang ae : ey (PY is Mb a > Pat vg eine 1 ae0R O° id (A) becomes (since l—rz=rcos¢+rsing. TELS ¢ &c.) reg (cos m0 -+- Ni —1] sin m0) 2” ~ (cos m+1¢—V—Isinm-+1 gp) ent (cos mo—N —] sin m6) a” (cosm-+-1¢ +7—lsinm+1 ) a COS |Z +] sin at ———_—__——-——=cos (4 +7) + —1 sin (uty); cosypV¥—l1 sin y uence the Aces is “Bpmti t {cos (mo+m-+1 m+1¢)+V—1 sin (do. do.) + cos (do, do.) —V —1 sin (do, do.) + 242 DIFFERENTIAL AND INTEGRAL CALCULUS. A"b.cos (m0-+m+1$).2” STIS ee eee eT (1—2x cosO+2") 2— or Hence, making m successively 0, 1, 2, &., and adding the results, we have the following THEOREM: i 9 ff =O aie Oe ia ae sstan(G aa) nae ( sin *) a (= cos *) —sin = Cos rharinsnichngy i r then b+b, cos 9.2+5,cos 20.a*-+-b; cos 30 X8-+-. +e oe 2 Rt cos p— +Abcos (94 2¢) <+ A cos (20+ 36) 3 + A% cos (30-446) G4... For instance, let b=},=).==....2=1; whence Ab=0, A*D=0, &e. ; we have then cos +r, or (1—z cos 0)—7°, for the sum of the series}; or k—2 cos 9 {—22 cos a ee 20.a°+c0s 30.2°+.. «5 which may be verified by page 125. The transformed expression may be discontinuous, for ¢, or tan” {x sin 0-(1—acos@)} has an infinite number of values, one of which may apply for one value of 6, and another for another. We have shown that no discontinuity can be produced by a change in the value of a (page 233.) As long as our conclusion preserves its present form, we are warned 0' the circumstances which may produce discontinuity by the explici' appearance of the ambiguous symbol tan~. But if we take a case il which the ambiguous symbol disappears, we may be led to a fals« result, if we do not take care to retain all the ambiguity of the origina form. Suppose, for instance, =1; then sin 0-—(1—cos 0) is cot 3 4 or tan (}r—46); and @ is therefore tan“'tan(47— $0); that is any one ofthe angles which has the same tangent as 4 7— 30. All thes’ angles are included in the formula mr+4nr—%4 0, where m is at whole number positive or negative ; whence we have, by substitution i the expression for the transformed series, (since r= +2 sin} 0, whe to=1). b+b, cos 0+, cos 20+; cos 30-4+.... _b cos(mn +4 (7—9)) is Ab cos (Qm7-+7) +2sin$0 4 sin? 40 A’h cos (3m+1ar+4 (7 +8)) Sa aT PTE. +, e@eerg +8 sinti@ in which A‘d is multiplied by cos @-+1 m+47+72—1 $0). Now } will be found on investigation, that these cosines, beginning from th first, are FURTHER APPLICATION TO ALGEBRA. 243 Fsin(—}$6), —1, +tsin30, +cos0, Xsin20, —cos 20, &e.; he upper sign being used when mm is even, and the lower when m is odd. We have then an ambiguity of sign in both numerators and denomi- iators of the alternate terms: but returning to the original equations of ondition (which become 1— cos 0=r cogs ¢, Sin 6=r sin @, in the case of = 1) we see that ifr be positive, sin@ and sing have like signs, and 0S is positive; that is, d hes between O and io, if @ lies between 0 nd w, and ¢ lies between O and —47,if 0 lies between 7 and 27. All hese conditions are satisfied by making m=O, or any even number, and he final result is as follows: b Ab A%bsin$@ , A%b.cosO Atbsingé BOCs 0+... = = Ss > t+. 7 see 2 4sin*Z@~ 8sin*46 16sin‘t0 32sin't0 An easy verification presents itself when 6=7; the preceding then lecomes 6 Ab Ad A%M Ard b—b +b.—....2——— Sa eee ae Toon wo Oat 6 6 ¥ 4 Mies 24 8 16° 39 ; vhich is a case of (B) in page 225. An analysis of precisely the same kind, it being remembered that A—1.sinmd= z"—z~", shows that we may substitute sines for cosines i the series obtained; or that (r and ¢ being as before) , ; 1 ; x 6, sin 6.2-+4+6,8in 26.¢+....=bsing AD sin (0+ 20) = wy + A% sin (20-+3¢) s+. eR | a before, we have sin @--7 or xsin 6-+r* for the um of the series; or x sin 0 ates Bi 1 2 gy 3 a itr eee 20 #*-+-sin 30a°+....65 nd, in the case of c=1, we may find A’b cos £6 Ssin® Se b b,sin@+6,sin 20+... 5 cot 40— A*dsin@d | Atbcos 26 16sin*4O° 32sin'4Q °°"? 1¢ terms of which, after the first pair, are positive and negative in tternate pairs. An instance of verification, though not so simple as ie former one, may be found in the case of 2=47. This gives b A*> A*d , Atb\ , (A% Ad =) mee ton... “h -AteAD oS 16° 32 @eer gy hich we leave to the student to verify by means of the separation of the ymbols of operation and quantity. He might, however, be perplexed y the reduction, if I did not call his attention to the equation Ae A\_ A THrorem. If drm=a)+a,0+ dyt®+ Get? eves R 2 244 DIFFERENTIAL AND INTEGRAL CALCULUS. Then 5 {9 (ex) +0( =) = aya cos8.24-ay00920.0"+. 4h. Pod 2 1 x ie : ; =" (x)—0(=)}= a, Sin 9 ,@-+a, sin 20 .a°-+.... ay @ where ze’. The student can easily prove this for himself, and also the following :* 1 1 b'x.cos 20 , ¢!!xcos36 ~b(r@tez ua +- j=dritd’'c. 9 +-——_—— —— + ———-+.. gO so( ete PEE EE OE ES aeagie an Rt HERES 1 a) b"v.sin 20 , O'x.sin 36 rete" a ae a Rat WOR Ret PER Whe aia er oes spe loet ) (242) + ¢!xsin6+—G a er Let ¢x=log a, and let the upper signs be used: we have then 1 ] 1 5 18 (22) +5 log (43 =log w+ ee cos@ cos 20. cos30 e mT ee eee 00 g Q2° 8x8 i (2°?+2xcosO0+1)% cos cos20 cos3 cos 40 toate x Bete 2x" 3H Art bie 1 v+z sin@ sin 20 sin36 sin 40 and | ——— 7 Ry tape ey Bae Sey) ee ate __¢t+cos 64+V—I sin 6 cos o+V—1 sin P_ ——_ But —_= ee ae ——— ap-eneval v+cosd—V —1 sind cosp—V—I1 sind govt, : : ‘ sin 0 in which 2-+-cos@=rcos¢, sind==rsing, or é=tan—! | ———— } and x-+cos 8 (pege 126) log eV Te og —14+2n0V—1, n being any whole num- ber, positive or negative. This gives nt =—- — —— +— — se. x«+cos @ + 9 Bs i r on: If e=—1, then ¢ is tan-! (—cot 36), or tan7' tan (tx—4 6); so that — eee ( sin 0 _siné sin20_ sin 30 sin 20 sin30 2 3 m being =< any whole number. This simply amounts to r—40+mr+nr=—sin 6— eoeoeg 3 in20 sin 30 4 0-+-mr=sin += = - The meaning of the undetermined quantity mm may easily be shown. The second side of the equation is periodic, giving the same values for @, 0+27r, 0+47, &c. It also vanishes with 0, and becomes 1-4+4-+ ++ or }7, when 0=437, and changes sign with 0; and it becomes 0 again when 0=7. This requires that m should =0, where @ lies between —7 and +7; but that in all other cases m should have such a value as will make 0-+-m7 lie between —7 and +7. I now proceed to some developments and examples, part worked at * These theorems are due, I believe, to M. Poisson. MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 245 length, part merely sketched out, and part proposed for exercise with their answers. In considering these, the student should read again carefully those parts of the preceding chapters which are cited. a CHapTer XIII. MISCELLANEOUS EXAMPLES* AND DEVELOPMENTS. 1. Required the successive diff. co. of Pe*, P being a function of 2. Ans. The first is e*(P+P’), the second e* (P+2P’+P”), the third &(P+3P/+3P’+P""), and so on: the nth is ¢ (P +7 P!4+n, P+ Mg P°+....+P™), where 1, N, Ms, Nz, &e. are the coefficients of the ; aL ae I 9s 2 several terms in (1+2)”, or 1, n,n 2 Maar er &e, 2. Find the diff. co. of PQ the product of two functions of x. Ans, The nth diff. co. is PQM Tne tee b Qe-O 4 4 po. 3. Diff. co. of P™Q” is P™-Qr? {mQP’+nPQ’}. m m—1 4. Diff. co. of - is on {mQP’—nPQ’}. 5. Diff. co. of e*.Q is eP {Q’4 QP’}. It will be worth while to retain the three preceding results in nemory. 6. (Page 63.) What is the diff. equation of y=r¢(cx)? This sives C ly q 2 2 “4 4 (cr) + cx d! (cx) 7, p! 1 ta r=74 ca T t a ¢ } ? 3 dx Li Lv x ie vhere fx means r+e@ ¢’f "x, and $x is that function which gives pr —-x. d P ay 4% Y 1. y=xe™ gives a (1 + log — Ge ae, 8. Eliminate the functions from z= (y+axr)+ we (y—ar) vy neans of partial diff. co, J Iz dz 5 ap" (y tax) —ars' (y—az), ae b” (ytar) +a? ye"(y — ax) : ax — PD (ytar)+ w'( ORE Boel (yan); 7 PY ar)+ %'(y—az), ae (ypar)+ uw! (y—az); J Day ln» 1erefore GP a reine e da? dy? * Many theorems of primary importance are deductions of so immediate a ‘aracter from the principles before laid down, that they are here introduced, con- ary to the usual practice, as examples, They areso far developed that no student ho has found himself able to follow the preceding portion of the work, will find 1Y great difficulty in completing what is left undone. DIFFERENTIAL AND INTEGRAL CALCULUS. du ,,( du du au _ (du, eu, ape 5): ? dz) da*” 246 dy * de)’ dedy’? dedy” du du & 2 therefore at” de dx dy eC 10. (Page 65 and Chapter V.) What relation exists between the two diff. co. of u, when % is that function of # and y which is obtained | by eliminating @ between u=ar+pa.y+ya ~O= 2+¢’a.yt'a, or cee da | du du da du du da 1 pgs Pee RRB AE ay & oy ye du du ant a) : 11. Eliminate the functions from z=¢ (y+az).% (y—az). d? log Ma d’ log z ze fz 1 ( + et 7 )=0- 5? [tae aa By (8.) dx dy? 3 or dix? a dy? 2 dt dy? dz dz — a be 2 4 wae mea == 12. 2=(@+y)* > (vy) gives at age : i : dz ae dz 1 dz dz 14. ~oee e 1 MEO dae SF = —— ae AL a dudy zx dx dy a? ( 1 1 dzdz A | “fess (px) gives sedi dx dy logz/ 2 da dy . 2 15. Required the expansion of tan in powers of 2, by Maclaurin’s theorem. du au du Let u=tanz; then ak 1+u*, ao Wott Qu* Pu d*u a 2+ 8u?+6u4, Aa 16u+ 402? + 24u° Bu ——== 16+ 136u?+ 240u*+ 120u° dx> du ri oe 272u+ 1232u>+ 16805 + 720u7 Ou 999.4. 3968u'+ OU 21936 es : Fae eooee ax u+ eeee Jy BOF oo saat ee A Ade ember MOT PR Tt TS 6 -“ MISCELLANEOUS EXAMPLES AND DEVELOPMENTs, 247 A] 16. Required the expansion of o—- This expansion (which is of great importance) may be facilitated by the following process :— tH Let pe | ; then ¢r—p (—x)=—2; consequently ox can have ao odd power of x except the first: for every odd power which appears in the expansion of @r must appear in Pr—d(—«) ; and every even power in Gv+@(—2z). Letuw=r; then ue*=a-fu, &(utw)=14+u, &(u+2u’+u)=u", e* (ut+3u'4+3u/+u%=u", & (utnu’+....)=u™, Make r=0, and let the values of the function and its diff. co. then become U, U’, U", &c. The preceding equations then become U=U, U=1, U+2U’=0, U+30'4+3U0"=0, U+4U'+60"+4U"=0, &e. Or, generally, n— n—1 2 2 the labour of using which is diminished by our having proved separately that U'’”=0, U'=0, U“"=0, &c. Let n=2m-+1, which gives U+nU’-+n : UE Ss en Gere pre M014 4 CAR 2n—- Yer m, 1 Uen-3)— om In=-1 2n-2 %m-3 7 (2m—4) RES 3 3 4 5 U U aa pe See Soe mu U 2nm+] Phis series exhibits the dependence of the terms on one another, after U"; but the series (A) is more easily used. It gives ‘ } U+2U’=0, or U=—33 U+30U0’'+3U"=0, or Uses U+40’4+6U0"+40"=0, or UO” =0; U+5U’+10U0"45U"=0, tol & Gel me 30: . ; this I U4+70'4+210"4+350"+ 10"=0; or abcess U+9U/+36U" + 126U"+84U"+9U"=0; or Ute-—; U+11U’+55U"+330U" 4. 462U"+ 165U""+11U*=0; or Uses 2) 691 ect 53 (ae 3617 wvitt_ 49967 at o7s0) Gets 540° 708 tar Hence, if [n] denote 1.2.3......7, we have v i 1-28. ] "pe eval i e—1~'—3°t6 230 [4] 22/6] sore)" °"° The values of U, U’, &c. are called the numbers of Bernoullé ; 248 DIFFERENTIAL AND INTEGRAL CALCULUS. and though they do not follow a visibly regular law, yet the con- ; nexion between them is simple. We shall in future call them Bg, B,, Bz, Bg &c.: thus 1 1 iDot 43 B,=—- I 1 P 17. Required the development of ea by Bernoulli’s numbers, — — a x 22 a pee SE a ey —+&c.—(B 224 &e. s* 41 pany | er —] Bot B,x+B, 9 + we ( ot B, f+ Ca 1 ue - a “oe ieee RR EO Re oe) Be ag 1g ae a 3 wast 4 TO 29 Rte OT Gt ae 18. Required the development of tan x by Bernoulli's numbers. : erV a1 eta 1 eerVi__y an c= ———_,S- S9s- a SS oe a eV gen A ae he ] ( 2 ) =—— + 1— ey iy V1 . pe 1 : Ox 1 (Qx Vit 7 —_——— 1 2B B ‘_ B Raa: tice Ve @eee Fay (28 6B S42 RSTO) tonne) Pina x te — — 24 29*— up . 2° — 1 pepe nt Ramat Yes, NY | an ow ( DB ray? ( ) Boe in which the law of the terms is sufficiently obvious. Reduced, this | becomes . Ropaee x OB ll. beeen oO 8 “QRS oo) oeao ane 19. Required the development of cot # by Bernoulli’s numbers. — cotazzV¥—1 (1+ pes. gel 1 on oar Cl rs fla 2 aie Da 1 ve a 2x5 as 2x —e 8 45 945 4725 93555 “°°° 2 0. Required the devel t of —U. 2 quired the development o faves u us" +- we *= 2 MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 949 oe” \ € ‘% Ted, n—|] nth diff. co. we’ =e" (1 +nu'-n - u'+... :) od La Se n— | Sass nth diff. co. ue “*== Fe“ ( u—nw! +n—-—_ yl... » ” odd, 9 +, 7 even, ed ‘ n—l] nr—I1 n—2 n—3 min (16.) U-+-¥n ——U"+7,-_ __* iv nS (16.) go Un eer d=, which is true for even values of n, and there ca | n be no odd powers of x in this development, eal: U+U"=t or US 1 3; U+6U0"+U"=0, or Lee U"=—61, U“"=1385, Ut= —50521, and so on; Yes a 5at 6la® 13852° 505217 ope 3 tiga tase bs 2 ia) [elt [sy [10] 21. From the last it readily follows that bat 6lz® 13852* 5052127" sec =] -+-— + —-{ 2 "(4)" te) * (ey * pio) : 22 22, Required the development of Sete [Why do we not attempt to develope 1--(e"—e) by Maclaurin’s theorem ?] UE AUSF cz Dy e* (uu!) +e-* (u—w!) = 2 Ver, P n—1 n—ln—2n—3_.. After which U +7 —— U"-++-n—— ne oy on Se Oe WHC 2 2 3 4 s derived from odd values of n, and gives the even diff.co. No others ‘an enter, for reason in (20.) 1 ™ U+3U"=0, or ea a3 U+10 U’+5U"=0, or Urea 31 ae SBE Li ae ee a y 2] ‘ 45 aoe eit te he OF ts TE ey AB” B81) x ee «38 215 [4] 216] 43 [s] | 23. From the last it follows that 1 biden es | ae se aes 4 cosec T=—-+ Me: bs ed Mi ba (oder eee Le a 32 15([4] 21 [6] 45 [8] (24.) What is the best formula for approximating to an arc of a ivele by means of its chord and the chord of its half, and what is the iror, nearly ; the arc being supposed not very great ? Let 6 be the angle (in theoretical units) subtended by the are S, and | the radius (unknown): let C be the chord of the arc, and C’ that of a) is half. Then S=aé, C=2asin 5 6, C/=2asin rae DIFFERENTIAL AND INTEGRAL CALCULUS. 250 Let pC-+qC’ be the formula required; then pC+qC'=2a (» sin 5 é+qsin 7°) SP 9 \e hind eee ee \ =oa{(2+9)o (G+5)3 Hei Tao hA8} n saint ee Gee ne ae ee 8C’—C_ o eS s(1 7680) nearly. Ans. The third part of the excess of eight times the chord of the half over the chord of the whole is nearly the arc: the result is too small by a proportion of the whole, which varies nearly as the fourth | power of the arc, arf is about 1-7680" for an arc subtending an’ angle of 572°. 95. If C” be the chord of the quarter of the arc, then C4560" 400s fy, 45 “ fQ—sin?ay ~ /(—sin? x) should have been taken negatively when cos. x becomes negative. Con- sequently, after e477, we have OE Ode Se the constant 7 being introduced because #==7 when s==0. Denoting the series (A) by A, our final result is that when « lies between —$7 and +47, =A; but that when @ lies between 4¢ and $7, c—r—Aj; when between $7 and iv, r=2r+A, &.; which may be all included in the following : 1 When z lies between (n -~ 5) 7 and (n+ 5) T, c=nr+(—-l1)’A, 28. If c=4-7 or s=1, we conclude that Lapp by hd 113.51 See oA ee We proceed to ascertain whether this series is convergent or divergent. 29, Granting, as will afterwards be proved, that ‘2 ayes nN ox nts er has the limit unity when m is increased -without limit; required an expression which may be made as nearly equal as we please to 1,3.5.,,,2n—1, on the same supposition, 252 DIFFERENTIAL AND INTEGRAL CALCULUS. het LQTS ia. nan or nrte en, on, so that 7 has the limit unity when 7 increases without limit. Then 1.2.3. .2nc21,3.5..(2n—1).2.4,6. .2n=1.3.5..(2n-1).1.2.3..0.2%3 Von Lomi Bie -2n Beye Sm ge p2y Oo" Jon n™*? a bn gn? and 2"*# n" e-* is the expression required. 30. The series at the end of (28.) has for its (2+1)th term °1.3.5....(2n—1)=—— 1.3.5....(2n—1) 1 ontd nt em p2n—pn 1 tena RESET TP oo? OL a 2.4.6. ee 27 2n+ LV Dye ge toe nits 6 ".on QIn+1? 1 Ll Qn 1 le ne (Pn)? 2n+V’ which (since $2n and ¢n have the limit unity) has always a finite or 3 ratio to m 2, Consequently, the series is of the same character as 38 =n %, and is convergent. (Page 235.) But it may be shown ima similar manner that the original series is divergent when s>1, in which case zis impossible. Here, as in many other cases, a series becomes divergent at the moment when its algebraical expression becomes im- possible. 3]. When z hes between mz and (x+1) 7 1 Bite o's cos’ x } 23 24 5 Prove on both from the preceding, and also ingen” 32. Required x in terms of tan a {tan w=); d.tan x ac ae rag —? + #8 “~~ e008 v yy (di—# di+#dt ) F ibid ae Sie’ vs C= 1S aye ey Gk der te the constant being nothing, since v and ¢ vanish together. This is true’ 1 1 from c= — aH to w= 3, oF from t=—o« to t=+a, though the | ae 1 1 peti series 1s convergent only when z@ lies between rget and Bory a, the | former exclusive, the latter inclusive. Generally, when 2 lies between | (ns) and (n45)x, c=na7 + tan 2 tan’ a--.. a | 33. ‘The fopomiNg series may be so easily deduced from some of those. which precede, that they are left to the student: ene Sete ae A 2 ae loo sinjasslog a — i ee es ee ¥ 4 3.2 45 4.045 60... 412508 0 oeo 7are 3 4‘ MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 258 2 pera 6 17 28 62 x” log cos = — — — a a ae Strasser eee. 2 a 4 15 0 $815 8 2835 10 : sin 2x Verify these by log —y = log sin e+log cos x. I now give some examples of finite differences, (Chapter IV.) 1 ‘wakl 34, Asinv=2 cos (e+ Ar) Sin > Ag ; aed I earl e A cos r= —sin (e4+4 An). sin ei Acs 35. Let Ar=26 A’ sin v= —4 sin (4+ 20), sin? A* cos = —4 cos (a+ 20) sin? A’ sin r= — 8 cos (v7+30) sin? @ A*cost@= 8sin (2+ 30) sin? 6 Afsinz= 16 sin (x+40).sin‘ 6 Atcosr= 16 cos («+ 46) sin‘ @ 36. Let n be any whole number; A sing « 2” Bein (e+ 4n8) sin*” 6 Aer'sinv= 241 cos (9 t4n+ 10) sin***!6 At? sin y= — 242 gin (e+ 4n-+ 26) sin? 6 A sin a — 2!"+2 egg (v-+ 4n-+ 36) sin” g ee SCOsige OPE) Gog (v7-+-4n8) sin*#" 9 A** cos = — 24! gin (e+ 4n-+ 10) ane 2 A™ cos == —2*"** cog (x + 4n-+ 26) sin*"t? 9 A®™ cos a= 28 sin (r+ 4n-+ 30) sin*"t* 6° 37. Required the successive differences of the first term of the series, ete Sd i ns positive wh. no.) i=l AO=1 AO 0 ALO= 0 Mapa) AAS) = (eae =2 A.O’=1 AX.0°= 2 A?,.o°= 0 At 02s 0: Ab IO%=0, &e, me 4.0°=—] AO= 6 A*.0=6 aAt.o— 0 A, 08==0 &e. =4 A.O‘=1 A*®.0'=14 A’.0*=36 A*.0'=24 A.0'=0 &C. | 98. The following table contains the differences for the first ten Iwers, and the same divided by 2, 2.3, &c. Mee | AP AS AS | As | Ar | at ls As |) aw | 1 | 1 |2022|55980 |318520/5103000 |16435440 | 99635200 |30240000| 16329600 .. 3628800} 1 3 | 1 | 510/18150 |196480| 834120 | 1905120] 2398480| *1451590 362880 1} 2 8} 1 | 254! 5796 | 40324] 126000 | 191590! 1411201... 40390 3 1| 3 7} 1] 126| 1806 | 8400] 16800 15120) ...5040 6 7 1) 4 6) 1] 62) 540 | 1560! 1800 |.....700 10 25 15 1} 5 5} 1] 30) 150] 240]....120 15 65 90 31 1) 6 Sil} 14] 36 |....94 21 140 350 301 63 Tet feel}. 6l....6| 98] 966 1050; 1701 966 197 1} 8 2] 14...2] 36] 469] 2646 6951} 7770} = 3025] 255 1| 9 1j}..1} 45] 750 | 5880! 929897 42525) 34105 9330 511 1| 10 mjAN) AP} Ae | Av | AS Veh pe 3% A$ A’ A i aera neeremnpsereenseensesnnens sayrmenrapeeeet! neceneeconoes 4. O54 DIFFERENTIAL AND INTEGRAL CALCULUS. The upper half of this table, including the dotted lines and all above them, gives, the differences as marked at the top, of the powers as marked on the left. The lower half shows those same differences (read as in the bottom line, the powers being on the right) divided by 2, 2.3, 2.3.4, &. Thus A'0%= 126000, and A®.0%®—2.3.4.5= 1050. The following will not be found in the table: A?.0?+2, A®.0°—-2.3, A‘ 04-+-2.3.4, &c.; but (page 83) each of them must be unity; for x" being a rational and integral function of the mth dimension, we must have A".2"=2.3....7, for all values of x. And for the same reason A".0™=0 when 7 is greater than m. r—1 ,, <0" o@eeee 3 AS. 08 + 39. (Page 19.) a”=0"4+2.4.0"+2 Fel erat «(@—-1) era+3r (2—-1)+ «(a—]1) (@—-2) ata+te (7—1)+ 62 («—1) (e—2)+2(x—1) (t-2) (w-3). AO. We leave the following notation, much used by the German mathematicians,* to be explained by the student : aael* a(a@-aleel® ax(eta)(t42a=2°"* &e. gael a2 (a—ler!? x(@—-1)(@—2)=27!"" &e. 1.2.83218" 1.2.3.4=14" 1.2.8....n=1%"" 163.5.7.92P £14.7.10.18. 16=1°" gm "=e (an) (x+2n)....(m factors)....(a-+m—I1n) x l*@—a2 xX (rtny™ Vrtrttaeyt ye (m1). 41. (Page 84.) 074+ 1"42"+4....+(e—1)” is 22”, x—-l r—lr—2 Pet vO .0” —— A? 0"-4+ «2.6 Seca +a = AO tas gal Wich Hen ek 1 1 ze => x (w—1), rr=s a (#—1) +52 (t—1) (@—2) 1 Laas e (e— 1) +e (7-1) (a2)434 (x1) (w—2) (28) 1 7 6 i Sate ge g8l be — ath ag?! Sees eee eR 1 25 zr=5 | pee og 4 grt gah ea on _ 42, Calling such expressions as « (7+) (x+2a), v"!", &c. factorials, tis required to deduce 2*, 2°, &c, to factorials, without the use of amy general theorem. 1. Let x, e—1, e—2, &c. be the factors; then ema (t—l)+a : #=2* (2—-1)4+2 * The only English work of which we know, in which the student can find in- stances of the use of this notation (which has not found favour anywhere bat im Germany) is Nicholson’s “ Essays on the Combinatorial Aualysis,” London, 1818. MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 255 | =a (t—2) (@—-1) 4.22 (@—-1)42=27432 (t—1) +2 (2—1) (x— 2) a= a(x-1)+a°= 22-2) (v-1)+22°(a-1)+2°=2(«-3)(a-2)(x-1) +32(2-2)(x-1) 4- 2a(a-2)(a-1)+ 4a(a-1) +04 32x(v-1) + «(a-1)(x-2)=2+ Tx(x-1) + 6x(2-1)(x-2) +2x(a-1)(a-2)(2-3). 2. Let 2, 2+-a, x+2a, &c. be the factors, then w—=a(2+a)-ar, P= 22+ a) —ax"=2(v+2a)(r+a) —2a2x(a+a)—ax(x+ a)+axr =2(c+ a) (2+ 2a) —8ax(2+ a)+a°r aim 7414 Gq72 144.72 72le_ g8y P= 9°!4_ 10qrt!* 1 25 q2 x14 1548 vr let qty, If a be negative, all the terms will become positive. 43. Required the law of the table for A” 0", (page 253.) n— 2 —2 BN" tS 2" — (n-1)(n- 1)"™"" + (m—-1) = (n— 2)". 1"-}, A’ .0"%=n"—n (n—1)"+7 1 (n—2)™ — sae Me tn, 1° 0? But the first series is » times the second, and by the nature of differ- ences A". 1" ig A*1 "1 An gmt 3 80 that we have the following simple law Ae Oo" — n Fer’ O™—-1 te An 0”-") for the upper part of the table; and for the lower A” 4 Oo” A"-? s om-2 A” ‘ o"- ON ER AGW fap ema Ye. ae oy ay This we may verify from the tables as follows : 240=4(36+24) — 1800=5(1204+240) — 126= 262-41) 350=4.65+90 301=3,.90+31 63=2.314+1 44. To form the differences, and the divided differences, of 0". Taking those of 0" from the table, we have A .0Mv= 1 A?.0’= 1022 Ae OM=—= 55980 A’. 0° = 1022 A ®.0"= 55980 A‘ 0”’= 818520 1023 57002 _ "874500 : 2 3 4 > A®.0%=2046 A®*,0"=171006 Ato" 3498000 and so on up to A” 0"=— 3628800 Au ov— 0 “3628800 LE A".0" = 39916800 Let the divided differences be signified by attaching accents instead xf mambers to the letter A, Thus A’” 0" means A’ O"—-2.3, AYO" is MO"2.3.4, &. Then - 256 DIFFERENTIAL AND INTEGRAL CALCULUS. AM o"= Aled) Qv-} 4. nA™ gn! xn j aN i fk mh 6 A” 0°= 9330 Bw a= t022 SAO 2190 4A” 0°=136420 A” 0%=1023 A” 0" =28501 A® 0"=145750 and so on up to AX ONs=1 11A8O"=0 Axigu ply 45, To find the law of the series for 2”, expressed in factorials of x, x—a, x—2a, &e. In (39.) substitute za for x, and multiply both sides by a”, v= a" 44+ AY o™ g- a? | LAM oO”, qn 3 4g?! Oy Ao 0” qr at | anc ae =z" es Ae) 0”, ax") | “ae tN) Qo” a2 yn? | we Sas 46. Let the terms of a series be, a@(a+6) (a+2b)....(a+2b) the first, (a + b) (a-+ 2b)....(a+ (@+1)b) the second,..... and (a-+(n—1) b) (a+nb)....(a+(at+n—1) bd) the nth; required the differences of any term, and the sum of any number of terms of the series. Jap Soom Aa (a-+b) =(a+b) (4+ 2b) —a (a+b)=26 (a+0) Aa (a+b) (a+2b)=3b (a+6) (a+2b). Aa (a+b) (a-+2b) (4+3b)=4b (a+b) (@+26) (4+30) Aa (a—b) (a—2b) (a—3b)=4ba (a—b) (a— 20). Thus, denoting by [a, a+b] the product of a, a+b, d+2),++4. a--xb, we have, on the supposition that successive terms are made by, changing a into a+, A[a, at+2b]=(@+1) b[a+), a+ 2x0] ND: un v, aa ree) y, Ge of all the factor) factors of factors. except the lowest. A [a+yb, a+ab)=(e—y+1) bfa+ (+1) b, a+ xb] A‘ [a+yb, a+ab]=(@—y+1) (ay) [at (y+2) 6, a+2b] A? [a+yb, a+ 2b]=(#@—y+1)(e—y) (a@—-y— 1) [a+(y +8) b, @+26] 47. What is the sum of the series [a, a+yb]-+[a+b, a+ (y+) b]+....+[a+xb, a+ (y+2) b]. This (page 82) is the function whose difference, when @ is changed) intox+l,is [a+(¢+1) b,a+(yt+«+1) 0], and whether x be changed into r-+1 or ainto a@+6 the result is the same in any single term. It’ is also denoted by = [a+ (#+1) bd, a+(y+a+1)b]. Now (y+2) b[a+(a+1)b, at+(y+e+1)b]=A [at+2b, at(y+e+1)b), pte+l)l or Spat (41); BaHeyte pL) Ble O-p et ett ) 4], MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 257 but by the hypothesis, © [a, a+ yb]=0, since there are no terms pre- ceding [@, a+-y)]: whence making r= —1, we have O — oleh at+y0) : (+2) 6 so that the final result is as follows: [a+2b,a+(y+ta+ 1)d] : eee b ss b rn [a,a+yb]+...+[at+xd, a+(y+x)b] G42) b Py la—b, a+yd] (y+2) 6 48. The following instances should be completely solved by the - preceding process, as well as by its resulting formula. a meee .9.6.7—1.2.3.4 BB a- FB r4-bi-4 h.G : a 204 4.1 be Gof a1 De 2.84+3.444.545.622— "8 _ gg x(v+1)(2+2)(x 1.2.3 1.2342.3.4+...42. (241) (42) = 7S CH IOHS) 0.1.2.5 x (e+) 14+2434+4... pa EE) 22(224+2)(27+4)(27+6) (274-8 PAGS+.+20.(2042)(22+4)2a46)=— ET ICHDC Cr+ 8) ey. Required 1"+-9"4.,.. , +2", or > (x+1)". BY 94+ +1), S41 st | a4) x (a+1) (2x+1 But ©.1°=0, C—0, and 5 (e+ pa St Grr) P A20™ Again, (39.) (v7+1)"= AO" (x7 +1 J+ @+De A20” tO3 A? 0” 23 (+1) a (¢e¢—1) (a-+1)2(a—1)+.... AQ™ 2(¢+1)"= Ts (t+1) a+ 3 Am A + 5-3 gt) 2(e-1)(a-2)4.. ompare this with (41.) 50. Required the successiye ‘differences of 1--[a, a+b]. As an stance, take 1 1 a@ (a+b) (a+2b)? (a+b) (a+2b) (a+3b) ° ] 1 I ee te) Be a(a+b) (a+2b) (a+b) (a+ 26) (a@4+3b) @(a+6) (a+ 2b) S DIFFERENTIAL AND INTEGRAL CALCULUS 3D ~@ (a+b) (a+ 2b) (a+3b) Similarly, if 7, Ws Us, &c. be in arithmetical progression, 6 being the bt cn 8 common difference, Uy — Unti 1 1 hd Uy Ugiiestyn Ugly. ce Uig, UjpUgee Uy Uj Ug Ug. eUnti nb 4 1 1 1 — ——___——_; whence 2———_—____ = — +C Uy) Uge ee o Unt U, Ug. eo Unt nb * Uy Uge ee Un 1 1 1 —= —- ——_—— —— +-C, Ui, Ughtigt ay (n—1) 6 Uj Ug. ee Un-d 51) meneited 1 A 1 + an’ ] 51. Required ———-+>——<+ -00e + To mr q 2.3.4'3.4.5 2 (@+1) (@+2) i 1 1 te ae 1 wo pe ly SE ee dle otal gh | = GED G42) ES) 2 GED HR) 22.3 2 @FI@H) for C must be such that © vanishes when v= 1. 1 1 1 52, eed he ee ges ee td 52. Required 7s-gat5¢.4513.4.56° °°" inf, The sum of z terms of the preceding is 3 ress pe abn i (c+1)(27+2)(2+3)(a@+4)) 31.2.3 3 (a+ 1)(a+2)(@+3) which, when z is infinite, becomes = : 31328 Verification. PATS =( 1 sa pol i ¢ : 31.2.3 \3.1.2.3 rae 3.4 3.2.3.4 3.3.4,.57 Sn 1 Od BES a +5575 +... oo Se Tae Lt : nip to x terms is 43.5.7 “UBS Oye T9011 oe 1 Beit alive a0" b"-* (at-bx) (atbx+b) 1 iar A" 0" b"-*(a-+ bx --b) (a+ bx) (at bu+b)+.... 1 . 1 4 AO” b"~* (a —b) a—— Al! om pr-8 (a—2b) (a—b) a—.... oO Thus for 1"-+4+3"+....+(2p—1)", (a=1, b=2, z=p—1), we get 1 ae SRT ey ok 3 AQ" 2"-* (2p—1.2p+ += AY0" 2" (2p — 3.2p—1.29+4+1)4+... 1 1 =p Aoman*x(-1x Dae’ oman’ X(-3x—Ix1l)—... + 59. From the preceding 124+3°+....4+(2p—1)? is? ou Bund gna Peas v1. Hepa Ee na which may be thus more simply deduced : (2p+1)°=(2p+1) 2p-+2p+1= Cp —12p-+1) +2 (2p+1) FOp+epe ee ic. This must vanish when p=0, that is C—O. Again, (2p+ 1)°=(2p4+1)2.2p+ (2p -+1)?=(2p—1)(2p+1).2p+2.2p(2p+ 1) + (2p —1)(2p +1) +2 Qp+1)=(2p—3)(2p—1) (2p +1) +3 2p—1)(2p+1)+2.(p—1)@p+1) +2 (2p+1)+(2p—1)@p+1)+2 p+1) = (2p—3)(2p—1)(2p+1) +6 (2p—1) 2p +1) +4 2p-+1), ‘the sum of w hich, made to vanish when p=0, is (p= )(2p—A—VYEP+V , 6 CP—NAp—VEp+) 4.2 % 8,2 et PDP) ot ee 8" Both of these expressions give the same results as before. 56. Required expressions for A“ 0+? (that is for A" 0"4?2.3. .2) in terms of x. We have (48.) AM orte—_A@-)D Orbe 7 AM Ori te, Let A” 0*+2 be ¢ (n, p); we have the S52 260 DIFFERENTIAL AND INTEGRAL CALCULUS. Ad (n—1, p)=ned (a, p — 1), where A refers to. This gives Ad (n, p)=(n-+ 1) d(r+1,p—1). Now p=0 gives A“) 0", which (page 84) is =1, whence Ad (n, 1)=(m+1) X1, or OM, =sn (n+1)+C. But A.0'*? is always =1, whence all these expressions become unity when n=1. Hence C=0. Ad (n, 2)=(n+1) 6 (r+ 1, d=. (n+ 1)?(n+2) n v(a+ 1) 1)(1+2) (u—1) n (n+ 1)\(+2) soak diate A SLAM oth itl lnc ars: n8) [u—1,n+3] | [n—2, n+3] a eta ow Pea 57. From the last it appears that ¢ (2, p), or A 0"*?, when divided by the product of all numbers from n to n+p, both inclusive, consists of p terms of the following form : A® ort Timer ie (n—1) +A, (n—1) (2—2)+4.. +A,-, [n—1,n—p-+1]. Required the law of the coefficients Ay, A,, &c. These may be easily expressed by means of the following p quantities, A") 0? (or 1), A® 07+", &c......up to A” 0”, Assume 7 in succession} CODE 1S 2s o +, Ps owe hae ‘then AS) olte A® 02+? a, ee A® Q2te A® Otte ——_—--—-== A), ——~—— 1 ——— [http (23,245) pants [2,2+p] [1,147] A® o8+P A® 93+? DA@ Q2+P AM olte Bop) ee Ree se ‘ [3, ,B+p] [2, [2,2+p] ge ‘1+p) 3.2.4 A® otte A® 08 te A® (2+P AW Olt e « — pate tI re ate aa (4,445) [8,38+p] [2,;2+p] [1,1+p] giving a law in which the coefficients of the binomial theorem will be always found. We have then (k not >p) Oly RET Ay pe tec, AOE hn 1 AO OF P) Kei [netp] teLpl-. 2 58. Apply the preceding to express A™ 0"*4, AM) grt4 P| ne +4] ae 1) + Ag (n—1)(n—2)+ A,(n—1)(n—2)(n -3) 1 sb io ae f 31 1 a4 ee ss* 2.3.4.5) 9.8.4,5.6 01.2.3.4.5 1/9/8,4000 MISCELLANEOUS EXAMPLES AND DEVELOPMENTs, Laas ae Bh) =o l cee 1% ete. t 2.3-4,5.6°" 10908, 4 ge et Dp. 7] 1701 301 4 3.2As= 7 8 5g tS Reagan! 5.6.7.8 3,4.5.6.7 °° 2.3.4.5:6 1.9.3.4.5 630 ids 105 Plg.45.0.0o °* 2.5418 67S ae 4 n—l AMort4— IE at J 336 +1400 (n—1) +840 {7 aft 1o5 n—2 HL [1,8] n—3 { 59. In the preceding, it is found, that if U,=A,+Ain+Agn(n—1)+.... Dera [n,x—p+2], then [] 3 ky A,= U; aeg provided k2: . whence n n+l +2 a b= rigs 0 pr Wg ot ‘i ete J ee 0” 2. ey. 2.3....n+] 3....2+2 er + v7 * © © * n—1n—2 61. Required V, =f} n — = .«+-(p terms)....dn, 1 ] Sidn=1 J vndn=s Sin an= —“T3" These are easily found by multiplication and integration : : thus = 1 4—8 Sn iA = dn== f (n! -3n? ti 2n) dn=-~ £( a nN +n) which, taken from n=0 to n= l, is DIFFERENTIAL AND INTEGRAL CALCULUS. Bai Nh _t 5(G—1+1)—§ 0-040, or But at every step the difficulty of the reductions increases, and the following method is given to show the manner in which the process of finding a large number of successive results may be shortened. fa +27)" dn=(1+2)"~+log (1+ 2) l+a 1 x 1 ”, Pees Rc em CER aa Died Pa Stay Sea NG gk el ve (1+2) Bee oe hein) log (1+2) log (1 +2) — —lLn—-2 (Q+a)"=ltnrtn-— ees = — Pirlemnick 5 a Be (l+2) dn=1+V,t+Vo27+V32°+....; or V,, is the coefficient of 2” in the development of log (1+ 2), or of 1 1 1 1+(1-52+ ay Sree Le 3 1 Taare a Ne Slt Vi aV, +s P+ ee Clear this equation of the fraction, and make (1+V,xa-+...-.) 1 ae fs ae (5 Bits). :) identical with 1, which gives 2 Vi-==0 ys 5 Vis Vitq=0 V.=— vray ae v= V5 Vet pling tees Vi=- = fected tub te tad) os all Vig Vt Vi 5 Vt ce ie Bi Vita Vi=- a5 and so on. 62. Required A"¢x, in terms of diff. co. of Ar, the series from which the differences are derived being dx, 6 (2+-h), (x +2h), &c. 1 may be shown, as in page 166, Aha A"pzx can really be expanded in a series of the form ab™z sh" +a, hr, ACTI+ ...., where a. ts &c. are independent of the function chosen. It therefore only remains to assume the function by which they may be most readily found. Assume . Arhr=agnn. hap Ms bh +agpeta Ate .. Let da=e*, then Adasz ett! — et (<'— 1) er: A’dr= ( th oe 1)(e"t*—€") MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 263 =(e"—1)?e"; and so on: whence A"ba=(e'—1)" 6”. And Px xz =O" 7, 4005 S67: whence (e°—1)"=ah" +a, hh" +a, h724,.., : or (60), An ; ort A” a-+2 A’g2= bx. h?- —— — gy poi? beh a, ot a. a ? 5 [l,n+1]° ibis, [l,n+2]" os bil Agr='e« h+6" zx fe FM £. ‘ ‘ rll 4 2.3 oe 9 ” Moy 7,3 Ths Lor=G" «.h?+hxz.h + Ae an “Fide “See ig ne BAT Nice DHF Aidr=h"z. LF +¢ 2. + px F + eee 3hs A'px=h''ex ht +h* x. 2h5 + p"v : = 4+ ..0. nh*t? (3n? +n) hr*? n = b() A” (n+1) (eee ae (n+2) ———— Clee ate A’Gr= hx A +42 : + px aa 63. Required the inversion of the preceding process, or the expan- sion of 4x in terms of Adz, A®ox, &e. As in page 166, it may be shown that a series may be assumed of the following form, in which a, a,, &c. are independent of the function chosen : ht pMr=mah porta, A 'ox-+ a, A" *oa+.... Let ¢z=e*, and we have, as in the last, h*=a (e"—1)"+a, (e'—1)"! +a, (e*—1)*t? 4. ,.. Let e*—l=z, or h=log (1+2) ; whence {log (1-2) = a2"-++ a, 2°F page"... ., whence a, a@,, &c. are the coefficients of the expansion of the nth power 9 ji of log (1+-2), or of the nth power of et erg 64. Required the expansion of (1+-ba-+ca®-++eat+fat-+... Let w=P", then P a or if w=1+Be+C2?+E2'+.... dx dz we have (1+ bx+ca?+....)(B4+2Cr+... =n (1+Br+Ca?+....) (b+ 2cr+..., =: Develope the mutiplications, and equate the coefficients of corresponding powers of x, which gives B=nb 2C+ Bb=2ne+nbB; C=ne+n n— | 2 b, Proceeding in the same manner, and making DIFFERENTIAL AND INTEGRAL CALCULUS n—Il n—1l n—2 n-——=n, Nn 2 My 3 =z, &c. &c., we have B=nb C=nc+n,.6° E=ne+n,.2bce+n,. 6° F =nf tng (2be +0”) +n,.3b°c-+n, bt G=ng +n, (2bf+ 2ce) +n, (3b’e + 8bc*) +n,. 4c +n, b H=nh-+neg (2bg+2cf+e*) +n; (3b?f+ Gbce-+-c*) +n, (4b%e + 6b°c*) +n,.5b4c+n, 6°. Though this is a good exercise in the method of indeterminate coefficients, yet the preceding coefficients (as far as Ha*) will be found more easily by actual development of Ltn (bate...) +2 (br+.. ; ] or [a \n 65. Required (14; +5 of eatt ae es ( ) i | 1 Oe aes — jee = he 1 B fia 1 1 3 5 Co 5 Mb GM n ‘ geil 1 Negi n (n+2)(n+3) loeae Bo 48 1 13 ] 15! +150n°-+- 485n° + 02n | F=—n Sa oe ee 5” 36 mtg en i 5760 CR te n an n : ? met nm at ie sTa5s ye! 87 59 q ¥ 1 ba =F n+ oT Nat Tan 35 8+ 54 rue +365 g "st eq” Changing the sign of x, we have . log (1+2)\" (et) =—1—Br+C2?—-Ex?-+Fert— ..... Verify the results of (61.) by making n= —l1. 66. From (63.) and (65.), 1 1 1 1 ‘co ch =Ax —~W’x4+— Aex—— At‘xr+— Abr—..... PME ET | Wracok AY» ne getty ae 5 137 "ye BoMar— Aex+—- Atr—— Adxr+— Atr—..... or lant T oe x 6 t+ 76 0% 3 : "2 B= = Ma—s Art ne get MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 265 iresitette2ats4 ger any Tne 67. Required Tae y, dx, in terins of Vos Yok Uses aon So yd fey, dat fe» y,at-+., AS eye, dx Let c=h0+ v0 fey, deaf Hier oy Odv reed | Yro4ve= Ya tVAYg+ v nt pa DY. + #6) £109 the differences being taken from the series 7 ae 0 Yrorve dv or - os dz tet AYig— poets 1 ima At Yrob - ts 0, CINE -» By (61.) Applying this result to the several intervals into which n0 is divided above, we have i 1 19 af? yz evi eas. A’y, —7a9 SY + 2 .)0 1 19 Siy.dr=( y+3 Aye 75 A*Yot = AY 790 Atyvtos..)6 e 5S ° e ° ° ° e ° ° ° ° ° e ° ° °° @¢@ ° e e i 1 ae VE d2&=Y apt 5 AY(n—1)9— ceee The addition of which gives 1 1 rome } no = = A paren er xA 2 eet 3, lied Se yade 2Yne + 5 BAY ng re) ak ry DA Yno— .. (0 ZA Yn Yot Ayot eoes +A Y(n-1)9== Yno— Yo aA*y, ng — ap A’yst.. eet agp tans &e. 0 Li Yu P= IY re 5 Yo) = F5 (AYno— Ay) +5; q (AY A°y,) — 68. As an example of the preceding, let y,==., 1 LYng=O-+O+20+....+(n—1) Ang n (n—1) 0 Yno—Yo=n9—0=78 ; Ay,.=0, Ay,=8, A’y,,=0, &c. 1 i 1 ey, di=5 n(n—1) itu n+ 0=5 n? 6°, which may easily be verified. 69. Required Zy, in terms of y, and its differential coefficients. From (67.), making nO=2 and 6=1, it is obvious that if the values of Ay,, &c. be substituted from (62.), a series will be obtained which may be reduced to the following form : 2y= fo yx de-+P (ys—yy) + Py (y/-—Y',) + Ps (Ye IY") vv ees where y',== diff. co. yz and y’, is its value when z=0, &c. And P,, Ps. 266 DIFFERENTIAL A &c. are independent of the value of x and the form of y,. ) INTEGRAL CALCULUS. therefore choose a function by which they may be determined. Let y,=é", then Pe edmond WO BL 5 Fe SLR ey, Go vB * 6 dz —=—_—_—. p a yf =a (7-1); yf" 'y Ya ett 9 Ys —y" = a? (se as gle—V)t an ® e¢— Sheed | 1 Yee Ec ho ee rad (<1 —1l), &c. Substitute, multiply by @, and divide by ¢**—1, which gives e* 16.)=1 1 Vs ly‘ at 1 Jak Sipiege een ery. 30 [4] 42 [6] Hence 1 YY .—Yy', 1 yl yl! oS sy dt 5 Quire Eee ge a are ok A Se aS OP ee ae 42°2.3....0. 30 200. 8 - 662.3 10 is This is ee series alluded to in page 165, and it might be obtained from A7!u==(e™—1)7 u. ya 1 a aap We a ee — —— a Lea Laat 3 oN Vee N es ene ae 2r+1 2 E (Qep1 SP)" (art 1-1) tee. —=14PatP, a+ P,a’-+P,a!+P,a°+P;ace+..e. The following are examples on the subject of Chapter V. 71. 2=6 (2, y), y, which y is of z and a: function of x and y, and of z dz _ dp dp da dy va(ee dp dz dx dy We must | y==P (2,2), or z is the same function of x and required all the diff. co. of this system. There are three variables, and two equations; consequently there is one independent variable. First, let the independent variable be x; let @ and ¢, denote the dy dv i (29 dp db +a) 8 dx and @. o = dp, dz dz dz dx dd, d¢. yp Vos ee dz dy p, dd dz dy Secondly, let the independent variable be y. _adpdx dz dy dx dy a dp a dQ, de dp, dx dx dy MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 267 da =(1 _ ah, dp _ (db, dp — do, dy dz 5) (3 dz eS) de (4b 4 tht) (drab a dy dz ~ dx aes du 2 ts Thirdly, let the independent variable be =. yee dB dd dy dy __ dd, oy dx ~ dx dz dy dz? dz dz ' dx dz ds_ (tht) . (tb db a dz. dz dy hy dx dy =) dy (th: 4 a8 do (dp apd dade dz dels dx dy BY 12, s=2*+y, eae dz du di dz ie Sar Laas 3. P at wat9;, fees dz * Cee dp See dz 2e+1 dy _ 4rz1+]} dx 1—2z (rn a dz dx dz dx 2. — == 2r — l=322— — dy dy * ‘ dy ia dz 241 de 1—2z dy 1+42rz dy 14-422’ dx | dy dy dx 3, Lor 7 ~=<9 bite TE 2 etre: dt__ 1—2z dy _1+42z dz r+] dz Qy+]! This example is given at length to illustrate the fact, that when there is only one independent variable, whatever the system of equa- tions may be, the algebraical character of the diff. co. pointed out in page 54 remains: thus in the present instance dz .dx - dz dy _y By Ons Gphiae ly Us da dy 73. (2x, y, 2)=0, which requires that z should be a certain func- tion of x and y, implied in the preceding equation: required the first and second diff. co. of z with respect to # and y, When there are (as in this case) two independent variables, and two only, the notation of Lagrange is sometimes convenient: or G2 soihhe ON a LE d?z het tl '— da’ '™ di? dal 3 abe dap ody? but when a function has several variables, as ¢ (x, Ys 2) the partial diff. 0. are expressed by Lagrange thus, 9'(2), ¢'(y), and '(a), which is objectionable. The notation used by Arbogast is as follows : — DIFFERENTIAL AND {INTEGRAL CALCULUS. Dy for 2 Dey for 2 Dy for $4, &e. ait When there arc several variables, this may be modified* as follows : D.@ for SP, D2,@ for ce D? for 2 &e. Leaving the student to employ either of these notations as an exercise, | I will suppose dg=Mdx+Ndy+Pdz=0 dz M dz N Jd (ou —- = — — —_ ge dx Pp dy p Care he ae ai d.P d.M eas paee) = — Pp? dx dv'P dx. ae dx ) : — — M dP a)? dM Paci = \\ is (M & ep dz d "Ga * dz ie) ea dP .«M dP. dM M faites p2 ' dg» Po-dz ia & ae : dM i =| dP ue re pe MP Pic MP Me ANY, ] 1 or ee we Sp dz — — OP. 2 dpdd dd (F o\? dp dp? ad (4) ey: dzdxe \dz) da \dx/ dz dz CERN: SEU) BOS He drdy dyP dzrP At ou eta ea d°p _ ay dp d’y -(f) a ee dp dz\.dedydz' dydudz) da dydz? \dz/) dx da dy dz te aN ye a dy Pp wy = {2% dp db @&h dp 2 dg =) at (‘s 8 z dz dy dz dzdy dz dy? dy 74. Show from the preceding that (SF Oz a2 ‘bax dp 4 re dy? (55) =x zi) + x(3) tie P . aes ie d * The system alluded to in page 198 (note) consists in writing d,y for = Ke. dx’ The confusion thereby introduced as to the fundamental meaning of the symbol d is reason enough against this system: had the capital letter D been substituted, as by Arbogast, it would have had some claims to be used coordinately with the ‘old system. I should recommend the student always to use the common system in expressing results and reasoning on principles ; employing the one now explained _ ie shorten mere processes, when the common notation becomes of troublesome — eng MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 269 r ox U4 dp dp oy! ;dp dz o7! dp dd dy dz dz, dx t > ede dy a ae ( i ) x f% @b ah ad dy” dz? dy dz dz dz" dz da dy ~ dy dz’ dx? y-2% a -(<3 yet? Cb adh ap dz? dz dz dx dx dy” dy d: dz dzdx dy? — 7 _&h Up -( dd i 7! — ap dd ap dd ~ ag? dy? dx dy dy dz'dzdx dx dy dz? 73. Show that R= rid (+05 gives ag = + Qt Ping Cokes dx dy dz 76. Given ¢ (p, g, r)=0, where each of the three, p, g, and 7, is dz dz a function of all the three, x, y, and z; required 7 and — ax di y 2 .-(% dp ihe dq es dy dr _(% dp ipa dq ,dddr dx ‘dp dx bay dx ' dr dx \ dp dz sida dz‘ dr dz)’ in which x may be changed into y throughout. The following are examples of methods subservient to integration. 747 What is the value of the diff. co. of (vx—a)”. dx, when Ce os Px eid its diff. co. being then finite, and m being a whole number. D* standing for the kth diff. cO., we have D" {(a—a)”".¢a}=2.D* (w—a)" + kd'z. D'! (2—a)"+ HAP! M.D (a—a)" 44x, (t—a)”. When r=a, D(x —a)” is =0, whether v be >m or Stee whence Anio=[m+v] x 7 =[o+1, v+m].d@a. hn ¢ for h write t—a, and we have («—a)", gz; and h=0 when r=a. 270 DIFFERENTIAL AND INTEGRAL CALCULUS. "8. For (2 — a)? G2; Ac 0, Aye 0; Ap 0), Aj= 1.2.38 A,=2.3.4.¢'a,; A,=3.4.5 9"a, A,—4.5. 69a, &c. 79. Required the values of the successive diff. co. (v=a) of wa=f{A +A, (v—a) +A, (t@-a)*+ 206. +A, (w@—a)"+.... } pa. Apply the preceding rule to the several terms of the form A,, (v—a)” $2, and we have wa =A, da wa =Ay Pd a+1A,¢a wv" a=A, pg" a+2Aif’ +1.2A, da y"a= Ay Pl"a+ 3A, ¢"4a4+2.3A,¢/a +1.2.3A; ha wita= Ap a+4A, P"a4+3.4 A, p!a+2.3:4A,¢/a+1.2.3.4A, da, and so on, the law being very obvious. 80. Having wa and ¢z, two rational and integral functions in which wa is of a lower dimension than (t—a)"¢z, it is required to expand t-+-(a—a)" be into a set of m+1 fractions of the form Wa Ao Ay Ae Pa Fe = + (t—a)" pe (@-a)y” (@— Gyms L—A ox : This equation, cleared of fractions, is woe={AytA,(a—a)+....+An.(2—a)""} dot+fe.(z—a)"5 and every diff. co. of the last term fx (#—a)", which is under the mth order, vanishes when w=a. Differentiate m—1 times following, and make x=a in the results, and we have thus m equations for the deter- | mination of Ay....An-1, precisely of the form obtained in the last, example; namely, a wa =A, da, or Aa a u/a=A, PatA, oa or A= wa = dia, &c. One differentiation more gives wma={A, d™MatmA, pr VMat i... 149 [33s Lenya, whence fa is finite or nothing. Consequently fx is an integral and | rational function of w; for it is the difference of two such functions divided by (a—a)”, which cannot be finite or nothing unless the numera- tor be divisible by the denominator. And fx may be found by the opera- tion just indicated. 81. It is required to perform the preceding process upon the fraction e+1)+-(a—1)*.(2@’+1). Here yese+l, érma'+l;, al, m=4, a a 1 Ag A, Ag Ag f@ GoIy@ ah ~G@=it TGs T Gab Taal Teel MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 271 2 Aso eye t 3= Ay. 2+A,.2 A=- } G=A, 2+A,.4+A,.4 As=5 G2 A,.6+A,.12+A,.12 Ag 1 . 1 ] x’ +1=(a+1) {l+5@—-D+5 ed ao (= 1) + fe (x— 1) Bie ck eee bee tae Dine t Pee: @-1I¥@41 ~@=1) 2 Gay +2 Gy I te ee = 1) Bar re 82. To perform the same process on (u*— 62") > (wx—2)*(a—1)°(a@—3) (2-5). The labour of such a process, which is considerable when there are many factors in the denominator, may be lessened by previous reduction, as follows : L xrt—62° P A B Et oO Ai IEi Oa ae ee, ta Sarg (a—2)?(a#—1)*(2—3)(a@—5) (@—2)?(a@—1)? #-3 2-5 Multiply both sides by c—5, and then make w= 9, which gives 125x—I1 125 ae a aa 3 . seals 81 Multiply both sides by z—3, and make z= 3, which gives A=—, 125 81 P (x—3)(a—5)= -62'+(a—2)"(@—-1)"} = (@-8)—> (2—5)}, and the first two diff. co. of the latter term vanish when Fh P Ao ae Ay af As ot je _ Pomoc oP ae te ee et ae (#=2)%(2—1)* (a—2)' "@—2)! *a—8 TG@—1» then since we need only two diff. co. to determine A,, A,, and A,, we ‘May use zt — 62" instead of the second side. To determine A,, &c., we shall have to differentiate P twice, and make w=23; we have then, neglecting the terms which must vanish, P (t@—3) (x#—5)= 2 —62°4+-.... P’ (x—3) (a@—5) +P (22-8) = 4a°9—18a°+.... P! (w—3)(a—5) +2P! (2e—8)+P.2=120°—360-+..0. Or making r=2, 3P=— 32, 3P’—4P= —40 Assume 5; or P= —— pix 2008 Pr OTe 9Pp— __ 9, . 3P 8P +2P=— 24 Q4 > 272 DIFFERENTIAL AND INTEGRAL CALCULUS. Hence, to determine A,, &c , we have We=P, or=(a#—-1)’, (10) Pea, — ae == A,.2+ A, A= — Re Ay 2A, At 2A, ee 32 j y Ce ra = SE eee QJ) Again, assume “ n= nid ne aaadeee (@-2)(a@—1 (a1)? “( =1) * @-2)" When r=1, the latter term of P and of its first diff. co. vanish; and proceeding, as before, we have (when t=0) Ya=P, gx=(a—2)%, P .8=—5 i pests P’,.8—P.6= -14 Piss a5 9) 5 (79.) nigel Bo=— a cAI 131 — 39° Bo 3 +B, (— 1) 5, = 39 P 5 1 L314 a: (22)? (z—-1)* 28. (9—1)2 7, 828 21 Geer ++ (hi). But since (f) and (f,) are identical, the form makes it obvious* that the indeterminate functional part of each is the determined part of the other: putting these determined parts together, with the two fractions which were separated at the commencement of the process, we have, as a final result, Ga VP82 FA. B6duisd (2—2)*(a@—1)*(x-3)(@—5)”——s 83 (a —2)® = 9 (a — 2)” 380 1 sae | _ 131 1 81 1 125 1 27 1808 (esd ia ao ee Bee eo 83. Given or=(x-—a)(x—b)(a—c)...., where no two of a, J, 2,+e0+ are equal, required ~x-—-Pv in the following form, Wx A B C — =r + — +-—— ~-- -——— -4- eeees px t—a «£—-b x—c ax being the integral part, if ya be of higher dimension than ¢z. If de be also given in its expanded form, an+pa""+...., common division will ascertain wz better than any other method ; but ifspax be no otherwise known than as the product of c—a, e—b, &c., the process * That is, when Wz is in the first instance of a lower dimension than the deno- minator. Were it otherwise, /v-—+-(x—1)? and fix (a—2)* would each contain the integral portion, besides the fractional portion of the other. This integral portion, if any, may be found as in the next example. MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 273 ~ of involution* will be more convenient. If wero Mo"+M, xh M,2""*-+...., division by «—k will give Ma™"'+ (Mk +M,) c™-?+ (MA*4+-M,k+M,) Pree xis Which gives the following rule for suc- cessive division by x—a, xr—b, &e. M M M M M, Ma+M,=N, Mb+N,=P, Mc+P,=Q, M, N,a+M,=N, P,b+N,=P, Qic+P,=Q, Go on in this way until the divisors are exhausted, taking only so many terms in each column as there are coefficients in the quotient to be determined. Thus, to find the integral portion of «°—2°—w divided successively by v—1, e—2, and x—3, we have ] 1 1 1 Ans. 2°+ 6x°4+252 + 89: the first column 0 1 3 6 contains the given coefficients ; the second,+ those By) l| 7) 25° after division by x—1; the third after division —] O| 14) 89 by e—2; and the fourth after division by «—3, — ——|-——_ The blanks show where work is needless, 0 5 —1 0 For the fractional portion, multiply both sides by r—a, and then make x=a, which gives baa bdr Babe SNL a Re ssd hl n(aetee ak At od pers EER y Tease are Aye ~ (e—a)(e—b)....” 84. Required the decomposition of (27 — 4x® 4. 32°— 2x) divided by the product of <—1, «—3, e+5,2+7, «Cc. 1 | a yi 5 | —7 1 i l 0 —5 | —192 0 25 | 109 —2 |~127 |—~so9 ¥% (1) oes ‘Ww (8) 81 (I-39) +5)(1+7) 48’ B—DBLDSETD) =~ 80 * See the Penny Cyclopedia, article Invoturion, + It is an advantage of this process, that the use of the divisors in a different der will serve for verification, fy 274 DIFFERENTIAL AND INTEGRAL CALCULUS. w(—5) 75525. % (—7) 674681 | (-5-1)(-5-3)(-5+7) = 48 *(=7-1)(-7=3)(-74+5) 80.” whence the final result* is ai —Axo + 325 — 22+ s 1 ] Oe ® — 127? + 109 2 — 890 - — — DGG GLiy ee | a 81 1 75625 1 674681 1 80 x—-3 48 a#+5 80 «+7: 85. Such an example as that in (82.) may be reduced to a succes- sion of such operations as the preceding, in the following manner.t First, “oki! ap gon eeiRen, he et 6 ania IR, FB (c—1)(a—2)(x—3)(2—5) 8xa—-1l1 34-2 42-3 TRB ack - 24 2-5 Divide both sides by (v—1) (e—2)?, and take the resulting fractions separately. | : 1 1 1 First. AE) peer: PET 1 59 ie! 1 ae? 1 1 (@—1N)(a—2)? @—2)) G@—N(#—2) (@—2)? a—2 “aed Secondly. 1 ne 1 1 1 Galp@ad)' “@-2)@-) @—-)G—-) ' @ 1 Re «| 1 1 1 1 1 (play Vad (lod ge ed * The calculations of ¥ (3), ~(—5), ~ (—7) should be performed by involution: and the safest plan is to put down every step of the work. Thus, for ~ (—7); the | complete calculation is as follows : . 1 x—7 —7—-4=—11 xX- 7 77+3=-80 x—7 : — 560 -2=—562 —7 4.3934 —7 —27538 —7 4.192766 At J (—7) = — 1349362 } 4 ‘This example, though prolix, is introduced as a suecession of simple examples of the preceding case. ; MISCELLANEOUS EXAMPLES AND DEVELOPMENTS 275 5 1 5 1 we #45 1 nm 8 (@—1)(@—2)' "8 (@—2y 1328 +8 Gop tga : 1 1 ] 1 a.) SNE, G> eo tee 32 1 32 I ete 1 $2.7 om I Be NGI Gays Gas Pea aT Fourthly. 1 1 ] 1 (1) (a@—2)(@—8) @—2)'@—3)_@—De@—8) *@oDaray 1 ra4 i] 1 al ee Ce Cee) a CRS SY CSO a a 1 l 1 1 ] 1 (2—2)"@—3) ~~ (@— 2) * G—3)@—8)— Gaye Be ] 1 | Bea | a i (@—1)(@—2)*(@—8) _ @—2)' 192-8 3 rT 81 1 Swe h | Meal 81. | 4 @-NG— @o3) 4 G2) * 8 a8 8 al Fifthly. l 1 ] 1 @-1)@-2)"@—5) (@—2)'@—5) @—D@—5) t GoD Gx) ] rk; 1 1 1 ; ete | Du] (@ -2)(x—5) ~ Ba—5 3 a—2 (w—1)(w-5) 42-5 42—1 1 | eee eS | (@=2)(@—5)~ "8 @—2}* 73 GB) GB) 1 ] } +] 1 ale TB)! Ogee tO a6 1 1 1 ae | i Set Jn ¥ (2-1) («—2)*(@—5) 8 @ 9) T9 pte 96S TI _ 12 1 __ 125 1 ea ee Oa («—1)(~—2)%( (e—5) 72 (a@—2)? 1082-2 125 1 125 0 864a7—5 864¢—1° For a moment let P,, Pes Ps, and P; stand for r—1, x—2, &c., and collect the five results, which gives for the original fraction 5 39-180. 39 = gp Bist Pte P- tae Pebuuraaiensy cho 81_, "81 125 125 125 125 —— P24 — Se ee Dee 7 PH SPAS nae 72 °* jos” * 864-51 96 7? T 2 DIFFERENTIAL AND INTEGRAL CALCULUS. nj 38, | 36, 880 5 ea ee eee A Ping Pigg Bat g Pit gg Pity Pera ‘© the same as before. ~ 86. It is required to decompose war+-(2"—1), Wa being a function of a lower than the nth degree. Let « be a primitive mth root of unity, (page 130,) then all the roots are 1, a, w....a", and by the preceding method (with page 130) xv Tom awa 1 ures «1 ep ae yo vl 1 jaye 1 oy 1 elon a sot wen ccnep. 2) Sh Ee n—1L xz"— nm «—1 Nn &—-a nm w-e n L—a Let wr=C,+C,a+....+C,.2", and let cos p+V—1 sin p and cos Ned sin p be two of the nth roots of unity, being a multiple of Ont-n: call these roots 7 andr’, The two factors belonging to these roots are then marr ail ope AEE 1 lL (ryer tr’ yr') err er eer) _ a nm «@—?r n «x—-r on a (r+r') xtrr’ 2(C, cos p+. .-+C,_1 cos np) t—(Co+C,cosu+... +C,_1cos (7—1) p) n x —2cosp.r+] 87. Required (2+2°)+(a°—1).{27+6 is in degrees 60°}, 1 Ci=2..C,-—1, cos G07— coro .60°= > cos 2.60°=cos 4.60° = ~5 Pe ie Samuaehe Ce bee ie ia ['tart —_— — Sg Dede Depl Davee) Me epee 88. Every thing being as before, except that the denominator is Us a”+1, » must be one of the odd multiples of a and we have Be yr oye 1 _pyB 1 A Lie z+] nn «2-4 fi Jonas ss ee where a, 6..--v are the m nth roots of —1. These nth roots are odd powers of any one of the primitive roots: for instance, if T maggots ea PS ac= COs —-++ ee sin -, n n the other roots of —1 are a3, @....a""". 89. Every corresponding pair of roots of the form cos utv—1 sin ps give in the decomposition a fraction of the same form as the last in (86), with its sign changed: thus (j denoting 7-7) is 2 cos (m+1) p.v@—cosmp _2co0s (2m+3) p.t—cos 3mp 1l+s"s 28 x*—2cosp.r+1 n v—2cos3u.a+1 the number of such fractions being half of n, when 7 is even, and of m—1 when 7 is odd: but in the latter case there is the additional frac- tion (—1)"t'n (+1) arising from the real root —1, eoeg 624/32] 2 l+2° 2-/3.a+1 * MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 90. 277 V3 .0-+1 P+ + /8.e+) The following are exercises in the methods of integration.* 91 1 (4(a+b2) _ 1 el ; (atbr)* 6) (a+bz)" (nb (a+bzx)"" le eae 1 1 ° dx 1 . —_——— == —_-__- —— =-loe ba). | (3—22)* ~ 10 (3-—-2n)° | Whine oS tae 92. To integrate 2"*(a+bx)"dx, in all c obvious method. —_—— — —— ases which present an First, let 2 be a positive whole number, as in 2 (a+ br) dx, ai x obias (a+ bxr)* dx =m A (a3 xt + 3a? bxr>+ 3ab? 26 + b° v7) dx 1 abr Sabea™ $378 a 2 7 8 5 8 i 2 3a°x3 = Bax? = 83 r2 J[# Gaya ——— + —_ ) + He Jrrd+e2)3de= ——. 3 Da ag +3logv+a, Secondly, let m be a positive whole number, as in x assume a+bx=y; then fy eee 3 d 1 x*(a+bzr)-° ta—(U*) ye a a (y—a)?y-* dy a c By 3 y { a fz (a+br) dr g Bay +3a De Ua _ (a+bx)4 Bee hey) & nay TP Ce °(a+ bx) dr: a> a BE Sata LAstess RAY A 8 | Rate ee ., fo 2 pues A Ben S2enNr—a iad h a Cael) a wa eraser ta) Ke 2 2ax*” S8a’r 16a? 7 105 105 } Thirdly, when both m and n are negative whole numbers, as in @?(a+bxr)“‘*dr. Assume x=1~+y, which gives 2 dq yeti—2 a x? (atbe)-! day? (4) raw — Y y ay+b y (ay--b)” which falls under the second case, since p and q, and therefore p+q—2 are positive whole numbers, Pe orde Bot dy v(a+br) —— _— 1 touts) 1 } @ weet aes ——loo (aa =-— log —_-——_ ay+d q oy a -atbs * Throughout these examples, merely the primitive function (p without any reference to the limits of the integration, age 100) is found, 278 DIFFERENTIAL AND INTEGRAL CALCULUS. sia — lB ; log at bar a(atbe)? a(atbr) a °\ a J 93. To investigate methods of reduction for the following formula : ifs a” (ax? + ba)” dx, or “bs ate (a+-bat-"?)" dz, the form of which is yA x” (aba) da. Here r and s may be supposed whole numbers; or if not, assume ay", k being such that rk and sk are whole numbers. Thus for ans (a+bat)? dx, let x=z", which gives _2 3 a (a+ bat)? daz 12z (a+b2°)? dz, a form in which 7 and s are whole numbers. Let ¢ be the fraction y+-5 3 assume a+ba°=v?, and we have y ad: y dye! O vi—a\ a (a+ ba’)? dr=2* .v’.<—— du — "21 ——__— Sine 8 sbx*—" sb b which is integrable by common expansion, if (r-+1)—~s be a positive whole number; and this whatever r and s and ¢ may be. Again, 2” (a+ bx’)'=2"*" (av~* +)‘ dx, which by a similar process may be shown to be integrable whenever (7r+st+1)+(—s) is a positive whole number; that is when r+1 ‘ io see —ztis a positive whole number. -1 Pog dv, The following functions, therefore, are immediately integrable, whatever s and ¢ may be, provided & be a positive whole number : ge (a+ bz’) dx and a *@+)-! (a+ ba’) dx. 94. Any function Yadx+ 2x, in which wx and ¢z are rational and integral, can be integrated in a finite form. 1. If wax be of higher dimension than ¢2, divide the first by the second, and let Q be the rational and integral quotient, and R the remainder of the same kind, which is of a lower dimension than ¢2. Then Wa R Tica R —_= —- —— dr= — dx Q+ abs x= f{Qde+ [xan and the difficulty is reduced to that of integrating the last term, in which the numerator is lower in degree than the denominator. 2. Let yx be of a lower degree than gz, and let the roots of dr=0 be a, 6, c, &c.: whence Px=A (wx—a) (x—b) (x—c)...., where Ais the coefficient of its highest power of z We have then various cases according as the roots are all unequal, or there are one or more sets of equal roots. After the decomposition is made, as in (82.) and (85.), the difficulty of integration is overcome, since each of the decom- posed fractions ean be readily integrated. Thus, let it be required to find f{ Pdx, where P=(a?+ 2+ 1)+(«—1)*(#—2), e+ta+l 3 6 ” =e Ss \_ sf (@—1)(@—2) (@—1)? a1 MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 279 a@+t+atl pfs (e=1)(e 2) 4 = Za 8 log @—1) +7 log (2—2) re f At+B _ Aa-+B Ab+B ree rae log (w—a) ++ a. (c— a)(a—b) — ab log (w—b). 96, It is, generally speaking, most convenient to integrate simple rational functions by transformations which a little practice will suggest. The following is an example, the fundamental integrals in Chapter VI. being assumed : *(Ar+ B) da =|2 (a+ a) da (B—Aa) dx (w+a)?+h? ~~) (a+a)?+b° (v-+a)2-+ 5? A oe B—A =z log (a+a +6?) + tan ae Az+B aig ’ Jama =5 log (2°—2 cosw.r+1) B+A r— at COS 4 nai v ges A sIn py SIN yu md 97. Required i i (m1, is integrable, after reduction by common division. Thus, the first integral in each of the following lines is found by ascending from the last, through the intermediate ones, by means of the preceding — formula, (P=1~-(a’?+ 2*)), MISCELLANEOUS EXAMPLES AND DEVELOPMENT. 281 ; fx Dy hs fx P® dx, of x® P® dz, if; xP7dx fuk P) as fa" P* dz, fx" P* drs fol Pdx fa Pedr, fx Pdz, Jat Pé dz, fa P*dz, ih P* dz. 101. The following formulae of reduction involve a large number of general cases. Let P=Aa*+Bz', oe Af at Prd Multiply the equation P"=(Az*+ Bz’) Pp by a”, and integrate, which gives 4 ee nt are i BY sh eS (1 ) Integrate V,,,,, by parts, which gives nti p” gmt a ES — nP*" { Aax*+ Bho’! dx : m+ 1 m+] : Bs j : gt) pe Na nb v a es haere m+a,n—17 Dv m—1® 06 2 ® . mtl m+] albert. m+] aor = Eliminate BV,,,,,,-, from (1.) and (2.), which gives | ar! ps nb —na + FA Ae, m—le ee 806 (3.), mm" m+l+nd “ m+1+nd ; which is a formula of reduction when n is positive. By it, for instance, we reduce the integration of 2” P? dx to that of yt Pi dx, and the latter to that of at? P? dr, To turn this into a formula of reduction when 7 is negative, proceed thus: id gee De m+1+nb Vata, =o AN i Ghat A. m,n For m write m—a, and for n write — (n—1), which gives Pe BEES gui OF th 1) ) ~ (6—a)(n—1) A (6—a)(n—1) A By this we can make the integral of at P73 dz depend upon that of Ds, = 4) Bia —(n—1) oeopeae (4.). a"P-idzx, this one again upon a*-**P-3dz, and the latter upon a*-** P-2 dr, _ To make a formula of reduction for the diminution of m, 2 remain- ing the same, eliminate V,,, from (1.) and (2.), which gives ott Pp=(mt1+na) AVinwa aat(m+l 1D Mey 's #' ss 9(5:)'3 for n write n-+-1, and for m write m—a, which gives y aor per m—a+1+(n+1)b m"(m+1-+na) A (m+1-+na) A which may be made a formula of reduction when m is positive by taking a as the greater exponent, and vice versa. Vi, Ba n 102. The preceding results can be stated as follows: P=Ar- De, -V,, = sé oe da, Pod 282 DIFFERENTIAL AND INTEGRAL CALCULUS. (m+1+nb) Venn tn (@—6) AVinga, n= 0 PP eee eee o. eea. (Am (m+1 +74) Vinnt? (b—a) BV ings, n—1 = ital ale Os eS ee ar Se (B.) (m+1+na) AVinn t+ (m—a+b+4nb+1) BVn—ap5,n= pea PEt (ae (m+1-+ nb) BV nat Gn-b+atnatl) AVn-s+2, nn Ants < pa Glee 103.* Let P=A+Bz, or a=0, b=1; and for m write —n. (m—N+1) Vin, n+ NAV m, (npr) = 0” that Be aan bei 7 ite m—n-+ | a” dx (A+Bz)"! nA (A+Bz2)” nA (A+ Br)" (M41) Vin, 2 — BV ni, —agy =a" Po a" dx 1 ger’ m+1 x” dx (A+Bay" 2B (A+Bay | 7B J (A+Ba)" (m+1) AVn, a+ (m—n-+ 2) BV mth p-(n-1) wer! dae: 5 1 get a oC 1) ah a” dx (A+Ba)" (m—n+2)B (A+Ba)"* (m—n+2)B) (A+Ba) (m—n+1) BVy, nF MAV ma, n= 2” Pry a Oe 1 4 it mA a de {aah “GoD GtEa Gee E) Gee The two last formule are really the same. For negative values of m, we have, writing —m for m in the third, dz ia 1 1 | rary or Far (m+n—2)B dx (mI) AS { x? (A+Bz)" 104, Let P=A+Br+Cz’; required a reduction for fa" Pdi =V ae Vin nz AVinn-it BV me, 4-1 C¥mnisn-1(from P*== P*""(a + Bx ca*))g get P* ‘ ; and, by parts, | acacia ~ | sre PP! (B+2Cz) nN” nB 2nC ah m+1 ne is 1 nPEREE Oa m+2,n—I* Eliminate V,,49,n-1. and we have gett Pp” OnA mM, % a: cpecbethala 7 Cy uae aes Viv wer ta V l le Qn-m+1 Q+-m+l1 ™ nek and: Oe Again, eliminate Vinny Which gives | (m-+1) AV n, n-it(m+n-+1) BV n+, ano (m-+-2n+ 1) CV nts, n-12 ee { write m—2 for m, and n+1 for », and we have, as a formula of reduction when m is positive, | OE Sgn et eeese) (m—1) A ™— Ga anFlyC Gn+2Qn+1)C ">" (m+2nt1)C ™*" * The results of this and the following articles should be separately deduced by the student, MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 283 In the last formula but one, write —m for m, and 2+1 for n, and we have as a formula of reduction, when 2n is negative, | atts (m—n—2) B Nom Guan) Aa! Gy A Vm (m—2n—38) C ae Ay (m—-1)A Vi cma), £05. Let. Viens {A sin” 8 cos" 6 do; required a formula of reduction. Since multiplication by sin?@+ cos?@ does not affect the expression to be integrated, we have ASP ra— Vai, at Vn a+2 Cg re Rapes) m+] m 7) Writing c and s for cos@ and sin@. This gives (dc=—sd0) dV,,,,,==cos"—6 sin” 6 d sin 0; ee Ty s™t! o"-8 de; ret ge tine f OT ee ei breed cv v.) =—_—-_-—._ + ——__ 2, 2-2 ——_-——— —-— om wens WL} tcl bine a m+1 TSE WO ta pic, cr grt a] 2 ie —— Vin, n= tah Aa (ee ene ae The last but one is a complete formula of reduction when m js negative and 7 positive: and the last is another as to n. By proceed- ing in the same manner with OV ea (—c) we find she t! om — I] Ra ey ne . Aid ee g@-1 cert m— l Vine ear wpe Vian > m+n m+n the first of which is complete when 7 is negative and m positive; and the second reduces m when positive. Combining the two results, we obtain cr! gtr r gee | gl gn m—1 7. — 6 ror es eee rer es ee Ves 3} m+n m+n mt+tn—2 m-+n—2 y ac at? (n—1) c™} s™-! (m—1)(n—1) é mm m+n (m+n)(m+n—2) * (m+ n)(m+n—2) ">"? ora m—1)c"-1s5"-} m—1)(n—1) Pa ( pS key ea + A Cons 1 ilo da Vis n—29 m+n (m+ n)(m +n—2) (m+n)(m+ n— 2) Which are complete formule of reduction when m and n are both positive. But when m and n are both negative, write —m+2 and —n-+ 2 for m and n, which gives (m + na— 2) ea (1) g—(m—8) ea (@—-) gm) oe (m—1)(n—1) 1 (m+n —2)(m Ta (m—1)(n—1) Vv Vis Game tei) DIFFERENTIAL AND INTEGRAL CALCULUS. (m+-n—2) 67-9) sim) E~O—D gD saeemeemmerreees (sy Mr sah TP (m+n—2)(m+n—4) (m—1)\(@@—1) ee (m—2), —(n--2)* 106. We now write the preceding, and particular cases of them, in the usual form: the student should deduce all the latter separately. s”-1 "1 S{m—1—(m+n—2) c®}* de at (m+n)(m+n—2) : (m—1)(n—1) | AD) i pg g o2 Jo (m+n) (m+n—2) a - m1 (mtn 2) 6 (m+n-2)(m+n-4) —~(m—1)(n—1)s Soa n—1 “. (m1) —l) sm? on a0. aon EO m—1 (*s"-* dé fe i) om { ory? iL Wy Con, n—1 (‘c"* dd {= oe pene. nA jo J tan” 2) do=" J tan"~? 6 dé, 107. When m or 2 is 0, proceed as follows: {= ae dé mile Aa dO S oi a a = mee att 324-7 err Le dé c m—2 dé Similarly, {= ™ (m—1) 8" ee oa 1 {= m= 2° From c”d0=c"" ds and s"d@=s”"7' d(—c) it is found that feediee s+ f(n—1)s*c"* dé=c""*s+(n— 1) f (a—c’) c*"* 0, pee. 8 Mie lies AN oy fe d0= - fe 2 dd. 7L gm! g@—2 d Q e 2 dg dé mee ad Kuen c c Similarly, J s" d0= 108. In cs” dd let tand=t: from thence deduce vce mdo= | , {k=m+n+2}. +t)? Call the last Tn, i, and deduce * This factor is also (m-+-n—2) s°~—(n—1). MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 285 tt gk k . ee = Ts k+2 m—k-+1 m—k+1 Cer Soy a JAB k—2 k—2 he k— fie. k—23 109. An integral is thus made to depend upon the integration of a more simple form, that again upon one still more simple, and go on, until we come at last to an integral which cannot be simplified by con- tinuing the process of reduction. This may be called the ultimate integral, and may be found, some- times directly, sometimes by a further reduction in a different form. The following table exhibits a large number of integrals, such as are dis- cussed in the preceding articles, with an exhibition of their ultimate forms. ‘To save room, denominators are written as ratios with the symbol (:), a plan which the student should not adopt in copying them. The first column contains the function to be integrated, the second the ultimate form, with its integral; or else a transformation of the integral, which reduces it toa preceding form. An ultimate form enclosed in { f means that it has been already given in the preceding part of the table. a"dr: atbs Sdx : a-+ br = log (a+br):b adx:(atba)" fde:(a+br)"=-1: (n—1)b(a+br)" dz} x"(a+bz)" a=1 iy gives —y™*"“"dy : (b+ay)" dx : (a+ bx*)" Jdzv: a+bx? = tany* (aJb°: fa) : (ab) 1 Jat a/b 2 @ Ay 2\n ° — 72 — ia o Aa Le dz} (a—bz*) Sdx : a—bzx Dab) log [eee 1 Ja—xrJb e Rist n 1; : ee Sa eo —__—______. dz: (bz’—a) Sdx bx? —a 37 (ab) log Urea: z"dx : a+b? { fdxia+bz}, fedriatbx* =log (a+ bx*) 2b a™dx: (a+bx*)" {e"dz : a+bx*}, page 281, formula (4.) dav} a"(a+bx*)” r=1:y gives —y™"*dy : (b+ ay*)" 2 + yer} dv: (a+br+ cx’) fdx > a+bet+cx eo ldnas By “Wane | 2cx+th—,/(b*—4ac) Mdxiatbhr+cx* V(@—4ac) den +b+ JP —4ac) ] fade: a+ ba -+ cx*—= = log (a+ bx+ cx’) ih ee > atbet+ cx? 2c Mdx:(a+brtcx*)” {dx: (a+ba+cx*)"} Inix™(a+brtex)" a=liy gives —y™*"—* dy : (c-+-by+ay?)" 1 n (dz : a+ bx" a=*/ (aib).y gives nfs dx: b+3* DIFFERENTIAL AND INTEGRAL CALCULUS. gis We ee l+e2° 81l+e 31Fe+2 oe a lee Lied! SOM 1 Oe —3"3>-ets ge TT tea 1 1-2 21—2 2142? l4at 414/2.a4+2° 41—/2.2+0 ade: J(atbr) fda: J(a+br)=2,/(a+bz):d. dx: x" /(a+br) be :y gives — aes J (ay? + by) obi Ia nanan (a+b) — - Ja fax : xf(a+be) “a ae Vy ani: wf (baa) =. 7, COs Ascot adr % (a-+b2) 1 @+br=z gives an , form, (page 277) Qn+1 : (a+ba*) 2 ipa J (a+ ba*)=log {a/b +,/(a+ bx") }—,/b a”dx : cone Sdz: f(a— ie gi (xJb: Ja): Jb fda: (a+ Bacay. af (a+ba*) ) fade : J(a+ba*) =,/(a+ba*) 2b dx: a"f(a+b2*) w=1:y gives —y""dy: J/(b+ay’) fae: sn (abba) = 7 log gv lade ae Bad! Sida: af (be? —a)=cos (Ja : se : fa Qn4+1 Qn+1 dx a™(a+ba*) 2? a«w=l:y gives —y"t*" dy: (b+ay*) 23 a : a _ tf (at ba?) | a@ dz @ a | (a+ba*) dx S (a+ b2") da : ts Tad ba) | N(a+ba*) dx a" w=1:y gives —y"*J/(b+ay") “i, 2y 4 & 2 : SN (at+ba°) de : oma | 8 +) Cat bel) | AV (a+b2") dx # Ce Pe BAM Win | 2 SN atba)de:v?= i ues Tapa) ade: (ax+ba?) fda: J (ar+ba*)=2 log { f(a+bxr) +,/(b2)}: fb | fda : a[ (ax — 62°) =vers“'(2bx : a) : Jd’ | dx: x"J(ax+bx*) w=1: y gives —y""'dy: J (ay+b) x" /(axv + bx’) dx SN (aa-+b2*) A Peete ee bere a v 86) J(ar-+ bx") N(ax+ bx") da: a" xv=1:y gives ~y"/(ay+b) dy MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 287 a*"(a-+be-+ort)* 2 said,’ Let a+ba+cr=X a+tbe—cr*= xX! S dx > /X “ log (2ca+b-+2,/(cX)), fda: es i occa ji “WF 4ac) xpere — asthe 4 4a 4ac—b? ( dx 8c } WX s ve “ dx 2 Wada: (Xa a a vx Ey f Xade= a2 IN sde== — > f'/X.de - dx 1 2a+ bx — —2/ (aX) 2 ‘ log a oe Gar SEN ape a 3 dx 1, bare 2a i sIn -_—_—_—_——— ————_ __., J af (cr +br—a) a x) (b?+4ac) ee Daag 2 2 (Qcx+b) | a 2 (a+b0-+ca%)= ane cee b BJ (a+ ba-e2*) sin*”6 cos*”6 d0 Jsin 6d0@= —cos 0, J cos 6d0=sin 0 8—sin 8 cos 0 6+sin Ocos@ ri fe (ep ca ade §6d0 = J sin 5 , {co os*6d 5 dé ah fi 0 dé man as 6 sind > her cos. ap) dé da —cot 0 ——=tan 0. sin? aan cos’ @ 110. The following miscellaneous forms will occasionally be found useful : 7 X/daf Vde dX ina! ~— -1 sae weed GS tes aes ie a: sin X.dx = sin X fVde | Fax ¥ ae | X'dxf Vda | rend _— —l JS V cos X, dx =cos X {Vda + Vax X'dxf Vdx —l —] eee ee ee SV tan X dz =tan Xf Vado ~ ~ oe ‘dxf Vd fv cot X .dx =cot X [ Vde “ i at [Vie =A /Vaenf Ob afin “X/de. f Vd: PV wgX. de=logX fVae— [AL Vas 288 DIFFERENTIAL AND INTEGRAL CALCULUS... (log x)" 2 mt+l1 bs ym . \n pS 2 a a a hah log n=l fa (log x)" dz = mS Taide (log x)"~'dx a""'\Cog vy** m+ a orl ee PY m ] n+l d. n+l n+ 1 AC . J 6". sin 0 dd = —6" cos 0+nf 6" cos 6 dd = — 6" cos 0-76" sin d—n (n—1) for sin 6 dé JS 6".cosé d@ =6" sin o—n for sin 6 dd = 6" sind +76" cos 0—n (n — 1) f 6" cos 6 dd. 111. The number of forms which can be completely integrated is — comparatively small; and the various methods by which functions are transformed into others more easily integrable may be classified under very few heads. (a:) Integration by parts. (6.) Rationalization of numerators. (c.) Combination with other integrals. (d.) Substitution of a function. of another variable for the in-= dependent variable of integration. (e.) Resolution of the function into an infinite series. We shall now take some examples, particularly of the three last. + x a’ dx a* dx 12. fv +a de =| sah ae =| sa a) Vera) ba dx SA (a+ ba + ca’) dx = | reste + Narberer) cu? dx +) Yabba Fea) The second sides are in both cases more easily integrated than the first. 1s. ig A aman =| Wer Tee ae Qc gee pe Scena) RA eg el Ma e Bitte heh V(a+be+ex*) 2c J (a+ bx -+c2)’ the first term of which is directly integrable, and the second can be integrated (page 116) a adr Ie hoe 2 (2cx+b) dx J(a+br-+cx") ) 2c.) J(atbr+c2*) ~ 2¢ { aes aX ‘bf ade aa =5 - [aR ws) (a+ba+cer=X) = Wea Se Hie rdx = —aJX—— f JX. di— x MISCELLANEOUS EXAMPLES AND DEVELOPMENTs, 289 an Medi} ale dx _ 4 (‘ede edx b (Cxdr JX Tr ay F c f/X Cc VX JX 2c VX 9 adr =o eS dx =f l EX vervh ah VX ¢ Guy APR” © 26: tae WX) 0 Ge he wdxr i a 3 20 aE dx VX (= ie Wx4(S ~5) (% ; é 114. Required bm ee a+6cos ¢ If cos 0=2z, this becomes f (—dr : (@+br)/(1—.2*), which can be integrated in the same way as dv: vu J/(a+ bv + cv’) by making at+br=v. The following process, however, will illustr ate more clearly the advantage of substitution. h oy eine 2) oN oe Ric +a cos te iaieniie. 49s (a’—b*) sin 8 iy fs. (a>—b ) stip a+b cosé (a+ cos 6) (a+b cos 6)? do ] dv b ere nee SS See eae Saks atbcosO0 J (@—b) J—v)’ ¥ dd ed 1 pt 4 b-+acos “| | a+ bcos0 ~~ /(a—6?) ne cos 0 dé 1 dv NST ey OEY F.brae xe 1 Fs pes Cos 0+ /U'—a’) sin ° J ices Es oo ae a a+6cos 0 do 1 do 1 0 ened — S| ——— —— tan —. eee a+acos@? 24 30° a i 2 cos’— 115, ae ee Monk te Ty l+¢s* g~7-+4 ] ¢ 7+] 116, S dex $ (log x) depends upon f«* prdz Oi ditih Cet ys ig, 6 GPS : PE a x zd JS dx ¢ (sin x) of Sgt cea &e. 117. Any function containing irrational functions of @-dzr only may v€ rationalized by simple substitution: thus 3 "ae dx bv" dv , : ——; becomes | — fey a v—l L— 28 la) ; 6 5h . : | ess ; becomes i if —— f atbrev’, (a+ br)? —(a4-br)3 ; U 290 DIFFERENTIAL AND INTEGRAL CALCULUS. 118. As examples of integration in series, we have already 27, 31, 32, 33. The following will be readily ascertained by integration by parts : d¢ Let [Par= re f Prdr= Ps f(s CE a A ky m=, &c. J PQdr= QP, — f Q’P.dc=QP, —~Q'P, + fQ"P.dx =QP,—Q’P,+ Q’P,— ef ee + Q@- Po PyQ° Pp. dx. John Bernoulli’s theorem (page 168) is a particular case of this, obtained by making P=1. If Q be a rational and integral function, the preceding series terminates. S & Qdx=e {Q—Q'+Q’—...}5 fe? Qdzr=e7{-Q-Q'-Q’—.. +f J cosx.Qdr= Qsin 1+ Q’ cos t—Q” sin e—Q” cosx+.... fin v2. Qdr=—Q cos x +Q’sin 7+ Q" cos x—Q” sin r—.... 119. The following method, which is a generalization of integration by parts, has been successfully applied to the formation of approximat- — ing series, in a particular case, by Laplace. Let of Qde= FE, SP, Qi dx=P,, SP:Qdz=P; &c., Q, Q,, Q,, &c. being any functions which may be found convenient. The order of processes, in passing from one to the next, is multeplica- tion before integration. Again, let y 1 dV, 1 dW ~—=V,, — —=V,, ——-=Vs, &e. Qi ty Qiide we Qiedolem the order of processes being division after differentiation. Then "4 ee 8 1 dV fyde= @ Q=V, PofPi gi d@=UPR= | > —. P,Qude * 1 ~dV. Vi P,—V. 12 -+- fo cana r as dx iQ. dz tra dV. —V,P,-—V.P,+V;P:— | ies dx =< ONE =V,P,—V.P.+V;P,—..-.£V,P, = | ee, dx. If Q, Q,, &c. be properly chosen, a convergent series may be frequently obtained. =a Q=— 0,0... 3 ter 42 PSU ee dy) bes dx dx d sn) dx d dx d fey Fenn Voeeapi— ie apes yp hy — SS ( ene doh ar dy dx (y dy)’ ae dy dx J dy dx Ge ’ Let dx ee {1 du d ( du a | d ( du \ ety ——u, fyda=yus 1——+— | u— J— = ju| u— oe sah is dy SY J ee ae OS Je. Sa which is the case given by Laplace. We shall have occasion to use it in treating on definite integrals. Let the student obtain this particular case in a more simple manner. MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 291 120. The common formule of trigonometry frequently expedite the performance of integration : thus 8 sin’ 6=cos 40 —4 cos 29 +3 gives ; in 46 8 fsinto da— 4 — 2 sin 20+30. 121. Required J cos (a0+5) cos (a'04+-d’) do cos (@9+4) cos (a’0 + b’) =5 cos (a+a/0+b+6/) bs Cos (a-a0+b—)’) Scos (a8 +8) cos (a/0-+b") doin at a0+d40/) 2 (a+a’) sin (a—a/0+b—b’) 2(a—a’) ; If in this we write esr m for b, we have JS sin (a0-+ b) cos (a'0+b’) do= 2(a+a’') cos (a—a’0+b—B’) 2 (a—a’') : and if we also write ones: for b’, we have ' { 2.4.6....28 \ 1 BA Gees 2Bwy? i] weil dla ll ey err eA E FL 2 1.3.5....26-—-1) 9e7] ca 2B—1" This remarkable result, which was first given by Wallis, should be verified by the student ina few instances. Thus $m being 1°570796, we find 2.4.6.8\2] 2.4.6.8\? 1] —— |} -=1°48 —— | -=}]: Ga 9 ? Goaee) et Since the two expressions for 4 can be made as near as we please by making 8 sufficiently great, and since 1+2f lies between 1+-(26+1) and 1+-(26—1), we find that, as P increases, the following equations approach without limit to truth: Owes S28 2 Decree wep ——— =a : Bey ———_—_—-_ - —_—_—_—_ = 9-8 rB=( ee Y, 1.3.5... 98 y= 2 126. It is obvious that 1.2.3....” divided by x must diminish without limit when 2 increases without limit, being only a fraction of ir. Let 1.2.3....2=0" fr, and (x being very great) we haye cD BRA @ _ (1.2.3... .)?.27 a (fx)? 2" 1.3.5....2m—l 1,2.3....20 ~ (2x) f(z) LBS Dia Cie)? \fiCae) ~_. =a pz 2-2: whence —=--_ ; hin hae” Qre /(Qr.2r)? But e . ee whence fr~,/(27rx) satisfies the equation (yx)*=y (2v). The most general solution of this equation is ¢*, where éz has the property of not changing its value when 2 is changed into 2x; or & (2x)=ér. But we may show, as follows, that in this case Ex must be a constant. Since, when z is great, 1.2.3, ...0=2".,/(Qrx).e" very nearly, we have 1.2.3....c+1=(e+1)"4/20 (e+ Ly) sep UESED x41)" ‘a1 and vp1sSto— ./| ay gP, v koe ka? 1’ = x {where P=(x+1)£ (x-+1)—2ér}; or 1=(1 oe ) J) er, The last equation must approach without limit to truth when x is increased without limit. But the limit of (l+1:.2)* is ¢, that of Vi(@+1): 2x} is 1: so that the limit of the expression is : gi tlimit of ((@4+1)Z@+1)—22z) =] ipeseeal or the limit of (x+1)£(7+1)—2éx is —1. But Ex cannot diminish nor increase without limit, nor can &(x+1)—ér; for tEx=é (27) E(4r), &c., and £ (v+-1)—ixr=F (22 +2) —F (2x), &e. Unless, therefore, £(x+1) =£17, we sce that x (§ (w+ 1)—Er) +£ (+1) will crease without limit, positively or negatively. But if £(#+1) = £2, then £(x+ 2)=£& (+1), &c., and &x is the same for all whole values of. The limiting equation in question is satisfied by &(#+1)=—1, and we then have 294 DIFFERENTIAL AND INTEGRAL CALCULUS. 1.213 OY) . ween/ (ara) ate, oF J (2). gtk e—*, nearly. 127. Required an approximation to the coefficient of «* in (1+.2)", k and n being both large numbers, but / very much less than 2: that is, required n(n—1)....(r—k+1) sid 1. 29Be%A att Lo € (1.2.3... Bl di dete. eR) t / (27) nite af (277) REE ek (Qa) Cnhy ery ! Jn nee: 9. Nee ay (21) JR (n—k)) (3) (*) ; ‘ which is nearly 128. The subject of definite integration will be treated ina future chapter ; we shall now give an instance of the manner in which it may happen that an integral may be found in a finite form between two specified limits, which cannot be generally found in the same way. Required ik e—2 dx from r=0 to x=. It is easily proved, either by expansion, or as in page 175, that (144A: )®" continually approaches to ¢** when n is increased without limit. If, then, we can find fa-# :n)" dx from x=0 to x= n, we afterwards find ff e—-«’ dx from 2==0 to a=, by increasing n without limit. Assume v—,/n.cos 0, or (1—a*: 7)" dx= (sin? 0)"(—,/7.sin 6 d@) jn 1 tie \ 2 0 aw { ( jem dx=—,f/n sin®”*' dw=an | sin?"*'9 dé 0 fe > 0 2 3 . ip ie ce ae an phere if a apegedl all res ee ) n+] : On4+-1° \1.3.5...2n—1} iss) ol bo load i tras b 91 vw 4A waete 4s — 2 A a The greater 1 is made, the more nearly does the factor in brackets approach to ./(7n), or the whole to ,/.2: (2n+1), the limit of which ish Jr. Hence fy ¢-* da=},/7. 129. From the definition of an integral, an approximation of any degree of nearness may be made to ‘t >dxdx, by the summation of terms of the form ¢x Az, where Ax remains the same throughout, and x is intermediate between a and b. We may express this by saying that the integral is the sum of an infinite number of infinitely small elements, each of the form @x dx. Again, the result shows that /ipx da is of the form ¢,b—%,a@, where ¢,2 has @v for its diff. co, From each of these considerations, let the student deduce the following theorems : I. ev pa Beebo px dx + fir px dx ; tj px dr=— fF px de. Il. If dr be a function which is unchanged when vw becomes —2, then +4 hy decaf px dx; Me px doce ee pr dy, Jo *badax= — fF" ba da. MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 295 Ill. If yx be a function which changes sign only, and not value, when x becomes —., then aan du=—O0, Ds yu dx= — [te ur dr, ee Wa Ee Mae wa dx. 130. Let a function which does not change, when x becomes —z, be called an even function, (it can be expanded only in even powers of 15) and one which changes sign only, and not value, an odd function; then ~x+h(—z) is evidently even, and @ («)—¢ (—2) is odd. And every function is either even or odd, or the sum of an even and odd function, as appears from _9@ +6 (—2) | de—o(-2) ee Also, if dz be even and we odd, . Li (prv+wWe) dg {te pudi= ys (da— x) dx. 131. The product of two functions of the same name is even, and of different names odd. The diff. co. of an even function is odd, and wice versa. Every even function fe is of the form ¢2+¢ (—.2), and gv is $fx+ any odd function: and every odd function fr is of the form pxr— (—2), where pxe=t fe + any even function. “i Pxdx is necessarily either an odd function of a, or =0, what- ever ox may be. 132. If ox be even and possible, (av —1) iS possible, and if $x be odd, b (eV —1) is impossible, and of the form /—1 x a possible function. This is easily proved, when it is remembered that every function of ,/(—1) and 2 is reducible to the form Fx + fx cay where Fx and fx are possible. 133. Show that fig «-@ di=,/r, and that f+t4sin 2° dv=0. 134. SI that ** cos x dx lad “ee cos 2a dx erdaihd 1 Mara fe, (1+2°)(cos e—sin x) 135. To reduce a ° dxdr to the form re +4 bx dx. Take a function of z, which becomes +a or +6, according as x is —c or +c, say of the form A+Br: then A—Bc=a, A+Bc=6), and we have A b—a (*t ees b—a or dr—-——. -— +-———__ x | dx. J iba dx a {-9( 3 -- 7 ») de b— is —f b—a 135. Show that Be Px fe pai (“ es a5 é v) da: Bey dee! a = (b—a) [3 oy) (a+b—a w)dx, =(b—a) Sco (brnbs—e s*) a dx, I now proceed to examples on some of the subjects in Chapter VIIT. 136. Required a discussion of the function (a-+bx)"e*. Its diff, co. is €* (a+bx)"" {nb—a—bzx}, the sign of which is to be considered. First, let n be an even positive or negative whole number, then the sign : the preceding depends upon that of (a@+6x) {nb—a—ba}, or on that of 296 DIFFERENTIAL AND INTEGRAL CALCULUS, —b (+5) (o—{n—F}) which is always negative, except when 2 lies between —a:b and n—a:b. There is then a minimum when « is the less of the pre- ceding, and a maximum when @ is the greater: and the function never increases with a except when x lies between —a:b and n—a:b. Thus, if the function be (1+ a:n)"¢~*, we have a=1, b=1:n, and there is a minimum when e=—n, and a maximum when «=O, if n be positive: or a minimum when 2=0, anda maximum when 2=—n2, if n be negative. But if n= 0, then (l+a:n)"=1 for all values of x. If n be a positive or negative odd number, the sign of the diff. co. depends upon that of nb—a—ba, or of —b{x—(n—a:b)}, which changes from the sign of 6 to that of —b when z increases through m—a:b. There is, therefore, a maximum or minimum at this point according as 6 is positive or negative. A rational numerical fraction, reduced to its lowest terms, has one of the following forms: 2n In+1 2Qn+1 : ———_, —, —-——, (mand n being wh. no.) 2m+1 2m. 2m-+ 1 The first case presents results resembling that of an even whole number ; the third, of an odd whole number; and the second is altogether different from either, since it gives two real values to the function for every positive value of a+da, and none for negative values of the same. 137. Required the discussion of y=(a+bz)"é, when mis a fraction which in its ‘lowest terms has an even denominator. Its diff. co. has the sign of (a+5x)""' (nb—a—be), the first factor of which, like its primitive, is impossible when a+6z2 is negative, and has the sign of ¥ when a+ bz is positive. Consequently, the sign of the diff. co. depends on that of y (nb—a—bz) or of —by{x—(n—a:b)}. If, then, a=n—a:b gives a+bx negative, that is, if bn be negative, there is no change of sign in the diff. co. throughout the whole range of the possible values of y; and the diff. co. has the sign of —é for all positive values of y, and of +b for all negative values. If bn be =O, the increase or decrease of the function (whether it be that b=0 or n=0) depends solely on that of e~*. But if bn be positive, then the diff. co. changes from the sign of by to that of —by when « increases through n—a:b; that is, if b be positive there is a maximum for the positive values of y, and a minimum for the negative, at that value of 2, and vice versa. 138. Required the discussion of the function cosv-+asinx, This function being evidently periodic, it will be sufficient to consider one complete cycle, namely, from x=0 to x=2r. The diff. co. is —sinz+acosx, which becomes =0 when tana=a, to which there are two solutions, one less and one greater than 7. Let « be the less, then the diff. co. is —sin y+ tanx cos 7, or sin (k—a) : cos x, while the original function is cos(e—2):cosx. If, then, .<47, or if a be posi- tive, the diff. co. is positive from #=0 to r=«, negative from w=« to x=7-+«, and positive from a=7+x« to v=27; or there is a maximum MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 297 when w=« anda minimum when x=r7-+«. The maximum is 1: cos ¢: or /(1+ a"); the minimum is —J/(1+a?). Butif«>47, or if a be negative, the words positive or negative, and maximum and minimum must be inverted in the preceding. And the function itself is (a bemg +-) positive from r=0 to r—=x-++ tf, negative from v=K+hr to t—=k+$r, and positive again from L=K+ mr tor=2r. But (a being —) the function is positive from r=0 to t=x—47, negative from t=k— om tow=ce+4rq, and posi- tive from t=x+4z7 to r=2z7. Both of these may be thus stated in one: cos#-+-asinz has the sign of a only when 2 lies between K—d ar and k+3 7. 139. Required the variations of sign in a formula of the form cos (ax+ b) cos (a’x+b’) cos (a"@+-b").... Every cosine changes its sign only when its angle passes through an odd number of right angles; so that we must examine the several equations ax+ b=} (2n4+1) 2, dr4b’=3 (2n41) x, a’e+ b"=3 (2n+1) x, &., ascertaining every value of x between 0 and 2z which can be given by a whole value of 7, positive or negative. Arrange all these values of x in order of magnitude: then the sign at the outset being that of cos b. cos b’.cos b”...., there is a change of sign whenever v attains one of these values; but if two of the values of zx coincide, there is no change of sign, if three coincide, there is a change of sign, &c. For if a number of factors change sign at once, there is or is not a change of sign accord- ing as that number is odd or even. But if there should be a sine among the preceding factors, as sin (kx+/), either write this cos (kxr-+/—% 7), or examine the equa- tion kv-+-l=nr,. 140. Required the variations of sign in y=cos (32+30°) cos (27 + 230°) cos (18° —4r) sin (v+15°). 1. As to 32+30°. The limits of the value (within the cycle from t=0 to r=360°) are 30° and 12.90°+ 30°, within which are contained 90°, 3.90°, 5.90°, 7.90°, 9.90°, 11.90°, to which the values of x are 20°, 80°, 140°, 200°, 260°, 320°, 2. As to 27+ 230°, or 27+ 2.9()°+50°. The limits are 2.90+50° and 10.90°+50, between which are 3.90°, 5.90°, 7.90°, and 9.90°, and the values of 2 are 20°, 110°, 200°, 290°. 3. As to 18°—2u. The limits are 18° and —(8.90°—18°), between which lie —90°, —3.90°, —5.90°, —7.90°, and the values of ware 54°, 144°, 234°, and 324°. 4. Astox+15° The limits are 15° and 4.90-+15°, between which lie 2.90° and 4,90°, to which the values of x are 165° and 345°. Arranging these in order, and bracketing those which occur twice, we have (20°, 20°) 54°, 80°, 110°, 140°, 144°, 165°, (200°, 200°) 234°, 260°, 290°, 320°, 324°, 345°. Now when z=0, y=cos 30°.cos 230°. cos 18°, sin 15°, which is nega- tive: consequently from #=0 to = 54° (neglecting 20°) y is negative, ko 298 DIFFERENTIAL AND INTEGRAL CALCULUS. from «=54° to x=80°, y is positive, and so on; finally from #=345° to v=360° y is negative, as in the following table: Lim. of x. | y Lim. of z. y Lim, of z. y Lim, of x. @0. 54° | — | 110° 140°} + | 165° 234° |} — | 290° 320° | + | 140° 144° | — | 234° 260°] + 320263842 — | 144° 165° | + | 260° 290° 4 — 4324? 345° 345° 360° | y + 54° 80° ~ S0° 110° 141. Every expression of the form Acos (a0+«)-+A’ cos (a'0+¢!) +.,... must have at least two values of 0, which make it vanish, if a, a’, @’.... be none of them evanescent. For if not, the preceding expression can never change sign, and in that case its integral (A: @) sin(a0-+a)+.... always increases or always diminishes. But the latter expression has at least one maximum and one minimum, since it has a value for every value of 0, and that value must lie between certain limits. Consequently, its diff. co. has at least two values of @ at which it changes sign, and at which it must become nothing, since it cannot be infinite. 142. Required the discussion of sin‘ w.cos* x, the diff. co. of which is sin? a . cos?.z (4cos’r—3 sin? x), the sign of which depends upon sin w (4—tan’ x), or sin x (tan? 49° 6’—tan’ x). Here is then a minimum when r=0, a maximum when 749° 6’, a minimum when a= 130° 54’, a maximum when r=180°, a minimum when 2=229° 6’, a maximum when 7=310° 54’, and a minimum when «=360°. When «=0, the function =0; whence it increases till =49° 6’, when it becomes °09161, from which it decreases till 130° 54’, when it becomes —‘09161. It thence increases till z=180°, when it becomes 0 again, after which it diminishes till z=229° 6’, when it is again. —*09161. It then increases until a=310°54’, when it is ‘09161, and thence diminishes till a=360°, when it again vanishes. 143. Required the discussion of («—1)*(3—2)°, the diff. co. of which is (w—1)7 (83—z2)5 (80—142), the sign of which depends on that of (a—1) (27—%4) (v3), when x<1, the function is decreasing as 2 increases, when w lies between 1 and #2 it is increasing; when vz lies between 3% and 3 it is decreasing, and when 2 is greater than 3 it increases. There is then a minimum when #=1, a maximum when r=%%, a minimum again when x==3, and the progress of the function from x=—c«c tow~=—-+4+c may be described as follows. When 2 is infinite and negative the function is infinitely great, from thence it diminishes till e=1, when it is =0; from thence it increases till v=%°, when it becomes 2%.3°:7; from thence it diminishes till a=3, when it is =0: and ever afterwards it increases. The questions of maxima and minima which present themselves are, with some exceptions, only of interest in particular problems: I give @ few of the most remarkable. 144. The base of a triangle is @, and the sum of its sides 6; required MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 299 the greatest triangle which can be drawn under these conditions. If «be one of the sides and S the area, we have b?§— a2 b2~ q? S? = ; = bxr— a? — a, 3 4 4 and the sign of the diff. co. of this is that of b—2x ; which, x increasing, changes sign from + to — when w=%b. There is, therefore, (page 133) a maximum when the triangle is isosceles, and the greatest area is ta (b?—a’). 145. A four-sided figure has a for the base, and b for each of the other sides: what is the greatest area which it can have ? Let 6 and # be opposite angles, the former being at the base: then the area is 3 4b sin 6+46* sin >; which is not, however, a function of two indepen- dent variables, since a®+4°—2ab cos 6=2h?—26? cos @. The latter equation gives dé ‘ dS 10 asin é, ape? sing, and ips? b (« cos 0 Bi +b cos 6) 5 being the area: whence we find cae (cos Gee ose b)=3 pee). do e sin @ sin @ Now it is easy to see that @ and ¢ increase together, as long as the figure is convex: whence, 6 being r, the only term independent of z is that obtained from B,_, 2", which gives 2 (n—1)... 2.18B,_,, when n= or >r, and0Q whennn, while D'~ (y" Dé") vanishes when r is equal tom. We then evidently have (when z=0) d" a / dt" de ge PPOs pay i 2 F 5.) Pe SY hy be (n) ——|— v’)j= ry dz‘ ) oat 4 Bs ait az ae ) dy"; which is the theorem above stated. For instance, let x=2°—1, y=a—1, which both vanish when e=1, and vanish in the ratio of 2 to 1. Let P=, we have then MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 305 t=r+1=J/(1+2)4+1; P=(1+y)*=(1+2)*; d?P ols d an d un ; ee ey)” os a (Ge )=S (alts) ‘YA (1+2+1}) 1 J(+2)-++1 V(L+2). ’ when «=1, and y=0, and z=0, the first becomes 2a (2a—1), and the second 4a (a—1)+2a, which are evidently equal. =a(a—1)(1 +2) {V1 +241 44 (1+2)* 153, Required the expansion of wx in powers of pr. Let abe one of the roots of dr, and let y=¢x, z-x—a. Consequently, y and vanish together, and in the ratio (page 173) of ¢/a to 1, which is finite, unless there be two or more roots equal to a, or unless ¢'q@ is infinite : exclude these cases. Again, since z=2—a, we have dA dAdz dA @A ddA dz. @A & Sa AP ee + eS, Ee. da\* “dade G2? de® de dz) dy de” tg diya Gwe y? Dx \ ys yonyas (TE) 94 (Ge ) 2 +( ay ) 3. the bracketed diff. co. standing for the values when y=0, or when z=a. But d"Wx = a (dba r—a\” diy) dz?" \ dz Ge ) i in which ¢ is «+z: which is not altered by writing « for z in the symbols of differentiation. We have then pamper FE ba (VEO) (ny dx (px)? a8 , ie eS (f (z—a)*\ (dr) dix (px)? Bina e being made =a in the coefficients of dz, (px)?, &c. Observe, that these coefficients are results independent of x, though written so as to show how they are obtained from 2. m8 69. '§ 154. Show that the preceding becomes Taylor’s theorem when $x=x—a, and also that Lagrange’s theorem may be deduced from Burmann’s, by making z<=a2—a, y=(«x—«a) : pa. 155. Required the development of wz in powers of x, d~'x being the inverse function of gz, or @ (dx) =x. Write ox for x in the preceding, and we have oo pa) bd ec a? oF raya ES ) de. ay) Z, Wian(x—a)?\ 2 (pxr)* 2.3 L—a d /a—a\? x? d? (/x—a\? 23 —!) CHL pares Ti ig os - 5 nao enee ® +( gz ) +3( px ) 2 al Px ) 2.3 »,4 a 53 dz? 306 . DIFFERENTIAL AND INTEGRAL CALCULUS. For example, let ¢r=(e—a) &~, then to find ¢~'x is the same as find- ing y in the equation v= (y—a) &: and the theorem gives 4 2.3.4 Y x? pres a Dene 8 2 -3a 3 4a yrrape r+ 9 +3 “ir Bee € SO 156. If x and @z vanish together, we have sran(2 é +2(4 rey Bai EL Noe dr cn 2" det \da) 2.8 °°" making v==0 in the coefficients. Let 6v=ar+ bx? + cr er*+ sony so that the determination of ¢~'x is equivalent to finding # in terms of u from u=ar+bree+...., as in page 157. We have then (2—9ryY= (a+br+....)~", which can be expanded in positive powers of 2, unless a be =0 (an excluded case.) The value of the (n—1)th diff. co. of | (a+tbr4...)™, when x0, evidently results from the term which | contains 2"-1, (say A,-i2"~"), and is (n—1)(n—2)....1LAu. Dividing this by 1.2.3....m, and multiplying by 2", we have An a"+n for the general term of dx. Now in (64.) we have found the development of the powers of a+6xv+.... when a=1, whence if im that development we write —n for n, b: a, c:a, &c. for 6, c, &G, | and multiply the whole by a, we shall have the development of — (at+b2+....)7. Let Ps denote the coefficient of x” in the develop- ment of (a+br+....)~”, and we have (64.) 3c . 6b° 1D yeas OSES id ae Dy 2 Pys=— ee we te 20bc 200° avi 5f 15 (2be+c?) 105b%c TOb* ° Pee ah ae gn Psa = — BESS ee te | Ge 21 (2bf+2ce) 56 (3be+3be%) 126 (4b%c) 252 H — Fobra cyqitty vot gh, Slee ae ti, | a a a pee Th 4 28 (2bgt+2cf+e*) 84 (30°f+6bce +c’) ais Sd YY Th Te) ae oiou a) ee a® ae a 4, 210 (Ab%e+6 be ct) _ 462 (5 540) 924 a a a But ¢7r=P Ip lp myth a : Bu oo orth Letts 20 DTP. e+e Pas @ 435 ee whence we have the following result: if u=art bu?+ca®+ ert +fa°+ gui tha'+.... _wu ii, u* ut Then t=——b at (20° — ae) = — (5b'—5abe + a'e)—_ Bon Oe SS 5 + (14b'—2lab e+ 3?2be+o —a°f) — a? cas aa ee Pi be — (42b'— 84ab*c + 28a? be +be?—Ta’ bf+ce+a‘*g) Pn "4 (13208 — 330abe + 30a? Ab%e + 6b°c*— 12a8 BUF + Ghee +e MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 307 FS 7 + 4a! 2g Qf -Ee—ah) — _— &e. + &e. — &e. This agrees with page 158, as far as the latter goes. 157. Returning to Burmann’s theorem, let Y=$2, z=yx, pu and yx having a common root a, and vanishing in a finite ratio. It is required to expand we in powers of dr. Transform z= xv into r=y—z, then wx and da made functions of x are wx 'z and dy z, And dus: * 2 3 3 ye (2) +( ST) y+ (SEP) +(e) y is dy? dys 2.3 eeee “ disy—'z 4 ) d oo Ae um y (dx)? — 1 —— + —— || ait I ——— SS SES a ATT oe a gz (hx *) +( dz ox 'z Pete dz dz hey 2 Tr 158. We proceed to some exercises on the separation of the symbols of operation and quantity, (page 163.) If atar+taa+..., =¢r, by A.b we mean to represent ab-+- a, Ab-+-a, A°b+....., where Ab, A*b, &c. are differences formed from 6, 6,, b,, &c. ‘Thus A*s means b;—3b.+ 3b, —b, (page 77.) (4+7)(a—z)=a?— x: required the exhibition of the meaning and proof of the theorem (a+A)(a—A) b=a%—A%, By (a—A)b we mean that the operation performed on 0 is the subtraction of its differ- ence from its ath multiple’ which gives ab—Ab or ab—b,+b. On this the operation a@+A is to be performed, which gives a(ab—6,+b)+ (ab,—b,+,)— (ab—b,+5), or a*b— (b,—2b,+b), which is a@b—Azb, or (a’?— A?) b, 159. fA.0" represents a finite number of operations; being a+ a, AO" +a, A?0"+.... 44, A" eke Pear UI os Oe in which (38.) all the terms after g” A" 0” vanish. ’ 160. Herschel’s Theorem.* Let it be required to develope f(¢) in powers of «. This might be done by Maclaurin’s theorem, or by making @r=log a and a=1, in (153.) But it is the object of the present theorem to exhibit the coefficients in terms of the differences of the powers of nothing, operated on in a manner depending on the form of the function f. By Taylor’s theorem ferafl+f (e+ SS" 4 pn CV (60.) =f1+f"l (> ax-b a att : ao (+ “ apo att ; .) ie = OT west. 2 5. .)+ ire from which, if we pick out the coefficient of x", we find ae {fl O"4-71 a afl a8 +.... tf) a m2.3....7 nh n * Given by Sir John Herschel in his Examples of the Calculus of Differences, page 66, X 2 308 - DIFFERENTIAL AND INTEGRAL CALCULUS. Carry on the series in brackets ad infinitum, and no difference is made, since A"+" 0*=0 in all cases. In this case the operations performed on 0” are 2 \f1 +f'1.A+f"l + 9a oy abbreviated into f(1+A).0", whence. feHfltf +) 02+ fU+4) 0. 5 +f0+4) 0°, sae he This theorem may be used either to discover unknown series by means of the differences of nothing, or to establish relations between those differ- ences by means of known series. 161. The following method of demonstration® exhibits the preceding theorem in a very striking point of view. The several terms ne x®,..., considered as particular cases of 2%, may be represented by x, (1+A) a’, (1+A4)? 2, &c. Hence Maclaurin’s theorem becomes ba=go.2°+¢'0.(1 +4) O45 00.14 A)? 2+ os l ={90490-040) 45,000 4A)'+... Las, which may be abbreviated into $ (1+A).2°. 1 , ; Now 2*=a°+ logz.a +5 (log r)?.a®+ ..., on which, if the operation ¢ (1+A) be performed, a being then made =0, we have (log x)” 2 pr=h (1+A).0°+¢ (1 +A) 0'.logr+¢ (1+A4).0. —+ cess in which, if we write ¢* for x, we have the theorem of the last article. yt ‘ re] = i a§ 162. Show that Paar \ se Ge -) =f(1+A).0" when =I. 163. Required the expression of Bernoulli’s numbers in terms of the differences of nothing. By definition, B,, the nth such number, is the coeflicient of 2"--[n] in the development of @ : (s"—1); and (17.) the coefficient of a"—-[n] in that of 1 : (¢*+1) 1s — Bi (2"'—1) : ae But, fe” being 1: (e*+1), the same coeficient is f(1 +A) 0" or $1 : (2+A)} 0", whence we have B yar Oe J o"= n+ Goi Die en A" 0" Roh aw, gry" 294A" =a orth] 2 i ie 4 -- 8 i oes arti A since A"t!0", A"t20", &c. are all equal to nothing. It is necessary to. retain 0": 2, for though it vanishes when n is >0, yet when n=0, 0°=1, which makes the preceding series perfectly general. And since — B10, whenever n+1 is an odd number greater than 1, or when-— ever 2 is an even number, we must have AQ™" LA? 02" fag Q?" A Q?" 9 ‘mw A aba. ~ gen ae O (n > 0 >: To verify this, when 27==6, we have * Given by Sir W. Hamilton in the Trans, Roy, Trish Acad. MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. —_309 1 62 540 1560 1800 720_ Behe nS 16 744 Siichn Gees: For the value of B, (n=7) we have 8 l1 126 1806 g8s400 IGG 15120 5040) Rye l — —— or — 4° 8 16 32 64. 1286 “age D380" 164. Show from a: (e«*—1), or log *: (e*—1), that AO” A? 0” A” 0” B,=0"—— = el sh ete ‘ 2 T 3 aE a bola. 36 9:24 1 For instance By=—5 aia ia pias aa 165. Required the development of cos(as"), Here Jt= cos az, J(1+4)=cos (a+aA)=cos a.cos aA—sin a.sin ak, or a? L? af A* a (1+A =oosa (1 +———.,.., ee Pewee y aks) @he ad As ) ae [3] [5] cos (as”) =cos @ + (cosa.0—asina) v 2 ao We + (cos a.0?—a’?—a sin a)—>+ gia at —sin @ (os This may be readily verified by Maclaurin’s theorem; but the deve- lopment is easier by this method, with the table in (38.), than by the direct use of that theorem. 166. If fe"=2", it may be shown that {log(1-++A)}"0"=0 in all cases, except when n=a, in which case it is =1.2.3....@. Also, if jr=2", it follows that (14+A)*0"=a" for all values of a, which was known before in the case of whole and positive values. Thus (144) 0"=0"— Ad*+ A?0"—.... +A" 0"=(—1)” (1+4)* 0"=0"— 2A0"+ 3A? 0"—.,.. £(n+4+1) A? 0"=(— 2)". 167. The preceding result is even true when the exponent is incom- mensurable or impossible. Thus, the second of each of the following | pairs verifies the first. —I1 (14A)Y7 0? = 08/7 20% 7 A? 0? Wt) = Ji+y7¥ fs =.2 may epee nine RAE nel 9 78 (1+ A)" 0°=0?-+(1+V7—1) 40°-+- (1 al =P) ork A* 0* (47 —1)=14/—147—1- 1. { 168. The following propositions may be easily proved by con- sidering the functions of ¢", in which the operations set down will be coefficients. " DIFFERENTIAL AND INTEGRAL CALCULUS. {(log 1A)" .fA} O'=a (a—1)....(a—n+1) { fA}.0°™ ff. (1-4A)"} 0° n* { f (1 +4)} 0° LfFAFA)+FAFAT}FOMM=O — {FA+A)—f +4) OM =. Thus, in the second instance, the first side is the coefficient of «* : [a] in the expansion of fe’, which is n* X the same coefficient in that of fe. 169. To express a function of differences as a function of diff. co, Let wu be a function of a, and let u=g2, w=9 (t1+h), w= (44+ 2A), &c., from which let differences be taken, namely Au=u,—u, A*u= o—2u,+u, &c. Let fA.w be the function in question, that is, fA being at+aA+aA°+...-.5 fA.u means au+aAu+aA°u+.... Then, Mu being (c’”»—1) wu (page 165) we have i 2 2 pr.uap (1) .u={fO4f4.0.AD+fA.0 +. O98 hu d Hence au+a Auta tut+....=f0.utfr.0 = h fA.0° du fA.0? Bu A —_ fP 4" —h?+..... TNS) de 8 de fo=a, fA.0=a,A0, fd.0°=a,A0°+ 4,470", &e. 170. To determine @u,+ @,t,4,+ Gol,19+.... in terms of differences and diff. co. of uv, Here the total operation performed on w, 1s ata, (1+A)+a,(1+A)?+...., or f(1+A), or fe”. Hence i du, + Usb... fl utp. duh WU, ee ess by | du fA+A).0% du fU+d)0° du =fl.utfOta).0— +5 oe thee Wha ee 171. Let yo = be 4+ 6(@4+1).a+¢(@4+ 2).@4......: then we={ita(l+A)+....}¢r=1:Cd—a—ad) pe. ke om adh he Let A=a: (1—a@), then yan -ialy ¥. Ge iat “a eat I. Bic: ont | mae Oh ale wyus AbtH 5A 0.be\s— aR OF: 3 ++ 0. 373 1—AA P+ A®A? 0? (—a) po pr+AA0.d/e+ ELEN) FF yy 2 ADO FATALE AAO sig 2.3 172. The coefficients in yw are these in the expansion of 1 : (L—aé*), and if a@==1 the expression fails to give a series in a finite form. To find the sum of the terminating series ¢?a+@(a+1).a+..--- + (x+y—1) a’, we have evidently yxr—y (w+y).a’, and (1—a) {wae—y (2t+y).@}=(pr—a'’py) + AAO (p'a—a" p! (a+y)) AA0?-+ A?A0? $a ERS (lye g" (tM) toe pa MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. dll 1 Dia. But if a=—1, or Asp the expression in (17.) gives Pe SR > 15B 635 Da a lp et gy OE Ws 5 5 7) [4] by [6] px ] 1 1d"2 3¢6'r = 5 Pt—— p'r4+— — 1, 3? 4? 2 [4] 2 [6] gu-$(x+1)+...+¢ (@+y-D=5 (ort o(xt+y)}- jigeky! (r+y)} A ete Gry) 3brt9' ~+y) 2 2.3.4 079 .OAl Goh the counterpart of (69.) : verify it from what precedes. 173. If « be a great number, show from the last that @ EHD) eas: (a+ 2y) Bs — (~+1)... eed ae at nearly. Pe, tae Zoi" a+2y — iy @ . » we ees ee >, ES ] 7 +1 £4+3 x+5 a+Qytl VV «+2y42 nearly 174. If the value of ya in (171.) be reduced to a simple function of a and 2, it will become or+¢ (@7+1) 44-6 (4+2) @4+... a bo+ a 2 ata "x at4e+ad"«& atlla+lleta dx Gd—-a)? 2 " @aayt” B33 (12a) 3 4 a+ 26a +66a°+26at+a> fxr Vinay AL, 3.3.4.5 cece eens 175. In the result of (69.) we may observe that the series contains a part which does not depend on a, but only on the specific value x=0, and which is in fact an arbitrary constant of an infinite number of terms, depending on the beginning of the series. Calling it C, we have 1 Baa, Loe — dz—— , a iS Le eee ee 8 2H O+ fy. 27°76 2 302.3.4 : where the constant of the integral is also contained in C. We shall now show how to use this series, which is most available in cases where the diff. co. of y, diminish rapidly. It must be remembered that Ly, ends with y,_,. ; 1 176. Required hehe, 1 : Poa 8 Fes a the » 2 x" 12 gt 120 yrs “eee oe Add 1: 2* to both sides, and we have I ] l n n (n+1)(n+2) 1+ as te © + ne (aes to 7 Dy + 520a"+8 except only when n=1, in which case the two first terms are C+ log a. ra=04f2 oe Reeds n(n+1)(n+2) 1 3 312 DIFFERENTIAL AND INTEGRAL CALCULUS. To determine C we must calculate one value of both sides of the equation in some particular case: thus, if »=1, and if we take the case of x10, we shall find by calculation 2‘9289683 for the first side; and therefore . 1 1 1 2 9289683 =C-+ log 10+ 55 7200 tT300000° eres which gives C=*5772157 (log 10 being 2°3025851) 1 1 1 1 1 = ~=°5772157 +logr+— ——— +——_-. pai bevy at 577 157-Floga+— 928 tin Thus we see that the series of reciprocals of whole numbers, when @ is considerable, increases with the (Naperian) logarithm of the last number, nearly. 177. Let the series be log 1+log 24....+log (w—1)=2 logaz. 1 1 bok ov ea dz—— log thee }— = Conti « Slog x C+f log «dx 5 0g tao, 360 at log 1+...+logr=C+(1 2)-+5 log 2-5 a5 og -.- Flog v=O+ Clog xr.@ +5 oS tS 360. ee-+ In this case we have already shown (126.) that the preceding approaches 1 to log (/(272).2*¢ 7) or log,/ (27) +5log ate log x-a: consequently C=log,/27, and 1 1 Lo Bi Su ws ait Ce ue gar 360 ast’ ** 178. Show that a,Au,—a,A®u,-+a,A°u, — oo. nme | a, AuU,-;— AQ, A’u, e+ Aa, M?u,-3—- @ oe 3 4 ty h* hI h* and also that ade+a,p'a.h+ad t>+ashp ree ee : 2 ~ =ap (a«t+h)+ Aad! (x+h).h+Nad" («+h) a iF 179. To expand A" y, by means of differences which can be obtained without using Yr415 Yrros &C. A A At y=" (1+A)*.y=C+Ay Fae eee fy Poe Se peng, | A ' n+1 Ae =A" (1+4) ite iat oka ayayt Yo n+l 2 mA" yg ENA" yn EN Att? inig huis 180. In (61.) it is shown that hi 1 1 MTR, wees See t V v be V. t 8 e@e 4 —— = oe oe i. RTE 1+V, 74+ Vov?4+ Vz 0° + We 5 Ve 2 &¢ For x write 2: (1—), whence the first side becomes as £1, ab Vi, Gh Be oh h (1—2) log (i=z)y’ ‘e jhe > whence IVa Vat... =(—2) 14, +. 4 | aap =1—2r+Vi2+V, (+ a°+....)4-V; (2° faye + 325+ ....) + V,(0*4 325+ 625+ mrad (a° + 42°-+ 1027-++....) =1=—(1—V,) 24+V.2 oy (Vs+ Ve) 2°+(V + 2V,4V,) x +(V,+3V.+3V3+V_) 8+ (V,+4V,+ 6V,+4V5+V_) 22--. 0s. whence Vet Ve=—V;, V,+2V;4+V,=V,, &c. —l Viaet nV tn—— Vat. vee +2Ve+ V2 (—1)"V.,. 1 aol. In (67.), the y,dr was expanded in a series, the variable part of which was (Yo=Y25 Yoa—1eZ2Y 2-1 Sicis’. « at LYe+ Vi Yst Vo Ays+ Vg A? ys t Vz A? Yet voces Which (179.) is Sy,+V, y.-+ Vo (Ayr-1+ A? ype tA® y, a+. 00.) +V; (A? 2 Yo9t 24° Yee SANs 4b a es) EV, (48 Yo-s-+3A* y,_4+ 60° vena =2y,+ Vv, ¥2+Vs AYyz-1 + (Vst V2) A® Yr—et (V,+2V,+V,) L* Yr-st oes = Ly.t V, Yat Ve Ay,.— Vs LA? Yn 9+ V, A* Pig, A‘ Y iD, Rake. « Joining to this the constant part, the same as in (69.), we have 1 ays VE emt Yro+ Vi (Yno—Yo) +V, (AYne——AYo) -V; (A? Yno~so tr” Yo) +V, ip gh a 3 —A’y,) —V, (AS Ynp—4o + At Yo)+-- i If the limits of the integral be a and a+n0, we ete i similar reasoning, 1 Q — fatrty, AL=Y a+ Yaro os oe FYar~@—pot Vi (Yatns— Ya) Sg os. 9 The use of this theorem in approximating to the values of definite integrals, is called the method of quadratures, from its most obvious application being the determination of the area of a curve in square units, which is the arithmetical problem answering to the quadrature of a curve, or the determination of a ehibte which is equal toits area. The two first terms, V, being $, make up 3 yatYapot-- ++ +4 Yatnoy and the theorem may be thus exhibited : efetet Y2 da= (4 ON a. a: coee TYu ane Ba 0 0 —_— 19 (AYetn—9— Ay—3, (A? Ya+no—go-t O° Ya) — = 2 (A% Jor: n6=307 7 — A? Ya) 863 6 : r-7 S (ks Yo+ns—19 + A! Ya)— (AS Dasa Uap eoevesn i 60480 182, As an example of the preceding, in a case which can easily be verified, we propose to find flog x dx from x=11 to#=20. We have then a=11, n0==9, let n=9, 0=1. Taking a table of hyperbolic logarithms, we find the following logarithms and differences : DIFFERENTIAL AND INTEGRAL CALCULUS. No. Log. A+. A?2—. Af+, : Los A3-+-. 11 |2°39789527'0: 08701138 0: 006968670: 00103393)0-00021429/0° 00005540 12 |2°48490665/0° 0800427 1/0+ 00593474)0+00081964)0* 00015889)0 + 00003859 13 |12°56494936/0+07410797/0° 00511510/0° 00066075|0+ 00012030 0+ 00002755 14 |2*63905733)0+ 06899287|0- 00445435)0+ 00054045)0 + 000092750 + 00002055 °15 |2+70805020)0: 06453852) 0*00391390/0+ 00044770)0+00007270)0* 00001497 | 16 |2°77258872)0° 06062462)\0+ 00346620/0+ 00037500\0° 00005773 | 17 |2°83321324)0°05715842/0+ 00309120 0+ 00031727 | 18 |2°89037176)0°05406722)/0: 00277393 19 |2°94443898)/0° 05129329 | 20 |2°99573227 | Slog 11+log 12+.....+log 19+ log 20 = 24°53439011 | — {0°05129329—0+08701138}—+12 =a -00297651 — { —+00277393— * 00696867 | +24 = *00040594 — 19 {*00031727—- 00103393 }—720 =+ °00001891 — 3 {—+00005773— *00021429}—160 =-4- *00000510 — 863 { 00001497 — -00005540}~+60480 =-+ +00000058 | | 24°53779715 Now ,f loge drz=x log —a, and fj} log x dx=20 log 20—11 log11—9 = 20% 2°995732274 — 11 x 2°397895273 —9=24°53779748 ; or the preceding approximation is true to six places of decimals. | } 183. The smaller the value of 6 in the preceding example, 70 being | given, the more nearly 4 y,+Yapo +++. +4 Yotns approximates to the value of the integral. If, for imstance, we were to divide 0 into ten parts, and it 0=10X, then | SYet Yotypt cane bE (Ya+10a> or Yoro) +Yaruat tees +4 Yariona is much more near to the required integral. The following questions will illustrate this, and at the same time introduce a useful theorem. 184. Required the development of u=x:{(1+.)"—1} in powers of xz. Here u(l+ey=ar+u; vw A+ay+nu0 +e) '=14+v, u™ (14a)"+-knu"®-) (14+2)"7+...4[n, n-k+1l]udte2yt=u™. Let v=0, and let U, U’, &c. be the values of w, w’, &c.; then U/+2U=14+U' U=- 2nU/+n(n—-1) U=0 tyes ee pee Qn 3nU"+3n(n—1) Uta (n—1(n—2)U=0 Us Seo | n | (n- Intl) 4nU" +6n(n—-1)U"+4[n,n-2]U/+[n,n—3]U=0 U"=- 7 | n : | MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 315 5nU"+10n(n 1) U"+....+[n,n—4] U=0 pre Dt NGI—n) 30n 6nU*+ 1l5n (n—1) U"+....+[n,n—5] U=0 Ue (2— 1)(n+1)(9—n?) wa An ay TnU"+21n (n—1) U'+....+[n, n—6] U=0 i (—1)(n+1)(863—145n? + 2n*) U = 84n Applying Maclaurin’s theorem, we have Ped med td ntl wi -Id9—n) |, (i-+7r)"—l 7 2n 2.67 2.3.41 2.3.4.30n (n'—1)(9—n*) | (n?—1)(863—145n? + 2n') : 2.3.4.5.4n PERRY TY, Rn a Verify this series (1.) by making n=2, when it ought to become the development of 1:(2+ 27); (2.) by multiplying by 7, and diminishing m without limit, when it ought to coincide with the development (61.) of w:log(1+2); (3.) by writing x:n for n, multiplying by n, and in- creasing 7 without limit, when it ought to become the development (16.) of x: (e*—1). 185. Let y%, ¥,....y, be the terms of a series, being the several values of a function of x, corresponding to r=0, r=0, c=20, &e. Between each of these terms let n —1 terms be interposed following the same law, so that, in fact, if the function were Gz, and if four terms were interposed, the terms @(a) and ¢(a+6) with their interposed terms would be $ (a), 6(4+48), P(a+20), (+20), 6 (4449), $ (a+9). Required the total sum of yo, y,-+..Yr—1 together with all the inter- posed terms, including those interposed between y,_, and y,, by means of >y,, the simple sum of y+y,+....+Y, 1 and differences taken from the original series, as if the terms had never been interposed. The following process contains the most difficult instance which has yet.occurred of the separation of the symbols of operation and quantity. { shall, therefore, follow it by another* demonstration, independent of that principle, and, the student who can comprehend the first will see that it is an abridgement of the second. The function y, is (1+A)’.y, and this whether x is whole or fractional. Hence the sum of all the terms, primitive and interposed, is {14+ (14+A)"+..+(1+4)4+(1+4) #4. .4+(144)%4+-.. LE(EAy = by bit Ay? A (14A)*—1 or eee SL ei Yo, OF oo SE Shel Varese Yo: (1+A)"—-1 (1+A)7%—1 . * Being that given by Mr. Lubbock, to whom this theorem is due, (Camb. Phil. Trans, vol. iii. p. 322.) 316 DIFFERENTIAL AND INTEGRAL CALCULUS. Now the operation (1+4)*—1 performed on y gives y,—Yo, and A™ is the same as ©. Write 1:7 instead of m in the development obtained in the last article, and substitute the expanded operation instead of the condensed one, which gives ye | Tr—!] n?—] ; ee woe fo ay On ee \" = As 12n + 24n me Ye (Yz—Yo) n?—1 2 24n lye Oe —1 n—] =nByet~S— Ye W— Fa, Aye Bye) + n?—1)(19n? — 1) (n? —1)(9n?—1) Se (A® y,— A’ Yo) + SPR TE Bay (At y, —A* y) (n?—1) (863n*— 1457? + 2) a Mees 2 ASaiene Te Ad y,— Ab ye) base 60480n° (Ary. — A yo) 4 Here Sy, meaning yo-k....+Y,2, stands for A™'(y,—Yy,): this transformation is obtained as follows. The meaning of A™ (y,—Yo) is that function which gives AA™ (y¥,—Y)=Yz— Yo: Where yp) is not a constant with reference to the operation A, as abundantly appears in the preceding process, in which we have Ay, not =0, but =y,—y. If, then, A~'y, stand for the sum of all terms up to y,_,, (as in page 82,) then A (y,—Yo), or A ¥,—A™" Yo, is the preceding diminished by the sum of all the terms preceding y, that is, Yot..+.+Yy.a- The | truth is, that A’ y, should stand for | Yoos PYr—a toe EY +Yotyrtyot-.. ad. inf. 5 this being the only series which satisfies AA~'y,=y,. Or the symbol Ly, beginning from y,,, and ending at ¥,_1, is A (Y,— Ym): 186. The second demonstration is as follows. Let 1: ==, then Yor Yorir Yorsior e+ Yos(n—i make up y,, followed by the terms interposed | between y, and y,4:, Using the theorem Ri—] 2 J 9 A Ee eese 9 w Yt Yo* ke Ayy+hki and summing the results, we have for the 2 terms beginning with y,, hed poy Qian ny, b fit Qit...(n—1) i} Aust 4é +2154 eer een bata bee | Apply this to every term, from ¥, to y, inclusive, and we have for the required sum . ih oe ndyo+ G+2i+ ++ +(n—1) i) Say ti Sabet a cee LT) Baty pee But LAy,=Ayy. ees FAY =Ye—Yo3 DA’ y,=Ay,—AYy, &e., and the coefficients are evidently those of the powers of # in | MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 317 (1+a)"—1 i xv (tay? .d 42-1 since mt=1. ‘Taking these coefficients, and writing 1: for 7, we have the same result as before. 14+ (1+2)'+(+2)"4+...4(1f2)° 4, or 187. It has here sufficiently appeared, that instead of Sy, being made an undetermined symbol, by not having a specified beginning, it would have been more agreeable to analogy that it should have begun from —c, or should have signified y,.+y,.+.... FYyotyst.... ad infinitum. In such a case A and 2 would have been really convertible operations; for AYy,—(y,+...-)—-Grit--..)=y,, and PaaS Bram yi t Ay, ot... . Sys Yea tye Ysa +... HY, That I may not, however, depart from established notation, I shall in future use A~'y, as meaning the preceding series : so that ee ond a —I Sy, AT y,—Ay,, or Ag (y, —Ym)s where m may be anything whatever. 188. [fin y+y:t.-.. +ytyruit-..++Yy.-; we multiply by 2 or 1:n, and increase n without limit, we approach (page 100) to i oat Let this be done with the preceding series, and we shall obviously approach without limit to the series obtained in (67.), as it becomes when 0=1, m=z. 189. If we add the term y, to both sides, we find for the sum of Yoo Yi» ee +Yz, and all the interposed terms v— n—Il ? i M(Yor It oo ++ +Yc) ~~ Yet Yo) — Fy AYs AY.) Five eves 190. In the series obtained by writing 1:n for n in (184.) write a:(1—2) for x, and then multiply by 1—a. This gives {A,=n, A,=4 (n—1), &c.} 9 : x : aaa Mana} =A,+(A,—A,) v— A, 2+ (A,— A,) 2? —(A,- 2A,+ Ae) xt (A,—3A, +3A,—A,) xt ST te | Oe But the first side may also be obtained by changing the sign of nm and | of x, and then changing the sign of the whole. The first and third Operations compensate each other in every term but the second, and we haye av v —. : =(1—2) {Act AL rere (i—v) »—] “i 1 Sey =A,—4t (n-+1) a A,w’—A; w—A, at— eos (l—x) *—1 whence A,—A,=—A,, A,—2A,+A.=A, kh—J — k Ape RAga bk A.— eevee 3 kA,;% Ag==(—1) Arte 191. The series in (185.) requires terms following y,, in order to . . } construct the necessary differences. But it may be reduced to another, . . . 7 e rp requiring only preceding terms, by the same process asin (181.) The scries in question is “318 DIFFERENTIAL AND INTEGRAL CALCULUS. Ag 2y,+ A, (Y. —Y)—Ag (Ay,—AYo) +A, (A° a A? Yo) — ie oy For Ay,, A’ y,, &c., substitute Ay,_;-+ A? Yet... 2, A’ Yr gt 2A° Yn—g +...., &c., which gives Ao 2Ye+ AvYo— Ao(Ay,1 + A” ah ae )+A,(A’y, 2+ 2A*y,-st+.. )— ee —A, YotAs AY —A; A’ yo Kies =A Sy,-+ Ay.—Achy,_:+(As— Ae) A’y,_9- (Ay—2A;+A,) A*y, 3+ tee —AyYtAgAy — As A’y, + A, A®y, aes =f; Ly.+ A, yA, (Ay,_,—Ayo) — A; (A? Yong tt A° Yo) —A, (A®y,-s— A® Yo) — # Tagore Or, making the alteration as in (189.), we find that the sum of the terMS Yo, Y/,++-+Yxy With the interposed terms, is n—] nv—] : M(YotYLt sees Bat E watarp al ek hear ss (Ay,~1— Ayo) (v?—1)(19n?—1) Se ABO worl ewe: (A? y2-s— A*y,) (n° —1)(863 nt — 14572 + 2) 60480n5 ‘ w—l, 9 i cage (A* y,-2+ A” Yo) = (v? —1)(9n? — 1) A80n3 a (At yng + A* yo) — We shall now proceed to some methods of obtaining the sums of | series connected with the roots of unity. The mth roots of unity are 1, O, A. 0..a" ', Where @ is Qa, wil a 2 a Bee e" —', or cos +01 sin —. (page 127, &c.) 192. Let Se” stand for the sum of the mth powers of these roots, then Sa«”=0 in all cases, except when m=O, or n, or a multiple of n, in which cases Sa”=n. Soa™=1+a"+em+t.... Sogn ae = : at — 1” but the numerator =O in all cases for o’""=(a")"=1"=1. But the | denominator is never =0, unless m=O, or n, or a multiple of m | Except in these cases, then, Sa”=0; and in these cases every term of — the series is unity, or the series ism. This theorem is equally true of negative powers, since «"=1 gives « "=1. 193. Given the equivalent function of a+a,0+ a,2°+...., required that of Gn 0" 4 Oingn DF Amson OP +... (m being the nth roots of unity,) or o°""”", &c. as may be most convenient, write ax for x. Do the same with 6, y, &c.; we have then eae pax aa” 4 a, Polat a+ > one +n a” xv Am+1 ant Fa 2 90) ee tae pbr=ap"-" + a, Get a+ ean +n Cy "ty 16 sh emt) o2 &c. &e. &e. Adding these together, every term vanishes except those which contain — m a”, «"*", &c., and we have MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 319 ] S (a "bar)=n4,, Oh NA man get. hes x S (ayo Pax) =A au +4 Anin gett oe ee x x8 riz 194, What is 1-4+-—— +— 4—_ at is woe tis +Tait: .. The preceding theorem gives 41S +e-*+2 cos x}; 1, —l, aa and sist aT being the fourth roots of unity. erin x =i fe 92-3" 1 3 net Te] ene e”-+.2¢ cos( 5 3.0 )p a 2a v 95. ir = $$$ 195. Required gz ae) a) ENCE Be where [6,6-+c] stands as before for 6(6+1)....(b+c). Multiply this series by x’-':[b—1], and we have a+b-l at b-1 2a+b—1 a x ‘CET haa ES an ee T Poa Pboiy as Huds deh Ao which, with intermediate terms, is eh ao? oa 1 Sina? ae@e iene chee ts Call tetagdiyrutpes) Let «, PB, &c. be the ath roots of unity; multiply the last by a*’t!, Br’, &c., and substitute az, Bx, &c. for x. The results added together give the series required in a finite form; and this multiplied by [6—1], and divided by 2’~', gives the original series. 196. The nth roots of —1 are &, a, o5....a"—!, where L, a, a®.... o#”— are all the 2nth roots of +1. And we have for the sum of the mth powers of these roots of —1, a” + a3" Peas + of—Dm, or a” Qnm_ l ll ~~. 1 The numerator, being (o")"—1 is =0 when mis a whole number, positive or negative; so is the denominator when m is 0, or n, ora multiple of n. But when m is an even multiple of 7, each term of the series is 1, and when an odd multiple of m, —1: consequently the sum of the mth powers of the nth roots of —1, isn, —n, or 0; the first when m is an even multiple of » (0 included,) the second when an odd multiple, the third in any other case. 197. Given xr=a+a,x+a,22+...., required a,, L— Opin BO" HP Onton 0 — 1. (MKD). Let «, 3, y, &c. be the 2nth roots of —1, multiply dx separately by 2n—m | av”, p-™, &c., and change ze into ar, Bx, &c. The results added toge- ther will give (rejecting terms which disappear) Sa2*—™ hax = S22". dnt” + Sa Omin a shel APA: ] = Sa" Dad = Ay BP — Bis BO be ng an UT — oss n 198. Required a, 7—a,2z*+-a,27—...., px being a+a,a+...- ~ The cube roots of —1 are —1, $4+4,/(-3)=«, $—4.,/(—3) =f, and the required result is one third of 320 DIFFERENTIAL AND INTEGRAL CALCULUS. $(—2) , $1G+4v (3) eh, ef G—hw(—8)) ef peek Pek ara ee a—3/(—3) ¢ eu or ; {@ (ar) + (Pr) — {bh (ax) —d¢ (Bx)} -; b(-2). aS me pees wc" J/3 ‘ads | be - — —_—_—- —_—_ — — 2 —— — ——EF& Aus x Cini te {eos a+4/3 sin 5 =o & 199. From the preceding it can be shown that if ata,xr+.... can be expressed in a finite form, #2, then also that series can be expressed in a finite form, which is made by allowing the first m terms to stand, changing the sign of the next m terms, and soon; changing the sign of every alfernate set of m terms. And this can also be done, if only every nth term of the original series be taken, and the result separated into parcels of m terms each, changing the signs of the alternate sets. And the same is true if the terms of the resulting series be multiplied by 6, b,, 6,, &c., b, being any integral and rational function of n. So that, for instance, if a+a,xr+.... be expressible in finite terms, the following has the same property : Gn BL py 0, 0"? ell toy Og BT? — in agy Dg BT? Fp — te 200. (Chapter X.) If dx and wa have the same limit, or if both increase without limit, or both diminish without limit, then of course the final tendency of 6r may be found from that of ya, or vice versa. And in the case of a finite limit, we may say that dx: ww has the limit unity, but we may not say the same if both increase or both diminish without limit. Thus, if 2 diminish without limit, a+ 2 and a+a®* have the limit a, and (a+2”): (a+) has the limit 1: but if a=0, x and 2 both diminish without limit, but 2: also diminishes without limit. Thus the tendency of ¢x: ya, if both functions vanish when r=a, can always be discovered from that of d’v: yx, or Pa: y''x, &c., but it is only when $v: Wz has a finite limit, as @ approaches towards 4a, that we can say that {#lv: we}: {ox: yr}, or (Pa yr) : (Ye hr) has the limit unity. 201. To avoid circumlocution, let us in futuré use the algebraical symbols of the limits of magnitude, interpreting them in the language of limits. Thus ¢(« )=c means that the function x increases without limit when x increases without limit, and nothing else. Also dae meaus that dx increases without limit as x approaches to a: @(0)=:«% means that dr increases without limit as v diminishes with- out limit. Sometimes when it is necessary to recall this caution to the student’s mind, we shall write the single word (limit) in parentheses, for that purpose. 202. If ¢a=0 and Ya=0, then x and wx may have two distinet relations. If da:(wa)’=c (limit), then still more does Ga: (ya) =a, k being positive; and if ¢a:(ya)’=0 (limit), then still more does da: (va) *=0, k being positive. But da: (a) is certainly =0, and we have the two following cases. 1. da: (va)’ (limit) may be =0 for all values of e, positive and ‘ | negative. Thus, for all values of e, ¢ *:a‘ diminishes without limit when x diminishes without limit. ' MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 321 2. There may be a eritical value of e, such that for every greater value da: (a)'=a, and for every less value =0.. This critical value must be nothing or positive ; and when e has it, the function Ga : (ara)’, may be finite, and may be nothing or infinite. Thus (as we shall see) (w= 1) loga:(#— POF once < according as e<= or >1 (a=a) aw": (e-*)' =0, 0, or & es Shekel cxptaruanie e on > 0, 203. In the ordinary functions of algebra, dr: (we)* is usually finite when e¢ has the critical value. The other cases have attracted but little attention; and as I have, in the preceding part of the work, made two errors from neglect of the distinction, I shall now proceed to correct them. Since da : (va)’=0 when e¢ is 0 or negative, it must, as e increases, either remain =0, or must, for some specific value of e, become finite, or for the first time infinite. When the latter happens, the critical value is finite; but when the function —0 for all values of e, we may say that the critical value is infinite. And, e itself having the critical value, Pa: (wa) =a, ga: (wa)"=0. Turorem. If ¢a=0, Ya=0, the critical value of e in da: (wa) is fava: pay'a Let R=¢e: (yr)’, and as we speak only numerically of the limit towards which it approaches, let dx and wz be positive. We have then / / / Inn 1 diff. co. log Rete se hes {eee el, gr we wr lwo de First, let x be increasing towards a, and therefore px and bx diminish, or begin to diminish before ra. (In this way all assertions about increase and diminution are to be understood.) Consequently ¢/x and y’x are negative, while ¢/x war: Pry'x is positive, and yin: wr is negative. Let k be the limit of ¢/rwer: dx vx; then diff. co. log R Must at last take the sign of —(k—e), or of e—k. If, then, e be the critical value; that is, if the substitution of e+e’ for e (however small e’) would make R a function increasing without limit, or diff. . co. log R positive, and if e—e’ for e would make R a function diminishing without limit, or diff. co. log R negative; it follows that e+e’—kh is positive, and e—e’—k negative, for all values of ¢ however small. This cannot be unlesse=k. But if R diminish without limit for all values of e, then diff. co. log R must become negative, or e—(P/r wu: Wie dr) must become negative for all values of e. Consequently, ¢’a wa: d'a wa (limit) must be greater than any value of e, or infinite ; that is to say, the same expression which gives the critical value, when there is one, becomes infinite when no value of e is great enough to fulfil the con- ditions of a critical value. Thus, adopting the usual phraseology, the critical value is infinite. Next, let @ be diminishing towards a, so that the diff. co. of a ‘ree? function is veaavee Moreover, let ox and Wx be positive, as before. Then ¢/r and w'v are positive, and so is $'2 wae: prw'e, Therefore diff. co. log R takes the sign of k—e. If, then, e be the critical value; that is, if the substitution of e+e’ for e (however small é) would make R a function increasing without limit, or diff. co. log R megative ; and if e—e’ for e would make R a function diminishing Without limit, or diff. co. logR positive: it follows that k—e—e’ is x 322 DIFFERENTIAL AND INTEGRAL CALCULUS. negative, and k—e+é' positive, for all values of e’, however small. This cannot be unless ek. But if R diminish without limit for all the values of ¢, then diff. co. log R must become positive for all values of e. Consequently, d'aya : day'a must be greater than any value of e, or infinite; and the conclusions are as before. Corottary 1. If dae, Ya=o, ga: (#a)' is the eth power of 1 TD no ; AE Hy —_+( — Je, and, e being positive, both are nothing, finite, or infinite, ya” \a together. But, by the theorem, since (fa)~=—0, (wea)-'=0, the critical value of 1 ; ¢ is diff. co. (ha). (fa) Wia.pa $$$ (ya) diff. co. (fpa)™ ae wa.pla Hence the critical value of e is #/a.ya : da.w'a, precisely as before. But since the reciprocals of ¢a and a took their places in the reason- ing, (and this can be shown independently,) it follows that, e being the critical value, da : (wa)*t’=0, and a : (wa) = o&; also, that when - ga : (vay is always infinite (at which it begins, if we begin with e | negative, or nothing,) the limit of ¢’a wa: pa y’a is infinite. | vs COROLLARY 2. If da be finite when r=a, and when %wa==0 or cc, it is obvious that e==0 is the critical value. “ But as the preceding demonstration did not apply to this case, though it might be adapted to — do so, consider the function in a form to which the theorem applies, namely, | I sed , which gives aie +1 for the critical value of e+1: (yay ba Wa but this value is <1, as is obvious from the function; whence | flawa:pay/a=0. And by such an inversion as that in the first | corollary, it follows that when yr is finite, flava: payla=—oa, if pa be 0 or «. | Corouuary 3. If one of the two be =0, and the other =o, then | ga: {(va)-1}-* can be treated by the theorem, and gives a positive | value for —e, or a negative value for e. And it readily follows that when e is less than this critical value, 6x: (yx)° has the same limit as %x, and the contrary. But if —e be infinite, or e infinite and negative, gar :(va)* has always the limit contrary to that of wax; that is, 0 or | wo when a has the limit « or 0. All these are, in fact, cases already described. 204, Allthat precedes may be collected into one theorem, as follows. When wa is finite, the character of the limit of pa: (y¥a)" (whether 0, © finite, or cc) is that of da: in every other case, e being p’a ya: pa y'a, the limit has the character of ya when 7 is less than e, or of (a) when 7 is greater than e; or has the character of (ya). The preceding demonstration has been purposely derived from first principles, and shows clearly what takes place when ¢ is infinite. The following, of a much more simple mechanism, is perfectly satisfactory only when ¢ is finite. We know that log A An log oe A= Bes 8, whence sia == {wa losve , (yay MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 323 When wa is 0 or «, log Ya=«a ; if then log ga: log wa be finite, we must have logda=e, and the value of logda:logwa is that of (Pa: pa)—(W/a: wa), or pa ya:ywlada. Hence da: (wa)" has the character of (ya)*~", as asserted. 205. If da: (wa)’ be finite, then e is the critical value, which is therefore finite: but the converse is not true ; that is, da: (wa) may be infinite or nothing, the critical value e being finite. Thus, if Gx=xlogx, wr=x, we have ¢’z we : We bv=1+(logx)-!; which =1 when z is infinite: but in that case Px : Wx is evidently infinite. This leads to an extension of the theory of algebraical dimension, as follows. If we take two powers of 2, x*, and a°***, and make « infinite, then, however small & may be, the second is infinitely greater* than the first ; and if a+/ lie betwen aand a4t-k,then xt! ig infinitely greater than «*, and infinitely less than x***, These three are of different dimension. Let us now make a definition of dimension, not attached to the notion of exponents, but to the necessary character of difference of dimension. Of two functions which simultaneously increase without limit, let the dimension be said to be the same if they be always to one another in a ratio which approaches to a finite limit, But if one increase without limit with respect to the other, let the first be said to be of a higher dimension than the second. Abbreviate as follows: when two func- tions are infinite they are of the same dimension if they have a finite ratio; but if one be infinitely greater than the other, the first is of a higher dimension. The following consequences are evident. ‘Two functions which have the same dimension with a third have the same dimension with one mother; and if A have a higher dimension than B, and B than C, A has a higher dimension than C, Usually 2* is the dimetiené function of algebra; we must come to the consideration of transcendental quantities before we find a function which is not of the same order as x, for some value or other of a: and hen between x* and a*t* may be found an infinite number of functions, nigher in dimension than the first, and lower than the second, however mall k may be. Find the critical value of e in (log x)’: x’, and we shall find e=0. That is, (log x)’: 2° is =O when v is infinite, for all dositive values of e. Therefore, b being positive, x*(log x) is of a aigher dimension than 2*, and of a lower than a***, however small may be, or however great 6 may be. Similarly, (log x)’ (log log x)’ is if a dimension between that of (log x)’ and (log v)’**, however small & nay be. Denote logx, logloga, &c. by Aw, ea, &c., then, however mall may be, the function in each line of the second column lies tween that of the first and third in dimension. xe a (dx) gtk a Az)’ x* (Av)’ 2x)" w* (da)? a Cray (Mx)? | a (Az) (2x) 8x)? | 2" (Ar)? (aoe) &e. &c. | &e. Ve have then an infinite number of iterpositions of dimensions * We intend to use this language in abbreviation of that of limits, See INFINITE nd Liwrr in the Penny Cyclopedia, Y2 324 DIFFERENTIAL AND INTEGRAL CALCULUS. between those of x” and v***; and between each of the dimensions 80 obtained, an infinite number may still be interpolated. Thus, write Av in the form ¢4", and it will be found, m being >0 and 1,| and many others might be given. We shall here confine ourselves to! the cases in which the several sub-dimensions are finite. Let us now find the critical value of n in @r.a77 (Az): Q2x)". Tf we call it c, we find | I c = limit of \°v jhe (: ad) a) —b \ px J Proceed in this way, and we come to the following theorem. Loe Reem ci Let cer 4 ae and let P,==d) when 2 is infinite. | P,\=A2(Po—a,) - - © Pata + 2 we ee Pes ig (P\—aq) ° ° ® Ve aie °°. 8 @ e ° | ° e Then so long as no one of a, a, @, &c. is infinite, the dimension of @r may be asserted to lie between those of [a,—k] and [a,--k], of fa, a,—k] and [a,a,—R], of [a, a, A,—h] and [dp, a, d+], &e., however small & may be: and if any one of the set pu:a”, px: x (dx)", &c. have a finite value when 2 is infinite, then da has: absolutely the dimension [@,] or [@, aq], &c. But when any one of the set, dy, @,, &c. is infinite and positive, say az, then Pz is of a dimension higher than that of a” (Xx) (A2x)2 (Sz), and lower than that of a (Az) (N20) however great m may be, or however small &. But if the first of the MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 325 set, say a3, which becomes infinite is infinite and negative, then Gx is of a dimension lower than that of xv (Ax)™ Or) OA8r)-™, and higher than that of « (Ax)™ CAE: however great m may be, and however small k. And it is useless to attempt to make any terminable scale of dimensions, since between any two different dimensions an infinite number of intermediate dimensions may be interposed. 207, The preceding contains only dimensions of the same, or a lower order than those of powers of x. The same theorem holds if Pics Px.wr: parw'r, provided Awa, wa, &c. be substituted for Ax, Pr, &e. By this means the dimensions of functions higher than any power of may be obtained; but there cannot be any method of ascending, or of obtaining the exponents of lower dimensions first. 208. We shall now proceed to apply the preceding theorem to the tule (page 237) for the determination of the convergency or divergency of a series; which is correct in every point but this, namely, that what in the preceding articles would be called a dimension greater than that of x'~*, and less than that of «'**, is there confounded with the absolute dimension of x. The rule, then, may be wrong when ape: pr= 1. Tueorem. If dx diminish without limit when x increases without limit, and do not become infinite after r= a, then, of the two expres- sions ) (a@)+¢ (a+1)4+6(a4+2)+.... ad infinitum and _f %. gu dx, either both are finite, or both are infinite. There must be, by hypothesis, some finite value of x, from and after which @x continually decreases; and this value may be chosen for a. Then, from z=a to r=a+1, fa>du> (a+1), whence Jc bade> fet! padx> fet d(a+1)dz; or pa> [2 oxda>¢(a+1). Similarly, it may be shown that fats pxdx lies between ¢(a+1) and p~(a@+2), and thus that pean pxdx, however great 7 may be, lies between da+¢(a4+1)+...+¢ (a@+n—1) and d(4+1)+¢ (a+ 2)+ ---.+P(a+n). But these last differ by ¢(a)—¢ (a+n): con- sequently the limit of the integral, and the sum of the series, do not differ by so much as (a)—@(c), or da. Hence iat px dx, and gat (a+l1)+.... do not differ by so much as ga. _ Hence it follows that the series 1 &.a.rNa... "a. (NA) SS Ls = yee sere ae + *2e#e08e0 t (a+ 1)AG@+1).2(a@+1)....4° "(a4 1) {a (a+) he (beginning at a value of «@ so great that all the factors of the first term are possible) is convergent when e is greater than, unity, and divergent when ¢ is unity or less than unity. For 1 TP ¢ 1, the series is convergent; if <1, divergent. But if a=1],) find a,, the limit of P, or Av (P)—a,); then if a, >1 the series 1s con-| vergent, if <1, divergent. But if aq=1, find a, the limit of P., or zx (P,—a,); then if a.>1, the series is convergent, if <1, divergent. | But if a.=1 examine P,, &. &c. The demonstration is as follows. If a)>1, then $2, being of a higher dimension than 2°-*, however small k may be, can be made of a higher dimension than x°, where e is greater than 1. But 22~* has in that case been shown to be convergent. Similarly, if a<1, ox, which is of a lower dimension than 2”**, can be shown to be lower than 2, where e1, (and this includes the case in which it is infinite,) #2 is of a higher dimension than «. (Av), and can therefore be shown to be of a higher dimension than x (Az)’, where e>1. But in this case x! (Ax)~ has been shown to be convergent; and so on. 9209. If a function could be shown for which a, a, &c. ad inf. are severally =1, this criterion does not determine whether the series is convergent or divergent. But if in such a case there be convergency; it must be less than that of Sa~°?, for any value of k, however small ; indeed, between the series just named and that in question, can be inter- posed an infinite number of series more convergent than the latter. 210. If we substitute,yz, the term of the series, for ¢z its reciprocal, we have Pp=—aw'ax : a, the rest being as_before. MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 327 o 1 1 Page 236, Example f., (using n for a.) yn=a2*—1, Pj=2" rv: n (c*—1)=1, when n=. 1 é 2 Le P\=Xn (Pp 1) =An{ x" r\x—n (am —1)}:n (a —1), the denominator is Ax when n= ce , and the numerator, expanded, gives 1 1 1 9 2 be —— PUPAL) AE AL)? An Anz" {An—n(1l—a# *)} =a eS GD cle ale BS 2 n 2.3 n? which =0 when n=: or the series is divergent. In page 237, Example V., for the words ‘unity or less,” must be read “ less than unity.” 211. The same error is made in pages 180-182, the whole of which* must be read with reference only to those functions in which ¢z is finite, when the critical value of e in gv: (v—a)’ is =0. It is possible, how- ever, that such functions may have the same dimension as {A (w—a) fo these functions cannot be expanded in positive powers of «—a, but require both positive and negative powers. The pages in question, there- fore, include all that can be included under Taylor’s theorem: what they omit is the notice of a particular class (little, if at all, noticed hitherto) of exceptions. We shall proceed to some considerations on series containing both positive and negative powers of «. 212. There is no difficulty in exhibiting any function in a double series, containing both positive and negative powers of 2. For example, «itself. From among the infinite number of equivalents for x, choose one, for example The first may be expanded into r—1+a'—wav "+a %—...., and the second into r—a*?+.23—&c. The sum of these two series then is an equivalent to 2, and an infinite number of such equivalents might be found. We are not then to say that two such developments must be identical, term for term, because they are developed from the same function: for one function may give an infinite number of different developments of this kind. Nor is the divergency of one part of the series, which will generally be found to happen, any impediment to the equation of the development and the function from which it was derived. For both developments may be made by Maclaurin’s theorem (as will immediately be shown) and Lagrange’s theorem on the value of the limits may be used, to represent the remnant, from and after any term, in a finite form. we Ll Nike Fa gt Eas For example, log (l-+ax)=ax—; a © +34 tee. Fo (1+ 6az)" log( 1+° —* ie ENT 208 XG gD : Oy ah Caialae. ee Digs Sgt —n («+6'a)" 6 and @’ being both <1. The second is obtained by writing l:a@ instead of x in the first. Consequently, by subtraction, * Beginning from page 180, the fifth line from the bottom. 328 DIFFERENTIAL AND INTEGRAL CALCULUS, x(1+az) 1 1 1 l log) ————— ]= —— }——a*| v—— J+.... og et, ) a(« a Su Or “ if N Me =( a —mn \Q+60ar)" (2+ aay) This series, carried ad infinitum, is convergent, if az and a: 2 be both <1. If, however, a=1, it becomes Seaiee UamnS: BONS Uae” TY Bee ACR rae Veet. If this be carried ad infinitum, it is the well known development of log z in positive and negative powers of x, and is never convergent. That log w cannot be developed in positive powers alone, nor in negative powers alone, is sufficiently evident if we consider that it becomes infinite both when 7 is =O and also when z=. 213. There is, however, a great difference between double series of this kind made by arbitrary transformations, and those in which the mixture of positive and negative powers arises from logarithmic develop- ments. This difference, however, has not yet been established by demonstration, though it is found in a very remarkable theorem,* as follows. Let wax be a function which has a root a, so that pw= (x —a) pr. Then If, then, log dx can be expanded in positive powers of «, and log (wa :x) in positive and negative powers of 2, (both which can gene- rally be done,) and if the identity of the two sides of the equation be then assumed, it follows that —a=coeff. of x~' on the first side. 214. We shall conclude this chapter of developments by giving a process which will successively introduce the student to a notion of the calculus of derivations, the combinatorial analysts, and the calculus of generating functions. We have already seen successive derivation, and its use, in the successive diff. co. of a function and the theorems by which they are employed in development. When possible. required the development} of db (a) +a,x+a,2°+..-) in powers of «. When it is required to represent complicated results, let d=, a=), do=c, &c., the indices of the different letters being _ G9 MBIA 012 ae NP ga ade eae ATID a ae oir iig sea ae ii 7 hs Cp en 1p ata * This theorem was given by Mr. Murphy, in the fourth volume of the Cambridge Philosophical Transactions, and, independently of the defect of absolute proof, is one of the most general and interesting contributions which analysis has received for many years. It is derived from the assumption, certainly not generally true, that two double series which are developed from the same function, are identical, term for term. Yet almost every general theorem of development can be obtained from the use of this theorem, and it has not shown any case of failure. See the volume just cited, and also Mr. Murphy’s treatise on Algebraic Equations in the Library of Useful Knowledge (page 77). + This investigation is a deduction of the method of derivation from a more analytical principle than that of Arbogast, though it terminates of course in the same process, or rather in the decomposition of the process of Arbogast into ifs most simple elements, | MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 399 Let P (Qo +a, 0+, 2°+....)=A +A, r+ A, at+.... Differentiate both sides with respect to a,,; we have then _ dA, , dA, tees eon. da,, . dd, dan x" Pp! (Qotad, e+....) but ¢! (an+....)A,+2A,0+... -, and contains no negative power of x; consequently, for all values of m, BSBA, ute de. d =O} ¢sjus Up to. —-_-=-(); : : da L mm or a,, does not appear before the coefficient A,, appears ; and we have ~ p (Q+a,0+. ara ty Ae 4 Ants np fAmas da,, TA, da 5 is Sai ~ m But this series is the same thing whatever value of m is employed ; namely, A,+2A,x+.... Consequently the coefficients of the same power of x with different values of m are equal, or GA hen GA Arse ONS ek, wae Fda - adie er AD that is, any A being differentiated with respect to any a, gives the same result as an A which is p terms before or behind the first mentioned, differentiated with respect to an a which is as many terms before or behind the first mentioned a. Or dA, dA, cE Aet cine i A HP Nee) dar Paa Oi dated) Fis Sen dag.ai” Kidagys 4. '0G 1A, dA First, Aj=¢a), and wae! =—"='a), whence A,=a, $’a)+C, where Doe edd. C is no function of a. But nothing higher than a, can enter A,, there- fore C is a function of a, only. But, in fact, C=0, for as it is in- dependent of a, a, a5, &c., it is the same as if they were all =0, or as in the development of @(a), in which A,=0, or C=0. The same con- sideration shows that in the remainder of the investigation no in- dependent constants can enter. Next, it is clear that the form of A,, with respect to a, is Po bin G+ Pi Pin1 Ao see see $Pry Pi Ay where P,, &c. are independent of @ and Py doy Pm, &¢. do not mean the simple diff. co., but those coefficients divided by 1.2.3.... m, 1.2.3....m—l, &c.: ¢’a and ¢,a being of course the same things. This follows obviously from the development by Taylor’s theorem, which is O(4, + Ay0+. .)=PAyt+G dy. (A, + Agt+. .)+9.4,.2°(A pat... P+. ees And it is clear that ¢,,a, enters for the first time in A,,, with the co- efficient a,”. Consequently, leaving blanks (numbered) for coefficients to be discovered, we have the following table of the general form of A,A,, &c. 330 DIFFERENTIAL AND INTEGRAL CALCULUS. A P ay A, = UP 1% A, = (1 )Qyao+ ay Polly As = ( 2 ) dat 4 ) Pt+ a’ Psy A, = (3 )daqt 5 ) Pato + 6 ) Psp + a Py Ay &c. &c. &c. The blanks are filled up by an easy process, which may be called derivation. ‘This is somewhat different from the derivation of Arbogast, which will appear hereafter. It follows immediately from the equations (A) that each blank must be so filled up as, on being differentiated with respect: to any letter, to yield the same as the next higher coefficient in the same column differentiated with respect to the next preceding letter. To fulfil this condition, the process is very simple; as follows. Suppose be-+ce-+Of fills up one of the blanks, what is to fill the one under it? | From be by b-diff”. (or differentiation with respect to b) comes e, but this must come by c-diff". from the next, therefore ce is in the next, and bf also, since 6 comes from e-diff" in the present term, and should come from f-diff" in the next. Again, ce would give ee by the same rule, but this must be divided by 2, for c-diff" of the present term gives e, and e-diff™ of ee would give 2e Also cf isaterm fromce. Again, from bf first would come ef, but this term has already occurred, and if of came twice, c-diff" of the next would give results from both, and would give 2f, whereas b-diff" of the present one gives only f from the term 6f, Obviously, whatever conditions a new term is required to fulfil, they are fulfilled if that term has already occurred, and would be repeated twice over if the term were allowed to enter twice. Finally, bg must enter in the new coefficient. Consequently, the derivative of be+ce+df is ce+bf+4e?+cf+bg. And the rules of derivation are as follows. | 1. Differentiate as if all the letters were functions of a common variable, and instead of the diff. co. of each letter write the next. (Thus) db | dt 2. Whenever, by the preceding process, a newly entering letter increases the exponent of one which is already in the term, divide the term as it stands after derivation by the exponent as increased. 3. When aterm newly obtained has been obtained before in the same derivation, throw it away. The successive derivations may be denoted by D, D’, in this particular problem. We give as an example some derivations from 54, A term m brackets means that it is either altered or thrown away: if altered, the alteration is written immediately after. When altered, and then thrown away, both are in brackets. D.bt=4b% D*.b¢=[128c.c] 6b2c?-+- 40%e= 6c? + 45% D?. bt=[12dc.c?] 4bc? + 126°ce+ [12b%ce] + 4b°f = 4b +12b’ce-+4b'f Dt .b¢=[4c.c*] c*+ 12bc%e + [24bc.ce, 12bc?e] + [126*e.e] 667° + 12b°cf 4 [120%f'] + 4% =c'-+ 12bc%e + 667? + 12b°cf+ 46%q. de. if ¢ be the common variahle, — e gives ce, b= gives bf, &c.) MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 331 This being done, and the results tabulated to a sufficient extent, we have An =D"'b fia FD" 2. oat oo. + DE" ohy ab" b,, a, and the total result may be represented by dh (a+ b2+ca*+ ea®+....)= >D"" OF dasa, the symbol = extending to every whole value of m, (0 included, ) and simultaneously to every value of p which does not exceed m : g, mean- ing the diff. co. 6™a divided by 2.3... .p. 215. The following is the table requisite for the formation of A, up to Ay inclusive : DéSe D*b=e D b°?=2be D%=f | D2b*=2be +e D‘tb=g | D*b*=2bf +2ce D°b=h | Dth?=2bg + 2cf +e? D%b=k | D'H’=2bh + 2cgz + 2cf Dib=! | Dl? =2bk +2ch + eg + f? D®%>=m | D'b?=2b1 +2ck + 2ch+ Ife D%=n | DX? =2bm+2cl + 2k + 2fh-+ ge? D }8=360?c D?h?= 3b%e + 3dc? D*b’=3b°f + 6bce +c° DY 3b°g + 6bcf +3be? +3c%e Dh*= 3h + 6beq + bbef +3c°f + 3ce? D°b* = 30°k + 6bch+6beg +3c°¢+3bf? +6cef +e D7b*= 3b" + 6bck+ 6beh +3c°h + 6bfg +6ceq + 3cf?+ 3c? D b4= 40% D*h*=4b3e + 6b7c° D*b+=4b*f + 12b?ce + 4bc° D*h‘=4b°g + 12b*cf + 6b’e? + 12bc%e +c D°b*=46°h + 126’cg +.12b°ef +12bc?f + 12bec* + 4c%e [+ 6c%e? D*b+=40°k + 12b°ch + 12b’eg + 12be? 2+ 6b°f? +24bcef +4c*f +4be D b'= 5btc D?h5=— 5bdte + 105%c? D*h5=5b*f + 206%ee + 107c° D‘h'=5b*g + 20b°cf +10 be? + 306%c’?e+ 5hc* Deb '=5b*h + 20b*cg + 20b%ef + 30b%c2f + 30b°ce* + 20be%e +c DIFFERENTIAL AND INTEGRAL CALCULUS. D &=6b'c D%B'=6b'e + 15d*e° D*h°= 60° f + 30b*ce + 20b%c° D1h'=6h5g¢ + 30b'cf + 15b*e + 60b%c%e + 15b%c* D b7=7b*c D2b7= 7b%e + 21 b5c? D*h7=7b°f + 42b5ce + 35b*c° D b8=8b’c D°b°= 8be + 285%’ Db°=9b*e. 216. Prove from the preceding, that . (= )F 9+ det (dies) v + Git 2bab be) +(),+3¢.+3624 Py) To saey where ¢, is the value of the divided nth diff. co. of da, when a=1. Also verify the developments in (61.), (64.), (156.) 217. If we form the successive derivatives of da, we shall find Dda='a. b ONS) D*ga='a. Dr gane —= =—¢,a.Db+¢,a.6° | tae: By b° Diga=dia Dd+9"a (2) gral =9$.¢ Db +40. Dd°+-h,0..8° 5 from which we should suppose that D"pax=D"7b. da + D2 0. Pat vee eT". Gndee sees (D) | The proof can be easily completed, as follows. Let the preceding be — true, then Dd,a, or Dd™a: 2.3....p is Pt%a.b: 2,3,...p, OF | by»n4X (p+1) 6. Consequently, subject to rejection of repetitions, D"'da=D"b.¢,a+ (20D"~'b + D™"'b") g.a 4 (83bD" 70? + DD") 6,04 - + (mbDb""*4- Db”) ba + 0". Pn ye a Now since any repetition of terms, however often it may occur, is followed by an “immediate rejection of the repeated terms, and since 12 © other respects the formule of differentiation will apply, we have (as in Ex. 2, p. 245) D” d'*+}, or D” (0'.6)=bD” b*+-mDb.D™"'b'4+.... all the terms therefore of BD” d* are found in D” b**!, and therefore in | * This term is rejected, the two being the same. MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 333 the formula above written for D”+ ‘ga, the first term of each coefficient in brackets may be rejected, as being no more than a repetition of terms contained in the second coefficient. We have then D"*'ga=D"b pya+D"-"8? pa... + DE" G,0-+6"" ba; or the theorem (D) is true for the m+ 1th derivation, if true for the mth. Being true for the first, as shown, it is therefore true for all. 218. We have then $(atbrtea+.... J=$a+Doa.27+D°ba.22+D'ba.2°+.... which shows that this method of derivation is a generalization, one particular case of which is divided differentiation, as follows. Let a be a function of ¢, and let _l da ul db ae de ia de hoe Sh detiwin wine aan Te des. if, then, at, we have dl (fpyt) ~ (a+ba+cr2+, ...)=h (vs OO) OG me rh iia dda Pha x Fi SRC ers ae a Consequently, D" da= nes begs 7, 1 d¢a ep 2 LdaDda 34,2 d-D'da & MG e chee ord ak 8. de? 219. The preceding affords a ready mode of finding any diff. co. which may be wanted of ¢a with respect to ¢. Suppose, for example, we would express the fifth diff. co. We first take out D®.ga, which is et hal RIN oa pa —- ——+ Db* ——_ + B° os ee ei ins Perit a greg: This, multiplied by 2.3.4.5, gives the diff. co. when the substitutions are properly made in the derivatives of the powers of b: take out the preceding derivatives from the table, after the multiplication just » alluded to, and we have (writing the index of each letter) 1202,6/a+60( 2b, f,+2c.es)¢""a+20 (3bie,+30,c3)6""at+5 .4bic." at big'a. Denote the diff. co. of a with respect to ¢ by a’, a’, &c. Then, for 6 Write a’, for c write a’: 2; for e, a’': 2.3; for f, a :2.3.4; for g, a@:2.3.4.5. The most commodious way of doing this is under b, CAE, f, and g, to write indices 1, 2, 3, 4, and 5, and to let these indices be guides to the divisors which are to be introduced. The result is a’d’'a+ (saa +10a"a!) 6”a+ (10a"al" 4+ 15a'al”) 6a + 10a*%a"h*a+a'*o"a, which may be verified by common methods. 220. The theorem in (217.) may be made to give higher derivatives from those already formed. Thus D+b. 6’a+D°*d? DIFFERENTIAL AND INTEGRAL CALCULUS. pe b'=D"c.7b" + D""c*. ae abi oy a [7, Teg st 1] b'-@ 4 orth, oe —m] br-™—1, [mm] [m+ 1] If, then, all the derivatives of 5" up to the mth be formed, those of ¢ can be found by changing b into c, c into e, &c.; whence the (m+1)th derivative of 5" can be found. I think, however, that the method in (215.) is the more easy, though the present one may serve for verifica- tion. Thus, D® 5‘, as found in the table, is, when arranged in powers of b, + Dc”. h.4b° + (2cg + 2ef) 6b°-+ (3c?f + 3ce*) 4b-+ 4c*e, or Dc. 46° + D*c*?. 667+ D*c°. 46+ Det. 1 +c5.0. This is the method employed by Arbogast himself, in whose work D”.b” stands for what in the present notation would be 2.3...m.D”.0". To exhibit the actual formation of D‘ b* by this method, we have D*c. 46° = 4% 2 4. D? Cc. 6b2°= 6b? ie ye 12b°cf Dé b* +e*.1 = 607° [+p c?.4b =12bc*e + ice 1 se. The five resulting terms put together make the value of D*d‘ in the table. 221. Having wr=ar+ba*+ca*+...., required an application of the preceding theory to the determination of %7*z, or to the reversion of the series aw+bx°+.... In (156.) it is shown that the development of wx is Po, c+4P,,. 2°+&c., where P,,,,, means the coefficient of 2 in the development of (at+bx+....)~". We want from this P,_;, Let da=a~": we have, then, for the coefficient of x”, n(n+1l)y Dv'ga=Dr0( — as 4D “o( ahr je i aE | The sign + being used when 7 is odd, and — when it iseven. The deve- _ lopment required is then obtained by writing the cases of the preceding expression instead of those of P,_,,, in the form obtained from (156.) Suppose it required to verify the coefficient of uw’ in the article cited. We have then to find the value of the preceding when n=7, and to divide it by 7. This gives —D*).a~°+4D*b?.a°—12D°b*.a-" 4+ 30D! .a7-"! —66D25,a-” + 132b°.a-™. Bring all to the common denominator a”, and take the derivatives from the table. This gives for the numerator the following, the order of the terms being inverted. 1326°—330ab*c + 30a? (46% + 66°c?) —12a° (3b°f-+ 6bhee-+-c*) +4a‘ (2b¢+2cf+e)—a°h. [Compare this with page 306.] MISCELLANEOUS EXAMPLES AND DEVELOPMENTS. 335 222. Required the expansion of (14+-ba+cx2+... -)=*. The diff. co. of a, when a1, are —1, 2, — 2.3, 2.3.4, &c., and divided, they are, —1, +1, —1, &c. Del =—D™ B+ D8—D=-B+.,. bbe {he meren, —,m odd, Q+br+....)7=1—ber+ (6°— Db) «®— (b?>—DB?-+D*) 2®+-.... The materials for finding this to the tenth power of x are in the table. Hence we have a simple form for the quotient of a! + BEL +. .0., divided by 1+ b2+ca?+....; namely, a’—{a'b—b'} «+ {a' (°—Db)—b/b+e'} 2? —{a' (’—Db’+D%)—b! (b?—Db) + cb—e'} w® +... 223. The combinatorial analysis mainly consists in the analysis of complicated developments by means of @ priori consideration and collection of the different combinations of terms which can enter the coefficients. The first theorem of the kind which the student usually meets with is the well known development of (1-+.2)", when 2 is a whole number, depending upon the obvious fact, that in (1 +2)(1+2) ---.(” factors) 2” must appear once for every manner in which m xes out of m factors can be combined by multiplication with the units of the n—m remaining factors. If we multiply together a+b+c+....,a/ 40! tol+....,a/4b/4 ev+...., &c. (n factors), the product consists of a number of products containing a term for every combination of n factors, one out of each of the polynomial factors. But if we multiply together a,+a,v+a,0°-+... : bo +- 5, 2+ b,x?-+ ....(n factors); the coefficient of 2” will consist of such combinations above described only, as have the sum of their distinctive indices equal to m. Thus, if we want the coefficient of x*, there being four factors, we must ask in how may ways 5 can be composed of four numbers, 0 included. Thus we have 0005 gives dy bo Cy €,5 Ao by Co, &C. | O113 gives a,b,c, e,, a, 5, C3 1, &c. 0014 gives dy By Cy es, Ap bye, @, &e. | 0122 gives ay by Cy ey Ay Cy bo Coy KC. 0023 gives a by C2 eg Ap bo Cao, &C. | 1112 Gives @, b,c es, a, b, e, Cx, &e. Collections of tables of the different methods in which numbers may be constructed by additions of lower numbers, under various conditions, make the fundamental tables of this method, just as those of the deriva- tives of powers of } are the fundamental tables of reference in the method of Arbogast. 224. Required the development of (a)+a,x--a,22+....)", 2 being 1whole number. To find the coefficient of 2” we must find every way in which m numbers (0 included) can be put together to make m. Let us suppose that the 10th power is. the one in question, and let n=4. Firstly; take 10 in four different numbers, as 1, 2, 3,4. Hence % a, a, a, is a part of the coefficient of 2. But a, may come from ither of the four factors, a, from either of the remaining three, &c., so +hat if we write first the number which comes out of the first factor, &c., we have, in the coefficient of w, Ay Ug Ay Ay + Ay Ay Az Ag +A, Ay Ae A,4+Ke., ‘peated as many times as there can be made different arrangements of our quantities. Hence 4.3.2.1 a; ds a, is a part of the coefficient. 336 DIFFERENTIAL AND INTEGRAL CALCULUS. Secondly, take four numbers to make 10, which are not all different, as 2, 2,3, 3. The number of ways in which 4g, dg, 4; ds, can be written | is not so many as before, for a, from the first factor and a, from the second is the same selection as a, from the second and a, from the first. In fact, by a well known rule of common algebra the number of different arrangements of dy, ds, Az, a, 1s (4.3.2. 1)+(1.2x1.2). Generalizing | this reasoning, we find the following method of finding the coefficient of the mth power of a in the development of the mth power of Qt+aa+.... Let M+kh'+.....=m, in which k+hk’+.... =n, and ‘find every possible’ way in which these equations can be solved, h, kh’, &c., 1, U’, &c. being positive whole numbers (0 included), Then the coefficient required, which call P,,,,, is Pp =3( a Bae Re pre ren a UB ee eee BING yet Ue eens 225. Required the development of ¢(a+bxr+cu+... .). Thi by Taylor’s theorem, 1s I] oatd¢'a.x(b+cx-+e2?+.. jew (b4-cxperrti....)Pperee5 i whence it is evident that, making b=q, c=a,, &c. in the last problem, the coefficient of a” is , pla gra ba { Pee ‘a Pan os a e@ecee P m— PAE UPN ER ma) ae . Li Pat Ena 2 i visit 57a" 0) 80 on Tables may be provided to facilitate the formation of these coefficients, but in Arbogast’s method they are already formed.* Comparing the | preceding expression with (214.), we see that Po =D" 6, Pe Das Pee 226.” Wejhave, however, gained by the preceding a method of form-| ing or of verifying any derivative of a power of b mdependently of the: rest. Take as an instance D6‘. We have, therefore, to examine every way in which four numbers (0 included) can be put together to make 5. The different ways are | 0005 0014 0023 0113 0122 1112. The letters which should have the indices 0, 1, 2, 3, 4, 5 are b,c, e, fs 8 h. Observing what indices are repeated, we have for the terms of D?d’ 1.2.3.4 1.2.3.4 Me PAs is 1.2.3. rae haa erat oe E58. a mcantil. ccd: ieee GUL vo Pa QeMiat which computed and put together give the same as in the table. 227. The most simple form of the development of (a+bx+ex hi 4 sian) tals * As far as I have compared the methods of Arbogast with those of Hindenburg, this is always the case. ‘The tables of reference of the former method are one step more towards the solution than those of the latter. In other respects their powers are much the game, as far as developments are concerned. MISCELLANEOUS EXAMPLES AND DEVELOPMENTS, 33% a+ Da*.2+D2a". 22+)? OU" rates oe where, when 7 is integer, the derivatives of a” may be formed directly from the table of b, by substituting a for b, b for c, c for e, &. From this it may be shown, that D4" may be described as the coefficient of a" y" in the development of 1 : (l—yx), dx standing for b+catex?+..., 228. The last article has left us in possession of a result which belongs to the calculus of generating functions, which should be con- sidered as a sort of inverse method to the combinatorial analysis, though neither was originally set forth in connexion with the other, and either may have developments to which the corresponding parts of the other have not yet been investigated. Every mathematical method has its inverse, as truly, and for the same reason, as it is impossible to make a road from one town to another, without at the same time making one from the second to the first. The combinatorial analysis is analysis by means of combinations; the calculus of generating functions is combina- tion by means of analysis. Thus, having observed (and the observation is common to both methods) that in (l+)(1l+2)....n factors, the coefficient of 27 must be the number of combinations of 7 out of n, the combinatorial analysis requires us to find that number, and thence to fer the coefficient of 27; the calculus of generating functions requires us to expand (1+.2)" by purely algebraical considerations, and from the 2oefficient of 2” infers the number of ways in which 7 can be taken out of n. 229. Let t, expanded in powers of ¢, give a@+a,t+tal+t.... Chen ¢¢ being given, and also n, the coefficient of t” is implicitly given, id is therefore a function of n. The function @t is then called the generating function of a@,, which is a function of n. Thus m: d—ij)= n+mt+mt?+.... or m:(1—2) is the generating function of the con- tant m: again m:(1—2)=m+me?+mit+.. -., and is the gene- ating function of a function of n, which is =m for every even value of ', and =0 for every odd value. This function is m (1+ gl 'eo The €nerating function of n itself is #:(1—t)2; the generating function of n6, is made by adding or subtracting the generating functions of a, nd 6,. If Pt generate a,, tpt generates a,_,; for in ¢*¢t the coefficient of 2” 3 that of ¢"-* in #t. Similarly,* ¢“é generates a, ;. If dt generate @,, and wt generate b,, t X wt generates Ab, +a, b, 1+ -e-+2,6,. If, then, 6,=1, or wi= 1: (1—#), we find that ot : (1—2) fnerates d+a,+....+d,, and tpt: 1—t¢ generates ad) + ad,+..-+aQ,_) thea,. 230. The last remark enables us to pass to the generating function in 1infinite number of cases. Let us, for abbreviation, express a+ a,¢ “€,0+&c. by (a,a,a,....). Then, for instance, 1+¢+2 generates fet, 0,0....), consequently (1+¢+¢°)¢:(1—d2) generates (0,0-+1, FI+i, 0414141, 041414140....), or (0, 1, 2, 3,3,3....). gain, 1+ ¢ generates (1, 1, 0,0....), (1+2):(1—2) generates * The student should now look through the various developments which have ém made, and should describe each in the language of generating functions. Z 338 DIFFERENTIAL AND INTEGRAL CALCULUS. (121252522 a0e%') 3 therefore (1-+¢) : (1—#)” generates (1, 3,:5,.7,e | and (1-+¢) : (1—#)® generates (1,4, 05 10,5 %n))- | If pt generate a,, dt: (1—#?) generates d,+G,21 +++%5 ending | with a) when m is even, and with a, when nm is odd. Find what | ot: (1—2") generates. 931. If Pt generate a,, whatever function of a, wt xt generates, ‘tis obvious that wtx (wt.Pt) generates the same function of the new coefficients. If, then, we find that a certain operation on a, 1s gene-_ rated by wét.dt, we know that the same operation repeated on the | results, and so on, until it has been repeated m times, will be generated by (yt). dé. . This may be exemplified as follows. Let the operation in question be @,41—4ns which call Aa,, and let Aa,,,—Aa, be A’a,, as , usual. The generating function Of Gn4i— Gy is (-'—1)- dt, whence that | of Ata, is ((?—1)* ot. But (1) bt=t* bt — a) bth {-F) bt— soos; of which t“é generates Griz, kt-*™ ot generates kOnpx—15 and 80 on. | But when two functions are identical they must generate the same function, since no function of ¢ can be expanded in whole and positive powers of ¢ in two different ways. Hence —1 2 k py A Ont Ong ~— ROngerbh On+-k—2 eoeeg as already known. Again fra(1tit— Dt=l1 tk (tl) +h ee ue Multiply by ¢f, infer the equality of the generated from that of the generating functions, and we have | I k—1 = 4, + kbd, +k —— ise an+ bie aie.5 an, +k which is also known. Let 1:éey, and assume y=2+2xXy;5 then, at in p.170, yra2i+yz. ket a+ e+e, OF substituting values for y and# lid t*—1\? . aad 2 fok—-l 4 bos ; ee ((xz)?. kz") » ia fees | } t Pee A (yeidee ry) caso xe Let z=1, multiply by ¢¢, and let'P,, Ps, &c. be the values of xz-ha &c., when z=1. Again, let (xt) ht, (XE) pt, &e. generat Kis ahs, oy wee. 32 then, inferring as before, we have | ee 1 nh On + Py AX, n + 9 P3 A?X, nT 5-3 P, A’ xy at eseve For instance, let yy=y’, then m—1L é fo kath [mr +h-1, mrp k-m+ 1) ker ae eel dz™-} P and (t-")~™ .pt generates Gn mre Consequently (21)? MISCELLANEOUS EXAMPLES AND‘ DEVELOPMENTS. 339 2r+hk—] On+4—=Ay of kAa,_, “f- k TT fone A°G,_ 9 3r+k—1 38r+k—2 Des, 3 According to analogy A-'a, denotes 2da,- But the generating function of A~' a, should be (¢~!—1)—' pt, and we have already shown that this is the generating function of da, +k AG Regi 232. To show the application of the calculus of generating functions to a question of combinations, we propose the following question ; in how many different ways* may the number p be made up of lesser numbers, no one of which falls short of n. If we take the quantity 2” 2"! +... adinf., and raise it to the kth power, it is plain that x enters once for every way in which p can be made up of k numbers, no one of which is less than n. If, then, we take tot... tate. 5D it ol Coe ae Sa +... ad inf, “ enters once for every way in which p can be made up of 1, 2,3, &c. numbers, no one of which is less than m. But A+A?+..... =A:(1—A), consequently the number required is the coefficient of x? in the development of ge A aa Rs xv": (1—a) a Sasa Ue OP Fea dic wot “aes SPN) 1—a”": (1—z)’ 1—z— x" Qe” x” yn yan But —— ° 142 ~ Tr tien) a Ciaey as Wak ic the kth term of which is a’ (1—2x)-*, and when developed contains a as long as kn is less than (or not greater than) p. The co- efficient of 2” in the development of 2” (l—a)~* is that of 2-™ in (1—x)~™, or Lt, k+p—kn—1] eS fer as Let then p:n give a quotient g, (neglecting the remainder,) and he answer required is, g terms of the following series, [1, p—n] [2, p—2n+1] [3, p—3n-+ 2] [p—n] ~ [p—2n] [p—3n] eae r ieee ye Oe) (ere HD) ch My ‘or example, in how many ways can 11 be made out of numbers, no ne of which is less than 2? Here p=ll, n=2, g=5, and the hnswer is 6.7 Gd.5.6iege 3.4.5 2 7 ep ee ‘hese 55 ways are 11; 942, 8-+3, 7+4, 6+5, each in two ways ; , or 55, * This counts different orders as different ways: thus 3+3+44 and 34443 are, (this problem, different ways of making 10, Z2 340 DIFFERENTIAL AND INTEGRAL CALCULUS. 71919,51343,3+4+4, each in three ways; 6+3+2, 5+4+ 2, each in 6 ways ; 2-+24+3+4in 12 ways; 243+3+3 and 2+2+2-+5, each in 4 ways; 2+2+2+42+5, in 5 ways; 55 in all. 933. It is sufficiently evident that two functions which are the same | in different forms must generate the same function, it may he also in different forms. Thus (¢+4t?+2°):(1—#)* generates n®, or the co- efficient of ¢” is n®. If we decompose the preceding fraction into three, the first will be found to generate [n,2+2]:2.3, the second 4[n—I, | n+1]:2.3, and the third [n—2, n]: 2.3, the sum of which is 7’. But the converse is not necessarily true, unless it happen that all the — different forms of the generating function are made to commence from the same power of ¢. For though we call q+a,t+a,0@+..-- the generating function of @,, yet a_, t+ aypta,t-a,t?+...-. is also the generating function of the same, with one more term, and a,f + dsl*-+ «+++ with one term less. When, therefore, the equality of two generated _ functions is asserted, that of the generating functions can only be inferred when they are made to begin with the same power of t. ‘The — following problem will illustrate this. Required the function a,, which has the property of being equal to Qny+Q,-» If ot be the generating function of a,, (beginning with Ap, a tpt is that of @,_,, and @@t that of a,,, whence tpt +t is that of Gn_\+G,_», but it begins with a,¢+(@+a@)Cl+.... Hence we have | dt—a,—a, t=tht + Upt—agt, or =f SPT Pola = 1 ean Sins, HE eee pepe ore eae ler eae TERME or: 29 copra (gage te BODE ot by a process similar to that in the last article, the coefficient of 2” in this development, or the value of @,, will be found to be ‘} fl, 2a—2). [2,0—3) [3,74] - \ [n—2] [n—4] [n—6] slat + [1,2—1] | [2,n—2] tate pes tot the number of terms in the coefficient of a being 4n or $(n—1), : according as 7 is even or odd, and the number in that of a, being Sn or 4(n4+1). And a) and a, may be taken at pleasure. Also, if in the preceding notation [0] appears in the denominator, the whole term 18” unity. For example, a, should be A ae peas PED IRALOE ho 5 % 1709.8 TISt™ 11.9,3.4 2 [= Mor is which is easily verified, since the terms are a, qd, Ag+, Gg | 2a,+d, 4,=8a,+2a), 6,= 5a, +3a. fi 34] Cuapter XIV. APPLICATION TO GEOMETRY* OF TWO DIMENSIONS. Tue applications of the Differential and Integral Calculus to geometry are twofold in character. Those of the first kind are such as simply require the algebraical treatment of a geometrical question, and make use of the Differential Calculus in aid of the algebraical treatment. Thus a question of geometry might give ¢ (a+h) as the answer, and ¢a being already known, and h small, it may be convenient to calculate an approximate result by applying our rules, (not so much to the geometry of the question as to the algebra which it is found convenient to employ in the solution,) and by using ¢a+¢/a.h. All the geometrical questions of maxima and minima in pages 296—303 fall under this head: and in this sense all the applications of our science hitherto made to algebra are also applications to every science in which algebra can be made useful. The second, and more direct application of the science of geometry, consists in the formation of a body of general rules, by which the differential relations of space are treated ; and in which, though the application is made through algebra, it is not the formation of isolated results, but of general precepts, which is the main object of the appli- cation. In this point of view we have to consider successively geometry of two and of three dimensions. I suppose the student to be familiar with the method of coordinates, the distinction of positive and negative coordinates, the equations of the straight line and of the conic sections. But as the general relations of sign are imperfectly treated in elementary works, and as the perception of the universality of the results and precepts to which we shall come depends upon a thorough acquaintance with this part of the subject, I propose to begin this chapter by supplying the necessary considerations. bie The directions OX and OX’ are the positive and negative directions of the abscissa; OY and OY’ of the ordinate. The positive direction of evolution round OP is from OX to OX again, through OY, OX’, Oy/, is marked by the arrows in the left hand diagram. Take any point Bs he line OP has no sign in itself, but according as one or the other sign * It is not my intention in this chapter to dwell on any matter which belongs to he simple application of algebra to geometry, and which can be found in the reatise on that subject. This treatise will be referred to by the initial letters A, G, ; hus, (A, G. 100) means the 100th article. 342 DIFFERENTIAL AND INTEGRAL CALCULUS. is given to OP, all lines passing through P divide into two directions | with different signs. And the rule for assigning the signs is this: if P were to move along a line drawn through OP, in one direction of motion | | OP would revolve positively, and in the other negatively; when OP is | positivé, the positive direction is that in which OP revolves positively | when P moves in that direction; when PO is negative the positive | direction is that in which OP revolves negatively when P moves in that direction. Or, the positive direction on any line is that m which OP | and the direction of revolution have the same names; the negative | direction, that in which they have different names. The preceding | diagrams contain various instances, all on the supposition that OP is | ositive. | If a line move parallel to itself, its directions retain their signs until it” crosses the origin O, when, if OP retain the same sign, the signs of the | directions change. But if OP change sign when the lines travel | through the origin, the directions do not change sign. At the moment | when the change of sign takes place, there is, as before, no sign except. an arbitrary one. : The angle made by a line with an axis is in all cases to be found by | drawing through the origin a parallel to its positive direction, and | measuring the angle made by that parallel with the axis in the positive | direction of revolution. Thus, if OP be positive, the angle* made by the | line drawn through P is XOM, greater than two | right angles; but if OP be negative it is XON. — The angle made by two lines may be con-| sidered as positive or negative, according as one or | the other is mentioned first. Thus,if OA and OB | make angles a and / with the positive side of the’ axis of X,then a—j3 should be called the angle! made by OA with OB, and B—« the angle made, by OB with OA. It is, however, possible to make the distinction between the angle of OB with OA, and that of OA with OB, as follows. Let the angle made by OA with OB be that | made by passing from OA to OB by revolution in the negative direction. | In this manner the angle of OB with OA, made by passing from OB to, OA in the negative direction is 27—O, if that of OA with OB be 0: : and 2r—06 has all the properties of —0. If, however, we allow the | second method, it must be kept in mind that results may be greater by Qn than they would be in the first. I shall use the first method. We gain by the preceding definitions not only the power of repre-| senting the relations of direction by simple and universal theorems,f | * It may be useful to notice that when a line cuts a triangle out of the first quarter of space, the angle it makes with the axis of a lies between one and two right angles ; out of the second, between two and three; out of the third, between three and four; and out of the fourth, between four and five, (or an angle less than one right angle. ) ‘-+ Since the angle made by a line is that made by the positive side of it with the axis of x, conversely, negative radii are to be measured in the direction opposite to the lines bounding the angles which belong to them; that 1s, if r= 94 be the polar, equation to a curve, whenever 4 is negative, the line which has traced out the’ angle 4 is not the direction of 7, but the opposite. Owing to the neglect of this | extension, the spiral of Archimedes has only half it8 convolutions, and r=a-+bé-+eF would frequently loose a loop. The reciprocal spiral also has only half its convolu- tions ; as it is usually given, it presents the anomaly of a curve which has a linear. asymptote, with only one branch approximating to it ; and what is still more strange, APPLICATION TO GEOMETRY OF TWO DIMENSIONS, 343 but also that of giving demonstrations as general as the theorems them- selves. I shall first show, by one or two separate cases, the universality of a certain theorem, and shall then prove it generally. A straight line, YX, making with the axis of x an angle 3, is cut by OP, making an angle 6 with the same. Again, YX makes with OP an angle ». Required the relation which exists between B, 0, and p. Two positions of the line XY are given, the first cutting a triangle out of the first quarter of space, the second out of the fourth. In the first, OP falls within the triangle cut out, but not in the second. In the first case, 2 is XOA, and @ is XOP, while » the angle of OA with OP is B XOA—XOP, or B—9, or p= -8. Oe Ye Again, in the second case, f is XOB, ts Wie and @ is XOP, (greater than two right i. ’ angles,) while », the angle of OB with As. OP, is XOB—XOP or B—8, as before, Ve XN. being now negative. The general proposition, which in fact Wr answers to that in Euclid relative to the P sum of all the angles of a polygon, is as ‘ follows. If A, B, C,D....M,N represent the m sides of any polygon, then the sum of the angles made by A with B, B with C....M with N, and N with A, is equal to nothing, provided that the above con- ventions with regard to the angles be strictly observed. Forif «, By y -+--pt, v be the angles made by the sides with the axis of X severally, then by definition the angles above described are a—~, B—y.... fs—v, y-—a, the sum of which is obviously equal to nothing. If, then, In the above we denote YX by T, OP by R, and OX by X, and if by AB we mean the angle made by A with B, we have B=TX, 0=RX, p=TR, XT+TR+RX=0, XT= —TX,=—B, whence — 3+ p+ 0=0, or p=B—8. We shall always, unless where the contrary is specified, consider OP as having a positive sign. We now proceed to establish those differential relations between the different coordinates of a point, on which much of the subject depends. The coordinates ON and NP of the point P are wand y¥, its radius vector OP is r, and the angle of OP and x (which in our figure is PON) is 0. The line PT, usually the tangent of a curve passing through P, makes an angle (3 with the axis of a, and « with OP. In our figure ( is equal to PIN, and pis equal to OPT. Let x stand for 1 7,, the reciprocal of r. And as we are at first only con- sidering mathematical consequences, without refer- ence to the geometrical considerations. from which the premises are derived, we shall introduce several suppositions which here merely denote abbreviations, and point out at a future time why these particular abbreviations become useful. the curve whose equation is Al (x+y?) tan“ '(y :x”)=1, has an infinite number of folds which are not found in r4=1, 344 DIFFERENTIAL AND INTEGRAL CALCULUS. Let # and y be both functions of some variable ¢, (in mechanics it stands for the time at which the pomt is at P, or the number of seconds measured from some given epoch,) and let all differentiations be made relatively to ¢. Instead of diff. co. write differentials: thus, when I say ds is to stand for ,/(dz?+-dy*), I mean that s is to be such another function of ¢ that ds The Est ds dy? | —_-_ = EL ay a ak l . di / € “aep ” dx iE ( Tae} : the latter if y be expressed in terms of c. Again, let p be the abbrevia- d. ds dp os tion of te or se Finally, let a perpendicular from O upon | be called p, and let it make with the axis of x an angle a. Let p,. which being drawn through O has no sign but an arbitrary one, have a alt . __dy dy dx positive sign. Also let PT be so drawn that tanp= a Ot as zi | Our symbols, then, are as follows : | x, one coordinate of P. 6, an angle so taken that tan f= y, the other coordinate. dy % | t, an implied independent varia- ie also the angle PIT x, =| ble, of which w and y are a Pete | functions. py the angle PT r. : r, the radius vector OP. p, the perpendicular* from O on u, the reciprocal of r, PT. 3 6, the angle of ra. a, the angle px, s, derived from ds=,/(dx°+dy’), oy dj” 7 a pi et NS ee p, abbreviation of ae The following equations follow immediately : Me Sg pitiek ; p=p-8@ w=6+ >" (rejecting 27 if necessary.) : The first has been already proved; the second follows thus: | pota.PT+PT p=0, or px—-PTr—p PT =0, “A “A A 3r pxr=PTa+p PT, or ape tee! To find the internal angle POK of the triangle POK, we have, when the - angles are measured by our conventions, pr=pe—rv=o—. And | the angle, as to magnitude and independently of sign, must be either | POK of the triangle, or the difference between the latter and four right | angles. In all these cases cos POK in the triangle is the same as COs | (a—6). Hence we have p=rcos(m—O). If we now collect these | equations, and add to them some others which are very evident, and | * It will be found that according to the conventions laid down pPT is always 7 2 h . 3 T . es | _ three night angles, =, or —5-, and not 9 8 might be supposed. APPLICATION TO GEOMETRY OF TWO DIMENSIONS.. 345 also those by which s, 3, and 9 are introduced, we have the following list. (Iii e==2 cos: 6. (6.) p=B—8@,. (2:). y=r sin 0: 3 Se (4.) tano—”, (Sy tan eee Ee da (5.) p=rcos (w—6) | (9.) ds=,/(dz*+ dy?) Sida tee dG We now proceed to find differential relations, all with respect to ¢, “MGS. d?x meaning 7 by dz, as by d*x, &. = ay ae (1+ tan’ 6) d= » or Pdb=ady—ydr (11.) dr cath dx (1+ tan’6) dB = A , or ds* dB=dzx Py —dy Px ig ca erey 5 ds* 1 ds* (dx? +dy’)* -. ods. dp ~ dx @y—dy d?x ~ dn y — dy dx a) 2 _ da’ ale 9__ ay _ COs p=, a as (14.) eat tan Pasay _tdy—yde r°dé cae do (15.) I+tanPtan@ adr+ydy \ rdr dr dz=cos 6dr—r sin 6 dd, 16 @x=cos 6 d?r —2 sin 6 d0 dr—r cos 6 dé? —rsin 6d (16.) dy=sin 0dr+r cos 6 dé, 17 d’y=sin 6 d’r +2.cos6d0 dr—r sin 6 d?-+-r cos 6 d’6 (17.) rdr=rde+ydy rd@r+dr=xdo+ y@y+da+ dy? (18.) “rom (9.), (16.), and (17.) ds®==dr®+- 7° do? (19.) ‘rom (19.) and (18.) rd’r—rd@=ad'x+yd’y (20.) ds d’s=dx d’x+ dy @y=dr &r +rdr dé? -+12d6 a6 (21.) ’ 4 ay ¢aG* , dy” ‘rom (15.) and (19.) sin? v=?" Ger 098 pa (22.) 1 3 ‘ rom (6.) and (7.) B—O=[r+u, cos(o—O)=sin p (23.) ' 4 d0 _xdy—ydx rom (5.), (22.), and (23.) Rete aa ae ( 24.) ] ] lL dr* J tom (19.) and (24.) tet Het (25.) Fe ae ata 346 DIFFERENTIAL AND INTEGRAL CALCULUS. ds (a& ad?x)—(axd da) ds d?s From (24.) dp= SS a ld cle _ (da? + dy’) (ad’y—y Pa) — (2dy —yda) (da Hat dy PY)\ (96 ds® RPK AeA DINED 4 8 8 : ds® fay p dp pe es d 2 2 Se Diag ee ch ited Gas Oi OO amare | P—p=r—r sin’ wT Cos p= ss > aan a ont | dp? | But, (7.) dp=dow, or dw’= er : For r write 1: u, and we have the following transformations : | Mes 2a | drat era — = +5 By (28.) ; | iy : | (18.) becomes xdx+ydy= ca (29.) | (19.) becomes ds*=u-* ed ude") (30.) : wh uw (25.) becomes 4 int ej’) aus (31) dp du dé du—du d’0 The last gives re =udu+— ream Tn {au (ud? + dod*u —du d°0) P de® z | or dp=— | } Divide rdr, or ~du:u*, by dp, putting for p® its value from (31.), or ! | (u? de® + du’) * dé; and ea perder (u? de? 4-du?)? dp u® (wd? + ddd'u—du 0) The preceding equations will admit of any quantity being taken as the independent variable, and are given in order that the complete relations may be first exhibited. They are also useful in their most general form: thus, in dynamics, where a material point is in motion,’ acted.on by forces, the question always is, at what ¢ime from the begin- ning of the motion will the moving point have a given position. Here the object is to express every coordinate as a function of that time; if, then, ¢ be the time from the commencement of the motion, equation (20.) would be expressed by diff. co. thus, : 4 ] i (32.) shit. dy dr dé "de! de de AG) The independent variables most commonly used in purely geometrica’ questions are 2, 0, ands. Ifthe first be used; that i eg if t=2, we fine | | ro | dee 0,-oY =e and this gives APPLICATION TO GEOMETRY OF TWO DIMENSIONS, 347 a (dx? + dy?)® ve] i dx Gp idy STAY ters 15: from (13.) If 6 be the independent variable, we have d*0=0, and : du” _ Cdut-pu'de*)* se) bk rs ‘ fret (ud0°+ dOd*) nea “(Fs ) 5» trom ( 2.) a ag” If s be the independent variable, we have from (21.) dxd’x+dy d*y =0, or a2 2 2 79 da dy — dy Pa=dea d*y seth ds dy a dx PE SUS ayn Rae ae N's Bae. dy ds -. dxv.ds” ds” ds ~ dy.ds ~ ds? ds San, Pye diy ex isms sabe ds. These differential relations are those which will be of most use n our future operations: and the more the student considers them by themselves, as simple deductions from the relations which exist between he coordinates, the better will he distinguish between the analytical yart of a problem, and the geometrical or mechanical considerations to vhich the analysis is applied. Thus he will afterwards learn that s is he arc of a curve, or he may remember the result of page 140; but, in he mean time, it will be clear that the function » may be considered imply as a function of x and y, the expression of which by a distinct ymbol will facilitate the formation of simple relations. The equation of a curve is generally written in the form y= r, but he more general form ¥ (2, y)=0 is frequently used, and requires some onsideration. The circumstance which needs notice is this, that the quation %=0 mav in reality belong to two or more distinct curves, Ossessing no property in common. If P=0, Q=0, R=0, be the quations of distinct curves, then PQR=0 is satisfied by either of the aree, and belongs therefore to all three. Thus y?—2°=0 is either +z2=0, or y—2=0, and belongs to either of two straight lines. tut y°—x°*=a" is the equation of an hyperbola, of which the preceding Taight lines are asymptotes, and as a diminishes, the hyperbola ap- roaches without limit to coincidence with the asymptotes, in which it is ually lost when a=0. See page 215. Similarly, the equation PQR=a elongs to a continuous curve having different branches, which branches, hen a diminishes without limit, approach without limit to coincidence ith the curves denoted by P=0, Q=0,"R=0. But even when we msider the equation PQR=0, we can trace the properties of either mye, or, as we should say with reference to this equation, of either 348 DIFFERENTIAL AND INTEGRAL CALCULUS. branch of the curve: bearing in mind (page 52) that when an incre- : ment is given to 2, the ordinates corresponding to x and x-++Az must be taken upon the same branch. As an instance, let us propose the equation (y—#)(y’—2x) =0, which belongs to a straight line passing through the origin, and equally inclined to. and y, and also to a parabola whose latus rectum is the linear unit. The developed equation is y*—a«y’—ay+2°=0, in which, unless we’ knew of the derivation, we should never suppose that two distinct curves were involved. From it we find og OY he Nah ala Ge Ny dy yty—2n oy pened a my ye Patan ag or Tiga Say ae which is ambiguous in value, since y is ambiguous in value. Put y=4, and the diff. co. becomes 2? —a2—(a?—2), or 1, as should follow from yaa. Put y’=2, and it becomes (-+./a—2)+(+ 22/r-+ 2x), whichis +1+2,/x, as should follow from y’=2. The only difficulty that can arise, is when the point in question lies. on the imtersection of two different branches: but of this, as we shall immediately proceed to show, we are warned by the appearance of the diff. co. in the form 0-0. Let PQ=0 be the equation of such a two-fold system. This gives dQ. dP 5 gO.qg8 14.9 dP dP qy\ _ 9 dy | : dx! ~’dx dit > dy dz, —— dx! dy de) de ee . , P — =. dy dy which, if P=0 and Q=0 at the same time, takes the form 0+-0, We shall presently see more of this point. What then, it may be asked, is it which distinguishes one curve from’ another, since an equation between coordinates may belong to any and all’ of twenty curves? In reply to this, we must first ask what is meant by’ one curve and another in the question? The eye will not distinguish) with certainty, nor do common notions drawn from inspection of curves always prove sufficient. A person accustomed to consider only the conic sections would always regard a complete oval as a finished curve: nevertheless, it often happens that one equation of the form > (a, y)=0. which cannot be separated into factors, yet belongs to two ovals, or more. The proper answer to the question is, that, as far as the eye is concerned, all distinct branches must be reckoned as different curves: thus the twe branches of an hyperbola are considered as distinct, and we know that) before the application of analysis they were not called opposite branches) of one hyperbola, but opposite hyperbolas. But if we reply with reference to analytical considerations, we answer, that, by convention.’ PQ =0 is only to be considered as representing one curve, when P and Q are really obtained by performing the same operations, the difference’ arising from the different results which ambiguous operations afford, 11 being understood that the operations which are ambiguous are ultimate forms, or not reducible algebraically. Thus y?=.2” gives y= +./2" ane) y=—,/a*; but these are considered as different curves,* since the sigh) of ambiguity may be made to disappear, giving y= +a and = ee * The term curve, in analysis, means a continuous line or collection of lines. Thu, the straight line is included under the term. APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 349 But y°=2 gives y= +,/z and y= —d/x, which are not further reducible; and the equations are considered as representing different branches of the same curve. I now proceed to consider the circumstances which attend the con- tacts and intersections of curves. The terms contact and intersection convey distinct and well-known notions, and the word coincidence may stand for both, Say that there is a coincidence when two curves have a point in common: let y=r and y= Wx be the equations of these curves, and let the coincidence take place when x=a, or let pa=wWa, Let the point of coincidence be a singular point on neither curve, and et x become a+h, giving @(a@+h) and w& (a+h) as the ordinates, and b(at+h)—w (a@+h) as the deflection (QR) of one curve from the ther, measured parallel to y, at the departure h (or NH) from the soincidence, measured parallel tox. This deflection we have expressed as meant to be positive when the curve ¢ falls above Ww, as expressed in both cases of the figure drawn, First, let no diff. co. be infinite: then the deflection may be written (pa—Ya, or O)+(Pa—wa)h +{o"(a+eh)—W"(atb)}—, there @ and ¢ are less than 1. If ga and wa be not equal, this eflection, when h is diminished without limit, bears to the departure a tio which approximates without limit to that of d’a—w'a to 1; that is, l€ ratio of QR to NH hasa finite limit. And since the first significant ™ of the deflection may be made greater than the second, by ficiently diminishing A, it follows that the sign of the deflection and that ‘A change together ; so that if @ were above y% when h was positive, p ill be below & when h is negative. This coincidence, then, is inter- ‘ction, and intersection without contact; the term contact being served to signify coincidence, whether with or without intersection, in hich the ratio of QR to NH diminishes without limit. Now let ¢’a=w'a: the deflection may then be represented by (pa—wa, or 0)+ (d/a—w'a, or O)A+ (pa—w"a) a | 72 | +10" (a4+6h) —¥""" (a+ch) | at nence, if "a and wa be unequal, it appears that the deflection pre- tyes a finite ratio to the (departure)*, and diminishes without limit as mpared with the departure: also that the deflection does not change sign, so that there is no intersection, but only a common geometrical atact. This is called a contact of the first order. Similarly, if 3 203° flection preserves a finite ratio to (departure)*, and diminishes without ult, as compared with (dep.) and (dep.)* And here, though the Mcidence is of a closer order than in the preceding case, there is an tsection: this is called a contact of the second order. Proceeding this way, we find that when two curves have a point of coincidence ia@=w"a, the first term of the deflection is (9”a—wl"a) and the 350 DIFFERENTIAL AND INTEGRAL CALCULUS. for which n (and no more) diff. co. of the ordinates are the same, the | deflection has a finite ratio to (departure)"*’, and diminishes without | limit as compared with all lower powers; and this is called a contact of | the nth order. In contact of an even order only, there is intersection, | And if two curves have contact of different orders with a third, then | that which has the higher order of contact approaches infinitely nearer | to the third than that which has the lower. I leave the following theorems for exercise, as they will be very easily. proved. If two curves, (A) and (B), have contact of the nth order with | (C), they have at least that contact with each other. If (A) and (B) | have contacts of the mth and nth order with (C), they have with each other at least the lowest of these two orders of contact. Next, let us | suppose that two curves have a coincidence at which m diff. co. are finite, and are the same in both, but let w“*¥a@ be infinite. Then | (page 182 and 327) for a large class of cases ia Dicnels auteh WU (ath)=watwplah+ ...o+ pa + h’x (a+h), where p lies between n andn+1. Hence, if Taylor’s theorem can be applied to x (a+/h), the deflection is het 1 n+2 h n+l it ih ox p n-+-2 DAT Te cemmvaeg eae 73 PGT OR) oe ano — x! (a+ch) he, in which h? is the lowest power of p, and the contact ‘might, by analogy, be said to be of the order p—1, a fraction between nandn—l. It isnot. necessary here todo more than hint at the peculiarities of the contacts | which take place at the singular points of curves. | Returning to the case of points which present no singularity, we see at. once that no curve can pass between two others, all three having @| common coincidence, unless the intermediate curve make with each of) the others a contact of at least the same order as they have with one, another. Weare thus enabled to find the closest line of a given species which can be drawn through a given point of a given curve. Whatever arbitrary constants exist in the equation of the given species, take their | values so as to make as many diff. co. as possible the same in the two. curves, taking care first to satisfy the condition that the two curves coincide in one point. What is the closest straight line which can be drawn coinciding with a curve whose equation is y=z, at the point whose coordinates are @| and da? | The general equation of the straight line is y=px+q, and the coincidence requires pa=pa+q or y—pa=p (x—a). Now v= which must be the same both in the line and curve: whence y—9a= | $'a («—a) is the equation of the line. This'line makes with the axis of | ; d a x an angle whose tangent is ¢’a, or the value of at the given point:: whence we sce that the line deduced in page 137 as being best caleu- lated to mark the direction of the curve at any point, is also the closest straight line which can be drawn. We also see that the contact cal only be of the first order, generally speaking. This line is the ¢wngent of APPLICATION TO GEOMETRY OF TWO DIMENSIONS. AY tah Meas : the curve. If, however, it should happen that = is infinite at the given z point, the preceding proof is not complete. investigation so that the axis of y (that was) shall be the new axis of L, ‘and vice versa. It will then appear that the closest line is parallel to the new axis of x; that is, perpendicular to the old one. Relatively to this change of axes, the investigation of the following generalization will be a useful exercise. Let the axes be changed so that the new axis of with the old one, and let wv’ and y' be the new coor whose old coordinates were wv and y. Then In such a case, change the « makes an angle w dinates of the point r=2' cosw—y’ sinw w=y sinw+2z cos w y=2' snw+y’ cos w y'=y cos w—#sin w oo “= 1, or (co w-b A sin ») (co: w — sin v= l =(Fo+tn ») +(1 te tan ») e =(3 —tan v) +(1 -- 2 tan ») my dé’ dy’ . NE aye! day 3 dy , 4 I? Tl +-( os i) —— eae SMW }, dx? as —— se w+ ne sin v) : It being proved that, generally speaking, the tangent has no more han a contact of the first order with the curve, required the insulated points, if any, at which a higher order of contact is possible. The suc- essive diff. co, in the straight line after the first are =0; consequently, it a point in the curve at which ¢”x=0 there is a contact of at least the econd order with the tangent; when "'x=0 of at least the third order, md so on. | For example, it is required to draw the tang llipse, and to ascertain those points at which rder than the first. Taking the centre iameter as the axes of w (a and b being t ent at a given point of an the contact is of a higher as the origin and the principal he semiaxes) we have Sema * x y dy 1 1 dy? y dy = til, —+~ —=0 -— - = a0, te 4 EEG, ad? GN de wi Gh) abt dae bs dz? ? dy: Bx - b xv Dy cab de) dy Naudfate.-ae@r to dx ay a Al ae aa (a°—~22): here — or + is used according as + or — is used in forming the due of y. The first shows the tangent of the angle at which the tan- pnt is to be inclined to the axis of x, und the second, which never tmishes, shows that there is no point in an ellipse at which the tangent 's a contact of a higher order than the first. If & and 7 be the co- dinates of any point in the tangent, the equation of the tangent is Me aty (€—zx), or oe bet As Go. Dita) ‘We have here changed our notation. In what precedes, a and da 392 DIFFERENTIAL AND INTEGRAL CALCULUS. were the coordinates of a given point in the curve, and x and y the co- ordinates of an arbitrary point in the tangent. In future, x and y are the coordinates of a given point of contact in the curve, and & and 9 those of an arbitrary point in the tangent. To exhibit the equation of the tangent, that of the curve being w(a,y)=c. We know that ‘dx ee a whence Pa ate! Cond becomes dy, dy dy | dy dx a Ara Ane : If $ be a homogeneous function of # and y of the mth degree, we have (pages 194, 205) nw or ne for the second side of the equation: butif ¥ be made up of several homogeneous functions, M of the mth degree, N of the nth degree, &c. write mM+nN-+.... for the second side. Thus for the cissoid of Diocles (A. G. 304.) 2ay?—(ay*+-2°)=0, in which is a | function of the second and of the third degree: the equation of the | tangent is | | . | — (y+ 32") &+ (day—2ay) n=4ay?—8 (ry? + 2°) : | =—2ay’. | If there be only two functions; that is, if M+N=c, we have mMinN=(m—n)M-+ne. The following are instances: | Curve. Ay’ + Bay+Ca2?+Dy+Er+F=0. | Tangent. (By+2Ca+E) + (2Ay+ Ba+D) n+Dy+Er+F=0, © : Curve. (A. G. 319.) oy? +ta—5az*y’=0. : Tangent. (5x*—10axy?) £+(5y4—10aa* y) n=5a2* y’. | The normal is a line perpendicular to the tangent, passing through : the point of contact. Its equation, therefore, is | vl es 3 dy | DU ay a> Sheet tig, me ante : | | | | | which in the manner already shown may be made dy, dy dy dy, de dy —-y) 5. Gee aaa Cast Tiago er ae the equation of the curve being given in the form (x, y)=0. | The angle PTN having ¢’x for its tangent, y=¢? being the equation of the curve, the value, as t0| magnitude, of the subtangent TN and the sub- normal NG are PN: tan PTN and PN x tan PTN, or Or: 'xandgo«xX¢'a. As to sign, if we call them positive when they occupy such positions as in the: corresponding diagram, we have this rule :-—the subtangent and subnormal have always the sy sion: positive, when dx and d'x are of the same sign; negative, when of different signs. The parts of the axis intercepted by the tangent are, a5) to magnitude, OT=2 —(¢r: ¢'r) and OU=OT x tan PIN=2'2— G0 But the latter being here negative, should be represented by pxr—apa, APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 353 and this expression will always represent OU, both in sign and magnitude. And if in the equation of the tangent we make 7»=(, and £=0, we find the same, after writing gv and ¢/x for y and dy: dz. Similarly, OH=$2+-2: $/a, OG=2+ $2 ¢’z, if UO and GP meet in H. The following expressions will often save trouble : 1 iff. co. log y’ ies ‘ Subtangent = Subnormal a diff. co. y*. Hence, in the exponential curve y=e, there is a constant subtangent ; in the parabola, y=cx, a constant subnormal, What is the curve in which the subnormal varies as a given power of the subtangent. Suppose uy PIN" en lm ny eae yBae(y i) 5) then rm ae yrs r= C med yn +C, ntl jC ee are n+l ; or y= ) c2?(4—C) 2 A straight line moves in such a way that OU is a given function of OT; to what curve is that straight line constantly a tangent? If UO be one function of OT, UO:OT or tan PTN is another 3 let this be called p, then, p being a function of OT, OT is a function of p, and so is UO. Let UO, with its proper sign, be fp; then y=px-/fp is the equation of the straight line: or, if we let £'and n be the coordinates of any point in it, n»=pi+fp. Compare this with the equation of the tangent to the curve, which it is always supposed to touch, and we have dydp dy) in a dy ES BEe a, 2 ee Spy 2. ~™ a Differentiate the last, and we have [ets So le fe Me eon t dz dx dx ds® Pe rf And the third then gives, substituting —f’p for a, eke OM at Eliminate p between these two, and we have an equation between x and 'y, the coordinates of a point in the required curve, which equation is therefore that of the curve. Or thus: the first and third equations give dy _ (dy RS dy a differential equation, already discussed in page 196. Its common Solution, y=ca+fc, would only give the straight line with which we began, which certainly falls within the conditions of the problem, for we have but to assign a value to p, and let it retain that value, and the Straight line so obtained is a tangent to itself at every point. The Singular solution derived from @¢-+ f'c=0 is precisely the equation to the curve in question, which is always touched by the moving straight line. 2A 354 DIFFERENTIAL AND INTEGRAL CALCULUS. A curve, whose equation is »=@(E,c), takes all the imaginable varieties which can be given to it by changes in the value of c. W hat is the curve to which it must always be a tangent? Let @ and y be the coordinates of the point of contact, when a@ is the value of c; then, since the point of contact is on both curves, y= (a,a). But this last equation is not true of every point of the curve of contact, but only of its point of contact with the variety of the original curve in which c=a, and which has the equation n=@(é,a). But if we were to allow the value of w to change with w, so that a should always represent the value of c in the individual curve which touches the curve of contact at the point (x,y), the equation y=¢ (2, a) would remain true throughout the curve of contact, and would be its equation: but a would be then a function of a. What function of 2 is it? To determine this, observe that since every variety of n= (&,c) is somewhere in contact with the curve of contact, the value of dy: dé from this equation must be, at the point of contact, the same as the value of dy: dx from y= (a, a). Let ¢’ (£,c)=dn: dé, then, giving é the value 2, which it is to have at the point of contact, and ¢ the valuea, which it has in the particular case in which the point of contact has # and y for its coordinates, we have, for that case and at that point, dy: dé=q! (a, a). To find dy: dx we must, in the equation y=¢ (2, a), suppose @ a function of « in the manner above described, which gives dy dp , do Aa 5 es hy pita ed da dx dx dx” da dx dp . Ae i d for - formed from @(a,@) gives precisely the same function as “a from n=¢ (é,c), since @ in the first case, and c in the iatter, are con- stants. Equate dy:d& (or rather the particular case described) and dy: dx, which gives dd da dp da ‘(a,a)—=¢! (a4,4)+ — —, or — —=—0. BAe aah omen Hither, then, dd: da, or da: dx=0; it cannot be the latter, since then @ would be a constant: consequently, d:da=0, which will give an equation between v and a, or will determine the function which a is of | x. Hence the following Turorem. The curve which touches every curve that can be represented by y= (@, c), whatever may be the value of c, is found by substituting instead of the constant ¢ a function of «, obtained by equating to nothing the diff. co. of # (a, c) with respect to c, and thence determining cin terms of x. But this is (page 189) precisely the mode of obtaining a singular solution to a differential equation whose ordinary solution is y= (a,c). Hence, the singular solution to a diff. equ. connecting v and y is the equation to a curve which touches every curve whose equation is a case of the general solution made by giving one oF another value to the constant of integration. The preceding demonstration will, I apprehend, be found diffi- cult; but as the principles which it involves are of the utmost © consequence in application, it is worth while to vary the form of the | problem. APPLICATION TO GEOMETRY OF TWO DIMENSIONS, gd5 Required the curve y=¥-« which cuts all the I ‘ y oli & L curves contained in n=¢ (é, c), made by giving - different values to c, in such a manner that, at each point of intersection, there exists between n dy —» >» and a, the relation dé dx’ ) O 59 Hs Ys XL, c |=, 0 BDNF HK “\d? da’ 2% Let AB, CD, &c. be varieties® of n= (6,¢), and let VW be the curve which makes the intersection in the manner required. Choose a case of y= (é, c), say GH, and for that case let c=a; that is, the equation of GH is y=¢ (g,a). Let P be the point in which VW cuts GH, and let zx and y be its coordinates. Then because P is on GH, y=$ (a, a), but this equation is not true of any other point of VW, for, @remaining the same, if the point P should move, its coordinates still satisfying y= (z, a@), it would move along PG or PH. But, if ca! give the curve EF, intersecting VW in Q, and if when P moves to Q, @ were to change into a’, the equation y=¢ (2, a’) would be true of the coordinates of Q, which is on VW. If, then, @ were to be such a function of x, that as y and x change on VW, a should always repre- sent the value of c which belongs to that case of n= (&, c) through which VW is passing at the moment, it follows that y= (2,a) would be true at every point of VW 3 that is, would be the equation of VW. What function of x, then, must @ be? The value of dn: dé is p! €,c), and in the curve GH, and at the point P of it, this is d’ (a, a), exactly what would be obtained by differentiating @ (2, @), @ varying and a being constant. But to make an equation to VW, we must write for a a certain function of x, and we then have dy db dda dz dx dada’ The required relation demands that r(w (2, a), 9! (230) 42 ae, p (2, a), 2, a)=0. gti Wp da dp da aay 5, St ore at Cre da dx where (2, a), ’ (2, a), and dp:da are known functions of x and a, and therefore this is an equation between a, x, and da: dx, or a com- mon differential equation. If it can be integrated, the problem can be solved. For example, required a curve which cuts the species of curves whose 2quation is 7=@ (&, c) always at the same angle, so that, at any point P, the angle of PL and PM, the tangents of the cutting curve and the curve of the species which passes through P, is a given angle a. If, then, 3 and 6! be the angles of these two tangents with the axis of 2, we have tan /3—tan f! dn dy dn dy B— p'=y, ————__ y—tan oe, or >. —— =tana( 1+—. — }, 1+ tan 3. tan 2 d—& dr dé dx ‘his gives, by the preceding process, * The equation of a curve is confounded with the curve itself in the language sed; thus the curve y=" means the curve whose equation is y=2°, Similarly, i¢ point x, y means the point whose coordinates are x and y . 2A2 aw £ 356 DIFFERENTIAL AND INTEGRAL CALCULUS. dg da) : Va {0 d+ THUR 40C O00 + Ge ae) or tana {1 +(P'@, ayy} +52 aC (x, a) tana+1}=0.! This equation cannot be integrated generally, but we may try our method on any particular case. Let the species be that containing all the straight lines drawn through the origin, having the equation y=ar. Here (2, a)=ar, 9! (%,a)=a, dp: da=x, and the preceding equation becomes a d t 1 tana {1 +a*s-p2 = fatana+1}=0, —tan ees etl Sa d l+@ log a tan a= —log ,/(1+a*).tana—tan™a+C. We cannot find a in finite terms from this expression, which we should do, in order to substitute a in y=az. But the same end will be gained by substituting a (=y: 2x) from the second in the first, which will give 3 A 2 log x.tana= se lg a—tan?=+QC, , x x ‘ or | log J (a -+y)= is lp tan 2 + : - tan a x tane Writing C for C: tan a, and using polar coordinates, we have 6 —-——+C e ° pee a raze ene’, which may be written r=Chk’, {k-"™*=} 5 for s° is merely an arbitrary constant. This is the equation of the | logarithmic spiral (A. G. 371.), which is now found to cut all the | radii at the same angle, and to be the only curve which does so. The preceding investigation would not have been altered in any respect if we had used polar coordinates. For it rests upon the suppo- : sition that # and y determine a point, and that an equation between them determines a curve; nor is there any reference made to the par- ticular naanner in which @ and y determine the point: so that the _ investigation applies to any kind of coordinates. If, however, we had proceeded to the solution of this problem with polar coordinates, writing 0 for 2, and 7 for y, in (f'), we should have met with a difficulty which it will be worth while to dwell upon. The equation of a species of curves is /=@(6',a), and a curve © r=W0 is to cut all the individuals at an angle a. Calling p and p’ the angles made by the tangents with the radius vector, we have, (page 345), B=p+6, P=p'+0, B—-P=p—p’, and tan p—tan p! do ,dé do d0\ ~ =o, ———_ Stan a, tr = SS oe BB 7 tan pe tan po! an EAE: fe Miers rr ere 2) or r ad BL ty th: Ni mk ag" ae *\ a6 art” J In this problem the angle « is taken with'a sign contrary to that. which it had in the last. Substituting for a the requisite function of 0 we obtain from (f) the condition (remembering that 7 and 7’ are the same at the point of intersection) — — APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 357 rg! Go ie, dr! ea, io stan a pa (G+ dr aa) rh dg! dd’ — da dé! dé’ \ de’ da de’ i If we apply this to the particular case where r’=¢ (6',a) is the equation of a straight line passing through the origin, we find 6’—==—— (a =e ) >, o— ass ab ENE yay Oe e* a ! whence, evidently, ( i + (+) ot ol ace aE) aera. We must not, in this subject, propose examples as if we had only to choose from an unlimited number capable of sufficiently easy solution ; for the fact is, that the elimination is generally of so difficult a character, that the few cases which are presented in elementary works contain all which the student should be invited to try. He may, perhaps, succeed with 7?= or &,a)=0, we are to eliminate £ between $=0, and db: dE=0. But there is an easier mode of obtain-~ ing a diff. equ., as follows. Differentiate ( J), which gives 2pq?— (1+ p? +p 3p°q) - (p-+p") 2 eat elas tap («Ph ue O-2e re. a) ee 5s 2 . " 2 H or pre +p gu (220. ey If we make the first factor =0 we merely recover the equation (¢), or rather the equation (*—£)°+(y—7)*=p*, & , and p being uncon- nected constants. In the other factor, made =0, is also to be found an equation which is true when (/) is true, or we have e! +P) p(+p*) J 1+ (abot =; or <—— —*— Lat See = Ps q ve dl P where f’-' means the inverse function of /”. Therefore ( J) gives gas (2), na aria el ae and if from the last two we eliminate g, we have pyt+2=pff (— 5) 497 = 5) veil f)3 a diff. equ. of the first order, and containing only one arbitrary con- stant in its solution. This equation cannot often be integrated by a separate method; but the preceding process gives a hint as to the method of deducing its general solution from the particular solu- tion of y—px=Fp, as found in page 196. The student who under- stands the preceding considerations will see that the following is merely an analytical translation of the process of finding the involute from the evolute. Required the general integral of “py+a=Fp, p being dy: dz. Assume _ ‘ n dp?= dé? + dy’, #=E—p ie Sidi ay atts 3+ +(p) p _—_— —_— dx d&—{di+pd (di:do)} dp di —di-d’p _ (d5*+- dn’) d?n—(dé dE + dn dn) dyn _ dé (di? +-dn’) @E— (dé E+ dn dn) dé dn 366 DIFFERENTIAL AND INTEGRAL CALCULUS. dy dé dy ee gt CRG 1? a5 Sn pgs dy’ and the original equation becomes bi BiB Hy dé dy me ( =) E ee (- 7) or 1— Fe | Fosee det a which are of the same form as y—pr=Fp, and can be integrated in the same way (page 196). If we take the general solution we find dy: do= const., whence dy: dx=const., which does not satisfy py+r—Fp: it must then be the particular solution of the preceding from which the relation between 7 and & is to be found;* and this being done, 9, or f- »/ (dé +dn*) contains an arbitrary constant, which remains in the relation between x and y, found by eliminationg between the second and third of the equations (p). Required the involute of the parabola 2n=@. Here fE = $2, '&—£, consequently f’é=£, and the equation to be integrated is PY Pe. D) p p ? PV phebent As there is no direct mode of integrating this, we must have recourse to the equations (0) ; this gives dp?= (142) dB p=hEY 14+8) +4 log (E+ A4+8))+C pee ee Jog eta Cos "5 — aay ' whence 2° 2 Jf (1+) V(1-+6") 1 ey 2 blog e+ +e) h. ). Ce a5 b+. 2 V-+8) VA+2) : The last equation is merely that of the tangent of the parabola, and from it € can be found in terms of x and y, and the elimination may be completed by either of the first two; but the result is so complicated that the expression of both coordinates by means of & is more con- venient. The constant C is the value given to the radius of curvature at the point of the involute answering to €=0, or the vertex of the para- bola. The result also gives the general integral of (p). Ine the case of the involute of the circle, we have &+7’=a’, the radius being a, whence, O being the angle of the radius vector R=a, we have do*=a* dO’, and p=C+a0. The equations of the involute are therefore z=acos0+a0sin0, y=asinO6—aO cos 0, assuming C=0. Show from ds’=da*-+dy’ that the length of the are of this involute measured from O=O (A in the page of errata A. G.) is one half of the arc of the circle which would be described by a radius equal to the arc of the eyolute, moving through the angle ©. The in- * This method must not be applied complete to finding the involute of a given evolute, as it would merely give between » and Z the equation of the evolute: the equations (¢) may then be used at once for elimination. APPLICATION TO GEOMETRY OF TWO DIMENSIONS, 367 volute of the circle being obviously an epicycloid in which the moving circle becomes a straight line, or has an infinite radius, the preceding equations should be deducible from those of epicycloid. . The equations of the latter curve are (A. G. 360) atb xv=(a+b) cos O—5 cos 0, y=(a-+0) sin0—bsin +", where a and 2 are the radii of the fixed and revolving circles. The first of these may be thus transposed : xw=acosO+b 108 O—cos (3 1) o} a a oe, b i ae ° 1 ame), e acos0+ 26 sin oy; 0.sin (itt )o If 6 increase without limit, the limit of 2bsin (a: 2b) © is a®, and the preceding becomes t=acosO + aOsinO, as above. The second equation may be treated in the same way. The equation (f) leads immediately to a conclusion respecting singular solutions which is worthy of notice. If we make f’! (35) =P, or p=—1:f’ P, that equation becomes ge] boty fp Pee rss CRY Let us inquire whether this equation has any singular solution. From it p might be expressed in terms of w and y; which being done, the singular solution, if any, is found by making the partial diff. co. dp: dy or dp: dx infinite. But since 1 dp 1 dP = = have — =——.f"”P.—: fro te pee. de: whence dp: dy and dP: dy become infinite together, unless f/ P=0 or f’P=« when dp: dy is infinite. Now, differentiating the above equation with respect to y, # being constant, we have dP dP dP 1 dp 1 eiaoas Vt > ea i — i, V/ 2s cheates = Ste oe —=- whence x=P is the equation which gives the singular solution, if any. Substitution in (P) gives y=fP or y=fr, the equation to the evolute again. But it will be obvious that the evolute is not the curve which touches all its involutes, but the one which passes through all their cusps. Hence, an equation presenting the. analytical characters* of the singular * I do not say ad/ the analytical characters; for if y= (a,c) were the primi- tive of P, we should not derive this singular solution from dg:dc—0, The fact is, that in page 190, we come only to those cases in which $ (x, e+ Ac) can be deve- loped by Taylor’s theorem. But if the intersection of the two contiguous curves approach without limit to a point at which this theorem fails, the method would not apply, and.the curve which passes through the limits of ail the intersections is not necessarily a tangent to all the genus of curves denoted by y=@(,¢). In order that this theorem may apply, in page 190, it is necessary that dp: de and ag: de® Should remain finite or nothing (not infinite) throughout the process. If, then, the 368 DIFFERENTIAL AND INTEGRAL CALCULUS. solution of a diff. equ. may belong to a curve, which instead of being a common tangent to all the curves denoted by the diff. equ., may be the locus of all their cusps, or other singular points. If our diff. co. of y are to be obtained from ¢ (2, y)=0, instead of y=dxr, we have (using the notation already explained) | dy dy " pee ae dx ob, dzx* 0b? __ Sid —28'b, 8 49%, (+0 | oe : BPP —2P D/P AP Gy haa Sa. a & BP G"=26'4, 840" 4, This form avoids all irrational quantities, if the original equation can be made free from them. Thus for the parabola in which y*—4cr=0 =? (a, y) we have a | (16c*+ 4y")? 16c*.2 ¢'=—Ae, $= 2y, o'=0, ¢' =9, $= 25 p= 1 Yee y+ 4ct ate : oe Wee ee ene ee : cya —ay, E=3x4+2c, y=Acr. ‘ Hence, by elimination from the last, the equation of the evolute of the | parabola is 2'7en°=4 ( —2c)*, which is the equation of what is calleda | semicubtcal parabola. | In all that has preceded, we have tacitly supposed, according to our | custom, that the diff. co. employed have finite values, It now remains | to consider the cases in which they cease to be finite; which will be | nothing more than a set of investigations connected with the singular | points of curves. Previously, however, to entering upon them, it will be | necessary to consider the general meaning of the diff. co.; the follow- | ing account of them is partly recapitulation, partly matter newly intro-_ duced. | dy ae y'} This function is the tangent of the angle 6, which the da’ curve’s tangent makes with the axis of v, the point of con- tact being (2, y). When positive, y and x are increasing or diminish- ing together; when negative, y diminishes as xv increases, or vice versd. | same substitution with respect to ¢ which makes dg:de==0, should also make | d”9 : de® infinite, the whole process will be vitiated. Now this may take place when | the limit of the intersections of the contiguous curves is at a cusp, as in the present | instance. _If we examine the equations of page 192, we shall find that if y=9 (#,c), the diff, co. of dy: dz or xv are ) dy 4). dx __ dp a 5d dp dep AR bee det dee ake ana a 4 : i} These are made infinite, not only by + =0, but also by Pop , and (at least the ae | first) by nothing else; hence the two sorts of singular solutions, or rather the two distinct cases which the test may present. APPLICATION TO GEOMETRY OF TWO DIMENSIONS. When 7’=0 the tangent is parallel to the }pendicular. When there is a change of sign, y is a maximum (M), ora ‘minimum (m), according as the change is from + to — or from — to | + (vincreasing). If the change of sign be made by y/ passing through 0, there is an ordinary maximum or minimum of y; but if by passing through « there is a maximum or minimum made at a cusp (C). But aif y pass through 0 or « without a change of sign, there is a point of ‘contrary flecure (F). These two last terms are better defined by jlooking at the figure than by words. In the figures the arcs along which y/ is positive are continued lines, those along which it is nega-~ tive are dotted. When y/=0 or a, being impossible immediately before or after, there is one or other of the cases marked on the right, between the characters of which it is left to the student to distinguish. 369 axis of a, when ¥/=«& , per- dr ,) The fundamental properties of these differential coefficients | do’ °* ">| areas follows. They must differ in sign, for r’-+u-* u/=0, | OR and they are connected with y! by the following equations We w'.} (page 345, equations 16, 177). , snd+resd wsind—ucos@ y= OZ rcosO—rsind w' cosO-+-wu sin 0 As long as 7’ is positive, 7 increases with 6, &c. When 7/0, ge —cot 4, or tan 6 tan@+1=0, whence 6 and 6 differ by a right angle, or she tangent is at right angles to the radius vector. There is then either 2 maximum or minimum value of 7, or a point of contrary flexure; but fr’ become impossible after passing through 0, there is a cusp. Again, fr’=c« , y/=tan 4, or the radius vector is itself the tangent. If +r’ con- ‘inue possible after passing through o, there is a cusp if there be a naximum or minimum, and a point of contrary flexure if there be none; wut if 7’ be afterwards impossible, there may or may not be a cusp. | uw’ is nothing or infinite with 7’, but when w’ is positive 7 is diminish- ‘ng as @ increases, &c. Fie at To give an idea of the geometrical meaning of y", re- face member (Chapter IV.) that if x increase successively by t, giving y the successive values Yrs Yo &e., y"' is the limit of Yo—2y,+-y livided by h?, and as h diminishes, y,—2y,-++-y must finally assume the ign of y”. This sign, therefore, is positive, when for any arcs, however mall, y.+y is algebraically greater than 2y,, or the mean of y, and y reater than y,; and negative when the same mean is the less. That 3, y” has the sign of VS—VQ, where NP, VQ, WR are the successive dinates y, y,, Yo: it is easily shown that NV being = ae is the 370 DIFFERENTIAL AND INTEGRAL CALCULUS. mean between NP and WR. In the convex curve VS—VQ is positive or negative with 7; but in the concave curve VS—VQ is of a dif- ferent sign from y. This will readily follow from giving VS and VQ their algebraical signs in the four figures adjoining, and finding that of VS— VQ. Hence, when a curve is convex to the axis of a, y/’ and y have the same signs, or yy" is positive: when the curve is concave y” and y have different signs, or yy" is negative. It may often be convenient to observe that this criterion may be altered as follows. If log y=z, the curve 1s convex when 2”-+2" is positive, and concave when the same is oats when y= 2, the curve is convex or concave, according as z (222! — z’*) is positive or negative. Thus y=é€" is always convex; for, in the first case,| 2!’-+2"=1; again, y=,/ (1—a’) is always concave; for, in the second) case, z (2z2/—z”) = —(1—a”) (44 82°). Again, if]: y==2z, the curve is| convex or concave according as 2z’*—zz” is positive or negative. Thus, in y=: (1+2), we have 2z"—z2"=2, and the curve is convex or con-| cave as y is positive or negative. The demonstrations of these theorems will be easy exercises for the student, and one or other of them will generally be found of more simple application than the fundamental theorem from which they are derived. We also learn from the preceding that A’y: (Ax)’=2 (VS— VQ): (NYV)?; so that, without attending to the sia y’ is the limit of 2QS: (NV)?. In the case of a point of contrary flexure, if y be finite, y” must, change sign; for it is the obvious character of such a point that the| curvature is convex on one side of it and concave on the other. But, when y changes sign at a point of contrary flexure, the characteristic of} the curvature is to be the same on both sides. Consequently y” must) also change sign; or, the criterion of a point of contrary flexure_is uni> versally a “change of sign in ¥”. We may give an easy geometrical proof of an important proposition as follows. Take an arc PR from a curve, let PA and PD be parallel to the axes of y and 2; bisect the chord PR.in S, and complete the figure asshown. Then 2QS is A’y, on the supposition that Av remains uniform; and 2ZS is A®z, on the supposition that Ay is uniform; but the two have different signs in the figure drawn, and if it were not so, it would be found that Ar and Ay would have different signs. But as the arc PR diminishes, the tangent at Z approaches without limit m' direction to the tangent at P; so that the limit of QS: SZ is the same! as that of QA: AP; or, allowing for the difference of signs, the equation A’y : Aas — Ay: Ax becomes nearer and nearer to the truth as | diminishes without limit. Put this in the form hey hd d’y dx dy’ | (Ax)® +e (5 xt) = =0, and the limit —— i tay Chas | is true; the same as was shown in page 153. . APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 371 d®r ; tees st OR) 14g : dé sical BE sas tied du d’r yet | du? aM d®u uw de 4 de de? 48 de? pede? ‘ de” or uw’. Let 6 be twice successively increased by Aé, and let the radii belonging | to the angles 6, 64 AQ, O-- 2A8 ber, 7,, and 7». Consequently, 7” is the limit of (7, — 27,+7) :(A6)?. Let OP, OQ, and OR be the values of + ; then the angle POR (or 2A) is bisected by OS. But if a and 6b be the sides, and C the contained angle, of a triangle, the length of the line bisecting the angle is 2ab cos$C:(a+ 0), whence 7, being r+ 2Ar-+A’r, we haye 2rr, 2rr, OS=— cos AQ=—? (12 sin? Ad) r-+Te T+Ts TT, AT Te— 27 7o me A —OS=7,—08S=——_— 2t Ad. OQ—OS=7,—0OS aed age ishhe ws The numerator of the first fraction will be found to be 2 (Ar)’?+ Ar.A*;r—rA’r, and if the whole be divided by (4@)2, and A@ be then diminished without limit, we shall have (remembering that in the second term the limit will be most evident when we write A@ as 2.4 Ae) ha OQ—OS ae dr? d’r NN Pl du limit of (20? ia Cae +7 aig FY ut ca) Tfr, and consequently u, be reckoned as positive, OQ—OS is positive when the curve turns concavity towards the pole O, and negative when nt turns convexity, and vice vers& when r is negative. Consequently, there is concavity when u+-u” has the same sign as u, and convexity When the two signs are different. And there is a point of contrary flexure when u—+-w’ changes sign. For instance, let us take the spiral called the dituus (A. G. 367.), the equation of which is wa’=6. If instead of d’u:d6® we use d°6:du we must (page 153) for | du d?9 de° 2a? u+— write u——:— ; in this case w——-__. de? du? du?’ 8a? u3 As long, then, as 4a‘ uw‘ is greater than 1 or 46°< 1, the curve is convex ‘owards the pole, and the contrary. ‘There is, then, a point of contrary dexure when 6=:5, which reduced to practical measurement is 28° 39’, dearly. Ina straight line, u+-w”=0; in either of the conic sections it S a constant, if the pole O be ata focus: the latter is one of the most Beportant propositions of the Newtonian theory of gravitation. hi _ If y’=0, the radius of curvature is 1 sy!s cand: nfs! s2Q) sit) is the reciprocal of u-+-w’ (page 347). If y’=0, the radius of curva- lure is infinite, or the circle of curvature becomes a straight line; his agrees with page 351. If w’=0, the radius of curvature is e+)? us, 2B2 372 DIFFERENTIAL AND INTEGRAL CALCULUS. The preceding cases are simple, but become more complicated when y' or wu’, or y" or w” are infinite. Let y' be not infinite, and y" infinite, or let wv’ be not infinite, and w” infinite: in such cases p is certainly =0. This means that no circle is small enough to be the circle of curvature ; but that every circle, however small, approaches nearer to the curve than all larger circles. This result may be illustrated as follows. Take one of the circles which has a contact of the first order only with the curve; that is, in page 360, use for the determination of the coordinates of its centre only the equation £—x+y/(y—y)=0, which merely implies that the centre of the circle must be on the normal of the curve. Let us now consider, as in page 349, the deflection of the curves from one another when x is changed into r-+h. Since the contact is only of the first order, these deflections have the same sign on both sides of the point of contact; that is, when the radius is greater than that of curva- ture, the circle lies between the curve and its tangent on both sides, but when the radius is less than that of curvature, the curve lies between | its tangent and the circle on both sides. But when the radius of curva- ture is nothing, every radius is greater than that of curvature, or all circles whose centres are on the normal lie (at least immediately on leaving the point of contact) between the curve and its tangent; but when the radius of curvature is infinite, every circle is less than that of | curvature, or the curve lies between its tangent and any circle whats | soever whose centre is on the normal. Next, let y’ be infinite, in which case y’ is infinite, and the radius of curvature is the limit of y*:y’. Returning to the theory of pages 321, | or take the limit of | &c., find the critical value of m in y":y", yl yl sy" .y", or of yy": y'”. If this be e, we know (page 322) that) y":y" has the same limit as y"~*, or the radius of curvature is 0 or ©, — according as y'*~* isO ore. But if e=3, it may* happen that the | radius of curvature is finite. The consideration of all singular points will require the examination of the critical value of » in y/:y", a subject on which some little detail f will be required. If p, q, 7, and s be four successive differential co- | efficients of y, it is obvious that the critical valne of » in q:p” is_ pr:q’, and that of ninr:q" is qs:7*. But if the first be of the form 0:0 or 0:0, we find for the value of pr:q’, ; Bee org 14 4 Sqr > OF = 1+ qe a4 If, then, e,, be the critical value of 2 in y("t) : {yf 1", we have 2e,— lL Cia fi+en mets or Cmti— = ° From the preceding, knowing e,, all the rest are found by substitution to be contained in oe (m+ 1) €y—™ oa mey—(m—1) Remember that if r:(wx)" can ever be finite when yr is 0° * The studerit must here avoid the mistake which, as already noticed, I have | twice fallen into in the course of this work. When z has the critical value, the value of ¢x*(~x)" may be nothing; finite, or infinite. APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 373 | or ©, it is when 7 has the critical value, and no other. (and perhaps not for that one.) The following scales of comparative dimension : among diff. co, are universal: we shall presently explain their meaning, | 2n—1 3n—2 4n—3 52 —4 mn Serene my —. &c. n 2n—]1 38n—2 4n—3 : 3 4. : 0 — oe 2 5 3 &e. : ] 1 1 1 1 &e. . That is, by means of the critical vaue of 7 in y’: y", if y be 0 or « , or |in y':(y—a)", if y be finite and =a at the point in question, we can Immediately ascertain the critical values in y':y", y":y!", &e., when- ever y’, y", &c. are all nothing or infinite. |. For example, let y=1:logz, which =o,when z=1. Its diff. co, iis —1:(logaz)*z, and the critical value* of m in ya is 2) Con | sequently, that of y”: y” is 14, which will be found to be true by writing —1: (log x)*?.2 or y’ for wr, and (2--log x) : (log x)* 2°, or y” for ozin ¢/x wx: bx w'x, and finding the value of this when r—=1. | If y’:(y—a)" be finite when y is =a or =x, andif n have the critical value e,, then y/: 9’, y"’:y', &c. are all finite when the several critical values are put for , provided those critical values be finite. Let these be called P,, P,, &c., then at the point_in question P, is 0:0 or a: a i and therefore its value is that of . n—1 " YW " P ers® ] n—} | es or pe Rel Bi a or P, and Alt hb or — P, P, 2: n (y—a)"—ty! sais tate n | * n(y':Po) ™ sy’ ny’ ™ 1 have the same limits. Hence nP," and P, have the same limits, or denoting the limits by 7, p,, &c. we have Piz=eo po’, similarly p,=e, p\'%, &. Returning to the preceding problem, we find that e,, the critical value of n in zy, is (2eg—1) : &, whence, 3—e, being (¢)+1):¢,, we find that, when y’ and y” are infinite, eo+l p is 0 or ©, according as (y—a) © is O orm ; and o is finite when e,=3 or e=—1, if ie (y—a@)~ or (y—a) y' be finite. For instance, let y=a-+./(r—b) . fv, where fe and its diff. co. are finite when r=), in which case y=a, and its diff. co. are infinite. If we then seek the critical value of » in y':(y—a)’, we find it in the value (x=b) of —. w —3(x-/ te rm fy 2 lt4+-(2r— 2 th y ) y , Or (x—b)? fr, 4(v—) ESA AD FECT y 13 (0—b)? fort (x—b)? fla} and (y—a) y! =(a—b)? fr {3(a—b)-? fr + (w—b)? fla} = (fb); and the radius of curvature is therefore finite ; it is in fact the second | * The value of 2 in gx: ()2)» can often be most easily calculated by finding the Walue of log ¢x: log x (page 322), 374 DIFFERENTIAL AND INTEGRAL CALCULUS. divided by the first, or —$( fb). This may easily be verified by common methods. No complete and general method has ever been given of treating those points of a curve at which y” and the succeeding diff. co. are infinite. I think a reason for this may be seen in the infinity of cases which must be considered, when all the possible dimensions of a function (page 324) are taken into account. We cannot evade investigating, in one manner or another, the order of infinitely small or great quantities to which the several differential coefficients belong; and this must be done by the consideration of their dimensions, the possible cases of which are not only infinite in number, but of an infinite number of different forms. No methods yet employed are competent to distinguish, for instance, between the singular points existing at =6 in the two curves y=(#—b) {log (x—b)}* and y=(x—b) log (a—b) {log log (v—b)}°. The deve- lopment of a function, when Taylor’s theorem does not apply, and the assignment of the character of the singular points of a curve, are the same problems; and if a method should be found which should be equivalent to trying how the diff. co. increase or decrease in comparison with every pussible case of x””*--, meaning x* (log x)* (log log r)*...., it would only serve to show how to interpolate as infinite a variety of new cases between each. Defining singularity at the point whose abscissa is a to consist in Taylor’s theorem not applying to develope @(a@+h), which is un- doubtedly the proper algebraical definition, we must divide singular points into those which exhibit perceptible differences from other points, and those which do not. The former are only those in which the singularity affects the first or second differential coefficient. A volume might be written on the infinite varieties of the forms of curves; it will here be sufficient to dwell on the peculiarities and uses of differential co- efficients with respect to them, remembering that the utility of the investigation depends more on the illustration which the curves give to the equations than on that which the equations give to the curves. Were it not for this nothing could be more serious trifling than the length at which, in many works, the courses of different lines are traced out, those lines being not of any use in application. But, when it is con- sidered that the curve whose equation is y=@z, is a lucid tabulation of all the changes of magnitude which $x undergoes when @ changes, it becomes evident, that under the semblance of investigating the course of the curve, we are not only making an inquiry of the most instructive algebraical kind, but also presenting the result of that inquiry in the most perspicuous form. The inquiry before us* will embrace the determination with respect to a curve of, 1. The most useful transformation, if any, of its equa- tion. 2. The points in which it cuts the axes, and the general character of the ordinates as to positive and negative. 3. The greatest and least ordinates, and the general character of the ordinate as to increase or decrease. 4. Its final tendency as 2 increases without limit positively or negatively, and the position of its asymptotes, if any. 5. The character of its curvature with respect to its axis, and its points of con- * The student will remember that he is supposed to have a good acquaintance with the purely algebraical branch of the inquiry, as set forth in the treatise on Algebraic Geometry. : APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 375 trary flexure. 6. Its abrupt terminations, or points d’arrét, as some late French writers have called them. 17. Its cusps, or points de rebroussement. 8. Its multiple points, whether of contact or inter- section. 9. Its conjugate points, or evanescent ovals. 10. Its pointed branches, or branches pointillées, &c. We shall take these questions in order. 1. As to the transformation of the equation. In some cases polar coordinates may be more convenient than rectangular. Thus, as to the spiral of Archimedes, r==a0 is more easily used than Mae +y*)= atan™'(y:@), and the curve (a?+7*)=a (2’—y’) is more easily traced by its polar equation 7*==a? cos 26. But here it must be observed that unless the proper signification be given to negative values of r (page 342), the polar equation will frequently not yield all the branches which would be given by the usual consideration of the rectangular equation. | Again, it may sometimes be convenient to consider the points of the curve as formed by the intersections of two others; thus y=Xr+4X, where Xis a function of a and y, may be considered as made out of the in- tersections of y=axz-+a, and a=X. If then the curve be drawn to which the first line is always a tangent, the intersections of the tangent of such a curve at any point with the curve w=X are points of the required curve. _ Next, when the curve has the form y=¢r+wzx, the most simple plan may be to describe separately the curves y=¢z and y=wa, and orm the required curve by the addition or subtraction of the ordinates. Thus y= (ar)+,/ (a@’—«°) is much more easily described by idding and subtracting the ordinates of the circle y=,/ (a*—2*) to amd from those of the parabola y=,/ (ar) than by attempting the com- plete equation. _ The same method may be sometimes advantageously applied to the ‘orm y=¢xr xX Wa, and often to that of y= (dx). Thus, by tracing /=(«#—1)(a—2)(a—3), we may easily trace Y=,/ (y). | But one of the most useful transformations is that of writing 1: y for /, giving a curve whose ordinates are the reciprocals of the ordinates of he given curve. Nothing is more easy, with a little practice, than to race out the general form of a curve, when the curve is given whose dinates are its reciprocals. _ 2. The points in which the curve cuts the axis of v or y are deter- mined by common algebra. The following observation may occasionally ve useful. If y=dr, =O when =a and when x=), and b>a, then he intervening branch of the curve, immediately following 2=a, has a vositive or negative ordinate, according as $’a is positive or negative ; nd that immediately preceding x=), has a positive or negative ordinate, ccording as ¢/b is negative or positive. | 3. On the method of ascertaining increase and decrease nothing more teed be said, nor on that of determining the maxima and minima. | There is no mode of discussing the property of the tangent in all ases (those for instance in which ¢ (@+A) contains an infinite number f positive and negative powers) unless we have recourse to a universal heory of dimensions. We shall now only consider the primary dimen- ‘ion of each of the diff. co. with respect to x, or the critical values of 7 in ess tf: 2", &c. . Let y= rx be the equation of the curve, the origin being removed to 376 DIFFERENTIAL AND INTEGRAL CALCULUS. the point under consideration, so that (O=0. Hence the critical values of nin y: 2”, y': a”, &c. are the limits (when 2=0) of dr pa Q=2 ha’ Qs az &e. Let the limits of Q, Q,, &c.' be g, gq, &e. Then q=q—1, m=9,—1, &e. This may first be shown when 2¢’2 diminishes without limit, and Q therefore approaches the form 0:0: for then we know (page 320) that Petagp'sz x rp! x qx Bre — —~, or 1+——, have the same limits. pv’ BD voy ae pa’ : But if ad’v should approach a finite limit, or be infinite, then da must increase without limit, and also Q, whence x: 2 (qx) ' approaches the form 0:0, and / v7 a has the same limit as 1 ‘(1 “ae or | (1% : whence Q has the same limit as Q:(Q—Q,). But as Q increases without limit, so must Q,, for in any other case the limit of the second would be unity. Hence the above equations are universally true. Let q be found, and let y=x' wa, then the limit of af’x: wr=R is readily found =0, and 7=qe*'wet+atw/r—at we {qtR}. But the critical value of m in Wwa:a" bring =0, wxr:2'~? takes the limit of x°-“-, or of 21-1; consequently the tangent is the axis of @ or the axis of y, according as gis >lor <1. Butif qg=0 or = @, 2” is not an adequate dzmetient of px, and (log x)” or e™ must be tried, if dx be sufficiently complicated to require it: the number of cases being infinite. If q=1, 7/ depends on yx, when c=0. Again, y’= a *we{qq—1+2qR+RR,}, R, being aw’: wa, which =—1 when w=0. Hence the sign of y’, near the origin, depends on that of g(q—1) 2** wa, and its magnitude at the origin upon 2?-*, except only when g=0, 1, or o, in the first and third cases of which other dimetients must be tried, and in the second of which awe R(2+R,)=y", the limit of which is that of at *wa.R, or at wn, or Ux, When g=2, 7! depends on wer. {1+a"™* a)+R)} 3 abe (q q—1)+2gR+RR) Ifq be greater than 2, this is infinite when c=0; if g=2, it is 0, finite, or ©, with (wr)~’; if q lie between 1 and 2 itis =0. If q=l, the radius of curvature depends upon the limit of {1+ (we)*(1 +R)*}2 ms ywaR(2+R,). This, if Wx have a finite limit, is 0, finite, or oc, with w:R or wr:y/x; if we diminish without limit, it depends on the limit of a: yx.R, or 1:u/x: but if wa increase without limit, it depends on (Yx)*:¥/z, When q<1, but not =0, the expression is 0, finite, or o, with a?’ (ya)*; that is, with 2*!—', in every case in which 2g—1 is finite, and with war, when 2g—1=0. 4. If, when 2 increases without limit, dx have the limit a, there is an asymptotic straight line parallel to the axis of a, and at the distance 4. But if y= cc when «=a, then the line parallel to the axis of y at the and The radius of curvature is APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 377 | distance a is itself an asymptote. The oblique asymptotes are readily | found: for with regard to any one of these it is obvious that if x increase without limit, the tangent perpetually approaches to the asymptote both in direction and position, so that the asymptote may be regarded asa _ tangent whose point of contact is at an infinite distance. Find then the | values of OT and OU (page 352), or of r—y:y' and y—zy'’, when _ x=, and the position of the asymptote or asymptotes will be thus | determimed. And if G and H be the points in which the normal cuts the axes, then OG=a+yy', OH=y-+«:y/, from which it may be _ found whether the normal drawn from a point at an infinite distance cuts the axes at finite distances ; and this may be proved to be impossible, which [I leave to the student with these two hints, 1. The preceding | expressions are halves of the diff. co. of 7* or 2’+y? with respect to @ | andy. 2. Any function in which the diff. co. has the limit a must be _ of the form ar+ Wa, where y/x diminishes without limit, or uw! o=0. All the curves which are asymptotic to ya are contained in the equation y=r-+ Wx, where yz may be any function such that Uv o=0 _ Cimit). A curve having the polar equation r=¢0, has an asymptotic | circle if 6 o=a, the radius of the circle being a. Generally speaking, the curve has two branches which approach the | asymptote, but it may have more even on the same side. Thus the axis of y is an asymptote to two distinct branches of the curve y (a+ v)—=a, _ and to four distinct branches of y (e+at)=a, A positive method of _ ascertaining how many branches of a curve belong to one asymptote is | as follows. Change the coordinates in such a way that the asymptote | may be the new axis of y: for y write 1: y, then for every branch of the curve which has the equation so altered, and which passes through the origin, there is a pair of branches to the asymptote; the two branches | which meet at a cusp (if two they are to be called) counting as one. It | will presently be shown how to determine the number of branches passing through the origin. 5. The general character of the curvature with respect to the axis, and pages 370, 371. the points of contrary flexure are discussed, for elementary purposes, in Generally speaking, the radius of curvature is infinite at a point of contrary flexure, and this is true when the tangent has a contact of the second order with the curve. But all our notions as to contact have as yet been founded upon the supposition that we are at a point of the curve at which ¢ (@+A) admits of development in whole powers of A (page 349). The following considerations are supple- mentary. When two curves have a contact of the mth order, the deflection is always finite when compared with h"*". But at a point for which $ (#-+-h) can be expanded into the series gxr+Ah*+Bhi+.. let us remove the origin of coordinates to that point; then x takes place 209 of h, and we have y=Aw*+Ba°+.... If, then, we take a straight line y=pz, (a, B, y,; &c. being increasing,) the deflection Aa*+.... —pzx will bear a finite ratio to vif @ >1, to #* if «<1, and if a1, to #°, bymaking A=p. In the second case, no line can be drawn between the axis of y and the curve, nor in the third case between that of x and _ the curve. If @ be a fraction which in its lowest terms has an odd _ denominator, there is certainly a point of contrary flexure if y be possible on both sides of the origin. The radius of curvature may be either 0 or cc at a point of contrary 378 DIFFERENTIAL AND INTEGRAL CALCULUS. flexure, but can never be finite. For (1 ty") : y", the numerator being positive in all cases, must change sign with 4”. 6. The abrupt termination, or point darrét, is in part a consequence of the imperfection of the theory of logarithms, as we shall see when we come to the tenth point. If y=2z* logs, it is certain that y diminishes without limit with v, and also that, according to the common theory of logarithms, y has only one value for one value of x, and no value when 2 is negative. ‘There is then an abrupt termination (or commencement) to the curve at the origin; just as there is to the spiral of Archimedes, if the negative values of the radius vector be not admitted. But, as we shall see, the abrupt termination is only the commencement of a pointed branch. 7. The cusp is a singular point which cannot be detected by any simple rule depending upon the differential calculus only. The follow- ing considerations are necessary for the elucidation of this case. Let we be a function which vanishes when r=a, and is impossible on one side of that value, having on the other side two equal values of contrary signs. Then w/a is either 0 or oc. For it is evident that the two values of Wx answering to the two values of @ differ in sign, and when the two values of ¥v coincide in one (0), either the two values of u'x must have the same limit, or ¥’a must have two values. But the last cannot be, if the function be continuous, and quantities of different signs approaching the same limit can only have the limits 0 or c. Let the preceding remain, and let y=@r+ Wz be the equation of a curve; this curve has, then, unless dr should destroy wa, no ordinates when r. Here is a cusp when r=0. And it will be found that the cusp is of similar curvatures. eeu Be eke Let y=2?+4.24, There is no cusp in this curve, the diametral curve of which is the parabola yn. But since a is greater than 2? when | x is less than unity, the two branches belonging to B the same branch of the parabola are on different sides of the axis until x=1, after which the con- trary takes place. The figure of the curve is as follows, BOAC being made from one branch of the Esa C parabola, and DOAG from the other. The appa- 0 rent cusps made by BOAG and DOAC are not | Ke ZA really cusps. © Let y=a2}+2%. There is now really a cusp at the origin, and the whole curve has the form of 1 . BOAG. If y=(2}+27) loga, there is a cusp at ~D the origin, and the curve has the form made by putting together OAC and the dotted branch. 8. Multiple points are those in which two or more branches of the ‘curve pass through the same point; according to the number of branches 'they are denominated double, triple, &c. In the case of a simple ‘double point, it is obvious that the diametral curve will pass through it, either touching or cutting both branches of the curve according as they itouch or cut one another. When the two branches touch, the only difference between the case and that of a cusp lies in the ordinate not - 7 380 DIFFERENTIAL AND INTEGRAL CALCULUS. becoming impossible before or after the cusp. Thus, in the curve 1 1 . . p 1 y= («?+.27) log z, the diametral curve has for its equation y= 4° log a, ie eve San Ra r : and the curve coincides with its diameter when a? loga=0; that 1s, when w==0 and when t=1. In the first case, the ordinate being impossible when x is negative, we have a cusp: in the second, a double , ae eros point, the values of y’ being 11. Similarly, in y=2?+2*, one diameter of which is y==0, we have coincidence with this diameter yes when +2%+24=0, or whenz=0orl. Inthe second case, y/=+$+4, $e Pag fa giving for the branches belonging to the ordinates -+2?—a* and —a? Lat, the values + and —1, which determine the tangents at the double point. The general test of a multiple point is a multiplicity of values in y/ for a single value of x and y. But if some of these values should be equal, that is, if some of the branches have a common tangent, it 1s not every test which will demonstrate the existence of these equal multiple values of y/. Theoretically speaking, the branches having then a con- tact of the first order, recourse should be had to the second diff. co. y”, which, unless some two or more branches have a contact of the second order, will have as many different values as there are branches. Pro- ceeding in this way, we see that if two branches have a contact of the mth order at most, the (1+1)th diff. co. of y is the first which will exhibit as many values as there are branches. Hence no absolute test of multiple points can be derived from the differential calculus, since the examination of all successive diff. co. isimpossible. Generally speaking, however, the equation itself will point out how many values of y may belong to one value of 2; and it is obvious that no more branches of a curve can pass through a point than there are values of y to a value of x closely preceding or following the multiple point: so that practically speaking the multiple point is detected with nearly as much ease as the point of contrary flexure. The most certain theoretical method of determining a multiple point, though not perfect and though rarely the best in practice, has been obtained in page 183. Let @ (x,y)=a be the equation of the curve, and let it be reduced to a form in which there is no ambiguity, by the destruction of all terms which have double values. Thus y=a2+ a? must be reduced as follows : (y—r)’?—x=0. Differentiate, say three times with respect to x, using Lagrange’s symbols throughout : P+h,y'=0, GO +26] y' +h, y2+9,y"=0 gl" +30," y' +3¢,/ y?+ Pi y+ (o/+¢, y’) of! +d, shad ‘ Now since #(a,y) is unambiguous,* so are $’ and 9, when finite 5 consequently there can be no double value of y’ unless when it takes 0 oc ; A the form Hy fee that is, when either ’=0, ¢,=0, or f’= a, ¢,= &. The second case can generally be avoided by a modification of the * This means, having but one value for one value of 2 combined with one value of ¥. APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 381 equation, and when ¢’=0, ¢,=0, then if @", d,, and @,, be finite, we have O" + 2b/ y' +h, y2=0 for the determination of the two values of y’. This answers well enough when @,, is finite, but when ¢,,=0, the common theory of algebra would instruct us to suppose one value of y' infinite; if, however, this be the case, the corresponding value of y” is infinite, and we have no longer any right to conclude that the term ¢,7” vanishes. We are only there- fore made perfectly certain, by the use of this method, that a double point exists when y’ is found to have two finite or zero values, Similarly, if ~", g/, and #, all vanish, we have the equation p+ 3o/' y! + 3p,/ yf? an hi ay Pe 0 for the determination of the three values of y! which may in this case exist, with the same reservation as before; and soon. And in any case one or more pairs of the values of y’ may be impossible. Let us take the curve ya at, already considered. An equation of this form can only* be reduced to another which perfectly includes all its cases, and is rational, by multiplying together all its forms. Thus the preceding must be rationalized by multiplying together (k=,/(—1)). y—Na-Ya, ytfo-Ya, yr-fetYx, ytiet+i2, y—Ja—kfe, ytsa—kfa, y—Jfatk2, ytifot hala, and equating the result to nothing. But if the possible factors only be multiplied together, and equated to 0, giving (y+2—Jx)’—4ry*=0, every possible branch of the curve is included by making this =0, and the resulting equation may, consistently with representing the whole curve, be made unambiguous by the understanding that /x shall have the positive value only. Pursuing this, we find for the first equation, ‘Sioa Grom "+ {4y (y?+a—,/x)—Saxy} y/=0. : 0 : ¢ In this y’ takes the form 5 when #==1, y==0, which is also found to satisfy the equation: here then there may be a double point. ‘To settle this, form the next equation, or Pe Le letite Cnepty 2 (y+ 2 — fx) i= — + 2 (1 Seiten ils a No Linlas U QJ ax cf v( ar) rey Y 1 12y* + 42—4,/4—82} y!? + {4y (y?+a—,/2)—82y} y"=0, when z=1, y=0, 4—8y"=0, and y’=+1 or —1. There is then a double point at (1,0). This method also indicates the double point which exists at (0,0), and for which both values of y’ are infinite. a I give the following as an exercise :—The curve y= (v—a) (b—2)? * Or by some process as general. The student might easily deduce (y?+-a)?— x (1+-2y)* from the equation; but he would find, on endeavouring to return to y=1,/rt4 zx, that the preceding is only satisfied by Y=/tt4/2, and not by y=—Jeri/e. 382 DIFFERENTIAL AND INTEGRAL CALCULUS. 4.2 (6—a)* has a double point when w==a, if ba. If abe made to vary, the curve to which every curve of the species is a tangent passes through all the double points. 9. I call the conjugate point an evanescent oval, because it never exists except where the equation is a degenerate variety of a wider class, each curve of which has an oval. The most simple case is that of (x—a)?+(y—6)’=0, which belongs to no point except (a,b). This conjugate point is the circle described with a radius =0, or an evanes- cent circle. Again, y=<=t,/ {#(#—a)(#—b)}, a and b being positive, and b>a, consists of an oval from =0 to =a, an unoccupied interval from «=a to x=), and infinite branches above and below the axis from x—=b upwards. As a diminishes, the oval becomes smaller, and finally when a=O the form of the equation becomes y=a,/ (v—b), which gives y=0 when x=0, or the origin is a point of the curve: but there is no further point until z=). It is useless to attempta test of a conjugate point by the differential calculus. 10. I now come to the consideration of the pointed branches, or branches potintilleés. ‘This is a curious question of analysis, in which some detail will become necessary, and strict recourse to definitions. If we define a curve to be the line made by the motion of a point according to a certain law, it is evident that if (a, 6) and (a’, b’) be two points at a little distance on the same branch of the curve, there is a point of the curve for every abscissa lying between a and a’. And such a branch of the curve, described by a continuous motion, is the only branch which falls within the definition. But if we define a curve to be the assemblage of all points whose coordinates satisfy a given equation, we no longer restrict ourselves to the consideration of branches described by the continuous motion of a point: for there may be points, the coordinates of each of which satisfy the equation, without any such points intervening. The simple conjugate point is an instance. Con- sider the curve whose equation is y= aa*+,/x.sinbr. The diametral curve is a parabola, from which, when z is positive, the curve alternately recedes and approaches, meeting it whenever sin br=0, or bx is a multiple of 7. But when @ is negative, y is impossible, except when sin br=0, in which case y is ax®: so that on the negative side there is an infinite number of conjugate points, each one situated on the parabola over against a double point of the curve, the successive abscissee being 7:6, 24:6, 3r:6, &c. The greater the value of 8, the more nearly do these points approach ; and if 6 were exceedingly great, they might be made, as nearly as we please, to resemble a continuous branch of the curye, Which of these two definitions we employ is purely a question of APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 383 analogy and convenience. If we were making a theory of curves, for the sake of the properties of space we should thereby gain, it might perhaps be thought that the first definition would best embody the objects under consideration. But if our theory of curves be carried to a greater extent than is practically necessary, solely for the clearness of illustration which it gives to the properties of algebraical functions, it then seems to me that the second definition is imperatively required. All who have sanctioned the introduction of the simple conjugate point have tacitly admitted this; though those among them who have rejected the pointed branch have refused to admit the legitimate consequences of their own definition. In the preceding example we have only a series of conjugate points, separated by finite intervals. If we admitted the symbol sin ( & 2) among the objects of analysis, we might appear to have a pointed branch which is not distinguishable from a continuous branch. If we never met with such a branch except upon the introduction of a new use of ec, we might well dispense with it altogether. But, as we shall now show, a pointed branch of a still more curious character meets us in the con- sideration of ordinary symbols of quantity. The expression a*, where 1 t—m:n, means that any one of the n values of a” is raised to the mth power. When we speak of arithmetical values only, we have the equation Ym 1 (a) (a) 5: and in all cases this equation is so far true, that each of the values of lym 1 lan) is one of the values of (a”)"; but the second may have values which the first has not, or may appear to have them. ‘Thus if 1‘=1, an indisputable arithmetical truth, we shall find —1 among the values 1 1 ha’ By; of 1°, or (1*)®; but it is not among the values of G*) . And since £ = ; : 5 1° and 1% are identical, and the second has only three values, the first must not have more; whence, if we allow ourselves to call 14 and 1 iden- tical, we may fall into error unless we remember that a® must stand for 1\m any value of es, , but only for (it may be) some of the values of 1 (a")". The safe method is, always to reduce the fraction m: 7 to its 1\m lowest terms, and then the m distinct values of Gea are severally equal 1 to the m distinct values of (a). A wider and better theory might be drawn from the general considerations of Chapter VII.; but the above will be sufficient for our present purpose. Between any two fractions, however near, may be interposed an infinite number of other fractions, (in their lowest terms,) either with even denominators or with odd denominators. Let y=a’ ; then when z is a fraction with an even denominator (being in its lowest terms) there are two possible values of y, numerically equal, but of different signs. But when 2 has an odd denominator, there is only one such value. Consequently, since fractions with even denominators may be made as nearly equal as we please, we have on the negative side of the ordinates a branch in all respects similar to that on the positive side, with this restriction, that we are not to be allowed to go upon it for 384 DIFFERENTIAL AND INTEGRAL CALCULUS. an ordinate, except when x is a commensurable fraction (an its lowest terms) with an even denominator.. Between any two points on it an in- finite number of allowable points can be found; and yet the branch is not traced out by the motion of a point, since between any two points an infinite number of unallowable points can be found. Similarly, a negative quantity must be allowed a possible logarithm, whenever the number, independent of its sign, is of the form ¢*, where x is a fraction with an even denominator.. Thus y=logz represents a curve which has a pointed branch, one point of which is found as follows. Let r=—,/e, then y==3. The abrupt termination, observable in the curve y=2 log z, and in many others, but all containing exponential or logarithmic functions, now appears* merely as the point in which a continuous branch meets a pointed branch. The general rule is; trace the curve on the suppo- sition that log (—#)=—logz, using the branch which arises from loga- rithms of negative quantities only when the negative quantity is of the form —™%/é?*", If we return to page 127, we find in the equation log (—x)=log er +(2m41)cV —1 no indication whatever of a possible logarithm of —« in any case. A further extension of the theory of logarithms must be now madet as follows. To find all the values of ¢, possible and impossible, we must put ¢ in the form ¢xe*"V@, in the same manner as, in page 127, the roots of unity were extracted by writing 1 in the form eV, If, then, we want to solve the first of the following equations in the most general manner, we must have recourse to the second (in which 7 is even or odd, according as 2 is positive or negative, e* being the numerical value of z). ez; { glt2mny (-1) Jem gttneVaT ar att] (—1) ~ 142mr,J/(—1) Now a is by definition the logarithm of z, and the preceding is the most general form of that logarithm, a being the ordinary alge- braical logarithm. If, then, a=p:q, p and q being whole numbers, we have r= {p+qnr)(—1)}: {9+ 2qmr/(—1)}; which is possible and equal to p:q, when p:q=n:2m. Now when 2 is an odd number, or z is negative, this equation can be always satisfied if g (p:q being in its lowest terms) be an even number. That is, one of the logarithms of or —7 sceP 3 | —7 > /s? is possible and =p: 4q, the same as appears from the common algebraical consideration of Yer * Those who object to the pointed branch as introducing discontinuity must choose between its discontinuity and that of an abrupt termination. It is also worthy of note that an asymptote which has an odd number of branches only approaching to it, is an abrupt termination. Such an asymptote can never occur, if pointed branches be admitted, and if, when polar coordinates are employed, the negative values of the radius vector be duly considered. _t See for the history of this question the article “ Negative and Impossible Quan- tities’ in the Penny Cyclopedia. APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 385 A great many curious modifications of the singular points of curves might be noticed, but they would require more space than I have here to give. I now proceed to some further uses of the equations in page 345. The area of a curve contained between the ordinates oa and $b, the interval of abscissze '—a, and the arc intercepted between the ordinates, is f gx dx, from x=a to v=b. (page 142). Let us now suppose it is required to find the area intercepted between two radii and the arc of the curve which these radii intercept; as BOA. Drawing a figure, in which the ordinate and abscissa shall increase together, such as the one W B V O M N annexed, it may be easily shown that AOB is half the excess of BWVA over BAMN. For we have BWVA=BWO+BOA—OVA BAMN =BON —BOA —AOM. Subtract, remembering that BWO=BON, OVA=AOM, and the pro- position asserted is evident. Now, if OM=a, ON=s, AM=d¢da, NB=¢8, we have BWVA= fx dy, from y= 9a to y=¢b, or fxd'xda from z=a to r=b: and MABN=/y dz from c=a to x=b. Con- sequently BOA=3 / (ady —ydz)=} f r*d0, (page 345, equation 11); in which the limits of @ in the last integral are from ZAOM to ZBOM. The student should now prove that the equation BOA=3 f (dy —ydz) ilways holds, if the signs of the integrals be attended to, whatever may be the disposition of the parts of the figure. This proposition may also be proved independently, as follows. If 0 vary by A@, the area con- ained between r and r+Ar lies between two sectors of circles whose reas are §7°AO and $(r+Ar)?A0. Consequently, proceeding as in sage 100, the whole area between any two limiting values of @ lies etween 4 Fr°AO and 4 27*A0+ 37 Ar AO+4 5 (Ar)? AO. But as Ao liminishes without limit, each of the elements of the second and third nentioned sums diminishes without limit as compared with the cor- sponding element of the first. The two preceding expressions have, herefore, the same limit, and the area of the curve, which ‘always lies etween them, has the same limit: this limit is, by definition, 4 [7°do. We have, then, the following four integrals, expressive of the rectan- ular area, the polar area, the arc derived from rectangular coordinates, nd the same derived from polar coordinates. Let 21, 1, r,, and @, be he coordinates of the point from which the area and arc begin, w, being 7» and x being r-', 2C 386 DIFFERENTIAL AND INTEGRAL CALCULUS. Page 142, rectangular area A= fi ydx beginning from 2=2; polar area® SH=} frido PAG a: | Page 140, arc (rectang. coord.) s= $A J (da?+dy?) . 2. + + . CP Page 345, arc (polar coord.) s=f,/(dr?+7r°d6*). . . - + + O=0 | = fut (dut+u'd).. + + » Ob We have also the following equations: H=/f (ady—ydx), A=} (ey—“1y)—$H. If either of the coordinates be expressed in terms of A or H, the othe may be sometimes expressed by simple differentiation. Thus : dA if x=wWA gives l=y/A,—- =wWA.y, or Essar If, then, A be eliminated between «=A and y= (y/A)~, we have at equation between wv and y, which is that of the curve. | But it is important to remember that though A or /yda can certainh, be found from a= (_fydx), it will generally happen that it is on}} one constant which can be appended to that integral; for it is manifesth not to be supposed that the equation r=¥ (, [ydz+C) can be mad: true for all values of C. It may easily be shown that this is a questial of a class we have not hitherto met with, involving an arbitrary constan which enters in a function in a manner depending on the form of th) function itself. 'To make the problem specific, we must suppose that th area measured from a given initial abscissa shall be a given function of th terminal abscissa. But (page 142) the equation , f,ydr= ex is incon gruous, and f,,yda=yar—wWa, is rational. If, then, we propose | Ts ydz= Yu —Yu,, or T= Ye { f,, ydo+wa,}, we have an equation in which the arbitrary constant enters in th) manner above described. It is required to find the curve in which a=log A, Here pA=log1) and y, or (W/A)~“=A; whence w=logy or y==e”. The area Syda } then e’-+C, © depending on the point from which it begins ; and i) order to satisfy the conditions we must have C=O, or the area begin from a point at an infinite distance on the negative side. In fact, th primitive equation A=¢’ is only intelligible as represénting the area of curve when written in the form A=e*—e-%. Difficulties of this sort will occur whenever z or y is given in terms ¢ a function which is necessarily dependent on an integral containing # 6 y itself. There is a large class of problems relating to curves in which such | property of the curve is given as implies a determinable differenti¢ equation. The solution of this differential equation, ordinary or singulai is therefore an equation of the curve: whence we see that two ver different curves may have the property in common, one being a Case ¢ the general solution, and the other being the singular solution. For example, it is required to find the curve in which the length the normal intercepted between the curve and the axis of w is a give * Certain usages of writers on mechanics make it more convenient to adopt | symbol H for twice the polar area, than for the polar area itself. APPLICATION TO GEOMETRY OF TWO DIMENSIONS. 387 function of the part cut from the axis of « by the normal: or which satisfies the equation This equation can be integrated generally; differentiate both sides, and we have pits WOE Tae Lt} Cty" tyy") 5 or ty! J +y"). 6! (otyy)} {1+y?-+yy"t=0. One of the factors of the last must then vanish. If 1+y'"+yy"=0, we have, by simple integration, (w—c)?-++ y’= ci, Which will be found to satisfy the equation (4), provided ci=(¢c)*; whence the general integral of (#) is the equation of a circle, namely, (#—c)?-++ y= (de)* ; so that there now remains only the vanishing of the factor y'— J+ y’*) $'(x+ yy’) to be explained. This it may be shown is satisfied by the singular solution of («#-—c)’+y?= (¢c)?. For, by page 192, that singular solution must make dy’: dx and dy’: dy infinite, these being partial diff. co. derived from y/ as expressed by the equation itself. If, ‘then, we differentiate yJ(1-+-y”) = (e+yy'), considering y! as a function of x only, we have yh tidyh owt i ov LTT de? ty yl ty ses dy! i p! (c+yy’) V1 shag Din dz y y'—d +9") dl ety’) Consequently y'—,/ (1 -y) .d! (w+ yy’) vanishes* when for y is put that value of x which is the singular solution of (#). The following theorems may be investigated by the advanced student as Exercises. 1. The equation which expresses that the radius of curvature is a given function of y’ may be integrated (assuming the integration of all functions of one variable) so as to give both « and y in terms of 7’: Whence the equation of the curve may be found by elimination. 2. A polar equation to the locus of the intersection of the tangent of A given curve with the perpendicular on the tangent may be found from equation 27, page 346, by substituting for r its value in terms of p, and mtegrating. we), 3. The method in page 355 may be applied to the determination of ptical caustics, both by reflection and refraction. or * The method of Clairaut in the integration of y—y’x=¢@y" might, therefore, be generalized, subject to close examination of the different cases, as follows. Let (a, y, y', y'’,.++-)=0, whence it follows that dp ; do dz * dy ; " d 3 - f each of the coefficients -. &c, have acommon factor M, the equation resulting from dx fs extermination (of one order higher than the given equation) may sometimes /@ more easily integrated than the original. If so, an equation between its con- tants may be obtained which shall make it satisfy the original equation, and the ‘ngular solution of this general solution satisfies M=0. An 2C2 388 DIFFERENTIAL AND INTEGRAL CALCULUS. 4. Trace the curves whose equations are y=log sin 2, y==sin log z, distinguishing both continuous and pointed branches. Show that the | logarithmic spiral has a pointed branch, and trace completely the curve whose polar equation is r=a+t,/(cos 0), a =—2,=4 2 ACs —=T, dx pa | =S, yy ° a If there be three independent variables, x, y, and 2z, it is very’ desirable to have a notation for use in the actual details of operation, to be taken up when they begin and laid down when they cease. The follow- ing will be perfectly distinct, and soon acquired. Let w bea function of x, y, and 2. du du du deu aru dent dy de at ay du du au du aa dady fine dy de? dzedx™ In making any integration with respect to one variable only, it musi be remembered that the constant to be added may be a function of the other, which though called variable with reference to what might have taken place, was by supposition a constant in the differentiation whick the required integration is to compensate. Thus 04 du : 1 1 TES amit) gives Ts = ary + dy, us ax*y + hy.a-- wy,’ | where dy and Wy are any functions of y whatsoever. Again 2 eS ‘ du 1 2 1 a abaya) Oras Tat Cees les ce ae where f¢rdx may be any function of 2, and wy any function of ¥ Such cases, in which no peculiar specification of Jimits is made, require no additional consideration; but if it should happen that the limits oi the first integration contain functions of the letter which will be ¢ variable in the second integration, the question takes a very differen’ character. For example, u,/=azy is to be integrated first with respec to y, and from y= to yx’, and then with respect to 7 from r=0 x=b, The first integration now gives | APPLICATION TO: GEOMETRY OF THREE DIMENSIONS. 389 4 ax (2°)?+o2—(har. x+x), or ta (25—2°), This integrated with respect tor from r==0 to r— b, gives ka (1 b5—1 4), The question now becomes, what is the use and meaning of the opera- tion we have performed? It has sufficiently appeared in Chapters VI. and VIII., that though we may look to the determination of a primitive function for the shortest mode of operation, we must find in the limit of a summation the readiest mode of conception of the result attained. Now the first process is really the limit of the following summation : jar.atax(r+0)+.... 4427 (x-++-m6)! 0, where mO=2*—wx. If we now assume nk=b—0, and add together the several values of the preceding answering to x=0, T=Kk,.... Up to =n, multiplying each by x, we shali have a succession of sums, the first, second, and last of which are as follows, if the value of 9 when t=ck be called 8,, {a0.0+a0(0+6,) +....+a0 (0+ 6,)} O).« + fax.xctak (e+6,)+.... +a (x-+m0,)} 0, .« +{ank.nk-+tank (nk-+6,) +....-ank (nx-+m0,)} 6.5 the limit of which, when m and 7 increase without limit, is the result obtained. And since every term is of the form ary Ax Ay, we may, as in page 99, call the preceding DAx (2ary Ay) or Laxy Ax Ay, and its limit [dx faxy dy, faxy dxdy, or faxy dx dy, if the two operations are to be represented. And since y is first made variable, we may denote this by writing dy last of the two, and the symbol of the in- tegral with the limits represented will stand thus: » Lif 2 axy dx dy. We may now give a geometrical illustration of the preceding, gene- ralizing the operation into /? /%*zdxdy, where z is a given function <) Hie = a of z and y. ' Draw the curves y=@x and y= Wa, and set off the abscisse a and b, OA and OB. Divide the interval AB into any number of equal parts m, and having drawn ordinates, divide the part of each ordinate intercepted between the curves into n equal parts. There will then be mz rectangles, which, as m and n are increased Without limit, have for the limit of their sum the area PQRS. This |» i © > is 390 DIFFERENTIAL AND INTEGRAL CALCULUS. limit, compared with the preceding process of summation, will be found to be represented by ye Pp ah ve dady. And this agrees with previous results; for writing the preceding in the manner first pointed out, we have /} dx f¥F dy, or S°. Qha—ge) dx, or Js waedx—fipxdx, or AQRB—APSB. But if we.want to form an idea of the meaning of ‘(fz dx dy, we may proceed in either of the following ways. | 1. Suppose the area PQRS to be everywhere of different and variable value per square unit, in such manner that at the point (x,y) the value of a square unit, if it were uniform, would be z. Then at the point (x,y), the sides of the adjacent rectangle being Aa and Ay, the value ol that rectangle is, not zAvAy, but (z+ ea (q3—qi) or 9 (pot pi) (4 bos qi)» which is precisely the area that would be obtained by taking the arc of the curve to be a straight line. The errors of this supposition, therefore, are all of the third order, and for our present purpose ABCD may be considered as a quadrilateral rectilinear figure, and even as a parallelo- gram: for, as far as terms of the second, order, by the values found, AP=AM+AN, or NP=AM; similarly, PC=MB+ND, whence BM=QC, and AB is equal and parallel to CD. If NR be jomed, | ABNR is also a parallelogram, and ABCD and CDNR together make | up ABNR=MBRP. But DCNR=DQPN; whence ABCD is the excess of BMPR over DNPQ, or BM.AN—DN.AM, or wi 2 a red Reg Av Au; ce dv du \du« dz or The sign of the result only indicates that the preceding expres- sion without its sign is negative in every disposition of the figure similar to that here adopted.. If we now take the equations y=¥ (a, v), y=¢ (#,u), and from them deduce y and « in terms of v and u, giving x=X, y=Y, X and Y being each a function of v and of uw, we may deduce the preceding factor by implicit differentiation, as follows. Substituting in the first pair the values derived from the second, we have identical equations, and this being implicitly supposed, we have dY _dy dX dY dy dX dy Oi He wide Tpaeae weil: d¥ db dX dp dY__ do dX dc ae ait is ee Ga dp sar AES dy 1 dX: x _we dys db dX dX du' W du? @& Wad” dv du du dv. ——s _ (de db\"_ dX dX: /dY¥ dX dY dX)" dx dx) dv du \dv du du do Ww dis dp _dY¥ dX dY dX dv du’ dv du du dv’ We have, then, for the integral required either of the following. Let z=f(«,y), and neglect the sign which depends on the diagram, and must be determined by each particular case; or rather, in most cases, that sign must be taken which makes the result positive. ' APPLICATION .TO GEOMETRY OF THREE DIMENSIONS. «© 395 bg yi (dp dy\ dye dé zdxdy= 2 x we Rep fee oe Ue ) Lf2dady=fr [ef @y) aay ae aye sh Hp dY dX dY dX Sod of & ¥ (3 du du =) Reais in the first of which there must be, swhsequently to the differentiations, that substitution of X for wand Y for y, which is made previously to differen- tiation in the second. This integral in geometry belongs to any function connected with the area contained, in the plane of wy, between the curves whose ordinates are ax, wx, Px, vx: (x, u) is a function which changes from a¢ to x, when wu changes from a to b, & (2, v) a function which changes from px to vz, when y» changes from m ton; and X and Y are the values of x and y in terms of v and w from y=9(a,w), Y=W (2, v). It is obvious that no part of the preceding investigation involves the limits of integration, except the manner in which > (2, vu) and & (a, v) are to be formed. But whatever these functions may be, if we call the differential last obtained Zdu du, we know that Z Av Au + terms of higher order than the second, is the element of the summation cor- responding to the element ABCD of the area; and though one particular supposition as to and % may require this summation to be made (as above) between limiting values of « and v which do not depend on one another, a second supposition may require that the limits of w shall be functions of v, or vice versd. Thus, if we integraté the preceding from v=Mu to v=Nu, (M and N beimg functional symbols,) and subse- quently from w=a to ub, we require that y=P(a,u) and y= uw («, Mu) should give y=pa by elimination of wu, and that y= (a, w) and y= (xv, Nu) should give y=va. Subsequently, we require that y= (a, a) should be equivalent to yaa, and y=¢ (a,b) to y= Ar. For example, it is required to find the area of a curve contained between two radii 7, and 7,, inclined to the axis of a at angles 0, and 0,,. In this case our bounding curves are y—tan 0,.0, y=tan 0,,.x, for ax and Ba: and y=0 and y=va, the latter being the equation of the curve. If we wish to express this area by means of polar coordmates 7 and 0, we have y=atan0, and y=./ (r?—a*), for @ and y. (6 and.r taking the place of wand v.) ‘These give dY dX. dY¥ dX _. a=r cos0=X, and y=rsind=Y, ae) Bi Cape ae? and f fr dr d@ is the transformation required. Let 7 be first taken as variable, and let M@ and N@ be the limits. The first limit is =O, the second is thus found: y=rtan@ and y= {(N6)?—2*} must give y=ve when @ is eliminated, which is satisfied if r-=N@ be the polar equation of the curve, derived from rsin@= vy (rcos 0). Again, y=xtan6 satisfies the equations at the limits; hence fede rdr. dé, or 4, f% (NO)? dé is the result, which agrees with page 385. But it is impossible, under these suppositions, to allow @ to be the first variable. If y=u ve and y=vz, and the area between the two radii be required, we have for its expression Py. (ur'a—v)~ vvx du dv, from v==tan@, to v=tan@,, and from u=0 to uw=1. In the preceding, the value of « must be substituted from wvxr—va. Let there be a cone, the vertex of which is at the origin, and the base 396 DIFFERENTIAL AND INTEGRAL CALCULUS. of which is parallel to the plane of zy, at the distance a. The equation of the conical surface has the form z=a2f(y:x), where f is such a function that a=x f(y: x) 1s the equation of the base projected upon the plane of zy. Between this base and its projection lies the content of a cylinder, made up of the conical solid, and a ring, wedge-like towards the interior part, the wedge terminating everywhere at the origin. This wedge has for its content / fz dx dy, which integral, according to the manner in which the limits are taken, may represent any part of the wedge. If r and @ be the polar coordinates of a point on the plane of ry, a transformation already given will reduce this integral to S Sof © .r dr db, or f fr-dr.cos 6f tan 0. dé. This may be first integrated with respect to 7, from r=0 to a= r cos 0. f tan 0, or r=a {cos 0. f tan 6}—". This gives} fa® {cos 6 ftan 0} dQ, or 5a.4 fR d6, where R is the value of r at the limit. This gives 3 @xX (area of the base) for the content of the ring; whence the remainder of the cylinder, or +a@~x (area of the base), is the content of the conical solid. Let there be any integral of the form {fo (x:y).dxdy. The pre- ceding transformation is frequently applicable, and simplifies the pro- cess. The integral then becomes Sie tan@.d0.rdr. For instance, a straight line setting out from the axis of x revolyes round the axis of z, in such a manner as to describe the angle at in t seconds, while it also moves up the axis of z, so as to describe /3¢ in ¢ seconds on that axis. Here at and (Gt are functional symbols: but if at=at, Gt=dt, the sur- face is that of a winding staircase (neglecting the irregularities of the steps). Its equation is derived from eliminating ¢ between z=/¢ and y=a.tan at: whence z is a function of y:z. In the simple surface just mentioned, we have z= (6: a).tan7! (y:2). The solid content bounded by the surface, and standing upon any part of the plane of ay is J fzdx dy, taken between limits depending on the form of the base. Making the transformation, we have m ff or do dr, where m=b:a. If we want to find the portion standing upon a circular sector whose radius is cand angle y, we must integrate from r=0 to rc, and from 6=0 to 6=y, which gives ¢mc’ y’ for the content. It will hereafter be shown that if z=@(a,y) be the equation of a surface, that part of the superficial area which stands over a portion of the plane of czy is SIN (1+2"+2,*) dx dy, between limits depending on the form of the base. Ifwe substitute r cos@ and 7 sin@ for x and Ys thus reducing $(2,y) to % (7,6), we may determine z’ and Z)y 28 follows : dz dy dr dy dé dz _ Ys dr dy do des dride 20 de dy de dy‘ d@ dy’ which equations are to be considered as derived by supposing % to contain w and y through 7 and 6, on the supposition that r=,/(2*+y*), O6=tan~' (yz—'). These give dr be . Hee Very) 0, ==sin @ DT. ols Non) dy J(@+y’) APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 397 AE? MY idl so 910.0) dd &——_cos 8 de #+y or dy” #+y% or dz _ dys _ dy sin 0 dz _ dye, , dw cos@ ° | dr d OD ays dy dr do or G2 yz. Oe, La Mee Tee int) dimes cirae if ta To apply this, take the helicotdal surface (helir, a screw) before described, in which z=m9. The integral which determines the surface is then SING +m'r~) rdrd0. This integrated with respect to 7 from r=0 to r=c, and with respect to 0 from 6=0 to 0=y gives the surface required; namely, belonging to the circular sector above-mentioned. ct J(m*+ =| un 7S I (mi+7*) .d0 dr = 57 fol mito) +m! log bet Ss Let the surface be one made by the revolution of a curve about the axis of z. Let the equation of this curve, when in the plane of z and 2, be z=@zx: whence z=¢ (,/(2°+’)) is that of the surface; or z=@r. We have then for the integrals determining the solidity and surface [for.rde dr and S/Vi+@'r)*} rdrd@. If we integrate through a whole revolution with respect to 9, we shall have 27 Sor.rdr and Qn f/{1+(4'r)*} rdr, expressions which we shall afterwards compare with others, which will be obtained for this particular case. If the generating curve be an ellipse, of which the centre is at the origin, and one of the principal diameters in the axis of z, we have, when the generating curve is in the plane of xz (a and b being the semi- diameters) , 2 2 eae ee ibe 7 +—=1, whence =a AV (@—r?r*) is the equation of the surface: and the integrals which determine the content and surface are (b°=a’*(1—e*)) b yuyu oeeG i — SIV (a’—r*) .rdr dé and ale ai rdr dd. Integrate first from 9=0 to 0=2r, and both integrals are then obtain- able from r—=0 to r=c. This gives the content and surface standing over a circle described on the plane of xy with the origin as a centre ; that is, intercepted by a cylinder on the same axis as the solid. The first integral obviously becomes 2 3 2 ces a {a—(a'—ey}}, or — mba’, when c=a. a 3 The latter is the whole content of the semisolid. In the second integral, after integration with respect to 0, for ,/(a’—7*) write (a: 6) z, which gives 2 9 5 2m J.J (a—e (a- < :')) x —— dz, or is f(h+@ e 2) dz. ) The integral of the latter beginning when 7=0 or 26 is 398 DIFFERENTIAL AND INTEGRAL CALCULUS. J bt +a2e z} 7 ab (1+e) ie b? a @° aez +) (b* +a? e 2°) Stopping at z=(b:a),/(a*—c’) or r=c, we have the surface required. If we go on to r=a or z==0, we have for the surface bound- ing the semisolid Ta -+ He log shee wa \a » which becomes 27a? when b=a, e=0. The last. result will immediately appear on expanding the logarithm in powers of e, and making e=0, b=a, after reduction. Doubling the semisolids, and remembering that 47a? is the surface of a sphere whose radius is @, the revolving semidiameter, it appears that the surface of an oblate* spheroid is less than that of a sphere described on the revolving diameter, by b? a(1+e) ( 5? ite Qn { at — log oe Neils fs. atti (a 3 log j ) or 27 a De log i) or 27 a’ e* nearly, when e is small. Let a surface of revolution be described by SS the revolution of a curve about the axis OB, Kj IR and let OA=az, AP=y, arc ending at P=s. Far Then AB, QR, and PQ are Az, Ay, and As. The portion added to the solid by changing x into x-+Az, made by the revolution of APQB, -: TH lies in magnitude between the cylinders gene- rated by ASQB and APRB, or between m (y+ Ay)’ Ax and ry’? Az, which differ by 7 (2y+ Ay) Ay Az, or «Aa, where « and Av diminish without limit together. Hence, proceeding as in page 142, the whole solid always lies between Lary? Ax and Sry? Ax + 2a Ax, of which the second term diminishes without limit as compared with the first. The content of the solid, then, is the limit towards which both of the preceding approach, namely, /7y?dx, taken between the proper limits. To find the surface, it is necessary, as in page 140, to assume an axiom; namely,t that the surfaces generated by the revolu- tion of the are PQ and the chord PQ may be made as nearly equal as we please by diminution of AB. The surface generated. by the chord PQ is the difference of two cones, the radii of whose bases are AP and BQ, and the difference of their slant sides, PQ. If z be the slant side of the former, we have 42.27y or wzy for its surface, and 7 (z+PQ) (y+Ay) for that of the other; whence: 7 (zAy+y.PQ +PQ.Ay) is the surface generated by PQ. But z:PQ: 1y: Ay; whence the preceding becomes a (2y.PQ+Ay.PQ), of which the second term diminishes without limit as compared with the first.. If the preceding, multiplied by 1+-a, give the surface generated by the arc PQ, by the axiom g and Az diminish without limit together, and the whole surface is 2 2ry.As(1+a)+27 Ay As(1+«@). From this the * Oblate, because 6?=a?(1—e?) has been supposed. The integral for the prolate spheroid takes a different form in integration. ; + This axiom might be deduced from others which would bear perhaps the appearance of a less amount of assumption ; but that they really have less might be disputed: see the end of this chapter, APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 399 surface cannot be found, since @ is an unknown function: allow Av to diminish without limit, and the preceding becomes fary ds or Qn fy ds, which must also be taken between the proper limits. To compare these formulz with those in page 397, observe that x must be changed into 2, and y into r, and also that the solid found in the page cited is not that contained within the curve, but that contained between the curve, and the cylinder generated by KP, or ry*x — firy*da, if we begin from 2=0; or, making the changes of notation, mr?z— Jo mredz. Butsince z=r, in page 397, we have Qn for.rdr=rr'z— fy mrdz, beginning from the same value of z. The integral for the surface, or 27 IV. +dz*: dr’) rdr is 27 f{r./(dr*+dz*), or passing to the notation last used, 20 fy ds, pre- cisely as just obtained. If one equation be given between a, y, and z, the coordinates of a point, that equation is the equation of a surface ; if two equations be given, they belong jointly to the intersection of two surfaces, or to a curve, plane or not, as the case may be. The equation of a plane is of the first degree, or of the form Av+By+Cz+H=0. The equations of a line are those of two planes. These, and many other results of the applica- tion of pure algebra to geometry of three dimensions, I shall presume to be known to the student. If two surfaces-have the equations ¢ (a, Y, 2, a)=0, ¥ (2, y, 2, a)=0, a@ being a constant, each equation defines a family of surfaces, not differ- ing from one another in general properties, but only in the value of a constant. ‘Thus (x—a)*+y?+2*=a’ defines a family pf-spheres, having their centres on the axis of , and every surface passing through the origin. If we take the two equations ¢=0, w=0, to exist simultaneously, we have the equations of a family of intersecting curves, in one of which each surface of the first family cuts that one of the second which has the same value of a, And if between 6=0 and J=0 we eliminate a, we have an equation which, though true of the points of every curve out of this family of intersections, is not restricted to any one value of a: that is, we have the equation of the surface which includes the whole family of intersections (page 359, note). For example, suppose we wish to get the most general notion of a surface formed by the motion of a straight line. The equations of a line are ytar+a, z=br+f. Let a, b, « B be functions of some variable v; there will then be an infinite number of straight lines, one for every value of » which makes a, 6, a, & all possible, and arranged according to some law depending on the manner in which a, b, a, and 6 depend on v. Eliminate v from between the two equations, and there results the equation of a surface passing through all the lines. It is also allowable to suppose one letter in each equation constant. A cylindrical surface, in the most general sense, is made by the motion of a line parallel to a given line, according to any law. Now y=ar+4v, 2=bx-+ Wy, are equations of an infinite number of lines parallel to the lines y=ax, z=b2, disposed according to a law depending on $v and wv. From these two equations, y—axr and z— bv are both functions of v: consequently, z—bz is a function of y—ax: or the general equation of a cylindrical surface is z—bxy=f(y—az). A similar process, choosing different forms for the equations, would give art+bhy+cz+h =f (a'e+by’+c'z+h'), but the second form is not really more general than the first. This is most easily shown by comparing the partial diff. equ. arising from the two forms, made as in page 64. These are 400 DIFFERENTIAL AND INTEGRAL CALCULUS. ° d Az tape and (b’c—bc’) + (da—ce’) awh which do not differ in form. A conical surface is made by the motion of a line which always passes through one point. If m, n, p’ be the coordinates of this point, the equations of two planes which pass through it are a («a—m) +b (y—n)+ce (z—p)=0, a! (2—m)+b'(y—n) +e (z—p)=0; and if a, a’, &c. be all functions of v, every value of v will give one line passing through the point m, n, p, and all these lines put together will constitute a cone of a species depending on the manner in which gq, a’, &c. depend on v. These may be considered as two equations between L—mM, Y—N, 2—p, and v, from which may be deduced oP adv, 2 =v; or ce =1(): t—mM L—m rL—m x—m the-partial, diff, equeis(@=m) <-#(y—n) == ep . equ. 18 (& FR n iy We D. A surface of revolution is one all whose sections perpendicular to a given line are circles. If we imagine a sphere to move with its centre on the given line and a variable radius, together with a plane which always passes through the centre of the sphere, and is perpendicular to the given line, all the intersections of the sphere and plane will make up a surface of revolution, of which the given line is the axis. Let its equations be y=az-+a, z=ba+ A, and let m, am+a, bm+8 be the co- ordinates of the centre of the sphere in any one position, and @m the square of its radius. ‘The equation of the sphere is then (a—m)*+(y—am —a)’?+(z—bm— By’ =dm. Now the equation of a plane passing through the origin perpendicular to the given line is r-+ay+6z=0; and that of such a plane passing through the centre of the sphere is wL—m + a (y—am—a)+b (z—bm—f)=—0. Eliminate m from these two equations, and we have the equation of the surface. If the axis of the surface be that of z, we have for the equations of the sphere and plane x +4°-+-(z—p)*=9p, and'z=p, giving a*-+y'=92z, or z=f(a"+y*) for the surface. The partial diff. equ. 18 : dz dz I de dy The preceding methods are the shortest by which the general definition of the class of surfaces can be made to lead to an equation which is neces- sary, and not more than sufficient, to express them.’ It leaves out of view the particular directrix of the cone, cylinder, or surface of revolution: whatever this may be, the equation of the surface must in each case take one or other of the forms above written, and some particular case of that APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 401 form, depending on the nature of the directrix. For instance, let it be required to find the equation of a cone whose vertex js the point (m,n, p), and whose generating straight lines always pass through the curve Whose equations are y=r,z=wWr. The equations of the generating line being y—n=a (t—m), z—p=b (v—m), we must haye, in order that the generating line and directrix may have a common point, n+a(t—m)=hr, p+b (T—m) = We. If we eliminate x from these two equations, we have a result of the form z— Syn b=f(a,m,n,p), or — =f Cc m, mp For any specific forms of ¢ and ¥, the specific form of f can be found. The ruled surface (or the surface réglée of the French writers) is made by a straight line, which moves in any manner whatever, accord- ing toa regular law ; thatis, a ruled surface (so called) is that which has the equation obtained by eliminating » from y= pu. r+yv, e=YWv.x+wv. The following are remarkable cases. Let the straight line be always parallel to the plane of zy. We have then z=wv, y=$v.x+xv, and elimination gives the form y= fe.xe+-fiz. The partial diff. equ. of this surface, which is of the second degree, since there are two functions to eliminate, is found by the following steps: 2! O=fz.2e+f!2.2'+fz, lax fx i20-[- fizz), ov feo — — 2 d ahr epee ytd PAA Se fiz .2'= — f'2.2,= —- ei of 2” 2,,—22! 2,2) 4-23 2"=0, or p’t—2pqst+qr=0. (See page 388). Let the straight line be always parallel to the plane of xy, and pass through the axis of z. Then z=wv, y=v.x, which gives the form s=f(x:y). The partial diff. equ. is pxr+qy=0. : et us now suppose a family of surfaces having the equation ¥ (2, y,z,@)=0, the different individuals being distinguished by the values of a. If we name the surfaces after their values of a, the two surfaces a and a+Aa, if they intersect at all, have an intersecting curve defined by the joint existence of the equations d & (t,y,2,a)=0, w (x,y, 2z,a+Aa)=0, or ye BGs. S08 | dy dw Aa nana oe be 4 eggs] APPAR Rat Oh q Lema da a da? 2 7 If Aa diminish without limit, it is clear that the equations y=0, dy:da=0 define a curve which can never be the intersection of the Surfaces a@ and a+Aa as long as Aa has any value, but to which the intersection approaches without limit* as Aa diminishes without limit. This curve is called the characteristic of the following surface. If we eliminate a between w=0 and dy : da=0, we have an equation which 1s true of all characteristics, and therefore belongs to the surface in which * The similar considerations applying to families of curves, page 354, &c., will ren- der it unnecessary to treat this point in detail. ale 402 DIFFERENTIAL AND INTEGRAL CALCULUS, all the characteristics lie. Using the language of infinitely small quan- tities, (which we shall often do in this chapter,) if all the surfaces of this family be described, each being infinitely near its predecessor and successor, the part of the surface a+da cut off by a and a+2da 1s bounded by the characteristics of (a, a+da) and (a+ da, a+2da), and is a strip of infinitely small breadth, forming part of the surface which contains all the characteristics. Perhaps the following diagrams may give some idea of this. The surface of which Aq isa part has the value @ in its equation, and becomes Bd when a is changed into a+da, Ce when a is changed into a+2da, &c. The characteristics are the curves ending at a,b, c, &c., and the strips which they inclose, parts of which make up af PQ, are portions of the surface which contains all the characteristics. Exampies. A sphere of a given radius # moves with its centre upon the curve whose equations are yar, zx. Required the character- istic of each position of the sphere, and the connecting surface* of all the spheres. This problem is chusen because the connecting surface is obviously a tube of the same diameter as the sphere, and having the given curve for its axis; the characteristic of two consecutive spheres is a circle of the tube. The equation of the sphere, when its centre ‘has the abscissa @, 1s («—a)?+(y—aa)’+ (2—a)?=h’, and we have for the equations of the | (wx—a)?+ (y—aa)y+ (y—Bbayr=h* characteristic (w—a) +(y—aa) d'a+ (2—Ba) p'a=0. These equations denote the intersection of the sphere with a plane, o1 acircle. We cannot eliminate a without giving specific forms to @ and 8, and even then the elimination will be generally tedious, and most frequently impossible in finite terms. If the axis be a straight lime, elimination will readily give the equation of a circular tube with a straight axis, or of a circular cylinder. If d (2, y, 2,4) =0 and & (a, y, z,a)=0 be the equations of a family of curves, and if we take the curves belonging to a and a-+-da, there will be an intersection if the four equations p(z,y,2a)=0, (27,y,2,a)=0; (x,y, %,a¢+Aa)=0, U (x,y, 2, a+ Aa) =0 * French writers (following Monge, to whom I need hardly say I am here indebted for every thing) call this connecting surface the enveduppe, (which it is very often,) and the family of connected surfaces enveloppées. These terms cause confusion when, as | t i \ ij i often happens, the envelope is itself enveloped by the surfaces to which it is nomi | nally the envelope. APPLICATION TO GEOMETRY OF THREE DIMENSIONS, 403 can be satisfied by the same values of x, y, and z. With four equations this cannot be generally true: but there may be a simultaneous existence of the four, independently of any particular value given to a, if three only of these equations be independent, and if the fourth be deducible from them. Similarly, if Aa be infinitely small, and the four equations become reducible to 6=0, ¥=0, dd:da=0, dy: da=0, as before, the two contiguous curves may have an intersection ina similar case. This ‘is precisely what happens when the family of curves is that of all the characteristics of a given surface, for if 6=0 and dé: da==0 be the two €quations, the four just noted are 99 o=0, esters g dP 5, Ae r Pe. da da da da* of which the second and third are the same. Consequently the three equations ¢=0, dp: da=0, d*d: da?=0, determine the values of a and z at an intersection of two consecutive and infinitely near character- istics. Form two equations by eliminating a, and we have the equations of a curve which passes through all the intersections of consecutive characteristics, and which may be called the connecting curve of the characteristics (the French call it the aréte de rebroussement). Let the connected surfaces be a family of planes, having for their equation 0; z= 2av-+a’y—a’, or 2—-2axr—a*y+-a?=0. Eliminate @ from the preceding, and —x—ay-+-a=0, which gives z=" : (1—y) for the connecting surface. The connecting curve of the characteristics has also the equation —y+-1=0, or is cut from the connecting surface by a plane parallel to that of xz at a unit’s distance. A developable surface is one which can be developed on a plane with- out any such alteration of parts as would be called rumpling, if it were a thin sheet of matter. In order that a surface may be developable, it must be the connecting surface of a family of planes, so as to admit of that mode of generation which we express by calling it an infinite number of infinitely thin plane strips. Each of these strips may then be supposed to turn round the line in which it joins the contiguous strip, intil all are in the same plane. The equation of a family of planes being z=axr+da.y+Wu, that of the connecting surface (which is levelopable) is obtained by eliminating @ from the preceding, and from r+ /ay+y/a=0. This gives (page 246) g=¢p and rt—s*=0, as partial diff. equ. belonging to this class of surfaces. Cylinders and sones are the most obvious of developable surfaces. Given ¢ (a, y, z)=0, the equation of a surface, required a method of inding whether a straight line can be drawn upon that surface. Let j=ar+a, 2=bx+8 be the equations of a straight line: its intersections vith the surface, if any, are found by finding » from the equation b(2, ax+a, br+8)=0, So many real values of x as this equation ives, so many distinct intersections are there of the straight line and urface. But if a, a,b, 8 can be so assigned that the preceding shall be rue per se, or for all values of «, the straight line everywhere coincides vith the surface. _Exampue. A surface is generated by the revolution of an hyperbola bout its minor axis (which place in the axis of 2); can a straight line e drawn upon it? (The common figure of a dice-box will sufficiently yell represent a part of this surface.) Let A and B be apn Gaeta, 404 DIFFERENTIAL AND INTEGRAL CALCULUS. when the revolving hyperbola is in the plane of xz, its equation is p? a? —a22’— a’ B®, and the equation of the surface is B® (2®-+4+y?)—a* 2 ats, Let vazta, y=bz+8A be the equations of a straight line: whence the intersections of this line and the surface are found from BY { (az +a)*+ (bz +B))}—A* SHA BY, which is made identical by B? (a?-+b") =A’, az+b8=0, and B*(a’+ 6’) =A*B*. These are equivalent toi 2 teh 4 G=tBa, a=+Ba, (a +8") eh 2. As here are only three equations with four quantities to determine, an infinite number of straight lines can be drawn on this surface. Take any point whose coordinates are 2, Yj, and z,, on the surface, then if the straight line be required to pass through this point, we have r—2, =a(z—z,) and y—y,=b(z—2z,) for its equations, or a=ax,— 42%, B=y,—bz,. Hence we find z, 0,4 By 2Y,+Ber id . Ae aes Dar aral B= tBa, og + Bd ses and the two first equations satisfy a’+-5°==A*: B®. Hence two straight lines can be drawn through each point of the surface. Show that any straight line drawn on this surface is parallel to a line drawn through the origin, making an angle with that axis which is the same for all the lines; and thence that this surface of revolution is the surface of revolu- tion formed by the revolution of a straight line which is not in the same plane with the axis of z. Required the equation of ‘a surface which passes through any number of curves whose equations are P,—0, Q,=0 of the first; P.=0, Q,=0 of the second, &c. Take P a function of P,, P., &c., which vanishes with any one of them, and Q a similar function of Q,, Q,, &c. Let F(P, Q) be a function which vanishes when P and Q both vanish: then | f (P, Q)=0 is the equation of a surface which satisfies the required — conditions; thus, if there be two straight lines, z=az+ a, y=bz+B, and e=a'z+q' and y=b'z+ 8’, the simplest equation of a surface pass= ing through both, is k (a—az—a)(a—a'z—e') +1 (y —bz—B) (y—b'z—f') = 0. I have entered into the preceding detail on the generation of surfaces _ that the student may, previously to studying the common theorems of | the differential calculus on this part of the subject, have a wider idea of the extent to which the generation of surfaces can be carried, than can be gained from the consideration of the few which occur in ele- | mentary geometry.* : * At the same time it must be remembered that I am not now teaching solid geometry by the differential calculus, but illustrating the differential caleulus by | geometry. The student who finds that his notions of solid space are not sufficiently practised, should make himself master of the Géométrie Descriptive of Monge, one of the most clear and elegant of elementary works. The synthetical part of the Eléments de Géométrie @ trois dimensions, Paris, 1817, by Hachette, might also be studied with advantage. Lest the student should imagine that any other work on | descriptive geometry would answer the purpose, he should understand that it is the peculiar simplicity of the style of Monge, and the general ideas which are given om APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 405 | © The coordinate planes* divide all space into eight compartments, which | may be distinguished by the signs of the coordinates of points in them. | Naming the coordinates in the order w, Y, 2, and choosing one com- ) partment in which the coordinates are to be positive, and proceeding in | the direction of positive revolution round the axis of z, we have what we may call the first, second, third, and fourth com- partments above, and the same below, the plane of xy. The student should remember to attach the idea of first, second, third, and fourth, to the order of signs +-+, —+,—~—, and +— in the two first places, and those of above and below to the signs -+- and — in the third place. Thus ——— should immediately suggest the third compartment below, and —+-++ the second above; and so on. Let a straight line (r) passing through the origin make with the positive sides of the ‘three axes in the positive directions of revolution, the angles rte, ry==B, and rrzy. Then the equations of the straight line may be represented by any two out of the three ane v y Z TRL Bey yy Dit Bi ‘cos @ cos cosy’ MaAGeD BIT Oot where a, 5, c are any quantities proportional to the three cosines. The ‘signs of a, 6, c as they stand, and when all are changed, show the com- partments through which the straight line runs. Thus a:3=y:—4 —2:—6 are the equations of a straight line passing through the origin into the compartments +—— and — +++, or the fourth below and the second above. The equation of a plane being Ax+By+Cz+H=0, the signs of A:H, B:H, and C:H, changed, show the compartment out of which the plane cuts a pyramid: thus 3x—2y—7z—1 cuts a pyramid out of —+-+ or the second above. And this plane has a por- tion in every compartment except -+-——, or the fourth below. But if @ plane pass through the origin, it then appears in six compartments only, those out of which parallels to it might cut pyramids being vacant. ‘Thus 3x—2y—%z=0 appears in every compartment except -- —-— and —+-+. Theangles ofa plane with the coordinate planes are those made by a perpendicular through the origin with the remaining axes: ‘Thus the angle of the planes P and cy is that which the line p, perpen- dicular to P through the origin, makes with the axis of z. And Ar+By+Cz+H=0 being the equation of a plane, those of the perpendicular through the origin are x: A=y: B=z:C. An equation in which one coordinate, say 2, does not appear, or > (z,y)=0, is the equation of a cylinder described on the curve ® (x, y)=0 in the plane of zy, by a line moving parallel to the axis of z. It is only when we tacitly suppose z=0 that this equation belongs to the curve just mentioned. In this last case (a, y)=0 may be called restricted. Required the equation of the tangent plane of the surface ® (2, 7, 2) the principal properties of solid space which are recommended to his attention; and not merely the processes of descriptive geometry, though these are very useful. * The student is here supposed to have read pp. 197—260 of the treatise on Algebraical Geometry. 406 DIFFERENTIAL AND INTEGRAL CALCULUS, =0, which gives z=(2,y). By definition, the tangent plane is that between which and the surface no other plane can be drawn. Let (2, y, 2) be the point of contact, and let 2, n, ¢ be the coordinates of an arbitrary point in the plane. Let a new point be taken, of which the horizontal* coordinates are z-+Az, y+Ay, and let the equation of the tangent- plane be ¢—-z=A(E—r)+B(n—y). Hence the vertical coordinate on the tangent plane is found from ¢—z=AAzr~+ BAy, when the horizontal coordinates are r+Az and y+Ay; while the vertical coordinate of the surface for the same point is z--+-pAr+ qAy+4 {r(Az)? +2s Ar Ay +t (Ay)?}+ &c. (pages 163 and 388). If, then, we assume | the deflection as positive when the coordinate of the surface is greater than that of the tangent plane, we have for the deflection (p—A) Av+(q—B) Ay+3 ir (Av)? + 2s Ar Ay +t (Ay)*} +... | Let ‘the line which joins the point x, y and w+Az, y+Ay, make an angle 6 with the axis of wv, and let Az and Ay diminish so as not to alter | this direction. Then Ay=Avz.tan 6, and the preceding becomes (Az) {(p—A)+(q—B) tan 6} Av+ {r+ 2s tan 6+¢ tan’ 6} ot | If p differ from A, and q from B, one or both, this deflection has | : j always a finite ratio to Ax, which has for a limit the ratio of p—A | +(q—B) tan 6 to 1, except only in the case in which Ay and Az are so | taken that tan €= —(p—A):(q—B), in which case the deflection | diminishes without limit as compared with Av. Consequently, there is one direction in which the plane deflects less from the surface than in any | other. Butif p=A and g=B, or if the plane have the equation G—z=p (E—2)+q (n—y)..--...(T); the deflection has to Az the ratio of }(r+2stan6+¢tan?€) Av+.... | to 1, which ratio always diminishes without limit. Hence the deflection of this plane (T) always becomes less than that of any other plane (P) | in whatever direction we proceed, except only for one direction in each | plane (P). But we shall now show that all these isolated directions, | one in each plane (P), are no other than those indicated by the lines in which the planes (P) cut the plane (T). The two equations -z=A (€—2)+B (y—y) and n —y=tan 6 (é-2) jointly belong to a straight line, which, lying entirely in the plane which has the first equation, is projected upon the plane of ry into a line pass- ing through the point (v, y), and making an angle 6 with the axis of a. If we assume tan 6=—(p—A):(q —B), and if we eliminate one of the two A and B from the equations, say A, we obtain an equation belonging to a surface which contains all the lines in question that can be drawn upon all planes whose equations only differ in their values of A. But it so happens that in eliminating A we eliminate B also, and obtain the equation T. For the second equation becomes (p—A) (é—2) +(q-—B)(—y)=0, or A(E—2)+B(y—y)=p (E—x) +9 (n—y), which, with the first equation, gives (—z=p (E—x)+q(n—y). Con- sequently, the plane (T) has a deflection from the surface less than that of any other plane drawn through (2, y,z), in every direction but one, * From the usuai manner in which diagrams are drawn, it will be convenient to call x and y the horizontal coordinates, and z the vertical coordinates. APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 407 namely, that of the line in which the two planes coincide. Hence no plane cau be drawn between this tangent plane and the surface. If @(2,y,2)=c be the equation of the surface, we find, as in page 352, je tt UO dO pane CID dR, 25 ear ORAM Rar de rAAlrT which will transform the equation of the tangent plane into qo, dd db, dd dd de dx” * dy" Ged We ny et ee which (as in page 352) if ® be a homogeneous function of 2, y, and z, has ne for its second side, m being the degree of the function. All the considerations used in the page just cited apply here. The equations of the normal, or perpendicular to the tangent plane through the point of contact, are either é—e+p(S—z)=0, n—y+9q (C—2z)=0, or any two of the three d® —x):—=(n—-Yy) : — =(l—2z) : —. G-2): SF =0-9):S=C-2): The line of greatest declivity (ligne de la plus grande pente) with respect to (ry) is that drawn in the tangent plane from the point of contact perpendicular to the intersection of the tangent plane and (ry). Its projection on the plane of zy is therefore perpendicular to that intersection. Now, making ¢=0, we have for the equation of the intersection —z—=p (E—2) +g (n—y), and the equation of a perpendicular to this, drawn through the point (2, y), 1s d® d® -yy—q(E-2)= — (n—y)—— (E—2x)=—0. p(a—y)—4 E—2)=0, or — (1-9) dy © t)=0 This, and the equation of the tangent plane, are the equations of the line of greatest declivity to the plane of zy. The projection of this line on (zy) is also that of the normal. Let the surface be an ellipsoid, and let A, B, C be the reciprocals of the squares of its principal semidiameters, the lines of these semi- diameters being the axes of coordinates. ‘Then the equation of the surface is Aa*+By?+Cz?=1, that of the tangent plane and those of the normal are : : : | egw y ote Azv§4+Byn+Czf=1; er By: ae fy A curve is the intersection of two surfaces; and its tangent line at any one point is the intersection of the two tangent planes of the two surfaces. If, as is most common, the curve be assigned by its projections on two of the coordinate planes (z7 and yx) ; that is, if y=av and z=fxr be the equations of the cylinders of projection, we find for the equations of the tangent planes, derived from y—av=0, z—fr=—V, 408 DIFFERENTIAL AND INTEGRAL CALCULUS. —ix (E—x) +1 (n—y) +0 Gn Ora B. ete C= a@)avn4 —f'x (E—x) +0 (n—y) +1 (C—2z)=0 f—2=f'x (E—2) which equations are jointly those of the tangent required ; severally, and restricted to the planes of the coordinates they include, they are the equations of the tangents of the projections, which are therefore the projections of the tangent. A curve has an infinite number of normals, or lines perpendicular to the tangent, which all lie in a plane called the normal plane. Again, of all the planes which can be drawn through a point of a curve, there may be (generally is) one which is closer to the curve than any of the others: this is called the osculating plane. Previously to considering these, it will be desirable to treat the subject of curve lines generally in a manner which does not refer to projections on two coordinate planes to the exclusion of the third. Let v be a variable, of which x, y, and z are severally functions, so that v=2,, y=-y,, 2=2,, where az, is an abbreviation of “ the function of v which z is.” Hence, by elimination of v, two equations between z, y, and z may be obtained in an infinite number of ways, and each pair contains the equations of a pair of surfaces, intersecting each other in the same curve. And 2’, x’, &c. mean diff. co., taken with reference to v3; and dy:dz, as obtained after elimination of » from the first and second equation above written, is the same as dy : dv--dx:dv, &c. The equations of the tangent of the curve above mentioned may then be reduced to any two of the three i dz dy | dz (€—<2): His Nee ‘ae aA ae ery whence the equation of a plane perpendicular to this line passing through the point of contact, or of the normal plane, is dx Lye rows dz (é-2) ao ig) Sig t Node eeeeee -(N). From this supposition we can easily pass to either of the more limited ones. Thus, if y and z be expressed in terms of 2, we have v==z and dx: dv=1, whence the equation of the normal plane is Lar yt ad dz (E—2)+(n—y) at (C—2)F=0. Let a plane be drawn through the point (a, y, x) of a curve, having the equation P(€—2x)+Q (n—y) +R (f—z)=0, and let us consider the deflection from this plane, in a direction parallel to the line y=aé, ¢€==bé, and at the point of the curve whose coordinates are t+ Aa, ytAy, z+Az. The equations of the line on which the deflection is measured are then n— (y+Ay)=a {i—(@+Ar)}, 6—(z+Az)=) {E—(a@+ Az)} and the intersection of the line and plane, (PAr+ QAy+RAz) : (P+Qa+Rb) being V, is made at the points whose coordinates are §=a+Ar—V, m=ytAy—aV, f=24+Azc—bV. i Now the coordinates of the two extremities of the deflection are Ess Nie ¢., on the plane, and w+Az, &c. on the curve: whence the length of ' APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 409 the deflection is the square root of the sum of the squares of €,—(r+Ar), &c., or VJ +a°+ 6°), or /(1+0?-40°). (PAr+ QAy + RAz) : (P+Qa+ Rd). To make the plane osculate, as the phrase is, with the curve, we must make PAr+QAy+RAz depend upon the highest possible powers of small quantities. Let the increments arise from v recelving the incre- /ment 2; whence Ar=ah+32"h?+...., &c. Make the coefficients of h and h? vanish, or let Pa’ + Qy' + Rz’=0, Pa!’ + Qy"”+Rz"=0, which requires that P,Q, and R should be in the proportion of y’2!’—2!y", al —2'2!, and a!y!—y!2", Consequently the plane (y'2"—2ly") (E—2) +(e! a"=2'2")(q-y) + (v'y"-y'2") (E-2)=0.... (O) is so placed that all deflections from the curve, in whatever direction measured, depend upon the third power of hf, while in every other plane the same deflection depends upon the second or the first power of h. This plane, then, is closer than any other to the curve, and is the oscu- lating plane. Those planes in which the deflection depends on the second power of h have Px’ +Qy’-+ Rz’=0: show that this condition is satisfied by all planes which pass through the tangent of the curve at the point (z,y,z). These might be supposed (as passing through the closest line) to be closer than other planes; and the preceding shows that such is the case. If the line on which deflection is measured be taken perpendicular to the osculating plane, we have for the parallel to it drawn through the Origin, €:2,—=n:y,=:2,,, where P=2,,=y7/2"— z'y", &. Hence @=Y,,:%,, b=z,,:x,, and substitution in V,/(1+4a®+2*) gives hs 6 (x, wy yl zy a) cal (1,7?+ Yj; EA) for the first term of the deflection. A plane passing through a given point (2, y,z2), and having the equation P (€é—2)+Q (n—y)-+R (f—2)=0, may be called the plane (P,Q,R). Hence the normal plane is (v’,7/,2') and the osculating plane is (x,,,y,,%,): and these two planes are perpendicular, since We j+Y y,,+2' z,,=0. +y,7+2,/), which is =0. This means that if we had written Ax for dr, and used the expansion of Az, &c., we should have found for the preceding shortest _ distance a quantity depending only on squares and higher powers of Av. The locus of all the centres of circular curvature is not made by the perpetual intersection of tiormals infinitely near, drawn in the osculating planes; so that this locus is not an evolute to the curve. Let us now further consider the polar surface, made by eliminating v from the two equations of the polar line, the intersection of two normal planes. These equations are a! (E—2)+&ce.=0 (N); a" (E—2)+&e.=2"?4+y"42?= 5" (N’). " If we now differentiate each of these with respect to v, reasoning as in page 403, we find only one more new equation, and the three jointly belong to the intersection of two infinitely near polar lines, or a point of the connecting curve of the polar lines. This new equation is ; 7 gl! (E—a) + &e. = 80/2" +4 By!y" +32 2= 35/6! (N”). Solving these three equations, we find for £—a, »—y, and £—z, three fractions having the numerators 3y,s's”— w',s”, 3y, s's—y', 8”, 32, 5/s/—2!;, 8", and the common denominator a, a!”+-y,,y!"+2,,2!", where Ta hie EL Pee AW, ct acer: Rl Rt. efi. Nf v4 Jett Na ee 2) m Bey 2 y=ry : ' If we take s, or the arc of the curve, for v, we find for /{(é—2) + (n—y)*+(S—z)*}, considered independently of ‘sign, the following expression, (s’ being =1, and s”=0), ¢ a 2 Pie yeas 2 WR». 2 ONS A ty pte; N (¢ +y $22 (a)? 4 yl? 4. 2ll2)?) | pe ea te aftt RAG | jp a ag) ET, 2 scare seca arse | By Ok YyY eye Ly ba pay cbeelly APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 413 The preceding equations (N), (N’), and (N”) are such as would be derived from the equation of a sphere, (§—a)’+ (y—b)*+ (6 —c)=r?, by three differentiations with respect to the common variable contained in &, », 2, if after differentiation we made Ra Ota ee See The only difference then would be, that where we had £, n, and € we should now have a, 6, and c. That is to say, 6, 7, and Z, as last found, are the coordinates of the centre of a sphere which passes through the point _ (,y, =), and has with the curve at that point a contact of the third order. Or if such a sphere be drawn, the curve runs so near its surface before and after contact that the deflection of the curve from the surface has always a finite ratio to the fourth power of the departure from the point of contact. The connecting curve of all the polar lines is then the locus of all the centres of spherical curvature: it is not an evolute of the given curve, because all its tangents are on the polar surface. I shall now proceed to the consideration of the two flexures from which a curve of double curvature derives its name. If we begin with a straight line, we have a line whose osculating surface is indeterminate, since an infinite number of planes can pass through it: and all its consecutive normal planes are parallel and make no angle. ‘Turn the straight line into a plane curve, and its osculating planes are all in one plane, which is the osculating surface. But the normal planes make angles depending on the flexure of the different points ; these infinitely small angles it has been customary to call angles of contingence. The normal planes being all perpendicular to the single osculating plane, the polar lines are the same, and the polar surface is cylindrical, having the evolute for a base. Now let the curve become one which is not all in one plane, and the successive osculating planes make infinitely small angles which may be called angles of Jlecure. The two planes (A,B,C) and (A,, B,,C,) make an angle, the cosine of which is (AA,+BB,+CC,) divided by the product of J (A?+B’+C*) and ./(A?+B?+C%), or the (sine)® of which is (AB,—BA,)’+ (BC,—CB,)?+ (CA, — AC,)? divided by the square of the preceding denominator. Hence, if 6 be the infinitely small angle made by (A, B, C) and (A+dA, B+ dB, C+dC), we have 6°= {(AdB— BdA)?+ (BdC—CdA)?+(CdA- AdC)*} : (A?+ B’+C?)*. If we apply this to two consecutive normal planes, in which A=v’, | dA='dv, &c., we find for the angle of contingence dvJ(2,°+y,; +2z,):s; and if the arc ds or s/dv be taken to subtend this angle, we have s’°:\\/(x,?+&c.) for the requisite radius, which is precisely the radius of circular curvature above determined. But if we consider two Successive osculating planes, in which A=2,, dA=w',, dv, &c., we have for the angle of flexure 2 Lo es 2 2 hee dv,/{ (yyy Yuk (Y= ZY! e+ (2,2! 72,2} (ay 44) 42) or dv Jf(2?+y"+2"). (2, xy yl! + By) oP yi t 2A) 5 the first two factors of which being =ds, we have (2,2-+y)?+<,2): (x, 2!" + yy!" +2,2'") for what we may call the radius of flexure. We have not yet found an evolute of the curve, or a second curve whose tangents are normals of the first. The two loci of circular and Spherical curvature are not of this character. If any evolutes exist they 414 DIFFERENTIAL AND INTEGRAL CALCULUS. must lie on the polar surface, and not elsewhere, for all normals lie in nermal planes, whence the intersection of two consecutive normals must lie in the intersection of two consecutive normal planes, or on a polar line; that is, on the polar surface. And we can {obviously make an infinite number of evolutes on the polar surface: thus, let P,Q, R,S be consecutive points of the curve, infinitely near, through which draw normal planes giving 14V part of the polar surface: joi P with any point 1 of its polar line, draw Q1 and produce it to meet the succeeding polar line in 2, and so on. We have then as many small arcs of an evolute, 1, 2, 3,4, as we can take points in the first polar line to join with P. Or, through every point of the polar surface one evolute passes, and only one. The question of finding an evolute is, therefore, reduced to that of drawing a curve on the polar surface, whose tangent shall always pass through the given curve. But since every tangent plane of the polar surface cuts the curve somewhere, one condition is satisfied by the mere circumstance of the curve lying on the polar surface, which makes its tangent lie in a plane cutting the curve. | If one only of the equations of this tangent be then that of a line passing through the curve another condition is satisfied; and but two are necessary. As, however, this reasoning (which is that of Monge) may be rather too refined, we will suppose the evolute drawn, and the coordinates of a point in it expressed in terms of v, the same variable as that in which the coordinates of the corresponding point of the curve are expressed. Let X, Y, and Z be the coordinates of an arbitrary point in the tangent of the evolute, whence (X—£):&=(Y—n) :n/=(Z—Z):Z' are the equations of the tangent: which being to pass through the point (a,y,2) of the curve, we have (w—é):7=(y—n):n =(e—Z):2'. But since the point (é, 7, 2) is on the polar line of (2, y, 2), we have (E—2) a’ + &c.=0, (E—2) a"+.... =s’*, so that we have four equations between é, y, Z, and v, which we can immediately show to be reducible to three. For if we differentiate the equation of the normal plane generally, or pass to a point of a con- tiguous normal plane without considering whether (£,,2) is on the polar line or not, we have Ba! tay! yl ol a! (Ea) 2" + (n—y) y+ (G—2) 2" —8?=0; or, if the pot be on the polar surface during the differentiation, Ele! +n y'+-¢'2/=0. This is true whether the line drawn on the polar surface pass through the curve or not, so is (—2) a/+(n—y)y +(¢—z)2'=0. But these last two equations with the equations to the tangent of the evolute at (v,y, 2) are not four distinct equations, but only three, for the latter equations with, (§—2) a +&c.=0 give S——————————————— ee Eee ee eS EE. aa — - APPLICATION TO GEOMETRY OF THREE DIMENSIONS, § 415 nf ry (§—2v) 2’ + Sree) Yt. (0-2) 2'=0, or) a Ely! n! +2! 2’=0. G c If, then, we take the equations (N) and (N’), and one of the equations of the tangent, say the first, and eliminate v, we have two equations | which we may so obtain that one of them, from (N) and (N’), belonging to the polar surface, shall be of the form @(&,n,¢)=0, and the other & (2, , 2) =x (é,n, £).n’. Substitute in the second the value of & from the first, and we have, remembering that 7 : /=dy: d=, a common . , diff. equ., the integral of which, and @(é,»,Z)=0 are the equations of the curve required, the arbitrary constant of the differential equation giving the multiplicity of evolutes which have been shown to exist. Let R be the distance between (w,y,z) and its corresponding point (€,7,¢) on an evolute. Then R?=(—2r)?+&c. and RR’/=(§—.2) (f’—2')+&c., of which (E—x)'‘x’+ &c.=0, so that (€—2)é’+&c.=RR’. Substitute in the last values of »—y and ¢—z from the equations (—2) :§'=(n—y) :n'=(6—z) : 2’, which gives tl RR’ n! RR’ a RR’ gé— C= 9 SI) uiege: Harare cay yaar 6—2=> = EP ple gi? E24 yf? 4 71? E24 fh Zh? the sum of the squares of €—2, &c. equated to R® gives R?= +n? +¢"=0", where o is the length of the arc of the evolute. Consequently R’=o', or dR=ds, and reasoning as in page 364, we find that the | difference between any two values of R is the arc of the evolute inter- ‘cepted between them. Exampie. Among curves of double curvature, the screw has that ‘priority which the circle has among plane curves. The straight line ma ¥ gp g y be described by making any length of it take a motion of translation in the direction of the line: no point of the length mentioned will ever be off the straight line. The circle may be equally described by giving any arc of it a motion of rotation alout its centre, and in its plane. The screw may also be described by giving any arc a motion both of translation and rotation, provided the two velocities remain uniform, or else always vary in the same ratio. Let the axis of 2 meet the screw, and let that of z be the axis of its cylinder. The screw is then the intersection of the cylinder, whose equation is 2°-+y*=a*, with an _helicoidal sur- face ‘(page 396), whose equation is z=btan™'(y:r). We may ‘reduce these two equations to three, expressive of 7, y, and z, in terms of v, as follows, @2=aCcosv, y=asinv, z—bv, where v is the angle of revolution of the describing point about the axis of z. We have then t= Na cos x aé=—asinv | v’=—acosv y= asine y= ~acosv | y/=—asinv pee oD Phot massa gia: oo A) w= asin’ | '2,c=° alsin vy | x'),=ad cose y"=—acosv | y,=—abcosv | y/,,=absinv TF an ates 2 T , Weta J ales Pe z,=0 ; 416 (DIFFERENTIAL AND INTEGRAL CALCULUS. ae! -- yy! +22'= bv aL), YY), + 22, =a%bv | s?=a* +0? 2 ee 2 Q 8 oo A wt ne A fl eee 4 V fee Ly bY +2, =4 (a+b?) | aye by, y! F2,2"=a'b | 80. The equations of the tangent are (§—a cos v) :—asin v= (9 —4 sin v) : acos v=(¢—bv) :b, from which it would be practicable to eliminate »v, and to get the equation of the osculating surface. This surface, then, is found by eliminating v from (é—acosv) b=—(f—bv) asin v, (yn -—asinv) b=(F— bv) acos v. | But if 20, or we ask for the curve in which the osculating surface cuts the plane of zy, we find for this curve the involute of the circular base, defined by £=acosv-+avsinv, n=asin v—av cosv (page 366). And it is obvious that the cylinder is the polar surface of the involute of the circle. In fact, the other evolutes (besides the circle) of the in-_ volute of a circle are all the screws which can be described upon a right cylinder having that circle for its base, and which meet the involute. The equation of the normal plane, and the ame differentiated with respect to v, are —iasinv-+ynacosv+ fbb", —acosvu—nasin v=b*. These equations jointly belong to the polar line: to find a point in the connecting curve of the | polar lines we must annex the equation asin v—yacosv=0, or n:&==tanv, whence the preceding equations become —a,/(é+ 77)=b*, C=bv, or F472 bt: a2, C=—btan-'(y:£). So that the locus of the | centres of spherical curvature is another screw, generated by the same | helicoidal surface, but having a cylinder whose radius is 6’: a. ‘The two screws, however, are in opposite positions; for if in the first two | equations we make £=0, thereby obtaining the equations of the curve | in which the polar surface cuts the plane of (xy), we find that é and 9 are the values of the coordinates of the involute of the circle whose radius is b?: a, with their signs changed. ‘The polar surface is then the - osculating surface of this new screw: and if d=a, the osculating and polar surfaces of the given screw are the same, the latter having only | made a half revolution about the axis of z. For the coordinates of the centre of circular curvature, we ‘find 2'y — yz, —ab® cosv —a* cos v, y'Z,,—v'y,,=0, vz, —2'e,= —a* smd —ab’sinv, whence if X, Y, Z be the coordinates of this centre, we have 2 i? X—a cos v= —a COS Bee cosv, Y—asinv=—asin Tete sin v, Z—bv=0; . giving the equations of the same screw which is the locus of the centres of | spherical curvature. Looking now to the coordinates of the latter, we find s’=0, and —a!,,s?=—ab (a’?+b*)cosv, —y',s"=—ab (a? +6") sinv, —2/,,s?=0, giving for the values of X,, Y,, Z, the coordinates of | the centre of spherical curvature, precisely the same as for the coordinates of the centre of circular curvature. And the radius of spherical curva- ture is found to be a+0*;:a, and the radius of circular curvature the APPLICATION TO GEOMETRY OF THREE DIMENSIONS, 417 same. The radius of flexure is b+a?:b. To find the evolutes of a screw, we must eliminate v between three of the four equations (acosv—f) :5’=(asinven) in’ (bv—Z): 2 —asinv.é+acosv.n+bl=b*v, —acosv.é—asinv.n=b*. The following may be a useful exercise for the student, though it does not give a result simple enough to be of muchuse. Eliminate v between the first and fourth equations by finding sinv and cos v, and expressing sin’ v-+cos?v=1: the result is @ (n= 88') = (Sry — nb PE +? + 26°) + bt (E24 9”), Let r and 6 be the polar coordinates of (é,n) on the plane of xy, the preceding then becomes, by the equations in page 345, (a’ 7? — 4) d ; ve Sde (a* 7° —b*) 2? = (7° + ?r) 26", or a ny Pee Let r=1:u, and the last result becomes . dd Jf (a’—d‘u®) a® +b? 63 du 1 ut Ft) (bw) J@—awy’ Let 6°w=acos\, and we have dé a*-+- 6? * (@+62) d.2r do oe ee ey. dr b*? + a? cos” rT “di: 267 -+-a*-+- a’ cos 2A P Integrate by the formula in page 289, and we have J (a? + 6°) oat = (2b?-+4+ a’) cos 2b 2b?-+ a?+-a® cos 2A Be /(a’+0’) ; = (26° -+a*?)—a@ | 1 C987 See A oa pew ar 2b a (7? + b*) Which is the polar equation of the projection of the evolute of a screw upon the plane of ry. If we take the cosine of both sides we can give the equation the form cos (0-++C)=¢r, where or is a finite and rational algebraical function only when ,/(a’+ 0”) : 20 is a whole number or when a?=(4m?—1) 87, m being integer. | I now proceed to extensions of the theory of curved suriaces. That of curved lines has been made to precede, as containing functions of one variable only. If we take the various ways in which the equation of a surface may be conveniently expressed, we have 1. z=¢(a,y). The diff. co. of z may be expressed by p, q, 7, 5, and ¢, as explained in page 388, Higher diff. co. than the second are useless in this inquiry. | 2. (2,y,z)=0. If we look at page 268, No. 73, where the diff. Co. of x are expressed in terms of ¢, we shall see that it is useless to Investigate formule deduced from this form, unless we contrive a more simple notation for the diff. co. of ¢ Let U=0 be the equation $ (7, y,z)=0, and let partial diff. co. of U be denoted by simply writing the characteristic letters of the differentiations as subscript indices ; thus dU :dr=U,, &c., and the diff. co. which we shall have occasion to use are U,, U,, U., Use, Uy, Uz; Usy, Uy, Us. Let powers be denoted 2H 6+C=r\— ° > ——COS- 418 DIFFERENTIAL AND INTEGRAL CALCULUS. as usual; thus U2 signifies the square of dU: dx, &c. We have, then, by the article cited, dz dz RS pp tea ee $ .g-=—U U, 7 or Up —UR su: dy or U.q= y Us mi 2 or Ur —(U? Uz, —2U; U, Uz, 4 U2 Ux) Us ini or v3 cas Ue (U, Uy.+ U, Us.) Fae (Uz U.,+U, U, U.) dz U dy’ or 5 Od Aca —(U? U,,—2U, ATS U,.+ Lin | BE > whence it follows (page 268, No. 74) that if we make X=Uy US Ue Oo RAUL ULE) A ea Ue = Om Ne, U,—Uetis VU, UU, Uy, Z=UpU.U. Oe Ut (nét—s?) =XU2+ YU?2+ ZU24 2X! U, U,4+ 2Y' U, U,4 22’ U, U,; expressions, the symmetry* of which makes their use both less difficult and more safe. 3. Leta, y, and x be severally expressed as functions of v and w: our method will then be analogous to that pursued in treating of curves. The expression of the second diff. co. of z in this system is so extremely complicated, that I shall confine myself to using it in those cases only in which first diff. co. are sufficient. 4. Let z=(2,y,¢), & (x, y,a@)=0, where % is the diff. co. of P with respect to a, or $,. We have then, the notation being as before, and a, meaning da: dx derived from the second equation, P= Vet Vee e q= by + Pa Py= Fy T= Pas + Pea Aa» S=P,y+ Drady= Pry + Pay A, ESQ Way Cy- Now $, 20 gives by, +QPra 4220, Pay +A Pia 4y= 03; substitute the values of a, and a,, thence obtained, and we have Daa ie ee Di Pans aa $= Daa Dry TR Dar Days Pua i= Daa Dyy — Diy Daa (ri—s*) es Pan (Dix Digs By ee} (Dix Day ae 2dhry Dax Day + Pyy Dix) : Much depends in the theory of surfaces on a knowledge of the pro- perties of the expression ax?-+ by?+ cz?-+ 2a, yz 4+ 2b, 2x + 2¢, ry, which may be always positive or always negative, or sometimes one and sometimes the other. We know that an expression of the form Av’+2Bvw+Cw* is of one sign, whatever v and w may be, when AC — B* is positive, and then only. Writing the preceding expression in the form ax’?+2(b\2+¢,y) + by?+2a,yz+cz,, we infer that it always retains one sign (that of a) when a (by’+ &c.)—(b,2+cy)* 8 — always positive, or when * Jn all general problems, then, expressions must be carefully written in a sym- metrical form. The risk of error in complex operations, whether of alteration, omission, or redundance, is materially lessened, since each error must either be | made three times in exactly the same way, or the operator is warned of the exist- © ence of an error by the want of symmetry in the results, + | APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 419 (ab — ct) y° +2 (aa,—b,c,) yz + (ac—h,)? z* is always positive. Hence ab—c} must be positive, and (ab—c}) (ac—bj)—(aa,—b,c,)? must be positive; that is a@ (abe+ 2a,b,c,— aa? — bb}—cc}) must be positive. Hence, since the expression can be arranged in powers of y or 2, and similar results obtained, we find that ax?+&c. is always of one sign (that of a, 6, and c), when ab—ci, be—a?, and ca—b? are all positive, and abc+ 2a,b,c,—aa?—bb?— cc; has the common sign of a, b, and ec. The equation of the tangent plane of a surface, the point of contact being («, y,z), has been exhibited in the forms s—2—p 6—2) +9 (n—y), Uz (§—2)+U, Q—y) +U, (2) 50: U=0 being the equation of the surface: The sign of the deflection from the tangent plane, called positive when the ordinate 2 of the surface increases (algebraically) faster than that of the plane, has been shown to be the sign of r(Av)?+2s Ar Ay+t (Ay). There are, then, three distinct modes of contact between a curve and its tangent plane; which we shall call (for reasons afterwards to appear) the elliptic, hyper- bolic, and parabolic contacts. The following diagrams will give an idea of them. 1. Let rt—s° be positive. Then the deflection always has the same sign: or in the immediate neighbourhood of the point of contact the surface is entirely on one side of the tangent plane. This is the elliptic contact, and is shown in the manner in which a sphere or an ellipsoid meets its tangent plane. 2. Let rt—s°=0; then r(Ar)’+&c. is a perfect square, or one _ taken negatively, and the deflection is always of one sign, except when | Sy: Ar=—s:t, in which case the terms of the second order are col- lectively =O. In this case, then, there appears no obvious difference between the contact and that last described, except that in one particular line the contact is of a closer order than elsewhere. But, as we shall presently see, if the tangent plane meet the surface in a curve, (as, for instance, a table meets a ring laid upon it ina circle,) all the points of that curve have a contact of this species with the tangent plane. 3. Let rt—s* be negative. If Ay: Av=tan €, that is, if the direction in the plane of xy in which we pass under a new point of the surface make an angle 6 with the axis of x, the sign of the deflection at the new point depends on that of r+2s tan 6+/tan?€, which is of the same Sign as r, except when ttan€ lies between —s+(s’—rt) and —s+,/(s*—rt). There are, then, two opposite angles in which the deflection has one sign, having the other in the two adjacent angles. But when ¢,tan€ is equal to either of the aboveementioned quantities, 2H 2 \ 420 DIFFERENTIAL AND INTEGRAL CALCULUS. the approach to the tangent plane is of a closer order. ‘This contact is such as takes place at every point of a single hyperboloid. When a surface is described as the locus of all the points of a family of curves, made by giving different values to a constant, the two equa- tions of the curve, which jointly, and for one value of a, represent one single curve, belong to all the curves, or to the surface, if a be considered as having any value: and the elimination of a actually gives the equa- tion of the surface. Conversely, we can at pleasure subject any given surface to an infinite number of modes of generation, by introducing a new variable. Thus 2+ 7?-+2’=c’, the equation of a sphere, is obtained by eliminating a between a*+y’=a’, and z= V(c?’—a’), which answers to generating the sphere by circles parallel to a given plane, or considering it as the locus of all the circles which are perpendicular to a given line. Again 2®—y’=a’, 2y?-+ 2*=c?—a® shows that the sphere is the locus of a family of curves formed by the intersection of hyperbolic cylinders, generated by lines parallel to the axis of z, with elliptic cylin- ders generated by lines parallel to the axis of . We shall now con- sider a wide class of surfaces, namely, of those generated by the motion of a straight line, as well for the exercise of the student in general con- siderations as to show the connexion of the theory of surfaces with that of partial diff. equ. Let a straight line move so as always to be upon three given curves. That we have here conditions no more than sufficient to make the line describe one implicitly given surface may be thus shown. Ifa cone be taken which has its vertex in the first curve, and the second curve for its base, this indefinitely extended surface can meet the third curve only in determined points: unless it should happen that the third curve hes entirely in the cone. If, taking every point of the first curve in succes- sion, we describe cones on the second curve as directrix, we shall have an infinite number of cones, with an infinite number* of points, in which they cut the third curve. Our results contain, 1. An infinite number of consecutive positions of a straight line upon the three curves, made from consecutive cones, and forming the surface required. 2. All the cones, if any, in which either of the curves is entirely upon a cone which has @ point upon another for its vertex, and the third for the directrix. If our resulting equation contain distinct factors, (page 347), should it be, for instance, ef the form PQR=0, we may be sure beforehand that of the three equations P=0, Q=0, R=0, which satisfy it, two belong to cones. Let the coordinates of the several curves be expressed as functions of V5 Ve, and v3. Let the joining line, being a line of the required surface, have in one of its positions the equations w=az-+a, y=bz+ 6. Then since some one point of this line is on each curve, if r=, U,, y= Yi Uy 2=x,v, be the equations of the first curve, we have, by substitution, two equations between a, a, 6, 8, and the value of v, belonging to the point in which the line meets the first curve. These two equations, by elimi- nating v,, give a relation between a, a,b, 6, and the same thing being true of the other two curves, we have three equations between these four quantities, and can therefore express any three of them as functions of * It might so happen that the third curve was placed in such a manner as never to come near any cone described with a point in the first as a vertex, and the second as a directrix, If so, we shall be reasoning on a problem, the final equations of which will be incongruous, or else will contain impossible quantities. _ APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 421 the fourth : or we can express all four as functions of some one quantity, say v. We have, then, for every value of v which gives possible values to a, &c., the equations of one position of the straight line, in the form and the elimination of v will give the equation of the required surface. If such a surface were approximately described, by constructing the positions of its straight lines answering to .... v=—2A, v= —A, v=0, v=4, v= 2A, &c., A being so small that any two consecutive lines should be very near each other at their shortest distance, we should form as good a notion of a surface from the cullection as we do of a curve line from a polygon of a large number of small sides. And on this surface we should be able to draw a line, at and near which the generating lines seem to come closer together, each to its neighbours, than in other parts, and from which they appear to diverge. If we now suppose A to diminish without limit, this line, which is the limit of all the lines pass- ing through the points of nearest approach, may be called the curve of greatest density. When the surface is developable, that is, when the shortest distance of consecutive lines diminishes without limit compared with A, this curve of greatest density is the connecting curve of con- secutive lines. If for v we write v +Av, we have the equation of a consecutive line: it remains now to find the coordinates of the point of the first which is nearest to the second. Resuming the problem in page 411, let there be two straight lines whose equations are (7—p) : P=(y—q) : Q=(z—r) : R and (r—p,):P, =&c. Introduce two new variables w and w,, and write these equa- tions in the form z=p+wP, y=q+uQ, z=r+wR; c=p,+w,P,, y=&e. Every value of w belongs to one point of the first line, and of w, to one point of the second line. Let w and w, belong to the extremities of the shortest distance between the two lines, so that the equation of the line joining these two points is gra weke yis, 6 yt OQ) rt wR) (A) DA+wPi—(ptwP) Ht+wQ—G+tuQ) n4+w,Ri-(@+wR)” If these denominators be A, B, and C, we know that AP+BQ+CR =0, and AP, +BQ,+CR,=0; form and reduce these equations, which gives for the determination of w and w,, P (1-p)+Q (m1 —-QM+R (7.—r) +(PP,+ QQ,+RR,) wW, —(P?+ Q’+ R’) w=0, Pi(p.— p) + Qi(Gi—@) + Bir. — 1) + (PI + Qi+ Ri) w - (PP, +QQ:4+RR,) w=0. Let P= QR,—RQ,, Q,=RP,—PR, R,=PQi—QP,, and we have w= { (Q R,—R Q,) (pi—p) (R Bi P R,) (n1-D + CR Q i-Q P,,) (7.—7)§ . (Py + Q,/ +R) v= 1(Q,R,,—RiQ,,)( Pi—Pp) 3% (R,P,,— PLR) (a—-@ a" (P,Q, QP ),) (rn —r)} 2 (P/7+Q/+R8,/). 422 DIFFERENTIAL AND INTEGRAL CALCULUS. If these belong to consecutive lines, so that p,=p+dp, P\=P+dP, &c., we find P,=QdR—RdQ, Q,=RdP—PdR, R,=PdQ—QaP; and Q,R,—R,Q,, differs only by a quantity of the second order from QR,,— RQ,, &c. If we now take the case before us, in which the equations have the form (omitting v) (e—®): d=(y—W) :w=(z2—-0): 1, we have p=, q=¥, r=0, P=¢, Q=¥,"R=1, P,=—wdv, Q,=P'dv, R= (P'—¥P) dv, and YP + P+ (Py —yug') ; and €=@+ wo, 7n7=v-+wyw, (=w, are the coordinates of a point in the curve of greatest density. And the equations (A), when the proper values of w and w, are substituted (not neglecting their difference) will, multiplied by dv, give two equations, from which, by eliminating v, may be obtained a new surface, described by the motion of the straight line in which the infinitely small perpendicular distance of two consecutive lines on the first surface is always found. The shortest distance of the consecutive lines, found in page 411 by an easier process, is (neglecting the sign) P,,(p,—p) +&c. divided by V(P)7+ >.-+)3 or, making the substitutions, dv(— w’'@/+ ¢/v'): NOE? +h"+(oY'—vW9')’). Consequently it is the condition of a deve- lopable surface that 6’%’=y’6'; a result which we shall presently verify. If the reader ask for the particular use of the theory we are now upon, I should reply that the notions of space which the student can and must previously acquire will give a conception of the meaning of diff. equ. which could not otherwise be attained, and will also enable him to single out from the infinite mass of equations which might be proposed, those which admit of being most easily comprehended. These notions of space are difficult in themselves, and so are the diff. equ.; but the difficulties of each being first considered by themselves, the former by geometry and the latter by analysis, the juxta-position of the results throws light upon both. I shall now deduce some results connected with this class of ruled surfaces (page 401) from geometry, and shall then proceed to the consideration of the equations. If through a point (2, y,z) of a surface (S), two planes (A) and (B) be drawn, these planes will make two sections, (AS) and (BS). If at (a, y, 2) two tangent lines be drawn to (AS) and (BS), the plane of these tangents will be the tangent plane of (S) at (x,y,z). For we have shown, page 406, that the tangent plane is in every direction the plane of nearest approach to the surface, and must, therefore, pass through the tangents of all sections; while two straight limes determine a plane. If, then, we can show that a plane passes through the tangents of two sections which meet in a given point, we show it to be the tangent plane to the surface at that point. Let all the generating lines (L) of a ruled surface (S) be pro- jected on a given plane (P). Then there is a curve (C) on (P) to which all these projections are tangents. On (C) as a base, with generating lines perpendicular to (P), draw a cylinder (K), which will, therefore, meet the surface in a curve (KS). And any tangent plane of APPLICATION TO GEOMETRY OF THREE DIMENSIONS, 493 this cylinder will contain, passing through the point at which it meets the surface (S) ; 1. one of the lines (L); 2. one tangent of the curve (KS). Any tangent plane of the cylinder, therefore, is tangent to two sections of the surface passing through the same point ; namely, through that point of (S) which is projected on (C) by a generating line of the cylinder; itis, therefore, a tangent plane of the surface. Next, any plane whatever (A) which passes through one of the lines (L) is the tangent plane of (S) at a point somewhere or other in that line (L). For, if a plane (P) be drawn perpendicular to (A), and the process of the last paragraph be performed, the plane (A), being the projecting plane of (L) on (P), will be a tangent to (S) at the point where (KS) meets (A). Otherwise thus: every such plane (A) meets the surface not only in the generating line (L), but also in another line (M): for the plane (A) must somewhere or other meet the other gene- rating lines, except in these isolated cases in which a generating line happens to be parallel to (A). And at the point where (A) and (M) meet, the plane (A) contains tangents to two sections, and is therefore a tangent plane at that point. We shall now consider some of the preceding points analytically. Take the equations (S), implicitly considering v as a function of @ and y obtained by eliminating z: let z and v be functions of the two independent variables x and y. For convenience’, let z, denote dz: dz, &c. Then we haye 1=d¢v 5 z,+G'v. 2v,+ 0'y oVxn5 0=Y%v oat wiv. 2U,+ by »V2x5 O= Gv. zy4 Pv.zvy+O'v.v,, L=y.2,+Y'v.2v, +0 ae Eliminate v, and v,, and we have, (dropping v), and making z¢’-+ 6’=G, zw’ -+/=H, - “4 H th G ee Bao. py Ne S22) +2(9-9) becomes (£—2z)(¢.H —uv.G)=H (é—¢.2z—6)—G (n—¥. z—), er H(—$.2-6)=G(q—-¥.f—¥) ; which is the equation of the tangent plane at the point (z, y z), and it is _ obviously satisfied as long as (é,, 2) is on the generating line which | passes through (a, y,z). And if Av-+-By+Cz+E=0 be the equation of a plane, this plane is a tangent plane to the surface, if \ and v can be so found that A=AH, B=—AG, C=dA (Gy -H¢), E=A(G¥-Ho), Let a plane be drawn passing through the generating line (L,), whose value of v is v,; whence 2, is a fixed constant throughout this process. The equations of (L,) are, therefore, =f9,+0,, n=l, +¥,, where P, means $v,, &c. Then, because the plane passes through the line just described, its equation must have the form A (£—Z¢,—®,) +B (n—fy,-V¥,)=0, or AE+ Bn—(Ad,+ BY) (—(A®,+ BY,)=0. This, with the equation of the surface, obtained by eliminating v (the arbitrary quantity) from the equations (S), gives the two equations to,the intersection of the surface and the plane, one branch of which is of course the straight line (L,). If we were to make v=v,, the two equations * This will often be useful in mere operations: but the student should read z* as _“d,z, by, d,x,” in the usual way. 424 DIFFERENTIAL AND INTEGRAL CALCULUS. (S) would jointly satisfy the equation of the plane; but if, instead of that, we make v approach without limit to v,, we shall make the point for which the three equations are true approach nearer and nearer to the line (L,), and shall finally obtain that point in which the other branch of the intersection meets (L,). Obtain the ‘value of Z from the three equations E= (644, n=oe+¥P, AE+-Bn—(A¢,+ BY) ¢—(A@,+BY,)=0, which gives = -{A(®—®,)+B(¥—¥,)}: {A(@—G) +B (Y—)}. If v==v,, this takes the form 0:0, indicating that ¢ may have any value, as is the case, since all the line (L,) is part of the intersection required. Butif v approach without limit to v,, we find, dividing the numerator and denominator of the preceding by v—v,, and taking the limits, that the limit of Zis —{A®"+BwW’,}: {Ad +Bw’,}, where ©’, means ®'v,,&c. Let this value of & be called z,; then A (¢\z,+ 6’) +B (Wiz, +¥,)=0, and if from this we substitute the value of A: B in the first form of the equation to the plane, we find (y, 2+) (E —pf—%)=(9',2,+ ©',) (n—¥o—W). Compare this with the general equation of the tangent plane, and it is evident that we have before us the equation of the tangent plane at a point of contact on the generating line which has v=v,, and whose ver- tical ordinate is z,. Thatis to say, if any plane be drawn intersecting the surface in a generating line (L,), and in another branch (M), that plane is a tangent plane to the surface, and the point of contact is the intersection of (L,) and (M). This is one of the theorems which has been proved by geometrical considerations. The preceding illustrations have been drawn from geometry, and applied to a partial diff. equ. of the first order. I shall now show (in the manner of Monge) how similar considerations not only explain the meaning of equations of higher orders, but furnish the readiest mode of obtaining them. If we look at the equations (S), we see, to all appear- ance, four arbitrary functions, >, w, ®, and ¥, and might therefore con- clude that the first partial diff. equ. which is free from these functions will be of the fourth order. This, however, would not be correct; for if gv be called V, we can thence find vin terms of V, and shall have in the equations the quantity V, (which will be eliminated between the ‘equations in forming the equation of the surface,) and three arbitrary functions of it. There are then only three arbitrary functions in the general equations, and the partial diff. equ. is of the third order. To find the partial diff. equ. in the case before us, take any point (x, y, x) inthe surface, and a set of contiguous points made by increasing v, y, and z, respectively by Aa, Ay, and Az, atevery step. It is then the property of the surface that for one set of values of Az, Ay, and Az, or rather for one set of relative values, the points (a+Az, &c.) (v+ 242, &c.) all continue on the surface. If, then, x be the vertical ordinate, we have for values in a certain proportion (say Ar=mh, Ay=mk, Az=ml) the equation s--ml=2+(s,.mh+2,mk) +h (225m h? + 22,,m* hk + zyme?) + aoe for all values of m. Take z from both sides, divide by m, and make | both sides identical, (which they must be since they are true for all | values of m,) and we have APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 425 = « ray © nest ay . 2 Ss P, dae 2pht-bitg los Om 2, h? 4 22, hk+z,, ke, oe ‘ O=2,,, h?-+-32,,, Wk + S2ryy AP + zy, hk, &e. Eliminate h, k, and lin the simplest manner from these, and we have the partial diff. equ. of the class of surfaces. This can be done from the second and third, for the second gives k Lee te ery rf NM (zy 2a Rui) - Cees at er ait 5 SUP DORE this z-—, h yy : The third then gives es Bi Acne o>. BSstrr ks. yy F220 3Az%y “avy +3A 2 yy Ray b Bale SOR PET Ge which is the partial diff. equ. required, and is of the third order, Again, since l1:h=2x,+ zyk:h, and since there is a relation (it matters not what) between /:h and k:h, because there is only one set of proportions of increments at a given point for which the preceding equations are true, /:A must be, on any one surface, a function of k:h. This gives a partial diff. equ. of the second order, which also belongs to the surface, It contaims one arbitrary function. Returning to the equations r=vz + Ov, y=Yr.2+Wv, (in which we write v for dv, since we have shown that one arbitrary function is superfluous,) we see that k:h is dy: dz on the supposition that we pass from point to point on the generating line, v being constant. We have then k:h= Wo: v, which therefore =A: ae Consequently v, bv, wv, and Vv are all functions of A: Z yyy OY We have two more partial diff. equ. of the second order, A A A A t= (= 24m(=), y=x (=).24 *)....@), Zyy) ey “yy yy But these, though they belong to the class of surfaces, do not belong to that class only, since, when integrated, they would each have four arbi- trary functions. To transform them into others containing one only a piece, eliminate z between the first equations, which gives WwW voVo—wyr Oy A A Y- — 2£=———_ 5 OMY art gt eh, St (A): v v myy “yy Also dz: dr=z,+z, dy: dz, or 1 :v==2z,+2,Wv:v 3 whence iby s A A Zz sid, OO ie gives s—(ate—)e=B(=).... OH SF v aa z yy The equations (2), (4), and (5) are the first integrals of the equation (1); to make one more step, eliminate A: z,, between each two of the three, and three equations are obtained, each containing z, and z, only, but with two arbitrary functions. Finally, the pair (3) of diff. equ. of the second degree, and the elimination of A:z,, between them, gives the primitive integral of (1) containing three distinct arbitrary _ functions. ; To verify all these results by actual elimination would be a tedious 426 DIFFERENTIAL AND INTEGRAL CALCULUS. process; I shall here confine myself to one of the same sort, which will verify the condition above obtained as that under which the ruled surface is developable. The condition of these surfaces being =0, we must obtain this function. We have (page 423) 1 1 @ zp’ +0" —=o—¥Z, —= 7 AA en Xe ey 2Us +YW Differentiate each with respect to x and y, and divide —2,,:2; by —z,,4:22, &c. This gives Zoe (B—WD) Ve YDy ty YP T) 1 AGL Ly zy (P WL) ry —YLy yy WPL) yA PLZ, But when the surface is developable these are equal, or (p'—wW'Z) v,—WZ, Bs, (¢'—V'Z) v.92" Z, (pl —WIZ) %y— eZ, (b'—'Z) v,— GL" Z, which gives (f!—¥/Z) (#¥—OZ"*) (Z, y—Z, v,)=0. Now if we equate the second factor to nothing, z, and z, will both be infinite. If we make the third factor vanish, this shows (page 187) that «and y only enter Z through v, whence z is a function of v, and @ and y are functions of v. In the first case (page 193) x aud y must be constants, or it is not a surface, but a right line perpendicular to (vy) which satisfies the condition: in the second case, it is not a surface but a curve, which satisfies the condition. Consequently, ¢’—y’Z=0 is the only condition of a developable surface: this gives dizt+o’ qi Uz ry a “a ~auy xe Syy w/b’ =P'P", as before. If upon any surface we draw a curve line, and through every point of that line draw a normal to the surface, all these normals will constitute a ruled surface: and since every tangent plane of the ruled surface passes through a normal of the surface, it is perpendicular to a tangent plane of the surface. The ruled surface may, therefore, be called a normal surface to the given surface; and it is obvious that the number of normal surfaces which a given surface admits of is infinite, since the number of curves which can be drawn upon the surface is infinite. Every normal surface of a sphere is a cone (or plane); in a right circular cylinder, the normal surface has the axis of the cylinder for its line of greatest density. And since a normal surface may or may not be developable, it will be a matter of interest to inquire whether any and what surface has developable normal surfaces, and how their directing curves are to be drawn. Let z= (a,y) be the equation of a surface, and let yoyo be the equation of the right cylinder which cuts off a curve from it. We have, then, at the point of contact (2, y,7), 6—z=p(§—a) +9 (n—y) for the tangent plane, (€—@#) :p=(n—y) :¢=—(¢—z) for thenormal. These last may be written ; E=—petpe+trz, n=—getoqzt+y; in which -=¢ (z,y), y=Wa, imply that y and z, and therefore p and q, _ may be made functions of a. Let dy: d«=y', and the condition of the APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 427 ruled surface whose equations (x taking the place of v, and & &c. of a, &c.) have just been exhibited being developable, is d.p d d.qg d Fae (qzt+y)= me at ah (pz+x), or — Orb sy) (sz tzy' op + Py + y= — (s+ty' (ret szy'+p*+pqy!+1) y? A+¢s—pgt)-y A+p?t—-1+g2r)+ (pqr—1 +p? s)=0. If the roots of this equation be always possible, a developable normal surface, or rather two, can be drawn through each point of any surface: for if y‘=A +,/B be the solution of the last, we find for y' two functions of z, which being integrated give two forms of y=wWe, which, by the arbitrary constant, may be made to belong to curves passing through the projection of any point of the surface. Representing the preceding equation by Ry”—Sy’+T=0, the possibility of the roots depends on the sign of S‘\—4RT. An artifice of an easy character will save us the investigation of this quantity in its present complicated form. Whatever may be the point of the surface under consideration, the possibility or impossibility of a developable normal surface passing through it does not depend on the coordinate planes chosen: if one or the other case can be shown for any one set of axes, the question is solved. Let us, then, take a plane of zy parallel to the tangent plane at the point in question ; this gives p=0, g=0, and the values of r, s, and ¢, on the supposition made, being 7,, s,, and ¢,, we have sy" —(4-7) y/—s,=0, of which the roots are both possible, since the first and third terms have different signs. Again, the values of y/ are tangents of the angles made by the tangent lines of the projections with the axis of x: let these be € and €,, then it follows from the preceding that tan €.tan €,= —1, or € and €, differ by a right angle. But in the simplified case, the normal is the continuation of the ordinate z; and the normal planes drawn through the tangents of the curves make angles € and 6, with the plane of zz: that is, since € and 6, differ by a right angle, these normal planes are at right angles to one another. If, then, through any point of a surface the two curves be drawn, the normal surfaces of which are develop- able, the tangents of these curves are at right angles to one another, and also the normal planes drawn through those tangents, I defer further cousideration of these normal developable surfaces until after the establishment of their most important use, which arises dut of their connection with the curvature of surfaces. We have already considered the contact of a tangent plane with the surface ; we shall now pursue this subject a little further. It has been shown that when rt—s* is negative, the tangent plane cuts the surface. Consequently, at any point so circumstanced, the tangent plane must neet the surface in a line: we now ask under what conditions does the angent plane not only meet the surface in a line, but continue to be the angent plane at every point of that line (a table, for instance, is a tan- gent plane toaring placed upon it at every point of the circle of coin- idence.) This obviously requires that we can, by going from point to * A solution of this problem, in an elegant and general form, may be found in Ol. ii. page 22,o0f the Cambridge Mathematical Journal, (Whittaker and Co.,) a rork which I strongly recommend to the student of analysis 428 ‘DIFFERENTIAL AND INTEGRAL CALCULUS. point of the surface in a particular way, keep the equation -z=p (f-2) -+q(7—y) representing the same plane; or p,q, and z—px ~qy must remain the same. Taking a point (2, y, z) on the curve of intersection, let (w+dza, &c.) be the contiguous point, and let y= Wear be the equation of the projection of the curve on (xy). ‘Then it is the condition of the curve that p and g remain unaltered as long as dy=w¥x.dx. But. dp=rdx+sdy, dq=sdx-+tdy; whence 0=rdx+swW/edzx, O=sdx+tb'x dr, or rt—s’=0, which, 7, s, and ¢ being functions of x and y, gives a relation of the form y=wa, which is the equation of the projection of the curve, if such a curve there be. Again, dz=pdx-+qdy, and if p and q can be made constant, we have z=px-+qy+C, whenever y is taken such a function of 2 as makes p and q constant. The only question remaining is, does it follow conversely that » and q are constant when y is so taken in terms of x that rt—s*=0? Assume this last, and add together the squares of dp and dq as above obtained, putting ré¢ for s? wherever it occurs. This gives dp’-+-dq?=(r-+t) {rda°+2s dxdy+tdy*}. Now this must be =0, for going in the direction required, there is no deflection from the tangent plane, and the terms of the deflection which are of any given order, must collectively be =0, and rdz*+2s dx dy +tdy* among the rest. Hence dp?+dq°=0, which requires dp=0, dq=0, or else shows that the curve is impossible. Consequently, when rt—s*=O0 gives y=Wa in such a way that there is a real intersection, that intersection is a plane curve, and its plane is the tangent plane to the surface at every point of the curve. Accordingly, we see that in developable surfaces, the tangent plane is everywhere tangent at all the | points in which it meets the surface. We might next ask, by analogy, what is the closest sphere which can | be drawn to the surface at a given point: but here we shall immediately see that though we can find-an infinite number of spheres having a con- tact of the first order, it can only be at certain points, if ever, that a | sphere can be made to have a complete contact of the second order. For there are but four constants in the equation of the sphere, while up | to the second order inclusive there are five diff. co. If, therefore, we dispose our constants so as to make the sphere pass through a given — point, and to make p, g, and r the same in both surface and sphere, we shall have no arbitrary quantities left to which to assign values which © shall make s and the same in both. There must then at least be six constants in the equation of any surface which can certainly be made to have a contact of the second order with any point of a given surface. Abandoning, therefore, the idea of estimating the curvature of a’ surface at any one point entirely by that of another surface, let a normal | be drawn through the point in question, and let a plane revolve about | this normal as an axis. This plane will make with the surface an infinite number of sections, one in each of its positions. Let these be called normal sections. We shall estimate the curvature of the surface by finding relations between the curvatures of the normal sections. And as our present object is to find absolute properties, independently of any position with respect to coordinates, let us take the point under examina- tion for the origin, and the tangent plane for the plane of zy. Let P, Q, | | APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 429 &c. be the values of p, y, &c. at the origin; then, because the tangent plane at the origin is that of ry, its equation (or =p£+ Qn) is Ga! or P=0, Q=0, whence the equation of the surface is 1 l sb CBee as ag ET) bo (terms with 2°, a*°y, &e.)+.... Lct R, S, and T, &c. be finite, whence the terms of the third order diminish without limit compared with those of the second, as x and y diminish. Let O be the origin, OX, OY, and OZ the axes, OAPBa portion of the surface, OPM a plane passing through the normal OZ, and making an angle MOK=6€ with the plane of xz. Let OP bea part of the normal section of this plane, OG, GM, and MP the coordinates of Hi a point in the section. If, then, OM be called x,, we have, for the curve OP, r=7, cos 6, y=x,sin€6, and substitution gives for an equation between 2, and z the coordinates of P in the plane ZOM, = (Reos’6+28 cos€ sinE-+T sin? 6) x? Avi+Bai+ &c., where A, B, &c. need not be calculated, Now if the equation of a curve be z— 4 ar, + Axt+ -+++, we have at the origin 2/=0, 2’'=a, whence he radius of curvature at the origin is l:a@. This theorem is often proved by supposing OP to be an infinitely small are of a circle, so that he rectangle of PM and the rest of the diameter is the square on OM, m the diameter is x°:z, when 2, is infinitely small, which is 2:a. Whichever way we prove it, the radius of curvature of the section OP 8 1:4, or, calling it p, we have — i ° PR con? 6+25 cos €sin6+T sin? 6’ x the curvature, which is inversely as the radius of curvature, varies with Rcos*€+&c. We shall use this latter phraseology, the student emembering that the greatest curvature has the least radius of curva- ure, and soon. And though we have drawn a figure corresponding to urvature in which all deflections from the tangent plane are made on ne side, yet it must be borne in mind that if the tangent plane cut the urface, z, and with it the radius of curvature, will be negative when the leflections are negative. The expression on which the curvature depends may be easily hanged into the form A cos? (€—g¢)+Bsin? (€—«): for if we expand 430 DIFFERENTIAL AND INTEGRAL CALCULUS. cos (€—a) and sin (6—«), and develope their squares, we find that the result is made identical with R cos*€+ &c., by assuming Acos’¢+B sin?a=R, (A—B) cosa.sina=S, Asin’e+B cos’ a= T, which give R—T= (A—B)cos2a, and tan2¢=28: (R—T). This gives for 2a two values differing by two right angles, and therefore for two values differing by a right angle, and one of these is less than a right angle; let it be the one chosen. Therefore sin 2¢— 28 :(+,/{48? +(R—T)*}, which must be positive, since a Rv? +28 xy + Ty’ is the equation (or more nearly so the smaller x and y are taken) of the projection KHL of the section APB of the surface and plane (BL, PM, &c. being 5). But this is the equation of acurve of the second order, whose centre is at the origin; and if 23 be changed into 1, it will remain the equation of a curve similar in all Tespects, but larger in linear dimension in the proportion of /(26) to 1. Now if the axes of w and y revolve through an angle @, being the least of those determined by tan 22=2S: (R—T), the equation of the curve will then be 1=A2®+ By’, where A and B are precisely as before. If, then, 9 be the angle made by a radius vector r with the new axis of x, we 'shall have 1:7?=Acos?@+Bsin?@. The lines of the second degree which have a centre are the ellipse, hyperbola, and (not the parabola, but) that extreme variety of the parabola which consists of two parallel straight lines. Hence the following theorem: if at a given point of a surface a plane be drawn parallel to and very near the tangent plane, cutting the surface, the parts of the section closely contiguous to the point of contact will be very nearly parts of a small curve of the second degree, and the"more nearly the closer the intersecting’ plane to the tan- gent plane. And if a curve of the same kind be drawn on the tangent plane about the point of contact as a centre, similar to the small curve, and similarly placed, but so much larger that ,/(25) in the smaller shall be 1 in the larger, the square of the radius vector on this curve (numerically considered) will be the radius of curvature of the normal section which is touched by that radius vector. Remember, that in the hyperbola, though the radius vector is impossible in one pair of Opposite 432. DIFFERENTIAL AND INTEGRAL CALCULUS. asymptotal angles, its square is not impossible, but negative, and is the square of the radius vector of the conjugate hyperbola taken negatively. The following method of using this theorem will perhaps explain the theorem itself. Given the magnitude and sign of the principal radii of curvature, and their directions, required the radius of curvature in any other direction. First, if both be infinite, all radii are infinite, and the tangent plane has a complete contact of the second order with the surface. Next let OB and OA be the principal directions, and let the radius it the direction OB be infinite, that in OA being OA. Let OK=.,/OA, take OL=OK, and through K and L draw lines parallel to OB. If the curvature be finite in both directions, take OK and OM=,/OA and OB, without reference to sign, and with OK and OM as principal axes describe an ellipse, if OA and OB agree in sign, and a pair of con- jugate hyperbolas if they differ. Put these figures on the tangent plane, O at the point of contact, OA and OB in the principal directions of cur- vature. Then, for every point Z, the square of OZ is the radius of cur- yvature of the normal section which cuts the tangent planein OZ. In the first figure this is to be taken of the same sign as OA, in the second of the same sign as OA or OB, and in the third it is to have the sign of OA or OB. according as the hyperbola on which it is passes through (K, L) or (M,N). As yet we have only considered sections made by planes passing through the normal; we shall now suppose a section which declines from the normal by an angle y. As the theorem we are now going to prove is isolated, I shall give a demonstration of it which assumes the infinitely small arcs of the sections to be parts of the circles of curvature, leaving the student to try if he can express the equations of the sections, and thence determine the curvatures in the usual manner. Let OX be a line in the tangent plane, and take it as the axis of # let OM be the normal section passing through that tangent, and let PO be an oblique section in the plane PNOA, making with, ZOMN an angle AOZ=y. Let OQ be the projection of the section OP on the plane of XY. Then, since the equation of the surface 1s Q2= Ra’ +28 cy+Ty+ Ke. ; and since ON=a, we have INM=R2z’*+ &c. (since y=0 for all points in APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 433 OM.) Again, since ON is tangent to OQ, NQ diminishes without limit compared with ON; so that 2S xy and Ty? are of the third and fourth order, or 2PQ=Ra?+.... Consequently the limit of PQ: MN is unity, or PQMN approaches without limit to the form of a rectangle. Taking OP and OM for small arcs of circles, their diameters are the limits of ON?: NP and ON®: NM, and diam. of OP: diam. of OM is limit of NM: NP, which as PMN approaches to a right angle, has cos PNM, or cosy for its limit. Hence, if OZ be the diameter of curva- ture of the normal section, and ZAO a circle with OZ for diameter, OA is the diameter of curvature of the oblique section. Or, all the sections made by planes drawn through one tangent have for their diameters of curvature the chords of a circle which has the diameter of the normal Section of that tangent for its diameter. And if the given tangent be made the axis of wv, and the circle be drawn in the plane of yz, any chord, with the common tangent, determines the plane of the section which has that chord for its diameter of curvature. I shall now show that the two normal sections, perpendicular to each other, of greatest and least curvature, are in those directions already obtained, in which the consecutive normals intersect the normal at O; so that the principal normal planes are tangents to the developable normal Surfaces which pass through the point O. Taking z=3(Ra’?+2S ay +Ty*)+&c., (remember that +&c. throughout refers to terms which diminish without limit as compared with those which precede,) we find for the equation to the normal at the point (2, Y, 2) ae ee el vsiektl 7 o =—(f—z); Ra+Sy-+ &e. ~ Se+Ty+&c. whence (Rx+ Sy) n»—(Sr+Ty) &=S (y?—2*) + (R—T) xy, neglecting terms which have no effect on the limit, is the equation of the projection of this normal on the plane of zy. Here, then, are two straight lines, the axis of z, (z), and the new normal (v) projected on the plane of (ry) into (v,), of which the equation has just been found. Hence it may easily be shown that the perpendicular let fall from O upon (»,) is equal and parallel to the shortest distance between (v) and (). But if ay—bx=c he the equation of a straight line, the perpendicular let fall on it from the origin is c:,/(a°+b?), giving {(R—T) ry—S (@®—y)} Jf { (Re +Sy)?+ (Sr+Ty)*} 2¥ 434 DIFFERENTIAL AND INTEGRAL CALCULUS. for the shortest distance between (x) and (v). But if two consecutive normals, infinitely near to one another, are to meet, (page 412,) this shortest distance must diminish without limit as compared with @ or y when the latter diminish without limit. Let the point (a,y) move towards the origin, and let y=. tan €, whence the preceding expression becomes xz{(R—T).tan€—S (1 —tan? €)t: J{(R+S tan 6)?+ (S+T tan €)*}, which cannot diminish without limit in comparison with 2, unless (R—T) tan€6—S (1—tan®6)+(the terms of a higher order neglected) diminishes without limit; and this cannot be unless (R—T) tan 6— S (1—tan®€)=0, or tan 26=2S:(R—T). But this is the formula by which the angles of the principal sections of curvature were obtained ; whence the theorem above stated. a It appears, then, that every surface may be traversed by an mfinite number of curves, two of which pass through every point, indicating by their tangents the directions of least and greatest curvature. And it Is the property of each of these lines that normals to the surface drawn through the several points of any one of them, lie on a developable surface, and are tangents to a common connecting curve.” If a moving point were obliged to seek its course so as always to take the most or least bent track, it would move on one of these curves. With this general knowledge of the subject, we shall now look for the means of finding the curvatures, &c. with any origin and any axes. The equations of the normal at a point (a, y, z) being (E—x) +p (—2)=0, (n—y) +g (6-2) =0; if we take an adjacent point, (7+dz, &c.), at which the normal is in the same plane with the one just given, there will be a point of inter- section (X, Y, Z) which is on both normals, or will satisfy (X-2)+p(Z—2)=0, (Y—-y)+q(Z—2)=0, (X—a—dx)+(p+dp)(Z—z—dz)=0, (Y —y—dy) + (q+ dq)(Z—z—dz)=0. Subtract the first set from the second, rejecting from the latter terms of the second order, and we have dp (Z—2z) -pdz—da=0, dq (Z—z)—qdz- dy=0. The elimination of Z—z gives dp (qdz+dy) =dq (pdz+dz), an equa- tion already obtained, and which gave (page 427) yo dy —. -— rornnn Te G+ ¢s—pq-)—+ (La ptt—1+q'r) +pq.r—ltp's=0.. .(y)3 é aL and the first two equations may be written (dy: dx being 7’) eat AEs i pe ee gig ers aim pape (2) +P eh) whence G 2 y ? ; y’ {t(Z—2z)—(1 +9") } +5 (Z—2z) —pq=0 * One sound writer on this‘subject (and perhaps more) has attempted to translate the words aréte de rebroussement into English by edge of regression, which seems to me a closer imitation of the words than of the meaning. Many words might be suggested, such as the ligature of the normals, or their osculatrix, or their omnl- tangential curve. Also with reference to the developable surface, the aréée, &e. might be called the generatrix, or the curve of greatest density, &c. APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 435 {{(Z—2) -(1+q))} {r (Z~2)—(1tp)}-{s (Z—2)—pq}’=0, ot Z; (rt—s°)—Z, (1+ q°.r— 2pq.s+l1 +p .t)+(1 4+-p*--9")==0, where Z,=Z—z. If we make I+p'=R, pg=8, 1+q?=T, the equa- tions which produce the above and their results, take the following symmetrical forms, sy’+r Sy'+R ty'+s Ty’ +S’ (sT—tS) y®—(tR-rT) y+ (7S—sR) <0 (rt—s*) Z?—(rT — 258+ 4R) Z, + (RT—S*)=0. Let V=p: /(1+p*+ q’), W=q: V(1+p?+9?), then Vy=Vps+V t=(+p°+¢)? {(1+¢@) s—path, ke. ; whence (y’) becomes V,, y”—(W,—V,) y’—W,=0, which when V and W are turned into functions of x and y by the substitution of the values of p and g, will be an easy form for ‘calculation, Putting RT— S$? for 1 ++-p?+ q’, we find V./(RT—S8")*= (1 +9") r— pqs Tr—Ss V,/(RT—S*)}*= (1+ 9°) s— pyt=Ts—St W,/(RT-S*)*= — pqr+(1--p*) s= —Sr+Rs W, /(RT—S’)'=—pqs+ (1 +p*) t=—Ss+Rt; whence (RT—S*)°(V, W,—V, W,) =rt—s*, V, y” —(W,—V,) y—W,=-0 (V, Wy—V, W.) Z?—(V,+W,) (1+ p?+4°)?.Z,+ (1+ p?-+q) 120. If X—r=X,, Y—y=Y,, we have for the square of the radius of curvature X°+Y/°+Z,*, or (rad.)?=Z? (1+ p?-+q") ; whence the values of this radius are determined from (V, W,—V, W,) (rad.)?— (V,+W,) (vad.)+1=0. Hence 2V,y/=W,—V.+,/(W,—V.+4V, W,) 1 2 (V,W, —V,W.) a =W,+V.5/{W,—-V.+4V,W,}=W,+V,$/H. rad, (rZ,—R)(Z,—T) = (sZ, —8)* rad. ={W,+V. +, (W,—V.+4V, W,)} It is important to determine which signs are to be used together. Let Z, and Z, be the two values of Z,, and y’, and y’, those of y/; then R+Sy’ .. (Sr—Rs)(y',—y’s) 7 gives Z,—Z.= = eat r-+7Ts (yi +ys) +5 Yi¥e In the denominator, substitute for y'+y’, and y', y's their values (W,-—V,) :V, and —W,:V,, and substitute for W,, &e. their values. This will be found to reduce the preceding fraction to (y/:—y's) Vy V(1 + p?+9°)8: (srt). Now, dividing the expression for rad. by (1+ p*+9°) to give Z, and looking at the difference of the values, we see that we shall get by substitution yf Y= tH: V, and Z,—Z,=+,/(H)/( +p°+q°)*:(s*—rt), so that (Z,—Z,) : (y/,— y's) 2F2 r ae Z= 436 DIFFERENTIAL AND INTEGRAL CALCULUS. is £/(1+p"+ @°) V, : (s?—rt) the upper or lower sign being used accord- ing as y' and Z, have radicals of the same or different signs. Con- sequently, since /(1-+p*+q°) was taken positively throughout, we can only make the latter form of the ratio agree with that directly deduced by giving the same signs to the radicals inthe corresponding values of Z,and y'. The most embarrassing part of this subject is the representation of the results to the eye: and I here digress to describe the best method of doing this. The perspective employed should be the orthographic, in which the eye is at an infinite distance from the plane of the picture ; or, to avoid the physically impossible character of this supposition, say at a very great distance compared with the linear dimensions of the picture. The properties of this projection are, 1. All lines or planes perpendicular to the plane of the picture are projected into points or lines. 2. All parallels are projected into parallels. 3. Equal lines, when in the same line or parallel, are projected into equal lines. 4. Equal lines, not parallel, are projected into lines proportional to the cosines of the angles they make with the plane of the picture, or the sines of the angles they make with lines drawn to the eye. If the line drawn through the eye make equal angles with the three axes, the projection is called zsometrt- cal:* it is inconvenient when there are any lines in the figure nearly equally inclined to the axes, and generally, the line drawn to the eye should not make small angles with any of the ‘principal lines of the figure. The following proposition will complete the theory of this perspective, so far as its application to rectangular coordinates is con- cerned. Let OA, OB, OC be the pro- jections of the three axes ; from any point D in OC produced draw EF perpendicular to CD, and draw FG perpendicular to EO produced ; join EG. Then will GEF be the projection of a triangle parallel to the plane of projection, so that EG, GF, FE are not altered by projection: and OK, OF, and OG will be the projections of lengths which are severally mean propor- ; tionals between EO and EH, FO and FK, GO and GD. Equal lines, therefore,t can be readily laid down on the three axes, and thence lines in any proportion. * The isometrical perspective was first thought of as the most convenient mode of representing machinery, &c. by the late Professor Farish: there are now, I believe, several treatises on it. + Ishould recommend those who wish to draw with tolerable correctness to have several cards or pieces of wood made as follows, to as many different species of pro- jection as may be wanted. The card or block COBVW admits of the three axes being immediately laid down by placing it on the paper and running a pencil along the edges CO, OB, and into the slit OA. Scales of parts answering to the projections of equal parts are laid down along the three axes, and repeated on the unoccupied sides. The position of a point B whose coordinates are given is then immediately found by taking off the coordinates on the axes. and using a parallelruler. The best way of laying down the different scales of equal parts is by observing that their units on OG, OK, and OF must be as the square roots of the sines of double the angles at G, E, and F: also the angle at G is the supplement of EOF, &c. See the Cam- bridge Mathematical Journal, vol. ii. p. 92. APPLICATION TO GEOMETRY OF THREE DIMENSIONS, 437 The diagram before us represents in three positions the projection of the lines of curvature of an elliptic paraboloid, to which we shall pre- sently come. In the middle figure, O (hidden by the solid) is the origin, and the line drawn to the eye is meant to make equal angles with OX and OY, and a much larger angle with OZ. This figure contains one quarter of the frustum of the paraboloid. On the right we see two quarters projected on the plane of ZX; the axis of y passes through the eye and is invisible, and the point Q of the last figure is now confounded with Z. On the left we also see two quarters projected on the plane of ZY, the axis of x is now invisible, and P and Z are confounded. Let 22=a2°+ by’ be the equation of the surface: that is, let it be an elliptic or hyperbolic paraboloid, according as @ and b have the same or different signs, the axis of z containing the foci of the principal parabolic sections (A. G. 422—500). We have then par, q=by, r=a, s=0, t=, rl—s’=ab; whence the equation for determining y is ab’ xy .y” + (b—a+a? bz*— ab? y’) y —a°b xy=0, or making (b—a) :ab°=B, a: b=A, ry .y°—(y’—Aax?—B) y’—Ary=0...... (7). This equation (and many others of a higher degree than the first) is most easily integrated by forming the diff. equ. of the next order: if this - last can then be completely integrated, it will have two new constants, between which an attempt to verify the given equation will give a relation which assigns one in terms of the other. Make a transforma- tion of the preceding equation, differentiate, and eliminate B as follows: (yy' + Ax) (ay! —y)+By'=0, (yy! ty? +A) (ry’—y) + (yy! +Ax) zy" +By"=0, yyy" Fy" + A) (zy! —y) + (yy! +Ax) ay'y"—(yy'+A2)(ay/ —y) y"=0, or (YP TA) {@y—y) ¥ Fry} = ; the first factor, 7/?+4 A, being made =0, may give a real* singular solu- * It will be found, however, on examination, that y= A (—A).2+,/B is the Singular solution, and it will be readily seen that —A and B cannot be positive together. 438 DIFFERENTIAL AND INTEGRAL CALCULUS. tion, if A be negative: if we equate the second facior to 0, observing that it is the diff. co. of (zy’—y) y, we find (vy'—y) y= C’ for a step in the solution, and if y:#=v, this is vw ay=C', or ve B= C’. This gives v2=—C'a *°-+C or y?= Cxz?—C! for the complete solution. Hence yy'=Cx; substitute these in the given equation after multiplying it by y, and we have C? a8 (Cx? — C/—Axv®?—B) Ca— Az (C2*?—C')=0, which is identically true if CC’+BC+AC'=9, or C’=—(BC) : (C+A). Hence (b—a) C ab? C+a% is, for every value of C for which y can be real, the equation of the projection upon zy of a line of greatest or least curvature of the paraboloid: and it is generally the equation of an ellipse or hyperbola, according as C is negative or positive; but its theaning will require exaimination. First, we do not seem to have drawn any distinction between lines of one and the other curvature, since (y’) has been completely integrated in (C). But if we now require a curve (C) which shall pass through a given point (X, Y,4aX’+36Y"), we find that C must be determined by an equation of the second degree, which, reduced, is ab? X?2 C-++ (b—a-+a? b X?— ab? Y2) C—a’ b Y*=0..... CC eek yf Ge .(C) There are always two roots to this equation, one positive and the other negative, when a and 6 have the same sign, and both positive or both negative, when a and 6 have different signs. Consequently, in the elliptic paraboloid, the projections of the lines of one sort of curvature are ellipses, and of the other sort hyperbolas; but in the hyperbolic para- boloid they are both hyperbolas. First, let @ and b have the same sign, which may be positive, and let b>>a, or let the parabola in the plane of zy have a greater curvature at the origin than that m 2a. Now one value of C is =0 when Y=0; that is, the section of the surface with the plane of 2 is itself one line of curvature. Again, C has one value infinite when XO; or the section in the plane of zy is a line of curvature, When C is negative, y, in (C) is impossible, unless ab? C+a°b be negative, or unless C be numerically ereater than a:b. If from 9z=a2°+ by and (C) we form the equations of the projections of these curves upon zx and zy we have the parabolas ae of ee CO SAYC BG ie eM Oe Milas athe: 2e=(a+b0) a+ Tae 2a=( b+ a )Y Rae We have, as already stated, only to consider the values of C from 0 to oc, and from —a:4 to —o, When C diminishes from @ to 0, remembering that C= oc gives x=0, C1?=0, we see that the projections on zx vary in their equations from 22=(b—a): ab to Q2z=aw", indi- cating, as seen in the right-hand figure, évery sort of parabola between the limit UZ (which is a straight line) to OP itself. But on the plane of zy wé,sée that 22=by* and Q2=>+(b +a): ab are the limitsy andin every parabola 2 is negative when’ ¥ is’ 0, giving, as’ in’ the left-hand figure, all kinds of paraholas, drawn about vertices from z=0 to APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 439 2=—(b—a):ab. And the projections on ay are a family of hyperbolas, of which we may get a good idea by imagining the ascending parabolas in the right-hand figure to be the bases of cylinders, which obviously cut the surface in curves which project on the plane of zy into pairs of curves with two infinite branches each. If we now suppose C to vary from — o to —a:b, we find the equations of the projections on zz varying from 2z= — c.v*+(b—a):ab to 2z— cc, while the inter- mediate form is 2z= (neg. qu.) z*-+(pos. qu.) We have, then, as in the right-hand figure, a succession ‘of parabolas turned the other way, having for one limit the line UO, and rising ad infinitum. On the plane of zy, the equation varies from 2z=by* to 2z= o, and its inter- mediate forms are 22—=(pos. qu.) y'—(neg. qu.), belonging to parabolas turned upwards. We have, then, the other set of parabolas in the left- hand figure, beginning with Q’/OQ. The equations of the projection on the plane of cy now belong to ellipses, and if we were to form parabolic cylinders from the parabolas just described in the right, they would obviously cut the surface in curves which would project on the plane of zy into figures resembling ellipses. We shall now consider the case in which @ and 6 have different signs, or the hyperbolic paraboloid. Let 6 be negative; then the parabola OQ must be turned round the axis of y until it is below the plane of xy in the plane of zy, and a parabola equal to OQ moving parallel to the plane of zy with its vertex on OP, will describe the surface. If for 5 we write — 6, the equations of the projections become (+a) C aprC—~ ah’ * a », Ota 2e=(E—D)y Si a If C be negative, the first equation is impossible: in fact, it will be seen from the equation (C, X, Y) that when a is positive and b negative the values of C are both positive. As C varies from 0 to oc, a change takes place in the character of the projections when it “passes through a:b. When Ca 2b. ‘First, let C change from 0 to a@:6; the equation of the second projection, then, varies from 2z=az* to 2z=— x, the intermediate form being 2z= (pos. qu.) «*—(pos. qu.) ; while that of the third varies from ae cy —(b+a):ab to 2z=c, the intermediate form being 22:= (pos. qu.) ¥° + (neg. qu.). These parabolas are seen in the next diagram with their branches going upwards, though in the projection on ZOY, a part on each side of the vertex does not belong to the projection. When C varies from a:b to cc, the projection on zz varies from 2z== & to Jem — oC t+ (b+ a) :ab, the intermediate form being 22—(neg. qu.) a — (neg. qu.) 5 while that on zy varies from 2z—=cc to 22 — by’, the intermediate form being 2z= (neg. qu.) y+ (pos.qu.) way We now pass to the consideration of the coordinates of the centres of curvature (X; Y,Z). We have, (page 434,) 7 being Cai y, (6+a)C A pale) y= Co’— nat Se J a?—ab C? 440 DIFFERENTIAL AND INTEGRAL CALCULUS. = R+Sy7 _14+p°+pqy’ _ (1+a*2*) tab Ca” r+sy' r+ sy! a rad.= a7 (1-0? 2?-++-ab Ca*) JL +e 2? +b’) ; where the two values of C are to be determined from (C, X, Y) for each point. Having drawn all the lines of curvature, we proceed to distinguish those of greatest and least curvature, which we shall do in the elliptic para- boloid, leaving the other to the student. Taking the projection upon the plane of zr, let it be remembered that for the ascending curves, C is positive, being nothing on OP, and infinite on UZ: while in the equa- tions of the descending curves C is negative, being infinite on UO, and continually diminishing (numerically) towards —a:6. And the co- ordinates of the point U are r=0, y=,J {(6—a) : abt, z= (b—a) : 2ab. When a and b are both positive, the equation (C, X, Y) shows that C has one positive and one negative value: and the expression above given for the radius of curvature is the greater of the two when C is positive, and the less of the two when C is negative. Hence the projections just described as having positive values of C belong to the curves of least curvature, and the others to curves of the greatest curvature. Hence the curve QUOU’Q’ (seen laterally in the figure on the left) is a line of greatest curvature from U to U’, and of least curvature everywhere else. Therefore the difference of the radii of curvature changes sign at U and U’, on the supposition that a point moves along the curve QOQ’: that is, this difference becomes nothing at U and U’, or the radii of curvature are then equal. A point of this kind, which is so situated upon a line of curvature that the arcs on the two sides of it are of different species of curvature, is called an umbilicus, or umbilical point: though it must be noted that the term is extended to every point at which the two curva- tures are equal. Since C is infinite at every point of the curve QUOU/Q,, and vis nothing, the term Ca2® in the expression of the radii is ambiguous. Return then to the equation by which Z—z, or Z,, is determined, and we find ab Z2—{A ey?) a+ +ea*) Zt +a eto y')=0. The values of Z, are the projections of the radii of curvature upon the axis of z, and will be equal when the radii are equal. Apply the test for equal roots to this equation, and it will be found, after reduction, that there are equal roots when {b—a—ab (by?+az°)\?4-4ab (ba) ar°=0; ~ tw APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 44] an equation which (b>) can only be satisfied by x=0, y= (b-a): ab’; that is, only at the points U and U’. The following problems may be easily solved from the preceding equations, 1. Neither radius of curvature is ever equal to nothing, unless at a point for which rt—s? is infinite, or infiuite, unless at a point at which ré—s*=0. And one of the radii of curvature is infinite, at every point of a developable surface, and the converse. _ 2. When the radii of curvature are equal in magnitude, but different in sign, (1-+4*) r—2pqs+(1+p*) t=0; and this, when true at every point of a surface, is the equation of a surface at every point of which the radii are equal and contrary in sign. 3. The last equation is satisfied by that of a plane: in what sense can this surface be said to have the property which it implies ? 4. The points at which the radii of curvature are equal, and of the same sign, are determined by the equation {(1+4q°) r—2pqs-+ (1+ p*) (P= 4 (rt—s)(1 + p*+4q°), or {Tr—28s+Ri?=4 (rt—s*?) (RT—S"), or (Tr—Rt)’+ 4 (St— Ts)(Sr—Rs)=0; which is satisfied by R:r==S:s=T:¢, and by nothing else.* I shall now briefly give the manner in which Monge shows that R:r=S:s=T:t, or Ts—St=0, Rs—Sr=0, can only belong to a sphere. From the equations in page 435, these give Ni aa0 aWies 0} whence V can only be a function of z, and W of y; that is, p=or.JA+p?+q), qavy./t+p+q), or p=hr{1—(pr)*—(wy)*}?, gq wy {1—(92)*- (Ha2)’} But dp: dy=dq: dx, which it is found will require ¢’xr=w'y to be true, independently of any relation between y and x. This cannot be unless ¢’x and yy are both constants, giving ¢r=ca+h, wyscyt+h,. * Solve the preceding equation with respect to S, and a result will be found, the reality of which depends on that of ,/ (s?—rt). But from the equation preceding that which was solved, since RT—S* or 1+p?+g* is necessarily positive, it follows that 7¢—s* is positive or s*—7¢ is negative. Hence no real relation can exist except the pair of equations which make the given equation identical. There is in the Application, &c. of Monge (page 171, edition of 1807) one of the most curious chapters which ever appeared on the subject: the remarkable part being the manner in which he has allowed the gradual correction of a false impres- Sion to appear, which most persons would have avoided by rewriting the whole Section. He is obviously, up to the chapter in question, under the impression that there exist other surfaces besides the sphere of which all the points are umbilical ; as appear both from his previous allusions to the coming chapter, and from the Manner in which he opens it. Setting out under this assumption, he proceeds to integrate the equation, in which he succeeds, but in a manner which gives two equations between a, y, and z, instead of one, from which he infers that the equation only belongs to a curve, instead of a surface. This extraordinary result, as he calls it, (still ne\ er looking to see whether the duplicity of the conditions was not implied in the fundamental equation,) he proceeds to verify, by attempting to construct a surface of the given kind in the form of a connecting surface ofa family of spheres, The result of this investigation is that the radius of the moving sphere is always 4 which reduces the surface again to a curve. 442 DIFFERENTIAL AND INTEGRAL CALCULUS. Let these be substituted, and the method in page 197 followed, and it will be found that (cz+h,)?+ (cy+hk) + (ce+k)'=1, which is the equation of a sphere. I now give a professedly incomplete demonstration of the method of drawing the shortest line between two points of a given surface: that is to say, incomplete, inasmuch as the considerations here laid down must be much developed and made more rigorous in form, before conviction could be brought by them to the mind of a beginner. The subject will be more fully treated in the next chapter. First; if a tangent be drawn through a given point of a curve, and also a very small chord, the plane of the chord and tangent may be brought as near as we please to the osculating plane. For if the curve had not two curvatures (page 413) that plane would be the osculating plane itself; and the smaller the arc taken, the smaller is the effect of the second curvature, or the more nearly does the ,plane of the tangent and chord coincide with the osculating plane. Secondly ; ifa very small chord be drawn to a curve which hes ona eiven surface, the shortest line which can join the ends of that chord on the surface must be that which is nearest to the chord itself, the latter being the absolute least distance between the two points. The smaller the chord, the more nearly is* this line situated in a plane which passes through the normal of the surface. Thirdly ; if the shortest line be drawn from A to B on a surface, and if C and D be any intermediate points, however near, then CD must be the shortest line on the surface between C and D: for if a shorter line could be drawn between C and D, it is obvious that a shorter path could be made from A to B. Hence, if the are CD be made infinitely small, the plane of its chord and tangent, which by the second consideration is normal to the surface, is by the first the osculating plane of the curve: or the osculating planes of the shortest line between two points are at all points perpendicular to the tangent planes of the surfaces drawn through those points. Thus much being admitted, the equations of the shortest line readily follow. Let s, the arc of the curve, be the variable in terms of which x, y, and z are expressed, so that a’=dx:ds,&c. Let ®(a,y, 2)=0 be the equation of the surface, ®,, &c. being the partial diff. co. of ®. Then, since the curve is on the surface, we must have ,. a’ +0,.9/ --®,.2/=0, while the expression of the tangent plane of the surface at the point (2, y, ) being perpendicular to the osculating plane of the curve is obviously ®,.2,,+0,.4,,+®,.2,=0, (page 407 and 409, and A.G. p. 219), or (o, 2" —®, y") a! + (@, a —, 2") y'+(@, y"” —4@, x”) z'=0. But since ©,.2/+&c.=0 and x.’ +&c.=0, it follows that a’, y’, and 2’ are in the proportion of ,2—®,y", &c. If, then, 6,2” —@,y" =r’, we must have ©, 2’—, 2""=oy! and ©, y!/—®, #”=az', whence the last: equation. gives «(v?-+y?+2")=0, or «X1=0, or,¢=—0. ‘That is, the diff. equ. of the shortest line drawn from one point to another * I have introduced this here that the student may try to see it: it is not demon- strated. APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 443 on the surface © (+, y,2)=0, exhibited in an unabbreviated form, are any two of the three db dz dé dy do dx _d6 de db dy d& dr a oT) iy St us 7 br elle ory ls aotearoa dy ds* dz ds dz ds* dx ds?’ dx ds dy ds’ I say any two of the three, because either of the preceding is a necessary consequence of the other two. These may be reduced af d=0, give 2= (x, y)) to the form mo SS exe), — + 0 ) Set yy el ds’ wha ds* ds? re asic vee ds? i When the surface is one of revolution about the axis of z, we have z= (x°+-y"), or py—gqr=0: and substitution in the third equation gives d’y ax r— +9 730, or edy —ydx=cds, or r°d0=cds ; rand @ being the polar coordinates, in the plane of xy, of the point (v,y). Hence, if the shortest line between two points on a surface of revolution about the axis of z be projected on the plane of xy, and if a point moving along it described equal arcs in equal times, the radius of the projection of that point would describe equal areas in equal times. Let the surface be a sphere, so that the shortest line between two points is an arc of a circle, and its projection is an arc of an ellipse concentric with the circle. [leave to the student to show from what well known properties of the ellipse the preceding assertion may be verified.* He may also show that, m every surface of revolution, the angle made by the shortest path between two points with the generating curve has a sine which is always inversely as the radius of the projected point. I shall conclude this chapter with the consideration of the ex- pressions for the arc of a curve, the volume inclosed by a surface, and the area of a surface, for which we have employed the expressions (say s, V, and S) s= f/(d2?+dy? +dz’), V= ffzdx dy, S= ffJA+p?+q) dr dy. That some connecting axiom must intervene between our con- sideration of purely algebraical formulee, and their application to space- magnitude, is sufficiently clear from the total difference of the subject- matters of arithmetic and geometry: but whether any new axioms are necessary to the application of the differential calculus, or whether those which are employed in the previous application of arithmetic and algebra will be sufficient, is now the real object of inquiry. Looking at Chapter VIiI., we might he led to suppose that one or the other suppo- sition might prove correct, according to the nature of the question: thus * Very simple mechanical considerations would give a general verification. Granting that a material point, acted on by no forces but those which constrain it to move on a given surface, must move uniformly, and must describe the shortest line between any two poiiits in’ its ¢ourses then, the whole constraining | pressure being lrinal, ahd the normal always passing through the axis of 2; it follows that the coniponent of the constraining force’in the plane of xy always passes through the origin’; or the projection of (2, 9, x) ot the plane of ay describes equal ares in 6qual times. Ad4 DIFFERENTIAL AND INTEGRAL CALCULUS. the consideration of area (page 141, 142) requires no new arithmetical relation of geometrical magnitudes to be assumed; while that of length (page .140) requires the assumption that the arc PQ (page 136) is greater than the chord PQ, and less than the sum of PT and TQ. What is the reason of this difference in the character of the two investiga- tions ? Area (and also volume, or solid content) is a magnitude of such a kind that portions of it, even when curvilinear, can be taken, such as have been considered in elementary geometry. Thus the area of a curve (page 141): can be subdivided into a succession of rectangles, and another succession of curvilinear triangles each of which is as much unknown, so far as an algebraic expression for it is concerned, as the whole area itself. But by continuing the subdivision, the sum of all the curvilinear triangles: diminishes without limit, while the sum of the rectangles does not. The rationale then of the method by which the difficulty is avoided is as’ follows: the result required is compounded of 2A, which can be attained, and >B, which cannot; it is in our power to make a supposition by which &B diminishes without limit, consequently the limit of 2A is the result required. But when we come to consider the arc of a curve, or the area ofa curved surface, the case is entirely altered. No subdivision of either of these is of a more simple kind than the whole: a small arc is still an are, as different in species from a straight line as a large arc; and the same of asmall curved area with respect toa plane. A new axiom,* therefore, becomes requisite, and the following will be found sufficiently easy, and perfectly adequate. If two finite and variable lines or surfaces perpetually approach to coincidence, the lengths or areas perpetually approach to a ratio of equality. To understand what is meant by approach to coincidence, through every point of each line or surface imagine a line drawn parallel to agiven plane and meeting the other. If, then, the lnes or surfaces remain finite throughout the variation, perpetual approach to coincidence’ means that all the parts of these parallels intercepted between the lines or surfaces diminish without limit. But if the lines or surfaces diminish without limit, approach to coincidence requires that the parts of the parallels should diminish without limit in their ratio to the lengths of the lines or the lengths of the boundaries of the figures. The plane to which * Some writers hasten forward to the actual investigation, with what looks like é feeling of unwillingness to state their axiom: some are explicit on the easier cases and abandon the harder ones with an “in the same manner it may be proved, &e.’ Others make assumptions which require long trains of investigation to produce the most simple consequences. Others again consider that they remove the difficult) by adopting the language and hypotheses of the infinitesimal calculus, forgetting that such language is not admissible instead of axioms, but that ‘on the contrary it is to the distinct conception of axioms and their consequences that the infinite: simal phraseology owes its title to be used in an accurate treatise. It would be invidious to produce instances of the first manner above mentioned for the second, compare Lagrange, Théorie des Fonctions, pp. 218 and 300: for thi third, see Lacroix, vol. ii. p. 198, (note): and for the fourth, see the text of the samt note. It is not professed that the axiom proposed in the text contains less of assump tion than is involved in those of preceding works: its recommendation is the univer sality of its application and, the distinctness with which the whole point assumed 1 seen. I apprehend that the same amount of assumption and no more will be foune in Newton’s first section. ' APLLICATION TO GEOMETRY OF THREE DIMENSIONS, 445 the parallels are drawn need not be fixed, but may preserve a fixed relation to one of the lines or areas. The axiom is most undeniably true when the lines or figures remain finite ; its truth, of course, eludes the senses when the figures diminish without limit. But here it may be made perfectly clear that the defini- tion of approximate coincidence, as applied to diminishing lines or figures, is a necessary consequence of the same in the case of those which remain finite, provided we admit that, however small a figure may be, we can conceive figures of any size, perfectly similar in form. With such an admission, suppose that while ‘the lines or figures diminish without limit, other lines or figures are formed which, always remaining similar to the diminishing lines or figures, do not diminish without limit. If, then, for example, p be the length of one of the lines (diminishing) and @ one of the intercepts between the two lines, drawn as above, and if P be the corresponding length in the finite picture of the diminishing system, and If the corresponding intercept, approach to coincidence, if it take place in the finite figures, requires that IL: P should diminish without limit. But by the similarity of the figures Il: P=a: p, whence 7:p must diminish without limit. And in the notion of the similarity # the figures, distinctly conceived, it is implied that if the axiom be admitted as to the finite, it must be admitted as to the diminishing, figures. From the preceding it immediately follows that the arc of a curve tends 0 a ratio of equality with its chord, even supposing that no arc of the curve, however small, is plane. Let AB be a small Qc arc, AC a portion of its tangent at A, and BC a line drawn parallel to a given plane. ‘Through every point c Rp - of the curve draw a plane PQR parallel to that plane, meeting the tangent and cherd in Q and R. By the way in which the tangent is drawn, both PQ and QR* may be nade as small as we please with respect to AR and to AB, by beginning vith an arc sufficiently small. Hence, when B approaches without imit to A, there is a continual approximation to coincidence between AB, the arc AB, and AC. If, then, we take a, so that the arc AB, As, hall be -ABx (l+a), we see that « and AB diminish without limit ogether, whence LAs or ¥,/(Ax?+ Ay?-+ Az’). (14 a) has the same mit as YA (Aa? + Ay?-+ Az?) 5 or s== fi (da?+ dy” + dz*). Q Next, let P be a point in a surface, and PA and PB being parallel to the axes of x and y, let PA and PB be Arand Ay. Hence PRQS is the por- tion of the surface which stands over, and is pro- jected upon the rectangle on the plane of xy, whose area is Ar.Ay. The corresponding portion Prgs of the tangent plane obviously approaches to coin- cidence with PRQS; for if lines be drawn through every point of PRQS perpendicular to the plane of ry, the intercepted deflections (as they were called) as PA and PB diminish, diminish without limit as * This must be proved: that is, it must be shown that a line passing through the ints (2, y,z) and (4-+-Az, y+Ay, z=+Az) approaches without limit to the tan- ent as Ax, &c, are diminished without limit, 446 DIFFERENTIAL AND INTEGRAL CALCULUS. compared with PA or PB, and therewith with Prand Ps. If, then, we say, let PRQS=Prqs (1+), a must diminish without ~ limit, or = (PRQS) and = (prqs) have the same limit, the first being, when the summation is made between the given limits, the required area of the surface. Let @ be the angle made by the tangent plane with that of ay; then, by a well known theorem, (A. G. p. 200,) Pars cos 9@= PBCA —Av.Ay; and, the equation of the tangent plane bemg —2z=p (¢—a) +q(n—y), we have cos 0=(1+-p’+ q’)?, neglecting the sign. Hence Pors= JO +p?+q?). Av Ay; area required = f'f,/(1+p°+¢) dz dy, the expression already used. The expression for the volume contained by a portion of the surface, the plane of xy, and all the planes which project the boundary of the former on the latter, has been already shown to be f fzdr dy. It may also be represented thus, af’ f {dx dydz. Ufupon the elementary rect: angle Av Ay we erect ordinates at the four corners, we have a figure which would be a prism if the upper surface were not curved. If 2 be divided into any number of parts, each Az, we have in this prismatic figure a number of right solids,* each having the content of Ax Ay di cubic units, together with a figure which, as z diminishes without limit diminishes without limit as compared with the sum of the preceding Hence the expression above given for the solidity is derived. Previously to entering upon the application of our subject to mechanic it will be desirable to treat of the Calculus of Variations, to which } accordingly proceed. CuHarTrEerR XVI. ON THE CALCULUS OF VARIATIONS. A cuaprER on this subject must be introduced b@fore anything like: general view of the application of the differential calculus to mechanic can be given. It must be remembered that hitherto we have considere only differentiations of one species. It is true that in functions of mor variables than one, we have treated together of differentiations made wit: respect to the different variables. Thus wlogy has two diff. co., logy and a: ¥y, according as we suppose wor y to vary. But we have neve yet supposed two increments independently given to a, arising fron different circumstances of variation, and requiring the simultaneous con sideration of differentials dx and 6x, essentially differing in the notion from which they are derived. If, indeed, we consider a as a function © two variables, v and w, and represent by dx and dw the differentials of . taken from the variation of v only in the first case and w in the secone we might make a science closely resembling the calculus of variations But the problems which will require consideration under this head a1 those in which dx and dv are purely arbitrary, and independent of a functional connexion between @ and other variables. * I use this term in preference to the longer one, rectangular parallelopiped. S PARALLELOPIPED, in the Penny Cyclopedia. ON THE CALCULUS OF VARIATIONS. 447 With regard to the term calculus of variations, it is obviously improper’as distinctive of this particular branch of the subject, since all ‘that has preceded is certainly « calculus of variations. It is only when by variation we agree to understand a new and distinct sort of differ- ential, that the word is significantly introduced: and it would be more proper to say that the differential calculus is a calculus of variations, but that the particular part of it now under consideration is a calculus of essentially different and independent species of variations, in which the same quantity is considered as an independent variable im two or more distinct points of view. For example, in every problem of equilibrium there is no change of place consequent upon mere lapse of time; nevertheless such problems are solved by consideration of the variations which a system would undergo, if an infinitely small change of place were made, such as the connexion of the parts will allow. This small change of place need not be supposed to be made in time; it would do equally well if it were instantaneous: and if the impenetrability of matter did not forbid, it might be simply supposed that a second system, perfectly similar to the first, was placed infinitely near to it, without any notion of the one system moving into the place of the other. Again, in dynamics, the actual motion of a system is the subject-matter of the problem; that is to say, the ageregate of actual successive infinitely small variations of place which occur in the successive lapses of infinitely small portions of time, accumulated by the integral calculus. But every problem of motion, of which the circumstances are known, may be reduced, as we shall see, to one of equilibrium: that is to say, the properties of the actual variations which do take place may be investigated by means of the simple changes of place, without reference to time, which might be Made in a system at rest. Here, then, enters a science of com parison of different species of variations, or acalculus of variations, technically so called. This calculus is essentially one of: differentials,* not of differential Coefficients. The latter do not change with the species of variation, as long as the connecting relation of the variables remains the same. ie for instance, y= 2°, and it be convenient in one point of view to increase @ by the infinitely small quantity dx, and in another point of view by éx, and if dy and dy be the corresponding infinitely small variations of y; it follows that dy=2rdyr and dy=2y dx, and dy: dr= cy: r= 22. Similarly, if a function of c', x, w3, &C. be increased by P, dx,+P, dz, +T--.., when w,, 2, &c. become 2,+dz,, Xea+dzx,, &e., it will be Increased by P,éz,+P,d27,+... +, when 2, 2, &c. become x,+ 6x, f+Sr,&e. ~ To form a primary notion of the distinction between differentials and variations, let y=@r bea relation existing between y and 2, and let the curve be drawn, of which it is the equation. If x increase, and if the continuance of this relation be the condition by which the corresponding increase of y is determined, the ratio of the changes of y and is deter- Mined by common differentiation ; or dy=¢'r.dv. By an increase of #and y, then, we move from point to point of the curve whose equation is xr. Next, let us consider another species of change, in which, when * The most rigid opponents of differentials haye never attempted to present the notation of the calculus of variations in a manner conformable to their own general principles. 448 DIFFERENTIAL AND INTEGRAL CALCULUS. x is increased by 6x, the value of y is altered by an infinitely small quantity Sy which, though it be a function of # and ox, isnot determined | hy dy=¢'x.d2, but by a totally different relation, in such a manner that x+dv and y+coy must be coordinates of another given curve, infinitely near to that of y= ou. Let: PC be the curve of y= x, and Ve the last mentioned curve, and let p and q be the points of the second curve corresponding to P and Q of the first. We have, then, the following relations between the variations and the differ- enteals of x and y: PR=dr, PA=d2, QR=dy, Aq=cy. By 6 dz is meant the change which dz undergoes when P and R are changed by variation to p and 7: or pr—PR. And by déz is meant the change produced in dx by changing the position of P on the curve y=¢x; or QH—PA- But QH—PA —RB—PA=AB—PR=pr—PR; or ddx=doz. Similarly, ddy is gr—QR, and ddy 1s gH —pA, whence djSy=cdy. And the same may be proved of any function of 2 which remains unaltered: thus dpz =d¢/r.ox, and didr=o"r.dxdx+$'x.dox, and dbx=9'x.dx, while dd dior dadr+¢'x cdr; whence dd pr=dé oa. It easily follows that S fyda= fo (ydx). Let Sy dr=z; whence yde=dz and § (ydr)=ddz=dez. Integrating both sides, we have fé (ydx)=tz=3 f yde. We have here but repeated theorems which we have already proved in pages 161 and 197. The whole of this subject may be connected with the calculus of several variables pre- | viously explained in the following manner. Let r= a (é,v), y=B (E, v), | where a and f are such functions as will, when v=a, give the required relation y= $2 by elimination of ¢. Thus, let z=a (é,a) give =o" (a, a); it is required, then, that Beth a),a} shall be identical with $2. If € only vary, 2 andy will therefore, when v=a, vary in such manner that dy=¢'v.dx: but if v vary, and become a+da, x and y will vary in a totally different manner. To compare this view of the subject with the preceding, we have _ da pilw. _ dp x _ da _ dp da=7y- ds oe Tae dé; or J. da, oy =o da da lor —— _ E ddr —- dé dadé, Sdx Ti da dé da, &c. This latter view of the subject, though instructive, is unnecessary in its details, partly because it is really but another way of expressing the complete independence of dv and dx, and partly because the present state of the calculus of variations will require us only to consider the first orders of variation (da, cy, &c., and not 6°, dy, &c.) There are, in truth, but two great problems in this subject, one general, the other mostly applied in mechanics. We pass on to further details. Let it be required to express dy’, éy!’, dy’, &c., y/', y”, &c. being | diff. co. of « with respect toy. Let P be a function of wv, and P’ its diff, co.; we have then ON THE CALCULUS OF VARIATIONS. 449 :p— a() _odP dP.ddv_d.sP_ dP daz the) Nady dz2:1 dr da dr d (3 dP \ @P. —— PT ir) -4- a Tae a OF : dx xv dx? ; or oP’—P"ox=D (8P—P'dn), where D stands for the diff. co. of the function to which it is prefixed. Apply this successively to y’, y”, &c., and we have by’ —y" dr=D (dy —y! 52) dy"! —y!"0x=D (dy! —y" 32) =D? (dy—y’éx) oy!” — "dx =D (dy" —y"dx) =D? (oy—y'dx), &c.; from which dy’, dy'’, &c. may be easily expressed. We may thus find the variation of any function of the form PAD, ofa eal by substi- tution in wed... dd 7 aS ey op de Ov dy nS Be ayiee SNe sate 3 which may be made to depend upon az, dy—y'dx, which call w, and the diff. co. of w. It remains to express in the most simple form 6fp.dx, being such a function as that just described. Sf da= fo (pdzr)= f (oh dx+oédr) =f do dx+ fb dir=Hoxr+ f (66 dx—dé dr). Let $, which is a given function of 2, y, yy", &c. be completely eee dp do df differentiated, and let the partial diff. co. ge ag ay &e: be XY, Ys dx dy’ dy’ Y,, &c.; then, remembering that the same* relation exists between the Variations as the ditterentials, we have dp=Xdx+Vdy+Y, dy'+Y,,dy"+Y,, OH ites Rae cp=Xox + Voy +Y, dy’ +Y,, dy" +Y,, Oy + 46. op dx— dh dr=Y (dy da—dy 62) + Y, (dy/ dx—dy' ox)+.... But dy=y' dz, dy/=y" dz, &c., whence op dx —dp dx=Y (Cy —y'dr) dx+ Y, (0y’—y" 6x) dr+.... =(Yw+ Y,o’+Y,,0"”+ Y pole, yada: therefore 6 f pdx=oor+ f (Yo+ Y,w+Y,,0"+....) dx. The expression remaining under the integral sign is now to be integrated as far as it is practicable, while the relation of y to x remains indeterminate. This may be facilitated by the following theorem, which follows immediately from successive integration by parts, and of which John Bernoulli’s theorem (page 168) is a limited case. Let ,=/rdr, v= fr, dx, &c., no constants being addedt+ in integration after v,, then * That is, because the manner in which 9 dependson 2, y, &c. remains unaltered : but it must be carefully remembered that the manner in which y depends on a, yand therefore the form of y/, y/’, &c., undergoes an alteration, which gives infinitely small alterations of value. + The student may endeavour to explain how all the constants would really be reduced to one only, if they were added. Ifw be a rational funchieate v be x DIFFERENTIAL AND INTEGRAL CALCULUS. fu dv=uv — fu' vdr=uv — wu’ v+ fu" v, dx =—uv—wu v,+u" va— full! Ve AX =yvo—u! v,+ull ve—.-.. Eu Vat fuer v, dt. Thus Ny w! dx=Vo— fY/' wdz XG; w' dx=Y,,0' —Y,/ +/Y,/" wdx JY oN dt=Y 0" —Y,,/0' +Y,,'0 —fY,,!" wax | SY w dr—y,, wo! — Y;,!0" + Yio! Sa w— fYin wdx ; and so on: in which it is to be understood that by Y/, for instance, or dY,:dx is not meant a partial diff. co. of Y,, but one formed on the implicit supposition that # enters both directly and through y. Substituting these, we have i ' f § foda=port(Y,—Y,/+Y,,"—-.--) w+ (V,—Yy/+¥w!—- ++) © 4 (YYW YG! age) oY + OY OEY yal" pee bh fVHV/4¥/" -Vil’ +... ode. This we may denote as follows: 8 fodr= dx + f(Y), ode+(Y).o+(Y).0'+(Y)s0"+.--- If ¢ be also a function of z, 2’, 2”, &c., x being another function of 2, the consequence will be that terms s milar to those depending on y, 9’, &c. will be added to the variation of /¢ddx, so that if Zdz+Z, dei... be the terms introduced into dd, and if (Z),, &c. be fermed from them in the same manner as (Y),, &c. from y, we have, making C=dz—2'62, 6 foda=hoet+ f(Y)o wdt+ f(Z))ldx+(Y)ot(Z)io+--.-3 and in the same way for any number of functions. [Let* ¢, besides 2, y, y’, &c. be a function of an integral v=f da, where % is another function of x, y, y’, &c. If, then, d6:dv=V, we have d>=(its former value) + Vdv, whence 6 f ddzx receives the accession of the term [V (dv dr—dvéx). But dv ord fydr=wort f (dyda | —dvoxr), and dv=wWdx, whence the accession is [iVvdaf (oy dx } —dw dr)}, or, integrating by parts, [Vaz f (Gy dx—dy dr) — f{ fV dx. (dy dx—dyp oz) }. Lett dw = Pdy+P,dy'+P,,dy”+..... , and let f Vdr.P=j JV dz. P,=IL, JVdzx.P,=W,, &c.: then, resuming the preceding | process with each of the terms just written down, and forming (P)o, (P) ae &c., (11), (1D), &c., we have dfpde=pdr+f (VY), wdr+(Y)w+(Y)ow'+.... + fVdr.f(P), ode+ [Vdx.(P), otf Vdr.(P).0' +... — fll), odz— (ID, o— (ID.o'—.... If y% itself contained another integral function, the process might be — COS ax, sinaz, &c., this theorem gives the readiest mode of actually performing the integration. * The student may omit the pages in brackets, at the first reading. Me) peat the term arising from dy: da, if there be one, since it does not appear in e result. ON THE CALCULUS OF VARIATIONS. 451 repeated : and the terms mig] tec it easily be written down which arise from containing z, 2’ &, The following may serve to throw light upon the general method, though in any complicated case the reductions required would be practically impossible. In finding 6U from U= /¢ dz, we have presumed that U is the solu- tion of a diff. equ. dU: dx=. Let us now suppose that U is connected with y and x by the more complicated diff. equ. P, UM lis (i Meat Se eae +P, U'+P,U+¢=0, P., &c. and # being functions of 2, y,y’, &. If this be Y=0, we have oY=0 upon the supposition that the dependence of U upon Yy, 7, &e. remains unchanged. If we take one of the terms, for example, P, U”, we have 0(P, U") =U" oP,+P,3U", or Ul" §P,+P, D? (8U—U! Ox) +P,U’ dx; that is, one term containing oU, namely P, D?dU, and others containing dz, dP,, &c., but not dU. We may then exhibit oY in the following form, P, D’.6U+P,_,D*™".sU+. ae -+P, D.0U+P,6U+6=0, ® not containing dU. Let the preceding be multiplied by A, a function of all but oU: then if we integrate, as in page 208, (a process which has been in fact already repeated,) we find S{A®+ (AP)— (AP,)!-+ (AP)! Trae) OU} ar rig(A Bye (APs (APS) ,2 21.) 8U + (AP,— (AP,)'+. .--)D.0U+....+AP, D*".SU=0. Let X be so taken that AP,—(AP,)'+ &c.=0, a diff. equ. of the nth degree. If its complete integral can be exhibited, with n arbitrary constants, then m particular solutions can be formed, each containing one constant only, and each one a sufficient factor for the preceding purpose. We have then m results of the form A,. D™" dU +A, 2D" *3U+.... +A, SU+ SAO dr=0; from which the n—1 diff. co. can be eliminated, and oU found from the resulting equation, with the n arbitrary constants which it ought to have. For instance, let the diff. equ. be P, U’-+P,U+¢=0, of the first degree. We have then P, DOU +P, 6U+ U’ (8P,—P, Dor) + UdP,-+6d—0. To find X we have AP,—(AP,)/=0, which gives A= Prt 4“, A being ae Multiplication and integration gives eM SUS Py" {U! (8P,—P, Daz) + USP, +36} de=C; which being reduced by the process already given will express 6U, though only by means of U itself. We shall now proceed to express of fo dx dy, @ being a function of z and its diff. co., both with respect to vand y. Let the diff. co. of z be p and q of the first order, 7, s, and ¢ of the second, u,v, W, m of the third, the following table showing what differentiations are made, and how often, in each. 2G2 DIFFERENTIAL AND INTEGRAL CALCULUS. pil|zir|2c |u| aa,ez ‘ Bape) Saat Ve, cen oe Bitte Pea. le ly,y || wl] 2yy dx dy? i | iit Mm) Ys Ys ¥ a Let > be a function of 2, ¥, 2, Ps 4% &c., and let db:dx=X, &c., so that dp=Xdr+ Ydy+ Zdz+Pdp+Qdq+Radr+ Sds+Tdit+.... Also z=pdxt+qdy, dp=rdx+ sdy, dq=sdxr-+ tdy, dr=udv +rdy, ds=vdr+udy, di=uwdr+ mdy, &c. The development of 6f {@ dx dy is made as follows : Sf fodrdy=ffd (pde dy)= ff Gp da dy + ody dors dx doy) alae sh do Papeba bey WL: = [fo¢dxdy +Saayer- [| 5 éxdady + fodxdy {| iy oy dx dy Lilind th. d. = {pdy bet fp dx iy | | (Gree ay iy) dx dy. It is here assumed that dx depends on z only, and dy on y only, a supposition which will be sufficient for our purpose. To point out the method of performing one of the integrations, take ff pdy dor, which is Say Sf ¢d Oa, OF Pia wpe. fdy {psa— ford}, or fob dy dx— | {FZ dx dx dy. In d.@:dv and d.@:dy remember the implicit supposition that @ is a function of x and y through z and its diff. co., as well as directly. Now from d¢, as above given, a 2 SPX 4 Up tPrt Qs -e. Goat Zat Pst Qt «os. op Xdv4+ Yoy+ Ziz+Podp+Qdq+..-. whence op ae io éy= Z (6x —pdx—qey) +P (Sp—roxr—sody) +Q (dqg—sda—téy) + R (er —udx—vdy)+S8 (ds—vdxr—woy) + .-.- Now let V be a function of x and y, and V,, V,, &c. its diff. co. We have then p dV ddV dV dox Siar tiet a2 de ide dV, ,aVy dx oF dx it being supposed that dy is not a function of =. This gives oy; - e (8V—V,d2—V, dy) + d 8V,—Ven 32 —Vay 8¥=—- (BV—V, dx — Vy by). ON THE CALCULUS OF VARIATIONS. 453 Let éz—pdx—qdy=w; we have then, by reasoning similar to that in page 449, op—rox —s6d easy aes toy = — P sar Ai 0gd—soxr— Fae or —udxr—vo 2 is ’) Ou oy= a &e. ; Bi dated fee ee ta whence, by substitution, Of f pdx dy=fodzdytfo dy dx+ ff & dx dy 2 2 2 ®=Zw+P OL eR es gon aaa) | dx dy dx? dady dy? To perform, as far as practicable, independently of a'l relation between z, z, and y, the integrations in Sfedx dy, let Vw™ be the term* which contains dx” dy” in the denominator of a diff. co. of w: we have then (V’, V,, V”, &. being diff. co. of V) [Vom dy=Vor™, —Vi wet Vj ome—..., EVE wef Ve w”™ dy. Multiply each term by dz, and integrate with respect to x, which gives Jf Vordxdy=Vo"2—V'ur= Watr—;, Vr aL ay Vues dx =, wt + Vo — Vor. yet: Vo, e+ LV Ao, % dx PEA i el tee tae deh snie aC. ae PEN oe pain Van TH elpisha 8 ota ar Agee, aah yes widx F fdy Ng Greens Vee Gites tee tVIO WF [VE wddx} 3 that is to say, Li Vue drdy is a collection of terms of the form EV; wr {X' for every possible combination of values of & and Z from 0 up to m and n, both inclusive, negative erpone nts reckoning as inlegrals of the whole terms; the sign + being applied when k+/ is even, and — when it is odd. To find Sf[® dxdy, let [m,n] stand for the coefficient of w” in ©: if then we wish to select the coefficients of wt, we must in every allowable way make m—1—h=p, n—1l—l=g, or m—k=p+1 and n—/=q-+1, and neither m nor n must be <—1, nor Rknor/<0. The admissible values of 2 and / being 0, 1, 2, &., we find P+1,p+2, &c. for those of m and q+1, ¢+2, &c. for those of n, and any value of m may be combined with any value of nm. Hence the following expression is the coefficient of w”: [Lp+l,q+1]}}—[p+1,q+20+[pt+1,q+3]—[p+1, q+4 + ..... —[p+2, 9+1],+[p+2, q+2]}i—[p+2,qg+3]:+ ..... +[p+3,q4+1]i—[p+3,q+2]i+..... —[pt4,qg+1]}4+..... The meaning of the symbol [a,b]? may be described, from its origin, as follows: a+b, [a, 6]\=diff. co. of @ with respect to qa* dy * We here use exponents without brackets, for simplicity, to denote differentiations with respect to x. 454 DIFFERENTIAL AND INTEGRAL CALCULUS, d’+"C a, b]°) implicitly ; (a, b]} containing x and y directly, and [a, b|,= ——_— i } ely tone ie a iu & da” dy” through 2Z, 2’, Z;5 2''5 2,5 2 WC Hence off dx dy contains, l. The integrals I¢ dx oyt+ fo dy ox. 2. Terms completely free of the integral sign, namely* £119 — 21312? 4 312422! 4-139—.... Fwf+{21}—31;—22N +... } of 4.{129—22! 1394... Pelt 1318 — 4132+... .} 05 4+4225—32}5—23}+ ...- t wi +{13?—23}—141+... ot WE a eles 3. Terms depending on single integrals, (p or q being —1,) it being remembered that the negative exponent of w denotes the integration of the whole term, £O1°9—11'— 0294+ 212+ 121+033—..} w'+ {02)—12)—O03i+ ... For! +{03°—13}—04,+....} or + Pia’ 2 Bah aS rp ihe es + £10°—20}—118+302+ 21} + 13)— ..}w2,+{20}—30)—21i+.- for +-{30°—40}—31i +. boty See 2 Gn ae ie 4. One double integral term (p and q both —1). £00°—10{—01}-+-20}+ 11}+02;—.... } wy. The preceding may serve as an exercise in that adaptation of symbols by which complicated selections and arrangements are reduced to a mechanical process: for all useful applications it will be sufficient to , suppose that @ is a function of x, y, 2, p, g, 7, 8, and ft, including no diff. co. of a higher order than the second. If then we take db=Xdx-+ Ydy ~ + Zdz+Pdp+Qdq+Rdr-+ Sds+ Tdi, we have 00°=Z, 10°=P, 01°9=Q, 20°=R, 1N=S, 02=T: all the rest being =0. This gives for 3/f¢ de dy, w being dz — pdx — oy, Sut fdr oy +f dy ox iS Wa eds ( Chee) Sad gon + [(a-F a Joae+ {(P— ceRr —_ Gow - > dw dw) (2 ?.R d.5 dT +f | dx ih dr a: Ber dy w dx dy. We have not limited the result by proceeding as if dr were a function of x only, and éy of y only, for it might be shown that the wider suppo- sition of dr and dy being both functions of x and y would lead to pre- cisely the same result: but a complete elucidation} of this point would be beyond an elementary work. | The applications of the calculus of variations which are of most importance are 1. Given any number of points (2, 1, 21), (a, Yo, 22), &c., and any number of equations V,=0, V.=0, &c. between their coordinates, * To save room I have omitted brackets and commas, thus 239 stands for [ 2, 3]$. + The advanced student should read on this point the Memoir of Poisson on the Calculus of Variations, in the twelfth volume of the Memoirs of the Institute, ON THE CALCULUS OF VARIATIONS. 455 required the relations which must exist between PTY Ree es Vos Ai, &c. given functions of the coordinates, in order that the equation X, 0m, + Y, 84, +Z, d2,+X,_ 52,4 Y, dy,+ Ze d2z,+....=0 may be true for every possible value which dz, dy, 62, &c. can have, consistently with V,=0, V.=0, &c. being true both of (a, y,, 2,), &c., and (27,+ dx, y,+0y,, 2, +62,), &c. 2. Given any integral, in which the integration cannot be performed because it contains variables which are related to one another, but between which no relation is assigned, required that relation, or those relations, which being substituted, and the integral taken between given limits, the result is the greatest or least which is possible; thatis, greater or less when the required relation obtains than it could be under any other possible relation. To give a simple instance of the first class of questions, suppose two points in a plane, (z,, y,) and (2,, Y2), which always preserve the same distance a: under what relations between 2,, &c. will the following equation be always true? 1, OX, + Y, OY: Le O1,+ Ys dY2=0......(1). The equation (7, —.22)?+ (y:—y2)°= @ gives (x, —22) (62,625) + (Yi—Ys) (07, 0),) = 05a. (2). In the first substitute the value, say of dy, from the second; the result, cleared of fractions, is (2; Yi—Zz Yo) 6x + (yi— ys) oy,-+ (2. Y,—2, Y2) Cree and this, which is to be true independently of 6x,, dy,, and dx, requires that TY, —X, Yo= 0, eh LY, —2, Yz=0......(3); which are satisfied by y,=y., 2,=2,, or by 7=—Yo T=—aLy. The first is Inconsistent with (a,—.2)?+(y,—y2)?=a’, but the second is not, and gives 4a}+4yi—=a’®. The answer then is that the two points may be the opposite extremities of any diameter of a circle whose centre is the origin, and whose radius is 4a. The following method is: particularly connected with this class of problems, as well as with some varieties of the other. There is an equation between 2, &c., dx, &c., say U=0, whichis not to be absolutely true, but only for such values of dx, &c. as make V=0. This we can express by one equation,* U-+ AV=0, where A may be any function of x,y, &c. independent of dr, &c. For the preceding equation expresses that U is or is not =0, according as V is or is not =0. If then we make each coefficient in U+AU separately =0, we have one more equation than we had before, but at the same time we haye one more undetermined quantity A. The elimination of A will reduce the number of equations by one, and will give precisely the results of the common mode of operation. If we multiply (2) by A, add it to (1), and then equate each coefficient to 0, we have * Many other forms might be given, but all are either reducible to the one here given. or else they introduce (3r)®, &c. while, since da, &c. are all to be taken as diminishing without limit, these terms of the second, &c. orders are useless. 456 DIFFERENTIAL AND INTEGRAL CALCULUS. 2,+A(a,-22)=0, v.-A(2,-7,)=9, ytA(m-y)=9, Yo—-A(yi-Y2) =, and elimination of A will give equations (3). To generalize the preceding process, let there be m points, and 3” coordinates, one equation U=0, or X,62,+Y,6y%,4+...-=0, and p relations between a, y, &c., V,=0, V.=0.... Then we have 6V,=0, which gives, say &,dx,4+71,¢yit0/,02,+....=05 also oVs=0, .or El dx, +n", dy,+...-=0, and so on. And U+A,V,+A,Vet..-. expresses that U=0 when V,, V2, &c. are each =0. KEquate the co- efficients of or,, &c. separately to 0, which gives Ki +A, 8, +A, 6%, Aart aals Yi, +A. 9'1+ Asn", + ....=0, 7 MET NE (te i gd BOR) ay CL and so on: giving 3n equations between 3n+p quantities. Elimination of A,, As, &c. will reduce these to 3n—p equations between 3n quantities, and the p equations V,=0, &c. finally leave us with 3n equations between 3n quantities, unless it should happen, which it frequently will, that all the 3 firial equations are not independent, in which case the problem is not determinate.* [The following PRoBLEM contains a most material portion of the purely mathematical part of the statics and dynamics of a rigid body. Let there be a number, 7, of points (a, y\5 21), (%25 Yo) Z2), &C. mmMove- ably connected with one another; that is, the distance between any two remains unchanged during variation. Supposing the whole system to undergo an infinitely small change of place, required the relations which must exist between P,, Q,, R,, &c. and 2, y,, 2, &c., m order that for every such infinitely small displacement we may have Pe OP ore hE) dy, Qs tyed «cos PR Of, Peer ne ee Take a new origin of coordinates, (a, b,c,) and a new set of co- ordinate planes attached to the system of points just mentioned, and moving with it. Let &,, ¢ be the coordinates of (a, y, z) with respect to the new planes, and (A.G., p. 224) let the new and old coordinates be so related that a=ateaée+bnt+yl, ybt+eEt+Anty'~%, z=ctalE+ ph nty'. Consequently, since &, n, £, &c. do not vary with the system, (for the new coordinate planes move with it,) ér=da+ 6da+ ndB+ Coy, &e., and substitution obviously gives ({P meaning P,+P,+...., &.) YP .da+2Pé.d¢+2Pn.d8+2PZ.dy+2Q.0b4+ 2QE. da’ + TQn. 5H’ + QZ .dy'+ ER.dc+ DRE. 8a"+ LRy. dB" + 2RZ.dy"=0. Now a, &, &c. are connected together by six equations,t A a+ a? + of? =1 By+B'y'+8"y"=0 | A’ B P+ pee pre 1 ya tylol + yal! =0 B’ C Vt+yt+y'"%=1 af +o! B! + a'/B"=0 Cc’ * The student may omit the part in brackets at the first reading. + It must be remembered that these are also equivalent to i a +82 +7 =] at! al! + Bl Bll + of of =0 al? +, +y"? =] elle +22 +yly =0 eZ BI 4 9/2 — ] a a! +£ p! +y y= 7 ON THE CALCULUS OF VARIATIONS. 457 Take the variations of these equations, and add them to the equation preceding, after multiplying them by the arbitrary multipliers written opposite to them. This gives 2>P.6a+3Q.8b+ RK. de +{ZPE+Aa+C'B+B'y} a+ {2Pn+ BB+ A’y+C’a} 38 +{2P6+Cy+B’e+A/p} Syt&e.= : where, in the terms included under +&c., we must change P into Q, and write 2’ for ~, &c. for a second set, and change P into R, and write &” for a, &c. for a third set. If we now equate each of the cvefficients to 0, we pave 2P=—0, 3Q=0, SR=0, and nine other equations, in which are the six multipliers and the nine quantities «, 3, &c.: but between these there are six equations; altogether. then, fifteen equations with fifteen arbitrary quantities, So that it should seem at first as if we might satisfy these fifteen equations by values given to the arbitrary quantities without any new relation between the data of the question, Pi, Pare ba &c., and I have introduced this example to show how little we must depend upon conclusions drawn from the mere number of equations to which a question can be reduced. without examination of their structure, The fact is that the fifteen equations cannot be rendered simultaneously true, unless three other equations between the data of the question only are satisfied. Let (6x) be the abbreviation of < coefficient of dg,’ in the preceding equation. From (¢%)=0, (86) =0, (oy)=0 deduce a’ (x) + B’ (83) +7 (dy) =0, or 2 iP @E+Bn+y'0)}+Aaa’ +BBB'+ Cyy' +A! (yi +6’) +BY (ay + ya") +C! (Ba! bass!) =0. Now form @ (60) +3 (88) + (oy’)=0, and we shall have 2 {Q (a6 + by+yf)+Aac +BBB'+... -(as before)=0, in which last, the accented letters were in the coefficients, and the un- accented letters are from the multipliers. Consequently, =P (E+ Bn + y'Z) —=EQ (a6 + n+ yZ)=0, or 2P (y-b) = SQ («—a), or SPy—b3 P= 2Qzr-a2Q, or SPy= Qr: and similarly it may be shown that 2Q2=2Ry, 2Rr= Pr. . These Six equations, /P=0, &c., 2Py=XQ-r, &e. are therefore necessary : it remaius to show that they are sufficient. For this purpose, remark that r=a+ gé+ &c., &. give, by the aid of the equations of condition, =a (r—a) +a (y—b) +a" (2——c), 7=P (2—a)+- 2%, C=y(r#—-a)+.... Form « (da)+a! (da') +2" (dx) =0, which gives, by aid of the equations of condition, A+ dé (Pa+Qe+R-2)=0, whence A is obtained, and B’ and C’ can be also obtained from ¥ (6a) + y! (da) +7" (62’)=0 and 2 (da)+&c.=0. By the three corresponding equa- tions 3 (68)+&c.=0, y (83)+&e.=0, (68) +&c.=0, B, A’, and C’ can be determined, and C, B’, and A’ from the three equations corre- sponding to (dy), &c. Thus A’, B’, and C’ are determined twice over ; 458 DIFFERENTIAL AND INTEGRAL CALCULUS. the equations which give them are therefore incongruous unless the two | values of A’ agree, and likewise.those of.B/! and of C’. If for , n, and | ] Z be substituted their values above, it will be found that the six equa- tions SP=—0, &c., >Py==Qzr, &e. will make these values agree, and that no other relations will do so, as long asthe equations of condition between «, a’, &c. exist. ] In order to explain the second class of problems, it will be advisable, | dropping for a time the progress made in pages 449, 450, in finding the variations of integral forms, to take a simple question and go through the whole process from the beginning. Let the question be as follows: to draw the shortest line from one curve to another, without assuming that a straight line is the shortest distance between two points. | When we consider the variation of an algebraical function, V, we know ' that its arithmetical minimum is 0, if any value of its variable can be ’ found which makes V=0. But this is not necessarily an algebraical minimum, since, if the value of V, in passing through 0, change its sign, * it is increasing or diminishing both before and after passing through @.' Now it is to be borne in mind, throughout the following investigations, that the results sought are algebraical, and not arithmetical, maxima and | minima. For example, let the two curves be as marked in the figure, the arrow points denoting that the branches there discontinued go on ad infinitum. Arithmetically speaking, there are absolute minima at P, Q; and R, and no maxima; for between any two points, one on each curve, a line of any length, however great, may be drawn. Algebraieally speaking,. P, Q, and R are not minima: for if we agree to measure arcs of intercepted curves from, say PAQBR itself, then such lengths when they pass through P,Q, or R change their signs. The: lower curves in the figure are so placed that certain straight lines AB and DC can be drawn, one of which would seem to be a minimum, while, in the upper curves it may be made obvious that AC and BD are not minima. It may seem certain that CD, in the lower curves, is a minimum: that is, the points C and D being (however little) displaced on their; curves, no line, straight or curved, so short as CD, can be drawn between their new positions. The fact is that these problems of the calculus of variations: mvolve two questions; the first completely and satisfactorily answered, the second left in a very imperfect state. These questions are, as to the instance before us, 1. What is the character of the line which is the shortest distance between two curves, when there is such a shortest distance? Is it straight* or curved, and if the latter, what is its law of, curvature ? 2. Two curves being given, can such a minimum distance be found or can two points be found, one on each curve, such that the lime whose, character is shown when the first question is answered, being duly drawn from one of these points to the other, is really the algebraical minimum distance of the two. We now proceed to the problem. . j * It is no doubt partly proved and partly assumed, long before the reader comes to this pvint of his studies, that the line in question is straight: but we will suppost that this is not proved, and has not been assumed, in order to avail ourselves of this very simple problem as an illustration, ' ON THE CALCULUS OF VARIATIONS. 459 Let AB and CD be the two curves between which the shortest line is to be drawn. Draw a curve cutting the two, and let x and y be the coordinates of any point init. At V let z=, y=Yo, at W let z=2,, y=y,, and lety,= Volos Yi= VW, be the equations of AB and CD. We want then to find a relation between x and y, together with the position of V and W, so that VW may be the shortest line; or to make JV Oty”) dex from t=2X to r=2, the least possible. Pass to a new curve, vw, by changing x and y for every point of VW into vor and ytoy. Let Q be the point corresponding to. P; and let v’w’ be a curve made by changing x and y into e—or and y— oy. Itis to be remembered that at the limiting curves we must have Yot CY = Yo (Xo+ Ox,) and yitoy, = th (2%, +-02,); also y,—dys=¥, (a—dx,) and Yi— oy, =W, (a,—da,). These last four equations are not compatible with each other, strictly speaking, on any but a straight line; if, however, oro, &c. be infinitely small, they are true together as far as small quantities of the first order. Let y/ become y’+ cy’, then substituting in /,/(1+y") dz, it becomes, by Taylor’s theorem, ( P oy’ 1 61)? ‘ [Watyy+ et tn cy : rine (an-+dbx) : Vi+y? 2 (i+y%)3 which, between the given limits, is the length of vw, and its excess over VW is, to terms of the second order, i ah ae y'dy'dx { y'oy'dor (dy’)? dx 1+y”) ddx4+——=—~ > + | ¢ x +s ts ind dy=y'dx gives dy’dr=ddy—y'ddz. Now, since VW is the least sossible, vw—VW must be positive, as must also v'w! —NW, and v'w! is btaimed by changing the signs of dx and cy, and consequently of déx ind doy: whence éy'ddx and (éy’)? retain the same sign in both casés. Moreover, since every element in the first integral is of the same rder as ddr, and inthe second as oy’dox, the second integral must, when iz and oy are diminished without limit, diminish without limit as com- yared with the first. If, then, the preceding be K,+K, for vw, it is —K,+K, for v’w’: and since K, is greater in numerical magnitude than Xs, the latter must have different signs, whereas they should be both sitive. The only way of avoiding this is by supposing that coefficients anish in K,, so as to make it identically =0, independently of dy and %. Both vw—VW and v'w/— VW then become = K,, and if this be sitive, when taken between the given limits, the required condition is ttained. ‘This reasoning, which applies in every case, is the ordinary easoning in problems of maxima and minima. Substitute as above for dy'dx, and K, becomes y'doy y”déx ne ? J f2 13 dg ONE Ed Teac See Jivats or Mee) iby) { dox 4 y' doy ! sx $1; J+y") JU+y") 460 DIFFERENTIAL AND INTEGRAL CALCULUS. which, integrated by parts, gives ba y'ey {{ day tt A y’ K,=——— 4 oe — 988 on 89 oa | Ja+y) | Va+y") da Ja+y) | 4 dz Jaty™)5 ox+y/oy { yy! ig Se are erg nd eer eee UO ae Ox) dx; Ja+y") a+y) Fhe met which is to be taken from rx, tov==7. Let (1+y”) ?=¢, and let. y'o9 Tos Yr» %1, &C. denote the values of y’, o, &c. at the limits: we have then finally for K,=0 om (6a,+Y'; Yi) — %% (dr +y"o Sy t faye (oy —y'ox) dz—0. The first terms depend only on the values of y’, dr, and oy, at the limits, but the integral depends among other things on the values of dy and Sr at every point of VW, and contains in fact two arbitrary, though infinitely small, functions of w and y ; namely, ov and dy. It is impos- sible, then, that the last term should a/ways (for all forms of dx and oy, for, the line required is to be shorter than any other line) make the preceding, equation true: nor can this equation be true unless the arbitrary term 1s made to vanish by a supposition not affecting dv or éy. The only sup- position on which this condition is fulfilled is y’=0, which amounts to, supposing VW to be a straight lme, since it gives y=ar+b, y'=4, nee om (l1+a’)"?. We have then to satisfy Before we proceed, however, it will he necessary to remember that our: only reason fur equating the terms of the first order to 0, by means of! coefficients, is to prevent our having a term which, being the largest off all, may be made to take either sign, whereas im the case of a minimum it must be always positive. The necessity of this supposition as to the! indeterminate integral is easily shown; for in /y’o* (dy—y'dx) da) there are the arbitrary functions éy and dr, which are altogether in our power except at the limits, so that the integral, if positive in one case, may be made negative in another. Nor can the other terms prevent this term, if allowed to exist, making the terms of the first order some times negative: for when the varied curve begins and ends at the original curve VW, (as in one of the dotted lines of the diagram,) we have d%q, dy, S%, and ody, each =0, so that if y” have any finite value, we may make the whole of the terms of the first order, in certain cases, negative) Hence y'/=0 is a necessary condition. But if we look at the part o, (on, +y' 6y:) —&c., which becomes (y”=0, y/=a) | (14a)? {62,+ady1— (Sto tady,)}, it is not obvious that this portion, unless made =O, may have any sign we please, for da) and ody» are connected by an equation, and also Ox; and oy,3 since (x+0xre, Yotdoy,), &c. are two points each on a given curve’ All that is necessary, then, is that the preceding should be positive: and if we add Ky, we find for the complete variation as far as terms 0} the second order, 62+ ady,—(OT+ Ady) {7 ada dor (da)? dx \, J (1+?) - : ——_______ V(1+a*) 2(1+a%)! ON THE CALCULUS OF VARIATIONS. 461 where 6a is constant or variable, according as vw is a straight line or a curve (VW being made a straight line, since dita (9 f The preceding must be positive. Nuw suppose that our axis of x had in the first instance been made parallel to the straight line we wish to consider, which can always be done. Then a=0, and the preceding becomes OX, —OLyp+ Lf (6a)? dz; where dz, and dx, are independent, and f (da)? dx independent of both. If 2)<.x,, as we have supposed, the last term is essentially positive, and the whole will be positive if dr) must be negative and éx, positive. The only cases in which this is true are represented in the following diagram, in which the straight line drawn being parallel to the Ow Wh ae, WN V W axis of r, and x being measured positively towards the right, we sce that, (%, Yo) being V, and (7, y,) being W, dx. must be negative if we pass ‘0 an adjacent point, and 6x, must be positive. Consequently, a line isa minimum distance between two curves when two perpendiculars being lrawn at its extremities, neither perpendicular passes through its curve 30 as to have the curve on both sides of it. Another case (answerlug to CD, p. 458) need not be discussed: the object being merely to show the msufficiency of the common method, and also its tendency to redundancy. The application of the preceding reasoning generally to d/¢dvr is endered extremely difficult by the complexity of the terms of the second mder. The only cases in which we can easily proceed are those in which we know beforehand that there is a maximum aud no minimum, a minimum and no maximum. ‘Then, taking 8/gdr=irt+ f (Y), wdr+(Y), 0+ (Y).0/+(Y),0"+.... -+terms of second order+...., ve may make (Y),=0, for a reason similar to that shown in the last roblem, and we then know from the nature of the case of what sign the erms of the second order must be. It remains to ascertain how the line ‘etermined by (Y),>=0 must be placed, in order that the value of the ategrated part of the expression taken between the limits, or ” 6x, — Po or, + (CY)i), wi— (CY): )o Wo+ (CY)o)i Ww 18 (CY 2))o w+ see ay always have the same sign consistently with every variation which 1€ conditions at the limits will admit, and that sign belonging to the 1aximum or minimum, as required. Before proceeding to some examples, let us examine the equation Y),.=0, or =Y/+Y,/'- ...2=0, where df/=Xdzxr+ Ydy+Y,dy'+Y,dy"+... If X=0, or a function of vw; that is, if » either do not contain 2 at l, or in such a manner that it has the form ¢(y,7’..-. -) ‘+r; then, age 208, it is obvious that Y= Y/—Y,/’+.... is precisely the con- tion necessary, in order that Xdr+Ydy+...., or (Xdvr+Yy/+ 1y'+....) dx shall be integrable per se, so that we have il! ' 462 DIFFERENTIAL AND INTEGRAL CALCULUS. ga fXdat+(¥,-Y¥j/+-- +) uF Vy HV +e yl ee DS so that the diff. equ. (Y)).=90 admits of one integration. For example, | let @ contain only y and y/, then X, Y,,, Y,,, &c. are severally =O, and | we have Y—Y!/=0, for (Y),=0; or, by the preceding, #=C+Y,y, | Now, Y, containing y', Y/ contains y”, and Y—Y/=0 is of the second order; but P=C+-Y, 7‘ 1s of the first. | Let @ contain y, y’, y, and y'”, with v in an independent term, Then p= f Xda t(V—V/+¥u! yt Ya) ft Yay" Let it be required to find the curve on which a material point, acted on by gravity, and descending freely, shall fall in the shortest time from a given point to a given curve. If x be horizontal, and y vertical, this | amounts, by the principles of mechanics, to making f{/(1+y") :/y} de aminimum.* We have then | Lista 1 ih +y'”) y! = /(—* |, Ye-2 S 4 -, Y=, fe /( “Bets 2 yt “ Ayd+y") Ley? y" —Y y'+C gives J (=) =——_*— _ -+-C, eae ian: y yaty)}" or 1=C*y (1+y") dx Toe Js "aK ae = /(q) w= 2K vers 5. ¢ /(@Ky—y) + L: 2K being 1:C®, and La new undetermined constant. Let us suppose the fixed point from which the descent begins to be the origin; then, since « and y vanish together in the curve, L=0, and we have the equation of a cycloid, whose cusp is at the origin 0, —A and the radius of whose generating circle, which rolls, Ve on the axis of v, is K. According as the upper or, ee : lower sign is taken the cycloid is placed with its ordi- | eae nates negative (as in OA) or positive (as in OB). ea We have also (Y),=Y,, (Y).=0, &c., whence the 'Q integrated part of of ¢dx is | oa Sheesh Sy oe CRT AS Oe J( y OUP TOR es | ea Taking this between the limits, we have, x, and y, being coordinates of the required point in the given curve PQ at which the descent is tc end, and oz, and dy, being each =0, yy 2 +y/2)-2 (0x,+ y' oy); hee 2K — or yn Ftatyy 4 (aa4(./ sat . iy: ‘ Yi putting for y/, its value in the last factor: it being remembered that y/’ is to be taken as positive when the point comes in the first half of the * From the nature of the problem, a maximum is impossible: by making thi curve sufficiently near to a level at its commencement, the time might be augmente( without limit. | ON THE CALCULUS OF VARIATIONS. 463 eycloid, and negative in the second. Let Yi=Wa, be the equation of the curve to which the cycloid is to be drawn : the sign of the preceding then depends on (l+y', Wx.) d2,, so that in every case in which dr; can be either positive or negative, we must have 1+y’, W’x7,—0, or the eycloid must cut the curve at right angles. But if there bea cusp so situated that dz, and dy, are necessarily positive, and that the cycloid drawn from the origin to the cusp meets the cusp In a point of its first half, that cycloid is a line of shortest descent: and also if it beso situated that Sx, is positive and cy, negative, and that the cycloid meets the cusp in its second half. Sometimes a further integration may be made in (P): thus, if ¢ contain only y’ and y”, we have Y,'—Y,"=0 gives Y,—Y_/,/=const.=C, whence, if X=0, (~) becomes g=c+Cy' + Y,,y". For example,* let ep=Ut+y?)?:y", the limits being two fixed points in the axis of x, and one of them the origin. We have then (ty) (+y")_ 9 9 2C+ty) J Maco cy OAH, og 2LLY y in which, since c and C are arbitrary, 2 may be struck out. Let y =tan B, y" dx =(1+tan 2) dB, and we have dr=(c cos’ 8 +C cos ( sin 8) df, dy = (ce cos 3 sin 34+-C sin? 3) dB 4v=2cB 4+-csin 2B—C cos 28+K 4y=2C8 —C sin 26—ce cos 26 +L; Which may be shown to belong to a cycloid. The integrated part of the variation is poxr+ CY ae Y,,', which gives Y)9w's—Y,, wy, since 6x and w or dy —yoxr vanish at both limits. “And w/= oy’ —y" dx gives Y 2 dy’,—Y,, dy’, for the above. If f, and therefore y’, be given at the limits, this vanishes of itself, and the arbitrary character of the Constants ¢,C, K, and L, is no more than sufficient to enable us to make the cycloid pass through the given points with given tangents at those points. But if y’ be undetermined at the limits, we have c+Cy'; | ot sees l+yi)? Ye Bee Ye oy, = ne (1+7')) 3,45 ee (1 +y% of, > 2 1 im which the power of giving different signs can only be avoided by making the coefficients of of, and of, severally =0. That is, the radii of curvature at the extreme points are both =0; which in the cycloid only happens at the cusps. Hence if A and B be the given points, ee every such figure as that in the diagram gives an NS, algebraical minimum: that is to say, any slight " "Variation of the upper curves with a corresponding ariation of the lower evolutes would increase the area contained. * Let the student show that this answers to the following problem: between Wo given points to draw a curve which with its extreme radii of curvature, and heir intercepted arc of the evolute, contains the least area. And let him show hat the problem may be susceptible of a minimum, but not of a maximum. 464 DIFFERENTIAL AND INTEGRAL CALCULUS. There is no absolute arithmetical minimum; for by sufficiently in-_ creasing the number of revolutions of the generating circle we might | dimiuish the whole area without limit. Let it be required to draw on a surface the shortest line from one curve to another, both curves being on the given surface. Let dz=pdx+qdy be the differentiated equation of the curve surface; the function to be made a minimum is then ox fii l+y?+(p+ay')} dr= [ods ; p and q being both functions of x and y. We have then f if f r no am ee dCs a 8 aes (ew =o. @ da Make o itself the independent variable, and for p+qy' write (dz: do) x (do: dx), remembering that do:dx=¢. We have then dz = ( dx dy d (FZ z) da Adulte None 1 dey Oe ce * F 2 9 ie a pata _@y , dz, de( de, dy) * de do) de® Vado?" do * de do / d°y d?z Pe - aye 1d gee as in page 443, and 468 spies might be deduced by combining this with the equation of the surface, or else by altering Sodx into f (ee +14 (pr' +? dy, and repeating) the process. on the supposition that a is a function of y. The integrated part, Por+Y,w. is subject to the remarks already made in the pro-) blem of page 459. If 2, y, and z be expressed as functions of v, the) preceding equations (page 158) become (a’ being dx: dv, &c.) o (y/'+92")—0" (y'+q2)=0, 0! ("+ p2")—o" (2' +72) =0, or (y" +2"): (a pz" Vay +492): + pe) 5 which is nothing more than the expression of the property that the osculating plane of the curve must be everywhere perpendicular to the tangent plane of the surface, partly proved in page 442. Hitherto we have not supposed the function ¢ to contain the limits of integration directly, as constants. If this be the case, and if 7, 7,5 Yor Wied a Yrs. cc. ve the values of z, y, y’, &c. at the limits, we shall have to add to the variation of Soda the series of terms | | (¢ Roe Of, +... .) ae, OY O2y ( ae, dain | dy ite . ae w \dX dx, ax adr, remembering that dro, dr,, &c. are constant throughout the integration. The general form of (Y))=O0 is, therefore, not affected, and the only change which is required is the consideration of the new terms annexed to the integrated part. Also if the quantity to be made a maximum or minimum were of the form K+ fod r, K being a function of limiting values, the only alteration requisite would be the addition of dK to the integrated part. | Thus, if in the question of the brachyslochron, or line of quickest descent, page 462, we suppose the line is required to be drawn from one > or | | ON THE CALCULUS OF VARIATIONS. 465 curve (7%, 2) to another (y,,7,), the velocity at any point depends upon the height of the point on the first curve from which it fell, and the expression to be minimized is lty\h . “\e f = dz, instead of [ (AX ars ee ; Y—Y% y in which, as it happens, d¢: dy,»= —dd:dy=—Y= —Y,, since Y—Y/=0. Hence Sy f (dp: dy.) dr=—Y, Oyo, or —(Y,,—Y,) OYo between the limits. Consequently the integrated part is now . (Ya a Yo) Yt, 61,— Om+Y, (cy, —y; 2,)— Y 9 (dy—y"o OX»). But since d =Y,y'+C at all points, the preceding becomes Coxm—Cdrm—YV_ YH +Y,, dY;. If y,=¥,2, and YoY. Xo be the equations of the curves; substitution gives (C+Y,, Ww ®,) ov,—(C+Y,, Woo Xp) O29 3 and assuming each coefficient =0, we deduce yw’, 2=W' xX, or the points at which the cycloid passes through the curves have their tangents ‘parallel; while from the former process it appears that the cycloid has its cusp on the higher curve, and cuts the lower one at right angles.* A cusp on one of the curves might offer an exception, as before. Let it now be proposed to find, not the independent maximum or minimum of an integral, ‘ode, but that which exists under the con- dition that if ywdzx shall remain constant, as in the following question : Of all curves of a given length, what is the curve of quickest descent from one given point to another? In this case we do not require ofgdx to be always positive, or always negative, but only in such cases as also satisfy )/ydv=—0. Let dp=Zdx+Hdy+H,dy/+...., and, consequently, as in page 450, of pda=Pdx+f (Y)owdz+(Y), o+...., Sf ydr=yoa+f (H))vde+(H),o+....; whence the following conditions: 1. (H),)=0 must make(Y),=0. 2, Wy, 02; — Yo oto +...-.=0 must make d 02, — $y d+... Pegative 10 the meese of a Oe To satisfy the first condition, it is sufficient that there should be any one constant quantity a, such that of (P+a) dx=0; for then, since ofedx+ad/ydxr=0, dfywdr=0 gives 6 {pdr=0. To satisfy the second condition it is sufficient that for the same quantity @ we should have ¢, 62,—¢, dx, +.... +a (Ws, O2,— Wy OX +..--. ) always positive or always negative. Hence it follows that if we proceed as in making (P+aw) dr a maximum or minimum, and then determine @, so that Swdx may have a given value c, we shall give Soda the greatest or least value which it can have consistently with the condition /ydr=c, * Having in the first three questions taken notice of the limitations and excep- tions which sometimes occur, I shall, in the remaining problems, simply ascertain the conditions under which the variation of the integral is nothing. Buf the student must remember that the results require further examination, except when a maximum or minimum resembling that indicated by the result is known to exist a@ priori. 2H — op nO eC OT OF amy a a —_ = Fo gt 466 DIFFERENTIAL AND INTEGRAL CALCULUS. For.example, it is required, on a given line AB=A, with ACB a curve” of given length to inclose the greatest possible | ‘e area: here the maximum obviously exists, and | 5 there is no minimum. Here fjydzx is to be i Se ONES maximized, while fea (l+y%)dz=c, or we A 3 must proceed as in making S fytafty)$ dx= f pdx, a maximum. We have then 6=Y,y’+C, which gives ay’? yta/(1 Re aE RV Me or (y—C)/(1 +y°)= —a, Leite ae) y—C aby A a Ce) or the curve is circular. By properly assuming the three constants, we may find the circular arc which passes through A and B, and has the length c: this are is the curve required. The integrated part vanishes of itself, since the limits are fixed. A curve of given length is to be drawn between two given curves in such a way that its centre of gravity may be at the least possible dis- tance from the axis of «, This distance is fyds:S, S or fads being the whole length: consequently | fyds is to be a minimum, fe ds being constant, or we must proceed as in making JS (y+) ds a minimum ; or , (2—Ky+(y—C)*=@ ; fod=fy+OlA+y”) de, YW=ytay Vat) P=Viy' tC gives ypa=CJ/(l+y"), Y= Cy’ dv C =, (et Ke Clog (y+a a)?—C® atK 2+K 2(ytayaec+Cre ©. the equation of a catenary, or of the curve in which the string would hang if the axis of x were horizontal. Now take the integrated part, derived from $d2+Y,o, or (6—Y,y’) ox +Y,dy, or Céxr-+Y, oy, which gives Cox, + Cy! dy, —Cb.29 — Cy'y dy 5 and this is to be always positive, or nothing. Substituting By, = Ys’, 2, OF; | and dy=¥', 2) d%, we find, to make the preceding =O independently - of ox, and 6x, the equations ! l+y',w',2,=90, L+y!o¥/oro=0 ; or the catenary must be perpendicular to both curves. But (C being positive) let there be a pair of cusps, one in each curve, so that dz, must be positive, dz) negative, and y', dy, positive, and y', dy, negative, as in the figure. There will then be the minimum required, if the string hang in a catenary from these \ l) points. If the distance were required to be a maximum, i the process would appear to be the same, and to __ determine the same curve. But it must be re ON THE CALCULUS OF VARIATIONS. 467 membered that K is arbitrary, and that by so assuming it that K:C=L:C+72,/(-1), the equation of the catenary takes the form z2+L a+L 2(y+a)=—e° —C2e7 cc. I have placed the curve downwards in the diagram, as the problem obviously requires, and it would have been placed the other way, if the maximum had been required. Such circumstances as these must be determined by the apparent necessity of each case, until the integrals answering to K, in page 459 can be satisfactorily examined. This has not yet been done in any manner which is sufficiently complete and elementary for the learner.* I shall now give some examples of the more extensive methods in pages 450, &c. The following was solved by James Bernoulli, in the early days of the differential calculus. On ~ a given line AB to draw a curve of given length ACB, in such manner that NP, the a 6 ordinate of another curve, being a given function of the arc AS, the area APB shall be the greatest possible. A % Let AN=2, NS=y, AS=v, let PN=S 3 (a function of v), and SSda is to be a C maximum, while /4/(1+y”) dz is constant, between the fixed limits. We have then (page 465) Sodz=f (Staf+y")) dz, fywde= SVA+y”) de=v, (page 450) P=0; Ray @ +y')-4, =O) Nisayl dd +y)-4, dS V= a? I= [Vde.y’ él +y") 2, v Sf bdzr—s gO EY WON sie Oe oy drmpiat | ( ary) ot Tasty)” PEEP BNI Si 9 Yo 208s +S vary J ( Facaan) det Tm | y! \/ y! — yi At ae In Pp, {( JNe2 Jay)” eons from the formula in page 450, which gives (Y)=Y—Y/=—Y//, &. Taking all the integrated part between the fixed limits, all those terms disappear which contain dz or w free of the integral sign. Also f Vda is, relatively to the integration of arbitrary variations, an undetermined Constant, which we may call H. We have then sea |i(— Tatts) (gat) +( ae ates) hae * The student who requires more problems may consult Woodhouse’s most valuable treatise on Isoperimetrical Problems, which is, in fact, a richly exemplified and referenced history of this calculus from the time of the Jsoperimetrical pro- blems, as they were called, to our own. Also the tract by Mr. Airy, in his Mathe- matical Tracts, and Mr. Abbatt’s Treatise on the Calculus of Variations. | 2H 2 468 DIFFERENTIAL AND INTEGRAL CALCULUS. and, equating the coefficient of wdx to 0, and integrating, we have ay +(H—f Vdz) y=C J/(1+y”)...- CH) ‘ Cy" a+-H—/ Varad Wy eae ae iff y VC y ) y? f/O+y”) aS Ca" dz C porte Vdv, or 7 v= po or ee y'=K-s6 Cdv (K—S) dv WKS OS) from which y and 2 are to be found in terms of S, and by elimination y in terms of «. There are four arbitrary constants, C, K, and the two introduced in integration; three are expended in making the curve ACB of the given length, and passing through the given points A and B. But the fourth constant is undetermined, a circumstance to be explained as follows. The curve APB would remain a maximum if all its ordi- nates were lengthened by Aa, as in apb: that is, no curve of the same length (ending ata and 6) can inclose so great an area as AapbB. Hence the problem is so far indefinite that the function S and S—K (K being any constant) must give the same form of the required curve. The preceding result expresses the degree of indeterminateness which is thus admissible into the function S, by presenting S always accom- panied by an arbitrary constant. The given conditions must then be satisfied by the three remaining constants, and K allowed to remain : the resulting curve APB will make J G—-&) dx a maximum. | Before exemplifying the remaining method (page 451), it may be | shown, in the manner of Lagrange, that all the unconnected methods | given in this chapter may he reduced to one only. Let @ be a function of a, y, y', y', &C., 2, 2’, 2", &c., &c. We have then as before (w being | oy —y'dx and ¢ being dz— 2’0z), | df pde=Poa+f {(Y).o+ (Z)o 5} de+ (Y)iot (Z)iS +(Y),0'+(Z).o’+..-. In order that df@da may be always of one sign, we must have, as already explained, (Y )) w+ (Z),€=0; and if y and z be independent, (Y,)=0, (Z),=0. But if y and x be connected by an equation, say L=0, we find that it is sufficient that there should be any one function r, for which 6 ip dda+o6 if ALdxr is always of one sign, since then the condition L=0, /Ldx=const., 6(const.)=0, shows that the per manence of sign of df/@dz is only simultaneous with L=0. UH, then, AL be a function of the same quantities, and Tiveen ts vA Zi. &c. denote its partial diff. co. with respect to y, y/, &c., 2, 2’, &c., and if (Y)os (Z)o represent abbreviations similar to CY). (Z)o, &c., we have | af GHNL) dr=(P4AL) B+ J'{ Dot Da} wd ; +f {(Z)ot+ (Z)o} Cdet {Yt M%)} o+{(Zit (Zit C4 oe eceeee If, then, we eliminate \ between (YH (Y),=0, (Z),4+-(Z),=0; ON THE CALCULUS OF VARIATIONS. 469 (which are necessary, since w and £ are now independent), we have an equation between y, 2, and a, which with L—0 will determine both y and z in terms of a, if the integration can be effected. But the preceding process may be materially simplified by showing that the ultimate use of L=0 will allow us to proceed as if XX were a function of x only, and not of y, yy", &. For we have of\Ldx= f (LX ddr+r.dL.dzr+L.dny. dx) = Libr" (dLdz—dLéaz) X+ J (&dd2— dx) L; of which the first term finally becomes nothing, and the third constant, when L=0. So far as the integral part is concerned, L=0, and o\dx—dddr=0, produce the same effect on the result, but the latter would happen identically if X were a function of x alone. In the simple case in which L is a function of x, y, and z only, we have Y,=0, Z,/=0, &., so that (Y),=AY, (Z))=)Z, and (Y)>+AY=0, (Z).+AZ=0, give* ele 2 ee di, aly Js (Y)oZ—(Z)o (Y)=0, or (Ye F (Z)=0. Let z= f wdz, v% being a function of a, Y, y', &c.3; or let the equa- tion L be z’/—yw=0, whence AL=dz’—dw. Consequently Z=0, eo), Z,,=2, &c., and Y, Y,, &c. are all derived from —Aw. Hence (Z),.=—)’. Again, if 6 be a function of z only, and not of 2, 2”, &., (which is the case in page 450,) we have (Z),=Z, whence (Z),+ (Z)o =0 becomes Z—)’=0, or =f Zdr—H, H being a constant. Sub- stitute this in (Y),+ CY 0, which then becomes Y-Y/+..... +(H—fZdz) P—{(H— fZdr) P}f+.... =0; a form similar to which might be deduced from page 450, in the manner of the example in page 467; dy being Pdy +P, dy'+.... The following problem will illustrate every part of the preceding method. i Required the curve of quickest descent, from one given limiting curve to another, in a resisting medium, the resistance being R, a function of the velocity. Let x be measured positively downwards in the direction of the action of gravity, we have then, by the principles of mechanics, v being the velocity, f /(1+y”) .dx: v to be minimized, and dv __ ds hy Away ten ta ®. sre RR or 2+ 2R,/(1+y")—29=0, where z=v? and R, being a function of v, is a function of zx. Here, then, 12 go= ioruak AL=)z! + 2AR,/(1-++-y"”)—2rAg...... (1), z ae : 1 /(+y”) hy Y=0, nerdulde Ty ee a Us L=—5 TT » Uncen * Let the student deduce from these equations that of the shortest line between two points on a given surface. DIFFERENTIAL AND INTEGRAL CALCULUS. — =f 2rRz! ro, Yer J <1+y") Z=ARV(1+y%), Z=r, Z,,=0, &e. Here R’ stands for dR: dz. We have then Yee 0) k&c., Ae ar an Oe 2Ry! _ ar Oot Oe=| FG ys at = ee (2) (Z)o+(Z)o= NO+Y) . aR +y2)—-N=0- a(S) ~ “ from which three equations, A, z, and y, must be obtained in terms of x. Now (2) gives 272-44 2\R= Ay’ (1 +y%)3, and (dR: dx=R’2’) {—}2842nR'} e+ 2RN'= — Ay"y/ (1+y2) 4; or (3), (1), NL”)? @g—2R (1+y")2) + 2RN Ay ge (Iho) a or Qed (1+) t= —Ay"y' (Ly?) 4, or 2gN= — Ay'y!*; 3 2 1 / 12 whence 2g¢\=Ay’"+B, and z 3+ ae pest ey (4). y From this equation, R being a function of z, z can be obtained in terms of y/, say z=fy’. Then (1) gives fly -y"+2R/(1+y%—2g=0...... (5) a diff. equ. from which y is to be found in terms ofz. If there be no resistance, or R=0, the equation of the cycloid (page 462) can easily be found. As to the equations at the limits, we have (P+AL) ox, or Pox (since L=0)=273 (1 +y")?. 82 (Y) 22-7 y! A+y!) 2, (Y= 2ARy (1-+y2)=?, (Z),=0, (Z)r=A; whence the part to be taken between the limits is 273 (L+y!)? dot (2 F+42AR) y! (1-+y") 2 (8y—y"d2) + d (82 — 2/82) 5 Be 2 2(1+y?* or+A(sy—y'dr)+r0z — {2Ag—2AR (1 +y)3} OL: or Ay’ (1+y”) 02+ Ady—Ay'dx +rOz — Qgexr ; or (Az/* — 2dg) dx + Ady + doz. There are four arbitrary constants, A, B, and the two introduced in | integration of (5). Two of these are expended in making the curve pass through the proper points of the limiting curves; by a third we may make the initial velocity what we please, say a given function F of the coordinates of the limiting curve at the commencement; but the fourth seems superfluous.* We shall, however, find that it is deter- * Many problems in this calculus present more constants than can at first sight be made determinate by the conditions, and until the theory is generalized (which ON THE CALCULUS OF VARIATIONS. 471 mined by the following circumstance. At the first limit, dz, is Fox, +F, oy, F’ and F, being partial diff. co.; but at the second, oz, must be determined from z,=fy’,, giving dz,=f'y',.dy’. Now dy’, is in- determinate, since it depends on the alteration of the angle at which the curve cuts the second limiting curve, an alteration which in no way depends on the variation of the coordinates at the limits. Hence 62, is indeterminate, and therefore when the whole is made = 0, independently of variations, we have \,=0, or Ay'y'+ B=0, whence 2gXr =A (y'"—y'y"), and one arbitrary constant is lost. Let Yom Wo Loy and y,=¥, 2, be the equations of the limiting curves; we have then, writing Ay'y' for Ay’"'—2g, and writing for dx, ox, &c., and oz, their values, the following conditions necessary to the complete vanish- ing of the variation, independently of dx and 62, Ay’ FAY, tot (E+E, Wy) =0 Ay'y'+ Aw, 2,=0. The second shows that the curve must cut the second limit at right angles. If y= (a, A,B,C,,C,) be the integral of (5), we have the two equations just obtained, with Yo = (xX, A, &.), W, t= (x, A, &.), / lz pore err) Yo Yo five in all, to determine x, x,, A, C,, and C,; while B is already determined in terms of A. Let us suppose a given velocity at the outset, independent of the position at starting: we have then F=const., B’=0, Ie=07 and yr +, 2%=0; from which, and yy +’, 2,=0, we deduce Wx, =v’, x,, or the tangents of the limiting curves at the extremities of the line of quickest descent are parallel. But if we suppose that the initial velocity is, whatever the point of starting may be, to be that acquired in fallg from a given height, say from the axis of y, we have z= 2g, =F, whence F’=2g, F,=0; and Ay FAW, ro+2¢\)=0, or Aw’, a+Ay'>'=0; whence the curve also cuts the first limiting curve at right angles. All these conditions are independent of the law of resistance, and are true if R=0 ; we have already seen some of them in this case, (page 462.) I shall now take an instance in which there are two independent variables. Looking back to the formula in page 454 we may see that if }fpdxdy is to preserve the same sign independently of w, the Coefficient inside the double integral sign ThE must vanish : for in every other part of the expression an integration has been made, either with respect tox or y; those other parts are therefore to be taken between limits, and w, dw: dx, &c., have only the restricted values derived from it never will be until great progress is made in the solution of diff. equ.) the meaning of the superfluous constants must be collected from the circumstances of each pro- blem. Lagrange merely says that 32! is indeterminate, but does not give any reason ; if he meant that it may be mude indeterminate because another condition will be thereby introduced to determine the fourth constant, his reasoning is not sound. It is remarkable, that Woodhouse and Lacroix both omit this part of the problem in silence. th pT - ee > agent Se TL OT . = 472 DIFFERENTIAL AND INTEGRAL CALCULUS. the conditions of the limits. But the term with the double sign f/f depends upon all the values of w intermediate to the limits, and may be made to change its sign by changing the sign of w, as in page 459. The nature of the function which makes 6 if odxdy=0, dp being Xda+Ydy+Zdz + Pdp+Qdq+ Rdr+ Sds+Tdé, is to satisfy the diff. equ. ds Poetd Qe rd? Re sida SF Sees che eid ee oh ds yin ye z being implicitly a function of «andy. But the conditions relative to the limits have had no progress* made in their solution which it would be worth while to present. What is the nature of the surface which under a given volume con- tains the least possible superficial content, the volume being contained by the surface itself, by cylinders whose projections are given on the plane of xy, and by the plane of zy, in the same manner as in pages 390, &c. We have then to make SING +p?+q°) dx dy a minimum, on the supposition that Lfz dx dy remains constant. Hence we must proceed as in minimizing Ni CJ +p?+ 9°) +42) dx dy= ff odx dy, Bem 50 I Saree +p+q) 4, Q=4q (+p°+q)?, Rea-Oreee a? (pr-+qs) 1+p+¢@)?; = =r(1+p*+9’) d.Q dy whence Z—(d.P:dx)—(d.Q:dy)=0 gives (r+t)A4+p?4+ 4) —(p*r-+2pgs+q@i)=a +p + gq’)? te (1+4°) r—2pqst+(1+p*) t=a(1+p°+q?). Substitute this value of (1+q°) r-+é&c. in the equation (page 435) by which the radii of curvature of the surface are determined, and then, p being one of these radii, we have, (rt—s*) p®-a+p'+q)yptUtp'+7y=0. Let p, and p,, be the radii of curvature, derived from the preceding equation, we have then p,+p,—ap,9,, or in every surface which under a given volume contains the least area, the sum of the radii of curvature is in a constant ratio to their product, or the sum of the curvatures is constant. This property is evidently true of the sphere. Again, if ri—s°=0, or if the surface be developable, (that is, if p, be infinite,) we find —ap,+1=0, or p,, is constant: so that the common circular cylinder is another surface which satisfies the equation. If we make the conditions independent of a given volume; that is, if we ask for the surface which under a given contour contains the least possible area, we simply minimize //,/(1+p*+q°) dxdy, or make a=0 in the preceding. We find then the equations —i 2 ak 2 =t(1tp+9°) a (pstqt) tp tq) 5 * The paper of Poisson already cited may be referred to on this point; but after all, it is very little which has been done. ON THE CALCULUS OF VARIATIONS. 473 (1+q°) r—2pqs+(1+p’) t=0, (rt—s?) o+(1+p?+¢@)2=0. Consequently the surface of least area must have its radii of curvature equal in length and of contrary signs, except only in the case of a plane in which the equation is satisfied by r, s, and ¢ severally vanishing. The following method will frequently integrate an equation of the preceding form Rr+Ss+Tt=0, where R, S, and T are functions of p and q. Assume a and y to be each a function of two new variables v and w. We have then (2, meaning dz: dv, &c.) y= Pty+ Qyr5 w= PLy+ yy 3 or if p=—X:Z, gq=—Y:Z, these become X2,+Yy,+Zz,=0, X2,+ YYo+ Z2,,.=0 ; - which are satisfied by X=y, eo Yw Zu» Yea &y Xp ce 2 Lys Z —2, Yo—Ly Yo: Again Zuv =(r2, of SY, ) Ly + (s vy ee ty, ) Yr Pl vp + 9Y a So — (r2z,+ SYw) vy Se (sx, =e LY) Yr hP Mois + VY ww Zo (TR yoASYy) Ly+ (sds + CY») Yw FPlowt GY wipe Substitute —X:Z and —Y:Z for p and q, and let XXy+YVYy+Zz,= (VV), X2,+ &e.= (VW), Xz,,.+&c.= (WW). We have then rai +2sx,y, + ty’, == ViVi) Zant Ty Le TS (Ly Yt Ly Yo) + tY 5 Y= (VW) .Z TL WH yp Yo + ty? =(WW).Z-: from which, by solution or verification, may be proved rT={ yo( VV) — 2YoYo(VW)+ ys (WW)}.Z-* —8= {Lu Yo (VV) — (24 Yo + i, y)(VW) +2, 4, WW)t. 27? t={ a2 (VV) — 22, (VW)+ a3 (WW)}Z-. These, substituted in Rr-+Ss+Tt=0, give Ry, — Sa. Yo tT ain} (VV) — {2Ry, Yo—S (2, Yoo+ 20 Yo) Ty 25} (VW) +{Ry;—Sza, y,+T2?} (WW)=0. In this equation, something is left arbitrary, since an infinite number of ways can be assigned of producing any given relation between 2, Y> and z, from three equations of the form z=¢(v,w), Egos AA GSI 70) y=F (v,w). Two of these, then, may be assumed in any manner which will simplify the resulting equation. Suppose, for example, as in the given equation, that R=1+q?, S= — 2pq, T=1+p*, or RZ?= Y*+Z?, SZ*=—2XY, TZ?=X*+Z*, (Ry, ms Sx, Yot T23,) Li (Yy.+ X2,,)°4-Z? (yo+ x) =Z? (vi, byt Z,). | Proceeding thus, and substituting in the preceding equation, we find (DoF Yo +z) (VV) —2 (ay ty +4, Yo + 2%) (VW) +(%+y5+20)(WW)=0. ee ee a — —— = oa 474 DIFFERENTIAL AND INTEGRAL CALCULUS. Now (VW)=0 is satisfied by t.=0, Yrw=9, %=O0, oF T=Pvth.W, YHWPvpPyew, *=KXi vt KW; and the remaining terms of the equation vanish identically if (¢", w)? + (Y'nw)?+ (y/o w)7=0, (PL PACH YH OX 1) = 95 or wan —l fl@Giw+wew) dw, yv=V—1 fy (oie +¥iv) de- But since ¥,v is a function of ¢,%, &c., we do not restrict our solution by writing v and w for d,v and d.w, whence if ga=vtw, y=w,v+%, w, it follows that 2_N—1 fYA+¥i0) dot V—1f JA+yew ) de, the elimination of v and w will give the equation of the surface required. © Since there are two arbitrary functions, this is the most general solution. From its form it would appear to be impossible; but it must be remembered that the elimination between equations involving /(—1) | does not necessarily give that symbol in the result. ‘The preceding | equations are useful as showing the nature of the problem, namely, that it cannot be completely solved without elimination between equations containing indefinite results of integration. It is required to ascertain whether any surface of revolution can have | the radii of curvature at every point equal, and contrary in sign. Let | the axis of z be that of revolution, and z= (a+ y”) the equation of the | surface; we have then p=2ry’, q=2yP', r=42° p' +24’, s=A4rvyp", t=4y°b"’+2¢'. Substitute these in (1-++q”) r—2pqs+ (1+p’*) t=0, and we find (2? +y’) ol +¢/+2 (2+ y*) o?=0. Write w for z*+y’, y for ¢, and we have. dy | dy ( dy? ds -ledr o—~ +—~+2r| = — —-- —=2. . Bact r 3) “ah dy?. « dy? Changing the independent variable, as in page 153. Multiply by 1:a, and let dx: edy=v, which gives 4 ; mat Ee or 2da—x? vdv, and v=,/(c-2) dy @ x d 2 y= TGs ay I=TE log (/(Ca—4) +,/(Cx)) +0". Subtract the constant (2:,/C) log.,/C, and make 2:,/C=a, y=alog (/(x- a") +2) +O; whence the only surface of revolution which satisfies the conditions is z=alog fi(e+yt+a)t/(a*+y)} +C'. The equation of the generating curve is zalog{zr+/(w—a’)} +C; APPLICATION TO MECHANICS, 475 which is that of the catenary, the axis of revolution being the well- known directrix, the property of which is that the abscissa of any point is the length of the chain whose weight represents the tension at that point. | Cuapter XVII. APPLICATION TO MECHANICS. Our object is here not to deduce any laws of matter from experiment, nor to inquire into the truth or falsehood of any propositions relating to material bodies, but only to show the mode of applying the prin- ciples of the differential calculus upon the supposition of laws previously established. The aim of the science of mechanics is the discovery of the relations which exist between motions and_ their producing causes. These Causes of motion might never have been considered separately from the motions themselves, except* in a purely mathematical point of view, if it had not happened that any cause of motion, prevented from pro- ducing its effect by direct human agency, gives to the individual agent the notion of pressure or resistance. Hence in pressure we have a cer- tain antecedent of motion, which will begin to take place the moment the opposition to the pressure is removed: and the pressure being one thing, and motion another and a distinct thing, the investigation of the manner in which the former produces or affects the latter is one science, under the name of dynamics; and the investigation of the method in which pressures may act upon a material system so as to counter- balance each other and produce no motion is another, under the name of statics. There is a real distinction between the two: for in the second it is not necessary to consider any laws of connexion between pressure and motion; whereas in the first, such connexion must be made, and its laws either laid down hypothetically for future verification, or deduced from actual experiments. Any one pressure may be caused or counterbalanced by the weight of a body: hence weight is made the measure of pressure ; and pressure, force, resistance, attraction, repulsion, tension, &c. are all terms of the same meaning, with differences expressive of the source from whence pressure is derived, or the manner in which it is communicated. And whereas bodies of very different bulks are found to possess the same Weights, it is assumed that the bulk of the larger body is the con- sequence of a wider distribution of the actual matter contained in it, so that bodies of the same weight contain the same quantities of matter. The fundamental laws of motion are three in number, as follows :— 1. A material point, moving with a certain velocity, will not change its velocity nor the direction of its motion, without some cause external to itself. 2. If two causes of motion act in two directions upon a material doint, neither cause in any way alters effect of the other. That is, * That is to say, we probably should not, but for our sensations of pressure, lave considered ourselves as treating of cause and effect, in investigating the elations of diff. co. and their functions: which is what we do in mechanics. 476 DIFFERENTIAL AND INTEGRAL CALCULUS. if the point A be acted upon by one pressure in the direction AB, such as would in a given time cause it to describe AB, and by another in the direction AC, which would in the same time cause it to describe AC, it will between the two be found at the end of the time in the position D, at the opposite corner of the parallelogram formed by AB and AC. 3. Action and reaction are equal and contrary. Action is a relative term to be explained as follows. When pressure, continued for a cer- tain time, produces a certain velocity ina mass of matter, it is found that, for the same mass, the velocity produced is greater or less in the same proportion as the pressure is greater or less: but the same pres- sure acting on different masses, produces velocities which are inversely as the masses; that is, less or greater in the same proportion as the. masses are greater or less. If then P and P’, two pressures, acting for the same time upon masses M and M’, produce velocities v and wv; that is, if, upon the sudden discontinuance of the pressures at the end of the time, the masses then proceed with the uniform velocities v and v,, ‘we may prove that P: P’::Mv:M’v’, as follows. Since P’ acting on M’ produces the velocity v’, it would in the same time have produced in M the velocity v}M': M, and P would produce a velocity which is to the preceding as P:P’. But P produces v, whence v: (v/M’: M) ::P:P’ or vM:0/M'::P:P’. Now oM is called the momentum of the mass M moving with the velocity v, and this word momentum is but a synonyme for action in the preceding principle, which may be thus stated: momentum is never produced in one mass by the action, of matter upon it, without the destruction elsewhere of as much mo- mentum in that same direction, or the creation of as much in the cun- trary direction. We may then write the equation P=cMz, where, as long as the, units of mass, velocity, and pressure, remain the same, c is a constant. The value of this fundamental constant is determined by measuring the motion produced by the species of pressure with which we are best acquainted, namely, weight. And since the mass of a body is pro- portional to its weight, we must have M=kW, W being the weight of the mass, and & a constant depending on the units employed. Hence P=ckWv; that is, if such a mass as at the earth would weigh W (pounds, ounces, or whatever the unit may be) were deprived of its' weight, and subjected to the action of a pressure P, such as would, in a given time, produce init the velocity v, the equation P=ckWv would be true for certain values of c and &, which depend only on the units employed, and not on the numbers of units in P, v, and W. Butif P be the weight itself, and if the number of feet per second measure the velocity, and if one second be taken as the time during which the weight acts, it is found that v, the velocity produced, is 32°1908, which we call g. Hence W=ck Wg, or ch=1: 8, whence P=Wo:¢.° The following, however, is the more usual mode of stating the equation. Let one given substance (usually pure water at a given’ temperature) be assumed as the standard, and let the density of every substance be the number of times or parts of times by which the weight of a cubic unit of it contains a cubic unit of water. Let the unit of mass be a cubic unit of water, then 1s the mass of a cubic unit of the substance whose density is %, and if V be the volume or number of APPLICATION TO MECHANICS, 477 cubic units in a mass, kV is the number of units of mass, or M=RY. Hence P=ckVv, where ¢ depends upon the unit of P. Let the unit of pressure be that, which acting uniformly upon one cubic unit of the substance whose density is 1, would produce a velocity of one linear unit in one second. Then 1=cx1x%1X1, or c=1, and P=kVv, or Mv. This is the tacit supposition as to units, upon which the common equation P=Mv must rest. If the pressure be the weight itself, we have W= Meg, but only upon a supposition similar to the preceding, The application of our science to mechanics does not consist in the ‘solution of isolated problems,* but in the Investigation of general methods. The most convenient foundation is the well-known pro- position of the parallelogram of forces, namely, that any two pressures acting upon a point, and represented in magnitude and direction by the sides of a parallelogram, are equivalent to a third pressure represented by the diagonal of that parallelogram, both in magnitude and direction, From which it is easily proved, in the usual way, that three pressures acting upon a point, represented in magnitude and direction by three straight lines not in the same plane, are equivalent to a pressure repre- sented in magnitude and direction by the diagonal of the parallelopiped constructed upon those straight lines. ~_ Let P represent a pressure exerted on a material point whose coor- dinates are 2, y, z, and directed towards another point whose coordinates are a,b,c. Let the distance from (a, y,z) to (a,b, c), or /{ (w—a)? +(y—6)*+(z—c)*}, be called p. Now o is the diagonal of a rect- angular parallelopiped whose sides are ©r—a,y—b, and z—c, con- sequently P is the equivalent (or resultant, as it is called) of three forces applied to the point (2, Yy, =), in the directions of the three axes, and expressed by P (w—a@):0, P (y—d) :p, and P(z—c):o. And these formulze will express the sign as well as magnitude of the com- ponents, if we agree that a pressure is to be considered as positive when it tends to move the point in-the direction in which the coordinates are measured positively, and the contrary. Again, the value of p gives do _x—a@ dp _y—b do Peta on —- — dx PU MOSeMa wiahea do d d whence P —, P ia and P~? are the components above deduced. If, dz dy dz then, we suppose a number of forces P,, P;, &c., applied to the point (x,y, z), and severally tending to the points (Geos, Ci), (ase. C2), &c. distant by 01, psx, &c. from (a, y, 2), it follows that all these forces together are equivalent to one whose components in the directions of @, y, and z are lp. 1 , dos d Ip. PP, ae, Pap, Ps ge, ppp, Hs ge, dx dx dy dy dz dz Call these X,Y, and Z. The resultant is then of the magnitude V(X°+Y?+Z"): andif P,, P,, &c. equilibrate each other, so that the resultant is =0, we must have X—0, Y=0, Z=—0. Consequently Xdr+ Ydy+Zdz=0, independently of the proportions of dx, dy, and dz, This gives * No student who is totally ignorant of the common elements of mechanics should attempt to read this chapter.’ 478 DIFFERENTIAL AND INTEGRAL CALCULUS. d dp, d P, (= de+a dy+7— de) +&0.=0, or P, do, +P. dp2+&e.=0. This last equation expresses the simplest case of what is called the principle of virtual velocities. If, taking forces acting in one plane to simplify the figure, we suppose one of them to be directed towards A, and if the point P on which the force acts B be removed to Q, the distance PA is A. Shortened by PK, QK being an infinitely Q small arc of a circle, or a perpendicular let Ly fall from Q upon AP. If PB be the PK direction of another force, BP is shortened by PL. Hence if PA=p,, PB=p,, we have do,= — PK, dp, =—-PL. Here if P be supposed to move to Q over PQ=ds in the time dt, and with a velocity ds: dt, then do,: dt is the velocity with which that part of the motion takes place which is directly towards A, and dp,:dt the velocity with which the point begins to move towards B. As the point does not actually move, but a different position is taken for it, simply to examine geome- trical, not mechanical, consequences of the change, this motion is called virtual, and the velocity with which the point begins to move from or towards each point of direction of a force, is called the virtual velocity of the point with respect to that force; or, for abbreviation, the virtual velocity of the force. Again, P, being a force, and do: dé its virtual | velocity, the product P, x (dp,: dt) is called the moment of that force. | Each moment, according to our preceding equations, is positive when its virtual velocity is positive, or when the virtual velocity is opposed in direc ’ tion to the force, and negative in the contrary case: but it would do equally well to make the moment positive when the force and its virtual velocity, conspire in direction, and the contrary. When the terms virtual velocity and moment are fully understood, the equation P, do,t+ Pedos.t....=0, or Pi BPnaps dos dt dt may be expressed as follows: if any number of forces applied to a point equilibrate one another, then for every possible small motion. which can be given to the point, the sum of the moments of all the! forces is equal to nothing. Let us now suppose a second point, acted only by forces Qu, Q,, &e: in directions also tending towards fixed points, distant from the second point by o, 02, &c. Moreover, let the distance between the first and second points be 7) 9, and let a force T;,, be applied to the first point, tending towards the second, and let another of the same magnitude be applied to the second point tending towards the first. If these pomts: be both in equilibrio, we have the equations (2Pdp meaning P, dpi’ a) hy) 2Pdp+T 2 d, 7,,.=0, 2Qdo+Ti,¢ dsT1,93 where by d,7;,2 we mean such a variation of 7;,, as takes place when the first point only varies its position, and by dsr, the same when the second point only varies. If both vary together, we have dr,,.=d,71,2. +-d3r,,, so that from the preceding equations we have LPdo+2Qde+Ti,s dy, ga 9! ii ee APPLICATION TO MECHANICS. 479 The same reasoning might be applied to any number of points, and the result is that if 2Pdo represent the sum of the moments of all the forces applied independently, and if T,, , represent the action of the mth point upon the mth, (or of the nth upon the mth,) and 7,,,,, the distance from the mth point to the nth, we have 2Pdp nF pod We n AT, n3 the second > referring to every combination of values of m and n which refer to points supposed to be connected. If the distances be invariable in the system, and if such motions only be supposed as are consistent with the invariability, we have dr,,,—=0, in every case in which it appears in the equation, whence }Pdp=0, or the sum of the moments of the forces of any invariable system is =0, whence we see that the principle of virtual velocities applies also in this case. It will be desirable to collect together the principal theorems by which the differential calculus is made useful in the application of this prin- ciple, whether to statics or dynamics. If L=0 be the equation of a surface, L being a function of x, y, and #, then if from a point (a, y,z) on the surface we pass to another point (v+oxr, ytoy, 242) infinitely near to the former, but not on the surface, the perpendicular distance from the new point to the surface will be 6L:,/(L?+1L3+L%), L, being dL:dr, &c. The equation of the tangent plane being L, (E—x) + &c.=0, we employ the general theorem, that if to the plane Av-+ By-+Cz+H=0 we drop a perpendicular from the point (a’, y’, 2’), the length of that perpendicular is (Aa! + &c.) :,/(A°+B’+C*). Applying this, knowing that at the given point §—x=062, &., we find (L,6r+&c.) :/(L2+ &c.), or dL: J (Li+ &c.). The perpendicular drawn on the tangent plane can only differ from that drawn to the surface by quantities of the second and higher orders. A rigid system makes an infinitely small rotation, 44, about an axis, of which the equations are (§—a): A=(n—d) :B=(€—c):C. It is required to find the variations of the coordinates of the point (x, y, 2). First suppose the axis of rotation to pass through the origin, giving S-A=7:B=2:C. Through (2, y,z) draw a plane perpendicular to this axis, the equation of which is A (€—27)+B (n—y) +C (€—2z)=0. ‘This plane meets the axis in a point whose coordinates are determined from aoe] ae owner bet C2 Abie bee er ye Ce” which gives (£—wx)(A?+ B°+C*)=A (By + Cz) —(B’+C’) a, with similar equations for n—y and ¢—z. Add the squares of these together, and let » be the perpendicular distance from (v, y,z) to the axis, or ./((E—x)?+ &c.), and, dividing by A?+B’+C’, we have p (A*+ B*+ C*?)=(Bzr— Ay)? +(Cy—Bz)?+(Az—Cz)’. Let (2, y, z), in consequence of the rotation, come to (x+0z, y+ dy, #+0z); whence since its second position is in the plane A (é —.) + &e. =0, we have Aéx+ Boy+Coz=0: also the distance from the origin, or 7, remaining unaltered, we have ror=0, or vox +Yoy +202=0 : from which equations it follows that dz, éy, and dz are in the proportion of 480 DIFFERENTIAL AND INTEGRAL CALCULUS. Cy —Bz, Az—Cz, and Br—Ay. But the sum of the squares of oz, &c. is the square of the infinitely small arc of rotation, or p op. From this it follows that (Cy— Bz) op ~ _. (Az—Cr) dh Pure (Br—Ay) op Jue cy “YVR B +O) fA B+C) Conversely, if da, &c. be in the proportion of Cy— Bz, &c., the motion of (x, Y, 2) 1s an infinitely small rotation about the axis whose equation is pag sad eb beet Ge Cs If the axis do not pass through the origin, let its equations be (é—a) : A=(n—6) : B=(Z—c): C, then x—a, &c. must be substi- tuted for 7, &c. throughout the preceding process, and —a, &c. for €. Every infinitely small motion of a rigid system may be compounded of one motion of translation, in which all the points move through equal | and parallel straight lines, and one motion of rotation about an axis. The axis of rotation may, by properly assuming the motion of translation, be made to pass through any given point of the system. Suppose, for instance, that the whole motion brings the points P, Q, &c. into the positions P’, Q’, &c. Assume that the axis of rotation shall pass through P; and first give the whole system the motion of translation PP’, s@ that all the motions shall be equal and parallel to that of P. Let P, Q, &c. thus be removed to P’, Q”, &c. Then there must be another motion by which, P’ remaining fixed, Ql’, &c. may be simultaneously brought into the positions Q’, &c. Take the points Q” and R’” into which Q and R are brought by the translation, and through the lines Q’Q’ and R’R’ draw a pair of parallel planes. The axis of rotation, must be perpendicular to these planes, and must pass through P’; hence a line drawn through P’ perpendicular to these planes is the axis of rotation. As the conception of the theorem that every small motion} of a system in which there is one fixed point is a motion of rotation about an axis passing through that point, is not by any means easy to) the beginner, the following mode of illustration is given. Let P’, the fixed pot, be made the centre of a sphere, immoyeably connected with the system. It follows then that we show the existence of an axis of rotation, if we show that for every possible motion of the sphere about its centre, there is one point of it, A, which does not move ; for if P’ and A be both fixed, the line P’A is fixed. Let a small motion take place which removes the circle BFC into the posi- H tion DFE: either then F has remained fixed, and P/F was an axis, or the circle BGC has slipped as well as revolved, so that G has come to F. This last supposition implies that the sphere has had a motion of rotation about HK, the axis of BC, as well as about’ P/F. Let LM be the axis of DE: then since! \ GK moves into the position FM, the point A MK does not move at all, or P/A is an axis Ol rotation, | The existence and position of this axis of rotation may now be shown algebraically, as follows. Let the original axes of coordinates be fixed in space, and let there be another set attached to the system, and moving with it. Let a, y, z and &, 7, be the coordinates of a point with respect to these systems; the latter being unaltered by the motion. 62 = iets i APPLICATION TO MECHANICS. 481 . Let (X, Y, Z) be the origin of the new system, referred to the old one, and let w=aE+B ynt+y 0+X ya E+B' n+y'6+Y (1) zaa'li Bnty"f4Z a a is the cosine of the angle made by € and x, &. We have also a + +y° aad at! at!’ +! Bl" +! o’=0 a +B ty? =1, ala +B"B +y"y =0 (2) gf Bes. or reel Be oe a! +3 [! +y 7 —0 Let the system move so that (X, Y, Z) becomes (X+68X, &c.), and a becomes g+0a, &c.; in consequence, of which the point (a, y, 2) becomes (x-+dx, &c.). We have then | Otaloa +0 + loy +0X oy fda! +0! + ody +6Y (3) 62 = bb g!! + 0/3" + Coy" + 0Z Now, looking at the equations (2), which give LS ata t BiB+yoy=0, oda! + BOB" + yoy" =— (ada! +08! +y"dy'), &c., &c., let al ba" + 36" 4-7 by" — (ala! +f!98' + Y'oy! =e 8a +PhOB +y"oy = —(a bal +P OB" +y by) =«! a da +f op! +y by = —(a' da +808 +y' dy )=e". To express £, &c. in terms of (vx—X), &c., we have E=a (v—X) +a! (y—Y) +a! (2 -Z) n= (e2—X) +8! (y—Y) +P" (2—Z) (4) Sy @—X) ty Yy—-Y) +7" @—Z) Substitute these in (3), and we shall have 0 (a—X)=K' (2-—Z)—«"(y—Y) 0 (y—Y)=x"(@-X)—k (z—Z) (5) 0 (z—-Z)=« (y—Y)—« («—X); which show (page 480) that the real excess of the motion above the motions of translation 6X, oY, 6Z, common to all the points, is, for every point (2, y,z),a motion of rotation about an axis passing through (X, Y, Z), and inclined to the original axes at angles whose cosines are proportional to x, x’, x”. st If we wish to find, in the most simple manner, the position of the axis of rotation with respect to £, n, , we must remember that the points of this axis have only the motion of translation, or for every one of them, 0xr=0X, dy=dY, 6z=0Z. Hence equations (3) give foatnoB+fdy=0, Edel +&e.=0, fal’ +&c.=0 (6). But between a, a’, &c., we have the equations DIFFERENTIAL AND INTEGRAL CALCULUS. éta*to?=l1, ab+ df B'+a"B"=0 Bert ph=1, — By+B'y +Pty’=0 (7) vy mi vf? ao yi =1, eee ya! ai. yal! =. () Whence ada ta!da! +a da"=0 dB +a 6! + of'5B" = — (Bocae+ B'da! + Bl'da", &e.) Assume Boy+ Boy! + poy"=— (yoB+ yop! + 0B") = —pdl | vba yltal -+y/"ball=— (aby bald tal'y"=—adt (8) dB + ofl dB! + ol 3B"= — (Bda+ Bde! +h 0a!) = —rdt dt being the time in which the small motion is made. Multiply equations (6) by a, a’, «”, and add, which gives 77 —q¢=0. Multiply | by 4, &c., and by y, &c., and we thus have three equations, gé—pn=0, rn—gqe=—0, pl—ré=0; which are the equations of a straight line inclined to £, 7, and ¢ at angles whose cosines are proportional to p, q, and 7. The following equations may be deduced from (2), as in page 497 | following. a == Bl y!!—»! B", al =p" y—y"B, Ql! = py! — yb p=y'e!—a'y", p! VA —allly, Bp" = yal —ay! (9). y=a'p"—fla!, y/=a"B —f"a, yy! =af'— Bol. To the order of which the following is the key, (7 WO ony. (aB, By> ya). | Properly speaking, the preceding should be +a Bly’—B", &e.,| the sign depending on the manner of measuring ¢, &c. positively and, negatively, with reference to the manner of measuring «. ‘Take a point | on the axis of £, so that y=0, ¢=0. We have then, if both sets have; the same origin, s=aé, y=a/t, z= c"6; so that, ¢ being positive, a, a’, and @’ must have the signs of 2, y, and z. And it can be shown that, according as « is f’y/—y'6" or y'B’—fply"”, so B 1s y'all —aly" or aly""—y'a", and y is &B'!'—B'a! or B'a!—a' B", &c. Hence, by proper selection between the two ways of measuring & 7, and ¢, the equations (9) may always be made true as above written. The quantities «, a, &c. are nine in number, connected by six! equations (for the set (7) is deducible from (2)). They can, there-) fore, be expressed by means of three quantities only, and the most simple way of doing this is as follows. Through the origin of «, y,# draw lines parallel to the axes of &, n, ¢ Draw a sphere with the origin as a centre, and let X, Y, Z and X’, Y’, Z’ be the points at which the several axes emerge from the sphere, and let N be the point at which the great circle in the plane of 9 cuts that in the plane of ay. Let X’, Y’; Z! be each joined with X, Y, and Z, and let the angles subtended by ZZ’, NX, and NX’ at the centre be 6, ys, and g. Then, making arcs the symbols of angles subtended at the centre, and denoting by [a, 6, ¢] the cosine of the third APPLICATION TO MECHANICS. 483 side of a spherical triangle whose other two sides are a, 6, and their included angle c, we have (remembering that Z and’ Z/ are the poles of XY and X’Y’, whence a9 {=—sin 0 cosh 2 2°2 y" =cos Z/Z, ea. cog. 0, From these we easily get dy. =) -B OP+ a! Ot ol! sin Uw 8 0B =—«a dp+f' dw+ A" sin y 30 oy = Y owt y" sin y 69 da! = B dp—a de+a" cosy 30 op! =—a! dp6—f Ow +B cos Ys 60 oy! = —%¥ OW fy! COS Us 68 Sa! = B"dh —y'' sin p 60 OB" = — odd —v'' cos @ 60 ys — — sin 600 Boy + Bld! + Bl dy"= (By! —yB'dye+ {y(Bsimbs + B'cos %) — B"sin 6 100 aby + dy! + al"dy" =(ay/—a'y) d+ iy (asin: + ¢'cos %)—a"sin 0430 Boa + B'Sa! +B Sa! = 06+ (Ba' — Bx) OUs + {a (Asin &+ A! cos) — fy" sin p} 00. Write — pdt, qdt, and rdt for the first sides, and, after using equations (9), substitute the values of a’, B’, and y", with those of B sin Ww +’ cos w and g sin w+! cos Ww, which will bz found to be cos@cos¢ and cos@sing. This gives, after reduction, pdt=sin ¢ sin 6 dus—cos $ 60 gdi=cos $ sin @ ou +sin 60 rdt= op—cos 0 d& The preceding results are of such fundamental importance in the 212 484 DIFFERENTIAL AND INTEGRAL CALCULUS. application of our subject to dynamics, that it will be worth our while to explain them at length. A simple rotatory motion is easily conceived ; an axis remains fixed, and all the invariably connected points describe circles about that axis, with an angular velocity which, however it may vary from moment to moment, is the same for all the points at any one moment. But any number of rotatory motions may be given to a system at once. Suppose A, B, the pivots of the first axis, to rest in a frame which is itself supported by another axis aN CD. If, then, the spheroid in the diagram rad be made to revolve about AB at the same \ | time that the frame revolves about CD, the | points of the spheroid will take a motion compounded of both rotations, the nature of which we have now to investigate. Again, if CD were attached to a frame, which itself was connected with a third axis, a third motion of rotation might be given, and so on. At the first instant, these rotations, however many, produce the effect of one rotation, if the axes all pass through the same point ; and the axis, or the insfantaneous axis as it is called, may be found as follows. First, let two rotations be made round two axes which meet at O, as OA and OB. Then, both axes being in the plane of the paper, all it points in that plane begin to move per- a ,C pendicular to it, from both rotations. | Also, in one of the angles made by BO and BA, each point aforesaid will be elevated by both rotations, in the oppo- Me A x site angle they will be depressed by both, while in the remaining two angles they will be elevated by one and depressed by the other. Let BOA be one of this last pair of angles, and let the points in it be elevated by the rotation about OA, and depressed by the rotation about OB: also let a and £ be the angular velocities of these rotations. Then any point P, distant by PM and PN from OA and OB, would by ‘the several rota- tions be elevated by PM.adt, and depressed by PNAdé, in the first in- finitely small time dé of the motion. Take PM.¢=PN.A, and the point P is therefore not moved at all, or the double rotation (O being also unmoved) produces one single rotation about OP as an axis. Take OA and OB proportional to the angular velocities « and 8, and describe the parallelogram OABC: it is then easily* proved that for any point P in the diagonal OC (or OC produced) PM.OA=PN.OB, or PM.a=PN.f. Again, since the point B (which is on the axis of one rotation, and therefore only affected by the other) only receives the elevation BQ.adt, let 0 be the angular velocity with which the system begins to revolve round OC; whence BQ.adi=BR.6dt, or BQ.a —BR.9@. But BQ.OA=BR.OC, whence «:6::0A:OC, or OC represents the angular velocity about OC. That is to say; if upon two axes of rotation lines be laid down representing the angular velocities, |. * If with any point as a vertex, triangles be formed which have for their bases the conterminous sides and diagonal of a parallelogram, the greater of the three triangles is equal to the sum of the other two. When the point is on a side or on the diagonal, one triangle vanishes, and the remaining two become equal. APPLICATION TO MECHANICS, 485 in such manner that the intervening points shall begin to move. in contrary directions: the resulting motion, at the first instant, will be one of rotation about the diagonal line of the parallelogram formed on the first lines as an axis, with an angular velocity represented by the length of that diagonal. Moreover, the resulting rotation will be in such a direction that points intervening between the diagonal and the axis of elevating rotation will be depressed, and vice vers. From this it may easily be proved, in a manner similar to that employed in com- pounding motions of translation, that three such motions of rotation may be compounded into one, by laying down on the three axes lines propor- tional to the angular velocities, and finding the diagonal of the paral- Jelopiped constructed on these three lines, which diagonal will be in the axis of the compound rotation, and will represent its angular velocity. Hence any rotation about a line drawn through the origin of a, Yy & may be decomposed into three rotations, one about each axis. Let a positive rotation about the axis of x be that which tends to move the positive part of the axis of y towards that of z; similarly, let positive rotations about the axes of y and z be those which move the positive parts of x towards those of x, and of 2x towards y: all which may be easily remembered by zyz, yza, xry. Then a rotation about the line which makes angles «, 4, y with the axes, the angular velocity being A, may be decomposed into Acose@, Acosf, Acosy round the several axes of 7, y, z, or else into —A cosa, —Acosf, —A cos >. according to the direction of the rotation A. Secondly; let the axes of rotation be parallel to one another, and per- pendicular to the plane of the paper, and let them pass through A and B. Let them be said to be in the same Soe direction when A and B begin to eit gee ad Ns move in contrary directions, and A i vice versa. If then the rotations be of equal angular velocity, and contrary in direction, the result of the two motions of rotation will be one motion of translation, in the direction perpendicular to AB. For each of the points A and B only moves in virtue of the rotation round the other: but the angular velocities being equal, and the directions contrary, the initial velocities of A and B are equal and in the same direction, whence AB is carried without change of direction in the direction perpendicular to AB. In any other case, take infinitely small lines described by A and B in the time dé, each of which is therefore proportional to the angular velocity round the other axis. Thus, let Aa=AB.Ad/, Bb=BA.cdé, whence a and b will represent the ‘positions of A and B at the end of the time dé, The point O, which remains at rest, and js therefore a point in the axis of the compound rotation, is determined by OA: OB : AB Adt, AB. ad, or OA.az=OB. 8. 486 DIFFERENTIAL AND INTEGRAL CALCULUS. 1. When the rotations are in contrary directions, that round A being | the greater, the axis of compound rotation is on the side of A, and OA.a= (OA+AB), or OA=ABB: (2—Af), OB=ABa: (a —8). The angular velocity gives the angle Aa: OA, or AB.fdt:(AB.8:(@—8)), | or (a—f) dé in the time dé, and is a—f. 2. By similar reasoning, if the directions be contrary, that round B being the greater, we have OA=AB./: (S—a), OB=AB.a: (b—a@), | and 6—« for the angular velocity. 3. If the directions be the same, we have OA=AB§:(a+8), OB=ABa: (a-+f), and «+f for the angular velocity. If three rotations be communicated round axes parallel to one | another, two of them must be compounded by the preceding rules, and | the result compounded with the third. Thirdly ; let the two axes of rotation neither meet nor be parallel, | the result is a motion of translation and one of rotation combined. Let | the axes be AK and BL, and let AK and BL be proportional to the | angular velocities. Take any point O, and axes passing through it parallel to AK and BL. About OM impress two equal and opposite motions of rota- tion, of the same magnitude as that about AK: and K L ii the same in magnitude as that about BL. The motion of the system is not altered by this intro- | O duction of.new motions which destroy each other. And the motion about AK with the equal and con- that about BL combined with the contrary motion about OP. ‘The whole motion, then, is equivalent to two translations and two rotations about axes passing through O: of which each pair may be compounded into one of its kind. The same reasoning may be extended to cases of more rotations than two: and hence follows the theorem already alge- hraically proved, namely, that any” motions whatever, translations or rotations, how many soever, are at every instant equivalent to one motion of translation and one of rotation: also that the axis of rotation may be | made to pass through any point. When a rotation is made round one of the coordinate axes, it is con- | | trary motion about OM produces a motion of translation only: as does | | iB about OP impress two others equal and opposite, and | | | : i i a 4 venient to call it positive or negative, as previously described ; but when | the axis of rotation. passes obliquely through the origin, though two | rotations may be made round this axis, in opposite directions, and there- fore relatively to each other positive and negative, yet there is no reason | for assigning + to either rather than to the other. ‘This ambiguity pre- | sents itself in formule by the appearance of a square root with an undetermined sign. | | If we now return to page 480, and call d, p, v the angles made with the axes by the line €: A=: B=2:C. We have then dr=(z cos p—y cosy) df, dy=(x cos v—z cos X) of, oz=(y cos \X—z cos p) Od. The signs here are not the same as in page 480, being changed to suit the hypothesis as to positive and negative rotation laid down in page | 485. Thus, if the whole rotation were about the axis of z, we should © have A=}7, p=4r, v=0, or dr=— yd, dy=axdp, dz=0, If op. be APPLICATION TO MECHANICS. 487 positive, or has the sign contrary to that of y, and oy has the sign of x, Hence, as may readily be seen, this positive value of 66 moves the posi- tive part of the axis of 2 towards that of y: which was required to be the case. Let = be the angular velocity of rotation, and a,, ®,, @,, the three rotations round the axes of x, y, and z, of which the rotation about the given axis may be compounded. We have then 3¢ =adl, o,=@.cos\X, &c., whence 6r=(@,.z—,.y) dt, dy=(@,.a—o,.z) dt, dz= (@,.y—@,.2) dé. If the coordinates £, 7, and Z had been employed, we should have obtained similar equations. In page 481, equations (6), suppose that .we consider a point which is not on the axis. We have then 0 (@—X) =foa+ 0B + Ldy, &e. ; which equations, multiplied by a, «’, and »’, and added, give a0(2—X) + a! d(y—Y) + a0(z—Z)=(ql—rn) dt, &e. We have supposed the axes of &, , ¢ to move with the system.. But if we now suppose a set of axes, coinciding with these at the com- mencement, to remain immoveable, so that the coordinates of a point attached to this system vary, we shall have (page 481, equations 4) S= 0d (©—X)+ a6 (y—Y) +5 (z—Z), whence the preceding equa- tions give of=(q6—rn) dt, dn=(réi—ph) dt, df= (pn—qgé) dt, which, compared with the preceding, show us that p,q, and 7 are a,, @,, and w,, the angular velocities of the three rotations about the fixed axes of £, 7, ¢, into which the single rotation of the system and its moving axes about the axis £:p=n:q=é:r, may be resolved. The values of p, g, and 7 have (page 483) been deduced in terms of dp: dt, &c.: a geometrical confirmation of this connexion may easily be given, now that we know the most simple meaning of p,q, and 7, as follows. A change in ¢ only, or NX’, @and w, or ZXNX’ and NX remaining the “ same, would obviously be nothing but a small y rotation about the axis which emerges at Z/, yea or the axisof 2. Hence d¢ is wholly a part r i of rdt. If 0 alone were increased by 60, X’ / \,.. and Y’ would move perpendicularly to NX‘Y’ «See Se through arcs, the angles of which are sin ¢.d0@ ree e, and sin (47+) 60, or sing0@ and cos¢@ 00. % ~ These angles, since X/Y’ is a quadrant, belong to corresponding rotations about the axes of Y’ or 7, and of X’ or é; but the second must be called negative, since its effect is to move Y’ from Z' (page 485). Hence —cos@d0 and +sinp 00 are the terms arising in pdt and qdé from the change of 6. Finally, let % be 4, , 488 DIFFERENTIAL AND INTEGRAL CALCULUS. increased by oy, and @ remaining the same; and let na’y’ be the new position of NX’Y’. Then, since the angles XNX’ and Xnz’ are equal, the internal angles at m and N are together equal to two right angles: but this, when true of the angles of a spherical triangle, is true of their opposite sides ; therefore NK+ Kn is two right angles, or KN and Kn are both infinitely near to one right angle. Hence X’v=Nié.cos¢ and y'w=Neésin@ are either true, or only erroneous by small quantities of the second order; it being remembered that since rK=a2’y’, we have Ky/=nz'=¢. Hence we see, 1. A rotation about Z’' of the magnitude nt, or cos 0.0%, and negative, since Y’ ‘is moved towards X’. 2. The rotation X’v about Y’, which is Nt.cos@, or cos@ sin 6 ow, and positive, since Z’ is brought towards X’. 3. A rotation wY’ round X’, which is Nésind, or sing@sin@ dy, and positive, since Y’ is moved towards Z’. Hence arise the terms of pdé, qdt, and rdt, which depend on oy. The preceding formule are adapted to one position of the figure, which is that adopted by all writers as the principal case. As in other problems, every modification of the figure requires modifications of the signs of the letters whose values determine the relative positions of the parts. The preceding results relate entirely to what takes place at the first instant after the system has been abandoned to the effect of two or more rotations. Let us now suppose the combined rotations to continue, it being supposed that each axis takes the motion of rotation round the other axis. The axes themselves are, therefore, continually changing their positions ; and the instantaneous axis of rotation, the position of - which is always given relatively to the other axes when the rotations are uniform, changes with them. It is difficult at first to see what can be meant by a line of rest which changes its place, but a description in other words will make it clear. ‘The motion of any system about a fixed point, however many the rotations of which it is compounded, must always have some one axis at rest for the instant, and as the motion proceeds, one axis after another becomes quiescent, the quiescence not continuing any finite time.* And instead of saying that axis after axis is successively brought to a state of rest, we say that the axis of rest, or the instantaneous axis, changes its place. That the student may more clearly comprehend the necessity of there being always an axis at rest, I shall show that any change of place which a system can undergo, one point only remaining stationary, 1s capable of being made by one rotation about one axis: or that, for any given finite change of position whatsoever, some one point remaining at rest, some one axis mayt remain at rest. Or thus, one point remaining fixed, it is impossible to give change of place to all the limes of a system at once. This may be proved either geometrically or algebraically, as follows. About the fixed point as a centre, describe a sphere, and let the motion bring PQ, an arc on this sphere, into the position P/Q’. Through V/ and VY, the bisections of PP’ and QQ’, draw great circles * When a ball is thrown up into the air, there is an instant at which it can neither be said to be rising nor falling, and it is then said to be brought to rest ; but it does not rest any finite time, however small. + Not must: the following proposition is a parallel. Any given change of place of a point may be made by moving it along a straight line; but it may also be made along an infinite number of different curves. APPLICATION TO MECHANICS. 489 : VR and V’R, perpendicular to QQ’ and PP’, meeting re in R. Then we have RP=RP’ and RQ=RQ’, so i that if the angles P’RP and Q’RQ be equal, a rota- tion round a diameter passing through R would bring Pay PQ into the position P/Q’. But these angles are ye equal: for the triangles PRQ and P’RQ’, having their - sides severally equal, have their angles equal ; whence mPROQ’ = Z PRO gatid QRP. to: both, and ZP’RP= ZQ’RQ.. A similar proof may be given for every one of the varied alterations of position which the figure will admit of. Hence, since every change of place may involve a quiescent axis, every infinitely small change may be considered as actually doing so: but it does not follow that the quiescent axes of two successive infinitely small changes are the same. The algebraical proof of the proposition will be as follows. Let 2, y, 2, be coordinates fixed in space, and é, 7, 2, coordinates fixed in the system, and let r=Af+Bn+Cl, y=A'i+k&c., &c. be the relations existing at the first position, and w=aé+bn+cl, y=a'é+&c, &. those at the second position. If, then, there be a line of the system which belongs to both positions, 2, y, and z will in that line remain unchanged when the system has been removed from one position to another. Consequently we shall have (A—a)é+(B—b) n+(C—c) f=0, (A’—a’)E+&c.=09, (A” —a") €+&c.=0. Eliminate 7: and @: , and we have (A— a)(B’—b')(C"” —e") + (B— b)(C'—c')(A"”—@") +(C=0)(A'—a)(B"— 8") ie —(C~c)(B/—b/)(A"—al") — (A—a)(C/—e/)(B" 0") [~ —(B--5)(A! a!) (C44!) | which must be universally true, if the ‘proposition asserted be so. The terms resulting from these products may be classed as those which contain three capital letters, three small letters, one capital only, and one small letter only. Also (page 482) we have A=B’C"—C’B", &c., or A=C'B"” —B’'C", &c., the sign being indifferent, provided the proper order be ob- served. The terms of the first class give A (B'C’”—C’B) + B(C’A”- A’C”) +C(A‘'B’—B’A”), or A2+B?+C?, or 1: those of the second give —a (b'c!!—clb"') —b (da! —a'c")—e (a'b"—b/a"), or —a’—b?—c*, or —1: all these terms then disappear.* The terms containing A with two small letters, make A (b'c’—c' 6”), or Aa; that containing @ with two capital letters is —a@(B’C’—C’B”), or —Aa: these terms, therefore, destroy each other. Ina similar way the remaining terms of the third and fourth classes destroy each other, and the identity of the equation is proved.t * It may be asked, why not adopt the order AB, BC, CA, in expressing A, &e., in terms of the rest, and ba, ac, cb, in expressing a, &c., which may certainly be done, consistently with the equations of condition ? The answer is, that if this were done, it would be equivalent to supposing a, &c. after the char before, but with the signs changed, so that we should have (A-+a)%+&¢.=0, &e., which would give the same results as in the text. : + The ease of this demonstration will illustrate the advantage of symmetry in mathematical processes. Euler, (Theor. Mot. Corp. Rigid.,) having proved the age, to be the same as 490 DIFFERENTIAL AND INTEGRAL CALCULUS. We have shown, page 483, how to express «, 6, &c. in terms of three. angles; the following method of determining six of them in terms of the | remaining three is due to Monge,* and will give an easy method of deter- : mining the axis of rotation just shown to exist. | Let the three data be the angles made by @ and é, by y and », and by) z and ¢, or their cosines a, f/ and y’. These being given, the| position of the axes of &, n, and @, with respect to w, y, and 2, is also| given. We have then ~vol—y?—y?=1—oe?&— B, or & +0 =y"?+y'” =]—o?— 2+"; whence 6?+a?=1—o?— f+ y'”. But y’=af'— Ba’, or 2Bo/=2ah’—2y", whence we have (B+eN=(—y'Y— (PY, b+aé=f/A+a- p'—y").J/A—ae+b'—y"), (8B—a')’= (1+ y")?—(a@-+8')’, B—d=J/A+tath'+y").J/A—e—b'+y"): | whence § and @’ are found in terms of the data. Proceed in this way,| and the conclusions are as follows. Let : T=1lto+f'+y", t=l+a-f!-y", U=1-atp'-y", t!=1-a-f'+y") B+a'=,/(tt’) a! + y=,/ (it), y+ p= Jt") | B—ol= (Tt) el —-y=J/(Tt), y—h"=J(T) 5 whence the remaining six are determined in terms of a, 6’, and vy". The ambiguity of the signs will always put a serious practical difficulty’ in the way of using these results for particular purposes. Let it be required to find the axis round which the system must, revolve, so that the axes of v, y, z may come into the position of &, n, @. We have then r=, &c. for every point in that axis, or z= ax-+ By+y2,i &c. This givest | (a—1)#+f6y+ yz=0 axe+(B'—1) y+y'z=0 al"e+ Bly + (y"—1) z=0, equations of which the coexistence has been proved. Taking the first pair, we find that x, y, and z, must be in the proportion of By'—yB'+y, yal —ay'+y', and («—1)(f’—1) — Bo’, or a+, BY 4-y, and 1+y''—a—B’, or J (tt!’), Jt"), and t’, or Jt, Jt’, and ,/t’. Hence there is this restriction upon the data, that ¢, d, and ¢’’ must be all positive or all negative; but ¢+¢/+é’, or 3—a—f!, —y'’ cannot be negative, whence a, 6’, and y” must be so taken that one more than either must be greater than the sum of the remaining property in question geometrically, professes himself unable to give an algebraical demonstration: Nemo facile stupendum hunc laborem in se suscipere volet are his words (as cited by Sr. Piola). In vol. xxii. of the Memoirs of the Italian Society of Modena, Sr. Gabrio Piola has conquered Euler’s difficulty in sixteen quarto pages of calculation and description: the whole difficulty arising from the loss of the view of general properties consequent upon preferring simplicity to symmetry. * Or rather the results to Monge and the demonstration to Lacroix. + ‘These are the unsymmetrical equations referred to in the preceding note, APPLICATION TO MECHANICS. 491 ones. Hence the cosines of the angles made by the required axis of rotation with those of x, y, and z are Nag? /JGt “a lh Oot Rar ae / pte) Resuming the equations in pages 481, &c., let all the rotations which are to take place simultaneously be reduced to p, g, and r, round the axes of ,, and ¢, which moye with the system. However these rotations may vary, either as to amount or position of their axes, we have seen that their effects may at any one instant be confounded with those of an in- finitely small rotation round each fixed axis. Given the position of the system, and the values of p,q, and r ata given instant, required the velocities of a given point, parallel to the axes in space, and to the axes in the system. We must first express ea, 08, &c. in terms of p, g, andr. To do this we have (page 482) Boat Boa'+h"Ca’=rdt, yoat&e.=—qdt, a«de+&e.=0. Multiply by 4, y, and a, and add, which gives (page 481, equations (2)) ea=(rh-qy)dt; multiply by 6’, y’ and a’, and by 8", y," «’, and we get similar expressions for dg’ and dq’. Proceeding in this way with the other equations (7) and (8), (page 482), we find the following set : da =(rB—qy) dt, da’ =(rB'—qy') dt, dal = (rB'—qy’’) dt dB=(py—ra) dt, op’=(py'—re') dt, o8"=(py"—ra") dt dy=(qa—pBh)dt, Sy'=(qa'—pf') dt, dsy/=(qa" — pB"’) dt. dx da. dp dy i dy tae! yh Nar Toetpe eeaA — = t+ &e., &e, dt. a. peu Se 86 Again, Hence the velocities in the direction of x are expressed in terms of £, &c. To find them in terms of a, &c., substitute S=ar+oalyta''z, &e., . a 2 Be a aT? fpit which will give, making use of g=f'y—y'B", &c., (page 482), dx qe Pe 9B! ry!) 2— (pal +g" try!) y dy a = (pal +qB" +ry")t—(pa +98 +ry Jz dz Bp Pe +98 try )y—(pal +48! +ry')a; whence it appears (page 480) that rotations p, &c. round &, &c. are, for the instant, equivalent to pat+tqg6+ry, &c. round a, &c.: a result which may easily be shown to agree with that in page 481. Lastly, to find the velocities in the momentary directions of &, &c., we must suppose @, &c. to remain constant, and é, &c. to vary, which gives dx 1 az od dz Sa tal ota pa! = + be, &C. d dt dt’ dt dé pee da ¢ , (da! a wW dg" e re Et&e. )-+e (qr bree +a dt f+ &e. =o —Tn. 492 DIFFERENTIAL AND INTEGRAL CALCULUS. And thus we get Za 3 dyn _ dg : qe Oh a Epes pas As an instance, let us suppose p, q, and r to be constants. To find «, B, and y we have to integrate the simultaneous equations da wee Ch ie ) dy _ era ine hb AH a nee Pig! ph. Differentiate the first, substituting from the second and third, and we! have ad x Wp =P (qB+ry) —(q?+7°) a. But pda+qdB +rdy=0, whence g8 + ry=K—pea. ‘Let P+ge+r= he, and da ee de nen : pK aa t* a=pk, ama cos ki- A sin kt-+— ane t gk Similarly, B=b coskt+B sin kt+ a K y=ccoski+C sin ht Here are seven constants, where from the original equations it appears that three only should enter. But pa+qS+ry=K, and* a@ + B?+7°=1, which will be found to require the five equations | pa+qb+rc=0, pA+qB+rC=0, aA+bB+cC=0, 72 ; eto tOSM BY Cal. | These jive equations between seven constants leave only two con- stants arbitrary; whereas the complete solution of the equations would; require three. But it must be remembered that in assuming g?+2) +y’=1, we have already obtained, and given a definitive value to, one. of the constants ; since a+ 6°+ y°=L will equally satisfy the diff. equ., L being arbitrary. In a similar manner, we may find «=a! cos kt+ A’ sin kt-+-pK’:R’, &c., with similar relations between the constants. This shows how to. express a, &c. as functions of the time: but since pe+q6+ry, &c. are constants, being K, &c., the preceding values of dx: dt, &c., with page 491, show us that the system does nothing but revolve about an axis \ fixed in space, making angles with the fixed axes whuse cosines are pro-: |i portional to K, K’, and K”. There are, however, some important cautions to be given connected with the subject of rotation. If we suppose the system always to have the. velocities of rotation p, q, 7, about axes which are perpetually varying | in consequence of those motions, the effect is not the same im a given time as if we suppose the whole rotation belonging to that time first communicated about one axis, then about the second as it stands after * Let it be particularly noted that this is a consequence of the equations them- | selves, which give ade-+6dB-+ ydy=0, and therefore «*+/?+ 4?=const. APPLICATION TO MECHANICS. 493 the first, and then about the third as it stands after the second rotation. For the actual motion in space depends not only on the rotation but on the position of the axis, and the effect of an infinite number of infinitely small motions, made round an axis which changes its position at the end of each, is not the same as it would have been if the axis had preserved its position. Again, if a motion of rotation round a fixed axis passing through the yrigin be continued for an infinitely small time dé, with an angular velocity P, a point at the distance o from the axis will describe an arc which belongs to the circular sector $p*Pdé. The rotation may be ‘esolved into three others, round the axes of 2, y, and z, and the area ust mentioned may be projected into three others, on the planes of Y2, sr,and xy. But the projected areas are not necessarily the areas made dy the resolved rotations, and must not be confounded with them.* I now come to another subject, namely, the consideration of those ntegrals depending solely on the constitution and arrangement of the yarts of a system, which are required in the investigation of its motion. Let the whole system be divided by planes parallel to the coordinate dlanes, as follows: parallel to the plane of wy, and distant from each ther by dz, let an infinite number of planes be drawn, and the same yarallel to the plane of yz, distant from each other by dz, and to the lane of zz, distant from each other by dy. The whole system is then livided into an infinite number of parallelopipeds, each having the rolume dx dy dz. If, then, p be the density at the point (2, y, 2), which nay be a function of 2, y, and z, the mass of an clement contiguous to 2, y, 2) is pda dy dz, and the whole mass is ff fodx dy dx, taken over he whole extent of the solid. It is usual to write odz dy dz as dm, hus making the common symbol of a differential of the first dimension ttand for one of the third: in this manner frdm is made to denote a riple integration, since it stands for Sf fro dx dy dz. If the system were to consist of a finite number of material points,+ laving the masses 7,, 7%, m3, &c., andif x, y,, x, be the coordinates of he first, &c., the sum m,2,+m,2,+.... or Exm must be substituted or fxdm in all equations connected with the motion of the system. In act, Dam and {adm only differ in the supposition as to the distribution if the system, the first becoming the second when the number of masses s infinitely great, each being infinitely small, and the whole forming one ontinuous mass. If we change the coordinates, an integral of the form ff [P dx dy dz akes the form ff fu dé dndf; and it is important to show that in the hange from rectangular to other, rectangular coordinates no other hange is requisite except substituting in P for a, y, and z their values a terms of ¢,7, and Z, and changing dxrdydz into didn df. Now rst observe that a complete change of coordinates may be made by three uccessive changes, at each of which one axis remains unchanged. * On the subject of rotation generally there is an excellent pamphlet by [, Poinsot, of which the title is “ Théorie Nouvelle de la Rotation des Corps,’” aris, Bachelier, 1834. Nothing but the press of matter more closely connected! ith the application of the differential calculus has prevented my inserting the hole of that pamphlet in the present chapter. t The material poirt, a common supposition of physical writers, should rather be a infinitely small mass of matter: though there is no mathematical impropriety i Supposing a point to be endowed with the weight of a given mass, or with any her property, the conception of which does not depend on that of bulk. 494 DIFFERENTIAL AND INTEGRAL CALCULUS. First, let the axes of w and y revolve round the axis of z until the plane of zx includes the axis of §; in which case the axis of y becomes per-| pendicular to that of g. Secondly, the axis of y retaining its new position, let those of z and 2 revolve round it until the axis of x coin-/ cides with that of €: the axes of 4, 2, y, and z will then be all in the same plane. ‘Thirdly, the axis of « remaining in coincidence with that of £, let the axis of y revolve until it coincides with that of y, in which! case the axis of z will also coincide with that of ¢. If, then, we can show that the theorem is true of one of these changes, it follows that it remains, true after any number of them. Now the axis of z remaining fixed, let those of # and y revolve through an angle 6, and let a’, y', and zx’ be the coordinates of the point) whose coordinates were «, y, and z. We have then z=2z’, y=a’ sin6) +y! cos 0, x= 2! cos6—y!sin 6. If we now write f ff P dr dy dz in the, form* fdz { f f P dx dy}, it being remembered that dz, dy, and dz are independent, and return to page 394, we see that 2! and y! stand in place of wu and v, and that to transpose 5 f Pdxdy into the form) ff P’ dz’ dy', we must substitute for « and y their values in P, while) tor dx dy we must write : dy doi, dy dx re ‘ : + ae cy! weak ani da’ dy’, or + (sin? 6+ cos? 6) da’ dy’, or da! dyj,) taking the positive sign. Hence ff Pdady=f f P’ da! dy’, and put-! ting dz’ for dz, we have f { f P’ da! dy’ dz! for the integral expressed in terms of the new coordinates: no other changes being required than those expressed in the enunciation of the theorem. The same is still true after the second.and third changes are made, which are requisite ta! bring the axes of 2, y, z into coincidence with those of &, n, Z. There is a pot in every system which takes the name of the centre of gravity, from the remarkable preperties which it possesses in con- nexion with the conditions of equilibrium, when the weight or gravity of| the system is one of the acting forces. This point possesses properties as remarkable in connexion with the laws of motion of the system, inso- much that if it were allowable to attempt to disturb any established term, the present would be a most legitimate occasion for the use of such permission. Retaining however the established phrase, I proceed to point out the geometrical properties of this point, by means of which its mechanical properties are found. Let there be points, 7 in number, (a, y1, 21), (@a,Ye%), &c. Takea point (X, Y, Z), whose distance from each of the coordinate planes is the mean distance of all the m points from such planes, or assume nie | Het nth Mie Ti | 7 7 v7) { t The point thus obtained has the property that its distance from apy other plane whatsoever is the mean distance of the points from that plane. Let the new plane, whatever it may be, be taken as a new plane of ay, so that the distances of the points from that plane are the * For actual integration this form would be useless unless the limits of z were the’ same for all values of « and y; but it must not be forgotten that a perfect con-" ception of the summations of infinitely small elements, in the order which the form’ given implies, is attainable in every case. : | } ‘ iD i APPLICATION TO MECHANICS. 495 new coordinates of z. Let the point (2, y, z) be (2’, y' 2’) in the new system, and let (X, Y, Z) be (X’, Y’, Z’). If then v= as! + By! + ye2', &., we have 2=yr+y/y+y"z, &c., and) Z/=yX+yV+y"Z. Con- sequently, the mean value of 2’ or ¥s,! s Ss Sy ~ e PAS * wit Lot be! ea isy— + hee nd th or yX+7/Y "7, r FI Nidhi a aNig.2 6 dein oitetaty Rae co The preceding supposes that the new plane passes through the origin : if, however, it should subsequently move, remaining parallel to its first position, no alteration would be made in the truth of the theorem, since each 2’ and also Z’ would alter by the same length: so that the altered value of Z’ would still be of the mean of the altered values of z’. _ If the plane just supposed pass through the point (X, Y, Z), we have Z’=0, or >Z=0, or the sum of the distances of the points on one side of the plane is the same as that on the other. Now let any number fk, of those points be supposed to coincide at (%, Y1, 2), also kz at (X2, Ye, 22), &c. Then, counting (x, y, z) as a col- lection of k, points, &c., the centre of mean distances (n being 2k) has the coordinates Ska: 2h, Sky: Dk, and Skz: Sk. Next, let each of these points be supposed to have the mass p: then wt the first point is collected the mass k,,.(=m,), at the second Rejt(—m,), &c. Multiply the numerators and denominators of the preceding coordinates by p, and we have y Dmx ZY | O oa, Diez 3 ~ — 3 sai ee =m >m >~m for the coordinates of the centre of mean distance, on the supposition that each point counts for a number of points proportional to the mass here collected. The centre of mean distance, on this hypothesis, is what is called the centre of gravity. If the system be one of which he mass is continuous, we have yam, _fzdm a fam ‘ fdm ; fdm ( lm standing for pdx dy dz. There are six other integrals, of which it will be necessary to consider he connexion; namely, fatdm, fy%dm, fzdm, fyzdm, fzxdm, faydm; r eee, Cee eA Ze He INL, PANE a iccording as the system is continuous or discontinuous. Of these it nay be shown that the theory is so intimately connected with that of the llipsoid, that a competent knowledge of the properties of that surface should* be an indispensable preliminary to the study of dynamics. Let 2, 2), Py, Ci, Ys Yo Ysly KC... 2, 2/, 2, &C. be three independent ets of quantities, positive or negative. Let * By this I mean that the long, isolated, and inelegant investigations which sually fill up the chapters of works on dynamics which treat of rotatory motions aight be almost entirely avoided, if the student were supposed to have that know- 2dge of the ellipsoid which he is supposed to have of the ellipse before he reads on he theory of gravitation. | 496 DIFFERENTIAL AND INTEGRAL CALCULUS. | Ase+aetaer+..., BeytyPtyets.., CHP tah betes! Al=yzty Zit yur eee, Bl=ze42, U4 2 By + .0-; C= ry try yt ru Yu-bevee Lemma 1. The three quantities AB—C”, BC—A”, CA—B”, are) necessarily positive. The first, AB—C” or Sa? Ly’—(Lzy)”, is the sum of every possible variety of terms of the form a,-yn—(TY)m:(2Y)as where (zy), denotes 2, y,, and m and m denote numbers of subscript accents. When mand are equal, these terms destroy one another; and all the cases in which m and m are unequal can be collected im couples of the form | a2, y2 —(2y) mn (LY) an 22 2, — (ty) n (LYDing OF (Lin Ya — Tn Ym) Hence AB—C” being © (2p, Yn—@, Ym)* is necessarily positive ; and the same of the other two. Lemma 2. The expression following is necessarily positive : ABC+2A'B/C’— AA”?—BB”?—CC”. This expression is a collection of all possible terms of the form | LnYnep F 2(Y2) (22) aC LY)p — Tm(Y2)n(Y2Z)p — Yn 22) (22 )p — Fin LY) nLY Doe Each term in which m, 7, and p are equal vanishes; and so do the terms which, when two are equal, arise from the term above with the. same accents varied in position. Thus 2 Ce AY 2 2 Lin Yn ent LC. Ym ent SCF IY; tn + &C.=0. But if m, n, and p be all different, and if the term be called {mnp, and if we collect the six terms answering to the preceding’ with the order of m, 7, p varied, and nothing else; that is, if we form Smnp\-+-{nmp|-+inpm}+{mpn}+ipmnt+{pnm}, | we shall find the result to be a perfect square, namely, Aan Py Yp— Ly Zn 1 he Lm Yn ep %m Yn Vy = ee en Uy Ym ln Zt 3 whence the expression given is the sum of squares, and is positive. These results are equally true if for 2 we write ,/m.., for x, ./m.2, | &e., or if A= Xm’, &., A’=Smyz, &. And being independent of the number of quantities, and of the magnitude of m, they are still true if A=faedm, &c., A'= fyzdm, &c. I now proceed to point out the method of establishing those pro-. perties of the ellipsoid* which will be required. The coordinates being rectangular, let the equation of a surface be Aa’+ By? + C2*-+2A'y2z+ 2B’2zr+2C/ry=M...... (1). | Retaining the origin, change the directions of the coordinates, and, if possible, let a, 8, &c. be so taken that A’, B’, and C’, in the new equation, shall vanish. Let this new equation be Ké?+K’7?+K"@ | = M, and let €=ar+6yt+ yz, n=e/a+&c., f=a"r+&c. Substituting” | * For the general treatment of the surface of the second degree, in the same manner, the advanced student may consult amemoir on the general equation of surfaces of the second degree, published in the fifth volume of the Transactions of the Cambridge Philosophical Society. APPLICATION TO MECHANICS. 497 hese values in the last equation, and making the result identical with L), we have A=Kat+Ka*+K'a"s, = KBy+K'B'y/ +K'B"7" B=KA*+K’B?+K"s", Bo=Kya+ Kya +K’ ya" ..6.. (2). Co Ky+ K'y?+ K"y’’, C— Kap le K’ o’ B’ me K" "Bp" fultiply the first by a, the last by 8, and the last but one by y, which ‘ives Aa+C£+B'y=Ka, or (A—K) a+C’B +B'y=0. ind by similar processes we obtain C’z-+(B— K) 6+-A’y=0 B'a+ A’B 4-(C—K) ye0. The truth of these equations will remain unaltered if we accent all ne four, K, x, 8, y, once, or twice. Eliminate 8: and y:@ from these aree equations, and there results A-K)(B-K)(C-K)+2A’B'C’- (A-K) A?-(B—K) B?-(C- K) C= 0, hile the same equation, with K’ or K” substituted for K, would sult from eliminating 6’: a, &c. or 6”: a’ from the second and third +t just mentioned. Hence it follows that K, K’, and K” are the roots ‘the equation K°—(A+B+C) K?+ (BC—A?4+CA—B?+AB—C?) A ps —(ABC+2A’B'C —AA?—BB?—CC*)=0 The roots of this equation are all possible, as will be presently proved. 1 the mean time, we may determine «, £, &c. in terms of K, K’, and “, as follows. The equations ac +ff'+yy'=0, a2”+ 6B" + yy =0 iow us that w, 8, and y are in the proportion of f’y— yf", y'a’—a y’, id ap" —f'a", But e+f*+y=1, and the sum of the squares of i¢ last quantities will be found to be (a*tBety*) (a? +B" 4+ y?)—(a'a" + fh" 4+ y'y)*, or 1. ence q@ is either By’ —y‘h" or yb"—f'y’, &c. It does not signify hich we now assume, as our present investigations will only contain juares or products of these quantities. By help of these theorems, € may obtain from (2), by actual calculation, the following equations, BC- A?=K’K"o? + K’Ka®+KK'y" B+C=(K’+K”") @ +(K"+K) ¢?+(K+K’ «” BC—AA’=K'K" By +K’K By +KK’ B'y” — A’=(K’+K") By+ (K"+K) By +(K+K’) B’y’; hich, with ¢?4+e?4+¢7=1, By+f'y'+"y’=0, give _ BC—A®—(B+C) K+ K’ pute BOAA FAK PROG ew hall eat) oe (tok) (Kono) which a? and f’y’ may be found by interchanging K and K’, and ¢” d B’y’ by interchanging K and K”. By similar equations may also found ,_ CA—B?—(C+A) K+K° {GABE EE PKK )(KOK) 1° KKK AK” 2K = 2 498 DIFFERENTIAL AND INTEGRAL CALCULUS. .- __AB—-C®?—(A+B)K+K* ,_ A’B/—CO'4 CK | YS nn (KBR = KY) ae Oe eee from which 6”, &c. may be found by similar interchanges. One of the roots of (3) must be possible, let it be K, and if it can be let K’ and K” be impossible; that is, of the forms A+y./(—1) an) A—p/(—1). Then it will be found that @ is possible, while a’ and a are of the forms just written; whence @/@” is the sum of two squares It may be similarly proved that 6’6" and yy’ are each the sum of tw squares: whence ¢’a’+/’h"++'y" is the sum of six squares. But: is =0, which contradicts what has just followed necessarily from tw of the roots being impossible. Hence this last is not true, or all th roots are possible. : If, in (3), A+&c., BC—&c., and ABC+&c. be all positive, th three roots are obviously positive; and this, M being positive, shows th original equation to belong to an ellipsoid, since it can be reducedt K2+K72+K"2=M. Here M:K, M: K’, and M: K” are the square of the semiaxes, which can be found from (3): and their position ca; be ascertained from the equations last given. | Let there now be a given system, continuous or discontinuous, so the f az? dm, &c., or 2mx’, &c. are quantities, the value of which is detei mined as soon as the position of the axes is given. Let A= f x dm, &e; A'= fyzdm, &c., and let M=1. Let X, Y, and Z be the coordinaté of any point in a surface determined by the following equation, | fa'dm.X?+ J yedm. Y*+ fz2dm. 2? +2fyzdm.YZ +2fzrdm.ZX+2feydm.XY=1. : Now with reference to any one fixed point of the surface jul described, the integration being made over the whole of the system fror! which f x dm, &c. are obtained, we may treat X, Y, and Z as constant) and the preceding obviously becomes : f (@X+yY +2Z) dm=1. The surface must be by an ellipsoid, for A, B, C, are positive, whenc A+B-4C is so, and the lemmas in page 496 establish that BCA +é&c. and ABC+2A/B’'C’—&c. are positive. Let R and r be the di tances of the points (X, Y, Z) aud (2, y, z) from the origin, and let 0T the angle made by R andr: also let (Re), &c., (rx), &c. be the angle made by R and 7 with the axis of xz, &c. We have then e=rcos (ra) &c., X=Rcos Rr, &c., whence | aX+yY¥+2Z=rR {cos (rz). cos (Rx) + &c.}=rR cos 0; whence fr? R? cos? 0 dm=1, or R? {7° cos? 0 dm=1. This new integral {7° cos’@dm is the sum of all the elements of thi mass, each multiplied by the square of rcos@, the projection of its diy tance from the origin upon the line on which R is measured. If th. line were a new axis of x, this would be the new value of fi x’ dm, if | were a new axis of y or z, it would be the new value of fy? dm ¢ z?dm. And the equation f{ (rcos 0)” dm=R expresses the follow mg remarkable theorem. If any system be given, and also a poll) through which axes are drawn, and if any one axis whatsoever be calle! the axis of p, (meaning of a, y, or z, as the case may be,) there mus APPLICATION TO MECHANICS. 499 lways exist, in a fixed position with respect to that system, an ellipsoid, hich has the property that fp?dm=R-’, R being the radius vector of ie ellipsoid drawn from the origin to the surface upon the line py. And 1e magnitude and position of this ellipsoid, the latter with respect to ee” depends solely upon the values of the six integrals A, B, C, vis’, C’. Ifin the equation fa? dm.X?+4&c.=1 we substitute X=aX!+a!Y’ -@'Z', Y=BX'+&c., &., we shall find that it is reduced to fie (aX! +a'Y! Fal'Z')+y (BX'+ &e.)--2 (yX’/+ &.)¥=1, Sf (art By+yz)dm.X?+f (a'x+&c.)? dm. Y?+&e., &.=1. Let 2’, y’, 2’ be the coordinates of the point (a2, y, 2) in the new stem: we have then v=a@r+ By+yz, &c. Hence the last equation is fz? dm.X”?-+ fy? dm. Y?+&e.=1; * the equation of the ellipsoid contains integrals of the same form in the me manner, whatever axes may be taken. The integrals fi dm, &c. are not so much used as others derived from em, which are called moments of inertia. By the moment of inertia any system with respect to an axis is meant fo? dm, where o is the ‘rpendicular distance of the element dm from that axis. If R be the dius vector of the ellipsoid measured on the axis, and 7 and 0 as before, 2 have p*=7* sin* =r* —7° cos’, and f p> dm=f7r?dm—R~. Now mdm is a given quantity, depending on the system only and the int chosen through which to draw axes, since the distance of a point ym the origin is independent of the position of the axes of coordinates. ence the moment of rotation with respect to any axis can be readily ‘termined from the ellipsoid. It is obvious that if R be, for instance, on the axis of x, we have =y+2° and f 9? dm= f (y?+2°) dm. If we had started with the ‘uations f (+2°) dm.X?2+4 &e.4+ &e.—2 f yz dm. YZ—&ce.—&e.= 1, » should by the same reasoning have found ff {(@@Y—yX)°4+ (yZ—2zY)?+ (2X —2Z)*} dm=1; d the same substitutions as before would have given f R®7*sin?@dm=1 Jf o?dm=R-*. It might also have been shown that in this case we ve an ellipsoid, having its principal axes in the same directions as se of the former one. But the first ellipsoid is more conveniently tived, and equally useful in the exposition of results.* I shail in ure call the first of the two the momental ellipsoid, as being that by vans of which we prefer to deduce the properties of moments of ‘tia, though the name would apply more directly to the second, if it re employed for the same purpose. Let the axes in which the principal diameters of the momental ellip- dlie be called the principal axes. Let a, b, and c be the principal ' The second ellipsoid may be geometrically deduced from the first by the follow- ' theorem. If there be two surfaces in which the sum of the reciprocals of | Squares of the radii drawn from a given point in the same direction is constant, . if either be an ellipsoid, having its centre in the given point, the other is the ne. ZK 2 500 DIFFERENTIAL AND INTEGRAL CALCULUS. | semidiameters, whence, the principal axes being the axes of coordinates we have for the equation, g 72 2 Nd Be which, compared with hits dm. X?+ &e.=1, gives fyzdm=0, fz cdm=0, frydm=0. The disappearance of thes integrals, at the origin chosen, can only take place for this one set «| (rectangular) axes, since there i is no other for which the equation of th ellipsoid assumes the preceding form. Let a be the greatest of the semiaxes, 5 the mean, and c the leas The moments of | inertia for the three axes are (es r dm—a~, fr r’ dm — b~ r?dm—c~—, of which the first is the greatest, and the Jast the least, fe r?dm—R-* increases with R. And the axes of greatest and lea) moment of all those which pass through a given point are the princip) axes on which the greatest and least semiaxes of the ellipsoid ai found. Let a new axis make with the principal axes angles a, A, and? Then, R being the radius of the ellipsoid on this axis, and fr'dm being ( cos’a@ .cos’B cos’ y 1 a . b? A ini Ste G (cos® ¢-+ cos’ 8 + cos* y)=G and calling M,, M,, M., and Mz the moments of the principal axes ar) of the new axis, we have, by subtracting the first from the second, Mr=M, cos’ a+M, cos? 8+M, cos’ y, which may easily be verified from M,=f(y’+2°)dm, &e. | The locus of axes of equal moment passing through a given point is. cone whose vertex is the given point, and whose generating lines pa’ through the intersection of the ellipsoid with a sphere of which the give) point is the centre, and the radius of which depends upon the value | the moment common toall the axes. Ifthe momental ellipsoid be one | revolution, all axes equally inclined to the axis of revolution hay equal moments: if it be a sphere, all axes whatsoever have the sam’ moments. Let us now consider the moments of two axes parallel to one anothe| Let axes of a’, y’, 2’ be taken parallel to those of a, y, 2 a having the origin im did point (g,h,k). Then a=2'+¢, y=y +h, z=2 sy and we have f (a? +y") dm= f (x? +y") dm+2¢ fs a dm+2h f ydm-+ (g*° +h’) f dal If (2, y', 2’) be the centre of gravity, this is reduced to | Sf @+y) dm= f (a?+y”) dm+(e+h) fdm. Now the first integral is the moment of rotation about the axis of | (which may stand for any axis;) the second is that about an ax parallel to it passing through the centre of gravity: and g*+/h? is tl square of the distance between the two axes. Hence, of all axes parall to one another, that which passes through the centre of gravity has tl least moment, that of an axis distant from. it by p, having a momel greater by o* M, where M is the whole mass of the system. Having seen that every motion of a system is, for any one ical compounded of one motion of translation and one of rotation, it become APPLICATION TO MECHANICS. 501 ‘xpedient to ascertain in what manner the efficiency of a pressure is to ye estimated, in causing one or the other species of motion. The former ias been already done, (page 476,) and it appears that a pressure which nay be represented by a weight W acting upon a mass which belongs othe weight W’, will create in one second a velocity Wg: W’, heine }2°1908 feet. In order to consider the latter, let there be a system vhich, if it move at all, can only revolve about a fixed axis passing hrough O, and perpendicular to the plane of the paper. Any pressure pplied to a point of this system is wholly ineffective in producing rota- ion, if applied parallel to the axis, or in a line passing through the axis, floreover, if the point of application of the pressure be altered by a imple revolution about the axis, the line of direction of the pressure evolving also, no alteration is produced in the effect of the pressure. a At the point A, distant by OA from the axis, Jet ~ the force AP=P be applied perpendicularly to OA, and let OA=a. No difference in the effect of the force will be caused if we apply it at B | instead of A, in the direction BP, B being any O A pointin AP or AP produced. Let Z BOA=8, and applying P at B, decompose it into two forces, one P sin @ in the direction BO, the other P cos @ in the direction perpendicular to BO. Let the perpendicu- p. lardrawn from O to the direction of a force be called the arm at which the force acts: then since the part i the direction BO has no tendency to produce rotation, and since P sin 0 ad P cos @ are together in all respects equivalent to P, we see that P sting at the arm a is of the same rotatory power as P cos @ at the arm 'B, or a:cos@. And since P x a=P cos 6 x (a: cos 9), we see that two irces are of the same rotatory power when the product of the forces and ms are the same. The product of any force, and its arm of rotation, called the moment of rotation of the force. This investigation may rve to explain the manner in which the product just mentioned squires the importance which it is soon seen to possess in all problems mnected with rotation. The principle of virtual velocities, like all other fundamental theorems, is had no proof given of it in the admission of which all writers agree. rom its universality and simplicity it may be supposed to be rather the pression of some axiomatic truth than the proper consequence of first ‘inciples by means of a long course of regular deduction. I have here, however, only to suppose the truth of the principle, and ‘show how to use it. In page 479, when it was proved in the case of rigid system, we supposed every force to tend towards a point, and esti- ated the virtual velocity by means of the approach to or recess from at point, of the point to which the force is applied. This, however, not absolutely necessary, since if A, the point of application of a force in the direction AK, move to B, AC B may be considered as the part of the a ao motion which is in the direction of the ecsan srs ener force, as well as the differential of AK. The principle may then be stated as lows: if any number of forces P,, P,, &c. act upon a system, and if y infinitely small motion which can be given to the system (such as € connexion ofits parts will allow) give to the points of application the 502 DIFFERENTIAL AND INTEGRAL CALCULUS. | motions 67, 2, &c., in the lines of direction of the forces, then if th| system be in equilibrium, 2Pdp=0, provided that dp be im ever) case called positive or negative, according as it is in the direction of ij force, or in the opposite direction. And conversely, if 2Pdp=0 fi every possible small motion of the system, it must be in equilibrium. | Let us first suppose a rigid system ; that is, one of which the distance( any two points remains unaltered. It is the characteristic of the motio, of such a system, that it may always be reduced to one motion of trans lation and one of rotation. Let a motion be given to the system, and ki) it amount to moying the point (X, Y, Z) to (X+0X, Y+0Y, Z+0Z) and at the same time giving a rotation dp about an axis which pass¢ through (X, Y, Z), and makes angles A, pw, and y with the axes. W) have then for the motion of the point (a, y, z), asin page 481, dv 0X + {cos pw (z—Z) —cos v (y—Y)} 5h dy=oY + {cos vy (2—X) —cos A (z—Z)} dh oz=dZ+ {cos (y—Y) — cos p (2 —X)} oh . ,) dp dp dp . For op write ra tide bitin dz, and Pdp becomes, when we put fi da, &c., their values dp dp dp ey ba 5) Wifey est a — 6: : Pp riavati bids a(aeddane d | we Sales hadi | +1y Yj P zie slew op.cosr | : dp dp +{(s-Z) PP @—x) PPh a8 cos : d d mH +{@—x) PP ¥) PEI ap. cos, Whence, remembering that X, Y, and Z enter in the same manner?) every term, we have, writing P,, P,, and P, for P (dp: dz), &c., | =P, -OX—(YEP,—ZEP,) 3¢ cos A4D (yP,—zP,) - dp cos 2 (Pop)=< + =P, .dSY—(ZEP,—X=P.) oo cos p+ E(zP,—aP,). do c08' +=P,.6Z—(X2P,—Y =P.) oo cos v +5 (vP,—yP,) . dp C08) Now in order that we may have © (Pop) =0, independently of 6) oY; and oZ, opcosr, opcosp, and dpcosy, which are six arbitrary quantities, we must obvicusly have 2P,=0, Z£P,=0, 2P,=0, 2 (yP,—z2zP)/)=0, > (2P;—2P) =O = (#P,—yP,)=0. | If the direction of P make the angles ~, 6, and y with the axes, W have, from page 477, P,—P cos, Py=Pcosf, P,=P cosy, and tt preceding are the six well-known equations of equilibrium of a rigi body. The full development of the meaning of these equations belong * Though cosa, cosw, and cos» are connected by an equation, yet the multip’ cation by 89, which is arbitrary, gives three arbitrary products. { APPLICATION TO MECHANICS. 503 9 professed treatises on the subject. I shall here only give one instance f the manner in which conditions which restrict the motion of the ystem are shown to be equivalent to the introduction of other forces, Let one point of the system be obliged to be always upon a point of a iven surface, which amounts to supposing that the surface can always xercise in either direction the force necessary to prevent the point from saving it either way. Let L=0 be the equation of the surface; whence ‘is only requisite that 2 (Pdp) should be =0 for such motions of the ystem as are consistent with (L=0 being true of the changes of coordi- ates of the given point. This (page 455) is equivalent to the suppo- tion that for some one quantity T, which may be a function of all the ariables of the problem, we have 2Pép+TdL=0, for any motion of the ystem, the given point being no longer restricted to move on the surface. or the preceding fully satisfies the condition that when dL=0, ‘Pop=0. Let a small distance perpendicular to the given surface, mtained between the surface and the point whose coordinates are +ex, &c., be or; we have then (page 479) 0L=,/(L?+L?+12).é7, being dL: dx, &c., and we have SPip+TyY(L2+ L2+-L!).3r=0. ‘ow this is precisely the equation which we should have, if, in addition i the other forces, we had a new force T,/(Li+&c.) acting perpendicu- aly (as pointed out by the direction of Sr) to the surface, the com- ments in the directions of x, y, and z being TL,, TL,, and TL,. The science of dynamics opens a wider field for the application of the ifferential calculus than that of statics. The first problem in it will &;—given the motion of a system, that is, the curve described by every article, and the velocity of the particle at every point of its curve, ‘quired the forces which will produce, and no more than produce, that lotion of the system,in such manner that every mass may be acted pon by the forces which are just sufficient to produce the motion, with- it any communication to, or reception from, the other masses of the stem. Let us consider one of the particles, at which say a mass m is ‘lected. Let the equations of the curve which it describes be implied 1 the expression of the three coordinates of any point in terms of a fourth ariable w: and let v, the velocity at any point, be known in terms of x, , and z; that is, in terms of w. Let (@, y, 2) be the pomt of the irve at which the moving point is found at the end of the time ¢ apsed from an arbitrary epoch, (usually the commencement of the ‘otion.) The reasoning of pages 143—46 may be thus briefly con- snsed, using the language of infinitesimals. Looking at the motion t the direction of x, we see that at the end of the time ¢+d, the Iscissa will be x+d.2, and at the end of a further time dé, or at the end *t+-2dt, the abscissa will be 7+2dr+d?’r: the increments described i the successive times dé and dt, are dx and dr+d*x, and the velocities ce dx: dt and dx: dt+d?x:dt. There is then, in the second infinitely nall time dt, another velocity than in the first, differing by d'n2dts ad if this acceleration of velocity were to take place in every df iroughout a second, (if seconds be the units of time,) the whole acce- ration in a second would be d’x:di®. Let W be the weight of m, removed to the earth’s surface,) then (page 476), the pressure in-the rection of x, which is actually applied to the mass m, at the moment at | ood DIFFERENTIAL AND INTEGRAL CALCULUS. | which we are speaking, is (W : g) X (d’x: dé*). ‘To suppose any less pres sure is to suppose an effect without a cause: and any greater pressure a cause without an effect.* Upon proper suppositions as to the units, w may make m itself the representative of W:g, and m (d’x : dé*) that ¢ the pressure in the direction of x. This supposes us to choose units ¢ mass and pressure in such manner that a unit of pressure acting durin) one unit of time upon a unit of mass, would produce a unit of velocity (page 477). If, then, more pressure were actually applied in th system of which m is a part, the surplus must have been removed, b the connexion of the parts of the system, and carried to other masses if less, the mass in ‘question must have received pressure from othe masses. And m (d*x: di*) is called the effective force in the direction ¢| x: being that from which, and noother, the motion actually taking plac’ is produced. Similarly, m (d*y: dé?) and m (d*z:d¢*) are called th| effective forces in the directions of y and z, and d’x:dt*, &c., may b called the effected accelerations.t To find these effected accelerations when the motion is fully given remember that x, y, and z, as well as v (which is ds: dé) are expressei in terms of w; let dr: du=a2', &c., whence wa’, xv", y’ y”, &c. are givel functions of u. We have then (s’=,/(a’*+ y”+2")) | dz. dvds... dz. ve a eS Ie he ee dé, (ds dt ds ny dx d dx\ ds sche dz Die Peyiiay: va'\! dt ds’\dt/ dé du\dt/' du s! s! v(s’c!'—a's")+vv's's! —v? (Ss a" —a's's’) _ vv'a STi eee TT on pee OT ga Change 2 into y or x, and we have the effected accelerations in thos directions. Each effected acceleration is made up of two parts, thi| separate consideration of which will be worth while. The first tern’ obviously contains that part which is necessary to the mere maintenance of v at its present value; for if v’ were =O, that is, if v were constant it would be the only term. Now if the curve were a straight line, m pressure would be required to maintain v at its present value, since thi constitution of matter gives it the power (if it be right to call it a power, of maintaining its velocity in a straight line. It is then, we mus' suppose, in the maintenance of the velocity in the curve that the part 0} the effective force which produces this acceleration is expended, which would make us suspect that it must depend for its value upon the curvature: and this will turn out to be the case. If-for s* and s’s” we write 2?-++y?+2" and aa” +y'y"+2'z", we find for the three effected accelerations, (so far as they are now considered, ) ve Pe yz,) v (v2) — ZL, v (y'x, re vy,) e tI ats ee 7 ’ 7 5) gs? s* s* * The student must not take these words as a reason, but only as reminding him of a reason already proved by experiment, the results of which are enunciated in pages 475, &e. " + It is usual to call md*x:dt? the moving force, and d®x: dt® the accelerating force. The word force, when used to signify both the pressure which produces acceleration, and the acceleration itself, has always been a stumbling-block to beginners. | APPLICATION TO MECHANICS, 505 where r,=y2z —zy', &c., as in page 409. Now (page 410) if &, n, and ¢ be the coordinates of the centre of curvature, and o the radius, we have s? (zy —yz) Ee po &, &., p=/(E—2)? + &e.) = 13 J (2,7 +y,2-+2"%) 9? 7 ie 2 2 av, ay, aor , ’ s* ~ ° whence =y,—y ey (E—wv), &c., and the effected accelerations here 2 s considered are 2 Cp ae v aS fa. (E—2); iar (n—y), piety e p p ee which being proportional to E—a, &c. have a resultant in the direction of the radius of curvature, the value of which being the square root of the sums of the squares of the preceding, is v’:o. Hence the pressure mv :p, directed towards the centre of curvature, is all that is necessary to the maintenance of uniform velocity in a curve: and is that force which is required to oppose the tendency of matter to maintain its velocity in a straight line. If we now look at the remaining parts of the effected accelerations, we see Vos UU sey LOE? 8, proportional to 2’, y’, z'; whence the pressure that is required to pro- duce them is in the direction of the tangent of the curve, and is the square root of the sum of the squares of the preceding, or vv':s’. Now ds d's _dv __dv duds vy dt’ df dt” duds dt $" Whence m (d?s:dt?) is the effective pressure which produces the requisite alteration in the velocity, depending upon the function which the arc is of the time according to precisely the same law as if the arc were a straight line: the first considered force providing (if we may so speak) all that is necessary on account of the curvature. _ If the system consist only of a single point P, at which the mass 7 1s collected, the impressed pressures are altogether, effective in producing | motion, since there is no other mass in connexion with the one to which | they are applied. If, then, A, B, and C be the pressures applied in the direction of 2, 7, and z, the accelerations produced im these several directions will be A:m, B:m, C:m, which, being wholly effective, we have (calling the latter X, Y, and Z) pipe gy ¢ per, Se (1), dt? dt dt three equations between 27, y, z, and ¢, from which, if they can be integrated, x, y, and z may be found in terms of ¢. ‘This integration will introduce six constants, and so many are necessary to the complete determination of the problem. For one starting point must be given, and the three velocities at that point in the direction of the three axes: that is, at one given time, 2, y, 2, dv: dt, dy: dt, and dz: dt must be known. The six constants are then expended in giving the required _ values to these quantities for a given value of ¢. 506 DIFFERENTIAL AND INTEGRAL,CALCULUS. The preceding equations give (v being the velocity) RRR = \= nia Preah de =2 (Xdxr+Ydy+Zdz) ; the first side of which is integrable, without reference to the depend- ence of v,y, and z on ¢t. If, then, Xdx+Ydy+Zdz be integrable, (say =d.o (a, y, z)), we can determine the velocity without knowing anything of the manner in which a, &c. are functions of ¢: and we have v?— V2=26 (a, y, 2) —2@ (a, b,c)... 6... (2); it being supposed known that at the point (a, d, ¢) the velocity is V. Hence it appears that, when Xdvx+ Ydy-+ Zdz is integrable per se, and the velocity at the starting point is given, the velocity at any other point is a function of the initial and terminal coordinates only, and of the initial velocity, and does not depend at all upon the manner in which the point moves from one to the other, » But this is not necessarily the case when the preceding function is not integrable. If we substitute in (1) the values of d*x: dt’, &c. from page 504, we have three equations of the form vs’? a! +us' (v's’ — ys") a = XK, &es... ..(3) 5 andif these be multiplied by x,, y,, and z,, and added together, the result is (since vav,+&c.=0, 22, +&c.=0, as in page 409) Xe + Yy,+Z2z,=0......(4); whicn is one of the equations of the point’s path. Again, if we remem- ber that the equation of the resultant of X, Y, and Zis (€—ax):X =(n—y): Y=(—z): Z, and that the equation of the osculating plane is ({—2)2,+&c.=0, we may see that the preceding equation expresses the following theorem :—the resultant of all the forces at any point lies in the osculating plane of the curve at that point. Hence, since the osculating plane always passes through the tangent, we see that at every point of the motion, the osculating plane passes through the tangent, and the resultant of the forces acting at that point.* If Xdz+&c. be integrable, so that (2) can be obtained, v® can be expressed as a function of x, y, and z, so that any two of the equations (3) will be two equations of the path of the curve. Four constants will be introduced in the integration ; a fifth, V, has already entered, and the sixth will appear in finding ¢ from dt=ds:v. But if Xdv+&c. be not integrable, we must, from any two of the equations (3) find v? and vv’; then since the second is half the diff. co. of the first, we equate the value of 2vv' to the diff. co. of the value of v*. This gives an equation of the third degree of differentiation ; and the last, and (4), are two equations to the path of the curve. Their integration introduces five constants; and the sixth is found in integrating dt=ds: v. It thus appears that the elimination of ¢ between the three equations * Hence, if a point move upon a surface unacted on by any forces except the reaction of the surface, which is normal to it, the osculating plane must always pass through the normal of the surface.. Consequently (page 442) the curve in which the point passes from one point to another is the shortest Jine which can be drawn on the surface between those two points, APPLICATION TO MECHANICS. 507 (1) is always possible: but there are very few cases in which we can completely integrate the resulting diff. equ. I now show the process by which the equations most convenient for astronomical purposes are obtained. Let r and 6 be the polar coordinates in the plane of ay of the pro- jection of (2, y, x) on that plane, and let wu be the reciprocal of 7, We have then x=rcos6,y=rsin@. Let the forces X and Y, which act in the plane of xy, be each decomposed into two, one directed towards the axis of z, and the second perpendicular to the first. If these forces be P and T, we have (P and T being supposed positive when their effect is to increase r and 0) a ; i d2x ay P=X cos 0+ Ysin@=—- ({ r— hat hale r G TY ae 2 2 Tee ea Yn Ak (ost | T dt? di? d?r de? (Page 345, equ. 20) Poa! aa d dy dx d dé do. dy =— SEE Sa 5: RO OC ee re Bl at a Depeied ipo) Sa (> di } Let r°dé: dt=H, then dH: dt=Tr and the preceding give HdH=Tr' do, or H's h' +2 f Tr’ do; h being the value of H at the commencement of the integral. Also €9:di=H :r°’=Hu*. dy 1 du _ ay dé du du dt w dt wv dt* do do ar d?u dé _ dH du _ aa pe) itn ad DAG aT: d@ °° dé dt dt’ do “id? outnette ar’ ae? we ( du tau Se ee ra gale es —— —=pP, at’ dt By de? +) wu do | edd bea or q Bs Pot! eres eee est BE 0, (wu): 2 re BY [+2 a ao) bf Eb Th a diff. equation which is here exhibited in a useful form for approxima- tion when Tis small. Take the third of the equations (1), and let o be the tangent of the angle which the line joining (2, y, z) withthe origin makes with its projection on the plane of zy; whence z=ro=o: 1. We have then dz 1 ded@ «a du dé} (uo 955) dt wu do dt wu? dd dt ‘dQ dé dz _ di ( do du dé do au dé dt ie lgar N was whee di “ae” ae 508 DIFFERENTIAL AND INTEGRAL GALCULUS. T du do d’u We do komt 2 958 25° 2 whence Z=T Lema Tot 0 a oH? u aT ih oa Le YG 2,2 i ea ate ay bb al” wf From (2), Hu 70 +e 7 de H*w—P; ad’ d whence H? wu? (4. +Pot+T =Z Ps—Z. T de 2 7 +18 do o u> de n} (0). or +ot+ r arp ia nde eho espe Be (a4 2 { = a) If (uw) and (c) can be integrated, exactly or approximately, we have ne means of determining two equations between 2, y, and z from the expressions of wu and o in terms of @: since u==(2+y?)-3, tand=y:2, one (22+ y%)72. The path is thus determined, and the time at which the moying point is at (2, y, z) is found by integrating do dé Hu? uw /(h?+2f Tu-*d0) © at yi Absolute velocities are rarely required for any astronomical purpose, and angular velocities supply their places, And d@: dt is Hu’, while | do do dé _ do din do” Gbieaen All that precedes, excepting only the equation (2), page 506, 1s equally true, whether Xdx-+ Ydy-+ Zdz be an exact differential independently of relation between 2, y, and z, or not. But in all problems of physics, the former is the case; and the consequence is that a great degree of simplification is introduced into the details of operation as far as regards the mode of expressing decompositions of the acting forces. ‘The follow- ing investigations will show in what manner. Let Q be the function of 2, y, and x, of which Xdx+ Ydy+Zdz is the differential. Hence (dQ: dx being written Q,, &c.) we have Q,=X, Q,=Y,Q,=Z. Let anew set of axes be taken, such that r= aa’ + By +y2, y=aa'+&c. &., and let R, the resultant of X, Y, and Z, make with the axes angles whose cosines are («), (a), and («”). Then the cosine of the angle made by R and «’ is («).a+(a’) & +(a’).a’, which multiplied by R gives aX+aY+«’Z, which is the component of R in the direction of 2. But di (t). pv dQ daz dQ dy ,dQdz _ dQ Xa+ Va + Za ~ de dix’ dy dx! et ae Siti whence, if in Q be substituted for 2, &c., their values in terms of a’; &@e and if the resulting functions of a’, &c. be differentiated with respect to a’, the diff. co. Q,, is the component of R in the direction of a: and similarly of the other coordinates. And if (2, y, 2) change to («+ dz, y+dy, z+ dz), the resulting differential dQ is the moment of the force R which is used in the principle of virtual velocities. Next, let rcos@ and rsin@ be substituted for v and y, r being the + APPLICATION TO MECHANICS. 509 projected radius vector, and @ the angle it makes with x. We have then dQ _dQ dr . dQ dy _ dr dx dr ‘dy dr dQ dQ dx dQ dy do de do" dy do 7 %9- Q, cos 0+ Q, sin @ It will be found that if R be decomposed into three forces, one parallel to x, one perpendicular to z passing through the axis, and one perpendicular to the two former, (the Z, P, and T of the preceding problem) ; Q, is the second, and Q, the moment of the third to turn the system about the axis of z, or Tr. But if at the same time we put o:u for z, cos@:u for x, and sin@:u for y, we have dQ dQ dx dQ dy dQ dz du dx’ du dy du ‘dz du chtak, o _ cos? dQ sind dQ « dQj tw a} a dr ut dy w dz sin 0 cos 0, ~.T 1 Ve Q=—Q,.—— +Q, TER Q-= ges Na Pp fi. ] Hence ae Q--Q.,, —==-3Q, Z=uQ.; u> u USER 1 which, substituted in the equations (1), (¢), and (é), give 1 du Oo Cu “we Fai sae eases Q- eee -- u eoxatiy p ] dé hi? + 9 7 Q, d@ do f poo ohokny ence arg BNE Ka — + 6+-——_—___———_ = 0 a8 (+ 9 = Q, a0) u® H a dé aeey) uu a/ (14 ys am Q, ao) These are the equations used by Laplace in his theory of the moon: the function Q will be hereafter noticed. . I now come to the equations connected with the motion of a system. If the connexion of the parts of a system were given, with the curve described by each* of its points, together with the velocity at each point of each curve, and the time at which the system is in some one position, the whole motion would be completely given: and the accelerations actually taking place at each point, at any one moment of time, being calculated asin page 504, the pressures simply sufficient to produce such * The equations of the curves of three of its points would be sufficient if the system were rigid. r. 510 DIFFERENTIAL AND INTEGRAL CALCULUS. accelerations on the masses supposed to be collected at the different points might also be calculated. Thus what are called the effective forces might be found. But the forces impressed at the moment in question may be very different from the effective forces: for if to the latter we add any number of mutually destroying forces, which will pro- duce no effect, the combination of these with the effective forces may produce an infinite number of systems of forces, which being only the effective forces combined with other of no effect, may be the forces actually employed to produce the effect. Thus the problem, “ given the motion, to find the forces which produce it,’’ is altogether indeterminate ; though the following, “ given the motion, to find the forces which will just produce it, without any forces superfluous and mutually destructive of each other,” is determinate, and has been solved. It is to the inverse problem, “ given the forces impressed, required the motion produced,” that our attention is now to be turned. The system and the connexion of its parts being given, let the masses collected at A,, As, &c. be m,, m,, &c., at which act such pressures, in the directions of x, y, and z, as would, if allowed to act uniformly for one second, produce velocities X,, Y,, Z,, Xe, Yo, Ze, &c. in the several masses and in the three directions. Then m, is acted on by pressures which may be represented by m, X,, m, Y,, ™,, Z,, on condition that the unit of pressure is in all cases that which would produce in the unit of mass a unit of velocity, if allowed to act uniformly for one second. The effected accelerations d*2,: dt’, d’y,:d?, &c. are now unknown quan- tities, as are m,d’.x,:di?, &c. the effective forces. This only is known, that the impressed forces may be resolved into 1. The effective forces. 2. A system of forces which destroys itself, or would if applied alone to the system at rest not disturb the equilibrium. Any other supposition would lead to the result that the forces proper to produce a motion, being applied, do not produce that motion. For the effective forces are so called because, being deduced from the actual motion, they would of themselves produce that motion: if the remaining forces could produce any motion they would, so that the motion of the system would be that which it is, and that due to the forces just called remaining besides: which is absurd. Hence the impressed forces (I) may be resolved into the effective forces (E), and an equilibrating system (Q). If, then, the velocity of all the parts of the system were instantaneously destroyed, and at the same moment were applied systems (F’) and (Q’), consisting of forces severally equal and opposite to those of (E) and (Q), the state of rest thus arbitrarily created would continue: for (E) and (Q) balance (E’) and (Q’), and (I) is equivalent to (E) and (Q). Hence (I) balances (E’) and (Q’): of which (Q’) balances itself, so that (I) balances (E’): or, a system of forces composed of the impressed forces, and the effective forces ‘with all their directions diametrically changed, must be in equilibrium. This is known by the name of D’ Alembert’s principle, and reduces every problem of motion to one of equilibrium (page 447). The force impressed on m, in the direction of x is m, _X,, and the opposite of the effective force is —m, (d*x,: dt”), and soon. Hence the forces applied to m, when (I) and (KE) are applied are da? x d? y,> d?z Mm, («=F mi(Y.— ie) mM, (a-F) &c. APPLICATION TO MECHANICS. 511 If, then, we give the system any small motion, (either the one which it was going to take when the velocity was destroyed, or any other which is consistent with the connexion of its parts,) and apply the principle of virtual velocities, we have, supposing that from the motion, whether actual or virtual,* x, becomes X,+62,, &e., dx \ ' hee d?z \ = > (m(Se—X ) sap +z n(3 —Y ) dy~ +> 4m EZ) ozp==0; in which, for convenience, the sign of every term has been changed. In this, remember that d* x, : di”, &c. are all supposed to be obtained from the actual motion. Let us now suppose the system to be rigid; the six equations deduced in page 502 become dr dx »2 ae wees of ome > — a =m B x) 0, or Sm aa rmX, &e d’y (thay } > —_— -— — j — — pz {mz (3 Y) my é x) 0, d? 2 or Sm (2 ee 7G =em (zY —yX), &c. Let x, y, 2 be the coordinates of the centre of gravity, and let «,, ¥, 2, be the coordinates of (x, y, x) referred to the centre of gravity as an origin, and axes parallel to the former ones. We have then (page 495) o.2m=Zmz, y..2m=Imy, z,.25m= >mz, L=C+1, Y=Yot Yn t= ZT 2, j es ie SPs The first set gives a m= rm TE? &c., whence we find da, 2mX dy, mY d*z, _ 2mZ ar omen nee en ae SIA” or, the actual motion of the centre of gravity is that which a point would _ have, if all the masses were collected in it, and all the impressed pres- _ Sures constantly applied to it. Again 2 ad? ? da? d? m (tot) (FE ae He min Jon my dt’ dt? aly fe CPB Ff: d° Yo , d° Y, +mz, de + Mx, de If these be summed, remembering which terms are common, we have, writing for d* y,: dé’ its value, : z=mY ig! Se hy MES be MRE Le Wey vig | 2M... 41,2 (m ae )t Smz,. Sm +2 | mz, de But c=21,+2, gives Smxe=x,. m+ Emz, and since Yme=x, Um, we * Actual, that which was about to take place ; virtual, any other which we may require to be supposed in the application of the principle of virtual velocities. 512 “DIFFERENTIAL AND INTEGRAL CALCULUS. have Smr=0. Similarly, Smy=0 and Sm (d?y,:dt’)=0. The middle terms of the preceding, therefore, disappear, and if we inter- change x and y, and subtract the result, we have, as before shown, an expression equal to 2m («#Y —yX), or d? 1 ax - £,LmY --y, Lm X+ 2m (« ——Y, aa Xm (ro+2,) Y-yoty/X); from which we get the first of the following equations, and correspond- ing processes give the others, a? ya 2m ( iy, a = =m (17/Y—y,X), az Cte, =m (< Tate ae ==" (z, X—a,Z). These are the equations which would be obtained, if the centre of gravity were a fixed point, so that its translation should be impossible: that is to say, the motion of the system about its centre of gravity is altogether independent of the motion of translation of that centre,* the forces which act being the same. Since any axes may be chosen, let us take, at the end of the time @, the system of axes of &, n, ¢, which moves with the system: but during each time dé, let a set of such axes remain in its position, while other axes move with the system, the angular velocities of rotation being p, q, and 7, On this supposition, in page 487, we obtained De ke es eel oil : dt 1° ™; ape Ps Hh oe” pie e sre (A). In these equations we do not see dp, dq, or dv, because the motion of the system during the first dé is round an instantaneous axis of rotation, with velocities which change only by small quantities of the second order. But if we consider a second dé, this instantaneous axis under- goes an infinitely small change of position, generally speaking, and p, &c. become p+dp, &c. Hence in forming d*§:d°, &c., we must consider p, &c. as varying, as well as &, &c. And of all the axes which can pass through the given point the most convenient are the principal axes, for which Xmén=0, Zmyl=0, YmZE=0, using the symbol Y belonging to a discontinuous system. We have then * If the centre of the earth were suddenly to be fixed, this principle shows that the rotation would continue as before. But the precession of the equinoxes would not continue of the same magnitude, for the sun, &c. not acquiring the same posi- tions relatively to the earth which would have been acquired, the forces which cause the precession would not be the same as they would have been if the motion of the centre had continued, and different amounts of precession and nutation would ke created in any given time. But if, when the centre of the earth was fixed, the actual motions of the heavenly bodies were altered, so that, relatively to the earth, they should move in the same manner as they do when the earth moves, all phe- nomena connected with the earth’s rotation would be unaltered. This principle simplifies all problems connected with the motions of bodies about their centres of gravity, by requiring us only to consider the motion of translation so far as it affects the magnitude of the impressed forces, } } | | f | APPLICATION TO MECHANICS. 513 dn de dé dg ee ip i igh dbs dial cake M dr Rd =9rs+pqs—(p*-+r*) n+é ‘iad 7 ad’, re Let ; z dP er) Op ee Fmé ao qr dm EF + pq Sm — (p+ id =m Be sme—T =m EL aye y =pq Sm ats Ymée. Interchange £ and y, p and q, observing that the first two equations A) are not then interchanged, unless », g, and r be made to change “ So) 3 9 d 5 ign, and we have 724 : d S “ dr as ey Be es 2 =m WFP mn Ti =mn dn ae : eS Fah ire geh e eL dm ( E —79 =, J=pq (2me? — mn?) + — (Sm2+ dmz7?). ‘ap gp JH Py (ame iar eral e+ 2imn*) Let Mz, Mx, M, be the moments of inertia (page 499) with respect 0 these principal axes, or M;= =m (72+ o*), M,—>m (EC +2), M.= Sm (22-+7°) ; md let Nz, N,, Nz be the values of Ym ({H—7n), &c., the impressed ressures on the point (&, n, 2) being m¥, mi, mZ, in the directions of he axes. We have then the first of the followimg equations, and the thers are obtained by similar processes. dr M, 4 (M,—My r= Nz. coe GB); dp My5-+(Mz—Mz) qr=Ny As the impressed forces can gencrally be made functions of, the dosition of the system, we may consider Nz, &c. as functions of «, /, v¢., or (page 482) of 0, d, and y%. If we were to substitute from page ‘83 the values of p, g, andr, in terms of 6, &c., we should have here hree equations between 8, $, Ww, and Zt, each of the second order: these eing integrated, the values of 9, @, and y are obtained in terms of ¢. ix arbitrary constants are introduced in integration ; three of which are Xpended in giving the system the initial position assigned to it by the onditions of the problem, and three more in giving it the initial motion elonging to three given initial values of p,g,andr. Thus the problem f finding the motion of any system, acted on by any forces whatever, is educed to that of the integration of three simultaneous diff. equ.: but hese can seldom be completely integrated. It must be observed that all that precedes is both necessary and ujjicient for the determination of the motion of a rigid system, or one he position of which is given when that of three points not in the same ine is given: and necessary, but not sufficient, to the we eee of 2L 514 DIFFERENTIAL AND INTEGRAL CALCULUS. the motion of any other system. For if a system be not rigid, the equi- | librium of the counter-impressed and effective forces must still be true: and in applying the laws of equilibrium every virtual motion which is possible in a rigid system is possible in one which is not rigid, and other motions besides. So that among the conditions which express that | =SPdp=0 for every motion which a system of variable form may take, must be found all those which express the same for every motion which the system could take without varying its form. The two most useful cases are the extremes; namely, a rigid system, in which variation of form is altogether impossible, and a system of separate masses, supposed to be collected in points, and wholly uncon- | nected with each other, except by an attraction or repulsion existing between every pair, which either attract or repel each other with equal - forces. If our object here were mechanical, and not mathematical, it | would be easy to show that the first is an extreme case of the second: but it will now be sufficient to point out some common properties of the | two systems. Let each of the two masses m, and mz, attract the other | according to a law depending on 7;,., the distance between the points at | which they are supposed to be collected. Let the attraction of each on the other be as its mass, and let the two attractive pressures be equal. Then m, mz 67,2 must represent the attractive pressure of each on the. other, $7;,2 being that function of the distance on which the mutual | attraction depends: for of no other function. of m, and mg is it true that any alteration of mm, or m, would alter the function in the same propor- | tion. “Now on the suppositions which make pressure=mass X accelera= | tion (page 477), this pressure, allowed to act without alteration for one second upon m,, would produce the velocity m_ Prj,2, and upon mz, the t velocity 7, $7;,9: so that each mass would produce in the other, ina | given time, a velocity altogether independent of the other mass, and dependent only upon its own. If there be a system of such masses, each one acting on all the rest, ' and acted on by it, it is obvious that the impressed forces would be ’ mutually destructive if the system were made rigid. Hence we have | the following equations, which belong equally to the rigid system acted | on by no forces, and to the system before us. Coe a d2z zm qo”? =m a0, =m qe 0. Cte. bY \ res Eas shih Ye dy . da wat (y Gade =07" 27t ( e-?ae = =m beer ao, 72)=0 These equations might also be readily obtained by the formation of =EmX, &., 2m (e#Y—yX), &c., which would all be found to vanish. It appears from the ‘first three that the centre of gravity (’9, Yor 20)’ moves in a straight line, or is at rest: for they give d’x,: d?=0, we., or: ro=at+b, y=at+l, z,=a't+b", the equations of a straight line, or, of a point, if a=0, a‘=0, a’=0. ‘To see the meaning of the second set. of equations, let 7 be the distance of (x, y, z) from the origin, and let 7, be the projection of r upon the plane of zy. Let 0, be the angle made by this projection with the axis of 2, we have then (page 345) : Dyed Gye Or Ee INGO, tae ae = aE ( (sie : ear. at? at APPLICATION TO MECHANICS. 515 Substitute and integrate, and we have Sh balitbldy a : 2m C2 in [rt a0. y= CaO and similar equations for the other planes. Now r2d0,: dé represents the areal velocity; that is, the area which would be swept over by 7, in one second, at the rate at which the radius vector is proceeding, its length being taken into account. And 7°d9,: dé is to be reckoned as positive or negative, according as 0, is increasing or decreasing. Hence, since the preceding property is independent of the origin and coor- dinate planes, we have the principle, which is somewhat improperly called that of the conservation of areas, namely, that if any point be taken, and a plane passing through it, and if all the radii drawn from a point to the different moving points of the system be projected upon this plane throughout the motion, the sum of the areal velocities, each taken with its proper sign and multiplied by the mass of the moving point to which it belongs, will be always of the same value. Let the constants above described belonging to the planes of yz, za, and xy be called A, B, and C. Take a new sct of coordinates é, n, Z, with the same origin, (but also fixed in space,) and let w=ab+Pntyé, y=aeé+&c., &c. Calculate &dy—ndé, or (artayt a'z)(Pde+ P'dy+B"dz)—(Bxtf'y+b'2)(ade-+eldyt+ea''dz), which, by common development, is (oB'— Ba’) (xdly — yd) + (By! = yB')(yde-2dy) + (ya!-ay/)(2de-ade). Whence (page 482) (£dy—ndé):dt is ¢’A+A"B+y"C. This is the value of the function ©. (areal vel.) for the plane of £); those for the planes of 7f and Zé are a@A+6B+yC and c/A+6'B+y'C. Now by assuming the latter two equal to nothing, we find that A, B, and C are in the proportion of By’—yf’, ye —ay', and of’—Be', or a, B” and y’, whence, since ¢’?+A’"-+ y= 1, we have a!'= A p'= B Lie _— C VAR+B +0)? © (AEB EC CARE B +O’ aA + B"B+y"C=,/(A2+ B+C). And (@A+ &c.)?+ (@/A+&o.)2+(eA+ &e.)*2 is always = A?+B?+(C? If, then, we take for a new axis of z the line whose equations are «:A=y:B=z:C, the projected areal velocities on any plane passing through this line, always give Ym (areal vel.)=0, and they give V (A+ B?+C?) for the plane perpendicular to this line. To dwell upon the numerous applications of these principles which are requisite for the complete elucidation of their physical bearings would be to write a treatise on mechanics: in the preceding, we see the manner in which the differential calculus is applied to general problems. I now go on to the general treatment of the fundamental equation in page 511, which was reduced to a system by Lagrange. One important step, lately supplied by Sir W. Hamilton,* renders the theoretical ex- pression of a large class of dynamical problems in terms of the differential calculus perfectly complete, and leaves only purely mathematical diffi- * In a paper headed “On a general method in Dynamics,” Phil, Trans. for 1834, — 2L2 516 DIFFERENTIAL AND INTEGRAL CALCULUS. culties, namely, those involved in the determination of one particular function depending upon the data of the problem. The equation in page 511 may be thus written : ax d’y d?z e Sona 6 Sad ee ce |= . Zoz eeee oe 1 ° >.m & t+Ts dy+ TE is) D.m (Xdx+ Yoy+Zoz) (1) In all the cases which occur in practice, the second side is a complete | differential, say SU. If the variations dx, &c. be actual, or those which | the motion of the system is itself about to produce (page 511) so that | orazdx, &c., the first side becomes sm( at a tke. ) or yma. (F+ Se } or d.(42mv") ; dt ‘dt v being the actual velocity of the point (a, y, 2). The second side is dU, whence integration gives L>mv’=U +H, and 32mv?—$2mrj=U — Ui sieles Caos v, being the value of v at the beginning of the motion, and U, the value | of U. This equation answers to (2) in page 506. The expression ©.mv®, the sum of the products of each mass, and the | square of its velocity, is called the vis viva,* or living force, of the system. If no forces act, that is, if X—0, Y=0, &c., we have} U—U,=0, or Smv’=Lm?; that is, the living force of the system always remains the same. This is called the principle of the conserva- tion of living force. In all physical problems, the values of X, Y, Z depend entirely upon | the positions of the particles acted upon, and not upon the time at which | those positions are attained. Hence U isa function of coordinates only, | and not of the time; that is, not directly, but only through coordinates ; | the coordinates themselves are, from the nature of the question, functions | of the time. From this it follows that &.v?, the living force at the expiration of the time ¢ from the commencement of the motion, is a} function of the initial living force, and of the initial and terminal coor- dinates of the system. If, then, any position he given to the system, | such as, consistently with the connexion of its parts, it can occupy, the living force belonging to that position can be found, whether the system | could ever arrive there or not, under the given circumstances. For, the: initial position and velocities being given, U, and 2.2nv; are given, and) for any other assigned position (possible or not) U can be calculated: hence Smv? or Smv?+2 (U—U,) can be found ; being the living force which the system must have if it pass through the assigned position: and there is nothing in the preceding mode of calculating 2.2v* to point out whether the system can pass through the assigned position or not. Consistently with preceding nomenclature, the value of 2.mov" belonging to any position which the system does take, might be called the actual living force; that belonging to any other position, the virtual living force. This distinction must be remembered, whether it be con-) veyed in words assigned to the purpose or not. If the living force mv® of the particle whose mass is m continue * The meaning of this funetion, =m, is of the greatest importance in a mechanical point of view: here, however, we have only to consider it as a pure result of calculation, ) | ) APPLICATION TO MECHANICS. 517 uniform during the time ¢, the product mv*t is called the action of the particle during that time. But if v vary, then mv*d¢ is the action during the time dé; and mfv'dt, taken between any limits, is the action during the interval between those limits; and ¥.mfv%d¢ is the action of the whole system during the same time. But it is more useful to consider the action over a given portion of the ‘motion, without any but indirect reference to the time. For dé write ds:v, ds being the element of the path of the particle m, which gives Y.mfvds; and this, taken between any limiting positions, is the action of the system in passing from one position to the other. And if we dis- tinguish the path which the system does describe from any other, we may calculate the action in either, and distinguish the actual action from the virtual, in the same manner as we have distinguished the actual living force from the virtual. Let us now suppose the initial position of the system to be altered, and also the initial velocities, in the manner pursued in the calculus of variations. Let the final positions be altered in a similar manner, and let the intermediate path be varied, so that 2.m/fvds is altered by 02.mfvds, or Z.mdfrds. For each particle, dfvds is f(dv.ds-+-vdés), which, ds being vdt, and ds.dés being dx ddx+ &c., gives ¢ AT OY dz O VOLS 100.8 =" OY 6 ° dfvds= f (wae dt + = doa, dey +— des) Make the integration by parts, take the integrated part between the limits, and, 2’, &c. being dx: dt, &c., let 2’, &c. be the initial values of a’, &c. Hence dfvds=al dx + y!dy +2/d2—2', da, — y,'0y,— 2162, + f (woe — a!0a—y"Sy—2!!0z) de. Multiply by m, perform the same operations for every other particle, add the results, and observe that equation (2) gives Lmvov= Zmv,ov, +OU—sU, ; whence L.mofvds==.m (a da+y'dy +262) —Z.m (a 62, + yoy + 2/1021) + f{Xmv,dv,—6U, 4+ 6U—Zm (ada ty! Sy + 2!/62z)} dé. In the integral part the last two terms vanish by equation (1), and the preceding pair being independent of ¢, we find that 6.2m fvds is com- pletely integrated, as follows,* dL .mfvds= z.m (a'dx+ &c.)—Z.m (ad2,+ &e.) + (2.mv,dv,—SU,).t.....-(3). One case of this equation has been long known; namely, that in which the virtual path of the system (or that supposed to be made by the variation) begins and ends in the same positions as the actual path, * This equation was first noticed by Sir W. Hamilton, (in the paper cited,) who proposes to call the relation which it enunciates the /aw of varying action. He also calls Xm/fuds the characteristic function of the motion, and U the force-function. He has also altered the phrase “ principle of ¢east action” into the more correct one “principle of stationary action?’ and has used the English term “ living force” imstead of the Latin “ vis viva,” 518 DIFFERENTIAL AND INTEGRAL CALCULUS. the initial velocities being the same in both. This gives 4r=0, &c., da,220, &c., dv,=0, &e., whence 6U,=0, and every term on the second | side disappears. Hence 6.2m fvds=0, and this, which may indicate } that the real action between any two positions of the real path is a maxi- mum or minimum, was assumed always to indicate such a conclusion ; an | error* of generalization perfectly similar to those already considered in | pages 458, &c. Hence the result was called the principle of least action; a maximum being apparently impossible from the nature of the question. The true statement is, that if a path be made between two positions, varying infinitely little from the real path, and begining and } ending with the given positions, the variation of 2m frds will be an infinitely small quantity of a higher order than the variations of the coordinates. R The object of this chapter being to show the student how to gene- ralize those notions with which the study of elementary problems is pre-| sumed to have made him familiar, I proceed to the general treatment of the fundamental equation (1). Let there be m distinct particles, having: the masses 7,, m,..+.™m,, and let the points at which the particles are at the end of the time ¢ from some fixed epoch be (#1 Y 21)---- (5 Yu Zn). And since the repetition of the same functions of a, y, and z'is unnecessary, let © stand for summation with respect to coordinates) as well as masses: thus Sx means m, (%-+y, +21) + Ms (124+ Yot 29) -+&c. The equation (1) then becomes 2m (x!'—X) d2=0, which is) to be true, not fer every value of each ox, necessarily, but for every set, of values which is consistent with the mutual connection of the parts of) the system. Suppose, for instance, that m, is attached to a surface on which it moves freely, but which it cannot leave: let L=0O be the| equation of this surface, whence L—0 must be true of 2, y;, and %,, and} L,, 6a, 4+ L,, oy, +L,, Sz,;=0 must be true of d2,, dy, and dz,.. Hence} dx, and oy, are arbitrary, if we please, provided dz, be made to depend | upon them in the manner preceding. Substitute in (1) for oz, its value, | and there will remain 32—1 variations of coordinates; and if for z, be) substituted its value from L=0, there will be 3u—-1 coordinates remain-| ing. If the coefficient of each variation be then made to vanish, we have 3n—1 diff. equ., each of the second order, to be mtegrated. If there had been p conditions, L,=0, L,=0....L,==0, we might in the same way have eliminated p variations, leaving 3n—p distinct and. arbitrary variations in the equation (1), and as many distmet coordinates in the coefficients. Hence, making each coefficient vanish, we have 3n—p diff. equ. between 3n—p coordinates and ¢, by means of which, when integration is possible, these coordinates can be expressed in terms * The assumption that A is a maximum or minimum when dA=0 has occasioned many errors, and the greatest writers have their full share of them, Among other things, it is frequently stated that a system acted on by gravity only, is never in equilibrium except when the centre of gravity is highest or lowest. This is not correct; it being sufficient to make any position one of equilibrium, that the ten- dency of the centre of gravity should be to move horizontally, or that the tangent of its path should be horizontal. Thus a system of which the centre of gravity describes a curve which has a cusp or point of contrary flexure with a horizontal tangent, has a corresponding position of equilibrium. With regard to the point on which this note is written, it must be noted that in most, if not all, of the cases which actually occur, the value of the integral between two positions of the system is really less, for the actual path, than for any other, APPLICATION TO MECHANICS. 519 of ¢: and the same can be done with the remaining p coordinates,* by means of the p conditions, L,=0, L,=0, &e. If, however, we prefer the process described in pages 455, 456, we must alter the equation (1) into Ym (a"—X) 504 P, dL, +P, dLy,+....+P,dL,20.... (4), which contains 3n arbitrary variations, and 37 +p quantities to be deter- mined, namely, the 3n coordinates, and P,, P,....P,. The elimination of the p last-named quantities (the diff. co. of which do not occur) between the 37 equations leaves 3n —p diff. equ., from which, with the p conditions, L,=0, &c., the 37 coordinates can be determined in terms of ¢. In whichever way we take it, a system of nm particles, moving under given forces, and subject to p conditions, leads to 3n—p diff. equ. of the second order, which introduce 2(3n—p) arbitrary constants in integration. The manner in which these constants are found for any particular case is as follows: since there are p conditions between 37 coordinates, only 37 —p of them are independent; this number of them may, at the commencement of the motion, be made to have given values, and made to begin with given first diff. co. It happens, however, for the most part, that the coordinates by means of which the fundamental equations are most readily expressed, are not those which it is desirable to use in the resulting equations. There must be 3n—p independent quantities; and it may be desirable that all the 3n coordinates, or any functions of them, should be expressed in terms of 3n—p quantities, which may be either simple coordinates, or any other magnitudes determining positions. Of these it will be only neces- sary to specify one, say €: so that when we say that x, &c. are functions of,é, &c., it is meant that each of the 3n quantities x, y,, 2, 2, Yo 225 &c. is a function of one or more (it may be all) of the 3n—p quantities ees, &c. The following theorem will now be necessary. Let the function f(a, y, &c., 2’, y’, &c., wv’, y”, &c.), 2’, x”, &c. being diff. co. of x with respect to ¢, &c. be changed into ¢ (&, n, &c., &,1/, &c., &’, 7’, &c.), by substituting for each of x, y, &c. its value in terms of En, &c. Let dff.dé and 6f.dt be found by the main process of the calculus of variations, between corresponding limits: that is to say, if a= (é, &c.), and we find f¢.dt from E=£, to £=£,, we then take f-dt between r==2, and r==7,, x, being =¥% (&, &c.), and x, being W& (&, &c.). Let the results be L+fP.dé, and A+ fIldt, abbrevia- ‘tions of the results corresponding to those in page 450. Then the theorem in question is that L=A and P=TII, subject to the relations between x, &, &c. That is to say, P would become identically =II if Ww (€, &c.), were substituted for a, &c. It is certain that L+ fPdt=A+ f Ide ort f¢(P+T) dit=A—L: the second side of this last, as far as variations are concerned, depends only on limiting values, while the first side also depends on the manner in which dz, dé, &c. are connected with #, x, &, &c. between the limits. Consequently, the value at the limits, and therefore, the second side, remaining of one value, the value of the first can be altered ad hbitum. * It is necessary that the p conditions should contain more than p coordinates: for otherwise they would either be contradictory, or else sufficient to determine some coordinates absolutely, without reference to the rest. + In these equations suppose for x, &c. their values in terms of 2, &c. to be sub- | Stituted: they must then become identically true. 520 DIFFERENTIAL AND INTEGRAL CALCULUS. The equation last written, then, cannot be true if P—II and L—A | have any values: but it must be true; therefore A=L and P=W. Let the function to which this is to be applied be / > ai SEA / ad T=4m, (city ite) thm (23 ty24+2') +&c.= sim’, > denoting summation both with respect to coordinates and particles. In page 449, if p=4my”, we have, using the notation there explained, X=0, Y=0, Yj=my’, Y,=0, &c., whence the indeterminate part of fodz is f (O—(my')) wdz, where w= dy — y'ox. To adapt this to the present case, we must write x for y and ¢ for «, and (since ¢ is not varied, or 5¢==0) dx for w. The preceding then becomes f(—mz''dr)db, and by applying the same reasoning to every term of 24mx"*, we find that the indeterminate integral part of Of Zima”. dé is — fImx''dx.dt, or f Pat. But if we now consider wz, &c. as functions of &, &c., then Bi | &c. will become functiens of £, &c. and é’, &c. ; so that the indeterminate | integral part of fXdmzx".dt will consist of as many parts as there are quantities in the set &, &c. Let Lhmx"=T, after the substitutions; we have then for the indeterminate integral part | (\(ge- d iz) tet cr @ Ti) bit panne or fIde. EAP aE, dé, dt déi Equating P and II, Ditties d aT dT x dé dit” dé 7 The equation (1) then becomes, after substitution in U, aa Aa Oa. : rs (5 dé ae ~ =O. eee »(5). If, then, we suppose &,, &, &c. to be independent of each other, we have the equations igor Mab aieie E AG d dY dT dU } | } | f Ue ona: (ile dey > dt Gee F ticg wakbae as many in number as there are independent coordinates. | For example, let there be one particle, moving freely, acted on by. forces X, Y, and Z in the directions of the three coordinates. Let the! mass be unity, and let Xéx-+ Yoy+Zdz=0U. Let the transformatior, required be as follows: z remaining the same, v and y are to be ex pressed in terms of r and @, as in page 507; we have then w=rcos@| yarsin0; dv?+dy?+dz?=dr’?+r’ d+ dz’; whence | T=} (224+ y2422)=3 (7? 47° 62-42") | == Oy See ieha'si ia age { , He AS dT dT dT dT dT ey! a9", eG — ae! le dr’ i dr ane? degre cre dg 0, dz’ sine dz ‘ d dU a», : CUA aa §+— .sind=X cos 0+ Y sin # | dr dx dy : dU aU ss dU . — = — Pe pe td ey pi f a0 ie rsin 0+ ia rcos =r (Y cos 0—X sin 6) APPLICATION TO MECHANICS. 521 These last results were P and rT in page 507. The final equations are (dr’: dt being r”, &c., and T having the meaning of page 507,) r'—ro?—P=0, (r°6)'—Tr=0, 2’—Z—0, as in page 507. This method of deducing the equations (5) and (6) is the second of those given by Lagrange, and is the most general mode of treating the question. The following, the first of the two, is more simple in prin- ciple, as avoiding the formal calculus of variations. It readily appears that x ox+ y"dy + s"a=— (a'dat yoy +2/dz) —36 (ec? +y2 +2”). If the transformation into terms of, say £, y, &c. give dv=Adé+Bdys +é&c., &c., we have v#=Aé’+By'+&c., and dr=Ad&é+ Bdy-+ &c. Again, since x'dx-+ &c. is symmetrical with respect to dv and dx, &c., the equivalent of this function must take the form FEOE+G (fow~+ woe) + Hy! dw+ &e.=P, and changing 6 into d, and dividing by 2dt, we change a’dv-+ &c. int 4(7?+&c.) This last then is LFE"4 Gi’ + SHY" + &€.=Q. If we now form 6Q and P’; we shall have AOF .E? + Fe'32! 4+ Gly! + Gw'dl +0G. fy’ + 40H. py? + &c.=5Q (FE )0E + FE/3E! + (GE) Ove + GEO! 4 (Gay’)'0E + Gy’ dé! + &e. =P". The first subtracted from the second gives a/3x+y"dy+2"02= (Fe)'0E—46F . f° + (GE')'dy — 0G. Ey! + (GW) 0E OH. We" + &e, ; in which F, G, &c. being functions of & &c., and not of é, &c., it follows that dé’, &c. do not appear in oF, &c. Now the last result may be obtained from 6Q, as appears from observation 1. By changing the sign of every term of 5Q in which o precedes unaccented letters. 2. By obliterating the accent wherever 0 precedes an accented letter, and differentiating all the rest of the term with respect to ¢, or accenting it. Thus in $Q we see 30F.£?, and in P’—dQ we see —}0F.£?; in the former we see Fé’0é’, and (F&)’0é in the latter. But CE Ma a Ne eae (a) a Che Stile ee = tr he i decal dy iaw osdyl tee make the changes just mentioned, and we have . d. dQ VedQng . dF) dG. hs “sk eed, (OS SMR Oe AS bai 2 RCA 8 ON 2 — >, —— -—— ]0 - &¢. ¥ bet ho. de! Re Ge Jer, dul dys ga Multiply both sides by m, repeat the process for every term of T, and add the results, which shows that (5) follows from (1). It is thus shown that the expressions T and U, transformed into terms of any coordinates, may be immediately made to give those equations of motion of a system which depend upon the coordinates 522 DIFFERENTIAL AND INTEGRAL CALCULUS. used. This completes the theory of the mathematical expression of dynamical conditions; and the complete solution of every problem is reduced to that of diff. equ. of the second order. But it can also be shown* that the determination of 2.m zy, vds from the beginning of the motion through any time ¢, in terms of the initial and final coordinates and of H, the initial value of T—U, leads to a complete solution of the equations. Let £, &c. be the independent coordinates, x in number, in terms of which x, &c. can be expressed. Let subscript units denote initial values, as before ; let ©.m (2/32 +&c.) be changed into 2.m (Pcé+ &c.), and let 2.m i vds be called V. The equation (3), page , 517 then becomes OV=D.m (PdE+ &e.)—Z.m (P,0k, + &e.) + 10H. In which each of P, &c. is a known function of &, &c. and é&’, &c., the relations between 2, &c. and £, &c. being known. If then V be given or determined in terms of é, &c., &, &c., and H, we have the equations d l VEG (E, 80 f, &e. H), Va de4 he. + 86,4 8. + OH; dé dé, dH . where dj: dé, &c., and d#:dH are given functions of ¢, &c., &, &c., and H, as obtained by differentiation. The two values of 0V must be identical, and we thus have d d K n equations of each of the forms oP -mP, e —mP,...(Aand A,), 5 51 p d one equation more oe ae = oi(kd pe Now we are to remember that @ contains the initial values of &, &c., but not of &’, &c.; it has also been supposed that 2, &c. can be expressed in terms of £, &c., without the initial values of @, &c.; which is but saying that the dependence of the coordinates on each other is wholly independent of time and velocity. Hence neither (A) nor (B) contain the initial values of &, &c.; and if between these 2+ 1 equations we eliminate H, and remember that (B) introduces ¢, we have m equations between é, &e., é’, &c., and é, containing n constants é,, &c.; which are 2 first integrals of the equations of motion. But if we eliminate H between (A,) and (B), remembering that the equations (A,) do not contain é', &c., we get m equations between é, &c. and ¢, containing 2 arbitrary constants ,, &c. and &',, &c. Hence each of é, &c. may be ex- pressed in terms of ¢ and constants, or the problem is completely solved ; the solution of a dynamical question being the expression of everything which varies with the time, in terms of the time and of constants depend- ing on initial position. Consequently the solution of the problem of the motion of a system under given forces is reduced to differentiation and elimination, as soon as V, or 2.m f vds, or what has been called the action of the system, is expressed in terms of initial coordinates, variable coordinates, and the initial value of the living force.t From what precedes it appears that the integration of simultaneous * This is the step made by Sir W. Hamilton, alluded to in page 515. } Since H=T,—U;, and U; is a function of 2', &c., any function of %, &c., 21, &e., and H, is also a function of 2, &c., 2), &c., and T). APPLICATION TO MECHANICS. 523 a diff. equ. of the second order is the sole difficulty which we meet with in the solution of dynamical problems of which the data are known with accuracy. In many most interesting questions, the absolute solution of the equations has not been attained, and approximation must be had recourse to: fortunately it happens that most of the problems connected with the theory of the solar system have circumstances connected with them which facilitate approximation to the required integrations. The theory of this process has been generalized and methodized by Lagrange, and it is now my object to present the peculiar manner in which the resources of the differential calculus are applied to the approximate development of the alterations which must be made in a solution, in consequence of certain minute alterations in the data of the question. The principles on which we are to proceed have been already Jaid down in a particular case (page 155). As in page 189, (a,c), a function of w and c, may be changed into any function of x and C, by substituting instead of c the proper function of # and C. If, then, y=# (a,c) be the solution of any one diff. equ. (A), it may be changed by substitution into that of any other, (B). Itis always open to us, then, to solve (B) by investigating what substitution for one of the constants in the solu- tion of (A) will give that of (B): and, in certain cases, as in page 155, this is the most direct road to a complete solution; in others, to an approximate solution. For instance, let there be a couple of simultaneous diff. equ. of the second order, U,=0, U,=0, between 2, y, and ¢. In the complete solution four arbitrary constants enter, say @, b,c, e; let the complete solution be t=¢ (t,a,b,c,e), y= (t,a,5,c,e). Let there be two other equations, U;=Q,, U,=Q,, U, and U, being the same as before, and Q, and Q, functions which are always smallin value. If a, 6, c, and e be made variable, we may, by taking proper values of them in terms of ¢ and other constants (say their initial values) make a=®@ (¢, a, &c.) and y= (t, a, &c.) become the solutions of U,=Q, and U,=Q,. Moreover, since the suppositions Q,=0, Q,—=0 destroy the variable parts of a, &c., we may predict that a, &c. will vary slowly when Q, and Q, are small, That is, if A, &c. be the initial values of a, &c., and if a=A+a(t, A, B, &.), b=B+84 (¢,4,B, &c.), &e., the functions g, 8, &c. will vary slowly in comparison with ¢. This circumstance is the main point of the approximation. The object of investigation is now, the manner in which a, &c. must be made to depend upon ¢ and initial values, in order that r= ¢ (1, a, &c.), y= (t,a,&c.), which satisfy Us=0, U.=0, when a, &c. are con- stant, may satisfy U,=Q,, Us.=,, when a, &c. are variable. From r= (t, a, &c.) we find dx dp dp da dp db dp dc | dp de, dt:..di-s dawt.,.dbo.dt 'dce.dt de. dt’ from which we might find d?x:dé; and similarly we might find dy: dt and d?y:dt2. In these expressions da: dt, da: dl’, &c. are unknown, and dp: dt, db: da, &c. are known functions of ¢, a, &c., since U,=0 and U,=0 are supposed to have been completely solved. Substitute the values of x and y and their diff. co. nm U,=Q, and U,=Q,, and we shall thus have two equations between four undetermined functions a, b, c,e and. the first two diff. co. of each. So far then it might seem 5 24 DIFFERENTIAL AND INTEGRAL CALCULUS. as if we had made no’ progress, having merely converted a pair of simultaneous equations of the second order into another pair of the same kind. But since in the new pair we have four undetermined functions, with only two conditions to satisfy, we can choose any two others which may be most convenient: and thus we can reduce the question to the solution of four simultaneous equations of the first order. Let the additional conditions which we are at liberty to introduce be that the parts of dv:dt and dy: dé which arise from supposing a@, &c. to vary, shall vanish by themselves. This gives dp da , dp db a du da , dy db ud at, at ear Shame! ne Wb aR apt fe.» + (A); . at dy dx dp dy dy, ets, reducing a and a to SAT oAe and START a which it must be ob- served that since in dp: dé and dy: dé, ¢ varies without a, &c., the forms of da:dé and dy:dt are precisely what they were in the solutions of U,=0 and U,=0. Again da @’p _ &e da ty i db Pp de i ' de de. de 'dtda dt 'dtdb' dt dtde'dt détde dt with a similar equation for d’y:dt. Here dd: dt?, @pd: dtda, &e. are known functions, so that on substituting values of x and y and of their diff. co. in U,—Q, and U,=Q,, we have, with the equations marked (A), four equations between a, &c., their first diff. co., and ¢. It is also to be noted that if any other variable be more convenient than ¢, the same process may still be applied. In language borrowed from the planetary theory, to which this method was first applied, U,=0 and U,=0 are called the undisturbed equations, U,=Q, and U,=Q, the disturbed equations, and Qy and Q, the disturb- ing functions. Thus the results above obtained may be enuntiated by saying that the disturbed equations may be ‘solved so as to allow both the coordinates and their first diff. co. to retain their undisturbed forms, provided that the elements (as the quantities a, ‘&c. are called) which are constant in the solution of the undisturbed equations, vary in that of the disturbed equations in such manner as to satisfy the four simulta- neous diff. equ. above deduced. The preceding process is equally a preparation for exact solution (when possible) or for approximation: in the latter the method of successive substitution alluded to in page 223 must be employed. I shall first give a simple example of this method, and then, after giving an example of the application of the whole method of variation of elements, shall proceed to Lagrange’s generalization of this method. du Let do? equation (pages 155, 210) is w=C cos (/—p).8+E), C and E being arbitrary constants. But +u=pu, p being a small quantity. The solution of this 0 6 J(1—p).0+E=0+ E— tig jo... =O+E—V, V being (4utiwe+...) 0 Again APPLICATION TO MECHANICS. cr bo Ger 2 C cos (0+ E— V)=C cos (0+)(1—++ ieige “at +C sin (0+ E) (v- = +60. ) Expand the powers of V in powers of p, and we shall have u=C cos (0+E)+4C 6sin (0+E) e+ Ap? + Bui + &c. ; A, B, &c. being functions of 0, C, and E. Suppose* now that we could not find the complete solution of the given equation, but that we knew it can be developed in a series of powers of yz. Suppose also that we can integrate it completely when 4=0. Perform this last process, which gives u=C cos(9+E). If we substitute this value of u in the term jw on the second side of the equation, we leave out of wu terms having fH, p’, &c., or out of pw terms having p®, n°, &c. We can therefore make no error in terms of the first order by so doing. But (page 155) du Gort MH EC cos (@+E) gives u=C'cos (0+ BE) + Csin 0 {cos 0.cos (9+-E) dd —p Ccos0 f sin 0 cos (0 + E) dd =C' cos (0 +E’) +3» Ccos (0+ E)+5,Cé@sin (0+ E); where C’ and E’ are new constants: but as two, C and H, have already been introduced, and no more are allowable, we must examine this result further. Taking the result u==C’ cos (0+ EH’) +4 pC cos (9+ E)+3 pCO sin (0+E), which absolutely satisfies the equation whose second side is pp C cos (0+5), we have du f . ‘ 7 ae eed C cos (9-+-E)—p C’cos (06+ FE’) —tp’ Cocos (O+E)—4 pC Osin (0+E). if then E=E’, and if C be either equal to C’, or differ from it by a quantity of the first order, so that ~»C—pC’ is of the second order, the second side of the preceding is entirely of the second order, or the given equation is satisfied as far as terms of the first order inclusive. If C—C’=1,C, the preceding value of « becomes precisely the first two terms of the real value, as found by the exact solution. If we substitute this value of uw, exact to terms of the first order, in pw, the error will be of the third order, and repeating the process of solution upon the equation du dé” we shall get a result which is exact to terms of the second order inclu- sive. We may then repeat the process with the new value of w, and so on. It appears, however, that we must, at the end of every process, know independently how to determine the values of the new constants. Let the undisturbed state of a system be as follows: a particle of matter is attracted towards a fixed pomt by a force which varies inversely as the square of the distance from that point. Let the disturbance be a +u=C cos (0+E)+4,? C 6sin (0+E) * These are the conditions under which equations usually present themselves in our present subject, 526 DIFFERENTIAL AND INTEGRAL CALCULUS. small additional force directed towards the same centre. If it were not for this disturbing force, and if the particle were in the first instance projected in any direction except directly to or from the centre of attraction, it would describe a conic section. It is required to apply the preceding principles to the determination of its actual motion. It might easily be shown* that the particle must always move in the plane which contains its first direction of motion and the attracting centre; let the coordinates be taken in that plane, and let w be the reciprocal of the distance of the particle from the centre of attraction at the end of the time ¢ from the beginning of the motion, and let wu’+TL be the acceleration} belonging to the attraction at the distance 7, where, je being constant, pu’ varies inversely as 7°, and II is the acceleration arising from the small disturbing force. Returning to page 507, we have here a particular case of the problem there proposed, in which T=0, P=—(pu’+), since the force is supposed to be directed towards the centre, o=0, 20, since the moving particle is always in the plane of ey. The equations of motion become then aM, Bae Il Ae: dé ‘ de” hn hu? hae and («), page 508, is satisfied identically. The second of these equations can be integrated when II=0, and gives | Hh; w= +B cos (0-—f); B and B being arbitrary constants introduced in integration, and depend- ing upon the initial position and velocity of the ‘particle. Again, since rd0:di=h, the constant & is determined by the initial value of redo:dé. The equation last obtained is that of a conic section, the centre of attraction being the focus; and if we suppose it to be an ellipse, of which a is the semiaxis major, and e the eccentricity, we have pe 1 e eu Ie “ad ey Te) wast Pk and f is the value of 6 when the particle is at its least distance from the focus. We are nowt to apply these results to the integration of the disturbed equation To tae) (Te Il (1) ; dé? he + hafhe ape ti tp: Be ye the disturbing function being TT: h?* w’. The integral of the undisturbed equation being * This might be shown directly from the theorem relative to the osculating plane in page 506. + Meaning, that if the attraction, such as it is at the distance », were to act without alteration upon the particle during one second, at the beginning of which it was at rest, it would at the end of that second be moving at the rate of wu*?-+ II per second. t The greater part of the preceding paragraph is a recapitulation of results with which the student is supposed to be familiar from the ordinary elements of analyti- cal dynamics which he is presumed to have read, APPLICATION TO MECHANICS, Sa | bt Cp u=—~ +-—— cos (@—8)...... U) 3 h? h? OS ( B) ( ) 3 in which e and £ are constants, let e and B now be such functions of @ as will make the preceding satisfy the disturbed equation. We have then du dé in which, there being two new indeterminate functions e and 6, with only one condition to {he satisfied by them, we may (page 524) create another condition by supposing the part of dw:d0 which arises from the variation of e and 8, to vanish by itself. This gives eal Sea tocingy 4H £9 5 fll as te Leonia EB sin (0 B) +, cos (0 B) 75 +4ssin (9—Bf) one le du d cos (0—f) ate sin (8—£) a, Ave = sin (0@—f) Pu ep Bow des. é be dp dp. 7 O° (@—B)— pom (9¢—f) 7 + 73 098 (0—f) qa For “t cos (@ — f) write was whence (II) gives dp iW i de —sin (@—8) 76 -+ecos (@—f) BP aie ; which, with the condition previously created, gives de dé If II be a known function of w and 0, substitution of the value of from (w) in the preceding will give two equations between é, B, and 0, from which, if by integration e and f can be determined in terms of @, the substitution of e and £ in (w) will give an equation between wu and @ which is that of the path of the particle. The equation (wz) is that ofa conic section when e and £ are constant; that is to say, pairs of values of w and @ which satisfy the equation are all coordinates of points in the Same conic section. And even if e and # should be functions of 0, it is still true that every point of the curve is a point of a conic section deter- mined by (w), though two different points are not on the same conic Section: thus, if e=@ and 6=6?, the equation uw=1 +6 cos (6 —6") is not that of a conic section; but if@=a and u=bd satisfy it, the point (a, b) is one of the points of the conic section whose equation is u=1-+a cos (9—a’). We may then say that the path of the particle is such as would be traced out by a point moving on a conic section, which conic Section itself changes its dimensions, varying its eccentricity and the place of its vertex in the manner indicated by the functions which e and B are of 0, and its semiaxis major in the manner indicated by a (1 ~e*) Bh? : ps, | It is only in this sense that planets and satellites can be said to move in ellipses about their primaries ; that is to say, the ellipse must be con- sidered as continually varying its form and position, At any one moment it is called the instantaneous ellipse. The adyantage of this supposition will be more clearly seen by a com- thong i =—-— sm (6—8), Bees A) ae arg COO (0— 6). pu pu 28 DIFFERENTIAL AND INTEGRAL CALCULUS. cr parison with a more simple case. When a point moves in a curve, we talk of the ‘different directions of its motion, as if it could at each moment be said to be moving ina straight line. The straight line chosen is the tangent of the curve, in which, however, the point can never be said to move, unless this tangent move also, and vary its point of contact with the curve. Any other line passing through the particle might be chosen, and the particle might be said to move on that line, if the line itself be also supposed to change its position. The geometrical advantage of choosing the tangent in preference to any other line is shown in page 136: the mechanical advantage lies in this, that the tangent at any point is the line in which the particle would continue to move, if all the forces were instantaneously withdrawn when the particle reaches that point. This amounts to considering the tangent as the line of undisturbed motion, and all the forces as disturbing forces:* and the tangent might be called the instantaneous straight line. In the preceding problem we have a similar geometrical and me- chanical advantage which arises from the introduction of the inmstan- taneous ellipse. Since first diff. co. are the same in both the ellipse and the curve, the former is always a tangent to the latter, and since velocities depend only on first diff. co., the actual velocity possessed by the particle at any one point of its path is exactly that which it would have if it had come to that point in revolving round the instantaneous ellipse. If at the point we speak of, the disturbing forces were instantly removed, the particle would continue its course, not in the disturbed orbit, but in the instantaneous ellipse, allowed to remain as it was at the moment when the disturbing forces were removed. The mathematical advantages of this use of the instantaneous ellipse are increased by the circumstance of the disturbing forces being always small, the consequence of which is that the elements of the instantaneous ellipse vary very slowly, so that the supposition of the orbits of planets and satellites being absolute ellipses is not far from the truth. To take a particular case of the example last discussed, let the dis- turbing force vary as the inverse cube of the distance, and let the whole force be —(pu?-+ku*). We have then AT iiaku ok an =F {1+e cos (0—A)}=1 {1 +e cos (0 ~ B)}; k:h? being called 2. Let it also be supposed that / is less than unity. The path of the particle, on these suppositions, can easily be determined by direct integration; for which purpose I have chosen it as an exercise in the methed of the variation of elements. Let 0—f=¢; we have then | * Let Y and X be the accelerations in the directions of y and 2, so that a//=X, y!=Y. The integrals of the undisturbed equations a/=0, y”=0 are v=at+, y—=Ai+B, from which ¢ being eliminated, we have the equation of a straight line. ‘Treat this by the general method in page 524, and we find for the diff. equ. of the disturbed motion, dit+=0, AN+B/=0, a/'=X, AY; X and Y being each a given function of af-+6 and At+B. If these four equations can be integrated, we find how a, A, 6, and B, the e/ements of the straight line of undisturbed motion, must vary, in order that a=at+b, y=At+B may be the equations of the line of disturbed motion, or of the line to what the straight line of undisturbed motion is always tangent. APPLICATION TO MECHANICS. 529 = —Isin P(1+ecos¢), e 0-f)=r cos P (1+ecos¢),. Eliminate d0, which gives de —le sin (1+e cos ¢) RS dp — e—icos d (1-fe cos dy’ OY ede=1(1+e cos ~) d.ecosg; whence e*=/ (1+ecosd)?+L. Let ecos =z, whence de” aie e de Sore p74 / (>. (1+2), its Aes a dz S eape e ——_ — l Fee a SO ae NC +2)°— 241}, do VU+L42iz—G—) 2 (1—l) z—-21 JAE + AL + DA—T} (ase 28. For /{4?+4 (L+1) (L—/)}, which, L being arbitrary, is merely an arbitrary constant, write 2M (1—Z), which gives 6/(1—) +C=cos™ = tM cos {0,/(1--2)+C}. In w or (uw: h?)(1+2) write the value Just obtained for x, and for 7 put back its value k: h2, which gives = ath, +E cos | ai } Lar heap pee a/(2 a) +e ‘ ‘or the equation of the particle’s path. This may be easily obtained by 7ommon methods from the substitution of ku® for II in (II), page 528, ind integration.* When w has been found in terms of @, the time of describing any ingle is found by integrating di=d0:hu®. It is also to be noticed that n the preceding example we might express the infinitely small variations if the elements in terms of dt, by substitution. Thus OL Son h dé ae (9—f/), gan =— cos (0—6), —=hw a ; fe dt df 5 @ system of three equations, the integration of which will give 6, e, B, nd thence w, in terms of ¢. From page 518 I have been endeavouring to give notions preliminary ) the introduction of the method of Lagrange for the variation of lements, to which I now proceed, taking up the subject from the deter- nation of the equations (6) in page 520? To avoid indices let £, %, ~, &c. be the independent coordinates, * This is the problem of the ninth section of the Principia. The result is that te path described is that obtained by waking the particle revolve in a given ellipse hile that ellipse revolves about the focus with an angular velocity which always *ars a given ratio to the angular velocity of the particle in the ellipse. At may be orth while to remind the readers of the Principia, that the ellipse of the ninth ction is not the instantaneous ellipse of the orbit. 2M 530 DIFFERENTIAL AND INTEGRAL CALCULUS. insteail of £,, £:, &c-, and let T+ U=Z. Remember that T is a function — both of coordinates and their diff. co., while U is a function of coor-_ dinates only. Hence Z and T have the same diff. co. with respect to_ Ew', &c., whence the equations (6) become | db dente ae SOL, dae ade Se ee Th, dues, dines Wha kis (6). When we integrate these equations, we express E, yw, &c. each in- terms of ¢ and a number of arbitrary constants (elements, as they are frequently called) a, b, c, &c. twice as many in number as there are equations. Now Z and its diff. co. are all known functions of &, 2, &em, and only unknown in the same sense as, and so long as, é, &, &c. are un- | known in terms of ¢. If, after the integration, we substitute for &, &, | &c., their (now) known values, then dZ:dé, &c. and dZ:dé, &e.| become known, the first can be explicitly differentiated with respect to | t, and the preceding equations then become identically .true, and in-| dependently of the values of the elements a, b, &c. If, then, these, elements be changed into a+ Aa, b+-Ab, &c., the equations still remain true, and if we denote by Aé, At, A(dZ: dé’), &c. the changes which | take place in consequence of the variations of these elements, we have _ d dZ, dZ d dZ, dZ, 7 ( dE A den” 7 a) A ie tase &c. If other variations be made, by which a, 6, &c. are changed into! a+da, b+ db, &c., equations of the same form may be made by changing A into 3. Multiply the $-equations by Ag, Ay, &c., and the A-equa- tions by of, ‘oy, &c., subtract the second results from the first, and) add all the results together, which gives UZ NN te ead, dZ, dZ\_.. al afar (o%B) age (age) Bese re ge =O > referring to aggregation of the same functions of different coordinates. Now d aL et dZ de aa eet EC verona py By alata We mie SSN UL we? (2 dé ) dt (a: : B) eh : ae or Form similar results by interchanging A and 6, and substitute, which gives d dZ adZ, >> ‘= (45.2 ape =) dZ AL VO ge w= : = {A6.3 TAD Tate {abd ae tea Se f=0 which, for a moment, we call $,—S,+5. If Aa, &c., da, &c. be infinitely small variations, each of the terms is of the second order ; bui it may be shown that in —S,+S, all the terms of the second order vanish, leaving, as a differential equation, 5S,=0. To show this observe that (using our abbreviated notation for partial differentiation; we have, Z, and Zy, Z, and Zy, &c., being each a function of every ont of the sets &, y, &c., &, wy’, &c., } APPLICATION TO MECHANICS. 531 OL, = Ze E+ Z,, 4 &e. + Ley 08 + Zey OY! +&e. OL, = Ly 06+ Z,, de + &e, + Zyy dé! + Ly OW’ +&e. &e, OL y = Lye 06 + Ly OYs + &C. + Zeydel Lay OYs' + &e. Ly = Ly db + LyOUs + &e. + Zid! + Lyyo'+&e. &e. Hence S, is entirely composed of terms of the followin g forms: Ze Ab OE, Z,, (Ay d&+AE SW), Zy (Ae! ow + Aw dé’), Lis (Aw! 38! + Ae OW!), Zur AE OE! in fact, S, is made by puiting together all such functions of single coordinates as are shown in the first and last of the preceding terms, and all such functions of every combination of two coordinates as are shown in the intermediate terms. But in no one of these terms would any change be made by using 4 for A, and A for 0; now S, is converted into 8, by this change; whence S.=S,; or S,=0. But §, is a diff. co. with respect to ¢; the quantity differentiated is therefore independent of t, or \ GAO CdA ys) < < ee (av ae 4 7B) is independent of ¢. (A,é). This conclusion is one which it may be worth while to verify in a particular case. Let there bea particle moving in a given plane, acted on by pressures in the directions of x and y, the accelerations of which arey and x. We have then dU=yde+ady, U=xy, T=} (a? +y"), Z=T+U, 7 Aaa Ra qo? dye and ay, y"/=a2 are the equations of motion, (a result we might have looked for) which give x"— 2, y"=y, or (page 21 1) x=As'+Be‘+Ccost+Esint y=Ae'+ Be“—C cost—Esint Ar=AA.«+AB.e*+ AC. cost +AE.sin t; or=dA.e-+ &e. dx = OA.e'— 6B.¢—*— SC. sin t+ dE.cos t; Av=AA.2+ &c. And we want Az ee —dxzA ts or Avda’ —ozr Ar’; form this by actual x v multiplication from the preceding, and we shall get Ax d2r'—dx Ad’ =2 (8A AB—AA SB) + (AC d6E—6éC AE) + (6A AC—AA OC) (cos ¢+sin ¢) + (AA SE-dAAE) (cos t-sind) & + (ABcE - dBAE) (cos t+sin £)<~*4(ABdC - SBAC) (cos t-sin ¢) e~*, Now observe that to change 2 into y we have only to alter the signs of © and E, which will change those of AC, &c._ If this be done, and the ‘esult added to the preceding, we find that all the portion depending on ‘disappears, and part of the independent portion, giving Ar $2! — dx Ax’ + Ay dy! — dy Ay’/=4 (SA AB — AA OB) ; ‘ result independent of t, which verifies the theorem. 2M 2 532 DIFFERENTIAL AND INTEGRAL CALCULUS. This very remarkable result, which is perhaps the most characteristic specimen of the genius of Lagrange which could be given, is the most general theorem which has yet been attained in the mathematics of mechanics, not excepting the principle of virtual velocities, or that of D’Alembert ;* for while the former gives a relation between the effects of one virtual alteration only, this theorem of Lagrange assigns a relation between the effects of two distinct and independent virtual alterations. Returning to the equations (6)’, page 530, let us now suppose such disturbing forces to be introduced as add the disturbing function Q,to U, Q being, as shown, a function of & y, &., but not of &, w’, &e. Hence dZ: dé’, &c. remain as before, but dZ: dé, &c. must be increased by dQ: dé, &c.; so that allowing Z to represent T+ U as in the undis- turbed question, the equations of the disturbed motion are found by writing Z+© for Z, which gives d dZ dZ dQ °d-dZ dZ dQ eee =—, &e. (Q). dt dé dé tdi’ dt dy’ dy dis Let us, moreover, suppose that the formule for the disturbed motion are to be those of the undisturbed motion, except that the arbitrary con- stants become functions of the time, and let 0&, &c., which are variations arising from variations of elements only, be those variations which actually take place in the time dé; while Aé, &c. arise from arbitrary | and virtual variations. The theorem of Lagrange still remains true, but not in the words hitherto used; for (A,d) (page 531) now becomes a function of the time; but this is only through the elements which it. contains, which were the arbitrary constants of the undisturbed motion; and (A, ) is now to be said to be not a function of the time, except through these elements. Moreover, as previously explained, the number of elements by proper determination of which we make the undisturbed formule represent the disturbed motion being double of the number of equations to be satisfied, leaves it in our power to make it a condition of | this determination that df, dw, &c. shall all vanish, the effect of which Ee oe Le ee upon (A, 0) being observed, we now see that 2 (45 é ae is independent ty of the time, except through the elements. Again, if we examine the first of equations (Q), or d. Ly —Z,.dt —©,.dt, it is plain that d.Z, must consist of two parts: first, that which arises from making ¢ vary where it enters explicitly ; secondly, that arising from making the elements (formerly arbitrary constants) vary so as to make the whole satisfy the disturbed equation. But the first is the d.Zy of the undisturbed question, and, therefore, page 530, equations (6)’, is equal to Z,.dt: the second must, from the hypothesis above made as to the meaning of 0, be denoted by 6Z,. Hence the pre-’ * The principle of D’Alembert is perhaps rather of a metaphysical than a mechanical character; by which I mean that its evidence depends rather on our: general notion of cause and effect, than on any conception particularly derived from the cause which we call force, or its effect, velocity, or the counteraction of effects called equilibrium. Assuming that a cause must produce its effect unless hindered by the effect of some different cause, it follows that if a set of causes A produce only the effect of another set of causes B, A and B can only differ in that A contains besides B, a set of causes the effects of which neutralize each other; these bemg removed, all that is left of A is B. APPLICATION TO MECHANICS, 533 ceding equation becomes 0Z,=Q,.di, or, in common notation, and extending the same reasoning, we have aZ* “dQ dG dB dé! ade dt, ") dys! ~ dis dl, &e. (7) ..dZ dQ. dQ 2 (43 r 7E) =(F et ae Ay + &e. ) dt=AO dt: whence AQ.dé is a function which is independent of ¢, except as ¢ enters through the now variable elements. Or rather,.if in the expres- sion > (AZ 0 Zy—o£ A Zy), which is certainly independent of ¢, we intro- duce the conditions &&=0, dw=0, &c., we then find an equivalent to AQ dt, which is, therefore, independent of ¢. But we are* not to suppose that if we were merely to find Q in the most direct manner, and thence AQ dé, that we should produce this function in the form in which it is independent of ¢. The theorem may be thus stated: the expression AQ, di—.dé AZ, may be made independent of ¢ directly, by substitution in AQ of the values of Q,, &c., furnished by the equations of motion, (this is the reversal of the last process,) and this form, which is independent of ¢, is in value an equivalent to AQ dt, if the equations 0&=0, O%=0, &c. be also satisfied by the variations of the elements. Let a, 8, &c. be the values of &, yw, &c. when {=0,and X, pty &e. those of Z,, Z,, &c., in the undisturbed question. These quantities, twice as many in number as the coordinates, may be taken as the con- stants of integration; since whatever constants integration may intro- duce, they may be determined in terms of «, &c. and A, &c. But since 2 (AE 6Z,—3£ SZ») is independent of ¢, it might, in the undisturbed question, be determined by making t=0, since the value which it then has, it must retain. But its initial value is © (Aw.d\—da Ad), whence, remembering that the value of the preceding is also AQ.d¢, and, Be tituting for €, Y, &c. in © their values in terms of ft, @, A, &c., we ave d dt (Fae AB+&e.+ 2 Arx+ be, )=Agih=2a M+ ABdu- he, a dp in which Ag, AP, &c. are altogether indeterminate. Hence, then, . dQ dQ, dQ dQ oA=— aa —— dt, du=—dt, (b=——dt, &e. (8), r 7 dt, oa or f, Om dp iC i » &c. (8) * For example, (A¢-+B)a—(at+) A is independent of ¢, unless as contained in A,a, &e. But should it happen that at+5=0, we do not become immediately cognizant of this theorem by looking at (A¢+B)a, though we may deduce it either by using the term (af+5)A, or by eliminating ¢ from (A¢+B)a by means of at+b=0. Thestudent who examines the Mécanique Analytique, pp, 333—337, will see that Lagrange, when he has proved the equation AQ dt==> (AE. 3Z x1 — 0 AZy), adds “On voit que /e second membre de Yéquation précédente est la méme fonction que nous avons vu devoir éire indépendente du tems fee Et he does not venture to add that therefore the first side is independent of ¢, and he cautiously abstains from _ any use of that first side, except by means of the second. The fact is, that though itis possible to write © in such a form that AQd¢é shall be independent of , yet, after the present step, he does not find it necessary to use or refer to that form : and it is in fact never used in practice. The difficulty arises from the particulariza- tion of the meaning of 3 being made a little too early in the process, which is avoided in the second proof of the resulting equations presently given (page 535). 534 DIFFERENTIAL AND INTEGRAL CALCULUS. and so on. Andsince 6A is supposed to arise from a change of ¢ into t-+-dt, as soon as we pass to the disturbed question and suppose a, A, | &c. functions* of ¢, it is not necessary to distinguish it further from dA, a differential relative to the time. We have thus a number of simulta- | neous differential equations sufficient, if they can be integrated, to deter- mine a, A, &c. in terms of ¢. Neither is it necessary that ¢ should enter directly in these equations; for since AQ.dé may be exhibited in a form which does not contain ¢, and this absolutely independently of the values of Aa, &c., the same thing is true if all but Aa vanish, im which case AQ. dt is (dQ:d«) Aadt, so that (dQ:da) dt will not contain t, if derived from the proper form of Q. The equations (8) are only particular cases of a more general form, © from which it may be advisable to derive them. In the general equa- tion . > (Az 6Z, -— & AZ) => (Aa dA—da Ad), which merely expresses that the first side, not containing ¢ directly, has always its initial form, substitute for AZ, &c., AZ», &. their developed values, the elements, by variation of which the variation A arises, being a, A, &c. We have then for the first side dé dé ; (dhe dZu . . > | 7, bat GAN + ke. )oZe of (3 Aa + aN dn +&e. . equate this to the second side, then since the equation must be true for all values of Ag, &c., we have a set of equations of the form aay OLy+ io OZy+ &e. ths e 0& Ar ous — &e. _dé dye yal es da= OLy+ ae OZy+ &e.— } ome a. ow—&e, Without making any particular supposition as to the derivation of 0, | repeat the process by substituting for 6Z,, &c., their developed forms in terms of da, OA, &c., which must make the preceding equations identical. The consequence is, that if p and q represent any two whatever of the set «, 3, A, w, &c., we find dé dZy dé di dp dq dq’ dp to be either +1, —1, or 0; +1, if p and gq be wand), or B and p, &c.; —1l if p and q be X and a, or p and f, &c.; 0, in every other case. But if in the preceding equations we take ¢ to arise from the simple | change in ¢, and make oa, &c. so that d£=0, d~=0, &c., we then find as * In the undisturbed question a, a, &c. are found by making ¢0. But the | student must not therefore imagine that ¢=0 in them when they become functions of ¢. In fact the question relative to them is this: the values of a, &c. are certain , functions of the elements of the undisturbed orbits; according to what law do these | functions change when the undisturbed orbit varies its dimensions perpetually, in | such manner that a body moving in the disturbed orbit may also be always in some point of the undisturbed orbit? And a, a, &c. are those functions of the elements | which %, Zy, &c. are when ¢—0, altered subsequently to this supposition by making | the elements take their proper forms in terms of f¢. APPLICATION TO MECHANICS. 535 before, from considering the fundamental equations, that dZy—= (dQ: dé) dt, &c., whence di d® dw da dQ, d\=| —.— 4+—4 —— = Ack Gp peaches die +8. a = dt, &c.; _and thus we verify the equations (8). Next, let the arbitrary constants be, not «, A, &c., but certain functions of any or all of them, namely, a, b, c, &c. We have then da da da da dx da dQ da dQ 2 referring to the change of g and X, into B and py y and y, &. succes- sively. But this is da (dQ da . dQ db da (dQ da dQ db \, aE (F da at) Odea ele a: cb Rte.) by development of which, and application of the same process to , ¢, &c., we get the following result. Let p and q be any two whatsoever of the set a, 6, c, &c., and let _dp dq _dp dq ,dpdq_ dp dq a +c. ; SO ar ary MEN du dB dB du ' dp dQ, dQ, dQ, then will uo (p, @) hadi (p, >) on +(p,c) = + &e.; in which for p we may write either a, or 6, or c, &c.: it being remem- bered, however, that dQ: dp does not appear in dp: dt, since (p, p) =0; and also that (p,q) =—(q, p). This is the generalization of the problem of which a particular case occurs in page 528, and we thus see that if the undisturbed question be solved, and the values of &, &c. in terms of ¢ and constants be substi- tuted in ©, we can immediately form the differential equations by which these constants must depend on ¢, in order to make the undisturbed formula represent the solution of the disturbed question. Up to this point we have nothing but what is common to all dynamical problems, and the results, though exhibited in a manner which is most practically _ useful when Q is always small in value, are yet true whatever may be the nature of Q. To proceed further would require that we should propose a specific problem, and enter into its details, which it is not either within the scope or limits of this work to do. I have placed the student at the very threshold of the most important problems of the theory of gravita- lion: and each of these, as he is probably aware, is matter for a treatise, not for a portion of a chapter. I shall conclude the present chapter by treating some remarkable points connected with disturbing functions as they actually occur. ee The gravitation of one particle of matter towards another is inversely as the square of the distance between them: that is, if m and m, be the Masses or quantities of matter in two particles whose distance 1s 7, the particle m exerts on m, an attractive force which would, were it allowed to act uniformly for one second, create the velocity cmr—, ¢ being, as in page 476, a constant depending on the units employed. It is usually said 536 DIFFERENTIAL AND INTEGRAL CALCULUS. that this force is mr~’, but that is only on the supposition that if 7 were — —1, the velocity created in one second would be m, which requires that _ such a unit of mass should be taken that the number of linear units in the rate of velocity created by the action of m continued uniformly for one unit of time such as it is at the distance of a unit, should be the | same as the number of units of mass in m. In physical problems, it is | only necessary to compare the ratios of different masses of the same kind, and this renders it absolutely indifferent what units are used, and makes it even unnecessary that they should be assigned. But the | student cannot safely proceed without a precise notion as to the method of actually determining the force of attraction in any particular case if required; and this is done as follows. Suppose f=cmr~ to be the formula, as above described. Let the unit of length be a foot, that of time a second, that of mass may, as we shall see, be left indeterminate. Suppose the earth a sphere of the mass E, and of a radius of A feet: we know that the action of this sphere creates in one second on a mass at its _ surface a velocity of 32°1908 feet. But a sphere acts on a particle at its surface precisely as it would do if all the mass were removed to the - centre, and there collected into one particle; which in this case would amount to a particle of the mass E acting at the distance A. Hence 32°1908==cEA~’, from which c may be obtained, and this being sub- stituted in the preceding, gives m A EP? in which the existence of the ratios m: E and A:7 renders it indifferent what units of mass are employed, or of distance, provided it be remem- | bered that the velocity which the result expresses 1s measured in feet per second. If we adapt the units so that f=mr—, and if the coordinates of the | particle acted on be (a, y, 2), and if the force tend towards a point at the | distance r, whose coordinates are (a, b,c), we have 7°=(«—a)’+(y—b)? | + (z—c)*, and the resolved accelerations in the directions of a, y, and z are f=32°1908 Mm-a—-z m b—-y m c—z a—x Sag fies > = 3 orm—, &e.; edo g Toe 7 th ar rs the condition under which all forces are represented being that they shall be called positive* when their effect is to increase the coordinates of | their directions, and the contrary. But d ] dz J{(«—a)*+ (y—b)? + (z—c)*} rt—a a—2 ~~ {@-ay+y-+e-oy ® whence it appears that the above forces in the directions of 7, y, and 2_ are the diff. co. of mr~! with respect to z, y, and z. If there be- another particle m, placed at the point (a, 5,,¢,), and if 7,= /((a—a)’ +&c.), in a similar manner the acceleration of 2m, on a particle at — 9 CC. 5 * In page 477, x—a, &c. are inadvertently written for a—«, &c. APPLICATION TO MECHANICS. 537 (x,y,z) has for its components the diff. co. of mr; with respect to z,y,2. Hence, if a number of particles so act, the whole accelerations on a particle placed at (a, y, z) are the diff. co. of > (mr—"). Suppose, then, that a continuous mass acts upon a particle at (a, y, z). At the point (a,b,c) let pdadbdc be the element of the mass, as in page 493, and let this be called dm. If, then, we compute v= [= ead — J {(2—a)? + (y—b)?+ (z—c)?)? a lah sa pdadbdce _ Ai (a—a)? + (y—6)?+ (2 =c)*} throughout the whole extent of the attracting mass, the whole attraction of the mass upon the particle at (#,y,z) in the direction of a is (dV: dev); and similarly for y and z. In the function 7* or (v—a)*+&c. preceding we have dr z—a dr 1 dr t—a —-=——, &.; -———— Se dx aa dx r dx Teale ae 1 x—a dr 1 (t—a)? dx? ry r* oi eae oe Ses, meer dt dtr 3 (r—a)?+ (y—b)? + (z—c)? == ———S{S oa —— 3 ——. ———— 2 dx? a dy? dz r 1s == 0; rs This simple result may be easily proved to be true of each of the terms of 2mr~', how many soever; it is, therefore, true of the integral V above noticed, or we have Nees Str VN. a to at ae dx* dy* dz a= fs a result which we are now to express when the variables are r, 0, and 7, r being now the distance of (2, y, z) from the origin, 0 being the angle which r makes with x, and @ being the angle made by the projection of r on the plane of yz with y; so that r=rcos0, y=rsin @coso, s=rsiné.sinw. Using the abbreviated notation, we have iP Ae r.+V, 0,+ as W, maces V,, ie -+- Veo 03+ EN ooes BD; + 2-6 Vz 0, + ZV ces 6, Wp,+ AV os Ww, rT, + V, Tx = V, 0..+ Veg WD x2 > and similar formule for V,, and V,,; the second side proceeding on the supposition that each of r, @, and @ expressed as a function of a, Y; and z; thus r=,/(2*+y'? +2"), cosd= z tan wo ==—. 4 av V(ae+yi+ 2?) Now let it be observed that z becomes y if ’@ become w+41, and that z becomes a if @ become 3m and 0 become 0+47 at the same 538 DIFFERENTIAL AND INTEGRAL CALCULUS. iy Hence, by determining 7,, T.:, &c., we can easily deduce* 7y, 7,y, Vx Ci ci dr 2 j ' dr. dr —_ =—==sin@sinw, —-=sin9cosw, -—-=c0s0 SE ioe tah b dy dz 16 doe a adr LZ dé Lz cos 0 sinw —sin@ —=—-— — SSCS eS : dz r. dz rs? dz rsn@ r d@ + cos0cosm dé sin 0 whence er Se dy r daz T dw : n cos & da sin @W da ————— COS 40. ae 3 wae Tot oes oS _> dz y rsindg dy r sind dx sy Pee dw P d@ cos?@ — cos? @sin’o ———sin 6 cos o —-}+cos 0 sin © — = dz? dz dz r r dr sin’@ | cos?@cos’o ar even’ 6 hie r r a es ey @0 cos@sinw dr cosOcosa dw sin@Osinw dé dz” 1 dz r dz tr dz 2cos@sin@sin?w — cos0cos?w ie ve r’? sin 0 ad’6 2cosAsinA@cos?*w cosdsin?w dy” r2 r* sin 6 ; : d’0 2 cos 6 sin @ nits a r | da cos@m dr sinw dw cos @ dé | — — —— ———__ cos 9 — dz? rsin@ dz rsin@ dz r sin? @ dz “4 | coSmsinw cosWsinw coswsinwcos 9 isfy r2 7? sin? 0 r? sin? 0 da cosmsinw cosWsinw coswsin ww cos?9 —-—— _- dy? r 7? sin? @ r* sin? @ Hence, > referring to summation of results of rectangular coordinates, (as in 27,=7,+7y+7;), we obtain 1 SL gama Ee 205= 5 3; 23> Er, 6203-20, 0, =03 Fo,7,=0; 2 cos 0 7.2 —; 26,,.2——.; 20,,-9. Taide ae oe Si at * Though this is what is here done, it is desirable that the student should deduce all these differentia] coefficients independently of each other. APPLICATION TO MECHANICS. 539 Substitute these in 2V,,=V,, = 72+V,, 262+ &c., obtained from the values of V,,, &c., and we have, since >V,,=0, d?V 1 @&V 1 iV 13 av cos@ dV dr? r? d@ ° 7*sin?0 dw 'r dr '7*sind gee Multiply by 7*, and the first and fourth terms together then make rd? (rV):dr*; also let cos9=,, which gives TU ese GAO CaeY dV ae r= int: on a in ee Hom en ag tog ay TE) gee Substitute, and the second and fifth terms (after multiplication by r?) are dV av dV d ' _ ee eal Ch ote ee ee yy A inte fapt MY) op ar ie dy (1 Me, 3 whence the final equation (in the form employed by Laplace) is Bea) Cea a0 Ne Maye pg OND = du (l—p dp Ly eae we bss gm eres. CV): _ If the attracting surface be a homogeneous sphere, or spherical shell of any thickness, with the origin for its centre, it is obvious a priore that the attraction is altogether independent of everything but r; whence if dr be the whole attraction, (which must be towards the centre,) we have dv ‘dz whence V is a function of r only; so that dV: du and dV: da vanish, while dV: dr is the whole attraction. Hence the preceding equation gives —$r—, &c., or dV=—®¢dr.dr ; d’ (rV) B dV B oe =0, or NE a 7 a th that is, the whole attraction of such a sphere or shell upon any evternal* particle is directed towards the centre, and is inversely as the square of the distance. Moreover, B is the mass of the sphere; for if the dis- tance r be very great compared with the radius of the sphere, the sphere must act nearly as an isolated particle, and the more nearly the greater ris. But B being a constant, cannot approximate to the mass of the sphere as r increases, and the preceding condition cannot be true unless B be the mass of the sphere itself. So that at all distances the attraction of the sphere is as its mass directly and the square of the distance r inversely : or the sphere acts as if it were all collected} at the centre. The equation (V), and another analogous to it, are employed by * If the particle attracted were within the limits of the sphere, the denominator in the function, which integrated gives V, would become infinite within the limits of integration, and could not be relied on. And, in fact, the laws of attraction are different for internal and external particles. + If our object here were the particular discussion of this problem we should give a better proof of this for the beginner. 540 DIFFERENTIAL AND INTEGRAL CALCULUS. Laplace in the deduction of the properties of some very remarkable functions, which it is usual to call Laplace’s coefficients.* Let there now be any number of particles, attracted by and attracting each other, but otherwise moving freely. Let one of them, having the mass M, be the one to which all the others are referred, (the sunt in the case of a planet, the primary in the case of a satellite,) and let X, Y,Z be its coordinates. Let the other particles have the masses m, m,, m,, &c.; let their situations at the end of a time ¢ from the beginning of the motion be at the points (2,y, 2), (% Y,2,), &c., when the origin is (X, Y, Z): that is, let their actual coordinates in space be X+a, Y+y, Z+z,X+a, &c. Let their distances from M be 7,7, r,,, &c., and let r,,, Tepresent the distance between the particles m, and m,. It is required to exhibit the diff. equ. by whicha,y,z, 2, &c. are to be determined. First, as to the motion of M, it is obvious that the accelerations with which it tends towards the several particles are mr~*, m,r,-”, &c., which, decomposed in the directions of x, y, and z, give OX: ee Me) Ne TY ek debe HO dB mea de the signs of the second side being marked as positive: for m, for instance, can only draw M towards the origin when it is in the direction of x nearer to the plane of yz than M, that is, when X +2 is less than X, or when w is negative. This will make mz:7r° negative, which is according to the conditions laid down for estimating the signs of the accelerations. Again, since the effect of M upon m is contrary in direction to that of m upon M, the components of the latter are —Me:r?, —My:7°, —Mz:7°, and, similarly, for m, &c. Also, the attraction of m, on m, in the direction of wv is (as before explained) L,—L, 1 d mm, mm, —————__———_—_-——;, or {(7,—%)?+ (Ya —y,)'4+ (%— 2)" }2 Consequently, putting together all the accelerations on m, in the direction of x, we have qt, a te Pee Lod (ms eae aiid bagi wo. soe (m), m, ar, t To,b Ti, b 72, b which contains every value of a except a=b. Suppose now we form the function A= {m, m, (ra, 5)~*}, in which every pair of masses enters in some one term: we have still | dn Pas, Ge ] Whole acceleration on m, in direction z=— —; m, dx, * The English reader may find the discussion of these functions in Murphy on Electricity, Cambridge, 1833, and in O’Brien’s Mathematical Tracts, Cambridge, 1840. + The planets are spoken of and treated as particles ; being spheres, or very nearly so, they attract each other very nearly as if they had their masses collected at their centres. The small irregularities arising from their non-spherical forms are usually treated subsequently to the main case of the problem. t | ) | | | ) | APPLICATION TO MECHANICS, 541 for with regard to terms already in (m,) they are alsoin X: and the terms which are not in (m,) vanish from dd: dz,, since they are not functions of x, Thus x,, is not in the term which has m,m,, and which only contains z,and z,. But since all the particles enter in the same way into the expression A, this function applies equally to the case of the action of any one particle on any other: and reasoning similar to the above shows it to apply also to the directions of y and z. Collecting the accelerations on the particle m in the direction of a, and equating them to their effect, we have d(X+. Nie IN OX aX a (X42) _ NP inde +— zit but ae => loamy dt® RT CaN Bi dt? = dx (M+m)x > me 1 dy- peers iat Bice ay Dede ear ee é ; m: m x0, mM, @ mae. in which > fh means —,~+ +—~+ &c., or all but —, this last term 7 T rs tT having been taken into the preceding. But this last is the diff. co. with respect to x of hy she 2Z > (m, SHWE), 7; in which m, is successively made m, m,, &c. The terms containing y and x, which disappear in the differentiation, are introduced that the same function may apply to the accelerations in the directions of these coordinates. . It will be remembered that r, containsonly 2,, Yo) and 2,. If, then, we make shee X R=E (m, MES) 7 m a function which, represented at length, is as follows, nr + yy, +22, LU, A YY, +22, re, +&c. a ae S\ pias tas 2) mM, 2 &e. : WAtat yz") he (27 +y7+2, ) us N (Qi +&e) J mm, m 7 (2,2 +Y,- y+ 2)"5 m,m +———_——_——_.._——_-~ 2 mH + &e. \ V2 —2,P + (YY) += 2)" the differential equations of the motion of m are (M+ being 1) 2. 2 2 pe pote eC Ee pe ata ce ek i 7 dt? ot dy dt > = dz which are the fundamental equations of the motion of a planet. These €quations may be approximately solved either by the direct application of successive substitution, (after transformation somewhat resembling those in page 507,) or by the method of variation of elements, described in the preceding part of this chapter. But, as before observed, this solution would require a treatise of itself. dR Tie = Cuapter XVIII. ON INTERPOLATION AND SUMMATION. Tue present chapter is intended to exhibit some developments of the general methods derived from differences, which are useful in practice. By interpolation is ‘meant the insertion of intermediate values of a function, corresponding to intermediate values of the variable. If the function itself be given, any value may be calculated without reference to other values; and the question of finding $x for a given value of @ is not one of interpolation. Let us then suppose that all we know of the function is that when z=a,, dx=A,, when r=a,, Gr=A,, &e. It is required to investigate, as far as that can be done, the function itself, so as to be able to find any values of it. The question proposed is indeterminate; that is, an infinite number of functions can be found, which satisfy the proposed conditions: For instance, suppose three values of the function to be given, Ao, Ay, As answering to three given values of v, namely a), a, ds. Let A,2, fot, Yt, &c. be any functions which vanish when x=a,, and do not become infinite when z is a, or @,; also let A,z, &c. and yz, &c. be functions similarly related to r=a, and r=a,: and let wa be any function of « which vanishes when w=a), and also when v=a, or a,. ‘Then AyD dev A [lp « oP A VoE VL Ab ye satisfies the conditions, and contains no less than seven arbitrary functions. If, for instance, z=a, Yr vanishes, and also pox and vor ; whence the last three terms vanish, and the first obviously becomes A,. If we want the most simple algebraical function which will satisfy the conditions, we must take \jt=p.x%x=—&c.=x —d, and so on: also wae=0. This gives @—a)a 1) (27—az) A, 1, Se tees) (t—ay)(2—a@) (a —a)(c — a) (ay — a) (a, — a) (@,— ze) ; (a@,.—y) (d2a—G,) If a, @, 42, &c. be themselves the values of a function of ¢ correspond- ing to t=0, ¢=1, &c., and if wt be this function, and ¢z the required —— AA, 1 SS AA, Ato fo ° fed Vo@e ¥\ A, Ag. function ae x, we have capt and @r=Pyt: whence Aj, A,, &c. are the values of Pyt answering to ¢=0, t=1, &c. Consequently (page 719) t— $yt=A,-+tAA, +t MATIC = = beAyt be. satisfies the conditions ; which, since a ¥ gives Lb ¢—] pum Ay tyr. AAg tye a A*A,+ &e. Thus, if it should happen that A,=1, AA,=2, A®A,=3, &c., or if Ag, A,, &c. be 1, 3, 8, 20, &c., we have (page 240, Ex. III.) 4 pr=2v* (14+ hy2). ON INTERPOLATION AND SUMMATION. 543 But this is only one out of an infinite number of proper forms: for since sin zt vanishes when ¢ is any whole number, and also y (sin 7), provided that yz and x vanish together, we may add yx (sin. r¥'z) to the preceding value of $2, without disturbing the values of ¢x when T=, or a, or a, &c. But this change would evidently alter the yalues of x corresponding to intermediate values of 2. Having said thus much to show the indeterminate character of the problem, we shall proceed to notice the particular cases upon which arithmetical interpolation is practically attainable; that is to say, in which we determine intermediate values by means of given values alone. I refer to the next chapter for an instance of another and very distinct sort of interpolation, which we may call interpolation of form. To simplify the mode of speaking, let us suppose that Ay, A,, &c. are the ordinates of a curve, to the abscissz a, a,, &c. Through any two points we can draw a ‘straight lime y=p+qz; through any three a parabola y=p+qr+ra*; through any four a parabola of the third order, y=p+quv+ra°+szx*: and so on. Again, if we take » points near one another, and having their abscisse in arithmetical progression, with a small, or at least not very large common difference, and their ordinates also not very unequal, as in the adjoining figure ; the parabola of the (x—1)th order,which can be drawn through these 7 points will very nearly coincide with any regular curve of the same general appearance, at least between the extreme points. Leta, a+A, a+2h, &c. represent the values of x to which those of y are Ay, A,, A, &c., then L-Ga t—a—h ,, ren ears APA,+.... will be the equation of the parabola which passes through all the points. If all the ditferences of Ay vanish, from and after A”A,, it shows that a parabola of the mth order ean pass through all the points, how many soever there may be. If, then, all the differences of A, diminish rapidly, so that from and after A”A, they are not worth taking into account in practice, it denotes that a parabola of the mth order will be a sufficient representation of the curve from z=a until x becomes so much greater or less than a, that the coefficients of A”A,, &c, become large enough to make those terms of sensible value. If z=a-+mnh, we have n, n(n—1):2, &c. for these coefficients, from which it may without much difficulty be estimated, in any particular case, how many terms of the series will be wanted to insure a given amount of accuracy. The pre- ceding is also, generally speaking, the most convenient form, though it does not differ essentially from the one proposed at the beginning of the chapter. Suppose, for example, that according as x is a, a+h, or a+2h, y is Ay, A,, or Az Let r=a+nh, then n is in these cases 0, 1, or 2, By the first method of this chapter we have for the simplest function which satisfies the condition (n—1)(n—2) n (n—2) n(n—1) (0—1) (0—2) Ati (aay 45 (2—1) for A, and A, write A,»+AA, and A,+2A4A,+A’A,, and the preceding may then easily be reduced to Os y=Aot+ — AAy+ Q¢ 544. DIFFERENTIAL AND INTEGRAL CALCULUS. n—— 2 Also it is to be observed that @(a-+-nh) and A,+7AA,+.... are identical if ¢a=Ay, ¢(a+h)=A,, &c. This follows most easily by observing the laws of operation: the first expression is €”""da@ or (e>")"A,, or (t+ A)"A,, which leads to the operations indicated in the second series. Andif in ¢a+¢’a.nh+ &c. we substituted for ¢’a, &c., their values in terms of ga or Ay, and its differences, as found in Chapter XIII., the result would be found identical with Aj +nAA,+&c. As an example, suppose that we have the following values; according as xis 5,7,9, 11, or 13, y is 672971, 553676, 456387, 376889, or 311805. What is the value of ywhenz=10? If#=5+2n, n is 25 when t==10; also we have Aj==672971, AAp>=—119295, A’A,= 22006, A*A,= - 4215, AtA,= 838, in which the differences ‘diminish with sufficient rapidity. The value required is A, +ndAy+n : A®A.. 1+5 1:5 °5 672971 —2°5 x 119295 +2°5 x ——— x 22006 —2°5 —>- — X4215 | a gateeauea salle 3 4+2°5-—- = —— x 838 = 672911 —29823 441261 —1317—33=414644.* Examples may be made at pleasure and verified from a table of logarithms ; as follows. Take out the logarithms of a, a+h, a4+2h, &c., and difference them, as it is called; that is, take the successive differ- ences of loga@ until the differences become very small. Let it then be required to find the logarithm of a+k, k bemg a whole number between ph and (p+1)h. Let z=a-+mnh, whence, in the case required, nh=k, or n=k:h, a fraction between p and p+1. Then calculate loga+nAloga+é&c. as far as the differences have been taken, and verify the result by the tables. Suppose Ay. A,, &c. to represent a number of results of observation or calculation, for instance, the right ascensions of the moon at intervals of twelve hours from a given date; thus A) is that at noon, A, that at midnight, A, that at the next noon, and soon. If, then, we wish to calculate the right ascension at a time between the noon and midnight at which it is A, and A,, let m be the fraction of twelve hours which has elapsed, and we may compute the right ascension required by the for- mule A,-+nAA,+3n(n—1) APA,+&c. But we might also compute it by A,+(1+n) AA,4+3 (142) nA?A,+ &c., or by Ay+ (2+4+n) AA, +&c.,andso on. These results-would be slightly different, owing to _ the necessary error of the process. And it is sufficiently obvious that most reliance is to be placed on that result in which the differences used" come from the places which are nearest to the interval in which the required right ascension lies. Thus if we are to go only as far as fifth differences, which will require six right ascensions to be used, it is better that they should be Ay, As, A,, Ay, A,, A, than A,, Az, A,, Ay, Atos * See this example in the Penny Cyclopedia, article InrzrPoLarion, in which also some other methods may be found which are convenient in particular cases. ON INTERPOLATION AND SUMMATION. 545 and A,,, the required interval lying between A, and A,. On this considera- tion it is generally thought desirable to use an odd number of differ- ences, and to let the values employed be distributed equally on one side and the other of the interval}in which the result lies. Tet it now be required to express A,, symmetrically, by means of A,, As, &c. following Ay, and A_,, A_s, &c., (which we write dh, Az, &C.) preceding it ;—thus, gees, 2,” Oy Mya, . (a, or A,) A, Ay A, A, Aves 5 3 On examining the manner in which differences are formed. we see that Aa,+ Aa, involves a,, Ay, A,, and is symmetrical; A®u,-+ A®a, in- volves from a, to Ay; A®d,+A%a, involves from a, to Aj, and so on: while A°a, involves @,, Ay, Ay; A‘as involves from ay to Ag; A’a, from a, to A,,and soon. If, then, we can expand A, in a series, every term of which is either of the form P (A**'a,+ A***!a,.:) or PA**a,, the object is gained as far as the symmetrical imtroduction of terms preceding and following ay is concerned. Now A, is (1 PA) Ay, and, ay is. (ie Aya also APHa,+ Aa, = At {(1-bA)-*4+4(1+A)"} A, Al(a&\ i A y — ——-— - iit ———— | Ae {a+ 144s oes RP Eee mA, whence (1+A)* is to be expanded in a formula involving powers of A’:(1+A). This is done by the method of generating functions, (page 337). The generating function of (1+ A)? is 1—E"'t 1—(E+E“")t+e ] -——_-——.,, or (1-+-A being called E Barca): or (1-+-A being called E) mut A: (1+ A)=E+E-'— 9, say =I", whence the preceding denomi- nator becomes (1—¢)*—Ft. The reciprocal of this is l Ft wh EF??? a a Beis (1—¢)2 (1—t)* CEs ce eee dp Now, x being greater than a, the coefficient of ¢* in Hin fats (ky ae that of ¢*-* in F*+-(1—t)***2, or 9 a ae pe (2a+2)(2a-+3)... /(Qa+2+2 a al or F* i =. 2 aa (7—a) ts iatcng Lea 0 tet El [r—a] Hence, successively making a=0, 1, 2, &c., and simplifying and summing the results, we have for the coefficient of é* in the aboye- named reciprocal, , wo = @+2)(r4+3 w(t+1)(4+2) |, (w—1) x (@4+-1)(@+2)(r+ ) 4 &e. ee ae o> ek hp Oy But x (r+2)=(r+1)*—1, (a—1)(@+3)=(#+1)?—4, and so on; whence the preceding becomes (v+1){(r+1)—-1} _ (2+1){(@+1)?-1}{ @4+1)?-4} PiL&e : a haekadiade. 1.2.3.4.5 tS Call this Piiit+Q.4i:F+&c. Now the coefficient of ¢? in E~'t: {(a—f)? ~ Ft} is E™ multiplied by that of #?-' in the simple reciprdcal : whence 2N r+1+ 546 DIFFERENTIAL AND INTEGRAL CALCULUS. coeff. oa 1—E't in J (—t)?—Fé and this coefficient we also know to be (1+-A)* or E”. From this I - Jeave the student to deduce the following: | A.=Pyay Ao Qeir MA 4+ Rear AtA ot &.—P, AQ, A’ Aa Sty | in which P,=2, Q.=2 (2-1) :1.2, RL=@ (a®—1)(a2—4) : [5], &e. This may sometimes be useful, but it has not the symmetrical form required: this form, as before seen, introduces a series of terms, not | containing E7' F”, but fA+A(] +A)7"} F", or (E—E™”) F”. Taking | the series obtained, | E'=P,,,+Q,4,F+&c.-E' {P,+Q,.F+ &c.}. For x write —1, and multiply by E, which gives E’°=E {P,+Q, F+&c.} ia {P,a+ Q.-1 F-+ &ce. } The sum of these expansions gives 2B =P, — Pet (Qei— Qe-1) F4+ &e. + (E—E™) (P+ &e ) Taking the numerators of P,4;—Po1» &e., we find (w7+1)—(a—1)=2 av (2+1)(a+2) —(@—2) (a@—1) r=2.3 as [v-1,0+3]-[¢-3,e+1]=(@—-1)a (x+1){ (w+2)(@+3)—(#-2)(#-3)} —2.5 x? (®—1) | [a—2, 0+ 4]—[#—4,2+2]=[2—2, a+2]{ (#+3)(«+4)—(@—3) (v~—4)}) 2.7 2 (a®—1)(a’—4), &e. | Substitute these results, and divide by 2; then perform upon Ap or a all the operations indicated on both sides of the equation, remembering | as shown at the outset, that F™ A, means 4” {(1-+A)~"Ao}, or AAR or A®"a,,, and that (Z—E™’) F"A, means Ame APC la. 3 an that E"A, means A,. This gives the formula required, namely, ) ) x (v’—1) a? (a®—1) (2° —4) iar a8 A Cee eo. is Puck Pos. Qi 41 F—Q, EE F+ &c. > A,= a+ - Aa,+ Aaa + &G, eink 1 x (a’—1) 1 3 +57 (Aa, + Aa) +5 Dg) | (A®a, +A as) 1 2 (a’®—1)(a*—-4 9 ee (Aéa,-+A’as)+&c. Change # intow+1, and form Avi A,, or MA,. This can easily b. done with the coefficients in the second line, which are, constant excepted, 2, [e—l, x+1], [w—2, 242], &e., A [a—a, t+a]=[e—(a—-1), r+4a+ 1]—[a—a,r+a] =(2a+1) [e—(a—1), x-+-a], (as im p. 256). 9 In the first line, the same coefficients, constants excepted, are mac ON INTERPOLATION AND SUMMATION. 547 by multiplying those of the second line by a Now, Aw being 1, A.wP is P+(v7+1) AP, or Aj«[se—a,c+a]!=[2—a, r+ a|+(24+1)(#+1) [e<—(a—1), e+a] =(4—(a—1), eta] {v—a4+(2a+1)(e+1), or (a+1)(2x+1)} . [w-a,v+a] _ [@—(a—1), e+a] or, ——— Rn a [2a+1] [2a] ; Az [w—a,x+a] bil. [w~—(a—1), r+a](Qx+1) [2a+2] 2 [2a+1] The Substitute these values, and having thus found AA,, write B, for it. We have then A”A,=A”'B,; also let B_,, B_., &c. be denoted by b,, bs, &c. 1 1 Qe4+1) 2 (@+1) —— 9 = / a 3 Bi=5 (20+ 1) Ad, | 5 irr iat Os ih (2x-+1) [a—1,#+2] 2 B64 Beis I v (t+ $5 bot b) +5 AT) (A%b, 44%.) A’ b,+ ce. 1 {a—l, x+2] 5 ogg At be Ath) +&e.: in which it is to be observed that as the former formula was symmetri- cal with respect to values preceding and following Aj, so this one is the same with respect to infervals preceding and following that of 6, and 0,. Thus, up to third differences inclusive, this formula will be found to require the use of b,, b,, b,, or By, and B,, or of one interval on each side of b,, bo. The method by which these formule are bs Ab found is instructive, but they give nothing > aA except the original formula in a different form. Gar B,} A®b, > bs For instance, taking the set of terms written at °AB ; ae ot 2 ellbamana at B, ‘ the side, let the origin be taken at 5, instead of By, whence r=v— 2, if v=0 give the term by. ‘For x write »—2, for b, and b, write b,+-2Ah,-+ Ad, and b,+Ad,, &e. We then have, up to third differences, 1 ‘i 1 (2vu—3)(v—2)(v— 1) : B,=5 (2v—3) (Ab, + d° b)t5 ( ee ey bs l 1 (w—2)(0—-L T 5 (2+ 2Ab, + A%b, +b, + Abs) +5 eo ) (24%), +A%,) »—1 —l v—2 =),+ vAb, +0 ——— A*b, +0 — — —— Athy a8 will be found by actual reduction. In astronomical interpolations, when third differences are used, it is common to proceed as follows. Let p, g, 7, s be the terms, the quantity be interpolated lying between g andr. If v be the value of the variable, p being the origin, we have 2N2 548 DIFFERENTIAL AND INTEGRAL CALCULUS. —] v—l v—2 ATR Terms for the interpolated quantity. This may easily be transformed into v AS : p p+vaptv y—2 v—2 v—3 g-+(o—1) Ag+ (1) “= 8 gt @-l)—- 7 A’P by writing for p, Ap, and A’p, their values g—Ag+A*q— A®p, Ag—A’q+ A’p, and A’g—A*’p. It is usual, however, to write this in the following form: | g+(w—l) Ag+ @—1) —— +(v—1) ; F 2 2 to which it may easily be reduced. This formula may be more con: venient than the preceding when it is required to bisect the interval 0} p and q, in which case v=14, 2v—3=0, and the last term vanishes | But in every other case the second involves more calculation than the first. As an example of its most advantageous application, let us fin the logarithm of 2°15 by means of those of 2°0, 2°1, 2°2, and 2°3| We have then v—2 A’g+A%p (@=2) 20—3 yy, | ey oli, v-1=45 p= *3010300 -0211893 pres al 9= "3222198 9999934 — 0009859 .yoggg7g @-D = yr 'B424297 |) ga5n3 — 70008983 »-2.20-8 il $= ° 3617278 cD aaa | Hence the formula, when A’g alone is used, gives the first line, anc when 3 (A’p-+A’q), the second, 32221934 *0101017 + *0001123+4 *0000055 = *3324388 | * 32221934 °0101017+4 °0001178+0= ‘3324388. | Extensive interpolations may be facilitated by tables, not only of thr values of «, }x(a—1), &c., but also by multiplication tables, in whicl these values are the multipliers. But when an interpolation is ofter wanted, for the same fraction of an interval, it may be better to construc) a formula in terms of the given values themselves than of their differ) ences. Thus the following method, deduced from that in page 542, may be applied. Let c, b, a, A, B, C be values of a function answering to the following values of the variable, m, m+1, m+2, &c.: it is required, using fift] differences inclusive, to interpose four values between a and A answering to m+24, m+22, m+2%, m4+24. For symmetry, let m+ r=m+2: 44y, which amounts to reckoning 4v from the middle value of 4) between those ofa and A. Hence v=2a2—5, and the values at whiel the interpolation is to be made are v= — 4, or —3, or +4, OF +3. Again, if we represent the function required by re+ gb +pat+PA+QB+RC, gq, &c. being functions of v, and the whole a function of v of the fiftl degree, (which is implied when we speak of rejecting all difference after the fifth,) it is obvious that we satisfy one condition by supposint ON INTERPOLATION AND SUMMATION. 549 that when e=0 or v= —5, we have r=1, g=0, p=0, &c.; or all but rare divisible byv+5. Similarly, all but q are divisible by v3, all but p by v+1, all but P by v—], all but Q by v—3, and all but R by v—5. These conditions are satisfied by f R=(v—3)(©—1) (0+ 1) (© +3) (v +5), Q=(v —5)@ —1)(v+ DWF: 3) +5), P =(v—5)(v—8)(v+ 1)(v+3)(v-45) ; p= (v—5) (\—3) (v—1) (V+ 3) (v5), g== (v—5)(v—3)(v—1 Yo+tl)(v+5), r=(v—5)(v—3)(v—1) (v4 1) (v8) ; and each of these must be divided by a coefficient, so that r may become 1 when v=—5, q when v=—3, &c. These coefficients are, then, Meten, 2. 4.6:8.10 For p, — 6.—4.-2. 2. 4 Baca 9.45648 une es gli g. iio .4bp P, —4.—2.2.4. 6 -. 7, —10,—8.—-6.—4,-—2 whence the required function is i (v'—1)(v'—9)(v—5) a (v*— 1) (v—3) (v0? — 25) ; St foS. ADK es 2.2.4.6.8 (v— 1)(»” — 9) (v?— 25) a 4.2.2.4.6 fi (v+1) (v'—9)(v*— 25) A UD O+3) (0? 25) B 6.4.2.2.4 8.6.4.2.2 (e%—1)(v'—9 (45) Ri OLaroraiay an expression which may be thus simplified : (1—»’) (9 — 2”) (25—v’) c C DGS Bie TOage ec lOw. 2.4.6.8.10 | 3 S+v 5—v 38+v 38-v l+v I1—v a form which exhibits the law of the result, and shows us that a change of sign in v is merely equivalent to an interchange of the large and small letters. Hence having calculated the coefficients for v= ++ and v= +4, we have immediately the same for v=—+}andv=—#. The general theorem is as follows: If we take any even number 2n of terms z,y,....a,A.... Y,Z, and if the variable 4v be the independent variable of the function measured from the middle of the middle interval of the terms, and if 1, c,.. ..c, be the coefficients of the development of (1+) up to the first middle term inclusive, the function made by rejecting all differences after the (2n—1)th is (1—v*)(9—2")... .Qn—l?—v’) Da A bite ota (AE 2) Cc, 1. c,A cae ey ahs hath ee Z \ ltv 1l—v 3+v 3-0 2n— l-+v Dip wee bes To find the value of the function answering to the mean of the values Which give a and A, we must make v=0, and this gives 550 DIFFERENTIAL AND INTEGRAL CALCULUS. | 9(A+a)—(B+)) 150 (A+a)—25(B+))4+3(C+9 } 16 ; 256 | according as we stop at third, fifth, &c. differences. | In the preceding process there is nothing which need necessarily confine the values of z to the form m, m+1, m+2, &c., and it may. therefore be made to produce a more general result, though not so. simple. But at the same time another and more elementary method may apply when the values of x are wholly unrelated to each other. Let A, B, C, &c. be the values of a function when «=a, b,c, &c., and) suppose it required to interpolate for intermediate values on the hypothesis that ail differences (made from uniformly increasing values of x) after the fifth are to be neglected. That is, we suppose that within the limits of the observed values, the function may, with- out sensible inaccuracy, take the form L+Maz+Na*+ Px?+-Qzat+ Rzt, Taking six of the observed values, we may then deduce six equations of the form A=L+Ma+Na’+ &c., from which the six quantities. L, M, N, &c. may be determined. This is, in fact, the fundamental method of all interpolation, nor is the common and easy case anything but an indirect method of obtaining the solutions of these equations. To. illustrate this, suppose three values only and second differences, and let the values of x be a, a+1, a+2, so that those of ¢ are 0,1, 2. We have then (the function being L+ Mt+N?’) | A=L, B=L+M+N, C=L+2M-+4N; whence 9M=—=4B-3A—C, 2N=C—2B+A, and A+ _——— — | | | SB rane 4 So eA (B-A) +4 (C—2B+A), a } which is the common formula as far as second differences. This being) the case, it is to be asked whether we cannot, by a similar formula | methodize the solution of the above equations when the values of x d¢ not increase in arithmetical progression. 7 | Let A, or Ay, &c. be the values corresponding to s=a,, or dy, &e. and assume for the required function the form pr=Po+P,(# — a) +P2(4—a) (4— a) + P3(x—a) (a — ,)(@— dg) +&e: This theorem requires the use of an extended method of taking differ, ences, or rather divided differences, as follows: let the symbol 0 operation be 0, ‘Ric _ Aim Ao | 4 OTE a, — & ee _ 9A, —OAo 5 A,—A, ~° a—a, eA. — eA | 6A,=—_— A ,—=—- : | a 6A, —O Az ON INTERPOLATION AND SUMMATION. 951 and so on; the law of relation being 6"A, = (6"'A,,,—6""A,) : (@,1,—,). From these we find A,=Ay+(@,—@) 0@Ay, As=A,+(a,—a,) OA, =A,+ (@i~ a) 9A, (d2— a) { OA) + (a— dy) 6?Ag} = Ay + (de— &) 9Ay + (a2 — 4,) (@g— a) OA, A,=A,+ (a3— a) 9A, + (@s—2)(a3—a,) OA, =Ao+(a:—) OAy+ (4,—4,) {9A + (az—ay) 62Ag} ++ (@s—z) (as—@,) {0,A 9+ (4,— ay) 6° Aot = Apt(dg — A) OAy+ (d3— a) (Ag — 2) 0° Ay+- (3 — ae) (dg—a) (tg— A) 0° Ay and so on; whence we find for all the values of # specified, A, = Ao+(@—ap) OAy+ (%—a)y)(x — &) OA, +(@—-a)(a— ay) (a@—ay) 6° Ao-+ &e. 5 which may be used as an approximation to any value of the function. In observations of a comet, for example, which cannot be made at stated intervals, but must be taken when opportunity offers, this method or some other equivalent must be employed to interpolate, and also to find the required function in a series of powers of w. If the preceding be called M,+ M, «+ M,2?+&c., we have M,= Ayp—a@ BAg+ dy A, 6? Ay — ay & Ay 0? Ay + &e. M,==@Ap—(ao-+ a) 6 Ap (dy G+ G, Ag+ de My) 6 Ayp— M,=6? Ay — (dp +4,+ a.) 67 Ap+ (a) a + &c.) 6* Ay — &e. I leave it to the student to show how these formule are reducible to the common ones, on the supposition that do, a,, &c. are in arithmetical pro- gression. The method is, in fact, an extended method of differences, rendered laborious by the number of symbols which occur. We may simplify it by writing (mn) to stand for @,,—a,, and in actually working the foregoing theorem even the parentheses may be omitted, since there are no numbers with which mn will then be confounded. Thus 21 may represent a,—qa,, and 10 may represent a@,—d. This notation, like that for diff. co., described in page 388, and also that of page 454, is only for the actual process, and the result should be then written at length. Thus, proceeding one step further in the theorem, we find A,=A,+410A,+42.41 @A,+43.42.41 @A, =A,+100A,+ 41 (6A,+206°A,)+42.41 (67A,+300°A,) + 43.42.41 (69A,+406*A,) Ao = (10+ 41) OL Aot+ 4] (20-+-42) 0° As +-42.41 (30443) 6°A,+ 43.42.41. 40 0*A,. But 10+41=40, 20+42=—40, 30+43=—40, A,=A,+ 409A, +40.4167A,+40.41.420°A,-+40.41.42.436A, ; and now, writing a,—a, for 40, &c., we have a new case of the theorem. By this simplification of notation, we may easily give a general proof of the theorem, showing that if 1 be true up to z=a,, it is true for r=a,,,. For if 52 DIFFERENTIAL AND INTEGRAL CALCULUS. A, A,+70.0A,+ 70.71 6?Ap~P&c., then A,,,=A,+(n+1) 10A,+&c. =A, +10.0A,+ (n+1) 1 {0A,+ 2067A,} +(n+1)1.(n+1) 2 {@A,+ &c.} + &e. —Ay+{10+(n+1) 1} OAp+(n+1) 1. {20+ (n+ 1) 2} BAl+ &e. But 10+ (m4+1) l=(m+1) 0, 20+(r+1) 2=(n+1) 0, &e, or A, eho (Qn4:—Ap) 9A,+ (An 41-0) (@n41— 1) OA +&e. Or The divided differences @A), 602A), &c. may be expressed in a manner which will throw some new light on the binomial theorem. For we | find A, ~'A,. Ay cA ats 10 ‘ol A iA eka Ay 2 dagen 20 6A aiet pe (SHY ty et alee 0A T1g,,\0 FOL? © °s,80.01 410.19 Olas LAS Wi p18 top la yale bak a leh IE 10: TORO MOLL AM TO ee 3 A; Ag A, Ao +5 +a + pe ta! SE Lite hi ee ee ae "30.31.32 | 20.21.23 ' 10.12.13 | 01.02.03 » Now, if ag=0, @,2=1, a,=2, &c., then 6"A,=—A*A, 2:3... 7%, ane mn, as here used, means m—n; then, from what we know of the law of the coefficients of A”A,, it appears that the coefficient of a” in the development of (1 —2)” has the form bit RS recs Bet Sen ee PEER BN Foy a ‘caine aie Se —_.,, leavi ut m—m. (m—0)(m—1)(m—2)....(m—n+1)(m—ny’ 5 I now proceed to some practical rules connected with the summation of series, a subject already considered in pages 82 and 311. We shall have to consider separately series in which all the terms are / of one sign, and those in which the signs are alternate. Let the series — for consideration be A,+A,+A,+....+A,+&c., and A,—A,+Ag _ —....3 A, being agiven function of x, and the series being convergent. — It is then to be remembered that A, and all its diff. co. diminish with- out limit as x 1s increased without limit. i When the series is of the first class, and its analytical equivalent not — known, the limit of the sum must be found either by actual summation, | or by transformation of the series into another and more convenient one, _ if possible one of the second class, which is often easier than one | of the first. If, for example, the series have the form },+6,+5,:2 | +6,:(2.3)+&c., we see (page 240) that bs ae “7 + &o= «(b+ Abj+ Ages A? bo +b,+ . +See.) b 2 2.3 and the required transformation is made if the differences of 6 are, or — finaliy become, alternately positive and negative. In the series 1+2™ 4+3-"+&c., we have, calling the limit S, » being >1, and s being 1—2°"4+3-"—&c., n—l1 . Sms4+Q'" S, or eer a ba Ss. ON INTERPOLATION AND SUMMATION. 553 ¢ 1 —n a 7 1 2" ee. i. Also I i zs 3) a5 &c.=S GO icae” 4 aaiee On—l l 3. From page 311, the sum of all the terms up to a, mclusive, or Da,+a,, which call Sa,, is ] : 3 Me—U-+ Ja.0¢-+—a\+-— —= — 8 ag K x Sar WC, ; Ja Ba \ Dye da MBBS 4 Medea poe where, making a little alteration in the notation of page 248, we mean by B,, B,, &c., the numbers of Bernoulli, as follows: B 1 B ] B 1 ] 5 pe a s—F5 a —- 6 od Doh 2 URN RS ie Renae The constant C, which depends on the lower limit of the integral, may be made to represent the sum of the series ad infinitum, by sup- posing that [az dx is made to vanish when w=c: for [Ue dx must be finite when r= cc, if the series be convergent, and we may so take C that it shall then be =0. But a, and all its diff. co. vanish when r= ©; sv that, C being as above, we have only C left on the second side of the equation when x= c, or Sa,=C. ‘This is an important step in the summation of series, since we may now generally reduce infinite summation to the summation of a finite number of terms of the given series, and the approximation to a much more convergent series whose terms are alternately positive and negative : thus C or Sa, =Sa,— fa, da—=a,—= a’ +&e., it being remembered that the series 3a,+4B, a’,—&c. may be of the species discussed in page 226, as will appear in the next chapter. As an example, let it be required to sum 1+27°+372+4-+&c. ad. inf. Let z=10, we have then, taking the reduced series from page 31], observing that Yom dx in its common form vanishes when w= co, S( 0 )=8 107 1 1 2 eae bog a 70 ~ 200 * 12000 ~ 72000000 1-?=1-* 00000000 (10)“'= - 10000000 2—= -25000000 (6000)—'= - 00016667 ak i eet Lg Oo —_--—_— 4*= + *06250000 -10016667 5= ++04000000 Grfe=6 00777778 (200)-'= +00500000 T= °02040816 (3000000)-'=: 00000033 8-*= -01562500 —--—— “== +01234568 —+00500033 16-*= +01000000 + *10016667 S 10°=1°54976773 + °09516634 + °09516634 S (eo )-? 1:64493407 And this answer is correct to the last place, other methods giving 1'6449340668.... To obtain as correct a result by actual summa- 554 DIFFERENTIAL AND INTEGRAL CALCULUS. tion would require at least 10,000 terms of the series. The following | table may either serve for exercise or reference: the meaning of the first © line must be collected from page 312, Let 1+2"4+3"+&c, pene €.0O) ny S(a)™ | -00000 19082 127166 -(00000 09539 620339 | -00000 04769 329868 -00000 02384 505027 | -00000 01192 199260 | °00000 00596 081891 | ‘00000 00298 035036 | ‘00000 00149 015548 | -00000 00074 507118) *00000 00037 253340 ‘00000 00018 626597 | ‘60000 00009 313274 ‘00000 60004 656629 -00000 00002 328312) ‘00000 00001 164155. 90000 00000 582077 ‘00000 00000 291038 S Coc) -57721 56649 015329+log (cc) 64493 40668 482264 -20205 69031 595943 -08232 32337 111382 -03692 77551 433700 01734 30619 844491 00834 92773 819227 -00407 73561 979443 -00200 83928 260822 ‘00099 45751 278180 -00049 41886 041194 “00024 60865 533080 ‘00012 27133 475785 -00006 12481 350587 -00003 05882 363070 -00001 52822 594086 00000 76371 976379 -00000 38172 932650 There is no other general method of any note or utility for the direct ab- breviation of the actual summation: though recourse is frequently had to, transformations, either into a finite algebraical quantity, ora definite. integral, as in the next chapter. If, however, it should be found more! convenient to sum a@)+4,+@,+&c., the sum of a+a,+ &c. may he! found from the formula in page 318, making @=Y, @,=Yr, XC. Then since a,, vanishes when @ is infinite, and also its difierences, we have, making @+4,+ &c. ad. inf.=A, ~~ ~ Oma Tour Wh = Pa ee a el eel ee ee ee — =) pee eee n—\I n?—1 ne—] Opt Aao>————_ A? -+ &e. 5 &e. = nA— —— | aria ets eH Ee 12n “B4n where Ad), A%d, &c. are taken for the series d, Gq, Gens &c. Butit| would rarely happen that this method is preferable to the preceding. _ We now pass to series whose terms are alternately positive and nega-| tive, included under the general form d—d,+a@,—.... The symbolic) representation of this is fJ—(1+A)4+(1+A)*—. « }, do, or (2+ A)™ Glo; or (1+°)7'% (pages 164, 248). Hence | 2 3 4 Aa, Aa, A®a A’a dy — 0, + &0.= Te, Oa Sy +&c. (see also p, 240) Lo (eB Wo. (1) B, “-—(—1) B a Fs —_g.— — — — >) — — aS eee a 5? ve °9.3.4 59°3.4.5.6 The last follows from page 248, by the principles in pages 164, &e., altering the notation of Bernoulli’s numbers as ubove: @, a”, &e standing for the values of the diff. co. of a, when e=0. In using this| last series it would be advisable in most cases to sum a few terms, anc then to make a, the first term not included in the summation. This ON INTERPOLATION AND SUMMATION. 555 series might also be obtained from §172, p. 311, by making y infinite, or from $174, by making a=—1. Previously to using these series, [ set down both the series for ota,+ &e., and a—a,+&c., with reduced coefficients.* Let Ata + &e. = (dot SN a a,) — a,dx —1 a,—P, a’, + BS Fahl — Pe ay &C. Gy — A+ &c.=(Q—.. £a,1) +4a,+Q, a’ .+Q; a, +Q, a: 4 &C. 1 a a he —— 7/2 P, 6 12] Poe)? 720 i , : [4] ; 30 Ps 11 230240 wt : 16 BARE 302 re 4oe [ | P= 1:1209600 39 =~: [8] ‘ 30 5 P,= 1:47900160 59/2 [10] P,, =691:1307674368000=2 2 : [19] Nera e wv j ~~ 94730 ° = ij 7 Py= 1:74724249600 =—:[14] 1 rears ee 2 Wy Q. 1:4 6 Xo [2] Q,== 1:48 wg 1834] A Rm . 30 sa | 1 ee Q,= 1: 480 ony X63 °1.6] Q) ==, 17 : 80640 = Q,= 31:1451520 ==; x 1023: [10] 691 4 Qu= 691:319334400 => x 4095 : [12] aid 7 Q,,= 5461 : 24908083200 =—— x 16383: [14]. Let it be proposed to determine 1—2-°+37*— &e. ad, wf. Let the terms be first summed as far as +97, whence, a, being (v+1)~™, we have Q. a.=—2(24+1)%, w= —[4 (#41, e@=—[6] @t+1)™%, ae. MOU ye da et OS cme ea de. (1-...+9-*)—-510*—2 10 3,10 aT ane | os * Enough are here given, as I suppose, for every purpose; but 1f more be ‘ . P “ aes ri ee s a] ales required, they must be calculated from the numbers of Bernoulli. These, up to B,,, will be found in the Penny Cyclopedia, article Numbers of Bernoullr. DIFFERENTIAL AND INTEGRAL CALCULUS. 1=1°0000 0000 0000 2-2— +2500 0000 0000 3-°= +1111 1111 1111 4= +0625 0000 0000 5= +0400 0000 0000 6°= +0277 7777 7778 H-= +0204 0816 3265 8~"= +0156 2500 0000 9"= ‘0123 4567 9012 | eT — +3559 0277 1778 | 1°1838 6495 3388 +1°1838 6495 3388 — S 977=°8279 6217 5610 10° +2=-+0050 0000 0000 10° X 3+-6= 0005 0000 0000 1OW: X63-+42—° 0000 0015 6000 | 10-" x 5 x 1023 +66 =*0000 0000 0775 | 10-° xX 7X 16383—6=*0000 0060 0019 ~~_———— = —°0055 0015 0794 ES Tac od °8279 6217 5610 107° X 15=+30= *0000 0500 0000 107° x 255—30=:0000 0000 8500 | 10—* x 691 x 4095~2730= 0000 0000 0104 | —— oe +°8279 6718 4214 —°0055 0015 0794 °8224 6703 3420 By the theorem in page 552, n being =2, it appears that 1— 2-°+ &c, | =3 (1+27*+&c.) Halve the eet given for 14+2-°+&c. in the | table, and we have *822467033424, so that the preceding result is | wrong only in the last place. This process is much less ee than that for a +a,+&c., owing to the entrance of the multipliers 3, | EDs Ua, Lee | We shall now try the same series by the formula $a,—tAa,-+ &e. | (page 554). Ifwe first sum the series up to -ta,, the remainder is | then +4a,,:+44a,,,=&c. ‘Taking the series as summed up to | +9-*, we find by taking 10~° and nine following terms, the results here | written: it is not worth while to write down the process. 10°+ 2=:°010000000000— 2=+005000000000 —A10°+> 4=:'001735537190+ 4=-000433884298 ?107°+ 8=:°000415518824— = * 000051939853 —A®107°— 16=-:000122785139— 16= 000007674071 A4107°+ 32=:+000042217188— 32=:+000001319287 —A5107°— 64=-°000016292396— 64=:+000000254569 AS10~°—~ 128= -000006890116— 128==-000000053829 —A7107°— 256=:000003139573— 256= -000000012263 A? 107% 512=°000001522337— 512= -000000002973 — A107 a 1024= *000000778115+1024= * 000090000759 a ee — ° 005495141902 Sum up to9 .. . . .-. +°827962175610 Approximate sum ad infinitum *82246'1033708 ON INTERPOLATION AND SUMMATION. 557 The result is only true to eight places, and involves much more calculation than the preceding, which is true to eleven places: never- theless the second method will be found preferable to the first, when the differences diminish more rapidly than in the preceding instance. Dr. Hutton (Tracts, vol. i. p. 176) gave a remarkable method of exhibiting the results of the preceding process, and added a process by which its power is much increased. If we take the successive sums 0, 4, b—A,, A—a,+a,, &e., and substitute values of a, a,, &c. by means of the differences of ty, We shall find 0, &, —Aad, a+Aa,+A%a,, — 2Aa,—2A*a,— A®a,, &ce. Leave out the symbol a, for brevity, and take a succession of means between each of the consecutive pairs, and repeat the same process, which gives % 20-4), £(01+4%), $0—A—A—AY), &e. It thus appears that the first terms of the several rows are the successive approximations 1 1 J e : 2X5 3A)—tAd,, $Ay—fZAQ+EA Qo &C. If instead of means we take simple sums, neglecting the division by 2, we must divide the several first terms at the end of the process by 2, 4,8, &c., or rather we need only divide the one which is correct enough for the purpose: the following exhibits the process in a more general form. Let the operation 1+A be called E; then the results of the sum- mations give the performance upon a, of the several operations following, 0, JI, I—E, 1—E+E’*, 1—E+E*—FE’, &c.; 1+K 1—EK? 1+E* 1—E Peis ee tors PLB ” and these results are alternately less and greater than (1+ E)~}, the sum of the whole series. Omit the common inverse operation (1-+ E)~', to be replaced at the end of the process; the first, second, third, &c. succession of sums are then, (1+ E being 2++.A), PA SRA, 24 RAL 2 BA, D4 BA Be, 4—A®, 44EA% 4—RB°A2 44E°A% &e. S+A?, S—EA’, 8+E%A’, &c. Consequently, when the rows have been divided by 2, 4, 8, &c., and (1+E)~' is restored, the sth in the rth row is obtained from a by an operation signified by {] + ( ae i eal E:-! Att Q. a By ee or (L4E)-' apt (—1)' 2 EA (1 Ey, or Sens 558 DIFFERENTIAL AND INTEGRAL CALCULUS. The first term of this represents the whole sum in question, and OTA Et (Lt Bi) dpe 2 A CE i Hs nine) a —2-" (A’a,.,.—A'a, + &¢.) If, then, the terms and their differences diminish without limit, we thus approach without limit to the sum of the series, whether by increasing r or s, or both. And the same thing might happen, and be due solely to the diminution of 27”. The results in each row are alternately greater and less than the sum, If the differences A, A®, &c. be all of one sign, then the first terms of the several rows give results alternately greater and less than the whole sum. But if the differences be alternately positive and negative, this is | only the case with oblique columns taken in the other direction; as, for instance, 2+E*A, 4+ E°A’, &c. And the errors of any such oblique — column (the nth, for instance, 2—EA and 4—A’ being the first) depend upon E"A, E”"'A?....A"*?, which by the formula finally depend — on | 2 (Aa,—&e.), 27? (A®a,_,— &e.),. 66.27? (A ay — &.) Now it may happen that these increase or decrease from the begin- ning to the end, or come to a maximum or minimum in the middle. This point can only be tested by the actual operation; the advantage of | this method being that we can always find a set of results which are | alternately greater and less than the truth, and the degree of approxima= tion of these results to each other determines, of course, a quantity greater than the error of either. This method succeeds very well when the series is not too convergent: for it is remarkable, that the easiest series of all to treat by it is one | which has no convergency whatever, cr @)—a,+a—a+é&c. This | follows from the method representing the results of 4a,—14Aa,+ &e,, | which, if a==a@=a, &c., is reduced to 4a. And by means of the | property proved in page 226, it even ascertains, exactly or approximately, the algebraical equivalent of a divergent series: thus Dr. Hutton has _ verified by it the known value of 1—1+1.2—1.2.34+&c. But if a series converge too rapidly, this method will give approximations but slowly. All that has been said will be illustrated by applying it to the series already considered, 1—2-°+3-*—&c. The first column contains the sums 1, 1—27*, 1—2-°+37°, &c.: all the remaining columns | exhibit the sums of the several pairs, in the manner above described, the | Roman numerals which mark the columns being followed by the figures common to every row in the column. Decimal points are omitted. L—1 Il.—3 IIl.—6 1000000000000 ( 750000000000 aD OEOOD OUND 361111111111 611111111111 ‘ , 631944444444 861111111111 Z : 2'70833333333 | : 659722222229 333 | SE TTTTTTTTT 798611111111 296944444444 ‘ | 6317222222222 583611111111 838611111111 ; 286666666666 | ‘ 649444444444 . - | 578185941041 810833333333 SHAC 291519274375 Aitegn 642074829931] 580452097502 831241496598 HQ02108 288932823127 . ; : 646857993196 , E 579369488531 815616496598 643578672208 290436665404 579939688832 is : 9895 934 ' 827962175610 64592435 1220 289503023428 | 817962175610 ON INTERPOLATION AND SUMMATION. IV.—131 V.—263 VI.—526 VII.—1052 99722222992 | _. | | Me ecan| DLELIIIITIL, | nae nna cs 51388888888 | 13185941040 | 64297052151 | gros cogacag 61797052152 | So ussoq9g9, | 33621031735 20435090695 | aogounie 72515747006 58638038543 | Je 459604576 | 38894715271 | Fee a5 43 59821586033 | j9)a9"6330g | 37390387972 | SO*°2 tea 59309177363 | | VITI.—2105 70433830892 49000850249 [X.—4211 19434681141 The differences being alternately positive and negative, the last numbers of the several columns, divided by 2, 4, 8, &c., will give a succession of results alternately greater and less than the truth, and it will be seen that the nearest approximation is in the middle of the set. If we had commenced with 1—2-°+....—10>*, and proceeded with the summations up to 19°, not only would the approximation have been more rapid, but the final termination would have been the most correct result of all. 1645924351220— 3289503023428— 6579939688832 13159309177363— 26319130763396— 32=822472836356 52637590387972— 64—822462349812 105276485 103243— 128= 822472539869 210549000850249-256= 822457034571 421119434681 141~—512=822498895862 2= 822962175610 4= 822375755857 8= 822492461104 16= 822456823585 Of these the fifth is the nearest to the truth. If these results be taken, and used in the same manner as the original sums, a close approximation will sometimes result, particularly when the original series was divergent. No rule, however, can be given as a guide when to expect additional advantage from carrying on the process. As a more simple instance, take 1-1++—...., beginning with the | Sum of six terms, which is *744012, and taking means of the sums to show more clearly the degree of approximation. *744012 | | » 9 5 820935 | (82474 | ogs037 | | ead iS7601 fe 135339 liwopab 154268 "83680 785641 785434 785387 785396 Brsoer ). 3.) 785927 |... |. 785405 ne in eee . 180775 POX ¢ 185375 { 160459 | - 4060 (89522 | < ilelsy fo Rai Sealed The result to six places of decimals is *785398, and the greater rapidity of approximation in this example, as compared with the last, arises from the slower convergency of the series treated. Any given result might be attained by one process, as follows. If § 5,, &c. represent the several sums a, @)—4,, &C., it is easily shown that the (m-+1)th mean of the cth column is DIFFERENTIAL AND INTEGRAL CALCULUS c—l : (satctetntoa te se Sie t Severe ie oh Sm 2 ° Substitute the values of s,,;,, &c., and it will be seen that a,, enters all, Am4, all but the last, &c.: also the sum of all the coefficients is 2°. Let c—l rome C,=2°—1, oat 2 —l—c, C,=2°—1l—c—e 9 5 &C. 5 nd the (m+ 1)th mean of the cth column is {Cy (ao—-Q+ oe ee Pe th) ee C; On4it Ce Qn+9-- eete OU, Ame} —=-Cos 4 = C, Onar— Ca ve euienes = + C, AOm+e os Quct + + <2 Fe toe aay. hrc () eed ne Ane 0 I have confined myself in this chapter to purely arithmetical con- | siderations, but in the next, and also in the one which follows, on 3 definite integrals, the reasons of the marked difference which exists jj between @+a,+&c. and a—a,+&c. will more fully appear. CuaprerR XIX. ON THE TRANSFORMATION OF DIVERGENT DEVELOPMENTS. Tue theory of series is intimately connected with that of definite in- tegrals, insomuch that previously to proceeding with the latter subject, | it may be advisable to resume the former. We have hitherto considered | series, pages 222—244, with reference to the actual arithmetical sum of § an infinite number of terms, and have given, page 326, the test for distin - a guishing between a convergent and divergent algebraical series. And though we have deduced series which are sometimes divergent, it has § been hitherto a matter of trial merely: nor have we attempted to draw | any conclusions by means of divergent series. When, indeed, it happens | that the divergent series is known to arise from development of a given 1 function, we may safely use it, since we have the means of avoiding the | divergency by using Lagrange’s theorem on the limits of Maclaurin’s. | In such case we may use the terms of the diverging series freely, since | those which we neglect might have been from the beginning expressed | in a finite form (page 73). But when it happens that we do not know § the original function from which a diverging series was produced, the | use of such a series has been considered unauthorized by many eminent |) mathematicians, whose opinions should be carefully weighed, whatever [ conclusion may be adopted. | In general, a series of the form a)+@,%+4,2?+ &c. is convergent for | all values of x less than a certain value (page 222), and divergent for } all greater values. And here a, is a function of , which we may call — gn, so that the series is (0) +$(1).2+¢(2).a°+.... Let us | consider ourselves as led to this series by the performance of a number | of operations which obviously lead to terms having the law in question, | though they end in a series which cannot be arithmetically summed: | TRANSFORMATION OF DIVERGENT DEVELOPMENTS. 561 and let us ask whether we might not, by putting the operations in another form, have obtained a convergent series ? In the answer to this question there is a marked difference between the case in which @n may become infinite for a finite value of n, and that in which it cannot. Let us suppose the latter case; the transformation is then rendered very easy by representing the whole series as one of operations performed on a, which gives aota,e+...=f14(14A) 04 vse} op a 1 1 Hae > Uti)? Gaye Gaye oe a iy Tibia ark or as follows, %(0)+0(1).c+6(2).2+.... =—{?(—1).27'+¢ (—2).2°%+¢ (—3) + &e.}. The same result might be obtained by taking the series 0 $(0)+@(1).2-b... ee + Gort. Fe (0) 4 AP (0).2 poet hE HAL? ot....=—y (-1)—y (—2)—.... The most condensed form of entirely algebraical sense, as meaning that the same operations which give w(0)+%(1)+.... would, differently conducted, have led to / term ¢ (m).2” is mn (1— (—1)") 2", and —¥(—1)—¥ (—2)—.... For instance, let us take log Lt 3 = )=2(etS+ ms .) Here the l-—z oy ae —(-1)".log (—1 Lak a when n=0, is ee Sree ) or — log (—1), and ¥(1)+¥(2)+....=—y (0) —y (—1)—...., or 1+2 eee te sed. I log (2S ane Co ae (Z T3 33 Ts aot ve ds which may easily be verified. But if we had taken the general term of the series to be 2(2n-+4+1)—! 2*"t!, we should have —2 (a +to Ft ,, ") for the inverted form, which is not true. But here observe, that in pass- ing from n=0 to n=—1, we pass through a value of n, —%, which 20 562 DIFFERENTIAL AND INTEGRAL CALCULUS, makes the term infinite. As another instance, take tan r=e 8 5 x — = rp mes one form of the general term of which is (n+1)e att T ef eG Vii SA SA which =— 5 When n= — 1, giving + Fogse N ets z 1 dak ERLE Whe aac NG I Te ee ees which holds in one case: for c—t+a°+&c. is that value of tana which lies between —3a7 and +42; in which case tan‘e-+tan™ a” = —4r, if tan—w lie-between 0 and —4z. No great stress is to be laid on these examples, because the method of supplying the function proper to make the even terms vanish, ,as 1—(-1)", &c. is arbitrary, and might be varied: and though I have taken these instances to show that when the proper function is used, true results follow, yet the determination of that proper function is not at present always attainable, nor can a test be supplied for distinguishing it from others. But in the case in which ¢ (7) is always finite, the theorem may be freely used, as showing, without reference to the arithmetical value of a series, a variation of development which might have been given to its algebraic invelopment. For example, let the series be Lat 3a24 Ta®+ 17at+ 4105 +990°+....2 24,2"; of which the law of the coefficients is that A,—=2A,_,+A,_s, whence A,» A,—2A,-, and A_,=A_,,.—2A_n41, giving A= —1, Apo A_,=—'l, and the rest of these coefficients are 17, —41, 99, &c. Hence the same series is Lies amet Gey Se dibiagr wih, wat i i a Now the original series is the development of (1—.2) : (1—2r—a"), and if w=v-, this becomes (v—v”) : (1+ 2v—v*), which developed by common division gives v—3v" + Tv'—&c., which verifies the preceding, As another example, take 1+ x cos 0+ x cos 20+ &c., which, by the theorem = —.2a'cos@—2~? cos 20—&c., which can be verified from page 242. If ¢ (n) be an even function, or if ¢(—n)=9 (n), we obviously have 1 1 y+ ay (2+5)4 a,( 2+ =) 1.0, or dy + 2a, cos 0-+ 2a, cos 20+&c.=0., making «-+a27!=2cos0. Thus if én=1, we have 1+2cos0+2 cos 20 +....=0, a well known result. If (n)=cos n@, we seem to have 1+ 2 cos? 6+2 cos’ 20+ 2 cos? 30+-....=0, a result which will require the following considerations. Divergent series are mostly developments, which though arithmeti- cally false, as presenting infinite arithmetical values for finite functions, — yet present specific cases in which the function actually does become TRANSFORMATION OF DIVERGENT DEVELOPMENTS. 563 infinite as well as the series. Thus, though 14+27+32°+...., or the development of (1—x)-?, is divergent when w>1, the invelopment is not therefore infinite: except only in the isolated case in which r=1, when (l—x)~ and 142+3-4.... agree in arithmetical value. In this case we must guard ourselves from the fallacy of making an arithmetical infinite the subject of reasoning, and must stop at the first step in which it appears. This fallacy, in its broadest form, is as fol- lows: there are many cases in which infinity is equally positive and negative ; that is, dw being = oc, (a+h) is (h being small) great and positive, and @ (a—h) is great and negative. If we then say that c= — o, wehave 2X o=0, a result which isa sufficient caution against the use of cc, that is, infinite in value, in the manner in which rational considerations entitle us to use that which appears infinite in value by divergent or (as those who reject divergent series say) wrong develop- ment. All I assert in the first instance is, that 1-++ cos’@. x-++ cos?20.a2-+ &e. is the development (whether right or wrong matters not here) of a function which may also be developed into — cos? 0..0-!—cos? 20.4-2—... Now the first series may be easily shown to arise from the develop- ment of bags: 5! cos 26.27—2? L+5 5 21—xr 2 1—2cos20.2+2 1 l i cos 20. 7 '—] bien So " 2 (ear 2 «"—2 cos 20.27!+1 Develope the second form in negative powers of wv, and we have 1—L(l+ta%4+ar74+.... )—4 (1+ 0s 20.2 + cos 40,.x27-+....) r — cos? 0,277'— cos? 20.4 — &e. 3 is asserted. In the particular case r=1, the original function becomes nfinite; consequently, though we may say that whenever we meet with 1+ cos?6+...., we might by a different process have obtained —cos* 6 — cos’ 20— &c., yet we may not say 1+2cos?0+2 cos? 20 ....=0, for by so doing we really commit the fallacy “oc—=— oc, herefore o-+ oc=0.’? But the student must not imagine that it is any oint connected with series that I have cautioned him upon: for the ‘ame care should equally be taken with finite expressions, as to these articular cases in which they become infinite, The real difficulty is, hat in using a general divergent series, and passing to a particular case, ve may light upon a divergent series which really represents infinity, nd we cannot as readily know whether this be the case or not as we ould if we had only finite expressions. If a, or ¢ (7) be an odd function, or if @ (—n)=—¢ (n), we readily btain (since then a,—=0 or oc, and by hypothesis we are not speaking of he latter case) a,(v—x-') +a, (#—2-*),.-+-....c=0; or 4,sin 0+ 28in20+....=0. And if E, andO, represent an even and an odd metion, and if (remembering that ‘every function is the sum of n even and odd function, if 0 be included among functions) we make n—E,,+0,, we have pa; (@+2) +45 (2? +a) +..= 0, (2+) +0, (2 +0) +... IT a; (@— 2) +, (a? — 0") + ,. = E+E, (2-07) +E, (a*- 2) +,., 202 364 DIFFERENTIAL AND INTEGRAL CALCULUS. This sets in’the clearest point of view the remark in page 327, that it ‘5 not allowable to make two series of the form Da, (a" +2") identical because they are derived from the same function. The two forms of $(0)+¢(1).a+.... cannot generally be both couvergent, though both may be divergent. ‘T’o prove this, let ~ (9) +y(1)+.... and —y%(-1)-Y(—2)—.... be the two forms. The convergency or divergency depends in the first instance upon the values of —n (log yn)! and —n {log w (—n)}', when n is infinite. These are —nw/n: wn and ny/(—n):¥(—x), which have different signs whenever w/n: wn and y'(—n) :y(—n) have the same limit as 2 increases without limit. This is the case whenever Yn is an algebraical function of 7, or one multiplied by a”; and since convergency requires that the function here treated should not be less than +-1, this necessary (though not sufficient) condition cannot be true for both forms, in any of the cases specified. But it is possible that both may be divergent: for instance, in 1+447+99a?+...., and its other development —a™ —4'~—.... But extreme divergence in one form is frequently attended by as great convergence in the other; for instance, in 1+2'2 +3° e+ o.,.,and —ax'— 2277-37 et. ww Since we have @—@,%+a,22—....=@_12 —O_¢ i now see the confirmation of a fact which every algebraist observes, namely, that in every series the terms of which follow a law expressible by common methods, and in which the terms are alternately positive and negative, the function so developed diminishes without limit when 2 increases without limit. This will yet more fully appear in the next chapter. a a3 When a series has the form a@+a@, 2+ a, i rare + ...0, Wher a_, can be assigned, the present theorem fails from our not being able to assign the value of the function from which 2.3....2 1s derived, m the case in which m is negative. It will, however, appear in the next chapter that these inverse values are not finite. In algebraical series, the values of a@,, a, &c. being those of diff. co. generally contain J, 1.2, &c., in the numerators. But in several remarkable cases the theorem will not apply, owing to our ignorance of the method of inversion, in the development of (1+.)" for instance. There are, however, cases i which we may invert the process and infer negative values by means of independent developments. Thus, 2 being a whole number, | i ie a ; wees (1) fr" nat wees | (l—2)"=1+n2r-+n hence, a, being the coefficient of x” in the first series, we may infer that | a_,—=0, d9220,..-G-ay=0, d_,=—(-—1)", a. 1=—(—1)"-0, eae I leave the following to the student: | ata cta,e+...=a_,(1-z)'+Aa_, (1-2) *+A%a_s (l-x)*. 0: In most of the cases in which the general term of the series is of the — form a, 2": (1.2.3....n), the denominator insures a high degree of | convergency. To examine this point, remember that (page 293) 1.2.3....nhas always a finite ratio to n"*? e-", as nm increases without » limit, so that (page 234) we need only examine the convergency of the TRANSFORMATION OF DIVERGENT DEVELOPMENTS. 565 ; ; 1 : series whose general term is a,a"e":n""?, Let this be wn, and we have + ee l ad A vit wn a SE 2 The only case in which this series can be divergent is that in which —na',:a, is —c when n=, in such manner that the limit of the first two terms is at least as small as +3. If, for instance, a,—n", which is a function increasing faster than 1.2.3... .n, we have for the preceding 4—n log (xe), whence the series 2 8 “3 x Xv a 1+74+2?—+43'— 44+ 2 2.93 2.3.4 is convergent whenever w is <7. The following methods will often convert a divergent series into a convergent one. Let pr=a,+a, +a, Fie oer and let ab) +a, d, t+ a,b, x?+ vane be the series in question: then, as in page 240, this series is obtained by a train of operations on b,, of which the symbol is (tE).6,,; where E stands for1+A. Assume E=m-+F, which gives p!'(mx) «x? gai Now E=m-+F means Eb,=mb,+F,, or Pb,=b,4:—mb,, which gives F6,=b,—mb,, F2b,=b,—2mb,+m'b,, &c.; Teese G,bo+a, bat... = (mr) bo +0'(me).2Fb)+ F2b,+... the process obviously being an extension of the method of differences, by substitution of the operations b,—7b,, b,—mb,, &c. for b,—bp, b.—b,,&c. We thus get Wats: (b6,—mb,) x © (b,—2mb,+m?*b,) « ett l1—mxz (l—mr)? (l1—mz) * 2 “2 b,+6, r+, 5+ Ao ee {but (b,—mb,) x+(b,—2mb,+m? bo) =f ete \ in which m may be any finite quantity, positive or negative. Let m= —1 in the first, and we have b b +b.) x bo + 2b,4+56,) x? Ae eae 0 oy Gurk Oe bic leuC l+a2 (1+ 2)? (1+ 2) If b,, 6, &c. be increasing, this series is convergent whenever b,+ 26,v+4b,v°+.... is convergent, v bemgar:(1+27). If5,4,:5, =k when n=, this last is convergent whenever v <(2k)™, or m<1:(2k—1). If 2k= or <1, the second side is convergent for all positive values of x. ’ If instead of E we write €”, by the theorem in page 307, we have e®ete bi! ay by + abye-f. = $v. bj +9 {a(1+A)}.0.0,+9{2 FA} OSH, where b’,, b”,, &c. are written for Db,, D%b,, &c. This, expanded, the table in page 253 being used, gives 566 DIFFERENTIAL AND INTEGRAL CALCULUS, na A, bya, ba... =b, gab p'r.2+ nae thy ah ty (¢'x.0+ 3o"x.a?+o!"'n. 2°) (ya. x+o"2. 2") iv 2.3.4 a result which might easily be verified from page 239 by help of page -- (g'a.a+iglc. 2+ 602.024 "a .a*) +o... 5 263. The remnant @,b,@°+ani,b4:0"" +.... may often be ren dered more convergent by use of this form of development. This chapter may serve to throw some light on the character of divergent series. Further considerations will offer themselves in the next chapter, previous to which it is hardly right to invite the atten- tion of the student to any final opinion upon the use of divergent series. This much, however, may here be said: the history of algebra shows us that nothing is more unsound than the rejection of any method which naturally arises, on account of one or more apparently valid cases in which such method leads to erroneous results. Such cases should indeed teach caution, but not rejection: if the latter had been preferred to the former, negative quantities, and still more their square roots, would have been an effectual bar to the progress of algebra, which would have been confined to that universal arithmetic of which Newton wished it to bear the name: and those immense fields of analysis over which even the rejectors of divergent series now range without fear, would have been not so much as discovered, much less cultivated and settled. CHAPTER XX. ~ ON DEFINITE INTEGRALS. ¥ In commencing with a title which may induce the student to think that he is already master of the principles on which the following pages rest, a conclusion which would not be altogether correct, it will be necessary to pomt out the extension of views with which the subject must be looked at, before the objects of the present chapter can become intel- ligible. The subject of definite integrals becomes daily of more import- ance: and, to judge from appearances, any very decided increase of the power of the mathematical sciences can only arise from successful m- vestigation of the methods of obtaining their general properties, and computing their numerical values. A definite integral is distinguished from an indefinite one by the sup- position that both its limits are specified; and the consequence is, that the former is no longer a function of the variable, but only of the limits and of such constants as enter into the function integrated previous to im- tegration. If, therefore, all indefinite integration could be successfully performed, all definite integration would necessarily follow. Thus when we know that 2v is the diff. co. of x*, we therefore know that {> 2rdz | is b°—a”, whatever 6 and @ may be. But we know that indefinite integration cannot always be performed; and, as in pages 103—109, ON DEFINITE INTEGRALS. 567 (which the student should here review attentively,) we may see that the lifficulty arises from a deficiency of means of expression. To carry on he same mode of illustration, remember that geometrical recollections ntroduced the circle and its properties into algebra before the differential salculus was invented. As algebra was applied to trigonometry, the sine, cosine, &c. of the latter science were made fundamental modes of ‘xxpression in the former. The consequence was, that at last a broad listinction was drawn between the two series 1—$a°+4412'—-&c.,, r—yzt2°+&c., and all others. The student finds, on his first intro- luction to these series, that he is already master of their properties by the qundred, is provided with tables to find their numerical values, and snows how to make them of continual use. But if he had been com- pelled to be a pure algebraist, without permission to draw suggestions rom any other science, he would have had no more occasion to investi- zate the properties of these series than those of many others of equal simplicity. And on the other hand, if the suggestions of geometry had seen more extensive,* he might have been familiar with many results which are now to be presented for the first time, and might have had sommon and well-known names for results of calculation which are now mly expressed by symbols, and have no distinct appellatives. In reometry, the previous treatment of the curve y=,/(a*—a®) made f./(@—a?) dx expressible in known functions as soon as fadx: had the same science directed attention to, and been made the means of leveloping the properties of, the curve y=¢—2", the integral fe—«” dr, 0 the consideration of which we shall come, would ‘perhaps have been dready known, named, and tabulated. If all the cases of f ov dx were written down, when $e stands for a ‘unction in common use, the greater number of these integrals would be nexpressible, except by infinite series. If all infinite series were con- vergent, the difficulty of computation would, not be insuperable; and if she,general properties of an infinite series, for which no finite equivalent s known, were as easily determined as those of a finite expression, we might satisfy the wants of any application of our science with compara- ‘ive ease, though the labour of computation might be considerable. But tis not always readily practicable to reduce integrals to convergent series, md it frequently happens that the form of a series does not throw any ight upon its properties. Atthesame time, nothing is more certain than what the results of most of the problems in which the higher mathematics ave necessary, must from their nature require integration. Do we then ind in what precedes premises requiring a conclusion that most, or at east many, of such problems must remain insoluble ? This question is to be answered in the negative, and the reason is as follows. Every particular case of an integral can be found by common irithmetic, whatever the function may be. It may easily be that ft ox dx may not be expressible in terms of a and 4, with such modes of 2Xpression as we now have; but specify the values of a and 4, say 42 and b=3, and by the definition of the symbol the equation Sion dn=- ‘9 (2)+¢ (247) +9(2+=) bist. +0(242)| * If the hyperbola had received as much attention as the circle, its area might have suggested the notion and properties of logarithms, and the attention thereby excited might have led to the calculation of tables, 568 DIFFERENTIAL AND INTEGRAL CALCULUS. may be made as nearly true as we please, by taking m sufficiently great, This symbol, then, for an isolated and specified value of a and 8, is merely the limit of a simple arithmetical conception, and every case of it may he calculated, quam proximé, by a person who knows only how to calculate the value of an algebraical expression in any particular case. The more artificial and rapid method of page 314 may be substituted : and it must be observed that in calling every definite integration practi- cable, we speak of possibility only. Should the actual computation of an integral occupy twenty computers for a year, it might well be a question (and one by no means always to be answered in the negative)whether it were worth while to employ them: but this does not affect my asser- tion. It is, then, admitted to be possible in every case to construct a table of the values of an integral which may be used like a table of logarithms, so that reference and interpolation shall give any value we please, with sufficient accuracy. ach integral so calculated is a fundamental table of reference, and the question is to choose such integrals as will admit of the largest number of uses, and to find out as many uses as possible for those which have heen calculated; previously using the shortest and most convenient method in the actual construction of the table. So much for the numerical attainment of results which can only be exhibited in an integral form: but this is by no means the only use of definite integrals. It frequently happens that one particular set of limits have an importance which distinguishes them from all others, and renders the case in which they are used perhaps the only one which it is ofany usetoexamine. Thus, in the theory of probabilities, f2"(1—ay* dr is of the most frequent occurrence, but only between the limits c=0 and xt=1, and also between limits which le near the value of 2 which makes 2"(1—«#)" a maximum: it would be only wasting time, so far _ as the most important cases which occur in that science are concerned, to examine any other limits. In such a case, we learn to look upon, the variable z, the most prominent symbol in the ordinary integration, as subordinate in importance to m and n; the first being necessary only in the conception of the manner of attaining a result which depends for its magnitude only upon m and n. It frequently also happens that the isolated cases which it is most important to examine are also those which can be most easily attained; and that we may thus arrive at a particular value of a function, the general form of which must be presumed to be an inexpressible transcendental. This happens, for example, in f$¢—°dt, which, when a is infinite, is 4/2, (page 294); but cannot be finitely expressed in terms of a. Another important branch of the calculus of definite integrals is, then, the determinatiom of useful isolated cases of general integral forms, of the complete solution of which no hope can be given. Again, an integral of the form f d (x, a) dx, between specified limits, whether those limits be functions of a or not, is, generally speaking, a function of a, and of the limiting values of z. If these limits be numerically specified, (say they are z=0 and =1,) fj ¢ (a, a) dvisa function of a. Say that this integral can be found, and that it is wa. We have then a mode of expressing wa, which may lead us to proper- ties of that function which would not othewise suggest themselves. There may be an infinite number of ways in which wa may be thrown into the form of a definite integral; and each of them may be the easiest ON DEFINITE INTEGRALS, 569 mode of expression for some one particular purpose, or for the develop- ment of some one particular property. Lastly, by looking at a definite integral as the mode of using a variable x, between given limits, to obtain an expression for a function of a, we may not only learn new properties of this function of a, but may even extend our views beyond what would be possible when the function retains its usual form. Thus, if 1.2.3....2 be considered as a function of m, we can form no rational idea of its existence except when nm is a whole number; but when we come to observe that 1, 1.2, 1.2.3, &c. are values of fie x"e-* dx answering to n=0,n=1, n=2, &c., we see no difficulty either in the conception or calculation of this integral when n is a fraction, and we have thus the means of interpolating values: between 1, 1.2, &c. answering to fractional values of n. The mode of obtaining a definite integral supposes that in {4*" da dz, px must not become infinite between a=a and x=a+A: not that the value of the integral is then necessarily infinite, but that we haye no obvious means of testing whether it be so or not. The diminution of w (page 99) may-more than compensate any increase of the terms of the sum, ‘To the criterion for determining the result in this case we first turn our attention: say that 5 is the value of x, intermediate between a and a+h, at which ¢@x becomes infinite ; it is required to ascertain the conditions under which, in pare px dx, or ip px da+ fit" dx dx, each of the two portions is finite. Since figx dx=$b.b—ga.a— ft xg/z dr=ga(b—a) +b (¢b—¢a) - firg'edaz =a (b—a) +b ib gx da— ft rpc dr= pa (b—a)+ f? (b—x) fx da, whenever $d and a are finite, this last result is true when b is any quan- tity (however little) less than that which makes $d infinite ; and supposing b to increase towards that value, it always remains true, and (page 22) is therefore true when w=) makes ox infinite; da, 6, and a being supposed finite. Let y=@x give t=G"'y; then, since y=¢a and y= © correspond to r=a and z=), we have fi prdxr=pa (b—a)+ foi (b— by) dy. Now (page 325) the last integral is found to be finite or infinite, precisely in the manner which determines whether the series whose general term is b—@~'y is convergent or divergent; that is to say, let wy =b—P"y, and find / —ly? Poe yy EASON, and dy, its value when y= &: asin. de Be according as a >1, or <1, the integral is finite or infinite. But when a)=1, find a, the limit of log y.(Po—1) when y= oc, and the integral is finite or infinite, according asad, > or <1. But when a,=1, find a, the limit of log log y.(Pi—1), &c. This seems to involve the necessity of inverting Gx, but it does not so in reality, for y= by gives y'=9' (Py). (P"')Y'y-y’, or (P")'YH1: P'2 ; whence Py»=¢2: d/x (b—2), and ay is its limit when v=0, If it be z=a which makes $2 infinite, the same result applies, substi- 570 DIFFERENTIAL AND INTEGRAL CALCULUS. tuting a for b, since fidx d= —fidrdzr, and —fpxdx and jdrdr are finite or infinite together. . Example 1. f'(logrdx:2). Here c=0 makes Or= cc, and seagate} log x Pe ——— 2 =], when r==0 (doubtful g@a(0—2) logv—-l ”’ ; ( ) log x Plog ( whence, since —1<1, this integral is infinite. This may easily be verified, since the indefinite integral is 4 (log x)’. Example 2. ‘F So far, then, the result is doubtful, and this case is more easily solved by inversion. We have (27 tan « dr= foydy: (1+y’), y being tan 2, which falls under another rule. For the preceding rule does not apply when b= cc. It is obvious that fs dy dy is infinite if dy be finite when —cc. But here y:(1+y*?)=0 when y=cc, so that the rule to be applied is that which determines whether 2y: (1 + y*) is convergent or divergent. Here /, py | P= - Ree og log a—log # ) (—D= =-—1 when «=0; x 1 EOF rea Nie tan @ = ol. (1+tan?)(4r—z) ° T fan 202s ke ° whence the required integral is infinite. Example 3. iv e~* x” dz, n being positive. Po=(¢-+n)™, q=1im This integral is then finite when x<1, and infinite when >1. In the doubtful case, or when a1, we have Pi lor te ea) faq} nd (x-+log x) 9 Levene or the integral is infinite, Example 4. {j¢~*dardx. Here the method of Ex. 2 also applies, and P,=2(a—¢'x: px). The integral is therefore finite whenever 'x: px diminishes without limit, or tends to any finite limit Pcosvdx. Consequently, at the final limit, or when P is a con- stant, we must write {> Pcoszdxr=0, or f;coscdr=0. Again, since {Psinadxr=—Pcosz +f P’cosada, we have fo P sin 2 dx = (P) “+ {3 P’ cosa dx, (P) being the value of P when 2=0. By the same reasoning, any supposition which diminishes P’ without limit brings the truth nearer to fee sinadx=(P). This is, then, the final limit when P is made constant, or P=(P); and it gives fo sin carl. he instance, (a being positive, ) sin ©—acos x Ve ™ cont a0=28 Se), cose Ot a 1+a? 1+@¢@ : cosr+tasin x : 1 fe-® sine dr=—e-* 5 foe sin x dr=-——.. l+a l+a? For every positive value of a, however small, these equations are arithmetically true, and might be verified to any extent by actual sum- mation: when a=0, they become 0 and 1, and ¢“* is reduced to a constant and =1. It may diminish any regret which the ambitious student may feel at being desired to lay aside, for the present, all idea of considering definite integrals in which the subject of integration becomes infinite between the limits, if we show explicitly that even those considerations on which we propose to enter necessarily require the algebra of discontinuous functions; and that those which we throw aside would probably intro- duce the same sort of difficulty in a more complicated form. Let fo sin az dx be proposed, which it should seem must be a function of a, and the more so, since it changes sign with a, on account of sin (az) =—sin(—azr): and when a=0 it is obviously reduced to C—C or 0: that is, it changes sign, passing through 0, when a changes sign. Nor is it one of the excluded integrals; for sinax:7 is finite when w=0, being then =a. But [ed oe oie sin ax sin ax sin v "Ik senor d(ar)== | —d, a ae heh: since writing v for ax does not alter the limits. The last result must be independent of a, so that we have a constant, not a function of a, which as 0 when a=0, and changes sign with a. Unless, then, this integral be always =0, itis a discontinuous constant. But it is not always =0, is will be afterwards shown. It must then be a discontinuous constant; and thus, even in such definite integrals as we do consider, we cannot always procure general algebraical expressions of the results. Our sole restriction being that in f@x dx, px must not become infinite between the limits, unless we can show, as in page 570, that the result is arithmetically finite, we are at liberty to substitute for w any function whatsoever which does not invade this restriction. Thus even if the function substituted should be impossible in form, the truth of the results is not affected, For example, take f tan~"@d0 from 0=0 to6= 7,” being positive. Here ¢9= cc when 0=0, and we therefore examine a oo ON DEFINITE INTEGRALS. 573 ¢0—~'0 (0—@), or the more convenient form {(log #6)! (0 — 6)}-". This gives “ 1-++- tan? 0 nQ@ ————. tan 0 al ) , which =~", when 0=0: so that the integral is finite when x is less than unity; this we must therefore suppose, the case of n=1 being left doubtful, as unnecessary for our present purpose (it gives the integral infinite). For tan~"@ write its value 63 74 ]— eV 2 Sse} 46 — te Smet which is, say ==(—1) 2(A,+A, Hosa aS a Yaa or 8 rela: BA ih 3 Vo and —1 being «*‘~', we find for integration (A, enbe VA A e~Grnt2e) V1 A Pn GrnH4eya/— 1 vee) dO; of which the impossible part must vanish by itself, since the required integral must be possible and finite. The possible part is { Ay cos (44n) + A, cos (47+ 20) + Ay cos (472+ 46)+....} do. Now fcos (c-+2k0) d0, from 96=0 to 0=4}n, is (2k)~' {sin (c+hr) —sin,c}: whence this integral vanishes when & is even, and becomes —k~*sin c when nis odd. This gives for the integral required the series im A, cos ($7n) — sin (4n7) (Act a +4 Vita. \ We might, however, obtain a finite result,* as follows. We have £ > “J > (—1)2 f tan“ 0 do= f(A. +A, + Agen V4...) de, and (—1)? is cos$ rn+,/(—1).sin}rn. Now integrate, and equate the possible parts on both sides to each other: the possible parts on the second side being all of the form A, fcos 2k0.d0, must vanish when taken from x=0 to r=47z, and we find (A, being =1, as appears from the function to be developed) 1 die 1 27 Pe wis cos sen f tan 6 dd=4r, or | tan-” 0 do =—= Mea ae. ; ‘ --——= COS 47 A further examination (or simple substitution of 47-4 for 0) will show that this integral is true for negative values of m also (if between 0 and —1). Let tan?6=2”, m being a positive integer, which gives rl. 2 ae Let 3m (1—n)—l=r, or $4 (1—n)=a7(r+1):m, and cos (41n) =sin(r(r+1):m). Hence m >(l—n}-1 Ls dx dr tan-" 0 do=— ( (n>—1<+1). ey a] 1+ 2” cos 477 * For this proof, which is much shorter than the one usually given, I am indebted to a writer who signs S,S, in the Cambridge Mathematical Journal, (vol. i, p. 17.) DIFFERENTIAL AND INTEGRAL CALCULUS. 2 adr . (r+l us He par ae {xin )s| Nepalis It will, however, soon be observed that there is a liability to fallacy in an incautious use of the preceding method. If, having deduced A+B /(—1)=P4+Q,/(—1), we infer A=B, P=Q, we are justi- fied only when we know A, B, P, and Q to be real. Now if either of these quantities be an infinite series, and divergent, it may represent an impossible quantity, as does x+4a°+.... whena>1. And even if we have a series which is real before integration, it may become im- possible after it; thus 1+a+a°+.... is real when e>l, while its integral, beginning at 2 =O, represents an impossible quantity. We shall, therefore, add the common proof of this result, which, though employing impossible quantities, does in a manner free from the pre- ceding objection. If we denote the » roots of the equation 2"+1=0 by a, A, &c., we find, as in page 276, (n L ta” ne | elm tt 8__e-(mthe sin} (m+1) #: nh i rdx T or eo ae aa Se er a kee ae nsin < (m+ 1)} a result of great importance. If we examine the limits within which it is true, we find that, as far as the lower limit 0 is concerned, m must be >—1, while, for the higher limit, m must be <2—1. The preceding, though it employs impossible quantities, 1s yet pre- cisely the same in its processes as the longer method which would be followed if x” (1-+-2")~' were integrated from the rational form found in page 276, § 89, by collecting the impossible factors of the preceding process in pairs. It would seem as if hitherto I had given nothing fet cautions, and this I have purposely done to impress upon the student the idea of the very slippery character of the subject ; or, which is the same thing, of the very imperfect manner in which it is understood. Some further hints of this kind will still be necessary. Every integral of the form 1¢ px dx may be thus expressed : fo dx dx= fi bu dat fii ghadx+ [2 gourdat+....ad infinitum ; Go, Qi, dz, &C. being a series increasing without limit. Every such integral, then, is really an infinite series, of which it is found that the divergent case is net so well understood as that of ordinary divergent series. Let us divide series into four classes, simple divergent and convergent series, in which all the terms are positive, and alternately divergent or convergent series, In which the terms are alternately positive and negative. Besides these we have the intermediate series, of which the terms are or become of the form a+a+a-+.... and @—at+a—.... When the above infinite series of integrals is of the simple diver- gent kind, we have rejected the consideration of {jx dx as being infinite ; t though it might penny be asked why such a diverging Series of integrals should be called infinite, when a diverging series of simple terms is only called at most a wrong development of a finite quantity. About converging series of either kind there is no question ; while diverging alternating series will be readily admitted, even by those who reject them, to stand upon a different footing from simple diverging series. But having thus pointed out that integrals taken from 0 to @ must have a general resemblance to series in their properties, or at least a similar classification, I now show that there is decided danger of error in any attempt to apply these conclusions to series in general, which are demonstrated in algebra to be true of series of powers of the same variable. ee 576 DIFFERENTIAL AND INTEGRAL CALCULUS. For example, take {cos xdx=0 from «=0 tow=c. We see (page 572) from what this springs; if we write bx for x, which does not alter the limits, we have b {cos bxr.dx=0, or feos bx .dxr=0. Now itis a funda- mental property of any integral, that if the limits remain the same, d dP — G @ [ios ——. ea eevee ig . re fPdp {i dp (page 197) CP) provided always that dP : dq does not become infinite between the limits, in which case the second side of the equation may not be within our present conventional boundary. This proposition is easily proved, independently of the page referred to: for since (returning to the definition in page 99) i 2b ep > a Ap, for any number of terms, © the limiting proposition must be true, or (P) must be true, Take, then, 1 cosbr.dx=0, and differentiate twice with respect to 6, which gives —fcosbr.atdx=0, or fcosbr.a%dx=0. We may readily find, as in page 572, that c d= ate aa ey "OO ox ; ees, 2S re E> cos br dx Se foe “cos ba. a°dr= 7h (x 5) which verifies the preceding when c=0 Also, if we differentiate twice » with respect to c, we have a conclusion of the same kind, verifiable | in the same manner. Differentiate again twice, and so on, which gives {> .cos ba.a°*"dr=0, | by making c=0. Various other methods coincide in the same result; surely, then, we should say fo cos bx (1 —a’-+a*—....) dx=0, orf “cos ba 9 (+2? This result is, nevertheless, not true, and we may see that we have here made an assertion which need not necessarily be true, in saying that f cos bx dx+ [cos br.adx+....==0, because each of its terms is so. If | each of the terms [¢cosbxr.dx, [cos bx.2°dz, &c. diminish without limit when a increases without limit, it by no means follows that their sum ad infinitum does the same. If we assume this in the case of © a+bxr+ca-+....; it is because we never have to use such a series, unless as the development of a function; and this function may always — have (as in page 73) all the terms after a given term expressed in a finite form, from which it easily follows that the series is comminuent — with z. But if it ever should happen that we find a series such as — a+bx+.... always divergent, no matter how small 2 may be, and not having any assignable mode of invelopment, I then say that we have no right whatever to assume that such a series is comminuent with 2. To prove the preceding assertion, assume an 2 Ey 2 } p= f cosbrdz d’P__ ra cos bv.w sane pie bode Pee! 0 | ier akg whence P=Ce?+-C, €~. me Ae al 3 Now C=0, for otherwise this integral, which is always finite, being | necessarily not greater than if 9 (dv: (1+.°)), or 47, would increase ee _ON DEFINITE INTEGRALS. 577 without limit with b. And C, must be the value of the integral when b=0, or 37. Hence are deduced the following results, being the above and what arises from differentiation with respect to b, * (“cosbadx ey “sinbs.adxr i = =~¢ ———__— =~ ¢-?, Gea, TRS i? 0 1+a* 2 If we suppose the sign of 8 to change, cos (br) remains the same, and the integral, while its equivalent becomes dre**, The result is evidently not allowable, since it would be then C,, which is =0, and C which is =4Jr, Consequently, this integral is represented by 4s~? when bd is positive, and: by ve’ when 6 is negative. Similar circum- stances frequently occur, and they arise from the difference of treatment of series and definite integrals. If we had rejected divergent series, we should have called x+xr+a22+... -(w>1), a mistake which is to be corrected by writing -1—2-'—xz~"—... Both series have the proper- ties of e(1—2x)-', An extended theory of definite integrals will, I con- fidently expect, at some future time contain the same distinction : ex- hibiting results in a form which points out numerical values when they exist, and algebraical equivalents when the numerical values are infinite : though I admit that there are some circumstances which appear to create a marked distinction between integrals and series. Many definite integrals of the form fe ov dv from v=0 to v== o have received particular attention. The most celebrated of all is fe’ v*dv, which, being 1.2.3....x when 2 is a whole number, supplies an expres- sion which is intelligible and calculable when x is a fraction ; andis the same extension of the notion of 1.2.3....2, which a fractional expo- nent is of that of a whole one. This function fev" dv is generally denoted by I (x+1), or | cy ae v*“'dy. This last integral is finite (page 570) whenever x is >0, andI (v+1)=aYIvz is a functional equa- tion which its values satisfy. For Ter vdr= — eve fen vw! dv, which, taken from 0 to cc, gives I'(x-+1)=2I 2, since v® e-’ vanishes at both limits. And it is perfectly possible that this equation may be true of fractional values, or any other of the same kind. Thus if oe stand for x terms of the series 1" -+4+2-"+ .... +27", we have before us a function which, when z is a whole number, satisfies @(x+1)=—¢x +(2%+1)~, and as to which the mode of derivation entirely fails when @ is not a whole number. Nevertheless, there may be a continuous function which satisfies the above equation for all values of x. Thus Pr=14+24+3+.... +2 gives (e+ 1) = r+ (v+1), and the deriva- tion is unintelligible when 2 is a fraction; but pxr=hx (x+1) satisfies the equation for fractional and even negative and impossible values of x. Let us now take ¢x from FO+SA)+£(2)+ .... $f (x), which satisfies @ (r+ 1)=¢r+f(@+1): required, if possible, the expansion of dx in powers of x. Let wr=f(@+1)4+f(a+2)+.... ad inf. when @ is a whole number, and let wx in all cases satisfy Wae—W (2+ 1) =f(*#+1). Then wr-+ pry (@7+1)46 (241) or Yx+¢x is con- stant. Now when z is a whole number, %r+ zis obviously the sum of the series f(0)-+-f(1)+.... ad inf., say => ; whence in all cases ¥a-+-per==>. We have then dr= 2—Ya, or T—Y (0) —W’ (0) .2- Ke. But since yx=f™(x+1) +f (@+2) + eveery we have wo 2P 578 DIFFERENTIAL AND INTEGRAL CALCULUS. =fO(1)+4+fPQ)+.--- This equation is not derived from differentia- | ting with respect to va function in which 2 is a whole number only, | but as follows: since wa in all cases satisfies Yr—¥Y (a+ HN=f (t+), we have wx —WO(e + l=HfO (at 1), or yOr=fO@ED tS @+2)4+¥° @+2) =f (@E IF fO (@t2) FSO +3) FH OT 3) + KE 5 & wOr=fO(r+1)+ eroorad inf. + ( oc ) (as in page VIS and all the series being supposed convergent, we have wrC or )=O0. Hence if f(1) +&c.= 2%, we have 2 pr S—TO—E..2—L© > PC ae i 2 & Observe, that it matters nothing if 2 be divergent, provided >, &e. be convergent, since [— 2 is simply > (0). To apply this, consider F (1+2)=aPx; we have then log Y (1+) =loga+logIz, but since both 2 logx and S27? are divergent, differ- entiate both sides, and let ¢a be the diff. co. of log Ix or Ia -Pa4 Required the development of 6 (1+.2) in powers of «, having ¢ (1 +2) =z +or. Let wo= (+l) *+(a+2) 0 +--+6, oO we—w (e+l) =(e+2)—(a+1), and $ (v@+1)+Yax=const. ; whence ¢™(xv+1) = we, Now vr=(a+1)'+¥ @+)) gives ; J iae > th i DS ow it ae eee : | oe toate (a+) = Gal + Gyan (v7 +2), or (0) :2.3.... n= 49-43-44 .,.,), which call S41 | d(atl=¢ (1) +8, 2—8, 2 +8, 0—.. +, a series which converges when v1. We can thus calculate log I'(v7+1), and thence loglr, which is log T(a+-1) —log x. The former function, which, since '(1)=1, vanishes when x=0, is what may be called the general function of log 1] +log 2+ +++. + log x, being the function of which log 1, log 1+ log 2, log 1+ log 2 +log 3, &c. “are the values when a=0, 1, 2, &c. We proceed to some properties of the function I (a+ 1), the general function of 1.2.3...2. Turing back to page 388, we see that J {ov.yw.dv.dw, if the limits of each variable be independent of the other, is fov dv X J ww dw. Hence P(e+1)xT(y+1) = {oe udu x Joe wedw= [> [* ey? w’ dv dw. If we assume w=tv, we may perform this integration by first integrating with respect*to » from 0 to x , and then with respect to ¢, also from 0 to ©. For, to change v and w into v, and vt, we have v=», w=, t, and dv dw dv dw alee ne =v,, wh Sf e~-1'y *+9EY » dy. dt ae de ai ing 1xv,—0xt=v,, whence ae i 1 QV, is the integral above given; while 0 and o are limiting values of v, and t, answering to those of v and w. Now, integrating first with respect to v,, we have 9 hy e—niltyetytigy =P (v+y+ 2) F (pater? Since fe-@ v* dv=a-*" fe v* da, feo bd 2P2 rf 580 DIFFERENTIAL AND INTEGRAL CALCULUS. and 0 and ce are the limits both of v, and », (1+4). Multiplying by | W#dé, and integrating, the original form compared with the transposed expression gives ev Aina ee . | . {{Ga) (ar) a+os | Let s=t (1+4)7, which gives 0 and 1 for the limits of z, and we have finally ; dt \ | U(#@+1).Cyt)=C (w+ y+2)| ar) aa i : m - if 2 (1-2) dz= AGE ae y gh) ; i (wt+y+2) which requires, to be finite, that a and 4 should both be >—1. Thus i Es Y an extensive class of integrals is made to depend on the general factorial function, as I'v is called. [f r=—y, which requires y and 2 to be numerically <1, we have, P (2) being 1X I'(1), or 1, P(l+2).0 (l—z)=f} 2? (1—z)* dz. Again, let t+y=— 1, or, for y and a, write —+-+2 and —}—a, 4 | which gives . Latte meee ee 4 git? (1 — 2) 4 dz es) tS (4—2). This integral admits of being found; for if z=sin?@, it is reduced. (page 573) to aft tan” @ dQ or r:cos (rx); Which may also be written thus, by writing 3-2 for 2, ~ THRU Dee -— @S0-<1). sin @5 Let oe}, then P (4)=,/7, a result found in page 294, though in a very different form. In the integral [{*2-“ dé, let “=v, which does not alter the limits if n be positive We have then : . ] RP 1 1 it 7 fi e—i" dt=— fg em? yn dv=— 1% G)er (+ 1) (n>0) ji foe-P di=sP (4)=4,/7, as in page 294. Returning to the series in page 579, we have log i A+9)=—-yt+% S,P—1S8, a+45,r°—...- 4 log f (1-27) = yi thS.x° +35, P41 S atte 5 ; but PU +a).0 GQ—az)j=are. Pa —z)= 7x: sin tx, whence log aes =S, 2448, c'+2S, a4... log F (1-+2)=$ log wr—Llogsin rr— yx —FS,0°—4$5,V— see Now (8 +2). F@-—9Y=EG+2EG—2) i (4+z).FG-2) i eubais (ake =U ) cos Te” : ON DEFINITE INTEGRALS. 581 and we can thus calculate Pd+$+2 ), or oe ee zis >} by means of I" (1+4$—z), or \(1+2r) where r A, A — A; A =—_— Awkt = » sind” Pay silt: sin 30 ° * 8” sin 40 T A; A==—— Al csinlaniiDG Lekite ta AA =r) DA, AA =Qn)H2A, A, AA =20.3! A, nine equations between eleven quantities; so that all can be determined by means of two only. It might appear at first as if we might carry the main theorem one step further, and form an equation (A, A, A, Ay, A,); but if we do so, we should find that the new equation is really contained in the others. The importance of this function Fa can hardly be over-estimated, and the progress of the mathematical sciences will probably render its use as frequent as that of its particular case 1.2.3....(¢—1) has been hitherto. Legendre has given a table of the values of com. log [' (1+-2) for every thousand part of a unit from v=0 tov=1. This is all that is necessary, if the table be carried to a sufficient number of figures; for Te= (#—1) P(@-1)=(-2)(a- 1) F(w-2), &c., which can be continued until I'(2—n) falls between 1 and 2; whence Iz can be found from I'(a—n). Again, Pr=a7'P (1+), which gives Tx when z is less than unity. The table presently given is an abridgment of Legendre’s, and the last column will enable any one to reconstruct as much more of the original as he wants. The value of I'v, considered as fer v*— dv, is finite as long as r>0, but infinite for v= or <0. But if I'v be considered as a solution of $(x+1)=-2x¢z, it does not become infinite when 2 is negative, except when z is a whole number. Thus J=1(1)=0.TO=—0 (—1).F(-—1)=0.(—1)(—2).T (—2), &e. ; whence (0). (—1), &c. must be infinite. But x beng >0<1, Te=(2 —1) F (a@—1)=(a—1) (a@—2) I (@—2) = (#—1)(@—2)(«—3) F(@--3), &e.: 584 DIFFERENTIAL ANDZINTEGRAL CALCULUS. so that T (w—1l), I (a~—2), &c. are not infinite. It must be remem- bered that many of the properties of Ia have been derived from the equation, not from the integral; and negative values given to 2, and used in the series for log I'(1+-2) give results perfectly coinciding with the formule just given. This point requires further examination. Tx, the integral, satisfies 6(a+1)—axpa, and so does ér.Ta, é2 being any function which satisfies £(r7+1)==a2; for instance, v may —cos2rz. The series for log (x-+1) was derived entirely from the equation; how then do we know that this series represents Dv, and not cos 2ra.I'x, or any other solution of the equation ? We should answer this, if we remember that the condition Fa I’ —) —-7 : sin ra is derived from the integral alone, if we could show, 1. That no other solution of the equation will satisfy this condition; 2. That the series obtained does satisfy this condition. If possible, let Ex. I'v satisfy the condition; then since I'x also satis- fies it, we have ér& (1 —x2)=1, an equation which can only be satisfied by the form P*—", wkere P is a symmetrical function of « and 1-2, ora function of 7+1—a and of x(1—2), or of «(1—z) simply; so that changing # into 1—z does not alter P, and changes 2x—1 into 1—2x. Let log P= (a—2*); then since ée=é (1+1), we have — (22-1) 6 (@—a*)= (2441) ¢(—a—2"). Change the signs, and both sides become integrable, giving Pd, (w—2*) —¢,(—2—2"), which, if it can be solved, determines ¢,?, and thence @x, and thence (2r7—1)@(ae—2"), or log.P*—'. The calculus of functions does not give any reason for supposing that this equation cannot be solved, though no solution has been attained; and therefore, so far as we have yet gone, we fail in showing that the series for a is that particular solution of ¢ («+1) = vpz, which Legendre and others have assumed it to be. There are plenty of solutions which coincide with f e-*y* dv, when x is a whole number, but not when @ is a fraction, For example, 1+cos’? 2rx 2+sin? 2rx ferv da, (l—cos 2na+cos?* 27x) fer?v'' da, &e.; any one of which may, for anything to the contrary shown in the method quoted from Legendre, be the function whose values have been tabulated for those of fe~? vu" dv. By the following method, however, I find that the series for log Y (1+) may be deduced entirely from the integral, without any reference to the equation @(a+1l)=apr. Take (a+ ef er” v’ dv, (the limits 0 and oc always understood,) and remember that v” is the limit to which (1 —e~%”)*: a® approaches when a is diminished without limit. If, then, we find fer (1—e~”)* dv, and then divide by a”, and diminish a@ with- out limit, we see I'(v-+1) in the limit attained. Let ¢~’=y, which changes the limits to 0 and 1, giving (page 580) “t = T(w+1).P ae © 1 —av\2r 1 0 x saad a 436 (1—s-") du=— f?a-y) (—y*,. dy)= —— he at( 24741) a Let 1: a=6b, whence (@+1)=F (a@+1).T06': PF (@+4+)) is an ON DEFINITE INTEGRALS. 585 equation which approaches without limit to truth as b is increased without limit; or [.b°t': Y(a@+6+1) has the limit unity. If, then, 6 be a whole number, we have (t+) @+b—1I)(@+b—2).. -@tEHNT +) ed bom Liebman) (1. ae i ee or log [ (1+2)=2 log b—log (1+.7) —log ease —log s(14 4p) eee continued ad nape Use the logarithmic series, and we have 1 1 | 2 log I (12) =( log 6-1-5 — ee 15) et 9 (1+ Stake a3) b l 1 a ast = ie Time ey gts tee provided 6 be increased without limit. This gives (y being as in page ~h 578) has the limit unity : log i} (1 +x)=—yr +4 Se r—t Se a+ eooee Ad before. We also find, when d is considerable, the means of calculating ap- proximately (7+1)(v+2)....(~@+0) for all values of x from eg a by means of pre +1) a mL C . (t+1) @+2)....(04+50= PG@dd) arly It will be convenient here to introduce some theorems by w hich the preceding results will be confirmed. It is required to expand ¢7-+¢-* and ¢—¢~* into products of an infinite number of factors. Let w=7:n, and it is known that 2n—1 a" + a= {2° —2axr.cos [4] 4+ af ea | ease. vr v | xr" —a°"= {2°—2Qax [cos w]+a"} [20]. [8v]....[n—1l.0] x (v’—a’) ; where by [$w], [3w], &c. we mean the repetition of the first factor with §w, §w, &c. instead of $0, &c. For x write 14+ 4:2”, and for a write l—xr:2n, and we easily find (+8)-2(048) (8) ee (8) =2 (1—cos 0) Wa a) remembering that (1+4cos6):(1—cos 6)=cot’?$ 0. For n write +: w, and we readily obtain oh dt Bate P : O=aw gives —-- =— 5; where P,=(!aw)’: (tan $aw)?. An” a1 2 ‘ oS PY iv , ve And for #2— «a write ( 1+— }—{ 1—— }, or 2—- 2n 2n n Substitution gives 586 DIFFERENTIAL AND INTEGRAL CALCULUS. 2n—1 2” }1—cos [io]\BolEe].. se » | x ria mr ( LA a 2 we) \"8)" beth CBC) eel po wb 3° || 5%] °° _J@n-)2] ‘n which one factor of each set is written down, and the part which is altered in the other factors being in brackets, the alterations necessary to make the other factors are adjoined, also in brackets. This notation, with which I do not feel quite satisfied, is here used merely to show how much some such notation is wanted. We have also Dy queen agi se fol} el Bel eae (wa) Coa) “sh es] HIE) Ls : 2n—1 Let v=0 in the first; 2=2"{1—cos [3]} [$e]... E o |. Divide the second by 2, and make 7=—0, which gives 1 lea 2"! $1 —cos [w]} [20] [80]... .[r—1.o]. Substitute, which makes the first and second become Ag* Mere P P P 2{ hel pe =| a] aS eee | 3 5? (2n-1)° a” P, Ps P3 2h Prost 2 —|— — — |....| = |I- aber Fal | Ea : mata Increase 2, and diminish w, without limit, and equate the limits of | equal quantities which gives an infinite number of factors in both pro- | ducts, and the results, restoring the common notation, are as follows : serial 3h) (aa2 lore AO tae a 3) pee) 252° 49n2)* °° ‘ag oo a La a ey —E =20(14 5) (145) (14g tiga) For x write 2,/(—1), and we deduce 42° 4x* 42° Aa? E COs G— (2 =) (1-35) G -55) ( -ipe) coe : x a x x” sinv=o(1-) (2 -7) € - 5) (a -=,). eee 5 results which can be easily proved by the theory of equations, provided | it be first shown that sina and cosa have no impossible roots, to intro-_ duce other factors. This can be readily shown, for if sma had an im- | possible root, <” -e~* would have either a possible root, (which, except - x==0, it cannot have,) or an impossible root of the form a+6 V(-). which it cannot have, @ and b being finite. I know of no results better ON DEFINITE INTEGRALS, 587 calculated to establish confidence in widely extended chains of algebrai- cal deduction than these formule, which can be verified to any extent by actual calculation. Take the logarithms of both sides, and expand by the common logarithmic series, which readily gives (s, being 14+37-"45-” ies ; ee Find log cos r= — 2? s, —— 2* 5, — — 2's — 2% 5,——.,. © 7 * Or § 376 8 AB z 2 nt § 8 L x x x x lo =— IJ=S,, +5.-——+8 +5 Fesee 8 sin @ a Qrt* 8 Br © 8478 Write zz for x in the second series, which then agrees with that in page 580, deduced from log f' (1+): compare the first with page 253. The values of Cv are found from the following table : a, CommonlogF(1+a). A(—). 1, and the terms of the second (which gives the integral in finite terms when n is a whole number) become alternately positive and negative if n be fractional ; so that, if ¢ be great enough, the principle in page 226 may be applied. One pine tl of reducing the latter integral to a continued fraction is as ollows. Assume te ey" du=ze-* v" V, Ib ey" dv=VT (14-n)—e-’ v'" V. Differentiation gives e~? v"=s —ne ve" 1 V +E ° vo" V—e Pu" Vi, or vV'=(v—n) V—v. . Consider the equation vV'=(v—a,) V—v+4,V%, divide by V%, anc ; make 1: V=1+h, V,:v, which gives V,-V V ; i ag Tes . M=@-a) (14h “)—v+0,(14h 2): qy2 i om or vV,=(+a,4+1) v= oth, Vi. 1 Let ki =b,—a,, b,=h,, dga=—(a +1), and we have ON DEFINITE INTEGRALS. 59] VV =(v—ay) V, ~v + by V3, an equation resembling the preceding, in which if we make 1: V, =1+42V_:v, we shall get another equation of the same form by making k2= by — de, bs=k, dg3= —(a,4-1). Go on in this way, and it is obvious that we have Vv 1 1 kv} if? Rope sheer key ey Ree oe ees ee ee ee sy using a recognised notation for the continued fraction; that which follows each + in any denominator being printed as a factor, to save room. ‘To determine the law of kh, ke, &e., remember that a;=n, b,=0, whence we have 1 2 + 4 5 6 7 8 &e. a nm —(mt+l1) n —(m+1) n —(4+1) » —(m+)) &e. b 0 —n 1 1—n 2, 2—n S 3—n &c. k| —n 1 l—n 2 2—n a 3—-n 4 &e. 9 ae wie bso J yar nv-' vw (l—n) v7 or. f.6 wv dv=e-"v s Stieber twee aa 1+ 1+ &e. which converges rapidly when »v is large. I leave the following* to the student : rel 2q 3 4 Ee a ih gh ya Cnc Pease Qa T+ 1+ 1+ 14+ 14+&c. 2a? = =} Oy-? Ay By By i ef? log » dv=log v +— er eeak SLY Chel Raia en TE RE eS ee ee es . Before proceeding further, I touch upon the general question which the consideration of Px has raised, namely, that of the interpolation of form, as, according to the suggestion in page 543, it might be called. When any process is constructed by successive operations, 2 in number, the result is a function of ; that is, depends for its value on n, and changes value with n. Nevertheless, this function is not imaginable when z is fractional, for there is no such thing as going through a process more than 7 times, and fewer than n+1 times. Students, how- ever, are apt to confound going through a process with a fraction, and going through a fraction of a process: and many figures of speech favour the misunderstanding. Thus it would not bea violent use of language to speak of multiplication by 10 as being the operation of multiplication by 4 performed twice and a half; whereas three multiplications are performed, two of them using 4, and the third 3 of 4; this third multi- plication is not the less a multiplication because its multiplier is one half of preceding ones ; just as a house is not the less a house because it has only half the size of another. * The values of the first of these integrals, though all important in the theory of probabilities, are of little use for general purposes, They will be found (reprinted from Kramp) in my article on that subject in the Encyclopedia Metropolitana, 592 DIFFERENTIAL AND INTEGRAL CALCULUS. Let there be a function of x which is 1 when a=1, 1+2 when e=2, 1-+2+3 when v=3, and so on: what is it when w==33? Here av means, when x is a whole number, the number of terms in a series 5 we have no right whatever to say.that 1--2 +343} is the value of the fanction when «=34, for the additional term is not the less a term because we make it 34 instead of 4. There is not then any direct mode of deciding upon the value of dx when a is a fraction, because oxrzl1t24+34+..6-4F2 when z is a whole number. If, however, we write $r(a+1) for 14+243+....+2, we sce that the new form is intelligible when @ is a fraction. The question now js, how far we are justified in asserting that pr=}x(a+1) must be true when & isa fraction, because it is true when z is an integer ? The sole condition necessary to determine Oris (w+ 1)=¢r+(e+1), nor even this universally, but only when wis integer. If, then, yr and yx be two functions the first of which is always unity and the second zero, Whenever & is integer, we have px=te («+1).yr+Pxz, : where P may be any function whatsoever, provided that Pyx and ya vanish together. or instance, dx (a+1).cos2ra+P sin 27x satisfies every condition. Nor is this the most general form, for the following will do equally well : draf (4a (a+), x), provided that f(z,7)=* when w is a whole number. For instance, pum f{har (a1) }*.ynrt Px, where Y%, is of the same kind as yx above described. Again, if Pr =1—24+3—4+.... +2 when z is a whole number, we have for one solution Pr=1 {1—(2x-+1) cos rx} =z, and for a general solution y=/f (z, 7), where f(z, v)=2z when « is integer. The general problem of interpolation of form 1s therefore doubly indefinite, every solution involving two distinct sorts of arbitrary functions. The ends of mathematical analysis are best answered by selecting from among this mass of interpolated forms certain of them for particular consideration. ‘The first limitation is made by requiring that the form selected shall not only satisfy the functional equation when x is a whole number, but also when @ is a fraction. This reduces the two arbitrary functions to one: thus, in the first example, taking 6 @+ l=) @& +ar+1, and assuming drv=—$r (e+1)+¥7, substitution gives ¥ («+ 1) =we as the sole condition for determining ya. The most general answer which the present state of algebra will allow of is wr=/f (cos 272), where fx is any function of which the operations do not require the inversion of cos 27x; any function, in fact, which remains periodic as long as its subject is periodic. It seems, then, that every solution of such an equation as @ (@+1) —dxr+ar, av being a given function of 2, may be separated into two terms, one not generally periodic, the finding of which is the only difficulty, and the other periodic, its period being a unit: the latter may, without hurting the solution, be changed into any other of the same laind. This non-periodic part of the solution is some= times treated as if it were the only solution; that is to say, all series OF developments derived from the equation are considered as equivalent forms of the non-periodic solution, which may or may not be the case. ON DEFINITE 1NTEGRALS. 593 Let this non-periodic solution be called the principal solution. It must, however, be remembered that this principal solution altered by any con- Stant does not cease to be a principal solution ; so that nothing but the accession of the variable and periodic term can deprive it of that character. If then P and Q can be shown independently to be principal solutions of @ (x+1)=—¢r+az, we may not affirm that P=Q, but that P=Q+C, where C is a constant. The function « (1)+a(2)+....+¢@ (c—1), is Z.ax (page 82) Which may be considered as the general representation of the function which, when 2 is a whole number, and then only, represents the sum of the series above given: it is a principal solution of the equation O(e+1)=¢r+ax; and we consider Se as a common functional symbol. Itis then easily shown that (Sa)/v is a principal solution of ? (t+1)=¢x+e'r, and so on. Having shown then that [v= ¢~ v*' dv is aprincipal solution of & (x+1)=axgz, we now know that og ['v is a principal solution of (v+1)=¢2+ log x, and must there- fore be the general form of > log (x). Similarly, log Tx being written Ar, we find that A’x is the general form of S2~, — Ax of Sa~, SA’ @ of Xz, and generally (—1)"*! (En). A@a of 2a~", -n being any positive whole number. Let us now consider S27 independently. It is easily proved by expansion and integration, that (x beimg a whole number) «—1 ee 2 dv, i 174274354... .4@—-1lt= : 0 1 =—0 and the integral is intelligible when is fractional. ‘This integral is a principal solution of ¢(#+1)=¢xr+a, and sois Ir: Ir or A'a, whence we have %} l —4* A’ (1+ epos ( i e 0 dv+C. seer) To determine C, make «0, which gives, by the series in page 580, ai)= — y, and the integral obviously becomes nothing, whence we have 1 st Ata) = [ Bae dv—vy (y ='5772156649....); > l—v which affords a ready mode of finding the last mentioned integral, since A‘A+.) can be found from the table by means of the differences ; it being remembered, however, that as the logarithms of the table are tommon ones, the result must be divided by the modulus *43429545... Integrate the last equation with respect to # from e=0, "Lt ioa l—v 1 A (1+.2) =| -- ——— ——. dvu—yx. 0 l—v ema. log v Make t=1, and A (1 +2)=log 1 (2)=0, whence ah eae ] — +; } ae, “f fies log v ‘form frequently used. I leave the following to the student : ; 1 a—l 1 a-—l «-2 SS oaiadeas ia Yao ar a 3 2Q 594 DIFFERENTIAL AND INTEGRAL CALCULUS. Since A/(x) is a principal solution of d(a+1)=bar+a™, it follows that —A"x, AMe:2, —A™@: 2.3, &c. are principal solutions of | o(x+1)=or+a™ for n==2, n=3, n=4, &c. But 2.0” is a prin- cipal solution. of this equation ; whence we find the general function =z" by the equation A Sat (—1)"*' ————__ + C. ots 33 ole Write 1+a for xv, and for A(1+2) write its value —yr+4 58,2 —18, 2°+&c., which gives | 1 5 (1 $2) P=C—S,4 08.120 Sepa. (MP1) Make r==0, then since 2 1=—0, we have C=S,, or — li 1 n+2 D142 =H= 08a 20 _ Sno C?-+N — is S,.49.0° i os am 8 iy. ag aR hig rN Let v=1, which gives, 22 being |, nt+1. , = n+1 n+2 2 Sn+2 LRN a eeiehiae eee J=n8,.1—-7 But 2-"=1—n+ In (n+1)—&e., whence n-P 1 2 (Sipe) +... 2" =n (Svan These last two equations may be verified in various ways. From the: integral form for A’(1+ a) we also obtain ' yy log vdv “ L (v* (log vy? de’ y(1+2)7°= s+ | — 2(1 say 7=8.—5 | vi Gee a 0 — and so on. The series for A’(1+2), &c. may be verified in a par, ticular way, as follows. Let Sax be the general form of the function which when z is a whole number becomes a (2)--a (e+1)+4 &e. ad infinitum, This functior is then a principal solution of ¥ (a+1)=yr—e2; again, Loa beim a solution of ¢ («+l)=¢r+ea, we find that (a +)4+¥ @+l —phxe-+yr has Sux+Saxr for one of its principal solutions. But this equation being of the form é (a+ 1)=éa, can have no principal solutior except a constant, all its variable solutions being periodic, We have then Sex-+ Sar=C, and C may be readily determined when a (0)+ a(1)+.... is convergent, by making a any whole number ; in whicl case Daxr+Sax becomes a(0)+a(1)+--.. ad infinitum: so that representing this series by Sa (0), we have | Sax+ Sax=Sa (0). | But when the series is not convergent, still 2a and Sax may be finit functions: thus when ara, Sa(0) may be the constant y (pag 5418) which occupies the place of 1 +h4i4+.... ad infinitum, and look like a sort of algebraical equivalent of it. This point may be furthe clucidated as follows. Let us take | ON DEFINITE INTEGRALS. 59 1 1 +-—5 + Reels Be eg The arithmetical value of the second side is unquestionably infinite, whatever the value of x may be. Now let 2 be less than unity, and expand each of the terms in powers of x, we have then SC +2)7s14+4414... o—S,0+S8, v—.... The first term of which is infinite, but all the others finite ; and even (if <1) forming a convergent series. Now since S2— altered by any constant is still a solution of y% (#+ 1)=Yyx—a-", and since the value of that constant is altogether immaterial, strike off the constant i+}+...., and it appears that —S,v+S,22—.... is alsoa solution, whence an S (l+2)7= se » 2 (l+2)"—S,27+58,2°— @eers a Cy And since ¥ 1-'=0, we find by making w=0 that C=0, or > d+2)7"=S,7-S, at Sy v—, Bee If, however, we choose A’(1+2) for the principal solution of ? (e+1)=¢2+ 27, we have A'(1+-2)= Za —y (page 593), whence we get A’ (1+2)+y—S,2+ S307—... -=0, Al (1+27)+8 (l+a2)"=—y; In which, if the distinction between principal solutions differing by a constant be forgotten, we might imagine* we see > (1+2)7+S(1+2)7 =—y; that is, —y in the place of 1-S+5-+. ne, Let it now be required to generalize the function a+bx @ atb a+26 a+b (t—1) p+q p p+q p+2q. **** p+q(@—l)’ Supposed to vanish with x. This is obviously the integral of av?! + (a+b) vt to .. + {a+b (a—1)} vPt?@Y-1 from y=0 to v=. This last series being summed, gives for the function ysp—l 959% BO i de ele af ae q(a+1) af ye (] —v! tb | vi—av™ + (a—1) v peaiity, 3 0 —_— NC ae 4 1—v? (1—v*)? For v! write v, which, g being positive, does not alter the limits, and we have, writing @ for 1: q, 1 ao [ Leg yP dy + be 0 ve l—v Ty — ay + (2—1) vt? 0 (Lx)? The multiplier of dv in the second integral is easily found to be pr’X diff. co. of (v—v’):(1—v); integrate by parts, taking the in- grated term between the limits, and we have vy! dv: & * I think Legendre has very obviously fallen into this misconception ( Fonctions Eliiptiques, vol. ii. p- 429), but it has led him to no false results. Indeed it is bvious that confounding ‘A may be written for B’ with ‘A is equal to BY though ‘must affect the logic, may not affect the result, of a process A 2 596 DIFFERENTIAL AND INTEGRAL CALCULUS. 1l—v } x Diet) She pI—l — 2 SET ee os) . LO (x +00 f vt dy bpot | ye! dv If we consider S27 as a known function, we have . (ey ay Last 1 1—1" I Lines ys 1 : 1 : ly ——— dy— y= Ss —-_ - > ——. | >» L—v % yes i i ig oe n+l eet Apply this, and the preceding becomes 1 1 “ 1 1 b9(¢+—1 dain ye Se ag £8 7 at yng ree weep 2) ey SEY” @ )+a0( S55 pe P (= po+ax 2) For ¥ (p@+1)~ write its value = (p0)7 + (p89), which gives finally 5 Ute +qt 1 Ly —=}0x +0 (a—); of, pipes h: b0% +0 (a—bp6) 2 pore py as (There is a remark which it is here essential to make, to prevent the — student from transforming expressions of the form Sac, generally con-— sidered, in the same manner which he would have done when they stood for no more than simple summations. If we consider &.pzx and p 22, we see that both mean the same thing if 2. pe merely stand for p.1+p.2+ ..eetp(a—1). In this case 2 ‘s the index of the extent of summa- | tion, and p a multiplier in each term. But if S.pzx be a case of 2a, and if px be a whole number, the symbol means 14+2+.... +(pr—1), which is altogether a different thing. We might easily make them dis- tinct either by appending the index of the extent of summation to the symbol >, which would make >, pr=p2.% evidently true, and _ Ye pL=p dav evidently false, or else by putting the index of summation | in parentheses. Thus, x and a being whole numbers, 1 1 , 1 1 jig st pares ie < (a+) mepuarn © ase | | Ee l 1 ‘ . = 5 — which then gives 2 a+ (a) “(a+x) 2 (a) Both methods have inconveniences ; a third is to use a specific: symbol for each form of , as we have done in making A (14+) or log (1+2) the representative of the eeneralized function of log] +log2+....+log (e—1). Thus Abel uses La to signify the func tion which, when 2 is a whole number, becomes 17*-+-27'+.- +(v-1)>. We have, however, 2 symbol for this function in A’r—y.) If in the last equation we make a=1, 6=0, we have sitions ae AY (E42) —— a (£ : p+q(t) Gg WI oe a S| When wa satisies $@w+l=—ortar, it is obvious that [yr dr satisfies @ (w+ l)=gdat fan de. Consequently, multiplying by q, and integrating, we have Y (og pr1@}+oes(Lte)—an toa PaO ON DEFINITE INTEGRALS, 997 in which the second side is corrected for the supposition that the value of the general function may become C, when z=0. It may here appear difficult to see why the constant is retained on the first side, while the second is corrected; but remember that the last equation was not obtained from the preceding by integration only ; but that there are two distinct introductions of arbitrary constants. If we satisfy d(x+1) =dx+ax,then fr xdz, thatis, , c+C,, satisfies ’(t#+l=—dr+o,274+€. Here C, may be determined without reference to C; for it disappears entirely when ¥,x-+ C, is substituted for gx, while C remains dependent upon the manner in which Wr and gr were obtained. Now, remem- bering that ZC in its most general form is C (w—1), the preceding gives for the function which becomes p (p+q)++-(p+q(a—1)) when % is a whole number, the value gore, -- (CHE ae D (piqte2)_ U(piqte) aR: 2) D(p:@) For since the first is to be p when v1, and P(p+q) when #=2, Be ea A lotus Cone basa iin. € E(Lai=p@tgso C, og 08 73 whence C,4+-C=0, crab —log gq, from which the asserted result is easily obtained. This conclusion might apparently have been obtained more easily, as follows. Let x be a whole number, then BES ay iy Ap ald Spa P(p+q)...-{ptq(e—l}=q my (Bat). (Ce 1) van ge EL PEGHE2) rt ['(p:q) © Why not then assume that the second side, which is always intelligible When 2 is fractional, is the function which gives the first side when 4sa whole number? With our present knowledge of the function I’, ‘and applying the doctrine of principal solutions to the equation oo) (t+1)=¢2r+log (p+ qr), I doubt if there would lie any solid ob- jection against such a proceeding ; but I prefer, in the first instance, the actual deduction of a definite integral which represents the function required when x is a whole number, for I think the habit of making the passage from whole to fractional values a purely arbitrary process is likely to lead the beginner to do it when he should not. DUR 2 The most striking use of the interpolation of form is in its appli- cation to the symbol of differentiation. I do not intend to go fully into this unsettled subject, but only to supply some general considera- tions which may be useful to the student of this work in reading the dis- cussions which have been written on the subject. Let an equation d(n+1,2)=D¢(n, x) exist, where D means the operation of differentiation with respect to w, and let the equation be true for all values of n. For instance, 598 DIFFERENTIAL AND INTEGRAL CALCULUS, ' as db (n, *) =a", p(n, c)=Fn. (—1)"2 ”.,. @ (7, 2) Coe (2 +7 5) | Let 2 be a whole number; then ¢(1,x) = Dd (0,2), (2,2) ~ — Dd, z) = DA @, «), and so on; whence (x being integer) b(n, 2) =D"d (0, 2), or P(N, x) is nothing but the nth diff. co. of & (0, x) with respect to 2. Are we then to infer that it would be proper to define the solution of @ (n+1,2)=D¢ (n, 2) to be for all values of x, the differential coefficient of (0,2); are we, for instance, to take 1 1 st 4 at D' e*=n'e", D ‘cos o=cos( en =) &c. ? On the answer to this question there has been some difference of opinion, such as we have seen might arise if different solutions of the same functional equation were represented by one symbol. Let «, (n, 2), a(n, 2), &c. be solutions of o (m+ 1,z)=De(, 2), and let £n, én, &¢. be periodic functions satisfying 6 (++ 1)=én, and — vanishing when 2 is a whole number (such as sin2mn). Let x(n, 2) be another solution, and let Bn be a similar periodic function, which always becomes 1 when 7 is a whole number, such as cos2mn. If then we examine x (2, ). Hint oy (n, v).£, n+ % (n, at) En oveeee(X)s we readily see that a change of m into n+1 is equivalent to differentia- tion with respect to x, or the preceding satisfies the functional equation. | Also, if 2 be a whole number, the preceding is always reduced to — 4 (n, 2), Or D’ x (0,2). Which of the infinite number of cases con- — tained in the preceding solution is entitled to be called D" y (0, x) when nis fractional? are all to have that title, or some only, or none? But the preceding (x) may not even be the widest form of the solution, though fettered by the condition that $ (O, x) is to bea given function | x (0,7). Let (7, x), one solution, have been found, and let it be asked whether A,,,x (7,v) cannot bea solution, where A,, . is a func- — tion of 2 and x, subject to the condition Ao, ,==15 such, for instance, as | 1+ox. xn, where x(0)=20. We have then to solve Aga eX M41, 2)=D fA, xX SHA, 2X A) + As eX +1); the accent meaning differentiation with respect to x: whence x Anh) ¢,% (n, 2) a Sees 1 hy s— (Age, eT. WR x) an equation which in all probability has an infinite number of solutions, | containing arbitrary functions and constants, the proper values of which | may make Ay,,=L. | The essential properties of the symbol D are D’.D"u= D".D'u —D"*y, and D*(w+v)=D"u+D"v, and these relations should be. required to remain true for all values of the symboln. It may happen that many solutions of the form (x) fulfil these conditions: and cer- tainly no function can be absolutely asserted to be the general diff. co. of x (m,x), unless it can be shown that no other solution of (x) whatsoever satisfies these conditions. ON DEFINITE INTEGRALS, 599 Several modes, reducible to two, have been proposed ;* the first pro- ceeding upon the fundamental properties of ¢*, the second upon those of w”. Weshall take the second of these first in order. Let P (m) represent D” 2” for all values of x; and since D’2"=Ma"-" for all whole values of m, let us extend this to fractional values. If we perform the operation D”~" upon both sides (assuming for the present that D”™ M=0) we have D® «"=MD”-" v"—, or M=P (m): P(m-n), whence n 11) samme B ( ) 11 = B ( 48 1) m—n Date P (m—n)*” ‘= r (m—n-+ 1) % } The question then is, what is P(m). When m is a whole number it is m(m—1)...3.2.1, say =F (m+1), ora solution of (m+1)=mdm. Let it stand for this same solution when m is fractional or negative, and let us choose the solution which contains no periodic function, which is found when m is positive by [ (m)= 4! 9 & 'v™— dy, and extended to the Case where m is negative, as in page 583. We have then the second expression given above. The first mentioned mode is as follows. Since D'e”’=m"e"™ for every whole value of m, let this expression be generalized and made to hold good when n is fractional, as the definition of D* e”. Now when ® is positive, we have oof 0" du=az{"T(m), or (-1)" for vt dyu=Fm D* 2, whenever 7 is a whole number. If this formula be generalized, we have aia F (m+ n) ; (— 1 aim Tm 4 D—” 5 ae op or Dp” Fibro ( iF nt) aon : a formula which has been assertedt to be universal, and demonstrated ina manner to which, on the assumptions laid down, I am not prepared to offer any objection. But according to the first system D"2~” is P(—m+1) a": T' (—m—n+1). Now as both these expressions are certainly true when 7 is a whole number, the one becomes the other after multiplication by a factor similar to Hn (page 598) ; namely, which becomes unity when mis a whole number. Both these systems, then, may very possibly be parts of a more general system; but at ‘present I incline (and incline only, in deference to the well-known ability of the supporters of the opponent systems) to the conclusion that neither system has any claim to be considered as giving the form of D" x”, though either may be a form. The following considerations may help to explain my meaning. In common numerical interpolation, we proceed without the introduction of * The subject has been mentioned by Leibnitz, Euler, &c., and has been system- atized by M. Liouville, in the Journal de ?Ecole Polytechnique for 1832, and after him by S. S., in the first and third numbers of the Cambridge Mathematical Journal: and still later by Professor Kelland, in vol. xiv. of the Transactions of the Royal Society of Edinburgh. Professor Peacock has proposed another and a dis- tinet system in his well-known report on the state of analysis (Proceedings of the British Association, third meeting). To avoid perpetual reiteration of names, we may here state that the system of MM. Liouville, &c. takes <* as the funda- mental function, and Dr. Peacock takes x", + By Mr. Kelland in the memoir cited. 600 DIFFERENTIAL AND INTEGRAL CALCULUS. any periodic function (as represented in page 543). When we know that the given values are those of a given function, as log x, our reason for this is, that we absolutely know the function to be of a uniformly increasing or decreasing character, without the wndulations of a periodic function. But if a question were to arise, in which, from the nature of the case, we could only make our first approach by limiting the value of toa whole number, and if the result of this first approach always gave log a, we should have no assurance whatever that logv was the function required ; it might be log #.cos 272, or log 2-+-sin 2rz, or log #.cos 27x 4+-sin2ra; all of which are reduced to loga whenever @ is a whole number. ‘Those, therefore, who would prefer either of these to the others, or to all the rest of the infinite number of cases which might be cited, must show some very cogent and direct reason why they take the one which they prefer. Again, when we interpolate between values derived from observed phenomena, we exclude functions of intervening undulation, because we know in the first place that it must be impossible that times of observa- tion arbitrarily chosen should always fall precisely at the epochs of dis- appearance, &c. of the undulating terms. But even this must be taken with restriction. Suppose, for example, that a planet had been observed only at one place, and could only be observed when on the meridian, the general laws of planetary motion being unknown. It might be satis= factorily deduced that the planet always was in a great circle, or in- sensibly near to it, at those times, but it would not at all follow that it was in that great circle at intervening times. How would it be known but that the place of the planet was connected with the earth’s diurnal | motion by a law which allows of periodic departures from the great | circle on one side and the other, the whole period being the interval . between two transits, and the time of coincidence with the great circle | being precisely that of transit.* I now quit the subject of interpolation of form, and proceed to modes | of determining the value of definite integrals by approximation. Among these one of the most celebrated is that of Laplace, which applies when an integral contains a large number of factors or large exponents. Let V be a function of 0, and @ a function of @, it is required to find the successive differential coefficients of V with respect to ¢. These might be easily expressed by means of the derivation in pages 331, &c.; but by a direct process, denoting differentiation with respect to ¢ by accentuation, and with respeet to 6 by subscript numerals, we have v'—0'V,, V"=—9?V.4+ Cr. Vv= O2V a+ 36’! Vo+0"V~, Viv 0!V + 6070"V,+ (3024-400) Vet OV, * The question of the interpolation of differential forms is embarrassed with con- siderations of a nature precisely similar to the preceding ones. I am not willing positively to assert that there exists no reason for one system in preference to the other, nor even that such reason has not been shown by the assertors of one or the other system. I can only say for certain that I cannot see the reason in thei writings; and I am sure that they themselves will admit that the doubt I haye raised is one that requires solution. The reason why it shonld strike me more forcibly than them is perhaps that 1 have written on the calculus of functions, and have had my attention particularly drawn to the wide character of the solutions o! even the most simple functional equations. ON DEFINITE INTEGRALS. 601 V"=6°V,-+ 1006" V.4 (150024 10020) V,+ (1090 +500") V.+0'°V; V"=6°V—.+15946"V 5+ (45976"2 + 2000") V, + (1503-4 60060" + 15676") V,+ (150"6" + 1002+ 66/6") V+ 6"V, V"=07V, +2100" V4 (1050"6"2-+ 35040") V, + (1059042100706 +3500") V.+ (1050"'0'" 4.7000"? +. 105000" 421676") V, +- (3506 + 2106" + 79/6") V+ 0"'V,, The law of this apparently complicated process (which should be per- formed by common differentiation, and verified as now explained) is as follows. Suppose we would form the coefficient of V; in Vs Investi- gate every way in which 7 can be subdivided into three parts; which will be found to be 14+1+5, 14244, 1+3+3, 2+2+3. The terms in the required coefficient have then 6/6'6", 6'0’0", 0/60", and "60". And the coefficient of each of these is the number of distinct ways in which seven counters differently marked can be so parcelled into (1, 1, and 5), (1, 2, and 4), &c., that all the parcels shall not be the same in any two modes. ‘This coefficient is found as follows. Let m=a+b+c+....3 then the number of ways in which m counters can be parcelled into a set of a,a set of b, &c. is ed O. B48 or Ae OS ..m—l.m Bee 2. 6 EyPCuigs .cby(E. SF aay Ifa, b, &e. be all different, then P=1: but if there be f parcels of one and the same number in each, g parcels of another, / of another, &c., then P=(1.2 ..f) (1.2...9)(1.2...4)... Thus 7 being 14+3+8, the coefficient of 6’ 6/2 is LPR 5s 3-52 . : —~" ++, or 70, as in the formula. | hoe (hoo ay (ted 187-1} Given $9 =Ae-#, required the expansion of 6 in powers of t, This equation cannot exist unless @0 be a maximum when {=—0; and we shall suppose that ¢’@ is =O (and not cc) in that case. Let us more- over suppose that 0=—0 at the maximum; whence ¢(0)=A, and ~(0)=0. Let logd@=V; then V—logA+#=0, V’4+ 2t= 0, W"4+2=0, V"=0, V"=0, &c., from which, as obtained by the pre- ceding process, we are to calculate the values of 6, 0”, &c. for substitu- ton in t2 3 6=(9)+().t+ (0) ++") i ahaa parentheses denoting values when ¢=9. Let v, 1, vy, &c. be the values a : . Shere of V, V,, Vs, &c. when t=0, then v=log A, v,=0, since Vi= 0: po, and ¢/0 vanishes with 0; that is, with ¢. Vv" +9—0 cives vs 6-4 2=-0, or CG yesal € — D5" ") q) Se ty - S aay PN 3 y"=0 gives 0 7,4-30'0/v,—0, (O)=- = 3 0 Q4 / 5 G he \ i a ta en ee ae batons V ==) gives (0 J= 314 vA 9 ie 602 DIFFERENTIAL AND INTEGRAL CALCULUS. and so on. Hence 0=Ae—* gives 9 a 3) 5v3— 302 V, tye Vo = i - —f+—— + De ES Bas Ra Rt Vs bik v? a 18v5 9 + 2 where Up, Va, Vs &C. are values (for 0==0) of the second, third, fourth, &e. diff. co. of log #6 with respect to 0: the conditions being that #0 is a function which is a maximum when #0 = A, and that v, is not =0. If v0, and generally if m diff. co. of log $8 vanish when 6=0, (and mm must be an odd number, or there could not be then a maximum, ) we must use the equation #@=Ae—?"*" in the same manner. It would be a work of great labour to calculate as far as the sixth power of ¢ by the preceding method, and an expression of the general term of the series would be altogether out of the question. The powerful method of Arbogast, however, (pages 328—335), will enable us to give the general term with very little trouble, and to deduce more coefficients than those given above. xa Having given t= (log A—V), where v=log A, 1.=9, it Is required to expand 6, of which V is a function, in powers of t. We have then, by Burmann’s theorem, page 305, the marks { } denoting that ¢, and therefore 0, =0, {9/14 (OR Fain ee RT ra nf Fh eels ei 2 a6 (+)} Like vs B,, 4”; whence B =| G ,; SAY Feet be fu TIEN aban & *93...m- Write a for —v,-2, 5 for —v,--2.3, &c., and we have 6:10: /(log A—V)=0: f(a +b0+....)={atbO+... A eck 25) Develope (a+50+... .) 2 by Arbogast’s method into 2.P,, 6", which gives for P,,_, the following series of terms, m:2 being 7, mx 2 m—3 22 ’ Li) m—1 Brot, on-rLADE ttios, fons en Se aaa a” 2 art? [m—1] aa If (a+b0+....)7~" be differentiated m—l1 times, and 6 be then made =:0, the result will be P,,, X1.2....(@m-—1), which, divided by 2.3....m gives the mth part of the expression (A) for B,, the co- efficient of ¢” in the development required. We have then the following, the first of which is independently obtained (title, page 331) : a—itL ae fe eke 1 Dd 15 8 at (5b?—4ac) Biot Besa be —ino a to 4g Boa 1 | 2D*% 3Db ar 4b3— 6abc+ 2a’e Bo )—- —> + er Ss (= f a a’ a? Aa’ — eee i -—c — —_— — "3a "24 a 246 TR TG 8 ail? _@67.9.11b'—1.9.8.3ab’c+-7.6.8a°(2he-+c?) —4.6.8a%f} ee 2.4.6.8a7 1 5D) 57D 579 Di 57911 J) ON DEFINITE INTEGRALS. 603 Taw wr) a anline aes > ° ; . write —v2:2, —v3:2.3, —1,:2.3.4, &c. for a, b, c, &c., and we lave \ € hy 2 \ i hag 5vs—3v5 0, v 3 sexteiien Byeres 4S iBeette tonks te esicct 9) i Beads e 4 ? Ve / vs 18v5 2 By= — (4003 — 45v, 03 v, + 903.) 27003 B,= — {385r5— 630v.03v,+2103(8r,v,+-5v%) — 24r%v, n/(-3) : 2160.v%. The use of this method is as follows. Let it be required to develope fydx between any given limits C=H, X=y, y being a function with factors having exponents of considerable magnitude, such as y=p”™ q", where m and n are considerable, and p and q are functions of «. Let there be a value r=a which makes p”g” a maximum, and let and y lie between two roots of y preceding and following the value of x which makes y a maximum. If, then, A be the maximum value of y and if we assume y=As—?’, there are real values of ¢ for every value of xv, from t=— cc, which gives += the first root, to ’=-++ oc, which gives x= the second root; and when t=0, we have y=A, or aa. Let «=X, a=9, be the values for which y vanishes, so that r; Hy ¥, are in order of magnitude, X being the least. Let e— pe and a=v give t=q, (=; let r=a+6 and y=¢(x). Then 6 and { vanish together, and #(a+0)—Ase—# gives 0=B, t+Bo@2+.... as just determined. From this find d6, which is dr, and we have Lyda=A {B, ffe—Pdt+2B, fe e—Ptdt+ 3B, (?e-Pedi+....2. y ‘ ‘ 5 If we examine B,, B., &c., we shall find that if each of the set Voy Vay &c. had a large numerical multiplier 7, these coefficients would severally a Bd BE have the multipliers n~2, mi ', 2%, &c., which would make the series convergent enough for use if m were considerable. A further reduction may be made as follows. Let ven f di=G; fe? pa s—? l : a SEP dt, C= fe-?.t* dt=— e~i? 3t 3 2 = = ot ee ~a @+1) Gs : (045) 436 e—?t? 2 tem? J G.= — ss +5 Ge G,=— _ Take limits, and substitute, which gives Dasa i: Bee Sayde=A( B43 Be+5-,B + 2° 1) fen dt 1 3 ez e~2 A 12B,-+0.3B,+ (a1) 4B,+( 0+ 5”) 5B y's | —5e- FA {2B,-+8.3B,+ (6?+-1) 4B+(6°+56 ) 5Bs+ .. ih If the limits be and y, two values at which y vanishes, one preceding and the other succeeding that at which y is a maximum, (that is, if B= v=p,) it follows that @ and f are — o and + , these being the only values at which {Ae—#* vanishes. But ft2e-? dt=2ffe-? dt (page 294)=,/7 ; whence (as in this case, the two last lines vanish) 604 DIFFERENTIAL AND INTEGRAL CALCULUS. 3 Be fi yda=A/r (Bap BASS Bt =) ; For instance, let it be required to find an approximate expression for 1.2.3...., where » is a large number, or generally for PF (n+1), where n is a large number, whole or fractional. In foe? x” dx, it will be observed that ¢7*a" vanishes when w is 0 or oc, and that there is an intermediate value of 2, namely v=n, at which e~’ a” is a maximum. We have then yea", w=0, v=o, a=n, A=E*n", Le V=log(e-*2")= -(n+0) +n log (n+ 6)=n log ra lat mba atts a= in, b= an ic Cees Cosi ass feiny; &C. 1 1 1 1 j Lice 2 Bv.=-= >s-. Bis Sears &e. } BREN OM i Sony gt oat Rall a | PNM shill ytd T(nt DHA (2rn).e*n"| lt+s5 toggete es J} ge a result which may be made to agree, as far as it goes, with page 312. The student is now prepared for the higher class of investigations connected with the theory of probabilities. The integrals which are of most importance in this science are F(2), already treated, f e— dt, and fx" (1—2)"dx. It will be worth while to make the calculations neces-_ sary in the latter case, m and n being considerable numbers. Here y=2” (1—.)", which vanishes when r=0 or I, and is a maxi- mum when c=m:(m+n), 1—a=n: (m+n): let the first be w and the second ». We have then log y==V =m log (W +0) +n log (p—8@) lfm n l1/fm n Slog {@" 9"S—5 +7) 045 Ome mm” 6° “Mite yee a= (ma-+np-’), b= —1 (ma ?—np™), c=} (mo *+np~“), &e. | k k ; Or SS (@a*+p), b=-— i (aw *—p~), —— (mo *+p~*), &e. 5 where m+n=k. Hence we find by actual reduction ear 20 (1—w) 4 1307-13041) as k : *~9,/i 20 (1—w)keV? an 77 3a? — fue ay dea, / 20 ei trea c 120 (1—o) k very nearly: which might be verified by applying the value of F (7) just found to the result in page 580. | When y has high exponents, but does not arrive at a maximum — between the limits ; or rather when it is not required that either of the — limits of integration should be near to that value of w which makes y @ | maximum, the formula in page 290 will give a convergent series, and even for the indefinite integral. ON DEFINITE INTEGRALS. 605 eee fyde=yu )1 des i, at = {ue ve ha ; day apa vey } de Vile \ ah dx dx dx wan)? For example, let y=x", or u=x:n. We have then x" +1 gt 1 l l - =— {1 Onin uated oie INO \ which is easily verified. n+] n 2 n° The preceding is taken from x==0 on both sides, but any limits may be taken in the usual way. Thus 3 ipo ak 2 7 3 Dba Tas perma tify a tA 8 Fc 2atn (2a+n)? (2a+n)4 (24+n)8 Me 2 5 4 (sinx)"dr== —_ oe + : 2 5 | n* n a —a n+l ; 2 9 & “a n n + 2na [codes ‘ -- —....} (>a). i (na)? © (n—a)t I now {proceed to the doctrine of periodic series, one of the most important applications of definite integrals ; the results are of a new and extraordinary character, on which account this part of the subject will be treated in detail, and by two distinct methods. From page 291, § 121, the following is easily proved: if a and a’ be two whole numbers, and m and n two other whole numbers, positive or negative, nr nr : ‘ : T { cos a9. cos a0. dé, and [ sin@@.sin a’0.d@ are =0 or (n—m) ty mr Mir namely, 0 when a@ and a’ are unequal, + (n—m) x when a and a’ are equal. This property is applied to the expansion of ordinary algebraical quantities in series of periodic terms, a subject which will require a close examination of its first principles. If we take such a series as A, sina+ A,sin 27+ Asin 3v+.... ad infinitum, we see that, whatever its algebraical equivalent may be, it must go through a succession of values from #=0 to e=2z, which suc- cession is repeated from xr—2r7 to w=4r,and soon. It might seem, then, as if we could affirm dG prior: that any function which admits the preceding development must itself be periodic and trigonometrical: but we should be mistaken if we drew any such conclusion; at least we can only drawn such a conclusion with some extension of the term trigono- metrical. d If we integrate both sides of (1+ a0’) = 1—a’?+a? 6*—.. 4.4, we nd 1 —1 1 6° @ 2 0? 3 ° +s CGT) (/a.0)=C+0— rs at —a perizeen, Aint eleeis'-s Ja " 3 2 y the first side of which is indefinite, since tan~* (a given quantity) has an infinite number of values ; the second side is also indefinite, con- taining an arbitrary constant. Nor do we avoid this indefiniteness by mtegrating from a given commencement, say from 0=0, which gives a 4 a A ' ; ‘ 606 DIFFERENTIAL AND INTEGRAL CALCULUS. 1 ‘| Bee ee cd ee Tee grape b ia (Ja.0)—tan 0} So Set ee es for tan7! 0 is ma, where m is any whole number positive or negative. For given values of @ and a, the second side has one value only, the first side (but for restrictions imposed by the equation itself) has an infinite number. Consequently, whatever value may be taken for tan (/w.0), such a value of tan~!0 must be taken as will give that value of the first side which is equal to the second side. Let a=1, and let @ be tan ¢, then one of the values of tan (tan t) is ¢, whence t—tan-! (0)=tan t—+ tan®é+ 7 tan’ é—...- Now as we begin from tan £=0, let us choose mz for the correspond- ing angle. We must not then carry our series of integer (of 1x d.tan ¢, tan? é.d tan ¢, &c.) up to a limit higher than t=m2-+41, nor might we have begun before t=ma—4z, since tant is infinite in both cases. But between ma+4x and mx—4$r the equality of the two sides is unobjectionably deduced, and the answer to the question, what value of tan7!0 must be taken is, mz, whenever ¢ lies between mr+hnr: so that tané—}tan’t+..-- is, for a given value of tané, that one of the corresponding angles which lies between —hr and +47. Let us now consider the equations (pages 242, 243) 1—w2z Cos 0 1—22 narrate cos 0+." cos 20+ &e. | e@ee® A). x sin 0 oe ( Toe a Oat oe sin 0+ sin 20+ Se. Here the periodic character of one side is a counterpart to that of the | other, and when r cos m ; (x—v) becomes cos (- m : (e+) or cos m= (a-+0). Add and subtract the second equation, thus altered, to and from the first, and we have (extracting the constants from the sign of integra- tion) grat Sipe dy 2 {Sices pode. cos _l 2 . mrv ee or= 2S fe gv dv.sin al. If l=, we have the theorems already proved, with something-more, as follows. When a=0, the preceding series (¢) are each =3¢0, so that their sum is ¢0, and their difference 0. But when v=/, each is equal to $¢/, and their sum is ¢/ and their difference 0. Hence the series for $x in cosines is true when z=0 and e=/; while in that for sines the series becomes 0 both when v=0 and when z=/, and con- sequently will not then represent ¢x unless GO=0 and #/=0. Thus we can now infer from page 614, that ee, aie ieee he Ah ich may be-yeritieds ed at eo “i ; ee 618 DIFFERENTIAL AND INTEGRAL CALCULUS, wm em™+e* 1 a a a — —- ee as eerie aa eal Tapa e+" 1 a a a cot ar 7 2a~ obetat Vbaoten) Sale eet | Returning to the original formula, let «m:?—w, whence in passing from term to term by alteration of m, we have r:/=Aw. We have then 1 1 p=z fiiov dot ~2 { f+ cos w(a—v). gv dv Aw}....(A); which being true for all values of J is true at the limit when / is infinite. Now i ov dv in the first term may increase without limit with J, and ub gv dv: 21 may in such case either increase without limit,t have a finite limit, or diminish without limit. If the latter be the case, which it cer- tainly will be whenever {t¢dv dv is finite, then, observing that w increases by continually diminishing gradations from 0 to o, we have, by the definition of a definite integral, mpuz fe {Sts cos w (x—v) gv dv} dw= ft [tcos w (a—v) ov dw dv; a result which is usually called Fourter’s Theorem. We shall presently have to consider the proposed limitation further ; in the mean while we shall see an apparent neglect of a corresponding limitation in every one of three methods which have been employed to verify it, or else an in- version of the order of integration. It is to be remembered that the theorem was obtained by integrating first with respect to v. 1. Consider f f cos w(a—v).e~’ gv dwdv. We easily find ‘ ie k ** _kovde fo cos w (@—0).€ P Ria ee Oe ie | eras is to be determined. Now since k is to be diminished without limit in | the result, we may, by reasoning similar to that of page 615, consider only that portion of the integral at which v is nearly =<. Let v=a—2, then the preceding becomes kpa. dz hole. zdz ae a {es — | eer tee or dx tan a taking this from — o to + &, or from —ato +A, as before explained, we find rz, which verifies the theorem, apparently without limitation. But what are we to say to this verification in those numerous cases In which | * The student must particularly observe that the theorem in Chapter xix. does not necessarily apply to series deduced from discontinuous expressions, or from any considerations in which discontinuity is involved. + The reasoning of Poisson neglects this limitation, though obvious enough, and Fourier makes a similar apparent error. Poisson makes J gv dv:22 always vanish — when J is infinite: Fourier has missed this term by writing a series P, cosa | +P, cos2a+..+.-., which should have been P +P, cos2+P, cos 2r+.--- Both are certainly wrong in expression, though the remarks to which I shall presently | come remove the limitation, and show the theorem to be universal. ON DEFINITE INTEGRALS. 619 pv dv: {k®+(v—2x)*} is infinite, taken from v—=—a to v=+ A? his question requires more answer than it can receive from the pre- ceding reasoning. sin @ (r—v +? sina (a—v 2. [cos w(x—-v) Pe ae at and [ a, ) wy dv —V as Up is to be determined, @ being made infinite in the result. Let e=v—za~, which gives +? sin z Ze +? sin 2 ey) ee) 2, or px dz, or rpx, ge Als a) ab. as will be afterwards shown. It is here assumed that since a@ is to be made infinite in the result, all finite values of z produce no effect, while the infinite ones are compensated by the infinitely small value of sin z: z. But it is well known that z~* does not diminish fast enough to compen- sate the increase of any function whatsoever. This verification I hold to be decidedly unsound, though its results are true, unless that meaning of sin cc should be admitted which has been already hinted at, and will hereafter be further discussed. (2—v)? « e-) kip 1 T Bs fo cos w (a—v) e~* ‘dw=; / #8 te 5, as will be shown. Multiply by dudv, and make v=x+z. Then since & is to diminish without limit, it is easily shown that the function to be integrated diminishes without limit, except when z is infinitely small; and reasoning as before, we have 1 uy gee mages di? na —s2 ; 4k Bo /-| e*** & (a+2) dz, or Pra/t err a Get 2k); —O or rx, since fe—# dt from t= — cc to t=+4 c is Jn. This seems to be subject to the same objections as before, for if dv increase without limit with v, when the latter increases positively or negatively, it may be that the conversion of (+2) into ¢z is not allowable. I now go on to point out what I conceive to be the manner in which the theorem is to be proved; and I do not regret the space apparently wasted upon the incautious phraseology of some of the analysts* to whose brilliant labours we owe these truly remarkable views, because the preceding considerations will serve the better to enable the student to see this new point of the integral calculus, nothing approaching to which has appeared in the preceding part of this work. Returning to the expression (A) (page 619), first observe that fA, da.x,+ f As da.%,+..., or Zz (fAda.xr) is identical with f(Aya +A,.2,+....)daor f(2Ax).da, provided only that 2, x., &c. are independent of a. Write the expression (A) in the form * The greatest writers on mathematical subjects have a genius which saves them from their own slips, and guides them to true results through inaccuracies of ex- pression, and sometimes through absolute error (see that of Legendre, page 595). But their humbler followers must not permit themselves such license, and those above all who write for students must correct that as an error of reasoning, which, in the guide they follow, was little more than an error of the pen, ten 620 DIFFERENTIAL AND INTEGRAL CALCULUS. 1 hats =f {LAw cos. 0 (v—v) + Aw cos. Aw (r—v) T =i + Aw cos. 2Aw(r-v) +....}¢vdv=¢u. This expression is absolutely true for —/ (2) /, whatever the values of J may be, and the series it contains is the limit of a set of convergent series made by diminishing & without limit in 1 Aw cos. 0 (v—v). e~*4"-4+ Aw cos. Aw (a—v). ¢*4” + Aw COs. 9Aw (v—v) e ek Aw 4 eee Let & have any positive value, however small, and let the precedin yt I § be multiplied by @v and integrated with respect to v, from v=—/ to v=+/; that is, from v=—(7: Aw) to v=+(r: Aw); and, if a he | between these limits, the result will be as near as we please to Pz, if k be taken small enough. Since the series is convergent, this might be veri- fied by actual arithmetical operation. Now since the individual terms of the preceding diminish without limit with Aw, any one or more of them, in fact any finite and fixed number, might be erased or altered m any finite ratio, without affecting the result. If, then, in the first term _ we change 4 into 1, (or if we erased the first term altogether,) the limit _ of the result, when Aw is diminished without limit, is strictly +730 si : al i cos. w (x—v) .e-™ gv dv dw=gatay -r:0 0 where g and k are comminuent. Diminish & without limit, and we have | Fourier’s theorem as given. Now for the first verification (page 618). If we begin by integrating | with respect to w, we have, as before, f cosw(«—v)e"“’dw=k: | (k?-+-(a—v)?), which vanishes with k, or is =0. Consequently, com- | pleting the process, it (might appear that we must have f 0.¢vdv (from | —co to -+o), and divided by =, or 0, for the result, even though | fv dv were infinite. But here it must be observed that if an integra — tion with respect to v is to follow our last conclusion, we are not entitled — to say that k:(k°+(v—z)*) always vanishes with k. Among the coming cases to which this conclusion is to be applied is the case of vx; in this case the preceding fraction, instead of vanishing, becomes | infinite. But this we have gained, namely, that we have a right to use the results of R=0 as to every value of v except v=a, or infinitely near tov. And we might have applied all this process to the series before Aw diminished without limit, or / increased without limit, as is actually _ done in page 615. Hence we have no occasion to consider more of fish dv dv: (k2+(«—v)*) than is involved in those values of v which are infinitely near to x. The rest of the verification need not now be repeated, In this theorem of Fourier, as well as in the formula from which it » was derived, x and dv may be discontinuous. The same thing may be said of the formule in page 617, or of their particular cases in page 614. We shall now ask what these last formule represent for,other values of not included between 0 and /? If we write them thus, ON DEFINITE INTEGRALS. 621 l 2Qr. > $t=h Bo+B, cos — +B, cos + pe A ym); l a; _ Qrex > oe= Aysin +A, sin ae Ee 0 (2) =, MEU ENG HE Se Bed} gv COS Ric dv, Je el ( uv sin ohh dv. we see that while v changes from 0 to 21, rv:1 changes from 0 to 2r, or completes a whole revolution; and the same while v changes from 2¢ to 4/, from 4/ to 6/, &c., or from —2/ to 0, from —4/ to —2/, &c. From the periodic character of the series, it is plain that the values of one interval recur in all the rest; now in half the interval, from 0 to Z, lx: 2 is the value of the series; what is it in the other half, from @ to 2/? Since sin (2mr—z)=—sinz and cos (2mr—z)=cos z, it is obvious that if we make either series the coordinate of a curve, and measure equal abscissee from the beginning of the interval 2/ forwards, and from the end backwards, the ordinates will be altogether equal in the series of cosines, and equal with different signs in the series of sines. For MT i so that all the terms of the cosine-series remain the same, and all the terms of the sine-series only change sign. If, then, OL=LL,, &c.=/, : 19 x v Mr pa | (2h 0) = sin v, cos m —- (2/—v)= cos 73 and if ipx: 2 be from v=0 to x=1, the discontinuous curve ABCD, the cosine-series is always the ordinate of the upper figure, and the sine series that of the lower. According to the last investigations, however, (page 617,) the points A, D, E, F, &c. in the lower figure do not belong tothe series, but the conjugate points O, L, L,, &c. take their places. But if we took for ordinates successively A, sin 0, A; sin @+ A, sin 20, &. (0=2x7:/), we should have a set of curves which perpetually 622 DIFFERENTIAL AND INTEGRAL CALCULUS. approach to the continued line ABCDLDEL,E, &c., and all the lines DLD, EL,E, &c. form parts of the limiting figure. | Let it be required, for instance, to find the equation to a set of simple — isosceles slopes, as dotted in the upper figure. From 2=0 to sh let y==ax ; then from e=41 to a=l, y=—a (4—J), or « ((—2). We are then to find 31 d Bl av COS — av [ ju 2) cos — dv 0 al? ./ mr = 2 cos ——1—cos mr mr? 9 i pai i 2 which is —4g@l2:m? 22 when m is of the form 44-+2 and O in every other case; except only when m=O, in which case it should be dal*, _ Hence, multiplying by 2:/, and putting 4B, for the first term, we have | for x the ordinate required, al»: Sal /y1 Deter ph 67x or 2 \ cos j ar, j Hee Verify this when a=0, w=3/, and x=l; and also verify the differ- ential coefficient by page 608, showing it to be « from z=0 to z=), and —q@ from r=4l to al. By the same process which gave Fourier’s theorem, the equations in page 617 may be made to give D) ie] io?) ort =— cos wv.cos wx. Pv dv dw . ad 06) 1,0 y) oe oe =—- i { sin wv sin wa Gv dv dw; TJ od o | the first of which is true when v=0, but not the second, unless ¢0=0. | Both are true for all positive values of «: and if ¢x be even, the first is. also true for negative values, and if pz be odd, the same may be said’ of the second. | Poisson has applied the fundamental equations +1 +1 ee oom ov doe z{ eee ov do} —Il(ax)l 2l sal l —l Y a 1 +1 1 +1 a + Holto(—Dh=z[ godve 34 { cos— — > gw av\ —l in many remarkable ways, from which we select two. Let the function employed, which for any thing to the contrary im- plied in the demonstration, may contain J as well as 2, be ¢ (x+1+ 2h), k being 0 ora positive integer. Let v+ 1+ 2ki=z, then the limits of 2 (answering to v= —/ and v=-+/) are 2kl and (2k+2) /, and we have | 1 *2k14-21 b (a+ lt 2h) => dz dz Qk 1 2k l+2l pay +7 2 tf (eran Oe ei ee az —l(a)l; 2kl 4 as l 3? ON DEFINITE INTEGRALS. 623 . L—V l+-Q2kl+a— ni since oa) ee TES) Okt) mee “ val Take k successively =0, 1, 2, &c., and add the results, which gives P@AD +4 (243) +4 (@+5)+.... buhay 1 . im (t— =a { dz dz+— > if CHP Gog Ce pz az}. Qbeyrs d P 2 But if r=/ or —I, the preceding expression represents 3 (60+ $2/) +4 (2/4041) +....5 or 460+ 4621+4/+..., For / write 4/, and we have 2mrz 7 dz\ (A). ball ¢0+4/+4+42/+ ... =\90+— fo $2 de + Ds {Si e0s By using ¢(v+2k) instead of ¢(a+/+42hl), the following may be deduced from the previous results of Poisson, r+ (7+ 2/)+... =e f2r62 dz +3 2 1700s uae pz dz} which, when z= —/ or +1, becomes $9 (—/) + ¢/+42/+...., or— (aw +1) +9 (a@4+2l)-.... pales oz dz +2 2 { ftscos = =) gz de } +=. > x Wi ci eee oz dz}. If in the expression for ¢0-+91+ &c. above given, we change the sign of J, and add, we have 290 +¢/+¢ (-)+¢6 (2/)+¢(—2) +... =¢0, or ...¢ (—/)+90+¢/+ ...=0, which verifies the theorem in Chapter xix. And if in the last result we change the sign of / and add, we have 1b (B=) +2b0—-$ (CHD +O CHM). ede +4 {fa 2 ea or see (2-1) + O2—h (etl) +6 (r+2l)—....=0, which is another verification of the same. To verify one of these formule, take that for 60-+¢/+....and let gz=e". Then fe cos (Qmrz:1) dz dz=l?: (2+ 4m? x’) gives ie ees Pee i tea tee 4 et ee paves pe ny (ars pea t ) write (1—e~)~ for the first side, and show that this agrees with the series deduced in page 612. Again, multiply the formula for 604+ @1+.... by 2, then for Z write 2/, and subtract the original equation. This gives eye) 624 DIFFERENTIAL AND INTEGRAL CALCULUS. p0—Pl+G2l—... = 540 4 +3 {f (cos ™F—c0s ae) pede} (Bye | 0 The second application may be made to have reference to the following — point. We have now gone through a number of new and strange ex-— pressions, involving the remarkable new form of an integral which has only instantaneous values, a term, the meaning of which the student | will sce if he understand the preceding pages. The following must bee | made to furnish verification, or something to show that these unusual | expressions have some affinity with others. I shall now point out, for this purpose, not only how to recover the theorems in pages 266 and 311, but to complete the conception of them, by giving values for all the rest of the series they contain, from and after any given term. In (A) make ¢ (ml+2z) the subject of the equation, whence we find | for the value of the series (71) + {(n+1) L$ 4+6{(m4+2) U6 +.0.8 | the following: 2 oo J pnt So l-+2) dz-+ + ES cos ~ (nl + 2) act 1 te Fs 2 » 2m (z—nl) or > bnl-+—- Siu bz dz +72 | Sacos real he dz} in which remember that 2mz(z—nl):1 and 2mrz:1 have the same cosine. Subtract the preceding from (A), which gives GOEL veee $OG—II=S (G0-Gnl)-+ fii be de QrMz l Now integrating by parts, we find | k—G0 1 * cos dz. Pz Ae ene! oe 4% cos az. Gz dz, a” Pha +4 SAS cos pz dh. ! if ka be amultiple of 2x. Carry this on, meaning 21m:/ by a, and nl by &, remembering that m and 7 are whole numbers, which gives jk—'0 pl R—P"0 (2e—-1) Jo (2e-1)y fi cos az bz dz= eyrer Fo eae 1 Fo fi cos az px dz. Substitute 1, 2, 3, &c. ad infinitum successively for m, write for a and k& their values, and add, making S,=17-"4+2™"+4+3-"+.... This gives ! 1 S27 BOE GLE... +O (NI) aT Sipe deg (nl 40) +555 alg) Ss » Uh phe ps | Sl ai (p / nl— 0) obs, wip igen Ge (P&Mnl — p&-Y0) | i tyabaes 1 Qrmz a pet, ps Stage pz dz \ ON DEFINITE INTEGRALS. 625 For S,, S,, &c. write the values deduced in page 581, and we then see the theorem in page 266, § 69; that is, if in that theorem we make Y.=Pn, we have what the preceding becomes when /=1. And we here see more, namely, that all the rest of the series, from and after any term, can be represented by a definite integral ; and from that definite integral, that the error made by stopping at the term which contains S,, (inclusive) is not, generally speaking, so great as that term itself. For that error is the definite integral last mentioned: throw out cos (27mz:l), and we certainly get a greater result ; for by so doing we not only increase all the elements of the integral, but we make them all of one sign (that is, if oz be of one sign, from z=0 to z=nl, as almost always happens). Hence the error in question is less than 2ce—1 Efe: ee e 2)» ] Sse! § fCe-Uin] — Be-Do2 “Copar perpen ae ~ por-see> or than — 9 Ge erUG or-1 ne me fo ra < D2e—1 xe he p sf) which is the last term included. And even though ¢@°? should change sign between the limits, yet if Ay be a constant quantity numerically greater than any value it has between the limits, it is easily shown that the error is less than f-* 2 2e oa. ‘a “Adz, or than Qre—1 ne ‘ ee Oee— 1 r Again, the above series gives the definite integral fs’ dz dz in terms of 1(590+¢l+....4+9(nm—1)l+4¢nl) and diff. co. of gz, so that approximation may be made by it to a definite integral in a manner re- sembling that of page 314. The series (B), page 624, might in like manner be made to give the series in page 311, but most easily by writing for the integral its equiva- lent form . marx Qnr2z "mre Pa 4 cos ———cos pzedz= | cos—— | Oz- > desis: i i l ’ l Cee Bi, I here finish the account of the manner in which periodic integrals are made to connect the mathematics of continuous and discontinuous quantity ; but it is still necessary to make a few remarks upon the very new species of results at which we have arrived. The impression which ordinary algebra leaves upon the mind of the student is that he has been studying the science of continuous quantity, represented by expressions which always vary according to regular laws. And he learns to imagine that every equation which is true for all values of a variable within certain limits must be true for all other values. The first exception to this rule occurs in the passage from the arithmetical to the algebraical view of series, in which we see that a series, as Tex? + x? +. ..., may be the representative of the arithmetical value of a function, (l1—«)~', when w lies between —1 and +1, and infinite in every other case. We soon learn, however, that the series still retains all the algebraical properties of the expression to which, when finite, it is an arithmetical equivalent; so that the use of the series for the finite function is allowable. A series of the form a+ba-+ca?+.... seems, if I may use the expression, to escape discontinuity by having recourse to divergency (pages 230—234): and eyenin series of other forms, those which can become divergent, or as near divergency as we please, never 28 *y 626 DIFFERENTIAL AND INTEGRAL CALCULUS. are discontinuously connected with different functions ; that is, never represent one function for a value of x between one pair of limits, and another for values between another pair. And by a series as near divergency as we please, I mean one which cannot diverge, but of which any given number of terms may diverge, such as the development of «%. But if we take a series which never diverges, nor appears to diverge, we almost universally find (as in page 230) that discontinuity is the result.* Sometimes, however, discontinuity is more apparent than real, and of this character is all that arises from the introduction of (—1)".. Thus an odd number of terms of 1—a7+a°—.... +a") is (1+a"):(1-++x), and an even number is (l—a"): (+72): both are represented by (1—(—1)" 2"):(1+2). There is here no real discon- tinuity: if we suppose 2 to vary continuously, and write the preceding expression with the numerator 1—cos 7.2", we find a perfectly con- tinuous change ; for instance, from 1—2* to 142°, when n changes from 4 to 5. In the theorems we have just left, however, we see the most complete discontinuity, not obtained by any new or arbitrary process, but fairly derived from the limits of continuous expressions. Some notion of the manner in which this arises is given in page 615, but as it is most essential that the student should fully see the meaning of such expres- sions as we have obtained, I now enter more at length into that matter. Through any given number of points (page 231) a purely algebraical curve can be drawn; from which it is possible to draw a curve which shall (page 621) from w=0 to w=/, resemble as nearly as we please the discontinuous line ABCD. The reason why it is more convenient to take a series of sines or cosines appears in page 610, in which it is shown that the actual determination of the equation of a line passing through the contiguous points is easy when compared with the cor- | responding purely algebraical process. And if by a finite number of | terms in the ordinate, we can make a curve as nearly coinciding with | ABCD as we please, it follows that by increasing the number of terms without limit the infinite series thus attained actually represents the ordinate of ABCD. This series is one of sines or of cosines, at pleasure, and having noted that hitherto series which are always convergent seem | to be those which are discontinuous, it may be interesting to showt that all the series of sines and cosines to which we have come must be con- yergent. Their coefficients are all of the form ficos ra bax dx and f ~sinra pe dx, and these must diminish as r increases, For if these integrals were so taken that the negative elements should be made positive and all added together, still each would be less than I gaxdx, since cos rx and sin rx never exceed, and are gertrally less than, unity. But in the actual integration, there are successive positive and negative por- tions, the balance of which is the integral required : moreover, the larger r is, that is, the more rapidly rz goes through a revolution, the more) nearly equal is each portion, numerically speaking, to those which are contiguous. Hence the integral is in each case of the form A,—A,++++ +A,, in which A,+A.+....+A, cannot exceed Hb oa da, and A,, As &c. all diminish, approaching to equality, as m increases. Hence * The preceding sentences contain matter of observation, not of demonstration. + This is a mere sketch of a proof, and requires some enlargement, but matters’ of more importance prevent me from giving the requisite space. ON DEFINITE INTEGRALS. 627 A, +A,+&c. and As+A,+.... are finite quantities, always remaining finite, and ultimately equal, or A,—A,+.... diminishes without limit. With regard to the signs of these integrals, it is obvious that when r is even, rv goes through a complete number of revolutions from v0 to x=n; and when 7 is odd, through a complete number of revolutions and half a revolution besides. There is no reason to assume, then, that fcos rz.gxdx and fos (r+1) a.¢dxdr must have the same signs when r is great; but by the law of continuity fcosra.pxdax and f cos (r-+2) x.gx dx are obtained in the same manner, and must at last ave the same signs. Consequently the only series we need examine are of the forms A, cos v-+ A,cos2r-+... and A, cos c—A, cos 2r+ .. ey and the same series with sines, it being supposed that the coefficients Aj, A., &c. begin to diminish without limit, sooner or later. Take any case of these kinds, and suppose x any quantity commensurable with 7, say mr:n, and owing to the recurrence of the values of sin rx and cos ra, it will be found that each series can be subdivided into two other series, each consisting of alternately positive and negative diminishing terms. If we now take the curve whose ordinate is (1—p*) {1—2p cos (w—v) +p?\— to the abscissa v, x being a fixed quantity and p<1, we shall find it to consist of a series of similar undulations on the positive side of the axis of v, the least ordinates, answering to v=a+(2m-+1) 2, being each = (1 —p):(1+p), and the greatest ordinates, answering to v=xt2mz, being each =(1+p):(1—p), as im the figure, in which OX=2. If 1—p be exceedingly small, the ordinate is everywhere small except when cos (w—v) is very nearly =1, in which case the denominator is also very small, and much smaller than the nume- rator. If we find the area of this curve from v=x— 7 to v=a2-+7, or indeed from v=a2—k to v=2+h, provided k be sen- sibly less than 27, we see that (1—>p being very small) no portion of the abscissa contributes sensibly to the area, except for values of v very near tox. Let1-p diminish without limit, and the curve becomes more and more hear to the axis of v in every part except where v= nearly, so that the limit of the curve is the axis of v and the positive parts of a set of straight lines perpendicular to it, at distances y=-2mz from the axis of y, m being any whole number, 0 included. The whole area seems to vanish, but it is not so in the formula, for on examining, as in page 615, the value of fyda, it is found that the diminution in breadth of the parts adjacent to v—xr+t2mr7 is compensated by the increase of the ordinates, sO that 2r Square units are left as the limit of each portion, one portion being from v=2+2mr—7 to v=«#+2mr+7. If a new curve be formed by multiplying every ordinate of the preceding by gv, the nature of the final limit will not be altered as long as ¢v is finite, and the limit of each portion of the area above described will be 2% x square units. Hence the theorem in page 615, and also the reason why extension of the limits gives sums in page 623. When we suppose z to vary, we pass in thought from one Such system of undulations to another, and there is no reason why 2 should vary continuously, or why ¢ should be a continuous function. We are thus able to lay down the formula for any ordinate varying con- tinuously or discontinuously, within the limits z—w and a+7. By using + (v—x):/, we are able to introduce the limits r—/ and w-+/. 282 623 DIFFERENTIAL AND INTEGRAL CALCULUS. Finally, by increasing 2 without limit, we arrive at Fourier’s theorem, an expression for any ordinate varying continuously or discontinuously, in any manner whatever, from e=—c@ tow=+ cc. I now show how that theorem furnishes a complete and natural expression of discon- tinuity of any kiud. We have gr=x'fp dw { ftecos w (v—a) pv dv}, where $v may undergo any number of changes of law, and $x would be found by actual calculation todo the same. Let us suppose ¢v=0 from v=—c to v=a; dv=—Wy, a continuous function, from v=a to v =08 and dy=-90 fromv=b tov=oc. Obviously, then, that part of the first inte- gration which is made from — o toagives nothing, and the same of that from b to & ; whence a and6 may be substituted for — oc and + ©, and we see in rf} dw { [2 cos w (v—a). $v dv} a function which is ¥v from v=a to v=b, and 0 everywhere else. But at v=a and v=8, it only gives 4wa and 4y¥b. Thus, if yu=1, we find that = {du} { cost o—2).do}, ie -{= (b—2x) w wef sin (a—2) Dap ee w i i w is 0 from a==— ce to r=a, 4 when x=a, 1 from «=a to x=), 2 when a=b, and 0 from w=) to c= : a prolixity of expression which might be more briefly, and sometimes usefully, represented by — & (0) 4@) a(1)5(4)6(0)e. And if we would express that the function is 1 at, as well as between, the limits a and b, we might write it thus, — @ (0) $a (1)b} (0) ; or perhaps —cc (0) (a,1,b) (0) might be prefer- able: the value of the function being in all cases in the middle of a parenthesis, and limits being written outside or inside the parenthesis according as they are included or excluded in the description. The preceding expression may be actually verified, either absolutely by analysis or approximately by computation, for both the integrals are finite and convergent. We shall presently arrive at the result fj sin kw dw:w= +47, or —4x, according as k is positive or negative. Now, a being the less of the two quantities, & is positive or negative in both the preceding integrals, according as @ is 6: these © integrals, then, destroy one another. But if #>a<6, the first is $3 and the second —4zx, so that we have + '{3r-+4z} or]. And when wa, the second vanishes, and the first is '.}0 or $; when a=), the first vanishes, and the second is —+~'(—4z), or also $, whence the result is verified.* Observing that in + "fj dw { [icosw(v—x) .$v dv} we can always construct the expression when $2, a, and 6 are given, we may denote it * It will thus appear that the verification (2) in page 619 shows the force of the theorem exceedingly well. It was first seen by the late M. Deflers, professor of the Bourbon College; and Poisson has shown his opinion of this verification by citing it whenever he proves Fourier’s theorem, which he does in four or five different places. But the defect alluded to in page 619 cannot be denied, and I have no doubt that sin z:z should be said to make the function integrated vanish, not merely because («0 )— vanishes, but because sin (0 ) is of the same dimension as ¢e-?, ON DEFINITE INTEGRALS, 629 by Fi oz, and $a Fi 1 is an equivalent of this. If, then, we wish to ex- press a function which is $x from a to b, wx from b to c, yx from ¢ to e, &c., &c., we have it in PF? ¢x+Fj wae+Fiyr+...., with this excep- tion only, that v=w gives 3a, c=b gives +(bb+%b), r=c gives * (We+yc), and so on, . To take another example: suppose it required to find a function of x which is =r from t=O to r=1, and =0 everywhere else. First we have cos Ww (v— 2x) we is Do, J cos w(v— 2) .vdv=— sin w (v—a) ++ j Fae a : from which ml dw Lf cos W (v—2) vdv } TJ 0 0 1 (°fsmw(1—a) . cosw(l—2x)—cos wer _—— = +—_______-_—_— } dw; ae ine w ew” cos (l—2) w—cos rw cos (l—2x) w—cos cw and 7 ( ) ape BA Sn) I 008 20 w* w (1-2) sin (1-27) w—2 sin rw _— —— and the first term vanishes at both w=0 and w=. Hence if P, denote a'fsin kwdw:w, we find for the function in the second line (which Fourier’s theorem shows to be that required, and which we are now verifying) P,_.- (1 i) P,_,+2P,, Or & (P,+P,_,). If <0, P,=—4, and P,_,=4, or the preceding vanishes; if x=0, it also vanishes ; if r>0<1, P,=P,_,=3, or it becomes =z; if 2>1, P,_.=—+}, P,=2%, or it vanishes again; when z=1, P,=3, P,_,—-0, or it becomes 42 or %. The geometrical explanation of this is as follows: if we take the curve whose equation is, for any point (2, y), } {} haa eos : —— { e—*” dw | af COs W (v—a).vdv}, Be}. 6 0 k being a small and positive quantity, we should find it to have a form resembling 1 2O0CB34: the smaller k becomes, the more nearly does OCB coincide with OB, and B3 with BA, while the undulations preceding and following diminish without limit m every ordinate. I*inally, when k=0, the limit of the curve is the dark line 10BA4, but when z=OA=1, the formula does not become indeterminate, but gives only AB, whereas every point on AB is in the limit of the curve. This is by no means the only instance where, when one side of an equation takes an indefinite value, the other gives the mean of all the values denoted by the first, 630 DIFFERENTIAL AND INTEGRAL CALCULUS. I now proceed to another branch of the subject, namely, the transform- ation of integrals which arises from giving impossible values to con- stants containedin them. It is a matter of some difficulty to say how far this practice may be carried, it being most certain that there is an exten- sive class of cases in which it is allowable, and as extensive a class in which either the transformation, or neglect of some essential modifica- tion incident to the manner of doing it, leads to positive error. It is also certain that the line which separates the first and second class has not been distinctly drawn. ‘The best plan will be to examine some cases of the transformation, both in their results and in the verification of those results, taking those instances which are valuable in themselves as the subjects of examination. Let us take [Se cosbaa" "dx and fpe™ sinbx x*dx, where a and m are both positive, and b is a real quantity: these integrals must then be finite. Now {je a"' dx=p™ Fn gives as follows ; let 7=,/(@+6*), O=tan' (6: a), then feet ng Nee eto Calpe =r" {cos nO F sin nO J(—1)} Fa; whence, adding and subtracting the two equations here written, and dividing by 2 and 2,/(—1), we find I'n.cos {n tan (b: a)} @ ar n—] pk | foe cos ba x See CNY | | Te ae I'n.sin {nx tan (6: a) } {iC b) ye These results can be obtained without the introduction of ,/(—1), by a process similar to that in page 576, and can each be verified in two distinct ways by differentiation. Let the first of these be C,, and the second S,, which gives foer@ sin ba a" 'de= b dC,, dC,, dS,, dS,, i db Tae? ida ape ab ans ahem We might verify either of these, but the following will be better. For a and bwrite rcos@ and rsin®@, and taking r positive, then cos 9 must be positive, since 7cos9=a. We have then cos cos © -—7c0OBO.a ~. ct yn = pan : fe 3 sin (rsind.a).a°o dr=7 tin sin (nO) n dé {sin 6 cos (n-+-1) 0—cos @ sin (n+ 1) 0} =r sin 6C,,,,—r cos. 08,4,= ri (n+1) yrth =—n7Inr "sin nd, the same as from the second side of the equation. In a similar way, dC,:dr, dS,:d0, and dS,:dr might be verified. Consequently, if the two sides of the preceding equation differ at all; it must be by a function of m and constants not depending on r and 0: but this cannot be, for in such a case C, and S, would not be reduced to f e—* x"—'dx and 0, orr-"F'n and 0, by making 02=0; to these they ON DEFINITE INTEGRALS. 631 are reduced as the equation stands, but would not be if a function of 7 were added to the second side. What value of @ is to be taken, of the infinite number which satisfy rcos@=a,rsind=b? It must be of the form 2kr+,, 0, lying between —1q and +4, for otherwise cos@ would not be positive. When 7 is. integer, it matters nothing what value of k is taken, the second side not being altered by any change of k from integer to integer; when 7 is fractional, the case is different. But the integrals must be reduced to 7" Fn and 0 by sin 9@=0, whether 7 be whole or fractional, but in the latter case 7" Fncos (2knx+7n9,), which becomes r~ F'n cos 2knz, is not so reduced unless kn be a whole number, in which case 2knxe may be suppressed. Consequently, 6, is the value required, or 9 must lie between —}$x and +4r. The following are remarkably particular cases, and deductions from them: b is supposed positive. — — ee ial S . 1 fo cos bx. a" *da=b™ Fn cos yn, fosin br.a*“dx=b™ Pnsin far ;) Pa es 1 (5 cos 62" sa" de=— 6 '™ ( ) eos (GE) m 2m ) : cn ) sin w |. 2m . Write F'n sin 3nx in the form F (n+1) {sin $n7: 7}, and let m diminish without limit. 2 AS b { Neca ee Fe sn OF da=4nx* (pages 572 and 628). ve 0 v 0 9 ant n-+ : | ee f 5sin b2".a°dr=——b = F m n 1 n 7 Let n=1, which gives fj cosbv.dr=0, fosinbrda=b", results already noticed. If all the preceding process be carefully examined, it will be seen that there is nothing in the change of possible into impossible quantities which either makes the subject of integration become infinite between the limits, or prevents us from expanding the possible form f et? eh "dz into an infinite series, then making b become b,/(— 1), and concluding that the result is identical with the impossible form. But if the change should make the subject of integration infinite between the limits, it is by no means to be inferred that the results of the change are true. Again, if the change should turn a convergent series into a divergent one, in the subject of integration, it is not to be inferred that the results will agree after integration ; for it has happenedt that ‘discontinuity is introduced by the integration of divergent series, and there are no means of knowing when this happens, and when it does not: Thus fi ¢rdr=(fi —fr) $x dx= fi {o (xt+a)—o @+d)} de. Write kx for x in the last, which does not affect its limits, and we have ’bada=kfy {> (kr+a)—$ (ka+b) } de. * It is obvious that a change of sign in 6 changes the sign of the result. + One of Poisson’s objections to divergent series (Journ. Ke. Polytech. Cah. 19, p. 484) turns upon this point. It seems to me that the objection here is not to the divergent series, as such, but to inferences drawn from its integration. Yd 632 DIFFERENTIAL AND INTEGRAL CALCULUS, © Let k=,/(—1), and first let $v=e*, we have fo CetevO — VY) da=(e"—e') { fF cosadr+i(— 1) fp sin adv} =,/(—1)(e*—e’), and this multiplied by /(—1) gives e’—¢*, the obvious result of ee pu dv from r=ato«=b. So if we take dx mm . at 1 l i oe se xr =/( Df \(a+a/(—1))? re ee (Le, we should find a@~'—b"" as the result of both sides. But let us now apply k=,/(—1) to the theorem [> $2 da=k {> 4 (kx) dx, where $z is, say (1+). We have then cid (l+a*)"'dr=,/(— 1) {¢ (L—a*)“"'dz, an equation which we cannot either affirm or deny, since the subject of integration in the second side becomes infinite between the limits. I now proceed to give some account of the methed of considering such integrals proposed by M. Cauchy. Let (1--x?)'=V, then fi Vda=h log (2Q—k)—4 log hk, a calculable result, however small k may be; and fri Vdr=$ log l—4 log (2+), also a calculable result, Hence [Vd from 0 to ©, with the exception of the part from 1 — to 1+/is }log (/:k) —4log {(2+/):(2—A)}, of which the latter term diminishes without limit with / and 1; but the former entirely depends on the ratio in which J and k vanish. If we now take the part from 1—k to 1+, we find it to be $log(—k:1)+4log {(2+): (2—k)}, which, if 2 and k are diminished so that k:l has the limit e, has 3 log (—«) for its limit. If a=1, this becomes }log(—1), or} (2n+1) 2 V(—1); and if we multiply by ,/(—1), which gives —(n-+4) 7, one of the values so obtained (for m= — 1) certainly is'f> (1 +a°)— dr, or 4r. But, at the same time, we cannot form a distinct idea of atk Vdx by summation, as in page 100, because V becomes infinite when r—1. If $x become infinite when z=a, and if (e—a) dx be then finite and =A, the value of {4%} da da, or r atl As o " bx (t—a) must approach to A | a—k X a : a+l dx >] a—k U— A : ie o — —— ae a A i ) as k and J diminish without limit: that is, assuming the ordinary rule of integration, in spite of the infinite intermediate value of (w—a)m. In the same way, if % (#—a).¢(«#—a) be finite and =A when Le ya being the dimetient function (page 324), which satisfies this con- dition, Af (¥x)"'dr is the limit towards which f¢xdx approaches, under the same extension. Many results may thus be obtained, and many incontestably true, but all labouring under the same difficulty, namely, the want of definition for f .pxdx, when dx becomes infinite — between the limits. It will certainly not do to define it as p,b—gya, — where @’,v=¢2, for such a definition would give the same result, no | matter how many times w becomes infinite between @ and b, which, in the developed theory to which we have alluded, is not* always the case: and the summative definition of page 100 is unintelligible. | There are, however, some results obtained with reference to this subject by M. Cauchy, which, though not quite complete in their * M. Cauchy has shown, as in the results we shall presently obtain, that every place in which the subject of integration becomes infinite gives a term to the result, generally speaking, ON DEFINITE INTEGRALS. 633 fundamental explanation, ought not to be omitted. A function of the form 9 (a+) —¢% (a—8) is continuous, and vanishes with 0, when da is finite: but if Ga= oo, there may be an evident discontinuity. Thus log (a+ 6)—log (a—@) vanishes with @, except when a0, in which case it is Jog(—1) for all values of 9. If, then, we have § hse Snr, which represents the area of an hyperbola from w= —~m io r=n, we find log n—log (—m), which can represent no area. But if we remove the portion f+} a—'dz, @ being infinitely small, we also remove that dis- continuity which, though essential to the function, has no geometrical interpretation. We thus get log x —log (—m) —log (—1), or lug (n:m), which is algebraically intelligible. Thus, if n==m,we have 0 for the area, which is visibly true, since its positive and negative portions are then absolutely equal. But if, instead of removing the portion from —@ to +9, we had removed [t¥,a7' dx, p and y being any given finite quan- tities, we should have had log (vn: um), which we may make any- thing we please. It seems, then, that if we wish to accommodate our notions of foe dz, when ¢x= cc between the limits, to those which we derive from applications, we must consider fox dx as divested of the part j a px dx, where fa=a. And if (w—a) dx be finite and =aA, when x2=a, we find, as before, Alog (—1) for the effect of discon- Muity which is to be removed. When this result of discontinuity has deen removed, M. Cauchy calls the remainder the principal value of the integral. Now, if the limits of the integral be x, and 2,, and if from fadrdr, we remove the portion f7t) px da, there remains [35° pa dx + f2., gxdx. If the portion removed, namely, eon, px dx, diminish without limit with 0, then the limit of the remaining part is [71 px da. But if the part removed have the limit L, then Sa uvdx—lL, and not 2) Px dx, is the value of the portion of area of the curve ype. Leaving for a moment the case in which the subject of integration Jecomes infinite, take the identical equation Ge sae d*z nian 12 d dz fee tp r= 8 (pS)=2 (pF) dx dy dedy dx dy CUNO er md integrate both sides with respect to x and y, namely, from x, to x, wd from 'y, to y,. Let z=¥%(a,y), and let w/ and ue, denote results of differentiation with respect to x and y. Sa iS (@, 1) w'(a, 1) — fi (2, Yo) ¥ yp! (2, Yo) j da =JISY Gr Yi, Yb Cay) ¥, (@,y)} dy. This equation* is absolutely identical, whether the function be pos- ible or impossible, for any degree of approximation may be made to it, is In page 289, and the first side represents the limit of a process which ‘onsists Jn summing rows and adding the results, each one in the row hus becoming a column, while the second consists in summing the same ‘olumns, and adding the results, each number in a column thus ‘ecorning one of the first rows. Thus, if (a, y)=a-+y,/(—1), we ave (k=,/(—1)) fz tf (aty,k)—f (a+y,h)} dx=k ft f (a+yk)-f (atyh)} dy (1). * This should be called Cauchy’s theorem, on account of the results which that minent mathematician has deduced from it, 634 DIFFERENTIAL AND INTEGRAL CALCULUS. For instance, let fr= e~™, or f (a+ yk) =e antax” (cos Zary—k sin 2ary) | f eayt (23 {cos 2axy,—k sin 2ary,} e~“ dx | —eay3 (21 {cos 2ary,—k sin 2axy,$ a") de | —=hke-azy (% {cos 2ary —k sin any} e dx —he—ar? (1 feos 2ax,y—k sin Qaayy} e” dy. | Let #,=-+; the first term of the second side vanishes and the equation of possible and impossible parts gives eay} if +? 2-1 cos Qaxy, dv—Eay J ty & "COS 2ATY, dx = — E— 425 f ve" sin Qax,y dy gay? (i? e— sin Qary, dv —erys fz e~*” sin Qaxy, dx = eax} [yr e™ cos 2ax,y dy. : Let 'a,2=0, yo==0 5 we have then (page 619, verification 3) Seen cos 2any,da=e—ayi foe da=h/a.a* e-ayi | 4 {oe sin Qary, da=e—ayi J e™ dy. | Many other such tranformations may be made, and with the utmos! certainty, as long as fx does not become infinite between the limits But let us now suppose that f (e+yk) becomes infinite once only betweer the limits, namely, when x=a, y=b. Avoid the point by integrating from r=, to c=a—Q9O, and from w=a+é to r=, also from y=y, tt) y=y, in both cases. We have then | oo {f(@+y kf (et yok)} dx =k fis {f(a—O+yk) —f (e+ yk)} dy Saro{f(ety.k)—f@tyok)} dx hf { f@rtyk)—f (a+0+yk) } dy | If we add these together, and then diminish @ without limit, the firs) side presents no singularity, since neither f(a+y,k) nor f(at+tyoh becomes infinite from +=.2, to x=2,; so that the limit is the complet integral from x, to z,: but on the second side we see KASS Gry) —f a tyk)} dy | hfK {fat O+yk)—fa-O+yk)} dy. | The first term being what we should get in an ordinary case, and thi second an integral which would vanish with 0,if f(w+y v— 1) did no become infinite, but which may have a finite value when 0=0, as in th, instance given (page 633). Again, since all parts of the mtegral jus named must vanish (when 6=0) for any limits which do not include ele ments adjacent to y=b, we may, without altering the value of the limit take y from —c to +o if 6 lie between y, and y,; but if y,=b, W must only allow those adjacent elements to enter in which y>0, m4 which we may go on to y=«, so that y, and w may be the limit a ae — (2). } t ; | ON DEFINITE INTEGRALS. 635 Similarly, if b=y,, we must take —a and y, for the limits.* Con- sequently, the correction for discontinuity described in page 633 is the subtraction of kf {f (a+0+yk) —f (a—0 +yk)* dy, with limits as just shown. Let (z—a—bk) fe=wz be finite and =A when za-+bk, then since only values infinitely near to z=a+bdk affect the preceding integral, we may write instead of it, first, at (a+O+yk) Sania Ae a p O+(y—b)k —0+(y—b)k Now f¥z.yrdz, between limits infinitely near to p, cannot, if wp be finite, differ from yp fyxdx; hence we may in the preceding write A for ¥ (a+0+ yk), and for y%(a—O+yk), and the result is, making y—b=z, dz dz } “26 dz 2 eee Berle la When this is taken from —cax to +o, it gives 27kA ; but when from —« to 0, or from 0 to ©, it gives tkA. And if there be any number of such roots of { f(#-++-yk)}~" between the limits, and if A be determined for each, the correction for discontinuity is the sum of the individual corrections, so that we have (k=,/(—1)) Sai f@+tyh—f (ety, dx Sh fii Sait yk)—f (a+ yk)} dy—Qrk ZA in which, however, $A is to be written for A in every term in which } 18 Yo OF Yi, =a and y=b being values for which f(@+yh) is infinite. It might also be shown that $A is to be written for A if a=a or a,=a. Now A is the value of (t—p) fx when =p and fp=«: let fx be px: yx, and let Yp=0, Pp being finite. The value of (w—p) fx is then (Chapter X.) that of dr: wr, when r=p. Let m=—a, y~=—+a, ¥=0, y=, and let f(@+yk) be a function which vanishes when x=— oo or + © independently of y, and When y= independently of x. We have then f(x+y,k) =0, F(@.tyh)=0, f («,+yk)=0, and the equation (3) becomes Ste fa da=2rk ZA (4) ; in which all the roots of fr must be taken which give positive coefficients of £ (0 included) since the limits of y are 0 and », but for every real root (6==0) 3A must be written for A, since 0 is one of the limits of y. Example 1. fr=¢xr:(1+2°), ¢v=c having no finite roots. Here the only admissible value of 6 is 1, the root of 1+. being k: the cor- responding value of A is #k: 2k, and we have t? dx dx J —» 142° (3) ; —rh (V—1) (5). * This is a new application of what may be called instantaneous integration, on which I do not think it necessary to dwell after what has been sail in pages 615 and 627, ad 636 DIFFERENTIAL AND INTEGRAL CALCULUS. uv Let dre", a being positive, & (w+yk)=(cosar-+-ksin ar) a, which yanishes when t=00 or —®, and when y=o (N. B. e7™ would not admit of the preceding demonstration being applied). Also op (k)=e~*, and we have " Conacdr ” sinardr { 1 = eMeD | Se I Gime —o +z : J —-2 1+ ar of which the first term is twice the same integral from 0 to ow, and the second vanishes, which gives the same result as in page 577 for focosax dz: (1+a*). But it must be noticed that if in (5) each of the portions of the integral, from —o to 0, and from 0 to ®, be infinite and of different signs, there may be, as in preceding instances, an effect of discontinuity, for the removal of which no provision has been made, Let gr=a") whence, if m<2, @v:(1+a") satisfies all the conditions. We have then | bi Pa"dn -w(—1)2 apa whence ener Sept =——-—. —__- as (-1)7F+(-D? T 32 cos {1 (Qk+ 1) mr\ where k may be any odd number. But since this integral cannot become infinite until m=1, we must have 2k-+1==1 or +: 2cos(4mz) is the value of the integral from 0 to , which agrees* with page 575. If. m==1, we have ada rw ode day be a ola ; _» lL +2? f i eae 1 The two first are correct ; the third is a singular value, and should be | <0. It can only be obtained by remembering that log,/(1+ Cauchy’s Réswmé des Legons sur le Calcul Infinitésimal, Paris, 1823, ON DEFINITE INTEGRALS, 637 —30(1) and $4(—1), and, both roots being real, we have rk SA or tk ip (—l)—¢ (1)} for the integral. Hence | fil Ord eV =1 (6 (-1)—6 (I. v Let Pt=2"5; reasoning as before, we have so wm x dat —= (—1)"—(1)” T “ 1+(—1)” oy Let x°=2", which, 2 being ‘positive, does not change the limits, we iave then Dy saeeadimmhad ¢ peetg Si ans Aone +1 Begs ee ay oe TR ry =- cot —— 7. nr 5 . n wt Me tie tan dmr. 2 0 1—2 Let it be remembered that by the symbol f%, when the function ategrated becomes infinite between the limits, say at wc, we mean othing but the limit of {rae bos when @ diminishes without limit. jut whether this is always the meaning of the symbol when it is at- uined in the usual way is another question.* Example 3. Let fr=¢zx: (1-2), where 6r= has no finite root. ‘he roots of #"+1=0 are cos mo +,/(—1) sin m0, where 0= 7: 2n, w all odd values of m from m=1 to m=2n—1; the value of A cor- sponding to each positive coefficient of (—1) is of the form we: 2nx"", or —xhxr:2n, where w=cosmO+J(—1).sinmd. We ave then 2 OOD Oe oe l leg a == /=1 zi" 1 { (cos mO+ V —Tsin m0) & (cos mo+V—1 sin m@) |}; L 1¢ summation being understood of odd values of m. Let gr=e¥VO ; e have (cos m0 +- /—Isin pee Ne me = e~aein? Sos (mO-+a cos m0) + ¥—Isin (m0+ a cos m6) }. If we pair the values of m thus, 1 and 2n—1, 3 and 2n—3, &c., we tall find, if n be odd, a middle term », giving $x for m9, and €~* for px; but if m be even, there is no middle term. And if the last be ant+Q,,../(—1), it will be found that P,,-+ Pam=0, Qn + Qon—m=2Qn; hence, summing, and multiplying by —2,/(—1):n, and proceeding as Example 1, page 636, we have + sd ~ cosardr - a 4 nodd,m= 1,3, baa of tte sin (mb racosm)} 5. sn —2; 4 5 AT n n even, m= 1 T . R ? 2 : soe _= {een sin (mO-+acosm9)} 3.5. . n—1, * We have seen that substitution of »é and sé for 4 in the two integrals would wea different result. Why is it that all the results of the method agree with lose already known when =», and not in any other case? To this question no ‘Swer has been given, as far as I have seen. . ‘ + These results agree with those of Poisson, (Journ. Ee. Polytech., cah. XV1e5 1229, &e.), allowing for the misprinting of — for ++ before = in his first formula. 638 DIFFERENTIAL AND INTEGRAL CALCULUS. Now return to the formula (3), and let the whole process be per- formed on the supposition that k=—,/(—1). If, then, we take the | function f(«—y/(—1)) so as to vanish when y= -+ ©, and construet SB, the sum of the corrections for discontinuity for all roots of the form | a—b,/(-1), where b is 0 or positive, we have, supposing f (a-+y/(-1)) | to vanish when z= or —o, the equation f +8 fe da=—2rk ZB. «e----- (5). Adding (4) and (5) together, f<3 fx du=rkz (A—B)..... (6). Now observe that =B and ZA both contain the same terms for every | real root, consequently the real roots vanish altogether as to their effects, and we have the following theorem. If fz be a function which vanishes | when # is --0 or —, independently of y, and when* y is +@ or —o, independently of «, and if for every pair of imaginary roots of fo=o, p=atbJ(—1), q=a—b,/(—1), be constructed the values A) and B of (a—p) fe and (7—q) fz, when x=p and q respectively, the) integral ft3 fa dx is = /(—1) 2 (A—B). "| Exampeir. Let fe=sinarz:snbe(1+a°). The imaginary roots in| question are z= +, and } ‘ _sin ax (¢%-+4~”) +-cos az (e-% —e"") 2h l | tS (e+ yh) bx (e’¥ +e—”) +008 bax (<4 — 29) - “T+ (e+yh) 4 This vanishes for y= +0, when a< or =6, and also (as we shall presently see) when z=-++0. Hence we easily deduce, v= £/ beimg) the imaginary roots of 1+a°=0, act sinar dr _ (ee 1 sin(—ak) 1 ie e*—e* sin bk 2k sin(—0k) —2k) eam a being < or =b. The same from 0 to o has evidently half the value. - Generally, let us have fe=¢x :(1+2°), with the same conditions, T° pdx _ a Lo oe ae We have hitherto supposed that («—p) fz is finite when =p and fp, but let us now suppose that (1x—p)"fx=Yyer is finite, and also its diff. co. Returning to the expression kf {f(a + O+yh) —f(a—60+yk)\ dy, substitute wax: (a—a—bk)” for fx, whence BLM %(@+Oryk) _ etd yo YO+(Y—b) ky" {—O+ (y—b) hy For y write z+, changing the limits into y,—6 and y,—d, and expan¢) wy (a+bk+zk+6) in powers of zk+6, writing p for a+bk, and m ant m for y,—6 and y,—6. This gives a —- ——=7T _. sinbda 1+2? 5{oW—1)+6(—V-1}- * M. Cauchy deduces that the function need only vanish for y=+o, but as i) happens that in all his examples the functions do vanish for y=—o as well, - suppose that this condition is inadvertently omitted, at some step of the demonstra tion, which is a very long one (Mém. Say. Etran., vol, i. p. 686—717). ON DEFINITE INTEGRALS. 639 f° {vo( kdz ix a ?( kdz kdz ; ny OGRE OY Ch") * YP \ Gre [Se Uhe first term of which, when integrated, has yp multiplied by 1l—m)* {(m k+0)-"— (1) k+0)'-"— (yn, R—O)-™ (7, k—@)'""} ; vhile the succeeding terms have 2—m, 3—m, &c. for l—m. Now vhen »—m is not =0, the preceding certainly diminishes without limit vith 6, however great the values of 7) OY n, May be. If, therefore, m be ) positive whole number, the coeflicient of y"—» p becomes indeter- minate. The value of A, treated as in page 635, will be the limit of kyr) p “ci dz dz w"") (a+bhk) 2.3...(m— 1) hkz+0 i =i ee aor. .(m—1) ubject to the same liability to be halved when y, or y,;=d. It might seem at first as if the preceding, applied to a fractional value fm, would always give 0 as the value of A. But when fV™dx is to be aken between limits which give different signs to V, m being fractional, yhere arises a difficulty as to which values of the mth powers of the posi- ive and negative quantities correspond to each other. Thus (—1)'** nd (+1)**” have each n values, but there can be none but a conven- ional test as to which value of (—1)'*” is to be used with, say, the value of (1)'*", If wand b be the limits, and if the change of sign take place t x=c, and if, moreover, Neg and se be finite, we can choose our own alues of the powers, and calculating each integral separately, we can ut the two results together. But when those separate integrals are infinite, I know of no attempt to ascertain the meaning of the complete itegral. The results of the preceding theorems, and of many others, have been aethodized by M. Cauchy into what he calls the Calcul des Résidus, rresidual calculus. The notation he uses requires a symbol for which new type must be cut, a necessity which, not liking the symbol itself, prefer to avoid. Let fe= cc when r=p, and let (~x—p)” fx be then mite. The residual of fx with respect to p means the coefficient of ™ (when there is such a term) in the development of f (p+), which an generally be expanded in negative powers of x if fp=cc. It is asily shown that this residual is what has been called A, when m is ity or any whole number. Let R® fx represent this residual for the oot p, and Ri fx the sum of all the residuals belonging to all roots etween p and q: also let R? fa represent the sum of all residuals elonging to roots of the form @+f./(—1), when « lies between p nd q, and £ between v and w. yo—b 1. The fundamental theorems of this method are, then, k being '—1) as before, Saf (etpbD—f @+yk)} de HKD S Gait yh) —f (Gotyk)} dy — 2ahkRey sh fe; rhich is universally true if r be written for 2x in every term in which 9 Or 2, is the possible part of the root, and y or y, the coefficient of the npossible part. Also 2. TE f (+ ot+yhkh)=0, f (a+ oh)=0, (42 fedr=QrkRt2% fr. 640 DIFFERENTIAL AND INTEGRAL CALCULUS at SUE oc +yk)=0, Ff (cet ey geod | te fe dv ak {RtE% fa—Rts. fr}. | A. Let f(e@+ chk)=0, H=0, Y=0, W=C, then fo (et+yk) dx= fp fx dv—k fof Yk) dy—2 rho" fx. | | 5. Let f(a@+ ek)=0, 2=0, Yor=0, Pi= C5 then | fz fe dx=2akRoy fr—k fo { f (aityk) —f (yk)} dy. | | | | | 6. Let fC o& +-yk)=0, m=— OC, N= + ©, y¥,=0; then +2 f(et-y, k)= fie fr dr—2QrkRi3;}} fz. 7. Let f(c +yk)=9, f(@+ ck)=0, n=0,m=C, Yyr= 0, Y= OG wey fo fe da=hfyf (ky) dy+2rkRo fz. 8. Let f(— ae +yk)=0, f(v+ ch) =0, H=—O, 4=0, w= 9 aie fru fx daz —h fpf (hy) dy — 2akR2E, oft. I shall conclude the subject of Cauchy’s formule (on which a grea! deal more might be said) by an example. Exampte 1. [+2 *(a-+ak)-", m being a whole number, and a and 6 being positive. The only root which makes fr= cc is LGR, which occurs m times. Now (a—ahk)"fx is (—k)™ ¢’, which, differ- entiated m—1 times, and divided by I'm, gives (—1)™ "6" e™, or he! po" g*§, which, multiplied by 2k, and ak being substituted for x, gives by the second theorem above (which applies here) Leer) e*V(-1) dx Orb”! ead ltfepp Fo) This theorem may be verified by differentiation with respect to 4, and it holds good when m is fractional and positive ; but it is not true when a is O or negative. The student may deduce the following for himself, using either the second or third theorem if dx UV (m+n—1) idee PM a des Beato ee 9 b 1—m—n eS She ; oe (a+ ak)"(b — ah)" oF ect 2) Em.0n If the second theorem he used, a==ak is the only root of fa=@ which applies; but if the third be used, c=ak and r= —bk both apply* a and 6 being positive quantities. | Before proceeding further, I shall finish what remarks are necessary) on the singular symbols sin c and cos cc. ‘The continental mathe ticians with one voice pronounce these symbols to be indeterminate in value, which is strictly-true as far as d priori considerations are con) cerned; for a periodic function of x cannot be said to be in one part ol) its period rather than another when @ is infinite. If, however, we assume da to stand for x terms of 1-1+1—...., we might equally conclude that gx is indeterminate when 2 is infinite, no reason existing to prefel| 0 to l or Lto 0: nevertheless, there exists no doubt that this series ON DEFINITE INTEGRALS. 645 ‘Tepresents halfa unit. And in many different ways (some of which are shown in page 571) sin @ and cos cc appear in formule which can only be made true by supposing them both to vanish. It must also be observed that every instance in which the case can be clearly tried by anything resembling an @ priori method confirms the conclusion that indeterminateness of value is to be removed by taking the mean of all ‘the results between which the doubt arises. Two remarkable classes of instances are as follows :— 1. Take, for example, @ + br + ca? + ax*+ brt+cex'+...., or (a+ bz+cx°):(1 — 2°). This, if a+b+c=0, becomes 0:0 when e=1, and its value is —1(b+ 2c), or atb+e—14(b+2c), or $ (3a+2b+c), the mean of a, a+b, anda+h+c. Now when r=1, the successive summation of terms of the series gives ad, a+b, atb+e, a,a+bh,atb+c, &e. 2. In applying Fourier’s theorem (page 629) to discontinuous func- tions, we find that at the point where the discontinuity takes place, and a function which generally can have but one value might be expected to have two, it takes neither, and gives only the mean between them. If we ask for the mean of all possible values of sin x or cos x, we find 0 in both cases, since every positive value is counterbalanced by a numerically equal negative value. This affords an additional confirma- tion of the general principle. But it would not be safe to apply this to (tan x or sec x, &c., or to any function in which oc is one of the values. Unquestionably the clearest way of considering such indeterminate results is to make them the limits of others which are determinate up to the limits, whatever they may be at the limits. Thus 1—-1+1-—... =} is the limit of 1—a+a2°—....=(1+.2)~, a result which is arithme- tically intelligible whenever « is (no matter how little) less than unity. It must not, however, be dissembled that this difficulty still remains, namely, that we can have no positive proof that every result of in- determinate form will give the same value whatever may be the function from which it is deduced asa limit. Thus, though we can show from Ay Aa,-& Ay—a, t+, x°—-....=— —- —— hg, Ife (+2) oe a el that 3 must be the limit of a~bw+ca*—...., whatever law a, d, ic, &c. may follow, provided they approach to equality when w approaches to unity, it is not demonstrable that in all cases sin cc, considered as the limit of, say f>dx.cosx.dx. (the limit of ¢x being unity,) is 0. Difficulties of this sort must occur as the ideas on which analysis is founded are widened, and there are so many on which we now look as completely removed, that the occurrence of new ones is matter of hope and not of discouragement. In the mean while it is of some importance that the student should, at the proper time, be made aware of their exist- ence. ' Those of the continental writers who reject divergent series seem to have no objection to retain those cases which separate divergency from convergency, such as 1—1+1—..... They sometimes express them- selves as being willing to consider this series as being l—x+a°—..., in which z is infinitely little less than unity. But this principle, taken alone, would scem to me to be very unsafe. For instance, « is the limit of €~*, x, when ¢ diminishes without limit. However small c may be, 2T 642 DIFFERENTIAL AND INTEGRAL CALCULUS. this function vanishes when @ is infinite; it must be said to do the same, then, when cis infinitely small. Whence 2 itself cannot be treated as <~*. 2, ¢ being infinitely small: and were it not for what we know of l—x+2°—...., when @ is greater than unity, I am inclined to assert that we should gain nothing by the fictitious representation of 1-+-1—IL -+-.... above alluded to. I now proceed to another class of questions depending on the funda- mental integrals in page 605. It will be observed that the use of these has been avoided in pages 610, &c., as likely to lead to the use of the unestablished proposition that a divergent™ series vanishes when all its terms vanish. If, however, we have a series of the form A )-+ A, cos 2 4+-...., where AytAi+.... 18 itself a convergent series, we may then | be sure that multiplication by cos mx and integration from, say 0 to 7, makes the whole series vanish, with the single exception of the term” A, f cos® ma dx. Now take the two equations (& being J(=1)) je Lip (othe) + (ethe™)j=oet dr hoosvt pre Ge Bi ek, h? sin 2v = $b (a@+he") —$ (@+ he) $= b'xhsinv+ px Stearns on which may be easily deduced, as in page 244, Let av and fv be any | functions which from 2==0 to v=- are the same as A,+A;cosv-+Ag cos2u-+.... and B,sinv+B,sin 2v+.... Multiply the first equa- tion by av A,+..., and the second by (bv=B,sinv+ ..., and integrate with respect to v from v=0 to v=, Every term then (page 605) vanishes, except those which are retained in the following results, which are only to be relied on when the series are convergent. as ae fo (athe) +6 (@t+ he) av dv ramets WA aA, geht aga + eats Za {9 (2+he)—9 (e-+he™) } Bo do =D; $l2.h+ Bag’ + i lb From which may easily be deduced (pages 242-3), making hb (x+ he*) —V.,,, and a lying between —1 and +1, 1—2acosv+ a? a | *(V,+ V_,)(1—a cos v) aD mg eriere aa : a (*(V,—V_.) sin vdv Se Ker es iain =% (r+ ha) —o2 (2). th.) . 1—2acosv+a?* Make a=0 in the first, which gives 2 {(V,+V_.) dv=2¢z, subtract, the half of this from the first equation itself, which gives * Poisson (Jo. Ec. Pol., tom. xii. p. 484) has made the errors which may arise, from such use of divergency an argument against all divergent series. There were two specific reasons why his particular use of divergent series should have there led. to error: the first noted in the text above, the second that previously mentioned in page 631 of this work. ON DEFINITE INTEGRALS. 643 * (V4V_,) dv — 2nh(e+ha) > L—2acosu+a? J]—a? Let dv=z', and make +=0 in the result, * cos cv. dv Ta » L—2acosv+a® 1—a* This equation is only true when c is a positive whole number, for itis only in that case that (w+he)* can be expressed in integer powers of e™ when 2==0. Let @r=e™, then V,-+ V_,—e?t"*" 9 cos (chsinv) and V,—V_, p Settetose 2k sin (chsinv). Make x==0, h=1, which affects neither | the convergency of the series nor the generality of the result, and we have, from (3) and (2), 1—a? (* °°" cos (csinv) dv » iL—2acosv+a? ca ? T 2a (7 €&°°*”sin (c sin v) sin v dv i. ae age | BA} se 1—2acosv+a Now (1—a?*):(1—2acosv+a?)=1+42acosv+2a®cos2vu+.... and asin v: (1—2acosv+a*)=asinv+a’sin2v+....: expand both equa- tions in powers of a, and equate the corresponding terms, which gives (n being integer) 2 : _ fi °°” cos (c sin v) cos nv dv vin nm ==; €°°°°" sin (c sin v) sin nv We aod yt except only when 2=0, in which case the first integral =2, and the second =0. These may be easily verified by differentiation with respect to c. The following result is obvious, fies (cos nv+-,/(—1) sin nx) dr=0, where n is any integer, positive or negative: but when n=O, we ob- viously have 2 for the integral. Making k=,/(—1) as before, we have then (27) f+ e* dx is 0 when n is any integer, and 1 when n is nothing. The following theorems are then obviously true, whenever the series which must be employed in producing them are convergent. 1 se ie 73 ] ee kx —knx di#= Ma pie (ate) do=dga, 5 Sid @te').e™ di=s 1 (‘**¢d(a+e").dzx ac | 1 —he* L (*G(ate").dz_ =¢ (a+h), on mye =¢a ; and all these theorems may be altered in form by turning f t* ox dx into f-{¢r+¢(—az)}dr. Again, if Pxr=A,+A,r7+...., and if Wrz=B,+B,7+...., we have = t= gets we de = A,B, +A, B, +A: Bet... T DIFFERENTIAL AND INTEGRAL CALCULUS. 1 =5- Saige” we + pe (pe } dz 1 “te het dx +x pe Ay+ A: +.+.- Sho 1[-e” Amat. = [ade l Tar ge”. e—nke dx A Ad Heine | oe Let yr=A,+A,c+...., and develope ye™ y (e" fe) .e M=V. We have then | ce AY ee A, we fe +A, 1 ae Chevy gh 1 jaghibens whence, remembering that fye*.<"* da from —7 to +7, and divided by 2z7, gives the value of y)a:2.3....2 when w=0, and that —n written for m would give 0, we find d 5: +" Vde= A, (We fr) + mes {wlan (fos) A; ( @ 3 A, d I 4 +B (Bite tor} $55 (Ga ve 94} OD: parentheses denoting that wv is made =0 after differentiation. Let o# be a function which has one root =0, and write v:@x for fr. It then appears, from Burmann’s theorem, page 305, thatif A,x=1, A.=4, As=3, &c., the preceding series is nothing but the value of yx—y0 for that value of « which gives @z=1, or solves the equation v=/z. But xv | being now 7+32°+.... is —log (1—2), whence we find that, « being some one of the roots of =z, the following equation is true, ] VO fit {we log (1 —e™ fe-™) } €™* dx. Let c—fr=2x, whence 1 —¢" fe“ =" fe, whence we find that ] Foy true log (2 he) .e~“ dr =yva—w0. The theorem* noted in page 328 may be now proved in an extended form, and without the objection there advanced. It is clear that the mode of developing log (e** ¢¢~"") assumed in the theorem is as follows. | The function ¢z entered in the form «—fx and 1—a~ fx was to have the logarithm developed into —a” fa —42~ (fx)?—...., without any process which can introduce the series which made the difficulty in page 327. This being done, the function to be integrated amounts t0 » writing é“" for v in —¥y/xlog (#x:x).x, which being done, and the | integration and division made, all the terms arising from powers of a * The first case of this theorem (namely, where ~x=a) was given by Parseval, — (Sav. Etr., vol. i. p. 570,) in 1805, and the definite integral just given was found by | Poisson, (Jo. Ec. Pol., vol. xii. p. 497.) Mr. Murphy found the whole theorem, = independently, (Camb. Phil. Trans., vol. iv. p. 125,) and has used it to an extent . which was not contemplated either by Parseval or Poisson, the latter of whom, it | may be noticed, though he deduced the integral, either did not see, or set no value | no, the deduction. | ON DEFINITE INTEGRALS. 645 must vanish, leaving only the coefficient of 2°, or the coefficient of a— in the development of —¥'v log (gz: x). If we make V,=y/e* y (e"™ fe) .<"", we find in the same manner l A q™ = 1.2V,a%—— | ——— 1 ware eee ne hs (dar iy eft}) A, q2*—! - By tT F5, dt {w'n (fx)*} Aree bial C¥sy None of these theorems are altered by changing & into —k, and if this alteration change V into W, we easily find that f" Vde= {7 (V+ W) dr, a result in which *& will not appear. And thus we may in many different ways find definite integrals which shall express given series. Choose forms for wx and fx, and let the series in (V) then become A, O,+ Ay Q,+3A;0;+...., in which Q,, Q,, &c. are known. We then find a definite integral for B,+B,+...., by making A,=B, OF, A,=B, Q5", &c., provided we can find a finite form for A, a+ A, 2 +...., or xt—Apy, when A,, Ay, &c. are thus assigned. Let fr=1+2, wx=z, we then find 1 eo Sa ix (tet) eM 4x (Le) ef da A\+2An4 BAs+.... For many curious applications of the theorem deduced from (V), the advanced student is referred to Mr. Murphy’s paper already cited. Much more might be said on the subject of integrals of the preceding form, but the object of this work is fulfilled, so far as they are con- cerned, when attention has been called to their leading properties. The student can hardly fail to have noticed the manner in which {¢v.e-*’ dv preponderates in importance over other forms, and par- ticularly when the limits are 0 and oc. In any case the result must be a function of x which diminishes without limit as w increases without limit ; and such functions can frequently (not always, witness ve~*) be expanded in negative powers of x. Let ®x be such a function, namely. of the form Av'+ Ba~*+....: required ¢v, so that [> gu e-"du= Oe. Take the equation {joy e-°™* dv=y:(a—y), supposing a>y and v the only variable. If then we write this as follows, oo v —xv aimee A B \ | y fo bye. iw=(= tate \(G+k+...) er «tdr—A Eas Fala a Y y together with a series of positive powers of y. If then we expand ®y.é” in positive and negative powers of y, and if we assume™ the identity of the two sides of the equation, we see that if gv be the coefficient of y~* in ®ye”, we have fov.e™ dv=®x as required. Thus, if for ®v we take w”, n being integer, we find ye” has v':(2.3....n—1) for the coefficient of y~', whence fy v"-'e-” dv =2.3....(n—1) x”, as is well known. * This assumption is by no means a satisfactory one; see page 327. ar 646 DIFFERENTIAL AND INTEGRAL CALCULUS, If dv can be developed into A, +A, v-+A,v°+...., we have ¥ Ach sate eee ¢0 0 . 0 Spy Sigieie alas ms : fopv.é qo Tie) oem {ere aba = cere yg and, by parts, 0 a @p t© | (n+1) v ee sisi eee eae +f pg SS dv eseece ~() F Xv a a& 9 # provided d've™, f've~™, &. vanish when v== oc. We have thus means of representing in a finite form many infinite series of the most divergent character. For example, let Gv=(1+v)~', which gives {Ws Adoii:2 cleQeBe, Bede = Ft > itv = pe) $5 at x The operation by which we pass from f«~" dv to fv «dv, between the same limits, can be represented as follows. Let guv=A,+A,0 +...., which gives fove dvu=A, fem dvo+A, fe vdv+.... d = Ax ‘perm du—A, jem G0 aw). eles whence, D standing for differentiation with respect to 2, A>—A,D +A, D?—...., or 6(—D) is the operation performed on fe~*’ dv, 80 that ian Sov e* dv=$(—D). fe dv=¢ log (x) fem dv. Now ¢log(1+A) can be developed in powers of A by Maclaurin’s | theorem, or as follows. Since @r==e*? dO is the representation of | Maclaurin’s theorem in the calculus of operations, we have, putting log (1+) for 7, 2 Plog 1 +0)=(. +2)? 60=60+DG0.2-+D (D--1) 40 = ge which, performing the operations, gives | 2 d log (1+2)=¢0+40.2+(¢'0—¢0) = 3 + (~"0—39"0 + 24/0) — aud And, similarly, writing —log (1+) for v7, we have 1 3 p log 3) =$0—g/0.2+(¢"0+9'0) = — ($0 +390 + 29/0) =. eo Substituting A for 2, and taking fe” dv from 0 to &, " / Abe pv Ene Tp — 0 A l 4 POTD Aas Bb x 7 ON DEFINITE INTEGRALS, 647 _¢0 , #0 p0+40 $0 + 390+ 29/0 ——— —_ (2), ew ox(@+1) r(@+1)@4+2) &(e+1)(@4+2)(e43) (2) This series must be the preceding series (1) in a different form, and from it we therefore learn thatif A,,,,, represent the sum of the products of every selection of m numbers out of 1, 2, 3,....7, ES aa | Ay As, nti o* [aetn] [ej etntl] [a,etn+2]° °°" I now proceed to some modes of calculating definite integrals by series. Integrals of the form f “cos (2"-++ax"-!+...,) dx (sometimes called Fresnel’s integrals) are useful in optical researches. If we call this f cos da.dax, and if we take two near limits, @ and a+h, we have* fit" cos ox.dx =f} cos ¢ (a+x).dx= f} cos {a+ ¢'a.x} dz, nearly, since x is always small. This gives : L ° fi" cos ox wae {sin (¢@-+ ¢'a.h) —sin ga}, nearly. a Thus, by proceeding from 0 to h, h to 2h, &c., we might approxi- mate to i hee cos dx. dx, provided ¢’x vanishes nowhere between x=0 and «=nh. But a better approximation would be obtained by writing +h = +2h h ile ‘cos ox dx in the form fii cos atest xv )dz, which gives, proceeding as above, and making a+3h=p, 1 h hy a+h — j oy ees Bh AS ers i iy cos px Sat i isin (s+ ' pe 7) sin (9 pp st 2 cos gy.sin (4 ¢/pu.h) ye This method, though of an enticing appearance, is not very safe, and is not in reality correct to more than terms of the second order, as the following, which is preferable, will show. Take $(a@+3h+.2), or P(ptr)=hut+o'p.c+...., and integrate from r= —$h tow=+5h, ‘which gives fot prdz=hp.h+o" h? ee h? 4 Passscaci uit DRS liens for x write cos Px, and we have . h3 fa** cos ha .dx=cos Pu. h— (cos bp (f'n)? +sin dp. PL) percep eu One If we now expand sin (4 ¢/y.h) in the preceding result, we shall find in it the term depending on cosgy and on cos pu. (P'p)*, but that depending on sin¢yu. will be missing. Two terms of this latter series, therefore, will be more correct than the method which preceded it. If the limits be 0 and o, a convergent series may be obtained as * See the Cambridge Mathematical Journal, vol. ii. p. 81. 648 DIFFERENTIAL AND INTEGRAL CALCULUS. follows, whenever $x is a rational and integral function of a. Let bu=ax"+ ba" +...., we have then q cos bx==cos ax" {1—4 (ba*-*+..2.)?+....} i reams EO —sin aa" | (b+, yee ) Be, OHM which, arranged in powers of x, shows that the result contains two series, arising from terms of the form A fcos ax” .x2” dx, and A fsin ax". 2" am Now, from the result in page 631, we have gh (an) Lae fe cos Gre ito Cle ae n 2n n 5 ee ees grt fcosin Some GN 8/s Sicmetad ry de sin 2 MS ni ec Ss n 2n n For instance, let $v=az*+ br, apply these formule, and we have (a—35=h) 3; cos az* cos ba dr=h cost 7.14 hb? cosdar.Fl , hé btcosé7.V 4 2 2.3.4 345 sin aa sinbadx=h bsintaz, F2 hb?singa7.F4 héb'sing.F2 “ | DBS said IOUS aOR ay aie | By subtraction, using the properties of the function FE’, we have bax Lye se Gr wee | oe 3 —-—] - — JI+— —-4+=- So cos(aa*+br)dr=< T= cos & ‘ 13°96. 3-3 2.314.516 | h 2 = fib? mbt 5 2 prov rere weal; 32.3.4°339.3...67° °F This series might be more briefly and symmetrically deduced, as fol- lows. Let it be required to find [> ¢—a7”"—la"dzr. We easily throw this into the form b? x 5? ae" Siero [l= bets DB 1 Now foe av?" da==— Ve eetanum dr=—a ™ ; Le m m 1 whence, a ™ being h, the required integral becomes mt Art" " Qn-+1 ht p>? 8n+1 h3"*b3 | +... } dn ( 1 {PhP ih Od en OC TS em 5 22. Se 1 m For a and b write a,/(—1) and b,/(—1), which gives | Osea BA celle 2 Apert) Vv=a m bP (—1) 2m 2 ON DEFINITE INTEGRALS. 649 wipes p(n—m)+1 zis oe eo {ecos! “4 ee: @—,/(—1) sin asdeem dale 6} art 3 Se @ being an odd multiple of 7, to be determined. Leth be as before, and we have, equating the possible and impossible parts of the integral, and dividing the latter by —/(—1), 0 l ] a) n+] Jo cos (ax"+ bu") da=— r eres eh A m m 2m m a—m +1 , h***5 Qn+1 2(m—m)+1 — Ah 92 } opearay! T= cos Eg : 2m ] t m i 2m 50 f 1 gee 6 I Josin (az"+ bx”) dr= — r = Sih A ie he m m 2m m n—m-+1 P Aes h yplttl A 2 (n—m) +1 9 h2nt1p2 sin — eee 2m 1 m 2m By. The value of 6 is found to be Tr; by making 6=0, and comparing the result with the formula already obtained for fcosax”.dx. If m3, n= 1, we find Jo cos (a23-+ b2) da = cos = ae -re - #5 ieee oe asbitit fo sin (a2a3+bx) dn=; sin = eons Be The series last subtracted, written at greater length to show its law, is I ie 1h°b?° 1.2 h°d° 2S ae on ER es rhe Siegen! - as 2 ae a. 3 2 Bio ae ta Bees a Cl 0 tid she oe LO ‘The last forms are more symmetrical, but the preceding ones are fitter for calculation. The series at which we arrive in the valuation of definite integrals are frequently of the kind considered in page 226, which have terms alter- nately positive and negative, and diminishing for a while, after which they increase. This very remarkable class of series has the property which is shown* in the page cited whenever Maclaurin’s or Taylor’s theorem can be applied, namely, that the successive approximations derived from the use of the converging terms are as good approximations as if the terms continued to diminish ad infinitum, notwithstanding the subsequent * Dr. Peacock refers to a proof by Erchinger, cited in Schrader’s Commentatio, &c., as relating only to some large classes of series, the chief of which is the well-known development of 2px, in terms of diff. co. of ga. Sucha proof is furnished by the formula in page 624, as there given. I presume from this reference that Dr. Pea- ‘ock would imply that he has never met with a general proof, which is sufficient ‘pology for my not making any search after one. ae ad 650 DIFFERENTIAL AND INTEGRAL CALCULUS. divergency. This property is proved in page 226 to belong to every development of a function of x which is made by Maclaurin’s theorem, as long as the diff. co. of that function retain the sign which they have when a=0, but I am not aware that a perfectly general proof has been given. It will require some examination to point out the cases in which this theorem is certainly true, and those in which, till proof is given, it may be imagined to be sometimes false. | Let ¢x be a function which is positive from rato m=O, and | diminishing from #—=a to a=atkh. Let wer be the algebraical expres- sion from which dr—¢ (c+1)+¢(2+2)—..-- is developed, and | which must therefore satisfy wet+%(a+1)=¢2. We have then | ya=ba—b (+1) +9 (a+2)—9 (G43) +06. Now, according to the theorem wa<¢a>ha— (a+1), &c. But waty(atlj=da, ya—y (a+ 2)=da—¢ (a+ 1), &e., which requires that wa, ¥ (a-+1), ¥ (@t 2), &c. should be positive. The rest of the theorem, however, may be made to follow as soon as it 1s proved that wa is necessarily less than pa. I see no prospect of a general proof of this theorem, and I think the | following consideration, while it establishes it in ordinary cases, may | throw a doubt upon others. As long as $x is positive, pat wy (a+ 1) must be positive : if, then, gx be always positive, which is the case | supposed in the series, yw can never continue negative through a whole | unit of variation of x, since in that case ¥x+y («+1) would have nega- tive values. Hence, if yx ever become 0 or &, and change sign, | becoming negative, there must be such another circumstance for a value | of x, not differing by a unit from the former value. Consequently the} theorem may be positively asserted whenever Wer is a function such that ywae+ys(e-+1) is always positive, pr having no pairs of vanishing or! infinite values corresponding to values of » which differ by less than a unit. Take as an instance the series a *—na "+7 (n+1) a" °—.. 0.) We may easily show that this series is < fe-*a" dz, from x to ©, 80 that in x zt" ger 1 yar eevee rae { etdx Vn DP etl) ,Pmt2)_ | Let [n.a—"=¢n, and the preceding becomes dn—@ (n+1)+.--.| The right-hand side has no finite roots at all, whence the theorem is ¢er-| tainly true of the preceding series, andif « be considerable, a few terms, will give a good approximation to the value of the integral. Thus we have the remarkable relation } SA or | papel ct tata Otte which, when n=1, has been found = *596347362324, lying, as migh have been expected, between ] and 1—1. Divergent series of this hypergeometrical character (such has beer the term given by Euler) may generally be immediately reduced ti definite integrals. ‘Thus | {= dx iat Tal € \ f | } | | ON DEFINITE INTEGRALS. 651 Siggy oy + Ee * dx 1-1.241.2.3.4—....=f2e"de— fre det. | Ae the value of which is -621449624236; and “e* adx fe 2.3-+1.2.3,4.5—....= fre *(s—o ta Ee. ee ( + ) ie a > the value of which is *343279002556. It is singular that the values of these series, such as are derived from the equivalent definite integrals, may be obtained from the divergent series themselves by continued ap- plications of Hutton’s method, page 557. Generally 1 [m, n]—[m,n+k]+[m,n+2k]—.. Soa ce ptr data thes on ir ny, Mertatde Seba Sa m—m and k being whole numbers. I shall give one more instance of the way of reducing factorial series to definite integrals. Let the series be Bone 21:0) (a+6)(a+2b) (@+2b)(a+35) 4 a (@+B) ~ (@-+f)(a+28) (a+ 2B) (a +38) ees Let a:b=m, a: B=p, and b* {m(m+1) 4 (mt 1) (m+ 2) \ u=— ) ———~ + Che Seda Ny Ble @tlh ~G@+) (4 + 2) Multiply both sides by 2"°*', and differentiate twice, observing, that in the reverse integration, we begin from x=0, mae.) (B* dx? cre B? {m (m+1) 2+ (m+ 1)(m+2) +... . Multiply by 2”, and integrate twice from r=0, PCAC an Se rg (foda)?. 42 ra ce j=a Ce +1 5 EE ie a b? ie. 2e< —_. 7-1 (2 : —m u= B ERO (5 de {| ae («" FE = To return to the theorem which gave rise to what precedes: a proof of it may be given, including every series A,—A,+A,—...., in which Ay, 2A,, 2.3A,, &c. are the values of 1, d/l, #1, &c, da being a function which does not change sign, nor any of its diff. co., from 7=0 to #=1. This follows from Bernoulli’s theorem, (page 168), since ¢1 Gl " aa a"dx I _ nt 2 tee oe —+/,¢ ? Siprda=$l ae Ray tr 3. n tsb oR from which, #1, ¢/1, &c. being positive, and the other suppositions just mentioned being made, it appears that the error arising from stopping at any term is of the sign of the first rejected term, which is, in other words, precisely the theorem to be proved. Again, from the theorem DIFFERENTIAL AND INTEGRAL CALCULUS, fSe-? gu dv=G04+ P04 ....+EO+ foe? gto dv we may easily see, that if dv, d’v, &c. be alternately positive and nega: tive when v0, and retain their. Pen from v==0 to v= o, the same) theorem is true of 60+@/0+.... But the preceding requires thal @™v.e— should vanish when v= c, for all values of 7. This theorem, being true in cases so extensive as those of page 226. and 624, and those obtained in the present chapter, might be suspected | to be universal, and is, in fact, treated assuch by some writers. [| believe it would be impossible to find an instance among those series to| which it has been applied, in which it is not true; but it must be re-| membered that most, if not all, of these are cases in which wa, a funetiall which. never vanishes for any positive value of 2, is developed into pa—d (x+1)+...., and in such cases the theorem can be proved. | It may not here be out of place to give what is perhaps the most direct and satisfactory mode of assigning the remnant of the series in| Taylor’s theorem. We obviously have b(a+h)=at fig! (ath) dh; for h write h—#, and let ¢ be the variable ; if Q! (ath) dh= — fi. d' (a +h—t).di= f% p! (a+h—t) dt. Successive integrations by parts then give fp (a+h)=pa+Pa.ht fio" (at+h—t).tdt | F | —dat+Pa.ht+d"a = +5 S30" (a+h—t) .tdt ; and so on: whence the value of all the terms after h” OW ete ole by MEG RC IES If C and ¢ be the greatest and least values of #**?x between r=¢ and «=a-+h, the last differential must lie between pt) C.2t" dé ant! pt c.t" dt, whence the integral must lie between @°*” C.h"+': (n-+1) and Porre. Paar (n+1), or must be P%t” (a + OA) A": (n+ 1), where @ is less than unity. But if we throw the integral into the forn| nen i pt) (a+ht).t* dt, and pursue the same reasoning, taking ( and 1 as the greatest and least values of ¢, it is found that all the term| after I) Pde | he oe het hee | Seas pe (n) | oe are equal to d (a+6h) ———__, | gp” a oe ft where 0 is also less than unity. | I now proceed to consider some more cases in which definite integall are expressed by series. And first let us take fre v" dv, which, ¢ being positive, is always finite. This is easily expanded into ‘the series | | n+ grt ants n+4 x x n+l] Rehan ~ 2(n+3) “by B(nL ay oe 4 in which C is to be determined. If n be >—1, we may make 20) and the first side becomes [(n+1), or C=P Tosi 1). And the serie fre v dv=C— ON DEFINITE INTEGRALS. 653 on the second side, 2"*": (n4+1)—a2": (2+ 2)+ &c., which it must be observed is always convergent, does not increase without limit as 2 increases, but approaches the limit I! (n+1); for the first side must =0 when z=c. All this might be proved by calculation* in any particular case, the restriction being —1(n)cc,and x being anything whatever, positive or negative. But let us now suppose n=—1], in which case z must be >0. We have then C5). cxag 2 3 4 et ap a a x it — =C—log r+r— — + “faites in which C cannot be determined by the same mode. A very simple process, however, will do what is required. When n>—1, we have, by the preceding series, 1 rth ght? Pe amie Ltt et) jie ee ae ABC ye if (n+1) n+] n+ J Wa Ge e@ese When n=—1, the third term is log z, the fourth term is 2, &c., so that it only remains to find the limit of the two first terms. Now (Chapter IX.) zP'2z—] wo Td+z)—1 Pz—27", or or is IY, or —y, (page 580,) when z=0. Hence we haye,t in the last series, C=—y. Now, let be a negative fraction, and <—1, say n= —m—k, m being a whole number, and a positive fraction less than unity. Integrating by parts, we have See an aan: 1 oy nt} Sy ae | (na Ia +—— fre or dy Be dv ~f ] rp eats ! | pont (m+k—1) a" -" —-(m+h-1)(m+h-2). kt! — I © =v ,.—k 5 Sane DRE es eo the last integral of which falls under the first of the preceding series. And if n be a negative whole number, and <—1, take m, so that k=1, in which case the integral here obtained will fall under the second of the preceding series. And if in this second series just mentioned, we use ax instead of x, we find 7 dv en a? a Mm a? x* =i Yi LO AX aL — EN pga ae a Tie QT 2.3 * The common series for cosa and sin x would af the study of analysis were made to end a little oftener in computation) have habituated the student to Series of this class, which are always convergent and calculable, and which do not lose that character by the increase of x In my“ Elements of Trigonometry’ (page 99), these series are actually verified when e=10. + These integrals have been fully considered by two excellent Italian analysts, Mascheroni and Bidone. The methods by which they have contrived to do without the use of the function I (which was not so well known then as now ) are, though prolix, very ingenious and successful, “Tw 654 DIFFERENTIAL AND INTEGRAL CALCULUS. * eride iy tds Also ores aa | ax Vv x which introduces no way of making the function integrated infinite, and, does not destroy the convergency of the series : for a write a,/(—1), equate possible and impossible parts on both sides, and we have, since) log ax becomes log ax + log /(—1), “cos av dv Oa atat i more Sm euler ress ht | *sinav.dv log V(=)) _ yg OT _ ae | SCN W(—1) 2.8 2.3.4.5 | Now log /(—1)=(/+4) tJ (—1), / being any integer; but from page 631 it appears that we must make /=0, or write $n foi log J (—1) :./(—D). eh Oikos a © —ay ; he ae ee de) Weems | Again —— = | ———_ == aoa 9 UtmM an v in am am EAD RE . sey aes ( ote |—y — log am-+-am — The v+m For m write successively —m,/(—1) and +m,/(—1), which gives fo the two integrals | emveO( —y—log am—log { —,/(—1)}—am,/(—1) | am am ,f(—1) a 3 1m ee a aD 1.) } \ f emay(-}) (= 1—-A0g am—log/(—1)+am J/(—1) | aac am J(—1) } | ecee For logy(—1) write 4rJ(—1), and for log(—¥(—1)) writ —171,/(—1), values which will be justified by subsequent verification) add and divide by 2; subtract and divide by 2m,/(—1). We the) have* | o —av | E vdv Lt Pes | ‘ FEWER rome 3 sin ma@— (y+log am) COS Ma 0 m a* mm a* ‘ m a | + cos ma Vouhao Bathe BIN ING {7a —— gitesss Reet oa osm + : (y+log am) sin —__ ¢ a+ — og am) sin ma om+tv® wm mt eee COs MGA mé a’ \ sin ma /m a m at | a + eee .) “Sar ~- een. aa — —_—._ {| ma —_—— ESE a SE, ee Se m 2.3? m 2? 2.3.4" _ Differentiate the second of these with respect to a, and it will give th * These results agree with those of Bidone, obtained by another method. ON DEFINITE INTEGRALS. 655 first with its sign changed, as it should do: the’ details of this verifica- tion will be found instructive. For @ write successively —a,/(—1) and +a,/(—1), subtract and add, dividing by 2,/(—1) and 2. We then have {2 av.vdv 1 gt _ gma g mte 2° 241) C=) if et gm T Eee *cosav.dy ww eMt game 1 etm Ate) . g = SS OT ane — PS F a 2 m 2.2/(—1) 2m g Tesults which are easily deduced from those in page 577. We also have “sinav.dy — s™*4_.—™4 gh a ‘ — =—_—— | ma+——4-.... > m+v? 2m 2. 3? ) - m? a m' at et me — & + +. +) _ (y+log ma) Mm 2.3.42 2m “cosav.vdv ¢™*#—.-™ ( m? as i cob arg #....)] one ( 5 peice 5 abs — (y+log ma). Let m=a=], and remember that, by common expansion, foedv:(1+v), foe?dv:(1+v*), and fpevdy: (140%), are severally F1—T'24+1T3—...., Pl1—F34PF5—...., and F2—14 +F6—....; so that we have 1—1+41.2—1.2.3- = +1 Ed a da : —Il+1.2—1.2.3+...=< Y ba + .... T ] I—1.241.2.3.4—...=cosl G-sgc .) ‘ l 1 —sin ] O-z T5347 sibel a) et 2 815.3. 4.5 oe ny fy ee oe voor MM .o+ . r e . eet TS Pall paeeent OL 9 +53 aLet y 1 1 — cos | Wet aaa at aas 3 the values* of which have been given in pages 650, 651. These series may also be expressed as follows : *e7* dv * sin vdv fae, 9 EP 9 i+v : J 0 I+v ’ * There is a misprint (sin for cos) in two places in Bidone, which might lead to @ supposition that it was an error in reduction, affecting the subsequent computed results. On examination, however, I find that the results are correct, to 656 DIFFERENTIAL AND INTEGRAL CALCULUS the two latter of which may be shown to coincide with their series by expansion of (1+), and by page 631. Again, if v—1 be written for) » in the two last, and the limits changed accordingly, and if cos (vw—1) be written cosv.cos 1+sinv sin 1, &c., the second side of the preceding: equations may be obtained by taking [(Fcosvdv:v and fysinvdv:» from the series in page 654. And all the preceding trigonometrical integrals, as well as the case in which m=a=1 might have been short-| ened by the same process: but the preceding is valuable as an instance of the legitimate passage from possible to impossible quantities. Various other ways of reducing definite integrals to series might be) proposed, but in the preceding will be found enough to give an idea of the most important of them. I have now given a sketch of the principal methods of definite integration, meaning by a method anything which) applies to a numerous class of mstances. There remain yet two! particular branches of the subject to be considered; first, the cases m which, owing to the impossibility of expressing a general integral, its values are arranged in tables; secondly, the large number of miscel: lancous definite integrals which have been found, each as it could be done, and out of which it may be advisable to make a small selection. The tabulated integrals with which it is most necessary that the mathematician should be familiar, may be divided into those which ar¢ generally useful, and those which have been computed for some particu lar purpose. Of the latter, it will merely be necessary to say that the student who reads this chapter will have no difficulty in mastering an} method hitherto proposed im works on mechanics, optics, &c. for the formation of a table of any definite integral. Of the former, that is, 0 integrals tabulated for general use, the most important and the mos) xcecessible are 1. Elliptic integrals, tabulated by Legendre. 2. Neon. dt, tabulated by Kramp. 3. Va, or [peo da, tabulated by Legendre. 4, Logarithmic transcendents, tabulated by Spence. Bie ats 2 | “_, tabulated by Soldner. 3 > log & 1. The subject of elliptic integrals, if entered into to the extent neces sary to explain methods of determining their values, would occupy mor space than we have to give. In accordance, then, with the plan pur) sued throughout this chapter, which is to enter on the discussion of ni integrals except those of which the actual numerical values are calcu) lated by algebraical formule, or are given in tables, I propose only t state in few words the nature of these functions, with references ti sources of information. Important as elliptic integrals are in certair classes of problems, and numerous as have been the properties of ther which have been investigated, it cannot yet be said that either thes problems or methods lie so close to the grand route on which a student’, elementary course should be marked out, as to require a detailed treatis on them to be inserted here. | An integral is called elliptic when it has, or can be made to have, th form f Pdx:Q,/R, where P and Q are rational and integral functions ¢ x, and R is a rational and integral function of the fourth degree, or ‘ the form a+bx+c.?+ex3-+ fe’. And it is shown that the actual calev lation of all such integrals is attainable as soon as tables of the foll ON DEFINITE INTEGRALS. integrals are constructed, dé a | J JG —c?sin®6)’ Séf1~ctsin* 6). | bs in which ¢ is less than unity, called elliptic functions of the first, second tables of the first two kinds have been g of approximating to the values of functi ‘* sy ae wat | 2. The values of Hf e—? dt, or 2 ( log je a » 1--n sin?" J- c’sin?0)’ and a does not exceed 3r. , and third species: extensive iven by Legendre, with methods ous of the third kind.* —- dx from x=0 to a—=€—a" may be calculated from pages 590-1, and the foll abridgment of Kramp’st table. important use of this function is eve V(m) of which any value ma fge-# dt, or 1— a es) 9 &— J y easily be obtained from the following, the value of a being in the first column, and that of fi¢—? dt in the second: But it must be noticed that the most best satisfied by tabulating re e—f di edi? "00 | *8862 7 -90 | -1800 7 1-80 | -0097 05 | +8363 | +95 | -1587 11:85 | -0079 "10 | -7866 § 1-00 | +1394 | 1-90 | -0064 gE By emt eS is: 1°05 | -1219 | 1°95 | -0052 "20 | °6889 | 1°10 | *1062 4 2°00 | *0041 -25 | °6413 | 1°15 | -0921 2°05 | +0033 "30 | °5950 | 1°20 | -0795 | 2-10 | -0026 "35 | *5500 | 1°25 | 0683 | 2°15 |. -0021 “40 | *5066 | 1°30 | °0585 2-208| £* OO17 "45 | -4648 | 1°35 | -0498 2°25 | +0013 "50 | °4249 | 1°40 | -0423 | 2-30 | -0010 "55 | *3870 | 1°45 | +0357 | 2°35 | -0008 "60 | *3511 | 1°50 | -0300 | 2°40 | ‘0006 OS) 3172) 1-55 | 0251 f 2°45 | +0005 “70 | °2855 | 1°60 | -0210 # 2°50 | -0004 "715 | +2560 | 1°65 | :0174 4 2°55 | +0003 "50 | °2286 § 1:70 | °0144 § 2°60 | -0002 "85 | +2032 J 1°75 | -0118 | * The newest and m ; Legendre, 7r vith three supplements, is follows. ost accessible sources of information on elliptic functions are aité des Fonctions Elliptiques, 2 vols., 4to., 1825 and 1826, (1828,) in which the subsequent discoveries of Abel and Jacobi are added. Abel’s papers were originally scattered through Crelle’s journal, Jut are now collected in the edition of his works, 2 vols., 4to., Christiania, 1839, lacobi’s work is Fundamenta nova Theorie Functionum Eliipticarum, Konigsberg, 1829. In English there is an account of Legendre’s earlier method, in Leybourn’s Repository, vols. ii. and iii.; the subject is also treated in Mr. Hymer’s Tntegral Jaleulus, and Mr. Moseley’s article on Elliptic Functions and Definite Integrals in he Encyclopedia Metropolitana. Ze ; _T Analyse des Refractions Astronomiques, Strasburg, 1799 ; reprinted in the ‘eyclopedia Metropolitana, in the article Thevry of Probabilities. In the latter tticle is found the second table alluded to in the text, a8 also in the treatise on *robabilities and Life Contingencies in the Cabinet Cyclopedia, and in the article a the same subject in the edition now publishing of the Encyclopedia Britannica, aU 658 DIFFERENTIAL AND INTEGRAL CALCULUS. 3. On the function Ix or f = --? yt! dx enough has been said, and a table has been given (pages 577—591.) I only add here a few words} on the faculties of numbers, as the German analysts call them, all the properties of which are really included in those of Pv. | The use of the term powers of x, to signify xx, rau, &C., suggested to Kramp the application of the kindred term faculties of x, to denote} x (x+a), «(a+a) (e+2a), &c., w being called the base of the faculty, a its difference, and the number of factors its exponent. Others have! called these functions factorials* of x. Besides the notation exemplified) in page 254, the following has also been used : } (x, -ayY=1, (#, +a=a; (2, +aP=2 (+a), &e. | (a, +a)*=a2 (41+a)(@7+2a).... (a+n—1la). Many properties of algebraical functions nave been expressed in| and even suggested by, these notations; and the extension of the! system to faculties or factorials with fractional or negative exponents has been made in several different ways, ending in the same results. These may all be obtained by generalizing the equation (x, +a)", or a(a+a)....(e+n—la)=a". = (E+ 1). ‘ (245-1) | z x =or(Z4n):0 (2); | a a i ° : ° ° ° . o Nie a result which admits of interpretation when 7 is fractional or negative, In all cases the notation r() a ede | a"! or (x, +a)" may be translated by a* ——~—— x 2 Thus EnV! , or (1, +1)"7. 4. The logarithmic transcendents of Spence are included under th formula Fh a. a 201 +7) tr—-— t— —-—t....; LEn=t9- oe Tes the first of which, or L(i+2), is obviously log (1+), and L’d +a =(1-+2)-*. We have then . Li a+a= | a vat)=/ SL A+), Go 0 l+o * dv ra+a)=[ Bete Sent ad, &e. a | 0 4 | Into the theory of these functions the authort has entered at gre * This term was suggested by Arbogast, and Kramp himself subsequent adopted it. + William Spence, (born 1777, died 1815,) of Greenock, was, at the time whr j he first studied, one of the very few men in Britain who acquired a knowledge | = ohh ON DEFINITE INTEGRALS. 659 length, and has deduced values of L?(1+-2) and L (1+2) for integer values of xv, from z=0 to x=99. He has also investigated the pro- perties of x x 7 C* (*)=2—-— += -— 4 &e 3 5 Mr. Spence has given two formule, by which z—2-" 2°++3-" a3— &c., or the function from which it is developed, can be calculated when diver- gent, by means of the case in which it is convergent. These formulee are as follows: let s,=1—2-"+3-"—.,. -, and, according as n is even or odd, we have 2 3 —2 a3 I. (m even) (e+ Neate) : .) (oo + —., n= yi} | (log x)? (log x)! (log x)" 1 | (log x)" ms +s ee ly ee ee gawd Sydah DiieaD ab fnare 253 swith RE \ : dee PF , x wate Oa ees PEM, ds ce Pate = II. (n odd) (« Gs Sen ) (@ ora pee oe (log 2)* (log v)"-? (log x)” 2 "nn i n—' Aaa oe e*¢ee % Nae ike ewe a ‘ a he ay eae Rea ane By a different method, which is simply making ‘use of the remnant of Taylor’s theorem as given in page 652, I have verified these formule, and found others analogous to them, as follows. Let S,=1 +27"°+3 +...., and according as 7 is even or odd, we have x x iS Fe ts» Sis ae a 14+ — +—+,,. |= III. (neven) (+3 -- mei, 494e +(« -+- iki eaa e ) (logz7)}? . “(log x)* Cora) (log x)” Sa-s ee Hee. $ S, —— 2 2( S.+ Ase. Parag res cba re cee, 2 Dayetedt wWe 2 3 z a is, ean Fak ly )= IV. (n odd) Ot oa arr yea mi! 2 Ton Toe e eal = a ts eo} - ee” 218.1 log 7+S,_s 3.3 +...+8, Wat tateo win By the same method the following are also found, Q, and q, represent- Mei +S "+574 .... andl—3“+57—.:.. When » is even (using Q,, Q,-», &c., and gn—15 Gna &C-) the works of the continental mathematicians. His essay on the various orders of logarithmic transcendents would have made his name better known if its subject iad been of more general interest to mathematicians. It is an original work, full % methods which any inquirer who is occupied in the investigation of the aumerical values of integrals would do well to consult for hints. The first edition vas published in 1809; the second (edited by Sir J. Herschel, with numerous idditions from Mr. Spence’s papers) was printed in 1820, but, owing to the impres- ion being almost entirely forwarded to the publishers in Scotland, ( Messrs. Jliver and Boyd, Edinburgh,) and other circumstances, it was never known in ingland as a work on sale till the year 1840, and was always spoken of as a book if the greatest scarceness, 2U 2 (ee 660 DIFFERENTIAL AND INTEGRAL CALCULUS. ) s —3 ] ~ n—2 | V. he aise techs \e(r at eae ) coms IO chars ay} | xv oe x 3 a ( (log a) 4 | VI. («5+ ia .)-(s a ot pee ee a a ase} | When n is odd, (using Q,-1, Qu-2, &¢., and ny Ya—2» &C.), x ee ty Bk (logx)*~ VII. (e454 ma .) (2 v3 Tas )=2|Qsloget + Q5 35 af | Tien i a delice cee i vitt. (2 at... (0 wt. )=2 Geet: 5 Gigs Conareat : In VI. the suceessive is of terms having ,, Yi» 5. The integral Af dz:log x, or —fe- dt:t, from t=-—loga to} t= oc, or fe'dt:t, from t=—cc to t=loga, has been tabulated by | Soldner in the first of the preceding forms, and is the key to so large a class of definite integrals, that it will be worth while to discuss it, and to add the table. In the first place, observe that when a>l, the subject | integrated becomes infinite between the limits of integration (at z=1);) in which case the principal value (as M. Cauchy calls it, page 633) is to be taken, or the limit of f t+ f «. When @ is diminished without limit.» Soldner uses the symbol li. a (from the initial letters of logarithm-integral) | to stand for [§dx:log.v; a notation which I propose to follow. | From the second form, and page 653, we have | et a 9 ef (loga)? (log a)? | c= | 5 =y-+log log a—log a+ 98 Br ey | t which applies when a>1. By expansion we have (@>1) 39 pat t a ] ?__ Az -| shee =loe(“% *) + 1oga—0-+5 a 2 : +.. rae Ba: 0 bar adc @? — a cen ——=—— ses i ia y tlog @ + 5 Add these together, and diminish @ without limit, which gives wae Se loga)? (log a)? [ hi. a=— © "y+ log log a-+-log ee 4 Eos +..0e] —loga t 2 Des i \ Observe, that if in the last we were to change @ into a™, the last, series would differ from li.a~? by log(—1), the correction for dis-| continuity described in page 633. Again, as in page 262, let : flog (1+a)}"= AeNa tie aet Vga bse po RAF he a*—] li. a-a= | foe tT a aaa —V, (a—1)+ Vz ak hae (<1) But (page 593) the value of y shows that li. a—log (1—a) approaches without limit to y as @ approaches to unity, and the same ol li. (1 —a)—log a, when a diminishes without limit. Hence we have | y=Vi- 4 Vath Va---.. li. (1—a)=y+loga—V,a+4V.a*—.... ON DEFINITE INTEGRALS. 661 ety fii! Si otual Lig, old Sate alae’ "iiss Dil GclDe 3°94 704 FOO SA AeO es a dxv ’ - dx Again, li. (1 +a=[ log (1-42) lim. of {fat ft} ‘lor (142) +a) _, log x 5 The first integral is found by making a=@ in the last series, and the second from the original development, which gives log a—log @ +V,(a—0)+4 V, (a?—67)+....: the addition of which gives for the limit Lao hb 2a® 401" 1808 ee ‘ a re Re aig teeing Fg yop The coefficients, as given by Soldner, which can be partly verified from page 262, are as follows, (a") meaning coefficient of a’. be ] A l Cea f Ko 19 me 3 oo) 9” CD mrt a) aes He erp? (a ae 863 275 6 = i sap Se FEATS ()=350880° (= igogay (a®)=+00116956705, — (a®) =: 00087695044 (a) =-00067858493, (a) = 00053855062 (a”) = +00043807461. From Taylor’s theorem, : x d(loga)' «® d(loga) a a Si ena st loci da 2 Adi) (Ho. 3 6 ane A particular case of Burmann’s theorem is also applied by Soldner, which may be useful in other cases; namely, a method of expanding F(a+2x) in powers of f(7+-a)—fa=s. If we assume F (a+z) =A,t+tA,s+4A,s°+...., we easily deduce Fa Re oc GAY Bada dA, Yiiimaebiecda.. . pede irl Be eS i Let Fr=li. xv, fr=log x, we have then Ave bo, Aye , Se. _ a _doga—l a __ {Clog a)’—2 log a+2} a ~ log a’ Cae Clore ss Fo (log a)® A,.1:=4 (log a)~" { (log a)"—n (log a)" + n (n— 1) (log a)""*—... . +n(n—1).... 1}. Let log (a+xr)—loga=y, and let the last factor of A,, all but its first term, be +(7—1)B,. We have then 1 {log a-B,} ay? { (log a)? + 2B,} ays li. (a2) =i, a+ —— y+ 0g 2 (log a)” 2.3 (log a)* ae . 5 y® og (a r7)—i0ga cote or since yt toate x ak (ott) ~108 Wiser, a 662 DIFFERENTIAL AND INTEGRAL CALCULUS. x li. (€+2) =i. a+ where B,=1, B,=B.—loga, B,=2B,+4 (log a)’, B,=3B, — (log a)’, and soon. This series is very convergent when a is considerable com- | pared with a. ute wert Lastly, take the equation li, —=— a e/ loga the integral into a continued fraction, as in page 591. Re: 1 li, —=—- - a aloga 1+ One or other of these methods will apply in every case, and by them the following table was constructed, for values of a less than | unity. a. lica.(—). ‘00 | *0000000 | ° “01 | *0018297 | "02 | °0042052 | - °03 | +0069137 4: “04 | -0098954 | - "05 | °0131194 | ° ‘06 | °0165667 f° "O07 | °0202248 § - "08 | 0240852 | - "09 | -0281416 | ° "10 | °0323898 § : "11 | *0368267 | ° -12 | 0414502 | - "13 | +0462592 | - °14 | -0512530 § ° "15 | °0564316 | - -16 | °0617955 f° “171 “0678455 7 -18 | +0730829 ff - °19 | :0790093 § : *20 | *0851265 § ° “21 | 0914368 | : “22 | °0979426 § - "23 | +1046467 | - -24 | +1115521 |: "25 | °1186621 | *26 | °1259803 | - + MgO LUA TES “28 | °1412566 -29 | -1492232 |: *30 | '1574149 ff - °31 | +1658366 | - °32 | °1744935 | °33 | *1833911 § - log a 9 (log a)? eS 1+ li.a.(—). *1925352 2019321 | *2115883 | *2215106 | 2317064 | 2421833 | "2529494 | °2640133 | *2'153841 | ‘2870714 | *2990852 § 3114326 | 3241357 | *3371959 | 3506294 | 3644496 § “3786711 | *3933088 | "4083791 | *4238992 | -4398875 | ‘4563637 § -4133487 | "4908650 | 5089366 § *5275895 | *5468515 | "5667522 | 5873242 | -6086021 | "6306240 | 6534306 | 6770666 § ‘7015805 | 3B, y’ 3.4 (log a)? yg log a in, 2Bs y This gives sin (log a) (Cog a) 2 Clog a)~ 2 (og a)™ Lag li.a.(—). * 7270254 *'7534596 *7809469 °8095577 -8393700 °8704701 °9029543 *9369300 °9725181 0098548 °0490943 ‘0904128 °1340120 *1801246 *2290215 °2810197 ‘3364941 -3958924 *4597547 *5287419 *6036733 6855829 *7758007 *8760780 ‘9887871 * 11492535 ‘2663481 4436226 °6617277 °944380] *3448241 4°0329587 infinite. WNNNNN RFE BR HE eR Re ee ee eee d | (w>1), and convert | ON DEFINITE INTEGRALS. 663 When a>1, li. @ continues negative until li. 1°4513692346, which is =0, after which it continues positive. Also li. *1=—0°0323897896 li, e~*= —0°2193839344 li.10= 6°1655995048 lke = 1°8951178164 The following is the table for values of a greater than unity :— a. lia(Z). a. li.a@(+). a. lia (+). infinite. f 8°8764646 140 38° 492841 1°6757728 # 18 | 9°2258743 | 150 | 40°502303 | 0°9337783 9°5686258 | 160 | 42°485178 0°4801779 ‘9053000 | 180 | 46:°380020 0°1449911 § 22 | 10°562353 — 200 | 50°192168 24 | 11°200316 220 | 53°9328'72 SED — «J ell el me wWNWe © EAN TTR SET SET ho © te) | i 5 | 0°1250650 | 26 | 11°821734 | 240] 57:°610933 "6 | 0°3537475 | 28 | 12°428628 260 | 61°233401 7 | 0°5537438 | 30 | 13°022632 | 280] 64°806034 8 | 0°7326370 | 32 | 13°605092 | 300] 68:333612 9 | 0°8953266 | 34 | 14°177131 320 | 71°820157 0 | 1°0451688 | 36 | 14°739697 360 | 78°683375 5 | 1°6672946 | 38 | 15-293602 § 400] 85:417888 -1635889 f 40 | 15°839544 | 440 | 92-040677 Lay 45 | 17°173366 °468696 * 2222224 ¢ 55 | 19°731245 "7570508 | 60 | 20°965412 * 2537182 65 | 22°174669 ° 1212387 70 | 23°361813 480 98°565102 520 | 105°00191 560 | 111°35993 600 | 117°64651 640 | 123°86784 720 | 1386°13526 °9675853 | °6345880 | OO =I OOP 0 DDD H&S eS ee SE war = or S&S — CO OonrNaTATOOATSHPARWNHONKHKHOCOCO 10 1655995 | 75 | 24°529138 800 | 148°19668 11 | 6°5919851 § 80 | 25-°678554 880 | 160°07861 12 ‘0005447 | 90 | 27°929887 | 960 | 171-80200 13 *3965480 | 100 | 30°126139 | 1040 | 183°38346 14 7808256 | 110 | 32°275096 | 1120 | 194:83783 In "1548249 | 120 | 34:382807 | 1200 | 206-17582 16 -5197165 § 130 | 36°454085 | 1220 | 217°40761 When the number is very nearly equal to 1, the table may be aban- doned in favour of the expressions for li. (1—a) and li. (1+a), in which a@will then be very small, and the series very convergent. Also the formula for li. (+2) must be used instead of interpolation, if values of li.a for large intermediate values of @ are required. The following integrals (and many others of more complicated forms) may be obtained by means of li. a, in which it is to be understood that the integration is indefinite, requiring to be taken between limits, or a constant to be added, as may be. b m a" dr et dae *t ; { eink 5 orf eed he Ut ee log x 2 log a 4 : SUT sees heh Gl w OR Reece | iia itt (at+br), ae €", =li.e J log(atbr) 6 oyrawa Pr Gat 664 DIFFERENTIAL AND INTEGRAL CALCULUS. . Lin e* dx } [= . f ee dah. es, f eas dr=—- li, gas?”, ce =e 1p alae | atx diz {acu =liloga, fli. fadr=ali. fa— eae log log I now come to those isolated instances which cannot be made to come. under any of the preceding methods: of these there is a considerable | number existing in various works, out of which a selection must be made, and it is a matter of no little difficulty to settle what parts of the. voluminous writings on definite integrals are most likely to be useful to, | the student. Having applied the remarkable property of cosines and sines in pagal | 291, it may be interesting to point out some other functions* which | have an analogous property, and which are of erent importance in some | questions of physical astronomy. Let w=z+4a(1—u?), which gives: by Lagrange’s theorem tae 170) 1 gee ae ink a | Ee ma 22 __ EX) Sane - Eh Behe se Fi <( ) liye Y Seong, 1: | du rd i ad? Bolts ZI-7)-247 waa plea ae (1-2)? = Hie a) —a) a ateee 8 dz | 1 1 du | But w= ae rye (14+ 227+27)3, ae (1+ Darbar). Let the last: series be Pi, +P, a+ Pe2?+.. 52% fe P,, is of the wth degree withl| respect to z, and if hl pis we have for a(n +1) fP, z* dz, oF fora ye zdz, the following terms, 2D "(1— 2") he Dd 2y se... ee] D129) every term of which vanishes when z= —I1 or z=+1: so that| fiP,2z*dz=0, if k be any integer less chain n. Hence, if m and n be} unequal, we must have JrrPiPa de=0, forlil abe the greater, then | P,, being rational and lower than P,, in dimension, we see that f Pls dz may be made to take the form > fA, fP 2" dz, k never being so great as 7. And each term of the last vanishes, whence the theorem is” evident. But if m=n, we have one term of the form A, fP, 2" dz, which integrated by parts as before leaves one term only, EA, F (n+1) D7? d— 2)": 2" (2+1), according as 7 is even or odd. Now (page. 580) : é FAT (n+1)8 | Tl (] — 2)" de— 2f! (1— 2?" dem fi a2 (l—a2y)'d a 2 NS ee SHa—24) Joh el Ceca ti? Gi) Ob tara =F(nt)): (1+) @—-PO-p..- bb. | Now A, is the coefficient of z" in P,, or in 2 D" (1—2®)": F (n+); we have then A, = +27". 2n (Q2n—1)....(m+1):F (n+1), (+, neven; —, noad). * These are certain functions, by aid of which Mr. Murphy (in his elements of electricity) has put Laplace's coefficients (page 540) in a very clear point of view | as to those primary properties on which their utility chiefly depends. | * ON DEFINITE INTEGRALS. 665 Hence +A,, is positive, and Hptdre* pq ~ 2°)" — re 2s EPL) M.S. cd 2 (n+3)(n—$)....2.4 P(m4+1) iD daete CAIN uae tet, 2 —~ (n+ b) eS PlyR- Gis) Ont bi If, then, (1+ 227+ 2°) be expanded into P,+P,2+.. ., the function P, has the property that f+} P, P,,dx is =O or 2:(2n+]), according as n and m are unequal or equal. Also —_ D*d—=*)” “a 1 d” ene MOUW nda) SiatYoo. in dk n Let dr= fe~* Wo do, the limits being anything whatever independent of x: if, then, we change x into 2+= —4 A’ (a log x), and so on, which gives TEUaES a Autre 1Oe 2s ee Edema Bo Te, age (—1)*fpeo? (e-"—1)* v Tim Tey be at dz. From page 575 we may deduce 666 DIFFERENTIAL AND INTEGRAL CALCULUS. tO a—l 1 ,a—1l —a Oe ae = Rise: Al) s(x 2, oy > lt+xv smar pute Ba ps The transformation is made as follows: ae 1 1 fi ox dx=f, ba dxt fi acne cae a pr+— 4G) hae » iter sinan i ae ieee (2-4. 6h) 5 “27 (1+ccos@.2) dx rce~* cos ad Pan pil deRe LLL Cae Ris NA sl nimisioes 1+2ccos0.x2+c? a? sin ar ORO ge sin at 0 if k=,/(—1). Make c=1; a corresponding subtraction gives i a* dx foe “de 7 sin ap ta _ or 8 ee ha e J ol +2cos0.a+a” > 1+2cos0.r+a sinaz sind’ q the second being obtained by changing w into 2" in the first. The | transformation already noted gives i ve +a 7 sinad ———_——___—_——— dir=-— an ae o L+2cos0.%+ 27 sinar sin6@ Multiply the two last by 2 sin 6 d0, and integrate from 0=0, sc) 1 x 2 ; ] 2 a1 aa { tos( Shue Cyl pe vl and. aie Sd dx e homie 1+2cos 0.242" x Mees 1— cos a asin ar ( ). In (A) write 1+logzv.a+.... for 2%, and let sina@:sinaz =A ,+A,a+.... Equate corresponding powers of a on both sides, | and we have (Cog ary*dx F 142 cos O.2+2° =m 6 Earle whence this integral vanishes whenever 7 is odd. In page 593, last equation but two, from the value of A (1+2), or log. (1+2), find AQ.+2)+A (1+y)—A(1+2+y), which gives *(1—v*)(l—v’) dv Pd+e2).Fd+y) 5 l—v log v I (1+a+y) Write y+2 for y, and subtract the first, which gives peo dW yo EOtety) TA+2+y) ‘ Lo layne P(lty) 0 (l+e+e+y) Subtract this from the preceding, with y changed into z, and we have * ON DEFINITE INTEGRALS. 667 ('Q-v)—v’)(1—v*) dv 4 l—v log v og POY LA+YL 49 PO4ety+2)_ 4 pet enhi ie Me htt Nd+e+y)PU+y+z2)0d+4+24+2) ’ and thus we might proceed until there are any number of factors in the numerator of the subject of integration. Many integrals may be deduced from this form. An equation in page 243 gives, with (—1, a, +1), eo . . ° adx sin x * sina.ardxr 4 * asin 2v.xrdx Aft > m+a? 1—2acosxr+a? » m2 ana a T l (page 655) => (e+e a+... J=> pet Change x into 2r, m into 2m, and then make a equal to +] and —1 successively. Sem Cota Haye “aetan 2.dxr € ; m? + x gem __ 1” : m+ 2 Sal eo 1 Change a into —a, and add; in the result write a for a’, ° adr sin x v e™ 1 >m +a? 1—Qacos2x+a® 2 &"—a' I+a ie ] xdz Te” a=1 gives a = posing m+2 @&—] In the first equation write nx for x, and nm for m, multiply by 2adn, and integrate with respect to n from n=O, 2 ox a ap faq he C—2e cosne-+a%) 20g (1—a) Jarre n dn — TO arnt ; em pak a ; TH d a which gives | aawee (1 —2a cos nea?) =— log (1—ae-™"). Differentiate this with respect to a, make n=1, and show that the result is the same as we should have got by beginning with the equation in page 242. For » write 2n, make a successively equal to —1 and +1, and subtract, which gives if log sin nx. dx 2 |-.s2m {> cos n& dx ? e a ea, 1g os Tie 4 m+ x 2m 8 ett — ie abs Eee — — log ee m+ 2° 1 7/ Wee acetates oe Ty i+ sli logtannr.dx = AE Baal A wats 3 0 The preceding two pages contain certain excursions (by Legendre) nto the field of definite integrals, not made with any fixed object, and zuided by the facilities which arise from being able to find some one 668 DIFFERENTIAL AND INTEGRAL CALCULUS. fundamental integral. Whenever we are able to find [yr.daxr.da, for all values of a, it is generally easy to find a function fr, which, ex- | panded in the form A,@ (2,4) +A,.P(a,x)+...., will give for fwc.fx.dx a series whose invelopment is known, But this method must always be of twofold application ; for since fwax.dxr.dx follows from fyx.pax.dr, a similar process can be instituted with functions | which can be expanded in the form A, (a, 7)+ A, & (a,0)+...4 | For instance, we have o . o sin nv.rdax T cos nxr.dx T —" —mn eo? — yas on nm sin 79 | $f Snih sha 3 sin $n0 ar ae a: x gb? gba soy n? 7 4 4b? x3 mere od Obes cos n0 ei er 4 nN cos $nO gt hy ae n? 77+ bx? ght gobs . n® * + 4b? x? Where = implies a series of alternately positive and negative terms, the Series on the left being summed for all znteger values of n, and those on the right for all odd values of n. Now make b=, whence 6=a: multiply the first and fourth by coscx, the second and third by sincz, and integrate with respect to x from 0 to cc, remembering that fe [==] . he cx. dx W 34 { siner.rdx rr —™ Sa ereS E NB, MCW EPSL n+ga® nl g 5 n+gz" Q¢ e O In this case 0=a, and we have (a<‘7) / e? — e- J coscx dx =sin a.e~°—sin 2a,e—" + sin 842°" —. .. uz —7Tr of Ee ( 243) e “sina sin @ DNS is 3 NSS oe Ey Gr a is pee lte"+2e°cosa +e °+2cosa © ax —ax c —c ; 3A e+e 1 eo—e b And similarly sin cr dz=— ———_—_._—_ 3 geet 2- & +e +2cosa@ fae gt ge ' (<#?—e-*) sin 4a — sin cx dz=——_____—— 9€ te e+e °+2 cosa jf ® 974 eT (+e) Cos 4a cos ex dx ——____-——_ e+2e°+2cosa uD —nTr tags +e I have left the completion of the processes as an exercise for the student: the following formule will be needed, DIFFERENTIAL AND INTEGRAL CALCULUS. sin 0 __sin 0 (a@—a*) | sa WUE Bos +2 cos 20.424 24 a4 cos 0 (x-+ 2°) e aK 30 e i eee —— 2 CORY aide nee ana 14+2 cos 20.27-++-2* sin 8.7—sin 36,.2°+. Let < fv dv, this, if fv can be expanded in positive powers of v, is only possible when gx can be expanded in negative powers of x, for we have from the preceding orx=f(0O)-0*+f" (0).2?+f" (0).a7+....; so that if ¢v=A,a"+A,2-°+...., we have fo=A,+A,v+d A, vo +32 A,v°+.... Such cases are generally those in which ¢z dimi- nishes without limit when 2 increases without limit; and these are the cases to which the following method will most often be applied. If we take any other limits, say —1 and 1], and if wz can only be deve- loped in whole powers of 2, development of <~* shows that we must have Sito" fo dv=(—1)" 9 (0). Assume for v a series of the form A, R,+A,R,+...., where R,=1, R= D (1—2"), &c. and R, =D" (1—v’)”, as in page 664. Since, then, tiv" Riim dv =0, we have A, fiiR, dv=¢ (0), A, fii R,vde+ A, ft} R, vedv=—¥ (0), &c.; from which A,, A,, &c. can be found. But the results of this method will give for the most part divergent series, which cannot be safely integrated. Assume x= f«~*’ fv dv, between fixed limits, but of what value is of no consequence. We have then i 70. aD @ (a+)-+h (x+2)+.. i= Peg (eC +e" +... =F Ee ; Ee 1 1 1 “* sin vt. dt Equation (c) gives UNS pe 4 | For Re Pp atid A CHD +9C+2+4...= | : : 5 fe fo de - C® sin ot.e ™ fu dv dt 7 | . et] ‘c dt Nah = fi ordr—} pr+2 / me Cf sin vt <~*" fo dv). e/ 0 b (wttJ/—1)= fer eV fu dv gives at+t,/—l)—¢ («—t.J/—1) f sin vt e—” fv PRIS Ait J ee ; r? dt g(atty-1)-9@-ty-1) p(@+1)+...=f: pudr—hox-2 | ata. we Ee ; 672 DIFFERENTIAL AND INTEGRAL CALCULUS. ! Let the last factor be ¥ (a, £); we have then | PA)+.-.-+¢ (e)= fj bu dx+§ (dr—G0) : | d +2 a {y (x, }—y (0, t)} Te e 0 sos BPO cy idé rT gx=(1+2)™ and r=c sive | , mn GE aL Let dzv=log (1+ 2), and deduce { Crees, oats, me re Ge fog (t me mS ek Ce? . Lee Jie 2 4 log F(1+2) Alia ania “tan“'t.dé 1 log (2r) jarcerpecse This theorem, though deduced from the supposition $= f[<— fv du, : may be proved independently of any such assumption. We evidently have by expansion and page 581 Wee goa highly ] i I Boa | —_ (2) {is — ee et 1 ," ! oun AN 62" An Be kets By =4n | eort l > 0 Substitute the values derived from this in page 266, in the value of | $x, making y,=$2, remembering that 1:6, 1:30, &c. are B,, Bz, &e. Lhr= fi px dxr—} (px — G0) TOS di : P28 \ t— Vee — DO, oreteta 42{ so i TEC ve ; the last factor being $ (+t. -1)—¢(a—tf—1)—{¢@J/-)) — (—t,/—1)}, all divided by 2,/—1. Subtraet ¢0 from both sides, and add gz, which turns 22 into ¢1+....+@zr, and makes the preceding correspond precisely with what was proved before. The following case arises from ¢a=sin ax or dx=cos aa, x disappearing of | itself, @ at .—at 1 1 a © tdt 1 ih TETRA ERE 1 Sigh r9 —, whence ——— == —, eed Hy OED gin ere the last is already known: a@=z furnishes another verification. In the value of O(7+1)+.... first found, add ¢z to both sides, and for mz write ¢27, putting 4a for x: the result is path (a+2)+....= fy, (22) dx+4 ga (ord ob Gta) =f Gara E) i; dt tx has b) — =f (fomak = fa prde+4da— | | ee ON DEFINITE INTEGRALS. 673 Multiply by 2, and subtract the value of a+ (a+l)+ anes. ee ga— (a+1)4+6(a4+2)—..... = A ole Da — (a+t,/—1)—¢ (a—t,J—1) =z pa 2f et, e Tae ema efor 1 _—_—_ $(4)+¢(@4+1)4+6(4+2)4+....2 Sidr dx+} pa—2 f fae EON ED. _ For gx write ¢ (—zx), and having determined ¢ (—a)+¢(—a—1) +...., write —a for a. The theorem in page 561 is then easily rerified. Moreover, whenever $ (a+t,/—1)—¢ (a—t,/—1) is posi- ive from ¢=0 to t=cc, the theorem in page 650 easily follows, since ba— Ph (a+1)+.... being then algebraically less than + $a, is less than da (if ha be positive). In the preceding theorems, the original supposition ¢xv= fe* fu dv las been rendered unnecessary by a demonstration which is independent fit. Resume this supposition, (which Abel takes as always possible,) nd take the known equations (from £=0 to t= cc ) = avt.di x sinavt.tdé © _, [‘sinart.dt TO ~w) AP Gi ingay aaaA 5 25 bajeraaea aso ne last f pitborthy od + viene bered that @ must 4 ast Irom ———_-— =— ——_ —__: i remempere 1at &@ Mus AWAD), catil ae 2 i € positive. Write a-+at,—1 and a—atJ—1 for x, which easily ives etre 2 (4—at, J—1) +4 (@+at,/—1) fe sc (avt) Rf alae ee ne para BVEk we ° ii “cos avt dt) low seer ~~ COS Bvt. jv dv} = tf em ee OY OD feces shy fi igen =e fest: v dv; hich last is a7 (x-+a): proceeding thus, we get the following \eorems, £if—] @(7—ai,f— Let E (a, ieee ee Oe BE hen ee a ee NT O (a,Jat)= 2 J-1 , F) dt T S tdt a zi vB (1, al). 5.4 e+e); S00 (@, at). ~=-5 6 (e+a) 1G, at). =F 2, f° Ce, a) — = 8 {9 +0) —Ga} ; Met tg gh J1 0 t(1+t)” 2 ; e fourth formula being obtained from the second and third. Different *ms may be obtained by making at=atan¢, and substituting. € shall presently cite an example, but we may, by means of the | 2X 674 DIFFERENTIAL AND INTEGRAL CALCULUS. preceding, refute the notion that every function of 2 can be expressed : by fe-” fv dv, between limits independent of w. | et gra, then E (a, at)=z, and if « can be expressed in the form fe fu dv, we have 2 | a ae (x+a), which is false unless a=0; | consequently it is not true that c= fe fo dv can be satisfied by any | form of fv which allows of integration. ! Remember that in the application of the preceding formula, | gx fe fu dv must not only be true numerically, but essentially true | in form, so that x+at,/—1 and a—at,/—1 may be substituted for @, | For instance, if we were to take ! sah eat eas ial all il i aiid - Baer exe an Wed Liss Sin and apply the first theorem, it would give { 3 e@ (20 Le ghar us ee ee > &”—2 cos 92.2% +1 "]42° Asina sin (v+a) | But this is not allowable; for the definite integral with which we commence is only true in a numerical and limited sense, from #=0 to) a=, both inclusive ; nor can it be permitted to substitute #+ al Nis} for z. Moreover, the result is false, it being easily shown that the. left side remains finite when « approaches s—a, whereas the right side. increases without limit. | The following theorem, however, will be afterwards shown, and may) be verified when # tat,/—1 is substituted for 2, -) | a a Agar ae dv==log (1+2x)—log z. Let then bx be log (1+) —log x, and apply the theorem, which gives ite 2° fo log V(e+et+eeyPt+ee dt —*} lt+a+a ote ea? t —) ta | : | fvten> (<= tdt By 14-c+a 3 bao PPL -2 8 ea he t di. «x 1l+a oi f LL, Bm rks ee wa Jagan (ee oo ™ t { x2 Again, fee err art du ett (tee Met doce x €% apply the theorems to ¢v=«*"'*, and the results may be easily shown to be false; and the same in every case in which the limits of integra- tion which give ¢x have different signs. Here, as in page 607, we must not use a result which is subsequently to enter into the subject 0) an integration, unless that result be true throughout the limits of mte_ eration. Now, in obtaining the first of Abel’s theorems, of which we are now speaking, we have to use the integral fj cos avt di: (1 +) 7 : ON DEFINITE INTEGRALS. 675 which enters into a subsequent integration with respect to v: as long as vis positive, this is 4 re” (a being positive), but when v is negative, it is}. It is easy, however, to extend Abel’s theorem to this case jn the following manner. Let ¢rz=fe fu dv, the limits being —y and +, negative and positive, and let this theorem be universally true. We have then ref wiht ———— gf Ee COR LLL O dv) i G + t? "4 . 1 re | = ftaten fray. { ("OEM so) Soar) ole fo Now in the first integral v is negative, and in the second positive ; mroceed accordingly with the included integral, and, on the same reason- ng as before, we haye, by this and similar processes, se dt Pa =) uJ —(«*—a)v £ Wi —(a+a) v fo E (a, at) => Je. shila f§ e-@49* fy dv 2 tdt T 0 —(e@—a) v ud B o—(ata) v fc O(#, at) ao ee y Rae: fodv—-> fie fo dw S50 @) SE fe fodv—= fre fod . Pe £ 2 ¥ Quem 2 12 Apply this to fuse”, gx =Jr.8”", E (a, 2at) = Jr.et™-*” cos azt, J (x, 2at)=Jr.8°-°° sin art. Then, remembering that e=—c, j=-+ oc, and 2 7 PUK VE (yy mem 4p? FP — (Apt)? Jo, cms Ape (PP mv? 7 fee me -* ct foe blgals ifm fire dv, ve have the following equations, 1S 92% (@—2a)2 (x+2a)2 : € cosazt.dit ae Fe ste pe on A Jee | =-¢ 4 mete “dotsé 1 itee dv meted 2 2 ’ —~2 p4-2a)2 J a8 cP ' sinartidt — fest as err a i 2 dy Ire — =— ee —-— Fong f 1+? 2 2 0 —t Re 2s invt.dt 7 7 T iam we | vz] Shining PE rl es du—; Sine le du=1{# « dv. t 2 ; ’ 0 The third, differentiated with respect to «, may be verified by page 34; the two first may be thus written, after reduction, with an obvious bbreviation, {= cos art dt ae fens (22—4 4. 60? fr gt Pi dv apie Fy 2 PP t tdt rig se, rr E ae ave fem toa et fe be “dv; e 2 0 de*second of which may be verified by differentiating the first with espect to z. If w=0, the first may be reduced to So waz 72 moka ep hy. “abel eo ily: Mey, ill these integrals can then be calculated by Kramp’s table (page 657). ¥ 2X: ore od 7 676 DIFFERENTIAL AND INTEGRAL CALCULUS, | If we throw the last result into the form fuga hie 5, dl not SORE [Ste /penartt we may see that differentiations with respect to a and b will enable us | to apply the table to the determination of fy«~** Pat, where P is any | function which has a rational and integral function of ¢ for its numera- tor, and an integer power of b+? for its denominator. Two integrals, each of which is infinite, may have a finite difference. | Thus, if in those of page 630, we make 2 diminish without limit, the | first increases without limit, while the second becomes : , 4 sin bx b f Bere te dr=tan™ (=). | ey a | Now let tan— (}: a)==8, tan7? (b': a’) =A’, and we have f__cos np cos nf’ “ f > (e—* cos br—e*” COS ba) ehdes Pin aby Go Ty 4 a Expand the second factor in powers of 7, which gives for the whole. product (I'n.n {4 log (a?+b") — 4 log (a +0°) +An+Bnit+.... ae | and In.n or M(n-+1)=1, when n=0. Consequently we have ? 4% cog br —e—*" cos D'X 1 al? +B! -————————— rors dz==— log ab i ° av 2 a+ 2? cheer” a! * cos br —cos bx : : —__——— dx=log —, — 7 =\lor 4 e a > x b | i ‘ The following integral can be found in finite terms : 8 em at — a2 a? Fp 7 (% om? 2? — 27? dat ere de=afye = by changing 2 into a:a. But the latter multiplied by —2 is dy : da,| whence —2a dy = + 2y=0, YroUC aba Mabe the constant being determined by making a=0. The following is a remarkable instance of discontinuity. Expand) and add log (l—as*¥~) and log (l—ae"*¥~"), which readily gives 2 3 log (1200082 ha) —2 ( acosn4-F cos dr4-S COS B2-f-eeee fy a series which is convergent from a=—l1to a=-+1. Integrate with, respect to 2, from #=0 to e=7, and we then have fi log (L—2a cos 2-+a').dx=0 (Sx oneedne i . | ON DEFINITE INTEGRALS. 677 3ut this process cannot be depended on when a>1 or <—1: let such e the case, then the preceding is true if for a we write a, which gives rom the preceding Slog (@®—2a cos x+1) dr=2 log a (a*> "or =1)); o that the first of these equations really involyes the second. Make ne step in integration by parts, and we haye * sinz.adzr T 3 1 ————_——_., = — loge i ioe faded i l-—2Qacosxz+a a ACAI ab og (1+ =) ecording as a*< or >1. Also s 1 f? log sin x. dv==— log ms l—becosx b ©” b s "sinz.cos*2.vdx = Ir a 1,, 6 penal 9 (l—bcosr)" ~ EF (n+1) db"’ b b s i the result of which b may again be changed into 1:b. Now differ- (tiate the second integral n times with respect to @, or 2n or 2n-+1 (nes with respect to b, which gives 678 DIFFERENTIAL AND INTEGRAL CALCULUS. r dd" ye fi e~*" cos bv. a™ dx =(—1)" vn — qute 4a vissimaat 9H 2n b2 eo — are 20, 7 — Jt d wEih eas fee cos bx,a" dx =(—1)" at tgp (a te *) | ee Jr de oo | foe-™ sin bx a" da=(—1)" 9 gpen ate a) in all of which, integration is made to depend upon differentiation. We also learn incidentally, that a~*e~”*** is a function which gives the same results, whether it be differentiated 2n times with respect to d, or | n times with respect toa. Let the student apply a similar process to | differentiations of f> 2° dy dw da dy dz = eee en eT es P*Q’R’S? T: PY Fh oe WE pare’ resto ete 444) Aa REET ai all elements being included in the integration in which () + (a) + Ge) +S) + Ge) does not exceed unity. For instance, the quarter of a circle is f dx dy, where 2°+7/? is not >a’; it is then a.a ¥3T3 wea ko ae dv..dz= 2 2 , or 7 (TH=s therefore TF (4)=,/r. Required the value of the total element of the first preceding integral in which the sum of the variables lies between ¢ andc+de. It will be sufficient to take three variables, x, y, and 2, and to suppose that the integral in question is 620 DIFFERENTIAL AND INTEGRAL CALCULUS. farty 2 da dy dz, subject to r+y+z, not >e, which must first be found. For a, y, 2 write cx, cy, cz, and the pre-_ ceding becomes or tery f gal y?! 27) dx dy dz, where z+y+< is not >1; and the integral is a constant already determined, call it C. Con- sequently the integral, «+y+z not exceeding c, is Cc*t?t”, and x+y+z not exceeding c+de, it is Cc*t?*7 + (a-+B-+ y) Co*t?t? des whence the latter term is that element of the integral which answers to the aggregate of values of x, y, and z, which satisfy the condition of © x-+y+z lying between c and c+dce. Next,* it is required to find {a ye 2°") fF (a+y+z2) dar dy dz, | on the supposition that 7+y-+<2 never exceeds 2, All the elements of | this integral answering to values of x lying between c and c+dec are aggregated in (a+/-+y) Ce*t*t?— fede. Consequently the integral required is (2a+B+y) Pal B Py vy) Cf! crt6t—! fo d wane eS cen: (atBh+y) ie fe tut, or T (e+h+y+1) TealBly P(@+h+y) iy a ehh | or far tye 2° f(at+-y+z) dx dy dz= and by a change similar to that already made, we find —1 ,,6—1~7—1 wo \P y , zy < 3 B of Ey eae ba _PQR niga oh ein are f' sige Meee | de pig aye ek bs oy NNS , ela tee eat) ? qd T 1 P q ~ r if (+) a (4) + (=) never exceed /, By this process can be immediately solved many problems connected _ t patatI—l f | fice fede, \ | with the eighth part of any solid whose equation is ax”+4 by”+c2"=], | among which are spheres, spheroids, and ellipsoids: including par-_ ticularly the determination of their solid contents and centres of gravity. And, similarly, of all curves whose equation is ax"+ by"=1, including circles and ellipses. Something of the same sort may be done, but not so easily, when the limits are 0 and oc. ‘Take, for instance, Sofood (ety) a ydxdy. Assume x=r cos*6, y=r sin? @; the pro- cess in page 395 gives f [¢r. (r cos® 0)* (7 sin® 0)? 2r sin @ cos 6 dr dO, from r=0 to r=, and from 6=0 to O=47 3; or 2f hr .1te dr. fsin®* @ cos** 6 dé, * This theorem is due to M. Liouville; all that precedes has been used by Laplace and others in problems of probability, but only in the case of whole | exponents: M. Lejeune Dirichlet appears to have first drawn attention to the general form of the theorem, There is a paper containing another demonstration, — by Mr. D, F. Gregory, in the Cambridge Mathematical Journal, vol. ii, p. 219, ON DIFFERENTIAL EQUATIONS. 681 a _ ne I — r (B+1) fegr pete! dp: P (a+6+2) ‘ the second integration being actually performed by making sin 0=v, and changing the functions and the limits accordingly. For 2 and y write ax” and by", for « and PB write (+1) :m—1 and (6 +1) n=l, / and we have or hi @ (ax + bx") 2° y? dx dy Cee nae pet peti am b m m n ati{ Pas, = f dr 7m 8 dr; . mn r= $e) m n the limits being 0 and cc for every variable. It would make no differ- ence if we wrote az"+ b2"+c and @(r+c). If we now ask for fo (e+y+2) xy’ 27 dx dy dz, first let « be constant: we have then I (B+1)T (y+1) aty+z) y’%2” dy dz=-——~ ——+ Bry an Jo (atyt+z2) y?2" dy T@tyea eer Multiply by a*dz, and integrate the second side by the same formula, which gives for the integral required Geta) b -F EPE Gee Le a ee ———$——— - — f,orert dr. P (atA+y+3) Proceeding in this way, the general theorem is, that o) (v,+2.+ eee ) Ge rat ee ere ea Gis. [oe | Lec Bie. atB-+euw—l J aa atte mabe ASC: pene a Hes Det Pace el) S¢ Qand c being the limits of every variable. A transformation may be made by writing @,2™, &c. for a, &c. This theorem, however, is nothing more than the last, since / may have any value: and in the proof just finished, the upper limit of r may be any whatever. But those of @ must be 0 and 37; or y:z or tan’@ must take every pos- sible value. To make c=rcos®0, y=r sin*@, and to assign 0 and / for the limits of r, and 0 and 4z for those of 0, is in fact to make x and y take all possible values in which e+ y does not exceed J. CuarTterR XXI. ON DIFFERENTIAL EQUATIONS,* AND EQUATIONS OF DIFFERENCKS. Hiruerto I have only considered the general theory of this subject, with a few applications to actual solution. The present chapter is * It would have been a more difficult task to have selected the matter for this chapter from the mass which has been written on the subject, had I not derived much assistance on this point from three very excellent French works which have 682 DIFFERENTIAL AND INTEGRAL CALCULUS. intended to exhibit those isolated modes of solution which may one day form part of a general theory. It will be most convenient to divide this chapter into articles, after the manner of Chapter XIII. By y/, y”,... y@, &c. are meant the first, second,...nth, &c. diff. co. of y with respect to w as usual. (1.) The equation y=¢ax is integrated as follows, Let Cy, Cy, C,...+C,-1 be the values which y,y,'y",....y are to have when r0. Then : C, a" C, x? y={ fede} drt oo i tet t gt Oat where ( f dz)” is the symbol of n successive integrations with respect to x. This successive integration may be reduced to single integrations by the following theorem, which, with its inverse, I leave to the student. Let I,=(fdx)" $2, P,= f2" de dz, —] T (n+1). 14,22" Pp—n2z"* Pin = 9 Pome oem MPa in Pia" L—nat Tn (n—1) 2? Ip... En (M— 1)... oD Ting (2.) If we take the equation adr+¢y.y/=0, we have the complete integral in afdrdx+ foy dy=C, provided that fx dx can be found. But if this integral should be an unknown transcendental, we are not to conclude that the equation cannot be integrated, for it may happen that a relation between y and a, independent of the transcendental, can be obtained from an equation involving this transcendental. Let wa and we be inverse functions of 2, in such manner that ~y'r=a, and wwe=e. Let Ox be another function of 2, and let us consider ywOus—'x, or the performance of two inverse operations separated by the performance of an intermediate operation @. It by no means follows that y6y—x contains we directly: for imstance, when % and 6 are con- vertible, or WOx=OWr, we have YOY" rv=—byy'w=—O2x. Now let wo= [ou dx, whence the preceding equation gives ape+yy=C, or y= (C—aypax) Ce mae a = +750 gives y=log' (C—alog x) =e° a lately made their appearance, and which I have thus been able to follow, to a con- siderable extent, in the choice of topics. They are 1. Cournot, Traité élémentaire de la Théorie des Fonctions et du Calcul In- finitésimal. Paris, Hachette, 1841. (2 vols. 8vo.) 2. Duhamel, Cours d’Analyse de l’Ecole Polytechnique. Paris, Bachelier. (vol. i. 1841, vol. ii. 1840.) 3. Navier, (suivi de notes par Liouville,) Résumé des Lecons d’Analyse donneés 4 lEcole Polytechnique. Paris, Carilian-Goury, 1840. (2 vols. 8vo.) Each and all of these works I can most cordially recommend to teachers and | students. There is also another work to which I may yet have to acknowledge my — obligations, but hitherto only the first volume has appeared, and too late for me to avail myself of its contents. Moigno, Legons de Calcul Différentiel et de Calcul Intégral rédigées d’aprés les pe et les ouvrages publiés ou inédits de M, A. L. Cauchy. Paris, Bachelier, 1840, ON DIFFERENTIAL EQUATIONS. 683 adx au dy or, when a=1, y=sinC.,/(1—2*)—cos C.a. In fact, the last result depends upon sin (a sin™ x) and cos (a sin2), which are simple algebraical functions whenever a is a whole number. Thus sin (2 sm™ z)=2 sin (sin~'x) cos (sing! v7) = 2x,/(1—x?). . When the transcendental introduced by integration F and its properties, are well known, the reduction of the integral to its simplest form is easy enough. And there are some cases in which the same determination can be obtained where the transcendental is unknown, of which the fol- lowing are historically remarkable : op sit Sri) dp nN /(1—e®sin?6) ~,/(1—e? sin? f) =0 gives y=sin (C—a@sin™ 2) ; dé dp Assume ara /(i—e’ sin? 0), whence a (1 —e® sin? d) d’0 dad ae ==—e’sin 0 cos 9, Sra e* sin p cos d, a’0 d*p we? 7p +7 e* sin (0-+¢).cos (@—4) de? dd? haere = —e* (sin® 6—sin’ 6) = —e’ sin (0+ o) «sin (0—¢@) do do do pa —0=0), —=—e’ sin cos O — ,— = ~—e? gj EN (p+ o,p Ys de silo 0; diel dt €' sino sino dc dd. dam. od : DRI. Ho sin 0 3 — COs 0 aTpre Gao sin os or J (1—e’ sin? 0) F./(1 —e* sin’ ¢)=C sin (0—4), 2 or JU —e’ sin’ 6) +,/(1 —e’ sin® 9) = —Gsin (O6+¢). dt”, dy = ‘ss | Ae atta Let i toy pu=,f/(a+ br + cu’ +ea* + fr‘) . az. ips. ; seer Assume Bat ap Ft a-y=o, t—y=o. 2 a: Proceeding as before, 2 oo =b+2car+ 8er?+ 4fr*, &. i ye coe Feet cette (OLA LGS +308") de do _ eee wee 4 a 2 A 3 52) t i’ Fn we teat te (30°+0°) +4 f (3+ oc”) } 1 ( do da nets & de dt dt . wh aids Multiply by 2de, and integrate, which gives 5 ap tert fo", or gaz py=(a—yW{Cte ety tfery)}. 684 DIFFERENTIAL AND INTEGRAL CALCULUS, In both these cases the evaded transcendental is ¢, an elliptic function, (page 656). (3.) Let x=f(y’), then dy=y'da gives y=yf (y)—JSf(y)- dy’: if this can be integrated, y’ must be elimmated between the values of x and y, and the primitive equation is obtained. (4.) Let y=f (y’), then (a’ being dx: dy) we have andi aad (>) dz’, and y=f(>) between which 2 is to be eliminated. (5.) Let y=a¢d (y')+% (y’), of which the equations in pages 196 and 365 are particular cases. Differentiation gives y'=9 (y +{roy)+y¥' (yt y” Cwrite = for ¥'), , p's dz is as o's dz dx eg « v ¥ chal 9 pm ee yor de es ats dz z—@z 2-2 z— Gz from page 195. Eliminate z between this and y=2x z+ Ye. (6.) The equation y= (2, y’) can be made to depend on one ofa linear form, and elimination. For y' write z, and differentiate with respect to x, which gives dz _dph dp z=P+Q Fy (p=%, 3G. . This equation is of the first order, and of the first degree with respect to dz:dz. If it can be integrated (say it gives z=W (z,c)) we have then y=p{2,u(a,c)}. Thus y=a+y"” gives z=1+2z2', or s=C+2z +2log(z—1), whence a=C+2/((y—2) +2 log @/(y—2)—1) is the primitive equation. ‘edz 2S C= 424 whence c=¥d""(y—ax—b)+C is the primitive equation, or y=ar+b+ x («a—C). (7.) The equation (ax+by+c)+ (Ar+By+C) y’ can be reduced to the homogeneous form by making x=v-+a, y=w-+8, and taking & and B, so that aa+b8+c=0, Aez+B8+C=0, in which case we have Again, y=ar+b+9y’ givesz=a+@'z.2,02= (av -+ bw) + (Av+ Bw) ~=0, integrable by page 194. There are two cases of exception, 1. When @ or > are infinite, or when re ; } A:B=a:b. 2. When they take the form on which case, besides the preceding, we have C:A=c:a. In the first case av+-by=z gives A sph fa Regt F taig 2 Ns stes(Aa40)(4 #2)no ON DIFFERENTIAL EQUATIONS. 685 from which the form dx-=Zdz can be obtained. In the second, the equation can be reduced to (av+by+c)(a+ Ay')=0, and if the first factor may be rejected (which, however, depends on the problem), we have a+Ay'=0 for the equation. (8.) y/+Py=Qy", P and Q being functions of x, is reduced by simple division hy y", and making y~"t'=z, to the form —(n—1)7 2! + Pz=Q (page 195). The exception when n=1 is obvious enough. (9.) The factor which will make an equation integrable per se (page 196) would, we might suppose, be the principal instrument in the integration of equations: but it is rendered almost practically useless by the difficulty of finding it. It can always be determined when the equation is integrated (that is, when it is no longer wanted), Reduce the equation to the form y'—x (w,y)=0, and let y=¢ (a,c) be the primitive, or c—=®(2,y). We have then db Pes. G and x (a, y)= dy °¥ dr > X PY Rai d@ do_ dx dy’ so that y’/—y(a,y) multiplied by d@:dy becomes d.®: dx, and is integrable. And if f® be any function of ® (2, y), the factor d® bes {/® — makes the equation integrable, “ dy If the form of the equation be P+ Qy’=0, the factor is e > y (10.) When the factor is a function of w only, or of y only, it can be found. Take the equation which determines the factor M (page 199), and since any solution is a sufficient factor, let there, if possible, be one in which M is not a function of y, so that dM:dy=0. The equation then becomes / 1 7dP dQ i dM eh: dP _ dQ Pot Maelo ea M dz Qvidy dz provided the second side be a function of & only. dz (11.) If an equation of the nth order be reduced to the form Bo +e (yy ,.. sy, 7)=0, or y +Y=0, and if y(y°-,....y, 7) =C be one of its immediately preceding equations of the (n—1)th order, the factor may be shown in the same manner to be fs (dy: dy“). And if ¥,=C,, ¥,=C., &c. be the m equations of the (7—1)th degree from either of which the given equation will follow, it may be shown that ds, dir, fi Gh) dye +fo (2) dyeot »«.» is an integrating factor ; Si, fz, &c. being any functions whatever. (12.) In the case of y”"=¢a, in which I (or any constant c) is a factor, x is also a factor, and ay"=x¢z gives yvx—y= fax dx, which . . . . . rm . is one of the corresponding equations of the first degree, ‘The other is y= fa de. a ae 686 DIFFERENTIAL AND INTEGRAL CALCULUS. (13.) When Pd2x+Qdy=0, where P and Q are homogeneous func- tions, the divisor Px+Qy gives the factor which makes the equation become integrable; for d P dP dQ ) Oe et a Beep eee OD dy Poly (Pz+Qy)*( Q re a! ae Q d Ap bogai we dQ dP ; te Peng Teer wr(P Be-0F PQ); and if P and Q be homogeneous functions of the mth degree, we have (pages 64, 194) dQ. dQ dP dP Ts aya ae ae ay =a dP dQ\ /,dQ dP (OG? ay I=? Ge) (14.) The functional equation dr+¢y=(xr+y) has a solution which is well known to be the only one, év=cz, and the proof * is given in Kuler’s celebrated proof of the binomial theorem. But a more simple proof is derived from differentiation. Consider y as constant, and the preceding gives ¢’x=@' (a+y); whence, y being arbitrary, g’x must be always the same, or Gxr==cr+c, Apply this to the equation, and we find c,=0. From this equation it may be immediately found that dz x ¢y=¢ (w+y) has no other solution than ¢x=c’, that ' pe+by=G (ay) has no other solution than @r=cloga, and that ox X oy =¢ (ty) has no other solution than dr=a*. It is important to observe that the limited character of the preceding solutions is entirely due to # and y having no dependence on each | other : take any instance of such dependence, and the case is much altered. For instance, let y=a, or 264= (22). This is solved by #r=ca, as before, and also by @r=ay (27 log x: log 2), where wx is any really | periodic function of sin x, cosa, &c. (15.) Any differential equation may be reduced to a set of simultas » neous diff. equ. of the first order. Thus, if in y’”/+Py”+Q7/+Ry | +S=0, we make y’=v, y’=w, we have the three simultaneous | equations v' +Pu'+Qy'+Ry4+S8=0, v=w, w=y’. Conversely, any simultaneous equations may be reduced to single diff. equ. between two variables. For example, let x, y, z be functions of f and let three equations contain diff. co. up to a’, y", and 2%. To obtain an equation between wv and ¢, differentiate each equation 6+7 times, giving 89-++3 equations involving 16, 19, and 20 diff. co. of the * In brief, that proof is as follows. The equation immediately gives 9 (max) =mor, m being any integer. Let be another integer, and let ma=nz, which ; n n ; ; ; gives mpu==nGz, or @ = rn gx, so that the preceding holds when m is fractional. — But from the equation, ¢x+¢0=¢x, or ¢0=0, and ¢r-+¢(—a2)=—¢0=0, whence —Pr=$(—2), 6(—mx)=—G(mxr)=—mGa, or the equation holds when m is negative. Hence ¢ (max) =méw is universal, and a=1 gives ¢m=mgl, so that m in gm can only enter as a simple factor; and the same of w in ga. ' , : | : ; | ON DIFFERENTIAL EQUATIONS. 687 several variables. Between these 42 equations eliminate y, Ms gaits @2',++++, 19+1+20-+1, or 41 quantities: the result is a diff. equ. of the 16th order between # and t. To generalize this, let there be the variables 2,, a,....7, and ¢, and » equations going up respectively to the kth, Ath,....k,th diff. co. of the several variables. Differentiate each equation k,-++-4,+....-+, times, which will give 1 (ko hg++.+.+hk,)+7 equations in all, These equations contain x, and (Ak, th.+... +R.) diff. co.; 2 and (Qk.+ks+....+k,) diff. co.; a3; and (k,+2hk,+... -) diff. co.; and soon. Exclusive of x, and diff. co. there are then (4, +4,+....+h, being x) 1+ («— ky + he) + 1+ (x—h, +h,) + eee. +1l+—h,+2,), or n—1+ (n—1) (kK—h,) + «—h,, or n(K—k,) +n—1 quantities ; with n (x—k,) +n equations, as before shown. The equations exceeding the quantities by one, all may be eliminated, leaving an equation of the xth order between 2, and ¢. For instance, let there be two equations of the form Pa!+ Qy’+Rer +Sy+T=0, between x, y, and¢, Differentiate each once, giving two new equations of the form Av" + By" + Ca! + Dy! +Exr+ Fy+G=0; between the four equations eliminate y, y’, and y"; there remains an equation of the second degree between w and f. This is the general theory of the reduction of such equations: but it would hardly be safe to say that the elimination is always practicable without any of the circumstances which sometimes require additional consideration in algebraical elimination. (16.) The only case in which there is anything like a method of integrating simultaneous equations without elimination is when they are linear. Suppose, for example, that x and y are functions of ¢ to be deter- mined from (a means dz: dt, &c.) Pie’ +Qi7/+Ric+Siy+T,=0, Pre’+Q.y/+Ric+S,y+T.=0, ‘where P,, Q,, P,, &c. are functions of ¢ only: this is the most general linear form. Reduce these by elimination to mt t+B,y+C, yA, 2+ Boy+ C,. Let 6 be a function of ¢ to be determined; add the second multiplied by 6 to the first, and assume z=2+6y, which gives x! y= (A,--A, 6) (2— Oy) + (B,+ B, 0) y+C,+ C29. Take @ so as to make the coefficient of y vanish, which requires Ca 6? 4. (A, = B,) d—B,, and gives 2’ =(A,+A,0)2z+C,+C,0. aay If the first can be integrated, the second, by substitution of 0, is made linear, and z can be found. Also, since the integral of the first equation 688 DIFFERENTIAL AND INTEGRAL CALCULUS. must contain a square root,* two distinct forms can be given to 0, and two forms of z, or e-+6y found. Hence # and y can be found in terms of ft. When A,, B,, As and B, are constants, it is sufficient that 6 should be a constant, and a root of A,6?+(A,—B,)9-—B,=0. Let « and v be the roots of this equation, then at pyce rts £ (C+ Cy pu) 7 Ortaet! dé apvysc|e@rtae! ((C, CO, y) erteat di. When p and » are equal, the values of v and y obtained from these take the form = 3 and the real values may be found by Chapter X. But in the particular case preceding, a more simple artifice will suffice. The two original equations give of +-6y'=(A,+0A:) 2+ (B, +6B,) y+C,4-0Co. Let 0 be so taken that B,+0B,=0(A,+6A,), then x+0y=z gives z'=(A,+0A,) z+C,+0€,, and the solution is as before. (17.) The same process may be applied to the case of three or more variables. Thus, let the equations be (# meaning dr:di, &c. as | before) waAc+BytC,2z+E, yA athe, “mA, r+&e. 5 A,, As, &c. being functions of ¢ only. Multiply the second by @, the | third by #, and add, making w=wv-+0y+¢z, which gives 4 u'—(A,+0A,4A,) u=E,+ 0E,+ Hs, if we assume 6’ =(A,+0A,+@A,) 90—(B,+0B,+¢B,) o'= (A,+ OAg+ PAz) p ams (C, + §C.+ PC) ° Thus the question is reduced to integrating a pair of simultancous equations between @, @, and ¢: if this can be done, substitution makes | the first of the three equations a common linear equation between u and | t. If all the coefficients be constant except E,, E,, and E,, it is | sufficient that 6 and @ should be the roots of the pair of equations got by writing 0 for 6 and @. If A,+0A,+$A,=a, we may reduce these to (a— B,) 0—B, p=B, (a—C,) P—C, 0=C, ; and the values of @ and 9 hence obtained, substituted in A,+A,.9@ +A, =a, give an equation of the third degree to determine a; from | which 9 and @ may be found by the two last. Each root of the equation of the third degree gives one form of uw —auz=i,+0E,+ dF, ; - and three final primitives are thus determined. (18.) Let «=A, 2+B,y+C,, andy’=A, +B, ¥+C2, where Aj, Ag | B,, B,, are constant, and C, and C, functions of ¢ only. Multiplication | on * Ag appears by instances, except when A,=0. But in the latter case | y'=Byy+C, can be integrated separately, and the value of y substituted in the other equation. ON DIFFERENTIAL EQUATIONS. 689 of the second by 9, addition, and assumption of B,+ B,0==9 (A,+A, 0), z=21-+ by give a= (A\+ A, 0) z+ Ci+ C, 9, which can be integrated, as in page 155, (19.) If the equations be linear and with constant coefficients, the solution always depends upon that of common algebraical equations. For instance, a" + aa! + by" +ca+ey=0, y" + fe! + gy! +hy=0. Assume z=", y=(3e“, which gives a +ac°+bBearte+ eB=0, bo +-fo+gaB+hp=0. Eliminate 3, and we have an equation of the fifth degree to determine a. Let the five values of « and B be a, a, &c., By, 2, &c, The complete mtegral is then got by adding all the particular integrals multiplied sy constants, and this gives the equations coms, Od ee C, Sony Cy eM oO yet) 4: C, §° y= C, B, g*l + C, By ert C, Bs enor C, B, erat C, B. eri (20.) Ifany of the roots be equal, a wider form must be taken; but he following (which might also be applied in page 211) is the best node of obtaining it. Let a, and a, be unequal (as yet), and put the wo first terms of x into the form ‘ts O; es 2 42 ayt (C,4-C, e (22-21) $i or Banat C, (cto 0) op seal Te .) Now let e&—, diminish without limit, by approach of a to a3 and Ss this process goes on, let C, increase, so that C, (a,—a,) may always be {,; while at the same time C, alters so that C,+C, is always K,. Then Y» (a@— a,)? or Ky (@2—a@,) diminishes without limit, and still more the ucceeding terms, so that e‘(K,+K, t) is the final substitute for the two rst terms when a becomes = q,. Similarly, 6, e™! (K,+ K, 2) must be jut for the first two terms of 7. (21.) Generally, let s=C,¢(a40+C,¢ (oo t)+.... be one of the lutions of a set of equations where qq, a, &c. are the roots of an lgebraical equation. If any of these roots become equal, some of the lutions merge into one only. Suppose, for example, four roots equal, ‘quired the general form of the solution, so that the number of con- ants shall remain the same as in the case of unequal roots. Let =at+%, a—a+93, «,=o1+9,, whence the solutions belonging to ese four roots may collectively be brought to the form (C,+C,+ C3+ C,) p (a, t) — (Cy 0,+ C,; 0; ++ C, 0,) q’ (a, t) fy {2 +(C, O:+C, &+C, 0) gp” (a, t) ‘a: £3 + (C84, O84 C, 0) 6" (Da tee hs 690 DIFFERENTIAL AND INTEGRAL CALCULUS. As 05, 43, 9, diminish, let C,, C,, Cs, C, be always determined so as to make the four first coefficients be K,, Ky, 2K;, 2.3 K,. Suppose also, which is allowable, that the above conditions are fulfilled in such way that C, 6%, C, 63, C, 6 shall have finite limits, or, say, shall be always — finite quantities L,, L,, Lu. This does but require that 05, 03, 0, shall | diminish without limit in such a way that Ty de Ue ng Ds G) Oge Dy 62 Lo aa +e shall always be finite and equal to 2K, and K,; which, as there are three | quantities diminishing, with only two conditions, is always possible. | Hence it follows that C,6{+C,03+C,0, &c. diminish without limit, | being L, 0.+ L, 0,+ L, 4, &c., and the final solution, belonging to the four equal roots, is Ki (mf) +Kod! (4 1). C+ Ks 0" (at) P+KG" att, and so on for any number of roots. (22.) Take the equation Ny’-+ Py’+Qy+R=0, and for y substitute V:(W+z). Multiply by (W+z2)*, and we have —NV2!+R2?-+(NV'+QV+2RW)z +N (WV —VW)+PV?+QVW+RW?=0, which has several integrable cases. First, when R=O, this equation is integrable whatever V and W may be; but in this case the original equation is easily reduced, for if y=", it becomes —N2’+P4Qz=0, and is linear. Hence the equation before us can be integrated (and thence the original one) whenever V and W can be found so as to give N (WV'—-VW’)+ PV!'+QWV+RW?=0.... (V,W); which, however, supposes (let the student show it) that a particula solution of the original equation can be found, but expresses this con dition in a useful form. Let V:W be a particular value of y, ascer tained by trial or other means, and = Y, whence the preceding conditiol is satisfied. Determine V from | Q+ery—! 3 NV’+QV +2RY“"V=0, or vers Vie. We have left then ~- NVz'+R2?=0, or z=—1: \ Acree ch; VEG < and y= TLV: is the complete solution. (23.) Thus, if P+Q+R should happen to be =0, in which case * is clear that y=1 is a particular solution, we have (making N=1 i simplicity) a complete integration in | yt} Reo ee dz+C} . BRE Sas’ dz a, (24.) Again, let V and W be determined by QV-+RW=0, whic ee (V,W) to N(WV'—VW’) +PV*=0. From these two ¥ 1aye . ON DIFFERENTIAL EQUATIONS. 691 WY LBe a We -7.Q gst Baap PONY SE eA Meena ie which equation is therefore necessary to the success of this artifice: and, this condition subsisting, QV+RW=0 alone, satisfies (V,W ). Now assume NV’-+QV+2RW==0, giving NV’/—QV=0, or Q Q *Rdz Ne Se V=—— SN ne | —— } x poe! R V, and 1 nyt Ci, as before. ‘The complete integral is y=V: {W +2}. / (25.) Assume PV-+QW=0, which shows that (3) men ~ is the necessary condition. And P NV’+QV+2RW=0 gives log V= HF = (aR 5-2) de, W=— 25 and, z being found as before, this case is integrable. Diack! (26.) Assume PV?+ RW°=0, which gives 2 Q =— — — to satisfy Ra a ee tk (V,W). Here NV’+QV+2RW=0 gives 1 J=pPR “ aa og Viex -{5 (Q+2V —PR) da, w=,/(- x) 3 ‘and, z being found as before, this case is also integrable. All these cases really depend on the same principle. (27.) From the preceding it may be shown that the complete integral of Ny'-+ Py?+Qy+R=0 must be of the form ree yr+eo’ e being an arbitrary constant, and @z, &c. not containing any arbitrary constant. (28.) Also by determining V from —NV=R, and W from NV’ +QV+2RW=0, the equation may always be reduced to the form y+y?+S=0. (29.) If in § (22.) we make N,=—NV, _P=R, Q=NV'+QV +2RW, R,=N (WV/—VW')+ PV?+QVW+RW?; 2 = if w ce 222 V,: (W,4+2,) we have N, 2/+P, 2?+Q, 2+R,=0, and if we make z=V,: (Wi+2,), we get another equation of the same form, and so on. Hence we reduce y to the continued fraction NeeTa Seo V2 STO Wp Wale ce 8 which may, in certain cases, exhibit its law with sufficient distinctness, when only a few of the first terms are found, Suppose, ee pe 692 DIFFERENTIAL AND INTEGRAL CALCULUS. we want a continued fraction for (1+-7r)~". We find that y=c (1+2)™ gives (1+) y¥ +my=0. Let V, V,, &c. be At*, Bx®, &c., and let W=W,=....=1. It is evident from the form of the fraction that we must have «=0, Aste; assume yo=c:(1+2), or V=c, W=1, which gives (N=1+2, P=0, Q=m, R=0) —(14.2) cz!-+me z-+mc=0, or —(1+2) 2 +-mze4+m=0. If 2 were Bx’, —(1+.2) 2!-+-mz-+-m would be —BBx*" + (m— BA) 2°-m, which vanishes with 2 when B=], B=m. Now when 2 is small, z= Ba’ nearly, as is evident from the fraction, so that it is only by | this supposition, namely, making Bx* approximate to a solution, that we can get a continued fraction of which all the terms after Ba®: (+... 4 become comparatively insignificant as x is diminished. Assume then z—mxr:(1+2,), or form the new equation with | N=—(1+2),,, P=0,.. Q=m,. R=, Vom, Wag which gives (L+2) maz',4-mzi+{m'e+ (1-2) m} 24:—(U+r) m+m’ e+m=0, or (142) xz; ei+ (ma—a+1) 2,+mr—ar=0. : If 2,=Cx’, it will be found that similar reasoning gives y=1, C=—}(m—1), and proceeding in this way it will be found that the | successive values of V are, after c and mz, _@m—l)e (m+))z —(m—2)r (m+2)2 (m—3)a 9 3 6 3 6 9 10 ; ra . Q+4a)"= 1 mx 4(m—1) ri (m4l1)a ¢(m—2)a Py (m+2) 2 “| ~ 14+ 1— 1+ L |e 5g 1—.... Find log (+2) by taking the limit of (1+2)"—1 divided by m | (m=0) i log(1+2)= ij BE aE ae ah eh rate 1 i+ 14:14 14 14 14.0 Find ¢” by taking the limit of (l--2:m)™ (m= ox) sb a 4 1 1 1 ae dr dx dr yor tyr set Pepe, Pe erariaoae C/V=0, LC/W=0, TCV’=.... FC VO%=O whence y= SCV gives y/=ZCV’, y’=ZCV"...., and y=ZOV" + >C’/V%-), whence the equation is satisfied by 2C’'V°-°=X, sinct the terms containing ©,, C., &c. all make y™+....+P,y=0. W' have then to determine C,’, &c. from ON DIFFERENTIAL EQUATIONS. 695 BONO) LOVier0; cl CN 0}. ae. VECV OM =X; by common algebra: and integration gives the values of C,, C,, &c. ae (34.) Apply the preceding to 2” y/’/—32"" y" 4+ 62" y! —6a"*y=X. 4 X=0, the complete solution is y=C,xr+C,2°+-C, 23, whence we ave Chg Coa + C,2°=0, Chav C’, +2C',2 +30’, °=0, Co= = Xue" 20%, +6C),c4=X2-", Cp—4Xa-t y=he Achat i +30 [2S re a 4 : 2° ante n “a 4 (35.) The equation (a+ bxr)*y™ + A, (a+ dr) t yO + wiaee +A, (a+bxr)*"y=0 has n particular solutions, and thence a general solution, found by assuming y=(a-+ bz)”, which gives m(m—1)...(m—n+1)+A,m (m—1)...(m—n+2) +... +A,=0, an equation of n dimensions: let its roots be m,, m.,....m,. The com- plete solution is then y=C, (a+ br)™-+C, (a+ bar)? + oo. +C, (a+br)™, subject to modifications already explained, (pages 211 and 689,) the solution for a pair of equal roots being (C,+C, log xz) (a+b2)™', &. If a+bx be made =¢*, this equation can be reduced to the common linear equation with constant coefficients. (36.) In theory it is permitted to suppose the solution of any alge- braical equation; but in practice the inability to do it in finite terms frequently makes a great difference. Suppose one differential coefficient given in terms of another, for instance y‘= (y/"). If y/”=z, we have 2’=@ (z), and if this can be integrated in the form z=wa, we have y= (fda)? vr. But suppose that (as is indeed generally the case) we can only obtain the form w=wez, inconvertible in finite terms. We must then take | = fy" dr= fy" yy” dy" yxy"; y! = fyda= fy! yy!" dy!” | Sey y= fy'da= fay" ly” dy" Soy", and 7” must then be eliminated between z= Wy’, and yoy". (37.) $(a,y',y")=0 is reduced to the first order by y' ==2, which gives $ (1, 2, 2')=0, or za, y= fx dx. But if r=wpz be the form, we must find y or fy'de, or fz ys'z dz, oY x2; and eliminate 2 between the two equations. “ And # (y, 7’, y’)=0 may be integrated in a similar manner by changing the independent variable, writmg 1:2 for y', and —x": 2° for y”: which brings the equation to the form ¥ (y, Ba" yO, Or thus: making 7/=z, we have “a dz z a a y ae and o(u 2 05 from which equation of the first order « is to be found in terms of y, 696 DIFFERENTIAL AND INTEGRAL CALCULUS. and a=fz-'dy. Or if y be found in terms of z, say y=¥z, then a= fz" w'z dz, and z must be eliminated. (38) Let the complete integral of ¢ (a, y,7/,....y™)==0 be known, and let it be y=w (x,a,b,c,....), a function of 2 and of n arbitrary constants. The equation 6=0, being identically true when is substi- tuted for x, gives d.$ 4 db dy dp dy | dp dy Wg oe dy da dy da’ *""* "dy" da dy dp dp du dp at being zw, or ae dy Us diy ae ae ot T@lar? or u=dw:da is a solution of this last linear equation, in which the coefficients of wu, wu’, &c. are functions of av, a, b, &c. By the same pro- cess it will be found that w=dw: db is a solution of the same, and so on. Hence the complete solution of the last equation is vi ds dus : uA iat? apt .e- A,B, &. being new constants. For example, the equation ryy”+yy’—ary?=0 has y=az’? for its complete solution. The new diff. equ. then is (ay +’) u+(y—2ry’) u4+acyu"=0; or, dividing by a,b? w+ (1—28) eu’ +- 2° u’=0, fy od Ya logx.x’, whence w=(A-+Blogez) x’ is the complete solution of the last, which shows that the equation deduced from § (35.) would have a pair of equal roots; as will be found to be the case. (39.) The equation ¢ (a, y, y“*?,....y%t™) can be reduced to the ath degree, as is shown by making y“=2z; when z is found, y is found by direct integration. But if x can only be found in terms of z, a process similar to that in § (36.) must be followed. (40.) The equation Py’+Qy”?=R is integrable, if P,Q, and R be functions of y. Divide by P, which leaves the form y'/+Qy?=R, multiply both sides by e/?%, and it will be found that the first side is the diff. co. with respect to x of y’e/°. We have then d dy d dy dy — (cy 4/)\)— Qsdy {Qi dy ad) ee SQdy LZ) QR cVUQdy dx SRE po dad 20 dx dx (« #1) oe dx g/24 dy “ V(2fRe” dy)’ By changing the independent variable, it will be found that y+ Py + Qy*=0 is integrable when P and Q are functions of x. To solve this directly, multiply by ¢/"“, which call W, and we then have dy? : (eroe) ee Des gy, clic x 3 d {dy dy” _ ia ae in 1:U being Wy’. Hence U=,/(2/QW~*dz), and ON DIFFERENTIAL EQUATIONS. 697 —fPdx Jp dx € alia { WV2SQW*dx) ~ J JQfQe*F* aay (41.) In the same way can be integrated y” + Py? + Qy”"=0, when P and Q are functions of y, and y+ Py’ +Qy"=0, when P and Q are functions of z The results are most easily obtained, that of the first from the second, that of the second from y'=2z, which gives 2'-+Pz +Qz2"=0. This last gives Wz)! Q We)’ Ws, Oe ——— = i ) +Q or ( Wz)" jaa 0, which is easily integrated. This case belongs to the general form ¢ (2, y',y"")=0, which is reduced, as in § (37.) preceding, (42.) The complete integration of y+ Py'+Qy+R=0, P, Q, and R being functions of 2x, requires only any particular solution of y’+Py'+Qy=0, other than y=0. Let y=Y be such a particular solution, and assume y=Yov for the general solution. The equation then becomes Yo! + 2Y'0' + Y"v4+ P (Yo'+ Y'v) + QYv+ R=0) or Yo"+(2Y'+PY) v/+R =; since Y"” + PY’+QY=0, by hypothesis. This, with respect to v’, is a linear equation of the first order, which gives a e J (F+P) e df. = ai (Ftr) dz= — fRYel? * da, RYE?” dx y= Yu= — x /{ =e soy dx. Reduce this, when R=0, to the form in § (33.). The negative sign may then be omitted, or replaced by any constant. Why ? (43.) If R=0, we find for the complete solution of dx (44.) If in § (42.) we suppress the condition that Y is to be a yarticular value of y, we have Yo"+(2Y’+ PY) vo’ +(¥"+PY’+QY) v+R=0; ind Y=e«-*#?* gives the form Yo"—+Y (P2+ 2P’—4Q) v+R=0. (45.) If Y be a particular value of y in 7’+Qy=0, the complete ralues of y in the following equations are as written, y+ Py' + Qy=0, y=cr | t see lade _ (fRY dx y +Qy=0, y=y [Fs y+ Qyt+R=0, yer [Ee a ae, (46.) The equation #’+Py’+Qy=0 is reduced by y= &"* to *+v°+Pv+Q=0. The solution of this last, § (27.), is of the form =$o+%:(x+C). I leave it to the student to reduce the value of Y, 8 derived from v, to the form CY-+C,Y, which it is known to haye. | 698 DIFFERENTIAL AND INTEGRAL CALCULUS. ! (47.) When an equation can be made homogeneous on any particular supposition as to the dimensions of the diff. co., substitutions invented accordingly will frequently reduce the order of the equation. For example, y?y"+a°y°=a%y"” is homogeneous if y, y's y” be of the dimension of 2%, 2', 2°. Assume y=a*u, y'=av, which gives w* v*-+-v" —7'. But 2ru4-2°u'=a2v, or 2Qudx+ardu=vda ; and u’v?+v°=a20v!-+9, or (u?v?-+-v°—v) dx=adv. Hence dx dv _ du tr wet —v v— Qu an equation of the first order between wu and v. The reduced equation may be as difficult as the original one, but there is always an advantage in knowing how to form an equation of a lower degree: and it may generally be taken, that if the reduced equation cannot be integrated by our present means, neither can the original one; or vice versa, that if the original equation can be integrated, methods can certainly be found for succeeding with the reduced equation. To generalise this process, let (2, y,y',y")=0 be homogeneous — when y, 7/, #/’ are of the dimensions n, n—1,n—2. Assume y =2"U, y'=2"'v, y/=a""*w, which gives an equation of the form ¥ (4, v, w) =:0, by hypothesis. Again, nw ytatu =a" v, or dr: a=du: (v—nu) | (n—1) 2" v4 2"! =2"*w, or dx: r=dv:(w—n—I1 v) ; du dv v—nu w—(n—1) v substitute for w its value, and we have the reduced equation required. or , and & (u,v, w)=0; (48.) When the equation is homogeneous with respect toy, yf, yan &c., the reduction of one unit of the order is always practicable, by | assuming ye". Thus yy? y= (a? y°+y") gives , eVedr a2 (2 +o!) meV (2-1 v?)2, or v® (v?-+0')=(@?+0°)*. i (49.) An equation may sometimes be reduced to an integrable form by a change of the independent variable. Let it be y+ Py’ +Qy+R=0; . and assume «=. We have then dy dx _, (du dy dy @x ‘(F) | pe ey aa AE eee hae dé Y Fi db”, we Gende: deoae dx & dx a d das dix’ x dy (e x z) Get Qapyt R=. dé dB \ de di) dé dé To destroy the second term, we must integrate i Onn Age take EP in P Te —qa= which gives £= fe“? dz. | But if we have P=0, and want to restore a second term in which the | coefficient is the function II of , we must integrate dx dx =I — ious 5: ee m— (e-sUds a dé dé which gives w= fe!" dé. | ON DIFFERENTIAL EQUATIONS. 699 (50.) The solutions of some equations, otherwise unattainable, have been expressed by definite integrals, but a general method of passing from any equation to such a solution has not yet been ascertained. The following are examples. Let y= fet (l—v*)" dv; we have then, differentiating with respect to x, and integrating by parts with respect to v, dy xv 2\% | ae i (1—v’)" vdv ax 2n+2 — os ear @! —vy*)} + . anv ee y2\ n+l md fe (l—v*)"#" do, Let the limits of integration be —1 and +1, the separate term ‘then vanishes at these limits, if 2-+-1 be positive, and we have dy ar ar ses +1 -axv Q\ 1%), +1 -arv ( 2\n 2.2 ee th et” (1—v*)" dv ——— fit e** (l—v’)" v'dv dz 2n+-2°° Qn+2/ ( ) ax er ory —= — Onto!” In+2 de® d’y | 2n+4+2 dy rr — — me dx? v dx a@y=0 gives y= fi} e™ (1—v*)" dv. A little examination will show that this integral undergoes no altera- tion when the sign of a is changed, and also that m must be >—1, or 2n-+ 2 positive. The preceding value of y may of course be multiplied by an arbitrary constant, but it is not yet complete. The following artifice will find another solution, and avoid the (in this case) com plicated form of § (43.) Assume y=2a*z, which gives Rete EA ohare eee re Ce RR OR 2. Fog” indy hipaa a yh a pel sp Seat ee ty Qn-+2 Yet alee tp hee 298 i k®—R+-(2n+2) ks r 2 y ee oes ey alae sabes y x x = oe y z x Assume k?—k+ (2n+2) k=0, or k= —2n—1, which reduces the pre- ceding to 3 Rese On 2! ~~ : — —— ——a'=0, and z= ftte**(1—v’)-"* dv Baalird Ade satisfies this if m be negative. But since 2n+2 is to be positive, n must lie between 0 and —1, or 27+2 between 2and0. Let 2n+2 =m; it then appears that, under the restriction 0 (m) 2, the complete solution of y/-+-m2~' y/—a’y=0 is y=C, ft e@” (1—v*)*#"- 1 dv+C, BT fy e” (l—v*)-™ dv; which this is not altered by changing @ into —a. Do this, add and divide by 2, and write a,/—1 for a, which gives for the complete solution of y+ ma of + a’y=0 y=C, fi cos axv. (1 —v*)#™"! dv+-C,a7"*? [t} cos arv (1—v*)-*" dv. When m=0, the whole process fails, since the separated term in{the | | ( 700 DIFFERENTIAL AND INTEGRAL CALCULUS. first integration does not vanish; but still the second solution is then of the form C,sin ax, which is a solution of y"+a°y=0. When m=2, the process of the second solution fails, and that of the first gives a solu- tion: but this case is best treated by observing that y”’+2v7"y/ ==(ay)’:a, whence the equation becomes (cy) +a° cy=0, and its solution is ry=C, cosar-+C, sin az. When m=1, the two solutions are no longer distinct, and we must proceed as in §(21.) Let m=1—6; the second solution without the constant arises from integrating with respect to v, cosazrv.(1—v)+ multiplied by y2 a? (1—v®)®, or 1 nee log .2* (Iv) 45 3 flog.a*(1—v") +4 oe ee 5 and for the first “solution, in place of the preceding we must put (1—v*)-® or 1—}5 log (1—v”)+.-.-. Hence the two together give a solution arising from integrating cos ave.(1—v*)~? multiplied by C,+C.+ 0C, log 27446 (C.—C,) log 1—v")+.... As 6 diminishes, let C,+C, have the limit K,, and let 6C, approxi- mate to K,. Then 6(C,—C,) has the limit K,—limit of 6 (K,—C,), or 2K,, since 0K, has 0 for its limit. And it is easily shown that the remaining terms diminish without limit, whence K,+K, log z+K, log (1 —v*) is the limit of the preceding, or the complete solution of y+a~'y’+a°*y=0 is y= K, fi cos axv (1—v*)~* dvu+Ky fticos arv (10°) log (@ 1-0") do. When m does not lie between 0 and 2, only one solution can be obtained by this method, namely, that one in which the exponent of 1—v? is greater than —l. (51.) Many equations can be reduced to one of the preceding forms: thus y==a"z turns y+ ay=n (n—1) vy into 2”+4 2nv™ 2'+ @2z=0, Again, Riccati’s equation, § (32.), can be made to depend upon y"==ax"y. Change the independent variable, and make c=", We dy p—l have, then, § (49.), a cae tay! pate ys). Of i Qe Oa Ht ng Lay A ta Pot) ae mane Been ae Again, let y”=ase".y. Assume =e", or r=2logé:b. This gives g y g 8 Pyislls gy sg ae ADORE dE Bhs + The student may try the following. If v be a function of t, and y of x, and if, moreover, y=y(v,t), t= (v,¢t), then the equation y + Py’ + Qy=0 gives Rv’ + Sv?+4+Tv?+ Ur'+ V=0, where Rex, Uy Be UW, Xto ~ = Xvv Migr Urey Not Py Wit Qyy? i= Kool re Woy Xt “ 2 (Xt Us — Wot Xv) +Pw, (2x, VAX: Wy) +3Qy v5 We UH Ku po Vu Xo + 2 (Xv YU — Uy Xt) + PY, (2x, V+ Xv Wi)+3Q Vi ys =X: — WexX: + Px; Wit Qyyi. ON DIFFERENTIAL EQUATIONS. 701 This method cannot reduce the equation y"+&c.=0 to the first degree, unless a solution be already known. Why? (52.) One or other of the solutions in § (50.) is integrable whenever m is an even number, positive or negative, since fs” dv dv can always be obtained when ¢ is a rational and integral function. But the follow- ing application* of the method of generating functions (page 337) will show us how to obtain the complete integral. Take the equation y" +mz-'y'+a*y=0, and let y be the generating function of a, to the variable 2—c; that is, let y have the form ...+a, (a—c)"+4,4, (a—c)"*? +....: call this Sa, (c—c)*. Then y’ is the generating function of (n+ 1) a,,,, or is S (2 +1) G4, (t~—c)”, and may! is that of m (n+2) G42, While y" is that of (n+2)(n+1) @,+2; and since every term of y”+m2x-y+a’y must vanish, we have {(m+2)(n+1)+m (n+ 2)} Qnreta® a,=0, Assume (n-+m—1)a,=(n+2) b,42, and therefore (n+ m +1) dais =(2-+4) 6,44, which give by substitution M4) (2+ M—=1) ba ps +A Oy 2=0, OF (2-++2)(-++m=3) Duro ta? d,=0 ; Whence x in 2+ (m—4) x7! 2/-+- a? z=0 is the generating function of >, Now n+2 h m—3 ; pe — Tas 2 woes — 27 ME fey ae ate ai+m-1 *t me 3 3 (2+4) Big But since b,1 is generated by 2: 2°, and (n+4) by, by 2/: a5, we find hat if we can integrate z’/+(m—4) 27 z'+-a°z=0, we can also to) ] ntegrate z”-+ ma! z'-+-a°z=0; and that we find y from z by the *quation 2 m—3 2! & Y= 2 es, Now we have integrated when m=O and when m=Q2, in finite rigonometrical terms; hence we can integrate, also in finite terms, vhen m=4, 8, 12, &c., or 6,10, 14, &c.: that is, when m is any sven number. The preceding reduction applies whatever may be the value of m, so hat all cases are integrable as soon as the integration is practicable for dl yalues of m between —2 and +2. (53.) Considering the nature of the preceding reasoning, it may be lesirable to give a verification of the result. This may be done as follows, tating only results. Starting with the last equation, differentiate both ides twice, but as fast as z” makes its appearance, substitute the ‘alue derived from 2”+(m—4) a z'+a’z=0. This gives : z (m—J})(m—3)) 2 y aa eS 9 Ber +{2*— eRe oe ire Ter Ses ra a * See a paper by Mr. R. L, Ellis, in the Cambridge Mathematical Journal, 1, ii. pp. 169 and 193. . 702 DIFFERENTIAL AND INTEGRAL CALCULUS. Z m(m—1)(m—3)) 2 | y= —{a? a?—m (m—1)} — —{2m—3) oo ST Glia | whence it readily follows that y//+- max y'-+a*y=0. (54.) The preceding gives no clue to the case in which m is a nega tive even number, but another transformation may be made which applies both to positive and negative even numbers. For a’ write a, or let the equation be y/-+-ma™ y’+ay=0, and let m=2p, p being, integer and positive. We have then, on the same suppositions as before, (n+2)(n-+2p+1) aryo+a.a,=0. | Assume a,2=b,:(n-+1)(n+3)....(m+2p—1), which readily gives (n+1)(n+2) b,42--ab,=0, or what we should have got at first if p had been =0. Hence Sd, 2" is to be Csin(f/a.2+C,), the complete integral of y’"+ay==0. Now, to take an instance of the mode of obtain- ing Sa, 2" from 86,2", observe that, if p=3, 2" aN ee "or ———— > Is xf pad dx{,b, x" dz, ies Ot Datayn+s) is xf ade f,cdrf,b, 2" dx or a-* (xf, dx) b, a” dar; signifying that the operations of multiplying by dz, mtegrating, and then multiplying by 2, are to be repeated three times in that order; the whole ending with division by 2°. Applying this to every term, we have for the complete solution of y+ 2pa~ y/+ ay=0, | y=Ca-” (2 f, dx)’ sin (fa.2+C,)). | ' The form of this may be usefully changed as follows. Since | fo Wa.2) d (Ya.a)=a fo (a2) de’; y= Ca” { Ja.vf,d (Ja.2)}? sin /a.2+C,) ; the power of @ introduced being immaterial, on account of the arbitrary character of C. Nowin [,¢(/a.x).d(/a.2), it is indifferent whether we suppose a or 2 to vary; let us then suppose a to vary, and a to bt constant ; we must then integrate from a=0. To show the sort 0 result we get, let us take p=3; at full length then we have Can Ja.afd (Ja.e).Ja.rfd (J/a.2x) Na.xfd (/a.v) sin (/a.c+ G) br, d iG x (Oat, wal saa | are" | a sin (Ja.2+C,) a ,sin G/a.e2+C,) , sin (,/a.7+C,) = Ja (fda) a a say =C (fda) Tey aa since C may be any function of a. And thus we have generally y" + 2pa~ y' + ay gives y=C (fda)? ig weete). Hence we might suppose by analogy that ON DIFFERENTIAL EQUATIONS, 703 z) sin (/a.a+C,) _ dx Ja . and this may easily be confirmed. Starting with this equation, we come by the process, as before, to (2+ 2)(n—2p+1) da42+aa,=0. Assume @,=(n —1)(n—3)....(n—2p+1)b,, which gives (n+ 2)(n+1) 6,4.+ab,=0 as the first would have been, had p been =0, Now we see that a, 2” is made from 6, «* by the following operation, y" —2pa y' + ay gives y=C ( \\P d 1 tio feo"? (+ =) (5, 2"), or y=Cx” (= 5 ) .sin (/a.z+C,), / 7 the operation being successive division by 2 and differentiation. ‘This can be reduced to the form d $e oN, y= C2” (; Clas) 3) sin (/a.a+C,) ; and if we now make ,/a the variable of differentiation, 2 being constant, we find that* Te —1 »,/ giam (aks jae d : con es y —2pu y' + ay=0 gives y=C ae a in G/a.v+C,) ‘i 5 It must, however, be carefully remembered, that the validity of the last operation, as in the corresponding integration, depends solely upon the function with which we start being a function of the product ,/a.a. (55.) We may now see how it arises that Riccati’s equation can only be integrated in finite terms in certain particular cases, By § (46), y'+y?=ax” depends upon y/=ax"y, and by §(51.), this depends upon an equation of the preceding form, in which 2p=m: (m+2). Hence m must have the form 4p:(1—2p), which will be found to agree with § (32.). (56.) Another method, proposed by Poisson, is as follows. Let y= fi 9 €—v"—ax" v~-" dy, a@ and n being positive, : l y n=-2 ['° n n n dv 2,2 ,2n—-2 [? n Nyy av = pe - a, See - cm yt mn ey? —_, dx Tr (n 1) ax ve E Vv ax v we +. 7d a av th i ve” 7 —e~—v" E— ar & Now the second integral is Re —d v sz—l tay > ) or, by parts, Nas mo) 1 n—1 dv — ev" — 02" y4 Pate fe-m aa" v—"dy a3 fe-er— ax" yan —, vax y* ax" nan” D The first term vanishes at both limits, and substitution gives simply * The preceding articles, (52.) and (54.), are taken, with some alteration of form, from the very ingenious paper already cited, which contains several generalizations of the process highly worthy of the attention of mathematicians. fy Ee ee | 704 DIFFERENTIAL AND INTEGRAL CALCULUS. y"=n ax" y. Let the preceding be f> Vdv; then Cp Vdu is a solu- tion of y=n?ax"*y. If n=2, the preceding is integrable, and all its solutions are contained in y=Ce*¥"*++C,¢°V"", Hence, for some values of C and C,, we have {eseenane dv= Oet¥ 4. C. en lV a. But since the first side must diminish without limit as v increases, we have (on the principle explained in page 576) C=0, and since 7=0) gives 4,/7m for the first side, we have | | {oer rae v2 do= V2 eNae ° 1 7 9 Change v into Ja.v, fp e—av—a? o* du=5. mf Me h I By successive differentiations with respect to a, it is easy to obtain from these results the value of fj ¢—v"— 42» v du, p being a positive or negative integer, and hence, by aggregation of results, can be obtained, i 5 E—v*—cx v— dy‘ dv, where Ov is a rational and integral function of » and v*. For our present purpose, however, let v'=2*in the first integral, so that we have 2 ahs foe anno du=- f, ee —ainz™ zn dz, n | This, then, is integrable whenever 2n7'—1=2p, p being a positive! or negative integer: that is, when 7 is of the form 2:(1+2p), orn—2' (the exponent of the equation) is of the form —4p:(1+2p); which agrees with preceding results. _ (5%.) The solution of y”=n? aay above obtained has only one arbitrary constant, consequently the solution of 227 = n?az™~) derived from it has none, and recourse must be had to the method 0) §(43.). To show how this arises, suppose that y’-+Py’+Qy=0 i completely solved in y=CV+C,W, then y=e* gives 2’-+2?+Ps +Q=0. But we have | sea y _ CV'+C,W! y CV4+C,W" | | . he | and the only arbitrary constant in z is C:C,; but this is still one arbitrary constant, and therefore the equation of the first order 1 completely solved. But if y=CV only had been gained, the value of z would have been simply V’: V, without any constant at all. | (58.) To form a proper notion of our state with respect to the solution of differential equations, I repeat the supposition of page 103 Let us suppose we had not been in possession of the operation inverse to involution ; so that all problems, the solution of which is reducible to, say r=,/a, would have presented the difficulty which those whe know better would call a want of adequate means of expression. ‘The first thing noted would be that such problems are soluble when a=0) 1, 4, 9, &c.; in fact, when a=nxXn. Other cases would have thei solutions obtained, by some in approximate fractions, by some in series ON DIFFERENTIAL EQUATIONS, 705 by some in continued fractions,* and go on. Finally, the acquisition of a distinct idea of, and notation for, the square root of a, would reduce all those problems to one class which had been practically divided into several. Thus it has stood hitherto with the equation of Riccati, y/+y’=ax”, or with y"=aa"y, from which it springs. Count Riccati first pointed out (Leipsic Acts, 1732, according to Dr. Peacock) that there were integrable cases: why those which remained were not integrable did not appear. ‘The various modes in which the remaining cases were after- wards integrated, by means of series, definite integrals, &c., were gene- rally themselves only partially applicable. At last, the general equa- tion y”+-ma- y'+ay=0, had its complete solution expressed by y=CD™*" {sin(/a.z+C,):,/at, in which D denotes differentiation with respect to a; a result} which is unintelligible whem m is anything but an even number, positive or negative. Any other supposition throws us upon the difficulties of fractional diff. co. (pages 598—600). But at the same time we sce that the difficulty arises from our not having well understood means of expression in which to convey the solution. It is a remarkable point in the history of this science, that most of the results which ordinary notations can express were obtained at an early period. Any stoppage has almost always, sooner or later, been found to arise, not from the defect of methods, but from the non- existence of the proper mode of expression. If we take any general form, and proceed to its differential equation, we shall always see that the €quation so obtained is one of those which admits of solution. For example, y=C@(x+C,) gives y’: y=! (2+C,): ¢ (x+C,), whence t+C, must be a function of y:y. Say a+C,=y(y':y); then we have ' py tee ' (A l=’ (“) AONE reducible to 7 xy (+), y/ y ye Ney an integrable diff. equ.; provided that the solution of all algebraical equations, or the inversion of all functions, be assumed. The following forms may be readily obtained : Wl 42, / = CAC: SOE A AOE iy! y=C$(C,2) gives TE = (2) * It may interest the historical reader to know that the continued fraction was used in the extraction of the square root long before the time of Lord Brounker, to whom the invention of this mode of expression 1s generally attributed. It was lately claimed by M. Libri for Pietro Antonio Cataldi, whose work on the square root (1613) is cited in support of the assertion. On examination of this work { find that there is no doubt of the fact, and the following sentence will be sufficient ‘0 show it. The author is speaking of ,/18 (page 70):—* Notisi, che no si potendo Omodam éte nella stampa formare i rotti, e rotti dirotti come andariano; cioé cos} 4, & 2 come ci siamo sforzati di fare in questo, epi de a inazi gli g P< hol 8&2 formaremo tutti a qsta similitudine 4.& > & Fy & 3? facendo 9 . * " we - . r \ aes = vn punto all’8 denominatore di ciascvn rotto, a significare, che : +) f° i ee ra 99 8 il sezuente rotto é rotto d’efso denominatore. bow ] + This result is stated to have been first given in the form of a question proposed or solution by Mr. Gaskin, in the Cambridge Examination Papers sang ie DIFFERENTIAL AND INTEGRAL CALCULUS. yap (e@+C)t+GQ gives y= x (y’) | y= (Cr) +Q, aA ese af ad KAYE) y= C (Gz)", or C.CM ws yy =y?+yy! xz y XY y= (a 4+Cr+C,) ayes y s y? =2Qyy. In all these cases, the solution may be obtained from the equation, if oz | be an ordinary function. ! (59.) The mode of deriving the singular solution of a’ differential equation from the primitive (page 190) may sometimes be insufficient, | as when y= (a,c) is first introduced in the form (#,y)=c- The | method may be thus extended, it being remembered that the object is | nothing more than to make ¢ such a function of # and y as will not alter | the form of y’. Let the primitive equation be ¢(a,y,c)=0, and | assume c to be a function of w and y. We have then, using the notation | of page 388, ; | 1 Pet Pele . aii EAA ACRE Cae Mesh M2 DG 97 and in order that y’ may not be affected by changing ¢ from a constant | into a variable, we must so choose the form of c that j I Pe _ Pat P. Cy or d, (x, Y; ® (2, y)) _Pe (a Ys Cc) +¢, (11; Cay | Dy Pyt Pe ey. dy (4, y, ® (a, y)) Py (29,6) + Pe (x,y, é) a where ¢ (x, y, c)=0 is supposed to give c=® (a, y), and the substitution | is made on the first side, in obedience to the well-known mode of form- ing y/ for the ordinary diff. equ. Observe also, that the first side of the’ equation is the same thing as ®, (a, y):®,(a,y). Here then is a | partial diff. equ., from which we might suspect that the form of ¢| required contains an arbitrary function. But it is not so, as follows. The complete solution of the preceding partial diff. equ. is @ (a, y, e} | =f (x,y), as may easily be verified; f being an arbitrary function. | Combine this with ¢ (a, y, c)=0, and we only get f® (2, y)=0, which, f being arbitrary, merely amounts to ®(7,y) = const., the original equation. Any other solutions of the proposed questions can then only. be obtained by other and particular considerations. First let it be pos-) sible to assign c so that $,(x,y,c)=0; it then appears that the two! forms become identical if c= (2, y), or (2, y, ¢)=0; so thate must) be derived from ¢,=0, for substitution in ¢=0: this is the common | mode, explained in the page above cited. But there may be others, and) the whole point will require the following elucidation. | (60.) An equation of two variables, such as 2 -a=(y—b) y’, 1s said: to be solved when a relation between w and y is found, which satisfies) it, and completely solved, when that relation introduces an arbitrary, constant. Thus x—a=y—b is a solution, but not complete: (v—@)* =(y—b)?+C is the complete solution. Nevertheless, a=a, yb satisfies the equation, and should therefore be called a solution, but not, a solution for which recourse must be had to the differential calculus : if would equally be a solution if y! stood for something else, and not for the | t | : 4 ON DIFFERENTIAL EQUATIONS. 707 diff. co. of y. Let the former be called differential solutions, and the latter extra-differential. A relation between w and y may even be extra- differential, as in (x—y)(x+yy')=0, which is satisfied by y==a, but without reference to the meaning of 7/. An equation of three variables may also have its differential and extra- differential solutions: thus (t—a) 2,+ (y—b) z,=x—a@ is satisfied by z=, and this is a differential solution, as it is only a solution when 2z, and z, are diff. co. of z. Again, z=a, y= is an extra-differential solu- tion, and e=a, z=v is a mixed solution, the meaning of z, being required, and not that of z,. Now it appears that the main question of the last article is reduced to the solution (of what sort matters nothing) . at . . 5 of a partial diff. equ.; and also that all the differential solutions lead to _ the constant value of c; all other forms of c must therefore be derived from the extra-differential solutions. One of these is obviously seen ; it is the pair of relations 6=0, ¢,=0: it remains to inquire if there be any others. The equation A=(B+Cm):(B,+ Cn) cannot be true independently of m and m, unless either C=0, or B and B, be infinite in the ratio of A:1 and C:B, be nothing. Applying this to the partial diff. equ., we find, then, that all its extra-differential solutions are con- tained in the determination of ¢ from the condition p —=0; or from ¢,=0. f y Dy arte Lip sat, Thus, if the original equation be c= (2, y), giving f=c—®, we find @.—1, and cannot be made =0: but ¢,: ~y=—1:4,, and ©, and 6, must be both infinite for any singular solution of the differential equation ; which agrees with page 191. ‘ The equation ¢(r,y,c)=0 implies that y is a function of xand c, such that dy:dc=—¢,:¢,, so that both the preceding cases come under dy: dc=0; and every different form under which y= (2, ¢) can be converted into ¢ (x, y, c)=0, gives the singular solution of the diff. equ. in its own way; some by ¢,=0, some by 4,=c. (61.) The manner in which Clairaut’s form is often solved (page 196) may be extended. The equation y=7/x+ fy’, being differentiated, gives (x+f’y’) y/=0, and y”=0 leads to the ordinary, and 7+/f'y’=0 |to the singular, solution. Now let ¢ (2, y, c)=0, and let ¢,+¢,.y/=0, : . 7 . . . ~T a ’ 4 derived from differentiation, give c=F (2, y,7/). Consequently the diff. equ. is ¢ (2, y, F)=0, which gives $2t by-y + or CS jh y+ Fy .y”) =0, or Dy Eo yt Py 0, which is satisfied either by FL+Fy,.y/+ By. y/=9, or dy=0. If the first can be generally solved, it leads to the form y=f (2, C1, C2), and the diff. equ. derived from @=0 may be satisfied by f, or rather only leads to a relation between C, and C,, which reduces these two con- Stants to one. But ¢y=0, combined with ¢(a,y, F)=0, gives the singular solution of this same diff. equ. in the usual manner. (62.) Given a solution of a diff. equ. y= x (x,y), not containing an arbitrary constant, it is required to ascertain whether it is a particular case of the general solution, or a singular solution. In the first place, ify=orx be this solution, try whether this last supposition makes x, 222 DIFFERENTIAL AND INTEGRAL CALCULUS. and x, infinite: if mot, it is certainly not the singular solution (page 193), and must therefore be a case of the ordinary solution: if it does, it must be, in the geometrical sense, the singular solution. But we must bear in mind that a solution which is in every property singular, for instance, which belongs to a curve touching all the curves denoted by the diff. equ., may also be itself only one case of the ordinary solu- tion, and therefore, in the distinctive sense, not singular.* (63.) The theory of the ‘singular solutions of equations of higher orders than the first has no very striking resuits, either in geometry or analysis; the following will be a sufficient specimen of it. Let V=0 be an equation between 2, y, c, and c,; and let V,+V, of Vi" Ard equ. of the second order is produced by eliminating c¢ and c, between V=0, V’=0, and V’,+V’,y’ or V’=0. Now suppose that c and ¢, are functions of « and y; it is required to determine them so that the diff. equ. of V=0, both of the first and second order, may remain the same as before. Let c’=c,+c,y', c=(4),+(¢),y’. Differentiation gives V,+V,y/+V.e'+V.,c1,=0; assume V,c'+V,,0=0, and we have the same equation as before for forming diff. equ. of the first order, The last equation then remains V’=0; differentiate again, and we have VitVW yy +Vi.e/+V!.,¢,=0; assume V’,c’+ V’,, c';=0, and we have again V’=0, as before, to be joined to the former two for obtaining the diff. equ. of the second order. The two assumptions give V.V14- Ve,V"e =0: with this, and V=O and V’=0. eliminate c and c,. The result is an equation between z, y, and y’, which is a first integral of the diff. equ. of the second order, but cannot be deduced from either of its ordinary first integrals by giving any particular value to the constants. If we integrate this singular integral of the first order generally, we — have an equation between x, y, and one constant, which is a singular primitive, but cannot be deduced from the complete primitive. A com- | plete example of this will be desirable. Let us have 3 Ch) vyeece’ HG ere Cerin (2) ay ace Cle pan ty y" =ce* +c, | Q,2) y=(ltaqe”*) y+ Qc.e7+ae*, y=—(14+ce’) y'+2ce7 +e eam (1, 2,3) 4y=y/?+4y"—y”. | Here are, the primitive equation, its two diff. equ. of the first order, and one of the second. Assuming c and ¢, to be functions of # and y, we must, to preserve the same resulting equation, have (e7+¢,) /+(e"+c)c,=0, & cb—e* c= 0, * A proof is frequently given which professes to show that when y=a makes xy infinite and x finite, that is, when y(x2,a+h) has a fractional power of hin its | development with an exponent less than unity, the solution ya cannot be deduced from the general solution by giving any particular value to its constant. At the | same time another proof is given that the curve which touches every curve that isa | solution of a diff. equ. is itself the singular solution. These propositions palpably contradict each other: for example, a given parabola moves with its vertexon a © fixed parabola of the same focal length, and so that the axis of the moving parabola | is normal to the fixed parabola. The fixed is, therefore, by the second proposition, the singular solution of the diff. equ. of all the moving parabolas, and by the first proposition it is not itself one of the moving parabolas: but it is evident that the fixed parabola is one of the moving parabolas. The defectis in the first proposition, — which applies the expansion of x (2, w-+A) in a very dubious manner. ae ON DIFFERENTIAL EQUATIONS. 709 which give ce*+-c, e~*== —2, and from this, and (1) and (2), we find (4) y?+4y+4=0 giving (5) y=—a?+Kr—1—1K?; (4) gives y”=—2, y!*—=—4y—A4, which satisfy (1, 2,3); and (5) also satisfies (1,2,3). But (4) is not a particular case of either of the equations (1,2), nor (5) of (1). Hence (4) is a singular solution of C1, 2,3) of the first order, and (5) a singular primitive of the same. But note that the stngular solution of (4), or y= —1, does not satisfy (1, 2,3). Also observe, that if we had deduced a singular solution from either of the equations (1, 2), by making ¢, or c variable, we should in either case have found the equation (4) 2 gain. The geometrical meaning of the preceding is as follows. The equation (1) belongs to an infinito-infinite number of curves, since any one value of c admits of an infinite number of curves, belonging to the different values of c,. Any relation whatever between c and c, amounts to a selection of a class of curves, every one of which is touched by another curve. Thus take c,=¢c, find the singular solution of y==ce*+ oce* —-ege, and we know that the curve thus found touches every one of the curves (1) which has its c, equal to the function ¢ of its c. But the curve (5) is, for every value of K, still more closely connected with a class chosen out of (1); it not only touches every one of them, but has the same curvature with each of them at the point of contact. Take any given value of 2 and y, and from (1) and from cé*+-¢, 6-7 = —2 determine c and ¢,, and from (5) determine K: then the curve (1), or its particular case thus determined, touches the particular case of (5) just determined, at the given point (2, y¥), and the two have the same radius of curvature at the point of contact. Moreover, for any one value of K, eliminate x and y between (1), ce*-+c,e"==—2, and (5), the result will be a relation between c, c,, and K, which expresses how to choose those curves which are all touched by that case of (5) which belongs to the value of K chosen. (64.) It is worth noting, that if y= (y*-”,....y, 7) be a diff. equ. of the mth order, its singular solution, if any, of the degree imme- diately preceding, makes the partial diff. co. dy™:dy"- become infinite. Thus, in the example above, we have Ae Vien dy J (y+ 4y+4)’ which is made infinite by y/®+4y+4=0. , y= -2t Jy? +4y+4), (65.) The equation Xdr+Ydy-+Zdz=0 does not of necessity arise from a relation of the form ¢ (2, y,z)=0; if it be the unaltered con- Sequence of such a supposition, we must have X,=Y,, Y,=Z,, Z,—X,. In this case the integration is an extension of that in page 197; suppose z a constant, or dz=0, integrate Xdxr-+Y dy on this supposition, as in the page cited, and let P be the integral, or rather P+C, where P is, or may be, a function of z, y, and z, but Cis a function of z only. Differentiate this last on the supposition that all three vary, then P, dx+P,dy+P,dz+C,dz must be identical with Xdzv+Ydy+Zdz. But P was so found that P, dx+P, dy should he Xdx+Ydy, whence (P,+C,) dz=Zdz, or, C being a function of z only, 710 DIFFERENTIAL AND INTEGRAL CALCULUS. 7, —P, must be the same, and Che f (Z—P,) dz. It will most frequently happen, unless a complicated instance be contrived for the purpose, or some peculiar artifice employed in integration, that we have P,=Z, or C is merely a constant. For example, let (y + 2) dx +(z+2) dy 4+ (x+y) dz=0, which fulfils the conditions. Make z a constant, or dz=-0, and P=ay+yz+z2e+C isthe integral, derived from integrating (y+2) dvt+(z+2) dy. But P,=2+y, or P,=Z; whence C is a constant. Now try another mode: make z a constant, and we have d d (y+2) dz+ (242) dy, or ae + ee or (¥+z)(a+2)+C=P=0 P,=(et+y+2z), Z-P,=—2z, C=—2?-+ const. (x+2)(y+2)+C=ry+yz+ 2x4 const., as before. (66.) Suppose that a factor M has disappeared from Xdx-+ &c. after differentiation. Then MXdxr+ &c. is a complete differential, or (MX),=(MY),, (MY),=(MZ),, (MZ),=(MX),. Develope these equations, and we have M (X,—Y.)=YM,—XM,, M(Y,—Z,)=—ZM,—YM, M (Z,—X,)=XM,—ZM.,, giving . Z(X,—Y.)+X(V.—Z,)+ Y (Z,—X,)=0. Unless this condition be fulfilled, no factor can make Xdr+ &c. integrable. If it be fulfilled, make z constant, or dz=0, integrate Xdxr+Ydy=0 as an equation between two variables, make the result- ing arbitrary constant a function of z, and proceed as before. The following instance will show the method. Let ay dz+ yz du+2udy+axyz (dx+dy+dz)=0; the equation of condition (divided by xyz) becomes (+2) (a@—y)+U+y) (@—2)+ (+2) (y—2) =0, which is true. z=0 gives (y+ary) dz+(r+2y) dy=0, or log (xy) +r7+y=Z, where Z is a function of z. Now consider z as variable, and for | yzdx+zx dy+xryz (dz + dy) write its value xyzdZ, which gives | ary dz+ xyz dz+aryz dZ=0, or ] (1+2) dz+2¢dZ=0, or Z=const.—log z—2z; whence log (xyz) +2+y+z=const., or ryze**’**= const. which is the primitive equation required. (67.) Next, let Xde+Ydy+Zdz=0 be neither integrable of itself, nor by the addition of a factor. Returning to our geometrical illustra- tration, it appears then that this is not the equation of any surface what- soever: that is, there is no surface on which any point (a, y, 2) beimg | assumed, and given infinitely small increments dw and dy, dz is always | expressed by —(Xdz+Ydy):Z. But on any one surface it may be © possible to draw a curve through any point, such that at every point of ON DIFFERENTIAL EQUATIONS. 711 that curve, transition from (2,y,2) to a point infinitely near it on the curve may satisfy the condition. To try this, let) U=0 be the equation of a surface, giving Pdr+Qdy+Rdz=0. Let M be an undetermined factor, multiply the first equation by it, and add the result to the second. We have then (P+MX) dr+(Q+MY) dy+(R+MZ) dz=0......(M), which is integrable, with or without a factor, by the preceding article, if M be determined from the partial diff. equ. fie ait 3 ate (R-+ MZ) (+ Pa MX Q+ My )- &c. = 0. dy dx ) Assuming then the possibility of integrating all partial diff. equ. of the first order, we can find M so that (M) shall be integrable: let it give V=0, then V=O and U=0 together give Xdx-+&c.—0, or the curve which is the intersection of the surfaces U=0 and V=0 satisfies the required condition. And since V=O contains an arbitrary function, an infinite number of curves may be made to pass through any given point of U=0, on each of which any point being supposed to move, its velo- cities in the directions of w, y, and z always satisfy Xdx: dt+Ydy: dt +Zdz:dt=0. Or any surface may in an infinite number of ways be supposed to be the locus of a family of curves, a motion on any one of which will give this relation always, but motion from any one curve across the rest, never. Another way of viewing the subject is this: assume y=¢@za, and sub- stitute, which gives (X+ Y ¢’x) dv+ Zdz=0, px being written for y in X, Y, and Z. Let the last give z=¥ (a,c), then the curve which is the intersection of the cylinders y=¢v, z=¥ (a, ¢c) satisfies the equation. Then an infinite number of curves can be drawn which satisfy the relation; but the preceding is more satisfactory, as showing that every surface may admit of having such curves drawn upon it. (68.) Equations of a higher order between dx, dy, and dz are not usually integrable per se; the following example, however, will be instructive. In dz*=dx*+dy? we see an equation which can have its /most general solution given in few words, as follows, This equation | denotes no general relation between 2, 7 and z; but, if y=¢z, z is the arc of the curve whose equation is y=. Jet us proceed to such an integration as that of the last article, without any reference to this pro- perty. One solution can be readily seen: let 6 be any constant, and if x sin 6+ y cos 0=A, then z=2 cos O0—ysin d+ B. Now let A and B be functions of 6, but such that «cos @—ysin =A’, —xsinO—ycos0+B/=0. The equation dz*=dz’?+dy* will still remain true, and we shall have B/=A. But «rcos0—ysind=B” and «sin 0+y cos 0=B’ give . + A at dd Ww - 2—B’'sin@+B"cos9, y=B’cosO—B"’sin#, z=B"+B. Take B any function whatever of 0, and if the first and second equations give the coordinates of a curve, the third gives the arc, measured from | some point to be determined: or rather, since xand y inyolve only diff. “ee 712 DIFFERENTIAL AND INTEGRAL CALCULUS. co. of B, it would no ways alter the question to add a constant to B, and to determine that constant so that z should vanish for a given value of 2. The solutions e=az+b, y=,/(1—a?) .2-+¢, treated in the same man- ner, will lead to the well-known determination of the arc by means of the involute (page 364). The student may also try to understand the following: the first solution above, when 0 is constant, amounts to sum- ming the elements of a tangent of the curve; when @ is variable, it amounts to summing the elements of the tangent supposed to roll over the curve, each element being taken into the sum as soon as it coincides for one instant with an element of the curve. (69.) In the preceding, integration is reduced to the solution of a functional diff. equ., thus. Let y=fz be the equation of a curve, and fJ(dx?+dy) is found, as soon as $0 is found so as to satisfy '0.cos 0 — 6"0.sin 9 = f(9'0.sin@ + $”0.cos9). The following is another instance of the same kind, which I leave to the student: show that Hf wr.dr=G'pr.pr—pper, if Px can be found so as to satisfy ox .9'x.dl"px =wWer. In both these cases, the converse is, generally speaking, the easier, namely, to satisfy the functional equation, or to | depress it, by the integration: a circumstance which points out the utility of noticing such relations, since it will generally happen that a mode of making the easier of two processes depend on the more difficult, is also a mode of making the more difficult depend on the more easy. (70.) The general process of page 203 has been extended (by Jacobi) as follows. Let there be any number of variables, say three, wu, v, W, each of which is a function of any number of independent variables, say two, v and y, and let there be three equations, as follows, w, meaning du: dr, &c., Xu Yuys U4) XU ch ¥ vss Vek wy Wie CL) where X, Y, U, V, W may each be a function of all the five variables. Grant that the simultaneous equations (4, or 3+-2—1 in number) du dv _dw dx _dy (2) U _ V ee WwW —— x, — Y° eereve can be integrated, and let P=const., Q=const., R=const., S=const. _ be the primitive system, where P, Q, R, S may each be a function of - the five variables. Then the system (1) is satisfied by the values of u, v, | w in terms of x and y, deduced from &(P,Q,R,8)=0, «(P,Q,R,S)=0, o(P,Q,R,S)=0......(3)anet where @, k,p are any functions whatsoever. Differentiate each of (3) with respect to v, and we have Dy tO, Ug Oy, Vz+O,, W,=0, Kp+&e.=0, 9,+&e.=0......(4), also o,dr+oa,dy+o,du+a,dv+oa, dw=0, or Xo,+Yo,+Uo0,+Vo,+ Wo,=0. yr -(5), by (2) + and similar equations from « and p. Let A, As, As be such quantities as will satisfy ON DIFFERENTIAL EQUATIONS. 713 Ay D+ Ags Ky trg ocr (6) and let (7) Ay ®yp + Ag Kip + Az 0,-== 0 hegle ; Ar Dupre ku As py ATT” j Multiply equations (6) by »,, w,, and (7) by u,, and add; which gives, by (4), ve Q, Wy + Ne Ky + As 0.) = Au, and — (Ar By +As ky As o,)=Au, by a similar process. Now multiply the equations (5) by Ay, As, As, and add, making use of the equations (6) and (7), and those just found, and we have —AXu,—AYu,+ AU=0, or Atiaots Yau =U, whence the first of (1) is satisfied: and similar processes may be applied to the second and third. (71.) The preceding theorem shows on what the integration of the general equation U=O (2, Y, Ur, Uy) depends. Let U,—=p, U,=q, and we have P—P2= hy Pot Pe Qe I — Py= Pp Py thr Qy Or P—os= by Pet Py Py 1 -—Py=bn G+, Gye oes (1), since py=q,. First, let us integrate these equations independently of the condition p,=q,. We are then first to integrate PPK te Aranda __ dy P—Pz g—oy, b> by let P=const., Q=const., R=const. be the integrals of this system : then w (P,Q, R)=0, «(P,Q,R)=0 are the integrals of the equations (1), independently of p,=g,. Now, considering @ and « as functions of p,q, 7, y, form the four equations of which the first is Dz+@, De +, q9.2—=0, by ordinary differentiation. Add the fifth equation p,=4q,, ind eliminate the four quantities Px Py Y2r Fy» from the five; the result is doa dk dkedw dod ad etn (2) dx dp dz dp dy jdgutyedg My kas yi hen any forms «=0, o=0, being taken which satisfy this equation, md p and q being obtained in terms of y, and substituted in the first value of u, the solution of the given equation is found. One mode of atisfying this equation is x=fw, f being any function: but this suppo- ition is equivalent to reducing w=0, c=0 to one equation only. (72.) Another general mode is as follows. However p and q may be Xpressed, the equation d.y:dy=d.q:dzx remains true, every mode in vhich p and q contain v and y being taken into the account. Let the artial diff. equ. be reduced to the form q=9 (p, 2, y, uw), and p being upposed a function of 2, y, vw, form the preceding relation. We have en dp dp id — g=g +g —O + +¢, : I= Pp dx %p du’ Pat PuP a 714 DIFFERENTIAL AND INTEGRAL CALCULUS. d dp .d or (q—@,-P) 2 $b, po rcaaa erase 0) the shorter notation expressing explicit differentiations from g=9. | Here p is a function of u, x, y, and the solution requires first the pre- vious solution of the simultaneous equations | AD oe PAs irk wy AOE i Pr+PuP p—Pp : P Pp If these can be integrated, we have, say M,=c,, M.=c., M,==cs, and f (My, Me, Ms) =9 for the solution of (1). Take any one solution in- volving an arbitrary constant, and having expressed p by means of it, it will frequently happen that z can be expressed, either by integrating q=, or dz=pdx+ qdy. Another arbitrary constant will thus enter, and a primary solution § (76.) is obtained, from which the general solu- tion must be got in the way presently pointed out. Of course those solutions should be taken in which p is expressed in terms of and y | only, or if w enter, it should destroy w in > after substitution ; or if not, w should enter only as a common factor in p and q. | 2 i ESS TET _ - (73.) Thus, let g=p" XYU, X, Y, and U being severally functions of a, of y, and of u. The differential equations then are Oe via pil cade te NCE. og er pXVUEpT RYU! p"XYU—np"XYU mp" *XYU~ + Thy Des and s vaitaek ets eri chiy tate nonlene aie From the first and second De ie du tery U du=0 From the second and third ; PR pp anu PW eee pdx, or dp+< d+ ata dius< 0; n—Il du= n 1 1 1 path log U=log a, or pX* Ut" =a 1 whence log p wip log X+ 1 1 du=U = (Pepi a” Yay) 1 geet ie Bae: dz+a"fY¥dy+. 1 1 n 1 gn" xyz—(ax7* tae) XYU=a° YU) Here is a primary solution. Make b=4a, and the general solution Is | determined by differentiation with respect to a and elimination. (74.) The singular solutions of partial diff. equ. have not been Aa vestigated in any manner which deserves the name of a general theory. The general solution, when it contains an arbitrary function, is itself) the singular solution of one which contains an arbitrary constant. Let, Xu,+Yu,=U, and let the equations dz: K=dy:Y=du:U be satisfied | by M=c, M,=a,, M and M, being functions of a, y, u, and ¢ and ¢. being constants. Each of these equations satisfies the given equation: for this given equation is in fact the same as XF,+YFy+ UF.=9;) ON DIFFERENTIAL EQUATIONS, 715 where F=0 is an equation involving 2, y, and u. ‘This follows from w= —F,: F,, uy= -F,:F, But M=c gives M,dr+ M, dy+M, dz =0, or, by the equations dr: X=dy: Y=du: U, we have XM,+YM, +UM,=0, whence M=c satisfies Xu,+ Yu,=U. Now the two solutions M=c, M,=c, answer to, and are involved in, AM+A,M,=A,, BM+B,M,=B,, where A, B, &c. are functions of any number of arbitrary constants: for these merely imply, and are implied in, M=c, M,=c, Hence AM+A,M,=A, satisfies the partial diff. equ. Now let its constants, instead of being constants, become functions of XL, YU, such that for every such function a, we have A, M+(A,).M,=(A,),; 30 that no differential relations of the first order are disturbed. There will be as many of such equations as of functions which were constants s and from them, a and all the rest may be deduced to be functions of M ind M,. Let the values of these functions be substituted in AM+A,M, =A,, and we have $(M,M,)=0, in which there is no restriction pon %, because A, &c. may be any functions. Here is the common general solution, which is therefore nothing but a singular solution of he most general form which satisfies dx: X=dy:Y=du:U (75.) Let a particular integral of any partial diff. equ. be found vhich'contains two arbitrary constants, say f(a, y, w, c,¢c,)=90. Let c, bea unction of c, then, if f,+/,, c’,=0, c may be supposed to be a function fx, y, and w, provided c be obtained in terms of x, y, and uw from the receding equation : which introduces an arbitrary function, since c, may ye any function of c. This illustrates the last article: but a singular olution may be often found, by making f,=0, f,,=0, finding the definite values of c and ec, which satisfy these, and substituting. When such a lution can be found the geometrical explanation is as follows. The ‘quation f=0 belongs to an infinito-infinite number of surfaces, cor- esponding to different values of c, and c. Every law of relation which onnects c and c, points out one peculiar family of these surfaces, which amily has a connecting surface: the solution which contains the arbi- rary function belongs to all these connecting surfaces. But these last urfaces may themselves have a connecting surface, which is related in he same manner to all: the solution without either arbitrary function or onstant belongs to the last. For instance, u=cv+c,ytar/(1+c’+c}) is the equation of every possible plane which has a for the perpendicular dropped on it from the Tigin. From such planes an infinite number of developable surfaces aay be formed; let c,.=¢c, and the equation of such a surface will be dund by eliminating ¢ between the preceding and a+c.yta {lt+e?+($c)*t-* (c+ $e. p'c) =0. All these developable surfaces have their tangent planes also touching 1€ sphere whose radius is a. Eliminate c and c, between the first quation and the two following, ata(l+c?+ ci) ?*.c=0, yta(l+e+c}) te,=0; nd we have 2*+ y?+-u°=a’, the equation of the sphere. (76.) It thus appears that we may distinguish the solutions of partial iff. equ. of the first order into three kinds. 1. One which contains two rbitrary constants more than were in the equation. 2, One which con- 716 DIFFERENTIAL AND INTEGRAL CALCULUS. tains an arbitrary function. 3. One which contains neither constant | nor function. Lagrange termed these severally the complete, general, | and singular solutions. To the third term there can be no objection, but the distinction of complete and general is not easily made. Thecom- | plete solution may be a very limited case of the general solution, as in | u==cx-+c, y, which is the (so called) complete solution of wu=au,+yUy The general solution is u=xp(y: 2), one form of which is cr+qy HOY? ACY? poe ee ad inf. It will much offend our ideas of | language to say that this last is completed by making co=0, =0, &e. It would be better to call the first solution primary,* the second general, — and the third singular. # Let (2, y, U, a, b)=0 be the primary equation, then the partial diff, equ. is obtained by eliminating a and b between $=0, $,+¢,U.=9, | dy th, Vy=0; or by considering @ and 6 in the first, as functions of a, : y, u, obtained from the second and third. Suppose that this substitution | made gives ¥ (2, y,U, P,q)=0, where p=u, and g=u, Then ¥=0 : is an equation identical in meaning with ¢=0, when @ and b are cun- | sidered as above. If, then, from y%=0 we find w in terms of a, y, p,q | and substitute it in @=0, we have (as in page 192) an equation abso- lutely identical, independently of all relations : and every diff. co. of @ so, : altered is identically =0. Differentiate then separately with respect to | p and q; the first operation gives Dy Upt ha (Au Up tay) + Pr Ou Up + oD =0e the implied suppositions are that @ contains p through wu, a, and 6, while wu, deduced from y%=0, contains p, and a and } contain p both directly and throngh wu. Now from ~=0, the proposed diff. equ., from which w is obtained for substitution in the preceding, we have ¥, + Ws, uy —(); substitute for u, in the preceding, go through a similar process relatively to g, and we have . We ‘ike Da a,+ Pp b, Ys, she Da a,+ D, b, Wy — hutda Ot by bu Ww, ~ Put Ga Aut hs by Now the singular solution is derived from ¢,=0, ¢,=9, and ¢=0 necessarily contains wv, so that ¢, is not =0: consequently, unless a, or b, are made infinite at the same time that , or ¢, vanishes, a singular) solution will give %,:¥,=0 and y,:%,—0. Singular solutions then may be sought among those relations which satisfy ¥,=0, ¥,=0, By being finite; or among those which make ¥, infinite, y, and Y, being finite. But it does not follow that these modes will give all the singular) solutions; for a, and b, may possibly become infinite when ¢, and $s. vanish. w i (77.) For example, take the surface on which the normal intercepted’ between the tangent plane and that of ry is always of the same length k: the equation of which, x, y, w being the coordinates of any point, is found to be w2(1+p°+q2)—#?=0. The singular solutions may be contained in 2u?p=0, 2u®qg=0; now w=0 does not satisfy the equa) * Even against this word lies the objection that there is an infinite number of primary solutions : thus y"~! u=ca"-pe, aly is, for all values of n, a primary solu: tion of the proposed equat’°n. BI ON DIFFERENTIAL EQUATIONS. 717 tion, but p=0 and g=0, implying w=const., do satisfy the equation, if that constant be tk: and 2°—f2—0 js the singular solution. It is evident enough that the two planes thus obtained are the envelopes of all surfaces\of the kind required. For the primary solution it is obvious that a sphere; with its centre on the plane of zy and a radius &, will answer, or (t%—a)*+(y—b)?*+u2=h®. Assume then b=¢a, and eliminate @ between the preceding and (x—a)+(y—¢a) ¢'a=0, and we have the general solution. The primary solution is thus a sphere of given radius, the general solution a tube (page 402) made by the motion of that sphere with its \centre on a given curve in the plane of xy, and the singular solution the pair of planes parallel to zy within which all such tubes are contained. (This tube is called surface-canal by the French writers.) (78.) In the same’ way it may be shown that u=pxr+qytf(p, q) nas for its primary the plane wu=ax+by+f (a,b) ; for its general solu- jon the result of eliminating a between this and r+ ga.ytfitfrgla =0, which gives a developable surface, and for its singular solution the result of eliminating a and 6 between the original and w+ 3 me /+f,=0. (79.) I now take some detached artifices which have been given for he integration of various partial diff. equ. of the first order. I (p,=0, or gq=¢p, the general equation of developable surfaces. dere du=pdx+ pp dy, u=px+dp.y—f(x+¢'p.y) dp, whence a+@Pp.y=say w'p, r u=pr+pp.y—vp, «+¢'p.y—w'p=0. iliminate p, and we have the general solution. This case is, under nother form, a repetition of that in the last article. (80.) z=f(p,q). It may be discovered from §(71.), that z= (y+-cx) must contain a solution of this equation : or, for some form of », we have 6 (y+cr)=f {ed! (y+cz), ¢' (yt+cr)}. For y+cx write z, nd for (y+-cx) write y, which gives y=f (cy’, y'), a common diff. equ. rom which can be found, say y=¥ (v7, c,). Hence z= (y+er,c,) is a rimary solution of z=f(p,q), from which the general solution can @found. For instance, let z=pq, then y=cy” is the diff. equ., which ives ba oy \2 1 rc > \ (<= te, ) > Or fags 7 CS zs +¢,) : et ¢=¢c and 2 pe) (2+ 29'c) : iminate c, and the general solution is found. Or, eliminate c from 2Je—F.- foot $o=0 ‘wii ntebtich gh tee —dq'c=0. 2efo 2c (81) O(p,2)=¥(q,y). Let o (p,7) =a, &(q,y) =a, whence =P (7,2), = ¥, (y, @), at 718 DIFFERENTIAL AND INTEGRAL CALCULUS. z= [id (aa) d2tyn (y, a) dys=he (2, a) + (y, a) +6, a primary solution. Assume b= @, and eliminate (for the general solution) a from do, .d =f. (mathGyd+xa 0H +4 yo (82.) If (2, y) be differentiated twice completely, it gives bz, da | +26,, dx dy+ oy dy®, say rdx®-+2s duvdy+t dy?: and the conditions | under which such an expression is completely integrable are 1 | sy=t, But it is seldom that a factor can make such an expression / integrable. Let Rdz?+4 28 dv dy +Tdy? be integrable, if possible, after multiplication by M; we have then (MR),=(MS),, or SM,—RM,=M (R,—S,) (MS),=(MT),, or TM, —SM,= M (S,—T.). oa ee ee ee From these find M,:M and M,:M, say A and B. Then, if A,=B,, | M is possible, and log M is found by integrating Ad7+Bdy. Hence it appears, 1. That when the expression is integrable already there is no- factor under which it will remain integrable, except when S°=RT, in which case there is an infinite number. 2. Whenthe expression is not integrable, there may be one factor, but generally only one, and most frequently none; except when S:T::R:8 st Ry Se ney bes in which case there is an infinite number. For example, ¥ da® +2 (ry+1) dx dy+a* dy is ngt integrable: to determine the factor, if any, we have (zy+1)M,—y*? My=My A=M,:M=y a? M,— (zy +1) M,=—Me B=M,:M=2 and Adz+Bdy is integrable, and gives zy, whence M=e’: multiply: and integrate, and we have ¢* itself for the primitive function. | (83.) Ifwe take a partial diff. equ. of the first order, containing 7) arbitrary constants, we may from it form one of the second order. ‘Thus, if @ (2, Y; Us Ps G4, b)=0, we may determine a@ and b from $,+4)7) +b, 5=0, dy tops +o, t=9, in terms of a, Y, U, P> J, 7 8, and b) These values substituted in #=0 give an equation of the second order. Again, assuming =a, we get precisely the same equation of the second, order if a and b be functions of 2, y, U, Ps q; provided that a be deter mined from ¢,+,-6,=0, or Pat Ps. X/'a=9. Hence we can get | solution of the first order having an arbitrary function, since x it arbitrary ; and if, therefore, we can integrate this equation of the first order, which integration will introduce another arbitrary function, Wt have the complete solution of the given equation, with its two arbitrar)| functions. But we must first extend the conclusions of page 64 to the extent of showing that two arbitrary functions cannot always be elimt) nated in the formation of the equation. | Let A and B be given functions of 2, y, and u, and f(x, y; %, bA, WB. =0(, an equation in which f is a given form, and @ and % any function ON DIFFERENTIAL EQUATIONS. 719 whatever. Differentiate with respect to 2, y, xx, ry, and yy, which gives altogether six equations, involving PA, WB, $A, w'B, dA, vB, with 2, y, U,P, q, 7, 8, t, and the known functions of them Ay tue Be, &c. Now>six quantities cannot generally be eliminated from six equations: therefore-the equation f=0 is not always the solution of an equation of the second order. It certainly very often happens that the process which eliminates five also eliminates the sixth. Therefore, although the preceding part of the process shows that every equation which has a primary of the first order containing two arbitrary constants has two arbitrary functions ; yet the converse is not true. Ifa represent the number of arbitrary functions, and r the number of complete orders of differentiation performed, the excess of the number of equations over that of the functions given and introduced by differentiation is 4 (r+ L)(@r7+2) —a(r+1). This can never be unity (which is required that one equation may be a necessary consequence of elimination) except when @—1, r=1. If a=5, then 4 (r+1) (r+2) first exceeds 5 (r+1) when r=9, and the difference is 5. Consequently, an equation of five arbitrary functions has five distinct equations of the ninth degree, in all Cases in which there is not some peculiarity in the elimination: and this is the first set in which all traces of the arbitrary functions vanish. (84.) It is not certain that every partial equation of the second order even has a solution. The most general case in which anything like a method has been proposed is as follows. Let Rr+Ss-+Tt=V, where R, &c. may be functions of 2, Y,*,p,q: this is the most general equa- tion of the second order and linear form. The principle of solution is that explained in page 203, and may be stated as follows. There are already three ordinary diff. equ. existing between the quantities 2, Ys Uy P, 9, 7, Ss, t, namely du=pdx+qdy, dp=rdx-+ sdy, dg=sdx+tdy, which are universal, or true when u is any function whatsoever of z and y. To make use of them then is not introducing any new condition into the question ; for that w should be a function of x and y 1s already mm implied condition. Consequently, the given equation is neither more nor less than R aoa Ss+ pbesdean.| 5g dx dy et Rdpdy+T dqdz—Vdy de=s (R dy?—S dx dy+T dz’). If we call this last c=se, we see that the equation includes among ts conditions that if o vanishes ¢ must vanish, and vice versd. This S not all the meaning of the equation, but a part of it, and, so it lappens, enough for our purpose. Proceeding in the same manner vith r and ¢, we find, making "(dp dx— dq dy) — 8 dp dy+Vdy'=p R dy?— Sdx dy+Tdx=y Ldp dy+-T dq dz—V dx dy to. du—p dx—q dy ua (dq dy —dp dx) —Sdq dv +V dat=r aat the given equation is equivalent to either of the following, p=rg, ‘=se,r=ta. Whence, wu must be such a function of x and y as will oe 720 DIFFERENTIAL AND INTEGRAL CALCULUS. vanish when any one of them vanishes. But make all the four, 0, 0 7, @, the equations Ro+Soe4Tr=Ve, adp=pdx + ody, adg=odr+rdy, be verified, show that the four equations p=0, o=0, n any two are satisfied. Hence we by far from a complete solution, owing system, containing, 1. Any which may easily r=0, «=0, are all satisfied whe satisfy the original equation, though when we find any primitive of the foll pair out of p=0, o= 0,7=0,0=0. 2. The equation v=0. Here are three equations between five variables a, y, U, P, 73 let A=a@ be one of the primitive equations, of which there may be three, two variables | being independent. We have then A, da+A,dy+A,dut+A, dp+A, dq=0. | n o=0 gives Rp dp+Tdq—Vp dr | —0, from which substitute for dq, and for du from v=0. The result contains only dx and dp, and, every necessary condition having been used, this must be true independently of dx and dp, which might be- made the two independent variables. Equating each coefficient to nothing, we have Let a=0 give dy=pdzx, and the Vp F Act A, pt As (pgp) + Aa pe =0> Ap— Aa apr =0- + ++ (a): T Let B=b be another primitive, which will give similar equations. - Then, as in page 203, the condition, not that the diff. equ. «=0, c=0, should be satisfied, but that one should be satisfied whenever the other | is, may be expressed by B=@A, among the cases of which we are | therefore to look for solutions of Rr+S8s+Tti=V. And on examination, | as in the page cited, we shall find that every form of ¢ satisfies it, as follows. Take the equation B, dr +B, dy+&.=@'A (Az dx+&c.), and for A, and A,, B, and B, write their values from (c,q@) and the corresponding equation for B. The first side becomes | Aa ve? f oie Vv —B, pde—B, (p+) dz—B, 57 de 'R +B, dy+B, (pac + qdy) +B, Ge dp + B, dg, or B (B, +B.) dy — pdz) + (Ry dp 4+Tdq—Vp dz), de which is @/AX a similar function of A, so that es es eee Rudp+Tdq—Vp dxr=w (dy—pdz), gives | which verifies the assertion above made relative to B=oA. For dp and dg write rda+ sdy and sdr+tdy, and dx and dy being independent, | we have { Ryur+Ts—Vpt op=0, Ryus+Ti—w=0 ; | Ryr-+(Rue-+T) s+ Tpt— Vy, or p (Rr+Ss+Tt-V)=0, | or ON DIFFERENTIAL EQUATIONS. 721 since Ru’—-Su+T=0. Hence the equation is satisfied by B=@A. It may be observed that 4 has two values, either of which may be chosen ; or it may happen that it may be convenient to use both. For example, let R, S, and T be constants, and V a function of x and y; whence « is a constant, and dy — pdx gives y=px-+a, which substi- tute in V. Then Ru dp+Tdq—Vydr=0 gives Rup+Tq—pfVdx =6. After integration of Vdr, put back y—yr for a, and the first integral of the given equation is Rup aL Tq—pf Vda=¢ (y—pr), with either value of ». If we proceed to integrate this equation by page 203, we must first integrate the system Ru dy —Tdx=0, or dy—j, dx=0, (1 being the other value of p, and pps, being T: R) and Ru du=pfVdx.dz+ (y—px). dz. The first gives y—p,2=a; substitute for y in the second, then since fo (a+ pe px). dz, ¢ being arbitrary, is simply (a+p,2—p2x), Which has the same appellation if divided by pR, we have for the integral of the second equation, putting back y—p & tor 6 after integra- tion, ] u=pJdafVde+¢ (y—px) +. Where the meaning of fdxfVdxe has much more than the notation ex- presses, nor does the operation’ occur often enough to require a distinct iotation. We begin with V=¥% (2, y), which we change into w (2, ur-+a), ind integrate, giving, say w, (2, a), which we re-convert into We, (2, y-p2). Phen we change this into wy, (a, a+} ¢—pxr), and integrate, giving, fay W%, (v, a), which we then re-convert into We (2, y--~, 2). From the oquation y—,, =a, and the last, we now have 1 Lu pi ars Vda +h (Y—pr) + (y— py, 2), p and y& being any functions whatever. Let us choose, for instance, e+6s+5t=x+y. We have then /2?—6u+5=0, or 5 and | are the falues of u. Proceed with V=.xr+y in the way pointed out, and we lave f(@+a+ 5x) dx=3.x" + ax, for which put 322+ (y—5z) a, or ry —2Q2x°, 2 : ; Last iaGge ; miegrate 2 (a+a) — 2u , which gives ax —zx°, for which put r(y— x) a?—42°, or dyz°—423. The solution is Uu=syX’—32° + (y—5rz) +4 (y—2). Verification. Taking the first two terms alone, r=y—bu, S=a, =0, r+6s+5t=x+y. Now (y— pax) gives r+ 6s +- 5¢ =(4°— 64-5) 6” (y—px), which vanishes when p= 1 or u=5. The quation would certainly be as well satisfied if to the preceding we ded Az+By+C, whence it might seem as if we had not the complete olution. But observe, that At-+By+C may be made to become 3A te 722 DIFFERENTIAL AND INTEGRAL CALCULUS. E(y—5r)+F (y—2)+C, if E+F=B, 5E+F=—A; so that the | preceding addition only amounts to an alteration of @ and . When the roots and p, are equal, first assume p,=p-+ 4, then show | by the method of § (21.), that the complete solution is | ier qlefVde+z9 (y—pr)+% (y—p2), which requires the assumption (obviously allowable) that an arbitrary — constant of any value may be a multiplier of either function. | from ordinary diff. equ., we might seem to have all but a demonstrative | right to infer that every partial diff. equ. of the second order has two of | the first order, each containing one arbitrary function: which two arise | from one primitive containing two arbitrary functions. All this is very | often true, no doubt; but there is not a single point of it which cannot : be refuted, if asserted universally, or at least shown to be hitherto ins | capable of general proof, and very unlikely in certain cases. First, in the equation osx, we have begun by presuming the existence of a solution which allows « to vanish, when of course o vanishes. The solution we thus obtain may be the most general of its kind: that is, of those which allow « and o to vanish; but how do we ascertain that there _ are no solutions in which this is impossible? or how do we know that there are not some in which, when @ vanishes, s necessarily becomes | infinite, and se remains finite ? | (85.) Looking at the preceding method, and generalizing by analogy | | But do not these objections equally apply to the solutions of equations | of the first order in page 203? Undoubtedly they do, and the proof of the perfect generality of such solutions is therefore not complete till page| 204. It may be thus further illustrated, with our knowledge of primary _ solutions. | Let f (2, y,u, PA)=0 be the solution of a partial diff. equ., f and A. being given functions, the latter of x, y, w; and @ the arbitrary” function introduced by the common method, which we may therefore | write cA-+c,yA. We have then one primary solution, with two in- dependent constants. If there be any other general solution, we cal obtain it in an infmite number of modes by making c and ¢, functions of x, y, and w; and we have a right to one relation between c and c,. Let | it be c,= we: and solve f=0 with respect to pA, giving, say cwA+c, XA =F (ax,y,u). If then we determine c from wA+¢c.xA=0, giving | for c and c, functions of A only, all differential relations of the first. order remain as they were when c and c, were constants; and the partial | diff. equ., from which f=0 arose, is satisfied: but cyA+c, XA Is still only an arbitrary function of A. | The process of page 204 might be extended to the proof in § (84.), and we might be compelled to admit, that when two arbitrary functions” appear, the most general solution is gained. But whether every diff. | equ. of the second order has two arbitrary functions ; and whether every | such equation has a solution; as also whether, if it havea solution, | there are diff. equ. of the first order belonging to it,—are all unsettled - questions. To take a well known instance illustrative of these doubts, © let r=q be the equation, or R=1, S=0, T=0, V=q. We have then p=qdy?, c= (dp-qdx) dy, r=dqdy-(dp-qdx) dx, c=dy’,which vanish simultaneously if dy=0, dx=0, or if dy=0, dp—qdx=—0. The first ON DIFFERENTIAL EQUATIONS. 723 fim would give the solution y=, which does not contain u, and must be rejected: the second cannot spring (with du=pdx+ qdy) from any relations between 2, y, ~, p, and q, all variable: and g=c can only give the solution t=5c0"+cy+c,x. But the following solutions can easily be verified, or (the two latter) obtained by indeterminate coefficients, u=C,e"% Um Yy 1 ren EMC y 1 C, es TMs Y 4. AN § V/ i iv y° vi y u= Gry d'z.yt+o" ac +g% 7s ee 2 2.3 a " ar 2 aie ited ye atoms In all three, two 2-differentiations give the same result as one -differentiation: which is all that the equation requires. The third seems to involve two arbitrary functions, and really does so with respect 0 y; but yet these two only amount to one with respect to z, as in the econd solution. For if PY=+ay+...., and wy = b+ by m--..3 if, after substitution, a+b, t+a, 27+), 23+ .... be called 1, the third is converted into the second. We shall see the complete itegration of this equation presently. / a / M=PYT WY t+DY.— + 'y (86.) The most important equation of the second degree, beyond all uestion, is aru , au d?u : du i e 73a, say —- =e —, or r=a?t: dx* dy” Y ae dx? i hanging the variables* for convenience, since, in mechanical pro- lems, one of them is usually the time (f). If an elastic fluid be con- lined in a tube of very small section, and if a be the velocity with hich sound travels in that fluid, then, the preceding equation being ved, du: dx will represent the velocity of the particles at the distance from an arbitrary origin in the tube, at the end of the time ¢, and. wu: dé will be always proportional to the compressing force. This juation has been already integrated; we have R=1, S, 0, T= —a’, =0, p°—-a@=0, p= +a, and ! u=¢ (t+at)+% («—al), here ¢ and ¥ are arbitrary functions, deducible from the state of the ‘be at any one moment. An independent integration may, however, be desirable, and we may tain it as follows. Supposing the equation to be rt, PHU, Q=U,, € equation gives p,=q,, and the property of all functions is p=; ‘e have then Pr+Q:=p.t de and p,—G=—(p,—q,)- Hence pt+q - @ function which satisfies (p+ 9):=(p+])2. and we must have +q=f (2+); and p—q satisfies (p—Q):=— (p—Q or we must Wwe p—q=f(«—t). The functions being arbitrary, we have at once P= (e+1)+¥ (x2), q=? (@+1)—% (a—t) U=9, (@+1)+4¥,(@-)+oy, u=d, (@+O+4,(c—t) +2; * The two different meanings of ¢ must be distinguished : both are so sanctioned /custom that the clashing of the two cannot always be avoided. 3A2 724 DIFFERENTIAL AND INTEGRAL CALCULUS. equations which can only agree when my and yx are constants, and therefore may be considered as included in the arbitrary functions. Or we might integrate by page 203 either pt+q=f(#+t) or p-q=f(@-t), and we should produce the same results. Change ¢ into at on both sides, and we obtain v,=a? w,, and its solution. | (87.) art2Qryst+y°t=0 gives u=ad(y:r)+% (y:2) gr—2pqstpt=0 gives u=h (cyuty) r r—t=2Qp:0 gives u=p (y +a) +4 (y—2)—2 {P (y+a)—w! (y—2) fe For Rr+Ss+Tt=0, when R, S, and T are functions of p and q, see | page 473. Apply § (84.) to this, and a=0 gives p a function of p and q; whence V=0 shows that o=0 can be reduced to an equation which can be integrated under the form f(p,q)=c. Also this and} dy= dex give du=yvdx, where p and y can be made functions of p and c only. Elimination of p gives an equation between da, dy, du, and ¢,' which may sometimes be integrable. From the preceding we gain this,| that some class of developable surfaces must be a solution of the given equation. i Rr=V and Tf=V, when the coefficients are functions of 2, y, and p only, and of x, y, and q only, are only ordinary diff. equ.: for y must be) constant throughout the first, and a throughout the second. Thus, take p for a variable in the first, and we have the form | d (2, YP) aif (t,Y,P), OF P=X (a, y; By) u=fx.da+ ay» @ and « being arbitrary functions. (88.) Let (7, s,0)=0 be the equation, not containing 2, y, u, p, 0 | . Let and y be each considered as a function of both s and #, ang dp=rde+sdy and dqg=sdxr-+-tdy then give P and adr-+yds and ads-+-ydt must be complete differentials. Assum then dv oe eye \ 7 —d eu —_——,; =—_— eoee e ; t ras+ydi=dv, or,« Pe inet) (v) | . The original equation gives ¢,dr+¢,ds+4,dé=9; from whic! cdr--yds becomes (and which must be a complete differential) a % p ) d (ibe od, Coes | —ar— dt—| «—-— st yon Lee eae e)| a fee t (« a y )ds, or 75 (« - Ti (2 - v) 4 But from ¢=0, 7 is a function of s and ¢, giving r= — $8 r= —¢;:9,, Whence in the last equation two terms disappear, ane have Pe 4 G. dt ih, dx sdy tae dy dD 5.) Lee Dean Tie RITTER Het a linear equation, similar to those already considered. If v can be foun from it in terms of s and.¢, we have w and y from (v), and thence p a q from (p, q), after which we find u from u=px-+qy—Jf (wdp+ydq)> M Pi 3 ON DIFFERENTIAL EQUATIONS. 725 ; d?y POIs diy For example, r{—s’=0 gives dt. — +28. —- (lL, = dt? di ds’ «tds? v= 14 (s:¢)-+¥ (s:t), from § (87.) Hence ai 5 ee fee S sarire a He 88 s .* ey ea Ce) Cae Ge Fe (7 )+4 ) ) io sb( +) -Seedr-+yds)= am fe Us (+)-« (+) g=th (+ )=SGds+yay= — fy! (+). d (+) u=prtqyt SP =) v' (= aes and we see that this merely amounts to supposing p any function of q,* adding to px+qy any other function of q; with the condition that, if u=yx+qy+Q, we must have cdp+ydq+dQ=0. The latter agrees with § (79.). This particular example, however, is thus most easily integrated. The equation, for dp and dq, combined with ri—s’=0, give 0: so that dp=rdn-+sdy=— (sda-+tdy) =— dq, whence p=fq necessarily (page 199). Afterwards, as in § (79.). d"u du d”u (89.) Let a, Tatu dv dy * snivie ted ayn? (2, y). Without going into the full investigation, the process may be described as follows. ‘First, when ¢ (2, y)=0, let py, po, &C. be the roots of the equation ap" a," '+....+a,=0. If all these roots be unequal, he solution is w=¥, (y +p. 0) + Wy (y+ pe ®) +o. - 0, Wy We, Ke. being irbitrary functions. But for every set of equal roots write w, (yyy 2) Bas (YE py 2) +2? We (YAM 2) +...., with as many terms as there re equal roots. Next, when ¢(2,y) is not =0, treat it successively with all the roots in the manner pointed out for two roots in § (84.), J+pr=c being the equation from which y is obtained. Divide the esult by a), and annex it to Ww,+%,+...., or whatever the preceding vart of the method gives. Suppose, for instance, that n=3, and that #+6y?+11n4+6=0 is the equation, the roots being —1, — 2, and —3. I write down without explanation all the substitutions, integra- ions, restitutions, &c. &c., P(x, y) being =ay. | Tees CBAs, ce? a YN a xy, «(2+c), 3 To ag hV?? ooh) oo hoetge ; 6 AEG FU an. ae at a ac’ lin eaee Byatt iGey 4s 56 Fas Ate! eb Ban ON Bi = To solve fx fr.dx—F (ffedx). Differentiate both sides, which gives a= "(/faedx), say ffx dx= xx; whence fr=+/x, and is therefore found. 726 DIFFERENTIAL AND INTEGRAL CALCULUS. ave Tg? a 2 = FEE es C Bee See 8 10 94 40° d?u du ee Ge dax® dx* dy pe oe pe ha Tote a od a x (32+ c) oe whence 4 5 gives w= Sot ¥i (y—2) + Ya Y— 22) + Ys (Y~ 32), the complete solution. (90.) If the linear equation with constant coefficients have also diff. | co. of lower order than the nth, assume w=¢""”. An equation is then | found between p and y, which, being solved, gives, say v= Pp. We} have then a very general solution in w= Ce” “+d where there may be || any number Of terms (even an infinite series) and two distinct arbitrary | constants in every term. For instance, let ar-+bst+cttep+fqtgew | =0. The supposition w= et" sives ap? tbuyteovteutfy+s=0; | é from which, if y=@p, we have a solution of the form i waa O, Mitte 4 C, eta Y 4 0, fer tey te C,, &c., pu, &c. being any constants whatsoever. An infinite number | of arbitrary constants is a circumstance of identical meaning with an) arbitrary function, for @(a+y), for instance, @ being arbitrary, and Co(aty)+C, (x+y)? +...-, Co, Ci, &c. being arbitrary, are cons) vertible; as are also P(x+y) and Cyo%(e+y)+C.% @+ty)+.-.--5)| Yo, YW, being forms of a given law. It may happen that two infinite) | trains of arbitrary constants, as in the last result, are equivalent to two) arbitrary functions: but this is by no means always the case. We must) ) now consider the question of the arbitrary functions which enter into results, as to the means of determining them. of (91.) It will be advisable to dwell on one particular instance, and view it in more than one light, since it is not practice in the operations, | so much as a clear view of the office of the arbitrary functions, which i8_| required. Let the equation be r=a’t, of which, beyond all question, the” complete solution is u= (y-+ax)+¥ (y—az), and the solutions of the” first order are p+-aq=2ap (y+azr), p—aq=—2ay! (y—az). If a, yy” w be coordinates, we have here the equation of a doubly infinite class of | surfaces, of which the third ordinate w is the sum of the ordinates of two} cylinders, whose generating lines are parallel to the plane of xy, and | make angles with the axis of «, of which the tangents are a and —@ Let there be a curve of which the equations are r=av, y= fv, u= yu! | the surface will pass through this curve if the forms of and %& be pro= | perly assumed. In order that it may do so, the equation yw } = (bv-+aav)+% (Gv—aav) must be identically true. This can be done in an infinite number of ways: let @v be the inverse function to— be +aav, so that fov-tacwv-=v. We have then you= pv + (Bov—aaov), and whatever y may be, ¢ can be found accordingly. | Let there be a second curve, v=a,v, y=A,v, u=y,v, and let 8, a, 0° +-aa,,v=v. We have then another equation like the preceding, and | subtraction gives ig ON DIFFERENTIAL EQUATIONS. 727 U (8, B, v—aa, B, v) — (Swov—aawv) =y,W,v—yov ; in which ¥ is the only unknown function: say the above is 40, v— UsOv = =— (93.) Let us now consider ,,—a? u,, as the equation which gives the jaw* of small oscillations in an infinitely thin tube of air. Here zw, repre- fents the velocity of any particle, and w,:@? is the compression, or the Hifference between the density of the particle and that of ordinary uir. The functions ¢ and y are determined as soon as the state of the pube at any one moment is known, say when ¢=0. At this epoch, let yx be the velocity of the particle distant by «x from an arbitrary origin jaken im the tube, and Bz its compression. Consequently, when ¢=0, Wwe have u,=azx, u,=-a* Bx, or, since u=P(e+at)ty (t—at), we have d/r-+ Yvan, $'x—W'n=aBx, whence ¢'x and uwa@ are found. irst, the tube being supposed of indefinite length, let the initial state of the system be, that it is all at rest and uncompressed, except only in the jaterval from x=c to a=c-+h, in which the velocity and compression pillow such laws that ér=6x and wa=WVe. We have then, using the fotation of page 616, dr=Is** Ox, wr=1st* x, where It* means a lonstant which is 1 whenever the subject of the function lies between ¢ ind c+-h, and 0 in all other cases. Calling v the velocity of a particle, ind s the compression, we have then [ct ®' (xtath+I*W' (c-at), asst" 6! (x2+-at)-Iet! wy! (val). It the end of the time ¢, then, the state of the tube will be this; all hose particles in which r-+-at lies between c and c+h will be affected ‘ith velocities represented by ®' (~+), let there be a fixed obstacle or closed end in the tube. Con- quently x=/ always gives v=0 for every value of ¢. By the equation * The demonstration may be found in any work on analytical mechanics, q 730 DIFFERENTIAL AND INTEGRAL CALCULUS. v=1' W (/—at), vis 0 until /—at=h, but from thence to l—at=0, there would be a succession of different velocities if it were not for the | obstacle. ‘The moment this begins to take effect, we have no longer any reason to suppose that the disturbed parts of the tube are affected by ¥ | only; but we must take the complete solution v= ®! (a+at)+W" (c—al) We have then ®/(/+at)+¥ (J—at)=0 for all values of ¢ greater / than (J—h):a; consequently, after this epoch, we have ® (J+ at) | — (J—a), by integration with respect to ¢. Write (t—/):a for tim and df= (2Ql—¢), or | j tf | u=¥ (Ql—2+at)+¥ (c—at), v= —¥' (Ql—-x tal) +! («—at) 3) in which, by the initial condition, Vz has value only when z lies” between 0 and hk. The obstacle, then, introduces a disturbance whic travels in the contrary direction to that of the one first given, and gives” velocities to the several particles contrary to those of the first. Tf Qi—(E+at)=r—at, or r+£—21, the particles distant by é and x from the origin will be similarly disturbed in everything but direction, | Hence it is easily shown, that the effect of the obstacle is simply to ‘he 4 back every disturbance which reaches it, and to make it travel in the | contrary direction; the effect being exactly the same as if a second disturbance similar to the first had begun to progress in the opposite direction from a point distant by / from the obstacle on the other side.™ | (94.) Finally, with regard to discontinuity itself, it may be observed that there is no difference between a continuous and discontinuous curve, except one which may be made as small as we please. As in page 610° we may find a curve which shall with any degree of accuracy represent | a succession of arcs of any different curves. Hence, were there any- thing solid in the refusal to admit curves (or functions) incapable o being represented by one and the same equation, it might be answere that even discontinuous curves may be so represented within any degre of approximation, and in finite terms: so that, in fact, a and a (discontinuous) may be made to stand on precisely the same footing as objects of algebraical calculation. 7 (95.) If we make £=y—azx, n=y+azr, which simply amounts fo) changing the coordinates in the plane of zy, with the same origin and 2 same axis of vw, we have u,=Uyb,+U, 1, =AUn— aU; 5 proceeding thus, * Ug ZO? (Unn— Wy Ug), — Uyy=Unn + Wey + Ug 5 aa mn — 2 = — _ ie a | whence Ung © Uyy ZIVES tye Ost ua Pk + Uy. ed On the planes of w,& and wu, construct two curves, uP, v= Wnt then, if £ and y be taken at pleasure, and if P and Q be the points of | those curves, a plane drawn through P, Q, and the origin, has P§ + Wy for | u, where & and 7 are the coordinates used in finding P and Q. If o and y be discontinuous, the mode of performing this construction is as | easy as before, and the discontinuous surface thus produced is readily shown to be a solution of the equation. ™ The elementary notions of the transference of waves contained in the artic | Acoustics in the Penny Cyclopedia, may be of use to those students who are new" to this subject. ON DIFFERENTIAL EQUATIONS, 731 4 (96.) An equation being proposed which involves any number of diff. co., the solution may be generally expressed in powers of 2, or in powers of y: but in the former case the arbitrary functions can be most ‘conveniently determined by knowing values of 1, 22,52¢.., &¢c. when a= 0, im the latter by values of w, Uy, Uyy, &C. when y=0. Suppose, for ‘instance, u+u,=u,,: assume u=A+Br+4C2*+4 a Ke*+...., and the equation obviously requires that A, B, &c. should be functions of Y> and that A,,—A+B, B,=B-+C, C,,=C+E, &e. Hence, from A only, all the rest can be determined, and we have u=A+(A,,—A) & es (A... —2A,+A)at+...,.. The value of A is that of w when 7=0, and this solution has only one arbitrary function of y. Now assume u= A+ By+43C7y°+...., A, B, &c. being functions of z. We have then A+A,=C, B+B,=B, &c., so that u=A+By+3 (A+A,)y? +33(B+B,)y°+..-.. Here A and B are arbitrary, and are deter- mined by the values of w and wu, when y=0. There are then two arbitrary functions of x. Nor is the second solution more general than the first; for either may be reduced to the other, as in the example of § (85.). - It appears then that the number of arbitrary functions depends upon the manner in which they are to be determined. “In an ordinary diff. equ. D(2,y,y',....)=0, the constants can only be determined by giving values, expressed or implied, to y and its diff. co., for some specific value of x; and we learn that the number of constants depends on the egree of the equation. The general theory of partial diff. equ. would seem to point out that there must be as many arbitrary functions as there are units in the highest degree of the equation: but it must be ‘emembered that that general theory does not succeed in integrating any bquations except those in which either one variable is not used at all in lifferentiation, as in §(87.), or in which there is the same order of lifferentiation with respect to both variables. The preceding instances ead us to conclude that when arbitrary functions are to be determined ”y values of w and diff. co. with respect to any one variable, their num- ver will be determined by the order of the equation with respect to that eee. It would also appear that the arbitrary functions in such an ’quation must either enter with their derived functions ad infinitum, or mmder the symbol of definite integration ; certainly not in an ordinary Wgebraical form. Take the equation Uyy=U+ Ur, and, if possible, let w=f (x,y. ¥U) be the solution, f and U being determined finite alge- oraical forms, and vu arbitrary. The first side must contain w'U, and he second cannot; it is therefore absurd to suppose that u=f can make ty, and u+-u, identical. But this absurdity disappears if YU, ¥U, ec. enter f ad infinitum, or if %U appear under the sign of definite ategration, in which it may happen that U+U, can be reduced to dentity with Uy, by integration by parts, which may introduce y”U ato w+u,. | (97.) I now proceed to point out the manner in which Laplace, ourier, Poisson, Cauchy, &c. have exhibited the solution of some partial iff. equ. by means of definite integrals. First take the equation u,,= au, being a function of ¢and 2. Assume w=A+B2+43C2°+...., from hich we readily find, as in § (96.), B= afdt.A, C =a’ ( fdt)*A, =a®(fdt)? A, &c. Let A=yt+e, and let y, t, yt, &c. be the suc- essive integrals without any constants added. We have then 732 DIFFERENTIAL AND INTEGRAL CALCULUS. B C aa ct? gs ct al? Sayittcttas a =valt > tal +P, “nett o 3 a7 5 +Bit+y; aa a x U=YWt+yy,t.ar+ Yet 9 Si c+a.ar-+B $+.) naan ax? ice a, at (cx ax 4 a ett a5 -F 9.3 ahelek +> 9 +%5 3 3 616 nee Let Gv=c+a.ar+...., and we have a {? 2 12 uytt ys tear tyat Soe A bet Geet pats bene no constants being added in either integrations: in fact, supplying the i constants in either set of integrations would only alter the arbitrary” function with which the other set commences. For a similar reason we © may make each integration begin where we please. But from § (1.) we — have T (n+1)(fiday"Pa=2" fF gv dv—nz"" fe vb'v dv+... = fi(w—v)"¢'vdo, and similarly for y'é. Substitute, and we have t—vya’a® (t—v)> aa? u=foyv 1+(t—v) Hoge y Cente 2 2.3? + fio'v+(#—v) at+.-.-)5 the second series merely interchanging # and ¢. But (page 292) 1.3.5...(2n—) * Dt Oia alainad 2n OH 1 oon See EE Be ot 2h d 1, ae WE PAT saute ce i ce eins ae " win fe Be es! sin? 0.2 ne sin Oe a | heh ipa Torget figs s 2 2.3.4. °° ae To oh is =- ft (2 ain OA) 1 eg BERENS) a6. toe fe sin"0 do= wit f' ey (e2sineV 0) a 4 g—asineV Gy) 2) wv dv do +5 vig Fie od V@—v) eres VG@—2) a) g'v dv dé re 0oJ0 ° Similarly, for w.,.= — au, we have Sufi fi cos (2 sin 6 V(@i—v) az) y'v dv dO + {5 f# cos (2sin ov (a—v) at) p'v dv dé. ; * * Théorie de la Chaleur, pages 150,151. In the first page, three lines from the bottom, supply @ in two of the values of 6; andin page 151, at and after the second value of y, for sin®a read sin a. r Hr 2 ON DIFFERENTIAL EQUATIONS. 733 and x—k, we should” have had in the final result ib , and if iy With a(x—k) and a (¢—A) in place of ax and at. If a=0, the preceding is obviously of the form u=dr+we, (98.) Next assume u,=a (u,,+u,,). Assume u==Ce*t+%, which satisfies the equation if y=a@(a?+ f°). We have then for a general solution Pak 2 2 2y 9 wma Cert tan7t by +ap +, etitbaait -B, yrapyt fas the constants being any whatever. Hence we have a right to assume as a solution u=f {¢ (a, B) et e+ dadB, with any limits, and any function ¢; for every element of this integral satisfies the equation independently. But we have tie Crit Gps | woe sree Gye | 8, duea ries, for c? write at. 2 and at. 6%, using different variables, and we have u=ffo (a, 2) .€*7.E”, A Rete Di Sia an oN, Canes), Cae tek fre steS fb Ca, B) eeteve ogetieva 8 e--™ dy din da: df. Make the integrations first with respect to « and (3, and it is obvious shat the indeterminate character of @ gives simply %(«#+2v,/at, y+ 2w ,/at), whence we have as a solution u=ftcstsy{e+2v/.at, y+2wJ.athe”-” dv dw. The same method would obviously apply, whatever might be the number of variables on the second side of the given equation. The solu- tion is also complete, for the equation is only of the first order with respect to ¢, and % must here be determined by making f=0, and Assigning w. | (99.) We now consider the other form of solution, of w=au,., derived from the series as in § (85.). at a it 1, 2 Nn Pee ’ ‘ aes sean TP digg tipgy tices th Me Eyl orig S O82 "2 | . «s 1 ie Qn 1 2” an Batik Be Ta dnc Tees (aa) ' ace dp Yh te Qre~° ; ieee J _.C+oy—-D™! Path) @—D)G—).. tyr’ 9" e° +0 erv-1 ce r L.3. ‘ .2n—1 OI _o(etovJ—l) tt? Che first part of u is therefore (c+v./—1 being called 27‘) c +0 2. 2 1 2. 2 . l xz 3 po te { J (sero tots ‘a +e a) tes) s ea T ° ° ge +o yt eV 7 dv aye OO {-2 Panto fd) (ctv /—1)' 734 DIFFERENTIAL AND INTEGRAL CALCULUS. ex APA c= gy —hidy figs JO Sacer 0) EO since the second series is obtained by differentiating the first with respect to a, multiplying by a, and then changing $'¢ into Yt. And dt and wt are the values of w and vw, when z=0. And c must be some > quantity >0, it does not matter what. ) (100.) When the complete integral requires two arbitrary functions, | the preceding method will generally give them. Let w,,4Uyyt+Uz=9, | and assume uw—=Ce*tt*, We find, then, a? + 6*?+y=0; or if a=A\J—1, B=pJ/—1, we have y=,/(’+p*), in all of which the | square roots may have any signs. Hence the following is a solution: “is {(e** Vine ee vo}) gy Volt (gAy Nol aga ae Yeh) enya} eV AM) 2 multiplied by any constant; and the same if —,/(?+ °) be used, | Hence, as in the last article, we have for uv : a SSCP Qs p).cos Aw. cos py EVV * dd dus + ff Qs, p).cos Ax. cos py.e-VOAFOO™ dd du: any limits being taken at either integration; for every element of either. integral satisfies the equations. Any transformation that we may make cannot prevent the whole integral from satisfying the equation, though - it may not yield separate elements which also satisfy it. Thus, let rsinO=A, rcosk=p, and proceed by page 395, which gives for w ¢ (7 sin 6, rcos 6) being really of the same effect as (7, 8), 4 ff9o (7, 9) cos (r sin 0.2) .cos (7 cos 0.y) &* rdr dO | +f fy (r, 6) cos (r sin @.2) cos (r cos 0.y) €" rdr dO ; 4 any limits being taken for r and 6, and not necessarily the same in both terms. : (101.) Let «=a? (u,,.—ma—u). Assume u=Pe?“, P being a function of x only. We have then Pe=P”—mar’*P. Letn(n-1)=m,) giving two values for n, except only when 4m=—1. We have then § (51.), P=2"Q, where Q"’+2nz—' Q’/—a Q=0, the complete integral of which is by § (50.). ) ak) OH Bak Haired! Gieccao pears 407 grant (+t evemr(] —o°)—" do. ae | Let p and q be the two values of m, then using p for m in the last, | —2p+1=—p+q, since p+q=l. Make v=cos w, then, remember- ing that the arbitrary constants may have any sign, we have | P=C, a? frerver” sin”? w dw +C, x {pervs sin™—y dw. : For ¢«”“ write 1? if +8 e P+ 2avv/-of dy, and Pe*” can then be expressed.’ | Let C\=da da, Coa=Wada, then, since any number of such terms may | be in the solution, we have Jarcuzca®f {5 frie a Vee. cos w+2avr/t)rfa da sin? .da dw dv+a'f, &e.5 " | the second term being a similar function of gand w%. Integrate first, | ON DIFFERENTIAL EQUATIONS. 735 with respect to a, which merely introduces an arbitrary function of x cos w+ 2av,/t, so that we have bx? |= “$9 (x cos w+ 2av Jt). sin®?-'20.e—” dw dv+a' fs, «ec. This solution can generally only be depended upon when p and q are positive, or 2p—] and 2q¢—-1 each >—1, for reasons shown in §(50.). If, however, p and q have the form AL pf —1, it will appear on substitution that X must be positive. The form sint“V—4w is but a transformation of cos (ulog sin w) +,/—1. sin (ulog sin w), which is never infinite. When 4m=—1, and p and q are each equal to 4, proceed as in §(50.): make g=4+6, reason in the manner cited, and it will be found that the result is u= Jr. {5 frip.e dw dv+,Jrfz ftsy.e log (x sin? w) dw dw, n which the subjects of @ and ¥% are omitted for abbreviation. (102.) The introduction of the arbitrary function in §(98.) and §(101.) lepends upon the following assertion : any function whatever of x can € represented by As”+ Be“+...., if the constants may be any what- soever, even infinitely small or great. In converting this series into (dv. dv, and making the result represent any function of a, we should ein fact making the mistake alluded to in pages 671 and 673, if we vere not to remember that >Ae** might be here written for fou." dv. f we would have the preceding give us 2”, for instance, we only give the imit of (e°"—1)": y”", when #=0. Knowing that whatever is true up 0 the limit is true at the limit, we may therefore write 2”, because we aay write (e""—1)™:u™ for any value* of «, however small. Also in egard to the results of the preceding articles generally, the student is eferred to the higher class of works on physics for the manner in which ney are used. What is now evident is, that whereas no general case that is, with any given values of the arbitrary functions) was actually ttamable before these transformations, any such case is now calculable ‘om the definite integrals: and that which is prolix is substituted for lat which was unattainable. | (103.) Whenever one of the variables x or y is missing from” the oefficients, and the equation is homogeneous with respect to u, u,, and y» the equation may be reduced to depend on a common diff. equ. ‘or example, let w+ Pu?—=Qu’, P and Q being functions of 7. Assume Sze", z being a function of x, which gives 2/?+ Pa? 22=Qz?, or z—Cef(Q—Pa?)tdr y=Cef(Q—Pa')tdatay, This is a primary solution, since there are the constants C and g; thence the general solution can be found, as in § (75.). As another xample, take witu;—u. (A, a~@*), and one case of 2v, being A™'v,, or vz ,+vpot+..-. ad wnf., We may throw the preceding solution into the form A,_,+A,..a+A,_,.@ ++...-, which, moreover, obviously satisfies the condition. Apply the calculus of operation, and this becomes the operation (1-+-A)™ (l+A)*.a+...., or (1 +A—a)7', or (1—a)— (l—a) "7? A4-... performed on A,, which gives for the complete solution Cat 4 A, AA, age Be 4 u,= a a se a aa aera. at ee (ee eS ere eeee I—a_ (1—a)? (l—a)® (l1-a)* ; which takes a finite form when A, is a rational and integral function. Thirdly ; proceeding by the formula in p. 311, § 174, we obtain for ithe complete solution (A’,, A”,, &c, being diff. co. with respect to x) 1 b, Wake bs All 1 Bae A AE za aii M23 (as, dea 2 danse iwhere b,.=1+ a, b,=1+4a+a2, hb =1+llat+lla+a, 16,=1+26 a+ 66 a?+ 26a? +a‘, bb=1457a+ 302 a? + 3020245 7atLas. If a be negative in the above example, say a= —c, the only circum- ) stance which requires consideration is the change of Ca* into C (—c)?, or C (—1)*.c*. Here C (—1)” implies a function which changes sign only and not value, when x becomes #-+1, and its plainest real form is cos rz. f (cos 272), where f (cos x):is truly periodie. (108.) If we take v,,.,—av,=2", we find for the solution =—C x 2 i Bic g aa ase): u,——Ca* +a Faas os Soa ORD Gy Ge : which is intelligible only when z is integer, unless it be thrown into the ‘form of a definite integral, (the only finite form known for it,) in which pease it becomes generally intelligible. If a>1, the following is the form : | ‘ ee | m=Car par | eaves dv. 2 v—-l Having, in Chapter XX., fully considered the method of transforming ‘finite series into definite integrals, and of making the definite integrals so found apply to cases in which the finite series become incon- ceivable, from the letter which expresses a number of terms becoming fractional, we have nothing to do in this chapter except to consider the ‘method of finding solutions to equations of differences in the manner preceding, namely, in the form of a finite series for integer values of a. The subsequent attainment of a definite integral by means of this series isa subject apart. Laplace has shown how, in a few instances, to pass from the equation at once to the definite integral, but the cases in which the application is practicable are mostly those in which it could be dis- pensed with. And, moreover, it does not apply to equations which have t term independent of w,. "742 DIFFERENTIAL AND INTEGRAL CALCULUS, ' (109.) The general reduction of a continued fraction to another form Sau upon equations of differences. If N,=a,: (6,+ (Gri: Oey +....))), we obviously have N,=a,:(6,+Nz241), or N. Niiit, N, =a,. This equation may he reduced to a linear form by assuming N= Uri! Uz Which gives U,i9+6, Up41:=4,Uz But even if this equa- tion could be integrated, two periodic functions would enter into the complete integral, “and it would not be easy either to distinguish the cases In which these functions are only simple constants from the rest, or to choose the proper periodic function in those cases which require if, In fact, a continued fraction ranks with a divergent series whenever a: Dzy Ar41: Ors, &C. are or permanently become severally greater than” unity: so that the continued fraction can only be known from its — inveloped: form. ‘To show the difficulty more closely, take the inverse method derived from the above, and assume 6,=1. We have then: = (Nant), or N,(Nanit) Nev CNeist 1) Nits (Neis+1) aoe i- 1 4Ge. ee a(a@+2) +1) @4+3) (442) @t4) N,2=2 gives g=———= + -—____—., {+ 1+ 1+&c. We might now, perhaps, be inclined to say, that if this divergent ~ development represent any thing, it is @: but, if we take it as an object © of inquiry, we find that the continued fraction last written might be — derived equally from N,, any solution of N, N,ij+N,=2 (a+ 9). li the fraction were convergent, we might decide by the common approxi- 4 mative process, in particular cases, “whether it is or is not equal to a: but as this cannot be done, and as in common algebra a divergent series produced from a function of ambiguous value can frequently be shown ~ to be an analytical representation ‘of any one of the values, I chinks it would not be safe to say anything else of a divergent continued fraction.” (110.) The general equation of the second order u,,.+P,Uz41+Q, Use -+-Z,=0 can be solved as soon asa particular solution of v,494+P, Upp +Q,u,=0 is found: that is, it can then be reduced to the solution of a general equation of the first order. y Let u,=®, be such a particular solution, and let v,=a@, v, be they solution of the complete equation. We have then Dato (Vze+2Av, + A? v,) +P, Br 41 (%+ Av,) +Q, @,0,4+Z,=0; | which, since @, 42+ Px 41+ Q,@,=0, gives (Av, being called z,) 3 Bap2 A? V+ (20.42+P, @24,) Av, +Z,=0, Re ro] Beis esit Ceiig hes Sec le, cel eos from which, z, being found, w,=@, 22, Two constants enter, one in 2,, and one in the summation. If Z,=0, we find for the general solu- tion of Upset P, U.4,+Q, u,==0, s @, ) EER S)... (14 PsZ=) 40 jem Oo | ON DIFFERENTIAL EQUATIONS. which may be reduced to u,—Co, {a 4 MO Qs, 5 A 4% Q ne el 1, TO x5 (6 2 Ws We Wa Gd ey 2 where C and (, are functions of cos 2rz. In this manner Unga 2Us41 +u,=0, which is satisfied by ,==C, is also found to have for its com- plete solution u,= Ciz--C. (111.) The general linear equation of the nth degree, Ursin t P, Ugin=1 +Q, Untn—2 a wae e's + Z,=0 era gets Ear if completely integrated when n arbitrary constants (or functions of cos 27x) enter the solution, is integrable when n distinct particular solu- tions can be found, satisfying the equation deprived of the term Z,. Let those particular solutions be u,=@,, U,=k,,++..U,==,: then the equation, deprived of its last term, will be completely satisfied by U,=Aw,+ Br,+ Co, +....+My,. To pass to the integral of (1), assume instead of A, B, C, &c. functions . of cos 2rz, A,, B,, C,, &c. any requisite functions of wx, which bemg n in number, we have a right to choose n—1 assumptions. Let 8 be the symbol of summation with reference to the various solutions, so that w,=S(A,@,) or S.A,a, Then w,,,=8.A, Bri, + S.AA, O43 assume S.@,4,AA,=0. Again, ti. =8.A, 0,42 +S. AA, Wi; p assume S.@,,.A4A,=0. Proceed in this way until we come to v,,,-) =5S.A,,in-1, by which time we shall have made n—1 assumptions, namely SA tO sGince A Ole TOs) AAG 0. | Finally, w,,,=S.A,@,4n+S.@24,4A,, and PS LBS Uyin—i = WON + ad S = A, (Wrint BR, Dy in—t + 91.678 )58 ) +8.a,,, 4A,+Zs3 of which S.A, (@,,n.+.-.-) vanishes in every term contained under §S, | because w,, &c. are particular solutions of the equation deprived of its last term. We have only then to add to the »—1 assumptions the equation S.a@,,, AA,+Z,=0, and thus we have m linear equations to determine AA,, AB,, &c. from: after which A,, B,, &c. must be deter- mined by integration or summation, each having an arbitrary constant, or function of cos 272. For example, w,49-+ P, siitQ,:V2=09, being satisfied by v,=o, and U,=K,, has u,=Aw,+Bx, for its general solution. Assume u,=A,@, +B, «, for the solution of u,49+ P, Urii+ Q: Ue+ Z,=0, and we have Ue A, Mr41 +B, koi if we assume @,,, AA,+«,4, AB,=0 Uz9=AyDri2+ Bs kep2+ Pr42 AArt Kets AB, Usiet Pr Urrit+ Qs Us +Z,= As (rp: +P, F241 +Q Fe) +B, (Kepe+ Pe Kui t+Q, ke) FOr42 AA, +k424B,+Z, ; whence the complete equation is satisfied, since ©,,2.+P, ®,41+Q,0, =0, Kya e+ P, Krai t Q, K,=20, by DIFFERENTIAL AND INTEGRAL CALCULUS. Wri AA, iW Ky+9 AB, + Vd rane 0, if Watt AAl-FK41 ab = 0 5) | " Ky+} Ly Way ZL. | or by AA, = = —_—_—, AB.= — —__—_——_ ; | Wr41 Kepa— Krq1 Wate Wi. Kep9—Kr41 Pate | K 4h Ze TD +1 7, | whence v,=0, [{—-———_——__ --k,, 2 Pulin gg : Wet Ky42— Ry+y Drie Wryiy Kyte Ky41 Werig (112.) If it should happen that two or more of the solutions become : the same, in any particular case, a process resembling that of § (21.) | must be employed. Suppose, for example, that wein+ Pi Urin-rtesee | +Y,u,=0 has for its general solution u,=Aw,+Br,+Co,+.... | Suppose, moreover, that @, contains the given constant a, that «, con- | tains b, p, contains ¢, &c., and first, when d=a, let o,=x,, so that in | that case the compound solution Aw,+Bx, is only (A+B) @,, and | contains one arbitrary constant only. Let b=a+h, and let accentua- | tion refer to differentiation with respect to a; we have then | Aw, +Be,= (A+B) o,+Bhol,+iBl.o",4.... | As h diminishes without lmit, let B increase without limit, so that | Bh=B,, and at the same time suppose A to increase without limit with | a contrary sign to B, in such manner that A+B=A,. The next term, | or 4B, ho", diminishes without hmit, and still more those which | follow; so that A,o,+B,0’, is the part of the complete solution which | must be substituted for Aw,+Br, when h=0, or a=d. Again, if — a=c makes 0,=@,, we have, making c=a+Ah, a Ay @,+B, w’,+Co,= (A, +C) @,+(B,+C) ko’a+iCha",4+. eee 3 : | Ke | ; 9 , sy La | whence it may be shown in the same manner that A,w,+B,0',+C,@, | is the part of the general solution which must take the place of Aw,+Br,+Cp, when b=c=a: and so on. a y q (113.) The theory of the linear equation w,4,-+Pusinakeeee -Yu,=0, where P, Q, &c. are constants, closely resembles that of differential equations of the same kind. Assume w,=c”*, and let Dy x, 9, &c. be the roots (supposed unequal) of c®+Pc™*+....+Y=0. | Then the general solution is vu, =A@*+Bk’+Cp’+.... Ifany num- ber of roots, say four, are equal, @ being one of them, the part of the general solution corresponding is, by the last article, a “sa a: Awt+A, am*'-+ A,r (a—1) 0° +A, e (t—1) (1—2) 0”; | which is of the same form as ©” (A+ A, 7+A,a°+ A; 2). b _ The general solution of u,4.-+Pu,i.+Qu,+Z,=0, P and Q being | constant, and @ and x the roots of c?-+ Pc-+Q=0, is e. Aa Te” Le K™ ie. uJ U,—= Aw* + Br* + Dire ae fern se Tae Pens K—-D @ Kt - K | except when @=«, in which case it is be Lye Zr uu," (A+Br)+0* 2 ees — To" > ate [ ON DIFFERENTIAL EQUATIONS. 745 (114.) The following mode is in theory applicable to equations of any order. Let us take one of the third order, u,43+ P, U.i9+ Q, ay +R,u,+Z,=0. Assume v,4,4+),%U.+9.=0, and let «, and B, be two undetermined functions of x We have then Urist Prte Urp2t Vere t be (lst Doar Urtit Qe41) +), aris + Px U,+ CP) = 0, which becomes the given equation if Peiata.—P,, > %: Pri +8. Q,, Bs pr2—R,, Gaye %2' Orb Peder Ly 5 or @,=P,—Ppris, Bs=Q:—Ps Dorr tpt Pras R,=Q, pe— Ps Ps Poti t Pe Poti Pr+2- Though this last equation be not of the first degree, it is of an order inferior by a unit to the given equation; and if only a particular solu- tion of it can be found, the value of p, thus obtained will produce corresponding values for a, and 6, with which the complete value of q, imust be found from the equation for Z,, containing two constants. Then the equation u,,,++p,v,+q- must be integrated, from which w, may be found with three arbitrary constants. If we apply this method to an equation of the second degree, Uriet P, U,4,+Q, u.+ Z,=0, we find Ursgt Dati Uerit Uetit Oe (Usp + Ps Uet 92) =0, Di cyerity Ds ay Dk), Frit b, fem a,= Pi—Prr Ge?) PiPx Pa+i From this it appears that when P, is =0 the equation is always theoreti- ‘cally integrable, since logp,=¢, enables us to determine ¢, from fii +t,—log (—Q,). (115.) The equation u,4, + pr Petron + Dz Do Qe tray haere EP, De-\. ++ + Deni Ys Uy 18 reduced by the assumption U,= Do Py Po. se i. ?, tO Vint P, Vein. +) + Ys Us—0. (116.) I shall enter no further into the subject of simultaneous ‘equations of differences than to show how to integrate the pair Ay Usp + Bi Veit Au,+Bv,=®, ay Urait b, Ohare au,+ bv, == Pry 6, and ¢, being functions of «, and A,, a, &c. being constant. Multiply the second by a constant 0, and add it to the first, which gives B +5..0 1 B+ 50 = ut SBA ve 6) 4u,+—. v, (= ®, Hat a A Pa, 0 Vsti | +(A-+-a ) Ut aap v +o Assume 9, so that (B,+0,0):(Ai+a@,9)= (B+50):(A+a0)=p. This gives two values for @ (0, and 9,) and two values for ps (4, and pr), [f v,+p,v,—=w’,, and Ut fla V2 Wy WE have (A\+ 4 Q,) W iit (A+ a6,) w’ ,=O,+¢, 9, (A, +9, @) wit (A+ 492) w=, + d, 42; (A, +, 9) on 5 746 DIFFERENTIAL AND INTEGRAL CALCULUS. found from those which precede. Or as follows. From the two equations given, and the two which are found by changing z into a+ 1, eliminate »v,, v,,, and v.42: the result is a linear equation of the second degree between 2,42, U,41, and wp. This is a method which will apply to linear equations of any order, and _ any number of variables, on considerations similar to those in § (15). and when w’, and w”, are found from these equations, wu, and v, can be | (117.) The solution of linear equations with constant coefficients — : may be effected even when there are more variables than one, by means of the theory of generating functions of which the first principles are — | explained in page 337. Let the equation first be of one variable, ay Unan th bn—) Usin—\ ee oe +a, Ura t QL PT iiond § is | For the complete solution of this, we must have either the set of - Values, Uo, Uy) Ua) + » + Uy—1, or the means of determining them. Let be | such a function that wu, is the coefficient of f, in it; or let baat +u,l+....: that is, let dt be the generating function of w, for al a positive values of . Then the first side of the preceding equation hap) for its generating function ay, Qn—1 ; m | ra +: rr coe he +4, ) Pt, | which function accordingly is, as far as positive powers of ¢ are con cerned, identical with 0+0.f+0.#+....or0. But, from the form of pt, it is obvious that negative powers of ¢ up to -” may enter the above product. Assume then : A,tA,it+A,s@+....+Ae ; 3 OH Oni tt rnelt.... + 4,0 +a it ae ean: | let A,...A, be so determined as to make the first » terms of this development become w)+2, t-tu,@+.... -+u,—t"-', and the rest of | the development will then give wv, é"+un,.,U"'-+.... of itself, If any_ of the various modes in Chapter XX. of expressing the coefficient of # | in the development by a definite integral be adopted, there will result a solution of the equation. But, as far as we have yet gone, the metho will be more powerful in making the solution of a linear equation giv the general term of a development than in making the latter give the | former. e; For example, required the development of 1—2t~—2¢#2, divided by | 1+t¢+?--@. First, find the solution of 1,43-+ Ur p97 Urs, + U,=0, for which we must have the roots of c?+c?+-c+1, which are —1, v¥—-h and —,/—1. Hence the solution required is tz =A (—1)°+B(/—1)?+C (—/—1)*. ‘ which gives df= Now the first three terms of the development are 1—3¢-+-0f, or m=, U=—3, uUe=0. Hence we haye the equations A-+B+C=],_ —A+B/—1—C/—1=—3, A—B—C=0, from which Bh ON DIFFERENTIAL EQUATIONS. a l —(/— “I ce ~I =5(—1)'4-7 28 Ni, —1)’ 1 ] cao OHO hele = ee sin =) from which we find for the coefficients the cycle 1, —3, 0, 2, 1,—3, i 2, &c. In this way an expression may always be found for the general term of any algebraical development. (118.) Let w,,, be a function of x and y, and let any equation of the form au,,,=0 be proposed; for instance, such as MUy, yr btn 41, yt Cuz, yti7h CUz+29, a + 2s 8.8) .8 a0. eee 2B. “Assume w,, ,=A*B’, whence it appears that any values of A and B give a solution, which are connected by the equation a+bA+cB+eA?+...,=0......(2). Say this gives B=@A, consequently 2A” (pA)” is a solution, k being sconstant. This, as in §(98.), we may make equivalent to fe (pA)* WA dA, for any limiting values of A. Or, if the equation (2) give n values of B in terms of ‘A, namely, 9, A, d, A, &c., we have for a solu- ition Uy, y= fA? (Pi A)’, A dA+ fA” (Gb, A)’ WA dA+...4, jcontaining arbitrary functions. Analogy might lead us to suspect that we have here the most general solution, even though finding B in terms of A might give a solution with a different number of arbitrary functions, since the same sort of thing occurs in partial diff. equ., §(96.) But such a conclusion would be unsafe, for we have no infor- mation on the genesis of partial equations of finite differences which warrants it. Suppose t,, y=AUbr4y, yt OUs, ypr tCUr+, y41, Which gives l=aA-+bB +cAB, or | aA \! geod Ms\ (ae WA dA. If b and c be both finite, this may be brought into either of the forms Y, fA°pwAdA+Y, fA pAdA+. or mie fA’ wAdA+Y, peer WwAdA+... where Y,, &c., are functions of y (not the same in both expressions). Now, attending to the remark in § (102.), it is seen that fA°wAdsA is merely an “arbitrary function of 7. soc that) ¥, px a ip pb («#+£1) mY, (ct+2)+... results. If} or c vanish, the series may be made finite, and the form may easily be altered into X, oy +X} p(iyt Babe Which may be made finite if a or c vanish. Again, Uz, yo AUz +1, yt Uz, yt. may QIVe Uy, eave PAS (l—aA)’ WA dA. Assume wA=k A*+t/AA+mA” WERT vhence 748 DIFFERENTIAL AND INTEGRAL CALCULUS. a-vbu, wekparte(t—a yaa pep act(t—a )aa4 sma a) (aA) aha wal (a@+e+1) PF Gy+1) from A=0 (page 679) =ha*-— Sree ta vod to A= aul ka-*- TD (ew tet 1) (y+1) ee P(atay ded 2). 4k | i eee De | P(r+y+A+2) ie | in which ka~*~', la~*—*, &c. are merely arbitrary constants. | | OD) GO ie (119.) Such equations as the preceding occur in the theory of pro- babilities, and Laplace treated them by the method of generating | functions, as follows. Let the most general solution of the equation (1) be adopted in the particular values woo, to,15 41,0, &c., and let p (¢, 0) be the function which can be developed into | Uo, oA U1, of Uo, 1 VA Ue, oO 4,1 V+ Uo, 2 UP + Reduce the equation (1) to the form au,, yt buy_1, y+ Cuz, yi TEUz gy a } PEG at OF e A ° (3), al A which can always be done: thus u,, y—bt,41, y—CUz, yi 0 Is trans- formed as follows. Let w,,,=U_,,.,; substitute and change the sign : of x and y, and we have U,,,—0U,_,, y—cU,, »1.=0. When U. sie found, u,,, is therefore found. F requently the change is more simply made ; thus w,, y+Uzi1, y41==0 is, writing x—1 for x, and y—l1 for gy reduced to uz, y+%.—-1,y-1.=0. Let¢ (, 2 be the generating function of, Uy, y above written, OF U,o+U,o¢+.... 3 then the generating function | of the first side of (3) 1s (a+ b¢+cv+e2+ oi (tv), which must be a function of ¢ and v, to be dulce ined by such conditions as the problem requires, and must give O for every term P,,,/*v*, whichis | such that w,, , can be the first term of (3). Subject to this condition we iy, | must have y | el ge Ee a v) | Lp) M 2 EDS atbt-+cutef+.... | For example, let w,, ,—bu,-1, y—CUz, y1==0, which gives @ (t, ”) = W(t, v): (1—bi—cv). Now the terms between which this equation | cannot establish relations, if only positive values of x and y be contem=_ plated, are all the cases of ™,, and w,,9. Let it be required that Uy¢ | shall be €,, and that a,, shall be n,, it being understood that &=m. This is not assigning too much, for it gives u,,,=by,+cé,, Us, = ba +o, cen a= by, + CUy,1, &C., not more than enough to proceed with It is then requir ed pack @ v): Cee —bt-cv) shall be (Ey or 9) +2, E+ bob +... tm vt+nove+....+ terms in which ¢and v both occur ; which condition being fulfilled as to the simple powers of ¢ and 2, the deve- lopment will in other terms generate the coefficients required by the: equation. s For instance, let w,, >=0, %, ,=1, Gf y>0); we then require that w (t,v):(l—bt-— cv) =v 4 v0? 440° + ....=0: (1—2) ON DIFFERENTIAL EQUATIONS. 749 hall be true without interfering with terms containing powers of 1, vhich gives simply v (l—cv) l—v ’ v bt bt? li aot 1+ oh “pee of | . I (l—cv) (1l—cv)? - v(1—cev) Nia g Wet Wg le AI, H(t, 0) = nd the coefficient of ¢*v’ is that of v’ in b*v(1—v)7' (1—cv)~. Now it is easily found that the coefficient of v’ in b* (utve+ev?+... )(I+2.cof2 BHT toed Ary s eas * ea 2 2 vhich is, therefore, w,,, required. It is not.easy to see that it satisfies 'z, o== 0, which is a case resembling in difficulty that of (1), when I'a s known only from 1.2.3....(@—1). If it be required that wo, , and w,,) be any given functions of y and a, ind T, and V, the generating functions of w,, > and 2, ,, or let 3 =U |1+acte 2 ayite 2 T= Uo thot +Uso bea 3 Vo Uo, oF Mor V+%M,00 Fee ’ . (l—bdt) T,4+ —ev) V,—{T, or V he generating function of w,, , 18 ee (120.) When we make the solution take such a form as that given hove, a change of sign in # and y produces an unintelligible result, so hat we cannot immediately pass to the solution of w,, y—Or41,y— Ce, y41 =0. In fact, an equation of this kind, in which there is not a highest ‘erm with respect to both « and y, presents difficulties. The application of the method of generating functions is complicated, nd it is best to have recourse to that of definite integrals, as in § 118. (121.) As another instance of this method, let us take Ur, y— OUiz9, ym CUlr, yng — Ua, y= 0. n order to solve this completely, we must know o,y5 U1, 45 Uz,05 and w,,4. set the generating functions of these be Yu, Yv, gt, and gt. The enerating function of w,, ,, or @ (é, v), is of the form «: (1- bl?—cv?—elv), nd having four conditions to satisfy in @, let us assume a=P,4+Q,+R,v+5, ¢. The values of w,,) and wo, , require that #(é,0) and 9 (0, v) should be ot and wv, whence we have * The student who kuows a little of the theory of probabilities will see that this $a solution of the following question. B and C want severally xz and y points of he game, their chances of making a point at each trial are 6 and ¢ (6-+-c=1), equired the chance which B has of winning. This chance is w,,y, as found above. DIFFERENTIAL AND INTEGRAL CALCULUS. P,+Q,+8,¢=(1—62) Ot and P)+Q,+R, v= U—cr*) wo, | or a==(1—bé?) d+ (1—co®) Yo+R,v +S, t—P,—Q,—R, v— Sot. | Again, the value of dp (t;v): dt; when é=0, is wv; and that of dp (t, v) : dv when v=0 is ¢,¢. These give (since P)+Q,=¢0=¥0) S,= (1 —cv’) y, v—ev Wo — Rv + S,—f/0 R,=(1—b¢) gt — et dt —S, t+R,—w’0. Whence the form of « is found: R+S% is ¥/, O—ew0, or 4/0 —ef0, which are the same, and we have : a= (1—be) (bt+ 9, t.v) +(l—ev’) Cho +, v.t) —evt (pt+ yr) | —vt (Ry +8',) -9/0.t—-W/0 .v—g0. | For example, let d==1, c=], e=2, and let wi y=1, u, >=1, 1 = : Us peels tt 1 1, Yo, a= Vpand ta all Gther Cases let t,o) Wo,ys Wi, 4) tate | vanish. We have then Yv=l+v+v%, ¥v=1l4y, dt=14t+P,G,t=144, fi, S/d | The generating function «: {1—(é+v)*} can then be reduced to i | | i+t t+) a t papinae oR Unis nia nist ‘ Expanding the last term, which gives 4v° (¢+-v)™ for a general term, It is obvious that é # never occurs except in the term 4v” P (to) which has no existence unless «+y be even. Consequently, the solu= | tion of U,, y=Uz-9, yb 2Uy, ya + Ue, y-a 18 | 47° t? i | | i a a | (@+y—4) (@ty—5)...-@—) - | U,, yA i — aE (w+y even), 4 t,, y=0 (a ty of) 5 ‘a provided that w,.,(a+ty= or <2) =1, uv, 5=0, m,=0 Cn othem | cases). . (122.) The verification of such a result as the preceding may be rey by actual solution ; that is, by forming a table of double entry for — Ly, yy Putting the given values in their proper places, and calculating the — es from them by the equation. This is done to some extent in following table :— hae at phage Be Aol § 0 0 0 0 0 Bs OF ane 0 0 0 0 0 Ty O'S A db 0 4 0 4 aN ig Ue Gee Ws 0 24 0 ON OS 4 ae ear. 0 60 Oe a doa Oh Ad ae G emotes! O 224 0 0. % Ow) 4a 0 deh 0 280 0 840 0).« 0).-0 2d 0 224 0 1008 0 OG HE See BS 0 840 0 3696 ON DIFFERENTIAL EQUATIONS. “I cr — Specimens of the mode of forming the tetms from the equation are 280=60+2x80+60, 224-8042 60424. It may also be observed that Uo,0> U1, Uo,, are useless in the forma- ion of the remaining terms, as might have been made to appear from he function x. (123.) The principles of the calculus of operations* have lately been nade to throw a very instructive light upon the connexion of linear perations with those of common algebra. The following theorems are he connecting steps. Let D be the symbol of differentiation with €spect to x; so that Dér=@'r. Let @ be a constant, and deduce the heorem D (<* dr)=c" (Ddx+agze), or (D+a)¢x. Repeat this m times, which gives D™ (e* hx) =e (D+ a)” $2, where (D+ O= f0dz=C}, Bre) "Ue (fary*. 0.” (0,40, e+. os. 40, | anh. et D, and D, be the symbols of differentiation with respect to 2 and y, ré have then (D, +4)" (D, +0)" ¢ (@, y=" De Ds (4 $ (@,y)). By similar reasoning 4 (a* ¢x)= ad (@+1)-a’ 6r=a"{a+aAh-1} 2, rif the operation 1-++A be called E, we have (aE—1)"¢r2=a~ A" (a* Gr), (E—a)" br=a™*" A” (a~ G2). imilarly, if E, denote the operation of changing # into «+1, and E, nat of changing y into y-+-1, we have (E,— a)" (By—))" > (a; y) =a" bot" BP AS (a? 6 (2, y)). hese may be extended to the cases of negative integer values of m and » Thus (E—a)7'.0=a* A*0=Ca*", or Ca*, which is the same in am, © being arbitrary. This function Ca” is the quantity which /anishes, or becomes 0, when the operation E—a is performed upon it; m (KE — a).Ca* = CKa’—Ca .a* = Ca**!— Ca.a’=0. Similarly, E—a)~” .0=a*” A-"0: now A~”0, the function whose mth differ- A nce vanishes, is C,+C,¢+....-4+C,-,0"". (124.) It is shown (see the references in the note below) that all the perations of algebra may be applied to the symbols of operation used * See pp. 163—168; Penny Cyclopedia, Ny Operation ” and s Be peed) ambridge Mathematical Journal, vol. i., pp. 22, o4, 123, 173, 212, 278, 280 ; Nitto, ditto, vol. ii., pp. 74, 144; Gregory’s Examples of the Differential Calculus. have been indebted to most of the places cited. 752 DIFFERENTIAL AND INTEGRAL CALCULUS. in the last article, as long as they are not mixed up with any operations — depending on the variables employed. And the theorems may be > generalized into WD. (e* px) =e". & (D+a). Or, w% (D., Dy). (e“*™” o (a, y)) =e" *".& (D, +a, D, +). 9 (2, y) wWA.(a* dx) =a". (aE—1). x, - us (A,, A,). (a bY p (a, y)) =a’ bY. (@E,—1, bE, —1).(2, y). These properties are particular cases of a more general set, which owe their simplicity, in the case of «”, to the identity of the operations of | differentiation and multiplication by a constant. Let there be any number of functions of x, Vy, Vs, &c., and let D be the general symbol of differentiation, while D, is that symbol for V, only, D, for Vo, and 80 on; so that D,V,=DV, or Vj, D.Vi=0, Dy 0. D.V.=DV.=V,,, | D,V.=0, D;V.2=0, and so on. We have then (D,(V;, Vs) being V,D,V,, &c.) D (V,V.V,....)=(D, FD, +D,+....).(ViViVs WD .(V,Vs... -)=¥ (D, + De+..+-).(ViVes-.- oe If DV,=aV,, we have ¥D (V,V,)=¥ (a+D.,) (V, V2), or Vi (oe D Og since % (a@+D,), so to speak, only acts upon Vo: inw (a+D,) Ve is simply %(a+D) V;, since the distinction is now useless. Again, 1 A,, Az, &c. refer severally to V,, Vz, &c., we have - A (VEE OSfB Blea ao) WA (V,V3.... = {EE (125.) Let a linear diff. equ. of one variable be given, namely a, D’ y+a,_, D*? y+ oe Pe ne +a, Dy+aq,y=V; V being a function of 2 The operation performed upon y on the left | is a, D’-+-a,_; D"*+-...., which may be reduced to the form a, (D—a@) | (D—/3)...., where a, 6, &c. are the roots of the algebraical equation — Ay Oo + Oy} vo to =(). If these roots be all unequal, then, making | A=(a—P) (a—y)...., B'=(—«a) (0@—y)...., &., we have NEI i dalddiobae bas WA — peat =I ak —) De eA ot B Ooty § (123)=-Ac™ fe Vdz-+Be™ fe Vde+. 0. +A, e+ 3B, & ee A,, B,, &c. being arbitrary constants. The effect of the inverse process on V may be best represented by remembering that V+0 may be written for V, and the process performed on V and on 0 separately. The latter gives all that arises in integration from the introduction of arbitrary constants, and must never be neglected, Sometimes it may be desirable | to take one mode of operation for V, and another for 0. For instance, | let V be a rational function of w of the kth degree: let (a9 +a,D+..++ | +a, D,)~’ be expanded into b,+0, D+...., then we have be y= {bo V+ V' +... ros ony e+ Bye cons Ges - ON DIFFERENTIAL EQUATIONS. 753 since V°*?, VC, &c. vanish. If there should be Z roots equal to a, we find among the fractions into which (D,+....)7} is decomposed, the following set : L, (D—2)+L, (D—2)" +... 4, (D—a) 7. [Now (D—a)? V=e* (fdr). Vte* (Ct. 2+...+C,,0"); ‘whence we find, for the part of the value of @,y depending on these J equal roots, e* {L, (fdx)'.e-"* V+L, (fda rie V+...+L,, fdr." V} spe {Co 4+-C, 24+-C,2?+....4+C,_, ie oN ; I am here only giving a sketch of a method; but abundant examples will be found in the citations* above made. (126.) Let aD,u+6D,u=V, a function of x and y. We have then —1 b b uz (aD,-+bD,)" V=a"! (0.47 D, v= ce! fea Vee. Now if V=¢ (2, y), &?” V is 6 (2, y-+mzr): hence we are directed, p:a@ being m, to find [¢(x,y+mer) dx by the symbol fers Vda; nfter which, by the symbol «~”*’», we are further directed to write v—me for y in the result. But, writing V+0 for V, we have dy for ihe integral, @ being arbitrary, and a~'¢ (y—mz) for the result: hence watery fe™?u Vdx+a > (y—ma). For example, let aD, u+0D, u=122°y. Integrate 122° (y+mz) with respect to x, and we have 4x°y+3mz2x*; put back y—mz for 2, and we lave 42° y—mr* 5 whence ) 4x°y ae ba* = =, + (ay —b2) ; hince a7! (y—mzr), being arbitrary, is @(ay—bx). This use of a symbol of cperation, D,, as a constant with reference to another symbol nf operation, D,, isone of the severest trials to which the calculus of )perations can be put, though following readily from the first principles of the science.t fi21,) Let a, Diu-+a,.D7"D,ut....+aDjiu=V. Ifa, £, ve. be the roots of @, v"+a,_,v0" '+....==0, and A, B, &c. be as in } (125.), we have a, D2+....=a,(D,—2D,) (D.—AD,), &c., whence ‘ve have ad, u= A (D,—aD,)7* V+ eee +A (D,—2@D,).0+ oene But (D,—aD,)7 Vers fe-* Py Vda ey f0.dx; * Page 751, note: particularly in Mr. Gregory’s examples, which should be in the hands of every student who wishes to have materials for self-exercise in the lighest processes and newest forms of the differential and integral calculus. | | > Penny Cyclopedia, ‘* Operation.” 3C 754 DIFFERENTIAL AND INTEGRAL CALCULUS. and the second term is ®(y+ax), ®y being any function: while, if V=¢ (a, y), the first is found by changing y into Ytax in the result of f P(x, y-ax) dx, taken with respect to x. This process is somewhat more easy than that of § (89.), inasmuch as the result for one root will give those for the others. (128.) Let D,uw==aD?u. We have then u=(D,—aD*)~.0, or ead? [0 dt, or ead; px, being arbitrary. This gives, by development, i a ¢ iv u=gr+atp er a ”) Ut eo.ers as already seen. For the symbol ea? write [tse "tv P= du: Ja, and we have usat (tee ev De he doa t (teh 4+20Jat).e™ dy, which agrees with § (98.) (129.) Let Gy Usin + On) Uranaifto++.+aU,=V,, whence w= : (a+a,E+....+a,E")"V. Ifall the rootsofa,tav+....+a,0" | be unequal, let them be a, 6, y, &c., whence a, U,=A (E— a)" V,+B(E—B)*V.+...- § (123.) =a Aa" Zam V,.+Bp >b-* Vit eee +A, a’ +B, Bb’ + fers | which gives at once the law of the result, where § (111.) only gives the process. This symbol 2 is here put for A~’; the only difference being that whereas Aq’ V, strictly stands for V,..+V,.+.... ad infinitum, 2V, stands for C+ V,_,4+ V,o+....+V,., 2 and a being supposed to : differ by an integer. Allafter V, is supposed to be included in C, and in the preceding case, the values of C in the different summations may be > supposed to be included in A,, B,, &c. . : | me (130.) The proper symbol for A™* or (E—1)- is E™+nE™? | +in(u+1) B*+.... ad inf. or, A A= A, + nA, ae ae : This is the only result which satisfies both A” A A,=A, and A~” A”! ‘ =A,. But 2"A, is generally taken in a manner which satisfies only A" >" At=A, and not >*A"A,=A,. For instance, let 2 be an integer, and let PA, —A, +....+A,. Then 52A, means >A, _,+ 2A. »++e+2ZA,, and ZA,=0. This gives 2?A,=A,.+2A, +. +(#—1) Ay AXYA,=A, +A, + ...-¢A,, A? >? A, =A. Bae LAVA, is APA, +....4+(¢—1) A? Ay, or A,—vA,+(2—1) Ap Nevertheless, in the solution of equations of the usual kind, =" may | written for A~", since the verification of the solution involves om repetitions of E, which requires only repetitions of A, performed upon % | and never introduces 2 performed upon A. And we have (1*t*A,=0, ‘a 5 A,, y is the series Again, the complete meaning of A>” A;* A,, ,; or (E,—1)-" (By =n a ON DIFFERENTIAL EQUATIONS. m+] A x—m, Gon PATA gon 1, yn ENA om, y—n—l nf m hans ahs Aas, y—n a oo hed “I Cr Or sontinued ad infinitum ; while, defining SA, to end with Ay, the ex- ression for 27 27 A,, only involves those terms of S CHM, Ratese, te psa) n which e—m—p and y—n—g are not negative. Here S means nerely collection of cases, and differs from ¥ in not being a symbol of yperation. (131.) Let there be 7 roots equal to & in the equation of § (129.), id let the resulting fractions be Tg ia) sl-+ ly (Be) 9s 55 ob deg (Bart Chis operation performed upon V, gives mo A (a V,) +L, oA (a V,) +. FL OAM (aV,) + 1y oe? “A (0) +L, "HV A-EY (0)-+4..... FL 7A! (0), vhich, since > may be written for A as far as the solution of the quation is concerned, gives, for the part of the solution arising from hese roots, ee dig ba? Vial Ba Vs i veo, Bh eR +a” (Co+ Ci e+ eoee +CL, got 4o, C,, &c. being arbitrary constants. (132.) Let @,Usin, y On Ustn—i,yti bees $M tis, Ses he pin i M . 4 oe A) jn—l TF vhich case 1,, y 1s the inverse operation of a, E?-+a,_, Ey E,+-.... Fa, Ey), or of a” (H,—aE,) (E,—AE,)...., performed upon V,, ,; vhence On Uy, y= A (E,—aE,)~ Vz,» +B (E,—BE,)— Va, yteess Now (E,—aE,)“ V,, ,=a"" Ez" A>" (a* E;” V,, ,) i? Vy ye ie aps A= (a E; x V,, * ta ae) Viet Seen? ae) Voss, yoeeerri evs Che operation Ej~’ performed upon this changes y. into y-+-2—1, so hat =f tT P Fr 27 (E,—cE,) ; V vy cl G1. y + a v2, yt \ t—3, yt27 e, 9'e. 0.9 vhich might readily be ascertained directly, but the object is here to how the conformity of the condensed notation above written with the ictual result of development. Again Pe 0=Yy, ae Sse a 0O= 2" Us (y+a—1), ¥- wel ¢ xv n which the arbitrary character of % allows us to change Gg” sade a. Phe solution might be readily written down, and the modification which t undergoes in the case of equal roots ; but we have instances enough of hese generalizations. If V,,,==0, the solution is simply ey (y+z) 3C2 756 DIFFERENTIAL AND INTEGRAL CALCULUS. ‘ +h? wv, (y+ta)+...., and the case of / roots equal to @ gives — a fw (yta)teyy, (2 Coe ta", (y+2)} for the contribu- tion of these roots; Ys, Y¥,, &c. being arbitrary functions. (133.) An equation of mixed differences is one in which operations — of differences and differentials both occur. For example, let | ee Ur d* u, » Y+1 oY ta, y~— +, dx” aa" or A, Uz, y= A (D,—aE,)™ V;, y +B (D,—BE,)™ V;, y+... ' OE Gg ea Nora in which (D,—aE,)™ V,, yoo" yf dpe ty Vid &C. each of these is a complicated form of the element of the solution required, In the first side we easily see fdx.V.,,+a(fdz)?. Vayu + (fdz)*V., yis+.... The second side shows how to obtain the same without repeated integrations. We have i x go aky Ve: seme Yor —axV,, eaters Ve FES serene Integrate this with respect to x, giving say W,,» then W,, ytarW,, gat” +....1is the development of (D,—aEH,)™ V.,,. Now invert the ordeal of the processes, and let a, 6, &c. be the roots of ayv"+a,v""'+. | —0. We have then (neither «, 6, &c., nor A, B, &c. being the same us as before) : a EX +a, Ej D,+....-+a, Di=a, (E,—eD,) (E,-&D,).... Dytld gate Wig 2D) 5 Ve gi, (ig Wg) ce ees (E,— aD 3, ett DIS AS? DF? Ve) in which it must be remembered that D%~’ and Aj" are not convertible” operations. If V,,,==0, the Pree dne becomes a? Dy! Ay (a7. Dz? Om Now D7’0 is Woy topyt.... +a" yay. In the operation Ay no term higher than 2’~ will appear, except in the arbitrary function of a which, so far as solution of the equation is concerned, may be added: hence Dy will make all disappear except what arises from this | function. Hence the preceding is o’~! D¥-! wa, so that the solution is — ye Aa’? DAS (a Dz? V,, y) +B AY DAZ (8-9 D5* Vz, tes | ad De ya +p Diy, o+. If there be J roots equal to «, the corresponding part of the solution may be found, as before, from a set of fractions made by giving & all values from 1 to /, both inclusive, in the following, 3 (E, see) “Ne oer ee Peal Crna De Nays which, when V,,j==9, is of" Di“ A, (a D7" 0), As. betors.s = contains a¥~, so that nothing above ae will appear in the result of the operation Ay“, and this will be destroyed by the subsequent opera= tion Dy. * All then that is left after A," to any effective purpose, is amd Wie (Peto e. yt..es HH42.y'"), the arbitrary functional being introduced in the summation, “f ON DIFFERENTIAL EQUATIONS. 757 For instance, let a? Di u., y — 20D 1D), Uz, yb Uzp-y42=0, or TR (E,—«D,)~*.0. The solution then is y Us, y= a Di (do a+, 2.4). rhis is thus verified: substitution gives for the first side of the equation i! D? (Port, x.y) - 20 DY (d,x+¢, 2 (y+1)) +2 DY (do e+¢, 2 (y+2)) =a" DY {hy x (l—2+1)+¢,2 (y— 2y~2+y7+42)}=0. (134.) The same processes may sometimes be applied when the quations are not homogeneous with respect to the indices of u,,,. For ‘xample, let us take the equation Alu, ,=a°A? Us OF aes ] Uz, y= (A, —@ A?) 0 i gad Vo. 2a \A,—aA, A,+ad,J Ye must first investigate (A,—aA,)7 0, or {E,—(1+aA,)}-.0. his is tad.) AT" {(1+a4,)-" 0}, or a {b, - b}y Ay {a-¥ (E,—b)-4 0}, there b=1—~—a™. Now a (E.— by" 0=a bY AT O=b* (P+Pie+.... +P, 2), Yo, &c. being arbitrary functions of y. The operation Ay" performed on us merely alters the arbitrary functions, and does not contain any lower of x above x*; but it introduces an arbitrary function of x, wa. f we now perform upon this result the operation (E.—b)’-!, or *t9-) AY B-* all vanishes except what is given by the arbitrary “ction Wx, so that the final result is a’? 6**9~" AY! wa, on which it tay easily be shown that the final operation (2aA,)~—' has no effect Xcept a change of the arbitrary function. Another simple change will pduce the result to (@—1)¥ (1—a™)* Av wax, which is one term of w vw y° he other is got by simply changing the sign of a, and taking a new rbitrary function, and the result is | Us, y= (a—1)" 1—a")* At wr (—1)" (at+1"(1 +a)? Av ya, twhich we may interchange x and y when we interchange a and a7}, ~ /0 verify one of these solutions, say the first, we have yy Ue, y= (1—a7")* {(a—1)"#? AYt2—2 (a— 1) AML (a—1)’ AY} wa = (a—1)" (1—a™")* { (a— 1)? AY**—2 (a—1) APL AY wa Aru, y= a (a—1)" {(1-- a") ALY (0+2)—2 (l-a™)* AY (24-1) +(—a™")? AY wat *(a—1)’ (1—a~)" { (a—1)* (AY DAY 4 AY) ~ 90 (a—1) (AUEAYY) + a* Ad} wa =(a—1)*(1—a™)* {(a—1) Ap — {2a (a=1)—-2 (a—1)*} Ay? - (a—1—a)* ay} War 5 758 DIFFERENTIAL AND INTEGRAL CALCULUS. whence it is readily shown that the two sides of the equation are identical. The preceding appears to fail when a=1; but if we return to the process, the step which is first affected by the supposition of O=1, or, 0-038 : a’ (E,—b)! Az} {a7 (E,—b)~.0}, which becomes E2~' 4,".0; or EX we, or ¥ (a+y—1), or ¥ (a+y), which is not affected by the : final operation Az!. Hence ¥ (w+y)+(—1)’ 2° Ai xx is the com) plete solution of A? u,, y= A} ts, y Nee ae (135.) Let Att, yv=Abtety OF Meh x FEY -0 * = (E,—E,)~ {1+ Es (Ey—Ez)7} 0 | E;' (E,— E;")7.0= EH," E;?t! AP BY 0=¥ (a@—y) 5 | (E,—E,)7 % @—y) =ED* AF BL (2—y) | =EY AS yi (w —Qy) = EL y (e@- 2y)=x (4-y—D) : (E,—E,)7 0O= Et Aj EY 0=oa (a+y—1). Hence the solution is of the form >(v+y)+¥ (a—y), ¢ and ¥ being arbitrary. | | (136.) Among other results of the preceding theory may be noted the : ease with which the intermediate diff. equations or equations of differs ences may be found. Thus, if a,D%u+a,_,Di*u+....=V, OF a, (D,—«) (D,—)....u=V, the equations of the (n—1)th order aré a, (D,—8) (D, — y).-..u=(D,—a)™ u, a, (D. — «) (D. — y).-. Ue =(D,—f)~'u, &c. Those of the order n—2 are a,(D,—y)...9@ =(D,—2)71(D,—)" V, &c. I do not however consider it desirable to enter more in detail upon a method which has not yet advanced beyond its elements, though I fully agree with those who have considered it as one which is likely to prove a very powerful instrument in analysis. Ai (13'.) In the equations preceding, it has been required that m | should be satisfied for one value only of Av, which has been taken =1. If we had proposed such an equation as U,.a,—P, u,—=Q,, Ax being | anything whatsoever, it would have been equivalent to requiring that) Uz+ az Should be different from 2,, and yet not a function of Aa, which is | absurd. The last equation could only be satisfied on the supposition EK | P, and Q, are given functions of Av, and then only in particular cases.) Nevertheless, when such an equation does occur, it may sometimes be re duced immediately to a common diff.equ. Suppose, forexample, f(x, Aa,U; u', ul", &c., Au, Au’, Au”, &c.)=0 is to be true for all values of Av; @; w'’, &c. being, as usual, the diff. co. of w. Take x, any given value of G and let 2,, 2’), &c. be the corresponding values of wu, u’, &c. Then; # passing from a, to x through the difference e—2», we have a ie F (Gy L— Xp; Up; Up, KC. U—Up, UL’ —1U'p, &C.)= 0... +. (A). ie | > Form a new diff. equ. by elimination of xo, 1%, uo, &c., and we have a ‘general equation belonging to the class of curves in question, Mm) 3 ON DIFFERENTIAL EQUATIONS. 759 dependent of the particular values of a’, 2), &c.; and the class of curves which has the required property, expressed by f=0, is that repre- sented by the general integral of the equation last obtained. Or, if the equation (A) be integrated, the class of curves required exists when the constants introduced by integration have the effect of rendering it in- different what value » is made to begin with in verifying the original equation f(x, Av, &c.)=0. For instance, having given a point S, required a curve such that if any two points P and Q be taken (the reader can easily construct the figure) the tangents at which meet in T, the line ST bisects the angle PSQ. Let AS be the line from which @ is measured, ASP=@, ASQ=0+ 40, SP=r, SQ=r+Ar. Produce TP to Z, and as in Chapter XIV., let SPZ=p, then SQT=p+Ap. Equate the two values of T'S in the triangles SPT, SQT, and we have r sin pu _(7+47) sin (u+ Ap) r+Ar=r, sin(u—340) — sin(u+Au+4 Ae) pt App, Ar tan tan po, cot 4 AO=r tan w+7, tan py. For r write 1: u, and remember tan p=rd0@:dr=—u: w', which gives Ar i; uw — cot 4 AS=——_ +——_,_ Aucot 4 AO=2u'+ Aw’, TT; i tan py, tan pu f an equation of differences which is to be universally satisfied ; that is, : ‘ / 4 for all values of AO. The first found diff. equ. (A) then becomes Oo (u%—U,) cot SA As Mo, egy CA) « Differentiate, multiply by 2 sin? 4 (6 —0,) ; differentiate again, and divide by 2sin? 4 (@—6,), and the result will be w/”+-u’=0, or w’+-u==const. the equation of the conic sections, Every conic section, therefore, has one position of SP, for which every position of SQ has the required pro- s perty. But, more than this, verification will show that the equation of 4 . t differences is satisfied by every position of SP. ‘Take the complete integral w=a+b cos (0+c), and substitute in the equation of differences, which gives b {cos (0+ A9+c) —cos (6+c)} cot 4 A@ =—bsin (0+c)—b sin (04+A0+c), an equation which is easily proved identical. It would do equally well to integrate the equation (A) directly. As another example, it is required to find the curve in which the ordinate let fall from the intersection of two tangents is equally distant } from the ordinates of the points of contact. The general equation and + y/"=0, or y=Ca?+C,x+C,: integrated directly, it gives y=C (7-2,)? the equation (A) here become (2y'+ Ay’) Ar=2Ay, and (o/+y',) (@—a)=2 (y—-y) 5 the latter of which, all constants being elimimated, gives the diff. equ. 769 DIFFERENTIAL AND INTEGRAL CALCULUS. +4) (x—2,) +Yyo, in which y, and y/) come out as they are defined ; namely, the values of y and y’ when w=a). The general equation is satisfied, and the property is that of any parabola whose axis is parallel to that of y. But we may easily imagine it possible that such a pro- perty might be given that y, yo, &c., being defined as above in mean- ing, the integral of the differential equation (A) does not allow them to have that meaning. In such a case the property is self-contradictory. Again, the property given may be true if one fixed abscissa and ordinate be started from, but not true if the starting point be changed: in such a case the integral of (A) gives the curve required, but the general equation of differences cannot be true except in a particular case. For instance, let the equation of differences be Ay+axAy'=h ; the diff. equ. (A) is y—Yyo + % (y/—Yy'o) =h, of which the integral is Yyytary th+Ce which for r=a gives y= Yo+% y/o th+Ce™, whence C=~<¢ (2, y',+h). Again, y'=—Ce~*' 8: 2, whence y/9=—Ce*: a= (2,9 0+): %, or we must have h=0. This last condition, it now appears, is necessary to the self-consistence of the property which the curve is required to have. If then there be any curve which satisfies the condition Ay+arAy'=0, it is Y=Yot Xo y'o Gena), Try this on Ay+zrAy’=0, and it will be found to satisfy the conditions” only when the differences begin with the point (a, yo), unless 7/,=0. The property announced cannot then belong to any two points of any curve. This may be proved independently, for if (7, y), (M4), &. be a succession of points, the equation gives y—yte yim y)=9, Yo-yo te (y'2—y')=0, p—-N +1, (y'2—y',) =O 4 4 from the first and second of which we deduce y,—y:+2 (y/2—4,)=0, which is inconsistent with the third, unless y’ be a constant, which does not satisfy Ay+vAy’=0, unless y be a constant. The last equation, then, required to be generally true, is equivalent to Ay=0. (138.) Any such equation as the preceding might have an infinite — number of solutions given to it of a discontinuous character, and for one riven value of Az, as follows. To take a simple instance, suppose Fa ’ » SUpp Ay'=$¢ (a, y, Sv, Ay) is the equation. Assume a value for Az, and divide it into n parts, so that nodv=Aa, and oz is very small. Assume — values for y, and x,, and for y, or y+ Ay; a, or e+Az, being determined — from Av. Join the points (x, Yo) and (2, y,) by any curve, and calcu- — lating Ay’, from the equation, and thence y',: lay down a straight line at (21, y:) accordingly. An ordinate to this line at the abscissa x, + dx is a new point in the curve, quam proxime. Repeat the process with the point (%+ 67%, YotoYo) and that just obtained, and so on until the — curve, or rather representative polygon, extends over the abscissa Lyo+2Ax; after which it is to be repeated again with the last obtained portion as a guide, The smaller or is made, the more nearly will a ] | | oF. ee ON DIFFERENTIAL EQUATIONS. 761 curve be obtained satisfying the given equation of differences. This method will aid in the formation of a complete conception of the possi- bility of satisfying any such equation, for any one value of Ax. And the same method will not only apply to ordinary diff. equ., but will furnish a strong presumption that no more constants can enter than there are units in the order of the equation: as follows: Suppose the diff. equ. to be "= (y",y/, y, x), and proceed to con- sider A® y= (Ax)? ys (A? y, Ay, y, x, Av), of which it is the limit. Take Az very small, and any ordinates ¥,, y,, yz, at pleasure, to the abscissee Zo Lot AX, To+2Ax. Having thus given y,, Ay, and A’ yo, calculate “$y, from the equation, whence y,, the ordinate to the abscissa To + 3A, is obtained. With y,, y,, ys, and A® y, from the equation, calculate y,, and so on. We have thus a polygon by joining the several points obtained each to the next: the coordinates of the angular points of the polygon satisfy the equation of differences, and the smaller Ar is taken, the more nearly does the polygon become a curve which satisfies the diff. equ. Through the three points thus assumed only one curve can be drawn, as is evidently pointed out in the course of the method: as also that the manner of choosing yo, Ayo, and A’ y, as the limit is approached deter- mines Yo, ¥'o, and y”,. Hence for one value of y, y/,, and y,", only one limiting curve can be obtained, from whence it may be presumed that only three constants can enter the solution of the diff. equ. The diff. equ. can only have such solutions as are limits of those of the equation of differences. I call this only a strong presumption, for reasons which [ will leave to the student, who will find them on close examination, (139.) In all the preceding equations, the coefficients employed have been continuous functions, though such continuity is not necessary, in the manner in which they have been used. If, for instance, we suppose @ an integer, and propose the equation w,4.+au,,,+2" U,+2=0, it is evidently not necessary that the functions of 2 should preserve the same form when 2 is fractional, since the equation, its solution, and the pro- cess of verification, are all wholly free from the consideration of such values. But it is not even necessary that the coefficients should preserve one form when a is integer, and results may be obtained in a finite form when they circulate* through any number of different forms as x changes its values. For example, let w,,,—P,u,=Q,, where P, is the constant a or b, according as 2 is even or odd, and Q, is a’ or b', accord- mg as zis even or odd. Hence a b i P.=$(1+(-1I))4+50-(-), w= qd] H-D)+5 (1—(—])’); and the method in § (106.) might be applied without much difficulty. But the process will be facilitated by assuming u,—v,+w, (—1)’, and, after substitution, equating the parts which are independent of (—1)*, and also the coefficients of those which depend upon it. We have then * Sir J. Herschel, Examples of the Calculus of Finite Differences, section xi, DIFFERENTIAL AND INTEGRAL CALCULUS. V241—3B (a+6) Vprriey (a—b) wW,=4 (a’+b’) —wW,4,—4 (a+b) w,—t (a—b) v,=3 (e—D'). As an example of the second method in §(116.), change # into 2+], giving a third and fourth equation: multiply the second and third severally by A, and pu, and add the first, second, and third, making (a+b)\+a—b=0, 24+(a—b) p=0. We thus get the first of the following equations, and by similar processes the second. V,49—abv,=4 (ab+ab' +a +b), W,42—abw,=4 (a'b—ab'—a' +b/) v,=4(d/b+ab!+a'+') (l-ab)+’ (Ki + K, (—1)*) w,=t (a’b—ab!—ad' +6’) (A—ab)"+ a’ (Li +L, (-1)"), a being ,/(ab). Hence / , / bh! Se Mapa I Sei | b! —1\* _(a'b-+-ab'+a! +b')-+(a'b—ab!—a' +b’) (-1) soe’ (My-+Me(- 19) ST TT Ooms M, and M, denoting arbitrary constants. One relation between M, and — M, must be expected, since the original equation is only of the first order ; this will be seen in attempting to verify the equation, The preceding value of w, gives alb+-b' 1—ab ab! +-a/ 1—ab ab’ +a! rae Sener yet 2x (Galle Fa (M, M3) +a” (M,+ Ms) 5 (veven) v,= (z odd) u,= +a* (Mi—M,) ; (weven) Uri —- Pu alb+b! 1—ab =a +e’ (M, a—M, a—M, a—M, (a) alb +0! 1—ab —aca’ (M,+M,) ay +a’°** (M,+M,) (xv odd) t4,;—P, Uz= ab! +a! 1—ab = b’/-+¢7 (M, e—M, b+M.a¢+M,)). Substitute for @ its value (ab), and the multipliers of a” have the common factor M, (/a+/b)+M,G/a—./d). The value of uv,, then, completely satisfies the conditions, and has one constant arbitrary, if Mz (fa+b)+M, (/a—./d)=9. Las —ba* (M,—M.) (140.) To generalize the preceding method, let m, stand for a function of x which is =1 when x=0, m, or a multiple of m, and which vanishes in every other case. If a, f,.... be the m mth roots of 1, such a function is seen in the mth part of a’+A°+.... If, then, we take — Cy m+ Cy my Coty of oe ee + Cm Meme We have a function which ~ ye oe ON DIFFERENTIAL EQUATIONS, 763 is Co, C,,..++On1, according as a:m leaves a remainder 0, 1,...., m—1. This has been termed by Sir J. Herschel a circulating function of the mth order. If P., Q,, &c. be circulating functions of this kind, we have, for all znteger values of x (the reader must be careful not to generalize this equation) PL Pen 0:5 $ ea? (P,Q). a »).m,+f (P,Q). ee ») Mr 1+» ee. 3 for f(P,,Q,....) is itself a circulating function which goes through the cycle of values f (Po, Q,-..-), (Pi, Qi..«.), &e. Circulating functions may be doubled, trebled, &c. in order, by assuming new circulating functions with doubled, trebled, &c. cycles of values, Thus a3,+03,_,+¢3,_s is altogether identical with a6,-+ 66,_, +c6,_.+a6,,+06,_,+c6,_;. A simple process will reduce the solu- tion of any equation whose coefficients circulate to that of a set of ordinary equations, as follows, Let @ (t), Ur41,++ ++ Ps, Q,...)=0, where P,, Q,, &c. are circula- tors of the pth, gth, &c. orders. Reduce them all to circulators of the same order, namely, that of the least common multiple of p, q, &c., say m. Assume wu, to be a circulator of the mth order, 7, m,+8, mM, 1+ «+. Then BR aes bss OD Ta Sans ch ge ss) Me P (Sentyse sbise a) Mpeg ae Determine 7,, 5,. ++ by the m simultaneous equations ¢ (7,...P,.. .)=0, p (s,..+P,..+)=0, and the conditions are completely satisfied, or may be satisfied by assuming relations enough to reduce supernumerary con- stants. Thus, suppose U,,.+P.U.ii4+Q, utR,=9, where P, is a circulator of the second order (a,, b,, d,, b,, &c.), Q, also of the second order (a’,, b’,,&c.), and R, of the third order (a’,,b",,¢., &c.) Reduce these to circulators of the sixth order, and assume one of the sixth order (7,5, ¢,V,W:Yzx) for u, We have then six equations derived from (12426249 4+ S242 6241 + tere 6, + Ve42 65-1 + Wr42 62-9 + Yx+2 62-3) a0 { de 6,+6, 6:1 +4, 6.-2+ b., 6,-3-+ Gr 62-414 5, 62-5 }4{Te41 624145241 6, +24; 6,-1.4 Ve41 6,-2+ Wry 62-3 FY 41 6,-«} + (a, 6,+ 0’, 621+ a's 6,_s LBL Gia +. a’, 6,2, 4-0’, 65-5} {7,654 8. 6,1 t, Beaks 6,-5-+ wi 62-4 +, 6,5} = G2 6, + 8", Orang FC! 62-9 + Gx Or-3 + Oe 6 sat C's 6455 remembering that 6,,.=6,_, and 6,4:=6.—5. These equations are Ym eS a be T Oy Spay FO Pe FO" =0, Yous t Oe Wert Oe Vet O = 0 r+2 v °a+l et 4 an Y mu Vise t Oy tery $0! 8:0 =0, Treats Your ts W, +6",=0 e l LP Wrst ay Ve41 a a’, be + bio 0, Seta b, ? e+1 38 b xv Yur c Fh ee 0. The actual solution of this problem would require us to change x successively intov+1....,up to v+10, which would give 66 equations between 65 quantities besides 7,410) Tx+09° ees The elimination of the 65 other quantities would give a final equation to determine re? the equation for s, would be found by changing 7, into s,, @, mto b,, a, into b, &c. As a more simple instance let us take the problem already solved in § (139.), namely (Pea 2e41+ S241 22) — (a2,+b2,-1) (1% 22452 2,,)=a' 2,+6'2,_,, which gives Spay, =a, 1 241 —08,=b', or (/ab being «) DIFFERENTIAL AND INTEGRAL CALCULUS, See — @bs,4,=ab'+a’, 7,,,—abr,,,=ba' +0 } ’ bal b Geno ee Oe Cray ee a l—ab Now 2,=3{1°4(-D*}, 2428 {F4(—D}=3 {1-4}, and substitution in u,=7,2,+5,2,, gives the same result as before, the superior simplicity of this process arising entirely from using a circulator for 2/,. +L, «+ Ly (—a)*. (141.) Required the sum of x terms of the series a)+b)+ Cotat+h +c+&c. This is obviously Au,=P,, where P, is Cis, Onc, Ch Gea (say A,, B,, or C,,) according as w is of the form 3m, 3m+1, or 3m+2. We have then ‘ Tory So 41 F'Sr44 ie ol se (, B24 SeSeli + te Ores) = A x Se ae B, 31-1 + C, 32-2 LPS. Re ad Oma C., Sie Ags tp 41-—8,2= Bz; Tr43-—T,; = AL “fo Bein sa Cotas Sz43—$,—= B, nig (oA + Ax 495 tis Sas She C, a Daa + B.+2 * Let 1, 2, £, be the three cube roots of unity, A,+Bri: tC... As, &e. r= DA, +ha* So 8,44 6 DAA, + K, 3.4K. 3,,4 K; 3,95 for it is evident that C,(1)*+C,a7+(C, 7 is a circulator. The three _ results put together by u,=r,3,+5,3,1+1, 3,2 will give the expres- sion required, if the resulting circulator derived from the constants, say L3,+M3,,+N3,_., have L=0, M=a, N=a.+6,. Let the student verify this in some instances. (142.)* A merchant begins with £A in the stocks at 7 per pound per bi annum, and £B in trade, which returns 7’ per pound every two years. a | He spends £a per annum, and invests half the returns of his trade, as _ they come in, im the increase of his trading capital, funding everything else. What has he in the funds and in his business at the end of & bs years ? At the end of wv years let him have F, in the funds and T, in trade. oa Then, if x be even, F,,, is (1-+7) F,—a, since the business makes no return at the end of the (w+1)th year. But if w be odd, F,,, is | (1+r) F,-a+4r'T,_,, F.4j= (1+7r) F,—a+4T,_,7' 2,_). Again, if x be even, T,,,—T,, but if x be odd, Ti41=T,+4T, 2", or f erat ad Was ey ted Assume T,=V, 2,+W, 2,_,, and we have * Herschel, Examples, &c., p. 161. ON DIFFERENTIAL EQUATIONS. 765 Veg: 22a Wags 252 V, 2:4 W, 22, by! (V,27-+--W,, 2,1) 2,4 Gest 0, 22 | Vou Wet +47’), W.iii=V., V.4a= (1+4r) V. (JO +3r)=0'), V.=e’. {C,+C.(—1)*}, Wes {0,+-0,(-1)*7}. There is only one condition to determine two constants, C, and C,, namely, that V.=B when z=0. But in the value of T,, these two constants are reduced to one ; for, since {C,+ C,(—1)’} {14+ (—1)*}=(C, +0.) {1+ (-)*} TL=V,2.+ W, 2.1. (p" 2,49! 2,_1) (C,+-C,), and C,+C,=B. From this we have (1+-r=o) F,=pF,—a+ 47’ 2,_, B (p21 +9" 2,), or Fri pF.= +47 Bo 2,.,—a. . Let F,=G,2,+H,2,.; then, a being a2,+-22,_,, we have, 47/B being denoted by B,, G,4,:=0H,—a+ Brac, H,.,==pG,—a, G,.2—p° Cb, o*—a (1 +p) Ge eB ae a : pose a B, pp" £ x H, yea chan an {K+K, (-1)**}. And «=0 gives G, ial 2 pe eae +5--—+K+K), which is A, x Goa Oe 0 whence i © | [%3* rh p= 97! F,=Ap “ionag as = =a 2et Bip lo p 22 (143.) There are various equations of differences which are sug- gested by their solutions, and for which no direct inverse method can be given. For example, w,,,==2u2—1l. Let w,cosv,, then cosv,4, =cos 2v,, OF V,4,;==2v,+2mr, m being any positive or negative integer. Hence v,=2”.C,—2mrz, or u,==cos (C, 2”). But we may also take Vz 4:=2mr—2v,, which gives v,== (—2)* C,-+2mr, or u,=cos {C, (— 2)” +2m7r}. Here are two distinct solutions, showing that the ordinary theory is insufficient, for each has an arbitrary constant, which may be ‘converted into an arbitrary function of the form f(cos27r). And, 2x being an integer, there is an infinite number of other solutions, for since m need only be integer, we may write a,v"+a@ z+ ....+a, for it, where a, @,, &c. are whole numbers, as also 7. : A Let wy Uy, dy (Urey: —Uz)+1=0. Assume u, = tanv,,' and we ave tan Av,=ar', v,=2> (tan at'+mr). 766 DIFFERENTIAL AND INTEGRAL CALCULUS. du., Us Y Ls . . . 7 Let rag lay A, Us, This equation is satisfied by d d’ da Un, y= jlo ean (144.) Such instances are not without their use, since they serve to show that the solutions of most equations are unattainable for want of means of expression. Until, for example, we have a perfect compre- | hension of fractional diff. co., the last equation is unintelligible except when y is integer. The converse, however, is not to be assumed; that, is, it is not to be concluded that when an equation is integrated in al unintelligible mode, or by a formula which cannot be interpreted, that | therefore no other mode is assignable. For example, the complete | integral of ty Usy==Ur Uy, has been shown to be y¢%x, where div means ty the operation @ performed y times following on 2, and is for the most, part unintelligible, except when y is integer; so that the process of the ~ . diff. equ. cannot be performed. But, notwithstanding this, x (w2—y) net is the complete integral, when x and @ are any functions whatever. Thus if A,wW2x, which is a function of x, must have x changed into a-+1, it is needless to write A.u: ¥(a+1), and A, % (#+ 1) will be sufficient, . d or differentiated with ; | (145.) In the preceding equations, and wherever D, or A, 1s used, it should be remembered that a is not a symbol of value, but of distinction. “8 | | APPENDIX. (Page 68.) The fundamental theorem admits of a proof which, though less elementary than the one in the text, is not so complicated. Grant- ing that a diff. co. is positive or negative, according as the function and the variable alter in the same or different directions, as seen in page 132, let C and c be the greatest and least values taken by vx: w’a in the terval from v=atow=a+th. Hence dx: wa —C and d/x: Wla—ce are of different signs throughout the whole interval, whence, yz retain- ing one sign, by hypothesis, o/x— Cy’ and ¢’x—cw’/x are also of different signs. From this it follows, that of dvu—Cww and pr — ca one must continually increase, and the other continually decrease, from t=a to x=a-+h: that is, P (ath) —da—C (& (ath) —Wa) and fp (a+h)—da—c ( (a+h) —wa) must have different signs. Divide both by w& (a+h)-—wWa, and the same thing remains true: this is the fundamental part of the theorem in the text. (Page 103.) The language and notions of infinitesimals may here be used, as is shown by the result. We have fr.dz, where r=wWet, and di=y't. dt, whence fist. yt. dt is to be integrated. (Page 163-168.) I have throughout this work made free use of what used to be called the separation of the symbols of operation and quantity, ander the name of the calculus of operations. The student who wishes really to understand algebra must make himself acquainted with What has been done of late years in the generalization of that science, after which the calculus of operations will cease to present any other difficulty than that of the differential calculus in general. The state- ment of principles partially laid down in page 164 may be completed as follows. In any science which proceeds by rules, these rules may be collected ind separately taught. ‘They depend upon the meanings of the symbols ‘mployed; that is to say, the meanings of symbols being given, the rules for the use of those symbols may be investigated. But there is an mverse question: having given a set of rules, derived from one particu- lar set of meanings, is this the only set of meanings from which that set of rules would follow? _ The answer is by no means in the affirm- itive: ‘A gives B, therefore B, when it comes, must come from A’ is ot good logic. Now algebra, in its most general sense, is every science which proceeds by the fundamental rules of general arithmetic, Whether the meanings of its symbols be those of general arithmetic or Mot. Technical algebra is the art (only an art, not a science) of apply- ing those rules to symbols, without reference to their meaning: logical 768 APPENDIX. algebra is any science in which those rules are used with any of the meanings which are allowable. The technical definition of a symbol is contained in the rules which — are laid down for its use: the logical definition, or explanation, pre=— cedes the branch of logical algebra in which the symbol is used. But) when, some symbols haying been explained, and it being understood — that all explanations are to be so given that the rules of general arith- _ metic shall be applicable, we wait until results shall indicate the mean- _ ings of the rest, the process of finding such meanings is interpretation.* — The science of general arithmetic, the rules of which are those of every algebra, has simple number, and operations upon it, for its subject” matter. Its symbols of quantity are, numbers represented by letters, _ and the rules are as follows :— | 1. In every combination of + and —, like signs sive + and unlike — slens —. | 29 Additions and subtractions are convertible in order ; thus _ a+b—b+a, and a—b+c=a+c—b. 3. Multiplications and divisions are convertible in order; thus axb=bxa and axXb—c=a—+cx b. a 4, Multiplications and divisions may be distributed over additions — and subtractions: thus (b-—-c)xa=bxaxkexa; and (bc) +a =—b—atc—a. ; 5. The rules for the use of powers are a’ X a’=a"** and (a) =a" To these rules all operations may be reduced ; though some may be of opinion that there are more, and some fewer. This, however, does” not matter much to our, present purpose ; be their number more or fewer, no one doubts that the processes of arithmetic are reducible to a small and fixed number of fundamental rules; and any one may add to. or take away from the preceding, as he thinks necessary. | Again, the rules in this science, as in any other, are to be understood — as applicable only to intelligible data. Thus, 6—10 being unintelli-— gible, cannot be the object of their application. The signs -- and = mean here simple addition and subtraction, and nothing else. ie In the next step, the common algebra of positive and negative quan- tities, we consider the symbols as implying numbers representing quantities, with the implied addition of an understanding as to the sense in which the quantities are to be taken. If --a represent a quantity of one sort, —a represents one of the some magnitude, but of a directly Li * This process seems to be peculiar to mathematics: to go on using a word ora sign without any knowledge of it, except that it is a word or a sign, to be used ma certain way, until the results of that use point out the meaning which the word oF | sign ought to have had, is a strange idea when presented for the first time. But, nevertheless, it has been used out of mathematics: in logic, for example. Wallis, the first mathematician, I believe, who formally introduced interpretation into algebra, had previously made use of it in logic. Ina disputation, (at Emanuel College, Cambridge, in 1631, ) whether a singular proposition is to be held universal | or particular, his thesis (printed at the end of his logic) decides the question by — interpretation, as follows. A singular proposition, such as ‘ Virgil was a Roman,’ is to be so taken that the rules of logic may be applied to it. From the premises ¢ Virgil was a Roman,’ an@ <‘ Homer was not a Roman,’ it certainly follows that ‘ Virgil was not Homer.’ Now - if the two premises be particular propositions, there can be no conclusion: from ‘some A’s are B’s’? and ‘some C’s are not B’s’ nothing can be inferred, Con- sequently the premises must be considered as universal propositions. ¥ . The preceding process answers precisely to interpretation in al gebra. TAL EEIN TS 2 769 opposite kind. And A+B means the junction of quantities equal to A and B in magnitude, and of the same kind as A and B, while A—B / means the junction of A and the magnitude of a kind contrary to B. The third species of algebra, which includes the form of the greatest _ extent in which the symbols represent magnitudes, rests upon geometri- cal definitions. The symbols imply lines, in which direction as well as length is signified, so that two lines which are jn different directions, but of the same length, or in the same direction, with different lengths, are tepresented by distinct symbols. This species of explanations leaves no symbol unintelligible ; and ,/—1 is as much the representative of a line of one unit in length, inclined at a right angle to the line signified by 1, /as —] is in common algebra that of a unit of length placed opposite to ‘the line 1. I do not propose here to enter upon the details of this algebra,* intending only to point out to the student that even the algebra of quantities is a gradual ascent from one generalization to another. But the symbols are not necessarily restricted to quantities; as long as the five rules, or those which any one else may substitute for them, ican be made true of the meanings, those meanings may be any what- ‘ever. For instance, dr, a function of “, may be the subject of opera- tion, just as the unit is that of ordinary arithmetic, and A, B, Cae: may be indications of operations to be performed on @r. As yet, the only fundamental species of operation which has been reduced to an algebra of operations, is that of changing x into #-+-a, a being a con- stant. ‘T'his system is only a commencement, and many of its results fare as yet incapable of interpretation; but, as in the history of the old algebra, the results are always found to be true whenever they are intelligible. The following are the explanations of this system. 1. The subject of operation, answering to the unit of arithmetic, is any given function of a variable w; and except under the symbol of this function, 2 must never appear. 2. The other symbols employed are those of operations performed upon $2, which are either multiplication by a constant, or change of x into x-+ a constant, or some combination of these, or the limit of some combination,. obtained by increasing or decreasing a constant without limit. 3. If we signify @(«+1) by Eger, or agree that the change of x into w+1 shall be an operation Whose symbol is E, then E" bz signifies («-+m) for all integer values pf m, positive or negative. 4. The signs +- and — preserve their usual meanings : thus (E+ E’) dr means Eda +E? hp or 6 @+1)+¢ (r+ pp i pnd (3E—4) Or means 3¢ (x +1)—4 px, &c. On this foundation the ruth of the five rules is easily established, and many results imme- Miately follow, as the student will see in the course of the work. I will now give some idea of the difficulties which yet embarrass this subject, and which may stand, with respect to this algebra of operations, n the place of such symbols as 4/—1 in the old algebra. The symbol 1 E*dzx is the result of an operation, which, repeated n times, gives Efe : : PVR ries mw O(e+1), Onesuch operation is b( 247) but if a ke any one of | * See the Articles Negative and Impossible Quantities and Relation in the Penny Dyclopzedia ; Dr. Peacock’s Algebra; or Mr. Warren’s work on the meaning of im. bossible quantities, 3D 770 APPENDIX. — the nth roots of unity, ep (o42 is an operation of similar effect. If by express convention we exclude all values of « except a=1, which is 1 % a a what is actually done, we may produce true results as far as we go, but . we have ascended to no higher place in the calculus of operations than a that which common arithmetic holds among the varieties of algebra, We cannot yet venture upon the unrestricted use of results which involye fractional exponents of operation. The next difficulty is one which is not peculiar to this calculus. Let us suppose that from and after, say 7= 0, we have a succession of values of a function, giving ¢ (0) when z=0, (1) when a=1, and so on fora | every positive iteger. Let us waive the difficulty of interpolation | (page 543), and say we have reason to know that @r would be the | | function of x for every positive and fractional value of w: there still > | remains an impossibility of deciding as to whether @r is the function | required when a is negative, if the case be one in which discontinuity | may occur. From among a number of similar cases we may choose 2 (sin rv dv papo{2 (“82291 } ye T 0 vO where Wr is any function we may name. This gives P=¢e for every positive value of x, and P=¢r—2wza for every negative value. e ie ma. O ii Now, suppose we consider the operation E~*@x, meaning that on | which, if the operation E be performed, bx resulis; or EE“ dr=Oa | One satisfactory answer is E" g2=¢ (#—1): but unless the question be one in which it is either proved, or justifiably assumed, that there 1s no discontinuity, there cannot be perfect assurance that E71 ¢dt=¢ (a-1) is always allowable. The data generally involve the assumption, that there is no discontinuity from and after a certain point: thus, i con, sidering the series P (a) +o (a+1)+...., we mean to lay it down that from and afier ca, pe is the sole object of consideration; but when we pass to preceding terms in the course of operations upon this series, it by no means always follows that the general term $z applies cone tinuously for all values of 2 which are = oe one —~+U——0 | Ae + aut 2 gy Oe or BS nitieraii er eal here $(2,y,z,u)=0 is the complete solution; the first equation is mediately reducible to the second (page 96). Let the simultaneous iquations dz dy dz du X = =7 = 0 give £(2,y,2,u)=a, (2, y, 2, u)==b, 6 (4, y,2,u)=c, v (a, y, 2, u) =e. | é lé 3 dé Tas Ve have then — dx. dy Fie dz+—- du=0, which is co-existently dz dy dz du fue with the simultaneous diff. equ,, and thence | 3D2 APPENDIX. xe yes ze ue=o; dy dz dx du — is a particular solution of the partial diff. equ. And the same or & is true of 7=b, ¢=c, and v=e. But the equation f (én, ¢,v)=9, whatever function f may be, also satisfies the partial diff. equ. ; for the receding equations give, when multiplied by df: dé, df: dn, &c., and added together, df dé df dy df dé df dv —_—_ —— — -—— ke os ibe: sips Bal dibion de dz id Hin ‘ 1 Aires 5 a xt iyi Tuto; dx dy dz du whence f (& 1) 4; v) is the solution of the equation. (Page 206.) “ But four of these twelve contain c, only, and are identical, and the same Of C2 and ¢;.’ This is an error; two contain C; only, and are identical, and the same of c, and c,: hence three distinct differential equations of the second order. In the remaining six, two contain both c, and c,, two more both c, and cz, two more ¢, and ¢. But no one of these six is a diff. equ. of the second order to the given primitive, because in no one does more than one of the constants of the primitive disappear. (Page 213.) The difficulty which arises about the constants in this and the next page is entirely a consequence of the discontinuous mode of effecting the solution, and might be remedied as follows, by merely integrating the generalized form of the value of y/, instead of its particular cases separately. For example, let y°—3Py"+Qy'—R=0, P, Q, and R being functions of x. Itis well known that the value of y' takes the form P+aV+0? W, where @ is any one of the cube roots of unity. Let fPdz= P,, &c.,whence y= P,+¢V,+ W,-+C, and the question now 1s simply to rationalize this equation, and to show that the same rational form is produced whatever may be the cube root of unity chosen. | Observe, that (y—C) and all its powers must be of the form P,+¢V, | +02 W,, since aaa, eo, &c.; assume then (y-C)?*=Q+Re+S%, (y -—CP=X+ Ya+Ze*, and let A and p besuch functions of w as are found from pV,+AR+Y=0, and pW,+AS+Z=0: we have then (y—C)+-r (y—CP +p (Fy -CN=X+AQT HP, which is the complete integral of the equation, whatever value of @ may be used. It is the same as that obtained by the method in the page cited. I — aes LL a 0 Ee (Page 222.) By neglect I have omitted to insert some account of | Fourier's theorem on the roots of equations, in conjunction with that of Sturm. The former is more connected with the Differential Calculus than the latter. . _ Since (px)? must be a minimum when ¢r=0, @e.¢'x must change) sion from — to -+- when px passes through 0 by increase of ©: oY if} ga=0, then ¢(a—da) and ¢ (a—da) must have different signs, and APPENDIX. 773 @ (a+da) and ¢! (a+da) the same sign. If zx be arational and integral function of x, as dy a+ aa" -+-...., and if da, dz, ox, &e. be taken, and if the succession of signs of these functions be called the criterion, it follows from Inspection that when 7= — the criterion shows nothing but changes of sign, and nothing but permanences when x=-+- 0. Consequently, in the passage from w=— cc to r=-+ co. the criterion loses n changes of sign: and, as there are 7 roots, real or imaginary, we may attach to each root one of these changes of sign, so as to say that every change has a root, real or imaginary, belonging to it. Now if we examine cases in which diff. co. of @x vanish, with or without @z, we find that a change of sign is lost for every real root, and that except at a root, changes of sign are always lost in even numbers. And since there are only n changes of sign to be lost, every pair which is lost by the vanishing of diff. co. unaccompanied by that of x takes away the possibility of a pair of real roots, or proves the existence of a pair of imaginary ones, Moreover, since signs can only be lost in even numbers, except when xz vanishes, the loss of an odd number of signs in passing from w=a the less, to x=0 the greater, shows that there must be one real root between « and 8, at least. There may be as many real roots in that interval as there are changes of sign lost; but if no change be lost, there cannot be any real root in the interval. The following are instances of the manner in which the changes of sign are lost, it being remembered that every function which vanishes is to differ in sign from its diff. co. before vanishing, and to agree with it afterwards :* ® One real | 6! df Two equal | d d' " fp" Three equal ®=a-h TF + root: one +z + real roots: A st real_roots: L=a 0 + change |0 0 + two changes}0 0 0 + three changes t=ath ++ ost. nt BE lost. Ne eA lost. hee pt FS BU il p' 6" $” Two imaginary | pp! pl No root. Phe me ce roots: two qa ae ae ay eee ra sit? Cheat changes | ee) ATT Idee © Boh By koe lost. rear yet os p"” p” p dd" Four imaginary | 6!’ db” pd’ d" Two ima- @—a-h + F+7+ = roots: four + fF t F + ginary roots: 2a Bl Ou 0 Og: changes + 0 0 O + two changes e—oth t+ttt+ + lost. ice ees lost. (Page 253.) Stirling (Meth. Diff:, p. 8, Introduction) is the first I can find who used the differences of nothing, though not under that Name or definition. He uses the divided form, and obtains them as the Coefficients of the development of ACB al) (8-2). 0285 giving a theorem which we should now express by 1 9 2 kh At 0" AE o*t! A‘ Ort? nl Le in iia hk Baar dT ede ak ee oe ee (n—1)(n—2)....(n—h) RiP Ging net? * For a more full account of this theorem, which is here given merely to show how the differential calculus has been applied in the subject of equations, see the article Sturm’s Theorem in the Penny Cyclopedia ; Young or Hymers, on Equa- tions ; or Peacock’s Report on Analysis to the British Association. 774 7 APPENDIX. which I leave to the student to prove. Stirling also uses the table of the coefficients of (v—1)(#—2).-- +5 and I leave the following also to the student. If A,,. be the sum of the products of every selection of m numbers out of 1, 2,3,--+ +7, then An, Pa Ana? (n + 1) Aiko nd from which a table of coefficients for (v—1)(a—2)....(a@—m) may be rapidly found. (Page 305.) Burmann’s theorem is nothing but Lagrange’s, as follows. If c=atyfax, Lagrange’s theorem is d 4 yar (pa) t (p's fr). y+(5 {y's (fr)’$ ) a +($3 {ya (ft ) wht he. where the external parentheses denote that r=a after the differentia- tions. This is expanding Wa in powers of (w—a): fr. Let y ory (a—a) :fr=r, whence (ex and x—a vanish together; substitute pa for y, and (x—a): Oz for fa, and we have Burmann’s development. (Pave 313.) The symbol fy, dr, or D- y,, is found from 1+4= ee and we have 0 noth baie " ~ log (L+4) —0(AT+V,+V,A+V,4°+.-- “) is A A? aU es Vv, — - BNP See Ee @eee f¥X ( A ORs ai ee V4 a ) since 1:log(1+A) is not altered by changing its sign, and writing —A:(1+A) for 4. Take the value at the upper limit, from the | second expression, and that at the lower limit from the first, and the | expression in the page cited is readily obtained. (Page 330.) The student must observe that the instance taken, be + ce + Of, though it serves well] enough to show the method, could never occur in any example, since it is not itself the derivative of any- thing. The mode of forming the derivatives given in the page cited, though | advantageous for the beginner, as saving him from error by presenting most of the terms several times, formed in several different ways, admits of simplificauon. ‘The process need only be performed on the last letter which enters, except where the last but one is that which comes immediately before the last in the series a, b, c, e, &c., in which case operate also upon the last but one. This will prevent the third rule in page 330 from ever being wanted. Thus, in forming D® d* from D*o*, oes 4b*h gives only 4b°h | 12bc2 f gives only 12bc* § 12b°cg gives only 120° ch 12bce® gives 24bcef+ Abe 12b' ef gives 12beg +65" f? Ac’e gives Ac®f+6c° é APPENDIX, 775 In the following tables the method of Arbogast is applied to the forms which more frequently occur. Let x a x2 $(atbete Ste s+ vee RAGE A DH Ae Soe, Then A,,=D"~" 6.f'a+D"“*b? h"a+.... + Db" pa +b" o™a, where D” 5” (which does not mean the same thing as in the text) is to be taken from the following table: : Dac D°?b=e, D B?=3be oa Mae D*b’= 4be +-3c?, ny Db?= 60’e D=g, Dv=5of+10ce, Ds=10s%+415be, Db*=10b% D*b =h, D‘b’=6bg + 15cf+ 10e?, D*b*= 1567+ 60bce +15 D°b*= 206% + 4567c?, DB=15d‘c D% =k, D°s*= Tbh + 21eg + 35ef D*b?= 216°g + 105bcf+ 7104e? + 105c*e D*b*= 356° f+ 210b%ce + 105dc8 D*b5=35b4e+ 1050°’, Dit= 21 bic Db =1, D°b? = 8bk + 28ch + 56eg + 35f? D°b?= 28b*h + 168beg + 280bef+ 210c°f 4 280ce? Dib*= 56b*g + 420b°cf+ 2806°c? + 840bc%e + 105c* D*b'= 70b*f +-5606%ce + 4206°c° D*b'=56b5e + 210b4c*, Db?=28b'e Yo =m, D7b°= 9b1 + 36ck + 84eh + 126f D°b®=360°k + 252bch+ 504beg + 315bf/?+378c7g + 1260cef+ 280¢° _ D°t*=84b°h+ 756b°cq + 1260b°ef-+-1890bc° f+ 2520bce* + 1260c%e | D*h5=126b4¢ + 1260b%cf + 840b%e? + 37800%%e-+ 945bc% D%°=12655f + 1260b%ce + 12606%° D*b7=84b%e +-3'78b5c°, Dbh?=360'c D%) =n, D*t?=10bm+45cl-+ 120cek + 210fh+ 1268" D7b°= 450°/ + 360bck + 840beh + 1260bfg + 6300°h + 2520ceg + 1575cf? 4+2100e%f D*%b*= 120° + 1260b?ch + 2520b'e¢ + 1575b°f?+ 3780bc7g-+ 12600bcef 4 2800be* + 3150c°f + 6300c%e? » D°*=210b*h +. 2520b%cg + 4200b%ef + 9450b°c%f + 12600b°ce -+-12600bc%e +-945c5 D*h§= 25 2b5¢ + 3150b*cf'+ 21005*e? + 12600b%c*e +4725 b°c* | D°b7=210b°f + 2520b%ce-+3150b'c* | Db'=120b7e+ 630b°c", Db = 456°. 776 APPENDIX. a Thus we have # atbates eine Seu -)= pa+bhd/a.c+(cpa+ b°p!’a) s + (eplat+ 3h f"a+b'p""a) aa x + ( fp! a+ (4be+3c*) da 60% 6a +b" a) — aa + &c. up to the tenth power of 2. The student may apply this to the verifica- tion of the series in pages 262, 264, and 315. The numerical coefficients above given have been carefully verified on a heat log (142 +5 on ora ee and in the literal part the terms all agree with those of page 330. (Page 410.) The following theorem will be yery useful in this part of the subject: (a? B+ 0°)(p? +g? -+ r°)—(ap + bq-tery =(aq—bp)?+ (br —cq)?+ (ep —ar)’. (Page 559.) Dr. Hutton’s method is not quite so convenient as the | following. Find Aa,, A’a, &c. in the usual way, and let A”a, be the last which is employed: Take half A’a, from A”a), half the result | from A"~a,, half the result from A"~%a,, and so on, until half a result has been taken from a); then halve this last result, which gives the approximate value of @—a,+...- This leads to the same result as Dr. Hutton’s mode, and saves the summations required at the begin- | ning of the latter, and most of the divisions by 2. cies 621.) To avoid confusion, I have omitted all notice of another mode of development, which may be obtained as follows. Add the two | series in page 621, which gives i oD hy ceo RelA cp tyes ees a =Il¢r O(a)l 4By+ B, cos ee P rant ee hi Lei B= if ¢(v+rl) cos = — + ae, Als fi o(v+l) Sh wana silat ; ten 3B,+B’, cos “et »...+ A’, sin a+ ae Petes 0) —i(x) On =16 (x+l) O(a)l Write x—l for x in the last, and we have 3B’,—B‘; cos ind A .».—A’, sin + ei eo O(r)l d ‘ =Ipz (x) 2l Add the first and third series, and we have APPENDIX. 797 4 QV + (By +B) + (B,—B’,) cos *+(B, +B) Cosh : PRS, Gimigi ame ein): +(A,— A’) sin >-+ (As+ Als) sin a It is (0, a, 27) and not 0(x)/(r) 2/, as might at first be supposed, because when x=0, or J, or 21, both the double series give L/px, and their sum gives /Ox. Qnrv l Qnrv d And Best Bien {4 {ov-+¢ (v+2)} cos. Goa (2 @v cos dy Bony Bien fo {ov—¢ (v+/)} cos dv (2n-+1) rv l 2n+ 1 = fi pv aa nt ae Lea (os Deck ans) OED . 9 ) = [2 dv cos EE ap ; , NEV m eve Also A, tA,= fo pv sin or an iB n ae by similar reasoning. Hence our final conclusion is TL on. Tv ai a Qrv lor=4 {5 ov dv-+cos 7 fii ov cos —- du-t cos —— fi) v.cos — dub... Mase foil eh as, . Qe . Qarv +sin— fo gv sin 5 dv-+sin ae [i ov sin — dv-+... Hence we have two distinct ways of expanding Jz in a series of both sines and cosines: namely Lage dot 35( figu eos a0.cos™™ 4 35( J. pu sin dv.sin : ; NEV Nee Ray ise, _ nev _ NTL | ft yvdv+B3( f¥pveosedv.cos j J+ 35 SiposinFavsin™™), but the first is only true from 2=0 to z=], and vanishes from »=1 to x= 2l, becoming 3/px when x=0, or /, or 2/: while the second is true from #=0 to a= 2/, both inclusive. ERRATA. In the following columns, the first denotes the page, the second the line; thus (10 means the tenth line from the top, and 10) the tenth line from the bottom of the page, not reckoning notes, if any. The | third column contains the erratum, and the fourth the correction. The | numerical tables in pages 253, 554, 587, 590, 657, and 662 have been | carefully compared with the authorities. 13 18 20 21 22 24 25 28 35 40 46 55 58 60 61 63 2) ‘0001 | “001. | Note | The assertion as to Peyrard refers to his smaller (or | octavo) translation of Euclid: the author was not | then aware of the existence of the larger one. Omit the word that. same . same time. were we are. a—2G 28 — a. (8, 10 | two hundredth, 200, 8 hun dredth, 100, 4. (27 absolutely absolute. (17 in a second in the fraction & of a second. 5, 6) (1 (px)? 13) D0 dy 13 37°; — ( d dx (10 sin? 1) V(2—1)22 | C7 cosec® % cosec? wu. du_ ie du dx dx F nhenmett9 , dgv_ Ri tenors (y a2) @C and ¢e do. do. do. do. lo. S| pz between GCand¢c | Gx between ¢’C and gc. ; P and Q. | A" a=0,) [alla o; Og, (t—a)* ; 2.3.4. | (n+1) ‘ Sager ' A? Chet? cht! | he Crantt fb cht 7 23..0 2.3.0 23.0 (23.0 23.0 23.NF1 | nm (n—1) n(n—1)a"~*. | di e — log @# u=1l+a da 08 *: ERRATA, 779 73 4) r+? her, 74 10) *501 *508. 75 17) px (C pac (C 76 (17 2°'11728 2°71828 78 | In Tab. Aru, Aa) A’u A és of Aus Af ‘ Leu, At Diff. Asy we A’, Uy Saeed! ; n—J ia 16) FN Us E nus + uy +n Ug -NUg+ Uy. — 3) | Strike out = between the columns. 79 | 4) 0, 6, 80 (6 Au-+v Au+ Av. 81 | (13, 14 y U. 83 6) vw e-+l. ay 2) w a, b, c, &e. n—l n—2 8 21 apie ‘ ( 3 3 au d*u 2 —— ao, 4 ( dx dy dy dx oS 1) (Az)? (42,)*. 89 (11 z u. du du 91 (11 dy Ax ay A 93 Gt of values of values of. 95 (18 objectional objectionable. 98 (5 greatest greatest and least. 99 14) being the being a the. 100 (20 a+nw or ath a+(n—1) w or a+h—w. 104 (6 z+ Latte — C7 a+] Ame tap = (16 —1:0001 —1+ 0001. 107 | 3, 4) diff. co. differential. 108 a) (on ae U2 dz. 111 (19 cos” 6 sin”~2 0 cos’ @ sin”~* dé. d.sin @ d.sin 0 i. a 1—sin?0 } —sin? 0” 6 a a. —— ) b — V b oe 114 (12 —c —2 in denom. Noce+a — 2) that than 115 (4 V1 8+ 4ac Vi=—4ac. ae ye 6} Ql Nar a | T wT y 5) a yy 117 5) | These results are subject to any error which may arise from integrating a function which becomes infinite between the limits of integration. log (—1) A’y .2m Remove the negative sion to the first. 2k [ and 2k vers“! Omit the words in pa- rentheses. 1 3 an! x is aN 2—aJ0 k t—At, and t” seconds scribed between the end of 10 and 20 seconds a’ ul —u! ox to y only BrERNOUILLI’S fractions 0 x Oo a becomes 0 ERRATA. Log (—1). 4th, cube n~- Na, sign from the second expres- k af and & vers7?. 2 3° Vax is V2ar—/2a0. the density. t—At, t, and. t feet. what is the length de- | what is the number of seconds in which the point moyes from 10 to 20 feet. matter manner. proportion proposition. could not be dx could not be ¢/z. to a to A. a as — — and+—_— — and — ee OTs ay ee ‘ Cuaprter III. Cuarrer IX. (a, 2). x (a, w). d?x Ora du du? Cen Ge K2 K” The letters A,, As, Az, &c. have been inadvertently used for different things in these two pages; in the first they stand for (2), (a), &c., and in the second for (u’), (w’’),&c. i iv__a/ 2 +g ag!!! —ay" al’, or to y only. BERNOULIW’S. exponents. Ovex +00. 0. becomes 1. I , and. 1 | | ERRATA. 2.3 * 64 g—) Oo=—1 a(x,y) ¢, ¢ includes annexed to y Ww here 2°=—2Z, ag ye” c(hn —- keg=1)x W, dW and: —-— as dx dx primitive diff. equ. series Qn Aa op" + 2 PL==VvL Pax 1 2.3.6 | On an error of reasoning contained in these pages look forward to page 327. 1— | evry e seems to include. annexed to when. J ‘Pdz. given diff. equ. series of Vian 18 Vs a—b-+-c. Hon+2° as of its. | A? ay. De 10: 2.3.4. 2.3, it. Wwr= VE .WaL, Mm. 781 A and B should be at the extremities of the continued curve line. For the completion of this test of convergency, see page 326. 6L Aa, x +rsin o tan7 tan (v3 0) 750 6+ m7 6l. 4a, 2. —rsin ¢. Tv tan7! tan €é + Ss 2 °° r—é . 2 ERRATA. 8—2)}Omit this paragraph altogether, as it is rendered | saat F jalge by the preceding mistake. (12 PQ @) fole(e) | @)ele®) (21 Poe. ae he same reason as. tS —— he — sin sin? the : A@opr AMO 2, S 1 k—2 —k and +k — ~(k-1), 4-1) at —— {[k—1, p] and [k— Ay p| [k—1,k—1+ 7] and [k—2, k—2-+ 9]. soe a . A®r, &c. Adz, A’dz, &c. qo Ly2— fryed x Sys icy pda. 7) oe nothing and unless inseré when x=a. (6 Ci) (a—1). (8 omit (when r=0). (9 (x—2)?. P25 el 5) 66 7aT" 4) ; (t—1)7!, (x—1)~ &e. 278 (11 »—a. 279 |\(2 and 9 (Av +B) dz. 293 ——-~— 7 The student should ascertain for himself that this error is of no con- sequence, and that its results in (5 and (10 are true. expressions. limits. positive negative. a and } c and 8. alter thus - { a(1+2)*" G/l+e+ ys} doby—z yyote, »cos a.0°—aq? cos a (0” —a®). u Une by o(@+ A°A0? and @’’y A’ A*0? and i ERRATA. 783 log A lo 329 1) A—B igs rea 329 (10 A, +2A.ur+... (A,+2A,¢+...) divided by (a,+ 2agtt+...). 331 6) | 4c*f+4be 4c®f+4be°. 336 (19 m—2 m—1, 337 16) m im. 341 (15 science of science to. me 3) Oy’ Oy!. 352 (9 ) Us. — (21 FE 2F 363 | (16, 20 ais > or Esq. Major Sir William Lloyd. Yarmouth—C. E. Rumbold, Esq. Dawson Turner, Esq. York—Rev. J. Kenrick, M.A. John Phillips Esq., F.K.S. F.G 2. THOMAS COATES, Esq., © ecretary, 99, Lincoln’s-inn Fields, LONDON : Printed by W. Cuowes and Sons, Stamford Street. ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS. Tue Differential and Integral Calculus, or, as it was formerly called in this country, the Doctrine of Fluxions, ‘has always been supposed to pre- sent remarkable obstacles to the beginner. It is matter of common ob- servation, that any one who commences this study, even with the best ele- mentary works, finds himself in the dark as to the real meaning of the processes which he learns, until, at a certain stage of his progress, depending upon his capacity, some accidental combination of his own ideas throws light upon the subject. The reason of this may be, that it is usual to in- troduce him at the same time to new principles, processes, and symbols, thus preventing his attention from being exclusively directed to one new thing at a time. It is our belief that this should be avoided; and we propose, therefore, to try the experiment, whether by undertaking the solution of some problems by common algebraical methods, without call- ing for the reception of more than one new symbol at once, or lessening the immediate evidence of each investigation by reference to general rules, the study of more methodical treatises may not be somewhat facilitated, We would not, nevertheless, that the student should imagine we can re- move all obstacles ; we must introduce notions, the consideration of which has not hitherto occupied his mind; and shall therefore consider our object as gained, if we can succeed in so placing the subject before him, that two independent difficulties shall never occupy his mind at once. The ratio or proportion of two magnitudes, is best conceived by ex- pressing them in numbers of some unit when they are commensurable ; ‘or, when this is not the case, the same may still be done as nearly as we please by means of numbers. ‘Thus, the ratio of the diagonal of a square to its side is that of ,/ 2 to 1, whichis very nearly that of 14142 to 10000, and is certainly between this and that of 14143 to 10000. Again, ally ratio, whatever numbers express it, may be the ratio of two mag nitudes, each of which is as small as we please; by which we mean, that if we take any given magnitude, however small, such as the line A, we may find two other lines B and C, each less than A, whose ratio shall be what- ever we please. Let the given ratio be that of the numbers m and n. Then, P being a line, mP and nP are in the proportion of m ton; and it '§ evident, that let m, m, and A be what they may, P can be so taken that mP shall beless than A. This is only saying that P can be taken less ‘han’ the m™ part of A, which is obvious, since A, however small it may be, las its tenth, its hundredth, its thousandth part, &c., as certainly as if it Were Jarger. We are not, therefore, entitled to say that because two 2 ELEMENTARY ILLUSTRATIONS OF magnitudes are diminished, their ratio is diminished; it is possible that B, which we will suppose to be at first a hundredth part of C, may, after a diminution of both, be its tenth or thousandth, or may still remain its hundredth, as the foliowing example will show :-— C 3600 1800 36 90 B 36 15 foo 9 Bn =e Bee Be i yecpbe fe 100 1000 100 10 Here the values of B and C in the second, third, and fourth column, are less than those in the first; nevertheless, the ratio of B to C is less in the second column than it was in the first, remains the same in the third, and is greater in the fourth. In estimating the approach to, or departure from equality, which two magnitudes undergo in conse- quence of a change in their values, we must not look at their diffe- rences, but at the proportions which those differences. bear to the whole magnitudes. For example, if a geometrical figure, two of whose sides are 3 and 4 inches now, be altered in dimensions, so that the corre- sponding sides are 100 and 101 inches, they are nearer to equality in the second case than in the first; because, though the difference is the same in both, namely one inch, it is one-third of the least side in the first case, and only one-hundredth in the second. This corresponds to the common usage, which rejects quantities, not merely because they are small, but because they are small in proportion to those of which they are con- sidered as parts. Thus, twenty miles would be a material error in talking | of a day’s journey, but would not be considered worth mentioning in one. of three months, and would be called totally insensible in stating the distance between the earth and sun. More generally, if in the two quan: | tities 2 and 2 +4, an increase of m be given to 2, the two resulting quantities 7 + m and 2 + m+ a@are nearer to equality as to their ratio | than wand # + a, though they continue the same as to their difference; | ey and ot of which m7] a rem x+-m +m a for is less than aid and therefore 1 + is nearer to unity than 1+ | x a +m In future, when we talk of an approach towards equality, we mean that the ratio is made more nearly equal to unity, not that the difference is more nearly equal to nothing. The second may follow from the first, but! not necessarily ; still less does the first follow from the second, It is conceivable that two magnitudes should decrease simultaneously *,. so asto vanish or become nothing, together, For example, let a point A move on a circle towards a fixed point B. The are AB will then dt minish, as also the chord A B, and by bringing the point A sufficiently near to B, we may obtain an are and its chord, both of which shall be smaller than a giyen line, however small this last may he. But while the magnitudes diminish, we may not assume either that their ratio increases,| diminishes, or remains the same, for we have shown that a diminution o!| * In introducing the notion of time, we consult only simplicity. It would do equally) well to write any number of successive values of the two quantities, and place them in two columns, . THE DIFFERENTIAL AND INTEGRAL CALCULUS. 3 two magnitudes is consistent with either of these. We must, therefore, look to each particular case for the change, if any, which is made in the ratio by the diminution of its terms. And two suppositions are possible in every increase or diminution of the ratio, as follows: Let M and N be two quantities which we suppose in a state of decrease. The first possible case is that the ratio of M to N may decrease without limit, that is, M may be a smaller fraction of N after a decrease than it was before, and a still smaller after a further decrease, and so on; in sucha way, that there is no fraction so small, to which ‘e shall not be equal or inferior, if the decrease of M and N be carried sufficiently far, As an instance, form two sets of numbers as in the adjoining table :— M Spe eee bare ’ 20 400 8000 160000 N ee kee) ae as Le th 2 4 8 16 Ratio of MtoN 1 25 i Sa hie 10 100 1000 10000 Here both M and N decrease at every step, but M loses at each step a larger fraction of itself than N, and their ratio continually diminishes. To show that this decrease is without limit, observe that M is at first equal to N, next it is one tenth, then one hundredth, then one thousandth of N, and so on; by continuing the values of M and N according to the same law, we should arrive at a value of M which is a smaller part of N than any which we choose to name; for example *000003. The second value of M beyond our table is only one-millionth of its corresponding value of N; the ratio is therefore expressed by *000001 which is less than °000003. In the same law of formation, the ratio of N to M is also increased without limit. The second possible case is that in which the ratio of M to N, though it increases or decreases, does not increase or decrease without limit, that is, continually approaches to some ratio, which it never will exactly reach, however far the diminution of M and N may be carried. The following is an example ; — 1 1 1 ] ] a ut ey eee ee N 1 a a jar geaisebe i> ak &e. 4 9 16 ..25. -86 +49 ; 4 Oe a a5 36 49 &e Ratio of M toN 1 3 6 To ii D1 98 a 9 | 4 The ratio here increases at each step, for - is greater than I, “a than =" and so on. The difference between this case and the last, is that the ratio of M to N, though perpetually increasing, does not increase without limit; it is never so great as 2, though it may be brought as near hs 2 as we please. To show this, observe that in the successive values of M, the denominator of the second is 1+2, that of the third 1+2-+3, and so on; whence the denominator of the a value of M is B 2 — ELEMENTARY ILLUSTRATIONS OF 2- l aha (400 a Be Olax. # Therefore the a” value of M is aean and it is evident that the 2 value Ce Ci a | re : : ‘ el at ah 2 Qu iat o the 2 value of the ratio _-_—=—_—__—__, or ___., of N is > which gives thew” valu RCE RS WA v % 2. Ifa«be made sufficiently great, ; may be brought as or xtl x é ; 1 ‘ihe 1 near as we please to 1, since, being 1 — pray it differs from 1 by ae x x+l which may be made as small as we please. But as , however great is always léss than 2. Therefore az may be, is always less than 1, 3 a ai I., continually increases ; II., may be brought as near to 2 as we please ; III. can never be greater than 2. This is what we mean by saying that M is an increasing ratio, the limit of which is 2. Similarly of = which is the reciprocal of 2 we may shew, I., that it continually decreases; II., that it can be brought as near as we please to; III., that it can never ‘ Ne ; ; be less than3. ‘This we express by saying that 4 is a decreasing ratio, whose limit is 4. To the fractions here introduced, there are intermediate fractions, which we have not considered. Thus, in the last instance, M passed from 1 to 4 without any intermediate change. In geometry and mechanics, it is ne- cessary to consider quantities as increasing or decreasing continuously ; that is, a magnitude does not pass from one value to another without passing through every intermediate value. Thus if one point move towards another on a circle, both the are and its chord decrease conti- nuously. Let AB be an arc of a circle, the centre of whichis O. Let A rr remain fixed, but let B, and with it the radius O B, move towards A, the point B always re- N maining on the circle. At every position of B, suppose the following figure. Draw A T touch- ing the circle at A, produce O B to meet AT Higa parallel toO A, and join B A. Bisect the are o M -“ and bisecting it. The right-angled triangles ODA and BMA having a common angle, and also right angles, are similar, as are alsoBOM and TBN., in T, draw B M and BN perpendicular and ~ A B in C, and draw O C meeting the chord in D © If now we suppose B to move | THE DIFFERENTIAL AND INTEGRAL CALCULUS, 5 towards A, before B reaches A, we shall have the following results: The are and chord BA, BM,MA, BT, TN, the angles BO A, COA, MBA, and 'T BN, will diminish without limit ; that is, assign a line and an angle, however small, B can be placed so near to A that the lines and angles above alluded to shall be severally less than the assigned line and angle, Again, O T diminishes and O M increases, but neither without limit, for the first is never less, or the second ereater, than the radius. The angles OBM, MAB, and B TN, increase, but not without limit, each being always less than the right-angle, but capable of being made as near to it as we please, by bringing B sufficiently near to A. So much for the magnitudes which compose the figure: we proceed to consider their ratios, premising that the arc A B is greater than the chord A B, and less than BN+N A. The triangle B M A being always similar to OD A, their sides change always in the same proportion; and the sides of the first decrease without limit, which is the case with only one side of the second. And since O A and O D differ by D C, which diminishes without limit as compared with O A, the ratio O D +O A is an increasing ratio _ whose limit is 1. But OD+OA=BM-—BA;; we can therefore bring B so near to A that B M and B A shall differ by as small a fraction of either of them as we pledse. ‘To illustrate this result from the trigo- nometrical tables, observe that if the radius B A be the linear unit, and ZBOA=86, BM and BA are respectively sin. 6 and 2 sin. 4 0. Let 6= 1°; then sin. 0= °0174524 and 2 sin 4 0 = -°0174530; whence 2 sin. + 0 +8in. 9 = 1°00003 very nearly, so that BM differs from BA by Jess than four of its own hundred-thousandth parts. If Z BOA=4/, the same ratio is 1:0000002, differing from unity by less than the hundredth part of the difference in the last example. Again, since D A di- minishes continually and without limit, which is not the case either with OD or OA, the ratios OD—~ DA and OA—DA increase without limit. These are respectively equal to BM—~MA and BA~MA; whence it appears that, let a number be ever so great, B can be brought so near to A, that B M and B A shall each contain M A more times than there are units in that number. Thus if Z BOA=1°,BM—MA = 114°589 and BA + MA=114°593 very nearly; that is, BM and BA both contain MA more than 114 times. If Z BOA = 4, BM—MA = 1718°8732, and BA —~ MA = 1718-8375 very nearly ; or B M and B A both contain M A more than 1718 times. No difficulty can arise in conceiving this result, if the student recollect that the degree of greatness or smallness of two magnitudes determines nothing as to their ratio; since every quantity N, however small, can be divided into as many parts as we please, and has therefore another small quantity which is its millionth or hundred-millionth part, as certainly as if it had been greater. There is another instance in the line T N, which, since T B N is similar to B O M, decreases continually with respect to TB, in the same manner as does B M with respect to O B. The are BA always lies between BA BA ,. } f MA; henc are © “lies between 1 and and BN+NA,orBM-+MA SESE os sewriy BM se Me me ™M has been shown to approach continually BA BA BA MA to decrease without limit; hence _are B AL con- towards 1, and chord BA 6 ELEMENTARY ILLUSTRATIONS OF tinually approaches towards 1. If 2 BOA =I, are BA x chord BA -0174533 + 0174530 = 1°00002, very nearly. If 7 BOA= 4’, it is less than 1:0000001. We now proceed to illustrate the various phrases which have been used in enunciating these and similar propositions. It appears that it is possible for two quantities m and m -r 7 to decrease together in such a way, that 7 continually decreases with respect to m, that is, becomes a less and less part of m, so that ” also decreases m when m2 and m decrease. Leibnitz * in introducing the Differential Calculus, presumed that in such a case, ” might be taken so small as to be utterly inconsiderable when compared with m, so that m+n might be put for m, or vice versa, without any error at all. In this case he used the phrase that 7 is infinitely small with respect to m. The following example will illustrate this term. Since (@ + h)PP=a+2ah+h, it appears that if a@ be increased by h, a2 is increased by 2ah+ h?. But if h be taken very small, #? is very small with respect to h, for since 1: hit h th, as many times as | contains A, so many times does h contain h?; so that by taking / sufficiently small, A may be made to be as many times #? as we please. Hence, in the words of Leibnitz, if 2 be taken infinitely small, h* is infinitely small with respect to , and there= fore 2ah-Lh’ is the same as 2a@h; or if a be increased by an infinitely small quantity 4, a is increased by another infinitely small quantity 2 af, which is tof in the proportion of 2a tol. In this reasoning there is evidently an absolute error ; for it is impossible that 4 can be so small, that 2ah -+ h? and 2 ah shall be the same. The word small itself has no precise meaning; though the word smaller, or less, as applied in com- pariig one of two magnitudes with another, 1s perfectly intelligible. Nothing is either small or great in itself, these terms only implying a relation to some other magnitude of the same kind, and even then varying their | meaning with the subject in talking of which the magnitude occurs, so that both terms may be applied to the same magnitude: thus a large field is a very small part of the earth. Even in such cases there is no natural point at which smallness or greatness commences. The thousandth part — of an inch may be called a small distance, a mile moderate, and a thousand leagues great, but no one can fix, even for himself, the precise mean between any of these two, at which the one quality ceases and the other begins. These terms are not therefore a fit subject for mathematical discussion, until some more precise sense can be given to them; which shall prevent the | danger of carrying away with the words, some of the confusion attending their use in ordinary language. It has been usual to say that when h decreases from any given value towards nothing, h? will become small as compared with h, because, let a number be ever so great, # will, before it becomes nothing, contain h® more than that number oftimes. Here all * Leibnitz was a native of Leipsic, and died in 1716, aged 70... His dispute with | Newton, or rather with the English mathematicians in general, about the invention of | Flusions, and the virulence with which it was carried on, are well known. The decision | of modern times appears to be that both Newton and Leibnitz were independent inventors of this method. It has, perhaps, not been sufficiently remarked how nearly several of their predecessors approached the same ground ; and it is a question worthy of discussion, whether either Newton or Leibnitz might not have found broader hints in writings acces- sible to both, than the latter was ever asserted to have received from the former, THE DIFFERENTIAL AND INTEGRAL CALCULUS. 7 dispute about a standard of smallness is avoided, because, be the standard whatever it may, the proportion of h* to kh may be brought under it. It is indifferent whether the thousandth, ten-thousandth, or hundred-millionth part of a quantity is to be considered small enough to be rejected by the side of the whole, for let hk be zh ee or Stil eae 1000 © 10,000 100,000,000 unit, and / will contain h*, 1000, 10,000, or 100,000,000 of times. The proposition, therefore, that 2 can be taken so small that 2ah+h? and2ah are rigorously equal, though not true, and therefore entailing error upon all its subsequent consequences, yet is of this character, that, by taking & sufficiently small, all errors may be made as small as we please. The desire of combining simplicity with the appearance of rigorous demon- stration, probably introduced the notion of infinitely small quantities 5 which was further established by observing that their careful use never led to any error. ‘The method of stating the above-mentioned proposition in strict and rational terms is as follows :—If @ be increased by h, a is in- creased by 2ah + h?, which, whatever may be the value of A, is to h in the proportion of 2a@+hto 1. The smaller h is made, the more near does this proportion diminish towards that of 2@ to 1, to which it» may be made to approach within any quantity, if it be allowable to take h as small as we please. Hence the ratio, increment of a? increment of a, is a decreasing ratio, whose limit is 2a. In further illustration of the language of Leibnitz, we observe, that according to his phraseology, if A B be an infinitely small arc, the chord and are A B are equal, or the circle is a polygon of an infinite number of infinitely small rectilinear sides. This should be considered as an abbreviation of the proposition proved (page 5), and of the following :—If a polygon be inscribed in a circle, the greater the number of its sides, and the smialler their lengths, the more nearly will the perimeters of the polygon and circle be equal to one another; and further, ifany straight line be given, however small, the difference between the perimeters of the polygon and citcle may be made less than that line, by sufficient increase of the number of sides and dimi- nution of their lengths. Again, it would be said that if A B be infinitely small, M A is infinitely less than BM. What we have proved is, that M A may be made as small a part of B M as we please, by sufficiently diminishing the arc B A. of the An algebraical expression which contains x in any way, is called a a+ 2x a— & function of x. Such are a+ a’, , log (a + y), sin 27. An expression may be a function of more quautities than one, but it is usual only to name those quantities, of which it is necessary to consider a change in the value. Thus if in 2?+ a@*, xv only is considered as changing its value, this is called a function of 2; if z and @ both change, it is called a function of x and a. Quantities which change their values during a pro- cess, are called variables, and those which remain the same, constants ; and variables which we change at pleasure are called independent, while those whose changes necessarily follow from the changes of others are called dependent. Thus in fig. (1), the length of the radius O B isa constant, the are A B is the independent variable, while BM, MA, the chord AB, &c., are dependent. And, as in Algebra we reason on numbers by means of general symbols, each of which may afterwards be particu- 8 ELEMENTARY ILLUSTRATIONS OF larized as standing for any number we please, unless specially prevented by the conditions of the problem, so, in treating of functions, we use general symbols, which may, under the restrictions of the problem, stand for any whatever. The symbols used are the letters F, f ?,9,%; $ (2) and ¥& (2), or da and wa, may represent any functions of «, just as v may represent any number. Here it must be borne in mind that ¢ and & do not represent num- bers which multiply 2, but are the abbreviated directions to perform certain operations with 2 and constant quantities. Thus, if év=ar+2’, P is equiva- lent toa direction to add z to its square, and the whole x stands for the result of this operation. Thus, in this case, @ (1) =2; 6 (2)=6;3 pa=at+a’; d(ath)=aexth+(e+hy; osinz=sin a + (sin cz)’, Itmay be easily conceived that this notion is useless, unless there are propositions which are generally true of all functions, and which may be made the foundation of general reasoning. To exercise the student in this notation, we proceed to explain one of these, of most extensive application, known by the name of Taylor’s Theorem. If in dz, any function of x, the value of x be increased by h, or 2 + h be substituted instead of «, the result is denoted by («+ h). It will generally* happen that this is either greater or less than G2, and h is called the increment of x, and (x + h) — a is called the increment of dx, which is negative when (a +h) < gz. It may be proved that @ (a + h) can generally be expanded in a series of the form pu + ph + qh? + rh? + &e., ad infinitum, which contains none but whole and positive powers of h. It will happen, however, in many functions, that one or more values can be given to 2 for which it is impossible to expand f (# + /) without introducing negative or fractional powers. ‘These cases are considered by themselves, and the values of 2 which produce them are called singular values. As the notion of a series which has no end of its terms, may be new to the student, we will now proceed to shew that there may be series so constructed, that the addition of any number of their terms, however great, will always give a result less than some determinate quantity. ‘Take the series Llt+at+a2?+ a+ at + &e.,, in which z is supposed to be less than unity. The first two terms of this series may be obtained by dividing 1 — aw by 1— 2; the first three by dividing 1 — # by 1 — x; and the first m terms by dividing 1 — a” by 1 —wv.. If be less than unity, its successive powers decrease without limit}+ ; that is, there is no quantity so small, that a power of x cannot be found which shall be smaller. Hence by taking m sufficiently great, 1 — 2” 1 a or — ——— may be brought as near to I—a 1-«# l—w@ —r it than which, however, it must always be less, since —-— can never en-_ —2r as we please, * This word is used in making assertions which are for the most part true, but admit of exceptions, few in number when compared with the other cases. Thus it generally happens that 22 — 10a -+ 40 is greater than 15, with the exception only of the case where x =5. It is generally true that a line which meets a circlein a given point meets it again, with the exception only of the tangent. + This may be proved by means of the proposition established in the Study of Mathe- matics, page 81. For” yx ™, is formed (if m be less than n) by dividing ” into 12 12 nm a parts, and taking away n— mof them, r THE DIFFERENTIAL AND INTEGRAL CALCULUS. 9 tirely vanish, whatever value 2 may have, and therefore there is always 1 wen It follows, nevertheless, that — = l1+et+ 2°+ &c., if we are at liberty to take as many terms as we something subtracted from please, can be brought as near as we please to , and in this sense we sao v 1 say that Te S=l+a2r+ 2°+ 2 + &e., ad infinitum. A series is said to be convergent when the sum of its terms tends towards some limit; that is, when, by taking any number of terms, however wreat, we shall never exceed some certain quantity. On the other hand, a series is said to be divergent when the sum of a number of terms may be made to Surpass any quantity, however great. Thus of the two series, ew aie oT Ti ke pt hinge t+ 7+ = + ke and 1+2+ 4+ 8 + &e. the first is convergent, by what has been shown, and the second is evidently divergent. A series cannot be convergent, unless itS separate terms decrease, so as, at last, to become less than any given quantity. And the terms of a series may at first increase and afterwards de- crease, being apparently divergent for a finite number of terms, and convergent afterwards, It will only be necessary to consider the latter part of the series. Let the following series consist of terms decreasing without limit: atb+c+td+....+k+l4+tm+4.... which may be put under the form ee ales cau oft Sed OP ae a b @ @ aby a the same change of form may be made, beginning from any term of the series, thus: k+l + m+ &. =k(L+ 4. T+ 8, b : We have introduced the new terms —, — &e., or the ratios which the Ry several terms of the original series bear to those immediately preceding. ; 5 RO It may be shown, I., that if the terms of the series —, = at ES a C come at last to be Jess than unity, and afterwards either continue to ap- proximate to a limit which is less than unity, or decrease without limit, the series a + b + c+ &c., is convergent; II., if the limit of the terms > Cc it a the series is divergent. 1. Let ’ be the first which is less than unity, and let the succeeding ratios v , &¢c., is either greater than unity, or if they increase without limit, . ar l m ‘ &c., decrease, either with or without limit, so that 9 > aie 3 _, &e.; whence it follows, that of the two series, m2 ee ee ee — a a a ee ee ee Re i lan Oe a ELEMENTARY ILLUSTRATIONS OF l | ae | bishet 1 em ee Me ets et EI Oi ay elem ra ae ra ) l a Ll_mn 1 3 pean han gy ees Mea ner &e. A CE hares j 4 he a Ook c.) the first is greater than the second, But since - is less than unity, the | 1 ? : first can never surpass k X emer or , and is convergent; the 1 sakes —— k second is therefore convergent. But the second is no other than & +1 + m + &c.; therefore the series a + b + ¢ + &c., is convergent from the term &. 2. Let s be less than unity, and let the successive ratios f o &e., increase, never surpassing a limit A, which is less than unity. Hence of the two series, KIi+tA+A A+ A A A + &.) l Lom Lm n A aa ierare TVET ETN o the first is the greater. But since A is less than unity, the first is con- vergent; whence, as before, a + b +c + &c., converges from the term &. The second theorem on the divergence of series we leave to the student’s consideration, as it is not immediately connected with our object. We now proceed to the series ph + gh? + rh? + sh* + &e., in which we are at liberty to suppose # as small as we please. The successive j yee ‘ git: ger ea ar ratios of the terms to those immediately preceding are +— or —h, or _A, phe payee gy 4 ws or ~_h, &c. If, then, the terms wiBs ee 5) &e., are always less than ena DV Geet ’ a finite limit A, or become so after a definite number of terms, ak h, ey qd &c., will always be, or will at length become, less than Ah. And since h _ may be what we please, it may be so chosen that Af shall be less than unity, for which # must be less than x In this case, by the last theorem, . the'series is convergent ; it follows, therefore, that a value of & can always _ be found so small that ph + qh? + rh® + &c., shall be convergent, at least unless the coefficients p, q,7, &c., be such that the ratio of any one to the preceding increases without limit, as we take more distant terms of the series. This never happens in the developments which we shall be re=_ quired to consider in the Differential Calculus. : We nowreturn to ¢ (+h), which we have asserted can be expanded (with the exception.of some particular values of x) in a series of the form px + ph + gh? + &c. The following are some instances of this deve- lopment derived from the Differential Calculus, most of which are also to be found in the treatise on Algebra :— THE DIFFERENTIAL AND INTEGRAL CALCULUS. 11 (@+h)"=2" 4+nz™hin.n—-1 2 be Bo l.nr—2 2" ibe &e. | 2 2.3 a =a -+ kath +- k?q* Lage heat a &c,* 2 2.3 log (#+h)=loga+ 1 2~ he Gio fe vat Median oe ee a 2 Yh 3 2 3 sin (v+h)=sin w+cos 2 h= sin v .— COS @& ges &c.t 2 2.3 2 3 cos (vx+h)=cos x-sina h- COS & a+ sin & i &e. It appears, then, that the development of ¢ (w + h) consists of certain functions of 2, the first of which is da itself, and the remainder of which 2 3 4 are multiplied by A, na an ag Bie bese the coefficients of these divided powers of h by 9a, Oz, G''x, &c., where p’, p”, &c., are merely functional symbols, as is ¢ itself; but it must be recollected that f’x, px, &c., are rarely, if ever, employed to signify any- 2 thing except the coefficients of h, = &c., in the development of (v+h). ,andsoon,. It is usual to denote Hence this development is usually expressed as follows: Ota hy =H dee ae kh Las Be Beige ve &e. 223 T Thus, when $7 = a”, @/a = na", d/e# =n. n=1 a, &e., when Or=sinz, ~/« =cosa, O’« = — sinw, &ce. In the first case Pweth=n(et+h), P@th)=n.n—1 (w+ hj)”; and in the second (a +h) = cos («+ h), O(a +h) = — sin (@ + A). The following relation exists between Ga, ¢/x, Q”x, &c. In the Same manner as @'a is the coefficient of A in the development of @ (x + h), so fz is the coefficient of & in the development of 9’ (w + h), and ~’x is the coefficient of A in the development of @’(x-+ h); x is the coefficient of # in the development of @'”(@ + h), and so -on. ‘The proof of this is equivalent to Taylor's Theorem already ‘alluded to; and the fact may be verified in the examples already given. When da = a”, fa =ka’, and f' (e«+h) = hak (a*+kha".h+&e:) The coefficient of # is here k°a”, which is the sameas @”x. (See the ex- ample.) Again, f!(@ + h) =ha®™" = Kk? (a* + ka*h + &e.), in which the coefficient of & is k’a®, the same as @’’x. Again, if dx = log. a, 1 eh Bid h ' a. = im and d’(a +h) = Senge Larne? + &c., as appears by Aye 1 hehe common division. Here the coefficient of A is — - which is the same 1 Vy j " ja —- —— = -—- (¢ + hy as @”x in the example. Also $/’(x + /) Gay ( )-2, * Here & is the Naperian or hyperbolic logarithm of a; that is, the common logarithm of a divided by + 434294482. ; se re + In this and the following series the terms are positive and negative in pairs, 12 ELEMENTARY ILLUSTRATIONS OF which by the binomial Theorem is — (vw? — 22-5h + &c.). The coeffi- cient of k is 2a~% or —, which is f/x in the example. It appears, then, w 3 that if we are able to obtain the coefficient of h in the development of any function whatever of 2 + /, we can obtain all the other coefficients, since we can thus deduce ¢’x from Ga, $x from ¢’x, and so on. Itis usual to call d’z the first differential coefficient of px, x the second differential — coefficient of dz, or the first differential coefficient of d’x; ¢/’x the third differential coefficient of @xr,-or the second of @/a, or the first of Oa; and so on. The nameis derived from a method of obtaining 9’a, &c., which we now proceed to explain. Let there be any function of 2, which we call dz, in which 2 is increased by an increment 3; the function then becomes h® / i ! hs gdu+Pah+o Eien OME ie tn The original value wv is increased by the increment h? h® lo. ht Oa — + Gx — + &e; ? 2 ‘ 2.3 e whence (/ being the increment of x) A Oe increment of Ox __ 7a oro kt pila ae hae increment of x 2% which is an expression for the ratio which the increment of a function bears — to the increment ofits variable. It consists of two parts; the one ¢’a, into which h does not enter, depends on x only; the remainder is a series, every _ term of which is multiplied by some power of /, and which therefore di- : minishes as /# diminishes, and may be made as small as we please by making | h sufficiently small. To make this last assertion clear, observe that all | the ratio, except its first term #’v, may be written as follows : | 1 h 7 vines wl Re. ees Ie Nati poy + &c.) fw @ the second factor of which (page 9) is aconvergent series whenever / is taken — less than < where A is the limit towards which we approximate by taking | the coefficients @’x X = pln X — &c., and forming the ratio ofeach — to the one immediately preceding. ‘This limit, as has been observed, is _ finite in every series which we have occasion to use; and therefore a value | for h can be chosen so small, that for it the series in the last-named formula | is convergent; still more will it be so for every smaller value of h. Let : the series be called P: if P bea finite quantity, which decreases when f | decreases, Ph can be made as small as we please by sufficiently diminish- | ing h; whence 9’x + PA can be brought as near as we please to 9/z. | Hence the ratio of the increments of dv.and w, produced by changing 2 into # + h, though never equal to ¢’x, approaches towards it as h is di- minished, and may be brought as near as we please to it, by sufficiently diminishing A. Therefore to find the coefficient of 2 in the development of d(x + h), find d(a + h) — x, divide it by , and find the limit towards | which it tends as /# is diminished. | In any series such as a | a OR ChE ees ORE EI nh ee, THE DIFFERENTIAL AND INTEGRAL CALCULUS, 13 which is such that some given value of h will make it convergent, it may be shown that h can be taken so small that any one term shall contain all the succeeding ones as often as we please. Take any one term, as kh". It is evident that, be & what it may, kh' : ih" +m? + &e, tk tlh + mh? + &e. the last term of which is h(l-+mh + &c). By reasoning similar to that in the last paragraph, we can show that this may be made as small as we please, since one factor is a series which is always finite when A is less than y and the other factor h can be made as small as we please. Hence, since i is a given quantity, independent of h, and which therefore remains the same during all the changes of h, the series h(l + mh + &c.) can be made as small a part of & as we please, since the first diminishes without limit, and the second remains the same. By the proportion above esta- blished, it follows then that lh"! +. mh"** +. &e., can be made as smalla part as we please of kh”. It follows, therefore, that if, instead of the full deve lopment of d(x + h), we use only its two first terms dx + 'x.h, the error - thereby introduced may, by taking h sufficiently small, be made as small a portion as we please of the small term ¢/v.h. ; The first step usually made in the Differential Calculus is the deter- mination of @’x for all possible values of @2x, and the construction of general rules for that purpose. Without entering into these we proceed to explain the notation which is used, and to apply the printiples already established to the solution of some of those problems which are the pecu- liar province of the Differential Calculus, When any quantity is increased by an increment, which, consistently with the conditions of the problem, may be supposed as small as we please, this increment is denoted, not by a separate letter, but by prefixing the letter d, either followed by a full stop or not, to that already used to signify the quantity. For example, the increment of # is denoted under these circum- stances by dx; that of dx by d.g@x; that of x" by d.x”. If instead of an increment a decrement be used, the sign of dx, &c., must be changed in all expressions which have been obtained on the supposition of an in- crement; and if an increment obtained by calculation proves to be Negative, it is a sign that a quantity which we imagined was increased by our previous changes, was in fact diminished. Thus, if 2 becomes x+dz, x’ becomes x + d.2*. But this is also (w + dzr)® or v + 2x dr + (din)*s whence d.a? = 2x dx + (dx)%. Care must be taken not to confound AI ‘the increment of 2, with (d.x)’, or, as it is often written, dz®, the square of one 1 1 1? the increment of x. Again, if « becomes x + da, i becomes math ates Oy wv Lens ] 1 dx ; : and the change of — is ———_- — —or — ————- ; showing x «+ dz x x + adzx that an increment of x produces a decrement in = It must not be imagined that because x occurs in the symbol dz, the value of the latter in any way depends upon that of the former: both the first value of #, and the quantity by which it is made to differ from its first value, are at our pleasure, and the letter d must merely be regarded as an abbreviation of the words “ difference of.” In the first example, if we divide both 1. a2 sides of the resulting equation by dx, we have — = 2x + dx. The 14 ELEMENTARY ILLUSTRATIONS OF , « smaller dx is supposed to be, the more nearly will this equation assume 2 the form “* — 2x, and the ratio of 2 to I is the limit of the ratio of x the increment of 2? to that of 2; to which this ratio may be made to ap- proximate as nearly as we please, but which it can never actually reach. Tn the Differential Calculus, the limit of the ratio only is retained, to the exclusion of the rest, which may be explained in either of the two following ways. 1. The fraction x , may be considered as standing, not for any value which it can actually have as long as dz has a real value, but for the limit of © all those values which it assumes while dx diminishes. In this sensethe | 2 equation = 2v is strictly true. But here it must be observed that the algebraical meaning of the sign of division is altered, in such a way that it is no longer allowable to use the numerator and denominator sepa-__ rately, or even at all to consider them as quantities. If ue stands, not for z the ratio of two quantities, but for the limit of that ratio, which cannot be dy obtained by taking any real value of dx, however small, the whole +4 F may, by convention, have a meaning, but the separate parts dy and dz have none, and can no more be considered as separate quantities whose | ratio is ef than the two loops of the figure 8 can be considered as separate : v | numbers whose sum is eight. This would be productive of no great in- : convenience if it were never required to separate the two ; but since all. books on the Differential Calculus and its applications are full of examples” in which deductions equivalent to assuming dy = 2rdx are drawn from | such an equation as = 2x, it becomes necessary that the first should & @ : be explained, independently of the meaning first given to the second. | It may be said, indeed, that if y = 2°, it follows that = = 2¢ + dz, in| uh ' | which, f we make dx = 0, the result is i — 2x. But if dr = 0, dy also’ 2 | == 0, and this equation should be written _ = 22; as is actually done in’ | some treatises on the differential Calculus, to the great confusion of the | learner. Passing over the difficulties* of the fraction * still the former objection recurs, that the equation dy = 2rdzx cannot be used (and it 1 used even by those who adopt this explanation) without supposing that 9, | which merely implies an absence of all magnitude, can be used in different | senses, so that one 0 may be contained in another a certain number of times, This, even if it can be considered as intelligible, is a notion of much too refined a nature for a beginner. : | ) * See Study of Mathematics, page 42. | ’ t THE DIFFERENTIAL AND INTEGRAL CALCULUS, 15 2. The presence of the letter d isan indication, not only ofan increment, but of an increment which we are at liberty to suppose as small as we please. The processes of the Differential Calculus are intended to deduce relations, not between the ratios of different increments, but between the limits to which those ratios approximate, when the increments are de- creased. And it may be true of some parts of an equation, that though the taking of them away would alter the relation between dy and dz, it would not alter the limit towards which their ratio approximates, when dx and dy are diminished. For example, dy = 2xdx + (dx). If v=4and dx = °01, then dy = -0801 and vs = S'Ol. If dx = -0001 dy = v *00080001 and a == 8°0001. The limit of this ratio, to which we dx shall come still nearer by making dz still smaller, is 8. The term (dx)?, though its presence affects the value of dy and the ratio S does not affect ae the limit of the latter, for in a or 2x -+ dx, the latter term dz, which x arose from the term (dz)*, diminishes continually and without limit. If, then, we throw away the term (dz)?, the consequence is that, take dx what we may, we never obtain dy as it would be if correctly deduced from the equation y = 2°, but we obtain the limit of the ratio of dy to dx. If we throw away all powers of dx above the first, and use the equations so obtained, all ratios formed from these last, or their consequences, are themselves the limiting ratios of which we are in search. The equations which we thus use are not absolutely true in any case, but may be brought as near as we please to the truth, by making dy and dx sufficiently small. If the student at first, instead of using dy = 2adx, were to write it thus, dy = 2xdx + &e., the &e. would remind him that there are other terms ; necessary, if the value of dy corresponding to any value of dx is to be obtained; unnecessary, if the limit of the ratio of dy to dx is all that is required. We must adopt the first of these explanations when dy and dz appear in a fraction, and the second when they are on opposite sides of an equation. If two straight lines be drawn at right angles to one another, thus di- _Viding the whole of their plane into four parts, one lying in each right | to denote any variable distance ‘Measured on or parallel to O A by the letter v, For a.similar reason, OB angle, the situation of any point is determined when we know, I., in which angle it lies; IL., its perpendicular distances from the two right lines. Thus the point P, lying in the angle A O B, is known when P Mand PN, or when O M and P M are known; for, though there is an infinite number of points whose distance from O A only is the same as that of P, and an infinite number of Others, whose distance from OB is the same as that of P, there is ho other point whose distances from both lines are the same as those of P. The line O A is called p the axis of xz, because it is usual “ Fig: 2. | MM A 16 ELEMENTARY ILLUSTRATIONS OF is called the axis of y. The co-ordinates* or perpendicular distances of a point P which is supposed to vary its position, are thus denoted by v and y; hence OM or PN is a, and PM or ON is y. Leta linear unit be chosen, so that any number may be represented by a straight line. Let the point M, setting out from O, move in the direc- tion O A, always carrying with it the indefinitely extended line MP per- pendicular to OA. While this goes on, let P move upon the line M P in such a way, that MP or y is always equal to a given function of OM or 2; for example, let y = 2°, or let the number of units in PM be the square of the number of units in OQ M. As O moves towards A, the point P will, by its motion on M P, compounded with the motion of the line MP itself, describe a curve OP, in which P M is less than, equal to, or greater than OM, according as O M is less than, equal to, or greater than the linear unit. It only remains to show how the other branch of this curve is de- duced from the equation y = 2”. It is shewn in algebra, that if, through misapprehension of a problem, we measure in one direction, a line which ought to lie in the exactly op- posite direction, or if such a mistake be a consequence of some previous misconstruction of the figure, any attempt to deduce the length of that line by algebraical reasoning, will give a negative quantity as the result. And conversely it may be proved by any number of examples, that when an equation in which @ occurs, has been deduced strictly on the supposition that @ is a line measured in one direction, a change of signin @ will turn the equation into that which would have been deduced by the same rea- soning, had we begun by measuring the line @ in the contrary direction. Hence the change of + @ into — a, or of —a into + a, corresponds in eeometry to a change of direction of the line represented by a, and vice vers. In illustration of this general fact, the following problem may be useful. Having a circle of given radius, whose centre is in the intersection of the axes of z and y, and also a straight line cutting the axes in two given points, required the co-ordinates of the points (if any) in which the straight line cuts the circle. Jet OA, the radius of the circle=7, O E=a, OF=6, | and let the co-ordinates of P, one of the SN’ points of intersection required, be O M=a, NI MP=y. The point P being in the circle whose radius is 7, we have from the right- angled triangle OM P, 2° + y*? = 7°, which equation belongs to the co-ordinates of every point in the circle, and is called the equation of the circle. Again, EM:MP:: EO: . OF by similar triangles; or@—xi yi: a : b, whence ay + b2 = ab, which is true, by similar reasoning, for every point of the line EF. But for a point P’ lying in EF produced, we have EM’ . M’P’:: E.0O;-8 OcF 3+ or ot. Biaivay Yo. tae whence ay — be = ab, an equation which may be obtained from the former by changing the sign of 2; and it is evident that the direction of a, in the second case, is opposite to that in the first. Again, for a point P’ in FE produced, we have EM”): M’P":: 7a > OF, or r—aiy :: a: 6, whence be — ay = ab, which may * The distances O M and M P are called the co-ordinates of the point P. It is mores over usual to call the co-ordinate QO M, the abscissa, and MP, the ordinate, of the point Ps THE DIFFERENTIAL AND INTEGRAL CALCULUS, 17 be deduced from the first by changing the sign of y; and it is evident that y is measured in different directions in the first and third cases, - Hence the equation ay + br = ab belongs to all parts of the straight line EF, if we agree to consider M’ P’ as negative, when M P is positive, and OM’ as negative when OM is positive. Thus, if OE = 4, and OF =5 and OM=1, we can determine MP from the equation ay + bx = ab, or 4y + 5 = 20, which gives y or MP= 33. But if O M’ be | in length, we can determine M’P’ either by calling M P, 1, -and using the equation ay — bx = ab, or calling MP, — 1, and using the equation ay + bx = ab, as before. Lither gives M’/P/= 64. The -latter method is preferable, inasmuch as it enables us to contain, in one Investigation, all the different cases of a problem. We shall proceed to show that this may be done in the present instance. We have to deter- Mine the co-ordinates of the point P, from the following equations,— ay -+- br = ab, Ct ay? mem 78 substituting in the second the value of y derived from the first, or } aa | a we have . as 2 a b? CoD =7 or (a? + b°) a — 2ab’x + a? (2-1) = 0; a and proceeding in a similar manner to find y, we have (a + b*) y? — 2a*by + LD? (@ —r*) = 0, which give pao ht VEER yt EGP Paae a’ + 6? a’ + b? The upper or the lower sign, is to be taken in both. Hence when (a + b*) r? >a°b*, that is, when 7 is greater Deepens perpendicular let a Va? + b? points of intersection. When (a + 6%) 7° = a’b?, the two values of x become equal, and also those of y, and there is only one point in which ‘the straight line meets the circle; in this case EF is a tangent to the circle. And if (@?-++ 62) r? < a’b?, the values of # and y are impossible; and the straight line does not meet the circle. Of these three cases, we ‘confine ourselves to the first, in which there are two points of intersection. : j a oe b? ‘The product of the values of 2, with their proper sign, is * a? reg ax and fall from O upon E F (which perpendicular is » there are two of y, b? a re G the signs of which are the same as those of b? — 7°, and a + a —7, Ifb anda be both >7, the two values of x have the same sign ; and it will appear from the figure, that the lines they represent are mea- sured in the same direction. And this whether 6 and a be positive or Negative, since 6» — 7? and a — 7° are both positive when a and 6 eS numerically greater than 7, whatever their signs may be. Rok is, x ps rule, connecting the signs of algebraical and the directions : Soe rica magnitudes, be true, let the directions of O E and O F be altere a ay Way, so long as O E and O F are both greater than O A, the two values o OM will have the same direction, and also those of MP. This result may easily be verified from the figure. Again, the values of x and y having the * See Study of Mathematics, page 45. C 18 ELEMENTARY ILLUSTRATIONS OF same sign, that sign will be (see the equations) the same as that of 2ab® for az, and of 2a%b for y, or the same as that of a for w and of 6 for y. Thatis, when O Eand OF are both greater than O A, the direction of each set of co-ordinates will be the same as those of OE and OF, which may also be readily verified from the figure. Many other verifications might thus be obtained of the same principle, viz.—that any equation which corresponds to, and is true for, all points in the angle AO B, may be used without error for: all points lying in the other three angles, by substituting the proper nume- rical values, with a negative sign, for those co-ordinates whose directions are opposite to those of the co-ordinates i the angle AO B. In this manner, if four points be taken similarly situated in the four angles, the numerical values of whose co-ordinates are x = 4 and y = 6, and if the co-ordinates of that point which lies in the angle A O B, are called + 4 and + 6; those of the points lying in the angle B O C will be — Aand + 6; in the angle COD — 4 and — 6; and in the angle DOE + 4 and — 6. To return to figure (2), if, after having completed the branch of the curve which lies on the right of BC, and whose equation is y= Bias we seek that which lies on the left of BC, we must, by the principles established, substitute — ¢ instead of #, when the numerical value ob- tained for (— x)? will be that ofy, and the sign will show whether y is to be measured in a similar or contrary direction to that of MP. Since (— 2)?= a*, the direction and value of y, for a given value of x, remains the same as on the right of BC; whence the remaining branch of the curve is similar and equal in all respects to O P, only lying in the ahgle BOD. And thus, if y be any function of z, we can obtain a geome- trical representation of the same, by making y the ordinate, and x the abscissa of a curve, every ordinate of which shall be the linear repre- sentation of the numerical value of the given function corresponding to the numerical value of the abscissa, the linear unit being a given line. If the point P, setting out from O, move along the branch O P, it will continually change the direction of its motion, never moving, at one point, : in the direction which it had ‘at any previous point. Let the moving — point have reached P, and let OM = a, MP=y. Let @ receive the increment M M/ = daz, in consequence of which y or MP becomes M’ P’, and receives the increment Q P/ = dy; so that w + dx and y + dy are the co-ordinates of the moving point P, when it arrives at P’, Join P Pea : : . P/Q dy which makes, with P Q or OM, an angle, whose tangent 1s PO or on fe Since the relation y = 2” is true for the co-ordinates of every point in the | curve, we have y + dy = («+ da)’, the subtraction of the former equa- tion from which gives dy = 2adx + (dx)’, or et = 2x + dx. If the Ms point P’ be now supposed to move backwards towards P, the chord PP’: will diminish without limit, and the inclination of PP’ to PQ will also. diminish, but not without limit, since the tangent of the angle P’PQ, or: d | =e, is always greater than the limit 22. If, therefore, a line P V be drawn through P, making with PQ, an angle whose tangent is 22, the chord P P! will, as P’ approaches towards P, or as dx is diminished, continually ap proximate towards PV, so that the angle P/PV may be made smaller | than any given angle, by sufficiently diminishing dz. And the line PV. cannot again meet the curve on the side of P P’, nor can any straight line : THE DIFFERENTIAL AND INTEGRAL CALCULUS. 19 be drawn between it and the curve, the proof of which we leave to the student. Again, if P’ be placed on the other side of P, so that its co- ordinates are « — dx and y — dy, we have y — dy = (v — dx)’, which, subtracted from y = 2°, gives dy = 2x4dx — (dx)*, or = = 2x —dx. By similar reasoning, if the straight line P'T be drawn in continuation of PV, ‘making with PN an angle, whose tangent is 2x, the chord P P’ will con- tinually approach to this line, as befure. The line T P V indicates the direction in which the point P is proceeding, and is called the tangent of the curve at the point P. Ifthe curve were the interior of a small solid tube, in which an atom of matter were made to move, being projected into it at O, and if all the tube above P were removed, the line P V is in the direction which the atom would take on emerging at P, and is the line which it would describe. The angle which the tangent makes with the axis of x in any curve, may be found by giving z an increment, finding the ratio which the corresponding increment of y bears to that of w, and determining the limit of that ratio, or the differential coefficient. This limit is the trigonometrical tangent * of the angle which the geometrical tangent makes with the axis of cz. If y = dz, x is this trigonometrical tangent. Thus, if the curve be such that the ordinates are the Naperian 1 logarithms + of the abscissa, or y = log a, and y + dy = log a + 2L Be 1 — 575 de®, &e., the geometrical tangent of any point whose abscissa fol V is z, makes with the axis an anole whose trizonometrical tangent is —. <=] 5 =) xv This problem, of drawing a tangent to any curve, was one, the considera- tion of which gave rise to the methods of the Differential Calculus. As the peculiar language of the theory of infinitely small quantities is extensively used, especially in works of natural philosophy, it has ap- peared right to us to introduce it, in order to show how the terms which are used may be made to refer to some natural and rational mode of explanation. In applying this language to figure (2), it would be said that the curve O P is a polygon consisting of an infinite number of infinitely small sides, each of which produced is a tangent to the curve; also that if M M’ be taken infinitely,small, the chord and are P P’ coin- cide with one of these rectilinear elements ; and that an infinitely small “are coincides with its chord. All which must be interpreted to mean thaf, the chord and are being diminished, approach more and more nearly toa Yatio of equality as to their lengths ; and also that the greatest separation between an arc and its chord may be made as small a part as we please of the whole chord or arc, by sufficiently diminishing the chord. We Shall proceed to a strict proof of this ; but in the mean while, as a familiar illustration, imagine a small arc to be cut off from a curve, and its extre- Mities joined by a chord, thus forming an arch, of which the chord is the base. From the middle point of the chord, erect a perpendicular to it, * Lhere is some confusion between these different uses of the word tangent. The geo- thetrical tangent\is, as already defined, the line between which and a curve no straight line ean be drawn ; the trigonometrical tangent has reference to an angle, and is the tatio which, in any right-angled triangle, the side opposite the angle bears to that which 18 adjacent, 7 It may be well to notice that in analysis the Naperian logarithms are the only ones used; while in piactice the common, or Briggs’ logarithms, are always preferred, i? 20 ELEMENTARY ILLUSTRATIONS OF meeting the arc, which will thus represent the height of the arch. Ima- gine this figure to be magnified, without distortion or alteration of its pro- portions, so that the larger ficure may be, as it is expressed, a true picture of the smaller one. However the original arc may be diminished, let the magnified base continue of a given length. This is possible, since on any line a figure may be constructed similar to a given figure. If the original curve could be such, that the height of the arch could never be reduced below a certain part of the chord, say one-thousandth, the height of the magnified arch could never be reduced below one-thousandth of the mag- nified chord, since the proportions of the two figures are the same. But if, in the original curve, an arc can be taken so small, that the height of the arch is as small a part as we please of the chord, it will follow that in the magnified figure, where the chord is always of one length, the height of the arch can be made as small as we please, seeing that it can be made as small a part as we please of a given line. It is possible in this way to conceive a whole curve so magnified, that a given arc, however small, shall be represented by an are of any given length, however great; and the proposition amounts to this, that let the dimensions of the magnified curve be any given number of times the original, however great, an arch can be taken upon the original curve so small, that the height of the cor- responding arch in the magnified figure shall be as small as we please. Let PP’ be a part of a curve, whose equation is y = ® (x), that is, P M may always be found by substituting the numerical Fig. d, yalue of O M ina given function of xv, Let OM=2 receive the increment MM’ = dz, which we may afterwards suppose as small as we please, but which, in order to render the figure more distinct, is here con- siderable. ‘The value of PM or y is $2, and that | of P’M’ or y+ dy is 6(a-+ dr). Draw PV, the | M M7 tangent at P, which, as has been shown, makes, with | PQ, an angle, whose trigonometrical tangent is the limit of the ratio d ma when zx is decreased, or $x. Draw the chord P P’, and from any point in it, for example, its middle point p, draw pv parallel to P M, cut- | ting the curve ina. ‘The value of P’Q, or dy, or d (v7 dz) — pris 2 p\3 PQ= Pr. dr-+ "x Sus + pe = + &e. But div. dristanVPQ.PQ= VQ. Hence VQ is the first term of this series, and P’ V the aggregate of the rest. But it has been shown that dx can be taken so small, that any one term of the above series shall contain the rest, as often as we please. Hence PQ can be taken so small that VQ shall contain V P’ as often as we please, or the ratio of V Qto VP’ shall be as great as we please. And the ratio V Qto PQ continues finite, being always @/x, hence P/V also decreases without limit, as compared with PQ. The chord PP’ or A (dx)?-+-(dy)’, or 2 2 dx J) -- (*) is to PQ in the ratio of i 1+ ) : 1, which, dx dx as PQ is diminished, continually approximates to that of /1+(#'2)* : 1, which is the ratio of PV : PQ. Hencethe ratio of PP’: PV continually approaches to unity, or P Q may be taken so small that the difference of P P’ and P V shall be as small a part of either of them as we please. The THE DIFFERENTIAL AND INTEGRAL CALCULUS. 21 are PP’ is preater than the chord PP’ and less than PV + VP’, ane 4 PY¥ VP’ Hence chord PP’ lies between 1 and PP’ Pp” the former of which two fractions can be brought as near as we please to unity, and the latter can be made as small as we please; for since P’V can be made as small a part of PQ as we please, still more can it be made as small a _ part as we please of P P’, which is creater than PQ. Therefore the arc and chord P P’ may be made to have a ratio as nearly equal to unity as we please. And because pa is less than pv, and therefore less than P’ V, it follows that pa may be made as small a part as we please of PQ, and | still more of PP’, In these propositions is contained the rational expla- nation of the proposition of Leibnitz, that ‘an infinitely small arc is equal to, and coincides with, its chord,’ Let there be any number of series, arranged in powers of h, so that the lowest power is first ; let them contain none but whole powers, and let them all be such, that each will be convergent, on giving to / a suffi- ciently small value :—as follows, Ah + Bh?+ CA? + Dat + EAS+ &e. (1) Be + Ch? + D/At + E/AE+ &e. (2) C’h? + Dht + EWA + &e. (3) D"ht +E”AP+ &e. (4) &e. &e. As h is diminished, all these expressions decrease without limit: but the first 272creases with respect to the second, that is, contains it more times after a decrease of h, than it did before. For the ratio of (1) to (2) is that of A+ Bh + Ch? + &c. to Bh + Clh? + &e., the ratio of the two not being changed by dividing both by h. The first term of the latter ratio approximates continually to A, as h is diminished, and the second can be made as small as we please, and therefore can be contained in the first as often as we please. Hence the ratio of (1) to (2) can be made as great as we please. By similar reasoning, the ratio (2) to (3), of (3) to (4), &c., can be made as great as we please. We have, then, a series of quantities, each of which, by making / sufficiently small, can be made as small as we please. Nevertheless this decrease increases the ratio of the first to the second, of the second to the third, and so on, and the increase is without limit. Again, if we take (1) and h, the ratio of (1) to h is that of A+ Bh+ Ch? + &c. to 1, which, by a sufficient decrease of h, /may be brought as near as we please to that of Ato 1. But if we take (1) and 2, the ratio of (1) to h? is that of A-++ BA + &c. to h, which, by previous reasoning, may be increased without limit ; and the same for any higher power of k. Hence (1) is said to be comparable to the first power of h, or of the first order, since this is the only power of A whose ratio to (1) tends towards a finite limit. By the same reasoning, the ratio of (2) to h?, which is that of B' + Ch + &c. to 1, continually approaches that of B’ to 1; but the ratio (2) to A, which is that of Bh + Ch? -+ &c. to 1, diminishes without limit, as 2 is decreased, while the ratio of (2) to h*, or of B/+ C’h + &c. to h, increases without limit. Hence (2) is said to be comparable to the second power of h, or of the second order, since this is the only power of whose ratio to (2) tends towards a finite limit. In the language of Leibnitz, if 2 be an infinitely “small quantity, (1) is an infinitely small quantity of the first order, (2) is an infinitely small quantity of the second order, andso on. We may also add that the ratio of two series of the same order continually approximates 99 ELEMENTARY ILLUSTRATIONS OF to the ratio of their lowest terms. For example, the ratio of AA? + Bhi+&e. to A/h? + B/ht+ &c. is that of A+ Bh+&c. to A’+ B/h+ &c., which, as his diminished, continually approximates to the ratio of A to A’, which is also that of AA® to A’A%, or the ratio of the lowest terms. In fig. A, PQ or dz 2 being put in place of h, QP’, or ple .dx-+ "x Ss. , &e., is of the first 2 order, as are P V, and the chord PP’; while P’V, or Dx Sat + &c., is of the second order. The converse proposition is readily shown, that if the ratio of two series arranged in powers of / continually approaches to some finite limit as fA is diminished, the two series are of the same order, or the exponent of the lowest power of f is the same in both. Let Ah* and Bh’ be the Jowest powers of h, whose ratio, as has just been shown, continually approximates to the actual ratio of the two series, as his di- minished. 'The hypothesis is that the ratio of the two series, and therefore that of Ah“ to Bh», has a finite limit. This cannot be if a >), for then the ratio of Ah“ to Bh” is that of Ak*-’ to B, which diminishes without limit; neither can it be when a < 6, for then the same ratio is that of A to Bh®-*, which increases without limit; hence @ must be equal to 0. We leave it to the student to prove strictly a proposition assumed in the preceding, viz., that if the ratio of P to Q has‘unity for its limit, when h is diminished, the limiting ratio of P to R will be the same as the limiting ratio of Q to R. We proceed further to illustrate the Differential Calculus as applied to Geometry. : Let OC and OD be two axes at right angles to one another, and let a line AB of given length be placed with one extremity in each axis, Let this line move from its first position into that of A/B/ on one side, and afterwards into that of A’'BY on the other side, always preserving its first length. The motion of a ladder, one end of which is against a wall, and the other on the ground, is an instance. Let A’ B! and A’ B! intersect A Bin P’ and P’. If A” BY” were eradually moved from its present position into that of A’ B’, the point P” would also move eradually from its present position into that of P’, passing, in its course, through every point in the‘line P’P”. But here it is necessary to remark that A B is itself one of the positions intermediate between A/ B/ and A” BY”, and when two lines are, by the motion of one of them, brought into one and the same straight line, they intersect one another (if this phrase can be here applied at all) in every point, and all idea of one distinct point of intersection is lost. Never- theless P” describes one part of P’P’ before A” B’ has come into the position AB, and the rest afterwards, when it is between A B and A' BY. Let P be the point of separation; then every point of P’ P’, except P, is a real point of intersection of A B, with one of the positions of A’ BY”, and when A’ BY’ has rnoved very near to A B, the point P” will be very near to P: and there is no point so near to P, that it may not be made the inter- section of A’ B’ and AB, by bringing the former sufficiently near to the latter. This point P is, therefore, the limit of the intersections of A” B” and A.B, and cannot be found by the ordinary application of Algebra to weometry, but may be made the subject of an inquiry similar to those which have hitherto occupied us, in the following manner :—Let OA=@, OB=}, AB=A BH A’BY = 1. Let AA’ = da, BB’ = db, whence OA/ =a+da, OB/=4-—db. We have then a +2 =P, o WARK C THE DIFFERENTIAL AND INTEGRAL CALCULUS. 23 and (a + da)? +(b —db?=P; subtracting the former of which from the development of the latter, we have lb = 2a+da * 2a da + (da)? — 2b db + (db?=0 op = # (da) ee) Ga aean ab As A’ B’ moves towards AB, da and db are diminished without limit, @ Ris aT ae Zee a and b remaining the same; hence the limit of the ratio — is op Grae Let the co-ordinates * of P’be OM’ = wand M/P/=y. Then (page 16) the co-ordinates of any point in A B have the equation ay + bx= ab (2). The point P’ is in this line, and also in the one which cuts off a + da and 6 — db from the axes, whence (a + da) y + (6 — db) « = (a + da) (b — db) (3) subtract (2) from (3) after developing the latter, which oives yda — «db =bda — adb — dadb (4) If we now suppose A’ B’ to move towards A B, equation (4) gives no result, since each of its terms diminishes without limit. If, however, we b divide (4) by da, and substitute in the result the value of - obtained from (1) we have BEGGS 50M 7 aera — ab (5); 26— db 2b —db from this and (2) we might deduce the values of y and 2, for the point P’, as the figure actually stands. Then by diminishing db and da without limit, and observing the limit towards which x and y tend, we might deduce the co-ordinates of P, the limit of the intersections. The same result may be more simply obtained, by diminishing da and db in equation (5), before obtaining the values of y and xv. This gives y—-a a a bare o=b— = or by—ac=—@ (6). b From (6) and (2) we find (fig. 6) om=—*. 22 wayoups— 2 F + ie Tle -KbeaAT ow hud LB TR This limit of the intersections is different fot every dif- ferent position of the line AB, but may be determined, + in every case, by the following simple construction. : Since BP: PN, on OM :: BA: AQO,wehaveBP= N > Hig. & BA e a b? OM cA as Gay i and, similarly, PA = 7H Q Let OQ be drawn perpendicular to BA; then since OA is a mean proportional between AQ and AB, we have Q Mt A. 2 b2 AQ= + , and similarly BQ = Tz" Hence BP = AQ andAP=BQ, or the point P is as far from either extremity of AB as Q is from the other. ; ry We proceed to solve the same problem, using the principles of Leibnitz, that is, supposing magnitudes can be taken so small, that those proportions may be regarded as absolutely correct, which are not so in reality, but which only approach more nearly to the truth, the smaller the magnitudes * The lines O M’ and M’ P’ are omitted, to avoid crowding the figure. 24 ELEMENTARY ILLUSTRATIONS OF are taken. The inaccuracy of this supposition has been already pointed out; yet it must be confessed, that this once got over, the results are de- duced with a degree of simplicity and consequent clearness, not to be found in any other method. The following cannot be rewarded as a de- monstration, except by a mind so accustomed to the subject, that it can readily convert the various inaccuracies into their corresponding truths, and see, at one glance, how far any proposition will affect the final result. The beginner will be struck with the extraordinary assertions 7 which follow, given in their most naked form, without any attempt ata less startling mode of expression. Let A’ B’ be a position of A B infi- Fi nitely near to it; that is, let A’ P A be an infinitely BY small angle. With the centre P, and the radit P A’ and PB, describe the infinitely small ares a Ala, Bb. An infinitely small arc of a circle is a straight line perpendicular to its radius; hence 0 oN A'aA and BOdB’ are right-angled triangles, | the first similar to BOA, the two having the angle A in common, and the second similar to B’O A’. Again, since the angles of BOA, which are finite, only differ from those of B’O A’ by the infinitely small angle A’ P A, they may be regarded as equal ; whence A’aA and B'} B are similar to BO A, and to one another. Also P is the point of which we are in search, or infinitely near to it; and since BA = B’ A’, of which BP = OP andaP= A’P, the remainders B’/b and Aaare equal. Moreover, Bd and A/a being arcs of circles subtending equal angles, are in the proportion of the radii BP and P A’. Hence we have the following proportions,— B Aa: Alia: OA: OB:: 4:0 Bb: Bb:: OA: OB: a3 6. The composition of which gives, since Aa = BB’), Bibce BAU, ethers Oe Also Bobo Ala sc BPs Bia, whence BPR 2g 2262 a: and BP BG she ii Oe But Pa@ only differs from P A by the infinitely small quantity A a, and BP+PA=l, and @& +08 =; whence 2 Pa PIA er Clas wes or PA= which js the result already obtained. In this reasoning we observe four independent errors, from which others follow,—l. that B 6 and A’ a are straight lines at right-angles to Pa; 2. that BOA and B/O A’ are similar triangles; 3. that P is really the point of which we are in search ; 4. that P A and Pa@are equal. But at the same time we observe, that every one of these assumptions approaches the truth, as we diminish the angle A’ P A, so that there is no magnitude, line or angle, so small, that the linear or angular errors, arising from the above-mentioned suppositions, may not be made smaller. We now proceed to put the same demonstra- tion in a stricter form, so as to neglect no quantity during the process. This should always be done by the beginner, until he is so far master of the subject, as to be able to annex to the inaccurate terms, the ideas necessary for their rational explanation. To the former ficure add BB and Aa, the real perpendiculars, with which the arcs have been confounded. Let ZA/PA = do, PA=p, Aa=dp, BP=q Bb = dg; and THE DIFFERENTIAL AND INTEGRAL CALCULUS, 25 OA=a,O0B=},andAB=i. Then* Ala — (p—dp) d0, Bb = qdo, and the triangles A’Aw and B/Bf are similar to BO A and B/O A’. The perpendiculars A’a and Bf are equal to P A’ sin. dé, and PB sin. do, or (p — dp) sin dé, and q sin do. Let aa =pand b8=v. These (page 5) will diminish without limit as compared with A/a and BG; and since the ratios of A’a to aA and BB to BB’ continue finite, (these being sides of triangles similar to AOB and A'O B’,) aa and 06 will diminish indefinitely with respect to aA and BB’. Hence the ratio Aa to BB’ or dp + p to dq + v will continually approximate to that of dp to dq, ora ratio of equality. ‘The exact proportions, to which those in the last page are approximations, are as follows :— dp+m : (p—dp)sin d0:: a eos qsin dé; dq + v >: @—da : b+db; by composition of which, recollecting that dp == dq (which is rigorously true,) and dividing the two first terms of the resulting proportion by dp, we have bb y 1 ae pS rey, / paletbaslee. Pa bt ‘ : ca+ +) (p D+ 2) 32 a (a da) : b (b + db) If d@ be diminished without limit, the quantities da, db, and dp, and also the ratios and = as above-mentioned, are diminished without limit, so that the limit of the proportion just obtained, or the proportion which gives the limits of the lines into which P divides A B, is Gia pir. a? 0.3 hence Qq$+p=H=lips @+LYVH=EF: dB, the same as before. We proceed to apply the preceding principles to dynamics, or the theory of motion. Suppose a point moving along a straight line uniformly, that is, if the whole length described be divided into any number of equal parts, however great, each of those parts is described in the same time. Thus, whatever length is described in the first second of time, or in any part of the first second, the same is described in any other second, or in the same part of any other second. The number of units of Jeneth de- scribed in a unit of time is called the velocity ; thus a velocity of 3°01 feet in a second, means that the point describes three feet and one- hundredth in each second, and a proportional part of the same in any part 'ofasecond. Hence, if v be the velocity, and ¢ the units of time elapsed from the beginning of the motion, v ¢ is the length described ; and if any length described be known, the velocity can be determined by dividing that length by the time of describing it. Thus, a point which moves uni- formly through 3 feet in 14 second, moves with a velocity of 3 — 14, or 2 feet per second. Let the point not move uniformly, that is, let different parts of the line, having the same length, be described in different times ; at the same time let the motion be continuous, that is, not suddenly increased or decreased, as it would be if the point were composed of some hard matter, and received a blow while it was moving. ‘his will be the case if its motion be represented by some algebraical function of the time, or if, ¢ being the number of units of time during which the point has moved, the number of * For the Unit employed in measuring an angle, see Study of Mathematics, page 90, " 26 ELEMENTARY ILLUSTRATIONS OF units of length described can be represented by gt. This, for example, we will suppose to be t-+-??, the unit of time being one second, and the unit of length one inch; so that + + 4, or $ of an inch, is described in the first half second ; 1 + 1, or two inches, in the first second; 2-4, or six inches, in the first two seconds ; and so on. Here we have no longer an evident measure of the velocity of the point ; we can only:say that it obviously increases from the beginning of the mo- tion to the end, and is different at every two different points. Let the time ¢ elapse, during which the point will describe the distance i+ #; let a further time dé elapse, during which the point will increase its distance to ¢-+ dé-+ (¢ + dt)”, which, diminished by ¢-+-#, gives dt + 2¢ dt + (dt)? for the length described during the increment of time dt, ‘This varies with the value of ¢; thus, in the interval dé after the first second, the length described is 3dé + dé; after the second second, it is hdt + (dt)®, and so on. Nor can we, as in the case of uniform motion, divide the length described by the time, and call the result the velocity with which that length is described; for no length, however small, is here uniformly described. If we were to divide a length by the time in which it is described, and also its first and second halves by the times in which they are respectively described, the three results would be all different from one another. Here a difficulty arises, similar to that already noticed, when a point moves along a curve; in which, as we have seen, it is improper to say that it is moving in any one direction through any are, however small. Nevertheless a straight line was found at every point, which did, more nearly than any other straight line, repre= sent the direction of the motion. So, in this case, though it is incorrect to say that there is any uniform velocity with which the point continues to move for any portion of time, however small, we can, at the end of every time, assign a uniform velocity, which shall represent, more nearly than any other, the rate at which the point is moving. If we say that, at the end of the time ¢, the point is moving with a velocity v, we must not now say that the length vdt is described in the succeeding interval of time dt; but we mean that dé may be taken so small, that vdét shall bear to the distance actually described a ratio as near to equality as we please. Let the point have moved during the time ¢, after which let successive intervals of time elapse, each equal to dt. At the end of the times, # ¢@ + dé, ¢ + Qdt, -t + 3d, &e., the whole lengths described will be €+@, ¢+ dé+ @+ dt), t-- 2dt + '@ + 2dt)’, t+ 3dt + (£+3dt)*, &e. ; the differences of which, or dé + 2tdé + (dt)?, dt + 2tdt + 3 (di)?, dé + 2idt + 5 (dt)?, &c., are the lengths described in the first, second, third, &e., intervals dt. These are not equal to one another, as would be the case if the velocity were uniform; but by making dé sufficiently small, their ratio may be brought as near to equality as we please, since the terms (dé), 3 (dt)?, &c., by which they all differ from the common part (1 + 2¢) dé, may be made as small as we please, in comparison of this common part. If we divide the above-mentioned lengths by d?, which does ot alter their ratio, they become 1 + 2¢ + dé, 1 + 2¢ + 3dt, 1+ 2¢--5dt, &c., which may be brought as near as we please to equality, by sufficient diminution of dt. Hence 1 + 2¢ is said to be the velocity of | the point after the time ¢; and if we take a succession of equal intervals — of time, each equal to dé, and sufficiently small, the lengths described in | those intervals will bear to (1 + 2¢) dé, the length which would be de- THE DIFFERENTIAL AND INTEGRAL CALCULUS, 27 scribed in the same interval with the uniform velocity 1 + 22, a ratio as near to equality as we please. And observe, that if dtis t+ 2, gt is 1 + 2¢, or the coefficient of A in (+h) ++ h)*. In the Same way it may be shown, that if the point moves so that ¢ always represents the length described in the time ¢, the differential coefficient of pt or P’'t, is the velocity with which the point is moving at the end of the _time?. For the time ¢ having elapsed, the whole lengths described at the end of the times ¢ and ¢-++ dé are $¢ and ¢ (¢ + dt); whence the length described during the time d¢ is p(t-+ dt) + dt, or Pit. dt + dt es + &e. Similarly, the length described in the next interval dé is Pp (t+ 2dt) —b+ dt); or 2dt)2 ot + Pit. 2dt + pt sn + &e. — (Pi + Pit dt + pt ae + &c.) which is pit. dt + 3f"t oo + &e; 2 the length described in the third interval dé is #/t . dt + 5@"t ioe 2 + &c. &e. It has been shown for each of these, that the first term can be made to contain the aggregate of all the rest as often as we please, by making dé sufficiently small ; this first term is d't. dé in all, or the length which would be described in the time dé by the velocity @’¢ con- tinued uniformly : it is possible, therefore, to take dt so small, that the lengths actually described in a succession of intervals equal to dé, shall be as nearly as we please in a ratio of equality with those described in the same intervals of time by the velocity #/t. For example, it is observed in bodies which fall to the earth from a height above it, when the resist- ance of the air is removed, that if the time be taken in seconds, and the distance in feet, the number of feet fallen through in @ seconds is always ai®, where a= 1654 very nearly; what is the velocity of a body which has fallen 27 vacuo for four seconds? Here ¢¢ being ai’, we find, by substituting ¢ + h, or t + dt, instead of #, that @’t is 2aé, or 2X 1655 x t, or 324.¢; which, at the end of four seconds, is 321 x 4, or 1282 feet. ‘That is, at the end of four seconds a falling body moves at the rate of 1282 feet per second. By which we do not mean that it continues ;to move with this velocity for any appreciable time, since the rate is 'always varying; but that the length described in the interval dé after the fourth second, may be made as nearly as we’ please in a ratio of equality with 1282 x dé, by taking dé sufficiently small. This velocity 2a¢ is said to be uniformly accelerated ; since in each second the same velocity 2a is gained. And since, when a is the space described, 9’¢ is the limit of 3 the velocity is also this limit; that is, when a point does not move ‘uniformly, the velocity is not represented by any increment of length di- vided by its increment of time, but by the limit to which that ratio con- ‘tinually tends, as the increment of time is diminished. We now propose ithe following problem :—A point moves uniformly round a circle ; with what velocities do the abscissa and ordinate increase or decrease, at any given point? Let the point P, setting out from A, describe the are A P, '&c., with the uniform velocity of @ inches per second. Let OA =7, mm AOP=6,2P0 P’= d,OM=2,M P= y, M M’= dz, QP/=dy. ‘From the first Principles of Trigonometry ee ee 28 ELEMENTARY ILLUSTRATIONS OF r=r cos 0 x—dx=r cos (0+ d0)=r cos 6 cos do—r sin 6 sin dO y=r sin 6 y+dy=r sin (0+d9)=r sin 8 cos dé--r cosé@ sin dé ; subtracting the second from the first, and the third from the fourth, we have dz — r sin@ sind@ + r cos 8 (1 — cos dé) (1) dy = rcos@ sind@ +r sin 0 (1 — cos dé) (2) but if dé be taken sufficiently small, sin dé, and Hig8. d0, may be made as nearly in a ratio of equality as we please, and 1 — cos d@ may be made as small a part as we please, either of d@ or sin dé. These follow from fig. 1, in which it was shown that BM and the are B A, or if OA=randAOB= dé,) r sin d@ and rd0, may be brought as near to a ratio of equality as we please; which is therefore true of sind@ and dé. Again, it was shown that Oo M M te eA M, or 7 — rcos d9, can be made as small a part as we please, either of B M or the arc BA, that is, either of 7 sin d6, or rd0; the same is therefore true of 1 — cos dé, and either sin dO or dé, Hence, if we write equations (1) and (2) thus, de =r sin 6 d6(1), dy = r cos 6 dé (2), we have equations, which, though never exactly true, are such that by making dé sufficiently small, the errors may be made as small parts of dé as we please. Again, since the arc A P is uniformly described, so also is the angle PO A; and since an are @ is described in one second, the angle Wix . : ; wie . ” is described in the same time; this is, therefore, the angular velocity™. Ca If we divide equations (1) and (2) by dé, we have SESE ee BY. cos @ : dt dt bes oh ris Mae these become more nearly true as dé and dé are diminished, so that if for | dx 4 ‘ ——, &c., the limits of these ratios be substituted, the equations will become dt rigorously true. But these limits are the velocities of x, y, and @, the last . hae oe a of which is also —; hence r velocity ofv =r sin 6 X =a sin 8, a P { a velocity of y= rcos 9 X om = i COS 8; that is the point M moves towards O with a variable velocity, which 1s always such a part of the velocity of P, as sin @ is of unity, or as | PM is of OB; and the distance P M increases, or the point N moves from O, with a velocity which is such a part of the velocity of Pas cos @ is of unity, or as OM is of OA. In the language of Leibnitz, the foregoing results would be expressed * The same considerations of velocity which have been applied to the motion of a point along.a line may also be applied to the motion of a line round a point. If the angle so described be always increased by equal angles-in equal portions of time, the angular velocity is said to be uniform, and is measured by the number of angular units described in a unit of time. By similar reasoning to that already described, if the velocity with which the angle increases be not uniform, so that at the end of the time?¢ the angle de- dé scribed is é = gf, the angular velocity is ¢’/, or the limit of the ratio : ¢ THE DIFFERENTIAL AND INTEGRAL CALCULUS. 29 thus:—Ifa point move, but not uniformly, it may still be considered as moving uniformly for any infinitely small time; and the velocity with which it moves is the infinitely small space thus described, divided by the infinitely small time. The foregoing process contains the method employed by Newton, known by the name of the Method of Fluzions. If we suppose y to _be any function of z, and that x increases with a given velocity, y will also increase or decrease with a velocity depending,—l. upon the velocity of «; 2. upon the function which y is of x These velocities Newton called the fluxions of y and x, and denoted them by 'yand 2. Thus, ify = a, and if in the interval of time dt, « becomes 2+ dz, and y becomes y + dy, we have y + dy = («+ dz)’, and y dx dx may — 22. dz - (dx)? or —— == 27 — + — dr. If we diminish dd, dt dt dt the term Ta dx will diminish without limit, since one factor continually approaches to a given quantity, viz., the velocity of x, and the other dimi- nishes without limit. Hence we obtain the velocity of y = 22 x the velocity of x, or y= 2x a, which is used in the method of fluxions instead of dy = 2r dx considered in the manner already described. The processes are the same in both methods, since the ratio of the velocities is the limiting ratio of the corresponding increments, or, according: to Leibnitz, the ratio of the infinitely small increments. We shall hereafter notice the common objection to the Method of Fluxions. When the velocity of a material point is suddenly increased, an wmpulse _is said to be given to it, and the magnitude of the impulse or impulsive force, is in proportion to the velocity created by it. Thus, an impulse which changes the velocity from 50 to 70 feet per second, is twice as creat as one which changes it from 50 to 60 feet. When the velocity of the | point is altered, not suddenly but continuously, so that before the velocity ' can change from 50 to 70 feet, it goes through all possible intermediate velocities, the point is said to be acted on by an accelerating force. Force is a name given to that which causes a change in the velocity of a body. It is said to act uniformly, when the velocity acquired by the point in any one interval of time is the same as that acquired in any other interval of equal duration. Itis plain that we cannot, by supposing any succession of impulses, however small, and however quickly repeated, arrive at a _ uniformly accelerated motion ; because the length described between any _ two impulses will be uniformly described, which is inconsistent with the idea of continually accelerated velocity. Nevertheless, by diminishing the magnitude of the impulses, and increasing their number, we may come as i near as we please to such a continued motion, in the same way as, by diminishing the magnitudes of the sides of a polygon, and increasing: their | number, we may approximate as near as we please to a continuous curve. Let a point, setting out from a state of rest, increase its velocity uniformly, | so that in the time ¢, it may acquire the velocity v—what length will have _ been described during that time ¢? Let the time ¢ and the velocity v be _ both divided into n equal parts, each of which is ¢ and v’; so that n= a and nv’ =v. Let the velocity v’ be communicated to the point at rest ; after an interval of ¢ let another velocity v’ be communicated, so that during the second interval ¢’ the point has a velocity 2v’; during the third Interval let the point have the velocity 3v’, and so on; so that in the last on 2 fh interval the point has the velocity nv’. The space described in the 30 ELEMENTARY ILLUSTRATIONS OF first interval is, therefore, vd’; in the second, Qu’; in the third, 30't'; and so on, till in the mn interval it is nv’d’. The whole space described is, therefore, vl + Ql +B +... tn— led + mil _, my't! + no'l! 2 In this substitute v for nv’, and ¢ for nt’, which gives for the space de- scribed 4v (£-+- v/). The smaller we suppose t’, the more nearly will this approach to 4vé. But the smaller we suppose i’, the greater must be 7, the number of parts into which ¢ is divided; and the more nearly do we render the motion of the point uniformly accelerated. Hence the limit to which we approximate by diminishing ¢ without limit, is the length de- scribed in the time ¢, by a uniformly accelerated velocity, which shall in- crease from 0 to v in that time. This is 4vé, or half the length which would have been described by the velocity v continued uniformly from the beginning of the motion. It is usual to measure the accelerating force by the velocity acquired in one second. Let this be g; then since the same velocity is acquired in every other second, the velocity acquired in € seconds will be gt, or v == gt. Hence the space described is $gt x #, or 1ot?. If the point, instead of being at rest at the beginning of the acce- leration, had had the velocity @, the lengths described in the successive intervals would have been at! + v’t’, al! + 2v't’, &c.; so that to the space described by the accelerated motion would have been added nat’, or at, and the whole length would have been aé + 42°. By similar reason- ing, had the force been a uniformly retarding force, that is, one which diminished the initial velocity a@ equally in equal times, the length de- scribed in the time ¢ would have been at — }gt?. Now let the point move in such a way, that the velocity is accelerated or retarded, but not uni- formly, that is, in different times of equal duration, let different velocities be lost or gained. For example, let the point, setting out from a state of rest, move in such a way that the number of inches passed over in @ seconds is always 3. Here ¢t = @, and the velocity acquired by the body at the end of the time Z, is the coefficient of dé in (¢ + dé)’, or 30 inches per second, Let the point be at A at the end of the time ¢; and let AB, BC, C D, &c., be lengths described in or (1 + 2-+3...5.n— 14) ytetn EE v't! Fig-9. “successive equal intervals of time, each of which is di: Then the velocities at A, B, C, &c., are o ON pall 32, 3 (t+dt)*, 3 (é+ 2dt)3, &c., and the lengths AB, BC, CD, &c., are (¢ + dé)? — &, (¢+2dt)? — G+ dt)’, (¢+ 3dt)? — (+ 2dt)*, &c. Velocity at Length of ot i AB 38@dt+ 3¢(dt)?+ (dt)? B 38+ G6tdt+ 3 (dt): BC 3édt+ 9¢(dt)?+ 7 (dt) C 32+12¢dt+12 (di)° CD 3édt+15t (dt)?+19 (dt)? If we could, without error, reject the terms containing (dt)? in the velo- cities, aud those containing (d¢)* in the lengths, we should then reduce the motion of the point to the case already considered, the initial velocity being 3é, and the accelerating force 6¢. For we have already shown that a being the initial velocity, and g the accelerating force, the space described in the time ¢ is aé-+ 3gi% Hence, 3¢* being the initial velocity, and 6¢ the accelerating force, the space in the time dé is 30°dt + 3¢ (dl)’, THE DIFFERENTIAL AND INTEGRAL CALCULUS. 31 which is the same as A B after (dt)? is rejected. The velocity acquired is gt, and the whole velocity is, therefore, a ++ gt; or making the same substitutions, 3° + 6fdt. This is the velocity at B, after the term 3 (dt)? is rejected. Again, the velocity being 3¢2 + 6tdé, and the force 6t, the space described in the time dé is (30 + 6¢ dt) dt-+ 3t (dé), or 38dt -|.- 9¢ (dt). This is what the space BC becomes after 7 (dé)? is rejected. The velocity acquired is 6tdt; and the whole velocity is 3¢°-++ 6¢ dé +- Gt dt, or 3¢? + 12¢ dt; which is the velocity at C after 3 (dt)? is rejected. But as the terms involving (dé) in the velocities, &c., cannot be rejected without error, the above supposition of a uniform force cannot be made. Nevertheless, as we may take dé so small, that these terms shall be as | small parts as we please of those which precede, the results of the erro-= neous and correct suppositions may be brought as near to equality as we please ; hence we conclude, that though there is no force, which, con- tinued uniformly, would preserve the motion of the point A, so that O A _ should always be @ in inches, yet an interval of time may be taken so small, that the length actually described by A in that time, and the one which would be described if the force 6¢ were continued uniformly, shall have a ratio as near to equality as we please. Hence, on a principle similar to that by which we called 32é? the velocity at A, though, in truth, no space, however small, is described with that velocity, we call 6¢ the accelerating force at A. And it must be observed that 6¢ is the differential coefficient of 32°, or the coefficient of dt, in the development of 3 (¢ + di)?. Generally, let the point move so that the length described in any time ¢is pt. Hence the length described at the end of the time ¢ +d is p (t+dt), and that described in the interval dt is @ (é + dt) — $t, or 2 3 pit. dt-+ ft Eo) + fp" fe “t- &G, in which dé may be taken so small, that either of the first two terms shall contain the ageregate of all the rest, as often as we please. ‘These two first terms are ft. dt-+40"t . (dt)*, and represent the length described during dé, with a uniform velocity #’/t, and an accelerating force dt. ‘The interval dé may then generally be taken so small, that this supposition shall represent the motion during that interval as nearly as we please. We have hitherto considered the limiting ratio of quantities only as to their state of decrease: we now proceed to some cases in which the limit ing ratio of different magnitudes which increase Without limit is investi- gated. It is easy to show that the increase of two magnitudes may cause a decrease of their ratio; so that, as the two increase without limit, their ratio may diminish without limit. The limit of any ratio may be found by rejecting any terms or aggregate of terms (Q) which are con- nected with another term (P) by the sign of addition or subtraction, pro- vided that by increasing x, Q may be made as smalla part of P as we ‘ 2 please. For example, to find the-limit soa ith when @# is increased 2x" -+- Ox without limit. By increasing 2 we can, as will be shown immediately, cause 2x +- 3 and 5z to be contained in z? and 22°, as often as we please ; 2 . . av Tejecting these terms, we have ey OF 1 for the limit. ‘The demonstration . Ay 1 ~~ wh is as follows :—Divide both numerator and denominator by 2, which gives 2 3 5 . ema oe gt Swe please ; that is, may ‘ 1000 1 he fraction may be brought as near (OTT 001 Still more then may * Observe that this conclusion depends upon the number of quantities «, 6, &c,, being eterminate. If there be ten quantities, each of which can be brought as near to unity as fe please, their sum can be brought as near to 10 as we please ; for, take any fraction t, and make each of those quantities differ from unity by less than the tenth part of A, ten will the sum differ from 10 by less than A, This argument fails, if the number f quantities be unlimited. D 34 ELEMENTARY ILLUSTRATIONS OF 8 z be made less than 1 flicks a ick str i aI Daa AP Se pa (2— lye SE 1001 eek 1000" or 2* may be less than the thousandth part of the sum of all the preceding terms. In the same way it may be shown that a term may be taken in | any one of the series, which shall be less than any given part of the sum of all the preceding terms. It is also true that the difference of any two succeeding terms may be made as small a part of either as we please. | For (x + 1)" — a, when developed, will only contain exponents less m—1 ' a2 + &e.; and we have shown: . =1 than m, being ma2"~*-+-m. a (page 32) that the sum of such a series may be made less than any given part of a”. Itisalso evident that, whatever number of terms we may sum, if a sufficient number of succeeding terms be taken, the sum of the latter “i shall exceed that of the former in any ratio we please. : : : a ah Sar ee a Let there be a series of fractions ———;, oT ED Se TER &e., in pa-+b pal’ a’ Y which a, a’, &c. 0, b', &c., increase without limit ; but in which the ratio. of b to a, b' toa’, &c., diminishes without limit. If it be allowable to” begin by supposing 6 as small as we please with respect to a, or ~~ : | small as we please, the first, and all the succeeding fractions, will be as near | 1 Be Senne as we please to ip? which is evident from the equations a a 1 re Sal i &e bay b’ pdt Bil sent vd pa + je ghd pa'-- agg lt a a’ }) Form a new fraction by summing the numerators and denominators of the ata'+a’+ &e. preceding, sudh ase a op AY ee? the &e p(a-+ a+ a+ &.) + b+ b+ bY + &e. q extending to any given number of terms. ‘This may also be brought as a a : 1 8 ’ ° ate i near to — as we please. For this fraction is the same as 1 divided oY P b+ b'+ &e. : Fy alt ipl-s-fo sR r p+ ee and it can be shown™ that peas: must : : ; ft lie between the least and greatest of the fractions —, —, &e. If, then a a | each of these latter fractions can be made as small as we please, so alse 4 / can aa oy = No difference will be made in this result, if we use the following fraction, | A+(a+tad+a'"+ &c.) i : Btp(a-tad fal + &.) + o+0' + o + &e. oo | A and B being given quantities 5 provided that we can take a number of the original fractions sufficient to make a+ a/+-a" + &c., as great as we please, compared with A and B. ‘This will appear on dividing the num - rator and denominator of (1) by a-+ a’ + a@”-++ &c. Let the fractions _&@ + 1)° (x + 2)’ (x + 3)° walt Ms / @+iy—e? @ Fey -@+D" @+3)*= @ + 2)" * See Study of Mathematics, page 88. i 4 THE DIFFERENTIAL AND INTEGRAL CALCULUS. [ (v+1)° ea f Che first of which, or Te ster may, as we have shown, be within any U iad MC, 1 riven difference of 7 and the others stil] nearer, by taking’ a value fe sufficiently great. Let us suppose each of these fractions to be 1 vithin ———— of —, The fraction formed by summine the numerators 100000 4A y a nd denominators of these fractions (72 in number) will be within the ame degree of nearness to +, But this is : n \4 v-+n)* — w* to 2 being (1 -+ +), can be madeas creat as we lease, fw) x S it be permitted to take for m a number containing w as often as e please. Hence, by the preceding reasoning, the fraction, with its umerator and denominator thus increased, or PHP +.... + 2+(@+)l?-+.... +(e +n)? (3) jay be brought to lie within the same degree of nearness to 1 as (2); id since this degree of nearness could be named at pleasure, it follows at (3) can be brought as near to 4 as we please. Hence the limit of le ratio of (1? + 2? +....4 «) to xv, as wis increased without limit, is 5 and, in a similar manner, it may be proved that the limit of the ratio a ss 1 ei fa , (x a i ial (174-24... . +2”) to 2+ is the same as that of @t pti bs 1 This result will be of use when we come to the first prin im ples of the integral calculus. Tt may also be noticed that the limits of the ; : t—\ e—lax-—2 tios which aw —-, wv — ——, &c., bear to 2% 2° &e., are severally 2 y) 2 3 > ? ? i? ’ ’ rs 5-3? &e.; the limit being that to which the ratios approximate as & a Grrlaji ks Bel? t—1 m—2 creases without limit. For 2 EE oi ae ae 2 2x 2 8 e~-l1 x-2 oe e—l e—2 a= ——, &c.,, and the limits of . , are severally : Q¢r 3x ax r ual to unity. We now resume the elementary principles of the Diffe- itial Calculus. The following is a recapitulation of the principal results which have herto been noticed in the general theory of functions :—I, That if in the uation y = ¢@ («), the variable w receives an increment dx, y is ine pased by the series 2 iy din. dx + px ea + A/a Set + &e. D2 36 ELEMENTARY ILLUSTRATIONS OF Il. That #!z is derived in the same manner from 92, that d'x is from 92, viz., that in like manner as 9x is the coefficient of dx in the development of d (x-++-dz), so $x is the coefficient of dx in the development of db! (x + dx) ; similarly @!"x is the coefficient of dx in the development of | o" (a+dzx), andso on. IIT. That db’ is the limit of eo or the quantity to which the latter will approach, and to which it may be brought as near as): we please, when dz is diminished. It is called the differential coefficient, of y. IV. That in every case which occurs in practice, dx may be taken. so small, that any term of the series above written may be made to contail the aggregate of those which follow, as often as we please ; whence, though g’x . dx is not the actual increment produced by changing x into «-+ dx im the function 2, yet, by taking dz sufficiently small, it may be brought as near as we please to a ratio of equality with the actual increment. | The last of the above-mentioned principles is of the greatest utility,’ since, by means of it, @/a . dx may be made as nearly as we please the’ actual increment; and it will generally happen in practice, that dx . dz may be used for the increment of @x without sensible error ; that is, if in Gx, x be changed into e-+- dx, dx being very small, Oz is changed into dr-+q'x . dx, very nearly. Suppose that w being the correct value of the variable, 7+ and #-+k have been successively substituted for it, or the errors h and k have been committed in the valuation of x, h and & being very small. Hence ¢ (c++ h) and ¢(@-+ ) will be erroneously: used for dr. But these are nearly dx-+9'x. h and px + pln . kk, and: the errors committed in taking @zx are d/x.h and ¢’a . k, very nearly, These last are in the proportion of h to k, and hence results a proposition of the utmost importance in every practical application of mathematics, viz., that if two different, but small, errors be committed in the valuation of any quantity, the errors arising therefrom at the end of any process, in which both the supposed values of 2 are successively adopted, are very nearly in the proportion of the errors committed at the beginning’ For example, let there be a right-angled triangle, whose base is 3, ane whose other side should be 4, so that the hypothenuse should be ./3? + 4 or 5. But suppose that the other side has been twice erroneously mea sured, the first measurement giving 4°001, and the second 4°002, the errors being ‘001 and ‘002. The two values of the hypothenuse thu) obtained are /32 + 4-0012, or V25:008001, and V3? + 4-0022, or ¥25-016004, which are very nearly 5:0008 and 5:0016. ‘The errors of the hypothe: nuse are then °0008 and ‘0016 nearly ; and these last are in the pro portion of *001 and °002. It also follows, that if # increase by successivi equal steps, any function of x will, for a few steps, increase so nearly the same manner, that the supposition of such an increase will not by materially wrong. For, if h, 2h, 3h, &c., be successive small increment given to a, the successive increments of dx will be d'x .h, f'a . Qh Q'x . 8h, &c. nearly; which being proportional to hk, 2h, 3h, &c., the in crease of the function is nearly doubled, trebled, &c., if the increase of be doubled, trebled, &c. This result may be rendered conspicuow by reference to any astronomical ephemeris, in which the position of an heavenly body are given from day to day. ‘he intervals 0 time at which the positions are given differ by 24 hours, or near!’ zizth part of the whole year. And even for this interval, though it ¢al THE DIFFERENTIAL AND INTEGRAL CALCULUS. 37 nardly be called small in an astronomical point of view, the increments yr decrements will be found so nearly the same for four or five days ogether, as to enable the student to form an idea how much more near hey would be to equality, if the interval had been less, say one hour instead of twenty-four. For example, the sun’s longitude on the follow- ng days at noon is written underneath, with the increments from day to lay. Proportion which the differences 1 834 Sun’s longitude of the increments bear to the — at noon, Increments. whole increments, n nes: = FQ0 1 QR" september ] 158° 30’ 35 9! Ol 2 159 28 44 ee Jane x 58 12 3 a8 Y 3 160 26 56 : ices 4 161 25 9 : wate 58 14 3493 5 162 23 23 (he sun’s longitude is a function of the time; that is, the number of fears and days from a given epoch being given, and called x, the sun’s ongitude can be found by an algebraical expression which may be called ox. If we date from the first of January, 1834, 2 is 666, which is the lecimal part of a year between the first days of January and September. “he increment is one day, or nearly ‘0027 of a year. Here a is suc- essively made equal to °666, ‘666 + 0027, ‘666 +2 x "0027, &c.; and he intervals of the corresponding values of dz, if we consider only ainutes, are the same; but if we take in the seconds, they differ from me another, though only by very small parts of themselves, as the last olumn shows. This property is also used * in finding logarithms inter- oediate to those given in the tables; and may be applied to find a #earer solution to an equation, than one already found. For example, uppose it required to find the value of a in the equation dx = 0, a being hear approximation to the required value. Let @ + h be the real value, a which # will be a small quantity. It follows that d(@ + ht) =.0, or, which is nearly true, Ga + @’a.h = 0. Hence the real value of h is a pa , a, early — Va or the value a — pla is a nearer approximation to the value a a fa. For example, let 2® + 2 —4=0 be the equation. Here dv = ptoe—4, andd(@+h)=(e@+thPtath-4e=e+ue—44 2e-+1)h-+h*; so that d’a= 2e+1. A near value of x is 1°57; Q pt this be a. Then da = ‘0349, and d/a = 4°14. Hence — o = 00843. Hence 1+57 — *00843, or 1 °56157, is anearer value of x. If ‘e proceed in the same way with 1°5616, we shall find a still nearer value Fx, viz.. 1°561553. We have here chosen an equation of the second egree, in order that the student may be able to verify the result in the ommon way; it is, however, obvious that the same method may be ap- lied to equations of higher degrees, and even to those which are not ) be treated by common algebraical methods, such as tan vw = az. We have already observed, that in a function of more quantities than ne, those only are mentioned which are considered as variable; so that il which we have said upon functions of one variable, applies equally to ictions of several variables, so far as a change in one only is concerned. jake for example a°y + 2a2y°. If x be changed into w + da, y remaining le same, this function is increased by 2vy dx + 2y*dx + &c., in which, * See Sludy of Mathematics, page 58. & enue ae. wee er - .. 38 ELEMENTARY ILLUSTRATIONS OF : as in page 15, no terms are contained in the &c. except those which, by dimi- nishing dv, can be made to bear as small a proportion as we please to the | first terms. Again, ify be changed into y+ dy, x remaining the same, | the function receives the increment a®dy + 6ry%dy + &c.; and if # be changed into x + dx, y being at the same time changed into y + dy, the increment of the function is (2ry + 2y*) dx + (a? + 6xy*) dy + &e, Tf, then, wu = ay + 2ry’, and du denote the increment of w, we have the | three following equations, answering to the various suppositions above- | mentioned, (1) when @ only varies, du == (2xy + 2y°) dx + &e. (2) when y only varies, du = (2° + Gry?) dy + &c. | (3) when both wand y vary, du = (2ry +2y°) dx+ (2° +6ry?) dy+&e, | in which, however, it must be remembered, that dw does not stand for the same thing in any two of the three equations: it is true that it always represents an increment of u, but as far as we have yet gone, we have used it indifferently, whether the increment of w was the result of a change in v only, or y only, or both together. To distinguish the different incre=| ments of uw, we must therefore seek an additional notation, which, without sacrificing the dw that serves to remind us that it was 2% which received an increment, may also point out from what supposition the increment arose. For this purpose we might use d,w and d,w, and d,,,,¥, to dis- tinguish the three; and this will appear to the learner more simple than the one in common use, which we shall proceed to explain. We must however, remind the student, that though in matters of reasoning, he has. aright to expect a solution of every difficulty, in all that relates to nota: tion, he must trust entirely to his instructor; since he cannot judge be tween the convenience or inconvenience of two symbols without a degree of experience, which he evidently cannot have had. Instead of the nota tion above described, the increments arising from a change in @ and y art d severally denoted by — dx and i dy, on the following principle:—I there be a number of results obtained by the same species of process, bu on different suppositions with regard to the quantities used; if, for ex. ample, p be derived from some supposition with regard to a, in the sam) manner as are g and r with regard to 6 and c, and if it be inconvenien | and unsymmetrical to use separate letters p, g, and 7, for the three result they may be distinguished by using the same letter p for all, and writin - b, £ c. Each of these, in commoi| aleebra, is equal to p, but the letter p does not stand for the same thin in the three expressions. The first is the p, so to speak, which belong, to a, the second that which belongs to 6, the third that which belongs to ¢ P Pi never be separated from its denominator, because the value of the forme depends, in part, upon the latter; and one p cannot be distinguishe from another without its denominator. The numerator by itself only it dicates what operation is to be performed, and on what quantity ; the de nominator shows what quantity is to be made use of in performing i ” the three results thus, ahs a, a Therefore the numerator of each of the fractions La, and 4, mus a ; b Neither are we allowed to say that P divided by £. is — ; for this suf a a | poses that p means the same thing in both quantities, In the ey THE DIFFERENTIAL AND INTEGRAL CALCULUS. 4 du du ; , | pressions = dx, and a dy, each denotes that «w has received an incre x 1” ment; but the first points out that #, and the second that y, was sup- posed to increase, in order to produce that increment; while dw by itself, or sometimes d.w, is employed to express the increment derived from _ both suppositions at once. And since, as we have already remarked, it -is not the ratios of the increments themselves, but the limits of those ratios, which are the objects of investigation in the Differential Calculus, : du du : here, as in page 15, Zz dx, and — dy, are generally considered as re- dy presenting those terms which are of use in obtaining the limiting ratios, _and do not include those terms, which, from their containing higher | powers of dz or dy than the first, may be made as small as we please with respect to dx or dy. Hence in the example just given, where u=2°y+2zy’, ' we have d : Li — dx = (2xy + 2y*) dz, or = = 2ry + 2y° l = dy = (2% + 6xy*) dy, or - — ce + 6xy? F di. du du or d.u rm dz + di dy. . The last equation gives a striking illustration of the method of notation, —_ | Treated according to the common rules of algebra, it is du = du + du, } which is absurd, but which appears rational when we recollect that the —— Se second dw arises from a change in @ only, the third from a change in y only, and the first from a change in both. The same equation may be | proved to be generally true for all functions of x and y, if we bear in ' mind that no term is retained, or need be retained, as far as the limit is ) concerned, which, when dz or dy is diminished, diminishes without limit du ‘ ; u , : A }as compared with them. In using — and — as differential coefficients dx dy | of w with respect to x and y, the objection (page 14) against considering | these as the limits of the ratios, and not the ratios themselves, does not hold, since the numerator is not to be separated from its denominator. Let wu be a function of # and y, represented * by ¢ (a, y). It is indift ferent whether x and y be changed at once into w+ dx and y + dy, or whether x be first changed into # + dz, and y be changed into y + dy in the result. Thus, a*y + y° will become (# + dx)? (y + dy) + (y + dy)® ineithercase. If«xbe changed into x + dx, uw becomes wu + wu’ dx + &e., where wz’ is what we have called the differential coefficient of uw with respect to 2, and is itself a function of « and y; and the correspond- ing increment of w is u/dx + &c. If in this result y be changed into y + dy, wu will assume the form w+ uw, dy + &c., where w, is the diffe- -rential coefficient of « with respect toy; and the increment which uw *The symbol ¢(a, y) must not be confounded with g(ry). The former represents any function of a and y ; the latter a function in which 2 and y only enter so far as they are _ contained in their product. The second is therefore a particular case of the first; but the ‘first is not necessarily represented by the second. For example, take the function _ represented in the same way, since ether functions besides the product are contained in it. zy -- sin ay, which, though it contains both x and y, yet can only be altered by such a change in # and y as will alter their product, and if the product be called p, will be p+sin p. This may properly be represented by ¢(ay); whereas x + vy? cannot be 40 ELEMENTARY ILLUSTRATIONS OF receives will be u,dy + &c. Again, when y is changed into y + dy, w, which is a function of 2 and y, will assume the form ul + pdy + &e.;5 aud u + u'dx + &c. becomes u-+udy + &c.-+ (u' + pdy + &c.) dx+&c., or utu, dy+udx + pdx dy + &c., in which the term pdx dy is useless in finding the limit. For since dy can be made as small as we please, pdx dy can be made as small a_ part of pdx as we please, and therefore can be made as small a part of dx as we please. Hence on the three suppositions already made, we have the following results :— 1. when @ only is changed _ into @ + dz, qu'dx + &e. 2, when y only is changed]. : w receives the t u &C. MERLIN PA increment centr ioe 3, when x becomes 2-+ dx j WwW bi C. and y becomes y + dy Lulde + ujdy 1% at once, the &c. in each case containing those terms only which can be made as small as we please, with respect to the preceding terms. In the language of Leibnitz, we should say that if # and y receive infinitely small incre- ments, the sum of the infinitely small increments of w obtained by making these changes separately, is equal to the infinitely small increment ob- tained by making them both at once. As before, we may correct this in- accurate method of speaking. The several increments in 1, 2, and 3, may be expressed by u! dv +P, u, dy+Q, land wda + u,dy +R; where P, Q, and R can be made such parts of dx or dy as we please, by taking dw or dy sufficiently small. The sum of the two first is wdx +udy + P + Q, which differs from the third by P+ Q — R3 which, since each of its terms can be made as small a part of dx or dy as we please, can itself be made less than any given part of dx or dy. This theorem is not confined to functions of two variables only, but may be extended to those of any number whatever. Thus, if 2 be a function of — PP, GT, and s, we have dz dz dz dz i es oe ria oe 1 & djxordz = a dp+ 2 dq+ zn dr =. ds + &e. d : : : : : in which = dp + &c. is the increment which a change in p only gives to z, and so on. The &c. is the representative of an infinite series of terms, the aggregate of which diminishes continually with respect to dp, dq, &c., as the Jatter are diminished, and which, therefore, has no effect on the limit of the ratio of d.zto any other quantity. We proceed to an im- portant practical use of this theorem. If the increments dp, dq, &c., be small, this last-mentioned equation, the terms included in the &e. being omitted, though not actually true, is sufficiently near the truth for all practical purposes ; which renders the proposition, from its simplicity, of the highest use in the applications of mathematics. For if any result be obtained from a set of data, no one of which is exactly correct, the error in the result would be a very complicated function of the errors in the data, if the latter were considerable. When they are small, the error in the results is very nearly the sum of the errors which would arise from the error in each datum, if all the others were correct. For if p,q, 7 and s, are the presumed values of the data, which give a certain value z to the function required to be found; and if p + dp, q + dg, &c., be the correct ce + 5$2,5- Hence sin 16°°1’ = sin 16°-+- °9612617 x ahs - = 12756374 + +0002797 = ‘2759171, nearly. The tables give -2759170. These examples may serve to show how nearly the real ratio of two increments approaches to their limit, when the increments them- selves are small. When the differential coefficient of a function of w has been found, the result, being a function of #, may be also differentiated, which gives the differential coefficient of the differential coefficient, or, as it is called, the second differential coefficient. Similarly the differential coefficient of the second differential coefficient is called the third differential coefficient, and so on. We have already had occasion to notice these successive differen- tial coefficients in page 12, where it appears that dx being the first dif- ferential coefficient of Or, Oa is the coefficient of h in the development. d'(a + h), and is therefore the differential coefficient of @’x, or what we have called the second differential coefficient of pz. Similarly fa is the third differential coefficient of dv. If we were strictly to adhere to our system of notation, we should denote the several differential coefficients of px or y by dy dy ne 1 dx dz &c, dx da dx in order to avoid so cumbrous a system of notation, the following symbols are usually preferred, * See Study of Mathematics, page 90. THE DIFFERENTIAL AND INTEGRAL CALCULUS. dy d?y d’y dx dx? dx” We proceed to explain the manner in which this notation is connected with our previous ideas on the subject. When in any function of x, an increase is given to wv, which is not supposed to be as small as we please, it is usual to denote it by Aw instead of dx, and the corresponding incre- ment of y or pz, by Ay or Adz, instead of dy or dfx. The symbol Ax is called the difference of x, being the difference between the value of the variable x, before and after its increase. Let x increase at successive steps by the same difference, that is, let a variable, whose first value is x, successively become «+ Av, r+2Aa,x+3 Ax, &., and let the stic- cessive values of dx corresponding to these values of & be Y, Yrs Yor Yas &e., that is, Gr is called y, P(@ + A 2) is Yo P(e-+ 2A2) is y,, &e., and, generally, P(e + mA) is y,. Then, by our previous definition Yi y is Ay, Yy— Yi iS AY, Ys — Yo is Ay,, &c., the letter A before a quah- tity always denoting the increment it would receive if #+- Aw were sub- stituted for x, Thus y, or $(w-++ 3A 2) becomes O(a + Ax + 8Ax), or P(e +4 Ax), when 2 is changed into # -+ Aa, and receives the incre- ment O(v + 4A 2) — O(@ + 3Aa), ory, — y;,. If y be a function which decreases when @ is increased, y, — y, or Ay is negative. It must be observed, as in page 13, that Aw does not depend upon a, because wv occurs in it; the symbol merely signifies an increment given to 2, which increment is not necessarily dependent upon the value of 2. For in« stance, in the present case we suppose it a given quantity; that is, when z+ Axis changed into r+Ax+ Aa, or x + 2Aa, zw is changed, and Awvisnot. In this way we get the two first of the columns under- neath, in which each term of the second column is formed by subtract- ing the term which immediately precedes it in the first column from the one which immediately follows. Thus Ay is y,—y, Ay, is ys—y,, &e, P(x) or y Ay ORR ORs gt eat) ae OIE re A*ty | A% @(wetb2Aa) s..3° Ys ud A*y, 4 A*y Ye AY P(@+3Az)_...:°Y, : p(w t4AX) 2. Y% &e. In the first column is to be found a series of successive values of the same function gz, that is, it contains terms produced by substituting successively in gx the quantities 7, r+Aa, x+2Aa, &c., instead of x. ‘The second column contains the successive values of another function p(a + Ax) — G2, or AGzx, made by the same substitutions; if, for ex- ample, we substitute 7+ 2 A @ for x, we obtain d(e+3 Av) —d(#+2Aa), Or Y;— Ys, Or Ay, If, then, we form the successive differences of the terms in the second column, we obtain a new series, which we might call the differences of the differences of the first column, but which are called the second differences of the first column. And as we have denoted the operation which deduces the second column from the first by A, so that which deduces the third from the second may be denoted by A A, which is abbreviated into A®. Hence as y, — y was written Ay, Ay, — Ay is written A Ay, or A’y. And the student must recollect, that in like manner as A is not the symbol of a number, but of an operation, so A®* does not denote a number multiplied by itself, but an operation repeated upon its own result; just as the logarithm of the logarithm of « might be written log *x; (log «)? being reserved to sig- &e, 44 ELEMENTARY ILLUSTRATIONS OF nify the square of the logarithm of x. We do not enlarge on this nota- tion, as the subject has been already discussed in the treatise on Alge- braical Expressions, No. 105, the first six pages of which we particularly recommend to the student’s attention, in relation to this point. Similarly the terms of the fourth column, or the differences of the second differences, have the prefix A A A abbreviated into A®, so that A2y,— A*’y= A*%y, &c. When we have occasion to examine the results which arise from supposing Aw to diminish without limit, we use da instead Aa, dy in- stead of Ay, d’y instead of A*y, and so on. If we suppose this case, we can show that the ratio which the term in any column bears to its cor- responding term in any preceding column, diminishes without limit. Take, for example, d’y and dy. The latter is f(x + dx) — x, which, as we have often noticed already, is of the form pdx -+ q (dx)* + &c., in which Pp, q, &¢., are also functions of z. To obtain d2y, we must, in this series, change 2 into v + dz, and subtract pdx + q (dx)? +- &c. from the result. But since p, g, &c., are functions of a, this change gives them the form p+ pide+ &, ¢ + q/dx + &e.; so that dy is (p--p!dx+&e.) de+ (q+q'dx+ &e.) (da)?+ &e.— (pdx-+q(dx)?+ &c.) in which the first power of dx is destroyed. Hence (page 21), the ratio of d?y to dx diminishes without limit, while that of d’y to (dx)? has a finite limit, except in those particular cases in which the second power of dx is destroyed in the previous subtraction, as well as the first. In the same way it may be shewn that the ratio of d’y to dx and (dx)* decreases without limit, while that of d°y to (dx)® remains finite; andsoon. Hence dy Wy dy dx dx”. da finite limits when dz is diminished. We now proceed to show that in the development of d(#-+ h), which has been shown to be of the form h® dr + Suh+ ole — + Hers -+ &e. we have a succession of ratios &e., which tend towards : Paid’ d in the same manner as @’a is the limit of = (page 12), so @’a is the # d*1 F d3 limit of — o''x is that of ey and so on. From the manner in which the preceding table was formed, the following relations are seen immediately: y=ytAy Ayn=Ayt AY Ay= Atyt Aly &e. Yo=Yit AN Ay,= Ayt A% A*y,= A*y, + A®*y, &e., Hence Y, Yo &e., can be expressed in terms of 9, Agta 7a, aura For y= yt Ays w= wt 4m = Yt Ay) + (Ay + A*Yy) = y+ 2Ay+ Ary; in the same way AYy,= Ayt2Aty + A*’y; hence y= Yo Ays= (Cy + 2Ay + A’y) + Ay + 2A%y + Ary) =ytBAyt3A*y + A*y. Proceeding in this way we have Yin Feng. “5 Ay Yo=yte2Aytr AY Y=yt+saAy+ sAyr.. AY yo=yts4Ay+ GAly+ 4 Alyy Aly Ys = yt bAy+lOA*'y + 10 At’y + 5A*y + A*y, Ke. from the whole of which it appears that y, or P(a@-+ 7A) is a series consisting of y, Ay, &c., up to A”y, severally multiplied by the coefli- cients which occur in the expansion of (1 +- a)", or ~* THE DIFFERENTIAL AND INTEGRAL CALCULUS. 45 m—l. n—1 n—2 in Y¥.=P(etnAzr)=ytnday+n. Lytn.——. a A’y + &e. Let us now suppose that 2 becomes «+h by 2 equal steps; that is, a, 2 nh . a ++ Rp? Ctr» &e..... 2+ — or e+h, are the successive values of 2, ! n n so thatnAr=h. Since the product of a number of factors is not altered by multiplying one of them, provided we divide another of them by the Same quantity, multiply every factor which contains 2 by “Aw, and divide the accompanying difference of y by Aw as often as there are factors which contain 7, substituting h for n Aw, which gives Ay nAxv—Ar A2y nAv—Av nAv—QAr A8y @-nAv) =7 —_— oe asd Sky) a res : =C. ¢ (v-+-nAzr) =y-+nAr oF +nAx 5 (ans +nAr 5 3 (Ar) + &e Ay N— Av Ay h—Av h—2QAvr A8y or ¢(#+h) = ——— - &e. ¢ + ) yYth bi, Test 2 (Ax)2 + 9 3 (Ax )3 By Ifk remain the same, the more steps we make between @ and 2 - h, the smaller will each of those steps be, and the number of steps may be increased, until each of them is as small as we please. We can therefore suppose Av to decrease without limit, without affecting the truth of the Series just deduced. Write dx for Aa, &c., and recollect that — dz, h — 2dz, &c., continually approximate to k. The series then becomes . yh? &y + dy AND FUN AHS she ox Ope aie : 4 ae d ‘ in which, according to the view taken of the symbols os &e. in page 14, Bs da é dy . dy, cy stands for the limit of the ratio of the increments, cy is a, ss 1S dz dx dx px, &e. According to the method proposed in page 15, the series written above is the first term of the development of @(7-+ h), the remaining terms (which we might include under an additional ++ &c.) being such as to diminish without limit in comparison with the first, when dz is di- : dzy , < minished without limit. And we may show that the limit of = ais the dif- ferential coefficient of the limit of. ; or if by these fractions themselves x dy , ; yp dy are understood their limits, that 4 is the differential coefficient of sa: for dx : since dy, or d(x + dx) — dr, becomes dy + d’y, when v is changed into dy _. « + dx; and since dx does not change in this process, fae will d dl i become = +, or its increment is —. The ratio of this to dz is xv L v ee the limit of which, in the definition of page 12, is the differential co- Ly baat © ny ; mt efficient of Tn Similarly the limit Or at is the differential coefficient of dx oh d? the limit of st and so on. dx? We now proceed to apply the principles laid down to some cases in which the variable enters into its function in a less direct and more com- 46 ELEMENTARY ILLUSTRATIONS OF plicated manner. For example, let 2 be a given function of x and y, and let 4 be another given function of #; so that z contains @ both directly and indirectly; the latter as it contains y, which is a function of 7 This will be the case if z=wa log y, where y= sin a. If we were to substitute for y its value in terms of 2, the value of z would then be a function of # only; in the instance just given it would be # log sina, But if it be not con- venient to combine the two equations at the beginning of the process, let us first consider zas a function of x and y, In which the two variables are independent. In this case, if @ and y respectively re- ceive the increments dz and dy, the whole increment of 2, OF G2: (or at least that part which gives the limit of the ratios) is represented by dz az . : — dx + Ta dy. fy be now considered as a function of 2, the conse- y dx quence is that dy, instead of being independent of dz, is a series of the form pdz + q (da)? -+- &e.,, in which p is the differential coefficient of y Gee dz dz ! dz dz with respect tox, Hence d.2 = ys de + ape pdx or Kg eee Ee -+ i D, d.z dae sins ". in which the difference between as and cof this, that in the second, 2 is only considered as varying where it is directly contained in Z, or 2 is considered in the form in which it first appeared, as a function of » and y, > . . 2 Mey where y is independent of x; in the first, or 5 the total variation of z is denoted, that is, y is now considered as a function of w, by which means if v become #+dz, 2 will receive a different increment from that which it would have received, had y been independent of 2. In the instance above cited, where z= log y and y= sin a, if the first equation be taken, and w becomes #-+dz, y remaining the same, z becomes # log yt+log y dz d or “ is log y. Ify only varies, since (page 11) « will then become dy 232 ly . ; x log y--@ = — &¢., i, is 7 And — is cos @ when y=sin w (page 11). dz dz dz dz dy a : geen LOE. If we substitute this instead of da, and divide by dx, taking the limit of d would have been expressed by = swering to that increment. IV. arising from @ containing x * It may be right to warn the student that this phraseology is new, to the best of our knowledge. The nomenclature of the Differential Calculus has by no means kept pace with its wants; indeed the same may be said of Algebra generally, 48 ELEMENTARY ILLUSTRATIONS OF the ratios, we have the result first given. For example, let z=a°ya’, y=2, anda=a*y. Taking the first equation only, and substituting w+ dx for x z d &e., we find a = aye’, = = aa, and _ = 32°ya?. From the second da dx —— d err 2 = 2x, and from the third — = 32°y, and fe =a. Substituting these diz in the value of —, we find dx 2 gare ie dz dy dz da dy d 2 da 2 Ore ee ee. aa ee dx dix dy . dz da dy dx da Lvs = Qrya + aa X 2x + 3a°ya’ x a xX 2a + B2°ya® x 3x°y = 2rya*? + 2a°a° + 62° ya? + 9aty?e® If for y and @ in the first equation we substitute their values a and 2’y, or xv, we have z= 2", the differential coefficient of which is 192". This is the same as arises from the formula just obtained, after a” and a have been substituted for y and @; for this formula then becomes Qe% + 22% + 6a + 9 a or 19 2”. In saying that z is a function of w and y, and that y is a function of 2, we have first supposed 2 to vary, y remaining the same. The student must not imagine that y is then a function of «; for if so, it would vary when w varied. There are two parts of the total differential coefficient, arising from the direct and indirect manner in which z contains #. ‘That these two parts may be obtained separately, and that their sum constitutes the complete differential coefficient, is the theorem we have proved. ‘The d first part = is what would have been obtained if y had noé been a func- tion of a; and on this supposition we therefore proceed to find it. The d other part eg is the product of—I. c*': which would have resulted dy dx dy 3 d from a variation of y only, not considered as a function of ar al: : the coefficient which arises from considering y as a function of 2. ‘These partial suppositions, however useful in obtaining the total differential co- efficient, cannot be separately admitted or used, except for this purpose ; since if y be a function of 2, « and y must vary together. If z be a function of 2 in various ways, the theorem obtained may be stated as follows :—Find the differential coefficient belonging to each of the ways in which z will contain z, as if it were the only way ; the sum of these results (with their proper signs) will be the total differential coefhi- : A. dz, d cient. Thus, if z only contains a indirectly through y, ~ is ae If 2 ‘ ‘ dz dz da db contains @, which contains 6, which contains 7, — = —- —- —. dx da db dz This theorem is useful in the differentiation of complicated functions ; for example, let z = log (a?+a’). If we make y=a"+ a’, we have z=log y, Gg 3 vik y dz and —- = —; while from the first equation — = 22. Hence — oF dy dx dx 4 dz dy. 2x 2x es ee Ea or 2 e dy dx y a -+a% If z = log log sin a, or the logarithm of the | | THE DIFFERENTIAL AND INTEGRAL CALCULUS. 49 logarithm of sin xv, let sin «= y and log y= a; whence z = loge a, . . . J @ and contains 2, because @ contains y, which contains 2 Hence 7 eae 2 dz da dy sk ted dz ; LE ee | ee nt si = ne = ,=- 3 da “dy dx nees Of tt, da q> since @ og ¥ ee: , dj d d. 1 1 and since y=sin 2, “T _ cosz. Hence ~ = Me nae =—— cos # dx dx dady dx ay cos & ; Mite ap a ee NOW pltasome, rules: dn; the steerer log sin w . sin x applications of this theorem, though they may be deduced more simply. I. Let z = ab, where a and D are functions of 2. The general formula, since z contains indirectly through @ and 3, is (in this case as well as in those which follow,) dz dz da _. dz db Gr tard h ie da db ‘ We must leave a and — as we find them, until we know what func v dx tions @ and 8 are of x; but as we know what function z is of @ and 8, ; dz dz A we substitute for ) and i Since z= ab, if a becomes a + da, z da dz : becomes ab + dda, whence es 6. Jn this case, and part of the follow- a ing, the limiting ratio of the increments is the same as that of the incre- hae dz ments themselves. Similarly ao whence dz da db from z = ab follows a b qa +a ae Il. Letz = -. If @ become a+ da, z becomes . =< or - a wert a a adb and Was: If b become 6+ dd, z becomes Fat ds pire, Sgr “ie. dex. a whence ~*~ ae Hence a db f; a 1 dz __1 .da adb sr te ae Baer a Qe ete dx b °° dx 2dr b2 Tit. Let z= a’. Here (a + da)'= a’ + ba’! da + &c., (page 11,) whence a = ba". Again, a” = a’ a® = a (1 + log a db + &c.) a hence = =a’ log a. Therefore 2 dz da... d P. apa a b=1 B A rom z= a’ follows ria ba ms + @ log a 7 If y be a function of x, such as y = x, we may, by solution of the iquation, determine x in terms of y, or produce another equation of the ormx=y. For example, when y = 22, «= y}. It is not necessary hat we should be able to solve the equation y = x in finite terms, that Ss, So as to give a value of w without infinite series; it is sufficient that x K 50 ELEMENTARY ILLUSTRATIONS OF can be so expressed that the value of « corresponding to any value of ¥ may be found as near as we please from 2 = Jy, in the same manner as the value of y corresponding to any value of @ is found from y = 92. The equations y = (2, and «== Wy, are connected, being, in fact, the same relation in different forms; and if the value of y from the first be substituted in the second, the second becomes v= (2), or as it is more commonly written, Lox. That is, the effect of the operation or set of operations denoted by y is destroyed by the effect of those denoted by #; as in the instances (x°)3, (2°)8, 8", angle whose sine is (sin 2), &c., each of which is equal tow. By differentiating the first equation y = hz, we obtain 4 — gx, and from the second iy =)/y. But what- ever values of and y together satisfy the first equation, satisfy the second also; hence, if when x becomes @ + dx in the first, y becomes y -+- dy; the same y + dy substituted for y in the second, will give the dx ; d same x + dz. Hence oF as deduced from the second, and as deduced from the first, are reciprocals for every value of dx. The limit of one 1s therefore the reciprocal of the limit of the other; the student may easily es aks 1 ; prove that if @ is always equal to ry and if @ continually approaches to the limit a, while 5 at the same time approaches the limit f, « is equal dx to rE But re or w'y, deduced from x = Vy, is expressed in terms of y, . dy . , while in? glx, deduced from y = 2 1s expressed in terms of x. There- v | fore )'y and ¢’2 are reciprocals for all such values of v and y as satisfy either of the two first equations. For example let y = e’, from which | d ec=log y. From the first (page 11) “ = «?; from the second a2 wile : dy Yy and it is evident that e* ane are reciprocals, whenever y = €*. ’ ' ; dy? Caan If we differentiate the above equations twice, we get oy = pa, and dx* dx . d”4 an — «yx. There is no very obvious analogy between sa and +3 dy dx? dy indeed no such appears from the method in which these coefficients were first 3 formed. Turn to the table in page 43, and substitute d for A throughout, | to indicate that the increments may be taken as small as we please. We) there substitute in Ox what we will call a set of equidistant values of 2, or values in arithmetical progression, viz., v, e+ dr, x + Qdzx, &e. The} resulting values of y, or ¥, Yi, &c., are not equidistant, except in one func tion only, when y=az-+b, where @ and b are constant. Therefore dy, dy | &c., are not equal ; whence arises the next column of second differences, 2 or dy d’y,, &e. The limiting ratio of d?y to (dx), expressed by ae is the second differential coefficient of y with respect to 2, If from y = pr we deduce w= Wy, and take a set of equidistant values of y, viz. ¥, Y =F dy, y + 2dy, &e., to which the corresponding values of x are 2, @,, %q, &C., @ similar table may be formed, which will give dx, dz,, &c., da, d?2,, Kes THE DIFFERENTIAL AND INTEGRAL CALCULUS, -51 eae, Pe , and the limit of the ratio of d2r to (dy)? or — 1s the second differential coefficient of x with respect to y. These tions, dx being given in the first table, second dy is given and dv varies, are entirely different supposi- and dy varying; while in the We may show how to deduce one from the other as follows i——-When, as before, y= dr andv= Wy, we have dy —o een y fe i Rigs: . ey 2 vy — a if u'y be called p. Calling this w, and considering it as a function of x from containing py, which contains y, which contains a, dudpdy,.. .. ; ; we have oF Bd foe tts differential coefficient with respect toz. But since ip dy dx anes lL du 1 dp oe ip =- Pp > since p = w'y, dy = yy: and wy is the differential dy \2 or (d’x)? or ce dx ‘ : : ay ‘ : Hence the differential coefficient of x or —, with respect to #, which is dy dy\? dx dy zy Bx 2 2 4¥ . HZ Eve: coeflicient of y/y, and is Te Also — is dy? l p Cy’y)? —,isalso — If y= e* whence a=log y, dx’ dx) dy? dz \aer dy? , d?: le 1 a’. we have —% — e and —/ = &.' But — = = and ieee oe ee z : dx dx dy y dy? yf? ly\? dd: : i! = Therefore — ) eeiche ie or — or af which is e*, the de} dy? y? oy ne : d*4 ‘ g . . value just found for Te In the same way a might be expressed in dx® dx dx d°x terms of —, —; and —; and so on. dy d d The variable which appears in the denominator of the differential co- efficients is called the independent variable. In any function, one quantity at least is changed at pleasure; and the changes of the rest, with the limiting ratio of the changes, follow from the form of the function. The number of independent variables depends upon the number of quan- tities which enter into the equations, and upon the number of equations which connect them ; if there be only one equation, all the variables ex- cept one are independent, or may be changed at pleasure, without ceasing to satisfy the equation; for in such a case the common rules of algebra tell us, that as long as one quantity is left to be determined from the rest, it can be determined by one equation; that is, the values of all but one are at our pleasure, it being still in our power to satisfy one equation, by giving a proper yalue to the remaining one, Similarly, if there be two equations, all variables except two are independent, and so on. If there be two equations with two unknown quantities only, there are no vari- ables; for by algebra, a finite number of values, and a finite number only, can satisfy these equations; whereas it is the nature of a variable to receive any value, or at least any value which will not give impossible values for other variables. If then there be m equations containing 7 vari- ables, (7 must be greater than m,) we have n — m independent variables, to each of which we may give what values we please, and by the equations, deduce the values of the rest. We have thus various sets of differential E 2 52 ELEMENTARY ILLUSTRATIONS OF coefficients, arising out of the various choices which we may make of in- dependent variables. If for example, a, b, 2, y, and 2, being variables, we have (a, b, 2, y, 2,) =9 ws (a, b, v7, Y, 2,) = 0 x (a, b, 2, Ys 2) = 9, we have two independent variables, which may be either and y, 7 and 2, aand 6, or any other combination. If we choose vand y, we should deter- mine a, b, and z in terms of # and y from the three equations ; in which case we can obtain ola an 18 Oe Seem egeTe lh rere ; ae dt dy dz When y is a function of x, asin y = 2, it is called an erplicit function of z. This equation tells us not only that y is a function of x, but also what function it is. The value of x being given, nothing more is necessary to determine the corresponding value of y, than the substitution of the value of vin the several terms of dz. But it may happen that though y is a func- tion of z, the relation between them is contained in a form from which y must be deduced by the solution of an equation. For example, in e—-aytyoa, when vz is known, y must be determined by the solu= tion of an equation of the second degree. Here, though we know that y must be a function of a, we do not know, without further investigation, what function it is. In this case y is said to be implicitly a function of a, or an implicit function. By bringing all the terms on one side of the equation, we may always reduce it to the form A(a, y) = 0. Thus, in the case just cited, we have 2° — ay +4? — a= 0. We now want to sda deduce the differential coefficient = from an equation of the form d(x, y) = 0. If we take the equation u = d(x, y), in which when x and y become x + dx and y + dy, w becomes w + du, we have, by our former principles, du —- wdx + u,dy + &c., (page 40), in which w! and u, can be directly obtained from the equation, as in page 39. Here x and y are independent, as also dx and dy; whatever values are given to them, it is sufficient that « and dw satisfy the two last equa- tions. But if 2 and y must be always so taken that % may == 0, (which is implied in the equation p(x, y) = 0,) we have w= 0, and du=0; and this, whatever may be the values of dx and dy. Hence dx and dy are connected by the equation 0 = wdr + udy + &e., and their limiting ratio mnst be obtained by the equation dy ul udx +udy=0, or =A = — 7 y and « are no longer independent ; for, one of them being given, the other | must be so taken that the equation (7, y) = 0 may be satisfied. The. ae du quantities w/ and w, we have denoted by and -. so that Y du dy ae dx ae" ) dy, We must again call attention to “the different meanings of the same symbol dw in the numerator and denominator of the last fraction. Had | du dx and dy been common algebraical quantities, the first meaning the” same thing throughout, the last equation would not have been true until | the negative sign had been removed. We will give an instance in which | - : } | THE DIFFERENTIAL AND INTEGRAL CALCULUS. 53 du shall mean the same thing in both. Let x= p(x), and let w= yy, in which two equations is implied a third dv = vy; and y is a function of x. Here, x being given, u is known from the first equation ; and being known, y is known from the second, Again, 2 and dz being given, du, which is 6(v + dx )—¢zx is known, and being substituted in the result of the second equation, we have du=(y-+ dy) — wy, which dy must be so taken as to satisfy. From the first equation we deduce du = @'x dx + &c. and from the second du = yy dy + &c., whence pln dz + &. = Wy dy + &e; the &c. only containing terms which disappear in finding the limiting ratios. Hence | du eek (2) dx uly i a result in accordance with common algebra. But the equation (1) was obtained from u = (a, y), On the supposition that wv and y were always so taken that w should = 0, while (2) was obtained from u— d(2) and u = Wy, in which no new supposition can be made; since one more equation between wu, x and y would give three equations connecting these three quantities, in which case they would cease to be variable (page 51). As an example of (1) let zy—a = 1, or ty —-x2—-1=0. From du d y= ry — x — 1 we deduce (page 39) =y-l, a = «v3; whence, by equation (1), dy y—- 1 1 By solution of cy — #= 1, we find y= 1 + vat and ] 1 dx 1 Hence = (meaning the limit) is — ost which will also be the result of C Gi ie (3) if 1 + Ee be substituted for y. x To follow this subject farther would lead us beyond our limits; we will therefore proceed to some observations on the differential coefficient, which, at this stage of his progress, may be of use to the student, who should never take it for granted that because he has made some progress in a science, he understands the first principles, which are often, if not always, the last to be learned well. Ifthe mind were so constituted as to receive with facility any perfectly new idea, as soon as the same was legitimately applied in ma- thematical demonstration, it would doubtless be an advantage not to have any notion upon a mathematical subject, previous to the time when it is to become a subject of consideration after a strictly mathematical method, This not being the case, it is a cause of embarrassment to the student, that he is introduced at once to a definition so refined as that of the limiting ratio which the increment ofa function bears to the increment of its variable. Of this he has not had that previous experience, which is the case in regard to the words force, velocity, or length. Nevertheless, he can easily conceive a mathematical quantity ina state of continuous increase or des 54 ELEMENTARY ILLUSTRATIONS OF crease, such as the distance between two points, one of which is in motion. The number which represents this line (reference being made to a given linear unit) is ina corresponding state of increase or decrease, and so is every function of this number, or every algebraical expression in the formation of which it is required. And the nature of the change which takes place in the function, that is, whether the function will increase or decrease when the variable increases ; whether that increase or decrease corresponding to a given change in the variable will be smaller or greater, &c., depends on the manner in which the variable enters as a component part of its fanction. Here we want a new word, which has not been invented for the world at large, since none but mathematicians consider the subject; which word, if the change considered were change of place, depending upon change of time, would be velocity. Newton adopted this word, and the corresponding idea, expressing many numbers in suc- cession, instead of at once, by supposing a point to generate a straight line by its motion, which line would at different instants contain any dif ferent numbers of linear units. To this it was objected that the idea of time is introduced, which is foreign to the subject: We may answer that the notion of time is only necessary, inasmuch as we are not able to con- sider more than one thing at a time. Imagine the diameter of a circle divided into a million of equal parts, from each of which a perpendicular is drawn meeting the circle. A mind which could at a view take in every one of these lines, and compare the differences between every two con- tiguous perpéidiculars with one another, could, by subdividing the dia- meter still further, prove those propositions which arise from supposing a point to move uniformly along the diameter, carrying with it a perpendi- cular which lengthens or shortens itself so as always to have one extremity on the circle. But we, who cannot consider all these perpendiculars at once, are obliged to take one after another. If one perpendicular only were considered, and the differential coefficient of that perpendicular de- duced, we might certainly appear to avoid the idea of time ; but if all the states of a function are to be considered, corresponding to the different states of its variable, we have no alternative, with our bounded faculties, but to consider them in succession; and succession, disguise it as we may, is the identical idea of time introduced in Newton’s Method of Fluxions. The differential coefficient corresponding to a particular value of the variable, is, if we may use the phrase, the index of the change which the function would receive if the value of the variable were increased. Every value of the variable, gives not only a different value to the function, but a different quantity of increase or decrease in passing to what we may call contiguous values, obtained by a givenincrease of the variable. If, for example, we take the common logarithm of a, and let x be 100, we have C.log 100=2. If x be increased by 2, this gives C. log 102=2 0086002, the ratio of the increment of the function to that of the variable being that of (0086002 to 2, or -0043001. In passing from 1000 to 1003, we have the logarithms 3 and 3:0013009, the above-mentioned ratio being ‘0004336, little more thah a tenth of the former. We do not take the increments themselves, but the proportion they bear to the changes in the variable which gave rise to them; so in estimating the rate of motion of two points, we either consider lengths described in the same time, or if that cannot be done, we judge, not by the lengths described in different times, but by the proportion of those lengths to the times, or the THE DIFFERENTIAL AND INTEGRAL CALCULUS. 55 proportions of the units which express them. The above rough process, though from it some might draw the conclusion that the logarithm of @ is increasing faster when = 100 than when w = 1000, is defective ; for, in passing from 100 to 102, the change of the logarithm is not a sufficient index of the change which is taking place when ¢ is 100; since, for an thing we can be supposed to know to the contrary, the logarithm might be decreasing when « = 100, and might afterwards begin to increase between « = 100 and ¢ = 102, so as, on the whole, to cause the increase above-mentioned. The same objection would remain good, however small the increment might be, which we suppose x to have; if, for example, we Suppose @ to change from a = 100 to x= 100°00001, which increases the logarithm from 2 to 2°00000004343, we cannot yet say but that the logarithm may be decreasing when ¢= 100, and may begin to increase between « = 100 and « = 100°00001. In the same way, if a point is moving, so that at the end of 1 second it is at 3 feet from a fixed point, and at the end of 2 seconds it is at 5 feet from the fixed point, we cannot say which way it is moving at the end of one second. On the whole, it increases its distance from the fixed point in the second second ; but it is possible that at the end of the first second it may be moving back towards the fixed point, and may turn the contrary way during the second second. And the same argument holds, if we attempt to ascertain the way in which the point is moving by supposing any finite portion to elapse after the first second. But if on adding any interval, however small, to the first second, the moving point does, during that interval, increase its distance from the fixed point, we can then certainly say that at the end of the first second the point is moving from the fixed point. On the same principle, we cannot say whether the logarithin of z is increasing or decreasing when « increases and becomes 100, unless we can be sure that any Increment, however small, added to a, will increase the logarithm. Neither does the ratio of the increment of the function to the increment of its variable fur- nish any distinct idea of the change which is taking place when the vari- able has attained or is passing through a given value. For example, when w passes from 100 to 102, the difference between log 102 and log 100 is the united effect of all the changes which have taken place between a= 100 and ¢ = 1003,; x= 100), and « = 100,2;, and so on. Again, the change which takes place between « = 100 anda = 100-4, may be further compounded of those which take place between 2 = 100 and «= 100745; x= 100;4, and «= 100+2,, and so on. The ob- jection becomes of less force as the increment diminishes, but always exists unless we take the limit of the ratio of the increments, instead of that ratio. How well this answers to our previously formed ideas on such subjects as direction, velocity, and force, has already appeared. We now proceed to the Integral Calculus, which is the inverse of the Differential Calculus, as will afterwards appear. We have already shown, that when two functions increase or decrease without limit, their ratio may either increase or decrease without limit, or may tend to some finite limit. Which of these will be the case depends upon the manner in which the functions are related to their variable and to one another. This same proposition may be put in another form, as follows :—If there be two functions, the first of which decreases without limit, on the same supposition which makes the second increase without limit, the product of ithe two may either remain finite, and never exceed a certain finite limit ; or it may increase without limit, or diminish without 56 ELEMENTARY ILLUSTRATIONS OF limit. For example, take cos 9 and tan 6. As the angle 0 approaches a right angle, cos @ diminishes without limit; it is nothing when @ is a right angle; and any fraction being named, 6 can be taken so near to a right angle that cos @ shall be smaller. Again, as 0 approaches to a right angle, tan 0 increases without limit ; it is called infinite when 6 is a right angle, by which we mean that, let any number be named, however great, @ can be taken so near a right angle that tan @ shall be greater. Never- theless the product cos 0 x tan 0, of which the first factor diminishes without limit, while the second increases without limit, is always finite, aud tends towards the limit 1; for cos @ x tan @ is always sin 9, which last approaches to 1 as @ approaches to a right angle, and is 1 when @ is aright angle. Generally, if A diminishes without limit at the same time as B increases without limit, the product AB may, and often will, tend towards a finite limit. This product AB is the representative of A di- 1 1 ceaentiee vided by 77 or the ratio of A to BR: If B increases without limit, 7 decreases without limit; and as A also decreases without limit, the ratio of A to Bp may have a finite limit. But it may also diminish without limit; asin the instance of cos? 6 x tan 0, when @ approaches to a right angle. Here cos? @ diminishes without limit, and tan @ increases without limit; but cos? 6 x tan @ being cos 9 X sin 6, or a diminishing magni- tude multiplied by one which remains finite, diminishes without limit, Or it may increase without limit, as in the case of cos 9 X tan? 6, which is also sin @ X tan 9; which last has one factor finite, and the other in- creasing without limit. We shall soon see an instance of this. If we take any numbers, such as | and 2, it is evident that between the two we may interpose any nuraber of fractions, however great, either in arithmetical progression, or according to any other law. Suppose, for ex~- ample, we wish to interpose 9 fractions in arithmetical progression between 1 and 2. These are 154, 1%, &c., up to 14%; and, generally, if m fractions in arithmetical progression be interposed between @ and a + h, the complete series is h oh mh a, Gtr ay ato &e....uptoa@+ ay, @+ crit eh he sum of these can evidently be made as great as we please, since no one is less than the given quantity @, and the-number is as great as we please. Again, if we take dx, any function of v, and let the values just written be successively substituted for x, we shall have the series h 2h pa, (4 + an) pa a at) &e., up to d(a -+ A) (2); the sum of which may, in many cases, also be made as great as we please by sufficiently increasing the number of fractions interposed, that is, by sufficiently increasing m. But though the two sums increase without limit when m increases without limit, it does not therefore follow that their ratio increases without limit; indeed we can show that this cannot be the case when all the separate terms of (2) remain finite. For let A be greater than any term in (2), whence, as there are (m -} 2) terms, (m+ 2) A is greater than their sum. Again, every term of (1), except the first, being greater than @, and the terms being m + 2 in pomp (m + 2)a : 2)A is less than the sum of the terms in (1). Consequently oe is greater THE DIFFERENTIAL AND INTEGRAL CALCULUS. 57 sum of termsin(2) : Hae sum of terms in (1)? since its numerator is greater than the Jast numerator, and its denominator less than the last denominator, But (m+ 2) A A : cate o : P ; (m+2) a a ae which is independent of m, and is a finite quantity, Hence the ratio of the sums of the terms is always finite, be the number of terms, at Jeast unless the terms in (2) in limit. than the ratio whatever may crease without As the number of interposed values increases, the int between them diminishes ; if, therefore, sum of the values, or form mat {80+ 0(@ +559) + 6 (e+ 57h)... oetmp) we have a product, one term of which diminishes, and the other increases, when mm is increased. The product may therefore pass a certain limit, when m is increased without li that this 7s the case. As an example, be a, and let the intermediate values of « erval or difference we multiply this difference by the remain finite, or never mit, and we shall show let the given function of x be interposed between a = q h : : 2 to 7 te. Tet v0= ey t whence the above-mentioned product is n v fat (atv) -p (@-+ 2v)? ...s, + (at@m+l)v >} = (m+2) vae+ 2av{ 142434, ; +(m+1) bro 12-+2?4-324.0 0 4 (m+1)°} : of which, ] + 2+.,,. +(m + 1)=4 (m+ 1) (m+ 2) and (page 35), P+ 22+....4+(m+1)2 approaches without limit with $(m-+1), when m igs increased without limit may be put under the form 5 (m+1)3 (l+¢a limit when m is increased’ without limit. to a ratio of equality - Hence this last sum ), Where « diminishes without Making these substitutions, and putting for v its value the above expression becomes m+) Mt2, . mM+2., h3 m+] Pe as OBL Sabe G bop aH has the limit 1 when m Increases without limit, and l+t<¢ m+ 1 in which has also the limit 1, since, in that case, « diminishes without limit, Therefore the limit of the last expression is e € 3 Ta ha? -+- ha + a or (ere z This result may be stated as follows :-—If the variable z, setting out from a value a, becomes successively a+ dr, a + 2dx, &e., until the total in- crement is /, the smaller dz is taken, the more nearly will the sum of all the values of a*dx, or adr + (a + dr) dx + (a +. 2dx)'dx + &e., ol Rit ae be equal to ee : » and to this the aforesaid sum may be brought within any given degree of nearness, by taking ‘This result is called the integral of 2%dx, betwee and is written /2*dx, when it is not necessary t dx sufficiently small, n the limits @ and gq + h, O specify the limits, and = ne a ——— - pocementie ae rt ee ELEMENTARY ILLUSTRATIONS OF 58 ‘ j xemath. } ft eda, Or. vi rdrim, or of x2da +” in the contrary case. We we a now proceed to show the connexion of this process with the principles of the Differential Calculus, Let x have the successive values a, a+ dz,a+ Qdz, &e.,...+ Up a + mdz, or a+ h, h being a given quantity, and dz the m™ part of h, so that as mm is increased without limit, da is diminished without limit. Develope the successive values of dx, or Pa, P(a + da)... . (page 7, ba = pa d(a+dz) = oat dpladx + pa ee + da he + &e: Dyk 2 Dd. 3 h(a + 2dx) = pa + dia 2dx + ga + fl = + &e ‘2 : x 3 d(a + 3dx) = oa + d'a 3dx + gig pa eer + &e. e 2 . a ° * wa aan ee 8 d(a + mdz) = oa + dia mda + git SOE + pa ee + &c. rv and add the results, we have a series from the different columns, If we multiply each development by d made up of the following terms, arising’ Da ik mda GO RU F's 98 Fe - .+m ) (dx)? 3 ia x i+ 24884...) SS (dx)* and as in the last example, we may represent (page 35), 1 2B ie ar by $m (1+ 2) P+ at g2+..... +m ey. 4m 1+ 8) To 2% as BF .. . =t tm(l1+y) &e. where a, B, y, &c., diminish without limit, when m is increased without limit. If we substitute these values, have, for the sum of the terms, he h? h* / Aaa mn" pah + Pa e GQ+a+ 9a 53 (+6) + oe 5 94 (1+) + &e. ; which, when m is increased without limit, in consequence of which a, P= &¢e,, diminish without limit, continually approaches to 4 h2 h3 h / tt : Wy . dah-+ dla 5 + pa 574 +o asa] + &e. which is the limit arising from supposing @ to increase from @ through ~ a + dx, a+ 2dz, &c., up to ath, multiplying every value of Px sO the objections which may be raised — * This notation fudagr appears to me to avoid ld require that f°a7da* should stand against rh, Z ‘ye ae as contrary to analogy, which wou for the second integral of #da. frdag dys". There is as yet no general agreement on this point of notation. | : ;| | id id © { It will be found convenient in such integrals ag io, he, and also put m instead of dx, we | THE DIFFERENTIAL AND INTEGRAL CALCULUS, 59 obtained by dz, summing the results, and decreasing dx without limit. This is the integral of dx dx from x = a tox=a-+h. It is evident that this series bears a great resemblance to the development in page 1], deprived of its first term. Let us suppose that Wa is the function of which ¢a is the differential coefficient, that is, that u'a = ha. These two functions being the same, their differentia] coeflicients will be the same, that is, w/a = Pa. Similarly w/"a@ = da, and so on. Substituting these, the above series becomes 4 h2 hs f tN | ") lv oi ahaa ee tS a gt which is (page 11) the same as ¥(@+h)— wa. That is, the integral of du dx between the limits a and at+h, is Ww(at+h) -wa, where we is the func- tion, which, when differentiated, gives gv. Fora+h we may write b, so that wb — wa is the integral of dr dx from x=ato v= b. Orwe may make the second limit indefinite by writing @w instead of b, which gives %x — wa, which is said to be the integral of x da, beginning when # = a, the summation being supposed to be continued from ¢ = @ until w has the value which it may be convenient to give it. Hence results a new branch of the inquiry, the reverse of the Differential Calculus, the object of which is, not to find the differential coefficient, having given the function, but to find the function, having given the diffe- rential coefficient. This is called the Integral Calculus. From the defi- nition given, it is obvious that the value of an integral is not to be deter- mined, unless we know the values of y corresponding to the beginning and end of the summation, whose limit furnishes the integral. We might; instead of defining the integral in the manner above stated, have made the word mean merely the converse of the differential coefficient ; thus, if px be the differential coefficient of Ya, Wx might have been called the integral of pe dx. We should then have had to show that the integral, thus defined, is equivalent to the limit of the summation already explained. We have preferred bringing the former method before the student first, as it is most analogous to the manner in which he will deduce integrals in questions of geometry or mechanics. With the last-mentioned definition, it is also obvious that every function has an unlimited number of integrals, For whatever differential coefficient pe gives, C + ex will give the same, if C be a constant, that is, not varying when x varies. In this case, if x become «+h, C+ zx becomes C + Wx + W'e 2h + &e., from which the subtraction of the original form C +- wa gives yx. h + &c.; whence, by the process in page 12, w/z is the differential coefficient of C + we as well as of Ya. As many values, therefore, positive or negative, as can be ziven to C, so many different integrals can be found for w’x; and these mswer to the various limits between which the summation in our ori- sinal definition may be made. To make this problem definite, uot onl /’x, the function to be intevrated, must be given, but also that value of 3 rom which the summation is to begin. If this be a, the integral of wa is, is before determined, wa —wa, and C = — Ya. We may afterwards end it any value of x which we please. Ifv=a, Wvar—wWwa = 0, as is evident Iso from the formation of the integral. We may thus, having given an ategral in terms of 2, find the value at which it began, by equating the ntegral to zero, and finding the value of z. Thus, since a, when diffe- entiated, gives 2x, x is the integral of 2z, beginning at a= 0; and * — 4 is the integral beginning at «= 2, : + &e, 60 ELEMENTARY ILLUSTRATIONS OF an integral would be the sum of an infinite In the language of Leibnitz, number of infinitely small quantities, which are the differentials or infinitely o, according to him, a small increments of a function. Thus, a circle being rectilinear polygon of an infinite number of infinitely small sides, the sum As before (pages 7, of these would be the circumference of the ficure. 20, 24,) we proceed to interpret this inaccuracy of language. If, in a circle, we successively describe regular polygons of 3, A, 5, 6, &c., sides, we may, by this means, at last attain to a polygon whose side shall differ from the are of which it is the chord, by as small a fraction, either of the chord or are, as we please, (pages 4,5.) That is, A being the are, C the chord, and D their difference, there is no fraction so small that D cannot be made a smaller part of C. Hence, if m be the number of sides of the polygon, mC + mD or mA ‘s the real circumference ; and since mD Is the same part of mC which D is of C, mD may be made as small a part of mC as we please; so that mC, or the sum of all the sides of the poly- eon, can be made as nearly equal to the circumference as we please. As in other cases, the expressions of Leibnitz are the most convenient and the shortest, for all who can immediately put a rational construction upon them; this, and the fact that, good or bad, they have been, and are, used in the works of Lagrange, Laplace, Euler, and many others, which the student who really desires to know the present state of physical science, cannot dispense with, must be our excuse for continually bringing before him modes of speech, which, taken quite literally, are absurd. We will now suppose such a part of a curve, each ordinate of which is a given function of the corresponding abscissa, as lies between. two given ordinates; for example, MP P/M. Divide the line M M’ into a number of equal parts, which we may suppose as great as we please, and coustruct fig. 10. Let O be the origin of co-ordinates, and let O M, the value of 2, at which we begin, be a; and OM’, the value at which we end, be b. Though we have only divided M M’ into four equal parts in the figure, the reasoning to which we proceed would apply equally had we divided it into four million of parts. The sum of the parallelo- wrams Mr, mr’, mr’, and mR, is less than the area MP P’ M,, the value of which it is our object to investi- “ qd P gate, by the sum of the curvilinear trian- {ire 7 ble oles Prp, pr’p', p'r'p", and p"RP’. ‘The ae ‘ sun of these triangles is less than the sum of the parallelograms Qr, qr’, 0" and q’”R; but these parallelograms are Tah 0 together equal to the parallelogram q’w, as appears by inspection of the figure, since the base of each of the abovementioned parallelograms is equal to m//M’, or ¢’P’, and the altitude P/w 1 , is equal to the sum of the altitudes of 4 M m m m’ NM’ the same parallelograms. Hence the sum of the parallelograms Mr, mr’, m’r”, and mR, differs from the cur- vilinear area MP P/M’ by less than the parallelogram gw. But this last parallelogram may be made as small as we please by sufficiently increase ing the number of parts into which MM/ is divided ; for since one side of it, P’w, is always less than P'M’, and the other side P’q", or m''M/, is as small 1 part as we please of MM’, the number of square units in 9/2, 18 the product of the number of linear units in P’w and P’q’, the first of | THE DIFFERENTIAL AND INTEGRAL CALCULUS. 61 which numbers being finite, and the second as small as we please, the product is as small as we please. Hence the curvilinear area MP P’M’ is the limit towards which we continually approach, but which we never reach, by dividing M M/ into a greater and greater number of equal parts, and adding the parallelograms Mr, mr’, &c., so obtained. If each of the equal parts into which MM’ is divided be called dv, we have OM = a, Om =a-+ dx, Om = a+ edr, &e. And MP, mp, m’n’, &e., are the values of the function which expresses the ordinates, corresponding: to a, a-+ dz, a+ 2dx, &e., and may therefore be represented by qa, ~ (a + dr), O(a + 2dx), &e. These are the altitudes of a set of paral- lelograms, the base of each of which is dx; hence the sum of their areas is pa dx+ o(a + dz) dx + f(a + 2dr) dr + &e. the limit of this, to which we approach by diminishing dr, is the area required. This limit is what we have defined to be the integral of dx dx froma=ator=b:; or if x be the function, which, when differentiated, gives dz, it is wb — va. Hence, y being the ordinate, the area included between the axis of x, any two values of y, and the portion of the curve they cut off, is [ydz, beginning at the one ordinate and ending at the other. Suppose, for example, that the curve is a part of a parabola of which O is the vertex, and whose equation * is therefore y’ = px where p is the double ordinate which passes through the focus. Here ae) Lage Y= p? «7, and we must find the integral of p#2%dx, or the function whose differential coefficient is p?a®, p being a constant. If we take the function cx”, ¢ being independent of x, and substitute vw + h for a, we have for the development cz" + cna" h + &c. Hence the differential coefficient of cx” is enz"-'; and as ¢ and n may be any numbers or frac- 1 trons we please, we may take them such that en shall = p? and n— I=, in which case n = 2 and c = 2p?. Therefore the differential coefficient of 2y2 x9 is pia, and conversely, the integral of prada is LHzxe, The area MP P/M! of the parabola is therefore 22 bz — Lprq If we begin the integral at the vertex O, in which case @ = 0, we have for the area OM’P’, 3p, where b= OM’. This js Sytem xX) By) which. (Ounce pb? = M’P’ is 2 P/M x OM’, or two-thirds of the rectangle + contained by OM’ and M’P’. We may mention, in illustration of the preceding problem, a method of establishing the principles of the integral calculus, which generally goes by the name of the Method of Indivisibles. A line is considered ag the sum of an infinite number of points, a surface of an infinite number of lines, and a solid of an infinite number of surfaces. One line twice as Jong as another would be said to contain twice as many points, though the number of points in each is unlimited. To this there are two objec- tions ;—first, that the word infinite, in this absolute sense, really has no meaning, since it will be admitted that the mind has no conception of a number greater than any number. 'The word infinite { can only be justi- * If the student has not any acquaintance with the Conic Sections, he must never- theless be aware that there is some curve whose abscissa and ordinate are connected by the equation y2 = pz. This, to him, must be the definition of parabola ; by which word he must understand, a curve whose equation is y2—= pa, t This proposition is famous as having been discovered by Archimedes at a time when such a step was one of no small magnitude, t See Study of Mathematics, page 41, 62 ELEMENTARY ILLUSTRATIONS OF fiably used as an abbreviation of a distinct and intelligible proposition ; for As; Re Se — is equal to @ when @ is infinite, we example, when we say that @ +- 5 ; 1 only mean that as # 1s increased, a-++ —- becomes nearer to a, and may x be made as near to it as we please, if 2 may be as great as we please. The second objection is, that the notion of a line being the sum of a number of points is not true, nor does it approach nearer the truth as we snerease the number of points. If twenty points be taken on a straight line, the sum of the twenty-one lines which lie between point and point is equal to the whole line ; which cannot be if the points by themselves con- stitute any part of the line, however small. Nor will the sum of the points be a part of the line, if twenty thousand be taken instead of twenty. There is then, in this method, neither the rigor of geometry, nor that ap- proach to truth, which, in the method of Leibnitz, may be carried to any extent we please, short of absolute correctness. We would there- fore recommend to the student not to regard any proposition derived from this method as true on that account; for falsehoods, as well as truths, may be deduced from it. Indeed the primary notion, that the number of points in a line is proportional to its length, is manifestly incorrect. Sup- pose (fig. 6, page 23,) that the point Q moves from A to P. It is evident that in whatever number of points O Q cuts AP, it cuts MP in the same number. But PM and PA are not equal. A defender of the system of indivisibles, if there were such a person, would say something equivalent to supposing that the points on the two lines are of different sizes, which would, in fact, be an abandonment of the method, and an adoption of the idea of Leibnitz, using the word point to stand for the infinitely small line. This notion of indivisibles, or at least a way of speaking which looks like it, prevails in many works on Mechanics. Though a point is not treated as a length, or as any part of space whatever, it is considered as having weight ; and two points are spoken of as having different weights. The same is said of a line and a surface, neither of which can correctly be supposed to possess weight. If a solid be of the same density throughout, that is, if the weight of a cubic inch of it be the same from whatever part it is cut, it is plain that the weight may be found by finding the number of cubic inches in the whole, and multiplying this number by the weight of one cubic inch. But if the weight of every two cubic inches is different, we can only find the weight of the whole by the integral cal- culus, Let AB be a line possessing weight, or a very thin parallelopiped of matter, which is such, that if we were to divide it into any number of equal parts, as in the figure, the weight of the several parts would be dif- ferent. We suppose the weight to vary continuously, that is, if two con- Fig. 11. tizuous parts of equal length be taken, as pg and qr, the ratio of the 5) : A, pag ee B weights of these two parts may, by taking them sufficiently small, be as near to equality as we please. The density of a body is a mathematical term, which may be explained as follows -—A cubic inch of gold weighs more than a cubic inch of water; hence gold is denser than water. If the first weighs 19 times as much as the second, gold is said to be 19 times more dense than water, or the density of gold is 19 times that of water. Hence we might define the density by the weight of a cubic inch of the THE DIFFERENTIAL AND INTEGRAL CALCULUS. 63 substance, but it is usual to take, not this weight, but the proportion which it bears to the same weight of water, Thus, when we say the density, or specific gravity (these terms are used indifferently), of cast iron is 7°207, we mean that if any vessel of pure water were emptied and filled with cast iron, the iron would weigh 7207 times as much as the water. If the density of a body were uniform throughout, we might easily determine it by di- viding the weight of any bulk of the body, by the weight of an equal bulk of water. In the same manner (pages 25, 26) we could, from our defi- nition of velocity, determine any uniform velocity by dividing the length described by the time. But if the density vary continuously, no such mea- sure can be adopted. For if by the side of AB (which we will suppose to be of iron) we placed a similar body of water similarly divided, and if we divided the weight of the part pq of iron by the weight of the Same part of water, we should get different densities, according as the part pq is longer or shorter. The water is supposed to be homogeneous, that is, any part of it pr, being twice the length of pq, is twice the weight of pg, and so on, The iron, on the contrary, being supposed to vary in density, the doubling the length gives either more or less than twice the weight. But if we suppose q to move towards p, both on the iron and the water, the limit of the ratio pq of iron to pg of water, may be chosen as a measure of the density of p, on the same principle as in page 26, the limit of the ratio of the length described to the time of describing it, was called the velocity. If we call & this limit, and if the weight varies continuously, though no part pq, however small, of iron, would be exactly & times the same part of water in weight, we may never- theless take pq so small that these weights shall be as nearly as we please in the ratio of k to 1. Let us now suppose that this density, expressed by the limiting ratio aforesaid, is always a° at any point whose distance from A is x feet ; that is, the density at g, 2 feet distance from A, is 4, and soon. Let the whole distance AB =a. If we divide @ into m equal parts, each of which is dz, so that ndzx = a, and if we call 6 the area of the section of the parallelopiped, (b being a fraction of a Square foot,) the solid content of each of the parts will be édz in cubic feet; and if w be the weight of a cubic foot of water, the weight of the same bulk of water will be wbdz. If the solid AB were homogeneous in the immediate neighbourhood of the point p, the density being then x*, would give a X bwdx for the weight of the same part of the substance. This is not true, but can be brought as near to the truth as we please, by taking dx sufficiently small, or dividing AB into a sufficient number of parts. Hence the real weight of pg may be represented by bwa°dx + a, where « may be made as small a part as we please of the term which precedes it. In the sum of any number of these terms, the sum arising from the term « diminishes without limit as compared with the sum arising from the term bwwx'dr; for if a be less than the thousandth part of p, a’ less than the thousandth part of p, &e., then a + a’ + &e. will be less than the thousandth part of p + p! + &c.: which is also true of any num- ber of quantities, and of any fraction, however small, which each term of One set is of its corresponding term in the other. Hence the taking of the integral of bw a%dx dispenses with the necessity of considering the term a; for in taking the integral, we find a limit which supposes dx to have decreased without limit, and the integral which would arise from «a has therefore diminished without limit. The integral of bw 22dz is tbwer*, which taken from wv =0 to w= a is $4 bwa*, This is therefore the weight t 64 ELEMENTARY ILLUSTRATIONS, &. in pounds of the bar whose length is a feet, and whose section is 6 square feet, when the density at any point distant by a feet from the beginning is a2; w being the weight in pounds of a cubic foot of water. We would recommend it to the student, in pursuing any problem of the Integral Calculus, never for one moment to lose sight of the manner in which he would do it, if a rough solution for practical purposes only were required. Thus, if he has the area of a curve to find, instead of merely saying that y, the ordinate, being a certain function of the abscissa a, ydx within the given limits would be the area required; and then pro- ceeding to the mechanical solution of the question : let him remark that if an approximate solution only were required, it might be obtained by di- viding the curvilinear area into a number of four-sided figures, as in fig. 10, one side of which only is curvilinear, and embracing so small an are that it may, without visible error, be considered as rectilinear. The mathematical method begins with the same principle, investigating upon this supposition, not the sum of these rectilinear areas, but the limit towards which this sum approaches, as the subdivision is rendered more minute. ‘This limit is shown to be that of which we are in search, since it is proved that the error diminishes without limit, as the subdivision is indefinitely continued. We now leave our reader to any elementary work which may fall in his way, having done our best to place before him those considerations, something equivalent to which he must turn over in his mind before he can understand the subject. The method so generally followed in our elementary works, of leading the student at once into the mechanical processes of the science, postponing entirely all other considerations, is to many students a source of obscurity at least, if not an absolute impediment to their progress; since they cannot ima= eine what is the object of that which they are required to do. That they shall understand every thing contained in these treatises, on the first or second reading, we cannot promise; but that the want of illustration and the preponderance of technical reasoning are the great causes of the difficulties which students experience, is the opinion of many who have had experience in teaching this subject.