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Wy Fi ibe uli ede Ut a i ete Hy int iid iil! Pe 1) melita deel t eee ii; fects Pees TS nr uit lie erabe iain ie iit, ene if Api ere ses sae ; fidedi ay, me itil tf Hy He heleh x Fs Pod Pol Bh dels eepeepes sag : ‘ i i 4 i 3 iy 372 at Aya cide ; : A Mtr nn AT milan eee +y4 Aj y fy if3¢ par Fe jaya } ip HF J is iF Ay fi up Fd 3 fit Ty AiG iil : iPod E pitts = re her us ee i i); A niversity, CHAMPAIGN, ILLINOIS. MATHEMATICS LIRPAPY NOTICE: Return or renew all Library Materials! The Minimum Fee for each Lost Book is $50.00. The person charging this material is responsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below.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 discipli- nary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN L161—O-1096 A TREATISE DIFFERENTIAL CALCULUS. BY WILLIAM WALTON, M.A. TRINITY COLLEGE, CAMBRIDGE, CAMBRIDGE: DEIGHTONS. LONDON: WHITTAKER & CO.; SIMPKIN, MARSHALL, & CO. MDCCCXLYI CAMBRIDGE : PRINTED BY METCALFE AND PALMER, TRINITY STREET. ae WAL NEMA ve aids PREFACE. THE method of expansions, developed by Lagrange in his Théorie des Fonctions Analytiques, has been for many years almost exclusively adopted in this University for the demon- stration of the formule of the differential calculus. The great name of the originator of this system gave a certain permanency to the method which probably it would not have possessed had it emanated from one less illustrious. Not to insist upon the doubts which have been thrown by several recent writers on the validity of conclusions deduced from the comparison of infinite series, it is certain that the absence of any notion of limits in the algebraical theory of derived functions, gives rise to an entire want of homogeneity between its fundamental conceptions and those which present themselves in its most interesting applications. Within the last few years ue to re-establish the system of limits, has been Several elementary writers in France, among whom may. . be ‘mentioned Moigno, Duhamel, and Cournot, and a Professors De Morgan and O’Brien in England. Fro my own strong conviction of the marked advantage which the method of limits possesses over that of derived functions, both abstractedly and in its applications, and, trusting to the ul PREFACE. valuable opinion of many writers of the present day on the comparative merits of the two systems, I have been induced to enter upon this treatise. My object has been to present to the English student a work from which he may acquire a thorough and systematic knowledge of the abstract theory of limits, and of its applications to certain branches of coordinate geometry. How far I may have succeeded in this attempt, and of the liability to failure in a work of such a nature I am fully sensible, will be determined by the judgment of the reader. In the composition of this work, which was commenced in the April of this year, I have derived great assistance from Moigno’s Legons de Calcul Différentiel et de Calcul Intégral, Duhamel’s Cowrs d’ Analyse, and Cournot’s Théorie des Fonc- tions: from the last of which treatises I have made considerable extracts in Chapter VII. on the Development of Functions. Cambridge, November, 1845. Article 1-9 10 CONTENTS. FIRST Paw, CHAPTER I. Functions. Functions CHAPTER If. Principles of Differentiation. Definition of a differential coefficient Differentiation of a constant cps Differentiation of the sum of a function and a constant Differentiation of the product of a function ‘and a constant Differentiation of a sum of functions Differentiation of the product of two functions Differentiation of the ratio of two functions Differentiation of the product of any number of functions Relation between inverse differential coefficients Differentiation of a function of a function Differentiation of a function of two functions Differentiation of a function of any number of functions of a single variable Differentiation of an implicit Aiken of a sinale ane : General theory of the differentiation of implicit functions of a single variable Total differentiation of a function of caren of ic epeatent variables Page V1 Article 25 26 27 28 29 30 3] 32 33 34 35 36 37 38 39 40 4] 42 43 44 45 45’ CONTENTS. Partial differentiation of an explicit function of three variables, one of which is a function of the other two Partial differentiation of an explicit function of n +7 warilee r independent and x dependent Partial differentiation of an implicit function of two Goniepennent variables Partial differentiation of eaplicne eather: of any namne? of independent variables Simple functions Differential coefficient of: a Differential coefficient of log, x Differential coefficient of a* Differential coefficient of sin z Differential coefficient of cos x Differential coefficient of tan x Differential coefficient of cot x Differential coefficient of sec z Differential coefficient of cosec x Differential coefficient of sin'z Differential coefficient of cos”'z Differential coefficient of tan”'z Differential coefficient of cot-'z Differential coefficient of sec-'z Differential coefficient of cosec™ Differentiation of simple aoe of y with fegard to «x Illustrative examples CHAPTER III. Successive Differentiation. Theory of the independent variable Change of the independent variable Order of partial differentiations indifferent Successive differentiation of an explicit function of t two dee. tions of a single variable : Successive differentiation of an implicit fencsion of a siugle variable Successive total differentials : Successive differentiation of an explicit euntdion of three vari- ables, one of which is a function of the other two Change of variables : : Transformation of one system of aitenan ens variables into another Page 44 46 50 52 05 57 58 60 6] Article 55 56 57 CONTENTS. CHAPTER IV. Elimination of Constants and Functions. Elimination of constants . : : 1 Partial elimination of constants Elimination of irrational, logarithmic, PeeN ea ay ereurtt functions of known functions 58, 59 Elimination of an arbitrary function of a ana aay 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77, 78 79 80 81 Elimination of any number of arbitrary functions of known functions ‘ Elimination of arbitrary amanane of aHECewe functions Elimination of arbitrary functions when the number of indepen- dent variables exceeds two ‘ CHAPTER V. Evaluation of Indeterminate Functions. Indeterminateness of explicit functions of a single variable Evaluation of functions of the form 2 = Failure of the method of differentials for the eetaenae of in- determinate functions : : Evaluation of indeterminate functions of aver iienenient variables vil Page 67 68 69 72 74 77 81 84 89 92 93 Evaluation of ardeterainars raptiatt Paceticns of a Rae sarahle 97 CHAPTER VI. Maxima and Minima. Definition of a maximum and minimum Lemma : : Rule for finding maxima and minima Abbreviation of operation : Alternation of maxima and minima , Modified method of finding maxima and minima Abbreviation of operation ‘ Maxima and, minima of implicit Ammetone of a amele marisa Maxima and minina of a function of a function Maxima and minima of a function of two independent mahted Maxima and minima of functions of any number of independent variables Maxima and minima cofrestentding to indeterminate differen tial coefficients , Application of indeterminate Lantatete to eee of maxima and minima Vill Article 82, 83 84 85 86 87, 88 89 90 91 92 93 94 95 96 97 98 ay 100 101 102 103 104 105 106 107 108 CONTENTS. CHAPTER VII. Development of Functions. Taylor’s theorem Another demonstration of Taylor’ s theotert Cauchy’s expression for R, Examples of Taylor’s theorem Failure of Taylor’s theorem . Lagrange’s theory of Functions Stirling’s theorem . ; Examples of the application Bf Stirling’ s theorem Extension of Taylor’s theorem to functions of two variables Failure of the development of f(v+h, y+k) by Taylor’s theorem Limits and remainders of the development of f(a + A, y + k) Example of the application of Taylor’s theorem for two variables Stirling’s theorem applied to functions of two variables Lagrange’s formula for the development of implicit functions Laplace’s formula for the development of implicit functions SECOND PART. CHAPTER T. Tangency. Definition of a tangent and of a normal Inclinations of the tangent and the normal at any Sent of a curve to the coordinate axes ‘ Equations to the tangent and the normal at any point of a curve ; : Distance of the origin of coordinates from the tangent Intercepts of the tangent . Subtangent Length of the tangent Normal and subnormal Form of the equation to the apes to curves ‘of eaten the equations involve only rational functions of 2 and y Oblique axes 168 CONTENTS. 1x CHAPTER II. Asymptotes. Article Page 109 Definition of an asymptote. Method of finding asymptotes 170 110,111 Asymptotes of algebraic curves : ! : 171 112 Examples of asymptotes : 173 1138 Algebraical method of finding Bitreititeat and weenilinent asymptotes : : : : 175 CHAPTER ITf. Multiple Points, Conjugate Points, Cusps, &c. 114 Definition of multiple points, conjugate points, and cusps 178 115 Analytical property of multiple points in algebraical curves 179 116 Analytical property of cusps in algebraical curves ; 181 117. Analytical property of conjugate points in algebraical curves — 118 Determination of the multiplicity and of the directions of the tangents at a multiple point : : 182 119 Multiplicity of a multiple point at the origin. ’ 185 120 Point of osculation ; , ; 186 121 Remark on the theory of nlite points ‘ 187 122 Points d’arrét or points de rupture : : 189 123 Points saillants : } — 124 Branches pointillées . 5 2p : 19] CHAPTER IV. Concavity and Convexity of Curves and Points of Inflection. 125 Conditions for concavity and convexity : : 192 126 Condition for a point of inflection . . : . 194 127 Symmetrical investigation of points of inflection : 197 CHAPTER V. On the Index of Curvature, the Radius of Curvature, and the Centre of Curvature, of a Plane Curve. 128 Index of curvature , : “ : . 208 129 Radius and centre of Retire : ; ; —_— 130 Expression for p when z is the independent SRS aE! 131 Expressions for p when s is the independent variable. 206 132 Expression for p in terms of dz, dy, d’x, d*y 3 We 2g 133 Expression for p in terms of partial differential coefficients a= 134 Another method of finding the radius of curvature : 209 line 7 from bottom 1 7 from bottom from bottom 9 from bottom 3 18 18 ERRATA. Error p at m S(t, y)t+A F(a, y) negative positive 3 ae v=ta Correction. tf m a z S(&, y+ AF (@, Y) positive negative — 3 ale eel = 3) a v=ta DIFFERENTIAL CALCULUS. FIRST PART. GENERAL PRINCIPLES AND ANALYTICAL APPLICATIONS. CHAPTER I. FUNCTIONS. 1. Ir any two quantities are so connected that any variation in the magnitude of the one implies a corresponding variation in the magnitude of the other, either of these quantities is said to be a function of the other. Such a connection is expressed algebraically by means of an equation involving the symbols of the two quantities and any other symbols of invariable magni- tudes. Thus z and y are functions the one of the other in the equation a 2 omc se ec e rn weer aes Gl); ie A OE. 9 ae reed eo ceee ees eoceee (2), where a and 4 are supposed to represent invariable magnitudes. 1 . If y be expressed directly in terms of z, as in (2), it 1s said to be an explicit function of x; if it be merely connected B 2 FUNCTIONS. with z by an unsolved equation, as in (1), it is said to be an implicit function of z. 3. The connection between 2 and y may be expressed generally by an equation where F(z, y) denotes any expression whatever involving z, y, and constants. These constants are usually called parameters, a term borrowed from the theory of conic sections, where the word parameter is used to denote a certain fixed line. If we wish to represent in the most general manner that y is an explicit function of x, we may write F(z) denoting any algebraical expression which involves x and ° parameters. ‘The equation (4) is easily seen to be a particular form of the equation (8); for we may write it thus, y-f (x) =9, y — f(x) being merely a particular instance of the general form F(z, y). 4. Functions may be termed mathematical or empirical ; mathematical, if the functionality is established by definition ; empirical, if discovered by observation. As an instance of the latter functionality, let y denote the attraction of the Sun upon the Earth, and z the distance between these two masses; then it is known by observation that pe ay ee? uw being a constant quantity: y is in this case an empirical function of z. 5. A function f(x) is said to be continwows when, as x in- creases continuously, f(x) passes continuously from one possible value to another through all intervening values: the function is said to be discontinuous whenever this condition is violated. Take for instance 1 {= ee ae FUNCTIONS. S then, as « keeps increasing continuously from 0 to a value a—h, where h is a positive quantity less than any assignable magnitude, it is plain that y also keeps continuously changing ° 1 through every gradation of value from — - to —- ©: but when a x changes from a-—h to a+h, y leaps from —-@ to +a without passing through the intervening values. Thus we see that in this case y is generally a continuous function, but that it experiences a dissolution of continuity when z becomes equal to a. If we take ? 1 nite (Case then, although, when z =a, y assumes the value o, yet this value of x does not correspond to a discontinuous state of the function, since, as # passes from a-h to a+h, there is no gap in the range of values of y. 6. Suppose y= f(x) to be the equation to a curve; then, if the function f(z) is continuous for a certain range of values of z, every two points of the locus will be joined by a con- tinuous curve: on the other hand, if there is a dissolution of continuity at any point, and if the function be possible before and after z has passed its critical value, there will be a gap between two points of the curve corresponding to con- secutive values of z. Thus, in the instance of the curve 1 Ve ’ z-a the asymptote, of which the equation is z= a, is touched at opposite ends by the curve for two consecutive values of z, one greater and the other less than @ by an indefinitely small magnitude. 7. Functions of z, which are expressed by the ordinary signs of algebra and trigonometry, are usually continuous, if we disregard certain dissolutions of continuity corresponding to peculiar and detached values of z There are however excep- tions to this principle. For example, if am (- ay, B2 4 FUNCTIONS. a being a positive quantity, it is pla that y will be imaginary whenever z is of the form 2+ 1 2m A and mw being integers. Now between any two values of z, however little they may differ from each other, we may inter- calate an infinite number of fractions of the above form. ‘Thus we see that it is impossible to join by a continuous curve two points of the locus of the equation, corresponding to two systems of real values of x and y, however near they may be to each other. ‘These anomalous functions are inapplicable to questions of natural philosophy, and have attracted but little attention even in pure analysis. In this treatise we shall direct our attention entirely to continuous functions. 8. Functions are distinguished also by the names of alge- braical and transcendental. If y be connected with x by an equation involving only the ordinary operations of addition and subtraction, multiplication and division, evolution and involution of assigned degrees, y is said to be an algebraical function of z. If the equation of connection does not satisfy this condition, involving for instance exponential, logarithmic, or circular functions of z and y, then y is said to be a trans- cendental function of x. Thus, for examples, in the equations 2 2 1 eens (Za (Bye, atid ar Ae AN y is an algebraical function of x; and a transcendental one in the equations eae ck, log (zy)=x+y, xsiny + y sin x = sin (zy). 9. It frequently happens that, if Lay a be the equation to a curve, y will for a certain value of z experience a dissolution of continuity by becoming impossible, although the curve is itself continuous at the point. Thus if y = 4mz, FUNCTIONS. 5) or y =+ 2mizi, y will continuously vary from one value to another as z de- creases from any assigned positive value down to zero, but, the moment x becomes negative, y becomes impossible. The two branches corresponding to the double sign, each of which terminates abruptly at the origin, join together at this point and thus form a continuous curve. CHAPTER II. PRINCIPLES OF DIFFERENTIATION. SECTION I. GENERAL FUNCTIONS. Definition of a Differential Coefficient. 10. Let y be a certain function of z, and let y' be the value assumed by y when z becomes z’. Then, as z' keeps continuously approaching to the value of z, the fraction Via L-x will continuously tend towards a certain value from which it will ultimately differ by a quantity less than any assignable magnitude, or, in other words, to which it will be ultimately equal. The indefinitely small values of the differences z' — zx, y'—y, are usually denoted by the symbols éz, dy, and the ultimate value of the fraction Yost Tey CY or a -—2 ox is ordinarily represented by dy dx In this expression dz and dy are any quantities whatever, either finite or infinitesimal, which are in the ratio of the ultimate values of dz and dy. The fraction a is called the @ differential coefficient of y with regard to 2, the quantities dx and dy being called the differentials of x and y. The object of the Differential Calculus is to investigate the pro- PRINCIPLES OF DIFFERENTIATION. 7 perties of differentials and differential coefficients, and to de- velop the general principles of their application to the theory of coordinate geometry and other branches of pure mathematics, and to the estimation of the phenomena of nature. Ex. Let y =2°: then y'=2": whence y-y=e*-ea(e—z)(2"+ezr+2), 8 age ve? + ax 4+ x’. oz When 2’ approaches indefinitely near to z, the left-hand member of the equation becomes dy , and the right-hand dx member assumes its limiting value 3z*: thus dy —- = 82° dz ‘ or dy = 3x'dz, that is, the differential coefficient of x’ with respect to x is 32”, and its differential is 3z*dz. Differentiation of a Constant. 11. If y=c, where e¢ denotes any constant quantity, that 1s, any quantity which does not experience variation in conse- quence of a variation in the value of z, then d = 50 For we have Ua Com yomec: whence cy=y'-y=0, oY = 0, and therefore, proceeding to the limit, ies Ge tb 2 or the differential coefficient of a constant quantity is always Zero. ; 8 PRINCIPLES OF DIFFERENTIATION. Differentiation of the Sum of a Function and a Constant. 12. If w=y +c, where y represents any function of z, and e denotes a constant quantity, then Let 2’, y', uw’, be simultaneous values of x, y, w; then U= Y + C, Uther oC, whence, proceeding to the limit, du dy dx dx’ or the differential coefficient of the sum of a function and a constant is the same as that of the function alone. Differentiation of the Product of a Function and a Constant. 13. If w= cy, where c represents a constant quantity and y a function of z; then du __—s dy ste ies Cc gee NY dx dz Let 2’, y’, u', be simultaneous values of z, y, uw; then u= cy, n= cy, UY ny ou fy g—-«x «wv-x &x’ and therefore, in the limit, or the differential coefficient of the product of a function and a constant is equal to the product of the constant and the dif- ferential coefficient of the function. PRINCIPLES OF DIFFERENTIATION. 9 Differentiation of a Sum of Functions. 14, Tf = 9, + Y, + Yyt . 0. +Y,> WHET Y,5 Yo, Yor ese Y,> denote any functions whatever of z, then du _ dy, _ dy, | Wy, dy, —_ so di dele dat dren a tae ee dre In fact, taking any simultaneous values 2’, wv’, y'15 Y'y5 Y'q9 + Y's ere i ret ryote fy We Lave CO Ue ee a ore oat Ys UY RY GY ge oe FY as ata cai B aU aera mee sey Hate Yn es Bae ag See og So ee pe ou OY, OY, OYs oY, i Se «Or de Seenhetd nae Sad and therefore, proceeding to the limit, du midy, dy, dy, dy, dx dx w dx s dz 5 dz Hence the differential coefficient of the sum ane any number of functions is equal to the sum of the differential coefficients of the functions taken separately. Differentiation of the Product of two Functions. 15. If w=y,y,, where y, and y, are any functions of z, then ey, oh. dz “'de ~? dz We have 2’, wv, y',, y', denoting simultaneous values of Ly Uy Yrs Yo Aa, U=YY2, U=Y,Y,> w= YY, — YiYo = Is (Yn — Yo) + Yo Yr — Yd t (Y's — Yi) (Y'2- Yo) or 10 PRINCIPLES OF DIFFERENTIATION. whence, proceeding to the limit, that is, equating oz to zero, and therefore, y, and y, being supposed to be continuous func- tions of x, putting also dy, and éy, each equal to zero, we have du __—s dy, dy, dz adi ax TY, dz’ oF du = y, dy, + y, dy, Hence the differential coefficient of the product of two functions is equal to the sum of the products of each function multiplied by the differential coefficient of the other. Differentiation of the Ratio of two Functions. 16. If w=“, then Sd L(y, He - y, %) =: —- — — FETE 2 de Oe N Taking wz’, uw’, y,, y,, to denote simultaneous values of Z, tt, Y,, Y,. we have 1 Y, Y l ‘ w—-u=2>-4=-—_(yy, - YY) Y, Y2 YY at igh 1 ue i sera y,(Y, - ia DOR a) J? ah {y, (Yy - 9.) - 4, (Ys - ¥2)} fu _ ( OY, e whence and therefore, proceeding to the limit, that is, equating 2 - x or 6z to zero, we get, observing that y,’ becomes y,, du 1, an, UW, de y?\" de ™ dz)’ 1 of du = y,? (y, dy, - y,dy,)- Hence, to differentiate the Ratio of two functions, we have the following rule: Multiply the denominator by the differential coefficient of the numerator, and the numerator by the differen- tial coefficient of the denominator: subtract the latter product PRINCIPLES OF DIFFERENTIATION. 1] from the former: this difference divided by the square of the denominator is the differential coefficient of the Ratio Differentiation of the Product of any number of Functions. 17. Ifu =y,.y,-y,---Y,, the product of x functions of x, then 1 du, id, i dy il dy . 1 dy TR CER LTE LOR Ee y, de eda Since wu = u n m7 1: Y,2 we have, by Art. (15), uh n-1 ay, Ax n dx n-l dx. 3 1 du 1 du 1 d whence Sonal pata "Pe Mirae caked ten Yn. omer 1 1- du the similarly dU, Mb ake et aah OF a1 lrdu. ia du, 2 dy, | Ge de ty Bar ye dae adding these equations together, cancelling terms which are common to both sides of the resulting equation, and observing that w, = y,, we have 1 du, 1 dy, 1 dy, i dy, 1 dy, Saya at en ah, sah SER Fass woeet— Sf, MD YA My ee SY. dt y. de me Bon _ Ys aa LeU a vance “. Y, Y2 Y, Y, Relation between Inverse Differential Coefficients. 18. If y be a function of x, in which case z will also be a function of y, then dy dz dx’ dy 12 PRINCIPLES OF DIFFERENTIATION. Let z', y', be simultaneous values of z, y; then it is evident that Y-¥Y £-—2 oy ox ; ie pores mee art) . 3 wx y-y du” }y and that, consequently, proceeding to the limit, dy dx | dx* dy Differentiation of a Function of a Function. 19. If w be a function of y, and y a function of z, then du _ du dy dz dy * dx For we have, by common algebra, taking z’, y’, wv’, as simul- taneous values of z, y, uw, w-u_ w-u Vaud em“ yy 2 a and therefore, proceeding to the limit, Cor. If wu be a function of y,, y, of y,, y, of y,,....and y, of x, it is manifest that we may prove in the same way that du du dy, dy, dy, dy, dz ~ dy,” A TOy Wed ay ty ar Take for instance three y’s ; then wh wu } Yy ~ Y Yo — Yo Vs tea “- x Tred; Tuk Usa u- @ and, in the limit, du du dy, dy, dy, dz dy,” dy,” dy,’ da” Differentiation of a Function of two Functions. 20. Let u=f(y,, y,), where f(y,, y,) denotes any function whatever of y, and y,, each of the quantities y, and y, being a PRINCIPLES OF DIFFERENTIATION. 13 function of a third quantity z. Let y,, y,, #, become y,', y,, wv’, when zx becomes z’: then U=LY,2 Yn)» UW =LYs Yo) wu =f» Y.) ew AC Le Yo) =F» 42) —F.Ys2 Ya) + IY Yo) —F Yrs Yo)s Wat LO rY)-FO I) ah , KY 9)- LHe) YoYo we yi -%; ‘aa yea al ae " Now in the limit, when 2’ differs from z less than by any assignable magnitude, eee ng on ey a Us Yue OY, ’ p z-—-x dx mee dee fake da? LY» Ye) ~ FI Yo) _— Fr» Ya) _ dee 4-9 dy, dy,’ and, first replacing y,’ by y,, and then y,' by y,, Tits Yat — JS Yi > Yo) AG Yo) Aas, Yo) a TTY, Yo) = du - Yo — Y Je ado dy, dy, hence EBAY TF etl 9G ets: Sataeasess Gli dx dy, dx dy, dz In this equation it is very important to observe that the numerators of the two fractions although represented by the same symbol dw, are essentially different, the numerator of the former corresponding to the ultimate value of the increment I's Y.) —F (Ys Yo)s and the numerator of the latter to the ultimate value of the increment FY.3 4.) -F rs Ya): The numerator of the fraction Du dx” 14 PRINCIPLES OF DIFFERENTIATION. which corresponds to the ultimate value of the increment FYss Yo) ~-F Yrs Yo)s and which we have represented by a distinct symbol Dvw, is evidently different from either of the numerators of the fractions du du ae ree dy, dy, . . In order to obviate all possibility of confusion, we might use du du the symbols d, wu, d,,w, to denote the numerators of — , —, gman ae dy, dy « ° id the suffixes serving to point out the origin of the two differentials. Such a notation, however, although remarkably clear, would frequently be very embarrassing, especially in long operations. It will be sufficient for distinctness if we remember to regard du du — and — dy, dy, rably attached to the numerators, the symbols dy, and dy,, which express the denominators, thus serving to indicate the true nature of the du in the numerators. If, however, as will be sometimes convenient, we do put d,,u, as fractions the denominators of which are insepa- ws : Le , we shall then dy, dy, be at liberty to treat these differential coefficients as ordinary algebraical fractions : thus Du _dyu dy, H dyu dy, dy wt, instead of du, in the expressions —e=a os. 2 Peete te «eka 2), dz dy, dx dy, dz @) may be written, multiplying both sides of the equation by dz, Dig hy Mit Uy Ui i ca egespa vids oo (3). The quantity d,,u denotes the differential of wu taken with regard to y,, as if y, were constant, d,,w the differential of u taken with regard to y, as if y, were constant, and Du the differential of w due to the simultaneous variations of y, and y, dependent upon the variation of zx. The quantities d,,u, d,,v, are called the partial differentials of u with regard to y,, y,, respectively, and Duw its total differential. 'The quantities dy,U dy, TRS mets PRINCIPLES OF DIFFERENTIATION. 15 in the equation (2), or their equivalents dus du dy,” dy,” in the equation (1), are called the partial differential coefficients of u with regard to y,, y,, respectively. Finally, tte is called dx the total differential coefficient of u. The equation (8) shews that the total differential of u is equal to the sum of its partial differentials. pe ure akon of a Function of any number of Functions of a single Variable. 21. Let w=f(y,, y,5¥,), a function of three variables y,,y,, ¥,, each of which is a function of z. Then if w’, y,', y,, y,, be simultaneous values of w, ¥,, 7, Y,, We have u =S(Y,, Yoo Ys)» wu =f (Y;» Yo Ys)» we -u =f; Yo Ys) SV ACEE Yo. Ys)s wi — u FY» Yo» ys) —=F(Yrs Yor Ys) Yes A a — 2 — @& Y, — Ys z- 2 RACE Yos Ys) —S(Y,» Yoo Y Ys) Yo Jo) 92 Gong, wx FY r2 Yar Ys) ~LS Yas Yo) Ys = Ys Ys — Ys ux Proceeding to the limit we get, as in the case of two functions, Du _ du dy, , du LY du dy, a Lyk he OGRE aa dy, . tye sihatateters (1). u du du In this equation —, —, —, are the partial differential ‘ dy, dy, dy, i D coefficients of « with regard to y,, y,, y,, respectively ; and Be da the total differential coefficient of w with regard to z. If we adopt the suffix notation, the equation (1) may be written Du me dy, dite dy, duu dy, CEL. ole ale apae an 2). dz dy, dx dy,’ de” dy,” de @) 16 PRINCIPLES OF DIFFERENTIATION. Multiplying the equation (2) by dz, and putting Ay U d,.u dy Yq : an d Y2 ; of git he dy, dy ] yyy dy, dy 2 yt, dy, 3 we have Du = dye 4. dy Ab LEW on) oakley st ate (3). ay hg Uh The same theorem may evidently be extended to any number of functions ; so that, if U=F(Y1> Yor Yorr + + Yn) Yi> Yoo Ygor ++ +, being any » functions of z, then Du _ du dy, du dy, du dy, du dy — . 4 tt de dy, dx dy, dx dy,’ de ""*" dy,” de’ or, replacing du in the expressions du du du de dy,’ dy,’ dy,’ """ dy,” by d, u, du, d, Uy. .d, u respectively, and clearing the equation of fractional rs Du= dur dowd HO. . + as yy Yo Y, Yn It may therefore be stated as a general proposition, that the total differential of a function of any number of functions of a variable is equal to the sum of its partial differentials taken on the hypothesis of the separate variation of each of the several subordinate functions of the variable. Cor. If any one of the quantities y,, y,, Y,...++ Y¥,» Y, for instance, be equal to z, which is the most simple form of functionality, then from the above demonstration it is plain that we may replace the corresponding term du dy, du. dy, ; BE by a ; thus if Uu Sys Ys > Y5> Y 49 OF Orr JULIE Du _ du du dy, du dy, du dy, then —— =——+— +—. 34 + —— dz dz dy, dx dy, de °""" dy.” dx’ or, transforming the equation from differential coefficients to differentials, Du = du t+ dy + dy + occas. + dy U. PRINCIPLES OF DIFFERENTIATION. | 17 Differentiation of an implicit Function of a single Variable. 22. Let v= 9(#, 4) = 0, y being therefore an implicit function of z: then wv’, 2’, y’, being corresponding values of v, z, y, =p(z,y)=90, v=9(2',y) = 0, v—-v= (2, y')-9(% y)= 9, or Pee eG) SAY) VAY 9) z-2 “u-— 2 Yoord a Now in the limit, when z' approaches indefinitely near to z, and therefore y' to y, we have Uae gi dy fio, On Le aw—2x dx’ zx—2 dz’ b(z', y) - 6 (% y) GEA do x -2 dz? and, first replacing z by 2, and then y’ ‘e Y> $(2',y')- $(@s4)_b@y)-$@_Apley) _ de, yy y-Yy dy dy hence the equation (1) becomes Do _ dv _@ dy | (2) de de” dy” de pay: B : dy . This result gives us the expression for A in terms of the br partial differential coefficients of » with regard to x and y taken successively as separately varying. If we replace the symbol dv in the numerators of the fractions dv dv dz’? dy’ by the expressive forms dv, dv, we have, transforming the equation (2) from differential coefficients to differentials, Dv=dv+d,v= 0. This result shews that if any function of z and y be always C 18 PRINCIPLES OF DIFFERENTIATION. zero, its total differential or the sum of its partial differentials is always zero. General Theory of the Differentiation of implicit Functions of a single Variable. aie ARES SW Oy Ay OR SAU gg SOk Hath Where 0,, 0,, 0,5. 4: 9.5 ale 7% fUNChIONS Of 2, 45 'Y/, Msc ee ee the variables ¥,, y,, Y,5... -Y,» being therefore implicit functions of z, then era ey by SR OD, 3 AO NY eee Og dy oe a dy we Ot i ee Or ee ee ae di, dz i together with 2 — 1 additional equations involving 2,, v,, 0,+++%,, precisely as v, is involved in this. Let wx’, ,'5 Yrs Yyo.+++Y, 9 %> be corresponding values of SS Aa hip 6 OCR te =f, Y.2 Yo Ysr++Y,) = 9, =F (L's y's Yoo YarrrYn) = 95 hence O = 0-0, = (Es Y, > Yor Voor =o sVu)— J E> is Yas Yas a 2 Ga) =f (Ls Yu2 Yor Yoo ++ Yu) —~ Fs Yio Yar Yare ++ Yu) + Is y's Yas Yor 06 Yn) ~F L's Yas Yor Yore e+ Yn) + Is Y's Yo's Yo oe + Yn) ~ IE's W's Yor Yor ++ Yn) eeecee8ee#eseseereee#eeeeeseeteenreeneeeeeeweenteeesee#te#e#eete eee eee0eeeeeeeeeeseeeeeeteeeeeeseeeeeeseeaeeeeee eee whence, 0,, 6,5 Oy. +++y,9 Genoting the partial increments of the functions to which they are prefixed with regard to z, y,, y,, ...Y,, respectively, and A the total increment of v,, we have gp AM _ OL Yas Yor YorYn) , On S's IYr2 Yor Yorn) SY; Oz ox °y; = jy, ' Oz a Wat + O), AZ » Yi5 Ya sYanV ans Y,.). oY, | OYn Ox PRINCIPLES OF DIFFERENTIATION. 19 Proceeding to the limit, we have, z, y,, y,, Y,)..-+-¥,, being the ultimate values of 2’, y,', y,'5 Yy5. 66 -Yy's 9 - 2a _ dv, . dy, , a, dy, , dv, dy, Gee de (iyi ae dy, dx. dy, dz dv, dy Ct ew 2 eoeere l e i dy, dx @) The analogous equations in regard to v,, v,, 0,,....Uny may be established in the same way. Multiplying the equation (1) by dz, we have dv, dv dv dv dv = D ——— ee | . ——— ae! e d aw . d eee is e d 3 0 v, Tr dx + oe dy, + a) Y + Wt Yq teoet i 4 or, if we express the different partial differentials of v, by suggestive suffixes, n 0 = Do, = dp, + dyv, + dyv, + GyW, +... .+ do 7 ¥31 Yn-1° From the equation (1) together with the (mz -— 1) analogous equations, making in all m linear equations, we may determine the » differential coefficients dy , %Y, ay, TY . Tie Og wa OER a Oe in terms of the (mz + 1) partial differential coefficients of “SO eee Total Differentiation of a Function of Functions of independent Variables. 24. We have now fully considered the principle of differen- tiating a function of functions, the subordinate functions being dependent each of them upon one and the same variable. Suppose, however, that b= S(Y,s Yar Yorr> « Y,,)s and that y,, ¥,, Y3,.-+-¥,, are not all of them dependent upon a single variable, but that they are functions of several inde- pendent variables. Let w', y,', y,, Y,5.+.-Y,» be corresponding values of %, ¥,, Y.5 Yo. -- -Y,2 then c2 20 PRINCIPLES OF DIFFERENTIATION. thee tt FY, Ys Uses ee ue AY Ya aehe eae =F (Ys Yos Yore s+ Yn) ~F (Yrs Yoo Yas ++ Yu) + FY; Yo Yoo 00+ -Y,,) -fy,; Yoo Yaar 0 VR, +FY,> Yas Pb ee Y,) —f (iy, yo Yoaseee of a = + IL Yas Yor nev Yn) ~ Fs Yas YorrY nvr Yu) A=, FY y2 Yor Yar"*Yu) + 8, LYr 2 Yo2 a2 Yu) + Oy LY 9 Yo's Yao" Yn) Fee FOL NG Yas Yor 0+ Yur? Yn): Now when these increments are diminished indefinitely by the continuous approximation of y,', ¥,, Y,5++.-Yn, to the values Y1> Yes Y39+++-Y,, and in correspondency with any restrictions to which these variables may be subject, let quantities finite or infinitesimal which are proportional to these vanishing incre- ments be called the differentials of the corresponding incre- mented quantities: this definition of differentials is merely an extension of the definition for the case of one independent variable to that of any number. Then, by the notation of differentials, we have 7 Du=du+du+duts...4d u, Vy Yo Y3 y Ex. Suppose that : U=F(Yy> Yor Ys) = Ya + Yo + Yas peach ete WIR ERe Chee Uae esata se then Oy, = CY, 8,6 = OY, Oye = Ys; d,u=dy,, du=dy,, d,u=dy,, Du = dy, + dy, + dy,. BY, OC te OS, 5 GO, ee Dyker etary ea = (Lit 2 ).00, tied Ors, Of 2 Lene lee = (204 rte One (Be Be a Noma: hence, by the definition, dy, = dz,+dx,, dy, = 2x,dz,+ 22,dz,, dy, = 3x, dz, + 32,’ dz,; Du= dz, + dz, + 2x,dx, + 2x,du, + 3x,'dx + 32x,’ dz, = (1 + 2u, + 3%,’) dz, + (1 + 2a, + 32,”) dz,. PRINCIPLES OF DIFFERENTIATION. 21 Partial Differentiation of an explicit Function of three Variables one of which vs a Function of the other two. 25. Suppose that Se MUP z being some function of two independent variables x and y. Since x and y are supposed to vary independently of each other, the variation of z being dependent upon the variations of zand y, we may assume y to remain unchanged while z and therefore 2 varies: then, the expression Du dx being taken to denote the total differential coefficient of u, as far as wu is affected, both immediately by the variation of z and indirectly by the variation of z as consequent upon that of z, we have, by Art. (21), Cor., Du du du dz ee = Ee Bp “ye eee o> ee oe eae Cb): In like manner, y being supposed variable and z constant, Du _ du du dz _—_—- = _ dy dy dz dy (2) In these equations a and ai are the partial differential coefh- o cients of z with regard to z and y respectively ; th and = are ys dx the partial differential coefficients of « with regard to z and y. du ; In the equation (1), FE represents the value of the ultimate Z ratio of the increment of w to the increment of z, when z recelves an Increment in consequence of the variation of z: in : du : : the equation (2), ra represents the value of the ultimate ratio z of the increment of « to the increment of z, when z receives an increment in consequence of the variation of y. It is important however to observe that in both cases the actual value of gt ae must be the same, the origin of the variation of z evidently not affecting the ultimate ratio in question. We are at liberty 22 PRINCIPLES OF DIFFERENTIATION. therefore to consider dz as the total differential of z in the du dz accordingly also the same in both. ‘The equations written in the most expressive form would accordingly be denominator of — in both equations, the value of du being dz dx Dz dz Du du du dz = 42,4 eeoeeeereee ee @ Dui du Hl du dz (3), Owing to the complexity of the notation in (8) and (4), it will be desirable to adhere to the form of expression which we have given in (1) and (2). No danger of confusion can arise from the several meanings of Du, du, dz, provided that we remember to regard as monads the expressions | Du Du du du du dz dad dz’ dy” dz’ dy’ dz% dz’ dy’ the denominators of these indissoluble fractions sufficing to suggest the significations of their numerators. Multiplying (38) and (4) by de and dy respectively, and adding, we have Du+ Du=du+ du oe (dz + dz): but, by Art. (24), Dze=dz+az;3 hence Du+ Du=dur+du+du: but, by Art. (24), we have also Du=dur+du+ du; hence Du= Du+ Duadurdu+ du. Partial Differentiation of an explicit Function of n+r Variables, r endependent and n dependent. BOs Reba Wires ois teeters isan et een where 4,5 Y> Ys «+ +Yn, are each of them functions of 7 inde- pendent variables z,, 7,, %,,....%,. PRINCIPLES OF DIFFERENTIATION. 93 Then, differentiating successively with regard to 2,, 2, ,2,,++-%,, each of these quantities being taken in turn as the only variable among them, we have, by Art. (21), Du _du du dy, du dy, du dy, _ ui dy, dz dv," dy,’ dx," dy, de,” dy,’ de, euegece dy. * da,’ dg, Ade. wdy; adxs\ dy, tas) dy; dz, °°. dy dais? dremel duel dy. i dg.edy (de. dy. dt. x .dy de? Dudu, du dy, du dy, du dy, du dy, dz, dz, dy dz, dy, dx, dy; da, dy, dz, Partial Differentiation of an implicit Function of two independent Variables. 27. Let z be an implicit function of two independent variables # and y by virtue of an equation iad 2,1 sie) 20. Then supposing, as we are evidently at liberty to do, that y remains constant while x and consequently z varies, we have, by Art. (22), Du du du dz de de” dz’ de Again, supposing y variable and z constant, we shall have also EO tee Cd. dy dy dz dy Partial Differentiation of implicit Functions of any number of independent Varvrables. 28. Let Ga= On si0a= 07100, =.0,5.5 .0,'= 0; where 2,, 0, 0,5:+-0,, are ” functions of n +7 variables z,, x,, £,, “sey Yi Yoo Yaorrr¥,,: then each of the variables ¥,, Y,, Yss++-Yns 24 PRINCIPLES OF DIFFERENTIATION. may be regarded as a function of r independent variables Ly Loy Lyy....%,- We are therefore at liberty to consider Z,, Lz5e 00%, a8 all constant; mand: fo werard ty. 5 ys5 Yie wcteu ns as functions of a variable z,. We have therefore, by Art. (23), A Dv, _ de, _@% dy, , a, dy, _ dd, dy, _ ® AY n Z dx, y dx, dy, dz, dy, dz, dy, dz, ce dyn dz, Similarly, z,, z,,... .@,, successively taking the place of z;, cee ai ADD Melia AN PSE Fs caches UKE praca “da dx dy. da, dys da dy eae, dg ae pe CAPRI. RE Nilay alle aie Ge ae ge he TE dg, (dz dy. dz ay, "da, (dy. az, dy, dz,” Do, dv, dv, dy, dv, dy, dv, dy, dv res Se eee ee dz, dz, dy, dx, dy, dx, dy, dx, dy, dx, There will evidently be also 7 analogous equations in relation to each of the functions v,, 2, v,,.... 0, We thus have nr differential equations, and may thence determine expressions for the ur partial differential coefficients of the dependent Vvariables"¥,,"4/,5 Yhi's was Ys Vide SEARLE NE dy, dari Uc ee ae ie eres dy, dy, dy, dy, Ogi die. eta ee a dz,’ dy, dy, dy, dy, die deal de Cees dn” iy dy yy, OE, MAL ae CE samy Ws dz,” in terms of the n(m+~r) partial differential coefficients of V,9 Uz Ug) «+» O,, taken with regard to the variables z,, z,, 2,5 --» X, 13.599 Y> Yoo Y3> aia Yn PRINCIPLES OF DIFFERENTIATION. 20 SECTION II. SIMPLE FUNCTIONS. 29. In the preceding section we have shewn how to reduce the differentiation of a function of functions to that of the differentiation of its subordinate functions. In this section we shall investigate the differentials of what may be called simple functions, as being the constituent elements or subordinate functions of all the complex functions of algebra. The essen- tial characteristic of a simple function consists in its not being susceptible of resolution into elements more simple than itself, except by the aid of infinite series: the number of simple functions might therefore, as may easily be imagined, be mul- tiplied indefinitely. The algebraical expressions ordinarily adopted as simple functions are the following : x”, m being any real quantity whatever, a, sin 2, cos x, tan @, cot z, sec 2%, Cosec Zz, and the inverse functions ] i Ee | 5 | -l -l -l -l og, , SIN” #, cos #, tan” x, cot” x, sec” Z, cosec™ 2. These expressions have been selected as the elementary functions of ordinary analysis, in consequence of their peculiar utility in the various applications of the science. To find the Differential Coefficient of x® with respect to x, n being any rational quantity whatever. 30. Put y = 2"; then, 2’, y’, being corresponding values of x and y, we have e n ” Ys Oias —a@; in ” m oy _ 2 -Zz eal = > 6x2 2-2 poe fl z being such a quantity that 2 = zz. Our object is now to find the limiting value of the fraction z”— 1] tas when z approaches indefinitely near to unity. Now whatever 26 PRINCIPLES OF DIFFERENTIATION. be the value of », positive, integral, fractional, or negative, we may always express it under the form Piz Y 3 r where 7, g, 7, are positive integers. Hence Pp-q 2 — 1 2%.—1. ota - . z-—1 7 z-1 * “A » putting z=, _1 @=1)-@-=1) ot deat ; Hence, dividing v? - 1, o7- 1, v — 1, by v — 1, observing that P>% 1, are positive integers, we have 27-1 1 (l+ot¢e+.... 407 )- (1+ 044... .t0%) po ae L+ovtor.... +0 Now, by making v approach more nearly to unity than by any assignable difference, z will also be made to do so: hence ae 2" -— 1 Dp — limit of ~——— = his hme - r d Hence we see that a = na”, ZL To find the Differential Coefficient of loga x with regard to x. 31. Put Tp yg ae then . OP UO RET Pin ce peer x dy 1 log,(1+8) 1 : Ls cs a eee ene eee | 1 ox 60 7 Be ae PRINCIPLES OF DIFFERENTIATION. 27 Our object is to determine the value assumed by the expres- sion 1\" log, (1 + *) ; when 2 = o, a value of m consequent upon the evanescent state of dz. Now whether x be a continuous or a discontinuous vari- able, yet, provided that it become greater than any assignable magnitude, dz will become less than any assignable magnitude, which is the only condition to be fulfilled by éz in the ultimate state of the hypothesis. We will assume then x to represent a positive integer, and proceed to ascertain the limiting value of the function 1 nm log, (1 + ") . when the integer becomes great without limit. By the binomial theorem we have the following expansion, Je 1 awe, consisting of m + 1 terms, for (1 + | viz: n + ———__—* , — + pres Teo n {Ror Ss n> BAe Fe ee is VY) A ( ~) = nm 1 n(nm-1) 1 m(m-1)(m-2) 1 1 n +. 12. Sie on ee V, ny _ m(n-1)(m— 2).... m= 2-1) 1 Lease n "gn? Ns 1 if 1 1 1 2 ek ee ee eee epee (OT ls | pee or(1 +) ste 5 ( -) a5 ( n n 2 = ea ee) Co) ee Ce *\ [2 Ons deat n n n 2 — 1 a + 1 a4 Mi -*)(3 — = o( 2 ae is Jn(1 ia *), 1.2.3...v(v +1)...” n n n n Now we may take z and yr so large that, if we stop the series at the (v + 1)" term, the sum of all the remaining terms will be less than any assignable magnitude. In fact, this sum is less than 1 1 1 a ef 0 fo 1.2.3...v(v4+1) 1.2.3..v(v+1)(v+4 2) W.2:3-.y (vitil )..7 28 - PRINCIPLES OF DIFFERENTIATION. and @ fortiort than x + a t's evale > wm , and a fortiort than mat saa By ad Ne gv ' gual : gr-l © gn © ont : which series is equal to ae and becomes therefore indefinitely small when v is increased without limit. If then we take v indefinitely large, and neglect accordingly all terms after the (v + 1), and if we then take , which is of course always larger than vy, an indefinitely large number of a higher order of magnitude than vy, so that in fact the ratio of m to v shall be indefinitely great, we shall have ee et lg : + 5 Rae ACTS A ST n}) ey kere a morc ier: * [i258 ees aan an approximation true without limit as vy increases without limit ; that 1s, in the limit, (isi) rste eee oe . ad infinitum n 1h? 1 20e 6223 = e, the base of Napier’s logarithms. Hence we conclude that ‘ dy 1 1 To find the Differential Coefficient of a* with regard to x. 39. Put y= as yf aigni= Teas then y —y=a’.(a* — 1), by ya aa Si Guieh Ox Hh Put eee dz log, a = log (142); n’ de 6 n) oe oY _ log, a ee log, a a . 1 . PE aE ET a poe n log, (1 + 3) log, ( + 5) PRINCIPLES OF DIFFERENTIATION. 29 Now, to proceed to the limit, putting » = an indefinitely large positive integer, and thereby rendering dz less than any assign- able quantity, we have, De a”. ae a’. log, a. dx log, e To find the Differential Coefficient of sin x with regard to x. 33. Put TEAS Hoke then dy = sin (% + oz) - sin & whence = . COS ox Nxt Now by the seventh Lemma of the first section of Newton’s Principia we know that the arc and the chord of any curve vanish in a ratio of equality: whence it follows that the ratio between the sine and the circular measure of an angle is ulti- mately unity. Hence, in the limit, Od sin ry Be 3 >. dy and therefore er dy = cos x. dz. He To find the Differential Coefficient of cos x. 34. Put y=cosz, y' =cos (% + 62); then dy = cos (% + 6x) — cos : oz\ . ox =—2sin| 2+ — }). sin —, 2 2 oy 2) BEI Fone 44 | ie OL 2 ox 30 PRINCIPLES OF DIFFERENTIATION. whence, in the limit, dy ee See 3 dz or dy =—sin x. dz. To find the Differential Coefficient of tan x. 35. Put y=tanz, y' =tan (7+ 62); then dy = y' —y = tan (x + dz) -tanz _ sin (@ + 62) cos x — sin x cos (« + 62) - cos x cos (x + 62) sin 0z cos .cos (x + 6x)’ oy _ 1 sin 0. dz coszcos(x+ox) oz proceeding to the limit, when sin 62 _ ; Nob ae dy _ we see that eh nae =(sec 2), or dy = (sec zy dz. To find the Differential Coefficient of cot x. 36. Put «y=cot2, y+ dy=cot (@ +éz); then dy = cot (a% + dx) - cot x _ sin x cos (x + 6x) — cos x. sin (x + dz) sin x. sin (x + 62) sin. Ox ~ sin z.sin (x + 8x)’ Oy an Me 1 sin 62 | dz sinz.sin(x+6x)° 82 proceeding to the limit, since ultimately sin 6z = ] Ox ; PRINCIPLES OF DIFFERENTIATION. By re es 2 we have 2B CYC ES (cosec x), or dy = — (cosec x). dx. To find the Differential Coefficient of sec x. 37. Putting y=secz, y+ dy =sec(# + dz), we get dy = sec (x + 8x) — sec x _ cos £ — cos (# + 62) cos x. cos (a + 62) cos Z. Cos (x + 62) in{ 2 ox fe Sy $1 ( aera re 8x cosa.cos(z +z) ot 2 proceeding to the limit, when Ow sin 2 = ] bz j 2 d: in 2 we have oo) yee tants (g60 2 dx (cos x) or dy, = tan 2. 8CC, 2. dx, To find the Differential Coefficient of cosec x. 38. Putting y=cosecz, y+ dy=cosec (e+ 62), we have dy = cosec (a + dx) — cosec x sin x — sin (x + 62) sin x. sin (@ + dz) . ox ( >) 2 sin —-. cost « + — 2 2 sin x. sin (x + 62) 2 82 PRINCIPLES OF DIFFERENTIATION. ( >) Soe cos{ 2+ — si he 2 ae. oz sin z. sin (@ + O£) bx 2 . Ox sin'\— in the limit eS mT Ox 2 Behe dy COs & and therefore —4=-— —~—~ = — cot Zz. cosec 2, z (sin 2) or dy =—- cot x. cosec x. dz. To find the Differential Coefficient of sin” x with regard to x. 39. Put y-=sin'z, y+ dy =sin" (zx + 82), then sin y=, sin (y + dy) = x + Oz, sin (y + dy) — sin y = 62, 1 . or 2 008 (y+ . sin — = 62; 2) 2 and therefore oy eee Now, proceeding to'the limit, when 5z and dy assume values less than any assignable magnitudes, we have oy a = 1, cos (y +H) = eos y: sin — dy Wie (ace yal ter ge ha hence dz cosy {1 - (sin yy Cita 2 d. or dj< PRINCIPLES OF DIFFERENTIATION. 33 To find the Differential Coefficient of cos” x. 40. Put y=cos'z, y+ dy=cos' (x + 2), whence cosy=2, cos(y + dy)=x + 62, cos (y + dy) - cos y = oz, or -2sin(y+ Y) sin Y ~ be, oy whence = = — wm : proceeding to the limit, we have ne DE a ceca ee dx siny {1-(cosyy}# (1-2*)’ dz | or Mee oe ae To find the Differential Coefficient of tan x. . 41. Proceeding in the same way as in the investigation of the differentials of sin" z and cos” z, we have y=tan’z, y+ dy=tan" (x+ dz), tany=2, tan (y+ dy)=2+ oz, éz = tan (y + dy) - tan y _ sin (y + 8y).cos y — sin y.cos (y + dy) a cos y . cos (y + dy) © Ros: be ~ cos y.cos (y + dy)” Ts Peg lI a — . cos ¥ . cos (y + oy), dy _ ” 1 1 ee is l+(tamyy 142°’ d. or dy = i — 34 PRINCIPLES OF DIFFERENTIATION. To find the Differential Coefficient of cot” x. 42, y= cot x, y + dy = cot’ (%& + dn), coty=2z, cot(y+ dy)=2 + oz, Oz = cot (y + dy) — cot y sin y cos (y + dy) — cos y sin (y + dy) i sin y. sin (y + dy) | — sin dy ~ sin y. sin (y + 6y)’ 2 = — ate . sin y. sin (y + dy), dy _ ey Oa 1 x 1 dz oe ae 14(eot yf 1a? d= ~—- de Tee on To find the Differential Coefficient of sec” x. 43. y= 5CC 12.) (oi + Oy sec (2 +02), BCC Vier, sec(y + dy)= 2% + Oz, dz = sec (y + dy) — sec y cos y — cos (y + dy) ~ “cosy. cos (y + dy) asin Hsin (y +2) 2 2 cos y . cos (y + dy) oy Sy 2 ~~ cos y cos (y + by) ee sin y sin (ys s) 2 2 dy (cos y) 1 1 dz siny (sec y) sin y 3 1 sec y ~ (sec y)’ * {(sec y= 1} a 1 ay dx ~ ¢(e?- 1) —— PRINCIPLES OF DIFFERENTIATION. St) To find the Differential Coefficient of cosec” x. 44, y=cosec’z, y+ dy =cosec” (x + 62), cosecy=2, cosec(y + dy) =x + Oz, dx = cosec (y + dy) — cosec _ sin y ~ sin (y + dy) sin y . sin (y + dy) 2 sin cos (y+), sin y.sin (y + dy) oy OF 2 sin y sin(y + dy) fies cos mercy : 2 ad, 9 QUie (BUSY ul ase 1 dz cosy —_— (cosec y) cos y qoemes cosec Y ~ (cosec yy ° {(cosec yy — 1}! iF Sorel Se ere il dy oe eee ae x(x -—1) Differentiation of Simple Functions of y with regard to x, y being a function of zx. 45. Letu=y”. Now, by Art. (19), du du dy dz dy dx’ f and, by Art. (30), ai = my"; : | m du ae anal dy hence, if «= y", Tee ee D2 30 PRINCIPLES OF DIFFERENTIATION. Similarly it may be shewn that, du 1 dy | if w = log, y, Te yoga as lice = a; Wat log, a. du dy fu= — = peer if 4) 81, Fe CORY 7 : du dy - if w= Cos y, Seay oe if w= tan y, S =! -i(séty/)s a it 20 =" CObY; = = — (cosec y)’. ae if w= sec ¥ ae tan y . sec Y dy ap 2 ob FF ‘ . ea if w = cosec « a cot y . cosec 4 we i 3 ee y J Ue hate ip. e = any “ by -y) dz : - du _ 1 dy Geet COS met) s ee Cea Fp ‘4 du 1 d ifw=tan'y, Fa rata Pree it (2 !COta 7, eee : du 1 dy a 1 SEES Leeman ide re LY 5 Seale if w= sec” ¥, oe AGS ee if w= cosec” y, cabpoge Teale Sy e\ee All these formule the student must carefully commit to memory. PRINCIPLES OF DIFFERENTIATION. i SECTION III. ILLUSTRATIVE EXAMPLES. 45’. We shall devote this section to the exemplification of the principles which have been established in Sections (1) and (2). The illustrations here given are not numerous: in order to acquire a practical familiarity with the processes of differenti- ation, as well as with the application of the general theorems which we shall develop in the subsequent pages of this work, it will be necessary for the student to have recourse to Peacock’s or Gregory’s Examples of the Differential Calculus. Ex. 1. Let y=sma+sin 8+siny, a, B, y, not involving x; then, the sum of the three sines being a constant quantity, we have, by Art. (11), dy dx Ex. 2. Let Y= 2 4 a, a being a constant quantity; then, by Arts. (12) and (30), we have Ex. 3. Let Pe UNOg ee, a and 6 being constants; then, by Arts. (13) and (31), we have CA is Aaa dz zlog,a’ Ex. 4. Let Y= u*+ a’, a being a constant; then, by Arts. (14), (380), and (32), es = az*’ + a®* log, a. Ex. 5. Let = Sue 2 COS 5 then, by Arts. (15), (23), (34), dy _ in p CLOSE cos Un? dx dx dx = sin x(— sin 2) + cos %. cos & = (cos zy — (sin 2y. 38 PRINCIPLES OF DIFFERENTIATION, 1k. Oe Let y= —— then, by Arts. (16) and (35), there is tama dtan x dx ae (2 — tan dx LEE = 4 {x (sec zy — tan x}. Hix. 7; (Let #y sia logs vagy, seo Si wi then, by Arts. (17), (80), (31), (82), (33), (89), dy u d(x”) d log, # , d(a®) dsing : d sin" x y ane log, rus sin x pin” o* n dx dx da dz =—— + ——__——_ + log, a dx + —— + -— = ee et looms lot aa. tanz sin’ z(l1 - 2} Ex. 8. Let it be proposed to find a in terms of 2, having ay given that y= sin By Art. (83), we have < = COs 2; as ly dx nd, by Art. (18), nena and, by Art. (18) FT; hence cos @. ue =1, Ws = Sec &. y dy Ex. 9. If w=y’*, and y=cot z; let it be proposed to find . du . We have, by Art. (30), Ae 3y’, Y and, by Art. (36), 2 = — (cosec z): hence, by Art. (19), . = = — 3y° (cosec 2) = — 3 (cot x) (cosec x)’. PRINCIPLES OF DIFFERENTIATION. oY Ex. 10. Given u=sin 2, <=siny, Y¥ =61D z, to find is . We have, by Art. (38), Lee Ce ey dy = COs/'2: dz > dy F > dx ; but, by Art. (19), Cor. dz dz dy dz U hence — = COS 2 COS y COS &. dz Ex. 11. Given u@ = sin (ay) + sin (Bz) + tan” (yz), J = SEC, 2 = COSEC Z, Du ashi ee, dz Du _ du dy . du dz B ECON ee eat aa 2) dz ~ dy dx’ dz dz By Arts. (14) and (45), d(ay), 1 aye) dy 1+(yzy dy - = cos (ay). = a. sumilarly = 2 cos (Sz) + ar > a3 Also, by Arts. a and (388), LU ener Zs al COL 2%. COSEC Z: xz dx Hence we have wae a cos (ay) + “ tan x sec x de Woe Taya : : pet {6 cos (32) + fy el cot # cosec &. 40 PRINCIPLES OF DIFFERENTIATION. ¥; Ex. 12. Given that t=) Males and that yea Yh eee re Du as We have, by Art. (21), Du _ du dy, , du dy, . du dy, to find dx dy, dx * dy, de #dy a2 ae ea : But ——=— .y%.y%°, by Art. (80): dy, Y, dike y ‘ du d (y,") vl saat fee Ye 3 b ce 9 dy, log, Y, dy, y,2°, by Art. (32) = OP 54, eye a yt, by Art. (30); 2 du v, a (y,”) Datel be log, i y 28 soe ’ b Art. (32), dy, ae dy, i = log, y,. y,”*. log, y,. y,’4, by Art. (32), = log, y, - log, y, 9,2. y,”"- Hence Du i. y Y,"s Ys 2 Vs £¥!.y 2° + 2z.log, ¥,-Y — -y,” +82z' log.y,.log,y,.y,"s.y," AAs yo y SeViYs Y, 2 oped Y,3. ye. (; + 2x log, y, + 8z° log, y, . log, | : Ex. 18. Given that u=(a-—2x+sin y=0, dy to find ”. ofind | du du dy By Art. (22 sa a 9 LEE Meet?) dz” dy dz But, by Art. (45), when we differentiate considering y constant, BN bea VO ae) =}(a* — 2)*.(- 22) = - « (a - 2’). 1S Se saith ager faan)s PRINCIPLES OF DIFFERENTIATION. Also - = COS y. Hence we have ly — x(a’ — z*)4 + cos 2 Mone (8 ( ) ae whence Ya a —. dz cosy(a@— 2x’) Ex. 14. Given that u= 2", z being a function of z and y by virtue of the equation 2 SIN 2), let it be proposed to find du du du Du Du FES GAR” GEE Oe By Art. (30), oe zo: by Art. (45), Fp 7108. 2 A = 2. log, 2.2” and “= log, _ git MANES aes ye Also, by Art. (45), dz d (ay) _ ay 7 008 (xy) = ea co (xy), = =1CO8 {Z7/) oe es Z Cos (xy). But, by Art. (25), dz dx dz dz’ Du _ du . du dz dy dy" dz dy’ Hence we see that —=2—.2"4y log, z.2”. y cos (zy) re = ys) * log A 4 Le YL ety og, & COs (2) 3 42 PRINCIPLES OF DIFFERENTIATION. Du =zlog,2.2”" +y log, z.x”. x cos (zy) =lOG eid") oir acos ry) re = yx" . +y log, Z.cos (ay) | dz +log,x.a”.{z+xy cos(ay)} dy. Ex. 15. Given that u= sin (ry + yz + 24) = 0, | lz dz to find ~ als o find 7 and 7 By Art. (45), d (xy + yz + 22) du — = COS (wy + YZ + 24). dz dz = (y + 2) cos (ty + yz + 22): ie du : similarly oF = (2 + ©) Cos (y + yz + 22), du Ein (a + ¥) cos (xy + Yz + 22). Hence, by the formule of Art. (27), viz. du du dz de * de de” du , du ae aes dy dz dy ° we have fee eee ee dations dz Z+H24+(e+y)—=0; dy "¢ ae a . rs Lo ! LY ate P eh. A MI OF DIFFERENTIATION. 43 rar a0 iw . : PRINCIPLES damman yee. ide Gosek a de u+y? dy “+y’ a sr, transforming partial differential coefficients into differentials, hd oty ory Y as ee ——— = 7 il at o. wh ii ne by, 53 os a oe | . iy i PS pee aca iit) i ee oe Bi 8 al tone ane Cali, eS CHAPTER IIL SUCCESSIVE DIFFERENTIATION. Theory of the Independent Variable. 46. Let d(z, y) = 0, where $(z, y) denotes any function of x and y whatever. When for z we substitute the successive values z + dz, x + 26x, x + 86z,... .let the corresponding values of y be ¥,, ¥,5 Ys». +.» Lhen, dy denoting the increment of y due to the increment 6z of z, we have Uies as oy ‘ hence, putting y + dy for y and y, for y, in this equation, which corresponds to the change in the equation due to giving z another increment 6éz, y,=y + Oy + Oy + dy), or, putting ddy = 6’y, as an abbreviation of notation, Y,=y + 2y + Sy. - Similarly, z receiving a third increment 6z, y,=y + Oy + 28(y + dy) + O&(y + dy) =y t+ ddy + 30'y + dy. Proceeding in the same way, we shall finally get, the law of the coeficients being evidently the same as in the binomial theorem, n(n 1) 65 n 2 Py +,...4+ — Oy + Oy. a5 y + tr O° BY rey n Yoayts dy + Thus we see that as x keeps increasing by equal increments 6z, y generally increases by unequal increments: in fact the incre- ment of y, corresponding to an increment 6z of z, is dy, and, for an increment 2dz of z, it is not ndy, but Me (n — 1) 1 i Ey Oy +... 2+ 7 oly + Oy. ae SUCCESSIVE DIFFERENTIATION. 45 The quantity 2, which is supposed to increase by equal augments, is called the endependent variable, while y, the increments of which are dependent upon those of z, and which are generally variable, is called the dependent variable. Such is the definition of an independent and a dependent variable in the calculus of finite differences. Suppose now the difference dz to be inde- finitely diminished, then we may replace CaO EOYs Onis sce ace by the differentials DLR OU SOY LY ysis os. which are proportional to them. We may then say, to adapt our definitions to the differential calculus, that if y be a function of z, x will be the independent and y the dependent variable, if, while z varies, its differential dz remains constant: in ac- cordance with this definition not only y but also dy will generally vary with the variation of z. Ex. 1. Let y=sin z; then, z being the independent variable, dy = cos x.dz: differentiating again, z and dy being variable, and dz constant, d’y = d(cos x). dz (- sin x. dz). dz Isic ae , lI ll where dz’, for simplicity of writing, is put mstead of (dz)’. Proceeding in the same way, we see that n dy = (-):.sin z. dz", n being even ; n—1 d"y =(-)* .cos«. dz", n being odd. These expressions may be written also thus : fe = (-. sin z, n being even: d” a on =(-) * .cos x, m being odd. Ex. 2. Suppose that yY=a,+a0+.a,0° + a2 +....6. + a,x", 46 SUCCESSIVE DIFFERENTIATION. a rational function of z of ” dimensions: then, differentiating successively 7 times, we have, x being the independent variable, ly =e = ‘lia + 2.0). 05 3.0,. Pike i tees dy n-2 ato 1g 2a ch 2 Paes ie + (mw -— 1) naz”, d° n--3 ee 1.2). 3 Gite ae + (% -— 2)(m—- 1) na 2””, ad" “a a 132; Baa wore na da” i) The differential coefficients of higher orders than the z", viz. d” “y d”**y d”**y dz ? dar? 9 dx oe baile is will all be zero. Ex. 3. To find the x differential coefficient of —. . Ce Put y= Pes a5 | l he 1 x -a LO NO OED 1 a : prea het) —~(a+a)"}. Then 1 (1) {(@- @?-@ + @)*}, pt ee Fe ag VO DI@- of-@+ ay}, eb Ar SW FA Oey Lc {(x _ oe a (x 4. hare § Change of the Independent Variable. 47. Suppose that we have an equation 2, \ Bay, 2, ot )=o, 2 9 dx: dx ~ SUCCESSIVE DIFFERENTIATION. 47 involying z, y, and successive differentials of y taken on the hypothesis that dz is constant. [t is frequently desirable in researches in the differential calculus to transform this differen- tial equation into an equivalent one in which, instead of z, some quantity 6 of which z# is a function, shall be the independent variable. On the new hypothesis dz will no longer generally be constant. . Suppose that y = f(z), and put Tf (%) op dz = f(x); Jf (z) being another function of z: adopting the same notation, put — Uf (#) _ ¢ ee x az nt ay gndsoon. Lhe quantities f (2), f(x), f (2),...... are called the first, second, third,....derived functions or derivatives of f(«), and are certain algebraical expressions constituting the results of the operations upon the function f(x) designated by the differential coefficients Uf) &f(e) &f(a) FER EEN UE hie Then, taking dy, d’y, d’y,.... to represent the differentials of y on the hypothesis that dz is constant, and d'y, dy, d”y,.... its differentials, supposing dz to vary, we have dy = f'(z).dz\ d'y = f'(a).daf atolls e ful su sWisterl oe 1s ats Cl}: ONE TINGS a fas ee eer | fle VO (2): Lyf (a) da’ | (3): d”y = f'"(2) dz? + 8f'(a) dxd’x + f(x) daJ * and so on to any order of differentiation. Now, by the aid of the relation subsisting between x and 6, dz, d*x, d°x,....may be found in terms of 9 and dé, and there- fore, from (1), (2), (3),.... we can obtain dy, d’y, d’y,.... m terms of d'y, d’y, dy... . , 0, dO. 48 SUCCESSIVE DIFFERENT IATION From (1) we see that dy=a'y.: - d' y % eae i . ¥ from (1) and (2) d?’y = Pr y+ ae Cie de d'y—@ady ay = ——_~____+ / dx from (1), (2), (3), we may get also ae dx (dxd"y — d’xd'y) -— 8d*x niy © ag and so on for dy, d-Y,. sss. The second equations of the systems (1), a). ae teeapitia written in a form which may serve to suggest to the memory . that @ is the independent variable corresponding to the differen- finds dyid Y; Cd -Yy tes Oley, ave. d' eee, a = f'(2) 2 (oR ees Ns, Sel ROE Sic Nagi Ne (7), of Eden e! ao WH Ree 7 Re PORN eA PAL 13 15 .(8), j , dz d’ EHF OGL OF tl Go, 7 o If we substitute in the differential equation dy aia’y ty ~~, —5....|=9, (2 Y ae? de ) > the values of dy d’y d’y dz” dat de ae that is, the-values of, "7(2z),*f-(@), ff (zw)... obtained from the equations (7), (8), (9),....Wwe shall have transformed the equation into an equivalent one ny, &, ee dy ay ay Ys TH Geer 2 gy qger c+ | =O SSIVE DIFFERENTIATION. 49 ..by the aid of the relation ae ‘h t our object is to change the independent xz to ne : then, by the formule (4), (5), (6), con- a and therefore equating dy 5 de Y mee tO d'y d*x dx — dx dx d'y + 8(d’xyd' by 3 = ee EES ey dx’ and so on. 2 Ex. 1. Given (1 - 2%) SY — 2B + n'y = 0: to change the independent variable from z to 6, when z = cos 0. In this case d'y = f(x) dx = — f(x) sin 9d0; differentiating again, considering d@ constant, we have d”y = — f'(x) dz sin 0d0 - f(x) cos 0d . = f(z) sin’ 0 d@’ — f(x) cos 0d, ; a’y dy * ae? -- 2) S42, ‘ and therefore the transformed equation is Ry dy By = ‘ | ae? + ny = 0. Ex. 2. To change the ie variable in | a | a Se (@+2) 75 f+ 3(a+ ay SY a+ (a+ 2) oY + by = 0, Sy : a ° . from z to 9, a given 6 = log (a + 2): 00 SUCCESSIVE DIFFERENTIATION. In this case ay= fiz) dz fie) eras d”y = f'(x) dx e’ dd + f(a) e db = f(x) ed? + f(a) e® dé, d®y = f(x) dx &? d& + 2f (x) e d0’ + f (x) da e° d+ f(x) e° d& = f -(z).e% d# + 3f"(x) e* ee 4 a (x) ef ue Ty | , ay or aoe 7 6% + &) Bh Cee which reduces the proposed equation to the form dy by = 0. 1) Lam. Order of Partial Differentiations indifferent. 48. The following is a theorem of great importance in suc- cessive differentiation: if u=f(Y,5 Y,)> then dy dy, = dy dy. For Oy,% =f (Y, + OY,» Yo) —FYr> Yo)s and therefore by. 8u% = {FY + OY, Yo + 8Y2) - FYr> Yo + BYo)} — {fy, + 8Y,, 9.) -S Gr» Yodt =F (Yt 8Y, » Yot 8Yo) — FY, 2 Yo 8Y2) — FY BY, 5 Yo) + FYy 9 Yo): In precisely the same way it may be shewn that by, Oya f (y+ OY,» Yo* dy.) =f4i+ OY, » ¥)S(Y; Yor dy )+fY,; Y): The right-hand members of these last two equations being identical, we must have also Oy) OA tit Oy Og 0h, ¥2 ~Y1 This relation is true whatever be the magnitudes of dy, and dy, : if we proceed to the limit, by taking dy, and dy, less than any assignable magnitudes, and replace infinitesimal differences by differentials, we get dy, dy,u = dy, dy. SUCCESSIVE DIFFERENTIATION. 51 Expressing the theorem by partial differential coefficients instead p § yP of partial differentials, we have dy, dy, dy, AyQt dy,dy, dy,dy,° or, as these partial differential coefficients are ordinarily ex- pressed for the sake of brevity, duis du dy,dy, dy, Uy, Cor. 1. By virtue of the theorem dy, dy,u a dy, dy,U, it is evident that the symbols dy,, dy,, of partial differentiation, may be permuted in every possible way: thus dy, due = dy, dy, dig = dy, dy, dye = Ady, dy, dij = An Butts or, in the language of partial differential coefficients, Cie an Gen Ot dy,dy, dy,dy,dy, dy,dy,” Cor. 2. The theorem which we have established in relation to partial differentiation of functions of two variables, may evi- dently be extended to the general case of a function of any number of variables: thus, in an expression Upiaily a? Oye. ohare. Us, u being a function of y,, ¥,, y,,--- the symbols d,,, d,,, dy... ... may be permuted znter se in the same way as the symbols of quantity, A,, A,, A,,....in an algebraical product yakuas oe Bt} ys Be Sn Ex. Let w= y,”: then du A dy Ti Y, Y, 2 du : 1 a(y,-1) = ye oo (y ) cy irl, 2? dy, dy, Y, 45 Y2 0s (y,) Y, dy, = 9)" + y,- log (y,) yy". E 2 2 SUCCESSIVE DIFFERENTIATION. é d : Again Dy =t LOS) atts d*u ==. y+ log (y,) -y,- 9 dy,dy, y, ”" eet iy (ait Ys lO CY errs Thus we see that the results are the same for both orders of differentiation. Ex. 2. Let wu =sin (ry). ‘Then a = y Cos (zy), a °F) = y" sin (zy), d°u dydat = ~ 2 8 (2y) ~ wy? c08 (xy). Also as y cos (zy), PGE = cos (xy) — xy sin (zy), aN — y sin (zy) -— y sin (zy) — xy’ cos (24 dady SESRR Pa oe y-y y y y) = — 2y sin (zy) — zy’ cos (xy). Thus we see that d*u 1 aa dydz’ dxdydz’ or, in the language of differentials, ddiu = ddd. Successiwe Differentiation of an explicit Function of two Functions of a single Variable. 49. Letu=f(y,, ¥,), y, and y, being each of them a function of z: then, by Art. (20), Du du yy, dx dy, dz , de SUCCESSLVE DIFFERENTIATION. 53 Differentiating again, x being considered the independent variable, and observing that, for convenience of writing, we may put, V being any expression functional of z, DV D ‘de dx’ we have Du _ D (s. Zi) + D (s. dy, dx d. dy, ax dz dy, dx } _D (du dy, du D (dy, tal year eae 7 ($2). Se Se 2 (ee aa \ dy.) dt dy, dz \dx } f But, since y, and y, are functions of z only, and not of any functions of z, it follows that D dy ¢ GY Ty ag P Hit MW _E%. —_— — ae ee Cl Bede deds (dt, de dn «de dz de. hence Big) hy 2 (S|. a dy, du NE hte BY, dz dz\dy,/ dx dzx\dy, * dy dx’ dy, dx Now = is equivalent to a function of y, and y, only, not 1 involving the differential dy,: thus, for instance, if u = y,’y,’, then ie. 2y,y,, Where dy, does not appear. It follows there- dy, fore that in the expression D (du dx \ dy, we may regard dy, constant without affecting results. Hence, a now occupying the place of uw in (1), this formula gives ; D (du\_ d (du\ dy, | d_ (du dy, dx dy, ~ dy, dy, ) dx dy, \ Ly, | dz du Ly, Ae Cu dy, 54 SUCCESSIVE DIFFERENTIATION. Similarly, we must have D du\ _ d’u WY, du dy, dx \ dy, er fige y, ax dy,dy, dx Hence we obtain Dui du dy , du dy, _ dy, is d*u dy, dae dy? da i dy,dy, dz’ dx dy2° dx du dy, du d’y, dy, dx’ yu dy, da’ "ub du du du di d pe TL .d a dy he ee dy, Y, + dy,dy, Y\ dy,” aye Y,* dy, Y>" We might proceed in the same way to find the expressions for D*’u, D*u,......3 the formule however rapidly rise into tedious polynomials. We have confined our attention to the successive differentiation of a function of two functions; the extension however of the theory to a function of any number of functions is too obvious to present any difficulty to the student. ‘Thus, supposing that ne =F (YY: Yoo Ys)s the student will easily find that Diu _ du dy? du dy; ' d*u dy,” dx* dy? dz’ dy? dz’ dy? dx 1 du dy, Ys g Fh du ay, yy 4 Fu du dy, dy, dy,dy,, dx dx dy,dy: ‘da dz dy, dy dy, dx dx du d’y, i du dy, H. du d’y, dy, dx dy, dx dy, dx’ Deep (Gi OP ae OA eR We lead havin Then on + Y,)s aay 3/0) + Y,), dy, 1 2 dy, 1 2. Oe eG + ¥,), Og =-— sin (y ey oe ae + Y,): dy,’ ] 2 de 1 2 dy,” 1 2 SUCCESSIVE DIFFERENTIATION. 590 hence Diu =— sin (y, + y,) (dy, +2 dy, dy + dy,) + cos (y, + y,).(d’y, + Py,). But dy. = dx, ay, = 0, dy, = 2xdz, d’y, = 2dx’ hence Du = — sin (@ + 2°). (da? + 4a dx’ + 42° dx”) + 2 cos (x + x*) dz’, LE. “3 =~ sin (2 + 2°). (1 + 4% + 4x”) + 2 cos (x + 2”). We might of course have obtained this result by first giving y, and y, their values in the expression for « and then differen- tiating : thus = sin (24+ °2"), - = cos (4 +-47) (1 + 22), at dx’ We may remark that, when wis expressed entirely in z, as in the latter method of differentiating w, the expressions =—sin (4 + 2’). (1 + 2x)’ + 2 cos (x + 2°). du du dz” da” are equivalent to Du Du. ii ie Ee whereas, when we put w=sin(y, + ¥,)5 du du on and apa? Oe both zero, being in fact the first and second partial differential coefficients of « with respect to z, a letter not appearing in sin (y, + ¥,). Successive Differentiation of an implicit Function of a single Variable. 50. Suppose that v= (2, y) = 0, y being thus an implicit function of z. 06 SUCCESSIVE DIFFERENTIATION. By Art. (22), we have DON Gee LRRD Teel Cte an dz dz dy” dix Differentiating again, z being considered the independent variable, we have, wee: now taking the place of », dx Dy D/(d +2(2). dy ,mw@D dy\ _ dé de\de)” dz\dy)° dz” dy de \dz} | But, by Art. (21), Cor. D(do\_ da (do +2(2)¢ dz\dz} dzx\de) dy\dzx} dz ited Ome Gf ES METRE GE d’o d’o dy Sere BPI cine: ne z(a)" d (Z)+5 (s) dy dz \dy} dx\dy)° dy \dy/ dz d’v rel v dy ~ dady " dy? dx Also de (2). ay de\dz}) dx’ Do dv Ati Uys Cop. apa: Hence Fv 2 dy de® i ig ate (2), oer. d*v G0 Gee or Do = i dx* + aa aie f iy dy. From (1) we may get a in terms of the first order of partial dx differential coefficients of v, and therefore, from (2), we may get a in terms of the first and second order of these coefh- las cients. The partial differential coefficients of » may be obtained : dy @& ‘ in terms of x and y: hence oe : a may be found in terms a ax of these two letters. We might proceed in the same way, by SUCCESSIVE DIFFERENTIATION. OT successive differentiation, to determine the third, fourth, fifth, &c. differential coefficients of y. The principles of this article the student will have no difficulty in extending to the succes- sive differentiation of the system of equations considered in Art. (28). Successwe Total Differentials. 51. Letu=f(y,, y,), y, and y, being independent variables: then, by Art. (24), Ey ice Oy Uierarclesie ier farts weer )e Differentiating again we have Dj= Ddg i HI Dd yh. veces erertis sllele cs (2). But, d,,w being a function of y,, y,, and a constant dy,, we have, by virtue of (1), dy, occupying the place of u, Du = dy, dy + Aydyu = d'yu + dy dyu. Similarly Day = Ayub + dy,dyt. Hence, from (2), Deu = dy + 2Wdy dy, U + Eyl. Proceeding in the same way, we easily see that Du = d*yu + 8 d’ydyu + 8 dy,duu + A yt, and so on, the law of the symbols of differentiation correspond- ing to the development of the binomial theorem: thus n(n —1) n : Du=d"v,u+ ; dy,” dy dy”? But +004 Ny, dy,” U+ dy," This relation may be expressed symbolically, thus : Du = (dy, + dy)” U. Cor. If 2 =f (Y,5 Yo» Yyo+++¥,,)) 1b may be proved in a similar way, viz. by induction, that Dot = (dy, + Uy, + Aygo. ce ot dy)” U. We may however establish this proposition by the following reasoning. By Art. (24), Du = (dy, + dig + Uys +. ovens + dy) U3 08 SUCCESSIVE DIFFERENTIATION. which shews that the symbol of operation D is equal to the sum of the operative symbols d,,, dy, dys... .d,,,; hence y1? ym Deu = D (dy, + dy, + dig +....+ d,) u = (dy, + dy, + dy, ++..dy,) (dy, + An, + dy, +...+ dy) U...(3). But the symbols dy,, dv, du, .... dy, are subject eter se to all the laws of combination which belong to symbols of quantity: thus for instance dy dy, a dy,dy,, dy, dy, = dy, dy, ae dy, == dy, + dy, 5 hence the product of the operative polynomials in (3) is equi- valent to (dv, + du, + dugt....t+ dy): proceeding successively to the higher orders to total differentials, we get the general formula Diu = (dy, + Ai, + dug +... 1+ Gy, YU. Successive Differentiation of an explicit Function of three Variables one of which rs a Function of the other two. 52. Suppose that i= J (an Ye); z being a function of z and y, two independent variables. Then, by Art. (25), Du _du du dz de a> A Pa PO Du du du dz apes 2B i PA OM oper Se (2) Differentiating (1) with regard to z, we have Du D (du, D {(du\ dz du D (ad wm le) tale) ete zz) -etess oe (3). But, Cite : aR being a function of z, y, z, we have, by virtue of (1); d. putting a in place of w, “aes du- du dz- dz \dz) dx” dzdx’ dx’ D (=) _ du ‘dude imilar] a Ee similiar Vi dz dz dadz dz dx ‘ SUCCESSIVE DIFFERENTIATION. 59 Also, since z is a function of z and y alone, and since the expression D (dz az (ae): ; f : dz : denotes the total differential coefficient of ig a function of zx x and y, only so far as the variation of 2 is affected by the varia- Zz tion of z when y remains constant, it is plain that D (z : dz dz \de) dx’ 2 Zz. neta é where ie 38 the second partial differential coefficient of z with es regard to z. Hence, from (3), dn adr. dzdz dz dz dx’ dz dx’ In like manner, from (2) there is dy’ dy’ dydz dy dz dy’ dz dy’ Again, from (2), Dui D (=z) ? D (=) dz r du D (=) dady dz dy) dx\dz})° dy dz’ dx \dy But, from (1), putting i f , successively for w, D (du _ du . d'u dz dz \dy) dazdy dzdy dx’ Dia sae ie dx\dz} dadz dz dx hence Du @u du dz Wu dz Wu dz dz du dz dedy dedy* dydz’ da dedz’ dy d2 dz’ dy dz’ dady We have therefore determined formule for the values of Du Du Da 60 SUCCESSIVE DIFFERENTIATION. we might proceed to find the differential coefficients of higher orders by a continuation of precisely the same kind of processes. Cor.: Suppose that» w= f(z, y,2)= 0; there being no other equation connecting z andy: z will thus be a function of z and y. ‘Then, from the equations of Art. (27), IB TER OR HAS rr Es —=—+—.— =), di Gar dz dz dy dy dz dy we shall obtain, by the simple repetition of the preceding reasonings, Dui du PETE GER BVT ieee WILT BERS? dé da” dedz' de dé da’ dz dx dy? dy dydz dy dz dy dz dy dzdy dzxdy dydz dx dzxdz dy dz’ dx dy dz dxdy From these five equations we can determine TERA Ae tee ified Site dz’ dy ? dv ? dady ’ dy’ the partial differential coefficients of the implicit function z, in terms of the partial differential coefficients of «, and therefore in terms of the variables z, y, 2. ea. Change of Variables. 53. Let it be proposed to change the variables of an equation dy dy dy ei fav, ys dx? Toh Boas oth Uy ae are (1) from x and y into two variables s and ¢, ¢ being, in the trans- formed equation, and 2 in the proposed equation, the inde- pendent variable. We suppose s and ¢ to be connected with z and y, which by virtue of the equation (1) are functional of each other, by two equations $ (&; Y, 8 t) = 0) W(a,y, 5,0) = 0) SUCCESSIVE DIFFERENTIATION. 61 Differentiating these equations successively m times each, d”y da” in (1), and regarding z, y, s, as implicit functions of ¢, we shall get 22 equations which we will denote by supposing to be the differential coefficient of highest order p” = OP yr" = 0. From these equations, in conjunction with the equations (2), we may obtain expressions for the 27 +2 quantities OD 02 -0°e d"x ay ay dry d'"y Men det aa eae ar de? are ds d's d's d"s GL edie Cl iedcas But, by Art. (47), we are enabled to obtain expressions for dys Gy .@y 4 ey dx’? dx” dz’? da’ in terms of Soa. in terms of de ras. ar ar ay dy d’y d'"y Te EES ORE LIE ETI EY a Listenin MORN Le Hence we are able to obtain expressions for y and its n differential coefficients with regard to z, in terms of s, ¢, and the n differential coefficients of s with regard tot. The equation (1) may therefore be transformed by substitution into an equivalent gine pale ds. d's d's d"s mcs ander eg ar Transformation of one system of dependent Variables into another. 54. Let z be a function of two independent variables x and y. We propose to express the partial differential coef- 62 SUCCESSIVE DIFFERENTIATION. ficients of z, taken with regard to z and y, in terms of those of another function 7, taken with regard to two other inde- pendent variables 6 and ¢. The six variables z, y, z, 7, 0, $, are supposed to be connected together by three equations. ERD, Ye, 08D, TU FF (@, y, 2, 9, b,.7) = | Fi (2; Ys &s 0, d, r)= 0 any four of the six variables, since z is by the supposition some function of z and y, being thus functions of the two remaining ones, which will be entirely arbitrary. It is evident that r may be regarded either as a function of x and y alone, or as a function of @ and @ alone, 0 and d being in the latter case regarded each of them as a function of x and y alone. Hence we see that, first considering y and next x as constant, dr dr de ars add | dx dO d eran dax | ide etna cee dy dO dy db” dy ) are dy) ae Vde dp d Mey. ; oe iy ie dy’ and es : o : being the partial differential coefficients of r, 0, and ¢, with regard to x and y, when vr, 8, and ¢, are expressed entirely in terms of 2 and y; - oe 5° being the partial differential coefficients of 7, with regard to @ and ¢, when 7 is expressed entirely in terms of 6 and ¢. Again, from the equations (1), we have, y being considered constant, DE dk dO oF db . dF dr ak | dF dz ‘dz dO dz 2 db da. dr de’ de dz de DES s dF, do aes as INE aa Es, (3) dx dé d dbs dae warwags -da > dz da aki as DF,_dF,d0 dF, db dk,dr dF, dF, de de dO dx dp dx dr de dx dz de SUCCESSIVE DIFFERENTIATION. 63 Considering z constant, we have from (1), DF _ dF dé _ ak dpb dk dr dF dF dz ‘dy dO dy db dy dr dy dy de dy DF, dF, d0 dF. db dF dr dF, dF dz Pea inded, © Hie a Were di kdsidy DF, _ dk, d0 dk, dd dk, dr | dF, , dk, d dy dd dy db dy dr dy dy dz dy From the former of the equations (2) combined with (3), we may determine a db dr de dx’ dx’? dx? dx’ in terms of il be el dO? db and, from the latter of equations (2) combined with (4), we may find dd db dr dz dy° dy” dy” dy’ dr dr also in terms of io’ db The above conclusions enable us to transform an equation dee Ue es ee he ()3 f(z Y; zy dz > Z| into an equivalent one ara ar 0 a S, ( 3 dp, r,s dé ’ ie Proceeding to the second order of partial differentiation we shall have, from (2), expressions for d’r dr d*r DEE UGB eg Fe draw, ee, dine day: d’r Gy Bod 1009100 ADan aD « wd co ao Wwe da dy dx’ dx dy” dy" ip dp de a6 Ws da? dy’? dx’ dzdy’ di’ 0. in terms of 64 SUCCESSIVE DIFFERENTIATION. From (8) and (4) we shall have nine equations, 2 2 2 Lee oa sole thes DF, _ 0, dx’ dx dx DF DF DF = 1<= 2 = 0, Remi Ui 9 dz dy ee dy Se rE: dy @) 2 2 2 DF : DF. et DF, ay week Sy Uh, ; dy’ dy” dy’ Hee 2 ERY tikes d*z d’z involving inate. daa da) eas ih ORE Dai s BS ola dz’ dy’ dx’ dxdy’ df’ dd dO dd add dO dx’? dy’ dz’ dxdy’ dy’’ ip db db a we dus dy Gada ary aaa The three equations for A Ce ge , obtained from (2), AEAY EER together with the equations (5), twelve equations in all, will enable us to express the quantities d*z d’z d*z dx’ dx dy’ dy?’ d’r Tee i. dx?’ dxdy’ dy’ rao da’ dgdy. dy gah Ne 5 cls dat da dy gy dt OEM rata 1785.1 Ga do’ db’ d&’ ddd’ dd’ dz dz dr .dr d0 a0. db do dx’ dy’ dz’ dy’ dx’ dy’ dz’ dy’ in terms of and SUCCESSIVE DIFFERENTIATION. 65 the last eight of these quantities being, as we have before shewn, expressible in terms of dr dr dO’ db From the above conclusions we are enabled to transform an equation Gi 2 es dz dz a2 We dz = (0) Ys &s AES dy’ dx ’ dx dy’ dy? mare, into an equivalent one ar art. af. d*r d*r 0, EU Niayiwh Goa ce Sepp Pe Seppe aera ee vA $31 76? db’? de? d0dd ae We might proceed, by a continuation of the process of suc- cessive differentiation, to the transformation of partial differential equations in z, y, 2, of any order whatever to equivalent ones in 0,¢,7. It is easily seen also that the same method of transformation may be extended to differential equations involy- ing any number of independent variables whatever. Ex. Transform the differential equation d*z ‘ dz dz* dy where z and y are the independent variables, into one in which @ and ¢ shall be independent variables, having given that 0, Ne ay Mes y =r sin 0. We have, considering y constant, dz dz de dz dr dx d0 dx dr dx ; . ee But, since ey to. and. / — tan 0, x , IIE there is sects C08) 0, diz 7 dO ’ 0 and =, sec’ 0 =- 4, LLG 66 SUCCESSIVE DIFFERENTIATION. dz dz hence ee POS i dx pth d: — a dx r AG de. aes. ae "7? de do dz 0 ( dd dz =| ’ e dé SURI hee OS x” ar d*z 4 a0 “a2 | 2y dr dz FE d0 dr de dr dx Wh dary d’z 2y cos @ dz TP btlG Hatake Seem ride |e? Romar y sin 0 dz pay die d’z 2 7+ cos 0 S ao gp + 008 9 Ga sin’ @ i 2sin@cosO0 dz 2 sin 6 cos 8 dz RO le 7 dr do * r d@ sin’ 0 dz soisaaeg d*z r adr dr? Putting $a -@ for @, the expression for = will be changed dx 2 2 into that for diz : hence dy” d’z _ cos’ 0 d’z , 2 sin 8 cos @ d*z 2sin@cos@ dz TEP eG pares he r dr dO - dé CosH a2. mee ane nt gp 0 Ta? and therefore diz dz 1 da thle (he de dy dh r dr dr a a, CEE AS Ey EEL ve. ELIMINATION OF CONSTANTS AND FUNCTIONS. Elimination of Constants. 55. Let the equation DUET oot oh a aie th oad i eRe (1) involve ” constants together with two variables z andy. If we differentiate this equation successively times, we shall obtain n differential equations Du=0, Du=0, Du=0,....D%u=0, involving the m constants, the variables z, y, and the 2n dif- ferentials Py eae a Bae Mae d’z, CEERI R OIE Bie he d”y. If we suppose dz to be constant, then dz, d*°x....d"x, will disappear from the equations. We shall thus have 2 + 1 equa- tions involving ” constants; and therefore, eliminating the con- stants, we shall arrive at a differential equation of the x order in regard to the differentials of z and y, or, if x be the inde- pendent variable, of the »‘” order in regard to the differentials of y. Ex. Let (c-ayt+(y- bf=ae’, a, 6, c, being constants: then (c-a) dx +(y- 6) dy=9, (cx -a)d’x+(y - 6b) d’y + dz’ + dy’ = 9, (2 - a) d°x + (y — b) d°’y + 3dxd*x + 3dy d*y =0. From the first two of these differential equations there is _ dy (dx? + dy’) _ dx (dy’ + dz’) dx d*y—dyd*x’ — dyd’x~ dxd’y’ KY y-b Ae LES 68 ELIMINATION OF CONSTANTS AND FUNCTIONS. and therefore, from the third, (da° + dy’) (d’xdy — dz d*y) + 8 (dx d*x+ dy d’y) (dxd’y - d’x dy)=0. If z be taken as the independent variable, ‘ d’x=0, d*x=0, and the result is reduced to — (dx? + dy’) dz d’y + 3 dx dy (d’yy = 0, dy’\ d°y dy (d’y\ . 1 SRA ete ApS 8 oe —— = bi ( i a) dx’ dx Zh) : We may also express the constants a, 6, c, in terms of the second differentials of the variables. In fact _ dy (dz + dy’) i dx (dy* + dz’) da dy — dy @x’ "dy Pa ded’y’ ate gs ee AY) © * (dx d’y — dy dx) and Partial elimination of the Constants. 56. Instead of differentiating the equation dea) 1S ete eects ee ee n times, suppose that we differentiate it successively only m times, m being some number less than 2. ‘Then we shall have m + 1 equations involving ” constants: we may between these equa- tions eliminate m constants, and shall thus obtain an equation of the m order of differentials containing » — m arbitrary con- stants. Since the m constants which we eliminate may be chosen arbitrary, it is evident that we may form as many equa- tions of the order m, containing ” — m constants, as there are combinations of 7 things taken m at a time: we may therefore obtain of such equations a number n(n —1)(n-2)....(%7-m+1) 180 een nr ae If we differentiate any one of these differential equations 2 — m times in succession, we shall have altogether m - m+ 1 differen- ELIMINATION OF CONSTANTS AND FUNCTIONS. 69 tial equations involving x — m constants. By the elimination of these » — m constants we shall arrive at a differential equation of the vn order. It is important to remark that this final differential equation will always coincide identically with that obtained by the process of Art. (55). That such must be the case will be evident when it is considered that neither method of elimination involves any limitation of the generality of the variables and their differentials, and that accordingly the results must in both cases be perfectly general. Ex. Let (2-a)+(y-by=ec’: then (= Gade EO) y= Ore. tease ws Ci); (x —- a) d’x+(y —- 6b) d’y + dv’ + dy’ =0.... (2). Eliminating a between (1) and (2), we have (y — b) (dxd’y — d’xdy) + dx (dx’ + dy’) = 0....(8). Differentiating (3), we have (y — b)(dxd’y — d’xdy) + dy (dad’y — d’xdy) + d’x (dx? + dy’) + 2dz (dxd*x + dyd’y) = 0, or (y — 6) (dxd’y — @xdy) + 8dx (ded*x + dyd’y) = 0. .(4). Eliminating 5 between (8) and (4), we get (da* + dy’) (d’ady — dad’y) + 3 (dad'x + dyd’y) (dud’y — @ady) = 0, a result coinciding with that obtamed by the method of Art. (55). Elimination of irrational, logarithmic, exponential, and circular Functions of known Functions. ov. Let TPES AES OER oy Aoi Ad ia At be an equation between two variables z and y; where ¢,, ¢,, c,, ....¢€,, are ” irrational, logarithmic, exponential, or circular Minetions Of S|, s,, 8.,...,$,, respectively, 8, , 8), %5....8,, being known functions of z and y. If we differentiate this equation successively m times, we shall obtain » differential equations tie a el) 0,4) Du = Oyo ae Dy = 0, 70 ELIMINATION OF CONSTANTS AND FUNCTIONS. involving the differentials of 2 and y up to the n™ order,-together with the 2” expressions dG) OC UAC, d"c, d8.2 AE ae mee a a 2 3 p) Ca tral , .) 92 2g ge es @ @ = > ds, ds, ds, ds, 2 p G6. 0 CMa C, Gs cy EI she Py 8 8 ae FORE See el ae ay ae ds, ds, ds, ds, doe A Cenc at n n n n ee 5) “> 9 98 eee . 8) nS Sm se ist n n n Now the first rank of these expressions can be expressed in terms of c,, the second in terms of c,, the third in terms of ¢,, and so on: hence we shall have, in all, 2+ 1 equations containing the m functions ¢,, ¢,, ¢,,....¢,. We may therefore eliminate these functions, and shall thus arrive at a differential equation of the m™ order in x and y. Ex. 1. To eliminate the functions te) G) ene where m and nv may be supposed to denote any numerical fractions. Differentiating (1) we obtain, dz being considered constant, from the equation dy m n At a pee ES | _- atl dz - qa” L uy a” Mies ¥ dy _m(m—-1) n,,nm-1) 4.2 dx us q™ & f qQ” x é da xr m xv n whence BP isnt VEN Reinet enea ae paee ee Ve) eked dz a a fa'9 Nail a a aa mim — (3) +n(n- v(Z orem ay) = oe eee a 4 S.centtian eat 5 “KLIMINATION OF CONSTANTS AND FUNCTIONS. Tt From (1) and (2) we have ny — ot —~ =(n — m) (y's a also, from (1) and (3), ‘ d’y a\m n(n — yt 74> {n(n -1)-m(m-—1)} (=) =(n —-m)(m+n- 1) (ey a from these last two equations we get d’y dy, n(n-l)y— 2 a =(m+n- 1) (my - ot), , 2 dy ai x dy = 0, or mny —-(m+n—1)2 dan ) dx dx* a differential equation of the second order. Ex. 2. To eliminate log x from the equation pee ENG aa Differentiating, we have dy i log x + 1, whence Z o = 210M ee et ce Ex. 3. To eliminate sin (x + y) from the equation y =sin (& + Y). Putting ae = y', for the sake of brevity, we get, by differen- tiation, y =cos(«+y).(1+y), . Obs (+yy But, from the proposed equation, y” = sin(e+ y): hence + pappeaiee Fro tae Classy") yd+yyryr=At+yy. "COS. (2) 1. 7). or 72 ELIMINATION OF CONSTANTS AND FUNCTIONS. Ex. 4. To eliminate the exponential functions from the equation ae’ + be” = fe + ge™*. Differentiating we get (ae’ — be’) y' = fe" - ge’, (ae’ — be”) y' + (ae + be”) y” = fe + ge’; whence (fe*’-ge*)y' +y" (fet+ge*)=y' (fe + ge”) ... (1). Differentiating (1), (fe + gery" + (fe- ger) y" + yy (fe + ge") + y" (fe — ge*) =(fe’- ge*)y +(fe+ge)y, or ayy" (fe + ge*)=(y' — y° — ¥") (fe - ge*): but, from (1), y' (fe - ge*)=(y'- y") (fe + ge*): multiplying together these last two equations, we get mt 3y'y" = (1-y”) (y’- y®-y”). Elimination of an arbitrary Function of a known Function. 58. Suppose that u=(v), where w, v, are known functions of three variables xz, y, z, and ¢(v) a perfectly arbitrary function of v. Differentiating this equation, first with regard to z and next with regard to y, we have Du , Dov Du De FE ah eae ia ae Du De_ Du Dr dz dy dy dx’ dx dz dz} \dy dz dy _ (dt) (do do dy dz dy)’\dx dz dx)’ and therefore owe ELIMINATION OF CONSTANTS AND FUNCTIONS. 73 eens aDTIRenD Rane diena denaien de dz dy Foie) dz dz dx dz} dy _du do du de ~ dy dx dx dy” Thus we see that, although the equation w = ¢ (v) varies in an infinite number of ways, with the variation in the form of the function ¢ (v) of v, yet that all this family of equations possesses one common partial differential equation. Ex. Eliminate the arbitrary function from the equation Differentiating, first with regard to z and then with regard to y, we have a Res ag ec Ag cl Gb) dz. 2—a)- dz ooo ,{2-c¢)\ - 9 (Z*). aS -{@ - a) Z - @- 0}, 1 a a)'2-a’ dy 1 dz Plz and eee (; Eliminating ¢' (== "| between these two equations, we see —a that z-a dz dz 1 if dz __¢)| y-b dx’ dy ~ (y— by by \y eile GEO} {@-9 Fe f Js whence (@-a) E+ y-B)En2-e 59. Ifthe arbitrary function ¢(v) involves y only, then it is sufficient to differentiate with respect to z: in fact Du Le me du F du dz a dz dz dz ELIMINATION OF CONSTANTS AND FUNCTIONS. 74 the arbitrary function being thus eliminated by one differen- tiation. Ex. Let z= zy ¢(y), whence z med p(y): then a r&_2)\=0, Brea ap OD Fn dx Elimination of any number of arbitrary Functions of known Functions. 60. Let there be an equation u =f 1, Y> &% P, (¢,), ?, (¢,), p, (¢,), eps ee involving m arbitrary functions , (¢,), , (¢,), p, (C5), Tihs See a ?,, (c,,)s c,, are all known functions of z, y, z. ?,, (c,)} = 0, WHELe cach 0. caer nine If we differentiate the proposed equation ” times we shall have, in all, the following equations, = 0), dz dye Du _ Du _ Du | A dx’ Uda emery men oe Du Du Du Du SETS Se Sere a 0, 727. 9) rma Six temo dx® dxdy dxdy™ dy Du Du . Du A 6 Du ” 0 Du au i Du A deta! i ae dy” "<3 dadyP © + ae dx’ da*'dy the number of which is 1 oe Bs Matas em eas +(m+1)=$(m +1) (m+ 2). or ELIMINATION OF CONSTANTS AND FUNCTIONS. These equations will involve the following functions, $,(¢,), b{C,), (C5), APE ee ,(C,,)s $,(¢,), b, (¢,), $, (¢,); sin aiso eae Piles), De (Cyan i (Cs. Geen Baten, P,"(¢,)5 ?," (¢,)s P,(C,) th ge Ptacbaly ed one.) the number of which is m(m +1). We have therefore m (nm + 1) functions and (+ 1) (m+ 2) equations: in order to eliminate these functions it is sufficient that S(m7+1)(m+2)>m(n + 1), m+2> 2m, n> 2m - 2, or m= 2m — 1. The number of quantities to be eliminated will therefore be m(n+1)= 2m’; and the number of equations involving them d(n +1) (2+ 2)=3.2m(2m4+ 1) = 2m? +m: we shall therefore arrive at, as the result of our elimination, m partial differential equations between the variables z, y, z, of the order 2m — 1. If m=1, then 2m-1=1; if m=2, then 2m—1=3; ifm=83, then 2m-1= 5, and soon. That is, if there be one arbitrary function in the.proposed equation, there will be one final equa- tion of the first order; if two functions, two final equations of the third order; if three functions, three final equations of the fifth order, and so on. This is the general theory of such eliminations: it frequently happens however, for particular forms of the proposed equation, that the elimination may be effected without proceeding to so high an order of differentiation, and arriving at so many final equations, as would be implied by these general considerations. So that we must consider the general theory as defining the number of sufficient, but not in all cases the number of neces- sary differentiations. 76 ELIMINATION OF CONSTANTS AND FUNCTIONS. Ex. 1. Eliminate the arbitrary functions from the equation z=2£o(2)+y x). Putting ? = = Oz), a x2) _ v,(2), we have 1 = hz) + rio hence O= {@b,(2) + 9x ae + Pl2), Us {xo, (z) ar YX (2 9 ah a X(2) : and therefore, putting $2) = x,(z) .f oe dz da _ fo. dy differentiating this equation, first with regard to x and next with regard to y, we get de de de Wz _ dz (dz dx’ dy da dxdy I@). da (%)> dz dz dz Wz _ dz dady “dy dx dy I). € ar eliminating f(z), we have a ee Age i hee fey dy} dx dz dy dxdy \dz} dy? Thus we see that, mstead of two final equations of the third order, we have a single equation of the second order. Ex. 2. Suppose that z= (et+y)+ vy) (@-y). ‘The elimination of the arbitrary functions in this case can be effected only by proceeding to partial differentials of the third order, there being two final equations of this order, a result in harmony with the general theory which has been laid down. For the discussion of this example the student is referred to Lacroix, Traité du Calcul Différentiel, tom. 1. p. 284. ELIMINATION OF CONSTANTS AND FUNCTIONS. qi; Ix. 3. To eliminate the arbitrary functions from the equation m oa aS y" a te y cae (3 ~ ,) ed (3 rH): ‘The two functions are readily reduced to one: thus ’ 2 = a function of 4 : x U! a function of dy, x hence | z= (2) ’ g@ being a symbol of arbitrary functionality. Differentiating, first with regard to 2 and then with regard to y, we have dz ¢(4).4, dz ee and therefore Z—+yYy This result would have been obtained by differentiating the proposed equation, without modification: the operation would however have been more tedious. Elimination of arbitrary Functions of unknown Functions. 61. Suppose that we have two equations u =f {2; Ys &, Cy p(e), x(C),. woes sy = 0, ad OAL Page TCA iy ay ewan oP Nee Os e and z being therefore implicit functions of x and y. The minetions p(¢); x (C), 2... are supposed to be m arbitrary 78 ELIMINATION OF CONSTANTS AND FUNCTIONS. functions of c; whence it follows that ¢ is an arbitrary function of z and y: for, supposing z to be eliminated between the two equations, we shall obtain an equation between z, y, c, of which the form is arbitrary. We propose to eliminate by differen- tiation the m arbitrary functions of ¢ and the function c itself. If we differentiate the proposed equations ” times succes- sively, we shall obtain the following equations, clheall Ph Sa st 0 | dx dy ie De Eh ote 0| dg) 2 dy 6 2 2 2 pea ee A dx dady dy’ oe ae Y ; fA = 0, De = 0, Do = 0 dx dxdy dy’ j 3 3, 3, p ge iis eee Pty ms Sa aM oe) dz dx*dy dady ay DI) Ry BN GeV DO a es ties Oke da Met) dedg lan Cin, away oe) the whole number of these equations, together with the two original equations, being 2{1+24+3+....+(@4+1)} =(m+1) (m+ 2). These (7 + 1) (2 + 2) equations will involve the quantities C, dc de d*c d’c d*e cn d°c dc d°c d°c d"c d"c dc dc dc de AA gee eer ore er ear mK Ire a 727, dx dz'dy — dxz"*dy dx dy n-2 ? n-1 9 dady dy"’ ELIMINATION OF CONSTANTS AND FUNCTIONS. 9 the number of which is [GSA ie aot i a ae +(n+1)=3(%+ 1) (m+ 2), and also the functions PC), PECL APi(e), te oh p"(¢), X(, XO) XO ecaes x'(e)s of which the number is m (7 + 1). Thus we shall have (7 + 1) (7 + 2) equations, between which it is proposed to eliminate quantities of which the number is equal to Tm+1)(n+2)+m(n+1): for this purpose it is sufficient that (2+ 1)(n+2)>$(m+1)™@+2)4+m(n+ 1), L(in+1)(m+2)>m(n+1), (n+ 2)>m, n>2m-2; or that m= 2m —1. We shall have then, for the number of the equations, (m7 +1) (2+ 2) = 2m (2m + 1), and, for the number of the quantities to be eliminated, lin+1)(m+2)+m(n4+1)=m (2m+4+1)4+ 2m’ = 4m? +m. It follows that, when the elimination is completed, we shall arrive at m partial differential equations, of the (2m — 1) order, which will be satisfied by all the equations which are compre- hended under the general forms of the two proposed equations. Tt frequently happens however that the order of the partial differential equations necessary for the elimination of the m+ 1 Puantities c, p(¢)) x¥.(¢),.....- is lower than the (2m —.1). Suppose for example that there are three arbitrary functions d(c), x(c), P(e): in this case m = 3, n = 2m —-1=5: we see 80 ELIMINATION OF CONSTANTS AND FUNCTIONS. then that generally to effect the proposed elimination we should have to proceed as far as the partial differentials of z of the fifth order, and should arrive at three partial differential equa- tions of this order, between z, y, z. But if we establish the Bia x(0) = $(¢); $0) = $), that is, if the equations are of the form SF {x y, 2, ¢, Ke), P(e), P(e) = 0; F {x, y, 2, ¢, (ce), P(e), P()} = 0: then, if we proceed as far as differentials of the second order, we shall have, in all, twelve equations, between which we may eliminate the eleven quantities ,, 0 do d'c de ae ° dz? dy? dx” dxdy’ dy’ P(e), d(C), P(e), P(e), P(e), the result being a single equation between z, y, z, involving partial differentials of z only to the second order. Ex. Eliminate the arbitrary functions from the equations LD {C) to 6) ee Gielen eran as een Lye EDC) NIC) en (6) ime) yt me te ‘ens Differentiating (1), with respect to z, we have {ed +y XO +2 ¥O}S +$O+Z¥O=0, whence, by the aid of (2), dz \ —_- = d(c) + Te UC) = One Ca ee (8). Similarly, differentiating (1) with respect to y, dz v(e) + Ty LAG) s Aaa eters < (4). From (3) and (4), putting PO _ £(0), XD - Fr), (c) Lc) ELIMINATION OF CONSTANTS AND FUNCTIONS. ! 81 dz we have —= av FAG + ey (Ver etiaica rakes att si Mee pala ty (5), F'(c) + i EUS Rs re Oe My Sore Fala kum Pa (6) From (5), we get So) a + ne = 0, xr de d*z 2 2 BAt therefore Ge Mill alas a Ca aC AR dy dz’ dx dzdy In like manner, from (6), de dz de dz dz dy dy dzxdy ie tale ean From (7) and (8), we get, as our final result, dx” dy’ \dady/)~ Elimination of arbitrary Functions when the number of independent Variables exceeds two. 62. In the preceding considerations respecting the elimi- nation of arbitrary functions, we have always supposed that there are only two independent variables. We will now pro- ceed to develop the theory of elimination when the number of independent variables is any whatever. For the sake of simplicity we shall confine ourselves to the case where it is not necessary to proceed to partial differentials beyond the first order. Let FARCE Eee Bee UC ees ME ee won.» being (m + 1) independent variables, and w the dependent variable. The quantity c is supposed to be an arbitrary function of a, 6, y,.... which are m known functions Gey; 2......and % §2 ELIMINATION OF CONSTANTS AND FUNCTIONS. Taking partial differentials with regard to z, y, z,...... in succession, we get Df . of df du df de da | dc dp , de dy de de’ du dx de\da d * dp dx dy dx Df ie df du. df da. de dB dc dy dy dy" du dy de \da dy ” Wp dy | dy dy Df of _Fu Of de da , de dB, de dy dz dz du dz de\da dz * dp dz dy dz Il Oo v Taking these differential equations conjointly with the pro- posed equation, we shall have altogether m+ 2 equations in- volving the m + 1 arbitrary functions de de de Se. CIE ER These arbitrary functions may therefore be elimimated, when we shall arrive at a partial differential equation of the first order of partial differentials. C, Ex. To eliminate the arbitrary function from the equation Cae ES ans (e, =). x“ tA beg | Clade Putting tat B, we have (m—1)u+ = = 2" db(a, 9), whence (m-1)— +2 = ma" $ (a, B) | va 1 $a, B). E45, (a, B). =I, ELIMINATION OF CONSTANTS AND FUNCTIONS. ees Multiplying these differential equations in order by 2, y, z, and adding, we get, attending to the proposed equation, ae du du ~ du\ zy % (m= 1(2 Bey Gress ne $ (a, 3) | =m(m-1)u+ 4, Pence la GT RAN aed 4 dz 4 ly dz a . cv Eat) CHAPTER V. EVALUATION OF INDETERMINATE FUNCTIONS. Indeterminateness of explicit Functions of a single Variable. _f@) 63. Suppose that d (x) = F@)’ and that, when a particular value z, is assigned to z, f(x) and F(z) both become zero: the value of ¢(x) will, for such a value of z, present itself under the indeterminate form 9. We proceed to investigate a rule which is often useful for the deter- mination of the true value of $(z,). We have generally Se J (xz) + 6f (2) PC Oe) ar RS TCA" and, for the particular value of z, of z, of(@) _ Ff @)_ _oe p (x, + Ox,) dF (2) ae SF (2)’ ox z and 6z being supposed to be replaced by z, and 6z, in the expressions for of(z) $F (zx) BLN s Bor vie which are generally functions of z and dz. Passing to the limit, when 62, becomes less than any assignable quantity, we have _ f() $=, it being supposed that, in the expressions for the functions f(z) EVALUATION OF INDETERMINATE FUNCTIONS. 85 and #" (x), z, is substituted for z. In other words, for any value of z which makes f(z) = 0 and F(z) = 0, the value of i) F(x)’ I @) F(x) If, for this same value of xz, f(z) = 0 and F(z) = 0; then, by the application of the same principle, we see that, for this is ae same as that of value of z, S@) _f@ F@) F(z)’ 4 fe) _f@) F(z) F(z)’ and soon. If f*(x) and F%«) are the lowest differential coefh- cients of f(x) and F(x), of which both do not vanish for the particular value z, of z, then the true value of $(z,) will be Sd) F(x,)’ an expression representing the value of S'@) » when z = %,. F(z) £) Ex. 1. To find the value of P(e) = we when z = 0. Here Fa@ae-—C, p(@) = 4 hence f () = log a . a® - log b . b’ = log a — log 6, when z = 0: f(z) =1: nO | a thus Fo) = log ae or $(0) = log + 86 EVALUATION OF INDETERMINATE FUNCTIONS. Ex. 2. To find the value of $(x) = — » Whe = 0. Here f(@) =x -sin z, GS eer hence FA2) ple Cos 0; dk CY eh eA UE: Jf (= sine o= 0; E(@)= .6e =0: of AL) =e COS Gea = 1s Be a) nO2 Hence (0) rare = , ; Ex. 3. To find the value of o(z)=(1 - #) tan = ; when 7 = 1. Putting the equation under the form (1 —- x) sin = $(2)- ———, cos — 2 we see that I (#) = (1 - x) sin - : F(x) = cos ue hence f(a) Sain See (is Bice = = & 1; 2 2 2 _ — Le oe — LS Ta) sin = a | Ly TEN ed and therefore p(1) = FQ) = > EVALUATION OF INDETERMINATE FUNCTIONS. 87 It would have been however more simple, observing that eg sin a =1 , when z = 1, to have sought the value of 1-2z p(x) = meets COs o eae (1 - #) sin instead of b(2) = COS oh We should then have I (@) =l1-42, F(x) = cos hia 2 F(2) =— 1, py eran en F(z) 5 sin | re and, as before, (1) = aa T Ex. 4. To evaluate (x) = (a - 2). cot {z (< = = , when z = a. a+2 By an obvious transformation we have sores 6 (C25)) ern yan ae (77 aa ~ Sie eee AM = (2a). 2) ; when w= a. leer Put J (a) = (a - xh, F(z) = sin " (< =a f(x) and F(z) being both zero when z = a. 88 . EVALUATION OF INDETERMINATE FUNCTIONS, Then, when z = a, J@)- ie; 3 Gai (a+2).(a-2) 2(2a) (a-2) and ate rol S(@) _ £@ _¢ 2a)" and b(x) = (2a). ae a The value of ¢(z) may be obtained also, and in fact more simply, without the aid of differentials: thus = (e-) (2) = = (a+ 2) o0s {7 (SN . 2\a+2 2 ata] ‘ x /a—xz\) 2\a+2 but, when z = a, ides) 2\a+2 age eee < g? When 9 = 0 Sinn \2 =) 2, nlogb jp," etd Sages by Art. (64), Ex. 3. CO We shall fall imto the same difficulty at each succeeding operation. In fact, the method of differentiation must not be considered as a universal rule for evaluating indeterminate functions, but merely as an instrument frequently of great use for this purpose. Evaluation of Indeterminate Functions of several Independent Variables. 66. Suppose that, to take the case of two independent variables : ya SI (2; Y) 79 p (2, Y) ae F(z, y) at 0 ? when x, y, receive respectively particular values z,, y,; the variables 2 and y being subject to no equation of connection. Generally Lf, 9) + APG, y) (a + oz, Pat esr pte RG ah Af («, y) and A F(z, y) denoting the total differences of f(z, y) and F(z, y). Suppose that z=z,, y=y,: then : _ AL y) (2, ; 6x, Yo mun oY,) it A F(a, y)? supposing that, in the expressions for A/f(z,y), AF‘(z, y), which are certain functions of z, y, dz, dy, we finally substitute 94 EVALUATION OF INDETERMINATE FUNCTIONS. Loy Yor Lys OY» for x, y, dx, dy. Proceeding to the limit we have Df z, y) DF(z, y)’ p (Xs Yo) “ Lys Y,» Ax,, dy,, being finally substituted for z, y, dz, dy. Bite Dyk, y= ZF dvs Tay, dk dk DTA 7) aa dx + a dy : dz dy hence $(2,, Y)) = Teil e Fa ‘77 dz+— di eA ioe dy f Since the ratio of dz to dy is undefined, it appears that the value of ¢(z,, y,) is generally indeterminate: suppose however that, when z=, and y=y,, either df dF rE 0, anand ie 0, or ae =0, and ws = 0. dy d: df In the former case $(2,, y,) = rr Dy of dz in the latter, O(2,5 Wie rio dx in both these cases the indeterminateness disappears. If the four quantities fa Ole dianed i dz’ dz’ dy’ dy’ EVALUATION OF INDETERMINATE FUNCTIONS. 95 be simultaneously zero, we must proceed to second differentials, when ae af d*f 2 CIR 3 dy” Fev DG, —) mado ii dudgeen capt 0? Yo DF (z, Y) Ce TALS ee oF ae dx? eae dy dy i an expression generally indeterminate by reason of the inde- finiteness of the ratio of dz to dy. If all the partial differentials of f and F of the second order are zero, we must proceed to the third order, and so on. The extension of the preceding considerations to indeter- minate functions of any number of independent variables is obvious. We have considered only the case of indetermination 0 Siaall of the form x the application of the method, however, to that of the form ~ may be established just as in the instance oO of a single independent variable. Ex. 1. Let _ log + log y | AGED perry rate t%=1, Y% Here ik Lace OE Bees dg x dy y te Greg dy whence o(%,, Y,) = dx + 2dy Panfiioas where a is an arbitrary quantity. ‘Thus 9¢(z,, y,) may have any value from - © to + @. Ex. 2. Let we (c- 1+ y' Ri (2* - 1) pz, y =1, y=1. role ree 96 EVALUATION OF INDETERMINATE FUNCTIONS. df 3 . df 3 : —=__ = _ — — = 3 = es Here om 3(~-1)Y=0, iB 5Y% = 5 dP er ae dF —— = _ = —$S == — ‘| ae 3z(z2- 1) = 0, ag hence ¢(2,, y,) = - 3, a determinate value. onal ACH EN SW esa Ex et p(x, ¥) ay ee Y, df df ay — 0 —_ = ‘ = 0, Then ie 2{f +4) = 0; aT; 2(x + y) dk dF — = = 0 — = Jy = 0 PB dap 3 y The partial differentials of the first order being zero, we must proceed to differentials of the second order. MAD ile di ha Dre dx dy — apa 2 2 2 Ly KB minis, dx dx dy dy _ 2dx* + 4dxdy + 2dy’ Hence ) oly Yo) = ode’ + 2dy 2 1 + a) ie ee 2 1+ a” Le i a a being an arbitrary quantity. The value of 9(z,, y,) is there- fore indeterminate, within certain limits; its greatest and least values corresponding to the least positive and least negative ] ‘values of a+—. Suppose that a 1 a+-= B, a then 4a° — 43a + 2" = 2B’ - 4, Mey nn (soaps. hence + 2 and — 2 are the least positive and negative values of EVALUATION OF INDETERMINATE FUNCTIONS. 97 3 or a + =. It appears therefore that $(z,, y,) may have any value whatever from 0 to + 2. Evaluation of indeterminate implicit Functions of a single Variable. 67. Suppose that an equation is satisfied identically by a certain value z, of x, whatever be the value of y. The function y will for this value of x appear to be indeterminate. Differentiating the proposed equation, we get df df Ne as digs i, cae, A ~ But since, when z = z,, f(z, y) has a constant value zero for all values of y whatever, it follows that in this case also AE, dy hence we have, when z = 2,, Oty on df | | =, de = 0, or dp Ot ttt tte es (8). The value of y, must be determined from the equation (3). In case the equation (8) be satisfied identically for all values of y, of we must, the function — now occupying the place of the original dz function f, proceed to determine y, from the equation an i 0, dx” and so on, until the indeterminateness is eradicated. Ex. 1. Suppose that F(a, y) = mz -~ z+ log (1+ ay)=0, a, = 0. H 98 EVALUATION OF INDETERMINATE FUNCTIONS. Then SIS oe We vi Bg dz 1+ zy whence —-1+ “ = 035 (Olid) mene Ex. 2. Let S@y=ay¥-1e¢-y {logd+z)pP=0, 2, =0. apie Here the equation ae) eine 4 dz gives us, for the determination of y,, ) 2y log (1 + &) _ 0 Ps ; 2(y’-1)2 or Fi(a,y)=(y' - 1) (@’+2z)-y log 1+27)=0: but this equation is identically satisfied by x=0: we must therefore differentiate again with regard to z. dF The equation mate 0 gives us, for finding y,, he 1) es eee Ye oh) NOt gL) ear whence Y, ipl — Yn=20, saree V5 +1 a quadratic giving two values oe for y,. G0) CHAPTER VI. MAXIMA AND MINIMA. Definition of a Maximum and Minimum. 68. Ler y= f(z), and suppose that, as x gradually increases, through a particular value z,, from a value z,- h to a value z,+h, h being an indefinitely small positive quantity, y in- creases, as x increases from z,-h to x,, and decreases, as 2 increases from x, to z+. In this case y is said to have a maximum value whenz=z,. If y decreases, as x increases from z,-h to x,, and increases, as z increases from z, to 2, +h, then y is said to have a mimmum value when x=2,. The words zncrease and decrease are here used in algebraical senses to indicate progress from -— 0 towards +o, and from +0 towards - o, respectively. Preliminary Lemma. 69. Before proceeding to investigate a rule for determining the maximum or minimum values of y, it will be necessary to premise the following lemma. Lemma. If w be a function of x; then, accordingly as xu is Increasing or decreasing as 2 increases, zs will be respec- tively positive or negative, and conversely. Suppose that, when x increases to a value ~+ dz, dz being very small, ~ becomes w+ du; then, if « be increasing with the increase of z, du must be positive, and if w be decreasing, du must be negative: hence a must be positive in the former Ox H 2 100 MAXIMA AND MINIMA. and negative in the latter case; and this must be true ultimately when 6z is diminished without limit. Hence =P must be x positive in the former and negative in the latter case. OF whe, © Ep e e Conversely, since — is the limiting value of 35? it is evident z dz du . Wee. yee se that, accordingly as a, 8 positive or negative, fe must also Me x be positive or negative when oz is sufficiently small, and that consequently « must be increasing or decreasing with the : du . a increase of z accordingly as = 1s positive or negative. AL In other words, that « may be increasing or decreasing with an increase of 2, it is sufficient and necessary that Tr be dz positive in the former and negative in the latter case. Rule for finding Maxima and Minima. 70. By the definition of a maximum or minimum given above, and by virtue of the Lemma of the preceding article, we see that, for a maximum value of y, it is sufficient and d: necessary that — change sign from + to — as 2 passes from z,—h to +h, being positive for the former range of values of x and negative for the latter. From these sufficient and necessary conditions it follows that, when z= ~z,, 2 must a become either zero or infinity, since a function of z can change sign only in passing through one or other of these values. Similarly, that z = z, may correspond to a minimum value of y, dy it appears that FE must pass from — to + as x passes from 2, — h to x, +h and become, as in the case of a maximum, either zero or infinity when v= ~2,. | We may now enunciate a general rule for finding the maximum and minimum values of y. MAXIMA AND MINIMA. 101]. Ruiz. Obtain all the values of « which satisfy either of the two equations I'@=9% f@)=%: if any one of these values of 2 be such that, as x increases through it, f'(z) changes sigh from + to —, it will correspond to a maximum value of y; if it be such that, as 2 increases through it, f’(x) changes sign from — to +, it will correspond toa minimum value of y; and if it be such that, as 2 increases through it, f’(z) does not change sign, it will correspond neither to a maximum nor to a minimum value of y. Ex. 1. To find whether, ~ being a positive integer, EE AC 3s has a maximum or minimum value. . dy is n-k 5 Here ae n(a- ay: equating to zero the value of 2 , we have 2 z-a= 0), B=a. Putting in the expression for a first a- A and next a+h, ; dy, athe we see that, if m be even, <% will be negative in the former dx case and positive in the latter; and that, if be odd, 2 will dx have the same sign in both cases. Hence, if x be even, «=a gives zero as a minimum value of y, and, if 2 be odd, y has neither a maximum nor a minimum value. Ex. 2. Let Y= + 38442: dy then Pe a Oe a= — 3. Sar gate dy . Fr If z= ~-3-A, it is evident that Jp 18 negative, and positive when z=-3+h. Hence z=-3 renders y a minimum, its value being - 3) + 3(-§)+ 2=-}. 102 MAXIMA AND MINIMA. Ex. 3. Let set, (7 + 2) He dy 2(w@+ 8) @ + 5) dx (a + 2)’ ; aed Putting “i = 0, we have ec zxr=- 3, OF 5 Die ==4p s ss putting “4 = , we have 2=- 2. I cape da Now ea ifz=]-2Fh; dx dy ! Bo if z=—8 Fh; hee dy uf and Se ee if 2=.—5° fh. dx Hence, i ¢a—-2,17 = 0, ja Maximum, if c=-— 38, y=0, aminimun, and if ¢=-—5, y=2%, a maximum. Abbreviation of Operation. 71. Suppose ¢(x#) to be any function of z which for all possible values of x has the same sign as f’(x). Then it is evident that in the rule which we have enunciated for finding the maximum and minimum values of f(x), we may replace f'(z) by ¢(az). This change will frequently abbreviate the processes of investigation. Thus if, for instance, J '(&) = ¥ (a). 9), where ¥(z) is a function of z essentially positive, we may reject ¥(z) and take ¢ (x) in place of f'(z). Ex. 1. To find the maximum or minimum values of y, when we zx- 1 OAT ey MAXIMA AND MINIMA. 103 In this case 2 Yo Fae OIG Pome): now ¥(z) cannot change sign: put therefore o(@)=1+22-—2°=0: we have then z=1-V2, or t=1+4+/YV2. It is easily seen that o(a)=F, if r=1-V2FA, and. g(~jy=+, f t=1+V2Fh: hence «= 1-2 gives a minimum and x=1+ V2 gives a maximum value of y. By the rejection of the factor 1 . = ———— , () (a +1f’ it will be observed that we escape the trouble of examining the consequences of putting f'(z) = 0. | jibes ual Ex. 2. Let y = waits di F: E. Then & or f'(e) -Ce P(x). (2), where ¢ (e) = (e - 1) (5 - 2). Putting ¢(x) = 0, we see that Piast =) Ol ae —= "0, Also Of) ae ited alee: fh. and d(e)=4,9 ifo2= 5 Fh: Hence, if z=1, y=0, aminimun, and if z=5, y=, amaximum. Alternation of Maxima and Minima. 72. Supposing y to be a function f(x) of z, which has several maxima and minima, then, as x keeps continuously 104 MAXIMA AND MINIMA. increasing, the maximum and minimum values of y will occur alternately. ‘This will easily be seen when we consider that whenever the sign of f(z) changes from + to -, y is a maximum, and, whenever it changes from - to +, a minimum ; and that a change from + to — can be succeeded only by a change from — to +, and conversely. Modified method of finding Maxima and Minima. 73. Suppose that f(x) has a maximum value when 2 = 2, and that none of the derived functions PAZ), ie daeicas paiedunete become infinite when z=2z,. Then, since /f'(w) decreases from +, through 0, to -, as z passes from z,-A, through z,, to 2, +h, it appears, by the Lemma of Art. (69), that f'(z) must have the sign — for this range of values of z. If f’(x) =-0, when 2 = x,, the symbol — 0 being used to denote zero regarded as a limiting state of negative magnitude, then, when z = 2, it is evident that f’(z) has a maximum value: from this it follows that, since f(x) now occupies the place of f(x), f’"(x) will change sign from +, through 0, to —, and f(z) will have the sign -, as z ranges from z,-htom+h. If f(x) =-0, when z=), then, f(x) now occupying the place of f’(x), we see- that f’(x) will change sign from +, through 0, to -, and f(z) will have the sign -, as z ranges from 2, - fh to x + h. We may proceed with this reasoning from step to step until we arrive at a derived function of an even order which does not vanish when x=2,. Our final conclusion is evidently that, for a maximum value of f(x), we must have f’(z)=0, and that, of the differential coefficients of f(x), the first which, for a corresponding value of zx, does not vanish, must be of an even order, and must be negative. By precisely the same form of reasoning, mutatis mutandis, we may see that, for a minimum value of f(z), the sufficient and necessary conditions are that Ff (@) = 0, i, MAXIMA AND MINIMA. 105 and that of the derived functions f’(x), f’(),.... the first which, for a corresponding value of z, does not vanish, shall be of an even order and shall be positive. Hence, to find the maxima and minima of a function f(z), we must equate its first differential coefficient to zero, and thence obtain corresponding values of x: we must then keep differentiating the function until, for each of these values of z, we arrive at a differential coefficient which does not vanish: if, for any one of these values of x, this final differential coefficient is of an even order, the corresponding value of f(x) will be a maximum or a minimum accordingly as the final differential coefficient is negative or positive. If the final differential coefficient is of an odd order, the corresponding value of f(z) will be neither a maximum nor a minimum. Ex. 1. Let Uiceayae then S = z a , or J'(@) = ¥(%). C1 - x), (x) being an essentially positive factor. Take then, instead of f'(2), d(z)=1-2=0: then z=1, or z=-1: Line eel @ (“) = —- 24 =-; tee tear ¢(4)=- 2r=+4+. Hence x =+ 1 makes y a maximum, and x=- 1 makes it a minimum. Ex. 2. Let y=2'- 82° + 222° - 247 + 12. Then OY a 403 — 240° + dda ~ 24 = 0: a the roots of this equation are 1, 2, 3. Now dy Z . 199 — 484 + 44: da 106 MAXIMA AND MINIMA. hence fe =+, when v=1, os ae when z = 2, ue =+, when z= 8. Thus we see that for the values 1, 2, 3, of z, y is respectively a minimum, a maximum, a minimum, Ex. 8. Let y=e+2cosxz+e™*: then, when z = 0, Y = ¢—2sinz-0*=0, CY -¢ 20082 +6" =0, TY = 7+ 2sine -e*=0, CY =o 5 200m term 4 Hence 2 = 0 gives for y a minimum value 4. Ex. 4. Let y=b+e(x-a)': then @ = 3e(2 - a)? =\0( te 0) Pir). where ¥(z) = 2(x — a) *, a quantity essentially positive. Instead, therefore, of a or f(x), we may take ¢ (x) =c(@-a)=0; whence we get z = a; also GAL) 1 If therefore c be a positive quantity, z = a makes y a minimum ; if c be a negative quantity, z = a@ makes y a maximum. ec MAXIMA AND MINIMA. 107 Ex. 5. Let Ys fae: ii : (7 7.2) ‘Then dy < x(x + 8) dz (x+ 2) =e (Qo ae Coe ' where ¥(z) = Soy 2 an essentially positive quantity. dy Hence, instead of ae or f (x), take x o(@) =2(2+2)=0: then L=0, Ole = — 2): fom 0, O(2)=¢+2=+, yo4, ca minimum: Meee 25 (2) 6 2 = f= oO, a maximum. From this and the preceding example it appears that, although either f(z), or its derived functions of sufficiently high orders, may become infinite for a value of zx which makes f(x) a maximum or a minimum, yet, if we replace f’(x) by an appro- priate function ¢(z) which has always the same sign as f'(), we may often apply with advantage the rule of Art. (73) for finding such a value of z. Abbreviation of Operation. 74. Suppose that, for a certain function f(z) = y, u being a factor which vanishes when z = a, while v remains finite. Then, if a be the first differential coefficient of « which does not vanish when z = a, it is easy to see that d" du want 9 ei , when z= a, ha Oe de" and that all the differential coefficients of y of lower orders than 108 MAXIMA AND MINIMA. the 7 will be equal to zero. ‘This consideration enables us to : i ascertain whether a particular value a of z, which makes oe = 0, Fe corresponds to a maximum or a minimum value of y, without being driven to the necessity of obtaining the general expression d'y et Ex. 1. Suppose that y is such a function of z that for the value of Y = (e-1) (© - 2) @- 8) (@- 4), and suppose that we desire to know whether z = 1, which makes dy di We have, if x = 1, TY = (@-2)(@- 8) (@- 4) -QOO=-: = 0, corresponds to a maximum or to a minimum yalue of y. which shews that z = 1 corresponds to a maximum value of y. Ex. 2. Suppose that dy _ 3 : Gp 6% - 1) @- 2) @- 8) @- 4); 2 3 then T4 = 0, “4 = 0, if z= 1, and d*y Tat 7 1:28 @ - 2) @- 8) (@- 4) = -: which shews that z = 1 corresponds to a maximum value of y. Maxima and Minima of implicit Functions of a single Variable. 75. In the preceding articles we have investigated the method of finding the maxima and minima of an explicit function of a single variable. We proceed now to the con- sideration of those cases in which the function is involved implicitly with its variable. Let y be a function of x by virtue of an equation MAG, 1/) e100. as tee =e eae rekely oa MAXIMA AND MINIMA. 109 Differentiating this equation, we get du du dy — a ig eee) | 0 eax 6 @ 6506 0 @ O26 (8 8 6 . dx’ dy dz 2 dy From (2), since eG 0 or = ©, is an essential condition for a on maximum or minimum value of y, we have du dz — a = 0 or = @ SCoPE sey Pa cred Go du dy From (1) and (3) we may obtain those systems of values of « and y which alone can correspond to maximum or minimum values of y. In order to find whether a value z, of x so discovered does really give a maximum or minimum value of y, we must substitute, in the expression du _ dx du’ dy z,-h, x, +h, successively for z, and the corresponding values of y, and see whether the expression changes sign from + to F, the additional conditions for a maximum and a minimum value respectively. Kies d For those values of z which do not render = , or any of the dx higher order of differential coefficients of y, equal to infinity, we may proceed to find, by implicit differentiation, the values of dy a : ae Rey eae a : _ >... until we arrive at one which does not vanish for Ze dz a value of z which makes 2 =0. Jf the order of this final differential coefficient be even, the value of x gives a maximum or a minimum value of y, accordingly as the sign of the dif- ferential coefficient is negative or positive. Ex. Let MST ey One g hele xo ve ws. (L): 110 MAXIMA AND MINIMA. then Day OY ee ee dx Supposing that Y = 0, we have Toi sQy SOs Fen Ahn Vise OR OTe Le Eliminating y between (1) and (3), we get zg — Qa°z* = 0, whence z=0, or c=a/V2, and therefore respectively y=0, or ¥ =aV4. Differentiating (2)' we have, taking the values a /2, a//4, 10Y2; 7, 22 + (y' ~ az) TH = 0, rg fe 2aV2+a@.aV2. 47 0, x 2 2 : . =——, a negative quantity: Ax a hence a /4 is a maximum value of y. If z=0 and y= 0, then e derived from (2) takes the inde- terminate form §: we may however extricate ourselves from this difficulty by differentiating (2) successively until we obtain . °. e d . . an equation in which no longer presents itself under this Es form. Thus, after one differentiation, 2 (y” - az) TY 4 Fc ea AR aS mip dx’ dx and, after a second differentiation, : A), dy d*y dy fe alts th alia, A vad / a (y ea Neri OS a) at? 7,42 0 When z = 0 and y = 0, this equation becomes d*y —3 2=0 a Ae a 9 dy 2 Pe Sore ; -, a positive quantity. whence i Lae ADE Se MAXIMA AND MINIMA. kid Thus we see that the system of values x = 0, y = 0, corres- ponds to a minimum yalue of y. We will discuss this example also by examining directly da whether = changes sign as x passes through 0 and a 7/2. LX First, taking the value 0 of x, put / for x in (1), A being very small: then h* — 8ahy + y’ = 0, or approximately, — sahy + y= 0, whence WoO, ae OY =: (8a). hi. Hence, if y = 0, we see that, yg ane 1. h OL i ty lta li ae dx + ah 2 when z=t+h, ee hence z= 0 gives y = 0 as a minimum value of y. The values + (3a). h* of y must be rejected because they are impossible when — / is put for + h. We will next take the system z=av2, ava. Putting z=a/2+h,y=a4+ k,n (1), we have nearly 20° + 8a" (2). h — 8a {a (2) + h} . (a V4 +b) + 4a° + 8a’ On foes whence, approximately, 3a” (2)'. h- 3a {a (4) h+a (2); k} + 3a’ (4)! fa 20, whence eas 0: Thus © SACS ed eek A? “a (4a? (ay _ _ 8at (4). A + a = +, if h be negative, = -, if h be positive : hence w = a ‘2 gives a,\/4 as a maximum value of y. 112 MAXIMA AND MINIMA. If we take y’ = az, we shall get = 0, y = 0, which we have : da already considered, and 2 =a */4, y =a 2, which makes oe =O! 2 we may shew, in precisely the same kind of way, that these values of the variables do not correspond to a maximum or minimum value of y. Maxima and Minima of a Function of a Function. 76. Suppose that r= f(#), and x = (0), f(x) denoting a certain function of z, and ¥(@) a function of 0; and let it be proposed to determine the maxima and minima of ras depending upon the variation of 6. We know that dr dr dz — SS = iO ( 4 . 0 ° d0 dx dd ahaa: From this relation it appears that, in order that + may have a maximum or minimum value, the expression F(a). ¥() must experience a change of sign as 8 varies from §, - a to 8,4 a, @, being the value of @ which makes 7 a maximum or minimum, and a being an indefinitely small positive quantity. In order that such a change of sign may take place it is necessary that either f(x) or W(0) change sign, but that both do not change sign at once. In other words, that y may have a maximum or minimum value, it is necessary that either f(z) have a maximum or minimum value as dependent upon the variation of z, or that (0) have a maximum or minimum value as dependent upon the variation of #, and that f(z) and ~(@) have not maximum or minimum values simultaneously. If f(@) have a maximum or minimum value in relation to x, then 7 will have respectively a maximum or minimum value also in relation to @: for, if '(@) be positive, then as @ varies from 0,- a to 0,+ a, x will vary from z, — h to z, +h, and therefore, f'(x) changing sign from + to -, f(x). W'(@) will also change sign from + to —; and, if Y/(@) be negative, then as @ varies from 9) — a to 0) + a, x will vary from 2) +h to x -—h, and therefore, f(z) changing sign from MAXIMA AND MINIMA. 113 —to+, f(x). (6) will change sign from + to -: that is, a maximum value of f(z) with regard to zx corresponds to a maximum value of r with regard to 9: we may shew in like manner that a minimum value of f(x) with regard to z cor- responds to a minimum value of r with regard to 9. Also, if (0) have a maximum value in relation to 0,7 will have a maximum or a minimum value in relation to 6 accordingly as f(z) is positive or negative ; and if £(0) have a minimum value in relation to 0, y will have a maximum or a minimum value in relation to @ accordingly as f(x) is negative or positive. Ex. 1. Let it be proposed to find the maximum or minimum value of r, when | 2 Ts + Ly v= m(tan 5) dr — =], dx a quantity insusceptible of a change of sign: hence 7 has no maximum or minimum in regard to z. Again Here dx : thus — changes sign from — to + as @ varies from — 0 to + 0; dé and from + to — as § varies from 7 — a to 7 + 4: hence if@=0, zx=0; yr =m, a minimum value of 7; and if@=7, x=+0;. r=+0,a maximum value ofr. This is the solution of the problem “ to find the maximum or minimum values of the radius vector of a parabola, the focus being the pole.” Ex. 2. To find the maximum or minimum values of 4 bx ¥ 9a— 2° a‘ a (1 - é’) having given that “L= : 85 1 +ecos 9 d 2a We have é da” (2a - a)” 114 MAXIMA AND MINIMA. an essentially positive quantity, which shews that 7 has no max- imum or minimum in relation to z taken absolutely. Td de tl eo dQ (1 +e cos 0Y . sin 6, Again ; dx : : which shews that — changes sign from - to + as 9 increases dd through zero, and from + to — as 9 increases through 7. Hence iS 1+e Lis eae ; . 6’, a maximum value of 7. —e . é e . if8@=0,z=a(1—e); r= _ 0, a minimum value of r; and, if 0=7,2=a(1+e); r= This is the solution of the problem “to find the maximum value of the perpendicular drawn from the focus of an ellipse upon the tangent,” 7 being the square of this distance and z the radius vector. For additional examples the reader is referred to a paper in the Cambridge Mathematical Journal, for February, 1843, entitled “On certain cases of Geometrical Maxima and Minima.” Maxima and Minima of a Function of two Independent Varrables. 77. Let z= f(x,y), x and y being two independent variables. We are at liberty to assume that y = ~(z) provided that (2) denotes an arbitrary function of xz. Differentiating z on this hypothesis, we have p, ds Z eke ete eee dx dz. dy 4 ae dy where y’ is used to represent —. dx Now that z may have a maximum or minimum yalue for any system of values 2) and y, of z and y, it is sufficient and necessary that, as x increases through z,, the total differential 4 AE coefficient Te? and therefore the expression Fe | dz dz dz” dy 9’ MAXIMA AND MINIMA. 115 shall change sign, whatever be the form of ¥ (x), the only restric- tion to the arbitrariness of this function being that it shall be equal to y) when z = 2, and whatever therefore be the value of y'. A change of sign from plus to minus corresponds to a maximum, and, from minus to plus, to a minimum. 78. In what follows we shall confine our attention to those maxima and minima corresponding to which the partial differen- tial coefficients of z do not assume either infinite or indetermi- nate values. Under this supposition we must have, for amaximum or minimum value of z, De _ de | di dx dx dy y and therefore, y’ being a perfectly arbitrary quantity, dz peer) 3 A pair of values of 2 and y, deducible from the equations (1), will certainly correspond to a maximum value of z, if Dz are oe vee ee eo Sih areal bo and, to a minimum, if D*z dx > 0 eeseeeere cee ere esses .(3). But, proceeding to the second total differential coefficient of z, F d* . we have, putting y’” for aba dx’ Diz _ dz, d*z G2 yr & . da? dat ~ dedy'" * dy” ~ dy"? or, by virtue of (1), when the values of z and y correspond to a maximum or minimum value of z, Dz dz d*z d*z —_ = — ) > : ° te sa ee ev Bd «6 4 he! dv dx is dxdy G5 dy® J (4) In order that, in conformity with the inequalities (2) and (3), 12 116 MAXIMA AND MINIMA. 9 + Z lige —— may never be zero whatever value be assigned to y’, 1t 1s dz’ sufficient and necessary that the values of y' deducible from the quadratic equation dz dia bt ? dx ~ dady'” * dp f be impossible. Hence we must have 12 dz dz d*z.\3 "7 3 ° ey aD eS iy ae oro ae 6 8 6 te 6 6 we Oe 2s (5); dx dy dxdy re Giz d*z a condition which requires that —; and —, have the same sign. dz dy’ From (4) we see that dy’ dx dz dy’ seay } he) eee ae dy® / dady } ° the right-hand member of which equation is, by virtue of the Dz ; must L inequality (5), an essentially positive quantity. Hence 2 dz and therefore as —. Hence, to a dy? dx recapitulate, we see that, for a maximum or minimum value of z, it is necessary that dz, & dee a diy have the same sign as and that any pair of values of x and y, which satisfy these equations, and also the inequality d*z dz (5 ) —- . —— > pe A dx’ dy’ \dxdy : F ; Pri d*z will certainly correspond to a maximum value of z, if Te and Ti zr y on ae d’z a be negative, and to a minimum, if —; and —, be positive. TES dy Be 33 In the above reasoning we have supposed that a 38 not E equal to zero for values of z and y which satisfy the equations eS MAXIMA AND MINIMA. Lis 9 ps D OOLts (1): if however ae = 0, then it will be sufficient that also 2 3 4 eae = 0, and that es dz that the first total differential coefficient of z which does not vanish be of an even order. > 0 or < 0; and so on, it being necessary at 4 re . is the first of these coefficients Suppose for instance that dx* which does not vanish. Then, for all values of y', we must have Dz dz d’z Bee Le AL oe, OREO ede aie, er and = 0), 1D a hee d*z d*z Ae awe ys 7.3 =p e 3. wi da da ~ dx*dy he dxdy’ oe dy’ i! (Se Vy @ Gwen dxdy dy® Yy y Y ly : Y ae > 3 ; and therefore, in addition to the relation (1), zr we must have, y’, y”, y'", being perfectly arbitrary quantities, az ay, d*z ule, d*z de” dady ys dy’ d°*z d°z . d°z . d°z dx? : dx*dy es dxdy’ aie dy” where Oe 4 Since —* is supposed not to be zero, whatever be the value dx* of y', it follows that the values of y' deducible from the biquad- ratic which results from the equation 1D: ies viz. d*z d*z d'z d*z d*z reser sot fyi Gyn Fig ee 4 dx* dx'dy dx*dy’ 4 dxdy” d dy" must be all impossible. We may proceed in the same way to the consideration of cases in which the first total differential coefficient of z which does not vanish is of a higher order than the fourth. 118 MAXIMA AND MINIMA. Ex. 1. To find the maxima or minima of z=n2'+y° — sazy. Put dz = 82° - 8ay = 0] dz ees TANS aT Hiaie enka, (1). ana 2a = 0| The equations (1) are satisfied by either of the systems Differentiating again, we get d*z d’z d*z dx dxdy dy If z = 0 and y = 0, then the expression Des Gc GaN dz’ dy’ \dzxdy has not a positive value, or this system of values does not cor- respond to either a maximum or a minimum value of wz. If « =a and y =a, then the expression is equal to a positive ad 205d < dgedy = ositive, — a is a minimum value of w, and, if a be negative, Pd 3 3 quantity, and are both equal to 6a: hence, if a be — q@ is a maximum value of wu. Ex. 2. To ascertain whether z=0, y= 0, which make a = 0 and ed = 0, when dz dy Co eh Pita render z a Maximum or a minimum. We have dz — = 47° + Qz7* = 0 dx y : eee 22°y + 4y° = 0, sR = 19277 + 2 = 0, ie i i te ee i a a MAXIMA AND MINIMA. 119 dre = 47y = 0 dedy % d*z —{ = 22° + 12y’= 0, dy’ 3 os = 242 = 0, Q d*z ——_—_—- = 4 = dedy. 2” d°z pena dady’ 1 ~ d°*z d‘z diz d‘z d*z d‘z ——_ = 24 pe eee ——,; = 4, = hh ——-= 24, dx* > dx*dy 3 dx*dy” dady’ Hence the equation Dee d‘z oy d‘z rae d‘z yi be dz* — da*dy ‘ diedy” dady’ 4 dy* becomes 24 + 24y" + 24y" = 0, the roots of which are evidently impossible. Also 4 is positive: hence it follows that z= 0, y = 0, cor- respond to a minimum value of z. Maxima and Minima of Functions of any number of Independent Variables. No muppose that) lee fils, 27.5 ox ) where Z, y, z,.... are independent variables. Assume each of the variables y, z,.... to be an arbitrary function of z. ‘Then, by the general theory of maxima and minima of a function of a single variable, it is necessary that the first of the total differen- tial coefficients Du Du Du Crider de oy 120 MAXIMA AND MINIMA. which does not vanish shall be of an even order, positive for a minimum value of w, and negative for a maximum. If the n of these differential coefficients be the first which does not vanish, it is evident that, by adopting the same course of reason- ing as in the case of two variables, we should ascertain that all the partial differential coefficients of « below the x must be equal to zero. In addition to this, we should find it to be necessary that a certain equation of m dimensions in y’, 2,.... the differential coefficients of y, z,....with regard to z of which they are arbitrary functions, be incapable of being satisfied without assigning impossible values to at least some of the quantities y’, 2’... .: the conditions arising from this considera- tion are generally of the utmost complexity. An instance of Maxima and Minima corresponding to Indeter- minate Differential Coefficients. 80. Let it be proposed to find the maximum or minimum value of hes Gis yy. We have dz 2e dz _ 2y at 3 (2 if y’)? dy 3 (2 se y?)? When «= 0 and y=0, the partial differential coefficients Ge ; ke dx dy both assume the form 2: we are unable therefore, by the tests of Art. (78), to ascertain whether this system of values cor- responds to a maximum or minimum yalue of z. We shall 3 : D have to consider whether =. , the total differential coefficient z of z, has a change of sign, as x increases through zero, for all values of y’. Now RN i Nad ayy hn ODS dz dx dy 7 igen, Suppose that z increases from — 0 to 0: then, if at the same time y increases from — 0 to 0, y' must, by Art. (69), have always the sign +: while, if y decreases from + 0 to 0, y' must MAXIMA AND MINIMA. 12t have always the sign -: that is, if z=-— 0, then z+ yy'=a negative quantity. Next, let us suppose that 2 increases from 0 to+ 0: then, if, at the same time, y increases from 0 to + 0, y must have always the sign +: while, if y decreases from 0 to — 0, y must have always the sign —-: that is, if e=+ 0, then “+yy =a positive quantity. We have shewn therefore that, Dz as x increases through 0, ar always changes sign from — to +, x whatever be the value of y’. Hence z = 0, y = 0, correspond to a minimum value of z, namely zero. We might have treated this example more easily in the following manner. Since (z’ + y*)’ is essentially positive, we may instead of ~* take dz Lae! y dx YY > De dz as sss . Putting dx being a function of x, y, y', which has always the same sign We. =x=0 Lis eae. and cae y= 0, y d’v d’v d*v eae ee ceva eer == His we see that ie 1, ead diy Hence aan gag > Se | dx* dy’ \dxdy/)’ 2 2 and oe oe , have each a positive sign. Hence x=0, y=0, & Y correspond to a minimum value of z. The existence of maxima and minima of functions of two ‘ : : 0 independent variables corresponding to values i of the first partial differential coefficients of the function, was I believe first pointed out by M. J. Bertrand in Liouyille’s Jowznal de Mathematiques, tom. vit. an. 1848, p. 155. 122 MAXIMA AND MINIMA. Application of Indeterminate Multipliers to Problems of Maxima and Minima. 81. Let TN SE RES ba one a function of m variables z, y, z,.... connected together by m equations NACA pes see aes Y> %; wey) kone (CRORE AP edih Urea ea we Wy ACA CHA IES there will accordingly be x — m independent variables. ‘That wu may be a maximum or minimum, we must have Du = 0, sup- posing that we confine ourselves to those maxima and minima which do not correspond to infinite or indeterminate values of the partial differential coefficients. Hence o DSic7 tes a pool ae RR Oh re DEAS re, aint eine Ooi GEE: The eae connecting x, y, 2,.... give also df, df, df, EN EAN, [eR GLa a d= FE eR oot 2+ 0 CE CE a FE aes dy + EBay ay Sect ~ (2), ETI ne tee dx After eliminating m of the differentials dz, dy, between (1) and (2), m+ 1 differential equations in all, we must equate separately to zero the multipliers of the n-m remaining differentials which are entirely arbitrary; the »—- m equations thus obtained, together with the m equations con- necting z,y,z2,.... will enable us to find the systems of values of z,y,2,.... which will render « a maximum or minimum. It will then be necessary to ascertain whether Du dx’ s, by the substitution of these values, negative MAXIMA AND MINIMA. 123 in the former and positive in the latter case, whatever relations be supposed to subsist among the »-—m arbitrary differentials. Leos Dixie ; This examination of the sign of = will usually involve very dx” embarrassing computations: it frequently happens, however, es- pecially in the applications of the theory of maxima and minima to questions of geometry and natural philosophy, that the exist- ence of maxima and minima is certain from the nature of each particular problem, our only object being to ascertain the precise circumstances of such critical values. Under these 2 dx circumstances, the examination of the sign of , or, more Du 0 generally, so as not to exclude cases where —- = o or -., the dx 0 Gr Du : investigation whether “s necessarily changes sign as 2 passes through the critical value, becomes unnecessary. The elimination of the arbitrary differentials may be ef- fected very elegantly by the method of arbitrary multipliers. Multiply the equations (2) im order by the arbitrary quantities rN,» A,» Ay ---- A,,, add the resulting equations to the equa- tion (1); and then equate to zero the coefficients of all the differentials in the final equation. The legitimacy of this process will be evident when it is considered that by equating to zero the coefficients of the first m differentials we subject the m arbitrary factors 2,, A,, Az.... A,,, to only m conditions, the coefficients of the remaining » - m differentials being also necessarily zero by reason of the independency of these diffe- rentials. We thus get equations IEE ent | Yu _ 0, de “de ‘* dz eae Dy Tay Gee. eh van 6, d dy dy "ay Oh Lara te .+ Go SS | co ~ ~ XN Ww a“ a] ca) ~ | i) v"| | “ 124 MAXIMA AND MINIMA. We have therefore in all ~+m equations involving the m variables x, y, 2,...... and the m arbitrary multipliers Ay Ags Age +s Ags Whence w, /,'z,....., and therefore wingm be determined. Ex. 1. Suppose that TIES OES) pick apeeale BA Pn oe ; and that the variables 2, y, z, ... are connected by the equation az + by +cz+....=h, a,b,c, ....k, being constant quantities. Then, by the theory laid down, we have C4 ON i0, it ONO.) vet CN t= Ot ies which equations are equivalent to x y 2 =——-—- == = euis on ee ace we om 1 ° = (1) Hence, by the ordinary theories of proportion, ny ee w+ypretie.e. u Se a es ee ge ge. Se ee ee Thee ata ae OC en Le GE eee 2 2 2 2 9 2 Zz Ee ae Sock eee ee Bi cialso hose ee ee ee yen: GL DY Be az+by+cz+.... & 2 Uu uU hence ea Lenalare te = — Cie Dit iC en ual u is or [= ~—_, —_, ______. ee hae ae ; We may easily satisfy ourselves that this is really a minimum value of w In fact there is, identically, @+ Peri...) Qtr e+ s..s) =(az+ byt+cz+.... + (bz - ay’ + (cv -az)'+...., and therefore Kk? + (ba — ayy + (cx - azyt+.... ee 2 2 2 DS EAD ate Corben inl, « b which shews that the inequality ke Tie) 2) Sst oe ae eae ee Cee eC ee MAXIMA AND MINIMA. | 125 is verified for all values of z, y,z,.... which do not satisfy the relations (1). Ex. 2. To find the maximum or minimum of the function Un OEY! AE heey where p, g, 7,.... are all positive quantities, the variables 2, Y,2,.... being subject to the equation ax + by+cz+....=h. ; D In this case we have, putting D log uw = balsas (i u ds AUR ASR OS bt) ic y Zz and adx+bdy+cdz+....=09, and therefore Pe iN Gy a0 Lea Ne Os a aNce OF yes z. y Z whence 2 chat RIN 5 98 hg ul ee BSL SR bat Gi by ace ki Be Paki acer ne Qa p+rqt+rt k Tea > PpPr+qtrt r k 2s: : C ptgtrt eee ese ene ew Se Fe Ge @ © @ 8 @ *eesveeree ese se @ee* Feee ee eee We have also Du ~pdz. gdy rdz acs Y enon Yay batedias Mab fe u ir y Z and therefore, considering dz, dy, dz,...... constant, while “,Y, Z,.... are supposed to vary, as we are at liberty to do in this example because there is no equation connecting the general values of 2, y, z,....and their differentials, Du Dia =) - ot) -r(S) - u Hk. eases exe Z y Z Pai oat ee 5 126 MAXIMA AND MINIMA. or, since Du = 0, Du _ dz\ dy \* de \ | Si AE anit pe ee which shews that D*w is essentially negative. The values of L,Y, 2,.... therefore, which we have found above, correspond to a maximum value of w. Ex. 8. To find the minimum value of TR PLT IRA ene or Cle x, Y, 2, being subject to the conditions WED Y Ce ale ee ec eee (2), CL + bys Cm hl Boel We fe ae (3). It is evident that there is a minimum value of w, for if we were to suppose w= 0, then we should have c= 0, y=0, z= 0, which values of the variables are incompatible with the equations (2) and (8). Differentiating (1), (2), (3), and putting Du =0, we get xdx +ydy + zdz=0, adz + bdy + cdz=0, ddz+b'dy+cdz=0: multiplying these three equations in order by 1, A, A’, adding, and equating to zero the coefficients of the differentials, we have RTA LEN Ed) tore eee ea (4), Uy NOSE A Dee Dilek nape (5), CA-AC TEN Cha) ace a tee (6). Multiplying (4), (5), (6), by 2, y, z, respectively, and adding, we get by virtue of (1), (2), (3), TREO eet WOE Neal tel poe Cie multiplying them by a, b, c, and adding, we get 1+A(a’ + 0? + 6°) + Ai (aa' + bb + cc’) = 0... (8): multiplying them by a’, b’, c, and adding, we get 1+ A(ad' 4+ bb + ce’) + '(a" + 07 + 67) =904.. (0) MAXIMA AND MINIMA. eon From (8) and (9) we have 0= aa + bb'+ cc -(a + 8 +c’) + X'{(aa’ + bb' + cc’)? — (a? + 8 +’) (a" + 8? +”) and 0 = aa’ + bb’ + cc’ —- (a” + 6” +c”) + W{(aa' + bb' + ec'y - (a? + B+ c*)(a" + 6" + c”)}, whence 0 = -(a-a’)-(b6- 0B) -(e-¢y + (A+ WV) {-(ad' - ab)’ - (be' — bey - (ea' - ea)’, and therefore, from (7), ee (a-ay+(b-BY¥4+(e-cy ~— (ab' — aby + (be — B'cy + (ea - cay” ent Ciera CHAPTER VII. DEVELOPMENT OF FUNCTIONS. Taylor’s Theorem. 82. Let y= f(x), and when for x we substitute successively BOLL A LOL, CE LO OLS let the corresponding values of y be denoted by ¥,, ¥,, Ys..---+> Then, by Art. (46), we have y, = f(a + nox) = ¥ + — by POD) by 4. Borys dy, and therefore, putting n6z = h, i of (a SS I Giese ier sid 10 30 i’ 1\ f. 2\ Sf(e) gs es) oe a ses (+4) (-2) (8) (0-2) 0 Suppose now the increment 6z to be indefinitely diminished, the number ~ being at the same time indefinitely increased, so that the product #62 = 4 may remain equal to a finite quantity. The number of the terms in the right-hand member of the equation will be indefinitely increased, and the ratios Sfx) Sf(x) SF (x) ee a Bat ils aa DEVELOPMENT OF FUNCTIONS. 129 will converge to the limits df(a) @f(e) &f() gee” a ge ae me ea or FED OF ake 5 AEA he Ge moreover the fractions 6-8) (0-2) Jom“) (where r is any positive integer less than m, which remains constant while z increases,) will all converge to unity. Hence, whatever be the magnitude of r, and whatever value be assigned to h, m may be taken so large that the sum of the first 7 + 1 terms of the second member of the equation (1) will approach without limit to the expression fO)+FSO+ EI O+ Sf" Or 8. +g 5S O--2). We suppose that all the functions f(z), f'(x), f' (©), f'"' (@)y. oo « jf'(z), have finite values, an hypothesis without which the pre- ceding reasoning would be inconclusive : in fact the product of the expression hr _ ee 22. Se" < *T2.3. in (1) and the defect of its coefficient from unity, would assume the form 0 x ©, a quantity not necessarily zero. We are not at libérty, from the conclusions already estab- lished, to regard the function f(# + 4) as equivalent to the sum K 130 DEVELOPMENT OF FUNCTIONS. of the series (2) continued to an infinite number of terms. For the (v + 1)™ term in the series (1), supposing v to be a number, between 7 and ”, much greater than 7, and, in fact, comparable with , will not converge to the expression hr ————— fir(x RR because the factor Ceara tesla Gee", will not converge to unity, but, in fact, for values of v very nearly in a ratio of equality to m, will be reduced to zero. We may however always At ae fler+M=fO+tf'@+2s'@+ Fa s"@ +... hr ee (2) + RB, R, being an unknown function of z and h. If we determine the conditions that R, may lie between limits, which, as 7 in- creases indefinitely, become less than any assignable quantities, we shall have thereby ascertained the conditions under which the function f(z +h) may be considered to be equivalent to the series (2) continued to infinity. 83. With this object in view, we remark that, if a function o(h) is equal to zero when f = 0, and if its derivative $'(A) retains always the same sign between the limits 0, A, without passing through infinity, the function ¢(h) will have always for this interval the same sign as ¢'(h). For (h) = the limit of ae ; and therefore 6¢(h), when S/ is indefinitely small, must have always the same sign as ¢'(h) 6h, that is, 6(h) must, with the continuous increase of /, be continuously increasing or continu- ously diminishing accordingly as ¢(/) is positive or negative, but, by hypothesis, ¢ (h)=0 when A= 0; it follows therefore that $(h) must always have the same sign as ¢'(h). DEVELOPMENT OF FUNCTIONS. 131 Suppose now that we assume, as we are evidently at liberty to do, that Art ae a Res (r +1) .w (a, h), y being a symbol of unknown functionality. It is manifest then that we shall have determined inferior and superior limits to the value of R,, if we ascertain two quantities P, Q, such that, for all values of h, feb -lf@+ifO+GI@+ : eee ee ae ae 1.2.3 teh) ; is -«4(8). Fe+{f@ +2 /O+ SI'O+ aal'"@ hr : jr | hat ge asp hs st AO hie died Let ¢,(2), ¢,(2), represent the former members of these two inequalities: the functions ¢,(2), ¢,(h), both vanish when h = 0; hence, from what has been said above, we know that the inequa- lities will be satisfied if we have, for all values of h, p, (2) > 0, $, (2) < 0, Ce AF@ £2 p'@)s Ze). 79.3 “ FO Tar Ph>o, 7 wa 4). fi@+h)- a eit uel Oa At? TUR Cae 5 Ota 7 O0, 9, (h) <9, 132 DEVELOPMENT OF FUNCTIONS. or f'(« + h) {F@) + 2 f(a) + ete he Ve * 7.9.8...(r —2) AGED: asec Poo Si(e + hy - \S@ + : fee) ee Nis : Tit a 12,057 2) HB Pose 1). Qh < o Continuing this reasoning, it is evident that after +1 suc- cessive differentiations, we shall arrive at the inequalities ee ee f''(@+h)-Q<0 Now these final inequalities, and consequently the primitive inequalities (3), will be satisfied, if we assign to P and Q respectively the least and the greatest value assumed by the derived function in I) (8). for the values of z which lie between z and «+h. Hence pro a r+1 h } fi 1.2.8 <2G?41) Serene § denoting an unknown numerical quantity lying between 0 and 1. Hence we ig nae phn Se + h)= Bee “f+ f(@)+ Ape t)+. 1.2.3 hm r+ big® Pre emai (x + Oh) ... (7). Now, if the Rarenel values of the function f*"(z) never exceed a certain finite magnitude 2, for all values of z from z=x to z=xz+h, and for all values of 7 the index of dif- ferentiation, it is plain that R - Ar ; xr C Deo Reng 1 0T) and then, whatever be the value of h, we may always take 7 so large that R, may become less than any assignable magnitude. Hence, in this case, the function f(x + h) will be equal to the series (2) continued to infinity, or DEVELOPMENT OF FUNCTIONS. ao as h h’ h’ md Ae oF! eu 75 te oe S(@ t+ h)=f(2) + ee au (x) + ee (x) + This formula is commonly called Taylor’s theorem, after the name of its discoverer, Brook ‘Taylor; it was published in the year 1715, in his Methodus Incrementorum. Another Demonstration of Taylor’s Theorem. 84. By the application of the elementary principles of inte- gration, a demonstration of ‘Taylor’s formula may be given, which has the advantage of determining not only the limits of F,, but also its actual value expressed by a definite integral. The student may pass over this demonstration until he has become acquainted with the Integral Calculus. Whatever be the function f, provided that it remains con- tinuous between the values of the variable designated by z and «+h, we shall have fe+W-f@=| f'@)-de In order to express more clearly the distinction between the general value of z under the sign of integration, and the value of z which enters into the expression of the limits, we may write ath f(x +h) - f(a) = | f'(a’). da’, or, putting z =x+h-2z, and therefore dz’ = - dz, fle+h)-fle)=-| fe+h-2)d: = [re +h- 2) dz. But, integrating by parts, h [FP @+h- 2 deaf @+ | ef erh-2a&; hence f(a +h) =f(«%) + Sfi@+| efe+k — z) dz. Again, integrating by parts, h h [ef@+h-ga-ep@+s | 2s"@ th -2) dz; 134 DEVELOPMENT OF FUNCTIONS. hence fle+h=f)+2¢'@+=f'@+ af 2 f(a +h —2) de. Similarly we may shew that h? u "W fle+h)=f@+2f'@+ Af @+ ‘inal (2) 2° W sper "Bf (2+h-2z) dz, and generally, by a series of successive “en we see that Fle +I) = fle) + ee ee Op Os h tee = f'@) + 5 [ere@sh-a ae Now, supposing P and Q to represent the least and the greatest value of the function SI’ (a@+h-2), for the series of values of z comprised between 0 and A, it is evident that we shall have h g h +1 r rel ae f Oy -—1 | ef (Phi 2) de> Pf de — h r+ and <{of Pappas }. ; r+ It appears therefore that the value of R, falls between Phe and Qh" (28. @ 41) Fe OFA): and that moreover eas r+i A ol Te rosea A: Cif toa. z) dz, which gives the actual value of &,. Cauchy’s Expression for R,. 85. Putting z for z, and then replacing 4 by z-z in the formula (7), we get a al DEVELOPMENT OF FUNCTIONS. 19) f@)-f@)+ = f@+S2 ro:... ie i. “ = = pee ww l | (z) + (2) ee ti stt eros (8), where (2) = eo ST {2+ 0 (xe - 2z)}. When r= 0, the formula (8) gives F(@)=f@)+@- 2) f'j2+ 9 @-2)}, §, denoting a number between 0 and 1, not generally the same as @: hence, replacing f by ¢, and observing that, by (8), @(z) = 0, we have 0=9(2)+(@-z)¢ {z2+0,(e-2)}...... (9). Again, by differentiating (8) with regard to z, we have, can- celling terms in the result which destroy each other, oa CAT Fe) + 9) 1.2.3 whence, putting z+ 6, (# - z) for z, g'{2 + 0,(e- 2} =~ EOE - Fg fo 0, we - 2)}, Consequently we have, from (9), since ¢(z) = 0, L.=— OY (a zy" ei and therefore we may replace, in the formula (7), the expression for the remainder &,, viz. hr +1 1.2.8 s0(7 41) pineal): by this equivalent expression ube Me he maine pm Oe bat), 186 DEVELOPMENT OF FUNCTIONS. Examples of Taylor's Theorem. 86, Ex. 1. Let Sizy= loge: Then df(a)_1 @f(e) 1 @f@)_2 af@)__ 28 dc rg nda a de ae da td a and therefore fle +h) =log(@ +h) = log 2+ 8-4 o h 22° 3a 4a" Kix. 2. Let f(x) = sm z: then af (a) _ Df (2) _ df (a) _ gg 8 O08 Ey) AS BIN Bye COS 4 EI) S in a dx* and therefore : h fae OE F(@r+h)=sin(#+h)=sin x + risi utetrwrvctacmerppyy © o) + sin z — &c. 1.2.3.4 Failure of Taylor’s Theorem. 87. In the demonstration of Taylor’s theorem we have sup- posed that the function f(z) and its derivatives are all of them finite. If, for a particular value a of z, this should not be the case, the theorem then becomes inapplicable, or is said to fail. Suppose for instance that f(x) involves a term of either of the following forms Cd ae Seed Ra een te eS A (ao, ON Ea TON Crete UE a BRP Oe Pe ena Fy (2), m and x being positive integers, and w a proper positive frac- tion. In the case of the form (1), f(z) itself becomes infinite when xz = a, and, in the case of the form (2), all its derived functions after the n't. ‘These two forms comprehend the only cases of DEVELOPMENT OF FUNCTIONS. 137 failure: which can present themselves in ordinary algebraical functions. Suppose that z=a+h; then p(#).(@-ay™= (a +h). h™ = \? (a) + : p (a) + a g(a) +... ‘ hm, which shews that a failure of the theorem, due to a term in f(z) of the form (1), corresponds to the existence of negative powers of h in the true development of f(a + h). Putting x =a +h in (2), we see that He Aes at he p(a t ces i he . +W = \o) + ; p (a) + a6 p (a) +. sae Deeg which shews that a failure, arising from a term of the form (2), corresponds to the existence of fractional powers of / in the true development of f(a + h). 88. When the first derived function of JF (%) which becomes infinite for the particular value a of z, is of the (7 + 2)" order, we may employ Taylor’s development provided that we do not carry it beyond the term involving h’, and that we take care to preserve the remainder &, which may be evaluated by the formula of Art. (84), a formula which is always applicable when the (r + 1) derivative, as we are now supposing, remains finite between the limits. Lagrange’s Theory of Functions. 89. The extreme generality of Taylor’s series had for a long time attracted the attention of analysts, when Lagrange con- ceived the idea of adopting it as the basis of a theory of func- tions, the object of which was to arrive at the conclusions of the differential calculus without introducing the idea of limits or infinitesimal quantities. We will give a brief sketch of the general system of Lagrange, which has been, until within the last few years, so generally adopted in elementary treatises. 138 DEVELOPMENT OF FUNCTIONS. Lagrange assumes, in the first place, that any algebraical function f(x + 4) can be developed in a series of the form SF (@) + O(a). he + p(x). hh + ofz).h +.... He then proceeds to shew that, from the algebraical nature of functions, these powers of / can neither be fractional nor negative, so long as # and A remain general in form. He remarks that, if the series were to involve a fractional power of h, this term would have several values, and that accordingly, for a single system of values of 2 and f(x), the function f(z + h) would have several distinct values ; a conclusion which cannot be true for all values of x and f(x), precisely as, to borrow an illustration from algebraic geometry, it is impossible that all the points of a curve should be multiple points. Again, if the series were to involve negative powers of h, he observes that the corresponding terms would become infinite when A = 0, and that consequently f(z) would be always infinite, which is im- possible, except for particular values of z. From the above reasoning, then, he concludes that the development of f(x +h) is expressed in a general form by the equation S(er+h=flayrh ofx) +h? ox) +h’ ofa)t+.... The coefficient ¢(x) of h, Lagrange calls the derived function or derwative of f(x), and represents by the expression f(z). Having thus defined the method of derivation, he determines without difficulty the law by which the functions $x), $,(@),.. . are derived from f(x), and thus arrives at the formula of Taylor, Bes Rane x eee, AACA OYE BCA Kear (BED on ret ALC ena (2) eres each of the successive functions f(z), f(x), f’'(@),....being derived from the preceding just as f(x) is derived from f(z). These functions are termed the first, second, third, &c. deriva- tives of f(z), the order of derivation being indicated by the dashes. If then in the elementary functions x”, log z, sin z, &c. we substitute z + A for z, and then expand, by the ordinary pro- | | DEVELOPMENT OF FUNCTIONS. 139 cesses of algebra, (« + h)”, log (2 + A), sin (a + h), &c., in series ascending by positive integral powers of h, the coefficients of the first power of / in these developments will be the first derivatives of z”, log z, sin x, &c. The second derivatives may in the same way be derived from the first, and so on indefinitely to higher orders of derivation. ‘The derivatives of composite functions, which depend upon those of the elementary functions, may then be determined. The theory of functions therefore, according to the method of Lagrange, resolves itself into a mere algebraical system, to the exclusion of what he considers to be the extraneous idea of infinitesimals or, in the language of Newton, fluxions. For the complete development of this theory the reader is referred to Lagrange’s two systematic Treatises, entitled Théorie des fonctions analytiques and Legons sur le calcul des fonctions. Within the last few years the logical value of Lagrange’s system has been called into question by all writers of authority both in France and in England. The chief objections to his method may be arranged under four heads. (1) All inductions drawn from developments in the form of divergent series are devoid of solidity, and frequently, as may be ascertained from actual instances, lead to erroneous results. It would appear therefore that Lagrange’s method, as involving the consideration of series without regard to convergency, is not entitled to the reputation, which it originally possessed, of being rigorously logical. (2) The hypothesis that f(«+h) may be expanded im a series by positive integral powers of h, restricts the application of Lagrange’s system to the ordinary functions of algebra, whereas the general theory of limits embraces all continuous functions whatever, and involves a code of doctrine which exists inde- pendently of its application to any subordinate science. (3) In the application of the theory of functions to geometry and natural philosophy, the idea of limits cannot really be avoided, although it may be disguised by the artifices of algebra. The doctiine of limits on the other hand, as explicitly involving in an abstract form essential principles of our conceptions of 140 DEVELOPMENT OF FUNCTIONS. curvature and of motion, lies in immediate contact with its most interesting applications. (4) When the remainder &, keeps indefinitely diminishing without limit as r increases, Taylor’s series is convergent ; the converse proposition is not, however, generally true. ‘he theorem therefore, regarded as an indefinite expansion, fails to express the true value of f(z +h), not only whenever the series is divergent, but also, which constitutes a still greater limitation, whenever #, does not diminish indefinitely with the increase of 1. Stirling’s Theorem. 90. Ifin the formula of eh we ee x = 0, we get fi) = FO) +2. f0)+ = f'@+— fF" or, putting z in place of h, SO) =f) 45S O+ GP Ot of" This theorem enables us to expand any function of z in a series ascending by positive integral powers of z. Its demonstration was given by Stirling in a work entitled Methodus Differentials, London, 1730, p. 102. The theorem was afterwards given by Maclaurin in his Treatise of Fluxions, Edinburgh, 1742, p. 610; and is now generally called Mac- laurin’s theorem. It is in fact, as we see from the demonstration given above, merely a particular case of Taylor’s theorem. Stirling’s series, as well as ‘Taylor’s, from which it has been deduced, ought to be completed by a remainder. Let &,' represent the remainder, when we stop at the (7 + 1) term of the Sakae Bed then F(@)=f(0)+ 2 f'0)+ = FO) + Zo f"O+ oes r i BREA al (ip pS ef es Le ee ot; lak) Ete = DEVELOPMENT OF FUNCTIONS. 141 the value of #,’ being given in the form of a definite integral by the equation R? A RL, ie aly PE (2 vd z) dz, AR Ge aes A a formula derived from the expression for R’ in Art. (84), by first putting x = 0, and then replacing / by z in the result. Again, the limits between which &,’ always lies, are Pz" Oe —____—_——. and ————_,____. , Ws Ana ea V2: ae eet Pie Ly) P’, Q', denoting respectively the least and the greatest values of f""' (xz — z), or, which amounts to the same thing, of f(z), between the limits z = 0, z= 2. If 0, 6,, represent certain unknown numbers comprised between 0 and 1, we shall have also for R, the two expressions r+l x a = (1 ms 0.y es rel Tate ee pe ee rer mer wale RS Lisi ee eo Examples of the Application of Stirling’s Theorem. 91; Ex. 1. Suppose that (f(x) = a". Then ile lO dt. G8) (2) (log ay. at, f(z) =(lopta)y sae. hence, putting z = 0, Fol nlog a.) 1f1(0) = (ogeay, 27, ((0) =. (log'a)y,.. o2.3/. We have then, by Stirling’s theorem, tre i z 1.2.3 CO ae Wine Z) 2 a= 1+ loga+— (log af + ee ogome cries _{zlog(@yr a” L2 eek ee) ; The remainder 142 DEVELOPMENT OF FUNCTIONS. which is equal to zloga xloga zxloga x log a : Set 57 has u 2 3 y+ 1 will evidently approach without limit to zero, when 7 becomes indefinitely great, whatever be the value of x Hence we are at liberty to put x a gat 7 = aa ee ae ie l y seeeee#e ’ a MOB 8 te MOB GIT a OB). the series being supposed to be carried on to infinity. Ex. 2. Let J (@) = log (x + 1): then F(x) =e tA), fe) het Tae Fe yy ey ee arid: generally '?f"6z) ="(AY1. 2380 Te 1) whence FO)a0. CON Het 0) arf 10) = (hi 2a POs HC). 28 a(n 1). Ce ee x" Hence. -loe (744) = eee Phy phy, 2h ted. 8 ( ) 1 2 3 (-) r r+2 1 am a) a", r+i1 + (-) If z be positive, then the expression #2’, will diminish without limit with the increase of 7, provided that 7 2 1 + Oz 1 EG hey ee eat QU | YR re ean) 1 14-30 but 1 is evidently the least value of : hence, if z have any positive value between 0 and 1, the series being prolonged to infinity. If z be negative, let — z represent its value: then TAS ce Ol A Pe tole — (=) : DEVELOPMENT OF FUNCTIONS. 143 and, in order that we may be sure of the propriety of regarding the series for log (x + 1) as infinite, we must have z 1 ——— <"}, 2<1 — Oz) .2< —; 1 -— 02 ; 1+ 60° hence we are obliged to suppose that z is < }, or that the nega- tive values of x are comprised between — } and 0. If however we take the other formula for R’,, we see that Ty, Pe (-)? (1 bate 6,y. (1 “ eye: ght} “Cy? (= 0 (ZF) + Oz . . fos 6, r+ = cya = 0)" (SERRE, if we put z=-z. Then the expression z-@z 1 Be will be less than unity, provided that z-@Oz2<1-6z, orz< 1. Thus we see that the latter formula for F’, allows a greater range of negative values for z than we could have inferred from the former. If then z have any value comprised between the limits - 1 and + 1, we know that 2 oe at x LO (ai Le ae eee g ( ) Magis fb anlad, : supposing this series to be continued to infinity. Ex. 8. Let f(@) = sin z. Mneneet (2)—.cos 2. 7. (2) icin’ a, f(2z)'= = cos 2, ... whence. f(0)=0, "f'(0)='1," f° (0)=0, f"(0)=—-1,... and therefore ore a f(@) == - + —— -.... +, 1.2.3 9:3 1222S 450 144 DEVELOPMENT OF FUNCTIONS. r+l r+} x where he, = a a CaP | > (-) 2 sin (Oz) ; a” rl or (-)? cos (0x), Peis gee pies accordingly as 7 is odd or even. From these expressions it is plain that, when 7 = © , R' = 0, whatever be the value of zx. Hence, generally, whatever be the value of 2, : x x x sin 1 93 —— + eS Oe 1 Lee 1.2.3.4.5 2 the series being regarded as infinite. We might prove in the same way that, whatever z be, xv a cos Z = 1 + —— 4+———_ — ... 123.019 122::3.4 a series likewise infinite. Ex. 4. Let Fayelarnz). Then f'() =m(a+ xy", f"(@) =m(n - 1) (a+ 2)", F''(2) = mm — 1) (m — 2) (a + zy", .... J '(@) =m(m - 1) (m - 2).... (m-r41).(@+%)™: whence f'(0) = a", f'(0) =ma™", f"(0) = m(m — 1) a”, ef O) = aL) ee, Oe ee JF’ (0) = mm - 1) (m- 2)... (m=r4+ 1). a". We have therefore m(m "te 1) m-2 2 m CDR phe get 2p 1.2 m(m — 1) (m — 2) 3% 3 + eae eece 1.2.3 Cy sete NG — LAW eed) ee I ee Le _m(n = 1)(Mm = 2). =P HV ger rg Be Lee ie ne ae DEVELOPMENT OF FUNCTIONS. 145 see a A SD yy ee 1S Seren tee) a rth , _ m(m — 1) (m — 2)... .(m-— 71) a oes 1.2.3." sree 1) a+ Ox Suppose that z=a positive quantity: then, in order that R, may become zero, when 7 is made indefinitely great, since, as may easily be seen, m(m—1)(m—-2)....(m—-7r) T2340 Give) is not infinite, it is sufficient that x eB a AE oR ES a+ Ox a 1-6? and therefore, a fortiorz, that 2 < a, If x be negative, it will be convenient to have recourse to the second formula of Art. (90) for #,’, which gives weg UL) I, 3) He ole 2. 7”) r m-7- a+ uf Lee ar RIA ANIL 5 ON or, putting — z for z, m(m—1)(m—2).. ..(m—- 1) qs ep eas a ote il Rie 1 6 ae ee +l i {( i Dred (a - *) : Hence &,' will be reduced to zero, when r becomes indefinitely great, if z(1 - 6,) a-O@z Hence, for all values of z comprised between - @ and + a, m(m—1 = Ha (a+z)"=a" + oa” 2 4 UBT) i See AY) UN a 1 1.2 ep -e: i ee De ad infinitum. (0) +5 ~ 9'(0) + ee Oe a) Now p(a) = £ fle +ah', y+ak').h 4+ Zt +ah',y+ak').k, or, to adopt a more concise notation, / ae) of, 1 By p(ay=h ae es a differentiating again, " h” ae ' WH 12 ay, 2h'k’ LK sas o"(a) = Le a for a third il phsee 2 Ji. hye GA, yo th 2 1 BY AW A icobialie dt) — : da'dy " dady’ dy and so on ee the law of derivation being obviously in accordance with the binomial theorem: we thus have, generally, ins ete m(M- 1) pinay) dnf. is }'” ah, = ly 1 h'*}:”? 1 a Cie da dy 1.2 da*dy pn(a)= he Ss From the expressions for ¢(a) and ¢"(a) it is plain that 9(0)=/f, where f is used to represent f(x, y), and AUC PE a Regie i Rt ee ay (AG ered ete df "0O)= fh» + De A In-2 J,!2 at dz” i 1 (me dx" dy i ge: hae da *dyt In af +h" —-., dy” Hence, from (1), substituting for (0), ¢'(0), (0), ...... their values, and replacing ah’, ak’, by h, k, respectively, we have F(@+h, y +k) =fth & +k ef dy 1 af d°f gh h? 2 (ed. onde ake s) VA erat id af ie he 2 2 mee Sarl da” ak + Oe oT + BS LOCC A ee, a ee die et a Se ae Ry eo hd ahs (2), which is the required formula. L2 148 DEVELOPMENT OF FUNCTIONS. The formula (2) may be expressed more briefly by the aid of the separation of the symbols of differentiation from those of the function upon which they operate. Thus ba, bP (aes bo) Ss d. dx dy d°f d*f 2 af | | Raves ; = v4 sees h aaa + 2hk eds +k dy? we ee eh dt the pice of these parbelical expressions depending upon the principles of Art. (48), which shew that the laws of the eo d d combination of oF and ay are the same 7nter se as those of two od y symbols of magnitude. We accordingly see that ferhy+D=f+(ho rk \ dy 1 d da» 1 bie ag teas 25 7 cs hoe acne | pe f ues & ° ool dz x) t* crn dix” a) : Be Yay fe eye Sona eee (3) Since 4 and & are any quantities whatever, we may put h = dz, k = dy, in which case whence there is UR CE RLY IPSUM) parti We BS Biron ee ries eb: or, since d, + d,= D, D denoting total differentiation, SF @ edz, dy) Sea fi ae dee Cor. The method of development which we have applied to a function of two variables may obviously be extended to a function of any number of variables whatever. Thus dod ANd Ale ihry ache Fee ete, eae whence also f(t het, y + dy, 24+ dz,. 5...) = cP ttetin Ff, DEVELOPMENT OF FUNCTIONS. 149 Failure of the Development of f(x +h, y + k) by Taylor’s Theorem. 93. The development of f(x +h, y + *), given in the pre- ceding article, fails for particular values of x and y, whenever f or any of its partial differential coefficients becomes infinite ; this failure being consequent upon the failure of Stirling’s theorem applied to the expansion of g(a). It will likewise cease to be applicable for particular values of x and y, which render f or its partial differential coefficients essentially in- determinate. Limits and Remainders of the Development of {(x+h, y +k). 94. Let R, be the value of the remainder which must complete the series (1) of Art. 92, supposing this series to be stopped at the end of the (7 + 1)" term; then, by Art. (90), r+l ee oe R ' = a r+1 ( r+ if Droste) ts Cer rg ana; seh e 6, 6,, denoting certain unknown numbers comprised between Oand 1. Hence, if we stop at the term d ad\ ys ( ay a) 1.2.3 ...7 of the series for the development of f(#+h, y+), we must add the eae remainder p,, where 1 # me f+ Oh, y + Ok) = (45 URL 2sares (Ly ii . : " f@+Oh, y+ 6,4) or okt ema Ns kia taka The numerical fractions 0, 0,, which enter into these formule, being unknown, we cannot employ the formule for the actual computation of p,: they serve only in fact to determine limits between which the value of p, must lie. 150 DEVELOPMENT OF FUNCTIONS. The value of R,' is equal to the definite integral iaaae J, 9-8 Bran, whence, for the actual determination of p,, we have ad (TART) Peele BM y+(a-B)} Bap ) os es dz oy =f, (4 athe) Sf{xth-hB, y+k-kB}.6" dp, RAE a Fae d. or, putting 6 = - : 1 h d d +l k'\ 2" dp’ Sirota. vas aye : h- 3 k— |, a5 eel (i dz” f a ples Balt nae as 1 Hood Bin nue 7 d kp TiS al +k — h-, j yes la 1.2.3...7 mal (a du” at Sle pai dhe 7). B' dp. By analogous reasoning we might shew also, as an equiva- lent formula, that 1 1 d The hp Oke: 1 eons es ake CARS pl, (az be) fle Pe a p).B eee In precisely the same way we might investigate symbolical formule for the remainder in the development of a function f(at+h, y+k, 2+4,...), @ y, 2,... being any number of variables. Example of the Application of Taylor’s Theorem for two Variables. 95. Let f(z,y)=0 be the equation to a curve, f(z, y) being a rational function of x and y; and let it be proposed to transform this equation into an equivalent one for a new origin (a, 3). Putting a+z, B+y, for x, y, we have, for the transformed equation, 0 = f(a +2, 3 + Y)s DEVELOPMENT OF FUNCTIONS. 151 or, by Taylor’s theorem, the dimensions of the proposed equa- tion in x and y = ae N, f U OF ot) 0 ache eR gree a5 7g Waka Bay a dp : DiS Speier Lhe halt oe homes af Deer Gar 1a Y da dp” 1.2 I da dp toe + Yy” “x . Let, for instance, the equation be 0= Av’ + By’ + 2Cry + 2A’e + 2By + C': then f= Aa’ + BR + 2CaB + 2A'a + 2BB+C’, + Ze =2Aa+ 206+ 2A’, i ~~. = 2 2Ca + 2B, FT Bp + 2Ca + af af af edad, 2A = 2) ; ATC 5 . da da dB C de: 25 the transformed equation will therefore be 0 = da’? + BP’ + 2CaB + 2A'a+ 2BB+ C' + 24 (Ada +CB + A')+ 2y(BB + Ca + B’) + Ax’ + By’ + 2Cxy. Stirling’s Theorem applied to Functions of two Variables. 96. If, in the development of f(z +h, y +h) by Taylor’s theorem, we substitute 0 and 0’ in place of z and y, where 0 and 0’ are used to denote zero values respectively of z and y, and then replace h, k, by z, y, respectively, we have S(% y) = f(0, 0’) + (2 se naa f aa) FO 0°) i ‘a oory a) LO 0’) 1 d ' "12.8 (ety) FO. - + &c., 152 DEVELOPMENT OF FUNCTIONS. which constitutes an extension of Stirling’s theorem to functions of two variables. The expressions for the limits and remainder may be ob- tained at once from those for the development of f(x+h, y+), by first putting z=0, y= 0’, and then replacing h, hk, respec- tively by z, y. Lagrange’s Formula for the Development of Implicit Functions. 97. Suppose that U = fty) © je ene 8) ste) eleva siete. 4 (1), y being an implicit function of z and z by virtue of the equation OD femirg ated OY CA DANE lle cca (2), The object of Lagrange’s formula, which we proceed to investigate, is to enable us to develop w in a series arranged by ascending powers of 2, and which does not involve y. If ¥(y) be any function of y, y being a function of z and z, then ait Y) At = \ea). Aen (8) for it is plain that each of these Sage is equal to es GD oO AO acces ae bk a Differentiating (2), considering z constant, we have = pyre oy), d. {1-2$'Y)} Ba GY)... (4); and, considering constant, ow =1+2rd(y). {1-2 $y} - a cle Ras sabe (5). DEVELOPMENT OF FUNCTIONS. 153 From (4) and (5) there is dy dy dz at p(y) . dz »aesnsese es © & o @ (6) From (1), differentiating on the eas that z is constant, du a fy). ; d = f'(y). oly). yt , from (6). Z Hence, f'(y). $(y) ae the place of ¥(y) in (3), we have ae atl (y). $(y)- au af» a =[7@- tor. 2]. Differentiating again, we have dus a , 5 -5(7o-eor-Z], bye, [ro is@my- 2], by. Proceeding in the same way it is evident that we shall have, generally, au. a etsy BP PP |S (y)- 1Py)}"- | By Stirling’s theorem, u = (2) ae ae on et ees fee 5 sd A 1 dx a=0 1.2 da® he 1.2.3 da’ 2-0 But from (2) and (5) it appears that y = z, and wy = 1, when e= (0). 154 DEVELOPMENT OF FUNCTIONS. Hence () ne oe Lf (z) . {b(z)}"], Uu =f(@)+* - .[f'(z). d(z)] + as or (2). {P(2)}] 1.2 ° dz + SU O16O} +2, SU OAG@H+ 0. In the particular case, when “u=2+2(u), or, when f(y)=y, we have f'(z)=1, and, instead of the formula (7), we have Vo ua e+ [6 @]+ > « Hoe} 4 172 “dz ME @F 1.2.3 5 + Soe} ]+.... @) This theorem was given by Lagrange, in the Mémovres de Berlin for the year 1770, (see also his Equations Numeriques, Note x1.), as a generalization of a particular development obtained by Lambert for the expression of the roots of certain algebraic equations: Lambert’s results were published in the year 1758. A demonstration of Lagrange’s theorem, due to Cauchy, which involves some important reflections respecting the convergency of the series, may be seen in Moigno’s Legons de Calcul Différentiel et de Calcul Intégral, tom. 1., pp. 162-172. An expression for the limits of error committed in stopping at any term of the series, has been given by Murphy in the fourth volume of the Cambridge Philosophical Transactions. xX et uU=24+2S8iN uU. The determination of uw in terms of z and z is a celebrated problem in astronomy called Kepler’s problem: the variable z denotes a quantity which varies as the time, the coefficient z represents the eccentricity of the elliptical orbit of a planet, and the variable uw is an angle called the Eccentric Anomaly. o(2) = sin 2, DEVELOPMENT OF FUNCTIONS. 135 d PEN OY, AGO bois ae: =, te @Y = = iin zy} = 2 sin 2 cos z= sin 22, 2 G PRR er ay. ne aye zt ae {9 (z)}° = ay (sin z) ee .@ sin z—jsin 8z)= —#sin z+ sin 82. Hence, as far as the term involving z’*, we have, by formula (8), 3 2 uw=2+sinzg.—+sin 22. —+4(8 sin 3z- sin 2). Ae 1 1.2 [eory} Ex, 25 Let Ue te Dabtey oan ye =i 81N14/;, Then lz) siete (2) Sines, af (2) = Cos: 2 f' (2). ¢(@) = sin 2 cos z =} sin 2z, f'(z). {¢ (®}’ = cos 2. sin® z =i sin 2z. sin 2 = 1 (cos z -- cos 82), £ [f'@). {9 @}1=4.G sin 82 - sin 2), Tf (a) Ag) =cos 2 . sin®z =} sin 22. (1 — cos 22) =1(2 sin 22 - sin 42), Sf (z) . {p(z)}*] = 3 (cos 22 - cos 42), {ure z). { (z)}*] = (2 sin 42 — sin 22). Hence, as far as the term involving 2’, we have by formula (7) : 1 x 1 ° . x u=sinz+4sin 2z.— +4 (8 sin 3z — sin 2). a5 + (2 sin 4z- sin 2z). e a en & 1.2.3 Laplace’s Formula for the Development of Implicit Functions. 98. Suppose that u=fly). Bel) TENE TREE BETTS d owes owe « (2) and 156 DEVELOPMENT OF FUNCTIONS. Differentiating (2), considering z constant, we have dy a ; dy \ LaF {2+29@}-\oy)+ 2d). 7), U / d / [l-F{e+agy}-2¢' Ql 5 =F {2+ 24)}-0)-@) and, considering « constant, dy _ "Sy ' dy \ =x =F'{z+zo(y)}. " +2 ply). dz{’ dy dz [1-F {2+ 2poy)}.2g'y)]. 5 = F'{z+xoly)j --.- @) From (3) and (4) we see that dup, dy pee WY) hak =F'(y)- oy). =, from (5). Differentiating again, and ae in mind the theorem estab- lished in the oe article, viz. = {ew aa aio Zh we have ze a: ad (y)- oy). 7} ah 4 bier yey Differentiating again, we have tna toe 1G: oS Rvaea Ei (y) -19(W)} | z eS lw): {oy} a -F [roto or ¥]. DEVELOPMENT OF FUNCTIONS. ay Proceeding in the same way we have, generally, anu ot =F [Fo foo 2]. From (2) and (4) it appears that, y = F(z), ie = F'(z), when zx = 0, and therefore d”u ad”* ——| =——{ | Gray) yn : (4) - SPO} ro Fe] Hence, by aes theorem, viz. ( a + du fh x ( de v ae du as cee Da eee UR Gaee smtacu aN ee Nhe wal we have, putting for the sake of brevity f{F'(z)} =A(z), and p {F(z)} = ae w= f+. KO -a@l+ = £KO. (0% + SRO AAO 1+ which is Bie s formula, first given by him in the Mémozres de l’ Académe des Sciences, 1777, p. 99. Lagrange’s formula is evidently a particular case of Laplace’s, from which it is at once derived by putting F(z) = z. Sa awn uty 9 ; eS et « er i Pez, int — a PA Sather , ’ ¥, s o> > a P y 7 CEP ACA ce Bie CRAs ak RED i syeginin rat Yao, 7 , . J ne ae | ~ ; . hare > - ma , ay 4 7 a: f i ; ; nS ay a i : ate hee | wy ts ier Bs ee er 7 i : i rs Pen babii ee : 7 . ‘S { my ee Pe hs 5 ’ 1 i ’ ' Peo “ey ‘ Ay } - i’ ar 4 a ANG i i ' , | i , « } “ , ' bs ip | : % | . ¥ n 7 » LJ = -t ; a mi Sy | NaS r 4 , : i. Z 5 m4 ia. | \ 5 »] oo hoe th ey +s tf “Wn theo te ; i | , - mt a = 4 t ; , , yt ¢ ‘ ' ¥ - i =. | ir pf * + * py » ie’ Jade 5 . 4 { < F y Pw 2 if i oe i - ~ by i ad ; i - ‘ | r $26 aM | ; ; 4 SeiSee St elena i Ve Ey i : ; Le ‘ > - t ‘ rt >. ae 7 : 5 + on a |< f - - d))*, i ms by = e qi) j \ * “< ‘ (-= < rs ele a za “ nt 7, oee . : hi “ F ® ? : : i ‘ ¢ pa ' > : . i aw > LY ‘ ; - i ‘ bs ihe, ‘ PLA Py vf ; 4 < - rene J ga? 7 b , Ps cA ee a 5 . $ é % fa i se . er, : ? ‘ ry Une ad “Susy, ; a. ve Sal ‘ Pe! , ot) Pe f 4 A Bad 298 | ; trl) ©t = a Pp 3 #7 N en \ . f . : ; f ’ rh ’ as ba Ah Lt ¥ , miss 2 . 7 ' > 7 Ss AAD nes ayo On Garay el Peay pacts e ei be DIFFERENTIAL CALCULUS. SECOND PART. GEOMETRICAL APPLICATIONS. Bide, Wed PAS Ee I TANGENCY. Definition of a Tangent and of a Normal. 99. Let P, Q, be two points of a curve AB, (fig. 1), and suppose that an indefinite straight line H’ A" is drawn through these two points. Conceive the point Q to move towards P; then the secant H’K’ will keep tending towards a certain limit- ing position, and ultimately, that is, just as Q is on the point of coalescing with P, is said to be a tangent to the curve at P. An 160 TANGENCY. indefinite straight line UV drawn through P, at right angles to HK, the limiting position of H' K’, that is, at right angles to the tangent at P, is called a normal to the curve at the point P. Inclinations of the Tangent and the Normal at any point of a Curve to the coordinate Axes. 100. Let Oz, Oy, be rectangular axes of coordinates, (fig. 1), AB being a curve contained in the plane zOy. Let x, y, be the coordinates OM, MP, of P, and 2’, y’, the coordinates of Q. Let ¢ denote the length of the chord PQ. Then the cosine, sine, and tangent of the angle AH’ 7’ z, T” being the point in which H' K' cuts Oz, are respectively @ C x2 When x approaches indefinitely near to z, in consequence of the approach of Q towards P, then ultimately, a denoting the angle AHTz, where 7J'is the point in which H& cuts Oz, s representing the arc AP, and s’ the arc AQ, / ‘ U . . aes . . u a Ss Pay 8s cos a = the limit of 7 —* = the limit of = sa : s = Ss Cc e . ° a ; TL ° . j ae s' baa! 8 sin a = the limit of 44 = the limit of y ein ; C s'-s8 c U tan a = the limit of 2—Y . H aa 2: But, by Newton’s Seventh Lemma, in the first section of the Principia, we know that es Seg the limit of = 1: dx dy dy hence cos a = — ink er = ee fannie ds as ~ dx Again, 8 denoting the angle PG'O, G being the point in which the normal UV cuts the axis of z, dy : : dx dz cos = S1Tee== = sin = COs a = — tan = “COL a =i =25 te dee s ds’ p pars a TANGENCY. 161 We may express cos a and sin a in terms of dx and dy alone, without ds: thus, adding together the squares of the two equations COSA Oe sin a = ey - ds? see dh Ray, we get jee AA Aa ® ds ds or ds” = dx* + dy’. da? z dy* PEN ee a be COSY oe pein cs "ds dy” da? + dy’ Again, supposing the equation to the curve to be u= 0, we have, by differentiation, du du — dz + — dy =0 dx Bye” du du and therefore dx: dy —t—, dy dz du : dx dy’ merativeds: tein’ 2) COs a Sas ey - dz’ + dy du’ du’’ 2 as 2 dx dy du? 2 GEE cos’ 3 = sin’ a = ay >= de r dx’ +dy’ du’ dw da ay du cot ? = tan PRES da du dy Ex. To find the inclination of the tangent at any point of an ellipse 2 2 a ke fy =] to the coordinate axes. M 162 TANGENCY. x Here wate I, du_ 2x du_ 2 Er oa PN i rH) og B Bf Hence cos’ a = - = ——, , 22 BY he es ee oe + | 75 orks ob =) x kaa: 4 sin’ a = 2 PO ae 27 =] P ( ey 2 \ 2 a” 5 22 tan a = — ae =— Oe TEA 6? The negative sign in the expression for tan a, supposing and y to be positive, shews that the angle P 7 is obtuse, instead of acute, as in fig. (1). Equations to the Tangent and the Normal at any Point of a Curve. 101. Let R, fig. (2), be any point whatever in the tangent ALK at the point P of the curve AB. ee Let (x, y), (#', y'), be the coordinates of P, R, respectively. Then, 7 denoting the distance between P and R, TANGENCY: 163 j dz Z-X=P7 cosa=P7 —, ds ion eit er dy y—-y=rsma=r—"; / Pe 9 and therefore CP Te od, Ax da y a form of the equation to the tangent at P. An equation, usually more convenient, may be obtained in the following manner. Differentiating the equation u = 0 to the curve, we have multiplying the former and the latter term of this equation by the two equal quantities respectively, we obtain, for the equation to the tangent, i OU du L-X)—+ == —=0. Cl rl Mo 2 ay The equation to a line through the point z, y, at right angles to the tangent, that is; the equation to the normal at P, will therefore be, as we know by the condition of perpendicularity given in treatises on algebraic geometry, (z' — x) dx + (y' - y) dy =0, C-x y—-y “du du dx dy Ex. To find the equations to the tangent and normal at any point of an ellipse 2 2 ad ~ M 2 164 TANGENCY. Putting for ag Ee their values oa ; dx’ dy’ a equation to the tangent, “i , we get, for the (v' — x) [+ -y) e=0, or, by virtue of the laa to the ellipse, and, for the equation to the normal, (a! - 2) — Eat - y=. Distance of the Origin of Coordinates from the Tangent. 102. Let 6 denote the distance of the origin from the tangent, e being the inclination of 6 to the axis of z. Then the equation to the tangent must coincide with the equation / PAN oso z cose+y' sine =6. Comparing this equation with each of the equations to the tangent, viz. , du , du du du ee a “bh fetes we see that du 1 § CGS Eni dz BONG NG Gee lL) Wd Roadie BE | eiige ol EE ce nobis a . ¥ | dz” dy du 1 BeBe, dy TANGENCY. 165 and consequently du’ du’ di+dy 1 dé dy (ady —-ydzy & ([ du du\*’ w—+y — dx dy which determines the value of é. Intercepts of the Tangent. 103. Let z, be the value of z' in the equation to the tangent at any point of a curve when y'=0. The length z,, viz. OT in fig. (1), is called the entercept of the tangent on the axis of z. It is easily seen from the two forms of the equation to the tangent that du du xdy - ydx e de” dy. in the same way, y, denoting the etercept of the tangent on the axis of y, x gy +4 ie yde-ady_, "de * dy ede: ‘ du i dy Subtangent. 104. The portion MT, fig. (1), of the axis of z, contained ‘between its intersections with the ordinate and the tangent at P, is ordinarily called the subtangent at the point P. It is evident, from the equations to the tangent, that JZ7' is equal to du Pie Ese kA a7 eal 166 TANGENCY. Length of the Tangent. 105. The word tangent is sometimes used to denote the finite line PT, fig. (1), included between the point P of contact of the indefinite tangent 7A with the curve and the point 7' in which the indefinite tangent cuts the axis of z. In this sense of _ the word it is plain that the length of the tangent is equal to {y’ +(e — 2} igh du? — ee 7 dx ” dy dx Normal and Subnormal. 106. The finite lmes PG, GM, fig. (1), G being the inter- section of the indefinite normal UV and the axis of z, are frequently called the normal and subnormal respectively. It is plain that the subnormal is equal to the value of z' — x deduced from either of the equations to the normal, when y'= 0; or du dy , di iss: MF == — Ff (me Y de da’ dy and therefore du 3 , dy’ , dx PPOs) ss G +9 ga] ~ PG > i" +y a dy’ Form of the Equation to the Tangent to Curves of which the equations involve only rational functions of x and y. 107. Let «=0 be the equation to a curve; and suppose that UaU,+U,+U,t+Uut.... +4, u, denoting a homogeneous function of x and y of r dimensions. Then the equation to the tangent at any point 2, y, will be TANGENCY. 167 (OG , Uy, My , _ u%, 4 (du, du, du, , UM, igh athe aay ae Asay dy dy + nu, +(m-1)u,+(m-2)u,+ (m- 38) uU,+....+2U,,4+u,,=0. Lemma. Let v be any rational function of 2 and y, of ry dimensions: th a imensions: then Poser yey, where c is a constant coefficient, and a+$=7, the term cx’ y?® being a type of all the terms. Then = E(caa™y?), dv mes ere a 9h} T= Bry), and therefore es com: y do _ > (caz*y®) ai >> (c[3. x*y*) * de dy = X {e(a+ PB) ay*} = 7 D(cexty?) = re. By virtue of this Lemma we see that dus du pe a U,+ 2u, + 8U,+...+(m—1)u_,+nU,: me Y but 0 = NU, + NU, + MU, + NU, +... + NU, + NU, hence du —s- du n-2 n-1? po ae »— (m- 1)u,-(n-2)u, -(m— 8) U,— + -— 24, ,- U and therefore, from the equation 2: du du" —s du du ice anti deel: wat (ge set) cane ah Medea dos wy dizi dx epee eee +o) 4 dy dy dy. dy + nu, + (m—-1)u,+(m- 2)u,+ (- 3) u, +... + 24, +4, = 0. 168 TANGENCY. Ex. To find the equation to the tangent at any point of the conic section ax’ + by’ + 2cry + 2ax+ 2y +c = 0. By the above formula we see at once that 2 .(a' + cy +az)+y'.(b' +cxr+ by) +4 ax + by = 0. Oblique Azes. 108. The forms of the equations to the tangent are not altered if we suppose the axes to be oblique instead of rect- angular. Let P, P,, (fig. 3.) be the two points in which the secant HK" cuts the curve AB, referred to oblique axes Ox, Oy. Draw PM, P,M,, P’'M, parallel to yO and cutting Ox in M, M,, M’, P' being any point whatever in the secant H'K’. Draw Pp,p’, parallel to Oz, and cutting PM, PM’, in p,, p’. et; OM= 2. Pie 70M =a. iad, OM m2 gil Then, by the similarity of the triangles P'p'P, P,p,P, we have PtP, Lp ep, NG eae ere or In the limit, when P, moves indefinitely near to P, x, y,, become 2, y, respectively, and ¥,7—Y a, - 2X TANGENCY. 169 becomes ie The secant Ht’ also becomes a tangent to the c curve at P. ‘Thus the equation to the tangent at P is y-y _ dy z-x dx’ Ce Le ata Bite .. de EP whence also, as in Art. (101), we have, as another form of the equation, du : du ret lNerrsa ts Mee Lira ar (canlgOnn) CHAPTER “IT. ASYMPTOTES. Definition of an Asymptote. Method of finding Asymptotes. 109. AN asymptote is a tangent line to a curve, such that, although the distance between the origin and the point of contact is infinite, the perpendicular distance of the origin from the line is finite. | Ex. 1. ‘Take for instance the curve “U=@-y=05 where @ is supposed to be a number greater than unity: then, by the formula du 9, , de ,_ au, du dz dy ae dx dy : we have, for the equation to the tangent at any point, log a.a*.z -y =loga.a’.x-y = wloeig sd =a Suppose now that z=- 0: then a” = 0, and _ Z ate OE will become less than any assignable quantity, and therefore the equation will ultimately become Pee Cri ty elt n(n — 1) mB du ste die ba dx" dy eartien it 7 aia aak dx” dy” a a3 in du els aby ay This equation, which is homogeneous in 2’ and y’, is equiva- lent to m linear equations in 2’ and y’, which will represent the tangents to the several branches of the curve, 2 in number, at the point (z, y), the point (x, y) being considered the origin of coordinates. ‘Thus the degree of plurality of a multiple point is defined by the order of the lowest partial differential coefficients of « which do not vanish. 184. MULTIPLE POINTS, CONJUGATE POINTS, CUSPS, &°©. Ex. 1. To determine the multiplicity of the point z=0, y=0, in the curve u=2'+y' - azy’ = 0. At the point in question ae 4z° — ay’ = 0, iy 4y° — 2axy = 0, Be 122 = 0, oe =- 2ay=0, —e 12y’ — 2az = 0, ae 242 = 0, ee Os pa 2a, S5 = By = 0. Hence, for the determination of the tangents at the multiple point, which we see is atriple point, we have, substituting for the partial differential coefficients in the equation 19 OU ptr hey ee iat 1, VU a + 82" Paya 3z'y SPER er, a0, — 6az'y” = 0, which is equivalent to 2 = 0, and y”=0: the former of which equations shews that the axis of y touches one branch of the curve, and the latter, that the tangents to two branches coincide with the axis of z. The form of the curve is exhibited in the diagram: as | Ex. 2. To determine the multiple points of the curve (y° — 19 = 2° (Qe 4+°3). In this case w=(y’- 1) - 2 (244+ 3)=0, and, as conditions for a multiple point, GUN AD TS ee ey. dz du , —=4y(y -1)=0. dy MULTIPLE POINTS, CONJUGATE POINTS, CUSPS, &C. 185 These three equations are satisfied by each of the following systems of values, ne Ne ae Proceeding to second differentials, we have du d*u du pe AE GB ee he ee EY A) dx’ Tete dx dy dy’ y The equation SOLES Sl Boy AOR RL ANE di’ Y dady y dy” ae becomes, for the first system of values of x and-y, 3a" — 2y” = 0, and, for each of the two second systems, — 8v" + 4y” = 0. Thus we see that there are three double points. The figure is subjoined. P vay Multiplicity of a Multiple Point at the Origin. 119. The existence and multiplicity of a multiple point at the origin may be ascertained more simply by inspection. Let the equation to a curve, arranged in groups of terms of diffe- rent dimensions, be , le TN ot) en a ae x 1856 MULTIPLE POINTS, CONJUGATE POINTS, CUSPS, &C. u, denoting generally a series of terms of r dimensions, and w, denoting those of lowest degree in the equation. ‘Then, in the immediate neighbourhood of the origin, we may neglect terms of higher compared with those of lower orders, so that the equation will become u, = 9, the dimensions of this equation determining the degree of the multiplicity, and the simple factors, into which it may be de- composed, defining, when equated each of them to zero, the directions of the branches. This method of finding the mul- tiplicity of a multiple point, may be readily deduced from the general equation given in the preceding Article. Ex. 1. Taking the first example of the preceding Article, we have, retaining only the term of the third dimension, Lif (e which shews that the axis of y touches at the origin one branch, and that of z two branches, of the curve. Ex. 2. Take the curve x - 2ax*y — 22°y’ + ay’ + y* = 0. Then, retaining terms of the lowest dimensions, we have — 2ax’y + ay’ = 0, whence, for the equations to the tangents at the origin, y=0, y=-24, or y=+ tz; so that there is a triple point at the origin. Point of Osculation. 120. A point of osculation is a multiple point in which the several branches of the curve have a common tangent. ‘Thus cusps are a species of points of osculation. Suppose that there are only two branches at the point, then the roots of the equation 2 i, UU Bp hey 2 aU +y" — = Sci ray = eat vy aad dy Oo Settee see MULTIPLE POINTS, CONJUGATE POINTS, CUSPS, &c. 187 must be equal: hence, as a necessary condition for a point of osculation, du \’ du d*u ddy} dx dy If this condition be not satisfied for any point of a curve corresponding to the three equations ui ey ax Yy then we must have either ( d’u \* “ du d’u dady}~ dx dy*’ in which case there will be a double point with two distinct tangents ; or dx dy dz dy’ when the equation (1) will give impossible relations between 2’ and y’, and (x, y) will be the coordinates of a conjugate point. Remark on the Theory of Multiple Points. 121. If there be a multiple point in a curve, its position and its multiplicity may be ascertained by investigating the pairs of values of z and y which satisfy the three equations du du dz f 2 and by determining the order of the lowest partial differential co- efficients of « which do not all vanish at the point. The directions of the tangents will be ascertained by the formula of Art. (118). If the relations between 2’ and y’', expressed by this formula, be impossible, the existence of a conjugate point is at once indi- cated. We cannot however be sure that the point is not a conjugate instead of a multiple point or cusp, even when all the relations between 2’ and y’ are possible. Additional considera- tions are necessary in order to ascertain this to a certainty: an examination of the general nature of the curve in the neighbour- “= 0, 188 MULTIPLE POINTS, CONJUGATE POINTS, CUSPS, &C. hood of the point, by an algebraical discussion of its equation, is sufficient for the purpose. Ex. ‘Take the curve (y - ex*} = (x - a). (2 - by, a being supposed to be less than 6. Then, putting u=(y—c2x*)!-(«-a). («- bY =0, du ; . eae bez’ (y — cxz®) - 6 (x — a) (x - bY - 5 (x@ - a) (x - DY = 0,7 du poe (y= cr.) =O dy d ) f we see that oD Se al Ge du du d*u (Chen also 2 180-0 ee a ee dx’ * dady Gop a and therefore, from the formula a d*u pe d*u by? ae ‘ dein Wake dx dy Y Days there is y” — bacz'y' + 9ate’x” = 0, (y' — 8a°ex'f = 0, which might seem to indicate a point of osculation, when the equation to the common tangent of the two branches would be y = 8a°cx’. It is easily seen, however, that the point is really a conjugate point. For 5 y = cx’ +(x - ay (a - Bb), which shews that y is impossible when z differs very slightly from a. For further information on the subject of this Article, the reader is referred to a paper on the General ‘Theory of Multiple Points in the Cambridge Mathematical Journal for November, 1840. ' _ MULTIPLE POINTS, CONJUGATE POINTS, CUSPS, &C. 189 Points d’ Arrét or Points de Rupture. 122. An algebraic curve never stops abruptly in its course, that is, it never possesses singular points of the kind called by French writers points d’ arrét or de rupture. Such points are however of frequent occurrence in transcendental curves. For instance, in the curve belonging to the equation 1 Yt Cx. the form of which is there is a point d’ arrét at P. The impossibility of the existence of such points in curves represented by algebraical equations depends upon the fact that impossible roots enter algebraical equations, involving one un- known letter, by pairs. Suppose in fact that, when z is equal to a —h, y has an impossible value for each small value of h how- ever small, and that y has a possible value when z is equal to a+h. Then, when z passes from a —h to a + h, one value of y, and therefore, by the nature of algebraical equations, two values of y, the two values being of the forms a + V(— (9), must change into possible ones, which will evidently, in consequence of the correspondency of their values, be equal to each other when h/ is indefinitely diminished. ‘The existence of two equal values of y, corresponding to the value a of xz, shews that there is no abrupt termination of the curve at the point of which the ab- scissa Is d. Points Saillants. 123. A point saillant is a point of a curve where two branches of the curve stop abruptly and have tangents inclined to each 190 MULTIPLE POINTS, CONJUGATE POINTS, CUSPS, &C. other at a finite angle. Such points are frequently to be met with in transcendental curves, but can have no existence in curves corresponding to algebraical equations. Ex. 1. Take the curve of which the equation is z bey 1 +e Then wv. ++. FT l+te w.e*.(l+ey Suppose that z= +4 0: then 0 y=—=0, ee) and SP dz Suppose that z= - 0: then a eet 1 dy d euies an Ts The origin of coordinates is therefore a point sailiant: the branch corresponding to positive values of the abscisse touches the axis of z, while the tangent to the other branch at the same point is inclined at an angle of 45° to this axis. Ex. 2. Take the curve 1 y=xtan” —. x There will be a pownt sadllant at the origin of coordinates: the directions of the two branches at this point being defined by the values $m and — 47a of ve We have observed that an algebraical curve does not admit of points saidlants. ‘This may readily be shewn. Suppose, in fact, that between the equation to the curve u=0Q, du wh du | ey dx dy ie and its derivative 0, MULTIPLE POINTS, CONJUGATE POINTS, CUSPS, &c. 191 we eliminate y: we may then obtain an algebraical equation between z and y’, free from radicals and fractional forms. If then we conceive a curve to be constructed, of which the abscissa shall be always equal to z and the ordinate to y’, this curve can have no point d’arrét, which would necessarily be the case if the primitive curve had any point saillant. Branches Pointillées. 124. We occasionally meet with equations, the geometrical loci of which consist, either entirely or in part, of a series of conjugate points, forming branches pointillées. Ex. Take the curve YN ee SER ae (2) = tos {sin (E)} It is evident that y will be always impossible when sin - has any negative value or any positive value less than unity ; and therefore, unity being the greatest value of the sine of an angle, we must have Se ANE; y = 0, a 2 » being an integer. Thus we see that the geometrical locus of the equation consists of a series of conjugate points arranged along the axis of z at a common interval equal to 2zra, the axis of abscisse being thus a branche pointillée. CHAPTER IV: CONCAVITY AND CONVEXITY OF CURVES AND POINTS OF INFLECTION. Conditions for Concavity and Convexity. 125. THE object of this chapter is to investigate the con- dition that a curve, of which the equation is given, may at any assigned point turn its concavity or convexity towards either of the coordinate axes, and to determine those peculiar points of the curve, called points of inflection or of contrary flexure, at which concavity is succeeded by convexity, or conversely. In fig. (1) the curve AB is concave at P towards the axis of x, and convex towards that of y; in fig. (2) there is a point of inflection at P, the curve being concave towards the axis of x at each point in the arc AP, and convex at each point of the arc PB. . Let ¥, ’, be the inclinations of the tangents of the curve at P, P’, in fig. (1) to the axis of z, P’ being a point near to P: then, as we pass through P from A towards B, it is evident that tan ¥ will keep continuously decreasing: hence, z, 2’, being the coordinates of P, P’, tan W' — tan ¥ = a negative quantity: eas 8 q y CURVES AND POINTS OF INFLECTION. 193 and this will be true however near P’ may be to P; and there- fore, proceeding to the limit, we see that, as the condition of saat a negative quantity. dx If the curve were convex at P towards the axis of z, it is evident from like reasoning that dtan + =a positive quantity. mrss I q y concavity, But tan pv = o : hence, that the curve may be concave towards d*y L dx” and, that it may be convex, that oe be positive. the axis of x at a point z, y, it is necessary that be negative, It is easily seen that the conditions which we have shewn to be necessary are also sufficient conditions for concavity and con- vexity. For it is evident that the curve will be concave or convex towards the axis of x at a point (x, y) accordingly as is negative or positive for all indefinitely small values of wW’ 2 dy 1s nearly equal to yr, that is, proceeding to the limit, as negative or positive. If y be negative, or the point P be on the opposite side of the axis of x, then, as may be seen simply by changing the sign of y throughout, we must have, for concavity, qd’ . , . a = a positive quantity ; hy and, for convexity, J d*y : : is a negative quantity. Without regarding the sign of y, we may state generally that the sufficient and necessary condition for concavity is that y wae a negative quantity dx - 2 O 194 CONCAVITY AND CONVEXITY OF and, for convexity, that d*y LEP Ex. To find where the Witch of Agnesi, of which the equa- tion is = a positive quantity. 2a-2z y oct 4a a x is concave or convex towards the axis of z. Differentiating, we have dy 4a’ Vag aoa dy dy 8a or, multiplying by y’ and replacing y’ 2 by its value in terms dx? of x, 3 16a° 8a 2a — Pai He dz x x d’y 16a whence jgeeeitlacd ye ee TT ira Arey fb se Semin As ) d*y This result shews that y is negative when z is greater, and dx’ *,° : 3a 3a positive when z is less than “ap Thus from z = 0 to # = i the ; : 3a curve is convex towards the axis of z, and concaye from z = a to z= 2a. There must accordingly be two points of inflection corresponding to the value 2a of z. Condition for a Point of Inflection. 126. Since a point of inflection separates two portions of a curve, one of which is concave and the other convex towards the axis of z, it follows that, as x keeps continuously increasing, tan ~ or e must either keep increasing as we approach the a point and decreasing as we recede from it, or conversely. CURVES AND POINTS OF INFLECTION. 195 Hence, that there may be inflection at any proposed point, it is : d : sufficient and necessary that a have a maximum value, or that d’ 4 i experience a change of sign as we pass along the curve from one side of the point to the other. In order then to determine points of inflection, we have only : : dy . to ascertain those values of x and y which render s either 4 a maximum or a minimum. Ex. 1. ‘To find the point of inflection in the curve at ie ax zh a’y a 0. Differentiating twice, we get a a 4a° — Qa®e + a? SY = 0, bp 3 a" 122" — 24° +a a = 30). 2 Putting le = 0, in order to get the values i x which cor- respond to the maximum or minimum values ee — , we have = pie gis. iO oa and therefore y= =. 36 Differentiating a third time, we see that “aA, Ago ae Os dz* d*y ; d*y which shews that —< has a finite value when —4 = 0. Hence dz dz a dy ; eee the values + a of « make ae a maximum or minimum, and therefore correspond to two points of inflection. Ex. 2. To find the points of inflection of the curve gt = ay’ + ary”. 196 CONCAVITY, AND CONVEXITY OF We have pa oa (oa) eh A law a 2 ate (a + a’) ay a’ (2a* — 2”) dz a, ant (2 +a’) 2 As far as signs are concerned, we may take » instead of 3 “ where v= + (20 - 2”). When v = 0, we have z =+ a v2, and dv : : — = + 2x =a finite quantity. Ae q ¥ Hence the values + @ V2 of x correspond to maximum or dy da minimum values of —, and therefore to points of inflection. Ex. 38. To ascertain whether the Lemniscata of which the equation 1s (x° a yy = a’ (xe i y’) has a point of inflection at the origin of coordinates. Differentiating twice, we have a@ry)(ery B)=a(e-y 2), dy\ d’y dy’ t(e+y 2) 2@en(i ian 75] hol Watt hie From the last equation we see that, when z = 0 and y = 0, dy —=4+1 daesiinin d’y iy fieedas ; d’y the value of —*% remaining indeterminate. In order to find —4, dx dx we must differentiate again. CURVES AND POINTS OF INFLECTION. 197 Thus dy d*y dy’ (ary B) (r+ y 9+ % dy d*y 2 2 ie! jae ee y ae ” de dx* d*y — 2 ee ay Fs d*y When z = 0 and y= 0, we see that = 0. In order to dx* d*y determine the corresponding value of —*, we must proceed to dx another differentiation: we then have, omitting all those terms which vanish when z and y are both zero, dy” was 2 dy d*y 16 (1+ 3) - 4a° Bre d°y 16 dx = whence We have shewn therefore that a has a zero value, and if ‘Zz a finite one, at the origin of coordinates, whether we take @ be equal to + 1 or to — 1. Hence both branches of the curve have inflection at the origin. Symmetrical Investigation of Points of Inflection. 127. A point of inflection being an absolute peculiarity in a curve, and not, like a point of concavity or convexity, having especial reference to either axis in particular, it is desirable to develop also a method of determining such a point which shall be symmetrically related to both axes. 198 CONCAVITY .AND CONVEXITY OF Let* the equation to an algebraical curve, cleared of radicals and negative indices, be represented by 0 ek Ab = Read | ks ee A where F is a rational function of x and y. Let ds denote an element of the arc of the curve at the point x, y, and let s be taken as the independent variable. Let = ge = ue dx dy PK. Wb a? a’F Sore ~ dx dy’ dy’ ’ _ dx _ dy ,_ db tie dm iodse as eas ~ ds Then, differentiating (1), we have ESET AIA Ve AIA TS Hee (2); differentiating (2), Pu + 2lnw+mo+lU0+m'V=o0... .... (8). Again, it is clear that 7’ + m* = 1, and therefore Le WAT Se a geen ee From (2) and (4), we get “UV=amU; hence, multiplying (3) by U, we obtain. U (Pu + 2lmw + mv) +l (U? + V*) =0, or, by virtue of (2), AES ara (Viu- 2UPw + Ur) +0 (0+ V%)=0.... (5). Jn a similar way we may shew that 2 a 2UVw + U'v) +m (U" + V*)=0.... (6). * This method of determining multiple points was published in the Cambridge Mathematical Journal for November, 1843, CURVES AND POINTS OF INFLECTION. 199 Again, by the relation 7’ + m? = 1 and the. equation (2), we get hence (5) and (6) give us U(V*u-2UVw+ Ur) +0(W'?+ Vy =0.... (9), V (Vu - 2UVw +07) +m (0+ V*Y=0.... (8). Now at a point of inflection it is evident that 7 and m must be the one a maximum and the other a minimum: hence, as we pass in the neighbourhood of the point along the curve from one side of the point to the other, we know by the theory of maxima and minima that 7’ and m’ must each of them change sign. It is evident then, from (7) and (8), that U (V*u-2UVw + Ue), and V (Vu - 2UVw + U0), must both change sign. Suppose first that neither U nor V changes sign as we pass through the point; then the sufficient and necessary condition for a change of sign in the value of /’ and m’, is that Vru-2UVwiUr...... Tea ele (9) change sign as we pass through the point: this condition evi- dently involves the fact that at the point itself Lg TION BR COMET Bc SO Ep aa (10). Next suppose that U changes sign; then it is evident that the expression (9) must not change sign, for otherwise /’, as will be evident from the formule (7), could not change sign. But from (8) we see that for a change of sign in the value of m’, one and one only of the quantities V and (9) must change sign; hence V only must change sign. ‘hus we see that if either of the quantities U and V change sign, both must do so, and that (9) must not change sign. If U and V both change sign it is clear that at the point itself Ux. 0.40 20; which are the conditions for multiplicity of branches at the point. 200 CONCAVITY, AND CONVEXITY OF The general rule, therefore, for finding points of inflection may be thus enunciated. First ascertain the values of x and y which will satisfy simultaneously the equations F=0, Vu-2U0Vw+ U'v= 0, and reject all the pairs of values thus obtained which do not, as we pass through the corresponding points, correspond to a change of sign in the expression Vru-—-2U0Vw + U2, or which do correspond to a change of sign in either U or V: the pairs of values of z and y, which are retained, correspond to points of inflection. Secondly, ascertain those pairs of wee which satisfy simul- taneously TEE Os te Wand KE 03 and reject all of these pairs which do not correspond to a change of sign in both U and V as we pass through the corresponding points along one or other of the branches, or which do corres- pond to a change of sign in the expression Vu —-2U0Vw + Ue. In the preceding investigation we have supposed F' to be a rational function of z and y. Should this not be the case it will be evident, from what has been said, that in addition to the values of 2 and y, which may be obtained by the rule which we have enunciated, we must likewise take those which will render in the first case, F=0, V*u-2U0Vw+Uv=0@ and, in the second case, Five. 05, Uiecny Sipe son the conditions depending on change of sign being the same as before. Ex. 1. Let the curve be FF = ax’ + by’ - c= 0. ‘hen Ce 307%, i Vie 8by’, u=6axz, w=0, v= bby, CURVES AND POINTS OF INFLECTION. 201 Hence, from the formula (10) there is, if we cast out constant factors, zy (ax’ + by’) = 0; or, by the equation to the curve, xy = 0. Thus <= 0, or y=0, and in both cases neither U nor V changes sign, while the formula (9) does change sign. Hence we have two points of inflection, C z=0, 4 arenes 5° 4 C and Sees RIKI Poca a Ex. 2. Take the curve Fa(e+y) - v2’ + by’ = 0, and suppose that we wish to find whether there be a point of inflection at the origin. Then TF = 22 (22° + By" —.a°), V = 2y (22’ + 2y’* + 6°), u= 122 + 4y’ - 2a’, v= 42° + 12y’ + 20°" w= 8ry. From these results it is evident that U and V both change sign if we change x and y each of them from + 0 to + 0. More- over it is clear that neither V*u, U°v, nor 2UVw, experience ° any change of sign when we put + 2, +y, for +z, + y, respec- tively. Hence the expression (9) does not change sign. If we had kept y positive or negative throughout, while we changed x from + 0 to ¥ 0, the expression (9) would have changed sign, and flexure would not have taken place. Hence we see that the branch which passes through the origin from below to above the axis of z, or that which passes from above to below, will have an inflection at the origin. 202 CONCAVITY AND CONVEXITY OF CURVES, &C. Ex. 3. Let the curve be Then Ye ey Se 38a \a 3b \b Hence the expression V-u-2U0Vw + Uo will vary as Putting this expression =o, we get x= 0, or y= 0; and therefore, by the equation to the curve, y=), z=a, respectively. It is evident, then, that as x passes through 0, U and V do not change sign while the expression (9) does; and similarly for y: hence there are two points of inflection, viz. x=0, y=6, and zr=a, y=9%. (07808\x) CHAPTER V. ON THE INDEX OF CURVATURE, THE RADIUS OF CURVATURE, AND THE CENTRE OF CURVATURE, OF A PLANE CURVE. Index of Curvature. 128. Ler ww, w+ dy, be the inclinations of the tangents PT, QS, at points P, Q, of a curve AB, very near to each . RY other, to the axis of z. ‘Then it is evident that the greater be the angle dW between the two tangents, for a given small arc PQ, or the smaller be the are PQ for an assigned value of the small angle dy, the greater will be the curvature of the curve in the vicinity of P. Hence, proceeding to the limit, when the arc PQ or 6s becomes less than any assignable mag- dy nitude, we see that rh is a measure of the curvature at P. Is The angle dy is called the angle of contingence, and Me the s index of curvature at P. “ ftadius and Centre of Curvature. 129. From P and Q draw two normals PC, QC, meeting in C. ‘These two normals will evidently include an angle dy. 204 INDEX OF CURVATURE, Let PC=p, QC= ’, the chord of PQ =’e, and let 57 -—a denote the angle between the chord of PQ and the normal QC. Then it is plain that c sin (ba —a)=p sin oy, sin dy sin ($7 - a), et p but, proceeding to the limit, when dy, c, and a, become less than any assignable quantities, sin dy and ¢ vanish in a ratio of equality with oY and 6s respectively: hence C Now, p and p’ being ultimately in a ratio of equality, it follows that a circle described with C as a centre, and touching the line PT in P, will ultimately touch the lnme QS in Q. The angle of contingence will accordingly be the same ultimately for the circle as for the curve: also the arc PQ in the circle will be ultimately in a ratio of equality with the arc PQ in the curve, since each of these arcs, by the 7th Lemma of Newton’s Principia, vanishes in a ratio of equality with their common chord. Hence a circle so described has the same curvature as the curve at the point P. This circle is called the osculating circle, or the circle of curvature at the point P, p the radius, and C the centre of curvature. . ‘The equation (1) shews that the index of curvature at any point of a curve is equal to the reciprocal of the radius of the - osculating circle. Expression for p when x is the Independent Variable. 130. Differentiating the equation (Art. 100) dy t isihey 19 iy dx considering x the independent variable, we have ay RADIUS OF CURVATURE, CENTRE OF CURVATURE. 205 but, by Art. (100), (ez) (&) hence (‘S) = \ da | ms dae’ ds ds* dy’ \3” vata 1p ee da’ ( bi vs) 1 2\3 and therefore Py cpeccl ls From this formula we perceive that the index of curvature will become zero, and the radius of curvature infinite, whenever a’ e e . . a is equal to zero: the osculating circle will then degenerate Z into a straight line and coalesce with the tangent. Such will be 2 the case, for instance, at points of inflection, where ce = 0 and 2 dy : — is not infinite. If, at any point of the curve, _ becomes Er dz infinite, while & is either zero or of finite magnitude, the index of curvature will become infinite, and the radius of curva- e and. es taneously infinite, the expression for p* will assume the form ture will vanish. If the quantities become simul- CO ts : i its real value must then be estimated by the rules for the evaluation of indeterminate functions. If one of the functions ay, da : : : s : ae becomes discontinuous, and experiences an abrupt change of value, such will also be the nature of the mdex of curvature: such a peculiarity will present itself, for example, at a point sadllant. Ex. 1: To find the radius of curvature at any point of the curve y? = 4me. 206 INDEX OF CURVATURE, We obtain, by differentiating the equation d i Et d’ lace ~-1\3 3 sel nz yas ae 4m) Hence p= — + = ——_— . 1mz m If x=0, then p’=4m’, p =2m; which shews that the radius of curvature at the vertex of a common parabola is equal to half its latus rectum. Expressions for p when s is the Independent Variable. 131. Since, by Art. (100), cos~=— and sin Y=, we see that, s being the independent variable, sin y Se SS and cos ~ ay = a : hence, squaring and adding these two last equations, Whe EE aD ase ene ad is ie ds Also SY AE AEE ST ds sin W dy ds dy whence = — as : a*x RADLUS OF CURVATURE, CENTRE OF CURVATURE. 207 similarly we may shew that Expression for p in terms of dx, dy, d’x, d’y. 132. Since tan y = a » we have and therefore dy = 1 df (dzxd’y - dy d’xy fds dea dy Expression for p in terms of Partial Differential Coefficients. 133. Let u=0 be the equation to a curve. Differentiating this equation twice we get du du 7 ont ifs Oty cere: ieee < Gu); OD) NTE Gls d*u d*u Se e = d: 2 Say d. aR 5 eee 2 ° -(¢ d’x +S ey) qt OH EGR adatcrg' dy’ ... (2) Multiplying (2) by dy, dz, successively, and in each case availing ourselves of the relation (1), we get d*u d*u de dy + Fe ay"), du a ae d*u a, ae dy dy We) = dy (Fe dx® + 2 — a fire dx dy dy” adding together the squares of these two equations, we obtain ae ‘ i) (da dy — dy day oe (dy dia — de dy) = de (Fe det +2 gi aires ay") A dx’ d*u du a’u 1. a 2 ha” ‘ : = (dz + dy) (Fe dz’ + 2 dy Yt Gg ay’) 208 INDEX OF CURVATURE, whence, by the formula of Art. (132), the ah “Ub d°u a 2 dz d dy’ , (aes dx dy % I Oy /') pi 2 2\2 du’ duu’ \ Y (de + dy?) aaa d therefore rt placthperLaaeiy a hirmeieintart aaa aie. ame and therefore, replacing dz, dy, by the quanti prince ts. which, by virtue of (1), they are proportional, we get 1 \dy* dz’ dx dy dxdy dx dy) p° due du?’ we i 7) Cor. If the function w consist of two parts, of which one contains x alone, and the other y alone, du x dzdy (S du ip du’ id) and therefore ‘a e dy" da da’ dy e ; p DEN casey de ay Ex. To find the radius of curvature at any point of the curve ER An tg et ae: a p3 1=0. du 2a du 2y Here ae oe Hi mete: du AUP au 4 du gees de Wa de dy dys Lh Ee vasncdin Nee 1 i ee Bi bas ui 1 RADIUS OF CURVATURE, CENTRE OF CURVATURE. 209 Another method of finding the Radius of Curvature. 134. Let 2 denote the distance of Q, in the figure, from the line CP, and 6 its distance from the tangent at P. ‘Then, since ultimately the circle of curvature touches PT, QS, at P, Q, we have, by the nature of a circle, limit of 2” = limit of (2p — 6). 6, or, since 6 vanishes in the limit when compared with p, rhe 25 Ex. 1. To find the radius of curvature at the vertex of p = limit of a parabola y? = Ame. 2 Here p = limit of YL _ 9m. 2x Ex. 2. To find the radii of curvature at the extremities of the axes of an ellipse. If a and & be its semi-axes, and if the major axis and the tangent at one of its extremities be taken as axes of coordinates, its equation will be }? Gee 7 (2az - x’). 2 2 2 Hence p = limit of L == (a-Ja)=—. Similarly it may be shown that the radius of curvature at an . . une a extremity of the minor axis 1s equal to a P 66210.9 CALA Sets tteev te ANALYTICAL DETERMINATION OF THE CENTRE OF CURVATURE. THEORY OF EVOLUTES AND INVOLUTES. Determination of the Coordinates of the Centre of Curvature. 135. Let a, (3, be the coordinates of C, the centre of the osculating circle of a curve AB at the point P. Then, z, y, being the coordinates of P, the inclination of the tangent PT at P to the axis of x, it is plain that and 8B —y = pcasah = ag on dz _ dy But COB a De ER dy\ dxd’y — dy dz | and dp =d tan™ (4 )- ZERO dz’ + dy’ hence a-%=- ge as Wades and Be ae Ley, dz d’y — dy d*x” EVOLUTES AND INVOLUTES. 911 These two formule will enable us to determine a and 8 for any assigned point of the curve. Suppose s to be the independent variable; then differen- tiating the equation di? + dy? = ds’, we have dx d*x + dy d*y = 0, and therefore the formule for a and 8 are reduced to dx + dy’ ; oe = ; *E qa 19 le 2 7e 39 5 SA aes ayy dx + dy” Formule for the Coordinates of the Centre of Curvature in terms of Partial Differential Coefficients of u. 136. From Art. (135) we see that (a - 2) da+(B-~y) dy=0.0....505, (1), and (a- 2) d’x&+(8-y) d’y - dx’ — dy’ =0.... (2). Between these two equations, together with the equation to the curve and its first and second differentials, viz. CAN easter, oghec deine iE eh eae (3), du du Cae LO ae, ate du Tuy er eee sea es, 2 Je da pr * = eee ea tle 3 dat FPRT y+ aad 0... (5), we may eliminate the six quantities z, y, dz, dy, d’x, d’y. In fact, from (1) and (4) we have du du CLO IP raies Ver. Banat caecum ee and therefore, from (2), A | Ae ’ dz P.2 212 CENTRE OF CURVATURE. whence, by (5), we have (a-2) 1,9 au Pu EU oy Oy ae Ai? + dy? + mama ah as ? le dy dx dy + —, diy dy* \= 0, dz and therefore, by (4) and (6), due dw’ fone cy SRE NUL cece Fe du du du’ d’u du du du duvduo ” di dy dy’ dx* ~ dx dy dz dy ” de dy’ Locus of the Centre of Curvature. 137. Between the equation w= 0 and the equations (7) of the preceding Article, we may eliminate z and y: we shall thus obtain an equation between a and [3 alone, which will be the equation to the geometrical locus of C, the centre of curvature, when the point P of the curve is supposed to be variable in position. ‘The locus of C, for a reason shortly to be explained, is called the evolute of the curve AB, which is itself called the envolute. Ex. To find the equation to the locus of the centre of curvature of the ellipse a 2%, Wty Here n=3(S40-1)=0, de @? dzedy °° -dy B’ due du | du du d’u tea iis 1 (= ‘|= 1 ee —— hence (a-2)% =(p - Non -b (24 +o) \@ EVOLUTES AND INVOLUTES. 218 and therefore we get, for the equation to the evolute, (aa) + (68)! = (a? - BY. To shew that the Normal at any point of the Involute is a Tangent at the corresponding point of the Evolute. 138. By the formule (7) of Art. (186) and the equation u= 0, we may obtain aand (3 in terms of x; thus a and [3, as well as y, are functions of x: hence as z varies, a and (3 as well as y must simultaneously vary. Differentiating the equation (a- x) dx+(P-y) dy=0, we shall accordingly obtain (a-z) d*x+(B-y) d’y + da dz +d dy - dz’ - dy* = 0, and therefore, by virtue of the equation (a-2) d*x+(—y) d’y- dz’ - dy’ =0, we have da dx + df3 dy = 0. This equation shews that the tangent to the involute at the point (%, y) is at right angles to the tangent to the evolute at the corresponding point (a, (3). Hence the normal at (2, y), which passes through C, must be a tangent to the evolute at (a, (3). 914 CENTRE OF CURVATURE. Generation of the Involute by the end of a thread unwound From the Evolute. 139. Since a-Z=~-psiny, 3 -y=p cosy, we have faa 45 = yy = io. cee OL and therefore (a — x) (da — dx) +(8-y) (dB - dy)=pdp.... (2). Also we know that (a—z) dx+(B—y)dy=0.......... (8), and du dd + UBdy sO iio. oe... 4). From (2) and (8) we see that (a-z)da+(B-y)dB=pdp........ (5), and, from (3) and (4), (a-2z)dG-(B-y)da=0.......... (6). Adding together the squares of (5) and (6), we obtain {(a-2)' + (B -y)} (de? + dB?) = p'dp’, and therefore, by (1), da’ + Af?’ = dp’. Let o denote an arc of the evolute originating at any pro- posed point and terminating at (a, (3): then do* = da’ + d[3° = dp?, and therefore, the positive or negative sign being chosen ac- cordingly as p decreases or increases with the increase of a, do+dp=0, aitp=c, c being some constant quantity. We proceed now to give the geometrical interpretation of this result. First take o+p=c. Let Cand P be any two corresponding points in the evolute and involute respectively. Let A’CB be an arc of the evolute. Join CP, which will be the radius of EVOLUTES AND INVOLUTES. 915 curvature to the involute at the point P, and a tangent to the evolute at the point C. Let A’C=o, and A’B’=c, Then A'C+CP=c=A'B. Hence it is obvious that, if a thread, of which the length is ce, be fixed with one end at A’, so as to touch the curve at this point, and be wound about the curve A’B’ by a hand taking hold of the string at P, its extremity P will trace out the involute AB. Next take o-p=c. Let B’B'=c, B’C=zc, the origin of the arcs being now some fixed point B’. Then BOG Cha a= BB; a result which points out the very same geometrical property as when we adopted the positive sign. From the geometrical property which we have established have arisen the names evolute and involute. To find the length of any Are of the Evolute of a Curve. 140. By the preceding Article we know that o+p=c. Let o,, p,, and o,, p,, be corresponding values of o, p: then Oo, +p, =C=9,+ Py, or O, ~ 5, = P, ~ Pe Hence, to find the length of an arc of the evolute correspond- ing to any proposed arc of the involute, we must take the difference between the radii of curvature at the two extremities 216 EVOLUTE AND INVOLUTE. of the latter arc, and this will be the length of the former ; provided that, for the whole interval, p either always decreases or always increases as o increases. Ex. To find the length of the whole evolute of an ellipse xv y° oy is Se = j a AT The radii of curvature at the extremities of the axes major 2 2 and minor are 2: and = hence the length of a quarter of the whole evolute is equal to or the length of the whole evolute is equal to a — & ab 4 ( @17 ) CIA PEP RaawEr. CONTACT OF CURVES. —______. Definition of Order of Contact. 141. Let y =f(v’), y' = F (2), be the equations to two curves. Suppose (z, y) to be a point common to both curves. ‘The two curves are said to have a contact of the first order at the point (, y), if F(a) =f (@); of the second order, if F(2)=f'@), F'(a)=f"(@) of the third order, if Fi(@j=f'(z), F’@=f'@), F"@=f"@), and so on; the contact being of the 7* order if F'(«) =f' (a), F" (2) =f" (x), F(x) =f" (a), ....F’(a)=f" (2). The higher the order of Contact, the closer the Contact. 142. Let the curve y' = ¢(z’) have, at the point (2, y), a contact of the m‘ order with the curve y' = F(z’), and of the n> order with the curve y' = f(z’), and suppose m to be greater than n. ‘Then, by the theory of vanishing fractions, when h becomes less than any assignablé magnitude, F(at+h)-go(xt+h) Fi(x+h)-9¢'(e£+h) fle+h)-¢@+h)~ f@+h-9 a+!) Bi (e+h)- path) _ ~ Ff" @+h)— g"erh) SLANE SONG OS le fea h)— grerh) fr(@)-gr@) 218 CONTACT OF CURVES. or, corresponding to a small increment A of z, the difference between the ordinates of the curves y'= F(z’), y'=9(£), 1s indefinitely small in comparison with the difference between the ordinates of the curves y'=f(z'), y'=9(#). Thus the’ contact is infinitely closer when of the m” than when of the n™ order. Order of Contact dependent upon the number of Parameters. 143. Let the general equation to a family of curves be fe 0 ee es eee C1); uw being a function of 2’, y', the coordinates of any point whatever in any one of the curves, and of m parameters @,, @,, d,).... @,. Differentiating the equation to the curve 7-1 times successively with regard to z' as the independent variable, we get Du a” Du ae” ef — = 0,) cert cree eee (2); x aaa dani =" Ft nm - 1 equations involving z’, y’, and the m — 1 differential coefhi- cients dy’ d’y' ay’ oy. da dc mame oe agree Since we have » equations, (1) and (2), involving 2 parameters and 2 + 1 quantities Bef dy gy wy any OPP SIE OL Ure ie aa it is evident that we may assign any values we please to these n+ 1 quantities, the values of the » parameters being deter- CONTACT OF CURVES. 219 mined accordingly. We may therefore obtain the equation to an individual of the family of curves denoted by the equation (1) which shall have a contact of the ( — 1)" order with any proposed curve at any point (x, y), by assigning to the quan- tities (3) the values of the corresponding quantities in the pro- posed curve, and obtaining the values of the » parameters accordingly. ‘Thus suppose that wv = F(z’, y’, @,,0,) @,) .. 2+ G and that f(x) is the ordinate in the proposed curve at the point of contact: then v,, v,, v,,....,, denoting certain functions of z, we shall have a, =< > a, = Vy) (oa. mb, and the equation EOL, ys V5 Vo9 Vo5 6: @. 6 Oy © vO = 0, will represent an individual of the family of curves represented by the equation (1), which shall have a contact of the (” - 1)® order with the proposed curve at the point (2, y). Ex. 1. The general equation to a circle is (2 - af + (y' - BY = 0, a, (3, p, being disposable constants, upon the particular mag- nitudes of which the dimensions and position of the circle depend. Let it be proposed to determine the values of a, (3, p, that the circle may have acontact of the second order with any proposed curve y' = $(v’) at a point (z, y) of the curve. Differentiating the equation to the circle twice with regard to x, we have dy’ 2’ -a+(y -B) 4 = ay. dy” 1+ - 6) 54+ B= 0 hey If then for SPOR a ; ty ; we substitute the quantities dy dy x; ¢ we JF 99 o> den? dxt 220 CONTACT OF CURVES. " ay are supposed to denote the values of side and dx* at the point (x, y) of the curve, we shall have for the dy where and dz AED, ae determination of a, (3, p, the three equations (c- a) +(y- BY =p’, z-a+(y-p) 2 =0, 2 2 1+(y- A) 54+ Bao. Now the equations (1) and (2) of Art. (136), supposing d’z to be zero, or z to be the independent variable, coincide with the last two of these equations. We see therefore that the coordi- nates of the centre of the circle, which has a contact of the second order with any proposed curve at any proposed point, coincide with those of the centre of the osculating circle at the same point; or that the osculating circle is identical with the circle which has a contact of the second order. Ex. 2. To determine the parabola which has a contact of the second order with an ellipse at an extremity of the latus rectum of the ellipse ; the equation to the ellipse being 24g 2y eA be 112 and the axis of the parabola being parallel to the major axis of the ellipse. Let the equation to the parabola be (7 EES) = On (ee) aac kt ee bE d then (y + (3) 4 Sec ASTUS wha alate te tata lo de dete Teter (2), dy’ Ty _ ae t ¥Y+P) aa = 0 bikshs i cste sce te gee tree (3). From the equation to the ellipse there is CONTACT OF CURVES. ry | dy’ ad’ 1+ 2 ait y ae = 0 nis Atos eheien tae react OO): Now the coordinates of an extremity of the latus rectum of the a a ellipse are —, ra hence, from (4), V2 Substituting these values of e and De , In (3), we get —2 }+@a+p).—=0, B=-}a hence also, from (2), (a- 4a). = 2m, 4m = — and thence, by (1), eda) Ga-jar=-5% (Fa), a a v2 5a | a i= mena? Of ks ——. TE aaah enn: 42 Hence the equation to the required parabola will be et “) WO ss 5a 2 ad act oko 4/2) When the Radius of Curvature 1s a Maximum or Minimum, the Contact is of the third order. 144. Let the equation to the curve be y" = ¢ (x"), and the equation to the circle of curvature, at a point (z, ¥) of the curve, (gee wee Cs (hep) aes 2. vy noe QL). 222 CONTACT OF CURVES. dy oy: d’y' Then, putting oe Ache q; 78 r’, we may get from(1), x by two differentiations, (14 py p - Re ed p) q 2 log p = 3 log (1 + p”) - 2 log q’, and therefore, differentiating again, p being invariable as we pass from one point of the circle to another, or y= SPY Cette ai ce (2). dy" d’y' / d*y" Again, putting p,q, 7, for the values of de"? de®? def? the point (xz, y), we know that °° (1 er py => 2 9 2 log p = 3 log (1 + p’) - 2 log g. abe = 0, and therefore But, if p be a maximum or minimum, Ta L at such a point we have ped 6 hs NK ee eG ay 2 ? ina 2 Litre) ey fe Now, by the nature of the contact between a curve and its circle of curvature, the values of p’, g’, at the point (z, y), are p> 73 hence, by (2) and (8), we see that the values of 7’ at the point (z, y) is the same as that of r. Thus the contact between the circle and the curve must be of the third order. CHAPTER VIII. ENVELOPS. Case of a single Parameter. 145. Ler the equation to a family of curves be TTA OO ices aka op z', y’, being the coordinates of any point in any one of the curves, and a being a parameter, the particular values of which determine the individual curves of the family. Suppose that a becomes @ + da, da being an indefinitely small increment of a. Then the equation (1) becomes W Ua) af fa Oa. 00) Lge see eee C2) Let x,y, be the values of z’, y’, at the intersection of the curves (1) and (2); that is, of any two consecutive individuals of the family of curves. Then GAC DMO ie ane eon Gay and. F(a, y, a+ 6a) = 0. Hence J, y, 4+ 0a) — f (2, Ys @) 6 oa ; and therefore, when 6a is diminished without limit, we have ultimately Uf (x, y, @) EEE | 4). ee (4) Between the two equations (3) and (4) we may eliminate the parameter a, and we shall thus obtain an equation Cena? eeUN GMAT Aish are ney iors CO), 994 ENVELOPS. expressing the relation between the coordinates of the point of intersection of any and every two consecutive individuals of the family of curves: this equation will therefore represent a curve, which is the locus of such consecutive intersections. It is easy to see that the curve (5) touches each of the individuals of the family (1). In fact, differentiating the equa- tion (1), we get d d os : / 4 SSS - 3 d: ‘= 0 es e 2 76 6 . ag d »Y,a)dx + pi y, a) dy (6) Again, since, by virtue of (3) and (4), a as well as y is a function of z, we have, differentiating (3), d d d ie tel “ ma . Da 2 2Y> = 0, apd y, a) dx + ys © y, a).dy + ag @ y, a) da or, by the aid of (4), d d EE e sae a = 0 a = love e ay Y, a) dx + ai F(a, y, @) dy (7) Now, when 2’, y’, are replaced by 2, y, d Rites d dg J Y,a)= pd (@ Y, @); d d d ger ! t meee ae an al 9D dy) Y, &); hence, from (6) and (7), we see that the ratio of dy’ to dz’ is the same in the curve (1) as that of dy to dz in the curve (5) at their common point (2, y). The locus of the consecutive intersections of the individuals of a family of curves has been called their envelop in conse- quence of this property. Ex. 1. To find the nature of the curve which shall touch all the curves represented by the equation y=art-—a’, whatever be the value of a. Differentiating with regard to a, we have 0=2- 2a; ENVELOPS. 225 and therefore, eliminating a between these two equations, we obtain, for the equation to the required envelop, 4y = 2°, which represents a parabola. Ex. 2. Straight lines are drawn at right angles to the tangents of a parabola at the points where they meet a given straight line perpendicular to the axis: to find the envelop of these straight lines. The equation to a tangent to the parabola y’ = 4mz will be ya ace tL a ed ore OE): a let that to the given line be EC aah eee. oat as gr The coordinates of the intersection of (1) and (2) will be EC = a 5) “L=-C;: hence the equation to the perpendicular to the tangent will be or 1 Bea Ou Or Tr wes CRS 3 in Aaa (8). Differentiating (3) with respect to the variable parameter a, we get Eliminating a between (3) and (4) we have, for the envelop of (3), the parabola yA d Vane ireee OVW teat ign CO) From the form of this equation it appears that, to arrive at its vertex we must proceed from the origin for a space m — ¢ along the axis of z, and to arrive at the focus we must afterwards proceed for a space ¢: hence we shall have proceeded in all for a space m to arrive at the focus. Thus (5) represents a parabola confocal with y’ = 4mz. Q 226 ENVELOPS. General case of any number of Parameters. 146. Let the equation to a family of curves be U=J(L5 Ye a Ulery = ee (a); z', y', being the coordinates of any point in any one of the curves, and 4,, @,, d,,... @,, being ” parameters, the particular values of which determine the individual curves of the family. We suppose these parameters to be connected together by m —1 equations, so that any » — 1 of them will be functional of the v* remaining one. Let the m—-1 equations be CRS CSE nS a ee Wb Ain (2). Conceive @,, d,, a)... @,, to become a, +8a,, a+ 6a,, A, + OAs, . @, +0a,; 8a,, da,, da,,... da,, being indefinitely small im- crements of the » parameters consistent with the simultaneous equations (2). ‘Then the equation (1) becomes u'+du'=f(c, y', a,4+00,, 4,4 60,, a,+ da,,+--d, + 6a,) = 0...(8). Let x,y, be the values of z’, y’, at the intersection of the curves (1) and (8), that is, of any two consecutive individuals of the family of curves. ‘Then, putting for the sake of brevity, a’, a, a+. a, in place of a,+6a,, a+ 0a,, a, + da,, + a,+0a,, we have, uw and w+ du representing the values of w’ and w'+ du’, when 2’, y’, are replaced by z, y, Theat BORD UNS UR ihe Sx, He P U 8 Se ay CE and + OU = F(T, 5 ha, Cay nO 2) = On from these two equations we see that the numerator of the left-hand member of this equation denoting the total increment of the function w due directly and indirectly to an increment 6a, of a, Hence, proceeding to the limit, when 6a, becomes less than any assignable magnitude, we have Du —=0, or Du=0: da, 3 : ENVELOPS. Q07 | or, expressing the total in terms of the partial differentials, du a, ee ys Set ce = OPS da, da, da, da, Differentiating the m -—1 equations (2), we get also the 2-1 differential equations d d dd, do, at Os + Get a, + da, +n + FE da, = 0 Pe ie eel eee Gata bs EAE Pi A fe a, petG): Eos een o th pega ts ime: al da, = 0 da, da, da, dp -1 dp -1 do,,. dp, n a Pe eur caus, = 0 i da, + ‘dE da, + ae dd, + ++. + an da, Adding the equations (6) multiplied in order by (” -- 1) arbitrary quantities X,, X,, A,»..-- »,4, to the equation (5), and equating to zero the coefficients of the differentials of the resulting equation, we obtain the m equations amy do, 4 KEES 5% dbs PN. @h,1 _ 9 ) aa Od CMe dated added ec: du do De ne do, ae ett 2 $4 $A the 0 da," s da, tage da, sap da, RS 2 da, 2 du dp do dd, do, fe Gh at corel ! g see peat = 0 Ba valde tl dahl naaaa ners di: du do d¢ do —— +X, 40, 2 +A, 4+... 4A. =e = Come Ud a he dae asda yr: Between the equations (2), (4), (7), 2% in number, we may eliminate the 27 —1 quantities @,, @,, 5)... @,) Nyy Noy Ngo *2* Nays and we shall then arrive at an equation TE aah) =O representing the required envelop of the family of curves. Q2 298 ENVELOPS. Ex. 1. To find the equation to a curve which shall touch each of the family of straight lines defined by the equation gee Le A reg Marco Se (2) Differentiating (1) and (2) with regard to a and (3, we have ada yd ie)", ae we bee 3), naira (3) a” da Ui 0 ate. cee cee (4). From (8) and (4) there is Az ees Cig AS i ise: Panes Wake puke (5). dy Multiplying the two equations (5) by a, 8, adding and attend- ing to (1) and (2), we see that ete: & hence, from (5), Teaie ett oh ae yte Areak Tp and therefore, from (2), n nel ml antl mete Bs) hte 8 os Se the equation to the required envelop. Ex. 2. An ellipse moves with its centre in the arc of an equal similar ellipse, and has its axes parallel to the axes of the fixed ellipse: to find the curve which envelops the moveable one. Let a, 6, be the semi-axes of either ellipse; the equation to the moveable ellipse will be OE) ESS ih Nee rane ENVELOPS. B29 where a, (3, are subject to the condition p pe! ir WGN sm 2 (2). Differentiating with regard to the parameters, we have 2 a a By (1), (2), (8), we have a(w-a), BY 8) N= P ; and » (ee) BOO a, and therefore Mel, A=4 1. If \ = 1, we have, from the equations (3), eat a P| ik ys and therefore, from (2), we have for the equation to the envelop 2 x y et es tp bar Eee ee which is the equation to an ellipse similar in form to either of the original ones, its axes being of twice the magnitude. Again, if \ = — 1, we have from (38), z=0, y=0, which are the equations to a point, viz. the centre of the fixed ellipse, through which it is evident that all the moveable ellipses pass. 230 ENVELOPS. Intersection of Consecutive Normals to a Curve. 147. The equation to the normal at any point z, y, of a curve f(z’, y') = 0, will be af du ; 2) ae AY Yee os win set 1), @'-) FE =U- DE (1) where w= f(z,y). Differentiating (1), considering % as a vari- able parameter, of which y, 2 and. Zo are functional, we get & y d*u du CB ese: erie du du From (1) and (2), and the differential of the equation «= 0, we see that Piatt hey dz’ dy’ Gi du Par du du d’u— du du’ de dy dy da ~ da dy dedy dx dy the values of z', y’, the coordinates of the intersection of two consecutive normals, coinciding with their values obtained otherwise in Art. (146). Geren’ CHAPTER IX. DIFFERENTIALS OF AREAS, VOLUMES, ARUS, AND SURFACES, Differential of an Area. 148. Let AB be any portion of a curve referred to rectangular axes Ox, Oy. Let PM, QU, be the ordinates of two points P, Q, of the curve, very near to each other. Let AC be the ordinates of A. Draw PR and QS, at right angles to QN and MP produced. Let OM=2, PM=y, ON =2 + &, QN = y + dy; let A =the area ACMP, and A+ 6A = the area ACNQ. Then, since the area PQNWM is evidently less than SQNM and greater than PRNM, it is plain that you < 0A <(y + Oy) Oz, 6A 5 or y <5 ) (H) {AY - (H)"} = 0; and, from (2), Cu) {5.(r)* = (H)*} (F) = @) 15 Wt - 04} (st) «o. Hence we have the four systems (A) = 0 (u) = 0 | (X) = (u) ejb 1&4 1@)-@)-+ (A) + (#) = 0 (2)-(&)-o) Hence, from the equation O)¥ -@)2 = 0)(F)-w(F). 258 ON THE METHODS OF TRACING we obtain four asymptotes to the curve, represented by the equations z=0, y = 0, y ~~ ae =, y+ 2 =0. Again, putting y = tz, we see that : a’ : at Shit Ce en ene and therefore, observing that the ratio of y to z must be always of the same sign as ¢, we have the following table of cor- responding values : t x y +0 + 0 + 0 +0 —- 2 — 0 1-0 + 0 + 0 1-0 — © - © — © + 0 — 0 — 2 —0 + —~(1+ 0) + oO = — (1 + 0) —- © + 0 THE FORMS OF CURVES FROM THEIR EQUATIONS. 259 Ex. 2. To trace the curve ay — 2x*y* + sn SO weg me fF The Cyclord. 164. As an example of deducing the equation to a curve from its geometrical definition, which is exactly the converse of tracing a curve from its equation, we will investigate the equation to the cycloid from the nature of its generation. & Let C be the centre of acircle in contact at A with the straight line H#. Let O be the extremity of the diameter through 4. Suppose this circle to roll, without sliding, along HF; the point O of the circumference will then trace out a curve OPH, which is called the Cycloid. Let OAz be taken as the axis of z, and Oy, at right angles to OA, as the axis of y. Suppose that, when O has arrived at a point P of the cycloid, the circle has revolved about its centre through an angle 6; then its centre must have advanced, parallel to 260 ON THE METHODS OF TRACING HK, through a space a@, a being the radius of the circle: for, since the circle rolls without sliding, it follows that the velocity of its point of contact, parallel to AH, due to its rotation about C, must be equal to the velocity of its point af contact, parallel to HA, due to the translation of C. From C draw CQ, making 4 OCQ = @: draw MQ at right angles to AO, and produce MQ to P, making QP equal to a9. Then Q will be the position into which O would be carried by the rotation alone, QP being its additional progress due to the translation of C. Let OM=2, PM=y. Then z= OC- CM=a-acos 0=a(1 — cos 8), and y=PQ+QM=a0+asin 0=a(0 + sin 8): eliminating 6, we shall get 1% 4 y =a vers” — + (2axr - 2’), a which is the equation to the cycloid. The curve will evidently be symmetrical on both sides of the axis of x: for if we put — 6 for @, we, see that y retains the same magnitude with an opposite sign, and z remains entirely unchanged. If@=n,y=AK=n7a=AH. Also are OQ =al = PQ. If for @ we write 2A7+ 0, X being any integer whatever, the expression a(1 — cos #) remains unchanged, while a (@ + sin @) receives an increment 2Am7a. This shews that the two equa- tions between 6, x, y, represent a series of similar, equal, and similarly situated cycloids, with their vertices arranged along the axis of y, both in the positive and negative directions, at intervals of 27a. Tangent and Normal to the Cyclord. 165. The general equation to the tangent of a curve is zidy -— y'dx = xdy — ydz: this equation becomes, for the cycloid, x (1+cos 0)-y' sin 0 = (1 ~ cos 0) (1 + cos 8) — (8 + sin 8) sin 6 =-— @ sin 0. THE FORMS OF CURVES FROM THEIR EQUATIONS. 961 If » be the inclination of the tangent to the axis of x, then ~ da sin@ 1-—cos0° a result which shews that the tangent at P is parallel to the chord OQ, and that consequently the normal at P is parallel to the radius CQ. Are of the Cyclord. 166. Differentiating the formule for z and y, we get ds* = dx’ + dy’ = a’ sin’ 0 d@’ + a (1 + cos OY d&? = 2a°d@’ :1 + cos 8) 1 — cos 0 2adz°* or s’ = 8azx. Let c = the chord of the arc OQ; then, by the nature of the circle, ce’ =2ax: hence s’ = 4c’, or arc OP = 2 chord OQ. Radius of Curvature of the Cyclord. 167. By Art. (132), we have apa’ dx + dy’y " (ded’y ~ dyl'ay Taking @ as the independent variable, dz = asin 0d0, d’z = a cos 0d&’, dy = a(1 + cos 8) dé, d’°y=—asin 0d#@: 262 ON THE METHODS OF TRACING hence dx® + dy’ = 2a°d@ (1 + cos 8), dxd*y — dyd*z = - ad? (1 + cos 4); and therefore p = 8a’ (1 + cos 0) = 8a (2a - 2). Evolute of the Cyclord. 168. If a, 8, be the coordinates of any point of the evolute, then, by Art. (136), adz + Bdy = xdx + ydy, and ad*z + Bd*y = ad*x + yd’y + dx’ + dy’. From these two equations there is a (dad*y — dyd*x) = x (dad’y — dyd’x) — dy (dx’ + dy’)...(1), and (3 (dyd*x — dxd’y) = y (dyd’x — dxd’y) — dx (dy’ + dx’)...(2). From (1) we have, expressing z and y in terms of 0, —a(1+cos 0) =- a(1- cos 8) (1+ cos 8) — a (1+ cos 0).2 (1+ cos 8), hence a=a(l1-—cos 0) + 2a(1+ cos #)=a(8 + cos 9), or = D0 =O (124-6080) mares beast): From (2), we have 3 (1+ cos 6) = a (0 + sin 9) (1 + cos #) - sin #. 2a (1 + cos 8), or $B =a(0+sin 0) - 2 sin 0=a (60 - sin 8)....(4). Putting 0 = @ + 7 in (8) and (4), we have G20 Ba {hoe COS Dye ete ters cc eet (5), and B=a(otmw+ sin o), or 3 gra ae (SPIN h) Fos bh oe oe LODE If we now change the origin to a point 2a, + wa, by putting a=a + 2a, (3 = B' + wa, in (5), (6), we get SP CAs (COS): Oa ee ee (7), and Di ate + sins) 82ee ELEN (8). THE FORMS OF CURVES FROM THEIR EQUATIONS. 263 These results shew that the evolute of the cycloid HOF is a portion of the locus consisting of the infinite series of cycloids lyn denoted by the equations (7) and (8). ‘Two of this series have their vertices at H and 4K, and their point of junction O', where they form a cusp, in the line OA produced, 4O' being equal to AO. It is evident that each of the cycloidal evolutes is similar, and equal to the original cycloid, and similarly situated. HO'A is evidently the portion of the locus of (7) and (8) which con- stitutes the evolute of HOK. THE END. CAMBRIDGE: PRINTED BY METCALFE AND PALMER, TRINITY-STREET, ; f ' ee , OT tk lines Steals og LE TLS Eee a Ret Cambridge, November, 1871. EDUCATION AL BOOKS PUBLISHED BY MESSRS. DEIGHTON, BELL, ’ AND CO. Agents to the nibersity. AND SOLD BY BELL AND DALDY, Cambrivae Greek ww Latin Certs. CAREFULLY REPRINTED FROM THE BEST EDITIONS. 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