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Theft, mutilation and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN St iin bu U L161—O-1096 Digitized by the Internet Archive in 2022 with funding from University of Illinois Urbana-Champaign https://archive.org/details/elementsofdifferOObuck_0 ) . A ‘aay Se » a an . = i . 7 % 7 6 * * g to Nes : ‘ © he a w — ‘ “4 4 : - . s ’ ; ¢ a A rp _ ae ¥ 7». -_ < io a hao S Ww a9! - #, J om Pisa Tong. — Jat © a df ELEMENTS OF THE LOTERBERENEPMLOAND L-NTEGRAL CSE CULES: By A New METHOD, FOUNDED ON THE TRUE SYSTEM OF str IsAAc NEWTON, WITHOUT THE USE OF INFINITESIMALS OR LIMITs. By CG. P. BUCKINGHAM, AUTHOR OF THE PRINCIPLES OF ARITHMETIC} FORMERLY ASSISTANT PROFESSOR OF NATURAL PHILOSOPHY IN THE U.S. MILITARY ACADEMY, AND PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY IN KENYON COLLEGE, OHIO, CHICAGO: “JLr SRC RIGGS & COMPANY a Be eh, a ese en Fo, © 4 en ~ a —< < a ‘i . > : Entered according to act of Congress, in the year 1875, by Tae a S. C. GRIGGS & CO., aa : : in the office of the Librarian of Congress at Washington, District of Columbia. 4 z > et : : 4 ae [PUBLISHING PRINTING COJ \ : | ; = a | MATHEMATICS Lig 5/5 RARY Sm ae — BSD “The student of mathematics, on passing from the lower branches of the science to the infinitesimal analysis, finds himself in a strange and almost wholly foreign department of thought. _He has not risen, by easy and gradual steps, from a lower to a higher, purer, and more beautiful region of scientific truth. On the contrary, he is painfully impressed with the conviction, that the continuity of the science has been broken, and its unity destroyed, by the influx of prin- ciples which are as unintelligible as they are novel. He finds himself surrounded by enigmas and obscurities, which only serve to perplex his understanding and darken his aspi- rations after knowledge.” * He finds himself required to ignore the principles and axioms that have hitherto guided his studies and sustained his convictions, and to receive in their stead a set of notions that are utterly repugnant to all his preconceived ideas of truth. When he is told that one quantity may be added to another without increasing it, or subtracted from another without diminishing it — that one quantity may be infinitely small, and another infinitely smaller, and another infinitely * Bledsoe — Philosophy of Mathematics, lV PREFACE. smaller still, and so on ad ¢njinitum — that a quantity may be so small that it cannot be divided, and yet may contain another an indefinite, and even an zzfimite, number of times — that zero is not always nothing, but may not only be some- thing or nothing as occasion may require, but may be doh at the same time in the same equation— that two curves may intersect each other and yet both be tangent to a third curve at their point of intersection,* it is not surprising that he should become bewildered and disheartened. Nevertheless, if he study the text books that are considered orthodox in this country and in Europe he will find some of these notions set forth in them all; not, indeed, in their naked deformity, as they are here stated, but softened and made as palatable as possible by associating them with, or concealing them beneath, propositions that are undoubtedly true. It is, indeed, strange that a science so exact in its results, should have its principles interwoven with so much that is false and absurd in theory; especially as all these absurdi- ties have been so often exposed, and charged against the claims of the calculus as atrue science. It can be accounted: for only by the influence of the great names that first adopted them, and the indisposition of mathematicians to depart from the simple ideas of the ancients in reference to the attributes of quantity. They regard it as merely inert, elther fixed in value or subject only to such changes as may be arbitrarily imposed upon it. But when they attempt to carry this conception into the operations of the calculus, and to account for the results by some theory consistent with * As in the case of exvelopes, PREFACE, Vv this idea of quantity they are inevitably entangled in some of the absurd notions that have been mentioned. Many efforts have indeed been made to escape such glaring incon- sistencies, but they have only resulted in a partial success in concealing them. To clear the way for a logical and rational consideration of the subject, we must begin with the fundamental idea of the conditions under which quantity may exist. We must, for the purposes of the calculus, consider it not only as ca- pable of being increased or diminished; but also as being actually in a state of change. It must (so to speak) be z7¢al- zze@, so that it shall be endowed with éendencies to change its value; and the’ rate and direction of these tendencies will be found to constitute the ground work of the whole system. The differential calculus is the SCIENCE OF RATES, and its peculiar subject is QUANTITY IN A STATE OF CHANGE. It is an error, therefore, to suppose, as has often been said, that the “veductto ad absurdum,” or “ method of ex- haustion,” of the ancient mathematicians, contained the germ of the differential calculus. This hallucination has arisen from the same source as the false notions before mentioned. That peculiar attribute of quantity upon which the transcendental analysis was built, never found a place among the ideas of the Greek Philosophers; and even Leib- nitz, the competitor of Newton for the honor of the inven- tion, and who was the first to construct a system of rules for the analytical machinery of the science, never got beyond the ancient conception of the conditions of quantity, and, therefore, gave, says Comté, “an explanation entirely erro- vi PREFACE. neous ”— he never comprehended the true philosophical basis of his own system. The only original birth-place of the fundamental idea of quantity which forms the true germ of the calculus, was in the mind of the immortal Newton. Starting with this idea, the results of the calculus follow logically and directly through the beaten track of mathematical thought, with that clearness of evidence which has ever been the boast of mathematics, and which leaves neither doubt nor distrust in the mind of the student. To develop this idea is the object of this work. C. P. BUCKINGHAM CHICAGO, 0/271.-1,0LO7 51 CoNnTENTS. PE NTIEROED UG ELON: PAGE, Objects of Mathematical Study among the Ancient Philosophers... 11 WEGENER INC ERGOT LCS Se ra se oe EEE 32, 2 eek a ok st cle 12 Pee ieTeN aA AICIINS «om ho roe eit Sg ak ce te eee betel ec 14 wo wrenetal wlethodes.3 f22tee* 2S TO he SE Cd een, Lie ie ed 16 Boreal Merbndy 2S 2 aa cose eee ee Sue Ee oe 18 Resiicarertrie while the Method 1s-Falsé: 3. sae = Soa es eke 20 Debord Oe Laniice & se oat os ten ye eee ns OG GRE eee ee eee o 22 Newtow 5 Defense Of his Leminia tt ese Fae ee 23 Opinions of Comtés Lbagratige and, Berkeley... + 5. £io lol ces 24 Dusveaments error or both oystems. — 20. - 2. ona sous shes ces 31 Lier prueivrernodion. Newton: cater oon. cece lew ea aa oe ceoe 33 Die MOUnGALOI Of tual: Method =~ . 2c setee se. Son See SLL ae esses 34 Explanatory Letter of the Author---__- I eke es Oe mk fe: PA Riis ok: DIFFERENTIAL CALCULUS. SuCtions ls. qLetnitions and birst. Principles 2 -__o Son aecann oe se 49 Warinnieamenneds 2eo.c- ent). en se Jo es eee 49 Prater Gtey ALICMON tries meso. Veter ye © Aira kc oe ota rohe S 50 Peretti ets ye en enn pe a re vem ee rae Oo ree 51 RO RtLiSe ee oe rs eae. Oe Sal oe ee Pee hs, feet Se 52 Pa IIee Sey Sees ker ere ee Oe os 53 mec ries uietCOration. Of MUNCHONS 4. senso. ee on 2d 5 3 56 SI eAeCUCREMCCT Attia shoe se tetas ect Ft 56 Differential of a Function Consisting of Terms ------------ 58 MrewiscOriceoraiGg Pernice kuere mad. Be. ces 60 Differential. of a Product of Two Variables.---....-------- 63 meriet Cala tie rAlOn Sw ee ee. fe ycte son At elo 65 jirerentiais of ractions «i024 02-2 Pn ho Eee eh ee lard 68 vill . CONTENTS. PAGE. Differential of the Power of*a Variabley.o. vane e ees eee 71 # + sRoot me Mh. Peo, eae see 73 ‘ “Function of Another Function ......:-. 74 SECTION III. “Suecessive Differentials $22 eee ee Fae ras A Maclaurin’s Theorem..-::<.- ---.¢slccn eaeee eee eee 85 Taylor's @ heorem=s)0o 8 22 ee er :. See 89 Identity of Principle in Both Theorems (note)_------------ 93 SECTION [V., Maxima and Minima .---Cs22. 2522.22. 95 Methodzof Finding by Substitution.22. 2322. 2>-2_ eee 96 Meahing-222232 eee 142 LN “__ -Versed Sine. ofan Arcs eee eee 143 A fe ATO 34 2252S Shs ee ee 143 Signification of the Differentials of Circular Functions __-__- 144 Values of Trigonometrical Liness2s- eee sa eee eee 148 SECTION VII. Tangent and Normal Lines to Algebraic Curves.__ 150 Length of ‘Subtangent to any Curve.... 3-2. o ree 152 4 ‘« Tangent ‘> OP ite dno 153 a wks ou TOs Ines = SU Reet hes oe ee 163 “Normal A Ff oo nie oS Line ae ee 154 AppHeation of the formulas 22.5 oco0 c= oe ae 154-156 SECTION VIII. . Differentials of Curves’. 2222-52 - oe oe 157 Differential Plane Surfaces Bounded by Curves._-_.-------- 158 «6 Surfaces. of Revolutioniz->.cs.ee. eee ee 162 CONTENTS. 1X : PAGE, Witterential Solids of. Revulotion 2 }<2 02 osc e cs ont pee =~ TOs DUC LION ¢Lacuwe t Olats GUiEVeSUcee. oc re oe Dee eens ot See ae 168 mangzents:to Polar-Curmes= i. Sass tet soe. «o~ =~ - == 168 Differential’ of ‘the Arc of/a Polar. Gurvers2. -—<.28_ J. =. L7I Puibtanpent tie: P Ole Curve 22 oe a eae oe naa wet rye Tangent : * CR ee nee wt renee r7r Subnormal ‘“ es De Se Cae hee ek yas ee aa 172 Normal . $ Ee a AP ae ne RS 172 Surrice: bounded by.a- Polar Curve 2222 2-2 oe 2 see 173 SY ATCT ia So ASE A Raced NAA BEI Sepa RO eee Te a 173 SPINOR ORURCUCS cin rea ns ain ae eee disciga's ah 174 Peper tie pin tns.~ 2 tata cea ee ens orn i a ee 176 Omri tia ae as ae ee a eee dees a ae <2 es SAS 179 MeOTION de ~ Asymntotes roi o at See a ee 182 Laka g Ko We he Ye Wie Canc hk ape ah et te aie aie, SAM Ae 5, Se 183 TORRID IOS ese ts erg crete pets kA eel Sa A eer 183-186 SECTION XI. Signification of the Second Differential-._.-..----- 187 Siegen of the Second Differential Coefficient £-- 4.2 (Ss. <.1--- 188 ~Value ‘“ . ~ Re he ey Ee ee, = a 190 DERTION: Sti eervatnre OL Mantes es esate ee ot tL oo als cous 192 Dboeurercn Curvature arenes eas een hee Se ee 192 (APRAOCOS KOT VES Speke ewe Sean keys Pee fas 193 Comerants.in the. Hquationof a:Curve S....-.--2--2-.t--<- 195 WSC retGlee LC) a) ENG ce ae ee Sete ren Oe 199 Isai ieGHitna ite eee pee toe eee 3 ks oets con 202 SEC TEON & Lic eth VOL Case eine a oer oe oa 207 Pleperiies Of ties Myo iter aa wen oe ee een tS 207 PorPme-the Equation sof: Mie Wyoute co. 5. 20.202. fect e 210 De TIONS Ci aenVelopes. - S ere e t o e e wS 216 Peseta Ol, ean VOLO Pe: Lomeenad (ooo oe ea ek 218 Bre wato a italieis Maton --bneees oo ee Fa oe las 216 SECTION XV. Application of the Calculus to the Discussion of Curves 229 fie Cyeloids sea gs s ee te ee ee aie cae dy sls uk 229 Ce periieer On tue VelOIG 22 06 — soa a= doin nce oe Stee 230 Domai iiiae Cine ie "Le Sees o ek sue 238 Pe One ey Leen Siloigi LOMG2e5 205 toe sacs 2k el 558. 243 Rigedinia CaM Wit tebe yoo a oe eo we oe See 243-248 Saha) Omen Te eae RA Feros, SER 2 Bo ae ok a oe eS 248 Mie Ot Sena. oo os See « den = Uy Seta} 253 Di ites Gin ts eres fer ed Se Serge ee es wee 254 xe CONTENTS. PARs wes INTEGRAL CALCULUS. PAGE SEETION J. . “Principles -of "Integration <. 22 oo eee ee 261 Integration of Compound Differential Functions....--.---- 263 FSS Monomial a SS es Po ee ee 263 3 Particular Binomial Differentials --.-....--- 266 xs Rational Fractious! se 52.8 wee koe eee 207 sd Between, limits =~ ei eee oo ee 277 is by Sertes Stee pee ee Se ee ee 278 of Differentials-ofCircnlan Arcelor 279 SECTION II. Integration of Binomial Differentials .......------_- 282 Integration’ of Particular omnes sss) — eee. eee 283-288 z by..Parts sare ari ered aes See ee 291 Formulasdor Reducing Eixponeniss 22. eee 293-303 SECTION III. Application of the Calculus to the Measurement of Geometrical “Mapnitudesc. 22) see. eee eee 304 Recwheation of Ciurves:. oS oe eo a ee ae 305 Ouddtature of (Curves espe ao. Pe ee ee 212 Suraces of: Revolution.c<.. <-eoeoe eaea> cee een eee ee 325 Cubature of Solids 22 see ok cae eae an ee Soe 329 APP a NgLD Lek, GEOMETRICAL FLUXIONS. Principles of the Calculus Applied Directly to Geometrical Magnitudes 337 To: Find the Area of a) Circle. 3. 22 sot eee - ee 338 So US (Convex Surlace Otta, CONG se sae a. era ene 339 “> se S* NMolume ofa Cone 2 cates oo eee eee 340 tern <® -Avea of the Surface’ofca Sphere-<- ae ee 341 “ce 66 6c Volume of a Spheteseis si sccec ee -te wee eee eee 342 INTRODUCTION. THE PHILOSOPHY OF THE CALCULUS. Among the ancient philosophers, the objects of mathemat- ical study were confined exclusively to the solution of deter- minate problems — that is, every quantity concerned had a determinate value, either known or unknown. Devoting themselves principally to Geometry, they sought to deter- mine the exact measurement of lines, surfaces, solids and angles, in terms of fixed and known quantities. The later methods of the algebraists did not change the ultimate object of their researches. All their problems were still determinate. Their conditions were definite, and the result certain. This, which may properly be character- ized as the s/at#c phase of mathematics, continued for two thousand years to guide, control and circumscribe the labors of the mathematical student. There was but little advance in the discovery of mathematical truth; none had the bold- ness to strike out a new method of investigation, or apply themselves to the solution of any but determinate problems. Algebra had indeed been successfully applied to geometry, but it was only the analytical method of stating arguments that had been used in ordinary language for centuries. X1l INTRODUCTION. Such was the condition of the science up to the time when the brilliant genius of Descartes seized upon a new idea, and boldly followed its lead until he developed a sys- tem whose results have astonished and delighted the world. Breaking away from the idea of determinate values and . absolute conditions, he adopted that of dependent condi- tions and relative values, which no longer fixed unchangea- bly the quantities sought, but gave them a wide range, so that within certain limits they could have all possible values. Hence they were called varzables, while those quantities whose values were fixed were called constants. In every equation containing a single unknown quantity the value of that quantity is absolutely fixed by the condi- tions expressed in the equation. If we have two unknown quantities, and two equations, or sets of conditions, both values are still fixed. If the higher power of the unknown quantity is involved, the number of values is greater, but they are equally fixed and certain. This idea of fixedness of value underlies all algebraic operations of an ordinary kind. Now suppose we have “wo unknown quantities in ove equa- tion, with no other conditions given than those expressed by the equation itself. In that case the values of both quanti- ties are absolutely indeterminate. But if we know or assume any specific value for one we can at once determine the corresponding value of the other; so that while the equation will give the independent value of neither quan- tity, it will give the szmultaneous values of both ; and these values will have a certain range or locus, which is in fact the true solution of the equation — the path, so to speak, through — which the simultaneous values range. In some equations the range of values‘is limited for both variables, so that if a value be assigned to one beyond the limit, that of the other becomes imaginary; in other cases INTRODUCTION. Xiil the value of one only is limited, while in others again the values of both are absolutely unlimited; any value of one giving a corresponding real value for the other. Since the values of these variables are thus dependent on each other, the equation expressing this dependence may be considered as containing the /azw of their mutual relations, and the fundamental ideas of Descartes was to exhibit in his equation the conditions or law which confined the two vari- ables to their prescribed range of values. This idea was something new, distinct and well defined, and a clear:ad-_ vance beyond the methods of the ancients. But the labors of Descartes would have been of little value had he proceeded no farther than we have indicated. In fact this was but a part of his invention, of which the specific object was a method of investigating questions of Geometry. To complete this purpose, he devised a new and beautiful method of representing magnitudes, to which his algebraic equations could be applied. In algebra, all _values are estimated by their remoteness from zero. In order to make a general application of algebraic symbols to geometry, it was necessary that the value of every line rep- resented by his variables should be estimated from a com- mon origin corresponding with zero; and as every point in a plane surface requires two values to fix its position, two such origins became necessary to his system, in order to represent plain figures; and these were found in two right lines, lying in the plane of the object to be represented, and intersecting in a known angle — generally a right angle. From these two lines all values or distances to points were estimated ; the positive on one side and the negative on the other of each line; while for points zz the lines, one of the values would of course be zero. Having then a method of representing the position of a point by algebraic symbols, it was easy to apply his analysis X1V INTRODUCTION. to the representation of lines, by making the locus or range of simultaneous values of the variables to correspond with the locus of the points in the line —that is, with the line itself Thus the method of Descartes was two-fold — the alge- braic idea of two variables in one equation with a range of simultaneous vaiues, and the geometrical idea of coordinate representation, and these two being adapted to each other, united to form a method of investigating, in an easy and sim- ple manner, questions of geometry which had taxed the utmost powers of the ancients. Upon the foundation thus laid by Descartes has arisen the Differential Calculus. Not that the calculus in its purely abstract conception is especially related to geometry. On the contrary its analysis is adapted to investigations in all those questions where the quantities are variable and the conditions can be analytically expressed. But it was in con- nection with problems of geometry that its methods were first discovered, and fora long time applied through the Cartesian system; and even now geometry is the principal arena upon which the triumphs of the calculus are displayed. The invention itself has many peculiarities both in its history and substance. It was not a result produced by means of ordinary scientific investigation — by a discovery of fundamental principles and a careful elaboration of those principles until they grew into a perfect science. On the contrary these results appeared rather as remarkable phe- nomena, discovered more by accident than by- logical deduc- tion. Newton seems sie to have had an indistinct per- ception of the principles lying at the foundation, but he has nowhere given aclear and satisfactory account of them; while the explanation given by Leibnitz proved his utter ignorance of the true theory of his own system. Thus was a most important branch of mathematics invented INTRODUCTION. XV a Bf =>. almost simultaneously by two of the most distinguished@rven of the age, without any clear and fundamental principle for its foundation. Its operations were accepted as undoubtedly reliable, zof because its principles were sound, but because ats results were undeniably true, ‘Vhis could not be disputed, and hence mathematicians were not so eager, at first, to establish a logical basis for the new science, as to extend its Operations into new fields of discovery. Attempts were, at length, made to assign a rational principle which would account for these extraordinary results, but although each theory has had for its advocates many of the most distin- guished mathematicians, yet each one has had as many and as distinguished opponents. No one has secured the universal approval of the scientific world, and, therefore no cne was founded in mathematical truth; for no pro- position is worthy of the name that does not command the unqualified assent cf every mind by which it is fully comprehended. | The conceptions of the calculus were so subtle, its pro- cesses so mysterious, and its results so astounding that mathematicians began to lock upon its ideas as not subject to the ordinary laws of thought and the rigid rules of logic. The inconsistencies and absurdities which were often propounded were regarded as only mysterious and zncomprehensible ; when quantities refused to obey the laws that had always hitherto controlled them, they were called infinitesimals, and thus released from all subjection to establish axioms. Of course theories were not wanting, but they did little more than give variety to what was, after all, the same compound of false premises and illogical conclu- sions. We are told by M. Comte that the science is as yet in a “ provisional state,” and that it is necessary to study all the principal methods in order to have even an approximate understanding of it. Xvl INTRODUCTION. I shall, however, confine myself to the consideration of the two principal conceptions attributed to its inventors, and to some more recent modifications of them. _The advocates of these two methods have approached the subject from the same direction, but the theories involved in their demonstrations are fundamentally different. These I propose to examine, and to show that as theories they are fatally defective; that a fundamental error underlies every form in which they have been proposed, and muws¢ vitiate any theory based upon the leading idea through which they approach the subject. By the invention of Descartes very many geometrical problems were beautifully solved, but there were some for which equations between the direct functions of the varia- bles would not suffice; such as the length of a curve, the amount of its curvature, and others of a similar kind. To form equations in which such values. could be introduced, it became necessary to represent the variables zzdrectly, using, instead of the actual value, a /wmction of that value. This function is what is called the a@fferential of the variable, and the true philosophical relation which it sustains to its actual value has been the subject of controversy from the © beginning. The particular application of the calculus, which will most clearly and exactly illustrate the various systems, is Zo determine from the equation of the curve the direction of tts tangent at any point. This process involves the fundamental principles of the science, and an examination of it will afford the best means of investigating the different theories that have been advanced to account for the exact truth of the results obtained. Let APC (Fig. A) be a curve, and let AM and AN be the coordinate axes to which it is referred. Suppose the line SD to be drawn tangent to the curve at the point P. The problem is to find from the equation of the curve an ex- INTRODUCTION. 2 XVil pression for the value of the angle PSB. Now the tangent of this angle is PB ‘SB S being the point of intersec- tion of the tangent line with the axis of abscissas, and B the foot of the ordinate through - Fig. A. the point of contact. Let BB’ be an increment added to the abscissa, and B’P’ the ordinate corresponding to the abscissa thus increased. Draw PE parallel to AM, intersecting B’P’ in E; then P’E is the corresponding increment of the ordi- nate BP. Draw the cord P’P, and let D be the point where the tangent intersects the ordinate B’P’. Since PY seit; ' “SB” PE the problem is reduced to finding -the ratio of PE to DE. Now we can easily find from the equation of the curve, the line P’E corresponding to any increment BB’ or PE of the abscissa AB But DE is the line we need, and to pass from P’E to it is the specific part of the operation which involves the fundamental principles of the Calculus. The two prin- cipal methods of doing this we will now examine. THE INFINITESIMAL METHOD. The fundamental propositions or principles of this method te: First. “ That we may take indifferently the one for the other, two quantities which differ from each other by an infi- nitely small quantity, or (what is the same thing) that a quantity which is increased or diminished by another infinity less than itself can be considered as remaining the same. Second. That a curved line may be considered as an assem- 2 xvlil INTRODUCTION. blage of an infinity of right lines each infinitely small, or (what is the same thing) as a polygon with an infinite num- ber of sides infinitely small, which determine, by the angle which they make with each other, the curvature of the line.” | In order to understand clearly how these propositions apply to the solution of the problem we will consider it analytically. Suppose the equation of the curve to be yas? (1) If we take AB=x we have BP=y; and if we add to x an increment, BB’ (which we will call 2), the corresponding ordinate will be P’B’ , which we will designate by y’, and P’E will be equal to y’—y. Having added 2 to x, the new state of the Sek will be y =(x+h)? =x? +2hx+h? mes, Subtracting (1) ge (2) we have y —y=2hx+h? or dividing by 2 ff é - la . 5 “ =2x+h as Now as # diminishes in value, this ratio approaches the value of 2x; and if Zis made infinitely small, it may by the first proposition above stated, be set aside as not affect- ing the value of the second member of the equation, which then becomes 2x. In the meantime the curve PP’ will have become infinitely small, and therefore by the second propo- sition may be considered a straight line, coincident with the tangent, that is with PD, so that P’E has become the same as DE, and the equation f PE =2x-+h has become DE PE —2X INTRODUCTION, X1X and thus the angle made by the tangent line with the axis of abscissas is determined. Such is the infinitesimal theory. The propositions upon which this theory is founded can- not be admitted as true, neither is the demonstration con- clusive. To whatever extent Z may be reduced, even to an infinitesimal (if it is possible to conceive such a thing), the two expressions 2x and 2x-+/ can never be equal so long as his anything. We must either admit this or abandon the use of our reason. If % becomes nothing in one number of the equation, it must do so in the other, that is; PE=Z must also become zero, and so must P’E, and we have instead of DE PE an expression which certainly amounts to nothing, unless we can show that the tangent of the angle DPE or PSB is Oo =2x simply one fe) equal to x Again it is impossible that a curve should ever be consid- ered as a polygon. The very definition of a curve is sim- ply that which distinguishes it from a straight line. Had the ancient geometers been willing to admit this principle, how easily could they have avoided the tedious and labori- ous “ reductio ad absurdum.” But they were too conscien- tious and exact to admit the possibility of establishing truth by even a doubtful principle. ‘In more modern times the greatest mathematicians and philosophers have, indeed, emphatically condemned the notion, that a curve is or can, be made up of right lines, however small. Berkeley, the celebrated Bishop of Cloyne, and his great antagonist, Mac- laurin, both unite in rejecting this notion as false and unten- able. Carnot, D’Alembert, Legrange, Cauchy, and a host of other illustrious mathematicians, deny that the circum- ference of a circle, or any other curve, can be identical with xX INTRODUCTION. the periphery of any polygon whatever.” So repugnant is this proposition to the fixed and fundamental conceptions of geometry, that it has been doubted and denied in all ages — by the most competent thinkers and judges. But notwithstanding all this, what shall we say when we find that the equation DE PE 2” is not merely an approximation to the truth, but that it is perfectly and exactly true. We have said that errors have manifestly been committed in arriving at this result. The advocates of the system point to the result and say behold the proof that we are right. The explanation of this seem- ing mystery was made long ago by Bishop Berkeley. “ For- asmuch,” says he, ‘‘as it may perhaps seem an unaccount- able paradox that mathematicians should deduce true prop- ositions from false principles, be right in the conclusion, and yet err in the premises, I shall endeavor particularly to explain why this may come to pass, and show how error may bring forth truth, though it cannot bring forth science.” He then proceeds to give an illustration, for which we will sub- stitute a similar one adapted to the figure we have chosen. It will be perceived that the curve we have taken is a par- abola whose axis is that of ordinates, and whose i aa is 1. Nowif we consider the curve PP’ as infinitely small and a straight line, the angle P’PE would be the Rar we are seeking; but since the curvecan never become a straight line, the increment P’E must always be too great, and the point P’ must always be above the tangent line; so that the error arising from taking PP’ as a part of the tangent line will be equal to P’D, or that part of the increment of the ordinate which lies between thetangent and the curve. But we have found mm yy —yH2xh+h* =P'E INTRODUCTION. xxl and since from the nature of the parabola x SB=— 2 we have from similar triangles DEaPH ra. 55 or DE:2:: ete ces f SAiyi Zee el gy hence DE=2xh and P'D=P’E—DE=24h+h? —2xh=h® Here then we find that by throwing out 2” we exactly cor- rect the error arising from taking the curve PP’ asa straight line; and reduce the line P’E=y'—y to DE the line we are : DE est ec. seeking to fix the value of PE? which is the tangent of the angle which the tangent line makes with the axis of abscissas. Thus the infinitesimal method arrives at the true result, not because its principles are true, nor because its errors are small, but because they are, whether great or small, exactly equal, and exactly cancel and destroy cach other. This theory then is not only false but unnecessary. The false proposi- tions with which it sets out are as unnecessary as they are absurd. The errors may as well be great as small. The system is but a mere artifice, which “ by means of signs and symbols and false hypotheses, has been transformed into the sublime mystery of the transcendental analysis.” We dis- miss it, therefore, with the remark, that to admit and accept an error, even infinitesimal, in mathematics, strips the sci- ence of its chief glory, and introduces darkness and doubt where only the pure light of truth should prevail — in fact it opens the ‘door for anarchy in all science, and unsettles the very foundations of all knowledge. Xxil INTRODUCTION. THE METHOD OF LIMITS. The philosophical principle on which this method is based, is thus stated by Sir Isaac Newton, in the enunciation of the first lemma in the first book of the Principia. “ Quantities and ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given differ- ence, become ultimately equal.” The principle here stated would be applied to the solution -of our problem in the following manner. Th we BAW : f ; € ratio PES considered as the ultimate value of the ratio A meee PE which approaches it as BB’ decreases, and coincides with it when B’ has reached the point B, for then the points U P’ and Dwillhavecome together. Again é ae =2x-+h ap- proaches the value of 2x as #, or B’B, diminishes, and reaches that value when 4 becomes zero. PE yy : Now since Soe and since by the lemma of New- DE ton >Re becomes ultimately equal to EE? and since by the same lemma = = becomes ultimately equal to 2x, at the : Dita same time and for the same reason, it follows’ that PE 3S equal to 2x. It will here be seen that the point on which this demon- , stration turns is, that since the ratio PE approaches the DE 2 . ratio pp? 3s BB’ decreases, nearer than by any given dif- INTRODUCTION. XXIil ference, they become, according to the lemma, ultimately PE y'—y DE equal; and the equation Si oy Oe becomes pr 2” which is its limit. . The objection that lies immediately under the surface of this demonstration is, that when BB’ or % has become zero, and we have for a result Date hs. ear DE 03 A —=2x instea O PE —=2x Thus while we arrive at the desired value of our ratio, the ratio itself has lost all meaning, or at least all the attributes of quantity. The mind of Newton was too acute not to perceive the apparent absurdity involved in this application of his prin- ple, and he therefore gives the following explanation and defense of it: . “ Perhaps it may be objected that there is no ultimate pro- portion of evanescent quantities; because the proportion before the quantities have vanished is not ultimate, and when they have vanished is none. But by the same argu- ment it may be alleged, that a body arriving at a certain place, and there stopping, has no ultimate velocity; because the velocity before the body comes to the place is not its ultimate velocity; when it has arrived is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place, and the motion ceases, nor after, but at the very instant it arrives: that is, the velocity with which it arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities, -is to be understood the ratio of the quantities, not before they vanish nor afterwards, but with which they vanish.” The illustration here given by Newton unfortunately throws no additional light upon the subject. It is certainly XXIV INTRODUCTION, no easier to conceive a body coming to a stop zw7/h any velocity, than it is to conceive of quantities vanishing wth aratio. Sir Isaac Newton is quite right when he says that the same argument may be alleged against both propositions, for both involve notions that are equally repugnant to our reason and consciousness; one that there may be a ratio without quantities, and the other that there may be velocity without motion. In fact the illustration is, if anything, the more objectionable notion of the two, for the very reason why a body stops in its motion is because its velocity has expired and is gone. The argument based on Newton’s lemma has been, by no means, universally received even among those mathemati- cians who reject the philosophy of Leibnitz. M. Comté, who views the method of limits with consider- able favor, says of it: “It is very far from offering such powerful resources for the’solution of problems as the infin- itesimal method. The obligation it imposes of never con- sidering the increments of magnitudes separately and by themselves, nor even by their ratios, retards considerably the operations of the mind in the formation of auxiliary equa- tions. We may even say that it greatly embarrasses. the purely analytical transformations.” Again in speaking of the course adopted by certain geom- dy eters, he says: “ In designating by 7 “. 7, that which logically ought, in the theory of limits, to be denoted by Lo (that Increment of ¥ is, Limit Tee i} and in extending to all the other rement of Xx analytical conceptions this displacement of signs, they in- tended, undoubtedly, to combine the special advantages of the two methods, but, in reality, they have only succeeded in causing a vicious confusion between them, a familiarity INTRODUCTION. XXV with which hinders the formation of clear and emict ideas of either.” ~ Says Lagrange: “That method has the great inconven- ience of considering quantities in the state in which they cease, so to speak, to be quantities; for although we can always well conceive the ratio of two quantities, as long as they remain finite, that ratio offers to the mind no clear and precise idea, as soon as its terms na¥e become the one and the other nothing at the same time.’ But the objection to the lemma in question as a funda- mental principle of the calculus lies deeper than in its weak- ness and inefficiency. The proposition carried to its legiti- mate results, overthrows the very system it 1s supposed to establish. Says the acute and candid Bishop Berkeley, in reply to Jurin: “ Fora fluxionist writing about momentums, to argue that quantities must be equal because they have no assignable difference, seems the most injudicious step that could be taken; it is directly demolishing the very doctririe you would defend. For it will thence follow that all hom- ogenous momentums are equal, and consequently the veloc- ities, mutations or fluxions proportional to these are likewise equal. There is, therefore, only one proportion of equality throughout which at once overthrows the whole system you undertake to defend.” This is conclusive. If quantities that during any finite time constantly approach each other, and before the end of that time approach nearer than any given difference are 4 xe | ultimately equal, then not only >> PE and, =; jz become ulti- Migtervacdttan butler, eb). bh P’B. and es all become ultimately equal, for they all fulfill the conditions required by D ° lemma. Hence instead of pr 2%, or even [=24, we have as the logical sequence of this proposition 2v=1, XXV1 INTRODUCTION. ‘ ‘Although the method of limits has generally been attrib- uted to Sir Isaac Newton, who was the author of the prop- Osition which has served for its foundation, it is certain that this method as applied to the Differential Calculus, or method of fluxions, was not his. He has laid for that a very differ- ent foundation, as we shall see in due time, which has cer- tainly this merit, that there is nothing false nor absurd about it; and if it doesnot clearly unravel the mysteries of the calculus, it places in our hands the only clue by which we can do it for ourselves. METHOD OF LIMITS APPLIED TO THE DOCTRINE OF RATES. Prof. Loomis, of New Haven, has undoubtedly adopted the true conception of the nature of a differential. But he has unfortunately attempted to combine it with the method of limits, and has, therefore, become entangled in the same inconsistencies that we have already found to be inseparably attached to that method. It will be observed that in the following demonstration taken from his calculus, the question hinges on the ratio of the differential or rate of change of a variable to that of its square. And hence the demonstration although not arising directly in a question of tangency, must yet be tested by the same principles as those we have examined. “Tf the side of a square be represented by ~ its area will be represented by x?. When the side of the square is increased by 4, and become x+%, the area will become (x+/)?, which is equal to x*+2xh+h* : While the side has increased by 4, the area has increased by 2xh+h?, If then we employ 2 to denote the rate at which x increases, 2x2+h* would have denoted the rate at which > a INTRODUCTION. eva the area increased had that rate been uniform; in which case we should have had the following proportion, rate of increase of the side:rate of increase of the area 2:2: 2xh +h? ::1:2x+/ but since the area of the square increases each instant more and more rapidly, the quantity 2x+4 is greater than the increment which would have resulted had that rate been uniform; and the smaller Z is supposed to be, the nearer does the increment 2x+ approach to that which would have resulted, had the rate at which the square was increasing, when its side became x, continued uniform. When Zis equal to zerv this ratio becomes that of 20 which 1s, therefore, the ratio of the rate of increase of the side to that of the area of a square when the side is equal tie This demonstration is plausible, but will not bear close examination. We are told that when Z is equal to zero, the ratio of the rate of increase of the side of the square to that of its area is that of 1 to 2x. But by hypothesis / repre- sents the rate of increase of the side of the square; if then A becomes zero, the side (and of course the area) of the square wel have no rate of increase, and hence instead of rate of increase of the side: rate of increase of thearea:: 1: 2% we shall have O:O0::1: 2% or f=2x a result that we are already familiar with. A COMBINATION OF METHODS. Another text-book of very high authority sets forth the following method, which seems to be partly derived from Lagranye’s idea of derived functions. We give the author’s explanation in his own words: XXVI1 INTRODUCTION, “ To explain what is meant by the differential of a quantity or function, let us take the simple expression u=ax® (1) in which z is a function of x. Suppose x to be increased by another variable “%; the original function then becomes a(x+h)* ; calling the new state of the function #’ we have uw’ =a(x+h)*=ax? +2axh+ah? From this subtracting equation (1) member from member we have u —u=2axh+ah? The second member of this equation is the difference be- tween the primitive and new state of the function ax, while his the difference between the two corresponding states of the independent variable x. As &% is entirely arbitrary, an infinite number of values may be assigned to it. Let one of these values, zich zs to remain the same while x ts inde- pendent, be denoted by dx, and called differential of x, to distinguish it from all other values of %. This particular value being substituted in equation (1) gives for the corres- ponding difference between the two states of w or x* uw’ —u=2ax dx+ta(dx)? (2) Now the first term of this particular difference 7s called the dif- JSerential of u, and is written du=2ax .ax The coefficient (2ax) of the differential of x, in this expres- sion, is called the differential coefficient of the function u, and is evidently obtained by dividing the differential of the func- tion by the. differential of the variable, and 1s in general written ” oe eax (3) Now will any inquiring student be satisfied with this “explanation?” Will he infer from it anything of the nature INTRODUCTION, XX1xX and office of a differential, and what is its philosophical relation to the function to which it belongs ? But perhaps this is not intended as a full explanation, for the author proceeds: “ Resuming this expression ui —u=2axh+ah* and dividing by 2, we have wu! —U h In the first member of this equation the denominator is the variable increment of the variable x, and the numerator is the corresponding increment of the function w,; the second member is then the value of the ratio of these two incre- ments. As # is diminished, this value diminishes and be- comes nearer and nearer equal to 2ax, and finally when =o it becomes equal to 22x. From this we see, that as these increments decrease, their ratio approaches nearer and nearer to the expression 2ax, and that by: giving to 4 very small values, this ratio may be made to differ from 2ax by as small a quantity as we please. This expression is then properly, the limit of this ratio, and is at once obtained from the value of the ratio by making the increment =o. It will also be seen that this limit is precisely the same expression as the one which we have called the differential coefficient of the func-’ tion wz.” Now we have here a term arbitrarily selected without explanation or apparent reason, in which /% has a fixed value; and this term is called the differential of the function. But afterwards in order to find the value of this term it becomes necessary to reduce % to zero. The question arises here, if h must be zero in the one case why must it not be zero in the other? It will not answer to say that 2ax, the differential coefficient, does not contain # or dx, and is therefore not affected by its value; because @x des occur in the first mem- =2ax+ah Xxx INTRODUCTION. ber of equation (3) as its denominator, and hence as to do : Lu d with the value of 2ax. We have then 2ax= Ts where ¢x 1s } u—tu 2 a particular value of 4, and 2ax =F where Z is equal to zero. If the case be so that the value of the fraction (Li hat, ed aad : Tz OF 7% 18 independent of that of the denominator, the author nowhere tells us why it is so. The object of the author in this setting forth of his method is evidently to avoid the apparent absurdity of making @v=o while it performs so important an office in his subsequent analysis; thus escaping one of the inconsistencies of the method of limits. But as he practically uses that method in the first part of his work, there seems to be after all some confusion of ideas, and it is difficult to regard dx as having a fixed value, while its representative 2 is continually re- duced to zero. The author himself seems to have become . wearied with this indirect and misty method, and when he comes to the practical application of the Calculus to Geom- etry, comes squarely down on to the infinitesimal system which, with all its inconsistencies, is far more direct and fruitful in its results than the method of limits. The most striking circumstance in connection with every - modification of the method of limits is, that the value of the differential coefficient is the objective point sought after. Whether this is obtained by a sound, logical demonstration ° or not, it is the only thing found which has a real value, of which we can form a definite conception. Now the differ- ential itself is quite as important as the differential coeffi- cient. It is true we do not regard its actual measured value, but we do regard its relative value as compared with that of other differentials; and for this purpose we need some value that the mind can grasp and upon which the imagination can a a INTRODUCTION. XXxXI rest with satisfaction. But none of the systems we have examined present the idea of a differential as consisting of any such quantity. The infinitesimal system does indeed profess to give a sort of value to the differential. It is the “last value of ‘the variable before it becomes zero,” or “ the difference between two consecutive values of the variable ”— words that con- vey no more definite idea of quantity than a sort of attenu- ated essence of one about to vanish; while the system of limits leaves us nothing whatever but the ghosts of those that have departed entirely. Who would not desire to be relieved from the constant strain upon the imagination, and the severe draft on the faith required to attain the results of the calculus by such feeble means? What a relief to have placed in our grasp a principle that has substance and vitality; that is adapted to our conceptions and meets the demands of our reason, whose meaning is not dim nor doubtful, but clear as the noonday sun, shining by the light of its own self-evident truth. We have said that a fundamental error runs through all the systems of infinitesimals and limits, arising from the method of approaching the subject. To understand clearly what this error is we must have a clear conception of the true nature of a differential, and of the symbol which rep- resents it. As we have already stated, it is truly defined by Prof. Loomis as “ the rate of variation of a function or of any variable guantity,’ and further, “by the rate of increase at any instant we understand what would have been the absolute increase if this increase had been uniform.” And the dif- ferential coefficient is the ratio between the rate of varia- tion of any variable and the consequent rate of variation of the function into which it enters. Now this ratio is really what is sought after in both the systems that we have exam- ined. In the system of Leibnitz the infinitesimal increments “~ XXXil INTRODUCTION. represent the rates of increase, and in the method of limits the ratio of the rates is obtained from that of the actual increments by reducing them to their vanishing point. Not that the authors of these systems were conscious of any such meaning in their methods, but this was, nevertheless, the real, though unrecognized, philosophy on which those methods were based. Now the error which gaye rise to all the absurdities, sophisms and obscurities of their system was this — “hey endeavored to arrive at the ratio of the rates of change by means of the actual changes. ‘That is, they gave to the variable an increment, and to the function a correspond- ing one, and from these attempted to derive what is really the ratio of the rates, or the differential coefficient. This can- not logically be done, except in the case of uniform varia- tion ; for in all other cases the rate changes as the value of the function changes; so that before the rate can be meas- ured by any actual change, it will itself have changed. Take a familiar illustration. It is a well-established fact in Natural Philosophy, that the velocity of a body falling 7x vacuo to the earth cannot possibly be measured for any one instant by the actual movement of the body subsequent to that instant, for no such subsequent movement will be made with the same velocity. Now if no actual change can rep- resent a variable rate of change, the ratio of the actual changes cannot truly represent the ratio of the rates how- ever small they may be made. It is this effort to do what is, in the nature of things, im- possible, that has introduced all the difficulties, enigmas and mysteries that have beset the differential calculus from the beginning. Now these are as unnecessary as they are objectionable. The true principles of the science are as clear and consonant with reason as the elements of Euclid, and the science itself flows from them as directly as the light from the sun. Not only so, but while the methods we INTRODUCTION, XXXIll | have examined have produced a study hard and unattractive, consisting almost entirely of manipulations of the mere machinery of analysis, the subject is really full of beauty, abounding in ideas of the most novel and interesting kind, and furnishing a field for the exercise of the imagination that will tax all its powers —it is, in fact, the poetry of mathematics. THE TRUE METHOD OF NEWTON. The method of arriving at the differential coefficient by means of the ultimate ratios of the increments, or, in other words, the method of limits, has generally been ascribed to Sir Isaac Newton; but this is evidently an error. The theory on which that method is founded is certainly his, and it is but just that he should be held responsible for the re- sults that legitimately flow from it. But it is not the theory on which he formed 4zs method of fluxions. Zhafis con- tained in the second lemma of the second book of his Prin- cipia. In a scholium to that lemma he says: “In a letter of mine to Mr. J. Collins, dated Dec. 10, 1672, having described a method of tangents——which at that time was made public, I subjoined these words. TZhes zs one particular or rather corollary, of a general method, which extends itself, without any troublesome calculation, not only to the drawing of tangents to any curved lines, whether geometrical or mechanical, or any how resolving other abstruse kinds of problems about the crookedness |curvature| areas, lengths, centers of gravity of curves, etc., nor ts tt limited to equations which are free fron surad quantities. This method I have interwoven with that other of working equations, by reducing them to infinite series. So far that letter. And these last words relate to a treatise I composed on that subject in the year 1671. The founda- XXXIV INTRODUCTION. tion of that general method is contained in the preceding lemma.” Here it is distinctly stated by Newton himself that he had invented a general method which was applicable not only to the drawing of tangents, but to all the higher and more del- icate problems which appear in the differential calculus, and that this general method has dhe lemma in question for tts FOUNDATION. We have then but to examine this lemma to ascertain the real basis on which the “method of Newton” was con- structed. Forthis purpose we give the lemma in the author’s own words. LEMMA II. “ The moment of any genttum ts equal to the moments of each of the generating sides drawn into the indices of the powers of those sides, and into thetr coefficients continually. “T call any quantity a genitum which is not made by the addition or subduction of divers parts, but is generated or produced in arithmetic by the multiplication, division or ex- traction of the root of any terms whatsoever; in geometry by the invention of contents and sides, or the extremes and means of proportionals. Quantities of this kind are pro- ducts, quotients, roots, rectangles squares, cubes, square and cubic sides and the like. “These quantities I here consider as variable and inde- termined, and increasing or decreasing as it were by a per- petual motion or flux; and I understand their momentane- ous increments or decrements by the name of moments; so that the increments may be esteemed as additive or affirm- ative moments, and the decrements as subducted or nega- tive ones. But take care not to look upon finite particles as such. Finite particles are not moments, but the very quan- tities generated by the moments, We are to conceive them INTRODUCTION. XXXV as the just nascent principles of finite magnitudes. Nor do we in this lemma regard the magnitudes of the moments, but their first proportion as nascent. It will be the same thing, if, instead of moments, we use either the velocities of the increments and decrements (which may be called the motions, mutations and fluxions of quantities), or any finite quantities proportional to those velocities. The coefficient of any generating side is the quantity which arises by applying the genitum to that side. “Wherefore the sense of the lemma is, that if the mo- ments of any quantities A, B, C, etc., increasing or decreas- ing by a perpetual flux or the velocities of the mutations which are proportional to them, be called a, 4, c¢, etc., the moment or mutation of the generated rectangle AB will be aB+6A; the moment of the generated content ABC will be aBC+/AC+cAB; and the moments of the generated pow- ers A®, AS, At, AP, A? AS AS A-1, A-*, A~® will be — tA, *, aA? —ah—*, al 2aA, 3aA*, 4aA%, 4aA *, 3aA® : PES. : ; —2aA~*,>—taA * respectively; and in general that the 7D nm moment of any power A” will be “aA -m Also that the moment of the generated quantity A?B will be 2aAB+dA®; the moment of the generated quantity A®B4C? will be 3aA*B*tC? +-40A3 BC? + 2cA2B4AC ; and the moment of the 3 generated quantity = or A®B-?, will be 3¢A*B-*— 26A°B-%,andsoon. The lemma is thus demonstrated. “Case 1. Any rectangle, as AB, augmented by a perpet- ual flux, when as yet there wanted of both sides A and B, half the moments $a and 44, was A—4a into B—4é, or AB—4aB—46A+4ad ; but as soon as the sides A and B are augmented by the other half moments, the rectangle be- comes A+-}a into B+34, or AB-+-4aB+30A+ j0d, From XXXVi INTRODUCTION. this rectangle subduct the former rectangle, and there re- mains the excess a@B+0A. ‘Therefore with the whole incre- ments a and & of the sides, the increment aB+0A of the rectangle is generated Q. E. D.” “Case 2. Suppose AB always equal to G, and then the moment of the constant ABC or GC (by case 1) will be gC+cG, that is (putting AB and aB+éA for G and g) aBC+é4AC+cAB. And the reasoning is the same for con- tents under ever so many sides. Q. E. D.” It is unnecessary to quote the demonstrations of the other cases, as they all flow naturally and logically from these which form the key to the whole system. We must concede that this demonstration is not as clear and complete as could be desired. Let us, however, endeavor to extract from it the real, though perhaps some- what vague conception of the subject which occupied the mind of Newton. It is to be remarked, however, that the doctrine of “mts is nowhere hinted at, but the results are direct, positive and substantial. The first question suggested by the lemma is, what is really meant by the term “ moment.” It might at first seem that the “ moments” of Newton were in fact the same thing as the differentials of Leibnitz, for he speaks of them as something (though not finite quantities) to be added or sub- tracted. But a very little examination of the lemma will dispel the notion. Their magnitudes are not to be regarded. But the magnitudes of the differentials of Leibnitz are to be regarded as infinitely small. Again, “finite particles ” are not “moments,” but the “very quantities generated by the moments.” Now the differentials of Leibnitz never gener- ate anything; they are the infinitesimal remains of incre- ments that have been added and then taken away. Again, moments are the “nascent principles of finite magnitudes.” But the “principles” which generate “ finite magnitudes ” INTRODUCTION. XXXVII or increments can be nothing else than the Zaws which con- trol the changes in the “ genitum;” that is, THE RATE OF CHANGE. This interpretation is confirmed by the further statement that we may use instead of them “the velocities ” or any finite quantities proportional thereto. Hence we infer that a, 4, c, which are called moments, are intended as “ symbols to represent the rates of change, being finite quan- tities proportional to those rates, and as the quantities A, B, C, etc., are increasing or decreasing by a “ perpetual flux,” that is by a uniform rate of change, the actual incre- ments or decrements a, 4, ¢ will represent those rates. So that the difference between A—4a and A+4a (equal to a) represents the rate of increase of A, and the difference be- tween B—46 and B+44é (equal to 4) accruing during the same time represents the corresponding rate of increase of B; and the ratio of a to 4 represents the ratio of those rates whatever may be their magnitude as symbols. But while these symbols or suppositive increments (being produced at a uniform rate) represent the respective rates of increase of A and B, we are told that the corresponding increment of their product (2B+--a Ans. 2bxdx—3y* dy ae 3. ax” —bx? +x Ans. Ex. 4: (¢c+d)(y?— +a) a b—2y* Ne me NV oax+x® 2 VJ — x? Be xt 1— x? (aaa) E a2 —x2 ah 1 +x? Vira— Vi-« Ae eA lr (a + bcnym Ans. Ans. Ans: Ans. Ans, Ans, Ans. Ans. Ans. Ans, Ans. Ans. Ans. Ans. Ans, Ans. Ans. Ans. Ans. Ans, Ans. Ans. Ans. Ans. DIFFERENTIATION OF FUNCTIONS. v9: Ex. 20; (6—c)(x—y)® Ans. 02-80. AV 22 +a n/x Ans. £x. 31 becomes with «=o, that is 3%, or half the SUCCESSIVE DIFFERENTIALS. 85 becomes one-sixth of the third differential coefficient of z or Ou = 6ax3 Hence, indicating by a vinculum that x has been made a : 6x equal to zero, we have for the three coefficients 3x, 5 and 6 : F 3-3 Which were true of the cube before any increment was made, ) 2, 3 Chas ersas) 3B, . also true of the cube at the same time and the different parts of the cube increased will be represented as follows : The cube 4Z by (wz) or 23. The three solids Da, Df and Dg by (S)x or 3h? x. 9 w : Zt The three solids De’, De" and Dé by (< Jac? OF 34%; 2ax* aru ees Ped Ae 8 a Ole Ory, ; The solid Da” by ( and these make up the value of (4+.)’, hence wor (its)? =(u)+(Fe) « +(SGs) a? +(=— a) 8 or (u+x)3 =h8 + 3h? x+ 3hx? +23 This illustrates to some extent the law which connects together the parts which go to make up the change in the function of a variable arising from that of the variable itself. A more complete and general demonstration of this law is contained in the following theorem. MACLAURIN’S THEOREM. (24) A function of a single variable may often be ex- panded into a series by the following method. 86 DIFFERENTIAL CALCULUS. Representing the function by wz and the variable by x we shall have : “i Pia) When this function can be developed, the only quantities that can appearin the development, besides the powers of x, will be constant terms and constant coefficients of those powers. Hencethe developed function may be put into the following form: Uu=At+BoetCxr? +-Dx3 +Fxt+ etc. (1) in which 4A, #, C, D, Z, etc., are independent of x. The problem is to find the value of this constant term 4, and the values of the constant coefficients B, C, D, #, etc. For this purpose we differentiate equation (1) successively, and divide each result by dx, the successive differential coeffi- cients will then be ai we B+ 2Cut 3 Da? + 4Exe + ete. (2) au Tae =2C+2.3.Dxt3.4.Ex® + ete. (3) aru Tye =2-3-D42.3.4. Ext ete, (4) Since x is an independent variable, these equations are true for all values of x, and, of course, when x=o. Reducing x to zero in equation (1), 4 becomes equal to the original function with x reduced to zero. We will rep- resent that state of the function by (w),; and also indicate by brackets around the differential coefficients that «=o in their values also. Then from equations (2), (3), (4), etc., we have a=(=), c=(5*), 28 p=(“%) and so on to the end of the development, if it can be com- pleted; if not, then in an unlimited series. Substituting these values of 4A, B, C, D, etc., in equation (1) we have SUCCESSIVE DIFFERENTIALS. 87 which is Maclaurin’s theorem. + etc. 2 EXAMPLES, Ex.1. Expand (a+)” into a series. Represent (2a+.x)” by z and we shall have u=(atx)"” =A+ Bet+Cxr?+Dx3 + etc. (1) Differentiating we have Tu =nla+x)"-1 VT ee le 1) (a+x)"-? aru beaks dtu err: =n(n—1)(2z—2)(n—3(a+x)"—4 =n(n—1)(z—2)(atx)"-8 from which, when «=o, we have A=a” Te ie oe re mn) -, Pe nin—1)(~—2) P On2 hy re aN 5 ty \ 2 Pu n(n—1)(nz—2)(2z—3)a ee ore Substituting these values in equation (1) we have (n —1) n—1) (n— u=(atx)” =a" +na"-1x Se Aa BE ft—3.8 and so on; the same result as by the binomial theorem. ih. : x. 2. - Develop. <-, into'a series. We have by differentiation 88 DIFFERENTIAL CALCULUS. MU I ax (a+x)® ee eee dx® ~(a+x)3 au 2a dx® ~~ (a+x)4 and so on. Making «=o in the values of w and the differential coef- ficients we have = (=e (G=2 (=F Substituting these values in Maclaurin’s formula we have Tory Ely in eee Sis poe to atx a Set qi 1) ie Pie eg EVElOp ine LENCO ss ae : ; into a series. Ans, 1 pate t+yr*+txr?+xr4+4+ etc. Ex. 4. Develop the function ee into a series. Ans. pe gt ye 5 qt Py? 7 maa ee I ; | , wig Wevelop Gaye aseries. Ans. #ix.6. Develop 34/(a+-x)2 into a series. Ans. fine Develop A a? +x into a series. Ans. Xo LION EID We = into aseries. Ans. Ex. 9. Develop (a?—x?)~ Ohrid series: Ans. The formula of Maclaurin apples in general to all the functions of a single variable that are capable of successive differentiations. But there are cases in which the function or some of its differential coefficients become infinite when x=o ; in such cases the formula will not apply. The func- ho: =n, & : tion, c+ax* is an example of this kind; for if we represent it by « we have SUCCESSIVE DIFFERENTIALS. 89 and If in this we make x=o for the value of the coefficient B, we have B=—=% In general, any function of x in which x is not connected with a constant term under the same exponent, cannot be developed by this theorem; for the differential coefficients will be such as to reduce to zero or infinity in every case, when x is made equal to zero. TAYLOR'S THEOREM. (25) Zhe object of this theorem ts to obtain a formula for the development of a function of the sum or difference of two vari-— ables. The principle on which this theorem is based is the fol- lowing: The differential coefficient of a function of the sum or difference of two variables, will be the same whether the function is differentiated with respect to one variable alone, or to the other variable alone. Thus the differential coefficient of (x+y)” will be w(x+y)"—-! if we differentiate with res- pect to either variable alone, the other being considered as constant. A function of the sum or difference of two variables is one in which both are subject to the same conditions, so that the value of their sum or difference might be expressed by a single variable without otherwise changing the form of the function; and hence we may regard this sum or difference as itself a single variable. Now any rate of change in one of the two component parts (the. other being regarded as go DIFFERENTIAL CALCULUS. constant) will produce the same rate in the compound vari- able (so to speak) as it has itself; thus x+y will increase at the same rate as x if y be constant, and at the same rate as y if x be constant. So that changing froin one to the other is merely changing the rate of the single variable that would represent the value of their sumordifference. But such change in the rate will not change the va#o which it bears to the corresponding rate of the function (Art. 6); that is it will not affect the differential coefficient. (26) To apply this principle let us take any function of the sum of two variables, as (x+y), which we will repre- sent by w. If it can be developed into a series, the terms of the series may always be arranged according to the power of y,; the coefficients being functions of x and the constants ; hence it may be made to take the following form : u=F(xty)=A+By+Cy?+Dyi+LZLy4+ etc. (1) im which 4, B,C, ), 2, etc, are, indéependentoiy sat functions of «x. If we differentiate equation (1) regarding y as constant, and divide by ax, we shall have Lu Ae OB ea. aD , ax SE RETO erg Dot tes If we regard x as constant, and divide by dy, we have V2 ue +42y?+ etc. . du - and since =, 1s equal to the second members of these a equations are equal; and since this equality exists for every value of y, and since the coefficients are independent of that value, the corresponding terms containing the same powers of y must be equal each to each; hence SUCCESSIVE DIFFERENTIALS. gI aA OR baited (2) CB ioe (3) HAC reer (4) rete Boe AL (5) If now we make y=a, then /(x+y) becomes F(x), which we will represent by z. Under this supposition equation (1) will become wz (now become z) = 4 Substituting this value of 4 in equation (2) we have a Substituting this value of B in equation (3) we have whence similarly we have eee: arg pV le 23 ae and and so on. Substituting these Aas in equation (1) we have gz Wee veF (et )=245 re LEP Ms EE ees Sie in which the first term is what the nett becomes when y=o, and all the coefficients of the powers of y are derived from it on the same supposition. Q2 DIFFERENTIAL CALCULUS. This is the formula of Taylor. : A function of «—y is developed by the same formula by changing y into —y, thus: @3 as ae WS ENE TY) PX a Dae a etc EXAMPLES. Ex. 1. Let it be required to develop (x+y)”. Representing this function by z we have u=(x-+y") and z=x" then by differentiation 26 3 & =x, TS =aln— i) come nln 1) (2—2)xr-8 Substituting these values in the formula we have =! oer ey n—1, 1") n-9,9 1 m(n—1) (n—2) » 3.9 w(x) 8x" Ae ye ay Eg By 4 etc., the same as by the binomial theorem. Ex. 2. Develop the function / x+y. sae o> 1 1 3 5 > > 5 1 —2 jh AP Bote aie: CLT ty eee oo ee His. (ob) SRO ae ee ey etc, Ex. 3. Develop ¥/ x+y a5 x %y3— etc, +.,.-% tee re ee Develop the function (x—y)”. Aus. 4 .5. Develop the function (x—y)®. Aas. 6 : 1 Develop the function 55. Azs. Ex. 7. Develop the function of eee Ans (27) Although a function of the sum or difference of two variables can generally be developed -by this formula, yet there are cases in which the coefficients (which are functions of one of the variables) may, by giving certain values #0 the variable they contain, become infinite. In such cases the formula cannot be applied; for in general such values for SUCCESSIVE DIFFERENTIALS. 93 that variable, would not reduce the function itself to infin- ity, although it would have that effect on its development. Thus in the function u=a+ (x+y) we have az I igs: I i 3 4(—x)* 1 z=at+(b—x)?, Sot Seas ere 7 = ax 2 (b—x)? ax and so on. If now we make x=4J, all the coefficients will become infi- nite, and we should have ij u=at+(b—x+y)? =a+ 0 by the formula instead of having as we ought u=a ates : which cannot be, for the value of y is not dependent on that of x, and hence wz is not necessarily infinite when x=; but for all other values of x the formula will give the true development of the function. And herein is the difference between the formulas of Tay- lor and Maclaurin; when that of the former fails it is for only one value of the variable ; while that of the latter when it fails at all, fails for every value of it. Note.— In fact, the theorems of both Taylor and Maclaurin are founded on the principle illustrated in Art. 21. The real object of both is to find from the rate of change of a function what will be its new state arising from a given change in the value of the variable. The general method of doing this is to find the successive differentials of the func- tion in its first state, and then to multiply the successive differential coefficients by the successive powers (properly divided) of the actual change in the variable, this will give the actual successive partial changes in the function which together make up the entire change, and thus develop the function in the new state. For this purpose the variable must have two points of value; one where the furc- tion is to be differentiated, and the other, the new value produced by the change ; and to this end the variable, in algebraic functions, is made to consist of two parts, either by making it a binomial or something that may be reduced to that form. In Maclaurin’s theorem the variable consists of a constant and a variable combined together, so that their united value is a variable one, and the constant part is simply one point in that variable value. This is the point at which the differentiation of the 94 DIFFERENTIAL CALCULUS, function is made ; but as a constant cannot be differentiated, the variable is attached to it long enough for that purpose and then made zero, In Taylor’s theorem the varia- ble is the sum or difference of two others, and the poimt of differentiation is when the variable has reachcd the value of one of its variable parts. This being a variable, the function can be differentiated directly, and the other variable may be made zero defore the operation. Hence the theorems of Maclaurin’s and Taylor are alike in this: both have a compound variable having two points of value, both are differentiated at the same point, and the successive differential coefficients, which are precisely alike in form and value, are multiplied by the successive powers of the change in value. The only difference is that in one the differentiation at the required point is made 7xd/rectly, and the variable change made zero afterwards ; while in the other the differential is made directly, the variable change being made zero beforehand. Hence a function of a binomial variable may be expamded by either method. By Taylor’s, considering both terms variable and reducing one to zero de/ore differentiation ; or, by Maclaurin’s, by considering one term as constant and reducing the other to zero after differentiation. Thus in the case of the function iz +y) Ls the differential coefficient will be pre- cisely the same if we reduce y to zero and differentiate 2 by Taylor’s method, or con- sider + as constant and reduce y to zero after differentiation, by Maclaurin’s method. In order thata binomial may represent a single variable, both terms must be subject to the same conditions, so that each term may be considered asa part of the same compound variable ; and the failing cases in Maclaurin’s and Taylor’s methods are simply those in which the binomial variable becomes a monomial, by giving the variable 1 ; 3 Pet se : ; : a certain value. Thus the case cited in Art. 24,¢-+ QX*, isnot a true binomial vari- able, since the terms are not subjected to the same conditions. If we make it a (ctax)? we have a true binomial variable, and the differential coefficient 4c 0) 2 will ot reduce to infinity when +=o. Similarly the case cited in Art. ie 27, namely, 7—q +(s—x +y) * is one in which when x=4, the variable in the function reduces to y, and the function itself to a+y*, which does not contain a binomial variable of the required form. The same principle will apply to transcendental functions; which, in order to be developed, must have two points of value in the compound variable — one for the dif- ferential and the other for the development. Thus @”% may be expanded by Mac- laurin’s theorem, since it has two points of value, one at %, the point of differentiation where +=a, and the other the full value produced by x. Ce LOIN eb VW" MAXIMA AND MINIMA. (28) We have seen that when a variable changes its value at a uniform rate, the value of its function will in general vary at a rate that is not uniform. It may increase at a diminishing rate, until at a certain point it ceases to increase and begins to diminish, in which case the turning point is the one of greatest value, and is called a maximum. Or it may decrease to a certain point and, having attained its min- imum, begin to increase. The problem is to find whether there zs a Maximum or minimum value for a function, while its variable is uniformly increasing, and if there is, to find the corresponding value of the variable and its function. (29) While a positive function is increasing as the varia- ble increases, its rate of change or differential will be posi- tive; and negattve when it decreases (Art. 3). Hence when a function is passing through a maximum or minimum value, the sign of the differential coefficient must change from minus to plus or from plus to minus — the former in case of a minimum, and the latter in case of a maximum. But such change can only take place while the differential coefficient is passing through zero or infinity. Our first inquiry then is whether there is any finite value of the vari- able that will reduce the first differential coefficient to either 95 96 DIFFERENTIAL CALCULUS. -of these values. For this purpose we solve the equation formed by placing the first differential coefficient equal to zero, and thus find the corresponding value of the variable. Here we have one of three results. First. There may be no real value for the variable. In this case there is neither maximum nor minimum. Second. There may be a real finite value for the variable that will reduce the differential coefficient to zero. In this case there will probably be a maximum or minimum. Third. There may be no finite value of the variable that will reduce the differential coefficient to zero, but at the same time there may be one that will reduce it to zzfinzty. In this case we form the equation by placing the differential coeffi- cient equal to infinity, and the root that satisfies the equa- tion will indicate a frodable maximum or minimum. In order to determine in the two latter cases whether there 7s a Maximum or minimum value of the function, and if so which of the two it is, we may substitute in the function, in place of the variable a quantity a little less, and one a little greater than that derived from the equation. If the result in both cases 1s less than when the true value is substituted there is a maximum; if greater, there 1s a minimum value of the function for the true value of the variable. We may also determine the same thing by substituting these approximate values in the differential coefficient, which the true value reduces to zero. If they cause the result to change the sign from plus to minus by substituting first the less, and then the greater quantity, there is a maximum, for the function is passing from an increasing to a decreasing state. If the change is from minus to plus, there is a min- imum, for the function is passing from a decreasing to an increasing state at that point. MAXIMA AND MINIMA. 97 EXAMPLE. Find the value of « which will render z a maximum or minimum in the equation, u=x>—ox* 244 —7 Differentiating and placing the differential coefficient equal to zero we have = 30? —182+24=3(x?—6x+8)=o0 from which we find e=4 and #=2 If we substitute in the function and in the different coeffi- clent I, 2, 3, 4, 5, etc., successively, we shall have for au Sie er th tee 5 au __ Ma eee 3 ee 7 -O fe: = du __ Mere autae, | Ciet al) a. og F avis *% du __ gots Aad OW a X—=5 Sas) a OLA is st ag a = du __ R= Ole PUR 26. «oi 24 Indicating that for x=2 the value of the function is a maximum, the differential coefficient passing from plus to minus; and for x=4 the value of the function is a mini- mum — the differential coefficient passing from minus to plus. (30) It must be understood that by maximum and mini- mum is not meant the absolutely greatest or least value of the function, but the ‘urning point, from an increase toa decrease, or wice versa. Hence there may be as many max- ima or minima of the function as there are values of the variable that will reduce the first differential coefficient to zero or infinity. It is also to be understood that in the discussion of ¢hzs subject, when a function is stated to be an zucreasing one, it 6 a 98 DIFFERENTIAL CALCULUS. is meant that it is either positive and becoming greater, or negative and becoming less. If it is said to be decreasing, it is either positive and becoming less or negative and becom- ing greater. Thus if we take the function _ ; u=x* — 25 and make «x, successively, equal to TRO a hae the successive values of the function will be 24,2 1s 105-010; otek See praet and it is said to be increasing throughout the whole change, although at first its numerical negative values decrease. This is also indicated by the sign of the differential coeffi- cient which is positive as long as x is positive. The terms. “increasing” and “decreasing” then, in this case, refer merely to the a@rection in which the function is changing, no matter on what side of zero its value may be. (31) There is another method of ascertaining whether the first differential coefficient changes its sign on passing through zero or infinity, for this is the unfailing test of a maximum or minimum. Having found that value of the variable which reduces the first differential coefficient to zero, substitute that value in the second differential coeffi- cient, if it contain the variable, then first, Tf it reduces the second differential coefficient to a negative quantity, it indicates that when the first is at zero it must be a decreasing function, which can only be at that point by its passing from a positive toa negative state, and hence the function itself must be passing from a state of increase to a state of decrease, and hence is at a maximum. Second. If it reduces the second differential coefficient to a postive quantity, it indicates that the first when at zero is an increasing function, and must, therefore, be passing from a negative toa positive state, hence the function is passing from a MAXIMA AND MINIMA. ai 99 a decreasing to an increasing state, and is, thx fore, at a minimum. | Third. If it reduces the second differential coefficient zero, We may resort to the third; and if the same value of the variable reduces that to a real finite quantity, either pos- itive or negative, it shows that the second, at zero, is chang- ing its sign, and, therefore, the first is changing from an increasing to a diminishing function, or we versa, and, therefore, does not at the zero point change tts sign. Hence there is neither maximum nor minimum in the value of the function. Fourth. Tf it reduces the éAzrd differential coefficient to zero we may resort to the fourth. If it reduces this to a real finite value, it indicates that at zero the third changes its sign, for it can only increase on both sides of zero by pass- ing from negative to positive, or diminish on both sides by passing from positive to negative. This will show that the second coefficient does zo¢ change its sign, for if it increases on one side of zero’ and decreases on the other, or wice versa, it can only approach the zero point until it éowches 77, and then must recede without changing its sign. This proves that the first coefficient does change its sign, for since the second does not change the first must be passing ‘Arough from one side to the other. There will, therefore, be a maximum or mini- mum —the first if the fourth differential coefficient has a negative value, and the second if it is positive. We may continue thus and show thatif the first differen- tial coefficient that is reduced to areal value, by substituting that value of the variable that reduces the first to zero, is of an even order and postive, there will be a minimum ; if it is negative, there will be a maximum; and if it is of an odd order there will be nether maximum nor minimum. Fifth. If any value of the variable reduces the first dif- ferential coefficient to infinity, it will probably reduce all the Too DIFFERENTIAL CALCULUS. succeeding ones to infinity, also. It will, therefore, be best in such a case to substitute values for the variable a little less and then a little greater than the one found. If the value of the first differential coefficient changes from plus to minus there is a maximum, and the second will be plus on both sides of infinity ; for the first must be an zzcreasing pos- itive function in order to become positively infinite, and if negative on the other side must be a decreasing function, for it cannot be an increasing negative function on leaving infinity. Hence (Art. 3) the second must be positive in both cases. Sixth. If the first differential coefficient in the last case changes from minus to plus there will be a minimum, and the second coefficient will be minus on both sides of infinity. Thus we see that when any value of the variable reduces the first differential coefficient to zero, and is: substituted in the second, a mnus result indicates a maximum in the function, and a plus result a menimum. When any value reduces the first coefficient to zzfinity, a plus sign for the second indicates a maximum, and a minus sign a minimum. EXAMPLES. fix. t. In order to illustrate the first case in this article we take the function u=16x—x? (1) from which we obtain by differentiation Tut Tp 16 28 (2) a*u i (3) We find that x«=8 will reduce the first differential coefficient to zero, while the sign of the second is minus. Hence at x«=8 the first must be a decreasing function, and, therefore, MAXIMA AND MINIMA, IOI passing through zero from plus to minus, the function will, therefore be an increasing one to that point and then a diminishing one; hence a maximum. If we substitute in the function and the first differential coefficient for « values a little less and a little greater than 8, we have for du x=7 uw=63 ig 2 C= 5 uz—64 eo —_ saat du x—9 eee u=63 a hac ap If we represent the values of the function by the ordi- nates of the curve ABC (Fig. 3), the curve itself will cor- respond to the range or locus of values of the function, while the variable increases uniformly in passing from 7 to 9. From A to B the function increases, B but at a decreasing rate, and hence “ C the first differential coefficient is positive but decreasing until it reaches zeroat B. The function then decreases at an increasing rate, and hence the first differential coefficient 7 P 9 must be negative and increasing. Fig. 3. But when this coefficient (or any variable) is positive and decreasing, or negative and increasing, z¢s rate of change, 7.¢., the second differential of the function, must (Art. 3) be neg- ative throughout, which corresponds with the result found in equation (3). fx. 2. To illustrate the second case we take u=x*—16x+70 (1) from which Lu Pei AS (2) £ Gite Watts (3) 102 DIFFERENTIAL CALCULUS. ° . du - . . We infer from equation (3) that > is an increasing func- tion for all values of x, and hence it is so when x=8, which du . : du + : reduces 7, to zero. From which we infer that > 1s passing from a negative to a positive state, and the function itself from a decrease to an increase. Hence a minimum. If we substitute 7, 8 and g successively for x in equations (1) and (2), we have for ask os Gitte a, ee cee aes Solas du Vo sao ee, du which corresponds with our deductions. If we let the ordinates of the curve A BC (Fig. 4) rep- resent the values of w, we see that from A A to B the value of wz diminishes, as is B : shown by the sign of “ oo and at a dimin- ishing rate as is shown by the positive . pee) 7 on nis. sign of [a * (Art. 3). From. Bto-C zw Fig. 4. increases, as is shown by the positive sign of = =, and at an Date . : increasing rate, as is shown by the sign of ee, which is still plus. Hence the shape of the curve. Ex. 3. To illustrate the third case we make uw=9+2(x—3)8 (1) whence Au w= 6(x—3)? (2) au =12(x—3) (3) Here we find «=3 reduces - to zero, and hence if there axe is a Maximum or minimum it will be for that value of «x. MAXIMA AND MINIMA. 103 9° ww au But it also reduces 7g Tee to zero also, hence we resort to the aru f ; ; value of xe? which we find to be 12. We infer from this a*u ee : 2 ee that when Fee =O it 1s passing from negative to positive, hence § = ~ is passing from a decreasing to an increasing func- tion at ae zero point, and, therefore, does zot pass through it. It is, therefore, all the time positive, and the function is at all times an increasing one, so that there is neither max- imum nor minimum. We may, also, learn the same thing du from inspection, for since the value of > 1s a square it must always be positive. If we substitute in the given function and in the values du au of ow and re the numbers 2, 3 and 4 successively for x, we have for du a? u SEARS Capel hay ed 0 OC aa Lu ae x= “= > = FiO 3 Pana aa? du Ce Cee AS Bs Tha re =6 Seba te If we let the ordinates of the curve A B C (Fig. 5) repre- sent the values of wz, we see that from 2 to 3 the function eeeesgaes as is shown by the positive C> value of ~, asis indicated by the negative sign of au ere (Art. 3), between those points but at a decreasing rate, or while x is less than 3. From 3 to 4 the function is still increasing, as is 104 DIFFERENTIAL CALCULUS. shown by the positive sign of 7 and at an increasing rate, eA e a* u as is shown by the positive sign of aya when the value of du xis greater than 3. Hence at B where the value of 7 is zero, the function having increased at a decreasing rate up to that point, ceases for an inappreciable moment, and begins again to increase at an increasing rate. ix. 4. To illustrate the fourth case take u=53+(e—7)* (x) whence F = 4(4—7)3 (2) Tt x47)? (3) Sraee— (4) Chang (5) Here we see that the fourth differential coefficient is the first that has a real finite value for x=7, which reduces all the preceding ones to zero. Hence, according to our rule, there should be a minimum value for the function at that point. In fact, the sign of this coefficient shows that the third changes at zero from minus to plus; and this shows that the second does zof change its sign at zero, but after being a decreasing function to that point, becomes an in- creasing one, and is, therefore, positive both before and after. And this again shows that the first des change its sign at zero, since it is an increasing function on both sides of zero, which can only be by passing from a negative to a positive value. Hence the function will decrease until x=7, after which it will increase, showing a minimum at that point. If we substitute in the given function and in the differen- MAXIMA AND MINIMA. 105 tial coefficients the numbers 5, 6, 7, 8, 9 successively for x we shall have for Lu ape Wate ee aol Lei ig 2a era =48 ear x=6 w=6 “=—4 “ =12 “ =—24 x=7 u=5 “=o “ =o “* =o C6 . =. Oe — “ =12. “ =24 x=9g uw=11 “ =32 “ =48 %“ =48 which illustrates the conclusions we have drawn. The general proposition enunciated in the fourth case may be demonstrated analytically as follows. Let us suppose u= F(x) and let the variable x be first increased and then diminished by another variable %; and let these new states of the function be represented by z’ and w’, then we have ul’ =F (x+h) u" =F x—h) Developing these by Taylor’s theorem we have, after sub- tracting the original function z, P SrOaIey Ce een i Le SH aay (in mee ary, Tg OE Bi ie etc: P é ae au Ups Ou hs u mete rl reat f + etc. Bete oar Since the powers of / increase in each successive term of this development, we may reduce the value of 4 to such an extent that the value of any one term shall be greater than that of all the succeeding terms added together. Such in fact will be the case if % is less than one-half in the series AES /L? GUC. Let us suppose # to be so reduced, then if w is a maxi- mum, it must be greater than # or wz”, and the second mem- bers of both these equations must be negative; if it is a minimum it must be less than z’ or zw”, and in this case the second members of both equations must be positive. 106 DIFFERENTIAL CALCULUS. Hence, in case of a maximum or minimum, the second mem- bers of both equations must have the same sign, and the first term of each (which controls the value of all the rest) : : 2 OIE having contrary signs must reduce to zero; that is, — must > ax be zero, since “is not. This then is a necessary condition to a maximum or minimum. If there is a real value for the a" u ax* since #? is positive), will now control that of the whole sec- ond member, and will determine whether zw is a maximum or second term in each equation, its sign (or that of Lu minimum. If the second term (or x2 ) become zero, there can be no maximum nor minimum unless the third term (or Dead axes it has contrary signs in the two equations. We see then that the conditions of a maximum or mimimum are: jirsé, that the first differential coefficient should become zero; and, second, that the first succeeding differential coefficient that has a real value should be one of an even order, since the even terms have the same sign in both equations. If that is negative, the whole of the second member of the equa- tion is negative, and there is a maximum; if it is positive, there is a minimum. Which agrees with the rule already found. “x.5. To illustrate the fifth case we take ) which is now the controlling term, is also zero, since 2 u=10—(x—3)% (1) whence au —2 we 1 (2) *.~ 3(e=3)8 a2 u 2 ax? (3) MAXIMA AND MINIMA. 107 Here we find that «=3 will reduce o to infinity. Refer- : au : ring to the value ot — 3 we find that it reduces that also to infinity, but we see by inspection that any other value for 9 ecere x, whether greater or less than 3, will make that of ee positive. We see also that = ~ will be positive when ~ is less than 3, and negative when it is greater. From all this we infer that the function is an increasing one before x=3, and a decreasing one afterwards. Hence there is a maximum at that point. If we substitute for x, in equations (r), (2) and (3), the numbers 2. 3. 4 successively, we have for _ Z, tif w bite toe Seopa et yet Ls es Qu ae ee heat O Beet OS org = #9 Cie Mn Coon See ye es Nee If we let the ordinates of the curve A B C represent the successive values of w (Fig. 6) we see that from 2 to 3 the function in- creases, as is shown by the positive value of “ *, and at an increasing rate as is shown by the positive value of (hte fomai: m 3 to 4 the negative sign Tne From 3 to4 the negative sig Baer of = indicates a decrease of the function, while ale positive mien sign of 2 = (which has not changed) shows that this decrease lsat a Fans rate. Hence the form of the curve. Lx. 6. To illustrate the sixth case we take 108 DIFFERENTIAL CALCULUS. 2 u=(3x%—9)* (1) whence Lub 2 (2) i ae 4 4 (3~—9)* es 2 (3) Hite OO 4 3 (3-9) In this case, as before, we find that x=3 reduces ~ and au : : F , Cu ee ' infinity. We see also by inspection that a changes from minus to plus as x passes from less than 3 to greater, 9 4 axe” infer that z is a decreasing function until «=3, and an in- creasing one afterwards. Hence a minimum at that point. If we substitute for x in equations (1), (2), (3), the num- bers 2.3.4. successively we shall have for while is negative on both sides of infinity. Hence we ge UU Demme ale 2 x=2 “u=} > =— 3 SS eee / 9 ax R/ 3 ax” A/ I bet of aa Oe x=3 u=o ie Te wie - Ses ae Zeus 2 Cig tee a V Weary x/ 53 28 SOR cars If -we let the ordinates of the curve A B C (Fig. 7) rep resent the successive values of z, A | we see that from 2 to 3 the function ; decreases, as is shown by the nega- tive value of =, and at an increas- ing rate, as is shown by the negative | a* 4 2 3 4 l anes value of 7x2 > from 3 to 4 the pos Fig. 7. we du - e e = ° : itive value of 7 indicates an increasing function, while the MAXIMA AND MINIMA, Iog . . a . . . negative value of FEES shows that increase to be at a dimin- ishing rate. Hence the form of the curve. (32) We have seen (Art. 30) that there may be as many maxima and minima of a function as there are roots for the equation formed by making the value of =e. To illustrate a case of this kind we take ERE gs u=x* —20x%3—132x%?—320x 4286 (1) whence wt Tn 4k? — 60x? + 2644— 320 (2) au Tuk 12K? — 1200+ 264 (3) Placing the second member of equation (2) equal to zero, we find for x three values as follows : a= 2 By a —o Substituting these values in equation (3) we have for Gas X=2 “3-7? Cu eae ax® oe lect axe 1? from which we infer that for x=2 and x=8 there is a mini- mum, and for x=5 there is a maximum. This will be seen by substituting for x in equations (1), (2), (3), successive values, as follows I10 DIFFERENTIAL CALCULUS. Lu aul = 0 W286 Wen oto. yt ee c= 1 @eleg -lio= tiie eee we Wa eR Ole dae OU Phere x= 30 uw 55 NS yo ae 4S 4. w= 64 1S Sl ee Pee x= 5 w=111 “ = o 6 =—36 x= 6 u= 94 “ =— 32 “ =—24 ee Reg SO eee Ot a) eS 2 (8) y= 300 eee Rest Kg v9. PS are rb KAO 2 OO eee S20 eo If we let the ordinates of resent the successive values wz corresponding to numbers substituted for x at the foot of each, we see that the function decreases at a diminishing rate until «=2, when it ceases to decrease and begins to increase at an increasing raté, as 1s the curve ABC (Fig. 8) rep- fs of | 8B Ht fo 2°S Fee LOT eae Fig. 8. Wi ; — and the continued shown by the change of sign in oy posi- a u tive sign of Wee" But at «=4, although still increasing, as fy th ee, eee is shown by the positive sign of yu? it is at a diminishing a u, Tae 8 how negative, and thus continues until at 5, rate, for au aes becomes zero, the function ceases to increase and be- gins to diminish at an increasing rate, as is shown by the au ax? Cu negative signs of ye and at x=6, But at x=7 we a+ MAXIMA AND MINIMA. LE. ” Sue 76h he the function has ceased to increase and begins to diminish, until at «=8 it has become zero, when the changes at x=2 are repeated. We notice that between x=3 and x=4 the 9 ~ ax” have positive, showing that the rate of diminution of sign of changes from vlus to minus, showing that be- ut tween those two points the rate of on has changed from an increase to a decrease, that is, the function has changed from increasing at an increasing rate to increasing at a diminishing rate. The exact point where this change takes (Seoga a, Reoeee place is where the value of eae Gs This will give x=s—1/3 and x=5+Yv 3 which last value corresponds to a point between «=6 and x=7, where the same change is repeated, only in a contrary direction. | From all these cases we deduce the following rule for ascertaining the values of the variable that will produce a maximum or minimum value for the function, if there be any. Place the first differential coefficient equal to zero; and substitute each of the roots of this equation for the varia- ble in the second differential coefficient. Each one that reduces it to areal negative quantity will produce a maxi- mum value for the function; while a similar positive result will indicate a minimum. Should any real root thus found reduce the second differential coefficient to zero, substitute it in the third, fourth, etc., successively, until a real finite value is found for some one of the coefficients. If the first thus found be of an even order and positive, there will be a minimum; if negative there will bea maximum. If the first that is reduced to a real finite quantity is of an odd order, I1I2 DIFFERENTIAL CALCULUS. whether positive or negative, there will be neither maximum nor minimum. The first differential coefficient may also be placed equal to infinity, and if there be any real finite roots, they may be treated in the same manner as those obtained by placing it equal to zero. In this case, however, a positive sign of the second differential coefficient indicates a maximum and a negative sign a minimum. If a given function contain two variables there must be an equation, and one of the variables must be considered as dependent on the other. The problem will be to find the maximum or minimum value of the dependent variable; for which purpose it must be considered as an implicit function of the other, and the differential coefficients will be found as in other cases. Nore.— It may be objected that, herein, the subject of maxima and minima has been treated in too prolix a manner, and the reasoning has been unnecessarily repeated. I reply, it is of the highest importance that the student should have not only a clear and correct, but a falar, conception of the Zaws which govern the relations of the different orders of rates or differentials, because these are among the fundamental ideas of the calculus, and essential to a complete comprehension of the subject. Now unless these ideas are presented sufficiently often to render them familiar ; if the student on every new occasion is obliged to draw afresh upon his powers of imagination, and go through the mental labor of forming his conceptions anew, the study will prove not only more difficult, but far less attractive. He will be like a traveler in the dark, who, instead carrying a constantly shining lamp to guide his footsteps, must light his candle anew for every fresh obstacle, Hence the importance of a full and elaborate explana- tion, even at the expense of some, otherwise unnecessary, repetition, EXAMPLES. Find the value of « for the maximum or minimum value of z in the following equations : Ex. 8. “w=x>+18x" +1052. Ans. Lx, 9.. “=a—bze-+x*. Ans. Lx.10. uw=at+h3x—cx?. Ans. Lx. 11. u=3a*x?—btx+c?. Ans. Lex. 12. u=a* +bx*? —cx3, Ans. MAXIMA AND MINIMA. Il3 APPLICATION TO PRACTICAL PROBLEMS. (33) In order to apply the rules for determining maxima and minima of functions to the solution of practical prob- lems, it is necessary to obtain an algebraic expression of the function, whose maximum or minimum is to be determined, in such terms that it shall contain but one variable. No specific rules can be given for this purpose, but care must be taken to express the function in terms of a variable that shall have a range of values deyond that which may be required to produce a maximum or minimum, for if it does not, although there may be a kind of maximum or minimum, it will not be one in the meaning of the term as used in the calculus, as there will be no ¢urning potntin the value of the function, nor any change of sign in the value of the first differential coefficient. A few examples will indicate the nature of the process more clearly. Ex. 1. Divide the quantity a into two such parts that their product shall be a maximum. Let x be one of these parts, then the other will be a—x, and the function will be x(a—x)=ax—x? which is to be amaximum. Representing it by z we have Lu pn ae (1) aD: FM rae (2) . au Placing the value of oe equal to zero we have = @ eee 9 ~ fhe Hence when a guanttty ts divided into two parts thetr product ts a maximum when they are equal. which the negative sign of shows to be a maximum. TI4 DIFFERENTIAL CALCULUS. £x. 2. To find the greatest cylinder that can be inscribed in a given right cone. Let the height SC (Fig. 9) of the cone be represented by a, and the radius of the base AC by 4, and let x represent the distance SD from the vertex of the cone to the upper base of the cylinder. From the trian- gles SAC and SED we have rn eID ED Orngs mia ED, hence ae But the area of the upper base of the cylinder is b* x? tH Ds a Multiplying this by DC=a—x, the height of the cylinder, we have the volume or capacity which we will represent by V, and hence | af cae V=— 3 x" (a—2) Now any value of x that will render «*(a@—x) a maximum mb” will render any multiple of it also a maximum, hence being a constant factor may be disregarded in the operation. Differentiating twice and representing x*?(a—x) by w, we have Lu ; Me 2k 3X" 2a , Making the value of 2 equal to zero we have Qa pra x=F MAXIMA AND MINIMA. IT5 The first cannot be amaximum since it reduces the value of a” u oak oe Aid xe tO 24; which being positive indicates a minimum. In fact, when x=o the cylinder is reduced to the axis of the cone, and vanishes with x. The other value x=4a will ° 7 ¢ solve the problem, since it reduces the value of S=> ue 19 — 24; a negative quantity which indicates a maximum ee for the function. Hence The maximum cylinder that can be tnscribed in a right cone ts one in which the height of the cylinder ts one-third of the height of the cone. The radius of the base will also be equal to two-thirds that of the base of the cone. The volume of the cylinder will be to that of the cone in the ratio of their bases, or as 4 is to 9. Lx. 3. Required to determine the dimensions of a cylin- drical vase, that will contain a given quantity of water with the least amount of surface in contact with it. Let v represent the given volume of water, x the radius of the base of the cylindrical vase, and y its alee Then we shall have V=Tx"*y from which 7pé 4 iam > mm ane ve Now the convex surface of the cylinder is equal to 27xy, and substituting the value of y we shall have the convex sur- face equal to 2TXU 20 Tx? x If to this we add the surface of the base =zx? we have the whole surface in contact with the water. Calling this surface S we have 116 DIFFERENTIAL CALCULUS. 2U ——— 4 x2 esree geo (1) as 2U ——_ = ——,- +27 2 ax ve art (2) 20 LAD aes (3) Placing the value of = * equal to zero we have 27x =2y whence eA Jal xX — This value answers to a minimum, since it renders the value 9 o Ss pes ; : : of x2 Positive. If we substitute this value of x in the expression we found for the value of y, namely we have hence the minimum surface will be in contact with the water whet thers cht of the cylinder ts equal to the radius of the base. Ex. 4. \ It is required to inscribe in a sphere acone which shall have the greatest convex surface. Suppose the semi-circumference AMB (Fig. 10) torevolve “about the axis AB, it will describe M the surface of a sphere, and the chord AM will describe the con- vex surface of a right cone in- scribed in the sphere, and AP will be its height, and PM the radius of the base. The convex surface of the cone, which we will call S, “9 will be iets MAXIMA AND MINIMA. ERT S=27PM .4AM=zPM.AM (1) We have now to determine PM or AM, either of which will determine the other. het AB=z2a and AP=x then rai PM =AP. PB=x(2a—2) and PM=V a(2a—2) Again AM=\/ 20x Substituting these values in equation (1) we have =n7zPM. AM =27v/ 20x —x2 . / 2ax=7/ 4a? x? —20x8 Differentiating this function we have ee eae ak AGT — 300 == a 2 Lx A/aarx2 —2ax3 ~/ 4a2—2ax ( ) = segs ds Placing this value of |, equal to zero we have — 4a ears: In order to determine whether this value of x corresponds to a maximum or minimum of the function, it will be neces- sary to find the sign of the second differential coefficient. Before doing this we will examine a method by which the Operation may be somewhat abridged. We have already seen that when the value of a function is reduced to zero by giving a particular value to the varia- ble, it does not follow that its differential will also be reduced to zero by the same value of the variable (Art. 31), for the function in passing through zero may, and probably will, be passing from negative to positive, or we versa, and, there- fore, may have a differential or rate of change at that point, the same as at any other. Thus the latitude of a vessel on the equator is zero, but it may be changing as rapidly there as anywhere else. 118 DIFFERENTIAL CALCULUS. Now if we wish to obtain the second differential coefficient for a particular value of the variable, we may take advantage of this circumstance. Suppose we find the first differential coefficient to be the product of two or more factors; either of these being reduced to zero will reduce the coefficient to zero. In this case we may obtain the value of the second differential coefficient for the corresponding value of the variable, without differentiating the entire coefficient. For suppose we have - Ss in which y. y’ and y” are functions of x ; this product will be reduced to zero by any value of x that will reduce either factor to zero. Suppose that for x=a we have y=o ; if we differentiate the function we have Uae) a? u_ayy'y")_ sy" dy yy a! yy ae th aa Dee TR ae. op eee ine and since x=o reduces y to zero, the two last terms of this expression become equal to zero, and we have etn Vt ae ax® ax ae hence to obtain the value Ot as 3 Jor that value of x that reduces one factor of the first Se res coefficient to zero, we have only to multiply z¢s differential coefficient by those factors which do zof become zero, and then substitute the value of x. If for example we have a x(x? —a?)=x(x+a)(x—a) we may reduce it to zero by making B= 0,0 CeO ee a1 If we wish the value Wereae De® SF the first value of x, we have au ———s — 72 2— 2 9 —X*" —a” ——a aX” »—0 MAXIMA AND MINIMA. 119 If for the second value we have au ft : Ti pag to0) =20 If for the third value we have au —— a —q)=2q? SEE Ns x(x—a)=2a Resuming now equation (2) aS. 40" — 3ax a ' ax x/4aX®—2ax WW 4a2 — 20x 4a—3) 4a —. and hence which becomes zero by making 3x=4a or x=3, Jor that value of x we shall have CBee a ae Nal ere ae ax® — a/4a® — 20x ° ax ir a/4a® — 2ax . . ° 4a which being negative shows that x=, corresponds to a max- imum of the function. Hence, zf a right cone be inscribed in a given sphere tt will have the greatest possible convex surface when the axis of the cone ts equal to two-thirds of the diameter of the sphere. Ex.5. A point o (Fig. 11) being given within the right angle B AC, through which a line is tobe drawn meeting p the axes AB and AC, it is 0 required to find the distance Az such that the length of the line between the points a a al! of its intersection with the Fig. 11. axes shall be a minimum. Let Am=a, Om=6 and mma=x. Then the right angled triangles Om and pAzn give the proportion mz: Om:: An: Ap or C pO LEAS! Sa ae OS whence I20 DIFFERENTIAL CALCULUS, But ; hp en — oa whence _ 2 b(atx)?* gn EE yee whence a+x pa= Tere +42 which is the function of which we are to find the minimum value. Representing this function by wz, and considering it as the product of two factors, we have Xx fe Ronit = a ————— ie ax K APSR ee TV. DS Hat ore Reducing to a common denominator we have dé (ae) x* ab? fa"), ee ee ax x 4/ $2 4x2 £8 N/R +2 Making this equal to zero we have x= h/ab2 To find if this is a minimum we differentiate again, but as the numerator of the differential coefficient is equal to zero for this value of x (Ex. 4), we multiply its differential by I X4/ 52 +H? which gives eh FOO): Rae 7 2 dx? — x 4/2 4x2 4/52 4x2 a result that is essentially positive whatever may be the value of x. Hence x=X/as? corresponds to a minimum length of the line gv, If a and 4 are equal, we have MAXIMA AND MINIMA. r21 x=6 or mnu=Om whence Ap=An £x.6. To find the maximum rectangle that can be in- scribed in a given parabola. Let A C B (Fig. 12) be the parabola of which 1). is the, axis’ Let DE be the height of the ae inscribed rectangle, and let CD=a and CE=x, then FE? = 2fx and FG=2,/2px. ‘ The area of the rec- tangle is Fig. 12. FG xX ED or 24/24x(a—-=tax *—3x* ax ¥ which being made equal to zero gives x= a te hence che altitude of the rectangle 1s equal to two-thirds that of the parabola. To show that this is a maximum we differentiate again, and find oo a fa i ge ee which is negative for every positive value of x, and therefore ao for x=. Ex.7. What is the length of the axis of the maximum parabola that can be cut4rom a given right cone ? 122 DIFFERENTIAL CALCULUS. Let ABC (Fig. 13) be the given cone, C and FDH the parabola cut from it. \ Let DE be the axis of the parabola, y and AB the diameter of the base of _the cone. Represent AB by a, AC by 6é,and BE by «. Then AE=a—z, / FE=V/ax—x?. Also ADAG? EBink Dora .v. we pap rf B hence eae ED ne 13. But the area of the parabola is eee to $FH x DE or § — . 20/ax—x? which is, pea to be amaximum. Dropping the con- stant factor 5 = (Ex. 2), representing the function by wz, and cneeeecn ne we have U=A/ ax3 —x4 (1) Gt AAI ee At I BYP Sere ples ax 24/ Axe — x4 apy rs Tare . (3ax? 4% ) (2) Placing the second member of equation (2) equal to zero =". and differentiating (2) for this last value of x we have (Ex. 4) hes I ax* ~~ 2r/ax8—x4 * Substituting this value of x in the second member of this equation it is reduced to fee he ee 2/5 we have x=o andx (6ax—12x") (3) flence the axts of the maximum parabola ts three-fourths of slant height of the cone. “ix. 8. It is required to determine the proportion of a cylinder, that shall have a given capacity, and whose entire surface shall be a minimum. MAXIMA AND MINIMA. 123 Let a? be the capacity of the cylinder, x =4AB (Fig. 14) the radius of the base, and y=AC be the height; then the two bases c D taken together will be equal to 27x?, and the convex surface to 27xy, so that am(x*?+xy) is to bea minimum. Now 3 a : zx*y=a>, whence Sense: Substitu- ting this value of y, representing the 4, - function by wz, and differentiating we have Fig. 14. 2a a x pie Bin uso? +— (1) au ane Te TARE — a (2) a? u Aan ax —4 x4 (3) Placing the value of o equal to zero we have atxs =ae=rx*y whence y=2x or the height of the cylinder 1s equal to the diameter of the base and om V on This value of x corresponds to a minimum value of the function, as is shown by substituting it in equation (3), which gives a positive quantity. Ex.9. ‘To divide a right line into two parts such that one part multiplied by the cube of the other shall be a max- imum. Ans. The part cubed is three-fourths of the given line. 124 DIFFERENTIAL CALCULUS. x. 10. To find the greatest right angled triangle that can be constructed on a given line as a hypothenuse. Ans. The triangle must be isosceles. Ex. 11. It is required to circumscribe about a given par- abola, a minimum, isosceles triangle. What is the length of its axis? Ans. ¥Four-thirds the axis of the parabola. Ex. 12. What is the altitude of the maximum cylinder that can be inscribed in a paraboloid. Ans, Half that of the paraboloid. £x. 13. The whole surface of a cylinder being given, how do the base and altitude compare with each other when the volume 1s a maximum ? Ans. fx. 14. Required the minimum triangle formed by the axis, the produced ordinate of the extreme point, and the tangent to the curve of a parabola. Ans, Py Hea OuNes Ve. PRPLICASION OF VI HE DIFFERENTIAL CALCULUS TO PI tee Ore y ROL Ge URL SIGNIFICATION OF THE FIRST DIFFERENTIAL COEFFICIENT. (34) In order to form such a conception of a line as will be adapted to the methods of the differential calculus, we must consider it to be “le path of a flowing point. The Zaw which governs the mcvement of the point deter- mines the nature of the line, as to forrn and position; and this law is expressed in the Cartesian system by the eguaton which shows the relation between the co-ordinates of the generating point in every position it may occupy throughout its movement. The drection in which the point is moving is determined by the relative rates at which the coordinates are changing their values at the same moment. If the rate of change should be constantly the same in each of the coordinates, whether negative or positive, the generating point would move constantly in the same direction, describing a straight line; and this direction would be determined by the razo of these rates, which in this case would be measured by the simultaneous increments or decrements of the coordinates themselves. 125 126 DIFFERENTIAL CALCULUS. If, for instance, the coordinates AB and BC (Fig. 15) have each a constant rate of increase, the ratio of the increments CO and OC’ will be constant, and the gen- erating point C will move in a straight line, whose direction will be determined by the relative rates with which those increments are produced ; or since the rates, being uniform, may be represented by the simultaneous increments, the direction will be determined by the ratio ae But if, while the rate of change in one of the coordinates is constant, that of the other should constantly vary, the ratio of their simultaneous increments would be constantly changing and the point would describe a curve, whose char- acter would be determined by the Zew which should govern these varying rates of change; and /Azs law would be expressed in the equation of the curve. But if the varying rate of change in the coordinate should, at any point in the curve, cease to vary, and should continue afterwards constantly the same as at that point, the generating point would cease to describe a curve, and would move in a straight line in the direction to which it was chen tending; and this direction would be determined by the ratio between the rates of change in the coordinates as they existed at the instant they both became untfor me. Low the tangent to any curve ts the line which would be des- cribed by the generating point if tt were to move in the direction to which tt ts tending on tts arrival at the point of tangency ; just as a stone, when it leaves a sling, describes a line tan- gent to the curve in which it was moving at the instant. Hence the ratio of the rates of change in the coordinates of THEORY OF CURVES. 127 any point of a curve determine the direction of the line tangent to the curve at that point. Suppose, for example, that AB (Fig. 16) has a uniform rate of increase, while BC has a rate of increase that is constantly diminishing, the point C would describe a curve. Now let us suppose that the generating point on arriving at the point C of the curve should con- tinue to move in the direction 5 ie ea Bt towards which it was then tending, Aas and should with a uniform motion, at the same rate as it had at C, describe the right line CD, then this line would be tan- gent to the curve at the point C. From D draw a line DB’ parallel to the ordinate, and meeting the axes of abscissas at B’; and from C draw a line parallel to the axis of abscissas meeting DB’ in O. The triangle CDO will have important properties which will require careful investigation. From whatever point in the line CD the line DB’ is drawn, the ratio between the lines CD, CO and DO will be the same, and hence as CD is described at a uniform rate, equal to that with which:the generating point 1s moving at the point C, the lines CO and DO will also be described at a uniform rate equal to that with which AB and BC were in- creasing at the same instant. Hence the three lines CD, CO and OD are the increments that would take place in the arc, the abscissa and the ordinate in the same unit of time, at their several rates of change existing when the generating point of the curve is at C; and, therefore, CO and OD may be taken as symbols representing the raé of increase of the abscissa and ordinate of the curve, while CD will represent the rate of increase of the curve itself at the point C; and is at the same time tangent to it at that point. 128 DIFFERENTIAL CALCULUS. So that if we designate the length of the curve by s, and consider CD as representing ds, we shall have CO=dx and. OD=dy for the point C of the curve. The tangent of the angle which the tangent line CD makes with the axis of abscissas is equal to aes and hence, calling this angle z, qo tang. v (1) and since CD» =CO” +0D’ we have ds* =ax* +ay* (2) We shall have frequent occasion to use these two equa- tions in investigating the properties of curves. (35) The usual method of obtaining equation (1) is, to sup- pose an actual increment BB’ (Fig. 16) given to the abscissa, and to find the corresponding increment C’O of the ordinate from the equation of the curve. The ratio of these two in- crements will not give the tangent of the angle v, which is DO CoO” and when they become infinitesimal or vanish, there is no ; DO C/O difference between ao ada: This manner of reasoning we have already discussed in the introduction to this work, and its defects have been shown. We commit an error In giving an actual increment to the abscissa and ordinate, for their rates of increase are not obtained from any actual increase in value (except where the rate is uniform), but from the /aw of change derived from the equation of the curve; and the suppositive incre- ments which we give are not real, but symbolical, repre- senting what they would be, by the operation of the Zaw con- equal to but will approachit as the increments decrease, trolling them at that instant, and are, therefore, a symbolical expression of that law. The truth is that CO and OD, so far from being infinitely small may have any value whatever assigned to them; so THEORY OF CURVES. 129 that we may consider them as varzadles whose simultaneous values always correspond to some point in the tangent line. In fact, if we differentiate the equation of a curve, and con- sider x and y as constants for the point of tangency, dx and dy may be considered as the variable coordinates of the tan- gent line, with the origin at the point of tangency, and the axes parallel to the primitive axes. Under these conditions the differential equation of the curve becomes the equation of its tangent line. This can easily be shown by a few examples. Ex.1. The equation of the circle with the origin at A (Fig 17) is y® =2Rx—x?* which being differentiated be- comes ydy =(R—x)dx If now we consider «x and y as constants for the point P, and dx(=PE) and a(=ET) as variables, we will replace the first by and x, and the latter by x and y, and we have Pig-o2: my=(R—n)x (1) which is the equation of the tangent line PT with the origin at P, while PE and TE are the abscissa and ordinate of the line, and # and z are coordinates of the new origin referred to the primitive one, or AB and BP. Suppose now we transfer the origin to O, the center of the circle. The formulas for transferring to a new origin ina system of parallel axes is y=bty’ and x=a+2' where a and & are the coordinates of the new origin. In this case a is equal to BO=AO—AB=R-—~z, and 4 is equal to PB=—wm, and hence by substitution, 130 DIFFERENTIAL CALCULUS, m(—m+y’)=(R—x)(R—-n2+x’) (2) Calling x” and _y” the coordinates of the point of tangency for the new origin, we have x” =—R+n or R—-n=—2" also y —BP—y Substituting these values in equation (2) we have y"(=y" $y) =—0"(—2" +2) yy al x" Hy"2 + y"2®=RE or dropping the accents yy +a" =R? which is the equation of the tangent to the circle, the origin being at the center and x” and y’ the coordinates of the point of tangency. Ex. 2. If we differentiate the equation of the ellipse whence referred to its center and 35 axes, we have ; A®’ydy + B*xdx=o ' Making | ae 0 y(=BP) and x(=OB) Fig..28. (Fig. 18) constant and dy(=TE) and @(=PE) variables, and replacing y by y” and x by x", dx by x and dy by y, we have A®y"'y+B®x"x=0 which is the equation of the tangent line to the ellipse, with the origin at P, the point of tangency, and the axes parallel to the primitive ones; x” and y” representing the coordinates of the new origin referred to the primitive one, and x and y the variable coordinates of the tangent line referred to the new origin. If we transfer the origin back to the primitive one we shall have x=a+x' and y=s+y’ THEORY OF CURVES. 131 where a and # are the coordinates of the new origin — that is, the center of the ellipse. oo a=—x" and 6=—y" (for a is essentially positive while x” is essentially negative, and 3 is essentially negative while y” is essentially positive), and substituting these values for x and y we have A®y"y’ + Bex" x ‘=Aty "24B2y"2=A2B3 or dropping the accents A®y"y+ Bx x=A?B? £x. 3. Differentiating the equation of the parabola we have ply =pax Representing x, y, a and dy by x", y", xand y respectively, we have by substitutior YY = px for the equation of the tangent line to the parabola, with the origin at the point of tangency. If we transfer it to the vertex by making y=é-+y’ and «=a+.’, in which 6=—y" and a=—.x", we have —y" + yy" =px!’—px" in which x” and y” are the coordinates of the point of tan- gency for the origin at A. Hence y"?=24x", and substitut- ing we have aa & Sap — 2px! by y! =px'—px" or, dropping the accents, yy" =pxt px" =p(x+x") which is the equation of the tangent to the parabola at the point whose coordinates are x” and y", the origin being at the vertex. Ex. 4. Lastly we will take the equation of the Hyper- bola referred to its center and asymptotes aA A2 +B? Heme from which axdy +ya@x=0 I 32 DIFFERENTIAL CALCULUS. Replacing y=PB (Fig. 19) by y’, x=AB by x’, dy=ET by y and zx=EP by x, we have . y LEX Y= in which x” and y” are the coordinates of the new origin referred to the primitive one, and x and y are the variable coordinates of the tangent line TP; the origin being at P and the coordinate axes, PM and PN, parallel to the assymp- totes. To transfer the origin to A the center of the hyperbola Fig. 19. make y=é-+y’, and «=a+x’; 6 being equal to —y” and - and substitute these values for x and y. This J Phew a Byres Y lf y / S\N wae i Ee ePaper Note or, dropping the accents MY Q g which is the equation of the tangent line to the hyperbola referred to its center and asymptotes. Thus it clearly appears that differentials are not infinitely small quantities, but are symdols to express the razs or laws of variation, which are, in fact, VARIABLE FUNCTIONS OF THE GIVEN VARIABLES. SIGN OF THE FIRST DIFFERENTIAL COEFFICIENT. (36) If x and y represent the coordinates of any curve, and while x increases uniformly y should have a positive value and, also, zzcrease, its differential will be positive and the curve will tend to Zave the axes of abscissas in a posi- tive direction; but if y should be decreasing while its value THEORY OF CURVES. 4 is positive, its differential will be negative and the curve will approach the axis of abscissas on the positive side. Again if y has a negative value and zzcreasing, its differen- tial will be zega#ve, and the curve will be receding from the axis of abscissas on the negative side; while if it is decreas- ing (being still negative) its differential will be fosztive, and the curve will be approaching the axis of abscissas on the negative side (see definition of a differential, Art. 3). Hence the following rule: When the ordinate and tts first differential have the SAM¥. sign the curve ts receding from the axis of abscissas, and when they have DIFFERENT SIGNS ¢he curve 1s approaching that axis. Nore.— The differential of the independent variable is supposed to be constant and positive,and hence the sign of the differential coefictent is the same as that of the differential itself of the function. This rule may be illustrated by means of the circle (Fig. 20) whose equation (the origin being at A) is y*® =2Rx—x? from which we obtain hae eet ee We see here that from A to C, y and its differential have the same sign, and the curve recedes | c from the axis of abscissas. Thesame is true of the curve from A to D. ieretcome (2 to: 4; and, from:D to B, where x is greater than R, the sign of y will be contrary to that of dy, and the curve approaches the axis of abscissas on both sides. ar We arrive at the same result if we Fig. 20. : ay . : consider - as representing the tangent of the angle which the tangent line makes with the axis of abscissas. From A to C and from A to D, where the curve leaves the axis of 134 DIFFERENTIAL CALCULUS. abscissas, the sign of the tangent and of y are alike; while from C to R and from D to B their signs are contrary dy_R-«# oie becomes At the point A where y—a, the value of infinite, and the curve departs at right angles from the axis dy ax becomes zero, and the curve neither approaches nor recedes from the axis of abscissas. And this corresponds with the value of the tangent of the angle made by the tangent line of abscissas. While at the points C. and D the value with the axis of abscissas; at A and B, se oo and the angle is aright angle; at C and D, aa and the angle is zero; the tangent line being parallel to the axis of abscissas. —— oH € LONG Vil: DIFFERENTIALS OF TRANSCENDENTAL FUNCTIONS. PROPOSITION I. (37) To find the differential of a constant quantity raised to a power having a variable exponent. Let the constant quantity be represented by @ and the variable exponent by v,; then the function will be Qa If we add an increment to v which we will call m we shall have QetM=qrqgm (1) Differentiating this equation we have daaet*M™=qN"7qQ? (2 ) and dividing equation (2) by equation (1) we have agerm akg gotm — gd (3) This equation being true, irrespective of any particular value of w, it will be true for any value we may assign to it; hence the differential of a constant quantity raised to a power denoted by a variable exponent, divided by the power itself is a constant quantity, or da” | a° But we have seen (Art. 6) that the differential of the vari- 135 136 DIFFERENTIAL CALCULUS. able is always a factor in the differential of the function. Hence av will be a factor of C. Calling the other factor k we have C=ha7 whence aa® a 7? =kiv or da” =arkdv The problem now is to find the value of & For this pur- pose we expand a” by Maclaurin’s theorem and have a%—A-+ Bo+Cv* +Dz?.+ Lut + ete. in which da” a* ae? an” dy ~ 2av® ~~ 2. 3% dys Tete. when v is made equal to zero. But we have found A=, B= Wary o7 Woe Gok hence LOT) > sy to BB al a ) =aa k=a k* dv or ae 1 7 Be aye —° and similarly Pen ae eet a from which making v=o we have ve B3 pA A=1, B=kh, C= E pe ok 3° Lies a etc. ° ° ss ° : Substituting these values in equation (1) we have ky? fyi = ptyt meee 2 eae 2. 3.04. ° i ° and making v=; this becomes etc. . i TRANSCENDENTAL FUNCTIONS. 137 1 a®=1+1+4+2 ak a gat etC—2.715282-- If we represent this number by e we have 1 a*=e or axe If ¢ is made the base of a system of logarithms & would be the logarithm of a to that base. This was done by Napier, the inventor of logarithms, and the system having that base is called the Naperian system. We shall indicate the logarithms of that system by the nota- tion 4g., while the logarithms of other systems will be noted by Zog. We have, therefore, da* =a" log. adv that is, The differential of a quantity ratsed to a power denoted by a variable exponent, ts equal to the power multiplied by the Nafpe- rian logarithm of the constant quantity into the differential of the exponent. Proposition II. (38) To find the differential of the logarithm of a varia- ble quantity. Let the quantity be represented by 7 and its logarithm by v, the base of the system being represented by a. Thenwe have : r=@ anddr=arkdv whence adr I ae du= oR or @ Log. ey Representing = by JZ we have @ Log. r=M = = in which J7is the reciprocal of ‘te Naperian logarithm Pa the base a, and is called the modulus of the system of loga- rithms of which @ is the base. Hence 138 DIFFERENTIAL CALCULUS. The differential of the logarithm of a variable quantity ts equal to the modulus of the system to which the logarithm be- longs, into the differential of the quantity divided by the quantity. In the Naperian system the modulus is, of course, ome. Hence in that system he a log. r=— from which we learn that in the Naperian system the rate of increase of the natural number, whatever may be its value, divided by the rate of increase of its logarithm, is always equal to the number itself. . Nortre.— This principle was used by Napier himself in constructing his table of logarithms, and explains his selection of his peculiar base. Hence he is one of the first discoverers of the Jrzzciple of the differential calculus, although he never applied it otherwise than to logarithms. (39) If we call e the base of the Naperian system of logarithms, athe base of any other system, 7 the logarithm of ~ to that base, z the Naperian logarithm of f, and s the Naperian logarithm of a@ we shall have pHa" pH=er a=e hence wherefore = pa Te S 72— S71 OY 71— 3 amnte but 4 is the modulus of the system, and hence The logarithm of a number tn any system is equal to the Na- perian logarithm of that number multiplied by the modulus of the system. This property is not peculiar to the Naperian system. The logarithm of a number in azy system is equal to the logarithm of the same number in the common system mul- tiplied by the reciprocal of the common logarithm of the base of the new system. In fact, in any two different systems, the ratio between the TRANSCENDENTAL FUNCTIONS. 139 logarithms of the same number is constant. Thus let @ and 6 be the bases of two systems, # and z two numbers, and «+ and y their logarithms in the first, and z and v their loga- rithms in the second system; then m=ae n= m= n=l? whence ae =$" and a =o whence hed 2 a=b® and a=bY whence . u_o or the ratio between the logarithms of the same number in two different systems is constant and equal to the ratio be- tween the logarithms of any other number taken in the same systems. Hence log. a: com. Log. a:: log. 10: com. log, 1o=1 whence a FOG. G2 com. Log. a= jog. 104 “og. @ as we have seen. Proposition IIT. (40) To find the differential of the sine of an arc. Let APD (Fig. 21) be a circle whose center is at O. Let POA be the given angle, then PB Y will be the sine of the arc AP, and also an ordinate of the circle to the axes OX and OY; while OB will be the cosine of the same angle, and also the abscissa of the point P of the curve, and AB the versed sine of the angle. From the equation of the circle with the origin at the center we obtain Fig. 21. 140 DIFFERENTIAL CALCULUS. xax=—ydy and eae dx? =* - a If we represent the arc AP by s we have (Art. 34) ds* =ax* +dy* whence wee 2 2 ds? => 2 met 1s ) yy2 But x +y? =R? whence Ready? ads? = ms x from which we obtain x ay =e or ; cos. SIDS R as (1) that is, The differential of the sine of an arc ts equal to the cosine of the arc tnto the differential of the arc divided by radius. PROPOSITION IV. (41) To find the differential of the cosine of an arc. From the equation COS. S=4/R?—sin.? 5 we have —sin. s.d@sin. s /R*—sin.? s Substituting the value of @sin. s (Art. 40) and replacing the denominator by cos. s, we have i, COS Go TRANSCENDENTAL FUNCTIONS. I4I z —sin.scos.s.@s —sin. s (2) COS Sa ee 2 IN GOSs-S R that is, The differential of the cosine of an arc ts equal to minus the sine of the arc into the differential of the arc, divided by radius. PROPOSITION V. (42) To find the differential of the tangent of an arc. From the equation ' R sin. s tan? 797s ee cos. $ we obtain (Art. 14) R cos. s, @sin. s—R sin. s.d cos. § COSs2" S Substituting for dsin. s and dcos.s their values (Art. 40 and 41) we have d@ tang. s= R cos.” s.as+R sin.® s.ds Plane cape eta R cos.* s or (cOS.£Scp sin. sj)ae oR? @ tang. s= cos.” $ an pery (3) that is The differential of the tangent of an are is equal to the square of the radius into the differential of the arc divided by the square of the cosine. PROPOSITION VI. (43) To find the differential of the cotangent of an arc. From the equation 9 ~ Ot Gre or a tang. § we obtain —R*d tang. s’ COUR EE cee ey erat: tang.” § 142 DIFFERENTIAL CALCULUS. Substituting for @tang.s its value from equation (3) we have a 28 —R*ds ast Are 5 Cos.? 5 tang.2 5 sin.” Ase (4) that is : , Lhe differential of the cotangent of an arc is negative, and equal to the differential of the arc multiplied by the square of the radius, and divided by the square of the sine. Proposition: VII. (44) To find the differential of the secant of an are. From the equation R2 sec. s=— COse's we have —R?dcos. s ES tf an meer ne cos.” § Substituting the value of @cos. s from equation (2) we have 5 Eas sin, s SeC. S=O og , a (5) that is The differential of the secant of an arc ts egual to the differ- ential of the arc multiplied by the radius into the sine, divided by the square of the cosine. Proposition VIII. (45) To find the differential of the cosecant of an arc. From the equation R2 coset. s=— = sin. ss we obtain —R?d/sin. s 20S sin, *s TRANSCENDENTAL FUNCTIONS. 143 Substituting the value of @ sin. s from equation (1) we have Cases 2 COSCC. Si — 5 eG (6) Silt f that is The differential of the cosecant of an arc ts equal to minus the differential of the arc multiplied by radius into the cosine, divided by the square of the sine. PROPOSITION IX. (46) To find the differential of the versed sine of an arc. From the equation ver. sin. s=R—cos. s we have ad ver. sin. s=—d COS. $ Substituting for 7 cos. s its value from equation (2) we have f sin. § : a Verrsile s = R as (7) that is, The differential of the versed sine of an arc ts equal to the differential of the arc multiplied by the sine and divided by radius. (47) In these equations the arc is supposed to be the independent variable; and the generating point 1s supposed to flow around the circumference at a uniform rate. The differential of the arc may easily be found, consider- ing it as a dependent variable and either the sine, cosine, tangent, etc., as the independent one varying uniformly. If we take the sine as the independent variable we have from equation (1). as @ sin, $= ee ry sin. $ (8) cos. § V R*® —sin.? s If we take the cosine we have from (2) R i A COs. De etna ae COS. S. (9) 144 DIFFERENTIAL CALCULUS. If we take the tangent we have from (3) 2 2g COS ee, Pain as ae Shee tang. eaeod fet tang. s but sec.*s=R* -+tang.’s hence aoe tang.s Dae +tang.*s (10) Lastly, if we take the versed sine we have from equation (7) (h= @ Vet. sin. wacin es but since sin.ts = /(2R — ver. sin. 5) ver. sin. 5 we have R d ver. sin. s as— (1 1) a/ (2R —ver. sin. s) ver. sin. s If, in equations (8), (9), (10) and (rr), we represent sin. s by z, cos. s by x, tang. s by y, and ver. sin. s by z, and con- sider R=1, we shall have ey ines, 7 (12) pens (13) fe (14) ee (15) From these equations we can find the rate of change in the arc when we know that of either of the four trigonomet- rical lines. SIGNIFICATION OF THE DIFFERENTIAL EQUATIONS OF THE TRIGONOMETRICAL LINES. (48) Describe the circle ADBE (Fig. 22) from the center O. Draw the diameter AB and the tangent TT’ at its TRANSCENDENTAL FUNCTIONS. 145 extremity. Let sc, s’c’,s"c", and sc" be the sines of the arcs As, ADs’, ADBs” and ADBEs”, and OT and OT’ (negative for ADs’ and ADBs") be the secants of the same arcs; then AT will be the tangent of the arcs As and ADBs", and AT’ will be the tangent of the arcs ADs’ and ADBEs”’; ac, a’, oc’ and oc” will be the cosines of the same arcs, and Ac, Ac’, Ac" and Ac” will be their versed sines. Suppose now the generating point of the circle to move from A around through D, B and E back to A with a uni- form rate of motion, then From equation (1) we find that at the beginning where cos.=R the rate of increase of the sine is the same as that of the arc and is positive. As the sine increases the rate immediately begins to decrease, being always in proportion to the cosine, until the generating point arrives at D and the cosine be- comes zero. ‘The sine then ceases p to increase, being at a maximum, and its rate of increase is zero. In the second quadrant, the-sine although still positive, decreases, and hence its rate of change is negative as shown by the cos. s, which is negative in that quadrant, Fig. 22. at the point B the rate of decrease has become equal to the rate of increase of the arc, although the value of the sine itself has become zero, and hence at that point @ sin. s=—das. In the third quadrant the sine is negative and increasing, hence its rate is also negative, as is shown by the cosine which is negative. At the point E the negative increase ceases and the rate becomes zero as does the value of the cosine. In the fourth quadrant, while the sine is negative, it is diminishing, and hence its rate of change is positive 146 DIFFERENTIAL CALCULUS. which is also indicated by the cosine which is positive for that quadrant. From equation (2) we learn that in the first and second quadrants where the sine is positive, the rate of change in the cosine is negative, and in the third and fourth quadrants where the sine is negative the rate change in the cosine is positive. We learn the same thing from the figure, for from A to B the cosine decreases, being positive, or increases, being negative; while from B to A, for the third and fourth quadrants, it decreases, being negative, or increases, being positive; the rate of change being at all times in direct pro- portion to the value of the sine. From equation (3) we learn that the rate of change in the tangent is at all times positive, and we learn the same thing from the figure, for as the arc increases from any point what- ever, the extremity of the secant which limits the tangent will move upward in a positive direction, and the tangent will increase in the first and third angles from A to positive infinity, and decrease in the second and fourth from nega- tive infinity to A; so that it will always increase positively or decrease negatively, and hence its rate of change is always positive (Art. 3). At A and B the rate is the same as that of the arc, and of the sine, being equal to as, while at D and E it is infinite. Similarly we learn from equation (4) that the rate of change in the cotangent is always negative, as it either decreases being positive, or increases being negative, for all points of the circle. From equation (s) we learn that the rate of change in the secant has the same sign as the sine, and hence is positive in the first and second quadrants, and negative in the third and fourth. By inspecting the figure we see that in the first quadrant the positive secant OT increases; in the second, the negative secant OT’ decreases; in the third, the nega- TRANSCENDENTAL FUNCTIONS. 147 tive secant OT increases; and in the fourth, the positive secant OT’ decreases. At the point A the rate of increase of the secant is zero for sin. s=o, while the secant itself equals R (a minimum). We see also that the rate of the secant is equal to that of the tangent multiplied by the sine; and since the sine is (except at two points) always less than 1, the tangent increases faster than the secant until the arc equals go’, when the sine is 1 and the rates become equal being infinite. | Equation (6) will give a similar result for the cosecant in connection with the cosine and contangent. We learn from equation (7) that the rate of increase of the versed sine corresponds at all times to the value of the sine, and is therefore positive in the first and second quad- rants, and negative in the third and fourth. The figure shows that the versed sine increases positively from A to B, the rate of increase being an increasing one (corresponding to the sine) in the first quadrant, and then a decreasing one in the second, but still positive. At B it ceases to increase and begins to decrease, first at an increasing rate in the third quadrant, and then at a decreasing rate in the fourth, until at A the versed sine and the rate both become zero. Hence in the two last quadrants the rate is negative corres- ponding to the sine, while the versed sine is always positive. At A and B the rate of change is nothing, as it should be, since at those points the generating point of the circle tends to move ina direction perpendicular to the line on which the versed sine is laid off, and, therefore, does not tend to alter its value; while at D and E, the generating point moves parallel to the line of the versed sine, and, therefore, at those points they should have the same rate, and thus we find to be the case, for at D sin. § R We¥ere sm. s= Wsacas 148 DIFFERENTIAL CALCULUS. and at E : sin. S Wer sites — R as =—ds because sin. s at that point is negative. VALUES OF TRIGONOMETRICAL LINES. (49) We are enabled by Maclaurin’s theorem to develop the sine and cosine of an arc in terms of the arc itself; for let s be the arc and z its sine, we shall have (Art. 24) w—=A+Bs+Cs?+Ds2-+4 etc. and (Art. 40, 41) making R=1 we have au au ; a> 4 ei: =COS. S$, Wee SID. S, 7 a S7aCtC: making s=o we have ) Lu au au (zZ)==0, = (eae Se Ws Is ete. whence re 5 w—=sIn, S=S—F— atita’ ty gic aA? CtG: If we represent by zw the cosine s then Mu ) au a*u- . Ws SID: 5) Te = C08. S$, 7g Sm. S, etc. Making s=o we have Lu \ au a*u (“)=1, Gee Te ets aya OE whence 34 5# sf 2#—=COS. Sl Ty aghig ag ee 5 eg These series are very converging, and for small arcs will give the length of the sine and cosine quite accurately. In order to apply these formulas, we take the length of a ~ quadrant, which is iB the radius being 1; and this divided TRANSCENDENTAL FUNCTIONS. 149 by go and then by 60, will give the length of one minute of arc, from which we can obtain the length of any number of minutes or degrees. Substituting the value of the arc thus found in the formulas, we obtain the length of the natural sine or cosine. If we wish these values for any other radius we shall have for sin. s=z Wit. COS, § “du Sines aru COS; S : —= Eee — ete as Re. as* Rieti? as? R38 whence 2 rn o a] ASIN § —5— Fe 7 ager a 5 5. RS ete. (50) We may in a similar way develop an arc in terms of its sine and cosine. Let s be an arc whose aah. is #, then (Art. 47) au / 1—u2’ du =u(1-1") : ons 5 Tas =(1-4?)~ 2 —3u?(1-u") — 2 making w=o , A=(z)=0, p=(@) =, c=1(—*) =o, p= Lu etc., hence wu ve 30° i Se be Ne 2.3 2.4.5 If we make w equal to the sine of 30° =4 we have for the value of the arc I SU s=arc whose sine is #=u+ 3 ed Poe en ea uy the sum of which is 0.52359 nearly; and multiplying this by 6 we have the length of the arc of a semi-circle, thus 180° =x =3.14154 nearly which is also the approximate ratio of the diameter of a circle to its circumference. SOROMIB MERRICK L. OF TANGENT AND NORMAL LINES TO ALGEBRAIC CURVES. (51) We have seen (Art. 34) that when x and y represent the abscissa and ordinate of the curve, oy will represent the tangent of the angle made by the tangent line of the curve with the axis of abscissas. Now the equation of a line drawn through any given point is IV =ax—x') in which y’ and x’ are the coordinates of the given point, and athe tangent of the angle made by the line with the axis of abscissas. Hence for any curve in which ~’ and y’ are the coordinates of the point of tangency, the equation of the tangent line through that point will be Oe ED f =F le % ) The value of oY will, of course, be obtained from the equation of the curve, and by substituting that value we obtain the equation for the tangent line of that curve. EXAMPLES. £x.1. From the equation of the circle we have dy’ x’ a =——_— yl 150 TANGENT AND NORMAL LINES. I51I and hence yy = ae: =E)) Ws or yy +xx' =R? becomes the equation of the tangent line to a circle. Lx. 2. Inthe case of the parabola we have ay _p ax’ =e whence FP Pp , —y =a(x—-x IY a ) or yy =p(x+x") becomes the equation of the line tangent to a parabola. Ex. 3. The equation of the ellipse gives ay Bex dx Ay’ whence Rab. F B dow 2 = TAns at) or A® yy’ +B? xx’ =A*B? becomes the equation of the line tangent to the ellipse. fx. 4. From the equation of the hyperbola referred to its center and asymptotes, we have Ti ores Je ax x’ whence DAS ' Vis reat) or A2 + B2 2 yx xy’ = 152 DIFFERENTIAL CALCULUS. becomes the equation of the line tangent to the hyperbola, referred to its center and asymptotes as coordinate axes — as in Art. 35. Since the normal line is perpendicular to the tangent, if a represent the tangent of z¢s angle with the axis of abscis- sas, then it will be equal to a where @ represents the tan- gent of the angle of inclination of the zamgen¢ line. Hence and substituting this value in the equation IY =a (x—2') we have ax —y =——(x—-2’ afeas ays ) for the general equation for the normal line, and it may be found for any particular curve by obtaining the value of —F from the equation of the curve, and making the sub- stitution as in the case of a tangent line. PROPOSITION I. (52) To find the general expression for the length of the subtangent to any curve. Let AP (Fig. 23) be any curve of which PT is the tangent at the point P, TB the subtangent, PN the normal, and PB the ordinate; then from the triangle TPB we have ; PB=TB. tang. PTB whence = + = A B N TB ~ tang. PTB Fig. 23. but ay tang. PTB Bear and PB=y TANGENT AND NORMAL LINES, 153 hence that is, The subtangent to any curve 1s equal to the ordinate into the differential of the abscissa divided by the differential of the ordinate. Proposition II. (53) To find the general expression for the length of the tangent to a curve. From the triangle PTB (Fig. 23) we have fo pik ax* PT= 2 : pereies =/9 ae ay ay” Ve T dy” that as; The length of the tangent to any curve ts equal to the ordinate into the square root of one plus the square of the differential coefficient of the abscissa. whence Nore.— By the ‘‘ length of the tangent” is meant that part of the tangent line between the point where it intersects the axis of abscissas and the point of tangency on the curve. Proposition III. (54) To find the length of the subnormal to any curve. Since the triangle PBN (Fig. 23) is similar to the triangle PBT, we have the angle BPN=BTP, and hence BN=PB. tang. BPN or ay BN= me that is, 154 DIFFERENTIAL CALCULUS. The subnormal ts equal to the ordinate into the differential coefficient of the ordinate. PROPOSITION IV. (55) To find the length of the normal to any curve. Since py” (Fig. 23) is equal to pp? +BN’, we have $ dy? dy” PN=A/ 9° 49° a =IN/ 14 oe that is, The length of the normal line ts equal to the ordinate into the square root of one plus the square of the differential coefficient of the ordinate. Note.— By the “‘ length of the normal” is meant that part of it which lies be- tween the point of its intersection with the axis of abscissas and the point of the curve to which it is drawn, (56) The following examples will show the application of these formulas to particular cases. £x. 1. From the equation of the circle we have ax y ; ay oe hence the subtangent (Fig. 24) is ax we PB Mire ar et 10) a result that we also obtain from geometry. P Age. vilhevienoth *onetveecy. tangent to the circle is AB 0 Fig. 24. We have also by geometry TR roa TANGENT AND NORMAL LINES, 155 whence £x. 3. The normal line of the circle is . oP aime PO=y\/1 $y 1 eyes Ex. 4. The subnormal of the circle is BO J dx Gg Pe Ex.5. From the equation of the parabola we have ave dy ?p hence the subtangent (Fig. 25) is Len OE SEs AE mm) gene ape —2x*—2A5 fae ae a result which we have also from geometry. ° fix. 6. The tangent of che ee: parabola is Ae B N Fig. 25. au? ie y* Lace Sees TRay\/ te IN 1+ =N/ 9? tor HV Fa We have just seen (Ex. 5) that TB=2AB, hence 7B” =4aB* =4x? whence TP=V y?+4x?=V pp? +R? as is evident from the figure. Ex.7. The subnormal to the parabola is YY p SOS eas gee roy! as we find from geometry. 156 DIFFERENTIAL CALCULUS. fx. 8. The normal to the parabola is ady* ine Sead PN =y\/'x SSN eae a +2?=V ve +5y° which is evident from the figure. £x.9. From the equation of the ellipse we have ay Bix Ea gis and the subtangent (Fig. 26) is Be ial aNd ft Bk Nhe ays Bx x This value for the subtangent does not contain B, and hence P is the same for all ellipses hav- ing the same major axis, the abscissa being the same. Hence 0 B T the tangent to the circle at P Fig. 26, will intersect the axis of abscissas at T, and PB OP —OB A?—, PR 2. etc, when the exponent 1s always equal to the number of divis- ions of the measuring arc, and is therefore represented by the arc itself corresponding to the radius vector, whence a”* =r or v=Log. 7 to the base a. If we differentiate the equation of this curve we have adr av = | | (2) whence (Art. 76) rdv rMar tang. PFT= 51 RVI = that is The tangent of the angle made by the tangent line with the radius vector ts constant and equal to the modulus of the system of logarithms to which tite system belongs. Jf the system is the Naperian, M=1 and the angle PDT is equal to 45°. The formula for the subtangent of a polar curve (Art. 78) 1s r* do Rar and substituting in this the value of eS from equation (2) we have (R being 1) 9 / [ —==7 MM. 4 Suptan— If M=r1 then subtang. =>. For the value of the tangent we have (Art. 79) tang. = 72 +72M?=7V 14+M? If M=r1 then ‘a tang=rv/ 2 POLAR CURVES. I8t we For the subnormal we have (Art. 80) arte te subnormal ory aT If M=1 then subnormal =r For the value of the normal (Art. 81) we have normal=A/7? + “a =rN/1 +373 If M=1 then normal=71/ 2 These values show that these lines are all in direct pro- portion to the radius vector. The same result flows from the constancy of the angle made by the radius vector with the tangent line. For all the triangles formed by the radius vector, the tangent, and the subtangent will be similar to each other, at whatever point of the curve the tangent may be drawn. The same may be said of the triangles found by radius vector, normal and subnormal. Hence these lines will always be in proportion to the radius vector. To construct a logarithmic spiral for a gzven base, des- cribe a circle with a radius equal to a unit of the radius veotor, PA, and lay off the arc Ad equal to a unit of the measuring arc. Draw the radius vector PB equal to the given base; A and B will be points of the curve. Other points may be found as already described. That part of the curve below the line PA corresponds to the negative value of v, and for that we have mit —@ in which when 7=a, v will be infinite. Hence the curve is unlimited in both directions. SH CPLO NX ASYMPTOTES. o (88) An asymptote to a curve is a line, which the curve continually approaches, but never meets. Such a line is said to be tangent to the curve at an infinite distance, by which we are to understand that the point of contact to which the lines approach is beyond any finite limit. That this may be the case it is necessary that, at least, one of the coordinates of the curve may have an unlimited value. Hence when we are seeking an asymptote to a curve, our first inquiry must be, whether the equation of the curve will admit of such values for the coordinates or either of them. If not, there can be no asymptotes. If it willdo so for either coordinate, we must substitute that value in the equation and ascertain the resulting value for the other coordinate. If this resulting value is finite, there is an asymptote parallel to the axis of the infinite coordinate; if zero then the axis of the infinite coordinate is itself the asymptote. But if it should be infinite, then we must resort to the following method. Find from the equation the values of the coordinates at the points where the tangent line intersects the axis, that is, 182 ASYMPTOTES. 183 the distances from the origin. These points may be found as follows : Let A (Fig. 37) be the origin of coordinates for the curve SO, and let PB be tangent to the curve at the point P, of which the coordi- nates are x’ and y’. The equa- tion of this tangent line is @ / y—y' =a («—2') ax If we make y=o we have , fire x=xX —y i =AB If we make x=o we have dy’ Pon yi Gg oty =AD If EC be an asymptote, and the values of x’ and y’ are made such as to remove the point of tangency to an infinite distance, then AB and AD will become AC and AE. If in such case we have finite values for these distances, then there will be one or more asymptotes; if there is but one finite value, there will be one asymptote parallel to the axis of the infinite coordinate. If one be zero then the axis of the infinite coordinate is itself the asymptote. If both be zero then the asymptote passes through the origin; but if both be infinite there is no asymptote. EXAMPLES, £x.1. The equation of the hyperbola referred to its cen- ter and asymptotes is xy=M in which if « is made infinite y becomes zero; and if y is 184 DIFFERENTIAL CALCULUS. made infinite « becomes zero; hence both axes are asymp- totes Ex 2. If we consider the hyperbola as referred to its center and axes, its equation is A2y? =B2 x? — AB? where either x or y may be made infinite, and such value makes the other infinite also. Hence we take the formulas for the points of intersection of the tangent with the axes, which give A2y! Ay A®y’? —B2x’2 Ae cme eee pee PRN Me X—X ipo ae Bex x! and ees tea Sawai peta a oe AYy AYy of both of which values becomes zero, when x’ and y’ are made infinite. Hence the asymptotes pass through the origin. Ex.3. The equation of the parabola y* =2px shows that « and y both become infinite together, and hence we take ’ ax’ ! ie , and las? ety ade GaP ey A Va ax roar eis — 5 both of which values become infinite when x’ and y’ are infinite, and hence there is no asymptote to the parabola. Ex. 4. If we take the ellipse whose equation is A®y? + B22 =A2B? we see that neither x nor y can ever be infinite; in fact y can never exceed B nor x exceed A; hence there is no asymptote to the ellipse. £x.5. The equation of the logarithmic curve is x=log. y ASYMPTOTES. ‘ 185 It may be constructed by laying off on the axis of abscis- sas (Fig. 38) the distances AB, AC, AD, etc., in arithmeti- cal progression, and, on the corresponding ordinates, the distances Aa, Bd, Cz, Dd, etc., in geometrical progression, and drawing a curve through the points thus found. We see from the equation that if either x or y is infinite on the PIB S Cae Fe positive side, the other will be Fig. 38. infinite also. If we apply the formula for the intersection of the tangent line with the axis we have (Art. 38) SNES ners ae Vice, paler yo ap (+z) (1) and be Ue ae ae x=x —y eae —y are —M (2) We see from these values, that when ’ is infinite x will be infinite positively, and y negatively. Hence there is no asymptote on the positive side of x. But if « be made infinite negatively, y’ will become zero; for the logarithm of o is negative infinity, which shows that the axis of abscissas is an asymptote on the negative side. ‘The value of y how- ever in equation (1) becomes —o%, which is indefinite. We learn from equation (2) that the tangent always inter- sects the axis of abscissas at a distance equal to M on the negative side of the ordinate of the point of tangency. Hence the subtangent is constant and equal to the modulus of the system to which the curve belongs. If x«’=M, then x and y both become zero, and the tangent passes through the origin. If we put the equation into the form ya" | DIFFERENTIAL CALCULUS. and make x negative it becomes . ee Daa which makes y=o when += 0 ; whence we infer that the axis of abscissas is an asymptote to the curve on the nega- tive side, as already shown. SECTION XI, SIGNIFICATION OF THE SECOND DIFFERENTIAL COLTFICIEN TF. SIGN OF THE SECOND DIFFERENTIAL COEFFICIENT. (89) We have seen (Art. 36) that the first differential of the ordinate indicates by its s¢gz whether the curve is leav- ing or approaching the axis of abscissas; and by its value it determines the vate of such approach or departure; that is, the tangent of the angle made by the tangent line with the axis of abscissas. As the point of tangency moves along the curve, the rate of its approach to, or departure from, the axis of abscissas is constantly changing, and upon the rate of this change will depend the direction and amount of curvature of the curve. Wherever the curve is situated with reference to the axis of abscissas, if its rate of departure is an increasing rate, or its rate of approach is a decreasing rate, then the curve is convex toward the axis of abscissas; while if its rate of departure is decreasing, or its rate of approach is increasing, it will be concave toward that axis. (90) Now the second differential of the ordinate will determine by its sign whether the first is an increasing or decreasing function. If the latter is positive and increas- ing, or negative and decreasing, its rate of change (that is 187 188 DIFFERENTIAL CALCULUS. the second differential of the ordinate) will be positive (Art. 3); but if it is positive and decreasing, or negative and increasing, its rate of change is negative. Note.— It will be remembered that the sign of the differential and that of its coeffi- cient are always the same, since the differential of the independent variable is always -uniform and positive. (91) If, therefore, the second differential coefficient should be positive, the first must be either an increasing positive or a decreasing negative function (Art. 3). If the curve is on the positive side of the axis of abscissas, it is convex to that axis ; if on the negative side it 1s concave. (92) If the second differential coefficient is negative, the first must be either an increasing negative function, or a de- creasing positive one. Hence the curve, if on the positive side of the axis of abscissas will be concave, and on the negative side convex to that axis. (93) To illustrate these principles let us suppose the second differential coefficient to be positive, then the first must be a positive increasing ora negative decreasing func- 99 0 Al tion. The curves in Fig. 4o , MIL NG eA and 41 answer to these con- p B ditions, for from C to D the first differential coefficient is Cele ‘ Fy 5 negative (Art. 36) and de- 40 0 creasing, while from D to E it is positive and increasing in both cases. If the second differential coefficient is negative, then the first must be positive and decreasing, or negative and in- creasing, and we find the curves in Fig. 39 and 42 to answer these conditions; for from C to D the first differential coeff- cient (Art. 36) is positive and decreasing, while from D to E it is negative and increasing in both cases. By inspecting these figures we see that for 39 and 4o the SECOND DIFFERENTIAL COEFFICIENT. 189 second differential coefficient has in each case a sign con- trary to that of the ordinate, and that both curves are con- cave to the axis AB; while in curves 41 and 42 the sign is the same as that of the ordinate, and the curves convex to Loera nts se Lience When the signs of the second differential coefficient and of the ordinate are contrary io each other, the curve will be concave toward the axis of abscissas; when these signs are althe the curve will be convex toward that axts. It will be noticed that in all these cases the first differen- tial coefficient changes its sign at D where it becomes zero, but this does affect the sign nor the value of the second differential, for the first may be changing as rapidly, and in either direction at the zero point as at any other. (94) To illustrate these rules let us take the general equation of the circle (xa)? +(y—0)?=R? in which a is the abscissa and @ the ordinate of the center. Differentiating we have dy x—a ax y—b and ay oR ee Bs (y—5)8 Fig. 43. From which we learn that so long as y is greater than 3 the second differential coefficient will be negative, while it is positive where y is less than 4, or where it is negative. Wesee also from the figure (Fig. 43) that above the line DE where y is greater than 4 the curve is concave toward the axis of abscissas, while between DE and the axis of abscis- sas, where y is positive and less than 4, the curve is convex toward that axis. Below the axis of abscissas where y is negative the second differential is still positive, while the 190 DIFFERENTIAL CALCULUS. curve is concave toward the axis. -All of which corresponds with the rule. : In the case of the parabola referred to its vertex and axis we have oy a pe hoe eee a fraction whose sign is always contrary to that of y,; hence the curve is always concave towards the axis of abscissas. The same may be said of the ellipse referred to its center and axes from whose equation we have Tea) MBs adx® ~ A®y3 In the case of the hyperbola referred to its center and asymptotes we have eye Iey, ; aa a fraction whose sign is always the same as that of y. Hence the curve is everywhere convex toward the axis. VALUE OF THE SECOND DIFFERENTIAL. COEFFICIENT. (95) The curvature of a curve at any point is the én- dency of the tangent line at that point to change its direc- tion, as the point of tangency is moving along the curve, in obedience to the daw of change derived from the conditions which govern the movement of the generating point. Note.— The curvature then ofa curve is zof ‘‘ its deviation from the tangent,’** nor ‘‘its departure from the tangent drawn to the curve at that point,’’t nor is it ‘‘the angular space between the curve and its tangent,’’} nor isit any acfwadZ change in the direction of the tangent line as the point of tangency moves along the curve ; nor does it depend on any such change, but upon the LAw which governs the movement of the generating point ; for itis this law which fixes the ¢ezdency of the tangent to change its direction and this tendency is the curvature. Hence in estimating the curvature of a curve at any point, we consider that point a/ove and seek, zof any actual movement of the generating point, but the Zaw which controls Zt, *Loomis. tDavies. {Church. SECOND DIFFERENTIAL COEFFICIENT. I9t Hence if several curves as CD, C’D’, C"D" (Fig. 44) have coincident tangeuts AB at the point A, and if we suppose the point of tangency to be at any instant moving along the curve, Carrying with it its own tangent _ A B line, that one whose tangent line at — > the moment of coincidence is chang- Ly. N ing its direction most rapidly will Co Cie (ee Be 7 have the greatest curvature at that point. For the raé of change in the «Fig. 44. direction of the tangent is the measure of its zendency to change. Since the first differential coefficient indicates the drec- tion of the tangent to a curve, by means of the tangent of the angle made by it with the axis of abscissas; the second, which is simply the rate of change in the first, will indicate the rate at which the tangent of that angle is changing its value.. Now as between two curves at common tangent point, that curve in which the tangent line tends to change its direction most rapidly, will be the one in which the tangent of the angle made by that line with the axis of abscissas will also tend to change z¢s va/ue most rapidly, and will, therefore, have the greatest curvature, while if these tendencies are equal the curvatures are equal, and this will be indicated by the equality of the second differential coefficients. SECT LONE CURVATURE OF LINES. THEOREM. (96) Zhe curvatures of different circles are inversely propor- tional to their radit. The curvature of a circle is the same at all points of the circumference, and all circles having the same radii have the same curvature. Since the change in the direction of the tangent, as the point of tangency moves around the curve is constant, its actual change of direction for any given movement of the point of tangency, will always be in proportion to its “z- dency to change, multiplied by the length of the are over which the movement 1s made, and may, therefore, be repre- sented by that product; and hence the zendency to change or curvature will be equal to the actual change divided by the length of the arc. Now the change in the direction of the tangent is equal to the angle contained between its two positions, which is the same as that contained between the two radii drawn to the extremities of the arc. Calling this angle v and the length of the arc a, we shall have v curvature aes 192 CURVATURE OF LINES, 193 If now we have two circles, which we will call 0 and o’, whose radii are 7 and 7’, and the angles at the center for the same length of arc a are v and 7, we shall have 7) < curvature of (Sects t o curvature of 0’ Bye hence curvature of 0: curvature of 0’ ::v: v7 (1) but ; 2000s Ger and Time lathe Fer ey vat whence y.2nr=v . arr’ or Substituting this ratio in proportion (1) we have curvature of 0: curvature of 0 ::7':7 Ore ew Le CONTAGE OF CURVES, (97) When two curves have a common point, the coordi- nates of that point must satisfy both their equations. This will generally be a point of zztersection, and not a point of contact; and is all that can be secured by having but one_ condition common to the two curves. If they are at the same time tangent to each other, at the common point, then another common condition is imposed and there is a contact of the first order. The condition required in this case is, that, for the point of contact, the first differential coefficients shall be the same for the equations of both curves. For since the curves are tangent to each other, they have a common tangent line, and 194 DIFFERENTIAL CALCULUS. the first differential coefficient, which determines the angle made by this line with the axis of abscissas, must be the same for both equations. If, besides this, the curves are required to have the same curvature at the point of contact, this will introduce a third condition, which is, that the second differential coefficients shall be the same for both equations (Art. 95). For the second differential is the rate of change in the first, which gives the direction of the tangent line, and the rate of change in this direction is the curvature. This isa contact of the second order. ; If now it is required, in addition, that the rate of change in the curvature should be the same in both curves at the point of contact; we must introduce a fourth condition, viz., that the ¢izrd differential coefficient should be the same in both equations. ‘This would be a contact of the third order. And thus the order of contact would become higher for every new condition introduced common to both curves, and every new agreement between the successive differential coefficients. If then we wish to find the order of contact of two given curves, we first combine their equations, and determine their common point if they have one. For this point the varia- bles will have the same value in both equations. If the values thus found being substituted in the first differential coefficient of each equation, reduce them to the same value, there is a contact of the first order; that is, they have a . common tangent line at the common point. If they also reduce#the second differential coefficients of the two equations to the same value they have a contact of the second order, and so on for the successive differential coefficients ; the order of contact being determined by the number of coefficients that successively become equal by the substitution of the values of the common coordinates. CURVATURE OF LINES. 195 EXAMPLE, (98) To illustrate this rule let us take the two equations 4y=x?—4 (1) and Tiler SR Pret (2) from which we obtain by combination y=—I and «=o | indicating that both the curves pass through the point of which these are the coordinates. We have also by differen- tiating twice — for equation (1) aes ary dz 2 M0 age t and for equation (2) ‘ay x a*y I xe da ya ey Substituting in these differential coefficients the values of x and y just found, we have the first differential coefficients ay es ay 6 > = _=o and 5G-=— =o Cane ax Neen! and the second differential coefficients a*y a*y I a 4} and —5 = FT i let le ae Mam (YT eee from which we infer that at the point whose coordinates are x=o and y=—1, the curves have a contact of the second order. We also see from the value of the first differential coefficient that at that point the tangent to both curves Is parallel to the axis of abscissas. A little investigation would show that the first curve is a parabola, and the sec- ond a circle tangent to the first at its vertex. (99) The constants which enter into the equation of a curve determine the conditions.which govern the movement of the generating point for that kind of curve; which must fulfil as many conditions as it has constants. . Thus the cir- 196 DIFFERENTIAL CALCULUS. cle whose general equation contains three constants, must fulfil three conditions, namely, two in the coordinates of the center, and one in the length of the radius. The ellipse must fulfil four conditions, namely, the coordinates of the center and the lengths of the two axes. x. (100) Now if one curve be given complete by its equation with fixed values for its constants, and another with con- stants which are indeterminate, and capable of being adjusted to any given conditions, we may easily assign such values to them as will cause the curve to fulfil such conditions as may be required of it. We may, for instance, require the curve to pass through a given point in a given curve. This will require that the same variable coordinates shall satisfy the equations of both curves for that point. We may also require them to have a common tangent at that point; this will require the constants to be so adjusted that the first differential coefficients of the two equations shall be equal. If there are three or more constants in each equation we may require such values as will cause the second differential coefficients to become equal also, thus producing an equality of curvature, or a contact of the second order, at the com- mon point. And thus we may continue until the order of contact is one less than the number of constants to be dis- posed of. (101) In order to make this adaptation of the second curve to the first we must consider its constants, or as many of them as will be required for the purpose as unknown quan- tities (Art. 4) and construct as many equations as may be required to determine them. These equations are derived from the conditions to be fulfilled by the constants. Thus the first which requires that the second curve shall pass through a point of the first will generally be met by the proper adjustment of a single constant; and an equation formed by substituting in that of CURVATURE OF LINES, 197 the curve to be adjusted the values of the coordinates of the designated point, and also the values of the known con- stants, will determine the value of the unknown constant. If itis required that the two curves be tangent to each other, we must adapt the values of “wo constants to this condition, and this is done by substituting the same values of the common coordinates, and of the remaining constants in the fs¢ differential coefficients of the two equations, and placing them equal to each other, thus forming a second equation. A contact of the second order may be secured by fixing the value of a third constant in a similar way by means of the second differential coefficients of the two equa- tions. The values of these constants thus determined being substituted in the general equation of the required curve, will produce an equation of one that will fulfill the required conditions ; that is, one that will intersect at a given point, or have a contact of a required order. EXAMPLE. (102) To illustrate these principles let us take the equa- tion of the ellipse referred to its center and axis A2y?+B2x? =A*B? and the general equation of the circle (a a)* y= )ee=R2 (1) in which the constants are arbitrary and may be adapted to any prescribed conditions. Suppose we say that the cir- cumference shall pass through the upper extremity of the conjugate axis where x=o and y=B This being but one condition will require the adaptation of but one constant. Let this be a, while we make R=A and d=o. 198 DIFFERENTIAL CALCULUS. Then substituting these values in equation (1) we have (o—a)* +(B—o)? =A? or a*+B?=A?*® a=LV A? —B? and the equation of the circle becomes (2 V A? —B2)? +y2 =A? the center being in one of the foci—the plus value of the radical corresponding with the focus on the positive side of the center. If we add another condition, namely, that the curves shall be tangent: to each other at the same point, we must adapt the value of fwo-constants to these two conditions. Let these constants be @ and 4, and make R=2B. Then we must construct an equation between the first differential coefficients of the curves; that is B*’x x—a Apes oy (2) Substituting the values of x and y as before we have whence B*o o-—a A?B B—é hence a=o and substituting these values in equation (1), we have (B—)? =4B? whence b=—B and the equation of the required circle becomes Hes tN Pt the center being at the lower extremity of the conjugate axis where a=o and J=—B. If now we add a still further condition there shall be a contact of the second order at the same point we must adapt CURVATURE OF LINES. 199 the values of ##ree constants to that condition, by forming a third equation, between the second differential coefficients, thus ¥ ay” (%:=a)* B4 Tax? TG? A? y3—_ y—d Sy, (3) Substituting, as before, the values of x=0 and y=B in equations (1), (2), (3), we have three equations from which to determine the values of the three constants; thus (o—a)? +(B—2)? =R? et aan. c A’B B—d (o—a)* Be 11 (B32 AZ Beebo From the second we obtain a=o From the third we have DAste a a a ae and substituting these values in the first we obtain A2 race and the equation of the circle becomes [eee ey any! et ins the radius being equal to half the parameter of the conju- gate axis of the ellipse, and the center being in that axis prolonged in a negative direction. (103) In this last case we have the highest order of con- tact of which the circle is capable, and hence the circle is called the osculatrix to the ellipse; or is said to be oscula- tory to it. An osculatrix toa curve is one which has the highest order of 200 DIFFERENTIAL CALCULUS. contact wrth tt, that any curve of the same kind as the osculatrix can have. ; Since the number of constants limits the number of con- ditions that can be assigned to a curve, and since the pass- ing of the curves through the same point is one condition, the order of contact can only be equal to the remaining number of possible conditions ; namely, the number of con- stants, less one, which enter into the general equation ; and this will be the same as the order of its highest differential. EXAMPLES. (104) x. 1. To find the equation of the circle oscula- tory to the parabola, whose equation is y" = 4o0 (1) at the point where the coordinates are x=1 and y=2. Differentiating this equation we have dy :- 2 a*y 4 wa hes 38 d _— dx y *P° “axe y whence 2s ei ye ye and x—ay* ieee hal pe 4 or I—a pet rae (2) and I—a\* fee =) (3) CURVATURE OF LINES. 201 Also from the general equation of the circle we have (1—a)? +(2—0)? =R? (1) and from these we find R*=32, a=5, b=—2 and the equation of the circle osculatory to the ERE at the given point is (x—5)? +(y+2)?=32 Fx, 2. ‘To find the circle osculatory to an equilateral hyperbola whose equation is xy=8 at a point whose coordinates are y=4 and *x=2. By differentiating we have dy J Net ax aie and DE Ye 27, ax® x2 : and from the general equation of the circle we have (2 SecA) ae (1) =a “ig ee (2) 2—a\* iii tga bots (3) 4—b from which we obtain a 4 2 and the equation of the required circle will be : 13 We es Capes) = Fx, 3. Find the equation of the circle osculatory to the curve whose equation is 202 DIFFERENTIAL CALCULUS. 4y= x? —4 at a point whose coordinates are x=0 y=—-I RADIUS OF CURVATURE. (105) Since the curvature of a curve at any point is the same as that of its osculatory circle at that point, we call the radius of the osculatory circle the radius of curvature of the curve. And since the formulas for the equation of the osculatory circle may be applied to any point of a given curve, we may consider them as expressing the general con- ditions required of the osculatory circle. These formulas, as we have seen, are (x—a)’ +(y—0)? =R? (1) ay (x—a dx y—b (2) dy” Bose LT aa? | axe” y= | (3) the two last may be written ay sare Focarrmee a Us Foot , (2) and ax* +dy* aa ae a ary (3) If we represent the coordinates of any given point in a curve by x’ and y’, then for the osculatory circle we must have x=x", yay, ve Coden y i eUEN ee ACS VEY The quantities @ and 4 represent the coordinates of the center of the osculatory circle, and R is its radius. If we substitute in equation (2) the value of y—d, we have CURVATURE OF LINES. 203 ae au e)) ax a*y whence equation (1) becomes x—a= dy* “2 ee Ce tory" =R ax* a ay iz wes ar from which we have 3 ie (dx? +dy?)? ee dx *y (5) which ts the general expression for the value of the radius of curvature in terms of quantities belonging lo a given curve. If we denote the length of the curve by w we shall have aus decd y (106) Since the curve and its osculatory circle have a common tangent, they will also have a common normal; and as-the normal to the circle passes through the center, the normal to any curve at any point will pass through the center of the circle osculatory to it at that point. This is also shown from equation (2) which is Le Ee ST x and y being coordinates both to the given curve and to the osculatory circle at the point of contact, and a and # the ceordinates of the center of the circle. : ay. For since ope Us the tangent of the angle made by the tangent line with the axis of abscissas, we shall have a rool See for the tangent of the angle made by the normal line with the same axis. But when astraight line passes through two points — «x and y being the coordinates of one, and a and the coordinates of the other—the tangent of the angle 204 DIFFERENTIAL CALCULU». made by that line with the axis of abscissas will be =o x—a through the first point will also pass through the second — that is, the center of the osculatory circle. And since from equation (3) we have 8 ax* and hence the normal to the curve, since it passes Poe pn (y—0)2 =~ (14 the value of the first member of the equation will be essentially Dy ~~ negative, and hence we infer that y—4 and on must have ove ye 8 hegative, 6 will be less than y, and, if positive, it will be greater. In the first case the curve will be concave toward the axis of abscissas, and 6 will be between the curve and that axis; while in the other case the curve will be convex toward the axis of abscissas, and 4 will be deyond it. Hence the center of the osculatory circle will be on the concave side of the curve. (107) To find the general expression for the radius of curvature of the parabola, we differentiate its equation twice and obtain contrary signs. So that if ydy=padx and yd*y tay? =o whence pax ay =—— ATs and acy janie iy ee cf y yp Substituting these values in the formula we have 3 2Ixey z VR od i ESS 3. 3 eS) pate tel _coetpn! — prdx? > — p*dx* Pe —p* —dUx ys CURVATURE OF LINES. 205 or, the cube of the normal (Art. 56) divided by the square of half the parameter. If we make x=o we have R=p or half the parameter for the radius of curvature at the vertex. If we make x=} we have R=pvV/ 8 for the radius of curvature at the point where the ordinate through the focus meets the curve. As every other value of R is greater than that where xo it follows that the greatest curvature of the parabola is at the vertex. (108) From the equation of the circle we have Ba ady=— : y and R&dx* a? y=——_ >? y and substituting these values in the formula we have 3. (2 span} ‘ CO ee 213 sel Ve Ot? ext | peg Re rai aaa ye the radius of the circle as it should be. (109) From the equation of the ellipse we have B*xdx TS Or and Atay? +Bedx? De es TERE ha or substituting in the last equation the value of zy” we have Baa A®ys “ These values being substituted in the formula Wi 206 DIFFERENTIAL CALCULUS. 3 Beatin (A eae ‘3 ye A4 Bie me aa he bee A2y3 ake which is equal to the cube of the normal divided by the square of half the parameter as in the parabola. If we make x=A we have y=o and B2 ae If y=B then x=o, and we have A2 US tre iy: Hence the radius of curvature of the ellipse at the princi- pal vertex is half the parameter of the transverse axis — that is the ordinate through the focus. At the vertex of the conjugate axis, the radius is half the parameter of that axis (Art. 102). (110) The equation of the hyperbola referred to its center and asymptotes gives ) ae aye oN Sy eager and r 2axay Sa oo Substituting these values in the formula we have after reducing. . aol ages _(atty?)#_ala?+ty?) 205) AL In the equilateral hyperbola, this value becomes equal to the cube of the radius vector divided by the square of the semi-axis. R SECT LON XII SE. PICOL EF L465 (fff) If we suppose a circle to roll along the concave side of a curve, being always tangent to it, and at the same time varying the length of its radius so as to be osculatory also, its center will describe a curve which is called the evolute of the given curve; and its variables will be the coordinates of that variable center. In other words, the evolute of any curve is the /ocuws of the centers of all the circles that can be drawn osculatory to that curve. The relation between the variables of the evolute can be determined and its equation found from the equation of the given curve, and the first and second differential coefficients derived from that equation; since these determine the posi- tion and length of the radius of curvature, and consequently the place of the center of the osculatory circle. Since the coordinates of the point of tangency and the first and second differential coefficients are the same for the given curve and for the osculatory circle, we can at once determine two of the properties of the evolute. . (112) The first of these properties is, the radius of the osculatory circle is tangent to the evolute. Let AC (Fig. 45) be any curve, and let ¢c be the center of the oscula- tory circle for the point A, while’, c’,c’’ are the centers of the oscula- tory circles corresponding to the 207 208 DIFFERENTIAL CALCULUS. points A’, A”, A”. Then the curve cc” passing through these centers. will be the evolute, and any radius as A’c’ will be tan- gent to it at the point ¢’, the center of the osculatory circle. The equations of conditions (Art. 105) may be put into the following form (x—a)? +(y—4)? =R? (1) (x—a)dx+(y—b)dy=o (2) (y—6)d®y +dy® +ax® =o (3) and in this case a, 4, R, x, y are variables; x being indepen- dent and dx a constant quantity; while x and y are coor- dinates of the given curve, and of the osculatory circle at the point of contact, and @ and 6 coordinates of the varia- ble center of the osculatory circle, that is, of the evolute, and are functions of x and y. From these equations, as we have seen (Art. 105), R may be determined for any point in the given curve by eliminating a and 4 considered as constants. But for the evolute curve we must consider them as variable coordinates; and hence under that supposition if we differentiate equations (1) and (2) we have («—a)dx+( y—b)dy—(x—a)da—(y—b)db=RAR_ (4) and ax? +dy? +( y—b)d* y—da.dx—db . dy=o (5) Subtracting equation (2) from (4) we have —(x—a)da—( y—b)db=RAR (6) and subtracting (3) from (5) we have , —da.dax—db .dy=o (7) whence db = ax da ay (8) joey, but peiaye® the tangent of the angle made by the normal line to the curve, at the point whose coordinates are x and y, ; db with the axis of abscissas: and “om the tangent of the an- EVOLUTES. 209 gle made by the tangent line to the curve at the point whose coordinates are a@ and 4 with the same axis. But x and y are coordinates of the given curve, and a and 2& are coordi- nates of the evolute, and, of course, of the center of the osculatory corresponding to the point (*.y) on the curve, and through this center the normal line must pass (Art. 106); and since both the normal to the curve (or radius of curvature) and the tangent to the evolute pass through the same point, and make the same angle with the axis of abscissas, they must be one and the same line; and hence the proposition. (113) The other property referred to in Art. 111 is The difference between the length of the evolute curve and the radius of curvature, measured from the same potnt ts etther zero or a constant quantity. From equations (2) and (8), of the preceding article, we have ada x—a=— (y—8) (9) and substituting this value of x—a in equation (1) we have aa” da” +dab* BIST Cape tae 9 bed oa eet) hermegere vem LS) From equation (9) and (6) we have aa” da? +d? en at) U) 0 )av— Ra Rae 7-0) which being squared gives 1a" a0" )* (yo EY RoR and this being divided by equation (10) gives da® + ab? =a R* If we designate the length of the evolute by w we shall have du? =da* +db* whence adu*® =a R® 210 DIFFERENTIAL CALCULUS. or du=dR or €R—du=o=a(R—xz) hence R—vwz is a constant quantity and R=u-+c If wz=o0 we have R=c and hence ¢ is equal to the radius of curvature at the begin- ning of the curve, and R is at all times equal to the length of the evolute to the point where. R is tangent plus the con- stant ¢. If, therefore, we suppose a cord to be fastened at B (Fig. 45) and drawn tight around the curve AB and then unwound from A, the end of the cord will describe the curve AC of which the curve AB is the evolute. For the cord will be at all times tangent to the curve from which it is unwound, and also the momentary radius of the curve AC for the point at its own extremity, and consequently normal to the curve at that point; while the length of the cord from the point of tangency to its extremity in the curve AC is equal to the distance from the same point to the origin at A measured along the curve AB. ({14) To find the equation of the evolute, we must com- bine the equation of the osculatory circle with that of the involute in such a manner that x.y and R shall disappear and leave an equation containing only a and @ as variables. This will require four equations, and these are obtained from the equation of the involute, the general equation of the circle, and those formed by placing the first and second differential coefficients of each of these equations respec- tively equal. Thus if we take the equations of condition (Art. 105) (2a) eR (1) ay ay x«-—a@ 7 ay ey A (2) EVOLUTES. 21ft ° wo a ax* +dy* my Ie ax? a zy Oe yas ee y—b (3) and then differentiating the equation of the involute twice, we find the values of the same differential coefficients and make them equal to the second members of equations (2) (3); then eliminate x, y and R, the resulting equation is that of the evolute. Since R is contained in only one equation, we omit that, eae aad as the remaining three are sufficient for eliminating x and y, and for the resulting equation. (115) To find the equation of the evolute to the parabola. The equation of the parabola is Vy = 2px (1) from which dy p ae. (2) and ax premier (3) Placing these differential coefficients equal to those of the general equation of the circle, we have Pe XE nie beer (4) and dy* es Pom ye (5) Dividing equation (5) by equation (4), and substituting for y? its value ffom equation (1), and reducing, we have a=3x+f (6) and substituting the values of @ and y in equation (4) we have after reducing 212 DIFFBRENTIAL CALCULUS. and substituting in this the value of x from equation (6), and squaring, we have 3 po PY Sop) 3°P iP which is the equation of the evolute of the parabola. If we make =o we have a=, which is the center of the osculatory circle for the vertex. If we transfer the origin to that point we have a=ptad and d=0 hence Since every value of a’ gives two equal values for J with contrary signs, the curve of the evolute ACE (Fig. 46) is symmetrical about the axis of abscissas. If a’ ‘is negative then J’ is imaginary, and hence the curve commences at C,a point inthe axis of abscissas at a distance from “ A equal to — that is, at double the Fig. 46. distance of the focus, or half the parameter. (116) To find the equation of the evolute of the ellipse. For this case we have A®y? + B242 =A2B? (1) (See Bin ea cog A®y yb (2) ay” oo ae nie SEE?) ax? A®y3 ~~ y— (3) From equations (2) and (3) we have dy” Qaro ye _ A®y(x—a) 2s yr ax? pa Bea BA EVOLUTES, 213 whence Btx? X—a +E) Wid Coa aay, x Be ei Be whence AA +Btx? B2(x—a)=2( Ad ) whence AB? x—AtB*a=Atay®? +Btx3 whence A? x(A®2B?—A®y?)=A4B2a+B4x3 whence A®B? xe =A4B*atBitx?3 whence prey Serekag x8 (4) Substituting this value of @ in equation (2) we have Aeoeo e O) Re Kaa en A4 A’y whence A*—(A?—B? ae? (y—6)B* iN: y whence Aty—A2®x*y+B*x*?y=A*B*y—A?B?O whence OR ato Mele a o Substituting for x? its value from equation (1) we have A2B2—A2y? UG ie eae ea eam d whence A?—B? F b= — pane) (5) Making A*—B* =c? we have 2 fea c a=ix® and ects a a¥ BA 214 DIFFERENTIAL CALCULUS. eetits a and making Fai and BR” we have or 1 3 a\3 y by3 Kalp)) ond penal) Writing the equation of the ellipse under the form 2 2 x fe ea AE eB te mene 2 and substituting the values of cy and = just found, we have 2 ay b ask (7) +(;) = (6 which is the equation of the evolute of the ellipse in which a and # are the variable coordinates, and m and z the con- stants. If we make a=o we have J=-+z, and if 6=o we have a=-+m, which shows that the form of the evolute is symmetrical with both axes of the ellipse. But ak eri se Amat eae aes and subtracting this from A we have the radius of curvature B? at the principal vertex equal to A as we have already seen (Art. tog). Similarly we find the radius of curvature at the nS vertex of the conjugate axis to be 3: If we differentiate equation (6) twice we have hae ih aC) +-(<) yr 8 whence EVOLUTES. 215 =i (2) eal | bare gol ab m\m m2 ( aa 3 da -4 n\bm and ay ( ee 20 FS ae ada n ae n\n n whence me —4 Uta A eh IY De AESPUG) eee ex da” Aes de Since the numerator of the second differential coefficient is always positive, it will have the same sign as the denominator, which is the same as that of J, and hence the curve is everywhere curvex toward the axis of abscissas. The first differential coefficient becomes zero when ao, and infinite when =o, hence both axes are tangent to the curve, asin Fig. 47. If we make A=B, then c=o, and also m=o and n=a, hence a and 2 in equation (7) will also become zero as they should, since in case of the circle the evolute is reduced to a point — the center. SECTION XL yve ENVELOPES. (117) Suppose two lines, AB and AC (Fig. 48), be drawn at right angles to each other, and a third line ed to move in such a manner that its extremities @ and e shall be con- stantly in these axes, while its length remains unchanged; so that while the extremity e¢ arrives c successively at the points ¢,¢", the 4g extremity @ will arrive at the corres- 4g ponding points a’, a”. di’ During this movement those points of the line near the extremity @ will move in the direction more nearly A ae B parallel to the axis AC than the lne Fig. 48. itself 1s, and will consequently fall zzthzn its first position, while the points near the extremity e will move in a direc- tion more nearly parallel to the axis AB than the line is, and will consequently fall z¢houtits first position. But between these extreme points there is one that ends te move in the direction of the line ttself. ‘ This point does not, of course, remain fixed on the line, but moves from one extremity to the other as the line changes its position and direction, always occupying that place in the line which at the moment does not tend to move out of it 216 ENVELOPES, 217 towards either side. Zhe curve described by this point ts the envelope of the curve. Again let AB (Fig. 49) be the transverse axis of an ellipse, and CD its conjugate axis; and suppose these axes to vary to any extent under the condition that the area of the ellipse shall remain constant. 7 Then as AB decreases CD will Boe increase at arate corresponding with this condition. When the curve ) Seam thus commences to change its shape, | C | " a point near the extremity A will —(4 Fj tend to move in a direction more NAL fp LY nearly parallel to AB than the tan- re gent to the curve at that point is; ip” while a point near the extremity C Fig. 40. will tend to move in a direction more nearly parallel with CD than the corresponding tangent line is. Now between A and C there is a point in the curve that tends (as the axes are changing) to move exactly in the direction of the tangent to the ellipse at that point. As the curve changes its shape and position this point will also change z¢s place on the ellipse, keeping always where its zeudency 1s in the direction of the tangent to the ellipse as it is at the moment. The movement of the point will be continuous, and it will generate a curve which will be the envelope of the ellipse. (118) Since the point on the given curve which describes the envelope always tends to move in the direction of the momentary position of the tangent to the curve at that point, and since any generating point always tends to move in the direction of the tangent to its own curve, it follows that the given curve and its envelope will have a common tangent line wherever the generating point may be at the moment during the formation of the curve. Thus in the 218 DIFFERENTIAL CALCULUS. last illustration, the ellipse, in every stage of its change, will be tangent to the envelope at that point of the curve just then generated. (119) Av envelope to any line, ts another line generated by that point of the given line, which lends to move in the direction of the tangent, whenever tts position or shape ts made to change by chang- ing the constants of tts equation, or any of them, tnto variables. An envelope is not always produced by this change of the constants, for it may be that no point of the given line will tend to move in the direction of its tangent; as in the case of an ellipse where both axes are increased. In general, there will be an envelope only where the suc- cessive positions of the line corresponding with minute changes in the constants, will zzéersect each other; for while the generating point of the envelope tends to move in the direction of the tangent, the points on each side of it will tend to move away from the tangent in opposite directions, hence the next position of the changing line will cross the previous one near the generating point of the envelope. (120) If in any equation of a line the constants are made to vary in value, it is evident that while the curve or line remains the same in kind, its shape and position may assume every possible form and place within the lhmits determined hy the law of variation imposed upon the constants of the equation. If we take for example the ellipse, and consider A and B in its equation as independent variables, then A®y? +B2x? =A*B2 will represent an infinite number of ellipses of every possi- ble size and proportions subject to but two conditions; namely, the axes must both coincide with the axes of coordi- nates. If we make A and B dependent on each other we limit the system of ellipses by the condition thus introduced, but still their number is infinite. If we introduce the still ENVELOPES. 219 further condition that the values of x and y shall be confined to those points of the system which tend to move in the direction of the tangent, while A and B tend to change their values, the first differential coefficient will not be affected by such tendency in A and B, and hence will be the same a¢ those points whether they are considered as variables or con- stants. So then if we take the differential of the equation with respect to them only as variables, and make it equal to zero, and incorporate it with the original equation, we put this limit on the values of « and y, which will then only apply to points in the envelope. The equation will, therefore, be that of the envelope itself —that is, zzstead of representing every point tn one ellipse, tt will represent one point an each quadrant of every ellipse thatcan be formed under the given conditions. | To find the equation then of an envelope we differentiate the equation of the given line with reference to such only of the constants as are considered variable for the time being, and place that differential equal to zero. ‘The values of the constants determined from this equation, and the conditions of relation among themselves, being substituted in the given equation, will produce one that will be inde- pendent of the variable constants, and this will be the equa- tion of the envelope. EXAMPLES, (121) For the first example, let us take the general equa- tion of the circle in which R and @ are constants, while a is considered as a variable. Now since the values of x and y are to be confined to those points of the circle which tend to move in the direction of the tangent while. a varies, it will make no difference whether we differentiate with re- spect to x and y only, or with respect to a also. Differenti- ating in both these ways we have 220 DIFFERENTIAL CALCULUS, («—a)dx+(y—b)dy=o and ) («—a)dx—(x—a)dat+(y—b)dy=o making these differentials equal, and cancelling like terms we have —(x—a)da=o (1) which we should have obtained at once by differentiating with respect to a alone, considering all the rest as constants. From equation (1) we have x=a and this value substituted in the general equation gives y—b=£R or y=dER If we take the positive value for R, this is the equation of a line DE (Fig. 50) | parallel to the axis of ab- ©} = scissas at a distance equal to that of the centers of the system of circles plus the radius, and hence tan- , | p/ E’ gent to them all on the Reger upper side, and is genera- Fig. 50. ted by the highest point of the circle as it moves from D to E, as @ varies in value; that point being the one that tends to move (and in this case does move) in the direction of the tangent to the circle drawn through it. If we take the neg- ative value of R, the equation represents the line D’E’ tan- gent to the system of circles on the lower side. (122) If we take the same equation and consider a and é both as variables, we must establish a relation between them in order to make them both functions of « and y. Let this relation be expressed by the equation a®* +6? =? (1) then the two equations will represent a system of circles ENVELOPES. (Fig. 51) whose centers lie in the circumference of another circle whose radius is equal to ¢, and its center is at the origin. Differentiating the general equa- | tion of the circle with respect to a \ and 4 only we have Lb = (x= a) =(y-8)G=0 whence Fig. 51. GO. Sts E da —-y—b We may now substitute for 4 its value obtained from equa- tion (1); or we may consider it as a function of @ in that equation and substitute the value of the differential coeffi- cient derived from it. This will give us ab Get emed dab yb Baha p= ple e—pt — Substituting this value of (v—a)?in the general equation of the circle, we have (c? —8°)(y—)* from which we obtain whence Cae oaks mbuadsst TTR and similarly CX oO ctR Substituting these values of @ and 4 in the general equation of the circle we have (see) +0-ceR) 222 DIFFERENTIAL CALCULUS. whence x? +y? =(c+R)? the equation of the envelope showing it to be twofold. The positive value of R gives a circle with a radius equal toc+R circumscribing the system, and the negative value for R gives one that is inscribed within it. (123) Let there be an ellipse in which the axes vary in length under the condition that the area of the ellipse shall be constant. This condition will be expressed by the equa- tion AB=c? (1) To find the envelope of this curve we put its equation under the form x? 2 Abt pet (2) and differentiating with respect to A and B only we have ci das FEE A’ TBs ZA or I ae a? Dea 1A A’ eamBie vag Be But from equation (1) we obtain io lel bese OA Se aR whence ve 2 Ara pot whence A=x/2 and B=yr/2 Substituting these values in equation (1) we have 2xy=AB=c? and _. ow C xy — 2 ENVELOPES. 222 which is the equation of a hyperbola referred to its center and asymptotes. The curve EF (Fig. 49) is then a hyper- bola, and the axis of the ellipse are its asymptotes. (124) Let AB (Fig. 48) and AC be the coordinate axes, and let the line ze of a given length move in such a manner that its extremities shall be at all times in the axis. What is the equation of the envelope described by that hne? Call the length of the line c, and the distance Ad and Aé’ respectively and a. Let Am=x and mn=y, then the general equation of the line will be ane (:) we have also a® +b" =c? (2) Differentiating these equations with respect to @ and éas variables we have aN a ia bay or y_ Ox eo? Substituting this value in equation (1) we have ie Gao C ie a? whence we obtain a a=V (2x : sees au and similarly b=V cry A B Nu e e! e'" ae +? =(c2x)8 + (c2y) 8c? =(c3)3 whence from which : 2 2 which is the equation of the envelope. 224 DIFFERENTIAL CALCULUS. The first differential coefficient of this equation is from which we learn that the curve is tangent to both coordinates. | ([25) Suppose the line DC (Fig. 52) to revolve about the point D in the axis of abscissas, c varying in length so that the extremity C shall be at all times in the axis of ordinates, required & the envelope described by the a line DE perpendicular to DC at the point C in the axis of ordi- A D B nates. Fig. 52. Representing the distance AD by, and the tangent of Tp, : : the angle CDB by —7, its equation will be li Saa \eet) in whichif we make «=o we have nic ales tor the distance from the origin at which the line DC inter- sects the axis of ordinates. And since the perpendicular passes through the same point, its equation will be C pa PUES 5 (x) If we consider a in this equation as an independent variable, it will represent all the perpendiculars that can be drawn under the given condition. Differentiating it with respect to @ we have G Kaas —oO a? ENVELOPES. 225 whence and substituting this value of @ in equation (1) we have é y = 26 whence py" =46x which shows the envelope to be a parabola of which D is the focus. It also demonstrates a well known property of the parabola, namely, if lines be drawn from the focus perpen- dicular to the tangent they will intersect it on the perpen- dicular to the axis through the vertex. (126) Let AB and EO (Fig. 53) be the coordinate axes, and let CD be a line revolving between the lines AH and BK in such a manner that its ex- tremities C and D shall always be in those lines, and the pro- ¢ duct of the distances CA and A 0 8 DB from the axis shall be a Fig. 53. constant quantity. Required the equation of the envelope generated. Let OA=OB=m, and AC.BD=c?. Then producing the line DC until it meets the axis of abscissas at S, and mak- ing the tangent of BSD =a, we have es Bite) 2G) bt or F OF pe Ry ie OF a a whence BD=OF +am and similarly AC=OF—am 226 DIFFERENTIAL CALCULUS. whence BD .AC=c* =OF —a*m? or f OF =a2m? +c* But the equation of the line CD is y=ax+b in which @ is the distance from the origin to the point where the line cuts the axis of ordinates, that is, the distance OF. Hence y=axt+(a*m* tec z (1) is the equation which, when a is variable, represents the line CD in every position it can assume under the given condi- tions. Differentiating with respect to a, we have mada 00 ee ae O (a®m® + c2)2 whence af lat mc")? m* c x ak ” (m?—x?)? which being substituted in equation (1) gives C xe ae . dane udea es) (m® —x?)? whence a ‘8 (mm? — x*)? = 762 bem whence ale my(m* —x*)® =c(m*? —x*) whence mm” y* = mre? —¢* x44 ENVELOPES. 227 or my? +7 x*% =m? which is the equation of an ellipse referred to 1s center as the origin, and whose semi-axes are mand ¢. (127) The equation of the normal line to the parabola is 4 y—y =F (2-2) (x) in which x’ and y’ are the coordinates of the point in the curve from which the normal is drawn, and x and y are the variable coordinates of the normal itself. If we consider x’ and y’ as variables, equation (1) will represent the entire system of normals which can be drawn to the parabola. To find this envelope of tius system we find the relation between x’ and y’ from the equation of the parabola J = 2px (2) and substitute in equation (1) the value of x’, which gives , 132 AD whence 2p*(y—y')=—2pxy’ +y'8 (3) Differentiating this equation with respect to y’ only we have eT mt VE BA whence . Y =V 3 pax —p*) Substituting this value in equation (3) we have pees SLs ee eg ee eye ap y— 2p" /2( px—p?) = — 2pxrv/ 2(pu—p?) +1 3(px—p’) |? whence 1. a: apry +2( px—p*)(3(px—p*))* =[5(pa—p*) | whence 2°y=—[8( px—p")]* =—(p)*(%—s)* whence pty = erp? (x—p) DIFFERENTIAL CALCULUS. which as we have seen (Art. 116) is the equation of the evolute of the parabola. Hence all normal lines to the parabola are tangent to the evolute | : Se BOe Lb LO) Neexs.V 3 ArritcCAd LON Or PHEWDIFREREN TIAL CALCOLGS FO THE DISCUSSION OF CURVES. Ais CY CUO). ({28) The cycloid is a curve described by any point in the circumference of a circle as it rolls along a straight line If for example, the circle EFD (Fig. 54) should roll along the straight line AB, the point F, starting frome thee point. “A, would describe the cycloid AD’B, and the distance from A to B where the gen- Fig. 54. erating point again meets, the line AB will be exactly equal to the circumference of the generating circle. If we place the origin at A we shall nave AG=«x and FG=y The arc FE will be equal to the line AE, and HE will be the versed sine of the same arc. Making DE=2r we shall have 3 FH =DH.HE=y(27—y) 229 230 DIFFERENTIAL CALCULUS. hence FH=GE=arc FE—x=,/p;y—y? whence x =Ver. sin. ey ee (1) which is the equation of the cycloid. The line AB is called the base of the cycloid, and ne line D’'E’ perpendicular to the base at its middle point is the axis, and is equal to 27”. , Since every negative value for y gives an imaginary value for x, the curve has no point below the base. If we make y=2r we have wit x=ver sin. 2r=7r and every value for y greater than 27 gives an imaginary value for x ; hence the greatest value of y is the diameter of the generating circle; and for all values of y between 27 and zero there will be a real value for x. (129) We will now proceed, with the aid of the differen- tial calculus, to investigate the properties of this curve in reference to its tangent, subtangent, normal, subnormal, curvature, invclute, etc. Differentiating equation (1) we have ray riy—jty Vary Vane Vay (2) Substituting this value of @v in the general formula for the subtangent (Art. 52) we have te oe gra V ary — y® and for the tangent (Art. 53) ob For the subnormal (Art. 54) GE=¥V ary—y? TG= DISCUSSION OF CURVES. 231 and for the normal (Art. 55) ier ey ee tS FE=y, ee —V/ Ve ae y® 27y Since GE the subnormal is equal to 4/27y —y?, which is equal to 4/DH.HE, the point E of the subnormal for the point F of the curve, must be at the intersection of the ver- tical diameter of the corresponding generating circle with the base; and the normal line =V/ 27y=1/ DE. EH must be a chord of that ctrcle joining these two points. The tangent being perpendicular to the normal will of course be the supplementary chord of the same circle. Hence to obtain the normal and tangent lines for any given point of the cycloid, construct the generating circle for the diameter D’E’ erected at the middle of the base, and through the given point draw the line FH’ parallel to the base intersecting the circle at F’. Join this point with the extremities of the diameter D’E’, and the line F’E’ will be parallel to the normal, and F’D’ will be parallel to the tan- gent. Hence lines parallel to these, through the given point will be the lines required. If it is required to draw a tangent parallel to a given line, first draw a chord from D’ parallel to the given line, and through the point where it meets the circumference of the circle draw a line parallel to the base. The intersection of this line with the curve of the Bs will be the point of tangency. (130) From equation (2) we hove dy NV ary—y' 2r bE ay fa which becomes zero Send y=2r, “ane the tangent at the extremity of the axis is parallel to the base. If we make y=o we have ay Gas oe 232 DIFFERENTIAL CALCULUS. hence the tangent at the base is perpendicular to it. Differentiating equation (3) we have ordy ray a9 yO) Taya ee eee 2x or eh Piet oo ve ie —w_y Te y 3 hence aerial fea ax Pe eye This second differential coefficient being essentially neg- ative, shows that the curve is everywhere concave toward the base. (131) The formula for the radius of curvature (Art. 105) gives in this case, 3 2rax? 3 rordx?\? tenn Ve nats h ONC aes 5 axe) ; naar R= rax? wax se Be y* or R=2,/ary But we have found (Art. 129) the normal to be equal to ‘/ 2ry; hence the radius of curvature at any point is equal to twice the normal at that point. Thus at A the radius of curvature is nothing, while at D’ it is equal to 2D’ E’=,;. (132) The equation of the evolute will be found by the rule given in Art. 114. In the equations of condition (Art. 105) ay st Foci aaa A (2) and e ax + dy" ; a yf Bring h®, Substitute the values) Guam e = just found from ax ax* 4 DISCUSSION OF CURVES. 233 the equation of its cycloid, and then, by means of that equation eliminate x and y. Thus ee a e—a=— = (y—B) J and dx? +-dx?(-~—1) baa ioe dnt yr yp whence y—b=2y and *—-a=—2V/ syy—y? or y=—b and x=a—2V/ —o7p—f2 Substituting these values of x and y in equation (1) (Art. 128), we have a—2/ — 27h —62 =Ver. sin.~* —b—4/— 27h—0? or a=ver. sin.~1—d-+/ — 975—52 (4) which is the equation of the evolute. (133) For all values of J that are positive a is imaginary, hence no part of the curve is above the base of the invo- lute. For all negative values of 4 greater than 27, @ is also imaginary, hence if we draw A’B’ (Fig. 55) parallel to the base at a distance below it equal to 2r, the evolute will lie between that line and the base. Ifwe make b=—2r, a be- comes equal to the arc whose versed sine is —J, 234 DIFFERENTIAL CALCULUS. that is half the circumference of the generating circle. Hence the point G where the evolute meets the line A’B’ is in the prolonged axis of the involute. If we make b=0, a also becomes equal to zero, and hence the evolute passes through the origin at A, and also the extremity of the base at B. For ver. sin.~!0 may be zero, or it may be a whole circumference. If we differentiate equation (4) we have rab rdb+bdb _— (ar-+b)adb /—e2rb—b? V—2rb—-B? V —27rb— ada= or db V —2rb—b? Veena 2ar+b showing that at the points C and B where =o the base of the involute is tangent to the evolute. Also since V/ —2rb—8? b g.."yn 2F eam eg if we make 6=—27 we have ab ee showing the tangent to the evolute at G is perpendicular to the line A’B’. ab rey Squaring the value of ae and differentiating, we have ab® b LEK ey rv da® ~~ art b Mo Wa? ~~ (ar+d)? which is essentially negative, and since every real value of bis also negative, the curve is everywhere convex to the base of the cycloid. (134) These circumstances, together with the form of the equation of the evolute, lead us to suppose it to be an equal cycloid, but for certainty we will transfer the origin to G, and the coordinate axes to EG and GF respectively par- DISCUSSION OF CURVES. 235 allel to the first. Calling the new coordinates x’ and y’ we have a=x'+m and b=)'/+n m and z being the coordinates of the new origin referred to the original axes. Then m=ver.sin.-12r and z=—a2r whence a=x'+ver. sin.~t27 and 6=—(2r—y’) Substituting these values of a and 2 in equation (4) we have x’+ver. sin.—127=ver. sin.—1(27—y") +4/ 2x(2r—y')—(2r— y)2 but ver. sin.~!27—ver. sin.~1(27—y’)=ver. sin.—1y’ hence x =—ver. sin.~1y' +4/ o7y/ 4/2 which is the equation of the curve CG, the values of x’ be- ing the same as those of x in equation (1) (Art. 128), except that they are negative as they should be, since the values of x are reckoned in a contrary direction from those of x ; and the curve CG is equal to the curve CF, but reversed in position with reference to the origin. | Since the curve CG is equal to FG (Art. 113) the length of the cycloid is equal to four times the diameter of the generating circle. (135) The character of the evolute of the cycloid may be demonstrated geometrically thus: Let us suppose two right lines AB and A’B’ (Fig. 55) to be drawn parallel to each other, and two circles to be des- cribed on the diameters DE and CE, each equal to the distance between the two parallel lines and tangent to each other at the point C. If now we suppose each circle to roll along the line on which it stands, at the same rate, so that they are at all times tangent to each other, then the point C of the upper circle will describe the first half of a cycloid CPF, while the same point C of the lower circle will des- cribe the last half of an equal cycloid CP’G. 236 DIFFERENTIAL CALCULUS. Suppose the two circles to have arrived at the point C’ in the line AB, and that P is a point in the upper curve. The diameter DC of the upper circle will have assumed the position PO, and the diameter CE of the lower circle will have assumed the position O'P’ parallel to it; and P’ will be the generating point of the lower cycloid. Draw the chord PC’ and it will be normal to the upper cycloid (Art. 129). Draw also the chord C’P’, and it will be tangent to the lower cycloid at the point P’ (Art. 129). Now since PO and O’P’ are parallel, these two chords and the corresponding arcs are equal, and hence the angles PC’D’ and P’C’E’ are equal; and since D’E’ is a straight line P’C’P is a straight line also, normal to the upper curve and tangent to the lower one. Hence the lower cycloid is the evolute of the upper one. (136) The equation of the evolute may also be obtained by considering it as the envelope of the normals drawn to the curve. The general equation of the normal to the cycloid is f Ne ectivine 5 1O (1) in which «’ and y’ are the coordinates of that point of the cycloid to which the normal is drawn; and x and y the gen- eral coordinates of the normal line. If we make x’ andy’ variables, still retaining their relative values, as in the equa- tion of the cycloid, the equation (1) will represent the whole system of normals that can be drawn tothe curve. Ifnowwe eliminate one and make the differentials of the equation with respect to the other equal to zero, then (Art. 120) by eliminating that we shall have an equation which will be that of the envelope of the normals, and also the evolute of the cycloid. ‘Substituting for x’ in equation (1) its value taken from the equation of the curve (Art. 128) we have DISCUSSION OF CURVES. 237 t y— y= ea ——= > (w— ver. sin.—1y/+V ary’ —y'*) V ary’ — yf? or eae L-VVCL: Slee yn bs aN sag aaa ee whence yV ary’ — y'®—y'ver. sin.—1y’ +y' =o (2) Differentiating this equation with respect to y’ we have r—y fs ee Vs sin. cork a 2 AY Spee ae: Sime? V ary! —y* Substituting in this equation for ver. sin.—1y’ its value taken from the equation of the cycloid, and multivlying by V ary’ —y/*, we have V(r) (2! +V Say aryl ag THAW aryl yl =O or Wt HV aryl =yl® tary —y"* ty 1 —2V/ ary! —y/? or VW r—y)=—(x—2')V/ ary’ —y® +377 waist but peer x—x/ = hence Wr—y")= ae Claas Oo Mehta ens f clearing of fractions and multiplying we have i VY E27 yy —27y *— yee a eanyeey, ® whence Ty aioe (Sg Of Vay Substituting this value of y’ in equation (2) we have yV —ary—y? +y ver. sin.-1—y—xy=o or x«=Vver. sin.~! —y+V —2ry— y? 238 DIFFERENTIAL CALCULUS. which is the equation of the envelope of the normals, and also of the evolute of the cycloid, as in Art. 132; for sub- stituting the variables @ and 4 for x and y, the equations are identical. THE LOGARITHMIC CURVE. (137) The logarithmic curve is one in which one of the coordinates is the logarithm of the other. Its equation is x=Log. y If we represent the base of the system by a the equa- 9 ) | tion may be written =5- 4-322-JuA Places y=at Fig. 56. The curve may be constructed by laying off on AB (Fig. 56) the axis of logarithms, the numbers 1, 2, 3, 4, etc., on both sides of the origin, and laying off on the corresponding ordinates, or on AC the axis of numbers, the corresponding powers of a. | When x=o then y=1, whajever may be the value of a, and hence all logarithmic curves will intersect the axis of numbers at a distance from the origin equal to 1. If ais greater than 1, and x positive, y will increase as x increases, and there will be a real value of y for every value of x as in the curve DE. If x is negative, then the value of y is fractional, and decreases as x increases negatively, but y will not become zero until x=— o. If y is negative, there is no corresponding value of x, and hence the curve can never pass below the axis AB. If ais less than 1, then y will diminish as x increases DISCUSSION OF CURVES. * 239 positively, and becomes zero when x=; but y increases for negative values of x, and the curve has a position the reverse of the first as DD” in the figure. (138) If we differentiate the equation Var we have 1h I I aN avy. a = y— ax m1 m and aay I I me ee Ba) aa Axe m*” m7 ay If we make y=o we have pha hence the tangent for that value of y is the axis of abscissas; and since yo gives x—=— o the axis of abscissas is an asymptote to the curve : ; a. (Art. 88). But since y=oo gives x= 0, and also ee 00, the curve has no tangent parallel to the axis of ordinates except at an infinite distance. The sgn of the second dif- ferential coefficient shows that the curve is at all times con- vex toward the axis of abscissas. The subtangent PT =F y7=M; hence the subtangent is constant and equal to the modulus of the system of logar- ithms, to which the curve belongs. In the Naperian system the modulus is 1, and in this case PT and DA are equal. (139) We will now investigate the curve whose equation is y=x log. x Every value for x gives a single value for y. If x is less than 1 the value of y is negative. (1) If x is greater than 1, y is positive. (2) lige Ome 4-21) —0: (3) If x is negative, y is imaginary. (4) 240 DIFFERENTIAL CALCULUS. - If we differentiate the equation we have ay Te OS +1 (5) zi py eT ax" — x (6) a Making ae we have I I — —— —1 = = log. x TSOL =F San ae (7) which corresponds to a minimum as shown by the positive 9 tates | J value of Tee” (3) ae When «=o, ie tea (9) 4) When «=1, 7 =1. (10) Since y is negative between x=o and x=1 and then pos- fs Lid ait i ne itive, while gt is always positive, the curve 1s concave toward the axis of abscissas between «=o and «=1, and afterwards convex. (11) Hence the curve begins at the origin (Fig. 57) and intersects the axis of abscissas at D, making AD=1 (3).- The tangent to the curve at D makes an angle of 45° with the axis of abscissas (10), while at A the axis of ordinates is tangent (9g). Atthe point E, whose abscissa is , the tangent to the curve is parallel to I 2e7 Loz the axis of abscissas (7), and the ordinate is at a minimum (or negative maximum) (8). Between A and D the curve is below the axis of abscissas (1), and concave to it (11); and DISCUSSION OF CURVES. 241 beyond D the curve lies entirely above the axis of abscissas : / : I : and is convex toit. Sincex= PPR PGS CoS a have FE=AF (140) We will next take the equation 1 ee ae oT Damian Ener a Every value of x gives a real positive value for y, and hence there can be no negative value of y. (x) If =o, y=o, and hence the curve passes through the origin. (2) If x is negative we have y=e”, in which if «=o, y will become infinite. (3) So that x=o gives two values for y, according as x ap- proaches zero from the positive or negative side. (4) If « be negative and increase in value, that of y will approach more nearly to 1, which it will reach when H=— Or, (5) If x be positive, and increasing the value of y approaches more nearly to 1, which it reaches when x= ©. (6) Hence the curve will be as in Fig. 58, in which AB and AC are the axes of coordinates, and DE a line parallel to AB ata distance from it equal to 1. It will pass through the origin A (2), extend indefinitely in a positive and nega- tive direction, and the line DE will Fig. 58. be an asymptote to both branches (5), (6). The axis of ordinates will also be an asymptote in the positive direction (3) (Art. 88). As the branch DC of the curve extends to an infinite distance in both directions, it has no connection with 242 DIFFERENTIAL CALCULUS. the branch AE, which commences at the origin and is infi- nite at the other extremity. There are, in fact, two curves, one answering to the positive, and the other to the negative value of x. 1 If we differentiate the equation y=e * we have 1 MeN AP ia ax x and _1 aPy Se (tT —22) ax ee Since ant’ e * AT a atas r wer we shall have ian oa a when either x=o or x=, hence the axis of abscissas is tangent at the origin, and parallel to the tangent at an infi- nite distance in either direction; in which case y=r1 (5) (6). For all negative values of x, = is positive, and hence (oe the branch DC is convex to the axis of abscissas. For all fe Cy 45 positive values of x less than $, a is also a positive, show- ing that between A and H the curve is convex to the axis of abscissas, while at the point H, where x=4, the value of ae a ; : a changes from positive to negative passing through zero, showing that at P the curve ceases to be convex, and be- comes concave toward the axis of abscissas. ‘This is called an inflexion. SECTION XVI. winwiat Ane eOLVe LS. (141) Singular points of a curve are those at which there exists some remarkable property not common to other points of it. Such, for example, as the maximum or minimum value of the ordinates or abscissas, points of inflexion, con- jugate points, cusps, etc. In many cases these points are easily discovered by the aid of the differential calculus, as will be seen by the fol- lowing examples. MAXIMA AND MINIMA. (142) If we differentiate the equation =3+2(*—4)* we shall have 2 =8(x—4)9 Tf a4(0—4)? se =48(«—4) TJ _ 48 Here we find that x=, will reduce the first differential 243 244 DIFFERENTIAL CALCULUS. coefficient to zero, showing that the tangent to the curve Is parallel to the axis of abscissas (Art. 36),and hence the value of the ordinate way be amaximum or minimum. But since the second differential coefficient is always positive except when it is zero, the first must be an increasing function, and hence at zero must p be passing from negative to positive, and the value of y must be changing from a diminish- — ing to an increasing one. So that there is a minimum when x«=4, as shown in Fig. 59. We infer the same thing from the sign of the fourth differen- tial coefficient (Art. 209). If we take the equation C B Fig. 59. y=2—2(x—2)4 we shall have a me aad (hae T} = o4(a—2)? ae y ed —48(x—2) d4 axt 2 Since x=2 reduces the first differential coefficient to zero the tangent at that point is parallel to the axis 9 of abscissas; and since the fourth differential ae coefficient (the first that has real value for x=2) is negative, the value of y at that point cane must be a maximum as in the figure. Since Fig. as 9 w De ; : Tue 1s at all times negative, except when x=z2, the curve will be concave toward the axis of abscissas for all positive values of y (Art. gr). a SINGULAR POINTS. 245 POINTS OF INFLEXION. (143) A point of inflexion is one in which the radius of curvature changes from one side to the other of the curve so that it will be convex on one side of the point of inflex- ion, and concave on the other towards any line not passing through the point itself, and this will, of course, be true for the axis of abscissas, and hence at such a point the second ° differential coefficient will change its sign. If the point of inflexion should be in the axis of abscissas, both parts of the curve would be convex or concave to it, but the second differential coefficient will still change its sign (Art. 93). Now in order that the sign should be changed, the function must pass through zero or infinity, and hence the equations as ary yiiae and eee Cs will give all the points of inflexion in any curve in which there may be such points. (144) Let us now take the equation pene whence ay Te Ne 2)? and if 2 <5 =18(2—2) ay eee In this case every value of y gives one for x, and wice versa, hence the curve has no limit. When x=a2, then a Po — and y=1, so that if wemake AC=2 and CD=1 (Fig. 61), the tangent at D will be parallel to the axis AB. But x=2 re- a” duces se to zero, also indicating that 246 DIFFERENTIAL CALCULUS. there may be a point of inflexion, hence we resort to the ge ree et: : value of =e which is a positive constant. From this we Sele dos . ; learn that at zero yae iS an increasing function, hence it must pass from negative to positive, showing that at the eee, ; same point x Passes from a decreasing to an increasing function, and hence does not change its sign, but remains positive both before and after the zero point; and this shows that the value of y is an increasing function both before and after the same point. There is, therefore, no maximum nor minimum for it at that point. 9 @ y ee = Since ae changes its sign at x=2 from negative to pos- itive, the curve will be concave toward the axis of abscissas when «<2, and convex when x>2, so that at the point D where «=z the curvature changes its direction and there is an inflexion. (145) If we take the same equation, and make the last term negative, we shall have ay “if } D ax =—9(x—2) a 29 —_ ie A C B ax* 18(x 2) Fig. 62. The point D where x=2 will still be the point of in- ay . flexion, but since <4 is negative for all values of x except «=2, the curve will approach the axis of abscissas me ; Aa for all positive values of y, except y=1, and since ey is positive for r<2, and negative for x>2, the curve will be ‘convex toward AB between A andC, and concave afterwards, as in Fig. 62. SINGULAR POINTS. 247 The first differential coefficient being zero when «=z, it follows that the tangent at that point will be parallel to the axis of abscissas (Art. 36), and hence the curve will pass fromone side of the tangent to the other at the point of tan- gency, and will be convex to the tangent on both sides of it. (146) If we take the equation petal =2)e we have w_ 6 s(x—2)8 dey T2 —— 7 ie 25(x—2)* If we make x=2 we have dy @? efile Cm oe 2 aa and hence at the point D where «=2 the tangent will be perpendicular to the axis of abscissas (Fig. 63), ‘ and since s is positive for other values of x, the curve will leave the axis of abscissas, for all j——¢__g positive values of y as x increases. And since Fig. 63. oe ant a. Da hes : =e changes its sign from positive to negative 1n passing through infinity where «=z, the curve will be convex toward the axis of abscissas for «<2, and concave for «> 2, and at x—=2 there will be an inflexion. (147) If in the same equation we make the last term negative we have - 3 y=2—2(x—2)? and YY : we 5(x—2)8 Vay) or Mt 248 DIFFERENTIAL CALCULUS. and the conditions will be changed so that the curve will be reversed. It will now approach the axis of abscissas, and the second differential coeffi- cient will change its sign from negative to positive in passing through infinity where x=2; the curve will be concave for x 2. The point D (Fig. 64) will still be a point of inflexion, and the tangent will be perpendicular to AB. (148) If we take the equation Fig. 64. y=(x—2) we have ay Wn a 3he—2)* and wey Wee =6(x—2) which all reduce to zero when x=2. This shows that the curve meets the axis of abscissas at the point where x=2, and that this axis is tangent to it there. And since the second differential coefficient will have the same sign as y (both being the same as that of x—z2), it will change from negative to posi- Fig. 65. tive at the point where «=z, showing an inflexion there, and that the curve is convex to the axis of abscissas on both sides of it. Cc B CUSPS. (148) A cusp is a curve consisting of two branches start- ing from a common point in the same direction, and imme- diately diverging from a common tangent. They are of two kinds, namely: Those in which the branches are on differ- SINGULAR POINTS. 249 ent sides of the tangent, which are cusps of the first order; and those in which the branches are both on the same side of the tangent, which are cusps of the second order. The following are examples of the first order. Let y=1+3(e-1)8 then Oa a(x—1) 4 and CDR 2 @)* x(2—3)8 dy J ee ae Ten and Pee mek For every value of «<1, es is negative, and positive for every value greater. The curve, therefore, approaches the axis of abscissas, in the first case, and recedes from it in the second (y being positive), which indicates a minimum, while the tangent at that point is perpendicular to ! ekiean ye D the axis of abscissas; and since ee always negative the curve is concave toward thesame A CB Every value of 7 less than 1 gives an imaginary value for x, while every value greater than 1 gives two values for x, one less and one as much greater than 1. Hence the curve has two equal branches commencing at D (Fig. 66), where they have a common tangent. (150) If we make the last term negative, the signs of dy eae k ea and Sa will be reversed, and the first will change from 250 DIFFERENTIAL CALCULUS. positive to negative as x passes from x<1 to x>1; while at x=1 the tangent is still perpen- D dicular to the axis of abscissas. Any value ofy | __ AN greater than 1 will give an imaginary value for Al C R Fig. 67. x, while every value less than 1 will give two real values for x equally distant from the point C where 9 w a x=1 (Fig. 67). The sign of a being now always posi- tive (except at x=1) shows that both branches of the curve are convex toward the axis of abscissas. ‘These are then cusps of the first order. (151) If we differentiate the equation y=ok(e—1)h (x) we have ay 1 aoe +3(a—1)? gy an We see from equation (1) that when x=1, y=2, and when “<1, y is imaginary, while when «>1, y has two values, one greater than 2 and the other as much less; so that DC (Fig. 68) be drawn perpendicular to AB, making DC=2, the curve will commence at D and be symmetrical about the line DE, =+23(x—1)# Z} and since = =o for the point D, the line DE will be tangent to both branches. Since for every other value of dy : i Xn has one negative and one equal positive value, one branch of the curve will approach the A ¢ B axis of abscissas, and the other recede from itat Fis. 68. 2 ° ra an equal rate. And since for every value of x> Sears has two equal values with contrary signs, the positive cor- responding with the greatest value of y, we infer that the SINGULAR POINTS. 251 upper branch of the curve is convex, and the lower branch concave, to the axis of abscissas, and that the curve is a cusp of the first order. ({52) If we change the sign of the last term and make the equation a yH24(1—x)* we have 1 2 = +3(1 —x)* A i a 2 i aa) and the curve will be similar, but reversed in position as in Fig. 69. | If «>1, y will be imaginary. D If x=o, y=3 and y=1. 9 C4 EY If y=o, x=1—4/4. Since sis both pos- A} C 8B itive and negative when «<1 there is no maxi- Fig. 69. imum nor minimum value for y. (153) The curve represented by the equation (y—2? P= x? contains a cusp of the second order, as well as some other singular properties. Solving this equation we have 5 you tx* (1) and by differentiation we have dy 3 —= 5 yk We ee Ee and A A an” 4 From equation (1) we find that the curve passes through 252 DIFFERENTIAL CALCULUS: the origin, and does not extend to the negative side of the axis of ordinates. / Every positive value for «<1 gives two real positive values for y, while x=1 | Dp gives one positive value for y and one A | ZE Se equal to zero. Hence the curve has E C¢ two branches, both of which pass packs through the origin, and one intersects the axis of abscissas at a distance from the origin equal to 1. wy If we make Pe ede have x=o and x=3%. Hence there are two points in which the tangent to the curve is parallel to the axis of abscissas; at the origin where the axis itself is tangent and at the point D (Fig. 70) whose abscissa is «=$$; and as the value of x at this point derived from the equation aoe corresponds to the minus sign in equation (1), the point of tangency is on the lower branch of the curve. : : . Loa The second differential coefficient has two values 2-2 xt wk and pee hed of which the first belongs to the upper branch 4 of the curve and is always positive, while the second is pos- : Wee eer ; : itive so long as a is less than 2; that is so long as x is d* : less than 34, which makes Sa te After that it becomes negative; showing that the lower branch of the curve is convex to the axis of abscissas, as far as the point whose abscissa is AE=°4,, and at this point there is an inflexion, the curve becoming concave to the axis of abscissas as long as y is positive and convex afterward. Hence at the origin there is a cusp of the second species. SINGULAR POINTS. 253 CONJUGATE POINTS. (154) Conjugate points are those single points which are isolated from the curve, but will satisfy the equation. — If we differentiate the equation a a? (x —5) yar, /t1e—4) VA a (1) we have Ly 3x20 ax 2 a(x—b) aey 3x— 4b dx? = If we make x=o in equation (1) we have y=o, but any other value of x less than 4 will make y imaginary. Hence while the origin will satisfy the equation, that point is iso- lated, having no connection with the curve. We also see that «=o will give ay b Tia al which is imaginary as it should be, since at that point the curve can have no tangent. If we make «=4, we have showing that the tangent at that point is perpendicular to the axis of abscissas, while the value of y is zero. As every positive value of x>6 gives two equal values for y with opposite signs, the curve is symmetrical about the axis of abscissas, and as the dL value of os has the same sign as y, the Fig. 71. curve departs from that axis in bothdirections. If we make x negative the value of y becomes imaginary ; showing that 254 DIFFERENTIAL CALCULUS. the curve does not extend to the negative side of the axis of ordinates. Ley lf we make ——>=0, we have ax? ? 46 3 showing that at the points C and C’ (Fig. 71) which lie in the ordinate drawn through D at a distance from the origin equal to 42, the curve has an inflexion in each branch, since for that value of « we have x— Ah aus s If we make x<—,, the second differential coefficient will °) 46 have a sign contrary to thatof y. If ae the signs will be thesame. Hence between H and D the curve is concave toward the axis of abscissas, and convex beyond D, which also shows an inflexion. a If we make seas we have 3x==24, or 2b CS 3 This value being substituted for x in equation (1) gives an imaginary value for y, showing that there is no point in the curve where the tangent is parallel to the axis of abscissas. MULTIPLE POINTS. (155) A multiple point is one in which two or more branches of a curve intersect each other. At such a point the curve will always have as many tangents as there are why branches, and hence Ye Must have the same number of val- ues for that point. SINGULAR POINTS. 255 Let us take the equation y=bt(x—a)\/x—c where a>c (1) then by differentiating we have ay ae WRC ha Fe a fee For «=a and x=c in equation (1) we have y=d ; hence H and H’ (Fig. 72) corresponding to these values of x and y are points in the curve. For all values of *c, except x=a, y has two Fig. 72. values, one greater, and the other as much less than @. Hence the curve is symmetrical about HH’. For x«=c we ad : : have = oo, hence the tangent at H is perpendicular to . . J the axis of abscissas. For «=a we have two values of 3 or equal to each other with contrary signs, namely, «1/2 —c and —/x—c. Hence at H’ there are two tangents making sup- plementary angles with the axis of abscissas, so that the two branches of the curve cross each other at that point in direc- : dy tions symmetrical with HH’. If we make Te =o We have a-+2¢ 3 which shows that at the point corresponding with the ordi- nate at E where AE equals one-third of (2AC+AB), the tan- “a= OV =—— (156) We will close the discussion of algebraic curves with that of the equation ay® —x° +(b—c)x* +bcx=0 Solving this equation with reference to y we have 256 DIFFERENTIAL CALCULUS. 5 eee a (1) and ay, 3x*—2n(b—c)—be dx ~~ 2N/ ax(x—b)(x+c) If in equation (1) we make x=o0, x=4, or x=—c gives an imaginary value for y, hence the curve has no point on the negative side of H, since AH=c. Every negative value of «b gives two equal values for y with contrary signs. Hence on the positive side of H’ the curve is symmetrical about Fig. 73. the axis of abscissas, and the entire curve consists of two parts having no connection with each other by a common point. Each of the values of x that reduce y to zero also reduce a ere: : = to infinity; hence at the points H, A and H’ the tangent is perpendicular to the axis of abscissas, and one of these tangents is the axis of ordinates. If we solve the equation 3x? — 2x(b—c)—bc=o we shall have ltt V 3bc+(b=0)? 3 but /3dc+(d—c)*<é+c,; hence if we take the positive SINGULAR POINTS. 257 value of the radical part the result will be less than b—e-+b--e ; he tee ; ——.——, that is, less than $4, hence it will give no point of the curve. If we take the negative value, the result will be numerically less than —%c,; hence there will be two points where the tangent will be parallel to the axis of abscissas, corresponding to the point on that axis where _b=—c—V 3bc+(b—c)? 3 If c=o the equation becomes XxX ay*® =x3 —bx* in which case the oval HA is contracted into a conjugate point at A as in Art. 154. If d=o the equation becomes ay* =x> +cx? x ® tex" y= 4£\/— Ay 3k eee dx ~~ 2/ gx?(x¢-+4c) In this case the curve takes the form in Fig. 74. There are two equal values for y with opposite signs for every value of x on the positive side of H where or and mea x=—c. Atthat point =~=o, and the ax tangent is perpendicular to the axis of Bie te: : ay 2¢ abscissas. If we make Vin ats have x=o and a ae ac hence the tangent at A and at T and T’ where eee are parallel to the axis of abscissas. 258 DIFFERENTIAL CALCULUS, If we make both 4 and ¢ equal to zero we have whence and In this case the curve assumes the form in Fig. 75. There is no negative value for x, and all positive values of x give two equal values fory with contrary A, wv signs. At the origin we have ee and hence the axis of abscissas is tangent to Fig. 75. both branches of the curve which is a cusp of the first species PART II. oe INTEGRAL CALCULUS. INTEGRAL CaALcuLus. SECTION I. PRINCIPLES OF INTEGRA TION. (157) The problem of the differential calculus is to obtain the differential or rate of change in a function arising from that of the variable, or variables, which enter into it. The corresponding problem of the integral calculus is to pass from a given differential of a function to the function - itself, The first of these operations can always be performed directly by rules founded on philosophical principles. The second can only be performed by empirical rules founded on actual experiment. We cannot proceed a@recdly from the dif- ferential to the function, but, as it were, backwards; that is, we show that a function is the integral of a given differen- tial by showing that the latter would be produced'by differ- entiating the former. Thus we know that x? is the integral of 2xdx, because 2xd¢x has been shown to be the differential of «*, Hence the rules for integration are merely the rules for differentiation inverted. While rules have been obtained for differentiating every algebraic function, it by no means follows that every differ- a 261 = 262 INTEGRAL CALCULUS. ential can be integrated. The number of simple algebraic functions is very small, and each one has its specific form of differential. Should any function be complicated, it can be analyzed and differentiated in detail, applying only the rules for simple forms. But before a differential can be integrated, it must be reduced to one of the forms arising from differentiating a simple function; and this can be done in comparatively few of the infinite number of forms that differentials may assume. The transformations available for this purpose form one of the chief subjects that demand the attention of the student of the integral calculus. The dif: ficulty of integration is very much increased when the dif: ferential is a function of two independent variables, for the rate of change in such a function can give but little indica. tion, generally, what the function is. There is still another difficulty in obtaining the exact integral of any given differential. We have seen that the constant terms in any function disappear when it is differ- entiated, and, of course, when we come to integrate an iso- lated differential expression, we cannot know what constants, if any, should belong to it. Insuch a case, then, we pay no attention to the question of constants. If. however, the function should occur in an eguation we can generally find from the conditions expressed by it what value would belong to the constant. Until this is done we indicate by adding the symbol.C to the integral that a constant is to be supplied if needed to render the integral definzte. Until then it is sald to be zudefintte. The notation indicating the integral of any differential is the letter s elongated, thus /xd@x would be read “the integral of xdx.” This notation was originally adopted by Leibnitz to indicate the sz of the infinitely small differentials or dif- ferences of which he supposed the function to be made up, and is still retained as a matter of convenience even by PRINCIPLES OF INTEGRATION. 265 those who reject its original meaning, as employed in the system of Leibnitz. The following rules for integration are derived from those for differentiating ; being in fact the same rules inverted. (158) Lf the differential have a constant coefficient it may be placed without the sign of integration. For we have seen (Art. 10) that the differential of a vari- able having a constant coefficient is equal to the constant multiplied by the differential of the variable; that is to say, the coefficient of the variable will also be the coeffi- cient of its differential; hence the coefficient of the differ- ential will also be the coefficient of its integral, that is, of the variable; and may be placed outside the sign of integ- ration. Thus dax)=adx hence fadx=afdx=ax (159) Zhe integral of a differential function, consisting of any number of terms connected together by the signs plus and minus, 1s equal to the algebraic sum of the integrals of the terms taken separately. For we found (Art. 9) that the differential of a polynom- ial is found by differentiating each term separately, hence to return from the differential to the polynomial, which is the integral, we must integrate each term separately. Thus Ax+y—sz)=dx+dy—dz hence | S (dx t+dy—adz) =fdext/fady—fdz=x+y—2 (160) Zhe integral of a monomial differential consisting of a variable, multiplied by the differential of the variable ts equal to the variable raised to a power with an exponent increased by one, and divided by the increased exponent and the differential of the variable. We have in (Art. 15) the rule for obtaining the differen- tial of the power of a variable. In other words we have 264 INTEGRAL CALCULUS. given the steps by which we pass from the power to its dif- ferential; and hence to pass back from the differential to its integral, that is, the power, we must retrace each step. Thus in the first case we diminish the exponent by one; in the latter we increase it by one. In the former we multiply by the differential of the variable; in the latter we divide by it. In the former we multiply by the exponent before reduc- ing it; in the latter we divide by the exponent after increas- ing it. Thus DEBE heer because ax” =ux" "qx (161) Ifthe function consist of the power of a polynom- ial multiplied by its differential, the same rule will apply. Thus let the differential be (ax+x*)"(at2x)dx=(ax+x*)"dax+x?) make ax+x* =u then (ax+x?)" (at2x)dx=u" du and é ynrti (ax+x?)r+1 Si du= a PEE ey HAS So ie EXAMPLES. i : x? ax x Lx. 1. What is the integral of : ? Ans. FF | 4 , ; ak ax Ex, 2. Whatis the integraliofix*adxr « Ans. i ax Ex. 3. What is the integral of yee Ans, 2/ x ; : ax : I Lx. 4. What is the integral of act Ans, oS eee PRINCIPLES OF INTEGRATION. 265 , : hy Ex. 5. What is the integral of ax?dx+ nye? ax? 7 Ans. 3 + x (162) If the exponent of the variable in the case pro- vided for in Art. 160 should be —1, the rule will not apply. For by this rule ey emt ee 00 rs Seats and this arises from the fact that a differential with such an exponent can never occur under the rule given in Art. 15; for then the variable must have been x°, a constant quan- tity, that cannot be differentiated. Such differentials, how- ever, do frequently occur, but the rule for their integration must be drawn from a different source. We have found : : ax (Art. 38) that the differential of log. a oe c-l7x~, and hence a differential of this kind must be integrated by the rule derived from that given for differentiating logarithms. That is to say, the integral of any fraction, in which the numerator is the differential of the denominator, is the Naperian logarithm of the denominator. EXAMPLES, ; ' adx £x.1. What is the integral of —\~? Ans. alog.x $ ; 2bxdx Ex, 2. What is the integral of rary a: ? Ans. log. (a+édx?) : ; adx Qt. Ex. 3. What is the integral of |? Ans. log. x An” oie Ex. 4. What is the integral of 3 ? Ans. alog. x ax /x.5. What is the integral of perce? Ans. log. (a+ +hx* aax até até ax—x9*— x " 2(a—x)' (OS aad and by integration i, a Fix g*x— x5 atb 4 at+é dx=alog. x——— log. (a—x)— log. (a +.) 270 INTEGRAL CALCULUS. which may be reduced to a log. «—(a+d) log. (a? —x?) Note.— The second term of the integral must be negative ; for since d(a—-~) is ax ax c —dx,we shall have d(log. GS and hence 7—x must be the differential a of —log. (a—~+). (169) Let us now take the fraction Lae x* +4ax—b* To find the factors of the denominator we must make it equal to zero, and solve the equation which gives L=—AEV 4a? +3? and hence the factors of the denominator will be LTAaTV ga?+5 and XTaA— a/4q? +52 To: simplify the expression we will represent the constant part of each factor by E and F and we shall have x* +4ax—b® =(x+E)(x+F) and we may make x Anges B —Av--Al Bs Bk + 4gax—b? x+ ae +F («+E)(«+F) making the numerators equal we have Ax+AF+Bx+BE=x whence A+B=r1 and ? AF+BE=o from which E F A= ae and B= ea Substituting these values es A and B we ae —— XIX ees A if can ae ax x? + sax— 2 ie c+E = EEF xtF PRINCIPLES OF INTEGRATION. 271 which becomes by integrating E F E_F |S. (w+E)—p 7 log. («+F) or by substituting the values of E and F XMX atr/4 + 6? ——_—. cer a eT log. («ta+V 4a? +23) a—V 4a? + 6 2V/ 4a? +5? (170) In all these cases the factors of the denominator are unequal. If a part or all of them are equal the rule will not apply. For suppose we have Pxt+Ox2+Rxv*?*4+Sae+T (@—a)(x—0)(a—c)(@—d (eZ) which we make Api Bey aC. See cup aa ar + ied, xX—C'X if some of these factors are equal, say a=4=c, we should have : log. («-+a—V 4a? +67) AE Pxt+t etc. evn dein ees oD) i (x—a)?(x—d)(x—c) xa x—a" x—e Thus in reducing the second member to a common de- nominator, A-+B-+C would have to be considered as a sin- gle constant A’, and the three constants A’, D and E would not be sufficient to establish the five equations of condition which are required in making equal the coefficients of the like powers of x. Inorder to avoid this difficulty we decom- pose the original fraction and make Px*+Qx3+ etc. Set Bx+ Cx? D E (x—a)*(x—d)(x—c) ss (x —)8 x—a! x—e which contains the necessary number of constants, and at the same time, when reduced to a common denominator, will produce a numerator containing «x to the fourth power; thus giving a sufficient number of equacdions between the coefficients of the like powers of x, 272 INTEGRAL CALCULUS. In the meantime the expression A+Bx+Cx? \~—a)? may be put into the form A’ B’ G (x—a) (ea) Tea in which A’, B’, C’ are determinate constants. For let x—a=u then x=u-+a and A-+-Ba+Ce* At Ba-+-CUe* - Be s-2Ca7- ae (x—a)> u Cree ise’ a u® hod Py, and replacing the value of wz we have A Bas- Cae A+ Ba Cas ae Sr (x—a)® ——— (x—a) (x—a)? *x—a and since these numerators are constant we may represent them by A’, B’, C’, which gives Asp Bae Cay wa en B’ es (G—4)2) 7 (aa) 3b aye ee which is the required form. As this demonstration may be applied to an expression containing any power of x, we make the proposition a gen- eral one, that Dig = (20012 rare tes sec ca (x—a)™ o A A’ AY (w—a)"™ (~—ayn—t Te ayn® Hence to integrate the expression + etc. __ Pxt+Qx*-betc. (x—a)? Ca) Ie = we write (2s ee) (x—a)?* (xa)? tx—a tad PRINCIPLES OF INTEGRATION. 273 and reduce these fractions to a common denominator and find the values of A, A’, A”, D, E, in the manner already stated. We shall then have to find the integrals of the fol- lowing expressions, E D A’ A’ | A ieee te ee ee ee x—e x—ad x—a~? (x—a)*”~” (x—a) the three first we can integrate by the rule for logarithms and the others as follows. Since dx is the differential of x—a we will make x—a=z ; then we have Adx Adz A A ieee =| oe = JAS ae era ais aye and oe A’dz A’ A’ =e p= {Grey =a Hence f Pxt+Qx3 Pete. A ACO) (x—a)3 EEN ee ~ 2(x—a)® x—a { +A" log. (x—a)+D log. (x—d)+E log. (x—e) (171) Let us take for example Eas 4 x3 —ax*—a*x—a® the denominator of this fracticn may be resolved into the factors | (x? —a?)(x—a)=(x—a)(x+a)(x—a) or (x—a)?(x+a) Making then a A A’ (x—a)? 1)? (aa) (x—a)? oT ea beta CR) and reducing the second member to a common denominator, we have x? A(w+ta) +A'(e®—a*) + B(a—a)? (x—a)*(x+a) (wa)? (w+) 274 INTEGRAL CALCULUS. Developing the numerator and making the coefficients of the same powers of x equal, we have A+B=1, A—2Ba=o, Aa—A’a?+Ba?=o (2) from which we obtain | A=ta, A’=?, B=} hence 6 ch Ch adx 30x 5 py If all the pntae of the denominator are ana the expression will be of the form mlx (a—a)” and we may integrate this more directly. For let x—a=z, then de=dz and x=z-++a, hence pe ree eae ( a= a) " gin Expanding (z+a)”~1! by the binomial theorem, we have flzeices oats Rela 1 ag" "az ) | m gin ———4- etc. each of which terms may be integrated separately as in Art. (160) (162). Let us tak* for example a ae (xa)? Making x—a=z, then x=s+a and dx=d2, we shall have xa (2z-+a) eee eee (x—a)>— 8 age anes 2 and paue 20 eo. 2a a" 2° g = log.z——-— 2 —log. (SE Feo ae epee 5 24 PRINCIPLES OF INTEGRATION. 275 ({73) When two differential functions are equal to each other it does not necessarily follow that their integrals are equal; but if they are not equal their difference will be a constant quantity for all values of the variable. In other words, if two functions have the same rate of change they will either be equal to each other constantly, or else their difference will be constantly the same. Thus the ages of any two persons increase at the same rate, and they will be therefore of the same age, or else the difference of their ages will always bethe same. ‘Two persons traveling in the same direction, at the same rate, will either be constantly together, or else there will be constantly the same distance between them. Hence in integrating the members of a differential equa- tion, it becomes an important part of the problem to ascer- tain if there be any difference between the integrals, and if so, what it is. To do this we add the indeterminate constant C to one member of the integral equation, which shall represent this difference if any. This is called the indefinite integral. Then, since the difference is constant for all values of the variable, we assign to the variable in one member of the equa- tion some value which will correspond to. a 4zown value of the other entire member, and thence obtain the value of C. That value having been substituted in place of C in the integral equation, will satisfy it for every value of the vari- able since it does for ove value. To illustrate these principles, let us take the triangle ABC (Fig. 30). The differential of the surface of this triangle is axdx (Art. 62); a being the tangent of the angle CAB, and A the origin of coordinates. Hence the equation @S =axdx and integrating each member, and adding C to the second, we have 276 INTEGRAL CALCULUS. ax? ge Tae (1) <= Now to determine the value of C we give to x a value cor- responding to a known value of S. But we know that at the origin in A where xo, we have also S=o, and by sub- stituting these values in equation (1) we have o=o+C hence C=o and is the definite integral. If now we wish to know the value of any specific part of the triangle, such as ADD’, we make x=x’=AD, and we have Pat Ge « ADA DD: 2 2 2 This is the specific integral. (174) We are not bound, however, to make the value of S commence at the origin where x=o. We may if we choose estimate it from any line as DD’. In this case (making x=AD=2’') we should have / ro E ax ® . eines TREE rt D whence . ax ® AD a D 3 C=— 2 Og 2 Fig. 30. and substituting this value in equation (1) we have : 2 ied AD 5 =a 2 2 This is again the definite integral. For any portion of the triangle estimated from DD’ we give the corresponding value of x, say x=AE=.", which gives PRINCIPLES OF INTEGRATION, 277 Wiens for the value of the area DD’E’E. (175) There is another method of disposing of the inde- terminate constant, which consists in giving to the variable two definite values, and then subtracting one integral from the other. This is called integrating between limits. Thus ir the case last noted, if we make x successively equal to x’ ==AD and x” =AE, we shall have "9 "9 ax -+G@ and 9.—=—_ -+¢ 2 2 aX Si and subtracting the first equation from the second we have ABA AD? ee ene ome PONT SS ee )=a the constant C having disappeared in the subtraction. The notation for this kind of integration consists in plac- ing the two values of the variable at the extremities of the sign of integration; thus on" AXNQN a! indicates that the integral is to be taken between the two values of x represented by x” and x’; the subtractive one being at the lower extremity of the sign; and the integral — .would be ox"? ax’? 2 2 When the integral is to be taken for any particular value of x, as x’, it would be written { p= CHEK which indicates that the integral is to be taken where x=2’. 278 INTEGRAL CALCULUS. EXAMPLES, fix. 1. . Integrate 2xdx between the values of «=a and x=0. Ans. p*—@? . . b Ex. 2. What is the integral of tf Bx" ax, Ans, 0% ae a d= m4 2 Ex. 3. Integrate ieee Ans. Gla—a?) : | Ex. 4. Integrate i 2(e+x)dx. Ans. 6? +2e(b—a)—a? a b 4,5, autegrate f 3(etnx*)* 2nxdx. = a Ans. (e+nb*)§ —(e+na?)* 0 ax e+b “x, 6. . Integrate i essen Ans. log. (176) INTEGRATION BY SERIES. If it be required to integrate a differential of the form F(x)¢@x in which F(x) can be developed into a series, the approximate integral may often be found by (Art. 164), and if the series is rapidly converging, its true value may be nearly reached. Let the differential be ax Stine =(1+*)-1lax developing by the binomial theorem we have (1+%?)-1=1—x? +4— (218) Zo find the length of an Arc of an Ellipse. We have found (Art. 57) that the differential of an arc of an ellipse is pee TN, A4—(A?—B?)x? au = Tai cae If we take ¢ to represent the distance from the center to the focus of the ellipse we have B2=A2—-2 hence I Ee CAS Guia Ae, gare Ses g s A data bet If now we represent the eccentricity of the ellipse by e we have c=Ae, and hence Bao A4—A2e2 x? du= aN] ee 8 spam 308 INTEGRAL CALCULUS. . or, dividing by A* under the radical and multiplying by A® without it we have giant i LL A? du BN eee A*—x Bee A Developing (1 — A2 ) by the binomial theorem we have vo Bie Pre Bese Bee xe (e= rep, “i gAP an a A Ned oy en ne hence (= Adx e? Max e# xed | | VA? — x? 2A VAP? 2.4A3" A232 gow xO6ax rey ms Sree Making Awe 5 oC ane Rt hoe Aiea Rae V RE GE LAY etc we have 22 es 206 Jdu=Xy— aXe—F gas iw GAB x6 —ete. (1) Now by (Art. 199) X)= the arc of a circle of which A is the radius and ~ the sine, and (Art. 200) 6 Ae eR ee 2 eS Xo Se Rey Be also and If we make x=o and estimate from the extremity of the conjugate axis, we have x=o and C=o. If we make x=A we shall have w= a quadrant, and since MEASUREMENT OF GEOMETRICAL MAGNITUDES. 309 we have A 3A? 3A Se 5k X,=FXo. Seal Se, pet a ae 6x0 and substituting these values in equation (1) we have ro SU Dip hss hat ARE Aine es TS Age yh Saath AAs | ate 4 Oe 6 ete.) for one-fourth of the circumference of an ellipse; X, being one-fourth of the circumference of a circle of which the diameter is equal to the major axis of the ellipse. Hence the whole circumference is equal to A ae Weave sie eee ie Seen ates 8 SNE AN Pet Dees AN Ales Rel aA SAO AO It will be seen that as the eccentricity diminishes the cir- cumference of the ellipse approaches the value. of 2z which it reaches when eo, and the curve becomes a circle. (219) Zo find the length of the Arc of a Cycloid. We have found (Art. 129) that the differential equation of a cycloid is Vy = V ary—y? By substituting this value of @x in @x in the f formula we have du=V dx® +dy? = LONe 2ry—y* or 225: a If we estimate the arc from D (Fig. 55) where y=2r.we shall have w=o and C=o, and hence making y=FG “=D! F =—2V ar (27—y) (1) We see from the figure that D'E’=27 and D’'H’==27—y hence 310 INTEGRAL CALCULUS. V 2r(2r—y)=V D'E’ . D'H’ =D'F’ so that the arc of a cycloid is equal to twice the corrv-_- ponding chord of the generating circle. If we take the arc D’A, the corresponding chord of the generating circle becomes the diameter D’E’, and half the arc of the cycloid is equal to twice the diameter of the gen- erating circle, or the entire arc is equal to four times that diameter. Thus, if we make y=o we have u=4r or D FA=2D' Gand ADB=s bere (220) Zo find the length of the Arc of a Logarithmic Spiral. We have found (Art. 77) that the differential of an arc of a polar curve is du=V ae bar® and the equation of the logarithmic spiral is v-=Log. rv Hence Mar M2ar? 2 a — and av? = Substituting this value of dv? we have du=NV Mar*® + dr? =drV M*? +1 In the Naperian system M=1 and au=arn/ 2 hence “t—ra/ BC If we estimate the arc from the pole where =o we shall have C=o and u=rr/ 2 That is, the length of an arc of a Naperian logarithmic spiral estimated from the pole is equal to the diagonal of a square of which the radius vector is the side. MEASUREMENT OF GEOMETRICAL MAGNITUDES. 3 Us (221) Zo find the length of an Arc of the Spiral of Archimedes. The equation of the spiral (Art. 84) is r=av : : I : : in which a= a and v= the arc of the measuring circle whose 4b radius is the value of ~ after one revolution. Hence 2 Rn Pe AY ED the integral of which may be found in (Art. 210). Substi- tuting 1 for a and v for x, thus Bas ua—a +tlog. (v+V 1+7'?) ) +C Estimating the arc from the pole where v=o we shall have C—oaand . f= -ai eee eaves ee *+log.(v+V 1-+0 )| (222) Zo find the length of an Arc of a Hyperbolic Spiral. The equation of this spiral is (Art. 86) ab rv=ab or See Differentiating we have abv 2 adr = — v whence ab*dv® ab? , g ae OP Ot aN fet du=\f— + ae mNee u=abfv-* avn y? +1 Integrating by formula B (Art. 206); making in the formula and m2 Goat at 312 INTEGRAL CALCULUS. we have Jo? aov v® +1 =—0-'(1 +0°)8-+ 2fao(t +v?) or (Art. 221) =—y-1(1 Busy paises +4 log. (otvV I $2") | hence u=abl—v-1(1 +y2)t +o itv? + log. (v+tV1+v77)|+C Estimating the arc from the point where v=o we have u=ab(t Fao : which is as it should be, since from the equation of the curve, when v=o the radius vector is infinite. As v is infi- nite when =o we shall have w=o at the same time. Hence the curve is unlimited in but one direction. We may, however, find the length of any intermediate portion by sub- stituting the two corresponding values of v in the integral function and taking the difference of the results. (223) Quadrature of Curves. The quadrature of a curve is the process of finding the measure of a plain surface bounded wholly, or in part, by a curve. To find the area of such a surface we must find its dif- ferential in a function of one variable, which being integrated will give an expression for an zzdefinite portion of the area, from which any sfecéfic portion may be obtained by assign- ing corresponding values to the variable. (224) To find the area of a Semt-Parabola. We have (Art. 65) for the differential of the surface of a parabola + AS =ypdx=/ 2px" ax of which the integral is MEASUREMENT OF GEOMETRICAL MAGNITUDES. 313 SHB yx" = hay ype=Fay+C But when x=o we have S=o, and hence Co; so that the surface of a parabola bounded by the curve, the axis and an ordinate is equal to two-thirds of the rectangle described on the ordinate and corresponding abscissa. (225) To find the area of any Parabola. The general equation of the parabola is y” =ax from which we obtain n—1/7, (ee 2 a hence ; ny Cay ee? S El “(n+1)a_ = vtec If we estimate the curve from the origin where S=o we have x=o, and hence C=o, and n Ser awe ye That is, the area of that portion of any parabola, bounded by the curve, the axis and the ordinate, is equal to the rec- tangle described upon the ordinate and corresponding : ee 5 7 ; abscissa, multiplied by the ratio es If 2==2, as in the common parabola, we have S=xy If ~=%, as in the cubic parabola, we have S= xy If z=1, the figure becomes a triangle and we have S=tey or half the base into the height. (226) To find the area of a Circle. We have (Art. 63) for the circle ydx=AxXV/ R2—x? 15 314 INTEGRAL CALCULUS. Making R=1 we have << t ad S=ydx=dxV 1—x* =dx(1—x*)? Developing the binomial and multiplying each term by av we have BERS ONAAS OO AR ee 53 =ax-— goa gran he Reels from which we obtain by integrating each term separately Ere Ore tata 5x9 Di iatp ene ouNGR Tena oe Estimating the area from the center where x=o we have S=o, and, therefore, C=o, so that the series expresses the area of any segment between the ordinate at the center where «=o and the ordinate corresponding to any other value of x. Hence if we make x=1 we have the area of a quadrant equal to eet cere ee it tires weg eee which by taking enough terms may be reduced to -78539 Hence the entire area of the circle will be equal to | 3.14156 equal to z where radius is 1. _ (227) We may also find the area of a circle by consider- ing it as being described by the revolution of the radius about the center. In this case the radius of the circle be- comes the radius vector and we have (Art. 82) where v represents the arc of the measuring circle and R its radius. Integrating the terms of this equation we have, since 7 1s constant, TSRaR Estimating the area from the beginning where S=o we have v=o, and hence C=o, and MEASUREMENT OF GEOMETRICAL MAGNITUDES. 315 r°y Oran (1) is the measure of a sector of a circle of which v=the meas- uring arc. Making R=, we shall have v=the arc of the given circle, and equation (1) becomes rv 5s=— 2 that is, the measure of a sector of a circle is half the pro- duct of the radius into the arc of the sector; and hence for the entire circle, the area is equal to half the product of the radius into the circumference. If we make v= the entire circumference we have v=27R and substituting this value in equation (1) we have rey that is the area of a circle is equal to the square of the radius multiplied by the ratio between the diameter and the circumference. (228) To find the area of an Ellipse. We have in the case of an ellipse (Art. 64) as =yav= Fas a hence S=5 /(At—x\hee Integrating by formula B (Art. 209), and substituting } o for m A® for @ —1 for 2 for 4 for p we have > 316 INTEGRAL CALCULUS. x(A® =a SJ (M2 22 \tdx= ee A (A228) tae but (Art. 178) tks x S (A? —x*) %de= =e ae = =sin. oe hence B ———— SAD Sr ee +"Ssin.-15+C Estimating from the center where x=o we have S=o, and hence C=o. Making then x=A we have | ABS 1 oye Bes S=—"sin.c11=-= - = 2 23g for one-fourth of the area of the ellipse, since the arc whose sine is 1 is equal to one-fourth of the whole circumference; and we have for the area of the entire ellipse =7AB We may also observe that (Art. 63) AX A — x? is the differential of the area of a circle whose radius is A, Pet, ius : hence the area of an ellipse is AX the area of the circum- scribing circle which is zA®; and is, therefore, equal to = =~ .7A*=zAB If A=B the expression oe zA® or zR? for the area of a circle. (229) Zo find the area of a Segment of a Hyperbola. We have inthe case of a hyperbola 1 ee =——“/ ~2—A2 df ‘A “id whence MEASUREMENT OF GEOMETRICAL MAGNITUDES. 317 : Be ees dS=ydx= Aqev x8—AF Integrating by formula € (Art. 209) we have B Sa ee BE: sad a Dit St Ve Ns log. (v9 +V «2 —A?)+C To find the value of C we make x=A where S=o, and we have AB Omar log. A+C hence AB VER pe log. A and gem) A which represents a portion of the area between the curve and the ordinate lying on one side of the axis. Hence the area of the entire segment cut off by the double ordinate is (eats) B ae AB ay AA ert log. ( B joe BO AES AV x? —A®—AB log. but The ab Pal ese V2? a Me —=y hence a S=xy—AB log. (<+5) =xy—AB log. ( for the value of the area. - (230) We may also find the area of that part of the sur- face lying between the curve and the asymptotes, by using the equation of the hyperbola referred to its center and asymptotes, which is bociBie) AB xy=m but as the asymptotes are not usually at right angles to each other, we must introduce into the expression for the differ- ential of this area, the s¢we of the angle which they make 318 INTEGRAL CALCULUS. with each other (Art. 58) which we will call v We shall then have : : max @SSsin. v.ydx=sin. Coe and S=sin.v.m.log.x+C If we call the abscissa of the vertex 1, and estimate from the corresponding ordinate, we shall have at that point m=1, S=o, log. x=o and hence C=o And since sin. v may be considered as the modulus of a sys- tem of logarithms, we may make S=M. log. x=Log. x That is, the area between the curve and the asymptote, estimated from the ordinate of the vertex, is equal to the logarithm of the abscissa, taken in a system whose modulus is the sine of the angle made by the asymptotes with each other. (231) To find the area of a Cycloid. We have (Art. 129) ad = = 5 V 2ry—y hence 4 ads = ye =F V ary—y? Integrating this by formula ¢c (Art. 202) we have S=3r. ver. nt oe 2ry—y*® +C 2 Estimating the integral from A (Fig. 76) where y=o, we have S=o and hence C=o and taking the integral where y=2r=DE we have are S=8r. ver. sin.-127=3 2 MEASUREMENT OF GEOMETRICAL MAGNITUDES. 319 that is, the area ADE is equal to three times the semi-circle DF’E. Hence:the entire area of the @-_6 D cycloid is equal to three times the area of the generating circle. p78 (232) Another method of obtaining the area of a cycloid is, to consider that { portion of the rectangle ACDE which Fig. 76. lies outside the curve. If we make GF =27—y=v7 we shall have the differential of the area DCAF equal to wdx or 20S. = =ayV 2ry—y? 2ry—y* V ary—y Now if we take the equation of a circle with the origin at the extremity of the diameter we shall have yae= aden yeaah which is the differential of the segment of a circle of which xis the abscissa. Hence dy 27y—y? 1s the differential of a circle of which y is the abscissa, that is of the segment F’BE. Hence the two areas ACGF and F’BE have the same rate of change or differential for the same value of y ; and since they are both equal to zero when y=o, they are equal for every other value of y (Art. 173), and, of course, when y=27. Hence ACD =Dh Hees Lm aS! =(ar—y)dx=(2r—-y) 2 But the rectangle ACDE=7;7 .2r=277?, hence = re ADE=ACDE—ACD— as we found in (Art. 231). | (233) Zo find the area bounded by the coordinate axes and the logarithmic curve. We have had (Art. 137) for the logarithmic curve May x=Log. y and dk=—— 5 320 INTEGRAL CALCULUS. hence @S=ydx=Mady and S=My+C If we estimate the area from AD (Fig. 56) where y=1 we have o=M+C whence C=—-M and S=M(y—1) If we make y=o we have S=—M=area ADD’ If y=z=RO we have S=M=area ADRO So that although the axis of abscissas 1s an asymptote (Art. 88) to the curve on the negative side, and, therefore, will not meet it within a finite distance, yet the area erclosed be- tween them is limited and equal to ADRO. (234) If we take the curve represented by the equation I DLae to which the axes of coordinates are asymptotes, we shall find a case somewhat similar. Putting the equation into the form ©, » 5 we have a ik h== tee y and @S=yde=—~ == — AA ye hence MEASUREMENT OF GEOMETRICAL MAGNITUDES. 321 If we estimate the area from the line AC where y= o we have o=o0+C hence C=onand 4 cy, If we make y=1=FT we have S=2=ATDC that is, the area ATDC is equal to twice the square AHFT, and is, therefore, finite, although the curve FD does not meet the axis AC at a finite distance. If we take the area between the limits y=1 and y=o, we shall have S=2—2 that is, the area FEBT is infinite, although AB is likewise an asymptote to the curve. (235) Zo find the area described by the radius vector of the Spiral of Archimedes, - The differential of the area of a polar curve (Art. 82) is r°av 2R v being the arc of the fheasuring circle and R its radius. The equation of the Spiral of Archimedes (Art. 84) is r=—av hence ar=ado or ar ij eet a, a Hence, making R=1 we have ready re-dr s=f Fas ml ae If we make r=o we have 21 322 INTEGRAL CALCULUS, S=o and hence C=o : r and since as, we have fae 6 Making v=2z we have for the area described by one revo- lution of the radius vector mr? S that is, the area described by one revolution of the radius vector is one-third of the area of the circle described with a radius equal to the last value of the radius vector. If the radius vector make two revolutions we have v=47 and re 2rr Se : 3 where 7 =27 and 8zr? See % But in making two revolutions, the radius vector describes the first part of the area twice. This, therefore, must be subtracted, and we have the area enclosed by the curve and radius vector after two revolutions equal to Sap2 Age? =I gy? 3 3 3 and by subtracting the first again we have the zzcreased area described during the second revolution equal to tar® —trr*® Sarr? After m revolutions we have mr * 3 where 7 =mr, hence mrm2r® mrxr® ‘7 5 ae (7) Subtracting from this the area desgribed by the radius vec- tor during (#—1) revolutions we have MEASUREMENT OF GEOMETRICAL MAGNITUDES. 323 LS a + (m3 —(m—1)°) (2) Substituting (#+1) in place of # we have : ((m+1)%—m?) (3) for the area described by the radius vector during the (m-+1)c revolution. Taking the difference between equa- tions (2) and (3) we have the additional area described by the (m-+1)éé revolution of the radius vector, that is the area lying between the mh and (m+1)é& spires thus = ce — 118 ood ((m+1)?—2m? +(m— )3)==2m77 > eee 3 ~ We have found the additional area described by the radius vector during the second revolution equal to 27r*, hence the additional area described during the (#-+1)é/ revolution is equal to # times that described by the radius vector during the second revolution. That is, the zzcrease of the additional areas described by the radius vector during successive revo- lutions, is wazform and equal to twice the area of the circle described with a radius equal to the radius vector after one revolution. If the area ABP (Fig. 34) be required, that is, the add- tional area corresponding to the arc BC described after the first revolution, we shall have 27 pate and ; r r ele. and the required area will be Tv uns ree ree Sr Aas or 324 INTEGRAL CALCULUS. wa(n1) 7" (eae) ee rt sy gee TS 2 a = Shahi Developing (z+1)® we have tre \ 7 Steir Us +32 een or If BCD=+ circumference oes then ee and Tv ABP=77?(1 +4+ 4) (236) Zo find the area of the surface described by the radius vector of the Hyperbolic Spiral. The equation of the hyperbolic spiral (Art. 86) is ab rv=ab or Pane in which a is the radius of the measuring circle and is the unit of the measuring arc. Hence s=/[— “dv = {2 @. ab? ry yi ae which is infinite when v=o, and zero Pie yv—o. If we make v=S=AB (Fig. 38), and v=46=As, we shall have nO ee C0 eet Dies S=a——=—= 2, 2 2 (237) Zo find the area described by the radius vector of a Logarithmic Spiral. We have (Art. 87) for this spiral v=Log. 7 MEASUREMENT OF GEOMETRICAL MAGNITUDES. 325 and Mar dv=— 7 Substituting this value in the formula, and making R=1 and M=1, we have LV EEREES V Sos = 2 Re eh Vina eieriaess If we estimate from the pole where So we have r=o, and hence Co, and 9 ye 4 That is, the area described by the radius vector of the Nape- aian logarithmic spiral is equal to one-fourth of the square described upon the last value of the radius vector. = (238) Areas of Surfaces of Revolution. We have seen (Art. 66) that the differential of a surface of revolution, where the axis of revolution is the axis of abscissas, is S=/2tyvV/ dx? +dy* the radical part being the differential of an arc of the re- volving curve. To apply. this formula to a particular case we must obtain from the equation of the revolving curve, the value of one variable and its differential in terms of the other, so that, when substituted in the formula we may have the differen- tial of the surface in terms of one variable which can then be integrated. (239) To find the convex surface of a Cone. We have (Art. 67) for the differential of the convex sur- face of a cone 2S =27axdxn/ g? +4 in which « is the length of the axis and @ is the tangent ot 326 INTEGRAL CALCULUS. the angle made by the revolving element of the cone with the axis of revolution. Integrating we have S=rax*V/q=+14+C Estimating from the vertex where S=o we have x=o, and hence C=o and S=rax*/gr+1 But from the equation of the generating line we have y=ax and hence 2 9 es a=" 3 Borer S=zyx, 5 =ayV x? + or (Fig. 35) making x=AB S=rCDV AB? +CD? that is, the convex surface of a cone is equal to the circum- ference of the base multiplied by half the slant height. (240) For the convex surface of a cylinder we have y=R=radius of the base hence S=/anyV dx? +dy*? =fenRdx=27Rx that is, the convex surface of a cylinder is equal to the cir- cumference of its base into its altitude (241) In the case of the sphere we have (Art. 68) @S=27Rdx hence S=27Rx«+C Estimating from the center where x=o we have S=o and hence C=o, and the measure of an indefinite portion of the convex surface of a sphere is S=e2zRx the same as that of the circumscribing cylinder having the same altitude. MEASUREMENT OF GEOMETRICAL MAGNITUDES, 327 Making «=R we have S=27R? for the measure of the convex surface of half the sphere ; hence for the entire sphere we have S=47R* or four great circles. (242) Zo find the surface of an Ellipsoid described by an ellipse revolving about tts major axts. Making V dx? + dy? =u we have anyV dx® +dy? =27ylu But we have found (Art. 218) Adx Cre ee ea" 3e°x8 ft agg AT ep tie 6As ete.) hence | az Ayax eae eet Bit A ee eg as hale Hy Nee a ie Auer eo Ata) But Ew iy ohn VA2— x? hence e* x" e4x4 3e% x6 A Fae 22 © Sanaa mes ag Aa iach, Integrating each term separately we have Ayo te Bae Ceae peg sae ie icing Wee ey ge Rn ay eee ae Taking the integral between the limits x=o and «=A we have for half the surface of the ellipsoid ae gh 3e4 oeeh a(t a pememaan eee, 7 etc.) 328 INTEGRAL CALCULUS. and multiplying this expression by 2 we have the measure of the entire surface of the ellipsoid. If we make A=B, then ¢e=o, and we have for the surface of the sphere S=47k? as before. (243) Zo find the surface of a Paraboloid of revolution. We have (Art. 69) in the case of the paraboloid as=anyy/2 TE ay? =F 4 p2yt Integrating according to (Art. 165) we have S=F (y?-+p*)F+C Estimating from the vertex where S=o and y=o we have 3 == = Spree Post hence amp* 3 C=— and Gedy ear OY ace (244) Zo find the area of the surface described by a Cycloid revolving about tts base. We have (Art. 129) in the case of the cycloid V ary—y? hence V dx + ay? =\/24 = ary ate =O 53 MEASUREMENT OF GEOMETRICAL MAGNITUDES. 329 and by substitution aS=anyV dx? +dy* Sanya / 22 ere or A S=erV ar {== V ary—y? But we have found (Art. 205) yay 8r 2y ——_—_ Fy ees een: _e Dig cas 3 2r—y 3 2r—~y hence eS, 2y ,——. S=27 ee aS If we estimate the surface from the plane passing through the middle point of the base we shall have S=o when y=2r, hence C=o. Then making y=o we have for half the sur- face required ee S=orV rales —V 2r)=32nr2 and for the entire surrace sey that quantity. That is, the area of the surface described by the revolution of a cycloid about its base is equal to twenty-one and one-third times that of the generating circle. (245) The Cubature of Solids. The cubature of a solid ts to find the dimensions of an equiv- alent cube or other known volume. We have (Art. 71) ty® ax for the differential of a solid of revolution where y is the ordinate and x the abscissa of the bounding line of the revolving surface which generates the solid; and the axis of abscissas is the axis of revolution. Hence V=/xy*dx 330 INTEGRAL CALCULUS. To apply the formula to any particular solid or volume, we eliminate one of the variables by means of the equation of the bounding curve, thus producing a differential function of one variable which may be integrated. (246) To find the volume of a Right Cone. Making the vertex the origin we have y=ax for the equation of the bounding line; but a is the tangent of the angle made by this line with the axis of the cone, and : b is equal to 7 where #4 is the radius of the base and % the the length of the axis; hence b ay ya7x and y =Fa Xe whence b* OF eee V=/ry*de= Ty aaa =A SOCES Estimating from the origin we have V=o, x=o, and hence C=o, and making x= we have for the entire cone h V=76?— 3 that is, the volume of a cone is equal to one-third of the product of its base by its altitude, or equal to one-third of a cylinder of the same base and altitude. (247) To find the volume of a Sphere. From the equation of the circle we have y2=R2— 2? hence oe Vim fry dx) 7 RE 8) ae ae Bec) EC Estimating from the plane passed through the center, where MEASUREMENT OF GEOMETRICAL MAGNITUDES. 43% x=o, we have S=o and C=o, and making «=R we have for half the volume of the sphere V=&7R3 and for the entire sphere V=4r7R3 Since the surface of the sphere is equal to 47R® we have the volume equal to the surface multiplied by one-third of the radius. (248) To jind the volume of an Ellipsoid. Taking the origin at the extremity of the transverse axis we have for the equation of the bounding line or curve B2 y* a (2Ax—x*) hence Va Sayama f 25(2hx—a? dene g(a? — =)+c Estimating from the origin where xo we have she and hence C=o; = making x=2A we have Vare5(4A2— $A3)=74B2A=87B? . 2A that is, the volume of a prolate ellipsoid is equal to two- thirds of the volume of a cylinder having the minor axis for its diameter, and whose altitude is equal to the major axis. If the ellipse is made to revolve about its minor axis we should have V' =74A?B for the volume of an oblate ellipsoid, and hence V: V'::7$B?A: c4A2B:: BA that is, the volume of a frolate ellipsoid is to that of an oblate ellipsoid generated by the same ellipse, as the minor axis is to the major axis. 332 viel INTEGRAL CALCULUS. If A=B we have as before. (249) To find the volume of a Paraboloid. In this case we have y” =2px and 2 V=/ry*de= onp/xdx=a27p— =rpx? Estimating from the vertex where «=o we have V=o, and hence Co, and pace V=7px* = i or, the volume of a paraboloid is equal to half the volume of the circumscribing cylinder. (250) Zo find the volume of a Soltd described by the revolu- tion of a cycloid about its base. Since in the case of the cycloid we have X,=Xo—V ary—y? X, = are of a circle of which ~ is radius and y the versed sine. MEASUREMENT OF GEOMETRICAL MAGNITUDES. 333 Integrating between the limits y=o and y=2r7 we shall have half the volume required; but y=o gives V=o and C=o and y—2r7 gives Gras X,=X,)=7,r easy LES X_= 2 X= 2 and sr 2 7278 hence the entire volume is V=52"78 But the volume of the circumscribing cylinder is 4rr® , amr=8r*73 Hence the volume of the solid generated by the revolution of a cycloid about its base is five-eighths of that of the cir- cumscribing cylinder, Albee HN Dex, GEOMETRICAL FLUXIONS. It has been said that the “ reductio ad absurdum” or method of exhaustion of the ancient mathematicians contains the germ of the differential calculus. This is an error. There is nothing in that method that has any affinity to the true principle of the calculus. ‘The method of rates, in the sim- ple and obvious meaning of the term, is as remote as possi- ble from the method of exhaustion. Its demonstrations are direct, logical and conclusive. No absurd hypothesis are admissible, and therefore no ‘‘ reductio ad absurdum.” There is neither exhaustion nor limits, nor any idea to which these methods have any sort of affiliation. Moreover the true principles of the calculus are so sim- ple and so easily applied that if they had occurred to these men they could at once have seized and used them without the aid of Algebra, and thus have avoided the “tedious and operose reductio ad absurdum” altogether. The principles of this science have been so exclusively associated with the forms of analysis, that it has come to be considered as purely analytical in its character. This is indicated by the term “calculus” itself, as well as by the terms ‘transcendental analysis” and “ calculusof functions.” But the truth is these 15 = 337 338 APPENDIX. principles are wholly independent of analysis, and may be applied as directly to the geometry of Euclid and Archi- medes as to that of Descartes. Those propositions that require the “ zedious reductio ad absurdum,” or the absurd method of the infinitely sided polygon, may be easily and directly solved by them without resorting to the abstractions of analysis. | | These principles are contained in the method of rates, which is in fact their development. As applied to geometry they are two, viz. : first. The rate of increase of any geometrical magnitude, while being generated, may be measured by a supposttive increment arising from the uniform movement of the generatrix, at the existing rate, during a unit of time, in the direction to which tt may be then tending; and, therefore, such supposttive increase may be taken as a symbol to represent that rate. Second. Lf two magnitudes begin to be at the same momeni, and the ratio of thetr rates of increase ts constant, the ratio of the magnitudes themselves will be constantly the same as that of their rates. Thus if two persons set out at the same moment and place to travel in the same direction, at constant rates, the ratio of the distances traveled by each will constantly be the same as that of their rates of travel. If one travel twice as fast as the other he will always be twice as far from the starting point. Now to apply these principles to the measurement of geo- metrical magnitudes. PROPOSITION I. (252) Zo find the arca of a Cireéle. We will suppose the circle to be generated by the revolu- APPENDIX. 339 tion of radius CA about the center Cat a uniform rate. When the radius is in the position CA, and revolving toward B, every | point in it will zezd to move in a direction perpendicular to CA, and hence the point A will zexd to describe the line AB tangent Fig. 78. to the circle and perpendicular to CA; and if left to its ten- dency zwould describe that line at a uniform rate. The line, therefore, may be taken as the symbol representing the rate of increase or generation of the circumference of the circle. But while the point A tends to move in the direction AB, every point in the radius CA éends to move in.a direction parallel to it, and at rates proportional to their distances from the center C. Hence the radius itself, ¢f left to tts ten- dency when at CA, would be found at CB, when the point A is at B; and the triangle CAB would be generated at a uni- form rate during the same time that the line AB is generated. The triangle may, therefore, be taken as the symbol of the rate at which the area of the circle is generated, and the ratio of these symbols is also the ratio of the rates which they represent. But the triangle is equal to $CA. AB, that is the ratio between the rate of generation of the circumfer- ence and that of the area of the circle is half radius; and this being constant is the ratio between any part of the cir- cumference and the corresponding part of the circle through- out their generation, and, of course, when it is completed. Hence the area of the circle is equal to half the radius into the circumference. Proposition II. (253) Zo find the area of the convex surface of a Cone. Suppose the cone to be generated by the revolution of the triangle ADC (Fig. 79) about the axis DC, The hypothe- “340 APPENDIX. neese AD will generate the convex surface, and the point A will gener- ate the circumference of the base. When the triangle is in the position ADC and revolving towards E, the point A if left to its zendency at that instant would describe the line AE, perpendicular to AC, in some unit of time, and hence AE may be taken to Hig. 79- represent the raze at which the circumference of the base is generated. Now every point in the line AD tends to move in a direction parallel to AE, and at a rate proportional to its distance from the axis DC; hence if left to that tendency the line AD would describe the triangle ADE, and be found at DE in the same unit of time. Hence ADE (Art. 251) may be taken to represent the corresponding rate of generation of the convex surface of the cone. But ADE AD ; ADE=AE.3AD or —\y =", » that is, the ratio between the rates of generation of the convex surface of the cone and the circumference of its base is constant and equal to half its slant height. Hence the ratio between the magni- tudes generated will be the same (Art. 251), and their con- vex surface divided by the circumference of the base equals half the slant height, or, the convex surface equals the cir- cumference of the base multiplied by half the slant height. Proposition III. (254) Zo find the measure of the volume of a Cone. The cone being supposed to be generated by the revolu- tion of a right angled triangle about one of its sides, which becomes the axis of the cone, while the base is generated by the other side as its radius; let us suppose the generating triangle to have arrived at the position ADC (Fig. 79), the APPENDIX. 341 point A moving towards C. Then every point in the trian- gle zends to move in a direction perpendicular to its plane and at a rate proportional to its distance from the axis CD; so that if AE is taken to represent the line that would be described by the point A in a unit of time in consequence of that ¢endency, than at the end of the same unit of time the line AD would be found at ED, and the triangle ADC, at EDC, so that the pyramid DAEC would be the volume generated by the triangle, during the same unit of time and may therefore (Art. 251) be taken to represent the raz at which the cone is generated; while at the same time the triangle ACE would be described by the radius AC of the base and would, therefore, represent the rate at which the base of the cone was generated. But the volume of the pyramid DAEC is equal to its base ACE multiplied by one- third of its altitude DC. Hence the rate of generation of the cone divided by that of its base =a constant quantity. Therefore (Art. 251) the cone itself divided by its base is equal to the same quantity being one-third of its height — or the volume of the cone is equal to its base multiplied by one-third of its height. ProposiTion IV, (255) Zo find the area of the surface of a Sphere. Suppose the sphere to be generated by the revolution of the semicircle CBD (Fig. 80) about the diameter CD. Then the semi-circumference CBD will generate the surface of the sphere. Now every point in the curve CBD tends to move in a direction perpen- A dicular to its plane at a rate propor- tional to its distance from CD, the axis of revolution, and under this tendency it would in some unit of time 342 APPENDIX. assume the position of the semi-ellipse CED, generating at the same time the convex surface of the ungula CEDB, which is, therefore, the symbol (Art. 251) representing the rate of generation of the surface of the sphere, while the line EB described by the point B, in the middle of the CBD, and perpendicular to its plane is the symbol representing the cor- responding rate of generation of the circumference of a great circle. But the convex surface of ungula is equal toits ex- treme height EB, multiplied by the diameter CD, which is, therefore, the constant ratio between the rates of generation of the surface of the sphere and of the circumference of its great circle. The magnitudes themselves are, therefore (Art. 251), in the same ratio, and the surface of a sphere is equal to its diameter multiplied by the circumference of its great circle. PROPOSITION V. (256) Zo find the measure of the volume of a Sphere. The sphere being supposed to be generated by the revo- lution of the semicircle CBD about the diameter CD (Fig. 80), when it is revolving towards the point E, every point in it will zeza to move in a direction perpendicular to its plane, and at a rate proportional to its distance from CD the axis of revolution; and the point B in the middle of the arc CBD will tend to describe the line BE perpendicular to the plane of CBD, in some unit of time; and would do soif left to that tendency. The semicircle CBD, at the end of the same unit of time, would be found in the ellipse CED, having described or generated the ungula ECBD, which may, therefore (Art. 251), be taken asthe symbol! of the rate at which the volume of the sphere is generated (Art. 251), while EB is the symbol of the rate at which the circumfer- ence of its great circle is generated. But the volume of the APPENDIX. 343 ungula is equal to its extreme height EB multiplied by two- thirds of the square of the radius, which is, therefore, the’ ratio between the rates of generation of the volume of the sphere and of the circumference of its great circle. Hence the magnitudes themselves are in the same ratio (Art. 251), and the volume of the sphere is equal to the circumference of its great circle multiplied by two-thirds of the square of its radius — or the circumference, multiplied by the diameter (equal to the surface), multiplied by one-third of the radius. UNIVERSITY OF ILLINOIS-URBANA a = 515B854E C001 | me ELEMENTS OF THE DIFFERENTIAL AND INTEGRA fin