•1lii- «. yH^cAoAtyUlJ. ^j^AJ: LIBRARY UNIVERSITY OF CALIFORNIA. Received CTl^y^^ . i8g6^ ^Accessions ^o.O 0//0 . Class No. if CO -.J ;t) ..J AN ELEMENTAEY TEEATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS. CambrftfQc : Prtnteti at t|je SSntfacrsitg ^rcss. FOR MACMILLAN AND CO. aonOon: GEORGE BELL. IBublin: HODGES AND SMITH. ©Hinbutgf) : EDMONSTON AND DOUGLAS. ©lasfloto: JAMES MACLEHOSE. ©ifOtB: J. H. PARKER. AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS, FOR THE USE OF COLLEGES AND SCHOOLS. BY G. W. HEMMING, M.A., FELLOW OF ST. JOHN's COLLEGE, CAMBKIDQE. SECOND EDITION, WITH CORRECTIONS AND ADDITIONS. CambritiQe : MACMILLAN & CO. 1852. >* Of THB wUHIVBRSITTj *^g/"^-^ zing library 5T7/3 PREFACE TO THE SECOND EDITION. Many corrections and additions have been made in the present Edition, chiefly with the view of fitting it for younger students by bringing out the leading principles of the subject with greater clearness. Some additional illustrative examples are given, but for purposes of practice the student is still referred to Gregory's Collection of Examples. Lately Published, hy the same Author, 12mo. cloth, 45. Qd. First Book on Plane Trigonometry, comprising Geometrical Tri- gonometry, and its Application to Surveying, with numerous Examples for the use of Schools. ijITIVEESIT CONTENTS. CHAPTER I. Variables and Functions. Article Page 1 — 7. Definitions of variables and functions . . . . 1 8. Equations of one or more independent variables . . 7 CHAPTER H. Limiting Values of Functions. 9. Definition and explanation of a limiting value . , .8 10. Axioms respecting limiting values . . . ii arc PQ ''' ^'^^"cho^dPQ^^ 11 12. /^.=o^" = l 12 X x" — 1 13. lt^=o- — T=« ....... 12 X - I 1 14. lt,=^{l + xf=e 13 ^^' '-^ X loga 1^ 16. /^^o— r— = logfl ...... 14 CHAPTER in. Differentiation. 17. Definition of differentials and differential coefficients . .15 18. Differentials proportional to the rates of increase of the variables 17 1 9. Geometrical illustration of the definition . . .17 20 — 44. Differentiation of elementar}' functions . . . 19 45 — 51. Differentiation of a function of a function . . .27 52. Examples ....... 30 53. Summary of results . . . . . .31 Vlll CONTENTS. CHAPTER IV. Integration. Article Page 54. Definition of integration . . . . .34 56, 57. Addition of a constant in integration ... 35 58 — 65. Integration of elementary functions . . .36 66. Summary of results ..... 40 67. Integration by algebraical transformations . . .41 68, 69. Integration by parts ..... 42 70 — 76. Integration of rational fractions . . . .43 78. Rationalization ...... 49 79,80. Criteria of integrability of a;" (a + ii;")^ c?a; . . .49 81 — 87. Integration by redviction . . . . 51 88. Exceptional cases . . . . . .56 89. Integration of sin"' x cos" xdx in particvilar cases . 57 90, 91. Integration of the same fimction by reduction • . 57 CHAPTER V. Successive Differentiation. 92. Definition of an independent variable . . .60 93. Geometrical illustration . . . . . 61 94, 95. Successive diflferentiation . . . . .62 96 — 98. Relations between successive differentials and differential coeffi- cients, when the independent ^'ariable is general . 63 99. Form of the above relations when the quantity (a?), under the functional sign, is independent variable . - 64 100. Examples ..... 65 101 — 104. Formation of differential equations . . 67 105. Homogeneity of differential equations ... 70 106. Change of independent variable . . . . 70 107. To pass from an equation among differentials, with x as inde- pendent variable, to one among differential coefficients with respect to x, and the converse ... 70 108, 109. To pass from a general independent variable to a', and the converse 7] 110. To pass from a general independent variable to any function of a? 71 111 — 113. To change the independent variable from x to any function ofx 72 114.. To change the independent variable from x to y . 74 115, 116. Illustrations and examples ..... 74 CHAPTER VI. Differentiation of Functions of several Variables and Implicit Functions. 117, 118. The total differential of a function of two connected variables equal to the sum of the partial differentials . . 78 CONTENTS. IX Article Page 119—121. Examples ....... 80 122. Extension to functions of any number of variables . 82 123 — 125. Implicit functions of one independent vaiiable . . 82 126 — 128. Relations between corresponding implicit and explicit functions 84 129. Examples . . . , . . .86 130 — 132. Definition of the total differential of a function of two inde- pendent variables . . . . . 87 133, 134. Another form of the definition . . . .92 135. Geometrical illustration ..... 94 136. Successive total and partial differentiation . . .95 137. Order of successive partial differentiation immaterial . 96 138. Notation for successive differentials and differential coefficients 97 139 — 142. Relations between successive total and partial differentials and differential coefficients, with general independent variables 99 143. The same relations, with particular independent variables 101 144 — 146. Change of independent variable in equations involving partial differential coefficients ..... 101 CHAPTER VII. Development of Functions. 147 — 149. Taylor's theorem. Limits of the remainder. Examples 104 150, 151. Stirling's theorem. Examples . . . .107 152, 153. Failure of Taylor's theorem .... 108 154. Other methods of expansion . . . .110 155. Extension of Taylor's theorem to functions of two variables 111 15fi. d''f(^x,y) _ d'^f{x,y) ^^^ dx'^'dy' dy'dx^~' 157. Limits of remainder of the above series . . . 112 158,159. Other forms of Taylor's theorem . . . 113 160. Lagrange's theorem . . . . .113 161. Laplace's theorem . . . . . 115 162. Laplace's theorem deduced from Lagrange's . . 116 Maxima and Minima CHAPTER VIII. Limiting values of Indeterminate Functions, values of Functions. 163. Fractions of the form J; . . . . 117 164,165. Fractions of the form -§■ . . . . .117 166. Other indeteraiinate forms . . , . 119 167 — 1 71 . Maxima and Minima of Functions of one independent variable. Geometrical illustration . . . . .119 172 — 177. Maxima and Minima of Functions of two or more independent variables. Geometrical illustration . . . 123 178. Examples . . . . . . .129 CONTENTS. CHAPTER IX. Tangents and Asymptotes. Article Page 179. Definition of a tangent .... 133 BO, 181. Equations to tangent and normal, dx^ + dy'^ = ds^ 133 182. Length of tangent and normal 135 183. Concavity and convexity of cm^^es 135 184. Points of inflexion ..... 136 185. Asymptotes ...... 136 186. Examples ...... 137 187. Inclination of tangent to radius vector, dr- + r'dd' = ds^ . 140 188. Values of ^T and SF ..... 141 189. Asymptotes of polar curves .... 143 190. Concavity and convexity of polar curves. Points of inflexion 143 CHAPTER X. Contact of Curves. 191, 192. Definition of contact. Curves whose contact is of a higher order, indefinitely closer than those whose contact is of a lower order ...... 144 193. Curves whose contact is of an even order cut, otherwise not 145 194. Conditions of contact modified . . . 145 195. Curve of closest contact . . . . .146 196 Circle of curvature . . . . . 147 197. Radius of curvature and coordinates of centre of curvature 147 198,199. Various forms of these expressions . . . 149 200. Circle of curvature has a contact of third order when the radius is a maximum or minimum . . , .151 201. Equation to evolute . . . . . 152 202,203. Properties of evolute . . . . .153 204. Evolute of polar curves . . . . 156 CHAPTER XI Singidar Points. 205. Definitions . 206—208. 209. 210. 211. 212. Multiple points Cusps . . . . Conjugate points Other singular points Summary. Example . 157 157 161 162 164 164 CONTENTS. XI Article 213, 214. 215. 216. Explicit Curves Implicit Curves Polar Curves CHAPTER XII. Tracing of Curves. Page Kit) 167 170 CHAPTER XIII. Envelopes — Differentials of Areas, S)-e. 217,218. Envelope of a group of curves . 219. Ultimate intersection of contiguous normals dr 2^-0- ^ = 'Tp 221. Differential of an area 222. Differential of a sectorial area 223. Differential of the volume of a solid of revolution 224. Differential of the surface of a solid of revolution 172 175 175 176 176 177 177 CHAPTER XIV. Integration between Limits. Successive Integration. 225. Integral the limit of the sum of a series 226. Area of a curve .... 227- Length of a cun^e 228. Sectorial area of a curve 229. Volmne and surface of a solid of revolution 230. The same results differently obtained 231. Further application of Art. 225 232 — 238. Successive integration. Examples 239. Integral expressed by a series . 240. Remainder of Taylor's series as a definite integral 178 180 181 182 182 182 184 185 192 ral . 192 DIFFERENTIAL AND INTEGRAL CALCULUS. CHAPTER L VARIABLES AND FUNCTIONS. 1. In all analytical inquiries the symbols with which we are concerned are divisible into two classes — those which by the con- ditions of the problem admit of variation, and those which are in- capable of it. The former are called variables, the latter constants. Thus, the co-ordinates of any point of a given curve are variables, being capable of a change of value in passing from one point of the curve to another, while those quantities which enter into the equa- tion to determine the particular curve of which we treat, as the latus rectum of a parabola, the semi-axes of an ellipse, the radius of a circle, &c., are constants so long as the equation is restricted to one particular curve. It should however be borne in mind, that the same quantities may appear in one problem as variables and in another as constants. Thus, if X and 7/ represent the co-ordinates of any point in a given circle, they are connected by the equation where x and 1/ are variables and r constant. But if we have to determine the locus of the intersection of pairs of equal circles, having their centres in two fixed points whose distance is 2b, the equations to any pair of circles whose radii equal r may be put in the form (x - by + ?/' = r^ (x + by + y^ = r'. The equation to the required locus will then be a relation between X and r/, which holds for every possible value of r and will be found from the above equations by treating r as a variable. The distance 2i, between the centres is the same for every pair of circles. In this problem therefore x, y and r are variables and h constant. By H. D. c. \ Z VARIABLES AND F€NCTIO.NS. eliminating r between them we find the equation to the locus required, viz. j: = 0. Hence we obtain the following definitions. A variable is a quantity which, by the conditions of the problem that we are investigating, may assume a number of different values. A constant is a quantity which the conditions of the problem suppose to be invariable. Quantities, which in some problems appear as variables and in others as constants, as the radius of the circle in the above example, are sometimes termed parameters. 2. A continuous variable is one which is capable of receiving any value whatever ; a discontinuous variable, one which only ad- mits of a particular series of values separated by finite intervals. Thus, the co-ordinates of an indefinite straight line are continuous variables, while a symbol which represents any integer is a discon- tinuous variable. In the ordinary operations of Algebra and in the present subject, all the variables employed are supposed to be continuous. Results, however, obtained on the supposition of continuity are applicable to variables which are continuous within certain finite limits, and become discontinuous beyond them. Thus, the general equation to a straight line is applicable to the side of a triangle, provided we restrict the values of the co-ordinates to the limits imposed by the figure. Variables are usually represented by the final letters of the alphabet, as u, w, x, y, z, &c., and constants by the initial letters a, h, c, &c. ; a notation purely arbitrary, and adopted for convenience. 3. A function of any variable is a quantity whose value depends upon and changes with that of the variable, so that when an arbi- trary value is given to the vai-iable the function receives a definite corresponding value. Thus, the distance which a man walks in a given time is a func- tion of his rate of walking ; and the algebraical expressions x"", a", a + bx", sin x, &c., are all functions of x, and x a function of each of them. So also, the co-ordinates of a curve are mutually functions of each other, since when an arbitrary value is given to one of them, the other receives a definite corresponding value. VARIABLKrf AND i-UiXCTIOXS. 3 Functions of the variables x, y, &c. are writteny(j:), f{y), (p (t), When the same functional sign is prefixed to different variables, it denotes that the function is obtained from the variables in the same way. Thus, if /(x) = sin x, f{z), if it occurs in the same in- vestigation, must = sin 2. 4. A function of a continuous variable is said to be continuous when its values are obtained from those of the variable hy a Jixed law, such that a gradual change in the variable produces a corre- sponding gradual change in the function, otherwise it is discon- tinuous. The meaning of these two conditions of continuity becomes more apparent by the aid of a geometrical illustration. Suppose a curve to be traced whose equation is y =zf(x). Such a curve must always exist whatever be the form ofy, since we can fix as many points of it as we please by giving a series of arbitrary values to x, and determining from the equation the corresponding values o^ y. In order that the ordinate of this curve may be a con- tinuous function of the abscissa, according to the above definition, two conditions must be satisfied : (1) The change of j/ consequent on a gradual change of x, must be gradual. (2) It must follow a fixed law. The former of these conditions excludes the case of a series of disconnected points, or of lines terminating abruptly, and restricts us to curves which may be traced without raising the pen from the paper. Where however y has several values corresponding to each value oi X, the condition is satisfied if each branch may be thus con- tinuously drawn. The latter condition excludes from the class of continuous curves, figures composed partly of one curve and partly of another, as for instance a triangle, the co-ordinates of whose several sides are related by three distinct laws or equations. It is even possible to trace curves which satisfy the geometrical idea of continuity whose ordinates are nevertheless discontinuous functions of the abscissae in the sense of our analytical definition. A curve is considered continuous in geometry where it can be traced by the motion of a point, without any abrupt changes of direction (Newt. Prin. Bk. i. Lem. 5). A triangle is therefore geo- 1—2 '4 VARIABLES AND FUNCTIONS. metrically discontinuous, but the egg-shaped curve formed by con- necting the extremities of a semi-circle with those of a semi-ellipse of equal diameter satisfies the geometrical definition. The equation to the latter figure however does not give ?/ as a continuous function of X, since for some values of the abscissa it follows the law of the circle and for others that of the elhpse. But the equations to the distinct parts of such composite figures are continuous, and results obtained by treating them as such may be applied to the geometrical figure if the values given to the variables are confined within the limits appropriate to the particular equation employed. Thus in the last example the results deduced from the continuous equation to the circle are true of the semi-circular portion of the figure. In the above instances of discontinuity the change in value of the function was gradual, but the law was not fixed. In other cases the law remains fixed, but there is an abrupt change in value at particular points. Discontinuity of the latter kind occurs most fre- quently in the form of a sudden change from + co to -co. Thus for example, when x passes through the value - , sec x changes abruptly from + co to - co , the law by which the value of the func- tion is ascertained remaining the same on both sides of the point of discontinuity. Geometrical examples are found in many curves having asymptotes parallel to the axes of co-ordinates. Thus the equation to the hyperbola referred to its asymptotes is xi/ = c*, and when X passes through zero, 1/ suddenly changes from -co to + co . In such cases results obtained from the equation common to both portions of the curve, on the hypothesis of continuity are applicable to all points (however near to those where the discontinuity occurs), except the actual points of discontinuity themselves. After reading the following chapter, the student will be able to see that such general results may be said to have an application in the limit to these points of isolated discontinuity themselves. There are other forms of discontinuity, but the ordinary alge- braical functions as ax"', sin a, log a;, are continuous except for par- ticular values of the variables. In all the propositions which follow, the continuity of the functions treated of will be assumed, except when the contrary is expressly stated. Our results, therefore, cannot be applied, without a separate investigation, to discontinuous func- tions except within their continuous limits. loff I iv . sin I VARIABLES AND FUNCTIONS. 5 5. A function of two ov more variables is a quantity whose value depends upon and changes with the values of those variables, so that when arbitrary values are given to the variables, the func- tion receives a definite corresponding value. Thus, the distance •which a man walks is a function of his rate of walking, and of the time in which he performs the distance, and the algebraical ex- pressions {x+yY,a^, sin (-j are functions of x and y, and -©} a function of the three variables x, jj, and z. Functions of two or more variables are represented by the sym- bols F(x, y), (p(x, y, z), f{x, y, z, u), &c. Thus the equation z = F(x, y) signifies that s is a function of the two variables x and y. Those who are familiar with geometry of three dimensions, will recognize an illustration of such a function in the ordinate of a surface. 6. In the equation F {x, y) = 0, x and y are not functions of two variables, but each is simply a function of the other, since on giving an arbitrary value to either, the other receives a definite cor- responding value. Thus if x and y are co-ordinates of a circle, y is a function of x alone, whether the equation be written in the form y = Ja^ — x^ or x" + y' = a'. And in general if any two variables x and y are connected in any manner, so that the value of either depends on that of the other (as is the case with the co-ordinates of a curve), this relation may be expressed in either of the following forms, (1) F{x,y)=0, (2) y^f{x), (3) x = 4>{y), ■where the forms of the functions F,f, {PN), .•.t.nNPQ = t(m, We can therefore in general determine the value of tan NPQ for any assigned value of PN ; but if PN equals zero, QN or (p (PN) also vanishes ; and tan NPQ being the ratio of two lines, each of which is equal to zero, has no deter- minate actual value. Since however, when (p is known, we can obtain an actual value of tan NPQ for any assigned value of PN different from zero, how- ever small, we are able to determine the value to which tan NPQ approaches when PN approaches zero, i. e. the limiting value of the tangent. Let r be a point in any line MT which is drawn perpendi- cular to PN produced, so taken that TM ^ = Zfp^., (tan iVPQ); then PT is evidently the line to which the chord PQ continually approaches as Q approaches P, or, in other words, PT is the limit- ing position of the secant PQ when Q approaches P. This is called the tangent to the curve at the point P ; and it will be seen that the investigation of the inclination of tangents is one of the most important applications of the Differential Calculus. LIMITING VALUES OF FUNCTIONS. 11 10. From the preceding explanations the truth of the following axioms becomes evident. (1) If a function f(^x) has an actual determinate value when X = a, lt^f{.x)=f{a). (2) If an equation is true for all values of the variable up to the limiting values, it is true in the limit ; that is, if the equation f{x) = («)• We shall now proceed to determine the limiting values of certain quantities which will be required hereafter. 11. If PQ is a curve of continued curvature, arc PQ _ "^•^o chord pa " A curve of continued curvature is one in which the exterior angle between two tangents at P and Q gradually diminishes, and ultimately vanishes when Q moves up to and ultimately coincides with P. Let then tangents at the points P, Q, of such a curve intersect in T, and draw TN perpendicular to the chord PQ.. Then -^ chord PQ = PT. cos NPT+ QT . cos NQT. The angles NPT, NQT are always less than the exterior angle between the tangents ; and therefore when Q moves up to P, each of these angles vanishes, and each of the above cosines becomes equal to unity; PT+QT J PT . cos NPT+ QT. cos NQ T •'• "''<^=» chord PQ " ^«=» chord PQ = 1 , by tlie above equation. 12 LIMITING VALUES OF FUNCTIONS. But the chord PQ, the arc PQ, and the broken line PT+QT, are in order of magnitude always^ and therefore in the limit ; . arc PQ " "^^=» chord PQ"-*' 12. IL \, where x is the circular measure of the angle. Let PQ be the arc of a circle whose centre is C ; draw CNA perpendicular to the chord PCl, and let the angle PCQ ■— 2x : whence we have PN=^ PQ, AP = ^ arc PQ, and ACP = x, r sin X _ NP chord PQ ~V ^~AP^ arc PQ ' chord PQ '"'=' arc PQ It ^^^ = It Hence 33. //. 1 = 1. , tan X J sin x X X = 1, sm X - , x , ^^0 = 1> a"d cos (0) = 1. x-1 Whether ?i is integral or fractional, positive or negative, it may be expressed in the form 3 ^ where p, q, and r are positive integers ; x"-l a; " -1 x — 1 X — 1 z^"-' - 1 if s' — j: zi z'-l LIMITING VALUES OF FUNCTIONS. 13 Therefore, by dividing numerator and denominator by -s-1, we obtain x"-l _ 1 (z''-' + 2^-^+ ... + 1) - (zi-' + z''-" + ... + 1) l^'I^ fJ-' + z'-^+ ... + 1 Now, when x=l, 2 = 1, and the actual vahie of the right-hand side of the above equation, and therefore the limiting value of the other side, is - — ^ or «. r IL x~i 14. //f^o (1 + ^y = ^> where e is the base of the Napierian system of logarithms. Assume - = 7i. Then X lt^,{\ + x'f=K^^(i+l^\ Also by the binomial theorem, we have 1 n.n-1 1 1 + 71 . - + ^ ^ . -5 + &C. n 1.2 ir When 71 becomes = . U^, ^ _log,e- loff a For ^-^?^^^=\og^{l^x)% X = log„e by (14) 1 r^ = log a 14 LIMITING VALUES OF FUNCTIONS. 16. K^^=\op;a. Let a'' -1 = 3, .: x = \oga{l + z) and when x=0, z also vanishes, a'- 1 .-. lu, =K = log«, by (15). The reader will observe, that none of the above quantities have determinate actual values for the particular values assigned to x. CHAPTER III. DIFFERENTIATION. 17- Let X and y be two variables connected by the equation F{x,7/) = (1), and let the equivalent explicit equations be y=fi.^) (2), ^ = 0(y) (3). Let X and y, .r + Aa: and y + Aj/, be any two pairs of values of the variables which satisfy the above equations. The quantities A,r, Ay are called respectively the increments of x and y. Then, em- ploying equation (2), we have y + ^y =/(^ + Ao:) ; ■whence Ai/=f(x + Ax)-f(x) (4), If we suppose x and y to remain unaltered, while Ax and Ay vary, equation (4) gives Ay as an explicit function of Ax. In the same manner we might . have obtained, from equation (3), the equation Ax = (p{y + Ay)-(p(y) (5), giving Ax as an explicit function of A^, an equation which is obviously equivalent to (4). From the equations (4) or (5) we can . Ay express the ratio -^- as a function of Ax or Ay. This ratio will generally change in value as Ax and Ay diminish, and since Aa- and Ay vanish together, it will ultimately assume the indetermi- nate form - . Since, therefore, the above ratio ceases in this case to have a determinate actual value, we must have recourse to the idea of a limiting value, as explained in the previous Chapter. Let then (/x, dy be any two quantities whose ratio is equal to the limit- 1 6 DIFFERENTIATION. ing ratio of the increments Ax, Ay, when these increments approach zero, so that ■^ = lt ~~ when A.r and A^ approach zero, dx and dy are called the differentials of x and y. From equation (4) we have , A?/ - . , ^ , „ (f(x + Ax)-f(x)]* It —^ when Ax and Ay approach zero = «a=o \ jr- — ^^^^ \ • This limit must be a function of x dependent on the form of (/), and is therefore written /'(a:). Being the coefficient by which the diff'er- ential of x must be multiplied to give that of y, it is termed the differential coefficient of y with respect to x. Hence we have dy=f'{x)dx, and/(^) = Z/Ayi^^-^\ From equation (5) we might have obtained, in a similar manner, (p'{ij) being called the differential coefficient of x with respect to y, and being obviously the reciprocal off'(x). The object of the differential calculus is to investigate the ratio of the differentials, or, what is the same thing, the value of the differ- ential coefficients of variables connected by equations of different forms. From the preceding explanations the following definitions become intelligible. Def. When two variables x and y, connected by any equation, receive corresponding increments, any two quantities whose ratio equals the limiting ratio of these increments when they approach zero are called the differentials of the variables ; and the coefficient by which the differential of x must be multiplied to give that of^, is called the differential coefficient of y with respect to x ; and its reci- procal, by which the diff'erential ofy must be multiplied to give that of X, is called the differential coefficient of x with respect toy. As differentials are defined merely by their ratio to one another, their actual magnitude is perfectly arbitrary; this, however, does * By?- • Hence TV and PV therefore satisfy the definitions of dy and dx. Since this is equally true at whatever point of the tangent T is taken, TV and PV may have any magnitude we choose to assign to them, the only restriction imposed by the figure being that they shall be lines parallel to the axes and bounded by a point in the tangent to the curve. The differentials therefore are not indefinitely small (as they are sometimes represented) but of a purely arbitrary magnitude, their relative magnitude only being fixed. Their ratios, that is, the respective differential coefficients are evidently given by the equations, fix) = ^= tan TPV= cot TPv, PV - is gene- rally finite, and UAu^^ 0) equals zero, dti duo du, ' ' dx ' rfx ^ dx ' or dy = til du^ + u^ dtii . 24. This rule may be extended to the case •where y is the pro- duct of any number of functions, of x. Thus, let y = Ui.U3. . .ii„.i.u„. Then, by (23), dy = dui.{u2. . .u„} + 7iid{TC2. . .u„}, d {m, . . . M„} dui rf {mj . . . «„} M, . . . M„ «, «2 • • • "« ' £? {Mj . . . M„} JMj J {Wg . . . M„} U3. . .U„ M2 ?<3 . . . M„ By substituting this value in the former equation, and continually repeating the process, we obtain d (m, . . . u,,} du, diia dit„ M,. . .M„ M, Mg *'« a theorem of which the result of the preceding article is a par- ticular case. 25. If ?/ = Cf{x)^ where C is independent of x and y, Ay = CAf{x), DIFFERENTIATION. 21 ' ' Ax Ax ' which being true in the limit, we have dx dx ' or dy = Cdf(x) ; that is, if a function of a; is multiplied by a constant, its differential and differential coefficient are multiplied by the same constant. 26. Letv = -'. With the usual notation, we shall have . 7/j + Am, .'. Ay = Us + Alls > til + Am, Us + Aus u, > Us MbAm, - UiAu.2 U2 (Ms + Aus) ' Am, ""^Ax- A«2 - ''' A:. Ax Us {us + Amj) Therefore, in the limit, we have dui diis dy ^ dx ' dx dx u^ Usdui — Uidus dy = u„. 27. Let y = j:", n being any number whatever. Then Ay = {x + Ax)" - x", Ay _(x + Ax)" — x" ' ' Ax~ Ax _ n-l \ X ) 22 DIFFERENTIATION. Let 1 + — =2. Then when Ax approaches 0, 2 approaches 1 ; dx ^~' z-\ The value of this Hmit has been found in (13), and =n, dy n-X ••• ^="^ ' or dy = n x"~^ dx. From the results of this and former articles, we have, if 7/ = a + bx + ex- + px", dy — [b + 2cx + + n^o;""'} dx. 28. Let y = log^ x. Then Ay = log„ {x + Ax) - log^ x ; . x + Ax Ax Ax 1 log„{l +z} .„ Ax X Z X And when Ax approaches zero, z approaches zero; . ^y -^ It log« (^ + ^) dx X z This limit has been found in (15), and = log a ' %_ 1 dx xloga' 1 dx dy log; a X Hence, if «/ = log x, this becomes , dx log X meaning the Napierian logarithm of x. 29. Let y = a". Then A^ = a'+^"-fl'; Ay , a^^ - I ' Ax Ax ' a' It d.v~ '^^ Ax A=o DIFFERENTIATION. 23 The value of this limit has been found in (l6), and = log a. .-. -j^=loffa.rt', or dy = log a . a'dx. Hence if y=^^> dy = e'dx. 30. Let y - sin x. Then A?/ = sin {x + Aa;) - sin x = 2 cos {x + ^ Aa;) sin 5 Aj; ; ... ^=cos(x+iA^).'^^|^, • ^-^^^ cosfx+^Ax) '^"^^'' and //a^o cos (x + | Ax) = cos x, and / dy sin x dx cos^ X ' y sin X or dy = ^— . dx. "^ cos X 33. Let y = cosec x. Then Ay — cosec (,r + Ax) — cosec ^r sin {x + Aj) sin X _ 2 cos (j; + |Ar) sin ^Ax sin j: . sin {x + Ax) ' Ay _ _^ cos (x + ^Ax) sin 5 Ax Ax sin X . sin (x + Ax) ' 5 Ax ' , ,, cos (r + 5 Ax) cos X , , sin ^ Ax and ^^A=o -^ \ , .\ = -^T- , and Ia_. — 5-?-^= 1 by (12), sm X sin (x + Ax) sin* x -" lAx "J K'^^J> dy cos X •*• rfx"~ihP^' J cos X sill X 34. Let y = tan x. Then Ay = tan (x + Ax) - tan x _ sin (x + Ax) sin x cos (x + Ax) cos X sin Ax cos X . cos (x + Ax) ' Ay 1 sin Ax Ax cos X . cos (x + Ax) * Ax ' 1 ij 1 1 17, sin Ax ^"^ ^''='> cos(x-fAx) = c3^' ^"^ ^'^=^-A^=^^ ^y (^2)' dy_ dx = sec X, dy = sec* x dx. DIFFERENTIATION. 25 35. Let y = cot x. Then Ay = cot (x + ^x) - cot x _ cos {x + Aa;) cos a; sin {x + Ax) sin x sin Ax sin X . sin (x + Ax) ' Ay _ 1 sin Ax Ax sin ^ sin (x + Ax) Ax ' and U^^—L-^^ =^J_ and lt^_^%^= 1, by (12), sin (x + Ax) sin x Ax •> j \ j' • . jT = — cosec" X, or dy= — cosec^ x dx. 36, Let y = sin~' ^. Then x = sin _y, /. dx = cos y cify, by (30), .•. dy = ■ dx; "^ cos 7/ or, expressing the differential coefficient in terms of x, 37. If it is required to find the differential of the inverse sine without assuming that of the direct sine, we must proceed as follows : y = sin~' X, .'. sin y=-x, .: sin (y + Ay) - sin y = Ax, .-. 2 cos {y + ^Ay) sin ^Ay = Ax, , I . . sin \Ay Ax .'. cos (y + 5 Aw)— ,^ >'- = —-. and Itt^ cos (y + \Ay) = cos y, and It^^^}^^^ = 1, by (12), dx 26. DIFFERENTIATION. Since the steps in the determination of the differential of the inverse function are precisely the same as for the direct function, it is needless to repeat the independent proof for cos"' or, sec-'x, &c., as the reader will be able to supply it without diflBculty when re- quired. We shall therefore deduce the differentials of these func- tions from those already found. 38. Let y = cos~* x. Then x = cos y, .'. dx=-s\y\ydy^ by (31), .*. dti = — -. — dx, "^ sm y *=~7a^) dx. 39. Let y = sec~^ x. Then x = sec y, .'. dx = ^^dy, by (32), , cos^ V , 1 7 .-. dy = ■—. — - dx = dx, ^ sm y tan y sec y 40. Let y = cosec"^ x. Then x = cosec y, sin^y , 1 J .-. du = dx= 7 -— dx, ' ^ cos y cot y cosec y 41. 'Lety = tan~^x. Then x = tan y, .'. dx = sec'y dy, by (34), .'. dy = ■ — 5— dx, ^ sec* y or % = Y 1 + x' dx. DIFFERENTIATION. 27 42. Let I) = cor' X. Then x — cot y^ .'. dx = — cosec^^ di/, by {35), .'. dy = s— dx, cosec y 43. Let 3/ = versin"' x. Then a: = versin y = 1 — cos y, .-. dx = smydy, by (31), .*. (/?/ = -; dx, "^ sm y and sin^ = ^(l - cos-» = ^(1 - (1 - xj] - J{9.x - x% •'. dy = — r— ^ dx. 44. Let y = suversin"' x. Then a; — suversin j/ = 1 + cos r/, .-. dx= -siny dy, by (28), .•. dij = : dx, and sin y = Ji} - cos^ y) = J{i-{x- 1)'] = J(2x - x'), .'. dy = 7— rr dx. ^ J{2x - x^) 45. We have now shewn how to differentiate all the simple functions which ordinarily occur in analysis, as well as any func- tion formed by the addition or multiplication of any number of them. It remains to differentiate functions compounded of these simple functions, such as log sin x, d' '°s ^j"^ (.qs (a + bx)", &c. All such functions are included in the general forms /{«/> (a;)}, /Q0 {\j/^ (•»)}]> &c., where/, '{x)dx, which gives the required relation between dy and dx,/'(ii) and ^'(x) being determinable by our previous methods. 47. The correctness of the above result may not appear quite obvious, because we have in effect defined dy, not by the funda- mental definition dx^^^^-'A-x ^^^ but by compounding the two definitions £-"-^:' <^> ^•^tr''--'^ ('> This is however immaterial since the last three equations are not independent, any one of them being deducible from the other two. Thus from (2) and (3) we can obtain the same value of dy as that given by (1). Ay _/^ Am . ^""^ Ax~ A21 Ax' •• ^^^='>'^x~^'='Au^='Ax- By equations (2) and (3) this becomes Ay _ dy du _ dy^ which is identical with equation (1). DIFFERENTIATION. 29 48. The following proof of the above proposition is free from the apparent difficulty. As before, let u = {x) =f'{{oc)}2. For putting u = (j> (x), and ^ =/(«), we have z = F(i/) ; ___^ .-. dz = F'{y)Mtf dy =/ («) du, du = (p' (x) dx ; .-. dz = F'{y)f'{u)'{x)dx, or without introducing any new symbols, dz = F' [/{0 (x)}]/ { , sin X , .... sin x = sec X, dy = 5— ax, / (x) = 5— > •^ COS^X ^ V v cos^x , COS X J .... cos X = cosec X, aw = — ^-s— dx, / (x) = — r-^ — , "^ sin'^x ./ V y sin^x = tan X, dy — sec* x c?x, /X^) = ^ec^ x, = cot X, dy = - cosec* x dx, f (x) = — cosec' x, • 1 , dx /.// X 1 = cos-x, dy=-j0^^, /(x) = _^_li_, = sec-' X, dy = — -y^r-;^ — — , /(x) = xj{x'-iy -> ^^~ xj{_x'-\y = cosec-' X, dy = -~j-^ — -^, /'W = = tan-x, dy = ^,, /'(^) = rT^' = cot-x, ^i'^f^, /(^)=Ili' DIFFERENTIATION. 33 (j^j» 1 y =f{x) = F{u) and m = (x), <^y = F' {ii) cp'(x) dx, f (x) - F'(«) 'W- These results are expressed both in the notation of differentials and of differential coefficients, in order to familiarise the reader with both. Both notations express the same fact, namely, that the ratio ^ is equal to a certain function in each case, and, so far as we have carried the subject at present, may be used with equal convenience. H. D. c. CHAPTER IV. INTEGRATION. 54. In the preceding Chapter we have shewn how to differen- tiate any explicit function of a single variable, that is, having any equation of the form y =/(A we have shewn how to determine the value of /'(a;) in the equation dy =f\x) dx. Integration is the converse of differentiation, that is, it consists in finding from any differential equation dy -f'{x) dx the integral equation ^=/ (a), from which it has been derived. The symbol (/) by Avhich this operation is represented is the converse of the symbol {d) which represents differentiation, and is therefore defined by the equation jdy=y. Hence, i^/Xx)dx is the differential of /(^), |/'Gr)^.r=/(x), /(x) is called the integral of /'(^) dx^ or sometimes, though not very correctly, the integral o?f'{x). 55. From the nature of the rules which define a differential, the form of /'(j:) can always be determined where that of/(j:') is given. We have only to find — ^— j and determine its limiting value, and f'(x) is known. No such general method of integration can be found: for since the integral of any diffei'ential is defined to be the function from which it may be obtained by differentiation, we can integrate those functions only to which the differentiation of other functions has chanced to lead us. Thus, if we are required to integrate any function (p(x)dx, we cannot be sure that the process is even possible, that is, we cannot say that there is atiy function whose differential is cp {x) dx, unless we have observed that this quantity, or some other to which it is equivalent, has been obtained by the dif- ferentiation of some known function. It will, however, appear here- after that all functions of the form (p (x) dx can be integrated in the form of infinite series, although there are very many whose integrals cannot be expressed in finite terms by any of the ordinary symbols INTEGRATION. 35 of analysis. The method to be pursued in integration will therefore be, first to collect a number of integrals by examining the results of the differentiation of some simple functions, and then by various artifices to make the integrals of other functions depend upon those so determined by inspection. In this way large classes of functions have been integrated in finite terms, although many are incapable of such reduction. 56. Since (Art. 22) /(x) + C and /(x) have the same differential f'{x)dx, it follows that we may take as the integral o£ /' (x) dx either of the above quantities, giving to C any value we please in- dependent of X. Hence it appears that while there is only a single differential of a given function, there is an indefinite number of in- tegrals of a given differential, all of which are included in the general form/(x) + C, where C may have any value independent of a; which we choose to assign to it; C is called an arbitrary constant, and must be added to every integral in order to express it in its most general form. Where, however, the form of (/) is the only object of the in- vestigation, the particular integral f{x) is often spoken of intead of the general form f{x) + C. 57- By considering the differentials of /(^) and x as measures of the rates of increase of /(x) and x respectively (as explained in Art. 18), the problem of differentiation may be said to be — Having given a function of x, to compare its rate of iiicrease with that of x; and that of integration. Having given the ratio of the rates of in- crease of the function atid the variable, to determine the function. Now it is evident that, for a given function, the rates of increase can have but one definite ratio for a given value of the variable, whereas an indefinite number of functions, differing only by constant quantities, will have the same value of this ratio. Thus we may draw as many parallel straight lines as we please, in all of which the rates of increase of the co-ordinates have the same ratio, since they depend only on the inclination of the line and not at all upon its distance from the origin. If y = mx is the equa- tion to one of these lines, they are all included in the general form y = mx + C, where C is perfectly arbitrary, and all have the same value of the differential coefficient. The equation to each may there- fore be obtained by integrating the same differential equation dy — mdx. 3—2 36 INTEGRATION. To take a more general case. Referring to the figure of Art. 19-, it is clear that an infinite number of curves may be drawn having their tangents at the extremities of a common ordinate, equally in- clined to the axis of x\ The equation which includes the whole class of curves is i/ =f(x) + C\ where C is arbitrary, since the addition of a constant to every ordinate merely changes the distance of the curve from the axis of x, without altering its form or the inclination of its tangent. In all these the differentials therefore have the same ratio for the same value of x, and any one of the equations may be ob- tained by integrating the differential equation dy =f'{,x) dx which is common to them all. These observations shew the meaning of the proposition established in the last article. We shall now proceed in the empirical manner indicated in Art. 55., to determine the integrals of the differentials given in Art. 53., and some others immediately deducible from them. 58. Since d{A{^) +fM ... +/„(.^)} = f(/. W + 4/'.G^) + ••• dM^)' .'. /]{x) +Mx) +M^) = j{dMx) 4 dMx) + ... d/jx)}, or f{f/{x) +/,Xx) + . . ./,Xx)\ dx = j/\'{x) dx 4 j/,'(x) dx...+ [//(a-) dx, that is, the integral of the sum of any number of functions equals the sum of the integrals of the several functions. 59. Since d Cf{x) = Cdf(x), .'. C/{x) = jCd/ix) = {Cf{x) dx, or [Cf'{x)dx^C if'{x)dx; that is, if the quantity to be integrated is multiplied by a constant factor, the constant may be placed outside of the sign of integration, 60. ^mce d(x") = nx"~'^dx, where n is any number, integral or fractional, positive or negative, different from zero. 'dx = -+C, n dx:^ , +C, 11+ \ where n is any number different from - 1 ; that is, to integrate any power of x except - we must increase the index by unity and INTEGRATION. 37 divide by the index so increased, an arbitrary constant being added to obtain the general integral. 61. Since dloi^x = — . I dx , -, ^ — = log .r + const. == log C x, if we write the arbitrary constant in the form log C. This deter- mines the integral of x" in the case excepted from the previous article. Hence d e p f ax" + bx"~^ . . .+ c A h-5 + ...-^ X X X = a + 6 — . . . + cj? + ri log a: ... - -, - , „, ■ , + C. n + \ n ° X {7)1 - 1 ) X f dx f dx 62. From I — we can find / .. „ „, as follows. J X J J{x'' ± a-) Let «* ± a^ = U-, .', xdx = udu, dx dii dx + die ' ' n X X + 71 ' "' J Ji.^^ =^a^)~ 111 ~ j x + u ' = log {x + li) + const., r= log C^*^ ^^ , ^ a C writing the constant in the form log — to give the integral a more convenient form. 63. From the same integral we can also deduce the values of dx , f dx [ dx . f dx j?^'""'' j^^^'- a — a; 2a [a + x a - a: j and d (a — a) = — dx, and d(a + x) = dx ; f dx If dx 1 j' dx J a' — x" 2a J a + X 2a J a — x ' 1 fd(a + x) 1 fd(a-x) 2a J a + X 2a J a - x h = — loff (a + x) log (a - x) + const. ; dx 1 , ^,n + X 7, o ~ — log 6 . ' - 1^ 2a ^ a-x 38 INTEGRATION. f dx _ J_ [ dx _ J_ /• dx •'■ }x^-a''"^]x-a 2ajx + a' 1 Cd{x-a) 1 /• (/ (x + a) ~ 2a J x-a 2a J x + a * = — log (x -a)-x~ log {x + a) + const. ; ' ■ j x' - a'' 2a ^ x + a' f dx 64. From Art. 62, we can deduce the value of / — . 2^_y2\ For let x = -, .'. log a; = log a - log u ; dx du ' ' X ?< ' 1 1 C dx _ f du ail du _ 1 /• du therefore, by Art. 62, dx f dx 1 X j X Jl^a'^x') ~ a^"^^^ a + J{a' ^ar')' 65. Recurring to Art. 53, we find d (a^) = log a . a'^dx, .'. I a'^dx = , . a" + C. J log« H a = e, this becomes, / e^dx = e^ + O. INTEGRATION. 39 Again, d sin x = cos x dx, cos X dx ^ sin a: + C. /' And since d sin 7nx = cos mx d(inx) = in cos mx dx, f cos mx dx = — sin mx + €'. m Similarly, d cos inx = - m sin mx dx. ' sin mx dx = — cos mx + C- m Also, d tan mx = m sec^ 7nx dx ; I sec^ mx dx = — tan ?«x + C m And d cot mx = — m cosec^ mx dx ; } cosec^ mx dx = cot mx + C. m d(-\ / ■■= sin"' - + C. ^(a^ - x^) a o 1 . J -ix \aj dx bo also, since d cos - = • /; // ^ 2\ = - COS ' - + (7. J{ar-x') a Here we have obtained a second form of the same integral, but since . , X . X IT sin~ - + COS" - = - = const. a a 2 the two expressions include precisely the same system of values when all possible constant values are given to the arbitrary con- stants. . . 7 _i X ^ \a) a dx Again, 6?sec -= 77-5- — 7 = — 77-5 ^; a X fx' \ xj{x^-a-)' h dx 1 iX ^ -^ = - see" - + C. X J{x^ - a') a a 40 INTEGRATION. As in the preceding case, this integral might have been expressed in the form — cosec"' - + C, a a which may be shewn as before to be identical with that already given, _^x \a) adx bmce o tan - -. " 1 + / ^ 7, = - tan - + C. J a' + X a a As before, we might have obtained the equivalent form - -cot-'-+C. a a „. J . _, a: V«/ dx bince aversm f dx . , X _, •'• 1^177, j7 = versm-' - + C ; J{2ax - X-) a or = — suversin"^ - + C. a 66. Collecting the results of the preceding Articles, we have I x"dx = h C unless w = - 1, and then ./ 71 + 1 \-J = log ^ + C, -nr— -2 = - tan-' - + C, J a^ + x^ a a dx 1 - ^a + X = — loff 6 , 2a ® a-x' f dx I a' - x"" f dx 1 , ^x-a ~5 s =7— iogC; , ] X- - a' 2a ° X + a I —rr^ 2\ = sm-' - + C, or = - cos"' - + (7, J-/r^A4xJ< C{/dL I INTBGRATION. 41 = - sec ' - + C, X J(x^ - a') a a dx ■ I ^ ^ -= versin"'- + C, J(2ax - x^) a a'dx = ; 1- C, log a sin mx dx = cos mx + C, m cos mx dx = — sin mx + (7, m sec^ mj: = h -; rrr + r ttt-i • • • + r + F x-a {x-bj {x-by-'"' x-b x'-2ax + a''+f3' K, + L,x Ki + LiX (x'-2a'x+a"+/3'y x'-2a'x + a''+(S" Then real values of A, Br, ^._,. . .7?,, M, JV, K,. . .K„ L,.. .L„ can always be found to satisfy this equation. For by multiplying both sides of the equation by V, we obtain an equation U-/(x) = (2), " Those readers who are not familiar with the Theory of Equations are recom- mended to omit the remainder of this chapter. • 44< INTEGRATION. where £7" is a rational integral function o£ x of not more than (fi — 1) dimensions, and /(a:) a rational integral function of (« — 1) dimen- sions, in which the coefficients of the several powers of x are linear functions of the indeterminate quantities A, Br, &c. in equation (1). In order that the two sides of equation (l) may be identical, the coefficients of the several powers of x in equation (2) must vanish. This condition will give us n linear equations to determine A, Br, &c. Now the number of these coefficients is evidently equal to the number of roots of V = 0, that is, to 7i, and therefore the ?i linear equations will give real values of them which satisfy equation (l). Hence -„ may always be resolved into partial fractions of the forms assumed in equation (1). If the dimensions of U had been greater than n - 1, this method would have failed, because the coefficients of the higher powers of ^r would have contained none of the indeterminate quantities, and could not therefore have been made to vanish by assigning particular values to these indeterminate coefficients. If the equation V—0 contains more than one of each of the four classes of roots, we must add partial fractions of the forms above given for each of these roots, and it will appear, as above, that real values may be found for the indeterminate coefficients which will satisfy the assumption. By this method / -p dx, where -p is a rational fraction, may always be reduced to the sum of a number of integrals of the forms jAdx f Bdx r {M+Nx)dx f (K+Lx)dx The two first of these are integrable by inspection, the third can always be integrated by the algebraical artifices before considered, and the last may be reduced to the third form by the method of reduction which will be investigated below. The general method of determining the partial fractions above given is often very laborious, and may be simplified in practice in the manner indicated by the following examples. 72. Let V ~0 have neither equal nor impossible roots. We may then proceed in every case as in the following example. Let INTEGRATION. 45 Udx c ^dx rudx _ r xdx j ~V' ~ j (x - a) {x -b){x- c) X ABC 1 hen assume , r-~, j^r? r = 1 r h , {x - a){x — o){x - c) X — a x — b x — c .'. x = A{x-h){x-c)+B{x-a){x-c) + C{x-a){x-b). (i) Since this must hold for all values of x, let x — a, .'. a = A (a — h)(a — c), x = 6, .-. b = B{b-a)ib-c), X = c, .'. c = C (c — a) {c — i), which equations determine A, B, and C; and then ( xdx _ fAdx CBdx f Cdx j(x-a)(x-b){x-c)~ J x-a J x-b J x-c' = A\og(x-a)-{-B log {x -b) + C log (x - c). 73. If we had pursued the method of Art. 71, we should have determined the values of ^, B and C, by equating the coefficients of x^ and x and the constant terms of equation (1). This would have given us the equations = A+B + C, l=-A{b + c)-B(a + c)-C{a + b), = A be + B ac + C ab, which give A, B, and C though less readily than the equations of the last article. 74. Let V =0 contain equal i-oots but no impossible ones^ as in (x' + x) dx . f (x^ + X) dx the example j y-^ — r^rr t\ ■ ^ J(x- ay (x - b) x^ + x As Ai B Assume , r„ r jv = 7 xa "■ •" r > {x—a)-{x — b) {x — ay x — a x-b .-. x^ + X = A2 (x - b) + A^ {x - a) (x - b) + B (x - a)- . . . (a), Let x = a, .'. a- + a=Ar^(a-b) (l), therefore subtracting, {x^-a^)-¥{x-a)=A„_{x-a) + A,{x-a) {x - b) + B (x ~ a)', and dividing by x-a, X + a + I =^ A2 + Ai {x - b) + B (x - a) ((3). Let x = a, .: 2a + l=A^ + Ai{a-b) (2).* * This step is admissible although we have previously divided by x — a, because (a) and {ft) are not equations for the determination of <^', but identical equations. Thus when the proper values are given to A2, Ai and B both sides of equation {ft) become x + a+l, and must therefore have the same value when ,r is put equal to a. 46 INTEGRATION. B may be determined by putting x = b either in (a) or {ft) ; the former is preferable, as it gives B independently of ^2 by the equation b'+b.= B(b-ay (3). Equations (1), (2), and (3) give A2, ^,, and B, and r (a;^ + x)dx _ f -^2 dx fA, dx CB dx J(x — ay {x— b) J (x — ay j x — a j x — b = ^ +A,\og{x-a) + B log {x-b) + a The same method applies when there are more than two roots equal to a, by repeating the substitutions and subtractions. 75, Let V = contain unequal pairs of impossible roots, as in , , /■ (x — c)dx the example j ^^,^^,^^^,^^^^^,^ . x-c Ax + B Cx + D Assume t-z i^tt-^ — ; tt: = —^ ir + {x' + a'){x' + bx + b') x' + a^ x^+bx + b" .-. x-c = {Ax + B) (x' + bx + b') + (Cx + D)(x'+a') ... (a). Let x^ = — a^; an assumption which we are at liberty to make, since the above equation, being identical, must be true even when impossible values are given to x; .-. X -c = (Ax + B) {bx + b' - a^), = Abx' + {Aib'- a') + Bb} x+B(b'- a'), ■^{A(b'- a') + Bb}x-Aba' + B {b'- a:), which cannot hold unless i=A{b'-(e) + Bb (1), and -c = -Aba'+B (//- u') (2). Equations (l) and (2) determine A and B ; C and J) may be determined in exactly the same manner. Thus, in equation (a), let x^ = — {bx + b'), .'. x-c = (Cx + D) (a- -bx- b'), = -Cbx'+{C (a* - b') -Db}x+D (a' - b'), = [C (b' + a' - b') - Bh] x + Cbb'+D (fl^ - b'), which cannot hold unless l = C{b' + a'-b')-Db (3), -c=Cbb'+D((r-b') (4). INTEGRATION. 47 Equations (3) and (4) determine C and T> ; and \Cx + D) dx r (x-c)dx _ f {Ax + B)dx f{C J{x' + a')(x' + bx + b')~J x' + a"" "^ j "^ + bx + b' These integrals can be found for all values of a, b, b' by alge- braical methods. When only one partial fraction remains to be determined, as in this example after A and B have been found, it may be done by substituting for the other indeterminate quantities in (a) their values, and dividing by the coefficient of {^Cx + D). This is generally a simpler method of determining the last partial fraction correspond- ing to a pair of impossible roots ; we shall employ in the following example, C x^dx \{x-\y{x'+i)' . x^ A^ A. Cx+ D Assume ; ,.3. , rr- = 7 ~T5 + + — „ , , .: x^ = A, (x'+ 1) + A^(x-l){x'+l) + (Cx + D)(x-iy .., (a). Let x = l, .-. 1 = ^2.2, .-. A. 1. •2 ' therefore subtracting, x^~\=A^ {x^ -1) + A,{x- 1) {x' + 1) -t- (C'x + D){x- ly, .-. x^ + x + l^A^(x+l)+A^(x' + l) + {Cx + D) (x - 1). 3 — '^A Let x = l, .'. 3 = ^2.2 + >4,.2, .-. ^, = ^^1—^=1; substituting these values in (a), it becomes |'-x + l = (Ca:+Z))(x-iy, .-. Cx + D = ^, or C-0, Z) = A, r x^dx _ I f ^^ f ^^ 1 ( ^ ■'■ J{x- ly {x'+i)~ ^Jj^'^TYy '^jTZ^i + ^jxTl + log (jc - 1) + i tan~'^ + C X - 1 When there are more than two pairs of impossible roots, the partial fractions may be determined in succession by the method given in the first example of this article, the last fraction being most conveniently found by substitution, as in the last example. 48 INTEGRATION. 76. Let V = contain equal pairs of impossible roots, as in (2x^ — x) dx f (^x-'-xJdx the example li-^ — ^^g, ^ — rr ^ J {x- + 2y {x^ + 1) ^x'-x Ax + B A'x + B' Cx+ D '"^ (x' + 2y (x' + l) '(^^f^ x' + 2 "^ o;^ + 1 ' .-. 2x''-x = (Ax + B)(x'+l) + (A'x + B') (x^ + 2)(x' + 1) + (Cx + D) (x' + 2)- . . . («)• Let a:' = -2, .-. - 4 - x = (^x + 5) (- 1) ... (/3')> which cannot hold unless -1 = -A, or ^ = 1, - 4 = - 5, B = 4>. Subtracting (/3) from (a) we have 2j'+4-(^x + ^)(^' + 2) + (^'x + J5')(*^'+^)(-^'+l) + (^'*^ + -^)(-^' + 2)'; dividing by x^ + 2, 2 = (Ax + B) + (A'x + B') (x^ + 1) + (Cx + D) (x' 4 2) (7). Let x^ = - 2, .-. 2 = (^x + B) - (A'x + B') (S), .-. 0=A-A', .'. A'^1, 2 = B- B', B' = 2. The remaining numerator, Cx + Z), may be determined by put- ting a;* = - 1, as in Art. 75, or, being the only remaining numerator, more conveniently by substituting for A, B, A', B', their values in (a ). If there had been three or more equal pairs of impossible roots, we must have subtracted (3) from (7), divided by (x^ + 2), and then put x^ = —2, and repeated the process till all were determined. The partial fractions being determined, we shall have C (2x'-x)dx _ f(A x+B) dx f (A'x + B')dx f(C^+D)dx J(x' + 2y(x'+l)~J (x' + 2y~'^j x' + 2 ^] x' + l ' The first of these integrals may be determined by the method of reduction, and the others by known methods. 77- We have now considered every case except where roots of all the different kinds occur together, in which case the different methods must be applied in succession, as it is easily seen, by observ- ing the preceding examples, that the determination of any partial fraction is not at all affected by the number or nature of the rest. INTEGRATION. 49" Rationalisalion. 78. Many functions, which are not in the form of rational frac- tions, may be reduced to a rational form, or, as it is termed, ration- alized by different transformations. Rules for the rationalization of several classes of functions have been investigated, but the student will find that practice will enable him to discover the appropriate assumptions in most of the cases which ordinarily occur, without burdening his memory with a variety of rules for the purpose. , f x^dx pie, -i i Take as an exam Assume x = z^, 1 2 .r^ = ^, and dx = 6z^dz, f z' dz and the integral becomes 6 I — — —^ , which is in a rational form, and J z^+ a^ may be integrated by the methods already investigated. Integ7-ation of x'" (a + bx"ydx. 79. Functions of the above form are of very frequent occur- rence. We shall first shew how to integrate them when the indices 7«, n, and p satisfy certain conditions, and then investigate a general method of integration applicable to all cases. , . ,-„ 7rt + 1 . ... 11- (1) Where is a positive integer, m, n, and p being posi- tive or negative, integral or fractional. T ... Let p — - , where r and q are positive or negative integers. Assume {a + bx") = z'', z'' - a b" .-. x"'dx=^-^{rJ-a)" dr., nh " H. D. C. 4 50 INTEGRATION. and multiplying by z", we have f n f »i+i n h" Now when the above condition is satisfied 1 inust either n equal zero or a positive integer, and the function under the integral sign can be expressed in a finite series of integral powers of ~, and can therefore be integrated in finite terms. If had been negative or fractional, the expanded binomial would have contained an infinite number of terms, and the method would have failed to give us the integral in a finite form. The condition = a positive integer, is called the First Criterion, /r^\ iTTL tn + 1 , . (2) VViiere — + p = a negative mteger. Then x'" (a + bx'y - ar'"^"^ {6 + ax-'Y. The function in this latter form may be integrated by the , , .„ 7n + up + 1 . . . , ..„?« + I previous method if ' = a positive integer ; that is, if + /) = a negative integer. The assumption to be made in this case is therefore b + ax~" = z'', putting p = - as before. The condition + p = a negative integer, is called the >Seco7id Criterion. 80. The following are examples of the application of these methods. (0 f x^^dx J (ah + x^' Here VI + n 1 i+ 1 3, a positive integer ; therefore the first criterion is satisfied. Since p = —\t the proper assumption is fl5 -f x-i = Z-. INTEGKATION. 51 -T Tri = ^\{^-a^dz, = 4 [{z'-2a^z'' + a)dz, = f (rt^ + o;')^ - f a^ («^ + x--f- + 4«(a^ + o:^)^ + C. Here = , and the first criterion is not satisfied. n 2 But + p = — i + 5 = — 2, and the second criterion holds. .{a'-x')Ux ( {a'x-'-\)^dx Then I ^^ « = | ^^3 /• («^- j:^)'c/jr _ f{ and we must assume ax - — 1 = 2 , rt _ _ -z{z'-^\)dz _ (a^j;-^- l)i («^j:-^-l)t 5a* 3a* n^x" \ 5 } + C, (a'-x"-) H3 a' + 2x-) ^ ^ 15aV 81. When neither of the criteria is satisfied, we must employ the following method, which is applicable to all cases. It consists in making /j;'"(a + ia;")''(/j; depend on another integral of the same form but with smaller values of 7h or ;; ; by repeating which process, 4 — 2 52 INTEGRATION. we arrive at last at an integral which can be determined by methods already given. This is called the method of reduction. As the treatment of the integral depends on the signs of m and ;?, we shall use these letters to represent positive quantities only, which will give the following cases. (1) To reduce m in f "^ ^^„. - or jx'"{a + bx"y dx. . r dx {(a + bxy J (2) To reduce m in \^,^^-^. or j ^^^^ dx. (3) To reduce p in j x'" (a + bx"y dx or j^ — -^ dx. , ^ _ -, . f x'^dx f dx (4) To reduce p m -. j—^ or | ^ — , , „xp- ^ >' ' J {a + bxy J x^{a + bx"y The mode of treatment is exactly the same in each pair of integrals as above arranged, that is, the method of reducing m or p depends only on the sign of the index to be reduced. n is always supposed to be positive, as the function can always be reduced to that form by altering the value of m. The follow- ing are the methods to be employed in the four cases respectively. ••''•"" j -^ (« + bxy ~J \nb (p - 1) (a + bx"y-'j ' therefore, integrating by parts, 71 b [p 71 + 1 f x'"~"\a p-l)J (04 _ „,_„^i 1 7)1-11 + 1 f x'" "dx '"""''*' 7ib{p-l){a + bx"y-' ^ lib (p - 1 ) J {a + bx"y-' '"^ ^' j,...-n+i jfi -^ ji + I f ,r'"~" [rt + bx"] dr P,n = 7-. T^^ . „ ■ , + 7ib Q)-l){a + bx''jP-' nb {p - 1) J (a + bx")'' p _ J^'""'"^' 7)1 -71+1 f , , •'• "'~~ 7ib(p- 1) {a + bxy-' ^ 7ib{p-l) ^" '"-" "^ '"^ ' ,, a"'~"+' ?« — ?j + 1 „ . •. j[^ = -)- a I _ ' b {)ip -7)1—1) (a + bx^y-^ b {lip -7)1-1) '" " ' by which formula P,„ is made to depend on P,„_„, that is, f x'"dx C x"'~"dx J (a + bx"y °" J (fl 4 bx"y ' and by repeating the process, P,„_„ may be made to depend on ^ INTEGRATION. P,n-i,n and generally P„, may be made to depend on P„,_r„ where r is any integer. By employing the form (A) we make the integral depend on another of the same form, in which m is diminished by n, and p by unity, by the repetition of which m may be reduced by any multiple of n and p by the same multiple of unity. I x"'(a + bx"y is treated in exactly the same manner, and we obtain formulae by which yn alone may be reduced by any multiple of?/, or m reduced by a multiple of 71, and p increased by the same multiple of unity. dx 83. Let P„ J X"" (rt + bxy ' , \ a h then -—-, ,—^-, = -—, ,— tni; + and dx x'" (a + /jjc")P-' .1"' (« + bxy x'"-' (fl + bx"y ' j a;'" (a + 6a;"/-' ' "' 1 w6(p — 1) /* dx -1) r rfj 1 j^"'-"(a- (/« - 1 ) a;"^i (a + 6x")'-' w - 1 j a;"'"" (a + bxy by integrating by parts : . «p I n{p-l) + m-l , (^ _ ] ) j;-' (a + bxy-' m - 1 by which formula P„ is made to depend on P,„_„, and therefore P„ may be made to depend on an integral of the same form, in which m is reduced by any multiple of n. The other integral in (2) is treated in the same manner. 84. Let P^ .-. P„ = I x"' {a + bx"ydx ; ; - a [ «"• (a + bxy-' dx + b ( x""-" (a + bx"y-' dx ; and j x-^" (a + bx^-' dx = j x^^' d {^^^iT ~ } ' a'"^' (a + bx'J' OT + 1 r , , = ^^- ^ , — x'" (a + bx^yUx, nbp nop J ^ by integrating by parts : . p ^p a-'""' (a + bx-y m + 1 •• ^r = "Pr-r+ ^ ^r, 54! INTEGRATION. P - 5^ — + « Fp '' np + ?« + 1 np + in + \ by which formula P^ is iTiade to depend on P^_,, and therefore P may be made to depend on another integral of the same form in which p is reduced by any integer. The other integral in (3) is treated in the same manner. 85. Let P then and ''^1 {a + hxy ' x"" _ ax"' 6x"'+" (fT+^jx")^-' ^ {a + hx"Y "*" (« + bxy ' j (a + hxy J Xnb (p - 1) (o + bxy-'j 7ib{p-i)(a+bx"y-' "^ «6(p-l) j (a + bxy-' ' by integrating by parts : •'• ''^" ^n{p-l)ia + bxy-' ~ \?i (p - 1) j ^-•' by which formula Pp is made to depend on P^_,, and therefore P may be made to depend on an integral of the same form in which p is reduced by any integer. The other integral in (4) is treated in the same manner. 86. Of these four forms the first two reduce 7n alone by 7i at each step, and the last two reduce p alone by 1 at each step, while the form (A) which occurs in Art. 82 reduces both in and p at once, and a similar form reduces m and increases p in the other integral of Art. 81, (1). On account of the increase of p, this latter form is never more advantageous, and often much less so, than the final form of Art. 82. Where however it is equally applicable, it has the advantage of being more readily obtained. The mode of proceeding with an integral of the general form under consideration, will be to reduce m and p together by (A), where that is possible, until either 171 is less than 7i, or p equal to unity. If the function is not then integrable, we must reduce p in the former and ?h in the latter case, until we arrive at an integrable form. Where formula (A) is not applicable we must begin by reducing w or p, choosing that one by INTEGRATION. 55 the reduction of which we arrive most easily at an integrable form. That we shall always obtain at length an integrable form is evident, since we may always reduce p to unity, either in the numerator or denominator, when the function is integrable at once in the first case, and in the second is either a rational fraction or admits of im- mediate rationalization. The reduction of ?« is often preferable to that of p, because it proceeds by 7t instead of unity at each step, and in many cases, as where w^ = n^-l, leads to a function which is integrable by inspection. The methods employed in the four distinct cases, and Jiot the resttlling formulae, should be remembered and applied in any par- ticular case. /x dx (a* + x^ x'dx Here we can first reduce 7 by 2 and f by 1, by means of for- mula (A), and then reduce the index of x alone by 2, twice, which will bring us to I j which can be immediately integrated. J {a^ + x^Y Employing therefore the method of Art. 82, we have /■ x''dx _ (x^.xdx x^ „ [ x^dx = i + 6 i . (a" + xy J («' + xy As we have now to reduce the index of x twice, we will put it equal to m, that the same process may serve for both reductions. ■r 1 T^ [ ^'"^-^•^ Let then P,„ = — -j ; J (a- + xy .: (as in Art. 82) P,„ = a:'-' (a' + .r)' - (m - 1) jx'"-' (a' + x')' dx = x'"-' {a' + x-'f - (m - 1) {a' P„._„ + P,„}, ... P„ = i .r"- (a' + x')'' - "^ a' P,„_, ; 711 ^ ' VI .'. P, = ix'(a' + x')^^-U'Pz, an d P, = i x' (a' 4 x') ' - § a' P, , 56 INTEGRATION. and P, = I — ^— ^-^ = (rt- + x^)^ + const. J (a^ + X-)- {a^ + x") Hence r x'dx _ x" J (a' + x-)i ~ {a- + x-f^ + 6 lix' (a- + a--)' - ia^ {^x" (a' + .r )* - f «' (a^ + x^)^} + const.] 88. There are some particular values of the indices p, m, and 7i, in which the above methods become inapplicable on account of the coefficients which occur becoming equal to infinity. This happens in the following cases : (1) Where the index of x is (-1) the reduction of m is im- possible, as appears from Art. 8.3. The integral is then of the C n Y C fl Y form — ; ; — r- or — (« + bx")^. The most general form of p J x{a + bx")'' } X ^ ' T ... IS -, r and q being positive integers. Assume then a + hx" = z'', (z^ - af .-. X- , ; b" dx q z^-'dz " X n {z''-a)' and the integrals become 91 J z'^-a ?i J z^-a both of which are integrable as rational fractions. m + 1 . f x"'dx C(a + bx"Ydx (2) When p = in ^-— - or ^ — —^ , the reductions of m and p respectively become impossible, as appears from Articles 82 and 84. r x"'dx _ f. x"''"''dx ^^"' J (a + bxy " I {ax- + by f dx = I —7 r^ by the above condition. J x{ax-+by ^ INTEGRATION. 67 This is the same form as that just integrated, and may therefore r be solved by the assumption ax~" + J = 2' if p = - . (3) When the index of (« + bx") is (- 1) the reduction of p becomes impossible, as appears from Art. 85. In this case the in- tegral is / -. — or —, i—r^ , both of which are rational ^ J a + Ox" J x"' {a + 6j: ) fractions. Integration of sixfx co%"xdx. 89. This function may be integrated by methods similar to those applied to x"'(rt + bx'y. We will first consider some particular cases in Avhich the integral may be immediately obtained, and then proceed to the method of reduction. (1) Let one of the indices (as m) be an odd positive integer, then m =: 2r + 1, where r is a positive integer ; • •. I sin"'a: coa^Xilx = I cos"a; (1 — cos'o;)'" sin xdx. By expanding the binomial, the integral is reduced to a finite num- ber of integrals, each of which may be determined by inspection. If the index of cos x is an odd positive integer, and equal to 2r + 1, we have isivTx cos^'xdx = i sin"'a: (1 - sin'x)'' cos xdx, which is integrable as before. (2) Let {m + n) be an even negative integer =— 2r. Then I sin'"a: cos"a: dx = j tan'"j: cos"'+"x dx tan^j: (1 + tan-j;)''"' ^ec'x dx. i By expanding the binomial, the integral is reduced to the sum of a finite number of integrals, each of which may be determined by inspection. • 90. When neither of these conditions is satisfied we must pro- ceed by I'eduction. For this there are only two distinct modes of proceeding, that for the reduction of a positive index, and that for the reduction of a negative one. 58 INTEGRATION. (1) To reduce m in \s.m!" x co^" x dx, being positive, and n either positive or negative, let Then P^ = \ sin"'~' x cos" x sin x dx, sin"^'xcos"+'x m-l f . , . „^, , ... = ; + , /sin"'-='xcos"+'xrfx (A), n + 1 n + 1 J ^ '^ sin'^'xcos"+'x »«-l,/"- ^2 „ 7 Tj 1 = + { I sni'^^x cos"x dx - P„], fi + 1 7^ + 1 V „ sin"^' X cos"+' X »/ - 1 „ ^•^ p ^5 ^. jP ' ' " m +n in +n '""^ * If the index to be reduced had been that of cosx, the method would have been exactly analogous. By formula (A) both indices are altered ; that of sin x reduced, and that of cos x increased or reduced according as it is positive or negative. , ^ r^ 1 . {&m"'xdx (2) To reduce n in I — , ^ ^ J cos^» n being positive and m either positive or negative, „ rsin'"j;(/x let P„ = „ — ; } cos X /" sin*"x (cos^x + sin^x)c?a: J cos X r, f • .„xi ^mxdx = Pn-2 + sm'"+'x -— - — , ) COS X p sin"'''"*a: 7n+l fsm'"xdx "~^ (?J — 1 ) cos""' X 71 — 1 J COS""^X ' sin^+'x ( w + n „ = ? 7^ ^TT- + ^ I r r "r-2- (m - 1 ) COS X ( M — 1 J If the index to be reduced had been that of sin .r, the method * would have been exactly analogous. It will be seen that in every case the reduction is by 2 at each step. 91. Take as an exam , [ dx pie —5-. J cos* j; INTEGRATION. 59 LetP„=f-^^-, J cos a; (cos^^ + sin^x)dx cos'',r = -r„_a + / sin a: , J cos x _ p s in j; 1 f dx "~^ {n — \) cos""' X n — I J cos"~^ j; ' .. p sin a: , " - - p " (/J— l)cos"~'a; ?« — 1 " • *~4cos^a;"^4^' _ sin.r 1 '~2cos^r 2 '' and P. = f = lot? C cot d tt - la:) ; jcoso; ° ^* ^ '^' /■ (Ix sin j: 3 sin j;- 3 , ^ ^ / , , s — 5^ =1 — + o — 2" + ologCcot(V- 5^')- CHAPTER V. SUCCESSIVE DIFFERENTIATION. Theory of the Independent Variable. 92. In Chapter III. we have shewn how from any equation of the form y =f{^\ ox x = (p{y) (1), to determine the value oi f'{x) or ^'{y) in the corresponding dif- ferential equation dy=f'{x)dx, or dx = (p'(y)dy (2). It may now be asked^ are the differentials c?x, dy to be con- sidered as functions of the variables x and y ? The answer obviously is, that they may have any values consistent with equations (2), that is, any values whose ratio equals the differential coefficient. Now these values may evidently be such as to make them both functions of the variables; but we are at liberty also, if we please, to give to either of them any arbitrary constant value, when the other will receive such a corresponding value as to satisfy equations (2), and will therefore be a function of the variables. We have already said that a single equation between two variables is called an equation of one independent variable, without applying the term independent variable to one rather than the other ; we will now add the following definition. Def. In an equation between two variables, that one whose differential is assumed to be constant is called the independent variable. When no such assumption is made and the differentials are left in their original generality, the equation is said to have a general independent variable. When y is given explicitly in terms of cT, it is generally most convenient to make x independent variable, and vice versa, as thereby equations (2) are presented in a more advantageous form. It is sometimes, however, better to keep the independent variable general for the sake of symmetry in equations derived from equa- tions (2) in a manner which will presently be explained. SUCCESSIVE DIFFERENTIATION. 61 isr L, 93. The following geometrical illustration will make the last article more intelligible. Let P, Q be any two points of the curve corresponding to equa- tions (1); T, S, any points arbitra- rily chosen in the tangents at P and Q. Then, the remaining lines of the figure being drawn parallel to the axes respectively, PF and TF may be taken to represent the values of the differentials at P, and QX and SX to represent their values at Q. The positions of T and S being perfectly ai'bitrary and inde- pendent of one another, the magnitudes of the differentials at Q are quite independent of their magnitudes at P. But we may if we will so fix the positions of T and S that PF and QX shall be con- nected by any law we please. Thus we may impose the restriction that one of the differentials, as da;, shall at every point of the curve be some determinate function of the corresponding abscissa, or ordinate, when dy will also take a variable value dependent on the assumed value of dx and on the form of the curve. Suppose for instance we determine that dx x^ shall always have the value — , i. e. that T, S, &c. shall be so chosen that and that the same law may prevail throughout the curve. The corresponding values of dy will then be given by the equations TF = -^ . tan TP F, SX = -^^- tan SQX. By this assumption both dx and dy are made functions of the variables, their analytical values being dx = — ^i/ = -/W- In like manner we may determine the positions of the points T^ S, &c., throughout the curve, by any other rule, or we may leave them if we please, perfectly arbitrary and unfettered by any rule at 62 SUCCESSIVE DIFFERENTIATION. all, since the equations (2) are equally true whatever rule is adopted and whether any rule be imposed or not. One of the simplest rules for fixing the values of the differentials is to give PV the same value at all points of the curve, so that QX= PV wherever P and Q may be. In this case TF will not preserve a constant value, since TV = PV. tan TPF and the angle TPV varies from one point of the curve to another. In fact we shall have with the above rule dx=C, dy = Cf'{x). Another equally simple rule is to make TV constant throughout the curve, that is, to assume dy = C and thence dx = C(p'{y). When PV retains a constant value x is (as above stated) called the independent variable, and when TV is constant y is called the inde- pendent variable. Another form of restriction sometimes imposed upon the differen- tials is to take some function of one of the variables as >// (x) and make its differential ^|/'(x) dx constant, that is, to determine dx and dy by the equations , C , Cf{x) \}/-{x)' ^ ^^ (•■»■) When this is done, \//(a;) is said to be the independent variable. When no restriction is imposed on the values of PFand TV, the independent variable is said to be general. In this case therefore the differentials must be treated as arbitrary functions of the vari- ables, to which we can at any time assign such definite forms as may be found convenient. 94. Since dx, dy are in general (before any variable is made the independent variable) functions of the variables, they, like all other functions, may have differentials, and we shall meet with such quan- tities as d{dx), d{dy), d{d{dx)], d{d{di/)). Sec, which are written for consciseness d^x, d'y, d\v, d^y . . . d''x, d"y, and are called the second, third ... ?«'*' differentials of x and y re- spectively. Again, f'{x) being a function of x must have a differential co- efficient, and this a third, and so on. These are written f"{x),f"'{x)...f'%x), SUCCESSIVE DIFFERENTIATION. 63' and are called the second, third . . . n^^ differential coefficients of /(.r) or 1/ with respect to x. So also we have the second, third ... m"* differential coefficients of (p{i/) or x with respect toy. 95. It should be observed that the idea of a particular inde- pendent variable applies only to differentials and not to differential coefficients. The ratio -~- ov f\x) is the same whether dx or dy be variable or constant and is in no way affected by the assumption of a particular independent variable. So from the figure of Art. QS, it is evident that tan TPV is independent of the position of T. df'(x) Again, /"{x) v/hich = - ' . has the same value whatever be the inde- pendent variable, and the same is evidently true of all the successive differential coefficients. An equation among differential coefficients therefore, if it involves no differentials, does not imply the assumption of any particular independent variable and is equally true irrespective of any such assumption. It will be seen hereafter that an equation among differential coefficients with respect to x, is immediately con- vertible into a corresponding equation among differentials with x for indepe?ident variable. In consequence of this, x is sometimes incor- rectly called the independent variable in the former as well as in the latter equation. It is important that the student should avoid this error*, as some of the following propositions wholly rest upon the fact that an equation among differential coefficients is unaltered by a change of the independent variable. 96. When /'(x) or ^'(y) is known, the ratio o^ dy to dx is also known. So when /'(a;) and/" (a;) or (p'{r) and (p'\x) are known, we can find the relation between dx, d"x, dy and d'y ; and generally when the first w differential coefficients of one variable with respect to the other are known, we can find the corresponding I'elation among the differentials of the variables of the orders up to the «"'. These relations we shall now determine. " The phraseology referred to is called errorieous, with reference only to the defini- tions before given. Totally different definitions are sometimes employed wliich would justify that application of the term ' independent variable.' 64 SUCCESSIVE DIFFERENTIATION. 97. To find the relations between the successive differentials and differential coefficients when the independent variable is general. Let y =/(x), then dy=f'{x)dx (l); differentiating both sides of this equation, we obtain d'y^d{f'{x)dx} = df'{x) dx +/{x) d'x. Art. 23. .-. d'y ^f"{x) dx' +f(x) cPx (2). By differentiating this equation, we obtain (Art. 21 and 23), d^y = df"{x) da? + d (dx')/"{x) + df'{x) d'x + d^xf'{x), .'. d^y =f'\x) dx^ + 3f"{x) dx d'x +f{x) d^x (3), and similar equations may be found connecting the differentials and differential coefficients of higher orders. These equations give the successive differentials of y explicitly in terms of the differentials of x and the differential coefficients of y with respect to x. The differential coefficients may be found expli- citly in terms of the differentials from the above equations, but more readily as follows. 98. To express the differential coefficients in terms of the dif- ferentials when the independent variable is general. If y^fi'^). /'«4: ^*)' ... (An..6)/'W = i<^{g} = 5?^^?^^/ (5). Again,/-(:.) =-^ = d^^XTx"^ \tJ] dx (dxd^y - dy d^x) - 3d'x (dxd^y - dyd^x) ^ = d^' ~" ^^^' and similar equations may be found for the differential coefficients of higher orders, the equations (4), (5), (6), &c. being equivalent to (1)' (2)5 i.^)' ^^' ^^^^ deducible from them. 99. To find the relations between the successive differentials and differential coefficients when x is independent variable. SUCCESSIVE DIFFERENTIATION. 65 When X is made independent variable dx is constant, and therefore d^jc, d^x, &c., all equal to zero. Hence we obtain by differentiating the equation, dy =/'(.^) doe, .'. d^y =f"(x)dx', since dx is constant, d'y=f"{x)dx', equations which might have been obtained from those of Arts. 97 and 98 by equating d^x, d^x, &c. to zero. 100. Examples of successive differentiation. 1. To find the successive differential coefficients of 1/ where 2/ =/(/) = «'^"- We have by repeated applications of the rule of Art. 27, /'(x) = 11 . aa;""^ f"{x) = n.n-l. cx"-^ f"\x) = n. 71-1... 2.1 .a, • From the above equations we see that when x is independent variable the successive difFerential coefficients may be written as above — — ^...---^. These dx dx^ dx" expressions are sometimes used instead of /' (a-), f"{x), &c., when x is not inde- pendent variable, but in that case it is clear that the numerators will no longer represent the successive diff'erentials of y, and that these expressions in fact cease to be fractions, and become mere symbols equivalent to fix), f"{x)...f'-"l{x). Thus, if this practice were adopted, equation (2) of Art. 97 would become the d^y on the left of the equation, meaning the second differential of y, while in the numerator of -r-^ it has no such meaning. It is better to avoid such a use of the notation, but as it is often met with, it must be remembered that -r^, &c. are then mere symbols and not fractions. II. D. c. 5 66 SUCCESSIVE DIFFERENTIATION'. and all the difFerential coefficients after the Ji*-^ vanish, since /"(x) is constant. The successive differentials when x is independent variable follow at once from the differential coefficients: as appears generally in Art. 99. Thus we have di/ =f'{j'c) dx = n . ax"~Ulx, d^y =f"{x) dx^ = n.n-l. ax^'-dx". d^y =f"\x) dx" = n.n-l ...2.l.a dx''. If it is required to find the values of dy, d^i/, &c. when the inde- pendent variable is general, we must proceed exactly as in the general case of Art. 97- Thus, by differentiating repeatedly and considering dx variable, we obtain y = ax"", dy = n. ax"'^ dx, d'y = n.n-l. ax^-^dx" + n . ax""' d% d^i/ = n.n-l.n-2. ax^'^dx^ + n.n-l. ax''-\2dxd'x) + n .n-1. ax"~- dxd^x + n . ax"~^ d^x, = n.n-l .n-2. ax"-^dx^ + 3n.n-l . ax^-'dxd^x + n . ax"-^d% and by pursuing the same method the higher differentials may be found. 2. Let t/ =/{x) = e\ Then proceeding as before, /'C-^) = ^^» f\x) = e\ f'Xx) = e-, and when x is independent variable, di/ = e"dx, d^y = e'dx^, d^y = e^Jx". When the independent variable is general, we have dy = e^dx^ d^y = e'dx' + e^d^ SUCCESSIVE DIFFERENTIATION. 67 d^y = e'dx^ + e' . 2dxd'x + e^'dxcPx + efd^x, = e^dx^ + Se'dxd'x + e'd^x, Sec. = &c. Further examples of successive difFerentiation will be found in Gregory's Examples, Chap. ii. Sect. 1. The quantities there obtained are successive differential coefficients expressed by the notation men- tioned in the note to Art, QQ ; or if x is considered to be the inde- pendent variable " {z)dz% d"x = (p^"\z)d2" ; by substituting which values in the original equation we obtain the required equation between the differentials of ^ and z. By dividing by some power of dz, this may be at once trans- formed into the equation among the differential coefficients of ^ with respect to z. (Art. 107). 111. In the preceding cases, one of the two forms of the equa- tion, either that to which or that from which we have had to pass, has had a general independent variable. In those that follow, both equations have determinate independent variables. Such transfor- mations are of much more frequent occurrence than the former ones. 112. Having given an equation among the differentials of x and y when X is independent variable, to find the equivalent equation among those of i/ and z when z is independent variable, z being a function of one of them as x, or as it is commonly expressed, to change the independent variable J'rotn x to z. By dividing by some power of dx, the original equation may be transformed into the equation among the differential coefficients of _y with respect to x. (Art. 107). Then by differentiating /(x) = i/, remembering that, when z is independent variable, dx is variable, we obtain w/.^N _1 ^/M _ d'i/dx~d'xdi/ J ^''^~ d'x'^Kdx) ~ dx' ^"'( \ ^ rl\^ .(dy\[ _dx{dxd^y- dyd^x)-3d^x{dxd^y-dyd^x) J ^''^^di'^Ui \tx)]- ' d^' '' ' &c. = &c. Also, if the given relation between x and z is put in the form x = (p{z\ SUCCESSIVE DIFFERENTIATION". 73 we have dx = (f)'{z)d3. d"x = (}>^"\z) dz", when z is independent variable. By substituting these values of x and its differentials in the former equations, /' (x), /"(x), &c. may be expressed in terms of z, dz and the differentials of ?/, and the required transformation effected by the substitution of these values in the original equation. By dividing by some power of dz, this equation is transformed into the equation among the differential coefficients of ^ with respect to z. (Art. 107). 113. Instead of first expressing the differential coefficients at length in terras of the successive differentials of x and i/, and sub- stituting for those of x their values derived from the equation ^ = (z), it will be found more convenient in practice to invert the order of these operations, and first substitute for dx its value in the unex- panded forms off(x), f"{x), &c., and then perform the differenti- ations, remembering that dz is constant. As an example, we will change the independent variable from x to z, where x = e' in the equation x^ d^y + 9.xdy dx +y dx"^ = 0. This is equivalent to "" dx'^"^ dx^^ ' or to ^y (^) + 2x/(:r) +/(x) = 0, since x is independent variable.- Then, since x = e% dx = e''dz, "J ^^~dx~ e'dz' -^W- ,1^ -eUlz'^Vdz) e'^dz' ' since z is independent variable, and therefore dz constant. Hence the equation becomes d-y + dy dz + y dz^ = 0, 74 SUCCESSIVE DIFFERENTIATION. where -j- , j4 ^^"^ ^^ differential coefficients of y with respect to z. 114. Having given an equation among the differentials of a; and y when x is independent variable, to find the equivalent equation when y is independent variable, or to change the independent variable from X to y. By dividing by some power of dx, the given equation may be transformed into that among the differential coefficients of 3/ with respect to x. (Art. 107)« Then, differentiating successively, remembering that y is inde- pendent variable, we obtain from the equation , Sid-xY-d^xdx = ^•^^-^1? ' &c. = &c. ; and by substituting these values in the original equation the required transformation is effected. The resulting equation may be transformed into that among the differential coefficients of x with respect to y, by dividing by some power of c?^. (Art. 107)- 115. The methods of effecting these transformations will be easily remembered, if it is observed that (1) An equation with x as independent variable is immediately transformed into an equation involving only differential coefficients with respect to x, by division by some pov/er of dx. (2) The expressions for the differential coefficients when any variable is made independent, are obtained from the expressions in which the independent variable is general, viz. / 'W=l-/"W=i where d^F(x,y) and i^'(x) signify the differential and differential coefficient respectively of F (x, y) obtained on the hypothesis that y is constant and F (x, y) a function of x alone. Similarly // ^yFJ.^, y) _ dyF(x, y) _ "^-^ A^^ ~ dy -^ ^-^^ where dyF(x,y) and F'{y) are the differential and differential co- efficient respectively of F(x, y) obtained on the hypothesis that x is constant and F(x, y) a function of j/ alone; . // ^yF(x,y) _ F(x,y) Ay •• '^^-» A^ «A=o ^^ •"^=OAx _ (Zy F (.r, ?/) dy^p,, .di dy dx ^"^^ dx _ d,jF (x, J/) ^ Substituting these values in the above equation and multiplying by c?x, we obtain du = d,F(x,y) + d,F(x,y) (3), or du = F' (x) dx + F' (y) dy (4). F'(x), F'{y), being differential coefficients obtained by treating F(x, y) as a function of one variable only, may be found by the methods of Chap, iii., and equation (4) will then give one relation between the three differentials du, dx and dy. 80 DIFFERENTIATION OF By differentiating equation (2) we obtain cly=f{x)dx, by which equation together with (4), du may be expressed in terms either of dx or dy. d:,F(x,y) and d^ F(x, y) are sometimes written for conciseness dji dyU*, and are called the partial differentials of u with respect to X and 3/ respectively; F'{x), F'(ij) are called the partial differential coefficients of u with respect to x and y respectively ; du is what we have already defined (in Art. 17), as the differential of «: to dis- tinguish it from the partial differentials, it may be termed the total differential of u. It is evidently the same quantity which we should have obtained by eliminating one of the variables between equations (l) and (2) and differentiating the result. The theorem proved above may therefore be stated thus : The total differential of a function of two variables, which are themselves functions of one another, equals the sum of the partial differentials with respect to the separate variables, or (with the ab- breviated notation) du = d^u + dytt. 119. We will apply the theorem to the example of Art. 117- u = Jx' + f, where j/ = mx + c. * In using this notation, it must be borne in mind that u is merely an abbrevia- tion for the function F (.r, t/). The pair of equations (1) (2) may be transformed into a number of different pairs, each of which would give the same value of u in terms of either of the other variables, but all leading to different values of dxU dyU. Thus the pair of equations u = xy^ and j^ = c' is equivalent to the pair m = wye' and y = ^' % but the former give d^u = y'^dx = e^'dx, and dyU — Ixydy = 2xe'dy, while the latter give dxU={\-vx)ye'dx ={\ + x)e^'dx and dyU = xe'dy. Hence when dxU and dyu occur, u must be understood to stand for the same function of x and y in both. Another variation of notation must also be noticed. The partial differential co- d 'it d It efficients F'(a') F'(y) which with the notation of the text equal -^, -~- respectively, dll du , , . , . • 1 fni 1 are sometimes written — , — , the subscripts being omitted. Ihese latter quantities then cease to be ratios, and become mere symbols equivalent to F'{x), F'{y), and the partial differentials are then written -r- • dx, -7- . dy. This is the notation used in *^ dx dy Gregory's examples. We shall recur to the general subject of notation in Art. 138, infra. FUNCTIONS OF SEVERAL VARIABLES. 81 Here cim = , _ , a,.u = J^' + f' " J^' + f' , , , X dx + 11 dif .: an = dji + dji = — , ' . du If we wish to obtain -7- as a function of x we must employ the second equation to eliminate y and dy. Thus differentiating, we have dy = m dx, . and substituting in the value of du for y and dy their values, we find , X dx + hnx + c)m dx du = . / '^ - , Jx^ + (rnx + cy du (I + 7)1^') X + mc or -^ = ^ == , "^ sjx'^ + {mx + cf the same result which we should have obtained by pursuing the method indicated in Art. 11 7. In like manner -— may be obtained. dy ^ 120. The relation between x and ^ was expressed in Art. 118., for simplicity's sake, by the explicit equation y =f{x) ; but the theorem du = dji + dyti, is independent of that assumption (which nowhere enters into the proof) and is equally true whatever be the form of the relation between x and y provided they are in fact functions of one another. Thus the relation may be expressed by the implicit equation /(x, y) = 0, or by the pair of equations ^ = 0(2)^ y = ^{^)> or more generally still by a system of n equations involving j:,^and n — \ other variables, since by solution or elimination any of these forms will give ^ as a function of x. 121, As an example, let it be required to express du in terms of dz from the equations Here dju = cos H. D. c. 82 DIFFERENTIATION OF dx - = e" (/~, dy- = 22 dz, du = = dji + dyif, -. --©{"'/■ vdt/) --(i){— - 2e'z dz z* _ = r-3 cos (-\ (z-2 )dz. }' If we had first substituted for x and y their values, we should have had = sin f pj by differentiating which equation we may obtain the same result as before. In like manner, du may be expressed in terms o£ dx or dy. 122. If there are more than two variables under the functional sign, it may be proved in a precisely similar manner that the total differential equals the sum of the partial differentials with respect to these variables. Thus, if u =f{pc, y, z . . . v) where x, y, z ... v are connected by a sufficient number of equations to make u a function of one independent variable only*, du = d^u + dyU + d^^u ... + d^u. Drfferenliation of Implicit Funciiojis of one Independent Variable. 123. We are now able to find -^ where x and y are connected dx ^ by an implicit equation, without solving it with respect to either of them. Let x and y be connected by the equation F{x,y) = 0. Then if Ai-, Aj/ be corresponding increments of x and y, the equation must be satisfied by x + Aj; and y + Ay, .-. F(x + Ax, y + Ay) = 0, * This condition is essential. The very iirst step in Art. 118 depends upon it, since without it, an increment An' of x would not produce corresponding increments of y and u. FUNCTIONS OF SEVERAL VAKIADLES, p3 ^ ^F{x,y) _ F{x + ^x,y + ^y)-F{x,y) _^^ "Ax Ax which being true always is true in the limit; ... ^^j^'-y) = 0, or dF {x, y) = 0. But dF(x, y) is the sum of the partial differentials of F(x, y), .'. F'(x)dx + F'(y)dy = 0, whence -r--- t^,, i • dx F'(^) 124. In like manner, if F{x,y,z) = 0, where x, y and 2 are connected by a second equation so as to make them functions of one independent variable the variables will have corresponding increments Ax, Ay, Az, all of which vanish together. Then F(x + Ax, y + Ay, s + Az) = 0, AF(x, y, z) _ F{x + Ax, y + A^, z + Az) - F{x,y, g) _ ^ A^ Ax ~ ' .-. iE^^^lil) = 0, and dF{x, y,z) = 0; .-. F'(x) dx + F'{y) dy + F'{z) dz = 0, and generally if lil F(x,y,z...v) = 0, where the variables are connected by a sufficient number of equa- tions to make them functions of one independent variable, dF{x,y...v) = 0, and therefore (Art. 122) F'(x) dx + r(y) dy+... F'(v) dv = 0. 125. Hence if n variables x„ X2.. .x„ are connected by the « - 1 equations i^i (j:„ ^2 . . . a;,,) = 0, F^i^i, Xs.. .x„) = 0, F„.i(xi, irj...x„) = 0, in which case they are functions of one independent variable only, we shall have F/ (x.) rfx, + F/ (r,) dx,...+ F,' (xj dx, = 0, 6—2 hi DIFFERENTIATION OF F/ (x.) dxi + F/ (j-g) dxa...+ F,' (x„) dx„ = 0, F'„_i{x,)dx^ + F'„_,(x,)dx,... + F'„_,{x„)dx„ = 0. From these w — 1 equations, any n-2 of the diiferentials may be eliminated, leaving an equation to determine the ratio of the remain- ing two, that is, the differential coefficient of one of the corresponding variables with respect to the other. 126. Let the explicit equations corresponding to F(x,i/) = (1), be 7/=/{x), x = \yy 127. Again, let three variables be connected by the two equations ^tf Fix,y,u) = (1), F,{x,y,v) = (2), and let the explicit equations obtained by the solution of (l) be u=f{x,y), x = {u,y), y^y^{x,u) (S). Then the ratios of the differentials may be determined by combining the equation dFi = with the differential of either of the above forms of equation (1), that is, by combining dF^ = with either of the following equations : F'{x)dx + F'{y)dy + F'(n)du = (4), du=fix)dx+/(y)dy....^ (5), dx = (p'(y)dy + tp'(ii)du (6), dy = \lrXx)dx + \l/'(ti)du (7), and since the ratios of the differentials must be the same from which ever form they are derived, the above equations must be identical. FUNCTIONS OF SEVERAL VARIABLES. t5 Hence by dividing (4) by F' (?<) and comparing the coefficients with those of (5), we obtain So, from (4) and (()) we find in like manner. And from (4) and (7) ^^^^ = -F^y ^(") = -7^y ^W = -T^)- ^(«) = -p(7)- 128. The same results may be obtained somewhat differently, as follows. If we differentiate equation (1) on the hypothesis that _y is con- stant, we obtain an equation between the differentials of x and ti, which may be written in either of the forms P (x) dx + F' (u) dji = 0, or F'(x) d„x + F' (m) du ^ 0, the former expressing the fact that y is constant, by assuming it to be expressed as a function of x and y, and differentiated with respect to X alone, the latter by assuming x to be expressed as a function of u and y, and differentiated with respect to ii alone. The one form tacitly refers to the first, the other to the second of equations (3). Again, by differentiating equations (3) partially, we have d^u =/' (x) dx, d^x — (p'{ii) du, ■which being substituted in the above give F'{x) + F'{it)f(x)^0, r{x)4>'(jc)+F\u) = 0; •• F'{u) ^ ^^~(p'{u)' Similarly, by differentiating equation (1) on the hypotheses that £ is constant and that ii is constant respectively, we should have obtained F'{y) ^^' {x,y), we obtain by equating the differentials of each side, f{x)dx+f{y)dy = cp'{x)dx + 4>'{y)dy, the same equation which results from the differentation of « =/('^, y)-

(2)> x = (p{tc, y), (3), i/ = \}^(u, x), (4). Then it will be found impossible to attach any meaning to the differentials du, dx, dy. For the only definition yet given of a differential is that of Art. 1 1, which is in terms restricted to the case of functions of one independent variable^ and may be stated as follows. If X and y are functions of one another and an increment i\x be given to x, y will receive a corresponding increment Ay bearing a definite ratio to Aa; and vanishing with it. Such increments will have a limiting ratio^ and this limiting ratio is denoted by the ratio -f- . dx For the sake of perspicuity we have, in Art. 118, prefixed the term total to the differential of u, but we added nothing to the definition and the -j- of that article means nothing more than the A?< limiting value of -r— which might have been obtained by first elimi- nating y between the given equations. So the partial differentials of Art. 118 are merely the differentials of u, obtained from the fundamental definition on a particular hypothesis, and involve no extension of the definition. Now if we endeavour to apply the definition to an equation such as (1) or (2), we find that it gives no meaning to the term differ- ential (J. e. total differential) of n, because the hypothesis on which the definition rests, viz., that ?< is a function of some one other variable is no longer true. Thus, if X + /\x, y + Aj/, n + An, be values of the variables which satisfy equation (2) we have Au=fix + Ax, y + Ay)-f{x,y), and no other relation exist between the increments. When an arbi- FUNCTIONS OF SEVERAL VARIABLES. B9 trary value is given to Ax, Au does not assume a definite value, until Ay also has been arbitrarily determined. Hence Au does not bear a definite ratio either to Ax or Ay but depends on both of them, and no such thing exists as a definite limiting value of either of the ratios -.— , -r— , on which our definition of a differential was founded. Ax Ay In short, the definition of Art. 17, was framed exclusively with re- ference to functions of one independent variable and has no applica- tion to the case we are now considering. It is otherwise however with respect to the partial differentials and diffei-ential coefficients. The values of /'(*), /'(i/) rnay l^e ob- tained as before, since the supposition of only one variable receiving an increment, while the other remains constant, reduces the function, so far as these operations are concerned, to a function of one variable only, and brings it within the scope of the original definition. Thus, if ,r receives an increment Ax while y remains constant, u will re- ceive a corresponding increment A^u, bearing a definite ratio to Ax. A II. —^ will therefore have a limiting value when Ax approaches zero. Ax and we may define the differentials d^u and dx to be any quantities whose ratio equals that limiting ratio, whence we have (as before), dM ^ , Am ^, . , I'M! dji . A„?/ .,, ^ W = -^^A^o ^=/(^-), and similarly -^ = //,,=„ ^^ =f{y). Hence, partial differentials of functions of two independent vari- ables have the same meaning as when the variables are connected. Total differentials of such functions have no meaning whatever. If then, the term total differential is to be applied at all to func- tions of two independent variables, it must be by virtue of some new definition. As yet, it is undefined, and we are at liberty either to leave it undefined and unused, or to define it in any way which may be found convenient. In selecting a definition, we shall seek one, which, while it applies to functions of two independent variables, in- cludes within it the definition before given of a total differential of a function of one independent variable only. The advantage of sucli a definition is obvious as it will enable us to work with differentials, without enquiring in each case into the number of independent vari- " We cannot now use ll^_^ instead of ltjy,_^ becaitse Ac and i\y do not necessarily vanish toijetlier. 90 DIFFERENTIATION OF ables. Indeed, unless we attached to the term when extended to functions of two independent variables, a meaning analogous to that which it bears when there is only one, it would be better to leave it undefined and to confine its use exclusively to the latter case. In order to maintain this analogy we define the total differential as follows. Def. The total differential of a function of two independent variables is the sum of the partial differentials. When expressed in analytical language (with reference to equation 2,), the definition is as follows. The total differential, dit, of ?< is a quantity which satisfies the equation du = d^u + dyii, or du^f(x)dx+f'{y)dy (5). So {with reference to equations (3) and (4)} the total differentials of X and_y are defined by the equations dx = (p'(u) du + (p'{y) di/, (6), and dy=^-^'{u)du + y\/'{y) dy, (7), or still more generally with reference to the implicit equation (l), the definition assumes the following shape. If dx, dy, du are any three quantities which satisfy the equation F'{x)dx + F'{y)dy + F'{z)dz = 0, (8), dx, dy and du are called the total differentials of x, y and u. The several equations (5) (6) (7) and (8) are all identical, by virtue of the necessary relations which exist between the forms of F,f,(p and y^/, as appears fi-om Art. 127, so that the differentials on the right hand side of equations (5) (6) and (7) as well as those on the left, represent total differentials. When u is a function of more than two independent variables, the total differential is defined in like manner as the sum of the partial differentials. Thus, if F{x, y ... ti, v] = 0, one of the explicit equations being the total differentials of the variables are defined by the equation F'{x) dx + F'{y) dy ... +F' {ic) du + F'(v) dv = 0, FUNCTIONS OF SEVERAL VARIABLES. 91 or by any of the equivalent equations, as du =f'{x) dx +f'{y) dy+... f'{v) dv. These equations, therefore, having been deduced from the original definition of Art. 17, in the only case where that definition applies, and being themselves taken as the definition in all other cases, are universally true whatever may be the number of independent variables. 131. Hence, generally, if n variables are connected by r equa- tions, i^,(xi,A-2 ... j:„) = 0, F^ (j?! , Xa . . . X,^ = 0, the relations among the total differentials will in all cases be given by the equations F\(x^) dxi + F\{x;) dx^...+ F\{x„) dx„ = 0, F'g{xi) dxi + F\{x^ dx^ ...+ F's,(x„)dx„ = 0, F\{xi) dx^ + F'X^,) dx^...+ F',(x^) dx„ = 0. 132. Whatever be the number of equations between the varia- bles, there will be the same number between their differentials. Hence, when the variables are functions of one independent variable, the number of equations between the differentials will be one less than the number of differentials, and one differential may have an arbitrary value given to it when the others will take definite corre- sponding values. When there are two independent variables, arbi- trary values may be given to any two of the differentials, and in general as many differentials may be arbitrarily determined as there are independent variables in the original equations. Thus when the only equation between the variables is F{x, y, u) = 0, dx, dy, du are by the definition any quantities which satisfy the equation F'(x)dx + F'(y)dy+F'(u)du = 0. To any two of them, therefore, we may give what values we please, when the above equation determines the value of the other. So in all cases where there is but one equation given between the 0% . DIFFERENTIATION OF variables, all but one of the differentials may have arbitrary values assigned to them, ]33. The definitions of Art. 130 will not at once enable the student to attach any other idea to differentials of functions of several independent variables than that of quantities which satisfy a par- ticular equation. There is nothing in this notion analogous to the fundamental idea of differentials of a function of one independent variable, viz. quantities whose ratio equals the limiting ratio of the increments. Such an idea is however involved in the definition of Art. ISOj and it is important to consider it, on account of its value in geometrical and other applications of the subject. An alteration in the form of Art. 17 will help to make this clear. Let y —fix) be any equation connecting x and y ; Ax, Ay, corre- sponding increments of x and y. Conceive two quantities Ix, ly which do not vanish with Ax and A_y but which (as Ax and Ay vary) so change that ly always bears the same ratio to Zx which Ay bears to Ax. Then, instead of the definition of Art. 17, we might have defined the differentials to be the limiting values of Ix and ly. For this would have given us |! = ''^l?- =;,,_. J = ft.., ^ , as before. dx It^^o'l^ ^^ Ax The definition so modified may readily be extended to functions of any number of independent variables when it will be expressed as follows. Def. The total differentials of variables connected by any num- ber of equations, are the limiting values of any quantities bearing the same proportion to one another as the increments, but not vanishing Avith them. It remains to shew that this is identical with the analytical defini- tion of Art. 130, which may be done as follows. Let u =f{x, y). Let Ax, Ay and A?e be any increments consistent with the above equation. Then Ah =/(x + Ax, y + Ay) -/(x, y) = {/(x + Ax, y + Ay) -/(x + Ax, y)} + {/(x + A.v, y) -/(x, y)\ = Aj,/(x + Ax,y) + A,/(x,y) = P.Ax+Q.Ay, FUNCTIONS OF SEVERAL VARIABLES. 93 where P= ^^— , Q- ^^, Therefore if hi, Sx, By be any quantities bearing the same pro- portion to one another as Am, Ax, Ay, we have lu = P.lx + a.hj. In the limits when all the increments approach zero, hi, Ix and ly become, according to the above definition, du, dx and dy, and we have du = It A=oP ' dx + It^i^Q . dy. Also as in Art. 118, and/^.=.Q=.Z/.=„M(-^^.lI./(y); .-. du=f{x)dx+fiy)dy, which is the equation by which the differentials were defined in Art. 130. A similar proof may be applied to functions of any number of variables*. * The mode in which the definition of a differential is extended in the text to cases not contemplated by the original definition may occasion some little difficulty. The principle however is not peculiar to this subject and is precisely the same as that adopted with reference to the fundamental definitions of Algebra. One instance will be familiar to most readers : «"" is defined when m is a positive integer as the product obtained by multiplying a by itself m - 1 times. Hence result the equations a'".a'' = «"■+", (a™)" = a"'-". The above definition is obviously unmeaning when m is negative or fractional. We can attach no idea for instance to the multiplication of a by itself half a time. A new definition becomes necessary, and accordingly we define a" (when m is other than a positive integer) to be such a quantity as satisfies the above equations, which then hold universally. This is precisely analogous to what we did in Art 130 of the text. Having so defined a" in all cases, the next question is whether we caji attach any meaning to it beyond that of a quantity satisfying particular equations, and the result obtained is that a-"" means — and a™ means "Ja. The analogy between this explication of the definition of a" and the process of Art. 133 will be readily seen. The primary rules of algebra require to be extended beyond their original meaning in the same way as the definition of a", although such an extension is com- monly omitted in elementary works for the sake of evading a difficulty at the 94 DIFFEUENTIATION OP 134. It may be observed, that to any two of the increments Am, A.r, Ay, in the last Article, we may give arbitrary values. Hence two of the quantities Sm, ^x, hi/, which are in the same pro- portion, will also be arbitrary, and the same must be true of their limiting values du, dx, dy^ a conclusion somewhat differently arrived at in Art. 132. 135. The foUowing figure will be intelligible to those who are acquainted with Analytical Geometry of three dimensions, and will shew the geometrical meaning of the differentials of functions of two independent variables. PQRS represents a portion of the surface whose equation is 2=f(x,7/), the axis of a; being parallel to Pw?, that of _y parallel toPn, and that of s downwards (which makes the figure clearer than if s were measured, as usual, upwards). Let P be the point x, y, z ; PM = Ax, PN=Ay. Then the increment of z corresponding to an increment of x alone threshold of the subject, and the rules are in fact proved only for particular cases and then tacitly assumed to be universally true. In strictness they can only be gene- ralized by a method similar to that of the text and affording a useful illustration of it. On this subject the reader is referred to Ohm's Mathematical Analysis, trans- lated by Ellis, liondon, 1843. FUNCTIONS OF SEVERAL VARIABLES. 95 is MQ, that corresponding to an increment of y alone is NR, and that corresponding to increments of both variables is LS ; Also RF and QU being drawn parallel to x and ^, LU=/l,z, LV=A^z, VS = A,/(x, y + Aj/), US = Aj{x + Ax, y\ since VS is what MQ, becomes when y + Ay is put in the place of y, and US the value of NR when x + Ax is put for x. If Pm, Pii are lines of any length parallel to x and y respectively, Pqsr the tangent plane at P, and therefore Pq, Pr the tangents to the sections PQ, PR, we may put Pm = c?x, P?J = dy ; whence we have (as in Art. 19) mq = d^z = lu, nr = d^z = Iv. Also since qs is parallel to Pr, us = nr; .: Is = III + us = lu + nr = d^z + d^z^ .: ls = dz. Hence the significance of the definition of Art. 133 appears. For as PM and PiV diminish, the surface becomes more and more nearly identical with its tangent plane ; and since the increments are the co-ordinates from P of any point whatever {S) of the surface, the differentials (or the limiting values of quantities in the same pro- portion as the increments) must be the co-ordinates from P of any point whatever (s) of the tangent plane at P ; which agrees with the above construction. 136. Exactly as in the case of equations of one independent variable, so when there are more, those variables whose differentials are assumed to be constant are called the independent variables, and when none of the differentials are considered constant, the inde- pendent variables are said to be general. Since the partial differentials and differential coefficients are gene- rally variable, we may have second, third... n"" partial differential coefficients and partial differentials, as has already been seen with respect to total differentials of functions of one variable. The same will also be true of the total differentials of functions of several variables, these being generally variable. 96 DIFFERENTIATION OF Again, if u is a function of two variables x and y, the partial differential or differential coefficient of ti with respect to x will generally be a function of ^ as well as x, and may therefore be dif- ferentiated partially with respect to y. Hence Ave may have such quantities as dAd.{u)}, dMij-% -^y-, -^' These latter may be represented in the functional notation by the symbols f"{y, x), f"(x, y), these symbols being understood to mean that we are to differentiate u twice, first partially with respect to x, and then partially with respect to y, and in the reverse order re- spectively. We shall now shew that the order in which such dif- ferentiations are performed is immaterial. '^(f)_'^(g) 137. To shew that ,- j- dx ay By the definition of partial differentials, we have, if ti — /(.r, y'), dy^ 7, J /(.r, ;/ + Ay) -f(x, y) \ dy ^ ^'^-^ I A^ / • Now when x becomes ■r + A^, the above expression becomes i f(x + Ax,y + A y ) -f(x + Ax, y) \ and A- -.''— is the difference between these two values : dy (j/) ••• (2). When we first find the partial differential coefficient of ic with respect to x, and then with respect to y, the result has been repre- H. D. V. ' 98 DIFFERENTIATION OF sented by F''{x, ?/) ; a notation which cannot conveniently be ex- tended to the case where the function is differentiated m times with respect to x, and n times with respect to y. This might be expressed by the symbol i^ <'"+"' (,r,„, ?/„), but such a notation would be very in- convenient in practice. Since, as far as partial differentiation is concerned, u may be con- sidered as a function of one variable only, the values of the differ- ential coefficients in terms of the differentials will be, (Art. 98), when the independent variables are general, dji \dx J f. d^u ^\dyj \dy J d7' —37^'^'' tj' —dT'^^' -TT-'^'---^'^' which differ from the expressions in the case of one independent variable, only by the use of the subscripts, to denote the variable with respect to which the differentiations are performed. When x and y are both independent variables, these become (Art. 99). dji d/u d"ii d^u dfu d,"n d^dyU d/'dy"u . . ~d^' ~d^"'li^' liy' 'df-""tf' dxdy'" dx"'df"'^ ^' These notations (2), (3), (4-) are precise, but the functional nota- tion (2) cannot be conveniently extended to the general case ; (3) if used merely as a notation to represent differential coefficients, is excessively cumbersome, and (4) holds only when x and y are made independent variables. To obviate these objections a less expressive but more convenient notation is generally employed. In the first place the expressions (4) are used for the differential coefficients, not only when x and y are independent variables, but in all cases. When this is done, the numerators cease to represent the successive differentials as heretofore ; but this will occasion no error if we regard those expressions merely as symbols, and not as frac- tions. This inseparability of numerator and denominator is indi- cated by enclosing them in brackets ; and since the variable with respect to which the differentiations are performed is then sufficiently indicated by the inseparable denominator, the subscripts also are omitted. The partial differential coefficients are then represented by fdu\ f(fn\ fdUi\ fda\ f(fu\ (dJ^juS W' \dx"-)--W)' \dy)' \dfj-\dfj' ( dhi \ ( d-^-u \ \dxdy)'"\dx"'dy"J"'^ ''' FUNCTIONS OF SEVERAL VARIABLES. 99 Where the natui-e of the problem precludes any probability of error, even the brackets are frequently omitted. . The notation (2) is recommended in preference, where no differ- ential coefficients beyond the second occur ; in other cases (5) should be employed. The latter however is more generally adopted in all cases. For the sake of familiai-ising the reader with it, we shall employ it in the following propositions. 139. To express the successive total differentials of ?< or F(x, tj) in terms of those of x and y, and the partial differential coefficients of «, neither x nor y being independent variables. We have du = dji + dyU (1), •■•■''-''{(£)''4-4(|)M' Substituting these values, remembering that (j^J = (x"^) ' '^^ have (2) Equations (l) and (2) give du and d'u, and by proceeding in the same way the expressions for the differentials of higher orders may be obtained, but they increase rapidly in complexity as we advance, when X and y are not independent variables. 140. The expressions for the successive total differentials in terms of the successive partial differentials of u are easily obtained Thus dti = dxU + djjii, d'u = d, {dji + dyii] + dy {dji + d^u}, .-. d^u = d/u + d^ d,ji + dy dji + dyU ; 7—2 100 DIFFERENTIATION OF and similar equations for the differentials of higher orders. It will be observed that d^ d^u and dy d^u are not equal because x and y are not independent variables. Equations in the above form are of little use, but we may deduce from them the equations of the preceding Article, which we will now do as an illustration of the principles above explained. 141. From the equation d^u = d/ii + d^ dyii + dy dji + dy^ii, to express d^u in terms of the partial differential coefficients of m. similarly d.dyu = d^ { ©^-^l = (.Sy)^"'^^^Q^^^^' chdji ^ dy I (I;) dx } = Q^;;^ d. drj + g) dydx. Adding these quantities, we obtain since f/j, f/>r + dydx = d'x and c?^ f/y + dy dy — d'y. 142. The expressions which we have found for the differentials of II, hold not only when x and y are connected by some further equation, when one at least of the differentials dx, dy must be vari- able, but also when there is no other relation between the variables than the equation u = F(x, y), so long as the independent variables ai'e left general. On account of the complexity of those expressions this is seldom done, but the variables under the functional sign are almost always made the independent variables. FUNCTIONS OF SEVERAL VARIABLES. 101 143. To express the successive total differentials of u or F {x, y) in terms of the partial differential coefficients when x and y are independent variables. We have du = dji + dyU, Also ^M = djiu + dydu, since dx and dy are constant. By differentiating d'hi, we obtain, in like manner, In these equations it may be observed that the coefficients and the indices of the operations -y-, -j- and of the differentials dx, dy are the same as those of the binomia theorem. By assuming this for d'li, it may be proved for d"^hi, by differentiation, and being true for which may be found from the above equations. In like manner, we have /^\ _ fdti\ fdr\ fdu\ fdd\ \dy)-\dr)\dy)^\de)\Ty)' by substituting which values in the given equation the transforma- tion is effected. The expressions for the differential coefficients of higher orders are obtained in a similar manner, but the general expressions are somewhat complicated, and all that is necessary is to pursue the same method in any particular case. 145. When the equations connecting r and Q with x and y can- not be reduced to such a form as to give r and Q explicitly in terms of X and y, we must proceed as follows. Let x = (p (r, 0), y = ylr (r, 0). (S) = (s)(S)-(|)©' (l') = (S)(l)-(S)(|)- Eliminating successively [-/-) and ( t- ) , we obtain ^du\ (dy\ /du\ /dy\ \drj \ddj~\de) [drj © = ^dx\ /dy\ /dx'K ^di/y \dr) \ddj~\dd) W ^dic\ ^dx\ /dH\ fdx\ fdu\ fdx\ /du\ /dx\ /chA _ \dd) \dJ^) ~ \d?) \ddj \dy) ~ (dx\ fdy\ _ fdx\ (dj/\ " \dr) \dd) \dd) \dr) FUNCTIONS OF SEVERAL VARIABLES. 103 The partial differential coefficients of x and y are immediately de- ducible from the equations connecting x and y with r and 0, and the transformation is effected by substituting in the given equation the above values of (-7- j and (;7-). 146. We will take as an example the expression fdR\ /dR\ the new independent variables being determined by the equations r- = x^ + if. tan = ^ . ^ ' X T,, (dR\ (dR\ X (dR\ -y ^^^" \-d^)=Kd?)-r^\le)-^l^e' = (f)-s-(f) 'dR\ sin Ai fdR\ /dR\ y A/i?\ 1 ^^^° {-dt,)-{-dF)r'-{-de)jl^e' fdR\ . . fdR\ cos d /dR\ /dR\ z'dRs , ^ . ^, ••• ^ U) "-^ W; = Kdi) ^^ ''''^'■^ «"^ ^J f—\ \y cos - X sin 01 _ fdR\ \dd)\ r \-'\d^)- We will treat the same function by the method of Art. 145, em- ploying the equations x = r cos Q, y = r sin Q. -» (|)=(§);--(f)-«. •••(S)=(f)---(S)^. (f)Kf)--(f)^. whence, as before, the expression is reduced to the form Examples will be found in Gregory's Examples, Chap. i. 11. and III. CHAPTER VII. DEVELOPMENT OF FUNCTIONS. 147. To expand f(x + Ji) in ascending powers of h. Lemma, If (p (x) be any function of x, such that cp (a) = and (p'(x) is positive for all values of a; from a to a + h inclusive, then (p (.r) will (between the same limits) have the same sign as h ; and if cp' (x) be negative for all values of x within those limits, then

(.r) must by the assumed conditions pass through A, and (if the differential coefficient is positive from A to B) its form must be that of figure 1, and y will be positive between A and B. So if A is negative, and therefore OB' less than OA, y -will be negative. If ' is negative, the curve will be asin figure 2. DEVELOPMENT OF FUNCTIONS. 105 the above quantities -will be the values of (p/ (x) and cp^'ix), and (pi (x) and (p2 (x) will satisfy the conditions of the lemma, and will thex'efore both be positive or both negative within the given limits, . F{x) . F(x) ^ .: A - 7 — M- and 7 — Mr„ - B, {x — a) {x — a) have the same sign ; wherefore, giving to x one of the admissible values, viz. a + h, ■ — ^~ — '- lies between A and B. This may be expressed by the equation where 0, has some unknown value between and 1, since the quantity on the right of the equation may, by varying B-^ from to 1, be made to assume all values from B to A, some one of which must therefore satisfy the above equation. If not only F {a) but also F'(ci) equals zero, we may prove in the same way that F'{a + e,h) _ F"{a + dji) where 6^ has some unknown value between and 0, , and therefore a fortiori between and 1 . If all the quantities F(a), F'{a) .. . F'"-"(a), equal zero, we may obtain a series of equations similar to the above, of which the last is F^{a + P„,i/0 _ J^w (a + eh) ie„_,hy 1 where the value of 6 must lie between and 1 . Collecting these equations, Ave have (with this last set of con- ditions), F(a + h) _ J-f"' ( g + e/Q A" \n ^^^' where Q has some unknown value between and 1. We will now give a particular value to F(.i;), viz. By differentiating F {£) successively, we obtain. 106 DEVELOPMENT OF FUNCTIONS. F"ix) =r W- |/"(a) +/-(«) (^-«) + - +/""'(«) ^^^TT- } ' j«(x) =/t^) - { fK^) +/^^"(«) (X - a) + . . . +/"-"(«) ^^^V } ' 2r.-i)(^)=/-)(x) -{/■-' (a)}, , i^(x)=/"'(x). All the terms -within the brackets in the above expressions, ex- cept the first, are multiplied by some power of (x-a), and will therefore vanish when x is made equal to a, unless some of their coefficients are infinite. If, therefore, f(x) be such a function that none of the differential coefficients /(a) /"(a) .../""^(a) are infinite, we shall have F (a) = 0, FXa) = ... F^%a) = ... F"-"(a) = 0, and F^^x) =f'(x), F(x) therefore satisfies the conditions necessary to equation (1) and by substituting the values of F(x) and F'"\x) in that equation it becomes /(a + ^)=/(«)+/'(«)AV''(«)-|^••V''^''(«)J|^+/"(« + 0/O^, ~ (2). In this expression we may give to n any value we please (if by so doing we introduce no infinite differential coefficients) ; and if none of the differential coefficients o^ f{x) become infinite when x = a, n may be taken indefinitely large, and we have f{a-i.h)=f(a)^f'{a)h+f"{a)^ + &iC.ininf. (3). The series in this form is called Taylor's theorem. The last term in expression (2) is evidently the difference between this infinite series and its first (w) terms. Its value cannot be exactly determined, since 6 is unknown, but we can determine the greatest and least values which it can assume while Q lies between and 1. Since the remainder of the series after («) terms must lie between these values, they are called the limits of the remainder of Taylor's series. 148. The above series is true for all values of h, and therefore among others for the value dx. If u =f{x) and h = dx, the general DEVELOPMENT OP FUNCTIONS. 107 term of the series becomes •^^^—^'^-^, or (if x is taken for indepen- dent variable) -7— . In this case, therefore, we have /(a: + (/a,) =« + (0) ^^ +/"'(0x) ^; or /(^) =/(0) +/(0)x +/'(0) ^+ . . .. m inf. 108 DEVELOPMENT OF FUNCTIONS. This is called Stirling's or Maclaurin's theorem, and is, as we see, only a particular case of Taylor's theorem. The condition that none of the quantities /(O), f'(0), &c. should be infinite is necessary to its truth in the second form, the first re- quiring only that those up to the h"* order should be finite. 151. As an example of the application of Stirling's theorem, let it be required to expand sin x in powers of x. We have f(x) = sin x, .: /(O) = 0, f(x) = cosx, .-. /(0) = 1, /'Xx) = -smx, .•./"(0) = 0, fXx) = sin fx + « l) , .-. /'"'(O) = (- 1)''""" if n is odd, = if 71 is even. x^ x^ ^r'"*""'* ••• ^^^^ = ^--[3^ [5 +(-^^""'[2^31 i^inf. The limits of the remainder after the term involving a:""-' will be the greatest and least values of /'^'"'(0a-) -t^— , and f^-'Hex) -r— = sin {dx + mir) . — , = (- irsinr^x) -^. Hence the limits are and (- 1)"' sin x t^— • 152. To determine the nature of the expansion of /(a + li) when Taylor's theorem fails. This will happen when any of the functions f{a),f'{a),f"{a), &'c. become infinite. (1) Suppose /(a) = CO . Then f{x) can generally be expressed in the form where m has such a value that ^ (a) is finite. Hence Taylor's theorem holds with respect to (a + A), ... f(a + h) = ^^-^ = h-'" { . Hence we have, as in the proof of Taylor's theorem, F{a + h) _ F^'-\a + Oh) F(a + h) _ F^Xa + dh) h- ~ [l ' which equations being always true, are true in the limit j . F(a + /0_ F^-\a) ' ' "h=o ITZi \ 7— = y> and .^.£&i±5=^,co. Now there will generally be some value of m such that V ( n ■X-h\ ^fk=o — j^ — shall be neither zero nor infinite; and in order that the above equations may be true, m must lie between r — 1 and r. We shall therefore have where C is independent of h and P a quantity which vanishes with h, and which, when arranged in powers of h, must be of the forni C'A" + C"h^ + &c. .'. F{a + h) = C/r + CVr-^+p' + (7'7«''»+*' + &c. or, putting for F{a + Ji) its value, /(« + h) =f{a) +f{a)h +/'•-"(«) r^ + CA'" + 07^+" + &c. i. e. the expansion follows the law of Taylor's theorem so long as that gives finite coefficients, after which a series of fractional powers of h commences, the first of which lies between the last integral index and the integer next greater than it. 110 DEVELOPMENT OF FUNCTIONS. 153. It will be observed that the term generally has been used in these demonstrations. The fact is, that our assumption that /(a:) may be expi-essed in the form '{y)f^, •• di-\^x\i>'{yy'^^^^ dz' and therefore, if C7be any function of ?/, dU ^ , ^ dU (Q\ H. D. C. ° 114 DEVELOPMENT OF FUNCTIONS. Let U=u, therefore -r- = (p(y)-r' cix ^ ^^ ^ dz Also, since £=/(^)|, 0(^)^ = (/;/(i/)|. dV If therefore V be such a function of _?/ that -p = ^ {y)f'{y)> ^^y^d^.^Tz' dhc d dV d dV d {, ^dV\ , , , Now suppose 57.=7;,^|0(j/)"jJ> dx"^^ dx di d" fdV\ d^ilK dz" dx d" i , sdF„) , , , I. e. if the law of formation above assumed holds for -;— , it holds dx" for -^j ; and, as it has been proved for -j-^, it holds generally. Hence we have ?/„ =/(^), DEVELOPMENT OF FUNCTIONS. 115 and the expansion becomes 161. Laplace's theorem. From the equations ij = F{z + xcp{y)} (1), ^«=/(^) (2), to expand ti in ascending powers of x. By Stirling's theorem, fdu\ /d"u\ x" . . „ dx ^^^^ dz' dU ^, .dU From where U is any function of y, L of i^; Let £/■=?/, and assume (p{y) — = —- where F is some function ^ dN,_ d fdV\ _ dx* dx \dz) dz dx ' By the same process as in Lagrange's theorem, we may shew from this equation and equation (3) that the successive differential coefficients are given by the same law, d"u d"-' ( , .du\ ••• -d^^=d^^\^^y^dz\' Hence u^=f{F{z)), + 116 DEVELOPMENT OF FUNCTIONS. dz^-' Y^^^-'J^ dz j \n' 162. Laplace's theorem may be deduced from Lagrange's as follows. Let z+x(p{i/) = v, .'. y = F{y\ V = z + x(p {F(v)}. These equations are now in the form required by Lagrange's theorem, in which we have only to change/ and (p into /jP and (pP, when we obtain the expansion of u as given by Laplace's theorem. CHAPTER VIII. LIMITING VALUES OF INDETERMINATE FUNCTIONS- MAXIMA AND MINIMA VALUES OF FUNCTIONS. 163. To determine the limiting value of a fraction which assumes the form ^ for a particular value of the variable. Let -^-^ be a fraction whose numerator and denominator both vanish when x = a. By the definition of a differential coefficient, •' ^^=°(plx + Ax)-cp{a:) cpXx) In the above equation put x = a, f(a + Ax) _f'(al " "^='(p{a + /^x) cp\ay '^- («) -cp(x) r^^y 165. If all the differential coefficients of/(jr) and (p{x) have indeterminate ratios when x = a, this method fails to determine the limiting value of the fraction. In this case we can only obtain it by algebraical methods, as in the examples of Chapter II. We may proceed as follows : "'="«/) (a:) """=>(« + 70 ■ Let the expansions off(a + h), (p(a +k) in ascending powers of A, obtained algebraically (since, when the differential coefficients become infinite, Taylor's theorem fails) be /(a + h) ■= A/r + J5/i" + . . . fp(a + Ii)^A'k""+B'k"'+... where m and m' must be negative or fractional. Then // /(■^) ^ // ^A"' + Bh"+ ... Ah"^"" + Bh"-""+ ... - iff.=o — ^, :^B'h"'-"" +777 ' = if m is > m', A .„ = -77it m = m' A = CO if m < ?«'. LIMITING VALUES OF INDETERMINATE FUNCTIONS. 119 If we had employed this method when neither f(a), (p (fl) nor any of the differential coefficients are infinite, we should have obtained the results of Art. l63, as is evident from the fact that the expan- sions will be those given by Taylor's theorem. 166. There are some other indeterminate forms which a func- tion may assume, the limiting values of which may be made to de- pend on those of functions which assume the form -^. Such forms are 0", co", 1=^=. (1) Let ?< =/(x)y) (1)' = F{x,y) (2). This may be done by substituting in the former equation for y its value in terms of x, when u becomes an explicit function of x and may be treated by our former methods. Those maxima and minima values corresponding to the condition du = may however be more conveniently obtained as follows. When M is a maximum or minimum, du — 0; .:f{x)dx+f'(y)dy^O, also F' (x) dx + F' (y) dy = 0. By eliminating dy and dx between these equations, we obtain an equation which, together with (2), determines the values of x and y which make u a maximum or minimum. The sign of d^u will distin- guish them. If the same value of x which satisfies the above equations also makes d^ vanish, we must find the first differential, d^u, of u which does not vanish for that value of x, and observe whether n is even or MAXIMA AND MINIMA VALUES OF FUNCTIONS. 123 odd, and in the former case whether dTu is positive or negative. In doing this it will always be convenient, though not necessary, to make either x or y independent variable. 172. To find the maxima and minima values of a function of two independent variables. Def, If any values a, b, of the independent variables in the function /(x, y) make /(«, V) always greater than /(a + ^, h-vh\ whatever be the relative magnitudes and signs of h and h, provided they are taken sufficiently small, /(a, 6) is called a maximum value of/(x, y); and if/(«, b) is always less than /(a + 7^, 6 + A), it is called a minimum value of /(x, y). lif{a,h) is a maximum or minimum, the above conditions, which hold for all relative values of h and k, must hold for that particular system of values of h and k which satisfy the condition k = mh or y-h = m{x-a) ; and conversely, if they hold for all such systems of values, when m is varied in all possible ways, they must hold for the general values, since there is no pair of values of A and k which cannot be found in some one of the particular systems. But the supposition y-h =m{x-a) reduces /(x, y) to a function of one variable, and our definition to that of a maximum or minimum value of such a function. Let therefore ^ be a function of one independent variable deter- mined by the equations, ^=f{^,v) 0)' y — b = 7n(^x — a) (2). The conditions that z shall be a maximum or minimum are (by the previous articles). th=f'{x)dx+f'{y)dy = (3), and dy = m dx, whence dz = dx{f'{x) + mf{y)\ = (4). In order that f{x, y) may have a maximum or minimum value, this equation must hold for all values of ?«, which cannot happen unless both the equations, n^)-0,f{y) = 0, are satisfied. Hence if /(«, b) is a maximum or minimum value, a and b must be a pair of roots of these equations. 124 MAXIMA AND MINIMA VALUES OP FUNCTIONS. z will be a maximum or minimum according as the sign of rf^s is negative or positive for the above values of x and y. If we differen- tiate equations (1) and (2) a second time with x as independent variable *, we obtain from equation (2) (Fij = and then from equation (1), d'z^f"{x)dx^ + 2f"{x,y)dxdy+f"{y)df (5) = rf^''{/"(^) + 2/'(x,3/).«+/"(i/)"^'} (6). When x = o^ y = ^) the value of drz becomes, dx'' {/"(«) + 2/"(a, h) m +f"{b) m% U f{a, b) is a maximum the above quantity must be negative, and if a minimum it must be positive, for all assignable values of ??e. In both cases therefore, it must be incapable of changing its sign for any change in the value of ?«, which will be the case if the equation /" {a) + 2/" (fl, b) m +f" (b) m^ =0 (7), has impossible roots, that is, if r{a)f"(b) is > {/"(«, b)Y. This is called Lagrange's condition. If it is satisfied, /(o, b) will be either a maximum or minimum: to discriminate between them, we observe that the sign of the above expression, being always the same, must be the same as that which it has when m = Q that is the same as that of /"(a), which again must be the same as that of f "(b), since otherwise Lagrange's con- dition would be impossible. Hence /(a, b) will be a maximum or minimum, according as f"{a) and f"{b) are negative or positive. If Lagrange's condition is not Satisfied the equation (7) will have i'eal roots and the value of d'S, when x = a and y = b, will in general be negative for some values of m and positive for others. Hence, some values of m will make z a maximum and others a minimum, in which case f{a, b) will be neither a maximum nor a minimum value of f(x,y). To determine therefore, the maxima and minima values off(x,y)y we must find all the roots of the equations /'(^) = 0, and/'(i/) = 0, * .r is made independent variable for simplicity's sake, but we might have left the independent variable general, when two additional terms would have been added to (5) which would have vanished when a. b were substituted for .i- and y, leaving the subsequent steps the same as in the text. MAXIMA AND MINIMA VALUES OF FUNCTIONS. 125 which satisfy Lagrange's condition and then distinguish the maxima from the minima values by observingthe sign either of/"(«) or of/"(6). 173. There is one case in which f{x, y) may have a maximum or minimum value even though Lagrange's condition fails, viz. where equation (?) has equal roots, and therefore f"{a)f"{b) = {f"{a,h)Y. For if the roots be equal to ?«, , (Pz will retain a constant sign for all values of ?«, except my, and for that value cPz will equal zero. For the value /«,, of m the condition that z may be a maximum or minimum will be that the first differential, d"z, which does not vanish shall be of an even order, and then z will be a maximum or a minimum according as the sign of (Pz is negative or positive. If therefore, 71 is even and the sign o^ d^z when m = m^^ the same as that of d^z for all other values of m, s will be a maximum for all values of ?n including m, or a minimum for all such values. In this case therefore /(or, b) will be a maximum or minimum accordingly. But if n is odd, z will be neither maximum nor minimum when m = mi, and if « is even and the sign of d"z different from the sign of d^z for other values of m, z will have a maximum or minimum value when m - ??«, corresponding to minima or maxima values re- spectively for all other values of m. In neither of the latter cases therefore will /(a, b) be a maximum or minimum. Again, if d^z vanishes for all values of ?w, which will be the case if /"(a) = 0, /"(a, 6) =0, and /'(&) = 0, the condition that /(«, b) shall be a maximum or minimum value, will be that the first actual differential d"z shall be of an even order and retain a constant sign for all values of 7n. The condition which will then take the place of Lagrange's will be that the equation of n dimensions in m, d"z = 0, shall have no real roots, with further conditions in the case of pairs of equal roots analogous to those above found. Such conditions are too complicated to be of much practical value, and in the majority of cases, Lagrange's condition will be found to apply. 174. The hypothesis introduced in Art. 172, that m is an arbi- trai'y constant in the expressions (4) and (6), is evidently the same as the assumption that dx and dy are arbitrary constants in (3) and (5), in which case those expressions -become the first and second total 126 MAXIMA AND MINIMA VALUES OF FUNCTIONS. differentials o^ f(x, y) considered as a function of the two independ- ent variables x and y. The higher differentials of z are in like manner converted by the same hypotheses into the total differentials of/(x, y) with X and y as independent variables. The conditions for determining the maxima and minima values of f(^x, y) may therefore be stated as follows : (1) d/(x, y) = 0, which involves f(x) = 0, f {y) = 0. (2) d'f{x^ y) must retain a constant sign for all values of the differentials, the negative sign corresponding to maxima and the positive to minima values. (3) If d^ f{x, y) vanishes for the same values of x and y as d/(xy), the first actual differential must be of an even order, and must itself retain a constant sign for all values of dx and dy ; and if dy(x, y) vanishes only for particular relative values of dx and dy, retaining a constant sign for all others, the first actual differential for those particular values must be of an even order, and must have the same sign which dy(x, y) has for other values of dx and dy. (4) In every case a negative sign corresponds to a maximum and a positive to a minimum value. By a process similar to that of Art. 172, these conditions may be extended to functions of any number of independent variables. 175. The geometrical significance of Art. 172 appears thus. Let a surface be constructed whose equation is 2 =f(x, y)^ and let/(a, b) be a maximum or minimum ordinate of the surface. Let a plane be drawn through this ordinate, making an angle tan~'7« with the plane of xz. Its equation will be equation (2) of Art. 172, and it will intersect the surface in a curve determined by the pair of equations (l) and (2). By making the intersecting plane revolve round the ordinate f(a, h), i. e. by giving to m all possible values, •we obtain a system of curve sections in some one or other of which every point of the surface is included. An ordinate which is greater than any of the surrounding ordinates of the surface, must be greater than the neighbouring ordinates in any of the curve sections, and conversely if it exceeds the neighbouring ordinates in all the sec- tions, it must exceed all the surrounding ordinates of the surface ; that is, if /(a, b) is a maximum in all the sections it will be a maxi- mum also in the surface, and similarly, if it is a minimum in all the sections it will be one also in the surface ; but if the ordinate is a MAXIMA AND MINIMA VALUES OF FUNCTIONS. 127 maximum in some sections and a minimum in others^ it will be neither a maximum nor a minimum ordinate of the surface, since some of the surrounding ordinates will be greater and others less than it. Lagrange's condition is the analytical expression of these facts. When it is satisfied the ordinate will be a maximum or mini- mum in the surface, if it is so in any one of the sections. The most convenient sections to examine are those parallel to the co-ordinate planes, and this is what we have done, in employing the sign off" (a) or f '(b) as the test for the purpose. 176. In the preceding articles we have considered only those maxima and minima values of functions of several variables, which make the differentials vanish. There are others which make them infinite, but they are of comparatively little interest except with reference to the singular points of surfaces, which is a subject quite beyond the scope of this work. 177. To determine the maxima and minima values of a function of n variables connected by r equations. Let M = i^(j;,, x^ ... x,^ = xa2LJi. or min. , where the variables are connected by the equations =y 2 C-^i > X2 •■■ X„j, (0. o=Mxi, 01:2 '"^„)' Then, since u is a maximum or minimum, du = F'(Xi) dxi + F'(x2) dx, ... + F^x,,) dx„ = ^ also df/i =fi{xi) dxi +fi'{x^ dxn ... +fi'{x„) dx„ = I df, ^fr'ipc,) dx, +fj{x,) dx,... +//(x„) dx„ = From these (r + 1) equations we may eliminate r of the differ- entials, leaving a linear equation involving the remaining (n — r) differentials. This elimination may be conveniently effected by the method of indeterminate multipliers. From the system of equations (1), we obtain du + \,rf/"i + X^dfi ... + A,(//, = 0, which holds for any values of Aj, Aj .., A,. 128 MAXIMA AND MINIMA VALUES OF FUNCTIONS. If we give to du, df^ ... df\, their values from equations (1) and collect the coefficients of each differential, this equation becomes M, dxi + M2dxo ... + M„dx„ = 0, where each quantity M is of the form F'(^) + X,//W + \2/Aa:) ... + K//{a:). The multipliers Aj, A.^ ... A,, being indeterminate, we will give them such values as to satisfy the r equations 3f, = 0, ]\L = ... M, = (2), by which assumption the above equation is reduced to the form M^i dx^+i + M^+j tZx^+2 ... + M„ dx„ = 0. Now in the given system of equations there are (n — r) indepen- dent variables, and therefore the (« — r) differentials in this equation are perfectly arbitrary, and the equation cannot be satisfied unless M,^^ = 0, M,+8 = . . . M„ = 0. By substituting in these equations the values of A,, A, ... A,, obtained from (2), we have (« — r) equations, which, together with the (r) given relations among the variables, are sufficient to determine the values of x,, X2 ••• x„, which make u a maximum or minimum. Hence, whatever be the number of dependent and independent variables, we have the (« + r) equations /.(or,, x.,... x„) = 0, /^(x,, x,...x„) = 0, fr{Xi, Xo ...x„)=0, F'(x,) 4 A,//(x.) + Kf,'{x,) ... + Kf/(x,) = 0, F\x.;) + A,//(x,) + A,//(j:,) ... + A,//(x,) = 0, F'{x„) + A,//(x„) + A,/,'(x„) ... + Kf/(x„) = 0, involving the (n + r) quantities x,, Xa ... .?•„, A,, A, ... A,, from which we may determine the values of j-,, x„ ... x„, which make u a maxi- mum or minimum. The introduction of interminate multipliers, makes this investi- gation rather complex. The principle involved is however very simple. It is (as may be seen) merely this. First to apply the test of a maximum or minimum value, viz., du = 0, and then after elimi- MAXIMA AND MINIMA VALUES OF FUNCTIONS. 129 nating as many differentials as possible to make the coefficients of the remaining independent differentials separately vanish. In most examples the use of indeterminate multipliers will be found much simpler than any other mode of elimination. The investigation of the condition which in this case supplies the place of Lagrange's condition, and the application of the test to dis- tinguish maxima from minima values, would be extremely trouble- some. In most problems, however, it is easy to see from « priori considerations whether maxima or minima values exist, and to dis- criminate them from one another, the exact values of the variables only remaining to be determined. In such cases the investigations alluded to become superfluous. 178. Ex. 1. Find the maxima and minima values of x^(a-j;)'. 71 = A® (a — x)", — = x^ (a - x) {3a — 5x] = or co . The only values of a; which satisfy this condition are x = (}, x = rt, andx = f«, except X =00 , which renders u also infinite, and need not be consi- dered. When X passes through the value 0, — changes from ¥{0, + h) (3a + 5h) to h\a-h) (3a- 5h), both of which have the same sign when h is sufficiently small. Hence when x-0, u is neither a maximum nor a minimum. When X passes through the value a, -— changes from - (a- lifh (2a — Sh) ax to (a + hy h (2a + 5h), the former of which is negative and the latter positive, when h is sufficiently small. Hence when x = a, u has a minimum value equal to zero. When X passes through the value \a, -J- changes from (fa - hf (|« + h) h io-(\a+hf(la-h)h, H. D. C. 9 130 MAXIMA AND MINIMA VALUES OF FUNCTIONS. the foi'mei* of which is positive and the latter negative when h is sufficiently small. Hence, when .r=f«, 71 has a maximum value lOS equal to (f«y(|ayor^^a\ The same results may be obtained by the method of Art. I69. Since ■J- = x^(a — x) (Sa — 5x), dx' = ^^(a-.)(3«-5x)|^-^-^^}, g = 2(«-x)(3a-5x) + P, where P is a quantity which vanishes with x; Therefore when x = Q, -^-^ = 0, ^-3 = ba ; dx' dx that is, the first differential coefficient which does not vanish is of an odd order, and therefore u is neither a maximum nor a minimum. d^u When X = a, -r-j = 2a^, which is positive, therefore ti has a mini- mum value. When X = f a, y-^ = - 2 (f )'' a', which is negative, therefore u has a maximum value. Ex. 2. To find the maxima and minima ordinates of the lem- niscate. The equation is (^x' + yy=a''{x'-r) 0), ^^ ^''' and its form is that of the figure. By differentiating (1) we obtain, 2 (x- + y-) (x dx + 1/ dy) = a^ {x dx - y dy), , dii a^ — 2.r^-2w* J? whence -y- = -^ — —^ — r^ - • dx a^+ 2x^ + 2y y The equation -f- = ^ will be satisfied when MAXIMA AND MINIMA VALUES OF FUNCTIONS. 131 The values of x and 1/ obtained from this equation together with (1) are which may therefore correspond to maxima or minima values. To distinguish them we will apply the criterion of Art. I69. By differentiating -^ , we obtain dx + P, dx" y {o^ + 2a;^ + 9.y-\ when P includes only terms which vanish when equation (2) i? satisfied. Hence when j? — ± */ ^ a-, and y = + a/ - «, dx^ J^a ' which corresponds to a maximum value. \ If therefore OM=OM' = ^| a, and MP = M'P' = ^^ a, ' MP and M'P' are maxima ordinates, /3 1 A when x=^ ./ -a, and y=-x/ 5 «, ^^'^^ ~ x/2a ' or y has minima values at the points Q and Q,'. Since a maximum value of (- y) corresponds to a minimum value of y, the geometrical ordinates MQ, M'Q! are maxima. The values x = ± a and y = will make ^ infinite, but they do not correspond to maxima or minima values, since x cannot exceed a without rendering the equation (1) impossible. In the following example we shall employ the method of indeter- minate coefficients. 9—2 132 MAXIMA AND MINIMA VALUES OF FUNCTIONS. Ex. 3. To find the rectangular parallelopiped which shall con- tain a given volume under the least surface. Let the edges be x, y, z, c? the given volume. Then u ■= xy -v xz -vyz — min. xyz = a^^ .'. (^ -V z) dx -v {x -¥ z) dy + {x + y) dz = 0, yz dx + xz dt/ + xy dz = 0, .'. (y + z + Xyz) dx + (x + z + Xxz) dy + (x + y + Xxy) dz = 0. .-. y + z + Xyz = (1), x + z + Xxz = (2), X +y + Xxy = (3). (l);c-(2)^gives z{x-y) = 0, similarly x {jj — £;) = 0, and y (z - x) = 0^ These, together with xyz = a^, are satisfied by x = y = z = a; that is, the parallelopiped must be a cube. It is obvious that this form makes the surface a minimum^ since, by diminishing one side, we may increase the surface as much as we please. Examples will be found in Gregory's Examples, Chap. vii. CHAPTER IX. TANGENTS AND ASYMPTOTES. 179*. Def. Let APQ be any curve, T'PQ a straight line cutting it in the points P and Q. y Suppose Q to move along the curve towards P, then the direction of T'PQ will continually change, and it will approach a certain limiting position as Q approaches P. Let TP he this limiting position ; then TP is called the tangent to the curve APQ. at the point P. A line PG drawn through P perpendicular to the tangent at P, is called the normal to the curve at the point P. 180. To determine the inclination of the tangent to the co- ordinate axes. Let the equation to the curve referred to rectangular axes Ox Oy, be y =f{x), OM = .r, MP = y, co-ordinates of P, ON =x + Ax, NQ=y + Ay, co-ordinates of Q. Draw PR parallel to the axis of x; then PR = Ax, RQ==Ay, and since PT is the limiting position of QPT' when Q approaches P, that is, when Ax and Ay approach zero, ■■ " A-o • tanPra:-Z^A=otanPrx 'Ax' tanPrx = di dx' ^ is known from the equation to the curve, and the inclination of dx the tangent to the axis of x is determined. • The substance of this and the following article has already been given by way of illustration. It is repeated here in a more condensed shape, as it properly belongs to the subject of this chapter. 134 TANGENTS AND ASYMPTOTES. Hence^ if x' , y' are the current co-ordinates of the tangent^ its equation is y'-y = %'^^'~^^ v' —y x' — X or - — — = ■ . dy dx Since the normal is perpendicular to the tangent, tan PGx = — cot P jTj; = — T- ; dy therefore the equation to the normal is y-y=-Ty^''-'^^' or "^^1^ = 0. dy dx In the figure of Art. 179, ^T, MG are called the suhlangerd and subnormal. Now MT= MP cot PTx, MG = MP tan PTx, .'. subtangent = ?/ — , subnormal =y —-. 181. The sine and cosine of the angle PTx may be conveniently expi'essed as follows. Let the arc ^P = *, arcPQ = A*, chord PQ = c; then RQ sin PTx = Z/a=o sin PT'x — U^^-^ ^^ _ Ay As Now //a=o ■7r-= -t- and ll^^ — = 1, by Art. 11, L\s as c .'. sm PTx = ^; ds PR also, cos PTx = lt/:^-a cos PT'x — It^-o^^^ Ax As A* c ' .*. cosPij: = -r-; ds TANGENTS AND ASYMPTOTES. 135 hence (^J + (^J = cos^ PTx + sin^ PTx^l, or dx^ + dif = ds^. 182, The lengths of the portions of the tangent and normal between the point of contact and the axis of x, are given by the equations Pr= MP cosec pro; ds Jjdx'+df) j{i+/ixy} -^dy-^ dy y fix) ' PG= MP sec PTx The expressions for MT, MG, PT, PG, may also be immediately obtained from the equations to the tangent and normal. Thus, putting y'-O in the equations to the tangent and normal, we obtain in the two cases , dx x-x =y-j-, dy , dy Hence, in order that our expressions for the subtangent and subnormal may hold in all cases, we must reckon MT positive when T lies on the negative side of i\f, and MG positive when G lies on the positive side of M. In the figure of Art. 179 these quantities are positive. 183. To determine the concavity or convexity of a curve at any point. A curve is convex to the axis of x, so long as the inclination of the tangent to the axis of x increases with x, and concave when it decreases with an increase of x. In the former case, therefore, the differential coefficient of tan PTx must be positive, and in the latter negative. Now, tan P 7^0;= •'^ when j/ is positive, = — —- when y is negative ; dx "^ ^ 136 TANGENTS AND ASYMPTOTES. therefore the curve is convex or concave to the axis of x according as -T^ and y have the same or different signs. 184. To determine the position of points of inflexion. Def. a poinl of inflexion is a point where the direction of the curvature changes from convex to concave, or from concave to convex. At such a point the inclination of the tangent to the axis of x, after increasing up to the point of inflexion, begins to decrease, or vice versa. The inclination therefore, and consequently its tangent or fly -J- , must be a maximum or a minimum at a point of inflexion, the analytical condition of which is ^-^^Ooroo. If not only -—^ but also -— vanishes, we must have the addi- tional condition, that the first differential coefficient which does not vanish shall be of an odd order. The values of x which make -yz infinite, will not necessarily cor- respond to points of inflexion, since this condition may be satisfied without the inclination having a maximum value. When the tangent is parallel to the axis oi y,-~ , and therefore generally -r^, are infinite, Avhether the point is a point of inflexion d^x or not. In this case we may employ the condition ~r^i = or co , in- stead of the above, as it might have been obtained in a precisely similar manner, and is free from the ambiguity attaching to our former criterion. 185. To find the equation to the asymptote of any curve. Def. The asymptote of a given curve is a straight line or curve which the given curve continually approaches but never meets. If the equation to the given curve is algebraical, the asymptotes may in general be determined as follows. TANGENTS AND ASYMPTOTES 1S7 Let the equation to a curve be /{x, y) - 0; and let this be reduced to the form 3/ = Jx"+J5a;'"...+ G^+iyx-''+&c (1), the indices being arranged in descending order. Let the equation to another curve referred to the same axes be y' = Ax" + Bx'^...+ G (2), then {y-y') = Hx-^ + 8^c., i. e. the distance between the two curves is generally finite when x is so, but approaches zero when x approaches infinity. The curve (2) is therefore the asymptote to the curve (t). If 71 = 1, m=0, the equation (2) becomes y = Ax + B, and the asymptote is rectilinear. If in the equation to the curve, y is given explicitly in terms of X, the expanded form may be obtained by the ordinary algebraical methods. If t/ is only given as an implicit function of x, the ex- pansion must be found by means of indeterminate coefficients. The most important cases are those in which the asymptotes are rectilinear. It is evident that a rectilinear asymptote coincides with the limit- ing position of the tangent when one or both of the co-ordinates approach infinity. This is sometimes given as the definition of an asymptote, and in many cases the asymptotes may be readily determined from this consideration. Whenever there is a rectilinear asymptote, the perpendicular upon it from the origin, and therefore one at least of its intercepts on the co-ordinate axes, must be finite. Hence the test of the existence of an asymptote is, that the intercepts of the tangent shall not both become infinite when the co-ordinates of the point of contact are put equal to infinity. Asymptotes parallel to either of the co-ordinate axes may be at once discovered, by observing whether any finite value of one of the co-ordinates renders the other infinite*. 186. Examples on the preceding articles. (1) Let the equation to the curve be " Symmetrical investigations of points of inflexion and asymptotes will be found in the Muthemaliail Joxirnal for February 1(541 and November 1843, 138 TANGENTS AND ASYMPTOTES. • ^ xdx _ydy dy _h^ x a dx a' y and the equation to the tangent becomes x(x'-x) _ y{y'-y) a" ¥ ' ^-f- = lby(i). The equation to the normal is or a'yx' + h^xy' = (o^ + h^) xy. The equation to the asymptote may be found from that to the tangent by making x and y approach infinity. The equation to the tangent may be put into the form y = —i — x: . Now when x and y approach infinity — approaches zero, and by (l), Avhich approaches ± j when y approaches infinity. Therefore the equation to the asymptotes is 7/ =^-x'.- ^ a This may also be found b)'^ expanding y in descending powers of X. Thus =44'-^--}' Hence, by our definition, the equation to the asymptotes is y' -^ti. - x'. a Since y-y'=^^~ + TANGENTS AND ASYMPTOTES. 139 the sign of (y-j/'), when x is positive and sufficiently large, is nega- tive for the asymptote whose equation is ^' = + - x', and positive for that whose equation is ^' = x' ; that is, in the former case the curve lies below the asymptote when x is positive, and in the latter above it. Since the sign o? y—y' changes with x, the reverse will be the case when x is negative. This is easily seen to agree with the known form of the hyperbola. (2) To find the asymptotes of the curve {y' + bf-){x-a)-x'^0 (1). When X approaches «, the value of y^ + hy^, and therefore that of y, approaches infinity, and there is therefore an asymptote parallel to the axis of ^ at a distance a from it. To find any other asymptotes that may exist, we must expand y in a series of descending powers of x. Assume, therefore C y = Ax + B +— +&C., .-. y^ = A'x^ + 3A'Bx^+8iC., ... (^3:-a)y^ = A^x* + (3A'B-aA')x^ + &c., b(x-a)y'= bA'x'+8cc., -x'^-x\ Adding these quantities, we have, by (l), = x*(A'-1)+x'' (3A-B - aA' + hA') + &c (2), which must hold for all values of x; .-. ^'-1=0, .-. A = l, 3A'B-aA'+bA' = 0, ■.-. B = "^. o Therefore the expanded form of the equation to the curve is a-b G . j/ = x ^ — ~ — I 1- &c. . . . 3 X and consequently that to the asymptote, / r f^-b ^ 3 If it had been required to determine whether the curve lies above or below the asymptote, we must have collected the coefficient 140 TANGENTS AND ASYMPTOTES. of one more term in the expansion (2), by equating which to zero the value of C might have been determined. The relative position, however, of the curve and asymptote in such cases may generally be more easily determined by indirect con- siderations, as will be seen when we consider the methods of tracing curves. (3) To find the points of inflexion of the curve i/ = - — >,-^ . !/ = (x-«r Then dy 3 (x — a)' which vanishes when x = a ; and as the next differential coefficient does not vanish when x — a, this point is a point of inflexion. Since y and -^ also vanish when x = a, the tangent at the point of inflexion is the axis of J, and the form of the curve is that of the figure y Curves referred to polar co-ordhiates. 187. To determine the angle between the tangent and the radius vector of any curve. Let QPT' be a secant through two points P, Q of a curve APQ. FT the tangent at P; r, d co-ordinates of P, r + Ar, + A0 co-ordinates of Q, refer- red to a pole aS", and prime radius Sx. Draw PR at right angles to SQ.. Then, by the definition of a tangent, tan SPT=ltpQ=o tan SQT, PR TANGENTS AND ASYMPTOTES. 141 r sin A0 = ^^^=° r + Ar - r cos Ad ' rA0 s[n A0 1 = «A=o -^^ -^^g ~ln7iA0 ' 1-^2^ Ar - rA0 dd And "-^=0^,:-^^' sin Afl , sill tJ" , 2smHA0_. siniA0A0 ^^^ //a=o ^^^ "-^=9 iA(? Ar ^ If we call AP, s, and therefore PQ, A.v, and the chord PQ, c, we have cos SPT = lfr(i=o —7" 5 dr "ds' ^ [ ^ sin*iAa\ A* c since Also sin SP T = / or rfr^ + r^ja- = ds\ 188. If 5r, 5r are drawn at right angles to SP, Prrespec tively, we have ST = r tan SPT, -'■'t- ST is called the j^olar suhlangenl. 142 TANGENTS AND ASYMPTOTES. Also, if SY= = P, P = = ?• sin SPT, = ds This quantity may be conveniently expressed in terms of the reciprocal of r. If this be u, we have du dr^ UUf .-. ds' = du'' d&^ 1 ,ds^ = „ fdu\ It may also be expressed in terms of the rectangular co- ordinates as follows. If -Sis the origin and Sx the axis of x, we have e = tan->-^; x' .'. de xdy — ydx ~~ 1 . .2 p xdy — ydx r^dQ X dif —ydx ds ds The expression for ST has been found from a figure in which dr ..... r, 0, -ja have all positive signs, but it applies equally to all cases, if we interpret properly the sign with which it is affected. By con- structing the figui'es in the cases where any of these quantities are negative, it is easily seen that the direction of ST is always deter- mined by the following rule, when the positive direction of Q is taken to the left of the prime radius. Look along the positive direc- tion of the radius vector, and draw ST to the right or left, according as it is positive or negative. It may also be observed that the expression for tan SPT is, in all cases, positive when the tangent falls behind the radius vector, and negative when on the other side of it. TANGENTS AND ASYMPTOTES. 143 189. To determine the asymptotes of polar curves. When any value of renders r infinite, there may be a recti- linear asymptote. This will be the case if the polar subtangent re- mains finite when r becomes infinite. Since in this case the angle SPT vanishes, the asymptote will be parallel to the infinite radius, and at a perpendicular distance from it equal to the corresponding value of ST, and measured in the direction indicated by its sign. When the radius vector approaches a constant limiting value, as 6 approaches infinity, the curve will approximate to a circle which is called an exterior asymptotic circle when r increases in magnitude up to the limiting value, and an ifiterior asymptotic circle when r de- creases towards it, as in the former case the circle is without, and in the latter within the curve. 190. To determine the concavity or convexity of curves referred to polar co-ordinates and the position of points of inflexion. It is easily seen that when a curve is concave to the pole, p increases with r, and when convex p decreases with an increase of r. The curve is therefore convex or concave according as -j- is nega- tive or positive. At a point of inflexion, the curve changes from convex to concave or vice versa, and therefore -j- changes its sign. At such a point therefore we must have -f- = 0, or CO . ar Those values of r which satisfy this condition, and also produce a change in the sign of -,- , will correspond to points of inflexion. Examples will be found in Gregory's Examples, Chap. ix. CHAPTER X. CONTACT OF CURVES. 191. Let y =f{x), y = F{x) be the equations to two curves ; then if, when a certain value, o, is given to x, the values of the ordinates of the two curves are the same ; that is, if f{a) = T{a), the two curves have a common point whose co-ordinates are a, f{a), and are said to intersect. If at the point of intersection we have also f{a) = F'{a), the curves have a common tangent, and are then said to have a con- tact of the first order. And generally, if we have f{a)=^F{a), f{a) = F'{a), f"{a) = F"{a) ...f'^\a) = F^^a), the curves are said to have a contact of the h"* order at the point in question. The reason of this definition appears from the following pro- position. 192. If any curve has a contact of the nf^ order with a second curve, and a contact of the n*'^ order with a third at the same point, where m is greater than w, and an ordinate is drawn cutting the three curves, the portion intercepted between the curves whose con- tact is of the m}^ order bears to that intercepted between those whose contact is of the n^'^ order, a ratio which approaches zero as the ordi- nate approaches that through the point of contact ; or, as it may be expressed, the curves whose contact is of the higher order are, in the limit, infinitely closer than those whose contact is of the lower order. For, let the curve y =f(j>c) have a contact of the m*^ order, and y = (p(x) a contact of the n**" order, with y = F{x) at a point whose abscissa is a ; let an ordinate be drawn to the three curves whose abscissa is a + ^, and let A„,, A„ be the portions intercepted between y = F{x) and the two other curves. Then A,„ =/(« + h)-F{a + h\ and /(a + h) =f{a) +f(a)k ... +/-'(«) j^^ +/'-'(« + Ok) p^^^. CONTACT OF CURVES. 145 [m ^ '' [in + I * The first (?m + 1) terms of these series are equal by the conditions of contact of the ?«'*' order, 7m+l similarly, A„ = ——^ {0'"+"(« + ^^0 - ^<"+"(« + 0/«)}, ... ^ = /r "• j^^^ (^'"+')(a + 0A) - F'"^\a + Oh) ' When h approaches zero, the coefficient of /i'""" approaches a finite value, since otherwise the contacts would be of higher orders than those supposed ; and if m is greater than n, the limit of /t"*"" is zero. A_„ A" Ith^o -r^' - 0. 193. Curves which have a contact of an even order cut, and those which have a contact of an odd order touch without cutting at the point of contact. For if A„ is the excess of the ordinate of i/ =f(x) over that of y = F{x) with which it has a contact of the m^^ order, ^- = tItt ^^"""^'' "■ ^^'^ - ^'"'^'^ + ^^'^^- When h is made sufficiently small, the sign of the coefficient is that of/''"+^'(a) - jP>"'+'*(«) whether h is positive or negative, and the sign of /^'"+' changes with that of h^ when m is even and is unaltered by a change of sign in h when m is odd. In the former case therefore the ordinate of one curve is in excess on one side of the point of contact, and that of the other on the other side, while in the latter, the ordinate of the same curve is in excess on both sides. Hence, when the contact is of an even order the curves cut, and when of an odd order, touch without cutting. 194. If we suppose the same quantity to be independent vari- able in two curves which have a contact of the m"' order, the con- ditions are that at the point of contact the values of X, ;/, dx, dy (f "x, dJ'^y shall be the same in the two curves. H. D. c. 10 146 CONTACT OP CUEVES. For if any function of x is independent variable, the values of dx, d^x . . . d"'x must be the same in both curves when the same value is given to x, and dj/, d^y . . . d"y can be expressed in terms of these quantities and of the m first differential coefficients, which are also the same in the two curves by the conditions of contact of the ?«"" order. When any of the differential coefficients have infinite values, the proof in Art. I9I, which depends on Taylor's theorem, fails. If the infinite differential coefficients are of the second or higher orders, the curves are of peculiar forms (as will be subsequently shewn) at the point in question, such that the notion of orders of contact is inapplicable. If however the first differential coefficients are infinite, there may be no peculiarity except that of the tangent being parallel to the direction which has been chosen for the axis of ^. The difficulty is at once obviated by equating the differential coefficients of x with respect to y in the two curves, instead of those of ?/ with respect to x, since these conditions might have been ob- tained exactly as the others. When the differentials are equated, no difficulty can arise, since the conditions of contact in this form might have been obtained from either of the other systems. For this reason, and also for the sake of symmetry, it is often better to employ the differentials than the differential coefficients. 195. To determine the particular curve of a given group which has the closest possible contact with a given curve. Let j/ =/(x) be the equation to any curve y = F{x, ai, a^ ... a„), that to a group of curves having 71 parameters Oj Oj. . .«„, by assign- ing particular values to which, any curve of the group may be determined. Let now such values be given to a^ a„ . . . «„, as to satisfy the n equations, y'-y ^-^ d^^_i1r^y y ~y' dx~ dx dx-' ~ daf-' ^ ^' when a particular value, a, is given to x. This can evidently be done, since o, a2,...a„ enter into the values of y' and its differential co- efficients ; the n parameters will then be absolutely determined by CONTACT OF CURVES. 147 the assumed equations and cannot be made to satisfy any additional conditions. When these values ai'e given to the parameters, the par- ticular curve thus selected from the group will have a contact of the (n - 1)"" order with the curve 1/ =f{x) at the point whose abscissa is a. Hence the order of the closest contact which a curve of a given species can be made to have with a given curve at any point of it, is equal to the number of parameters in the general equation to curves of that species, diminished by unity. It may happen in particular cases, that the same values of the parameters which satisfy the 71 equations (1) will also make some of the higher differential coefficients equal in the two curves, and in that case the contact will be of a higher order; but in general the order of closest contact is less by unity than the number of parameters. The general equation to the circle contains thi'ee parameters ; a circle may therefore always be found which shall have a contact of the second order at any point of any given curve. 196. The curvature of a curve at any point may be conveniently measured by determining the radius of the circle which has the closest possible contact with the curve at that point. Hence the fol- lowing definition. The circle which has a contact of the second order with a curve at any point, is called the circle of curvature of that point, and its radius the radius of curvature. 197- To determine the radius of curvature and the co-ordinates of the centre of the circle of curvature at a point (x, y^ of a given curve. Let p be the radius, a, /?, the co-ordinates of the centre. Then the equation to the circle is (^-«r+(^-/5y=/ (1), ■whence {x — a)dx+ {y — ft)dy = (2), (x - a) d^x + {y-ft) dhj = - dx" -dy'- = - ds^ (3) ; where dx, dy, d^x, d^y, ds are the differentials of the co-ordinates of the circle. We have now to give to o, /:?, and p such values, that the circle and curve may have a contact of the second order at the point (x, y'), that is, such values that x, y, dx, dy, d-x, d'y may have the same values in the circle as in the curve. , If therefore we give to the 10—2 148 CONTACT OF CURVES. differentials in equations (l), (2), (3) their values derived from the equation to the curve, those three equations will give the required values of a, /3, and p. To find these explicitly, we have from (2) ^ X — a y — ft ^(x - a) d^x + {y — ft) cPy dy — dx dy d^x — dx d^y dyd'x-dxd'y' ^ ^^^ :c-a_ y-ft _ J{{x-ay + {y-ftr] dy - dx J{dx^ + dy^) = -£.by(0. Equating these values, we obtain dyd?x — dxd^y dy ds P rl,.rl^ ^ ^ rl-^rfi,,' X + ft=y- dyd^x — dxd^y dxds^ dy^x — dxd^y ' The magnitude of p without regard to sign being the object of investigation, we must employ the upper or lower sign in any par- ticular curve according as the former or the latter renders the above expression positive. The direction in which p is to be drawn will be determined by the values and signs of the expressions for a and ft. " The artifice of elimination here used, depends on the following algebraical proposition. If a _ c t hen -7 = -. = — ; jt tor all values of «i and n, o d nib + nd' " >J(,b^+d')' For ma+nc = jlmb +71 - c\=-{mb + nd), and VK + 0=| \/ [f^' + ^.c^yiy/Cb^ + d^). whence the above equations follow. By giving convenient values to m and n, as in the instance in the text, this often affords a ready method of elimination. COXTACT OF CURVES. 149 198. If we denote by t the angle which the tangent to the curve makes with the axis of x, we have T - tan"' -; : ax ' and r', which is the equation to the evolute. For examples on the present Chapter, see Gregory's Examples, Chapter xii. CHAPTER XI. SINGULAR POINTS. 205. Certain points of curves present peculiarities of different kinds inherent in the curves themselves, and not dependent on the position of the co-ordinate axes. Such points are called singular points: the most important are miultiple points, conjugate points, cusps, and points of inflexion. Multiple points are those through which several branches of a curve pass. Conjugate points are isolated points through which no continuous branch passes. Cusps are points at which the curve suddenly stops and returns in the opposite direction. At a cusp, therefore, the curve has two branches with a common tangent. Cusps are of two kinds : (1) When the two branches are on opposite sides of the tangent. (2) When they are on the same side of it. The two kinds are represented in the following figures. Points of inflexion have already been defined in Art. 184. In the following investigations of the analytical properties of singular points, the equations to the curves are supposed to be alge- braical and not transcendental or circular. 206. If a straight line is drawn intersecting any curve, it will meet it in a number of points which may equal but cannot exceed the order of the curve. (Hymers' Conic Sections, Sect. 238). If we suppose the line to move parallel to itself, the number of points in which it meets the curve may in the course of its motion undergo vai'ious changes. Thus in the accompanying ^ 158 SINGULAR POINTS. figure the line MPp meets the curve in two points. When by- moving parallel to itself it reaches either the point A where it be- comes a tangent or the multiple point B, the two points of intersection coincide. Beyond B again, the double intersection is restored. So if there had been more than two branches passing through B a greater number of intersections would have been reduced to a single one. This property Avill furnish an analytical test of the existence of a multiple point. For convenience, suppose the intersecting line to be parallel to the axis of i/. The ordinates MP^ Mp of the points of intersection for any assigned value of x will then be the roots of the equation to the curve, when solved with respect to y. Whenever two of these roots become equal the corresponding point must either be a multiple point as at B or must have its tangent parallel to the axis of j/ as at A. To distinguish multiple points from such points of contact, sup- pose another set of intersecting lines to be drawn parallel to the axis of X. Then the points of intersection for any assigned value of y will be given by the roots of the equation when solved with respect to x; and as before, when any two of these become equal there must either be a multiple point or a point whose tangent is parallel to the axis of x. If therefore for any values of x and y two or more roots of the equation become equal both when solved with respect to x and with respect to y, the corresponding point must be a multiple point, since the same tangent cannot be parallel to both axes at once. Hence the following proposition. 207. To determine the analytical properties of a multiple point, and to find the number and direction of the branches which pass through it. 'Let f\x,il)-0 (1), be the equation to a curve cleared of radicals; ^,, y^ co-ordinates of a multiple point. If in (1) we put ;r = x,, the equation must have two or more roots each equal to y^; and if we put y=^y^, it must have two or more roots each equal to x^. Since the equation is in a rational SINGULAR. POINTS. 159 form, the condition of its having equal roots is that the derived equation shall have a root of the same value*. The equations, /W = 0, /(y) = 0, ■will therefore be satisfied as well as equation (1) by those values of X and _y which correspond to a multiple point. To determine the values of -^ at the point in question, we have, by differentiating (1), /'0^)+/(J/)|^=0 (2), which gives only an indeterminate value of j^ when the conditions of multiplicity are satisfied. Differentiating (2), we have /"W+2/"(.,y)g+/'W(|y+/'(^)g=0 (3), and putting x = x^^ y ^y^' /'(.,) +2/"(x..,,)|u^"(y,)(|y=o (4). If any of the coefficients in this equation are finite, it will give two values of -i~: if these are real, two branches of the curve pass through the point {x^ , _?/,) at the inclinations determined by the two values of — : if they are impossible, no branch passes through the rtXj point, which is therefore a conjugate point, since its co-ordinates satisfy equation (1). If all the coefficients in (4) vanish, the values of the differential coefficient remain indeterminate. To determine them we must differentiate (3), and in the result make x and y equal to x, and j/, : to simplify the process we may observe that (4) might have been obtained from (2) by differentiating 7 72 it as if -T- were constant, since the term involving -j4 must be mul- dx ax tiplied by a coefficient which vanishes when .r, y are put equal to • See Hymers' Theory of Equations, Sect. IV. It will be observed that the proof of the property employed in the text depends upon the equation being cleared of radicals. 160 SINGULAR POINTS. Xi, 3/,; the same will be the case with the equation derived from (3), which will be a cubic equation giving three values of —^ . If these are all real, three branches pass through the point at inclinations determined by these values ; if only one root of the equation is real, there is a single branch of the curve through the point (x,,3/,). If this equation becomes indeterminate by its coefficients vanish- ing when X, y equal ar, , y^, we must proceed as before, differen- tiating as if -J- were constant, until we arrive at an equation which has finite coefficients. The number of real roots will indicate the degree of multiplicity of the point. If the equation for determining -j- has equal roots, two or more branches will touch. Points where this occurs are sometimes called points of osculation. The equation obtained after n differentiations is generally of the «'^ degree in -^ , but it may happen that the coefficients of some number (r) of the highest powers of ->— vanish while other coeffi- cients remain finite, thus reducing the degree of the equation to dx, . n — r. If, however, we had sought the values of -j- instead of those of ^, these would have been the coefficients of the r lowest instead ax, of the r highest powers, and there would have been r roots of the equation each equal to zero. In the former case, therefore, there must be r roots each equal to infinity, and r branches of the curve pai-allel to the axis of y, besides those determined by the other real roots of the equation. 208. In this method of determining the values of — it has been assumed that -j— vanishes when multiplied by a coefficient equal to zero; this may not be true, however, when the second differential coefficient becomes infinite at the point in question, a condition which, as will be seen, generally indicates some other SINGULAR POINTS. 161 peculiarity besides multiplicity, such as the existence of a point of inflexion or a cusp. Any error arising from this source may be obviated by transferring the origin to the point which we are ex- amining, and finding the limiting value of ^ when x and j/ approach zero, which will then be identical with the differential coefficient at that point. This is more laborious than the preceding method, except when the multiple point is originally at the origin, and as the error alluded to is of rare occurrence the former method is generally preferable. 209. To determine the analytical properties of cusps. Since a cusp is in fact a species of multiple point, it will appear, by the same reasoning as before, that the co-ordinates of such a point in a curve whose equation is /(or, ij) = must in general satisfy the equations That which distinguishes a cusp from a multiple point, is that the two branches of the curve lie only on one side of the point instead of passing through it. If, therefore, a, b are the co-ordinates of a cusp, and we give to X in the equation to the curve the values a + h, a - h, one of these must give two real values of y while the other renders y impossible, h being taken sufficiently small. This can only be the case, in algebraical curves, if the equation, when solved with respect to y, is of the form y = F(x) + (p{x) (x ~ a)", where w is odd and 7i even, since the radical has two real values when (x ~ a) is positive and is impossible when this quantity is negative. By repeatedly differentiating the above equation, we shall at length arrive at one of the differential coefficients of y which con- tains a term involving a negative power of (x ~ a) and which there- fore becomes infinite when x = a. Hence the distinguishing pro- perty of cusps is that their co-ordinates render some of the differ- ential coefficients of ^ infinite. H. D. c. 11 162 SINGULAR POINTS. There is one case to which the above reasoning does not apply, viz. that where the cusp is of the first kind and the tangent paral- lel to the axis of 3/, since in this case the values a-h, a + h of x will each give a single real value of y. Here, however, the first differential coefficient is infinite, and our general criterion that some of the differential coefficients must become infinite still holds, the other necessai-y conditions being those of 7/ having a maximum or minimum value at the point in question, which have been already investigated. If the tangent is parallel to the axis of 1/ and the cusp of the second kind, the substitution of a + h and a-h for x in the equa- tion to the curve will give respectively two and no real values of ^. To distinguish between the two kinds of cusps when the tan- gent is not parallel to the axis of 1/, we must observe whether the difference of the ordinates of the curve and the tangent at the cusp, corresponding' to an abscissa a ^ h, has the same sign for both the values of y, or whether it is affected by a double sign. This differ- ence =/(fl ± A) -{/(«) =^/(«) ^'}- If we give to h that sign which corresponds to the real values of 1/, and expand the above expression in ascending powers of h, the development must contain an even root of Ii, since a change in the sign of h renders y impossible. If this root appears in the first term of the series the cusp is of the first kind, otherwise it is of the second kind ; for in the former case the two values of the series have opposite signs, and in the latter the same sign, when h is taken so small that the sign of the first term is that of the whole series. If /"(a) is finite, the test assumes a simpler form; for then the difference becomes = -^/"(« ^ Gh), which ultimately has the same sign or signs as /"(a). If, therefore, /"{a) has a double sign the cusp is of the first kind, otherwise of the second. 210. To determine the analytical properties of conjugate points. Whenever the co-ordinates of a point of a curve render the first differential coefficient impossible, no branch of the curve can pass through the point, since at every point of a continuous branch the SINGULAR POINTS. 163 tangent of the angle which its tangent makes with the axis of x must have a real value. If the equation to the curve, f(x, y) = 0, is in a rational form, the value of -p obtained from the equation cannot be impossible for any values of x and y, since both /'(x) and fXy) niust be free from radicals. If, therefore, -^ have an impossible value, this equation must become indeterminate, or the co-ordinates of the conjugate point satisfy the equations f{x,i/) = 0, f(x) = Q, f{y) = 0. There may, however, be conjugate points in a curve without -J- becoming impossible, in which case the above equations are not satisfied. For the point (o, b) will be a conjugate point if y is real when x^a but impossible when x = a + h or a — h, h being taken suffi- ciently small. This can only occur, in algebraical curves, when the equation solved with respect to _?/ is of the form y = F(x) +

Qb <.a, ^ is — , y = dy_ _ /(ct-h\ dx Sj \ a )' x> a, y \% -V, X =TO y = co, dx (3) Negative values of ^: - x = 0, J/ = 00 , r/ = co, dy ~ X = (X> , dx = 1. Hence the form of the curve must be that of the figure. The curve may be verified and more exactly determined by seeking the positions of the maxima and minima ordinates, and the points of intersection of the curve and the asymptotes. 215. When the equation to the curve gives y only in an im- plicit form, we may often trace the curve by obtaining simple ap- proximate forms of the equation which hold when the co-ordinates approach zero and infinity respectively. Let, for example, the equation to the curve be y^ - 20" x^y + x^ = 0. 168 TRACING OF CURVES. (1) To find the asymptote, assume (J i/=Ax + B + — + &c., .-. 9/'=A'x' + 5A*Bx* + 5A3(AC + 2B')x^ + &c. " - Sa" x'lf = - 20," Ax^ + &c., x^ = x^, .: = (^^+1)0;''+ 5A'Bx' + A {5A'C + 10A'B'-2a')x'+ &c., .-. ^'+1=0, .-. ^=-1, 5A'B = 0, .-. B= 0, 5A^C+ 10 A'B'-Oa'=0, .'. C = -la\ Therefore the expanded form of the equation to the curve is 2 a^ y = — x — - — + &c. •^ 5 X The equation to the asymptote is therefore and the curve lies below or above it according as x is positive or negative. (2) Omit the term x^ in the equation, and consider whether the equation thus obtained is for any values of the co-ordinates an ap- proximate representation of the given curve. It is y^ = 2a Vy, or y* = 2a'x^ When this equation holds, the dimensions in x of each of the terms retained are t, while those of the neglected term are 5 ; if, therefore, we suppose x to be small, the term omitted is small com- pared with those retained; and the equation so obtained is there- fore, Avlien x is small, an approximate form of the original equation. Hence the given curve has a branch which approximates, when near the origin, to the form of the curve y^ = ^j2ax, which is represented in the accompany- y ing figure. (3) Omit the term y\ Then the equation becomes 2fl V^ = x\ " ^ 2a-' TRACING OF CURVES. 169 Here the dimensions in x of the terms retained are 5, and those of the term neg- lected 15. The above equation is therefore approximately true when x is small. Its form is that of the figure. By omitting 2a^x^y, we should only obtain the equation to the asymptote, which we have already found. This would give the asymptote accurately in the present example; but in general, when the asymptote does not pass through the origin, it would give only an approximate form of it, namely, a line through the origin parallel to it ; the former method of determining the asymptote is therefore in general necessary. From the above data we can construct the curve, which must be of the form represented. This may be verified by seeking the position of the points of in-* flexion and the maxima and minima values of the ordinates. It may be further verified by examining the nature of the mul- tiple point at the origin, which we will now do. y" - 2aV_y + x' = (1), .-. (5x^-4a^^j/) + (5y-2«y)^ = (2). There will be a multiple point when, together with equation (1), the equations 5x* — A:a^xy = 0, 5/ - 2a'x' = 0, are satisfied. The only values of x and y which satisfy these three equations are x = 0, y = o. Differentiating (2) as if -.- were constant, 170 ITUCING OP CURVES. (20a:« - A.a'y) - %a^x ^ + 9.0 f (^Y = 0, which becomes indeterminate when x and j/ are put equal to zero. Differentiating again, after dividing by 4, we obtain which becomes, when x = Q, y-0, f- = 0. ax There is therefore one branch parallel to the axis of x, and two parallel to that of y, corresponding to the two infinite roots due to the disappearance of the two highest powers of the cubic. This result accords with the figure. 216. When the equation to the curve is expressed in polar co-ordinates, the direction of the curve at any point is determined by the inclination () of the tangent to the radius vector where tan ^ = r-7-. It is also necessary to find the polar subtangent in order to determine the position of the asymptotes. Let, for example, the equation be ^ "TTTcosT' ^ ^^^"^ greater than unity; differentiating the logarithm of r, we have dr e sin 1 + e cos Q JA = -k J •'• tan rf> = ; ;; , r dQ 1 + e cos ^ e sm /ST = r tan = — %—^ . Let = 0^ 1+e' ( tan ^ = CO , > < a, r is +, a being the smallest value of cos~^ f — 1 , and therefore between and TT ; ( r = 0D, e sin a -H ^(e* - 1) TRACING OF CURVES. 171 > a < TT, 7" IS — , e + 1' tan (p =(X) . When = 6' ± 2w TT, the value of r is the same as when 6 = 6'. Hence all portions of the curve corresponding to values of 6, either negative or greater than ^ir, only repeat those already found between the limits and Stt. By observing the rule with respect to the sign of ST, the asymp- totes corresponding to = a, and = 27r — a respectively, will be Ppy Qq, and the curve will be represented by the figure. A being the point where 6=0 and a that where 6 = Stt. The asymptotes evidently intersect at C the bisection of Aa^ since sT=sr= and ST Additional examples will be found in Gregory's Examples, Chapter xi. '-r ?jyi^ /> . OS* CHAPTER XIII. ENVELOPES, DIFFERENTIALS OF AREAS, ETC. 217. To find the equation to the envelope of a group of curves. Let f{xy,a) = (1), be the equation to the curve whose parameter is (a), f{x,y,a + Aa) = (2), that to another curve of the same group, whose parameter is (a + Aa). These curves will intersect in the points whose co-ordinates are given in terms of a and Ac, by equations (1) and (2). As Aa varies, this point of intersection changes its position, and the limiting posi- tion to which it approaches when Aa approaches zero is called the point of ultimate intersection of the curve (1), and the contiguous curve of the group. At the point of intersection of (1) and (2) we shall have /(x, y,a + Aa) -/(x, t/, a) _ ^ . Aa ""' f(x, t/,a + Aa) -f{x, y,a) _f. and, therefore, at the point of ultimate intersection, + Aa) -/(j Aa ... <%^ = (3). da ' Equations (l) and (3) therefore determine the point. By eliminating a between (1) and (3), we obtain an equation between x and y, which holds at the point of ultimate intersection, and which is independent of a. This is therefore the equation to the locus of the points of ultimate intersection of curves of the group represented by equation (1). Let the equation to this locus be <^(^,^) = (4). Then it may be shewn that the curve (4) touches each of the curves of the group (l), and that every point of (4) is touched by some curve of the group. For this purpose it is only necessary to ENVELOPES DIFFERENTIALS OF AREAS, ETC. 173 bear in mind that any one of the equations (I) (3) and (4) is deriv- able from the other two. Thus, let a have a particular value in (1) so that that equation may represent some one curve of the group. Then at the point of intersection of (l) and (4.) equations (1) and (4) hold together, and therefore at that point equation (3) also holds. Now the value of -^ at the point of intersection, in the curve (l) ax is given by the equation /W+/Wj = (5), and that in the curve (4) by the pair of equations /'W+/w|-/'«| = ol (g), which are identical with (5), and therefore the curve (4) touches the particular curve (1). So also, if we take any point of (4) whose co-ordinates are (x,, y^), it will meet that curve of the group whose pai'ameter is given by equation (1) when x,, y^ are put for x,y. The parameter thus obtained will therefore also satisfy equation (iV (3)with the same values of x, y ; and the values of ^- at the point of contact will be determined by equations (5) and (6) respectively, which as before are identical. Hence, any point x,, y^ of the curve is touched by one of the curves of the group (1), For these reasons the locus of ultimate intersections of a group of curves is also called the envelope of the group. 218. If the equation to the group contains two or more parame- ters connected by equations, so that one only is independent, it may be reduced to the assumed form by eliminating all the parameters except one, by means of the equations connecting them. It is however often more convenient to differentiate the equation before such elimination, and afterwards eliminate the differential coefficients of the dependent parameters by means of the given equations between them. The operations in this case become more 174 ENVELOPES DIFFERENTIALS OP AREAS, ETC. symmetrical by the use of difterentials instead of differential coeffi- cients, when the elimination may generally be most readily effected by the method of indeterminate multipliers, explained in Art 177- As an example, we will find the envelope of the system of lines of a constant length whose extremities lie in two lines at right angles to each other. Let the two lines be taken for the axes of x and 1/ ; c the constant length of each line of the group. The general equation to any line of the group is M- (-)• where a, h are parameters connected by the equation a' + b'^^c^ (2). The equation to the envelope will be given by the elimination of a and b, between (1), (2), and the equations xda ydb _ ada + bdb = 0, which together are equivalent to equation (3) of Art. 217. Using an indeterminate multiplier X we have i^-Xa\da + \^^-Xb\db = Oy whence we may obtain X —,-Xa-O, a and p — a6 = 0, X y X y _ a b I •'• ^ " P " 6^ ~ a' + b^ " ? ' a WxJ ' b=\7if' therefore substituting in (1), the required equation is 2. z z For examples see Gregory's Examples, Chapter xiii. ENVELOPES DIFFERENTIALS OF AREAS, ETC. 175 219. To find the point of ultimate intersection of a normal to a given curve with the contiguous normal, and the locus of all such pointe. Let the equation to the curve be y=K^) (!)• Let j:', y' be the co-ordinates of the point of ultimate intersection of the normal at (x, 7/) with the contiguous normal, and let the dis- tance between {x, y) and (x', y') be denoted by p. Then {x'-xj^^y'-yy^p' (2). The equation to the normal is {x'-x)clx + {y'-y)dy = Q (3). And at the point of ultimate intersection the differential of this equa- tion with respect to the parameters x, y, must also vanish ; .-. {x'-x^d^x + {y'-y)d?y = dx^ + dxf (4). Equations (2), (3) and (4) will give x', y' and p in terms of x and y and their differentials, y and the differentials may then be eliminated by means of equation (l) and its derived equation, and j;', y' and p obtained in terms of a: alone. By eliminating x between the values of x' and y' we shall obtain the locus required. The equations for determining x\ y' and p are the same as those found in Art. 197 for the determination of a, /3 and p. The point of ultimate intersection of contiguous normals is there- fore the centre of curvature, and the locus of all such points the evolute of the given curve, a result which might have been foreseen from geometrical considerations. 220. This property furnishes a more convenient method of ex- pressing p in terms of ^ and r than the one given in Art. 199- Let P be any point of a curve, SP = r, SY = p, \^> the centre of curvature at P, and therefore , , . , PO = p. o/ Then SO' = SP' + P0'-2 SP. PO cos SPO = r^+p'-2pp. S Since is the point of ultimate intersection of contiguous nor- mals, a change in the position of P will in the limit produce no 176 ENVELOPES DIFFERENTIALS OF AREAS, ETC. change in the position of or the length of OP. The above equa- tion may therefore be differentiated with respect to r and p, con- sidering SO and p as constant. Hence 0^2rdr-2pdp; dr ''■P^'Tp- 221. To find the differential of the area included by a curve and two ordinates. Let Pd be two points of a curve whose co-ordinates are AM = x, MP = y, and AN=x + Ax, NQ. = y+Ay; and let the area between AM, MP, the curve CP, and some fixed ordinate equal ^. Then the area MPQN will with the same notation equal AA, AA being the increment of A corresponding to an in- crement Ax of X, This lies between the areas of the pa- rallelograms MR, MQ, i. e. between i/Ax, and (y + A?/) Ax ; and since the limiting ratio of these areas when Ax approaches zero is unity, that of AA and either of them must equal unity, AA 't/A. or dA =ydx. It ■AzzO = h 222. To find the differential of the area included by a curve and two radii vectores. Let P, Q be points of a curve whose polar co-ordinates are ?; 6, and r + Ar, 6 + A0, and let A be the area between the curve CP, the radius SP, and some other fixed radius. Then will SPQ equal AA, the increment of A corre- sponding to an increment Ad of d. AA lies between the areas of the circular sectors whose radii are SP, SQ respectively, and vertical angle equal to PSQ, i. e. between r -A0 2 and ^SZ^rlAe: 2 ENVELOPES DIFFERENTIALS OF AREAS, ETC. 177 and since the limiting ratio of these sectors when Ad approaches zero is unity, that of AJ. and either of them must equal unity ; AA lis=,- = h Ad .-. dA = -dQ. Q 223. To find the differential of the volume included by a sur- face of revolution and two planes perpendicular to its axis. Let V be the volume pi'oduced by the revolution round AN of the area bounded by MP, the curve CP and a fixed ordinate, (see fig. Art. 221). Then AV will be the volume produced by the revolution of TQ,, AV being the increment of V corresponding to an increment Ax of x; AV lies between the cylinders produced by the revolution of PR, QS respectively, i. e. between TTT/^Ax and Tr(i/ + Ayf Ax : and since the limiting ratio of these volumes when Ax approaches zero is unity, that of AFand either of them must also equal unity; AV f(s=o TxfAx : dV = 1, = ir'lfdx. 224. To find the differential of the surface of a solid of revolu- tion bounded by two planes perpendicvilar to its axis. Let S be the surface produced by the revolution of a curve CP about the axis of x, A M. Draw Pn, -m q Qm parallel to the axis of x and each p^ equal to PQ. Then A^", the incre- ment of S corresponding to an incre- ^ ment Ax of x, will be the surface pro- duced by the revolution of PQ, and ^ ~ '^ ip PQ = As if arc CP = s ; AS lies betwen the surfaces produced by the revolution of Qm and Pit, i. e. between IttjjAs and 27r(y + Ay) As. Since the limiting ratio of these surfaces when A^ approaches zero is unity, that of AS and either of them must equal unity ; A^ 'i.TryAs .'. dS =Q.-7r)/df, = S'/ry J(dx- + df). II. D c. 12 //v- = 1, CHAPTER XIV. INTEGRATION BETWEEN LIMITS AND SUCCESSIVE INTEGRATION. 225. To express an integral as the limit of the sum of a series. Let/'(a:) be any function of x, a any constant^ and let the interval X- ahe divided into n parts each equal to Ax, so that Ao; = , and .•. x = a -vn Ax. n Let (j) (x) =f(a) Ax +f'{a + Ax) Ax +f(a + 2 Ax) Ax +f'{a + (w - 1) Ax} Ax. The value of (p (x) depends on the number of terms (re) of this series, and the consequent magnitude of Ar. Now when « ap- proaches infinity. Ax and therefore each term of the above series approaches zero, and (p (x), whose actual value then becomes indeter- minate, will approach a certain limiting value ; let this be \}/- (x). To determine the value of xjy (x) cliange x into x + Ax in the above series, which is equivalent to adding to it the term / (a + n Ax) Ax, or f'{x) Ax. Hence, (p (x + Ax) - (p (x) =f'{x) Ax ; (p (x + Ax) — (p (x) Ax -fix). In this equation let Ax approach zero; then (pix) becomes in the limit V^(^), .:lt^.i^^^^^^=^^^^A/(x), .:^'{x)=f{x); therefore \\/ (x) must be one of the values contained in the general integral, /(x) + C, of/'(x). To determine the value to be given to C, we have y}.{x)=fix) + C, .: x|,(fl)-/(a) + (7=0. INTEGRATION BETWEEN HM1T8. l79 since (p (a) and therefore \lr {£) evidently vanishes when x is made equal to a ; ••• C = -/(a), and x/.(*)=/(a)-/(a) (1). The general form of the integral of /'(x), viz. /(x) + C, is written l/Q'')dx, the indeterminate constant being included in the integral sign. When such a value is given to C that the integral shall vanish when x equals any particular value a, the integral f{x) — /(«) is written I /'(x)dx ; and if in this we give to x any Ja particular value b, the resulting value of the integral, viz. /(&)—/(«) is written \f'{x')dx, \f'{x)dx is called the indefinite integral of f\x), \f'{x)dx the corrected integral of /'(x), Ja j f'(x)dx the dejinite integral off'(x), between the limits a and b. With this notation equation (l) becomes, giving \|/(x) its value, and writing f(x) instead off'(x), ^y(x) dx = ftA=o {/(«) ^^ +/(« + -^0 ^x ... +f(x - Ax) Ax} ; : j/(x) dx = Iti^ {/(a) Ax +f(a + Ax) Ax ... +f(b - Ax) Ax}. i- The indefinite integral must therefore equal the limit of the sum of any number of terms of a series formed like the above, commencing with any term whatever of the series, and termi- nating with that in which the quantity under the functional sign is X — Ax. The above equations may be written for the sake of brevity as follows : ^f{x)dx = //a=o 2/-^'{/(x)Aa:}, ^J{x)dx = U^^, ^t^' {fix) Ax}. 12 2 180 INTEGRATION BETWEEN LIMITS. 226. To determine the area included by a curve and two given ordinates. In Art. 221 it was proved that if A is the area included by a curve, a fixed ordinate, and an ordinate whose abscissa is x, clA = y c?x, = \ydx, = ^ (x) + C suppose. The area and the integral are alike indeterminate, the former on account of the indeterminate position of the fixed ordinate, and the latter on account of the arbitrary constant which enters into it. If the fixed ordinate is made definite and taken to be that whose abscissa is a, the arbitrary constant must have such a value given to it that the area shall vanish when x is made equal to a ; .-. (a) + C = 0, and A—f^ (x) —

y{-©>- 230. These expressions for areas, arcs, &c., may be found at once from the considerations of Art. 225. Thus, to find the area ABPN. Let 0A = a, ON=b, Om = x, mp = 1/. Divide AN into a number of parts each equal to Ax, and INTEGRATION BETWEEN LIMITS. 183 A m n jsr on these as bases construct parallelograms as in the figure. Then, assuming- that the area is equal to the limit of the sura of the parallelograms when Aa- approaches zero, we have area AB^P = It^ 'EJ'-^^yAx, since i/Ax is evidently the area of any of the parallelograms, as tip, .-. area ABNP= j 1/dx, Art. 225. The parallelograms or other portions into which the whole quantity is divided are called elements. By taking as our element the cylinder generated by the revolution of one of the pai'allelograms, and assuming that the volume of the solid of revolution is the limit of the sum of the cylinders when Ax approaches zero, we have = TT / y-dx. By taking the chord pq as our element and assuming that the arc BP equals the limit of the sum of the chords, when Ax approaches zero, we have s = lts^i:j'-''^J(Ax' + Af), =iV{-(g)'}- By taking as our element the surface generated by the revo- 184 INTEGRATION BETWEEN LIMITS. lution of the chord pg about the axis of .r, and assuming that the sur- face of revolution is the limit of the sum of these elements when Ax approaches zero, we have S = U^^^ 2/-^- 27rijJ{Ax' + Aif), =-/>y{-(S)v^- By taking as our element the circular sector whose radius p is Sp and vertical angle A0 and assuming that the sectorial area BSP between radii, for which the values of 6 are a and j3, is the limit of the sum of such elements when A0 approaches zero, we have area BSP - It^^, ^^'-^^^r'AO, < rcld. It will be observed that the assumptions made in each case are equivalent to the equations giving the differentials of the areas, arcs, &c., in Arts. 221 —224. In fact it is the same thing to say that the whole quantity equals the limit of the sum of the elements, as that the increment of the whole and the element have ultimately a ratio of equality, which is what those equations amount to when expressed in words instead of symbols. 231. This method of dividing a quantity into elements may be applied to other functions besides areas, volumes, &c. Thus let it be required to find the limit of the sum of every portion of a circular area, multiplied by the square of its distance from the centre, when the magnitude of each portion is indefi- nitely diminished. Let the radius of the circle be a. Take as an element the annulus whose inner radius is r and breadth Ar. INTEGRATION BETWEEN LIMITS. 185 The required function for this annulus = (area of annulus) r*, neglecting the difference of the values of r for different parts of the annulus, since when the limits are taken any error from this source vanishes. Therefore the required quantity = 27r / r^ dr = 2t- 4 - (area of circle) — . 2 Successive Integration. 232. Hitherto we have considered integration only as the con- verse of the simple differentiation of a function of one variable. It may however be extended in the same way as differentiation, and the converse of the operations of successive and partial differentiation will be called successive and partial integration. Such operations are denoted by merely prefixing the integral sign ( () to the differential, of whatever kind it may be, as many times as there are operations to be performed. Thus if u —f{x) we have, when x is independent variable, du =/' (x) dx, d'u=f"{x)dx\ the converse of which equations will be written, tl = If (j;) dx, u^jjr(x)dx^ So if u =/(x, 2/), djc =/' {x) dx, d^u =f" (or) dx\ dj,u = f" (.»;, y) dx dy. 186 INTEGRATION BETWEEN LIMITS. X and y being the independent variables^ whence we have n = j/ {x) dx = fjf" (x) dx- = J|r (^, y) d^ dy. The last integral is more often written \dx \A\)f''{x,y), a form which will be found convenient when the integration is between definite limits. An integral involving two or more signs of integration is called a double or multiple integral. 233. We have seen that in order to obtain the general value of \f'(x)dx it is necessary to add an arbitrary constant, because such a quantity would disappear in differentiation. The same must be done at each stage of the successive integration of a function of one variable. Thus ff"(x)dx=f(x) + C; ••• f ff"(^) d^' = f(/'^ + O) dx =f(x) + Cx+ C, where C and C' are arbitrary constants. So if we integrate n times in succession with respect to x we must add, since such a quantity would disappear in the corresponding differ- entiation. For the same reason, when we integrate a partial differential djf{x, if) we must add the most general quantity which would have disappeared in obtaining c?^/(a:, _?/). This may be any function ofy or a constant, both of which are considered as included in the general form (p {y) where

,{y), (pi and (p2 being arbitrary functions. Again, if we have to integrate first partially with respect to one variable and then partially with respect to the other we have jdyjdxf" ix, y) =jdy {/{y) +

// {y) is the integral of the arbitrary function

/XP r-Jd^-j:^ Also I dy = sja--x^, and j^ xdy = x Jcc" - x- ; Also j^ Jx j^ dy - j^ Jx ^a--x^ = \x Ja^- x- + 1 «' sin-^ Ji'^ ^' =!;«■• "3' 3 4m y ~ IT „ ~ 3^ "7 « 4 o nd -=— ^= — 4 238. The method of converting the limiting sums of series into multiple integrals, is equally applicable whatever may be the number of independent variables. In every case, the increments are changed into the corresponding differentials and the sign of summation into that of integration, the limits of the integration being the same as those of the series. The order of integration is very material. For in general the limits at each integration are functions of all the variables with respect to which the integration remains to be per- formed; their values therefore depend on the order of integration, 192 INTEGRATION BETWEEN LIMITS. and the latter cannot be varied after the limits are once fixed. The only exception to this, is when all the limits are constant, as some- times happens, in which case we may differentiate in any order we please. A careful consideration of the examples above given, will make the truth of these observations apparent. In working ex- amples great care is requisite to avoid errors in fixing the limits, which can only be done with accuracy by keeping constantly in view the series of which the integrals are the limiting values. Examples of the determination of areas, surfaces, &c. will be found in Gregory's Exainples, Part ii. chap. ix. Applications of in- tegration to summation, are met with in all physical subjects, and form the principal use of the Integral Calculus. 239. To express an integral by an infinite series of finite terms. By integrating by parts, we obtain \f{x) dx =/(*) X - \f'{x) X dx, jf(x)xdx=^fix)'^-ljr(x)x^dx, ff'ix) x^ dx =f" {x) I' - 1 jr'{x) x^ dx, &c. = &c. .-. fAx)dx^f(x)x-f{x)^...^f-^x)^^l^jf"^ix)x''dx. If in this series we make ?t infinite, we have f/(x)dx=fix)x-fix)~ +/"(x)-^-&c hi inf. The term t— | /"*"' (j?) x" dx is therefore the difference between \ny the infinite series and its first n terms. An arbitrary constant must be added to give the general value of the integral. In the first form of the expression the constant is included in the last term. 240. From the formula of integration by parts we may obtain a proof of Taylor's Theorem, which presents the remainder after n terms in the form of a definite integral. INTEGRATION BETWEEN LIMITS. 193 let x = a + k — z; .'. dx = — dz; and when x = a, z = h, X = a + k, z = ; .'. f{a + h) -/{a) = JV(« + k-z) dz. Now by repeatedly integrating by parts, as in Art. 239, jf(a + h-z)dz=f(a+h-z)z+/"(a + h-z)^...... +f"-\a + h-z)^^+^jf^(a+h-2)z''-^dz. Therefore taking the integral between the limits and h, /(« + /^) =/(«) +/(«.) h+f"(a)^ +/"-)(«) -^ + -. f V"' (a + h- 2) z"-' dz. The value of the remainder after 7i terms is, therefore, j^ />(»+/, -.).-&. This is easily shewn to lie between the limits of the remainder before found. For [ "/"" (a + h- z) z"-' dz = It^^ So"-A-~ {/(") {a + h- z) z"-' Az). The particular value ofy'"'(a + h - z), by which any term of this series is multiplied, must lie between the greatest and least values which that function can assume while z varies from to h. There must, therefore, be some intermediate value of z which gives to the function a value, which multiplied into every term, gives the same result as when each term is multiplied by the par- ticular value belonging to it. This vaUie of rr may be represented by (1 — 0) h, where lies between and 1. By substituting this, the series becomes f''{a + eh)lt^^,^:-'^-'z''-'dz, H. D. C. 13 194 INTEGRATION BETWEEN LIMITS. ••• (/*"' (a + h- z) s"-' dz =/<"> (« + eh) f z"-' dz Jo . Jo = i>O(a-f0/O. And the value of the remainder is, therefore, ^f'Ka + eh), the expression before obtained. The expression in the form of a definite integral is of course preferable, when the integi'ation can be effected, as it determines the exact value of the remainder and not merely the limits between which it lies. THE END. NEW AND IMPORTANT MATHEMATICAL EDUCATIONAL WORKS. A Treatise on Analytical Statics, with numerous Ex- amples. By I. ToDHUNTER, M.A., Fellow of St John's College, Cambridge. Preparing for immediate Publication. Solutions of Senate-House Problems for Four Years (1848 to 1851), By N. M. Ferrers, Fellow of Caius College, Cam- bridge, and Rev. J. S. Jackson, Caius College, Cambridge, 870. cloth, 15s. 6d. Solutions of the Senate-House Riders for Four Years (1848 to 1851). By the Rev. F. J. Jajieson, M.A., Fellow of Caius College, Cambridge. 8vo. cloth, 7s. 6d. 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