JBHHBHg | LIBRARY j j B. 0. M. DeBECK [ | ^.jAm j V ■*v. ; \ h JT i i **^^ ***•<**. ** \ AN ELEMENTAEY TREATISE ON THE DIFFERENTIAL CALCULUS FOURTH EDITION. AN ELEMENTARY TREATISE ON THE INTEGRAL CALCULUS, CONTAINING APPLICATIONS TO PLANE CURVES AND SURFACES. BY BENJAMIN WILLIAMSON, F.R.S. IN THE PRESS. AN ELEMENTARY TREATISE ON DYNAMICS. BY BENJAMIN WILLIAMSON, F.E.S., AND FRANCIS A. TARLETON, LL. D., Fellows of Trinity College, Dublin. AN ELEMENTARY TREATISE ON THE DIFFERENTIAL CALCULUS, CONTAINING THE THEORY OF PLANE CURVES, WITH NUMEROUS EXAMPLES. BY BENJAMIN WILLIAMSON, M.A., F.R.S., FELLOW OF TRINITY COLLEGE, AND PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF DUBLIN. NEW YORK: APPLETON AND COMPANY 1884. [all rights reserved.] 9A50+ ixri 242141 BOSTON COLLEGE LIBRARY CHESTNUT HILL, MASS. PREFACE. In the following Treatise I have adopted the method of Limiting Eatios as my basis ; at the same time the co- ordinate method of Infinitesimals or Differentials has been largely employed. In this latter respect I have followed in the steps of all the great writers on the Calculus, from Newton and Leibnitz, its inventors, down to Bertrand, the author of the latest great treatise on the subject. An ex- clusive adherence to the method of Differential Coefficients is by no means necessary for clearness and simplicity ; and, indeed, I have found by experience that many fundamental investigations in Mechanics and Greometry are made more intelligible to beginners by the method of Differentials than by that of Differential Coefficients. While in the more ad- vanced applications of the Calculus, which we find in such works as the Mecanique Celeste of Laplace, and the Meea- nique Analytique of Lagrange, the investigations are all conducted on the method of Infinitesimals. The principles on which this method is founded are given in a concise form in Arts. 38 and 39. In the portion of the Book devoted to the discussion of Curves, I have not confined myself exclusively to the ap- plication of the Differential Calculus to the subject ; but have availed myself of the methods of Pure and Analytic vi Preface. Greometry, whenever it appeared that simplicity would be gained thereby. In the discussion of Multiple Points I have adopted the simple and Greneral Method given by Dr. Salmon in his Higher Plane Curves. It is hoped that by this means the present treatise will be found to be a useful introduction to the more complete investigations contained in that work. As this Book is principally intended for the use of begin- ners, I have purposely omitted all metaphysical discussions, from a conviction that they are more calculated to perplex the beginner than to assist him in forming clear conceptions. The student of the Differential Calculus (or of any other branch of Mathematics) cannot expect to master at once all the difficulties which meet him at the outset ; indeed it is only after considerable acquaintance with the Science of Greometry that correct notions of angles, areas, and ratios are formed. Such notions in any science can be acquired only after practice in the application of its principles, and after patient study. The more advanced student may read with advantage the Reflexions sur la Metaphysique clu Calcul Infinitesimal of the illustrious Oarnot : in which, after giving a complete resume of the different points of view under which the principles of the Calculus may be regarded, he concludes as follows : — " Le merite essentiel, le sublime, on peut le dire, de la methode infinitesimale, est de reunir la facilite des procedes ordinaires d'un simple calcul d' approximation a 1' exactitude des resultats de l'analyse ordinaire. Cet avantage immense serait perdu, ou du moins fort diminue, si a cette methode pure et simple, telle que nous l'a donnee Leibnitz, on voulait, sous l'apparence d'une plus grande rigueur soutenue dans tout le cours de calcul, en substituer d'autres moins naturelles, Preface. vii moins commodes, moins conformes a la marche probable des inventeurs. Si cette methode est exacte dans les re- sultats, comme personne n'en doute aujourd'hui, si c'est tou- jours a elle qu'il faut en revenir dans les questions difficiles, comme il parait encore que tout le monde en convient, pourquoi recourir a des moyens detournes et compliques pour la suppleer? Pourquoi se contenter de l'appuyer sur des inductions et sur la conformite de ses resultats avec ceux que fournissent les autres methodes, lorsqu'on peut la demontrer directement et generalement, plus facilement peut-etre qu'aucune de ces methodes elles-memes ? Les objections que Ton a faites contre elle portent toutes sur cette fausse suppo- sition, que les erreurs commises dans le cours du calcul, en y negligeant les quantites infiniment petites, sont demeurees dans le resultat de ce calcul, quelque petites qu'on les sup- pose; or c'est ce qui n'est point: 1' elimination les emporte toutes necessairement, et il est singulier qu'on n'ait pas apercu d'abord dans cette condition indispensable de 1' elimi- nation le veritable caractere des quantites infinitesimales et la reponse dirimante a toutes les objections/' Many important portions of the Calculus have been omitted, as being of too advanced a character; however, within the limits proposed, I have endeavoured to make the "Work as complete as the nature of an elementary treatise would allow. I have illustrated each principle throughout by copious examples, chiefly selected from the Papers set at the various Examinations in Trinity College. In the Chapter on Eoulettes, in addition to the discussion of Cycloids and Epicycloids, I have given a tolerably com- plete treatment of the question of the Curvature of a Roulette, as also that of the Envelope of any Curve carried by a rolling viii Preface. Curve. This discussion is based on the beautiful and general results known as Savary's Theorems ; taken in conjunction with the properties of the Circle of Inflexions. I have also introduced the application of these theorems to the general ease of the motion of any plane area supposed to move on a fixed Plane. In this Edition I have made little alteration beyond the introduction of a short account of the principles of the deter- minant functions known under the name of Jacobians, which now hold so fundamental a place in analysis. Tklntty College, June, 1884. TABLE OF CONTENTS. CHAPTEB I. FIKST PRINCIPLES. DIFFERENTIATION. Page Dependent and Independent Variables, - I Increments, Differentials, Limiting Ratios, Derived Functions, . . 3 Differential Coefficients, .5 Geometrical Illustration, ........... 6 Navier, on the Fundamental Principles of the Differential Calculus, . . 8 On Limits, 10 Differentiation of a Product, . . . • . . . . 13 Differentiation of a Quotient, . . . . . . . • 15 Differentiation of a Power, 16 Differentiation of a Function of a Function, 17 Differentiation of Circular Functions, 19 Geometrical Illustration of Differentiation of Circular Functions, . . 22 Differentiation of a Logarithm, 24 Differentiation of an Exponential, ... ..... 26 Logarithmic Differentiation, 27 Examples, 30 CHAPTER II. SUCCESSIVE DIFFERENTIATION. Successive Differential Coefficients, . Infinitesimals, ....... Geometrical Illustrations of Infinitesimals, Fundamental Principle of the Infinitesimal Calculus, Subsidiary Principle, Approximations, ...... Derived Functions of x m , .... Differential Coefficients of an Exponential, Differential Coefficients of tan _1 #, and tan -1 -, . x Theorem of Leibnitz, Applications of Leibnitz's Theorem, Examples, ....... 34 36 37 40 4i 42 46 48 50 Si 53 57 Table of Contents. CHAPTER III. DEVELOPMENT OF FuTSTCTIOlSrS. Taylor's Expansion, Binomial Theorem, . .... Logarithmic Series, . . .... Maclaurin's Theorem, ...... Exponential Series, Expansions of sin x and cos x, . Huygens' Approximation to Length of Circular Arc, Expansions of tan -1 x and sin -1 x, .... Enler's Expressions for sin x and cos x, . John Bernoulli's Series, Symbolic Eorm of Taylor's Series, .... Convergent and Divergent Series, .... Lagrange's Theorem on the Limits of Taylor's Series, Geometrical Illustration, ..... Second Form of the Bemainder, .... General Form of Maclaurin's Series, Binomial Theorem for Fractional and Negative Indices, Expansions by aid of Differential Equations, . Expansion of sin mz and cos mz, .... Arbogast's Method of Derivation, . Examples, ........ Page 6l 63 63 64 65 66 66 68 69 70 70 73 76 78 79 81 82 85 87 CHAPTER IY. INDETERMINATE FORMS. Examples of Evaluating Indeterminate Forms without the Differential Cal cuius, .... Method of Differential Calculus, Form o x 00 , Form^, .... CO ' Forms o°, 00 °, i ±cc . Examples, .... CHAPTER Y. PARTIAL DIFFERENTIAL COEFFICIENTS. Partial Differentiation, ....... Total Differentiation of a Function of Two Variables, Total Differentiation of a Function of Three or more Variables, Differentiation of a Function of Differences, Implicit Functions, Differentiation of an Implicit Function, Euler's Theorem of Homogeneous Functions, . Examples in Plane Trigonometry, Landen's Transformation, . .... Examples in Spherical Trigonometry, .... Legendre's Theorem on the Comparison of Elliptic Functions, Examples, ......... Table of Contents. XI CHAPTER VI. SUCCESSIVE PARTIAL DIFFERENTIATION. The Order of Differentiation is indifferent in Independent Variables, Condition that Pdx + Qdy should he an exact Differential, Euler's Theorem of Homogeneous Functions, .... Successive Differential Coefficients of (too ™y* p dW dW dW Transformation <&—+—+ — ,. . . . Geometrical Illustration of Partial Differentiation, 399 403 404 405 407 DIFFEKENTIAL CALCULUS CHAPTEK I. FIRST PRINCIPLES — DIFFERENTIATION . i . Functions. — The student, from his previous acquaintance with Algebra and Trigonometry, is supposed to understand what is meant when one quantity is said to be a function of another. Thus, in trigonometry, the sine, cosine, tangent, &c, of an angle are said to be functions of the angle, having each a single value if the angle is given, and varying when the angle varies. In like manner any algebraic expression in x is said to be a function of x. Geometry also furnishes us with simple illustrations. For instance, the area of a square, or of any regular polygon of a given number of sides, is a function of its side ; and the volume of a sphere, of its radius. In general, whenever two quantities are so related, that any change made in the one produces a corresponding variation in the other, then the latter is said to be a function of the former. This relation between two quantities is usually represented by the letters F, f, (cc) is said to be a continuous function of x, between the limits a and b, when, to each value of x, between these limits, corresponds a finite value of the function, and when an infinitely small change in the value of x produces only an infinitely small change in the function. If these conditions be not fulfilled the function is discontinuous. It is easily seen that all algebraic expressions, such as &qX t Oi\X t • • • . dny and all circular expressions, sin x, tan x, &c, are, in general, continuous functions, as also e°, log x, &c. In such cases, accordingly, it follows that if x receive a very small change, the corresponding change in the function of x is also \&ery small. 5. Increments and Differentials. — In the Differen- tial Calculus we investigate the changes which any function undergoes when the variable on which it depends is made to pass through a series of different stages of magnitude. If the variable x be supposed to receive any change, such change is called an increment ; this increment of x is usually represented by the notation Ax. When the increment, or difference, is supposed infinitely small it is called a differential, and represented by dx, i.e. an infinitely small difference is called a differential. In like manner, if u be a function of x, and x becomes x + Ax, the corresponding value of u is represented by u + Au ; i. e. the increment of u is denoted by Au. 6. limiting Ratios, Derived Functions. — If u be a function of x, then for finite increments, it is obvious that the ratio of the increment of u to the corresponding increment of x has, in general, a finite value. Also when the increment of x is regarded as being infinitely small, we assume that the ratio above mentioned has still a definite limiting value. In the Differential Calculus we investigate the values of these limiting ratios for different forms of functions. The ratio of the increment of u to that of x in the limit, dif when both are infinitely small, is denoted by — . When ax B 2 4 First Principles — Differentiation. u =/(#), this limiting ratio is denoted by /'(#), and is called the first derived function* off(x). Thus ; let x "become x + h, where h = Ax, then u becomes f(x + h), i.e.u + Au =f(% + h), .'. Au =f(x + h) -f{x), Au f(x + h) -fix) Ax h The limiting value of this expression when h is infinitely small is called the first derived function of f(%), and represented Au Again, since the ratio — has/' (x) for its limiting value, i^X if we assume Au e must become evanescent along with Ax ; also — becomes Ax — at the same time ; hence we have dx This result may be stated otherwise, thus : — If u x denote the value of u when x becomes a l9 then the value of the ratio — , when a?i - x is evanescent, is called the first derived Xi ~~ x du function of u, and denoted by — . * The method of derived functions was introduced by Lagrange, and the different derived functions of/ (x) were defined by him to be, the coefficients of the powers of h in the expansion of /(# + h) : that this definition of the first derived function agrees with that given in the text will be seen subsequently. This agreement was also pointed out by Lagrange. See "Theorie des Fonctions Analytiques," N os . 3, 9. Algebraic Illustration. 5 If x L be greater than x, then u x is also greater than u, pro- vided — is positive ; and hence, in the limit, when x x - x Xi - x . du is evanescent, u x is greater or less than u according as — is ax positive or negative. Hence, if we suppose x to increase, then any function of x increases or diminishes at the same time, according as its derived function, taken with respect to x, is positive or negative. This principle is of great importance in tracing the different stages of a function of x, corresponding to a series of values of x. 7. Differential, and Differential Coefficient, of Let u =f(x) ; then since we have du = d{f(x)) = f ' (x) dx, where dx is regarded as being infinitely small. In this case dx is, as already stated, the differential of x, and du or f (x) dx, is called the corresponding differential of u. Also f (x) is called the differential coefficient of f(x), being the coefficient of dx in the differential of f{x). 8. Algebraic Illustration. — That a fraction whose numerator and denominator are both evanescent, or in- finitely small, may have a finite determinate value, is a na evident from algebra. For example, we have T = — what- no ever n may be. If n be regarded as an infinitely small number, the numerator and denominator of the fraction both become infinitely small magnitudes, while their ratio remains unaltered and equal to -. It will be observed that this agrees with our ordinary idea of a ratio ; for the value of a ratio depends on the relative, and not on the absolute magnitude of the terms which compose it. Again, ff u = na + n 2 a' nb -r n 2 tf' in which n is regarded as infinitely small, and a, b, a' and b' 6 First Principles — Differentiation. represent finite magnitudes, the terms of the fraction are both infinitely small, i j j i • j»« Co \ n& but tneir ratio is -„ b + no the limiting value of which, as n is diminished indefinitely, . a is -. Again, it we suppose n indefinitely increased, the limiting value of the fraction is y>. For a + dn d ab' - bd b + b'n V b f (b + b'n) ' but the fraction 77-77 — — — ■ diminishes indefinitely as n b (b + bn) J increases indefinitely, and may be made less than any assignable magnitude, however small. Accordingly the limiting value of the fraction in this case is —.. b 9. Trigonometrical Illustration. — To find the values of 7 — 77, and — 77-, when is regarded as infinitely small. Here - — ^ = cos0, and when = o, cos0 = 1. tan u Hence, in the limit, when 9 = o,* we have sin0 _ tan0 . .. 7 — 77 = 1, and, .*. —. — jz = 1, at the same time. tan 6 sin 6 Again, to find the value of -r— 5, when is infinitely small. From geometrical considerations it is evident that if 6 be the circular measure of an angle, we have tan > 6 > sin 0, tan0 or -7— 7j > -t—x > 1 ; sin sin * If a variable quantity be supposed to diminish gradually, till it be less than anything finite which can be assigned, it is said in that state to be indefinitely small or evanescent; for abbreviation, such a quantity is often denoted by cypher. A discussion of infinitesimals, or infinitely small quantities of different orders, will be found in the next Chapter. Geometrical Illustration. but in the limit, i.e. when is infinitely small, tan0 "sin~0 = *' and therefore, at the same time, we have e sin0 i. This shows that in a circle the ultimate ratio of an arc to its chord is unity, when they are both regarded as evanescent. 10. Geometrical Illustration. — Assuming that the relation y = f(x) may in all cases be represented by a curve, where , . expresses the equation connecting the co-ordinates (x, y) of each of its points ; then, if the axes be rectangular, and two points (x, y), (x x , yi) be taken on the curve, it is obvious that — represents the tangent of the angle which the Xi - x ° ° chord joining the points (a?, y), (x x , y^) makes with the axis of x. If, now, we suppose the points taken infinitely near to each other, so that x x - x becomes evanescent, then the chord becomes the tangent at the point (x, y), but — — - becomes -f- or f (x) in this case. Hence, f (x) represents the trigonometrical tangent of the angle which the line touching the curve at the point (x, y) makes with the axis of x. We see, accordingly, that to draw the tangent at any point to the curve y = f( x ) is the same as to find the derived function fix) of y with respect to x. Hence, also, the equation of the tangent to the curve at a point (x, y) is evidently y-Y -/(«) (x-X), (2) where X, Y are the current co-ordinates of any point on the 8 First Principles — Differentia Hon. tangent. At the points for which the tangent is parallel to the axis of x, we have f (x) = o ; at the points where the tangent is perpendicular to the axis, f (x) = co . For all other points f (x) has a determinate finite real value in general. This conclusion verifies the statement, that the ratio of the increment of the dependent variable to that of the independent variable has, in general, a finite determinate magnitude, when the increment becomes infinitely small. This has been so admirably expressed, and its con- nexion with the fundamental principles of the Differential Calculus so well explained, by M. Navier, that I cannot for- bear introducing the following extract from his "Lecons d'Analyse": — "Among the properties which the function y = fix), or the line which represents it, possesses, the most remarkable — in fact that which is the principal object of the Differential Calculus, and which is constantly introduced in all practical applications of the Calculus — is the degree of rapidity with which the function / (x) varies when the in- dependent variable x is made to vary from any assigned value. This degree of rapidity of the increment of the function, when x is altered, may differ, not only from one function to another, but also in the same function, ac- cording to the value attributed to the variable. In order to form a s ' ' precise notion on this point, let us attribute to a? a deter- mined value represented by ON, to which will correspond an equally determined value of y, represented by PN. Let us now suppose, starting from this value, that x increases by any quantity denoted by Ax, and represented by NM, the function y will vary in consequence by a certain quantity, denoted by Ay, and we shall have y + Ay = f(x + Ax), or Ay = f(x + Ax) -fix). The new value of y is represented in the figure by QM, and QL represents Ay, or the variation of the function. Geometrical Illustration. g The ratio — of the increment of the function to that of Ax the independent variable, of which the expression is f(x + Ax) -fix) Ax ' is represented by the trigonometrical tangent of the angle QPL made by the secant PQ with the axis of x. Ay " It is plain that this ratio — is the natural expression of the property referred to, that is, of the degree of rapidity with which the function y increases when we increase the independent variable x ; for the greater the value of this ratio, the greater will be the increment Ay when x is in- creased by a given quantity Ax. But it is very important Ay to remark, that the value of — (except in the case when Ax the line PQ becomes a right line) depends not only on the value attributed to x, that is to say, on the position of P on the curve, but also on the absolute value of the increment Ax. If we were to leave this increment arbitrary, it would be Axt impossible to assign to the ratio -~ any precise value, and it is accordingly necessary to adopt a convention which shall remove all uncertainty in this respect. " Suppose that after having given to Ax any value, to which will correspond a certain value Ay and a certain direction of the secant PQ, we diminish progressively the value of Ax, so that the increment ends by becoming evanescent ; the corresponding increment Ay will vary in consequence, and will equally tend to become evanescent. The point Q will tend to coincide with the point P, and the secant PQ with the tangent PT drawn to the curve at the Ay point P. The ratio — of the increments will equally LXX approach to a certain limit, represented by the trigonometrical tangent of the angle TPL made by the tangent with the axis of x. "We accordingly observe that when the increment Ax, io First Principles — Differentiation. and consequently Ay, diminish progressively and tend to vanish, the ratio — of these increments approaches in Ax general to a limit whose value is finite and determinate. A 7 ?/ Hence the value of — corresponding to this limit must be t\X considered as giving the true and precise measure of the rapidity with which the function f (x) varies when the independent variable x is made to vary from an assigned value ; for there does not remain anything arbitrary in the expression of this value, as it no longer depends on the absolute values of the increments Ax and Ay, nor on the figure of the curve at any finite distance at either side of the point P. It depends solely on the direction of the curve at this point, that is, on the inclination of the tangent to the axis of x. The ratio just determined expresses what Newton called the fluxion of the ordinate. As to the mode of finding its value in each particular case, it is sufficient to consider the general expression ., A x Ml . r Ay f(x + Ax) -j (x) Ax Ax and to see what is the limit to which this expression tends, as Ax takes smaller and smaller values and tends to vanish. This limit will be a certain function of the independent variable x, whose form depends on that of the given function f{x) We shall add one other remark; which is, that the differentials represented by dx and cly denote always quantities of the same nature as those denoted by the variables x and y. Thus in geometry, when x represents a line, an area, or a volume, the differential dx also represents a line, an area, or a volume. These differentials are always supposed to be less than any assigned magnitude, however small ; but this hypothesis does not alter the nature of these quantities : dx and dy are always homogeneous with x and y, that is to say, present always the same number of dimensions of the unit by means of which the values of these variables are expressed." 10a. I(x + h); . Vi-y = /(s + h) (j>(% + h) -f(x) (a?) " h h Now, in the limit, A = /(*)> (y), * The Student -will observe that this is. a case of the principle (Art. 10a) that the limit of the product of two quantities is equal to the product of their limits. Differentiation of sin x. 19 we have, in all cases, du du dy dx dy dx This result must still hold in the particular case when u = x, in which case it becomes dxdy dy dx Examples. 1. u = (a? - x 2 ) 5 . Let a 2 — x 1 — y, then u — y 5 , du , , dy — = svS and — = - 2X. dy dx Hence — = - iox (a 2 — a; 2 ) 4 . 2. u = {a + bx*)*. 3- u — (r + % 2 )i. 4- u = (1 + a")'". ^4«5. — = r25a; 2 (a + bx 2 )*. dx du x dx (1 + x 2 )^ du , . . . — = mnx n - l (i+x n ) m - 1 . dx We next proceed to determine the derived functions of the elementary trigonometrical and circular functions. 21. Differentiation of sin x. — Let y = sin#, y x = sin (x + h), . h ( h . , 7X . 2sin- cos x+ - yx- y _ sin (# + #)- sin x _ 2 \ 2 h h h . h sm- But by Art. 9, the limit of — t— = 1 ; moreover, the limit of X + - J is cos #. C 2 20 First Principles — Differentiation, Hence a(sinx) = cos a?. (12) dx 22. Differentiation of cos x. y = cos x, y x = cos (x + h), . h . ( h^ f , x 2 sin - sin [x + - Vi — y _ cos (x + h) - cos x 2 \ 2 J h h h Hence, in the limit, d cos x , x - 1 -=- w , K . (13) This result might be deduced from the preceding, by substi- tuting % f or x 3 and applying the principle of Art. 19. It may be noted that (12) and (13) admit also of being written in the following symmetrical form : — dsinx . ( ir dx = sm ^ + i'' d cos x ( 7r' = cos [x + - 1. dx \ 2 23. Differentiation of tan x. y = tan x, y\ = tan (x + h), sin (x + h) sin x yi-y tan (x + h) - tan x _ cos (x + h) cos x ~~h h h sin h hco&x cos (x + h) 9 which becomes — =- *& the limit. cos 2 # Differentiation of y - sin" 1 x. 21 -r-r d(tana?) 1 ■ . . Hence — ~ — - = — z- = sec 2 #. (14) dx gq&x v ' Otherwise thus, , sin # d sin a? . 6? cos x j,, x a . COS X ; — - - sm X ; — #(tan#) cos a? ax ■• . ax dx dx cos 2 a? cos 2 x + sin 2 x cos 2 x cos 2 x 24. Differentiation of cot x. — Proceed as in the last, , , d (cot x) 1 2 , A and we get — y — — - = — r-r- = - cosec 2 #. (15) dx snrx This result can also be derived from the preceding, by put- ting — z f or x, as in Art. 22. 25. Differentiation of sec x. 1 y = sec x = ; COS X dy max . . ,. .*. -r = — r— = tan x sec a?. (10) dx cos 2 x a. .-, 1 d cosec x . Similarly ; = - cot x cosec x. J dx 26. Differentiation of y = snr 1 ^. Here x = sin y, .*. -7- = cos y. dy Hence, by Art. 20, we get dy _ 1 1 dx cos y ~ «/ l -x* 22 First Principles. — Differentiation. The ambiguity of the sign in this case arises from the ambi- guity of the expression y = sin" 1 x ; for if y satisfy this equa- tion for a particular value of x, so also does it - y; as also 27r + y, &c. If, however, we assign always to y its least value, i. e. the acute angle whose sine is represented by x, then the sign of the differential coefficient is determinate, and is evi- dently positive ; since an angle increases with its sine, so long as it is acute. Accordingly, with the preceding limitation, d . sin -1 x dx y/j _ #3 In like manner we find d . cos*" 1 x i (17) dx y'j _ x * (18) with the same limitation. This latter result can be at once deduced from the preced- ing by aid of the elementary equation sin l x + cos -1 = -. 2 hence 27. Differentiation of tan -1 x. y = tan -1 x, .*. x = tan y ; dx 1 i n . » Similarly, dy cos 2 y d . tan' 1 x du . 1 , x = -r = cos i y= x. (19) dx dx 1 + x % x ' e? . cot -1 a? 1 db 1 + x 2 ' 28. Geometrical Demonstration. — The results ar- rived at in the preceding Articles admit also of easy demon- Geometrical Demonstration. 23 stration by geometrical construction. We shall illustrate this method by applying it to the case of sin 0. Suppose XP QY tohe a quadrant of a circle hav- ing as its centre, *and construct as in figure. Let denote the angle XOP expressed in circu- lar measure ; then Fig. 2. „ arc PX , , ~ arc PQ = — tt^t- , and h = At/ = OP OP Accordingly, sin (0 + h) - sin = -^ = ^ . -^ = cos P QR . ^ ; . sin(0 + A)-sinfl = cog pQR m h arc PQ PQ But we have seen, in Art. 9, that the limiting value of ^-x arc JrQ = 1 ; also PQR = 0, at the same time ; hence — ^ — = cos 0, as before. The student will find no difficulty in applying the pre- ceding construction to the differentiation of cos 0, sin -1 0, and cos" 1 0. The differential coefficients of tan 0, tan" 1 0, &c, can, in like manner, be easily obtained by geometrical construction. 1. y — sin (nx + a). 2. y = cos mx cos nx. 3. y = sin w #. Examples. — = n cos ma? + a), ax dy dz dy dx = - {m cos nx sin m# + w cos mx sin ft#), = n sin M-1 a; cos #. 24 First Principles — Differentiation. 4. y - sin (1 + a 2 ). -^ = 23 cos (1 + # 2 ). 5. Show that sin 3 x — - (sin m # sin mx) = m $m m + l x sin (w + j) x. _ d , . Here — (sm"*# sm mx) — m sin w_1 a; (cos x sin mx + sin x cos m#) = m sin" 8 - 1 x sin (m + 1) x : .*. &c. 6. «/ = O sin 8 x + b cos 2 #)». ~ = n(a-b) sin 22: (« sin 2 x + b cos 2 a;)"- 1 . 7. ?/ = sin (sin #). = sin#. Or «/ = sin u, where w = sin x. —- = cos x cos (sin 3). dx v ' 8. ar = sin -1 (# w ). v ' tf# (1 - a 2 ")* 9. y = sin- 1 (1 — # 2 )*. Here (1 - # 2 )^ = sin y ; .*. .r = cos y. dy dy \ i=-smy-?-; .-.— = ■ dx dx yr^ . b + a cos x dy . /Tjj I£ 10. y = cos- 1 — — : . ~- = v g 6 a + £cos# ), where tan ^/i + x* * T* 1 X * + aX + \S( X2 + aX Y ~ bX , U 4. dU ■ V A t 36. If u = log - 1 prove that — is of the form x* + ax - ^{x 2 + ax)i - bx dx Ax 4- B , and determine the values of A and B. Ans. A — 3, B = a. \/ (x 2 + ax) 2 — bx d f . \ A sin* 9 + .Bsin 2 + C 37. Prove that - ^ sin e cos e J x _ # sin 2 J = —^===^- , and determine the values of A, B, C. Ans. A = 3c 3 , B = - 2 (1 + c 2 ), C = 1. ■?8. I£u = x + H — f- — - — 7 —+...«(.) =/ ». Let l(l) bere P resentedb yS' then g =/"(,)• In like manner — f -^ ] is represented by — , and so on ; Successive Differentials. 35 henoe 2 =r{x) ' &o - • • • S =/(n) w- w The expressions ^ <^V <# 3 ^ d n y da? da?' dx z<> dx n are called the firsts second, third, . . . n th differential coef- ficients of y regarded as a function of x. These functions are sometimes represented by •, y", y"', . . . y«, a notation which will often be found convenient in abbre- viating the labour of forming the successive differential coefficients of a given expression. From the mode of arriving at them, the successive differential coefficients of a function are evidently the same as its successive derived functions considered in the preceding Article. 35. Successive Differentials. — The preceding result admits of being considered also in connexion with differen- tials ; for, since x is the independent variable, its increment, dx, may be always taken of the same infinitely small value. Hence, in the equation dy = f(x) dx (Art. 7), we may regard dx as constant, and we shall have, on proceeding to the next differentiation, d {dy) =dxd [/' (a?)] = (dx) 2 f"(x), since d [/' (a?)] =/" (x) dx. Again, representing d (dy) by d 2 y, we have d 2 y = f" (x) (dx) 2 ; if we differentiate again, we get d*y=f"(x)(dx*); and in general d n y=/W(x)(dx) n . Prom this point of view we see the reason why/( n ) (x) is called the n th differential coefficient oif(x). d2 36 Successive Differentiation. In the preceding results it may be observed that if dx be regarded as an infinitely small quantity, or an infinitesimal of the first order, {dx) 2 , being infinitely small in comparison with dx, may be called an infinitely small quantity or an infinitesimal of the second order ; as also d 2 y, if /" (x) be finite. In general, d n y, being of the same order as (dx) n , is called an infinitesimal of the n th order. 36. Infinitesimals. — We may premise that the expres- sions great and small, as well as infinitely great and infinitely small, are to be understood as relative terms. Thus, a magni- tude which is regarded as being infinitely great in comparison with definite magnitude is said to be infinitely great. Similarly, a magnitude which is infinitely small in comparison with a finite magnitude is said to be infinitely small. If any finite magnitude be conceived to be divided into an infinitely great number of equal parts, each part will be infinitely small with regard to the finite magnitude ; and may be called an infini- tesimal of the first order. Again, if one of these infinitesimals be conceived to be divided into an infinite number of equal parts, each of these parts is infinitely small in comparison with the former infinitesimal, and may be regarded as an infinitesimal of the second order, and so on. Since, in general, the number by which any measurable quantity is represented depends upon the unit with which the quantity is compared, it follows that a finite magnitude may be represented by a very great, or by a very small num- ber, according to the unit to which it is referred. For ex- ample, the diameter of the earth is very great in comparison with the length of one foot, but very small in comparison with the distance of the earth from the nearest fixed star, and it would, accordingly, be represented by a very large, or a very small number, according to which of these distances is assumed as the unit of comparison. Again, with respect to the latter distance taken as the unit, the diameter of the earth may be regarded as a very small magnitude of the first order, and the length of a foot as one of a higher order of smallness in comparison. Similar remarks apply to other magnitudes. Again, in the comparison of numbers, if the fraction (one million)^ or — - 6 , which is very small in comparison with Geometrical Illustration. 37 unity, be regarded as a small quantity of the first order, the fraction — -, being the same fractional part of — 6 that this is of i, must be regarded as a small quantity of the second order, and so on. '-Y If now, instead of the series — -, ( — - ) , io 6 Vio 6 / we consider the series -, — , — , n w w \IQr, in which n is supposed to be increased without limit, then each term in the series is infinitely small in comparison with the preceding one, being derived from it by multiplying by the infinitely small quantity -. Hence, if - be regarded as an infinitesimal of the first order, — „, —,...—-, may be regarded as infini- nr n z n r tesimals of the second, third, . . . r th orders. 37. Geometrical Illustration of Infinitesimals. — The following geometrical results will help to illustrate the theory of infinitesimals, and also will be found of importance in the application of the Differential Cal- culus to the theory of curves. Suppose two points, A, B, taken on the circumference of a circle ; join B to E 9 the other extremity of the diameter AE, and produce EB to meet the tangent at A in D. Then since the triangles ABB and EAB are equiangular, we have Fig. 3. AB BE BD AB AD ~ AE' AB ~ AE' Now suppose the point B to approach the point A and to become indefinitely near to it, then BE becomes ultimately AB equal to AE, and, therefore, at the same time, AD = 1. 38 Successive Differentiation. Again, -j=r becomes infinitely small along with -j=, i. e. BD becomes infinitely small in comparison with AD or AB. Hence BD is an infinitesimal of the second order when AB is taken as one of the first order. Moreover, since DE - AE < BD, it follows that, when one side of a right-angled triangle is regarded as an infinitely small quantity of the first order, the difference between the hypothenuse and the remaining side is an infinitely small quantity of the second order. Next, draw BN perpendicular to AD, and BF a tan- gent at B; then, since AB > AN", we get AD - AB arc AB, hence we infer that the difference between the length of the arc AB and its chord is an infinitely small quantity of the third order, when the arc is an infinitely small quantity of the first. In like manner it can be seen that BD - BN is an infinitesimal of the fourth order, and so on. Again, if AB represent an elementary portion of any continuous* curve, to which AF and BF are tangents, since the length of the arc AB is less than the sum of the tangents AF and BF, we may extend the result just arrived at to all such curves. * In this extension of the foregoing proof it is assumed that the ultimate ratio of the tangents drawn to a continuous curve at two indefinitely near points is, in general, a ratio of equality. This is easily shown in the case of an ellipse, since the ratio of the tangents is the same as that of the parallel diameters. Again, it can be seen without difficulty that an indefinite number of ellipses can be drawn touching a curve at two points arbitrarily assumed on the curve ; if now we suppose the points to approach one another indefinitely along the curve, the property in question follows immediately for any con- tinuous curve. Geometrical Illustration. 39 Hence, the difference between the length of an infinitely small portion of any continuous curve and its chord is an infi- nitely small quantity of the third order, i.e. the difference between them is ultimately an infinitely small quantity of the second order in comparison with the length of the chord. The same results might have been established from the expansions for sin a and cos a, when a is considered as infi- nitely small. If in the general case of any continuous curve we take two points A, B, on the curve, join them, and draw BE perpendicular to AB, meeting in E the normal drawn to the curve at the point A ; then all the results established above for the circle still hold. When the point B is taken infinitely near to A, the line AE becomes the diameter of the circle of curvature belonging to the point A ; for, it is evident that the circle which passes through A and B, and has the same tangent at A as the given curve, has a contact of the second order with it. See "Salmon's Conic Sections," Art. 239. Examples. 1. In a triangle, if the vertical angle be very small in comparison with either of the base angles, prove that the difference between the sides is very small in comparison with either of them ; and hence, that these sides may be regarded as ultimately equal. 2. In a triangle, if the external angle at the vertex be very small, show that the difference between the sum of the sides and the base is a very small quantity of the second order. 3. If the base of a triangle be an infinitesimal of the first order, as also its base angles, show that the difference between the sum of its sides and its base is an infinitesimal of the third order. This furnishes an additional proof that the difference between the length of an arc of a continuous curve and that of its chord is ultimately an infinitely small quantity of the third order. 4. If a right line be displaced, through an infinitely small angle, prove that the projections on it of the displacements of its extremities are equal. 5. If the side of a regular polygon inscribed in a circle be a very small magnitude of the first order in comparison with the radius of the circle, show that the difference between the circumference of the circle and the perimeter of the polygon is a very small magnitude of the second order. 4-0 Successive Differentiation. 38. Fundamental Principle of the Infinitesimal Calculus. — We shall now proceed to enunciate the funda- mental principle of the Infinitesimal Calculus as conceived by Leibnitz :* it may be stated as follows : — If the difference between two quantities be infinitely small in comparison with either of them, then the ratio of the quantities becomes unity in the limit, and either of them can be in general replaced by the other in any expression. For let a, j3, represent the quantities, and suppose a = /3 + *, or ,3=i + j3. Now the ratio ^ becomes evanescent whenever t is infinitely small in comparison with j3. This may take place in three different ways : (1) when j3 is finite, and i infinitely small : (2) when i is finite, and [5 infinitely great ; (3) when j3 is infinitely small, and i also infinitely small of a higher order : with /3. * This principle is stated for finite magnitudes by Leibnitz, as follows : — 1 i Cseterum aBqualia esse puto, non tantnm quorum differentia est omnino nulla, sed et quorum differentia est incomparabiliter parva." . . . " Scilicet eas tantum homogeneas quantitates comparabiles esse, cum Euc. Lib. 5, defin. 5, censeo, quarum una numero sed finito multiplicata, alteram superare potest ; et qua3 tali quantitate non differunt, sequalia esse statuo, quod etiam Arcbimedes sumsit, aliique post ipsum omnes." Leibnitii Opera, Tom. 3, p. 328. The foregoing can be identified "with tbe fundamental principle of Newton, as laid down in his Prime and Ultimate Eatios, Lemma I. : " Quantitates, ut et quantitatum rationes, quse ad sequalitatem tempore quovis finito constanter tendunt, et ante finem temporis illius proprius ad invicem accedunt quam pro data" quavis differentia, fiunt ultimo sequales." All applications of the infinitesimal method depend ultimately either on the limiting ratios of infinitely small quantities, or on the limiting value of the sum of an infinitely great number of infinitely small quantities ; and it may be observed that the difference between the method of infinitesimals and that of limits (when exclusively adopted) is, that in the latter method it is usual to retain evanescent quantities of higher orders until the em? of the calculation, and then to neglect them, on proceeding to the limit ; while in the infinitesimal method such quantities are neglected from the commencement, from the know- ledge that they cannot affect the final result, as they necessarily disappear in the limit. Principles of the Infinitesimal Calculus. 41 Accordingly, in any of the preceding cases, the fraction ^ becomes unity in the limit, and we can, in general, substi- tute a instead of j3 in any function containing them. Thus, an infinitely small quantity is neglected in comparison with a finite one, as their ratio is evanescent ; and similarly an infinitesimal of any order may be neglected in comparison with one of a lower order. Again, two infinitesimals a, j3, are said to be of the same order if the fraction — tends to a finite limit. If ¥. tends a a to a finite limit, j3 is called an infinitesimal of the n th order in comparison with a. As an example of this method, let it be proposed to determine the direction of the tangent at a point (x, y) on a curve whose equation is given in rectangular co-ordinates. Let x + a, y + j3, be the co-ordinates of a near point on the curve, and, by Art. 10, the direction of the tangent )3 depends on the limiting value of — . To find this, we substi- a tute x + a for x, and y + /3 for y in the equation, and neglect- ing all powers of a and ]3 beyond the first, we solve for — , a and thus obtain the required solution. For example, let the equation of the curve be x 3 + y % = $axy : then, substituting as above, we get x z + sx 2 a + y z + 3£/ 2 /3 = $axy + $ax$ + $aya : hence, on subtracting the given equation, we get the Umit of & = *—% a ax - y A 39. Subsidiary Principle. — If a x + a% + a 3 + . . . + a n represent the sum of a number of infinitely small quantities, which approaches to a finite limit when n is increased indefi- nitely, and if )3i, j3 2 , • . . /3» be another system of infinitely small quantities, such that /3x j3 2 (5 n . — = I + £l, — = I + fo, ... = I + t nt ai a 2 a n 42 Successive Differentiation. where e l9 e 2 , . . . £ n , are infinitely small quantities, then the limit of the sum of |3i, j3 2 , . . . j3 w is ultimately the same as that of ai, a 2 , . . . a^. For, from the preceding equations we have j3i + j3 a + . . • + 13» = ai + a 2 + . . . + a n + a x £i + a 2 £ 2 + . . . + a n £ n - Now, if t} he the greatest of the infinitely small quan- tities, ei, £ 2 , . . . s n , we have j3i + |3 2 + . . . + fi n - (ai + a 2 + . . . . + a n ) < rj (ai + a 2 . . . + a») ; hut the factor ai + a 2 + . . . + a n has a finite limit, hy hypo- thesis, and as rj is infinitely small, it follows that the limit of fii + j3 2 + . • . + fi n is the same as that of a x + a 2 + . . . + a n . This result can also he estahlished otherwise as follows : — n The rat io P. + ^,+ ... + P > ai + a 2 + . . . + a n hy an elementary algehraic principle, lies hetween the greatest and the least values of the fractions fil p2 fin m > > • • • J a\ a% a n it accordingly has unity for its limit under the supposed con- ditions : and hence the limiting value of /3i + fit + . . . + fi n is the same as that of a t + a 2 + . . . + a n . 40. Approximations. — The principles of the Infini- tesimal Calculus above estahlished lead to rigid and accurate results in the limit, and may he regarded as the fundamental principles of the Calculus, the former of the Differential, and the latter of the Integral. These principles are also of great importance in practical calculations, in which approximate results only are required. For instance, in calculating a result to seven decimal places, if — j he regarded as a small quantity a, then a 2 , a 3 , &c, may in general he neglected. Thus, for example, to find sin 30' and cos 30' to seven de- cimal places. The circular measure of 3 o' is -7-, or .008 7 266; 360 Approximations. 43 denoting this by a, and employing the formulae, ar er sin a = a - — , cos a = 1 , 6 2 it is easily seen that to seven decimal places we have a 2 a 3 — = .OOOO381, — = .OOOOOOI. 2 6 Hence sin 30' = .0087265 ; cos 30' = 9999619. In this manner the sine and the cosine of any small angle can be readily calculated. Again, to find the error in the calculated value of the sine of an angle arising from a small error in the observed value of the angle. Denoting the angle by a, and the small error by a, we have sin {a + a) = sin a cos a + cos a sin a = sin a + a cos a, neglecting higher powers of a. Hence the error is repre- sented by a cos a, approximately. In like manner we get to the same degree of approxima- tion tan (a + a) - tan a = eos 2 a Again, to the same degree of approximation we have a + a a ha - afi where a, j3 are supposed very small in comparison with a and b. As another example, the method leads to an easy mode of approximating to the roots of nearly square numbers ; thus \/a 2 + a = a + — ; ^/a 2 + a 2 = a + — = a, whenever a 2 may 2 Co 2,Ct be neglected. Likewise, l/a z + a = a + — =, &o. If b = a + a, where a is very small in comparison with a, i /— 7 /-= a a + b we nave */ab = yar + aa = a + - = . 44 Successive Differentiation, Again, in a plane triangle, we have the formula C C c 2 = a 2 + b 2 - 2ab cos C = (a + b) 2 sin 2 — + (a - b) 2 cos 2 — . Now if we suppose a and b nearly equal, and neglect (a-b) 2 in comparison with {a + b) z , we have / O C C c= Ua + b) 2 sin 2 — + (a - b) 2 cos 2 — = {a + b) sin — . This furnishes a simple approximation for the length of the base of a triangle when its sides are very nearly of equal length. Exaiuples. i. Find the value of (r + a) (i - 2a 2 ) (r + 3a 3 ), neglecting a* and higher powers of a. Ans. I + a — 2o 2 + a 3 . 2. Find the value of sin {a -t- o) sin (b + j6), neglecting terms of 2nd order in a and j8. Am. sin a sin 6 + a cos a sin 5 + fi sin a cos b. 3. If m = u — e sin w, e heing very small, find the value of tan § u. AVI Ans. (1 + e) tan — . tan - = tan I ho), where a = -smw; .\ &c. 2 \2 / 2 u m e . Here - = — 4- - sin u : 222 4. In a right-angled spherical triangle we have the relation cos c — cos a cos b; determine the corresponding formula in plane trigonometry. The circular measure of a is — , R being the radius of the sphere ; hence, jS a 2 substituting 1 for cos a, &c, and afterwards making R = 00, we get R 2 c 2 = a 2 + b 2 . 5. If a parallelogram be slightly distorted, find the relation connecting the changes of its diagonals. Ans. dAd + d'Ad* = o, where d, d' denote the diagonals, and Ad, Ad' the changes in their lengths. In the case of a rectangle the increments are equal, and of opposite signs. 6. Find the limiting value of j_ a m + £a m+1 + Ca m+2 + &c. aa n + ba n+1 + ca™ 2 + &c. when a becomes evanescent. . Aa m A In this case the true value is that of — — = — a w_n . aa n a A Hence the required value is zero, — , or infinity, according as m>, =, or < n, a Examples. 45 7. Find the value of x z x* 1-- + — 6 120 x 2 x^ 1 + — 2 24 neglecting powers of # beyond the 4th. -4ws. 1 H h - — . 8. Find the limiting values of - when y = o, x and y being connected by y the equation «/ 3 = 2xy — a; 2 . Here, dividing by y % we get # 2 # — - 2 - = - y. y* y If we solve for - we have 5-.±d-rtt Hence, in the limit, when «/ = o, we have - = 2, or - = o. y y 9. In fig. 3, Art. 37, HAB be regarded as a side of a regular inscribed polygon of a very great number of sides, show that, neglecting small quantities of the 4th order, the difference between the perimeter of the inscribed polygon and that of the circumscribed polygon of the same number of sides is represented by - BD. Let n be the number of sides, then the difference in question is n (AD — AB) ; , A ttAE ,^ ,„, v AE (AD - AB) but n= — ; .-. n(AD-AB)= 1— * arc AB AB T)J? A J? v = *AE ."T = w(DE-AE) = - BD, q. p. AE 2 This result shows how rapidly the perimeters of the circumscribed and in- scribed polygons approximate to equality, as the number of sides becomes very great. 10. Assuming the earth to be a sphere of 40,000,000 metres circumference, show that the difference between its circumference and the perimeter of a regular inscribed polygon of 1,000,000 sides is less than r§-th of a millimetre. 11. If one side b of a spherical triangle be small, find an expression for the difference between the other sides, as far as terms of the second order in b. Here cos c = cos a cos b + sin a sin b cos G. Let a denote the difference in question ; i. e. e = a — z ; then cos a cos z + sin a sin z = cos a cos b + sin a sin b cos G; ,\ sin z - sin b cos G = cot a (cos b — cos z). 4 6 Successive Differentiation. Since z and b are both small, we get, to terms of the second order, z - b cos G = (z 2 - b 2 ). 2 The first approximation gives z = b cos C. If this be substituted for z in the right-hand side, we get, for the second approximation, _ £ 2 sin 2 C cot a z = b cos C . 2 We now proceed to find the successive derived functions in some elementary examples. 41. Derived Functions of x m . Let y = x m , ., dy and similar reasoning applies to the other terms. The work can therefore be simplified by neglecting such terms as we proceed. The student will find no difficulty in applying the same mode of reasoning to the determination of the value of -=---, where y = x n ~ x log x. For, as in the last, we may neglect as we proceed all terms which do not contain log a? as a factor, and thus we get in this case, d n y _ (n - 1) . . . 2 . 1 _ \ n ~ I dx n x x 48 Successive Differentiation. 43. Derived Functions of sin mx. Let y = sin mx, then dy dx d 2 y — = m cos mx, dx dx = - m* sm mx, A * d 2n y and, in general, -j~ = (- i) n m 2n sin mx, dx ~\ d 2n+1 y dx 2n+ 7 = (- i) n m 2n+1 cos mx. • • i mi n ^^ $ W tt W developed by the -binomial Theorem, and — , — , . . . — ax ax ax substituted for ( — j u, ( — J u, [-f\ u, in the resulting ex- pansion. 50. In general, if (a) represent any expression in- volving only positive integral powers of a, we shall have For, let (j) ( — ), when expanded, be of the form d_\ n /dy- 1 dx) l \dx^ A [—)+A 1 [ — ) + . . . + A n , 54 Successive Differentiation. then the preceding formula holds for each of the component terms, and accordingly it holds for the sum of all the terms ; .'. &c. The result admits also of being written in the form This symbolic equation is of importance in the solution of differential equations with constant coefficients. See " Boole's Differential Equations," chap. xvi. 51. Iff y = sin -1 x, to prove that / 2^ dn+2 y f x dMl y .d n y Here 4~ = — ? or (1 - # 2 )i-^ = 1 ; dx )l - °' Again, by Leibnitz's Theorem, we have d\ n {, ^dry) r ^ dn+2 y d n+1 y . ,d n y — (1 -x 2 ) -rr-Jt = (1 -x 2 )-—£-- 2nx-~ ~n(n- i)~-. dx) ( v } dx 2 ) K J dx n+2 dx M1 v J dx n ( d\ n ( dy) d n * l y d n y Also — \as-f \=x-—^-^n-\. \dx) ( dx) dx n+1 dx n On subtracting the latter expression from the former, we obtain the required result by (14). If x = o in formula (13), it becomes \dx na ) \dx n J« ' Applications of Leibnitz's Theorem. 55 '*y\ *- iL ~iJ: ^ fry where f — j represents the value of —^ when x becomes cypher. Also, since (— j = 1, we get, when n is an odd integer, \daf» % Jo d 5 ' ' * " Again we have ( — J = o ; consequently, when n is an even integer, we have (^ = a Ml 52. If ^ = (1 + # 2 ) 2 sin (wtan" 1 ^), to prove that ^ + >dx~ 2 ~ 2 ( m ~ 0*;^ + "»(*»- 0y = o. (15) Here d?y --1 . --1 — = m#(i + a; 2 ) 2 sin(m tan*" 1 a?) + m(i + x 2 ) 2 cos (m tan" 1 a?), or (1 + a; 2 ) — -vnx(\ + a; 2 ) 2 sin(mtan~ 1 ^) + w(i+ii? 2 ) 2 cosm(tan~ 1 ^) m = mxy + m(i + x 2 ) 2 cos (m tan" 1 x) ; .'. (1 + x 2 ) 2 cos (m tan" 1 a?) = f - xy. x y v ' m dx * The required result is obtained by differentiating the last equation, and eliminating cos (m tan -1 x) and sin [m tan" 1 ^) by aid of the two former. Again, applying Leibnitz's Theorem as in the last Article, we get, in general — / 2N dn+z y r n dn+1 y t x / \&y 56 Successive Differentiation, Hence, when x = o, we have Moreover, as when x = o, we have y = o, and ~ = m ; it follows from the preceding that M) = O; MJo= (" 0"*(«-i) . . . (—** (16) For a complete discussion of this, and other analogous expressions, the student is referred to Bertrand, " Traite de Calcul Differ entiel,'' p. 144, &c. Examples, 57 Examples. 1. y = «* log a?, prove that ^ =- *=;• 2.y = x\ogx. „ ■t Z =(-i) w r • 3. y = x*, „ — = ^(i + log») 2 + ^ 1 . V^ 1 + jb 2 - 1 2* rf V 5# 5 . y = t an-i + tani— , „ _ = _____ 6. y = s«log(s§), „ ^5-=-- '• ' = log V ,-,^ + ^ ^ " ^ = " F^¥" where tan

17. If y = , prove that ■— = (1 -)» '- y v /y where d> = tan* 1 -. x This follows at once from Art. 46, since — f tan -1 - J = ... It can also be dx \ xj a 2 + x z proved otherwise, as follows : _i_ = _i_r_j l_1- a 2 + x 2 20 (- if \_x - (- i)£ a; + (- 1)* J ' ^»«/ _ 1 / d \ » 1 1 / 0* \ » 1 dx n ~ 2a (- i)i \dx) ' x — (— 1)* 2a (- 1)* V027 " a; + (- 1)* _ (- i)« I . 2 ...» [~ 1 1 "1 = 20 (- ijl |_(« -0(-i)^ + i " (a? + (- i)*) M+1 J _(-i)»|» |" (a; + (- i)*)"* 1 ) - (a; - (- i)*)™* "] " 27(=l)i L~ WTa^ J ' Examples. 59 Again, since - = tan . sin w+1 $ — 2 - (_ i)n tr , <&;« v ' « w+3 18. In like manner, if y = — -, a 1 + x~ dny \n . sin' l+1 <$> . cos (n + 1) prove that — - = (- i)» — . dx n a n+1 19. If u = xy, ,. , d n u d n y d n ~^y prove that -— = x — - + n dx n dx n dx n ~ 1 ' 20. If u — (sin" 1 x) 2 , prove that (1 - x 2 ) —r - a; — = 2. 21. Prove, from the preceding, that and (*5fl _W**\ . 22. If y = *«* sin bx, prove that — ^ - zct — + (a 2 + b 2 ) y = o. dx 2 dx ax + b d n y -z - 9 , nnd - — . x* — c 2 dx n 23. Given y = - , find — -. TT ax + b ac + b I etc - b I Here = 1 . x 2 — c 2 ic x - c ic x + c Hence *SL = t_^l- (JLtL + - — * ^ ^ W 2C \(X - C) n+ 1 (X + tf)» +1 / * ( 6o ) CHAPTEE III. DEVELOPMENT OF FUNCTIONS. 53. Lemma. — If u be a function of x + y which is finite and continuous for all values of x + y, between the limits a and b, then for all such values we shall have du du dx dy For, let u =f(x + y), then if x become x + h, — = limit of ^"*-^* 7 *) ~f( x + y) dx h when h is infinitely small. Similarly, if y become y + h, we have — = limit of f( x + y + h ) -f( x + y) dy h ' which is the same expression as before. -r-r du du Hence — = — . dx dy Otherwise thus : — Let z = x + y, then u =/(s), dz dz dx ' dy ' du du dz ,. . dx dz dx du dudz „,. . du -7- = -v ~r = J M = ~T' dy dz dy dx Taylor' } s Expansion. 61 54. If f(x + y) be a continuous function, which does not become infinite when y = o, its expansion in powers of y can contain no negative powers ; for, suppose it contains a term of the form My~ m , where M is independent of y, this term would become infinite when y = o ; but the given function in that case reduces to fix) ; hence we should have /(a?) = co, which is contrary to our hypothesis. Consequently the expansion of f{x + y) can contain only positive powers of y. Again, iif(x) and its successive derived functions be finite and continuous, the expansion of fix + y) can contain no fractional power of y. For, if it contain a term of the form Py n+ q, where - is a proper fraction, then its (n + i) th derived -* function with respect to y would contain y with a negative index, and, accordingly, would become infinite when y = o ; which is contrary to our hypothesis. Hence, with the conditions expressed above, the expan- sion of f(x + y) can contain only positive integral powers of y. 55. Taylor's Expansion of/ (x + y).* — Assuming that the function /(# + y) is capable of being expanded in powers of y, then by the preceding this equation must be of the form f(x + */) = P + P p I ^P 2 I d 3 /(tf) 3 cfo; i . 2 . 3 dx z i .;v" ( * ); and in general, i ^/(a>) i I . 2 ... 91 dx n 1.2... Accordingly, when /(a?) and its successive derived func- tions are finite and continuous we have /(•+») -/(•) + f /» + j^rw + • • • + £/« w+. • • (i) This expansion is called Taylor's Theorem, having been first published, in 17 15, by Dr. Brook Taylor in his Hethodus Incrementorum. It may also be written in the form x *t x yclfix) if d 2 f(x) y n d n f(x) y v *' ^ w 1 da> 1 . 2 ^ 2 | to daj» ' w or, if m = /(a?), and toi = f(% + to), toc?to to 2 d 2 u y n d n tt . , . i& 1.2 da? I to dk w To complete the preceding proof it will be necessary to obtain an expression for the limit of the sum of the series after n terms, in order to determine whether the series is convergent or divergent. We postpone this discussion for the present, and shall proceed to illustrate the Theorem by The Logarithmic Series. 63 showing that the expansions usually given in elementary- treatises on Algebra and Trigonometry are particular cases of it. 56. The Binomial Theorem. — Let u = (x + y) n ; here/(#) = x n , therefore, by Art. 41, f{x) = nx n ~\ . . . /W (w) = n(n - 1) . . . (n - r + 1) x n ~ r . Hence the expansion becomes (0 + ^)» = af» + - x n ~ l y + — x n ~Y + • • • I . 2 . . . r u If w be a positive integer this consists of a finite number of terms ; we shall subsequently examine the validity of the expansion when applied to the case where n is negative or fractional. 57. The logarithmic Series. — To expand log (x + y). Here /(*) = log .(*), /»=A /"M — i /» = i • • • /« w - (- «r '•'••• ( *- I > . Accordingly i / \ i y I p 2 I y z ^ y i o log (a? + y) = log x + £ - - J - +-V-T^ &o. ' « 2r 3 # 3 4 # 4 If a? = 1 this series becomes log( I +2/)=f-f + ^-...(-i)'- 1 J..&e. (5) When taken to the base 0, we get, by Art. 29, iog.(i + y)-.ar(|-£ + £-£ + &o.)- (6) 64 Development of Functions. 58. To expand sin (x + y). Here f{x) = sin#, f{%) = cos a?, f\x) = - sin x, f\x) = - cos x, &c. Hence sin (a; + y) = sin # 1 - -^— + &c. ± -. — v ' \ 1.2 1.2.3.4 \2n (y y z y 5 y 2n ' x \ + cos x[ J - - — + . ..± — . (7) \I 1.2.3 1.2.3.4.5 in As the preceding series is supposed to hold for all values, it must hold when x = o, in which case it becomes #3 n fi s i n y = V 1 — + 1 &c. (8) I 1.2.3 1.2.3.4.5 7T Similarly, if x = — , we get cos y = 1 — - — + &c. (9) 9 1.2 1.2.3.4 We thus arrive at the well-known expansions* for the sine and cosine of an angle, in terms of its circular measure. 59. Maclauriu's Theorem. — If we make x = o, in Taylor's Expansion, it becomes / (jf) =/(o) + f/(o) + -^-/"(o) + . . . £/«(<>) + ..'., (10) where /(o) . . ,/W(o) represent the values which f(x) and its successive derived functions assume when x = o. Substitute x for y in the preceding series and it becomes /(*) =/(o) + X - /'(o) + f- /"(o) + ...+£ / W (°) + &c - 1 1 . £ \iti * These expansions are due to Newton, and -were obtained by him by the method of reversion of series from the expansion of the arc in terms of its sine. This latter series he deduced from its derived function by a process analogous to integration (called by Newton the method of quadratures). See Opuscula, torn 1., pp. 19, 21. Ed. Cast. Compare Art. 64, p. 68. Exponential Series. 65 This result may be established otherwise thus ; adopting the same limitation as in the case of Taylor's Theorem : — Assume f(x) = A + Bx + Ox % + Bx z + Ex" + &c. then f (x) = B + 2 Ox + 3BX 2 + ^Ex z + &c. f" (x) = 2O + 3 . 2Bx + 4 . 3EX 2 + &o. f"(x) = 3 . 2 B + 4 . 3 . 2Ex + &c. Hence, making x = o in each of these equations, we get /(o) = A, /'(o) = 5, €M = C, • g^ = D, &o. whence we obtain the same series as before. The preceding expansion is usually called Maclaurin's* Theorem ; it was, however, previously given by Stirling, and is, as is shown already, but a particular case of Taylor's series. We proceed to illustrate it by a few examples. 60. Exponential Series. — Let y = a x . Here f(x) = a x , hence f(o) =1, f(x) =a x loga, „ f(o) = loga, f(x) =a*(loga)\ „ /»=log«) 2 , /(») (x) = a* (log a)\ „ /W (o) = (log a) n ; and the expansion is (xloga) (xloga) 2 (x log a) n n , . /» _ 1 . 2 . . . n If 6, the base of the Napierian system of Logarithms, be substituted for a, the preceding expansion becomes e 1 = 1 + - + + ...+ + ... (12) 11. 2 1 . 2 ... ra * Maclaurin laid no claim to the theorem which is known by his name, for, after proving it, he adds — "This theorem was given by Dr. Taylor, Method. Increm." See Maclaurin's Fluxions, vol. ii., Art. 751. F 66 Development of Functions. If x = i this gives for e the same value as that adopted in Art. 29, viz. : 111 1 e = 1 + — h + + + . . . 1 1.2 1.2.3 1.2.3.4 61. Expansion of sin x and cos x by Maclaurin's Theorem. Let /(a?) = sin a?, then /(o)=o, /(o)-i, /» = o, /"(o) = - I, &c, and we get sm x = + &c 1 1.2.3 1.2.3.4.5 In like manner COS X = I + 1.2.3.4 the same expansions as already arrived at in Art. 58. Since sin (- x) = - sin a?, we might have inferred at once that the expansion for sin x in terms of x can only consist of odd powers of x. Similarly, as cos (- x) = cos x, the expan- sion of cos x can only contain even powers. In general, if F(x) = F(- x), the development of F(x) can only consist of even powers of x. If F(— x) = - F(x) t the expansion can contain odd powers of x only. Thus, the expansions of tana?, sin -1 a?, tan -1 #, &c, can con- tain no even powers of x ; those of cos x, sec x, &c, no odd powers. 62. Iff ny gens' Approximation to length of Circular Arc.* — If A be the chord of any circular arc, and B that of half the arc ; then the length of the arc is equal to , q.p. ■J For, let R be the radius of the circle, and L the length of the arc : and we have A .LB . L R = 2m ise B = 2Bm ^ — — - — i * This important approximation is due to Huygens. The demonstration given above is that of Newton, and is introduced by him as an application of his expansion for the sine of an angle. Vid. " Epis. Prior ad Oldemburgium." Huygens' Approximation. 67 hence, by (8), X 3 L 5 A = L =r 9 + 7T- - &C. 2.3. 4. B? 2 .3 .4. 5 . 16. i2 4 T3 7" 5 8J5 = 4 Z — - + - - &o. 2 . 3 . 4 . i£ 2 2 . 3 . 4 . 5 . 64 . B* consequently, neglecting powers of — beyond the fourth, we get sb -a _/ x 4 \ •; Hence, for an arc equal in length to the radius the error in adopting Huygens' approximation in less than th part of the whole arc ; for an arc of half the length of the radius the proportionate error is one-sixteenth less ; and so on. In practice the approximation* is used in the form L = 2B + - (2B-A). 3 This simple mode of finding approximately the length of an arc of a circle is much employed in practice. It may also be applied to find the approximate length of a portion of any continuous curve, by dividing it into an even number of suitable intervals, and regarding the intervals as approxi- mately circular. See Eankine's Rules and Tables, Part I., Section 4. * To show the accuracy of this approximation, let us apply it to find the length of an arc of 30 in a circle whose radius is 100,000 feet. Here B = 2R sin 7 30', A = iR sin 15 ; hut, from the Tahles, sin 7 30' = .1305268, sin 15 = .2588190. „ „ 2B-A Hence 2JB + = 523^9.71. The true value, assuming ir = 3.1415926, is 52359.88 ; whence the error is but .17 of a foot, or about 2 inches. F 2 68 Development of Functions. 63. Expansion of tan _1 #. — Assume, according to Art. 61, the expansion of tan -1 a?to be Ax + Bx z + Cx 5 + Bx 1 + &c, where A, B, C, &c, are undetermined coefficients : then — '—= = A + $Bx 2 + 5O 4 + 7D0 6 + &c. ; ax 1 . 01 * tan x 1 o a n o but = = = i - x 2, + or - x s + &c, ax 1 + x z when x lies between the limits ± 1 . Comparing coefficients, we have A = i, B = --, C = \ D = --, &c. 3 5 7 Hence /Vi /ViO /y,D /yi&lVT 1 tan -1 # = + ... + (- i) n + . . . ; (14) 135 v ' 2n + 1 v ' when x is less than unity. This expansion can be also deduced directly from Mac- laurin's Theorem, by aid of the results given in Art. 46. This is left as an exercise for the student. 64. .Expansion of sin _1 #. — Assume, as before, sin -1 # = Ax + Bx 3 + Cx 5 + &c. ; then 7 ^ = A + sBx z + sCx* -^ &c. ; (1 - x~y but , V xl = (1 - ^ 2 )-* = 1 + -x* + — 4 + . . . (1 - a?) 4 v ' 2 2.4 1 . 3 . . . 2r - 1 _ + # 2r + . . . 2.4... 2r Hence, comparing coefficients, we get Finally, A=i, B=-.-, C = ±-^-.-,&c. 23 2.45 . , x ix 3 1 . 3 x 5 1 . 3 . . . 2r - 1 ar rH . x sin _1 a? = - + - .- + — -. — + .. . + — . +... (15) 1 23 2.45 2.4... 2r 2r+ 1 Eider's Expressions for Sine and Cosine. 69 Since we have assumed that sin -1 a? vanishes along with x we must in this expansion regard sin" 1 ^ as being the circular measure of the acute angle whose sine is x. There is no difficulty in determining the general formula for other values of sm~ l x, if requisite. A direct proof of the preceding result can be deduced from Maclaurin's expansion by aid of Art. 5 1 . We leave this as an exercise for the student. From the preceding expansion the value of ir can be exhibited in the following series : 7T 1 111.31. - = - + + + &c. 6 2 2.38 2. 4. 532 I 7T I For, since sin 30° = -, we have - = sin" 1 - ; .'. &c. 2 62 An approximate* value of it can be arrived at by the aid of this formula ; at the same time it may be observed that many other expansions are better adapted for this purpose. 65. Enler's Expressions for Sine and Cosine. — In the exponential series (12), iix*/ - 1 be substituted for x, we get /yi2 nA gXV-i _ j + . , + &0. . . . 1.2 I.2.3.4 + ^ + &G I 1.2.3 = cos x + v - 1 sin x ; by Art. 59. Similarly, e~ x,/ ~ l = cos x - */ - 1 sin x. Hence e x ^~ x + e"^" 1 = 2 cos x, ) o*"/-l ._ />-%>/ ~ 1 = 2 A /- (16) - #■**->■ = 2a/ - i sin x. A more complete development of these formulae will be found in treatises on Algebra and Trigonometry. * The expansion for snr 1 ^, and also this method of approximating to ir, were given by Newton. 7° Development of Functions. 66. John Bernoulli's Series. — If, in Taylor's Ex- pansion (i) we make y = - x, and transfer fix) to the other side of the equation, we get x f(x) =/(o) + xf(x) - ~ /"(co) + -^— f"(x) - &o. (17) 1.2 1.2.3 This is equivalent to the series known as Bernoulli's,* and published by him in Act. Lips., 1694. As an example of this expansion, let/ (x) = e x ; then /(o) = 1, fix) = e x , f"(x) = e\ &c, and we get e x = 1 + xe x e x + &c, 1 . 2 Or, dividing by e x , and transposing, a? 6* = I - X + &0. f \'2 which agrees with Art. 60. 67. Symbolic Form of Taylor's Theorem. — The expansion may be written in the form /(„„,= j,.„| + i(|)V...|(|)V..J /w , ( , s) in which the student will perceive that the terms within the brackets proceed according to the law of the exponential series (12) ; the equation may accordingly be written in the shape f{x + y) = e y r x f(x), (19) * In his Heduc. Quad, ad long, curv., John Bernoulli introduces this theorem again, adding — " Quam eandum seriem postea Taylorus, interjecto viginti annorum intervallo, in librum quern edidit, a.d. 1715, demethodo incrementorum, transferre dignatus est sub alio tanturn characterum habitu." The great in- justice of this statement need not be insisted on ; for while Taylor's Theorem is one of the most important in the entire range of analysis, that of Bernoulli is comparatively of little use ; and is, as shown above, but a simple case of Taylor's Expansion. Symbolic Form of Taylor's Theorem. 7 1 d where e * x is supposed to be expanded as in the exponential theorem, and V^ written for f- ( — ) fix), &c. \n dx n \n \dxj J x ' This form of Taylor's Theorem is of extensive application in the Calculus of Finite Differences. 68. Other Forms derived from Taylor's Series. — In the expansion (3), Art. 55, substitute h for y, ,, hdu h 2 d 2 u h n d n u then u ± = u + - — + — + . . . ;— + &c. 1 dx 1.2 dx 2 1 . 2 ... n ax n If now h be diminished indefinitely, it may be represented by dx, and the series becomes du dx d 2 u dx 2 d n u dx n u, = u + - + -r- — + . . . + — - dx 1 dx 2 1 . 2 dx n 1 . 2 . . . n or u,-u= "^ dx + J ^- dx 2 + £^&- dx" + &c, (20) 1 1.2 1-2.3 in which u x - u is the complete increment of ^, corresponding to the increment dx in #. Again, since each term in this expansion is infinitely small in comparison with the preceding one, if all the terms after the first be neglected (by Art. 38) as being infinitely small in comparison with it, we get du =f f (x) dx, the same result as given in Art. 7. Another form of the preceding expansion is du d 2 u d 3 u d n u . . Ui - U = — + I h . . . H + &C. (21) 1 1.2 1.2.3 i . 2 . . . n ' 69. Theorem. — If a function of x become infinite for any finite value of x then all its successive derived functions become infinite at the same time. If the function be algebraic, the only way that it can be- come infinite for a finite value of x is by its containing a P term of the form -=, in which Q vanishes for one or more 72 Development of Functions. values of x for which P remains finite. Accordingly, let dPPdQ _ P . clu _ dx Q dx ; this also becomes infinite when Q ' dx Q Q = o. ft if fJ 11 Similarly, -r- , — - , &c, each become infinite when Q = o. dx 1 dx z \ Again, certain transcendental functions, such as e , cosec (x - a), &c, become infinite when x = a; but it can be easily shown, by differentiation, that their derived functions also become infinite at the same time. Similar remarks apply in all other cases. The student who desires a more general investigation is referred to De Morgan's Calculus, page 179. 70. Remarks on Taylor's Expansion. — In the pre- ceding applications of Taylor's Theorem, the series arrived at (Art. 56 excepted) each consisted of an infinite number of terms ; and it has been assumed in our investigation that the sum of these infinite series has, in each case, & finite limiting value, represented by the original function, /(# + y), or fix). In other words, we have assumed that the remainder of the series after n terms, in each case, becomes infinitely small when n is taken sufficiently large — or, that the series is con- vergent. The meaning of this term will be explained in the next Article. 71. Convergent and ^Divergent Series. — A series, Mi, u 2 , u 3 , . . . u n , . . . consisting of an indefinite number of terms, which succeed each other according to some fixed law, is said to be convergent, when the sum of its first n terms approaches nearer and nearer to a finite limiting value, accord- ing as n is taken greater and greater ; and this limiting value is called the sum of the series, from which it can be made to differ by an amount less than any assigned quantity, on taking a sufficient number of terms. It is evident that in the case of a convergent series the terms become indefinitely small when n is taken indefinitely great. If the sum of the first n terms approximates to no finite limit the series is said to be divergent. Convergent and Divergent Series. 73 In general, a series consisting of real and positive terms is convergent whenever the snm of its first n terms does not increase indefinitely with n. For, if this sum do not become indefinitely great as n increases, it cannot be greater than a certain finite value, to which it constantly approaches as n is increased indefinitely. 72. Application to Geometrical Progression. — The preceding statements will be best understood by apply- ing them to the case of the ordinary progression I + X + x 2 + x s + . . . + x n + . . . I — x n The sum of the first n terms of this series is in all cases. 1 - x (1). Let#< 1 ; then the terms become smaller and smaller as n increases ; and if n be taken sufficiently great the value of x n can be made as small as we please. Hence, the sum of the first n terms tends to the limiting value ; also the remainder after n terms is represented 1 x fl n X by , which becomes smaller and smaller as n increases, I - — x and may be regarded as vanishing ultimately. (2). Let x > 1. The series is in this case an increasing one, and x n becomes infinitely great along with n. Hence T _ /V>W /Ylft T the sum of n terms, or , as well as the remainder 1 -x x - 1 after n terms, becomes infinite along with n. Accordingly the statement that the limit of the sum of the series 1 + x + x 2 + . . . + x n + . . . ad infinitum is holds only when x is less than unity, i. e. when the 1 x series is a convergent one. In like manner the sum of n terms of the series I - X + X* - X 3 + &c. I -(- l) n X n is — . I + X 74 Development of Functions. As before, when x < i, the limit of the sum is ; but i + x when x > i, x n becomes infinitely great along with n, and the limit of the sum of an even number of terms is - co ; while that of an odd number is + oo . Hence the series in this case has no limit. 73. Theorem. — If, in a series of positive terms repre- sented by Ui + u 2 + . . . + u n + &c, the ratio ~^— be less than a certain limit smaller than unity, for u n all values of n beyond a certain number ', the series is convergent, and has a finite limit. Suppose k to be a fraction less than unity, and greater than the greatest of the ratios -^ . . . (beyond the number u n n), then we have < A/, .*. Ufi+i < HUfi' ^ w+2 Z- • TA *C it, . . UinYl *"* "> lA/xi. Mn+l U n + r <~ K, . . Ufi+r "^- it Ufi' hi+r-i Hence, the limit of the remainder of the series after u n is less than the sum of the series ku n + k 2 u n + . . . + k r u n ... ad infinitum ; therefore, by Art. 72, less than - ; , since k < 1 . 1 -k Hence, since u n decreases as n increases, and becomes infi- nitely small ultimately, the remainder after n terms becomes also infinitely small when n is taken sufficiently great ; and consequently, the series is convergent, and has a finite limit. Again, if the ratio ~^- be > 1, for all values of n beyond u n Convergent and Divergent Series. 75 a certain number, the series is divergent, and has no finite limit. This can be established by a similar process; for, assuming k > 1, and less than the least of the fractions -^-, . . . then by Art. 72 the series u n + ku n + kht n + &c. ad infinitum has an infinite value ; but each term of the series Un + %i+i + U n +2 + &C. is greater than the corresponding term in the above geome- trical progression ; hence, its sum must be also infinite, &c. These results hold also if the terms of the series be alter- nately positive and negative ; for in this case k becomes negative, and the series will be convergent or divergent according as - k is < or > 1 ; as can be readily seen. In order to apply the preceding principles to Taylor's Theorem it will be necessary to determine a general expres- sion for the remainder after n terms in that expansion ; in order to do so, we commence with the following : — 74. liemiiaa. — If a continuous function (f>(x) vanish when x = a, and also when x = b, then its derived function §\x\ if also continuous, must vanish for some value of x between a and b. Suppose b greater than a; then if (b) = o, we have y = o, when x = a, and also when x = b ; therefore the curve cuts the axis of x at distances a and b from the origin ; and accordingly at some inter- I 76 Development of Functions. mediate point it must have its tangent parallel to the axis of x. Hence, hy Art. 10, we must have $ f (x) = o for some value of x between a and b. 75. Lagrange's Theorem on the ILimits of Tay- lor's Series. — Suppose R n to represent the remainder after n terms in Taylor's expansion, then writing X for x + y in (1), we shall have + ^^P /<"-'> (*) + p», (22) in which f(x), f'{x) /( n ) (a?) are supposed finite and continuous for all values of the variable between X and x. From the form of the terms included in B n it evidently may be written in the shape (x - x y \n where P is some function of X and x. Consequently we have + — 1 — - -P I = o. (23) Now, let g be substituted for x in every term in the pre- ceding, with the exception of P, and let F(%) represent the resulting expression : we shall have F{z) =/(X) - [/(,) + { *^f («) + ... + ^^" P j, (24) in which P has the same value as before. Again, the right-hand side in this equation vanishes when 2 = X; .: F(X) = o. Also, from (23), the right-hand side vanishes when z=x; .-. P(^) = o. J Limits of Taylor's Series. 7 7 Accordingly, since the function F (z) vanishes when z = X 9 and also when z = x 9 it follows from Art. 74 that its derived function F f (z) also vanishes for some value of z between the limits X and x. Proceeding to obtain F\z) by differentiation from equa- tion (24), it can be easily seen that the terms destroy each other in pairs, with the exception of the two last. Thus we shall have W - I IW Consequently, for some value of z between x and X we must have /W («) = P. Again, if be a positive quantity less than unity it is easily seen that the expression x + (X - x), by assigning a suitable value to 0, can be made equal to any number intermediate between x and X. Hence, finally, P=/W {0+0 (X-a>)}, where is some quantity > o and < 1 . Consequently, the remainder after n terms of Taylor's series can be represented by A = £j^>> {» + «(*-*)}. (as) Making this substitution, the equation (22) becomes /(X) =/(x) + £^V (*) + ^f^-V W + . . • " "' ' -/<•*) W +^f-^/W {* + (X-a)}. (26) + w — 1 \n The preceding demonstration is taken, with some slight modifications, from Bertrand's " Traite de Calcul Differentiel" (273). 78 Development of Functions. Again, if h be substituted for X - x, the series becomes f(x + h) =ffa) +hf{x) +&c. h 71 - 1 „ h n + f{n-i) ^ + f(n) (^ + 0/^ # ( 27 ) » — I W In this expression n may be any positive integer. If n = 1 the result becomes fix + h) =f(x) -r hf (X + Oh). (28) When n = 2, /(» + A) -/(«.) + hf ( ) + -£-/" ( + 0/*). (29) The student should observe that has in general different values in each of these functions, but that they are all subject to the same condition, viz., > o and < 1. It will be a useful exercise on the preceding method for the student to investigate the formulse (28) and (29) inde- pendently, by aid of the Lemma of Art. 74. The preceding investigation may be regarded as furnish- ing a complete and rigorous proof of Taylor's Theorem, and formula (27) as representing its most general expression. 76. Geometrical Illustration. — The equation f(X) =f(x) + (X-x)f {x+0(X-x)} admits of a simple geometrical verification; for, let y =ffa) represent a curve referred to rectangular axes, and suppose (X, Y), (x, y) to be two points P x , P 2 on it : then f(X)-f(x) r-y X - x X - x' Y - v But = — - is the tangent of the angle which the chord Pi P 2 ■A — X makes with the axis of x ; also, since the curve cuts the chord in the points Pi, P 2 , it is obvious that, when the point on the curve and the direction of the tangent alter continuously, the tangent to the curve at some point between P x and P 2 must be parallel to the chord P x P 2 ; but by Art. 10, f fa) is the tri- gonometrical tangent of the angle which the tangent at the Second Form of Remainder. 79 point (a>i, y x ) makes with the axis of x. Hence, for some value, x lf between X and x, we must have ff , y-y /m-/w ; W"X-«" x-* ' or, writing x x in the form a? + 9 (X - x), /(X) =/(*) + (X - x)f [x + 9 (X-x)}. 77. Second Form of Remainder. — The remainder after n terms in Taylor's Series may also be written in the form W— I For it is evident that R n may be written in the form (X -x)P 1 ; .-.AX) =/{x) +(x-x) f\x) + . . . + ^rT" 1 /^ w w - I + (X - 0) Pl Substitute g for x, as before, in every term except P 1 ; and the same reasoning is applicable, word for word, as that employed in Art. 75. The value of F f (z) becomes, however, in this case n - 1 and, as F'(z) must vanish for some value of % between x and X, we must have, representing that value by x + 9 (X - x), = (X X)^ (l 9Y- 1 f[n) ^ + q (x _ x y^ n (30) where 9, as before, is > o and < 1 . If h be introduced instead of X-x, the preceding result becomes R n = il S!! 1 #»/(») (x + Oh), n- 1 (31) which is of the required form. 80 Development of Functions. Hence, Taylor's Theorem admits of being written in the form /(.+ *) =/ (0) + \f\x) + -*!_/»+... + -^/M (•) n— i ■ -^— (1 - O^fW (x + Oh). (32) w - 1 The same remarks are applicable to this form* as were made with respect to (27). From these f ormulse we see that the essential conditions for the application of Taylor's Theorem to the expansion of any function in a series consisting of an infinite number of terms are, that none of its derived functions shall become infinite, and that the quantity £/«(* + 0A) shall become infinitely small, when n is taken sufficiently large ; as otherwise the series does not admit of a finite limit. 78. Limit of when n is indefinitely great. 1 . 2 ,,n Let u n = , then — = ; .*. — becomes smaller 1 . 2 . . n u n n + 1 u n and smaller as n increases ; hence, when n is taken sufficiently great, the series u Ml , u n+2 , . . . &c, diminishes rapidly, and the terms become ultimately infinitely small. Consequently, whenever the n th derived function fW (x) continues to be finite for all values of n, however great, the remainder after n terms in Taylor's Expansion becomes infinitely small, and the series has a finite limit. * This second form is in some cases more advantageous than that in (27). An example of this will be found in Art. 83. Remainder in the Expansion of sin x. 8 1 79. General Form of Maclanrin's Series. — The expansion (27) becomes, on making x = o, and substituting x afterwards instead of h, /(•) -/(o) +7/(0) +-^- 2 /"(°) + • • • + i|zi/ (M) (°) + -/W(fe). (33) Hence the remainder after n terms is represented by -m (Ox) ; where 6 is > o and < 1. This remainder becomes infinitely small for any function x n f(x) whenever r-f^ (Ox) becomes evanescent for infinitely \n great values of n. We shall now proceed to examine the remainders in the different elementary expansions which were given in the commencement of this chapter. 80. Remainder in the Expansion of a x . — Our for- mula gives for R n in this case ~ CLosaYa 9 *. \n Now, a Qx is finite, being less than a x ; and it has been proved ( x I02* a\ n in Art. 78 that - — s becomes infinitely small for large values of n. Hence the remainder in this case becomes evanescent when n is taken sufficiently large. Accordingly the series is a convergent one, and the expansion by Taylor's Theorem is always applicable. 8 1 . Remainder in the Expansion of sin x. — In this case D x n . frnr Q \ Rn= \ — sm — + ux . \n \ 2 G 82 Development of Functions. This value of R n ultimately vanishes by Art. 78, and the series is accordingly convergent. The same remarks apply to the expansion of cos x. Accordingly, both of these series hold for all values of x. 82. Remainder in the Expansion of log (1 + x). — The series /y» /ys-* /ViO /yi* + + &c, 1234 when x is > 1, is no longer convergent ; for the ratio of any term to the preceding one tends to the limit - x ; conse- quently the terms form an increasing series, and become ultimately infinitely great. Hence the expansion is inappli- cable in this case. 1.2. n 1^ Again, since f n (x) = (- iV* -1 * * * ' x - — -, the remainder 6 ' J v ' v ; (1 + x) n R n is denoted by - — — ( tt- ) |; hence, if x be positive and x less than unity, — —-r- is a proper fraction, and the value of I ~r UX R n evidently tends to become infinitely small for large values of n ; accordingly the series is convergent, and the expansion holds in this case. 83. Binomial Theorem for Fractional and Nega- tive Indices. — In the expansion m m (m - 1) _ ( I + x) m ■ = I + — X + — -x 2 + . . . . I 1.2 •m(m- 1) . . . (m-n+ i)x n + — * - — + &c. 1 . 2 . . . n if u n denote the n th term, we have u n+1 m-n+ 1 x, u n n the value of which, when n increases indefinitely, tends to become - x ; the series, accordingly, is convergent if x < 1, but is not convergent if x > 1 . Binomial Theorem. 83 Accordingly, the Binomial Expansion does not hold when x is greater than unity. Again, as /(») (x) = m (m - 1) . . . (m - n + 1) (1 + #) m_w , the remainder, by formula (25), is m (m- 1) . . . (m-n + 1) '. n . m _ in i . 2 . . . n or m (m- 1) . . . (m-n + 1) x n 1 . 2 . . . n (1 + 6x) n - m ' Now, suppose x positive and less than unity ; then, when n is very great, the expression m(m— 1) . . . (m-n+ 1) i . 2 . . . n becomes indefinitely small ; also m is less than unity; ( I "T UX) hence, the expansion by the Binomial Theorem holds in this case. Again, suppose x negative and less than unity. We employ the form for the remainder given in Art. 77, which becomes in this case . . m(m- 1) . . . (m-n+ i)x n , n7„"T/ n \™ n (- i) n — \ r — '- — (1 - O)"- 1 (1 - Qx) m ~ n ; ' 1 . 2 . . . [n- i) ' x ' or . .m(m- 1) . . . (m- n+ 1) (1 - 0) m ~ 1 af i ' _I) 1.2 ... (n- 1) -0 X 1 — Also, since x< 1, Qx< 0; .*. 1 - Ox > 1 - ; hence pr- 1 - ux is a proper fraction ; .*. any integral power of it is less than unity ; hence, by the preceding, the remainder, when n is sufficiently great, tends ultimately to vanish. g 2 84 Development of Functions, In general (x + y) m may be written in either of the forms Wi+^j ory w fi+- now, if the index m be fractional or negative, and x > y, or y - a proper fraction, the Binomial Expansion holds for the x series (x + y) m = x m l 1 + ^ ) = #"» + — oj^y + — ^ x m ' 2 y 2 + &c, but does not hold for the series / v / x\ m m , m(m-i) (x+y) m = y m li +-J =y m + - y m ~ 1 x + — K - * ^" 2 ^ 2 + &c, since the former series is convergent and the latter divergent. We conclude that in all cases one or other of the expan- sions of the Binomial series holds ; but never both, except when m is a positive integer, in which case the number of terms is finite. 84. Remainder in the Expansion of tan" 1 ^. — The series /yi /ytO /yiD . - 1*/ w i/U * tan -1 # = + &c, 1 3 5 is evidently convergent or divergent, according as x < or > 1 . To find an expression for the remainder when x< 1, we have, ky ( 8 )> p. 50— / w W-(s)- tlHrt "-(- 1 )** \n - 1 . sin [n n tan~ ! # (i + x*)% Hence we have, in this case, # TO sin hi n tan~ 1 m J W(l + 0V)S ' which, when x lies between + 1 and - 1, evidently becomes infinitely small as n increases, and accordingly the series holds for such values of x. Expansion by aid of Differential Equations. 85 85. Expansion of sin" 1 a?. — Since the function sin" 1 a? is impossible unless x be < 1, it is easily seen that the Series given in Art. 64 is always convergent ; for its terms are each less than the corresponding terms in the geometrical pro- gression x + x z + x 5 + &c. Consequently, the limit of the series is always less than the limit of the preceding progression. A similar mode of demonstration is applicable to the expansion of tan -1 x when x < 1, as well as to other analogous series. In every case, the value of B ni the remainder after n terms, furnishes us with the degree of approximation in the evaluation of an expansion on taking its first n terms for its value. 86. Expansion by aid of Differential Equations. — In many cases we are enabled to find the relation between the coefficients in the expansion of a function of x by aid of differential* equations ; and thus to find the form of the series. For example, let y = e x , then dy — = ^ = y dx y * Now suppose that we have y = a + a x x + a 2 x 2 + . . . a n x n + . . . , then ~ = a x + 2d % x + . . . na n x n ~ l + &c. dx Accordingly we have «i + 2a z x + 3%# 3 + . . . = a + aix + a z x 2 + &c, * This method is indicated by Newton, and there can be little doubt that it was by aid of it he arrived at the expansion of sin (m sin -1 x), as well as other series. — Vide Ep. posterior ad Oldemburgium. It is worthy of observation that Newton's letters to Oldemburg were written for the purpose of transmission to Leibnitz. 86 Development of Functions. hence, equating coefficients, we have #1 = CC , Cl%— — = — , Clz = — = , &C. 2 2 3 2.3 Moreover, if we make x - o, we get a = 1, .'. e x = i + - + + + &c, I 1.2 I.2.3 the same series as before. Again, let y = sin (m sin -1 x) . Here, by Art. 47, we have d 2 y dy Now, if we suppose y developed in the form y = a + a x x + a 2 x 2 + . . . + a n x n + &c, du then — = a x + 2a 2 x + xa z x 2 + . . . + na n x n ~ x + &c, ax d 2 y -zr- = 2a 2 + 3 . 2a % x + . . . + n(n - i)a n x n ~ 2 + &c. dx 2 x 7 Substituting and equating the coefficients of x n we get ri~ - m 2 , (n + 1) (■« +2) Again, when = o we have y = o ; .*. # = o. Hence we see that the series consists only of odd powers of x ; a result which might have been anticipated from Art. 61. To find a x . When x = o, cos (m sin -1 ^) = 1 , hence l—\=m; accordingly a x = m ; m 2 - 1 m (m 2 - 1 ) • . ci% = — #1 = « 2.3 1.2.3 m 2 - 9 m (m 8 - 1 ) (w 2 - 9) m &§ = — #3 = • 4-5 1.2.3.4.5 Expansion of sin mz and cos mz. 87 hence we get • * / • , \ m m ( m2 ~ - 1 ) •, sin* (m sm"^ = — x x s 1 1.2.3 + mK-i).K-9)^ 5 _ &c> ( } I.2.3-4-5 In the preceding, we have assumed that sin -1 ^ is an acute angle, as otherwise both it, and also sin (m siirt?), would admit of an indefinite number of values. — See Art. 26. 87. Expansion of sin mz and cos mz. — If, in (35), 2 be substituted for sin -1 #, the formula becomes ( 1 m 2 - 1 . sin mz = m sin z I suns (1 1.2.3 + (nf- 1) (m» - 9) sin% . & ) (6) In a similar manner it can be proved that m 2 sin 2 s m 2 (m 2 - 4) . . D . . cos mz = 1 + — surs - &c. (37) 1.2 1.2.3.4 If m be an odd integer the expansion for sin mz consists of a finite number of terms, while that for cos mz contains an infinite number. If m be an even integer the number of terms in the series for cos mz is finite, while that in sin mz is infinite. The preceding series hold equally when m is a fraction. A more complete exposition of these important expansions will be found in Bertrand's " Calcul Differentiel." In general, in the expansion (36), the ratio of any term to that which precedes it is -, 7—, r sin 2 s, which, when x (n + 1) (n + 2) n is very great, approaches to suns. Hence, since sin z is less than unity, the series is convergent in all cases. Similar observations apply to expansion (37). * This expansion is erroneously attributed to Euler by M. Bertrand ; it was originally given by Newton. See preceding note. 88 Development of Functions. The expansion . , ax a 2 x 2 a {a 2 + i 2 ) _ a 2 (a* + 2 2 ) e asm-ix =I+ — + + _^ L x z + _^ £ ar + . . . I 1.2 1.2.3 I.2.3.4 can "be easily arrived at by a similar process. 88. Arbogast's Method of derivations. X X 2 x^ If u = a + b- + c + d + &c, 1 1.2 1.2.3 to find the coefficients in the expansion of

' (u) . u, f\x) = 0' {u) . n + " M • ^ • ^ + 0'" M (O 3 , f*(x) = $ (u) . & + " (u) O' u'" + 3 KO 2 ] + 6f'M • M*- **" + iv (w) . (V) 4 . Now, when x = o, w, w', ?/', w'", . . . obviously become a, b, c, d, . . . respectively. Arbogasfs Method of Derivations. 89 Accordingly, B = f (o) = 0' (a) . ft, C = /" (o) = 0' (a) . + f (a) . J», D = /'" (o) = 0' (a) . rf + 3tf>" (a) . fc + 0'" {a) . b\ E = /* (o) = 0' (a) . e 4 tf>" (of) (4-bd + 3c 2 ) + 60'" [a) . & 2 c + iv (a) . bK From the mode of formation of these terms, they are seen to be each deduced from the preceding one by an analogous law by that to which the derived functions are deduced one from the other ; and, as /'(#),/"(#) . . . are deduced from/(a?) by successive differentiation, so in like manner, B, C, D, . . . are deduced from (u) by successive derivation ; where, after differentiation, a, b, c, &c, are substituted for du dhc „ ' dx dx 2J If this process of derivation be denoted by the letter 8, then B = 3.A, C=$.B, D = 8.<7, &c. (38) From the preceding, we see that in forming the term 8 . 0(#<), we take the derived function 0'(#), and multiply it by the next letter b, and similarly in other cases. Thus 8 . 6 = c, 8 . c = d, . . . 8 . b m = mb m ~'c, 8 . c m = mc m -\l . . . Also 8 . 0' (a) b = , and are expressed in terms of the letters, b, c, d, &c. solely ; so that, if calculated once for all, they can be applied to the determination of the coefficients in every particular case, by finding the different derived functions $' (a), "(a), &c, for that case, and multiplying by the respective coef- ficients, determined as stated above. Examples. g i Examples. i. If it = f(ax+bu), then - — = t ^r- This furnishes the condition that J v ^" adx b dy a given function of x and y shou d be a function of ax + by. 2. Find, by Maclaurin's theorem, the first three terms in the expansion of tan x. X 3 2X 5 Ans. x + 1 . 3 i5 3. Find the first four terms in the expansion of sec x. x z c# 4 61 x 6 Ans. 1 + — + — + . 2 24 720 4. Find, by Maclaurin's theorem, as far as # 4 , the expansion of log (1 + sin x) in ascending powers of x. Let f(x) = log (1 + sin x), ,, cos a; 1 -sino; then/ (x) = : — = = sec x — tan x, i + sma; cos x f"(x) = sec x tan x — sec 2 x = - f'(x) sec x ; .'. f'"{x) = —f"(x) sec x —f'(x) sec a; tana;, /iv (x) — _/"'(#) sec x - 2f"(x) sec a; tan x -f'(x) (2 sec 3 # - sec x) ; .'./(o) = o, /'(o) = l, /"(o)=-i, /'"(<>) = 1, /iv(o) = -2; yy»2 /y»3 />*4 .\ log (1 + sin x) = # + -7 1- &c. N ' 2612 e x 5. Find six terms of the development of ■ in ascending powers of x cos X „ 2# 3 # 4 3a; 5 Ans. 1 + x + x 2 -i 1 — + — . • . 3 2 10 6. Apply the method of Art. 86, to find the expansions of sin x and cos x. 7. Prove that i / r\ j. » • sins ., . .„ sin 2s ._ . sin 32; tan" 1 (x + h) = tan"i x + h sin z (h sin zf V (h sin z) 3 cec, where a = cot -1 *. if d n z Here/(a;) = tan- 1 i» = — z; and by Art. 46, — = (- i)»J^-i sin w zsin«z; .\&c. 9 2 Examples. 8. Hence prove the expansion ir sin z sin 2z „ sin 3s - = 2 + COS 2 H COS 2 Z + C0S 3 Z + &C. 2123 Let h — — cot z — — x, &c. g. Prove that ir z sin 2 sin 22 sin 3z - = - + + + + &c. 2 2 1 2 3 Let h sin z = - 1 in Example 7 : then h + x — — : = - tan - ; . \ &c. sms 2 10. Prove the expansion it sin z 1 sin 2z 1 sin iz - = ■ + 5- + - — ~ + &C. 2 COS Z 2 COS 4 2 3 C0S d 2 Assume h — , then sin 2 cos z # + h = - tan2 = tan (ir - z) ; .: w — z ■= tan -1 (# + A), &c. Suhstituting in Example 7 , we get the result required. The preceding expansions were first given by Euler. 11. Prove the equations sin 9% = 9 sin x — 120 sin 3 # + 432 sin 5 a: - 576 sin'ta + 256 sin 9 a;, cos 6x = 32 cos 6 # — 48 cos 4 # + 18 cos 2 # — 1. These follow from the formulse of Article 78. 12. If m = 2, Newton's formula, Art. 87, gives ( . sin 3 # sin% . ) sin 2X = 2 sins; &c. > ; ( 2 2.4 ) verify this result by aid of the elementary equation sin 2% = 2 sin x cos x. 13. If

iv (v) prove that ■ , . = z -^r-r = &c. = constant ; '(o) = o, <}>'"(o) = o, &c. 14. If, in the last, ^-7-/ = a% '> prove that expand y in powers of x by the method of indeterminate coefficients. 27. Show that the series x x 2 x z # 4 jt» 2 m Z m A m is convergent when x < 1, and divergent when x > 1, for all values of m. 28. Prove the expansion f(x) = I /(ft) 1 ^ (/(«)) (# - a) m £ (a?) (# - «) m («) (a; - a) w-1 ^# (^(«) J , * MV(^)+&c I . 2 . (aj - a)™-* \da] \. e 2 ~* 4 ' " r (x x 2 x* \ x 2 (i x x 2 \~ x 3 (i x \ 3 L I_ \2 _ I + 4"/2 \2~3 + 4 7 ~2TiU" 3 + / ""J [a? ii# 2 7a; 3 I-- + i-7 .... 2 24 16 J 30. In Art. 76, if /(as) and/'(a:) be not both continuous between the points Pi, P2, show that there is not necessarily a tangent between those points, parallel to the chord. 31. Find the development of - in ascending powers of as, the coef- sin x sin 2x ficients being expressed in Bernoullian numbers. " Camb. Math. Trip., 1878." _. x sin "\x Since : = x cot x + x cot 2x, the expansion in question, by (22), sin x sin 2x ' r ^ > * \ '» is 3 22^2^2, , 2 4 j5 4 ^ 4 „ , N 2 6 B 6 x* e = I T7-(2+i) rf- (2' + l)-— nr- (2 5 + i)- &c. ( 96 ) CHAPTEE IY. INDETERMINATE FORMS. 89. Indeterminate Forms. — Algebraic expressions some- times become indeterminate for particular values of the variable on which they depend ; thus, if the same value a when substituted for x makes both the numerator and the f (x) f (a) denominator of the fraction —7—; vanish, then ~~- becomes of (x) (p{a) the form -, and its value is said to be indeterminate. o Similarly, the fraction becomes indeterminate if / (x) and (x) both become infinite for a particular value of x. We proceed to show how its true value is to be found in such cases. By its true value we mean the limiting value which the fraction assumes when x differs by an infinitely small amount from the particular value which renders the expression indeter- minate. It will be observed that the determination of the diffe- rential coefficient of any expression/^) may be regarded as a case of finding an indeterminate form, for it reduces to the determination of — '- — — — when h = o. h In many cases the true values of indeterminate forms can be best found by ordinary algebraical and trigonometrical processes. We shall illustrate this statement by a few examples. Examples. m , » . ax 2 - 2acx + ac 2 . „ ;1 . O , . 1. The fraction 5 becomes of the form - when x = o ; but since bx 2 — 2bcx + oc i o it can be written in the shape — 7 ^. its true value in all cases is -. r b{x -c) 2 b Examples. 97 x o 2. The fraction — z=== . becomes - when x — o. a/ a + x -a/ a — x ° To find its true value, multiply its numerator and denominator by the com- plementary surd, a/ a + x + a/ a — %, and the fraction becomes x\a/ a + x + a/ a — x) a/ a + x + a/ a — x —LL r i or : the true value of which is a/ a when x = o. - V a 2 + ax + x 2 — \/a 2 — ax + x 2 i- — — — , when x = o. a/ a + x — a/ a — x Multiply by the two complementary surd forms, and the fraction becomes 2ax {a/ a + x + a/ a - x} 2x {a/ a^ + ax + x 2 + a/ a 2 — ax+x 2 } a [a/ a + x 4- v a — x ) a/ a 2 + ax -J- x 2 + a/ a 2 — ax + a 2 the true value of which evidently is a/ a when x = o. From the preceding examples we infer that when an expression of a surd form becomes indeter- minate, its true value can usually be determined by multiplying by the com- plementary surd form or forms. or 2X — a/ §x 2 — a 2 , 1 4. when x — a. Am. -. x — V2a; 2 — a 2 2 a - a/ a 2 — x 2 , 1 5. when x = o. Am. — . x z 2a , a sm - sin a9 . o . o. -- 1 — - becomes - when = o. 0(cos - cos ad) o To find its true value, substitute their expansions for the sines and cosines, and the fraction becomes / 3 \ / n « 3 3 \ a + . . •) - lad + ... V 1-2.3 / V 1.2.3 / „ ' e 2 a 2 e 2 e{ + ...+ .. .} 1 1 . 2 1.2 J 3 g- (a 3 - a) + . . . or 3 H 98 Indeterminate Forms. Divide by 3 (a 2 - 1), and since all the terms after the first in the new numerator and denominator vanish when = o, the true value of the fraction is - in this 3 case. 7. The fraction A x m + Aicc m ~ l + A 2 x m ~ 2 + . . . At a x n + aix n ~ l + ...+«» becomes ~ when x = 00 : its true value can, however, be easily determined, for it is evidently equal to that of A\ A% A + — + — + . . . x x^ % + - + -5 + X X 6 Moreover, when x = 00, the fractions — , — ...— ..., all vanish : hence, x x* x the true value of the given fraction is that of A , %m- n — when x — 00. The value of this expression depends on the sign of m — n. (i.) If m > n, x m ' n = 00 -when x = 00 ; or the fraction is infinite in this case. An (2.) If m = n, the true value is — . a (3.) If m < n, then x m - n = o when x = 00 ; and the true value of the frac- tion is zero. Accordingly, the proposed expression, when x = 00, is infinite, finite, or zero, according as m is greater than, equal to, or less than n. Compare Art. 39. 8. u = *y x + a - y/ x + b, when x = 00. __ a — b . Here u = = o when x = 00. ■\/x + a + \/ x + b 9. -v/^ 3 + ax — x > when x = 00. -4ws. -. 10. w = «* sin f — K when a; = 00. (?) (1.) If a < i, a x — o when a; = 00, and therefore the true value of u is zero in this case. c . . (2.) If a > I, then a* becomes infinite along with # ; but as — is infinitely Or G C small at the same time, we have sin — = — . Hence, the true value a x a x of u is e in this case. Method of the Differential Calculus. 99 1 1. u = a/ a % — x l cot - A is of the form oxm when x = a. 2 \ a + x Here but, when a - x is infinitely small, la — x it la v a + x 2 \ a + *y a 2 — x 2 .a - x tan a + x IT tan 2 >/ a + x 2 v a + x \/ a? — x' 1 a + x 4a u = , - = = — when x = a. - x * t ir la l\a + x x sin sin #) — sin 2 # , 12. w = r^ , when x = o. Substitute the ordinary expansion for sin x, neglecting powers beyond the sixth, and u becomes ( . sin 3 a; sin 5 x) / # 3 x 5 \ x I sin x — - + - — [ - lx- r + — ) ( 3 5 ) V 3 5/ x z x 5 \ 2 - -4- i — i v. — |3 X % x z X s 1 / # 3 \ 3 # 5 / a; 3 x i \ 2 |3 |5 6\ J3 / \5 \ |3 |5/ x 6 Hence we get, on dividing by x 5 , the true value of the fraction to be — when I o X = O. (asin^ + gcos^)*-^ . 13- ■ a n-pn ' When a = &- ■ Ans - Sm $> Similar processes may be applied to other cases ; there are, however, many indeterminate forms in which such pro- cesses would either fail altogether, or else be very laborious. We now proceed to show how the Differential Calculus furnishes us with a general method for evaluating indetermi- nate forms. 90. — Method of the Differential Calculus. — Sup- f(x) . o pose -—-!- to be a fraction which becomes of the form - when (j>{x) o x = a; i. e.f(a) = o, and (a) = o ; h 2 ioo Indeterminate Forms. substitute a + h for x and the fraction becomes f(a + h) -f(a) f(a + h) h or (p(a + h) y (j)(a + h) - ${a) * ~ h but when h is infinitely small the numerator and denominator in this expression become f'(a) and (j>'(a) 9 respectively; hence, in this case, f(a + h) f(a) (p(a + h) '(a) = o, our new fraction -}t4 * s still of the indeterminate form -. Applying the preceding process of reasoning to it, it follows that f tr (ri\ its true value is that of „, ! . (a) If this fraction be also of the form -, we proceed to the next derived functions. In general, if the first derived functions which do not vanish he fW(a) and #(*%)> then the true value of " isthatof^ 4>W Examples, 101 Examples. if x sm x i. u= , when x = -. COS # 2 Here /(a;) = x sin # ,

'(x) = r(x — ay- 1 , \ a ) is o or oo, as r > or < i. Hence the true value of u is oo or o, according as r > or < i . This result can also be arrived at by writing the fraction in the form S gm(x-a) j "J. gma gmh j (x - a) r h r e ma , where h = x — a hence, expanding e mh , and making h = o, we evidently get the same result as before. x — sin x 3- — - — w nen a; = o. Here f'(x) — I - cos#, /'(o)=o $'(x) = sA '(o) = o. f"(x) = sin a;, /"(o)=o. (j>"(x) = 6x, *» = o. f'"(x) = COS#, /'"(o) = i. '"(o) = 6. io2 Indeterminate Forms. Hence, the true value is 7, as can also be immediately arrived at by substituting 6 # 3 x — — + &c. instead of sin x. o 4. when x = o. Ans. log a. x e*f(x)-e°f(a) ^_^ /(«) +/'(a) 5* r~; 7~x when # = #. ,, — — tttt. «*$>(#) - e a (j)(a) $(a)+ tf{a) It may be obseived that each of these examples can be exhibited in the form 00 — 00 , that is, as the difference of two functions each of which becomes in- finite for the particular value of x in question. 91. Form o x 00. — The expression/^) x ty(x) becomes indeterminate for any value of x which makes one of its fac- tors zero and the other infinite. The function in this case is easily reducible to the form - ; for suppose f{a) = o, and $ (a) ■Pi \ = 00, then the expression can be written , which is of the required form. Examples. ttX 1. Find the value of (1 — x) tan — when x = i. This expression becomes . the true value of which is - when x — r. r irx 7t cot — 2 I x sin x j , 2. Sec x ( x sin x - ) , when x = -. IT x sin x — 2 This becomes . a form already discussed, cos x 3. Tan (x - a) . log (x - a), when x = a. Ans. o. 4. Cosec 2 /3^ . log (cos «as), „ # = 0. a- »> — Tp3* 2/3 2 Method of the Differential Calculus. 103 92. Form ^-. As stated before, the fraction -^ also becomes indeterminate for the value x = a, if /(#) = 00, and (a) = 00. It can, however, be reduced to the form - by writing it in the shape 0(g) 1 A") The true value of the latter fraction, by Art. 90, is that of fix) Now, suppose .4 represents the limiting value of -j-\ (p (X) when x = a, then we have f\d) '(a) that is, the true value of the indeterminate form ^ is found in the same manner as that of the form -. o In the preceding demonstration, in dividing both sides of our equation by A 9 we have assumed that A is neither zero nor infinity ; so that the proof would fail in either of these cases. It can, however, be completed as follows : — ■pi \ Suppose the real limit of -W to be zero, then that of 0(«) . > is k, where k may be any constant ; but as the (x) logf(x). This latter product is indeterminate whenever one of its fac- tors becomes zero and the other infinite for the same value of x. (1.) Let (J)(x) = o, and log {/(x)} = ± 00 ; the latter re- quires either f{x) = 00, ot/(x) = o. Hence, {/(x))^ becomes indeterminate when it is of the form o°, or 00 °. (2.) Let 1, and n> in, b xK ~ m = 00 when x = 00. Consequently the value of wis of the form o 00 , or is zero in this case. Again, if m > n, b x11 m = o when x = oo } and the true value of u is 00. io8 Indeterminate Forms. a x u = r "when x = o. Let x = -, and this fraction is immediately reducible to the form discussed in z the previous Example. (i - cos#Wlog(i +x)} m . , i 6. ' } &v — . when x = o. -4rcs. — . (r + #)* -e 7. w = , when x = o. From Art. 29, this is of the form - ; to find its true value, proceed by the o method of Art. 90, and it becomes 1 ( X - (1 + x) log (1 + x)) Again, substituting for (1 + x) x its limiting value e, we get ( a; - (r + a;) log (1 + x) ) ^ e [ x*(i+x) y the true value of which is readily found to be — when x = o. Compare Ex. 29, p. 94. \ m x — 1 8. sin x (a sm x — smax) n . { — : > , when x = o. (x(cos# — cos«#)) The true value of — : , when x = o, is log m ; a sin a; — sin ax , and that of -. :, when x = o, #(cos# — cos##) has been found in Example 6, Art 89, to be-; hence the true value of the given 3 (a\ n - J log m. 8. Examples. 109 Examples. [ x + sin 7.x — 6 sin - J f 4 + cos x - 5 cos - J a: = o. x = n. 00 . (x)-4>(a)

'"(y) 6 ' 2. 00 . 2. 2 " I. I 2* 27 2m ■ (i-^)vr^m 2 tan .1 \/ 1 - m 2 m 1 + m cos d> - cos d>, m = 1. T 1 - w cos 3^) 28. log (1 + X + x 2 ) + log (I - X + X 2 ) sec x — cos x /ai x + a 2 * + . . . fl^X ~ ■ 9 - \ n ) ' x = o. a? = o. a\ a-i . , • «w Examples. 1 1 1 /log x\ x 30. { — — j, when a; =00. Ans. 1. . i ex (1 + x) x -e + — 31. - , # = 0. 33- * 2 (1 + ^j-^ 3 log (1 + £j, log(i + #) ' " — e~ x — ix '* tan # — a; ' «? tax 2 + bx + c\ dx \ a\x + b\ ) ' # =00. a; = 00. 24 sin # — log [e x cos a?) 1 32. Li 2, a; = o. 2 8* 1 — a; + log a; 34- 7 - > * = 1. - 1. I - \/ 2X — x z X 2 — X 35- ; » x=i. 00. I - # + log X x x -x 36. — , *=I. -2. I — # + log X cos # — log (1 + x) + sin x — 1 37- , ~s » x = o. o. e x -(i+x) e x + sin x — 1 3 8 » 1 — 1 — ; — 7-9 x = o 2. e x _ e -x _ 2a; 39- -z— - — -- # = o. 1. a a — x — a log I - J 41 . ■ ? 3. _ a . _ lm a — y 2ax — x* tan (# + #) — tan (« — x) 1 + a 2 42. - x = o tan -1 (a + x) — tan -1 (« — x) * cos 3 # ' x z — 3a; + 2 43- £4_ 6 *» + 8*- 3 ' * =I - 1 1 2 Examples. 44. (sin a;) 8 " 1 *, when x = o. Am. 1. 45. (sec#) Mseca! , # = o. 1. tan* Tj. 46. (sin x) , # = -• 1. 2 47. Find the value of (x-y)a n + (y -a)x n + (a- x)y n n.n- 1 an -2 # (# - y) {y -a) {a- x) * 1.2 when x — y — a. Substitute a + h f or #, and + & for y, and after some easy transformations we get the answer, on making h = o, and k = o. # + tan # — tan 2x . 7 4.3. , x = o. Ans. -p< T 2# + tan a; - tan 3# 20 a; + sin x — sin 2# — 7 49 # = 0. 20; + tan x — tan 30;' 52 \/x — \/a + '\/x — a 50. . , ,— ^/a;2 _ a 2 ^/ 20 % = a. 2 • r * # sin a; — tan ar 3 1 — 1 ci 1 2-0. — 5 1, #5 » 20 ( »3 ) CHAPTEE Y. PARTIAL DIFFERENTIAL COEFFICIENTS AND DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE VARIABLES. 95. Partial Differentiation. — In the preceding chap- ters we have regarded the functions under consideration as depending on one variable solely ; thus, such expressions as e ax , sin bx, x m , &c, have been treated as functions of x only ; the quantities a, b, m, . . ;... being regarded as constants. We may, however, conceive these quantities as also capable of change, and as receiving small increments ; then, if we regard x as constant, we can, by the methods already established, find the differen- tial coefficients of these expressions with regard to the quan- tities, a, b, m, &c, considered as variable. In this point of view, e ax is regarded as a function of a as well as of x, and its differential coefficient with regard to a d (e ax ) is represented by , or x e ax by Art. 30 ; in the derivation of which x is regarded as a constant. In like manner, sin {ax + by) may be considered as a function of the four quantities, x, y, a, b, and we can find its differential coefficient with respect to any one of them, the others being regarded as constants. Let these derived functions be denoted by du du du du dx' dy' da 9 db* respectively, where u stands for the expression under con- sideration, and we have du du . . — = a cos {ax + by), — = b cos (ax + by), du , _ x du . . . — = x cos (ax + by), — = y cos (ax + by). 1 1 4 Partial Differentiation. These expressions are called the partial differential coef- ficients of u with respect to x, y, a, b, respectively. More generally, if /0, y, *) denotes a function of three variables, x, y, z, its differential coefficient, when x alone is supposed to change, is called the partial differential coefficient of the function with respect to x; and similarly for the other variables y and %. If the function be represented by u, its partial differential coefficients are denoted by du du du dx 9 dy* dz ' and from the preceding it follows that the partial derived functions of any expression are formed by the same rules as the derived functions in the case of a single variable. Examples. 1. u = {ax* + by* + ez z )«. Here — = 2 nax (ax 2 + by 2 + cr £ V 1 " 1 , dx — = 2nby (ax 2 + by 2 + cz 2 ) n ~ l , dy — = 2ncz (ax 2 + by 2 + cz 2 ) n 'K y du 1 du — x a/V 2 ~ x<1 y y*y y 2 —% 2 3- du , du u = xv, — = yxv- 1 , — = xv dx dy 4- u = x-

{x, y + k) ft (x,y + k) -ft (x, y ) h k If now h and k be supposed to become infinitely small, by Art. 6 we have ft (x + h, y + h) - ft (x, y + k) d . ft (x, y + k) and h dx ft (x, y + k) - ftp, y) d . ft (x, y) k dy In the limit, when k is infinitely small, $ (x, y + k) becomes ft (x, y), and rf -»(W*) becomes (z), y) y *0- du du dz dx dz dx du du dz dy dz dy y 1 ydx — xdy -o ■••*-*€) t Again, multiplying the former of the two preceding equations by x, and the latter by y, and adding, we get du du x— + y— = o. dx dy Differentiation of a Function of Three or more Variables. 117 97. Differentiation of a Function of Three or more Variables. — Suppose u = $ (x, y, a), and let h, k, I represent infinitely small increments in x, y, z, respectively; then Au = (j> (x + h, y + Jc, z + I) - (j) (x y y, z) (j> (x + h, y + h, z + T) - (x, y + k, z + I) (j) (x,y + k,z+l) - (x,y,z + l) {x i y,z+l )-(x,y i z ) + - k k + f I, which becomes in the limit, by the same argument as before, when dx, dy, dz, are substituted for h, k, I, du du _ du T , . du = — dx + — dy + — dz. (2) dx dy dz Or, the infinitely small increment in u is the sum of its infinitely small increments arising from the variation of each variable considered separately. A similar process of reasoning can be easily extended to a function of any number of variables ; hence, in general, if u be a function of n variables, x x , x % , x z , . . . x n , 7 du 7 du du - . N du = — dxi + — dx 2 + . . . + — dx n . (3) ax\ aXi aXifi 98. If u = f(v, w), where v, w, are both functions of x ; then, from Art. 96, it is easily seen that du df(v, w) dv df(v, w) dw dx dv dx dw dx' This result is usually written in the form du du dv du dw . dx dv dx dw dx' * ' In general, if 1 1 8 Partial Differentiation. where y lf y 2 , . . . y n , are each functions of x, we have du du dyi du dy % da dy n dx dy l dx dy % dx dy n dx' Also, if y l9 y 2 , &c, y n9 be at the same time functions of another variable s, we have and so on. du _ du dy x du dy 2 du dy n dz dy x dz dy 2 dz dy n dz Examples. 1. Let u = (X,Y,Z), where X = ax + by + cz, Y = a'x + b'y + c'z, Z = a"x + b"y + c"z. "When these substitutions are made, u becomes a function of x, y, z, and we have du du , du „ du Tx = a dx + a dY +a dz> du y du du du- dy- =b dX +b dY +b 12 du du .du „ du dz dX^ dY dZ' Differentiation of a Function of Differences. 1 19 98*. ^Differentiation of a Function of diffe- rences. — If u be a function of the differences of the vari- ables, a, j3, y : to prove that du du du \- —^-\ = 0. da dp dy Let a - ft = x, j3 - y = y, y - a = z; then, u is a function of x } y } % \ and, accordingly, we may write u = (x, y, z). du du dx du du du dz du du da dx da dy da dz da dx dz du du du du du du djd dy dx' dy dz dy ' du du du ' da dj5 dy This result admits of obvious extension to a function of the differences of any number of variables. Hence Similarly, 1. If 2. If dA dA dA da dp dy * I, h I, I, A = a, a~, ft 7, 7 2 , 5, 5 3 , a 3 , 3 , 7 3 , 5 3 , dA dA — + — da dfi dA dy dA + ~d~8 = o. I, i, I, I, A = a, « 2 , ft 7> 7 2 , 5, s 2 , a 4 , &, 7 4 , 5 4 , h I, I, 1, d8 v a, a-, ft 2 , 7> 7 2 , 8, s 2 , a 3 , /3 3 , 7 3 , s 3 , , prove that , prove that 1 2 o Partial Differentiation. 99. Definition of an Implicit Function. — Suppose that y, instead of being given explicitly as a function of x } is determined by an equation of the form /(*> y) = °> then y is said to be an implicit function of x ; for its value, or values, are given implicitly when that of x is known. 100. Differentiation of an Implicit Function. — Let h denote the increment of y corresponding to the incre- ment h in x, and denote f(x, y) by u. Then, since the equation fix, y) = o is supposed to hold for all values of x and the corresponding values of y, we must have fix + h,y + k) = o. Hence du = o ; and accordingly, by Art. 96, we have, when h and h are infinitely small, du 7 du 7 — - h + — - k = o ; ax ay du . k dy dx . hence in the limit - = _=_^. (6) dy This result enables us to determine the differential coefficient of y with respect to x whenever the form of the equation f(x, y) = o is given. In the case of implicit functions we may regard x as being a function of y, or y a function of x, whichever we pl ease — { n the former case y is treated as the independent variable, and, in the latter, x : when y is taken as the inde- pendent variable, we have du dx dy _ 1 dy du dy QjJU (JjJU This is the extension of the result given in Art. 20, and might have been established in a similar manner. Differentiation of an Implicit Function. 121 Examples. 1 . If a 3 + y 3 — 2>axy — e, to find — . Here *!. « 3 (* - d(j> dx' dy du dx df_ df_ dx* dy df_ dy More generally, let u =

dcj) dx' dy' dz dfx df dfi dx' dy' dz df, dx' dfz dy' dh dz da dx dy' az This result easily admits of generalization. (8) Miller's Theorem of Homogeneous Functions. 123 102. Eider's Theorem of Homogeneous Func- tions. — If u = AaP y q + Bx*' y qf + Cap" y q " + &c, where p + q = p + q = p" + q" = &c. = n, to prove that du du , > x — + y-— = nu. (9) dx J dy K J (I'll Here x — = Apx p y q + Bp' x pf y qf + &c. ; ft ?/ y — = AqxP y q + Bq x?' y q ' + &c. ; .'. x— + y — = A(p + q)xPy q + B(p r + q') x p ' y q ' + &c. dx cty = nAx p y q + nBx pf y qf + &o. = nu. Hence, if u be any homogeneous expression of the n th degree in x and y, not involving fractions, we have • du du x -=-+ y — = nu. dx dy Again, suppose u to he a homogeneous function of a fractional form, represented by — ; where 0i, 2 , are homo- 02 geneous expressions of the n th and m th degrees, respectively, in x and y ; then, from the equation we have and du 0i u = — 02 d(j} 2 dx dx du (0 3 ) 2 C?0i ^Hy~* d(f> 2 dy (*f 124 Partial Differentiation, accordingly we get / dtyi d£A _ ( dfa dfc du du \ dx dy ) \ dx dy X Tx + y dy= ^p ' but, by the preceding, dd)i d(bi dd)2 dd)2 . du du ni d> 2 h6nCe 'H + '*" (^ = in -m) — = (n - m) u\ which proves the theorem for homogeneous expressions of a fractional form. This result admits of being established in a more general manner, as follows : It is easily seen that a homogeneous expression of the n th degree in x and y, since the sum of the indices of x and of y in each term is n, is capable of being represented in the general form of Accordingly, let u = x n $ I - J = x n v, where v =

x> + by ' » X dx + ^d~y = U ' . . x 2 — y 2 du du /x 3 — y 3 \ du du 103. Theorem. — If U = u + u x + u % . . . + u n , where u is a constant, and u x , u 2 , . . . u n , are homogeneous functions of x, y, z, &c, of the ist, 2nd, . . . n th degrees, respectively, then dU dU dU 2 ~a^ + 1/ 'ay +Z 'dz + ''' :=Ul + 2U2 + 3U3+ ''' + nUn ' (* ^ For, by Euler's Theorem, we have aUy a%vf aWiy ■ \- y + % dx dy dz since u r is homogeneous of the r th degree in the variables. Con. If TT = o, then dU dU dU . x X ~dx : + y d^ + Z lz''" := ~ ^ Un - 1 + 2% ~ 2 + ' * * + m ^' ^ This follows on subtracting nii + nu x + . . . + nu n = o from the preceding result. tv Ivy* Lb Ivy \Af tl'y q x — + y —- + z-=- + &c. = ru n Remarks on JEuler's Theorem. 1 2 7 104. Remarks on JEuler's Theorem. — In the appli- cation of Euler's Theorem the student should be careful to see that the functions to which it is applied are really homogeneous expressions. For instance, at first sight the expression sin -1 ( — — '— J might appear to be a homogeneous function in x and y ; but if the function be expanded, it is easily seen that the terms thus obtained are of different degrees, and, consequently, Euler's Theorem cannot be directly applied to it. However, if the equation be written /y> -J- if/ in the form -= r = sin u, we have, by Euler's formula, xi + yh ' ' J 9 d sin u cl sin u sin u x — = h y — = — = , dx dy 2 du du\ sin u or cos u \x — + y — = — ; \ dx dy J 2 . du du tan u \ x + y hence x — + y — = dx dy 2 2 + y)«" When, however, the degrees in the numerator and the denominator are the same, the function is of the degree zero, and in all such cases we have du du x—+y— = o. ■ dx dy 1 , . ,JL\ hx T2 , . xs + ya\ x + y — . For example, sm" 1 - x — —. , tan" 1 -, &, &c, may be treated as homogeneous expressions, whose degree of homo- geneity is zero. The same remark applies to all expressions which are reducible to the form $ ( - ) ; as already shown in Ex. 3, Art. 96. 105. If x = r cos 0, y = r sin 0, to prove that xdy - ydx = r 2 d0. (13) 1 2 8 Partial Differentiation, In Ex. i, Art. 96, we found dx = cos Odr - r sin QdQ ; similarly dy = sin Odr +■ r cos 06?0. Hence a?% = r cos 9 sin 0^r + r 2 cos 2 OdO, ydx = r cos sin 0<#r - r 2 sin 2 0^0 ; .*. xdy - ydx = r 2 dO. 106. If a? and y have the same values as in the last, to prove that (dx) 2 + (dy) 2 = (drf + r 2 (dO) 2 . (14) Square and add the expressions for dx, dy, found above, and the required result follows immediately. The two preceding formulae are of importance in the theory of plane curves, and admit of being easily established from geometrical considerations. 107. If u = ax 2 + by 2 + cz 2 + zfyz + 2gzx + ihxy, to find the condition among the constants that the same values of x, y, z should satisfy the three equations Here du — = 2gx + 2jy + icz = o. Hence, eliminating x, y, % between these three equations, the required condition is abc -of 2 - bg 2 - ch 2 + 2fgh = o ; or, in the determinant form, a h g h b f 9 f c du dx 0, du dy 0, du dz 0. du dx 2 ax + 2hy + 2gz = 0, du dy 2hx -I- 2by + 2jk* = 0, Remarks on Ruler's Theorem. 129 The preceding determinant is called the discriminant of the quadratic expression, and is an invariant of the function ; it also expresses the condition that the conic represented by the equation u = o should break up into two right lines. (Salmon's Conic Sections, Art. 76.) The foregoing result can be verified easily from the latter point of view ; for, suppose the quadratic expression, u, to be the product of two linear factors, X and Y ; or u = XY, where X = Ix + my + nz, Y = Vx + m'y + n'z ; then £ = X~ + T^ = I'X + IT, ax ax ax du ^.dY rrdX ,^. ^ — = A — + F — = mX + mY, dy dy dy au ._ a jl __ u>JL , __ ._ — = A — + F — = nX + nY. dz dz dz Here the expressions at the right-hand side become zero for the values of x, y, z, which satisfy the equations X = o, F= o, or Ix + my + nz = o, Vx + m'y + n'z = o. Hence in this case the equations du du du dx ' dy ' dz are also satisfied simultaneously by the same values. We shall next proceed to illustrate the principles of partial differentiation by applying them to a few elementary questions in plane and spherical triangles. In such cases we may regard any three* of the parts, a, b, c, A, JB, C, as being * The case of the three angles of a plane triangle is excepted, as they are equivalent to only two independent data. K 130 Partial Differentiation. independent variables, and each of the others as a function of the three so chosen. 108. Equation connecting the Variations of the three Sides and one Angle. — If two sides, a, b, and the contained angle, C, in a plane triangle, receive indefinitely small increments, to find the corresponding increment in the third side c, we have & = a 2 + ¥ - lab cos C; .*. cdc = (a - b cos C) da + (b - a cos C) db + ab sin CdC; but a = b cos C + c cos B, b = a cos + c cos A. Hence, dividing by c, and substituting c sin B for b sin C, we get do = cos B da + cos A db + a sin B dC. (15) Otherwise thus, geometrically. By equation (2), Art. 97, we have dc 7 tfc 77 die 7 ~ dc = — da + — do + — ^ du. da do Oj\j dc Now, in the determination of — we must regard b and C as aa constants ; accordingly, let us sup- pose the side CB, or a, to receive a small increment, BB' or Aa, as in the figure. Join AB", and draw B'D perpendicular to AB, produced if necessary; then, by Art. 37, AB r » ] = AB when BB' is infinitely small, , neglecting infinitely small quanti- ties of the second order. Hence Ac = AB' -AB = AD-AB = BB; dc .. ., , Ac BD „ .-. — = limit of — = r—-, = cos B. da Aa BB Fig. 4. Examples in Plane Trigonometry. 131 HP Similarly, — = cos A ; which results agree with those arrived at before by differentiation. dc Again, to find -j~. Suppose the angle C to receive a small increment AC, represented by £ BCB' in the accompanying figure; take CB' = CB, join AB', and draw BB perpendicular to AB ' . Then Ac = AB' -AB = B'B (in the limit) = BB' cos AB'B = BF sin^(7(q.p.). Fig. 5. Also, in the limit, BB' = #(7 sin BOB" = a AC. dc Ac Hence -t~ = limiting value of — ^ = # sin B ; the same result as that arrived at by differentiation. In the investigation in Fig. 5 it has been assumed that AB - AB is infinitely small in comparison with BB; or that AB-AD the fraction =p=r — vanishes in the limit. For the proof of this the student is referred to Art. 37. When the base of a plane triangle is calculated from the observed lengths of its sides and the magnitude of its vertical angle, the result in (15) shows how the error in the computed value of the base can be approximately found in terms of the small errors in observation of the sides and of the contained angle. dC ioo. To find -— when a and are considered y dA Constant. — In the preceding figure, BAB' represents the change in the angle A arising from the change AC in C; moreover, as the angle A is diminished in this case, we must denote BAB' by - AA, and we have BB' - AB/ ^ A ABAA cAA sm AB'B~ cos B cosB' k 2 132 Partial Differentiation, Also, BB' = a&C; dO AC ,. , .. ... c ,. ^ = ^(m the limit) — j^. (16) This result admits of another easy proof by differentiation. For # sin B = b sin A ; hence, when # and b are constants, we have a cos B dB = b cos -4 *L4 ; also, since A + B + C = it, we have dA + dB + dC = o. Substitute for <£Z? in the former its value deduced from the latter equation, and we get (a cos B + b cos A) dA = - a cos B dO; or c dA = - a cos B dC, as before. no. Equation connecting the Variations of two Sides and the opposite Angles. — In general, if we take the logarithmic differential of the equation a sin B = b sin A, regarding a, b, A, B, as variables, we get da dB db dA . a tan B b tan A' * in. I^anden's Transformation. — The result in equa- tion (16) admits of being transformed into dA __dO m a cos B c but c = */a? + b % - 20b cos C, and a cos B = */a % - b* sin 2 ^l; hence we get dA dO *s/a* - ¥ sin % A \/c? + b z - zab cos C* Examples in Spherical Trigonometry. 133 If C be denoted by 180 - 20i, the angle at A by <£, and b - by h, the preceding equation becomes a d(J> 2d(pi 2d(f)i *y 1 - ¥ sin 2 $ \/ \ + 2kco8 2i + k 2 ^/{i +k) 2 - 4&sin 2 $i 2 *fyi ~ (1 + &) a/ 1 - ^i 2 sin 3 0i ' ^ ' where Id = , . 1 + h Also, the equation a sin 5 = 5 sin ^ becomes sin (20! - (j)) = h sin $. The result just established furnishes a proof of Landen's* transformation in Elliptic Functions. We shall next investigate some analogous formulse in Spherical Trigonometry. 112. Relation connecting the Variations of Three Sides and One Angle. — Differentiating the well-known relation cos c = cos a cos b + sin a sin b cos C, regarding a and b as constants, we get dc sin a sin b sin C dC sin c sin a sin B. dc Again, the value of — , when b and C are constants, can be easily determined geometrically as follows :— • This transformation is often attributed to Lagrange ; it had, however, been previously arrived at by Landen. (See Philosophical Transactions, 177 i and 17750 *34 Partial Differentiation, In the spherical triangle ABO, making a construction similar to that of Fig. 4, Art. 108, we have ^^ = A#; /. -7- = limit 01 — - = -=r=^ (in the limit) = cos B. Similarly, when a and C are con- stants. — = cos A. do Hence, finally, lg * ' de = cos B da + cos Adb + sin sin i? <#(7. (19) This result can also be obtained by a process of diffe- rentiation. This method is left as an exercise for the student. As, in the corresponding case of plane triangles, we have assumed that AB' = AB in the limit ; i.e., that AB' — AD — — — is infinitely small in comparison with AD in the limit ; this assumption may be stated otherwise, thus : — If the angle A of a right-angled spherical triangle be very small, then the ratio — — - becomes very small at the j± same time, where e and b have their usual significations. This result is easily established, for by Napier's rules we have tan b sin b cos c cos A = = ; — ; — ; tan c cos sin c 1 - cos A sin c cos b - cos c sin b sin (c - b) \ or sin 1 + cos A sin c cos b + cos c sin b sin (c + b)' , o-A • / 7\ sm (c-b) . , ,v, A (c-b) = tan 2 — sm (c+b); .*. v ' = sin(c + o)tan— . tan — <; 2 But the right-hand side of this equation becomes very small along with A, and consequently c-b becomes at the same time very small in comparison with that angle. Examples in Spherical Trigonometry. 135 The formula (19) can also be written in the form 7 ^ dc da db dC = = — ^ - 7 5 - • 7 . r ( 20 ) sm a sm B sin a tan i* sin tan ^4 x ' The corresponding formulae for the differentials of A and B are obtained by an interchange of letters. Again, from any equation in Spherical Trigonometry another can be derived by aid of the polar triangle. Thus, by this transformation, formula (19) becomes dO = - cos b dA - cos a dB + sin A sin b dc. (2 1 ) These, and the analogous f ormulae, are of importance in Astronomy in determining the errors in a computed angular distance arising from small errors in observation. They also enable us to determine the most favourable positions for making certain observations ; viz., those in which small errors in observation produce the least error in the required result. 113. Remarks on Partial Differentials. — The be- ginner must be careful to attach their proper significations to the expressions — , — , &c, in each case. Thus when a and aa cl\y dc are constants, we have -^ = sin a sin B ; but when A and a are constants, we have -77^ = ■: ^: these are quite different dC tanC x dc quantities represented by the same expression — -. The reason is, that in the former case we investigate the ultimate ratio of the simultaneous increments of a side and its opposite angle, when the other two sides are considered as constant ; while in the latter we investigate the similar ratio when one side and its opposite angle are constant. Similar remarks apply in all cases of partial differentia- tion. When our formulae are applied to the case of small errors in the sides and angles of a triangle, it is usual to designate these errors by Aa, Ab, Ac, A A, AB, AC; and when these expressions are substituted for da, db, &c, in our formulae, they give approximate results. 136 Partial Differentiation. For instance (19) becomes in this case Ac = Aa cos B + Ab cos A + AC sin a sin B ; (22) and similarly in other cases. It is easily seen that the error arising in the application of these formulae to sucli cases is a small quantity of the second order ; that is, it involves the squares and products of the small quantities Aa, Ab, Ac, &c. This will also appear more fully from the results arrived at in a subsequent chapter. 114. Theorem. — If the base c, and the vertical angle C, of a spherical triangle be constant, formula (19) becomes da db + =. = o. cos A cos B Now, writing $ instead of a, \p instead of b, and k for —. — , this equation becomes sin c . 1 sin ^4 since k - -= \ Bina sin^X sin b J d(p 1 dip \/ 1 - k 2 sin 2 ' */ 1 - W sin 2 i// where $ and \p are connected by the following* relation : — cos c = cos $ cos \p + sin

W = li - Also, since y is independent of x, we have du du dx' du A du du dx dx' dx dx" dy dyT TT du du dx' du dy' du ~ du dt dx' dt dy' dt dx dy Partial Differentiation. 139 In like manner, if x', y', z', be substituted for x + at, y + fit, z + yt, in the equation u = W = /3 ' M m f' t-t du du ^du du . x Hence _- a _ + 0_ + y _ ( 2 8) tt't' Uw W-w Cf* This result can be easily extended to any numberof variables. 140 Examples. Examples. • -, (%\ . , fy\ j, L , d* &y 1. If u = sin" 1 ( - ) + sir 1 ( 7- ) , prove that «w = , + — . W W A/a 2 -a; 2 ^/b^-y 2 3. Find the conditions that «, a function of a?, 2/, 2, should be a function of a? + y + s. . du du du Am. — - = — = — . dx dy dz 4. If f(ax + 3y) = e, find ^. „ - |. 5. If /(w) = dy dx dz dz dx dx " dfd$ _df_dj>' dz dy dy dz df dip df d

df d

(i%,y)+*/-ity{x,y) admits of being reduced to the form d

~dy> dz~> C '' being also functions x, y, z 9 &c, admit of being differen- tiated in the same manner as the original function ; and the partial differential coefficient of — , when x alone varies, is ■"■ dx denoted by d fdu\ d 2 u dx \dxf dx 29 as in the case of a single variable. du Similarly, the partial differential coefficient of — , when y ttX alone varies, is represented by d fdu\ d 2 u dy \dxj dydx 9 and, in general, - — — denotes that the function u is first ° dy m dx n differentiated n times in succession, supposing x alone to vary, and the resulting function afterwards differentiated m times in succession, where y alone is supposed to vary ; and similarly in all other cases. The Order of Differentiation is Indifferent. 145 We now proceed to show that the values of these partial derived functions are independent of the order in which the variables are supposed to change. 119. If u foe a Function ofx and y, to prove that d (du\ d fdu\ d 2 u d 2 u dy\dos) dx\dyj dydx dxdy* ' where x and y are independent of each other. du Let u = § (x, y), then — represents the limiting value of cix (x, y) h when h is infinitely small. This expression being regarded as a function of y, let y become y +% x remaining constant ; then -f-(-r) is the limiting value of (ft (x + h, y + h) - (p (x, y + k) -

(x, y) hk when both h and k become infinitely small, or evanescent, Q/tl In like manner — is the limiting value of (x,y + h) -(j>(x,y) k d 1 din \ when k is infinitely small ; hence — ( — ) is the limiting value of $ (x + h, y + k) - (x + h, y) - ^ (x, y + k) + (x, y) hk when both h and k are infinitely small. Since this function is the same as the preceding for all L 146 Successive Partial Differentiation. finite values of h and k, it will continue to be so in the limit; hence we have d fdu\ d fdu\ dec \dyj dy \dxj In like manner for by the preceding dhi dx 2 dy d 2 u dxdy dydx ' d 3 u dydx 2 9 d 2 u d f dhi dx \dxdy d_ dx d 2 u d d du d d du dydx dx' dy dx dy' dx dx similarly in all other cases. Hence, in general, d?**u cP*eu dx p dy q dy q dx p ' Again, in the case of functions of three or more variables, by similar reasoning it can be proved that d 3 u d 3 u . dzdxdy dxdydz' Hence we infer that the order of differentiation is in all cases indifferent, provided the variables are independent of each other. 1. If u = d>[ - J, 2. If u = tan -1 1 - J , \yl 3. If u = sin (ax n + by n ), Examples toe Verification. d?u verify that dydx d 3 u dy 2 dx d^u d 2 u dxdy d z u dxdy* 1 ' d*u dx^dy 2, dy 2 dx z ' 120. Condition that Pdx + Qdy shall he a total differential. — This implies that P dx + Q dy should be the exact differential of some function of x and y. Denoting this function by u, then du = P dx + Q dy, (2) Condition for a Total Differential 147 and, by (1), Art. 95, we must have _ du ~ du P = — , Q = — ; ax ay dP d 2 u dQ d 2 u rfy %^' dx dxdy Hence the required condition is dPdQ dp dx 121. If u be any Function of x and ?/, to prove that i(*<">SK(™S). (3) where x and ^ are independent variables. Here each side, on differentiation, becomes __ . v a u Tit / \ an dxdy ' dxdy' 122. more generally, to prove that d f dv\ d f dv\ dy \ dx) dx \ dy/ * ' where u and v are both functions of z, and z is a function of x and y. ._ <£ / cfo\ dud/o d 2 v For -T- w-r- = -T- t- + w but c?w e?w dz dv dv dz dy dz dy* dx dz dx ' d ( dv\ du dv dz dz d 2 v — U— =-——- — — + u dy \ dx) dz dz dx dy dydx ' and — ( u— ) has evidently the same value. dx\ dy } J l 2 148 Successive Partial Differentiation. 123. Euler's Theorem of Homogeneous Func- tions. — In Art. 102 it has been shown that du du x — + y — = nu, dx dy where u is a homogeneous function of the n th degree in v and y. Moreover, as — and — are homogeneous functions of the dx dy degree n - 1, we have, by the same theorem, dx x d fdu\ d fdu\ . du Ix \clxj dy \dxj ' dx* d fdu\ d fdu\ .du^ dx \dyj dy \dyj dy' multiplying the former of these equations by ss, and the latter by y, we get, after addition, „ d 2 u d 2 u „ d 2 u , s ( du du\ x* -— + 2xy -7—7 - + y' -=-= = {n-i)[a;— + y—\ dx 2 y dxdy * dy 2 v ; \ dx * dy) = (n-i)nu. (5) This result can be readily extended to homogeneous functions of any number of independent variables. A more complete investigation of Euler's Theorems will be found in Chapter VIII. 124. To find the Successive Differential Coeffi- cients with respect to t 9 of tbe Function (j>(x + at, y + fit), where x, y, a, fi, are independent of t, and of each other. By Ait. 1 1 7 we have in this case, where stands for the expression _ d(j> dt dx dy Differentiation qf(j>(x + at, y + j3£). 149 Hence g-;»(?Ws(£) dtf dt\dx) dt\dy) d fd(f\ R d (d<$\ ~ a dx\dtJ P dy\dt) dx ( dx dy) dy { dx dy) = a ^ + 2a R £*. + &*¥$. (6) dx 2 dxdy dy 2 ' ' This result can also "be written in the form d 2

(7) f d d\ 2 in which f a — + j3 — ) is supposed to be developed in the usual manner, and -7-f , &e., substituted for f — J 0, &c. d?<$> Again, to find — . ccz dt z ~ dt\df)~ dt\ a dx + "dj/J* dx dy) dt \ dx ' dy) \ dx ^ dy) \ dx ^ dy) ^' By induction from the preceding it can be readily shown that d n

- «y, y % A, B, G B, C, D and show that the left-hand side of this equation vanishes -when V is a perfect cube. 9. liu = (s 2 + y % + z 2 )i i? prove that dht d 2 u dhi 1 -j = o. dx 2 dy 2 dz 2 ( i5i ) CHAPTEB VII. lagrange's theorem. 125. Lagrange's Theorem. — Suppose that we are given the equation z = x+.y(z), (1) in which x and y are independent variables, and it is required to expand any function of z in ascending powers of y. Let the function be denoted by F(z), or by u, and, by Maclaurin's theorem, we have . y f^t\ j[_ ( d%u \ y n _ ( d * u \ & (y 1 \dyJo 1 . 2 \dy 2 Jo 1.2... n\dy n J "' (/Jf/\ nil — ) , &c, represent the values of u, -7-, &c, when dyjo . d V zero is substituted for y after differentiation. It is evident that u = F(x). To find the other terms, we get by differentiating (1) with respect to a?, and also with respect to y, dz ,. . dz dz . . ,. .dz . dz , .dz hence — = 6[z)-—. dy r dx Also, since u is a function of z, we have c?w ^w dz du du dz dx dz dx' dy dz dy* 152 Lagrange's Theorem. hence we obtain du f jdu , v Ty m *®& (3) Again, denoting $(s) by Z, we have by Art. 121, since Z is a function of u, ( „du\ d ( r,du\ d 2 u „ , N d 2 u d ( du\ . v dy z dx\ dx)' t-t , d 3 u d z ( m du\ Hence also -7-= = -=-=- \" t > since # and y are independent variables ; « i(*S-i(»S-£(**W» d 2 (^ du\ = fdV ( z% du\ m dxdy \ dx) \dx) \ dx) ' To prove that the law here indicated is general, suppose that *±J*X*(z&\l dy n \dx) \ dx) ' dy\ dx) dx\ dy ) dx\ dx/ d n f du\ _ d?_ ( yn+1 du\ t dx n ~ x dy\ dx) dx n \ dx)' andheace _=y^_]. (6) Lagrange's Theorem. 153 This shows that if the proposed law hold for any integer n, it holds for the integer n + 1 ; but it has been found to hold for n = 2 and n = 3 ; accordingly it holds for all integral values of n. du d ?/ It remains to find the values of — , -— , &c. when we ay ay 2 make y = o. Since on this hypothesis Z or 0(s) becomes 0(#), and -j- becomes J" ' or F'(x), it is evident from (3), (4), (5), (6), that the values of du d 2 u d 3 u d n ™u dy 9 dy 19 dy 3 ' become at the same time dy ,«+!> #(*) F'(x), L dx {*(*)*■*"(*) dx % {#(«.) j 'j"(«) d n ~ Consequently formula (2) becomes r ' [{x)Y F' (x) y F(z) = F{x) + *- (x)} n+1 F'(x) + &C. + &c. (7) This expansion is called Lagrange's Theorem. If it be merely required to expand g, we get, on making F(z)=z, z = x+ y -(x) +-£-4- {0W} 2 +&c 1 rw i.2dx XYy n f 1 .2 . . ,n dx' *,*{*(»)}* + **. (8) 154 Laplace 7 s Theorem. 126. Laplace's Theorem. — More generally, suppose that we are given z=f{% + yi — i) a 3n ~ 2 1- &c. I . 2 I.2.3 3. Given z = % + ye", find the expansion of z. Ans. z=x + ye x + y 2 e 2x + -z— xe 3x + — - — <&e ix + &c. 1.2 1.2.3 4. z — a + e sin 2, expand (1) 2, (2) sin 2. e* d e 3 f d\ 2 (1). -4ws. 2 = + sin # + — (sin 2 #) + 1 — 1 (sin%) + &c. 1 . 2 da 1.2.3 \daj (2). ,, sin 2 = sin a + e sin a cos a 4 — (sin 5 # cos a) + &c. 1 . 2 da 5. If 2 = a + - (2 2 — 1), prove that 2 %{a 2 - 1) # 2 rf /a 2 - i\ 2 z = « + - v 1 + — ' 1 2 i . 2 da + 1 .2 ... n 6. Hence prove that . 2 . . . w \«0/ \ 2 / + 1 ( 156 ) CHAPTER Yin. EXTENSION OF TAYLOR'S THEOREM TO FUNCTIONS OF TWO OR MORE VARIABLES. 127. Expansion of $ (x + h, y + k) . Suppose u to be a func- tion of two variables x and y, represented by the equation u = ${x, y) ; then substituting x + h for x, we get, by Taylor's Theorem, d h 2 d 2 (x + h,y) = ${x, y) + h— (0 (x, y)} + — — {(x, y)) + &c. Again, let y become y + k } and we get d <$>(x + h,y + k) = (j>(x, y + k) + h— {^(x, y + k)) h 2 d 2 + 7r2^ i ^^ 2/+ ^ )+&0, (l) But d k 2 d 2 , du W d 2 u - u + k— + — + &c. dy 1.2 ay* Also d du d 2 u hk 2 d 3 u „ dx W* 9 * '' ~ dx dxdy 1 . 2~dxdy 2 "' and JL *Lt k\\-——— -^- & 1 . 2 ffe 2 '^ ' ^ ' ' " 1 . 2 dx 2 1.2 d# 2 dy Extension of Taylor' } s Theorem. 157 Substituting these values in (1), we get , / 7 7\* . du 7 du h z d 2 u 7T d 2 u h % d 2 u . . ;i + M = + -i: 3 + &c - ( 2 ) 1 . 2 dx % dxdy 1 . 2 dy 1 128. This expansion can also be arrived at otherwise as follows : — Substitute x + at and y + fit for x andy, respectively, in the expression $ (x, y), then the new function $ (x + at, y + fit), in which x, y, a, fi, are constants with respect to t, may be regarded as a function of t, and represented by F(t) ; thus

d 2 x It 9 ~a¥ 9 C,, when t = o ; where stands for (f>(x + at, y + fit). Moreover, by Art. 1 1 7, we have d(f> d(x + h, y + k) =u + h— + k — + _-; + hk r v * ' dx dy 1 . 2 dxr k 2 d 2 u 1 /. d _ dy* 1 . a7 07\ / \ + iti^ + - + !^t(^ + ^J *(* + «*»* + «)• (5) 129. Expansion of (j> (x + h, y + k, z + /). — A function of three variables, x, y, z, admits of being treated in a similar manner, and accordingly the expression (x + h,y), since y is indepen- dent of x. Extension of Taylor's Theorem, 1 6 1 d In like manner, the operation e kdv , when applied to any function, changes y into y + k ; d d d_ .-. e k *» . e hTx (f> (x, y) = e kdy (j> (x + h, y) = (x + h, y + k), d_ £ or e hTy+hdx ^ (x, y) =

( X) y, z) = $(x + h, y + k, z + I). (8) 131. If in the development (2), dx be substituted for h, and dy for k, it becomes {x + dx,y + dy) = $ + —dx + -^ dy + - 1 - 2 (3^ +2 ^^ + 0^) +&c - <«> If the sum of all the terms of the degree n in dx and dy be denoted by d n $, the preceding result may be written in the form . 7 x d6 d 2 6 d 3 d> + — + — - + - — + . . . rv r 1 1.2 1.2.3 \n Since dx, dy, are infinitely small quantities of the first (1 ?/ n 1£ * That this is the case appears immediately from the equations — — =-t-j } d 3 u __ d 3 u dx^dy dydx 2 ' M 1 62 Extension of Taylor's Theorem. order, each term in the preceding expansion is infinitely small in comparison with the preceding one. Hence, since d 2 is infinitely small in comparison with dtp, if infinitely small quantities of the second and higher orders be neglected in comparison with those of the first, in accordance with Art. 38, we get dip = a\/ '2 or ) 2 ' {x + a) (x + b) x + c (2 + a — c) (z + b — c) Let x + c = z, and the fraction becomes z In order that this should have a real min. value, (a — e)(b — c) must be posi- tive ; i. e. the value of e must not lie between those of a and b, &c» 5. Find the least value of a tan 6 + b cot d. Am. 2 \/ ab. 6. Prove that the expression ; will always lie between two fixed r x% + bx + c~ finite limits if a 2 + c 2 > ab and b 2 < 4 c 2 ; that there will be two limits between which it cannot lie if a 2 + c 2 > ab and b 2 > 4 c 2 : and that it will be capable of all values if a 2 + c 2 < ab. 136. To find the Maximum and Minimum values of ax* + ibxy + cy % dx % + ib'xy + cy 1 ' Algebraic Examples of Maxima and Minima. 167 x Let u denote the proposed fraction, and substitute z for -; If then we get az 2 + ibz + c . v " = aV + aftW ' (I) or (a - a'u)z 2 + 2 (b - b f u)z + c- c'u = o. Solving for z, this gives (a - du)z + b-b'u = ± y y {b - b'u) 2 - (a - a'u) {c - c'u) . (2) There are three cases, according as the roots of the equation (b' 2 - a'c') u 2 + {ad + ca'- 2bb')u+b 2 -ac = o (3) are real and unequal, real and equal, or imaginary. (1). Let the roots be real and unequal, and denoted by a and (3 (of which (3 is the greater) ; then, if V 2 - a'c' > o, we shall have (a-a'u)z + b-b'u = ± y(b' 2 - a'c') (u-a) (u-[3). Here, so long as u is not greater than a, z is real ; but when u > a and < ]3, z becomes imaginary ; consequently, the lesser* root (a) is a maximum value of u. In like manner, it can be easily seen that the greater root (j3) is a minimum. Accordingly, when the roots of the denominator, a'x 2 + 2b' x + c f = o, are real and unequal, the fraction admits of all pos- sible, positive, or negative values, with the exception of those which lie between a and ]3. If either a' - o, or c' = o, the radical becomes b' y(u -a) (u- j3), and, as before, the greater root is a minimum, and the lesser a maximum, value of u. * In general, in seeking the maximum or minimum values of y from the equation, y = (%), if for all values of y between the limits a and j8, the corre- sponding values of x are imaginary, while x is real when y = a, or y = # ; then it is evident that the lesser of the quantities, o, j8, is a maximum, and the greater a minimum, value of y. This result also admits of a simple geometrical proof, by considering the curve whose equation is y = f(a + h), and/(a) >f{a-h) ; and, for a minimum, f{a) 40. TO I ^ 2 ^ If a = 40. we get when x = ir, — - = o. On proceeding to the next differentiation we have — -g = « (sin 2; + 2 sin 22), = o when # = tt. d% Again, — = a (cos x + 4 cos 2#) = 3^. Consequently the solution is a $># minimum m this case. Again, the solution (2) is impossible unless a be less than 40. In this case, d 2 u i. e. when a < 40, we easily find — ■ positive, and accordingly this gives a min. ctx value of u, viz. - — -J. 80 4. Find the value of x for which sec x — x is a maximum or a minimum. Ans. sins = . Application to Rational Algebraic Expressions. 173 139. Application to Rational Algebraic Expres- sions. — Suppose fix) a rational function containing no fractional power of at, and let the real roots of fix) = o, arranged in order of magnitude, be a, j3, y, &c. ; no two of which are supposed equal. Then fix) = (x - a) [x - j3) (x - y) . . . and /"(«)= (<*-0) (a-y) . . . But by hypothesis, a - j3, a-y, &c. are all positive ; hence /"(a) is also positive, and consequently a corresponds to a minimum value oif(x). Again, /"(/3) = (fi - «) (0 - y ) here j3 - a is negative, and the remaining factors are positive ; hence f'((3) is negative, and/(|3) a maximum. Similarly,/ (7) is a minimum, &c. 140. Maxima and Minima Values occur alter- nately. — We have seen that this principle holds in the case just considered. A general proof can easily be given as follows : — Suppose fix) a maximum when x = a, and also when x = b, where b is the greater ; then when x = a + h, the function is decreasing, and when x = b - h, it is increasing (where h is a small incre- ment) ; but in passing from a decreasing to an increasing state it must pass through a minimum value ; hence between two maxima one minimum at least must exist. In like manner it can be shown that between two minima one maximum must exist. 141. Case of Equal Roots. — Again, if the equation fix) = o has two roots each equal to a, it must be of the form fix) = ix - a) 2 ip (x). In this case /"(a) = o,f"(a) = 2\p(a), and accordingly, from Art. 138, a corresponds to neither a maximum nor a minimum value of the function /(#). In general,, if fix) have n roots equal to a, then f(x) = {x-a) n 4,(x). Here, when n is even, /(a) is neither a maximum nor a minimum solution : and when n is odd, f(a) is a maximum or a minimum according as \p(a) is negative or positive. 174 Maxima and Minima of Functions of a Single Variable. 14.2. Case where fix) = 00. The investigation in Art. 138 shows that a function in general changes its sign in passing through zero. In like manner it can be shown that a function changes its sign, in general, in passing through an infinite value ; i.e. if 0(a) = co, (a - h) and (a+h) have in general opposite signs, for small values of h. For, if u and - represent any function and its reciprocal, they have necessarily the same sign ; because if u be positive, - is positive, and if negative, negative. u Suppose Ui, u 2 , u z , three successive values of u, and — , — , — , the corresponding reciprocals. Ui U 2 U 3 r o r Then, if u 2 = o, by Art. 138, u x and u 6 have in general opposite signs. Hence, if — = co , — aud— have also opposite signs; and U-i U\ u$ we infer that the values of x which satisfy the equation fix) = 00 may furnish maxima and minima values of fix). 143. "We now return to the equation f(x) = (x-a) n xp(x), in which n is supposed to have any real value, positive, nega- tive, integral, or fractional. In this case, when x = a,f (x) is zero or infinity according as n is positive or negative. To determine whether the corresponding value of fix) is a real maximum or minimum, we shall investigate whether fix) changes its sign or not as x passes through a. When x = a + h, /{a + h) = h n \p (a + h), x = a-h, f(a-h) = (-h) n ip(a-h): J5 now, when h is infinitely small, \p(a +h) and \p (a - h) become each ultimately equal to \fj (a) : and therefore f\a + h) and f\a - h) have the same or opposite signs according as ( - 1)" is positive or negative. Examples. 175 (1). If n be an even integer, positive or negative, ./"(a?) does not change sign in passing through a, and accordingly a cor- responds to neither a maximum nor a minimum solution. (2). If n be an odd integer, positive or negative, f(a + h) and/"(« - h) have opposite signs, and a corresponds to a real maximum or minimum. ir + — 2r 2r p (3). If n be a fraction of the form ± — , then ( - 1) r + - = 1 p = i , and a corresponds to neither a maximum nor a minimum. 2r + i £ (4). If n be of the form ± -, then ( - 1) = (- 1) ; jj this is imaginary Up be even, but has a real value ( - 1) when p is odd. In the former case, f r (a — h) becomes imaginary ; in the latter, f'(a + h) aji&f'(a-h) have opposite signs, and f(a) is a real maximum or minimum. Thus in all cases of real maximum and minimum values the index n must be the quotient of two odd numbers. Examples. 1. f(x) = ax 2 + 2bx + c. b Here f(x) - i{ax + b) = o ; hence cs = , f'\x) = 2a. ft/* Jy» w And is a maximum or a minimum value of ax 2 + 2bx + c, according a as a is negative or positive. 2. f{x) = 22? - i$x 2 + 36a; + 10. Here /' (*) = 6(* 2 - 5^ + 6) = 6(* - 2) (s - 3). (r.) Let x = 2 ; then /"(#) is negative ; hence /(2) or 38 is a maximum. (2.) Let x = 3 ; then/ "(#) is positive; hence / (3) or 37 is a minimum. 176 Maxima and Minima of Functions of a Single Variable. It is evident that neither of these values is an absolute maximum or mini- mum ; for when x = 00 , f(x) = 00 , and when x - - 00 , f(x) = - 00 ; accord- ingly, the proposed function admits of all possible values, positive or negative. Again, neither + 00 nor — 00 is a proper maximum or minimum value, because for large values of #, f{x) constantly increases in one case, and constantly dimi- nishes in the other. It is easily seen that as x increases from - 00 to + 2, f(x) increases from - 00 to 38 ; as # increases from 2 to 3,/0) diminishes from 38 to 37 ; and as x in- creases from 3 to 00, f(x) increases from 37 to 00. When considered geome- trically, the preceding investigation shows that in the curve represented by the equation y = 2X 3 — I$X 2 + $6x + IO, the tangent is parallel to the axis of x at the points x = 2, y = 38 ; and x = 3, y — 37 ; and that the ordinate is a maximum in the former, and a minimum in the latter case, &c. 3. f{x) = a + b (x — c)i. Am. x = c. Neither a max. nor a min. 4. f{x) = b + e{x - «)§ + d{x - a)i. Substitute a + h f or x, and the equation becomes f(a + h) = b + c0 + dJ&\ also /(« - h) — b + c0 + dh? ; but when h is very small h* is very small in comparison with hi, and accordingly b is a minimum or a maximum value of /(#) according as c is positive or negative. 5. /(#) = 5# 6 + I2 % 5 - I S xi — 4 oa;3 + I S X% + 6o# + J 7' Ans. x = ± 1 gives neither a max. nor a min. ; x — — 2 gives a min. 6. - — - . Let x — 10 = z, and the fraction becomes x — 10 z z 36 The maximum and minimum values are given by the equation 1 = o; .*. z = + 6, and hence x = 16 or 4; the former gives a minimum, the latter a maximum value of the fraction. *, n O - 3 Hence f(m) = ^=^ (0 + 5). If # = 1, /(#) is neither a maximum nor a minimum ; if x = — 5, /(#) is a maximum. 7 vr. /. «a? + ^bccv + cy 2 177 ife. and Mm, of — o ~ j-. . ' ' ax* + 2b xy + cy 2 (x + i) 2 Again, the reciprocal function 7 rz is evidently a max. when % — - 1 ; {% - i) 6 for if we substitute for x, — 1 + h, and - 1 — h, successively, the resulting values are both negative ; and consequently the proposed function is a minimum in this case. This furnishes an example of a solution corresponding to f'{%) =0°. See Art. 142. 144. We shall now return to the fraction ax 2 + 2bxy + cy* a'x 2 + 2b' xy + c'y 2 ' the maximum and minimum values of which have been already considered in Art. 136. Write as before the equation in the form z 2 {a - a'u) + 2%(b - b'u) + (c - c'u) = o, x where 2 = -. y dt/ijfi Differentiate with respect to 2, and, as — = o for a maxi- az mum or a minimum, we have 2 (a - a'u) + (b - b'u) = o. Multiply this latter equation by 2, and subtract from the former, when we get z(b - b'u) + (c - c'u) - o. Hence, eliminating 2 between these equations, we obtain {a - a'u) (c - c'u) = (b - b'u) 2 9 or u 2 (a'c' - b' 2 ) - u(ac + ca' - 20b') + (ac - b 2 ) = o ; (3) the same equation (3) as before. The quadratic for 2, z 2 (ab' - bd) + z(ac' - ca') + be' - cb' = o, (4) is obtained by eliminating u from the two preceding linear equations. N 178 Maxima and Minima of Functions of a Single Variable. This equation can also be written in a determinant form, as follows : — I -z z 2 a b c a' V c' = o. It may be observed that the coefficients in (3) are in- variants of the quadratic expressions in the numerator and denominator of the proposed fraction, as is evident from the principle that its maximum and minimum values cannot be altered by linear transformations. This result can also be proved as follows : — T aX 2 +2bXY+cY 2 .Let u = a'X 2 + 2V XY + c'Y 2i where X, Y denote any functions of x and y ; then in seeking the maximum and minimum values of u we may substitute % for — , when it becomes az 2 + ibz + c az 2 + 2bz + c and we obviously get the same maximum and minimum values for u, whether we regard it as determined from the -original fraction or from the equivalent fraction in z. Again, let X, Y be linear functions of x and y, i. e. X = Ix + my, Y= I'x + ni'y, then u becomes of the form Ax 2 + 2JBxy + Cy 2 A'x 2 + 2&xy + C'y 2 ' where A, B, C, A', B', (7, denote the coefficients in the trans- formed expressions ; hence, since the quadratics which deter- mine the maximum and minimum values of u must have the same roots in both cases, we have AC - B 2 = X(ac - b 2 ), AC + OA! - 2BF = \(ac' + ca' - 2bb'), A'C -B> 2 = \{a'c' - V 2 ). Q.E.D. Application to Surfaces. 179 It can be seen without difficulty that A = {M - ml') 2 . We shall illustrate the use of the equations (3) and (4) by applying them to the following question, which occurs in the determination of the principal radii of curvature at any point on a curved surface. 145. To find the Maxima and Minima Values of r cos 2 a + 2s cos a cos/3 + ^ cos 2 j3, where cos a and cos ]3 are connected by the equation (1 + p 2 ) COS 2 a + 2pq COS a COS |3 + (1 + q 2 ) eos 2 /3 = I, and p, q, r, s, t are independent of a and j3. Denoting the proposed expression by u, and substituting „ cos a . zior ^TR> we get cosp rz 2 + 2sz + t u = (1 + p 2 )z 2 + zpqz + (1 + q 2 )' The maximum and minimum values of this fraction, by the preceding Article, are given by the quadratic u 2 [i +p 2 + q 2 )-u{(i +q 2 )r - 2pqs + (1 +p 2 )t} +rt -s 2 = o; (6) while the corresponding values of z or ^ are given by s 2 {(i +p 2 )s - pqr) + z[(i + p 2 )t - (1 + q 2 )r} + {pft - (1 + ^ 2 )s} = o.* (7) The student will observe that the roots of the denominator in the proposed fraction are imaginary, and, consequently, the values of the fraction lie between the roots of the quadratic (6), in accordance with Art. 136. * Lacroix, Dif. Cal., pp. 575, 576. Ts T 2 180 Maxima and Minima of Functions of a Single Variable, 146. To find the Maximum and Minimum Radius Vector of the Ellipse ax 2 + zbxy + cy 2 = 1. (1). Suppose the axes rectangular ; then r 2 - x 2 + y 2 is to be a maximum or a minimum. x Let - = z, and we get y 2 _ s 2 + 1 az 2 + 2bz + c Hence the quadratic which determines the maximum and minimum distances from the centre is r* (ae - b 2 ) - r 2 (a + c) + 1 = o. The other quadratic, viz. bx 2 - (a - c)xy - by 2 = o, gives the directions of the axes of the curve. (2.) If the axes of co-ordinates be inclined at an angle w 9 then r 2 = x 2 + y 2 + 2xy cos &> z 2 + 2% cos w + 1 — . • az 2 + 2&3 + c ' and the quadratic becomes in this case r* (ac - b 2 ) - r 2 (a + c - 2b cos 10) + sin 2 w = o, the coefficients in which are the invariants of the quadratic expressions forming the numerator and denominator in the expression for r 2 . The equation which determines the directions of the axes 1 the conic can also be easily written down in this case. Maximum and Minimum Section of a Eight Cone. 1 8 1 147. To investigate the Maximum and Minimum Values of ax 3 + ^bx 2 y + $cxy 2 + dy 3 dx 2 + $b r x % y + 3c' xy 2 + d'y 'a/3* x Substituting % for -, and denoting the fraction by u, we have if az 3 + $bz 2 + $cz + d ~ dz 3 + $b'z 2 + $c'z + d r Proceeding, as in Art. 144, we find that the values of u and 2 are given by aid of the two quadratics az 2 + 2bz + c = (dz 2 + 2b 'z + c')u, bz 2 + 2cz + d= (b f z 2 + 2cz + d')u. Eliminating u between these equations, we get the following biquadratic in z : — z^(ab r - bd) + 2z z (ac f - cd) + z 2 {ad f - dd + 3 (be' - cb')} + 2z(bd' - db') + (ed f - c'd) = o. (8) Eliminating z between the same equations, we obtain a biquadratic in u, whose roots are the maxima and minima values of the proposed fraction. Again, as in Art. 144, it can easily be shown that the coefficients in the equation in u are invariants of the cubics in the numerator and denominator of the fraction. 148. To eut the Maximum and Minimum Ellipse from a Right Cone which stands on a given circular base. — Let AD represent the axis of the cone, and suppose BP to be the axis major of the required section; its centre ; a, b, its semi-axes. Through and P draw LM and PR parallel to BO. Then BP = 20, b 2 = LO . OM (Euclid, Book 111., Pr. 35) ; but LO = ™ 0M=—; .-. b 2 =-.BC. PP. 2 2 4 Hence BP 2 . PR is to be a maximum or a minimum. s * ^ 1 82 Maxima and Minima of Functions of a Single Variable. Let L BAP = a, PBO = 9, BG = c. Then BP = BC sinBCF CC0Sa sin BPC cos (0 -a)' sin PJ9i2 _ c cos (0 + a) - -^ginp^ = C os (0 - «) ; COS (0 + a) . •"• w = — ttt\ ( is a maximum or a minimum. cos 3 (0 - a) TT dw sin 20- 2 sin 2a . n -tLence -^ = — — — = o ; .\ sin 20 = 2 sm 2a. dO cos 4 (0 - a) The solution becomes impossible when 2 sin ia > 1 ; i.e. if the vertical angle of the cone be > 30 . The problem admits of two solutions when a is less than 1 5°. For, if 0! be the least value of 9 derived from the 7T equation sin 20 ^= 2 sin 2 a ; then the value 0i evidently gives a second solution. Again, by differentiation, we get d 2 u 2 COS 20 1^5 = — iTTi \ (when sin 20 = 2 sm 2a). cftr cos 4 (0 - a) x y This is positive or negative according as cos 2 is positive or negative. Hence the greater value of corresponds to a maximum section, and the lesser to a minimum. ^ In the limiting case, when a = 15 , the two solutions coincide. However, it is easily shown that the corresponding section gives neither a maximum nor a minimum solution of the problem. For, we have in this case = 45 ° ; which value d if gives -r^ = o. On proceeding to the next differentiation, we find, when = 45 , d 3 u - 4 64 W = cos 4 (45°-a) = " "9" Hence the solution is neither a maximum nor a minimum. When a > 1 5 , both solutions are impossible. Geometrical Examples. 183 149. The principle, that when a function is a maximum or a minimum its reciprocal is at the same time a minimum or a maximum, is of frequent use in finding such solutions. There are other considerations by which the determina- tion of maxima and minima values is often facilitated. Thus, whenever u is a maximum or a minimum, so also is log (u), unless u vanishes along with — . Again, any constant may be added or subtracted, i.e. if fix) be a maximum, so also is/(#) ± c. Also, if any function, u, be a maximum, so will be any positive power of u, in general. 150. Again, if z = f(u), then dz = f'(u)du, and conse- quently s is a maximum or a minimum; either (1) when du = o, i.e. when u is a maximum or a minimum ; or (2) when f(u) = o. In many questions the values of u are restricted, by the conditions of the problem,* to lie between given limits; accordingly, in such cases, any root of fiu) = o does not furnish a real maximum or minimum solution unless it lies between the given limiting values of u. We shall illustrate this by one or two geometrical examples. (1). In an ellipse, to find when the rectangle under a pair of conjugate diameters is a maximum or a minimum. Let r be any semi-diameter of the ellipse, then the square of the conjugate semi-diameter is represented by a 2 + b 2 - r 2 , and we have u = r 2 (a 2 + b 2 - r 2 ) a maximum or a minimum. Here — = 2U 2 + b 2 - ir 2 ) r. dr Accordingly the maximum and minimum values are, (1) those for which r is a maximum or a minimum ; i.e. r = a, or r = b ; and, (2) those given by the equation r (a 2 + b 2 - 2r 2 ) = o ; * See Cambridge Mathematical Journal, vol. iii. p. 237. 1 84 Maxima and Minima of Functions of a Single Variable. 4 la 2 + b 2 or r = o, and r The solution r = o is inadmissible, since r must lie between the limits a and b : the other solution corresponds to the equiconjugate diameters. It is easily seen that the solution in (2) is the maximum, and that in (1) the minimum value of the rectangle in question. 151. As another example, we shall consider the following problem* : — Given in a plane triangle two sides [a, b) to find the maximum and minimum values of 1 A - . cos — , C 2 where A and c have the usual significations. Squaring the expression in question, and substituting x for c, we easily find for the quantity whose maximum and minimum values are required the following expression : 1 2b a 2 - b 2 2 s > tAs m/ U/ neglecting a constant multiplier. Accordingly, the solutions of the problem are — (1) the maximum and minimum values of x, i.e. a + b and a - b. du (2) the solutions of the equation — , i.e. of ax 1 \b 3 (a 2 - b 2 ) _ x l X s ar or x 2 + ^bx - 3 {a 2 - b 2 ) = o ; whence we get x = ystf + b 2 - 2b, neglecting the negative root, which is inadmissible. Again, if b > a, */$a 2 + b 2 - 2b is negative, and accord- ingly in this case the solution given by (2) is inadmissible. * This problem occurs in Astronomy, in finding "when a planet appears brightest, the orbits being supposed circular. Maxima and Minima Values of an Implicit Function. 1 85 If a > b, it remains to see whether y^fl 2 + b 2 - 2b lies between the limits a + b and a - b. It is easily seen that V^tf 2 + b 2 - 2bis> a - b: the remaining condition requires a + b > Via* + b 2 - 2b, or a + 36 > y^a 2 + b 2 , or « 2 + tab + gb 2 > 3a 2 + b 2 , i. e. 4& 2 + 3«& > fl 2 , oa 2 25a 2 7 3& 5# 10 10 44 or, finally, b > -. We see accordingly that this gives no real solution unless the lesser of the given sides exceeds one-fourth of the greater. When this condition is fulfilled, it is easily seen that the corresponding solution is a maximum, and that the solutions corresponding to x = a + b, and x = a - b, are both minima solutions. 152. Maxima and Minima Values of an Implicit Function. — Suppose it be required to find the maxima or minima values of y from the equation fix, y) = o. Differentiating, we get du du dy dx dy dx ' where u represents f(x, y). But the maxima and minima du values of y must satisfy the equation —- = o : accordingly the 1 86 Maxima and Minima of Functions of a Single Variable. maximum and minimum values are got by combining* the .. du n equations -7- = o, and u = o. dx 153. maximum and Minimum in case of a Func- tion of two dependent Variables. — To determine the maximum or minimum values of a function of two variables, x and y, which are connected by a relation of -the form fix, y) = o. Let the proposed function, (x, y) be represented by u ; then, by Art. 101, we have d

df dx dy dy dx ' furnish the solutions required. To determine whether the solution so determined is a maximum or a minimum, it d 2 u is necessary to investigate the sign of — . We add an ax example for illustration. 154. Given the four sides of a quadrilateral, to find when its area is a maximum. Let a, b, c } d be the lengths of the sides, the angle between a and b, xfj that between c and d. Then ab sin (j> + cd sin ^ is a maximum ; also a 2 + b 2 - 2ab cos = c 2 + d 2 - 2cd cos \p being each equal to the square of the diagonal. * This result is evident also from geometrical considerations. Maximum Quadrilateral of Given Sides. 187 Hence ab cos 6 + cd cos \L — = o r d§ for a maximum or a minimum ; also, ab sin 6 = cd sin. \p-f- ; .*. tan + tan ip = o, or <£ + \p = 1 8o°. Hence the quadrilateral is inscribable in a circle. That the solution arrived at is a maximum is evident from geometrical considerations ; it can also be proved to be so by aid of the preceding principles. For, substitute —=—. — ^7 instead of -?-. and we get cd sin \p d(j> du _ ab sin (0 + \p) d(p sin ip „ d 2 u ab cos (d> + \L) f d\L\ . . . Hence -r-z = ^~ — — 1 + -f- 1 + a term which d (2) for oblique. V ab — h 2 28. A triangle inscribed in a given circle has its base parallel to a given line, and its vertex at a given point ; find an expression for the cosine of its vertical angle when the area is a maximum. 29. Find when the base of a triangle is a minimu m, being given the ver- tical angle and the ratio of one side to the difference between the other and a fixed line. 30. Of all spherical triangles of equal area, that of the least perimeter is equilateral ? 31. Let u z + x z — $axu = o ; determine whether the value x = o gives u a maximum or minimum. Ans. Neither. 32. Show that the maximum and minimum values of the cubic expression ax 3 + $bx 2 -f $cx + d are the roots of the quadratic ah 2 - 2Gz- A = 0; where G = a 2 d — yibc + 2b s , and A = a 2 d 2 + qac 3 + qdb z — $b 2 c 2 — 6abcd. 33. Through a fixed point within a given angle draw a line so that the triangle formed shall be a minimum. The line is bisected in the given point. 34. Prove in general that the chord drawn through a given point so as to cut off the minimum area from a given curve is bisected at that point. 35. If the portion, AB, of the tangent to a given curve intercepted by two fixed lines OA, OB, be a minimum, prove that PA = NB, where JP is the point of contact of the tangent, and N the foot of the perpendicular let fall on the tangent from 0. 36. The portion of the tangent to an ellipse intercepted between the axes is a minimum : find its length. Ans. a + b. 37. Prove that the maximum and minimu m values of the expression, Art. 147, are roots of the biquadratic (a — ua') 2 (d - ud') 2 + 4 (a — ua') (c — uc'} 3 + 4 (d - ud') (b - ub') z — 3 (b - ub') 2 (c — uc') 2 — 6 (a — ua) (b — ub') (c — uc) (d - ud') = o. ( i9i ) CHAPTEE X. MAXIMA AND MINIMA OF FUNCTIONS OF TWO OR MORE IN- DEPENDENT VARIABLES. 155. Maxima and Minima for Two Variables. — In accordance with the principles established in the preceding chapter, if $ (%, y) be a maximum for the particular values x and y , of the independent variables x and y, then for all small positive or negative values of h and k f

(x + h,y + h); and for a minimum it must be less. Again, since x and y are independent, w$ may suppose either of them to vary, the other remaining constant; accordingly, as in Art. 138, it is necessary for a maximum or minimum value that du _ du ^ = o, and- = o; (,) omitting the case where either of these functions becomes infinite. 156. Lagrange's Condition. — We now proceed to consider whether the values found by this process correspond to real maxima or minima, or not. Suppose x , y to be values of x and y which satisfy the equations du . du -7- = o, and -7- = o, ax dy and let A, B, C be the values which — , — — , — - assume dx* dxdy dy % when % and y are substituted for x and y ; then we shall have $(#<>+ h, yo + k)- (j>(x Q , y )= (Ah?+ zBhk+Ck 2 ) + &c. (2) 192 Max. and Min.for two or more Independent Variables, But when h and k are very small, the remainder of the expansion becomes in general very small in comparison with the quantity AW + zBhk + Ck 2 ; accordingly the sign of (x + h, y + k) - AC the solution is neither a maximum nor a minimum. The necessity of the preceding condition was first estab- lished by Lagrange ;* by whom also the corresponding con- ditions in the case of a function of any number of variables were first discussed. Again, if A = o, B = o, C = o, then for a real maximum or minimum it is necessary that all the terms of the third degree in h and k in expansion (2) should vanish at the same time, while the quantity of the fourth degree in h and k should preserve the same sign for all values of these quan- tities. See Art. 138. The spirit of the method, as well as the processes em- ployed in its application, will be illustrated by the following examples. 157. To find the position of the point the sum of the squares of whose distances from n given points situated in the same plane shall be a minimum. * Theorie des Fonctions. Deuxieme Partie. Ch. onzieme. Maxima and Minima for Two or more Variables. 193 Let the co-ordinates of the given points referred to rectangular axes be («i, &i), (a 2 , b 2 ), (a 3f h) . . . (a n , b n ), respectively; (x, y) those of the point required ; then we have u = (x - a x ) 2 + (y - b x y + (x - a 2 ) 2 + (y - b 2 ) 2 + . . a ■ + (x - a n ) 2 + (y - b n y a minimum ; du , . .*. — = x-a 1 + x~o 2 + . . . -1- x - a n = nx-[a i +a 2 + . . ,+a n ) -o; dx v / ftlH -r=y-bi + y-b 2 +. ,.+y-b n =ny-(b l +b 2 +... + b n ) = o. tt a x + a 2 + . . . + a n bi + b 2 + . . . + b n Hence x = , y n n and the point required is the centre of mean position of the n given points. From the nature of the problem it is evident that this result corresponds to a minimum. This can also be established by aid of Lagrange's con- dition, for we have _ ab, the solution is neither a maximum nor a minimum. The geometrical interpretation of the preceding result is evident ; viz., if the co-ordinates of the centre be substituted for x and y in the equation of a conic, u = o, the resulting value of u is either a maximum or a minimum if the curve be an ellipse, but is neither a maximum nor a minimum for a hyperbola ; as is also evident from other considerations. 159. To find the Maxima and Minima Values of the Fraction ax 2 + by 2 + 2hxy + 2gx + ify + c dx 2j r b f y 2 + 2h'xy+ 2g'x+2f'y+c'' Let the numerator and denominator be represented by i d 2 2 d 2 u dxdy dxdy dxdy' d 2 6\ d 2 i = (Pfc _ , d 2 (jn dx 2 " ' dx z ' dxdy ' Hence i ** IS w -©)! ■ 4{(a - °' w)(J - 6%) - ( * - *w ' Accordingly, the sign of AC - B 2 is the same as that of the quadratic expression (ab - h 2 ) - (aV + Id - 2hh') u + (db f - h' 2 )u\ (7) where u is a root of the cubic (4) or (5). If A 2 represent the determinant in (4), the preceding quadratic expression may be written in the form — — 2 . Again, u l9 u 2 , u 3 representing the roots of the cubic (4) ; a, |3, those of the quadratic (7) ; if u x be a real maximum or minimum value of u, we must have (u x - a)(ui - ff)(db f - h' 2 ) a positive quantity. Accordingly, if db r - h' z be positive, u x must not lie be- tween the values a and /3. Similarly for the other roots. 198 Maxima and Minima for Two or more Variables. If all the roots of the cubic lie outside the limits a and ]3, they correspond to real maxima or minima, but any root which lies between a and j3 gives no maximum or minimum. In the particular case discussed in Art. 1 60 the roots of the cubic (6) are all real, and those of the quadratic a - u~ l , h = o are interposed between the roots of the h, b - w 1 cubic. (See Salmon's Higher Algebra, Art. 44). Accord- ingly, in this case the two extreme roots furnish real maxima and minima solutions, while the intermediate root gives neither. This agrees with what might have been anticipated from the properties of the Ellipsoid ; viz., the axes a and c are real maximum and minimum distances from the centre to the surface, while the mean axis b is neither. It would be unsuited to the elementary nature of this treatise to enter into further details on the subject here. 162. Maxima or Minima of Functions of three Variables. — Next, let u = (x + h, y Q + h, z + l) - {x ,y 0i Zo)=A— +B— ■ + C— — + FM+ Ghl + mk + &e. where A, B, C, F, G, H, are the values that d 2 u d 2 u d 2 u d 2 u d 2 u d 2 u dx %1 dy 29 dz 2 ' dydz dxdz* dxdy respectively assume when x , y , Zo are substituted for x, y, 2 in them. Maxima or Minima for Three Variables. 199 Now, in this, as in the case of two independent variables, it is necessary for a real maximum or minimum value that the preceding quadratic function should be either always positive or always negative for all small real values of A, k, and I. Substituting al for h, and (51 for k, and suppressing the positive factor P, the expression becomes Aa 2 + B(5 2 + C+2F(3 + 2Ga + 2Ha[3, (8) (sp + Gy or a" + 2a A + B(5 2 + 2F(5 + C. Completing the square in the first term, and multiplying by A, we get (Aa + S(5 + G) 2 + (AB-S 2 )(5 2 + 2{AF- GH)(5 +{AC- G 2 ). Moreover, since the first term is a perfect square, in order that the expression should preserve the same sign, it is neces- sary that the quadratic (AB - S 2 )(5 2 + 2 (AF- OH")j3 + AC - G % should be positive for all values of j3 : hence we must have AB-R 2 > o, (9) and (AB - E*)(AC - (?) > (AF- GE) 2 , or A(ABC + 2FGK - AF 2 - BG 2 - CH 2 ) > o, (10) i.e. A and A must have the same sign, A denoting the dis- criminant of the quadratic expression (8), as before. Accordingly, the conditions (9) and (10) are necessary that x , Vq, So should correspond to a real maximum or mini- mum value of the function u. When these conditions are fulfilled, if the sign of A be positive, the function in (8) is also positive, and the solution is a minimum ; if A be negative, the solution is a maximum. 163. Maxima and Minima for any number of Tariables. — The preceding theory admits of easy extension 200 Maxima and Minima for Two or more Variables. to functions of any number of independent variables. The values which give maxima and minima in that case are got by equating to zero the partial derived functions for each variable separately, and the quadratic function in the ex- pansion must preserve the same sign for all values ; i.e. it must be equivalent to a number of squares, multiplied by constant coefficients, having each the same sign. The number of independent conditions to be fulfilled in the case of n independent variables is simply w - i, and not 2 n — i, as stated by some writers on the Differential Calculus. A simple and general investigation of these conditions will be given in a note at the end of the Book. 164. To investigate the Maximum or Minimum Value of the Expression ax 2 + by 2 + cz 2 + 2hxy + zgzx + zfyz + ipx + 2qy + 2rz + d. Let u denote the function in question, then for its maxi- mum or minimum value we have du , s -— = 2 {ax + hy + gz +p) = o, — = 2{hx + by +fz + q) = o, — = 2(gx +fy + cz + r) = o ; hence, adopting the method of Art. 158, we get u = px + qy + rz + d. Eliminating x, y, z between these four equations, we obtain a h g p h b f q g f c r p q r d u a h g h b f g f c A . . d 2 u d 2 u 7 p Again, since — = 2a, — = 2b, &c, Maxima or Minima for two or more Variables. 201 the result is neither a maximum nor a minimum unless a h g is positive, and | h b f 9 f a h h b has the same sign as a. The student who is acquainted with the theory of surfaces of the second degree will find no difficulty in giving the geometrical interpretation of the preceding result. 165. To find a point snch that the sum of the squares of its distances from n given points shall be a Minimum. — Let (a, b, c), (of, b\ c'), &o., be the co-ordi- nates of the given points referred to rectangular axes ; x, y, z, the co-ordinates of the required point ; then (x - af + (y- b) 2 + (z - c) 2 is equal to the square of the distance between the points [a, b, c), and (x, y f z). Hence u = (x - a) 2 + (y - by + (z - cf + (x - aj + {y - bj + (* - c)' 2 + &c. = 2(0 - a) 2 + S(y - b) 2 + 2(* - c)\ where the summation is extended to each of the n points. For the maximum or minimum value, we have -=- = 22(# — a) = 2nx - 2z,a = o, ax — = 2^(y - b) = my - 22& = o, is -=- = 22 v s - c) - 2nz - 22c = o ; d% 2fl 2& 2c n 9 n n i.e. % , yoi So are the co-ordinates of the centre of mean posi- 202 Maxima and Minima of Independent Variables. tion of the given points. This is an extension of the result established in Art. 157. A . d 2 u d 2 u d?u d 2 u Agam - = 2n, — = zn, ^ = zn, — = o, &c. The expressions (10) and (11) are both positive in this case, and hence the solution is a minimum. It may be observed with reference to examples of maxima and minima, that in most cases the circumstances of the prob- lem indicate whether the solution is a maximum, a minimum, or neither, and accordingly enable us to dispense with the labour of investigating Lagrange's conditions. Examples. 203 Examples. Find the maximum and minimum values, if any such exist, of ax 4- by + c c ± «/ a 2 + b 2 + t? I. — — — -— — . Ans. - - . % + y £ + 1 ax + by + c „ ± \A 2 + b 2 + c*. y x 2 + y 2 + 1 3. z* + y* - x 2 + xy - y 2 . (a), x — o, y = o, a maximum. 03). x = y = + -, a minimum. — 2 / v v 3 • • ( - dd> _ dd> _ -rf- axi +~- dx 2 + -T- dxz + -p- dx± = o. axi ax 2 dx s dx± Moreover, the differentials are also connected by the rela- tions dF 1 . dF, 7 dF x , dF 1 . — — dx y + — dx 2 + -=— dx z + -=— ^4 = o, (M?j W5?2 W#?3 W#?4 ■rf^i _ y dF 1 . dF 2 \ ' fd(f> . dFx . dF 2 \ -~ + Ai — - + A 2 — )dxy + -^ + Ai — + A 2 — )dx 2 dx x dx x dxij \ax 2 dx 2 ax 2 ) (d _ dF x _ dF 2 \ _ / . dF x ^ dF 2 -£- + Ai -=- + A 2 — = o, ao?3 «a? 3 aa?3 ■/- + Ai -7- + A 2 -J- = o. «#4 «#4 UXi These, along with equations (1) and dd> - ^ . ^ -^- + Ai — + A 2 -j- = o, dxi dx x dxx d6 . dFx ^ ^2 -^- + Aj — + A 2 — = o 9 ax 2 ax% wXz are theoretically sufficient to determine the six unknown quantities, Xi, x 2 , x 3 , a? 4 , A x , A 2 ; and thus to furnish a solution of the problem in general. This method is especially applicable when the functions Fi 9 F 2 , &c, are homogeneous ; for if we multiply the preceding 206 Method of Undetermined Multipliers. differential equations by x l9 x 2 , x 3 , x±, respectively, and add, we can often find the result with facility by aid of Euler's Theorem of Art. 103. There is no difficulty in extending the method of undeter- mined multipliers to a function of n variables, x x , x 2 , x 3 , . . ,. x n , the variables being connected by m equations of condition. F 1 = o,F z = o,F 3 = o s . . . F m = o, m being less than n ; for if we differentiate as before, and multiply the differentials of the equations of condition by the arbitrary multipliers, Ai, A 3 , . . . \ m respectively; by the same method of reasoning as that given above, we shall have the n following equations, dd> . dFi . dF m ■j- + Ai— -+...+ \ m -=— = o, ax L dx x axi d(b dF x . dF m — + Ai —- + ...+ A m -5 — = o, ax 2 dx % dx 2 d

is a maximum. 22 r 4. Find the maximum value of (# + 1) («/ + 1) (z + 1) where a x bv c z = A. {log (Aabe)} z Ans. 27 log as . log 5 . log c 5. Find the volume of the greatest rectangular parallelepiped inscribed in the ellipsoid whose equation is x 2 y 2 z 2 8 abc a 2 £ 3 c 2 3^/^ 6. Find the maximum or the minimum values of u, being given that u = a 2 # 2 + 6 2 y 2 + c 2 z 2 , x 2 + y 2 + z 2 = 1, and fo? + my + wz = o. Proceeding by the method of Art. 1 67, we get a 2 x 4- Kx + fxl = o, £ 2 y + Ay + jum = 0, c 3 z + Az + /m = o. Again, multiplying by x, y, z, respectively, and adding, we get A = - u. .'. (u — a 2 ) x = /xl, (u — b 2 ) y = jxm, (u — e 2 ) z — /j.n. Hence, the required values of u are the roots of the quadratic I 2 m 2 n 2 + ^ +■ = o. u - a 2 u — b'~ u — c- Examples. 211 X 2 ifl z 2 7. Given — + — + — = I, and Ix + my + nz = o, find when x 2 + y 2 + z 2 is a a z # 2 c 3 maximum or minimum. Proceeding, as in the last example, we get the quadratic a 2 1 2 b 2 m 2 c 2 n 2 + T5 + ~ n = °. u — cfi u - b 2 u — c 2 This question can be at once reduced to the last by substituting in our equations ax, by, and ez, instead of x, y, z. 8. Of all triangular pyramids having a given triangle for base, and a given altitude above that base, find that whose surface is least. Am. Value of minimum surface is \/r 2 + p 2 , where a, b, c repre- 2 sent the sides of the triangular base ; r, the radius of its inscribed circle ; and p, the given altitude. 9. Divide the quadrant of a circle into three parts, such that the sum of the products of the sines of every two shall be a maximum or a minimum ; and determine which it is. 10. Of all polygons of a given number of sides circumscribed to a circle, the regular polygon is of minimum area? For, let l 4>2 tan — + tan 2 — + . . . + to V 2 2 where S-* W The points on the curve at which the tangents are parallel to the axis of x must satisfy the equation — = o ; QsX they are accordingly given by the intersection of the curve, du u = o, with the curve whose equation is — = o. The y co- ax ordinates at such points are evidently in general either maxima or minima. Similar remarks apply to the points at which the tangents are parallel to the axis of y. To find the tangents parallel to the line y = mx + n. The points of contact must evidently satisfy du du — + m— =0. dx dy The points of intersection of the curve represented by 214 Tangents and Normals to Curves. this equation with the given curve are the points of contact of the system of parallel tangents in question. The results in this and the preceding Article evidently apply to oblique as well as to rectangular axes. Examples. i. To find the equation of the tangent to the ellipse x 2 y 2 a 2 o 1 du 2x du iy Here Tx = #> di = W and the required equation is !(z-*) + £(r-,) = o, xX vY x 2 0r a 2 + £2 a 2 + b i *' 2. Find the equation of the tangent at any point on the curve x m ym Xx m - 1 Yy» 1 - 1 — + V-=i. Ans. + -f — = i. 3. If two curves, -whose equations are denoted by u = o, u' — o, intersect in a point (x, y), and if a be their angle of intersection, prove that du du' du' du dx dy dx dy tan w = T , ■ / .. du die du du dx dx dy dy 4. Hence, if the curves intersect at right angles, we must have du du' du du' dx dx dy dy 5. Apply this to find the condition that the curves x 2 y 2 x 2 y 2 a 2 + b 2=h a T2+ b i2==l should intersect at right angles. Ans. a 2 - b 2 = a' 2 - b' 2 . Equation of Normal. 215 170. Equation of formal. — Since the normal at any point on a curve is perpendicular to the tangent, its equation, when the co-ordinate axes are rectangular, is or (Y-y)^ + X-x = o, v y} dx du /T ^ v du . —■(Y-ij) =— (X-x). dx K JJ dy K J (3) The points at which normals are parallel to the line y = mx + n are given by aid of the equation of the curve u = o along with the equation du du dy dx' Examples. 1. Find the equation of the normal at any point (x, y) on the ellipse /2 x" y a 2 ¥ A a*X FY Am, = a 1 - P. x y 2. Find the equation of the normal at any point on the curve ym — ax n. j_ m . nYy + mXx = ny 2 -f mx 1 . 171. Subtangent and Subnormal. — In the accom- panying figure, let PT repre- y sent the tangent at the point P, PiV the normal; OM, PM the co-ordinates at P ; then the lines TM and MJST are called the subtangent and subnormal corresponding to the point P. Fig. 9. To find the expressions for their lengths, let

a 1 y yi intercepted by the axis of #. Ans. — . a 5. Find at what point the subtangent to the curve whose equation is xy 2 = a 2 (a — x) is a maximum. Ans. x = -, y — a. 172. Perpendicular on Tangent. — Let p be the length of the perpendicular from the origin on the tangent at any point on the curve F{x, y) = c, Length of Perpendicular on Tangent. 217 then the equation of the tangent may be written X cos w + T sin o> = p, where w is the angle which the perpendicular makes with the axis of x. Denoting F (x, y) by u, and comparing this form of the equation with that in (2), and representing the common value of the fraction by A, du du du du — — - x— + y ~r , dx dy dx ay _ . ° cos (v sin d) p »-(£MD' du du x— +y — -, dx dy t \ and p= J • (4) 4\dx) + \dk/J Cor. If .F(#, y) be a homogeneous expression of the n th degree in x andy, then by Euler's formula, Art. 102, we have du du x— + y — = nu = nc, dx dy and the expression for the length of the perpendicular becomes in this case nc } 173. In the curve duV fduY dx) \dy) x m y m to prove that m m m p™ 11 = (a cos w) J ^ r + (b sin a.)*" 1 " 1 . (5) 2 1 8 Tangents and Normals to Curves. By Ex. 2, Art. 169, the equation of the tangent is Xx m ' 1 Yy m ~ l comparing this with the form X cos id + Y sin 10 = p 9 we get or cos u> x m ~ l sin w y m ~ x p ~~a™' p = 1r i a cos wY^ 1 x (b sin oA m-1 y p J a' \ p J b' Hence, substituting in the equation of the curve, we obtain the result required. 174. l = — : substi- tuting these values for cos o> and sin w in (5), it becomes (X 2 + Y 2 ) m ' 1 = {aX) m ~ l + (b Y) m m-i since p 2 = X 2 + Y 2 . 175. Another Form of the Equation to a Tan- gent. — If the equation of a curve of the n th degree be written in the form d$ , A a -j- + p— + Un^i + 2W«_a + . . . + flU = O. (7) dy»i/ ay This represents a curve of the (n - i) th degree in x and y, and the points of its intersection with the given curve are the points of contact of all the tangents which can be drawn from the point (a, (5) to the curve. Moreover, as two curves of the degrees n and n - \ intersect in general in n (n - i) points, real or imaginary (Salmon's Conic Sections, Art. 214), it follows that there can in general be n{n - 1) real or imaginary tangents drawn from an external point to a curve of the n th degree. If the curve be of the second degree, equation (7) be- comes d(b r^dcb a- ! -+l5-j-- + U l + 2U = O, ax ay an equation of the first degree, which evidently represents the polar of (a, |3) with respect to the conic. In the curve of the third degree u z + u 2 + u x + u = o, equation (7) becomes d(h d6 which represents a conic that passes through the points of contact of the tangents to the curve from the point (a, |3). This conic is called the polar conic of the point. For the origin it becomes U % + 2th + 3^ = o. 220 Tangents and Normals to Curves. 177. Number of Normals which pass through a Given Point. — If a normal pass through the point (a, j3), we must have from (3), . x an . ~. x ate („-*)_= (0-j,)-. This represents a curve of the n th degree, which intersects the given curve in general in n 2 points, real or imaginary, the normals at which all pass through the point (a, j3). For example, the points on the ellipse x 2 y 2 a~* + ¥ =I ' at which the normals pass through a given point (a, j3), are determined by the intersection of the ellipse with the hyperbola xy(a 2 - b 2 ) = a 2 ay - b 2 (5x. For the modification in the results of this and the pre- ceding article arising from the existence of singular points on the curve, the student is referred to Salmon's Higher Plane Curves, Arts. 66, 67, in. 178. Differential of the Arc of a Plane Curve. Direction of the Tangent. — If the length of the arc of a curve, measured from a fixed point A on it, be denoted by s, then an infinitely small portion of it is represented by ds. Again, if $' represent the angle QPL (fig. 8), we have , PL , . , QL cos

(fig. 9). * In Art. 37 it has been proved that the difference between the length of an infinitely small arc and its chord is an infinitely small quantity of the second arc PQ - PQ . order in comparison with the length of the chord; i.e. — is infinitely small of the second order, and therefore this fraction vanishes in the limit. arc PQ Hence r - ^ = i> ultimately. ord PQ ' J Differential of the Arc of a Curve. 221 Hence dx dy COS0=-, on* ^ squaring and adding, we get fdx\ 2 (dyY_ \dsj \dsj Hence, also, we have ds 2 = dx 2 + dy 2 , (8) (9) and therefore ds (io) On account of the importance of these results, we shall give another proof, as follows : — Let, as before, PR be the tangent to the curve at the point P, OM=x, PM=y, MN=PL=Ax, QL = Ay. Z.PTX=, arc PQ= As, Then, if the curvature of the elementary portion PQ of the curve be continuous, we have evidently the line PQ<2ltgPQ Again, if P, Q be infinitely near points, denoting the lengths of the corresponding elements of the curve and of its inverse by ds and ds', the preceding result becomes ds= T -ds'. (18) 185. Direction of the Tangent to an Inverse Curve. — Let the points P, Q belong to one curve, and P', Q' to its inverse ; then when P and Q coincide, the lines PQ, P'Q' become the tangents at the inverse points P and P' : again, since the angle SPP' = the angle SQ'Q, it follows that the tangents at P and P' form an isosceles triangle with the line PP'. By aid of this property the tangent at any point on a curve can be drawn, whenever that at the corresponding point of the inverse curve is known. It follows immediately from the preceding result, that if two curves intersect at any angle, their inverse curves intersect at the same angle. Q 226 Tangents and Normals to Curves. 1 86. .Equation to the Inverse of a Given Curve. — Suppose the curve referred to rectangular axes drawn through, the pole 0, and that as and y are the co-ordinates of a point P on the curve, X and Y those of the inverse point P f ; then x_ _ op_ _ op . or _ k 2 . y_ k 2 X~ 0F~ OF 2 " X 2 + Y 2 ' similari ^ y" X 2 + Y 2 ; hence the equation of the inverse is got by substituting k 2 x , k 2 y and *—z x 2 + y 2 x* + y instead of x and y in the equation of the original curve Again, let the equation of the original curve, as in Art. 174, be U n + U n -i + u n _ 2 + . . . + u 2 + u x + u = o. When — and -r— — - are substituted for x and y. u n x 2 + y 2 x 2 + y 2 k 2n u becomes evidently —^ — ^— . J (x 2 + y 2 ) n Accordingly, the equation of the inverse curve is k 2n u n + k 2n - 2 u.n_ x (x 2 + y 2 ) + W^u^x 2 + y 2 ) 2 + . . . + u (x 2 + y 2 ) n = o. (19) For instance, the equation of any right line is of the form u L + u = o ; hence that of its inverse with respect to the origin is Wu x + u (x 2 + y 2 ) = o. This represents a circle passing through the pole, as is well known, except when u = o ; i.e. when the line passes through the pole 0. Again, the equation of the inverse of the circle x 2 + y 2 + u x + u = o, with respect to the origin, is (& 4 + k 2 Ui + u (x 2 + y 2 )) (x 2 + y 2 ) = o, which represents another circle^ along with the two imaginary right lines x 2 + y 2 = o. Pedal Curves. 227 Again, the general equation of a conic is of the form %h + u x + u = o ; hence that of its inverse with respect to the origin is &% 2 + k 2 ih(% 2 + y 2 ) + w (# 2 + y 2 ) 2 = o, which represents a curve of the fourth degree of the class called "bicircular quartics." If the origin be on the conic the absolute term u vanishes, and the inverse is the curve of the third degree represented by k 2 u 2 + Ui (x 2 + y 2 ) = o. This curve is called a " circular cubic." If the focus be the origin of inversion, the inverse is a curve called the Limacon of Pascal. The form of this curve will be given in a subsequent Chapter. 187. Pedal Curves. — If from any point as origin a per- pendicular be drawn to the tangent to a given curve, the locus of the foot of the perpendicular is called the pedal of the curve with respect to the assumed origin. In like manner, if perpendiculars be drawn to the tan- gents to the pedal, we get a new curve called the second pedal of the original, and so on. With respect to its pedal, the original curve is styled the first negative pedal, &c. 188. Tangent at any Point to the Pedal of a given Curve. — Let ON, ON' be the perpendiculars from the origin on the tangents drawn at two points P and Q on the given curve, and J 1 the intersec- tion of these tangents ; join NN'; then since the angles ONT and ON f T are right angles, the qua- drilateral ON N'T is inscribable in a circle, .-. lONN=lOTN In the limit when P and Q coincide, L OTN = L OPN, and NN' becomes the tangent to the locus of N; hence the q 2 228 Tangents and Normals to Curves. latter tangent makes the same angle with ON that the tangent at P makes with OP. This property enables us to draw the tangent at any point N on the pedal locus in question. Again, if p' represent the perpendicular on the tangent at N to the first pedal, from similar triangles we evidently have p 2 r = —. p Hence, if the equation of a curve he given in the form p 2 r =f{p), that of its first pedal is of the form — =f(p), in which p and p f are respectively analogous to r and p in the original curve. In like manner the equation of the next pedal can be determined, and so on. 189. Reciprocal Polars. — If on the perpendicular ON a point P' be taken, such that OP'. ON is constant (k 2 sup- pose), the point P' is evidently the pole of the line PN with respect to the circle of radius k and centre ; and if all the tangents to the curve be taken, the locus of their poles is a new curve. We shall denote these curves by the letters A and P, respectively. Again, by elementary geometry, the point of intersection of any two lines is the pole of the line joining the poles of the lines* Now, if the lines be taken as two infinitely near tangents to the curve A, the line joining their poles becomes a tangent to B ; accordingly, the tangent to the curve B has its pole on the curve A. Hence A is the locus of the poles of the tangents to B. In consequence of this reciprocal relation, the curves A and B are called reciprocal polar s of each other with respect to the circle whose radius is k. Since to every tangent to a curve corresponds a point on its reciprocal polar, it follows that to a number of points in directum on one curve correspond a number of tangents to its reciprocal polar, which pass through a common point. Again, it is evident that the reciprocal polar to any curve is the inverse to its pedal with respect to the origin. "We have seen in Art. 1 80 that the greatest number of tan- gents from a point to a curve of the n th degree is n(n - 1) ; * Townsend's Modern Geometry, vol. i., p. 219. Reciprocal Polars. 229 hence the greatest number of points in which its reciprocal polar can be cut by a line is n(n — 1), or the degree of the reciprocal polar is n (n- 1 ) . For the modification in this result, arising from singular points in the original curve, as well as for the complete discussion of reciprocal polars, the student is referred to Salmon's Higher Plane Curves. As an example of reciprocal polars we shall take the curve considered in Art. 173. If r denote the radius vector of the reciprocal polar cor- responding to the perpendicular p in the proposed curve, we have ¥ Substituting this value for^> in equation (5), we get m m m fk~\m~i m-l . m-l i—j = (a cos to) + (0 sin to) , 2m m m or I™ 1 * = (axY~ x + {byf^\ which is the equation of the reciprocal polar of the curve re- presented by the equation — + 7T- = I. a m b m In the particular case of the ellipse, x 2 if a % b % ' the reciprocal polar has for its equation ¥ = aV + 5y. The theory of reciprocal polars indicated above admits of easy generalization. Thus, if we take the poles with respect to any conic section ( U) of all the tangents to a given curve A, we shall get a new curve B ; and it can be easily seen, as before, that the poles of the tangents to B are situated on the curve A. Hence the curves are said to be reciprocal polars with respect to the conic TJ. It may be added, that if two curves have a common point, 230 Tangents and Normals to Curves. their reciprocal polars have a common tangent; and if the curves touch, their reciprocal polars also touch. For illustrations of the great importance of this " principle of duality," and of reciprocal polars as a method of investi- gation, the student is referred to Salmon's Conies, ch. xv. We next proceed to illustrate the preceding by discussing a few elementary properties of the curves which are comprised under the equation r m = a m cos mO. 190. Pedal and Reciprocal Polar ofr m = a m cos mO. We shall commence by finding the ^ angle between the radius vector and the perpendicular on the tangent. In the accompanying figure we have tan PON = cot OPN =-■%,. /-^ rdd Fig. 15- But m log r = m log a + log (cos mO) ; dr hence — Q = - tan mO, rclu and accordingly, LPON=mO. (20) 7* WJ+1 Again, p = ON = r eosmO = or r m+1 = a m p. (21) The equation of the pedal, with respect to 0, can be im- mediately found. For, let l A ON = w, and we have a) = (m +1)6. Also, from (21), [-)-[-) ■ Hence, the equation of the pedal is ■ »»'+i = a m+1 cos . {22) \m + 1 ' On the Curve r m = a m cos mQ. 231 Consequently, the equation of the pedal is got by substi- tuting instead of m in the equation of the curve. ° m + 1 By a like substitution the equation of the second pedal is easily seen to be mm f\ — • — mu ™2m+i _ /,2m+i = a""^ 1 cos ; 2m + 1 and that of the n th pedal m J>L_ m Q r mn+l = a mn+\ cog p / 2 ,\ WW + 1 Again, from Art. 1 84, it is plain that the inverse to the curve r m = a m cos md, with respect to a circle of radius a, is the curve r m cos mO = a m . Again, the reciprocal polar of the proposed, with respect to the same circle, being the inverse of its pedal, is the curve m n m — mu — ■ r ro+i cog = a m +K f 2 .\ m+i It may be observed that this equation is got by substitut- /yyi ins: for m in the original equation. m + 1 Accordingly we see that the pedals, inverse curves, and reciprocal polars of the proposed, are all curves whose equa- tions are of the same form as that of the proposed. In a subsequent chapter the student will find an additional discussion of this class of curves, along with illustrations of their shape for a few particular values of m. Examples. 1 . The equation of a parabola referred to its focus as pole is r (1 + cos 6) = 2a, to find the relation between r and p. a Here H cos - = ah, and consequently p 2 = ar, a well-known elementary property of the curve. 232 Tangents and Normals to Curves. 2. The equation r 2 cos 20 = a 2 represents an equilateral hyperbola ; prove that^r = a 2 . 3. Trie equation r 2 = a 2 cos 2O represents a Lemniscate of Bernoulli ; find the equation connecting p and r in this case. Ans. r 3 — a 2 p. 4. Find the equation connecting the radius vector and the perpendicular on the tangent in the Cardioid whose equation is r = a(i + cos 9). Ans. r 3 = 2ap 2 . It is evident that the Cardioid is the inverse of a parabola with respect to its focus ; and the Lemniscate that of an equilateral hyperbola with respect to its centre. Accordingly, we can easily draw the tangents at any point on either of these curves by aid of the Theorem of Art. 185. 5. Show, by the method of Art. 188, that the pedal of the parabola, p 2 = ar, with respect to its focus, is the right line p = a. 6. Show that the pedal of the equilateral hyperbola pr = a 1 is a Lemniscate. 7. Find the pedal of the circle r 2 = iap. Ans. A Cardioid, r 3 = 2ap 2 . 191. Expression for PN — To find the value of the intercept between the point of contact P'and the foot N of the perpendicular from the origin on the tangent at P. Let p=ON,u>=L NO A, ^ PN=t; then z_NTN'=LNON' = Aw, also SN'=TS sinSTN; SN' sin NON TS = : ^^ r/ ; hut in the CLYi limit, when PQ is infinitely small, -: — ^ftttf/ becomes -=-, and TS becomes PN or t ; Also OP 2 = ON 2 + PN 2 ,^f, [fj. (>6) 192. To prove that ds dt / v - r =p+ T . (27) Vectorial Co-ordinates. 233 On reference to the last figure we have ds .. ., £ PT+TQ dt .. ., £ QN'-PN — = limit of , — = limit 01 ; da) Aw dd> Aw but PT + TQ - QN f + PN = TJST - TN', , ds dt ,. ., „TN-TN' v ., ,SN niKT hence — = limit 01 = limit 01 - — = OJy=p ; dhi do) Aw Aw ds dt da) db) This result, which is due to Legendre, is of importance in the Integral Calculus, in connexion with the rectification of curves. If -~- be substituted for t, the preceding formula becomes doj dio doj z This shape of the result is of use in connexion with curva- ture, as will be seen in a subsequent chapter. 193. Direction of lormal in Tectorial Co-ordi- nates. — In some cases the equation of a curve can be expressed in terms of the distances from two or more fixed points or foci. Such distances are called vectorial co-ordi- nates. For instance, if r l9 r % denote the distances from two fixed points, the equation r x + r 2 - const, represents an ellipse, and r x — r % - const., a hyperbola. Again, the equation ri + mr- z = const. represents a curve called a Cartesian* oval. Also, the equation 7*1 r 2 = const. represents an oval of Cassini, and so on. The direction of the normal at any point of a curve, in such cases, can be readily obtained by a geometrical con- struction. * A discussion of the principal properties of Cartesian ovals will be found in Chapter XX. 234 Tangents and Normals to Curves. For, let F(n, r 2 ) = const. be the equation of the curve, where F 1 P = r 1 , F z P = r 2 , then we have dFdr v dFdr 2 di\ ds dr % ds Now, if PThe the tangent at P, then, by Art. 1 80, we have dr dv — = cos ^1, -^ = cos i/> 2 , where ^ = l TPF l9 ?// 2 = z. TPF 2 . TT dF . dF . ' xtence — cos \pi+ — cos 1/- 2 = o. (29) ctr-i Qj)"% Again, from any point R on the normal draw RL and R M respectively parallel to F 2 P and F X P 9 and we have PL : LR = sin RPM : sin RPL = cos ^ 2 : - cos & _dF m dF dr x ' dr 2 ' Accordingly, if we measure on PFi and PF% lengths PL and PM 9 which are in the proportion of — to — , then ar\ ar^ the diagonal of the parallelogram thus formed is the normal required. This result admits of the following generalization : Let the equation of the curve* be represented by F{n, r t9 n, • • . r n ) = const., * The theorem given above is taken from Poinsot's Elements de Statique, Neuvieme Edition, p. 435. The principle on which it was founded was, how- ever, given by Leibnitz (Journal des Savans, 1693), and was deduced from mechanical considerations. The term resultant is borrowed from Mechanics, and is obtained by the same construction as that for the resultant of a number of forces acting at the same point. Thus, to find the resultant of a number of lines Fa, Pb, Pc, Pd, . . . issuing from a point P, we draw through a a right line aB, equal and parallel to Pb, and in the same direction ; through B, a right line BC, equal and parallel to Pc, and so on, whatever be the number of lines: then the bine US, which closes the polygon, is the resultant in question. Normals in Vectorial Co-ordinates. 235 where r iy r 2 , . . . r n denote the distances from n fixed points. To draw the normal at any point, we connect the point with the n fixed points, and on the joining lines measure off lengths proportional to dF dF dF dF ,. , ^-' J"' -J-> • • • 7"> respectively; ai'i dr-i dr z clr n then the direction of the normal is the resultant of the lines thus determined. For, as before, we have dFdr x dFdr 2 dF dr n _ dr x ds dr 2 ds ' dr n ds it dF . dF . dF ' Hence — cos fa + — cos fa + - • • -7— cos ip w = o. (30) drx ar 2 ar^ dF , dF dF JNOW, — -COS^i, — COS \pz, ... — COS \l n , ar x ar 2 ar^i are evidently proportional to the projections on the tangent of the segments measured off in our construction. Moreover, in any polygon, the projection of one side on any right line is manifestly equal to the sum of the projections of all the other sides on the same line, taken with their proper signs. Consequently, from (30), the projection of the resultant on the tangent is zero ; and, accordingly, the resultant is normal to the curve, which establishes the theorem. It can be shown without difficulty that the normal at any point of a surface whose equation is given in terms of the distances from fixed points can be determined by the same construction. Examples. 1. A Cartesian oval is the locus of a point, P, such that its distances, Pltl, PM\ from the circumferences of two given circles are to each other in a constant ratio ; prove geometrically that the tangents to the oval at P, and to the circles at M and M ', meet in the same point. 2. The equation of an ellipse of Cassini is rr' = ab, where r and / are the distances of any point P on the curve, from two fixed points, A and B. If he the middle point of AB, and PN the normal at P, prove that L APO= L BPN. 3. In the curve represented hy the equation rj. 3 + r 2 z = a 3 , prove that the normal divides the distance hetween the foci in the ratio of r 2 to n. 236 Tangents and Normals to Curves. 1 94. In like manner, if the equation of a curve be given in terms of the angles 9 U 2 , . . . 9 n , which the vectors drawn to fixed points make respectively with a fixed right line, the direction of the tangent at any point is obtained by an analo- gous construction. For, let the equation be represented by F(0 l9 a , . . . 0„) = const. Then, by differentiation, we have dFdfa dFa% dF dOn _ dOi ds ddz ds ' dd n ds Hence, as before, from Art. 180, we get 1 dF . , 1 dF . , 1 dF . . , N --sinf + -- 5 -siinf 2 + . . - + — JX- sin \fr n = 0. (31) i\ ddx r 2 dVi r n dd n Accordingly, if we measure on the lines drawn to the fixed points segments proportional to 1 dF 1 dF 1 dF r x dOi r 2 ddz ' ' r n ddn and construct the resultant line as before, then this line will be the tangent required. The proof is identical with that of last Article. 195. Curves Symmetrical with respect to a l^ine, asad Centres of Curves. — It may be observed here, that if the equation of a curve be unaltered when y is changed into - y, then to every value of x correspond equal and oppo- site values of y; and, when the co-ordinate axes are rect- angular, the curve is symmetrical with respect to the axis of x. In like manner, a curve is symmetrical with respect to the axis of y, if its equation remains unaltered when the sign of x is changed. Again, if, when we change x and y into - x and - y, re- spectively, the equation of a curve remains unaltered, then every right line drawn through the origin and terminated by the curve is divided into equal parts at the origin. This takes place for a curve of an even degree when the sum of Symmetrical Curves and Centres. 237 the indices of x and y in each term is even ; and for a curve of an odd degree when the like sum is odd. Such a point is called the centre* of the curve. For instance, in conies, when the equation is of the form ax 2 + 2hxy + by 2 = c, the origin is a centre. Also, if the equation of a cubicf be reducible to the form U z + U x = O, the origin is a centre, and every line drawn through it is bi- sected at that point. Thus we see that when a cubic has a centre, that point lies on the curve. This property holds for all curves of an odd degree. It should be observed that curves of higher degrees than the second cannot generally have a centre, for it is evidently impossible by transformation of co-ordinates to eliminate the requisite number of terms from the equation of the curve. For instance, to seek whether a cubic has a centre, we substi- tute X + a for x, and T + j3 for y, in its equation, and equate to zero the coefficients of X 2 , XFand Y 2 , as well as the abso- lute term, in the new equation : as we have but two arbitrary constants (a and j3) to satisfy four equations, there will be two equations of condition among its constants in order that the cubic should have a centre. The number of conditions is obviously greater for curves of higher degrees. * For a general meaning of the word " centre," as applied to curves of higher degrees, see Chasles's Apercu Historique, p. 233, note. f This name has heen given to curves of the third degree by Dr. Salmon, in his Higher Plane Curves, and has heen generally adopted hy subsequent writers on the subject. 2 38 Examples. Examples. i . Find the lengths of the subtangent and subnormal at any point of the curve yn — a n ~^X. y 1 Ans. nx, — . nx 2. Find the subtangent to tbe curve %myii — a m+n. nx Ans. . m 3. Find the equation of the tangent to the curve x 5 = a?y*. Ans. =3- x y 4. Show that the points of contact of tangents from a point (a, /3) to the curve x m y n = a m+n are situated on the hyperbola (m + n)xy = nfix + may. 5. In the same curve prove that the portion of the tangent intercepted be- tween the axes is divided at its point of contact into segments which are to each other in a constant ratio. 6. Find the equation of the tangent at any point to the hypocycloid, #! + y% — a$ ; and prove that the portion of the tangent intercepted between the axes is of constant length. 7. In the curve X"- + y n = a n , find the length of the perpendicular drawn from the origin to the tangent at any point, and find also the intercept made by the axes on the tangent. a n . , . n 2n Ans. p = — ; intercept = v/^n-Z + yZn-l' px^yn-^ 8. If the co-ordinates of every point on a curve satisfy the equations x = c sin 20 (1 + cos 26), y = c cos 20 (1 — cos 20), prove that the tangent at any point makes the angle 6 with the axis of x. 9. The co-ordinates of any point in the cycloid satisfy the equations x = a(9 — sin0), y = a(i — cos 6): prove that the angle which the tangent at the point makes with the axis of y is -. 2 Examples. 239 Here -f = -7- = cot -. ax ax 2 dd 10. Prove that the locus of the foot of the perpendicular from the pole on the tangent to an equiangular spiral is the same curve turned through an angle. 11. Prove that the reciprocal polar, with respect to the origin, of an equi- angular spiral is another spiral equal to the original one. 12. An equiangular spiral touches two given lines at two given points ; prove that the locus of its pole is a circle. 13. Find the equation of the reciprocal polar of the curve rk cos - = aK 3 a with respect to the origin. Ans. The Cardioid H = ah cos -. 14. Find the equation of the inverse of a conic, the focus being the pole of inversion. 15. Apply Art. 184, to prove that the equation of the inverse of an ellipse with respect to any origin is of the form 2a P = OFi . pi + OF 2 . p 2 , where F\ and F% are the foci, and p, pi, 02 represent the distances of any point on the curve from the points 0, /1 and / 2 , respectively ; f\ and f% being the points inverse to the foci, F\ and F%. 16. The equation of a Cartesian oval is of the form r + Jcr' = a f where r and r' are the distances of any point on the curve from two fixed points, and a, k are constants. Prove that the equation of its inverse, with respect to any origin, is of the form api + j3/>2 + 7/>3 = o, where pi, p2, pz are the distances of any point on the curve from three fixed points, and a, 0, 7 are constants. 17. In general prove that the inverse of the curve api + #/} 2 + yps = o, with respect to any origin, is another curve whose equation is of similar form. 18. If the radius vector, OF, drawn from the origin to any point Pon a 240 Examples. curve be produced to Pi, until PPi be a constant length ; prove that the normal at Pi to the locus of Pi, the normal at P to the original curve, and the perpen- dicular at the origin to the line OP, all pass through the same point. This follows immediately from the value of the polar subnormal given in Art. 182. 19. If a constant length measured from the curve be taken on the normals along a given curve, prove that these lines are also normals to the new curve which is the locus of their extremities. 20. In the ellipse — + ^- = I, if # = # sin d>, a- o % prove that ds = a\/ 1 - e 2 sin 2 d>. d$ 21. If ds be the element of the arc of the inverse of an ellipse with respect to its centre, prove that , a \/i — e 2 sin 2 d> ' , a 2 - b 2 ds = & 2 — ^~- dtp, where n = — - — . b 2 if « sin 2 b* 22. If co be the angle which the normal at any point on the ellipse — \. — = 1 makes with the axis-major, prove that a 2 b 2 b 2 da ds = a (1 — £ 2 sin 3 w)2 23. Express the differential of an elliptic arc in terms of the semi -axis major, fi, of the confocal hyperbola which passes through the point. I a 2 - a 2 24. In the curve r m = a m cos mB, prove that T »n-l ds „ -— = a sec m m9. dO 25. If F(x, y) — o be the equation to any plane curve, and

an ^ so on. Method of finding Asymptotes in Cartesian Co-ordinates. 243 Accordingly, when the values of \x and v are determined so as to satisfy the two preceding equations, the correspond- ing line y = fix + v meets the curve in two points in infinity, and consequently is an asymptote. (Salmon's Conic Sections, Art. 154-) Hence, if fi x be a root of the equation f (fi) = o, the line y = n&- 777— x \5) Jo (fli) is in general an asymptote to the curve. If /i(ju) = o and/ (ju) = o have a common root (jui suppose), the corresponding asymptote in general passes through the origin, and is represented by the equation y = ti&. In this case u n and u n ^ evidently have a common factor. The exceptional case when'/ '(ju) vanishes at the same time will be considered in a subsequent Article. To each root of f (/i) = o corresponds an asymptote, and accordingly,* every curve of the n th degree has in general n asymptotes, real or imaginary. From the preceding it follows that every line parallel to an asymptote meets the curve in one point at infinity. This also is immediately apparent from the geometrical property that a system of parallel lines may be considered as meeting in the same point at infinity — a principle intro- duced, by Desargues in the beginning of the seventeenth century, and which must be regarded as one of the first important steps in the progress of modern geometry. Con. No line parallel to an asymptote can meet a curve of the n th degree in more than (n — i) points besides that at infinity. Since every equation of an odd degree has one real root, it follows that a curve of an odd degree has one real * Since /o(a0 is of the n th degree in jx, unless its highest coefficient vanishes, in which case, as we shall see, there is an additional asymptote parallel to the axis of'jf. ' • R 2 244 Asymptotes. asymptote, at least, and has accordingly an infinite branch or branches. Hence, no curve of an odd degree can be a closed curve. For instance, no curve of the third degree can be a finite or closed curve. The equation fo(fi) = o, when multiplied by x n , becomes u n = o ; consequently the n right lines, real or imaginary, represented by this equation, are, in general, parallel to the asymptotes of the curve under consideration. In the preceding investigation we have not considered the case in which a root of / (ju) = o either vanishes or is infinite; i.e., where the asymptotes are parallel to either co-ordinate axis. This case will be treated of separately in a subsequent Article. If all the roots of f (/.i) = o be imaginary the curve has no real asymptote, and consists of one or more closed branches. Examples. To find the asymptotes to the following curves : — 1. y % = ax 2 + %?. Substituting fix + v for y, and equating to zero the coefficients of x 3 and x\ separately, in the resulting equation, we obtain H z — i = o, and iixrv = a ; a hence the curve has but one real asymptote, viz., a 2. # 4 — a 4 + 2ax 2 y = b 2 x 2 . Here the equations for determining the asymptotes are ^-1=0, and 4[x z v + 2afi = o ; accordingly, the two real asymptotes are a , a y = x - r , and y + x + - = o. 3. x z + $x 2 y - xy 2 - zy 3 + x 2 - 2xy + w 2 + $x + 5 = o. x 3 I 3 Ans. y + - + - = o, y = x+-, y + x=-. 3 4 4 * Asymptotes Parallel to Co-ordinate Axes, 245 199. Case in which u n = o represents the n Asymp- totes. — If the equation of the curve contain no terms of the (n - i) th degree, that is, if it be of the form u n + ii n _ 2 + U n -z + &c . . . + Mi + u Q = o, the equations for determining the asymptotes become / (ju) = o, and vfo'(p) = o. The latter equation gives v = o, unless fo(fj) vanishes along with/ (ju), i.e., unless / (ju) has equal roots. Hence, in curves whose equations are of the above form, the n right lines represented by the equation u n = o are the n asymptotes, unless two of these lines are coincident. This exceptional case will be considered in Art. 202. The simplest example of the preceding is that of the hyperbola ax 2 + 2hxy + by 1 = c, in which the terms of the second degree represent the asymp- totes (Salmon's Conic Sections, Art. 195). Examples. Find the real asymptotes to the curves 1. xy 2 - x^y = ar(x + y) + b z . Ans. x = o, y = o, x -y -o. 2. y z — x 3 = a?x. „ y — x = O. 3. # 4 - y i = a 2 xy + b 2 y 2 . „ x + y = o, x - y = o. 200. Asymptotes parallel to the Co-ordinate Axes. — Suppose the equation of the curve arranged accord- ing to powers of x, thus a x n + {a x y + h)x n ~ l + &c. = o ; then, if a = o and a x y + b = o, or y = , two of the roots #1 of the equation in x become infinite ; and consequently the line a$ + b = o is an asymptote. 246 Asymptotes. In other words, whenever the highest power of x is wanting in the equation of a curve, the coefficient of the next highest power equated to zero represents an asymptote parallel to the axis of x. If a = o, and b = o, the axis of x is itself an asymptote. If x n and x n ~ x be both wanting, the coefficient of x n ~ 2 re- presents a pair of asymptotes, real or imaginary, parallel to the axis of x ; and so on. In like manner, the asymptotes parallel to the axis of y can be determined. Examples. Find the real asymptotes in the following curves : — 1. y 2 x — ay 2 = x 3 + ax 2 + b 3 . Am. x = a, y = x + a, y + x + a = o. 2. y{x 2 - $bx + 2b 2 ) = x 3 — 3«« 2 + a 3 , x = b, x = 2b, y + ia = x + $b. 3. x^y 2 = a 2 (x 2 + y 2 ). x = ±a, y = ±a. 4. x 2 y 2 = a 2 (x 2 — ?/ 2 ). y 4- a = o, y — a = o. 5. y 2 a — y 2 x = x 3 . x = a. 201. Parabolic Branches. — Suppose the equation /o(/0 = o has equal roots, then/ '(jUi) vanishes along with/o(At), and the corresponding value of v found from (5) becomes in- finite, unless /i(/u) vanish at the same time. Accordingly, the corresponding asymptote is, in general, situated altogether at infinity. The ordinary parabola, whose equation is of the form (ax + j3?/) 2 = lx + my + n, furnishes the simplest example of this case, having the line at infinity for an asymptote. (Salmon's Conic Sections, Art. 254.) Branches of this latter class belonging to a curve are called parabolic, while branches having a finite asymptote are called hyperbolic. 202. From the preceding investigation it appears that the asymptotes to a curve of the n th degree depend, in general, only on the terms of the 11 th and the (n - i) th degrees Parallel Asymptotes. 247 in its equation. Consequently, all carves which have the same terms of the two highest degrees have generally the same asymptotes. There are, however, exceptions to this rule, one of which will be considered in the next Article. 203. Parallel Asymptotes. — We shall now consider the case where fo(ji) ~ o has a pair of equal roots, each repre- sented by jui, and where /i(jui) = o, at the same time. In this case the coefficients of x n and x n ~ x in (2) both vanish independently of v, when jit = jUi ; we accordingly infer that all lines parallel to the line y = fxix meet the curve in two points at infinity, and consequently are, in a certain sense, asymptotes. There are, however, two lines which are more properly called by that name ; for, substituting jui for fi in (2), the two first terms vanish, as already stated, and the coefficient of x n ~ % becomes Hence, if vi and v% be the roots of the quadratic v 3 2 /o"W+v//W+/ 2 W=o), (6) the lines y = /mix + v i} and y = \x x x + v 2 , are a pair of parallel asymptotes, meeting the curve in three points at infinity. If the roots of the quadratic be imaginary, the corre- sponding asymptotes are also imaginary. Again, if the term u n ^ be wanting in the equation, and if / (jtt) = o have equal roots, the corresponding asymptotes are given by the quadratic — /o"(/*l) +/ 8 (jUi) = o. In order that these asymptotes should be real, it is necessary that/ 2 (jKi) and /o"(/*i) should have opposite signs. There is no difficulty in extending the preceding investi- gation to the case where fo(ji) = o has three or more equal roots. 248 Asymptotes. Examples. 1. (x + yf {x 2 + y 2 + xy) = a 2 y 2 + a 3 (x - y). Here Mix) = (1 + ^) 2 (i + fx + /*«), /^> = o, / 2 ( M )=-«V; .'. yui = — I, fo"(m) = 2, / 2 (mi) = -« 2 ; accordingly yi = «, y 2 = - a, and the corresponding asymptotes are y + x — a = o, and 3/ + x + a = o. The other asymptotes are evidently imaginary. 2. x 2 (x + y) 2 + 2ay 2 (x + y) + Sofixy + a 3 y = o. Here fM = (1 + tf, ffa) - 2^(1 + fi), M/i) = Sa^; .-. ^1 = - I, /o"0) = 2 > /iVl) = 2 «, /2(/*l) = - 8 «*> and the corresponding asymptotes are y + a; — 2.a = o, and y + # + 40 = o. 204. If the equation to a curve of the n th degree be of the form t (y + ax + |3) fa + 02 = o, where the highest terms containing x and y in fa are of the degree n - 1 , and those in $ 2 are of the degree w - 2 at most, the line y + ax + |3 = o is an asymptote to the curve. For, on substituting - ax - j3 instead of y in the equation, it is evident that the coefficients of x n and x n ~ l both vanish ; hence, by Art. 198, the line y + ax + f3 = o is an asymptote. Conversely, it can be readily seen that if y + ax + fi be an asymptote to a curve of the n th degree its equation admits of being thrown into the preceding form. In general, if the equation to a curve of the n th degree be of the form (y + a x x + (3i) {y + a 2 x + j3 2 ) . . . (y + a n x + fd n ) + fa = o, (7) ^ Asymptotes to a Cubic. 249 where

) (j)x = <£ 2 , where 2 is at least one degree lower than (pi in x and y. A 250 Asymptotes. Hence y + a x + /3 = - , and the perpendicular distance of any point (x , y ) on the curve from the line y + ax + f5 = o is y + ax + (5 1 (fa — , or vA + a 2 ^/i + a~ 2 \0iA where the suffix denotes that # and y are substituted for # and y in the functions fa and 2 . Now, when x and 2/0 are taken infinitely great, the value of the preceding fraction depends, in general, on the terms of the highest degree (in x and y) in fa and fa ; and since the degree of fa is one loiver than that of fa, it can be easily seen by the method of Ex. 7, Art. 89, that the fraction — 0i becomes, in general, infinitely small when x and y become infinitely great. Hence, the distance of the line y + ax + j3 from the curve becomes infinitely small at the same time. It is not considered necessary to go more fully into this discussion here. The subject of parabolic and other curvilinear asymptotes is omitted as being unsuited to an elementary treatise. Moreover, their discussion, unless in some elementary cases, is both indefinite and unsatisfactory, since it can be easily seen that if a curve has parabolic branches, the number of its parabolic asymptotes is generally infinite. The reader who desires full information on this point, as well as the discussion of the particular parabolas called osculating, is referred to a paper by M. Pliieker, in Liouville's Journal, vol. i., p. 229. 206. Asymptotes in Polar Co-ordinates. — If a curve be referred to polar co-ordinates, the directions of its. points at an infinite distance from the origin can be in gene- ral determined by making r = 00, or u = o, in its equation, and solving the resulting equation in 9. The position of the asymptote corresponding to any such value of is obtained by finding the length of the corresponding polar subtangent, i.e., by finding the value of — corresponding to u = o. WUi Asymptotes in Polar Qo-ordinates. 251 d9 It should be observed that when — is positive, the asymp- Cvtv tote lies above the corresponding radius vector, and when negative, below it ; as is easily seen from Art. 182. If we suppose the equation of the curve, when arranged in powers of r, to be *Vo(0) + OW) + • • • + rf**(P) + /»(&) = o, the transformed equation in u is u%(0) + t*"-%-!(0) + . . . + uMO) +/ o (0) = o : (9) consequently, the directions of the asymptotes are given by the equation /o(6) = O. (10) Again, if we differentiate (9) with respect to 0, it is easily seen that the values of -^ corresponding to u = o are given by the equation A(d)f d +/.'(«) = o, (11) provided that none of the functions MO), MB), . . . MO) become infinite for the values of which satisfy equation (10). Consequently, if a be a root of the equation / o (0) = o, the curve has an asymptote making the angle a with the prime vector, and whose perpendicular distance from the origin is represented by - -jt^t. /o(«) It is readily seen that the equation of the corresponding asymptote is r sm(a - V) +777—. = o. Jo [a) This method will be best explained by applying it to one or two elementary Examples. 252 Asymptotes. Examples. 1. Let the curve be represented by tbe equation Here u = r = a sec + b tan 0. COS0 a + b sin when = -, we nave u = o, and 2 de a + b Accordingly, the corresponding polar subtangent is a + b, and hence the line perpendicular to the prime vector at the distance a + b from the origin is an asymptote to the curve. 3"" Again, u vanishes also when = — , and the corresponding value of the 2 polar subtangent is a - b ; thus giving another asymptote. 2. r = a sec mQ + b tan md. cos md Here a + b sin md Txri ^ 1 . du — m When = — ■, we have u = o, and — = = , 2m h% + 3% *t- 3c x 2 -f dc\xy + l^y 1 + d Q x 3 + id\.x 2 y + id^xy 2 + 2 + Cixy + c 2 y* + &c. + &c. + l(& n + l x x n -hj + &g. + l n y n = o, where the terms are arranged according to their degrees in ascending order. When written in the abbreviated form of Art. 175, the preceding equation becomes U + U x + U% + . . . + %_i + U n = O. "We commence with the equation in its expanded shape, and suppose the axes rectangular. Transforming to polar Multiple Points. 257 co-ordinates, by substituting r cos and r sin instead of x and y, we get a + (#0 cos + #1 sin 0) r + (tf cos 2 + Ci cos sin + c 2 sin 2 0) r 2 + . . . + (l cos n + h cosT 1 6 sin0 + . . . + 4sin"0)r n = o. (1) If be considered a constant, the n roots of this equation in r represent the distances from the origin of the n points of intersection of the radius vector with the curve. If a = o, one of these roots is zero for all values of 0; as is also evident since the origin lies on the curve in this case. A second root will vanish, if, besides a = o, we have b cos + b x sin = o. The radius vector in this case meets the curve in two consecutive points* at the origin, and is consequently the tangent at that point. The direction of this tangent is determined by the equation b Q cos + h sin = o ; accordingly, the equation of the tangent at the origin is b x + biy = o. Hence we conclude that if the absolute term be wanting in the equation of a curve, it passes through the origin, and the linear part (u^ in its equation represents the tangent at that point. If b = o, the axis of x is a tangent ; if b x = o, the axis of y is a tangent. The preceding, as also the subsequent discussion, equally applies to oblique as to rectangular axes, provided we sub- stitute mr and nr for x and y ; where sin fa - 0) n sin m= : -, sman=- ; sm w 8inw to being the angle between the axes of co-ordinates. From the preceding, we infer at once that the equation of the tangent at the origin to the curve x* (x 2 + y 2 ) = a (x - y) * Two points which are infinitely close to each other on the same branch of a curve are said to he consecutive points on the curve. S 258 Multiple Points on Curves. is x - y = o, a line bisecting the internal angle between the co-ordinate axes. In like manner, the tangent at the origin can in all cases be immediately determined. 209. Equation of Tangent at any Point. — By aid of the preceding method the equation of the tangent at any point on a curve whose equation is algebraic and rational can be at once found. For, transferring the origin to that point, the linear part of the resulting equation represents the tangent in question. Thus, if f(x, y) = o be the equation of the curve, we sub- stitute X + x x for x, and Y + y x for y, where (#1, yi) is a point on the curve, and the equation becomes f(X + w S9 Y + y t ) = o. Hence the equation of the tangent referred to the new axes is \axji \dyj\ On substituting x - x x , and y - y x , instead of X and Y, we obtain the equation of the tangent referred to the original axes, viz. <'-*>(l), + fr-*>(f)> =a This agrees with the result arrived at in Art. 169. 210. Double Points. — If in the general equation of a curve we have a = o,, b Q =0, b x = o, the coefficient of r is zero for all values of 9, and it follows that all lines drawn through the origin meet the curve in two points, coincident with the origin. The origin in this case is called a double point. Moreover, if 9 be such as to satisfy the equation c cos 2 + ^cos 9 sin 9 + c % sin 2 = o, (2) the coefficient of r 2 will also disappear, and three roots of equation ( 1 ) will vanish. As there are two values of tan 9 satisfying equation (2), it follows that through a double point two lines can be drawn, each meeting the curve in three coincident points. Double Points. 259 The equation (2), when multiplied by r 2 , becomes c x 2 + c x xy + c 2 y 2 = o. Hence we infer that the lines represented by this equa- tion connect the double point with consecutive points on the curve, and are, consequently, tangents to the two branches of the curve passing through the double point. Accordingly, when the lowest terms in the equation of a curve are of the second degree (w 3 ), the origin is a double point, and the equation u 2 = o represents the pair of tangents at that point. For example, let us consider the Lemniscate, whose equa- tion is (x 2 + y 2 ) 2 = a 2 (x 2 - y 2 ). On transforming to polar co-ordinates its equation becomes r i = a 2 r * ( cos 2 _ s in 2 0), or, r 2 = a 2 cos 2O. Now, when 6 = o, r = ± a ; and, if we confine our atten- tion to the positive values of r, we see that as increases from o to 7T r diminishes from a to zero. When > 3*- Fig. 18. and < — , r is imaginary, &c, and it is evident that the figure of the curve is as annexed, having two branches intersecting at the origin, and that the tangents at that point bisect the angles between the axes. The equations of these tangents are x + y = o, and x - y = o, results which agree with the preceding theory. 211. Wodes, Cusps, and Conjugate Points.* — The pair of lines represented by u 2 = o will be real and distinct, coincident, or imaginary, according as the roots of equa- tion (2) are real and unequal, real and equal, or imaginary. * These have been respectively styled erunodes, spinodes, and acnodes, by Professor Cayley. See Salmon's Higher Plane Curves, Art. 38. S 2 26o Multiple Points on Curves. Fig. 19. Hence we conclude that there may be one of three kinds of singular point on a curve so far as the vanishing of u and u L is concerned. (1). For real and unequal roots, the tangents at the double point are real and distinct, and the point is called a node; arising from the intersection of two real branches of the curve, as in the annexed figure. (2). If the roots be equal, i.e. if u 2 be a perfect square, the tangents coin- cide, and the point is called a cusp : the two branches of the curve touching each other at the point, as in figure 20. (3). If the roots of u% be imaginary, the tangents are imaginary, and the double point is called a conjugate or isolated point ; the co-ordinates of the point satisfy the equation of the curve, but the curve has no real points consecutive to this point, which lies altogether outside the curve itself. It should be observed also that in some cases of singularities of a higher order, the origin is a conjugate point even when u 2 is a perfect square, as will be more fully explained in a sub- sequent chapter. We add a few elementary examples of these different classes for illustration. Fig. 20. 1. Examples. Here the origin is a node, the tangents bisecting the angles between the axes of co-ordinates. 2. ay 1 = x z . In this case the origin is a cusp. Again, solving for y -we get a* Hence, if a be positive, y becomes imaginary for negative values of x ; and, accordingly, no portion of the curve extends to the negative side of the axis of x. Moreover, for positive values of x, the corresponding values of y have opposite signs. This curve is called the semi-cubical parabola. The form of the curve near the origin is exhibited in Fig. 20. Double Points rn General. 261 3. y z - x 2 (x + a). Ans. The origin is a cusp. 4. b {x* + y 2 ) = xK Ans. The origin is a conjugate point. 5. # 3 - $axy + y z = o. Ans. The two branches at the origin touch the co-ordinate axes. 212. Double Points in Creneral. — In order to seek the double points on any algebraic curve, we transform the origin to a point (x ly yi) on the curve ; then, if we can deter- mine values of x L , y x for which the linear part disappears from the resulting equation, the new origin (x 1} y x ) is a double point on the curve. From Art. 209 it is evident that the preceding conditions give dxjx ' vW 1 ' moreover, since the point (x 1} y x ) is situated on the curve, we must have /Oi, tfi) = °- As we have but two variables, x i9 y if in order that they should satisfy these three equations simultaneously, a con- dition must evidently exist between the constants in the equation of the curve, viz., the condition arising from the elimination of a?i, y t between the three preceding equations. Again, when the curve has a double point (a?i, «/i), if the origin be transferred to it, the part of the second degree in the resulting equation is evidently d 2 u\ ( dhi \ 2 fdhi\ dx'), Jr2xy \dxd y y y Wk Accordingly, the lines represented by this quadratic are the tangents at the double point. The point consequently is a node, a cusp, or a conjugate point, according as / dhi V . fdht\ /WV \dxdyjx \dx 2 Ji \dy 2 Ji 262 Multiple Points on Curves. It may be remarked here that no cubic can have more than one double point ; for if it have two, the line joining them must be regarded as cutting the curve in four points, which is impossible. Again, every line passing through a double point on a cubic must meet the curve in one, and but one, other point ; ex- cept the line be a tangent to either branch of the cubic at the double point, in which case it cannot meet the curve else- where; the points of section being two consecutive on one branch, and one on the other branch. In many cases the existence of double points can be seen immediately from the equation of the curve. The following are some easy instances : — Examples. To find the position and nature of the double points in the following curves : — 1. (bx — cy) 2 = (x — a) ab The point x = a, y = — , is evidently a cusp, G at which bx — cy = o is the tangent, as in the accompanying figure 2. (y - c) 2 = (x — #)* {x - b). The point x = a, y = c, is a cusp if a > b, or if a = b ; but is a conjugate point if a < b. Fig. 21. 3. y z = x(x + a) 2 . The point y = o, x = — a is a. conjugate point. 4. x§ + y$ = «i. The points x = o, y = + a ; and y = o, x = + a, are easily seen to be cusps. 213. Parabolas of the Third Degree. — The follow- ing example* will assist the student towards seeing the dis- tinction, as well as the connexion, between the different kinds of double points. Let y 2 = (x - a) (x -b) (x - c) be the equation of a curve, where a< b < c. * Lacroix, Gal. Dif., pp. 395-7. Salmon's Higher Plane Curves, Art. 39. Parabolas of the Third Degree. 263 Here y vanishes when x = a, ovx=b, or x = c ; accordingly, if distances OA = a, OB = b, OC = c, be taken on the axis of x, the curve passes through the points A, B, and C. Moreover, when x < a, y 2 is negative, and therefore y is imaginary. „ x > a, and < b, y 2 is positive, and therefore y is real. „ x > b, and < c, y 2 is negative, and therefore y is imaginary. „ x > c, y 2 is positive, and therefore y is real ; and increases indefinitely along with x. Hence, since the curve is sym- metrical with respect to the axis of x, it evidently consists of an oval lying between A and B, and an infinite branch passing through (7, as in the annexed figure. It is easily shown that the oval is not symmetrical with respect to the perpendicular to AB at its middle point. Again, if b = c, the equation becomes y 2 = [x-a){x- b) 2 . Fig. 22. In this case the point B co- incides with C, the oval has joined the infinite branch, and B has become a double point, as in the annexed figure. A B Fig. 23. On the other hand, let b = a, and the equation becomes y 2 = (x - a) 2 (x - c) ; in this case the oval has shrunk into the point A, and the curve is of the annexed form, having A for a conjugate point. Next, let a = b = e, and the equation becomes A y 2 = (x - a) s ; tfig. 24. 264 Multiple Points on Curves. here the points A, B, C, have come together, and the curve has a cusp at the point A, as in the annexed figure. The curves considered in this Article are called parabolas Fig. 25. of the third degree. As an additional example, we shall investigate the fol- lowing problem : — 214. Given the three asymptotes of a cubic, to find its equa- tion, if it have a double point. Taking two of its asymptotes as axes of co-ordinates, and supposing the equation of the third to be ax + by + c = o, the equation of the cubic, by Art. 204, is of the form xy[ax + by + c) = Ix + my + n. Again, the co-ordinates of the double point must satisfy the equations du du dx ' dy ' or [lax + by + c) y = /, {ax + iby + c) x = m ; from which I and m can be determined when the co-ordinates of the double point are given. To find n, we multiply the former equation by x, and the latter by y, and subtract the sum from three times the equa- tion of the curve, and thus we get cxy = zlx + 2my + 311 ; from which n can be found. In the particular case where the double point is a cusp,* its co-ordinates must satisfy the additional condition dhi d~u ( d 2 u dx 2 dy* \dxdy / or (2 ax + iby + c) 2 = ^abxy, and consequently the cusp must lie on the conic represented by this equation. * It is essential to notice that the existence of a cusp involves one more relation among the coefficients of the equation of a curve than in the case of an ordinary double point or node. Double Points on a Cubic. 265 It can be easily seen that this conic* touches at their middle points the sides of the triangle formed by the asymp- totes. The preceding theorem is due to Pliicker,f and is stated by him as follows : — " The locus of the cusps of a system of curves of the third degree, which have three given lines for asymptotes, is the maximum ellipse inscribed in the triangle formed by the given asymptotes." It can be easily seen that the double point is a node or a conjugate point, according as it lies outside or inside the above-mentioned ellipse. 215. Multiple Points of Iliglaer Curves.— By follow- ing out the method of Art. 208, the conditions for the existence of multiple points of higher orders can be readily determined. Thus, if the lowest terms in the equation of a curve be of the third degree, the origin is a triple point, and the tangents to the three branches of the curve at the origin are given by the equation u % = o. The different kinds of triple points are distinguished, according as the lines represented by u 3 = o are real and distinct, coincident, or one real and two imaginary. In general, if the lowest terms in the equation of a curve be of the m th degree, the origin is a multiple point of the m th order, &c. Again, a point is a triple point on a curve provided that when the origin is transferred to it the terms below the third degree disappear from the equation. The co-ordinates of a triple point consequently must satisfy the equations die du d 2 u d 2 u d 2 u 9 dx ' dy ' dx 2 ' dxdy 9 dy 2 Hence in general, for the existence of a triple point on a curve, its coefficients must satisfy four conditions. The complete investigation of multiple points is effected * From the form of the equation we see that the lines x = o, y = o are tangents to the conic, and that 2ax + 2by + c = o represents the line joining the points of contact ; hut this line is parallel to the third asymptote ax + by + c = o, and evidently passes through the middle points of the intercepts made hy this asymptote on the two others. t Zioiwille's Journal, vol. ii. p. 14. 266 Multiple Points on Curves. more satisfactorily by introducing the method of trilinear co- ordinates. The discussion of curves from this point of view is beyond the limits proposed in this elementary Treatise. 215 (a). Cusps, in Greneral. — Thus far singular points have been considered with reference to the cases in which they occur most simply. In proceeding to curves of higher degrees they may admit of many complications ; for instance ordinary cusps, such as represented in Fig. 20, may be called cusps of the first species, the tangent lying between both branches : the cases in which both branches lie on the same side, as exhibited in the accompanying figure, may be called cusps of the second species. p . , Professor Cayley has shown how this is to be considered as consisting of several singularities happen- ing at a point (Salmon's Higher Plane Curves, Art. 58). Again, both of these classes may be called single cusps, as distinguished from double cusps extending on both sides of the point of contact. Double cusps are styled tacnodes by Professor Cayley. These points are sometimes called points of osculation ; however, as the two branches do not in general osculate each other, this nomenclature is objectionable. It should be observed that whenever we use the word cusp with- out limitation, we refer to the ordinary cusp of the first species. Cusps are calledpoints de rebroussement by French writers, and Riickkehrpunkte by Grermans, both expressing the turning backwards of the point which is supposed to trace out the curve; an idea which has its English equivalent in their name of stationary points. A fuller discussion of the different classes of cusps will be given in a subsequent place. We shall conclude this chapter with a few remarks on the multiple points of curves whose equations are given in polar co-ordi- nates. Examples. 1. (y — x 2 ) 2 = x 5 . Here the origin is a cusp ; also y = a; 2 + x% ; hence, when a; is less than unity, hoth values of y are positive, and consequently the cusp is of the second species. 2. Show that the origin is a double cusp in the curve x b + bx* — a 3 y 2 = o. Multiple Points with Polar Co-ordinates. 267 216. Multiple Points of Curves iu Polar Co-ordi- nates. — If a curve referred to polar co-ordinates pass through the origin, it is evident that the direction of the tangent at that point is found by making r = o in its equation ; in this case, if the equation of the curve reduce to f(Q) = o, the resulting value of gives the direction of the tangent in question. If the equation f(0) = o has two real roots in 0, less than 71-, the origin is a double point, the tangents being determined by these values of 0. If these values of 9 were equal, the origin would be a cusp ; and so on. In fact, it will be observed that the multiple points on algebraic curves have been discussed by reducing them to polar equations, so that the theory already given must apply to curves referred to polar, as well as to algebraic co-ordi- nates. It may be remarked, however, that the order of a multiple point cannot, generally, be determined unless with reference to Cartesian co-ordinates, in like manner as the degree of a curve in general is determined only by a similar reference. For example, in the equation r = a cos 2 - b sin 2 0, the tangents at the origin are determined by the equation tan 5= ± /-, and the origin would seem to be only a double point ; however, on transforming the equation to rectangular axes, it becomes (a? 2 + y 2 y - (ax 2 - by 2 ) 2 ; from which it appears that the origin is a multiple point of the fourth order, and the curve of the sixth degree. In fact, what is meant by the degree of a curve, or the multiplicity of a point, is the number of intersections of the curve with any right line, or the number of intersections which coincide for every line through such a point, and neither of these are at once evident unless the equation be expressed by line co-ordi- nates, such as Cartesian, or trilinear co-ordinates; whereas in polar co-ordinates one of the variables is a circular co- ordinate. 268 Examples. Examples. i. Determine the tangents at the origin to the'curve 2/ 2 = x 2 (i — x 2 ). Ans. x + y = o, x - y = o. 2. Show that the curve # 4 - $axy + 2/4 = o touches the axes of co-ordinates at the origin. 3. Find the nature of the origin on the curve x i - ax 2 y + by 3 = o. 4. Show that the origin is a conjugate point on the curve ay 2 — x z + bx 2 = o when a and b have the same sign ; and a node, when they have opposite signs. 5. Show that the origin is a conjugate point on the curve y 2 (x 2 - a 2 ) = a 4 . 6. Prove that the origin is a cusp on the curve 7 . In the curve (y — x 2 ) 2 = x n , show that the origin is a cusp of the first or second species, according as n is < or > 4. 8. Find the numher and the nature of the singular points on the curve a: 4 + ^ax 3 — 2ay 3 + ^arx 2 - ^a 2 y 2 + 4« 4 = o. 9. Show that the points of intersection of the curve &+ (ff with the axes are cusps. 10. Find the douhle points on the curve a; 4 - 4.CIX 3 + 4« 2 # 2 — b~y 2 + 2b 3 y — a 4 - 6 4 = O. Examples. 269 11. Prove that the four tangents from the origin to the curve Mi + u% + W3 = o are represented by the equation 4W1 u% = u\. 12. Show that to a double point on any curve corresponds an other double point, of the same kind, on the inverse curve with respect to any origin. 13. Prove that the origin in the curve a 4 - 2ax 2 y — axy 2 + cfiy 2 = o is a cusp of the second species. 14. Show that the cardioid r = «(i + cos df since -7- = o for the point on the envelope. da dn Consequently, the values of — are the same for the two tlX curves at their common point, and hence they have a common tangent at that point. One or two elementary examples will help to illustrate this theory. The equation x cos a + y sin a = p, in which a is a variable parameter, represents a system of lines situated at the same 272 Envelopes. perpendicular distance p from the origin, and consequently- all touching a circle. This result also follows from the preceding theory ; for we have fix, y, a) = x cos a + y sin a - p = o, df(x, y, a) — rr- — - = - x sin a + y COS a = o, da J ' and, on eliminating a between these equations, we get x 2 + y 2 = p 2 y which agrees with the result stated above. Again, to find the envelope of the line m y = ax + — , a where a is a variable parameter. Here f{x,y, a) = y - ax =0, a df(x, y, a) m - = — x -\ - = o ; da a" jm •*• a = J~~* \ x Substituting this value for a, we get for the envelope y 2 = ^mx, which represents a parabola. 2 1 8. Envelope of La 2 + 2Ma + JSf= o. Suppose L, If, iV, to be known functions of x and y, and a a parameter, then f(x, y, a) = La 2 + 2Ma + N= O, -^ = zLa + 2M= o; da accordingly, the envelope of the curve represented by the preceding expression is the curve LN=M 2 . Undetermined Multipliers applied to Envelopes. 273 Hence, when L, M, JSF are linear functions in x and y, this envelope is a conic touching the lines L, iV, and having M for the chord of contact. Conversely, the equation to any tangent to the conic LN - M % can be written in the form Za 2 + 2Ma + JSr=0* where a is an arbitrary parameter. 219. Undetermined Multipliers applied to Enve- lopes. — In many cases of envelopes the equation of the moving curve is given in the form f(x, y, a, ($) = e l9 (3) where the parameters a, ]8 are connected by an equation of the form (a, P) = C 2 . (4) In this case we may regard j3 in (3) as a function of a by reason of equation (4) ; hence, differentiating both equations, the points of intersection of two consecutive curves must satisfy the two following equations : df df d8 ., d d6 dB 7 + ^7 = 0, and -f + -± -r = o. da dp da da dp da d£ d£ ~ ,, da d(5 Consequently g-^. da d(5 If each of these fractions be equated to the undetermined quantity A, we get da da (5) £JT d$ djd dp * Salmon's Conies, Art. 248. T 274 Envelopes. and the required envelope is obtained by eliminating a, |3, and X between these and the two given equations. The advantage of this method is especially found when the given equations are homogeneous functions in a and |3 ; for suppose them to be of the forms fix, y, a, (5) = d, are connected by the equation - + — = 1, is the parabola ■ Of O 0H*)'- 2. One angle of a triangle is fixed in position, find the envelope of the opposite side when the area is given. Arts. A hyperbola. 3. Find the envelope of a right line when the sum of the squares of the perpendiculars on it from two given points is constant. 4. Find the envelope of a right line, when the rectangle under the perpen- diculars from two given points is constant. Arts. A conic having the two points as foci. 5. From a point F on the hypothenuse of a right-angled triangle, perpen- diculars PM, FIST are drawn to the sides ; find the envelope of the line MN. 6. Find the envelope of the system of circles whose diameters are the chords drawn parallel to the axis-minor of a given ellipse. 7. Find the envelope of the circle x 2 + y 2 - 2aex + a 2 - b 2 = o, where a is an arbitrary parameter ; and find when the contact between the circle and the envelope is real, and when imaginary. (a). Show from this example that the focus of an ellipse may be regarded as an infinitely small circle having double contact with the ellipse, the directrix being the chord joining the points of contact. 8. Show that the envelope of the system of conies £ V 2, _, a a — h where a is a variable parameter, is represented by the equation (x ± y/h) 2 + y 2 = o. Hence show that a system of conies having the same foci may be regarded as inscribed in the same imaginary quadrilateral. 9. Find the envelope of the line xa m + y& m = « m+1 , where the parameters o and /3 are connected by the equation a »» + j3 n = b n . n Am. #»-"» + y n ' m = I -j— 1 Examples. 277 10. On any radius vector of a curve as diameter a circle is described: prove geometrically that the envelope of all such circles is the first pedal of the curve with respect to the origin. 11. If circles be described on the focal radii vectores of a conic as diameters, prove that their envelope is the circle described on the axis major of the conic as diameter. 12. Prove that the envelope of the circles described on the central radii of an ellipse as diameters is a Lemniscate. 13. Find the envelope of semicircles described on the radii of the curve y» = a n cos nd as diameters. 14. If perpendiculars be drawn at each point on a curve to the radii vectores drawn from a given point, prove that their envelope is the reciprocal polar of the inverse of the given curve, with respect to the given point. 15. Find the envelope of a circle whose centre moves along the circum- ference of a fixed circle, and which touches a given right line. 16. Ellipses are described with coincident centre and axes, and having the sum of their semiaxes constant ; find their envelope. 17. Find the equation of the envelope of the line \x + /xy + v = o, where the parameters are connected by the equation ax* + bfi 2 + cv 1 + zffjLV + igvX + ihXfi = o. Ans. x, y, 1, o 18. At any point of a parabola a line is drawn making with the tangent an angle equal to the angle between the tangent and the ordinate at the point ; prove that the envelope of the line is the first negative pedal, with regard to the focus, of the parabola ; and hence that its equation is ri cos - 6 = ai, the focus 3 being pole. N.B. — This curve is the caustic by reflexion for rays perpendicular to the axis of the parabola. 19. Join the centre, 0, of an equilateral hyperbola to any point, P, on the curve, and at P draw a line, PQ, making with the tangent an angle equal to the angle between OP and the tangent. Show that the envelope of PQ is the first negative pedal of the curve r z = 2fl 2 sin - 6 sin - 6, 3 3 the centre being pole, and axis minor prime vector. N.B. — This gives the caustic by reflexion of the equilateral hyperbola, the centre being the radiant point. 20. A right line revolves with a uniform angular velocity, while one of its points moves uniformly along a fixed right line ; find its envelope. Ans. A cycloid. a, h, 9, X h, h f, y ff> f, c> i ( 2 7 8 ) OHAPTEE XVI. CONVEXITY AND CONCAVITY. POINTS OF INFLEXION. 221. Convexity and Concavity. — If the tangent be drawn at any point on a curve, the neighbouring portion of the curve generally lies altogether on one side of the tangent, and is convex with respect to all points lying at the opposite side of that line, and concave for points at the same side. Thus, in the accompanying figure, the portion QPQ is convex towards all points lying below the tangent, and concave for points above. If the curve be referred to the co-ordinate axes OX and OY, then whenever the ordinates of points near to P on the curve are greater than those of the points on N m N the tangent corresponding to lg * 27 ' the same abscissae, the curve is said to be concave towards the positive direction of P. Now, suppose y = (x) to be the equation of the curve, then that of the tangent at a point x, y, by Art. 168, is Let P be the point x, y, and MN =h = MN', QN = y ly TN = Pi, and we have y x = (x + h) = 00) + fy'(«) + — *'» + »"» +&c. Y x = y + h(j/(x) = (j>(x) + h(j)'(x) ; ... y x - Y l = — <}>"(x) + -~^—y\x) + &C. I . 2 1.2.3 (0 Points of Inflexion. 279 When h is very small, the sign of the right-hand side of this equation is the same in general as that of its first term, and accordingly the sign of y x - T Xy or of QT, is the same as that of "(x). Hence, for a point ahove the axis of x, the curve is convex towards that axis when "(x) is positive, and concave when negative. We accordingly see that the convexity or concavity at any point depends on the sign of tf\x) or -7^, at the point. ttx 222. Points of Inflexion. — If, however, "(%) =0 at the point P, we shall have ft - Tx = h 3 2 .3 *"» + U 2.3 .4 tf*{x) + &G. (2) Now, provided 0"'(a?) be not zero, ft - T x changes its sign with h, i.e. if MN' = MN= h, and if Q lies above T, the corresponding point Q' lies below T\ and the portions of the curve near to P lie at opposite sides of the tangent, as in the figure. Consequently, the tangent at such a point cuts the curve, as well as touches it, at its Fi S- 28 - point of contact. Such points on a curve are called points of inflexion. Again, if iy (x) be not zero at the point, y x - T x does not change sign with A, and accordingly the tangent does not intersect the curve at its point of contact. Generally, the tangent does or does not cut the curve at its point of contact, according as the first derived function which does not vanish is of an odd, or of an even order ; as can be easily seen by the preceding method. 2 So Points of Inflexion. From the foregoing discussion it follows that at a point of inflexion the curve changes from convex to concave with respect to the axis of x, or conversely. On this account such points are called points of contrary flexure or of inflexion. 223 The subject of inflexion admits also of being treated by the method of Art. 196, as follows : — The points of in- tersection of the line y = fix + v with the curve y = $(x) are evidently determined by the equation '(xi) = ju, where x x denotes the value of x belonging to the point of contact. Again, three of the roots become equal if we have in addition 0"(#i) = o ; in this case the tangent meets the curve in three consecutive points, and evidently cuts the curve at its point of contact ; for in our figure the portions PA and CD of the curve lie at opposite sides of the cutting line, but when the points A, B, C become coincident, the portions AB and BC become evanescent, and the curve is evidently cut as well as touched by the line. In like manner, if $ m {xi) also vanish, the tangent must be regarded as cutting the curve in four consecutive points : such a point is called a point of undulation. It may be observed, that if a right line cut a continuous branch of a curve in three points, A, B, C, as in our figure, the curve must change from convex to concave, or conversely, between the extreme points A and C, and consequently it must have a point of inflexion between these points ; and so on for additional points of section. Again, the tangent to a curve of the n th degree at a point of inflexion cannot intersect the curve in more than n — 3 other points: for the point of inflexion counts for three among the points of section. For example, the tangent to a curve Harmonic Polar of a Point of Inflexion on a Cubic. 281 of the third degree at a point of inflexion cannot meet the curve in any other point. Consequently, if a point of in- flexion on a cubic be taken as origin, and the tangent at it as axis of x, the equation of the curve must be of the form # 3 + y$ = °t where represents an expression of the second and lower degrees in x and y. For, when y = o, the three roots of the resulting equation in x must be each zero, as the axis of x meets the curve in three points coincident with the origin. The preceding equation is of the form U 3 + U 2 + Ui = o, or, when written in full, x z + y (ax 2 + 2hxy + by 2 ) + y (2gx + 2fy + c) = o. (4) Now, supposing tangents drawn from the origin to the curve, their points of contact, by Art. 176, lie on the curve u z + 2%h = o, i. e. on the curve (gx+fy + c)y*=o. The factor y = o corresponds to the tangent at the point of inflexion, and the other factor gx + fy + c = o passes through the points of contact of the three other tangents to the curve. Hence, we infer that from a 'point of inflexion on a cubic but three tangents can be drawn to the curve, and their three points of contact lie in a right line. It can be shown that this right line cuts harmonically every radius vector of the curve which passes through the point of inflexion. For, transforming equation (4) to polar co-ordinates, and dividing by r 9 it becomes of the form Ar 2 + Br + C = o. If /, /' be the roots of this quadratic, we have r + r" ~ C 282 ft Points of Inflexion. Now, if p be the harmonic mean between / and /', this gives 2 _ i i B 2g cos 9 + 2/ sin 9 p r r (J c Hence the equation of the locus of the extremities of the harmonic means is gx+fy + c = o. Q.E.D. This theorem is due to Maclaurin {Be Lin. Geom. Prop. Gen., Sec. in. Prop. 9). From this property the line is called the harmonic polar of the point of inflexion. This line holds a fundamental place in the general theory of cubics.* 224. Stationary Tangents. — Since the tangent at a point of inflexion may be regarded as meeting the curve in three consecutive points, it follows that at such a point the tangent does not alter its position as its point of contact passes to the consecutive point, and hence the tangent in this case is called a stationary tangent. d 2 y The equation — = o follows immediately from the last U'X consideration ; for when the tangent is stationary we must have -j- = o, where 0, as in Art. 171, denotes the angle ax du which the tangent makes with the axis of x ; but tan = — , ax hence -~ = o, which is the same condition for a point of ax inflexion as that before arrived at. * Chasles, Apergu Historique, note xx. ; Salmon's Higher Plane Curves, Art. 179. Examples. 283 Examples. 1. Show that the origin is a point of inflexion on the cuwe a 3 y = bxy + cx s + dx*. 2. The origin is a point of inflexion on the cubic #3 + u\ = o ? 3. In the curve a m ' x y = x m , prove that the origin is a point of inflexion if m be greater than 2. 4. In the system of curves y n = kx m , find under what circumstances the origin is (a) a point of inflexion, (b) a cusp. 5. Find the co-ordinates of the point of inflexion on the curve 2P x s — 3&s 2 + a 2 y = o. Ans. x =» b, y = —. Of 6. If a curve of an odd degree has a centre, prove that it is a point of inflexion on the curve. 7. Prove that the origin is a point of undulation on the curve U\ + Ui + «5 + &c, + u n = o. 8. Show that the points of inflexion on curves referred to polar co-ordinates are determined by aid of the equation dHt , 1 « + -r-r = o, where w = -. dd z r 9. In the curve rd m = a, prove that there is a point of inflexion when —^/m (1 — m). 10. In the curve y = c sin -, prove that the points in which the curve meets the axis of x are all points of inflexion. 11. Show geometrically that to a node on any curve corresponds a line touching its reciprocal polar in two distinct points ; and to a cusp corresponds a point of inflexion. 284 Examples. 12. If the origin be a point of inflexion on the curve U\ + u% + uz + . . . + u n = o, prove that u% must contain u\ as a factor. 13. Show that the points of inflexion of the cubical parabola y 2 = (x - a) 2 (x - b) lie on the line 3# + a = 4b : and hence prove that if the cubic has a node, it has no real point of inflexion ; but if it has a conjugate point, it has two real points of inflexion, besides that at infinity. 14. Prove that the points of inflexion on the curve y 2 = x 2 (x 2 + 2px + q) are determined by the equation 2x 3 4 6px 2 + 3 (p 2 4 q) x + 2pq = o. 15. If y 2 = f{%) be the equation of a curve, prove that the abscissae of its points of inflexion satisfy the equation {/'W} 2 = */(*) ./''(*)• 16. Show that the maximum and minimum ordinates of the curve 2/ = 2/(^r(^)-{/'^)} 2 correspond to the points of intersection of the curve y 2 =f(z) with the axis of a\ 17. When y 2 =f(%) represents a cubic, prove that the biquadratic in x which determines its points of inflexion has one, and but one, pair of real roots. Prove also that the lesser of these roots corresponds to no real point of inflexion, while the greater corresponds, in general, to two. 1 8. Prove that the point of inflexion of the cubic ay z + ^bxy 2 + T>cx 2 y + dx % + 30a; 2 = o lies in the right line ay + bx = o, and has for its co-ordinates ■\a 2 e _ xabe *-- -^-,and2/= — , where G is the same as in Example 32, p. 190. 19. Find the nature of the double point of the curve y 2 = {x - 2) 2 (x - 5), and show that the curve has two real points of inflexion, and that they subtend a right angle at the double point. 20. The co-ordinates of a point on a curve are given in terms of an angle 6 by the equations x = sec 3 6, y = tan 9 sec 2 6 ; prove that there are two finite points of inflexion on the curve, and find the values of 6 at these points. ( 285 ) CHAPTEE XVII. RADIUS OF CURVATURE. E VOLUTES. CONTACT. RADII OF CURVATURE AT A DOUBLE POINT. 225. Curvature. Angle of Contingence. — Every con- tinuous curve is regarded as having a determinate curvature at each point, this curvature being greater or less according as the curve deviates more or less rapidly from the tangent at the point. The total curvature of an arc of a plane curve is measured by the angle through which it is bent between its extremities — that is, by the external angle between the tangents at these points, assuming that the arc in question has no point of in- flexion on it. This angle is called the angle of contingence pf the arc. The curvature of a circle is evidently the same at each of its points. To compare the curvatures of different circles, let the arcs AB and ab of two circles be of equal length, then the total curvatures of these arcs are measured by the angles between their tangents, or by the angles ACB and acb at F . their centres : but lg * 3 °* .-,_ , aroAB arc«5 1 1 LACB: Lacb = — 777-:-- = -t^ : ""' AC ac AC ac Consequently, the curvatures of the two circles are to each other inversely as their radii; or the curvature of a circle varies inversely as its radius. Also if As represent any arc of a circle of radius r, and A0 the angle between the tangents at its extremities, we have As r = -— . 286 Radius of Curvature. The curvature of a curve at any point is found by deter- mining the circle which has the same curvature as that of an indefinitely small elementary arc of the curve taken at the point. 226. Radius of Curvature. — Let ds denote an infi- nitely small element of a curve at a point, d(p the corresponding ds angle of contingence expressed in circular measure, then — d d 2 y or sec 2 ^-^- = -t^ 2 . dx dx 7, d6 dd> dx d dy d 2 y ds dx* dx 7. At a point of inflexion -~ = o : accordingly the radius of ax curvature at such a point is infinite : this is otherwise evident since the tangent in this case meets the curve in three conse- cutive points. (Art. 222.) ^ Again, as the expression f 1 + \-j-\ ) has always two values, the one positive and the other negative, while the Expressions for Radius of Curvature. 287 curve can have in general but one definite circle of curvature at any point, it is necessary to agree upon which sign is to be taken. We shall adopt the positive sign, and regard p as d fj being positive when — \ is positive ; i. e. when the curve is ax convex at the point with respect to the axis of x. 227. Other Expressions for p. — It is easy to obtain other forms of expression for the radius of curvature ; thus by Art. 178 we have dx . dy Hence, if the arc be regarded as the independent variable, we get d d 2 y - sin 6 -f = -=— . cos 6 -f = -7^, y ds ds*' r ds ds 2> from which, if we squaf e and add, we obtain P » \ /Vo I \ /Vo* / \ Wo' / V / ds J \ds 2 J \ds Again, the equations dx = cos (pds, dy = sin ), P (ds) 2 (dsY d 2 x = cos (pd 2 s - sin $- — — , d 2 y = sin $d 2 s + cos - — -. (3) Whence, squaring and adding, we obtain (d 2 x) 2 + (d 2 y) 2 = (d 2 s) 2 + @£, P ds 2 9 , and \ve have <£c = - « sin d 2 x = — a cos , and re- garding

a cos 2 <£ sin d 2 (2 sin 2

, d 2 y = $a sin

— sin 2 (p), whence ( cos , and hence deduce an expression for the radius of curvature at any point on the curve. 2. In a parabola referred to its focus as pole, prove that p = m sec co, and hence show that p = 2m sec 3 a. 237. Evolutes and Involutes. — If the centre of cur- vature for each point on a curve be ■p l Vo taken, we get a new curve called the evolute of the original one. Also, the original curve, when considered with respect to its evolute, is called an in- volute. To investigate the connexion be- tween these curves, let P l9 P 2 , -Ps, &c., represent a series of infinitely near points on a curve; C X9 C 2 , C z , &c, the corresponding centres of curvature, then the lines P^C i9 P 2 C 2 , PsC 3 , &c, are normals to the curve, and the lines OiC 2 , C 2 C 39 (7 3 C4,&c.,mayberegardedin the limit as consecutive elements of the evolute ; also, since "2 p 2q8 Radius of Curvature. each of the normals P X C X , PzC2,PzCz,&o., passes through two consecutive points on the evolute, they are tangents to that curve in the limit. Again, if p x , p 2 , p 3 , p i} &c, denote the lengths of the radii of curvature at the points P x , P 2 , P3, Pi, &c, we have p x = P X C X , p 2 = PA pz = PzC s , p 4 = P4O4, &o. ; .*. p x — p% = P X X — x 2C2 = PiL> x — P2C2 = Cj.G 2 ; alSO p%— pz = C 2 C 3 , pz — p± = C364, . . . p n _! — p n = Cn- X Cn ', hence by addition we have P\ ~ Pn — GiG 2 + 6263 + C/3C/4 + . . . + (7 w _i t7«. This result still holds when the number n is increased indefinitely, and we infer that the length of any are of the evolute is equal, in general, to the difference between the radii of curvature at its extremities. It is evident that the curve may be generated from its evolute by the motion of the extremity of a stretched thread, supposed to be wound round the evolute and afterwards unrolled ; in this case each point on the string will describe a different involute of the curve. The names evolute and involute are given in consequence of the preceding property. It follows, also, that while a curve has but one evolute, it can have an infinite number of involutes ; for we may regard each point on the stretched string as generating a separate involute. The curves described by two different points on the moving line are said to be parallel; each being got from the other by cutting off a constant length on its normal measured from the curve. 238. E volutes regarded as Envelopes. — From the preceding it also follows that the determination of the evolute of a curve is the same as the finding the envelope of all its normals. "We have already, in Ex. 3, Art. 219, investigated the equation of the evolute of an ellipse from this point of view. 239. Evolute of a Parabola. — "We proceed to deter- mine the evolute of the parabola in the same manner. Evolute of Ellipse. 299 Let the equation of the curve be y 2 = 2 ma?, then that of its normal at a point (%, y) is m or [Y-y)- + X-x = o, is y z + 2my (m - X) - 2m 2 Y = o. The envelope of this line, where y is regarded as an arbi- trary parameter, is got by eliminating y between this equa- tion and its derived equation $y 2 + 2m (m - X) = o. Accordingly, the equation of the required envelope is obtained by Y instead of y substituting 2 m X Fig. 34. in the latter equation. Hence, we get for the required evolute, the semi-cubical parabola 2jmY 2 = 8 ( X - m) 3 . The form of this evolute is exhi- bited in the annexed figure, where VN=m = 2VF. ^ If P, P', repre- sent the points of intersection of the evolute with the curve, it is easily seen that VM = 4VJST = 4m. 240. Evolute of an Ellipse. — The form of the evolute of an ellipse, when e is greater than -J-v 2, is exhibited in the accompanying figure ; the points M, iV, H ', N', are evidently cusps on the curve, and are the centres of cur- vature corresponding to the four vertices of the ellipse. In general, if a curve be symmetrical at both sides of a point on it, the oscu- lating circle cannot intersect 3°° Radius of Curvature. the curve at the point ; accordingly, the radius of curvature is a maximum or a minimum at such a point, and the corre- sponding point on the evolute is a cusp. It can be easily seen geometrically that through any point four real normals, or only two, can be drawn to an ellipse, according as the point is inside or outside the evolute. It may be here observed that to a point of inflexion on any curve corresponds plainly an asymptote to its evolute. 241. Evolute of an Equiangular Spiral. — We shall next consider the equiangular or logarithmic spiral, r = a . Let P and Q be two points on the curve, its pole, PC, QC 'the normals at P and Q; join OC. Then by the fundamental property of the curve (Art. 181), the angles OP C and OQC are equal, and consequently the four points, 0, P, Q, C, lie on a circle : hence L QOC = L QPC; but in the limit when P and Q are coin- -p. 6 cident, the angle QPC becomes a right angle, and C becomes the centre of curvature belong- ing to the point P ; hence POC also becomes a right angle, and the point C is immediately determined. Again, L OCP = L OQP ; but, in the limit, the angle OQP is constant; .*. L OCP is also constant ; and since the line CP is a tangent to the evolute at C, it follows that the tangent makes a constant angle with the radius vector OC. From this property it follows that the evolute in question is another logarithmic spiral. Again, as the constant angle is the same for the curve and for its evolute, it follows that the latter curve is the same spiral turned round through a known angle (whose circular measure is log a M) . 241 (a). Involute of a Circle. — As an example of involutes, suppose APQ to represent a portion of an involute of the circle BA C, whose centre is 0. Let OC = a, L CO A = 0, and CA the length of the string unrolled ; then CP= CA = a. Points of Inflexion in Polar Co-ordinates. 301 Draw ON perpendicular to the tangent at P, and let ON = p, then we have *? •*. + \ p = a$. D^ Hence, since z£Oi\r= z C; from which it is easily seen that aft s = 242. Radius of Curvature, and Points of In- flexion, in Polar Co-ordinates. — "We shall first find an expression for p in terms of u (the reciprocal of the radius vector) and 0. By Article 1 8$ we have 1 _ fdu\" hence Also P 1 dp dhc p 3 du d6 z ' dr 1 du P = r T~ = "17' dp w dp 302 Radius of Curvature. consequently P (u + ^- 2 J = — 3 = j i + {-^ - fdu\ 2 ^ 1 + \u~dOJ (15) i , du i dr Again, since u = -, we have ^ = -^ ^, rf 2 « _ 2 /<&A 2 I rfV and rffl 5 = r 3 \rf9j~r 2 5^ ; p "- — rfv — 73*' (l6) This result can also be established in another manner, as follows: — On reference to the figure of Art. 1 8o, it is obvious that Q = + \p ; where is the angle the tangent at P makes with the prime vector OX. d6 d\p dd> ds d\L Hence dO =I+ T9> ° T ds-dd = I+ T9> dip i dtp dO p ds ds dd dr d r Again, denoting -^ and -^ by / and r", we have r tan \b = — ; and hence '2 Mn Jf M f 1 ,v,J f dd> , , r 2 -rr r 2 -n dd r f* + r* 1+ f ^-rf^r\ ^ds dd r 2 + r 2 dO Intrinsic Equation of a Curve. 303 9 ff /9 * r - rr + zr Hence, we get p = Or, replacing / and /' by their values, r%+i % 2\i d 2 r fdr\ 2 Again, since p = 00 at a point of inflexion, we infer that the points of intersection of the curve represented by the equation ^- r w +2 W ~ with the original curve, determine in general its points of inflexion. In some cases the points of inflexion can be easier found by aid of (15), which gives, when p = 00, d z u Examples. 1. Find the radius of curyature at any point in the spiral of Archimedes, (1 + e»)i r = ad. Ans. a 2 + 2 2. Find the radius of curvature of the logarithmic spiral r = a . Ans. r (1 + (loga) 2 )=. 3. Find the points of inflexion on the curve 9 r = 29 — 11 cos 26. Ans. cos 20 = — . 1 1 4. Prove that the circle r = 10 intersects the curve r = 11 — 2 cos 50 in its points of inflexion. 5. Prove that the curve r = a + b cos nd has no real points of inflexion unless a is > b and < (1 + w 2 ) b. "When a lies be- tween these limits, prove that all the points of inflexion lie on a circle ; and show how to determine the radius of the circle. 304 Radius of Curvature. 2\2 (a). Intrinsic Equation of a Curve. — In many cases the equation of a curve is most simply expressed in terms of the length, s, of the curve, measured from a fixed point on it, and the angle, 0, through which it is bent, i. e. the angle of deviation of the tangent at any point from the tangent at the fixed point, taken as origin. These are styled the intrinsic elements of the curve by Dr. Whewell,* to whom this method of discussing curves is due. The relation between the length s and the deviation for any curve is called its intrinsic equation. If this relation be represented by the equation • -/(*). then if p be the radius of curvature at any point, we have » = | =/'<*>• Also, if s y denote the length of the evolute, from Art. 237 it is easily seen that the equation of the evolute is of the form Si =f'(. Again, from Art. 241(a), the intrinsic equation of the involute of a circle is reducible to the form ad) 2 s = — — . 2 We shall meet with further examples of intrinsic equa- tions subsequently. 243. Contact of Different Orders. — As already stated, the tangent to a curve has a contact of the first order with the curve at its point of contact, and the osculating circle a contact of the second order. We now proceed to distinguish more fully the different orders of contact between two curves. * Cambridge Philosophical Transactions, Vols. viii. and ix. Contact of Different Orders. 305 Suppose the curves to be represented by the equations y=f(x), and y = 0(a?), and that x x is the abscissa of a point common to both curves, then we have f(x x ) -#'(*i). Again, substituting x x + h, instead of x in both equations, and supposing y x and y% the corresponding ordinates of the two curves, we have 7 2 yz = (x x + h) = (j> (xi) + h$'(x x ) + $"(^0 + & c - Subtracting, we get tfi - v> = * l/fa) - *' W ) + ^ IT M - #"(*0 ) + &o. (17) Now, suppose /'(#i) = 0'(^O> or that the curves have a common tangent at the point, then * - * - r-, {/'W - ♦* W ) + ^rr (TW - *">.) }+ &o. 1.2 1.2.3 In this case the curves have a contact of the first order ; and when h is small, the difference between the ordinates is a small quantity of the second order, and as y x - y 2 does not change sign with h, the curves do not cross each other at the point. If, in addition /»(«) = *>>), then y, - y, = {/">0 - f"(*i) I + &o. 1.2.3 In this case the difference between the ordinates is an in- finitely small magnitude of the third order when h is taken an infinitely small magnitude of the first; the curves are then said to have a contact of the second order ', and approach infinitely nearer to each other at the point of contact than in v 306 Radius of Curvature. the former case. Moreover, since y x - y 2 changes its sign with h, the curves cut each other at the point as well as touch. If we have in addition f"(x\) = 0'"(#i), the curves are said to have a contact of the third order : and, in general, if all the derived functions, up to the n th inclusive, be the same for both curves when x = x X} the curves have a contact of the n th or( j erj an( j we have Vl - y, = ^— {/(»«) (*,) - ^) («0 j + &e. (i 8) n + i Also, if the contact be of an even order, n + i is odd, and consequently h n+l changes its sign with h, and hence the curves cut eac other at their point of contact ; for whichever is the lower at one side of the point becomes the upper at the other side. If the curves have a contact of an odd order, they do not cut each other at their point of contact. From the preceding discussion the following results are immediately deduced : — (i) . If two curves have a contact of the n th order, no curve having with either of them a contact of a lower order can fall between the curves near their point of contact. (2). Two curves which have a contact of the n th order at a point are infinitely closer to one another near that point than two curves having a contact of an order lower than the n th . (3). If any number of curves have a contact of the second order at a point, they have the same osculating circle at the point. 244. Application to Circle. — It can be easily verified that the circle which has a contact of the second order with a curve at a point is the same as the osculating circle determined by the former method. For, let (X-a) 2 + (r-/3) 2 = i2 2 be the equation of a circle having contact of the second order at the point (x, y) with a given curve ; then, by the preceding, the values of — and -7-f must be the same for the circle and ax dx~ for the curve at the point in question. Application to Circle. 307 Differentiating the equation of the circle twice, and sub- stituting x and y for X and T, we get and x - a + {y - /3) ^ = o, \dx 1 + Hence y - )3 = - \dx = or < o. Moreover, if a = o, one root of the quadratic (30) is in- finite, and the other is -7^. The origin in this case is a double 2/3 cusp, and is also a point of inflexion on one branch. Such a point is called a point of oscul-infiexion by Cramer. If j3 = o in addition to a = o, the origin is a cusp of the first species, but having the radii of curvature infinite for both branches. It is easy to see from other considerations that the radii of curvature at a cusp of the first species are always either zero or infinite. For, since the two branches of the curve in this case u 1] turn their convexities in opposite directions, — -- must have ax opposite signs at both sides of the cusp, and consequently it must change its sign at that point ; but this can happen only in its passage through zero, or through infinity. It should be observed that the preceding discussion applies to the case of a curve referred to oblique axes of co-ordinates, provided that we substitute y instead of p ; where 7 is half the chord intercepted on the axis of y by the osculating circle at the origin. 253. Recapitulation. — The conclusions arrived at in the two preceding Articles may be briefly stated as follows : — (1). Whenever the equation of a curve can be transformed into the shape y 1 = ax 2, + terms of the third and higher degrees, the origin is a cusp of the first species ; both radii of curva- ture being zero at the point. (2). When the coefficient of x 2 vanishes,* the origin is * In this case, if v\ be the equation of the tangent at the cusp, the equation of the curve is of the form Vl 2 + V1V2 -f- v± + &c. = o. This is also evident from geometrical considerations. General Investigation of Cusps. 315 generally either a double cusp, a conjugate point, or a cusp of the second species. In the latter case the two branches of the curve have the same centre of curvature, and conse- quently have a contact of the second order with each other. (3). If the lowest term in x (independent of y) be of the 5 th degree, the origin is a point of oscul-inflexion. If, however, the coefficient of x 2 y also vanish, the origin is not only a cusp of the first species, but also a point of inflexion on both branches of the curve. 254. Creneral Investigation off Cusps. — The pre- ceding results admit of being established in a somewhat more general manner as follows : — By the method already given, the equation which deter- mines the form of an algebraic curve near to a cusp may be written in the following general shape : y 2 = iAx a y + JBx b + Cx°, (32) where 2Ax a is the lowest term in the coefficient of y, and Bx b , Cx c * are the lowest terms independent of y. By hypothesis, a, b, c are positive integers, and a > 1, b > 2, c > 3 ; now, solving for y, we obtain y = Ax a ± = or < b. (1). Let 2a = b + h, then b + h b y = Ax T ± x 2 2 : m being assumed to be greater 2 than unity. 10. Two plane closed curves have the same evolute : what is the difference between their perimeters ? Ans. 2ird, where d is the distance between the curves. 1 1 . Find the radius of curvature at the origin in the curve 3^ = 4#- i5# 2 -3a: 3 : find also at what points the radius of curvature is infinite. 12. Apply the principles of investigating maxima and minima to find the greatest and least distances of a point from a given curve ; and show that the problem is solved by drawing the normals to the curve from the given point. (a). Prove that the distance is a minimum, if the given point be nearer to the curve than the corresponding centre of curvature, and a maximum if it be lurther. 320 Examples, (b). If the given point be on the evolute, show that the solution arrived at is neither a maximum nor a minimum ; and hence show that the circle of curva- ture cuts as well as touches the curve at its point of contact. 13. Find an expression for the whole length of the evolute of an ellipse. a 3 - P Ans. 4 — . ab 14. Find the radii of curvature at the origin of the two branches of the curve c a x i —- ax 2 y — axy 2 + a 2 y 2 = o. Ans. a and -. 2 4 15. Prove that the evolute of the hypocycloid #§ + y% = «l is the hypocycloid (a + j8)i + (a - jB)l = Mi. 16. Find the radius of curvature at any point on the curve y + v x ( l — ®) — si 11 " 1 V x ' 17. If the angle between the radius vector and the normal to a curve has a maximum or a minimum value, prove that 7 = r ; where 7 is the semi-chord of curvature which passes through the origin. 18. If the co-ordinates of a point on a curve be given by the equations x = c sin 20(i + cos 20), y = e cos 20 (1 - cos 20), find the radius of curvature at the point. Ans. 4c cos 30 19. Show that the evolute of the curve r 2 - a 2 = mp 2 has for its equation r 2 — (1 - m) a 2 — mp 2 . 20. If a and j8 be the co-ordinates of the point on the evolute corresponding to the point (x, y) on a curve, prove that dy da dx dfi 21. If p be the radius of curvature at any point on a curve, prove that the radius of curvature at the corresponding point in the evolute is -7- ; where a> aco is the angle the radius of curvature makes with a fixed line. 22. In a curve, prove that 1 d (dy\ p dx \ds J ' Examples. 321 23. Find the equation of the evolute of an ellipse by means of the eccentric angle. 24. Prove that the determination of the equation of the evolute of the curve y = Jcx n reduces to the elimination of % between the equations n - 2 Fn 2 . . , . 2» - 1 i a = g a 2 "" 1 , and £ = #*» + n -i n-i ' w-i &«(*»- i)# M -* 25. In figure, Art. 239, if the tangent to the evolute at P meet the parabola in a point 2Z, prove that EN is perpendicular to the axis of the parabola. 26. If on the tangent at each point on a curve a constant length measured from the point of contact be taken, prove that the normal to the locus of the points so found passes through the centre of curvature of the proposed curve. 27. In general, if through each point of a curve a line of given length be drawn making a constant angle with the normal, the normal to the curve locus of the extremities of this line passes through the centre of curvature of the pro- posed. (Eertrand, Cal. Dif., p. 573.) This and the preceding theorem can be immediately established from geome- trical considerations. 28. If from the points of a curve perpendiculars be drawn to one of its tan- gents, and through the foot of each a line be drawn in a fixed direction, pro- portional to the length of the corresponding perpendicular ; the locus of the extremity of this line is a curve touching the proposed at their common point. Find the ratio of the radii of curvature of the curves at this point. 29. Find an expression for the radius of curvature in the curve p = \/ m 2 — r 2 ' p being the perpendicular on the tangent. 30. Being given any curve and its osculating circle at a point, prove that the portion of a parallel to their common tangent intercepted between the two curves is a small quantity of the second order, when the distances of the point of contact from the two points of intersection are of the first order. Prove that, under the same circumstances, the intercept on a line drawn parallel to the common normal is a small quantity of the third order. 31. In a curve referred to polar co-ordinates, if the origin be taken on the curve, with the tangent at the origin as prime vector, prove that the radius of r curvature at the origin is equal to one-half the value of - in the limit. d 32. Hence find the length of the radius of curvature at the origin in the A n(i curve r = a sin nd. Am. p = — 2 33. Find the co-ordinates of the centre of curvature of the catenary ; and show that the radius of curvature is equal, but opposite, to the normal. 34. If p, p' be the radii of curvature of a curve and of its pedal at corre- sponding points, show that p'(2r 2 —pp) =r 3 . Ind. Civ. Ser. Exam.y 1878. Y ( 3 2 2 ) CHAPTER XVIII. ON TRACING OF CURVES. 257. Tracing Algebraic Curves. — Before concluding the discussion of curves, it seems desirable to give a brief state- ment of the mode of tracing curves from their equations. The usual method in the case of algebraic curves consists in assigning a series of different values to one of the co-ordi- nates, and calculating the corresponding series of values of the other ; thus determining a definite number of points on the curve. By drawing a curve or curves of continuous cur- vature through these points, we are enabled to form a tolerably accurate idea of the shape of the curve under discussion. In curves of degrees beyond the second, the preceding process generally involves the solution of equations beyond the second degree : in such cases we can determine the series of points only approximately. 258. The following are the principal circumstances to be attended to : — (1). Observe whether from its equation the curve is sym- metrical with respect to either axis; or whether it can be made so by a transformation of axes. (2). Find the points in which the curve is met by the co-ordinate axes. (3). De- termine the positions of the asymptotes, if any, and at which side of an asymptote the corresponding branches lie. (4). De- termine the double points, or multiple points of higher orders, if any belong to the curve, and find the tangents at such points by the method of Art. 212. (5). The existence of ovals can be often found by determining for what values of either co-ordinate the other becomes imaginary. (6). If the curve has a multiple point, its tracing is usually simplified by taking that point as origin, and transforming to polar co-or- dinates : by assigning a series of values to 6 we can usually determine the corresponding values of r 9 &c. (7). The points On Tracing of Curves. 323 where the y ordinate is a maximum or a minimum are found du from the equation ~ = o : by this means the limits of the (XX curve can be often assigned. (8). Determine when possible the points of inflexion on the curve. 259. To trace the Curve y 2 = x 2 (x - a) ; a being sup- posed positive. In this case the origin is a conjugate point, and the curve cuts the axis of # at a distance OA = a. Again, when x is less than a, y is imaginary, consequently no portion of the curve lies to the left-hand side of A. The points of inflexion, I ' and I\ are easily determined d 2 y , Fi S- 38. from the equation -7-f = o ; the CLX corresponding value of x is — ; accordingly AN . Again, if TI be the tangent at the point of inflexion 7, it a AN can readily be seen that TA = — = . J 9 3. This curve has been already considered in Art. 213, and is a cubical parabola having a conjugate point. 260. Cubic with three Asymptotes. — We shall next consider the curve* y 2 x + ey = ax % + bx 2 + ex + d, (1) where a is supposed positive. The axis of y is an asymptote to the curve (Art. 200), and the directions of the two other asymptotes are given by the equation y 2 - ax 2 = o, or y = ± x \/a. * This investigation is principally taken from Newton's Enumeratio Zi- nearum Tertii Ordinis. Y 2 324 On Tracing of Curves. If the term bx z be wanting, these lines are asymptotes ; if b be not zero, we get for the equation of the asymptotes s- b ,- b y = x*/a + — — , y + x*/a + — — = o. 2\/a 2y a On multiplying the equations of the three asymptotes together, and subtracting the product from the equation of the curve, we get b* ey = (c ) x + d : this is the equation of the right line which passes through the three points in which the cubic meets its asymptotes. (Art. 204.) Again, if we multiply the proposed equation by x, and solve for xy, we get xy el 6 2 = — ± lax 4, + bx z + ex 2 + dx + - : (2) 2 \ 4 from which a series of points can be determined on the curve corresponding to any assigned series of values for x. It also follows that all chords drawn parallel to the axis e of y are bisected by the hyperbola xy + - = o : hence we infer that the middle points of all chords drawn parallel to an asymptote of the cubic lie on a hyperbola. The form of the curve depends on the roots of the bi- quadratic under the radical sign. (1). Suppose these roots to be all real, and denoted by a, j3, 7, 8, arranged in order of increasing magnitude, and we have xy = ± a/ a (x - a) (x - /3) (x - y)(x - §). Now when x is < a, y is real ; when x > a and < /3, y is imaginary ; when x > |3 and < 7, y is real ; when x> 7 and < d, y is imaginary ; when x> S, y is real. Asymptotes. 325 We infer that the curve consists of three branches, extending to infinity, together with an oval lying between the values (3 and 7 for x. The accompany- ing figure* repre- sents such a curve. Again, if either the two greatest roots or the two least roots become equal, the corres- ponding point be- comes a node. If the interme- diate roots become Fi g- 39- equal, the oval shrinks into a conjugate point on the curve. If three roots be equal, the corresponding point is a cusp. If two of the roots be impossible and the other two un- equal, the curve can have neither an oval nor a double point. If the sign of a be negative, the curve has but one real asymptote. 261. Asymptotes. — In the preceding figure the student will observe that to each asymptote correspond two infinite branches ; this is a general property of algebraic curves, of which we have a familiar instance in the common hyperbola. By the student who is acquainted with the elementary principles of conical projection the preceding will be readily apprehended ; for if we suppose any line drawn cutting a closed oval curve in two points at which tangents are drawn, and if the figure be so projected that the intersecting line is sent to infinity, then the tangents will be projected into asymptotes, and the oval becomes a curve in two portions, each having two infinite branches, a pair for each asymptote, as in the hyperbola. * The figure is a tracing of the curve yxy 1 + ic% = (x - 5) (x - II) (x - 12). 326 On Tracing of Curves. It should also be observed that the points of contact at infinity on the asymptote in the opposite directions along it must be regarded as being one and the same point, since they are the projection of the same point. That the points at infinity in the two opposite directions on any line must be regarded as a single point is also evident from the considera- tion that a right line is the limiting state of a circle of in- finite radius. The property admits also of an analytical proof; for if the asymptote be taken as the axis of x, the equation of the curve (Art. 204) is of the form y§\ + (j>2 = o, 02 Or y = - XL, 0i where 2 is at least one degree lower than

= ^ = 2 ^ = 48sm 5 = 2POs (5) or the radius of curvature is double the normal. From this value of p the evolute of the curve can be easily determined. For, produce PO until OP' = OP, then P r is the centre of curvature be- longing to the point P. Again, produce LO until 00' = OL, and describe a circle through 0, P / and 0' ; this circle evidently touches A A', and is equal to the generating circle LPO. Fig. 51. Also, the arc OP' = arc OP = AO; .-. arc O'P' = O'P'O - P'O = AD - AO = OB = P'0\ Hence the locus of P' is the cycloid got by the rolling of this new circle along the line B'G\ and accordingly the evo- lute of a cycloid is another cycloid. It is evident that the evolute of the cycloid ABA! is made up of the two semi- cycloids, AB' and B'A, as in figure 51. Conversely, the cycloid ABA' is an involute of the cycloid AB'A'. The position of the centre of curvature for a point P on a cycloid can also be readily de- termined geometrically, as fol- lows : — Suppose Oi a point on the circle infinitely near to 0, and take 00* = 00*, z LstP' 3} 8 Roulettes. be the centre of curvature required, and draw POl and P'0 2 . Now suppose the circle to roll until O x and 2 coincide, then C0 2 becomes perpendicular to AD, and PO x and P'0 2 will lie in directum (since P' is the point of intersection of two consecutive normals to the cycloid) . Hence z OCO, = L PO x Q = l OPO, + l oro 1 , since each side of the equation represents the angle through which the circle has turned. But L OCO l = 2 l OP Ol (Euclid, III. 20.) Hence lOPO, = l OP , 1 ; .-. P0 1 = P'O l] and consequently in the limit we have PO = P'O, as before. "We shall subsequently see that a similar method enables us to determine the centre of curvature for a point in any roulette. 276. length of Arc of Cycloid. — Since AP'B' (Fig. 51) is the evolute of the cycloid APB, it follows, from Art. 2 3 7, that the arc AP" of the cycloid is equal in length to the line PP\ or to twice P'O ; hence, as A is the highest point in the cycloid AP'B', it follows that the arc AP' measured from the highest point of a cycloid is double the intercept P'O, made on the tangent at the point by the tangent at the highest point of the curve. Hence, denoting the length of the arc AP' by s, we have s = 40 sin P'OD = 4# sin 0. (6) This gives the intrinsic equation of the cycloid (see Art. 24.2(a)). Hence, also, the whole arc AB' is four times the radius of the generating circle : and accordingly the entire length ABA' of a cycloid is eight times the radius of its generating circle. Again, if the distance of P' from AA! be represented by y, we shall have P'O 2 = 00' xy = lay. Hence s 2 = $FO~ = Say. fj) Epicycloids and Hypocycloids. 33Q Fig. S3- Their forms are exhibited in This relation is of importance in the applications of the cycloid in Mechanics. Again, since AO = arc OP\ if we represent AO by v, we have* v = 2a. (8) 277. Trochoids. — In general, if a circle roll on a right line, any point in the. plane of the circle carried round* with it describes a curve. Such curves are usually styled tro- choids. "When the tracing point is inside the circle, the locus is called a prolate tro- choid ; when outside, an oblate, the accompanying figure. Their equations are easily determined; for, let x, y be the co-ordinates of a tracing point P, referred to the axes AD, and AI (A being the position for which the moving radius CP is perpendicular to the fixed line). Then, if CO = a, CP = d, L OCP = 0, we have x = AN= AO- OJST=aO-d sin 0, ) (9) y = Pi\T = a -d cos 0. ) 278. Epicycloidsf and Hypocycloids. — The investi- * This is called, by Professor Casey, the tangential equation of the cycloid, and by aid of it he has arrived at some remarkable properties of the curve (" On a New Form of Tangential Equation," Philosophical Transactions, 1877). "In general, if a variable line, in any of its positions, make an intercept v on the axis of x, and an angle

, such as v =/(*), will be the tangential equation of a curve, which is the envelope of the line." For applications, the reader is referred to Professor Casey's Memoir. See also Dub. Exam. Papers, Graves, Lloyd Exhibition, 1847. f I have in this edition adopted the correct definition of these curves as given by Mr. Proctor in his Geometry of Cycloids. I have thus avoided the anomaly existing in the ordinary definition, according to which every epicycloid Z 2 340 Roulettes. gation of the properties of the cycloid naturally gave rise to the discussion of the more general case of a circle rolling on a fixed circle. In this case the curve generated by any point on the circumference of the rolling circle is called an epicycloid, or a hypocycloid, according as the rolling circle touches the outside, or the inside of the circumference of the fixed circle. We shall commence with the former case. Let P be the position of the generating point at any in- stant, A its position when on the fixed circle ; then the arc OA = arc OP. Again, let C and (7 be the centres of the circles, a and b their radii, £ACO = 0, lOC'P=&\ then, since arc OA = arc OP, we have aO = bO. Now, suppose C taken as the origin of rectangu- lar co-ordinates, and CA as the axis of x; draw PN and G f L perpendicular, Fi s- 54. and PM parallel, to CA, and we have x=CN=CL-JSfL =(a+b) cos - 6 cos (0 + 0'), y = PJSr=C'L- C'M= (a + b)smO-b sin (0 + 0'); a or, substituting - for 0', x = {a + b) cos - b cos —7— 0, a + b > y = (a + b) sin - b sin — =— 0. b J (10) is a hypocycloid, but only some hypoeycloids are epicycloids. While according to the correct definition no epicycloid is a hypocycloid, though each can he gene- rated in two ways, as will be proved in Art. 280. Epicycloids and Hype-cycloids. 341 "When the radius of the rolling circle is a submultiple of that of the fixed circle, the tracing point, after the circle has rolled once round the circumference of the fixed circle, evidently returns to the same position, and will trace the same curve in the next revolution. More generally, if the radii of the circles have a commensurable ratio, the tracing point, after a certain number of revolutions, will return to its original position : but if the ratio be incommensurable, the point will never return to the same position, but will describe an infinite series of distinct arcs. As, however, the suc- cessive portions of the curve are in every respect equal to each other, the path described by the tracing point, from the position in which it leaves the fixed circle until it returns to it again, is often taken instead of the complete epicycloid, and the middle point of this path is called the vertex of the curve. In the case of the hypocycloid, the generating circle rolls on the interior of the fixed circle, and it can be easily seen that the expressions for x and y are derived from those in (10) by changing the sign of b ; hence we have x = (a - b) cos + b cos — j~- 0, a-b > (II) y = (a - b) sin - b sin —7— 6. The properties of these curves are best investigated by aid of the simultaneous equations contained in formulas (10) and (11). It should be observed that the point A, in Fig. 54, is a cusp on the epicycloid ; and, generally, every point in which the tracing point P meets the fixed circle is a cusp on the roulette. From this it follows that if the radius of the rolling circle be the n th part of that of the fixed, the corresponding epi- or hypo-cycloid has n cusps : such curves are, accordingly, designated by the number of their cusps : such as the three- cusped, four-cusped, &c. epi- or hypo-cycloids. Again, as in the case of the cycloid, it is evident from Descartes' principle that the instantaneous path of the point P is an elementary portion of a circle having as centre ; ac- 34 2 Roulettes. cordingly, the tangent to the path at P is perpendicular to the line PO, and that line is the normal to the curve at P. These results can also be deduced, as in the case of the cycloid, by differentiation from the expressions for x and y. We leave this as an exercise for the student. To find an expression for an element ds of the curve at the point P; take 0', 0" ', two points infinitely near to on the circles, and such that 00' = 00"\ and suppose the gene- rating circle to roll until these points coincide :* then the lines CO and CO" will lie in directum, and the circle will have turned through an angle equal to the sum of the angles OCO f and OC'O f, \ hence, denoting these angles by dO and d&, respectively, we have ds = OP (dO + d&) = opfi + |) <*0; (12) since dO' = T dO. o 279. Radius of Curvature of an Epicycloid. — Suppose u) to be the angle OSN between the normal at P and the fixed line CA, then o»= (?OS-C'CS = ----0', .'. da, = -d0\i+^r 22 [ 20 Hence, if p be the radius of curvature corresponding to the point P, we get ,. * 0P ^>. (I3) r d(s) a + 2b Accordingly, the radius of curvature in an epicycloid is in a constant ratio to the chord OP, joining the generating point to the point of contact of the circles. * It may be observed that O'O" is infinitely small in comparison with 00' ; bence tbe space through which the point moves during a small displacement is infinitely small in comparison with the space through which Pmoves. It is in consequence of this property that may be regarded as being at rest for the instant, and every point connected with the rolling circle as having a circular motion around it. Double Generation of Epicycloids and Hypocycloids. 343 Fig. 55* 280. Double Generation of Epicycloids and Hypo- cycloids. — In an Epicycloid, it can be easily shown that the curve can be generated in a second manner. For, suppose the rolling circle in- closes the fixed circle, and join P, any position of the tracing point, to 0, the correspond- ing point of contact of the two circles; draw the diameter OED, and join O'E and PD ; connect C, the centre of the fixed circle, to O, and produce CO' to meet DP produced in D', and describe a circle round the triangle OPD'; this circle plainly touches the fixed circle ; also the segments standing on OP, OP, and 00 are obviously similar ; hence, since OP = 00' + O'P, we have arc OP = arc 00' + arc OP. If the arc 00' A be taken equal to the arc OP, we have arc Of A = arc OP ; accordingly, the point P describes the same curve, whether we regard it as on the circumference of the circle OPD rolling on the circle OOE, or on the circumference of OPD' rolling on the same circle ; provided the circles each start from the position in which the generating point coincides with the point A. Moreover, it is evident that the radius of the latter circle is the difference between the radii of the other two. Next, for the Hypocycloid, suppose the circle OPD to roll inside the circumference of OOE, and let C be the centre of the fixed circle ; join OP, and pro- duce it to meet the circum- ference of the fixed circle in O ; draw O'E and PD, join CO, intersecting PD in D', and de- scribe a circle round the triangle PD'O. It is evident, as be- fore, that this circle touches the Fig. 56. 344 Roulettes. larger circle, and that its radius is equal to the difference be- tween the radii of the two given circles. Also, for the same reason as in the former case, we have ar c 00' = arc OP + arc O'P. If the arc OA be taken equal to OP, we get are O'P = arc O'A ; consequently, the point P will describe the same hypocycloid on whichever circle we suppose it to be situated, provided the circles each set out from the position for which P coincides with A. The particular case, when the radius of the rolling circle is half that of the fixed circle, may be noticed. In this case the point D coincides with C, and P becomes the middle point of 00', and A that of the arc 00'. From this it follows im- mediately that the hypocycloid described by P becomes the diameter CA of the fixed circle. This result will be proved otherwise in Art. 285. The important results of this Article were given by Euler [Acta.Petrop., 1781). By aid of them all epicycloids can be generated by the rolling of a circle outside another circle; and all hypocycloids by the rolling of a circle whose radius is less than half that of the fixed circle. 281. Evolute of an Epicycloid. — The evolute of an epicycloid can be easily seen to be a similar epi- cycloid. For, let P be the trac- ing point in any position, A its position when on the fixed circle ; join P to 0, the point of contact of the circles, and produce PO .., -r^, ^^2a + 2b until PP f = OP T , a + 20 then P is the centre of curvature by (13) ; hence a or =op a + 20 Fig. 57. Next, draw P'O' perpendicular to P'O; circumscribe the Evolute of Epicycloid. 345 triangle OP'O' by a circle ; and describe a circle with C as centre, and CO' as radius : it evidently touches the circle OP'O'. Then 00' : OE = OP' : OP = a: a + 2b = CO : CE; .\ CO-00':CE--OE = CO:CE, or CO' :CO=CO : G# ; that is, the lines CE, CO, and CO' are in geometrical pro- portion. Again, join C to B', the vertex of the epicycloid ; let CB' meet the inner circle in D, and we have arc 0'B:slvgOB= CO': C0= CO : CE = O'O :EO = arc P'O': arc OQ. But arc OB = arc OQ : .-. arc 07) = arc P'C Accordingly, the path described by P' is that generated by a point on the circumference of the circle OP'O' rolling on the inner circle, and starting when P' is in contact at D. Hence the evolute of the original epicycloid is another epicycloid. The form of the evolute is exhibited in the figure. Again, since CO : OE = CO' : O'O, the ratio of the radii of the fixed and generating circles is the same for both epicy- cloids, and consequently the evolute is a similar epicycloid. Also, from the theory of evolutes (Art. 237), the line PP r is equal in length to the arc P'A of the interior epicy- cloid ; or the length of P'A, the arc measured from the vertex A of the curve, is equal to 2 J^3op' = 2 op'^ = 2or^ a Ur 2Ur CO Ur CO'' Hence, the length* of any portion of the curve measured from its vertex is to the corresponding chord of the generating circle as twice the sum of the radii of the circles to the radius of the fixed circle. * The length of the arc of an epicycloid, as also the investigation of its evolute, were given hy Newton (Principia, Lib. 1., Props. 49, 50): 346 Roulettes. "With reference to the outer epicycloid in Fig. 57, this gives arc P& = 2PE . ccr CO' (i4) The corresponding results for the hypocycloid can be found by changing the sign of the radius b of the rolling circle in the preceding formulse. The investigation of the properties of these curves is of importance in connexion with the proper form of toothed wheels in machinery. 282. Pedal of Epicycloid. — The equation of the pedal, with respect to the centre of the fixed circle, admits of a very simple expression. For let P be the generating point, and, as be- fore, take arc OA = arc OP, and make AB = 90 . Join CA, CB, CP, and draw CN perpendicular to DP. Let lPBO = $,l BCN = = (th) a + 2b CN= CBsin = (a + 2b) sin 7 . (18) u v J a + 2b v 7 The corresponding formulae for the hypocycloid are obtained by changing the sign of b in the preceding equa- tions. Again, it is plain that the envelope of the right line re- presented by equation (18) is an epicycloid. And, in general, the envelope of the right line x cos ay + y sin id = Jc sin mw, regarding w as an arbitrary parameter, is an epicycloid, or a hypocycloid, according as m is less or greater than unity. For examples of this method of determining the equations of epi- and hypo-cycloids the student is referred to Salmon's Higher Plane Curves, Art. 310. 284. JEpitrochoids and Hypotrochoids. — In general, when one circle rolls on another, every point connected with the rolling circle describes a distinct curve. These curves are called epitrochoids or hypotrochoids, according as the rolling circle touches the exterior or the interior of the fixed circle. If d be the constant distance of the generating point from the centre of the rolling circle, there is no difficulty in proving, as in Art. 278, that we have in the epitrochoid the equations x = (a + b) cos - d cos — - — 0, a + b > (19) y = (a + b) sin 9 - d sin — - — 9. 348 Roulettes. In the case of the hypotrochoid, changing the signs of b and d, we obtain x _ _ a - b n ~\ = (a- b) cos 6 + d cos —r~ u, . a — b „ y = (a - b) sm 6 - d sm — - — U. \ (20) J In the particular case in which a = 2b, i.e. when a circle rolls inside another of double its diameter, equations (20) become x = (b + d) cos 6, y = (b - d) sin ; and accordingly the equation of the roulette is f X' + - {b + dy (b-d) = 1 ; which represents an ellipse whose semi-axes are the sum and the difference of b and d. This result can also be established geometrically in the following manner : — 285. Circle rolling inside another of double its Diameter. — Join C x and to any point L on the circumference of the rolling circle, and let C y L meet the fixed circumference in A ; then since L OCL = 20C X A, and OC x = 2OC, we have arc OA = arc OL ; and, accord- ingly, as the inner circle rolls on the outer the point L moves along C X A. In like manner any other point on the circumference of the rolling circle describes, during the motion, a dia- meter of the fixed circle. Again, any point P, invariably connected with the rolling circle, describes an ellipse. For, if L and M be the points in which CP cuts the rolling circle, by what has been just shown, these points move along two fixed right lines C X A and &B, at right angles to each other. Accordingly, by a Epitrochoids and Hypotrochoids. 349 well-known property of the ellipse, any other point in the line LM describes an ellipse. The ease in which the outer circle rolls on the inner is also worthy of separate consideration. 286. Circle rolling on another inside it and of naif its Diameter. — In this case, any diameter of the rolling circle always passes through a fixed point, which lies on the circumference of the inner circle. For, let CiL and CJL be any two positions of the moving diameter, C x and 2 being the corresponding positions of the centre of the rolling circle : and 2 the corresponding posi- tions of the point of contact of the circles. Now, if the outer circle roll from the former to the latter position, the right lines (7i# 2 and C0 2 will coincide in direction, and accordingly the outer circle will have turned through the angle C2O2C1; consequently, the mov- ing diameter will have turned through the same angle ; and hence L CJLCx = lC z O % Ci', therefore the point L lies on the fixed circle, and the diameter always passes through the same point on this circle. Again, any right line connected with the rolling circle will alicays touch a fixed circle. For, let BE be the moving line in any position, and draw the parallel diameter AB\ let fall CiF and LM perpendicular to BE. Then, by the preceding, AB always passes through a fixed point L ; also LM = C X F= constant ; hence BE always touches a circle having its centre at L. Again, to find the roulette described by any carried point Pi. The right line PiC h as has been shown, always passes through a fixed point L ; consequently, since CJ? X is a con- stant length, the locus of Pi is a Limagon (Art. 269). In like manner, any other point invariably connected with the outer circle describes a Limacon ; unless the point be situated on the circumference of the rolling circle, in which case the locus becomes a cardioid. Fig. 60. 350 Examples — Roulettes. i. "When the radii of the fixed and the rolling circles hecome equal, prove geometrically that the epicycloid becomes a cardioid, and the epitrochoid a Limacon (Art. 269). 2. Prove that the equation of the reciprocal polar of an epicycloid, with respect to the fixed circle, is of the form r sin ma = const. 3. Prove that the radius of curvature of an epicycloid varies as the perpen- dicular on the tangent from the centre of the fixed circle. 4. If a = 45, prove that the equation of the hypocycloid becomes #§ + y% = flf . 5. Find the equation, in terms of r and^?, of the three-cusped hypocycloid ; i. e. "when a = 3b. Ans. r 2 = a 2 — Sp 2 . 6. Find the equation of the pedal in the same curve. Ans. p = b sin 3a. 7. In the case of a curve rolling on another which is equal to it in every respect, corresponding points being in contact, prove that the determination of the roulette of any point P is immediately reduced to finding the pedal of the rolling curve with respect to the point P. 8. Hence, if the curves be equal parabolas, show that the path of the focus is a right line, and that of the vertex a cissoid. 9. In like manner, if the curves be equal ellipses, show that the path of the focus is a circle, and that of any point is a bicircular quartic. 10. In Art. 285, prove that the locus of the foci of the ellipses described by the different points on any right line is an equilateral hyperbola. ir. A is a fixed point on the circumference of a circle ; the points L and M are taken such that arc AL — m arc AM, where m is a constant ; prove that the envelope of LM is an epicycloid or a hypocycloid, according as the arcs AL and AM axe measured in the same or opposite directions from the point A. 12. Prove that LM, in the case of an epicycloid, is divided internally in the ratio m : 1, at its point of contact with the envelope ; and, in the hypocycloid, externally in the same ratio. 13. Show also that the given circle is circumscribed to, or inscribed in, the envelope, according as it is an epicycloid or hypocycloid. 14. Prove, from equation (14), that the intrinsic equation of an epicycloid is ±b (a + b) . ad> s = sin a + 2b where s is measured from the vertex of the curve. 15. Hence the equation s — I sin nd> represents an epicycloid or a hypo- cycloid, according as n is less or greater than unity. Centre of Curvature of an Epitrochoid or Hypotrochoid. 35 16. In an epitrochoid, if the distance, d, of the moving point from the centre of the rolling circle be equal to the distance between the centres of the circles, prove that the polar equation of the locus becomes r = 2 (« + b) cos ad a + 2b 17. Hence show that the curve r = a sin mO is an epitrochoid when m < 1, and a hypotrochoid when m > 1. This class of curves was elaborately treated of by the Abbe Grandi in the Philosophical Transactions for 1723. He gave them the name of " Rhodonese," from a fancied resemblance to the petals of roses. See also Gregory's Examples on the Differential and Integral Calculus, p. 183. For illustrations of the beauty and variety of form of these curves, as well as of epitrochoids and hypotrochoids in general, the student is referred to the admi- rable figures in Mr. Proctor's Geometry of Cycloids. 287. Centre Mypotrochoid.- of Curvature of an Epitrochoid or -The position of the centre of curvature for any point of an epitrochoid can be easily- found from geometrical considerations. For, let Ci and C 2 be the centres of the rolling and the fixed circles, P 2 the centre of cur- vature of the roulette described by Pi ; and, as before, let O x and 2 be two points on the circles, infinitely near to 0, such that 00 x = 00 2 . Now, suppose the circle to roll until Ox and 2 coincide; then the lines C X X and C 2 2 will lie in directum, as also the lines P1O1 and P 2 2 (since P 2 is the point Fig. 61. of intersection of two consecutive normals to the roulette). Hence L 0C X X + L OCA = L 0P X X + L OP 2 2 , since each of these sums represents the angle through which the circle has turned. Again, let z C x OP x = 0, 00 x = 00 2 = ds ; then *Oftft-*p <«MV-^. (JJr i C/P2 35 2 Roulettes. consequently we have m + m' m *[m+Gp} (2I) Or, if OP x = r 1} OP 2 = r 2 , ii (\ i - + T = COS d> — + — « 6 r \n r 2 From this, equation r 2 , and consequently the radius of curva- ture of the roulette, can be obtained for any position of the generating point Pi. If we suppose Pi to be on the circumference of the rolling OP circle, we get cos = * ; whence it follows that 2 (J 1 OP, = — °—- OP l9 a + 2b which agrees with the result arrived at in Art. 279. 288. Centre of Curvature of any Roulette. — The preceding formula can be readily extended to any roulette : for if (7i and C 2 be respectively the centres of curvature of the rolling and fixed curves, corresponding to the point of contact 0, we may regard OOi and 00 2 as elementary arcs of the circles of curvature, and the preceding demonstration will still hold. Hence, denoting the radii of curvature 0& and 0C 2 by pi and p 2 , we shall have — + — = cos & I — + - ]. (22) Pi p% r \ri r 2 J It can be easily seen, without drawing a separate figure, that we must change the sign of p 2 in this formula when the centres of curvature lie at the same side of 0. It may be noted that P 1 is the centre of curvature of the roulette described by the point P 2 , if the lower curve be sup- posed to roll on the upper regarded as fixed. 289. Greometrical Construction* for the Centre of * This beautiful construction, and also the formula (22) on which it is based, were given by M. Savary, in his Lemons des Machines a V Ecole Poly technique. See also Leroy's Geome'trie Descriptive, Quatrieme Edition, p. 347. Construction for Centre of Curvature. 353 Curvature of a Roulette. — The formula (22) leads to a simple and elegant construction for the centre of curva- ture P 2 . We commence with the case when the base is a right line, as represented in the accom- panying figure. Join P, to Ci, the centre of curva- ture of the rolling curve, and draw OiV" perpendicular to OP„ meeting P,C, in JSf; through iV draw NM parallel to OC ly and the point P 2 in which it meets OP, is the centre of curvature required. For, equation (22) becomes in this case A oc 1 = cos 1'\oF> + op> whence we get PiP 2 __ OP,. OP, JSTP, JVC, . OP, ' OC, sin C, ON NC, sin C,NO P 1 P 2= NP 1 " op, jyc, 9 and, accordingly, the line NP, is parallel to OC,. Q. E. D. The construction in the general case is as follows : — Determine the point N as in the former case, and join it to C 2 , the centre of curva- ture of the fixed curve, then the point of intersection of NC, and P,0 is the required centre of curvature. This is readily established ; for, from the equation + JL_ 00B ^_L_ + _L. C,C, COS (f>P,P , oc, we get oc.oc, (he* OP^ OC, 'P,P, OP,. OP, OC, cos $ op, • 2 A Fig. 63. 354 But, as before, 00, cos

OE,' Hence, if the tracing point P, lie on the circle* OE,B„ * This theorem is due to La Hire, who showed that the element of the roulette traced hy any point is convex or concave with respect to the point of contact, 0, according as the tracing point is inside or outside this circle. (See Envelope of a Carried Curve. 355 the corresponding value of OP 2 is infinite, and consequently Pi is a point of inflexion on the roulette. In consequence of this property, the circle in question is called the circle of inflexions, as each point on it is a point of inflexion on the roulette which it describes. Again, it can be shown that the lines PiP 2 , P\0 and P x E x are in continued proportion ; as also C X C 2 , Ci0 9 and (7iA- For, from (23) we have P.P. 1 ? OP x . OP 2 OE x ' Hence P X P 2 :P 1 0= OP,: OE, .'. P 1 P 2 :P 1 = P,P 2 - OP 2 : P,0 - OE x = P x O : P 1 E 1 . (24) In the same manner it can be shown that C 1 C 2 :C l O = &O: Oft. (25) In the particular case where the base is a right line, the circle of inflexions becomes the circle described on the radius of curvature of the rolling curve as diameter. Again, if we take OD 2 = OD ly we shall have, by describing a circle on OD 2 as diameter, 2 Oi 1 Cs 2 (J = (J 2 (-s '. \j 2 jJ 2 5 and also P 2 P X : P 2 = P 2 : P 2 E 2 . (26) The importance of these results will be shown further on. 291. Envelope of a Carried Curve. — We shall next consider the envelope of a curve invariably connected with the rolling curve, and carried with it in its motion. Since the moving curve touches its envelope in each of its Memoires de V 'Academie des Sciences, 1706.) It is strange that this remarkable result remained almost unnoticed until recent years, when it was found to contain a key to the theory of curvature for roulettes, as well as for the envelopes of any carried curves. How little it is even as yet appreciated in this country will he apparent to any one who studies the most recent produc- tions on roulettes, even by distinguished British Mathematicians. 2 A 2 356 Roulettes. positions, the path of its point of contact at any instant must be tangential to the envelope ; hence the normal at their common point must pass through 0, the point of contact of the fixed and rolling curves. In the particular case in which the carried curve is a right line, its point of contact with its envelope is found by dropping a perpendicular on it from the point of contact 0. For example, suppose a circle to roll on any curve : to find the envelope* of any diameter PQ : — From draw OJV perpendicular to PQ, then JV, by the preceding, is Fi 65 a point on the envelope. On OC describe a semicircle; it will pass through N, and, as in Art. 286, the arc ON = arc OP = OA, if A be the point in which P was originally in contact with the fixed curve. Consequently, the envelope in question is the roulette traced by a point on the circumference of a circle of half the radius of the rolling circle, having the fixed curve AO for its base. For instance, if a circle roll on a right line, the envelope of any diameter is a cycloid, the radius of whose generating circle is half that of the rolling circle. Again, if a circle roll on another, the envelope of any diameter of the rolling circle is an epicycloid, or a hypocycloid. Moreover, it is obvious that if two carried right lines be parallel, their envelopes will be parallel curves. For ex- ample, the envelope of any right line, carried by a circle which rolls on a right line, is a parallel to a cycloid, i.e. the involute of a cycloid. These results admit of being stated in a somewhat different form, as follows : If one point, A, in a plane area move uniformly along a right line, while the area turns uniformly in its own plane, then the envelope of any carried right line is an involute to a cycloid. If the carried line passes through the moving point * The theorems of this Article are, I believe, due to Chasles : see bis Sistoire de La Geometrie, p. 69. Centre of Curvature of the Envelope of a Carried Curve. 357 A, its envelope is a cycloid. Again, if the point A move uniformly on the circumference of a fixed circle, while the area revolves uniformly, the envelope of any carried right line is an involute to either an epi- or hypo-cycloid. If the carried right line passes through A, its envelope is either an epi- or hypo-cycloid. 292. Centre of Curvature of the Envelope of a Carried Curve. — Let aj>i represent a portion of the carried curve, to which Om is normal at the point m ; then, by the preceding, m is the point of contact of a^b x with its envelope. Now, suppose a 2 b 2 to represent a por- tion of the envelope, and let P x be the centre of curvature of a } b lr for the point m, and P 2 the corresponding centre of cur- vature of a 2 b 2 . As before, take O x and 2 such that 00, = 00 2 , and join P x O x and P 2 2 . Again, suppose the curve to roll until Oi and 0% coincide; then the lines P1O1 and P 2 2 will come in directum, as also the lines OyC, and O z C 2 ; and, as in Art. 288, we shall have Fig. 66. z(7 1 + z<7 2 = zP 1 + lP 2 \ and consequently oa + m = oos * \m + m)- (27) From this equation the centre of curvature of the enve- lope, for any position, can be found. Moreover, it is obvious that the geometrical constructions of Arts. 289, 290, equally apply in this case. It may be remarked that these construc- tions hold in all cases, whatever be the directions of curvature of the curves. The case where the moving curve a x b x is a right line is worthy of especial notice. 358 Roulettes. In this case the normal Om is perpendicular to the moving line ; and, since the point Pi is infinitely distant, we have COS0 I ~oK = oa, + ok = ok (Art - 29 ° ); Fig. 67. whence, P 3 is situated on the lower circle of inflexions. Hence we infer that the dif- ferent centres of curvature of the curves en- veloped by all carried right lines, at any instant, lie on the circumference of a circle. As an example, suppose the right line OM to roll on a fixed circle, whose centre is C 2 , to find the envelope of any carried right line, LM. In this case the centre of cur- vature, P 3 , of the envelope of LM, lies, by the preceding, on the circle described on OC as diameter; and, accordingly, CP 2 is perpendicular to the normal PiP 2 . Hence, since L OLP x remains constant during the motion, the line CP 2 is of constant length ; and, if we describe a circle with C as centre, and CP 2 as radius, the envelope of the moving line LM will, in all positions, be an involute of a circle. The same reasoning applies to any other moving right line. We shall conclude with the statement of one or two other important particular cases of the general principle of this Article. (1). If the envelope a % b 2 of the moving curve a x b x be a right line, the centre of curvature P x lies on the corresponding circle of inflexions. (2). If the moving right line always passes through a fixed point, that point lies on the circle OD 2 E 2 . 2*92 (a). Expression for Radius of Curvature of Envelope of a Right Line. — The following expression for the radius of curvature of the envelope of a moving right Fig. 68. On the Motion of a Plane Figure in its Plane. 359 line is sometimes useful. Let p be the perpendicular distance of the moving line, in any position, from a fixed point in the plane, and w the angle that this perpendicular makes with a fixed line in the plane, and p the radius of curvature of the envelope at the point of contact; then, by Art. 206, we have p = P+—- (28) Whenever the conditions of the problem give^> in terms of u) (the angle through which the figure has turned), the value of p can be found from this equation. For example, the re- sult established in last Article (see Fig. 68) can be easily deduced from (28). This is left as an exercise for the student. 293. ©n the Motion of a Plane Figure in its Plane. — We shall now proceed to the consideration of a general method, due to Chasles, which is of fundamental importance in the treatment of roulettes, as also in the general investi- gation of the motion of a rigid body. We shall commence with the following theorem : — When an invariable plane figure moves in its plane, it can be brought from any one position to any other by a single rotation round a fixed point in its plane. For, let A and B be two points of the figure in its first position, and A x , B x their new positions after a displacement. Join AAi and BB X , and sup- pose the perpendiculars drawn at the middle points of AA X and BB X to intersect at ; then we have AO = A x O, and BO = B y O. Also, since the triangles AOB and A x OB x have their sides respectively 1S " 9 ' equal, we have A AOB = lA x OB x ; .\ /. AOA x = lBOB x . Accordingly, AB will be brought to the position A X B X by a rotation through the angle AOA x round 0. Consequently, any point C in the plane, which is rigidly connected with AB, will be brought from its original to its new position, C l9 by the same rotation. This latter result can also be proved otherwise thus : — Join OC and 0C X ; then the triangles OAC and OA x C x are equal, 360 Roulettes. "because OA = 0A X , AC = A x d, and the angle OAC, being the difference between OAB and BAC, is equal to OA x C h the difference between 0A X B X and B X A X C X ; therefore 0(7 = 00 l5 and lAOC=Z.A 1 OC 1 ; and hence ziOi! = L COd. Consequently the point C is brought to C x by a rotation round through the same angle A OA x . The same reasoning applies to any other point invariably connected with A and B. The preceding construction re- quires modification when the lines AA X and BB X are parallel. In this case the point, 0, of intersection of the lines BA and B X A X is easily seen to be the point of instantaneous rotation. For, since AB = A x B Xy and AA X , Fi ?0 BB X , are parallel, we have OA = 0A X , and 05 = Oi?i. Hence, the figure will be brought from its old to its new position by a rotation around through the angle AOA x . Next, let AA X) and BB X be both equal and parallel. In this case the point is at an infinite distance ; but it is obvious that each point in the plane moves through the same distance, equal and parallel to AA X ; and the motion is one of simple translation, without any rotation. In general if we suppose the two positions of the moving figure to be indefinitely near each other, then the line AA X , joining two infinitely near positions of the same point of the figure, becomes an element of the curve described by that point, and the line OA becomes at the same time a normal to the curve. Hence, the normals to the paths described by all the points of the moving figure pass through 0, which point is called the instan- taneous centre of rotation. The position of is determined whenever the directions of motion of any two points of the moving figure are known ; for it is the intersection of the normals to the curves described by those points. This furnishes a geometrical method of drawing tangents to many curves, as was observed by Chasles.* * This method is given by Chasles as a generalization of the method of Des- cartes (Art. 273, note). It is itself a particular case of a more general principle concerning homologous figures. See Chasles, Sistoirc de la Geometrie, pp. 54S-9 : also Bulletin Universel des Sciences, 1830. Chasles' Method of drawing Normals. 361 The following case is deserving of special consideration : — A right line always passes through a fixed point, while one of its points moves along a fixed line : to find the instantaneous centre of rotation. Let A be the fixed point, and AB any position of the moving line, and take B 'A! = BA ; then the centre of rotation, 0, is found as before, and is such that OA = OA', and OB = OB'. Accordingly, in the limit the centre of instantaneous rotation is the inter- section of BO drawn perpendicular to the fixed line, and AO drawn perpendicular to the moving line at the fixed point. In general, if ABhe any moving curve, and LM any fixed curve, the instantaneous centre of rotation is the point of inter- section of the normals to the fixed and to the moving curves, for any position. Also the normal to the curve described by any point in- variably connected with AB is obtained by joining the point to 0, the instantaneous centre. More generally, if a moving curve always touches a fixed curve A, while one point on the moving curve moves along a second fixed curve P, the instantaneous centre is the point of intersection of the normals to A and B at the corresponding points; and the line joining this centre to any describing point is normal to the path which it describes. We shall illustrate this method of drawing tangents by applying it to the conchoid and the limacon. 294. Application to Curves. — In the Conchoid (Fig. 49, P a g e 33 2 ) ? regarding AP as a moving right line, the instantaneous centre is the point of intersection of AO drawn perpendicular to AP, with BO drawn perpendicular to LM; and consequently, OP and OP x are the normals at P and Pi, respectively. For the same reason, the normal to the Limacon (Fig. 48, page 331) at any point Pis got by drawing OQ perpendicular to OP to meet the circle in Q, and joining PQ. 362 Roulettes. Examples. 1. If the radius vector, OP, drawn from the origin to any point P on a curve, be produced to Pi, until PPi be a constant length ; prove that the normal at Pi to the locus of Pi, the normal at P to the original curve, and the perpendicular at the origin to the line OP, all pass through the same point. 2. If a constant length measured from the curve be taken on the normals along a given curve, prove that these lines are also normals to the new curve which is the locus of their extremities. 3. An angle of constant magnitude moves in such a manner that its sides constantly touch a given plane curve ; prove that the normal to the curve de- scribed by its vertex, P, is got by joining Pto the centre of the circle passing through P and the points in which the sides of the moveable angle touch the given curve. 4. If on the tangent at each point on a curve a constant length measured from the point of contact be taken, prove that the normal to the locus of the points so found passes through the centre of curvature of the proposed curve. 5. In general, if through each point of a curve a line of given length be drawn making a constant angle with the normal, the normal to the curve locus of the extremities of this line passes through the centre of curvature of the pro- posed. 295. Motion of any Plane Fignre reduced to Roulettes. — Again, the most general motion of any figure in its plane may be regarded as consisting of a number of infinitely small rotations about the different instantaneous centres taken in succession. Let 0, 0', 0", 0'", &c, represent the successive centres of rotation, and consider the instant when / --t the figure turns through the angle X 00' o- /"''' 3 round the point 0. This rotation will ,/>-'' T 2 bring a certain point Oi of the figure to 9i*sz.~ coincide with the next centre 0'. The next ""^^^ T ^ rotation takes place around 0'; and suppose o \>-... ^ the point O z brought to coincide with the o\ centre of rotation 0' ' . In like manner, by n'V t' a third rotation the point 3 is brought to 1 \ coincide with 0"', and so on. By this ' > T » means the motion of the moveable figure F - 2 is equivalent to the rolling of the polygon 00i0 2 0z . . . invariably connected with the figure, on the polygon 00'0"0'" . . . fixed in the plane. In the limit, the polygons change into curves, of which one rolls, without Epicyclics. 363 Fig. 73- sliding, on the other ; and hence we conclude that the general movement of any plane figure in its own plane is equivalent to the rolling of one curve on another fixed curve. These curves are called by Beuleaux* the " centrodes" of the moving figures. For example, suppose two points A and B of the moving figure to slide along two fixed right lines CX and CY; then the instan- taneous centre is the point of inter- section of AO and BO, drawn perpen- dicular to the fixed lines. Moreover, as AB is a constant length, and the angle ACB is fixed, the length CO is constant ; consequently the locus of the instantaneous centre is the circle described with C as centre, and CO as radius. Again, if we describe a circle round CBOA, this circle is invariably connected with the line AB, and moves with it. Hence the motion of any figure invariably connected with AB is equivalent to the rolling of a circle inside another of double its radius (see Art. 285). Again, if we consider the angle XCY to move so that its legs pass through the fixed points A and B, respectively ; then the instantaneous centre is determined as before. More- over, the circle BOA becomes & fixed circle, along which the instantaneous centre moves. Also, since CO is of constant length, the outer circle becomes in this case the rolling curve. Hence the motion of any figure invariably connected with the moving lines CX and CY is equivalent to the rolling of the outer circle on the inner (compare Art. 286). 295 (a). Epicyclics. — As a further example, suppose one point in a plane area to move uniformly along the circum- ference of a fixed circle, while the area revolves with a uniform angular motion around the point, to find the position of the " centrodes." The directions of motion are indicated by the arrow heads. Let C be the centre of the fixed circle, P the position * See Kennedy's translation of Reuleaux's Kinematics of Machinery, pp. 65, &c. 364 Roulettes. of the moving point at any instant, Q a point in the moving figure such that CP = PQ. Now, to find the position of the instantaneous centre of rotations it is necessary to get the direction of motion of the point Q. Let Pi represent a con- secutive position of P, then the simultaneous position of Q is got by first supposing it to move through the infinitely small length QR, equal and parallel to PPi, and then to turn round P x , through the angle RP x Qi, which the area turns through while P moves to Pi. Moreover, by hypo- Fig. 74- thesis, the angles PCPi and RPiQi are in a constant ratio : if this ratio be denoted by m, we have (since PQ = PC) RQ X = mPP 1 = mQR. Join Q and Q l9 then QQ X represents the direction of mo- tion of Q. Hence the right line QO, drawn perpendicular to QQi, intersects CP in the instantaneous centre of rotation. Again, since the directions of PO, PQ, and QO are, re- spectively, perpendicular to QR, RQi, and QQi, the triangles QPO and QiRQ are similar; .-. PQ = mPO, i.e. CP = mPO. Accordingly, the instantaneous centre of rotation is got by cutting off (29) m Hence, if we describe two circles, one with centre C and radius CO, the other with centre P and radius PO; these circles are the required centrodes ; and the motion is equivalent to the rolling of the outer circle on the inner. Epicyclics. 365 Accordingly, any point on the circumference of the outer circle describes an epicycloid, and any point not on this cir- cumference describes an epitrochoid. When the angular motion of PQ is less than that of CP, i.e. when m< 1, the point lies in PC produced. Accordingly, in this case, the fixed circle lies inside the rolling circle ; and the curves traced by any point are still either epitrochoids or epi- cycloids. In the preceding we have supposed that the angular rotations take place in the same direction. If we suppose them to be in opposite directions, the construction has to be modified, as in the accompanying figure. In this case, the angle E r%p?% RPxQx must be measured in an opposite direction to that of PCPi ; and, proceeding as in the former case, the direc- tion of motion of Q is repre- sented by QQi; accordingly, the perpendicular QO will in- tersect CP produced, and, as before, we have PO = PC m Fig. 75. Hence the motion is equi- valent to the rolling of a circle ©f radius PO on the inside of a fixed circle, whose radius is CO. Accordingly, in this case, the path described by any point in the moving area not on the circumference of the rolling circle is a hypotrochoid. Also, from Art. 291, it is plain that the envelope of any right line which passes through the point P in the moving area is an epicycloid in the former case, and a hypocycloid in the latter. Again, if we suppose the point P, instead of moving in a circle, to move uniformly in a right line, the path of any point in the moving area becomes either a trochoid or a cycloid. Curves traced as above, that is, by a point which moves 366 Roulettes. uniformly round the circumference of a circle, whose centre moves uniformly on the circumference of a fixed circle in the same plane, are called epicyclics, and were invented by Ptolemy (about a.d. 140) for the purpose of explaining the planetary motions. In this system* the fixed circle is called the deferent, and that in which the tracing point moves is called the epicycle. The motion in the fixed circle may be supposed in all cases to take place in the same direction around C, that indicated by the arrows in our figures. Such motion is called direct. The case for which the motion in the epicycle is direct is exhibited in Fig. 74. Angular motion in the reverse direction is called retro- grade. This case is exhibited in Fig. 75. The corresponding epicyclics are called by Ptolemy direct and retrograde epicy- clics. The preceding investigation shows that every direct epi- cyclic is an epitrochoid, and every retrograde epicyclic a hypotrochoid. It is obvious that the greatest distance in an epicyclic from the centre C is equal to the sum of the radii of the circles, and the least to their difference. Such points on the epicyclic are called apocentres and pericentres, respectively. Again, if a represent the radius of the fixed circle or deferent, and j3 the radius of the revolving circle or epicycle ; then, if the curve be referred to rectangular axes, that of x passing through an apocentre, it is easily seen that we have for a direct epicyclic x = a cos + |3 cos mO, ) y = a sin t) + p sin mu. ) * The importance of the epicyclic method of Ptolemy, in representing ap- proximately the planetary paths relative to the earth at rest, has recently heen Drought prominently forward hy Mr. Proctor, to whose work on the Geometry of Cycloids the student is referred for fuller information on the subject. We owe also to Mr. Proctor the remark that the invention of cycloids, epi- cycloids, and epitrochoids, is properly attributable to Ptolemy and the ancient astronomers, who, in their treatment of epicyclics, first investigated some of the properties of such curves. It may, however, be doubted if Ptolemy had any idea of the shape of an epicyclic, as no trace 01 such is to be found in the entire of his great work, The Almagest. Example on the Construction of Circle of Inflexions. 367 The formulae for a retrograde epicyclic are obtained by changing the sign of m (compare Art. 284). It is easily seen that every epicyclic admits of a twofold generation. For, if we make mO = , equation (30) may be written x - B cos + a cos — , ' r m _ . . d> y = B sm + a sin — , which is equivalent to an interchange of the radii of the deferent circle and of the epicycle, and an alteration of m into — . This result can also be seen immediately geometri- m cally. It may be remarked that this contains Euler's theorem (Art. 280) under it as a particular case. 296. Properties of tlie Circle of Inflexions. — It should be especially observed that the results established in Art. 290, relative to the circle of inflexions, hold in all cases of the motion of a figure in its plane, and hence we infer that the distances of any moving point from the centre of curva- ture of its path, from the instantaneous centre of rotation, and from the circle of inflexions, are in continued proportion. Again, from Art. 292, we infer that if a moveable curve slide on a fixed curve, the distances of the centre of curvature of the moving, from that of the fixed curve, from the centre of in- stantaneous rotation, and from the circle of inflexions, are in continued proportion. The particular cases mentioned in these Articles obviously hold also in this case, and admit of similar enunciations. These principles are the key to the theory of the curvature of the paths of points carried by moving curves, as also to the curvature of the envelopes of carried curves. We shall illustrate this statement by a few applications. 297. Dxample on the Construction of Circle of Inflexions. — Suppose two curves a v b l and c x d h invariably con- nected with a moving plane figure, always to touch two fixed curves a 2 b 2 and c 2 d 2 , to find the centre of curvature of the roulette described by any point B x of the moving figure. 368 Roulettes. The instantaneous circle of inflexions is easily constructed in the following manner : — Let Pi and P 3 be the centres of cur- vature for the point of contact m for the curves aj)i and a 2 b 2 , respectively : and let Q l9 Q 2 , he the corresponding points for the curves c 1 d l and c 2 d%. Take „ _ PiO 2 , n _ Q 1 0* then, by Art. 290, the points ™- 6 JSi and Fi lie on the circle of inflexions. Accordingly, the circle which passes through 0, Ei and F ly is the circle of inflexions. Hence, if B x meet this circle in Gi, and we take R O 2 R X R 2 = -jT7t> ^he point R 2 (by the same theorem) is the centre of curvature of the roulette described by Pi. In the same case, by a like construction, the centre of cur- vature of the envelope of any carried curve can be found. The modifications when any of the curves a-J) ly a 2 b 2 , &c, becomes a right line, or reduces to a single point, can also be readily seen by aid of the principles already established for such cases. 298. Theorem of Bobillier.* — If two sides of amoving triangle always touch two fixed circles, the third side also always touches a fixed circle. Let ABC be the moving triangle ; the side AB touching at c a fixed circle whose centre is 7, and AC touching at b a circle with centre /3. Then the instantaneous centre is the point of intersection of bf5 and cy. Again, the angle j30y, being the supplement of the con- stant angle BAG, is given; and consequently the instanta- neous centre always lies on a fixed circle. * Cours de geometrie pour les ecoles des arts et metiers. See also Collignon, Traite de Mecanique Cinematique, p. 306. This theorem admits of a simple proof by elementary geometry, The in- vestigation above has however the advantage of connecting it with the general theory given in the preceding Articles, as well as of leading to the more general theorem stated at the end of this Article. Analytical Demonstration. 369 Also if Oa be drawn perpendicular to the third side BC, a is the point in which the side touches its envelope (Art. 291). Produce aO to meet the circle in a ; and since the angle aOj3 is equal to the angle ACB, it is constant ; and consequently the point a is a fixed point on the circle. Again, by (4) Art. 292, the circle j30y passes through the centre of curvature of the envelope of any carried right line ; and accordingly a is the centre of curvature of the enve- lope of BC; but a has already been proved to be a fixed point ; consequently BC in all positions touches a fixed circle whose centre is a. (Compare Art. 286.) This result can be readily extended to the case where the sides AB and AC slide on any curves ; for we can, for an in- finitely small motion, substitute for the curves the osculating circles at the points b and c, and the construction for the point a will giYe the centre of curvature of the envelope of the third side BC. 298 (a). Analytical Demonstration.— The result of the preceding Article can also be established analytically, as was shown by Mr. Ferrers, in the following manner : — Let a, b, c represent the lengths of the sides of the moving triangle, and p l9 p i} p z the perpendiculars from any point on the sides a, b, c, respectively ; then, by elementary geometry, we have ap x + bp 2 + cp z = 2 (area of triangle) = 2 A. Again, if pi, /o 2 , /o 3 be the radii of curvature of the enve- lopes of the three sides, and w the angle through which each of the perpendiculars has turned, we have by (28), api + bp 2 + cp 3 = 2 A. (31) Hence, if two of the radii of curvature be given the third can be determined. 2 B 37° Roulettes. We next proceed to consider the conchoid of Nicomedes. 299. Centre of Curvature for a Conchoid. — Let A be the pole, and LM the directrix of a conchoid. Construct the instantaneous centre 0, as before : and produce AO until OA x = AQ. It is easily seen that the circle circumscribing A x OR x is the instantaneous circle of inflexions : for the instantaneous centre always lies on this circle ; also R x lies on the circle by Art. 290, since it moves along a right line : again, A lies on the lower circle of inflexions of same Article, and conse- quently Ai lies on the circle of inflexions. Hence, to find the centre of curvature of the conchoid described by the moving point Pi, produce P x O to meet the circle of inflexions in F u and take P G l PiP3== p!p ; then ' by ^ 22 )' Fz is the centre of curvature belonging to the point Pj on the conchoid. In the same case, the centre of curvature of the curve described by any other point ft, which is inva- riably connected with the moving line, can be found. For, if we produce ftO to meet the circle of inflexions in E T9 and take Q\Q% Q O* = -^=- ; then, by the same theorem, Q 2 is the centre of curvature re- Fig. 78. quired. A similar construction holds in all other cases. 300. Snnericai l&ouiette§. — The method of reasoning adopted respecting the motion of a plane figure in its plane is applicable identically to the motion of a curve on the sur- face of a sphere, and leads to the following results, amongst others : — (1). A spherical curve can be brought from any one position on a sphere to any other by means of a single rotation around a diameter of the sphere. (2). The elementary motion of a moveable figure on a sphere may be regarded as an infinitely small rotation Motion of a Rigid Body about a Fixed Point. 371 around a certain diameter of the sphere. This diameter is called the instantaneous axis of rotation, and its points of intersection with the sphere are called the poles of rotation. (3). The great circles drawn, for any position, from the pole to each of the points of the moving curve are normals to the curves described by these points. (4) . When the instantaneous paths of any two points are given, the instantaneous poles are the points of intersection of the great circles drawn normal to the paths. (5). The continuous movement of a figure on a sphere may be reduced to the rolling of a curve fixed relatively to the moving figure on another curve fixed on the sphere. By aid of these principles the properties of spherical roulettes* can be discussed. 301. Motion of a Rigid Body about a Fixed Point. — We shall next consider the motion of any rigid body around a fixed point. Suppose a sphere described having its centre at the fixed point ; its surface will intersect the rigid body in a spherical curve A, which will be carried with the body during its motion. The elementary motion of this curve, by the preceding Article, is an infinitely small rotation around a diameter of the sphere ; and hence the motion of the solid consists in a rotation around an instan- taneous axis passing through the fixed point. Again, the continuous motion of A on the sphere by (5) (preceding Article) is reducible to the rolling of a curve i, connected with the figure A, on a curve A, traced on the sphere. But the rolling of L on X is equivalent to the rolling of the cone with vertex standing on Z, on the cone with the same vertex standing on X. Hence the most general motion of a rigid body having a fixed point is equivalent to the roiling of a conical surface, having the fixed point for its summit, and appertaining to the solid, on a cone fixed in space, having the same vertex. These results are of fundamental importance in the gene- ral theory of rotation. * On the Curvature of Spherical Epicycloids, see Resal ; Journal de lEcoh Poli/technique, 1858, pp. 235, &c. 2B2 372 Examples. Examples. i. If the radius of the generating circle be one-fourth that of the fixed, prove i mm ediately that the hypocycloid becomes the envelope of a right line of constant length whose extremities move on two rectangular lines. 2. Prove that the evolute of a cardioid is another cardioid in which the radius of the generating circle is one-third of that for the original circle. 3. Prove that the entire length of the cardioid is eight times the diameter of its generating circle. 4. Show that the points of inflexion in the trochoid are given by the equation cos + - = o ; hence find when they are real and when imaginary. u a 5. One leg of a right angle passes through a fixed point, whilst its vertex slides along a given curve ; show that the problem of finding the envelope of the other leg of the right angle is reducible to the investigation of a locus. 6. Show that the equation of the pedal of an epicycloid with respect to any origin is of the form , .. ad . r = (a+ 20) cos - - c cos (0 + a). a + 20 7. In figure 57, Art. 281, show that the points C, P' and Q are in directum. 8. Prove that the locus of the vertex of an angle of given magnitude, whose sides touch two given circles, is composed of two limacons. 9. The legs of a given angle slide on two given circles : show that the locus of any carried point is a limacon, and the envelope of any carried right line is a circle. 10. Find the equation to the tangent to the hypocycloid when the radius of the fixed circle is three times that of tbe rolling. Am. x cos ta + y sin w = b sin 3«. This is called the three-cusped hypocycloid. See Ex. 5, Art. 286. 11. Apply the method of envelopes to deduce the equation of the three- cusped hypocycloid. Substituting for sin 3© its value, and making t = cot a, the equation of the tangent becomes %$ + (y - 3#) t- + xt + b + y = o, in which t is an arbitrary parameter. If t be eliminated between this and its derived equation taken with respect to t, we shall get for the equation of the hypocycloid, (#2 + ^2)2 + !g£2 ^2 + y8 ) + 24bx°~y - % 3 = 27K Examples. 373 12. If two tangents to a cycloid intersect at a constant angle, prove that the length of the portion which they intercept on the tangent at the vertex of the cycloid is constant. 13. If two tangents to a hypocycloid intersect at a constant angle, prove that the arc which they intercept on the circle inscribed in the hypocycloid is of constant length. 14. The vertex of a right angle moves along a right line, and one of its legs passes through a fixed point : show geometrically that the other leg envelopes a parabola, having the fixed point for focus. 15. One angle of a given triangle moves along a fixed curve, while the opposite side passes through a fixed point : find, for any position, the centre of curvature of the envelope of either of the other sides, and also that of the curve described by any carried point. 16. If a right line move in any manner in a plane, prove that the locus of the centres of curvature of the paths of the different points on the line, at any instant, is a conic. — (Resal, Journal de V Ecole Polytechnique, 1858, p. 112). This, as well as the following, can be proved without difficulty from equa- tion (22), p. 352. 17. "When a conic rolls on any curve, the locus of the centres of curvature of the elements described simultaneously by all the points on the conic is a new conic, touching the other at the instantaneous centre of rotation. — (Mannheim, same Journal, p. 179.) 18. An ellipse rolls on a right line : prove that p, the radius of curvature of the path described by either focus, is given by the equation - = ; where par r is the distance of the focus from the point of contact, and a is the semi-axis major. — (Mannheim, Ibid.) 19. The extremities of a right line of given length move along two fixed right lines : give a geometrical construction for the centre of curvature of the envelope in any position. 20. Prove that the locus of the intersection of tangents to a cycloid which intersect at a constant angle is a prolate trochoid (La Hire, Mem. de V Acad, des Sciences, 1704). 21. More generally, prove that the corresponding locus for an epicycloid is an epitrochoid, and for a hypocycloid is a hypotrochoid. (Chasles, Hist, de la Geom., p. 125). 22. If a variable circle touch a given cycloid, and also touch the tangent at the vertex, the locus of its centre is a cycloid. (Professor Casey, Phil. Trans., 1877.) 23. Being given three fixed tangents to a variable cycloid, the envelope of the tangent at its vertex is a parabola. (Ibid.) 24. If two tangents to a cycloid cut at a constant angle, the locus of the centre of the circle described about the triangle, formed by the two tangents and the chord of contact, is a right line. (Ibid.) 25. If a curve (A) be such that the radius of curvature at each point is n times the normal intercepted between the point and a fixed straight line (B), 374 Examples. then when the curve rolls along another straight line, {B) will envelope a curve in which the radius of curvature is n + i times the normal. Thus, when n = - 2, {A) is a parabola, and (B) the directrix ; and when the parabola rolls along a straight line, its directrix envelopes a catenary (for which n = — 1 ), to which the straight line is directrix. When the catenary rolls along a straight line, its directrix passes through a fixed point, for which n = o. When the point moves along a straight line, the straight line which it car- ries with it envelopes a circle (n = 1), and (B) is a diameter. When the circle rolls along a straight line, its diameter envelopes a cycloid (n = 2), to which (B) is the base. When the cycloid rolls along a straight line its base envelopes a curve which is the involute of the four-cusped hypocycloid, passing through two cusps, and is in figure like an ellipse whose major axis is twice the minor. (Professor Wolstenholme.) The fundamental theorem given above follows immediately from equation (27), P- 357- 26. Prove the following extension of Bobillier's theorem : — If two sides of a moving triangle always touch the involutes to two circles, the third side will always touch the involute to a circle. 27. Investigate the conditions of equilibrium of a heavy body which rests on a fixed rough surface. In this case it is plain that, in the position of equilibrium, the centre of gravity G of the body must be vertically over the point of contact of the body with the fixed surface. Again, if we suppose the body to receive a slight displacement by rolling on the fixed surface, the equilibrium will be stable or unstable, from elementary mechanical considerations, according as the new position of G is higher or lower than its former position, i. e. according as G is situated inside or outside th# circle of inflexions (Art. 290). Hence, if pi and p% be the radii of curvature for the corresponding fixed and rolling curv 3, and h the distance of G from the point of contact of the surfaces, the equilibrium is stable or unstable according as h is < or > — . See Walton's pi + pz Problems, p. 190 ; also, for a complete investigation of the case where h = pi + pz 1 Minchin's Statics, pp. 320-2, 2nd Edition. 28. Apply the method of Art. 285 to prove the following construction for the axes of an ellipse, being given a pair of its conjugate semi-diameters OP, OQ, in magnitude and position. Prom P draw a perpendicular to OQ, and on it take PD = PQ ; join P to the centre of the circle described on OB as diameter by a right line, and let it cut the circumference in the points F and F ; then the right lines OH and OF are the axes of the ellipse, in position, and the segments PE and PF are the lengths of its semi-axes (Mannheim, Now. An. de Math. 1857, p. 188). 29. An involute to a circle rolls on a right line : prove that its centre describes a parabola. 30. A cycloid rolls on an equal cycloid, corresponding points being in con- tact : show that the locus of the centre of curvature of the ro llin g curve at the point of contact is a trochoid, whose generating circle is equal to that of either cycloid. ( 375 ) CHAPTER XX. ON THE CARTESIAN OVAL. 302. Equation of Cartesian ©val. — In this Chapter" it is proposed to give a short discussion of the principal pro- perties of the Cartesian Oval, treated geometrically. We commence by writing the equation of the curve in its usual form, viz., n ± fir 2 = a, where n and r 2 represent the distances of any point on the curve from two fixed points, or foci, F l and F 2i while fx and a are constants, of which we may assume that fi is less than unity. We also assume that a is greater than FiF 2 , the dis- tance between the fixed points. It is easily seen that the curve consists of two ovals, one lying inside the other ; the former corresponding to the equation n + fir 2 = a, and the latter to n - fir 2 = a. Now, with Fi as centre, and a as radius, describe a circle. Through F 2 draw any chord DF, join FiD and FiE; then, if P be the point in which FiD meets the inner oval, we have FD = a- ri = fj.r 2 = p.PF 2 , From this relation the point P can be readily found. Fig, 79- * This Chapter is taken, with slight modifications, from a Paper published hy me in Hermathena, No. rv., p. 509. 376 On the Cartesian Oval. Again, let Q be the corresponding point for the outer oval r x - jnr 2 = a; and we have, in like manner, BQ = fxF 2 Q ; .-. F 2 Q : F 2 P = QD:DP; consequently, F 2 D bisects the angle PF 2 Q. Produce QF 2 and PF 2 to intersect F X E, and let P x and Q x be the points of intersection. Then, since the triangles PF 2 B and PiF 2 E are equiangular, we have P X E = \xP x F 2 ; and consequently the point P x lies on the inner oval. In like manner it is plain that Q x lies on the outer. Again, by an elementary theorem in geometry, we have F 2 P . F 2 Q = PD.DQ + F 2 D 2 ; .*. (i - jtr) F 2 P . F 2 Q = F 2 D\ Also, by similar triangles, we get F 2 P : F 2 P X = F 2 D : F 2 E ; consequently ( i - fj 2 ) F 2 Q . F 2 P X = F 2 D . F 2 E = const. (2) Therefore the rectangle under F 2 Q and F 2 P X is constant; a theorem due to M. Quetelet. 303. Construction for Third Focus. — Next, draw QF 3 , making Z.F 2 QF 3 = lF 2 F x P x ; then, since the points P l9 F l9 Q, F s lie on the circumference of a circle, we get F X F 2 . F 2 F 3 = F 2 Q . F 2 P X = const. (3) Hence the point F s is determined. "We proceed to show that F 2 possesses the same properties relative to the curve as F x and F 2 ; in other words, that F 3 is a third focus* "For this purpose it is convenient to write the equation of the curve in the form mr x ± lr 2 = nc 3 , (4) in which c 3 represents F X F 2 , and l 9 m, n are constants. It may be observed that in this case we have n>m> I. * This fundamental property of the curve was discovered by Chasles. See Eistoire de la Ge'ometrie, note xxi., p. 352. Construction for Third Focus. 377 Now, since L F X F 3 Q = LF X P V F 2 = L F X PF 2 , the triangles F1PF2 and FiF 3 Q are equiangular ; but, by (4), we have mF.P + lF^P^nF.F,; accordingly we have mF l F 3 + IF,Q = nF 1 Q, or nFxQ - IF 3 Q = mF x F 3 ; i. e. denoting the distance from F 3 by r 3 and F X F 3 by c 2 , wri - lr 3 = ^c 2 . This shows that the distances of any point on the outer oval from F x and F 3 are connected by an equation similar in form to (4) ; and, consequently, F 3 is a third focus of the curve. 304. Equations of Curve, relative to eacb pair of Foci. — In like manner, since the triangles F X QF 2 and F X F 3 P are equiangular, the equation mF x Q - IF 2 Q = nF x F 2 gives mF x F 3 - IF 3 P = nF x P. Hence, for the inner oval, we have nr x + lr 3 = mc 2 . This, combined with the preceding result, shows that the con- jugate ovals of a Cartesian, referred to its two extreme foci, are represented by the equation nr x ± lr 3 - mc 2 . (5) In like manner, it is easily seen that the conjugate ovals re- ferred to the foci F 2 and F 3 are comprised under the equation nr % - mr 3 = + lc x , (6) where c x = F 2 F 3 . 305. Relation between the Constants. — The equa- tion connecting the constants /, m, n in a Cartesian, which has three points t[ 9 F 2 , F 3 for its foci, can be readily found. 378 On the Cartesian Oval. For, if we substitute in (3), c 3 for F X F 2) &c, the equation is easily reduced to the form Pc x + n% = m 2 c 2 , or PFJF t + m*F^F x + n 2 F x F 2 = o, (7) in which the lengths F 2 F S , &c, are taken with their proper signs, viz., F Z F X = - F X F 3 , &c. 306. Conjugate Ovals are Inverse Curves. — Next, since the four points F 2 , P, Q, F z , lie in a circle, we have F x P.F l Q = F 1 F z .F x F 3 = const. (8) Consequently the two conjugate ovals are inverse to each other with respect to a circle* whose centre is F l9 and whose radius is a mean proportional between F X F % and F x F z . It follows immediately from this, since F 2 lies inside both ovals, that F z lies outside both. It hence may be called the external focus. This is on the supposition that the constants! are connected by the relations n > m > I. Also we have L PF Z F 2 = lPQF 2 = l F 2 Q x P x = L F 2 F Z P X ; hence the lines F Z P and F z Pi are equally inclined to the axis F X F 3 . Consequently, if P 2 be the second point in which the line F 3 P meets the inner oval, it follows, from the sym- metry of the curve, that the points P 2 and Pi are the * It is easily seen that when I = o the Cartesian whose foci are F Xi F 2 , F z , reduces to this circle. Again, if n = o, the Cartesian hecomes another circle, whose centre is JF3, and which, as shall he presently seen, cuts orthogonally the system of Cartesians which have F\, F 2 , F$ for their foci. These circles are called hy Prof. Crofton (Transactions, London Mathematical Society, 1866), the Confocal Circles of the Cartesian system. f From the ahove discussion it will appear, that if the general equation of a Cartesian he written \r + fir' = vc, where c represents the distance hetween the foci ; then (1) if, of the constants, A, fi, v, the greatest he v, the curve is referred to its two internal foci ; (2) if v he intermediate hetween A and fx, the curve is referred to the two extreme foci ; (3) if v be the least of the three, the curve is referred to the external and middle focus ; (4) if X = fi, the curve is a conic ; (5) if v = A, or v — n, the curve is a limacon ; (6) if one of the constants A, yu, v vanish, the curve is a circle. Construction for Tangent at any Point. 379 reflexions of each other with respect to the axis FiF 2 , and the triangles F 1 P 2 F 2 and FiPiF 2 are equal in every respect. Again, since z F 2 PF 3 = L F 2 QF 3 = L F*F x Px = L F 2 F X P 2 , the four points P, P 2 , F x and F 2 lie on the circumference of a circle. From this we have F 3 P . F 3 P 2 = F d Fi . F 3 F 2 = constant. Hence, the rectangle wider the segments, made by the inner oval, on any transversal from the external focus, is constant. In like manner it can be shown that the same property holds for the segments made by the outer oval. If we suppose P and P 2 to coincide, the line F Z P becomes a tangent to the oval, and the length of this tangent becomes constant, being a mean proportional between F 3 F X and F 3 F 2 . Accordingly, the tangents drawn from the external focus to a system of triconfocai Cartesians are of equal length. This result may be otherwise stated, as follows : — A system of triconfocai Cartesians is cut orthogonally by the confocal circle whose centre is the external focus of the system (Prof. Crofton). This theorem is a particular case of another — also due, I believe, to Prof. Crofton — which shall be proved subsequently, viz., that if two triconfocai Cartesians intersect, they cut each other orthogonally. 307. Construction for Tangent at any Point. — We next proceed to give a geometrical method of drawing the tangent and the normal at any point on a Cartesian. Retaining the same notation as before, let R be the point in which the line F 2 D meets the circle which passes through the points P, F 2 , F 3 , Q ; then it can be shown that the lines PR and RQ are the normals at P and Q to the Cartesian oval which has F x and F 2 for its internal foci, and F 3 for its external. For, from equation (4), we have for the outer oval dn , dr 2 m -— - l~— = o. as as 3 8o On the Cartesian Oval. Hence, if w x and w 2 be the angles which the normal at Q makes with QF X and QF 2 respectively, we have m sin wi = I sin w 2 ; or sin an : sin W2 = I : m. Fig. 80. Again, we have seen at the commencement that l:m = J)Q:F 2 Q; also, by similar triangles, BQ : RFz = DQ:F 2 Q=l:m; BQ : BF 2 = sin BQP : sin BQF 2 ; sin BQF X : sin jKQi^ 2 = l:m. but hence (9) (10) Consequently, by (9), the line BQ is the normal at Q to the outer oval. In like manner it follows immediately that PB is normal to the inner oval. This theorem is given by Prof. Crofton in the following form : — The arc of a Cartesian oval makes equal angles with the right line drawn from the point to any focus and the circular arc drawn from it through the two other foci. This result furnishes an easy method of drawing the tangent at any point on a Cartesian whose three foci are given. Confocal Cartesians intersect Orthogonally. 381 The construction may be exhibited in the following form : — Let F x , F 2 , F 3 be the three foci, and P the point in question. Describe a circle through P and two foci F% and F 3i and let Q be the second point in which FiP meets this circle ; then the line joining P to R, the middle point of the arc cut off by PQ, is the normal. 308. Confocal Cartesians intersect Orthogonally. — It is plain, for the same reason, that the line drawn from P to Bi 9 the middle point of the other segment standing on PQ, is normal to a second Cartesian passing through P, and having the same three points as foci — F% and F d for its in- ternal foci, and F x for its external. Hence it follows that through any point two Cartesian ovals can be drawn having three given points — which are in directum — for foci. Also the two curves so described cut orthogonally. Again, if R C be drawn touching the circle PRQ 9 it is parallel to PQ, and hence F % C:F X C = F 2 R : RB = F 2 R 2 : F 2 R . RD ; but F 2 R . RB = RP 2 ; .-. F 2 C : F X C = F 2 R 2 : PR 2 = m? : l\ (1 1) Hence the point C is fixed. Again CR : F X B = RF 2 :BF 2 = m 2 :m 2 -l 2 ; .\ CR = — — -, (12) which determines the length of CR. Next, since RP =RQ, if with R as centre and RP as radius a circle be described, it will touch each of the ovals, from what has been shown above. Also, since C is a fixed point by (1 1), and CR a constant length by (12), it follows that the locus of the centre of a circle which touches both branches of a Cartesian is a circle (Quetelet, Nouv. Mem. de VAcad. Roy. de Brux. 1827). 3«2 On the Cartesian Oval. This construction is shown in the following figure, in which the form of two conjugate ovals, having the points F i9 F 2 , F S9 for foci, is exhibited. Again, since the ratio of F 2 R to RP is constant, we get the following theorem, which is also due to M. Quetelet : — A Cartesian oval is the envelope of a circle, whose centre moves on the circum- ference of a given circle, while its radius is in a constant ratio to the distance of its centre from a given point. 310. Cartesian Oval struction has been given as an Envelope. — This con- in a different form by Professor Casey, Transactions Royal Irish Academy, 1869. If a circle cut a given circle orthogonally, while its centre moves along another given* circle, its envelope is a Cartesian oval. This follows immediately ; for the rectangle under F X P and F X Q is constant (8), and therefore the length of the tan- gent from Fi to the circle is constant. This result is given by Prof. Casey as a particular case of the general and elegant property of bicircular quartics, viz. : if in the preceding construction the centre of the moving circle describe any conic, instead of a circle, its envelope is a bicir- cular quartic. * It is easily seen that the three foci of the Cartesian oval are : the centre of the orthogonal centre, and the limiting points of this and the other fixed circle. Examples. 383 Examples. 1. Find the polar equation of a Cartesian oval referred to a focus as pole. If the focus F\ be taken as pole, and the line F\F% as prime vector, we easily obtain, for the polar equation of the curve, (in 2 — l 2 )r 2 — 2cs (inn — I 2 cos 0) r + C3 2 (n 2 - I 2 ) — o. The equations with respect to the other foci, taken as poles, are obtained by a change of letters. 2. Hence any equation of the form r 2 — 2 (a + b cos 8) r + c 2 = o represents a Cartesian oval. 3. Hence deduce Quetelet's theorem of Art. 302. 4. If any chord meet a Cartesian in four points, the sum of their distances from any focus is constant ? For, if we eliminate 6 between the equation of the curve and the equation of an arbitrary line, we get a biquadratic in r, of which — 4a is the coefficient of the second term. 5. Show that the equation of a Cartesian may in general be brought to the form S 2 = WZ, where S represents a circle, and L a right line, and Jc is a constant. 6. Hence show that the curve is the envelope of the variable circle \ 2 kL + 2XS + h 2 = o. Compare Art. 309. 7. From this show that the curve has three foci ; i. e. three evanescent circles having double contact with the curve. 8. The base angles of a variable triangle move on two fixed circles, while the two sides pass through the centres of the circles, and the base passes through a fixed point on the line joining the centres ; prove that the locus of the vertex is a Cartesian. 9. Prove that the inverse of a Cartesian with respect to any point is a bi- circular quartic. (See Salmon, Higher Plane Curves, Arts. 280, 281.) 10. Prove that the Cartesian r 2 - 2 (a + b cos 6)r + c 2 = o has three real foci, or only one according as ^ a - b is > or < c. ( 384 ) CHAPTEE XXI. ELIMINATION OF CONSTANTS AND FUNCTIONS. 311. Elimination of Constants. — The process of dif- ferentiation is often applied for the elimination of constants and functions from an equation, so as to form differential equations independent of the particular constants and func- tions employed. We commence with the simple example y 1 = ax + b. By du differentiation we get 22/— = a, a result independent of b, ax A second differentiation gives dy\ d 2 y dx) +y dx*~°'' a differential equation containing neither a nor b, and which accordingly is satisfied by each of the individual equations which result from giving all possible values to a and b in the proposed. In general, let the proposed equation be of the form f(x, y, a) = o. By differentiation with respect to x 9 we get df dfdy dx dy dx The elimination of a between this and the equation/^, y,a) = o dii leads to a differential equation involving x, y and — , which ax holds for all the equations got by varying a in the proposed. Again, if the given equation in x and y contain two constants, a and b ; by two differentiations with respect to x y we obtain two differential equations, between which and the Examples. 385 original, when the constants a and b are eliminated, we get a differential equation containing x, y, — and -~. dx ax* In general, for an equation containing n constants, the resulting differential equation contains x, y, —-, —~ . . . — ? ; ax ax ax arising from the elimination of the n constants between the given equation and the n equations derived from it by suc- cessive differentiation. Examples. 1. Eliminate a from the equation y 2 — 2ay + x 2 = a 2 . Ans. (x 2 — 2y 2 ) I -f- } - Axy — - x 2 \— o. \dxj dx ' 2. Eliminate a and /8 from the equation 3 d 2 y + p — - = o. (y- «)3=^(^_ j8). Ans. 2\-^-\ \dxj 3. Eliminate the constants a and /3 from the equation d 2 ti y = a cos «# + # sin nx. Ans. — - + n 2 y = o. dx 2 4. Eliminate a and b from the equation (* - «) 2 + (y - *)» = o 2 . ^m. c 2 = ( V 2 ^{ ? ■ This agrees with the formula for the radius of curvature in Art. 226. 5. Eliminate a and /3 from the equation y = ax cos - + &). Ans. — £ -\ f =o. \x J dx 2 x 4. 6. Eliminate the constants #0, «i, . . . «« from the equation y = (a?) + tfo# n + aix n ~ l + ...«„. ^ 7. Eliminate the constants a and # from the equation y = ae™ + fab*. 8. Eliminate a and J from the equation xy = #0* 4- he~ x . 9. Eliminate, by differentiation, c, c' from the equation d ml y y = d>(x) + a x n + aix n ~ l + ...«„. .4m$. - — £ = which gives — = y ) . . We have met several 9 5rw & dx

and the dependent variable by z. It will also be found convenient to adopt the usual notation, and to represent the partial differential coefficients dz dz d 2 z d % z d 2 z dx dy y dx*' dxdy' dy 2 ' by the letters p, q, r, s and t } respectively. We proceed to show that in this case we are enabled by differentiation to eliminate functions whose forms are alto- gether arbitrary. In fact we have already met with examples of this process ; for instance, if z = x n

(% — az). ,, ap + bq = I. 3. x-a = (z-y)<}>l^-~-y „ {z-a)p + (y-0)q = z- 7 . 4. <£> (x* 1 + y m ) = z r . „ nx n - x q = my m - x p. 5. z* = xy + (-)■ „ xzp + yzq = xy. 6. £ + V# 2 + «/ 2 + s 2 = # 1_n (-) - >> z =P% + , we get T ^ cos - cos 2 v = sin cos + cos sin = sin (0 + ) ; which establishes the result required. We have here assumed that whenever equation (1) is satis- fied identically, V is expressible as a function of v : this can be easily shown as follows : — Since Fand v are supposed to be given functions of x and y, if one of these variables, y, be eliminated between them we can represent Fasa function of v and x. Accordingly, let V=f(x 9 v); dV df df dv dV df dv \ dx dx dv dx* dy dv dy ' therefore dV dv dV dv df dv dx dy dy dx dx dy Hence, since the left-hand side is zero by hypothesis, we must df have -7- = o ; i.e. the function fix, v) or V reduces to a func- dx tion of v simply ; which establishes the proposition. 39° Elimination of Constants and Functions. 315. More generally, let it be proposed to eliminate the arbitrary function $ from the equation where V and v are given functions of three variables, x, y, and s. Eegarding x and y as independent variables, we get by differentiation dV dV ,, Jdv dv\ ix- +p ^=^ v \dx +p d*y dV dV ,, , (dv dv\ eliminating '(v) we obtain dVdv dVdv fdVdv dv dV dx dy dy dx \dz dy dz dy fdVdv dv dV\ , . \dx dz dxdz J ' a result independent of the arbitrary function $. This equation can also be established as follows : — Differentiating the equation V ' = (3) dv , dv 7 «# 7 -7- Go? + — tfV + — as = o. a# dy dz J Condition that one Expression is a Function of another. 391 Moreover, introducing the condition that % depends on x and y, we have d% = pdx + qdy ; consequently, eliminating dx, dy, dz between this and the equations in (3), we get dV dV dV dx ' dy ' dz dv dv dv dx' dy* dz which agrees with the result in (2). = o: (4) Examples. Eliminate the arbitrary functions in the following cases : — 1. z = (p(a sin a; -f- b siny). dz dz Am. cosy- — a cos x — = o. dx dy 2. y z = e" (ax + by + cz). ah x dz . .dz Am. (dz — cy) — + (ex — az) — = ay — bx. CvOC Cvtf 39 2 Elimination of Constants and Functions. 316. Next, let it be required to eliminate the arbitrary function

(u), and (p\u). Accordingly, if (u) and '(u) be eliminated between these and the original equation, we shall have a resulting equation containing only x, y, z, p, and q. 317. Case of two or more Arbitrary Functions. — If the given equation contain more than one arbitrary func- tion, we have to proceed to partial differentiations of a higher degree in order to eliminate the functions : thus, in the case of two arbitrary functions, $(11) and \p(v) 9 the first differen- tiations with respect to x and y introduce the functions '(u) and 4> f (v). It is plainly impossible, in general, to eliminate the four arbitrary functions between three equations ; we accordingly must proceed to form the three partial differen- tials of the second order, introducing two new arbitrary functions ^\u) and \l/'(v). Here, again, it is in general impossible to eliminate the six functions between six equa- tions, so that it is necessary to proceed to differentials of the third order : in doing so we obtain four new equations, con- taining two additional functions, m (u) and ^/"(v). After the elimination of the eight arbitrary functions there would remain, in general, two resulting partial differential equations of the third order. 318. There is one case, however, in which we can always obtain a resulting partial differential equation of the second order — viz., where the arbitrary functions are functions of the same quantity, u. Case of Tico or more Arbitrary Functions. 393 Thus, suppose the given equation of the form F{x,y, z, (u), \fr{u)) = o, (5) where u is a known function of x, y, and s. By differentiation we get dF dF dFfdu du\ _ dx dz du \dx dz) dF dF dFfdu du\ _ dy dz du \dy dz) dF Eliminating — between these equations, we obtain au dF du dFdu fdFdu dFdu\ dx dy dy dx \dz dy dy dz) (dFdu dF du\ ... + q[ = o. (6) \dx dz dz dx) This equation contains only the original functions (u) y ^ (u) 9 along with x 9 y, z, p, and q. Again, if we apply the same method to it, we can form a new partial differential equation, involving the same functions (u) and \p [u) 9 between this last equation and equations (5) and (6), leads to the required partial differential equation of the second order. The result in (6) admits also of being arrived at by the method adopted in the seoond proof of Art. 315. For re- garding x 9 y 9 z 9 as all variables, we get from (5), on differen- tiation, dF 7 dF _ dF , dF (du 7 du _ du 7 \ -— dx + -r- d y + — dz + -_— — dx + — ay + — dz =0. (7) dx dy dz du \dx dy dz J x ' ^ dF dF „ . dF f „ x But ^raW) Hu)+ ^¥)^ {u)i 394 Elimination of Constants and Functions. and accordingly, since (7) must hold for all values of $'(11) and \p'(u) 9 we have and dF- dF 7 dF . — dx + -r-dy + -z— dz dx dy dz o 1 > (8) du , du 7 du _ dx + — dy + — dz = o. \ Eliminating between these equations and f/s = pdx + qdy, we get the following determinant : dF dF dF dx' dy' dz du du du dx* dy* dz = o; (9) which plainly is identical with (6). This admits also of the following statement : substitute c instead of u in the proposed equation : then regarding c as con- stant, differentiate the resulting equation, as also the equation u = c (on the same hypothesis) : on combining the resulting equations with dz = pdx + qdy 9 we get another equation connecting (c) and \p (c) ; and applying the same method to it, we obtain the result, on eliminating the arbitrary functions cp(c) and \p(c) between the original equation and the two others thus arrived at. These methods will be illustrated in the following ex- amples. Examples. 395 Examples. 1. s = x (z) + {a^'(a) + yf (a) } i?, q = $(z) + {x (c) dx + if/ (c) % = o ; also pdx + qdy = o ; therefore I = *® q xp(c) Differentiating again, we have qdp — pdq = O, or q{rdx + sdy} — p {sdx + tdy) = o, which, combined with pdx + qdy = o, leads to the same result as before. 2. z — %<$> {ax + by) + y-ty {ax + by) . Here p = {ax + by) + a {x$'{ax -J- by) + y-t/ (a%l+ by) } , q = ${ax + by) + b {x {ax + by) — aty {ax + by) ; hence br - as = a {b(f>' {ax + by) — a^'{ax + by)}, Is — at —b {bcpf {ax + by) - aty {ax + by)}; therefore b 2 r — zabs + a 2 t — o. 396 Elimination of Constants and Functions. Otherwise thus : let ax + by = c> then adx + bdy = o ; also, dz =

(xy), prove that /p2 y _ yi} ^ xp — yq — o. 11. If ary + bery = ce x + de~ x , prove that L^ 2 U*/ dx\ L U^v J ^ \^ 3 / ' 12. z = x n

nz = o. 13. Eliminate the arbitrary functions from the equation z — (p {x +/(«/)}• Ans. ps - qr = 0. 14. If the substitution of Ae ax for 2/ satisfies the differential equation with constant coefficients, d n y d n ~ 1 y dy prove that a must be a root of the equation z n + p lZ n-l + . . . +p n -iz+pn = O. 15. Eliminate the constants from the equation ax 2 + 2bxy + cy 2 + 2dx + 2ey + /=~o. Ans. 40r 3 — 45gr 2 s + qq 2 t = O, where * = g, j = g, r = g,&e. ( 399 ) CHAPTEE XXII. CHANGE OF THE INDEPENDENT VARIABLE. 320. Case of a Single Independent Variable. — We have already pointed out the distinction between indepen- dent and dependent variables in the formation of differen- tial coefficients. In applications of the Differential Calculus it is sometimes necessary to make our differential equations depend on new independent variables instead of those which had been origi- nally selected. To show how this transformation is effected we commence with the case of one independent variable, and suppose V to represent any function of x, y, — , — , &c. We proceed to CIX CIX (it J ft II show how the expressions for — , — , &c, are transformed, CIX (XX when, instead of x, any function of x is taken as the indepen- dent variable. Let this new function be denoted by t, and suppose that dx d x — , — -, &c, are represented by w, x\ &c, then in all cases we have du die dx . du dt dx dt dx 9 where u is any function of x ; 5 M '"S5W' (I) Hence ^ = 1% (2) dx x dt' 400 Change of the Independent Variable. , d 2 y d fdy\ d f\ dy\ i d f\ dy\ dx 2 dx \dxj dx \xdt ) x dt \x dt / {substituting --j- instead of u in (i) j; x at . d 2 y ..dy d 2 y X dt*~ X ~dt hence —? = JUL 2£ . U\ dx- ~ 3 x c Again, ' Vy ~*y\ fx&y-fy d z y d [ dtf dt id[ df dt dx % dx \ {xf J xdt\ a? d^u du d"}j {xf (4) and so on for differentiations of higher degrees. If y be taken as the independent variable, we obtain the corresponding values by making dy = i* d 2 y o, &c. dt dt 2 d % x dy i d % y dy 1 dx dx' dy dx 2 fdxV' \dy) Hence j„ - X"> m - ~ TXv*; (5) fd 2 xY dx d*x d z y = 3 \dy*)~ dy dy\ dx % (dx (6) K d y) and so on. The preceding results can also be arrived at otherwise, as follows. The essential distinction of an independent variable is, that its differential is regarded as constant; ac- du cordingly, in differentiating — when x is the independent ax Case of a Single Independent Variable. 401 variable we have d[-f) = ~i~- However, when x is no longer regarded as the independent variable, we must consider the numerator and the denominator of the fraction — as both ax variables, and by Art. 15, we get ■,(dy\ dxd 2 y -dyd 2 x d fdy\ dxd 2 y - dyd 2 x \dxj dx 2 ' dx\dxj dx 3 Differentiating again on the same hypothesis, we get d fd 2 y\ dx 2 d 3 y - dxdyd 3 y - $dxd 2 xd 2 y + 3 (d 2 x) 2 dy dx \dx 2 J dx 5 These results are perfectly general whatever function of x be taken as the independent variable. Their identity with the equations previously arrived at is manifest. Examples. 1. Being given that x = a(0 — sin0), y = a{\ - cos 0), find the value of d 2 y j -i Ans. dx z ' «(i - cos 0) 2 ' 2. Hence deduce the expression for the radius of curvature in a cycloid. 3. If x- (a + b) cos - b cos — - — 0, y = (a + b) sin $ - b sin — — 0, find the value of -r-i • dx 2 « + *„ cos — cos — - — „ dy b Here — = , * = tan dx . + sin — - — — sin GH d*y a + 2b 4. Change the independent variable from x to in the expression — f , sup- posing x = sin 0. • n dy gvri Q t_ dy _i_ dy m d^y_ = J_d_ f_s__ dy\ _ 1 d z y £0 6 dx ~ cos d0 i ' ' dx 7 - cos d0 \cos d0J cos 2 d0 % cos 3 2 1) 402 Change of the Independent Variable. 5. Transform the equation „ d 2 y dy . x 2 -^ + ax^- + by = o dx 1 dx into another in which 6 is the independent variable, being given x = e e . dy dy dx dy _ Here dl~ dxdd~ X dx' d (dy\ d I dy\ d 2 y _ d 2 y dy hence d~e \de) = x Tx \ x dx)> ox d¥~ x d^ +x dx ; d 2 y d 2 y dy therefore X d^ = W ~dO' and the transformed equation is d 2 y , , dy 6. Transform the equation „ d 2 y dy a 2 st ;2 — 5 + 2x-f+ . y = o dx 2 dx x 2 into another where z is the independent variable, being given x = -. z dy dy It is evident that in this case x — = - z — , and Q/$0 ctz d dx (•a ■ d 1 Z dz[ Z dy\ dz)' or dx* dy dx dy az therefore dx 2 dy + 2X-f = dx Z dz 2 ' and the transformed equation is d 2 y dz* l y = 0. 7. Change the independent variable from x to z in the equation * These transformations are useful in the Planetary Theory. Again, we have dV^dVdr dVdd-) dx dr dx dO dx I dV_ dV(fr dVdO [ dy dr dy dO dy , 2 D 2 404 Change of the Independent Variable. But from (8) we have dr x n dr . Q — = - = cos 6, -7- = sm V, dx r dy dO , n y sin dd — = - cos 2 v — =- dx COS0 therefore dV a — = cos d dx dr x" r dy dV sin OdV dd' dV . n dV cob ddV dy dr r du (14) (15) (16) (17) The two latter equations can also be derived by solving for — and -7— from the equations (11) and (12). dx dy d 2 V d 2 V X22. Transformation of — — - and -r-r- . — Since 1 for- dx 2 dy 2 inula (16) holds, whatever be the form of the function F, we have d . . n d / \ sin d , , dx dr r dd where stands for any function of x and y. On substituting dV — instead of 6, this equation becomes dx d_(dV\ = , b q± dx\dx ) dr n dV smOdV cos0- — dr r dd sin d ~r~"dd ' n dV sin OdV cosd- -^ dr r du , n d 2 V cosdsmQd 2 V cosdsmddV = cos 2 d — 7-t^ + dr 2 + sin0 r sin0 COS0 drdd d 2 V drdd dd 'cos OdV smOd'V r dd r dd 2 - sm 6 -r— dr J ]■ m ^ \. ■' d 2 F ^ 2 F Transformation oj -r-z- or <# 2 F ,nd 2 JP 2sin0cos0 = oos 2 0-^^ + cfe e?r y rd0~ drdd 405 sin 2 0dF sin 2 0tf 2 F + — + r dr r 2 dd 2 In like manner we get d 2 V . , n d 2 V 2sin0cos0ri^F d 2 Vl = surd dy dr 2 r dO drdd] cos 2 QdV cos 2 Od 2 V r dr r 2 dO 2 This result can be also readily deduced from the pre- ceding by substituting in it — for 0. If these equations be added we have d 2 V d 2 V d 2 V idV id 2 V _i_ _i_ -j. — dx 2 dy 1 dr 2 r dr r 2 dO 2 ' (18) d 2 V d 2 V d 2 V A 323. Transformation of — ^ + -3-5- + —7^- to polar dx 2 dy' dz* Coordinates. Let the polar transformation be represented by the equa- tions x = r sin cos 0, y = r sin $ sin 0, 2 = r cos ; also, assume p = r sin 0, and we have x = p cos 0, ?/ = jo sin ; hence, by (18), d 2 F d 2 V d 2 V idV 1 d 2 F dx 2 + + - + -= dy 2 dp 2 pdp p 2 dO 2 406 Change of the Independent Variable. Again, from the equations p = r sin * But by (17) we have dV _ . dV cos dV m dp dr r d Hence we get finally d*V d 2 V d 2 V d 2 V 1 d 2 Y dx 2 dy 2 dz 2 dr 2 r 2 sm 2 dV , x p 2 d(p 2 r dr r 2 d$ 324. Remarks on Partial Differ entials. — As already stated in Art. 113, the student must be careful to attach the correct meaning to the partial differential coefficients in each case. dx Thus in finding — in (10) we regard x as a function of r ar and 6, and differentiate on the supposition that 6 is constant ; dr in like manner the value of — in (14) is found on the suppo- ax sition that y is constant. Geometrical Illus tra tion . 407 The beginner, accordingly, must not fall into the con- fusion of supposing that in this case we have — x — = 1 . dx dr This caution is necessary, as even advanced students, from not paying proper attention to the meanings of partial de- rived functions, sometimes fall into the error referred to. 325. Geometrical Illustration. — The following geo- dr dT metrical method of determining the proper values of — and — dx dr under the preceding hypotheses may assist the beginner towards forming correct ideas on this important subject. Let P be the point whose coordinates are x and y ; then OM = x, PM = y, OP = r, POX = 9. Now, in finding -=-, regarding 9 as constant, dr we take on the radius vector OP produced a portion PQ = Ar, and draw QiV perpen- dicular to OX ; then Ax, the corresponding increment in x, is represented by MN or PL ; therefore Ax PL X~ = Wn = cos 0> Ar PQ dr or dx dr = cos 0. Again, to find — on the supposition that y is constant : let MN be Ax, the increment in x, and draw the parallelo- gram PLMN, and join OL, meeting in / a circle described with radius r and centre ; then LI represents the corre- sponding increment in r, and we have — = limit of — = limit of -=r— = cos 6, dx Ax PL dr dx so that in this case the values of — and — are each equal to cos 9 or -, as before. r dx dr 408 Change of the Independent Variable. dr dO The values of — ^, — , &c, can be also readily represented do dec geometrically in a similar manner. 326. Linear Transformations. — If we are given x = aX+ bY+ cZ, y=a'X + b'Y+c'Z, z=a"X+b"Y+ c"Z, (20) then any function V, of x, y and s, is transformed into a func- tion of X, Y y Z\ and, as in Ex. 2, Art. 98, we have dV_ dV ,dV „d_V dX dx dy dz dV_ h dV h ,dV h „dV dY dx dy dz ' dV_ dV ,dV „dV dZ dx dy dz Again, proceeding to second differentiation, we get f dV „dV\ ,d(dV ,dV „dV\ — + a — )+af—[a—-+a— + a — dy dz J dy\ dx dy dz J (PV_ d_( dV dX 2 dx \ dx „ d ( dV ,dV „dV + a — [a — + a -— + a dz \ dx dy dz ^d 2 V ,d 2 V „d 2 V , „d 2 V = a" T7 + 2aa ~—r- + 2aa -=—7 + 2a a - r - dx 2 dxdy dxdz dz & , 2 d>v „,d>r Similarly we have dY 2 dx 2 dy 2 dz 2 dxdy + iW -—— +2b b — - ; dxdz dydz Orthogonal Transformations. 409 < ZZ = ^^Z ^^Z+^JZ zcc' — dZ 2 dx 2 dy % d% 2 dxdy „d 2 V , „d 2 V + 2CC -—r + 2C C -—- . dxd% dyd% 327. Orthogonal Transformations. — If the transfor- mation be such that x % + y 2 + z 2 = X 2 + Y 2 + Z 2 , we have a 2 + d 2 + d' 2 = i, b 2 +b n + b" 2 =i, c* + c n + c"* = 1. (21) ab + db' + d'b" = o, ac + dc' + d'c" = o, bc + b'c'+b"c" = o. (22) Again, multiplying the first of equations (20) by a, the second by a', and the third by d\ we get on addition, by aid of (21) and (22), X = ax + dy + d'z. In like manner, if the equations (20) be respectively multiplied by b, &', b'\ we get Y = bx + b'y + b"z ; similarly Z = ex + c'y + c"z. If these equations be squared and added, we obtain a? + b 2 + c 2 = 1, a' 2 + b /2 + c' 2 = 1, a" 2 + b" 2 + c" 2 = 1 . (23) ad + bb' + cc f = o, m" + bb' f + cc" = o, dd r + b'b" + e'e" = o. (24) Hence in this case, if the equations of the last Article be added, we shall have d 2 V d 2 V d 2 V_d 2 V cPV (PV dx 2 + dy % + dz 2 " dX % + dY 2 + dZ 2 ' ^ 4-io Change of the Independent Variable. The transformations in this and the preceding Article are necessary when the axes of co-ordinates are changed in Analytic Greometry of three dimensions ; and equation (25) shows that, in transforming from one rectangular system to d 2 V d 2 V d 2 V another, the function — rT + ——r + - TT - is unaltered. dx* dy* dz" 328. General Case of Transformation for Two Independent Variables. — Suppose that we are given the equations * = *(*-, 0), t/ = +{r,9), (26) then any function V of x and y may be regarded as a function of r and 0, and we have, from (9), dV_dVdx dVdy dO 'dxdQ^d^dQ' dV_dVdx dVdy dr dx dr dy dr 9 where the values of -^, -^, — , ~ can be determined from da dv dr dr equations (26). Whenever these equations can be solved for r and 0, separately, we can determine, by direct differentiation, the values of — , —,—,—, and hence by substituting in (13) ax ay ax ay we can obtain the values of - T — and -7—. dx dy When, however, this process is impracticable we can ob- tain the values of t-> -7-, &c., by solving for — and — ax ay ax ay from the preoeding equations. Thus, we obtain dV dy dV dy dV aWdf'~dr"d9 , N — = .. • (27) dx dx dy dx dy dO dr dr dd Transformation for Two Independent Variables. 411 dVdx dV dx dV dO^'aVdO dy dx dy dxdy ^ ' dr dd dO dr d 2 V d 2 V The values of -r^-, -r— -, &c., can be deduced from these : dx 2 dy 2 but the general formulae are too complicated to be of much interest or utility. 329. Concomitant Functions. — We add one or two results in connexion with linear transformations, commencing with the case of two variables. We suppose x and y changed into aX + bY and a'X + b'Y, respectively, so that any func- tion (j)(x, y) is transformed into a function of X and Y; let the latter be denoted by 0i (X, Y), and we have {x,y) = 1 (X, Y). Again, let x' and y' be transformed by the same substitu- tions, i.e., x' =aX f +b Y\ y' = a'X' + V Y'; then since x + kx' = a(X + kX') + b^Y+kY'), and y + ky = d(X + kX') + b\Y+kY), it is evident that cf>(x + kx, y + ky') = fa(X + kX\ Y+kY'). Hence, expanding by the theorem of Art. 127, and equating like powers of k, we get x > d ± + /± =X '%+r%, (29) dx " dy dX d¥ v ; x ^ + 2x y^Ax>^ 2 TY'p^ + Y-% dx 2 J dxdy J dy 2 dX 2 dXdY dY 2 &c. &c. (30) 412 Change of the Independent Variable, Accordingly, if u represent any function of x and y, the expressions denoted by ( ,d d\ { , d dY [x — +y —)ii, [x -r + y — u, &c, V dx u dyj ' V dx y dy) ' ' are unaltered by linear transformation. Similar results obviously hold for linear transformations whatever be the number of variables (Salmon's Higher Algebra, Art. 125). Functions, such as the above, whose relations to a quantic are unaltered by linear transformation, have been called con- comitants by Professor Sylvester. 330. Transformation of Coordinate Axes. — When applied to transformation from one system of coordinate axes to another, the preceding leads to some important results, by applying Boole's method* (Salmon's Conies, Art. 159). For in the case of two dimensions when the origin is unaltered we have x 2 + 2x1/ cos w + y 2 = X' 2 + zX'Y' cos Q + Y r \ (31) where w and £2 denote the angle between the original axes and that between the transformed axes, respectively. Multiply (31) by A, and add to (30): then denoting

dxdz 9 dydz dydz d 2 u . = o. (34) But, as the transformed expression must also be the product of two linear factors, we have d 2 u x d 2 u d 2 u — 2 + A, dx 2 d 2 u d 2 u A dxdy 9 dxdz d 2 u dydx dy' dydz d 2 u d 2 u d 2 u , dxdz 9 dydz 9 dz 2 d*U . dX 2 + * 9 d 2 U d 2 U dXdY 9 dXdZ d 2 TJ d 2 U . d 2 U dXdY 9 dY 2 ' '•' dYdZ d 2 JJ d 2 U d 2 U dXdZ 9 dYdZ 9 dZ 2 + A (35) Orthogonal Transformation. 4i5 Equating the coefficients of like powers of A, we see that the expressions d 2 u d 2 u d 2 u ay" dx 2 dv 2 dz 2 ' d 2 u d 2 u dec 2 dy 2 and d 2 u\ 2 . . . x n9 the determinant dyi dxi • • - yn), the variables x l9 x %9 . . . x n being understood. Jacobians. 417 If the equations for y ly y 2} . . . y n be of the following form : yi =/(4 y% =M%i, a*), y% =/ 3 (^i, #2, #3), it is obvious that their Jacobian reduces to its leading term, viz., J= dyi dy* dyn . g . » This is a case of a more general theorem, which will be given subsequently (Art. 336). Examples. 1. Find the Jacobian of y\, y%, ... y n , being given y\ — 1 - »i, ^2 = a?i(i - #2), ^3 = #1 #2(1 - %i) ... ^n= #1 #2 • . • #»-i(i - fl?»). ^««. /= (- iJ'W 1 ^"- 2 . . . %n-i> 2. Find the Jacobian of x\,x%,... x n with respect to 0i, 2 , • • . 0«, being given #1 = cos 0i, #2 = sin 0i cos 02, %% = sin 0i sin 02 cos 03, . . . x n = sin 0i sin 02 sin 03 ... sin 0„-i cos n . d(xi, X2, . . • x n ) , . ■^ ws - ^/n a T- = (~ J ) n sm M 0i • sin*" 1 2 . . . sm0 M . »(01j 02, • • • 0«) 333. Case of tSae Functions not being Indepen- dent. — If the system y x , y 2 , . . . y n be connected by a re- lation, it is easily seen that their Jacobian is always zero. For, suppose the equation of connexion represented by f&i, y*> • • • y n ) = o ; 2 E 4 1 8 Jacolians. then, differentiating with respect to the variables x ly x 2 . . . x ny we get the following system of equations : — dF dy-i dF dy 2 ^ dF dy n _ dy x dx x dy 2 dxi dy n dx x dF dy x dF dy 2 dF dy n dy x dx 2 dy 2 dx 2 dy n dx 2 dF dy x dF dy 2 dF dy n dy x dx n dy 2 dx n dy n dx n ,. . L . dF dF dF whence, eliminating — , — , . . . — , we get dyx ay 2 ay n d(yi, y%> ■ . . y n ) = Q , , The converse of this result will he established in Art. 337 ; and we infer that whenever the Jacobian of a system of functions vanishes identically, the functions are not indepen- dent. This is an extension of the result arrived at in Art. 314. 334. Case of Functions of Functions. — If we sup- pose Ui, u 2 , u 3 to be functions of y ly y 2i y d , where y 1} y 2 , y d are given functions of x lf x 2 , x 3 ; then we have dui dui dy x dux dy 2 dux dy z dxx dyx dxi dy 2 dxx dy z dx x du x dux dyx du x dy 2 du x dy z dx 2 dy Y dx 2 dy 2 dx 2 dy 3 dx 2 dux dux dyx du x dy 2 du x dy 3 dx 3 dyx dx 3 dy 2 dx 3 dy % dx 3 &c. General Theorem on Jacobians. 419 Hence, by the ordinary rule for the multiplication of de- terminants, we get dux dux dux dx^ dx 2 dx z i du 2 du 2 du 2 ClXx CtX 2 dX 3 j \du 3 du 3 du 3 OjXx Ct/0S 2 0/X 3 dux dtix dux dyx dy 2 dy 3 du 2 du 2 du 2 dyx dy 2 dy 3 du 3 du 3 du 3 dy' dy 2 ' dy 3 • dy x dyx dy t OjXx CvX-2, CIX 3 dy* dy 2 dy 2 dx x dx 2 dx 3 dy 3 dy 3 dy 3 dxx dx 2 dx 3 (40) or d(ux, u 2 , u 3 ) _ d(ux, u 2 , u 3 ) d(yx, y 2 , y 3 ) d(x 1} x 2 , x 3 ) d{y x , y 2 , y 3 ) ' d(xx, x 2 , x 3 )' It follows as a particular case, that d{yi>y*> ^3) d(x h x 2 , x 3 ) d{xx, x 2 , x 3 ) d(yx, y 2 , y 3 ) = 1. (41) These results are readily generalized, and it can be shown by the method given above, that d(ux, u 2 , .. . u n ) ~ d(u u u 2 , ... u n ) d(y x , y 2i . . . y n ) d(x ly #2, • • • %n) d t (yx, y*, ... y n ) ' d(x if x 2 , . . . x n )' ^ ' This is a generalization of the elementary theorem (Art. 19), du du dy dx dy dx' Again, d(yx, 3/2, • . • y n ) d(xx, x 2 , . . . x n ) _ d t (xx, x 2 , . . . x n ) d(y l9 y 2 , . . . y n ) (43) This may be regarded as a generalization of the result dx 1 dy " dy' dx 2 E 2 420 Jacobians. 335. Jacobian of Implicit Functions. — Next, if u, v, w, instead of being given explicitly in terms of x, y, z } be connected with them by equations such as Fi{x,y,z,u,v,w) = o, F 2 (x,y,z,u,v,w) = o, F z (x,y,z y u 9 v,w) = o, then u, v, w may be regarded as implicit functions of x, y, z. In this case we have, by differentiation, dF x dF x du dFx dv dF x dw _ dx du dx dv dx dw dx dF x dF t du dFx dv dF x dw _ dy du dy dv dy dw dy dF 2 dFz du dF 2 dv dF 2 dio + — + — + -r— T- = O, dx du dx dv dx dw dx Hence we observe, from the ordinary rule for multipli- cation of determinants, that du dv dw dx' dx* dx du dv dw dy' W dy du dv dw dz' Tz* dz (44) dFx dFi dFx du ' dv ' dw dll dF* dF, du ' dv ' dw dF 3 dF s dF* du ' dv ' dw This result may be writtten d{Fx, -F a , F 3 ) d(u, v, w) d{u,v,tv) 'd(x,y,z) d(x,y f z) The preceding can be generalized, and it can be readily shown by a like demonstration that if y i9 y if y^ • • • y n dF\ dFx dFx dx ' dy ' dz dF % dF\ dl<\ dx ' dy ' dz dF\ dFs dJF\ dx ' dy ' dz d(Fx, F„ F z ) Jacobian of Implicit Functions. 421 are connected with x i9 x 2 , x z . . . x n by n equations of the form F l (a?!, x 2 . . . x n9 y l9 y 2 . . . y n ) - o, F 2 (x h x 2 ... x n , y l9 y 2 .. . y n ) = o, F n (x l9 x 2 . .. x n , y l9 y 2 . . . y n ) = o, we shall have the following relation between the Jacobians : d(F l9 F 29 ... F n ) d{y x y 29 ...y n ) d{F l9 F 29 ...F n ) d(yi> y*> • • • y n ) ' d[x 9 x 29 ... x n ) d(x l9 x 29 . . . x n )' Accordingly d(F l9 F 29 ... F n ) d{yi, y 2 , . . .y n ) , , n d(x l9 x 2 , . . . x n ) , , d(x i9 x 29 ...x n ) K } d(F l9 F 2 ,... F n ) ' l45; d(yi,yz,-- • y n ) 336. Again, if we suppose that the equations connecting the variables are transformed, by elimination or otherwise, to the following shape — 01 (x l9 x 29 . . . x m yi) = o, 2 (#2, #3, ... %n 9 2/ij y*) = O, ^3 (#3, x i9 ... x n9 y h y 29 y z ) = o, #»(#», yu ^2, • . . y») = o, then the Jacobian determinant d{(f>i 9 2i . . ■ i dn) ) • • • <^n) reduces to dfa dfa dxi dx 2 d(p n ClXfi Accord; ingly, in this case, the Jacobian d(f)i dfa dcj> d[yi> 2/2, • • a \X\ 9 x%) m . . a? n ) v dfa dfa Q/Xrfl d(p n dy x dy z dy n (46) 337. Case where J = o. — We can now prove that if the Jacobian vanishes, the functions y lf y i9 . . . y n are not indepen- dent of one another. For, if J(y lf y 2 , . . . y n ) = o, we must have dfa dfa d(p n dxi dx z ' dx n ' that is, we have -p 1 = o f or some value of i between 1 and n. Hence fa must not contain xi ; and accordingly the cor- responding equation is of the form fa (x i+1 , . . . % n , y ly y 2 , . . . yi) = o. Hence between this and the remaining equations, fan = O, 0i+ 2 = 0, ... (p n = O, the variables asu-u x t&j • • • x n can be eliminated so as to give a final equation between y iy y 2 , . . . y n alone. This establishes our theorem. Jacobian of Implicit Functions. 423 338. In the particular case where yx = Fi(xi, x 2 , ... x n ), y* = Fi{yi, x*, ... x n ), y n = F n {y l , y % , ... h'n-l, x n ), we have d{y u y 2 , . . . t/n) ^cfyi dy2 dyn Ct> («?ij X2, ... Xn) CtXi (X'X<2, (XtJOn (47) It may be observed that the theory of Jacobians is of fundamental importance in the transformation of Multiple Integrals (see Int. Calc, Art. 225). Examples. i. Find the Jacobian of yi, y%, ... y n with respect to r, 9i, 62, • ■ ■ n .i, being given the system of equations yx = r cos 0i, y 2 = r sin 0i cos 62, yz = r sin di sin 62 cos 03, • • • y n = r sin 0i sin 02 ... sin d n -\. If we square and add we get yi 2 + yz 2 + • • • yn 2 = ^ 2 - Assuming this instead of the last of the given equations we readily find J = r n ~ l sin M " 2 0i sin" -3 2 • • • sin 0„_ 2 . 2. Find the Jacobian of yi, t/2, • • • yn, being given y\ = %i (1 - #2), 2/2 = SO1X2 ( 1 - afo) . . . 2/n_l =#i#2 • ■'. #w_l(l - #»)> ^n = #1 #2 • . • # n . Here 2/1 + 2/2 + • • • yn = %i, and we get <%i, y2> • • • y») _, „ o «(#1, %2> ■ • ■ x n) 4 2 4 Jacobians. 339. If y lf y 2 , . . . y n , which are given functions of the n variables Xi, x 2 , . . . x m be conneoted by an independent re- lation F{yi, y 2 , . . • y n ) = o, (48) we may, in virtue of this relation, regard one of the variables, x n suppose, as a function of the remaining variables, and thus consider y i} y 2 , . . . y n ~\ as functions of x Xi o? a , . . . aw* In this case it can be shown that dF_ d(Vi> ^, ... yn-i) _djfa d(y l9 y 2 , ... y n ) d(xi, a? 2 , • • . ^Vi) d_F e?(a?i, x 2 , ... #»)" For, if we regard x n as a function of o?i, we have d . , _ #1 ctyi dxn d^ _dy 2 dy 2 dx n - W$?i ui^x ClXift (IX\ OjX\ dX\ (ZXji (XX\ Also, from equation (48), c?i^ e£F ^ f/i^ 7 dF <&„ „ h = O, + = O, CCC. dxi dx n dxi dx 2 dx n dx 2 dF dF dF « • n j -v CIX\ » QjX 2 \ ClXfl—l Again, let A 1= — „ A, = 3F . . . A„_, = ^ r ; CvXoi (JjJUiyi, LvX, n w^ n then — = - Ai, — = - A 2 , . . . -r — = - A M _i. W#?l W#?2 fl'^'Ti— 1 d dy x dy x d dy x dy x Hence — (2/1) = - — Ai -r- , t - (yi) =3 — A 3 -7—, &c. rfa?i <^i a# w a# 2 dx 2 dx n &G. accordingly, substituting in the Jacobian d(yi, p2 9 ♦ . . yn-i) CC [Xly X 2 ) . . . Xn^lJ Jacobian of Implicit Functions. 425 it becomes dy x _ dy x OjX\ QjXfi dyx A ^A OjX% Q/Xi)i dyx . e&/i dy% x dy 2 dy% x #2 dy % . <% 3 dyn-x x %«-i dyn-x ^ wj/^-i If this determinant be bordered by introducing an addi- tional column as in the following determinant, the other terms of the additional row being cyphers, its value is readily seen to be or dyx QjX\ dyx (X1X1 dyx dy* dxi dy 2 dx 2 dy* dy n -x dy n -x dy n -x dx x ' dx % ' (JvXift K A 2 , . . . 1 dyx dxi dyx dx% dyx Ci/Xfi I dy 2 (XX\ djh dx 2 dy% CtXfi dF • . (XXi)i dy^x dy n -x dy n -x CtX\ dx, ' iX/JUryi dF dF dF dx* (1X2 CvXm 426 Jacobians. Again, we have dF dF dy dx x dF dy 2 dtfx dx x dy % dxi dF dF dt/i dF dy z dx 2 dyi dx z dy 2 dx 2 + + dF dy n dy n dxi ' dF dy n (X'U 'n UiX'(x)\K 2. If y = -F(0) t=f(u), u = (x), find the value of — 5. Am. F(t)f{u) "(z) + W{x)Y{f'(u) F\t) + (f(u)) 2 F"(l)}. 3. Change the independent variable from x to % in the equation . d 2 y o % o ' 1 1 # 4 — — — 2nx 6 — + a'y = o, where x = -. . <^V , 2 (» + I) <&/ 4. Transform (1 - # 2 ) — — x — + a 2 y = o, being given # = sinz. . d 2 y Am. — — + a 2 y — o. dz 2 5. If Fbe a function of r, where r 2 = # 2 + y 2 , prove that d 2 V d 2 V _d*V 1 ^T rf« 2 dy 2 dr 2 r dr 6. If V be a function of r, where r 2 = x 2 + y 2 + z 2 , prove that d 2 V d 2 V d 2 V_^Z 2 ^L dx 2 dy'* dz 2 dr 2 r dr 7. If # = r sin cos d>, v = r sm sin d>, 2 = r cos 0, prove that — ■ = — , dr dx where in finding — , 6 and <\> are regarded as constants ; while in finding dr — , y and z are regarded as constants. dx 8. If z be a function of two independent variables, x and y, which are connected with two other variables, u and v, by the equations f\ (%, y, w, v) = o, f 2 (x, y,u,v) = o; dz dz dz dz show how to express — and — m terms of -7- and — . 428 Examples. 9. Transform the equation d 2 y 2x dy y dx 2 1 + x 2 dx (1 + x 1 ) 2 = into another in -which is the independent variable, supposing x = tan 0. d 2 y Am. MT + y = o. 10. If z be a function of x and 2/, and u = px + qy — z, prove that when p and q are taken as independent variables, we have du dht d 2 u dht ir = %, T=y, -J-; dp dq dp 1 rt — s 2 ' dp dq rt — s 2 ' dq 2 rt — where p, q, r, s, t, denote the partial differential coefficients of z, as in Art. 313. ir. If the equation d n y . . d n ~ l y xn-^ + AiX*- 1 - — f + dx 11 dx 7 *' 1 dy + An-lX— + A n = o dx be transformed to depend on 0, where x = e d , prove that the coefficients in the transformed differential equation are all constants. 12. In orthogonal transformations, prove that dV 2 dY 2 dV 2 _ dV 2 dV 2 dV 2 dx 2 ' + dy 2 + dlF ~ dX 2 + df 2 + ~dZ 2 ' n*), v(t) find the value of the Jacobian y\ — r sin d\ sin 2 , y% = r sin 0i cos 2 , yz = r cos 0i sin 3 , y± = r cos 0i cos 03, <*(yi> y^ yz> y±) Am. r z sin 0i cos Q\. d(r, 0i, 02, 3 ) Examples. 429 15. Find the Jacobian -V^ — - — ;, being given d(r, e, <*>)' x = r cos 6 cos Vi — w 2 sin 2 0, where wfi + n 2 = 1. Ans. r 2 (w 2 cos 2 4- w 2 cos 2 0) Vi — w 2 sin 2 (£ Vi — w 2 sin 2 16. Being given X% X3 X\ Xz Xi x% v\ = — , y% = , y$ = , Xl x z x z find the value of the Jacobian of yi, y%, yz. 17. In the Jacobian d(yi, y%, . . . y n ) d{X\, X%, . . . X n ) if we make prove that it becomes y n = Ans. 4. u, Ml, «2, • . - Un du dui dui du n d%\ dx\ dx\ dx\ I u» +1 du dx% du\ dx% du% dx% du n dx% du du\ ClXii du% CvXyi du n dx n This determinant is represented by the notation K(u, u\, . . . u n ). 18. If a homogeneous relation exists between u, u\, . . . u n , prove that E{u, «i, ... u n ) = o. 19. In the same case, if yi, y%, . . . y n possess a common factor, so that yi = UiU, &c, prove that J{yu yi> • • • y») = 2u n J{ui, « 2 , . . . «») - w"- 1 jt(w, «i, . . %). 43o Miscellaneous Examples. Miscellaneous Examples. i . If o, £, y be the roots of the cubic x z + px 2 + qx + r = o, dp dq dr da da da dp dq dr df? ~d& dp dp dq , dr dy' dy ' dy show that = ( 7 -jB)(j8-a)(a-7). 2. Being given the three simultaneous equations (pl{%l, %2, %3, #4) = O, $2(21, %% y #3, %l) = °> rove that x V V ^r + z — -will also be a solution of it. r dx dy dz 4. If x and y be not independent, prove that the equation — — - = does not hold, in general. axa V a ^ ax 5. Prove that the points of intersection of a curve of the fourth degree with its asymptotes lie on a conic ; and in general for a curve of the degree n they lie on a curve of the degree n — 2. 6. Prove that every curve of the third degree is capable of being projected into a central curve. (Chaales.) For if the harmonic polar of a point of inflexion be projected to infinity, the point of inflexion will be projected into a centre of the projected curve {see p. 282). 7. Two ellipses having the same foci are described infinitely near one another; how does the interval between them vary? (a). How will the interval vary if the ellipses be concentric, similar, and similarly placed ? 8. Eliminate the arbitrary functions from the equation z = ${x) . $(y). 9. Show that in order to eliminate n arbitrary functions^ from an equation containing two independent variables, it is, in general, requisite to proceed to differentials of the order 2n - 1. How many resulting equations would be ob- tained in this case ? Miscellaneous Examples. 431 10. In the Lemniscate r % = a % cos 20, show that the angle between the tan- gent and radius vector is - + 20. 2 ii. In transforming from rectangular to polar coordinates, prove that 12. Prove that the ellipses «V + b*x 2 = aW (i), a?x 2 sec 4 (p + % 2 cosec 4

( n ~ 2) (> 2 > &c. 1.2 / r \ /' 27. If x + iy = (a + i/3) w , where i = V- I, prove that dx 2 +dy 2 da 2 + d& 2 . = n i . x* + y 2 a 2 + j8 a 1 j-, > dd> /i — e 2 sin 2 ^zrrr prove that — + J a-^Ti ^/j c 3 # >/ 1 - c 2 srn 2 »J/ 1 „ . »

CM+1 ^2n 2 I.2.3. • • 2n ■ cos ( 2W + *) • COS 2 "" 1 ^) % 2 »» +1 1.2.3... { 2n + J ) sm ( 2 ^ + 2 ) • COS 2w+2 = \ */ ,v2«.+2 37. If u be a homogeneous function of the n th degree in x, y, z, and wi, #2, #3, denote its differential coefficients with regard to x, y, z, respectively, while #n, 2*12, &c., in like manner denote its second differential coefficients ; prove that #11, #12, #13, #1 #21, #22, #23, #2 #31, #32, #33, #3 #1, #2, #3, O n — 1 Mil, #12, #13 #21, #22, #23 #31, \ #32, #33 Miscellaneous Examples. 435 38. If u be a homogeneous function of the n th degree in x, y, z, w, show that for all values of the variables which satisfy the equation u = o we have w~ (n - if Wil, U12, U13, ttu W21 5 W22, 2<23, ^24 %lj W32, ^33, W34 %U, W42, W43, W44 Wli, W12, Wl3, Wi #21, #22, #23, #2 #31, #32, #33, #3 #1, #2, «3, O 39. Show that the equation d ( 9N dP) 1 ^p is satisfied if P is any of the quantities - - /jfi, (1 - /* 2 ) cos 20, (1 - fj) sin 20, fi^/i - fjfi cos 0, ^ Vi - fx 2 sin 0, or any linear function of them. 40. If x 4- A be substituted for x in the quantic , n(n — 1) a x n + naix n ~ 1 + — - - a%x n ~ 2 + &c. + a n , and if a'o, «'i, ....«'*••••• denote the corresponding coefficients in the new quantic ; prove that da'r , —- = ra r-i. It is easily seen that in this case we have r(r — 1) a' r = a r + ra r -\ X + 1 . 2 «r-2A 2 + &C. + a \ r ; &c. 41. If 4> be any function of the differences of the roots of the quantic in the preceding example, prove that Id d \ aai da% d_ daz + + na% d \ da n ) This result follows immediately, since any function of the differences of the roots remains unaltered when x + A is substituted for x, and accordingly d with respect to the circle described on the line joining the foci as diameter, has for its equation a 2 b 2 a> y* 46. If the second term be removed from the quantic («o, ai, fl2, . . . «n) (#, y) n bv the substitution of a; y, instead of a;, and if the new quantic be denoted by (Ao, o, A%, A3, . . . A n ) [x, y) ; show that the successive coefficients A2, A3, . . . An are obtained by the substitution of a\ for x and — oq for y in the series of quantics (a , ai, az) (x, y), {a 0) «i, a 2 , «s) (#, y), . . . (#0, «i, . • • «n) («, y)- 47. Distinguish the maxima and minima values of 1 4- 2x tan -1 a; 1+a; 2 ' a'a; 2 + 2&'a; + c ,_ . 1 ^y _ (ac - b 2 ) y 2 + (ae' + a'c - 2bb')y + a'c - b' 2 2 «# (ab') x 2 — (ca') x + {be) Miscellaneous Examples. 437 49. IflX + mY+nZ, l'X + m'Y+ n'Z, l"X + m"Y + n"Z, be substituted for x, y, z, in the quadratic expression ax 2 + by 2 + cz 2 + 2dyz + 2ezx + 2fxy ; and if a', b\ c', d\ e', /', be the respective coefficients in the new expression ; prove that e\ d\ c', = o whenever a, f, e f, h d e, d, c = 0* 50. If the transformation be orthogonal, i. e. if & + y i + S 2 = X 2 + Y 2 + £ 2 , prove that the preceding determinants are equal to one another. 5 1 . Prove that the maximum and minimum values of the expression ax^ + $bx z — 6cx 2 + \dx + e are the roots of the cubic a 3 s 3 - 3 {a 2 ! - 3# 2 ) z 2 + 3 («1 2 - I8J57) 2 - A = o, where II = ac — b 2 , I *= ae — ^bd + 3c 2 , / = a, b, c b, c } d e, d, e , and A = I 3 - 27J" 2 . By Art. 138 it is evident that the equation in z is obtained by substituting e — z instead of e in the discriminant of the biquadratic ; accordingly we have for the resulting equation (J- azf =27(7- zEfj since the discriminant of the biquadratic is J3 _ 27/2 _ o. In general, the equation in z whose roots are the n — 1 maximum and mini- mum values of a given function of n dimensions in x, can be got from tbe dis- criminant of the function, by substituting in it, instead of the absolute term, the absolute term minus z. It is evident that the discriminant of the function in x is, in all cases, the absolute term in the equation in z. 52. If A be the product of the squares of the differences of the roots of # 3 — px 2 -f qx — r = o, 43 8 Miscellaneous Examples. find an expression in terms of the roots for — , by solving from three equations of the form dA dA dp dA dq dA dr da dp da dq da dr da' Ans. 2 (£ + 7 - 2a) {y + a - 20) (a + - 27). 53. If X + Y*y - 1 be a function of x + y */ — 1, prove that X and Y satisfy the equations d*X d*X , d 2 Y d*Y H 5- = o, and 1 = o. dx 2 dy- dx % dy 1 54. If the three sides of a triangle are a, a + a, a + /3, where a and £ are infinitesimals, find the three angles, expressed in circular measure. . 7r a + fi it 2a — /J w 2fi — a Ans. z , - H Z.J - + -. 3 ciy/ ?> 3 a\/z ^ ay^s 55. If 2/ = # + ax 3 , where a is an infinitesimal, find the order of the error in taking x — y — ay 3 . 56. The sides a, b, c, of a right-angled triangle become a -t- a, b + /3, c + 7, where a, )8, 7 are infinitesimals ; find the change in the right angle. . cy — aa — bfi Ans. -. ab 57. If a curve be given by the equations 2x = \/t 2 + 2t + \/p - 2*, 2y = ), and

, prove that the curves represented by the transformed equations intersect at the same angle as the original curves. (Mr. W. Roberts, ZiouvilWs Journal, Tome 13, p. 209). Miscellaneous Examples. 441 This result follows immediately from the property that — is unaltered by the transformation in question. 76. A system of concentric and similarly situated equilateral hyperbolas is cut by another such system having the same centre, under a constant angle, which is double that under which the axes of the two systems intersect. Ibid., p. 210. 77. In a triangle formed by three arcs of equilateral hyperbolas, having the same centre (or by parabolas having the same focus), the sum of the angles is equal to two right angles. Ibid., p. 210. 78. Being given two hyperbolic tangents to a conic, the arc of any third hyperbolic tangent, which is intercepted by the two first, subtends a constant angle at the focus. Ibid., p. 212. An equilateral hyperbola which touches a conic, and is concentric with it, is called a hyperbolic tangent to the conic. 79. A system of confocal cassinoids is cut orthogonally by a system of equi- lateral hyperbolas passing through the foci and concentric with the cassinoids. Ibid., p. 214. The student will find a number of other remarkable theorems^ deduced by the same general method, in Mr. Eoberts' Memoir. This method is an exten- sion of the method of inversion. 80. If P n be the coefficient of x n in the expansion of (1 - 2ax + x 2 )-', prove the two following equations : dP n (a 2 — 1) — — = naP n — nP n -\, da nP n = {2n - 1) aP n .i - (n - 1) P M _ 2 . 81. If at each point on a curve a right line be drawn making a constant angle with the radius vector drawn to a fixed point, prove that the envelope of the line so drawn is a carve which is similar to the negative pedal of the given curve, taken with respect to the fixed point as pole. 82. If 2 U = ax 1 + 2bxy + eg 2 , 2 V = a'x 2 + 2b' xy + e'y 2 , = AU 2 + 2BUV+ CV 2 , find A, B, C. and dU dU dx' dy dV dV dx ' dy 83. Prove that the values of the diameters of curvature of the curve y 2 =f(x) where it meets the axis of x are /'(«)» /'(#)> .... if a, /8, ... be the roots of /(*)=o. Hence find the radii of curvature of y 2 = (x 2 - m 2 ) (x — a) at such points. 84. A constant length PQ is measured along the tangent at any point P on a curve ; give, by aid of Art. 290, a geometrical construction for the centre of curvature of the locus of the point Q. 44 2 Miscellaneous Examples. 85. In same case, if PQ[ be measured equal to PQ, in the opposite direction along the tangent, prove that the point P, and the centres of curvature of the loci of Q, and Q', lie in directum. 86. A framework is formed by four rods jointed together at their extremities ; prove that the distance between the middle points of either pair of opposite sides is a maximum or a minimum when the other rods are parallel,' being a maximum when the rods are uncrossed, and a minimum when they cross. 87. At each point of a closed curve are formed the rectangular hyperbola, and the parabola, of closest contact ; show that the arc of the curve described by the centre of the hyperbola will exceed the arc of the oval by twice the arc of the curve described by the focus of the parabola ; provided that no parabola has five-pointic contact with the curve. (Gamb. Math. Trip. 1875.) 88. A curve rolls on a straight line, determine the nature of the motion of one of its involutes. (Prof. Crofton.) 89. Prove the following properties of the three-cusped hypocycloid : — (1). The segment intercepted by any two of the three branches on any tangent to the third is of constant length. (2). The locus of the middle point of the segment is a circle. (3). The tangents to these branches at its extremities intersect at right angles on the inscribed circle. (4). The normals corresponding to the three tangents intersect in a common point, which lies on the circum- scribed circle. Definition. — The right line joining the feet of the perpendiculars drawn to the sides of a triangle from any point on its circumscribed circle is called the pedal line of the triangle relative to the point. 90. Prove that the envelope of the pedal line of a triangle is a three-cusped hypocycloid, having its centre at the centre of the nine-point circle of the triangle. (Steiner, JJeber eine besondere curve dritter klasse, und vierten grades, Crelle, 1857.) This is called Steiner'' s Envelope, and the theorem can be demonstrated, geometrically, as follows : — Let P be any point on the circumscribed circle of a triangle ABO, of which D is the intersection of the perpendiculars ; then it can be shown without difficulty, that the pedal line corresponding to P passes through the middle point of DP. Let Q denote this middle point, then Q lies on the nine-point circle of the triangle ABC. If be the centre of the nine-point circle, it is easily seen that, as Q moves round the circle, the angular motion of the pedal line is half that of OQ, and takes place in the opposite direction. Let R be the other point in which the pedal line cuts the nine-point circle, and, by drawing a consecutive position of the moving line, it can be seen immediately that the corresponding point T on the envelope is obtained by taking QT = QR. Hence it can be readily shown that the locus of T is a three-cusped hypocycloid. This can also be easily proved otherwise by the method of Art. 295 (a). 91. The envelope of the tangent at the vertex of a parabola which touches three given lines is a three-cusped hypocycloid. 92. The envelope of the parabola is the same hypocycloid. On the Failure of Taylor's Theorem. 443 For fuller information on Steiner's Envelope, and the general properties of the three-cusped hypocycloid, the student is referred, amongst other memoirs, to Cremona, Crelle, 1865. Townsend, Educ. Times. Reprint. 1866. Ferrers, Quar. Jour, of Math., 1866. Serret, Now. Ann., 1870. Painvin, ibid., 1870. Cahen, ibid., 1875. On the Failure of Taylor's Theorem. As no mention has been made in Chapter III. of the cases when Taylor's Series becomes inapplicable, or what is usually called the failure of Taylor's Theorem, the following extract from M. Navier's Legons d' Analyse is intro- duced for the purpose of elucidating this case : — On the Case when* for certain particular Values of the Variable, Taylor's iSeries does not give the Uevelopinent of the Function. — The existence of Taylor's Series supposes that the function f(x) and its differential coefficients f(x), f"(x), &c, do not become infinite for the value of x from which the increment h is counted. If the contrary takes place, the series will be inapplicable. Fix) Suppose, for example, that/(#) is of the form - — K —^-, m being any positive number, and Fix) a function of x which does not become either zero or infinite when x = a. Fix + h) If, conformably to our rules, - — y — V- be developed in a series of posi- ' J ' ix + h - a) m tive powers of h, all the terms would become infinite when we make x = a. At F(a + h) _ t the same time the function has then a determinate value, viz. : — — — . -But h m as the development of this value according to powers of h must necessarily con- tain negative powers of h, it cannot be given by Taylor's Series. Taylor's Series naturally gives indeterminate results when, the proposed function fix) containing radicals, the particular value attributed to x causes these radicals to disappear in the function and in its differential coefficients. In order to understand the reason, we remark that a radical of the form p ix - df, p and q denoting whole numbers, which forms part of a function /_(#), gives to this function q different values, real or imaginary. As this same radical is reproduced in the differential coefficients of the function, these coefficients also present a number, q, of values. But, if the particular value a be attributed to <%, the radical will disappear from all the terms of the series, while it remains p always in the function, where it becomes h q . Therefore the series no longer re- presents the function, because the latter has many values, while the series can have but one. The analysis solves this contradiction by giving infinite values to the terms of the series, which consequently does not any longer represent a determined result. The development of f(x) ought, in the case with which we are occupied, to p contain terms of the form h q . We should obtain the development by making x = a + h in the proposed function. 444 On the Failure of Taylor's Theorem. Fractional powers of h would appear in the latter development : for example, suppose f(x) = 2ax — x 2 — a y x 2 — a 2 ; this gives (too /» = 2 («-*) + /»=-2 + */x 2 -a 2 {x 2 -a 2 Y On making x — a, we have fix) = a 2 , and all the differential coefficients "become infinite. This circumstance indicates that the development of f(x + h) ought to contain fractional powers of h when x-a'.'vo. fact the function be- comes then f(a + h) =a 2 - h 2 -{- a^2ah-\- h 2 , of which the development according to powers of h would contain hi, h*, ifi, &c. It should be remarked that a radical contained in the function f(x) may- disappear in two different ways when a particular value is attributed to the variable x, that is, i°, when the quantity contained under the radical vanishes: 2 , when a factor with which the radical may be affected vanishes. In the former case the development according to Taylor's Theorem can never agree with the function f(x + h) for the particular value of x in question, for the reason already indicated. Eut it is not the same in the latter case, because the factor with which the radical is affected, and which becomes zero in the function, may cease to affect the radical in the differential coefficients of higher orders ; in fact it may not disappear at all, and the series may in consequence present the necessary number of values. For example, let the proposed function be f(x) = (x - a) m \'x - b, m being a positive integer. Here we have , , z m (x — a) m f(x) = m (x - a) m - 1 Nx - b + , 2 vx - b i mix — a)™- 1 (x f" (x) = m{m - i) {x - a)™- 2 Vx - b + v ' — Vx-b ${x-b)2 Each differentiation causes one of the factors of (x — a) m to disappear in the first term. After m differentiations these factors would entirely disappear; and consequently the supposition x = a, in causing the first m-derived functions to vanish, will leave the radical vx — b to remain in all the others. Conditions of Maxima and Minima in General. 445 On the Conditions foe a Maximum oe, Minimum of a Function OF ANT ISUMBEK OF VaEIABLES (Art. 1 63). The conditions for a maximum or a minimum in the case of two or of three variables have been given in Chapter X. It can be readily seen that the mode of investigation, and the form of the conditions there given, admit of extension to the case of any number of inde- pendent variables. "We shall commence with the case of four independent variables. Proceed- ing as in Art. 162, it is obvious that the problem reduces to the consideration of a quadratic expression in four variables which shall preserve the same sign for all real values of the variable. Let the quadratic be written in the form #11 #1 2 + #22#2 2 + #33#3 2 + #44#4 2 + 2^12^1 X% + 20,\'zX\ X<$ + 2auX\X± + 2#23#2#3, + 2«24#2#4 + 2 #34 XsXi, (I) in which «n, #12, 022, &c., represent the respective second differential coefficients of the function, as in Art. 162. We shall first investigate the conditions that this expression shall be always a positive quantity ; in this case «n evidently is necessarily positive : again, multiplying by an, the expression may be written in the following form : — (#1121 + o, (#11#22 ~ #12 2 )(#11#33 - #13 2 ) - (#11 #23 ~ ^IJJ#13) 2 > O, #11 #22 — #12 2 , #11 #23 — #12 #13) #11 #24 — #12 #14 #11#23 — #12#13, #11#33 — #13 2 , #11 #34 — #13#14 #11 #24 — #12 #14, #11 #34 — #13 #14, #11 #44 — #14 2 > O. (3) (4) (5) To express this determinant in a simpler form, we write it as follows :- #12, #13, #14 #11#22 — #12 2 , #11 #23 — #12 #13, #11 #24 — #12#14 #11 #23 — #12 #13, #11#33 — #13 3 , #11 #34 — #13#14 #11 #24 — #12 #14, #11 #34 — #13 #14, #11 #44 — #14 2 #11, I 0, #11 0, 0, (6) 446 Notes. Next, to form a new determinant, multiply the first row by 012, «i3, #14, suc- cessively, and add the resulting terms to the 2nd, 3rd, and 4th rows, respec- tively ; then, since each term in the rows after the first contains an as a factor, the determinant is evidently equivalent to an" #11, #12, ai3, #14 #12, #22, #23, #24 #13, #23} #33, #34 #14, #24, #34, #44 (7) In like manner the relation in (4) is at once reducible to the form #11 #11, #12, #13 #12, #22, #23 #13, #23, #33 >o. Hence we conclude that whenever the following conditions are fulfilled, viz. : 1 #11, #12 #11 > 0, #12, #22 >°, #11, #12, #13 #12, #22, #23 >o, #13, #23, #33 #11, #12, #13, #14 #12, #22, #23, #24 #13, #23, #33, #34 #14, #24, #34, #44 >o, (8) the quadratic expression (1) is positive for all real values of x\, #2, #3, #4- Accordingly the conditions are the same as in the case (Art. 162) of three variables, %\, %%, #3 ; with the addition that the determinant (7) shall be also positive. In like manner it can be readily seen that if the second and fourth of the preceding determinants be positive, and the two others negative, the quadratic, expression (1) is negative for all real values of the variables. The last determinant in (8) is called the discriminant of the quadratic func- tion, and the preceding determinant is derived from it by omitting the extreme row and column, and the other is derived from that in like manner. "When the discriminant vanishes, it can be seen without difficulty that the expression (1) is reducible to the sum of three squares. It can be easily proved by induction that the preceding principle holds in general, and that in the case of n variables the conditions can be deduced from the discriminant in the manner indicated above. According as the number of rows in a determinant is even or odd, the de- terminant is said to be one of an even or of an odd order. Conditions of Maxima and Minima in General. 447 If the notation already adopted be generalized, the coefficient of x r 2 is de- noted by cirr, and that of x r x m , by 2a rm . In this case the discriminant of the quadratic function in n variables is #11, #12, #13, #12, #22, #23, #13, #23, #33, #lra» #2m, #3h, #1? #3» (Inn (9) and the conditions that the quadratic expression shall be always positive are, that the determinant (9) and the series of determinants derived in succession by erasing the outside row and column shall be all positive. To establish this result, we multiply the quadratic function by #n, and it is evident that it may be written in the form (#11#1 + #12#2 + • • • #ln#n) 2 + (#11#22 - #12 2 ) #2 2 + • . . + (#ll#nn — #1m 2 ) X n % + 2 (#n#23 — #12#13)#2#3 + &C. -f (2#n# m — a\ r Ci\n)x r X n + • • • In order that this should be always positive, it is necessary that the part after the first term should be always positive. This is a quadratic function of the n — 1 variables x%, %%, . . . x n . Accordingly, assuming that the conditions in question hold for it, its discriminant must be positive, as also the series of deter- minants derived from it. But the discriminant is #11 #22 - #12 , #11 #23 — #12 #13, #11 #23 — #12 #13, #11 #33 — #13 2 , #11 #24 — #12 #14, #11 #34 — #13 #14, #1 1 #2w — #12 #1m, #11 #3m — #13 #1«, #ll#2w — #12#l»i #ll#3w — #13#lw #ll#4w — #14#lrc #11 #««— #in 2 (10) "Writing this as in (6), and proceeding as before, it is easily seen that the determinant becomes 011* #11) #12, #13, • • #lw #12, #22, #23, • • #2w #13) #23, #33) • • • #3» #l?l) #2m, #3M) . • • #WM (») i. e. the discriminant of the function multiplied by #n n_3 44 8 Notes. Hence we infer, that if the principle in question hold for n — I variables it holds for n. But it has been shown to hold in the cases of 3 and 4 variables ; consequently it holds for any number. "We conclude finally that the quadratic expression in n variables is always positive, whenever the series of determinants 011, 011, 012, - • • 01» 011, 012, 013 012, 022, • - 02w 011, 012 t 012, 022, 023 , • . 012, 022 013, 023, 033 • • • 01n, 02», • 0?m , (12) are all positive. Again, if the series of determinants of an even order be all positive, and those of an odd order, commencing with an, be all negative, the quadratic expression is negative for all real values of the variables. Hence we infer that the number of independent conditions for a maximum or a minimum in the case of n variables is n — 1, as stated in Art. 163. It is scarcely necessary to state that similar results hold if we interchange any two of the suffix numbers ; i.e. if any of the coefficients, #22, #33, . . a nn , be taken instead of an as the leading term in the series of determinants. If the determinants in (12) be denoted byAi, A2, A3, . . . A n , it can be proved without difficulty that, whenever none of these determinants vanishes, the qua- dratic expression under consideration may be written in the form Ai Ui* + - w + - U£ + . . . + — TTJ Ai A2 A,i-i (13) Hence, in general, when the quadratic is transformed into a sum of squares, the number of positive squares in the sum depends on the number of continua- tions of signs in the series of determinants in (12). It is easy to see independently that the series of conditions in (12) are neces- sary in order that the quadratic function under consideration should be always positive ; the preceding investigation proves, however, that they are not only necessary, but that they are sufficient. Again, since these results hold if any two or more of the suffix numbers be interchanged, we get the following theorem in the theory of numbers : that if the series of determinants given in (12) be all positive, then every determinant obtained from them by an interchange of the suffix numbers is also necessarily positive. Also, since, when a quadratic expression is reduced to a sum of squares, the number of positive and negative squares in the sum is fixed (Salmon's Higher Algebra, Art. 162), we infer that the number of- variations of sign in any series of determinants obtained from (12) by altering the suffix numbers is the same as the number of variations of sign in the series (12). As already stated, a quadratic expression can be transformed in an infinite number of ways by linear transformations into the sum of a number of squares multiplied by constant coefficients ; there is, however, one mode that is unique, viz., what is styled the orthogonal transformation. Conditions of Maxima and Minima in General. 449 In this case, if Xi, X 2 , Z3, . . . X n denote the new linear functions, we have V= xi* + x z 2 + . . • + Xr? = X1 3 + Z 2 2 + &c. + X M 2 ; and also, denoting the coefficients of the squares in the transformed expression by 01, «2, • • • 0w, V = a\\X^ + 022#2 2 + • • • + Onn&n* + • • • + 2a\%X\%z + 2air%l%r + . . . = «iZi 2 + «2-^2 2 + . . . thiXn 2 . Hence, equating the discriminants of U - \V for the two systems, we get 011 — A, 012, . . 01m 012, 022 — A, 02w «13) 023, • • 03» 017 02m 1 0MM ~ A = (01 - A) (a 3 - A) . . . (a n - A). (14) Accordingly, the coefficients 01, 02, . . . a n , are the roots of the determinant at the left-hand side of equation (14). Moreover, in order that the function U should be always positive or always negative for all real values of the variables x\, %%,... x n , the coefficients «i, 02, • • • 0«j must be all positive in the former case, and all negative in the latter; and consequently, in either case, the roots of the determinant in (14) must all have the same sign. The application of this result to the determination of the conditions of maxima and minima is easily seen ; however, as the conditions thus arrived at are clumsy and complicated in comparison with those given in (12), it is not con- sidered necessary to enter into their discussion here. 2 G INDEX. AcNODE, 259. Approximations, 42. further trigonometrical applica- tions of, 130-8. Arbogast's method of derivations, 88. Arc of plane curve, differential ex- pressions for, 220, 223. Archimedes, spiral of, 301, 303. Asymptotes, definition of, 242, 249. method of finding, 242, 245. number of, 243. parallel, 247. of cubic, 249,^ 325. in polar coordinates, 250. circular, 252. Bernoulli's numbers, 93. series, 70. Bertrand, on limits of Taylor's series, 77- Bobillier's theorem, 368, 374. Boole, on transformation of coordi- nates, 412. Brigg's logarithmic system, 26. Burnside, on covariants, 412. Cardioid, 297, 372. Cartesian oval, or Cartesian, 233, 375- third focus, 376. tangent to, 379. confocals intersect orthogonally, 381. Casey, on new form of tangential equation, 339. on cycloid, 373. on Cartesians, 382. Cassini, oval of, 233, 333. Catenary, 288, 321. Cayley, 259, 266. Centre of curve, 237. Centrode, 363. Change of single independent variable, 399- of two independent variables, 403 , 410. Chasles, on envelope of a carried right line, 356. construction for centre of instan- taneous rotation, 359. generalization of method of draw- ing normals to a roulette, 360. on epicycloids, 373. on Cartesian oval, 376. on cubics, 418. Circle of inflexions in motion of a plane area, 354, 358, 367, 374. Conchoid of Mcomedes, 332, 361. centre of curvature of, 370. Concomitant functions, 411. Condition that Pdx + Qdy is a total differential, 146. Conjugate points, 259. Contact, different orders of, 304. Convexity and concavity, 278. Crofton, on Cartesian oval, 378, 379, 380. Crunode, 259. Cubics, 262, 281, 323, 334. Curvature, radius of, 286/ 287, 295, 297, 301. chord of, 296. at a double point, 310. at a cusp, 311, 313. measure of, on a surface, 209. Cusps, 259, 266, 315. curvature at, 311. Cycloid, 335, 356. equation of, 335, 336. radius of curvature, and evolute, 337- ' length of arc, 338. 452 Index. Descartes, on normal to a roulette, 336. ovals of, 375. Differential coefficients, definition, 5. successive, 34. Differentiation, of a product, 13, 14. a quotient, 15. a power, 16, 17. a function of a function, 17. an inverse function, 18. trigonometrical functions, 19, 20. circular functions, 21, 22. logarithm, 25. exponential functions, 26. functions of two variables, 115. three or more variables, 117. an implicit function, 120. partial, 113, 406. of a function of two variables, US- of three or more variables, applications in plane trigono- metry, 130. in spherical trigonome- try, 133- successive, 144. of

5- trigonometrical illustration, 7. Limits, fundamental principles as to, 11. Maclaurin, series, 65, 81. on harmonic polar for a cubic, 282. Mannheim, construction for axes of an ellipse^ 374. Maxima or minima, 164. geometrical examples, 164, 183. algebraic examples, 166. „ ax 2 + 2bxy + a/ 2 of -r-= ^ ~, 166, 177. ax 2 + 2b' xy + cy 2 ' condition for, 169, 174. problem on area of section of a right cone, 181. for implicit functions, 185. quadrilateral of given sides, 186. for two variables, 191 ; La- grange's condition, 191, 197. for functions of three variables, 198. of n variables, 199, 447. application to surfaces, 200. undetermined multipliers applied to, 204. Multiple points on curves, 256, 265, 367. Multipliers, method of undetermined, 204. Napier, logarithmic system, 25. Navier, geometrical illustration of fundamental principles of the calculus, 8. on Taylor's theorem, 443. Newton's definition of fluxion, 10. prime and ultimate ratios, 40. expansions of sin x, cos x, sin" 1 x, &c, 64, 69. by differential equations, 85. method of investigating radius of curvature, 291. on evolute of epicycloid, 345. Nicomedes, conchoid of, 332. Node, 259. Normal, equation of, 215. number passing through a given point, 220. in vectorial coordinates, 233 . Orthogonal transformations, 409, 414, 449. Osc-node, 259. Osculating curves, 309. circle, 291, 306. conic, 317. Oscul-inflexion, point of, 314, 317. Parabola, of the third degree, 262, 288. osculating, 318. Parabolic branches of a curve, 246. Parameter, 270. 454 Index. Partial differentiation, 113, 406. Pascal, limacon of, 227. Pedal, 227. tangent to, 227. examples of, 230. negative, 227. Pliicker, on locus of cusps of cubics having given asymptotes, 265. Points, de rebroussement, 266. of inflexion, 279. Polar conic of a point, 219. Proctor, definition of epi- and hypo- cycloids, 399. epicyclics, 366. Ptolemy, epicyclics, 366. Quetelet, on Cartesian oval, 376, 381. Radius of curvature, 286. in Cartesian coordinates, 287, 289. in r, p coordinates, 295. in polar coordinates, 301. at singular points, 310. of envelope of a moving right line, 358. Eeauleaux, on centrodes of moving areas, 363. Eeciprocal polars, 228, 230. Remainder in series, Taylor's, 76, 79. Maclaurin's, 81. Resultant of concurrent lines, 234. Roberts, W., extension of method of inversion, 429. Rotation, of a plane area, 359. centre of instantaneous, 360, 364. of a rigid body, 371. Roulettes, 335. normal to, 336. centre of curvature, 352 ; Sa- vary's construction, 352. circle of inflexions of, 354. motion of a plane figure reduced to, 362. spherical, 370. Savary's construction for centre of curvature of roulette, 353. Series, Taylor's, 61, 70, 76. binomial, 63, 82. logarithmic, 63, 82. for sin# and cos;r, 64, 66, 81. Maclaurin's, 64, 81. exponential, 65, 81. Bernoulli's, 70. convergent and divergent, 72, 75. for sin -1 as, 68, 85. for tan~ 1 .r, 68, 84. for sin mas and cos mx, 87. Arbogast's, 88. Lagrange's, 151. Spinode, 259. Stationary, points, 266. tangents, 282. Subtangent and subnormal, 215. polar, 223. Symbols, separation of, 53. representation of Taylor's theo- rem by, 70, 160. Tacnode, 266. Tangent to curve, 212, 218, 258. number through a point, 219. expression for perpendicular on, 217, 224. expression for intercept on, 232. Taylor's series, 61. symbolic form of, 70. Lagrange on limits of, j6. extension to two variables, 1560 to three variables, 159. symbolic form of, 160. on inapplicability of, 443. Three-cusped hypocycloid, 350, 372, 430, 442. Tracing of curves, 322, 328. Transformations, linear, 408. orthogonal, 409, 449. Trisectrix, 332. Trochoids, 339. Ultimate intersection, locus of, 271. for consecutive normals, 290. Undetermined multipliers, application to maxima and minima, 204. applied to envelope, 273. Undulation, points of, 280. Variables, dependent and indepen- dent, 1. Variations of elements of a triangle, plane, 130; spherical, 133. Vectorial coordinates, 233. Whewell, on intrinsic equation, 304. THE END. i y i - ' •*. T ^A ''I!!l I > \S Date Due 5/-,/ : ' / 1AM 1 /I ! rt 0! vJrAl ■« r| icrdi l i • f t / BOSTON COLLEGE 242141 3 9031 01548938 8 60 row cout ef sc /£WC w« *„ ^ i ^ -$^MTH.S Boston College Library Chestnut Hill 67, Mass. Books may be kept for two weeks unless a shorter period is specified. Two cents a day is charged for each 2-week book kept overtime; 25 cents a day for each overnight book. If you cannot find what you want, inquire at the delivery desk for assistance. 9-51 • is WB& HI m SzaasSf isfinfiS p ess? iimrffl