11 m rvej&c : yMoAAu/: h & r ^/^/lC(^Vc<-y AN ELEMENTARY TREATISE THE DIFFERENTIAL CALCULUS, AN ELEMENTAKY TREATISE ON THE INTEGRAL CALCULUS, CONTAINING APPLICATIONS TO PLANE CURVES AND SURFACES. BY BENJAMIN WILLIAMSON, F.R.S. FIFTH EDITION. AN ELEMENTARY TREATISE ON DYNAMICS, CONTAINING APPLICATIONS TO THERMODYNAMICS. BY BENJAMIN WILLIAMSON, F.R.S., AND FRANCIS A. TARLETON, LL. D., Felloivs of Trinity College, Dublin. SECOND EDITION. AN ELEMENTARY TREATISE THE DIFFERENTIAL CALCULUS, CONTAINING THE THEORY OF PLANE CURVES, NUMEROUS EXAMPLES. BY BENJAMIN WILLIAMSON, M.A., F.R.S., FKLLOW OF TRINITY COLLEGE, AND PROFESSOR OF NATURAL PHILOSOPHY IN THK UNIVERSITY OF DUBLIN. %tbtxd\ (Kbitixw, %tb'ntis aiiir <&nhtgtb, NEW YORK: D. APPLETON AND COMPANY i [all rights reserved.] PREFACE. In the following Treatise I have adopted the method of Limiting Ratios 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 Geometry 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 Meca- nique Anahjtique 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. Geometry whenever it appeared that simplicity would be gained thereby. In the discussion of Multiple Points I have adopted the simple and General 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 Geometry 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 Hetaphysique du Calcul Infinitesimal of the illustrious Carnot ; 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 l'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 conform es 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- jonrs 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 avex ceux que fournissent les autres methodes, lorsqu'on peut la demontrer directement et general ement, plus facilement peut-etre qu'aucune de ces methodes elles-m ernes ? Les objections que l'on 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 l'elimi- nation le veritable caractere des quantites infinite simales 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 Roulettes, 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 case of the motion of any plane area supposed to move on a fixed Plane. I have also given short Chapters on Spherical Harmonic Analysis and on the System of Determinant Functions known as Jacobians, which now hold so fundamental a place in analysis. Trinity College, October, 1889. TABLE OF CONTENTS. CHAPTEE I. FIBST PRINCIPLES. DIFFERENTIATION. PACR Dependent and Independent Variables, ....... I Increments, Differentials, Limiting Eatios, Derived Functions, . . 3 Differential Coefficients, .......... 5 Geometrical Illustration, ......... 6 Navier, on the Fundamental Principles of the Differential Calculus, . 8 On Limits, io Differentiation of a Product, 13 Differentiation of a Quotient, ......... 15 Differentiation of a Power, . . . . . . . . . l6 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 CHAPTEE II. SUCCESSIVE DIFFERENTIATION. Successive Differential Coefficients, 34 Infinitesimals, ........... 36 Geometrical Illustrations of Infinitesimals, ...... 37 Fundamental Principle of the Infinitesimal Calculus, . . . .40 Subsidiary Principle, . . . . . . . . . . 41 Approximations, . . . ..... . . .42 Derived Functions of x m , ......... 46 Differential Coefficients of an Exponential, ...... 48 Differential Coefficients of tan" 1 x, and tan" 1 -, ...... 50 Theorem of Leilmitz, 51 Application of Leibnitz's Theorem, 53 Examples, 57 Table of Contents. CHAPTEE III. DEVELOPMENT OF FUNCTIONS. PAGE Taylor's Expansion, . . . . .'. . . . .61 Binomial Theorem, . . ........ 63 Logarithmic Series, .......... 63 Maclaurin's Theorem, .......... 64 Exponential Series, .......... 65 Expansions of sin x and cos x, ........ 66 Huygens' Approximation to Length of Circular Arc, . . . .66 Expansions of tan -1 x and sin -1 x, ........ 68 Euler's Expressions for sin x and cos x, ....... 69 John Bernoulli's Series, ......... 7° Symbolic Form of Taylor's Series, 70 Convergent and Divergent Series, ........ 73 Lagrange's Theorem on the Limits of Taylor's Series, . . . . 76 Geometrical Illustration, ......... 78 Second Form of the Remainder, . 79 General Form of Maclaurin's Series, . . . . . . .81 Binomial Theorem for Fractional and Negative Indices, . . . .82 Expansions by aid of Differential Equations, ...... 85 Expansion of sin mz and cos mz, ........ 87 Arbogast's Method of Derivation, ........ 88 Examples, 91 CHAPTEE IV. INDETERMINATE FORMS. Examples of Evaluating Indeterminate Forms without the Differential Calculus, ............ 96 Method of Differential Calculus, 99 Form Oxoo, 102 Form 2-, 103 Forms o"", 00 °, l tc0 105 Examples, 109 CHAPTEE V. 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 Fhme Trigonometry, ..... Landen's Transformation, ...... Examples in Spherical Trigonometry, .... Legt-ndre's Theorem on the Comparison of Elliptic Functions, Examples, "3 ll 5 117 119 120 123 130 *33 133 137 140 Table of Contents. xi CHAPTER VI. SUCCESSIVE PARTIAL DIFFERENTIATION. Page The Order of Differentiation is indifferent in Independent Variables, . 145 Condition that Tdx + Qdy should be an exact Differential, . . . 146 Euler's Theorem of Homogeneous Functions, . . . . .148 Successive Differential Coefficients of

(x + h, y + k, z + I), 159 Symbolic Forms, . . . . . . . . . . .ito Euler's Iheorcm, 162 CHAPTER IX. MAXIMA AND MINIMA FOR A SINGLE VARIABLE. Geometrical Examples of Maxima and Minima, . . . . .164 Algebraic Examples, .......... 165 Criterion for a Maximum or a Minimum, ...... 169 Maxima and Minima occur alternately, . . . . . . .173 Maxima or Minima of a Quadratic Fraction, ...... 177 Maximum or Minimum Section of a Right Cone, ..... 181 Maxima or Minima of an Implicit Function, ...... 185 Maximum or Minimum of a Function of Two Dependent Variables, . 186 Examples, 188 CHAPTER X. MAXIMA AND MINIMA OF FUNCTIONS OF TWO OR MORE VARIABLES. Maxima and Minima for Two Variables, ...... 191 Lagrange's Condition in the case of Two Independent Variables, . . 191 Xll Table of Contents. Maximum or Minimum of a Quadratic Fraction, . . . Application to Surfaces of Se< ond Degree, .... Maxima and Minima for Thi se Variables, .... Lagrange's Conditions in the case of Three Variables, Maximum or Minimum of a Quadratic Function of Three Variables Examples, .......... rAGH 194 196 198 199 200 203 CHAPTEB XI. METHOD OF UNDETEKMINED MULTIPLIERS APPLIED TO MAXIMA AND MINIMA. Method of Undetermined Multipliers, ....... 204 Application to find the principal Radii of Curvature on a Surface, . . 208 Examples, 210 CHAPTER XII. ON TANGENTS AND NORMALS TO CURVES Equation of Tangent, .... Equation of Normal, .... Subtan gent and Subnormal, . Number of Tangents from an External Point, Number of Normals passing through a given Point Differential of an Arc, .... Angle between Tangent and Radius Vector, Polar Subtangent and Subnormal, Inverse Curves, ..... Pedal Curves, Reciprocal Polars, .... Pedal and Reciprocal Polar of r m = a m cos m9, Intercept between point of Contact and foot of Perpendicular, Direction of Tangent and Normal in Vectorial Coordinates, Symmetrical Curves, and Central Curves, Examples, 212 215 215 219 220 220 222 223 225 227 228 230 232 233 236 238 CHAPTER XIII. ASYMPTOTES. Points of Intersection of a Curve and a Right Line, Method of Finding Asymptotes in Cartesian Coordinates, Case where Asymptotes all pass through the Origin, Asymptotes Parallel to Coordinate Axes, Parabolic and Hyperbolic Branches, .... Parallel Asymptotes, . . . . . The Points in which a Cubic is cut by its Asymptotes lie in a Right Line Asymptotes in Polar Curves, ..... Asymptotic Circles, ....... Examples, 240 242 2 4S 245 246 247 249 250 252 254 Table of Contents. xiii CHAPTER XIV. MULTIPLE POINTS ON CURVES. Pace Nodes, Cusps, Conjugate Points, 259 Method of Finding Double Foints in general, 26 1 Parabolas of the Third Degree, 262 Double Points on a Cubic having three given lines for its As) nrptotes, . 264 Multiple Points of higher Orders, 265 Cusps, iu general, .......... 266 Multiple Points on Curves in Polar Coordinates, 267 Examples, 268 CHAPTER XV. ENVELOPES. Method of Envelopes, 270 Envelope o(Za 2 + 2 Ma + N=o, 271 Undetermined Multipliers applied to Envelopes, . . . . .273 Examples, 276 CHAPTER XVI. CONVEXITY, CONCAVITY, POINTS OF INFLEXION. Convexity and Concavity, 278 Points of Inflexion, .......... 279 Harmonic Polar of a Point of Inflexion on a Cubic, . . . .281 Stationary Tangents, 282 Examples, 283 CHAPTER XVII. RADIUS OF CURVATURE, EVOLUTES, CONTACT. Curvature, Angle of Contingence, ........ 285 Radius of Curvature, .......... 286 Expressions for Eadius of Curvature, ....... 287 Newton's Method of considering Curvature, 291 Radii of Curvature of Inverse Curves, 295 Radius and Chord of Curvature in terms of r and p, .... 295 Chord of Curvature through Origin, 296 Evolutes and Involutes, ......... 297 E volute of Parabola, .......... 298 Evolute of Ellipse, 299 Evolute of Equiangular Spiral 300 xiv Table of Contents. PAGE Involute of a Circle, 300 Radius of Curvature and Points of Inflexion in Polar Coordinates, . .301 Intrinsic Equation of a Curve, ........ 304 Contact of Different Orders, 304 Centre of Curvature of an Ellipse, 307 Osculating Curves, . 309 Radii of Curvature at a Node, 310 Radii of Curvature at a Cusp, . . . . . . . . 311 At a Cusp of the Second Species the two Radii of Curvature are equal, . 312 General Discussion of Cusps, . . . . . . . • 3'5 Points on Evolute corresponding to Cusp? on Curve, .... 316 Equation of Osculating Conic, . 317 Examples, . . 319 CHAPTEK XVIII. ON TRACING OF CURVES. Tracing Algebraic Curves, . . . . . . . . .322 Cubic with three real Asymptotes, . 323 Each Asymptote corresponds to two Infinite Branches, .... 325 Tracing Curves in Polar Coordinates, . 328 On the Curves r m = a m cos mO, ........ 328 The Limacon, . . . . . . . . . . . 331 The Conchoid, • 332 Examples, 333 CHAPTER XIX. ROULETTES. Roulettes, Cycloid, .......... 335 Tangent to Cycloid, 336 Radius of Curvature, Evolute, ........ 337 Length of Cycloid, . 338 Trochoids, . 339 Epicycloids and Hypocyoloids, ........ 339 Radius of Curvature of Epicycloid, ....... 342 Double Generation of Epicycloids and Hypocycloids, .... 343 Evolute of Epicycloid, 344 Pedal of Epicycloid, 346 Epitrochoid and Ilypotrochoid, ........ 34/ Centre of Curvature of Epitrochoid, . . . . . . • 35 1 Savary's Theorem on Centre of Curvature of a Roulette, .... 35 2 Geometrical Construction for Centre of Curvature, ..... 35 2 Circle of Inflexions, .......... 354 Envelope of a Carried Curve, ........ 355 Centre of Curvature of the Envelope, 157 Table of Contents. xv PAGE Radius of Curvature of Envelope of a Right Line, 358 On the Motion of a Plane Figure in its Plane, 359 Chasles' Method of Drawing Normals, 360 Motion of a Plane Figure reduced to Roulettes, 362 Epicyelics, 363 Properties of Circle of Inflexions 367 Theorem of Bobilier, .......... 368 Centre of Curvature of Conchoid 370 Spherical Roulettes, 370 Examples, 372 CHAPTER XX. ON THE CARTESIAN OVAL. Equation of Cartesian Oval, 375 Construction for Third Focus, . . . . . . . . 376 Equation, referred to each pair of Foci, ....... 377 Conjugate Ovals are Inverse Curves, . 378 Construction for Tangent, ......... 379 Confocal Curves cut Orthogonally, .381 Cartesian Oval as an Envelope, ........ 382 Examples, 384. CHAPTER XXI. ELIMINATION OF CONSTANTS AND FUNCTIONS. Elimination of Constants, 384 Elimination of Transcendental Functions, ...... 386 Elimination of Arbitrary Functions, ....... 387 Condition that one expression should be a Function of another, . . 389 Elimination in the case of Arbitrary Functions of the same expression, . 393 Examples, ............ 397 CHAPTER XXII. CHANGE OF INDEPENDENT VARIABLE. Case of a Single Independent Variable, 399 Transformation from Rectangular to Polar Coordinates, .... 403 cPV d 2 V Transformation of — — - + -~-„ , 404 ax 1 dy 1 t d*V d-V cT-V Transformation of — — + -=-? + — -, 40? ax 1 dy* dz l Geometrical Illustration of Partial Differentiation, 407 XVI Table of Contents. Linear Transformations for Three Variables, .... Ca-°e of ( >rthogonal Transformations, . General Case of Transformation for Two Independent Variables, Functions unaltered by Linear Transformations, Application to Geometry of Two Dimensions, Application to Orthogonal Transformations, .... Examples, .......... PAGB 408 409 410 411 412 414 416 CHAPTER XXIII. SOLID HARMONIC ANALYSIS. iPV drV cT-V OntheEquahon— +--+^- = 0, Solid Harmonic, Functions, . Complete Solid Harmonics, . Spherical and Zonal Harmonics, Complete Spherical Harmonics, Laplace's Coefficients, . Examples, .... 418 419 421 423 427 429 432 CHAPTER XXIV. JACOBIANS. Jacobians, ............ 433 Case in which Functions are not Independent, . . . . -435 Jaeobian of Implicit Functions, .... .... 438 Case where /= o, .......... 441 Case where a Relation connects the Dependent Variables, . . . 442 Examples, 446 CHAPTER XXV. GENERAL CONDITIONS FOR MAXIMA AND MINIMA. Conditions for Four Independent Variables, ...... 447 Conditions for n Variables, ......... 449 Orthogonal Transformation, ......... 452 Miscellaneous Examples, 454 Note on Failure of Taylor's Theorem, ...... 467 The beginner is recommended to omit the following portions on the first reading: — Arts. 49, 50, 51, 52, 67-85, 88, ill, 114-116, 124, 125, Chap, vn., Chap. viii. ; Arts. 159-163, 249-254, 261-269, 296-301, Chaps, xxiii., XXIV., XXV. DIFFEEENTIAL CALCULUS. CHAPTER 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, /, +&>>*** ±1':: b~TY7i = v + b'(b + b'n) ; " \ i t ^C\ " but the fraction -i-z — — — 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 Tr b g. Trigonometrical Illustration. — To find the values of - — ^> an( i ~ a •> when 9 is regarded as infinitely small. Here - — - = cos0, and when 9 = o, cos0 = i. tant/ Hence, in the limit, when 9 = o* we have sin0 .. tan 9 . .. - — Tr = i, and, .•. . _ = i, at the same tune, tan 9 sin 9 a Again, to find the value of - — ^, when 9 is infinitely small. From geometrical considerations it is evident that if 9 be the circular measure of an angle, we have tan 9 > 9 > sin 9, tan0 9 or -7— n > ^— n > x 5 sin 9 sin 9 * 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. 7 but in the limit, i.e. when is infinitely small, tang _ sing " '* and therefore, at the same time, we have sing 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 . . . V - / («) 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 chord joining the points (x, y), (z lt 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 V\ — y , dii ., , . . .. . becomes -f- or / (x) in this case. Xi — x ax 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 f'{x) 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 =f'(x)(x-X), (2) where X, Y are the current co-ordinates of any point on the 8 First Principles — Differentiation. 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) = oo . For all other points / (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 = f{x), 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 Fie. i. the variable. In order to form a precise notion on this point, let us attribute to x 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 = fix + 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(z + Ax) -f(x) 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 impossible to assign to the ratio -~ any precise value, and LX*C 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 point P. The ratio — - of the increments will equally 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 Ay vanish, the ratio — of these increments approaches in general to a limit whose value is finite and determinate. Ay Hence the value of — corresponding to this limit must be considered as giving the true and precise measure of the rapidity with ichich 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 Ay /(, + *,)-/(«) 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 dy 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." io«. Limit of a Variable Magnitude. — As the con- ception of a limit is fundamental in the Calculus, it may be well to add a few remarks in further elucidation of its meaning : — Limit of a Variable Magnitude. 1 1 In general, when a variable magnitude tends continually to equality with a certain fixed magnitude, and approaches nearer to it than any assignable difference, however small, this fixed magni- tude is called the limit of the variable magnitude. For example, if we inscribe, or circumscribe, a polygon to any closed curve, and afterwards conceive each side indefinitely diminished, and consequently their number indefinitely increased, then the closed curve is said to be the limit of either polygon. By this means the total length of the curve is the limit of the perimeter either of the inscribed or circumscribed polygon. In like manner, the area of the curve is the limit to the area of either polygon. For instance, since the area of any polygon circumscribed to a circle is obviously equal to the rectangle under the radius of the circle and the semi-perimeter of the polygon, it follows that the area of a circle is repre- sented by the product of its radius and its semi-circumfe- rence. Again, since the length of the side of a regular polygon inscribed in a circle bears to that of the correspond- ing arc the same ratio as the perimeter of the polygon to the circumference of the circle, it follows that the ultimate ratio of the chord to the arc is one of equality, as shown in Art. 9. The like result follows immediately for any curve. The following principles concerning limits are of fre- quent application: — (1) The limit of the product of two quan- tities, which vary together, is the product of their limits; (2) The limit of the quotient of the quantities is the quotient of their limits. For, let P and Q represent the two quantities, and p and q their respective limits ; then if P=p + a, Q = q + (5, a and /3 denote quantities which diminish indefinitely as P and Q approach their limits, and which become evanescent in the limit. Again, we have PQ =pq +pfi + qa + a(3. Accordingly, in the limit, we have PQ = pq. 12 First Principles — Differentiation. Agam - r^-^jwr The numerator of the last fraction becomes evanescent in the limit, while the denominator becomes q 2 , and consequently the limit of — is -. Q q 1 1 . Differentiation. — The process of finding the derived function, or the differential coefficient of any expression, is called differentiating the expression. "We proceed to explain this process by applying it to a few elementary examples. Examples. i. y = x*. Substitute x + h for x, and denote the new value of y by yi, then yi = (x + A) 2 = x 2 + 2xh + h 2 ; j/i - y Ay .-. — ; — or — - = ix + n. h Ax If h be taken an infinitely small quantity, we get in the limit dx or if /(*) — £ 2 j we have/' (x) = ix. i 2. $/=-. Here yi x + h yi V x + h x x(x + h) ; y\-y nr Ay or h ' Ax x(x + hf which equation, when h is evanescent, becomes 4(1) dy i \x) _ i dx ~ a; 2 ' dx a? Differentiation of a Product. 13 12. Differentiation of the Algebraic Sum of a Finite Number of Functions. — Let y = « + v - to + &c. ; then, if x x = x + h, we get Pi m Ui + Vi - u\ + . . . ; j/i — y «i - u i\ - v it\ — id *'• h = ~T~ + ~ h h~ + • • •' which, beoomes in the limit, when h is infinitely small, dy du dv dw dx dx dx dx Hence, if a function consist of several terms, its derived function is the sum of the derived functions of its several parts, taken with their proper signs. It is evident that the differential of a constant is zero. 13. Differentiation of the Product of Two Func- tions. — Let y = uv, where u, v, are both functions of x ; and suppose Ay, Au, Av, to be the increments of y, u, v, corre- sponding to the increment Ax in x. Then Ay = (u + Au) (v + Av) - uv - = uAv + vAu + Au Av, Ay Av , , Ate or — = u h (v + Av) — . A.r A# Ax Now suppose Ax to be infinitely small, then Ay Av Au Ax' Ax' Ax' become in the limit dy dv , du dx' dx' dx ' also, since Av vanishes at the same time, the last term dis- appears from the equation, and thus we arrive at the result du dv du , , — =u — + v — . (3) dx dx dx 14 First Principles — Differentiation. Hence, to differentiate the product of two functions, multiply each of the factors by the differential coefficient of the ether, and add the products thus found. Otherwise thus : let fix), (x), denote the functions, and h the increment of x, then t/i =f(? + h) (x + h); " h h = /(« + *) -•% > * {x + h ) +/ (.) *(« + *)-»(*). Now, in the limit, f(x + h) -f{x) ^.ix A /^ and {x + h)-

'(x) + (x)f'(x), which agrees with the preceding result. When y = au, where a is a constant with respect to x, we have evidently dy du dx dx' 1 4. Differentiation of the Product of any Number of Functions. — First let 11 = xivw ; suppose vw - 2 > then y = uz, and, by Art. 13, we have dy = dx dz du z Tx' Differentiation of a Quotient. 15 but, by the same Article, hence dz dv dw dx dx dx ' du du dv dw -f- = VIC — + tCU — + ICV — . dx dx dx dx This process of reasoning can be easily extended to any number of functions. The preceding result admits of being written in the form 1 dy 1 du 1 dv 1 dw y dx u dx v dx w dx y and in general, if y = y x . y 3 . y* . . . . y n , it can be easily proved in like manner that £ dy = _i dy, + _i_ ^ £ dy» , y efe ^! (/# y 2 d# " y n dx' 15. Differentiation of a Quotient — Let y = -, then u = yv: v therefore, by Art. 13, ^-y-*,^ or rfv rftt C?0 ^M udv dx > v dx ... eft/ tffa* cfa? (5) This may be written in the following form, which is often useful: d fu\ 1 du u dv dx \v J v dx v 2 dx' 1 6 First Principles — Differentiation. Hence, to differentiate a fraction, multiply the denominator into the derived function of the numerator, and the numerator into the derived function of the denominator ; take the latter product from the former, and divide by the square of the denominator. In the particular case where u is a constant with respect to x (a suppose), we obviously have d fa\ a dv (6) dx } \v) v 2 dx' Examples. I. a — x a + x' du la Ans ' dx = ~ {aTx?' 2. u = (a + x) (b + x). du — = a + o + 2Z. ax 1 6. Differentiation of an Integral Power. — Let y = x n , where n is a positive integer. Suppose yi to be the value of y, when x becomes x x , then y 1 _-y = x x n - x n = ^ + ^ n _ 2 + _ + ^ Now, suppose Xi - x to be evanescent. In this case we may write x for x x in the right-hand side of the preceding equation, when it becomes nx 1l ~ l \ but the left-hand side, in the limit, is represented by —■ • Hence -/- = nx"' 1 , dx cl (x n ) or ; ' = nx n ~\ dx This result follows also from Art. 1 4 ; for, making yi = y* = yz - • • • = y n - «, we evidently get from (4), d(u n ) flu , s -V^ = ntt"- 1 — . (7) dx dx This reduces to the preceding on making u * .*. Differentiation of a Function of a Function. 1 7 17. Differentiation of a Fractional Power. — Let m y = w n > 1 d(y n ) d(u m ) then y n = u m , and ~^- L = -^-* ; dx ax hence, by (7), „ , dy m .du ax ax tn d (u n ) dy mu m ~ x du m --\du ,_. j» m v y = _ = - = yn . (8) dx dx n y n ~ l dx n dx' 18. Differentiation of a Negative Power. — Let y = tr™, then y = — , and by (6) we get du mu m " x — d , _,x dx du . N dx K ' u im dx v/ Combining the results established in (7), (8), and (9), we find that d (u m ) m , du \ ' = mu m ~ l — dx dx for all values of m, positive, negative, or fractional. When applied to the differentiation of any power of x we get the following rule : — Diminish the index by unity, and multiply the power of x thus obtained by the original index ; the result is the required differential coefficient, with respect to x. 1 9. Differentiation of a Function of a Function. — Let y = f(x) and u = (y), to find — . Suppose y y , u x , to be the values of y and u corresponding to the value x x for x ; then if Ay, Au, Ax, denote the corresponding increments, we have evidently Ui - u u x — u y Y - y Xx - x y x - y (Ci - x* or Au Au Ay Ax Ay Ax' 1 8 First Principles — Differentiation. As this relation holds for all corresponding increments, however small, it must hold in the limit,* when Ax is evanescent ; in which case it becomes du dudy . x — = -. (io) dx dy dx Hence the derived function with respect to x of u is the product of its derived with respect to y ; and the derived of y with respect to x. 20. Differ entiation of an Inverse Function. — To prove that dx 1 dy ~ d£ dx Suppose that from the equation V = /(*) («) the equation x = (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* - a; 2 ) 5 . Let a 2 - x 2 = y, then u = y 6 , — = 5/, and — - « - 2a?. Hence — = - io* (a 2 — * 2 ) 4 . ax du 2. u = (a + 5a;3)*. ^w*. — = i2&r 2 (0 + Ja^) s . 3. « = (1 + * 2 )». 4. u = (1 + r")*". — = wn^'fi +a^ , ) m * 1 . ' dcosx ( 7j° cos \X + - I. dx \ 2, 23. Differentiation of tan x. y = tan x, y Y = tan (x + h), sin (x + h) sin x y x - y tan {x + h) - tan x cos (x + h) cos x h = h = h sin h h cos a; cos {x + h)' which becomes — =— in the limit. COS^iT Differentiation of y = sin~ x x. 2 1 „ d(taxxx) 1 Hence — *-= = — =— = sec* a;. (14) ax cos 8 a; v Otherwise thus, , sin x d sin x . d cos x j,, v a . cos x ; sin x — = — a (tan 2) cos x dx dx dx dx cos 2 x cos" x + sin x COS 2 X COS 2 24. Differentiation of cot x. — Proceed as in the last, • , d (cot a?) 1 . . and we get — ^-= — - = — r-r- = - cosec 2 a\ (15) dx snrrr This result can also be derived from the preceding, by put- ting — 2 for x, as in Art. 22. 25. Differentiation of sec x. 1 y = sec x = ; cos X dy sin a; . .. ,\ -~ = — — = tan x sec 2. (16) air cos 2 x c,. .-, 1 d cosec x similarly -= = - cot x cosec x. 26. Differentiation ofy = sin -1 a;. XT • dX ±±i--t*3 a? = sin y, .*. — = cos y. Hence, by Art. 20, we get dy 1 1 )» 9. y = sin- 1 (1 — a; 2 )*. Here (1 — x 2 )i = sin y ; .*. .t = cos y. dy dy 1 I = - sm y — ; .*. — = . rfa; tfa; ^ .^ .£+«C03 2; (fy A /^2 I2 10. y = cos- 1 . — = V a - o' a + b cos a; <& a + I cos 2; 1 1. y = sec n ar. -y- = n sec" a; tan ar. 12. y = sec -1 (a;'). If x be a large number, it is evident, from the preceding, that the tabular difference (as given in Logarithmic Tables), i. e. the difference between logi (x + 1) and logio*, is — , ap- proximately. The student can readily verify this result by reference to the Tables. 30. Differentiation of a x . Let but . y = «* , then log y = X log a; , d (log y) _ dx log a; d (log y) __ d (log y) dy dx dy dx = idy^ y dx' d.a x dx dy 1 = a x log a. Also, since log e ■- = 1, we have d . e* , dx (23) (24) Logarithmic Differentiation. 27 Examples. I. y = log (sin a;). Let sin x — z, then y = log z. And since dy dy dz dx dz dx we get y- dy cos x dx sin x = log \/ a 2 - -■? _ 1 1 „ t a 2 ~2\ . d v — x x 4 log(a *»;, dx ^_^ !• y = (P*. dy Ans. — = lie'**, dx **£ + cos a; £ 2 sin-- 2 x — ^— — tan - ; , x dy 1 y = log tan -. Hence — = - — . 2 dx sin a; 31. Logarithmic Differentiation. — When the func- tion to be differentiated consists of products and quotients of functions, it is in general useful to take the logarithm of the function, and to differentiate it. This process is called logarithmic differentiation. Examples. 1. y «= y\ . y% .ys . . . y n , log y = log y\ + log y 2 + . . . + log y». Hence 1*1 = L d J± + L d t + .,, + ± d Jlt. y dx y\ dx y« dx y„ dx Tins furnishes another proof of formula (4), p. 15. sin m x Here, log y = m log sin x — n log cos x ; uua" a I dy co3 x sin x dy sin" 1-1 x . „ . . . - — - ss m -. + n — — ; .*.-—= — (mcoa* x + n sin- a;). y ax sin x co3 x dx cos' u l x 3- y = First Principles — Differentiation. (* - OS (*-2)*(*-3)r 5 3 7 Here log y = - log (x - i) — log (x — 2) — log (* - 3) ; 2 4 3 hence 1^=5 _i 3i 7 r _ 7^ + 30^-9: y tfa; 2 a; - 1 431 — 2 3 a; - 3 12 . (a; - 1) {x - 2) {x - 3) ' . (fy = _ (a?- Qi (7* 2 +3°^- 97) ' ' da; 12 . (a; - 2)J (a? - 3)y dy _ a* + a 2 x 1 - 4a; 4 4. t/=ar(«2 + a ; 2) v /a2-a:2. ^ - 7 ==— . 5. y = x x . Here log y = a; log x. 1 rfy d . x x HenCe y ds = (1 ° g " + *) ; •*• ~dx~ =x '( lJr lo S *>* 6. y = £**. Here log y = a;*, 1 dy d . x x , , . ydx = -dx- = XX{l+l0SX)i dy x „ .*. — = e* «* (1 + log a;), aa: 7. y = u e , where m and v are both functions of x. Here log y = v log m, 1 ^ _ ^ v v dw y dx dx u dx' dy /, dv v du\ , dp , dw .". — = « r log u — H — = «" log « — - + vW 1 — -. dx \ «a; m da;/ da; da; 32. The expression to be differentiated frequently admits of being transformed to a simpler shape. In such cases the student will find it an advantage to reduce the expression to its simplest form before proceeding to its differentiation. Examples. I. y ■» sin- • 1 + x i Here — = = sin y, or = sin 2 y ; hence x = tan y, /i + it 2 l + * /l+x 2 + */i- Here tan y = — — •v/i + x 2 - \/i - x 2 \/i + x 2 tan y + 1 . ^/7^2 ~ tan y - I * ^ = (r + tany) 2 - (i -tany)- _■ 2 tany _ ^ (1 + tan y) 2 + (1 - tany)* 1 + tan'y Hence — cos iy = *, ax dy dx cos iy y, _ x i W i+z + \/i-x 1, */ 1 + x + */ 1 - x y=log — — = - log /=• \ v^ 1 + * - v 1 -as V I + * - v I - * 1 1 + \/ 1 - at* 1. , / ;. 1 . = _ log 1 . = - log (1 + */ 1 - x-) - - log X. dy 1 dx 2X Vi - z2 r 1 + tan- 1 - 2X y — tan" x 1 - x- Let a; = tan z, and the student can easily prove that 5 , d'J 5 y = -z; hence - - = - a /x* - I i dy 2. y = x log x. — = i + log x. 3. y = log tan x. 4. y = log tan" 1 ar. ^ _ 2 <£r sin 2a;" dx (14 a; 2 ) tan- J x dy /- ay 3 y v da; 2^* , . _ , dy cos (log a;) 6. y = sin (log x). x 7. y = tan" 1 y = tan-i ■v/i - s»" V x + \/a dx 2; dy 1 dx v/i : -x 3 dy 1 I - -x- 1 }. _ = ___. \\+x\/2 + x i x\Zl dy 2^/2 °\Ji -x\/2 +x- Q I_a;2 ' 5x _ i + a; 4 ' 22. y = e* tan"' ar. -- = e* ( 1 + 2* tan">a: (1 + log x) dx \ 1 + x 1 I 23. Being given that y = a^(i-a; 2 ) ( J - - ) "> i* dy ex" 2 + c'x* + c"x 6 dx I „\W x-\k' determine the values of c, c', e". Ans. c = 3, c' = — 6, 0" = £. 24. y = log (log x). dy_ 1 ), dy = t where tan = -. 32 First Principles — Differentiation. , , / v £y i 29. y = log(y/x-a + V (* — o) (* — 6) 3 o. y = 2 tan-i(i-^) \ i ■n- r - x * - y ^y Here = tan- - ; ,\ a; = cos y; " i + a; 2 *' dx (I - # 2 )* 3i.y = a;*". ^ = «»"+«-» (» log * + 1). m , m- 1 — dii 32. y = (1 + x 2 ) 2 siD (m tan" 1 z). -- = m(i + x 2 ) 2 cos^wi-i^tan- 1 *}. /a cos a: 33. y = log.| A sin a; rfy — ab + b sin x dx a? cos 2 x — b 2 sin- a;" 34. Define the differential coefficient of a function of a variable quantity, •with respect to that quantity, and show that it measures the rate of increase of the function as compared with the rate of increase of the variable. 35 . If y — -, prove the relation dy dx V 1 + y 4, ■s/ 1 + #* s t/. 1 x 2 + ax + V (x- + ax) 2 — bx du 36. Ifw = log — — - , prove that — is of the form x 2 + ax — v (x- + ax) 2 — bx " x Ax 4- Z? , and determine the values of A and B. Ans. A — i,B-a. %/ (x 2 + ax) 2 — bx A sin 4 d + B sin 2 d + C d ( \ 37. Prove that - f sin g cos g y , _ # sin ,> g j = and determine the values of A, B, C. Ans. A = 3c 2 , J5 = — 2 (1 + c 2 ), C = 1. 1 x^ 1 . 7 #** 1 . ^ . c cfl 38. If m = x + h — h — — '— — + . . . ad inf. ; find the sum 23 2.45 2.4.67 of the series represented by — . Ans. (1 — a;-)"*. 39. Reduce to its simplest form the expression 3« 2 d x (x 2 4- 2«)i Ans. (x 2 + a)i (x 2 + 2«)J dx' (x 2 + a)l ' ' (x 2 + «)i (x 2 + 2a)i' •rr ■ • , x i dy sin 2 (a 4- v) 40. If amy = x sin (a + y), prove that — = : ■ . dx sin a Examples. 33 41. If x (1 4 y)» + y (1 + t)» = o, find 2f. In this case x 1 (1 + y) = y 2 (1 + a;) ; ,\ a; 2 - y 2 = yx (y - z), a; + y + xy = 0; .*. y = - 1 + x dx (1 + xf / x dy 1 * + « 42. y = log (x + V* - « 2 ) + sec-i -. - = - ,J— . 43. If x and y are given as functions of t by the equations x=f(t); y = F{t); dy « dy F' (t) find the value of -— m terms of *. — = 777-7 . dx dx f (t) y- I + £ i + x 2 Hence y — I + &c, ad infinitum. x* dy I + y" dx each of which is the derived function of the preceding. 34. Successive Differential Coefficients. If y = f 0) we have ■£ = fix) . Hence, differentiating both sides with regard to x, we get d fdy\ dx \dxj dx Let -7- ( -7- l be represented by -r^* dx\axj x dx 2 then g-/-w. In like manner — ( — - 1 - ] is represented by — , and so on ; dx\dx 2 J dx 3 Successive Differentials. 35 hence g =/"(*), &o ^ -/M (*). (1) The expressions dy d 2 y d 3 y d n y dx' da?' da?' ' ' ' da? are called the first, second, third, . . . n th differential coef- ficients of y regarded as a function of x. These functions are sometimes represented by y> y > y > f ( n ). 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 = /'(#) dx (Art. 7), we may regard dx as constant, and we shall have, on proceeding to the next differentiation, d (dy) =dxd [/' (x)-] = (&>)'/», since d [/' (a?)] =f"{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 . From this point of view we see the reason why/W (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 f" (a?) 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 & finite 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. If now, instead of the series — -, ( — - ) , ( — ; io 5 Vio 6 / Vio' we consider the series -, — , — , . . . in which n is n n 2 n z 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 . — , may be regarded as infini- of the first order, — „ — , 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, 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' AD ~ 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 beticeen 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 - BjV 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 axe tangents, since the length of the arc AB is less than the sum of the tangents AF and BF, we may extend thp 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 oi 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, By 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 bo 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. 40 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, /3, represent the quantities, and suppose a . a i Now the ratio 73 becomes evanescent whenever i is infinitely small in comparison with /3. This may take place in three different ways : ( 1 ) when )3 is finite, and i infinitely small : (2) when i is finite, and j3 infinitely great ; (3) when j3 is infinitely small, and i also infinitely small of a higher order : thus, if i = kQi 2 , then -r = k&, which becomes evanescent along P with /3. * This principle is stated for finite magnitudes by Leibnitz, as follows :— " Caeterum aequalia esse puto, non tan turn 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 quae tali quantitate non differunt, sequalia esse statuo, quod etiam Archimedes sumsit, aliique post ipsum omnes." Leibnitii Opera, Tom. 3, p. 328. The foregoing can be identifier! with the fundamental principle of Newton, as laid down in his Prime and Ultimate Ratios, Lemma I. : " Quantitates, ut et quantitatum rationes, quoe ad sequalitatem tempore quovis finito constanter tendunt, et ante finem temporis illius proprius ad invicem accedunt quam pro data quavis differentia, fiunt ultimo acquales." 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 end 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- P 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, /3, are said to be of the same order if the fraction — tends to a finite limit. If ±— tends a a n to a finite limit, /3 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 + (5, be the co-ordinates of a near point on the curve, and, by Art. 10, the direction of the tangent j3 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- 3 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 3 = ^axy : then, substituting as above, we get x 3 + 3x-a +f + $y-fi = $axy + $ax(3 + $aya : hence, on subtracting the given equation, we get the linit of e - ^-% a ax - y 39. Subsidiary Principle. — If ai + a 2 + 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 p\, /3 2 , • • • [3 n be another system of infinitely small quantities, such that ft 14. ft 14- ft T + — = I + £1, — = I + 6 2 , . . . = I + t n , <*i a 2 a n 42 Successive Differentiation. where £i, e 2 , . . . *«> axe infinitely small quantities, then the limit of the sum of j3i, j3 2 , . . . j3» is ultimately the same as that of eti, a 2 , . . . a n . For, from the preceding equations we have /3i + /3 2 + ••• + /3« = ai + o 2 + ...-•- a„ + etiEi + a 2 £ 2 + . . . + a„E». Now, if r\ be the greatest of the infinitely small quan- tities, fi, e 2 , . . . s n , we have j3i + /3 2 + . • . + (3 n - (ai + a 2 -»- -1- a n ) < jj (ai + a 2 . . . + a„) ; but the factor ai + a 2 + . . . + a n has a finite limit, by hypo- thesis, and as ?j is infinitely small, it follows that the limit of /3i + /3 2 + . . . + j3 M is the same as that of a x + a 2 + . . . + a n . This result can also be established otherwise as follows : — The ratio & + *+•• • + * ai + a 2 + . . . + a n by an elementary algebraic principle, lies between the greatest and the least values of the fractions > » • • • > eti a 2 a n it accordingly has unity for its limit under the supposed con- ditions : and hence the limiting value of /3i -f /3 2 + . . . + j3„ is the same as that of a x + a 2 + . . . + a„. 40. Approximations. — The principles of the Infini- tesimal Calculus above established lead to rigid and accurate results in the limit, and may be 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. Foi instance, in calculating a result to seven decimal places, if — : be regarded as a small r ' io 4 & quantity a, then a 2 , a 3 , &c, may in general be neglected. Thus, for example, to find sin 30' and cos 30' to seven de- cimal places. The circular measure of 30' is -— , or .0087266 : 360 Approximations. 43 denoting this by a, and employing the formulae, a 3 a 2 sin a = a — — , cos a = 1 , 6 2 it is easily seen that to seven decimal places we have a 3 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 = — —. cos a Again, to the same degree of approximation we have a + a a ba - a/3 &T/3 = b + b 2 ' 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 2 */a 2 + a = a + — ; \/ci- + a 2 = a + — = a, whenever a 2 may 2a 2a be neglected. Likewise, l/a z + a = a h -, &c. If b = a + a, where a is very small in comparison with a, we have */db = \/, =, or < n. Examples. 45 r _ - + 6 I20 X* *» ' 1 + — 2 24 7. Find the value of X~ IX neglecting powers of x beyond the 4th. Ans. 1 -\ 1 . x 8. Find the limiting values of - when y = o, x and y being connected by the equation y 3 = 2xy — x 2 . Here, dividing by y~ we get x 2 x -r - 2 - =* - y. y i y If we solve for - we have y ?=l+(l-2/)i. Hence, in the limit, when y = o, we have - = 2, or - = o. y y 9. In fig. 3, Art. 37, \£AB 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 - BB. 2 Let n be the number of sides, then the difference in question is n {AD — AB); ttAE , .„ M , -kAE(AB-AB) but w= Hi' ■•■ n(AB-AB)= y -— '■ arc AB AB = V AE J)E -f E _ „. i DE _ AE \ = 5 £jD a . p . Ah 2 * This result shows how rapidly the perimeteru of the circumscribed and in- scribed polygons approximate to equality, as tho 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 6 -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 C. Let z 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 C; .•. sin 2 - sin b cos C = cot a (cos b — cos z). 46 Successive Differentiation. Since z and b are both small, we get, to terms of the second order, s _jcosc= — (**-a»). 2 The first approximation gives z= b cos 0. If this be substituted for a in the right-hand side, we get, for the second approximation, _ ^sin'Ccota z = b cos G -. 2 We now proceed to find the successive derived functions in some elementary examples. 4 1 . Derived Functions of x m . Let y = x m > ,i dy , dry . then ~ = mx™" 1 , —^ = m(m- i) x m ~ 2 . dx ax 2 v ' d n y and in general, — = m (m - i) (m - 2) . . . (m - n + 1) x m ~". If m be a positive integer, we have d m (x m ) dx" 1 = 1 • 2 . . . m. and all the higher derived functions vanish. If m be a fractional, or a negative index, then none of the successive derived functions can vanish. Examples. 1. If u = ax* 1 + bx n ' 1 + cx n ' 2 + &c, prove that — =- n [n - 1) ax"-* + (n - 1) (n - 2) bx"' 3 + &c . d»u d"* l u also — - = 1 . 2 . . . . n . a, and ; = o. ax* dx"* 1 a 2 - y = —, prove th;.t * = -" **9 __ « i» + *) a dx a?» +1 ^, = (- i) n w 2n+1 cos m#. (0 It is easily seen that these may be combined in the single equation (Art. 22), d r (sin mx) daf In like manner we have = m r sin ( mx + r - ] • (2) d r cos ?»# / it = W r COS »2# + r - r/,/ r \ 2 44. Derived Functions of e"*. Let y = e"*, ,i eft/ ^ 2 y , ^ n y , v then -f = ae ax , -~ = rre ax , . . . — — = a n e ax . (%) dx dx 2 dx" vo/ Tliis result may be written in the form d\n — ) . e ax = a n e ax . (4) dx) ( d\ n where the symbol I — ) denotes that the process 0/ differentia- tion is applied n times in succession to the function e ax . Derived Functions of e ax cos bx. 49 In general, adopting the same notation, we have -^© v+ ^(ir«" + ^(ir«- +&o - =A a n e ax + A x a n -'e ax + A 2 a n - 2 e ax + &c. = [^ a n + Atf"- 1 + Aza"-* + &c. An] e ax . This result, if (x) denote the expression A&r + A^- 1 + . . . A ni may be written in the form , and « = */a? + 6 2 cos 0. Hence we get ~ = {a 2 + b 2 )*e a * cos (bx +0). 50 Successive Differentiation. Again, d z v -2 = (a' + 5 2 )i e «* [« cos (5a? + ) - b sin (6* + 0)] = (a 2 + b-) ef 1 * cos (bx + 2 ^ = * (sm y sm 2y) = ^ < sm y sm 2y ) = - sin 2 y — (sin 2 y sin iy) = - 1 . 2 . sin 3 y sin 3?/. (^c. 5, ^r*. 28.) Hence, also — = 1.2.3. s i n4 V s i Q W 5 and in general, — = (- i) n \n - 1 sin n y sin ny. Theorem of Leibnitz. 5 1 Again, since tan -1 x - — tan" 1 -, we have ££gp!J = (- i)™ \n - i sin»y sin ny, (7) where y = cot" 1 x, as before. This result can also be written in the form sin ( n tan -1 - d n (tan -1 x) , ._., V "ay /ON — ^- = (- i) n_1 w - 1 * j- 8 • (8) (1 + x?)l 47. If y = sin (m sin -1 ^), to prove that cPy dy . . (, -^S-*5 + ^- a (9) Here dy m cos (m sin" 1 x) dx ~ v^i -a; 2 ' .*. (1 - a; 2 ) ( -p J = m? cos 2 (m sin -1 a;) = m 2 (1 - y 2 ). Hence, differentiating a second time, and dividing by 2 — , we get the required result. 48. Theorem of Leibnitz. — To find the n th differen- tial coefficient of the product of two functions of x. Let y = uv; then, adopting the notation of Art. 34, we write , , , dy du , dv y , u , v , for — , — , and — , J ' ' dx' dx' dx' and similarly, y", u", v", &c, for the second and higher derived functions — thus, , , d n y , , d n u B F 2 52 Successive Differentiation. Now, if we differentiate the equation y = uv, we have y = uv' + vu', by Art. 13. The next differentiation gives y" = uv" + u'v' + v'u' + vu" = uv" + 2u'v + vu". The third differentiation gives f = UV'" + U'V" + 2U'V" + 2U"V' + VU" + Vu'" = uv'" + 3u'v" + 3 u"v' + vu"\ in which the coefficients are the same as those in the expan- sion of (a + b) 3 . Suppose that the same law holds for the n ih differential coefficient, and that v ip) = uvW + nu'vW + — i '- u"v C*" 2 ) + &c, + nu^'^v' + mW»; then, differentiating again, we get y{™) = W (»+i) + u ' v {n) + n (u'vW + U"V^- 1 ^) + n ( n ~ l ) ( u " v (n-l) + M '^(n-2)) + &o. . . . + UWV = uvW + (n + i) u'vW + (, w + 1 ) n u " v {n-i ) + & . . 1.2 in which it can be easily seen that the coefficients follow the law of the Binomial Expansion. Accordingly, if this law hold for any integer value of «, it holds for the next higher integer ; but we have shown that it holds when n = 3 ; therefore it holds for n = 4, &c. Hence it holds for all positive integer values of n. In the ordinary notation the preceding result becomes d n (uv) d n v dud n ~ 1 v n(n - 1) d 2 ud n ~ 2 v ———■ = u -— + n — ■ + — • -7- -7— - + &c. dx n dx n dx dx 7 '- 1 \ .2 dx 1 dx 71 ' 2 J n u , . + *d?' ^ I0 > Applications of Leibnitz's Theorem. 53 49. To prove that where n is a positive integer. Let v = e** in the preceding theorem ; then, since dv „ d*v „ nr d n v „ „ — = ae"*, — = a'e a * . . . — = a n e' dx dx 2 dx" we have d n u\ dx^r d\ , n „ s nr f „ n ,du n(n-i) „„d*u p — (e^u) = e° x a n u + na™ — + — a n ~ 2 -^-r- + &c dx) \ dx 1 . 2 dar which may be written in the form ( d\ n ( d\ n ( d\ n . where the symbolic expression ( a + — j is supposed to be developed by the Binomial Theorem, and — , - du d?u d r u dx' dx*' ' ' ' daf substituted for ( — j u, ( — j u, ( — ) u, in the resulting ex- pansion. 50. In general, if • This symbolic equation is of importance in the solution of differential equations with constant coefficients. See " Boole's Differential Equations," chap. xvi. 51. If y = sin -1 x, to prove that * d ™y t x d n "y d n y (1 - x 2 ) -7-r, - (2» + 1) x -r~ - n % — ^ = o. (1 *) v ' dx n+2 v ' dx n+1 dx* K 6) hence, by differentiation, x^- . ,, d 2 y dy («-^5— a-* ! «*) Again, by Leibnitz's Theorem, we have ^\"(/ ^ d ~U) r *d M2 y d Ml y d»y dx) ( v ' dx 2 ) v V# n+i! dx n+1 v ; ob» .. fd\ n (dy) d n * l y d^y \dxj \ dx\ dx"* 1 dx" On subtracting the latter expression from the former, we obtain the required result by (14). If x = o in formula (13), it becomes (d^y\ Jdy\_ W' +2 /o W'/> ' Applications of Leibnitz's Theorem. 55 (d?i/\ d n v — J represents the value of -^ when x becomes cypher. Also, since i—j = 1, we get, when n is an odd integer, n\ Again we have f — ) = o ; consequently, when n is an even integer, we have ( — - I = 6 ' \dapJo 52. If y = (1 +# 2 ) 2 sin (wtan -1 ^), to prove that / « d?V / v dy . . . , ^ + ar) ^~ 2 ( m ~ O^^ + w*^- 0y = o. (15) Here dii --1 . --1 y = m#(i + ar 8 ) 2 sin (m tan" 1 a?) +m(i + a: 2 ) 2 cos (?w tan -1 a?) , or dii - - (1 + a?) — =ma?(i + a; 2 ) 2 sin(^tan" 1 a;) + m(i+a! 2 ) 2 cosm(tan" 1 a;) = mxy + m (1 + ar*) 2 cos (m tan" 1 x) ; / ,v? / A , x 1 + x*dy .% (1 + ar r cob (w tan -1 a?) = p - aw. 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 — . d™y d™y d"y (l+ ^dx^ + 2 ( n ~ m+ *) x - d ^i + ( n ' m ) ( n ~ m + -^ = °- 56 Successive Differentiation. Hence, when x = o, we have Moreover, as when a; = o, we have y = o, and — = m ; it follows from the preceding that = ° ; \d^y ( ~ ')"»»(«»- i) . . . (m - 2n). (16) W n Jo For a complete discussion of this, and other analogous expressions, the student is referred to Bertrand, " Traite de Calcul Diff erentiel," p. 1 44, &c. Examples. 5 7 Examples. d*y 1 4 1. y = x* log *, prove that — - - ~ d n y , ,1.2. ..(n-2) 2.y = x\ogx, „ _£ = (_i)«i \ '. d 2 y 3. y = x>, „ — = x*(i+logz)* + z*-\ 4. y = log (sin z), V r + x - ' ,2* 5. y = tan- 1 + tan" 1 ^, * I — X' 6. y = x* log (*i), <*3y 2 C03 X «?^3 sin 3 x ' d 2 y S* f dx 1 (1 + x? d*y dx 5 2* x' d 2 y Z\/2. z* dx 3 (1 + x* r d n y e rx sin (a; - «4») \\+x\/2+x* t ,x\/2 d*y %*/ 2 . 8. « = e r * sin a?, „ dx n sm' l where tan <*> = -. r 9. Ify = e ax x r , prove that d n y , n(n - i)r (r - i) „ . "1 —l =e ax\ a"x r + nra n ~ l x*-i + -i i — i : a»-~ x r ' 2 + . . . , ax" i . 2 and (|)V^)= £)~ (|) W). 10. li y = a cos (log a;) + b sin (log x), xi x .. ^V dy prove that a^ V x , + y — o. <£r 2 rfa; 11. If y = ga'ln-lx^ 1 , o ^V &—* + ** — is satisfied by either of the following values of y : y = cos (a sin" 1 x), or y = e a _1 • ! *- ! «. 13. Being given that y = (x + */ x- - i) m , tPy dy prove that (x 2 — 1) --— +• a; - — «i 2 y = o. «te 2 aa: 14. If y = sin (sin x), d-y dy prove that — '- + — tan a; + 3/ cos*a; = o. 15. In Fig. 3, Art. 37, if AB he regarded as a side of a regular polygon of an indefinitely great number of sides, show that the difference between the circum- ference of the circle and the perimeter of the polygon is represented by - BD, 6 to the second order of infinitesimals. I d 1 \ 16. If y = A cos nx + B sin nx, prove that I h « 2 ) y = o. \dx 2 J T , I ,, . d n y , . I n . sin n+1 rf> sin (n + 1) d> 17. If y = — -, prove that — - = (- I)" L - y *■ ;v \ a where d> = tan" 1 -. x This follows at once from Art. 46, since — - f tan" 1 - ] = — -,. It can also be dx \ x) a- + x 2 proved otherwise, as follows : 1 = ' r ' - 1- a 2 + a: 2 ia(~ i)i \_x-a(- 1)* ar+a(-i)*J' i»y _ 1 / rf \ n ' 1 / , and x = «/ a 2 + x 2 cos

)» +1 n + l = (a 2 + ar2)~2~{cos(» + i)

i + 1)

dx" v ' a» + - 18. In like manner, if y = — -, a 2 + x 2 jn v \n . sin" +1 . cos (» + 1)

u ,d n u (I - x-) - — - - (2« + 1) x - — - - n 2 — = o ; dx"* 2 dx»* 1 dx n , (d n *-u\ I d"u\ and l^).-"U;).' _„ . , , ^w dy 22. If y = e"* sin ox, prove that 2a — -\ (a 2 + b 2 ) y = o. «z 3 dx ax + b „ , rf";/ dx" 23. Given y = — , find -— . _ ax + b ac + b 1 ac - b 1 Here — -. = + . x — , dx h when h is infinitely small. Similarly, if y become y + h, we have (l ^ = limit of /{x + j>_ + h )-x* + y\ dy h which is the same expression as before. TT du du Hence —-=—-. dx dy Otherwise thus : — Let z = x + y, then w =/(s), dz , dz du dudz ,. . dx dz dx ' du du dz , . du dy dz dy dx' Taylor's Expansion. 6 1 54. If a continuous function f(x + y) be supposed ex- panded in a series of powers of y, the expansion 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, for all values of x, when y = 0; but the given function in that case reduces to f(x) ; and since/ (x) cannot be infinite for all values of x, it follows that the expansion of f(x+ y) can contain only positive powers of y. Again, if f(x) and its successive derived functions be continuous, the expansion of/(# + y) can contain no fractional p power of y. For, if it contain a term of the form Py M q, where- is a proper fraction, then its (n + i) th derived func- tion with respect to y would contain y with a negative index, and, accordingly, it would become infinite when y = o ; but this is impossible for the same reason as in the former case ; hence, with the conditions expressed above, the expansion olf(x + y) can contain only positive integral powers of y. 55. Taylor's Expansion of/ (x + y).* — Assuming that the function f(x + y) is capable of being expanded in powers of y, then by the preceding this equation must be of the form f{x + y) = P + P x y + P 2 y 2 + &c. + P n y n + &c, in which P , P u . . . P n are supposed to be finite and con- tinuous functions of x. When y = o, this expansion reduces to/ (x) = P . Again, let u =f(x + y) ; then by differentiation we have du dP dP x .dP, dP n s — = -—- + y — — + y* — - -r . . . + y n —j- + &c. ; dx dx dx dx dx — = J\ + zP 2 y + 3P 3 if + &c. * The investigation in this Article is introduced for the purpose of showing the heginner, in a simple manner, how Taylor's series can be arrived at. It is based on the assumption that the function/(a; + y) is capable of being expanded in a series of powers of y, and that it is also a continuous function. It demon- strates that whenever the function represented by f{x + y) is capable of being expanded in a convergent series of positive ascending powers of y, the series must necessarily coincide with the form given in (i). An investigation of the conditions of convergency of the series, and of the applicability of the Theorem in general, will be introduced in a subsequent part of the Chapter. The parti- cular case of this Theorem when f(x) is a rational algebraic expression of the n" 1 degree in z is already familiar to the student who has read the Theory of Equations. 62 Development of Functions. Now, in order that these series should be identical for all values of y the coefficients of like powers must be equal. Accordingly, we must have I . 2 «# 1 . 2 dX' \ . 2 x _idP 9 i rf»/(a?) i ,„ , m 3 riip 1.2.3 "^ 1.2.3 and in general, , */») L_ /WM . 1.2...;? rix" 1 . 2 . . . w Accordingly, when /(a?) and its successive derived func- tions are finite and continuous we have /(»+*) -/w + f /» + ^/» + • • • + £/ w w+... (1) This expansion is called Taylor's Theorem, having been first published, in 17 15, by Dr. Brook Taylor in his Methodus Incrementorum. It may also be written in the form /( . + ,). /W+ »^ + j!^... + i £^ + ..; W J v *i •> \ * l dx 1.2 dx 2 \n dx" ' v y or, if f« »/(*), and «i =/(* + y), ?/ riw ?/ 2 ri 2 z< y" ri w M , , Ml =«+-—+ -*- -— + . . . + p— — + &c. (3) 1 dx 1.2 dxr n dx" 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/(#) = z", therefore, by Art. 41, f'(x) = nx"-\ . . ./fr? (x) - n (n - 1) . . .. (n - r + i)^"- r . Hence the expansion becomes / w . , w(« - 1) „ , , (x + y) n = z» + - a^y + — x n -y z + . . . nf„-i)...(n-r + i) 1 . 2 . . . r If n 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 f{x) = log (x), f{x)-K f\x) = 1 X X r-jrg-£ + £-{ + *a). (6) 64 Development of Functions. 58. To expand sin (x + y). Here f(x) = sin a?, /(») = cos x, f r {x) = - sin x, fix) = - cos x, &c. Hence / yi yi y-» sin (x + y) = sin x 1 + &c. ± '. — . . v \ 1.2 1.2.3.4 \in + cos a; + . . . ± ; • (7) \i 1.2.3 1.2.3.4.5 \ zn - 1 ) As tlie preceding series is supposed to hold for all values, it must hold when x = o, in which case it becomes sin y Jl £_ + 1 &c> (g) 1 1.2.3 1.2.3.4.5 7T Similarly, if x = — , we get ?/ 2 y* cos y = 1 ; — + &c. (9) J 1 . 2 1 . 2 . 3 . 4 vw We thus arrive at the well-known expansions* for the sine and cosine of an angle, in terms of its circular measure. 59. Maclaurin's Theorem. — If we make x = o, in Taylor's Expansion, it hecomes / (!/) =/(o) + ?/(o) + /-/>) + . . • £/W(o) + . . . , (10) where /(o) . . ./( n )(o) represent the values which /(a;) and its successive derived functions assume when x = o. Substitute # for y in the preceding series and it becomes /(*) -/(o) + ? /'(o) + ~ /"(o) + . . . + £ /CO (o) + &c. * These expansions are due to Newton, and were obtained by him by the method of reversion of scries from the expansion of the arc in terms of its sine. Tli i s latter series lie deduced from its derived function by a process analogous to integration (called by Newton the method of quadratures). See Opuscula, torn 1., pp. j 9, 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 + Cx 2 + Dx 3 + Ex i + &c. then /' (x) =JB + 2Cx + 3DX 2 + $Ex 3 + &c. /" (x) - 2O + 3 . zBx + 4 . sEx 2 - &o. f" (x) = 3 . 2D + 4 . 3 . 2Ex + &C. Hence, making x = o in each of these equations, we get f(o)=A, f'(o)=B, £^-<7, f^ = A&c. 1.2 1.2.3 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 /(o) =1, f(x) =a*loga, „ f{p) =loga, f"{x) = a*(log«)», „ r(o)=log«y, /W (*) = cf (log a)\ „ /W (o) = (log a)- ; and the expansion is (x log a) (x log a) 2 (x loff «) n . . . 1 1.2 1 . 2 . . . n If c, the base of the Napierian system of Logarithms, be substituted for a, the preceding expansion becomes ~xx 2 x n . (f = 1 + - + ■ + . . . + + . . . (12) 1 1 . 2 1 . 2 . . . n * Maclaurin laid no claim to the theorem which is known by his name, for, after proving it, he adds — "Tins theorem was given by Dr. Taylor, Method. hicrem." See Maclaurin's Fluxions, vol. ii., Art. 7.51. 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 + - + + 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 x, then /(o)=o, /(o) = i, /» = o, /'"(o)=-I,&C, and we get xx 3 x s . sin x = h ccc 1 1.2.3 1.2.3.4.5 In like manner X s x* COS X = I - 1.2.3.4 the same expansions as already arrived at in Art. 58. Since sin (—#)= — sin x, 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), the expansion can contain odd powers of x only. Thus, the expansions of tana;, sin~'#, tan -1 #, &c, can con- tain no even powers of x; those of cos x, sec x, &c, no odd powers. 62. Huygens' 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. For, let It he the radius of the circle, and L the length of the arc : and we have A .LB . L ir 2Sin iB' iT 2Sm 4^' • 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 Oldernburgium." Huygens' Approximation. 67 hence, by (8), X s I? A = L- — + — — - &o. 2.3.4.R* 2 .3 .4. 5 . i6.i2 4 7-3 TS SB = 4 L - — - + -— - &c. 2 . 3 . 4 . i2» 2 . 3 . 4 . 5 . 64 . JR* consequently, neglecting powers of -= beyond the fourth, we get 8B-A r( & \ Z hi^) (13) 3 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 U3 apply it to find the length of an arc of 30° in a circle whose radius is 100,000 feet. Here J5 = 2R sin 7 30', .4 = 2.S sin 15 ; but, from the Tables, sin 7° 30' = .1305268, sin 15 = .2588190. -,, 2B - A Hence iB + = 5 2 359-7 I - The true value, assuming w = 3.1415926, is 52359.88 ; whence the error is but . 1 7 of a foot, or about 2 inches. F 2 68 Development of Functions. 63. Expansion of tan" 1 ^. — Assume, according to Art. 6 1 , the expansion of tan -1 # to be Ax + Bx 3 + Cx* + Bx" + &c, where A, B, C, &c, are undetermined coefficients : then dA 2^ = A + lBx% + 5Cxi + lDx * + &0 - ; but ■ ; = . = 1 - x- + x* - x« + &c., ax 1 + x* when x lies between the limits ± 1. Comparing coefficients, we have A = i, B = --, C = -, B = -- i &o. 3 5 7 Hence tan _1 a; = + ... + (- i) n + . . . ; (14) 135 v ' 2n + 1 K ' 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 z + Cx 5 + &c. ; then , X ... =A + ^Bx* + sCx* ^ &c. ; (1 - x-p 1 1 1 . 3 but -. . = (1 - x 2 )-l = 1 + - x~ + x* + . . . (1 -ar)s v ' 2 2.4 1 . 3 . . . zr - 1 . + x" + . . . 2.4... 2/* Hence, comparing coefficients, we get Finally, A=i, B = -.-, C = .L^.-,&o. 23 2.45 a; 1 a;- 1 1 . 3 of 1 .3 ... 2r r 2r+i sin -I a; = - + - . — + — — .— + .. .4 ' . — +... (15) 123 2.4 5 2.4... zr ir+i Eider's Expressions for Sine and Cosine. 69 Since we have assumed that sin -1 # vanishes along with x we must in this expansion regard sin _1 a; 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 sin" 1 .?, 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 it can be exhibited in the following series : it 1 11 1 . 3 1 - = - + o + + &C. 6 2 2.38 2. 4. 5 32 For, since sin 30° = -, we have - = sin" 1 - ; .'. &c. 2 6 2 An approximate* value of tt 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. Euler's Expressions for Sine and Cosine. — In the exponential series (12), if x */ - 1 be substituted for x, we get — x 1 # 4 e 3 *' 1 = 1 — + + &c. . . . 1.2 1.2.3.4 x + & :c. 1 1.2.3 = cos x + v—i sin x ; by Art. 59. Similarly, e'^' 1 = cos % - & c -> eacn become infinite when Q dx dx 6 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. Meniarks 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, a finite limiting value, represented by the original function, /(# + y), or f(x). 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 tho next Article. 71. Convergent and Divergent Series. — A series, M l5 n 2 , «s, . . . 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 bo 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 sum 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"' + X 3 + . . . + 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"' 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 x n by , which becomes smaller and smaller as n increases, 1 — 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 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 I cc series is a convergent one. In like manner the sum of n terms of the series I - X + X~ - X z + &G. I -(- l) n X n IB . 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 - oo ; while that of an odd number is + co . Hence the series in this case has no limit. 73. Theorem. — If, in a series of positive terms repre- sented by Ml + U% + . . . + 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 < k, U» .*. w„ +1 < ku n . Wn+2 7 < k, Wf»H •"• Wn+2 < k 2 U n . ?'n+r 7 .'. w n +? < k u n < A., Hence, the limit of the remainder of the series after u n is less than the sum of the series ku n + k'Un + . . . + k r u n ... ad infinitum ; therefore, by Art. 72, less than kll n i -A , since k < 1. Hence, since u n decreases as n increases, and becomes infi- nitely small ultimately, the remainder after 11 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 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 + k z u n + &g. ad infinitum has an infinite value ; but each term of the series w» + w n+ i + M M+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. Lemma. — If a continuous function (22) in which /(.r), /' (a;) /( w ) (a*) are supposed finite and continuous for all values of the variable between X and x. From the form of the terms included in It n it evidently may be written in the shape (X - x) n 7? _ ^ ' p \n where P is some function of X and x. Consequently we have AX) - {/(*) + Qzllf («) + ... + 2^V^ W (X-z) n r>) + ^_|_LpJ-o. (23) Now, let s be substituted for x in every term in the pre- ceding, with the exception of P, and let F(z) represent the resulting expression : we shall have F(z) =f(X) - |/(b) + £^*)f>( z ) + .. . + ^^- n PJ, (24) in which P lias the same value as before. Again, the right-hand side in this equation vanishes when z = X ; .-. F(X) = o. Also, from (23), the right-hand side vanishes when z =x ; .'. F{x) = o. Limits of Taylor's Series. 7 7 Accordingly, since the function F (2) vanishes when z = X, and also when z = x, it follows from Art. 74 that its derived function F'(z) also vanishes for some value of z between the limits X and a?. 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 \n-i J" ( Z ) = _ (X Z ^ /W (a) + (X "' - P. Consequently, for some value of z between x and X we must have /W to - P. Again, if be a positive quantity less than unity it is easily seen that the expression x+0(X-x), by assigning a suitable value to 9, can be made equal to any number intermediate between x and X. Hence, finally, p=/(«) { x + e (x-x)}, where is some quantity > o and < 1. Consequently, the remainder after n terms of Taylor's series can be represented by A -^^/W {•+*(*-*)}. (2 5 ) Making this substitution, the equation (22) becomes ax) -/(*) + (^-V to + { ^fr to + (z - ^) w - 1 /(w _ 1) ( v + C?>f)_ M /(n) { x + (x-x)}. (26) 1 » - 1 » x ' The preceding demonstration is taken, with some slight modifications, from Bertrand's " Traite de Calcul Differentiel" ( 2 73). 78 Development of Functions. Again, if h be substituted for X - x, the series becomes f(x + h)=f(z)+hf'(x) + &o. + ^/ (n_1) (*) + £/ (n) (* + ^)- ( 2 7) In this expression n may be any positive integer. If n = i the result becomes f(x + h) -/(*) + V (* + Oh). (28) When m = 2, fix + A) -/(«) + //' (0) + ^/" (0 + 0/0. (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., 6 > o and < 1 . It will be a useful exercise on the preceding method for the student to investigate the formulae (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 =f(x) represent a curve referred to rectangular axes, and suppose (X, Y), (x, y) to be two points Pi, P 2 on it : then f(X)-f(x) = Y-y X ~ x X - x' Y-y But — — - is the tangent of the angle which the chord Pi P 2 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 t and P 2 must be parallel to the chord Pi P 2 ; but by Art. 10,/' (.r,) is the tri- gonometrical tangent of the angle which the tangent at the Second Form of Remainder. 79 point (a?i, 1/1) makes with the axis of x. Hence, for some value, x u between X and x, we must have X- x X-x or, writing x x in the form x + (X - x), f(X) -/(•) + (X - «)/ {* + (X- «)}• 77. Second Form of Remainder. — The remainder after n terms in Taylor's Series may also be written in the form R n = (' g ) W " 1 h n fW ^ + 0/^ «- 1 For it is evident that R n may be written in the form (X-x)P 1 ; .-./(X) -/(*) + (X-x) fix) + . . . + gz^? /0») (,) + (X-*)Pi. Substitute 2 for a?, as before, in every term except P Y ; and the same reasoning is applicable, word for word, as that employed in Art. 75. The value of F' (z) becomes, however, in this case F'(z) = - {X ~ z)n ~ 1 fW(z)+P 1 , and, as F'(z) must vanish for some value of z between x and X, we must have, representing that value by x + (X - x), ^ m (X-z)^-0)" /w {x+e{x _ x)l (3o) where 0, as before, is > o and < 1 . If h be introduced instead of X - x, the preceding result becomes (1 _ Q)n-i ^= in., h*fl*){x+Qh), (31) which is of the required form. 80 Development of Functions. Hence, Taylor's Theorem admits of being written in the form /(•+*) -/m + *rw + ^ rw + . • • + j^V (n_1) ( * } + r^- (i-^/w^ + W). (32) The same remarks are applicable to this form* as were made with respect to (27). From these formulae 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 fofW(x+0h) shall become infinitely small, when n is taken sufficiently large ; as otherwise the series does not admit of a finite limit. h n 78. Limit of when n is indefinitely great. 1 . 2 . . n .Let u n = , then — = ; .*. — becomes smaller 1 . 2 . . n u n n + I «„ and smaller as« increases ; hence, when n is taken sufficiently great, the series u n+u u n+2 , . . . &c, diminishes rapidly, and the terms become ultimately infinitely small. Consequently, whenever the n th derived function ,/M (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. 81 79. General Form of Haclaurin's Series. — The expansion (27) becomes, on making x = o, and substituting x afterwards instead of h, /(•) -/(o) +5/(o) +^/(o) + . . . + ^ / (fe) ; where is > o and < 1. This remainder becomes infinitely small for any function x" f{x) whenever ,— /( n ) (6x) becomes evanescent for infinitely 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 ^ (log «)»««*. Now, a 0x is finite, being less than a x ; and it has been proved in Art. 78 that - — 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. 81. Remainder in the Expansion of sin x. — In this case D x n . fnir a M n = — sm 1 — + Vx \n \2 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 x x 2 x 3 x* + + &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. Again, since/" (x) = f- iY 1-1 ' ' * * , -, the remainder B n is denoted by — — — f — ] j; hence, if x he positive and less than unity, ^- is a proper fraction, and the value of 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 1 + I 1.2 mini- 1) . . . (m-n+ i)x n + — — + &o. 1 . 2 . . . n if n n denote the n th term, we have u n+l m-n+ 1 x, 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 + a , ) m ~ n , the remainder, by formula (25), is m(m-i)...(>n-n + i) aj , + 1 . 2 . . . « or w (m- 1) . . . (w-w + 1) af 1 . 2 . . . w (1 + &r)"-'" Now, suppose a? positive and less than unity; then, when n is very great, the expression **(/»- • • . (w-w+i) ^ 1 . 2 . . . ra becomes indefinitely small ; also -, 77-^ — is less than unity; J (i + 0x) n - m J 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-i)...(m-n+i)x n . n . n . . n .__ (-i) n — ^7 r '- — (1 - B) n ~ l (1 -Ox)"™; ' 1 . 2 ... [n- i) ' v or , . n m(m- 1) . . . (m - n+ 1) (1 - 6) m ~V i-0 (_I) " 1.2 ..T(n- 1) 1 -Ox t — a Also, since x< 1, 0x< 0; .' . 1 - Ox > 1 - 6 : hence jr- 1 - Ox 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 i+-J or y m ii+- now, if the index m be fractional or negative, and x > y, or - a proper fraction, the Binomial Expansion holds for the x series / y\ m m , m(m-i) (x + y) m = x m ii +-) =x m + — x m - 1 y + — -x m - 2 y 2 + &c, but does not hold for the series ( x \ m m , m(m-i) (x + y) m = y m li +-) =y m + - y n ^ l x + — ± '- y m - 2 x* + &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 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, *>y ( 8 )> P- 50— /(n)(ir) = (IJ' tan ' 1 ^ ( - l)n " 1 \n - 1 . sin { n n tan" 1 ^ (1 + x 2 )Z Hence we have, in this case, x n sin In n tan' i?n=(-l)"- 1 "'(ft*)] n(i + 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 3 + x* + &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 a; when x < 1, as well as to other analogous series. In every case, the value of JR n , 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*, then dx * V ' Now suppose that we have y = a + a x x + a 2 x 2 + . . . a n x n + . . . , dy then -f- = «! + 2a t x + . . . m tt x n ~ l + &o. dx Accordingly we have «i + 2« 2 # + 3«3# 2 + . . . « & + #2# 2 + &o., * 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 ()»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 «i do a.% a p 22 3 2.3' Moreover, if we make x = o, we got a = 1, x x % x 3 .-. e* = 1 + - + ■ + + &c, 1 1.2 1.2.3 the same series as before. Again, let y = sin (msin" 1 ^). Here, by Art. 47, we have . ,. cPy dy Now, if we suppose y developed in the form y = a + a& + a 2 x 2 + . . . + a n x n + &c, dv then — = a x + 2a 2 x + xagr' + . . . + na n x n ~ l + &c, ax dry . . _ „ -r-r = 2a 2 + 3 . 2a 3 # + .,. + 11(11 — i)a n x n " + &c. Substituting and equating the coefficients of x n we get (11 + 1) (11 +2) Again, when x = 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 i-j-\=m; accordingly a y = m ; m 2 - 1 m(m 2 - 1) .'. a 3 = «! = - 1.2.3 s (m 7, - 1 ) 4-5 1.2.3.4.5 m 2 - m (m 2 - 1 ) (m* - a) a 5 = a* = — — — : Expansion of sin mz and cos mz. 87 hence we get • • / • -, \ m m ('* 2 - 3 sm* (m sm W = - « x 3 1 1.2.3 m(?» 2 - 1) Cm 2 - 9) „ . . + — - — x 5 - &o. (35) 1.2.3.4.5 In the preceding, we have assumed that sin _1 # is an acute angle, as otherwise both it, and also sin (m sin"" 1 ^), would admit of an indefinite number of values. — See Art. 26. 87. Expansion of sin mz and cos?»3. — If, in (35), zbe substituted for sin -1 #, the formula becomes (36) sin mz = m sin z 1 1 nv ~ * „:^2,. [1 1. 2-3 (m* + - — 1 - 1) (m 2 - .2.3.4, ■ Q) sin's - • 5 &c. In a similar manner it can be proved that cos mz = 1 - m 2 sin 2 , 3 m 2 (m 2 ■ - + — 4 ^ sin 4 z &c. 1.2 I.2.3.4 (37) 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 misa fraction. A more complete exposition of these important expansions will be found in Bertrand's " Calcul Diff erentiol." In general, in the expansion (36), the ratio of any term to that which precedes it is -. r—. sin 2 s, which, when 1 (n + 1) (n + 2) n is very great, approaches to sin 2 s. 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 wa3 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 1 ) . gOBin-i* = j + _ + + _J / x 3 + _^ / x i + > < # I 1.2 1-2.3 1-2.3.4 can be easily arrived at by a similar process. 88. Aroogast's Method of Derivations. If u = a + b- + c h d + &c, 1 1.2 1.2.3 to find the coefficients in the expansion of ^ (u) in ascending powers of x — Let f{x) =

dx~ 2 ' d?' C '' by successive differentiation of the equation /(#) =

'" («) («0 8 > f"(x) = $' (a) . u* + f (u) [>' «r + 3 (w") 2 ] + 6f\u) . (uj. u" +

'(a).c + "(a).b\ J) = f" (o) = $ (a) . d + 30" (a) . fc + 0'" (a) . 5 3 , E = /*» (o) = $' (a) . e + " {a) {$bd + 3c 2 ) + 6f (a) . b 2 c + iv (a) . 6 4 . From the mode of formation of these terms, they are seen to be each deduced from the preceding one by an analogous law to that by which the derived functions are deduced one from the other ; and, as /'(#), f"{%) • • . are deduced from fix) by successive differentiation, so in like manner, B, O, D, . . . are deduced from (p (u) by successive derivation ; where, after differentiation, a, b, c, &c, are substituted for du d 2 u „ W '^' ^'••• & °- If this process of derivation be denoted by the letter S, then B = 6.A, C=$.B, D = B.C,&c. (38) From the preceding, we see that in forming the term $ . '(a), and multiply it by the next letter b, and similarly in other cases. Thus $ .b =c, S . c = d, . . . 8 . b m = mb m ~ J c, $ . c m = mc m ~ l d . . . Also 8 . $' [a) b = "(x) "(x) and also tf>'"(°) = °> & -• 14. If, in the last, ^-~ = a 2 ; prove that d>[x) = e"* + " ( x ) If ■ — r-r- = — ar ; prove that * (x) = 2 cos (ax). #(z) ' Examples. 93 15. Apply Arbogast's method to find the first four terms in the expansion of (a + bx + ex 2 + dx 3 + &c.)». Am. a n + Ma" -1 bx + ( — ' b 2 + nac ) a"- 2 x 1 \ 1 .2 / + n | ( w ~ 0( w ~ 2 ) gn -3^3 + ( M _ 1) a n-2fo + «n-]^j ^ + & c . ^x _|_ I 16. Prove that the expansion of . x can contain no odd powers of x. For if the sign of a; be changed, the function remains unaltered. x 17. Hence, show that the expansion of contains no odd powers of x e x — 1 beyond the first. „ x x x e x + 1 . Here + - = - . ; .*. &c e* — 1 2 2 e" — 1 18. If w = , prove that e* - r * rfn^MX «(m - I) fd«*u\ (du\ ^)o + -7^Wo + - +M (4 +(M)o=0i n /d n ' l u and hence calculate the coefficients of the first five terms in the expansion of «. Here e x u = x + w, and by Art. 48, we have (du nln— 1) d 2 u d n u\ d n u u + n — + — — + . . . + — J = — -, .-. &c. dx 1.2 dx 1 dx n ) dx n % x B\ Bz 2?3 19. If = 1 - - + x* z 4 + — -, z* - . . . e x — I 2 1.2 1.2.3.4 1.2...0 prove that 1 „ 1 ^ 1 ,, B 2 m — , 1? 3 = -, 6 30 42 These are called Bernoulli's numbers, and are of importance in connexion with the expansion of a large number of functions. ao. Prove that x x Bix 2 . , . B 2 x* . t . B31* = ( 2 2_l) + __ 2*-i) 2G_ i)+ .. e* + 1 2 1 . 2 v ' 1 . 2 . 3 . 4 v ' 1 . 2 . . . 6 94 Examples. 21. Hence, prove that , e i = Six (22 - i) + f^( a 4-i)+ 3 - -(2«-l) + &C «"+i 3-4 3-4-5-6 v a; s 3 a; 5 „ = + &c 2 24 24O 22. Prove that 2'BlX* 2*i?2^ 2«# 3 2« a; cot a; = 1 7 — &c. 1.2 1.2.3.4 1 . 2 ... 23. Also, tan - = B x x (2^ - 1) + -^— (2* - 1) + &c. 24. Prove that xx _ a* _ a* i? 3 a; 8 - cot - = r - Hi Bz-r- — 22 (2 |4 |6 z This follows immediately by substituting - for x in Ex. 22. 2 j. Given u{u— x) = 1 ; find the four first terms in the expansion of w in terms of a;, by Maclaurin's Theorem. 26 ' If *^ + ^ +y = °' expand y in powers of x by the method of indeterminate coefficients. 27. Show that the series x 3? x 3 x* + — +— + —+... jfrt 2, m 7 n * 4,"* is convergent when x I, for all values of w. 28. Prove the expansion (a: - «)'» (ar) (a; - «)"» <£(«) (x - a)'"" 1 rfa (<£(«) J _i_ /f\ 2 (/Mj +& ,... I . 2 . (* - a)" 1 " 2 \da) {(*>(«)) 29. Find, by Maclaurin's Theorem, the first four terms in the expansion of ( r -f x) x in ascending powers of x. Let f{x) = (i + x)', Examples. 95 (1 log(i + x)\ x(l + X) X* } —/CO {J-f'+l**-* -}- rw=-/"w(J-^+^-^)+ j /'w(^-h fa } But, by Art. 29, /(o) = e; •••/'(o) = -p /» = ^ /'"(o) = -^- *• ex ilex 11 ne , Hence (1 + *) = * + ' c x + &c - v ' 2 24 16 This result can be verified by direct development, as follows: 1 let u = [i + x)', then log u = - log ( 1 + x) = 1 — + - + ...; x 234 2 3 2 3 1-! + -- -• •• -%*—-... f /a: a; 2 r 5 \ a; 2 / r x x 2 \ 2 x 3 1 1 a; \ 3 "1 — L" _ Ci" + T'-0 + ?U~ + 5— ) -io(i-i + --; •— J t« 1 1 x 2 7a; 3 "1 i-- + S • • • • 2 24 16 J 30. In Art. 76, if f(x) and/'(.r) be not both continuous between the points Pi, l'i, show that there is not necessarily a tangent between those points, parallel to the chord. cc sin toe 31. Find the development of H in ascending powers of a:, the coef- sin x sin 2a; ficients being expressed in Bernoullian numbers. " Camb. Math. Trip., 1878." Since — — - — - = x cot x + x cot 2a;, the expansion in question, by (22), sin x sin 2a; J is 3 r-rinX** . 2«7? 4 .T 4 2°i? G xG |-_ -(2+1) 5- («• + ») jj-Ca' + O-Ae, ( 96 ) CHAPTER 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 denominator of the fraction —7-^ vanish, then*— \~ becomes of 0(*) 0(«) the form -, and its value is said to be indeterminate. o Similarly, the fraction becomes indeterminate if / (x) and

its true value in all cases is 7. b (x - cy b Examples. 97 2. The fraction — ^^ ^^r becomes - when x = o. Va+£-v a—x ° To find its true value, multiply its numerator and denominator by the com- plementary surd, \/ a + x + \f a — x, and the fraction becomes n, x m -» = 00 when x = 00 ; or the fraction is infinite in this case. (2.) If m = n, the true value is — . (3.) If m < n, then x 1 *-* = 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. « = \/ x + a - \/x + b, when x = oc. Here u = = o when x = 00. V x + a + v x + b / A a 9. \/ x 2 + ax — x, when x = co. Ans. -, 10. u = a x sin ( — ] , when x = 1 (1.) If a < 1, a* = o when x = 00, and therefore the true value of u is eero in this case. (2.) If a > 1, then a* becomes infinite along with x ; but as — is infinitely small at the same time, we have sin — = — . Hence, the true value ' a* a* of u is c in this case. Method of the Differential Calculus. 99 1 r. « = va 2 - x 1 cot - A is of the form o x so when x = a. v 2 \a + x Here + *,/ a 2 — x- ir la - x tan - J 2 \a + x but, when a — x is infinitely small, it la — x ir la — x tan- ,J = - A — ■ — ; 2 y a + x zva + x \/ a 2 - x* a + x \a m = , — = — — = — when x = a. ir la - x * * 2 \a+ x 2 x sin (sin x) — sin's 12. m = ^ , when* = o. x 6 Substitute the ordinary expansion for sin a;, neglecting powers beyond the sixth, and it becomes x ] sin x sin'a; IF sin 5 *) (~ x 3 x*y X - X 3 X* "iI + F «- X 6 X 3 \ 3 X 5 - xl I a-2 x i \ 2 Hence we get, on dividing by x 5 , the true value of the fraction to be — when 1 o x = o. (a sin> + J3 cos 2 . a" — p" 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. — Hethod of the Differential Calculus. — Sup- f(x) . . o pose '-~-4 to be a fraction which becomes of the form - when 1 (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 <£(«+ A)' 0(« + h) - '(a), respectively; hence, in this case, /(* + *) = y>) '(a)' '(a) = o, the true value of tK 1S °°« f r (n\ (3.) If f'(a) = o, and #'(#) = o, our new fraction -77-T is \ a ) still of the indeterminate form -. Applying the preceding process of reasoning to it, it follows that its true value is that of when a; ~ a. (x - a y Here /(a;) = e™* — e ma , '{ 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 he arrived at hy writing the fraction in the form { fn(*-«) — I } e ma e mh _ r (x - a)' e">", where h — x — a ; hence, expanding e mh , and making h = o, we evidently get the same result as hefore. 3- X- - sin a X 3 ■ when x = o. Here f'(x) = i - cos a;, f'io) = 0. 4>'(x) = 3* 2 > '(o) = 0. f"(x) = sin a;, /"(o) = o. [a) 104 Indeterminate Forms. latter fraction has a finite limit, its value by the preceding method is cp\a) '%» + *' therefore '■ ,, , = o ; «*. e. when -4 is zero, -r,-4 is also zero, and 4>'(a) {a) vice versa. fix) Similarly, if the true value of -~- be infinity when x = a, then ^Vr is really zero ; we have, therefore, %rrx = o, by what ft t \ has been "just established ; .'. .. . = 00. Accordingly, in all cases the value of ' , , * determines that of — 7-r for either of the indeterminate forms - or ~. '{x) = , and the fraction . , . is still of the form — , but it can 2 ■K X 2 O be transformed into r~, which is of the form - : the true value of the latter cos 2 * o TT fraction can be easily shown to be — 00 when x = -. ' 2 In some instances an expression becomes indeterminate from an infinite value of x. The student can easily see, on substituting - for x, that our rules apply equally to this case. Indeterminate Expressions of the Form {/(#)}*(*). 105 93. Indeterminate Expressions of the Form {/(«)}+(*). Let u = {/(*)} *W, then log u = 0(a?) log/(ar). 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 (x) = o, and log {/(#)} = ± 00; the latter re- quires either f(x) = co, oxf{x) = o. Hence, {/(#)}*(*) becomes indeterminate when it is of the form o°, or oo°. (2.) Let m+» 2. — when x = . This fraction can be written in the form / — \ . The true value of ■£-, by the method of Art. 92, is that of — — ; but the value of the latter fraction is zero n when x = 00 ; hence the true value of the proposed fraction is also zero at the same time. 3. m = x n (log x) m , when x = o, and m and « are positive. Here u = (x m log x) m , log a; is of the form — when x = o ; its true value is that of x m Hence, the true value of the given expression is zero. This form is immediately reducible to the preceding, by assuming *•• = «t. m = when x =00, Here tt= f_^\-"' : but if b > 1, and n > m, }* n m = co when x = 00. Consequently the value of wit of the form o" , or is zero in this case. Again, if m > n, b* =0 when x = oo, and the true value of u is 00. io8 Indeterminate Forms. 5 . u = j- when x = o. b* m Let x = -, and this fraction is immediately reducible to the form discussed in z the previous Example. . (i - cosa;)"{log(l + x)} m , . i 6. i — - — — -!— , when a; = o. Am. —. x 2n*m 2*» 1 (i +x)* -e . 7. m = , when x = 0. x 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 ( S-(I 4-g)log(l +X) ) Again, substituting for (1 + x) x its limiting value e, we get ix - (1 + a;) log (I + x) { x 2 (i + x) the true value of which is readily found to be — when x = o. Compare Ex. 29, p. 94. I m x — 1 I la sin x — sin ax) n , 8. — < — ; > , when x = o. I sin a; | [x(cosx - cosaxj) tn x — 1 The true value of — : , when x = o, is log m ; sin a; a sin x — sin ax and that of -. :, when x = o, x (cos x — cos ax) a has been found in Example 6, Art 89, to be-; hence the true value of the given expression when x - o, is ( - J log Examples. 109 Examples. x> •{_!_£ — £_!_' when a; = a. ^fws. ~-f. («) (sin wx\ "» cos xd - cos M0 ■>. , x = n. 00 . 3 (a; 2 -« 2 )' ' V« + X — \/ 2X 4- — - — ^, * - a. 2;n+; _ a n+l 5. 1 « = - I. n+ I ' 9- log sin x' 12. n fx\ - - cot - ) , x \nj [ 3- x 1 + 2 cos a; - 2 X* 14. 1 a; + sin zx - o sin - 1 1 4 + cos a; - 5 cos - 1 (^-1)3 ' 3' i — sin x + cos X ir , *=•-. I. sin x + cos x — i 2 tan x - sin x I sin^a; 2 (a 2 - x 2 )i + (a - x)t \/™ (a»-*»)M(a-*)»* • I+»/: z* tan a; ;, X = o. I. («*- l) f a s ' nx —a w , a; = -. a Iok a. (?)' no Examples. V z + cos ix — sin x . t v ' 15. : , when x = -. _4«s. ar sin ix + x cos x 2 3*- , a*» sin wa — m° sin xa 16. , * = M. tan na — tan xa n<*~ 1 (n cos m« - sin na) cos 5 «a. x 2 tan nx — n tan a; 4 17. . ■. : , * = o. — . 1 — cos mx n sin x - sin nx m* (2 sin x — sin 2z) 2 8 18. -. —, x = o. . (secx — cos2x) J 125 -!- I 19. X' (« - y) WW + »'( y)} - 2 '"0) {•-»)* ' *' 6 ' at log (1 + *) _ ^ l 22. a; . e x , e* - er* 23- 1 — ; r» x = o. 2. log (I +*)' was — I w _ ^* 24 ' ~a?~ + («»** - 1) *» * 2 * log (tan 2a;) log (tan x) , e* + log(i - x) - 1 1 26. — > a; = o. - -. tan x - x 2 27. t ; — r=^ tan- 1 V ' w cos d> - — cos d>, m = 1. cos 2

{xy)m Differentiation of a Function of Two- Variables. 115 96. Differentiation of a Function of Two Vari- ables. — Let u =

(x + h, y + k) -

(x + h, y + k) - $ (x, y + k) + (x, y + k) - $ (x, y) (x,y + k) {v,y + k ) - (x + h, y + k) - (x, y +- k) d . (x, y+ k) h dx * and (x, y + k) - (x, y) ^ d . (x, y) k dy In the limit, when k is infinitely small,

' + *> becomes (x,y,z+l )-(x,y,z ) which becomes in thv limit, Ly the same argument as before, when dx, dy, dz, are substituted for h, k, I, du . du T du T , . du = — dx + — du + — dz. (2) dx dy J dz v ' 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 1} x 2 , x-, . . . x n , du du . du . ' du = — - dxi + -r- dx 2 + . . . + -— dx n . (3) dxi dx 2 dx n 98. If u =/(?, w), where v, w, are both functions of x ; then, from Art. 96, it is easily seen that du df(v,w) dv df{r, w) dw dx dv dx dw dx' This result is usually written in the form du du dv du dio . dx dv dx dw dx' In general, if « = 0(*/i, 2/2, • • • Vn)y 1 1 8 Partial Differentiation. where y lt y 2 , . . . y n , are each, functions of x, we have du du d//i du dy 2 du dy n dx dt/i dx dy-i dx ' ' ' dy n dx' Also, if y lf y 2 , &c, y n , be at the same time functions of another variable z, we have du _ du dj/i die dy 2 „ du dy„ dz dyi dz dy 2 dz ' dy n dz ' and so on. •SAMPLES. i. Let u = (X, Y), where X = ax + by, Y = a'x + b'y ; du du dX du dY dx = dXlx + OYdx' then du du dX du dY dy~ dXdTj* dfdy' . . dX dX T dY , dY L , but _ = a , — = b, — = a' , — » b\ dx dy dx dy -rr du du ,du Hence — = a 1- a — , dx dX dY' du , du ,.du — = b — • + V — . dy dX dY 2. More generally, let u = (x, y, z). du du dx du dy du dz du du da dx da dy da dz da dx dz _. .. . du du uu du du du Similarly, ^ = - - ^ , ^ J% - Ty \ du du du ' da dfi dy This result admits of obvious extension to a function of the differences of any number of variables. 1. If 2. If dA dA dA da dB dy Examples. h «i «• '» A = a, a 2 , 0, B\ 7, 7 2 , 5, 8 2 , , prove that « S , B f , T 3 , s 3 , dA dA dA da d@ dy dA = 0. I. ', h I, A = « 2 , B, b 2 , 7> 7 2 , 8, S 2 , , prove that ««, B*, 7 4 , 8 4 , h i, h h dA a, b, b 2 , 7, 7 2 , 8, 8 2 , « 3 , P, 7 3 , 8 3 , 1 20 Partial Differentiation. 99. Definition of an Implicit Function. — Suppose that y, instead of being given explicitly as a function of %, is determined by an equation of the form /(*> y) = o, 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 k denote the increment of y corresponding to the incre- ment h in x, and denote f(z, y) by u. Then, since the equation f(x, y) = o is supposed to hold for all values of x and the corresponding values of y, we must have f{x + h, y + k) = o. Hence du = o ; and accordingly, by Art. 96, we have, when h and k are infinitely small, du , du . — h + -7- k = o ; dx dy du hence in the limit 1 = ~r =- ~7~- (6) h dx du v ' dy This result enables us to dete»:iiine the differential coefficient of y with respect to x whenever the form of the equation /(», 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 please — in 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 dx dx 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. dy 1. If z 3 + j/ 3 - laxy = c, to find -~ . dx Here ^ = 3(^-^y), -j- = Z^-ax); See Art. 38. dy x 2 — ay dx ax — y 2 x m y™ , „ , dy 2. If — + i- = 1, to find ~. rfw wis" 1 " 1 dy dx dx dy dx' df df dti — + — — = o. dx dy dx Hence, eliminating —-, we get (fydf dfd

d

dz dx dx dy dx dz dx' df + dx df dy df dz dy dx dz dx o, df + dx Hence, we get df dy dfi dz dy dx dz dx d<{> d d(p dx' dy' dz df df dfi dx' dy* dz dfi df df = 0. du dx' dy' dz dx 4f df dy' dz df dfi df dz This result easily admit 3 of generalizat] on (8) Miller's Theorem of Homogeneous Functions. 123 102. Kuler's Theorem of Homogeneous Func- tions.— If u m Ax* y* + Bx& 1/ + W y*" + &c, where P ■*" ? = P + ?' = P" + 4' = & 0, = w i to prove that du du , . "S + '«-""■ (9) Here x — = ^a^ y? + 2?p' ^ ^'/ + & c . j y — = .4^ ;«/ + Bgf a? y q ' + &c. ; .\ x— + y — = A(p + (fixPyi + B(p' + /) x p ' y q ' + &c. (too ciy = nAxP y^ + nBa?' y q/ + &c. = nu. Hence, if u be any homogeneous expression of the n ln degree in x and y, not involving fractions, we have du du x—+ y ~ - nu. dx dy Again, suppose u to be a homogeneous function of a fractional form, represented by — ; where 6 U 6 2 , are homo- 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$ x d(j> 2 02"7 0i~F" • dx dx dx du dy (0*) 2 ' dtyl d(pn 92 dy l dy t 124 Partial Differentiation. accordingly we get du du r \ dx J dn J y \ dx * dy x Tx + y T y = ' faf but, by the preceding, _ du du n 2 - m6i f - Then -r- = nx n ~ x v + # n -7-, ote dx . du dv and — = x n — : dy dy multiply the former equation by x, and the latter by y, and add ; then du du ( dv dv\ x~r+y-r = nx n v + x \x— + y — ; dx y dy \ dx J dy) ' Euler's Theorem of Homogeneous Functions. 125 but (by Ex. 3, Art. 96), dv dv x — v v — = o : dx * dy ' du du hence x — + y -7- = nx n v = nu, dx dy which proves the theorem in general. In the case of three variables, x, y, 2, suppose u = Ax p y q z r , then we have du . du . du . x—=Apx p y q z r , t!—-=Aqx p y' 2 z r , z— -Arx v y q z r ; cioc cly d% du du du . , . . . ''' X dx + y dy + Z dz = A (P + Q + r ) xP ?/ q * r = {P + q + r)u; and the same method of proof can be extended to any homo- geneous function of three or more variables. Hence, if u be a homogeneous function of the n th degree in x } y, z, we have du du du , . X T x +V 7y + Z di = " U - (I0) It may be observed thac the preceding result holds also if n be a fractional or negative number, as can be easily seen. This result can also be proved in general, by the same method as in the case of two variables, from the considera- tion that a homogeneous function of the n th degree in x, y, z admits of being written in the general form u = x n d>[-,- \X X or in the form 1/ £» u = x" & (v. w), where v - -, and w = -. x x Proceeding, as in the former case, the student can show, 126 Partial Differentiation. without difficulty, that we shall have du du du x — + y — + z— = nu. dx dy dz Another proof will he found in a subsequent chapter, along with the extension of the theorem to differentiations of a Higher order. Examples. Verify Euler's Theorem in the following cases by direct differentiation : — x 3 + y 3 du du 5« *■ u ° / — ; — u » P rove x t + y T = —' (z + yp dz dy 2 z 3 + axhj + by 3 du du dz dy . . z z — tfl du du (z s — y 3 \ du du ++?)' ' Z Tz + y dy = U ' 103. Theorem. — If U = u a + u x + u 2 . . . + u n , where u is a constant, ^nd u lf u 2 , . . . u„, are homogeneous functions of x, y, z, &c, nf the ist, 2nd, . . . n th degrees, respectively, then dU dU dU , x x — + y — + z — + . . . = u x + 2U 2 + 3W3 + . . . + nu n . (1 1) ctOu ay ci% For, by Euler's Theorem, we h^ve du r du r du r x -—+ y —- + z—r + &c. = ru n dx dy dz since u r is homogeneous of the r th degree in the variables. Cor. If U = o, then dU dU dU » / n x— + y — + z— ... = - « n _! + 2w n _ 2 + . . . + nu ). (12) dx dy dz This follows on subtracting nu + nth + . . . + nu n = o from the preceding result. Remarks on Eider's Theorem. 12J 104. Remarks on Euler's Theorem. — In the appli- cation of Eider'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 (x + t/ \ —r — — j might appear to he 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 /»• l 4/ in the form —, — ^- = sin u, we have, by Euler's formula, a£ + y* d sin u d sin u sin u + y- 'u or cos u I x — + y dx dy ( du du\ sin u T55 dy . du du f an u \ x + y hence x — + y — =» dv d V 2 2 ^(xi + yiy-(x + yy When, however, the degrees in th-« numerator and the denominator are the same, the function is of the degree zero, and in all such cases wo b^« du t^j x—+y — = o. dx dy -j — —, j, tan -1 -, e y , &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

, and draw B'D perpendicular to AB, produced if necessary; then, by Art. 37, AB' d i = AD when .RC is infinitely small, , neglecting infinitely small quanti- ties of the second order. Hence Ac = AB' -AB = AD-AB = BD; dc .. ., , Ac BD _ ,\ — - limit of — = -7773 = cos B. da Aa BB Fig. 4. Examples in Plane Trigonometry. 131 dc Similarly, — = cos A ; which, results agree with those arrived at before by differentiation. dc Again, to find -r~. Suppose the angle C to receive a dL/ small increment AC, represented by BCB' in the accompanying figure; take CB' = CB, join ABf, and draw BD perpendicular to AB f . Then Ac - AB' - AB = B'D (in the limit) = BB' cos AB'B = BB' sin ABC(q.-p.). Fig. 5. Also, in the limit, BB' = EC sin BCB' = a AC. Hence -^7= = limiting value of —7= = a sin B : «6 T ° AC the same result as that arrived at by differentiation. In the investigation in Fig. 5 it has been assumed that AB - AD is infinitely small in comparison with BD ; or that AB - AD the fraction — — 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 100. To find — , when a and are considered 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 BR = ABAA _ ABAA _ _ cAA sin AB'B cos B cosB' k 2 132 Partial Differentiation. Also, BB' = a&C; dC AC ... • • 1~a " Tli ( m the linut ) = 5- ( l6 ) dA AA s a cos B v ' This result admits of another easy proof by differentiation. For a sin B = b sin A ; hence, when a and b are constants, we have a cos B dB = b cos -4 d4 ; also, since .4 + B + C = tt, we have dA + dB + dC= o. Substitute for di? in the former its value deduced from the latter equation, and we get (a cos B + b cos A) dA = - a cos B dC ; or c gL4 = - a cos 5 c?C, 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£ = T + tan -4' ^ 7 ' in. Landen's Transformation. — The result in equa- tion ( 1 6) admits of being transformed into dA dC a cos B c but c = */a 2 + b 2 - zab cos C, and a cos B = */a 2 - b 2 sin 2 .4; hence we get dA dC x/a 2 - b 2 sin 2 A */ a 2 + b 2 - zab cos C Examples in Spherical Trigonometry. 133 If C be denoted by 180 - 2fa, the angle at A by , and - by k, the preceding equation becomes a d 2d^i 2di (1 + k) \/i - k* sin 2 #i where fa = ; . 1 + K Also, the equation a sin B = b sin A becomes sin (20! - <£) = Jc sin^. The result just established furnishes a proof of Landen's* transformation in Elliptic Functions. We shall next investigate some analogous formulae 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 •=6infl sin B. dc Again, the value of — , when b and C are constants, can da be easily determined geometrically as follows : — * This transformation is often attributed to Lagrange ; it had, however, been previously arrived at by Landen. (Se6 Philosophical Transactions, 177 1 and "7750 (34 Partial Differentiation. In the spherical triangle ABC, making a construction similar to that of Fig. 4, Art. 108, we have BB=&a; .'. — = limit of — - = -^^ da Aa BB (in the limit) = cos B. Similarly, when a and C are con~ stants, — ; = cos A. do Hence, finally, Fi s- 6 - dc = cos Bda + cos Adb + sin a sin B dC. (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 = AD in the limit ; i. e., that =-7-=- — 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 c - b very small, then the ratio — — becomes very small at the same time, where c 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 = 7 = j—. — ; tan c cos b sin c 1 - cos A sin c cos b - cos c sin b sin (c - b) t 1 + cos A sin c cos b + cos c sin b sin (c + b) ' or • / i\ i-4 • / 7\ sin ( instead of a, 1// instead of b, and k for — — , this equation becomes sin c . T sin .4 smi?\ since k =» —. = — : — 7 \ sin a sin b) d(j> dip yi-k 2 sm 2

cos \p + sin sin \p */'i - F sin 2 c. 115. In a Spherical Triangle, to prove that da db do , . + IZrh + 7ZTFi= > ( 2 4) cos A cos B cos G sin C . when -: — Is constant. sin c * This mode of establishing the connexion between Elliptic Functions by aid of Spherical Trigonometry is due to Lagrange. Examples in Spherical Trigonometry. 137 Let smC = k sin c, and we get ._ k cose , sin A cose . dC = - 77 tfc = -: -x.de. cos 6 sm a cos C substitute this value for dC in (19), and it becomes . „ _ 7 cos c sin -4 sin B . dc = cos Adb + cos B da + ~ ac ; cos C / cos c sin A sin 2A or cos Ado + cos B da = [ 1 ~ dc \ cos C T / cos A cos J? . cos C since sin -4 sin B cos c = cos C + cos ^4 cos B. _ rfa <#£> dc Hence 7 + ~ + 7y = °- cos A cos i* cos U Again, since cos .4 = /i - sin 2 ^ = + cos B cos -4rfc = o, or (cos A - sin B sin C cos a) da + (cos B - sin A sin C cos 5) db + (cos (7 - sin A sin i? cos c) dc = o; .'. cos -4ffo + cos Bdb + cos Cdc = sini? sin Ce?(sin a) + sin .4 sin CV/ (sin b) + sin J. sin Bel (sin c) = k 2 { sin 6 sin cd (sin a) + sin a sin erf (sin &) + sin a sin 5rf (sin c) } = Z,-V/ (sin a sin J sin c) ; 138 Partial Differentiation. or -v/ 1 - k 2 sin 2 ada + */i - k 2 sin 2 bdb + */\ - k 2 sin 2 c dc = k 2 d (sin a sin b sin c) . (26) This furnishes a proof of Legendre's formula for the compa- rison of Elliptic Functions of the second species. The most important application of these results has place when one of the angles, C suppose, is obtuse ; in this case cos C is negative, and formula (25) becomes da db dc (x + at, y +fit, z + yt), it becomes u =

~di = ^ Tt=t' _ du du „ du du , n . Henoe Tt'-Tx^Ty^Tz- {2&) This result can be easily extended to any number of variables. 1 40 Examples. Examples. (x\ /y\ dx dy - J + sin" 1 I — J , prove that du = , + — , 2. If u = xy

* ™ + y ^ = 2«. 3. Find the conditions that «, a function of x, y, z, should he a function of x + y + 2. aw «« rfw ,4n». — = —-=_. ax ay dz 4. If /(«c + by) = c, find ■£. „ - ^. 5. If f(u) = are each functions of x and y, prove that du dv dv du dx dy dx dy' du du t>. Find the values of x — *■ y ~r > when dx dy . , axi -t- 3y* (a) M = r- ;, wix 2 + ny z 7. If m = sin ax + sin 5y + tan -1 I - J , prove that t r , z ^y _ yds du = a cos ax dx + cos oy dy -\ r — . y 2 + z 2 _, , . . du . du du r du — log* 8. IfM = loguS, find— and— . ,4ms. — = — , — = — — . dx dy dx xlogy dy y (log y) 2 q. If = tan -1 — , prove that y (z 2 + y 2 ) de = ydx - xtfy. 10. If u = y", prove that du = y**- 1 (xzdy + yz log yak + xy log yrfz). Examples. 1 4 1 If a + */ a* - y % = ye a , prove that dy -y 1-3. In a spherical triangle, when a, b are constant, prove that dA tan A , dC sin C -, and — - = — - dB tan.B dB sin .5 cos ^4 13. In a plane triangle, if the angles and sides receive small variations, prove that cAB + b cos AaO= o ; a, b heing constant, cos CAb + cos BAc = o ; a, A heing constant, tunAAb = bAC; a, B heing constant. 14. The hase c of a spherical triangle is measured, and the two adjacent base angles A, B are found hy observation. Suppose that small errors dA, dB are committed in the observations of A and B ; show that the corresponding error in the computed value of C is — cos adB — cos bdA. 15. If the hase c and the area of a spherical triangle be given, prove that a b sin* -dB + sin 2 - dA = o. 2 2 16. Given the base and the vertical angle of a spherical triangle, prove that the variation of the perpendicular p is connected with the variations of the sides by the relation sin Cdp = sin s'da + sin sdb, s and tf being the segments into which the perpendicular divides the vertical angle. 17. In a plane triangle, if the sides a, b be constant, prove that the variations of its base angles are connected by the equation dA dB *f a 2 — b 2 sin 2 A */ br - a 2 sxd?B' 18. Prove the following relation between the small increments in two sides and the opposite angles of a spherical triangle, da dB dA db tan a tan B tan A tan b' 19. In a right-angled spherical triangle, prove that, if A be invariable Bin 2cdb = sin 2bde ; and if c be invariable, tan ada + tan bdb = o. 142 Examples. 20. If a be one of the equal sides of an isosceles spherical triangle, whose vertical angle is very small, and represented by da>, prove that the quantity by which either base angle falls short of a right angle is - cos a da>. 2 2i. In a spherical triangle, if one angle be given, as well as the sum of the other angles, prove that da db sin a sin b 22. If all the parts of a spherical triangle vary, then will cos Ada + cos Bdb + cos Cde = kd (k sin a sin b sin e) ; sin A sin B sin O where k = — : — = - — r = - — . sin a sin b sin c da db dc „ /i\ Also — — t + _-- + -^;=tan^teni?tanC^I-) cos A cos B cos These theorems can be transformed by aid of the polar triangle? — M'CullagK, Fellowship Examination, 1837. These are more general than the theorems contained in Arts. 115 and 116, and can be deduced by the same method without difficulty. 23. If z =

' dz dy dy dz Examples. 143 26. Prove that any root of the following equation in y, y m + xy = i, satisfies the differential equation „ cPt/ , dy 3 dy 2 27. How can we ascertain whether an expression such as y) + */- »+(*f y) admits of heing reduced to the form /(* + yv /~i)? dd> dili dd> d\p dx dy dy dx 28. If IX+mY+nZ, I'X + m'Y+ n'Z, l"X + m"Y + n"Z, be substituted for x, y, z, in the quadratic expression of Art. 107 ; and if a', b', (x, y), then — represents the limiting value of {x, y) h when h is infinitely small. This expression being regarded as a function of y, let y become y + [k, x remaining constant ; then — f — j is the limiting value of (x + h,y + k)-(x + h, y) +(j>( x, y) hk when both h and k become infinitely small, or evanescent. In like manner — is the limiting value of {x,y + k) - 4>\x, y) when k is infinitely small ; hence — ( — ) is the limiting value (X00 u(y ' / of (x+h, y) - (x,y + k) + H ~ dy> dP_dhi_ dQ _ d 2 u dy dydx' dx dxdy Hence the required condition is dP = dQ dy dx' 121. If u be any Function of x and y, to prove that where x and y are independent variables. Here each side, on differentiation, becomes dxdy dxdy' 122. More generally, to prove that d ( dv\ d ( dv\ , . Ty\ U dxr H\ U Ty) (4) where u and v are both functions of 2, and z is a function of x and y. _, d ( dv\ du dv d 2 v For — u-r } = —— + u but dy \ dx) dy dx dydx* du du dz dv dv dz dy dz dy* dx dz dx ' d f dv\ du dv dz dz cPv dy \ dx) dz dz dx dy dydx ' and — ( u — j has evidently the same value. l 2 148 Successive Partial Differentiation. 123. Euler's Theorem of Homogeneous Func- tions. — In Art. 102 it has been shown that du du where u is a homogeneous function of the n th degree in v and y. Moreover, as -7- and — are homogeneous functions of the ax ay degree n - 1, we have, by the same theorem, d fdu\ d fdn\ du d fdu\ d fdu\ du # Tx\^jj +y ly'\dy[) = ^'^dj/'' dx multiplying the former of these equations by x, and the latter by y, we get, after addition, , d 2 u dhi , dhc . . / du du\ ar -r-r + 2x1/ -——- + y 1 -—7. = {n-i)\x— + y—\ dx 2 J dxdy J dy> v ; \ dx J dyj = {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, of the Function d(j> _ d(f> dt dx dy Differentiation of(x+ at, y + fit). 149 ma dx\dt) P dy \dt) dx \ dx dy) dy \ dx ^ dy) m *d* + laf *I* + P?p (6) This result can also be written in the form (P ( d n d) 2 . . f d d\ 2 in which ( a — + (5 -y- is supposed to be developed in the usual manner, and ~, &c, substituted for ( — ) 6, &o. dx- \dxj T d?$ Again, to find — . d 3

\ _ d( d d V !& ~ dt\df) ~ dtydx + "dyj* ( d dYdcp f a f dVf d

(z) becomes {x) + JL-{(x)}> + & . !/" d»- 1 . 2 . . . n da? ^ {*(*)}" + &o. (8) 154 Laplace's Theorem. 126. Laplace's Theorem. — More generally, suppose that we are given *=/{z + ytf>(z)}, (9) and that it is required to expand any function F(z) in ascend- ing powers of y. Let t = x + y(x + h, y) = 4>(%,y) + h—{(x,y)}+^ C —{{x, y + k) + h — (0(a>, y + k)\ h 2 d 2 But d k 2 d 2 4>(x,y+k) = $(x,y) +k — { (x -r h. y + k)* = u + h- r + fc — r v i9i dx dy h 2 d?u d/u k 2 dru 1 .2 dx 2 * dxdy 1 . 2 dy 2 128. This expansion can also be arrived at otherwise as follows : — Substitute x + at andy + (St for x and y, respectively, in the expression § (x, y), then the new function (x + at, y + (Si), in which x, y, a, (5, are constants with respect to t, may be regarded as a function of t, and represented by F(t) ; thus (x + at, y + (3t) =F(t). The latter function F(t), when expanded by Maclaurin's Theorem, becomes, by Art. 79, F{t)=F{o) + t -F\o) + ^- 2 F'\o)^... + j£*M(0O, (3) where F\o) is the value of F(t) when t = o, i. e. F(p) = $ (x, y) = u; also i* v (o), F"(o), &c. are the values of d ± ^ Rra dt' df' ' when t = o ; whert. y stands for $(x + at, y + fit). Moreover, by Art. 117, we have d(j> d

)-*-&+*4zai+PijP (4) &c. &o. &c. These equations may also be written in the symbolio form m-i'Ii + fiih Again, f a — J u = a r — , &a, since a, /3, are independent of a; and y : and hence the general term in the expansion of F(t) can be at once written down by aid of the Binomial Theorem. Extension of Taylor's Theorem. 159 Finally, we have, on substituting h for at, and k for fit, T . T du du h 2 (Pn „ d 2 u (x + at, y + fit, z + yt), when u is substituted for (p(x, y, z), becomes {x + h,y + k). (7) This is analogous to the form given for Taylor's Theorem in Art. 67, and may be deduced from it as follows : — d We have seen that the operation represented by e %dx when applied to any function is equivalent to changing x into x + h throughout in the function. d Accordingly, e dx § (x, y) =

(x + h, y + k), d £ or e kd ~** hdx (p (x, y) = (x + h, y+k,z + I). (8) 131. If in the development (2), dx be substituted for h, and dy for k, it becomes dy 2 ) + &Q. (o) 1 . 2 \da? dxdy y dif J J w/ If the sum of all the terms of the degree n in dx and dy be denoted by d n + — + — - + — + . . . rv r 1 1.2 1.2.3 + -—■ + &c. r Since dx, dy, are infinitely small quantities of the first * That this is the case appears immediately from the equations — — - = -—- , dxuy (Xyu.3/ d*u d 3 u c dx- dy dy dx 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 d, if infinitely small quantities of the second and higher orders he neglected in comparison with those of the first, in accordance with Art. 38, we get d = (x, y) --y-dx + -j- dy, which agrees with the result in Art. 97. 132. Euler's Theorems of Homogeneous Func- tions. — We now proceed to give another proof of Euler's Theorems in addition to those contained in Arts. 102 and 123. If we substitute gx for h and gy for k in the expansion (5), it becomes , x f du du\ (x + gx,y + gy)=u + glx— + y — J a 1 ( „ d*u d*u , d 2 u\ _ where u stands for (x + gx,y + gy) = (1 + g) n (x, y) = (1 + g) n u, or (1 +,)•* = * + ,(*- + y-j g 2 ( ,d 2 u d 2 u ~d 2 u\ + T7z{^ + 2X ' J cMy + *Vj + & °" where u is a homogeneous function of the n th degree in x and y. Euler's Theorems. 163 Since the preceding equation holds for all values of g, if we expand and equate like powers of g, we obtain du du ** + *-%-"* 2 d 2 u dhi dhi x 33" + 2x y~j—r + V -7-, = n(n - 1) u. dx 2 * dxdy * dy 2 v ' ' ,d 3 u d 3 u d 3 u d 3 u . . , , *S? + 3 * V I*dy + iX!/ d^> + * If = »(•"')(•- •) «> &c. &o. &c. The foregoing method of demonstration admits of "being easily extended to the case of a homogeneous function of three or more variables. Thus, substituting gx for h, gy for k, gz for I, in formula (6) Art. 129, and proceeding as before, we get du du du ■ — + y (- s — dx dy dz x ^z + y~. + z ^z- nu > d 2 u „d 2 u „d 2 u (Pu cPu x ti + V tt + z ~r% + 2x y -j~r + 2ZX -j—r dx 2 * dy 2 dz 2 ' dxdy dzdx d 2 u . . + zyz ~r- — = n[n - i)u. dydz v ' &o. &o. &c. These formulae are due to Euler, and are of importance in the general theory of curves and surfaces, as well as in other applications of analysis. The preceding method of proof is taken from Lagrange's Mecanique Analytique. m 2 ( i6 4 ) CHAPTEE IX. MAXIMA AND MINIMA OF FUNCTIONS OF A SINGLE VARIABLE. 133. Definition of a Maximum or a Minimum. — If any function increase continuously as the variable on which it de- pends increases up to a certain value, and diminish for higher values of the variable, then, in passing from its increasing to its decreasing stage, the function attains what is called a maximum value. In like manner, if the function decrease as the variable increases up to a certain value, and increase for higher values of the variable, the function passes through a minimum stage. Many cases of maxima and minima can be best determined without the aid of the Differential Calculus ; we shall com- mence with a few geometrical and algebraic examples of this class. 134. Geometrical Example. — To find the area of the greatest triangle which can be inscribed in a given ellipse. Sup- pose the ellipse projected orthogonally into a circle ; then any triangle inscribed in the ellipse is projected into a triangle inscribed in the circle, and the areas of the triangles are to one another in the ratio of the area of the ellipse to that of the circle (Salmon's Conies, Art. 368). Hence the triangle in the ellipse is a maximum when that in the circle is a maxi- mum ; but in the latter case the maximum triangle is evidently equilateral, and it is easily seen that its area is to that of the circle as -v/ 2 7 to 471-. Hence the area of the greatest triangle inscribed in the ellipse is 3 « b*A where a, b are the semiaxes. Moreover, the centre of the ellipse is evidently the point of intersection of the bisectors of the sides of the triangle. Algebraic Examples of Maxima and Minima. 165 Examples. 1. Prove that the area of the greatest ellipse inscribed in a given triangle i« T , — (area of the triangle). 1. Find the area of the least ellipse circumscribed to a given triangle. 3. Place a chord of a given length in an ellipse, so that its distance from the centre shall be a maximum. The lines joining its extremities to the centre must be conjugate diameters. 4. Show that the preceding construction is impossible when the length of the given chord is >a\/ 2 or x x and the max. value in question is . -^rrj . W a + vJ) (a + a) (x + b) A. -^ — — — — • x + c t x , , /. . -l (z + a-c) (z + b-e) Let x + e = 2, and the fraction becomes . z In order that this should have a real min. value, (a — c)(b — c) must be posi- tive ; i. e. the value of must not lie between those of a and b, &c. 5. Find the least value of a tan d + b cot d. Ana. i^/ ab, 6. Prove that the expression ; will always lie between two fixed x 2 + bx + c z finite limits if a s + c 2 > ab and b 2 < 4 c 2 ; that there will be two limits between which it cannot lie if a 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 1 + zbxy + c\f a'x 2 + zb'xy + c'y 2 ' Algebraic Examples of Maxima and Minima. 167 Let u denote the proposed fraction, and substitute s for -; then we get tfS 2 + 20Z + c U = a'z* + 2b'z+e' ; (I) or (a-a'u)z 2 + 2(b-b'u)z + c-c'u = o. Solving for z, this gives (a - a'u)z +b- b'u = ± ^/(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 + (ac f + ca f - 2 bb') 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 b' 2 - a'c > o, we shall have (a - a 'u)z + b- b'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 (ft) is a minimum. Accordingly, when the roots of the denominator, a'x 2 + 2b' x + c = 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 (5. If either d = o, or c' = o, the radical becomes V f (a + h) , and / (a) >f (a - h) ; and, for a minimum, f{a) + Oh), f(a - h) -/(«) = - hf\a) + ^/'(« - M). Now, when h is very small, and /"(«) finite, the second term in the right-hand side in each of these equations is very small in comparison with the first, and hence /(a + h) -f(a) and f(a- h) - f{a) cannot have the same sign -unless /(«) = o. Hence, the values of x which render f(x) a maximum or a minimum are in general roots of the derived equation f{x) = o. This result can also be arrived at from geometrical considerations ; for, let y = f(x) be the equation of a curve, then, at a point from which the ordinate y attains a maximum or a minimum value, the tangent to the curve is evidently parallel to the axis of x ; and, consequently fix) = o, by Art. 10. Moreover, if x be eliminated between the equations f(x) = u and fix) = o, the roots of the resulting equation in u are, in general, the maximum and minimum values of fix). This is the extension of the principle arrived at in Art. 134. Again, since /'(a) = o, we have f(a + h)-f(a) = —f"(a + dh), 1 f(a-h)-f{a)=-—f"{a-Q l h) [. (5) 1 . 2 * In the investigation of maxima and minima given above, Lagrange's form of Taylor's Theorem has been employed. For students who are unacquainted with this form of the Theorem, it may be observed that the conditions for a maximum or minimum can be readily established from the form of Taylor's Series given in Art. 54, viz., li 2 h 3 f(a + h) -/(«) = hf{a) + —f"(a) + — — /'"(«) + &c. ; 1.2 1.2.3 for when h is very small and the coefficients/^),/" (a), &c. finite, it is evident that the sign of the series at the right-hand side depends on that of its first term, and hence all the results arrived at in the above and the subsequent Articles can be readily established. Condition for a Maximum or a Minimum. 171 But the expressions at the left-hand side in these equations- are both positive for small values of h when f{a) is positive ; and negative, when f\a) is negative ; therefore f(a) is a maximum or a minimum according as f'{a) is negative or positive. If, however, /"(a) vanish along with f(a), we have, by Art. 75, f{a + h) -/(a) = _|- /"» + _£_/*(« + Qh) t /(a - h) -/(a) = -f^-ria) + h \ /*(* - QJi). Hence it follows that in this case, f(a) is neither a maximum nor a minimum unless f" f {a) also vanish; but if f"{a) = o, then /(a) is a maximum when / iv («) is negative, and a minimum when/ iv (a) is positive. In general, let /(")(«) be the first derived function that does not vanish ; then, if n be odd, f(a) is neither a maximum nor a minimum ; if n be even, f(a) is a maximum or a mini- mum according as /( n ) (a) is negative or positive. The student who is acquainted with the elements of the theory of plane curves will find no difficulty in giving the geometrical interpretation of the results arrived at in this and the subsequent Articles. Examples. 1. u = a sin x + b cos x. Here the maximum and minimum values are given by the equation du a — = a cos x — sin x = o, or tan x = T . ax b Hence, the max. value of u is *S a- + b n -, and the min. is - ^/a? + b' z . This is. also evident independently, since w may be written in the form V a 2 + b- sin [x + a), where tan a = — . a 2. u = x — sin x. - ... du 40. -. . , d"-u li a = 40, we get when x = tr, — = o. dx- On proceeding to the next differentiation we have dhi dx 3 —= = a (sin x + 2 sin 2x), = o when # = x. ^ 4 w . ..... Again, -— = a (cos x + 4 cos 2#) = 3a. Consequently the solution is a minimum in this case. Again, the solution (2) is impossible unless a te less than 4b. In this case, d 2 u i. e. when a < 40, we easily find — positive, and accordingly this gives a min. « 2 value of u, viz. — — — b. 80 4. Find the vnlue of x for which sec x — x is a maximum or a minimum. Ans. sinx = . Application to Rational Algebraic Expressions. 173 139. Application to Rational Algebraic Expres- sions. — Suppose f(x) a rational function containing no fractional power of x, and let the real roots of f(x) = o, arranged in order of magnitude, be a, j3, 7, &c. ; no two of which are supposed equal. Then f[x) - (x - o) [x - |3) (x - y) . . . and /"(a)=(a-/3)(a- 7 ) . . . But by hypothesis, a - j3, a - 7, &c. are all positive ; hence /"(a) is also positive, and consequently a corresponds to a minimum value of /(#). Again, /"(i3) = (j3-a)(j3-7) here ]3 - a is negative, and the remaining factors are positive ; hence /"(/3) is negative, and/(/3) a maximum. Similarly, f(y) is a minimum, &c. 140. Maxima and Minima Tallies 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 f\x) = o has two roots each equal to a, it must be of the form f{x) = {x-a)^{x). In this case /"(«) = o,/'"(a) = 21// (a), and accordingly, from Art. 138, a corresponds to neither a maximum nor a minimum value of the function /(#) . In general, if f'{x) have n roots equal to a, then f{x) = {x-aY^{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 $(a) is negative or positive. 174 Maxima and Minima of Functions of a Single Variable. 142. Case where f'(x) = 00. The investigation in Art. 138 shows that a function in general changes its sign in passing through zero. In like manner it can he shown that a function changes its sign, in general, in passing through an infinite value ; i.e. if (a) = 00, ^(a - h) and / 2 A'x 2 + 2B , xy + C'y 2> where A, B, C, A', JT, C, 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 = \{ac - b 2 ), AC + CA' ~ 2BB = \(ac' + ca' - ibV), A'C -B' 2 = \(a'c' - b' 2 ). Q.E.D. Application to Surfaces. 179 It can be seen without difficulty that A = (lm' - mf) 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 + t COS 2 /3, 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 ) cos 2 /3 = 1, and p, q, r, s, t are independent of a and j3. Denoting the proposed expression by u, and substituting - cos a , z for ri, we get cos/3 ° rz 2 + 2sz + t u = (1 + p 2 )z 2 + ipqz + (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 ) -w{(i +q 2 )r - 2pqs + (1 +p 2 )t} +rt -s 2 = o; (6) COS a while the corresponding values of z or -^ are given by 2 2 {(i + p 2 )s - pqr) + z{(i + p 2 )t - (1 + q 2 )r) + {PQt- (1 +q 2 )s} =0* (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. N 2 1 80 Maxima and Minima of Functions of a Single Variable. 146. To find the Maximum and Minimum Radius Vector of the Ellipse ax 7, + zbxy + cy 2 = 1. (1). Suppose the axes rectangular; then r 2 = x 1 + y 2 is to be a maximum or a minimum. Let - = z, and we get y Z 2 + I r = az 2 + zbz + c Hence the quadratic which determines the maximum and minimum distances from the centre is r 4 (ac - b 2 ) - r 2 {a + c) + 1 = o. The other quadratio, viz. bx 2 - (a - e) 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, then r* = x 2 + y 2 + 2xy cos a> Z 2 + 2Z COS w + I az 2 + 2bz + c ' and the quadratic becomes in this case r 4 (ac - b 2 ) - r 2 (a + c - 2b cos w) + 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 si the conic can also be easily written down in this case. Maximum and Minimum Section of a Right Cone. 1 8 1 147. To investigate the Maximum and Minimum Values of ax 3 + ^btfy + 3 CX !/ Z + dy 3 ax 2 + ib'tfy + 3c' xy 2 + d'y 3 ' Substituting z for -, and denoting the fraction by u, we have w = az 3 + 3J2 2 + s cz + d a'z 3 + 3# 'z 2 + $c'z + d'' Proceeding, as in Art. 1 44, we find that the values of u and z are given by aid of the two quadratics az 2 + zbz + c = (a'z 2 + 2b'z + (T)u t bz 2 + 2cz + d= (b'z 2 + ic'z + d')u. Eliminating u between these equations, we get the following biquadratic in z : — z\ab' - ba f ) + 2z 3 (ac r - cd) + z 2 {ad' - a'd + ^(bc - cb')) + 2z(bd' - db') + {cd' - c'd) = o. (8) Eliminating z between the same equations, we obtain a biquadratic in w, 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 cut tbe 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&nd. PR parallel to BC. Then BP = 2a, b 2 = LO . OM (Euclid, Book in., Pr. 35) ; but LO =™ } 0M=—; .-. b 2 =-.BC. PR. 2 2 4 Hence BP 2 . PR is to be a maximum or a minimum. Fig. 7. 1 82 Maxima and Minima of Functions of a Single Variable. Let L BAD - a, PBC = 0, BO = c. sin BCP c cos a Then BP = 5C PB = BP Bin BPC oos(fl-o)' sin P.BJ2 _ c cos (0 + a)_ slnP^ZP cos (0 - a) J cos (0 + a) . •*• w = — ttt; ( is a maximum or a minimum. COS 3 (0 - a) -n- du sin 20- 2 sin 2a . n Hence -^ = —- n r — = o : /. sin 20 = 2 sin 2a. dO cos 4 (0 - a) The solution becomes impossible when 2 sin 2 a > 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 X be the least value of derived from the equation sin 2 = 2 sin 2 a ; then the value 0i evidently gives a second solution. Again, by differentiation, we get d 2 u 2 cos 20 . . . 3715 = — TTTi v (when sin 20 = 2 sm 2a). dd 2 cos 4 (0 - a) v ' 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?u gives -jjp = o. On proceeding to the next differentiation, we find, when = 45 , d?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. 1 83 149. 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 he added or subtracted, i.e. if f(x) be a maximum, so also is/ (x) ± 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 z 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 f'{u) = 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 — = 2 (a 2 + b 2 - zr 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 - zr 2 ) = o ; • See Cambridge Mathematical Journal, vol. iii. p. 237. 1 84 Maxima and Minima of Functions of a Single Variable. la 2 + b 2 or r = o, and r = I . The solution r = o is inadmissible, since r must lie between the limits a and b : the other solution corresponds to the equicon jugate 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 : I 2b a% ~ h% X X X 3 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. (2) the solutions of the equation — , i.e. of 1 4& 3 (a? - b 2 ) — h • — ■ — — = o, X 2 X 3 X* or x 2 + 4.bx - 3 (a 2 - b z ) = o ; whence we get x = v^a? + b 2 - 2b, neglecting the negative root, which is inadmissible. Again, if b > a, \/yi 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. 185 If a > b, it remains to see whether a/ 3a 2 + b 2 - 2b lies between the limits a + b and a - b. It is easily seen that */2>a 2 + b 2 - lb is > a - b: the remaining condition requires a + b > \Zsa 2 + b 2 - 2b, or a + sb > */ yi 2 + b 2 , or a 2 + tab + gb 2 > 3a 2 + b\ i.e. 4 J 2 + 2>ab > a 2 , or 4b 2 + $ab + ?-r > ^; .-. 2b + — > —; 16 16 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 -j- = o : accordingly the 1 86 Maxima and Minima of Functions of a Single Variable. maximum and minimum values are got by combining* the equations — = o, and u = o. 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 f(x, y) = o. Let the proposed function,

df d(p df dx dy dy dx * furnish the solutions required. To determine whether the solution so determined is a maximum or a minimum, it is necessary to investigate the sign of — . We add an 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,

that between c and d. Then ab sin ^ + cd sin t// is a maximum ; also a 2 + b 2 - 2ab cos $ = & + d? - zed cos \fj 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 + cd cos \L-r- = o Y Y d(j> for a maximum or a minimum ; also, ao sin $ = ca sm i//— ; #0 •\ tan <£ + tan \p = o, OT^> + xp=i 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 —r—. — ^7 instead of — -, and we get cd sin d du ab sin (

sin xp „ d 2 u ab cos (6 + \L) ( d\L\ , . . Hence -—^ = —-, — — 1 + V 1 + a term which dtf sin Tp \ d(p) vanishes when + \L = 1 8o° ; and the value of -=—. becomes r T dtf in this case ab f ab 1 + — sin \ cd/ which being negative, the solution is a maximum. 1 88 Examples. Examples. r. Prove that a sec b + b cosec 6 is a minimum when tan = lj -. 2. Find when 42 s - 15** + i 2x - 1 is a maximum or a minimum. Am. x = J, a max. ; a; = 2, a min. 3. If a and * be such that /(«) = f{b), show that /(*) has, in general, ■maximum or a minimum value for some value of x between a and b. 4. Find the value of x which makes 6in x . cos * cos 2 (6o° — x) maximum - Am. x = 30 . 5- K 77^t — ^-rl fee a maximum, show immediately that "^is a minimum. is a maximum. V 5 — 4 cos a; -4ms. cos x = 5 ~ V *3 ■n- j -l x + 3 X • • 12 7. Jb ind when is a maximum. „ x — - ■■■. V4 + 5* 2 5 8. Apply the method of Ex. 5 to the expression — . 9. "What are the values of a; which make the expression ix 3 — 21X 1 + $6x — 20 a maximum or a minimum? and (2) what are the maximum and minimum values of the expression? Ans. x = I, a max. ; x = 6, a min. (a + x)(b + x) .— (a - x){b - x) — Examples. 1 89 14. Show that b + e [x - «)?, when x = a, is a minimum or a maximum according as = k sin if', and \p + ij/ = a, where a and k are constants, prove that cos if' cos (j> is a maximum when tan 2 <£ = tan ty tan \p'. 1 90 Examples. 27. Find the area of the ellipse ax 2 + 2hxy + by 1 = e in terms of the coefficients in its equation, by the method of Art. 146. (1) for rectangular axes. Ans. (2) for oblique. s/db - h? ire sin (x +h,y + k) - (x , y ) = (Ah 2 + zBhk + Ck?) + &c. (2) 1 92 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 Ah 2 + zBhk + Ck 2 ; accordingly the sign of 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. Thcorie 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 he (oi, 61), (a 2f h), (a 3) b s ) . . . (o„, b n ), respectively ; (x t y) those of the point required ; then we have u = (x - atf + (y - J1) 2 + (a? - «a) 2 + (y - b,)* + . . . + (a? - a„) a + (y - b n )* a minimum ; du t . ,_ «i + a, + . . . + a n bi + bt + . . . + 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 prohlem 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 a - ^ u - n - d* u _ ri _ d % u _ dx z ' dxdy * dy 2. In this case AC - B 2 is positive, and A also positive; and accordingly the result is a minimum. 158. To find the Maximum or Minimum Value of the expression ax 2 + by 2 + 2hxy + 2gx + ify + c. Denoting the expression by u, we have 1 du . -- = ax + hy + g = o, 1 du J . - — = hx + by +/ = o. 2dy y J 194 Maxima and Minima for Two or more Variables. Multiplying the first equation by x, the second by y, and subtracting their sum from the given expression, we get u = gx+fy + c; whence, eliminating x and y between the three equations, we obtain a h g u(ab-h 2 ) = h b f . (3) 9 f This result may also be written in the form e?A where A denotes the discriminant of the proposed expression. . . d 2 u d 2 u 7 d 2 u Again, _-2«, - = 2 J, j- y = 2h. Hence, if ab - h* be positive, the foregoing value of u is a maximum or a minimum according as the sign of a is negative or positive. If h 2 > 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- + by 2 + ilixy + 2gx + 2fy + c a'x 2 + b'y 2 + zlixy-v 2g'x+2f'y+c'' nd denominator be r< j the fraction by u, we (pi - u<}> 2 . (a) Let the numerator and denominator be represented by i and ^2 ; then, denoting the fraction by u, we get Examples for Two Variables. 195 Differentiate with respect to x and y separately, then d$ x du dx dx d2 d\ du d

2 d 2 u ltf =U ~oW + i dx 1 ' dxdy dxdy dxdy' d 2 (pi d 2 ^ dru ~dtf =U ~df ***&> d 2 ^ d 2 fa , (Pfa , g Hence ^|SS-0'}=4k-^-^)-(a-^). Accordingly, the sign of AC-B 2 is the same as that of the quadratic expression (ab - h 2 ) - (aV + 00*- ihh') u + {a'V - 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 he written in the form — — - 2 . Again, ih, 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 (ic x - a)(ih - fi)(a'b' - h' 2 ) a positive quantity. Accordingly, if a'V - H 2 be positive, u x must not lie be- tween the values a and j3. 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 (3 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 - w 1 , h = o are interposed between the roots of the h, b - u~ l 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 = o, (9) and {AB - E*)(AC - (?) > (AF - GH)\ or A{ ABC + 2FGH - AF 2 - BG 2 - CE 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 , y n , z Q 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 Variables. — 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 n - 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 + ikxy + zgzx + ifyz + 2px + 2qy + 2r% + d. Let u denote the function in question, then for its maxi- mum or minimum value we have — = 2 (ax + hy + gz + p) = o, du dy du dz = 2(hx + by +fz + q) = o, = z{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 d 2 u d 2 u in, since - = 2a, - a h g h b f g f c = 2b, &c, Maxima or Minima for two or more Variables. 201 the result is neither a maximum nor a minimum unless a h g \ a h \ is positive, and h b f 9 f 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 such tbat the sum of the squares of its distances from n given points shall be a Minimum. — Let {a, b, c), {a', b', c'), &c, be the co-ordi- nates of the given points referred to rectangular axes ; x, y, a, the co-ordinates of the required point ; then (x - a) 2 + (y - by + (z - c) 2 is equal to the square of the distance between the points {a, b, c), and (x, y, z). Hence u = (x - a) 2 + (y - b) 2 + (z-c)* + (tB-tTf+ (3/ - bj + {z -c)'* + &o. - 2 (a? - a) 2 + 2(y - b) 2 + 2(s - c)\ where the summation is extended to each of the n points. For the maximum or minimum value, we havo — = 22(2? - a) = 2nx - 2^a = o, cue — = 2"2(y - b) = iny - 225 = o, du 22 2 - c) = 2nz - 22c = o ; Wo = 2a 26 z = 2c n ' " n n i.e. x , y , 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. . . d 2 u d 2 u (Pu (Pu Again _ = 2M) _ = 2 „, _ = 2 „, — = 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 + by + e e + \/a 2 + J ! + e 3 I. -5— — f— - . ^n*. -= . x 2 + y* + 1 a as + £y + e V x 2 + y 2 + 1 3. x* + y* - x 2 + xy - y 2 . (a), x = o, y = 0, a maximum. ()3). a; = y = ± -, a minimum. 2 # » \^3 • • (y). x = - y = ± - — , a minimum. 4. ax 2 + Ixy + dz 2 + Ixz + myz. x = y = z = o, neither a maximum nor a minimum. a a 5. If « = ai?y 2 — x i y 2 - spy 3 , prove that x = -, y = - makes w a maximum. 6. Prove that the value of the minimum found in Art. 165 is the -th part of n the sura of the squares of the mutual distances between the n points, taken two and two. 7. Find the maximum value of (ax+by + cz)e . Ans. J- {- + - + -^ , 8. Find the values of x and y for which the expression (aix + hy + a) 7 + [aix + b%y + e 2 ) 2 + . . . + {c*x + b n y + «*)' becomes a minimum. ( 20 4 ) CHAPTER XI. METHOD OF UNDETERMINED MULTIPLIERS APPLIED TO THE INVESTIGATION OF MAXIMA AND MINIMA IN IMPLIJIT FUNCTIONS. 1 66. Method of Undetermined Multipliers. — In many cases of maxima and minima the variables which enter into the function are not independent of one another, but are con- nected by certain equations of condition. The most convenient process to adopt in such cases is what is styled the method of undetermined* multipliers. We shall illustrate this process by considering the case of a func- tion of four variables which are connected by two equations of condition. Thus, let u = 0(#i, x 2 , x 3 , , d(b . -r- dx 1 +—- dx 2 + — dx 3 + -±- dx A = o. dxi dx 2 dx 3 dx t Moreover, the differentials are also connected by the rela- tions dFx , dFi . dF x . dFi , — — dxi + -— dx 2 + -r— dx 3 + — — dxi = o, dx y dx 2 dx 3 dXi dF 2 , dF 2 , dF 2 . dF 2 J —r- dxi + -r- dx 2 + — — dx 3 + -7— dx t = o. axy dx 2 dx 3 dXi Multiplying the first of the two latter equations by the arbitrary * This method is also due to Lagrange. See Mee. Anal., tome 1., p. 74. Method of Undetermined Multipliers. 205 quantity Xi, the other by X 2 , and adding their sum to the pre- ceding equation, we get . dF t , dF 2 \ , + [j + A * 3- + * a 3- ^3+ -r 1 + Xi — + X 2 — )dx i = o. yfoj a# 3 ax 3 J \dXi. dxt, dx^J As Xi, X 2 are completely at our disposal, we may suppose them determined so as to make the coefficients of dx x and dx 2 vanish. Then we shall have x dFx > dF 2 \ ■f- + Xi — + X 2 — )dx 3 + [-f- + Xi — + X 2 -— )dx i = o. aa? 3 a# 3 a# 3 / \o# 4 a# 4 dx^j Again, since we may regard x 3 , x^ as independent variables, and Xx, x 2 as dependent on them in consequence of the equa- tions (1), it follows that the coefficients of dx 3 and dx^ in the last equation must be separately zero, for a maximum or a minimum ; consequently, we must have dd> . dF x , dF 2 -f- + \x -j- + X 2 — = o, dx 3 dx 3 dx% dd> . dFx ^ dF% ■j- + A! — + X 2 -j- = o. dxi dx t dXi These, along with equations (1) and dd> . dFx „ dF 2 j- + Ai — + A 2 — = o, a#i dxx dxx dd> . c?jPi . dF 2 -/- + Ax — + X 2 — = o, cto 2 «a- 2 a^ 2 are theoretically sufficient to determine the six unknown quantities, x lf x 2y x 3) x i} Xi, X 2 ; and thus to furnish a solution of the problem in general. This method is especially applicable when the functions F 1} F 2 , &c, are homogeneous ; for if we multiply the preceding 206 Method of Undetermined Multipliers. differential equations by x t , a^, x 3 , x if 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 u x 2 , x z , . . . x„, the variables being connected by m equations of condition. F, = o, F 2 = o, F 3 = o, . . . 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, X 2 , . . . \ m respectively ; by the same method of reasoning as that given above, we shall have the n following equations, d(b . dFy . dF m -j- + \ x — -+...+ \ m -j- = o, dxi axi dxi dd> . dFi , dF m -J- + A, T - + . . . + \ m - r - = o, dx 2 dx 2 dx 2 d N L, M, N, o and the coefficient of w 2 is U + M 2 + N 2 . Accordingly, the product of the roots of the quadratic (2) is equal to the frac- tion whose numerator is the latter determinant, and denomi- nator L 2 + M 2 + N 2 . From this can be immediately deduced an expression for the measure of curvature* at any point on a surface. • SaJmon'8 Geometry of Three Dimensions, Art. 295. 2io Examples. Examples. i. Find the minimum value of where x\, xj, . . . x n are subject to the condition a\x\ + a&% + . . . + etnXn = k. Am. 2. Find the maximum value of xPytz*, where the variables are subject to the condition ax + hy + ez = p + q -t- r. *» a* + a* Ana. mm- Q y* z 2 7. Given -= + — + -= = 1. and Ix + my + nz = o, find when x 2 + y 2 + z 2 is a a 2 b 2 and the angle 9, find when the ratio sin (

+ &<*-'>-* xX yT _ a; 2 y 2 _ 2. JFind the equation of the tangent at any point on the curve ggm ym Xx m ' 1 Y-U m ' 1 — + \- = i. Am. + -f — = I. Qin font Qtn frm 3. If two curves, -whose equations are denoted by u = o, w' = o, intersect in a point (x, y), and if « be their angle of intersection, prove that du du 1 du' du dx dy dx dy tan o = du du 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' 1 x 1 y % snould intersect at right angles. Ans. a 2 - b 3 = a' 2 - b'\ Equation of Normal. 215 1 70. Equation of \ormal. — 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 (7-,)| + X-* = o, du . N rf« /r v -(F-2,)=^(X-*). (3) The points at which normals are parallel to the line y = mx + n are given by aid of the equation of the ourve u = o along with the equation du dv dy dx Examples. 1 . Find the equation of the normal at any point (x, y) on the ellipse x i y* a 2 o* A cC-X FY Am. = a 1 - J 1 . x y 2. Find the equation of the normal at any point on the curve y» = ax*. Am. nYy + tnXx = ny 2 + mx*. 171. Subtangent and Subnormal. — In the accom- panying figure, let PT repre- y sent the tangent at the point P, PN the normal; OM, PM the co-ordinates at P ; then the lines TM and MN are called the subtangent and subnormal o" corresponding to the point P. Fig. 9. To find the expressions for their lengths, let $ = L PTM y then PM dy MN dy PM =i ™* = -dx> TM=%-, dy dx MN = y% dx 2 1 6 Tangents and Normals to Curves. The lengths of PT and PAT are sometimes called the lengths of the tangent and the normal at P : it is easily seen that dx Examples. i. To find the length of the subnormal in the ellipse x 2 y 2 „ dy h % Here y— = - — x; dx a 1 the negative sign signifies that MN is measured from M in the negative direction along the axis of x, i.e. the point 2V lies between M and the centre ; as is also evident from the shape of the curve. 2. Prove that the subtangent in the logarithmic curve, y = a', is of constant length. 3. Prove that the subnormal in the parabola, y 1 = mix, is equal to m. 4. Find the length of the part of the normal to the catenary / - x \ intercepted by the axis of *. Ana. — . a 5. Find at what point the subtangent to the curve whose equation is xy 1 = «*(« — x) . a is a maximum. Am. x = -, y = a. 2 172. 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 9 y) = c, Length of Perpendicular on Tangent. 217 then the equation of the tangent may be written X cos w + Y sin u> = p, where to 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 X, du du du du dx du dx ay x cos a; sm (o p Hence A' = (g' + g)', du du x— +y — and p = — - — • (4) jfduV Tdu <\dx) + \dy) Cor. If F(x, 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 ay and the expression for the length of the perpendicular becomes in this case j/duV /duV >j{dx) + \dy) 173- In the curve x m y m — 4 — = 1 to prove that pm-l = {a cos w) m-1 + (b sin CO) (5) 2 1 8 Tangents and Normals to Curves. By Ex. 2, Art. 169, the equation of the tangent is L ^ — f • comparing this with the form X cos to + Y sin and sin &> in (5), it becomes (X 2 + Y 2 ) m ~ l = {aX) m ~ l + (b Y) m ~\ since p 1 = X z + Y z . 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(p a -j- + [3-f- + w«_i + 2W„_2 + . . . + nu = o. (7) dx dy 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 - 1 intersect in general in n (n - 1) 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 dd> n d a / + P-7- + Ui+2U =O, 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 k 3 + u 2 + «i + u = o, equation (7) becomes dd> n d a 2 fl- at which the normals pass through a given point (a, /3), are determined by the intersection of the ellipse with the hyperbola xy {a 2 - b 2 ) = a 2 ay - b 2 (3x. 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, m. 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

becomes PTX, or

, and Ji + ( — ) Ax dx y ' \ \Ax) 222 Tangents and Normals to Curves. "becomes J i + ( — j or sec

pQ = "^^t the same time; , dr rdd . rdO . . or cos \L = — , sin xl = —- , tan xl = — — . (11) ds ds dr * These results can be easily established from Art. 37. Polar Subtangent and Subnormal. 223 Also, rdOY fdr\ % (") Hence, also, we can determine an expression for the differential of an arc iu polar co-ordinates ; for, since PQr _ P2P QM* " l + QM~ % ' we get, on proceeding to the limit, ds_ dr J = 1 + r 2 dd 2 dr or ds = r 2 dB 2 7 1 + — -r^- dr. dr (13) These results are of importance in the general theory of curves. 181. Application to the Logarithmic Spiral. — The curve whose equation is r = a e is called the logarithmio spiral. In this curve we have , rdO 1 tan \L *■ -7- = ; . dr log a Accordingly, the angle between the radius vector and the tangent is constant. On account of this property the curve is also called the equiangular spiral. 182. Polar Subtangent and Subnormal. — Through the origin let ST be drawn perpendi- g cular to OP, meeting the tangent in T, and the normal in S. The lines OT and OS are called the polar subtangent and subnormal, for the point P. To find their values, we have OT = OP tan OPT = r tan xL = T ~. dr dr dff dO du OS = OP tan OPS = r cot $ Also, if « = -, OT=- 224 Tangents and Normals to Curves. Again, if ON be drawn perpendicular to PT, we have PN = OP gob xp = r^. (15) 183. Expression for Perpendicular on Tangent. — As before, let p = ON, then . , r*dO p = r&m\p = — -; , 1 ds 2 _ dr^ + r^dB" 1 dr 1 1 hence -= ^ = ^ fla = -^ + -, £-* + ®" (i6) The equations in polar co-ordinates of the tangent and the normal at any point on a curve can be found without difficulty : they have, however, been omitted here, as they are of little or no practical advantage. Examples. 1. To find the length of the perpendicular from a focus on the tangent to an ellipse. The focal equation of the curve is 0(1 - e 7 ) 1 - e cosfl I - ecosfl 0(1 -e 2 ) du e sin hence -— = — r, ; dd a(i-e 2 )' 1 _ r + e 2 - ie cos0 _ 1 /2a _ \ *'V = «*(i-e 2 ^ «*(i -e 2 ) \~r ~ )' 2. Prove that the polar suhnormal is constant, in the curve r — a$ ; and the Bolar subtangent, in the r.nrvn *•« = a. Inverse Curves. 225 184. Inverse Curves. — If on any radius vector OP, drawn from a fixed origin 0, a point i y be taken such that the rectangle OP . OP' is constant, the point P' is called the inverse of the point P ; and if P describe any curve, P' describes another curve called the inverse of the former. The polar equation of the inverse is obtained immediately from that of the original curve by k z substituting - instead of r in its equation ; where h* is equal to the constant OP . OP / . Again, let P, Q be two points, and P, Q> the inverse points ; then since OP . OF = OQ . OQ', the four points P, Q, Q', P / , lie on a circle, and hence the triangles OQP and OP'Q' are equiangular ; PQ _ OP = OP . OQ = OP. OQ •'' P y Q' " OQ' ~ OQ . OQ' ¥ ' (I7j 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 d8 = T2 ds'. (18) 185. Direction of the Tangent to an Inverse Carve. — 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 ttco curves intersect at any angle, their inverse curves intersect at the same angle. Q 226 Tangents and Normals to Curves. 1 86. Equation to tbe Inverse of a Given Curve. — Suppose the curve referred to rectangular axes drawn through the pole 0, and that x and y are the co-ordinates of a point P on the curve, X and Y those of the inverse point i y ; then £ _ 9JL - 0P • 0jy - ** ••iii_ jl! X ~ OF ~ OF 2 ~ X*+Y 2 ' y F" X 2 + F 2 ; hence the equation of the inverse is got by substituting k 2 x . k 2 y and' 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 + Un-\ + Wn-2 + • • • + «2 + «l + Wo = O. When — and , ; are substituted for a? and y, u n x 2 + y 2 x l + y* k 2n u becomes evidently -r-= -tt-v («» + y 2 ) n Accordingly, the equation of the inverse curve is k 2n u n + k^u^ (x 2 + y 2 ) + F n - 4 M„_ 2 (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 y + u = o ; hence that of its inverse with respect to the origin is k 2 Ui + 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 1 + u ( > = o, with respect to the origin, is (k l + khii + 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 Carves. 227 Again, the general equation of a conic is of the form «2 + t«i - u = o ; hence that of its inverse with respect to the origin is k*u 2 + k 2 Ui (a? + y 3 ) + u (a? + y 1 ) 3 = 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 khi z + iii. (x 1 + y 3 ) = 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 T the intersec- tion of these tangents ; join NN'; then since the angles ONT and ON'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 y 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 r-t P" Hence, if the equation of a curve be given in the form r =f(p), that of its first pedal is of the f orm — =f(p), in which p and p' 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 PiVwith 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 B, 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 polars 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 76. 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 p = — . r Substituting this value for p in equation (5), we get m tn m fk 2 \m~i m-l . m-i ( — j = (a cos id) •*- (0 sm w) , 2m or A" 1 " 1 = (ax) m - 1 + (by) m -\ which is the equation of the reciprocal, polar of the curve re- presented by the equation — + f- = 1. In the particular case of the ellipse, & y 2 _ the reciprocal polar has for its equation W = a?x 2 4 t/y\ 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 y 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 U. 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 of r m = a"' cos md. "We shall commence by finding the N angle between the radius vector and the perpendicular on the tangent. In the accompanying figure we have tan PON = cot OPN - - ~. rdO Fig. 15- But m log r = m log a + log (cos mO) ; hence — - n = - tan m9, rdO and accordingly, LPON^mO. (20) Again, p = ON= r cosmfl ,m+i a" > 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 u - (m +1)6. m fr\ m fp\™** Also, from (21), (-1 =(-) . Hence, the equation of the pedal is p m+l = a m+l cos . (22) r \m + 1 / On the Curve r 1 * = a™ cos mB. 231 Consequently, the equation of the pedal is got by substi- tutingr 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 „2m+i _ „2m+i Ar mB m a"" T1 cos ; 2tn + 1 and that of the n th pedal m m a r mn+i _ a mn*k 0QS § / 2 ,\ mn + I 7 Again, from Art. 184, it is plain that the inverse to the curve r™ = a m cos mB, with respect to a circle of radius a, is the curve r 1 " cos mB = 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 r"^ 1 cos — — = tP*. (24) m + 1 It may be observed that this equation is got by substitut- ing 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. I. The equation of a parabola referred to its focua as pole is r (1 + cosO) sa ia, to find the relation between r and p. Q Here H cos - = a\, and consequently p* = or, a well-known elementary property of the curve. 232 Tangents and Normals to Curves. 2. The equation r* cos 20 = a 8 represents an equilateral hyperbola ; prove that pr = a 2 . • 3. The equation r 2 = a 2 cos i9 represents a Lemniscate of Bernoulli ; find the equation connecting^? and r in tnis case. Ant. r 3 = a z p. 4. Find the equation connecting the radius vector and the perpendicular on the tangent in the Cardioid whose equation is r = a(i 4- cos 0). Ant. r 3 = 2ap z . 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 1 = ar, with respect to its focus, is the right line p = a. 6. Show that the pedal of the equilateral hyperbola pr = a 2 is a Lemniscate. 7. Find the pedal of the circle r 2 = zap. Ana. A Cardioid, r 3 = zap 1 . 191. Expression for PN. — To find the value of the intercept between the point of n' contact P and the foot N of the perpendicular from the origin on the tangent at P. Let p = ON,«>=L NOA, PN= t; then l NTN'=lNON = Acu,also SN' = TS sin STN; SN ■'■ TS= . ^,7/ ? hut in the sin NON SN dp limit, when PQ is infinitely small, -: — . T/ , - T . becomes ~- t J am NON du> and TS becomes PN or t : ••■'-£ w> Also OP' = ON 2 + PN 2 ; 192. To prove tbai ds dt , s dio d(o Vectorial Co-ordinates. 233 On reference to the last figure we have ds .PT+TQ dt .. ., ,QN'-PN — = limit of , — = limit of ; da) Au du) Atu hut PT+ TQ- QN' + PW= TN- TN'\ , ds dt .. .TN-TN* .. ., ,£JV ___ hence 3 3- = limit of = limit of — = ON=p\ dui du) Aw Ato is dt d(o do> This result, which is due to Legendre, is of importance in the Integral Calculus, in connexion with the rectification of curves. dp If — be substituted for t, the preceding formula becomes ds d 2 p , .. ato aw This shape of the result is of use in connexion with curva- ture, as will be seen in a subsequent chapter. 193. Direction of Xormal 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 u r 2 denote the distances from two fixed points, the equation n + r 2 = const, represents an ellipse, and /*i - r 2 = const., a hyperbola. Again, the equation r v + mr 2 = const. represents a curve called a Cartesian* oval. Also, the equation r^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 oi the principal properties of Cartesian ovals will be found in Chapter XX. 234 Tangents and Normals to Curves For, let F(ri, r 2 ) = const. be the equation of the curve, where F 1 P = r 1 , FJP = r^ then we have dFdn dFdr, _ dri ds dr 2 ds Now, if PTbe the tangent at P, then, by Art. 1 80, we have — • = cos xpi, ~ = cos \p 2 , where \p t = l TPF lf xp 2 = L TPF 2 . tts cts TT dF . dF . Hence — cos \p x + — cos \f, 2 = o. (29) Again, from any point R on the normal draw RL and RM respectively parallel to F 2 P and F X P, and we have PL : LR = sin RPM : sin RPL = cos i£ 2 : - cos 1//1 _dF dF dr z ' dr 2 Accordingly, if we measure on PF r and PF 2 lengths d~P dT? PL and PM. which are in the proportion of -7— to -r-, then di\ dr 2 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(r u r 2 , r 3 , . . . r») = const., • The theorem given ahove 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 Pa, Pb, Pc, Pd, . . . issuing from a point P, we draw through a a right line aB, equal and parallel to Pd, and in the same direction ; through B, a right line BO, equal and parallel to Pc, and so on, whatever be the number of lines : then the line PR, which closes the polygon, is the resultant in question. Normals in Vectorial Co-ordinates. 235 where r„ r 2 , . . . r„ 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 L . . *? *? ^"••^' reSpeCtlVely; then the direction of the normal is the resultant of the lines thus determined. For, as before, we have dFdr, dFdn dF dr n _ dr-i ds dr 2 ds ' ' dr n ds tt dF . dF . dF . . . Hence — cos fa + — cos fa + . . . — cos fa = o. (30) JNow, — cos ^ — cos fa, ... — cos ?//„, dF , dF , dF -cos^, _ CO s^ 2 ,... - 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. r. A Cartesian oval is the locus of a point, P, such that its distances, P31, 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 r/ = ab, where r and / are the distances of any point P on the curve, from two fixed points, A and B. If be the middle point of AB, and PiV the normal at P, prove that L A1'0= L BPN. 3. In the curve represented by the equation ri 3 + r 2 3 = a 3 , prove that the normal divides the distance between the foci in the ratio of r-i to r\. 236 Tangents and Normals to Curves. 1 94. In like manner, if the equation of a curve be given in terms of the angles } , 2 , . . . 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 lt 2 , . . . 0„) = const. Then, by differentiation, we have dFdjh dFdfh dFdBn_ dOi ds d0 2 ds ' dd n ds Hence, as before, from Art. 180, we get 1 dF . , 1 dF . , 1 dF . , 7M* m + 1 + 7M Bm ^''- + 7 n aT n Bm +» = °- (3I) Accordingly, if we measure on the lines drawn to the fixed points segments proportional to i_dF T_dF i_dF r, dOS r 2 d6 2 ' .' ' ' r n dd n ' 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 Line, and 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 oix. 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* = c, the origin is a centre. Also, if the equation of a cubicf be reducible to the form tt3 + Wj = 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 Y + (5 for y, in its equation, and equate to zero the coefficients of X 2 , XYand Y 2 , as well as the abso- lute term, in the new equation : as we have but two arbitrary constants (a and /3) 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 Jlistoriqite, p. 233, note. t This name has heen given to curves of the third degree by Dr. Salmon, in his Higher Plane Curves, and has been generally adopted by subsequent writers on the subject. 2 38 Examples. Examples. i. Find the lengths of the subtangect ana subnormal at any point of the curve yn = a"- l z. Am. nx, — . nx 2. Find the subtangent to the curve nx x*'y n = a m * n . Am. . m 3. Find the equation of the tangent to the curve x s = a 3 «*. Am. = 3. x y 4. Show that the points of contact of tangents from a point (a, 0) to the curve are situated on the hyperbola (m + n) xy = n$x + 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, xi + yi = a* ; and prove that the portion of the tangent intercepted between the axes is of constant length. 7. In the curve x n + y" = a", 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 a 2n Jns. p = — ; intercept = ; -. , prove that — = a\/ 1 - e 2 sin 2 d>. d

, , » 2 - * 2 ds = ffl — ■—- df (ji) +**- l {vf (fl) +/ 1 (iu)} + ^ 2 { I - 2 ^/o / '(/u) + v/i'Ou) +/ 2 ( /U )| + &c. = o. (2) The roots of this equation determine the points of section in question. We add a few obvious conclusions from the results arrived at above : — i°. Every right line must intersect a curve of an odd de- gree in at least one real point ; for every equation of an odd degree has one real root. 2 . A tangent to a curve of the n th degree cannot meet it in more than 11-2 points besides its points of contact. 3 . Every tangent to a curve of an odd degree must meet it in one other real point besides its point of contact. 4°. Every tangent to a curve of the third degree meets the curve in one other real point. 197. Definition of an Asymptote. — An asymptote is a tangent to a curve in the limiting position when its point of contact is situated at an infinite distance. i°. No asymptote to a curve of the n th degree can meet it in more than n - 2 points distinct from that at infinity. 2°. Each asymptote to a curve of the third degree inter- sects the curve in one point besides that at infinity. 198. Method of finding the Asymptotes to a Curve of the n th Degree. — If one of the points of section of the line y = \ix + v with the curve be at an infinite distance, one root of equation (2) must be infinite, and accordingly we have in that* case /o(a0 - o. (3) Again, if two of the roots be infinite, we have in addition v/o'G*) +/1&O - o. (4) * This can be easily established by aid of the reciprocal equation ; for if we substitute - for x in equation (2), the resulting equation in z will have one root z zero wnen its absolute term vanishes, i.e., when/o(^t) = o ; it has two roots zero when we have in addition v/ an d so on. Method of finding Asymptotes in Cartesian Co-ordinates. 243 Accordingly, when the values of fi 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 fii be a root of the equation f (fi) = o, the line y = \x x x- jrr— (5) is in general an asymptote to the curve. lifiQj) = o and/ (jti) = o have a common root (/xi suppose), the corresponding asymptote in general passes through the origin, and is represented by the equation y = viz. In this case u n and w^i evidently have a common factor. The exceptional case when fo(fx) vanishes at the same time will be considered in a subsequent Article. To each root of / (/z) = o corresponds an asymptote, and accordingly,* every curve of the n™ 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. Cor. No line parallel to an asymptote can meet a curve of the n th degree in more than (n - 1) 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 fo{/j.) is of the n th degree in fi, unless its highest coefficient vanishes, in which case, as we shall see, there is an additional asymptote parallel to the axis of y. 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 f (fi) = o, when multiplied by #", 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 f (n) = 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 {fx) = 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 : — i . y 3 = ax 2 + x 3 . Substituting (juc + v for y, and equating to zero the coefficients of x 3 and «*, separately, in the resulting equation, we obtain jj? - i = o, and 3/x 2 v = a; a hence the curve has but one real asymptote, viz., a y - x + -. 3 2. ^ 4 — z 4 + 2ax 2 y = b 2 x z . Here the equations for determining the asymptotes are ft 4 - I = o, and 4/i 3 v + 2afi = o ; accordingly, the two real asymptotes are a a y = x , and y + x + - = o. 2 2 3. x 3 + %x*y - xy 2 - yj 3 + x 2 - zxy + yj' 1 + 4* + 5 = °- x 3 I 3 Am. y + - + - = o, y = x + -, y + x = -. 3 4 4 2 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 Wn + «n-2 + w„_3 + &c. . . . + Ui + u = o, the equations for determining the asymptotes become / (/u) = o, and vfo{fi) = o. The latter equation gives v = o, unless / '(/u) vanishes along with/ (/x), i.e., unless /o(ju) has equal roots. Hence, in curves whose equations are of the above form, the n right lines represented by the equation n 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 + ihxy + by 2 = 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 i - x*y = a 2 (x + y) + 4 s . Am. x ■ o, y = o, x — y = o. 2. y z — 3? — a 2 x. „ y — x — o. 3. x* - y* = a 2 xy + W-y 1 . „ 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 + {flxy + b) x n ~ l + &c. = o ; then, if a = o and a : y + b = o, or y = , two of the roots of the equation in x become infinite ; and consequently the line a x y + 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 Q = o, and b = o, the axis of x is itself an asymptote. If X" and x n ~ l be both wanting, the coefficient of x n ~* 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 . Ans. x = a, y = x + a, y + x + a — o. 2. y(x 2 - 3&S + 2b 2 ) = x 3 — T,ax 2 + a 3 , x = b, x = 2b, y + 3a = x + 3b. 3. x 2 y 2 = a 2 (x 2 + y 2 ). x = ± a, y = ± a. 4. x 2 y* = a 2 (x 2 - y 2 ). y + a = o, y - a = o. 5. y 2 a - y*x = x 3 . X = a. 201. Parabolic Branches. — Suppose the equation /o(ju) = o has equal roots, then/ '(jui) vanishes along with/ (ju), and the corresponding value of v found from (5) becomes in- finite, unless /1 (fj) 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 + fty) 2 = Ix + 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 n th and the (n - i) th degrees Parallel Asymptotes. 247 in its equation. Consequently, all curves which have the same terms of the two highest degrees have generally the samt 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 / n (/x) = o has a pair of equal roots, each repre- sented by jUi, and where fi(ni) = o, at the same time. In this case the coefficients of x n and # w_1 in (2) both vanish independently of v, when jx = /ii ; we accordingly infer that all lines parallel to the line y = fitf 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 ju x for fx in (2), the two first terms vanish, as already stated, and the coefficient of x n ~ 2 becomes T^/o"^) + v/ZOuO +/,0« x ). Hence, if vi and v% be the roots of the quadratic 7^/o"0"0 + /2(/"0 = - 8 « 2 » and the corresponding asymptotes are y + x — 2a = o, and y + x + 4a = o. 204. If the equation to a curve of the n th degree be of the form \{y + ax + |3)0i +

i and 2 . Now, when x and y 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 X and

cos B + h sin 9 = 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 cos 9 + bi sin 9 = o ; accordingly, the equation of the tangent at the origin is box + 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 (wi) in its equation represents the tangent at that point. If b = o, the axis of # is a tangent ; if 6 2 = 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 (w - 9) , sin 9 m = ■ -, and n = ; sin to sin a) w 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* (a? + y 2 ) = a (x - y) * Two points which are infinitely close to each other on the same branch of a curve are said to be 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 (x x , y x ) is a point on the curve, and the equation becomes f(X + x h Y+y,) = 0. Hence the equation of the tangent referred to the new axes is \dxji \dy \ fd f\ r ^ A ( X - X \dx\ +{y - y % On substituting x - x iy and y - y r , instead of X and Y, we obtain the equation of the tangent referred to the original axes, viz. = or < — dxdyjx \dx 2 )i \dy- 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 : — i. [bx — cy) 2 = (x — a) 5 . The point x — a, y = — , is evidently a cusp, at which bx — cy = o is the tangent, as in the accompanying figure 2. (y-c)*=(z-ay(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. 3. y 2 = x(x + a) 2 . The point y = o, x — - a is a, conjugate point. 4. x% + y$ = a$. 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~ = [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, 00 = e, be taken on the axis of x, the curve passes through the points A, B, and O. Moreover, when x < a, if is negative, and therefore y is imaginary. „ x > a, and < J, y 2 is positive, and therefore y is real. „ x > b, and < c, y 1 is negative, and therefore y is imaginary. „ x > c, y 1 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 f it evidently consists of an oval lying between A and B, and an infinite branch passing through C, 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 f -(«-a) (* -b)\ Fie. 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. Fig- 23- On the other hand, let b = a, and the equation becomes y 2 = (x-aY(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 = c, and the equation becomes y 1 = (x - a) 3 ; Fig. 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 ~ J~ 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 (2ax + by + c) y = I, (ax + iby + c) x = m ; from which / 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 + zmy + 3» ; 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 d 2 u dru f d 2 u dxr dxf \dxdy j or (lax + 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 Plucker,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 Higher 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 a 3 = o. The different kinds of triple points are distinguished, according as the lines represented by u z = 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 a 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 du die d 2 u d?u dru ' dx ' dy ' dx 2 ' dxdy ' dif 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 lino joining the points of contact ; hut this line is parallel to the third asymptote ax + !>>/+ c = o, and evidently passes through the middle points of the intercepts made by this asymptote on the two others. t Liouville'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 General. — 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. j,.^ 26 Professor Cay ley 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 called ^om^s de rebroussement by French writers, and Miickkehrpunkte by Crernians, 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 = # 5 . Here the origin is a cusp; also y = x 2 ± x% ; hence, when a; is less than unity, both 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 l — a^y 2 = o. Multiple Points with Polar Co-ordinates. 267 216. Multiple Points of Curves in 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{9) = o, the resulting value of 9 gives the direction of the tangent in question. If the equation f{9) = o has two real roots in 0, less than ir, the origin is a double point, the tangents being determined by these values of 9. 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 9 = ± J j, and the origin would seem to be only a double point ; however, on transforming the equation to rectangular axes, it becomes (x 2 + y~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 y 2 = x 1 (i — x 2 ). Ans. x+y = o, x — y = o. 2. Show that the curve z 4 — 2>axy + y* = o touches the axes of co-ordinates at the origin. 3. Find the nature of the origin on the curve x* — ax 2 !/ + by 3 = o. 4. Show that the origin is a conjugate point on the curve ay 2 — x 3 + 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 ) = x 4 . 6. Prove that the origin is a cusp on the curve (y — x 2 ) 2 = X s . 7. In the curve (y - * 2 ) 2 = «"f show that the origin is a cusp of the first or second species, according as n is < or > 4. 8. Find the number and the nature of the singular points on the curve x i + \ax 3 — 2ay 3 + £,a-x 2 - 3« 2 y 2 + 4 ft4 — °» 9. Show that the points of intersection of the curve i ® ,+ 0)- with the axes are cusps. 10. Find the double points on the curve x 4 - 4fl£ 3 + $a 2 x 2 - b 2 y 2 + 2b 3 y - a* - i 4 = O. Examples. 269 11. Prove that the four tangents from the origin to the curve Ml + U% + «3 = o are represented by the equation 4«i «3 = "J. 12. Show that to a double point on any curve corresponds another double point, of the same kind, on the inverse curve with respect to any origin. 13. Prove that the origin in the curve x 4 — iax-y — axy i + a 2 y 2 = o is a cusp of the second species. 14. Show that the cardioid r = 0(1 + cos d) has a cusp at the origin. 15. If the origin be situated on a curve, prove that its first pedal passes through the origin, and has a cusp at that point. 16. Find the nature of the origin in the following curves : — aQ- r 3 = a 3 sin 30, r" = a" sin nd, r = + c 17. Show that the origin is a conjugate point on the curve x* - ax 2 y + axy 2 + a 2 y z = o. 18. If the inverse of a conic be taken, show that the origin is a double point on the inverse curve ; also that the point is a conjugate point for an ellipse, a cusp for a parabola, and a node for a hyperbola. 19. Show that the condition that the cubic xy 2 + ax 3 + bx 2 + cx + d + 2ey = o may have a double point is the same as the condition that the equation ax* + bx 3 + ex 2 + dx - e 2 = o may have equal roots. 20. In the inverse of a curve of the n th degree, show that the origin is a multiple point of the n th order, and that the n tangents at that point are parallel to the asymptotes to the original curve. ( 270 ) CHAPTER XV. ENVELOPES. 217. method of Envelopes. — If we suppose a series of different values given to a in the equation fix, y, a) = o, (1) then for each value we get a distinct curve, and the above equation may be regarded as representing an indefinite number of curves, each of which is determined when the corresponding value of a is known, and varies as a varies. The quantity a is called a variable parameter, and the equation/ (x, y, a) = o is said to represent a family of curves; a single determinate curve corresponding to each distinct value of a ; provided a enters into the equation in a rational form only. If now we regard a as varying continuously, and suppose the two curves f fo V, a) « O, fix, y,a + Aa)=0 taken, then the co-ordinates of their points of intersection satisfy each of these equations, and therefore also satisfy the equation fix , y,a + Aa) -fix, y, a) Aa Now, in the limit, when Act is infinitely small, the latter equation becomes 7 = °' ( 2 ) da and accordingly the points of intersection of two infinitely near curves of the system satisfy each of the equations (1) and (2). Envelopes. 271 The locus of the points of ultimate intersection for the entire system of curves represented by / (x, y, a) - o, is ob- tained by eliminating a between the equations (1) and (2). This locus is called the envelope of the system, and it can be easily seen that it is touched by every curve of the system. For, if we consider three consecutive curves, and suppose Pi to be the point of intersection of the first and second, and P 2 that of the second and third, the line P x P 2 joins two infi- nitely near points on the envelope as well as on the inter- mediate of the three curves ; and hence is a tangent to each of these curves. This result appears also from analytical considerations, thus : — the direction of the tangent at the point x, y, to the curve /(a 1 , y, a) = o, is given by the equation dx dy dx ' in which a is considered a constant. Again, if the point x, y be on the envelope, since then a is given in terms of x and y by equation (2), the direction of the tangent to the envelope is given by the equation df df dy df Ida dad/j\ dx dy dx da \dx dy dx) df df dy or 7+77 = 0, dx clij dx If since — = o for the point on the envelope. da Consequently, the values of -j- are the same for the two 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 u + 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 f(x, y, a) = x cos a + y sin a - p = o, d/(x, y, a) — — — - • = - x sm a + y COS a = o, da and, on eliminating a between these equations, we get x 2 + y 2 =p 2 , 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, ■?/, a) m fm da a' \ x Substituting this value for a, we get for the envelope y 2 = 4tnx, which represents a parabola. 2 1 8. Envelope of La 2 + zMa + N= o. Suppose L, H, N, to be known functions of x and y, and a a parameter, then f( x > Vy «) = La 2 + 2Ma + N = O, 3- = 2La + 2M= o; da accordingly, the envelope of the curve represented by the preceding expression is the curve LN = M\ Undetermined Multipliers applied to Envelopes. 273 Hence, when L, M, iV are linear functions in x and y, this envelope is a conic touching the lines L, JV, and having M for the chord of contact. Conversely, the equation to any tangent to the conic LN = M 2 can be written in the form La 2 + 2J/a + iV=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 /(*» V> «> 0) = *i> (3) where the parameters a, /3 are connected by an equation of the form (a, /3) = c 2 . (4) In this case we may regard /3 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 dQ , dd> dch dS -J- + -Ja IT = °> and T- + -Tn TT = °- da d[3 da da dp da df d£ Consequently %=%• da d(5 If each of these fractions be equated to the undetermined quantity A, we get df =x d$^ da da I df _ dcp dj3 = d[3 * Salmon's Conies, Art. 248. T 274 Envelopes. and the required envelope is obtained by eliminating a, |3, and A 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 j3 ; for suppose them to be of the forms f(x, y, a, (5) = Ci, (a, (5) = (x) + htfix) + — $"(x) + — — '"(x) +&c. Y x = y + htf(x) = '(x)', •"• yi ~ Fl = & 2 ^ {x) + 7TT7-/' {x) + &c - (I) 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 3/1 - Yi, or of QT, is the same as that of (p"(x). Hence, for a point above 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 "(x) or —, at the point. 222. Points of Inflexion.- the point P, we shall have t/i - Pi = P 1.2.3 f'{x) + h If, however, $"(x) =0 at iv (z) +&c. (2) 2.3.4 Now, provided "'(x) be not zero, y x - Y x changes its sign with h, i.e. if MN' = HN= 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 g- 28 - point of contact. Such points on a curve are called points of inflexion. Again, if $"{x) as well as ix (x) be not zero at the point, iji - Y x does not change sign with h, 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. 280 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. 221 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 = fxx + v with the curve y = {x) =nx + v. (3) Suppose A, B, C, D, &c, to represent the points of section in question, and let #1, x 2 , . . . x n be the roots of equation (3) ; - ,<£ ^v. — .-^ ^^ — then the line becomes a /-p >v tangent, if two of these ^ roots are equal, i.e., if Fi s- 2 9- 0'(#i) = /x, 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 $"{xi) = 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, G 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 0"'(#i) also vanish, the tangent must be regarded as cutting the curve in four consecutive points : such a point is called el 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 cubio be taken as origin, and the tangent at it as axis of x, the equation of the curve must be of the form x 3 + y

) a cusp. 5. Find the co-ordinates of the point of inflexion on the curve 2* 3 X s — %bx 2 + a-y = o. Ans. x = b, y = — . 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 «i + «4 + W5 + &c., + «„ = o. 8. Show that the points of inflexion on curves referred to polar co-ordinates are determined by aid of the equation d 2 u 1 "'^dd*" ' e U = r' 9. In the curve r9 m = a, prove that there is a point of inflexion when =i/m (1 - m). x 10. In the curve y = c sin -, prove that the points in which the curve a 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 Ml + «2 + «3 + • . • + Mn = O, prove that «2 must contain u\ as a factor. 13. Show that the points of inflexion of the cubical parabola y 2 = (* - a)* (x - b) lie on the line ix + a = ifi : 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 z + ipx + q) are determined by the equation 2a: 3 -f 6px 2 + 3 (p 2 + q) x + 2pq = o. 15. If y 2 = /(z) be the equation of a curve, prove that the abscissae of its points of inflexion satisfy the equation {/'(*)}* = */(*) ./"(*)• 16. Show that the maximum and minimum ordinates of the curve y= 2/(*) /"(*)-{/'(*) } 2 correspond to the points of intersection of the curve y z =f(x) with the axis of*. 17. When y l =f(x) represents a cubic, prove that the biquadratic in * which determines its points of inflexion has one, and but one, pair of real roota. 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 3 + ibxy 2 + 2,cx 2 y + dz 3 + %ex 2 = o lies in the right line ay + bx = o, and has for its co-ordinates ■\a 2 e . %abe * = - -£pandy= — , 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 - s), 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 by the equations x = sec 3 6, y = tan Q sec 2 6 ; prove that there are two finite points of inflexion on the curve, and find the values of at these points. ( 28 5 ) CHAPTER XVII. RADIUS OF CURVATURE. EVOLUTES. CONTACT. CURVATURE AT A DOUBLE POINT. RADII OF 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 of 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 their centres : but Fig- 3°- A ACB: Lacb = arc AB arc ab ~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 A$ the angle between the tangents at its extremities, we have As r = — . A

evidently represents the radius of the circle which has the same curvature as that of the given curve at the point. This radius is called the radius of curvature for the point, and is usually denoted by the letter p. To find an expression for p, let the curve be referred to rectangular axes, and suppose x and y to be the co-ordinates of the point in question ; then if

a-**- 00 "**-"**!? dy ' dx Hence = — - ^± - * W J • (0 - tLence p d$ (Py (Py ds dx 2 dx 2 d 2 y At a point of inflexion -^ = o : accordingly the radius of 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 ( 1 + f -j- j 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 being positive when —^ is positive ; i. e. when the curve is 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 cos* = -, smtf>=- Hence, if the arc be regarded as the independent variable, we get dd> d 2 x dd> d 2 y y ds ds*' Y ds ds 2 ' from which, if we square and add, we obtain 1 fd- — — , d 2 y = sin (j>d 2 s + cos $ - — -. (3) P P Whence, squaring and adding, we obtain (d 2 x) 2 + {d 2 y) 2 = (d 2 s) 2 + £*£, P ds 2 01 P y/ \drxf + (dy) 2 - (d 2 sf W 288 Radius of Curvature. Again, if the former equation in (3) be multiplied by sin 0, and the latter by cos 0, we obtain on subtraction, ds 2 ds* cos d?y - sin (bcPx = — , or dxd 2 y - dyd 2 x = — . P P „ (dx 2 + dy 2 )% . . HenCe p ■ cfoVy - frdV (5) The independent variable is undetermined in formulae (4) and (5), and may be any quantity of which both x and y are functions. For example, in the motion of a particle along a curve, when the time is taken as the independent variable, we get from (4) an important result in Mechanics. Examples. 1. To find the radius of curvature at any point on the parabola x 2 = $my. -, dy d 2 y ldy\ 2 x 2 y Here 2tn-f- = x, zm-/- = 1, 1 + ( -f =1+ — -=i+ — ; dx ax- \dx/ 4>n i m 2(m 4 y)% •'• P= ml 2. Find the radius of curvature in the catenary r -=('*♦ .-*)■ Hero d J>,l(£-rA, *»_». ..., = -£ Hence the radius of curvature is equal to the part of the normal intercepted by the axis of x, but measured in tbe opposite direction (Ex. 4, Art. 171). 3. In the cubical parabola ^a 2 y = x 3 , we have ***.*, rf f». M| .(.4.fjsy)U (« t +* t ) | ; ...,-£±*a dx dx 2 { \dxj ) a e 2a * x General Expression for Radius of Curvature. 289 4. To find the radius of curvature in the ellipse — + — = i. ar 0* Let x = a cos , and we have dx = — a sin , rf'z = — a cos

2 — a sin

, 2 + 5 cos <£rf 2 4>. Hence by formula (5) we obtain (a 2 sin 2 + £ 2 cos 2 )3 P= ^ ' 5. In the hvpocycloid xi + .V* = «'i let £ = a cos 3 ^, then y = a sin 3 sin £), dy = 3a sin 2 <£ o,os, and dxd?y - dydrx = — 9a 2 sin 2 <£ cos 2 sin w = -=-r=. = — , in the limit. PN y' Henoe, in the case of oblique axes, we have p sin a) = — . (10) r 2Co v ' If bi and c have opposite signs, p is negative ; this indicates that the centre of curvature lies below the axis of x, towards the negative side of the axis of y. The preceding results show that the radius of curvature at the origin is the same as that of the parabola, b { y = c x 2 , at the same point ; and also that the system of curves obtained by varying all the coefficients in (9), except those of y and x 2 , have the same osculating circle, in oblique as well as in rectangular co-ordinates. Again, as in Art. 223, the osculating circle, since it meets the curve in three consecutive points, cuts the curve at the point, in general, as well as touches it. If c = o in the equation of the curve, and b x be not zero, the radius of curvature becomes infinite, and the origin is a point of inflexion. This is also evident from the form of the equation, since the axis of x meets the curve in this case in three consecutive points. 232. In general, the equation of a curve referred to any rectangular axes, when the origin is on the curve, may be written in the form zb it x + 2b x y = c x 2 + 2c { xy + c 2 y 2 + u 3 + &c. 294 Radius of Curvature. Here b x + b x y = o is the equation of the tangent at the origin ; and the length of the perpendicular PN from the point (x, y) on this tangent is b x + b x y y v + b? Also, OP 2 = x 2 + y\ and OP 2 = 2 9 . PN in the limit. Accordingly, we have, when x and y are infinitely small, 1 2PN 2b x + 2b x y P OP 2 (a* + y*)1 = a m p, by Art. 190. Hence ; also, 7 = If p and 10 have the (»» + i)r m " 1 (m + \)p ' m + 1 This result furnishes a simple geometrical method of finding the centre of cur- vature in all curves included under this equation. 236. To prove that p = p + —±- same signification as in Art. 192, the formula of that Art, becomes ds d 2 p , , Examples. 1. In a central ellipse prove that p = y^a 2 cos 2 o> + b 2 sin J o>, and hence deduce an expression for the radius of curvature at any point on the «urve. 2. In a parabola referred to its focus as pole, prove that^? = m seca>, and hence show that p = im sec 3 w. 237. Evolutes and Involutes. — If the centre of cur- vature for each point on a curve be p 1 Pa 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 Pi, P 2 , P3, &c., represent a series of infinitely near points on a curve; C x , C 2 , C 3 , &c, the corresponding centres of curvature, then the lines P^,, P 2 C 2 , P 3 C 3 , &c, are normals to the curve, and the lines C X C 2 y C 2 C 3 , C 3 C 4 ,&c.,maybe regarded in ■» P Fig- 33- the limit as consecutive elements of the evolute; also, since 298 Radius of Curvature. each of the normals P1C1, P 2 (7 2 ,P 3 C 3 , &c, passes through two consecutive points on the evolute, they are tangents to that curve in the limit. Again, if p l5 p 2 , p 3 , p 4 , &c, denote the lengths of the radii of curvature at the points Pi, P 2 , P 3 , P 4 , &c, we have Pi = P1C1, p 2 = P 2 C 2 , p 3 = P3C3, pi = P4C4, &c. ; •*• pi ~~ p% = P1C1 — P2C2 = P 2 Ci — P2C2 = C1C2 ; also p 2 - p 3 = C 2 C 3 , p 3 - p 4 = C 3 C 4 , . . . p»_i - p» = C,»_iC n ; hence by addition we have pi ~ Pn — C1C2 + C 2 C 3 + 6364 + . . . + Cn-i C». This result still holds when the numher « is increased indefinitely, and we infer that the length of any arc 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 he parallel; each being got from the other by cutting off a constant length on its normal measured from the curve. 238. Involutes 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 war, then that of its normal at a point (x, y) is m or (Y-y)- + X-x = o, y 3 + 2tny (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 3y 2 + 2m (m - X) =0. Accordingly, the equation of the required envelope is obtained by substituting =. instead of y 2 7)1 - X in the latter equation. Hence, we get for the required evolute, the semi-cubical parabola 2-jmY 2 = 8 (X - m)\ The form of this evolute is exhi- bited in the annexed figure, where VN = m - 2 VF. If P, F, repre- sent the points of intersection of the Fig. 34- evolute with the curve, it is easily seen that VM=4VN=4m. 240. Evolute of an Ellipse. — The form of the evolute of an ellipse, when e is greater than ^ a/ 2 > is exhibited in the accompanying figure ; the points M, N, M', 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 300 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 9 . Let P and Q be two points on the curve, its pole, PC, QCthe normals at Pand Q; join OC. Then by the fundamental property of the curve (Art. 181), the angles OPC 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 BAC, whose centre is 0. Let OC = a, L COA - = CP = a< from which it is easily seen that a(p 2 242. 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 6. By Article 183- we have ? =ws+ 8 hence i dp d 2 u v l du (W- ' Also dr 1 du dp ir dp 302 Radius of Curvature. consequently „(.+ £[)_-I-.| I + ( *L JJ | /rf«Y)S I , C?M I flfc Again, since « = -,we have _ = --_, c? 2 w 2 /f/rV i d 2 r r*d6 2 ' .drV)* d 2 r fdr\ 2 (.6) This result can also be established in another manner, as follows : — On reference to the figure of Art. 1 80, it is obvious that

is the angle the tangent at P makes with the prime vector OX. __ d(h diL d(h ds d\L Hence dd = I+ dd> 0T d^de = I+ -dd ; dxf, 1 _ dtp dd p ds ds_ dd dv d T Again, denoting y„- and -^ by r and r, we have tan \L = -. ; and hence r dxL , , r 2 - rr r 2 - rr dO r~ r + r 2 d\L r 2 - rr + 2'r 2 ds 1 + m = i ~2 — > also lQ = \ r+ r r- dO r + r 2 dd Intrinsic Equation of a Curve. 303 (r + r 2 ) 1 Hence, we get P = ^ _ rr + 2f a - Or, replacing r and r by their values, - + U p , d'r Idr r - r dtP + 2 {T8 Again, since p = 00 at a point of inflexion, we infer that the points of intersection of the curve represented by the equation . o?r fdr\ *- r de> + 2 \dd) =0 > 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 2 u w + u = °- Examples. 1. Find the radius of curvature at any point in the spiral of Archimedes, r = ad. Ans. a r— . 2 + 6~ 2. Find the radius of curvature of the logarithmic spiral r — a . Ans. r(i+ (log a) 2 )i. 3. Find the points of inflexion on the curve 9 r = 29 — n cos 2d. Ans. cos 20 = — . 1 1 4. Prove that the circle r = 10 intersects the curve r = 11 — 2 cos $9 in its points of inflexion. 5. Prove that the curve r = a + b cos nO has no real points of inflexion unless a is >b and <(l + w 2 )i. 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. 242(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, andy = tf>(*), and that x x is the abscissa of a point common to both curves, then we have /(*i) =0(«i). Again, substituting x Y + h, instead of a? in both equations, and supposing y y and y 2 the corresponding ordinates of the two curves, we have Vx =/(*i + h) -/(a*) + VW + ^/"W + &c, y 2 = ^ (#1 + ^) = ^ (#1) + h'(x\), or that the curves have a common tangent at the point, then 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 z = — ^— {/">,) - f >,) J + &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 x 306 Radius of Curvature. the former case. Moreover, since y x - y 7 ^anges its sign with h, the curves cut each other at the point as well as touch. If we have in addition /"'(#i) = ^'"(^O* 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 order, and we have Vl ~ y2 = bT7 {/(w+1) {Xl) ~ * (n+1) {Xl) } + &G - (l 8) Also, if the contact be of an even order, n + i is odd, and consequently h nn 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 y + {Y-py = R 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 -j- and -Hr must be the same for the circle and dx dx 2 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 Y, we get and V^y Hence y - /3 = - "(!)' a; - a = dy dx 1 + ~ dy\ dx .-. R^s-ay + (y-p) = 1 + dy dx 2 dy'^ dx (19) (20) (21) P = ji 1 «-— ^ • ( 2 4) Again, substituting the values of x and y given by these lations, in t of the evolute Oft" f/" equations, in the equation - + ^=i,we get for the equation {aa)% + ((5b)l = {a 2 - b 2 )h 246. It may be noticed that the osculating circle cuts the curve in general, as well as touches it. This follows from Article 243, since the circle has a contact of the second order at the point. At the points of maximum and minimum curvature the Osculating Curves. 309 osculating circle has a contact of the third order with the curve ; for example, at any of the four vertices of an ellipse the osculating circle has a contact of the third order, and does not cut the curve at its point of contact (Art. 240). 247. Osculating Curves. — When the equation of a curve contains a number, n, of arbitrary coefficients, we can in general determine their values so that the curve shall have a contact of the {n - i) ih order with a given curve at a given point ; for the n arbitrary constants can be determined so that the n quantities dy d 2 y d n ~ 1 p Vy die dx 2 ' ' ' 'daF v shall be the same at the point in the proposed as in the given curve, and thus the curves will have a contact of the (n - i) th order. The curve thus determined, which has with a given curve a contact of the highest possible order, is called an osculating curve, as having a closer contact than any other curve of the same species at the point. For instance, as the equation of a circle contains but three arbitrary constants, the osculating circle has a contact of the second order, and cannot, in general, have contact of a higher order; similarly, the osculating parabola has a contact of the third order ; and, since the general equation of a conic contains five arbitrary constants, the general osculating conic has a contact of the fourth order. In general, if the greatest number of constants which determine a curve of a given species be n, the osculating curve of that species has a contact of the (w - i) th order. 248. Geometrical method. — The subject of contact admits also of being considered in a geometrical point of view ; thus two curves have a contact of the first order, when they intersect in two consecutive points ; of the second, if they inter- sect in three ; of the n th , if in n + 1 . For a simple investi- gation of the subject in this point of view the student is referred to Salmon's Conic Sections, Art. 239. 249. Curvature at a Double Point. — We now pro- ceed to consider the method of finding the radii of curvature of the two branches of a curve at a double point. 310 Radius of Curvature. In this case the ordinary formula (8) becomes indetermi- nate, since du , du — = o, and — = o dx dy at a double point. The question admits, however, of being treated in a manner analogous to that already employed in Art. 230 : we commence with the case of a node. 250. Radii of Curvature at a Node. — Suppose the origin transferred to the node, and the tangents to the two branches of the curve taken as co-ordinate axes, w represent- ing the angle between them. By Art. 210, the equation of the curve is in this case of the form ihxy = ax 3 + fix 2 y + yxy* + $y 3 + w 4 + &o. : dividing by xy we obtain 2k = a — + Bx + yy + 8 — + — + &C. V x xy Now, let pi and p 2 be the radii of curvature at the origin for the branches of the curve which touch the axes of x and y, respectively; then, by Art. 231, we have 2pi sin id = — , and 2p 2 sin 10 = — , in the limit. y r x Again, it can be readily seen, as in the note to Art. 230, that the terms in — , &c, become evanescent along with x xy X" 11* and y, and accordingly the limiting values of — and — can y x be separately found, as in the Article referred to. Hence we obtain h h . Pi = — — > P2 = §—• • l 2 5) a sin «u osinw Also, if a = o, we get pi = 00, and the corresponding branch of the curve has a point of inflexion at the origin. Similarly, if § = o, p 2 = 00. Radii of Curvature at a Cusp. 3 1 1 If a = o, and $ = o, the origin is a point of inflexion on both branches. This appears also immediately from the consideration that in this case « 3 contains « 2 as a factor. If the equation of a curve when the origin is at a node contain no terms of the third degree, the origin is a point of inflexion on both branches. An example of this is seen in the Lemniscate, Art. 2 1 o. Examples. x. Find the radii of curvature at the origin of the two branches of the curve ax* - zbxy + cy z = x* + y*, the axes being rectangular. Ans. - and -. a e 2. Find the radii of curvature at the origin in the curve a (y 3 — x 2 ) = a?. Transforming the equation to the internal and external bisectors of the angle between the axes, it becomes $axy \/ 2 = (x - yf ; hence the radii of curvature are 2a \/ 2 and — 2a*y 2, respectively. 251. Radii of Curvature at a Cusp. — The preceding method fails when applied to a cusp, because the angle w vanishes in that case. It is easy, however, to supply an in- dependent investigation : for, if we take the tangent and normal at the cusp for the axes of x and ;/, respectively, the equation of the curve, by the method of Art. 2 1 o, may be written in the form y 1 = ax z + (5x"y + yxy 2 + Sf/ Z + ih -f &c. (26) Now in this, as in every case, the curvature at the origin depends on the form of the portion of the curve indefinitely near to that point ; consequently, in investigating this form we may neglect tfx, y* f &c, in comparison with if\ and x x x?y, &c, in comparison with x 3 . 3 1 2 Radius of Curvature. Accordingly, the curvature at the origin is the same, in general, as that of the cubic y 2 = ax 3 + fix 2 y. (27) Dividing by # 2 , we get ^ = ax + (3y. Hence, in immediate proximity to the origin, - be- comes very small, i. e. y is very small in comparison with x. Accordingly, the form of the curve near the origin is repre- sented by the equation y 1 = ax 3 . From this we infer that the form of any algebraic curve near a cusp is, in general, a semi-cubical parabola (see Ex. 2, Art. 211). Again, since we have, by Art. 230, x % x -**£ from which we see that p vanishes along with x, and accord- ingly the radii of curvature are zero for both branches at the origin. This result can also be arrived at by differentiation, by aid of formula ( 1 ) . 252. Case where the Coefficient of a? is wanting. — Next, suppose that the term containing x 3 disappears, or a = o, then the equation of the curve is of the form ?/ = (3x 2 y + yx?f + <$y 3 + ax* + &c. ; and proceeding as before, the curvature at the origin is the same as in the curve y 2 = fix 2 y + oV. (28) Radii of Curvature at a Cusp. 313 The two branches of this curve are determined "by the equation y = jz 2 ±-a//3 8 + 4o'. (29) The nature of the origin depends on the sign of j3 2 + 4a', and the discussion involves three cases. (1). If |3 2 + 40' be positive, it is evident that the curve extends at both sides of the origin, and that point is a double cusp (Art. 215(a)). On dividing equation (28) by y 1 , and substituting 2p for -, we get 1 = 2(3p + 4a> 2 . (30) The roots of this quadratic determine the radii of curva- ture of the two branches at the cusp. These branches evidently lie at the same, or at opposite sides of the axis of x, according as the radii of curvature have the same or opposite signs : i. e. according as a has a negative or positive sign. These results also appear immediately from the circum- stance, that in this case the form of the curve very near the origin becomes that of the two parabolas represented by equation (29). (2). If /3 2 + 4a' be negative, y becomes imaginary, and the origin is a conjugate point. (3). If /3 2 + 4a' = o, the equation (30) becomes a perfect square : we proceed to prove that in this case the origin is a cusp of the second species. To investigate the form of the curve near the origin, it is necessary in this case to take into account the terms of the fifth degree in x (1/ being regarded as of the second) : this gives (y - £ x 2 ) 2 = 7 x?f + /3Vy + oV = x ( y if + |3Vy + a'V) . (31) It will be observed that the right-hand side changes its sign with x ; accordingly the origin is a cusp. Also, the cusp is of the second species, for the two roots of the equation in y plainly have the same sign, viz., that of /3 ; and consequently both branches of the curve at the origin lie at the same side of the axis of x. 314 Radius of Curvature. Moreover, as equation (30) has equal roots in this case, the radii of curvature of the two branches are equal, and the branches have a contact of the second order. We conclude that when the term involving x 3 in equation (28) disappears, the origin is a double cusp, a cusp of the second species, or a conjugate point, according as /3 2 + 4a' > = or < o. Moreover, if a = o, one root of the quadratic (30) is in- finite, and the other is — r. The origin in this case is a double 2J3 cusp, and is also a point of inflexion on one branch. Such a point is called a point of oscul-in flexion 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 d 2 t/ turn their convexities in opposite directions, -- 2 must have 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 7 instead of p ; where 7 is half the chord intercepted on the axis of y by the osculating circle at the origin. 253. J&ecapitulation. — 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 if = ax 3 + 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 z 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 Vi 2 + 0it>2 + Vi + &C. = O. This is also evident from geometrical considerations. General Investigation of Cusps. 3 1 5 generally either a double cusp, a conjugate point, or a cusp of the second species. In the latter case tho 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'* degree, the origin is a point of oscul-inflexion. If, however, the coefficient of a?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. General Investigation of 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 1 = 2Ax n y + Bx h + Cx c , (32) where iAx a is the lowest term in the coefficient of y, and Bx b , Cx°* 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, w r e obtain y = Ax a ± */A 1 xr a + Bx h + Cx% which represents two parabolasf osculating the two branches at the origin. The discussion of the preceding form for y resolves itself into three cases, according as za is > = or < b. (1). Let 2a = b + h, then b + h b y = Ax 2 ± x' 2 : m being assumed to be greater 2 than unity. 10. Two plane closed curves have the same e volute : what is the difference between their perimeters ? Ans. 2ird, where d is the distance between the curves. 11. Find the radius of curvature at the origin in the curve 3y = 4*- I5* 2 - 3Z 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 further. 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 - 43 Ans. 4 — . ao 14. Find the radii of curvature at the origin of the two branches of the curve- x* ax 2 y — axy 2 + cfiy % = o. Ans. a and -. 2 4 15. Prove that the evolute of the hypocycloid x\ + y\ = at is the hypocycloid (o + 0)1 + (a - £)! = zai. 16. Find the radius of curvature at any point on the curve y + *y x ( 1 — x) = sin -1 \/ x. 17. If the angle between the radius vector and the normal to a curve has a maximum or a minimum value, prove that y = r ; where y 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 (1 + cos 20), y = e cos 2d (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* has for its equation r 2 — (1 - m) a 2 = mp*. 20. If a and fi be the co-ordinates of the point on the evolute corresponding to the point (x, y) on a curve, prove that du da — — + I = o. dx d$ 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 -J- ; where a> dot is the angle the radius of curvature makes with a fixed line. 22. In a curve, prove that p dx \ us ) ' 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 = kx n reduces to the elimination of x between the equations n - 2 k 2 rr . , , „ 2w - i 1 a = * * 2n_1 , and £ = A*» + kn(n - l)x n -* 25. In figure, Art. 239, if the tangent to the evolute at P meet the parahola in a point H, prove that HNis 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. (Bertrand, Cal. D'f., p. 573.) This and the preceding theorem can be immediately established from geome- trical considerations. 28. If from the points of a curve perpendiculars he 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- - 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. 32. Hence find the length of the radius of curvature at the origin in the curve r = a sin nO. Ans. p = — 2 33. Find the co-ordinatts 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- - pp) = r 3 . Ind. Civ. Scr. Exam., 1878. y ( 322 ) 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, &c. (7). The points On Tracing of Curves. 323 where the y ordinate is a maximum or a minimum are found from the equation — = o : by this means the limits of the curve can be often assigned. (8). Determine when possible the points of inflexion on the curve. 259. To trace the Curve y* = x* (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 /', are easily determined from the equation -^= o ; the corresponding value of * is — ; accordingly AN = . Again, if TI be the tangent at the point of inflexion I, it can readily be seen that TA = - = . . 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* Fig. 38. y*x + ey = ax 3 + hx % + ex + d, (0 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 1 - ax 2 = o, or y = ± x y«. * This investigation is principally taken from Newton's Enumeratio Li- nearum Tertii Ordtuis. Y 2 324 On Tracing of Curves. If the term bx 2 be wanting, these lines are asymptotes ; if b be not zero, we get for the equation of the asymptotes r b /- h y = x^/a + =, y + x >y a + — — = o. 2^/a 2\/ a On multiplying the equations of the three asymptotes together, and subtracting the product from the equation of the curve, we get e y={°-—j x + d: this is the equation of the right line which passes through the three points in which the cubic meets its asymptotes. (Art. 2 °4-) Again, if we multiply the proposed equation by x, and solve for xy, we get xy e I e 2 = — + lax* + bx 3 + ex* + dx + - : (2) 2 V 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 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, /3, 7, $, arranged in order of increasing magnitude, and we have xy - - ± a/ci (x - a)(x - j3)(a? - y)(x - $). 2 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 < S, 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 S- 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 9*y 2 + io8y = (x - 5) (x - 11) (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$ x + — v = o ; and v, y = (a + b) sin 9 - b sin — — - 9. b J (io) is a hypocycloid, but only some hypoeycloids are epicycloids. While according to the correct definition no epicycloid is a hypocycloid, though each can be gene- rated in two ways, as will be proved in Art. 280. Epicycloids and Hypocycloids. 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 —7— 0, y = (a - b) sin - b sin — j— 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", 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 C(y and CO' will lie in directum, and the circle will have turned through an angle equal to the sum of the angles OCO' and OG'O"; hence, denoting these angles by dO and dd\ respectively, we have ds = OP (dO + dff) = Op(i + ^\dd; (12) since dft = r dd. 2jg. Radius of Curvature of au Epicycloid. — Suppose w to be the angle OSN between the normal at P and the fixed line CA, then w = C'OS- C'CS = ----0; .'. tfa, = -d0Ji+4-|. 22 [2b) Hence, if p be the radius of curvature corresponding to the point P, we get ^ = 0P ±±3. (I3) da) 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 tbat CO" is infinitely small in comparison with Off ; hence the space tbrough which the point moves during a small displacement is infinitely small in comparison with the space through which P moves. 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. 5S« 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 (/, and produce CO to meet DP produced in D', and describe a circle rouDd the triangle OPD'; this circle plainly touches the fixed circle ; also the segments standing on OP, OP, and OO are obviously similar ; hence, since OP = OO + 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 OA = 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 OO'E, 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 O'E, 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 (X ; 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 arc 00' = arc OP + arc O'P. If the arc OA be taken equal to OP, we get arc C/P = arc OA ; 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 until PP= OP '-^4, a + 20 then P is the centre of curvature by (13) ; hence OP' = OP :. a + 20 Fig. si- 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 OO : OE= OF : OP = a: a + 20 = CO :CE; .-. C0-0O:CE-0E = C0:CE, or CO:C0=C0:CE; that is, the lines CE, CO, and CO' are in geometrical pro- portion. Again, join C to B', the vertex of the epicycloid ; let CI? meet the inner circle in D, and we have arc O'D : arc OB = CO' : C0= CO : CE=O0:E0 = arc P'O: arc OQ. But arc OB = arc OQ; .: arc O'D = arc P'O. Accordingly, the path described by P f 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 : OO, 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' 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' = 20P'^ = 2 op'^. a CO 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 oircle. * The length of the arc of an epicycloid, as also the investigation of its evolute, were given by Newton (Principia, Lib. i., Props. 49, 50). 346 Roulettes. With reference to the outer epicycloid in Fig. 57, this gives CO' txoPB-zPE.^. (14) The corresponding results for the hypocycloid can be found by changing the sign of the radius b of the rolling circle in the preceding formulae. 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 = go°. Join CA, CB, CP, and draw CN perpendicular to DP. Let l PDO = 0, l BCN = w,lACO = Q, CN = p. Then since AO = PO, we have a9 = 20$ ; .'. 6 = — (I9) y = (a + b) sin - d sin — - — 0. 348 Roulettes. In the case of the hypotrochoid, changing the signs of b and d, we obtain x = {a - b) cos + d cos — — 0, ] y - (a - b) sin - d sin — — - 0. <2<>) 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 (b + d)* (b-dy ' 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^L meet the fixed circumference in A ; then since L OCL = zOC.A, and 0C 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 CiB, 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 case in which the outer circle rolls on the inner is also worthy of separate consideration. 286. Circle rolling on another inside it and of hall" 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 C2L be any two positions of the moving diameter, Ci and C 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 C x 2 and C0 2 will coincide in direction, and accordingly the outer circle will have turned through the angle C 2 2 Ci ; consequently, the mov- ing diameter will have turned through the same angle ; and hence L CJjC, = l CaOaC, ; 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 icith the rolling circle will ahcays touch a fixed circle. For, let BE be the moving line iu any position, and draw the parallel diameter AB; let fall dF and LM perpendicular to BE. Then, by the preceding, AB always passes through a fixed point L ; also LM= CJ?= 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 { , as has been shown, always passes through a fixed point L ; consequently, since CiP x is a con- stant length, the locus of Pi is a Limacon (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 become 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 mw = 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 = 4.5, prove that the equation of the hypocycloid becomes $ + yi = al. 5. Find the equation, in terms of r and^?, of the three-cusped hypocycloid ; t. e. when a = 3b. Ans. r % = a? — &p 2 . 6. Find the equation of the pedal in the same curve. Ans. p = b sin 3«. 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 F 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. 11. 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 A L 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) . a

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 " Ehodoneae," 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 of Curvature of an Epitrochoid or Hypotrochoid. — The position of the centre of curvature for any point of an epitrochoid can be easily found from geometrical considerations. For, let Ci and 2 he 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 X and 2 be two points on the circles, infinitely near to 0, such that OOi - 00 2 . Now, suppose the circle to roll until Ox and 02 coincide; then the lines dOi and C 2 2 will lie in directum, as also the lines Pi 0i and P 2 2 (since P 2 is the point Fig. 61. of intersection of two consecutive normals to the roulette). Hence L OC.O, + L OC 2 2 = L 0P X X + L 0P 2 2 , since each of these sums represents the angle through which the circle has turned. Again, let L COP, = $, 00, = 00 2 = ds; ds ds then Z0d0i = ^-, lOC % % --qq, ,0Pi0i=^, Z0P 2 2 = ^: 35 2 Roulettes. consequently we have ^^ 2 =cos *(ok + ok) (2i > Or, if OP x = r u OP 2 = r 2 , ii (\ i - + T = COS — + - a o r \ri 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 P v . If we suppose Pi to be on the circumference of the rolling OP circle, we get cos = ' ■ ; whence it follows that 2 \J\j\ OP, = — °— h OP lt a + 20 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 d 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 00, and 00 2 as elementary arcs of the circles of curvature, and the preceding demonstration will still hold. Hence, denoting the radii of curvature Od. and OC 2 by pi and p 2 , we shall have (22) 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 x is the centre oi 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. Geometrical 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 Legons des Machines a V Ecole Foly technique. See also Leroy's Geometrie Descriptive, Quatrieme Edition, p. 347. I I 1 r \- ■ — = = COS hence Roulettes. CyN. OP, NP, PC, cos NO, OP, " NP X c,c 2 op, _ nc^ 0C 2 ' P,P 2 ~ NP,' Consequently, by the well-known property of a transversal cutting the sides of a triangle, the points C 2 , Pi, and N are in directum. The modification in the construction when the rolling curve is a right line can be readily supplied by the student. 290. Circle of Inflexions. — The following geometrical construction is in many cases more .q useful than the preceding. On the line OC, take OB, such that 1 1 1 onroCi* ocj and on OB, as diameter describe a circle. Let E, be its point of inter- section with OP,, then we have cos

OE,' Hence, if the tracing point P, lie on the circle* OE,B, f * This theorem is due to La Hire, who sho-\ved 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 P : 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 P, P 2 , P x O and Pi E x are in continued proportion ; as also C\C 2 , CiO, and CiD u For, from (23) we have PiP 2 1 op 1 .op 2 oe: Hence P,P 2 :P i O= OP 2 : OEr, .-. P l P 2 :P l O=P l P 2 -OP 2 :P i O-OE 1 = P 1 0:P l E l . (24) In the same manner it can be shown that C 1 C 2 :C l O = £7,0: C 1 D l . (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 i} we shall have, by describing a circle on OD 2 as diameter, dOi : C 2 = C 2 0: C 2 D 2 ; and also P 2 P, : 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 ivith it in its motion. Since the moving curve touches its envelope in each of its Mdnoires 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 tho envelopes of any carried curves. How little it is even as yet appreciated in this country will be 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 he 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 OH perpendicular to PQ, then N, by the preceding, is F - 6 a point on the envelope. On OC describe a semicircle ; it will pass through H, and, as in Art. 286, the arc OH = 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 A 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 his Histoirt dt La Gionutrie, p. 6q. 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 hj'po-cycloid. 292. Centre of Curvature of the Envelope of a Carried Curve. — Let a^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 x bi with its envelope. Now, suppose a 2 b 2 to represent a por- tion of the envelope, and let P a be the centre of curvature of tfi&i, 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.O, and P 2 2 . Again, suppose the curve to roll until Oi and 2 coincide ; then the lines Pi d and P 2 2 will come in directum, as also the lines dd and O z C 2 ; and, as in Art. 288, we shall have L & + L C 2 = L P, + L P 2 and consequently oc oc 2 COS 1 OP. op} (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 aibi 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 OP 2 oc 1 I oa = 7777 + 7777 = 7777 ( Art - 2 90); OA Fig. 67. whence, P 2 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 2 , 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>. 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, pj K gg_ and CP 2 as radius, the envelope of the moving lino 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 2 b> of the moving curve a x bi be a right line, the centre of curvature Pi 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 . 292 («). Expression for Radius of Curvature of Envelope of a Rigut Line. — The following expression for the radius of curvature of the envelope of a moving right 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 d 2 p = P + dZ>' (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. On 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 mores 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 A A x 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 x O. Also, since the triangles AOB and A x OB x have their sides respectively equal, we have LAOB = lA x OB 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 Cin the plane, which is rigidly connected yvithAB, will be brought from its original to its new position, d, 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, Fig. 69. lAOA, = lBOB x . 360 Roulettes. because OA = 0A X , AC = A1C1, and the angle OAC, being the difference between OAB and JBAC, is equal to OAiC x , the difference between OA l B i and B1A1C1 ; therefore OC = 0C X , and lAOC = lA,OC x \ and hence L AOA, = L COC x . Consequently the point C is brought to C x by a rotation round through the same angle A 0A 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 B ^ the point of instantaneous rotation. J For, since AB = A1B1, and AA X , F - #2?i, are parallel, we have OA = 0A ly and OB = 0B X . Hence, the figure will be brought from its old to its new position by a rotation around through the angle AOAi. Next, let A A „ 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 1 ; 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, Histoirc de la Geometric, pp. 548-9 : also Bulletin Universel des Sciences, 1830. B M 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 q any position of the moving line, and take y 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- F - section of BO drawn perpendicular to the fixed line, and A drawn perpendicular to the moving line at the fixed point. In general, if AB be any moving curve, and LM&ny 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 B, 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 OPi are the normals at P and Pj, respectively. For the same reason, the normal to the Limacon (Fig. 48, page 331) at any point P is 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 Figure 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 O y OO' q 1/ 3 round the point 0. This rotation will /:.-•''' T 2 bring a certain point O x of the figure to 9j*^-~ coincide with the next centre ' . The next "\~ T i rotation takes place around (/; andsuppose o \>-.. T the point 2 brought to coincide with the o '\-. centre of rotation 0". In like manner, by o'V T ' a third rotation the point 3 is brought to \\ coincide with (/", and so on. By this ' '*_, -. means the motion of the moveable figure F - is equivalent to the rolling of the polygon OOidOi . . . 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. 3°3 this 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 Eeuleaux* 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, 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 XG'^Fto 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 BCA becomes a 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. Fig. 74- 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 P, 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 Pi, through the angle RP x Qi, which the area turns through while P moves to Pj. Moreover, by hypo- thesis, the angles PCPi and RPiQi are in a constant ratio: if this ratio be denoted by m, we have (since PQ = PC) RQ, = wPPi = mQR. Join Q and Qi> then QQi 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 QQ\, the triangles QPO and QiRQ are similar; .-. PQ = mPO, i.e. CP = mPO. Accordingly, the instantaneous centre of rotation is got by cutting off CP PO = — . (29) 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. 36$ 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- oycloids. 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 R r^?Q, BPiQ\ 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 Hence the motion is equi- Fig. 75. valent to the rolling of a circle of 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 3 66 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 G 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 )3 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 9 + j3 cos mB, -an- a f (3 ° } y = a sin ij + p sin mil. * The importance of the epicyclic method of Ptolemy, in representing ap- proximately the planetary paths relative to the earth at rest, has recently been brought prominently forward by 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 oi' 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 epicyclie are obtained by changing the sign of m (compare Art. 284). It is easily seen that every epicyclie admits of a twofold generation. For, if we make mO = , respectively : and let Q h Q 2 , he the corresponding points for the curves Cidi and c 2 d 2 . Take P t O> Q t O~- then, by Art. 290, the points F x and F x lie on the circle of inflexions. Accordingly, the circle which passes through 0, Fi and F iy is the circle of inflexions. Hence, if i^O meet this circle in Gi, and we take RiO* RiR-i = p ~ , the point R> (by the same theorem) is the centre of curvature of the roulette described by R x . 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 aj)^ 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 AG touching at b a circle with centre (3. Then the instantaneous centre is the point of intersection of b(5 and cy. Again, the angle fiOy, being the supplement of the con- stant angle BAC, is given ; and consequently the instanta- neous centre always lies on a fixed circle. * Co/as de geomelrie pour les ecoles des arts el metiers. See also Collignon, Traite de Mecanique Cinemalique, 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 a 0/3 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 /30y 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 give 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 ly p 2 , p 3 the perpendiculars from any point on the sides a, b, c, respectively ; then, by elementary geometry, we have ap-i + bp 2 + cp 3 = 2 (area of triangle) = 2 A. Again, if /»,, p 2 , p 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. (3i) Hence, if two of the radii of curvature be given the third can be determined. 2 B 37Q Roulettes. "We next proceed to consider the conchoid of Nicomedes. 299. Centre of Curvature for a Concboid. — Let A be the pole, and LM the directrix of a conchoid. Construct the instantaneous centre 0, as before : and produce AO until OAy=AO. 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\ 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 A : lies on the circle of inflexions. Hence, to find the centre of curvature of the conchoid described by the moving point P x , produce P^O to meet the circle of inflexions in F lf and take P G i P 1 P 2 = W^ftJ then, by (22), P.F, is the centre of curvature belonging to the point P, on the conchoid. In the same case, the centre of curvature of the curve described by any other point Q:, which is inva- riably connected with the moving line, can be found. !b'or, if we produce QiO to meet the circle of inflexions in E,, and take QiQ? = -^-= ; 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. Spherical Houiettes. — 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 L, connected with the figure A, on a curve A, traced on the sphere. JBut the rolling of i on A is equivalent to the rolling of the cone with vertex standing on L, on the cone with the same vertex standing on A. 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 Eesal; Journal de lEcoU PoIi/technique, 1858, pp. 235, &c. 2 B 2 372 Examples. Examples. 1. If the radius of the generating circle be one-fourth that of the fixed, prove immediately 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 d equation cos + - = o ; hence find when they are real and when imaginary. 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 + ib) cos - c cos (6 + a). v ' a + zb v ' 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 limaQon, 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 the rolling. Am. x cos to + y sin w = b sin 3 — . Sec Walton's P1 + P2 Problems, p. 190 ; also, for a complete investigation of the case where h = — p\ ~ P2 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. From P draw a perpendicular to OQ, and on it take PP> — 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 E and F ; then the right lines OE and Oi^r.re the axes of tho ellipse, in position, and the segments PE and PFare the lengths of its semi-axes (Mannheim, Nouv. 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 rolling 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 Oval. — 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 2y while fx and a are constants, of which we may assume that n 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 r x + fxr 2 = a, and the latter to r x - fxr 2 = a. Now, with F x as centre, and a' as radius, describe a circle. Through F 2 draw any chord DF, join FiD and FiF; then, if P be the point in which F\D meets the inner oval, we have PD = a-r 1 = f xr 2 = fiPF 2 . From this relation the point P can be readily found. Fig. 79- * Thi3 Chapter is taken, with slight modifications, from a Paper published by me in Hermalhena, No. iv., p. 509. 376 On the Cartesian Oval. Again, let Q be the corresponding point for the outer oval r x - fir 2 = a; and we have, in like manner, DQ = fiF 2 Q ; .-. F 2 Q : F 2 P = QD : DP ; consequently, F 2 B bisects the angle PF 2 Q. Produce QF 2 and PF 2 to intersect F X E, and let Pi and Q, be the points of intersection. Then, since the triangles PF 2 D and P X F 2 E are equiangular, we have P X E = fiP 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* ; .'. (i - f S)F 2 P.F 2 Q = F 2 ]y. Also, by similar triangles, we get F 2 P : F^ = F 2 D : F 2 E ; consequently (i - ff) F 2 Q . F 2 P, = F 2 D . F 2 E = const. (2) Therefore the rectangle under F 2 Q and F^P^ is constant ; a theorem due to M. Quetelet. 303. Construction for Third Focus. — Next, draw QF 3 , making lF 2 QF 3 = LF 2 F 1 P l ; then, since the points P„ Fi, Q, F 3 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- A is determined. We proceed to show that F 3 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 /, 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 Histoire de la Gtometrie, note xxi., p. 352. Construction for Third Focus. 377 Now, since L F X F 3 Q = L F X P X F 2 = L F X PF 2 , the triangles F X PF 2 and F X F 3 Q are equiangular ; but, by (4), we have mF.P + IFJ> - nF t F 2 ; accordingly we have mFiFz + IF 3 Q = nF x Q, or nF x Q - IF 3 Q = mF x F 3 ; *. e. denoting the distance from F 3 by r 3 and FiF 3 by c 2y wr, - /r 3 = 7nc 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 mFxQ - IF 2 Q = nFxF 2 gives mF x F 3 - IFJ> = nFxP. 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 2 - mr 3 = ± lc u (6) where c, = F,F 3 . 305. Relation between the Constants. — The equa- tion connecting the constants /, m, n in a Cartesian, which has three points F u 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 y F 2 , &c, the equation is easily reduced to the form Pcy + n% = m 2 c 2 , or PFzF 3 + m 2 F 3 F t + n 2 F x F 2 = o, (7) in which the lengths F 2 F 3 , &c, are taken with their proper signs, viz., F 3 F X = - F x Fs> &c. 306. Conjugate Ovals are Inverse Curves. — Next, since the four points F 2 , P, Q, F 3 , lie in a circle, we have F 1 P.F 1 Q = F l F 2 .F l F 3 = const. (8) Consequently the two conjugate ovals are inverse to each other with respect to a circle* whose centre is F lf and whose radius is a mean proportional between F X F 2 andi^i^. It follows immediately from this, since F 2 lies inside both ovals, that F 3 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& = £PQF 2 = L F 2 Q X P X = l i^Pi ; hence the lines F 3 P and F 3 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 l—o the Cartesian whose foci are F\, F 2 , F 3 , reduces to this circle. Again, if n = o, the Cartesian becomes another circle, whose centre is F 3 , and which, as shall be presently seen, cuts orthogonally the system of Cartesians which have JFi, I' 2 , F 3 for their foci. These circles are called by Prof. Crofton {Transactions, London Mathematical Society, 1866), the Confocal Circles of the Cartesian system. t From the above discussion it will appear, that if the general equation of a Cartesian be written \r + fx.r' = vc, where c represents the distance between the foci; then (1) if, of the constants, A, p, v, the greatest be v, the curve is referred to its two internal foci ; (2) if y be intermediate between \ and /u, 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 A. = fx, the curve is a conic ; (5) if v = A, or v = /u, the curve is a iimac,on ; (6) if onu of the constants A, n, v vanish, the curve is a circle. Construction for Tangent at any Point. 370, reflexions of each other with respect to the axis FiF 2 , and the triangles F X P 2 F 2 and F x PiF 2 are equal in every respect. Again, since L F 2 PF 3 = L F 2 QF 3 = L F 2 F X P X = L F 2 F x P 2y the four points P, P 2 , Pi and P 2 He on the circumference of a circle. From this we have P 3 P . F Z P 2 - P3P1 • P3P2 = constant. Hence, the rectangle under 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 3 P becomes a tangent to the oval, and the length of this tangent becomes constant, being a mean proportional between P 3 Pi and F 3 F 2 . Accordingly, the tangents drawn from the external focus to a system of triconfocal Cartesians are of equal length. This result may be otherwise stated, as follows : — A system of triconfocal 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 triconfocal 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, P 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 Y and P 2 for its internal foci, and F 6 for its external. For, from equation (4), we have for the outer oval dn dr 2 m / — - = o. Us ds 38o On the Cartesian Oval. Hence, if wi and w 2 be the angles which the normal at Q makes with QF^ and QF 2 respectively, we have m sin wi = I sin w 2 ; or sin 2 = / : m. (9) Fig. 80. Again, we have seen at the commencement that / : m - DQ : F*Q ; also, by similar triangles, RQ: BF Z = DQ : F 2 Q = I : m; BQ : EF 2 = sin RQP : sin MQF, ; sin RQF X : sin RQF 2 = l:m. but hence (10) Consequently, by (9), the line RQ is the normal at Q to the outer oval. In like manner it follows immediately that PR 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 Fi, F 2 , F 3 be the three foci, and P the point in question. Describe a circle through P and two foci F 2 and F 3 , and let Q be the second point in which PiP meets this circle ; then the line joining P to P, the middle point of the arc cut oh* by PQ, is the normal. 308. Confocal Cartesians intersect Orthogonally. — It is plain, for the same reason, that the line drawn from P to Pi, 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 2 and F 3 for its in- ternal foci, and Pi 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 tico curves so described cut orthogonally. Again, if RC be drawn touching the circle PRQ, it is parallel to PQ, and hence F % C : F,C = F 2 R : RD = F 2 R* : F 2 R . RD ; but F 2 R . RD = RP 2 ; .-. PLC : PiC = F 2 R 2 : PR 2 = m 2 : I 2 . (11) Hence the point C is fixed. Again CR : F,D = RF 2 : DF 2 = m 2 :m 2 -I 2 ; ^CR = 4^j, (») m 2 - V v 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 V Acad. Roy. de Brux. 1827). 382 On the Cartesian Oval. This construction is shown which the form of two conjugate ovals, having the points F l} F 2 , F z , for foci, is exhibited. Again, since the ratio of F 2 R to HP 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 as struction has been driven in a in the following figure, in an Envelope. — This con- different form by Professor Casey, Transactions Royal Irish Academy, 1869. If a circle cut a given circle orthogonally, while its centre along another given* circle, its envelope is a Cartesian moves oval. This follows immediately ; for the rectangle under F X P and FiQ 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 a 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\Fi as prime vector, we easily obtain, for the polar equation of the curve, (*n 2 - P)r» - 2c 3 [mn - P cos 0) r + c 3 2 (n 2 - P) = 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 6) r 4 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 eqiiation 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 = &Z, where S represents a circle, and L a right line, and h is a constant. 6. Hence show that the curve is the envelope of the variable circle \ 2 kL 4 2\S + k 2 = o. Compare Art. 309. 7. From this show that the curve has three foci ; t. 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. Trove that the inverse of a Cartesian with respect to any point is a bi- circular quartic. (See Salmon, Higher Flane Curves, Arts. 280, 281. 1 10. Prove that the Cartesian r 2 - 2 (a 4 b cos Q)r 4 c z = o has three real foci, or only one according as a - b is > or < e. ( 384 ) CHAPTER 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 dy differentiation we get 2 y — = a, a result independent of b. A second differentiation gives dxj +y d? = °> 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 he of the form f(x, y, a) = o. By differentiation with respect to x, we get dx dy dx The elimination of a between this and the equation/^, y, a) = o leads to a differential equation involving x, y and — , which 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, 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 -7^. In general, for an equation containing n constants, the resulting differential equation contains x. 1/. — , -^ . . . — ; ax dx~ dx" 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 + z 2 = a i . Am. (z i -2y 2 )( — J - $xy — - x 2 = o. \dx / ax 2. Eliminate a and j8 from the equation (y-«)*«p(*-/i). ^, 2 (|) 3 + i ,g = o. 3. Eliminate the constants a and £ from the equation cPy y = acos nx + $ sin nx. Ans. — -f n % u = O. dx 2 4. Eliminate a and b from the equation i ( d A 2 ) 3 (x-af+(y-b) 2 = c 2 . Am. e 2 - ' m This agrees with the formula for the radius of curvature in Art. 226. 5. Eliminate a and /3 from the equation cPy ri A y In \ . ( - J is the most general form of s which satisfies the preceding partial differential equation, and con- sequently 2 = x n ( - J is said to be the solution of equation (1), 2 c 2 388 Elimination of Constants and Functions. where the function is perfectly arbitrary. This latter process, as in the case of ordinary differential equations, comes under the province of the Integral Calculus, and is mentioned here for the purpose of showing the connexion between the integration of differential equations and the formation of such equations by the method of elimination. As another simple example, let it be proposed to eliminate the arbitrary function from the equation z = f(x i + y 2 ). Here P = Jx = 2X ' f '^ + P ^' q=Z ^ = 2y f ,( ^ + ^ 5 hence we get yp - xq = o ; an equation which holds for all values of z whatever the form of the function (/) may be. Examples. 1.2 = (p(az + by). Am. aq = bp. 2. y — bz = (x n + y m ) = z-. 5. z*~ = xy+ (-). 6. x + ^z i + y i + z- = x Un (p (-). ap + bq = I. (x- a)p+(y-0)q = e-y. nx"- 1 q = my m - l p. xzp + yzq = xy. z=px + qy + n^Jx 2 + y 1 + z 2 . 314. Condition that one Expression should be a Function of another. — Let z = = sin 6 cos + cos sin = sin (0 + 0) : this establishes the result required. "We have here assumed that whenever equation (2) is satis- fied identically, V is expressible as a function of v : this can be easily shown as follows : — Since V and v are supposed to be given functions of x and y, if one of these variables, y, be eliminated between them we can represent V as a function of v and x. Accordingly, let V=f{x, v) ; d _Z _ d f_ d £(v) is perfectly arbitrary, this equa- tion must hold whatever be the form of the function <}>'(v), and hence we must have dV 7 dV j dV j dx- d * + df dl/+ dz- dZss0 > y (4) dv , dv . dv , — dx + — dy + — dz = o. dx di/ ' dz Condition that one Expression is a Function of another. 391 Moreover, introducing the condition that 2 depends on x and y, we have dz = pdx + qdy ; consequently, eliminating dx, dy, dz between this and the equations in (4), we get dV dV dV dx ' dy ' dz dv dv dv ■ o. (5) dx 1 dy* dz This agrees with the result in (3). Examples. Eliminate the arbitrary functions in the following cases : — 1. z = (J>(asina: + isiny). 2. s = e a (---)• z x \y xl ._ y 3 »(y) + « i-x(-)• 7. z=(z + y)''(* 2 -j/ 2 ). 8. x t + y 2 +z i = (ax+by+cz). . . dz dz Ant. ocosy- — a cos a* — =0. dx dy dz dz z dx dy a dz dz xy x hv — = — . dx dy z x- — + y 2 — = s z . dz dy dz dz . »— — - „ z— + y — = aVx*+y 2 . dx dy dz dz » V Tx +X d-y =nZ - Ant. (bz—ey) h (ex — az) — = ay - bx. dx dy 392 Elimination of Constants and Functions. 3 1 6. Next, let it be required to eliminate the arbitrary function from the equation F{x,y,z, (u) ,. Jdu du \ hence we obtain two partial differential equations involving x > Vi z j P> <7> #( M ) an d $'(11) ■ Accordingly, if (ic) and '(u) and ip'{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 $"(u) and 4f r {v). Here, again, it is in general impossible to eliminate the six functions between six equa- tions, so that it is necessary to proceed to differentials of the third order : in doing so we obtain four new equations, con- taining two additional functions, <\»"\u) and \j/"{v). After the elimination of the eight arbitrary functions there would remain, in general, two resulting partial differential equa- tions of the third order. 318. There is one case, however, in which we can always obtain a resulting partial differential equation of the second order — viz., where the arbitrary functions are functions of the same quantity, u. Case of Two or more Arbitrary Functions. 393 Thus, suppose the given equation of the form F{x,y y z,${u),ip{u)}=o, (6) where u is a known function of #, y and 2. 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 dFdu _ dFdu (dFdu _ dFdu\ dx dy dy dx \dz dy dy dz) (dFdu dFdu\ +q {dx-dz-dz-Txr°- (7) This equation contains only the original functions (w), -^(w), along with x, 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 i. dF dF „ . dF .,. . du d'(u) and i//(w), we have and dF , dF J dF J -> -r- dx + -— dy + -r- dz = o, dx dy dz >• du T du , dw . dx + — dy + — dz= o \ dx dy dz (9) Eliminating between these equations and dz = pdx + qdy, we get the following determinant : dF dF dF dx' dy' dz du du du dx' dy' dz (io) which, plainly, is identical with (7). 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, 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 0(c) and xp(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. t = x(z) + y^(z). Hera p = (ax + by) + y\p(ax + by). Here p = 4>(ax + by) + a{x

(6) \dy) 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- cordingly, in differentiating — > when x is the independent CliiC Case of a Single Independent Variable. 401 variable we have dl — J = — . However, when x is no longer regarded as the independent variable we must consider the d?/ numerator and the denominator of the fraction — as both dx variables, and, by Art. 15, we get ti(^\ - dxd?y - dy d 2 x d (dy\ _ dx d 2 y - dy = *d* + x T*> therefore x* — | = — f - — , + ay = °' 7. Change the independent variable from x to z in the equation . d*y 2dy dx* z dz * Two Independent Variables. 403 321. Two Independent Variables. — We will next consider the process of transformation for two independent variables, and commence with the transformations intro- duced by changing from rectangular to polar coordinates in analytic geometry. In this case we have x ■ r cos 0, y = r sin ; (7) and therefore r' = x % + y z , tan = -. (8) Accordingly, any function V of x and y may be regarded as a function of r and 0, and by Art. 98 we have dV_dVdx dVdy^ dd ~~dx dd + dy d6 I >• (9) dV_dVdx dVdy | dr dx dr dy dr J But, from (7), dx dx . n dy . f. dy . N --cose, -=- r ^0=-y, l = sm6, a —;(io) hence we obtain dV dV dV , s df =X dy-- y dx~> (I1) dV dV dV . , r-r- =« r +2/T-. (12) dr dx dy These transformations are useful in the Planetary Theory. Again, we have efa? rfr dx dd dx . . >> ( J 3) dV _dVdr_ dVdB I dy dr dy dO dy J 2 d 2 404 Change of the Independent Variable. But from (8) we have therefore dr x a dr • a -7- = - = cos 0, -r- ■ Bin 0, dx r dy (14) dO in y sin0 dB — = -cos 2 0— = , — = dx x % r dy cos 9 r ' (15) dV Q dV shiBdV dx dr r dO * (16) dV . r.dV cosOdV — = sin 0-7— + -r=-. dy dr r d\) (17) The two latter equations can also be derived from equa- tions (11) and (12) by solving for — and — . d 2 V d?V 322. Transformation of — — and — — -. — Since for- dx* dy 3 mula (16) holds, whatever be the form of the function V, we have d . . . d sin 9 d . . (*)-oos0— (*) — -g^), dx dr d9 where # stands for any function of x and y. On substituting dV — instead of v idv \_cpv dx 2 + dy 2 ~ dr 2 + r dr + r 2 dV 2 ' ( ' (PV d 2 V (PV 323. Transformation of y-z- + -r-r- + Coordinates. dx 2 dy 2 dz 2 to Polar Let the polar transformation be represented by the equa- tions x = r sin cos <£, y**r sin sin $, z = r cos ; also, assume p = r sin 9, and we have x = p cos <£, y = p sin <£ ; 1. / «n ^^ ^^ ^T irfF 1 d 2 V hence, by (18), — - + — - = -7-3- + - — + — -7-7. 406 Change of the Independent Variable. Again, from the equations p = r sin 9, z ■ r cos 0, we have in like manner cPV d 2 V_d 2 V idV i d*V dp* + dz 2 ~ dr 2 + rdr + p 2 dd 2 ' Accordingly, (PV cPV d 2 V_d 2 V idV j_d 2 V idV i d*V da? + dtf + dz 2 ~ dr 2 + P dp + p 2 dtf + r dr + r 2 d(T ' But by (17) we have dV _ . Q dV cos 9 dV m dp dr r dd ' ,, . itfF ia?F cot0c?P therefore - — - - — + — — -^-. p dp r dr r do Hence we get finally r h fr h f » dxdz dzdy* Orthogonal Transformation. 409 (T'V_ 2 (PV , 2 fflV „ 2 o?V , d 2 V dZ*~ C dx* +C dy 2 +C aV +2CC dxdy „d 2 V „, d 2 V + 2CC - — - + 2C C -r— r-. dxdz dzdy 327. Orthogonal Transformations. — If the transfor- mation be such that x 2 + y 2 + z 2 = X 2 + F 2 + Z% we have a 2 +a' 2 + a" 2 =i, b 2 + b' 2 + b" 2 = 1, c 2 + c' 2 + c" 2 = 1. (21) ab+ab' + a"b"=o, ac + ac + a"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 a", we get on addition, by aid of (21) and (22), X = ax + dy + a"z, In like manner, if the equations (20) be respectively multiplied by b t b\ b'\ we get F= bx + b'y + b"z ; similarly, Z= ex + c'y + c"%. If these equations be squared and added, we obtain a 8 + b 2 + c l = 1, a' 2 + b' 2 + c 2 - 1, a" 2 + b'" 1 + c" 2 = 1. (23) ad+bb'+cJ-o, aa"+ bb"+ cc"= o, (ta"+b'b"+c'c"=o. (24) Hence in this case, if the equations of the last Article be added, we shall have d*V d 2 V (PV_(PV d*V (PV . . ~dx T + dy T *dz 2 ~d~X 2 + dY 2 + dZ 2 ' ^ 5) 410 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 Geometry of three dimensions ; and equation (25) shows that, in transforming from one rectangular system to tt> & c -> can De deduced from these : dx" dy 1 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 ^ (x, y) is transformed into a function of X and Y : let the latter be denoted by i(X, Y), and we have {x,y) = l {X, Y). Again, let x' and / be transformed by the same substitu- tions, i. e., x' = aX'+b Y', y = a'X' + b' Y'; then since x + kx =a{X + kX') + b(Y+kY r ), and y + ky' = a'{X + kX') + b'( Y+ kY), it is evident that {x + kx, y+ki/) = ^{X^kX\ Y+kY'). Hence, expanding by the theorem of Art. 127, and equating like powers of k, we get x dx- +y dy =x dx +Y dr (29) * d^ +2 ^dxTy +y ay~ X dX> + 2XY dXdY +Y dY" &o. &c. (30) 4 1 2 Change of the Independent Variable* Accordingly, if u represent any function of x and y, the expressions denoted by ,d ,d\ ( ,d ,d\* M dx y dyj ' \ dx * 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 quantio 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 + zx'y' cos w + y' 2 = X' 2 + zX'T oosQ + Y'\ (31) where 0; and Q denote the angle between the original axes and that between the transformed axes, respectively. Multiply (31) by X, and add to (30): then denoting (x, y) by u, and i(X, Y) by U we get fd 2 u \ ^ f d 2 u \ (d 2 u d % u \ , ,( d 2 u \ , (d 2 u \ Now, suppose A assumed so as to make the first side of this equation a perfect square, it is obvious that the other side will be a perfect square also. The former condition gives d?u ^\fd 2 u _\ / (d* u d* u d* u \ or A* siira> + A I — + -y-r - 2 - — r cos w \dx 2 ay* dxdij ) tfudhi f cPu \»_ efo 2 % 2 \dxdyj Accordingly, we must have at the same time x, • ,~ ^( sin 2 £2 .(33) Consequently, if u be any function of the coordinates of a point, the expressions d 2 u (Pu f d 2 u Y d 2 u cPu (Fu dx 2 dy 2 \dxdyj .dx 2 dy 2 dxdy sin 2 W sin 2 w are unaltered token the axes of coordinates are changed in any manner, the origin remaining the same. In the particular case of rectangular axes, it follows that d 2 u dxdy* d 2 u dtf d 2 u + A, ~ d 2 u dxdz (x), find — • Ans. /W'(s) +/>){<*>'(*) }*• 2. If y = *■(*), < =/(«), « = (*>(*), find the value of -j^ • Ans. F'{t)f{u) *"(*) + {<*>'(*)} 2 {/"(«) J"(0 + C/») s -F "W }• 3. Change the independent variable from x to % in the equation x* — : — 2«# 3 — - + a 2 y = o, where a; = -. dx 2 dx z d 2 y 2 {71+ 1) dy « az* z az M*> y> M . v ) = o; ' m ) =m(m+ i)r m ~ 2 . (3) Hence, from (2), we have X? 2 (r m V) = r"'V 2 r+ m{m + i)r m ~ 2 V + zm / dV dV dV\ . . r m ~ 2 [x-z- + y—— + z — — 1. (4) V dx J dy dz J KH/ Moreover, if V be a homogeneous function of the n th de- gree in x, y, 2, we get, by Euler's theorem of Art. 98, y 2 ( r «» V) = r m V 2 V + m(m + in + 1 ) >-"' -2 F". (5) Solid Harmonic Functions. 4 1 9 333. Solid Harmonic Functions. — Any homogeneous function in x, y, z which satisfies equation ( 1 ) is called a solid spherical harmonic function, and frequently a solid harmonic. We shall denote a solid harmonic of the n th degree by V n ; in which the degree n may be positive or negative, in- teger or fractional, real or imaginary. It is evident that any constant multiple of an harmonic is also an harmonic of the same order. From (5) it follows that a solid harmonic of the n th degree satisfies the equation V 2 (r m V n ) = m [m + 2 n + 1 ) r m ~* V n . (6) Vn Hence we see that if V n be a solid harmonic, -=— r is also a solid harmonic, whose degree is - (n + 1). Again, from (2) we see that - is a solid harmonic of the . 1 degree - 1 . Also it can be readily shown that - is the only function of r that satisfies equation (1). For by (19), Art. 323, we can transform that equation into d ( dV\ 1 d ( . Q dV\ 1 d*V d?{ } ^J + ^eTe{ smd -dj) + ^e^ = - W Hence, if V be a function of r solely, we must have — ( r 2 -— ) = o. This gives V in the form - + b. dr \ dr J r In like manner, if V be a function of the angle solely, d 2 V it must satisfy the equation — — = o : this leads to V, = a$ + b. Hence we observe that tan -1 f - ) is a solid harmonic, of the degree zero. Again, if V be a function of solely, we have Te( sia6 T) = O. 1 J 2 E 2 420 Spherical Harmonic Analysis. Hence we see that log f tan - j satisfies the equation — — J is a solid harmonic. 7* -\~ CU f* *4" II In like manner, log and log are also solid har- ° r - x r - y monies. It is readily seen that

d m (?- 2 "- x ) , ... 4. Hence prove that - — — 1S a harmonic function. (* 3 + y 2 )* For, let Fo = tan -1 ( - ) ; then, since y = x tan , it can be shown, as in Art. 46, that £.( ta i)=<--)"L- sin n (*» + y-)- t**'* - sin t?d> Hence is a solid harmonic, as also any function derived from («- + y 2 ) 2 " it by differentiation. »•- r„ 5. Prove that w = — is a solution of the differential equation 2 (2ti + 3) V 2 « = F„. 334. Complete Solid Harmonies. — A solid harmonic that is finite and single valued for all finite values of the co- ordinates is said to be a complete harmonic. It can be proved, by aid of the Integral Calculus, that every complete solid harmonic is either a rational integral function of the coordi- nates, or is reducible to one by multiplication by some power of r. Assuming this, it follows that the number of indepen- dent complete harmonics of degree 11 is 211 + 1, when n is positive. For it is readily seen that the number of terms in V n , a rational homogeneous function of the n th degree in x, y, 2, is (n + 2) (n + 1) . . . , , J . ,_. . : and also the number of terms in V* V n is 2 11 (ft J J ; hence, since V 2 V n - o identically, we must have n (n — 1 ) , . . . T , linear equations connecting the coefficients in r H ; consequently the number of independent constants is (n + 2) (n + 1) n(n- 1) , or 2n + 1. 422 Spherical Harmonic Analysis. It can now be shown that every complete harmonic can be deduced from - by differentiation. For the solid harmonic r J dx>dy k dz l {x* + y 1 + z 2 )*' when the differentiations are performed, is readily seen to be a fraction of which the numerator is a homogeneous func- tion of the degree n, and whose denominator is -^ , where n = k +J + I. If this function be represented by -^, the numerator V n , by (6), is also a solid harmonic. We can now show that the number of independent har- monics of degree n that can be thus derived is in + i. For, since dz*\r)~ \dx 2 + dy 2 J\r)' we see that dz *m [j.) -{- 1 r \^- 2 + jj,) { r )> in which f — + — J can be expanded by the binomial theorem as if — and — were algebraic quantities, and the resulting ciOu (xy differentiations of - taken. r Hence, if I be even, we have d* m /i\ ' d>+ k /(P_ d*_ Wi\ dx> dy k dz l \r) m ^ *' dx> dy k \dx 2 + dy* ) \rf and, if / be odd, d j+k*i /,\ w d j*k /jz d 2 \ 2 d dx J dy k dz T \r)~^ l '* dxi dy k \dx z + d\f) dz \ r)' Spherical and Zonal Harmonics. 423 Accordingly, in the former case we get a number of terms dP+i /i\ each of the form . , . -■ I - , where p + q = n ; and in the dx p dy q \rj latter, terms of the form - J — ( - ] | , in which p + q = n - 1 . Now there are p + q + 1, or n + 1 terms in the former case, and n in the latter. Hence there are in + 1 independent forms, as was to be proved. 335. Spherical and Zonal Harmonics. — If a solid harmonic V n be divided by r n , the quotient may be regarded as a function of the two angular coordinates, or spherical surface coordinates, 9 and . Such a function is called a spherical surface harmonic of the degree n. Hence, if V n = r n Y n , then Y n is a spherical harmonic of the 11 th degree. It is obvious that the general spherical harmonic of the first degree is of the form a cos 6 + b sin cos + c sin 6 sin 0, where «, b, c are arbitrary constants. Also, the general expression for Y 2 can be written down readily (see Ex. 1, Art. 333). Again, by (19), Art. 323, we see that Y n satisfies the differential equation 1 df. Q dY n \ 1 d*Y n . ._ .. ■HITS an ^ + " T Tfl7J + nn+lF » s0 ' ( 8 ) Bin dif \ dU J sin 2 dp This equation admits of a useful transformation : for let H = cos 0, then, since we get d ( dY n \ 1 d 2 Y n . . _ , . TA (I -^ifc\ + -nr?w n{ ] " ■ (9) Again, if a spherical harmonic be a function of 6 solely, it is called a zonal harmonic. Hence, if P n be a zonal harmonic of the n' h order, it must satisfy the equation 5{('-»fl$j + »(» + 0A-a (,o) 424 Spherical Harmonic Analysis. When n is a positive integer, the value of P n can he readily represented hy a finite series. For since, by hypothesis, P« is a function of the n th degree in ^u, we may assume P„ = S (a mf i m ). dP Hence — — = 2 [ma m /t* m_1 ) ; .'. — (i-/u 2 ) -7- ? =2m(m-i)cf m /u m ^-Sm(m+i)a m /Li m . Substituting in (10), and equating the coefficient of fi m to zero (since the result must vanish identically), we get (m + i) (m + 2) fl w « = - (n - m) (m + n + 1) a m . Hence, observing that the highest power of /u is n, we have w(w-i) - 2(2H- i) and we may write - i) 2.4(2»-l)(2W-3) ^ (») 2(2»-l) 2.4(2^- 1)(2«-3) where « n is an arbitrary constant. This is the general form of a zonal harmonic of integer positive degree; and we see that two zonal harmonics of the same degree can only differ by a constant multiplier. It can be shown independently of the above that ( — J (/x 2 - i) n satisfies the equation (10). In order to prove this we shall assume u = n* - 1, and write the symbol D for — ; then we have to prove that D (uL nn (u n )) -n(n+i)2) n (n n ) - o. Zonal Harmonics. 425 (lit Now, observing that — = 2/*, we get, by Leibnitz's theo- rem of Art. 48, D nH (w n+1 ) = D"* 1 (u . u n ) = wD n+1 (u n ) + 2 (» + 1) fxD n (u n ) + n{n + i)D n - 1 (u n ). Again, since D(w fl+1 ) = 2(n + i)/xw n , we have also B»»(u» ¥1 ) = z(n + i)D H (fiu n ) = 2 {n + 1 ) ixB n (u n ) + 2n[n+i) D n ~ l (u n ) . Equating these values of D rt+1 (w w+1 )» we get wD n+1 (u n ) =n(n+ i)B n - x (u n ) ; hence D { uD Ml (u n ) }-n(n+i)D n (u n ) =0. (12) Consequently D n [u n ) satisfies the equation in question. Hence we infer that The student can verify, by direct differentiation, that this expression differs only by a constant factor from the value of P n found in (1 1). It is usual to assume that P„ is that value of the pre- ceding expression which becomes unity when p = 1. To find this value, we have 7 d \ n ' 1 f d \ n_2 " 2 Hw w , -i) ,rt +2»(»-i)fj-J (M 2 -ir, by Leibnitz's theorem. 426 Spherical Harmonic Analysis. Now it is readily seen that f — j (jx 2 - i ) n_1 = o when n = i ; hence, when fx = i, we have (|)V-o"-<|)""V-o- = 2'»(»-l)(|J"V-l)"-'.&0- Consequently, when n = i, and we have P » = ^(|)V-»». C3) The foregoing can be readily shown in another manner. ( d\ n For 2 |_w ( — ) (jj? - i) n is the coefficient of h n in the expansion of (i -2fxh + h 2 )~ l (see Ex. 6, p. 155). Again, ( (x - a) 2 + y 2 + z 2 f* = (r 2 - larp + a 2 )" 4 1/ r r 2 \~i . r in which we suppose « > r. But v 2 {( ; r-a) 2 +/+s 2 }" i = o; — ) (jti 2 - i) M satisfies equation (10), &o. hence The functions Pi, . . . P n are usually called Legendre's Coefficients. Complete Spherical Harmonics. 427 Examples. 1. If fi =» I, prove that P» = 1 for all values of n. 2. If /j. = - 1, prove that P„= (- i) n . 3. If n < 1, show that the series Pi + P> + . . . + P„ + . . . is convergent. 4. Prove the relations p 35m 4 - 3°M 2 + 3 p 6 3m 5 - 7Qm 3 +i5m P 4= , P 5 . 5. Prove the equations — (P„ + i - Pn-l) = (2» + l)P«, fl/x (» + I) P^i - (2fl + i) ilP. + «Pn-l = O. 336. Complete Spherical Harmonics. — From Art. 334 it follows that a complete spherical harmonic Y n of the n th order, when n is an integer, contains in + 1 arbitrary con- stants. Its value can be expressed by aid of the corresponding zonal harmonic P„, as we proceed to show. Since Y n is in this case a rational integer function of sin f) cos (j>, sin 9 sin cp and cos 0, we may suppose it expressed in a series of sines and cosines of multiples of , . , . , . . ,, that — = - s cos s0, we obtain, on equating to zero tne d

+ 2 P m + 2 (s + i) A *D* 1 P„ + * (s + 1) B>P m -n (» + 1) D*P m = o; but by Leibnitz's theorem the first three terms are equivalent to B Hl (uDP n ^ ; whence the equation becomes D» x {uDP m ) - n(n + i)D s P m = o. But this equation follows immediately from (10) by differentiating s times with respect to fx. Accordingly the expression satisfies equation (14), and hence oos^O* a -i)^0P w satisfies (8). t vi «P(sin *0) , . „ In like manner, as = - s 2 sin stf>, the expression ad> sins^-iW — \P n also satisfies the same equation. Accordingly, equation (8) is satisfied by the expression {A s cos s<}> + B s sin s) {ji* - 1) 2 1 — J (P„), (17) in which A„ and B s are arbitrary constants. Laplace's Coefficients. 429 This expression is called a Tesseral Surface Harmonic, and is said to be of the degree n and order s. If we give all integer values to s from 1 to n, the com- plete spherical harmonic T n can be written in terms of Tesseral harmonics as follows : — *=» i rf* p Y n = A P n +2 (J,eos*0 + £,sinty)(/t4 2 -i)*— 2, (18) in which ( — j (/x* - i) n may be substituted for P„ if neces- sary. This equation contains the proper number 211 + 1 of arbi- trary constants, and consequently may be regarded as a general expression for a complete spherical harmonic of in- teger positive degree. There is no difficulty in showing by differentiation that f — j (/x 2 - i) n differs only by a constant from (n - s) In - s - 1 ) 2(211-1) + (n-s)(n-s-i){n~s-2)(n-s-3) + & ^ 2 .4 .{211- 1) (2*1-3) Hence that part of T n depending on the angle scj> may be written ( I - fxf L n -° - 2( 2 "~_ S i) V"' 3 ' 2 + &C «) ( A * C0S S< P + B * sin S( P) • This agrees with the general expression given by Laplace {Mecanique Celeste, tome m., chap, ii., p. 46). 337. Laplace's Coefficients. — It is immediately seen that the expression — — — ; 7- — ; 7—: satisfies the {(x-x)-+(y-ijy+{z-zy)* general equation (1), as also the corresponding equation - 0'). If P and P' be the points whose coordinates are xyz and x'y'z\ respectively, then i i / . r r 2 \-i i / . r r 2 V* = _( I _ 2 A_ + _) = _( I _ 2 A- / + - r2 . PP' r\ r r 2 / r \ / r'\ Accordingly, if (i -2\h + h 2 )~^ = i +LJi + LJ?+ . . . +L n h n + . . ., we have i i LS LS' L n r'» pp> = r + ~S r+ ~r r + '" + ~r^~ + "' when r > r ' and i i L y r L 2 r z L n r n , , pp ? = 7 + "7 r + "7 r+,, - + 7^ 1+ "- when r < r ' Hence, since v 2 ( -pp, ) s o, we must have V 2 ( -^ ) s o, and also v 2 (i n r n ) ■ o. From this we see that L n is a spherical harmonic of the degree n, and that it satisfies the equation d I . „ dL n ) i d 2 Z n The functions L x , L 2 , . . . L n are called Laplace's Coef- ficients , after Laplace, to whom their introduction into analysis is due. The value of L n may be deduced from that of P n in ( 1 1 ) or (13), by substituting nfi + and 6 i dx dy ' dz (0 was styled by Jacobi a functional determinant. Such a determinant is now usually represented by the notation d (u, v, w) d(x, y, s)' and is called the Jacobian of the system u, v, to with respect to the variables x, y, z. In the particular case where 11, v, w are the partial diffe- rential coefficients of the same function of the variables x, y, z, their Jacobian becomes of the form given in Art. 331, and is called the Hessian of the primitive function. Thus the determinant in Art. 331 is called the Hessian of u, after Hesse, who first introduced such functions into analysis, and pointed out their importance in the general theory of curves and surfaces. 2 F 434 Jacobians. More generally, if y ti y 2y y 3 . . . y n be functions of x i} x 2) .r, . . . x ny the determinant dxi dyi dx 2 '' dyi ' dx n dy% dx? dy 2 dx 2 ' dy% djh dxx dyn dx 2 ' dy n ' dx n is called the Jacobian of the system of functions y u y a , . . . y n with respect to the variables x u x 2 , . . . x n ; and is denoted by d(yi, t/2, • • . y n ) . . d[X\, x 2y . . . x n ) Again, if y Y , y 2 , . . . y n be differential coeciffients of the same function the Jacobian is styled, as above, the Hessian of the function. A Jacobian is frequently represented by the notation <%i, 1/2, • • • y n ), the variables x x , x 2 , . . . x n being understood. If the equations for y ly y if . . . y n be of the following form, yi = Jz\%\) X 2) %t)y yn—Jn\ x \, #2> • • « %n)i Examples. 435 it is obvious that their Jacobian reduces to its leading term, viz. dxx dx-i ' ' ' dx n ' This is a case of the more general theorem, which will be given subsequently (Art. 343)- Examples. 1. Find the Jacobian of y\, y<>, . . . y,„ being given yi = i-xi, y 2 = x\{i - x 2 ), yz = x\xi{i-xz) . . . y n = X\ X2 . . . X n -l{l - Xn). AtlS. J= (- l)»Xi n - 1 X2 n ~' i . . . x,,-i. 2. Find the Jacobian of x\, xz, . . . x„ with respect to 0i, 02, • • . 0«, being given x\ = cos 0i, xz = sin 0i cos 02, #3 = sin 0i sin 02 cos 03, . . . x n = sin 0i sin 02 sin 03 . . . sin 0,,-i cos 0„. d(xi, X2, . . . x n ) . . . „ , „ . „ Am. -7— = (- i)» sin»0i . sin'' 1 2 . . . sin 0„. d(9 u 02, • • . 0„) 339. Case where the Functions are not Indepen- dent. — If 1/1, 1/2 • • • Vn be connected by a relation, it is easily seen that their Jacobian is always zero. For, suppose the equation of connexion represented by F{i/i, y%> . . . y n ) - o ; then, differentiating with respect to the variables x x , x,, . . . x,„ we get the following system of equations : — dFdfr dFdyt (!F_d]U = dt/i dxi df/2 dx r dij n dx x 2F2 43 6 Jacobians. dFdyi dFdfr dF dy n _ dy x dx 2 dy 2 dx 2 ' ' dy n dx 2 ' dFdy x dFdy 2 dF dy n _ dy x dx n dy 2 dx n ' ' ' dy n dx n ~ ' , .. . L . dF dF dF whence, eliminating -r-, — - . . . — — , we eret dy x dy 2 dy n d(?/» P*, • • • Vn) , , d[x x , x 2) • • • Xfi) The converse of this result will be established in Art. 344 ; 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. 3H- 340. Case of Functions of Functions. — If we sup- pose u Xf w 2 , u 3 to be functions of y lt y 2 , y 3 , where y x , y 2 , y 3 are functions of x x , x 2 , x 3 ; we have du x dui dy x du x dy 2 du x dy 3 dx x dy x dxi dy 2 dx x dy 3 dx x du x _ du x dy x du x dy 2 du x dy 3 dx 2 dy x dx 2 dy 2 dx 2 dy 3 dx 2 du x _ du x dy x du x dy 2 du x dij 3 dx s dy x dxi dy 2 dx 3 dy 3 dx 3 &c. Case of Functions of Functions. 437 Hence, by the ordinary rule for the multiplication of de- terminants, we get du x dui diii d%x dx 2 dx 3 du 2 du 2 du-i dxS lx 2 dx 3 du 3 du 3 du 3 dx' dx 2 dx 3 diii diii du x di/i dy 2 dy 3 du 2 du 2 du 2 dt/i dy 2 dy 3 du 3 du 3 du 2 di/i di/2 dij 3 ... x n ) d{yi, y 2 , . . . y n ) ' d (,r : , x 2 , *n) This is a generalization of the elementary theorem of Art. 19, viz. du du dy dx dy dx' Again, d{y x , y 2 , . . . ?/„) d(xi, x 2 , . . . x n ) d(xi, x 2 , . . . x n ) d(yi, ?/•>, . . . y n ) This may be regarded as a generalization of the result dx dy dy ' dx (8) 43« Jacobians. 341. The Jacobian is an Invariant. — In the parti- cular case of linear transformations we have a system of equations as follows: — t/i = «!#i + a 2 x 2 . + a n x ni y-i, = brfi + b 2 x 2 . + b n x n , y n = h x x + l 2 x 2 . . . • + 'n %n' In this case «! a 2 . . . a n (1(1/,, y 2 .... y n ) h b 2 ...b n d[X\, x 2 . . . . x n ) k 1% . . . l n This determinant is a constant, and is called the modulus of the transformation. Accordingly, in linear transformations the transformed Jacobian is equal to the original Jacobian multiplied by the modulus of the transformation. In the case of orthogonal transformation (see Art. 327) the modulus of the transformation is unity, and accordingly the Jacobian is unaltered by such a transformation. 342. Jacobian of Implicit Functions. — Next, if m, v, to, instead of being given explicitly in terms of x, y, s, be connected with them by equations such as Fx(x, y, s, m, v, w) - o, F 2 (x,y,z,u i v,w) = o, F 3 (x,y,z,u,v,w) = o, then it, v, w may be regarded as implicit functions of x, y, z. In this case we have, by differentiation, dF, dF\du dF x dv dF x dw _ dx du dx dv dx dw dx dFi dF x du dF x dv dF x dw dy du dy dv dy dw dy dF 2 dF 2 du dF 2 dv dF 2 dw dx du dx dv dx dw dx Jacobian of Implicit Functions. 439 Hence we observe, from the ordinary rule for multiplica- tion of determinants, that dF\ dF\ dF\ du ' dv ' die (IF, dF\ dF\ du' dv' die dF s dF 3 dF\ du ' dv ' dw This result may he written d(F,F 2 ,F 3 ) d{u,v,w) du dv dw dx' d? dx du dv dw dy' Jy' dy du dv die dl> d~z' 'dz' dF\ dF\ dF\ dx' dy' dz dF 2 dF 2 dF\ dx' dy' dz dF\ dF 3 dF* dx ' dy' dz d(F„F 2 ,F 3 ) (9) d(u, v, w) ' d{x, y, z) d(x, y, z) The preceding can be generalized, and it can be readily shown by a like demonstration that if y ly y 2y y 3 , . . . y n are connected with Xi, x 2 , x 3 . . . x n by » equations of the form F x (x t , x 2 . . . x n , p u y 2 . . . y n ) = o, F 2 (x x , x 2 . . . x n , y„ y 2 . . . y n ) = o, (10) F n (x 1} x 2 . . . x n , y x , y 2 . . . >/„) = o, ) we shall have the following relation between the Jacobians : d(F lt F 2 ,... F n ) d(y u y 2 , . . . y n ) *l(y» Va • • • Vn) Accordingly, d(y» y* -(-«; d(F ly F 2 ,. . .F n} d{x u X 2 , . . . Z n ) d(X\ y x 2) . . . x n ) d\Xi y x 2 , d{F, F 2 ,... F n ) d[F lt F 2 , d['Ji, !h, Fn) «h) :») 440 Jacobians. 343. Again, the equations connecting the variables are always capable by elimination of being transformed into the following shape fa (a?x, X2 t . . . x w y,) = o, \, #2, • • • i d(f>n d

n dyi dy 2 dy s ' ' ' dy n In like manner, d{ n dx n d(x u x 2 , . . . x„) v XJ d$ x d$ 2 dyi dy 2 d(j> n ' dy n (12) (13) 344. Case where Jaeobian vanishes. — We can now prove that if the Jacobian vanishes the functions y Xi y 2 , . . . y„ are not independent of one another. Case where Jacobian Vanishes. 441 For, as in the previous Article, the equations connecting the variables are always capable of being transformed into the shape given in ( 1 2), and accordingly, if J(y ly y 2 , . . . y n ) = o, we must have dty i d

i must not contain xi ; and accordingly the corresponding equation is of the form i(x i+1 , . . . x n , y u y 2 , . . . f/i) = o. Consequently, between this and the remaining equations, 01+1 = o, 0,- +2 = o, . . .

. . . y n alone. This establishes our theorem. Also, it follows that if the Jacobian does not vanish, the functions are independent. 345. In the particular case where y x = Fi(x lt Xo, . . . x n ), y? = F.i(x lf x if . . . x n ), we have y n = F n {y ly y,, . . . y„_i, x n ), d{yx, y 2> ■ ..y») = d]h d]fr <*!/» d(x l} a* 2 , .. . x n ) dx x ' dx 2 ' ' dx n ' (14) It may be observed that the theory of Jacobians is of fundamental importance in the transformation of Multiple Integrals (see Int. Calc. y Art. 225). 44 2 Jacobiam. Examples. i. Find the Jacobian of y h y 2 , . . . y„ with respect to r, 0i, 62, . . . 6n-\, being given the system of equations — y\ = r cos $i, yz = r sin 0i cos 2 , yz = r sin 0i sin 02 cos 3 , . . . yn — f sin 0i sin 62 • • • sin 0n-i. If we square and add we get yi 2 + y£ + . . . y» 2 = »- 2 . Assuming this instead of the last of the given equations, we readily find /= r*"i sin"" 2 0i sin"- 3 02 . . . sin 0„_2- 2. Find the Jacobian of y\, y2, ■ ■ • yn, being given yi = #1(1 - aj 2 ), yj = afi«8(i-*s) • • • y„_i = a;iar2 . . . #„-i(i -x„), yn = X\%2 • • ' Xn. Here, yi + y 2 + . . . y n = x\, and we get d(i/u yi, ■ ■ -y») d{x\, X2, . . . X n ) = Xl n - 1 X2 n ' i ■ . ■ Xn-\. 346. Case where a Relation connects the Depen- dent Variable*. — If y x , y 2 , . . . y n , which are given func- tions of the n variables cc u x 2 , . . . x n , be connected by an independent relation, F[yi> y*> . . • p n ) = o, (15) 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 x , y 2 , . . . y n _ x as functions of x 1} x 2 , . . . x n . x . In this case it can be shown that (IF_ d{i/i,?h, • • . y n -\) f j!/n d(y„ y 2 , . . . //„) ^ d{x lf x 2 , . . . x n ^) dF d(x 1} x 2 , . . . x n ) d,X n Case where a Relation connects the Dependent Variables. 443 For, if we regard *„asa function of x l} # 2 , &o. we have dx^ 1 ' dx, dx n dxS fl^i ' dx, + dx n aW™' Also, from equation (15), dF dFd^_ dF dFdx n _ dxi dx n dx! ' dx 2 dx n dx 2 ' dF dF dF_ Again, let A, - |, X, -£ . . . A M -^ aXfi aXn uXfi Hence /.(*) -& - A.&, ' W -&-X* Ac. &c. Accordingly, substituting in the Jacobian d(y„ y 2 , . . . y n _ x ) it becomes d(Xi, x 2 , . . . av^i)' otei * dxn dx 2 2 dxn ' ' dx n .i n ~ dx n flfoi ' c/.r n ' d# 2 " dir n ' ' dx n _ } dx n If this determinant be bordered by introducing an addi- tional column as in the following determinant, the other 444 Jacobians. terms of the additional row being cyphers, its value is readily seen to be or i dF dx n dy x dih dxx dx 2 d]h d]fr dxi dx 2 dy n -x dy^x dxi ' dx 2 dF dF dx^ dx 2 Again, we have dyx dx n dy* dx n dy*. dxx ' dx 2 ' dx n K A 2 , . . i dyi dx^ dyx dx 2 dyi dx n dy 2 dx? dy 2 dx 2 dy 2 dx n dy n -i dy n -i dtjn-i uX n dF dx n dF _ dF dyx dF dy 2 dF dy n dxx dyx dxx dy 2 dx dy n dx x ' dF dy n dF__dF_dy 1 dFdy 2 dx 2 dyx dx 2 dy 2 dx 2 dy n dx 2 * Substituting these values in the last row of the preceding Case where a Relation connects the Dependent Variables. 445 the theorem is established, since we readily find that the de- terminant is reducible to dyi dxi dy x dx 2 dyx dx n dF dijn dF dy 2 dx? dy-i dy 2 dx n dx n * dj/n dxx djjn dx 2 ' dy n dx n (16) It may be well to guard the student from the supposition that this latter determinant is zero, as in Arts. 339 and 344. The distinction is, that in the former cases the equation F{yi, y 2 , ■ . . y n ) = o, connecting the y functions, is deduced by the elimination of the variables x ly x 2 , . . . x n from the equations of connexion ; whereas in the case here considered it is an additional and independent relation. 446 Jacobians. i . Being given find the value of the Jacohian 2. Find the Jacohian Examples. yi = r sin 0i sin 02, t/2 = r sin 0i cos 62, j/3 = r cos 0\ sin 63, yi = r cos 0i cos 03, ^(.Vl, #2, */3, */4> <*(»-, ei, &2, e 3 )' d(x, y, z) Am. r 3 sin 0i cos (?i. rf(r, 0, ) , heing given a; = r cos cos V 1 - w 2 sin 2 0, where m 8 + »* = 1 . r 2 (w 2 cos 2 $/2| • • • «/n) .4««. 4. if we make d [X\, X2, ■ . . X„) «i 2/i =— » u «2 «,. j/2 = — , • . . y n - — prove that it hecomes u, Ml, M2, ... M M du du\ du,2 d't-u dxi dx\ dx\ ' ' dx\ 1 du du\ dU2 dlt n «,.+! di~2 dxz dx2 ' ' dx2 du du\ dua dii n dx,i dXn dXn ' ' dx n This determinant is represented by the notation K(u, u\, . . . w„). 5. If a homogeneous relation exists between 11, u\, . . . w,„ prove that K{ll, Ml, . . . n n ) = o. 6. In the same case if y\, 2/2, •• • yn possess a common factor, so that yi = mil, &c, prove that J(y, 2/2, •• • Vn) = 2U n J(l( h M 2 , . . . Kn) - M"" 1 E{u, Mj, M 2 , . . . M„). ( 447 ) CHAPTER XXV. GENERAL CONDITIONS FOR MAXIMA OR MINIMA. 347. Conditions for a Maximum or Minimum for Four Variable*;. — 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 independent variables. We shall commence with the case of four independent variables. Proceeding as in Art. 162, it is obvious that the problem reduces to the consideration of a quadratic expres- sion in four variables which shall preserve the same sign for all real values of the variable. Let the quadratic be written in the form It = tf niTi 2 + «22#2 2 + «33Z"3 2 + a^x? + 2a 12 XiX 2 + 2a u XiX A + 2a ii x 1 x i + 2a 23 x 2 x 3 , + 2d u X 2 Xi + 2d 3i X 3 X iy ( I ) in which a U) a n , a 22 , &c represent the respective second differentia] coefficients of the function, as in Art. 162. We shall first investigate the conditions that this ex- pression shall be always a positive quantity. In this case fl n , «22» #33, &c. evidently are necessarily positive : again, multiplying by a n , the expression may be written in the following form : — a u u=(a n x 1 +a 12 x 2 +a 13 x 3 +a H x i y + (a u a 22 -a r f) x 2 2 + [a n a3s-a lz z )xs + («ii«« - tfu 2 )^ 2 + 2 (fl u a 2i - a u a l3 )x 2 x 3 + 2 {a n a 2i - flu a u )x 2 Xi+ 2 (a n a u -a n a n ) x 3 Xi. (2) 448 General Conditions for Maxima or Minima. Also, in order that the part of this expression after the first term shall be always positive, we must have, by the Article referred to, the following conditions : — and #11 #22 ~ #12" > O, (rt„ Gr 22 - «i 2 2 ) (« n « 33 - d\z) - (#11 #23 - #12#1 3 ) 2 > O, «11«22-«12 2 , #11#23-#12#13) #11 #24 ~ #12#14 #11#23~" #12#13) #11#33~#13") #11 #34 — #13#14 #11 #24 — #12 #14} #11 #34 — #13 #14) #11 #44 — #14~ > O. (3) (4) (5) To express the determinant (5) in a simpler form, we write it as follows #11) #12) #13) #14 O, #n#22 — #12 ) #14#23 — #12#13) #11 #24~#12 #14 O, #11#23~"#12#13) #11#33 — #13 ) #11 #34 — #13 #14 O, #n#24-#12#14) #11#34-#23#14) #11 #44 ~ #14* (6) Next, to form a new determinant, multiply the first row by #12) #i3, #i4) successively, and add the resulting terms to the 2nd, 3rd, and 4th rows, respectively ; then, since each term in the rows after the first contains a n as a factor, the determinant is evidently equivalent to #11) #12) #13) #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) #12, #13 #12) #22) #23 #13) #23) #33 > O. Maxima and Minima. 449 Hence we conclude that whenever the following condi- tions are fulfilled, viz. (7 n >o, flu, a. fli2, #2 >o, a n , a. a i #12, #22) #23 #13, #23, #33 >o, #11) #12) #13} #14 #12) #22) #23} #24 #13) #23) #33) #34 >o, (8) #14) #24) #34) #44 the quadratic expression in (i) is positive for all real values of x u x 2 , .r 3 , .r 4 . Accordingly the conditions are the same as in the case (Art. 162) of three variables, x u x 2 , 2*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 values of the variables. The last determinant in (8) is called the discriminant of the quadratic function, and the preceding determinant is derived from it by omitting the extreme row and column, and the others are derived 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 now 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. 348. Conditions for n Variables. — If the notation already adopted be generalized, the coefficient of ay* is denoted by a rn and that of x,-x m by 2a rm . In this case the discriminant of the quadratic function in n variables is a id a 12> a 13) a in #12) #22) #23) • • • #2n #13) #23) #33) > • • #3« #1M) #2M) #3rt) • • • #nn 2 G (9) 45° General Conditions for Maxima or Minima. 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 func- tion by fl u , and it is evident that a n u = (a n x v +« 12 a? 2 + . . . ain%n) 2 + (a n a n -a l f)%?+ . . . + (#11 ttnn- a ln 2 ) Xn + 2 (fl u « 23 - « 12 « 13 ) X % X 3 + + 2 (fl n a rn ~ #ir #1«) %r %n + &C. In order that this should be always positive it is necessary that the part after the first term should always be positive. This is a quadratic function of the n - i variables x lt x 2 , . . . x n . Accordingly, assuming that the conditions in question hold for it, its discriminant must be positive, as also the series of determinants derived from it. But the discriminant is #11 #22 — #12 » #11 #23 — #12 #13, #U #23 — #12 #13j #11 #33 — #13 > #11 #21 ~ #12 #H, #11 #34 ~ #13 #11, #11#2?J — #12#lM> #ll#3» — #13#1/|> . . #11#2« "~ #12#ln • • #ll#3n — #13 #m . . #n#m — #u#i» . . a n a nn — a Xn 10 Writing this as in (6), and proceeding as in Art. 347, it is easily seen that it becomes #n" #n, #1 #13, #23, #33, #1« #2n #3>i> (»: i. e. the discriminant of the function multiplied by a n nJ *. Hence we infer, that if the principle in question hold for n - 1 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. Maxima and Minima. 45i We conclude finally that the quadratic expression in n variables ; s always positive whenever the series of determi- nants tfn, #11, #12 a,*, Oii #11, #12, #13 #12, #22, #23 , • " • #13, #23, #33 #12, #12, #22, #2» #l>i, #2n, , (12) are all positive. According as the number of rows in a determinant is even or odd, the determinant is said to be one of an even or of an odd order. If the determinants of an even order be all positive, and if those of an odd order, commencing with a u , 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 ; *'. e. if any of the coefficients, a 22 , a 33 , . . a nn , be taken instead of a n as the leading term in the series of determinants. If the determinants in ( 1 2) be denoted by Ai, A 2 , A 3 , . . . A», it can be seen without difficulty that, whenever none of these determinants vanishes, the quadratic expression under consideration may be written in the form a,^ 2 + ^- 2 u.? + ~ 3 m + . . . + -£2- u n \ (13) A H . 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 continuations of signs in the series of determinants in (12). It is easy to see independently that the series of conditions in (12) are necessary in order that the quadratic function under consideration should be always positive ; the preceding 2 g 2 452 General Conditions for Maxima or Minima. 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. 349. Orthogonal Transformation. — 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 (see Art. 341). In this case, if Xi, X 2 , X 3 , . . . X n denote the new linear functions, we have x 2 + x 2 2 + . . . + x* = Xx 2 + X., 2 + &c. + X n 2 = V; also, denoting the coefficients of the squares in the transformed expression by a u a 2 , . . . a n , JJ= «ii#i 2 + a-, 2 x 2 + . . . + a nn x r 2 + . . . + 2a u x x x 2 + za^XxXr + . . . = a l X 1 2 + a 2 X 2 2 + . . . + a n X, 2 . Hence, equating the discriminants of U - A V for the two systems, we get #11 — A, (?u, . . . a m #12, a 22 — A, ... a 2n #135 ^23, . . . «3fj A = 7 w»~A = {a 1 -X)(a 2 -X)...(a„-\). (14) Conditions of Maxima or Minima in General. 453 Accordingly, the coefficients a u a 2 , . . . a n are the roots of the determinant A. Moreover, in order that the function U should he always positive or always negative for all real values of the variables x u Xo, . . . a*„, the coefficients ai,a 2 , .. . a„, must be all positive in the former case, and all negative in the latter ; and con- sequently, in either case, the roots of the determinants in (14) must all have the same sign. For a general proof that the roots of the determinant A are always real, and also for the case when it has equal roots, the student is referred to Williamson and Tarleton's Dynamics, Chapter XIII. 454 Miscellaneous Examples. Miscellaneous Examples. r . If o, 0, y be the roots of the cubic s 3 + px 1 + qx + r = o, show that dp da dq da dr da dp d? dq dr dfi dp dy dj_^ dy dr dy = (7 - 0X0 - «)(« - 7)- 2. Being given the three simultaneous equations \{x U X% X 3 , Xi) = O, 2(X\, X 2f X 3 , Xi) = O, = o, develop y r in terms of a by Lagrange's Theorem, given x = r cos 6, y = r sin 9, transform 3 I + (I)} 1 d 2 y into a function of r and 6, where 6 is taken as the independent variable. ( Ant. Idr )■}' + 2 Ui 15. Apply the method of infinitesimals to find a point such that the sum of ite distances from three given points shall be a minimum. If pi, pi, p3 denote the three distances, we have dpi + dp% + dps = o : suppose dp\ = o, then d(pi + p3) = o, and it is easily seen that pi bisects the angle be- tween pz and p3, and similarly for the others ; therefore &c. 16. Eliminate the circular and exponential functions from the equation . -1 y = g«»> ». 17. 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 may be reduced to the investigation of a locus. 45t> Miscellaneous Examples. 18. If two pairs of conjugates, in a system of lines in involution, be given by the equations m = ax 2 + zbxy + cy"- = o, it = ax 3 + 2b' xy + c'y 1 = o, show that the double lines are given by the equation du du' du du «,,.,,. —-- — — - = o. (balmon s Comes, Art. 342 ) dxdy dydx v t oh 1 T . Xi x% X„-i 19. If Ml = -, W 2 = — , ll n -\ = %n Xn X n where x\, x%, . . . x„ are connected by the relation Xi 2 + X 2 2 + Xy + . . . + Xn 2 = 1, prove that the Jacobian d{it\, «2, • • • m»-i) _ 1 d{X\, Xi, . . . #„-]) Xn' 1 ' 1 ' 20. If the variables y\, y 2 , ... y,, are related to x\, x 2 , ... x„ by the equations y\ = a\Xi+ doxo + . . . + a n x„, yt = hx\ + i 2 ^2 -f . . . + 6,1^1,, and if we have also y n =hxi + hx 2 + . . . + l»x H , Xi 2 + *2 2 + . . . + *» 8 = I, ?/i 2 + y-i 2 -r . . . + y,, 2 = 1, prove that the Jacobian <*Q/i> yi, ■ • ■ y»-0 _ y» rf(a-i, *2, . . . *»-i) #„* 21. Prove that the equation ry 2 — 2sxy + tx 2 — px + qy - z cPz may be reduced to the form - + z = o by putting x = w cos v, y = u sin t\ 22. Investigate the nature of the singular point which occurs at the origin of coordinates in the curve x i — 2ax 2 y - nx ij 2 + a 2 y 2 = o. Miscellaneous Examples. 457 23. Investigate the form of the curve represented by the equation y = »"■' 24. How would you ascertain whether an expression, V, involving x, y, and z, is a function of two linear functions of these same variables ? Am. The given function must be homogeneous ; and the equations dV dV_ dV_ dx dy dz must be capable of being satisfied by the same values of x, y, z: i. e. the result of the elimination of x, y, and z between these equations must vanish identi- cally. 25. If y = (x 2 ), prove that ^ = (2x)» *>('•) {x 2 ) + »(« - 1) (2*)»- a 0(»- 1 ) (**) «(»- I) («-2) (» - 3) , . . . «,,,.. „ + — '— — (2x)"- i +I ) u = xn dx-« This can be easily arrived at from the preceding by the method of mathematical induction : that is, assuming that the theorem holds for any positive integer n, prove that it holds for the next higher integer (» + 1), &c. 32. Find - + — I - J in terms of r, when r 2 = a 2 cos 20. Am. 33. If u = (x 2 + y 2 + z 2 )l, prove that d*u d*u d*tt d i u d*u d l u dx 1 + dy* + 1z x + * dx- dy 2 + 2 dy 2 "dz 2 + 2 dz- dx 2 ~ °' 34. If z = — -, and

. cos"* 1 * d*-^ ^ ~ ' d?»z , I . 2 . 3 . . . 2« . cos {in + I) * . COS 2 " ' A — = (- 0" A — r » dy 2 " v ; x-"' 1 3« 4 d 2n " l z , s ^ 1 . 2 . 3 . . . (2w+ 1) sin (2«+ 2)(f> . cos 2 " ,; < Miscellaneous Examples. 459 35. If u be a homogeneous function of the n th degree in x, y, z, and «i , «2, «s, denote its differential coefficients with regard to x, y, z, respectively, while *n> Mil) &c- in like manner denote its second differential coefficients, prove that Kll, UlS, «13, Ml Mil, «12, M13 "21, «M, «23, «2 mm «31, «32, «33, M3 n — 1 «2i, M31, «22, "32, M«3 M33 «1» «2, «3, O 36. If u be a homogeneous function of the n th degree in x, y, z, tc, show that for all values of the variables which satisfy the equation u = o we have Mil, M12, MlS, Ml KM, M 22 , U22, tt 2 M31, M32, M33, M3 Ml, M2, M3, O (» " I) Mil, M12, M13, Uu «21, M22, M23, M24 «31, MJJ» M33, «3« ««, M42, M«, Uu 37. If x + A be substituted for x in the quantic , m(m-i) ., „ c n + naix"- 1 + — - a 2 x n - i + &c. + a„, aox" 1.2 and if a'o, a'i, . . . . «V . . . . denote the corresponding coefficients in the new quantic, prove that lb-™- 1 ' It is easily seen that in this case we have r(r-l) « r = «r + r« r -lA + ffr-2\ 2 +&C . . . + 0!oA r ; .*. &C. 38. If <£ be any function of the differences of the roots of the quantic in the preceding example, prove that Id d d d \ I «0 3— + 2«i — + 3«2 — + . . . + Mrtn-l -r— M> = o. \ da\ daz da 3 da n ) This result follows immediately, since any function of the differences of the roots remains unaltered when x + \ is substituted for x, and accordingly d

(x) is a rational function of x. 42. Show that the reciprocal polar to the evolute of the ellipse - + V - - 1 with respect to the circle described 011 the line joining the foci as diameter, has for its equation a* P -5 + -J - «• s 2 jr 43. If the second term he removed from the quantic (tfo, «i, «2, . . . a„)(x, y) n by the substitution of * y instead of x, and if the new quantic be denoted #0 by (Ao, o, Az, A3, . . . A H ) (x, y) ; show that the successive coefficients! A%, A3 . . . A„ are obtained by the substitution of a\ for x and - a for y in the series of quantics («o, «i, «2) (z, y), («o, ffi, «2, "3) (x, y), • • • («o, «i, . . . an) (*, y). 44. Distinguish the maxima and minima values of 1 + 22 tan -1 x I +x* a'x*~ + 2b' x + c' «• Kyas ■^T^TT7-' provethat 1 rfy (ae — i 2 ) y 2 -f (ac' + a'c — 2W) y + a'e' — V 2 dx (ab') x 1 — (ca') x + {be') Miscellaneous Examples, 461 46. If IX + mY + nZ, VX + m'Y + n'Z, l"X + m"Y + ri'Z, be substituted for x, y, z, in the quadratic expression ax 1 + by 1 + m 3 + 2rfyz + 2ezx + zj'xy ; and it* a', b , e', d', e', f be the respective coefficients in the new expression ; prove that »'j ft < f '» a, f, /, v, d', = whenever /> *, e\ ), 6 and (p being the angles at which PQ cuts the curve. 72. In the polar equations of two curves, F (r, w) = o, f (r, oi) = o, if P in be substituted for r, and tin for w, prove that the curves represented by the transformed equations intersect at the same angle as the original curves. W. Roberts, Liou/'ille' s Journal, Tome 13, p. 209. Miscellaneous Examples. 465 This result follows immediately from the property that — - is unaltered by the transformation in question. 73. 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. 74. 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. 75. 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. 76. 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. Koberts' Memoir. This method is an exten- sion of the method of inversion. 77. If on 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 curve which is similar to the negative pedal of the given curve, taken with respect to the fixed point as pole. 78. If 2 U = ax 2 + zbxy + eg 2 , 2f= a'x 2 + zb'xy + c'y 2 , and dU dU dx* dy dV dV dx* dy = AU 2 + 2BUF+ CV 2 , find A, B, C. 79. Prove that the values of the diameters of curvature of the curve y 1 =/{x) at the points where it meets the axis of x are/'(a),/'(/8), . . . . if a, j8, ... be the roots of f(x) = o. Hence find the radii of curvature of y 2 = (x 2 - m 2 ) (x — a) at such points. 80. 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. 8 1 . 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. 82. 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 2 H 466 Miscellaneous Examples. U 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. 83. 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. (Camb. Math. Trip., 1875.) 84. A curve rolls on a straight line : determine the nature of the motion of one of its involutes. (Prof. Crofton.) 85. 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. 86. 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, U'eber eine besondere curve drifter Masse und vierten grades, Crelle, 1857.) This is called Steiner'' s Envelope, and the theorem can be demonstrated, geometrically, as follows : — Let Pbe any point on the circumscribed circle of a triangle ABC, 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 Ton 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). 87. The envelope of the tangent at the vertex of a parabola which touches three given lines is a three-cusped hypocycloid. 88. The envelope of the parabola is the same hypocycloid. 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. Town send, Educ. Times. Reprint. 1866. Ferrers, Quar. Jour, of Math., 1866. Serret, Nouv. Ann., 1870. Painvin, ibid., 1870. Cahen, ibid., I075. On the Failure of Taylor's Theorem. 467 On the Failt/re 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 Lemons 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 Series does not give the Development of the Function. — The existence of Taylor's Series supposes that the function f[x) and its differential coefficients/'^), /"(#), &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. ry„\ Suppose, for example, that/(x) is of the form — , m being any positive number, and F(x) a function of x which does not become either zero or infinite when x — a. If, conformably to our rules, - - — be developed in a series of posi- tive powers of h, all the terms would become infinite when we make x = a. At F(a + h) 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 f(x) 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_ (x — a)»-i (x-a)"* f"(x) ss m{m -l)(x- a)" 1 " 2 Vx - b + ' -> '—, s/x-b 4(2; - b)* 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. 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. Bernouilli'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, ^33- Catenary, 288, 321. Cayley, 259, 266. Centre of curve, 237. Centrode, 363. Change of single independent variable, 399- Change of two independent variables 403, 410. Chasles, on envelope of a carried righ 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, 454. Circle of inflexions in motion of a pi ane area, 354, 358, 367, 374. Complete Solid Harmonics, 421. Conchoid of Nicomedes, 332, 361. centre of curvature of, 370. Concomitant functions, 411. Condition that Pdx -f Qdy should be 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. Descartes, on normal to a roulette, 336. ovals of, 375. Differential coefficients, definition, 5. successive, 34. 470 Index. 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, 1 20. partial, 113, 406. of a function of two variables, 115. of three or more variables, IX S- applications in plane trigono- metry, 130. in spherical trigonometry, J 33- successive, 144. of (x + h, y + k, z + I), 159. Family of curves, 270. Ferrers, on Bobillier's theorem, 369. on Steiner's envelope, 466. Folium of Descartes, 333. Functions, elementary forms of, 2. continuous, 3. derived, 3. successive, 34- examples of, 46. partial derived, 113. elliptic, illustrations of, 136, 138. Graves, on a new form of tangential equation, 339. Harmonic polar of point of inflexion on a cubic, 281. Iluygens, approximation to length of circular arc, 66. Hyperbolic branches of a curve, 246. Hypocycloid, see epicycloid. Hypotrochoid, see epithelioid. Indeterminate forms, 96. treated algebraically, 96-9. treated by the calculus, 99, ct seq. Infinitesimals, orders of, 36. geometrical illustration, 57. Inflexion, points of, 279, 281. in polar coordinates, 303. Intrinsic equation of a curve, 304. of a cycloid, 338. of an epicycloid, 350. of the involute of a circle, 301. Index. 47* Inverse curves, 225. tangent to, 225. radius of curvature, 295. conjugate Cartesians, as, 378. Involute, 297. of circle, 300, 358, 374. of cycloid, 356. of epicycloid, 357. Jacobians, 433-45- Lagrange, on derived functions, 4, note. on limits of Taylor's series, 76. on addition of elliptic integrals, r 36- . . . theorem on expansionin series, 151. on Euler's theorem, 163. condition for maxima and. minima, 191, 197, 199, 202. La Hire, circle of inflexions, 354. on cycloid, 373. Landen's transformation in elliptic functions, 133. Laplace's theorem on expansion in series, 154. coefficients, 429. Legendre, on elliptic functions, 137. on rectification of curves, 233. coefficients of, 426. Leibnitz, on the fundamental principle of the calculus, 40. theorem on the 11 th derived func- tion of a product, 51. on tangents to curves in vectorial coordinates, 234. Lemniscate, 259, 277, 296, 329, 333. Limaeon, is inverse to a conic, 227, . .331, 334, 349, 361, 372. Limiting ratios, algebraic illustration of, 5- trigonometrical illustration, 7. Limits, fundamental principles, II. 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 4- ci/" 1 of — ~ tt- T"5» l66 > r 77- ax- + 2b xy + pre- 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; Lagrange's condition, 191, 197. for functions of three variables, 198. of n variables, 199, 449. 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, 467. Newton, definition of fluxion, 10. prime and ultimate ratios, 40. expansions of sin a - , cos a;, sin" 1 *, &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, 452. 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. Partial differentiation, 113, 406. Pascal, lima