fT^K) ^*Y^A " f0/ i Digitized by tine Internet Archive in 2008 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/elementarytreatiOOedwaricli d-^ d . \f4C*^s/iCLx . Sihj, itjr-' ^TuC.s DIFFERENTIAL CALCULUS. AN ELEMENTARY TREATISE ON THE DIFFERENTIAL CALCULUS APPLICATIONS AND NUMEROUS EXAMPLES JOSEPH EDWARDS, M.A. KORMERLY FELLOW OF SIDNEY SUSSEX COLLEOK, CAMBRIDGE SECOND EDITION, REVISED AND ENLARGED OFTK f UNIVER- ^^UFpR> |C n b n MACMILLAN AND CO. AND NEW YORK 1892. [All Rights Resei-ved.] ^ ^i^ i^^*- ^ (- -a V 7 n ^ GLASGOW ; ydnteb at the Sttibereitg ^ress bs ROBERT MACLEH08E, 153 WEST NILE STREET. t \ PREFACE TO THE SECOND EDITION. In issuing a second edition of the present volume it has been found desirable to enlarge it considerably beyond its original limits. The necessity for this has arisen partly from the increased requirements of the class of students for whom the book was originally written, and partly from the expressed opinion of many teachers that its sphere of useful- ness might be thereby extended. Chapters have been added on Maxima and Minima of Several Independent Variables, on Elimination, on Lagrange's and Laplace's Theorems, on Changing the Independent Vari- able, and one giving a short account of the principal properties of the best-known curves, which may be con- venient for reference. A number of isolated theorems and processes, which do not find a convenient place elsewhere, have been put into a separate chapter entitled Miscel- laneous Theorems. Considerable additions have been made to some of the original articles, and others have been rewritten. Many additional sets of easy examples, specially illus- trative of the theorems and methods proved or explained in the immediately preceding book work, have been inserted, in the hope that a selection from these will firmly fix in the mind of the student the leading principles and pro- cesses to be adopted in their solution before attacking the generally more difficult problems at the ends of the chapters. In a text-book of this character there will not be found much that is new or original, the object being to present to the student as succinct an account as possible of the 1^7224 vi PREFACE. most important results and methods which are up to the present time known, and to afford sufficient scope for practice in their use. To attain this object many treatises on this and allied subjects have been consulted, and my acknowledgments of assistance are therefore due to many authors. More par- ticularly I am indebted for much information to the admirable works of Cramer, Gregory, De Morgan, I'Abb^ Moigno, Serret, Frenet, Bertrand, Frost, Todhunter, William- son, and Salmon, whose labours have done so much to develope and extend the principles and applications of the subject. I have consulted a large number of university and college examination papers set in Oxford, Cambridge, London, and elsewhere, and many of the examples given have been extracted from them. Such papers clearly define the extent of knowledge expected from students by the large body of distinguished scholars who from time to time are engaged in conducting these examinations, and the present work has been constructed to meet these requirements as far as possible. My thanks are due to several friends and correspondents who have kindly sent me valuable suggestions and lists of errata occurring in the first edition. JOSEPH EDWARDS. 80 Cambridge Gardens, North Kensington, W. February, 1892. CONTENTS. PRINCIPLES AND PROCESSES OF THE DIFFERENTIAL CALCULUS. CHAPTER I. Definitions. Limits. ARTS. PAGES 1-2 Object of the Calculus, 1 3-9 Definitions, 1-4 10-12 Limits. Illustrations and Fundamental Principles, . . . . 4-6 13-16 Undetermined Forms, 6-7 17-22 Four Important Undetermined Forms, ..... 8-9 23 Hyperbolic Functions, 10-11 24-34 Infinitesimals 11-17 CHAPTER IL Fundamental Propositions. 35-36 Tangent to a Curve 20-22 37-39 Differential Coefficients, Examples, Notation, . . . . 23-25 40-42 Aspect as a Rate-Measurer, ....... 26-27 43-51 Constant, Sum, Product, Quotient, 2^-31 52-55 Function of a Function, 32-34 56-59 Inverse Functions, 34-37 CHAPTER m. Standard Forms. 60-64 Differentiation of a;", a*, log a?, » . 38-39 65-70 The Circular Functions, 40-41 71-78 The Inverse Trigonometrical Functions, 42-44 79 Interrogative Character of the Integral Calculus, ... 45 80 Table of Results to be remembered, ...... 46 81-82 Cases of the Form m", 47 83 Hyperbolic Functions. Results, 48 84-85 Illustrations of Differentiation, 48-50 vii Vlll CONTENTS. ARTS. 86-87 88-92 93-95 96-99 100-105 106-108 109 CHAPTER IV. Successive Differentiation. PAGES Repeated Differentiations, 57-58 — - as an Operative Symbol, 58-61 dx Standard Results and Process, 61-65 Leibnitz's Theorem, 66-69 Some Important Symbolic Operations, .... 69-70 Successive Differentiation of -F(a;2), etc. , .... 71-72 Note on Partial Fractions, 72-74 CHAPTER V. Expansions. 110 Enumeration of Methods, ....... 78 111 Method I.— Algebraical and Trigonometrical Methods, 78-80 112-120 Method II.— Taylor and Maclaurin's Theorems, . 80-83 121 Method III. — Differentiation or Integration of known Series, 84-85 122 Method IV.— By a Differential Equation, .... 86-89 123-128 Continuity and Discontinuity, 91-93 129-133 Lagrange-Formula for Remainder after n Terms of Taylor's Series, • ^3-96 134-136 Formulae of Cauchy and Schlomilch and Roche, . . 96 137-138 Application to Maclaurin's Theorem and Special Cases of Taylor's Theorem, 97 139 Geometrical Illustration of Lagrange-Formula, . ' . . 97 140-142 Failure of Taylor's and Maclaurin's Theorems, . . . 98-99 143 Examples of Application of Lagrange-Formula, . . . 100 144-145 The Rule of Proportional Parts. Interpolation, . . 101-103 146 On the Expansion of ^, 103 147 71* Differential Coefficient of a Function of a Function; .104-105 148-149 Bernoulli's Numbers, 105-106 CHAPTER VI. Partial Differentiation. 150-152 Meaning of Partial Differentiation, 153-155 Geometrical Illustrations, 156-158 Differentials, 159-163 Total Differential and Total Differential Coefficient 164 Differentiation of an Implicit Function, 165-170 Order of Partial Differentiations Commutative, . 171 Second Differential Coefficient of an Implicit Function, 172 An Illustrative Process, 173-174 An Important Theorem, . . • . 175-180 Extensions of Taylor's and Maclaurin's Theorems, 181-188 Homogeneous Functions. Euler's Theorems, 189-190 Laplace's Equation. Conjugate Functions, 112-113 113-116 116-118 118-120 120-121 121-124 125 126 127 128-130 130-135 135-137 CONTENTS. IX APPLICATIONS TO PLANE CURVES. CHAPTER VII. Tangents and Normals. ARTS. 191-196 Equations of Tangent and Normal in various Forms 197 Tangents at the Origin, ...... 198-202 Geometrical Results. Cartesians and Polars, 203-205 Polar Subtangent, Subnormal, etc., 206-207 Polar Equations of Tangent and Normal, . 208-210 Number of Tangents and Normals from a given point Curve of the 7i*^ degree, 211 Polar Line, Conic, Cubic, etc., 212-215 Pedal Equation of a Curve, 216-219 Pedal Curves, 220 Tangential-Polar Equation, . 221-225 Important Geometrical Results, 226 Tangential Equation, . 227-232 Inversion, .... 233-235 Polar Reciprocals. to a PAGES 143-147 148 149-154 154-156 157 158-159 160 161-163 163-167 167 168-171 171-172 172-175 176-177 CHAPTER VIII. Asymptotes. 236-238 To find the Oblique Asymptotes, 182-183 239-241 Number of Asymptotes to a Curve of the »*■'• degree, . 184 242 Asymptotes parallel to the Co-ordinate Axes, . , . 185 243 Method of Partial Fractions for Asymptotes, . . 186 244 Particular Cases of the General Theorem, .... 186-187 245-246 Limiting Form of Curve at Infinity, . . . 188-189 247-248 Asymptotes by Inspection, 190-191 249 Curve through Points of Intersection of a given Curve with its Asymptotes, 191-192 250 Newton's Theorem, 192 251-253 Other Definitions of Asymptotes, 193-194 254-257 Curve in general on opposite sides of the Asymptote at opposite extremities. Exceptions, .... 194-195 258 Curvilinear Asymptotes, ....... 195 259-261 Linear Asymptote obtained by Expansion, .... 196-199 262-264 Parabolic Branches, . . ' 199-202 .265-269 Polar Equation to Asymptote, 203-205 270 Circular Asymptotes, 205-206 CHAPTER IX. Singular Points. 271-273 Concavity and Convexity, . 274-275 Points of Inflexion and Undulation, 276-285 Analytical Conditions, 211. 212 213-21» CONTENTS. ARTS. 286-287 288-291 292-294 295-296 297 298-299 300-301 302 303 304 305-310 311-313 314 315-316 318-319 320-321 322-324 325-328 329-332 333 334-335 336 337 338 339 340-345 346-349 350-352 353 354-356 357 358-359 360 361 362-363 364-367 368-371 372 373 374-375 Multiple Points, Double Points, To examine the Nature of a specified point on a Curve, To discriminate the Species of a Cusp, Singularities on the Keciprocal Curve, . . . . Singularities of Transcendental Curves, . . . . Maclaurin's Theorem, with regard to Cubics, Points of Inflexion on a Cubic are Collinear, Number of points necessary to define a curve of n^^ degree. Maximum Number of Double Points, Homogeneous Co-ordinates. Polar Curves, The Hessian. Number of Points of Inflexion, . Pliicker's Equations, ........ Deficiency. Unicursal Curves, ...... CHAPTER X. Curvature. Angle of Contingence. Average Curvature, Curvature of a Circle. Radius of Curvature Formula for Intrinsic Equations, Formulae for Cartesian Equations, Curvature at the Origin, Formula for Pedal Equations, Formulae for Polar Curves, Tangential- Polar Formula, . Conditions for a Point of Inflexion, List of Curvature Formulae, Co-ordinates of Centre of Curvature, Involutes and Evolutes, Intrinsic Equations, Contact. Analytical Conditions, Osculating Circle, Conic of Closest Contact, Tangent and Normal as Axes ; x and y in terms CHAPTER XL Envelopes. of s, PAGES 220-221 221-224 224-227 227-232 232-233 234-235 236-237 237 237 238 238-242 243-244 245 246-247 252-253 253-254 254-255 255-257 258-260 262 263 264 264 264-265 266-267 268-271 272-274 275-279 279-280 280-283 283-284 293 Families of Curves ; Parameter ; Envelope, The Envelope touches each of the Intersecting Members of the Family, 293-294 General Investigation of Equation to Envelope, . . . 294 Envelopeof .4\-4-2^X-fO = 0, 295 The c-Discriminant Singularities, 297-299 Several Parameters. Indeterminate Multipliers, . . 300-303 Converse Problem. Given the Family and the Envelope to find the Relation between the Parameters, . . . 303-304 Evolutes, 305 Pedal Curves, 305-308 CONTENTS. xi CHAPTER XII. Curve Tracing. ARTS. PAGES 376-379 Nature of the Problem ; Order of Procedure in Cartesians, 314-317 380-381 Examples, 317-322 382-383 Newton's Parallelogram, 323-325 384 Order of Procedure for Polar Curves, 326-327 385-387 Curves of the Classes r = a sin nd, r8mnd = a, . . . 327-329 388 Curves of the Class r" = a" cos w6'. Spirals, . \ . 330-332 CHAPTER XIII. On Some Well-Known Curves. 389 Introductory, 337 390-401 The Cycloid, 337-342 402-403 The Trochoids, 343 404-423 Epi- and Hypo-Cycloids and Epi- and Hypo-Trochoids, . 343-349 424-431 Limafon, Cardioide, Trisectrix 349-352/ 432-434 Curve of Sines, Harmonic Curve, Companion to the Cycloid, 352-353 435-439 Cissoid of Diodes, 353-355 440 Witch of Agnesi, ........ 355 441 Folium of Descartes, 355-356 442-443 Logarithmic and Probability Curves, . . . ih . ■ 356 444-445 Catenary and Tractrix, ....... 357-358 446-448 The Conchoid of Nicomedes, 359-360 449 Equiangular or Logarithmic Spiral, ..... 360-361 450 Parabolic Spirals, 361 451 Spiral of Archimedes, 361-362 452 Reciprocal or Hyperbolic Spiral, . . . ' . . . 362-363 453 TheLituus, 363-364 454 Cotes's Spirals, 364-365 455 Involute of a Circle, 365-366 456 Evolute of a Parabola, ....... 366 457 Evolute of an Ellipse, 366-367 458 Cassini's Ovals. Lemuiscate of Bernoulli. .... 367-368 459 Cartesian Ovals, - 368-369 460-463 The Quadratices of Dinostratus and Tschirnhausen, . . 369-370 APPLICATION TO THE EVALUATION OF SINGULAR FORMS, MAXIMA AND MINIMA VALUES, ETC. CHAPTER XIV. Undetermined Forms. 464-466 Forms to be discussed, 373 467 Algebraical Treatment, 374-376 468-471 Form^, 376-379 Xii CONTENTS. ARTS. PAGES 472 Form Ox 00, 319 473-475 Form^, 379-382 00 476 Form 00 -00, 382 477 Forms 0", oo", 1", 382-383 478 A Useful Example, 38a 479 p- of Doubtful Value at a Multiple Point, .... 383-384 CHAPTER XV. Maxima and Mii^ima — One Independent Variable. 480 Elementary Methods 388-391 481-483 The General Problem Definition, 391-393 484-487 Properties of Maxima and Minima Values. Criteria for Discovery and Discrimination, 393-396 488 Analytical Investigation, 401-404 489 Implicit Functions, 405-407 490-491 Several Dependent Variables, 407-409 492 Function of a Function, 409-411 493 Singularities, 412-413 CHAPTER XVI. Maxima and Minima— Several Independent Variables. 494-495 Preliminary Algebraical Lemma, 420-421 496 To search for Maximum and Minimum Values, . . . 421-422 497-498 To discriminate between them. Two Independent Vari- ables. Lagrange Condition, 422-423 499-500 Geometrical Explanation, 424-425 501 A Ridge of Maxima and Minima, 425-426 502-503 Discrimination in the Case of three or more Independent Variables, 426-427 504 Lagrange's Method of Undetermined Multipliers, . 427-429' CHAPTER XVII. Elimination. 606-507 Construction of a Differential Equation. Elimination of Constants, 432-436- 508 Elimination of Functions of Known Form, .... 436-437 509 Elimination of Functions of Arbitrary Form. Genesis of Partial Differential Equations, 438-43^ 510 Condition of Dependence of Functions of two Independent Variables, 439-440 511-514 Elimination of Arbitrary Functions of Known Functions, . 441-444 CHAPTER XVIII. Expansions (Continued from Chapter V.). 515 Arbogast's Rule, 448-449 516-517 Lagrange's Theorem, 449-453 CONTENTS. xiii ARTS. PAGES 518 Laplace's Generalization of Lagrange's Theorem, . . 453 519 Burmann's Theorem, 453-454 CHAPTER XLX. Change of the Independent Variable. 521-523 Total Diflferential Coefficients, 458-460 524 The Operator ^, 460-461 525-526 Transformation to Polars and vice versa^ . . \ . 462-463 527-530 Partial Differential Coefficients. Two Independent Vari- ables, 463-465 531 Transformation of ^%ud ^J^ to Polars, . ... 466 &c dy 532 Transformation of V- ''' to Polars, 466-467 633 Orthogonal Transformation of v*-F, 467-468 CHAPTER XX. Miscellaneous Theorems. 534-548 Jacobians, 475-482 549 The Operator ^, 483-484 550 The Operators A' and A 484 551 Secondary Form of Maclaurin's Theorem, .... 485 552-554 Results Established by Operative Symbols, . . . 486-487 555-560 Cauchy's Proof of Taylor's Theorem, 488-489 561-568 Roulettes, etc 490-493 PRINCIPLES AND PROCESSES OF THE DIFFERENTIAL CALCULUS. CHAPTER I. DEFINITIONS. LIMITS. 1. Primary Object of the Differential Calculus. When increasing or decreasing quantities are made the subject of mathematical investigation, it frequently becomes necessary to estimate their rates of growth. The primary object of the Differential Calculus is to describe an instrument for the measurement of such rates and to frame rules for its formation and use. 2. The whole machinery of the Diflferential Calculus will be completed in the first six chapters, and the student should make himself as proficient as possible in its manipulation. The remaining chapters simply consist of various applications of the methods and formulae here established. 3. We commence with an explanation of several technical terms which are of frequent occurrence in this subject, and with the meanings of which the student should be familiar from the outset. 4. Constants and Variables. A CONSTANT is a quantity which, during any set of mathe- matical ojperationSy retains the same value. A VARIABLE is a quantity which, during any set of mathe- matical operations, does not retain the same value, but is capable of assuming different values. Ex. The area of any triangle on a given base and between given parallels is a constant quantity ; so also the base, the distance between the parallel lines, the sum of the angles of the triangle are constant quantities. But the sep>arate angles, the sides, the position of the vertex are variables. E.n.c. A « ih 2 ■ . , ; : CH^^P'l'ER I. It has become conventional to make use of the letters a, b, c, ..., a, P, y, ..., from the beginning of the alphabet to denote constants ; and to retain later letters, such as u, v, w, x, y, z, and the Greek letters ^, rj, f, for variables. 5. Dependent and Independent Variables. . An INDEPENDENT VARIABLE is One which may take up any arbitrary value that may be assigned to it. A DEPENDENT VARIABLE is One wMch assumes its value in consequence of some second variable or system of variables talcing up any set of arbitrary values that may be assigned to them . 6. Functions. When one quantity depends upon another or upon a system of others, so that it assumes cC definite value when a system of definite values is given to the others, it is called a function of those others. The function itself is a dependent variable, and the variables to which values are given are independent variables. The usual notation to express that one variable y is a, func- tion of another x is y=f(x), or y = F(x), or y = (x) ; the letters /( ), F( ), /l. Prove LtJ^S-^^-i. x-\ [Put ^=1+3/.] /s. Prove Z^..„^^«"^=%--. ^(fu^f / 1 r a * "^ :^'' • 3. Prove Z^:,=o(l+a.5P)^ = e^ v i^^ ^4. Prove Z^.=o'4^^^=^. ^ ^^^"^ "" ' V^ 5. Prove Z^.=o^^-=^-i^^-^ = i(log «)^ r" — 1 . 6. Prove Ltx=\- =n without assuming the Binomial theorem ; con- X — 1 sidering the several cases, (i.) ti a positive integer, (ii.) n a positive fraction, (iii.) n negative, (iv.) n incommensurable. 23. Hyperbolic Functions. By analogy with the exponential values of the sine, cosine, tangent, etc., the exponential functions are respectively written sinh 0, cosh 0, tanh 0, etc., and called the hyperbolic sine, cosine, tangent, etc., of 0, and as a class are styled hyperbolic functions. ' o' • /I e'^-e-<-^ , ^ e^Q+e-^^ Since sin d = -, , and cos 6 = ^ , where i = \/ —1, it will be clear that sin lO — L sinh Q, cos S = cosh 0, and hence or from the definition (1) tanf0 = / — T-^ = itanhO; (2) cosh2e-sinh2e=l; (3) sin (0 + i(p) = sin cosh ^ + 1 cos sinh (p ; with many other formulae analogous to, and easily deducible from, the common formulae of Trigonometry. If x = sinh 0, we have = sinh-ia*, an inverse hyperbolic function of x analogous to the inverse trigonometrical function sin-^ir. ^•- DEFINITIONS. LIMITS. H This species of function however is merely logarithmic; for, since aj = 2 we 1 lave e^ =x-\- m^^ i + x\ and e==\oge{x+s/l-\-x^},' while corresponding results hold for cosh-^ic, tanh'^a*, etc. Examples. 1 . Prove the following formulae — (a) cosech^^=coth-^-l ; (6) sinh (0 + )= sinh 9 cosh ^ + cosh sinh <^ ; ^^ ^ ' l + tanh(9tanh' (rf) sinh ^+sinh cf>^2 sinh ?±^cosh ^ t 2. Show that the co-ordinates of any point on the rectangular hyperbola x^—y'^=a- may be denoted by a cosh ^, a sinh B. 3. Prove (a) sinh-^.r = tanh-^-, -; {h) 2tanh-iji;=log^t±f. \— X 4. If .r + ty = a tan(w 4- iv), show that the curves u = constant and ij = con- stant are circles whose radii are respectively acosec2?« and acosech:^y cutting each other orthogonally. 5. Show that sinh x and cosh x have an imaginary period 2i7r, and that tanh X has an imaginary period t7r. Infinitesimals. 24. All measurable quantities are estimated by the ratios which they bear to certain fixed but arbitrary units of their own kind. The whole measure of a quantity thus consists of two factors — the unit itself and an abstract number which re- presents the ratio of the measured quantity to the unit. The magnitude of the unit should be chosen as something com- parable with the quantity to be measured, otherwise the abstract number which measures the ratio of the quantity to the unit will be too large or too small to lie within the limits of comprehension. For instance, the radius of the earth is conveniently estimated in Tniles (roughly 4,000) ; the moon's distance in earth's radii (about 60); the sun's distance in 12 CHAPTER I. moon's distances (about 400) ; the distance of Sirius in svbu's distances (at least 200,000). Again, for such relatively small quantities as the wave-length of a particular kind of light, one ten-millionth of an inch is found to be a sufficiently large unit : the wave-length for light from the red end of the spectrum being about 266, that from the violet end 167 such units (Lloyd, "Wave Theory of Light," p. 18). 25. Any comparison of two quantities is equivalent to an estimate of how many times the one is contained in or contains the other ; that is, the one quantity is estimated in terms of the other as a unit, and according as the number expressing their ratio is very large compared with unity or a very small fraction, the one is said to be very large or very small in comparison with the other. The terms great and small are therefore purely relative. The standard of smallness is vague and arbitrary. An error of measurement which, centuries ago, would have been reckoned small would now be considered enormous. The accuracy of observation, and therefore the smallness of allow- able eiTors of observation, increases with the continual im- provement in the construction of instruments and methods of measurement. 26. Orders of Smallness. If we conceive any magnitude A divided up into any large A number of equal parts, say a billion (10^^), then each part -^,^2 is extremely small, and for all practical purposes negligible, in comparison with A. If this part be again subdivided into a A billion equal parts, each = Tjr^i, each of these last is extremely small in comparison with j-^, and so on. We thus obtain a AAA series of magnitudes. Ay ^p-rjg, fT^y fTm^ •••» ^^^^ ^^ which is excessively small in comparison with the one which precedes it, but very large compared with the one which follows it. This furnishes us with what we may designate a scale of smallness. DEFINITIONS. LIMITS. 13 More generally, if we agree to consider any given fraction / as being small in comparison with unity, then fA will be small in comparison with J., and we may term the expressions /A, pA, pA, . . ., small quantities of the first, second, third, etc., orders; and the numerical quantities /, /^, /^, ..., may be called small fractions of the first, second, third, etc., orders. Thus, supposing A to be any given finite magnitude, any given fraction of A is at our choice to designate a small quantity of the first order in comparison with A. When this is chosen, any quantity which has to this small quantity of the first order a ratio which is a small fraction of the first order, is itself a small quantity of the second order. Similarly, any quantity whose ratio to a small quantity of the second order is a small fraction of the first order is a small quantity of the third order, and so on. So that generally, if a small quantity be such that its ratio to a small quantity of the ^*^ order be a small fraction of the q^^ order, it is itself termed a small quantity of the {p-\-qy^ order. 27. Infinitesimals. If these small quantities Af Af^, Af^, ..., be all quantities whose limits are zero, then supposing / made smaller than any assignable quantity by sufficiently increasing its denominator, these small quantities of the first, second,, third, etc., orders are termed infinitesimals of the firsts second, third, etc., orders. From the nature of an infinitesimal it is clear that, i/ any equation contain finite quantities and infinitesimals, the in- finitesimals may he rejected. 28. Prop. In any equation between infinitesimals of differ- ent orders, none but those of the lowest order need be retained. Suppose, for instance the equation to be ^i+5,+ai+i)2+^2+^;+---=o, (i.) each letter denoting an infinitesimal of the order indicated by the suffix. Then, dividing by A^, ^+x+z-+x+?+J+-="' ^"-^ -CL, xLj^ xLj xlj Xi-Y Tf n T) W the limiting ratios -j^ and ~- are finite, while -^, -p, are in- A^ A^ A^ A^ 14 CHAPTER I. finitesimals of the first order, -~ is an infinitesimal of the second order, and so on. Hence, by Art. 27, equation (ii.) may- be replaced by and therefore equation (i.) by A, + B, + C\ = 0, which proves the statement. 29. Prop. In any equation connecting infinitesimals ive may substitute for any one of the quantities involved any other which differs from it by a quantity of higher order. For if ^i + 5i + 0i + i)2+...=0 be the equation, and if A^ = F^-\-f^, /g denoting an infinitesimal of higher order than F^, we have i.e., by the last proposition we may write i^i + 5i + ai = 0, which may therefore, if desirable, replace the equation ^i+^i + (7, = 0. 30. Illustrations. Since sin = — —+--.— ... o ! 5 ! and cos = 1 — — ^ + j-| — ... sin 0, 1 — cos 0, — sin are respectively of the first, second, and third orders of small quantities, when is of the first order ; also, 1 may be written instead of cos Q if second order quantities are to be rejected, and for sin 6 when cubes and higher powers are rejected. 31. Again, suppose AP the arc of a circle of centre and radius a. Suppose the angle AOP ( = 0) to be a small quantity of the first order. Let PN be the perpendicular from P upon OA and AQ the tangent at A, meeting OP produced in Q. Join P, A. DEFINITIONS. LIMITS. 1.' Then arc AP = aO and is of the first order, iVP = a sin do. do., AQ = at£in6 do. do., chord AP = 2a sin x do. do., NA = a(l — cos &) and is of the second order. So that OF — ON is a small quantity of the second order. Again, arc ^P — chord J.P = a^— 2a sin ^ a^ 4.3! — etc. and is of the third order. PQ-NA=NA{^^Q,Qi-\) I sin^- and similarly for others. cosO = (second order) (second order) = fourth order of small quantities, 32. Such results may also be established without the use of the series for sin 6 and cos 6. N A I O Fig. 3. '^ For example, let APB be a semicircle, P any point very near to A, so that the arc AP may be considered a small quantity of the first order. 16 CHAPTER I. Join AP, BF, and let BP produced cut the tangent at A in i2, and let the tangent at P cut AR in T, and draw the perpendicular PN upon AB. T will be the middle point of AR, and AT=TR= TP. (1) We ma J take it as axiovnatic that the length of the arc AP i& inter- mediate between the chord AP and the sum of the tangents AT, TP ; i.e., between chord AP and tangent AR. Hence chord AP, arc AP, tangent AR are in ascending order of magnitude, and therefore 1, -^ — ^ ?^^? . ^ are in ascending order of magnitude, chord AP XT /. ^^ T*^^ 1 -Now, ^^-r — J— ro= ^^^D= 1 y chord ^P BP , r J arc -4P , when ce Lt- ?--rT»= 1 j chord AP and therefore, if arc ^P be reckoned a small quantity of the first order, the chord AP and the tangent AR are also of the first order of smallness. AJV AP (2) Again, since - = — -, and since ^Pis of the first order of smallness, AP AJj AN is of the second order. PR BP (3) Also — - = -— ^ which is ultimately a ratio of equality, and therefore PR is also of the second order. (4) Similarly, since ^i2-^P=^-^^—-^=^ -—.-_, and since PB;^ is a small quantity of the fourth order, and AR + AP is a small quantity of the first order, we see that AR-AP is of the third order of small quantities. And similarly for other quantities the order of smallness may be geometrically investigated. 33. The base angles of a triangle being given to be small quantities of the first order, to find the order of the difference between the base and the sum of the sides. By what has gone before, (Art. 31) if APB be the triangle and Pif the perpendicular on AB, AP—AM and BP — BMare both small quantities of the second order as compared with AB. DEFINITIONS. LIMITS. 17 Hence AP+PB—AB is of the second order compared with AB. 1? AB itself be of the first order of small quantities, then AP+PB-AB is of the third order. 34. Degree of a'pjproxirriation in taking a small chord for a small arc in any curve. p A B Fig. 5. Let AB be an arc of a curve supposed continuous between A and B^ and so small as to be coDcave at each point through- out its length to the foot of the perpendicular from that point upon the chord. Let AP, BP be the tangents at A and B. Then, when A and B are taken sufficiently near together, the chord AB and the angles at A and B may each be considered small quantities of at least the first order, and therefore, by what has gone before, AP-\-PB — AB will be at least of the third order. Now we may take as an axiom that the length of the arc AB is intermediate between the length of the chord AB and the sum of the tangents AP, BP. Hence the differ- ence of the arc AB and the chord AB, which is less than that between AP-\-PB and the chord AB, must be at least of the third order. EXAMPLES. 1. Show that, in the figure of Art. 31, the area of the segment bounded by the chord AP and the arc JP is of the third order of small quantities. 2. In the same figure, if P3f be drawn perpendicular to AQ, show that the triangle PMQ is of the fifth order of smallness. 3. A straight line of constant length slides between two straight lines at right angles, viz., CAa, CbB ; AB and ab are two positions of the line and P their point of intersection. Show that, in the limit, when the two positions coincide, we have AaCB .PACB"- Bh~CA^ PB~ CA^' 4. From a point T in a radius OA of a circle, produced, a tangent TP is drawn to the circle, touching it in P ; PN is drawn perpen- E.D.C. B 18 CHAPTER I. dicular to the radius OA. Show that, in the limit, when P moves upto^, iv^=^r. 5. Tangents are drawn to a circular arc at its middle point and at its extremities. Show that the area of the triangle formed by the chord of the arc and the two tangents at the extremities is ultimately four times that of the triangle formed by the three tangents. [Frost's Newton.] 6. If, in the equation sin(a) - 0) = sin w cos a, 6 be very small, show that its approximate value is 2 tan (0 sin^Vl - tan^w sin^^Y [I. C. S.] 7. If G be the centre of gravity of the arc FQ of any uniform curve, and if FT be the tangent at P, prove that, when FQ is in- definitely diminished, the angles GFT and QFT vanish in the ratio of 2 to 3. [I. C. S.] 8. If a side of a regular polygon be a small quantity of the first order in comparison with the radius of its inscribed circle, prove that the difference between the perimeter of the polygon and the circum- ference of the circle is a small quantity of the second order. [I. C. S.] 9. Assuming the radius of the earth to be 4000 miles show that the difference between its circumference and the perimeter of a reg- ular inscribed polygon of ten thousand sides is less than a yard. 10. Show that the curved surface of any belt of a sphere contained between parallel planes is equal to the surface of the corresponding belt of the circumscribing cylinder whose axis is perpendicular to the planes. 11. The sides of a triangle are 5 and 6 feet, and the included angle exceeds 60° by 10". Calculating the third side for an angle of 60° find the correction to be appHed for the extra 10". 12. If a triangle be inscribed in a given circle prove that the algebraic sum of the small variations of its sides, each divided by the cosine of the angle opposite to it, is equal to zero. [Math. Tripos.] 13. If X, y, z be the diagonals and the join of their mid-points in a quadrilateral whose sides are given, and f, -7, ^ their respective increments when the quadrilateral receives a slight deformation, then will x^ -t- y-r] + 45;^ = 0. Also if the quadrilateral be a parallelogram ^£ + 2/^ = 0, and if cyclic y^ + a^v; = 0. DEFINITIONS. LIMITS. 19 14. A person at a distance q from a tower of height^, observes that a flagpole upon the top of it subtends an angle B at his eye. Neglecting his height show that if the observed angle be subject to a small error a, the corresponding error in the length of the pole has to the calculated length the ratio qa cosec dj^q cos 6 -p sin 6). 15. Prove that . _itan 29 -{■ tanh 2<^ , .^tan 6 - tanh <^ tan '26 - tanh 2<^ tan + tanh cfi = tan-i(cot 6 coth <^). [Math. Tripos, 1878.] 16. If ic + ly = c cos(f + irf), the curves rj = constant and ^ = constant are confocal ellipses and hyperbolae respectively. Prove that the square of the distance between the points (^, 77), (if v) is the same as the square of the distance between the corresponding points (£', r]), (^, t/'), viz., c'{cosh.{rj + rj') - cos(^ + f )}{cosh(7; - 7f') - cos(f - f )}. Prove also that a bisector of the angle between these distances makes with the o^axis an angle tan"^ ^^^i(g + g ) , [London, 1887.] ^ tanh^-(r; + 7]') 17. If cos .^• cosh w=l, x is called the Gudermannian of u and -svritten gd it. [Cayley, Elliptic Functions.] Prove (a) gd u = tan~^sinh u = sin~Hanh u. (b) J gd « = tan~^tanh -. (c) w = logtan(^ + Jgdw J. (d) singd(zi + «;) = - 18. Prove that t g(x+Sx), whence RQ = Sy — (j)(x + Sx) — (f)(x). XT T.PQ r, Sy J. d)(x + Sx)-d>{x) Hence we can express -^^p^ as Lt^x^o^ or ■L't^x=o —^ — r^^.- Hence, to draw the tangent at any point (x, y) on the curve y — (j)(x), we must draw a line through that point, making with the axis of x an angle whose tangent is Ltsx^o~- —w^ — ^^ ; ox and if this limit be called m, the equation of the tangent at P(x, y) will be Y— y = m(X — x), X, Y being the current co-ordinates of any point on the tan- gent ; for the line represented by this equation goes through the point (x, y), and makes with the axis of x an angle whose tangent is m. Examples. Find the equation of the tangent at the point (.r, y) on each of the following curves : — J 1. x'^+;^^=c\ 4. i/ = logx. 5. y=tan^. 6. y = tan-^x 3. y FUNDAMENTAL PROrOSITIONS. 23 37. Def. — Differential Coefficient. Let {x) with respect to x. The operation of finding this limit is called differentiating After reading Ohap. Y., it will be obvioufe why the above expression is styled a " coefficient," for it is shown there to be one of a series of coefficients occurring in the expansion of ^(x + h) in powers of A. The geometrical meaning of the above limit is indicated in the last article, where it is shown to be the tangent of the angle \Jr which the tangent at any definite point (x, y) on the curve y = \x), ^a,, 0', 26 CHAPTER II. or (p. Again, as the letters u, v, ^v, etc., are frequently used to * denote functions of x, we shall consequently have the differ- ential coefficient variously expressed, as -r-, u\ u^, or it, with a similar notation for those of v, w, etc. 40. Aspect of the Dijfferential Coefficient as a Rate-Measurer. When a particle is in motion in a given manner the space described is a function of the time of describing it. We may consider the time as an independent variable, and the space described in that time as the dependent variable. The rate of change of position of the particle is called its velocity. If uniform the velocity is measured by the space described in one second; if variable^ the velocity at any instant is measured by the space which would be described in one second if, for that second, the velocity remained unchanged. Suppose a space s to have been described in time t with varying velocity, and aa additional space Ss to be described in the additional time St. Let v^ and v^ be the greatest and least values of the velocity during the interval St ; then the spaces which would have been described with uniform velocities v^, v^, in time St are v^St and v^St, and are respectively greater and less than the actual space Ss. Hence v^, ^, and v^ ^^^ i°- descending order of magnitude. ot If then St be diminished indefinitely, we have in the limit v^=:V^ = the velocity at the instant considered, which is there- fore represented by Ltj- i.e., hy -77- 41. It appears therefore that we may give another interpre- ds tation to a differential coefficient, viz., that -jj means the rate dx dij of increase of s in point of time. Similarly -ti, -^y mean the rates of change of x and y respectively in point of time and measure the velocities, resolved parallel to the axes, of a moving particle whose co-ordinates at the instant under consideration are x, y. If x and y be given functions of t, and therefore the FUNDAMENTAL PROPOSITIONS. 27 path of the particle defined, and if Sx, Sy, St, be simultaneous infinitesimal increments of x, y, t, then §y §y ^=:Lt^ = Lt^= — dx Sx 6x dx It dt and therefore represents the ratio of the rate of change of y to that of X. The rate of change of x is arbitrary, and if we choose it to be unit velocity, then -,^ = -^^ = absolute rate of change of y. 42. Meaning of Sign of Differential Coefficient. If X be increasing with t, the cc-velocity is positive, whilst, if X be decreasing while t increases, that velocity is negative. Similarly for y. dy Moreover, since ;; =;t-» j^ is 'positive wlien x and y di increase or decrease together, hut negative when one increases as the other decreases. This is obvious also from the geometrical interpretation of -J^. For, if x and y are increasing together, -^ is the tangent of an acute angle and therefore positive, while if, as x increases y decreases, -~ represents the tangent of an obtuse angle and is negative. Examples. Find from the definition the differential coefficient of 3/ with respect to X in each of the'following cases : \J 1. y=x^. 8. y = tdin-^a^. J^ 9.. 2/=2^cuv. % y=logcosx 3. y=i>Jcf+W. 10. y = logtan.r. ^ 4. y=e*. 11. y=af. 5. y = e\/* 12. y=af^^'. 6. y =««'"*. 13. y=(sin^)''. 7. y = a^°«*. 14. y={B\\\xY^''. 28 CHAPTER II. 15. In the curve y = cec^ if -^ be the angle which the tangent at any point makes with the axis of x^ prove y = c tan ;/'. 16. In the curve y=c cosh -, prove y = c&QC xj/. x^ 17. In the curve lP-y = -^ — ax'^ find the points at which the tangent is jmrallel to the axis of x. [N.B. — This requires that tan \^=0.] 18. Find at what points of the ellipse ^4.^ =1 the tangent cuts off a equal intercepts from the axes. [N.B. — This requires that tan t/'= ±1.] 19. Prove that if a particle move so that the space described is propor- tional to the square of the time of description, the velocity will be pro- portional to the time, and the rate of increase of the velocity will be constant. 20. Show that if a particle move so that the space described is given by s cc sin /xt, where />i is a constant, the rate of increase of the velocity is proportional to the distance of the particle measured along its path from a fixed position. 43. It will often be convenient in proving standard results to denote by a small letter the function of x considered, and by the corresponding capital the same function of x + h, e.g., if u = (p(x), then U= h Vv and taking the limit du dv dy _ dx dx dx v^ 51. This proposition may also be deduced immediately from Prop. IV., thus : Let . 2/=~ ; whence dx dx "dx dy u dv ^ and therefore du udv dy dx V dx dx~ V du _dv _dx' dx' Examples. J 1. Deduce the result of Prop. II. from propositions I. and IV. J 2. Deduce from Prop. V. that d(c\_ c du dx\uJ u^ dx' 32 CHAPTEE II. V 3. Apply proposition lY. and the results of Art. 38 to show that -5- (^^sin x) — ^r^cos x+'2x sin cc. ax 4. Apply proposition V. to show that c? / sin^ Vcos ^ . ^- 2 sin ^ dx\ x^ )~ P 52. Prop. YI. To find the Dififerential Coefficient of a Func- tion of a Function. Let u=f{v) (1) and v=F{x) ..(2) Then, by elimination of v, we have a result which may be expressed as u = (p(x) (3) Suppose the independent variable x to change to X in (2) and let a value of v deduced from (2) he V. Let this be sub- stituted for V in (1), and let a value of u deduced from (1) be U. Then we have the following equations. ^-f(V) (4) and V=F(X) (0) and by the same process by which (3) was deduced from (1) and (2) we obtain from (4) and (5) U==— that is, a a u=bBm \a a I Then, by Ex. 1, Art. 38, ^=6 cos v. dv dv 1 dw a Jl- w" dw '■Lx dx a' du du 'do dw dw "di~ -.h cosv . 1 a ' 1 2x dx n/1- w' a b --co, Jh^. n-^n 1. 1 2x Ex. 2, ibid. Ex. 3, ibid. Hence The rule may he expressed thus : djlst Func.) ^ d(lst Func.) d{2nd Func.) d(Last Func.) dx d(2nd Func.) d{^rd Func.)'" dx or if y^={yl^{F{fx)}l ^ = 0TV^{^(^)}] X ir'{F{fx) } X F\fx) xfx 54. There is a diflBculty in Prop. VI. arising from the fact that for one value of X in (2) there may be several values of v^ and for any value of v in (1) there may be several values of u. In fact the/(v) and F{x) may one or both be many-valued functions (such, for example, as sin"^^, which denotes any one of the series of angles whose sines are equal to x). But it is clear that the «awie values of u and x will satisfy equation (3) as would simul- taneously satisfy (!) and (2), and that Lt ~^ when X—x is indefinitely A —X diminished is one value of the differential coefficient of u considered as a E.D.C. c 34 CHAPTER II. function of x ; and it is equally obvious that there may be a series of suck values for -^, as also for -- and for --^, so that in the theorem enunciated ax dv dx and proved above, in Art. 52, a 'proper selection of those values is assumed to he made. 55. If in the theorem -^- = ^=— . -^ (where y is written for v dx ay dx ^ in the result of Art. 52) we suppose u = x, then du dx J (x+h)—x Hence we have or dx~ ~dx~ = Lth=o- dy dx dx 'dy~ = 1, dy_ dx~ 1 ~ dx dy h = 1. bQ>. In this application of the general theorem of Prop. VI. y is assumed to be a function of x and consequently x is the inverse function of y. So that -p is the differential coefficient of y with respect to x when y is considered as a function of cc, dx and -7- is the differential coefficient of x with respect to y when X is considered as the inverse function of y : e.g., if y=8in^, then ^=sin-^y, ^=cos ^ (Ex. 1, Art. 38), dx and ^ = -7^=. (Ex. 2, ibid. ), J dv dx ^^„ ^ 1 cos :r - and f . --- =cos :?7 . — j^ — = ■ , — r= 1. c?^ d2/ n/1-^2 Vl-sin^^ 57. The same difficulty occurs in Arts. 55 and 56 as that discussed in Art. 54. If 2/=/W (1), and this equation be supposed solved for x, the result will be of the form x = F{y) (2). Now, if X be changed to X in (1) and F be a value deduced for y, then if Y be substituted for y in (2), X will be one of the values thence deduced for x. FUNDAMENTAL PROPOSITIONS. 35 X — x Hence Lty—- when Y—y is indefinitely diminished is a value of the differential coefficient of x with respect to y, as Y—v derived from equation (2), while Lt^ — ^ when X — x is in- definitely diminished is a value of the differential coefficient of y with respect to x as derived from equation (1). -And since F-v X-x . X-xY-y ' we have ^.^=1, ax ay when the limit is taken, the proper selection being made of the values deduced for -^ and -^. ax ay 58. This may be illustrated geometrically. Let the curve y=f(x) be drawn. Let the tangent to the Fig. n. curve at the point P, (x, y), make an angle i/r with the axis of X. Then, by Art. 37, ^ = tan ^ ; and in the same way it is dx dx obvious that ^ = tan (90 — V^) = cot xfr, so that -^'1. — = tan \[r . cot -^/r = 1. Suppose however that the ordinate through P cuts the curve again at Pp Pg' ^3' ••• Then, for a given value of x there are several values of y, and therefore also for a given increase 6x in the value of x there may be several values of Sy the increment of y. But if it be carefully noted that the Sy and Sx chosen are to refer to CHAPTER 11. the satne branch of the curve at the same point when we con- sider ^- as when we consider -r-, then, under these circum- stances, these expressions are respectively the tangent and Fig. 12. cotangent of the same angle, and therefore their product is unity. We say the same branch of the curve, for it may happen that more than one branch of the curve passes through a given point Fig. 13. P, as in Fig. 18, and then there are two or more tangents at P QOJ (1 '7' and therefore two or more values of -,- and -,— at P. But the dv dx ^^ ^y product of the -r and the -t~, which belong to any the same branch through P, is unity. 59. Differentiation of Inverse Functions. When the differential coefficient of any function of x is found, that of the corresponding inverse function is easily deduced by means of the theorem of Art. 55. FUNDAMENTAL PROPOSITIONS. 37 But therefore For let x=f(y), and therefore y=f~\x); then dx dx' dy EXAMPLES. 1. Differentiate by means of the definition and the foregoing rules : — y = a; log sin x. , (ii. (iii. (iv. (V. (vi. X-. C2 5 X vAtJ^' XK y = 2jaUj where u = a''°'. y = e"^". where u = log sin v, and v = (sin wy, and w = x^. The results of any preceding examples may be assumed. 2. If Wj, 2^2» ^3) • • • "^u ^2' ^3' • • • ^® functions of jc, prove that flte «j'y2'y3...'y. VjV^.. .V^ \ «*-'r = 1 U^ dx '^r=lV^ dx )' CHAPTEE III. STANDARD FORMS. 60. It is the object of the present Chapter to investigate and tabulate the results of differentiating the several standard forms referred to in Art. 38. We shall always consider angles to be measured in circular measure, and all logarithms to be Napierian, unless the contrary is expressly stated. It will be remembered that if u = (p{xy, then, by the defini- tion of a differential coefficient, du J. (f>(x -\-h) — (p{x) 61. Differential Coeflacient of a?". If 1l = (f>{x) = X'', then (l>{x-^h)^{x-{-hY, , du y. (x+h)'^—x'' = Lth=GX^~ h Now, since h is to be ultimately zero, we may consider - to be X less than unity, and we can therefore apply the Binomial Theorem to expand ( 1 + - j , whatever be the value of n; hence —Lth=onx'^-Hl + - X (a convergent series) |- = nx''-\ STANDARD FORMS. 39 62. If it be required to find the diflferential coefficient of a:** without the use of the Binomial Theorem we quote the result of Ex. 6 p. 10, viz. : and proceed as follows : ^-=Lt„.v^''^ —^ [as before] = Lty=xx'"-'^ where v = 1 + y-l L. xA 63. Differential Coefficient of a*. If u = (x-\-h) = co&{x+h), J du J- cos(x+h) — cosx ^"""^ s=^^^=° h 2 sm ^ sm (-1) h == -Lth=(y-—- sin {x-\-^ J 2 = — sin X. 67. Differential Coefficient of tan x. If u = (p(x) = tan X, u-\x-{-h). a? = tan 16, and a? + ^ = tan U; h = tan U— tan u, ^'^_T+ U—u_ j^ U—u d^ - ^^^-^IT' ~ ^^^="tan?7-tanu U—u TT B\n{U—u) 1 1 1 sec^w, l + tan% \-\-x^ STANDARD FORMS. 43 75. Differential If u = U-. Hence x = therefore h = and du dx'~ ^=Lt Coefficient of cot"^a:;. (l)(x) = coi-'^x, ■■ (p(x -i-h) = cot - \x + h). cotu, and x-\-h = cot U; cot U—cotu, U—'^_j. U—u U-u = -Ltu-. = — sin^i6= — %m{U-u) 1 sin ^ sin 21/ 1 1 cosecm 76. Differential Coefficient of 8ec"^a:?. If u = {x) = sec " ^x, U—(l>{x+h) = 8ec-\x-\-h). x = secu, and 03+^ = sec U; h = sec U—secu, du J. U—u J. U—u dx h sec U—secu ^^ U—u Tj = Ltn=u rf cos U cos U cos u — cos U f U—u \ cos u cos U H-cot^u l+aj2* Hence therefore and sm U^-u sin u sec2u>v/l — cos^u '4^i 77. Differential If n- U-. Hence X-- therefore h- and du di'- XiJx^—\ Coefficient of cosec~^a?. = <^{x) = cosec - ^oj, = (^{x 4- /i) = cosec - \x + Ih). - cosec Uy and x-^-h — cosec U ; = cosec C/"— cosec u, J. U—u ^. U—u = Ltu= = Lti/=- U-u ^sin u — sin U '^cosec L^— cosec u sin u sin U 44 CHAPTER in. U-u \ = —Ltjj^A sin% cosu 1 sin U- u sin u sin U cos 1 2 ^^1-^ cosec^ifcVl — sin% __ 1 78. From the importance of the results it has been thought preferable to deduce the differential coefficients of the inverse functions sin"^aj etc. immediately from the definition; but by- aid of Prop. VI. of the preceding chapter we can simplify the proofs considerably. Ex. (i.) If we have whence and therefore u = sin~^^, dx du du^ 1 _ 1 dx dx cos = C0S1 1 Jl — sin% iJ\—a^ and since we have Ex. (ii.) If we have whence and therefore and since we have Ex. (iii.) If we have whence and therefore whence also dco^~^x dx 1 u = tan~^.r, x=td^nu; dx 2 du du dx~ 1 sec^i* 1 1 l+tan% 1+^2 cot -^x= =|-tan 1 -^^, c?cot- -l£^ dx 'W=vers~"^, ^=vers'it=l — cos?^ dx du du 1 1 =sm^; dx sin^ Vl-cos-z^ sf'2.x-x'^ dcOVQTB'^X 1 dx sj^X - X^' STANDARD FORMS. 45 79. The Integral Calculus. Suppose any expression in terms of x given ; can we find a function of which that expression is the differential coefficient? The problem here suggested is inverse to that considered in the Differential Calculus. The discovery of such functions is the fundamental aim of the Integral Calculus. The function whose differential coefficient is the given expression is said to be the " integral " of that expression. For example, if {x)-{-G, J'(l>{x)dx being read " integral of (f>\x) with respect to x." Thus we have seen ^r-(sin x) = cos Xy dx ;T-(tan-ia;) = /r dx"- ^ l+ic2' etc., whence it follows immediately that ycos xdx = sin x, etc., where the arbitrary constant may be added in each case if desired. 80. We do not propose to enter upon any description of the various operations of the Integral Calculus, but it will be found that for integration we shall require to remember the same list of standard fonms that is established in the present chapter and tabulated below, and it is advantageous to learn each formula here in its double aspect. We have therefore tabulated the standard forms for Differentiation and Integration together. Moreover, we shall find it convenient to be able to use the standard forms of integration in several of our subsequent articles. 46 CHAPTER III. Table of Results to be committed to memory. • U — X' u = w^. u = e-^. u logaCO. U = logeX. ♦ u = sm X. f U = COS X. u = i u = tamx. cotaj. f u = sec X. u = cosec X. , u = sm~^x. u = cos~'^x tan - '^x. cot"^a3. , i6 = sec~^a;. ^6 = cosec" ^03. u = I u = vers "^07. covers " ^x. nx' 1 loga^. dx ax dx X du_l dx X du dx du dx du dx du dx du _ sin X dx cos'V du _ _ cos X dx sin^fl?* du 1 n+l n + 1 logea \og^. — cos X. = — sm X. = sec^x. — cosec^a?. fx'^dx JoJ^dx fe^dx rdx J X or =—^^-. logae ycos Qi^x = sin flj. ysin xdx — — cos x. j^QC?'xdx = tan cc. Jcosec^xdx = — cot a;. ^"^""dx sec ic. cos^aj 'cos a) (io: = — cosec x. dx J\-o? du _ _ 1 dx Jl—x du_ 1 (ioj 1 + x^' du_ _ 1 du_ 1 cZu_ _ 1 C^l(,_ 1 dx~ J2x^^ du_ _ 1 c^a? ^2x-x^ A sm^x dx ^l-x' y^dx = sin-^a;, = tan-ia?, or — cot"^a;. r dx _ _i^ J xJx^ — 1 \A dx J\x- or — cosec "^aj. = vers "-x, or — covers "^ic. STANDARD FORMS. 47 81. The Form u^. In functions of the form u'^ where both u and v are functions of X, it is generally advisable to take logarithms before proceed- ing to differentiate. Let y = u\ then log«2/ = v logeU ; therefore - -r- = zr - ^ogeV^ -\-v.- -j-, Arts. ^, 52, 64. y dx ax ^ u dx dy ,/, dv V du\ Three cases of this proposition present themselves. dv dx dv I. If V be a constant and u a function of a;, -^^ = and the above reduces to -/ = -y . u'' - S » ax ax as might be expected from Arts. 52, 61. dij II. If u he a constant and v a function of ic, -i- = and the general form proved above reduces to dv „, dv as might be expected from Arts. 52, 63. III. If u and V be both functions of x, it appears that the general formula dy , dv . ^du dx ^ djx dx is the sum of the two special forms in I. and II., and therefore we may, instead of taking logarithms in any particular example, consider first u constant and then v constant and add the results obtained on these suppositions. 82. We shall presently (Art. 162) see further that if y be any complex function of x, then, in whatever way the various simple functions of which y is composed be connected together, the complete differential coefficient of y is the algebraic sum of the differential coefficients obtained severally by considering all the functions but one to be constant. 83. Hyperbolic Functions. The differential coefficients of the direct and inverse hyper- bolic functions are now appended as additional formulae. Their 48 CHAPTER III. verification is very simple and is left as an exercise. They will be found useful by the more advanced student by reason of their close analogy of form with the results tabulated above for the direct and inverse trigonometrical functions. Results for Hyperbolic Functions. — - = cosh X. ybosh xdx = sinh x. CLx — = sinh X. f&mh xdx = cosh x. dx — = sech^a;. /sech^ajrfa; = tanh x. dx -^ = - cosech^a:. ycosech^arda: = - coth x. du sinh x rsinh x u- = sinh X ■■ 2 ' u- = cosh X ■■ 2 ■ u- = tanh X _ sinh X cosh x' u- = coth X ■■ _ cosh X sinh a;' u- = sech X ■■ 1 cosh x' u- = cosech -= ■ I ■ smha; rtt*^_smn^ rsinh^^^ =-8echa:. dx cosh^a: J cosh^a; du cosh X rcosh x dx sinh^a; J sinh'^a; /i -dx = - cosech x. .. = sinh-ia: = log(a: + VrT^). t = J^' /jf^^^ ='^"^"^- ,. = cosh-a: = log(a: + V^l). f^ = ^^- /;^ =-°«^"^^- , u = tanh-a: = ilogj4^. g=r^.(^l). /-^ =-coth-a:,.>,, w = sech-ia; = cosh-il ^ = - .i__ . A^S= =-sech-ia:. •^ a;V 1 - x^ X dx xjl-x^ du _ 1 X dx~ xj^^i -^ xslx^-^l w = cosech-ia: = sinh-il. ^ = J—. f—^= =-co8ech-ia:. ^W^\ Jx 84. Transformations. Algebraic or trigonometrical transformations are frequently useful to shorten the work of difierentiation. For instance, suppose y = tan~^- We observe that y=2tan-i^ ; , dy 2 whence -^ = .• dx l+x- 1 -\- X Again, suppose y = tan~^^— — Here y=tan-^^ + tan-il, and therefore ^ dx l+.r^' STANDARD FORMS. 49 As another example suppose Here =^-\cm-^a?; therefore dy X dx sl\-a^ 85. Examples of Differentiation. Ex. 1. Let y= sjz^ where « is a known function of x. Here y = z\ and % = ^-'-^ whence 1 dz 2 ^' dx' Tliis form occurs so ofli m tJiat it toill he found convenient to commit it to memoi'y. Ex. 2. Let y- =e\/cotJc. Let fcotx-- =z and cotA'=jD, so that y- = e*, where z= ^p. Now dy, dz = ^. (Art. 63.) dz 1 /T?^ 1 „1.^ \ cot dr. dp o -f-= — cosec-r, dx and (Art. 53) $=^ ,^j£=- cosec-".r . — L=.- . e^< dx dz dp ax ^ijcotx With a little practice these actual substitutions can be avoided and the following is what passes in the mind : — cg(ev/cot^) _ d{eV^^) ^ d('Jcotx) d(cotx) ^^ diJcotx) d{cotx) dx = gA/eSfi . — . ( — cosec^jr). 2v cot X Ex. 3. Let .y = (sin^)'°s*cot{e'(rt + 6^)}. Taking logarithms log y = log X . log sin x + log cot{ + 6r. ^^^^_,;^« ' 23. y^^. ''■ ^ = 7Fr^- 1+a; ^ > 24. y='Ja + x. 48. 2/ = cosfasin-i-Y . 25. y='^a^ + x'. , _,« + 6cosa; o/» / — i — 49. v = sin 1 . ' 26. 2/ = vcosh .r. ^ 6-r«cosa; . 27 2/ = log cosh a;. 50. i/ = e*^~^' log(sec2a;3). 28. 2/ = tan-i(tanh a;). 51. 2/ = e'^cos(6 tan'^a;). 29. 2/ = vers" V. 62. 2/ = tan~^(a''*.a;2). 30. 2/ = vers~Mog(cot a;). 53. y = sec(log„>/a^ + a;^). 31. 2/ = cot"i(cosec a;). 54. 2/ = tan"ia; + tanh-^a;. y. 52 CHAPTER III. ^ 56. y = \og([ogx). ^ b1. yr= log"(a;), where log" means log log log . . . (repeated n times). , Jb + a -\-Jb- a tan « 58. v = _J__log -. ^ V^ + ^^v^ "'^'''tan rt 59. 2/ = sin~i(a;^l -x - Jxjl -.r^). 60. 2/ = tan-i-i^, ^ , fl-^s/x Y'^'^ ( rx-^\n / Vi + 2./.y 61. y = \ogU'l^—-^j I. 4 75. v/ = (coscc)-*"'^ 62. 2/= 101^* 76. v/ = (cot-i.x)i. 63. y = e^\ ' 77. 3/= ("l +iy + xi4 /65. 2/ = ^< ■ 78. 2/ = aan-ig + tan-i|). 766. 2/ = ^^*. 79^ tan2/ = e''"^'"sina:. 67. 2/ = a^* + a^. ^ 80. a{K2+ 2^X2/ + 62/^=1. 68. 2/ = (sina;)''°"+(cosa;)''''^'. . g^ ^^ _ (^ + 60;")^ - a^ 69. 2/ = (cota;)'='^** + (cotha;y°*^^ " (6a;")* . . Ja: 82. (cos£c)^=(sinv/)*. 1 +a;t y-ac 83 £C = e**° a;^ • 71. 2/-si n-i(e--"-). 84*. . = 2/ legacy. 72. 2/ = J(l+cos^Vl-sin^)./85. 2/ = «^. 73. 2/ = tan~^V^\^a; + cos~^aj. 86. 2/ = a:< 87. y = x\og a + bx 88. aa;2 + 2hxy + 62/^ + 2^x + 2/y + c = 0. 89. .'i:V = («^ + 2/r''-- 90. 2/ = e*''^"~^''logsec2x'S. J 91. Differentiate logjocc with regard to x^. 92. Differentiate (x^ + ax + a^)" log cot - with regard to tan~i(a cos 603). 93. Differentiate log^ STANDARD FORMS. i + b tan - 1 with regard to 53 a-b tan - 9. 1 a^cos^-bhin^. 2 2 94. Differentiate af'" ' with resrard to sin'^a*. v% 95. Differentiate tan~^^- with re«rard to tan~^a;. X 96. Differentiate \L'^,^'^^^j ;£! with regard to JT^^. 97. Differentiate sec~^^o^_-| with regard to ^1 -^• 98. Differentiate tan-^ —- — — with regard to sec"^ ^/r^2 ^ 2a:2_l 99. Differentiate tan~ \ '^ with regard to sin-^— ::^ — . 1 - ar 1 + ar^ '^ 100. Differentiate a;*'Iog tan-^a; with regard to £i£^. ^ 101. \iy = xr prove (/a; 1 - y log a;' y 102. If 2/ = ^ a; prove $^ = 1 2a; I+T4.- ^^ 1+1" u.^ a- ^ + i + ...tocx, ■^r+^^^ 103. Ify = iK + - 1 1 prove^ = l x . a; + - 1 ^ (/.-B 2 - - 1 1 x+ ... to 00, x + - . a;+... V 104. Ify = • cos a; dy _ (1 + ?/)cos X + y sin aj I 4. sm a; ^ ^ 1 +-— — cos a; 1 4- 1 +... to 00, prove , , . aa; 1 + 2y + cos a; - sin a; V 105. If 2/ = \/sina; + Vsinaj + vsina + v/etc. to 00, dy cos X •^ 106. If >S'„ = the sum of a G. P. to n terms of which r is the common ratio, prove that {r - 1)^" = (n - 1)^„ - nS„_,. [Coll. Ex.] 54 CHAPTER III. 107. If^ = (i + i 1 T prove A('A=±1 Q % + -"^l 1 dx\Q) Q^ 108. Given (7=1 +r cos ^ + — — — + — k-j — + ••• and »S; = rsin6^ + — -~^ — + — — — + ..., show that dr dr G~-J^=^(C^ + S^) sin e. [Coll. Ex.] dr dr 109. If 2/ = sec 4:X, prove that di ^ m{l-t^) ^^^^^ ^ ^ ^^^ ^ dt {i-U^ + f^)' ' 110. If y = e'^^sec'^ix Jz) and z^ + x'^z = x^, find -^ in terms of a; and z. [Trinity Schol.] 111. Prove that if x be less than unity J_ + j£_+ *^ + _^ + ... ad inf. = -J_. [CcL. Ex.] i+x l+x^ l+x!^ l+x^ i-x 1 1 2. Prove that if x be less than unity 1 - 2x 2x - 4x^ 4cc3 _ 8a;7 1 + 2 + ... ad inf. = I -x + x^ 1 - a;- - a;"* 1 -xl^ + x^ l+x + v? 113. Given Euler's Theorem that r , XXX X sin x X^^^cos ^ cos - cos 23 • • • ^0^ 2" " "^ ' prove I tan 1 + A tan 1+ -tan ~ + ... ad inf. - - - cot a;, and 1 sec2 1 + ^ sec^ ^^ + ^, sec^ | + . . . ad inf. = cosec^o: - 1^, 114. Given the identity (2co8 2^-l)(2cos220-l)...(2cos2"^-l)=?|^|^^, ^.^„ 2"sin2^^ 2^-^isin2" ^i|9 _ 2 sin 2^ prove that 2,^1 2 cos 2»'6' - I ~ 2 cos 2"+^^ + 1 2 cos 2^+1' 116. Given sin sin(2a -t- ) sin (4a + <^) . . . sin {2{n - l)a + <^} = ^^^ where 2na = tt STANDARD FORMS. 55 prove that cot <^ + COt( 2a H-<^) + C0t(4a + } = n cot n4>, and that cosec2(/> + cosec2(2a + <^) + cosec2(4a + )+ ... + cosec2{2(?i - l)a + <^} = n^cosecH. 116. From the expression for sin 6 in factors prove , 2 2'^ and hence that tt coth tt = 1 + - — - + — _ + -JL- + ... ad inf , 1 + 12 1 + 22 1 + 32 and that |coth|= 1 + ^-1^^ + _!_, + ^_^+ ... ad inf. 117 Prove *^^ ^ - V"=" ^ ill, rrove --^- - 2^^^^ —-—---, and deduce ^ tanh. = ^ + ^^ + ^^^ ... ad inf., and -tanh| = ^^-.^^ + ^-^^-....adinf 118. Prove ^ coth a; = ^ + a2y"=*__JL_ . 119. Prove that 2?'7r - 1 1 , n-2 iK-rtCOS iK - rt cos 2~ af^-a" x-a ' x + a — <»^i a;^ - 'Zax cos + a^ 2 _ 2^ n if n be even, 2r7r _ „_i re -a cos but =j_+22::7" — £c2 - 2aa; cos - — + n'^ n if n be odd. 120. Prove that „ 1/ « . /l^ a; -a COS — wa;" ^{pir - a"cos fc') _ ^r=»i-i n x^ - 2a;'*a»cos + a'" ~ 2ur=o ~T~Z 2r7r + 6 , ' x^ - zax COS + a^ n 121. Determine the coefficients -4^, A^... A^so that ~[{x^ - A^x^-^ + A^x^-^ -.-. + (- lyAj^e'^^x^e^ m being a positive integer. [Univ. London, 1890.] 56 CHAPTER III. 122. Writing sgw for singdw, etc., establish the following results — (y) tff a? = . ^'^^ dx ^ cga; 123. The functions a;^, x^, x^...x^ being defined by the equations x^ — \/x\/x, aj^^i = \/x/i/xx^f find the differential coefficient of the function towards which cc„ tends when n increases indefinitely. [Frenet.] 124 If 6v denote the sum of the r*^ powers of the roots of the equation a" +p^x^~'^ -\-'p^~'^ + ... + j(?„ ^ 0, prove that if the coefficients be expressed in terms of s^, Sg* •^s • • • *«> then will ^-_-P^. [Brioschi.] 125. Defining the Bessel's function of the «" order as "^""^ 2»rt!l 2(2« + 2) "^ 2.4(2n + 2)(2re + 4) '■■]" prove (1) ^/„(x)=-/i(a=). (3)j-„+/,-/„-/„«=22z;'j^/. CHAPTER lY. SUCCESSIVE DIFFERENTIATION. 8G. Repeated Operations. The operation denoted by y- is defined in Art. 37 without any reference to the form of the function operated upon, the only assumption made being that the function is a function of the same independent variable as that referred to in the operative symbol, viz. x. It is moreover clear that the result of the operation is also a function of x, and as such is itself capable of being operated upon by the same symbol. That is to say, if 2/ be a function of x, -^ is also a function of x, and therefore we can have -p( -7^ j as a true mathematical quantity. And further, it will be thus seen that the operation -j- may be performed upon any given function of x any number of times. 87. Notation. The expression -[-(-J^) is generally abbreviated into (;t-) y or -—, and is called the "second derived function'' or "second differential coefficient " of y with respect to x. And, generally, if the operator -j- be applied n times, the result is denoted by \-T-) y or -^, and is called the n^^ derived function or nP^ differential coefficient of y with respect to x. 57 58 CHAPTER IV. It will be convenient to denote the operative symbol ^ by D, which, in addition to being simpler to write, makes no assumption that the independent variable is denoted by x ; and in maoy problems the independent variable is more con- veniently denoted by some other letter. For example, in dynamical problems the time which has elapsed since a given epoch is frequently taken as the independent variable and is denoted by t, while the letters x, y, z, are reserved to denote the co-ordinates at that time of the point whose motion is considered. It appears then that if we use indices to denote the number of times an operation has been performed, we may write " dx D.Dy = D^y = ^,, D.i)^-V=iyy-^- 88. Analogy between the operator -^ and symbols of quantity. The index notation employed above to denote the number of times an operation is repeated is exactly analogous to the index notation used in algebra to denote powers of symbols of quantity. If a be an algebraic quantity, the algebraical notation for a .a is a^, and for a .a .a is a^ and so on ; the index here denoting the number of factors each equal to a which are multiplied together. But, as defined above, there is no idea of multiplication in D . D or B^, but a simple repetition of an operation. In the same way D^ has no quantitative meaning in itself, but represents an operation consisting of employing the process of differentiation n times. For example, the difference between such quantities as D^y, (Dy)\ and D^y^ should be carefully noted. The index in the first case has reference only to the symbol of operation "D," which is there- fore to be applied twice to y. SUCCESSIVE DIFFERENTIATION. 59 In {DyY the index is a purely quantitative one ttsed in the algebraical sense to denote the product Dy x Dy. While in D^y'^ we are to understand that the square of y is to be differentiated twice. That the ultimate results are different may be easily seen by taking any simple case, e.g.,ii y = x\ then By = 2ic, and Dhj = 2 (1) Again (Dyf = 4!x\ (2) whilst y^ = x'^, and Dy^ = 4a;^ giving Z)y = 12a;2 (3) A comparison of the results (1), (2), (3), will at once satisfy the student of the truth of the above remarks. 89. The operator D satisfies the elementary rules of Algebra. We will next consider how far the analogy goes between symbols of quantity and the symbol of operation which we have denoted by D. The fundamental rules of algebra are three in number and are known as (1) The ''Distributive Law,'* (2) The " Commutative Law,'' and (3) llhQ '' Index Laivr These three laws form the basis of all subsequent algebraical formulae and investigations. (1) The Distributive Law is that denoted by m{a-\-b-\-c-\- ...)=ma-\-mb-\-mc+ .., Now, in Chap. II., Prop, iii., it is proved that D{u-\-v-\-iv-\-...) = Du-\-Dv-\-Dw+,.., so that the symbol D is distributive in its operation. (2) The Commutative Law in algebra is that expressed by ab = ba. Now, in Chap. II., Prop. II., it is proved that Dcy = cDy, so that the symbol D is commutative with regard to constants. But it is clear that the positions of the D and the y cannot be interchanged ; such an error would be similar to writing 60 CHAPTER IV. ^sin instead of sin 0. So that, while D is commutative with regard to constants, it is not so with regard to variables. (3) The Index Law in algebra is denoted by a'^.a'^ = a'"+^ m and n being supposed to be positive integers. Now, to differentiate a result m times which has already been operated upon n times is clearly the same as differentiating m+n times, i.e., D^ . D'^y = D^+'% So the operator D"* . D^ is equivalent to the operator Z)"^+» where m and n are positive integers. Hence the symbol D obeys the Index Law for a positive integral exponent. To sum up then, the operative symbol D satisfies all the elementally rules of combination of algebraical quantities, with the exception that it is not commutative with regard to variables. 90. It follows from the above remarks that any rational algebraical identity has a corresponding symbolical operative analogue. For example, (m + a)(m +b) = m^-\-{a+b)m-i- ot6, so also the operation {D-\-a)(I)+b) is exactly equivalent to the operation D^-{-{a-\-b)D+ab. Similarly, to the identity {m + ay = m^-\-2am + a^ corresponds the equivalence of the operations {D-\-aY and 91. It is clear that in cases like the above an ab initio proof may be given of the identity of the operations represented. For instance, suppose it be required to show that (D-\-a){D-{-b)y = [D^ + (a+b)D + ab]y, we have (D + b)y = Dy + by, and (D + a)(D + b)y = (D + a)(Dy + by) ^D(Dy + by)^a{Dy + by) = Bhj + bDy + aDy + aby = I)hj + (a + b)Dy+aby ^[D^-\-(a+h)D+ab]y, the result to be proved: and the process of proof is exactly the same as that employed in proving that (m + a)(m + 6) = m^ 4- (22*-ip-l)!}2. Examples. 1. If y=^e«* find y„. 3. If 2/=^a' find yn- 2. If y=a7-sincu: find y„. 4. If y=a?*'e"sin6ji; find y„. 5. Prove that the differential equation (l+.r%o+a^i=m2y is satisfied by y=8inh(m sinh"^^). Prove also that ( 1 + ^2)y„^2 + (2n + lKy„+i + {ri' - rtv^jn = 0, and find the value of ?/„ when ^=0. 100. Some Important Symbolic Operations. It has been proved, Art. 93, that if r be a positive integer, Let us define the operation Z)"*" to be such that Thus i)~^ represents an integration (Art. 79). We shall sup- pose moreover that no arbitrary constants are added. Now, since B'^a - ^e^^ = e«^ = D^'D - ''e<^, it follows that D-''e<^ = a-'-e^. Hence it is now clear that for all integral values of r 'positive or negative. 101. Let f{z) be any function of z capable of expansion in integer powers of z, positive or negative ( = I^A^z'^ say, Ar being independent of z). 70 CHAPTER IV. Then /(i)>^^ = (S^.D^e"'' 102. Next let 2/ = e'^*X, where X is any function of cc. Then since D'^e'^^ = a^e^^, we have by Leibnitz's Theorem which by analogy with the Binomial Theorem (Art. 91) may be written D^e^^X = e^'^iD + a)"Z, n being a positive integer. 103. Now let X = {D + aY Y, so that we may write T=(D+a)~'^X. Then i)"e«^F=e«^(i)+a)"F(Art. 102), or D''e^(D + a) - ^X = e^^X, and therefore i)-^e«=^X = e«^(i) + a)-'^X. Hence in a^^ cases for integral values of n "positive or negative j)n^axx = e^%D + ayX, 104. As in Art. 101 we shall have /(i))e«^X = 2{ArD'')e^X = S(^^Z)^e«^^X) = e«^2^,(i) + a)^X = 6«y(D + a)X. 105. Again D^ ''^^ ma; = ( - m^) ''^na;, and therefore D^** ^'^^ ma; = ( — rrv^y ^'^" ma;. Hence, as before (Arts. 101 and 104), it will follow that /(^') Ts ^^ =/( - ^') 7s ^^- Ex. /e"''sin 6^ da^ = i>-ie«*siii 6^ = e^%I) + a)-hm hx (Art. 103) -(a-D)sm hx (Art. 105). „,a sin hx-h cos 6^ =e«^(a2 + 62)-isin/'6jp-tan-^-] (compare Ex. 4, Art. 93). SUCCESSIVE DIFFERENTIATION. 71 106. Successive Differentiation of F{x^). [Lemma, ii n^2k = -^ — h ^ it is an elementary exercise to show that This is left to the student.] We shall establish inductively that the series continuing until a zero coefficient occurs; ^A^ being supposed unity, and indices of F denoting differentiations with regard to x^. For differentiating this, the coefficient of is n^2fc+2(7i-2A;+2)„^2A-2, i.e. n+iA^k, by the lemma. Hence we obtain so that if the law holds for n differentiations it holds for n-\-l. Moreover, the law is obvious for one and for two differentia- tions. Hence it is true for any positive integral value of n. Ex. If i^(:i-)=e«^, then since we obtain 107. Successive Differentiation of F{^x). [Lemma. If nBzk — - — ' — -j-. — ^^ then will A/I vPik + n^2/t - 22(n + A; — 1) = n+1^2A;- The verification is left to the student.] We shall establish inductively that |;^(Vx)=5:A(-i/(i^rv-(v.). the summation continuing until a zero coefficient occurs ; „5o being supposed unity, and indices of J^ denoting differentiations with regard to aJx. 72 CHAPTER IV. For differentiating, the coefficient of i.e. n+iB2k by the lemma. Hence we obtain SO that if the law holds for n differentiations it holds for n + 1. Moreover the law is obvious for one or two differentiations. Hence it is proved true for any positive integral value of oi. Ex. Prove that dx''^ ' V2 ^x) ^r=o \ V ! (91 - r - 1) ! (2a ^xj] [Math. Tripos, 1886.] 108. Function of a Function. A general expression for the ti*^ differential coefficient of a function of a function will be found in Chapter V. 109. Note on Partial Fractions. Since a number of examples on successive differentiation and on inte- gration depend on the ability of the student to put certain fractional forms into partial fractions, we give the methods to be pursued in a short note . Let -^-^^ be the fraction which is to be resolved into its partial fractions. <^{x) 1. If f{x) be not already of lower degree than the denominator, we can divide out until the numerator of the remaining fraction is of lower degree : (.r-l)(^-2) (^-lX^-2) Hence we shall consider only the case in which f{x) is of lower degree than <^(^). 2. If ^a?) contain a single factor {x — a\ not repeated, we proceed thus: suppose ^{x) ={x- ay\r{x\ ■and let *) =_^ + XW {x - a)yjr{x) x — a y{x) A being independent of x. Hence ^l = A + {x - a)^^. ir{x) 'fix) This is an identity and therefore true for all values of the variable x ; put x=a. Then, since yj/ix) does not vanish when x=a (for by hypothesis "^x) does not contain ^ — a as a factor), we have A = M f{a)' SUCCESSIVE DIFFERENTIATION. 73 Hence the rule to find A is, "Put x=aiii every portion of the fraction except in the factor x-a itself." Ex. (i.) ^;-c ^a-c 1 ^ b-c ^ 1 {x — u){x-b) a — h'x-a h — a'x — h' Ex (n ) ^+p^+<7 ^ a^^r pa-\-q _1_ H^+ph + q 1 ^ ''^ (^-aX-^-^X^-c) (a-&Xa-c)^-a (6-c)(6-a)^-6 (c — aXc-6) ;r— c* Ex. (iii.) . ,,, ^.., , = ^-i ?-+ ^ Ex. (iv.) (jc- 1X^-2X^-3) ^x-\) a:-2^2(jc-3)' {x — aX-^ — h) Here the numerator not being of lower degree than the denominator, we divide the numerator by the denominator. The result will then be A B expressible in the form 1 + 1 -^ where A and B are found as ^ x-a x—b before and are respectively =- and := . a—b b—a 3. Suppose the factor {x-a) in the denominator to be repeated r times so that <^x)={x—ayylr{x). Put x — a=y. Then A^)_f^±y)_ w ; supposing n to be a positive integer. 5. \iy = A sin mx + B cos mx^ prove that y^ + rr?y = 0. 6. I£y = Ae"" + Be-"^", prove that y^ - m'^y = 0. 7. If y = ax sin x, prove that x^y^ - 2xy^ + {x^ + 2)y = 0. 8. l{y = a cos(log x), prove that x^y2 + ^2/i + 2/ = 0. 9. If 2/ = aa^'*'*'^ + bx-^'y prove that ar^v/^ = n{n + l)y. 10. If 2/-2 = 1 + 2 n/2 cos 2£c, prove that y^ = y(3y^ + l){7y^ - 1). [Oxford, 1889.] SUCCESSIVE DIFFERENTIATIOK 75 11. Ify = x log r— , prove that a^^/o = {y- ocyi)^. a + ox 12. If 2/ = sin a; prove 4— ^^|^= 105 sin 4a;. [Oxford, 1890.] 13. Find the w*** differential coefficient of e'^jaV _ 2nax + n(n + 1)}. [I. C. S.] 14. If w = sin nx + cos nx, show that u^ = 'nr{\ + {-lYs,m2nxY. [I. C. S.] 15. If 2/ = sin "^x, prove that (1 - x^)y^ -xi/^ = 0; also that (1 - a;%„+2 - (2n + l)a:2/«+i " ^'2/n = 0. 16. lfy = A{x+ Va;2 + «2)« + ^(a;+ ^/a:2 + a2)-», then will (aj2 + a^)ij^^z + {2m+l )xy^+i + {m^ - n^)y,r, = 0. i -1 17. If 2/"* + 2/"* = 2a;, prove that {^' - l)yn+2 + (27^ + l)an/„^, + (^2 _ m^)y„ = 0. 18. If 2/ = e~* cos a;, prove that y^ + 42/ = 0. 21. If 2/ = (a;*" - 0~^> fi"^^ ym ^ being a positive integer. 22. If 2/ = a;**log a;, find y„. 23. If 2/ = (1 + a; + a;2 + ar^)-i and^^ = cot~ia;, show that 7/,, is J( - l)"n ! sin"+i^{sin(n + 1)^*- cos(n + 1)^ + (sin + cos (9)-"-^}. [Math. Tbipos.] -1 24. My = e**" * = «o + ^1^ + "2^"^ "•"•••» show that |.) (l+a)2)2/2+ (2a;- 1)2/1 = 0; p.) (l+a;%„,2+{2(n+l)a;-l}2/„^i + n(n + l)2/„ = (iS.) (n + 2)a„^2 + wa„ = a^+i. The last equation is to be found by substituting the series for y in equation (i.) and equating the coefficient of a;** to zero. 25. If 2/ = sin(m sin ~'^x) = a^ + a^x + a^^ + . . . , show that (i.) {l-x^-)y^ = xyj^-m^y; (ii.) (1 - x^)y„^, - (2n + l)xy„^, - (n^ - m^)y„ = ; and (iii.) (n+1 ){n + 2)a„+2 = (^^ - ^^)«n- -1 26. If e" "^'^ * = Oq + a^a; + ^2^2 4. _ , ^ prove (n + l)(n + 2)a„+2 = (n2 + a^)a„, 27. If (sin-ia;)2 = a^ + a^x + ag^^^ 4. ^gar^ + . . . , show that (n+l)(n-i-2)a„+2 = «V 76 CHAPTER lY. 28. If w, V, w De functions of t^ and if suffixes denote differentia- tions "with regard to t^ prove that dt Wp i^i, w^ = Wj, ^^1, 1^1 ^2, ^2' *<'2 ^2, ^2' "^2 ^3' ^3> ^3 W4, V4, t»4 29. If 1 -1 [Coll. Exam.] be differentiated i times, the denominator of the result will be (e*-l)*'^\ and the sum of the coefficients of the several powers of e* in the numerator will be ( - 1/1 . 2 . 3 ... t. [Caius Coll.] 30. Prove that d/'u d'^uv d^- dx"" dx"" dr-^ ( dv\ n(n-l) d^-^ / d^\ ''dx^-'X^dx) "^ 1.2 dx^'-'Vd^y .+(-l)^u dTv daf"' 31. Show that if a? = cot y n ! sin y{ sin y - „CiCOs y sin 2y + JJ^cos^y sin 3?/ - . . . } . dx"" l+£c2 32. Prove that if ac > b^ d"" b + cx dx"" a + 2bx + cx^ [Oxford, 1890.] 33. Show that tsmfy-- jsin nix = tanh my . cos mx ; tan~^(y- Jsinma; = tanh~^m2/. cos ma;; gdfy—\in mx = gd~\my)cos mx. [London, 1890.] [Oxford, 1888.] also and 34. Prove \dx) e'^a" = «*■-"»;**-'•( — 35. Prove that if aj + y = 1 d [Gregory's Examples.] ^>V) = n\ (2/" - JJ]y--'x + JJly-^x^ - . . . ) 1 d [Murphy, Electricity.] 1 /c?\2, 36. Prove that arloga; + a:(loga.)2 + i^ JL^^'^^^S^')'} +^,(0'{.^^(loga:)n + ...tow + l terms = —!,--/ 4-V{a;""'Yloga;r+n. {n+ \y\dxj ^ \ o / J [Math. Tripos, 1889.] SUCCESSIVE DrFFERENTIATION. 77 37. Find the n^ differential coefficients of sin x^ and cos x-. 38. Establish Rodrigues' Theorem* that if n be a positive integer sin?ia;= -(-. — (sin a;)-" \ 1 .3.5 ...{2n- i)\sinx axj 39. Prove that h T - 1.3.5...(2A;-1) / l-a; y wnere ^*-(2n- :^)(2n- 5) ... (2n- 2^- l)Vl +a;; ' [Frenet.] 40. If /(cc) = «() + «!»: + a^^ + . . . + aX + • • • > prove that 41. If (l)a;+ <^(2)a;2+ ... +(n)a;" = <^/'a;^^xl^. 42. Ify^ic) can be expanded in positive integral powers of a;, prove that AD){uv) = uf{D)v^Duf{D)v + ?^r{D)v,.P^^r{D)v-, ... . 43. Show that the Bessel's Function JJ^x) (Ex. 125, Chap. III.) satisfies the differential equation d^u \du / ^^ =0 dx^ X dx \ ojV 44. Prove that Legendre's function of the n^^ order, viz., satisfies the equation and may be expressed as (a) hvT + jC\ur-'v + J^X-H'^ + . . . + V"} ; where w = aj + 1 and v = x~\ ; (P) =0"+ ^"Oj'C^x-^x'' - 1) + I-c/c^x"- V - 1)' * M. Frenet has pointed out [Recueil )+... +^/^(a') + ... to infinity, (l^^ ^ ^ fe'iKv^ an expansion of/(a;+^) in powers of A. EXPANSIONS. 8X This result was lirst published by Taylor in 1715, in his "Methodus Incrementorum Directa et Inversa." In 1717 Stirling pointed out another form of Taylor's Theorem, viz., y(a5)=/(0)+a/XO)+f/XO)+|5r(0) + ... J^—f(0) + ... to infinity, which is easily deducihle from Taylor's Series by writing for X and x for h; the meaning of f%0) being that f(x) is to be differentiated r times with respect to x, and then x is to be put equal to zero in the result. The latter series gives a method of expanding any function of X in positive integral powers of x. Being a form of Taylor's Theorem it is subject to the same limitations. It is generally known as Maclaurin's Theorem, though its publication by Maclaurin was not made until twenty-five years after its first discovery by Stirling. 114. Taylors Theorem also deducihle from Maclaurin's. It has been shown that Maclaurin's series is deducihle from Taylor's form. Taylor's series is also deducihle from Maclaurin's. For, let f{x) = Fix-\-y), then fix) = F\x -f- 2/), e tc. , so that /(O) = F(y), /(O) = F\yl f(0) = F'\y\ etc. Hence Maclaurin's Theorem /(a,)=/(0)+^/(0)+g/'(0)+... becomes F{:y +x) = F{y) + xF\y) + f^F\y) + . . . , which is Taylor's form. Taylor's Theorem. 115. Prop. To prove that, iff(x+h) can be expanded in a convergent series of positive integral powers of h, that ex- pansion is f{x+h)=f(x)+hf(x)+'^'\x)+ ... to oo. Put x+h = X ; then since x and h are independent dX dh~ E.D.C. F 82 CHAPTER V. Hence <^/W = ^).^=/(Z). dh dX dh ■' Similarly ^^)=/"(X), etc. Now, assuming the possibility of such an expansion, let f(x+h) = A„+AMA^+ A -, + ■■■> (1) where Aq, 4i> ^2' ••• ^^'^ functions ofx alone, not containing A, and are to be determined. Differentiating with regard to h we have, by the preceding work, f'(x+h)^^fi^ = A, + AJi+Az^+-^!^, + (2) Differentiating again f"{x-\-h)='' ^^1^ ^ = A^+A^h + A,j^-{^A,~i-,.., (3) etc. Put h = 0, and we have at once from (1), (2), (3), ... A^=f(x), A^=f(x\ A^=fXx), etc., ... Substituting these values in (1) f(x+h)=f(x)+hfXx) + '^/'(x)+...+'^-,fix)+... 116. This theorem may be written and by analogy of form with the exponential theorem the operator may be represented shortly by Thus f(x-{-h) = e''''f(x). Stirling's or Maclaurin's Theorem. r 117. Prop. To prove that if f{x) can be expanded in a convergent series of positive integral powers of x, that ex- j)an8ion is /(a;)=/(())+<(0)+|^/"(0) + |^/"'(0) + ... to 00. Assuming the possibility of such an expansion, let X QCy /(a;) = ^„+^ia; + ^,2,+^33J+..., (1) EXPANSIONS. 83 where Aq, A^ A^,..., are constants to be determined, not containing x. Then differentiating we have f\x) = A, + A^ + A^^^,+A,^^-\- (2) f\x) = A, + A^. + A^^^ + A^-^ , (3) etc. Hence putting a; = in (1), (2), (3), ... , we have ^,=/(0), A,=f\0), ^2=r(0), etc., ... : and substituting these values in (1) fix) =/(0) +«,/(o)+|y (0)+ Jr(0)+ . . . + Jao)+ . . . 118. It will be noticed that in the above proofs there is nothing to indicate in what cases the expansions assumed in the equations numbered (1) in Arts. 115, 117 are illegitimate, and we shall have to refer the student to Arts. 130 to 142 for a fuller and more rigorous discussion. 119. It is important before proceeding further, that the student should satisfy himself that the well-known expansions of such functions as {x-\-hy\ e*, since, etc., are really all included in the general results of Arts. 115, 117. For example, if f{x) =af^, f{x+h)-={x+hY, f{x) =nx''-\ f'{x)=n{n - l):c""2, etc. Hence Taylor's Theorem, f{x+h)^f{x)+hf{x)+^^r{x)+... , gives the binomial expansion Again, suppose /(:i,-) = e*, then /'(.r)=e*, /"(•2^) = e*, etc., tlierefore /(O)=l,/'(O)=l,/"(^0 = l, ^'tc Hence Maclaurin's Theorem, f{x)=f{0)+xf\0)+tf'XQi)+... , gives ^=l+^+|!+|^+..., the result known as the Exponential Theorem. 120. We append a few examples which admit of expansion, and to which therefore the results of Arts. 115, 117 apply. [Oxford, 1888.] 84 CHAPTER V. Examples. Prove the following results : — 1. sma: = ^— +- — ... . 2. log(l+^)=^_'|V^-.... 3. tan~^^ = :r ! 3 ' 5 4. e*cos £c = l + 2^cos'^ • X + 2*cos?^ ^ + 2^cos?^ ^ + 4 4 2! 4 3! + 22COS-— — + .... 4 nl 5. cos^.cosh^=l — 4-r+-8-i uY 6. log(l + e-)=log2 + |^+^^2_^^.... 8. sin(^+A) = sin^+Acos^- -sin^ — — cos;r+.... -^ ' o! 9. sm-\^+A)=sin U'H , —-\ -. • — »-.... v/1-^2 (1-^2)1 2! (l-^2)t3i A «* cos 7* 10. logsin(^+A) = logsin:r+Acot:r-— cosec2^H ^-^4-.... 11. 8ec-i(^+A)=sec-i^+ — -L=-~?^^^^ — + .... Method III. 121. Expansion by Differentiation or Integration of a known series or equivalent process. The method of treatment is indicated in the following examples : Ex. 1. To expand tan~Kv in powers of x, assuming x to be numerically- less than unity. Gregorie's Series.* Suppose f{x) ■■= tan~^^ = «© + ^i^ + ^^^ + ^^ + . . . , then f{x) = ^ = «! + 2^2^ + 3a3^2 4. 4^^^ + . . . ; also (l+^2)-l = l_^2+^_^_,_^,_ Hence, comparing these expansions, we have • a2 = ct^ = (iQ = a^= ... =0, and «i = l, 3^3= —1, 5a5 = l, etc. Also ao=tan~^0 = W7r ; therefore tan"^:^; =mT-{-x- ---+ ;. + .... 3 5 / * Commercium Epistolicum, \). 98. EXPANSIONS. 35 This result may be obtained immediately by integration of the series the constant Oq being determined as before. Ex, 2. To expand sin-^x. Suppose f{x)=sm~^x—aQ+aiX-{-a^^ + a^+... ; therefore f{x) = — — = ai + Soaa: + Za^"^ + 4a4^ + . . . . Vl-^2 But _1^=1 + ^^2^1jl3^_^ -^l_^-2 2.4 Hence, comparing these series, we have a^=a^ = aQ= ... =0, 1 3 and ax = 1, 803 = ^, Sa^ = " -. . . . 2.4 Also «() = sin ~ ^0 = ?i7r. Hence sin-..=«.+xH.^+^-^^+'-|^ • ^+...; and, as before, this might have been obtained immediately by integration of the expansion of - — =. Ex. 3. Again, if a known series be given, we can obtain others from it hy differentiation. For example, borrowing the series for (sin~'^^)2 established in Ex. 2 of the next Art., viz, — we obtain at once by diflferentiation ^iZa^ ^3 ^3.5 ^3:y?7 ^ Examples. , , 1 t"3 1 3 r^ 1. Erove log(jc+vl +^^) = sinh~'.r=,r — ';- + -^ • ' - — .... 2. Prove t^nh-^x=-x+^ +^ -{-.... 3 5 Expand examples 3 to 9 in ascending integral powers of x. 3. tan~i.r + tanh~^r, 4. tan-i -^^,+ sinh-^ ^f-,,- 1 - .r^ 1 - ^-^ 5. tan-i?^::^+tanh-i?^:^. l-3a?2^ l + 2x^ 6. tan-i-."^ 'l-,r2 86 CHAPTER V. 1 /. sec 8. sinh-X3:i:+4^). 10. Deduce from Ex. 3, Art. 121, i^ 2u -1 ^ps 2a'5 2.4 .c^ (l-^>m-^=^-----^- — .^-.... And hence by putting a?=sin ^, prove a .a ^ sin2^ 2 sin*(9 2 . 4 sin^^ (7 cot t7=i — . — ^ — - — ^ .... 3 3 5 3.5 7 [Quarterly Journal, vol. vi,] 11. Given that sin log(l +^)=4T^ + 4f ^2 + ^3^+. ..+^^»+... coslog(l+:«;)=l+^j^+^^^2+^;^+... + ^';^+..., calculate the first eight coefficients of each expansion. [Math. Tripos, 1887.] 12. Prove that when x is between -— and4--, 2 2 -i cos a? - — cos 3.27 4-— cos 5.r- ... to infinity =^(^ -.r^ !» 33 53 8V 4 . [Math. Tripos, 1875.] Method IV. — By the Formation of a Differential Equation.* 122. This method may often be employed with advantage. Assume a series for the expansion (say aQ-\-a^x-\-a^^-\- ). Then form a differential equation in the way indicated in several of the examples in the preceding chapter. Substitute the series in the differential equation and equate the coefficients of like powers of x on each side of the equation. We shall thus obtain sufficient equations to find all the coefficients except one or two of the first which may be easily obtained from the values of /(O) and /'(O). * Professor Williamson has pointed out that some historical interest attaches to this method, as having probably been employed by Newton in his expansion of 8in(?» sin"^) and other expressions. EXPANSIONS. 87 Ex. 1. To apply this method to the expansion of (1 +ar)". Let ^=(l+xy=aQ+aiX+a2a;^+a3pC!^+ (1) Then i/i = n{l +xy-^ or (l+x)]/i=ni/ (2) But yi = ai + 2a^v + 2ajX^+ (3) Therefore substituting for (1) and (3) in the differential equation (2) (H-.'pX«i + 2a2^ + 3a3a:2+...) = 7i(ao+a]5: + a2^2+ ) Hence, comparing coefficients ai=naQ, 2a2+ai=naif 3a3 + 2a2 = nag, etc. , and by putting ^=0 in equation (1), giving «! = ??, n — l n(n — 1 ) w-r+1 _n{n-l)...{n — r+l) ar= «r-l = r r\ whence (1 4- .a?)" = 1 +na:+ ^ — ^a:^+ . Ex. 2. Lety=/(^)=(sin-*a:)2 yi = 2sin-^a: Differentiating, and dividing by 2yi, we have (l-^)g^2=-^i + 2 (1) Now, let y = ao + ai^ + ajfr^ + , . . 4- a„57" + a„+ 1^:"* ^ + a„+2^"+2 + , , ^ ^ therefore yi = cfi + 202^:+ . . . +7m„ar'*-i + (w+ l)an+i^"+ (w + 2)a„+2^''+^ + . . ., and y2=2a2+...+7i(w-l)a„a:"-24-(n + l)Ma„+ia7"-^+(»i+2)(n+l)a„+2^"+.. . Picking out the coefficient of of* in the equation (which may he dona without actual substitution) we have (72 + 2)(7i+l)a„+2-w(7i-l)a„ = 7ia„ ; n" *^^^^^^^^ «— (^-+lX.H^f" ^'^ Now, «o =/(0) =(sin-^0)2, and if we consider sin~*a: to be the smallest positive angle whose sine is a: sin-iO=0. Hence a^ = 0. Again, «i=/'(0) = 2sin-i0.— L^ = 0, vl —0 and «2=^/'(0)=i(^ + 0) = l. 88 CHAPTER V. Hence, from equation (2), a^j ©5, a^, ... , are each=0, and a,=_.a,=_ = -2, 42 22.42 22.42 _ etc. = etc. ; therefore (sm-i^)2=2^V222^ 22^2 22^42.62 ^ ^ 2! ^4! ^6! ^8! A different method of proceeding is indicated in the following example. ^2 a^ Ex. 3. Let ■3m{msm-^a;) = aQ + aia; + a^^ + a3~^ + (1) Then ^1 = cos(m ain-^zp)— , Vl— ;r2 whence ( 1 - x^)^/^^ = m%l- f). Differentiating again, and dividing by 2yi, we have {l-x^)7/2-x^i + m^i/ = (2) Differentiating this n times by Leibnitz's Theorem (l-^2)^n+2-(2?l+l>y„+l+(w2-?l2)y^==0 (3) Now, Uq = Q/)x=o = sin(m sin-^0) = 0, (assuming that sin~^jj7 is the smallest positive angle whose sine is x) «2=(.y2)^=o=o, etc. «n = (3/n)x=0. Hence, putting :r=0 in equation (3), a„+2 = — (m^ — n^)an. Hence ^4, ag, «§, ... , each = 0, and a3= - {m^ - \^)ax = - ^ifm2 - 12), ag = _ (m2 - 32)a3 = m^ni^ - V){m^ - Z^\ ay = - (m2 - 52)a5 = - 7n(m2 - l2)(7/i2 - 32)(m2 - 52), etc. Whence 3! 5! _ w(m2 - I2)(m2 - 32)(m2 - 52) ^^ ^ The corresponding series for cos(m sin-^^) is 2! 4! 6! If we write ^=sin ^ these series become sin mS^m sin ^_M^^^-1^) sin3^+^(^'-l'X^'-3^) sin^^- etc., 3! 5! .««^/3 1 ^^ • 2Z1 m2(wi2-22) . .^ m2(w2-22)(m2-42) . „^ . , cos W26' = l --^- sin2^-| ^i-— i- Bm^d ^^ —^ ^^sm^^+etc. EXPANSIONS. g9 Ex. 4. Expressions of the form {ta.n-^a;y/p\ may be easily expanded as follows : Taking tan~^^ to lie between - and — — we have 4 4 tan~^^=:r--^ +—-.... 3 5 We may therefore evidently assume expansions of the form y = (ta.n-'^xy/p ! = apXP - ap+^+^ + ap+^x^^'^ - ... = (tan-la;^- V( Jt? - 1) ! = 6^-1^^-' - 6^+i^^+H 6p+3^^+' - . . . . Then 2/i=2(^+.v^)-\ or papxP-^-{p + 2)ap+^p-'^-{-(p + 4)ap+^xP*^-... whence, equating coefficients, {p + 2)ap+2 = hp-i + bp+iy (jo+4)ap+4=6p_i + 6p+i + 6p+3, etc., and the law which connects the several coeflBcients is obvious. Thus starting with Gregorie's Series we successively deduce (tan-^.)^ ^x^/ IN^ /,^1 1N^6_A^1 1 iNg / 2! 2 V 374 V -i bJiS \ 3 5 7/8 3! 2 3 12 4V 3//5 12 4V d) 6\ 3 5/ J 7 etc. These results have been communicated to me by Professor Anglin of Queen's College, Cork. Examples. 1. Apply this method to find the known expansions of a*, log(l+;r), sin.?-, tan~^.r. 2. If y = sin-^a; = CTq + «i-^ + %^^ + ^s^^ + . . . , prove that a„+2=- — -Ji— — - a„, and in this manner deduce the expansion given in Ex. 2, Art. 121. 3. U €^''^''~^'=aQ + aiX+a-iX- + a^+...y prove /i\ n^ + a^ „ . (2) ."..-'-= i+<„+^+*^..+2X2^)^ 90 CHAPTER V. (3) Deduce from (2), by expanding the left side according to the exponential theorem and equating the coefficients of a, a^, ... the series for sin"^^, (sin-^^)^, ..,, and show that if in the development of (sin~i^y ^ J— ^ , VIZ., l"^2' 3"^2.4' 5"^""' every number which occurs be increased by unity, the result, viz., ^2 2^ 2^ ^ 2"^3 4'^3.5*'6 "* is equal to i^-^^)', 4. Prove that if logey=tan-^^ (1 +^%n= { 1 - 2(7* - 1)^ }y,._i -{n- l)in - 2)^^.,, and hence find the coefficient of a^ in the expansion of y by Maclaurin's Theorem. [I. C. S. Exam.] _ -p- (tan~^:y)^ _a»y^ a^x* ^ a^ 2! ~~2 T^~^ •••' prove that a2„ — Osn-a = • 6. If y satisfy the equation y,^ — m^y=0^ and if the first and second terms of its expansion be respectively A+B and {Am — Bm)a;^ show that the general term is i A +( - 1)*^}??^. Hence show that y^Ae'^+Be-'^. 7. If y satisfy the differential equation and the first terms of the expansion of y are \—kx-\ — x^+... A continue the expansion. 8. If sin-^^=2 -^ and ^^ — ^rr— ^ = S -^> show that a»+2=^^«n + 6«. Hence establish the expansion (sirr^_l a:^ 1 . 3/ 1 1 \^^ 1 . 3 . 5 / 1 1 1 \^^ 3! ~2' 3"^2.4U2"^3V5'^2.4.6U2"^32"'"52;7 .,. ( 8inh-^^ )2_^2 2 ^ 2^£^_ ^' 2! ~2 3' 4 "''3.5 6 •"* + .... (c)log(l+V2)=l-i. 1 1.3 3^2.4 1 5 ■... • ^^tM-^)-^ 3^3.5 1, 3 .2 .5 .3 .7 10. Establish the expansion '^'=1 + 1. ^-^ 8 ^2 3^3 3.5^4 I. ■ 3. , 2 . "5^ ,3 ,7' [Anglin.] [Anglin.] EXPANSIONS. 91 Continuity. 123. Def. a function 0a; is said to be continuous between any two values a, h of the independent variable involved if, as that variable is made to assume successively all intermediate values from a to 6 the function does not suddenly change its value, but is such that its Cartesian graph [y = (px] can be described by the motion of a particle travelling along it from the point (a, (pa) to the point (6, ^6) without moving off the curve. 124. Trace the curve y = (px between the ordinates AL{x — a) and BM(x — b). Then if we find that as x increases through some value, as ON (Fig. 14), the ordinate x cuts the axis at J. (x = a, y = 0) and EXPANSIONS. 93 B (x = b, y = 0), then it is obvious (Fig. 16) that if the curve y = '^-\z\ f'^{x + z) - R=ct>\z). All the functions ^(0), ^'(z)..., (l)\z) are finite and con- tinuous between the values and h of the variable z, and evidently 0(0), <^'(^)> "(^)'-' , (p'^'HO) are all zero. Also from equation (1) ^{h)=0. Therefore by Art. 126, " (^) = ^ ^0^ some value Qi^ of z between and \, (ji'Xz) = for some value (^,3) of z between and h^, and so on ; and finally fj^n, (0) = O for some value (A^) of 2; between and hn-i. EXPANSIONS. 95 Thus f\x-\-h^)-R = 0. Now since hn < /i„_i < /t„-2 ... —P^ a form suggested by the remaining terms of Taylor's series. Consider the function formed by writing z instead of x throughout the left hand member of equation (2) except in P. Let <^(2)= f{X)-f{z)-^l^f\z)-i^^nz)-...-i^"^^^ From equation (2) cf){x)=0, and it is evident that cf){X)=0. Also cf)(z) and +fe-?^ w whence F=f\z) for that value of z which makes (x)+z where z is written for =)+^^f(«>)+'^4>"'(s»)+ (1) Hence f(u) becomes f{u + z). Thus f{(/>{x-hh)}^f{u-hz). Expanding each side by Taylor's Theorem Substituting the series (1) for in the right hand member of equation (2) we obtain on equating coefficients of h^ + coefi: h- in ^-l^[^(a.+/t)_0(a;)]n-i/«-i(u) H-coefi-. h- in ^-l^^[0(a;4-/O-^(a5)?-y»-2W + etc., the result stated. EXPANSIONS. 105 Ex. 1. Suppose u=x^y and therefore u=f{x^). Here {x +h)- {x) = A( 2^ + A). We thus have to pick out the coefficient of A"* from the series ^(2.;+ A)7^:r2) + ^-il:i-j (2^+ A)«-^ thus obtaining =:=(2x)"/»(:^)+?(g--jl>(2xr-V^-X:r^+ '^"-lX^-^X«-3) (2x)"-y»-'(x^+... as inductively proved in Art. 106. Ex. 2. If u=a + bx+car^ and Ui = b + 2cx, prove that //♦■-J/** ^ = w(7l-l)...(7l-r+l)M»»-'-Mi'- r r(r-l) C2. r(r-l)(r-2)(r-3) cV ^ \ ■^l.(H-r+l)wi2-fi,2.(n-r+l)(7i-r+2) i*/ ••••]* [Lagrange.] Bernoulli's Numbers. 148. To expand u =/ (:c) = - — ^^- tn powers of x. Let u=f(x) and u'=f{0), ui=f{x) and u\=f {0)y z^=/'(:r)andw'2=/'(0), with a similar notation for higher differential coefficients. Then Mac- laurin'e Theorem gives X e*4-l , . / x^ , Changing the sign of x we see that the left hand member of this equation remains unaltered ; hence we have u = u' -xu'i+—u'2 — ... , 2 ! and by subtraction = 2^i + 2^i*'3 + 2|^<+..., whence, by equating to zero the coefficients of the several powers of x, we infer that w\ = w's = w '^ = . . . = 0, so that the expansion contains no odd powers of x* Again, since e*M = m + _- + a;^, we have, by differentiating, c*(m2 + 2^1 + ^) = -^^ + (^ + 2)^j A c*(t^3 + 3w2 + 3wi + w) = Uz + (^ + 3) ^ etc., * This artifice may often be advantageously employed. O 1 H UNIVERSITY 106 CHAPTER V. and putting ^=0 in these equations we obtain from the first, third, fifth, etc., w'=i+i 1u'q+25u\ + 2Uc'2+u' = 1, etc., giving u' = l,u'2= h ^U = - #(7> ^'e = T2 » ^^ = -wu^ etc. Hence ^ ^-!±l = i+l ^-1 ^+1 ^_J: ^+ 2 e*-l 6 2! 3041^426! 30 81 This series introduces a set of coefficients which are found of great importance in the higher branches of analysis. The series is frequently written in the form and the numbers ^i, B^, B^^ ..., which are calculated above are called Bernoulli's numbers, having been first discovered and used by James Bernoulli.* The coefficients of this expansion were investigated as far as the term containing ^2 ^y Rothe, and published in Crelle^s Journal. Professor Adams has recently calculated thirty-one more.f 149. Many impoi-tant expansions can be deduced from that of ^ "^ . e* I Q-x g2a: I I For example, ^coth^=.a7 — — =x — _L Writing ix for x, ix coth lx becomes x cot ^, and we have xcotx=l-B^'^-B^'^-... Again, tan x = cot ^ — 2 cot 2^ -X ^^2r"^^4r~-"l2^-^^2T-^^4! --J EXAMPLES. 1. Find the first three terms of expansion in powers of x of log(l + tan cc). Result. x-^x^ + lx^+ ,.. 2. Expand as far as the term containing a;* (1) log(l + cos a;) and (2) log(l + X sin x). f(l)log2-^'-^... Results. V ^ ^ 4 96 i(2)a;2_2^ + ... * Ars Conjectandif p. 97. t Encyclopaedia Britt, : Infinitesimal Calc. Proceedings of the British Assoc, 1877. EXPANSIONS. 1Q7 3. Provelogcosa;=-^^-2— -16--272^... ^ 2! 4! 6! 8! 4. Prove e"'='«"=l+a; + ^-^-iI^-^... 2 3 24 5 5. Prove e"«^*=l+a; + i£c2 + :^,^^ . Prove loer = + V-1 2 24 2880 7. Prove log{log(l + cc)^} = -? + ^-^+?^ ^'^ ^^ ^ ^ 2 24 8 2880 ••• 8. Prove Iog(l +a; + iB2+ ar»+ a:4) = ^ + ^ + ^ + ^_ 4^^.^^ ^^^ li t> 4 D 9. Prove (l+a;)'=l-fa^-K + l«^-f«^--- 10. If a„ be the coefficient of a;** in the expansion of e*sina;, show that . mr sm — _«^ a^_a„_3 ^ 2^^ " 1! 2! 3! n! [I. C. S. Exam.] 11. From 2/ = (a; + ^1 + aj^)** obtain a linear differential equation with rational algebraic coefficients, and by" means of it find the expansion of y in ascending powers of x, 1 2. From the relation y = i- — zL obtain a linear differential equa- tion with rational algebraic coefficients, and by means of it find the expansion of y in ascending powers of a;. [I. C. S. Exam.] 13. If tan 2/ = 1 + aa; + h:/?^ expand y in powers of x as far as ^. [I. C. S. Exam.] 14. If ^0' ^1' ®^^-» ^® *^® successive coefficients in the expansion [I. C. S. Exam.]- 15. If a„a3" + «„^.icc"+^ + «z„+2''«""^^ be three consecutive terms of the expansion of (1 - x^^%\vr^x in powers of ic, prove that _n-\ n + 2 also that all even terms vanish, and that the expansion is x-h^-^,x^- J:^ x'^-... 3 3.5 3.5.7 [QUAETERLY JoURNAL]. 108 CHAPTER V. 16. If x^ + 2x -K-S) and y=^aQ + a^x + ~H^+ ..., A 1 show that a„+2 + a„+i + na^ = 0. [Oxford, 1888. ] 17. Prove -^I^+M^Z^ =/(.c) + g/>) + J^^^^^^^ 18. If ^ = loga;, ,1 , du x^ d^u , , ^ du ^ (log 2)2 c^m , prove that « + x_H-^j^,+ ...=« + log2.^ + i-|j±^-^, + ... 19. Deduce from Taylor's Theorem, by putting h= -x, the series /(x) =/(0) + .m - J/"(x) H- J/"(x) - etc. ^^^^^^^^^^ 20. Prove tan~i(a3 + h) = tan~^a; + {h sin 6) sin ^ - ^ ^^^ ^ sin 2^ + ^^ — 5 — ^ sm 3^ - ^ — - — ^ sm 4^ + etc., where a? = cot 0. 21. Verify the following deductions from Ex. 20 : — n ^ + — - — sm 26 + — -— sm 3( by putting h= -x= -cot 6. (1) _=: 6^ + 008^. sm^ + — - — sm2^ + — ;r— sm3^ + _^sm4^+ ... 2 A O 4: (2) 5 = | + sin^ + Jsin2(9 + Jsin3(9 + Jsin4^+... by putting h= - sj\+x^=^ - ^j^. /Q\ T_sin^ 1 sin 2^ 1 sin 3^ 1 sin 4^ ^''' 2~cos^ 2*"cos2^ 3*cos^ 4 "^os*^ "" by putting ^= -x----^ - -r—^ -. [Euler.] ^ ° X sm ^ . cos ^ 22. If iSHr l>e a rational fraction in which the denominator has n F{x) factors, each equal to a; -a, and the remaining factors are x-h^ x-k^ etc., so that Fix) = {x~ aY<^{x) where <^{x) — {x-h)(x-k)..., prove that M 1 /(«) I 1 d lf(a) ■> J'(a;) (a;-a)".^(a) (K-a)""' rfa\<^(a)/ 2 !(» - a)"-' rfa^l^a) J a; - A (»t-l)!da"-'\(o)(a:-a)J x-h '" EXPANSIONS. 109 23. Establish the following approximations to the length of a circular arc : — Let C be the chord of the whole arc, H do. half the arc, Q do. quarter the arc. (1) Arc = — nearly. [Huyghens.] (2) Arc = ^±^51^^12^ nearly. 45 Examine the closeness of the approximation in each case. 24. Find by division the first six of Bernoulli's coefficients. rp, 1 1 1 1 5 691 iney are -, ^, -, — , — , ^^. 25. Prove by continuing the differentiations in Art. 148 that n+1 2 2! ^ 4! ^ ' a formula from which the values of the coefficients B^^ B^... can be successively deduced by putting n = 2, 4, 6, etc. [De Moivre.] 26. Expand ( ^— ^ ) in powers of $. [DiflFerentiate expansion of cot^, Art. 149.] 27. Prove -^ = 1 +2(2 - l)|i^2 + 2(2^- l)ff^+ ... sm^ 2! 4! [Use cosec ^ = cot - - cot 6 and Art. 149.] 28. Prove tanh x = ?^i^^Z^B,x - 2'(2^-l)^^ + . . . 29. By taking the logarithmic differential of the expression for sin in factors and comparison of the expansion of the result with that of ^ cot e (Art. 149), show that _ 2(2n)! 1 n(^l-_j [Eaabe.] where 11(1 - - j denotes the continued product of such factors as 1 - — for all integral prime values of r from 2 to oo . 30. Show that , ^sinhrr ^ 22^:2 2*0^4 2%« 110 CHAPTER V. 31. Expand log?^^ and log. tan a; by means of Bernoulli's num- aj bers. [Catalan.] 32. Show that log cosh .T- cos g; 22«-^^cos^ ?^.^-^0. ^ x^ 2 2n (2n)! [Math. Tripos, 1890.] 33. Expand sin(m tan~ia3)(l + x^)^ in powers of x. [Bertrand.] 34. Being given the two convergent series 2/ - CTq + a-^x + a^'^ + . . . + cf-„£c** + . . . log y-hQ + h^x-\- h^x^ + . . . + &„aj" + . . . prove na^ =- 6ia„_i + 262^^-2 + ^Mn^s + . • • + nh^a^. 35. Prove tan-ia: = - \\\ ^^l \ +...1 [Frenet.] 36. In the equation f{x-\-h)^f{x)^}if{x^Qh\ if d be expanded in powers of A, the first four terms will be ^-2 + ^4 7/^48 ^^^^ + 57-60^3 * +••• suffixes being used to denote differentiations. 37. In the equation f{x + h) =/(«) + hf(x) + . . . + _i^^/»-i(a, + SA) show that the limiting value of ^ as ^ is indefinitely diminished is — 38. If in a plane curve y =f{x), Fbe the midpoint of a chord AB drawn parallel to the tangent at any point P {x, y), prove that when AB approaches indefinitely near to the tangent at P, the angle which PF makes with the axis of x approximates to tan~i(p--^Y where p^ q and r are respectively the first, second and third differential coefficients of y with regard to x. [Oxford, 1890.] Show also that the angle which FY makes with the normal is ultimately tan-^jp- ^ 32 f-^ [Oxford, 1886.] * In a circle PF coincides with the normal. This angle therefore measures the deviation of the curve from the circular shape. Transon (Liouville, vol. VI. ) calls the angle the "deviation." Dr. Salmon names it the "Aberrancy of Curvature " (see Higher Plane Curves, p. 356). EXPANSIONS. Ill 39. If u = e'^ = ^^x^, ove that «...=Ji(»»+«»-.-^"ir=-V^-<1> 40. [Gregory's Examples.] Show that l\ £"[''"^'°^ "=)"] = 1 + ^1 log 0. + |(loga:)^ + ... + ^,{\ogxf where S^ = the sum of the products r at a time of the first n natural numbers. [Murphy.] 41. If F{z) and f{z) be two functions which are continuous and finite, as also their differential coefficients, between the values x and x + h of the variable z, and if f'{z) does not vanish between these limits, prove that F{x^h)-F{x) _ F'{x-\-eh) /(x + h)-f{x) f{x+eh) where 6 is some positive proper fraction. [Cauchy.J CHAPTER VI. PARTIAL DIFFERENTIATION. 150. Functions of several Independent Variables. Our attention has hitherto been confined to methods for the differentiation of functions of a single independent variable. In the present chapter we propose to discuss the case in which several such variables occur. Such functions are common; for instance, the area of a triangle depends upon two variables, viz., the base and the altitude ; while the volume of a rec- tangular box depends upon three, viz., its length, breadth, and depth ; and it is plain that each of these variables may vary independently of the others. 151. Partial Differentiation. If a differentiation of a function of several independent variables be performed with regard to any one of them just as if the others were constants, it is said to be a partial differen- tiation. The svmbols 7—, ;:r-, etc., are used to denote such differentia- •^ dx dy tions, and the expressions — , — , etc., are called partial differential coefficients with regard to x, y, etc., respectively. Thus if, for instance, u = 6*2/ sin Zy we have ^ = ye'^y sm 0, ^^ ^« • —-=:xe^y8inz, dy —-=e^ycoaz. dz 112 PARTIAL DIFFERENTIATION. 113 152. Analytical Meaning. The meanings of the differential coefficients thus formed are clear ; for if we denote u by f(x, y, z) the operation denoted by Zx may be expressed as T. f{^+h,y,z)-'f(x,y,z) J-'f'h= y > and similarly for — - or v^. "^ ?)y dz These partial differential coefficients are often conveniently written u^, Uy, Uz. 153. Geometrical Illustration. It will throw additional light upon the subject of partial differentiation if we explain the geometrical meaning of the process for the case of two independent variables. Let PQRS be an elementary portion of the surface ;:; =f(Xy y) cut oflf by the four planes Y=y, Y=y + Sy'\ [Capital letters representing X = x, X = x-\-Sx) current co-ordinates], so that the co-ordinates of the corners P, Q, R, S are forP i^,y,f{x,y), for a; 4- Sx, y, f{x + Sx, y), for S x,y+ Sy, f{x, y + Sy), and for i^ x + Sx, y-\-Sy,f(x+Sx, y-^Sy). c p_ "r p 2^ L r~ / N X / Pi / L, Nr Mi Fig. 20. If PLMN be a plane through P, parallel to the plane of xy, and cutting the ordinates of P, Q, R, S in P, L, M, F respec- tively, we have E.D.C. 114 CHAPTER VI. Lq=f{x^-Sx,y)-f{x,y), NS=f(x, y+Sy)-f(x, y), \ (1) MR=f(x+Sx,y+Sy)-f(x,y). Hence the partial differential coefficient — obtained by con- sidering y a constant is =Lt^ J^"'+^'''y^-^^^'y^ =Lt^ = LtUnLPQ. (2) = tangent of the angle which the tangent at P to the curved section PQ (parallel to the plane xz) makes with a line drawn parallel to the axis of x. 'dz Similarly —, which is obtained on the supposition that x is if constant = Ltt8inFPS, (3) = tangent of the angle which the tangent at P to a section parallel to the plane of yz makes with a parallel to the axis of y. It further appears from the figure that ^M y + Sy)-Lt,,=o ^ ^^ MR-NS = tangent of the angle which the tangent at S to the curve SR m^akes with a parallel to the X-axis. Now when Sy or PJ^ is diminished without limit the plane NSRM approaches indefinitely near to the plane PQL, and the tangent at S to the curve SR ultimately coincides with the tangent at P to the curve PQ. i-e. Lt^y=o-^f{x.y+^y) = ^f{x,y) and the order of proceeding to the limit when Sx and Sy vanish is immaterial. 154. If the tangent plane at P to the surface cut LQ, MR, N8 in Q\ R, S' respectively, LQ' = PLtB.nLPQ'=^.Sx, (4) NB' = PNi2.nNP8'=^ .Sy, (5) PARTIAL DIFFERENTIATION. 115 Also the section made on the tangent plane by the four bound- ing planes of the element is a parallelogram, and the height of its centre above the plane PLMN is given by \MR' and also by h{LQ'+]SrS'), which proves that 'dz. , -dz. .(6) The expressions proved in (4), (5), and (6) are first approxi- mations to the lengths LQ, IfS, and 2IR respectively, and differ from those lengths by small quantities of higher order than PL and PI^, and which are therefore negligible in the limit when Sx and Sy are taken very small. The investigation of the total values of LQ, NS, MR must be postponed until we have investigated the extension of Taylor's Theorem to func- tions of several variables. (Art. 175.) 155. We may state the rule established in the preceding article (equa- tion 6) thus : In the limit, the total variation in z = the variation due to the change in x +the variation due to the change in y, supposing that as each variation is estimated the other quantity is regarded as constant. This may be illustrated further. i Let P be any point (co-ordinates r, 6). \l.et a point travel from P to any contiguous position Q{r + Sr, 6 + 86) along any path whatever. Let x and x + Sx he the abscissae of P and Q. Let P and Q be so close that Sx, Sr, 86 are infinitesimals of the first order, so that in comparison with them their squares, products, and higher powers may be disregarded. Draw circular arcs whose centres are at the pole and radii OQ and OP cutting OP at P' and OQ at Q' respectively. O M N Fig. 21. Then PP=8r, P^^rBO, and to the first order P'Q[^r + 8r)8e] = r8e, chord PQ=RTcrQ = r8e. 116 CHAPTER VI. Also the angle QP'O differs from a right angle by an infinitesimal of the first order. Hence to this order the projection of P'Q on the initial line= -P'(^siii 9 = —r^OsmO. Also projection of P§ = algebraical sum of projections of PP'f P'Q. Thus we have the following equation among first order infinitesimals, viz. : 8x=Br cos e-rSd sin (1) It should be noticed that the projection of PP', viz. S?'cos^, is the variation in x due to a change 8r in the value of r, 6 remaining constant ; whilst the projection of P'Q or of PQ' is the variation in a: due to an increase S^ in the value of 0, r remaining constant. Moreover, since .r=rcos^, we have 7=r~ =cos 6^ p^y^= —r sin ; or od so that equation (1) may be written or od verifying equation 6 of Art. 154 in this case. Examples. 1. If A =xy^ explain geometrically the equation ox oy by reference to the area of a rectangle whose sides x^ y are allowed to increase to x-\-8x^ y-^^y ] the increments being infinitesimals of the first order. 2. If V=xyZj show geometrically that ox oy oz 156. Differentials. It is useful at this point to introduce a new notation, which will prove especially convenient from considerations of sym- metry. Let Dx, Dy, Dz be quantities either finite or infinitesimally small whose ratios to one another are the same as the limiting/ ratios of Sx, Sy, Sz, when these latter are ultimately diminished indefinitely. We shall call the quantities thus defined the differentials of x, y, z. Also, as we shall be merely concerned with the ratios of these quantities, and any equation into which they may enter will be homogeneous in them, it is imnecessary to define them further or to obtain absolute values for them. The student is warned again (see Art. 39) that the PARTIAL DIFFERENTIATION. II7 -^^ is to be considered as the result of performing the operation represented by -p upon y, an opera- tion described in Art. 37. The dy and dx of the symbol -^ cannot therefore be separated, and have separately no meaning, and hence have no connection with the differentials Dx and JDy as defined in the present article; but at the same time we have by definition Dy : Dx ^ Limit of the ratio Sy : Sx Sx dx and therefore Dy = -^J^^i Dv and j^ (which is a fraction) ~'ir~ (which is the result of the process ^ of Art. 37). We have used a capital in the differentials Dx, Dy, Dz for the purpose of explanation, and to avoid any confusion between the notation for differentials and for differential coefficients ; but when once understood there is no necessity for the continuance of the capital letter, and it is usual in the higher branches of mathematics to denote the same quantities by dx, dy, dz. Hence we shall in future adopt this notation. 157. Equation 6 of Art. 154 may now be written when Sx, Sy, Sz become infinitesimally small. This value of dz is termed the total differential of z with regard to x and y. The total differential of z is therefore equal to the sum of the 'partial differentials formed under the supposition that y and x are alternately constant. Ex. Consider the surface then ^- = ^ry"" and ?^ = 2^2y, ox oy ■whence dz = 'ixy'^dx + larydy. 118 CHAPTER VI. 158. It is easy to pass from a form in which differentials are used to the equivalent form in terms of differential coefficients. For instance, the equation dx ^y ^ may be at once written dz _'dz dx dz dy di~'dx di^dy dt' - where t is some fourth variable in terms of which each of the variables Xy y, z may be expressed ; for dz = -r7, dty dx = -^.dt, dy = -^.dt (Art. 156). Similarly the equation ds^ = dx^-\-dy^ may, by the same article, be written in the language of differ- ential coefficients as ©■+©"=■• ©■=©*-©■ ©■-+©' 159. Total Differential (Analytical). Two independent variables. We may investigate the total differential of the function {x+K y+k)--^(x, y-hk) , (x,y+k) h •pi becomes (by Art. 152) -^(pi^, y + k), Ox PARTIAL DIFFERENTIATIOK. 119 and ultimately when k also diminishes indefinitely ?^ or ^ (Art. 153). OX ox ^ Again ^(^>y+A:)-0(g;,2/) at the same time becomes ^^fe y) ^.., '^ -dy ^y And lastly in the notation of differentials (Art. 156) the ultimate values of the ratios Su:h:k may be expressed as duidx: dy. Hence equation (A) becomes 160. Several independent variables. We may readily extend this result to a function of three or of any number of variables. Let u = (/>(x^,x^,Xs), and let the increments of x^, x^, iCg, be respectively A^, li^, h^ and let the corresponding increment of uhe Svb\ then _ (l){x^+li^, ar^+Ag' ^z+h)-^(^v ^2-^K ^3+^3)? . " h^ ^^1 , {x^, x^, x^-\-h^)-^(x^, X,, x^) j^ . "*■ h, ''« ' whence on taking the limit and substituting the ratios du : dx^ : dx^ : dx^ instead of the ultimate ratios of Su : h^:\; A3, we have du = --^- dx. + ~dxo+ ;^-dxo, dx^ ^ a»2 ^^3 i.e., the total differential of u when x^, x^, x^, all vary is the sum of the partial differentials obtained under the supposition that when each one in turn varies the others are constant. 161. And in exactly the same way if u = (x^, X2,...Xn), we have du = ;,-—dx. + ^^dxo + ^^—dxo + . . . + ^^—dxn- dx^ 0^2 ^^3 ^^» 120 CHAPTER VI. 162. Total Differential Coefiacient. If U = ^n)» where iCp x^, ...,aJn> are known functions of aj, we obtain dii_du dxj^ du dx^ dv, dxn dx dXj^ ' dx dx^ ' dx '"dXn ' dx And further, if x^ x^, x^, ..., Xn be each known functions of several variables x, y, z, ...y -we shall have in the same way the series of relations 3i6_9u_ dx^ du dx^ du dXn 'dx^dx'j' "dx dx^' 'dx dXn'^* 'du_du dx^ du dx^ dw dXn dy~dx^ ' 'dy ScCg * 'dy ""dXn ' dy' etc. 163. An Important Case. The case in which u = ^(oj, y)^ y being a function of x, is from its frequent occurrence worthy of special notice. Here, by Art. 162, ^-'^=|«+|^ . J •^ dx dx dy dx dx - smce ^ = 1. 164. Differentiation of an Implicit Function. If we have ^(a?, y) = 0, then (t>(x+h,y+Jc) = 0. Hence (x+h, y+Jc)'-(x, y+k) , (x, y + 7c)-(l>(x, y) k_ k '^ k 'h~^' PARTIAL DIFFERENTIATION. 121 which, when h and h are indefinitely diminished, becomes (as explained in Art. 159) dx 'dy ' dx ' dy _ 'dx dx dtp 30 dx dy This is a very useful formula for the determination of -^ dx in cases in which the relation between x and y is an implicit one, of which the solution for y in terms of x is inconvenient or impossible. Ex. <^(jr,.v) = ^+y3_ 30^^ = 0; find ^. Here |_^=3(^2_ay) di/ _ a? — ay dx y^ — ax and ^i=3(3^2_«^.) 165. Order of Partial Differentiations Commutative. Suppose we have any relation y = (l>{x,a), where a is a constant, and that by differentiation we obtain Then since the processes of differentiation take no cognizance of the particular values of any of the constants involved it is obvious that the result of differentiating ^(a;, a) would be F{Xy a') ; that is, the operation of changing a to a' may be performed either before or after the differentiation, with the same result. We may put this statement into another form, thus : Let Ea be an operative symbol such that when applied to any function of a it will change a to a, i.e., such that then in operating upon the function (p(x, a) the operations Ea and -T- are commutative, that is, ^ag-^fe «) = ^-^a95>(aj, a)^F(x, ay 122 CHAPTER VI. Next, suppose z = (f>(x, y). The partial differential operations — and ~ have been defined to be such that when the operation with regard to either vari- able is performed the other variable is to be considered constant. We propose to show that these operations are commutative, ^.e., that ^^ —z=~-—z. ox dy dy dx Let By denote the operation of changing y to y-\-Sy in any function to which it is applied ; then By and the partial oper- ■pi ation — are com^mviative symbols. And ^ ^(pix, y) d(f>(x, y) 3 3 3^ 3'7' ^ ^^(». y)==LUy^. ^ , by Def., -my^o ^^ _ T. ^ Ey(f>{x,y)-(l>{x,y) -^^'y-'^x Ty 3 J Ey{x,y)-ct>{x,y) = ^r-Lt8y= dx"'"''-'' Sy =3i3^^(^'2/). 166. Another Proof. The ordinate /(O, 0) of the point in which any surface z=f{a;,2/) cuts the 2-axis is clearly independent of the particular path traced by any point moving from the arbitrary position (^, y, z) to the ultimate posi- tion {0, 0,/(0, 0)} ; notwithstanding that in some cases, in estimating the ultimate value /(O, 0), it may be necessary to evaluate an undetermined form. In other words, whenever it is necessary to evaluate /(A, k) for zero values of h and k, the order or manner of making h and k diminish to zero is indifferent, and it is allowable if we choose to suppose them to approach their ultimate values simultaneously. Thus LtH=o Ltk=of{h, k) = Z4=o Lh^ofih, h) = Ltk=jc=ofih, k). Again, it was pointed out in the previous article that processes of differentiation take no cognizance of the particular values of any constants involved. It therefore follows that if Z^.^o*^^+^>^l-^^^>^>=/-(^, a), h then will Z^>^o^^^+^- ^V ^'>=/-(^, ay, h PARTIAL DIFFERENTIATION. 123 that is, a may be changed to a' either before or after the limit is taken, with the same final result. Now, by definition, OX h and, since x and y are independent, we may regard y and its increment h as constants, while x is varying. Hence the value of — ^^-^-^, when y-^h ox ^ is written for y, is J {x+h,y + k )-{:r,y+k)_r {^+Ky)-<^x,y) 9^ xjOh='0 1 ■L/i'h=o J hk and as it has been established that the order of proceeding to the limit is indifferent, ^^ ^- <^ar, y) may be shewn equal to the same expression. ox oy 167. Extension of Rule. This rule admits of easy extension by its repeated applica- tion. Thus ■dy\dxJ ^ «-""■' (ir©v-©'(i)v Also if we have more than two independent variables, for instance if u = (^,y, !^) (D(|,)(l)*-(|;)(l)(s)" =(|)£)(l)-=-. so that the order in which the differentiations are performed is immaterial in the final result. 168. Notation. It is usual to adopt for 124 CHAPTER VI. the more convenient notation 3% 3% dP+^+^u etc. and the propositions above enunciated will then be written dxdy dydx dx^dy dydx^ dx'^dy'' dy^'dx' etc. We shall also sometimes find it useful to further abbreviate the expressions ^, ^^, — ^^ ®^c., into u^^, u^y, Uyy, etc. 169. The formulae here established may be easily verified in any particular example. Ex. Let ^=sin(.a7y), then Ux—^ cos(sn/), and Uyx=cos, xy — xy Bin xy (1) Again Uy=xcosxy^ and iixy=cosxy-xysmxy, (2) and the agreement of expressions (1) and (2) verities for this example the result of Arts. 165, 166. 170. It is convenient to use the letters p, g, r, s, t, to denote the partial differential coefficients ?^ ^, 2V ^V 5V dx dy dx^' dxdy dy'^' where ^ is a given function of the two variables x and y. Hence we have, if z — (^{x, y\ dz =pdx + qdy, Art. 159; and to obtain ^- from the implicit relation ^(x, 2/) = 0, we have dx q 171. To obtain the Second Differential Coefficient of an Implicit Function. d^v To obtain -t-^ we have only to differentiate the last result of PARTIAL DIFFERENTIATION. 125 the preceding article ; thus, d'p dq Now ^ = |E+§£^ = r+8r-^) = ^!:zP^ dx dx dy dx ^ 5^ 5' and ^=p+^±'k=s + t(-P) = 91^, dx ox dy dx ^2/ 9. ( qr-ps \_Jqs-pt\ giving ^y=- ^y q ) ^\ g ) _ _q^r — 2pqs+pH d^v Similarly -t~, etc., may be found, but the results are compli- cated. Examples. ^ 1. If U=X'^1/"j prove — =m Vn-^^ u X y and verify the formula ^^^^— ^ -^ oxoy oyox J 2. Verify the formula J^^ =7=r-^ ^^ ^^ch of the following cases : oxoy oyox (1) i^=sin-^'^- (3) u^Xo^Y- (4) M=a*. 3. If 2x + z show that 'dy'dz^ dz^'dy 4. If x=rcoB ^and ?/ = rsin prove dx = cos Odr - r sin 6d6, and dy = sin ddr + r cos ^c?^ ; and hence that dx^-\.dy^=d7^ + r'de\ and that xdy- -ydr=r'de. 126 CHAPTER VI. 5. If i* = l0g(^2 + y2 + 22), prove ^|^=^|^=,^9^ Sy32 'dzdx 'dxdy' 6. Prove that if ^+|'-%i ^= -^ and ^= --— . dx a^y daP' a?y^ 7. Show that if ^'^+/'' = a'", y 8. Show that at the point of the surface x^yy^z=c where x=y=z^ "Mdy = - { "^ ^<^^ ^-^ y^' [Oxford, 1889.] 9. If there be an equation between three variables jt?, ?, v, prove that (f) x(f) x(l^) =-!• \at /v const. \ a V /p const. \ {x+it, y+rjt), 128 CHAPTER VI. we shall nave — = ^ — + w — > etc. Taylor's Theorem. Extension. 175. To Expand ^{x-i-h, y-\-k)m powers of h and k By Taylor's Theorem we obtain and expanding each term we have ix+h, y+k) = 4.(x, y)+{hl+k^)^+l(h^+kl^y/+... 176. Since it is immaterial whether we first expand with regard to Jc and then with regard to h, or in the opposite order, we obtain by comparison of the coefficient of hJc in the two results the important theorem dxdy dydx already established in Arts. 165, 166. 177. Agreeably with the notation of Art. 116, we may write the result of Art. 175 as T.3 I 7 ' (x + h, y-\-k, z + l,...) = A(a+/3)x'^y^ = n'2Ax^yP = nu. It is clear that this theorem can be extended to the case of three or of any number of independent variables, and that if, for example, u=^A x'^y^zy + Bx°''yP'zy' + . . . where a + /3 + y = a'+/3'+7'=... = ti, then will x~ — h y^r- + s^p^- = nu. dx ^dy dz The functions thus described are called homogeneous functions of the n^'^ degree. 182. We now put the same theorem in a more general form. Def. a homogeneous function of the n^^ degree is one which can be pui in the form \x X / PARTIAL DIFFERENTIATION. Let -=-^ei'-> Put |=F, | = ^,etc.. whence dY y dZ z dx a?' ?>x~ x^"" BF 1 ZZ ^ ^ ;5— = , ^r-=0, etc. ^y x' -dy Now, since u.^x-F(ZZ,...l ^ .,..,,. r. . . JdF -dY .dF 131 = tia:-^^(F,Z,...)~^-2(2/|^+4j+-}> dy'^^dY' dy-"^ d'Y' 'du ?)F ^Z ^ ,dF zz^"" zz' -dz'"" ^r etc. = etc. Finally, multiplying by Xyy.z,.., respectively, and adding 183. If It be a homogeneous function of x and y of tlie ri}^^ degree, , — will be homogeneous functions of the (7^—1^ degree, and applying the result of Art. 182 to these we have ( 3 , a\3u , ^.du Multiplying by x and y we have on addition = n(n — I )u. Similarly we may proceed and finally by induction establish a general theorem of similar character, but of higher order; but it is better to adopt the method hereafter applied in Art. 186. 1S2 CHAPTER VI. 184. If V = Un + Un.i + Un-2-^'.' + U2-\rUi-\-Uo, where u„, Un-i,... are homogeneous functions of degrees n, 91 — 1, ... respectively, then = (a)|+...)u. + (a.^+...)u„., + etc. Hence if F=0 dV dV 185. Let u = (p(Hn), where H^ is a homogeneous function of the n^^ degree. Suppose we obtain from this equation then x^F(u)+y^~F{v,)+...=nH„, M^'^i+yt^-h-^^-)' or ^ ,^,.^^ . _^ -dy or X;r — hy:^ +...=n-~y-( (1) dx ^dy F^W In the particular case in which 71 = we therefore have du ^ 'du ^ „ .^. ^ar?+% + -=*' <2> Examples. Verify the following results by differentiation. 1, Let u—j^+y'^ + Zxyz. This is clearly homogeneous and of the 3rd degree, whence 'dx dy "dz ,_^*H-^_.A^ + (f)*. 2. Let u=- — ^=x^ x-^+y^ i+{iy This is a homogeneous expression of degree -2^, whence du , du 1 „, Here Art. 182 gives *^+i'|^=*'- PARTIAL DIFFERENTIATION. 133 4. Let w=taii~^- i-^+.V^ X -y Here Art. 185 gives x^ + y~ = sin 2m. 5. Find which of the following functions are homogeneotis, and in cases of homogeneity verify Euler's Theorem of the first degree : (a) xe~K (7) (^ - y) (log^- log y). (S) An-^^l^H. 6. Given z=x^+y and y=z^-¥x, find the differential coefficients of the first order, (1) when X is the independent variable, (2) when y is the independent variable, (3) when z is the independent variable. v. Given xyz — o?^ find all the differential coefficients of the first and jsecond orders, taking x and y for independent variables. 8. If t,=8in-i^^±^, prove that x^ + y^^ = i tan t^. ox " oy 9. If w = cwr^ + 6/ + cr* + 2/y2 + 2^2^+2Aj:y, show that, if it be possible to find values of r, y, z which will simultane- , ously satisfy Sm_3m_3m_ '^x~'by~'dz~ ' a, A, ^ then will h, h^ f =0. 10. If w be a homogeneous function of the n^^ degree of any number of variables, prove that 11. If u={x-^xt, y + yt, ...)=aj«(l+0-i^(|±|-J,...) so that the effect is simply that of multiplying the original function by (1 + 0^. Now, let Vm denote the symbol of operation obtained by expanding {xX+yY-\-zZ+ ...)'^ by the Multinomial Theorem, 7^ TS "r^ and after expansion writing — , — , :^, • • . in place of X, T, Z, if etc.; then we have, upon expansion of each side of the above equality, And on equating coefficients of like powers of t V^u = nUy V^u — n{n—l)u, Y^u = n{n — \){n — 2)u, etc. YrU=n{nii — l)...{n — r-{-l)'w. 187. When there are two independent variables x and y, these become etc. ; PARTIAL DIFFERENTIATIOJS. 135 and for the case of three independent variables dx ^dy dz = 71(92 — 1)11-, etc. 188. It may be observed that although the expressions are identical, care must be taken to distinguish between "'^^ + 2^2'^^ + 2^3/ and (a,|+y|)«. It is apparent that the latter = (^^ + 2/3^X^3^ + 2/^^) = V^^+2/§^A^3:.; + l^^+2/3^)V2^3^ / „B2^ , du . 3% \ = \x + (^2/^^+2/^+2/^3/) and therefore differs from the former expression by the addition of the two terms 3u dm ^3^' 2/^- 189. Laplace's Equation. 32 32 32 The operator ^2+^^^"^ (- V^) pl^js a fundamental part in the Higher Physical Analysis. The equation y^y—Q jg called Laplace's equation ; and any homogeneous function of x, 3/, z which satisfies it is called a Spherical Harmonic* It is customary to denote o(?-\-y'^-\-z'^ by r^. ♦See Thomson and Tait, Treatise on Natural Philosophy, vol. I., p. 171. Laplace, Mtcanique Celeste, bk. II. 136 CHAPTER VI. Ex. 1. Let V=r"'. Then and we have ox or ox 3217 and ^- = m(m - 2);'"*- V + mr^-\ "^217 Similarly, ..^— = m(m-2}r"^-^y^ + mr^-'\ _^= m(m - ^y^-^z^ + onr"'-\ oz^ . * . by addition, y^ r= w(m - 2)r"*-2 + 3wr"*-2 or y V" = m{m + 1 )r"»-2. Since this expression vanishes when m=-l, it appears that - is a spherical harmonic of degree — 1. ^ Ex. 2. If Vn he a spherical harmonic of degree w, then will V.„l'r'"^^ be a spherical harmonic of degree -n—\. Let u=VJr^"+\ Then ^1^=^«. _L_-(2n + l)F..^3, with similar expressions for ^^-^ and ;^-2- Adding these together, Hence, remembering that y2 F„=0, and x^^ + y? -" + ^^" = ^ F,,, we have y%=0. Ex. 3. Show that each of the functions . ii/ xz 1 , r + z tan -^-2^, -^ -J - log X x^ + y r r — z satisfies Laplace's equation. Ex. 4. If u and u' be functions of ^, y, z, each satisfying y2F=0, prove that y-(ww') = ^u^u'x + %^'y + w^m'z). 190. Conjugate Functions. When two real functions u and v oi x and y are defined by the equation u + s/ — \v =f{x + V — l^/) they are said to be conjugate. PARTIAL DIFFERENTIATION. I37 Differentiating, we have 'dxi I — r-^V / — -'druij 'dv ^y oy dx dx S = |."- •(!) and _=-— (2) dy dx Again differentiating, and B^u ^2^ Sa;2 ^^-y (3) whence ^ + ^i = ^ and similarly ^+-^j.= <> Ex. 1. If 14+ *y— ly be a homogeneous function of x, y, 2, of degree ^ + J -Iq, then 3m om 3w a„.l x3^+y| + .|=p.+5„. [Thomson and Tait, Natural Philosophy.] Ex. 2, If a and /? be conjugate functions of a and 6, whilst a and 6 are conjugate functions of a; and y, prove that a and (3 are conjugate functions of X and y. [Maxwell, Electricity.] Ex. 3. If the equation ^—. + ——=0 be satisfied when V is a given oar oy^ function of ^, y, it will also be satisfied when V is the same function of a and 6, where a = log ^^^' and 6 = tan-i|. ^.^^^^^^ ^^^^,^^^ EXAMPLES. 1. Verify the formula ^ = ^- in the following cases : ^ Zxdy ay ox (a) w = sin^' (j3) u = log{a; tan"! Jx^ + y^}. 138 CHAPTER VI. 2. Find ^ {a) \iax^-^2hxy + hy'^=l. (/?) iis^ + y^ = 6a^xy. (y) if (cos xf = (sin y)'. (8) ify' + o(^ = {x + yy'-''. 3. If w = sin"i- + tan~i^, show that x^ + y~ = 0. y X ox oy 4. If w, y, « be functions of x such that _ c? / c?3/\ _ d / dz\ dx\ dxj dx\ dx) prove that ^«(2,|-«|) = 0. 5. If u and v be both functions of the same function of x and y, prove that _-.-- = -_.—, and that ^{u^ = — w— • OflJ oy oy ox ox\ oy) oy\ ox) 6. If F=/(w, -y), u=f^{x, y), v=f^{x, y\ show how to find — - in . 9F , 9F '''* terms oi — - ana — — ox oy Ex. Given u = x^ + y^, v = Sicy, show that 7. Verify Euler's Theorem ox oy for the functions (a) tt = sm(^-^) • 8. If w = <^(2/ + ax) + xp{y - ax), prove ^ = a2_^- 9. If « = .^(|) . ,(|). prove .^g . 2^g- ',^g = 0. 10. If 1* {x^ + y^ xcf)(^j + i^(-\ prove that ^^SZq + ^^y^^Z^. + 2/^57.2 = (^ + 2/ ) 2m(2m-l) 2 + 2£C2/ + y — 11. If /(a, 2/) - 0, prove that /m2 2 2 21. If -o + ?ii + ^ = l, and lx + my + nz = Oj prove that a^ 6^ c^ c?a; _ dy dz ny mz fe na; ma; ^y 62 c2 C2- a2 a2 62 140 CHAPTER VI. 22- If "; + C + ^ = l. and ^-^ +^ + ^=1, prove that x(o^-c^) y(c^-a?) z{a?-¥) ^ ax ay az 23. If -^ + -^ + -^ = 1, prove that a^ + u y + u c^ + u ^/ + %^ + ^8^ = 2(£cw^ + 2/tfcy + 2;w,). [Oxford, 1888.] 24. If « and u be functions of x and 2/ defined by {z - <^(w)}^ = ic2(2/^ - ii'^), {z - (li{u)}cf>{u) = tix^y .-, . "dz dz prove that ,::- • ^- =xy. r„ dx dy ^ [Bertrand.] 25. If Pc?a; + Qdy be a perfect differential of some function of x, y, prove that ^^=— *. 02/ da; 26. If Pdx -t Qdy + Rdz can be made a perfect differential of some function of x, y, z by multiplying each term by a common factor^ ■'■■'■"'»"{lfD-{y-x)}, prove that {x, y)Y{xP{x + h,y + k)-xl^(x, y)]'. [Math. Tripos, 1888.] 29. If 1* be a homogeneous and symmetrical function of x and y of n dimensions, and if its expanded form is 2^^£c*"2/"~'", prove that 2{(2r - n)Q^) = 0. [Gregory's Examples.] 30. If /(«!, ccgj 2C3 . . . a;„) be any homogeneous function which becomes F(X-^j X^ ... X„) by any linear substitution for the variables ajj, jCg • • • ^^ terms of X^, Xg . . . and if ajj', x^, x^ ... ; X^', Xg', X3' . . . be simultaneous values of the two systems, prove that APPLICATIONS TO PLANE CURVES. CHAPTER VII. TANGENTS AND NORMALS. 191. Equation of TANGENT. It was shown in Art. 36 that the equation of the tangent at the point (x, 2/) on the curve y=/(ic) is Y-y='^X-x), ....(1) A'' and Y being the current co-ordinates of any point on the tangent. Suppose the equation of the curve to be given in the form It is shown in Art. 164 that dy dx -dy Substituting this expression for -^ in (1) we obtain T-y=-'^{X-x), (X-a,)g+(F-2/)| = (2) for the equation of the tangent. 192. Simplification for Algebraic Curves. If /(ic, y) be an algebraic function of x and y of degree -ti, suppose it made homogeneous in x, y, and z hy the introduction of a proper power of the linear unit z wherever necessary. 143 144 CHAPTER VII. Call the function thus altered f{x, y, z). Then f(x, y, z) is a homogeneous algebraic function of the n^^ degree; hence we have by Euler s Theorem (Art. 181) by virtue of the equation to the curve. Adding this to equation (2), the equation of the tangent takes the form X^^-\-Y^f+M-=0 (3) 'dx ^y 'dz ^ ^ where the z is to be put =1 after the differentiations have been performed. Ex. f{x,y) = x^-\-a^xy-^¥y-\-c'^=0. The equation, when made homogeneous in x^ y, z hy the introduction of a proper power of z, is /(•^» y? 2) = ^ + a^xyz^ + h^yz"^ + ch* = 0, and ^=^-{-a^yz% oy |/= 2a?xyz + 363^22 + ^ch\ uz Substituting these in Equation 3, and putting 2 = 1, we have for the equa- tion of the tangent to the curve at the point {x, y) X{4x^ + ahj) + Y{a"x + h^) + ^a^xy + 36^^ + 4c* = 0. With very little practice the introduction of the z can be performed imenially. It is generally Tfiore advantageous to use equation (3) than equation (2), because (3) gives the result in its simplest fo7'm, whereas if (2) be used it is often necessary to reduce by substitutions from the equation of the curve. 193. Application to General Rational Algebraic Curve. If the equation of the curve be written in the form (where Ur represents the sum of all the terms of the 7***^ degree), then when made homogeneous by the introduction where neces- sary of a proper power of z we shall have fix, y, Z) =Un + Un-lZ + Un.2Z'^+ ... and ;/ = U„_i + 2Un-22;+3u„_302+... oz -\-(n-2)Uc^2^-^+(n-l)u^z''-^-{-nuQZ''-\ TANGENTS AND NORMALS. 145 and therefore substituting in (3) and putting z = l, the equa- tion of the tangent is X^+F^+u„.i + 2^„_2+3u„_3+... + {n-2)u.^-\-(n-l)u^+nuQ=0 (4) 194. NORMAL. Def. The normal at any point of a curve is a straight line through that point and perpendicular to the tangent to the curve at that point Let the axes be assumed rectangular. The equation of the normal may then be at once written down. For if the equa- tion of the curve be y —f(x), the tangent at (a?, y) is F— ?/ = -p(Z— cc), and the normal is therefore If the equation of the curve be' given in the form the equation of the tangent is (X-.)|+(F-,)|=0. and therefore that of the normal is X-x _ Y-y dx dy Ex. 1. Consider the ellipse %.+^^=l. This requires z^ in the last term to make a homogeneous equation in Xy y, and z. We have then Hence the equation of the tangent is X. =0, where z is to be put = 1. Hence we get f-^f= = 1 for the tangent, and therefore E.D.C X-x_ X -^ K = — 11^ for the normal. 146 CHAPTER VIl. Ex. 2. Take the general equation of a conic ow;^ + 2 A^y + 6y2 + 2^^ + 2^ + c = 0. When made homogeneous this becomes The equation of the tangent is therefore Xiaa;-\-hy+g)+ Y{hx+hy+f)+gx+fy+c=0, and that of the normal is X-x ^ Y-y ax+hy+g hx+hy+f Ex. 3. Consider the curve ^=log sec— • a a Then $^=tan?, ax a and the equations of the tangent and normal are respectively r-v=tan-(Z-a;), a and ( r-y)tan- + {X-x)^ 0. Cb 195. If/(a5, 2/) = ^J^d -^(^» 2/) = ^ b^ ^wo curves intersecting at the point ic, 2/, their respective tangents at that point are and XF^+YFy+ZF, = 0. (Z is often written for z for the sake of symmetry.) The angle at which these lines cut is tan " ^ -^^^ "fy^^ ^ fxFx+fyFy Hence if the curves touch fxlFx=fylFy\ and if they cut orthogonally, fa^Fx+fyFy=0, Ex. If ^ + t'7 =y(-^+ty), the curves given by f = constant, and by 7/= constant, form two families such that each member of the first set cuts orthogonally each member of the second. Tor by Art. 190, ?_^=^ and ^= -^ ox oy oy ox whence ?l . ?!?+?l . ?5=0 'dx dx dy dy * which is the condition that the tangents at the points of intersection should include a right angle. TANGENTS AND NORMALS. I47 196. If the form of a curve be given by the equations the tangent at the point determined by the third variable t is by Equation 1, Art. 191, or Xyf/{t). - Y'{t)+Yir'{t)='{t)+^iAtW{t)- Examples. 1. Find the equations of the tangents and normals at the point {x, y) on each of the following curves : — (1) ^+/=c2 (5) x^y + xy'^^a\ (2) y^ = ^ax. (6) ci'=sin^. (3) xy=Jc'. (7) a^-Zaxy-{-f=0. (4) y = ccosh?. (8) {x'^-\-y'^f==a\x^-y'^). c 2. Write down the equations of the tangents and normals to the curve y{a^ + a'^)=ax^?Lt the points where y = -^' 3. Prove that - +f = 1 touches the curve y — he'^- at the point where the curve crosses the axis of y. 4. Find where the tangent is parallel to the axis of x and where it is perpendicular to that axis for the following curves : — (a) aa;2 + 2A^y + 63/2=1. (7) 3^=^(2a-^). 5. Find the tangent and normal at the point determined by on (a) The ellipse ;r = a cos Q'\ (/?) The cycloid ^=a(^+sin (9)\ ;r = acos t'Y y = 6 sin 0) ^=a(^+sin y=a(l-cos^). A (y) The epicycloid ;r = ^ cos ^ - 5 cos — ^ ^ A y=Aam6-Bain—6 Jo 148 CHAPTER VII. 6. If p=^ cos a +y sin a touch the curve fn TTi VI prove that p»^^= (a cos a)*"-^ + (b sin a)"*-^ Hence write down the polar equation of the locus of the foot of the perpendicular from the origin on the tangent to this curve. Examine the cases of an ellipse and of a rectangular hyperbola. 7. Find the condition that the conies ax^ + bi/'^ = l, a'x^ + b'y^ = l shall cut orthogonally. 8. Prove that, if the axes be oblique and inclined at an angle o>, the equation of the normal to y=f{x) at (^, y) is (r-y)(cos(o+^) + (Z-^)(n-cosw^)=0. 9. Show that the parabolas x^ = ay and y'^='2.ax intersect upon the Folium of Descartes o^-\-y^ = Zaxy ; and find the angles between each pair at the points of intersection. 197. Tangents at the Origin. It will be shown by a general method in a subsequent article (291) that in the case in which a curve, whose equation is given in the rational algebraic form, passes through the origin, the equation of the tangent or tangents at that point can be at once written down by inspection ; the rule being to equate to zero the terms of lowest degree in the equation of the curve. Ex. In the curve x^+y^ + ax+hy = 0, ax+hy = is the equation of the tangent at the origin ; and in the curve {x^-^y^Y=a\x^-y^\ x^—y^=0 is the equation of a pair of tangents at the origin. It is easy to deduce this result from the equation of the tangent established in Chapter II. That equation is Y—y — m{X-x) where m=-f' ax At the origin this becomes F=mX, where the limiting value or values of m are to be found. Let the equation of the curve be arranged in homogeneous sets of terms, and suppose the lowest set to be of the r*^ degree. The equation may be written x%{^\ + ^*-+y^+/^) + ... x''fn{A=-0. Dividing by af^ and putting y=mx, and then x=0 and y=0, the above reduces to the form /r(^^)=0, an equation which has r roots giving the directions in which the several branches of the curve pass through the origin. If Wi, mo, rrizj ... tw,. be the roots, the equations of the several tangents are y=^mix, y=m.2;)c, ,..y=m^. TANGENTS AND NORMALS. These are all contained in the one equation 149 ^©^O' and this is the result obtained by " equating to zero the terms of lowest degree " in the equation of the curve, thus proving the rule. Ex, Find the equations of the tangents at the origin in the following curves : — (a) {x^ +y^Y = a^x^ — b^y\ Geometrical Results. 198. Cartesians. Intercepts. From the equation F— y = -r-iX — x) it is clear that the intercepts which the tangent cuts off from the axes of a? and y are respectively X — -T- and y—x-^i ay ^ ax dx for these are respectively the values of X when Y= and of FwhenZ = 0. l^ P^-^*^ Y t^-r^ Ayi O : ^f G Fig. 22. Let PN, FT, FG be the ordinate, tangent, and normal to the curve, and let FT make an angle ^/r with the axis of x; then tan i/r=-^. Let the tangent cut the axis of y in t, and let OY, OY^ be perpendiculars from 0, the origin, on the tan- gent and normal. Then the above values of the intercepts are nlso obvious from the figure. 150 CHAPTER VII. 199. Subtangent, etc. Def. The line TN is called the suhtangent and the line FG is called the subnormal. From the figure Suhtangent = TN = 2/ cot ^ = y-. dx Subnormal — NO = y tan i/r = y-^. Normal ^PG^ysec-yfr^ yjl + tan^^ = y^ij^{^, Jl + (^X Tangent = TP = y cosec i/r = y'^^—~ — ^— ^ = y tan yp- ^ dy dx dy dy 0F= 0^ COS ^ y^'^dx y^^'dx Vl+tan2^ M%1 nvr nn 1 ON+NG ^'^^dx >v/l+tan2i/r V-+©' These and other results may of course also be obtained analytically from the equation of the tangent. Thus if the equation of the curve be given in the form the tangent * Xf^+ Yfy-\-Zfz = makes intercepts —fz/fx and —fz/fy upon the co-ordinate axes, and the perpendicular from the origin upon the tangent is and indeed, any lengths or angles desired may be written down by the ordinary methods and formulae of analytical geometry, Ex. 1. For the chainette v=ccosh-, we have 3/1= sinh-, c c Hence suhtangent = ^ = c coth -> 2/1 subnormal = vvi = c sinh cosh ^1 c c normal =y sli + 2/1^=^, etc. TANGENTS AND NORMALS. 151 Ex. 2. In the general equation show that if for a given abscissa each ordinate be divided by the corresponding subtangent the algebraic sum of the resulting quotients is constant. If yi>y2) 3^3) ••• ^6 ^^^ several ordinates and s^j 82 ... the several sub- tangents, 2yr= -(«o+^i^)> hence, differentiating, 2-^=-ai, and '==2/r dx 200. Values of ^, % etc. ax as Let P, Q be contiguous points on a curve. Let the co- ordinates of P be (x, y) and of Q {x-\-SXy y-\-Sy), Then the Fig. 23. perpendicular PR = Sx, and RQ = Sy. Let the arc AP measured from some fixed point A on the curve be called 8 and the arc AQ = S'\-Ss. Then arc PQ = Ss. When Q travels along the curve so as to come indefinitely near to P, the arc PQ and the chord PQ ultimately difier by a small quantity of higher order than the arc PQ itself (Art. 34). Hence, rejecting infinitesimals of order higher than the second, we have ^s2 = (chord PQf=(Sx^-{'Sy^\ or =«©+f)=(S"+©' 162 CHAPTER Vn. Similarly <.-<»%} or O--©' and in the same maimer (|)"='+(|)' If yjr be the angle which the tangent makes with the axis of X we have as in Art. 87, and also cos vr — Lt—, ^^/^ = -'-'(^ rvri == ^*ir = t' » ^ chord PQ arc PQ 6s as and sin o/r = Lt-r — ^j^ = Lt —^ = Lt-f- = ^' ^ chord Py arcPQ 6s as Examples. 1. Find the length of the perpendicular from the origin on the tangent at the point ^, y of the curve ;r^+y*=c*. X 2. Show that in the curve y=h^ the subtangent is of constant length. 3. Show that in the curve &y^=(^+a)3 the square of the subtangent varies as the subnormal. 4. For the parabola y'^=Aaa;, prove ds _ fa+x dx ^ X Prove that for the ellipse -3+^2= 1> i^ ^=asin^, ^=av/l-e2sin^<^. 6. For the cycloid ;r=avers^ ) y=a(^+sin^)/' prove dx ^ V 7. In the curve y = « log sec-j CL ds X ds X J I prove -^=sec-> =cosec-, and x=ay. ^ dx a dy a 8. Show that the portion of the tangent to the curve which is intercepted between the axes, is of constant length. Find the area of the triangle formed by the axes and the tangent. TANGENTS AND NORMALS. I53 9. Find for what value of n the length of the subnormal of the curve :j;^«=a**+i is constant. Also for what value of n the area of the triangle included between the axes and any tangent is constant. 10. Prove that for the catenary or chainette y=ccosh-, the length of the perpendicular from the foot of the ordinate on the tangent is of con- stant length. 11. In the tractory prove that the portion of the tangent intercepted between the point of contact and the axis of x is of constant length. 201. Polar Co-ordinates. If the equation of the curve be referred to polar co-ordinates, suppose to be the pole and P, Q two contiguous points on the curve. Let the co-ordinates of P and Q be (r, 0) and (r-|-(5r, ^4-^^) respectively. Let PN be the perpendicular on OQ, then NQ differs from Sr and NP from rSO by small quan- tities of a higher order than SO (Art. 31). Fig. 24. Let the arc measured from some fixed point ^ to P be called s, and from A to Q, s-{-S8. Then arc PQ = Ss. Hence, rejecting infinitesimals of order higher than the second, we have Ss^ = (chord PQ)2 = (NQ^ + PN^) = (Sr^ -f- r^SO'), and therefore according as we divide by Ss\ Sr\ or SS^ before proceeding to the limit. 154 CHAPTER Vn. 202. Inclination of the Radius Vector to the Tangent. Next, let (j) be the angle which the tangent at any point P makes with the radius vector, then , ^ de ^ dr . ^ rde tan0 = r^, cos^ = ^^, sin0 = ^. For, with the figure of the preceding article, since, when Q has moved along the curve so near to P that Q and P may be con- sidered as ultimately coincident, QP becomes the tangent at P and the angles OQT and OPT are each of them ultimately equal to 166 CHAPTER VII. therefore P~^'^{dW ^^^ =-'+y ^z where z is to be put equal to unity after the differentiation is performed. If this pass through the point A, h we have dx dy dz This is an equation of the (n — iy^ degree in x and y and represents a curve of the (n — 1 y^ degree passing through the points of contact of the tangents drawn from the point (h, k) to the curve f(x,y) = 0. These two curves have 71(72/ — 1) points of intersection, and therefore there are in general n(n — l) points of contact corresponding to n(n—l) tangents, real or imaginary, which can be drawn from a given point to a curve of the n^^ degree.* It appears then that if the degree of a curve be n, its class is n(n — l); for example, the classes of a conic, a cubic, a quartic are the second, sixth, twelfth respectively. *Poncelet, Annalcs de Gergonne, vol. VIII. ; Bobillier, ibid. vol. XIX. TANGENTS AND NORMALS. 159 209. Number of Normals which can be drawn to a Curve to pass through a given point. Let h, k be the point through which the normals are to pass. The equation of the normal to the curve f(x, y) = at the . . , . . X—x Y—v pomt {X, y) IS -^ = -^. If this pass through hy k, This equation is of the n^ degree in x and y and represents a curve which goes through the feet of all noiwials which can be drawn from the point 7i, k'io the curve. Combining this with f(Xy y) — 0, which is also of the n*^ degree, it appears that there are n^ points of intersection, and that therefore there can be n^ normals, real or imaginary, drawn to a given curve to pass through a given point. For example, if the curve be an ellipse, n=2, and the number of normals is 4. Let '^+^ = 1 be the equation of the curve, then is the curve which, with the ellipse, determines the feet of the normals drawn from the point {h, Jc). This is a rectangular hyperbola which passes through the origin and through the point {h, k). The student should consider how it is that an infinite number of normals can be drawn from the centre of a circle to the circumference. 210. The curves (;,_,)|+(,_,)|=0 (1) and (A_a,)|_(^._2/)|=0, (2) on which lie the points of contact of tangents and the feet of the normals respectively, which can be drawn to the curve f{x, y) = so as to pass through the point (h, k), are the same for the curve /(ic, y) = a. And, as equations (1) and (2) do not depend on a, they represent the loci of the points of contact and of the feet of the normals respectively for all values of a, that is, for all members of the family of curves obtained by Varying a in f(x, y) = am any manner. 160 CHAPTER VII. 211. Polar Curves. The curve }M+lM-+z%=^0 dx dy dz is called the " First Polar Curve " of the point h, h with regard to the curve f{Xy y) = 0; z being a linear unit introduced as explained previously to make /(cc, y) homogeneous in cc, y^ Zy and put equal to unity after the differentiation is performed. As this is a curve of the {n — Xf^ degree it is clear that the first polar of a point with regard to a conic is a straight line, the first polar with regard to a cubic is a conic, and so on. The first polar of the origin is given by ?=«• • dz If the curve be put in the form Un + Un-l + Un-2+ •"+U2-}-U^-hU0 = 0, the first polar of the origin is Un-l + 2Un-2 + SUn-z+."+(n — l)Ui + nUo = 0. In the particular case of the conic the polar line of the origin has for its equation Uj^+2Uq = 0. For the cubic u^+U2+Ui-}-Uq = the polar conic of the origin i^s Ex!a.mples. 1. Through the point h, h tangents are drawn to the curve show that the points of contact lie on a conic. 2. If from any point P normals be drawn to the curve whose equation is y^=max^, show that the feet of the normals lie on a conic, of which the straight line joining P to the origin is a diameter. Find the position of the axes of this conic. 3. The points of contact of tangents from the point A, h to the curve o(^-\-y^ = Zaocy lie on a conic which passes through the origin. 4. Through a given point h, k tangents are drawn to curves where the ordinate varies as the cube of the abscissa. Show that the locus of the points of contact is the rectangular hyperbola 2^y +kx- Zhy — 0, and the locus of the remaining point in which each tangent cuts the curve is the rectangular hyperbola xy-Akx+Zhy=0. TANGENTS AND NORMALS. 161 212. The p, r or Pedal Equation of a Curve. In many curves the relation between the perpendicular on the tangent and the radius vector of the point of contact from some given point is very simple, and when known it frequently forms a very useful equation to the curve ; especially indeed in investigating certain Statical and Dynamical properties. 213. Pedal Equation deduced from Cartesian. Suppose the curve to be given by its Cartesian Equation and the origin to be taken at the point with regard to which it is required to find the Pedal Equation of the curve. Let x, y be the co-ordinates of any point on the curve ; then, if F(x, y) — be the equation of the curve, that of the tangent is XF,+ YFy+zF, = 0, where z is as usual to be put equal unity after the differentia- tion is performed. If ^ be the perpendicular from the origin on the tangent at FJ^ (x, y) we have p^ = (1) FJ'+F^ Also r2 = a;2+2/2, (2) and F(x,y) = (3) If X and y be eliminated between these three equations the required relation between p and r is obtained. Ex. U F{x,i/)=Ohe we have and therefore or a^ f_ 1 a4 + fc4-p2 x^+y^=r^', 1 h^ 1 1 1 6*' p = 0, 1, 1, r2 a2^,2 5 ^1 !^+r2=a2 + 62. This result may be at once obtained by eliminating CD from the equations CP" ■\-CD' = ar + h'' and CD.p = ah^ CP and CD being conjugate semi-diameters. E.D.C. L 162 CHAPTER VII. 214 Pedal Equation deduced from Polar. Let the curve be given in Polar co-ordinates and the pole be taken at the point with regard to which it is required to find the pedal equation of the curve. Let r, be the co-ordinates of any point on the curve, and p the length of the perpendicular from the pole on the tangent at r, 0. If F(r,e) = (1) be the equation of the curve, then we have (see Fig. 25) ^ = r sin ^, (2) and tan0 = ^ (3) dr ^ ^ Eliminate and tp between the equations (1), (2), (3), and the required equation between p and r will be obtained. Ex. Given r'^ = a'^sin md, required its pedal equation. Taking logarithms and differentiating, m dr_ cosmO _ r dO~ sinmO' therefore cot<^=cotm^, or , yp- equation may be deduced at once from the equa- tion of the first positive pedal. If r=f(0) be the pedal curve, then, since i/r =— -f-0 (see Fig. 26, Art. 217), the equation between p and yjr is clearly P=f{i^-t)- Ex. 1. The jo, yfr equation of Ax^ + Bt/'^^I is ^2 = «i^+'^(Art.216). ^ A B Ex. 2. The pedal of — = 1 + cos ^ with regard to the origin is r cos ^= a, r and therefore its p, y}r equation is p sin yfr = a. 168 CHAPTER VII. 221. Relations between jp, t, p, etc. Let PY, QY' be tangents at the contiguous points P, Q on the curve, and let OY, OY' be perpendiculars from upon these tangents. Let OZ be drawn at right angles to Y'Y produced. Let the tangents at P and Q intersect at T, and let them cut the initial line OX in M and S. Let the normals at P and Q intersect in 0. Fig. 27. Let the co-ordinates of P be (r, 6), and let those of Q be (r + Sr, + SO). Let OY=p, OT = p + Sp, PRX = y[r, QSX = ylr+Sxl^. Then STR, PCQ, YOY each = ^i/.. Let PY=t, and arc PQ = Ss. Let OY' cut TY in V; then, since OFF is a right angle and YOV=S\lr a small angle of the first order, OV differs from OF by a quantity of higher order than the first (Art. 32). Hence VY' differs from Sp by a quantity of higher order than Sp, and TY' tan Syfr = VT, therefore TT^.^ = ^, dp and proceeding to the limit t dy/r' (1) Similarly, if PC be called yo we have &vcPQ = PG.Sxlr, neglecting infinitesimals of higher order than S^Jr, therefore p^^ arcPQ and proceeding to the limit, -$ • • (^) TANGENTS AND NORMALS. 169 Again St=rQ-YP = ( TT-^ TQ) - ( YV->t VT- PT) = (PT+ TQ) + ( FT- VT) - YV. Now YV=pia,nS\ly, and remembering that when Syjr is an infinitesimal of the first order, VT and FT, PT-i-TQ and Ss, tan S\fr and Sifr, each diflfer by quantities of order higher than the first, we have, upon dividing by 6\{r and proceeding to the limit, dt _ ds d^~~d^~^' or P=P+^^^ ^y W ^^^ (2) (3) '222, Perpendicular on Tangent to Pedal. From the same figure it is clear that since YOY'=YTY\ the points 0, Y, Y\ T are concyclic, and therefore OYZ=ir-OYT = OTT', and the triangles OFZ and OTY' are similar. Therefore jyy=-f)T' And in the limit when Q comes into coincidence with P, F comes into coincidence with F, and the limiting position of FF is the tangent to the pedal curve. Let the perpendicular on the tangent at F to the pedal curve be called jp^, then the above ratio becomes —=-, P ^ or pjr=p^. 223. Circle on Radius Vector for Diameter touches Pedal. It is clear also from the figure of Art. 221 that the circle on the radius vector as diameter touches the first positive pedal of the curve. For OT is in the limit a radius vector ; and the circle on OT as diameter passing through F and F, two con- tiguous points on the pedal, must in the limit have the same tangent at Fas the pedal curve, and must therefore touch it. 224. Pedal Equation of Pedal Curve. Let r=f{p) be the pedal equation of a given curve. Then, since p^r=p^, we have Pi = ^—y and therefore, writing r for p and p for p^, the pedal equation of the first positive pedal r2 curve IS 75=:;^— r. 170 CHAPTER VII. Ex, The first positive pedal of the rectangular hyperbola r= — is r which is the p, r equation of Bernoulli's Lemniscate, as is also obvious from Art. 218. Examples. 1. Write down the pedal equations of the first positive pedals of the curves given in the table of Art. 214. 2. From the origin is drawn a perpendicular OPi to the tangent at P, similarly OP2 is drawn perpendicular to the tangent at Pj to the locus of. Pi, and so on. Show that the figure PP1P2 ... is equiangular, but cannot be equilateral. [Oxford, 1888.] 3. Show that the k^^ pedal, positive or negative, oiP=f(r) is 225. We may also prove the results of Art. 221 as follows : — Let the tangent P^T make an angle i/r with the initial line. Then the perpendicular makes an angle a = \^ — ^ with the same line. Let OY='p. Let P^^. ^^ ^^® normal, and F^ its Fig. 28. point of intersection with the normal at the contiguous poino Q. Let OFi be the perpendicular from upon the normal. Call this 'py Let Pg^s ^® drawn at right angles to FJt*^, and let the length of OY^y the perpendicular upon it from 0, be 'p^. TANGENTS AND NORMALS. 171 The equation of P^T is clearly 'p = x cos a + 2/ sin a (1) The contiguous tangent at Q has for its equation p + ^^ = iCCOs(a + (5a) + 2/sin(a + ^a) (2) Hence subtracting and proceeding to the limit it appears that -d^ = _ ic sin a + 2/ cos a .' (3) is a straight line passing through the point of intersection of (1) and (2) ; also being perpendicular to (1) it is the equation of the normal PiP2- Similarly 72" — ^cos a — ysma (4) represents a straight line through the point of intersection of two contiguous positions of the line P1P2 ^"^ perpen- dicular to PiP2» ^^^-y ^^® ^^°® -^^Pz* ^^^ ^^ o^ for further differentiations. From this it is obvious that ^^^~da^~d^^' etc. Hence t = PJr=% and p=P,P,=OY+OT,=p+^,. 226. Tangential Equation of a Curve. Def. The tangential equation of a curve is the condition that the line lx-\-my+n = may touch the curve. Method 1. Let F{x, y) = Ohe the curve, then the tangent at aj,2/is XF^+YFy+ZF, = 0. Comparing this with IX -{-m F+ n = 0. Pr Fy F, ^ -7=-^= — = X, say. I m n "^ If X, y, \ be eliminated between these equations, and F{x, y) = 0, or lx+my-^n = 0, a relation between I, m, n will result. This is the equation required. 172 CHAPTER VII. Method 2. We may also proceed thus. Eliminate y between F(x,y) = and lx+iny-\-n = ; we obtain an equation in a?, say ^(ic) = 0. For tangency this equation must have a pair of equal roots. The condition for this will be found by eliminat- ing X between 0(a;) = and ^'(x) = 0. In following this method, instead of eliminating y it is often better to make a homogeneous equation between F{x, y) = and Zoj-f 7712/4-71 = 0, and then express that the resulting equation for the ratio y : x has a pair of. equal roots. Ex. Find the tangential equation of the conic CUV- + 2hx7/ + 6y^ + 2gx + 2fi/ + c = 0. The first process gives us Also lx + my + n=^0. The eliminant from these four equations is a, A, g, I A, 5, /, m 9, /, c, n I, tHj n, which may be written Al^+Bm^ + Cn^ + 2Fmn + 2Gnl+2mm==0, where Ay B, C, ... are the co-factors of the determinant 0, a, A, 9 h h, f 9, /, c Inversion. 227. Def. Let be the pole, and suppose any point P be given ; then if a second point Q be taken on OP, or OP produced, such that OP . 0Q = constant, k^ say, then Q is said to be the inverse of the point P with respect to a circle of radius 7c and centre 0, (or shortly, with respect to 0). If the point P move in any given manner, the path of Q is said to be' inverse to the path of P. If (r, 0) be tlie polar co-ordinates of the point P, and (r\ 0) those of the inverse TANGENTS AND NORMALS. 173 point Q, then rr'^lc^. Hence, if the locus of P be /(r, 0) = 0, that of Q will lbe/(^, 6) = 0. For example, the curves r*" = a"*cos mO and r"'cos 7nd = a^ are inverse to each other with regard to a circle of radius a. 228. Again, if {x, y) be the Cartesian co-ordinates of P, and (x\ y') those of Q, then IS ^^^ n 72'^' cos ^ ,„ x' x = r cos e=-7 cos = A;2-^72— = ^^^:j:^' and similarly y = ^^qp^2- Hence, if the locus of P be given in Cartesians as the locus of Q will be Ex. The inverse of the straight line x=a with regard to a circle of radius Jc and centre at the origin is -5 «=«, or x^-\-i/^=—Xy a circle which touches the axis of y at the origin. Examples. 1. Show that the inverse of the parabola y'^ = Aax with regard to a circle whose centre is at the origin and radius the semi-latus rectum is the pedal of the parabola y'^-\-Aax=0 with regard to the vertex. 2. Show that the inverse of the conic W2+Wi + Wo=0 with regard to the origin is the quartic curve i^u. + kHi{a^ +/) + Uq{x^ + y^f = 0. 3. Show that the inverse of the general curve of the n^ degree, viz., ^n + ^*n-l + W„_o+...+«i + Wo = 0, with regard to the origin is where r'^=x^+y^. 4. Show that the inverse of a conic with regard to the focus is a Lima- 9on (Equation r=a + 6 cos 6\ which becomes a cardioide if the conic be a parabola. 5. Show that the Equation of the inverse of a conic with regard to the centre is of the form r2 = ^ + 5cos2^, which becomes a Lemniscate of Bernoulli if the conic be a rectangular hyperbola. 174 CHAPTER VII. 229. Tangents to Curve and Inverse inclined to Radius Yector at Supplementary Angles If P, P' be two contiguous points on a curve, and Q, Q the inverse points, then, since OP . OQ=OP' . OQ', the points P, P\ Q\ Q are concyclic ; and since the angles OPT and OQ'T are therefore supplementary, it follows that in the limit when P' / Fig. 29. ultimately coincides with P and Q' with Q, the tangents at P and Q make supplementary angles with OPQ. The ultimate ratio of corresponding elementary arcs, viz., dSjPP' Lt OP OP OQ' OP.OQ k^ OQ' OQ OQ^ 230. It follows from the preceding article that when two curves intersect, their inverses intersect at the same angle ; and as particular cases, if two curves touch, their inverses touch, and if the original curves cut orthogonally their inverses cut orthogonally. Ex. 1. It is an obvious property of two confocal and co-axial parabolas whose concavities are turned in opposite directions that they cut at right angles. By inverting this proposition, the focus being the pole of inver- sion, it is clear that the curves which cut orthogonally each member of the family of cardioides r=a(l + cos 6) found by giving different values to a, are also cardioides. Ex. 2. Show by inverting a conic with regard to its focus that the circle .37^ -}-^^ = Z(e + cos a)^ + ? sin a . y touches the Lima9on r = l + le cos at the point given by ^=a. 231. If P, P' be any two points, and Q, Q' their inverse points, then as before (Art. 229) the triangles OPP\ OQ'Q are PF OP B QQf similar and Thus PF. oq OQ.OQ" QQ' OQ . oq' TANGENTS AND NORMALS. 175 Ex. 1. It a,b, c be points in a straight line in the order indicated, then ab-\-bc = ac. Suppose A, B, C to be the inverse points of «, 6, c with regard to any point 0. Then 0, A, B, C are concyclic and AB . „ BO ,„ AC P-. + F OB.OC~^OA.O(f OA.OB whence 00. AB+ OA . B0= OB . AC, the result known as Ptolemy's Theorem. Ex. 2. li 0, Ay Bj C ... J, K be points on a circle, prove AB . BC . . JK AK OA.OB^OB.OC + ...+ OJ .OK~OA.OK [Math. Tbipos, 1890.] Fig. 30. 232. Mechanical Construction of the Inverse of a Curve. In the accompanying figure AC, CB, BQ, QA, PA, PB is a system of freely jointed rods, of which AC=BG, and AQ = QB=BP = PA. At P and Q sockets are placed to carry tracing pencils. A pin fixes C to the drawing board. The system is then movable about C. It is clear from elementary geometry that C, Q, P are in a straight line, and that GP.CQ = CA^-'AQ\ and is therefore constant. Hence whatever curve P is made to trace out, Q will trace out its inverse, the point being the pole of inversion. In the figure P is represented as tracing a straight line, in which case Q will trace an arc of a circle, as shown in Art. 228. Peaucellier has utilized this construction for the conversion of circular into rectilinear motion. 176 CHAPTER VII. Polar Reciprocals. 233. Polar Reciprocal of a Curve with regard to a given Circle. Def. If OY be the perpendicular from the pole upon the tangent to a given curve, and if a point Z be taken on OF or OT produced such that OY. OZ is constant ( = k^ say), the locus of Z is called the pola7' recijprocal of the. given curve with regard to a circle of radius h and centre at 0. From the definition it is obvious that this curve is the inverse of the first positive "pedal curve, and therefore its equation can at once be found. Ex. Polar reciprocod of an ellipse loith regard to its centre. For the ellipse ^,+^=1, a^ 6^ ' the condition that jo=^ cos a+y sin a touches the curve is p^^a^co&^a + 6%in2a. Hence the polar equation of the pedal with regard to the origin is 'T = a^cos^^ -f 6-sin-^. Again, the inverse of this curve is ^=a2cos2(9+6lsin26', r or aV + 6y=y&S which is therefore the equation of the polar reciprocal of the ellipse with regard to a circle with centre at the origin and radius k. 234. The method may therefore be stated thus : — First find the condition that p = xcos a-\-y sin a will touch the given curve. Then write — for p and for a in that condition. The result is the required polar reciprocal with regard to a circle of radius A; and centre at the origin. 235. Polar Reciprocal with regard to a given Conic. Def. If S=0 be any curve and U—0 a given conic, the locus of the poles %vith regard to U of tangents to S is called the Polar Reciprocal of the curve S with regard to the conic U. Let the equation of a tangent to S be ^ = Xcos a+Fsin a, and the condition of tangency TANGENTS 'AND NORMALS. 177 If X, y be the pole of this tangent with regard to U= 0, the tangent must be coincident with the polar cos a ZZg sin a U^ therefore X 'J'^" ^ _ ^y P u; p If, Hence -2 = y-^g ^ and tan a = y^- Hence the equation of the Polar Reciprocal is i7J+£r,^=tr.y{/(tan-i^?)}' For further information on the subject of reciprocal polars and the methods of reciprocation the student is referred to Dr. Salmon's Treatise on Conic SectioTiSf chap. XV. EXAMPLES. 1. If the tangent at tCj, y^ to the curve oc^ + y^^a^ meet the curve again in (JT, F), show that Illustrate the result by means of a figure. [Oxford, 1889.] 2. In the four-cusped hypocycloid jb' + y » = a ', show that if x = a cos^a then y = a sin^a, and that the equation of the tangent at the point determined by a is a; sin a + 2/ cos a = a sin a cos a. Hence show that the locus of intersection of tangents at right angles 2 to one another is r^ = ~ cos^lO. 3. In the semicubical parabola ay^ = ot? the tangent at any point P cuts the axis of y in Jf and the curve in Q. is the origin and N the foot of the ordinate of P. Prove that MN and OQ are equally inclined to the axis of x. 4. At any point of a curve where the ordinate varies as the cube of the abscissa, a tangent is drawn ; where it cuts the curve another tangent is drawn ; where this cuts the curve a third is drawn, and so on. Prove that the abscissae of the points of contact form a geo- metrical progression, and also the ordinates. E.D.C. M 178 CHAPTER VII. 5. If jOj and p^ be the perpendiculars from the origin en the tangent and normal respectively at the point (aj, y\ and if tanT/r = -^, prove dx that jOj = ic sin i/' - 2/ cos i/', and jOg = £c cos ^ + 2/ sin ;/'. Hence prove that 'p^ = -^• 6. The tangent at a point P of the cissoid y\a -x) = x^ meets the curve again in Q and the tangent at Q meets the curve again in R. If be the origin, prove that cot ROQ - cot POQ = J cot POi?. [Oxford, 1885.] 7. The curve aj^ + ^/S = 3a£C2/ is cut in the points P, Q, other than the origin by two lines drawn through the origin which are harmonic conjugates of the axes. Prove that the tangents at P, Q will inter- sect on the curve. [Oxfoed, 1890.] 8. Show that, if the curves r=f(d), r = F{0) intersect at (r, 6), the angle between their tangents at the point of intersection is 9. Prove that the locus of the extremity of the polar subtangent ofthecurve?^+/((9) = 0is u.=ff'^ + o\ 10. Prove that the locus of the extremity of the polar subnormal of the curve r =f{0) is r =/'( 6. - ^V Hence show that the locus of the extremity of the polar subnormal in the equiangular spiral r = ae""^ is another equiangular spiral. 1 + tan- 11. In the curve r = y. m + n tan- 2 the locus of the extremity of the polar subtangent is a cardioide. [Professor Wolstenholme. ] 12. If the normals at the points (r^, 6^), (rg, 0^)^ (r^, 6^) on the cardioide r = a(l + cos 6) be concurrent, show that tan^ + tan^ + tan^ + 3tan|-tan^tan^ = 0. [Oxford, 1890.] 13. If in the last question r-^ + r^ + r^- 2a, show that the locus of the point of concourse of the normals is a circle passing through the pole. [Oxford, 1886.] TANGENTS AND NORMALS. 179 14. Show that the locus of intersection of the normals at the ends of a focal chord of a cardioide is a circle. 15. Show that tangents at the ends of a focal chord of the cardioide r = a(l+ cos 6) intersect at right angles on a circle of radius ~ and centre ( ^, oY 1 6. If Wj, Tig, n^, n^ be the lengths of the four normals and t^, t^, t^ the lengths of the three tangents drawn from any point to the semi- cubical parabola ay^ = x^, then will 27n^n^n^n^ = at^t^t^. [Math. Tripos, 1890.] 17. The polar equation of the pedal of the curve (x^ + 2/2 _ ^2)3 + 27a^xY = with respect to the point h, k may be written in the form r ^ a sin ^ cos ^ - {h cos6 + k sin 6). [Oxford, 1888.] 18. Determine the relation between j9 and r for the curve i/2(3a -x) = {x- af. [Oxford, 1889.] 19. Show that the polar reciprocal of the curve r^ = a**cos mO with regard to a circle whose centre is at the pole is of the form -2L_ m JUL. r'^+^cos ^(9 = 6'"+\ 20. Show that the polar reciprocal of the curve x'^if' ^ a**"*"** with regard to a circle whose centre is at the origin is another curve of the same kind. ,m+l 21. Show that the first positive pedal of the curve p is a and that its polar reciprocal with regard to a circle of radius a whose centre is at the origin is jo*""*"^ = oT'r. 22. Show that the inverse of the curve p =/{r) with regard to a circle whose radius is k and centre at the pole is r2 ./k^\ and that the polar reciprocal is ?-/©■ 23. Show that the pedal of the inverse of p=/(r) with regard to a circle whose radius is k and centre at the origin is kY ^ f(^P\ 180 CHAPTER VII. m+l 24. Show that the pedal of the inverse of jt? = — ^ with regard to a circle whose radius is k and centre at the origin is 25. Show that the polar reciprocal of the curve r*" = a'*cos7/i^ with regard to the hyperbola r^cos 26 = ^^1^ "* m -^ r'^+^cos T-^ = a"*"*'^ m + l 26. The locus of a point X is defined by the equation ^{Pv P2' Ps' ••• P«)="«» where p^, p^-, ... are the distances of X from n fixed points Pj, Pg* • • • ^^n- Show that the equation of its inverse with regard to any origin is ^VW' -R' "• -R-y' where /o/, pa'? • • • are the distances of X\ the inverse of X from the n fixed points ^i, ^2> • • . which are the respective inverses of Pj, Pg, . . . ; ^'i> ^2> ^3> • • • are the lengths of OPj, OF^^ . . . ; and i? ^ OX'. 27. Show that the inverse with regard to any pole of the Car- tesian oval whose equation is Ir + mr' ~ n, where r, r' are the distances of any point on the curve from two fixed points P^, Pg) is / . OPi . pi + m . OP2 . p2 = wp3> where p^, p^ are the distances of any point on the inverse curve from the points which are the inverses of P^, Pg, and p^ is the distance of the same point from the pole of inversion. 28. Show that the inverse of a Cassini's oval defined by the equation rr' = constant is of the form p^p^ = Ap^, the letters p^, p^, p^ denoting the distances of any point on the inverse curve from certain fixed points. 29. If all the normals be drawn from a given point P to any num- ber of given curves, and if P move so that the sum of the squares of the normals PQl + PQ^ 4- . . . + PQn = constant, the normal to the locus of P will always pass through the centre of mean position of the points ^1, Q^^ ^3, ... Q^- [Frenet.] 30. A straight line A OP of given length always passes through a fixed point 0, while A describes a given straight line AT -, show that if PT be the tangent at P to the locus of P, the projection of PT onAOP = AO. TANGENTS AND NORMALS. 131 31. The point P moves so that OP . O'P = constant, 0, 0' being fixed points. If OF, O'Y' be the perpendiculars from and O'on the tangent at P to the locus of P, prove that PY :PY':: OP^: O'P^. 32. Prove that the normal to the curve /(r^. r^) ^ 0, where r,, ra ai e the distances of any point on the curve from two fixed points, divides the line joining the fixed points in the ratio df df ^^dr ' ^^' [Math. Tripos, 1888.] 33. ^ and B are fixed points and P a variable one lying on a curve given by the relation /(^i, ^2) = between the angles PA£( = Oj) and PBA( = 6^). Prove that the tangent at P to the curve divides AB in the ratio sin=^^,^^ : sin=^,^^. [Oxford.] 34. and 0' are two fixed points, P any point in a curve defined by the equation ,--* r r c where r=OP, r =0'P, and c is constant. Prove that the distance between P and the consecutive curve obtained by changing c to c + Sc is ultimately 8c V , 3c2 aV where a = 00'. [Smith's Prize.] 35. In a system of curves defined by an equation containing a variable parameter investigate at any point the normal distance between two consecutive curves, and determine the form of the equation for a system of parallel curves. [Professor Cayley, Messenger of Mathematics, vol. V.] CHAPTER VIII. ASYMPTOTES. 236. Def. If a straight line cut a curve in two joints at an infinite distance from the origin and yet is not itself wholly at infinity, it is called an asymptote to the curve. 237. Equations of the Asymptotes. Let the equation of any curve of the n^^ degree be arranged in homogeneous sets of terms and expressed as a'"0»(|) + a="-Vn-i(|)+«:"-'0»-2(|)+- = O (A) To find where this curve is cut by any straight line whose equation is 2/ = M^+iS (b) By. substitute /)i+— for - in equation (a), and the resulting equation a;"^„(M+§+a;-V.-i(y«+f)+a:''"V»-2(M+f)- = 0...(c) gives the abscissae of the points of intersection. Applying Taylor's Theorem to expand each of these func- tional forms, equation (c) may be written -2!0Vm)+--- 0....(D) + ^n-2(M) This is an equation of the 7^*^ degree, proving that a straight line will in general intersect a curve of the n^^ degree in n points real or imaginary. The straight line y = iuLX-\-/3 is at our choice, and therefore the two constants jul and /5 may be chosen, so as to satisfy any 182 ASYMPTOTES. ]83 pair of consistent equations. Suppose we choose fx and ^8, so that 'n(juL) + (pn-iiiuL) = (f) The two highest powers of x now disappear from equation (d), and that equation has therefore two infinite roots. If^ then, juLi, yug, • . ., /jlu be the n values of yu deduced from equation (e) (which is of the n^^ degree in /x), the correspond- ing values of ^ will in general be given by O _ 0n-l(Mi) o _ 0n-l(M2) ^' ^P'nif^r ^'~ 0'nW '■*• and the n straight lines 2/ = /i2^ + i5: y = IUinX + ^r are the asyTnptotes of the curve. 238. Rule. Hence, in order to find the asymptotes of any given curve, we may either substitute julx + P for y in the equation of the curve, and then hy equating the coefficients of the two highest 'powers of x to zero find /x and p. Or we may assume the result of the preceding article, which may be enunciated in the following practical way : — In the highest degree terms 'put x = l and y — fx \the result of this is to form ^{^)=)x^-2.jj?- ti+'2 = 0\ therefore (/x-l)(/>i + l)(/x-2)=0 ; giving ft=l, -1, or 2. Again, J^fx) = 2-{-fx- fx^ and ^'siij) = 3/>t^ - 4/x - 1 ; therefore P=$^i 184? CHAPTER VIII. Hence if fi=l, i8 = l, if /*=-!,. /?=0, and if /a = 2, /5=0. Hence the asymptotes of the curve are Examples. 1. The asymptotes of y^-^xy^^-\\xSj-^aP--{-x-\-y=^ are y='^-, y — '^^t y—^x- 2. The asymptotes of y^— ^y+2y^+4y+.27=0 are y=0, y-^+l=0, y+^+l=0. 239. Number of Asymptotes to a Curve of the mP^ Degree. It is clear that since ^n(/w) = is in general of the n^^ degree in yw, and /5^'n(M)+0n-i(M) = O is of the first degree in /3, that n values of jm, and no more, can be found from the first equa- tion, while the n corresponding values of /3 can be found from the second. Hence n asymptotes, real or imaginary, can be found for a curve of the n^^ degree. 240. If the degree of an equation be odd it is proved in Theory of Equations that there must be one real root at least. Hence any curve of an odd degree must have at least one real asymptote, and therefore must extend to infinity. JS^o curve therefore of an odd degree can he closed. Neither can a curve of odd degree have an even number of real asymptotes, or a curve of even degree an odd number. 241. If, however, the term y'^ be missing from the terms of the n^^ degree in the equation of the curve, the term jul^ will also be missing from the equation ^„(ya) = 0, and there will therefore be an apparent loss of degree in this equation. It is clear, however, that in this case, since the coefficient of u^ is / . ... zero, one root of the equation (pnijud = is infinite, and there- fore the corresponding asymptote is at right angles to the axis of X ; i.e., parallel to that of y. This leads us to the special consideration of such asymptotes as may be parallel to either of the axes of co-ordinates. ASYMPTOTES. 185 242. Asymptotes Parallel to the Axes. Let the curve arranged as in equation (a), Art. 237, be + b^x''-'^ ^h^-^-^y +. -^-hny''-^ + c^'^-^ 4-... + ... = (aO If arranged in descending powers of x this is aoa;'»+(ai2/+6iK-H...=0 (b') Hence, if a^ vanish, and y be so chosen that a{y-\-h^ = 0, the coefficients of the two highest powers of x in equation (b') vanish, and therefore two of its roots are infinite. Hence the straight line a^y + 6^ = is an asymptote. In the same way, if a„ = 0, an-ix + hn = is an asymptote. Again, if a^ = 0, a^ = 0, h^ = 0, and if y be so chosen that three roots of equation (b') become infinite, and the lines repre- sented by a^y'^ -{-h^y+c^^O represent a pair of asymptotes, real or imaginary, parallel to the axis of x. Hence the rule to find those asymptotes which are parallel to the axes is, " eqvxite to zero the coefficients of the highest powers of X and y" Ex. Fimd the asymptotes of the curve Here the coefficient of x^ i&y'^—y and the coefficient of y^ i^ ar^—x. Hence 07=0, ^=l,y=0, and y=\ are asymptotes. Also, since the curve is one of the fourth degree, we have thus obtained all the asymptotes. Examples. =0 \. ■=±a/ 1. The asymptotes of y\x^ — a^) = x are y=Q X- 2. The co-ordinate axes are the asymptotes of xy'^+x^y=a'^. 3. The asymptotes of the curve x^y'^=c\x^+y'^) are the sides of a square. 136 CHAPTER Vni. 243. Partial Fractions Method. The values of ^, viz., _ 0n-l(M l) ^71-1(^2 ) Q^Q are exactly the constants required in putting 0n-l(O 0n(O i'/i^o partial fractions * This gives a very easy way of obtaining the asymptotes. For if 5 ^ 3 y + ^=5. .244. Particular Cases of the General Theorem. We return to a closer consideration of the equations 0n(M) = O, (E) l3'n(fl)-^n-l(fJ^)=0, (F) of Art. 237. It is proved in Theory of Equations that if an equation such as and putting «=^i, +by)Fn-2-\-fn-2 = 0, in proceeding to infinity in the direction ax+by = Oy we have {ax+byy+(ax+by) . Lt^-hLt^^==0 Jrn-2 -t n_2 190 CHAPTER VIII. when the limits are to be obtained by putting x=—-,y = j, and then diminishing t indefinitely. We thus obtain a pair of parallel asymptotes ax+by = a and ax-\-by = l3 where a and /8 are the roots of And other particular forms which the equation of the curve may assume can be treated similarly. Ex. 1. Tojind the 'pair of parallel asymptotes of the curve {2x-Zy+l)\x+y)-Sx + 'iy-Q=0. Here 2x-2y + \=±J~LEEM^, ^ x+y where x and y become infinite in the direction of the line '2.x=Zy. 3 2 Putting x=-y y = -, the right side becomes ±2. Hence the asymptotes t t required are 2x-3y = l and 2^-3y + 3=0. Ex. 2. Find the asymptotes of {X -y)2(^2 +^2) _ 10^^ _ y)^2 4- 12y2 _|. 2^ +y = Q. Here (^-yf-io{x-y)Lt.=y=^-^^+ 12Zi^.=,=„^,=0. or (x-yf-5{x-y) + 6 = 0, giving the parallel asymptotes x—y = 2 and x-y=2. 47. Asymptotes by Inspection. It is now clear that if the equation -^^ = break up into linear factors so as to represent a system of n straight lines no two of which are parallel, they will be the asymptotes of any curve of the form i^„ + J'^ _ 2 = 0. Ex.1. (x-y){x+y){x + 2y-l)=3x+4y + 5 is a cubic curve whose asymptotes are obviously x-y=0, x+y=Oy x+2y-l = 0. Ex.2. (x-yy{x+2y-l) = 3x + 4y + 5. Here x + 2y-l = is one asymptote. The other two asymptotes are parallel to y=x. Their equations are .,/7! 3 + 4+5^ j_.A ASYMPTOTES. 191 ^^248. Case in which all the Asymptotes pass through the Origin. If then, when the equation of a curve is arranged in homo- geneous set of terms, as it be found that there are no terms of degree n — ly and if also Un contain no repeated factor, the n straight lines passing through the origin, and whose equation is u^ — 0, are the n asymptotes. Examples. Find the asymptotes of the following curves : — 1. f=x%2a-x). 2. 7/^=a^a^ — x^y 3. a^+f=al 4. y{a^-\-x^)=a^x. _5. axy=a^ — a?. 6. y\2a-x)=a?'. 8. x'^y-\-'tp-x=o?. 9. xY={a+y)\h'i-y^). 10. x'f=aY-h'^x\ 11. xy{x-y)-a{x'^-y'^)=h^. 12. (a2-^)y2 = ^2(a2 + ^2), 13. xy^^^aX'Ha-x). U. y\a-x)=x(b-xy. 15. x^y=x^ + x+y. 16. xy'^ + a^y=xi^+7n^+nx+p. 17. x^ + 2x^-xy^-2y^ + Ay^+2xy-i-y-l=0. 18. x^-2a^y+xy^+x^-xy + 2 = 0. 19. y{x-yY=y{x-y) + 2. 20. x^ + 2x^y-4xy^-8y^-4x+8y = l. 21. {x+yf{x+2y+2)=x + 9y-2. 22. 3x^ + 17^y + 21;i:^ - 9^^ _ 20^2 _ jgo^ _ l8a/ - 3a^x+a^y=0. . — 249. Intersections of a Curve with its Asymptotes. If a curve of the ti*^ degree have n asymptotes, no two of which are parallel, we have seen in Art. 247 that the equations of the asymptotes and of the curve may be respectively written and Fn^-Fn-i^^. The n asymptotes therefore intersect the curve again at points lying upon the curve i^„_2 = 0. Now each asymptote cuts its curve in two points at infinity, and therefore in ti/— 2 other 192 CHAPTER Vin. points. Hence these n(n-^2) points lie on a certain curve of degree -n — 2. For example, 1. The asymptotes of a cubic will cut the curve again in three points lying in a straight line ; 2. The asymptotes of a quartic curve will cut the curve again in eight points lying on a conic section ; and so on with curves of higher degree. Examples. 1. Find the equation of a cubic which has the same asymptotes as the curve .r^ — Qx'^y + 1 \xy^ — Qy^ + ^+y + l=0, and which touches the axis of y at the origin, and goes through the point (3, 2). 2. Show that the asymptotes of the cubic a^y — xy^+xy+y'^+x—y^Q cut the curve again in three points which lie on the line x-\-y=0. 3. Find the equation of the conic on which lie the eight points of intersection of the quartic curve xy{x'^—y^) + a^y^-{-l)^x'^ = a^h^ with its asymptotes. 4. Show that the four asymptotes of the curve {aP- -y'^y^'^ - 4r2) - 6^ + bx^y + ^xy"^- 2f - ^24.3^^ -1=0 cut the curve again in eight points which lie on a circle. 5. Form the equation of the cubic curve which has ^=0, v=0, -+f = l a for asymptotes, and cuts its asymptotes in the three points where they intersect the line — + |^ = 1, and also passes through the point a, h. . a 6. Form the equation of a quartic curve which has ^=0, y=0, y=Xy y= —X for asymptotes, which passes through the point a. 6, and cuts its asymptotes again in eight points lying upon the circle x^-\-y^=c^. 250. Common Transversal of a Curve and its Asymptotes. The equation of the asymptotes and that of the curve coincide in the terms of the n)^^ and (71— 1)*^ degrees. Hence, if we put both equations into polars, the sums of the roots of the two equations for r are equal ; also, the origin is arbitrary. Hence, if through any point a line OP^P^P^... be drawn to cut the curve in the points P^, Pg, P3, . • . and the asymptotes in _Pp P2. Vz>' • • ^^^^ 2^^ = ^Ojp, whence, if 1!^0P = 0, it follows that ^Op==0, so that both systems of points have the same centre of mean position. Hence also the algebraical sum of the intercepts between the curve and the asymptote is zero. [Newton.] • ASYMPTOTES. 193 A well known case of this is that of the hyperbola, where, if be the middle point of P^P^j 0Pi+0P2 = 0, and therefore 0^1 + ^'p2 = ^' ^^^ therefore is also the middle point of ^i2^2» whence it follows that in that case P^Pi^VJ^^- 251. Other Definitions of " Asymptotes." Other definitions have been given of an asymptote, e.g.^ (a) That an asymptote is the limiting position of the tangent to a curve when the point of contact moves away along the curve to an infinite distance from the origin, while the tangent itself does not ultimately lie wholly at infinity; again, (13) That an asymptote is a straight line whose distance from a point on the curve diminishes indefinitely as the point moves away along the curve to an infinite distance from the origin. 252. To prove the Consistenoy of the Several Definitions. We propose to show that the results derived from these definitions are the same as those derived from our definition in Art. 236. Consider definition (a). Let the curve be U=Un+Un-\-¥Un-2+...-\-UQ=0. The equation of the tangent is "We shall now suppose the point of contact x^ y to move to oo along some branch of the curve. We shall therefore only retain the highest powers of X and y which occur, viz., those of the {n-Vf^ degree. Thus we must retain only -~ for -^, -~ for -^, and Un-\ for w„_i + 2m„_2 + ... +wmo. Hence in the limit we shall have lit or T=^X{ -Lt^ OUn and it is easy to see that this agrees with the equation of an asymptote found in Art. 237. 253. We next consider definition {/3) ; we have already shown that ax + by + c=0 is, according to our definition, in general an asymptote of the curve (ax + by + c)Fn-x + Fn-2= 0. The perpendicular from any point x, y of this curve upon the line ax+by + c = is ax+by + c ^ 1 F„-2 ^ B.D.C. N 194 CHAPTER Vm. and the limit of this expression is clearly zero when x and y become in- finite in the ratio —b:a, provided that the terms of degree n-l in F^.j do not contain ax+hy as a factor, for the degree of the denominator is higher than that of the numerator. Hence the distance between the curve and the asymptote is ultimately a vanishing quantity, and the line ax+hy + c—Q is such as to satisfy definition {fB). — 254. The Curve in General lies on Opposite Sides of the Asymptote at Opposite Extremities. Let the straight line acc-f % + c = be an asymptote of the curve, and suppose there is no other asymptote of the curve parallel to this. The equation of the curve is of the form {ax-\-hy-\-c)Fn-\-\-Fn-2 = ^', and, as in the last article, the perpendicular from any point x, y of the curve on this asympr tote is given by P=--^|zJ. When X and y become very large in the ratio given by y _ ^ X this may ultimately be written as where k is & constant, and it is therefore obvious that P changes sign with x. Hence in general the curve at the opposite extremities of this asymptote lies on opposite sides of it. 255. Exceptions. If, however, ax+hy be a factor of the terms of highest degree in Fn-2, we may write the equation of the curve {ax+hy + c)Fn-i+Fn-B:^0, so that the perpendicular on the asymptote is now given by j^^ ax+hy + c ^ 1 j^V^. J^^F+W Ja^+h^ Fn-i ' and when x and y become very large in the ratio given by y__a x" V this can be ultimately written This, however, though ultimately vanishing, does not change ASYMPTOTES. I95 sign with aj, so that in this case the curve at opposite extremities of the asymptote lies on the same side of it. 256. Again, if the equation of the curve be expressible in the form {ax-^by+c)^Pn-2+Fn.^=0, the expression for the length of the perpendicular is in the limit of the form /(-). This does not in general ultimately vanish, and therefore in general ax+hy-\-c = is not an asymp- tote, but is parallel to a pair of asymptotes. This case has been discussed in Art. 246. 257. If, however, the curve assumes the form {ax + hy-\-cfF,.2+Fn-s = 0, the length of the perpendicular is given by (Perpendicular)2=-^, ^l Hence, if the ratio of - be that of — r when x and y become X infinite, this may ultimately be written X'' \x/ and therefore Perpendicular = ± a/ - . /( - j> which ultimately vanishes, but x cannot change sign or the perpendicular will become imaginary at one extremity of the asymptote. Hence the line is only asymj^totic at one end and the curve approaches the asymptote on opposite sides. And in the same way other particular forms may be discussed. 258. Curvilinear Asymptotes. If there be two curves which continually approach each other so that for a common abscissa the limit of the difference of the ordinates is zero, or for a common ordinate the limit of the difference of the abscissae is zero when that common abscissa or common ordinate is infinite, these curves are said to be asymptotic to each other. For example, the curves y = Ax^+Bx+G-{--, X y = Ax^+Bx-{-G are asymptotic ; for the difference of their ordinates for any common abscissa a; is — , a quantity whose limit is zero when X X is infinite. 196 CHAPTER VIII. 259. Linear Asymptote obtained by Expansion. Stirling's Method.* If it be possible to express the equation of a given curve in G D the form y^Ax-\-B-\ 1 — ^+..., ^ X x^ then the line y = Ax-\-B is clearly asymptotic to the curve. This method of obtaining rectilinear asymptotes is frequently useful. ,^ ■ , ^^60. To find on which side of the Asymptote the Curve lies. The sign of G (Art. 259) is useful in determining on which side of the asymptote the curve lies. Let y be the ordinate of the curve, y' that of the asymptote, Q If X be taken sufficiently large, the sign of — governs the sign X of the whole of the right-hand side. Suppose X and y to be positive, i.e., m the first quadrant, then y — y' will have in the limit the same sign as G. If G be positive, y — y' will be positive, and the ordinate of the curve will be greater than that of the asymptote, and the curve will therefore approach the asymptote from above. Similarly, if G be negative, y — y' will be negative, and the curve will approach the asymptote from below. And in the same way for portions in the other quadrants. Ex. 1 . Find the asymptotes of the curve Here x^ — a^^O gives x=a and x= -a, two asymptotes parallel to the axis of y. Again, y=±x\ ^ * Lin. Tert. Ord. Newtoniance, p. 48. L^-'FORuW ASYMPTOTES. 197 Hence the asymptotes are and Again, considering it appears that if ^ be positive the ordinate of the curve is less than the ordinate of the asymptote, and therefore the curve approaches the line 2^ =^ in the positive quadrant from below. Similarly the curve approaches the asymptote y= -x ui the fourth quadrant from above^. The student should observe that the curve cuts the axes where :r= ±2a, and also at the origin where the tangents are y = ± 2a7. Also that y is imaginary when x^ lies between a^ and Aa^. There should now be no difficulty in drawing a graph of the curve. Ex. 2. Find the asymptotes of (y - x^x - Syiy - ;r)+ 2^=0, and examine how the curve is placed with reference to them. Here the coefficient ot y^ ia x-Z; therefore J7 = 3 is an asymptote. Also the curve may be written X and therefore, in the direction y=x a,t infinity, this ultimately takes the form (y-^)2-%-ar) + 2=0. Fig, 31. Hence y-x=l and y-x=2 are asymptotes. Put B y-x=A+ +...y X therefore the equation of the curve becomes (a + ^+...\'x-3(^x+A+^+...\U+?^ + ...\ + 2x=0, or x{A^-3A + 2) + {2AB-3{AHB)}-{-...=0. 198 CHAPTER VIIL Equating to zero the several coefficients ^2-3^ + 2=0, etc., whence A=l or 2, B=-3 or 12, etc. Hence the equation of the curve may be expressed in either of the ways y=£c+l-^..., y=^+2 + — .... X a; Hence to the right of the y-axis the curve lies below the asymptote y=.r+l, and above the asymptote 3/=^+ 2. On the left side of the y-axis the curve is above y=^ + l, and below y=:r+2. The student will easily verify (a) that neither of the cross asymptotes cuts the curve again in a point whose co-ordinates are finite ; (/?) that the asymptote ^=3 cuts the curve where y = 3f ; (y) that the product of the roots for y is -^^ — -i— ^, and is positive unless ' x — 3 X lies between and 3, but is then negative ; (8) that y is imaginary if x lies between and — 24 ; (e) that the tangent at the origin is ^=0. Figure 31 is a tracing of the curve. Examples. 1. Find the asymptotes of the curve y=x ^,^ ^^ . Find on which side X — ct of the oblique asymptote the curve lies in the positive quadrant. Show also that the hyperbola x{y -x) = 2a^ is asymptotic to this cubic curve. 2. Find the asymptotes of the curve y2=^2^X?j and find on which side the curve approaches these asymptotes. 3. Show that the curve x= ^ ~^ has a rectilinear asymptote y=0, and a parabolic asymptote 7/^= ax. 4. Show that the curve x^y =x*+x^+x^+x+l has a parabolic asymptote whose vertex is at the point ( - J, |), and whose latus rectum = 1 . 5. Show that the curve x^y=x^+x^+x + l has a hyperbolic asymptote whose eccentricity \/2-f^2 261. General Investigation. In order to express the general equation »>„g)+a;"-V»-.(|)+^"-V»-.(|)+-=0 (1) ASYMPTOTES. 199 in the form 2/ = Ma^+/^+^+-2+..., (2) substitute for y from (2) in (1) ; then, since the result must be an identity, the coefficient of each power of x will be zero. This will give sufficient equations to determine jul, fi, y, .... The result of this substitution is a^^0n(M)+^^'' which gives us the series of equations yf,(M)+...=0 0nW=O, (i.) ^0'„(M)4-0n-iW = O, (ii.) y0nW + fr*"n(M) + /30'n-l(/x) + ^„.2(M) = O, (iii-) Hence jm, ^, y ... are determined. 262. Parabolic Branches. In the case when ^n(/x) = has equal roots /i^, it follows as in Art. 244 that 'M-^AB\{lj) + -g (t>"'n{f^) + ^'ti - l(/x) = 0, A^ + B etc., which determine the hitherto unknown constants A,B,C..,. The parabola {y — juLX — By = A^x is then asymptotic to the curve, and the side of the parabola on which the curve lies is indicated by the sign of C. ASYMPTOTES. 201 It should be noticed that the first approximation is not in general asymptotic to the curve. 263. In practice it is found more convenient to adopt a method of successive approximation to obtain the ultimate form of a parabolic branch at an infinite distance from the origin. This method will be indicated best by an example. Ex. Obtain the rectilinear asymptote of the curve (y—^)2(y+;r)=2ar2^ and examine the parabolic branch. The rectilinear asymptote is parallel to y+x=0. We may write the equation y+x=2a ^ {y-xf = 2a — - to a first approximation =%■■ a) giving the equation of the asymptote. Proceeding to a second approximation, y+x='zar 2V 4^/ =^+«^ (2) This indicates that the curve lies above the asymptote on the right-hand side of the y-axis, but below on the left. To examine the parabolic branch. The axis of the asymptotic parabola is clearly in the direction y=x. For a first approximation to the shape at infinity, 2^-^=Vl^=^'^- (3) ' x-\-x For a second approximation, substitute this value of y and we obtain y-x=\/ - x-\-x-\ '•lax =s/«-i(l+jVi)-* .VS(i-iVf...). or y-x='Jax-^ (4) 202 CHAPTER VIII. To obtain a third approximation, use the value of y given by Equation (4) Thus y-x +n/< X->rOC y — x=s/ax—^a + 32 x^ .(5) It appears, therefore, that though the first approximation (3) indicates the ultimate shape of the curve at infinity, it is not asymptotic to the curve. The second approximation (4) is a true asymptotic parabola, for Equa- tion (5) shows that the limit of the diflference of its ordinate and that of the curve is zero. The third approximation (5) shows that the ordinate of the upper branch of the parabola is less than that of the curve, and that the ordinate of the lower branch of the parabola is greater than that of the curve, so that both branches of the curve approach the parabola from the outside. We add a tracing of the curve. Fig. 32. 264. For further information on the subject of curvilinear asymptotes the student is referred to Frost's Curve Tracing, chapters VII. and VIII. ASYMPTOTES. 203 26 S. Polar Co-ordinates. Let the equation of the curve be r"/„(0)+r'-y„.x(0) + ...+/„(e) = O (1) or _ u%{e)+u--'f,{e)+...+fn{e)=Q (2) To find the directions in which r = oo or u = we have /n(e) = (3) Let the roots of this equation be Let XOP — a. Then the radius OP, the curve, and the asymptote meet at infinity towards P. Let OY^^'p) be the p p" perpendicular upon the asymptote. Since OF is at right do angles to OP it is the polar subtangent, and p— — -v-. Let XO Y= a\ and let Q be any point whose co-ordinates are r, upon the asymptote. Then the equation of the asymptote is p = rcos (O — a) (4) It is clear from the figure that a=a — ^' do To find the value of — j when u — diflerentiate equation (2), and put u = and = a, and we obtain (J\y»-i(«)+/n(a)=0 (5) Substituting the value of i—j-) hence deduced for p i equation (4) we have fj^^rcos(0-a + ^) fn{a) \ 2/ = rsm(a — 0). in 204 CHAPTER VIIL Hence the equations of the asymptotes are etc. CoK. The case most often met with is that in which n = l, when the equation of the curve is rf^{0)+fQ(0) = O. Then /i(^) — ^ gives a, p, y, etc., and the asymptotes are rsin(a-0) = ^„ etc. / 1(«) 266. The equivalent Cartesian form will be found convenient to remember and somewhat easier to draw the asymptote from than the polar equation. 267. Rule for Drawing the Asymptote. After having found the value of i-r-) suppose we stand at the origin and look in the direction of that value of which makes u = 0. Draw a line at right angles to that direction through the origin and of length equal to the value of ( — ^ ) to the right hand or the left according as that value is positive or negative. Through the end of this line draw a perpendicular to it of indefinite length. This straight line will be the asymptote. 268. To deduce the Polar Asymptote from the Polar Tangent. The same results may be deduced from the equation of a tangent (Art. 206). The result u=Ucos(0 — a)+V''sm(0 — a) at once reduces to jp = T sin (0 — a), when U=0. Putting 1 _ /n-i(a) as found in the last article, we again obtain the equation r sm (a — 6) = ' .. / x • ASYMPTOTES. 205 Ex. Find the asymptotes of the curve r=a tan ^ or r cos ^ — asin ^=0. Here /i(^) = cos ^ and /o(^)= -asin . cos^=0 gives ^~^y i^"^"^' ^*^-' and /o(a) _—a sin a _ /'i(a.) - sin a Hence rsin(^-^j==a or 7'cos^ = a rsinf-^-^J = a or rcos^=-a are the asymptotes. Or, using the Cartesian formula of Art. 266, u= cot 6. a u=0 if ^=7i7r + J, and -^=asin2^=a 2 du for this angle. Hence the formula y cos a=x sin a + ( -y- ) KduJu^o becomes .r= ±a. 269. An Exceptional Case. In forming Equation 5, Article 265, it has been assumed that the value of ( -7^ ) there obtained is not indeterminate ; and, further, that none of \at7/u=Q the coefficients of the several powers of u become infinite in the limit when 6 is put equal to a. If on differentiating Equation 2 and putting -^ = and = a any term should occur which is indeterminate, it must be retained and -J-) evaluated, either in an elementary manner or au/u=o by the methods laid down for undetermined forms in Chap. XIV. Examples. 1. rO^=a. 5. r=2asin^tan^. 2. r6=a. 6. rsin26/=acos3^. 3. rsin?i^=a. 7. r=a + bcotn9. 4. r = acosec^+6. 8, r"sin?i^ = a**. 9. Show that all the asymptotes of the curve rtanw^=a touch the |276i Circular Asymptotes. In many polar equations when is increased indefinitely it happens that the equation takes the form of an equation in r, which represents one or more concentric circles. 206 CHAPTER VIII. For example, in the curve r=^a-^~-, u— 1 which may be written r=a , '-\ it is clear that if becomes very large the curve approaches indefinitely near the limiting circle r=a. Such a circle is called an asymptotic circle of the curve. EXAMPLES. 1. Find the asymptotes of the curves (i.) a^ -\- a^xy -- y^ = 0. (ii.) y^—x^=^a^xy. 2. Show that there is an infinite number of asymptotes of the curve y = (a- a;)tan — , viz., x= —a, X— ±. Za, x= ± 5a, etc. 3. Prove that any tangent to the curve Zxy'^ = c^ is divided by the asymptotes and the curve into segments which bear a constant ratio to each other. [Oxford, 1889.] 4. Find the asymptotes of the curve x\x + y){x - 3/)2 + ax%x -y)- a^y^ = 0. [Oxford, 1889.] 6. Find the asymptotes of the curve (x - yf{x - 2y){x - 3y) - 2a{x^ - y^) - 2a\x + y){x - 2y) = 0. [Oxford, 1888.] 6. Determine the asymptotes of the sextic (cc2 _ 22/2)2{2(a:2 + 22/2) - 3} = {^x^ + 2^/2) - 4)2. [Oxford, 1886.] a^2 7. If r = ^2 — T» *-^® curve has two rectilinear asymptotes at a dis- tance ^ from the pole, making angles ±1 with the prime radius. Also, there is a circular asymptote. 8. Find the asymptotes of the curve rd cos6 = a cos 26. [Oxford, 1889.] 9. Find the asymptotes of the curve rO cos = ae^. [Oxford, 1888.] 10. Show that there is an infinite series of parallel asymptotes to the curve . r = ^r—, — ^ + h and show that their distances from the pole are in Harmonical Pro- gression. Find the circular asymptote. ASYMPTOTES. 207 11. Show that the curve Q\ar - r^) = h^ has a circular asymptote. 12. If w =f{d) be the equation of a curve and f{d) = gives a root ^ = a, the corresponding asymptote is . sec a y = a;tana + _— -• / W Ex. For rS = a{&^ - tt^) the asymptote is y + a-K^ = 0. 13. Show that if y — xf{x) be the equation of a curve which admits of a rectilinear asymptote, then ••=^[4)L*[r/e)L is its equation. Apply this method to find the asymptote of a:^ + 2/^ = 2>axy. [Baily and Lund.] 14. Show that one of the asymptotes of the curve x\x - 2yf - Say^x -2y + 2a) - 2a^xy = touches it at a point whose co-ordinates are finite. [Oxford, 1890.] ] 5. Determine the asymptotes of the curve 4(x4 + y^) - ITrcy _ 43^(42^2 _ ^,2) + 2{x^ - 2) = 0, and show that they pass through the points of intersection of the curve with the ellipse x^ + 4y^ = 4. [Oxford, 1890.] 16. Prove that the mn intersections of two curves of the m^ and ^th (Jegrees, and the mn intersections of the asymptotes of each with those of the other lie on a curve of the (m + n- 2)"" degree. Examine the case of a number of the asymptotes being the same for both curves. [Math. Tripos, 1876.] 17. Determine completely the relation of the line ax + by = to the curve (ax + hyfv^.^ + {ax + hy)w^_^ + w„_s + . . . + Wq = where w,., v^, w^ are homogeneous functions of x and y of degree r. q2 52 Trace the curve — + — = 1, and determine the form it assumes x^ y^ when a is diminished indefinitely. [Math. Tripos, 1884.] 18. Obtain the rectilinear asymptotes of the curve y\x^ - 2/2) _ 2a?/3 + 2a^x = 0, and the parabolic asymptotes of y^ - 2xy\x + a) + (aj + a)o(^ = . [Oxford, 1887. ] 19. Form the equation of a quartic curve which has asymptotes x-y = and x + y = 0, the curve being supposed to approach each asymptote at one extremity only, but from both sides of that asymp- tote, and also to touch the axis of y at the origin. 208 CHAPTER VIII. 20. Form the equation of a quartic curve with asymptotes 2/ = 0, x + y = Oj x-y = 0, the curve being supposed to approach y = from opposite sides at the same extremity, but the other two asymptotes from the same side and at opposite extremities in each case. The curve is also to touch the axis of y at the origin and to pass through the point (2a, a). 21. Find the equation of a curve of the fourth degree which has two coincident asymptotes x + y = \y an asymptote x-y = \ and a fourth parallel to this, and of which the origin is a double point, the branches touching the axes of co-ordinates. [Math. Tripos, 1887.] 22. Find the equation of a quartic which has y=±x±\ for asymptotes, which cuts the a;-axis in four contiguous points at the origin, and the 3/ axis in three points (other than the origin), for which the product of the ordinates is - 1. 23. Obtain the asymptotes of the curve {y -h){y- c)x^ = ahj"^, and find upon which sides of the asymptotes the curve lies. ^3 24. Show that the curve y + x + a- --- = () is asymptotic to the folium of Descartes x^ + y^ = Zaxy. Hence find on which side of the linear asymptote the curve lies. 25. For the quartic ax^ - hy^ + c^xy = show that a^ (^ c^ y = x— + b^ 4.aH^x 42. 1 . 2aT6tic3 Draw the asymptotes, and determime on which sides the curve lies. [ViNCE, Fluxions ; Peacock.] 26. Find the asymptotes of the quartic (2/'+^-'){a2/ -!)'+«''}+ Ky + «) = 0, examining in the several cases on which side of the asymptotes the curve lies. [A. Beer.] 27. In the curve y^ = 6x^2/ + ^^ there are no rectilinear asymptotes, but the curve is asymptotic to the parabola y = x^-{- 2x. 28. Find the asymptotes of the curve y{y - xY(y + 2x) = 9cx^, showing that the parabola (y-x + 2c)2 = 3cx is asymptotic to the curve. [Frost, Curve Tracing.'} 29. Show that the curve (y - 2x)^y + x) + (y + 3x){y - x) + x = has a parabolic branch to which Sy^ - 12xy+ \2x^ + 5a; = is a first approximation, and to which the parabola 8(2/- 2x + ^Y + 5ic = is asymptotic. ASYMPTOTES. 209 30. Find the rectilineal asymptotes and the parabolic branches at infinity of the curve {y-xY + {y-xf2y + {y-x){Zx-y)-'2x-2y + l=0, and find the position of its points of intersection with (y-xf + x + y^Q. [Oxford, 188a ] 31. Find the asymptotes of the curve r(sin a - sin ^) = a sin a cos 6. Examine the case when a becomes a right angle. [WoLSTENHOLME, EducotioncU Times.] 32. Show that a cubic curve with a double point cannot have parallel asymptotes. A cubic has three given asymptotes which form an equilateral triangle. Show that if the curve possess a cusp it must lie on the inscribed circle of the triangle. [Math. Tripos, 1890.] 33. If the equation of a curve be written and if cf>„(fh) = 0, <^„'(/^,) = 0, <^„_i(/ti) = 0, and '„_! W = 0, show that there are two parallel asymptotes equidistant from the origin, whose equations are 34. Show that the first approximation to the difference of the ordinates of the curve «^<^„(|)+»"-*,.{|).x-^„_,g) + ...=0 and its rectilinear asymptote y = /xx + /3 for a point whose abscissa is X is H'Mf ' ' assuming that no other asymptote is parallel to this one. Show from this result that the curve at opposite extremities is in general also on opposite sides of the asymptote. 35. Prove that an algebraic curve of the n^ degree represented by the equation has two parallel asymptotes, provided ^oC/*)? ^o W' /i(/*) vanish for E.D.C. O 210 CHAPTER VIII. the same value of /x ; and that the approximations to the correspond- ing infinite branches of the curve are given by where v is a root of the equation J-'^'W + v/.X/x) +/,(/.) = 0. Find also an approximation for the case of equal roots. [Math. Tripos, 1888.] 36. Show that the asymptotes of the general curve of the n*^ degree will all pass through one point if Gq, «!, ttgJ •••» ^M-l = 0, aj, a^, %, ..., a„ bo, &i, 62, •••, K-. and that the co-ordinates of that point are a-fii — a^bf) OjbQ — Qfjb^ a^a^ — Obx ci{fi'i — ci\ [The notation {clq, a^, ..., a^^x, yY is used for the general binary quantic of the n*^ degree, viz. CHAPTER IX. SINGULAR POINTS. 271. Concavity. Convexity. In the treatment of plane curves the terms concavity and convexity with regard to a point are applied with their ordin- ary signification. Thus, for example, any arc of a circle is said to be concave to all points within the circle ; whilst to a point without the circle the portion lying between that point and the chord of contact of tangents drawn from the point is said to be convex and the remainder of the circumference concave. 272. In general the portion of a curve in the immediate neighbourhood of any specified point lies entirely on one side of the tangent at that point. This is clear from the definition of a tangent, which is considered as the limiting position of a Fig. 34. chord. There is an ultimately coincident cross and recross at the point of contact, as shown at the ultimately coincident points P, Q in fig. 34 ; so that the immediately neighbouring portions AP, QB must in general lie on the same side of the tangent PT. 273. We may thus give the following definition of concavity and convexity. Let P be any point of a curve in the midst of continuous curvature. Let A and B be two points near together on the same branch of the curve passing through P, but on opposite sides of P. Then in the limit when the arc 211 212 CHAPTER IX. AB is indefinitely diminished the curve is concave in the immediate neighbourhood of P to all points on the same side of the tangent as the arc APB and convex to all points on the opposite side. 274. Point of Inflexion. Stationary Tangent. The kind of point discussed in Art. 272 is an ordinary point on a curve. It may however happen that for some point on the curve the tangent, after its cross and recross, civsses the curve again at a third ultimately coincident point. Such a point can be seen magnified in Fig. 35. Fig. 35. In this case it is clear that two successive tangents coincide in position : viz., the limiting positions of the chords PQ, QR. The tangent at such a point is therefore said to be " stationary!' and the point is called a " point of contrary flexure " or a " point of inflexion " on the curve. The tangent on the whole crosses its curve at such a point, and the curve changes from being concave to points on one side of the tangent to being convex to the same set of points. 275. Point of Undulation. Agaiu, there may be a point on the curve for which the Fig. 36. tangent crosses its curve in four ultimately coincident points, P, Q, R, S, as seen magnified in Fig. 36, and the point is then called a *^ point of undulation " on the curve. There are now three contiguous tangents coincident, and the tangent on the whole does not cross its curve. And it is clear that singular- ities of similar character but of a higher order may arise. SINGULAR POINTS. 213 276. Analytical Tests. Concavity and Convexity. It is easy to apply analysis to the investigation of the form of a curve at any particular point. Let us examine the point x, y on the curve y — (p{x). Let P be the point to be considered, P^ au adjacent point on Y Let then N N, X Fig. 37. the curve. Let PN, PiiV^ be the ordinates of P and P^, and suppose PjiVi to cut the tangent at P in Q^. Then ON=x, NP = y = cf>{x). ON^=x-\-h, ^\Pi = cl>(x-\-h) = {x)+hcl>\x)+^f(x)-{-,.., (1) by Taylor's Theorem. Again, the equation of the tangent at Pis Y-'y = cpXx){X-x). Putting X — x + h we obtain Y=y+ h"\x) be positive, N-^P^ — N^Q^ changes from negative to positive in passing from negative to positive values of h : i.e., in passing through P the change is from concavity to convexity with regard to the foot of the ordinate. But if (f/'Xx) be negative, the change is from convexity to concavity, and this latter is the case represented in the figure. SINGULAR POINTS. 215 278. Point of Undulation. Again, if ^'"(aj) = at the same point, and (l)""{x) do not vanish, the first term in the expansion oi N^P^ — N^Q^ depends on h^, and therefore this expression does not change sign in passing through P. The tangent therefore on the whole does not cross its curve at P. The point is of the kind described in Art. 275 and called a, point of undulation. 279. Higher Degrees of Singularity. It will now appear that, if by two successive differentiations a result of the form d?v be deduced from the equation to the curve, although -— vanishes both at the points given hy x = a and hy x — h, yet it only undergoes a change of sign when it passes through x = h, the index of the factor x — h being odd. Hence at the points given hy x = a there is no ultimate change in the direction of flexure, while at those given hy x = h there is a change. The points given hy x — a look to the eye like ordinary points on a curve, while those given by 03 = 6 resemble points of inflexion, and indeed have been for distinction called by Cramer points of visible inflexion* although the singularity is of a higher order than that described in Art. 274, which is the case of m = 0. lin=\, the points given by (T = a are points of undula- tion, such as described in Art. 275. So that for an Inflexional Point the condition -j-^ = 0, though necessary, is not sufficient. The complete criterion is that -^-^ should change sign. If ^\ vanish, hut do not change sign, the curve at the point under consideration is undulatory. 280. Case when the Tangent is parallel to the ^/-axis. The test of concavity or convexity has been shown to depend upon the sign of -j-^. In the case, however, of an arc, the tan- CLX 7 gent to which is parallel to the axis of y, the value of -^- and * Dr. Salmon, Higher Plane Curves, p. 35. Cramer, Analyse des Lignes Courbes, Geneva. 216 CHAPTER IX. of all subsequent differential coefficients is infinite. But in this case it is obvious that it would be convenient to consider y instead of x for the independent variable, and then the sign of -Y- 2 will test the concavity or convexity to the foot of the ordinate drawn from the point under consideration to the axis of 2/. Similarly, at a point of inflexion at which the tangent is (J IT parallel to the axis of y, ~t-^ must change sign. And in other cases whenever it is more convenient to use y instead of x for our indepeudent variable, we are of course at liberty to do so with an interchange of the letters x and y in the formula quoted. Fig. 39. 281. The test for concavity or convexity may also be investi- gated as follows : — Let P be any point of the curve, co-ordinates x and y. Let the adjacent points on the curve P^ and P^ have co-ordinates, (x — h, 2/i) and (x + h, y^ respectively. Let the ordinate of P cut the chord P1P2 in Q. Then if h be made infinitesimally small, the portion of the curve in the immediate neighbourhood of P will be convex or concave to N, according as NP is < or > NQ, i.e., as Now 2/ is < or > ^i±^2. , jdy h^ d^y , SINGULAR POINTS. 217 so that the criterion depends upon whether and proceeding to the limit the curve is convex or concave to iV according as -r^ is positive or negative. Ex. 1. Consider the curve y = '^sjax. Is it convex or concave to the foot of the ordinate ? Here dx' ^da^ a x' and d?v Hence y~L is negative for all positive values of x (and negative values of X are not admissible), so that the curve in the neighbourhood of any specified point is concave to the foot of the ordinate of that point. Ex. 2. Consider the curve x=y^\Zy^. Has it a point of inflexion? Here '^=^+^^^ so that -^ changes sign as y passes through the value y=-\. Therefore y the point (2, — 1) is a point of inflexion on the curve. (282. Convexity and Concavity of a Polar Curve. Suppose the equation of a curve to be given in polar co- ordinates as u=f(0), and that it is required to find a test of convexity or concavity towards the pole. \B Fig. 40. Let be the pole, P the point of the curve to be examined. Let the co-ordinates of P be denoted by r, 0, and let A,B be two points on the curve adjacent to P, and one on each side of it whose co-ordinates are respectively (7'^, — S6) and (r^, + SO). Then the curve in the immediate neighbourhood of P will be concave or convex to 0, according as AAOP-\-ABOP is > or < AAOB 218 CHAPTER IX. when we proceed to the limit. That is, according as r^r sin SO+rr^sm SO > or < r-^r^sin 2S6, or T-^r-^-TT^ > or < 2r^T2Cos SO ; i.e., as U2 + u^> or < 2u cos SOy where we have written r, = — , etc. Now, by Taylor's Theorem, , du„^ , d^u SO^ , du ^ d^u SO^ and therefore J d^u S0\ \ whence we have concavity or convexity to the pole according as 2u + 2^2 --+... IS > or <2u(^l---+...j, and proceeding to the limit according as 2^+^2 is > or < 0. 283. Polar Condition for a Point of Inflexion. At a point of inflexion the curve changes from concavity to convexity, and therefore the necessary condition is that d^u u-\--T^ should change sign. Ex. Find the point of inflexion on the curve r=aO~^. Here . au=0^j therefore a-^755 = — - 9~^. Hence, putting ^4. =0 to find for what value of ^ a change of sign can occur, we have e=±h And the positive value only is admissible, giving 0=i J as the polar co-ordinates of the point of inflexion. SINGULAR POINTS. 219 284. Condition for Pedal Equations. It will be obvious from a figure that for an element of a curve which is concave towards the pole p and r increase or decrease together. But for convexity _p increases as r decreases and vice versa. Thus for concavity ^ is positive ; for con- vexity negative. At a point of inflexion — changes sign. 285. This condition is deducible at once from the polar condition, for since ■!P~^~**^"*"(;7^) d^ii 'T dp d6^ p^ dr whence the result follows immediately. Examples. 1. Show that the curve y=e* is at every point convex to the foot of the ordinate of that point. 2. Show that for the cubical parabola ah/ = {x-hf there is a point of inflexion whose abscissa is h. 3. Show that there are points of inflexion at the origin on each of the curves (a) y = ^ cos -. a (^)y=atan|. (y) y=x^\og{\-x). 4. Show that there is a point of inflexion on the curve at the point (8, c^). 6. Show that every point in which the curve of sines a cuts the axis of x is a point of inflexion on the curve. 6. Determine the nature of the point where x = h on the curve (y - a - xy=a{x— hf. 7. Show that the curve (y - a)3 = a^ - ^a^x+aa^ is always concave towards the foot of the ordinate. How is it situated with regard to points on the y-axis ? 8. Ascertain whether the spiral rcosh 6=a is convex or concave towards the pole. 220 CHAPTER IX. 9. Show that if the origin be a point of inflexion on the cnrve u^ will contain u^ for a factor. 10. Show that there is a point of inflexion at the origin oti the cubic y = axy + hy^ + cc(^. 11. Show that there is a point of undulation at the origin on the curve y = aa^ + bx^y^ + cy*. 12. Find the positions of the points of inflexion on the curve 12y=j^~l6a^ + 42x^+l2x+h 13. Prove that the curve 2/- ^he'O" has a point of inflexion given by ^=<^\ — V n 14. Prove that the point ( - 2, — ^ j is a point of inflexion on the curve y^xe'. Multiple Points and Tangents. 286. Nature of a Multiple Point. A singularity of different nature from those above described occurs on a curve at a point where two branches intersect, as at the point A in the accompanying figure. It will appear from an inspection of the figure that at such a point as the one drawn there are two tangents to the curve, one for each branch. Fig. 41. Each tangent cuts the curve in two ultimately coincident points, such as P, Q on one branch, and it incidentally intersects the other branch through J. in a third point R, ultimately also coinciding with A. Each tangent therefore at such a point intersects the curve in three ultimately coincident points at the point of contact ; and if the curve be of the n^^ degree, each tangent will cut the curve again in. n — Z points real or imagin- ary. In this respect the tangent at such a point resembles the tangent at a point of inflexion, for (Art. 274) the point of con- tact of a tangent at a point of inflexion counts for three of the n intersections of the line with the curve. SINGULAR POINTS. 221 287. Points through which more than one branch of a curve passes are called "noulti'ple 'points" on the curve. If two branches pass through the point A, as in the above figure, A is called a " double point" If three branches pass through any point, that point is called a " triple point " on the curve ; and generally, if through any point r branches of the curve pass, that point is referred to as a " multiple point of the r*^ order " on the curve. From what has been said with regard to the tangents at a double point it will be obvious that there are r tangents (real or imaginary) at a multiple point of the r^ order, one for each branch. At such a point each of these r tangents cuts its own branch in general in two points, and each of the other branches in one point : i.e., in r+1 points altogether, all ultimately coincident with the multiple point. Such a tangent therefore cuts the curve inn — r — 1 other points real or imagin- ary. But if at the multiple point there happen to be a point of inflexion on the branch considered, the tangent will cut that branch in three points instead of two at the point of contact, making r + 2 points of intersection with the curve at the mul- tiple point, and therefore reducing the remaining number of points of intersection to n^r— 2. 288. Species of Double Points. Consider the case of a double point. The tangents there may be real, coincident or imaginary. Ca'Se 1. If the tangents be real and not coincident, there are two real branches of the curve passing through the point, and the point is called a node or ci^unode.. Fig. 42. Case 2. If the tangents be imaginary, there are no real points on the curve in the immediate neighbourhood of the point con- sidered, and we are unable to travel along the curve from such a point in any real direction. Such a point is therefore simply an isolated point, whose co-ordinates satisfy the equation to the curve, and is called a " conjugate point " or " acnode." 222 CHAPTER IX. Case 3. If the tangents at the double point be coincident, the two branches of the curve will touch at the point con- sidered. The point is then in general of the character called a stationary point, cusp or spinode. 289. Two Species of Cusps. There are two kinds of cusps, as shown in the accompanying figures. ,,y Fig. 43. Fig. 44. (a) In fig. 43 the branches PA, QA lie on opposite sides of the tangent at A. This is referred to as a cusp of the first species or a keratoid cusp (i.e., cusp like horns). (j8) In Fig. 44 the branches PA, QA lie on the same side of the tangent at A. This is called a cusp of the second species or a ramphoid cusp (i.e., cusp like a beak). 290. A Multiple Point can be considered as a Combination of Double Points. A triple point may obviously be considered as a combination of three double points, for of the three branches intersecting at the point each pair form a double point at their point of inter- section. And in general a multiple point of the r*^ order may be considered as the result of the combination of ^ — - double points, since this is the number of ways of combining the r branches two at a time. 291. To examine the Nature of the Origin. If the equation of a curve be rational and algebraic, it may be written in the form a + c^x^+c^xy+c^y^ + ... ^k^x''-\-k^x^-^y + ...-{-kn+iy'' = (a) SINGULAR POINTS. 223 If this be put into polar co-ordinates it becomes a -t-r(&^cos 0+62^11^ ^) -{-^^(CiCos^^H-CgCos Q sin O+Cgsin^^) + ... + r^(A;iCos"0+A;2Cos'*-i0sin0-l-...-|-^n+isin'*0) = O (b) Let be the pole and OA the initial line. Then equation (b) gives the points P^, Pg, P3..., in which a 'radius vector o Fig. 45. OPJP^..., making a given angle with OA, cuts the curve. The roots of this equation are OP^, OP^, OP^ It is clearly of the n^^ degree, and therefore has n roots. These may, however, become imaginary in pairs. I. If a = it will be obvious from either the Cartesian equa- tion (a) or the Polar equation (b) that the curve passes through the origin 0. In this case one root of the equation (b) is zero, and in the figure OP^ = 0. II. In this case, if 6 be so chosen as to make fe^cos Q 4- ^gsin ^ = 0, a second root of the equation (b) vanishes, and therefore we infer that a straight line making an angle tan-^f — y^j with the initial line cuts the curve in two contiguous points at the origin, and therefore is the tangent there. The Cartesian equation of this line is obvious upon multiplying by r, viz., Hence if a curve pass through the origin, the terms of first degree (if any such exist) on being equated to zero form the equation of the tangent at the origin. (See Art. 197.) III. If a = 0, b^ = 0, and 63 = 0, then in general it is possible to choose & so that c^cos^O + c^cos sin 0-{-c^sm^6 = 0, and then three roots of equation (b) will vanish ; that is to say, of the pair of lines whose equation is c-^x^+c^y-{-c^^ = each 224 CHAPTER IX. cuts the curve at the origin in three contiguous points. There are therefore two branches of the curve intersecting at the origin, to each of which a tangent can be drawn, and of the three contiguous points in which it has been seen that each of these tangents cuts the curve two lie on one branch and the other on the remaining branch. The origin is in this case a double 'point on the curve, and the terms of lowest degree in the equation of the curve, viz., when equated to zero form the equation of the tangents at the origin. The tangent of the angle between these straight limes is given by tan = ^ — — i-^. If c^ > 4C1C3, the tangents are real and not coincident, and there is a node at the origin. If Cg^ = ^c-^c^y the tangents are coincident, and the two branches of the curve touch, and there is in general a cusp at the origin. If C2^<4ciC3, there are no real tangents at the origin, although the co-ordinates of the origin satisfy the equation of the curve ; there is then a conjugate point at the origin. If Cj^ 4- C3 = 0, the tangents at the origin intersect at right angles. IV. If ct = 0, &1 = 0, 62 = 0, CjL = 0, C2 = 0, Cg = 0, the origin is a triple point on the curve, and (as shown in III. for the tangents at a double point) the tangents at the origin are d-^x^ + d^^y-\-d^xy^+d^y^ = 0. V. And generally, if the lowest terms of an equation are of the r*^ degree, the origin is a " multiple point of the r*^^ order'' on the curve, and the terms of the r*^ degree equated to zero give the r tangents there. 292. To examine the Character of any Specified Point on a Curve. Results similar to those of the preceding article may be deduced for any point on the curve. Let the straight line —. — —- =p be drawn through a given point Qi, h) to cut the curve /(a;, y) — Q. Then x = h + lp, y = k+mp. SINGULAR POINTS. 225 The use of these equations is obviously equivalent to a double transformation of co-ordinates, the first to parallel axes through /i, k, the second to polars. Substituting for x and y in the equation of the curve we obtain /(h-^lp, k+mp) = to find the points P^, Pg, ..in which a radius vector through the point h, k cuts the curve. If this be expanded by the extended form of Taylor'ij Theorem, the equation becomes *lK«4)V+^^=«. which is exactly analogous to equation (b) of Art. 291, and corresponding results follow. I. If f(k, k) = 0, one root of the equation for p vanishes and the point h, k lies on the curve (which is otherwise obvious). II. In this case, if the ratio limhe now so chosen that then another root vanishes, and this relation gives the direction of the tangent, whose equation is therefore as found in Art. 191. III. But if ^ = and 1^ = 0, as well as f{h, A;) = 0, then all lines through h, k cut the curve in two contiguous points. But if the ratio Z : m be so chosen that we have in general, as in Art. 291, III., two directions in which a radius vector drawn through Qi, k) cuts the curve in three contiguous points. The point (h, k) is a double point on the curve, since two branches of the curve pass through this point ; and of the three contiguous points in which each of the above- 226 CHAPTER IX. mentioned radii vectores meets the curve, two lie on one branch and one on the other. The equation of the two tangents is IV. Further, if ^ = 0, -^ = 0, and -,{ = 0, in addition to ^ = 0, ^ = 0, and f(h, k) = 0, identically for the same values of h, k, and if on going to terms of the third order we find that all these do not identically vanish, the point (k, k) is a triple point on the curve. V. And generally the conditions for the existence of a mul- tiple point of the r*^ order at a given point h, k of the curve are that f(x, y) and all its differential coefiicients up to those of the (r— 1)*^ order inclusive should vanish when x = h and y = k; and then the equation of the r tangents at that point will be 293. Special Case of Double Point. Recurring to the case of a double point at a point (h, k), since the equation of the tangents is (.-/.)^g+2(.-;.)(,-/.)^+(2/-i)^g=0, the angle between these tangents is given by 2 /TWYZW"^ tan0 = ^™^£_2^L^ sy By and the point ^, A; is a node or conjugate point according as \dkdk) < dh^ syb2' and is in general a cusp if (dj\^_d^f sy KdhdkJ ~dh^ dk^' with the preliminary conditions in each case that /(;^,^) = O,|=O,and|=0. SINGULAR POINTS. 227 We say in general a cusp ; for it will be seen that in some cases when the above conditions hold the curve becomes imagin- ary in the neighbourhood of the point considered, which must therefore be classed as a conjugate point. In the case of the coincidence of tangents, further investigation is therefore neces- sary. The mode of procedure is indicated below in the method \ for the investigation of the character of a cusp. > It appears that \dxdyJ dx^ dy^ represents a curve which cuts f(x, y) = in all its cusps ; and is a curve which cuts f(x, y) = in all the double points at which the tangents are at right angles. 294. To search for Double Points. The rule therefore to search for double points on a curve f(x, y) = is as follows. Find ~- and ^ ; equate each to zero and solve. Test whether any of the solutions satisfy the equa- tion of the curve. If so, apply the tests for the character of each of the points denoted, i.e., try whether \dxdyJ < dx^ dy^' ^95. To discriminate the Species of a Cusp. Method I. Suppose the position of a cusp to have been found by the foregoing rules. Transfer the origin to the cusp. The transformed equation will be of the form {ax-\-byf-hu^+u^+... = 0, (1) where ax+by = is the tangent at the origin, and u^, u^, ... are homogeneous rational algebraical functions of x and y of the degrees indicated by their respective suflBxes. Let F be the length of the perpendicular drawn from a point x, y of the curve, very near the cusp, upon the tangent ax-^by = 0. ^'^ <^> 228 CHAPTER IX. If 2/ be eliminated between equations (1) and (2), an equation is obtained giving P in terms of x. It is our object to consider only the two small perpendiculars from points on the curve near the origin, and having a given small abscissa x ; hence in comparison with P^ we reject cubes and all higher powers of P and also all such terms as P^x, PV, . . . which may arise on substitution. Fig. 46. — Single cusp, first species. Fig. 47.— Single cusp, second species. Fig. 48. — Double cusp, first species. Fig. 49. — Double cusp, second species. Fig. 50.— Double cusp, change of species. Osculinflexion. We shall then have a quadratic to determine P. If, when x is made very small, the roots be imaginary, the branches of the curve through the origin are unreal, and therefore there is a conjugate point at the origin. If the roots be real, but of SINGULAR POINTS. 229 opposite signs, the two small perpendiculars lie on opposite sides of the tangent, and there is a cusp of the first species at the origin. If the roots be real and of like sign the perpendic- ulars lie on the same side and the cusp is of the second species^ and the sign of the roots determines on which side of the tan- gent the cusp lies. Complete information is also afforded by this method as to whether the cusp is single or double, i.e., as to whether the branches of the curve extend from the cusp towards one extremity only of the tangent, or towards both extremities as shown in the annexed figures. The reality of the roots of the quadratic for P will in some cases depend upon, and in others be independent of the sign of ;». In the former cases the cusp is single; in the latter, double. Moreover, if double, we can detect whether the cusp is of the same or of different species towards opposite extrem- ities of the tangent. When the cusp is of different species towards opposite extremities the point is called by Cramer a point of Osculinflexion. In adopting the above process it will clearly be sufficient to put P = ax-\-by, thus dropping the s/d^ -h 6^ for the sake of brevity; the effect of this being to consider a line whose length is proportional to that of the perpendicular instead of the per- pendicular itself. Ex. 1. Examine ike character of the origin on the curve Here the tangent at the origin is ,y=0. According to the rule puty=P. The quadratic for P is PX4: - 2x) - 4Pj!^ + .r* = 0. The roots of this equation are real or imaginary according as 4.V* is > or < ^*(4 - 2a;), i.e., according as x is positive or negative. Hence the cusp is "single" and lies to tlie right of the axis of y. Moreover the product of the roots 18 and is positive when x is very small, and the roots are therefore of the same sign. The origin is therefore a single cusp of the second species. Moreover the sum of the roots is positive, so that the two branches near the origin lie in the first (piadrant. 230 CHAPTER IX. Ex. 2. Examine the character of the curve at the origin. Here y=0 is a tangent at the origin. Put y — P. The quadratic for P is P'{^-Zx)-Zx'^P+x'=0. The roots are real or imaginary according as Qx^ - 4(9 - Zx)x^ is positive or negative, ^.e., as —Tlx^+l^sfi is positive or negative. Now, when x is very small, x^ is negligible in comparison with x^, and therefore the above expression is negative for very small positive or nega- tive values of x. The roots of the equation for P are therefore imaginary, and the origin is a conjugate point on the curve. Ex. 3. Examine the character of the curve 2W-1-1 7j^F{x)±{x-h) ■'- fix) (1) in the neighhoicrhood of the point x—h^ y = F{h), m and n being positive integers. By Taylor's Theorem we may write F{x + h) = F{h)-\-ax+bx^-\-... and [/(^+A)P = aj + 6j^+..., where a-^ being \_f{h)f is necessarily positive. Hence on transforming our origin to the point {A, F{h)} we obtain for the transformed equation 2w+l {y-ax-bx'^-...f=-x " (ai + 6i.r+...) (2) Examining the form of the curve at the origin, there are obviously coin- cident tangents if ^ be > 2. n Put y — ax=P^ then 2m+l P2_2P(6.r2 + ...)-}-62a;4_^^^ n _...=o. That the roots of this quadratic are real, if x be positive and small, is obvious from equation (2) ; also, that the roots are imaginary for small negative values of x. There is therefore a single (Msp extending to the right of the new axis ofy. 2WJ-J-1 Again, the product of the roots = b^xl^ — a\X ** — . . . . If ^"^ > 4, this product has the same sign as x^ when x is taken n sufficiently small, and therefore is positive, giving a cusp of the second 2m -fl ^^^^ If < 4, the term —axx " is the important term in the product and is negative, x being positive. There is therefore in this case a cusp of the_^s^ species. We have assumed that the coefficient b or ~F"{h) is not zero. If how- ever this coefficient vanish, it is easy to make the corresponding change in the subsequent investigation. SINGULAR POINTS. 231 Ex. 4. Examine the nature of the double point on the curve Here '^=3{a;+7/f+2^2(^-x+2)=0, |i=3(:r+y)2-2V2Cy-^+2)=0. These give x+^ = 0, and y — .27+2 or ^=1, :3 Now this point obviously lies upon the curve, and there is therefore a multiple point of some description there. A^ain, ^=6(^+y)-2V2= -2^2 at the point (1, -1), P=6(:r+y)-2V2=-2V2, ^=6(.4-y)-l-2V2 = 2V2. Hence at this point g |t=(^)'» and we have a double point at which the tangents are coincident. Next, transforming to the point (1, - 1) for origin, the equation becomes (a;+2/f-^2(7/-xy=0. According to the rule we put y-.r=P. Then rejecting terms in P^^and P2:i;wehave I^-6a^ ^2P-4a^ s/^=0. The roots are real if 18^ + 4 ^2^ > 0, which is the case if x be very small and positive. There is therefore a single cusp at the point (1, - 1). Again, the product of the roots = — 4r' fJ2, and is negative when x is small. This indicates that the cusp is one of the Jirst species. [This curve is obviously only a transformation of the semi-cubical para- bola y2=^,] Ex. 5. Search for a multiple point upon the curve x''-\-23(^ + 2ofiy-\-2x^-\-x^-{-2xy^y^+2x + 2y + l=0. Here '^ = l3fi+Qx^ + Qx'y-\-Qx'^^2x+2y + 2=0 (i.) '^- = 2a^ + 2x-^2y-\-2=0 (ii.) From the second equation y= -a^ — x—\. Substituting in (i. ) *ix^ - 6^ = 0, whence ^=0 or f, and therefore y=—\ or -|^. 232 CHAPTER IX. It is obvious that the latter solution cannot satisfy the equation to the curve. Transforming to the point (0, - 1 ), the equation becomes ^7 + 2ar* + 2.r3y + (^ + y f = 0, indicating that there is either a cusp at the new origin to which ^-f-y=0 is a tangent, or a conjugate point. Put x^-y = P, then P^ + 2P^ + jt^ = 0. The roots will be real if x^-x"^ is positive, which is true when x is positive and less than 1, and also when x is negative. Hence there is a double cusp. The product of the roots is j?^, which is positive or negative according as x is positive or negative. It is therefore ramphoid on the right-hand side of the new y-axis and keratoid on the left-hand side, and therefore there is an oscidinflexion. Also the sum of the roots is — 2x-'', and is therefore positive when x is negative ; hence on the left side of the new y-axis the upper portion of the curve deviates from the tangent more rapidly than the lower portion. 296. Method II. Another method of discrimination of the species of a cusp depends upon the test for concavity or convex- (Jul/ (mOC ity. Find the two values of ^ (or ^-g, see Art. 280). If these have opposite signs very near to the cusp, the two branches starting from the cusp are in general one concave and the other convex to the foot of the ordinate, and the cusp is of the first species. But if the signs he the same, the two branches are either both concave or both convex to the foot of the ordinate, and the cusp is of the second species. In the case however when the ir-axis is a tangent at the cusp, the cusp will be keratoid when both branches are convex to points on the axis near the cusp. But in this case the values of ~j-^ are of opposite sign. Hence the above test still holds. Ex. Discuss the form of the curve y=x'^x^ at the origin. Here y^=±Z\^^x. Hence only positive values of x are admissible and the two values of ^3 have opposite signs. The origin is therefore a single cusp of the first species. 297. Singularities on the Reciprocal Curve. Since to a tangent to a curve corresponds a point on its polar reciprocal, it will be evident that to the points in which a straight line cuts the one correspond the tangents which can be drawn from a given point to the othei". If the one has a multiple point of the p*^ order the other has a multiple tangent SINGULAR POINTS. 233 touching its curve at p distinct points ; to a double point on the one corresponds a double tangent or bi-tangent to the other ; to a stationary point on the one corresponds a stationary tan- gent on the other. These considerations tend to show that the multiple- tangent should be classed as a distinct singularity. Examples. 1. Show that for the semi-cubical parabola the origin is a cusp of the first species. 2. Show that the origin is a cusp of the first species on the curve a{y — x)' = x^'. 3. Show that the curves v-=J?-sin-, y-=x^tan- a a have cusps of the first kind at the origin. 4. Show that at the origin on the curve i/^=hxB\\\ - a there is a node or a conjugate point according as a and h have like or unlike signs. 5. Show that for the Cissoid y2= the origin is a cusp of the first species. 6. Examine the nature of the point on the curve y-2 = .r{\+.r + x^-) where it cuts the y-axis. 7. In the curve a^y^ — 2abary=a^ show that there is an osculinflexion at the origin. [Cramer.] 8. Search for the double point on {y-2y=x{x-\y\ and find the directions of the tangents there. 9. Determine the position and species of the cusps of the following curves : — (a.) (2y + .r ^ 1 )■•' = 4( 1 - xf^ (6.) (jf^-xf-{y-xf=\, (c. ) xy^ + 2a^y - or- — ^a^x - 3a^ = 0. 10. Examine the nature of the point { — a^a) on the curve X* — ay^ + 2aa^y + 4a^ + Ba^^/- + Aa?xy + 4m^x^ — c?y = 0. 11. Show that at the point ( - 1, - 2) there is a cusp of the first species on the curve ^ -H 2x'^ + 'Ixy — ?/ + 5.r - 2y = 0. 12. Show that at each of the four points of intersection of the curve {axf^{hyf={a'-h-f with the axes there is a cusp of the first species. 234 CHAPTER IX. 13. Show that the origin is a conjugate point on the curve ^ - axh) + axy^ + ay = q 14. Show that at the origin there is a single cusp of the second species on the curve ^ - 2a^y - axy^ + (ry^ = 0. 15. Show that the curve ?/^ = 2^^3/ + ^y+.r^ has a single cusp of the first species at the origin. 16. Show that the curve 2/^ = 2a;^;i/ + x^+a^ has a double keratoid cusp at the origin. 17. Show that the curve ?/^=2^3/+^-2^* has a conjugate point at the origin. 298. Singularities of Transcendental Curves. In addition to the siDgularities above discussed others occur occasionally in transcendental curves, due to discontinuities in the values of y, -^, etc. For instance, if the value of y be discontinuous at a certain point the curve suddenly stops there and the point is called a "point d' arret " or " stop point" 1 Consider the curve y^cn'' \ (« > 1). When .r= — 00, y = l, and as x increases from -co to zero y is always positive and decreases down to zero. As soon, however, as x becomes positive, being still indefinitely small, y suddenly becomes infinitely great, and as x increases to + oo y gradually diminishes down to unity. The origin is a 'point d arret on this curve, and the shape is that shown in the annexed figure. Fig. 51. Next suppose that the value of y is continuous, but that at a certain point -r- becomes discontinuous, so that tv^o branches of the curve meet at a certain angle at the same point and stop there. Such a point is called a "point saillant" SINGULAR POINTS. 235 299. Branch of Conjugate Points. It sometimes happens that a curve possesses an infinite series of conjugate points, satisfying the equation to the curve and forming a branch of isolated points. M. Vincent, in a memoir published in vol. XV. of Gergonne's "Annales des Math.," has discussed several such cases, and calls such discontinuous branches by the name branches pointillees. Ex. In tracing the curve y—x^^ it is clear that, when ^=00, y = oo ; and when J7=l, y=l. Also that as x decreases from 00 to \ y also decreases from 00 to 1. Between ^ = 1 and ^=0 y is less than 1; and when ^ = 0, y = \ (see Chap. XIV.). There is therefore a continuous branch of the curve, viz., 00 PB^ above the axis of x. Again, whenever :r is a fraction with an even denominator there are Fig. 52. two real values of y, differing only in sign ; e.g.^ whilst, whenever the denominator of x is odd, there is but one real value for y. There is therefore a set of conjugate points below the axis forming a discontinuous branch, of the same shape as the continuous branch above the axis. Next consider what happens when x is negative. Let the co-ordinates of any point P on the branch in the first quadrant be (.r, y\ then ON=x. Take On= —x along the negative portion of the axis of x, then, if jo be the corresponding point on the curve, we have pn = {-x)-% PN=x', and therefore pn . PN={ - 1)*, which may be =1, - 1, or imaginary, according to the particular value of 236 CHAPTER IX. X. Hence, when the ordinate pn is real, its magnitude is inverse to that of the corresponding ordinate FN. Hence on this curve wq have two infinite series of conjugate points, as shown in the figure. For an account of M. Vincent's memoir and criticisms upon it see Dr. Salmon's " Higher Plane Curves," 2nd ed., p. 275, or a paper by Mr. D. F. Gregor}^, "Camb. Math. Journal," vol. i., pp. 231, 264. 300. Maclaurin's Theorem with regard to Cubics. If a radius vector OPQ be drawn through a point of in- flexion (0) of a cubic, cutting the curve again in P and Q, to show that the locus of the extremities of the harmonic means between OP and OQ, is a straight line. If the origin be taken at the point of inflexion and the tan- gent at the point of inflexion as the axis of y, the equation of the cubic must assume the form y^ + xu = (1) where u is the most general expression of the second and lower degrees, viz., ax^-\-2hxy-\-by^-{-2gx-\-2fy-\-c, for it is clear that the axis of y cuts this curve in three points ultimately coincident with the origin. The equation (1) when put into polars takes the form Lr^-\-Mr-\-N=0, where L = am^O + (a cos^O -\-2hsm cos + b sin^O) cos ^, M= (2g cos 0+ 2/ sin 0) cos 6, iV=ccosO. If 7\, r^ be the roots of this quadratic, and p the harmonic mean between them, we have ^_1 "^ _ ^^_ 2.^ cos e +2/ sin /o~n ^2" N~ c which shows that the Cartesian Equation of the locus of the extremity of the harmonic mean is the straight line gx+fy-\-c = 0. 301, It is obvious from Art. 211 that the equation of the polar conic of the cubic (1) with regard to the origin is x{2gx+2fy) + 2cx = 0, or x(gx -\-fy + c) = 0. SINGULAR POINTS. 237 Hence the polar conic of a point of inflexion on a cubic breaks up into two straight lines, one of which is the tangent at the point of inflexion, and the other the locus of the extremities of the harmonic means of the radii vectores through the point of inflexion. It appears from this that only three tangents can be drawn from a point of inflexion on a cubic to the curve, viz., one to each of the points in which the line gx-\-fy-\-c = meets the curve, and consequently also that their three points of contact lie in a straight line. 302. If a Cubic have three real points of Inflexion they are Collinear. It follows immediately from Maclaurin's Theorem above proved that if A and B be two points of inflexion on a cubic, the line AB produced will cut the curve in a third point C, which is also a point of inflexion on the cubic. For if B, B^, B» be the three ultimately coincident points on the cubic, which lie in a straight line (B being a point of inflexion), let AB, AB^, AB^ cut the curve in (7, C^ G^, and let AH, AH^, AH^ be the harmonic means between AB, AG; AB^, -4(7^; AB.^, AC, respectively, then H, H^, H^^ lie in a straight line by Maclaurin's Theorem, and B, B^, B^ lie in a straight line; therefore by a theorem in conic sections C, G^, G^ also lie in a straight line, and they are ultimately coincident points. G is therefore a point of inflexion. 303. Number of points necessary to define a Curve of the 91*^ De^ee. The number of terms in the general equation of the n*^ degree J, .. . . . (71 + 1X^+2) , n(7i + 3) . , It therefore con tarns ^^ ^^ ^ — 1 or — ^^-^ — - indepen- dent constants. Hence in general a curve of the ti*^ degree may be drawn to pass through ^ arbitrarily chosen points. 238 CHAPTER IX. 304. Maximum Number of Double Points on a Curve of the 'n}'^ Degree. There cannot be more than ^{n — l){n — 2) double points on an n-iic curve. For if there could be ^- ~ ^+1 double points, a curve of degree 71 — 2 could be drawn to pass through them and through any n — 2> other arbitrary points on the curve, for and therefore these would make just sufficient points to com- pletely define the new curve. But the number of intersections ^ouldbe 2 f^-^f-'> +l}+(n-3). or n{n — 2)-\-\, which is one more than possible for curves of degrees n and ^-2. Examples. 1. Show that a cubic curve cannot have more than one double point, and cannot have a triple point. Examine the case of the curve and show that there are apparently two nodes at (1, 1) and at (2, 0) respectively. Explain this result. 2. Show that a quartic cannot have more than three double points, and cannot have a double point and a triple point. 3. The curve whose equation is x^ + 7/4 = ^a\x^ -^if) - a* has four double points. Find them ; account for this, and trace the curve. [Cramer.] 4. All curves of the third degree which pass through eight given points also pass through a ninth common point. 5. All the double points of a family of cubics determined by seven given points lie on a sextic. 305. Use of Homogeneous Co-ordinates. Let j{x, y, z) = be the equation of any curve of the n^^ degree, which may be considered expressed either in trilinears, areals or Cartesians made homogeneous by the introduction of a proper power of 2;( = 1) where requisite. SINGULAR POINTS. 239 Let (iCj, 2/1, 2;^) be the coordinates of any iixed point A, and let (X, Y, Z) be the current co-ordinates of any point P on the secant AP. Let AP cut the curve in the points Q^, Qg. •••• Let any of the points Q{x, y, z) divide AP in the ratio X : /x where X-f-yot = l. Then x = \X+ijlx^, Hence /{XX+fxx^, XF+m2/p X^+m2^i) = 0. This may be expanded in two ways by Taylor's Theorem; and to abbreviate the algebra let/(X, F, Z) be written /and /(^i» Vv ^1) ^^ written f^, also denote the operations ^nd (zJ^ + fA+z^Y \ dXi dy^ dz^/ by P and F/ respectively. Then we have ■\n-l,, ■xn-2,,2 >»»-»'/*»' ##»» =0 (1) or . =0 (2) Either of these equations gives the n values of the ratio Comparing the coefficients we have the series of identities -VH = f nl 1 etc., 240 CHAPTER IX. 306. Polar Curves. The several loci defined by the equations etc., are respectively called the polar line, the polar conic, the polar cubic ; and so on. The curve Y^-'^f^=zO^ or, which is the same thing, F/=0, has been called (Art. 211) the first polar of the point x^ y^ z^. Similarly the curves Fy=0, Fy=0, etc., are called the second, third, etc., polar curves. It is clear then that the n — V'^ polar curve is the polar line, the n — 2*^ polar curve is the polar conic ; and so on. 307. Geometrical Interpretations. The geometrical meanings of these equations will be obvious : — If l^i/i = 0, the sum of the roots of Equation (2) vanishes, i.e. 2^ = 0, or putting AQ^ = t^, etc., and AP=^R, r giving 2g-^) = 0. This property is due to Cotes, and the special case of it when the curve is a conic gives rise to the name polar line. IfF/A^O.wehaveE^-J|=0, which may be interpreted as before, and similarly for the higher polar curves. It appears that since each of these curves is completely defined by its geometrical property it is totally independent of any system of co-ordinates used in its description. SINGULAR POINTS. 241 308. Polar Curves of the Origin. Taking Cartesians, if the origin be chosen at the point A^ x^ = y^ = 0, and it appears that the polar line, polar conic, polar cubic, etc., of the origin respectively reduce to 3"-y_0 ^-V-o 3"-y-o etc B^^-"' ai^^-"' 3i^'-°' ^''- If the Cartesian equation be written UQ+u^^-^u^-hu^-h -.'+^11 = this becomes when the z is introduced and the equations of the several polars of the origin are =- u^ + {n — 1)! Ui = 0, ^u^^-{-(n-l)\u^z-\-(n-2)lu^ = 0, etc., i.e. nuQ+Uj^ = 0, 1.2 Uo + (n-l)uj^-\-U2 = Oy n(n-l)(n-2) {n-l)(n^2) , , ox . a 1%^ — ■'^0+^^ ^72 ^^1 4- (71 -2)^2 + ^3 = 0, etc. 309. General Conclusions. If the point A which has been taken for origin lie on the curve, then Uq = 0, and the polar curves all have u^ = for tangent at the origin. If also the first degree terms are absent from the equation of the curve, they are absent too from all the polars, and the terms of lowest degree throughout the whole system are u^. We therefore draw the following conclusions : — (a.) The polar curves at any point on the original curve all touch it at the point in question. (6.) The polar curves at any multiple point all have a mul- tiple point of the same order, with the same tangents as the multiple point on the original curve. E.D.C. Q 242 CHAPTER IX. (c.) The polar conic at a double point on a curve breaks up into two straight lines, viz., the tangents at the multiple point. (d.) The polar conic at a cusp breaks up into two straight lines coincident with the tangent at the cusp. (e.) The polar conic at a point of inflexion breaks up into two straight lines, one of which is the tangent at the inflexional point and the other does not in general pass through that point. [For in this case u^ must contain u^ for a factor, = u-^v-^^ say, so the polar conic becomes u^(v^ + n-l) = 0; the line v-^^-\-n — l=0 is called the Harmonic Polar of the point of Inflexion (see Art. 301).] 310. First Polar. Cases of Node or Cusp. If a curve have a node at any point let the origin be taken there and the tangents at the node for axes. The curve then takes the form The first polar of x^, y^, z^, viz. becomes x^{yz'^-^-{-...)-\-y^{xz''-^+...)+... = 0, the lowest degree terms only being retained. And since these terms are linear it appears that the first polar of any point x^, Vv ^1 S°®^ through the origin and therefore through all the other double points on the curve. If the curve have a cusp and the origin be taken there with the tangent at the cusp as a;-axis the equation of the curve takes the form u = 2^V-2+te,32;"-3+^^4^'*"*+...=0, and the first polar of any point x^, y^, z^ is y^{2yz^-^+...) + ... = the term of lowest degree only being retained. tience this curve also touches the a;-axis at the origin. Thus the first polar of any point goes through all the cusps and touches the curve at each. SINGULAR POINTS- 243 311. The Hessian. We have seen that at all double points and points of inflex- ion the polar conic degenerates into two straight lines. Hence its discriminant vanishes. Also, conversely. Now the equation of the polar conic of the curve u—f{x^, y^, 2;^) = ^ corresponding to the point x^, y^, z^ is F^^/j = 0, or + ...=0. Hence if x^, y^, z^ be a double point or a point of inflexion. we have ay -dyfix; By dxfiy^' B2// By that is, the curve Ua U. xy, dyfiz^ 32/ u. = 0; u yx> ^yy> u yz u zy U, = cuts the original curve u = in all its multiple points and points of inflexion. The determinant H(u) is called the Hessian of u from M. Otto Hesse, the discoverer of the relation between the curves u = 0, H(u) = 0. 312. Number of the Points of Inflexion. The degree of this curve is clearly S(n — 2). Hence it cannot have more than Sn{n^2) intersections with the original curve. Thus in a curve with no multiple points upon it there will be 3n(n — 2) points of inflexion real or imaginary. 313. Cases of Node and Cusp. If the curve has a node let the origin be taken there, and the tangents to the node for axes. The equation to the curve now becomes u = xy ^n-2 + u^z''-^ + u^z''-*-]-. 0. Hence 'i^xz = -H... Uyy = -H... CHAPTER IX. ; Uyz=(n-2)xz''-f -{-...; u,, = (n-'2){n-S)xyz''-^+...', u^y=z^-'^+ ... ; the lowest degree terms only in x and y being retained in each case. Hence in n. \U) = Uxx • '^yy • ^^zz + 2Uyz . Uzx • ^Jxy — '^xx • '^yz "- '^yy • '^zx ~~ ^zz • '''^.tj/ = 0, the lowest degree terms are of the form Axy. Hence the Hessian has a node also at the origin and the tangents to the node of the Hessian coincide with the tangents to the node on the original curve. It is easy to prove further that when the curve has a mul- tiple point of order k the Hessian has a multiple point of order 2Jc — 4 at the same point and that each of the tangents at the multiple point is a tangent to one or other of the 3^ — 4 branches of the Hessian. (See Dr. Salmon's Higher Plane GurveSy 2nd ed., page 58.) We next consider the case of a cusp. Let the origin be taken at the cusp and the tangent for the oj-axis. Then the equation to the curve becomes ^ = 2/2s'*-2-|-'i*32;'»-3+u^2^-*+... =0. Here Uyy=2z''-^-\-... u,, = (n-2)(n-S)yH''-'^-\- Uyz = 2{n-2)yz'^-^-\-.., the lowest degree terms only in x and y being retained. Hence in H{u) = the lowest degree terms in x and y are of the form A . ^~ . y\ So the Hessian has a triple point with two coincident tan- gents 2/ = and a third tangent — -^ = SINGULAR POINTS. 245 314. Pliicker's Equations. We are now in a position to establish Pliicker's Equations for the number of tangents which can be drawn from a given point to a curve of the n^^ degree and for the number of points of inflexion upon it. It was established in Art. 208 that the first polar cuts the curve Id n(n — 1) points. The first polar however goes through all the double points and in the case of a cusp touches the curve there. Hence a node counts as two and a cusp as three poiats of intersection. Thus if there be 8 nodes and k cusps the class of the curve, viz. n{n~'l), is diminished by 28 -{-Sk. Hence if m be the class m = n(n-l)-28-SK (1) Again, let i be the number of inflexions on the curve. Then it has been established that if there are no multiple points i = Sn(n^2). But it has been shown that the Hessian passes also through all the double points and has tangents coincident with those of the curve. Hence each node counts for six intersections of the Hessian with the curve. And since at each cusp on the curve the Hessian has a triple point, two tangents being the coincident tangents to the curve at the cusp, each cusp counts for 8 intersections (3 + 3 + 2). Thus the number of inflexions is diminished by 6<5 + 8k: and stands as i = 2n(n-2)-68-SK (2) By considering the reciprocal curve for which a stationary point gives rise to a stationary tangent, a double point gives rise to a double tangent, a stationary tangent gives rise to a stationary point, it follows that if r be the number of double or bi-tangents, i.e. tangents having contact at more than one point of their length, and m the degree of the reciprocal curve, i.e. the class of the original curve '^ = m(m-l)-2T-3t, (3) /c = 3m(m-2)-6T-8£ (4) These four equations are due to Pliicker. 246 CHAPTER IX. 315. Deficiency. The number ^{n-'l)(n — 2) — S — k, hy which the number of double points falls short of the maximum possible is called the deficiency of the curve. Examples. 1. Prove that the four equations established in Art. 314 are not inde- pendent. 2. Show that the geometrical property of the polar conic may be expressed as 7 — , 2— + 2 — =0. 3. If ^ be a point of inflexion on a curve and Aj Pi, F2, ..., P„-i be a secant cutting the Harmonic polar of the point of inflexion in Q, prove that —T7T = -7-n+^rTr+-'- + AQ AFi AP2 AFn-i 4. Form the Hessian of x^ + 9/^=3ax^, and find the number of points of inflexion. [Oxford, 1885.] 5. Establish the equations 2r=n{n-2){7i'-9)-2(ri'-n-6)(28 + 3K) + 48{8-l) + 128K + 9K(K-l), 28=m(w-2)(m2-9)-2(m2-w-6)(2T+3t) + 4r(T-l) + 12Tt + 9t(t-l). [Plucker.] 6. Prove that the deficiency of a curve is the same as that of its reciprocal. 316. Unicursal Curves. When a curve has its full number of double points, so that its deficiency is zero, the current co-ordinates can each be expressed as rational algebraic functions of some single para- meter. For supposing that there are ^^ -^ double points, a curve of the (n — 2)*^ degree may be made to pass through them and through n — S other points on the curve. Then since (^-l)(^-2) |^ ^_ (n-2)(n+l) ^^ the points now chosen are insufficient by one to completely determine the new curve. Its equation will therefore contain one arbitrary constant and may therefore be written with an undetermined parameter X. SINGULAR POINTS. 247 Eliminating y between this equation and that of the given curve, we have remaining an equation between x and X of degree n{n — 2) determining the abscissae of the points of intersection. Of the n{n — 2) roots all but one are known, being the abscissae of the ^(tz. — l)(7i — 2) double points each counted twice and the abscissae of the chosen 'Ji — 3 points for M^-2)-{2 <^-^f-'^ +^-3} = i: If then the corresponding factors be divided out we are left with X, the abscissa of an}?- other point on the original curve, expressed as a rational integral function of X. In the same way y may be similarly expressed. 317. Though it is impossible to compress into the limits of the present volume a complete account of the singularities of curves, it is hoped that the later articles of this chapter will form a fair introduction to a study of their general properties in Dr. Salmon's Treatise, to which the student is referred for more detailed information and to which also the Author desires to acknowledge his indebtedness. EXAMPLES. 1. Write down the equations of the tangents at the origin for each of the following curves : — (a) y-\-c = c cosh -• c (P) y = atan|. (y) 2/^ = a;log(l+a;). (8) a^ + y3 = 3a£cy. 2. Show that on the curve {ay - x^Y — ^^ there is a cusp of the first species at the origin, and a point of inflexion whose abscissa is -^h. 3. Show that the Trident curve axy + a^ = a^ has a point of inflexion at the point in which it cuts the axis of x, and show that the tangent at the poiut of inflexion makes with the axis of X an angle tan'^S. 248 CHAPTER IX. 4. Show that the curve h{ay - x^f- = a? has a cusp of the second species at the origin. 5. Show that, if n be greater than 2, the curve has a cusp at the origin of the first or second species according as n is less or greater than 4. 6. Find the two points of inflexion of the curve y _ x^ /x — aVh and draw figures showing the characters of the inflexions. 7. Show that the points of inflexion on the cubic _ a^x '^~x^ + a? are given by oj = and x= ± aj3. Show that these three points of inflexion lie on the straight line x = 4:y. 8. Show that the curve au = 6"' has a point of inflexion where n au= {n(l —n)}^. 9. Find by polars the points of inflexion on the curve 2x{x^ + y^) = a{2x^ + y^). 10. Show that the origin is a triple point on the curve a^ + 2/4 = axy^, and that there is a cusp of the first species there. 11. Show that the abscissae of the points of inflexion on the curve are roots of the equation 12. Show that the abscissae of the points of inflexion on the curve y = e'^" tan fxx are given by 2fi sec^ixx{fi tan fix- X) + A^tan fix = 0. 13. Show that the curve y = - — ^ ^ — x^ -a^ has a point of inflexion at the point whose abscissa is ^J/3 + 1 1-4. Show that there are two points of inflexion on the cubic at the points {a, 0), (0, a) respectively. SINGULAR POINTS. 249 15. In the curve ot? -^ y^ = aoip' show that there is a cusp of the first kind at the origin, and a point of inflexion where x = a. 1 6 . In the curve y'^ = {x- a)(x - h){x - c) show that if a = 6 there is a node, cusp, or conjugate point at cc = a according as a is >, =, or a there is a node at cc = 0, y= -a, and if 6 = a there is a cusp at the same point 29. The curve whose tangent is of an invariable magnitude is always convex towards the foot of the ordinate. 30. Examine the nature of the origin on the curve 2/5 + aa^ - b'^xy^ = 0. [Cramer. ] 31. Examine the nature of the origin on the curve od^ - ayx^ + by^ = 0. [Rolle. ] 32. Examine for multiple points the curve x^ - '2ay^ - 3aY - ^^^^^ + a* = 0. [Peacock.] 33. Examine the singularities of the curve a;4 _ 4^^3 _ 2ay^ + ia^x^ + 3aY -a^ = 0. There are nodes at the points (0, a), (a, 0), (2a, a). Find the direc- tions of the tangents at these points. 34. Show that the curve X^ - 2xhj -xy'^-2x^-2xy + y^-x+2y + l=0 has a single cusp of the second kind at the point (0, - 1). 35. Search for double points on the curve y4 _ 82/3 _ 1 2xy^ + 1 62/2 + 48xy +ix^-6ix=^ 0. [Rolle. ] 36. Show that there are two double points in all respects similar on the curve x'^ - 2ax^ J2 + 2a^x^ - ay^ - a^ = 0, and that there is an inflexion at each double point. [Cramer, Lignes Gourbes.} SINGULAR POINTS. 251 37. Determine the double points, distinguishing their species, on the sextic {x^ - 2y^Y{2{x^ + 2y^) - 3} = {d{x^ + 2y^) - 4}2. [Oxford, 1886.] 38. Determine the double points on (x^ - y^){x - l)(2a; - 3) + 4(x^ + y^- 2xf = 0. [Plucker.] 39. The points of contact of parallel tangents to a curve of the n^^ degree lie on a curve of the {n-l)^ degree. . [Serret.] 40. If il be any point on a curve, and AP^P^ ... P„_i be a secant cutting the curve in Pj, Pg) ••• Pn-i and the polar conic of ^ in Q, n-\ 1 1 1 1^^^^^ -AQ=AP,^AP,^"-^AP- 41. A nodal cubic intersects in the points P and F two lines which are harmonically conjugate with respect to the tangents at the node. Prove that the tangents at /*, F meet on the curve. 42. Prove that the locus of the cusp of a cubic with three given asymptotes is the maximum ellipse inscribed in the triangle formed by the asymptotes. [Plucker.] 43. If (jc, y, z) be a double point on a curve w = 0, and if be a tangent at the double point, then will I m ^ n and l^xu^ + rri^yu^ + n^zu^^ = 0. [Oxford, 1 886. ] 44. If the equation to a plane curve be <^ = 0, where <^ is a function of x and y which fulfils the condition — -$ + —% = 0, prove that if n ox^ oy^ branches of the curve meet in a multiple point their tangents will form 2n angles with each other, each equal to 46. Prove that the Hessian of the cubic ^ + 2/^ + ^ + ^frixyz - 18 ar + 2/3 +.23 _ yyy^ ^ Q, and show that the curve and its Hessian have the same points of inflexion. [Salmon, H. P. C] TT n [Smith's Prize, 1877.] CHAPTEE X. CURYATURE. 318. Angle of Contingence. Let PQ be an arc of a curve. Suppose that between P and Q there is no point of inflexion or other singularity, but that the bending is continuously in one direction. Let LPR and MQ be the tangents at P and Q, intersecting at T and cutting Fig. 53. a given? fixed straight line LZ in X and M. Then the angle RTQ is called the angle of contingence of the arc PQ. The angle of contingence of any arc is therefore the difference of the angles which the tangents at its extremities make with any given fixed straight line. It is also obviously the angle turned through by a line which rolls along the curve from one extremity of the arc to the other. 319. Measure of Curvature. It is clear that the whole bending or curvature which the curve undergoes between P and Q is greater or less according as the angle of contingence RTQ is greater or less. The 252 CURVATURE. 263 fraction — ^ ^^ — r—- is called the average bending or length of arc ^ ^ average curvature of the arc. We shall define the curvature of a curve in the immediate neighbourhood of a given point to be the rate of deflection from the tangent at that point. And we shall take as a measure of this rate of deflection at the . ^ ^, ,. .^ ^ ,, . angle of contingence given point the limit oi the expression — °.. » , — ^ — - — when the length of the arc measured from the given point and therefore also the angle of contingence are indefinitely diminished. 320. Curvature of a Circle. In the case of the circle the curvature is the same at every point and is measured by the reciprocal of the radius. p. Fig. 54. For let r be the radius, the centre. Then the angle being supposed measured in circular measure. Hence angle of contingence _ 1 length of arc r and this is true whether the limit be taken or not. Hence the " curvature " of a circle at any point is measured by the recipro- cal of the radius. 321. Circle of Curvature. If three contiguous points P, Q, R be taken on a curve, a circle may be drawn to pass through them. When the points are indefinitely close together, PQ and QR are ultimately tangents both to the curve and to the circle. Hence at the point of ultimate coincidence the curve and the circle have the 254 CHAPTER X. same angle of contingencey viz., the angle RQZ (see Fig. 55). Moreover, the arcs PR of the circle and the curve differ by a small quantity of order higher than their own, and therefore Tmiy he considered equal in the limit (see Art. 84). Hence the curvatures of this circle and of the curve at the point of contact are equal. It is therefore convenient to describe the curvature of a curve at a given point by reference to a circle thus drawn, the reciprocal of the radius being a correct measure Fig. 55. of the rate of bend. We shall therefore consider such a circle to exist for each point of a curve and shall speak of it as the circle of curvature of that point. Its radius and centre will be called the radius and centre of curvature respectively, and a chord of this circle drawn through the point of contact in any direction will be referred to as the chord of curvature in that direction. 322. Formula for Radius of Curvature. Referring to the figure of Art. 318, let the arc AP measured from some fixed point A on the curve up to P be called s, and AQ, s+Ss', let the angle PLZ= ^, and QMZ=\lr+S\l^. Then the angle of contingence RTQ = S-^ and the measure of the curvature = Xt-~=-/^. If therefore the radius of curvature 68 as be called /o, we have -=-j > ^^ P~~]j (-*■) 323. This formula may also be arrived at thus. Let PQ and QR (Fig. 55) be considered equal chords, and therefore when CURVATURE. 255 we proceed to the limit the elementary arcs FQ and QR may be considered equal. Call each 8s, and the angle RQZ= Syp-. Now the radius of the circum-circle of the triangle PQR is PR 2amFQR ^ J.. PR J.. 2Ss j-.Ss Sifr ds Hence p = U^^ ^in PQR^ ^^Y^^' ^% ' ^n^'d^' Also, it is clear that the lines which bisect at right angles the chords PQ, QR intersect at the circum-centre of PQR, i.e., in the limit the centre of curvature of any point on a curve may be con- sidered as the point of intersection of the normal at that point with the normal at a contiguous and ultimately coincident point 324. The formula (a) is useful in the case in which the equation of the curve is given in its intrinsic form, i.e. when the equation is given as a relation between s and yjr (Art. 316). For example, that relation for a catenary is 5=c tani/r, whence and the rate of its deflection at any point is measured by l_cos^_ c 325. Transformations. This formula however must be transformed so as to suit each of the systems of co-ordinates in which it is usual to express the equation of a curve. These transformations we proceed to perform. We have the equations , dx . . dy cos^/.=^, 8mV^ = ^. Hence, differentiating each of these with respect to 8, . ,(Z^^• d^x ,d\[r d^y _^ d^ , 1 ds^ ds^ whence -=—7 — =-7— (b) p dy dx ds ds and by squaring and adding h-o'-m («) These formulae (b) and (c) are only suitable for the case in which both x and y are known functions of s. 256 CHAPTER X. 326. Cartesian Formula. Explicit Functions. Again, since tan ^ = -— > we have sec2\/rv-= -=-% ^ ax dx^ by differentiating with regard to x. Now *^^d^ &^_1 . dx ds dx p cos \fr l_d^y P and sec^x/r = 1 +tan2i/. = 1 + (^^ f therefore sec^-i/r . — 7 2 {'*©T therefore = -^ 75 ^ (d) d^y dx^ This important form of the result is adapted to the evalua- tion of the radius of curvature when the equation of the curve is given in Cartesian co-ordinates, y being an explicit function of X. 327. Cartesians. Implicit Functions. We may throw this into another shape specially adapted to Cartesian curves, in which neither variable can be expressed explicitly as a function of the other. Thus if ^(x, 2/) = be the equation to the curve, we have and differentiating again dx'^ -dy' dx'^Kdx "^ dy ' dxJdx'^^Ux^ ' dij / dii\ ^ d^^ or *^^+2^=^s+Msy +^^3s^=*^- Hence substitutiner for ~- and Vo ^^ the formula ° ax ax^ p=- — df — we have CURVATURE. 267 p= ^ ^..+2.^4-J-^)+^4-J^) or =___M+^|)i___ (E) 328. A curve is frequently defined by giving the two Car- tesian co-ordinates x, y in terms of a third variable, e.g., the equation of a cycloid is most conveniently expressed as aj = a(0+sin0), y = a{\—co^O). Formula (d) is very easily modified to meet the requirements of this case. Let x = F{t)\ be the equations of the y=f{t) J curve. Then dy_di_m dt and f|=|.(^).|* dx^ dt \dxJ dx dry dx d?x dy _ di^"di~W dt /dx\^ \dt) f(t).F\t)-f(t).F\t) {FXt)}^ and formula (d) becomes ___ {(J) +(J)r _ {[Fx t )fHmf}^ .F^ ^~d^ dx_d^ di'fXt)'FXt)-f{t).F'Xt) ^^ dt^' ' dt dt^ ' dt Ex. In the above-mentionel case of the cycloid «|=„si„., and by formula (f) .3^ a((l +coa ef+^inW]^J'' '"' 2 ^ ^^ _ ^. ^ cos^(l+cos^) + sm^6' gcos^^ ^ 2 S.D.C. 258 CHAPTER X. 829. Curvature at the Origin. When the curve passes through the origin the values of -^(=p) and -T% = q) at the origin may be deduced by substi- tuting for y the expression px+^+... (the expansion of y by Maclaurin's Theorem) and equating coefficients of like powers of X in the identity obtained. The radius of curvature at the origin may then be at once deduced from the formula />= ±(1±^ [Formula (d)]. Ex. Let the curve be Putting we have therefore and giving "whence ^— — -jj^ — ^,, .. — yt—h- This result of course might be deduced at once from formula (e). ,880. It will be noticed that, if the lowest terms of the equation be of the second degree, we should get a quadratic equation giving two values for p, and consequently also two values for q. These indicate the two values of p corresponding to the two branches of the curve passing through the origin. Ex. Find the radii of curvature at the origin for the curve 3/2 _ 3a^ + 2^2 _ ' dr and it has been pointed out that the values of r and -i- are the same at the point P for the curve and for the circle. Hence for the curve itself we also have dr f'-'^d-p (H) Ex. In the equation p^=^Ai^-^B^ which represents any epi- or hypo- cycloid [p. 163, Ex. 6], we have A dr and therefore P ^P- The equiangular spiral, in which peer, is included as the case in which B=0. CURVATURE. 263 334 Polar Curves. We shall next reduce the formula to a shape suited for application to curves given by their polar equations. We proved in Art. 205 ^^ 1 dp /' , d^u\ du Now p = -^r— and r = d^u\ rdr , p = -^— and r = ^ dp u therefore 1 du or P = — 7 ^vT 335. This may easily be put in the r, form thus : — Since u = -, r we have du _ I dr de~~^d& , ^, « d'^u 2 fdr\^ 1 ciV and therefore -M' = Ade) "^ W Xv'^rKdQ) j therefore /> = 1 ri 2/^Y__i c^vi (J) 264 CHAPTER X. 336. Tangential-Polar Form. In Art. 220 it was proved that p=p+ir ^""^ giving us a formula for the radius of curvature suitable for py yp- equations. Ex. It is known that the general p, i/r equation of all epi- and hypo- cycloids can be written in the form ^=^ sin Bf (p. 163, Ex. 6). Hence p = ^ sin B"^ - A Bhin Byfr, and therefore P '^Pj thus again proving the result of the Example in Art. 333. 337. Point of Inflexion. At a point of inflexion the radius of curvature is infinite. This is geometrically obvious from the fact that it is the radius of a circle which passes through three collinear points. We may hence deduce various forms of the condition for a point of inflexion ; thus if p = ^ , we get "^y ~^ ^^^^ (^)' -y^ = from (d), ^^ f* u\u-\-n^ ^' P-r2+2ri2— n-j ^ '' ''=^+^- W Examples. 1. Apply formula (n) to the curves f'^ar, ap=r', P=-;:;iir 2. Apply formula (i) to the reciprocal spiral au = 0. 3. Apply the polar formula for radius of curvature to show that the radius of the circle r=a cos ^ is -• 2 4. Show that for the cardioide r=a(l +cos 6) p = _.cos-; I.e., ex: Jr. Also deduce the same result from the pedal, equation of the curve, viz., 5. Show that at the points in which the Archimedean spiral r = aB intersects the reciprocal spiral r^=a their curvatures are in the ratio 3 : 1 266 CHAPTER X. 6. For the equiangular spiral r=ae'^^ prove that the centre of curvature is at the point where the perpendicular to the radius vector through the pole intersects the normal. 7. Prove that for the curve r=a sec 2^, 8. For any curve prove the formula r where tan = ^-^ • dr Deduce the ordinary formula in terms of r and 0. 9. Show that the chord of curvature through the pole for the curve is given by chord = 2«--^= 247^?- dp f{r) 10. Show that the chord of curvature through the pole of the cardioide r=a(l + cos^) is -r. 11. Show that the chord of curvature through the pole of the equi- angular spiral r=ae'^^ is 2r. 12. Show that the chord of curvature through the pole of the curve 2/* m + 1 Examine the cases when ?7i= - 2, — 1, -^, ^, 1, 2. 13. Show that the radius of curvature of the curve at the origm is — • 14. For the curve r*^ = a*"cos w^, prove that p= Examine the particular cases of a rectangular hyperbola, lemniscate, parabola, cardioide, straight line, circle. 339. Centre of Curvature. The Cartesian co-ordinates of the centre of curvature may be found thus : — Let Q be the centre of curvature corresponding to the point P of the curve. Let OX be the axis of a? ; the origin ; a?, y CURVATURE. 267 the co-ordinates of P ; Xyy those of Q; i/r the angle the tan- gent makes with the axis of x. Draw PiV, QM perpendiculars T M N Fig. 58. upon the £C-axis and PR a perpendicular upon QM. Then x = OM=ON-RP = 0N-QP8m\lr = 07 — p sin yfr, y = MQ = NP+EQ = y-\-p COS\[r. dy dx sm y = and Now therefore and Also Hence 008^ = d^y dx^ x=x — dy dx {-©■)] 1 + ''=y+ d^ dx^ \dx) d^ dx'- .(a) m 268 CHAPTER X. Involutes and Evolutes. 340. Def. The locus of the centres of curvature of all points of a given plane curve is called the evolute of that curve. If the evolute itself be regarded as the original curve, a curve of which it is the evolute is called an involute. The equation of the evolute of a given curve may be found by eliminating x and y between equations (a), (/3) of the last article and the equation of the curve. Ex. Tojmd the locus of the centres of curvature of the parabola Here Hence dy _ X dx~2a 4a da^ 2a ..^JL=X-- dx\ \dx) dx" 4a2 Z^fl whence (y-2a)3: dx" 210^ 27aP (p .^: 64a3 4 Hence the equation of the evolute is 4(y-2a)3 = 27a^2. 341. Evolute touched by the Normals. Let Pp Pg, P3 be contiguous points on a given curve, and let the normals at P^, P^ and at Pg, P3 intersect at Q^, Q^ respectively. Then in the limit when Pg, P3 move along the Fig. 59. curve to ultimate coincidence with P^ the limiting positions of Qp Q2 ^^^ ^^® centres of curvature corresponding to the points Pj, Pg of the curve. Now Q^ and Q^ both lie on the normal at Po, and therefore it is clear that the normal is a tangent to the CURVATURE. 269 locus of such points as Q^, Q^, i.e., each of the normals of the original curve is a tangent to the evolute ; and it will be seen in the chapter on Envelopes that in general the best method of investigating the equation of the evolute of any proposed curve is to consider it as the envelope of the normals of that curve. 842. There is but one Evolute, but an infinite number of In- volutes. Let ABGD ... be the original curve on which the successive points A, B, C, D, ... are indefinitely close to each other. Let a,h,Cy ... be the successive points of intersection of normals at A, B, C, ... and therefore the centres of curvature of those points. Then looking at ABC... as the original curve, ahcd... is its evolute. And regarding abed... as the original curve, A BCD. . . is an involute. If we suppose any equal lengths AA\ BB\ CC, ... to be taken along each normal, as shown in the figure, then a new curve is formed, viz., A'B'C'..., which may be called a parallel to the original curve, having the same normals as the original curve and therefore having the same evolute. It is therefore clear that if any curve be given it can have but one evolute, but an infinite number of curves may have the same evolute, and therefore any curve may have an infinite number of involutes. The involutes of a given curve thus form a system oi parallel curves. 270 CHAPTER X. 343. Involutes traced out by the several points of a string unwound from a curve. Since a is the centre of the circle of curvature for the point ^ (Fig. 60), aA=aB = 6jB 4- elementary arc ah (Art. 34). Hence aA—hB = arc ah. Similarly hB—cC= arc he, cC—dD = axc cdy etc., fF—gG = 2^vcfg. Hence by addition a^— ^G = arca6+arc 6c+...+arc/^ = arc ag. Hence the difference hetween the radii of curvature at two points of a curve is equoZ to the length of the corresponding arc of the evolute. Also, if the evolute ahc... be regarded as a rigid curve and a string be unwound from it, being kept tight, then the points of the unwinding string describe a system of parallel curves, each of which is an involute of the curve ahcd..., one of them coinciding with the original curve ABC... It is from this property that the names involute and evolute are derived. 344. Radius of Curvature of the Evolute. It is easy to find an expression for the radius of curvature at that point of the evolute which corresponds to any given point of the original curve. Let (Fig. 60) be the centre of curvature for the point a of the evolute. The angle Syj^' between the normals at a, h = the angle between the tangents at a, h = the angle between the tangents at A, B to the original curve = 6xlr. And if s' be the arc of the evolute measured from some fixed point up to a, and p' the radius of curvature of the evolute at a, and p that of the original curve at A, we have, rejecting infinitesimals of order higher than the first, Ss' = arc ah = Sp, and therefore p' = i«^^=^«^^ = || = $. CURVATURE. 271 s being the arc of the original curve measured from some fixed point up to J., and i/r the angle which the tangent at A makes with some fixed straight line. 345. From Articles 337, 340, it will follow that to an inflexional or undulatory point on a curve will correspond an asymptote on the evolute. For an inflexional point the evolute will be asymptotic at opposite ends of the normal and on opposite sides. For an undulatory point it will be asymptotic on opposite sides at the same extremity. Examples. 1. For the parabola y- = 4ar, prove ^=2a + 3a;, p=2— -, SP being tlie focal distance of the point of the parabola whose co-ordinates are (x, y). 2. Show that the circles of curvature of the parabola y^=\ax for the ends of the latus rectum have for their equations ^ 4.y2 _ loo^ ± 4a3/ - 3a2 = 0, and that they cut the curve again 'in the points (9a, q:6a). 3. Show that the evolute of the parabola y^=4ar is the semicubical parabola 27ay2 = 4(^ - 2a)^, and that the length of the evolute from the cusp to the point where it meets the parabola = 2a(3\^ — 1 ). 4. Show that in a parabola the radius of curvature is twice the part of the normal intercepted between the curve and the directrix. 5. Prove that in an ellipse, centre (7, the radius of curvature at any pomt P IS given by p= ^g-=^=^j-' where a, h are the semi-axes, r, r' are the focal distances of P, p the per- pendicular from the centre on the tangent at P, and CD the semi-diameter conjugate to CP. 6. Show that in any conic (normal)^ P = (semi-latus-rectum)2 272 CHAPTER X. '7. For the ellipse ^4.|! = 1, prove ^=^__^^3 a* Hence show that the equation of the evolute is and prove that the whole length of the evolute a) 8. Show that the co-ordinates of the centre of curvature of any curve may be written ^d^ ^ ISj dy^ da^ Inteinsic Equation. 346. The relation between the length of the arc (s) of a given curve, measured from a given fixed point on the curve, and the angle between the tangents at its extremities (i/r) has been aptly styled by Dr. Whewell the Intrinsic Equation of the curve. For many curves this relation takes a very elegant form. The name seems specially suitable to a relation between such quantities as these, depending as it does upon no external system of co-ordinates. The method of obtaining the intrinsic equation from the Cartesian or polar relation is dependent in general upon processes of integration. If the equation of the curve be given as y=f(x), the axis of x being supposed a tangent at the origin, and the length of the arc being measured from the origin, we have tan^=/(aj), (1) and J = Vl+[/(^)? (2) If s be determined by integration from (2) and x eliminated between the result and equation (1), the required relation between s and -yjr will be obtained. CURVATURE. 273 Ex. 1. Intrinsic equation of a circle. If i/r be the angle between the initial tangent at A and the tangent at the point P, and a the radius of the circle, we have PUA=PTX=f, and therefore s = ayjr. Ex. 2. In the case of the catenary whose equation is y — c cosh - ^ c the intrinsic equation is s = c tan yfr. For tanV^ = ^=sinh?, dx c and $ =/v/l + 8inh2'- = cosh -, da; ^ c c and therefore «=C8inh-» c the constant of integration being chosen so that x and s vanish together, whence s = c tan yjr. Examples. 1 . Show that the cycloid x=a{9 + sin 6)^ y = a{l— cos 0)f has for its intrinsic equation 5= 4a sin yfr. 2. Show that the epi- or hypo-cycloid given by x=^{a + h)cose-bcos^e 2/=(a + b)s[ne-bam^0 has an intrinsic equation of the form s=A ain B-yfr. 347. Intrinsic Equation of the Evolute. Let s =/(V^) be the equation of the given curve. Let s' be the length of the arc of the evolute measured from some fixed point A to any other point Q. Let and P be the points on E.D.C. is 274 CHAPTER X. the original curve corresponding to the points A, Q on the evolute ; p^, p the radii of curvature at and P ; \jr' the angle the tangent QP makes with OA produced, and \f^ the angle the tangent PT makes with the tangent at 0. Then \f/ = i/r, and s'—p — pQ dxfr ^«' the intrinsic equation of the evolute. 348. Intrinsic Equation of an Involute. With the same figure, if the curve J.Q be given by the equation ^'=f{^')> P = ^'+pQ^ P = ,TJj and Vr = i/r', we have whence 349. Evolutes of Cycloids or Epi- and Hypo-Cycloids. If we apply the result of Art. 347 to the intrinsic equation 8 = A sin B\lr, we get for the equation of the evolute s' = AB cos Byf/ — p^, or, dropping the dashes, if s be supposed measured from the point where yfr = ^. This proves that the evolute of an epi- or hypo-cycloid is a similar epi- or hypo-cycloid. Also, the case in which B = l shows that the evolute of a cycloid is an equal cycloid. [For further information on Intrinsic Equations the student is referred to Boole, Differential Equations, p. 263, and to Camb. Phil. Trans.y vol. VTII., p. 689, and vol. IX., p. 150.] CURVATURE. 275 Examples. 1. If A be the area of the portion of a curve included between the curve, two radii of curvature, and the evolute, prove 2. Show that the evolute of an equiangular spiral is an equal equiangular spiral. 3. Show that the intrinsic equation of the evolute of a parabola is s = 2a(sec^ — 1). 4. Given the pedal equation of a curve, viz., p=f{r); show that the pedal equation of its evolute may be found by eliminating p and r between this equation and the equations r'^=p^ + r'-2pp, (a) p'' = r^-p' W Again, that if the equation p' = f{r') of a curve be given, the general differential equation of its involutes may be obtained by eliminating p', / between this equation and the equations (a), (/?). 5. Show that the curve whose equation is p^=^'r-' d r for , = cos CPT' = — sin 0, the arcs of the curve and of the conic being measured from the points and 0' up to P, and 2 = cos ; r ^ therefore -i^ = 3 tan 0, 0,9 and tan = - -^, where -^ is found for one of the conies. But since the conic and the curve have contact of the third d 7' dj /* order they have the game tangent, the same ,^, the same -^^g' d^v and the same j^^ at the point of contact. They therefore also have the same p and the same -f, for p depends on ,^ and -j Hence the value of found above is the same for all the conies, and depends only upon the' shape of the curve at the point of contact. The locus of all such centres is therefore a straight line through the point of contact inclined in front of the normal at an angle tan~^f ^ J^Y where ^^ is found from the curve. 355. This result may be established analytically as follows : — Eef erring the conic to the common tangent and normal as axes, its equation takes the form 2y = ai^ + 2kjn/ + 6^^ ^ If y be expanded in powers of jc, by Maclaurin's Theorem we have as in Art. 329 ; p, a. and r being the values of .•^, -=-'{. and -^ at the ax' dor da^ origin. Since there is contact of the third order the values of these are the same for the conic and for the curve and are therefore known quanti- ties. Moreover, since the tangent has been chosen for the a;-axis, we have p=0. 282 CHAPTER X. Substituting in the equation of the conic we have giving ' a=q, and thus determining a and h in terms of the known quantities q and »l Also the centre of the conic lies on the line or Zq^x+ry=0^ which is a straight line through the point of contact inclined in front of the normal at an angle tan~^(— — -^V Also since p = {\ +p^yq~'^ ^={3(1 +p^)^pq-' - (1 +i>2#r M J' which, when jp = and -7-=!, becomes -^= — -„. as (f Hence the above angle may be written as in the preceding article. 356. Osculating Conic. We can now pick out the particular conic which has fourth order contact with the given curve at the given point. Let be the centre of curvature of the point considered and G the required centre of the conic of closest contact. Let P^ Fig. 67. be a point on the curve adjacent to the given point P. Join CP, GP^ and draw P^JSf at right angles to GP. CURVATUKE. 283 Let OPG=ct>: O^^C=(p + Scl>, PC=R. Then PEP^ = PdE+(p, and also = P^CE+

. Also, neglecting infinitesimals of higher order than the first, PP^ = Ss, PdE^A P and r,CP--p^^—^. TT Ss SS COS d> , ^^ Hence — = — ^— ^ + Sd>, p It or, proceeding to the limit, cos _ 1 dcp where <^ = tan"^^ f And since the contact is of the fourth order, -^ is the same as for the curve as for the conic, and may therefore be supposed derived from the equation of the curve. These equations determine the position of C. 357. Tangent and Normal as Axes. Co-ordinates of a Point near the Origin in terms of the Arc. When the tangent and normal at any point of a curve are taken as the axes of x and y it is sometimes requisite to express the co-ordinates of a point on the curve near the origin in terms of the length of the arc measured from the origin up to that point. Assume oc = a-\-a-^s+a2-^^+a^^-\-..., the letters a, a^..., h, h^... denoting constants whose values are to be determined, and s being the length of the arc. Then, when 8 = 0, X and y both vanish, and therefore a = 6 = 0. 284 CHAPTER X. Again, by Maclaurin's Theorem ^^=©„=(^-^>-«' [the suffix zero denoting the values at the origin] «.=(S)r-(-*f).=-C^).=«i '.-©.-{•«4t).-C-7*).%' ) _ /(Fx^, _ /cos i/r sin i//- dp\ _ 1 ' 3~VcZsV( cZs/c etc. whence ijp' y §2 s^ dp 2p'~6p^'ds~"" EXAMPLES. 1. Determine the curvature of the curve [Coll. Exam.] at the origin. 2. Find the radii of curvature of the two branches of the curve (x - yy{x - 2y){x - By) - 2a{x^ - y^) - 2a\x + y)(x -2y) = at the origin. [Oxford, 1888.] 3. For the curve /^^=VJ dx~ " - y ^ prove that the radius of curvature is m times the normal. 4. Establish the formula ^ dsjl^ \dj ds^J 5. Find the equation of the circle of curvature at any point of the curve y/a = vers~^a;/a. 6. If p be the radius of curvature of a parabola at a point whose distance, measured along the curve, from a fixed point is s, prove '^''' ^''2-©'-^ = *^- [OXFOKB, 1889.] CURVATURE. 285 7. A curve is such that the normal at any point passes through the centre of curvature of the corresponding point on the pedal with respect to a given point. Show that the curve is an equiangular spiral. [Oxford, 1890.] 8. If p and p be the radii of curvature at corresponding points of curve and its evolute, and p, q, r are the first, second, and third differential coefficients of y with respect to a;, prove that p'lp={Spq^-r{l+p^)W. 9. The projections on the tc-axis of the radii of curvature at corre- sponding points of 2/ = log sec x and its evolute are equal. [Coll. Exam.] 10. Show that the radius of curvature of the point of the evolute of the curve r** = a"cos nd corresponding to r, 6 is -^j'secnOtamnd. ^^ ^ ^ ' (w+l)2 [Oxford, 1889.] 11. A tangent to the evolute of a parabola at the point where it meets the parabola is also a normal to the evolute at the point where it again meets the evolute. [Coll. Exam.] 12. If /)j be the radius of curvature. at any point of a parabola, p^ the radius of curvature of the corresponding point of its first nega- tive pedal with respect to the focus, show that 27p,^ = 32lp,^ where I is the latus rectum. [Oxford, 1889.] 13. P, Q, E, S, T SLve five points on a curve of continuous curva- ture whose abscissae are in arithmetical progression, the common difference being 8x ; show that as 8x diminishes without limit, PT, QS, and the tangent at E ultimately intersect in the same point, and that in the parabola y^ = mx the locus of this point is a parabola with the same vertex and axis. [Coll. Exam.] 14. The radius of curvature at the point t on the curve is given by the equation ^/)-i = 'Ir'-Q + r'rQ - rrO + rW^ ""^^'^ ^' = § '^ = %^^'- [Oxford, 1888.] 15. Show that the parabola whose axis is parallel to the axis of i/, and which has the closest possible contact with the curve at the point (a, a), has for its equation n{n -\)x^ = lay + 2n{n - 2)ax -{n-l){n- 2)a^. 286 CHAPTER X. 16. If £c, y be the co-ordinates of a point P of a curve OP passing through the origin 0, then the radius of curvature at hLt- x^ + 2/2 X sm a - 2/ cos a where y = x tan a is the equation of the tangent at the origin. Hence show that the radius of curvature of the curve a;4 + 2/2 =. 2a{x + y) at the origin is 2a^2. 17. Show that the arcs of the two curves xy = a^^ x^ = Sa^y turn through the same angle between any the same pair of ordin- ates. Also show that the ratio of the radii of curvature at points on the two curves which have the same abscissa varies as the square root of the ratio of the ordinates. [Oxford, 1887.] 18. The radius of curvature of the first negative pedal of/? -=f{r) at a point corresponding to (/?, r) on the original curve is 2/^ r* dp p^ p^ dr 19. Show that the curvature at any point of the pedal of an epi- or hypo-cycloid is ^-^ — ^^ K where a is the radius of the fixed circle and r and p refer to the pedal curve. [Sidney Coll., Camb.] 20. If r, p, p be respectively the radius vector, perpendicular from the origin on the tangent and the radius of curvature at any point of a curve, prove that the radius of curvature at the corresponding point of the reciprocal polar with regard to the origin is where k^ is the constant of reciprocation. Hence show that the reciprocal of a circle is a conic with the origin as focus. 21. If r, p, p be the same as in the last question, show that the radius of curvature at the corresponding point of the inverse with regard to the origin is - — ^> 2pp - r^ k^ being the constant of inversion. 22. Find the radii of curvature of the confocal orthogonal Lima9on8 r sin^a = a(cos - cos a), r sinh-j8 = a(cosh /3 - cos 0) at a point of intersection, in terms of a and p. [Math. Tripos, 1884.] CURVATURE. 287 23. Show that the intrinsic equation of the curve e « = sec IS - = ed V- a a 24. If A"i, /i"i be the ratios of any arc of the curve s = c tan xp measured from the point ^ = 0, to the corresponding arcs of the evolute, and of that involute which meets the curve at the point ^-^0, find the relation between A and /a. * [Oxford, 1888.] 25. An inextensible wire in the form of a plane curve is bent so that each point of the wire moves a distance u in the direction of the normal to the curve at that point ; prove that u\dsj p [Oxford, 1888.] 26. Show that the locus of the centre of the rectangular hyperbola, having contact of the third order with the conic has for its equation ^c^ + 2/^ = f j + -^jjAx^ + ByK 27. Show that the locus of the centres of the rectangular hyper- bolae, having contact of the third order with the parabola 2/2 ~ iax, is the equal parabola y^ + 4a(x + 2a) = 0. 28. If the equation to a curve passing through the origin be where u„ is a homogeneous function of x, y of n dimensions, show that the general equation to all conies having the same curvature at the origin as the given curve is Wj + «2 + (Ix + my)u^ = 0. Thence find the circle of curvature. 29. Show that the circle of curvature at the origin for the curve x + y = ax^ + by^ + cx^ is {a + b){x^ + y^) = 2x+2y. 30. Obtain the equation of the conic which osculates the curve ay = x^ + a^xy + a^'^ + b^a? + b-^x^y + b^^ + b^^ at the origin. PQ is the common chord of a parabola 2/2 = 4aa; and its osculating circle at P. Prove that the locus of the point of intersection of PQ with the perpendicular drawn on it from the vertex is the cissoid y2(3a - a) = a^. [Oxford, 1890.] 288 CHAPTER X. 31. Show that when the osculating circle has third order contact with its curve the curvature is measured by 32. A line is drawn through the origin meeting the cardioide r = a(l - cos 6) in the points P, Q, and the normals at P and Q meet in C. Show that the radii of curvature at P and Q are proportional to PC and QC. 33. If PQ be an arc not containing a point of maximum or mini- mum curvature, the circles of curvature at P, Q will lie one entirely within the other. [Math. Tripos.] 34. Determine the equation of the circle which touches the curve r=/(e) at the point (r^, 0^) and goes through another point (r^, 6^ on the curve ; and hence derive the expression for the radius of curvature in polars. [Math. Tripos, 1884.] 35. Show that the osculating conic of the catenary y = c cosh - c at a point whose ordinate is ^V 10 is a parabola. ..^ .^^^ 36. An equiangular spiral has contact of the second order with a given curve at a given point ; prove that its pole lies on a certain circle, and that, if the contact be the closest possible, the distance of the pole from the point of contact is P V l + /'4^^ ^dsj [Math. Tripos.] 37. If accented letters refer to a point on a curve and unaccented letters to the corresponding point on the involute, prove ,-^ dx as '-T- %' y=y'fp£r Show how, by means of these equations and s'Tp = iy the equation of an involute of a given curve may be found ; s' being supposed known in terms of the co-ordinates of the extremities of the arc. 38. If a right line move in any manner in a plane, the centres of curvature of the paths described by the different points in it in any position lie on a conic. CURVATURE. 289 39. If, on the tangent at each point of a curve, a constant length be measured from the point of contact, prove that the normal to the locus of the points so found passes through the corresponding centre of curvature of the given curve. [Bertrand.] 40. If through each point of a curve a line of given length be drawn, making a constant angle with the normal to the curve, the normal to the locus of the extremity of this line passes through the corresponding centre of curvature of the proposed curve. [Bertrand.] 41. If on the tangent at each point of a curve a constant length c be measured from the point of contact, show that the radius of curva- ture of the curve locus of its extremity is given by (p2 + c2# P = "^-^-"^ where p and x// refer to the corresponding point of the original curve. 42. If through each point of a curve a line of given length c be drawn, making a constant angle a with the normal at that point, the radius of curvature of the locus of its extremity is given by , (p^ + <^- 2pc cos a)^ p = }^ c i — __, p2 ^ c'l _ 2pc cos a - c sin a^\ where p and if/ refer to the corresponding point of the original curve. 43. If on each tangent to a given curve a length be measured from the point of contact equal to the radius of curvatui-e there, the centre of curvature at any point on the locus of the extremity of the mea- sured length is at the centre of curvature of the corresponding point of the original curve. 44. Show that the equation of the involute of the catenary y = c cosh - c which begins at the point where a; = 0, y = c, is the Tractrix x = c cosh~i- - Jc^ - y^. y 45. If a straight line be drawn through the pole perpendicular to the radius vector of a point on the equiangular spiral to meet the corresponding tangent, show that the distance between the point of intersection and the point of contact of the tangent is equal to the arc of the curve measured from the pole to the point of E.D.C. T 290 CHAPTER X. contact. Hence prove that the locus of this point of intersection is one of the involutes of the spiral, and show that it is an equal equi- angular spiral. 46. A fixed oval curve on a smooth horizontal plane is surrounded by a smooth endless string, and a particle is projected inside the string so as to move round, keeping the string stretched. If t and tf are the lengths of the straight portions of the string at any time; f/), d{x,y,c^Sc) = 0,] ^ be two consecutive members of the family. Expanding the latter we have ^(a;, y, c) + (5c- 0((k, 2/, c) + . . . = 0. Hence in the limit, when Sc is infinitesimally small, we obtain — 0(aj, 2/, c) = as the equation of a curve passing through the ultimate point of intersection of the curves (a). If we eliminate c between the equations ^(x,y,c) = and :^^(^, y, c)=-a:+|=0, therefore whence c=±V- i a _i ^ / ax. ± y'^ = iax, ENVELOPES. 205 a parabola, which is therefore the envelope. In other words, every straight line, obtained by giving any arbitrary special value to c in the equation y=cx + , touches the parabola y- = 4a.r. 362. The Envelope of A\^+2B\ + C=^ is B^ = AC. If A, B, G be any functions of x and y, and the equation of any curve be ^ XH 25X + C= 0, X being an arbitrary parameter, the envelope of all such curves is B'^AG. For we have to eliminate X between A\^+2B\-\-G=0 and 2^X4-25 = 0, and the result is clearly B^ = AG. The result of the example of Art. 361 may be obtained in this way ; foi- the equation y = ma;+^ m may be written m^x — my + a = 0, and therefore the envelope is y'^ = ^ax. 363. Another Mode of Establishing the Rule. The equation ^X^ + 2i?X + C=0 may be regarded as a quad- ratic equation to find the values of X for the two particular members of the family which pass through a given point {Xy y). Now, if {x, y) be supposed to be a point on the envelope, these members will be coincident. Hence for such values of ic, y the quadratic for X must have two equal roots, and the locus of such points is therefore B^ = AC. The envelope of the system (p{x^ y, c) = might be considered in a similar manner. And it is proved in Theory of Equations that if /(c) = is a rational algebraic equation for c, the con- dition that it should have a pair of equal roots is obtained by eliminating c between the equations /(c) = 0, /(o) = 0, a result agreeing with that of Art. 861. Examples. 1. Show that the envelope of the line - +^= 1, where ab=c^, a constant, is Aan/^c\ ' 296 CHAPTER XL 2. Find the equation of the curve whose tangent is of the form y=7tix + m^, m being independent of a: and y. 3. Find the envelope of the curves .r y a for diiferent vahies of Q. 4. Find the envelope of the family of trajectories y=;r tan B — \g- x^ being the arbitrary parameter. 5. Find the envelopes of straight lines drawn at right angles to tan- gents to a given parabola and passing through the points in which those tangents cut (1) the axis of the parabola, (2) a fixed line parallel to the directrix. 6. Find the envelope of straight lines drawn at right angles to normals to a given parabola and passing through the points in which those normals cut the axis of the parabola. 7. A series of circles have their centres on a given straight line, and their radii are proportional to the distances of their corresponding centres from a given point in that line. Find the envelope. 8. P is a point which moves along a given straight line. Pi/, FN are perpendiculars on the co-ordinate axes supposed rectangular. Find thd envelope of the line MN. 9. A straight line has its extremities on two fixed straight lines and forms with them a triangle of constant area. Find its envelope. 10. Show that the envelope of the lines whose equations are X sec^d+y cosec"^=c is a parabola touching the axes of co-ordinates. 11. Show that the system of conies obtained by varying A in the equation ^+ 2A.'^+f' = 1 - X^ ^ a^ ab ¥ have for their envelope the parallelogram whose sides are x='i.a, y= + b. 12. Show that the envelope of the line lx+my + 1—0, where the parameters I, m are connected by the quadratic relation aP+2hlm + bm^ + 2gl + 2fm -f c = 0, is the conic Ax^ + 2ffxy+By^- + ^Gx+2Fy + C=0, J, B, C\ F, Gj H being minors of the determinant a. h, 9 h h, f g^ /. c ENVELOPES. 297 364. The c-Discriminant. The function of x and y, whose vanishing expresses that 0(fl?, y, c) = has equal roots for c, is, when expressed in its simplest rational integral form, called the c-discriminant of ^, and may be denoted by Ac0. Hence the envelope for different values of c will be given in the equation Ac0 = 0. •365. Singularities. The equation A^^ = may contain loci other than the true envelope. Imagine a curve with a double point iV^ to be made to move in a given manner altering its shape as it travels but retaining the same general characteristics. Take a point P near the locus of the double point. First one and then the other of the branches which form the node pass through P, and when P is ultimately on the locus of the node the two positions of the curve in which a branch passes through P ultimately coincide. We can now generalize this idea. When fixed values are assigned to x and y the equation ^(a;, y, c) = may be regarded as giving the several values of c, corresponding to the several members of the family which pass through a specified point. If this equation be of the n^ degree in c, there will be n real or imaginary solutions and therefore n members of the family each passing through this point. When successive values of c give a locus of multiple points of the r^'^ order for the family ^(ic, y, c) = and the chosen point (x, y) happens to lie upon this locus, r of these members will coincide, and therefore the equation 0(aj, y, c) = will give for such a point r equal values of c. Hence it may be expected that the equation Ac0 = O will contain, besides the true envelope solution, the loci of any nodes, cusps or conjugate points which the members of the family may possess. 366. The more advanced student is referred for further information to Papers by Cayley, Messenger of Mathematics, vols. II. and XII. ; Henrici, vol. II., Proc. Lond. Math. Soc. ; and M. J. M. Hill, vol. XIX., Proc. Lond. Math. Soc. ; where it has been shown that the c-discriminant in general contains the 298 CHAPTER XI. envelope locus as a factor once, the node locus twice, and the cusp locus thrice. Ex. 1. c{y + cf=a^. Here differentiating witli regard to c (y + c)2 + 2c(y + c)=0, giving y + c=0, (i.) or y + 3c=0 (ii.) Substituting from (i.) in the curve we get x = Q (iv.) Substituting from (ii.) we get — ^=^ (v.) Kji these (iv.) is a cusp locus and (v.) is a true envelope Zx= -4^.y. This is exhibited in the accompanying figure. Fig. 69. Ex. 2. It may happen accidentally that the node or cusp locus is the true envelope locus. Thus in the family of semicubical parabolas ay'^={x-cf y <-^-^-<^^ Fig. 70. the c-discriminant is y=0 or the .r-axis, and as this line touches each mem- ber of the family it may be regarded as a true envelope. The cusps are now arranged as shown in Fig. 70. ENVELOPES. 299 Ex.3. c(i/ + cf=aiXa^+l). Here there is a node or conjugate point at (0, — c) according as c is positive or negative. Differentiating we have (y + c)(y + 3c)=0. Eliminating c we have the results Of these :r=0 is the node locus for the portion of the ^/-axis below the origin, and the conjugate point locus for the portion above the origin.^ The line x+ 1 ^0 is a true envelope solution, as also the cubic Ex. 4. Examine the cases of (i.) iy + cy=a^{x+l), (ii.) 2/=c(x+cy\ (ill.) y^=c{x-cf. 367. It may happen that the consecutive members of the family ^(a?, y, c) = do not all intersect in real points. In this case the curve Ac^ = does not touch all the members of the family at leal points. Ex. Let circles be described having for their diameters the double ordinates of the parabola 2r=Aax. Find their envelope. If 2c be the double ordinate, the typical equation of such a circle is c»\2 (^-4a) +^'="' or c'^-%(K?{x+^a)->r\Qa\x^-\-y^-) = 0, (1) and the envelope is {x + 2a)- = :r^ + y^ or y^=Aci{x + a\ (2) I.e., an equal parabola whose focus is at the vertex of the original curve. To find where the circle (1) touches the envelope solve for ;r between (l)and(2). We obtain c2=4a(:r + 2a) =y'^ + 4a-, which gives an imaginary ordinate for the point of contact if c<2a ; i.e., if the centre of the circle lies between the focus and the vertex of the original parabola. The student will be able to illustrate this result by a figure. 300 CHAPTER XL 368. Case of Two Parameters. Next, suppose the typical equation of the family of curves to involve two parameters a, (3 connected by a given equation. Then two courses are open to us. We may eliminate one of the parameters by means of the connecting equation and thus reduce the problem to that solved in Art. 361, or, as is frequently better from considerations of symmetry, consider one of the param^eters capable of independent variation and the other dependent upon it. We then proceed as follows. Let ^(x,y,a,/3)==0 (1) be the typical equation of the curves whose envelope is to be investigated, and f{a, I3) = (2) the relation connecting a and /5. Then, supposing a the independent parameter, we have K|-f- <«) l+|-f=« (*) We thus have four equations and three quantities to eliminate, viz., a, p, -J^. The result of elimination is the equation of the envelope. The parameters a, ^, connected by the relation f(a, /3) = 0, may be regarded as the co-ordinates of a parametric point which lies on the curve f(x, y) = 0. 369. Indeterminate Multipliers. The equations (3) and (4) may be written |^cZa + ||cZ/3 = (Art. 158), The result of eliminating da, d^ between these equations is da d/3 ENVELOPES. 301 Call each of these ratios \. We then have l^-=\f. (5) oa da ^ '^'t>-->?f (a) a^-^3^ w This quantity X is called an " Indeterminate Multiplier." It remains to eliminate a, ^, and X between .equations (1), (2), (5), and (6). This method is peculiarly adapted to the case in which (pix, y, a, ^) = i(x, 2/, a, P)-a^ = 0, and /(a,)8)^/i(a,/3)-a2 = 0, where 0^ and f^ are homogeneous in a and /8, and of the p^^^ and g*^ degrees respectively, a^ and a^ being absolute constants. Multiply equation (5) by a and (6) by ^, and add. Then by Euler's Theorem ])a^ = qa^, so that in such cases X is easily found. Ex. Fiiid the envelope of " +^ = 1, where a and h are connected hy the relation a^-\-b^= c^, c being an absolute constant ; i.e., the envelope of a line of constant length which slides with its extremities upon two fixed rods at right angles to each other. 3 4- Here ^da+'^db = 0, a^ ¥ ada + bdb=0. and therefore — =Xa, ^5' or ^ Multiplying by a and b respectively, and adding, or 1 = \c^. Hence a^=c^x\ ¥=:ch/ y and since a" + 6^ = c^ we have (c^x)^ 4- {c^yr = <^^ or ^»+;y*=c* 302 CHAPTER XI. Examples. 1. Find the envelopes of the line a under the following conditions : — (1) a^h=l, (2) a" + 6"=^« (3) a"*6'* = F*+", Ic being a constant in each case. 2. Find the envelopes of the systems of coaxial ellipses whose semiaxes a and h are connected by the equations (1) a+6 = ^, (3) ar + h^=k'^, (4) ab = k\ h being a constant in each case. 3. Find the envelopes of the parabolas which touch the co-ordinate axes and are such that the distances (a, /3) from the origin to the points of con- tact are connected by the relations (1) a + ^ = ^, (2) a^ + /5"* = ^"*, (3) a^==lc\ Jc being a constant in each case. 370. Case of Three Parameters jonnected by Two Equations. Next, suppose the equation of a curve to contain three para- meters connected by two equations. Let the equation of the curve be {x,y, a, p,y)==0, (1) and let /i(a, fty) = 0,| (2) /.(a,Ay)-Oj (8) be the two connecting equations. Then we have l^ + > + '^'^V=0. (4) M^„+|d^+|,Z^ = 0, (5) %da+^Adp+f^ly = (6) = (7) ENVELOPES. 303 The result of eliminating da, d^, dy between these three equa- tions is 90 90 90 I 9a' 9^' 9^ ^/i ^ % 9a' 9/3' 9y §^ ^^2 ^ 9a' 9/3' 9y If a, ;8, y be eliminated between the four equations (1), (2), (8) and (7), the result will be the equation of the envelope. It is to be noted that the same determinant would arise from the elimination of the " indeterminate multipliers " \ and Xg from the equations t^^^^-'' • («) ^^^X ^/i4.> ^/2_o /q\ 9^ ^9S 2^^^" ' ^^ l^+^.|+^4-5=« <^o> and it is often advantageous to use these latter equations in place of (4), (5), (6), involving da, d^, dy. The result of eliminating a, ^, y, X^ Xg between the six equations (1), (2), (3), (8), (9), (10) will then be the equation to the envelope. 371. The general investigation of the envelope of a curve whose equation contains r parameters connected by r— 1 equations proceeds in exactly the same way, and is the result of the elimination of the r parameters and r—\ indeterminate multipliers between 2r equations. 372. Converse Problem. Given the Family and the Envelope to find the relation between the Parameters. Suppose we are given the equation of a curve 0(^, 2/, a,/3) = (1) containing two parameters. Suppose also the envelope given, viz., F{x, 2/) = (2) Required the relation between a and ^. Eliminate y between (1) and (2). We obtain an equation of the form f{x, a, )8) = 0, (3) 304 • CHAPTER XI. giving the abscissa of the point of contact of the curve with its envelope. Since the curve touches its envelope, equation (8) must also be true for a contiguous value of x, viz., x + Sx (unless the tangent at the point of contact be parallel to the axis of y, in which case we could have eliminated x between (1) and (2) and proceeded in the same way with y). Hence f(x,a,h) = OA (4) f{x + Sx, a, h) = 0j (5) The latter may be expanded in powers of Sx, when it becomes f{x,a,h) + f^Sx-\-.., = 0, (6) and therefore in the limit l=« •(^) If, then, X be eliminated between f{x,a,l3) = 0, we obtain the relation sought. ^ It will be observed that this is precisely the same process as finding the envelope of We have to find the envelope of this line for difierent values of the parameter . Difierentiating with regard to <^, sin ^ /-. aoc — ^ + bi/-.- y=0, (2) cos^<^ "'sin^c^ ^ ' sin^d> , cos' and 0)'. Show that the pedal equation of the envelope of a line joining their extremities is (a> + w')V- = Aididijr + (w — {a'YaP. 30. The envelope of polars with respect to the circle x^ + y'^ = 2ax of points which lie on the circle x2 + 2/2 = 2hx is {{a-h)x + ahY = 1f-{(x-af ^y^}. 31. A square slides with two of its adjacent sides passing through fixed points. Show that its remaining sides touch a pair of fixed circles, one diagonal passes through a fixed point, and that the envelope of the other is a circle. 32. An equilateral triangle moves so that two of its sides pass through two fixed points. Prove that the envelope of the third side is a circle. 33. Prove that the envelope of the circles obtained by varying the arbitrary parameter a in the equation c^{y - of + {ex - a2)2 = (a2 + ^2)2 consists of a straight line and a circle. 312 CHAPTER XI. 34. Two points are taken on an ellipse on the same side of the major axis and such that the sum of their abscissae is equal to the semi-major axis. Show that the line joining them envelopes a para- bola which goes through the extremities of the minor axis and whose latus rectum is equal to that of the ellipse. 35. Given the centre and directrices of an ellipse, show that the envelope of the normals at the ends of the latera recta is 272/4 ± 25Qcx^ = 0. 36. Prove that the envelope of a circle which passes through a fixed point F and subtends a constant angle at another fixed point F' is a lima9on. 37. Find the envelope of a parabola of which the directrix and one point are given. 38. Show that the envelope of the common chords of the ellipse a:7a2 + 2/2/62=1 and its circles of curvature is the curve /^ 2/\i /^_2/\§_9 \a b) \a h) "' [Math. Tripos, 1884.] 39. Find the condition between a and b that the envelope of the line ^ + f = 1 a may be the curve x^y^ = k^'^''. ■ 40. iS' is a fixed point, and with any point P of a curve for centre and with radius PS + k^, circle is described. Show that the envelopes for different values of k consist of two sets of parallel curves, one set being circles ; and find what the original curve must be that both sets may be circles. 41. Eays emanate from a luminous point and are reflected at a plane curve. F is the perpendicular from on the tangent at any point P, and OT is produced to a point Q, such that YQ = OY. Show that the caustic curve is the evolute of the locus of Q. Show that the caustic curve may also be regarded as the evolute of the envelope of a circle whose centre is P and radius OP. [If a ray of light in the plane of a given bright curve be incident upon the curve, the reflected ray and the incident ray make equal angles with the normal to the curve at the point of incidence, and the reflected ray lies in the plane of the curve. If a given system of rays be incident upon the curve, the envelope of the reflected rays is called the caustic by reflection.] ENVELOPES. 313 42. Parallel rays are incident on a bright semicircular wire (radius a) and in its plane. Show that the caustic curve is the epicycloid formed by a point attached to a circle of radius - rolling upon the a ^ circumference of a circle of radius -• 2 43. Kays emanate from a point on the circumference of a reflecting circular arc. Show that the caustic after reflection is a cardioide. 44. Show that if rays emanate from the pole of an equiangular spiral and are reflected by the curve the caustic is a similar equi- angular spiral. 45. Eays of light parallel to the y-axis fall upon the reflecting curve y =f(x). Show that the equation of the reflected ray is {Y-y)2p + (l-p'^){X-x) = where p =f'{x). Also that the length of the reflected ray between the point of reflection and the caustic is one quarter of the chord of curvature parallel to the 2/-axis. 46. If rays of light emanating from the pole fall upon a reflecting curve, show that the length {I) of the reflected ray is given by 47. A straight line meets one of a system of confocal conies in P, Q, and RS is the line joining the feet of the other two normals drawn from the point of intersection of the normals at P and Q. Prove that the envelope of RS is a parabola touching the axes. [Math. Tripos, 1884.] 48. Show that the tangents to a system of conies inscribed in a given quadrilateral, at the points where a fixed straight line meets them, envelope a curve of the third class touching the given line and the sides of the given quadrilateral. [Math. Tripos, 1885.] 49. Show that the vanishing of the c-discriminant of the eliminant of /> from the equations xp'^ - 2yp + a = 0\ and cp^ - xp'^ + = [ gives exactly the same locus as the vanishing of the jo-discriminant of the first equation. Show that this is not a true envelope but a cusp locus. [Math. Tripos, 1878.] 50. Find the condition that every curve of the family /(a;, y,c) = may have a double point, i.e. that there may be a node locus. [Prof. M. J. M. Hill.] CHAPTER XII. CURVE TRACING. 376. Nature of the Problem. Cartesian Equations. If in the Cartesian equation of any algebraic curve, various values of x be assigned, we obtain a number of equations whose roots give the corresponding values of the ordinates. The real roots of these equations can always be either found exactly or approximated closely to by methods explained in the Theory of Equations. We can by this means, laborious though it will in most cases be, find as many points as we like which satisfy the given equation of the curve ; and by joining these points by a curved line drawn freely through them we can form a fairly good idea as to its shape. The experience, however, which we have gained in previous chapters will in general obviate any necessity of resort to the usually tedious process of approximating to the roots of equations of high degree; and we propose to give a list of suggestions for guidance in curve tracing which in most cases will enable us to form, without much difficulty, a sufficiently exact notion of the character of the curve represented by any specified equation. 377. Order of Procedure. 1. A glance will suffice to detect symmetry in a curve. (a) If no odd powers of y occur, the curve is symmetrical with respect to the axis of x. Similarly for symmetry about the axis of y. (h) If all the powers of both x and y which Occur be even, the curve is symmetrical about both axes, as, for instance, x^ ^3 in the case of the ellipse -g+p "= ^• 314 CURVE TRACING. 31 5 (c) Again, if on changing the signs of x and y the equation of the curve remain unchanged, there is symmetry in opposite quadrants, as in the case of the hyperbola xy = k^. The origin is then said to be a centre of the curve. {d) If the equation remain unchanged when x and y are inter- changed there is symmetry about the line y = x. If the curve be not symmetrical with regard to either axis, consider whether any obvious transformation of co-ordinates could make it so. 2. Notice whether the curve passes through the origin ; also the points where it crosses the co-ordinate axes; or, in fact, any points whose co-ordinates present themselves as obviously satisfying the equation to the curve. 3. What linear asyni'ptotes are there ? First find those parallel to the co-ordinate axes ; next, the oblique ones (Art. 237). These results point out in what directions the curve extends to infinity. Find also on which side of each asymptote the curve lies (Art. 260). If there be a parabolic branch it is useful to obtain a para- bolic asymptote and to ascertain on which side of this parabola the curve lies (Art. 263). 4. If the curve pass through the origin, equate to zero the terms of lowest degree. These terms will give the tangent or tangents at the origin (Art. 291), and thus tell the direction in which the curve passes through the origin. A more complete method of finding the shape of the curve near to and at a great distance from the origin is to follow in Art. 382. 5. If there be a node, cusp, or conjugate point at the origin, or a multiple point of higher order than the second, take note of the fact. If there be a cusp*, test its species (Art. 295). 6. Find what other multiple points the curve has (Art. 294), and ascertain the position and character of each. 7. Find -y-; and for what points it vanishes or becomes infinite. These results will indicate the points at which the tangent is parallel or perpendicular to the axis of x. The 316 CHAPTER XII. direction of the tangent at other points may also be ascertained if desirable. 8. Find, if convenient, the points of inflexion. 9. A straight line will cut a curve of the n^^ degree in n points real or imaginary, and imaginary intersections occur in fairs. These facts are often useful in detecting a false notion of the shape of a curve. 10. If we can solve the equation for one of the variables, say y, in terms of the other, x, it will be frequently found that radicals occur in the solution, and that the range of admissible values of ic which give real values for y is thereby limited. The existence of loops upon a curve is frequently detected thus. 11. It sometimes happens that the equation is much simpli- fied upon reduction to the polar form. This is especially the case when the origin is a multiple point on the curve. (a) y=x Straight Line (e) y ^^S Inflexion at (i) y'=:x* Two Parabolas V. y (b) y=x'' Parabola r V (f) y'=x Parabola (J) /-- Cusp atO (c) y=x^ Cubical Parabola Inflexion at O (g) y^x^ Pair ofSt.L ines (k) yi^x Cubical Parabola Inflexio7i at Fig. 72. (d) y=x* Undulation at O (h) y'-=x3 Semicubical Parabola Cusp at O (I) Semicubical Parabola Cusp at O 378. It is not necessary of course in every case to take all the steps indicated above, or to keep to the order laid down, but the student is advised in any curve he may attempt to CURVE TRACING. 317 trace to note down the result of each investigation he may make. For instance, he should remark, the absence just as much as the existence of symmetry, asymptotes, or singular points, and the total information gained will generally be sufficient to give a tolerably good diagram of the curve. 379. It will be useful to be able to draw at once a graph of any of the cases which come under the head Accordingly the student should carefully consider the figures of diagram 72 and verify the drawing in each case. 880. We add a few examples to illustrate the points enum- erated. Fig. 73. I. To trace the curve y = {x—\){x- 2X^ - 3). (a) This curve is not symmetrical about either axis ; but if the origin be transferred to the point (2, 0) the equation becomes y=.r(^-l), showing symmetry in opposite quadrants when referred to the new axes, and that the tangent at the new origin is inclined at an angle 135° to the axis of X. 318 CHAPTER XII. (y8) Eecurring to the original equation, If y=0, ^=1, 2, or 3; If ^=0, y=-6; If 07=00, 2/=®; If x= — 00 , y = — 00 . When X is >3 y is positive, ^<3 but >2 y is negative, x<'2 but >1 y is positive, x<\ y is negative, (y) The curve does not go through the origin, and, although extending to infinity, it has no rectilineal asymptote. (S) Since y=^c(?-^x^ + \\x-% we have which vanishes when dx 3^2_i2^+ll, 2± V3 (e) Also -y4= 6(^-2), which shows that there is a point of inflexion at dx^ the point where ^=2. The shape of the curve is therefore that shown in Fig. 73. Y /" o Fig. 74. X \ TT m ^ w ^(x-afslx — b II. To trace the curve y= ±.^ -'- a^ Case 1. Suppose a>h (Fig. 74). (a) The curve is symmetrical with regard to the axis of x. ()8) While xa, y is positive. When X lies between a and - a, y is negative. When ^ is <—a,y is positive until x passes the negative root above referred to, and then is negative afterwards. Fig. 76. (8) The asymptotes parallel to the axes are x=±.a. To find the oblique asymptote we have X x^ ft , a , a^\f-, , a^ , \ or '=x + a+ — + .., . X CURVE TRACING. 321 Hence y =^+ a is the oblique asymptote, and, if x be positive, the ordinate of the curve is obviously greater than that of the asymptote, and the curv^e lies above the oblique asymptote. If x be negative, the curve lies below it. X V dy _x{a^ — 2a^x — 4:0?) which gives -j^ = ^i when ;r = or when x^ -3a^x-4a^=^0j which clearly dx lias a positive root lying between x = 2a and x='ia, and which can be shown to have only this one real root. Also, -^ = 00 only when x=±a. {() A point of inflexion lies between x=—5a and x=—Qa (Ex. 13, p. 248). The shape is therefore that given in Fig, 76. IV. To trace the curve y^ + 2x^y+x'^ = 0. (a) The curve is not symmetrical about either axis and there are no asymptotes. (^) The curve passes through the origin, but cuts neither axis again, (y) There is a cusp at the origin, the equation of the tangent being v=0. Fig. 77. Proceeding according to Art. 295 the quadratic for P is P^' + 2Pj^+x'^=0, an equation whose roots are real if x be very small, positive or negative ; for the criterion for real roots is that x^ - x"^ should be >0. This condition is fulfilled until ^ is >1, when P or y becomes imaginary. Moreover, the product of the roots =a;^ and is positive or negative according as x is positive or negative. There is therefore a double cusp at the origin, and on the positive side of the axis of y it is of the second species, wliile on the negative side it is of the first species. The point is therefore a point of oscul-inflexion (Fig. 50). OFTHc UNIVERSITY 322 CHAPTER XII. (5) so that _/ = Qo if X- dx y= -x^+.x^\^l -Xj Also, one value of -^ is zero when x= — dx 49 The shape of the curve is now readily seen to be that shown in Fig. 77. 381. The following curve illustrates a particular artifice which may be occasionally employed, namely to express the ordinate of the curve as the sum or difference of the ordinates of two known or easily traceable curves. Fig. 78. V. To trace Here {x^ -hf - 2axy = Aax^la - x). y- = ^ax — x^it '2.sJ axsJ 'Hax — x^+ax = {J2ax — x^ ± sfaxf ; y— ±.ij'2.ax — x^±sfax^ '=±yi±'i therefore or where yi and 3/2 are corresponding ordinates of the circle x'^-\-y^='2ax and of the parabola y"^ = ax. Hence the ordinate of the curve is the sum or difference of the corresponding ordinates of these curves. The circle and the parabola are shown by dotted lines in the accompanying figure, and the resultant curve by the continuous line. Examples. 1. Trace the curve {x+y + \f={i- x)\ showing that there is a cusp of the first species at (1, — 2) ; also that all chords parallel to the axis of y are bisected by the line x+y+l = 0. 2. Trace the curve f =asec ^±acos ^, the radius vector being the sum or difference of the radii vectores of a straight line and a circle. CURVE TRACING. 323 /S82.. li'ewton's Diagram of Squares. ^When a curve whose equation is algebraic and rational passes through the origin, it is frequently desirable to ascertain the shape of the curve in the immediate neighbourhood of the origin more accurately than can be predicted from a mere knowledge of the direction of the tangents, and also to form some idea of the limiting form of the curve at a great distance from the origin. The following is a graphical method of determining what terms of an equation are to be retained or rejected in such cases : — Let AxPy'iy Bx^y' be any two terms of the equation of the curve ; and let us suppose them to be such that they are of the same order of magnitude. Take a pair of co-ordinate axes and mark down the positions of the points (p, q) (?', s), which we shall call P and R respectively. Then, since x^y^ and x^y' are of the same order of magnitude, xp~^ and y*"' are also of the same order, and therefore the order of aj is that of yP~^. S ■"" o Now — ^=tan*0, where is the ansfle which the line PR r—p ^ makes with OX. So that the order of x is that of y~*^^y and therefore the order of the term AxPy^ is that of y^ -p ^"^ ^. Now Y M ^ ^ ^ yA r^ ^ B ^ ^ ^ ^ R ^ ^ A ^ ^ P ^ ^ i T O X Fig. 79. ^f— j? tan = the intercept OA made by the line PR upon OY, so that the order of the terms Ax^y^ and Bx^y^ is that of y^^ and is measured by the intercept OA, 324 CHAPTER XII. Consider next any other term Cx'^y^ in the equation. Let its graphical point (m, n) be denoted by M in the figure. Then the order of this term is that of y n-m tan I or y^^, the line MB being drawn parallel to RP, cutting off the inter- cept OB on the axis of y. OB therefore graphically marks the order of this term, which may therefore be rejected in tracing near the origin in comparison with the terms denoted by the points P and R if OB be greater than OA ; and in tracing the curve at a great distance from the origin it may be rejected if OB be less than OA. Thus if all the terms of the equation be represented graphically by the series of points P, Q, R, S ... in the manner above described, and if when any two, say P and R, are chosen all the other points lie on the side of the line PR remote from the origin, they may all be rejected in tracing the portion of the curve in the immediate proximity of the origin ; but if they all lie on the origin side of the line PR they may all be rejected in tracing the curve at an infinite distance from the origin. Ex. If the equation be ^2y3 _|. 2jc1/ + 3^V + ^ V +3/-' = 0, the points A, B, C, Z>, B represent the 1st, 2nd, etc., terms respectively, Y L E Al D B C O X Fig. 80. and a glance at the diagram will show that the second and third "^ and the second and fifth / aie pairs which may be taken together in tracing near the origin, whilst tlie first and thirds and the first and fifth / are pairs which may be tiaken together in approximating to the form of the curve at an infinite distance from the origin. CURVE TRACING. 325 383. The above method is a modification of the one adopted in such cases by Newton, and is known as Newton's Parallelo- gram. A further slight variation on the same method is due to De Gua, and is known as De Gua's Analytical Triangle. [De Gua's Usage de V Analyse de Descartes, Paris, 1740.] VI. To trace a^+^^-5a^x^i/ = 0. (a) Newton's diagram shows at once that near the origin the first and third of these terms, or the second and third, may be taken together, Fig. 81. whilst at a great distance froin the origin the first and second may be taken together. This indicates that at the origin the curve assumes the parabolic forms y^=±ax^Jby x^ = ba^y^ and that at infinity it approximates to the straight line :p+y=0, which is obviously the only asymptote. {fi) Moreover, the equation may be written when, in the limit y= -^=a very large quantity. 326 CHAPTER XII. Hence again y = - ^ is an asymptote, but we gain the additional informa- tion that if X' be negative and very large the ordinate of the curve is greater than the ordinate of the asymptote. (y) Since when the signs of x and y are both changed the equation remains of the same form there is symmetry in opposite quadrants. (8) Since f^-, Art. 202. This will indicate the direction of the tangent at any point. The length of the polar subtangent is often useful, Art. 203. 4. Examine whether any values of 6 exist which give an infinite value of r. If so, find whether the curve has asymp- totes in such directions (Art. 265) and find their equations. CURVE TRACING. 327 5. Examine whether there be an asymptotic circle (Art. 270). 6. Find the positions of the points of inflexion (Art. 283). 7. It will frequently be obvious from the equation of the curve that the values of r or are confined between certain limits. If such exist they should be ascertained. E.g. J iir=a sin w^, it is clear that r must lie in magnitude between the limits and a, and the curve lie wholly within the circle r — a. 8. It may 1 '• useful to know too whether the curve is convex or concave to me pole at certain points. This can be tested by Art. 282. 385. Curves of the Class r = a sin u6. These curves were called Rhodoneae by the Abb^ Grandi from a fancied resemblance to rose-petals.* VII. To trace r=a sin 50. (a) "We have the following table of corresponding values of r and : — Values of t-H TT To H-l 10 Is II t-H 4ir 10 it il 1— 1 Corresponding Values of r Pos. and Incr. a Pos. and Deer. Neg. -a Neg. Pos. Values of 57r 10 67r 10 77r 10 Stt 10 etc. Corresponding Values of r a Pos. Neg. — a Neg. etc. (j8) r is never greater than a, and there is no asymptote. (y) tan ^ = ^ tan 5^, and therefore vanishes whenever r vanishes and = 00 whenever r='^a. The curve therefore consists of a series of similar loops as shown in Fig. 82, all being arranged symmetrically about the origin and lying entirely within a circle whose centre is at the pole and radius a. Any other curve of the class r = a sin nO may be traced in a similar manner. * Flores Oeometrici and Phil. Trans, for 1723, referred to by D. F. Gregory, Examples, p. 185. 328 CHAPTER XII. Fig. 82. We annex a figure of the curve r = a sin 60 (fig. 83). It will be noticed for this class of curves that if n he odd there are n loops, whilst if n be even there are 2n loops. This will 9 4 Fig. 83. be easily seen from the order of d^escription of the loops, which we have denoted by the numerals 1, 2, 3 ..., in the figures. 387. Curves of the class r sin n6 = a CURVE TEACING. .329 belong to a group of curves known as Cotes's Spirals and are inverse to the above species. Their forms are therefore obvious, going to 00 along an asymptote whenever the radius of the companion curve r = a sin nO vanishes, and touching r = a sin nd Fig. 84. at the extremity of each loop. Since the polar subtangents corresponding to the values of for which r becomes infinite (viz. n6 = Kir) are given by de _ a du n cos KIT the asymptotes are not radial but can at once be drawn. We give in illustration a tracing of the curves T = a sin 4^, a — r sin 4^, with the asymptotes of the latter in one figure (Fig. 84). 330 CHAPTER XII. 388. Class r"" = a"" cos 7i0. The class of curves of which is the type, embraces, as has been previously noticed, several important and well known curves. For instance, we get Bernoulli's lemniscate (n=2), the circle (7^ = l), the cardioide (n = J), the parabola {oi= —J), the straight line (n= —1), the rectangular hyperbola {n= — 2). VIII. To trace r'^ = 0^008 20 {Bernoulli's Lemniscate). (a) Negative values of cos 2$ give imaginary values of r. Hence the only real portions of the curve lie in the two quadrants bounded by e ^ and (9=+^, and by 6 = ^ and 6=^- and r=0 when ^=±- or — or — > 4 4 4 = ±a when 0=0 or tt. Fig. 85. (y) Since the only power of r occurring is even, the curve is symmetrical about the origin. Again, since the equation is unaltered by writing - for 0, the curve is obviously symmetrical about the initial line. Also, r increases from ^= - — to and decreases again from ^=0 to ^ and is nowhere infinite or in fact greater than a. The curve therefore consists of two similar loops as shown in Fig. 85. Other curves of this species may be treated in a similar manner. It will be easily seen that if n be fractional ( = ), the curve will have p portions arranged symmetrically about the origin. 3 S 3 For example, in the curve r" — a" cos _0 o we have the following scheme of values for r and : — 57r 6 IOtt 6 IStt 6 207r 6 257r 6 SOtt 6 etc. r a — a a — a etc. CURVE TRACING. 331 whence we obtain a figure with three equal loops, the whole lying within a circle whose radius is a and centre at the origin (Fig. 86). Fig. 86. Examples. 1. Trace the curves r=a cos 2^, r cos 2^= a, r=acos3^, r=a cos 4^. 2. Trace i^=a?(^o^W, r3cos3^=a', r' = a^cos \ 9, r^cos ^6 = ci"', r* = a*cos 1 6 J r*cos ^6=a^. 3. Trace the curve y2(^2 + «2) =, ^2(0^2 _ -^2)^ j-j^ q g^ ^ 1885. ] Show that the abscissa corresponding to any given central radius vector is equal to the corresponding radius vector in Bernoulli's Lemniscate, and hence that the curve consists of two loops passing through the origin and resembling those of the Lemniscate. ad IX. To trace i+e (a) By giving a set of values to we have the following table : — Values of 6 in Circular Measure 00 4 3 2 1 1 2 1 4 1 4 1 2 Values of r a 4a 5 3a 4 2a 3 a 2 a 3 a a 3 -a Values of 9 in Circular Measure 3 4 -1 5 4 4 3 3 2 -2 -3 -4 -10 — 00 Values of r -3a 00 5a 4a 3a 2a 3a 2 4a T 10a 9 a 332 CHAPTER XII. {jS) Since we may write the equation a r= J, when 6 becomes very large, either positively or negatively, the form of the curve approximates to that of an asymptotic circle r^a, which it approaches both from within and without. (y) Art. 265 shows that r sin(^+l)+a=0 is an asymptote to the curve. This line touches the asymptotic circle and is shown by the dotted straight line in the figure. Fig. 87. (8) The points of inflexion (Art. 283) are given by the equation ^•' + (92 + 2=0, an equation which has one real root which lies between ^= -1 and ^= — 2. The curve is therefore that shown in Fig. 87. Examples. 1. Trace .=^-J-, showing that it lies entirely within the circle r = a, which is an asymptotic circle ; also, that there is a cusp of the first species at the origin. 2. Trace ''^0^' .Show that there are two linear asymptotes and an asymptotic circle ; also a cusp of the first species at the origin and a point of inflexion when 6^=^. EXAMPLES. 1. Show that the curve y'^ = x^~- -„ a^ - x^ consists of two branches each passing through the origin and extend- CURVE TRACING. 333 ing to infinity, and that the whole curve is contained between two asymptotes parallel to the axis of y. 2. Show that the curve y- = oj^l— _ X- -a- has two infinite branches passing through the origin and lying between the asymptotes a;= ±a, and that there are in addition two other infinite branches resembling those of the hyperbola 3. Show that the curve dt?-\-y^ = d^ consists of one infinite branch running to the asymptote cc + 2/ = at each end and cutting the axes at right angles at the points (a, 0), (0, a) at which there are points of inflexion. 4. Show that the curve 7? + y^ = Boxy consists of one infinite branch running to the asymptote x + y + a = at each end and lying on the upper side of that line. Also, that the axes of co-ordinates are tangents at the origin, and that there is a loop in the first quadrant. This curve is called the Folium of Descartes. 5. Trace the curves (a) 7? + y^ = a^x. ^ (P) x^ + f = 2aa?. (y) ^2/^ = ^("^ - ^)« 6. Show that the curve has a cusp of the first species at the origin and an asymptote x-\-y^a cutting the curve at ( ^, ^Y Trace the curve. 7. Trace the curves (a) ay"^ - ^axy + a^ = 0, {fi) y^^axy^-h7? = ^, a and 6 both being positive quantities. 8. Trace xy'^ = ^a\1a - x). (The Witch.) 9. Trace the curve y'^(^a-x)=Qi?. (Cissoid of Diodes.) 10. Trace (^ = ^I:3l, \x + a/ a; + 2a and show that the oblique asymptote cuts the curve at an angle tan-18 334 CHAPTER XII. 11. Trace 2x{a? + y^) = a{2x^ + y'^) and find by polars the co-ordinates of the points of inflexion. 1 2. Trace y{a^ + x^) = a^ic, showing that there are points of contrary flexure where a; = or ±aj3y that the tangent is parallel to the axis of x where x=±aj and that the axis of x is an asymptote. 13. Trace x^y^^^a^ix^-y^), showing that the curve lies entirely between its asymptotes y= ia, and that its tangents at the origin are y= ±x. U. Trace the curv^e {x^ - a^)(y^ - 62) = ^252^ 15. Trace x^ = a%x^-y^). 16. Trace (y^ -a^f = a^{x^-2a^). 17. Trace axy = a^-a\ (The Trident.) 1 8. Trace the curve x* - 2mx'^y'^ + y^ = a* when m is respectively greater than, equal to, and less than unity, and also when m is zero. [London, 1880.] 19. Trace y^:=x^^^±^. x-a 20. Trace y' = «j'^^^. x-^ — a^ 2 1 . Trace x{x -\-yf = a(x- yf. [I. C. S. , 1879. ] 22. Trace x? = y{x-af. [Oxford, 1876.] 23. Trace f-^Y = ^-^' [H. C. S., 1881.] \y-aj y + a 24. Find the multiple points on the curve 2(.x4 + 2/4) + 5a;y 4- 4^* = 6a2(a;2 + y2^ and the directions of the tangents at those points. [H. C. S., 1S81.] Also trace the curve. 25. Trace the curve ic^ + 2/^ + 36^2/ = a^, and prove that as c diminishes to a the ultimate form of the loop is that of an ellipse whose eccentricity = ^|. [Math. Tripos.] 26. Trace {x-y)\x + y){2x + y)=-a^y^. [Camb., 1879.] 27. Trace the curve r = a(l+cos^). (Cardioide.) CURVE TRACING. 335 28. Trace r = a + b cos 6. (The Lima9on of Pascal.) 29. Trace r = a(2 cos 6'± 1). (The Trisectrix.) 30. Trace the following spirals : — (a) r = aO. (Spiral of Archimedes.) (P) rd = a. (The Hyperbolic or Reciprocal Spiral.) (y) rW = a^. (The Lituus.) (S) r = ae'^^. (The Logarithmic or Equiangular Spiral.) Show that in each case there is an infinite number of convolutions round the pole, and that r sin ^ = a is an asymptote to (P) and the initial line an asymptote to (7). 31. Trace the curves r = a cos 5^, r cos 5^ = a, r = «cos^^. 32. Trace the curves r§ = ascos|^, rs = assec§^, r^ = a^cos,^d. What is the relation between them ? [Camb., 1876.] 33. Trace the curve = , r-a showing that a line parallel to the initial line at a distance a above it is an asymptote. Show also that there is an asymptotic circle r = a. 34. Trace ^ = «7^ -• — .,• ^ - sm 6^ Show that this curve has an asymptotic circle ; also that as each branch of the curve comes from infinity it approaches the asymptotic circle from the outside on one side of the initial line and from the inside upon the other. 35. Trace r=2a!i^ (The Cissoid) cos 6 from the polar equation 36. Trace d~a 37. Trace r^^^^an^, from ^ = to e = 27r. [Oxford, 1876.] 38. Trace r^sin 3(^ - a) = sin - sin a. [Camb., 1879.] 39. Trace the " curve of sines " y = bsm -' a 40. Trace y = e~^' tan fxx. 41. Trace ^ = 77. — ? (7^ — I for positive values of 0. [Trtn. Coll. Camb., 1873.] 336 42. Trace 43. Trace 44. Trace 45. Trace 46. Trace 47. Trace 48. Trace or 49. Trace examining the cases CHAPTER XII. 1- sin 2(9 (x + aY{y - a) + {y + af{;x - a) = 0. {x - a)[x — b) x = a{\ - cos 6)^ y = ad ]' (The companion to the Cycloid.) y^c cosh -. (The Catenary.) c y -X^ + cosh X. xy = {a + yf{b^-y^), r = a cosec dH^h. (The Conchoid of Nicomedes.) {2/2 + (a + xf}{y^ +{a-xf}= a%\ [Oxford.] [Oxford.] (1) ab. (Cassini's Ovals.) 50. Trace y* + x^ ^- 2y^ - ar^ ■= 0. [Cramer.] 51. Trace r = a(cos a cos ^ - J cos 3a cos SO + l cos 5a cos 50 - ...). [Math. Tripos, 1878.] 52. Trace y = e'^. (The Probability Curve.) 53. Trace the curves (a) 2/4 - axy- + a^ = 0. (/?) ay-2abx'^y-x^ = 0. (y) y^ + ax:^~b^xy^ = 0. , [Cramer.] 54. Trace x^ - ax^y + by^ = 0. [DeGua.] 55. Trace (a) x^ + y^ = 2a^xy. (fS) x^ + y'^ = xy{a^x + bhj). [Frost.] CHAPTER XIII. ON SOME WELL-KNOWN CURVES. 389. The present chapter is devoted to a short description of some special curves whose properties have been investigated and which have acquired historical importance, being associated for the most part with the names of some of the greatest Geo- meters of past ages. It has been considered advisable to intro- duce at this point an enumeration of their principal properties for the sake of reference, though unnecessary to give in all cases full proofs of the results stated as the student will be readily able to supply them. In some cases several of these properties will be found to have been already proved or sug- gested for proof for the student in earlier pages. The Cycloid. 390. This curve appears to have been discovered in the fifteenth century, and is associated with the names of Galileo, Descartes, Wren, Pascal, Huyghens, and many others. It derives its principal interest from its importance in Mechanics. 391. Def. When a circle rolls in a plane along a given straight line, the locus traced out by any point on the circum- ference of the rolling circle is called a cycloid. 392. Description of the Curve. The nature of the motion shows that there is an infinite number of cusps arranged at equal distances along the given straight line. It is usual to confine the name cycloid to the portion of the curve lying between two consecutive cusps. E.D.C. . Y 337 338 CHAPTER XIII. Let A, B he two consecutive cusps, ACB the arc of the cycloid lying between them. The line AB along which the circle rolls is called the base. Let GPT be the rolling circle, G the point of contact, GT the diameter through G, and P the point attached to the circumference, which by its motion traces the cycloid. The circle GPT is called the generating circle. Let G be the point of the curve at greatest distance from AB ; this point is called the vertex. Let CX be the tangent at C, and GY the normal, obviousl}^ bisecting the base AB in the point D. We shall take these lines as co-ordinate axes. It is clear that the curve is symmetrical about GY. 393. Tangent and Normal. Since a circle may be considered as the limit of an inscribed regular polygon with an indefinitely large number of sides, the circle GPT may be supposed to be for the instant turning about an angular point of this polygon situated at G. Hence the motion of the point P is instantaneously perpendicular to the line PG, which is therefore the direction of the normal at P. Moreover, since this motion is in the direction of PT, PT is the tangent at P to the locus of P. 394. Equations of the Cycloid. Let DQG be the circle described upon BG for diameter and let be its centre. Draw PM, PiV perpendicular to GX and OF respectively, the latter cutting the circle DQG in Q. Join DQ, GQ, GQ. Now, since the circle rolls without sliding along the line AB, every point of the circle comes successively into contact with :} ^'^ ON SOME WELL-KNOWN CURVES. 339 the straight line, so that the length of AD is half of the circum- ference of the circle, and the portion GA=SircGP = a.rcI)Q. Hence the remainder DG = arc GQ. Now, PQGT, PQDG are parallelograms ; whence, if a be the radius of the generating circle and the angle GOQ, PQ = DG = arc CQ = ae. Hence, if x, y be co-ordinates of P, x = GM=NQ-\-QP = a{e-\-^me)] y=:GN = GO-NO=a(l-cose). From these equations the Cartesian equation may be at once obtained by eliminating ; the result being x = ayers~^--\-\/2ay — y^, (b) but from the form of the result the equation is not so useful as the two equations marked (a). 395. Length of the arc OP. Since x = a{6+ sin 6)} y = a{l — cosO)j' we obtain dx=a{l+ cos 6)d0^ dy — a&inOdO y squaring and adding, ds^ = dx^ -\- dy^ = 2a\l -f cos 6)d6'^ = 4ia^cos^pie\ or cZs = 2acos^c?0, and upon integration s = 4asin- (c) the constant of integration vanishing if s be measured from (7, so that s and Q vanish together. Again, since chord (7Q = 2a sin -, we have arc CP = 2 chord GQ (d) sin^-, 7^ = V8^. (E) Further, since y='la sm\ « = -ia. 340 CHAPTER XIII. 896. Geometrical Proofs. These results may be established by geometry as follows : — Let TPG be any position of the generating circle, Q being the point of contact, GT the diameter through G, and P the tracing point. Let the circle roll through an infinitesimal distance till the point of contact comes to G'. Let the circle in rolling turn through an infinitesimal angle equal to POQ, OQ being a radius of the circle, and let P come to P\ Then QP' is parallel and equal to GG\ and therefore to the arc QP. PP' is ultimately the tangent at P and therefore ultimately in a straight line with TP. Draw Qn at right angles to PP'; then Tn and TQ are ultimately equal, and Pn is therefore the increase in the chord TP in rolling from G to G\ Moreover PP' is ultimately the increase of arc, and since in the limit QP' = arc QP = chord QP, and Qn is drawn perpendicularly to PP\ n is the middle point of PP\ and therefore the rate of growth of the arc OP is double that of the chord TP, and they begin their growth together at C. Hence arc OP = 2 chord TP. 397. Intrinsic Equation. If in Fig. 88 PTX = \lr, we have \^ = | ; whence the intrinsic equation of the cycloid is s = 4a sin i/r. t.e 398. Radius of Curvature. The formula of Art. 322 gives p = -T-, = 4a cos yj/' = ^a cos - = 2PG, radius of curvature = 2 . normal. ON SOME WELL-KNOWN CURVES. 341 399. Evolute. By Art. 347 the intrinsic equation of the evolute of the curve s=/(^) is s=f(\j/). Applying this, we have for the evolute of the above cycloid s = 4a cos \[r, which clearly represents an equal cycloid (see Art. 349). 400. Geometrical Proofs. These results may also be established geometrically as follows : — Let AD he half the base and GB the axis of a given cycloid APG. Produce CD to F, making Di^ equal to CD, and through F draw FF parallel to DA. Through any point G on the base draw TGG' parallel to CD and cutting the tangent at C in T and the line FE in G\ On GT and GV as diameters describe circles, the former cutting the cycloid in the tracing point P. Join PT, PG and produce PG to meet the circle GPV in P' and join PV\ Then obviously the arc G'P'= arc PT=DG = FG\ and therefore the point P' lies on a cycloid, equal to the original cycloid, with cusp at F and vertex at A. Moreover P'G is a tangent to this cycloid and P'G' a normal. The cycloid FA is therefore the envelope of the normals of the cycloid ^(7 and therefore its evolute; and P' is the centre of curvature corresponding to the point P on the original cycloid. 342 CHAPTER XIII. If, therefore, a string of length equal to the arc FP'A have one extremity attached to a fixed point at F the other end, when the string is unwound from the curve FP'A, will trace out the cycloidal arc APG. Thus a heavy particle may be made to oscillate along a cycloidal arc, by allowing the sus- pending string to wrap alternately upon two rigid cycloidal cheeks such as FA, FB. Moreover, since PP' is obviously by its construction bisected at G, the radius of curvature at any point of a cycloid is double the length of the normal. 401. Area bounded by the Cycloid and its Base. Let PGP\ QG'Qi be two contiguous normals. Then Gy G' are their middle points, and therefore ultimately the element- ary area GPQG' is treble the elementary area P'GG'Q'. Hence, summing all such elements, the area APOD is treble the area F Fig. 91. ADFP'; i.e., the area of the cycloid is three-fourths of the circumscribing rectangle, for the area of ADFP' is equal to the area GXAP. Now the length of AB = half the circumference of the circle = ira. Hence the rectangle AXGD = '7ra.2a = ^ira^, and therefore the semicycloidal area APGD = | . 2ira^ = f Tra^ and the area bounded by the whole cycloid and its base = STrct^ and is therefore three times the area of the generating circle. . ON SOME WELL-KNOWN CURVES. 343 The Trochoids. 402. If the point P (in Art. 392) be attached to the rolling circle at a point not upon the circumference, but at a distance b from the centre, the curve traced is called a curtate or a pro- late cycloid according as b is greater or less than the radius a. These curves as a class are called Trochoids. It will be obvious from the mode of description that if 6>a the series of cusps which characterize the ordinary cycloid are replaced by a series of nodes and loops. 403. The equations of a trochoid referred to the same axes as the cycloid in Art. 394 will obviously be x = aO + b sin 0\ y = a — b cos J Epi- and Hypo-cycloids and Epi- and Hypo-Trochoids. 404. When a circle rolls without sliding upon the circum- ference of a fixed circle, the path of a point attached to the circumference of the rolling circle is called an epi- or a hypo- cycloid according as the moving circle rolls upon the exterior or the interior of the other. The path of any other carried point is called an epi- or a hypo-trochoid. Fig. 92. 405. The figure (92) represents the three-cusped epi- and 344 CHAPTER XIII. hypo-cycloids formed wlien the ratio of the radius of the rolling circle to that of the fixed one is 1 : 3. 406. Let the radii be respectively b and a. In the figure the rolling circle with its carried point P is represented as tracing the epi-cycloid. Let be the fixed centre, Q the point of contact, J. the point with which P is originally in contact, G the centre of the moving circle. Join 0(7, cutting the rolling circle in D. Join QP, GP, and DP, the latter cutting the initial radius OA, which we choose for ic-axis, in T. Then, as in Art. 893, PQ is the normal and PT the tangent to the path of P. Let QdA = and QGP = ^. Then, since iiYcQP = SircQA, we have b(p = aO. ' Hence CfiP = | = g and ^ = Pl'a>^e + 'l = '^d. (A) 407. Again, GP makes with the ic-axis the angle Hence the equations of the curve are x = (a + 6)cos — h cos , y=^(a-\- 6)sin d — h^in , 408. If the carried point P be not upon the circumference but at a distance mh from G it is plain that the corresponding equations for the epitrochoid will be x = {a+ 6)cos Q — mb cos — y— - [.. (b) y = (a + b)sin — mb sin — -. — 61 409. The path of the carried point when the moving circle rolls upon the interior of the circumference is obtained from equations (a) or (b) respectively by changing the sign of 6. ON SOME WELL-KNOWN CURVES. 345 410. If p be the perpendicular from upon the tangent PT to the epicycloid (Fig. 92) we have p = ODBmf = (a+26)sin^^. This furnishes us with the tangential-polar equation. 411. From the triangle OCP (or otherwise) r^ = {a + bf-\-h^--2(a + h)bcos . = (a + by -\-b^- 2(a + b)b(l - 2 sin^^) ^ =aH4(.+6)6(^J, the pedal equation. 412. Differentiating equations (a) r jn— —{ci'+b)sin 6-{-{a + b)sin r~d, ^1= (a-h6)cose-(a+6)cos'^-^a Hence, squaring, adding, and extracting the root, f^=±2(a+5)sin|5. Hence s = — ' cos ^rO, a 2b s being measured from the vertex, where = 7rb/a. 1 hus 8 = — ^^ cos — T—rrW' a a+^o is the intrinsic equation to the curve. This may also be obtained quickly by applying the formula These results will (as in Art. 409) all remain true for the hypocycloid when the sign of b is changed ; or they may be obtained independently. 413. Thus any epi- or hypo-cycloid may be represented by any of the equations, 2^ = ^ sin Byjj^, or A cos B^p-^ s = J. sin B^lry or A cos Byfry r^ = A+Bp\ the constants A and B being readily determinable in any par- ticular case by aid of the preceding Articles. 346 " CHAPTER XIII. 414 Any of these formulae give the radius of curvature. For example, taking p^Asin B\fr, we have i.e., the radius of curvature varies as the central perpendicular. 415. The e volute of any epi- or hypo-cycloid is a similar epi- or hypo-cycloid. (See Art. 849.) 416. The equations of the tangent and normal at any point on the curve where = a may be written . a+2h a + 2h . .,. . a ■ X sm -25-« - y cos -^^a = (^^ + 2Z>)sin ^a a+26 . . a4-2b a X cos — ^T — a + y sin^ — ^yr— a = a cos ^a 417. The polar equations of the tangent and normal with for pole and OA for initial line are therefore 7' sin(|^ + a - Oj = (a + 26)sin ^a r cosi V 4- a — t^ ) = a cos-^ a If the initial line were chosen to bisect the arc joining two consecutive cusps A, B, we should have to change a to a-\ and to 6'-\ . If this change be made, the equation a a ^ . of the normal becomes . faa which shows by comparison with the tangent that the normal touches another epicycloid formed by the rolling of a circle of radius B upon another of radius A where %.e., A= — ncT.a, B= — -^.o. a+26 a + 26 This also follows from Art. 241 and verifies the result of Art. 349. ON SOME WELL-KNOWN CURVES. 347 418. Double method of generating Hypocycloids. Changing the sign of b in Equations (a) the equations of the hypocycloid are x = (a — 6)cos 6 + hcos — y— 0^ (c) y = (a — 6)sin — 5 sin —7—^ Writing — ^~ for h and 0' for 6, we have a—c a-\-c^, , a-\-c a — c^\ x = —7r- cos^ 6 H — ET-cos u 2 c 2 . c a — c . a-\-c^, a-\-c . a — c^, and it is evident that a change in the sign of c does not alter these equations. It follows therefore that the same hypocycloid can be generated by the rolling of either of the circles whose radii are — ^ upon a circle of radius a. And if we write a+c for 6 and make the same change for as above, the equations of the hypocycloid become x = (a-\- c)cos 6 —c cos 6 , c y={a,-\- c)sin 0' — c sin &. These are the equations of an epicycloid. It appears then • that the hypocycloid formed when the radius of the rolling circle is greater than that of the fixed circle may be regarded as an epicycloid generated by the rolling of a circle whose radius is the difference of the original radii.* This can also be shown geometrically. 419. If the ratio of a:6 be commensurable, there will be a finite number of cusps, the curve returning into itself. The equations x = {a — 6)cos + 6 cos —r—0 y = {a — t))sin — 6 sin — t—Q J Peacock, Examples. Citing Euler, Acta Petrqp.j 1784. 348 CHAPTER XIII. of the hypocycloid become, when 6 = Ja. x = a cos 6\ y=o y indicating that the curve degenerates into a diameter of the fixed circle. This admits of easy geometrical proof. If 6 = la, we have iT = t(3 cos ^ + cos 30) = a cos^O y = ^(3 sin - sin SO) = a sin^O giving x^-{-y^ = a^ th e fou r-cusped hypocycloid. a = b the equations of the epicycloid reduce to X = a(2 cos — cos 20), y = a(2sin0 — sin20), and after elimination of 6 we obtain (^2^2/2-a2)2 = 4a2{(a;-a)2+2/2}. If now the origin be transferred to the point (a, 0) and the resulting equation transformed to polars, it will be apparent that the epicycloid becomes a cardioide (Art. 424). \ The trochoidal curves corresponding to this case become I lima9ons. I? follows from Art. 415 that the e volute of a cardioide is also a cardioide. 420. The portion of the tangent to fl3^ + 2/^ = a^ intercepted between the co-ordinate axes is of constant length. The portion of the tangent of the three-cusped hypocycloid intercepted by the curve itself is of constant length. 421. It may be observed that the envelope of any line whose equation can be thrown into the form X cos a + 2/ sin a = c sin na{ = p), being obtained by the elimination of a between this equation and — X sin a-\-y cos a = nG cos na, has for its pedal equation r^ — c^sin^a + n^c^cos^na or r^ = ( 1 — n^)p^ + n^c^, and is therefore an epi- or hypo-cycloid. ON SOME WELL-KNOWN CURVES. 349 422. The equation of the pedal of this curve is obviously r = csmn6, and therefore the polar reciprocal, which is the inverse of the pedal, is the Cotes's spiral r sin 71^ = constant (Art 454). 423. The student is referred to Dr. Heath's Optics, Arts. 100 to 103, where epicycloids are shown to occur in' certain cases as caustics by reflection from a bright circular arc. The LiMAgoN of Pascal, the Cardioide, and THE TrISECTRIX. 424. Take a circle OQD of which OD { = b) is the diameter and E the centre. Let a straight rod PP" of any length (2a) move in such a manner that its mid-point Q describes the given circle whilst the rod is constrained to pass through a fixed point on the circumference. Its ends trace out the Lima9on. Fig. 93. Obviously this rod can be constrained to move as described above by a simple mechanical contrivance. 350 CHAPTER XIII. Taking OD for initial line let r, 6 be the polar co-ordinates of P, then evidently r = QP+0Q = a+6 cose (1) Similarly OP' — r'^a — h cos d. This however is obtained at once from equation (1) by increasing by tt. Hence P and P' describe the same curve. Evidently also any " focal chord " PP' is of constant length. When a = 6, the limagon is called a cardioide from its heart- like shape. The curve then has a cusp at 0. Other lima9ons are of two classes according as a is > or < h. The outer curve in the figure typifies the class for which a is > 6. The dark curve on OB for diameter is the cardioide. The inner curve is a lima^on for which a is < 6. There is on this class a node at 0. The dotted curve is the circular locus of Q. The point P is shown in the figure as tracing the cardioide. The equations of the particular curves drawn in the figure are r = 2cose, r = l-h2cose, r = 2-|-2cose, r = 3Tf2cosa 425. Considering the motion of the rod the following facts will be clear : — {a) Since Q is moving along the tangent to the circle the instantaneous centre for the motion of the rod must lie somewhere in the normal QER. (h) The motion of the point of the rod which is just passing through can only be in the direction of the rod itself Therefore the instantaneous centre must lie somewhere in a line OE drawn at right angles to the rod. (c) The instantaneous centre must therefore be at P, the point on Q's circular locus which is diametrically opposite to Q. (d) The motion of P and of P' is therefore at right angles to RP and RP' respectively. These lines are therefore the normals at P and P'. (e) Thus in any lima9on the normals at the extremities of any focal chord intersect on a fixed circle. (/) In the case of the cardioide QP = QR = QP' and the normals at P and P' intersect at right angles. (g) The tangents at the ends of the focal chord PP' (of the cardioide) also intersect at right angles, ' the figure TPRP' forming a rectangle. ON SOME WELL-KNOWN CURVES. 351 (h) Also in this case, since RQ if produced passes through T and ^J'=3^i2 = constant, the locus of ortho-tomic tan- gents (or " orth-optic " points) is a circle whose centre is E and radius three times that of the Q-circle. 426. The cardioide and the lima9on may also be generated as an epic3^cloid or an epitrochoid by the rolling of one circle^ upon another of equal radius. (Art. 419.) 427. These curves are also the first positive pedals of a circle with regard to an arbitrary point. Take a circle, centre C and radius a. Let OP be a perpen- dicular from the pole upon the tangent at any point Q. Let 00 = 6, OP = r, POC=e. Draw CR at right angles to OP. Fig. 94. Then r=Oi^ 4- i?P = a + 6 cos a When lies upon the circumference of the circle we have a = h and the pedal becomes a cardioide. 428. The equation 7^ = a + 6cos0 shows that a limacjon is the inverse of a central conic with regard to the focus, and that a cardioide is the inverse of a parabola. 429. For some purposes it is a little more convenient to call the angle POA' — O (Fig. 93), and the equation of the cardioide then becomes r = a(l — cos 6). We have at once . ^ de 1-COS0 ^ tan d) = r^- = — ; — jc— = tan ^» ^ dr sin 2 ^ = -, i.e., 0Pt=\P0A\ 430. This property shows that the curves r = a(l — COS0), r=6(l+cos0). 352 CHAPTER XIII. whose axes are turned in opposite directions cut at right angles for all values of a and h. Thus the " Orthogonal Trajectories " {i.e., curves which cut at right angles) of the system of cardioides obtained by giving a different values in the lirst equation is the system of cardi- oides obtained by giving h different values in the second. This result may be obtained by inversion of the corresponding property for parabolas. 431. The particular limagon shown with a node in Fig. 93, and whose equation is r = 1 + 2 cos Q, is called the Trisectrix. With centre at (Fig. 93) and radius OE describe a circle. Lay off from OE any angle EOS less than four right angles, and let the bounding radius cut the circle (centre 0) at /Sf, and the chord ES cut the lima^on at J. Then it is easy to show that OJ trisects ^S>Sf.* The Cukve of Sines, Harmonic Curve, Companion to the Cycloid. 432. Figure 95 is a graph of the equation y = sin X. Fig. 95. There are points of inflexion whenever the curve cuts the a^-axis; also the curve lies entirely between the lines y=±l. 433. The curve y = msm.x (sometimes referred to as the Harmonic Curve) only differs from the above in that its ordinates are each m times the corresponding ordinates of the Curve of Sines. 434. The companion to the cycloid X = aO, 2/ = a(l — COS0), * Azemar and Gamier, Trisectlon de V Angle, Paris, 1809. Cited by Peacock, Examples, p. 173. ON SOME WELL-KNOWN CURVES. 353 differs from the cycloid in that, instead of producing the abscissa NQ to P (Fig. 88) to make the 'produced part QP = arcaQ, we make NP = arc CQ. The equation may be written . fx irX 3/-6. If a be < by there will be a node at and a loop below the initial line. If « = 5, there will be a cusp at 0. 360 CHAPTER XIII. If a > 6, the curve will be as shown by the dotted lines 3, 3 in the figure. As defined by the Cartesian equation, there would also be in this case a conjugate point at 0. 448. The curve was used for the trisection of an angle, and the insertion of two mean proportionals between two given straight lines. It admits, as shown by Nicomedes, of a simple mechanical construction.* For it is easy to make a mechani- cal contrivance which will constrain the motion of a given rod so as to pass always through a fixed point, whilst a given point of the rod performs a rectilineal path. By what precedes, any other point of the rod describes a conchoid. THE SPIRALS. The Equiangulau Spiral. 449. This curve possesses the characteristic property that the tangent makes a constant angle with the radius vector. T Fig. 102. Let be the pole, PT the tangent at P, OY the perpen- dicular, OT the polar subtangent cutting the normal in G. Let OPT=a. We have the following properties : — (1) p = OY=r sin a, dr (2) /o = r-T-=rcoseca = OP. Hence G is the centre of curvature. *Montucla, Histoire des Math., torn. I., p. 23G, referred to by Peacock ; and Newton, App. to Arith. Univ. ON SOME WELL-KNOWN CURVES. 361 (0) -^ = COS a. as Hence if the arc be measured from the pole, where r = 0, we have r = s cos a. But T = PT cos a. Therefore PT=s. (4) If YY' be the tangent to the first positive pedal curve Y'YO=YPO = a. Hence the first positive pedal is an equal equiangular spiral. Hence also all other pedals are equal spirals. (5) Since PC is a tangent at G to the evolute, and OGP = a, the first, and all other evolutes are equal spirals. (6) From similar considerations the inverse, and the polar reciprocal with regard to the pole are equal spirals. (7) Since -^r- = tan a we have — = cot a . dO, and the polar ^ '' dr r equation is of the form (8) If the spiral roll along a fixed line, the locus of the pole, and also of the centre of curvature of the point of con- tact is a straight line. 450. Of the system of " Parabolic Spirals " r = aQ^ the most remarkable are those for which 71 = 1 (the Archimedean Spiral). n= — 1 (the Hyperbolic or Reciprocal Spiral). 71= —J (the Lituus). The Spiral of Archimedes. 451. The equation of the curve is r = aO. This curve is due to Conon, who however died before he had completed his investigations of its properties. These inves- tigations were continued and completed by Archimedes who published them in a tract on spirals still extant. (1) If a circle of radius a be drawn with centre at the pole any radius vector of the curve is equal to the arc of this circle measured from the initial line to the point in which the radius vector cuts the circle. 362 CHAPTER XIII. (2) We have for this curve V= /4-r-o ; tan = ^ = 0; subtaiigent = ^- (3) The locus of the extremity of the polar subtangent is Fig. 103. For this curve the corresponding locus is = a{e, + '=ff/2U and so on. The n^^ locus thus formed is n+l / n\. These loci thus form a series of " Parabolic Spirals " of ascending order.* (4) The area bounded by any portion and its extreme radii vectores can easily be found by the Integral Calculus. The Reciprocal or Hyperbolic Spiral. 452. The polar equation is rO = a. This curve is the inverse of the Archimedean spiral. The name Hyperbolic is derived from the analogy between the form of its equation and that in Cartesians for a hyperbola referred to its asymptotes. * Peacock, Examples, p. 180. ON SOME WELL-KNOWN CURVES. 363 (1) If a circle be drawn with any radius and centre at the origin, the arc of this circle intercepted between the points where it is cut by the curve and by the initial line is of constant length. (2) We have i&n^ = r^=:r(^-^^= --0. (3) Subtangent t'^-j- = — a = constant. dr The asymptote is at a distance a from the initial line and above it. (4) The pedal equation is The Lituus. 453. The equation to the curve is The initial line is an asymptote. Fig. 105. 364 CHAPTER XIII. If any radius vector OP be taken and a circular sector OP A described bounded by the radius vector and the initial line, its area is Jr^0 = — » and is therefore constant. CoTEs's Spirals. 454. The group of curves included in the formula are called Cotes's Spirals. They occur as the path of a particle projected in any manner under the action of a central force varying as the inverse cube of the distance. There are five varieties. (1) If J5 = 0, — is constant, whence is constant and the curve is an equiangular spiral. (2) If^ = l, wehave giving u=^\/BO, being supposed measured from an initial line drawn parallel to the asymptote. This is the reciprocal spiral. More generally w or (sy=(^-i)-^+^- The right-hand side may be put into one of the forms according to the signs of ^ — 1 and B ; a and n being constants. we have —i . = ndO and u = asinh72d ON SOME WELL-KNOWN CURVES. 355 (4) If (^^y^n\u^-a% we have similarly u = a cosh nO. (5) If (JJ = .V^_..). u = a sin nO. (Art. 387.) Cases (3) and (4) present no difficulty in tracing. Involute of a Circle. 455. If a thread be unwound from a circle, any point of the unwiDding thread traces out an involute of a circle. Let PQ be any position of the thread, P the tracing point. Then PQ is a tangent to the circle and a normal to the involute. Let Fig. 106. be the centre of the circle and a its radius. Then clearly the pedal equation is T^=p^+a^. Also p = PQ = a.TcAQ = axjr, giving ^"^ 2 ' 8 being measured from the point A at which the involute meets the circle, and OA being the initial line. If F be the perpendicular from upon the tangent at P we have OY=a\l^ = a(YOX+'^\ 366 CHAPTER XIII. Hence the first positive pedal is the Archimedean spiral. The polar equation is at once obtainable. For or 0. J'^ — cos^-* The Evolute of a Pakabola. 456. The evolute of y'^=^ax may be shown to be the semi- cubical parabola 27a2/^ = ^{x — ^af. Fig. 107. The cusp is at the point (2a, 0), and the curve cuts the parabola again at a point whose abscissa is 8(X. The tangent to the evolute at this point cuts the parabola again upon the ordinate through the cusp. From points on the right-hand side of the evolute three real normals can be drawn to the parabola. From points on the left side only one real normal can be drawn. The Evolute of an Ellipse. 457. The equation of the evolute of x^/a^ + y^/h^=l has been shown to be (ax)^ + {hyf = (a^ - h'^f. ON SOME WELL-KNOWN CURVES. 367 There is a cusp at each point where the curve meets the co-ordinate axes. From points within the evolute four real normals can be drawn to the conic. From points outside two normals can be drawn. Fig. 108. Normals from the portion of the ellipse marked (1) touch the portion of the evolute marked (1), and the correspondence is similarly denoted by numerals for the other quadrants. The radii of curvature at A and B are respectively - and -j-- Thus Aq^- and BR=^. a (2 7 2\ h )* Cassini's Ovals. 458. Let r and r' be the distances of a moveable point P from two fixed points S and S\ The locus traced out by P when TV = constant ( = 6'^ say) is called an Oval of Cassini. Let SS' = 2a and take SS' for cc-axis and its mid-point for origin. The Cartesian equation is then [(aJ-a)2+2/2][(aJ + a)H2/'] = ?>^ (1) or in Polars (r^ -|- a^y — 4a Vcos^O = ¥, reducing to r^ + a^— 2rVcos 20 = h*. 368 Ifh CHAPTER XIII. a = c/^2 this further reduces to r2 = c2cos20 (2) This species of Cassini's oval is called the Lemniscate of Ber- noulli. This is shown by the thick line in the figure. It is the first positive pedal of a rectangular hyperbola with regard to the centre, and possesses the property that Fig. 109. In equation (1) when 6 is < a the curve consists of two ovals within the loops of the lemniscate. When 6 is > a the curve consists of one oval lying outside the lemniscate. The curve {x^ + y'^Y = ^^^ + ^^2/^ which is the pedal of a central conic with regard to the centre, has a similar shape, and becomes a Bernoulli's lemniscate when the conic is a rectangular hyperbola. Cartesian Ovals. 459. If r and r' be as defined in Art. 458, the loci indicated by the equation lT-\-rrir' = n, are called Cartesian Ovals. This equation in general gives rise to a quartic Cartesian equation. The following cases will be recognized : — If l = Tfh we have r-f-r' = constant ; an ellipse. If Z = — m we have r — r' = constant ; a hyperbola. If 71 = we have rw' — constant ; a circle. ON SOME WELL-KNOWN CURVES. 369 Also since ?-r- + m-7- = it will be evident that in all cases the as as normal divides the angle between r and r' in such manner that the sines of the portions are in the ratio m : I. The student is referred for further information to a chapter on Cartesians in Professor Williamson's Differential Calculus* where several interesting properties are investigated. The Quadratrices of Dinostratus and Tschirnhausen. 460. Let AFAi be a semicircle of which AA^ is a diameter and the centre. Let QiV be an ordinate of a point Q on the circle and P another point so related to Q that the ordinate QN travels at uniform rate from ^ to in the same time that OP rotates uniformly from OA through a right angle. Let OP and NQ intersect in R, then the locus of R is the Quadratrix of Hippias or Dinostratus. Fig. 110. Let N0P = e8i.nd OA = a, then a.vc AP=ae. Also AN ^ Single AOP AO right angle* Hence AN'= or ^ = ^ -, (0 being: the origin). But E.D.C. 2 : tan 6. * Sixth edition. 2a C- 9l ^ ro' ^r A 370 CHAPTER XIII. Hence the Cartesian equation of the locus is 2/ = a^cot— . The form of the equation shows that there is symmetry about the 2/-axis, and the curve may be seen to be as shown in the accompanying figure. 461. This curve if accurately traced could be used for the trisection of an angle. Lay off any angle AOP by a line OP cutting the quadratrix in R. Draw the perpendicular RN to OA. Trisect AN at X, M and erect perpendiculars to AN cutting the curve in X, Y. Then since O O IT the angle ^0Z = ^. o Similarly AOY=^, o and the angle is trisected. 462. Again, since the intercept OB made on the t/-axis is ttx \ J.. , TTX J.. ttx 2a ' 2a 2a Ltx=(^ cot ^r- = Lt cos -p: ; = — , 0.3 2a 2a . irx \ IT IT ^ ^ we could (if the curve could be accurately drawn) measure OE and hence deduce the value of tt. Hence the area of a circle could be found. It is from this property that the curve derives its name. 463. If a parallel to the a;-axis be drawn through P cutting MQ in /Sf, the locus of S is . ^^ . TT a — x y = a sin 6 = a sm -^ Zi a TTX or y = a cos -^i-' ^ 2a This cury^e is called the Quadratrix of Tschirnhausen. APPLICATION TO THE EVALUATION OF SINGULAR FORMS, MAXIMA AND MINIMA VALUES, ETC. CHAPTER XIV. * UNDETERMINED FORMS. 464. In Chap. I. it was explained that a function may involve an independent variable in such a manner that its value for a certain assigned value of the variable cannot be found by a direct substitution of that value. And in such cases the function is said to assume a " Singular" " Undeter- mined,'' ** Illusory" or " Indeterminate " form. 465. It is proposed in the present chapter to consider more fully the method of evaluation of the true limiting values of such quantities when the independent variable is made to approach indefinitely near its assigned value. 466. List of Forms occurring. Several cases are to be considered, viz., when upon substitu- tion of the assigned value of the independent variable, the function reduces to one of the forms %, Oxx, -, 00- X, 0^ QoO, or r. It is frequently easy to treat these cases by algebraical or trigonometrical methods without having recourse to the Differ- ential Calculus, though the latter is required for a general discussion of such forms. By far the most important case to consider is that in which the function takes the form ^ ; for, in the first place, it is the one which most frequently occurs ; and, secondly, any of the other forms may be made to depend upon this one by some special artifice. 373 374 CHAPTER XIV. 467. Algebraical Treatment. , Suppose the function to take the form - when the inde- pendent variable x ultimately coincides with its assigned value a. Put x — a-\-}i and expand both numerator and denomin- ator of the function. It will now become apparent that the reason why both numerator and denominator vanish is that some power of ^ is a common factor of each. This should now be divided out. Finally, put /t = so that x becomes = a, and the true limiting value of the function will be apparent. In the particular case in which x is to become zero the expansion of numerator and denominator in powers of x should be at once proceeded with without any preliminary substitution for x. In the case in which x is to become infinite, put x — -y so ^ y that when x becomes =00,?/ becomes = 0. The method thus explained will be better understood by examining the mode of solution of the following examples. Ex. 1. Find Lh^Q^—1^ X Here numerator and denominator both vanish if x be put equal to 0. We therefore expand a* and 6* by the exponential theorem. Hence ^tx=Q X {l+.rlogea + ^^(log,a)2+...} - {l +^log,6+|^(log,6y^ + ...} =X^.=o{log.a-log,6 + |(logeal^-i^2)+...| = Iog«a-loge6 = log«^- Ex. 2. Fi^d ^'-fc-S + 2- This is of the form - if we put x=\. Therefore we put x=\ + h and expand. We thus obtain J. x"^ -2x^+1 _ J.. ( 1+A7- 2( 1 4-^)^ + 1 *~° (l + 3A + 3A''^+...)-3(l + 2A+A'0+2 UNDETEKMINED FORMS. 375 3A + A2+... — Lth=Q ■3A + ... •3 + A + ... 3 + ... It will be seen from these examples that in the process of expansion it is only necessary in general to retain a few of the lowest powers ofh. Ex.3. Find Lt^=.J^^^^Y' tan X 1 sin x Since cos^c X we have Z«,=o^^^ = l. X 1 Hence the form assumed by i—^-^y is 1* when we put ^=0. Expand sin x and cos x in powers of x. This gives z^*_^)-^=zU^_^^^ ' =X^«=o(l +— + higher powers of xy where Z is a series in ascending powers of x whose first term (and there- fore whose limit when x = 0) is unity. Hence i;(„,(^)?=ii„.{(l +?^^y^}'' = «J, by Art. 20. 1 Ex. 4. Find Lt^^^oi?-*. This expression is of the form 1*. Put \—x=y^ and therefore, if ^ = 1, y=^ \ therefore Limit required =Z^y=o(l -y)y=e~'^ (Art. 20). 1 Ex.5. Ltx^^x{a^-\). This is of the form 00 x 0. Put x=\ y therefore, if ^ = 00 , y = 0? and Limit required = Z^y=o =logea(Art. 21). y 376 CHAPTER XIV. Examples. Find the values of the following limits : — 1. -^U^Oj^ =-• 1^- ^^^=07 i— 0-1 taii--^.r ^ /rt — i IK r^ sin~^x — sinh^ oj-^-l x^ X cos3^-log(l +^)-sin-i'|^ 3. Lu^f-:Z±. 16. X2^.=o . 1. V-1 J? 1 1-7-^ 2sin^ + tanh-^^ — 3^ 4. ujl±^^t:l. ^'- ^''" ? X . „ TO 7-^ ef'Qinx-x-ar f, J. x^+x^-x^-bx + 4: 1^- ^^^=033-rTTr7i — :a" 5. Lt^^i^- — ^+^'log(l-.2;) j:^-x^-x+1 ^ 3 6 Lf ,-«^-2^-4^^ + 9^-4 ^^ ^^ .^e4-sin%2 1. Lt^^^tzfl.. ■ 20. Z^*-^)^. a J. e'+e-'=-2 jL_ 22. Lt^J'^f 9. X. ^^ cos^-log(l+4 10. Z^.^o^^-^^g(^+4 ' "^ ^ 23. x^.=o(^i^y^. ,, 7-, .27-sin.rcos:j7 \ x J 11. -tytx=0 o • x^ . L sin~-^^— .r 24. Z^^.=of^H^^^. 12. Z^,=o--V-^^— ^- - ^\ X J x-'cos ;r 1 25. Ltx=oicoYers x)x. ■,o 7-/ cosh a; — cos :z? ^=° ^sin.^ 26. ZiJ,=|(cosec^)t--^ Application of the Differential Calculus. 468. John Bernoulli* was the first to make use of the pro- cesses of the Differential Calculus in the determination of the true values of functions assuming singular forms. We propose now to discuss each singularity in order. 469. I. Form 5. Consider a curve passing through the origin and defined by the equations x = \p'{t),\ * Acta Eruditorum, 1704. UNDETERMINED FORMS. 377 Let X, y be the co-ordinates of a point P on the curve very- near the origin, and suppose a to be the value of t correspond- o N X Fig. 111. ing to the origin, so that 0(a) = and '»/r(a) = 0. Then ultimately we have .y dy ^ Hence Zf ^ = Lt tan PON= Lt^=o^^ = Ltt=a 0W ylr\t)' and if v^ be not of the form ^ when t takes its assigned value a, we therefore obtain 0'(O , i>(t)_{a) = and yfria) = by supposition when x=^a (and therefore h = 0), we have (p\a-\-eh) _ci>{a) Hence in the limit Lt h=0. xlrXa + e,h) yfrXa) If it should happen that ^'{a) and \[r\a) are both zero, we can, as before, repeat the process of differentiating the numerator and denominator before substitution for x. Ex. 1. Here Lt. sine- 6 -0 ^ <^(6')=sin 6-0, and f{e) = 6>3, which both vanish when 6 vanishes. cf>'{e) = cosO-l, and f'(e) = 3e% and both of these expressions vanish with 0. Differentiating again "(6l)=-sin(9, and V^"(^) = 6(9, and still both expressions vanish with 0. We must therefore differentiate again <^"'(6')= -cos (9, and V^'"(6') = 6, whence <^"'(0)=-l, and •^"'(0)=6 ; ^, sin(9-6>_ 1 e^-e-^ +2 sin 0-40 therefore Ex. 2. £t. =Lt^ = Lt =Lt e^ + e-^ + 2 cos 6^-4 bO^ -2 sin ^=0 206*3 c^ + e-^-2cos^ 0=0 d=Q e^-e~^+2sin(9 120^ / + e ~^+2cos^ 120 Form 0] Form ^] Form ?1 oJ Fonn«] Form 0] 30* 471. The proposition of Art. 469 may also be treated as follows. Let ^(a) = and \[r{a) = 0, and let the _p*^ differential co- efficient of ^(x) and the q^"^ of \lr(x) be the first which do not vanish when x is put equal to a. Then by Taylor's Theorem, putting x = a-{-hy hp-"^ Kp .j>{x) = g, LU=ohP-'i = 0. If ^<^j X4=o/i^"^ =00. so that the limit is 0, ^A, { , or qc , according as p is > , = , or < q. 472. II. Form x x . Let 9!)(a) = and ^a) = 00 , so that (x)xl,(x) = Lt,=a^> W) and since -t/-\= — =0, YW ^ the limit may be supposed to take the form -, and may be treated like Form I. Ex. 1. Lt^ dcot e=Lt^ ^^-^=Lt^ -iws=l. .a .a sin - sm - Ex. 2. Lt:c=»^sin~=Lt^=^—r^ = Lta a f =a. a; L ' - X X 473. III. Form — . Let d)ia) = X , yh-ia) = x , so that —tA takes the form — when X approaches indefinitely near the value a. The artifice adopted in this case is to write 1 _0Oc)_i/.(a;) y^rix) _J._- 380 CHAPTER XIV. Then since , , . = — = 0, and , . = — = 0, we mav consider this as taking the form -, and therefore we may apply the preceding rule. 1 ^\x) Therefore X,.„|| = [i,=.||]x,=„g|. Hence, unless Ltx=.a .\ \ he zero or infinite, we have If, however, Lt^=,,-^~- be zero, then ^, 0(^) + xH^)_ ■ylr{x) and therefore, by the former case (the limit being neither zero nor infinite), =Lt^=a ^^ '^ ' . Hence, subtracting unity from each side, Tf ii^ ■ ^^"=V(a;) Finally, in the case in which ^^^=Y(aj) and therefore by the last case J. ^x) _ J \x) -Tf ^tM- "^^^=V(a))' therefore ^^x=a%zx = Ltx=aj)j:l' This result is therefore proved true in all cases. UNDETERMINED FORMS. 381 474. If any function become infinite for any finite value of the independent variable^ then all its differential coefficients will also become infinite for the same value. * An algebraical function only becomes infinite by the vanishing of some factor in the denominator. Now, the process of differentiating never removes such a factor, but raises it to a higher power in the denominator. Hence all differential coeflBcients of the given function will contain that vanishing factor in the denominator, and will therefore become infinite when such a value is given to the independent variable as will make that factor vanish. It is obvious too that the circular functions which admit of infinite values, viz., tan x, cot ic, sec ic, cosec x, are really frac- tional forms, and become infinite by the vanishing of a sine or cosine in the denominator, and therefore these follow the same rule as the above. The rule is also true for the logarithmic function \og(x — a) 1 when 33 = a, or for the exponential function 6*"** when x = a,b being supposed greater than unity.* 475. From the above remarks it will appear that if (x)-ir{x) = i.(x)\^^^-l}. From this method of writing the expression it is obvious that unless Ltx=afY~{==^ ^^® limit of u becomes \[y(a) x (a quantity which does not vanish) ; and therefore the limit sought is oo . But if LU=a^?~\ — 1, the problem is reduced to the evaluation of an expression which takes the form oo x 0, a form which has been already discussed (II.). Ex. Ltx=Ji - - cot^ ) =Ltx=o- (1 - ^ cot ^) X sm X \ 0/ =-Lt:, /which is of the same sin x+x cos X \ form still J. sin.a7+a;cos.r . 2cos^-^sm^ 477. V. Forms 0^, oo«, 1". Let y = y-^'", u and v being functions of a? ; then loge2/ = 'ylogeU. Now logel = 0, logeoo = 00 , loggO = — 00 ; and therefore when the UNDETERMINED FORMS. 383 expression u^ takes one of the forms 0^, x®, 1*, log 2/ takes the undetermined form x oo . The rule is therefore to take the logarithm and proceed as in Art. 472. Ex. 1. Find Ltx=o^, which takes the wndetermined form 0". \ Lt^^^og^ = Lt^J^ = Lt^^o-^ = Lt,=o( - •^) = 0, whence Lt^^^=^=\. Ex. 2. Find Lt iri&mxy^'. This takes tJieform 1". , T^ 4. A • T* log sin A- J. cotx and Lt -jrtdiXixlos sin x=Lt ir— ^-r — =Lt {x)]^^-\ given that {^)^{^) = '>n. We can write the above in the form . r J_-10(x).;/.(x) Lt,=a[_{^+K^)]^'^] which is clearly 6"* by Art. 20, Chap. I. It will be observed that many examples take this form, such, /tan ic\ ^ for example, as Ltx^^i ) on p. 375, and Exs. 20 to 26 on p. 376. 479. ^ of doubtful value at a Multiple Point. Since :^ = and —- = at any multiple point on the curve u = 0,\t will be apparent that at such a point the value of -^ as derived from the formula dy _ dx d^~~'d^ will be of the undetermined form -. 384 CHAPTER XIV. The rule of Art. 469 may be applied to find the true limiting values of ~ for such cases, but it is generally better to proceed otherwise. If the multiple point be at the origin, the equations of the tangents at that point can be at once written down by inspec- tion and the required values of -^ thus found. If the multiple point be not at the origin, the equation of the curve should be transformed to parallel axes through the mul- tiple point and the problem is then solved as before. Ex. Consider the value of-^- at the origin for the curve ax xf^ + ajp"y + hx'ip' +3/* = 0. The tangents at the origin are obviously ^ = 0, 3/=0, ax-\-hy = ^^ making with the axis of x angles whose tangents are respectively r. a CO, 0, -5, which are therefore the required values of -^• ax EXAMPLES. Investigate the following limiting forms : — log COS X "■ tan^Tra; . ^•^^-£-^Sf2- 5.XWog(2-gcot(.-«). 3. U._. ^-j^^^ . 6. Z^._,l2i^*-^^. ' n-N/2sina; * " V„ _^^„a; 2 cot B tan~^(m tan 9) — m cos^- sin^- 2 8. X^^^o(cos £cy<'*'*. 9. Lt^^^{l - x^y^). 10. Lt^^,{\ogxf^^'-'^\ 11. Lt^_^ —, z — — accordmsr as 71 IS >, =, or < m. ir UNDETERMINED FORMS. 385 12. Lt.x-'^, m being positive. 14. Lt^^Jl±^^^^', '^\\ — cosav 13. X«...(^jy. 15. i^,.„{cot(45°-^)}— . 1^ 1 1 1 16. Z^,_(^il±<±<±:ii±^)" 18. Z^^^i Vl+a--Vl4-a;2 19. Z<_a^sin^- / (i-) If « be >1. a' t(ii.) If a be <1. *=ao" OA 7- / cosec X - cot £c 21. Lt^^,. J a- + ax + x^ - ayJ 1 + - Ids: cos - e « sin 1 + log- - J( 1 - ) 22 x^ ^ g 2\ aj Va-a2-^.75J -^ 23.i...,g-cotg. 24. Z«. v/a^ + ax + x^ - Jd^ -ax + x^ J a + x- J a - X 25^ H _ log(l+a; + a:2)+log(l-a; + a:2) sec X - cos X o^ J ^sin(sin jc) - sin2aj ^_ ^^ n+x)*-e ^b. 2>«^^o 27. Z^o^ -i . 28. LtJ / 24, E.D.C. 2 B 386 CHAPTER XIV. 30. Ti ^ _ (-^ - y)^" + (y - ^*)^" + (^ - ^)y" """"" (^-2/)(2/-«^)(«-a') [Put £c = « + A, y = a + k, and expand in powers of ^ and k, and finally, after reduction, put A = 0, ^ = 0.] 31. Lt^J.2i^i+My. x + y-2 32. Show that generally, if a function of two independent variables take one of the singular forms -, etc., for certain values of the vari- ables, its value is truly indeterminate. 33. Given o(^ + y^ + a^ = 3axy, find the values of -^ when x = y = a. ax 34. Find the values of -^ at the origin for the curve x^ + y^ = 3axy. 35. For the curve x^y^ = (a^ - y^){b + yY find the values of -^ at the point (0, - b). ax 36. For the curve x'^ + ax^y = ay^ find the values of -^ when x = 0. ax nn T^ Ts «* - 1 /^ sin X - sin bxY /by. 37. Prove Lt^^o-^. — ( r- = d M^S ^• * "£c"sin x\ cos X - cos bx J \3/ 38. Prove Z^^^o^jr^ = ^' - ^'» where w = ^^^^ and a; = sin y. cos 2/ 39. Find Lt^ „^^, where y = -^—^ and (9 = cos-i(l -a;). ^=^dx^ sm ^ [I. C. S., 1884.] 40. If 2/ = (sin~ifl3)2, prove that dx"" 41 Prove that Lt^„ — is zero or infinite according as n is greater or less than m, a and b being both greater than unity. 45. Find 7j ^^sin^^-^' 'sinar« tan bx - tan ax UNDETERMINED FORMS. 337 (™\ tan — 2 43. Prove Lt^J^fT^^cotf^ E^\ = ^. {2^a + x) IT 44. Find Lt^^^{coB aa;)~«*'**. /(I) lix^O. \(2) If a = 6. .- 46. FindZ^ x^-\+(x-\)\ 47. yin^ r.f. J^-sJa+Jx-a Jx^-a^ 48. FindZif^^o(cosa;)*'°*^ 49. Prove that if, when x is infinite, (x) = 00 , then will Lt^^^ = Lt{<^{x+\)-4>(x)), and also that Lt^^{{x)Y = L&^^ '^ ^^ (x) [Todhunter's Diff. Calc] 50. Prove that Z<^ 1^1' = , [Todhunter's Diff. Calc] 51. Prove x^^l^^ 2-^ 3-^ ... ^n-^_2_, "^ 71"*+! 7/1+1 being positive. 52. Prove Ltj,^Ji{ar + a + h\^ + a + 2^)'" + . . . + a + (n - I)A)'*} m+ 1 where A= , and a, 6 are any given quantities. CHAPTER XY. • MAXIMA AND MINIMA— ONE INDEPENDENT VARIABLE. 480. Elementary Methods. • Examples frequently occur in algebra and geometry in which it is required to find whether any limitations exist to the admissible values of certain functions for real values of the variable or variables upon which they depend. These investi- gations can sometimes be conducted in an elementary manner. A few examples follow in illustration of this. Ex. 1. The function ^-4^ + 9 may be written ig the form (^-2)^ + 5, from which it is at once apparent that the least admissible value of the expression is 5, the value which it assumes when :r = 2. For the square of a real quantity is essentially positive, and therefore any value of x other than 2 will give a greater value than 5 to the expression considered. Ex. 2j Investigate whether any limitation exists to the values of the expression ^-^-^^ for real values of x. Putting a:2-.r+l we have x-{\—y) — x{\+y)-\-\—y=0^, an equation whose roots are real only when (l+3/)2>4(l-y)2, i.e., when (Sy - 1)(3 -y) is positive ; i.e., when y lies between the values 3 and \. It appears therefore that the given expression always lies in value between 3 and \. Its maximum value is therefore 3 and its minimum ^. 388 MAXIMA AND MINIMA. 389 Ex. 3. If a, 6, c, Xj .y, z be all real quantities such that a^ + b'^ + c^ and ^2^y2^z^ are both given, then ax+b7/ + cz will have its maximum value u X Ij Z when =:-f = -- a c For the identity (a2 + J2 + c2)(^2 +y2 + ^2) = (ot^ + 5^ + C2)2 + (60 - C2/)2 + (cT - 0^)2 + (a^/ - hxf shows that {ax ^hy •{■ czf will have its maximum value when the remaining three squares on the right-hand side have their minimum values. And being squares of real quantities they cannot be negative. Their minimum is therefore when each separately vanishes, which gives the result stated. Ex. 4. To determine geometrically the greatest triangle inscribed in a given ellipse. It is obvious from elementary considerations that if the ellipse be pro- jected ortliogonally into a circle a triangle of maximum area inscribed in the given ellipse must project into a triangle of maximum area inscribed in a circle ; and such a triangle is equilateral and the tangent to the circle at each angular point of the triangle is parallel to the opposite side. This property of parallelism is a projective property, and therefore holds for a maximum triangle inscribed in the given ellipse. T,, Area of a maximum triangle inscribed in the ellipse Moreover -. t-tt- — Area of ellipse Area of equilateral triangle inscribed in a circle ~ Area of the circle 47r Hence the area of the greatest triangle inscribed in an ellipse whose semi- axes are a, o is a" Ex. 5. li A, B, C ... he a number of fixed points and P any other point, and if G be the centroid of masses A at J, ^m at B, etc., then it is a geo- metrical proposition that (ZkPA^) = (2XGA^)+(2X) . PG^ Hence, since HkGA^ is a fixed quantity for all positions of P, H,XPA^ has its minimum value when P is at G. Ex. 6. In any triangle the maximum value of cos A cos B cos C is J. For 2 cos A cos B cos (7=cos^(cos^ — (7 — cos^), and therefore ks long as B and C are unequal we may increase the expres- sion by making them more nearly equal and keeping their sum constant. Thus cos A cos B cos does not attain its maximum value until 3 and then its value ={hy. 390 CHAPTER XV. Examples. 1. Show algebraically that the expression x + ~ cannot lie between 2 and — 2 for real values of ^. Illustrate this geometrically by tracing the hyperbola an/ — X!^ = l. 2. Prove that, if x be real, -^5 — 7^^,. must lie between 5 and i. 3. Show that, if x be real, ^i^ • ^^, cannot lie between the values x—a x—b 4. Show that the triangle of greatest area with given base and vertical angle is isosceles. 5. If v4, ^ be two given points on the same side of a given straight line and P be a point in the line, then AP+BP will be least when ^Pand BP are equally inclined to the straight line. 6. Show that the triangle of least perimeter inscribable in a given triangle is the pedal triangle. 7. Show that the greatest chord passing through a point of intersection of two given circles is that which is drawn parallel to the line joining the centres. 8. Determine the maximum triangle of given species whose sides pass through given points. 9. Find the least isosceles triangle which can be described about an ellipse with its base parallel to one of the axes, and show that it has its sides parallel to those of the greatest isosceles triangle which can be inscribed in the same ellipse with its vertex at one extremity of the other axis. [I. C. S., 1884.] 10. The diagonals of a maximum parallelogram inscribed in an ellipse are conjugate diameters of the ellipse. 11. If the sum of two varying positive quantities be constant, show that their product is greatest when the quantities are equal. Extend this to the case of any number of positive quantities. 12. If a^x^-\-h'^y^ = c\ find the maximum value of xy. [i. c. S., 1889.] 13. If A, B, G be the angular points of a triangle and Pany other point, then AP+BP + CP will be a minimum when each of the angles at P is 120°. [AP is a normal to the ellipse with foci B, (7 and passing through P.] 14. Find a point P within a triangle ABC such that AP^ + BP'^+CP^ has a minimum value. 15. Prove from statical considerations, or otherwise, that if P be a point within a triangle, then ^PHan A+BPh&n B+CP^ta.n C has its minimum value when P is the orthocentre. MAXIMA AND MINIMA. 391 16. If a triangle be inscribed in a circle of given radius i?, show that the maximum value of the sum of the squares of the sides is 9RK 17. If ^+^ = const., the maximum value of sin ^ sin <^ is attained when 18. Show trigonometrically that the greatest and least values of the expression a sin x+b cos x are sfcf+F^ and -'Ja^^bK 19. Show by trigonometry that the greatest and least values of the function a cos^O + 2A sin Ocosd+b sin^^ a+b _ 2 ■ are respectively ■^wr +A2. 20. Find the rectangle of maximum area whose sides pass through the angular points of a given rectangle. 21. PSP", QSQf are focal chords of a conic intersecting at right angles. Find the positions of the chords when PP + QQ has a maximum or mini- mum value. The General Problem. 481. Suppose X to be any independent variable capable of assuming any real value luhatever, and let ^(a;) be any given function of x. Let the curve y = (p{x) be represented in the adjoining figure, and let A, B, C, D, ,.. be those points on the curve at which the tangent is parallel to one of the co-ordinate axes. Fig. 112. Suppose an ordinate to travel from left to right along the axis of x. Then it will be seen that as the ordinate passes such 392 CHAPTER XV. points sisA,G, or E it ceases to increase and begins to decrease ; whilst when it passes through B, D, or F it ceases to decrease and begins to increase. At each of the former set of points the ordinate is said to have a maximum value, whilst at the latter it is said to have a minimum value. 482. Points of Inflexion. On inspection of Fig. 113 it will be at once obvious that at such points of inflexion as (r or iZ, where the tangent is par- allel to one of the co-ordinate axes, there is neither a maximum Fig. 113. nor a minimum ordinate. Near G, for instance, the ordinate increases up to a certain value NG, and then as it passes through G it continues to increase without any prior sensible decrease. This point may however be considered as a combination of two such points as A and B in Fig. 112, the ordinate increasing Nx ^= Fig. 114. up to a certain value N^G^, then decreasing through an inde- finitely small and negligible interval to Nfi^, and then increas- ing again as shown in the magnified figure (Fig. 114), the points (?i, (xg being ultimately coincident. MAXIMA AND MINIMA. , 393 483. We are thus led to the following definition :— Def. If, while the independent variable x increases contin- uously, a function dependent upon it, say ^(a?), increase through any finite interval however small until x = a and then decrease, (p(a) is said to he a maximum value of (x). And if {x +h)- (a+h)-{a -h)- {a)\ {a—h) — "{cb) also vanish, in which case the sign of the right side of each equation depends upon that of ^""(a). And, as before, if this be negative we have a maximum value and if positive a minimum. Similarly, if several successive differential coefficients vanish when X is put equal to a, it appears that for a maximum or minimum value it is essential that the first not vanishinor should be of an even order, and that if that differential co- efficient be negative when ic = a a maximum value of (j){x) is indicated, but li positive a minimum. Examples. 1. Determine for what values of x' the function '{a;) = eO{x*-Z:v^ + 2x^). Putting this =0 we obtain a:=0, ar=l, .r=2. Again \a:) = 60(4^-3 _ 9^2 + 4 -j.^. If x — l, {a; — A) — 2/) = 0, (1) S+|S=« (^) Now in searching for maxima and minima values of y those values of x are critical which make ^-~ zero or infinite. Thus dx we should examine the cases for which ^, or -^ change sign. Fig. 123. 406 CHAPTER XV. Taking, for instance, the case of maxima or minima deduced from the equations (j){x, y) = 0^ ?^=o| -^-^^ 'dx J we can proceed to their discrimination as follows : — Differentiating equation (2) we have 5V , ^0_ dy (Z^- 3V dy\d^ -d^ d?y , ^x^'^^xdy dx'^Xdx'dy'^Zy^ dxJdx'^^y dx^ ^^••••V / and, remembering that ;^ = 0, this reduces to ^=_2^ (o) dx^ d^ ^ ^ dy Substituting the values of x and y derived from equations (3) (If 7 J we can test the sign of -^, and thus discriminate between the maxima and minima values. The case in which this test fails, viz., when —^ = for the values of x and y deduced by equations (3), is complicated owing to the complex nature of the general formulae for d^y , d^y dx^ dx^' Ex. Find the maximum and minimum ordinates of the curve Here * {x^-ay)+{y''-ax)^=0, (1) and ^^ = gives x^ = ay. Combining this with the equation to the curve we obtain y^ = '2.axy\ i.e. J y = or y^ = 2ax. y=0 gives sc=Oj whilst /=2<«|gi^^ y*=4a3y, and ar = ay J which presents the additional solution x = al/2. MAXIMA AND MINIMA. 407 Hence the points at which maxima or minima ordinates may exist have for their co-ordinates (0, 0) and {a 'ij2, a X/4). Now '^'^-Ga; and ^^ - 3(y^ - a;r), dor 9y and therefore at the point ^dy y=aV4, a and is negative, and therefore at this point y has a maximum value. At the point ji7 = 0, y = 0, the formulae for -^ and -^ both become indeterminate, and we have to investigate their true values. Differentiating equation (1) we have And when x and y both vanish these give *=0 and ^=,2, dx ax- 3a showing that the ordinate y has for this point a minimum value. Several Dependent Variables. 490. Suppose the quantity u, whose maxima and minima values are the subject of investigation, to be a function of n variables x, y, z, etc., but that by virtue oi n — 1 relations between them there is but one variable independent, say x. We may now, from the n — 1 equations, theoretically find the n — 1 dependent variables y, Zj.,. in terms of x, and suppose that by substitution u is expressed as a function of the one independent variable x. The methods of the preceding articles can now be applied. It is often, however, inconvenient, even if possible, actually to eliminate the n — 1 dependent variables y, s, etc., and it is not necessary that this should be immediately done. Suppose, for instance, u = and therefore the radius of the semicircle is equal to the height of the rectangle. To test whether this result gives a maximum value to A we have dx^ dx dor , d^P ^ ^d-y therefore -^-^=7r + 2(-2-7r)= -77-4, and is therefore negative. Hence the relation found, viz., ^=.y, indicates a maximum value of the area. 491. In the solution of such questions as the foregoing it is frequently unnecessary to employ any test for the discrimina- tion between the maxima and minima, since it is often suf- ficiently obvious from geometrical or other considerations which results give the maxima values and which give the minima. 492. Function of a Function. Suppose z—f{x), where x is capable of assuming all possible values, and let y — F{z)\ then it appears that since the vanishing of either of the factors f{x) or F'{z) will give -7^=0, and therefore y may have maxima or minima either for solutions of F\z) = or for such values of x as make f{x) = 0, and which therefore make z a maximum or minimum. More- over, if z be not capable of assuming all possible values, it may happen that some of the roots of F'{z)=^ are excluded by reason of their not lying within the limits to which z is re- stricted. Several such problems have been discussed at length in the Cambridge Mathematical Journal, vol. III., p. 237. Ex. 1. To find the maxima and minima values of the perpendicular from the centre of an ellipse upon a tangent. 410 CHAPTER XV. If r and / be conjugate semi-diameters, a and h the semi-axes, and 'p the perpendicular from the centre on the tangent at the point whose radius vector is r, we have r^-\-r"^=d^-\-}p'^ pr' = a6, giving ^^=a2 + 62-r2, . Differentiating with respect to r, a^b^ dp _ p^ dr * and putting ^=0, dr we obtain r=0, a result which is inadmissible, since r is restricted to lie between the limits a and b. It appears therefore at first sight as if the ordinary criteria had failed to determine the true maxima and minima values of r. We should remember, however, that since r is restricted to lie between certain values it will not do for an independent variable, and we should therefore have substituted the value of r from the equation of the curve in terms of 6, which is susceptible of all values and therefore suitable for an independent variable. We should thus have a%^ dp _ dr ^dQ~'^dS and the vanishing of -j^ indicates that the maximum and minimum values du of p are to be sought at the same values of d for which the maximum and minimum values of r occur; i.e., obviously when r=a and when r=b. This result was of course apparent ab initio from the form of the relation between p and r. Ex. 2. The orbits of the earth and Venus being assumed circular and co-planar, to investigate in what position Venus appears brightest. The brightness of a planet varies directly as the area of its phase, and inversely as the square of the distance of the planet from the earth. Fig. 124. Let E and S be the earth and the sun and V the centre of Venus, the plane of the paper being the plane of motion. Let PVP'j QVQ' be diametral planes of the planet, perpendicular to the lines JUV And JSVf and let ZVZ' be the diameter perpendicular to the plane MAXIMA AND MINIMA. 411 of motion. Draw QN dit right angles to PP'. Let c be the planet's radius and X, a, r the lengths of E F, ES, and >S' F respectively. The hemispherical portion QPQ' is illuminated by the sun's rays, whilst PQP' is the portion exposed to view from the earth. The illuminated portion visible is there- fore bounded by the line ZQZ'PZ, whose projection upon the plane PZP'Z is a crescent-shaped area bounded by a semicircle and a semi-ellipse, the greatest breadth being PN. The area of this crescent is Jttc^ - \-KC . c cos N VQ, and therefore oc 1 - cos A^ VQ. The brightness therefore 1 - cos NVQ l+co^EVS °^ EV^ ^^ EV^ Now cos ^F>Sf= -+'"'"'*' 2xr ^ (-+rf-\ r} + ^,+^ whence brightness — • ^ • ^ This expression has its maximum and minimum values, (1) when ^ is a maximum or a minimum, i.e., when x=a±r; (2) when y^^?^-±^ = 0. This second relation gives or a:=\^a2+r2-2r, the negative root being inadmissible. We have now to inquire whether this value of x lies between the greatest and least of the admissible values of x, viz., a±r. Now V3a- + r--2r>a-r if r-- 4 For the inferior planets, Venus and Mercury, whose mean distances from the sun are respectively '7a and 'Sda roughly, r obviously lies within the prescribed limits. To distinguish between the maxima and minima, we observe that when the earth and planet are in conjunction, i.e., when x=a — r, the brightness =0, and is obviously a minimum. Hence x=^'Sa}^ + r^ — 2r gives a maximum and x = a + r a minimum. It is easy to deduce hence that, for the position of maximum brightness, V 2 tan ^= tan—. an equation due to Halley, and 3a cos2^-f 4r cos E-4a=0y which determines the angle E. [See Godfray's Astronomy, 2nd Ed., p. 287.] > ^ qr^ 412 CHAPTER XV. 493. Other Maxima and Minima; Singularities. The accompanying figure (Fig. 125) is intended to illustrate some points with regard to maxima and minima which we have not at present considered. Fig. 125. At S there is an asymptote parallel to the y-axis. The curve y = (j)(x) approaches the asymptote at each side towards the same extremity. Here y=co and -^ = oo , but 3- changes sign in crossing the asymptote, and there is an infinite maxi- mum ordinate at S. At T there is another asymptote parallel to the y-axis, but in crossing the asymptote the curve reappears at the opposite extremity and -— does not change sign ; there is therefore neither a maximum nor a minimum at T. At M there is a "point saillant " giving a discontinuity in the value of -^. The ordinate at such a point is a maximum or a minimum. In the case in the figure we have a maximum ordinate. At B the curve has a ''point dl arret" and a maximum ordinate, though -~- does not vanish or become infinite. dx At N there is a cus'p, but -J^ is neither zero nor infinite. Yet the ordinate at N is the smallest in its immediate neigh- bourhood, and therefore a minimum. It is to be noticed, MAXIMA AND MINIMA. 413 however, that in travelliDg along the branch MN the value of X does not 'pass through OTF, and therefore the ordinary theory does not apply. At such points as Q, -j- = ^ and changes sign, and yet obviously the value of y is not a maximum or minimum. As in the last case, it should be observed that in travelling along the branch NQR the value of x does not pass through the value OF, but recedes to it from TT to F and then increases again. We notice, however, that this result may be written as t- =0, dx and that -r- changes sign at Q, indicating a maximum or minimum value of the abscissa x. For further information upon this subject the student is referred to Professor de Morgan's Diff. and Int. Calculus, EXAMPLES. v/ 1. Show algebraically that the greatest value of x(a - x) . a? . is -, and illustrate the result geometrically. 2. Find algebraically the limits between which the expression ax->r — X must or must not lie for real values of x. Illustrate your result by a sketch of the curve y = ax + -. ^ X 3. Investigate algebraically the maximum and minimum values of ^i ' a^-4:X+'2 the expression ! — for real values of x. Illustrate your answer geometrically. 4. Find for what values of x the expression {x-iy{x+3)^ has maximum or minimum values. 5. Investigate the maximum and minimum values of the expression 2£c3-21a;2 + 60a;+30. 6. Find the minimum ordinate and the point of inflexion on the curve x^ - axy + 6^ = 0. 414 CHAPTER XV. 7. Find the maximum and minimum ordinates of the curve {y-cf==(x-af{x-b). 8. Show that the curve y = xe' has a minimum ordinate where x= -\. 9. Show that the values of x for which e*"'""" has maximum or minimum values may be determined graphically as the abscissae of the points of intersection of the straight line y= -a;, with the curve of tangents y = tan x. 10. Show that the expression a + (x-h)^ + ix- h)^ has a minimum value when x = h. 11. Find the minimum value of + sin^a; cos^aj 12. Show that sin^6>cos^^ attains a maximum value when 13. Show that ^e is a maximum value of ( -) • 14. Show that the function 33 sin a; + cos x + cos^a; continually diminishes as x increases from to ^. 15. If y = 2x-i2in~'^x-\og[x + Jl+x^}^ show that y continually increases as x changes from zero to positive infinity. 16. If 2;= _ + __, X y where £c + 2/ = a, show that z has a minimum value when a^ a + b And a rnax'irmiTTi wVipn a-b' 17. Given that X y T - + r = l< a b MAXIMA AND MINIMA. 415 show that the maximum value of xy is — - and that the minimum 4 vahie of oi? + y- is 18. Show that the area of the greatest rectangle inscribed in a given ellipse and having its sides parallel to the axes of the ellipse is to that of the ellipse as 2 : tt. 19. Show that the maximum and minimum values of x'^y^ where ax^ + Ihxy + hy^ = 1 are given by the roots of the quadratic (-i)('-i)-- Hence find the area of the conic denoted by the first equation. y 20. Divide a given number a into two parts, such that the product of the 1^^ power of one and the (f" power of the other shall be as great as possible. 21. Show that if a number be divided into two factors, such that the sum of their squares is a minimum, the factors are each equal to the square root of the given number. 22. Into how many equal parts must the number ne be divided so that their continued product may be a maximum; n being a positive integer and e the base of the Napierian Logarithms ? 23. What fraction exceeds its 'p^'^ power by the greatest number possible % 24. Given the length of an arc of a circle, find the radius of the circle when the corresponding segment has a maximum or minimum area. [Pappus Alexandrinus.] 25. The centres of two spheres, radii r^, rg, are at the extremities of a straight line of length 2a, on which a circle is described. Find a point in the circumference from which the greatest amount of spherical surface is visible. 26. In the line joining the centres of two spheres find a point such that the sum of the spherical surfaces visible therefrom may be a maximum. [Educational Times.] 27. AC and BD are parallel straight lines, and AD is drawn. Show how to draw a straight line COE^ cutting AD and BD in and E respectively, so that the sum of the triangles EOD, CO A may be a minimum. [Viviani.] 416 CHAPTER XV. 28. A person wishes to divide a triangular field into two equal parts by a straight fence. Show how it is to be done so that the fence may be of the least expense. 29. If four straight rods be freely hinged at their extremities the greatest quadrilateral they can form is inscribable in a circle. 30. A tree in the form of a frustum of a cone is n feet long, and its greater and less diameters are a and h feet respectively. Show that the greatest beam of square section that can be cut out of it is feet loner. 3(a -h) ^ 31. If the polar diameter of the earth be to the equatorial as 229 : 230, show that the greatest angle made by a body falling to the earth with a perpendicular to the surface is about 14' 59", and that the latitude is 45° 7' 29". 32. The resistance to a steamer's motion in still water varies as the n*^ power of the velocity. Find the rate at which the steamer must be propelled against a tide running at a knots an hour so as to consume the least amount of fuel in a given journey. 33. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semivertical angle a is ^TT^^tan^a. 27 34. Show that the height of the cone of greatest convex surface which can be inscribed in a given sphere is to the radius of the sphere as 4 : 3. 35. Two particles move uniformly along the axes of x and y with velocities u and v respectively. They are initially at distances a and h respectively from the origin, and the axes are inclined at an angle 0). Show that the least distance between the particles is {av - hu) sin w {u^ + v^ - 2uv cos a))i 36. For a maximum or minimum parabola circumscribing a given triangle ABC, show that the sum of the perpendiculars from ABC upon the axis is algebraically zero. 37. In a submarine telegraph cable the speed of signalling varies as cc^log- where x is the ratio of the radius of the core to that of the covering. Show that the greatest speed is attained when this ratio is l:Je. MAXIMA AND MINIMA. 417 38. A and £ are fixed points and P is a variable point on a fixed line ; show that X.AF + jix. BP will be a minimum if A cos ^ = /x cos <^, 6 and being the angles which AP and BP make with the fixed line. 39. /S is the focus of an ellipse of eccentricity c, and ^ is a fixed point on the major axis, and P is any point on the curs^e. Show that when PE is a minimum SP = e 40. Find the maximum value of / \2/ 7.\ f(l) when a > 6, (^-^)'(^-^M(2)whena<6. What happens \i a = h'\ Illustrate your answers by diagrams of the curve 2/ = (^ - a)~{x - h) in the three difi'erent cases. [I. C. S., 1879.] 41. An open tank is to be constructed with a square base and vertical sides so as to contain a given quantit}^ of water. Show that the expense of lining it with lead will be least if the depth is made half of the width. 42. If two variables x and y are connected by the relation ax^ + hy"^ = aby show that the maximum and minimum values of the function x^ + y^ + xy will be the values of u given by the equation i(u - a)(u -b) = ah. 43. If SP and SQ be two focal distances in an ellipse inclined to each other at the given angle 2a, find the greatest and least values of the area of the triangle PSQ. 44. SQ is a focal radius vector in a given ellipse inclined at a given angle a to SA, where A is the vertex nearest to the focus S. Find the angle ASP, where SP is another focal radius, such that the area of the triangle PSQ may be a maximum. 45. Find the point P on the parabola y^ = 4:ax such that the perpendicular on the tangent at P from a given point on the axis distant h from the vertex may be the least possible. What is the geometrical meaning of the result ? 46. Find the area and position of the maximum triangle having a given angle which can be inscribed in a given circle, and prove that the area cannot have a minimum value. 47. From a fixed point A on the circumference of a circle of radius c the perpendicular AY is let fall on the tangent at P. Prove that the maximum area of the triangle ^P}" is 3^2 E.D.C. s^V3. 418 CHAPTER XV. 48. If a parallelogram be inscribed in an ellipse the greatest possible value of its perimeter is equal to twice the diagonal of the rectangle described on the axes. 49. is a fixed point without a circle, A one of the extremities of the diameter through O^OQQ' el chord through 0. Find its position when the area of the triangle QAQ' is a maximum. Does it ever become a minimum ? 50. A length I of wire is cut into two portions which are bent into the shapes of a circle and a square respectively. Show that if the sum oi the areas be the least possible the side of the square is double the radius of the circle. 51. Obtain the maximum and minimum values of the volume of a right circular cone whose vertex is at a given point and whose base is a plane section of a given sphere; and point out the difference of the cases of the point being within or without the sphere. [Math. Tripos, 1876.] 52. Prove that a chord of constant inclination to the arc of a closed curve divides the area most unequally when it is a chord of curvature. 53. When the area of a triangle has a maximum or minimum value and all the parts vary, then cos A . da + cos B . db + cos C .dc = 0. [Oxford, 1888.] 54. Show that the normal chord to the parabola y'^ = iax which cuts off the least arc is normal where y = '—j- — -' and is inclined 2 ^ to the axis at an angle tan ^-^. 55. When the product of two perpendicular radii vectores of a curve is a maximum or a minimum, show that they make supple- mentar}?- angles with the tangents at their extremities. 56. Two perpendicular lines intersect on a parabola, one passing through the focus. Show that the triangle formed by them with the directrix has its least values when the focal distances of the right angle and the vertex of the parabola include an angle of 36° or of 108°. 57. A plane triangle ABC, right-angled at B, and of given peri- meter F, revolves either round an axis through A parallel to BC, or round an axis through C parallel to BA, and the solid generated is a maximum ; show that the three sides of the triangle are equal to ^(-3 + ^17), J(5-V17), J(7-V17). • [Smith's Prize, 1878.] MAXIMA AND MINIMA. 419 58. Show that when the angle between the tangent to a curve and the radius vector of the point of contact has a maximum or minimum value the radius of curvature at that point is given by /)- -. P 59. Show that the greatest distance which can be saved in a single voyage by sailing along a great circle instead of a parallel of latitude, 9 is «{2sin-i- + V7r2-4-7r}, * wliere a is the earth's radius. [Math. Tripos.] 60. Show how to find the co-ordinates of the points on a curve given in Cartesians at which the curvature is a maximum or a minimum. CHAPTER XVI. MAXIMA AND MINIMA— SEVERAL INDEPENDENT VARIABLES. 494. Preliminary Algebraical Lemma. The binary quadratic /g = ax^ + ^Ifixy + h'if' may be written -\{ax + hyf + {ah — l^)]f'\ and therefore retains the same sign as a for all real values of x and y \i ah — h? be positive. The ternary quadratic /g = ax^ + hy"^ + cz^ + 2fyz + 2gzx + S/iici/ may be written ^[{ax+liy+gzf + (ah-h'')y^^-2{af-gh)yz + {ac-gVl and therefore by what has gone before will retain the same sign as a for all real values of x, y, z if ah — h^ and (ah — h^)(aG — g^) — (a/— ghy be positive, i.e.y if ah — h^ and a(ahc + 2fgh — af^ — hg^ — ch^) be positive. That is to say, I^ and /g will both be positive if a, a, h } a, h, g h, h h h,f 9> f> G be all positive, and will both be negative if these expressions are alternately negative and positive. 495. These results may be generalized. For the general homogeneous quadratic function of n variables can be thrown into the form jOi(^l + ^2^2 + ^3^3 + . . . + anXnf +^2(^2 + h^A + . . . + KoOnf + P3(^^3+...+Cn^n)^ + ... 420 MAXIMA AND MINIMA. 421 since the number of arbitrary constants at our disposal is the same as the number of coefficients of the original quantic. And it is a known proposi- tion * that the values of jOj, jt?2, . . . , jo„ are where A^ is the discriminant of the quantic obtained from the original function by putting a^r+i, •^r+2, •••, etc., all zero. Now assuming that all the letters involved are real, it is clear that if Ai, Ag, A3, ... be all positive, we shall have jOj, ^g) i's* •••>i'n all positive, and therefore the quantic positive; and if A^, Ag, A3, ... be alternately negative and positive, jOj, jOg, JO3, ..., Pn will all be negative, and hence the quantic will also be negative. For an inductive proof of this result the student is referred to a note at the end of Dr. Williamson's Treatise on the Diferential Calculus. 496. To search for Maxima and Minima. Def. Let (l){x, y,Zy...)he any function of several independent variables x,y,z,..., supposed continuous and finite for all values of these variables in the neighbourhood of their values a,b,c,... respectively. Then the value of (p(a, h, c, ...) is said to be a maximum or a minimum value of ^(0:, y, z, ...) according as (a, b, ...) answer to the above definition we expand by the extended form of Taylor's theorem (Art. 178) ^{x+h,y+k, ...)-(p(x,y, ...) = h-;~-]rk^-\- ...-\' terms of the second and higher ordera dx dy ■ ^ Now by taking h, k, I, ... sufficiently small, the first degree terms can be made to govern the sign of the right-hand side, and therefore of the left side also, of the above equation ; therefore by changing the sign of h,k,l, ... the sign of the left- hand member would be changed. Hence as a first condition * Burnside and Panton, Theory of Equations^ p. 401. 422 CHAPTER XVI. for a maximum or minimum value we must have ft|^+ifc?^+z|^ + ... = 0, dx dy dz and therefore as these arbitrary increments are independent of each other, we must have 1^ = 0, 1^ = 0, ^ = 0,etc (1) dx dy dz ' If there be n independent variables, we have thus obtained n simultaneous equations which serve by their solution to find the admissible values of x, y, %, ... for which maxima and minima values may exist. The above equations therefore form essential conditions for the existence of maxima and minima, but we shall see that they are not in themselves sufficient, and we shall have to employ a further test for their discrimination. We shall now consider the cases of two and of three inde- pendent variables separately. Let one system of values of x,y,z ... satisfying equations (1) be a, b, c, ... respectively. 497. Case I. Two Independent Variables. The Lagrange- Condition. Let us put r, s, t for the values of — ^, ^ I^ , --^ when x = a ^ . dx^ dxdy dy^ and y = h, then (p{a + h,b + k)- {a-\-h, h-\-k) — (l)(a, b). And the biquadratic terms must collectively be of the same sign as r and t when t = ^' 498. In the case in which r, s, t are each of them zero, the quadratic terras are altogether absent, and the cubic terms would change sign with h and k, and therefore all the differen- tial coefficients of the third order must vanish separately when x=^a and y = b and the biquadratic terms must be such that they retain the same sign for all sufficiently small values of h, k. Ex. Let u=xy + - + -j X y -dy"'^ y^ So r and t are positive when x =y = a, and j ^' * ' = 2.2-1=3 \ s, t \ and is positive ; and therefore there is a minimum value of u, viz., u = 3a-. 424 CHAPTER XVI. 499. Geometrical Explanation. Let the reader imagine that the plane of xy is the horizontal plane at the sea level, and that z = (p{x, y) is the equation of the surface of a mountainous tract of country in which there are isolated hills, mountain chains, valleys, lakes and mountain passes. Let a map be constructed showing the various contour lines of the hills, lakes, etc., at different altitudes. Correspond- ing to an isolated hill or a lake these contour lines will form closed curves, dwindling to a point at the top of an isolated hill or at the deepest point of a lake. At a saddle-shaped moun- tain pass the contour lines at the highest point of the pass will intersect and form a node while, corresponding to the ridge of a chain of mountains of uniform height or the bottom of a V-shaped depression of uniform depth in a lake, the closed contour line degenerates into a single curved terminated line. Again, at a bar across a valley, as at a mountain pass, the con- tour lines form a node at the highest point of the bar. Now at all these several places the tangent plane to the country is horizontal and the preliminary conditions ^ = 0, ?^ = are satisfied (Art. 496). dy At the top of an isolated hill we have a true maximum value of 2; ; rt — s^ is positive whilst r and t are both negative. At the deepest point of a lake we have a true minimum ; rt — 8^ is positive whilst r and t are both positive. At a mountain pass rt — s^ is negative, and although the tra- veller over the pass arrives at a maximum height in the direction in which he travels, yet if he diverge from the path either to right or left he at once begins to ascend to higher ground. This therefore is not a point of maximum height on the sur- face. The same is true at the highest point of a bar separating two depressed regions. If rt = s^ then in the direction of h7^ + ks = the tangents to the contour lines through that point coincide. Further investi- gation is now necessary. If the contour lines open out and separate immediately after their contact, there is neither a maximum nor a minimum; but if they dwindle down to a single line all along their length, we have a row of what may MAXIMA AND MINIMA. 425 be called maxima or minima. This is the case of a Hue along the ridge of a mountain chain of uniform height or along the bottom of a V-shaped depression of uniform depth ; and a person travelling along such a line will move continually at a constant distance above or below the sea level. (See Greenhill's Diff. and Int. Gale.) 500. The effect of the variation in sign of rt — &^ will he more easily understood by the student of solid geometry. The equation giving the principal radii of curvature at any point is {rt - 5-V + ^{^(1 ^f) + K 1 + ?^) - 2i5g«)}/3 + ^ = 0, where ^=1+^2 + ^2^ P = ^'> 9'=o-* ox oy Hence the principal radii of curvature are of the same or of opposite sign according as rt — s^ is positive or negative; and one of them is infinite when rt-&^=0. In the latter case the corresponding line of curvature has either an inflexional or an undulatory point. 501. A ridge of Maxima or Minima. Suppose that ^ and -^ contain a common factor v. ^^ dx d2/ Let ^ Then dx '■ ' 'dx dv dw. dx ^ dx So that for the values of a; and y satisfying -^ = we have , ., dv dv ( dv\f dv\ ^ Suppose we solve v = ^ and find y=f(x). Substituting this in z = (f)(x, y) we have z a function of x only and d^_d(p d(/) dy dx~dx dy dx which vanishes for such values as satisfy v = 0, and therefore make ?*=?^ = dx dy Thus along a curve line on the surface, whose projection on the plane of xy \^ v — (),z is constant. * Smith's Solid Geometry, p. 218. d(f) dx'' dv 90 . div. ■ vw^. r- dx ^ dv dx dw^ 8- dy ' 1 + '^'^— 7 and also ' ^y dv , '^'^o t= dy 2 • ^y 426 CHAPTER XVI. This is the case of a locus of maxima or minima such, for instance, as would be produced b}^ the revolution round the 0-axis of any closed curve. The definition of maxima and minima according to this view needs a slight modification, and we must suppose a maximum value to be one which is not less than and a minimum to be one which is not greater than any other value which is immediately contiguous to it.* Ex. Consider the Anchor Eing or Tore formed by the revolution of a circle of radius h about a straight line in its own plane at a distance a from the centre. Taking the axis of revolution for the 0-axis and the plane through the centre perpendicular to the 2-axis for the plane of ^y, the equation is 2^ = 6^ — a^ - a^ - y^ + 2a\/x^ + y^. Here =— ^ + - 9^ Jx^+y^ 'dz , ay The vanishing of the common factor 1 - , gives both ,.^- and >^ = 0^ and the cylinder x'^-{-y'^ = a^ cwi^ the surface along the ridge formed by points which are all at the same distance h from the plane of xy and at greater distance from that plane than any other points of the surface which do not lie in that circular ridge. 502. Case II. Three Independent Variables. Let a set of the values oix, y, z determined from the equations dx dy dz be a, h, c, as explained in Art. 496. Let the corresponding T . d^d> 3V BV d^ — ^j ^ i > ~*^- dx^ ^2/^ ^^^ 9^92; ?)zdx 'bxdy be called A, B, G, F, G, B. Then we have (p{a+h, h + k, c + l) — (j){a, h, c) = hAh^-\-Bk^-{-Gl^ + 2Fkl + 2Glh + 2Hhk) + R^, A. where JK3 consists of terms of the third and higher orders of small quantities, and by taking li, k, and I sufficiently small the second degree terms can be made to govern the sign of the right-hand side and therefore of' the left also. If this group of terms form an expression of permanent sign for all such values * See Fran^ais Annales de Gergonne, vol. III. Gregory's Examples, p. 110. MAXIMA AND MINIMA. 427 of h, k, and I, we shall have a maximum or minimum value according as that sign is negative or positive. Hence by our Lemma, Art. 494, if the expressions H, B H, B, f\ G, F, G \ be all positive, we shall have a minimum value o£ 0(ic, y, z), and if they be alternately negative and positive we shall have a maximum, whilst if these conditions are not satisfied we shall in general have neither a maximum nor a minimum. 503. Similarly we might proceed by aid of the generaliza- tion in Art. 495 to consider the case of several independent variables. And according to that article we shall have a minimum when all the discriminants are positive and a maxi- mum if they are alternately negative and positive. 504. Several Independent Variables. Lagrange's Method of Undetermined Multipliers.* Let u = 428 CHAPTER XVI. Multiplying these lines respectively by 1, Xp Xg, ... X^ and adding, we get a result which may be written P^dx^ + P^dxc^-\-P^dx^+ . . . -^Pndxn = (2) The 7Yi quantities X^, Xg, . . . X^ are at our choice. Let us choose them so as to satisfy the m linear equations P —P —P = =P =0 The above equation is now reduced to It is indifferent which n — rti of the n variables are regarded as independent. Let them be Xm+i, ^m+2, ..., Xn. Then since the n — ni quantities dx^+i, <^^m+2, ..-, dxn are all independent their coefficients must be separately zero. Thus we obtain the additional n — m equations Thus the on + n equations Ji~ h~ Jz~ •'•~ Jm — "j and P^ = P2 = ^3=---=^n = 0, determine the m multipliers X^, Xg, ..., \m and values of the n variables x^, x^, ..., Xn for which maxima and minima values of u are possible. 505. If u be a homogeneous function of degree jp, and /i» /2' /s' '"■> fm be capable of being put into the forms Ua = A^ Uj, = B, Uc = G, ..., ujc = K; Ua, u^, etc., being homogeneous expressions of degrees a, h, etc., and A, B, etc. constants, there is a very useful relation between the quantities X. Multiply- ing the n equations of which Py = is a type by x^^, x^, ... Xn and adding, we have by Euler's theorem on homogeneous functions yu + X^aA+XJDB + X^cG-^- ...^-\mkK=(). Ex. Let us investigate the maximum and minimum radii vectores of the section of the " surface of elasticity " * (^2 +^2 + ^2)2 = ^2^2 + ^2^2 + ^2^2 made by the plane lx+my + nz = 0. We must make r^=x^+y'^+z^ a max. or min. * Gregory's Examples, p. 120, and Fresnel, Memoires de Vinstitut, vol. VII. MAXIMA AND MINIMA. 429 Then xdx+ydy + zdz = (1) a^xdx+hh/dy+chdz=0 (2) ldx-irmdy + ndz=0 (3) Whence multiplying (2) and (3) by Ai and Ao and adding, we have by Art. 604 x+X^d^x+Xil =0 (4) 3/ + Ai62y + A2»i = (5) z + X^ch ^X^n =0 (6) Multiplying by x, y, z respectively and adding, . r2 + Air4=0 or Ai=-^- and two similar equations. Whence multiplying by I, m, and n and adding, _^ _^^ ^^_ a quadratic which gives the values of r required. EXAMPLES. 1. Discuss the maxima or minima values of u in the following cases: — (a) u = a:?y\\ -x-y). (P) u = a^ + y^-2a^ + 4xy-2y^. (y) u = 2((rQcy - 3ax^y - ay^ + x^y + xy^. (5) u = axy'^z^ - 7?yH^ - xy^^ - xy'^7^. (c) u = sin x sin y sin(a; + y). (0 u = x^y'^ - bx^ - Sxy - oyK (r)) u = x^-\-y^ - 3axy. 2. Find the minimum value of x^ + y^ + z^y having given ax + by-{-cz=p. 3. Find the maximum value of x'^y^z^ with condition x + y + z = a. 4. In a plane triangle find the maximum value of cos A cos B cos C. 5. Find a plane triangle such that sin"'-^ sin"jB sin^'C has a maximum value. 430 CHAPTER XVI. 6. Divide a number n into three parts a?, 2/, z such that ayz + hzx + cxy shall have a maximum or minimum value and determine which it is. 7. Find the maximum or minimum values of xya^ + y^M + zyc^ when Ix + my + nz = and x^/a^ + y'^lb^ + z'^lc^ = l. [Oxford, 1888.] 8. Inscribe in an ellipsoid the maximum rectangular parallelo- piped. 9. Given a^'b^c'^A, find the maximum value of {x + l){y+l){z + l). Interpret the result. [Waring.] 10. Eequired the rectangular parallelopiped of given volume and least surface. 11. Find the minimum value of x^ + y^ + z^ witli the conditions ax + by + cz = ax + b'y + c'z = l. 1 2. Find the maxima and minima of x- +y^ + z^ subject to the following conditions : — (1) ax^ + by^ + cz^ = l. (2) ax^ + by^ + cz^ + 2/2/2; + ^gzx + ^hxy = 1. aaj2 + 5^/2 + cz^ — I and Ix + my + nz =0. ax'^ + by^ + cz^ + 2fyz + 2gzx + 2hxy = 1 and Ix + my -i- nz = 0. 13. Find the maximum or minimum of ax^ -{• by^ + cz"" with the condition x^ + y"' + z^ = k. 14. Find the maximum value of xyzjia + x){x + y){y + z){z + b). [L\grange. ] 15. Find the minimum value of x^ + y'^ + z^ + w^ + ... with condition ax + by + cz + dw+ ...= k. 16. Show that the point within a triangle for which the sum of the squares of its perpendicular distances from the sides is least is the centre of the Cosine-Circle. 17. Find a point within a triangle such that the sum of the squares on its distances from the three angles is a minimum. 18. Find a point within a triangle such that the sum of the distances from the angular points may be a minimum. [Fermat.] 19. Find the triangular pyramid of given base and altitude which has the least surface. [Gregory's Examples.] (3)[ MAXIMA AND MINIMA. 431 20. Find the minimum value of the continued product of the perpendiculars drawn from a point upon the faces of a given poly- hedron. [Coll. Exam.] 21. If a be a maximum or a minimum value of /(ar, y^ z) for points which lie on F{x, y, z) = 0, then the surfaces f{x, y^ z) = a and F{x, y, z)=^0 will touch. [Coll. Exam.] 22. Find the maximum and minimum values of p where rx^ + 2sxy + ty'^ = kjp^ ' having given that (I + jt>2)a;2 + 2pqxy + (1 + g2)/ = 1 . 23. If there are p tops of mountains on the earth and q bottoms of lakes and seas, prove that there must be ^ - 1 passes, or places where a level surface drawn through the point cuts off two elevated regions which meet at that point ; and also ^ - 1 bars, or places where the level surface cuts off two depressed regions which meet at that point. Show also that there must be at least two summits higher than any pass, and two bottoms lower than any bar. [Math. Tripos, 1870.] 24. A framework crossed or uncrossed is formed of two unequal rods joined together at their ends by two equal rods ; prove that the distance between the middle points of either pair of rods is a maxi- mum when the unequal rods are parallel and a minimum when the €qual rods are parallel; unless the two unequal rods are together less than the two equal rods, in which case the unequal rods are parallel in both the maximum and minimum positions. [Math. Tripos, 1875.] 25. If w be a function of n independent variables ccj, x^, ..., ic„, prove that, in order that u may have maximum or minimum values, the roots of the equation must all be of the same sign; ?7,., ?/„ denoting the particular values of -—2, , for certain values of £Cj, ct-g, ..., x^ which make du du ^" — ^' ^' '"' dx^~ ' [Math. Tripos, 1873.] CHAPTER XVII. ELIMINATION. 506. Construction of a Differential Equation. It has been seen that the equation f{x,y,a) = (1) is representative of a certain family of curves, for each individual of which the constant a receives a particular and definite value, the same for the same curve but difierent for different curves of the family. Problems sometimes occur in which it is necessary to treat of the whole family of curves together, as for instance in finding the family of curves which intersect each curve of the first system at right angles. And it is manifest that for sach operations the letter a ought not to appear as a constant in the functions operated upon, otherwise we should be treating one individual curve of the system instead of the whole collectively. Now the process of difiBrentiation can be easily applied to get rid of a. For by differentiation with regard to x, we have 'dx ?)y dx ' • V / and a may be eliminated between these two equations, if indeed it has not already disappeared. There will now result an equation between X, y, and g. which may be called the Differential Equation of the family of curves. 432 ELIMINATION. 433 For example, consider the family of straight lines obtained by giving special values to the arbitrary constant m in the equation y = mx. Differentiating, j^"*^* and therefore y=x-^^ CLX a differential equation which is true for each member of the family since the m has been eliminated. It is clear that the m would have disappeared at once upon differentia- tion if we had written the equation of the line X for, differentiating, we have — ^ — = 0, or yz=x^ ^ dx as before. This is then the differential equation of all straight lines passing through the origin and expresses the geometrical fact that the direction of the straight line is the same as that of the vector from the origin at all points of the same line. 507. Again, suppose the representative equation of the faniily of curves to be /(.x, y, a, h) = 0, containing two arbitrary constants a, h whose values particu- larize the several members of the family. Now a single differentiation with respect to x will either cause one of the constants to disappear or will result in a relation between x, y, ^, a and 6. From this relation and the original equation of the curve one of the two arbitrary constants may be eliminated, say a. Then we have a result of the form If we again differentiate with respect to x, we shall either cause the h to disappear or shall be able to eliminate h between the result and the last equation, thus obtaining a difierential equation of the second order between '''y'%^''^% E.D.C. 2e 434 CHAPTER XVII. Thus if a function with one independent variable contains one arbitrary constant, the result of eliminating it is a differ- ential equation of the first order. If it contain two arbitrary- constants, the result is a difierential equation of the second order. And our argument is general ; so that to eliminate n arbitrary constants we shall have to proceed to n differentia- tions, and the result is a differential equation connecting dy d^y ^''^'dx""'d^' and is therefore of the n^^ order. Again, the final result is independent of the order and of the manner in which the eliminations are effected. For suppose the arbitrary constants to be and let any particular values be assigned to these constants. Then we have made choice of some particular curve of the system. Next take any value of x; at the points thus deter- mined, y, -~, -j-^, . . . , -j-^ have each definite values dependent upon the chosen values of x, a^, a^, . . . , a„, thus fixing the inclination of the tangent to the axis of x, the measure of curvature, and peculiarities of shape of a higher order at the point in question. These peculiarities of shape intrinsically belong to the chosen curve, and cannot be de- pendent upon any particular algebraic process which it may be found necessary to employ in obtaining a numerical measure of them, but must depend solely upon the geometrical character of the curve. Hence, if for the whole family any general algebraic identity be discovered connecting these peculiarities, in which none of the particularizing constants are present, and which is therefore true at any point of any member of the family, it must amount to a statement of some geometrical property characteristic of the family, and be independent of the method of its discovery. And in obtaining the n differential coeflolcients of y with regard to x we have in all 91 + 1 equations, including that of the original curve, with n arbitrary constants to eliminate, leaving one single relation between dy d'^y ^'^'d^'""d^^' ELIMINATION. 435 Ex. 1. Form the general differential equation of all straight lines. The general equation of a straight line is y=mx-\-c. Hence y\—i^i (1) and ^2=^ (2) Equation (2) evidently then is the general differential equation sought. Its geometrical interpretation is clearly that the curvature vanishes at every point of each member of the family. Ex. 2. Eliminate a and c from the equation x^+y'^ = '2.ax+c. Differentiating, x-\-yyi = a. Differentiating again, 1 -\-yi^ -^-yy-i — 0. This is the differential equation of all circles whose centres lie on the X-axis. Ex. 3. Eliminate a, 6, and c from the equation (x-a)2 + (y-6)2 = c2 and thus form the general differential equation of all circles. We may write this equation ^ + y^ = 2aar -|- 26y + ^ ^ Eliminating h between the last two results (3yiy2 -¥yyz)y-. = (1 +^1^ +yy2K Referring to the result of Ex. 38, p. 110, the geometrical meaning of this equation is plainly that the aberrancy of curvature vanishes at any point of any circle. Ex. 4. To eliminate the constants from the equation of the general conic. Let the conic be ax^ + 'ihxy + hy^+ '^gx + 2/^ -f c = 0. We have by differentiating aa7+A(^i+y) + 6yyi+.9+/yi = (1) a + A(a^2 + 2yi) + %y2+yi')+/y2=0 (2) h{xy^+Zy2)+h{yyz-\-'^y^^)+fyz=0 (3) h{xy^ + A^/^) + h{2/y^ + 4.y^y^ + ^y^^)+fy^ = i) (4) ^(•^y5+sy4)+%y5+5yiy4+ioy2y3)+/y6=o (s) From the last three equations ^3+33/25 yyz+^yiy2 » yz ^4 + 4^3, yy^ + 4y lya + Zy^^ , y^ \ ^^6+5^4, yy6+''>yiy4+ 10^2^3, y^ =0, 436 CHAPTER XVII. which immediately reduces to 3y2, , ys =0, 4y3, 3y2, ^4 or 9.^2^5 - 45y2;/3y4 + ^O^s^ = 0. This general differential equation of all conies was discovered by Monge. Dr. Boole, in his Differential Equations, p. 20, remarked : " But here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms." A remarkable interpretation which calls for notice has however been recently offered by Mr. A. Mukhopadhyay, who has observed that the expression for the radius of curvature of the locus of the centre of the conic of five pointic contact with any curve (called the centre of aberrancy) contains as a factor the left-hand member of Monge's equation, and this differential equation therefore expresses that the " radius of curvature of the ' curve of aberrancy ' vanishes for any point of any conic." * Examples. 1. Eliminate a from the equation 2. Eliminate a and h from the equation a^^¥~ ' 3. Eliminate a and b from r=a + 6 cos 6. 4. Form the general differential equation of all parabolas whose axes are parallel to the axis of y. 5. Eliminate a and b from the equation y=ae'^+be-'^. 6. Eliminate a and b from the equation y = asm{nx+b). 508. Elimination of Functions of Known Form. We have already met with examples of elimination of various functions of known form from given equations by means of differentiation. For example, we found that if the function tan"* was eliminated by simply differentiating, giving the result (1 + 03^)2/1 = 1. And again, from the equation y = smm{sm~'^x) * Journal of the Asiatic Society of Bengal, vol. LVIII. Part I. ELIMINATION. 437 we eliminated the circular and inverse functions sin and sin~^, obtaining the differential equation {l—x^)y^ = xy^ — m^y. (Art. 122.) These were both made use of when series in ascending powers of cc were required for tan-^ic and sinm(sin-^a;) respec- tively. And it was seen that such differential equations are frequently useful in the expansion of certain classes of func- tions. But the chief interest in the processes by which, by differentiation and elimination, the differential equation is formed from its primitive equation rests in the light which they throw upon the converse problem of obtaining the primitive equation of the typical member when the differential equation of the family is given. This problem presents itself in numberless investigations and is the subject of special consideration in works on Differential Equations. Ex. 1. Eliminate the constant a and the logarithmic function from y = a log X. Here y, = ? X and xy^=a. Differentiating again yi -f- xy-i = 0. Ex. 2. Eliminate a, and the exponential and inverse circular functions from y = ae""sin ~ ^x. Here yi = a»nc""8in -'^x+ af>^ - and ya = am^e'^sin " ^x + ^am^^ — + ae*^— -^ — . i.e. , y2=m^y+ 2m (yi - my) + _^^i - my) or ( 1 - ar^)yi = {2m( l — x^)+ x}yi - m^y - nixy + m^x^y. Examples. 1. From the equation y^jrlogx eliminate the logarithmic function. 2. Given y = e« *»"■'*, eliminate the exponential and inverse functions. 3. Given the equation y=cos(log^), eliminate the circular and logarithmic functions. 438 CHAPTER XVII. 509. Genesis of Partial Diflferential Equations. When more than one independent variable enters into our primitive equations, partial differential coefficients occur upon differentiation. A differential equation containing partial differential coefficients is styled a "partial differential equa- tion" in distinction from an "ordinary differential equation" in which there is but one independent variable. In Chapter VI. we proved Euler's Theorem, that if . -=-'*/£ ^••> we nave a;-- + y^— + 0-- 4- . . . = nu, dx ^ dy dz thus eliminating a function of perfectly arbitrary form. We shall give other examples of elimination of arbitrary functions of unknown form obtaining as a final result in each case a partial differential equation. When only three variables x, y, z occur, two being inde- pendent, it is usual to take % for the dependent variable, and to use the abbreviations p, g, r, s, t to denote the partial differential coefficients ^z -dz -d^z 3% 9% ,. , ,. , _-. SS' V ^- 3^' -^f respectively. (Art. 170.) Ex. 1. Eliminate the arbitrary function ^ from the equation Here p = a<^'{ax-\-hy), q = bcfi'{ax+by), bp-aq=Oj the partial differential equation required. Ex. 2. Eliminate the arbitrary functions ^ and ^ from the equation z = xcf)( ^ j +yyjr(a;). Here ' p=(^)-^P-)+^^'{^\ 1 ,w X \xf ELIMINATION. 439 Hence z—px — qy= — xi/\}r'{.v). Also xs+yt=x\lr'{x\ z-px-qy + xi/s-^y^t = Oy a partial differential equation of the second order. Examples. 1. If z = (ax + by) +yylr{ax + hy\ prove h-r-%ahs-{-aH=0. 510. Prop. If u and v he explicit functions of x and y, and if u be a function of v, then will du dv _du dv_^^ 'dx' dy dy' dx ' and conversely y if this relation be identically satisfied, u will be a function of v.* For if u = F{v), we have ?!f: = iPY^)^^ ?)X ^ ^dx and ^^^nv)^' dy ^ 'dy and therefore eliminating F'(v), du dv __du ^_Q dx'dy dy'dx~ Conversely, suppose this condition satisfied, then will u be a function of v. For since u and v are known functions of x and y, we may eliminate one of these letters, say y. Then, unless x is simul- taneously eliminated, we obtain a relation of the form u=f(v,x), * Boole, Differential Equations, p. 24. If :^ = 0, f{Vy x) is independent of x, and therefore u is a 'dv . function of v. And ~ cannot in general vanish, since v in- 440 CHAPTER XVII. Now ^=^1.'^+% dx dv dx dx Su^B/ dv dy dv ' dy du dv _du dv _df dv dx'dy dy'dx dx' dy Hence §^.?^ = 0, dx dy i.e., either — - = 0, or ^ = 0. dy dx dx iion volves usually both x and y. dv If, however, -y be a function of x alone, ;^ = 0. If also u be ^y du a function of t?, 16 is a function of x alone, and ^^ = : hence in this case the relation dm dv dm dv _,. dx ' dy dy' dx is satisfied. And conversely, if this relation be identical!}^ dv satisfied, and if — - = 0, we must have dy die dv _ dy'dx' and therefore u must be independent of y, since we cannot assume v independent of x as well as of y. Hence, as u and v are both functions of the same variable, it is obvious that by eliminating it we can express u sls a, function of v. Examples. 1. If ^,=^ and v=^ ^lt/^-f\ \-xy (1+^)(1+/) prove that UxVy=UyVx, and interpret this result. 2. If u=xij\-y^-irysjl-x^ and v=xy-'Jl- x\'\ - y^, prove that w is a function of v. ELIMINATION. 441 3. If u and v be explicit functions of three variables .r, y, 2, and if 2* be a function of v, prove that =0. 511. Next suppose u a known function of x, y, and 0, and that {u) is an arbitrary function of u. We shall show how to eliminate the arbitrary function 0(i6) from an equation ol the form f{x,y,z.^{'^)} = ^ (1) Supposing X and y to be the independent variables, we have by differentiating 'dx^'^fz+^uK^-^^Tz) = '\ .(2) Hence, eliminating ^, 3/ .(3) an equation containing x, y, z^ p, q and J,u^\ (l>lu^) } = 0. If we form the equations we introduce the two new functions Proceeding to differentiations of the second order, we have the three additional equations 442 CHAPTER XVII. introducing again two new unknown functions We now have in all six equations with six quantities to eliminate. It is therefore in general necessary to proceed to differentiations of the third order, thereby obtaining the four new equations Jxxx^^) Jxxy — ^y Jxyy — ^h Jyyy = ^y and introducing at the same time the two additional unknown functions ^iX'^'i)- We now have ten equations with eight unknown functions to eliminate, leaving in general two independent resulting partial differential equations of the third order. 513. Generally, suppose an equation given of the form containing ^ + 1 variables, of which p are independent, and n arbitrary functions ^n('^n) of the n known functions u^, u^, ..., Un; to eliminate the n arbitrary functions* Suppose t the dependent variable ; then forming all differential coefficients up to those of the r*^ order inclusive, we have (a) f= ; one equation. {^) fx = 0,fy = 0, ...; p equations. (y) fxx = 0, U = 0, U = (),...; £(£+L) equations. [being the number of homogeneous products of p things of two dimensions] (k) and proceeding to r differentiations ; j,(p+l)(p + 2)...(p + r-l) ^^^^^.^^^^ making in all l+p+P(|+l) + ...+ KP+l)---(p + >--l) eq„^tions, containing partial differential coefficients up to those of the r*^ order inclusive. ELIMINATION. 443 The sum of this series is the coefficient of x^ in the product of the series for {\—x)~^ and (1 — a;)"^ = coefficient of x^ in (1 — ic)-(^+i) ^ (j?+l)(p + 2)...(j> + r) ^ (j) + r )! r\ 'p\ r! We have therefore thus obtained ^ ^ '' equations, containing differential coefficients up to those of the ?'*^ order. Moreover, there are (r+l)7i functions to eliminate, viz., 01, 02» •••» ^n, originally; d(\)^ c^02 ^^» /introduced among the differ- du( du^ * * ' ' dun lentiations of the 1st order ; J d^(f)^ d^ n(r + 1 ), and therefore r+2 > 27i, and r > 2ri — 2, Hence in general it is necessary to proceed to differentiations of the (2n — 1)*^ order at least, and there will in general be n independent results. 514. The case however in which the n arbitrary functions i{u), 02W' "'y ^nW ^o be eliminated are all functions of the same known function of x, y, z is exceptional.* We now have * See Todhunter's Diff. Calc, Arts. 251-254. 444 CHAPTER XVII. Proceeding as in Art. 511, equations (2) and (3) are still true, and we obtain 3/+ 3/ 3j*+«?!^ ?>x ^^z dx -^dz dy'^'^dz dy'^^dz 0, an equation containing ^, y, ^, P, q, (piiu), (fy^iu), ..., (j>n(u), for (piXu), 02X^^)» • • • ' ^1^ disappear as before, on the elimination of ^- . Treating this equation in like manner, we obtain a third equation, containing X, y, 0, _p, q, r, s, t, (p^(u), . . . , ^„(u). And the process may be repeated until we have in all n-\-l equations from which the n arbitrary functions may be in- volved, leaving as result a partial differential equation of the 71*^ order. EXAMPLES. 1. Eliminate a and 6 from the equation y = a sin nx + b cos nx. 2. Eliminate a and b from the equation y=.(a + bx)e^. 3. Eliminate a and b from the equation xy = ae" + 6e~^ 4. Eliminate the circular and exponential functions from the equation y = e'^cos x. 5. Eliminate the circular and exponential functions from the equation y = ae^cos 3x + 6e^sin 3x. 6. Eliminate the circular and exponential functions from the equation y = ae'"*sin nx + 6e*^cos nx. 7. Eliminate the hyperbolic functions from the equation ^ a cosh X + b sinh x = c cosh y + d sinh y. 8. Eliminate the constants from the equation ax^ + 2hxy + by^ = c. 9. Eliminate the circular and logarithmic functions from the equation y = sin log x. ELIMINATION. 445 10. Eliminate the circular and logarithmic functions from the equation y = log sin x. 11. Eliminate a and h from the equation 7 . cos mx y — a cos 7vx-\-o sin nx + — „ • w- - m^ 12. Eliminate a and h from the equation y = a cos nx + b sin nx + xsin nx. 13. Eliminate a, h, and c from the equation Wj, rig, W3 being the roots of the cubic ^3 +p2^ + qz-\-r = 0. 14. If 2/ = -46*" satisfies the linear differential equation prove that a is one of the roots of the equation 15. Show that for a given primitive equation involving x, y, and n In arbitrary constants, there are : — ^= — differential equations of the r*^ [r [n-r ^ order {r<:n), each involving n-r arbitrary constants, but that only r + 1 of these equations will be independent. [Boole.] 16. Eliminate a, b, and c from the equation y = (a + bx)e' + ce~*. 17. Eliminate a, b, c, and d from the equation y = (a + bx)cos nx + {c + c?a;)sin nx. 18. Eliminate the circular and logarithmic functions from the equation y--=A cos| ^log(a -\-bx)\ + B sinj Sog(a + 6a;) I- 19. Eliminate the function from the equation z = ysm-^- + {y). if 20. If z = - + (ay-bx), a prove ap + bq = l. 21. If 2: = Pn ^% Pi. P2, -^ PrT^ where A represents the operative symbol x-~ + y-y • 34. If {SHu-x), S^u-ij), .9i(^.-^)} = where S = x + y + z + u, show that <''-"l*<''->S-<''->s-»- I. 36. If .♦(5.r.s-- show that X 'du y du z 'du u a dx b dy c dz d 37. Eliminate the arbitrary functions from the equation .=/(|)f(«.). [Laqrangb.] CHAPTEK XVIII. EXPANSIONS. (Continued from Chapter V.) Theokems of Arbogast, Lagrange, Laplace and BURMANN. 515. Arbogast's Rule. Arbogast has given in his Galcul des Derivations (Stras- burg, 1800) a useful method for the expansion of (p(aQ-\-a-^x-{-a2X^-\-...) in a series of asceDding powers of x, (p being any arbitrary function. Taylor's theorem at once gives ^(Cto + tti^J + ^2'^^ + ^3^^ + • • • ) = 0(ao) + ^-^-^ (a^x + a^x^ + a^x^ + (i4X^ + • . • ) + ^-r^{a^x+a^x^ + a^x^-{-...y + ^^{a,x+a,x^-^.„y + ; or, expanding the several powers of this polynomial increment which occur, and arranging in powers of x, we have 448 EXPANSIONS. 449 Upon examination, it will appear that each of these co- efficients may be formed from the preceding one by differ- entiating each term with regard to the last letter contained and integrating with regard to the next letter, and then differentiating with regard to the letter next before the last and integrating with regard to the last. Professor Cayley (Messenger of Math., vol. V.) calls this the " rule of the last and the last but one." Arbogast estab- lishes it generally, but the proof is too complicated to find place here. ' 516. Maclaurin's theorem gives a method of expanding 2> in powers of x whenever the limiting values of z, -^, -y-^, ,•••, where x = 0, can be found. It is therefore specially adapted for the case in which z is expressed explicitly in terms of X. But in the case of the fundamental relation between z and X being implicit, the evaluation of high difierential coefficients is often tedious and difficult, and it is therefore advantageous to make use of theorems specially constructed to meet the requirements of this case. We therefore proceed to the investigation of Lagrange's and Laplace's Theorems. E.D.C. 2f 450 CHAPTER XVIII. 517. Lagrange's Theorem. Suppose to be a function of x and y defined by the equation z = y-\-X(f){z), (1) and let u be any function of z, say f{z) ; it is required to expand u in a series of powers oi x. Maclaurin's theorem gives u = u, where (^^- ) , (^r-n) , ... indicate that x is to be made zero Kdx/Q \dx^/o after the differentiations indicated are completed. The values of these expressions may all be calculated by successive differentiation, but the process may be much simplified, as we now proceed to show. It will be clear that |[^(«)S]=|[^(-)|] (^) where u is any function of x and y; for each side is equal to Differentiating equation (1) with regard to x and y, we obtain ^ = 0(0) + H\^)^ and --= i^xc^\z)~ Giving :g^=4,{z)l{\-x^'(z)], (3) 7)z ^= l/{l-x'{z)} (4) whence S = <^(*§ (5) If now u be any function of z, we have ?yii_du dz 'dx~dz' 'dx and 'du^dAJb ?>z_ 'dy'~ ^z 'by EXPANSIONS. 451 whence equation (5) becomes 9S = ^(">3^ ^^> We shall in a similar manner change the independent vari- able from X to y for each of the remaining expressions dx^* dx^' "" Thus we have ^, =^1(%] =^l^^'^^x} remembering that u is a function of z, and therefore, con- versely, z a function of u, and applying equation (2). Hence, substituting the value of — from equation (6), 3-=^^L<0(-)P9^J W Differentiating again, =3| D^(^)}^|] («) The general law indicated by equations (6), (7), (8), viz., may easily be proved by induction. For differentiating equa- tion (9) with respect to x, =^r^^^xU^^'^^%Ji =1= [|*»|-S} 452 CHAPTER XVIII. whence ^ ^ , ., follows the same law of formation as -—- : but equations (6), (7), (8) established the form for the special cases 71 = 1, 71 = 2, 71 = 3, and hence the form holds universally. In finding (^) » (^r^) > •••» ^ ^^ ^^ ^^ P"^ =^ after the differentiations are performed, but as all these differential coefficients are transformed into differentiations taken with regard to y, which is independent of x, it is permissible to make x = before effecting the differentiation with regard to y, and we shall therefore be able to write 2; = y and ^ —fXy) j ^^d ^^6n equation (9) gives (S)„=|^^[{^(2/)}"/(2/)]> and the development of u or f(z) by Maclaurin's theorem, viz., becomes /(^)=/(2/)+»<^(2/)/(2/)+ J ^[{^W}y'(2/)] + - Ex. Given ^ = 3,+f(22_i)^ (1) to expand z in powers of ^. Here f{z)=z a.nd 4>{z) = l{z'^-l), and therefore ^^[(3,)"/3/]=i ^,(3/^- ir, and Lagrange's theorem gives , From this result we may deduce an important expansion, viz., that of From Equation (1) ?=^~--\/l-2:pi/ + x^, (3) .27 .3? the negative sign being adopted, since when ;r=0 we are to have z=^. EXPANSIONS. 453 Differentiating the right-hand sides of (2) and (3) with regard to ?/, we have The coefficients P^, Pj, Pg, ... of the several powers of ;r in this expansion are called Laplace's Coefficients. "We thus have established p 1 d\y^-\Y ** 2«w! dy"" 518. Laplace's Generalization of Lagrange's Theorem. The result of the preceding article is due to Lagrange.* The proof is however due to Laplace, who has thrown the same theorem into a more general form, which is easily deducible from the foregoing. Suppose that instead of equation (1) of the preceding article we had z == F {y + X(l>{z)] , (1) and that it is required to expand any function of z, say f{z), in ascending powers of x. If we write y-hx{F(t)} taking the place of the simple functions f(z) and 2/, or reversing the order of the operations, = D{D-l){D-2) ... {D--n-\-l)y. Ex. The differential equation reduces at once to D{D-l){D-^)y + W{D-l)y + Wy + ^y = 0, or l)^y-D'^y-{-ZDy+Ay=0 by putting a? = e*. 462 CHAPTER XIX. 525. Transformation to Polars, and vice versa. It often happens that a result in Cartesians is much simpli- fied on reduction to Polars, or vice versa. In such cases we have x — r cos 6, 2/ = r sin 6. Suppose 6 to be the independent Polar Variable, then dy dy _dO dx ^ „ sm t' + r cos 6 dx dr do dO Similarly, d'y 1 d dx^' ~dxde dO cos — 0" sin -j^smO+rcosd do dr ^ • /I -Y7i cos d — r sin 6 do which easily simplifies down to 'dr + 2 dO dh; ^dO^ cosO dr do r sm >) 526. Suppose x and y to be expressed in terms of some third variable t, then it is easy to show that dx 'di ^-^+y-± dy dt dr '^^dt' dy dx _ ^dO " '^dt~^dt' ■(1) 'dt^dt~' dt' ^^^ -"" ©■+©*=©■+•'©■ ^' « Equation (1) is at once obvious by differentiating the equation x^-\-y^ = r^ with regard to t To prove (2). Let be the pole of the curve whose equation is obtained by the elimination of t between the expressions for x and y. Let P be a point on the curve whose co-ordinates are (x, y) or (r, 0), Q an adjacent point whose co-ordinates are by CHANGE OF THE INDEPENDENT VARIABLE. 463 Taylor's theorem ^* I or in Polars 7 The Cartesian and Polar expressions for the area of the triangle OPQ, when St is very small, are equivalent. Hence which gives in the limit ^^^~2/^ = '^'^7//' Formula (3) obviously represents the equivalence of the two expressions for ( ^ j . Arts. 200 and 201. All these formula may of course be established otherwise. Ex. Transform to Cartesians the formula t This we may write as dr dx , dy , dy dt dt '^ dt ^dof 527. Two Independent Variables. We shall now consider the case in which there are two independent variables x and y. Let U=^f{x,y) (1) Suppose x = ^{u,v)'\ y=^2(.'^> '^)/ be the proposed transformation; then we have du dx du dy du dU^W ^ dx -dU ^ dy dv~ dx dv dy dv. .(3) 464 CHAPTER XIX. These equations may be solved for ^-, ^-, giving dU ,^y dU dy -dU dw -dv dv du -dx dx /dy dx 'dv du -dU -dx dU dx dU dv du du dv dy dx ^dy dv dx 'dv dy dnjb .(5) and If, however, we could solve equations (2) for u and v in terms of X and y so as to express them thus, u = F^{x,y)\ v = Flx,y)\ we can find ^^, ;r-, ;:— , ::r-, and substitute in the formulae ox dy ox oy dU_dU_ ^ du dU ^ dv dx du dx dv dx aCr^BCT ^ 3it 3[7 ^ dv dy du dy dv dy 528. The differential coefficients on the right-hand sides of the equations of the preceding article are all partial. For dx instance, in finding the value of — from equations (2), v is diJ treated as a constant, while in finding — from equations (4), ox y is treated as a constant. The student should [therefore guard carefully against any such assumption as that ^ •;7~ = 1- For the truth of this equation was proved in Art. 55 on the assumption of a relation between u and x and no other vari- able, but this is not the case now considered. 529. The case of transformation from the Cartesian to the polar form deserves special notice. Here x = r cos Q, r = Jx?- ■\- y^^ y = r B,mOy O^tan"^*^, ^ X ■ CHANGE OF THE INDEPENDENT VARIABLE. 465 ;r- =COS0 = -, 3r r dr r = cos 6, r sin Now. dx W' 'dV_'dV dx dr dy dr dV_dV dr ~dx dV^dV dO dx X, dr dV dx'^dO di^ dV dy'^dO dx dO . ^dV , cosO dy dr r dV dd' dV 30' dV ^ dy_ '^]dV,y . '^y dr dy ' dr~ f^' dx^r jdy' dx dV dy_ _ dV dV ^dy ' de~ ^dx^^dy' dO ' dy dO Hence we have the following equivalence of Polar and Cartesian operations : — while d ^d sm d \ — = cos 6^ ^^, dx dr r d& d . . 3 cos 3 dy-''''^d^^^-r~ d& and either of these operators may be obtained from the other TT by changing Q into Also e. d 3 . 3 ^ r^^x~^y~^ dr 3 dx 3 and ^^ = ^^ 2/o • dd dy ^dx ) 530. It will be noticed here that dr dx dr ^ dy_ dy dr' dx ^ dO dO' dx dy W dO ' dy ^^ = cos20, dr sin20, sin20, cos^O, K.D.C. 2q 466 CHAPTER XIX. thus bearing out the observations of Art. 528 that such pro- ducts are not of the kind contemplated in Art. 55, and whose values are unity. 3217' 32y 581. To transform -^—^ and 7:-^ to Polar Co-ordinates. w I. ^'^ f^Wr f n'^ sine 3\2 Wehave -^^=y ^=H%-— W ^ dV siTiOdV^ sine dF 3F sinQ^F"] 2 a^F .^ sine cose 3^ , sin^e m^ sin^eBF 2sinecose3F ^^^ V ^y ^=H V.+^ ae) ^ . ^ar. .^F cose 3F"i cose ar . .3F coseaFn 3rL Sr r ^e J r 3eL Sr r se J _ • 2/3^^^. 2 sine cose 3^F cos^e B'^F -»in^3^2+ ^, "srse"^ r^ se^ cos^eaF 2sinecose BF r 3r r^ 3e" 532. Transformation of v^^- By addition we have It is easy to deduce from this result the corresponding trans- formation of the expression ^ "^'-dx^^dy'^dz"- to polar coordinates, the operator y^ standing for dx^'^dy^'^dz^' CHANGE OF THE INDEPENDENT VARIABLE. 4^7 The transformation formulae are now x = T sinOcos[' z = r cos I Let r sin = u, then x = u cos , 2/ = r sin 6 sin <^, ;s = r cos 6, , . 'dx 'dr prove (a) -=-, / ri\ dx n'dd <^) W^'\x ^^^ ddx d^dy «e)'HS)"Ht)*=©'H;5)"*y duV\ sin ^ 9<^/ 17. If F be a function of two independent variables x and y which are connected with two other variables r and 6 by the equations F^{x,y,r,e) = 0, F^{x,y,r,e)==0, show how to express -- and •— - m terms of — - and — -• ox oy or ou CHANGE OF THE INDEPENDENT VARIABLE. 471 18. If in the differential equation the independent variable be changed to 6, where x = e^, show that all the coefficients in the transformed equation are constants. 19. Show that by putting x- = s and y^ = t the equation is reduced to ''^ A'^.+ l (18 20. Transform the equation U^^ -€)■ ■t-ri(n + by putting X = cos 6. 21. If X^ + Z 2=1, show that the equation - -( ( 1 - a;^) — I + nln + 1 ) /^ = dxx. ax) becomes z{z^ - \f]~- + {2z^ _ 1)^ - n(n + 1)^/^ = 0. dz' dz 22. If ^ = i(~~ + j) show that the equation -— -j ( I - x'^)^-— \ + n{n + 1)P = dx L dx J becomes zHz^ - l)Vr + 2»3%- " ^(^+ 1)(-' " 1)^ = 0. 23. If a; = rcos^ I, and r = e', prove x^^ + 20^^-- + 3/^- = r -^r^- - 1 j. = _^- - 1 )., 24. If x + y = 2e cos <^, and x-y = lj -\ e^sin <^, show that + =4a;y— _-. B^^ 9<^2 -^^x^y [Oxford.] 472 CHAPTER XIX. 26. Prove that, if a; = r sin ^ cos <^, y = rsm 6 sin (/>, z = r cos 0, 9a;2 Si/^ '^^^ transforms into 3r andalsomto r-^^ ^ ^9^{(1 "/^V/ " 1^ ^?^^ ' where /a = cos 6. 26. Transform /*-,-^ +Q^, where P and ^ are functions of x only acc^ ax so that ^ may be the independent variable, where — = JP. 27. Transform the equation ry^ — '2sxy + tx^ =px + qy -z by putting x = y 28. If ^s be a function of x and ?/, and Z be written for px + qy- z, prove that if jo and q be taken as independent variables, = u cos v'y = -w sin v) dZ dZ d^Z t ^^Z s d'^Z r 3p 3^ B/y^ r^ - s2 9p3y r^ - s'-^ 3^^ r< - s^ , 29. Show that where A represents the operative symbol a?— + y— or r- in polars. 30. Prove generally that if x = e^ and y = e^, 31. If x-^e^, 2/ = e^, transform the expression CHANGE OF THE INDEPENDENT VARIABLE. 473 32. If u + LV=/{x+Ly\ where x and y are independent and w, v, Xj y are all real, prove that Hence establish that if ic = r cos ^, y = r sin 6, ^"^ 3^ "^ 3p ~ 9r^ "^ r Br "^ r2 3^2 ■ [Oxford, 1888. 33. Transform the equation 3^2 ^2/2 to polar co-ordinates. 34. Show by a change of rectangular axes that 9iC^ 9y2 ^^2 •'SyS^; 3;2;93; 3x9// may be transformed to ^^-1+5^+ C^- 07? oy^ 9^2 35. Under what condition can 9F. ,dV\ 97. .97 by a rectangular transformation be reduced to the forms A --, B~- .. 1 « dx ay respectively 1 36. If w=/(a;, y), x^ = ^i] and 2/^ = ^, change the independent vari- ables to ^, ^ in the equation «2^^ - 2a://^!^ + 2/2^ + 22/^ = 0, 37. If X, y be the rectangular, r, the polar co-ordinates of the same point, prove 'd^u 92i^_/92u\2_l 32u 92jA 1 9m92m_ 1/9^_1 9j^\2 9^" ' 9y2 V9(c32/ j r^ 3r2 ' 36'2 "^ r 3r 3r2 r2V3r3(9 r 3^J ' 38. The position of a point in a plane is defined by the length r of the tangent from it to a fixed circle of radius a and the inclination 474 CHAPTER XIX. of the tangent to a fixed line. Show that the continuity equation transforms into 39. Prove that ?)x^ dx^ dx.^ ' dx,^, w\dr\ or) doXu.ddJ'^ddXu.ddX 'dO„_\u^_,de„Jf where x^ = r sin ^^sin 6^... sin ^„_i, w^ = r^, ajg = r sin ^jsin 0^... cos ^„_i, Wg = r^sin^^^, ajg = r sin ^^sin ^2 • • • cos ^„_2, ^^3 = r^sin'^^jsin'-^^g' a?„_i = r sin ^^cos B^, x„ = r cos 6^, w^.i = r2sin2^jsin2^2 • • • sin^^^.g^ and Tr=r"-isin'*-2^iSin"-=*^2'--sin^n-2. [Math. Tripos, 1889.] 40. If Pi=fi{^, 2/» ^)i show how to change the independent variables from x, ?/. 2; to /Oj, pg, /93 in any partial differentials. If /)j, P2. P3 be a system of orthogonal surfaces show that the expres- sion ?!^+?!Z^+?T ^x^ dy- dz^ transforms into ' ^ 'lapiVAA api/ 3ft Wi 3^/* apsVAiA^ 3^3/'/ ".-(l')"*(l)'*©)' etc. [Math. Tripos, 1875.] CHAPTER XX, MISCELLANEOUS THEOREMS Jacobians. 534. Definition. Notation. If iip U2,...,Unhen functions of the n variables i the determinant dii^ du^ 'du^ du^ -dx^' ?m^ dx^' •••' dXn du^ du2 du^ 'dXi ^X^ ' ^Xn dUn dWn dUn 'dx-^ Sfljg' ' ^^» ^..Xr has been called by Dr. Salmon the Jacobian of u^, Ug, • . . , Un with regard to x^, x^, ... , ajn* This determinant is often denoted by d{x,,X,,...,xJ *^(^1' ^2— ^n;, or shortly J, when there can be no doubt as to the variables referred to. 535. The Jacobian of Three Curves. If u = 0, t; = 0, ^y = be the equations of three curves in any homogeneous co-ordinates, it has been shown that the polar lines of x, y, z with regard to these curves are respectively Xu^ + Yuy + Zuz = 0, Xv^+Yvy-hZvz=0, Xw^+Ywy-\-Zwz = 0. * Salmon, JSigher Algebra^ p. 78 and p. 292. 476 476 CHAPTER XX. These three lines are concurrent if lUrCy Wy, Wz I Thus the vanishing of the Jacobian of three curves indicates the locus of a point whose three polar lines are concurrent. Ex. Show that the Jacobian of three circles gives their orthotomic circle. 536. Prop. If any set of homogeneous equations he satisfied by a common system, of variables, the equation J—0 is also satisfied by the same system, and if the degrees are the same, the equations - = 0, — = 0, . . . will also he satisfied by the same , OdO Oil system. ^ For if u = 0, -^ = 0, %v — ^, ... be the equations, of degrees p, q, r, ... respectively, and x, y, z, ... the variables, Euler's theorem on homogeneous functions gives xu^ + yuy+zuz-\-. ..=pu, xvx+yvy-\- ...=qv, xwx+ytOy-jr ...=rw, etc., and solving for x we obtain xJ=p)uU-\-qvV-\-r%vW-\-... , (1) where U, V, W, ... are the co-factors of J corresponding to Ux, Vx, Wx, — Hence if any system of variables can be found to make u = v = w=...—0 simultaneously, that system will also make J=0. Again, differentiating equation (1) of the last article we have x~ + J=p[u^U+u-^^-j + ...=pu^U+qv^V -{-..., when u = v=...=0. Hence if the expressions u,v, ... were all of the same degree, we should have p = q= ... = n say, and and therefore for such a set of variables as simultaneously satisfy the equations u — 0,v = 0,tv = 0,... we have — = 0.* ox * The method of proof adopted is given by Dr. Salmon, Higher Algebra, p. 78. MISCELLANEOUS THEOREMS 477 Similarly, - = 0. 3^=0, etc. 537. If then the curves u = 0, i; = 0, 'm; = have a common point, the curve /= will go through that point, and further, if the curves be of like degree, we shall have dx dy so that J= will have a double point there. 538. Since the equation s=». Ux, Uy Vy u. Wz = is satisfied wlien u^ = Uy = u^ = 0, it goes through all the multiple points on the curve u = 0. Similarly, it passes through all the multiple points on any of the curves of the families u = a,v = b, w = c for any values of a, b, c. 539. The Hessian. The Jacobian of the first diflferential coefiicients u^, Uy, Uz of any function u is Ua U yz, Vyz Uzz and has been called the Hessian (Art. 311). 540. Prop. If J be the Jacobian of the system u, v with regard to x, y and J' the Jacobian of x, y with regard to u, v, then will JJ'= 1. Let u=f{x,y) and v = F{x,y), and suppose these solved for a? and y, giving X = (u, v) and y = \j^{u, v), we then have ■t _'du dx du dy^ dx du dy du ,^_du dx du dy dx dv dy dv. 0_dv Sa* 3v dy^ dx du dy du -, _dv Ba; B-y dy dx dv dy dv. 478 CHAPTER XX. Also T Tf_ ou ou du du dx dy X dx dy du du dv dv dx' dy dx dy dv' dv du dx du dy dx du dy du du dx du dy dx dv dy dv dv 9;:c , B'y dy^ dx du dy du dv dx dv dy dx dv dy dv 1, = 1. 0, 1 In the same way the theorem admits of proof if there be more functions and more variables than two. This theorem may be written d(u/v^_^ ^ d{x,y,...) ^ ^ d(x,y,...) d(u,v,...) 541. Prop. If U, V a^^e functions of u andv, where u and v -are themselves functions of x and- y, we shall have d(u, v) For let Now d(U,V) ^ d(U,V ) d(x, y) d(u, v) U=f(u, v), dU _dU du dx ~~ 9i/. ^ll_dU du dy du dx du dx dV dV du d{x, y) V=F{u,v), v = \Ir(x,y). dU dv dx dv dx dy dv dy dV dv dv dx + dy du dy dv dy d(u, v) d{x, y) dU^ du dV^ du dU dv X du dv dx dx dV du dv dv dy dy MISCELLANEOUS THEOREMS. 479 du dx dv dx 'du dx dv dx dU_du 'dUdv 'du "by dv dy dV du -dV ^ du dy dv dy ■dU ■dU zx! dy .-diujn^ d{x, y) ' and the same method of proof applies if there are several functions and the same number of variables. 542. The above propositions exhibit the curious analogy pointed out by Jacobi between these determinants and ordinary differential coefficients. 543. Prop. // u, v he connected implicitly ivith the inde- pendent variables {c, y hy the relations f^(x,y,u,v) = 0, f2(Xyy,u,v) = 0, d(x, y) 'd(u, V) 9(a;, y) we shall have For -dA %^ -^dv Zx 'dm 'dx 'dv 'dx M4.M?!^+^/i^^ = dy^du dy^dv dy ' 'df^ df^du df^dv^ 'dx 'du dx dv dx ' 'dy du dy 'dv dy Hence ?iIlJ^} d(u,v) d(u, v) -dixTy) df,du df,dv^ du 'dx dv 'dx df^du 'dl^'dv^ du dx dv dx '^'du^.df^dv du dy dv dy dfi du df^dv du dy dv 'dy _§A, ^% =Mi/2). dx dx dy dh dy d{x, y) 480 CHAPTER XX. 544. If there had been three independent relations with six variables u,v,w; x,y,z; it is plain that we should in a similar manner obtain 9(it, V, w) d(x, y, z) -^A -A 3/i dx ■dy ■dz _§/., ^%, 3/, dx 'dy ■dz _§/3, -%, _3/3 3a; -in, Zz = ( — \ y^il^J^-^ -dix, 2/, z) And in general, if there be n independent relations /i = 0, /, = 0, ..../n = involving 2n variables u-^, u^, ... ,Un and x^, x^, ... , shall have fn) 3(^1, U. Xn, then we Un) OyX-^, iCg* • • • f '^n) 545. Covariant. Definition. Let / be any quantic from which another function (p is derived in any manner, involving the constants and variables of the first. Let the variables of /and be changed by any linear transformation, the functions becoming F and #. Then if it be found that the ratio of $ to ^ is a power of the modulus of the transformation,

• • • > "^nj will vanish identically. Also conversely, if J is identically zero, some relation must subsist amongst the several functions. This result has already been established in the case of two functions of two variables in Art. 510. Consider the case of three functions u^, u^y Ug. Let the relation subsisting among them be f{u^,u^,u^ = 0. Then, for all values of the variables, 'du^ dx^ 3^2 dx^ Sitg dxj, ' df Bw-i , _§^ ^2 . ¥ ^3^0 'du^ 'dx^ du^ dx^ '^f '^\_^^ ?!L2 , ^/ ?^8^Q Sttj BiTg 'di62 SiCg 'du^ dx^ Hence eliminating -^ , -^, -^, we have ^ dUj^ dUc^ du^ dx^^ du^ -dx^ du^ du^ ^^3 dx^ dXo identically satisfied, i.e., 548. Conversely, let K.D.C. du^ dx^' dXc,' du^ dx^' SiCg' dx^ d{u^,u^,u^) d{x^, x^, x^) ' d (u^, u^, u^) 3(iCi, x^, x^) = 0. 2h 0. 482 CHAPTER XX. Between the equations connecting the u's and the remaining variables eliminate two of the latter (say 07^ and x^), and we obtain a relation between u^, Ug, u^, x^, say u^ = F{u^, u^, x^) (A) Now, OyX-j^y tZ/gj X^) 1, 0, 3(Ui, Ug, F) 0, 1, ZF ?)X^ SiCj 3a:2 ^^^3 3Uo St6„ Su« 0, 0, 1 Therefore for in forming the first determinant we are regarding u^, ttg, x^ as independent variables, and in the second x^, x^, x^ O^kC-ij t^Ot Xq) OXa 0\X-ty "^9/ Now the left-hand side by hypothesis vanishes, hence either (1) = 0. or (2) ?-(!^lL^ = o In the first case F is independent of x^, hence the quantity x^ has not appeared in equation (A) after the elimination of x^ and ajgj ^.nd therefore a relation between u^^ u^, Ug has been established. In the second case, viz., J: ^* — ^ = 0, d{x^, x^) no differential coefficients with regard to x^ occur, and there- fore x^ may be regarded as a constant. Hence by Art. 510 there is some relation between itj and Ug, which may however involve x^ as a constant. Let it be If fl?3 be eliminated between this equation and (A), there will result a relation between u^, u^, u^. By proceeding in similar manner the proof may be extended to any number of functions of the same number of variables. See Forsyth's Differential Equations, Art. 9. MISCELLANEOUS THEOREMS. 483 Some Important Operative Symbols. 549. The Operator ^. It has been shown in Art. 524 that the operator x^ be- comes -^- by the change in the variable x = eK Let this operator be denoted by the symbol ^. The fundamental properties of this symbol are (1) ^"aj'» = a"a;«, (2) 4>{^)x<' = ^(a)x<^, (4) 95)(^>c"u = ic"0(^ + ?i>. (1) The first of these is obvious — For ^x^ = x-f-af' = ax^, ax etc. etc. where n is any positive integer. For negative indices — Let ^-ia:« = 2/, therefore %y = x^= ^— , so that ^-iic* = a-^ij:^, supposing that no constants are added in the inverse operation. Hence also- %-'^x^ = a~ ^x^, so that the law (1) is true for any integral index. (2) If ^(2;) be any function of z, which is capable of expan- sion in integral powers of z, ^AnZ'^, say, @(0^)+|>Q(03)+.... which Dr. Boole * calls the secondary form of Maclaurin's theorem, and writes F{x) = F{D)e^ • '^. Examples. 1. ^-u^={E- iyu^=[E^ - nE*'-' + ^'':^E^-^ -...+(- l^K 1 . ^ = U^+n- nU^+n-i + ^^^^U^+n-2 -...+(- 1 ^U^ 2. ~=^u=log{l + A)u [fore^=l + A] cLv cue 2 3 * Finite Differences, p. 22. 486 CHAPTER XX. Similarly JL ^=[l„g(l+A)r»=^-./>.^,+,«i'.,-^,- .... rl dx"^ ° r! (r+1)! (r + 2)! (See Ex. 11, p. 80.) 3. Prove APx"'=(a;+py''-p{x+p-iy''+P^f^{j^+p-^)'^- .... 4. Prove F{e')==F{\)-^F{E)0 . a;+F{E)0'' . ^+ ... = F{Fy-'. [Herschel's Theorem.] 5. Deduce the secondary form of Maclaurin's Theorem from Herschel's Theorem. 552. Many other curious results may be established by means of these operators. For example, and writing for z the operator hJ) we have .;e)tT"(*^)-'-|+f;AO-t(W+§(W-.... and therefore 1+ ^'^ + i?^^ + . . . + .£:(«-i'^ = -r.,—4 ^[(Ai))-i-l + |v.Z>-|pZ))3+...](^«'^-l). Applying each side to the function <^'{x) we obtain '{x) + '{x + A) + 4>'{x + 2A) + . . . + '{x ■\-{n-\)h} ^{E^^-l)^^clix)-\^\x) + ^h<^'\x)-^^^ = 1[<^(^ + nh) - \x + nh) - {x)\. /I or 4ix + nh) - (x) = ^cfi^x) + (fi'{x+h) + ... + '{x + nh)] r=ao A2r + 2(^- ir^..-i(2^)j[<^''-(^ + ^^) - i>'%^)l [POISSON.] 553. Various Trigonometrical identities may be used to establish similar results. Ex. Taking the identity cos ^ — cos2^+cos3^-... to oo =^, we have {e^^ + e-'')-{e''' + e-'^') + {e'^^ + e- ■''')- ... = 1. d Writing for e'^ the operator e dx or E^ we get MISCELLANEOUS THEOREMS. 487- and applying this operator to (f>(x) we obtain (f)X=(f>{a;+h) - {x-Sh)-.... [Gbkgory. ] 654. The expansion of e" in powers of ze'^ by Burmann's Theorem (Ex. Art. 519), may be applied to establish a remarkable result due to Murphy, as follows : — Dividing by e' we have Replacing z by h we have the corresponding operative analogue i=.-^..+ V dxJ ^-2^ \ da:J ^-.H± 2! 3! and applying each side to the function /(:c) we obtain X^)=X^ - h)+^f{x - 2A)+ ^^-g^'g/'C^ - 3/0 + etc. Examples. L Establish the series |^^<^(^) = <^^+A)-i{x+h)-{x-h)'\-\[{x^2h)-{x-2h)'\ + i[<^(^ + 3A) - <^(a; - 3A)] - . . . = A<^'(^). [Gregory.] 3. Prove that h\z) = F\z) - Rf(z). Now,

{a+z)-i>(a)-z,t>'{a)-...-^£^.f-\a),...{D) then equations (A) are all satisfied. And if we -put f{z) = z^, equations (B) are satisfied. Also all the imposed conditions as to the continuity of F(z),f(z) and their first n differential coefficients are satisfied, and no differ- ential coefficient of f{z) up to the n^^ vanishes for a value of z intermediate between z = and z = h. Hence equation (C) is applicable ; and since F'\z) = (l>\a+z), Si.\\di f\z) = n\, it becomes F(h)= -(p\a-\-6k). Therefore by equation (D) ^{a+h) = , the perpendiculars on BO and PR from A are both ds cos

J . ri rv ^8 COS AOB=-T^ — , and APB = ^. B — 7' r Hence 1+i = |'^+<>OAt Ex. Show similarly that the radius of curvature of the envelope of a carried curve is given by the equation 1,1 _co8 <^ cos and let p be the perpendicular from A on the carried line. Then y^-.y^=i 2a sin ^, and x^ = h—p-i-acos\lr, X2 = h—p — acos\lr. Hence the tangential polar equation of the envelope is 2a sin xp- =fi(h — p -h a cos i/r) —/^(h —p — a cos >/r). 492 CHAPTER XX. Ex. 1. If the slots be straight, say y = Ax+B\ the result is of the form jo = X + /a cos i/^ + v sin t//-, where A,, /x, v are constants ; so that the locus of the foot of the perpendic- ular on the carried line is a limagon, and the envelope being its first negative pedal is therefore a circle. (See Art. 375.) Ex. 2. Suppose one slot elliptical and the other slot along the major axis, the distance between the pegs being the semi-minor axis. Show that the envelope of any line parallel to the minor axis is one of two circles, and that the minor axis itself passes through one of two fixed points. 566. Prop. Given three straight lines traced upon a lamina, and that two of them are made to touch two given curves. To find the envelope of the third. Let the three lines form a triangle ABC whose sides BC, CA AB make angles -i/r^, ^2> V^ respectively with a given straight line. Let p =fi{'^), p ^f^i^) ^^ ^^® tangential polar equations of the envelopes of BG and GA. Then ^^ = i/. + a| a and ^ being constants known in terms of the angles of the triangle. Also, if p^, p^, p be the perpendiculars from any fixed origin on the three given straight lines ap^ + bp^ + cp = 2Ay therefore the tangential polar equation of the envelope is cp = 2A^af,(ir + a)-bf,{xl. + /3). 567. Since ,^p-,^, and a^+h^l^^ = 0, we have by addition ap^^ + bp^ +cp = 2A. Ex. 1. It follows at once that if p^ and pg ^^^ constants p is also con- ijtant. Hence if two of the sides of the moving triangle envelope circles the third side also envelopes a circle. Ex. 2. Similarly if two of the sides touch respectively p = X\j/ + fx, jP = X't^+jw,'j the third will also touch a curve of the form ^ = A"V^+ft" These are the involutes of three concentric circles. Ex. 3. If two sides touch equiangular spirals with a common pole, the third side will touch an equiangular spiral with the same pole. Ex. 4. If two sides touch concentric epi- or hypo-cycloids, the third side will touch a parallel to an epi- or hypo-cycloid. MISCELLANEOUS THEOREMS. 493 Ex.' 5. If two curves fixed in a given lamina touch two given straight lines, show how to find the envelope of any straight line carried by the lamina. Hence show that the envelope of the axis of a parabola touching two perpendicular straight lines is the first negative pedal of a certain Cotea's spiral. 568. Many interesting results in this part of the subject will be found in Dr. Besant's " Notes on Roulettes and Glisettes,* to which the reader is referred for further information. MISCELLANEOUS EXAMPLES. 1. Sum the infinite series , . 1 1 _i J__ ib\ ^ 1 1 ^ ^ P + a;2 + 32 + a:2"^52 + a:2+-' and evaluate the results when a; = 0. 2. Prove that if JJipc) is the Bessel's function of the n"* order, [Math. Tripos, 1889.J 3. If 2/ = ("^ + ^')~^ prove that 3a^(a73)"+^//„ - (-l)"n! sin"+^6'|cosec"+'C(9 + '^') + 2"+-sin^T+T^-|^|, where x-\-a = aj3 sinf O + Z) I sin 6. ^ \ 6/ / [Prof. Anglin.] 4. If y = {a^ + a^x^ + x^)-\ prove that 2a"+-*siir"''^^2/« = ( - l)"n!|sin"+^^ sinfn +ld + '^\ + sin''+'<^ sinCn + 1<^ - l^j. where x = a cos( 9 + ^ )cosec 6 = a cos( 6 - ^ Jcosec . [Prof. Anglin.] 5. Prove that where P = x"- n{n - 1 )a;" "^ ^n{n-\ ){n - 2)(w - 3)ic"-* - . . . , and Q = nx''-'^ - n{n - l)(n - 2)x*'-^ + ... . [London, 1891.] 494 CHAPTER XX. Prove also 6. Prove that j- f ^°"^^" ^'' ) = e"^[P smjbx + n<}y) + ^ cos(6a; + nc^)]/a;"+\ dx\ X J where P = {rxy - n(rxY~'^cos + n{n - 1 )(ra;)"~^cos 2 \ xj dx\ ex+f J when aj = 00 . 17. If prove that Jf Vn- 2/ = sin(mcos '^Jx), 4n^ - m^ [London.] [Oxford, 1889.] 18. Show that if m, n, p, q be positive integers, the limiting value, when x-y -z — a of the fraction, x^i^y" -%'')-\- y^iz'' - a;") + z^x^ - y") mn{m - n) rn+n-p-g ~(m • 19. Find 20. If pq{p - q) Tt n 1.2.3...n -] [Math. Tripos, 1882.] [London, 1891.] show that ^.^^ + 2.y^-+f'S^ = n^?illzin-rJL, 'by? ' " dx'dy dy^ where the function F= (f>~^. 21. Prove ^'3 22. If prove that 3 9 3^3 3 •••' 9-. X^, X X„, X 2» ^3i ' V ^2> • 9 d dx^ d d ... , ^^ 0X„ 1 ^n-li ^ni '^V • ••J ^«-2 d 'dx„_, 3 '9-: 9 •^2' *^3' "^4' • .., (Ti dx^ 9^ 3 [OXFOI f e" = *^1> ^2' '^S' ••• ' ^« ^n» ^1> ^2' •" ' ^n-1 ) ^'2J ^3» "^4' ■.. . , a^i d^u ^^^+|^^ + ^^+...+^^ = (-irM^-l)!/(^i + a.,+ ...+^X 3£Ci' 9x/ 903, provided that r is not a multiple of n. = 0. [London.! MISCELLANEOUS THEOREMS. 497 23. Prove that the maxima and minima values of the fraction aoc^ -i-by^ +c + 2hxy 4- 2gx + 2/y a'x^ + 6'2/2 + c' + 2h'xy + 2g'x + 2f'y are given by the roots of the equation a-a'u, h-h'uj g-g'u h - h'u, b -h'uj /-/'u g-g'u, f-/'u, c -c'u 24. Show that if a triangle of minimum area be circumscribed about an ellipse, the normals at the points of contact meet in a point, and find the equation of its locus. [London, 1891.] 25. If g, 7, c are real quantities, the fraction — — ^ — -^ has ^' '^' ^ ' x^ + y^+ '2yx+c two critical values or none according as c is positive or negative, and interpret the result geometrically. [Oxford, 1890.] 26. Find the maximum area of a triangle which is such that the sum of the squares of the distances of the angular points from the centroid is constant. [Oxford, 1890.] 27. From a point P on an ellipse PS, PH are drawn to the foci and produced to meet the ellipse in Q and R ; PN is the ordinate of P. Show that when P moves up to one extremity of the major axis, ultimately QR : PN- 4e : (1 - e^). [Math. Tripos, 1882.] 28. A, B are two given points and KL a given straight line, find a point such that if OC be drawn perpendicular to KL, the sum of OA, OB, OC may be the least possible. [Coll. Exam.] 29. Given the volume of a paraboloid of revolution bounded by a plane perpendicular to the axis, find the maximum sphere that can be inscribed in it. [Coll. Exam.] 30. PF is a double ordinate of an ellipse, and from F is drawn a perpendicular FQ on the tangent at P. Find the positions of P for which the square of the area PQF is a maximum, and show that the value is really a maximum. [Oxford, 1889.] 31. With the foci of an ellipse as centres two fixed circles are described so as not to intersect the ellipse in real points ; show that the point on the perimeter of the latter at which the two circles subtend equal angles is that for which the sum of the four tangents from it to the circles is a maximum. [Oxford, 1888.] 32. If the equations of two curves are given in rectangular co-ordinates, show how to find the points on the first curve the normals at which will touch the second, and determine how many such points there are. [Math. Tripos, 1885.] E.D.C 2 1 498 CHAPTER XX. 33. Prove that for any constant value of fi the family of curves cosh X cosec 2/ — f^ cot y = constant cut the family /x coth x - cosech x cos y = constant at right angles. [London, 1890.] 34. In the curve whose equation is Xy2 -y=:OC^+ 2x^ + X + b the hyperbolic asymptotes are defined by the equations y = x +1+^, y=-x-l--~. [Hind.] 35. The equation of a curve is 2/2(a;2 _ 2/2) _ 2ax{x + 2y){x -y)- a\x + yf + 2^4 = ; show that the parabolic asymptote is (2/ - of = 2a{x - a\ and find on which side of the asymptote x = y the corresponding branch lies. [Math. Tripos, 1882.] 36. If the equation of the curve be .^(|)+..-y(|)+«"-x(|) where the equation <\>{z) = has two roots equal to /x, and /x is not a root of yl^{z) = 0, show that there are a doubly infinite number of parabolas meeting the curve in three points at infinity, and a singly infinite number meeting it in four points at infinity, and satisfying the condition of indefinite approach, and that the general equation of the latter is (y - i^Wip) + %{y - /xa;){3f (/x) - ^'"(/x)^(/x)/<^"(/x)} + 2,^(/x).; = c, where c is a constant. [Math. Tripos, 1891.] 37. Prove that when a curve is defined as the envelope of a line Ix + my = 1 moving subject to the condition <^(?, m) = the line is an asymptote approached by the curve at one end, but on both sides when the values of l^ m are those given by the equations 38. For any plane curve prove that [Math. Tripos, 1888.] 1 d'^x (Py as the old curve ; and that if p, p be corresponding radii of curvature nr r - = {n- l)sin <^. [London, 1887.] f. p 41. Show that the centre of curvature at any point of an ellipse is the pole of the tangent at the point with respect to the confocal hyperbola which passes through that point. 42. From F the centre of curvature at any point P of an ellipse, two other normals, UQ, ER are drawn. Prove that the locus of the point of intersection of QR with the normal at P is an ellipse, and that the line QR always touches the curve (xja)^ -\-{;ylb)^ = \. [Math. Tripos.] 43. Show that as we pass along a curve the tangent turns round more quickly than the radius vector, when logp changes its value more rapidly than log r. Prove that in all curves for which these lines turn round with equal speed the radius of curvature is propor- tional to either r or r^ : and hence show that these curves must be of one of the forms given by r = ce"^ or iHm 26 = c. [Math. Tripos, 1888.] 44. The envelope of a family of equilateral hyperbolas is a lemnis- cate if a vertex lie on the circle r = c cos 9 and the pole be the centre. [Coll. Exam.] 45. Find the equation to the envelope of a circle which rolls on an ellipse ; prove that the area between the two enveloping curves, formed by the circle rolling on the inside and outside of the ellipse respectively is twice the rectangle formed by the perimeter of the ellipse and the diameter of the circle. [Coll. Exam.] 46. A three-cusped hypocycloid moves without rotation in its own plane and always passes through a fixed point. Show that the tangent to the hypocycloid which is at right angles to the tangent at the fixed point envelopes another three-cusped hypocycloid, and determine its magnitude and position. [Math. Tripos, 1891.] 47. Prove that the envelope of the latera recta of all parabolas inscribed in the same triangle is a three-cusped hypocycloid. [Math. Tripos, 1887.] 500 CHAPTER XX. 48. Show that the axes of the conic of closest contact at any point of the curve whose intrinsic equation is (s-a)2,// = 62, are equally inclined to the tangent and normal at the point. [Math. Tkipos, 1887.] 49. Show that the equation of the conic of closest contact with the curve y =f{x) at the point whose abscissa is (x, y) is X'^-x\ 2{XY-xy), 72 _ y2^ 2{X-x), 2(Y-y) =0. 1 , 22/1 + 0^2/2, Vi+yy^y , 2/2 , 32/2 + ^yv 32/i2/2 + yy^^ ^ > 2/3 , 42/3 + 0:2/4, ^ViV^ + ^y2 + 2/2/4. , 2/4 50. Show that the locus of the centre of the conic of closest con- tact to the curve y^ - x^ is 322/^ = 5a;2. [Math. Tripos, 1891.] 51. Find the equation of the conic of closest contact at the point {x, y) of the curve y = x\ Show that the centre of aberrancy is at the point '2±±ix. -2"^1 .} . 2n - 1 ' n-2^ and show that its locus is similar to the original curve. 52. If p and q be positive integers such that q is not greater than /;, and f{z) any function of z which is continuous and finite, as also its differential coefficients up to the n^ inclusive, between the values x and x + h of the variable z, show that the remainder after n terms of the expansion of/{x + h) in powers of h may be written ^ ~ (»-!)!(/- + 1)! e^-- "^ ^■^ * ""'' 6 being a positive proper fraction. Deduce the forms of Schlomilch and Eoche, Lagrange and Cauchy. [Memoires de l'Academie . . , de Montpellier. *] 53. Show that sin(n + 1)- is the limiting value of — ^[ . _^ ) zi ux \sin X/ when X is zero. [Oxford, 1889.] 54. Show that one of the roots of the equation z^-2z^ + z-U^ = may be expanded in the form [Oxford, 1888.] * See Todhunter, Diff. Calc, p. 404. MISCELLANEOUS THEOREMS. 501 55. Prove that cos ax=l -ax sin bx — -—-- — 'a^cos 2bx + ^ " — ^ai^sin Sbx *J!i\ o! + ^itL — / a^^cos 46a:- ... . 4! [Math. Tripos, 1891.] 56. If ^^ + 2xy^'^ + 2(2/ - 2/3)|^ + x^yh = 0, then '^-l + 2u^^ + 2(i; - i;3)|^ + u^v^z = 0, where u = xy. v= - ,^ „ ^' y [Coll. Exam.] 57.*If the co-ordinates x and y be transformed orthogonally to f , t; and F be any function of «, y, then will dx^ ' V V^^y/ ~ 9p * V V^P^/ * 58. A curve PQ rolls on a straight line Ox, and P is the point of contact. If C be the centre of curvature corresponding to P and CT the tangent to the locus of C meet Ox in jT, prove tan era; = ^1, P where p = CP and p^ is the corresponding radius of curvature of the evolute of the rolling curve. Hence show that if for the rolling curve p = (8\ then the locus of the centre of curvature of the point of contact will be y = (x). 59. If an equiangular spiral roll along a straight line, show that the loci of the pole and of the centre of curvature of the point of contact are the same straight line. 60. If a catenary roll along a straight line its directrix always passes through a fixed point. 61. If any of the class of curves r*" = a*"sin mO roll along a straight line, the radius of curvature of the path of the pole =Vt±lr. m Examine the special cases m=-2, -I, 1 502 • CHAPTER XX. 62. The curve r'" = a"*sm tti^ rolls along a straight line. Show that the intrinsic equation to the e volute of the locus of the pole is s"* = a^Y 1 + — I sini/'. [Coll. Exam.] 63. If the curve r = 6sin-^ roll upon an ellipse whose axes are 2a, 26, and if the pole coincide originally with the extremity of the major axis, it will always lie on the major axis. 64. The equation of a curve is given in the form /(r^, r^ = 0, where rj, r^ are the lengths of the normals OP, OQ drawn from any point on the curve to two fixed curves. The perpendiculars drawn from the centres of curvature at the points P and Q of the fixed curves, at right angles to the normals at P and Q respectively, meet the normal at in N^ and N^- Prove that the radius of curv- ature o- of the locus of is given by (.in^-.i.,|^J/=|,...(^_l).|..,(^i.-; where a, )8 are the angles which the normal at makes with OP, OQ respectively and the diifferentiations on the left-hand side only affect/. [Math. Tripos, 1888.] ANSWERS TO THE EXAMPLES. ANSWERS TO THE EXAMPLES. CHAPTER I. Page 7. 1. (i.) CO ; (ii.) i ; (iii.) 00. 4. ±-. 7. 3a2. 2. (i.)i;(ii.)2. 5. i 8.(i.)^(ii.)f. 3. 00. 6. a. Page 17. 11. -0027 of an inch. 9. i CHAPTER II. Page 22. 1. X^+r^ = c2. 3. Y-y=y{X-x). i^. coB'x{Y-y) = X-x. 2. ^+§=1. 4. x{Y-y)=^X-x. 6. (l+^)(r-^)=A'-x Page 24. 1. sec*^. 2. -,. 3. - r^ . 4. 1+^^ sin^:r xJa^-\ Page 27. 1. 3:r. 4. e*. 7. ia'««^log,a. 10. -,4r-- X sin 2.r 2. y|. 5.g. 8.^. ll..^(log.xH.l). 3. J^_ . 6. a^'^'^cos.rlogea. 9. -tan.r. 12. a:'"»*|cosr.log^+^HL^\. 14. (!l!l£)^'(logx/^iri+a;cota:). I X J sjX 13. (sin a;)* { log sin .r+ A' cot A'}. 17. (0, 0) and ( 2a, - %^\ 18. ^^ ^' - ^' 505 606 ANSWERS TO THE EXAMPLES. 1. (i.) log sin .r + .r cot .r. (iv.) Page 37. -«3 /• • N a^-2a?2 l+Binx L+gJiix /"^a _ ^' (v.) a ^ cos^.logea. (iii. ) f e« ( 1 - f Y (vi. ) eV" J^ cot v (sin 'Z(?)"'(log sin w? + if; cot w\ X \ xj Ju 1. 2V^* 2. -k-i 3. 6 4. -^- 6. cosh^. CHAPTER III. Page 51. 8. 6cos(a + 6^). 15. Jl-x' 9. bnx''-'^QO^{a+hx''). 16. ~^- — }. ^[H-(log^)2] jQ eoSv^;r Jv'-^ 180 11. _g ^^. 18. log.r+1. 2^sin X 12. cos^_ jg^ ^\og{eaf). Aijx sm^x ^ 6. sinh^. 13. j9g'^«~^cos^sin^-^x'. 20. cose*, e*. log^ + ?H?^, X 7. {hxi-ba)IAc^^fxK 14. -S=. 21. logV^^t^ _ 2 tan- V sjl-a^ cosh.r sin 2^ 22. {x + af-\x + 6)"-^[(m + yi)^ + m6 + ?ia]. 23 ^^ + 2^-2 ^ 25. ?f(a2+^2)'if. 27. tanh^^r. ■ * {x + \Y n 24. l(a+^)^^ 26. -^f-. 28. sech2^. w 2^cosh:x? 29. 2 ^ 3Q^ _ 2cosec2^ /y2 — ;z?2 ^2 log cot X - (log cot ;r)2 31. ^^^.•^. . 32. -— !— . 33. ^ 1 + sin'-^^r l+x^ xjx^ - 1 3^^ ^^--2.y^-2;r^+l^ 35 sin'"-ia7cos"-iayi - (log xy^ - sin(e*log ^)ji:Z^^2 38. 1 40. ^-2a2^2 + 4«4 ( 1 - ;r)i(l + ^)^ (^2 _ a2)^(^2 _ 4^2)^ 39 _ 3^+^ 41. - 2 + 2^-^ (1 + .r2)f 2(1 - .r)^(l + .?; + .r2)t ANSWERS TO THE EXAMPLES. 507 42. ^(^-^>. 46 ^ :>|tan-^^+-^\. 43. log(^). cos~^a^ — Xijl — ar^ (l-:r2)^ 2 .o a sm(a cosec"^^;) 44 - . 48. ^ -. /^ \7i/ \ V ^71/ ;rj 6 + acos^ 50. 2,^-.«|!2|^+3^tan^|. 51. e«*Jacos(6taii-\r)- — ^sin(6t-an-^.r)l. 52. ara'^(2 + C3?loga«) gQ 2 0? log»e sin(logaVa=^ + ^) ^^i ^-1 * (a2 + ^2)cos2(logaVa' + ar^) ^-4 54 -A^. 62. 10*.10io'(logJO)2. 55.-1-,. 63. e'.c^ 66.^. 64.e^.x'(logx+l). 57. \ r-' 65. of' .e'Uogx-^-^X. X log X Xog^x Xog^x . . . log**- ^0? ( X ) lXs I 66. ar^..r'j(loga:)2+log.r+ll. 59. .J: - } . 67. o^'log ex - of hog -. 68. (sin &)~»*r ^^^ - sin x log sin :r "j - (cos a:y*'>*| ?HLf - cos x log cos x \ ^ sin 07 -^ Vcos^ ° / 69. - (cot xY*^ 'cosec^a; log c cot ^ - (coth x)'^^ *cosecli^^ log e coth x. 70. -^ a'^^»in« , (^,^,„,^ ^ ^-°- j ^ tan-^(a'^.r''^"^)-4f- - -rx-g- 71. ^ f . Vl-e2uu-i* 1+^ I sm — + COS— )( 1-sm— +C0S — )— ^ X X '\ X X ' X'' 72. 73. ^\^^-%^x . 4^xJ\ — x^J^x + cos-^:r(l + s^3C + cos-^:r) 508 ANSWERS TO THE EXAMPLES. 75. -ycot^(l + 2cosec2^1ogcos.^r). 76. -y|^-^§^„^ + - L_ _1 78. ^ ^^+/-ay ^ g^ ^^ « (^+r)sec^|-&^ ^-^ylog.r 79. cos :r cos 2^ cos V"''*. 86. U}^M 1+^log^logy^ ^log^ 1-^logy 80. -^^±M, 87. .yi («+M.y-M hx+by ' x(i/-x){a + bx)' SI ''^ ( ^ \^ 88 - ^^+%+ff ' 2^\a+6W • hx+by+f g2^ y tan a? + log sin y g^^ ,^ log cos .r - ^ cot y ' x «3. .r(3 + 2tanlog:r+tan21og:r). 90. 6^^(l+y^)tan^.6t-n-^. ^ 1 +3/2 - log SecV. gtan-ly 84. 3^(^~.y) . 91. l^gio^. x{x+y)' ' 2x^ S2. (1 + a^cos'^bxX^^ +ax+ a^y-^rn{2x + a)log cot - - cosec x{x^ +ax+ a^)] — ab sin bx 93. -^(a2cot|-62tan^Y 94 ^--- Vlog^+^:/IH?sin-i^). a24-52\ 2 2/ \ ^ / 95. 1 96. - > /l+-^ +v/l ^_ 97^ 2^ 98, _i 99, l. 1 00^ 2 ^^■'^ "^ x^)taiii-^x log tan~^^ +x J^ '{l+x^)ta,n-'-x{Jxco8^~^sinAjx) 110. -|I-r-^~1i^-(5^^ + 4^)sec-w4 121. Ar = m{m-l) ... (m-r+l). 123. Yzjx ^-1 (assuming ^<^>1). CHAPTER IV. Page 63. 3 . I ^,mr\ 3" ..?si„(.+«^'^)4si„(3.+-). 3. -JL|6"cos( 6^ + ^)-6.4''cos(4jp + !|1:U 15. 2"cos[ 2^+^)1. 4. -JL|5"cos(5^+^) +3«cos(3^+^2'^) -2cos(^+^|:) 1. ANSWERS TO THE EXAMPLES. 509 5.^|8'.cos(8.+!|r)-4.4^cos(4. + !|^)}. 9. - ^r(34f cos(5^-HJ taii-i|V(18f cosf 3^+^V2(107cosf ^-Hitan-il)1 Page 64. ■ a-b ((:c-a)"+^ (a;-6)"+M* 2 (-ir^lf 1 _ 3"-^^ ^ 7 l(a;-3)''+i (3jp-2)"+M* 3 (-ir^! f 1 _ 1 ) 2a ((a;-a)'»+i (A-fa)'*+M' 4 r-n--i?2' n^^+^)^^^+^) 4- ^<^+^>' + 4 _ 4 n ■^ ^ t 2(a;-l)"+=^ (^-l)''+='^(.r-l)'»+i (a;-2)"+U Page 65. 1 . Cz P"^!^sin(n + 1 )d sin"+^^, where x=acot 6. 2. (-l)""X^-l)? 8m nO sin"^, where x = a cot ft «»» 3. ( - 1)"?! ! cos(7i + \)d sin^+i^/a^+i, where ^ = a cot ^. 4 (n-l)! (( -l)"-i 1 ^ 2 \{a+xY {a-xf\' 6 ( - 1 r^^ !r sin(yi + 1)^ sm"^-^^ _ 8in(n + 1)<^ sin"+^<^ -] where x = h cot 6 = a cot ^. 7. 2( - l)"-i(?i - 1) ! sin nO sin"^, where .r = cot 0. 8. ( - 1 )"-i(7i - 2) ! sin"-^^ cos cos nd { n tan ^ - tan w(9 }, where x = cot ^. 9. (-l)"-X'i-l)!sinw^sin"^cosec"a, where cot ^=^'coseca- cota. 10. ( - l^i! I ^^l_^ + sec«+2|sin(9i+ l)^sm"+^^|, where .r=cosf ^+^ jeosec ^. where ^=^ ms(6l-|")=-A,cosf ?^, y^ = (). 19. y„=tli&L'/-_^L___J^\ 20. y„ = (-l)''-i^!/(^±2)M:^lj _^H^ 1 i . 21. If..heev.n ^ '^^-^>'^"~ (..- l)- + (^ri)^ " (^Z^prj- — H-a^ 2 «i / _, ^••=(-^)"^. r='|-i cos I ^ *•=! (^-2a^cos where and if m be odd cot(^,+cot?^=fcosec?!:^ m a m ' r yn={-iT n\ (^-a)-''-i + 22: ^=3' <^-i~ + ^^l- Tcos ^ + ^^=(J + i5)sin4^~,^(9. J 2/> !:iz> 2£> ^ JNormal, .vcos^±^e+ Yam^^e^iA- By-os'^^d. JFor an ellipse, ?'^'=a^cos^^+6-8in-'^. ^' \For a rectangular hyperbola, r-' = a-cos 2$. 7. 1--=- -— , t>., tliev must be confocal. a b a 9. The axes are tangents at the origin. Also at the point (2^a, 2^a) the tangents to the parabolas make angles tan~^2^, tan~^2"^ re- spectively with the tangent to the Folium. Page 149. (a) av = ± by. (ft) .r = and y = 0. {y) ax= ±yjb'-a\ Page 152. fA 1. - . -— — - . 8. Area = ^^«*;ry. 9. n=-±; n = \. sjx -ry Page 177. Ex. 18. p2=9a2(r2-a2)/(rHl5a2). CHAPTER VIII. Page 191. - 2a 1. x+y — — . 6. .r = 2a. 11. x=a, y=a^ x—y. «3 2. x+y=0. 7. x-{-y + a=(}. 12. x=±a. 3. x-\-y=0. 8. ^ = 0, y=0, x+y=0. 13. x=0. 4. ?/=0. 9. ?y = 0. 14. x=a. ,-,. .r = 0. 10. .r=±«. 15. .r=±l, y = .r. 1(5. .r=0, y=±(^+*jy 20.^-2y=0, .r+2y=±2. 17. x+2y = 0, .r+y=l, ^-y=-l. 21. .r+y=±2^/2, ^■ + 2y + 2=0. IS. .r=0, .r-3/=0, a?-;/ + l=0. 22. y = 3:c-2a, .c+3y=±a. 19. .y=0, x=yy x=y±l. E.D.C. 2 K ^^^ ANSWERS TO THE EXAMPLES. Page 192. 1. a^-6xh/ + Uxi/^-ef=-^v. 3. ^',+^2 = 1- ^' Ka'^b' JU 6 Jab a'^b' Page 198. 1. x=±a, y^x. Above. r?/=^ + a, y=-x-a, x=a. 2. |ln the first quadrant above the first. In the fourth quadrant below \ the second. Page 205. 1. ^=0. ^- '*^"^ ^=«. 3. wr sin( ^ - — ) =a sec Jcir, where ^ is any integer. 4. rsin (9=a. 5. rcos ^=2a. 6. 6I = |, ^ sin 6^=|. 7. r sin f ^ - ^ ^ = ^ j where k is any integer. \ n J n 8. nO = kTr, where ^ is any integer. Page 206. 1. (i.) x=y. (ii.) .^=3/. ^-+^=0. 5. .r=2y-14a, a;=3?/ + 13«, x-y = a, x-y = '^a. 6. x±ysl^=±^' 8. rsin^ = (X, rcos ^= -2«/(2?^ + lV. 9. rsin^ = a, rcos(9 = 2ac 2 /(2,i + l)7r. 13. a?+^ + a = 0, 18. (i.) y=0, ^-y-a=0, ^ + ^ + a=0. /•• \ / a. , 3a\2 a;xr 19. (x2_y2)2 = ^^ or ?-3=a3^^. 22. (^-/)'-4/+y=0. 20. 2?^2(^2 _ yi) = 3^3^. 23. x=± rt, y = 6, .?/ = c. 21. {x-yf{x+y-Vf-{x+yf=0. 28. 2?/-9c = 0, y+2^+|=0. 30. Linear asymptotes y=x+\, y^x-'i. Parabolic asymptotes {ij-x+ \f +^x=0. CHAPTER IX. Page 219. 8. Concave. 12. x=l and x==\. ANSWERS TO THE EXAMPLES. 5I5 Page 233. 6. A single ramphoid cusp. 8. A node at (1, 2). Directions of tangents ?/= ±x. 9. (a) Single keratoid cusp at (1, —1), (b) Two single keratoid cusps at (^, i.), ( - i^^ - 1.). (c) A single keratoid cusp at ( - a, a). 10. There is a triple point at which the tangents are parallel to the lines ;/ = 0, y= ±Xy/2. Page 247. 1. (a) i/ = 0. {/3) ax=hy. (y) \j^±x. (5) ^=0, y = 0. 6. x=a and ^=2a. 9. ^=±sin-Vf- 30. There is a single keratoid cusp and also a third branch having an inflexion at the origin, the latter touching the y-axis. The shape of the curve resembles the letter R. 31. The origin is a triple point, one branch touching the :r-axis and the others inclined to it at angles whose tangents are ± \% 32. The form of the curve is that of the •' Staffordshire Knot." The nodes are situated at («, 0), ( - a, 0), (0, - a) and the values of -^1 are respectively ±Vi ±Vi ±v/§. 'At (0, a), tani/r=±-? . , v^3 35 At.tr=2,y = 2wehave^=±l/2J2. 33. - At (a, 0), tani^= ±V§. dx At (2a, a\ tan i/r = + .l 36. At the origin and at (a^S, 0). 37. Two keratoid cusps at (0, +1) ; two nodes at (±^2, 0). Four conjugate points at (±-v/§, ±\^^)- 38. Three nodes at (0, 0) and (1, ±1). CHAPTER X. Page 260. 1. p = a; p=acoayJr; p=3asec^smyfr; p = asec>/r. 2. p = 2(a+x)ila^; p=f/c. 4. p=(ahm^e+b^cosW)^(ab. Page 265. 1. p = 2rt/a^; p=a/2 ; p=a"»/(»w + l)r^-\ 2. p = a{e^ + l)i/0*. Page 284. 1. Infinite. 2. p= -3a^2/2 or l5a^5lU. b.liy = ady />= -a(l+sin-^)^/cos^; ^/a = l -2cos ^+2sec ^ ; yja = 0- tan ^ - tan ^ sin^^. 2k2 516 ANSWERS TO THE EXAMPLES. 22. The radii of curvature are respectively a(cosh /? - cos a)^/sin a (2 cosh (^ — cos a)(cosh ^ + cos a) 2. and rt(cosh /? - cos a)"27siiih /3(cosh ^ - 2 cos a) (cosh ^ + cos a)i. 24. €^2^=1 +A2. CHAPTER XI. Page 296. 2. 256/ + 27.^4 ^ 0. 6. / + 4«(..^ _ 2a) = 0. 3. ^^ + ^=^1 7. Two straight lines. x'^ y^ a^ 4. i/ + 'hg%=^^' . 8. A parabola touching the axes. ({!) 4ar^ + 27af = 0: 9. A hyperbola. 1(2) r = 4/Ka + ^-^). Page 302. f(l) Jx-\-Jy = Jk. (2) jp»+i+y^^ = ;(;«+^ ,771 +»i (1) .r'+?/t = R (2) ^^+yf = yl'i _27)i_ _27«_ 2wt (3) .r"'+Hy'"+" = >t'"+2. (4) %vy=k\ Ul) x^+y^ = k^. O ' i» 111 m '^- 1(2) .»;-'" +Hy-"*+^=F'"+\ l(:i) 16^7/ = k Page 308. 1. 27«/ = 4(;r-2a)". 6. r2 = a2cos2(94.62sin2^. 3. x^+y^ = a^. 12. .ro+?/o=al 16. 3/2(^+16a)2 + 4{6/-(2a-.r)2}{y2_3«(2a_.^)}=0. 37. A parabola with the given point for focus. 39. aPh'^p^q''={p + qY'^''kP^'^- 40. A conic. CHAPTER XIV. Page 376. 1. logjo. 5. 4. 9. i 13. 1. 17. H. 21. 00. 2.f- 6. 4. 10. f. 14. 1. 18. -2 22. 1. 3. '5. 7. 2. 11. f. 15.1^.. 19. |. 23. C-*. 4.1. 8. 1. 12. i 2.5. 16. -J^. <'-^ 26. eK 20. 1. 24. 0. ANSWERS TO THE EXAMPLES. 517 Page 384 1. 2. 10. 1. 17. -1. 26. 1^- 2. i 'If n>m,=o. ^g Q 27. e 2 -tV- 3. 2. 11. - '^=='''^'^a 19.6,0. 28. 4. i ^ ?io, uiaxmium it a:=- 32. !!dl.la knots an hour. 40. J ^ . .^ . ^ n I a< 0, maximum if a; = a. {.a = b, gives a point of inflexion. /■re 1 ^ rt„«„+^c4. ^sinacosa y , ?^sin a cos a If cos a be >6, Greatest = ,- .„, Least = , --. (1-ecosa)^ (l+ecosa)2 43. i If cos a be < e, the above values are both minima, and there are two maxima each equal to — ^. 1 _ g2 44. The tangent at P must be parallel to *S'^. [If h< 2a, P is at the vertex. 45. 4 If A >2a, the abscissa of P is h-2a, and the perpendicular is there- in fore the normal at P. 46. Maximum area = 4r2sin a cos%, where r is the radius of the circle and 2a the given angle. CA 49. sin AOQ= ^^ ,, C being the centre. ANSWERS TO THE EXAMPLES. 519 CHAPTER XVI Page 429. 1. (a) Maximnm when x=^, 2/=h (/?) Minima when j:= ±^J2, y= ii^J'i. (y) Maxima when x= and when .r = — and minima when x= y= and when x = (8) Maximum value = 108aV7^. (e) A maximum when x=y = ^. o (D x=2/ = gives a maximum and x=i/= ±3 give minima. (l) x=y = agives a maximum or minimum according as a is negative or positive. 2. Minimum value =pV(<^^ + ^'^ + c^)- 3. Maximum value = m'"'7i"pPa'"-^"+^l(m + n +jt))**+ ♦•+''. 4. Maximum value =4. 5. A maximum when tan .l/m = tan B In =ta.n C/p. 6. A maximum value given by 0, c, 6, 2u n c, 0, a, 2u n ^ a, 0, 2u n 1, 1, 1, n =0, assuming that a, 6, c are such that a triangle could be constructed with these sides. 7. The results are the roots of the quadratic Z2aV(l -a2w) + wi26V(l -62i^) + 7iV/(l -c2t^)=0. 8. Volume = 8a6c/3V3. 9. {log(Ja6(?)P/log a? . log b^ . log c\ 10. If a^ be the given volume the parallelopiped is a cube of surface 6a^. 1, 1 '■f V^2 =0. 11. The root of 1, z.a'j zaa 1, 2aa', 2a'2 12. The solutions are respectively the roots of 0) a-.)a-«)(i-«)-- 520 ANSWERS TO THE EXAMPLES. (2) h, b f /, (3)y(«-i)-i-.y(6-i)^.y(.-i)=a (4) 1 h, fh K h-\ /; /; =0. 9^ /«. 1 I, m, n, 13. The values of jo, y, z are given by AV-7._?/g-"*_g''~" Ijpa mjqh nj/'C 14. a, X, y, £, 6 are to be in geometrical progression and the maximum value is (a* + 6+)"'*. 16. w = P/(a^ + 62 + c2 + ...). 17. The centroid. 18. It is such that each side subtends an angle of 120° there. 19. The faces should be equally inclined to the base. (3 jr\w/ - — ) {AiA^A-i ... J„) when F is the volume and Ai, A> ... An the n faces. 22. They are the roots of CHAPTER XVII. Page 436. .J c?V, ^^ /, dr 1. 2.ryi=y. 2. oi2i\-\-yy^=yy\' 1. xyi—x^+'iy. 1. ^2+^^=0. 2. y2-2wyi+^V=<>- 3. xyi + ^yx-xy=0. 5. y2-n^y=o. 6. y2+»iV=0- 3. ^•='3/2+^i+.y=0. 4. y3=0. Page 437. 2. {\+^)y^ = ay. Page 444. 4. y2-2yi + 2y=0. 5. y,-4ya + 13y=0. 6. yo-2myi+(m2 + ?i2)y=o. ANSWERS TO THE EXAMPLES. 521 7. {t/i +3^1' - ^1 Xyi" - 1 ) = %iy2^- 12. 3/2 + n^y = 2n cos w.r. 8. «/3(y-A'j/i) + 3r.?/,-=0. 13. y3+jt)y2 + ^ri + n/-=0. 9. xh/2+xy^+i/ = (^ 16. 2/s-y->-yi+2/ = 0. 10. ^2+3/1*'*+ 1 = 0. 17. y4 4-2^2^2 4- 7^42^=0. 11. i/2+nh/ = co^mx. 18. (a4-6^)^2 + K« + ^-^)yi + 'i"y = 0- 19. p=yl(y'^ — ic^)^. 37. {x^r—y'H)z+{z-pa:-qy){px—qi/)=0. CHAPTER XVIII. Page 455. whi<;h is true between 1 and - 1. If jc > 1 the series stands for — 16. ««* = 1 + «(2c'') + ('{a - 2hp^^'^~ + a{a - '^bf^^y +.... CHAPTER XIX. Page 46& dz^ \duj du^xdnj 6. ^+y = 0. 7. 4=^2 + 8. tan<^ = duy d$) ' 27. ^+4=0. W 'dtp 35. aia2 + 61^2=0. 37 2^"^''V-=0 PRINTKD AT THB UNIVKRSITY PRESS KY ROKERT MACLEHOSE, GLASOOW. 2. ^ 3. ^. RETURN T0»^ LUAN PERIOD 1 2 4 5 3546 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS DUE AS STAMPED RFinw iMT FnnonAPVinftN fiPR [ 0'-'"' uNiTorcRnrrBEW' FnPA>l^,r^ r^^^^ UNIVERSITY OF CALIFORNIA, BERKELEY FORM NO. DDO, 15m, 2/84 BERKELEY, CA 94720 ®i 3a \ :/ \idT ^.x'4 / if- '< ' UNIVERSITY OF CAUFORNIA LIBRARY L •♦ I fM^^^^^^