DIFFERENTIAL CALCULUS. o a B a Be -,~ O P O DIFFERENTIAL CALCULUS WITH APPLICATIONS AND NUMEROUS EXAMPLES: AN ELEMENTARY TREATISE. BY JOSEPH EDWARDS, M.A., FORMERLY FELLOW OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE. Xonbon: MACMILLAN AND CO., AND NEW YORK. 1886. [All Rights Reserved.] trOinteb iSt thSe LSniberSttp VReGss BY ROBERT M3ACLEBIOSE, ITWEST NILE STREET, GLASGOW. PREFACE. THE object of the present volume is to offer to the student a fairly complete account of the elementary portions of the Differential Calculus, unencumbered by such parts of the subject as are not usually read in colleges and schools. Where a choice of method exists, geometrical proofs and illustrations have been in most cases adopted in preference to purely analytical processes. It has been the constant endeavour of the author to impress upon the mind of the student the geometrical meaning of differentiation and its aspect as a means of measurement of rates of growth. The purely analytical character of the operator d- as a symbol and the laws of combination which it satisfies have also been fully considered. The applications of the Calculus to the treatment of vi PREFA CE. plane curves have been introduced at an earlier stage than usual, from the interesting and important nature of the problems to be discussed. At the same time, the chapters on Undetermined Forms and Maxima and Minima, which have been thereby postponed, may be read in their ordinary place if thought desirable. The direct and inverse hyperbolic functions have been freely used, and the convenient notation ', to denote partial differentiation, has been adopted. It is hoped that the frequent sets of easy illustrative examples introduced throughout the text will be found useful before attacking the more difficult problems in the copious selections at the ends of the chapters. Many of these examples have been selected from various university and college examination papers, others from papers set in the India and Home Civil Service and Woolwich examinations, and many are new. I have to thank the Rev. H. P. Gurney, M.A., formerly Senior Fellow of Clare College, Cambridge, for the kind interest he has taken in the preparation of this work, and for many useful suggestions. I have also been much assisted in the revision of proof sheets and in the verification of examples by J. Wilson, Esq., M.A., PREFACE. vii formerly Fellow of Christ's College, one of H.M. Inspectors of Schools, and also by H. G. Edwards, Esq., B.A., late Scholar of Queen's College. I hope therefore that the book will not be found to contain many serious errors. JOSEPH EDWARDS. 80 CAMBRIDGE GARDENS, NORTH KENSIN GTON, W., November, 1886. ERRATA. Page 82, line 1. - -For b"+ read bV.,, 258, Ex~. I (a).- Pior - read ~. ARTS. 1-2 3-9 10-12 13-17 18-24 25 26-36 CONTENTS. PRINCIPLES A ND PROCESSES OF THE DIFFERENTIAL CALCUL US. CHAPTER I. DEFINITIONS. LIMITS. Object of the Calculus,. Definitions,. Limits. Illustrations and Fundamental Principles, Undetermined Forms..... Four Important Undetermined Forms, Hyperbolic Functions,. Infinitesimals,. CHAPTER II. FUNDAMENTAL PROPOSITIONS. PAGES. 1 2-5 5-9 9-11. 12-15. 16-17. 17-24 37-38 Tangent to a Curve,... 26-30 39-41 Differential Coefficients, Examples, Notation,. 30-34 42-44 Aspect as a Rate-Measurer,...34-36 45-54 Constant, Sum, Product, Quotient,.. 37-41 55-57 Function of a Function,.. 42-44 58-62 Inverse Functions,.... 44-48 CHAPTER III. STANDARD FORMS. 63-67 68-73 74-81 82 Differentiation of x", ax, log x, The Circular Functions,... The Inverse Trigonometrical Functions,. Interrogative Character of the Integral Calculus,. 49-51. 51-53. 54-57 58 X CONTENTS. ARTS. 83 84 85 86-87 Table of Results to be remembered, Cases of the Form uv, Hyperbolic Functions. Results, Illustrations of Differentiation, PAGES. 59-60. 60-61. 61-62 62-65 CHAPTER IV. SUCCESSIVE DIFFERENTIATION. Repeated Differentiations, 88-89 90-94 95-97 98-100 101,72-73 d d as an Operative Symbol, Standard Results and Processes, Leibnitz's Theorem,. Note on Partial Fractions, *... 74-78..... 78-82 82-85 85-88 102 103 104-111 112 113 114-119 120 121-122 123-124 125 126-128 129 130 CHAPTER V. EXPANSIONS. Enumeration of Methods,.... 92 Method I.-Algebraical and Trigonometrical Methods, 92-94 Method II.-Taylor's and Maclaurin's Theorems,. 94-99 Method III.-Differentiation or Integration of known Series,..... 99-101 Method IV. -By a Differential Equation,..101-103 Continuity and Discontinuity,... 104-107 Lagrange-Formula for Remainder after n Terms of Taylor's Series,...107-109 Formulae of Cauchy and Schldmilch and Roche,.109-110 Application to Maclaurin's Theorem and Special Cases of Taylor's Theorem,. 110 Geometrical Illustration of Lagrange-Formula,. 110-111 Failure of Taylor's and Maclaurin's Theorems,.111-114 Examples of Application of Lagrange-Formula,. 114-115 Bernoulli's Numbers,....115-117 132-134 135-136 137-139 140-144 CHAPTER VI. PARTIAL DIFFERENTIATION. Meaning of Partial Differentiation,. Geometrical Illustrations, Differentials,. Total Differential and Total Differential Coefficient, 126-127 127-129 129-132 132-134 CONTENTS. xi ARTS. 145 146-150 151-152 153 154-155 156-160 161-168 PAGES. Differentiation of an Implicit Function,... 135 Order of Partial Differentiations Commutative,. 135-138 Second Differential Coefficient of an Implicit Function, 138-139 An Illustrative Process,... 141 An Important Theorem,. 141-143 Extensions of Taylor's and Maclaurin's Theorems,. 143-145 Homogeneous Functions. Euler's Theorems,.. 145-152 APPLICATIONS TO PLANE CUR VES. 169-171 172 173 174-178 179-181 182-183 184-186 187 188-190 191-193 194 195-199 200 201 —204 205-207 CHAPTER VII. TANGENTS AND NORMALS. Equation of Tangent in various Forms, Equation of Normal,. Tangents at the Origin, Geometrical Results. Cartesians and Polars, Polar Subtangent, Subnormal, etc.,. Polar Equations of Tangent and Normal, Number of Tangents and Normals from a point to a Curve of the n9th degree, Polar Line, Conic, Cubic, etc., Pedal Equation of a Curve, Pedal Curves, Tangential-Polar Equation, Important Geometrical Results, Tangential Equation,. Inversion, Polar Reciprocals,.159-161. 161-163. 164-165.165-169. 169-171.171-172 given.172-174. 174-175. 175-177. 177-181 181. 181-186. i86-187. 187-190. 190-192 208-210 211-213 214 215 216 217-218 219-220 CHAPTER VIII. ASYMPTOTES. To find the Oblique Asymptotes, Number of Asymptotes to a Curve of the nth degree, Asymptotes parallel to the Co-ordinate Axes,. Method of Partial Fractions for Asymptotes, Particular Cases of the General Theorem, Limiting Form of Curve at Infinity, Asymptotes by Inspection, 203-205 206 206-208 208-209 209-211 211-213 213-214 Xii CON TEN TS. ARTS. 21 222 223-225 226-229 230 231-233 234-235 236 Curve through points of intersection of a given curve with its Asymptotes, Newton's Theorem,. Other Definitions of Asymptotes, Curve in general on opposite sides of the Asymptote at opposite extremities. Exceptions, Curvilinear Asymptotes, Linear Asymptote obtained by Expansion, Polar Equation to Asymptote,. Circular Asymptotes, PAGES. 214 215 215-216 216-218 219 219-221 221-224 224 237 238-240 241-248 249-250 251-253 254-257 258-259 260-261 262-263 264 CHAPTER IX. SINGULAR POINTS. Concavity and Convexity, Points of Inflexion and Undulation,. Analytical Conditions, Multiple Points, Double Points, To exaniine the Nature of a specified point on a Curve, To discriminate the Species of a Cusp, Singularities of Transcendental Curves, Maclaurin's Theorem with regard to Cubics, Points of Inflexion on a Cubic are Collinear, 229 229-231 231-238 238-240 240-242 242-248 248-253 254-256 256-257 257-258 CHAPTER X. CURVATURE. 265-266 Angle of Contingence. Average Curvature, 267-268 Curvature of a Circle. Radius of Curvature,.. 265-266. 266-268 269-271 272-275 276-279 280 281-282 283 284 285 286-290 291-294 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 Infiexion, Co-ordinates of Centre of Curvature, Involutes and Evolutes, Intrinsic Equations,..268-269... 269-272. 273-275. 276-277. 277-278 278. 278-279. 279-281.281-285. 285-288 CONVTENTS. xiii ARTS. PAGES. 295-297 Contact. Analytical Conditions,.. 288-293 298 Osculating Circle,... 293-294 299-300 Conic of Closest Contact,.... 294-297 301 Tangent and Normal as Axes; x and y in terms of s, 297-298 CHAPTER XI. ENVELOPES. 302-303 Families of Curves; Parameter; Envelope,.. 311 304 The Envelope touches each of the Intersecting Members of the Family,......311-312 305 General Investigation of Equation to Envelope, 312-313 306-307 Envelope of AX22BX+ C = 0.. 313-314 308-311 Several Parameters. Indeterminate Multipliers,.315-318 312 Converse Problem. Given the Fanily and the Envelope to find the Relation between the Parameters, 318-320 313 Evolutes,......... 320 314 Pedal Curves,........ 321-322 CHAPTER XII. CURVE TRACING. 315-317 Nature of the Problem; Order of Procedure in. Cartesians,.... 330-333 318-319 Examples,.... 333-340 320-321 Newton's Parallelogram;... 340-344 322 Order of Procedure for Polar Curves,.344-345 323-325 Curves of the Classes r = a sin nO, r sin nO a,. 345-347 326 Curves of the Class r' = a" cos nO. Spirals,. 347-352 APPLICATION TO THE EVALUATION OF SINGULAR FORMS AND MAX1MA AND hMLIMA VALUES. CHAPTER XIII. UNDETERMINED FOPMS. 327-329 Forms to be discussed,.... 361-362 330 Algebraical Treatment,... 362-365 331-334 Form...... 365-369 335 Form 0x co,........ 369 xiv CONTEN TS. ARTS. 336-338 Form 0o.. 00 339 Form oo -.o, 340 Forms 0~, o~,..... 341 A Useful Example,. 342 dy of Doubtful Value at a Multiple Point, dx PAGES.. 369-373. 373 374 374 375 CHAPTER XIV. MAXIMA AND MINIMA —ONE INDEPENDENT VARIABLE. 343-344 345-347 348-349 350 351 352-353 354 355 Elementary Methods, The General Problem. Definition, Properties of Maxima and Minima Values. for Discovery and Discrimination, Analytical Investigation,. Implicit Functions, Several Dependent Variables, Function of a Function, Singularities,. 381-383.384-386 Criteria. 386-392. 393-395. 396-398. 398-400. 400-404. 404-405. 417-424 APPENDIX. 356-366 On the Properties of the Cycloid, ANSWERS TO THE EXAMPLES,.. 427-439 PRINCIPLES AND PROCESSES OF THE DIFFERENTIAL CALCULUS. CHAPTER I. DEFIN ITIONS. LIMITS. 1. Primary Object of the Differential Calculus. In Nature we frequently meet with quantities' which, if observed for some period of time, are found to undergo increase or decrease; for instance, the distance of a moving particle from a known fixed point in its path, the length of a moving ordinate of a given curve, the force exerted upon a piece of soft iron which is gradually made to approach one of the poles of a magnet. When such quantities are made the subject of mathematical investigation, it often becomes necessary to estimate their rates of growth. This is the primary object of the Differential Calculus. 2. In the first six chapters we shall be concerned with the description of an instrument for the measurement of such rates, and in framing rules for its formation and use, and the student must make himself as proficient as possible in its manipulation. These chapters contain the whole machinery of the Differential Calculus. The remaining chapters simply consist of various applications of the methods and formulae here established. A 2 DEFIVNITIONS. LIMITS. 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 wuhich, during any set of mathematical operations, retains the same value. A VARIABLE is a quantity which, during any set of mathematical 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 separate angles, the sides, the position of the vertex are variables. It has become conventional to make use of the letters a, b, c,.., a, 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,,,, 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 which assumes its value in~ consequence of some second variable or system oj variables taking up any set of arbitrary values that may be assigned to them. 6. Functions. When one quantity depends aupon another or upon a systemr of others, so that it assumes a definite valute DEFINI TIONS. LIMITS. 3 vwhen 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 function of another x is y=f(x); or?JyF(x), or y= ~(x); the letters f( ), F( ), ( ), X( ),. being generally retained to represent functions of arbitrary or unknown form. If u be an arbitrary or unknown function of several variables x, y, z, we may express the fact by the equation U =f(xy,, ). Ex. In any triangle, two of whose sides are x and y and the included angle 0, we have A =xy sin 0 to express the area. Here A is the dependent variable, and is a function of known form-of x, y, and 0, which are the independent variables. 7. It will be seen that we could write the same equation in other forms, 2A e.g., sin 0=, which may be regarded as an expression for sin 0 in terms of the area and two sides; so that now sin 0 may be regarded as the dependent variable, while A, x, y, are independent variables. And it is clear that if there be one equation between four variables, as above, it is sufficient to determine one in terms of the other three, so that any one variable may be regarded as dependent and the others as independent. 4 4DEFINITIONS. LIMITS. This may be extended. For, if there be one equation between n variables, it will suffice to find one of them in terms of the remaining (n-1), so that any one variable can be considered dependent and the remaining (n -1) independent. And, further, if there be r equations connecting n variables (n being greater than r) they will be enough to determine r of the variables in terms of the other n-r variables, so that any r of the variables can be considered dependent, while the remaining (n - r) are independent. 8. Explicit and Implicit Functions. A function is said to be EXPLICIT when expressed directly in terms of the independent variable or variables. For example, if z= x2, or z = sin 0, or z=./, or z = aye log x + (a +.r)": z is expressed directly in terms of the independent variables, and is therefore in each of the above cases said to be an explicit function of those variables. But, if the function be not expressed directly in terms of the independent variable (or variables), the function is said to be IMPLICIT. If, for example, ax' + yx - b =; or x+y3 + = 3 ax/; or ax2 + g2bxy + cy2 + 2dx + 2ey/ +f= 0; or..2y2 = (a2-?J2) (b + )2; y in each case is said to be an implicit function of x. Sometimes, however, we can solve the equation for y: e.g., the b - aIx2 first equation we can write as 7y= —, and in this form y is said to be an explicit function of x. It appears then that if the equation connecting the variables be solved for the dependent variable, that variable is reduced from being an implicit to being an explicit function of the remaining D EFI VITNIONS. IMI'TS. 0 variable or variables. Such solution is not, however, always possible or convenient. 9. Species of Known Functions. Functions which are made up of powers of variables and constants connected by the signs + - x - are classed as algebraic functions. If radical signs or fiactional indices occur in the function, it is said to be irratiional; if not, rational. All other functions are classed as transcendental functions. Of transcendental functions, sines, cosines, tangents, etc., are called trigonometrical or circular functions. Functions such as sin-1x, tan-'x, etc., are called inver.se triogonometrical functions. Functions such as e", a 2, in which the variable occurs in the index, are called exponential functions. While if logarithms are involved, as for instance in logex or log,,(a+bx), etc., the function is called logarithmic. Besides the above we have the hyperbolic functions, sinh x, cosh x, etc., of which a short description follows in Art. 25. 10. Limit of a function. DEF. When a function can be made to approach continually to equality with some fixed value or condition so as to differ from it by less than any assignable quantity, however small, by making the independent variable or variables approach some assigned value or values, that fixed value or condition is called the LIMIT of thie finetion~ for the value or values of the variable or variables referred to. 6 DEFINITIOTNS. LIMITS. 11. Illustrations. Ex. 1. If an equilateral polygon be inscribed in any closed curve, and the sides of the polygon be decreased indefinitely and at the same time increased in number indefinitely, the polygon continually approximates to the form of the curve, and ultimately differs from, it in area by less than any assignable magnitude, and the curve is said to be the limit of the polygon inscribed in it. 2x+ ' 3 Ex. 2. The limit of 2x+ when x is indefinitely x~1 2x +3 diminished is 3. For the difference between -- and.~~~x +1 3 is -; and by diminishing x indefinitely + can be made less than any assignable quantity however small. Hence it is said that the limit of 2 —+ when x is inx+1 definitely diminished is 3. 3 2+-X The expression can also be written -, which shows 1+ that if x be increased indefinitely it can be made to continually approach and to differ by less than any assignable quantity from 2, which is therefore its limit in that case. Ex. 3. The limits of some quantities are zero, e.g., ax2 + bx, sin x, when x is zero, 1- cos x, 1-sin, ) - xwhen x =cosX, ) 2 DEFINITIOJNS. LIMITS. 7; When the limit of a quantity is zero for any value or values of the independent variable or variables, the quantity is said to be a vanishing quantity for those values. It is useful to adopt the notation Lt,=a to denote the words "the limit when x= a of." Ex. 4. The sum of a? G.P. of which the first term is a, pfl- 1 common ratio r, and n the number of terms, is a _ ~ - a If r < 1, the sum to infinity is 1 -. For the difference is ar and since Lt=,;a,' =0 (when r < 1), this difference is a vanishing quantity. Ex. 5. We say '6=, by which we mean that by taking enough sixes we can make '666... differ by as little as we please from -. Ex. 6. The DEFINITION OF A TANGENT is another example. DEF. Let PQ be a chord joining P, Q, two adjacent points on a curve. Let Q travel along the curve towards P and come so close as ultimately to coincide with P. Then the limiting position of PQ, viz. PT, is called the tangent at P. P ---Fig. 1. Fig. 1. The angle QPT is a vanishing quantity; for it can be made less than any assignable quantity by making Q move along the curve sufficiently close to P. 8 DEFINITIONS. LIMITS. 12. There are several important principles with regard to limits which we shall continually require to use and which we may stay to enunciate here:(1) The limit of the stum of a finite number of quantities is equal to the sum of their limits. (2) The limit of the product of a finite number of quantities is in general equal to the product of their limits. (3) The limit of the ratio- of two quantities (whose limits are not zero or infinite) is equal to the ratio of their limits. (4) The limits of two quantities (whose limits are finite) are equal when the limit of their difference is zero. These statements are almost self-evident. We give however formal proofs of the most important. (1) Let u1, u2... be the variable quantities, vl, v2... their limits. Let uI = V1 + al,?'L'2 2 -+ a2, etc., where al, a2,... become less than any assignable quantities when the variables u,, uo, etc., approach their limits. Then i1+2b+ -... = (f t+al) + (v2+ a2) - ~ ~ =(v + v, +..-)+(a+a2+-...). Now, if a be the greatest of the quantities al, a,..., and if n be their number, a +a +... < na; and therefore, u1 + u2 +... differs from v +v2+... by less than na. But by hypothesis Lta = O; and therefore, if n be finite, Ltna = 0. Whence Lt(u1 + u. +...) = vv + v2 +... =Ltuz+ Ltu2+Lt3+ —.... (2) Again, with the same notation, ul2 = (vl +- al) (@2 + a2) = v1v2 + a2Vl + aV2 + ala2. DEFINITIONS. LAIMITS.. Now Lta~v1 =0; unless VI be infinite, LtaI1v2=0, indless 02 be infinite, Ltala2=0. Hence Ltu.,u2 =sum of the limits of the above terms = V?2= Ltu1 x LtU2. Similarly _Lt(qtIu2u3....u,") = Ltu.I Ltu,. Ltu3... Ltu??, sUpposinig none of these limits infinite. (3) Again 'a 2V2 + a2 V2 V2+ V02) VI ~ cV2- a2,V v2 V2(V2~ +2)':and if V1,v2, be finite Lt(alV2 - a2V1)=0; -and therefore also LaV 21~ v2(V2 + a2) provided v2 does not vanish. Hence Dlt0 - -I tu '02 V2 L[ttu2 The student will find no difficulty in establishing the fourth.Statement in za similar manner. Jn the same way may be proved (with certain exceptions) (5) Itl =(Ltu)" for positive, negative, etc., values of?i. (6) Ltaau - 0L (7) Lt logn,= log Ltu. 13. Indeterminate or Illusory Forms. When a function involves the independent variable (or variables) in such a manner that, for a certain assigned value of that variable, its value cannot be found by.simply substituting that value of the variable, the function is usually said to take an indletermninate for-m or to nassurne an 'indleterm'tnate -value. 14. The natne indetermiinate, though sanctioned by,commnon use, is open to objection, inasmuch as it will be found that the true values of such forms 10 DEFINITIONS. LIMITS. can in general be arrived at by means of certain processes which we shall hereafter discuss at length in a special chapter; whereas it would seem to be implied in the name indeterminnate that it would be impossible to obtain the value of a function to which that name was applied. " Undetermined " or " Illusory Forms " appear to be better designations for such cases. 15. One of the commonest forms occurring is when the function takes the form of a fraction whose Numeratorl and Denominator both vanish or both become infinite for the value (or values) of the variable (or variables) assigned. Several other indeterminate forms are treated fully in Chapter XIII. 16. The limit of the ratio of two vanishing quantities mnay be zero, finite, or infinite. 2 2 2 2 2x Ex. (iii.) Lt=o = Lto7 =c, infie. x2 22 Ex. (iv.) Ltx o - = Lt-(x + a) = a, finite. sin0 Ex. (v.) Lt,=o —I =1, finite. x-2 I Ex. (vi.) Ltx= 224 4 - = Lt=- c infinite. - 4x + -L_ 17. Two functions of the same independent variable are said to be ultimately equal when, as the independent DEFINITIONS. LIMITS. 11P variable approaches indefinitely near its assigned value, the limit of their ratio is UInity. sin 08 Thus Lt = 0 and therefore, when an angle is indefinitely diminished, its sine and its circular measure are ultimately equal. EXAMPLES. 1. Find the limit when x=O of -L X2' (i.) When y=mx. X2 (ii.) When Y=-X (iii.) When y= ax2+b. I + 2:x 2. Find Ltt + 2 (i.) when x=O; (ii.) when = oc. 2+x' 3. Find Ltx=OgL, when y2=2a - 4. Find Ltx=0' whenT -=. r q2 1 5. Find Ltx=o 4/+cV1 x 6. Find L tx=O, when y2=ax+ +bx2 +c, X3 - a3 i. Find Ltx=a X-a 8. Find Ltax + bx when (i.) =0; (ii.) = bX2 + ax 9 Find Lt=CL 4./( 4.! + x-.!). 10. Prove that p - qx and q -px tend to equality as x diniinishes to zero, but yet that their limits are not equal. 11. The opposite angles of a quadrilateral inscribed in a circle are together equal to two right angles. What does this become wheni in the limit two angular points coincide? 12. Find the ultimate position of the point of intersection of the' diagonals of a rhombus, when one of the angles diminishes indefinitely. 12 DEF.VNITIONS. LIMITS. 18. We now proceed to con'sider the limits of four very important indeterminate forms. 19. I. The proofs of the well-known results sin 0 Lt=o -— 1, Lto=o cos 0 = 1, tan 0 Lto=o - =1, can be found in any standard book on Plane Trigonometry. x — 20. II. 1itxlx Let x =1 +z. Then when x approaches the value unity z approaches zero, and we can therefore consider z to be less than 1, and therefore can apply the Binomial Theorem to expand (1 +)', whatever n, may be. xn "- (1 + )" - Hence Lt X —.=Ltz.=o ~. nz+ -... Ltz (n-1) } =Ltz=o2!.. 21. III. Lt. =x(l+= -=e, where e is the base of the Napierian system of logarithms. This number e is defined as the value of the series 1+ 1 + +... to, and it may easily be shown to be 2'7182818.... Since x is to be ultimately infinite, we may throughout DEFINITIONS. LIMITS. M3 - consider to be less than unity, and may therefore apply x the Binomial Theorem to the expansion of + We thus obtain (lI I x-1) 1 x(x-1)(x-2) 1 X X 1. 2 X2 1. 2. 3 - 3 + +13 + ~~~ 21 3 in the limit, when x is indefinitely increased. Cop. Lt (1~)= Ltx {(i a ax- 1 22. IV. Ltx = —0 =log'a.' Assume the expansion for ax, viz.: 2_ x"~X(logaSa) ax l+xl oga+- 2! 2! This is a convergent series, for the test fraction is x loga and can be made less than any assignable quantity by making 'a sufficiently large. We have therefore ___ av x(logea)2 __ =log'a + 2! + and the'limit of the right-hand side, when x is indefinitely diminished, is clearley loQa. 23. The following proof of II. is independent of the Binomial Theorem: DEFIVNTIONS. LIMITS. (i.) Let n be a positive integer. Then, by division. x_- I = x=n -1 + xn-.X2 - 3 +... + 2 + x. x-1 Putting x= 1, X1-1 Ltx= - = 1+1 +...+1+1+1, X-i there being n terms, (ii.) Let n be a positive fraction==-. q Let y =x, so that, if x=1, y= 1, and..?= p x,. " — 1 r. -1 Then Lt=a - = Lt -x1 -- x -- 1 -1 =- y. —1- 7lby (i.), and. =n. It,8g-l j XTn - 1 L - xn - 1 TJ~hen Lt,=X__i_ =L tx=l x- x-l x-1 x -=Ltx1(- ) X X= Ltx=l( - 41) x Lt.=x 1- -n by (i.) and.^tj x-1 (ii.), and.. =n. (iv.) Finally, if n be incommensurable, two numbers ni,,l, can be found, both commensurable, one on each side of n, such that their difference is less than any that can be assigned beforehand, however small. Hence xl x,, 12 are in order of magnitude; and therefore so also are Ltx=l- 1 Ltx x Lt x-1 x-1I DEFINITIONS. LIMITS. 15 and n1 LtZ=.,, n^; x-l but nl, n, are indefinitely nearly equal, and n lies between them,.Lt=x-1 =n in this case also. x-1 24. Also IV. can be deduced from III. thus: Let ax-1=, then a =1+-,.y.and therefore when x becomes zero y becomes infinite, and x=log(1 +1), 1 Lt.. tx t x = ----.,ylog(l +) log,.( +l1 1 l= og[L1(1 _ l=o- 1 [Arts. 12 (7) and 21.] Ioga[t{z, (+1 )] 10gae = logea. EXAMPLES. 1. Prove t= log x 1. x-1 [Put x==l-+.] 2 -a' _n..... 2. Prove.LtZ= Xn _ a" ~ 1.3. Prove Ltz=o(l +ax)X=e. 4. Prove LtX=o sin nx n.5. Prove Ltxoa - 1 - x logea (logea)2 X I~w 1 6 16 ~~DEFINITIONS. LIMITS. 253. 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 hyperbolicfitnction~s. e - e-10e0 ~ (",-o Since sin 0- and cos 0= whereI=V I it will be clear th at Sin tO t sinh 0, cos i0 cosh 0, atnd hence or from the definition sinh 0 (1) tan tO = t i- - = tanhO; cosh 0 (2) cosh2-1 -sinh120O==I (3) si ( u) = sin O coslb (p~cos O sinh q5; with many other formulae analogous to, and easily de (lucible fromt,the common formulae of Trigonometry. if x =sinh 0, wNe, have 0 = sinh-'x~, an Inverse~P hyperbolic function of x analogous to the inverse trigonometrical function sin-'x. This species of function howvever is merely logarithmic for, s~ince 2 we have eO=X +VA1~4~-i, and 0=ognx I +w) wh ile 'corresp onding results hold for cosh-'x, tanh-', etc. DEFINITIONS. LIMIT'S. 17 EXAMPLES. 1. Prove the following formulae(a) cosech28=coth28- 1; (b) sinh (O + ) = sinh 0 cosh q + cosh 8 sinh; (c) tanh ( + )= tanh 0 + tanh 1 + tanh 8 tanh q' (d) sinh 0 + sinh = 2 sinh cosh —. 2 2 2. Show that the co-ordinates of any point on the rectangular hyperbola x2 -y2 = a2 may be denoted by a cosh 0, a sinh 0. 3. Prove - (a) sinh-1x= tanh-1- x (b) 2 tanh-lx=log 1 + 1-x INFINITESIMALS. 26. 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 represents the ratio of the measured quantity to the unit. The magnitude of the unit should be chosen as something comparable 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 milts (roughly 4,000); the moon's distance in earth's radii (about 60); the sun's distance in moon's distances (about 400); the distance of Sirius in sun'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 B 18 DEFINITIONVS. LIMITS. end of the spectrum being about 266, that from the violet end 167 such units (Lloyd, "Wave Theory of Light," p. 18). 27. 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 allowable errors of observation, increases with the continual improvement in the construction of instruments and methods of measurement. 28. Orders of Smallness. If we conceive any magnitude A divided up into any large number of equal parts, say a billion (1012), then each A part 101 is extremely small, and for all practical purposes negligible, in comparison with A. If this part be again A subdivided into a billion equal parts, each = -- each of 1020 a A these last is extremely small in comparison with -,, and so on. We thus obtain a series of magnitudes, A, A A A ii02, *10, 10, "1, each of which is excessively small in DEFINITIONS. LIMI TS. 19 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. More generally, if we agree to consider any given fraction f as being small in comparison with unity, then JA will be small in comparison with A, and we may term the expressions fA, f2A, f3A,..., small quantities,of the first, second, third, etc., orders; and the numerical quantities f, f2, f3,..., may be called small fractions of the first, second, third, etc., orders. Thus, supposing A to be any given finite magnitude, any given firaction 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 pth order be a small fraction of the qth order, it is itself termed a small quantity of the (p+q)th order. 29. Infinitesimals. If these small quantities Af, Af2, Af3,..., be all quantities whose limits are zero, then, supposing f 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 first, second, third, etc., orders. From the nature of an infinitesimal it is clear that, if 20 2 DEFINITIONS. LIMITS. any equation contain finite quantities and infinitesimals, the infinitesimals may be rejected. 30. PROP. In any equation between infinitesimals of ditferent orders, none but those of the lowest order need be retained. Suppose, for instance, the equation to be Ax+B+C,+D+E+F+... 0,......... (i.) each letter denoting an infinitesimal of the order indicated by the suffix. Then, dividing by A1, B, C Ds E F' 1 + Bl+ A]+D+2+ + *..=..............(ii.) Al Al Al Al Al B C0 Dz Ei the limiting ratios A and A are finite, while A A2 A1 A- A) A1 are infinitesimals of the first order, 3 is an infinitesimal of the second order, and so on. Hence, by Art. 29, equation (ii.) may be replaced by B C 1+ A + A = and therefore equation (i.) by A1 +B +C = 0, which proves the statement. 31. PROP. In any equation connecting infinitesimals we may substitute for any one of the quantities involved any other which differs fromo it by a- quantity of higher order. For if A+B+ C,+ D +... = be the equation, and if A= Fl +f2,, fa denoting an infinitesimal of higher order than F~, we have F1 +B, + C +C f, + D.2... =0, DEFrlITIONS. LIMITS. 21 i.e., by the last proposition we may write F1+ + B,+ C, = 0, which may therefore, if desirable, replace the equation Ai+B,+C, =0. 32. Illustrations. 03 05 Since sin 0=0-. +5-. o! 5! 02 04 and cos0= 1-+4 -2! 4C. sin 0, 1 - cos 0, 0- sin 0 are respectively of the first, second, and third orders of small quantities, when 0 is of the first order; also, 1 may be written instead of cos 0 if second order quantities are to be rejected, and 0 for sin 0 when cubes and higher powers are rejected. 33. Again, suppose AP the arc of a circle of centre 0 and radius a. Suppose the angle A OP(= 0) to be a small. quantity of the first order. Let PNr be the perpendicular from P upon OA and A Q the tangent at A, meeting OP produced in Q. Join P, A. Q \ P O N A Fig. 2. Then arc AP = aO and is of the first order, NP= asin0 do. do., AQ=catan do. do., 0 chord AP = 2a sin do. do., LTA = ca(1 - cos 0) and is of the second order. 22 DEFINIITIONS. LIMITS. So that OP - ON is a small quantity of the second order. Again, arc AP- chord AP = - 2a sin, =0- 2a.j^- t+... =aO- 2at02 8.3!* (:03 4= 3 -- etc., and is of the third order. PQ - NA = NA(sec 0-1) 2 sin2 -=NA. ---_ cos 0 = (second order)(second order) =fourth order of small quantities, and similarly for others. 34. Such results may also be established without the use of the series for sin 0 and cos 0. PR / 1,1,^ B 0 N A 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. Join A P, BP, and let BP produced cut the tangent at A in R, and let the tangent at P cut AR in T, and draw the perpendicular PN upon AB. T will be the middle point of A R, and A T= TR= TP. (1) We may take it as axiomatic that the length of the arc AP is intermediate 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 A R are in ascending order of magnitude, and DEFINITIONS. LIMITS. 23 therefore 1, arc AP tangent AR therefore 1, are in ascending order of chord AP' chord A P magnitude. AR Now, Lt = Lt -1 chord A-1P Bt' whence Lt arc AP chord AP and therefore, if arc AP be reckoned a small quantity of the first order, the chord AP and the tangent A R are also of the first order of smallness. (2) Again, since AN AP and since AP is of the first order A P =A B) of smallness, ANV is of the second order. PR BP (3) Also P = BP, which is ultimately a ratio of equality, and therefore PR is also of the second order. (4) Similarly, since AR-AP= -A R2 -A P and since AR+AP AR+AP PR2 is a small quantity of the fourth order; and AR + AP is a small quantity of the first order, we see that A R-AP is of the third order of small quantities. And similarly for other quantities the order of smallness may be geometrically investigated.:35. The base angles of a triangle being given to be. small quantities of the first order, to find the order of the (,ifference between the base and the sum of the sides. P A M B Fig. 4. By what has gone before (Art. 33), if APB be the triangle and PM the perpendicular on AB, AP-AM and BP-BM are both small quantities of the second order as compared with AB. 24 DEFINITIONS. LIMITS. Hence AP+PB-AB is of the second order compared with AB. If AB itself be of the first order of small quantities, then AP+ PB- AB is of the third order. 36. Degree of approximation 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 concave at each point throughout 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 togetler, 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 difference 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. DEFINITIONS. LIMITS. 25 EXAMPLES. 1. Show that, in the figure of Art. 33, the area of the segment bounded by the chord AP and the arc AP is of the third order of small quantities. 2. In the same figure, if PM/ 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 A a CB PA CB2 and Bb CA PB CA 2 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 perpendicular to the radius OA. Show that, in the limit, when P moves up to A, NA -AT. 5. If, in the equation sin( - 0) = sin o cos a, 0 be very small, show that its approximate value is 2 tan W sin 1 - tan2 sin2). [I. C. S. EXAM.] 6. 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.] 7. If G be the centre of gravity of the arc PQ of any uniform curve, and if PT be the tangent at P, prove that, when PQ is indefinitely diminished, the angles GPT and QPT vanish in the ratio of 2 to 3. [I. C. S. EXAM.] CHAPTER II. FUNDAMENTAL PROPOSITIONS. 37. Direction of the Tangent of a Curve at a given point. Let AB be an arc of a curve traced in the plane of the the paper, OX a fixed straight line in the same plane. B A _P R 0 T AI N X Fig. 6. Let P, Q, be two points on the curve; PM, QN, perpendiculars on OX, and PR the perpendicular from P on QN. Join P, Q, and let QP be produced to cut OX at T. When Q, travelling along the curve, approaches indefinitely near to P, the chord QP becomes in the limit the tangent at P. QR and PR both ultimately vanish, but the limit of their ratio is in general finite; for LtR =- Lt tan RPQ = Lt tan XTP = tangent of the angle which the tangent at P to the curve makes with OX. FUNIDAMENTAL PROPOSITIONS. Ex. 1. Consider the straight line whose equation is y=mx-+c.. Y P R o M N X Fig. 7. Let OX, Y, be the axes, and let the co-ordinates of P be x, y. Then, taking the general construction of the preceding article, the intercept OA = c, for y= c when x=0O. Draw AK parallel to OX to meet AP in A'; then, from similar RtQ KP MiP- OA triangles, R K OA PR - ARK- OM y-c mx - =- -'n. x X Hence tan XTP = tan RPQ=m. Ex. 2. Consider the parabola referred to its usual axes, viz., the axis of the parabola and the tangent at the vertex. With the same Fig. 8. construction as before, we have PM2=4AS. AM, QN2= 4AS. AN, QN_ P2 = 4AS(AN-A M) = 4AS.PR. 28 FUNDAMENTAL PROPOSITIONS. But QN2 -PM2=(QN- PM)(QN+PM) =RQ. (QN PM),.. RQ(QN+PM)==4AS.PR, whence RQ A 4AS 4AS whence t 2Pwhen Q comes to coincidence with P, and therefore in the limit tan XTP=2A S Ex. 3. Consider the "curve of sines" whose equation is - sin - b a The same construction being made, if P be the point (x, y) on the,- -.- -- —.f / -W — ^- ' / Fig. 9. curve we have MP= b sin. a x+h Let fMN= h, then NQ b sin — +h. a Hence RQ=b sin +h- sin - 2sin cos2x- h, - a =bsia 2a 2a and therefore sin hcos 2x h (sin h 2RQ Lt 2=o Lta ob.l2a s2x+h Lt-Lt-h ---6 -=Lthbo-k- PR h 2a bo. = b cos~. a a Therefore in the limit tan XTP -cos. a a FUNDAMENliTAL PROPOSITIONS. 29 From the above examples it will now be obvious that the direction of the tangent at any point of any curve may be determined in a similar manner. 38. Equation of Tangent. Let us consider the general case in which the equation of the curve is y = (x). Y// R OT M N X Fig. 10. Let the co-ordinates of the points P, Q, on the curve be (x, y) (x+Sx, y+Sy) respectively, dx and Sy being used to denote increments of the variables x and y. Then, the construction being as before, OM= x, ON= x + Sx, therefore PR = MN= x; also, MP = y, NQ = y + Sy, therefore RQ = Sy. Again, since the point x + 6x, y + Sy, lies on the curve, y+ Sy= 0(x + x), whence RQ = Sy = 0(x + Sx) - ((x). Hence we can express Lt-R as Lta=o6 or Lt_ (X+ x) - 0(x) Hence, to draw the tangent at any point (x, y) on the curve y=o(x), we must draw a line through that 30 FFUNDAMENTAL PROPOSITIONS. point, making with the axis of x an angle whose tangent is Lta=05(x + x) -(x); and if this limit be called n, ex the equation of the tangent at P(x, y) will be Y- y = r(X- x), X, Y being the current co-ordinates of any point on the tangent; 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 in. EXAMPLES. Find the equation of the tangent at the point (x, y) on each of the following curves:1. x2+y2=c2. f 4. y=logx. x2 y2_l 5. y=tanx. a2 b2 6. y=tan-1x. 3. y=ex. 39. DEF.-DIFFERENTIAL COEFFICIENT. Let p(x) denote any function of x, and (x + h) the.sacne function of x+ h; then Lth=o(+ -h)- ~(x) s called the FIRST DERIVED FUNCTION or9 DIFFERENTIAL COEFFICIENT of q(x) with respect to x. The operation of finding this limit is called differentiating 0(x). After reading Chap. V., it will be obvious 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 h. The geometrical meaning of the above limit is indicated in the last article, where it is shown to be the tangent of the angle qr which the tangent at any definite point (x, y) on the curve y = ((x) makes with the axis of x. FUNDAMENTAL PROPOSITIONS. 31 40. We can now find the differential coefficient of any proposed function by investigating the value of the above limit; but it will be seen later on that, by means of certain rules and a knowledge of the differential coefficients of certain standard forms, we can always avoid the labour of an a priori evaluation. When an a priori investigation becomes necessary, it may often be conducted very simply by pure geometry. It is however usual to treat the more complicated functions algebraically. Several examples are appended. Ex. 1. To find geometrically the differential coefficient of sin x. Let the angle A OP=x, AOQ=x + h, and let a circle with centre 0(.and radius unity cut the lines OA, OP, OQ, in A, P, Q. Draw per/^ ^.. O IN I A Fig. 11. pendiculars PM, QN, to OA, and PR to QS.. Join PQ. Then MP = sin x, NQ = sin (x + h),. sin(x+ ) -sinx= RQ. / Again, h=angle POQ= arc PQ, the radius being unity. Hence Lt sisin( h)-_sin RQ = Lt It -h -are PQ chord P'( (for chord PQ and arc PQ are equal in the limit) = Lt cos RQP = cos OPR (since in the limit QPO is a right angle) = cos A OP = cos x.? 32 32 ~FUNDAMIENTAL PROPOSITIONS& Exi 2. To find geometricall~y the differential coefficient of -siir<'x. A In Fig. 11 let A OP-= sin- 1x and AOQ-siin'(x+/h). Then, with the same construction as before, 31P XI, NQ= +h, therefore RQ=h. Hence A' A sin'1(x+h)-sin'1x=L A OQ -A 01' h t A UQ A ~LtPO=L R2YUQ Lt 1 1 RQ RQ Cos R'QP Cos 0IPR cos A OP Vi-sin2AOP EXAMPLES. Find in a similar manner the differential coefficients of (1) tan x. (3) cosec x. (2) tan k1x. (4) coseck-ri. Ex. 3. Find from the definition the differential coefficient of X2 -, where a is a constant. a Here p -r =7 a a therefore Ltl,='O~" + )-/x Lthl= +A) A ~~~ha =Lth,0 hhLt,.(2x + ) ha a 2x a The geometrical interpretation of this result is that, if a tangent be drawn to the parabola a~y= X2 at the point (x, y,), it will be inclined -to the axis of xat the angle tan.'2x a FUNDAMENTAL PROPOSIT1IONS. 33 Ex. 4. 'Find from the definition the differential coefficient of log sin x, where a is a constant. a Here (x) = log sin a, al log sin +h _ log sin x and Lth=x + ) - () Lt=O a a h h sin - cos - + cos x sin o a a a a = Lth=o- log s X sin - a =Lth=o log (I + cot - higher powers of hA by substituting for sin and cos - their expansions in powers of [by surbstituting for ofA]h h v - cot' - higher powers of h Lth=oa a = hLt= --- h [by expanding the logarithm] 1 x =cot. a a Hence the tangent at any point on the curve -=log sin is a a inclined to the axis of x at an angle whose tangent is cot x; that is at an angle - -. 2 a' 41. Notation. The result of the operation expressed by Lt=oh(x+h-(X) d dy or by Lt =o6y is generally denoted by dy or d. It will be well to note distinctly once for all that in the notation thus introduced, dx and dy, as here used, are not separate small quantities as 6x and Sy are, but that c 34 FUNDAMELVTAL PROPOSITIONS. d A is a symbol of operation which, when applied to y, denotes the result of taking the limit of the ratio of the small quacntities ~y, 8x. Sometimes dxy is used to denote the same thing; or, if y = ((x), we often meet with the forms dr,() d (x) ox, O', or A. Again, as the letters u, v, w, etc., are frequently used to denote functions of x, we shall consequently have the differential coefficient variously du expressed, as d-, u', uu, or u, with a similar notation for those of v, w, etc. 42. Aspect of the Differential 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 an additional space ds to be described in the additional time st. Let v1 and v2 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 v1,t FUNDA MENTAL PROPOSITIONS. 35 and v,%t, and are respectively greater and less than the actual space 8s. Hence v1, -, and v2 are in descending order of magnitude. If then st be diminished indefinitely, we have in the limit v = == the velocity at the instant considered, which 68. ds is therefore represented by Ltte, i.e., by dt. 43. It appears therefore that we may give another interds pretation to a differential coefficient, viz., that t means the rate of increase of s in point of time. Similarly dt dy 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 path of the particle defined, and if dx, Sy, 6t, be simultaneous infinitesimal increments of x, y, t, then 6y dy dyo da dt dt dx - Sx dx 6t 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 dy di if we choose it to be unit velocity, then d- =- = absolute rate of change of y. 44. Meaning of Sign of Differential Coefficient. If x be increasing with t, the x-velocity is positive, 36 FUNDAMENTAL PROPOSITIONS. whilst, if x be decreasing while t increases, that velocity is negative. Similarly for y. dy dy dt dy. Moreover, since - = -, -d is positive when x and y dx dx' dx dtogether, but increase or decrease together, but negative when one increases as the other decreases. This is obvious also from the geometrical interpretation of d-. For, if x and y are increasing together, dy is the tangent of an acute angle and therefore positive, while, if dy as x increases y decreases, d- represents the tangent of an obtuse angle and is negative. EXAMPLES. Find from the definition the differential coefficient of y with respect to x in each of the following cases: 1. y= X3. 8. y = tan - x3. 2. y&a= 2 x. 9. y =log cos x. 3. y= Ja2 + x2. 10. y=logtallx. 4. y = e. 11. y=x. 5. y= eV/. 12. y=xsin. 6. = asi x. 13. y=(sin x). 7. y = aogx. 14. y=(sin x)/O. 15. In the curve y=cee, if V be the angle which the tangent at any point makes with the axis of x, prove y=c tan V. 16. In the curve y =c coshx, prove y =c sec V. 17. In the curve b2y=,3 -ax2 find the points at which the 3 tangent is parallel to the axis of x. [N.B.-This requires that tan '= 0.] FUNDAMEN TA L PROPOSITIONS. 37 18. Find at what points of the ellipse — + t-= 1 the tangent cuts off equal intercepts from the axes. [N.B.-This requires that tan i= ~ 1.] 19. Prove that if a particle move so that the space described is proportional to the square of the time of description, the velocity will be proportional to the time, and the rate of increase of the velocity will be constant. 20. Show that if s cc sin ut, where /A 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. 45. It will often be convenient in proving standard results to denote by a small letter the function ofx considered, and by the corresponding capital the same function of x+h, e.g., if u= p(x), then U =q((x+h), or if.m=a, then U= ax+h Accordingly we shall have d(lu? y U- u b Lth=O h' dv V-v dxLth_ o h etc. 46. We now proceed to the consideration of several important propositions. 47. PROP. I. The Differential Coefficient of any Constant is zero. This proposition will be obvious when we refer to the definition of a constant quantity. A constant is essentially a quantity of which there is no variation, so that if y = c, dy = absolute zero, whatever may be the value of ~x. Hence = 0 and d -0 when the limit is taken. 62 dr X 38 FUNDAMENTAL PROPOSITIONS. Or, geometrically; y =c is the equation of a straight line parallel to the axis of x. This makes an angle zero with that axis, and therefore tan r or d= 0. d d48. PROP. II. Product of Constant and Function. The differential coefficient of a product of a constant and a function of x is equal to the product of the constant and the differential coefficient of the function, or, stated algebraically, d du dx(CU) = Cd For, with the notation of Art. 45, d cU-cu U-u d(cu) = Ltho h = cLth-o h du dx 49. PROP. III. Differential Coefficient of a Sum. The differential coefficient of the sum of a set of functions of x is the sum of the differential coefficients of the several functions. Let u, v, w,..., be the functions of x, and y their sum. Let U, V, T,..., Y be what these expressions become when x is changed to x+h. Then y=-u+v+w+... Y= U+ V+ W+..., and therefore Y-y= (U-U) +( V-v) +( W-w)+...; dividing by h Y-y U- V-v W_-w h h + h + h + FUNDAMENTAL PROPOSITIONS. 39 and taking the limit dyd_1yd dv dw dW dx dx dx If some of the connecting signs had been - instead of + a corresponding result would immediately follow, e.g., if Y=U+v-w+... then dry dmd dv dw dx dx dx r 50. Pnop. IV. The Differential Coefficient of the product of two functions is (First Fmrnction) x (Diff Coeff. of Second) + (Second Function) x (Diff, Coeff. of First), or, stated algebraically, d(uv) dv du, dz= Udx - dx With the same notation as before, let y = my, and therefore Y= UV; whence Y-Y- UV-Uv =u(V-`v) + V(U-U); therefore Y-Y V- LT-It = l + TV --- h h h and taking the limit dy d v du dx dx dx 51. On division by uy the above result may be written Ildy _I du+ IdA y dr -Um X 21 X Hence it is clear that the rule may be extended to products of more functions than two. For example, if y = uvw; let vw = z, then y = uz. 40 40 ~FU2IVDAMENVTA.L PROPOSITIONS. Whence 1 dy 1 du 1 dz y dz dxA I dx' but ld ldv ld whence by substitution i.dy 1 dm Id l dw Generally, if y = vwt... l dy_1 du ld AIdw Ildt ydx~u&dx vdx wdx +t d+ and if we multiply by uvwt... we obtain dy_ din dv dw i.e., mulitipily the differential coefficient of each separate funnction by the prodmet of all the remaining funnetions and add mp all the results; the sum will be the differential coefficient of the product of all the functions. 52. PROP. V. The Differential Coefficient of a quotient of two functions i's (Diff Coqff. of Numr.) (Den r.) - (Duf. (Joeff. of Denw.) (Num-a.) ~Square of Denominator. or, stated algebraically, d With the same notation as before, let in U y=-, and therefore Y U in whence Y_ = I" UV-v-m FUNDAAIEiYTA4L PROPOSITIONS. 4 41 U-U V-v therefore Y- = v h? and taking the limit du dv dy dx dxh dix 53. This proposition may also be deduced immediately from Prop. IV., thus: Let V -whence - v — 4V-u ----.and therefore dX dx - dx =dy +uciv dyix vd cix__v d 54. We may also remark that Prop. II. is deducible from Propositions I. and IV. For by Prop. TV. ci cit dc dc and by Prop. I. - 0 dx Whence. 9Ca) = c -.The differential coefficient of -C is also of importance; and it fola lows immediately from Prop. V. that ci(c)= c cia 42 FUNDAMENTAL PROPOSITIONS. 55. PROP. VI. To find the Differential Coefficient of a Function 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) be V. Let this be substituted for v in (1), and let a value of u deduced from (1) be U. Then we have the following equations. U =f(V)........................... (4) and V= F(X )........................... (5) and by the same process by which (3) was deduced from (1) and (2) we obtain from (4) and (3) U= (X)........................... (6) This result proves that if x be changed to X in equation (3), then one of the values thence deduced for u will be U, and therefore Lt ---- when X-x is diminX-x ished indefinitely is a value of the differential coefficient of u with respect to x, reckoned as a direct function of x as expressed in equation (3), fU-u U-u V-v Now X-x V-v' X-x and Ltv_ =o --- is a value of the differential coefficient of u with respect to v derived from equation (1) and du V-v denoted by ( —; also, Ltx_-=o — is a value of the differential coefficient of v with respect to x derived FUNDA3IENTAL PROPOSITIONS. 43 43. from equation (2) and denoted by dv We thereforedx' have, when we proceed to the limit, du -du dv dx = dv* dx' a formula already established in a different manner and, with different letters in Art. 43. 56. It is obvious that the above result may be extended.. For, if u =qc(v), v =xfr(w), w =f(x), we have du dm dv dx -dv' dx' but dv dv dw. dx-dw' dx' and hereore du du dv dw and theefore x=d-v'dw-' dx' and a similar result holds however many functions theremay be. Ex. Let u=bsinV=~Sin1w, Ww=_ 5that is, a a /1 2 u=b sini -sin V'X~).. a a~ Then, by Ex. 3, Art. j79 ~ va -Co Ex. 3, ibid, dw 2x Hencedudc dw v 1 ~~ dw' dxa a'/1 -a b "Os (Isin -r2\ 1 a2 The rule may be expressed thu8: d(lst Func.) dflst Func.) d(2ndFunc.) d(LastFunc.) dx _d(9?nd.Vunc.)'d(8rd Func.)... dx 44 FUNDAMENTAL PROPOSITIONS. 57. There is a difficulty 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 f(v) and F(x) may one or both be many-valued functions (such, for example, as sin-lx, which denotes any one of the series of angles whose sines are equal to x). But it is clear that the same values of u and x will satisfy equation (3) as would U-u simultaneously satisfy (1) and (2), and that LtX — when X - x is indefinitely diminished is one value of the differential coefficient of u considered as a function of x; and it is equally obvious that there may be a series of du du dv such values for -, as also for - and for, so that in dx dv dx' the theorem enunciated and proved above, in Art. 55, a proper selection of those values is assumed to be mnade. dua du dy 58. If in the theorem -. (where y is written dx dy dx for v in the result of Art. 55) we suppose u=x, then du dx ( +h)-x -Lt (xh = 1. dxd= h-=O — h Hence we have dy dx dor dy I dx = dx dy 59. In this application of the general theorem of Prop. VI. y is assumed to be a function of x and consequently x is dy is the differential the inverse function of y. So that -y is the differential dx f UNDAMENTA L PROPOSITIONS. coefficient of y with respect to x when y is considered as dx a function of x, and - is the differential coefficient of x dy with respect to y when x is considered as the inverse function of y: e.g., if y=sinx, then x=sin-y, dy =cosx (Ex. 1, Art. 40), dx dx 1 and d- = 1 (Ex. 2, ibid), dy- J1-Wy2 dy dx Cos x and = cos. 1 COS X dx' dy /1 -y2 2 1 - sin2x 60. The same difficulty occurs in Arts. 58 and 59 as that discussed in Art. 57. If y =(x)...(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 Ybe 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. X-x Hence Lt-_ when Y-y is indefinitely diminished Y-y is a value of the differential coefficient of x with respect Y-y to y, as derived from equation (2), while Ltfxv when X -x is indefinitely diminished is avalue of the differential coefficient of y with respect to x as derived from equation Y-y X-x 1 (1). And since Y-. X — =1, 2 X-x Y-y1 dy dx we have dxdy d-' dy 46 FUNDAMENTAL PROPOSITIONS. when the limit is taken, the proper selection being made of the values deduced for - and d. dx dy 61. This may be illustrated geometrically. Let the curve y=f(x) be drawn. Let the tangent to Y ~0 - X Fig. 12. the curve at the point P, (x, y), make an angle r with dy the axis of x. Then, by Art. 39, d- tan ~; and in the dx same way it is obvious that -- tan'(90- )=cot r, so dy that dy dx tan. cot = 1. dx dy Suppose however that the ordinate through P cuts the curve again at P,, P2, P,... Then, for a given value of x there are several values of y, and therefore also for a given increase 8x in the value of x there may be several values of dy the increment of y. But if it be carefully noted that the Sy and Sx chosen are to refer to the same branch of the curve at the same point dy dx when we consider -y as when we consider, then, under these circumstances, these expressions are respec FUNDAMENTAL PROPOSITIONS. 47 tively the tangent and cotangent of the same angle, and Y:~3 P 0 N X Fig. 13. therefore their product is unity. We say the same branch of the carve, for it may happen that more than one branch of the curve passes through a Y ~C x' Fig. 14. given point P, as in Fig. 14, and then there are two or more tangents at P and therefore two or more values of dy dx dx and - at P. But the product of the and the dx dy dx dy' which belong to any the same bcranch through P, is unity. 62. Differentiation of Inverse Functions. When the differential coefficient of any function of x is 48 8 FUNDAMIIENTAL PROPOSITIONS. found, that of the corresponding inverse function is easily deduced by means of the theorem of Art. 58. For let x=f(y), and therefore y=f -'(x); then CIX y=f '(y). dy 1 But dxdx' dy d I I I therefore df() 1 1 dX ~ f'If-(xl EXAMPLES..Differentiate by means of the definition and the foregoing rules - 1. y =x log sin X. 2. y=x 11/a:i1- X2 C2 X 3. Y " 4. 5. y=2 V'au, where u as nx. 6. y = ev/u, where qt = log sin v, and v = (sin w)w, and w = a2. The results of any preceding examples may be assumed. CHAPTER III. STANDARD FORMS. 63. It is the object of the present Chapter to investigate and tabulate the results of differentiating the several standard forms referred to in Art. 40. 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= ((x), then, by the definition of a differential coefficient, du z 0(x + h)- 9(x) d x - ~=. ' 64. Differential Coefficient of xn. If U = (x) = a', then ((x + h) = (x +h), and du Lt=o(x + h)- x and = h (i + -1 =Lthoxn h N ow, since h is to be ultimately zero, we may consider - to be less than unity, and we can therefore D 60 50 ~~STANDARD FORMS. apply the Binomial Theorem to expand (i + whatever be the value of n; hence du _LX h n(n-1) h2 dx hhfx~ 1.2 -Lth=O(nx11-1~ powers of h) = nXn- 1. 65. If it be required to find the differential coefficient of Xnwithout the use of the Binomial Theorem we quote the result of Art., 23, viz.: Ly Y = ii, and proceed as follows: (1Lt [a) s before] ~Lthon -1 =Lt~y~1 nifl [here y= 1 -. 1 66. Differential Coefficient of ax. qO(x+h) =ax h. du ax+h -ax and 7=Lth~o h ah= axLtho h ~ =axlogea. [Art. 22.] COR. If u=ex) du = exloge =ex. STANDARD FOIISX. 51 67. Differential Coefficient of logax. If U = (x) = logax, p(x + h) = loga(x + h), and du Lth log(x + h) - logax and =Lth_ dx h = Lth=O 1 loga(l +) Let =z, so that if h= 0, z =; therefore da= Ltz xoga( zI ) ZI — lt=.log (l +1 loge. [Arts. 12 (7) and 21.] du 1 1 COR. If u=logx, d- = - loge =68. Differential Coefficient of sin x. If u = ((x)= sin x, (x+ h) = sin (x + h), and du Lt sin (x +h)-sinx and = = --- —---- dx h 2 sin h cos + h sin h (2) and 19 =cosx. [Arts. 12 (2) and 19.] 52 STANDARD FORRMS. 69. Differential Coefficient of cos x. If = p(X) = cos x, ((x + h) = cos (x + h), du cos ( + h) -cos x and - = th=o h = -Lt=o. h sin2 = - sin x. 70. Differential Coefficient of tanx. If == ((x) = tan x, p(x + h) = tan (x + h), dua L tan (x + h) -tan x and d = Lt=o h dx h t sin (x + h) cos x - cos(x + h) sin x h cosx cos (x+h) sinh 1 = Lth=o h * cos x cos (x+h) 1 2 -- = = sec x. COS2X cos Xk 71. Differential Coefficient of cot x. If = f(x)= cot x, p(x +h) = cot (x + h), nd du Lth cot (x+h)-cotx and =Lth=o-h dx - h Lhcos (x+ h) sin x- cos x sin (x +h) - =o h sin x. sin (x + h) STANDARD FORMS. 53 sin h 1 =-Lth=o h ' sin x sin (x + h) 12 =-.sin -- cosec2X. sin2x 72. Differential Coefficient of sec x. If = 0(x) = secx, p(x +h) = sec (x + h), dz. sec(x +h)-sec x and - = Lt^=o 7 dx h cos x- cos (x + h) = Lth Th s x+ Lth~oh X cos c ( + h) sin sin x h 'cosxcos(x+h) 2 sin x 2C0S2 73. Differential Coefficient of cosec x. If = (x) = cosec x, * (x+h) =cosec(x+h), du - L cosec (x+ h)-cosec x and d=thn=O dx o h sin x - si (x + h) = L^=h sin x sin (x + h) sin cos + -Ltho h sin xsin (x+h) 2 Cos, X sin2x 54 54 ~~STANDARD FORMS. 74. Inverse Trigonometrical Functions. For the inverse trigonometrical functions it seems useful to recur to the notation of Art. 45, and to denote q5(x+h) by U. 75. Differential Coefficient of sin 1x. U= q5Qc+ h) = sin' (x +h). Hence x =sinm, and x +h=sinU; therefore h =sin U -sin u, and ~du U-in U-un and ~ = Lth=o h Ltu=ui i UnU 1 2 1 -Cos Un 4/ sinkJ 4/I X2 76. Differential Coefficient of cos-Ix. U= p(x +h) - cos-'(x +h). Hence x=cosU, and x+h=cosU; therefore h = cos U- cos in, and ~din U-in U-in and ~ -=ELth= h~ = Ltu~U sin U-usinU sin UnCs STANDARD FORMS. 55 77. Differential Coefficient of tan-Ix. If u = (x) = tan-1x, U=O (x+h) =tan-'(x+h). Hence x = tan,' and x+h = tan U; therefore h = tan U- tan u, and -Lth= h du_ t U-u, _t U-, dxand =Lt= h - t-tan U-tanu = Ltu=, sin U-u cUos U Sin (U- Z) I 1 1 CO sec2,- I + tan2U 1 + 2' 78. Differential Coefficient of cot-1x. If u = (x) = cot- 1x, U= (x +h) = cot- (x +h). Hence x=cot, and x+ h= cot U; therefore h= cot U- cot u, du U-u, U-u and T=Lth:=o h =tu= tU t U-u = -LtU U — sin Usin U 1 2 1 1 =-sin =- =- - cosec2 - 1 + cot2 - 1 +x2 79. Differential Coefficient of sec-lx. If u = ~(x) = sec-lx, U= -(x+h) = sec-l(x+h). Hence x = sec u, and x+ h = sec U; therefore h = sec U- sec n, du 7-Un U-n and dx-=Lth=o h =Lt = -see U- see U-n d-= L h - s e UC-se U = _Ltrr=u -Cos U- cos U cos U LT"COS ~ t- COS C/ STANDARD FORMS. U-Lf Lt 2 cos u cos U -n== ~. U- U. U+U [smn — J smincos2w 1 1,sin zf COS2U -sin u~- sec2/i - cos2 _ 1 "^^-l" 80. Differential Coefficient of cosec -x. If zu =q () = cosec-1x, U= -(x + h) = cosec-(x + h). Hence x =cosec r, and x +h = cosec U; therefore h = cosec U- cosec u, dud U-uq U-v, and d =Lt=o h UL cosec U-cosec U - _ t7-. -sin si U - n-smu mT81 U 1^ U-in Lt - __ sin s2 sin in U u U U+t, si 2 cos 2 sin2u 1 cos u2 cosec2/ /l - sin2k, 1 1 x2~1 — x2 81. From the importance of the results it has been thought preferable to deduce the differential coefficients of the inverse functions sin-lx etc. immediately from the STANDARD FORMS. 57 definition; but by aid of Prop. VI. of the preceding chapter we can simplify the proofs considerably. Ex. (i.) If u=sin-1x, we have x = sin; dx whence -cos; du= d du 1 1 1 and therefore = - = cos - 1 sin dx dx cosu f 1-sinm2U l-2 du and since cos rr -sin-x, dcos-lx 1 we have = — c dx 7/1-x2" Ex. (ii.) If = tan-1x, we have x = tan u; dx whence = sec; du du 1 I 1 and therefore dx sec2I- 1+ tan 1 + x2' and since cot-_x = - tan -x, dcot-lx 1 we have - - dx 1 X' Ex. (iii.) If = vers- x, we have x = vers u= - 1-cos u; dx whence sin,; du I d- l 1 1 and therefore = dx- sin l 1- cos2 - /2x- x2 \ 58 STANADARD FORMS. d covers - 1 1 whence also d c =- - dx s/xx2x 82. 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) be the differential coefficient of ((x), p(x) is said to be the integral of +'(x). Moreover, since p'(x) is also the differential coefficient of +(x) +, C being any arbitrary constant disappearing upon differentiation, it is customary to state that the integral of 0'(x) is p(x)+ C, C being any arbitrary constant. The notation by which this is expressed is f(q'(x)dx = p(x) + C, fp'(x)dx being read " integral of p'(x) with respect to x." Thus we have seen d-(sin x) = cos x, -(tan -1) = - + dx 1+x2 etc., whence it follows immediately that fcos xdx = sin x, J1+ x dx=tan-lx, etc., STANDARD FORMS. where the arbitrary constant may be added in each case if desired. 83. 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 forms 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 ventured to tabulate 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. TABLE OF RESULTS TO BE COMMITTED TO MEMORY. dx u -Xn. = __ nx- 1 dtb dx zu -= ex. -- ex. dx du 1 U = log"x. d= logae. L = log1. dq -d 1 u=logex. dm I du = sin x. = cos x. dx du U==cosx. - s-sin X. dx fxndx = xn+1 /' axdx loga fexdx fdx X^ - ex. = logex. logex or lga logae fcos xdx = sin x. /sin xdx = -cos x. fsec2xdx = tan x. 60 STANDARD FORMS.? = cot x. d = - cec2. fcosec2 du sin x sin x -= sec '. = c2' --- c dxcos2x J cos2x du cos x ycos x u = cosec x. = -. 2 dx sinx' j sfinx du, I du I _ dz s/1-_S2 u = tan-'x. dr- 1 +2 +dx dx 1+x2 i+ mrsctcvz. C7 L2 1. f+x2 - du 1 Uf s x. ==- - = — r dx = cot-~x' dx- 1+x2' du 1 u= cosec x. dx — —. I m=ooes'cv. - a /2-_ 1c { du 1 uf = vers-x. - = — 2x-x. dx du ^/x-x2 J2x — cfu I U = covers-Ix. =- dz ^,2gx-x2) 'xdx = -cot x. Ix = sec x. x = -cosei = sin-lx, X2 CX. or -COS-1X. = tan-lr, or -cot'lx. -1 o -sec- x, or — cosec-]X. = vers-lx, - x2 or - covers'-x. 84. The Form u. In functions of the form zv, where both u and v are functions of x, it is generally advisable to take logarithms. before proceeding to differentiate. Let y= U, then logey = v loge,; therefore 1 dy dv. 1 du. 5 Yt e d- -.log +v.~ -~ cArts. 50, 55, 67, STANDARD FORMS. 61 dy L dv v du\ or -=vid v = Oge u.- +-d -j.+) dx ' G dx d x Three cases of this proposition present themselves. dv I. If v be a constant and u a function of x, = 0 and the above reduces to dy,._Vdu dx dx' as might be expected from Arts. 55, 64. II. If u be a constant and v a function of x, = 0 and the general form proved above reduces to dy c dv dx-= Ulog. dx' as might be expected from Arts. 55, 66. III. If u and v be both fmnctions of x, it appears that the general formula dy =, du vu(I ldU dx = dogIx+ 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. 85. Hyperbolic Functions. The differential coefficients of the direct and inverse hyperbolic functions are now appended as additional formulae. Their 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 62 STANDARD FORMS. form with the results tabulated above for the direct and inverse trigonometrical functions. RESULTS FOR HYPERBOLIC FUNCTIONS. u = sinh x = - e-x d- = cosh x. fcosh xdx = sinh x. 2 dx u = cosh x eX + = sinh x. fsinh xdx = cosh x. 2 dx sinh x du. sech2 xdx =tanh x. cosh x' dx = coth cosh x du = c cosech2dx - coth x. u X = coh x =' cosedxd x. Ismh x dx u=sec 1 dc sinhx sinhx x — secha= ^o. -= — 2*- d-x-^ =-secha. cosh dx ax cosh2 x / cosh s e 1 du_ cosh xz coshx u = cosech x = -. - I dX = - cosech x. sinh x dx sinh2 x J sinh2x du 1 dx. u = sinh-1 x = log(x VI + /x). d 1 =sinh-lx -- du 1 f Vdx,X = cosh- x = log(x+ -1). du 1 = = tanh-1 X - og + ( < 1). - = tanh -x(< 1) = coth-1 = x-g~. duZ= g(>).(- _ > - coth-1,2, dx- -x2- X1- 1,,, J ^ 1d/ dx, u = sech-1 x = cosh-1. d _ --- =1 _sech-x. x (dx x /1 x2 _ xs /1 _2 1* dux 1 / cd dxu = cosech- x = sinh-l. d- _= - cosech-lx. x dx x f + Nl X' -2-+1 86. Transformations. Algebraic or trigonometrical transformations are frequently useful to shorten the work of differentiation. For instance, suppose 2x y = tan-_2 We observe that y = 2tan-'x; whence dy 2 dx 1 + 2 STANDARD FORMS. 63 Again, suppose Here and therefore 1+x y = tan_-11 + x 1-x y = tan-'x + tan-l 1, dy_ 1 dx 1 +x2' 87. Examples of Differentiation. Ex. 1. Let y= /z, where z is a known function of x. Here y = z and dy - 1^ whence dy z (Art. 55.) dx dz ' dx'(.5 1 dz 2 /z dx' This form occurs so often that it will be found convenient to commit it to memory. Ex. 2. Let y=e eCt~. Let N/cotx=z and cot x=p, so that y= e, where z- ^p. Now dY=ez (Art. 66.) dz i d-=2 1 (Ex. 1 above.) dp 2j p dp= _ cosec2x, dx and (Art. 56) dy- dy. d- =-cosec2. e/Ct dx dz dp x 2,/cot X With a little practice these actual substitutions can be avoided and the following is what passes in the mind:d(e/_____) d(e/ct) d( cot ) d(cotx) dx d(/cot-x) d(cotx) dx =eot. 1. (- cosec2x). 2 v/cofc x 64 STANDARD FORMIS. Ex. 3. Let ~y =(sin x)l09 - cot fex(at+ bx)} Taking logarithms log y= log X. log sin X +log cot Iex(a +bx)} The differential coefficient of loge, is I dy y dz Again, log x, log sin x is a product, and when differentiated becomes (Art. 50) 4~og sin +1logx. 1 ---.cos x. Also, log cot {e(a+ 6x Ibecomes when differentiated co Ix~ )} [ -cosec2{ ex(a +bx)}] {ex(a + bx) +ex);. Y=(sinx)1os9x.,cotjex(a+bx)}[!1og sinx+cotx.logx - 2ex(a + b + 6x) cosec 2(exa + bx)] When logarithms are taken before differentiating, the compound process is called Logarithmic D?flerentiation. It is useful to adopt this method when variables occur in the index, or when the function to be differentiated consists of a product of several involved factors. Ex. 4. Let y- N/a2-b2cos2(log X). dy d 'a b2COS2(logxs) dja2 - b2cos2 (log X)} dx dja2 -b2COS2(log X)}I d fcos (log x)} d~cosj(og x)} d(logxs) X d(log x) X dx =f2- b2COS2(log X)} I x~ { - 2b2cos (log X)} X { -sin (logx)} X<I - b~sin2(log x) 2x Va2 - b2cos2 (log )5 Ex. 5. Differentiate x with regard to X2. Let X2 Z. dx Then dx5_dX5 X d.V _5X4 -dzd TX'T dz dzj alx =5,3 2 STANDARD FORMS. 65 Ex. 6. Given that x3 +y3-f 3ax, find the value of y/ dx' Here 3x2+ 3 2cn=3a( y+ x ) dy x2 ay giving2 - ac/ d,,v y2 - ax EXAMPLES. Find dyin the following cases: dx 1. y= Jx. 2. y=-. * a + bx X3 X5 6. y=x-..+. 10. y == sin x. 12. y = /sin \/t. 13. y = sinx5. 14. y=sin32. 15. y = (sin-'x)'- (co-X)2. 16. y= tan+ (log ). 17. y= sinx. 17. y==sina3~. 18. y=xlogx. 19. y=eXlogx 20. y = sin (ex) log x. 21. y=tan l(ex) log cot x. 22. y = (x + a)(x + b) 23. y=2 +X 24. y= a Za+x. 25. y = /a + x". 26. y = x/cosh x. 27. y= log cosh x. 28. y = tan-l(tanh x). 29. y = vers-x~2. 30. y = vers-'log (cot x). 31. y = cot- l(cosec x). 32. y= sin-' - 33. y=tan-. jx~- 1 34. y = tan-1 X x 35. y = sin"x cos"x. E 66 STANDARD FORMS. 36. y = (s8ij-l>)(cos-lx)". 37. y = sin(elogx). /1 -(logx)2. 38. y-_- _. 1 + x \J + X2 39. y- 6 x x - 4Ca2 40. y=./ - 4a' 41. y _ J1 l- 2 +x+X2' ^ - x _ + 1 a 43. y=logs. ' 44. y =cos-l(l - 22). 45. y I + log. 46. y=btan- -tan-I. a a X Cos-1X 47. y - osx. 48. y=cos(a sin-l). 49. y=sin-a + b cosx b + a cos x 50. y = etn -x log (sec23). 51. y = e'xcos (b tan- lx). 52. y= tan-l(a. x2). 53. y = sec (loga /a2 + x2). 54. y = tan- Ix + tanh1-x. 55. y =tanh-3x + tan x3 I 3- 3X2 I - 3a'56. y = log (log x). 57. y = log"(x), where log" means log log log... (repeated n times). 1 58. y= -/ -_a- 2 log V62 - a~ I/b - + /lb - a tan l/b - a- l/6b-a tan x 59. y= sin-l(x< J1 - - Jx/ - /lx2). 60. y= tan-l -4.65. y=e. 1 - 4xx (IxZ-2'~) 66. y=ee. 61. y = log exy-_2) 67. y=. l/ l { x4 + 2) } 67. y- 10xo. ~62. 2 +ri~ g68. y = (sin x)cox + (cos x)sinx. 62 ^- X -x +.69. y= (cotx)cotx + (coth x)cth. 63. y=. 64. =y =x. x sin JX 70. y=tan-l(acX8i1x) x. 1 + x STANDARD FORMS. 67 71. ysin-](6tan-l ). 78. y = b tan- + tan-mY. 72. y- (1+cos- 1(-sin ).79. tany-e=.xsinx. a 80. ax2 + 2hxy + by2= 1. 73. y = tan-/ x + cos-. e ( + b"). 72 81. ey-= (a + bxn)l - a' 74, y = 1 + 2 si e (bx 74 1+ ^ 2 82. (cos x)= (sin y)x 75. y=(cos )cot 83. a=e. 76. y = (cot-l). 84. x = y log xy. 77. y X +1 85. y-=x. 77. y l- 1+) +xc. - 6. X * 86, y=xy. 87. y=xlog Ya a+bx 88. ax2 + 2hxy + by2 + 2gx + 2fy + c = 0. 89. xmyn = (x+ y) n+f. 90. y = eta -l log sec2x. 91. Differentiate log,0x with regard to x2. 92. Differentiate (X2 + ax + a2)" log cot with regard to tan'(a cos bx). ('TC'\ a+b tan - 93. Differentiate loge - with regard to a- b tanJ 1 a2cos2X - b2sin2? 2 2 94. Differentiate x"i'l '" with regard to sin-lx. 95. Differentiate tan-'l X with regard to tan -x. x 68 STANDARD FORMS. 96. Differentiate /l + + + 1 x2 with regard to J1 - x4. J1 +X- 21 X22 1 ____ 97. Differentiate sec-'Z -- with regard to /1I- X2. x 1 98. Differentiate tan-1l with regard to secl-l —. /1 - x2 22 -99. Differentiate tan- 2 with regard to sin-' 2x _1 - X2 1 + x2 100. Differentiate x"logtan-lx with regard to sin. o.. T~ dy y2 101. If y = e prove xd = 1- ylog x =dx I - y log x' x dy _ 2x 102. If=y = - proved 1+- 2x 1+ +X x- X 0l. Ify ++ +... to oo,.+ 103. Ify=x+ 1 prove = +- pr1 dx 2 — 1 x+ - x3~ + x-... sin x 1 + sin co 1 +... to o. dyove ( + y) cos x + y sin x dx 1 + 2y + cos x - sin x 105. If y = /i s /in x + sin + /sin + /etc. to oo, dy_ cos x prove dx 2y- 1 106. If S3:= the sum of a G. P. to n terms of which r is the common ratio, prove that dS l) STANDA RD FORMS. 69 107. If =a+ 1 1 prove ( 1 1 dx*+\]- Q' Q 3+... +, r2cos 20 r3cos 30 108. Given C=1 + r cos + 2 - + 3- 3 + 2! 3! r2sin 20 r3sin 30 and S=rsin 0+ 2 ---+ 3 -. 2! 3! show that dC sdS c-j + sdr = (C2 + S2) cos 0; CdS- d= (C2 + S2) sin 0. dr dr 109. If y = sec 4x, prove that dy =16t( - t4) where t = tan x. dt (1 - 6t2 + t4)2' 110. If y=e-xsec-l(xz) and z4+ x2z=X, find dy in terms dx of x and z. 111. Prove that if x be less than unity 1 2x 4x3 8x7 1 -- + + + +... ad inf.-+ 1 +x 1 +x2 1 4-4 1 + X ' 1-x' 112. Prove that if x be less than unity 1 - 2x 2x- 4x3 4x- 8x7 1 + 2x........... + 1 +x - - +... ad inf. = - 1 - x+2 1 _ 2 + X4 1 - 8 x+ x 1 x + X2' 113. Given Euler's Theorem that x x x x sin x Lto cos - cos 2 cos -... os 2n- 2 222 Xx' 1 x 1 tanx 1 Gan X prove tan + tan + tan +... ad inf. =- cot x, 2 2 + 2 sec2 sec2 ad i 1 x 1 2x 1 x 2 and sec2 X+ sec + sec 2 +... ad inf. = cosecx -2. 9;2 2 )4 2 ~6 3" X 70 STANDARD FORMS. 114. Given the identity n 2cos 2n+lO + 1 (2cos 20 — 1)(2cos 220- 1)...(2cos 20 - 1)2- 2Cs 20 +1 2cos 20 +1I prove that r=n 2"sin20 _ 2n+lsin2n+l' 2sin2O r=i 2cos2-1 - 12cos2 2 cos 20n+ + ' 2cos 115. Given sinnq sin4 sin(2a + O) sin(4a + )*...sin{2(n - l)a + )} = 2nl where 2nza = r, prove that cot c + cot(2a + >) + cot(4a + ) +... + cot{2(n - 1)a + } = ncotn4, and that cosec2% + cosec2(2a + +) + cosec2(4a + ) +., + cosec2{2(n - 1)a + n} = n2cosec2nqb. 116. Given sin-0 (1 - -1 0)(... 22-\ " 32 1=00 1 prove Ocot0= 1 + 2 02 2 n27r and hence that rcothr = 1 2 + + 2 +... ad inf., 1 ~ 12 1+22 1~32 2 2 2 and that coth 1 + 2 +1 + 42 +.. ad inf. 117. Given coso = (1 - 42 21 - 42 52- 402) tan0 n=a= 1 prove 80 =- =,=1 (2n - 1)2vr2 -402 and deduce r = I 1 1 tanhw = 1 + 1 + +... ad inf., 8and 22t 122322+ 2+5 ad inf _ _rr 1 1 and 1 tanh= 1 ++ 1 ~ ++... adinf. 4 2 1~12 1+32 1+52 STANDARD FORMS. 71 118. Prove X othx=l + x2. 1 2 2 =I X2 + r2r2 119. Prove that 27r 1n- ~ 1 + n-2 x - acos —2 = - X + a +2 2 2rr 2 - - xa x~+ X2a - 2axcos2 + a n if n be even, 2r7r n- X -- acos 1 r 2 but + 2 n x-a =+ 2 2rr x" - 2axcos — + a n if n be odd. 120. Prove that 2r7r + 0 nx"( ao0) n- x- o _ -acosX2n - 2x"oa-cosO + a= _ r=- o 27r~ + 0 X2 - 2axcos + a2 nl CHAPTER IV. SUCCESSIVE DIFFERENTIATION. 88. Repeated Operations. The operation denoted by d is defined in Art. 39 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 y be a function of x, -y is also a function of x, and therefore we can have dx d ~(\ ~as a true mathematical quantity. And further, it d will be thus seen that the operation dx may be performed upon any given function of x any number of times. 89. Notation. The expression 7-(d) is generally abbreviated into SUCCESSIVE DIFFERENTIA TIOV. 73 ( d- y or d, and is called the "second derived function" d or " second differential coefficient " of y with respect to x. And, generally, if the operator - be applied n times, the dx result is denoted by ( d y or -, and is called the nth derived function or nth differential coefficient of y with respect to x. 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 many problems the independent variable is more conveniently 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 dy may write Dy = ddx2 D. Dy=D2y d, d3y D. D2y =D3y = dxy D. D-ly = Dy_ dx. 74 SUCCESSI VE DIFFERENTIATION. 90. 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 a2, and for a. a. a is a3, 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 )2, but a simple repetition of an operation. In the same way Dn 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 D2y, (Dy)2, and D2y2 should be carefully noted. The index in the first case has reference only to the symbol of operation, "D," which is therefore to be applied twice to y. In (Dy)2 the index is a purely quantitative one used in the algebraical sense to denote the product Dy x Dy. While in D2y2 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., if y 2 then Dy = 2x, and D2y=2........................... (1) Again, (Dy)2 4x2.......................... (2) whilst y2 = X4, and Dy2 = 43, SUCCESSIVE DIFFERENTIATION. 75& giving D2y2= 12x2........................ (3) A comparison of the results (1), (2), (3), will at once satisfy the student of the truth of the above remarks. 91. 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) The " Index Law." 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+w+...) = 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 0sin instead of sin0. So that, while D is commutative with regard to constants, it is not so with regard to variables. 76 SUCCESSIVE DIFFERE7NTIATION. (3) The Index Law in algebra is denoted by am. an = am+n, 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., Dm DnD = Dm+ny. So the operator D"m.Dn is equivalent to the operator Dm+n 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 elementary rules of combination of algebraical quantities, with the exception that it is not commutative with regard to variables. 92. It follows from the above remarks that any rational algebraical identity has a corresponding symbolical operative analogue. For example, (m + a)(mn + b) = m2+ (a + b)m + ab, so also the operation (D+a)(D+b) is exactly equivalent to the operation D2 + (a + b)D + ab. Similarly, to the identity (m a + a)2 = 2 + 2am + a2 corresponds the equivalence of the operations (D+a)2 and D2+ 2aD+a2. 93. It is clear that in cases like the above an a priori 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 = [D2 +(a+ b)D +ab]y, SUCCESSIVE DIFFERENTIA TiON.Y7 77 we have (D b)y=Dy+by, and (D + a)(D + b)y=(D + a)(Dy + by) D(Dy + by) ~ a(Dy + by) -D2y + bDy + aDy + aby D2y + (a + b)Dy + aby - [D2 + (a + b)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 + b) = m2 + (a + b)m + Ab. However, such proofs are unnecessary after the remarks of Art. 91, for they simply repeat in form the proof of the corresponding algebraical theorem. It will now be obvious, for instance, without further proof, that since (m +a)"= z~+namv ' ~ 1.2(n) a 2r +a~-..., we shall also have (D + c)"y (Dn+ naDn1+ n(n - 1) n(n - 1) =DYy+naD1y~ 12 2aDn- 2y+... +ay 94. Notation. The first derived function of y with respect to the independent variable is often denoted by yl, y', or y. This notation can, be conveniently extended, and we shall often find it convenient to denote Dy, Thy, Dhy,... Dny by y1, y2. y3,... or by y(l), y(2), y(3).. n or by y', y"/, y'I etc., or by VI,, y, etc. 78 SUCCESSIVE DIFFERENTIATION. It is clear however that the notation of dashes or dots as used in the last two systems is inconvenient for higher differential coefficients than the fourth or fifth by reason,of the number of dashes or dots which it would be necessary to use. The bracketed index notation is a somewhat dangerous one, from the liability of confusion with an algebraical index. The suffix notation appears to be free from objection in cases when there can be no misunderstanding as to which is the independent variable. 95. Standard Results and Processes. The nth differential coefficients of some functions are easy to find. Ex. 1. If y = e; y1 = aez; Y2 = a2e;.. y = caey. COR. (i.) If a =1 y = e, Y = ex... yn= ex. COR. (ii.) y = a = exl~gc; Y1 = (loga)exlog1 = (logea)a; 2 = (logea)2e loge = (logea)2ax; etc. = etc., yb = (logea)'ex'log,, = (loga)ax. Ex. 2. If y=log(x+ a); 1 1 (-1)( —2) Yl=+a; Y2- (x+a)2; Y/3 (x+a )3 (-1)(-2)( —3)...(-+ 1) Y~= (x+aY) (x +a ) COR. If Y= a y =+ -) SUCCESSIVE DIFFERENTIATION. Ex. 3. If y = sin(ax+ b); Y1= acos(ax+b) = asin(ax+ b+ y=a2sin(ax+b+ 2); Y3= 3sin(ax+b+ -); 9y=ansin(ax+b+ ~r). Similarly, if y = cos(ax + b), n = ancos(ax + b+ COR. If a=l and b=O; then, when y = si, y =sin(x+2); and, when y = cosx, y = cos( +- -. Ex. 4. If y = eaxsin(bx+ c); Yl = aea'sin(bx + c) + beaxcqs(bx + c). Let a=rcoso and b= rsinq, so that r2= a2 + b2 and tan = -; C6 and therefore y, = reaxsin(bx + c + q). Similarly Y2 = r2eaxsin(bx + c+ 2p), and finally y,, = rneaxsin(bx + c + nq) (a2+ b2)2eaxsin(bx+c +tan-). Similarly, if y = excos(bx + c), yn = (a2+ b2)2eaxcos(bx + c + tan-1 ~~ (b~a an 79 80 SUCCESSITVE DIFFERENTJIA TJON. As the above results are frequently wanted, it will be well for the student to be able to obtain them immediately. 96. Fractional expressions of the form f (x) (both functions being algebraic and rational) can be differentiated n, times by first putting them into partial fractions. (See p. 85".) Ex. 1. X2 ~~a2 1 Y = __ = + ~ 2_ (b-c)(b-a) x-b (c-a)(c —b) x-c' (see note on partial fractions); a2 (1)nn! + ~ n + c2 -l bn Ex. 2. Y=x)(x2 To put this into Partial Fractions let x = I1~ z; then I~ 1+2z+ Z2 then ~ ~ 3+z /1 5z 4 Z2 by diiion A+9 9 3+z)b ii 1 54 1 3z2 9Z 9.3+ Z 1 5 ~~~4 SUCCESSI VE DIFFERENTIA Tl ON. 81 (,+l)!(- 1) 5i!( —1)' whence= -(-l)- +9(x^ -1 y -- 3 — )n2+ 9(x- 1)n+l 4n!(-l)n 9( - 2)n+l 97. When quadratic factors (which are not resolvable into real linear factors) occur in the denominator, it is often convenient to make use of Demoivre's Theorem. Ex. Let 1 1 Y +2+ b2 (+ + + +{( x+a) + b}{(x+a)- b} 1I 1 1 then y 1(-)!{( a -b)L+ n-,( C b)7+1 _=(-l). (x+a+_b)l +l — (x+a+L b)61+ (7- )nn!{ (x + a + tb)n+l - (x + a -,b) '| - 2b [(x + c)2+ b2]w+1 Let x+ c=r cosO, and b =r sin 0; whence r2 (x + a)2 + b2, and tan 0= x3r- +'a x+(t (_-l)nn! 7+1 Hence y,,, =(-1)n! + Hence = 2, [(x + a)2+ b2]+1 x {(cos 0 + sin 0)n+l - (cos 0 - sin 0)n+l} _(-l)nn! 2tsin (n+l) fl+1 2,b [(x + )2+2] T....__ 1);+ sin (n+ 1)0 sinn+10, b where 0 = tan-l. x+a Co. If y=tan-l -, y= +b b F 82 ~SUXEYSSIVE DIFFERENATIATION. adtherefore =J(- 1)~n- 1n-1)! sin nO Sinno, and ~ ~ YA bn+1 where tan =__b xc+ a' EXAMPLES. Find the Path differential coefficients of y1 with respect to xin the foallowing cases..4 2.: --- I 3. $~/ 5h,)3X. 5. Y __eaSinj3bxr. 8. 1 =x2 + a2. 9. Y =tan-'_. a 10.y 1 (x2 +a2)(x2~ b2)' 12. y = ~a2 2~a4 — 98. LEIBNITZ's THEOREM. To find the nth difl rential coefficient of a prodmet 'of two fuanctions of xc in ter~Mns of the, differential coefficients~ of the separatde fmnctiov, 8.' Tt was proved in Chap. IT., Pr-op. iv., that dx dx dc It appears from this formula that the operative. symhol d -u or _D may he considered as the sum of two operative symhols D1 and D,, such that D1 only operates on u~ and differential coefficients of u, while D 2 operates solely SUCCESSIVE DIFFERENT1A TION. 83 upon v and differential coefficients of v. For with such du symbols D,(uv) v —, dv and D2(u'v)= Cdu du d whence D(uv) = vd+ + u,- = D1(uv) + D,(tv) dx = (D1 + D2)v. We may therefore write for D the compound symbol D + D2. Now, since D1 and D, are symbols which indicate differentiations, they each, like the original symbol D, obey the distributive and index laws and are conmmutative with regard to constants and each other. It therefore follows by formal analogy with the Binomial Theorem that the operations (D+D2)71 and Di + nD1? -1D, + D1-2-D2- 2 +.+ D are identical. dnW Now D n(Uv) = v dv d"-lu oD,-*-D2(Uv) = dC' d: —' etc. Hence d'-v) = Dn(uv) = (D, + D2)4(t) (D n +D n1D2+ 1n(n- I)-D-2 22 +... + D2)V dcn dcv (ln-ltu n(v-1) d2v dn-2 -vdx' dx'd dn-l 1. 2 dt2'd — 2 dv + dX+U n' 884 SUCCESSIVE DIFFERENTIA TION. It appears therefore from this formula that if all the differential coefficients of u and v be known up to the nth, inclusive, the nth differential coefficient of the product may at once be written down. 99. Another Proof. From the importance of the above result it is considered useful to add here an inductive proof of the same theorem. [Lemma. If nC denote the number of combinations of n things r at a time, then will Cr + in Cr+l = n+l Cg.+1 -This will form an easy exercise for the student.] Let y= uv, and let suffixes denote differentiations with regard to,x. Then yl =uxv+uv1, Y2 = u2v + 2u1v +uv2, by differentiation. Assume generally that yz = UC11 U- ' V -2v2 + C CUU _ + C2,_2V2 + + Cr +?nC.+1,.- n- r - IV,,+l +... + U........................................................ (a) Therefore, differentiating, y/n+l =U+lVi+UV { +n } + 1,, -1 V2{ }... + nC. } = U,2+]V +,+llC2Unvl + n+lC2_zn-.lv2 + +1+IC3U, - 2V3 +... + ni-Cr+iUn - rVr+ + *.. + uvc+i, by the Lemma; therefore if the law (a) hold for n differentiations it holds for n; + 1. But it was proved to hold for two differentiations, and therefore it holds for three; therefore for four; and so on; and therefore it is generally true, i.e., (Uv)7, = unv +,CC1Uu, - 1'z + nCUn - 2V2 +... + n2 X, - rVr +... + UVn. 100. Applications. Ex. 1. Let y=eaX, where X= any function of x. Since d (eaS) = reax, edxr SUCCESSIVE DIFFERENTIATION. 85 I~~(~~x+,crC cl cl"d eax X +,Cn-l X+ nan-2 X-+... + which may be written by analogy with the Binomial Theorem eax(a +d X (Art. 93). Ex. 2. y=x3sin ax. y = x3sin ( (ax + -7 +~ n3x2an-lsin (ax+ — 1 ) + ~(n1)3 3.2xa-2ssin ax+ -- 27r) 2! 2 n(~ - 1)(n - 2), in 3 +n( nl, )3.2. la'sin ax+ -- ). Ex. 3. Differentiate n times the equation,c12 y dy x 2X + + y = O. ((X2y2) = 2Y+2 + n. 2x. y,,+1 + 2! y d'n dxn I (zY^1) 3~xyf,+l + /.yn, cx = y-; therefore by addition x2yn+2 + (2n + )xy,+l + (2 + )y, = 0, or x2 + (2n + 1) + (%2+ 1) C7,Xn2$2 dxn+l 101. NOTE ON PARTIAL FRACTIONS. Since a number of examples on successive differentiation and on integration 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. 86 86 ~SUCCESSIV-E DIFFERENTIATION. Let -f(x be the f raction which is to be resolved into its partial O(X) fractious. 1. If f(x) be not already of lower degree than the denominator, we can divide out until the gnuerator of the remaining fraction is of lower degree; e.g., (x-1) (x -2) =+ x1x2) Hence we shall consider only the case in which f (x) is of lower degree than O(x). 2. If 95(x) contain a single factor (x - a), not repeated, we proceed thus:suppose O(x) =(x -- a I() and let f(X) 'A + X ( A being independent of X. Hence f +(x)a ~ x(X) );(-X), This is an identity and therefore true for all values of the variable x; put x =- a. Then, since ~t(x) does not vanish when x = a (for by hypothesis Vj{x) does not contain x - a as a factor), we have ~f(a) Hence the rul e to find A is, "Put x =a in every portion of the fraction except in the factor x- a itself." Ex. (i.) +- - - (x-a)(x-) a-b x-a+ b-aX-b E x. (ii.) X2 +pX+q -a2+paZ+q 1+b2~pb+q 1 (x.- a) (x -b) (x -c) (a -b) (a -c) x - (b -c) (b - a) x- b + c2~pc+q 1 (c -a)(c -b) x - c _ _ _ _ 1 2 3 Ex ii)(x- 1)(x- 2)(x- 3) 2Qx- 1) x -22_(~x -3)' Ex. (iv.) (x -a)(x- b)' Here the numerator not being of lower degree than the denomninator, we divide the numerator by the denominator. The result will then be SUCCESSIVE DIFFERENTIA TION. 8 87 expressible in the form 1~ A + B where A and B are found x -a ~-b6 as before and are respectively,- andb 3. Suppose the factor (x - a) in the denominator to be repeated r times so that P(x) = (x - a)r~b(X). Put x-a=y. Then f()fa~) or expanding each function by any means in ascending powers of Y, yr'(B + B y~+Bi2~) Divide out thus: Bo +Bly +..) A 0+ Aly +... (Co+ Cly + Cy2+.. etc., and let the division be continued until y( is a factor of the remainder-. Let the remainder be yr'x(y). He-neethe fraction=-CO+,+ 2 +... +C + -~ X2( — - CO C1 + C2 (x - a)r (x -- a)r1, (X - r2 -+ X(x -a) Hence the partial fractions corresponding to the factor (x - a)", are determined by a long division sum. Ex. Take (X-1)(X~ Put =. Hence the f raction =ly) 93(2 +?j) 2+y ( 2 8 2+y' 2~_ + y2 1g2 ~,+ I 3.13 88 SUCGESSJVE DIFFERENTIATION. Therefore the fraction 1 3 1 1 2y3 4dy2 8y 8(2~y) 1 3 1 1 2(x - 1.)3 +4q(x - 1)2 (- 8i. 4. If a factor, such as XI2 + ax + b, which is not resolvable into real linear factors occur in the denominator, the form of the correspondinm partial fraction is Ax+. For instance, if the expression partial XI+ax + b be1 b (x - a)(x - b)2(X2m +2) (X2 + b2)2 the proper assumption for the form in partial fractions would be A B C Dx+ E Fx+G 1Fx~K -— x -- b - -t- 3~ + ~P2-4s-O? + x - b~ +-z~ x-a + x~+xb2 2a~< mb+ x+~2 where A, B, and C can he found according to the preceding methods, and on reduction to a common denominator we can, hy equating coefficients of like powers in the two numerators, find the remaining letters D, E, F, C, H, K. Yariations upon these methods will suggest themselves to the student. EXAMPLES. 1. Ify=tan-xm, find d dx3 2. Ify= x2logX, find y 3. If y = sin mx cos nx, find dX2' ad3~ 4. Ify=xe '3'in 6dX3 15. If y = x, find dy distinguishing the cases in which dxr' r <, =, or > n. X X~dy _ dy 6. If y = x log, prove that dy a-bx dx'2 dYxd SUCCESSIVE DIFFERENTTIATION. 89 7. If y = ax sin x, prove that x2d - 2x.d + (x2 + 2) =0. dx2 dx 8. If y = a cos (log x), prove that x2d 2 + xy + y = O. dWX dx 9. If y = axs'+l + bx-n, prove that x2'd = n(n + 1 )y. dx2 10. If y =sin-x, prove that (1 - x2)d -- dx = 0 11. If y = (sin-lx)2, prove that (1 - x2)dY = xdk + 2. dx2 dx 12. If y = easin, prove that ( 2 _ d'y d_ 2 dy a 2 13. If y = A sin?nx + B cos mtx, prove that d2 + n2y = 0. 14. If y = Ae mx+ Be-x, prove that d - m2y = 0. 15. If y = A(x + /x2 + a2)', prove that (x2 + a2) +X A- -y2 = 0. (x2 d+x 16. If y = tan-'x, prove that (1 + xx2)~ + 2 dy 0. dx" dx d4y 17. If y = e-cos x, prove that d- + 4y = 0. CdX4 18. If y = Xeax find dy 19. If y = xsin ax, find y dx52 y2 dny^/ 20. If Y = (- x cb- b) find dxn 20. Ify= ___x2 find dy (x - 1)(x - b)' d' 21. Ify 1 find d 22. If y = x tan-x, find dny d" 23. If y = log xx + xnlog x, find d —. 90 SUCCESSIVE DIFFERENTIATION. 24. If y 1 s shopv thatd is 1 +x+x2 +x dxx ( - 1)rn! sinn+l0{sin(n + 1) - cos(n + 1)0 + (sin 0 + cos 0)-n-} where 0 = cot-Ix. [MATH. TRIPOS. 25. If (1 - 2 _ x dy =0, prove that dx2 dx 1)~,_~+.ol~_ dn (I — x2) -(2n+ l)d n 0. d,,xe2~ dxn^l dx'1 d2y dy 2 26. If (1 - x)y - xd = a2y, prove that (1-2) — +(2n l) - (n2 +2 d= (1&/+2 dxl+l dX. 27. If y = sin(m sin-'x), show that (1 - x2)d2 = xdy - m2y, dx2 dx and (1 - 2)d2 _ (2n l)xd+ - (n2 _- 2)-t 0. /Xn+2 dX n+Y1 - d- cx 28. If y=A(x + Jx2t+ ) + BB(x — Jx2+ a2)-', then will (x2 + a)y + dy - n2=. dx2 dx 2 2 (2 dm(ly 2)d Y 0. and (x2 + (2on + + (+.2 - d)= 0. 29. If y =etan'-l = ao + ax + a2x2 +..., show that (i.) (1 + ) (2 1)d 0 (ii.) (1 + x2)- + {2(n + l)x -1}d + n(n + 1)d 0 d'x i+2 dl "++ de (iii.) (n + 2)a,+2 + na, = a,+. The last equation is to be found by substituting the series for y in equation (i.) and equating the coefficients of xn to zero. 30. If sin(m sin-x) = ao + alx + a,2x +..., prove (n + 1)(n + 2),+~2 = (n2 - m2)a. 31. If esiu= ao cax + ax + a..2, prove (n + 1)(n + 2)ac_+2= (n2 + a2),,. S UCCESSIVE DIFFERENTIATION. 91 32. If (sin-lx)2 = a0 + a1x + a2x2 + a3x3 +..., show that (n + 1 )(n -+ 2)a =2= 2a,. 33. If f() can be expanded in a series of positive integral powers of z, show that f(d.) e =f(ca)et. 34. Show that f (deaXx = eaf (, + dX where X represents any function of x. 35. Show that /d2\ sir 2 sin t _-xl mx= o(-~ m) oma. f ( ^j)os f 2 cos2 36. Find the nth differential coefficient of eax{2x2 _ 2nax + n(n + 1)}. [I. C. S. EXAM.] 37. If u = sin nx + cos nx, show that dru = n 1 + (- 1) sin 2nx%}. [I. C. S. EXAM.] 38. If -- be differentiated i times, the denominator of the ex - 1 result will be (e - 1)'+1, and the sum of the coefficients of the several powers of ex in the numerator will be (- 1)i. 2. 3...i. [CAIUS COLL.] 39. Prove that cldlT dluv rz-1 / dv\ (n- 1) da-2 / d2vX dnx dd~f cln-l\ dx) + 1.2 dx d-2 dx2 -... +(- )-" den CHAPTER V. EXPANSIONS. 102. The student will have already met with several expansions of given explicit functions in ascending integral powers of the independent variable; for example, those for (x+a)n, ex, log (l +x), tan- x, sin x, cos x, which occur in ordinary Algebra and Trigonometry. The principal methods of development in common use may be briefly classified as follows: I. By purely Algebraical or Trigonometrical processes. II. By Taylor's or Maclaurin's Theorems. III. By Differentiation or Integration of a known series, or equivalent process. IV. By the use of a differential equation. These methods we proceed to explain and exemplify. 103. METHOD I. Algebraic and Trigonometrical Methods. Ex. 1. Find the first three terms of the expansion of log sec x in ascending powers of x. By Trigonometry 2 X4 X6 cos= - - +.... 2! 4! 6! Hence log sec x= -log cos x= -log (1 - z), X2 X4 X6 where 2= — -— +. 2! 4!-6... EXPA NVSIONS. 93 and expanding log (1- ) by the logarithmic theorem we obtain 2 3 log sec x-=z + ] + +-... L! 4!+6! 2L2! 4!+** 1 X2 -13 3 -...... 3L2! J 2 A 4 56 2 24 720( &,4 X6 +... 8 48 24.92.94 X6 hence log sec =- i- -..... 2 12 45 Ex. 2. Expand cos3x in powers of x. Since 4 cos3x = cos 3x + 3 cos x 32,x2 34X4 32nX22 1 2 -'.+ -+ () 2 4! (2n2)! +3 -4 (2+-.+ )!+... we obtain cos3,9= 1 (1+3)-(32+3)i+(34+3)X4-. + (-1)(3,3)(2 +.... Similarly si'x= 3 ( - 3(- (3- 3) + (37 - -7 '3! 5! '7! 32n,-1_ ) (n -1) 1) Ex. 3. Expand tan x in powers of x as far as the term involving,"v. A'3 X7#5 X%- 2?5 x-3! 5! * — Since tan:.= — -..T2 X4 - + 2! 4!.. we may by actual division show that v3 2 tan.=+ +. 15+... 3 15 EXPANSIONS. Ex. 4. Expand { log (1 + )} 2 in powers of x. Since (1 +x)Y eylog(l+), we have, by expanding each side of this identity, 1 +x++(y(? 1)x2 +y(Y2 - 1)(Y - 2)X3 +Y(Y - 1)O( - 2)(y - 3)x4 + 2! 3! 4! 1 +ylog(1+X)+) +{log(l+x)}2+... Hence, equating coefficients of y2, ~{log (l+X))_)} 1+ 2 1. 2+2. 3+ 3. 1 t 8(lo (1 $ x) 3~.2+ X3 + X4 etc. 2! 3! 4! a series which may be written in the form — 2 3 4+ _ EXAMPLES. 1. Prove ex sill ' = + X + + 4 i6... 57 02 X4 2. Prove coshx = 1+ + (3n - 2)4. 3. Prove that [log (1 + x)]i x}" __)2 _ _x+3 -! -= -r-! l( r+ l1)!+ + +2(r + 2)! '3(r + 3),! where,Pk denotes the sum of all products k at a time of the first r natural numbers. 104. METHOD II. Taylor's and Maclaurin's Theorems. It has been discovered that the Binomial, Exponential, and other well-known expansions are all particular cases of one general theorem known as Taylor's Theorem, which has for its object the expansion of f(x+ h) in ascending integrcl positive powers of h, f(x) being a function of x of any form whatever. It will be found that such an expansion is not always possible, but we reserve for later articles [120 to 128] a rigorous discussion of the limitations of the theorem. EXPANSIONS. 95 105. The theorem referred to is that under certain, circcumstances h2 h. f(x + h) =f(x) + hf'(x) + 2/f"() + f ( + +- n!P() +... to infinity, an expansion of f(x - h) in powers of h. This result was first published by Taylor in 1715, in his "Methodus Incrementorum Directa et Inversa." In 1717 Stirling pointed out another form of Taylor's Theorem, viz.,,2 3 f/(x)=f()+xf(())+! f"(0)+.)!,(o)+... +;f'l(0) +... to infinity, which is easily deducible from Taylor's Series by writing 0 for x and x for h; the meaning of f'(O) 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. 106. Taylor's Theorem also ceduciblefrom iMaclautrin's. It has been shown that Maclaurin's series is deducible from Taylor's form. Taylor's series is also deducible from Maclaurin's. For, let f(x) = F( + y), then f'(x)=F'(x+ y), etc., so that f() = Fy), f'(O)= F(y), f"(O) = F"(y), etc. 96 96 ~~~EXPAYSION~S. Hence Maclaurin's Theorem becomes F(y ~x) =F(y) +xF'(y) ~ -F,(Y) ~..., which is Taylor's form. TAYLOR'S THEOREM. 107. PROP. To prove that, if f(x +h) can be expanded in a convergent -series of positive integral powers of h, that expansion is f(x+h)=f(x)~hf'(x)+_f"(X)+... to 00. Put x + h =X.; theii since x and h are independent dhL Hence df(X)X- df(X) dX dh dX dh Similarly dh2 -f"(X), etc. Now, assnuning the possibility of such an expansion, h2 h 3 let f(x+h,)=A0+A~h+A -— +A _-+......(1) where AO, A1, A 2,... are functions of x alone, not contamning h, and are to be determined. Differentiating with regard to h we have, by the preceding work, df(x +h) h 2 h 3 f (x dhh=A4+ A2h~A -~+A 4-~.. (2) Differentiating again f(h)df'(x +h) /62 h3 (x + h) =A +A h+-i4_ ~A5~I+..., (3) etc. EXPANSIONS. 9.97 Put h = 0, and we have at once from (1), (2), (.3),. AO =f(x), Al'=f'(x), A2 =f"(X), etc.,... Substituting these values in (1) f(x +h) = f(x) hrf()+-,(X.. ~.. J -f'(x) MACLAU RIN's THEOREM. '108. PRtOP. To prove that if f(x) can be expanded iv a convergent series Of positive integral powers of x, that expansion is f(x) =f(O) + xf'(0 n) + x X!f/I/(o)~+ to W. AssUmn'ing the possibilit~y of such an expansion, let x 2 X3 f(x)=A0~A x~A -+-A y+.........(1) where AO, Al, AD... ' are constants to be determined, not containing x. Then differentiating we have x 2 x3 f'(x)=A,+A X+A3&+A4........(2) f(x)=A2~A x+A2-~A.........(:3) etc. Hence putting x = 0 in (1), (2), (3),..., we have A0 =f(0), A1l=f'(0), A2 =f"(), etc., and substituting these, values in (1) f~x =f0) xf(0) ~9X2f"(O)+ 109. 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 each of the last two G 98 EXPANSIONS. articles are illegitimate, and we shall have to refer the student to Arts. 120 to 128 for a fuller and more rigorous discussion. 110. It is important, before proceeding farther, that the student should satisfy himself that the well known expansions of such functions as (x+h)", ex, sinx, etc., are really all included in the general results of Arts. 107, 108. For example, if f(x) = xn, f(x + h) = (x + h)", f'(x) = nX-1, f"(x) = n(n - 1)x-2, etc. Hence Taylor's Theorem, f(+ h) =f(x)+ hf'(x) + (x) +..., gives the binomial expansion (x + h)n = X + rihXz-I + ( -I ) h2-2 +. 2! ' Again, suppose f(x) = e^, then f'(x) = ex, f"(x) = e", etc., therefore f(O) = 1,f'(O) = 1, f"(O) = 1, etc. Hence Maclaurin's Theorem, f(x) =f(O) + xf'(0) + f"(0) +..., 2 3 gives e = 1 + x+,+:+..., 2! 3! the result known as the Exponential Theorem. 111. We append a few examples which admit of expansion, and to which therefore the results of Arts. 107, 108 apply. EXAMPLES. Prove the following results:3 X 5 L. sinx=x —+ -'"' 3! 5! EXPANSIONS. 99 x2 3 2. log(l+x)=x-+ —....x3 X5 3 5 412 +2 27r x2 3 3w x3 4. excosx=1-+ 2 cos-. x+ 2icos2 + 2cos3 +. 4 4 2! 4 3! +22cos- +.... 4 n! 5. log(l+e))=log2+2x+~ 2- x. 192 6. esiuC=.1 x + x2 - 4 -... 7. sin(x +h)=sinx+hcosx — -sin x- - cosx+.... -x h2 I+2X2 h3 8. sin-l(x +A)=sin-1x+- _ + _x -2 1+x h3 1-x2 (1- x2)- 2 (1- 2) 3 hA2 A3 cosrh 9. log sin(x + ) = logsin x + / cot- -- cosec2x+3- co+. 2 3 sin3. h A 2x -1 A 10. sec-(x+h)=sec —lx + -- +... x /x-l x-(x2- 1) 2. METHOD III. 112. 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-lx in powers of x. Gregory's Series. Suppose f(x) =tan-lx = a + a + a2x2 + a33 +..., then f' (x) -- = al + 2a2x + 3a3x2 + 4a4x3 +.. '; 1 4 x also - - X2+X -x4 X-6+.... 1 +Xs Hence, comparing these expansions, we have Ca2= 4= a6 = ag =... = 0, and a,= 1, 3a3= -1, 5a,=1, etc. Also, ao=tan'O-=nTr; therefore tan-1Ix =nr + x - x- + - -.. 3 5 7 1 00 EXPANSIONS. This result may be obtained immediately by integration of the series for 1, viz., 1-2+ X4- X6+..., the constant a0 being determined as before. Ex. 2. To expand sin-l x. Suppose f(x) = sin-x = ao + alx + ax2 + ax +...; therefore f'(x) -1 = V, a + 2ax + 3a32 + 4a3 +.... 1 1-2 But 1 + x2 + 1 -I4 +.... 2.4 B/1 — X2 2.4 Hence, comparing these series, we have a2=a4-=a6=-... =0, and 'a=1,33=, 5a 1.... 'ai=l, 3 ~ 2, 5%=-. Also Co = sin'l- = nr. X3 1.3x5 1.3.5 xi -lence sin1-x= r+x+. 3 + l -4- +. 7 +... 3 2.45 2.4.6 7 and, as before, this might have been obtained immediately by integration of the expansion of 1 Vl -mx Ex. 3. Again, if a known series be given, we can obtain others from it by diff&rentiation. For example, borrowing the series for (sin-lx)2 established in Ex. 2 of the next Art., viz.xsin-X 2 x4 2.4 x 2.4.6 xs (i 2 34 3.5 6 3.5.7 8 we obtain at once by differentiation sin-ll 2 4x + 2 467 +.. 1 - X2 3 3.5 3.5.7 EXAMPLES. 1 x3 1.3 x5 1. Prove log(x + l + x2) =sinh-lx = - - +23 2.45 X3 5 2. Prove tanh-x-=x+ -+ +.... 3 5 EXPANSIONS. 101 3. Deduce from Ex. 3, Art. 112, (1 - x2)sin- = x3 2 2.4 7 3 35 3.5'7... And hence by putting x = sin, prove t sin20 2 sin40 2.4 sin60 0 cot 0= - - 3 3 5 3.5 7 [QUARTERLY JOURNAL, vol. vi.] IV.-A NEWTONIAN METHOD. 113. It remains to exemplify a fourth method of proceeding which may often be employed with advantage, and moreover is of historical interest, as having been employed by Newton. Assume a series for the expansion (say ao + ax + atx2 +......). 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 get equations enough to find all the coefficients except one or two of the first which may be easily obtained from the values off(0) and f'(O). Ex. 1. Expand ax in this manner. Let ax=ao+alx+ax2 + a3X3 +.........................(1) If y =, But d/ l= a] + 2a2. + 3a3x2 +..;............... (3) dx therefore, substituting from (1) and (3) in the differential equation (2), a, + 2a2x + 3a2 +... loga(ao + alx + a2 +...). Hence, comparing coefficients, a1- aologea, 22 =alllogea, 3c3= a21ogea, etc. 102 EXPANSIOVS. Now ao=f(O)=a~= l; ~therefore ~a,=log~a, a 2 (logea)2 - (logea)3 therefore al=ogea, a2=- -, l3!,... 22!! and the series is a = 1 + x loga + (loga)2 +.... Ex. 2. Let y =f(x) = (sin-1x)2. =2 siInl- 1.- 2 2x Y=2- + sin - ix 2.Xy 1 - x (1 - X2) -2 2~.. (1 - 2)Y = + 2.................................................. (1) Now, let.y = aO + al + ax2 2+... + anx_ + an+lxn+l + a2Ln+2X+... therefore y = a+ 2a2x+... + na,xl" -1 + (n + I)acG+lx +(n+2)a,+2xl1+l..., and YJ2= a2.... + n(n - l)a,,x-2 + (n + l)na,n+lx"-l + (z + 2) (n + )a,,2+ +.... Picking out the coefficient of xn in the equation (which may be done without actual substitution) we have (n + 2)(n + 1)an+2 - n(n - l)a - na; therefore a.+2 =a, *** -.(2) (n+1)(n+2)...... Now, ao =f() = (sin-10)2, and if we consider sin-1x to be the smallest positive angle whose sine is x, sin-l =O. Hence a =0. Again, a1 =f'(0)= 2 sin- 0. 1 =0,,-0 and a- " (o) =( 2 +o)=1. Hence, from equation (2), a3, a5, a7,..., are each =0. 22 22 22 and aa= and ~ a 4=-3.4' 3.4 41 42 22.42 22. 42 a6 —. a4= =- 6!. 2, 5.=6 3. 4.45.6 6! 2 etc. = etc.; t2 2 22 2 2. 4 2 92' 42 62 therefore (sin -Ix)2 = + 24 + 2x6 + 2 42 — 28 +.... 2! 4! 6! 8! EXPANSIONS. 301 A slightly different method of proceeding is indicated in the following example. X2 a2 Ex. 3. Let y=sin (msin - ) = a + ax + a2'+3+....(1) Then Yncos (m sin1x), J1 = COs (~ 'in-,~ 77 whence (1 - x2)y2 =m2(1 -_2). Differentiating again, and dividing by 2yl, we have (1 - x2)y2- xy + m2.y= 0..O.................(2) Differentiating this n times by Leibnitz's Theorem (1 - X2)y+2- (2n + l)xyn+l + (m2- n2)yn = 0.............(3) Now, a0 = (y)x=o = sin (m sin-10) = 0, (assuming that sin-ix is the smallest positive angle whose sine is x) al = (Yl)x=o = in, a2= (y2)X=0= 0, etc. an= (yn)x=o. Hence, putting x=0 in equation (3), an+2= - (n2 - 2)a,. Hence a4, a6, a,..., each =0, and a= - (m2 - 12)a1= - m(m2- 12), a= -(m2 - 32) a3= n(m2 - 12)()2~ - 32), 7= -(m2 -,52)a5 = - m(2 - 12)(m2 - 3a)(m2 - 52), etc. Whence * ( si - m(m2-12) + 2 n(m2- 12)(m2- 32), sin (m sin-'x) = mx - m ---- +5 3! 5! _ (n2 - 12)(n2 - 32)(m2 - 52)7 + 7! The corresponding series for cos (m sin- x) is cos (m sin-l )= 1 m2X2 IM2(m2 -22)x4 _n2(m2 - 22)(7n2- 42) 6 cos (m sin-lx) = I - — + -- ---- X — — '-x +... 2! 4! 6! If we write x=sin 0 these series become sin me=ms m((nt2 - 12) si3 + re(m2 - 12)(m2 - 32)sn5 - etc, sinmO=msinO- )! 5!0-etc., 3! 5 cos m = 1 -m2 sin2 + m2(m2 - 22) sin40 2! 4! - t(m2 - 22)(m2 - 42) sin6o + etc. 6! 104 EXPANSIONS. EXAMPLES. 1. IfY=(1+x)'" = ao ~ a,x +a2X2 a3X 3+..., prove that dI.+x)!y_ = ny, dx and hence that ar+iA=rn-rar. In this manner find all the coefficients of the Binomial Theorem. 2. Ify= sin-1x=a0+ajx+a2 2+a~x3+..., prove that an+2 = (n + 1)(n + 2)a and in this manner deduce the expansion given in Ex. 2, Art. 1.12. 3. If y = (tan -1X)2 = a0o + a1x +a2X2 + a3X3 +..., prove that (n+ 2)(n+ 1)a2+ 2n22a,, + (o. - 2)(n - 1)a,,-2=. CONTINUITY. 114. DEF. A function is said to be continuous between any two values of the independent variable involved if, as that variable is made to assume successively all intermediate values from the one assigned value to the other, M Q.iP 0 A N B N Fio. 15. the function does not suddenly change its value, but changes so that for any indefinitely small change in the variable there is never a change of finite magnitude in the value of the function. EXPANSIONS. 105 115. Suppose the function to be p(x). Trace the curve y = 0(x) between the ordinates AL(x =a) and BM(x= b). Then if we find that as x increases through some value, as ON (Fig. 1.5), the ordinate ((x) suddenly changes from NP to NQ without going through the intermediate values, the function is said to be discontinuous for the value x= ON of the independent variable. 116. Similarly, we may represent geometrically the discontinuity of a differential coefficient. For represents the tangent of the angle which the tangent line to.Y M P/ L,;T 0 A B X Fig. 16. the curve makes with the axis of x. If, therefore, as the point P travels along the curve the tangent suddenly changes its position (as, for example, from PT to PT' in the figure), without going through the intermediate positions, there is a discontinuity in the value of d. dx' 117. PROP. If any fanction of x, say p(x), vanish when x=a and wihen x =b and is finite and continuous, as 106 EXPAN SIOVNS. also its first differential coefficient q'(x) between those values, then will 9q'(x) vanish for at least one intermediate value. For if ('(x) were always positive or always negative between x = a and x = b, 5(x) would be continually increasing or continually decreasing between those values (Art. 44), and therefore could not vanish for both x = a and x = b which would be contrary to the hypothesis. Hence +'(x) must change sign and therefore vanish for some value of x intermediate between x = a and x = b. 118. The same thing is obvious at once from a figure. For, suppose the curve y= p(x) cuts the axis at A Fig. 17. Fig. 18. (x =a, y =0) and B (x=b, y = ), then it is obvious that if the curve y = (x) and the inclination of its tangent be EXPA NSIONVS. 107 continuous between A and B, the tangent line must be parallel to the axis of x at some intermediate point P. It is also clear that the tangent may be parallel to the axis of x at other points between A and B besides P as in Fig. 18, so that it does not follow that +'(x) vanishes only once between two contiguous roots of ((x)= 0. 119. The same proposition is thus enunciated in books on Theory of Equations: "A real root of the equation ='(x)=O lies between every adjacent two of the real roots of the equation (x) = 0"; and is known as Rolle's Theorem. 120. Remainder after the first n terms have been taken from Taylor's Series. There is much difficulty in giving a rigorous direct proof of Taylor's Series, as might be expected from the highly general character of the result to be established. It is therefore found easier to consider what is left after n terms of Taylor's Series have been taken from p(x+h). If the form of this remainder be such that it can be made smaller than any assignable quantity when sufficient terms of the series are taken, the difference between p(x + h) and Taylor's Series for (x + h) will be indefinitely small, and under these circumstances we shall be able to assert the truth of the theorem. The following investigations of an expression for the remainder are taken, with few changes, from Bertrand's "Traite de Calcul Differentiel et Integral." Let R denote the remainder after n terms of Taylor's Series have been taken from O(x+h); so that 108 108 ~~EXPANSIONS. Let x~h=X, hence X -x (X-X)2 1 4'(x)- 2! (X X 1(x)-.R-0........(2) (n-i )! 17 PutR=( - ~n a orm 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, which is therefore bnd~ependent of z. Call the function thus obtained F(z). Hence F(z) = (X) -21 z -XZ/( -(Xz) (n-I) nl(j) n! P..... We shall assume that (fi(z) and all its diffprential,coefficients up to the nth1 inclusive are finite and -continuou~s between the values x and X of the variable z. It is clear from equation (2) that F(x) = 0, also by putting z = X in (3) we have F(X) = 0; also F(z) and F'(z) are finite and continuous between these values of the variable z. Hence F'(z) vanishes for some value of z intermrediate between x and X, say for z = x~+ O(X - x), where 0 is a 'proper fraction. Differentiating equation (3) with respect to z, the terms alternately destroy each other except at the end of the series, and we have left Y~)= - - (\n- )n~(Z-) +( I)......(4) whence P=,p{1x+0(X-x)}, EXPA VSIONS. that value of z being taken which makes F'(z) vanish. Hence, remembering that X - x = h, the true value of R sought is.9(x + Oh)........................(5) The theorem may therefore be written h2 ~(x+h)= -(x)+hp'(x)+ o (x )+... + - (x) + i(x + ),....(6) where 0 is a proper fraction. If then the form of the function O(x) be such that by making n sufficiently great the expression -,.(x+ Oh) can be made less than any assignable quantity however small, we can make the true series for 9((x+h) differ by as little as we please from Taylor's form ~(x) + hp'(x) + 2 "(x) +...to so. The above form of the remainder is due to Lagrange, and the investigation is spoken of as Lagrange's Theorem on the Limits of Taylor's Theorem. 121. A different form of the remainder is due to Cauchy. In equation (2) put R=(X-x)P and proceed as before, then, instead of equation (4), we shall have F'(z) = (X-) (z) + which vanishes as before for some value of z between z = x and z = x + h, say for z = x+ Oh; whence p=(1 - O)n-lhn(I 0hrfoe 1-0 1 ~(x+ oh) and therefore R = (1 ~ hn(x + Oh). 110 EXPANSIONS. 122. Another form is obtained by Schlimilch and Roche by assuming a slightly different form for R, viz., -(X-x)P+ix p. p+l 'This gives, instead of equation (4), (X-Z)n -- F'(z)= (n-)_ - (z)+(X - )PP, whence P = (1 - )f- + h"), (n - 1)! and R = (1 —)(pl + ) (n - 1)! (p+ )+ O h)~ The last form includes the two former as particular cases; for putting p+l=n it reduces to Lagrange's result, and putting p= 0 it reduces to Cauchy's. 123. The corresponding forms of remainder for Maclaurin's Theorem are obtained by writing 0 for x and x for h, when the three expressions investigated above become respectively xn (1 - O)\-lxn (1 X_ )n -p-l X. n(Ox), )i (Ox), and - 1)! (p I (Ox). 124. The student should notice the special cases of equation (6), Art. 120, when n = 1, 2, 3, etc., viz., 0(x + h) = ()(x) + hp'(x+ 0,h), (x + h) = (f (x) + hl'(x) + h9,"(x + 02h), etc. All that is known with respect to the 0 in each case being that it is a proper fraction. 125. Geometrical Illustration. It is easy to give a geometrical illustration of the equation ((x + h) = 0(x) + hp'(x + Oh). EXPANSIONS. 111 For let x, ((x), be the co-ordinates of a point P on the curve y= 0(x), and let x+ h, ((x + h) be the co-ordinates of another point Q, also on the curve. And suppose the curve and the inclination of the tangent to the curve to the axis of x to be continuous and finite between P and Q; draw PM, QN perpendicular to OX and PL perpendicular to QN, then (Px+h) -(x)_ Q-MP LQ t L PQ tan LPQ. h MN PL o T I S N X 0 T M S N X Fig. 19. Also, x+Oh is the abscissa of some point R on the curve between P and Q, and ~'(x+Oh) is the tangent of the angle which the tangent line to the curve at R makes with the axis of x. Hence the assertion that (x +h )-m (x) +,) is equivalent to the obvious geometrical fact that there must be a point R somewhere between P and Q at which the tangent to the curve is parallel to the chord PQ. 126. The cases in which Taylor's Theorem is said to fail are those in which it happens (1) That )(x), or one of its differential coefficients, 112 EXPA NSIONS. becomes infinite between the values of the variable considered; (2) Or that ~(x), or one of its differential coefficients, becomes discontinuous between the same values; (3) Or that the remainder, h-,n((x+Oh), cannot be made to vanish in the limit when n is taken sufficiently large, so that the series does not approach a finite limit. Ex. If ~(x)= N/x 0(x +/h) = /Ix+, '(x)=2- etc. Hence Taylor's Theorem gives 0(+r )= ah = x + 2 ish +'" oweer e put x= becomes infiite while If, however, we put 0=O, 2-7 becomes infinite, while /x+~ h becomes s.h. Thus, as we might expect, we fail at the second term to expand N/h in a series of integral powers of h. 127. In Art. 107 the proof of Taylor's Theorem is not general, the assumption being made that a convergent expansion in ascending positive integral powers of x is possible. The above article points out clearly when this assumption is legitimate. For any continuous function in which the (p+l)th differential coefficient is the.first to become infinite or discontinuous for the value x of the variable, the theorem ~(xt+ h) = q(x) + h0'(x)+... +~ h P(x + Oh), which involves no differential coefficients of higher order than the pth, is rigorously true, although Taylor's Theorem, EXPANSIOVS. 113 O( +h) = 0(x) +h0/( +... +h ( + '-1P+(X) + fails to furnish us with an intelligible result. Ex. If ~(x)= (x- a)t, we have +'(x)=- 5( a), l"{(z)=4 (x - a), 15 1, etc., and Taylor's Theorem gives (x + h - )-=(X - )\ + 5( - a) + 15( a)2 15 1 h3 (x+h-a)r-=(x-a)'+5(-a)+ 15x - a) — q- +..., 2 4 2! 8 (x-a)+ 3! which fails at the fourth term when x=a. But Equation 6 of Art. 120 gives the result 5 5 3 15kh2 (x + h- a)= (x - a) + (x -.a)A~ + 4 2! + h - a)A, 2 4 2! which, in the case when x=a, reduces to ],= 150-, 8 = 64 or 0 —,225' and this obeys the only limitation necessary, viz., that 0 should be a proper fraction. 128. The remarks made with respect to the failure of Taylor's Theorem obviously also apply to the particular form of it, Maclaurin's Theorem, so that Maclaurin's Theorem is said to fail when any of the expressions O(0), O'(0), p"(0),... become infinite, or if there be a discontinuity in the function or any of its differential coefficients as x passes through the value zero, or if the Xn remainder —!(Ox) does not become infinitely small when H 1114 1 EXPANSIONIS.,n becomes infinitely large, for in this case the series is divergent and does not tend to any finite limit. 129. Examples of Expansions by Maclaurin's Theorem, with investigation of Remainder after n terms. Ex. 1. Let f(x) = ax, then fn(x) = ax(log~a)", and fn(O)= (logea)n. Hence the fornmula (X) ==fn-I Xn f( )=f(O)~jf'(O) + f"(O) +... 1)f-1(0)+ fn(O) 2! (n - 1a2! oives x2 n-I n0(19,' a"= 1 + x log~a + (logaI2 + +1 +nl (logeXa)nl + a g 2! (n- 1)! na! Now fa(log!) can be made smaller ttan any assignable quantity by sufficiently increasing n; hence the remainder, after n terms of Maclaurin's Theorem have been taken, ultimately vanishes when n is taken very large, and therefore Maclaurin's Theorem is applicable and gives ax = 1 + x log,a +-(logea)+'+-(logka)3+. to co. 2! 3! Ex. 2. Let f(x)=log(1+ X), I I ____ 1-1-x (1+ X)2 (1+ X)n ilence (O)=O, f'(o)=O, f"(o)P = -O1, f"'(O)=2 fn(O) = (- I)n-1 (nj - 1)!. And the Lagrange-formula for the remainder, after n terms of Maclaurin's Series have been subtracted from f(x), viz. Xft(Ox) becomes n! and if x be not greater than 1, and positive, x is a proper fraction, and therefore by making n sufficiently large the above remainder ultimately vanishes, and therefore Maclaurin's Theorem is applicable and gives 2 x3 x4 log(I +x)=x - - + +... to 00 2 3 4 -where x lies between 0 and 1 inclusive. EXPANSIONS. 115 It appears that if we consider f(x)=log(l -x) the remainder is 1 (1 SY. -1n x - n In this form it is not clear that the limit of the remainder is zero. But if we choose for this example Cauchy's form of remainder, Art. 123, it reduces to I-dx,1 -O x and if x be positive and less than unity, x — is also less than 1 -Ox unity, and therefore 1-0 (-1 ) can be made as small as we like by sufficiently increasing n. Hence Maclaurin's series is applicable T2 X3 X4 and gives log(l - ) = - - ---- - -... to oo. 2 3 4 BERNOULLI'S NUMBERS. 130. To expand u=f(x) =' e + in pooers of x. 2 ex-l Let u-=f(x) and u'=f(0), u1 =f'(x) and i', =f'(0), =t2=f(x) and '2/=f(O), with a similar notation for higher differential coefficients. Then Maclaurin's Theorem gives x~: e':+l X2 2 ex -1 2! Changing the sign of x we see that the left hand member of this equation remains unaltered; hence we have 2 U=q' - X?6 +-2 '- ~ ~, and by subtraction 0 = 2x 'l + 2J3! '3 +2+u1' + 2,., i! 5 whence, by equating to zero the coefficients of the several powers of x, we infer that u' = u'1= '6... =0, so that the expansion contains no odd powers of x. 116 1EXPANSIONS. X ex Again, since &"U= + +x e 2 2 we have, by differentiating, (Uju,+u)=qtj,+ I + TX I1e' e2 2 ex('U2~+2u + u) =u2 + ( + 2)ex 2 e(u3+ 3U2~ 3u,+u)=u3+(x + 3),27 etc., and putting x = 0 in these equations we obtain from the first, third, fifth, etc., Y 3u'2 +nU' = T 5tt'4 + 10OU'2 + t'= y~ 7u'6 + 35u'4+ 21U'2 + u' =2 etc., giving U U'2 = 6LI lt'4 U '6 = 7T, U'8- - etc' X 1 1 X2 1 X4 I X6 I X8 Hence 1+ -1 -+- -- 1 2 ex -1 62! 30 4! 42 6! 30 8! 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 tbe form X e+I X X X2 X4 X6 X 8.rex+1 or ( 1or + 1+Bx- B3$~BX B79- -~ 2 e1 - 2/ 2! 64! 5 8! and the numbers B,, B3, B5,..., which are calculated above are called Bernoulli's numbers, having been first discovered and used by James Bernoulli. The coefficients of this expansion have been investigated as far as tbe term containing X32 by Rothe, and published in Crelle's Journal. 131. Many important expansions can be deduced from that of x ex+1 2 ex 1 ex +- e-'x e"x + I For example, x coth x = x e - eX+-1 '~x - Be- X 22X2 4X 2! 3 4 EXPANSIONS. 117 Writing Lx for x, tx coth Lx becomes x cot x, and we have 22x2 24 4 xcotx=l-B1 2 -B - B4... Again, tan x = cot x- 2 cot 2x 1 22x 24X3 __ -B — B3..] = B22(22 1) x+ B24(24 - 1)X 4... 2! 4 EXAMPLES. C3X3 C5X5 9c,. n r 1. Prove sin ax =ax - + - + sin +..., and 3! n!5! 2 that the remainder after r terms may be expressed as arXr rr\ Tsin aOx + r r! I2' a 2x2 2 4X4 CaX' 2,71 2. Prove cosax=1l- - + c-... + os- C, +, and 2! 4! n! 2 that the remainder after r terms may be expressed as - cos aOx +-. r! ( 2 ) 3. Prove (1 -)-n = l + C+^(n + 1 ) ). + 2! n( + -)...(n +r- 2)xr (r- 1)! n(n+ l + 1)... (n+r.- x) r! (1- Ox)"+'r 4. Expand and find the general term of the expansion of e Xcos bx. RESULTS. 1 + ax + ( 2- b22 + a(a2 - 32) + 2! 3! General term (2+b2)2 cos n tan- ib. n!\ \ aJ 118 EXP4A NSIONS. 5. Expand sinh3x and cosh3x, giving the general term in each case. 6. Find the first three terms of the expansion in powers of x of log (1 + tan x). RESULT. - 1x2,2 +.... 7. Expand log (1 + xe) as far as the term containing x5. RESULT. X3 + X' +... 8. Expand as far as the term containing x4 (1) log (1 + cos x) and (2) log (1 + x sin x). X2 X4 RESULTS. {(1) o10g-. 6 ((2) -2 _ 24 +. - 2 4 26 8 9. Prove logcosx= — 2 - 16- 272.. 2! 4! 6 8! sinx X2 X4 10. Prove log = -6 ---. x 6 180 x2 7 11. Prove log x cot x = - - - -x... 3 90 sinh x x2 x4 12. Prove log- -. x 6 180"' tan-1' A 13 251 13. Prove log tan =-3 x+ -. 51 1-75X 3 90 5.7.92 x2 x'? 1 1x4 X'5 14. Prove exosx = I. +... 2 3 24 5 1 2x:' 15. Prove exse= 1 + x +-_22 2 3. xex x x2 34 16. Prove log = - -z + ex" 1 2 24 2880 p {o + } _x 5x2 x3 25lX4 17. Prove log {log (1l+x)4 = -+ - -- 0. 18. Prove X2 2 9x: X4 X5 2 6 X7 X8 log(1l -x+ x2)= -x +-4 -+ - 3 -7 +8 23 4 5378 EXPANSIONS. 119 19. Prove log (I + X + X -- X3 + X4)= X +X2 3 -a14 4x.X 2 3 4 x5 x6 log(1 +x+x2+x3~+ x4)_=x~+ ~ + + _ - - + " 20. Prove (1 + x)x=l+ 1 + + 2 - X5 6 4-... Expand Examples 21 to 30 in ascending integral powers of x. 21. tanlx + tanh-lx. 22. tan- 2x +sinh-1 2x 1 - x2 1 - 2' 23. tan1 --- +tanh-l+ x. 1 - 3x2 r-.1~ 3,x2 24. tan-P- q. q+px 25. tan l/+x2- 1 26. tan- J1-x2 1 - 2X2 28. si- 2x x + x, 29. cos- -. 30. sinh-1(3x + 43). 31. If y = esin-lz = ao + al + a2X2 + a3x +..., prove (1) (1 -2)dY2 dya2y; _ 2 + a2 (2) a+- (+ +.a+-) (n f + +(n 2)( 3 (3) e fmsi- (,lx= 1 + expadig atx + f a2(a2 + a2)2d4 2! 3! 4! a(a2 + 1)(c 2 + 32)x e..5 (4) Deduce from (3), by expanding the left side according 120 1EXPANSIONS. to the exponential theorem and equating the coefficients of a, a2,..., the series for sin-'x, (sin-lx)2,..., and show that if in the development of (sin1, viz., xi1 IX 1.3 X5 1 2 3 2.4' 5 every number which occurs he increased by unity, the X2 2 x4 2. 4x result, viz., 2+34+.5. is equal to (sin'X)2 LPROFESSOR CAYLEY.] 32. Prove that if logy = tan-lx (l+ X)dY n 1.2(l=t \(2\ y dX11 L x 1\ dxdn-2' and hence find the coefficient of x5 in the expansion of y by Maclaurin's Theorem. [I. C. S. EXAM.] 33. If y satisfy the equation dy n2y = 0, and if the first dX2 and second terms of its expansion be respectively A + B and (Am - Bin)x, show that the general term is {A + ( - l)kB Mk Hence show that y = Ae" + Be-". 34. If y satisfy the differential equation dsy+ 2k + (k2+6 b2)y = 0, dx2 dx and the first terms of the expansion of y are k~2 - b22 l-kx+ _-_ continue the expansion. 35. If aR, be the coefficient of ex in the expansion of e'sin x show that sin sin a G a _i+a - 2 an3+ 2 2!! 3!.. A [I. C. S. EXAM.]! EXPANSIONS. 121 36. From y =(x+ \/1 +x2) obtain a linear differential equation with rational algebraic coefficients, and by means of it find the expansion of y in ascending powers of x. 37. From the relation y = + x)- obtain a linear differential 1 -x equation with rational algebraic coefficients, and by means of it find the expansion of y in ascending powers of x. 38. If tan y =1 + ax + bx2, expand y in powers of x as far as x3. [I. 0. S. EXAM.] 39. If Ao, Al, etc., be the successive coefficients in the expansion of y = ecs mx+sinmx prove + m nmr. r r \} An+=1 { A + - An(r {os 2 -sin ). [I. C. S. EXAM.] 40. If a x4 + a+1xfl+l + an+2x"+2 be three consecutive terms of the expansion of (1 - x2)lsin-lx in powers of x, prove that n-1 a+2 n+2 n; also that all even terms vanish, and that the expansion is 1 2 2.4 X - r-X 5. X5_ -.. 3.5' 3.5.7 [QUARTERLY JOURNAL.] 41. Show that if a rational integral function of x vanish for n values between given limits, its first and second differential coefficients will vanish for at least (n - 1) and (n - 2) values of x respectively between the same limits. Illustrate these results geometrically. [I. C. S. EXAM.] 42. Prove that no more than one root of an equationf(x) = 0 can lie between any adjacent two of the roots of the equation f'() = 0. 43. Show that the following expressions are positive for all positive values of x: (i.) (x - 1)e + 1; (ii.) (x- 2)ex+x+ 2; 122.EXPA NSIONS. 2 (iii.) (x - 3)e + + 2x + 3; (iv.) x - log(1 +x). [N.B. —By Art. 44, if dy be positive, y is increasing when dx x is increasing. Hence, if y be positive when = 0, and if also dy be positive as x increases from 0 to o, it follows that y will be positive for all positive values of x.] 44. Show for what values of x and at what differential coefficient Taylor's Theorem will fail if; f(x>_ (my- t)5(x- b) (x - c) -(x d)l4 45. Can log x or tan-l1 be expanded by Maclaurin's Theorem in a series of ascending positive integral powers of x? 46. If f(x) =ex, how does Maclaurin's Theorem fail for an expansion in ascending powers of x? Is f(x) continuous as x passes through zero? 47. If f(x)- x, show that there is a discontinuity in 1 +ex df(x) as x passes through zero. dx 48. Prove x'(x + hl) +f(x - h) = + ) + h4 2 2!x 4! 49. If 0= log x, du, x2 d2U prove that u+x d+ x +... x + 2! dX2 + du (log2)2 du, =+log2.dO 2! dO2 + EXPAVSIONS. 12S 50. Deduce from Taylor's Theorem, by putting h =- x the series -2 3 f(x) =f/(O) + ') + f"(x) - etc. 3! [JOHN BERNOULLI.1, 51. Prove tan-l(x + h)= tan-l + (h sin 6) sin 0 - (Ahs —2 sin )2 + sisinn)3 0)4. +(h '3 _ sin 30 - (-Si-n 0) sin 40 + etc., 3 4 where x = cot 0. 52. Verify the following deductions from Ex. 51:I\ crr a=+co0.sn0 cos2O Cos30 cos40 (1) =0 + cos. sin + os2sin20 COsin30 si4+ snsin4.... 2 2 3 4 by putting h = - x = - cot 0. 0 0 (2) == + sin0 + sin20 + si 3+ sin40+... 2 2 2 3 1 by putting h= - V1 + 2 = - sinwr sin 1 sin 20 1 sin.30 1 sin 40 (3) 2- 3-21..4 2 cos 0 2 cos20 3 cos3' 4 cos40 1 1 by putting = - x - = z sin 0. cos 0 [EULER.T 53. If f(x) be a rational fraction in which the denominator P(x) has n factors, each equal to x - a, and the remaining factors are x - h, x - k, etc., so that F(x) = (x - a)'y(x) where (x) = (x - h)(.x - k)..., prove that f(x)_ I f(a)+ _ d f(a) t + 1 __I d2 H 2!(x -ac)-2 da2 { () -A+ h 1_ d-1 f(c{ ) H+ (n- 1)! da n-1 I <>(a)(; - a) J,- - ' 124 EXPA NSIONS. 54. 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 -8- nearly. [HUYGHENS.] (2) Arc C + 256Q - 40 nearly (2) Arc= -. --- " ---- nearly. 45 Examine the closeness of the approximation in each case. 55. In the equation f(x + h) =f(x) + hf'( + Oh), show that the limiting value of 0 as h is indefinitely diminished is. [Expand f'(x + Oh) in the above in powers of Oh, and also f(x + h) in powers of A, and compare the two series, remembering that 0 itself, being a function of x and h, may be written =A + Alh + A2h2+... where Ao, Al... are functions of x. The term Ao will be the limiting value of 0 when h = 0.] 56. In the equation f(X + A) =f(x) + f.'(x + ohA), if 0 be expanded in powers of A, the first four terms will be 1 + 1 f3 + 1 2f A -A22 + 33ff2 - 90AfA + 553/3 + 2 24 f/2 48 22 5760f3 suffixes being used to denote differentiations. 57. Find by division the first six of Bernoulli's coefficients. 1 1 1 1 5 691 They are 5 691 They are 6' 30' 42' 30' 66' 2730 58. Prove by continuing the differentiations in Art. 130 that ++ nn - 1)( - 2) +.. 1, n+12 2 I 4! a formula from which the values of the coefficients B1, B3... can be successively deduced by putting n = 2, 4, 6, etc. [DE MOIVRE.] EXPA NSIONS 2 125 59. Expand i in powers of 0. [Differentiate expansion of cot 0, "Art. 131.1 60. Prove = +2(2 - I)B82+ 2(23 - I)-f -~ =1 +2 2! 4! [Use cosec 0 = cot~- cotO0 and Art. 131.1 61. Prove tanlix = 22(22 1- 24(24 - 1) 2! B 4! 62. By taking the logarithmic differential of the expression for sin 0 in factors and comparison of the expansion of the result with that of 0 cot 0 (Art. 131), show that 2(2n)! I 2fl 2 - (2rr 2 2 ",2 32.1 2(2n)! 1 (2w)2f - \ )'r' [RAABE.] where II - _ denotes the continued product of such factors r2 r 63. Expand sin(m tan1x)(1 + X2)M in powers of x. CHAPTER VI. PARTIAL DIFFERENTIATION. 132. 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 sev-_al 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 rectangular 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. 133. 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 di'ferentitiotn. a - The symbols -,, etc., are used to denote such differentiations, and the expressions etc., are differentiations, and the expressions Yx y etc., are ax, ay' PARTIAL DIFFERENAT TIA T1OiN. 127 called partial differential coefficients with regard to x, y, etc., respectively. Thus if, for instance, u= exy sin z, we have Ot =yey sinz, ox =xeXy sin, Dy Zz 134. 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 - may be expressed as L f(x + h, y, z) -f(x, y, z) Lth=o 'h - - h and similarly for - or 'y Dz 135. 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 z=f(x; y) cut off by the four planes Y= y, Y= y + y [Capital letters representing X=x, X=x+dx current co-ordinates], so that the co-ordinates of the corners P, Q, R, S are for P x, y, f(x, y), 128 PARTIAL -DIFFERENTIA T10. for Q X + ex, yf(x +, y), forS8 X, y + 6y,f(x, y~+ 6y), and for B x + x, y + 6y, f(x +ex, y + y). 2 0 Y~~~~~~~~ Nir Ni Fig. 20. If PLMN be a plane through P, parallel to the plane of xy, and cutting the ordinates of P, Q, B, S in P, L, M, N respectively, we have LQ =f(x + x, y) -f(x, y), N 5 =f(x, Y + 6y) -f(x, A),. (1) MBR =f(x + ~e, y + 6y) -f(x, y). J Hence the partial differential coefficient — obtained by ax considering y a constant is A(X + ex, Y) -f(x, Y) L Q = Lt fx=O ~ x = LtiL- = Lt tanLPQ. (2) 8cc P1 = 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. Similarly which is obtained on the supposition i an that x is constant PA RTIA L DIFFERENTIATION. 129 = Lt tan NPS,............................................. (3) = tangent of the angle which the tangent at P to c section parallel to the plane of yz makes with a parallel to the axis of y. 136. If the tangent plane at P to the surface cut LQ, MR, NAS in Q,' R', S' respectively, Dz LQ'=PL tan LPQ'= a.dx,............(4) NS'= PN tan NPS'= y. y,..........(5) Also the section made on the tangent plane by the four bounding planes of the element is a parallelogram, and the height of its centre above the plane PLIMN is given by ~MR' and also by I(LQ'+NS'), which proves that MR = LQ' + NS' x + y......................(6) The expressions proved in (4), (5), and (6) are first approximationso o the lengths LQ, NS, and MR respectively, and differ from those lengths by small quantities of higher order than PL and PNA, and which are therefore negligible in the limit when Sx and 6y 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 functions of several variables. (Art. 156.) 137. Differentials. It is useful at this point to introduce a new notation, which will prove especially convenient from considerations of symmetry. I 130 PARTIAL DIFFERENSTIA TJON. 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, 8y, 6z, 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 unnecessary to define them farther or to obtain absolute values for them. The student is warned again (see Art. 41) that the differential coefficient dy is to be considered as the result of performing the dx d operation represented by - upon y, an operation described in Art. 39. The dy and dx of the symbol dy cannot therefore be separated, and have separately no meaning, and hence have no connection with the differentials Dx and Dy as defined in the present article; but at the same time we have by definition Dy: Dx = Limit of the ratio 6y: 6x = Lt3Y 1 6A dy and therefore Dy = dYDx, and D- (which is a fraction) DX = (which is the result of the process of Art. 39). cess of Art. 39). PARTIAL DIFFERENTIATION. 131 We have used a capital in the differentials Dx, Dy, Dz for the purpose of explanation, and for the avoidance of 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. 138. Equation 6 of Art. 136 may now be written dz = >zdx + dy ax 'ay when 6x, 8y, 8z 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 =z y and z-x, ax 'y whence dz = xd + ydx. 139. 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 Dz Dz dz =-dx + -dy Dx ay may be at once written dz _z dx za dy dt ax t ay dt' where t is some fourth variable in. terms of which 132 PARTIAL DIFFERENTIATION. each of the variables x, y, z may be expressed; for d, d dyd dz =. dt, dx = -- dt, dy =. dt (Art. 137). ctt dt dt Similarly the equation ds2 = dx2 + dy2 may, by the same article, be written in the language of differential coefficients as dx2 (dy\2 = 1 Od (lds) 01' )2 (dI X y)2 or -1~ Let m = d(]t, dt vabecomse ay+ ives, then the function iy(x, y) analytically as follows: Let u = O (x, y), and when x becomes x+h and y becomes y+k, let Vu become tb+&S, then u+&,=O(x+h, y+k) and Su = -(x + h, y + k)- q(x, y) h (,(x,y + k) - ((x, y) and when we proceed to the limit in which A and k become indefinitely small we have Ltho)( Jr h, y + k)- (x, y+k) + Yk) = 3u Un ~ — - k —o x aX)=x, PAR TIA L DIFFEREN TIA TlIONT. 133 and Ltko(xo' y +k) - (x, y) _ -=O...k - y' Also da: clx: dy = the ultimate ratios of 8&a: h: k, hence d-u = dc + — dy. ax ay 141. Several independent vacriables. We may readily extend this result to a function of three or of any number of variables. Let ut = (xr x2, x3), and let the increments of x,, x, x3, be respectively h,, h,, h:3, and let the corresponding increment of /i be &u; then dU= (x+ h,1, x+ h2, x.+ 3 h,3)- ((x1, x, x3) _ (x+h,, h + +,,x, X3+h 3) - (x, X2 +h23+ h3) + 9I(X, x2+ 2) t3+, ~3)-(X1, x2s X3+ h3) h, + (x.I, x3+h3,)-( (x1, x2, x 3)1 k3 whence, on taking the limit and substituting the ratios du: dxi: dx2: dx3 instead of the ultimate ratios of hSi: h: 2: h3, we have du = dx + - — Ix2 + x ----, ax3 az, 2 OX3 i.e., the total differential of in when xz, x2, 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. 142. And in exactly the same way if u= (X,, X2'... x1), in = ((x1, x^2... aun at D DtDin we have di = - dxc + - dx, d + -dx3 +..+ ~-dx,,.,a —VI ax,? 3 Xn, 134 PARTIAL DIFFERENTIATI0ON. 143. Total Differential Coefficient. If = 5(X1, x,) where x1 and x2 are known functions of a single variable 7 u l, 'au x, we have du = -dxl + dx2, Dx1 ax 2 2OX and remembering that dcu dx cZx, du = -- dx, dx = — ldx, dz = dx, dx ' dx '2 dx du Dau dx, au, dx9 we obtain. dx- ax' dx x2 d-x' And similarly, if u= =((x1,, x x. ), where x1 X2,..., x, are known functions of x, we obtain dm Dm dx au dx a+t dx,, du -xu dx. ~._2 +? _ dx~ ax,' dx 'ax' d " x,,' dx ' And further, if x1, x, x,,..., xn be each known functions of several variables x, y, z,..., we shall have in the same way the series of relations Du 3' 3 ax l, 3D ax2 aau 3x,, 3ax ax 3 ax, 3 x ax" 3'x Dauau ax au ax au, x,, ay DaxjDy x,2 Day.t*,, ay etc. 144. An Important Case. The case in which u= 9(x, y), y being a function of x, is from its frequent occurrence worthy of special notice. dD aD4 D05 dy Here, by Art. 143, dx - + ydx dx d since = 1. dx PARTIAL DIFFERENTIATION. 1315 145. Differentiation of an Implicit Function. If we have O(x, y) = 0, then 9(x+h, y+k) = 0, and the 8u of Art. 140 vanishes. Proceeding as in that article we obtain a'+ 9. dy = 0 'x 'y dx dy Dx ay This is a very useful formula for the determination of dy dx in cases in which the relation between x and y is an implicit one, of which the solution is inconvenient or impossible. Ex. p(x, y) x3+ y3- 3axy = 0; find dy. dx' Here ~-=3(X2 xy) |ox \,dy - a and =- 3(Y-a)f d -y?/2- ax 146. Order of Partial Differentiations Commutative. Suppose we have any relation y = ((x, a), where a is a constant, and that by differentiation we obtain F(d x a), dx it is obvious that the result of differentiating ((x, a') would be F(x, 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 136' PAR]?TIAL DIFFERIENTIA TION. operative symbol such that when applied to any function of a it will change a to a', i.e., such that -Eaf(ac) == f(ac')~ then in operating upon the function p(x, a) the operations d Ea, and are, commutative, that is, d ct Ea dq5(x, a) = I-Eq(x, a) = F(x, a'). Next, suppose z = q5(x, y). The partial differential operations and a have been Dx ay defined to be such that when the operation with regard to either variable is performed the other variable is to be considered constant. We propose to show that these operations are commutative, i.e., that a a -a D-x DyY. Dy ax Let E, denote the operation of changing y to y+ 3y in any function to which it is applied; then Ey and the partial operation ' are commutdative symbols. And Ox Dp(x; y) D~(x, y) a a Y ~~ ~~ax ax D (xD x LtDy=x,by Def., 'ay ax 6yr g- (y)t, _ _y__ax ax -Ltay=o O JIV, O, (Xt~C:/ Ra~ty=OaaX 6y 'a x - c(X ~;/). Dx ay PAR TIA L DIFFERENTIA TION. 137 147. Another Proof. The symbols - and - may also be shown to be Ox oy commutative as follows: By definition a(x, y) Lt (x -, y) - x, y) ---- == j^/, ----.- ax h- o a x and --- Lk-(-x, yLto (f(xh, y+k)-((x, y+ Lt) Lt o(x+h, y)-p(x, y) _ _h_ h = Ltk= ----oLt, (x+h, y +k)- (x, y+k)- (x+h, y) )+ (x, y) =.Lt7o --- —-- k=O h. lk And, similarly, + -ah(x, y) may be shown equal to ax ay the same expression. 148. Extension of Rule. This rule admits of easy extension by its repeated application. Thus (D)(D) = Similarly (y) ( y) (,)) ' Also if we have more than two independent variables for instance, if =U=(x, y, z) = (33 )(+ -)(,3 g'v= etc. 3 d)^" u 3 ^) _( ^ - 1.38 1.38 PARTIAL DIFFEREA7TIA4TION. so that the order in which the differentiations are performed is immaterial in the final result. 149. Notation. It is usual to adopt for qD)2y U ( f rU., etc., the more convenient notation a2U a2'itt ap+q+rU Dm2' Dx~' ~P~q~_ etc., and 'the propositions above enunciated will then be. written D? 2j DX2Dy -DyaX2 etc. 150. The formulae here established may be easily verified in any particular example. Ex. Let U =sin (XY), thenr and -=cty Cos (xy), Dax -~ n == O y -x inx............(1) ZD2/a Againiav = x COS xy,~ a2u and = O rD1~ il g and the agreement of equations (1) and example tbe result of Arts. 146, 147..............(2) (2) verifies for tbis 151. It. is convenient to use the letters p, q, r,, s, t, to denote the partial differential coefficients PARTIAL DIFFERENTIA TION. 139 ax' Dy' x' DXay' Dy2' where (p is a given function of the two variables x and y Hence we have, if z = O(x, y), dz = pcx + qdy, Art. 140; and to obtain from. the imp~licit relation p)(x, y) ==0, we have qy 152. To obtain the Second Differential Coefficient of an Implicit Function. To obtain &1we have only to differentiate the last result.-of the preceding article; thus, d y dx d 4x2~~~~ Now drD) _al- + P~_r+8_j q p dlx Dx -ay d x q/q and DqDqdy=+t( ~sp dx 'ax 'ay dxq q giving - - 2 _ q q2r - 2pqS + p2t Similarly d etc., may b~e found, but the results are complicated. 140 PARTIAL DIFFERENTIATION. EXAMPLES. 1. If u = x;y, prov du G dx dy pD{ro)ve -- =..n- + n "?t x? and verify the formula - 2. Verify the formul;a i = in each of the following, xy oqax cases:- (1) (2) u - s -Ya&2 z2'.,. (3) u log + 2 3?/ (4) u -=.. 3. If u=- /i2x +;' show that^ Dyz2 zd/ 4. If x = r cos 0 and y = r sin 0, prove cd = cos Odr - 9r sin OdO, and de = sin Odr + r cos cd; and hence that dx2 + dy = dr2 + '2d02, and that xcly - ycx = r2do. 5. If q= log(x2+y +2 +), D~zu D"~u D"~2u prove x —"= 6. Prove that if -2 +=-, C b2 dy b2x cld - b4 ----- allnd -t2= - -.. d.x ac? dxC a~y' 7. Show that if Xmq + yi = an" dxs21 (m- I)a a /72.,/ A>"-2) PARTIAL DIFFERENTIA TIO14 141 153. To find a fnd d- from the equations F1,x'y, z=0, F(x, y, z) =. Here, as in Art. 145, DF F + ct y+DF cl _z ax y ' dx Dz dx DF, + t dy F, dz _ Dx y ' dx Z ' dx Solving these equations we obtain dy dz dcx _ _dx_ _F, aF2 DF. DF DaF1 DF _DF DY, ____ - 1 2 2...... 2- 9 Dz Dx Dz ' Dx ax ' Dy ax ' y -F DF., DaF DFE 1__ Oy 'z O y ' Oz d y dz D which give the values of dy and dz. 6 dx dx' Ex. Given y = F(x. z), and z= F(x, y), DZ?+Ft DF:, dy_ x Dz ' ax prove dr 1 DF DF 154. Given' that O=(px+st, y + rt, z + t,...) vwhere x, y, z..., ~,..., and t fo rm a system of independent variables, to show that DT DV DV- DV vt =~ - + + +'"D 142 PAR TIA L DIFFERENTIA TION. Let x,=x+ t, Y1 = Y.+) t, etc., ax-, y, etc. so that =1, 1 et., ax 'ay axi ay, __1=- =,. =rl etc., at at -x =- 0, 1 = 0, etc.' ay ax then V= q(xl, y1, z,...), md ^ JaValr 'ax 'aV 'a and 0 = 3x — T. - x ~ yx+~... (Art. 143) ax a9^ ax 'ay, ax av,ax' aV 'av Similarly = etc., 'ay 'y, and aV'aV 'ax+=aV,ay +, 'at ax, 'at 'ay at+.. 'av 'aV 'a V aV+ nV + V 155. Hence we have the following identity of operators, viz.:'a 'a a a = c-t +) '+ +*' and as the variables are all. independent and the operators partial, 'Y ( a, 'a the development being made in formal analogy with the Multinomial Theorem. For example, in the case of V= ((X + t, y+,)t), PARTIAL DIFFERENTIATION. 143 DV aV aV we shall have - == -V + -+ V, 2V a22V D 2 V D2 V etc. TAYLOR'S THEOREM. EXTENSION. 156. To expand {(x+h, y+k) in powers of h and k. By Taylor's Theorem we obtain, (x+h, ) 1/2 320(x+h, ) 9(x+h, y+ic) = (x+h, y)+k- - -,... -y 2! aD/2 and expanding each term we have h2 a2~ 9(x+h, y+1c) = 0(x, y) + h +!2 - +... ay axay C2 320 + 7,O+.. +'I Dy2 1 920 0 0 102 72, \ = f(Xt,) y )a(h + 1753) + - (h2 _ + 2hk- + -2-) +.., as it may be y,y2 or, as it may be written symbolically, (x+h, y+k) = (, y) + (h + + (/ + ) +... 157. Since it is immaterial whether we first expand with regard to kl and then with regard to h, or in the opposite order, we obtain by comparison of the coefficient of hk in the two results the important theorem 32xy 3_y2 axay yaxes already established in Arts. 146, 147. 144 144 PARTIAL DIFFERENTJA TION. 158. Further Extension. Several Variables. The form of the general term in the preceding case and the further extension of Taylor's Theorem to the expansion of a function of several variables is more readily investigated as follows: Let q5(X+et, Y+nt,.. be called F(t). Then Maclaurin's Theorem gives F(t) =F(O) +tF'(O) ~ <FI(O) +... + Fn(Ot), and by Art. 155 Fr(t) -a4;. ~.) and since the variables x, Y,...,I are in-dependent of t, we may put t =0 either before or after the operation has been performed. Hence F'(0) = ~ 4~. x,...) We thus obtain Now+ e, pu+ ttin...~ k=.t, l= t C...w obtain + a +Dy "/T...I PARTIAL DIFFERENTIA TION. 145 159. Extension of Maclaurin's Theorem. Moreover, if we put x =0 and y=0, and then write x for h and y for k, we have an extension of Maclaurin's Theorem which, for two independent variables, may/be written $(x, y) = 0((, ) )+ x( ) +y o ax 0)-,,0) 0(y+,( +2,X2($2) + 2xy (X) + a 20 + etc. 160. If we now recur to Art. 136 we see that the true value of MR is f(x + Sx,.y + y) -f(x, y) -D+e D /f _ ____ a2V a'f~y)+etc., = x + ey + (2f8x2 + _ S3 x y + 8y2 + etc. 'x y 2!\aX2 axay ay" showing what error was made in that article in taking MR' as an approximation to the correct value. The student will find no difficulty in writing down the true values of the lengths of LQ or NS. EULER'S THEOREMS ON HOMOGENEOUS FUNCTIONS. 161. If u a= A xay + Bxa'yP'... = Axay, say, ywhere a +/3=a' +/3=....=, to show that x- + y- = u. ax ay By differentiation we obtain au D = ay3 K :146 PARTIAL DIFFERENTIATION. then + y = 2+y-=AaXay4 + Z xayxy ax ay = -A(a + )xayP = nAx ayf = 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 xayxz + Bxa'y'z'' +... where a+/3+y= a't'+y... n, 'zc 'tau.u then will x- +y- + z- = nu. Ox.y a z The functions thus described are called homogeneous functions of the nth degree. 162. We now put the same theorem in a more general form. DEF. A homogeneous function of the nth degree is one which can be put in the form xnFQ" ) u= y, -....). Put Yy, -=Z, etc., X X l Y y 'Z z whence DX X2' Dx - 2. DY 1 eZ a- = -, I- = 0, etc. xy X- xay Now, since u =x iF( Y, Z...), v...Da-2 F..Z =nx-n F(Y, Z, I) * * a yT+z2+., } 'a Y az f PARTIAL DIFFERENTIATION. 147 Dau DF DZY, F 7, __ _ = "-n -- Dz DZ' ay DZp etc. = etc. Finally, multiplying by x, y, z,... respectively, and adding xat D + aZ +... = nx:I( Z,... ) = 163. If u be a homogeneous function of x and y of the nth degree, D- - will be homogeneous functions of the x' -ay (n-1)th degree, and applying the result of Art. 162 to these we have ( + = (( -1) -, O+Yx}~ (nx - 1)/ a, \3t _ Du ax ay' V —~+ Y3-]ey= (-l) vy. Multiplying by x and y we have on addition 2.2:, 2. 1 au 'au\ _a2+2xy, +y2 = (n-I) x-+ -+ ) DX2 Dx~y ay x ay = n(n - 1)Z. 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. 166. 164. If V= uL, +,n + u -2 +... + Lt2+ U' + where un, un-,... are homogeneous functions of degrees n, n-1,... respectively. Then x ---+yk+... x ayetc. X(^+...) +( t+...)-i+etc. 148, PAR RTIAL DIFFERENTIATION. -=L,, + (n- 1)zn_ + (n - 2) u,_2 +... + 2 2 + tU =nV- {n,_l +2T_-2+39q_-3~+... + (9 - 1)u1 + L o} Hence if V= ( 1ax D' x —+y +...+u^~l+2yn 2+... +'a-=. 165. Let u= o(Hn), where H', is a homogeneous function of the nth degree. Suppose we obtain from this equation ffn = F(); a x then x-F(u) + y-yF() +... =. H or F'(u){ x +... = zF(), +zy aDt = F(u) or x+y +.=F'()..... (1) In the particular case in which n= 0 we therefore have --- +Y =+ 0.................. (2) EXAMPLES. Verify the following results by differentiation. 1. Let u= x3 +y3+ 3xyz. This is clearly homogeneous and of the 3rd degree, whence % + za= ~. 1+(Y)4 xD — +?- = x3u 2. Let u- ' -x x5 + Ya 1 + \X/ This is a homogeneous expression of degree -V, whence,x e D PARTIAL DiFFERENTIATION. 149 3. Let U= sin - + Here Art. 162 gives x au + a-= 0. Dr Dy 4. Let qt=tan-1 X3+Y X-Y Here Art. 165 gives D Dr-= sin 2u. 5. Find which of the following functions are homogeneous, and in cases of homogeneity verify Euler's Theorem of the first degree: (a) xe5. (A3 ye'Y. (y) (x-y)(log x - logy). ( sin-' II/ X +?f 6. Given z=x2+y and &= 2 +, 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. 7. Given v.ryz= a, find all the differential coefficients of the first a1nd second orders, taking x and y for independent variables. 8. If u=sin'1 x + prove that X-a+ y- tan u. Dx Da 9. If u = ax2 2+by2 + C,2 # 2fyz +2gzx + 2hxy, show that, 'if it be possible to find values of x, y, z which will simultaneously satisfy* Dn au aD Ox ay C -Z a, h, g then will A, b,,f g, f, c 150 1P0 ARTIA L D)IFFERENTIA TION.l; 10. If u be a homogeneous function of the nth degree of any iumiber of variables, prove that (x~a+<+a...) t X' +Y a + 71. If u O(x, y) and (x, y) 0=, prove that du Tx 'ay O, DxJ 166. General Proof of Euler's Theorems. We now proceed to give a more complete investigation of Euler's results. Let m,(x, y, z...) be any function expressible in the form It is observable that if x +xt, y+gt, z+ zt,... be written instead of x, Y, Z,... in any such function we obtain the result Ox+ xet, y +yt,.) = xl(l + t)iflF ~j Y' xV + Xt" eb(I +.)~' t)" X.... = x(1 ~-tlu so that the effect is simply that, of multiplying the original function by (I1~ t)n. Now, let 'Vr denote the symbol of operation obtained by expanding (xX+yY+ zZ+...)'n by the Multinomial rheorern, and after expansion writing in ax' ay', -z' place of X, Y, Z, etc.; then we have, upon expansion of each side of the above equality, PARTIAL DIFFERENATIA TION. 151 t t3 l) tr u+tlV,++ V. + + + ={+nt+ n(n - )t (n )(n -2)t3+. + n(- 1)- +.. - r r 1)tr }U. And on equating coefficients of like powers of t Vyu = nu, VU = n(n- 1)u, V3U = (n - 1)(n - 2)u, etc. V'U = n(n - 1)... (n - r + 1)U. 167. When there are two independent variables, x and y, these become x3- + y = -- x ff- 2 x"y +y2+ ==n(n - ), etc.; and for the case of three independent variables?u 'au au a2? a2) u 32U, 32.,t 32U a2tt a2 a Dy2 Z3 2 ayaz 3zax axay = J(n- 1)t, etc. 168. Care must be taken to distinguish between the expressions 2a2t a2u 32mu x n+ 2xYZy + y2 and + Y U1 (ax +ay y)ZG, 152 PARTIAL DIFFERENTIA TION'. which might at first sight be thought to be identical. However, it is apparent that the latter / ', \/ ' u\ - x+ Yy + + / 232% 3 \ 32 \ Dm a/ 32U 3u D2 a2\ + x +y ++~y2ay \2% 32 32D D2m2 D 3 =X2 +2XYD + 2~+ Y+^ = [ 2 ~2 + -x — + xyvy+x + and therefore differs from the first expression by the addition of the two terms 'au" 3u -- y-. ax ay EXAMPLES. 32u a2u. 1. Verify the formula _ -DU in the following cases: Dxy 'yax (a) u =sin -. 2. Find dy (a) if ax2 + 2hxy+ by2=. (/3) if X4 + y4=5a2xY. (y) if (cos x)Y= (sin y). (8) if y + ==(x + y)X+" (e) if xy. y =- Xcosy + yloga' PARTIAL DIFFERENTIA TION. 153 3. If u = sin-lx + tan-'Y, show that x-_ + yu = 0. y x ax+ ay=0 4. If u, y, z be functions of x such that d / dy\ d \ dz dx\ dx) dx\ d x, prove that -d u yd - zy) =0. da xdz ddx 5. If u and v be both functions of the same function of x and au, V a v /v \ 3 / V\ y, prove that - v=.u, and that - - =v (uv x ay x' Dx y Dy Dx 6. If V=f(u, v), u=fi(x, y), v,=f(x, y), show how to find DV. DV DV in terms of- - and Tx- a _nd ~. Ex. Given u = x2 + y2, v = 2xy, show that - -y - 2(u2 2- 2)- V 3x 'ay au 7. Verify Euler's Theorem au + yDa= Z for the functions (a) u= sin( — Y x +y () U = x3ogy - D2u 2D2u \32U a2U, 8. If u = 9(y + ax) + f(y - ax), prove 2 = a D2 -9. If u= (Y)+ y 2 prove x2 + 2x y +u y22 0. aX32 axay aY2 10. If u = (2 + x ) (s) + + t, prove that 2Du u 3 Du 2u ( 2o + y)m. xD + 2y-D- + Yy2 - = + y( 154 PARTIAL DIFFERENVTIA TION. 11. If V2 + + and r2_ X2 + y2 + 2, show that each of the functions tan-~l 1, z log r + satisfy the equation V',. = 0. 12. Prove V27"r= m(rT + 1)r'.-2. 13. If u and u' be functions of x, y, z, both satisfying V FV= 0, 2 au'. +7. +au, au prove that V2(uu ') = 2(- * +?. D a + D -u' Dv ' r\Dx Dx Dy Dy Dy^z Dz 14. If Vn be a homogeneous function of the nth degree, satisfyingV V = 0, then will A, also satisfy the same equation. 15. Iff(x, y)= 0, O(x, z)= 0, show that D3 af dy_af vD 3.x y 'dz 3ds 16. Find dy in terms of y and z from the equations: dz a sin x + b sin y - c. a cos x + b cos z = c. [I. C. S. EXAM.] 17. If x4 + y4 + 4a2y = 0, show that (yW + ax)3d = 26a2y(x2y2 + 3a4). 18. If ( + ( + = l, find d x ad. Also, find \a/ \b \c/ x Dyoz dy when the variables are connected by the two equations dx /zV A /xY' __ /yY1 xy " 1. C) a) - )' a b [H.C.S. EXAM.] au au a u. 19. If u= F(x- y, y- z,- x), prove +- +- = 0. Dx Dy Dz PAR TIA L DIFFERENTIA TION. 155 x2, 2 D2u Du ' 20. If u- x,, z, prove - + y +- z -0. 1, 1, 1'a a 21. If u = cosec-1 x/ +-, show that v x + yf" 2'a~2 __U ' 2'2U tan u/13 tan2u X ^?^21' + 2x I/- 2: 'acx Xa/a, -+aY Dy2 12\12 12 / a2Z a2Z 22. Find the value of the expression - + ---2 when a2x2 + b2y2 - c22 = 0. [I. C. S. EXAM.] 23. If F= Ax2 + 2Bxy + Cy2, prove 'DaY' 2 '2 2'V DV D2V l(D)2 D2V_ 'ax) 'a-2.. j y +. 8 V(A C - J2). nau _ Dv Du Dr 24. If u + /- l= f(+ + - ly), prove = - a = a_ v a2u u 2, 'a 2 a2v + - O= and 2 + 2=0. ax2 Dy2 'a2 o2x 25. If u + \/- lv be a homogeneous function of x, y, z, of au aqu auq degree p + /I- lq, then 'a +?y- + =pu - qv, av D v Dv and x-x + ya + z- = pv + qu. ax Dy az 26. If V= (1 - 2xy y2)-, prove that DYV 'aI 2Y'3 _ — -y —= y V. ax Dy Also that a (1 2 - ) + y23 = 0 V~' o,,; 9 % y o 156 PARTIAL DIFFERENTIA TION. '2 y2 z2 27. If 2- + b + - = 1, and lx + m+ nz = 0, prove that a2 b2 C2 dx dy dz ny mz -l nxmx ly 62 c2 c2 a2 a b2 28. If -+2+=1, and 2- + =1, prove a2 /rj2 ~_C2 2 + 1 prov that (. - a2) + (2 -- 2) += 0. cdx cly dz 29. If Pdx + Qdy be a perfect differential of some function of x, y, prove that DP = DQ Dy Dx 30. If Pdx + Qdy + Rdz can be made a perfect differential of some function of x, y, z by multiplying each term by a common factor, show that Q \ + QR DP + PR(P \ Q =. P( D D-ay La / Dy Dx, APPLICATIONS TO PLANE CURVES. CHAPTER VII. TANGENTS AND NORMALS. 169. Equation of TANGENT. It was shown in Art. 38 that the equation of the tangent at the point (x, y) on the curve y=f(x) is Y-y= dY(X - ),................. (1) X 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 f(x, y)= 0. It is shown in Art. 145 that Of dy _ x dx dy Substituting this expression for dy in (1) we obtain dx Y- y=-f(X-x_), ay 160 TAN'GENTS AND nO11RMALS. (X af ff or (X-x) +(Y- <-, (.............. 2) Ox a for the equation of the tangent. 170. Simplification for Algebraic Curves. If f(x, y) be an algebraic function of x and y of degree n, suppose it made homogeneous in x, y, and z by the introduction of a proper power of the linear unit a wherever necessary. Call the function thus altered f(x, y, z). Then f(x, y, z) is a homogeneous algebraic function of the nth degree; hence we have by Euler's Theorem (Art. 161) Z f + = nf(x Y, Z) = 0, 'ax 'ay a by -.virtue of the equation to the cur-ve. Adding this to equation (2), the equation of the tangent takes the form XI- + O. 0..................... (3) Ox ay az w'here the z is to be put = 1 after the differentiations have been performed. Ex. f(x, y) = +- x4 a2xy + b3y + e4=0. The equation, when made homogeneous in x, y, x by the introduction? of a prloper power of z, is f(x, y, -) X4 + a2Xyz2 + b~yZ3 + C4z4 = 0, and Df=4X3 ~ a2Y22, ax Df a2Xz2 + b36~ __ 2a2xyz + 3b3yz2 + 4c4z3. Substituting these in Equation 3, and putting z= 1, we have for the equation of the tangent to the curve at the point (x, y) X(4x3 'a2y) + Y(a2x+ b3) + 2a2xy + 3b3y 4c4 = 0. With very little practice the introduction of the z can TANGENTS AND NORMALS. 161 be performed mentally. It is generally more advantageous to use equation (3) than equation (2), because (3) gives the result in its simplest form, whereas if (2) be used it is often necessary to reduce by substitutions from the equation of the curve. 171. Application to General Rational Algebraic Curve. If the equation of the curve be written in the form f(x, y)- n+Un-l+Un-2+... 2 +uI+U+U= (where Ur represents the sum of all the terms of the rth degree), then when made homogeneous by the introduction where necessary of a proper power of z we shall have f(x, )y, )=Un+Un, Z+U_2z2 +.. + Un - 2 + nUzn -1 + UVoZ, and =,n- 1J 2un-z+32Z+3Un-32+... + (n - 2)2zn - 3 + (n -1 )n_ z2 - 2 + nvoZn -1, and therefore substituting in (3) and putting z=1, the equation of the tangent is X —f+ Ya+ 8U-_ + 2U_2 + 3Un-3 +.. ax ay + (- 2),2+ (~ — l)u, +, = 0...... (4) 172. NORMAL. DEFr- 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 equation of the curve be y=f(), L 162 162 ~TANGENTS 11ND NORMALS. the' tangent at (x, y) is Y-y "( ) and the normal is therefore (X-x)+(Y dcJ=O If the equation of the curve he given in the form fAx, Y) 0, the equation of the tangent is XDx ay ' and therefore that of the norm'al is X-x Y-y: Ex. 1. Consider the ellipse This requires z2 ithe last term to make a homogeneous equation in x,. and Z. We have' then. a2 b2 le-nce the equation of the tangent is.9 2XA+ 2yz2zO a2 Y ) Z.2 - ' where z is to be put =1. Hence we get +xJYIothtnet a2, =1fr h anet and therefore X - =.-Y. for the normal. a; Y Ex. 2. Take the general equation of a conic ax2+2hxy + by2+ 2gq +2fy + cO. When 'made homogeneous this becomes. ax2+2hxyi+by2+2gX z+2fyZ +cz2~=O. The equLation of the tangent is therefore TANGENTIS AND NORMALS. 163 X(ax + h' +g) + Y(hx + by +f) + g., +fy + c0=, and that of the normal is ax+hy+g hx+by+f Ex. 3. Consider the curve X p-log sec -. aL Ca Then d tana dxX a' and the equation of the tangent is Y-y =tanff(X-x), a and of the normal (Y-y)tan +(X- X) = 0. EXAMPLES. 1. Find the equations of the tangents and normals at the point (x, y) on each of the following curves: (1) X2+y2=c2. (5) ~ Xy2 ==a3a (2) y 2=4ax. (6) ey - sinx. (3) xy=k2. (7) X3-3axy +y3=0. (4) y = c cosh. (8) (X2 +y2)2 = a2(X2 2y2). 2. Write down the equations of the tangents and normals to the curve y(X2 +a2)= ax2 at the points where 3. Prove that x 1 touches the curve yYbe= - at the point 3. be~.C: where the curve crosses the axis of y. 4. If p = cos a +y sin a touch the. curve x~n in_ a' b~ an b yb prove that prn-l= (a cos a) '+ (b sin a)m-. 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. 164 TANGENTS AND iVORMALS. 5. Prove that, if the axes be oblique and inclined at an angle w, the equation of the normal to y=f(x) at (x, y) is (Yy)(cos w+co )+(Y-x)(l+ Cos (Yc-)=0. 173. Tangents at the Origin. It will be shown by a general method in a subsequent article (254) 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; the rule being to equate to zero the terms of lowest degree in the equation of the curve. Ex. In the curve x2y2 + ax + by = 0, ax + by = is the equation of the tangent at the origin; and in the curve (X2+y2)2=a2(X2-y2), x2 -_2=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 = (X - x) where = cmx. tdx At the origin this becomes Y= mX, where the limiting value or values of n 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 rth degree. The equation may be written X)rf() + xr+lf+l(.) + fn Dividing by xr, and putting y = mx, and then x = 0 and y = 0, the above reduces to the form f.(m) -= 0, an equation which has r r oots giving the directions in which the several branches of the curve pass through the origin. If mn1 m, 3n,... mr be the roots, the equations TANGENTS AND NORMALS. 165 of the several tangents are y = m, y = mx,... y = mr. These are all contained in the one equation f()= 0; 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. In this manner all the trouble of differentiation is avoided, and the result written down by inspection. GEOMETRICAL RESULTS. 174; Cartesians. Intercepts. dy From the equation Y- y a-(X-) it is clear that the intercepts which the tangent cuts off from the axes of x and y are respectively y dy x-_ Y and y-x xY, dy d dx dx for these are respectively the values of X when Y=0 and of Y when X =0. _ t\~ P A fTO]. Nt G X Fig. 21. Let PN, PT, PG be the ordinate, tangent, and normal to the curve, and let PT make an angle 0 with the axis 166 TANGENTS AND NORMALS. of x; then tan 0 =. Let the tangent cut the axis of y dx in t, and let OY, 0 Y be perpendiculars from, 0, the origin, on the tangent and normal. Then the.above values of the intercepts are also obvious from the figure. 175. Subtangent, etc. DEF. The line TN is called the subtangent and the line NG is called the subnormal. From the figure Subtangent = TN= y cot 0 =. dx Subnormal = NG = y tan 0 = d-. Normal = PG = y sec 0 = y/1i+ tan20yI + y2. {dy\" mn N/1+tan 2 d1'^ Tanget = TP = y cosec 0= y- t = ytan 0 =y dy dx.- dy dy 0 Y= Ot cos 0 =, = ( Vl + tan20 d/ 2 dy Y1=OGcos= tON~+NG dx_ 0 O =O G cos = =0 -I.+tan2O l d y\2 These iresults may of course also be obtained analytically from the equation of the tangent. TANGENTS AND NORMALS. 16 U ds dx 176. Values of j-, T etc. Let P, Q be contiguous points on a curve. Let the co-ordinates of P- be (x, y) and of Q (x + x, y + 6y). Then the perpendicular PB= rx, and RQ=dy. Let the P R A O ~ RI~1\ N X Fig. 22. arc AP measured from some fixed point A on the curve be called s and the arc AQ=8s+68s. Then arc PQ = 8s. When Q travels along the curve so as to come indefinitely near to P, the arc PQ and the chorid PQ ultimately differ by a small quantity of higher order than the arc PQ itself (Art. 36). Hence, rejecting infinitesimals of order higher than tim second, we have -2 =(chord PQ)2 = (8X2 ~ 8y2)' or i = L(X2 2+ y2) (dX)2+ (dy)2 82 6Y 2~y" Similarly Lt L4 62 2 X/ or (i-)2 =1+ dx' and in the same manner ids \2 Idx W90 11-I 168 TANGENTS AND NORMALS. If r be the angle which the tangent makes with the axis of x as in Art. 39, tRQ ty dy tan+= Lt-=Lt6y= - d PR PR 8x dx and also cos r=L -- =L --- =Lt cos = chord PQ are PQ s ds' RQ K.RQ &u dy and sin 'fr = Lt h Q =Ltr -t = Lt -s d chord PQ arc PQ 8s ds 177. Polar Co-ordinates. If the equation of the curve be referred to polar coordinates, suppose 0 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+Sr, 0+80) respectively. Let PN be the perpendicular on OQ, then NQ differs from 8r and NP from r80 by small quantities of a higher order than 60 (Art. 33). o Fig. 23. Let the arc measured from some fixed point A to P be called s, and from A to Q, s + s. Then arc PQ = s. Hence, rejecting infinitesimals of order higher than the second, we have 8s2 = (chord PQ)2 = (NQ2 + P\T2) = (8r2. r282), TANGENTS AND NORMALS. 169 and therefore dr2, d2 = 1, or,ds2 = 1 +,(d o (r (8,,! dds) d, ')\2 according as we divide by &s2, 8.2, or S02 before proceeding to the limit. 178. Inclination of the Radius Vector to the Tangent. Next, let 0 be the angle which the tangent at any point P makes with the radius vector, then dO dr rdO tan = rcr, cos =d-, sin = ds 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 considered as ultimately coincident, QP becomes the tangent at P and the angles OQT and O1T are each of them ultimately equal to p, and tan L = Lt tan NQP = LtP= Lt- = rdO cos =t cos =Lt Q Q Lt d dr cos (bt cos NQP = Lt - Lt --- =Lt- -d chord QP arc QP 6s ds' i rp 1 VP 8 0 rd0 sin = LtsinQP = = Lth = Lt. rC L= - Lt -d chord QP arcQP ds ds 179. Polar Subtangent, Subnormal, etc. DEF. Let OY be the perpendicular from the origin on the tangent at P. Let TOt be drawn through 0 perpendicular to OP and cutting the tangent in T and the normal in t. Then OT is called the "Polar Subtangent" and Ot is called the "Polar Subnormal." 170 TANGENTS AND NORMALS. dO It is clear that OT==OPtan=r2d......... (1) dr and that Ot=OPcot.d (2) and that Ot=OP cot o=.................. (2) X Fig. 24. 180. It is often found convenient when using polar 1 1 du dv co-ordinates to write - for r, and therefore - for - b UIU2 dO dO With this notation Polar Subtangent = r2dO dO dr= du: 181. Perpendicular from Pole on Tangent, etc. Let OY=p and PY=t. Then p=rsin b, and therefore p = 2 cOsec25 (1 + cot22) = l1 + d-; }; 1 1 l.1l/d\'2" therefore = +-4-)........... P2 r2 (\d2 Similea:r.2co........................2) Similarlv t = r cos; TANGENTS AND NORMALS. 171 therefore s = -1 sec2 = — (1 + tan20) t2- ~ e2 =2{+t2) 1 1 /d\2 2 therefore +................(3).= I2+ \4 ]............ (4) 182. Polar Equation of the Tangent. Let the polar co-ordinates of the point of contact be 1 a); and let U' be the value of d- for the curve at that point. The equation of any straight line may be written in the form u=A cos (0 a) + B sin (0-a),.... (1) A and B being the arbitrary constants. Let this straight line represent the required tangent. By differentiation d - _ A sin (- a) + B cos (0-a).....(2) dONow, since the tangent touches the curve, the value of du/, - at the point of contact is the same for the curve and d0 for the tangent. Hence, putting 0 a in equations (1) and (2), we have U=A and U'=B, whence the required equation will be u=Ucos (o -a)+ U'sin (O-a)....(3) 183. Polar Equation of the Normal. The equation, of any straight line at right angles to the 172 TANGENTS AND NORMALS. tangent given by equation (3) of the preceding article may be written in the form Cu = U'cos (0- a) - U sin (0- a), C being an arbitrary constant. This equation is to be satisfied by u = U, 0= a for the point of contact; therefore substituting we have CU= U', whence the required equation of the normal is UU= U'cos (0- a)- Usin (O-a). 184. Class of a Curve of the nth degree. DEF. The number of tangents which can be drawn from a given point to a rational algebraic curve is called its class. Let the equation of the curve be f(x, y)=0. The equation of the tangent at the point (x, y) is 'af + f + f wx ay aa where z is to be put equal to unity after the differentiation is performed. If this pass through the point h, k we have haf +af =. ax 'y?z This is an equation of the (n-l)th degree in x and y and represents a curve of the (n-l)th 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 n(n -1) points of intersection, and therefore there are n(n-1) 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 nth degree." It appears then that if the degree of a curve be n, its * Poncelet, "Annales de Gergonne," vol. VIII. TALNGElVTS AND VORMALS. 173 class is n(n.- 1); for example, the classes of a conic, a cubic, a quartic are the second, sixth, twelfth respectively. 185. 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)=0 at X-x Y-y the point (x, y) is = - -f — af Ax ay If this pass through h, ok, (h-x)3 —y = (/ - Y)-f' This equation is of the 7th degree in x and y and represents a curve which goes through the feet of all normals which can be drawn from the point h, k to the curve. Combining this with f(x, y)= 0, which is also of the nth degree, it appears that there are n2 points of intersection, and that therefore there can be n2 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 -2+2= be the equation of the curve, then (A-b(- a2 is the curve which, with the ellipse, determines the feet of the normals drawn from the point (A, k). This is a rectangular hyperbola which passes through the origin and through the point (h, k). 186. The curves (h - x)-f + ( - y)f = O................),ax 'ay 174 TAN GENTS AND NORMALS. and (h - c)- — (7-y)0 =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) = 0 so as to pass through the point (h, k), are the same for the curve f(x, y) =a. And, as equations (1) and (2) do not depend on a, they represent the loci of the points oJ contact and of the feet of the norrmals respectively for all values of a, that is, for all members of the family of curves obtained by varying a in f(x, y)=a in any arbitrary manner. 187. Polar Curves. The curve h3+f- + kf + zf0 o ax -ay z is called the "First Polar Curve"' of the point h, k with regard to the curve f(x, y)=0; z being a linear unit introduced as explained previously to make f(x, y) homogeneous in x, y, z, and put equal to unity after the differentiation is performed. As this is a curve of the (n- )th 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 0.f dz If the curve be put in the form 'an+TZ ~1+ l +-2 +.. + 2+u+ 0o = the first polar of the origin is UIl -+ 2un - 2 + 3a_3+. + ( e- 1)nL + ni= 0. In the particular case of the conic TANGENTS AND NORMALS. 175 the polar line of the origin has for its equation l1 + 20 = 0. For the cubic US + + + z 0u=O the polar conic of the origin is % + 2u + 3o= o0. 188. 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. 189. 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)=O be the equation of the curve, the equation of the tangent is 'F,. F.F X- + Y- + z- - =0 Dx ay Dz where z is as usual to be put equal unity after the differentiation is performed. If p be the perpendicular from the origin on the tangent at (x, y) we have * (2 p2= ~: /..,..,(~ a"m By' 176 TANGENTS AND NORMALS. A lso 2 x2 + y2.............................. (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. If F(, y)=Obe 2+?/2 x2 +2 1 we have a4 4 P2 and x +.2 = r2; 1 11 therefore 1 11 =0, a b4' b' or a2 + r2 = a2 + b2. This result might be at once obtained by eliminating CD from the equations CP2 + CD2 = a2 + b2 and CD. p = ab, CP and CD being conjugate semi-diameters. 190. 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, 0 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, 0)= 0........................(1) be the equation of the curve, then we have (see Fig. 24) p=r sin,..................(2) and tan = d.......................(3) Eliminate 0 and 0 between. the equations (1), (2), (3), TANGENTS AND NORMALS. 177 and the required equation between p and r will. be obtained. Ex. Given rm= a99sin mO, required its pedal equation. Taking logarithms and differientiating, m dr cosmO r do sinl M therefore cot 5 = cot mO, or 1 mO. Again, p=rsin-p=rsinmo =r. r a"e m+1 therefore r=-. a" The following special cases of this example are worthy of notice, and will furnish exercises for the student. Value Pedal 0 f r~. ~Equation. Name. Equation. -2 r2sin 206+ a2 =0 Rectangular Hyperbola p = a2 -I rsin6+a=0 Straight line p=a 2a -8 1~~~~ - Cos 0 Parabola 2 =CII) 1~~~~~ 2~ 3 (I - cosO) Cardioide a-r r=a sin 0 Circle pa= q2 r2=a2sin 20 Leminiscate of Bernoulli pa2 = r PEDAL CURVES. 191. DEF. If a perpendicular be drawn from a fixed point on a variable tangent to a curve, the locus of the M 178 TANGENTS AND NORMALS. foot of the perpendicular is called the " FIRST POSITIVE PEDAL" of the original curve with regard to the given point. To find the first positive pedal with regard to the origin of any curve whose Cartesian Equation is given. Let F(x, y)= 0....................... (1) be the equation of the curve. Suppose X cosa + Ysina =p touches this curve. By comparison of this equation with DF yaF DF aF OF DF,ax Zy ~z we have D =Xsay...........(2) cos a sin a -p If x, y, X be eliminated between the four equations (1) and (2) a result will remain which depends on p and a only. And since p, a are the polar co-ordinates of the foot of the perpendicular, if r be written for p and 0 for a, the polar equation of the locus required will be obtained. Ex. Find the first positive pedal of the curve Axm +Bym = 1. The tangent is A Xxm-l + B y"- = 1. Compare this with X cos a + Ysin a=p, Ax' _ C-O a and By'"-l =illa P P en. m Hence A( csa) + B(sa )m1=1., Ap (-Bp) Therefore the polar equation of the locus required is m- cosOm -1O sinm-:o r - -____ +Am-1 Lin-1 192. To find the Pedal with regard to the Pole of any curve whose Polar Equation is given. TANGENTS AND NORMALS. 179 Let F( 0r, 0)=0.......................... (1) be the equation of the curve. Let r', 0' be the polar co-ordinates of the point Y, which Is the foot of the perpendicular 0 Y drawn from the pole YP (r, ) o( A Fig. 25. on a tangent. Let OA be the initial line. Then 0=AOP=AOY+ YOP = e'+ -.................. (2) dO also tan rr...(:3) and r'= r sin,, 1 1 1/ c7,r\ (Art. 181).....(4) or / 2-7r+4\dOr If, 0, 0, be eliminated from equations 1, 2, 3, and 4 there will remain an equation in r', 0'. The dashes may then be dropped and the required equation will be obtained. Ex. To find the equation of the first positive pedal of the curve r, = amC6OS mo. Taking the logarithmic differential m d - m tan n r do therefore cot 0 = - tan m; therefore 05 = + em. But 0 =0'+ S2 - therefore 0 = '- mn, or 0= -- rm+ 1 180) 18 TANG-ENTS AND NORMALS. Again r/ = r sin P = r, cos m6O ==a cosrnmO. cos mtO =acos m )2 +1 Hence the equatiol of the pedal curve is m?n m r')n+fl+ Wfl+lCOS+l 0 193. DEF. If there be a series of curves which we may designate as A, Al, A2, A3,. such that each is the first positive pedal curve of the one which immediately precedes it; then A, A., etc., are respectively called the second,, third, etc., positive pedals of A. Also, any one of this series of curves may be regarded as the original curve, e.g., A3; then A2 is called the first negative pedal of A3, Al the second negative pedal, and so on. Ex. 1. Find the kth positive pedal of I'm= amcos m6. It has been shown that the first positive pedal is rrnl - aflzicos m10, where ni1 1+ Similarly the second positive pedal is -In2= a?"2cos m20, where In ntl r1 M 21+rn 1+212' and generally the ktM' positive pedal is 7rm - arnkcos mo,' where Wnk M, Ex. 2. Find the kth negative pedal of the curve rrn ==amZcos Mo. TA NGEN7TS A AND NORIMALS. 181 We have shown above that r"= a"'cos mO is the kth positive pedal 1 +n' This gives n= - I - km' Hence the kt1 negative pedal of r. == acos nm is ^ = a"cos no, where n - 1 - km' 194. Tangential-Polar, or p, Vr Equation of a Curve. If Ifr be the angle which the tangent to a curve makes with any fixed straight line, the relation between p and f often forms a very simple and elegant equation of the curve. This relation has been called by Dr. Ferrers the Tangential-Polar Equation. The p, r equation may be deduced at once from the equation of the first positive pedal. If r =f(0) be the pedal curve, then, since 5= +0 (see Fig. 25, Art. 192), the equation between p and r is clearly P=. (VEx. i. The p, + equation of Ax2 + By2= I is p2 _ Sin2, + Cos~ (Art. 191). A B Ex. 2. The pedal of 2a-l+cos0 with regard to the origin is r cos = a, and therefore its p, p equation is p sin f = a. 195. Relations between p, t, p, etc. Let PY, Q Y' be tangents at the contiguous points P, Q on the curve, and let 0 Y, 0 Y' be perpendiculars from 0 upon these tangents. Let OZ be drawn at right angles to 182 TANGENTS AND NORMALS. Y'Y produced. Let the tangents at P and.Q intersect at T, and let them cut the initial line OX in 1 and 5S. Let the normals at P and Q intersect in C. C ~~~~~~~1 —AZ-' 0 ~ R S X Fig. 26. Let the co-oidinates of P be (r, 0), and those of Q (r+dr, 0+M0). Let OY~p, OY'~3p+D, PBX=vf, A A A A QSX = Vvf + &ir. Then STR, PGQ, YO Y' each = &f. Let P Y=t, and arc PQ==&s. Let QY' cut TY in V; then, since 0YV is a right angle and YOTTV= &6/ a small angle of the first order, 0OV differs from 0Y by a quantity of higher order than the first (Art. 3:3). Hence VY' differs from 6p by a quantity of higher order than ap, and TY' tan Vf= VY', therefore TY' tan - h/ VY' d2k and proceeding to the limit t CIP...... (-c Similarly, if PC be called p we have arc P Q= PC. Nv, TANGENTS AND NORMALS. 183 neglecting infinitesimals of higher order than 6oV, therefore PC arcQ S&kf' and proceeding to the limit, ds P.................................. (2 ) Again t= Y'Q- YP =(Y'T+ TQ)-(YV+ VT-PT) = (PT+ TQ) + (Y'T- VT) - YV. Now YV==ptan 65, and remembering that when 6SA is an infinitesimal of the first order, VT and Y'T, PT+ TQ and Ss, tan l6 and 6Vr, each differ by quantities of order higher than the first, we have, upon dividing by 6S4 and proceeding to the limi, dt ds limit, d p d2p (3 or P-=+dl2, by (1) and (2)............(3) 196. Perpendicular on Tangent to Pedal. A A From the same figure it is clear that since YO Y'= YTY', the points 0, Y, Y', T are concyclic, and therefore OYZ 7r-OYY'=OTY'; and the triangles OYZ and OZ O17Y' OTY' are similar. Therefore -y-= O. And in the limit when Q comes into coincidence with P, Y' comes into coincidence with Y, and the limiting position of Y'Yis the tangent to the pedal curve. Let the perpendicular on the tangent at Y to the pedal curve 184 TANGENTS AND NORMALS. be called p2, then the above ratio becomes,p1 _ 2~ p r' or 2r =9p2. 197. Circle on Radius Vector for Diameter touches Pedal. It is clear also from the figure of Art. 195 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 Y and Y', two contiguous points on the pedal, must in the limit have the same tangent at Y as the pedal curve, and must therefore touch it. 198. Pedal Equation of Pedal Curve. Let v-=f(p) be the pedal equation of a given curve. _ 2 Then, since pr=p2, we have p= -, and therefore, writing r for p and p for p, the pedal equation of the 7,2 first positive pedal curve is p=f( ). a2 Ex. The first positive pedal of the rectangular hyperbola r =- is r2 9,3.whic which is the p, r equation of Bernoulli's Lemnliscate, as is also obvious from Art. 190. EXAMPLES. Write down the pedal equations of the first positive pedals of the curves given in the table of Art. 190. 199. We may also prove the results of Art. 195 as follows: TANGENTS AND NORMALS. 185 Let the tangent P1T make an angle r with the initial line. Then the perpendicular makes an angle a== - with the same line. Let Y=p. Let P1P2 be the normal, and P, its point of intersection with the normal at the contiguous point Q. Let 0 Y1 be the perpendicular from 0 upon the normal. Call this p,. Let P2P3 be drawn at right angles to PlP2, and let the length of OY,, the perpendicular upon it from 0, be p2. P24,. 4 T X Fig. 27. The equation of P1T is clearly p = cos a+y sin a..................(1) The contiguous tangent at Q has for its equation p+ 6p = x cos (a + da)+ ysin (a+ f a).......(2) Hence subtracting and proceeding to the limit it appears that dv a- = -xsin a+y cosa................. ca(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 PP2. 1e86 18 AN.-GENTS AND NORlNAAL LS. Similarly x Cosa -y -sin a................. (412 cla2 represents a straight line through the point of intersection of two contiguous positions of the line P,P2 and perpendicular to P,P2, viz., the line P2P., and so on for further differentiations. From this it is obvious that 0 y dyj dp d3iOc da d1fr' cia d2p d12? 2 da2 cliA' etc. Hence t= P Y=and d2P p =P1P =zO Y037,= p~ 12 200. Tangential Equation of a Curve. DEF. The tangential equation of a curve is the condition that the line lx + my + n = 0 may touch the curve. Method 1. Let F(x, y) = 0 be the curve, then the tangent at x, y is F aDF aDF X +Y + - -- = 0. Dx Dy Dz Comparing this with iX +m Y+ n = 0, F DF aDF ax ay az X, say. tm n If X, y, X be eliminated between these equations, and F(x, Y)= 0, or lx + my ~ n = 0, a relation between 1, m, n will result. This is the equation required. Methodl 2. We may also proceed thus. Eliminate y between F(x, y)=O and lx+my+n=0; we obtain an TANGENTS AND NORMALS. 187 equation in x, say ((x)= 0. For tangency this equation must have a pair of equal roots. The condition for this will be found by eliminating x between +((x) = 0 and +'(x) = 0. In following this method, instead of eliminating y it is often better to make a homogeneous equation between F(x, y)=0 and x + my + n = 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 ax2 +2hxy + by22 +2fy +c =. The first process gives us ax+hy +g=, ax + by +f= N2, X gx +.fy + c =- n. 2 Also Ix + my + n =0. The eliminant from these four equations is a, h, g, I'1 h, b, f, m = g, f, c, n 1, mn, n, 0 which may be written A +2 +Bm2 + Cn2 +2Fn + 2Gg + 2 Hl = 0, where A, B, C,... are the minors of the determinant a, A, g h, b, f g, f, c INVERSION. 201. DEF. Let 0 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. OQ= constant, k2 say, then Q is said to be the inverse of the point P with respect to a circle of radius k and centre 0. 1%88 TAN1GENTS A ND NORMALS. 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 the polar co-ordinates of the point P, and (r', 0) those of the inverse point Q, then rr'=lc2. Hence, if the locus of P be f(r, 0) =0, that of Q will be f\(, 0) =0. For example, the curves rIn = a cos m and r"' cosm6=a"'" are inverse to each other with regard to a circle of radius a. 202. Again, if (x, y) be the Cartesian co-ordinates of P, and (x', y') those of Q, then k2 = 9 COS 0 x2 X x=r COS = = OS 0 = 2 --- =2 h X12y' and similarly 2 + y,' Hence, if the locus of P be given in Cartesians as (x, y)= 0, the locus of Q will be 2X cy \1 F( y2 x +y2 =0. Ex. The inverse of the straight line x=a with regard to a circle radius k and centre at the origin is x2 +ya or x2 +y2 k= or. =?-t -,X a circle which touches the axis of y at the origin. 203. Tangents to Curve and Inverse inclined to Radius Vector 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 TANGENTS AND NORMALS. 189 OPT and OQ'T are therefore supplementary, it follows that in the limit when P' ultimately coincides with P OI Fig. 28. T K1 - and Q' with Q the tangents at P and Q make supplementary angles with OPQ. The ultimate ratio of corresponding elementary arcs, viz., d.s PP' OP OP OP. OQ.W, 2 -a'- t QQ'= LtOQ =~Q= OQ2 = I- =. I '' I/ Fig. 29. 204. Mechanical Construction of the Inverse of a Curve. In the accompanying figure AC, CB, BQ, QA, PA, PB 190 TANGENTS AND NORMALS. is a system of freely jointed rods, of which A C BC, and AQ= QB=BP= PA. At P and Q sockets are placed to carry tracing pencils. A pin fixes ( 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 P. CQ = 2 AQ2, and is therefore constant. Hence whatever curve P is made to trace out, Q will trace out its inverse, the point C 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. 202. Peaucellier has utilized this construction for the conversion of circular into rectilinear motion. POLAR RECIPROCALS. 205. Polar Reciprocal of a Curve with regard to a given Circle. DEF. If 0 Ybe the perpendicular from the pole upon the tangent to a given curve, and if a point Z be taken on 0 Y or 0 Y produced such that 0 Y. OZ is constant (=k2 say), the locus of Z is called the polar reciprocal of the given curve with regard to a circle of radius k 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 reciprocal of an ellipse with regard to its centre. x2 y2 For the ellipse + 1 = 1 the condition that p =x cos a +?sin a touches the curve is p2 = a2cos2a + b2sin2a. TANGENTS AND NOR~MALS. 191 Hence the polar equation of the pedal with regard to the origin is r2= a2cos20 + b2sin2. Again, the inverse of this curve is' = a2cos20 + b2sin20, r2 or a2x2 + b212 = k4, 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. 206. The method may therefore be stated thus:First find the condition that p = x cosa + y sina w ill touch the given curve. Then write 2 for p and 0 for a in that condition. The result is the required polar reciprocal with regard to a circle of radius k and centre at the origin. 207. 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 with 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 p = Xcos a+ Ysin a, and the condition of tangency p =/(a). If x, y be the pole of this tangent with regard to U= 0, the tangent must be coincident with the polar Xa + yaU +z ax ay az lDU?aU cos a x sin a ay therefore - - - p iU' p — D-z DP z az xz 1922 TANGENTS AND NORiVIALS'. Ua U\ 2 U\2 DU 1 ~I,/'V\;y!ja Hence a -and tan a= aU3" 2 la Oz ax Hence the equation of the Polar Reciprocal is (U2 \DX/\ \D/ DU~~a 2 U~O axl ~aJj CS taii - a; \ax 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 Sections, Chap. XV. EXAMPLES. 1. Find where the tangent is parallel to the axis of x and where it is perpendicular to that axis for the following curves:(a) aX2 + 2hxy ~- by2 = 1. X3 -a ax (Y) y3 =x2(2a-x). 2. Find the equations of the tangents at the origin in the following curves:(a) (x + y2)2 = a2x2 - b2y2. (y -a) 2X'2 =b2. (y) Y5 = 5aX2y2. 3. Find the length of the perpendicular from the origin on the tangent at the point x, 'y of the curve X4 +, y4 -4. TANGENTS AND NORMALS. 193 x 4. Show that in the curve y = be the subtangent is of constant length. 5. Show that in the curve by2= (x + a)3 the square of the subtangent varies as the subnormal. 6. For the parabola y2= 4ax, prove ds /a u+r dx __ 2 2 7. Prove that for the ellipse 2 + I2=, if = a sin 4, C2 = a J1 - e2sin2). 8. For the cycloid x= a vers 0 y = a(0 + sin ) f' prove ds 2a. dx 9. In the curve y - alog sec, ds x ds x prove - sec- = cosec-, and x= ah. dx a dy a 10. Show that the portion of the tangent to the curve j2 2 2 xS + y- = S, which is intercepted between the axes, is of constant length. Find the area of the portion included between the axes and the tangent. 11. Find for what value of n the length of the subnormal of the curve xyn=a'+l is constant. Also for what value of n the area of the triangle included between the axes and any tangent is constant. 12. Prove that for the catenary y,= c cosh t, the length of the normal = Y2. c N 194 1TANGENTS AND NORMALS. Prove also that the length of the perpendicular from the foot of the ordinate on the tangent is of constant length. 13. In the tractory y2' 1ogC c2 11/2 - 2 c~ 9 2 2 prove that the portion of the tangent intercepted between the point of contact and the axis of x is of constant length. 14. In the spiral r aeeota, prove dr r=cos a and p = r sin a. ds 15. For the involute of a circle, viz., - a2 r a qr prove Cos a 2at 16. In the parabola 1 -cos 0, prove the following results (a- w (/) P sin - 2 (y) p2=ar. (8) Polar subtangent = 2a cosec 0. 1 7. For the cardioide r = a(1 - cos 0), prove (a) (=k 2 ____) si2~cs n (j8) p a ii13 () 30 sin-2 (8) Polar subtangent 2aw 0 cos - 2 TANGENTS AND NORMALS. 195 18. If the curves fix, y) = 0, F (x, y) = 0 touch at the point Jaf -aF f -_a_ X) Y, prove T DT -': at the point of contact. 19. If the curves f(x, y)= 0, F(x, y)= 0, cut orthogonally, prove that at the point of intersection af DTY +-af aJ 0. ax?Y y by 20. If the form of a curve be given by the equations x:- - 0), Y = 0L~), prove that the equation of the tangent at the point determined by the third variable t is xXp(t) - Y~'(t) = S(t)(t) - 0)(F(05 and that the corresponding normal is X(~'(O.+ Ysb'(t) = C(t)I'(t) + OW(t>,b'(t). 21. Apply the preceding example to find the tangent and normal at the point determined by 0 on (a) The ellipse x - a cos 0 y= bsin 0 (/) The cycloid x a(0 + sin 0) y:=a(1 - cos )f (y) The epicycloid x = cos 0 - B cos -01 B y= asin 0-Bsn i0J 22. In the four-cusped hypocycloid 2 2 2 x7 +? = ct 3 show that if x = a cos~a then y = a sinka, aud that the equation of the tangent at the point determined by a is x siu a + y cos a = a sin a cos a. Hence show that the locus of intersection of tangents at right angles to one another is r2 Co 22~~ r =Cos 220. 196 TANGENTS AND NORMALS. 23. If p, and P2 be the perpendiculars from the origin on the tangent and normal respectively at the point (x, y), and if tan = y, prove that p, = x sin - y cos 1b, and p2 = x cos + y sin ~. Hence prove that 1 2- d1. 24. Through the point h, k tangents are drawn to the curve Ax3 + By3 = 1; show that the points of contact lie on a conic. 25. 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. 26. The points of contact of tangents from the point h, k to the curve x3 + y3 = 3axy lie on a conic which passes through the origin. 27. 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 2xy + kx- 3hy = 0, and the locus of the remaining point in which each tangent cuts the curve is the rectangular hyperbola xy - 4kx + 3y = 0. 28. Prove that the locus of the extremity of the polar subtangent of the curve u +f(0)= 0 is u=f( + a) Ex. Find this locus in the case of the conic 1 + e cos 0. 29. Prove that the locus of the extremity of the polar sub TANGENTS AND NORMALS. 197 normal of the curve r=f(O) is r =f'(0-). Hence show that the locus of the extremity of the polar subnormal in the equiangular spiral r = aemO is another equiangular spiral. 30. In the curve 1 +tan 0 2 m + n tan - 2 the locus of the extremity of the polar subtangent is a cardioide. [PROFESSOR WOLSTENHOLME.] 31. Show geometrically that the pedal equation of a circle with regard to a point on the circumference is pl = r2, d being the diameter of the circle. 32. Show that the pedal equation of the ellipse - q+ -1 a6b2 with regard to a focus is b2 2a6.1. P2 4 33. Show that the pedal equation of the parabola y2=4ax with regard to its vertex is a2(r2 -_p9)2 = p2(r-1 + 4a2)(p2 + 4a2). 34. Show that the pedal equation of the curve r== a is of the form p = mr where m is a constant. 35. Show that the pedal equation of the tetracuspidal hypocycloid x' + y_ = a'3 is r2 + 3p2 = a2. 36. Show that for the epicycloid given by x = (ca + b)cos 0 - b cosa- b y = (a + b) sin 0 - b sinl b 0 198 198 ~TANGENTS A ND NORMIAL/5. y) = (a + 2b~sin-0; =(a+ 2b)s 2b 2 b' -a + 26 and that the pedal equation is r2 a2 ~ (a +2b )2 37. Show that the first positive pedal of the parahola 'y2 4ax with regard to the vertex is the cissoid X(X2 + y2) + cty2 = 0. 38. Show that the first positive pedal of the curve x 3 + V3=- a3 is (x~~~~~~~~2 ~ y2)~ afx +) 39. Show that the first positive pedal of the curve x7 + 3= aI is r= ~ asin Ocos 0 Also that the tangential polar equation of thie curve is T - in2'b. p 2 40. Show that the -first lpositive pedal of the curve m~y n =am+1 is enn oarnn(1fl_~n)'m?,",co siO,,nj0 41. Show that the fourth negative pedal of the cardioide r: ---a(1 + cos 0) is a parahola. 42. Show that the fourth and fifth positive pedals of the 2 2 2 curve r9osOa are respectively a rectangular hyperhola and a Lemniscate. 43. Show that the nt1' positive pedal of the spiral r -ae0 cot ct is ~~~~r = asinnaenA(2aco co 44. Show that, if the curves r =f(), r=F(0) intersect at (,0), the angle hetween their tangents at the point of intersection is tan-'r OJ 0 OfO P'(O)f (0) + F (0)f(0)' TANGENTS AND NAORMALS. 199 45. Show that the inverse of the parabola y2 = 4ax with regard to a circle whose centre is at the origin and radius the semilatus rectum is the pedal of the parabola 2 + 4ax= 0 with regard to the vertex. 46. Show that the inverse of the conic u2 + u- + UO =0 with regard to the origin is the bicircular quartic curve k4,u + k21(x2 + y2) + u0(x2 + 2)2 = 0. 47. Show that the inverse of the general curve of the nth degree, viz., u. + uz- + u-2 +... + U1 + U = 0, with regard to the origin is k2U1, + k2n-2_n-r2 + k2z —4_u-2r4 +.. + 1 2u'2rz-2 + Uo r2 = 0, where r2 = x2 + Y2. 48. Show that the inverse of a conic with regard to the focus is a Limacon (Equation r = a + b cos 0), which becomes a cardioide if the conic be a parabola. 49. Show that the inverse of a conic with regard to the centre is an oval of Cassini (Equation r2 = a b cos 20), which becomes a Lemniscate of Bernoulli if the conic be a rectangular hyperbola. 50. If P1, P2 be two points whose inverses are Q1, Q2 with regard to any origin 0, prove that P P - O P1P2=-Q. Q1Q2. 51. The locus of a point X is defined by the equation ~(Pl, p, P3,... Pn) = a, where pi, P2,.. are the distances of X from n fixed points P1, P2,... P,,. Show that the equation of its inverse with regard to any origin 0 is w elP rP' 2 rnPn = a, where pi', P2',... are the distances of X', the inverse of X from the n fixed points Q1, Q2,... which are the respective inverses 200 TANWGENTS AND NORMALS. of P1, P2,...; ri, r2, r2,... are the lengths of OP, OP2,...; and R=OX'. 52. Show that the inverse with regard to any pole 0 of the Cartesian oval whose equation is Ir + mr' =n, where r, r' are the distances of any point on the curve from two fixed points F1, F2, is 1. OF1. P1 + M. 0 2. p2 = np,, where p, P2 are the distances of any point on the inverse curve from the points which are the inverses of F1, F2, and p3 is the distance of the same point from the pole of inversion. 53. Show that the inverse of a Cassini's oval defined by the equation rr = constant is of the form PiP = A pi, the letters p1, P2, p3 denoting the distances of any point on the inverse curve from certain fixed points. 54. Show that the inverses of two curves intersect at the same angle as the original curves; and as particular cases that if two curves touch their inverses also touch, and if two curves cut orthogonally their inverses cut orthogonally. 55. 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 inversion, show that the curves which cut orthogonally each member of the family of cardioides r = a(1 + cos 0) found by giving different values to a, are also cardioides. 56. Show by inverting a conic with regard to its focus that the circle x2 + y2 = l(e + cos a)x + I sin a. y touches the Limacon r = 1 + le cos 0 at the point given by 0 = a. 57. 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 rm mo m r+'cos__ 0 = bm+l. {m + 1 TANGENTS 4AND NORMALS. 201 58. Show that the polar reciprocal of the curve x"'y' = a"m" with regard to a circle whose centre is at the origin is another curve of the same kind. 1p+l 59. Show that the first positive pedal of the curve p = a is am p M+~1a = l2m}+1 and that its polar reciprocal with regard to a circle of radius a whose centre is at the origin is pm+l = anr. 60. Show that the inverse of the curve p=f(r) with regard to a circle whose radius is k and centre at the pole is =and that the polar reciprocal is k2 / 61. Show that the pedal of the inverse of p=f(r) with regard to a circle whose radius is k and centre at the origin is k2p9 fk k2)) 9p~ + 1 62. Show that the pedal of the inverse of p=- with al" regard to a circle whose radius is k and centre at the origin is a\ m 21n-2 1 yr nm-lr rn-I 63. Show that the polar reciprocal of the curve rm = amcos mO with regard to the hyperbola r2cos 20= a2 is rm+lcos - 0 -= C7~+1. +n 1 64. In the semicubical parabola ay2 x3 the tangent at any point P cuts the axis of y in A/ and the curve in Q. 0 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. 65. 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 202 TANVGEN7TS AND NVORMALS. third is drawn, and so on. Prove that the abscissae of the points of contact form a geometrical progression, and also the ordinates. 66. 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 on AOP=AO. 67. The point P moves so that OP. O'P= constant, O, 0' being fixed points. If OY, O'Y' be the perpendiculars from 0 and 0' on the tangent at P to the locus of P, prove that PY:PY':: OP: O'P2. 68. 0 and 0' are two fixed points, P any point in a curve defined by the equation -, r r c where r= OP, r' -O'P, and c is constant. Prove that the distance between P and the consecutive curve obtained by changing c to c + 8c is ultimately 8c 3c2 a2c4 1 r++r'3 where a= 00'. [SMITH'S PRIZE.] 69. 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. 208. DEF. If a straight line cut a curve in two points at an infinite distance from the origin and yet is not itself zthotly at infinity, it is called an csymptote to the curve. 209. Equations of the Asymptotes. Let the equation of any curve of the nth degree be arranged in homogeneous sets of terms and expressed as n)g,(Y)+ a -1 l()+ - 2 _ -2(Y) +...=0..........-0. (A) To find where this curve is cut by any straight line whose equation is Y = x +............................. (B) substite,+ for Y in equation (A), and the resulting equl1 1 i Q +x) +x qbX-202 2(t+ Sd)*e*_..0. (c) gives the abscissae of the points of intersection. Applying Taylor's Theorem to expand each of these 204 2A SYMfP TOTES. functional forms, equation (c) may be written Xon(/]+ z) + n-I 1 9 5n(( ) + Xn-2 (D)t If P+ (M)~X - i(g) +()~-2 / )2. =0.(.) +~ P-' -1(JU2 + 95n-2(Cu) This is an equation of the qnth degree, proving that a straight line will in general intersect a curve of the nth degree in n points real or imaginary. The straight line y= x+-~3 is at our choice, and therefore the two constants, and /3 may be chosen, so as to satisfy any pair of consistent equations. Suppose we choose /. and 3, so that n(tU)= O........................... (E) and f3n,'(g) + 'n- 1(i )= -0..................... (F) The two highest powers of x now disappear from equation (D), and that equation has therefore two infinite roots. If, then, U, /,2..., /UxI be the n values of /u deduced from equation (E) (which is of the nth degree in,/), the corresponding values of 8f will in general be given by B - n-l(,ua) i _ -1(R2) _~- (U]) _2 -'n(U2) and the n straight lines Y = tkX +~ / Y = laX + 02 are the asymptotes *....F. *of the curve. y= unX + /3n 210. Rule. Hence, in order to find the asymptotes of any given curve, we may either substitute x + f3 for y in the equation of the curve, and then by equating the coefficients of A SY MPTOTES. 205 the two highest powers of x to zero find,u and /3. 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= 1 and y =u [the result of this is to form pn(())] ancd equate to zero. Hence find tx. Form n -l(Iu) in a similar way fro m the terms of degree n-1, and differentiate qn(4), then the values of / are found by substituting the several values of ut in the formula = -_, (_() Ex. Find the asymptotes of the cubic 2X3 a- a;2 2xy2 + y3 4- 2 x2 + y y + y +1 = 0. Here 3(f) =u3 2L2_ - I+ 2 =0; therefore (A- 1)(+ 1)(/ -- 2) =0; giving \I,= 1, -1, or 2. Again, %2() = 2 + - 2, and 0'3()3) 3/2 - - 4 1; therefore P= 2 -- -2 32- - 4 1 Hence if, =., f=1, if F= -1, P=0, and if =2, 3=0. Hence the asymptotes of the curve are Y=x+ 1, y2x. p = 2-v. EXAMPLES. 1. The asymptotes of ya - 6x, + i lxy - 6x3 + x +y= O are y = x, y = 2x, y = 3x. 2. The asymptotes of?3 - x2y + 2?y2 + 4y + x =0 are y=0O, y -+l= 0, y+x+l==0. 206 ASY MP TO TES. 211. Number of Asymptotes to a Curve of the nth Degree. It is clear that since qn(t) 0 is in general of the nth degree in Au, and Pf'9n(/) + ~,_-l(u) =0 is of the first degree in 3, that n values of A, and no more, can be found from the first equation, while the n corresponding values of H can be found from the second. Hence n asymptotes, real or imaginary, can be found for a curve of the ntlh degree. 212. 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. No curve therefore of an odd degree can be closed. Neither can a curve of odd degree have an even number of real asymptotes, or a curve of even degree an odd number. 213. If, however, the term yfl be missing from the terms of the nth degree in the equation of the curve, the term tun will also be missing from the equation,(j) = 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 uil is zero, one root of the equation fi(g)=0 is infinite, and therefore 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. 214. Asymptotes Parallel to the Axes. Let the curve arranged as in equation (A), Art. 209, be ASYMJ'IPTOTES20 207 a(VIG + a1X" - 1y + a2xn - 2y2 ~... + an, - 1 + any it + bi X ~-1 +b2 xn- 2 +.................. +bn-IY +C2 Xn-2... ~~.(..................... (A') If arranged in descending powers of x this is ao +(a "1y +bb)x"-l+...==............... (B") Hence, if ao vanish, and y be so chosen that a"y b, = 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 acy + b, = 0 is an asymptote. In the same way, if an=O. a,,-jx+b,,i=() is an asymptote. Again, if ca0 =0, ar=O, b1 =0, and if y be so chosen that a2Y2 y+ b2Y + e2 = 0, three roots of equation (B') become infinite, and the lines represented by a2y2 + b2y + C2 = 0 represent a pair of asymptotes, real or imaginary, parallel to the axis of y. Hence the rule to find those asymptotes which are parallel to the axes is, " equate to 'zero the coefficients of the highest powers of x and y.." Ex. Find the asymptotes of the curve!2y2 -7 X2y X y2 + X +y~1-OI 0. Here the coefficient of 2 j5 p2- and the coefficient of p2 is x2 - x. Hence x=0, r= 1, y=O, and y= 1 are asymptotes. Also, since the curve is one of the fourth degree, we have thus obtained all the asymptotes. 208 A 28Y3IP TO TES. EXAMPLES. 1. The asymptotes of y2(X2 - al) =x are Y=O 2. The co-ordinate axes are the asymptotes of IXY" 3 + X3'y = a4. 3. The asymptotes of the curve ~y2 = C2(X2+y2) are the sides of a square. 215. Partial Fractions Method. The values of /, viz., 'O~n -I Gti) On,-l(u)z 'VIA),VIA) 'etc., are exactly the constants required in putting - n-I(t). into partial fractions.* This gives a very easy way of obtaining the asymptotes. For if q ~n(t) - t- jx t-1x2 t-M3 the asymptotes will be Y mix ~+ /31, Y =luX +,3,, etc. j Suppose the single factor t - ju, to occur in b,(t). Let On(t) (t - /-41)X(t) - Hence, differentiating Of n(t) =X(t) + (t - AI)XV), and putting t=1 41, 'l(Pi) == X (,,) But ifA be the partial fraction corresponding to the factor t - ul,. t - ~~ b1(L)(Art. 101). X(PAI) ASYiIPTOTS. 209 209 Ex. Find the asymptotes of the curve, (X2 -y2)(X+2y) +5(x2 +i2) +XX+ y=O. 25 5 Here -l(t) t2+1 3 3 5 95n(t) (2t e1)(t-1 )(t -t1) 2t +1t- ~ Hence the asymptotes are 2y+x= 25 y+ r=5. 216. Particular Cases of the General Theorem. We return to a closer consideration of the equations 0"'()= 0,...................(E) fPan'(M') + On-1.L). = 0, ~...........(F) of Art. 209. It is proved in Theory of Equations that if an equation such as pn(/A)= 0 have a pair of roots equal, say /, then q$,'(ux1)= 0. I. Let the roots of p,(1x) 0 be /v, M2'...) Mm supposed all different, so that On'(AU) does not vanish for any of these roots. Also, suppose pn(jA) and OnQx(M) to contain a common factor IA - IA, say, then O,, (,)=0, and therefore 3, =0. Hence the corresponding asymptote is y = IAxx and passes through the origin. II. Next, suppose two of the roots of the equation On(ft) = 0 to be eqcal, e.g., IA2 = Mx,, then On'(M,)= =0. In this case, if On -1(jA) do not contain,a - t, as one of its factors, the value / determined from equation (F) is infinite. The line y=jAxx+0j then does indeed cut the curve in two points at an infinite distance from the origin, but it makes an infinite interce7pt on the axis of y and there0 210 A SYMPTOTES. fore this line lies wholly at infinity. Such a straight line is not in general called an asymptote, but it will however count as one of the n theoretical asymptotes discussed in Art. 211. III. But if qn(u)= 0 have a pair of equal roots each =U, we have pn'(M,)=0, and if M, be also a root of 5- l(Mu)=0 the value of /3 cannot be determined from equation (F). We may however choose /3 so that the coefficient of xn-2 in equation (D) of Art. 209 vanishes, that is so that l2 n (M) + 0'n-1(M) + n-2(GL) = 0, from which two values of /3, real or imaginary, may be deduced. Let the roots of this equation be /1 and j3'. We thus obtain the equations of two parallel straight lines y = L1 + f31 y = M1X + 1, which each cut the curve in three points at an infinite distance from the origin. In this case there is a double point on the curve at infinity (see Art. 249). It is clear that in this case any straight line parallel to y=,x will cut the curve in two points at infinity. But of all this system of parallel straight lines the two whose equations we have just found are the only ones which cut the curve in three points at infinity, and therefore the name asymptote is confined to them. The one equation which includes both straight lines is obtained at once by substituting y - Mx for /3 in the equation to obtain /3 and is (y - MUX)2"n (U,) + 2(y- MUx) 'ln l(M,) + 2_n-2(u) = 0. AS YJMPTOTES. 211 Ex. Find the asymptotes of the cubic curve x3+ 2x2 +xy2- x2 2 = 0. Equating to zero the coefficient of y2 we obtain x=0, the only asymptote parallel to either axis. Putting Lx + p for y, x3 + 2x2(Ix + /) +x(x + xp)2 - x2- xO p) + 2 = 0, or rearranging x3(1 + 2/u +.I2) + 2(2p + 2/up - 1- /) + x(p2 - ) + 2 =0. 1+2gA+A2=0 gives two roots L= -1. 2/p+ 2j3- 1 -- =0 is an identity if u = - 1, and this fails to find /. Proceeding to the next coefficient, 2 - p=O gives P = 0 or 1. Hence the three asymptotes are x=0, and the pair of parallel lines y+x=O, y+x=1. 217. Form of the Curve at Infinity. Another Method for Oblique Asymptotes. Let P?., Fr be used to denote rational algebraical expressions which contain terms of the rth and lower, but of no higher degrees. Suppose the equation of a curve of the nth degree to be thrown into the form (ax+ by+ c)Pn._ +F.i = 0.............(1) Then any straight line parallel to ax + by =0 obviously cuts the curve in one point at infinity; and to find the particular member of this family of parallel straight lines which cuts the curve in a second point at infinity, let us examine what is the ultimate linear form to which the curve gradually approximates as we travel to infinity in the above direction, thus obtaining the ultimate direction of the curve and forming the equation of the tangent at infinity. To do this we make the x and y of the curve become large in the ratio given by x: y = -b:a, and we obtain the equation ax+ by +c+L _ ty=-X= ) 0. 212 A SYMPTO TES. If this limit be finite we have arrived at the equation of a straight line which at infinity represents the limiting form of the curve, and which satisfies the definition of an asymptote. To obtain the value of the limit it is advantageous to b a put x= - and y=, and then after simplification make t = 0. t=0. Ex. Find the asymptote of x3+ 3x2y+ 3xy2 + y3 =x2 +y2 + x. We may write this curve as ( +2y)(x2+ xy +y2) =x2 +y2 +, whence the equation of the asymptote is given by x+2,yLtx=_2y= 2+y+ x2 + 2 + xY2 - 2 and putting x=, y= - we have t t 4 1 2 t2 + t2 t 5 -2t 5 x + 2y = Ltt=o =Ltt=o — 4 2 1 3 3 t2 +.e., f+2y=.5 i.e., x+2y=-. EXAMPLE. Show that x+.?y=a is the only real asymptote of the curve (x+y)(4 +y4) = a(x4 +a4). 218. Next, suppose the equation of a curve put into the form (ax+by+c)F,_1 + F _2=0, then the line ax+by+c=0 cuts the curve in two points at infinity, for no terms of the fth or ( — l)th degrees remain in the equation determining the points of intersection. Hence in general the line ax+by+c=0 is an asymptote. We say in general, because if Fnl be of the ASYMPTOTES. 213 form (ax+ by + c)P.2, itself containing a factor ax+ by + c, there will, as in Art. 216, il., be a pair of asymptotes parallel to ax+by+c==0, each cutting the curve in three points at infinity. The equation of the curve then becomes (ax + by + C)2Pn_ - e-2= 0, and the equations of the parallel asymptotes are ax+by+c= ~ _/Lt- -2 where x and y in the limit on the right-hand side become X ) infinite in the ratio - y a And other particular forms which the equation of the curve may assume may be treated similarly. Ex. To find the pair of parallel asymptotes of the curve (2- 3y+ 1)2( +y)-8x+2y-9=0. Here 2x- 3y + 1 = 2 9 where x and y become infinite in the direction of the line 2x=3y. 3 2 Putting x=3, y= -, the right side becomes + 2. Hence the t t asymptotes required are 2x - 3 = 1 and 2x - 3y+ 3=0. 219. Asymptotes by Inspection. It is now clear that if the equation F,=0 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 Fn+Fn-2=0. Ex. 1. (x-y)(+)(x+)(+2y- 1)=3x+4y +5 is a cubic curve whose asymptotes are obviously x-y=0, x+ y=0. x+2y-1=0. 214 A SYMPTOTES. Ex. 2. (x -y)2(x+2y- 1)=-3x+4y+5. Here x+2y -1= 0 is one asymptote. The other two asymptotes are parallel to y=x. Their equations are -y= + Ltto3+4+5t= + 7 220. Case in which all the Asymptotes pass through the Origin. If then, when the equation of a curve is arranged in homogeneous set of terms, as Un+Un_2+ n-+... = 0, it be found that there are no terms of degree n -1, 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. 221. Intersections of a Curve with its Asymptotes. If a curve of the nth degree have n asymptotes, no two of which are parallel, we have seen in Art. 219 that the equations of the asymptotes and of the curve may be respectively written = 0, and Fn+Fn2=(. The n asymptotes therefore intersect the curve again at points lying upon the curve F,_2 =0. Now each asymptote cuts its curve in two points at infinity, and therefore in n-2 other 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. ASYMPTOTES. 222. Common Transversal of a Curve and its Asymptotes. The equation of the asymptotes and that of the curve coincide in the terms of the nth and (n-l)th 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 0 a line OP1P2P3... be drawn to cut the curve in the points P1, P2, P3,... and the asymptotes in P P2, p3 then OP = 0p, whence, if OP =, it follows that 1Op=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.] A well known case of this is that of the hyperbola, where, if 0 be the middle point of PP2, OP+ OP2=O, and therefore Op1,+0O)2=0, and therefore 0 is also the middle point of p1P2, whence it follows that in that case P1Pi =p2P2. 223. 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, (3) 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. 224. To prove the Consistency 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. 208. AS YMPTOTES. Consider definition (a). Let the curve be U- un + Un_ + Un-2+ +... + = -0. The equation of the tangent is X U+ Y a U+z,_ +2-,_2+..+ nuo = 0. ox ay 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- l)th 'an for 'U au, aoU degree. Thus we must retain only -n for, for -, and,a x 01 x oy 'ay un'1 for 1,_ + 2n-_2f +.. + nh0. I+ence in the limit we shall have lt{i D~+ D j -u } -0, ax ay 'aun or Y==X -LtA -Lt,_ i (2Y a{y an ' any and it is easy to see that this agrees with the equation of an asymptote found in Art. 209. 225. We next consider definition (8); we have already shown that ax+by+c=O is, according to our definition, in general an asymptote of the curve (ax + by + c) Fn-l + Fn-2 = 0. The perpendicular from any point x, y of this curve upon the line ax + by + c=0 ~is a~ax +by+c 1 F_2 Ja - b2 /a2 + 2 Fn_and the limit of this expression is clearly zero when x and y become infinite in the ratio - b: a, provided that the terms of degree n - 1 in Fn- do not contain ax+b by 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 + by + c=0 is such as to satisfy definition B. 226. The Curve in General lies on Opposite Sides of the Asymptote at Opposite Extremities. Let the straight line ax+by + c = be an asymptote of A S YMPTO TES. 217 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+by+c)Fn - + Fn-2=0; and, as in the last article, the perpendicular from any point x, y of the curve on this asymptote is given by p 1 IVan-2./a2+b2 Fn-i When x and y become very large in the ratio given by y a x - this may ultimately be written as where k is a 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. 227. Exceptions. If, however, ax+ by be a factor of the terms of highest degree in Fn_2, we may write the equation of the curve (ax+by+c)F.__+Fn-3= 0, so that the perpendicular on the asymptote is now given by p=ax+by+c_ F_ 3 /a2+b2 - Ja2+b2 Fn-_ and when x and y become very large in the ratio given by y _ x b' this can be ultimately written k MX. Xs 1 218 ASYMPTOTES. This, however, though ultimately vanishing, does not change sign with x, so that in this case the curve at opposite extremities of the asymptote lies on the same side of it. 228. Again, if the equation of the curve be expressible in the form (ax + by+c)2P,_2+ F-2=O, the expression for the length of the perpendicular is in the limit of the form f(Y). This does not in general ultimately vanish, and therefore in general ax-{-by+ c = O is not an asymptote, but is parallel to a pair of asymptotes. This case has been discussed in Art. 218. 229. If, however, the curve assumes the form (ax+ by + c)2Fn.2 +F _3 = 0, the length of the perpendicular is given by I Fn-3 (Perpendicular)2= -2 + b2 Fn_2' Hence, if the ratio of Y be that of - when x and x a become infinite, this may ultimately be written Xf \(x) and therefore Perpendicular= + x,. f ( ) 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 asymptotic at one end and the curve approaches the asymptote on opposite sides. ASYMPTOTES. 219 And in the same way other particular forms may be discussed. 230. 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=Ax2+Bx+C+D, y = Ax2 + Bx + C y=Ax2+Bx+C are asymptotic; for the difference of their ordinates for any common abscissa x is -, a quantity whose limit is x zero when x is infinite. 231. Linear Asymptote obtained by Expansion. If it be possible to express the equation of a given curve in the form C D y=Ax+B+C +...,+ then the line y=Ax+B is clearly asymptotic to the curve. This method of obtaining rectilinear asymptotes is frequently useful. 232. To find on which side of the Asymptote the Curve lies. The sign of C (Art. 231) is useful in determining on which side of the asymptote the curve lies. 220 AS YMPTOTES. Let y be the ordinate of the curve, y' that of the asymptote, then C D y-y =x ++.... xX C If x be taken sufficiently large, the sign of - governs the sign of the whole of the right-hand side. Suppose x and y to be positive, i.e., in the first quadrant, then y - y' will have in the limit the same sign as C. If C 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 C 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. Find the asymptotes of the curve 2(x2 - a2) =x2(X2 - 4a2). Here x2 - a2=O gives x=a and x= -a, two asymptotes parallel to the axis of y. Again, y= +~x - 1-,i =~x I-_4a2]) (1 _aa -j =+z 21-... a X2+ {+~I 3a2~ 3a2 2x = +__ xUi+^+... Hence the asymptotes are y= ~ and x= +a 3a2 Again, considering Y=x- +... and y= x, AS YMPTOTES. 221 it appears that if x be positive the ordinate of the curve is less than the ordinate of the asymptote, and therefore the curve approaches the line y=x in the positive quadrant from below. Similarly the curve approaches the asymptote y= -x in the fourth quadrant from above. 233. General Investigation. In order to express the general equation (n) +x- l f-() + 2 - ) + *' * = X0 (1) in the form y=- +f+ 5'3+++...,............ (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 /,, y,,.... The result of this substitution is xnOf n() + Xn- -l /'n(l) +n-2 +... 0 ]+ Ofn-l() + 2!O nA) + /3n - l(U) + On-2(/X) which gives us the series of equations n(/) =o0, 2/I3'n(/J) + /n-_l(pu) = 0, 70yn',() + -"(y) +IN' - 1() + 5n- 2(,() = 0, Hence, /, y... are determined. 234. Polar Co-ordinates. Let the equation of the curve be tn(0) + n-lfl(0)+... +fo(O) =0........ (1) 222 ASYMPTOTES. or?f,(o(0) +,1 - lf(0) +... +fA(O = o.......... (2) To find the directions in which r = o or u= 0 we have (0) =......................... (3) Let the roots of this equation be =a,/3,,.... P 0 X 2o~J x Fig. 30. A Let XOP = a. Then the radius OP, the curve, and the asymptote meet at infinity towards P. Let 0 Y(= p) be the perpendicular upon the asymptote. Since 0 Y is at right angles to OP it is the polar subtangent, and dO A p= -—. Let XOY=a', and let Q be any point whose co-ordinates are r, 0 upon the asymptote. Then the equation of the asymptote is p=r cos ( - a').................. (4) 77 -It is clear from the figure that a'= a - dO To find the value of - - when u =0 differentiate du equation (2), and put =0 and 0=a, and we obtain f l(a) +f'(a) =0............ (5) Vd_ = o ASYMPJTOTE2S. 223 Substituting the value of(- hence deduced for p in equation (4) we have Jj,-i(a) =rcos (-a~77) f'n(a) 2 = r sin(a-0). Hence the equations of the asymptotes are fn - iGa) r sin (a 0) ff4(a) r sin (3 -0) f) fn(18) etc. CoR. The case most often met with is that in which m =1, when the equation of the curve is rf,(0) +f,(0)= 0. Then f1(0)= 0 gives a, 3, y, etc., and the asymptotes are r sin (a - 0)=oa) etc. 235. To deduce the PolarAsymptote from the Polar Tangent. The same results may be deduced from the equation of a tangent (Art. 182). The result um= U cos (O - a) + UP'sin (0 - a) at once reduces to I =r sin (0-a), when U=0. Putting 1 fnit(a) U f'n(a)' as found in the last article, we again obtain the equation _fn-lica) r sin (a -0) f —i(a) f'n(a) Ex. Find the asympototes of the curve r=atanO or r-oes O-.wsino=o. Here fA(M) cosO0 and fo(6)= -asinO0. 7 3w cosO=O gives a= —, P etc., 224 A SYMP TO TES. and f(O) =- a in 0a f'1(6) - sin 0 Hence rsin(2-0)=a or rcosO=a r sin(3 - 0)==a or rcos= - a are the asymptotes. 236. Circular Asymptotes. In many polar equations when 0 is increased indefinitely it happens that the equation takes the form of an equation in r, which represents one or more concentric circles. For example, in the curve 0 r=a, 0-1 which may be written 1 r=a- 1 it is clear that if 0 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. Find the asymptotes of the following curves:1. y3=x2(2a-x). 2. y3= x(a2 _ X2). 3. X3 + y3 = a. 4. y(a2 + x2) = a2x. 5.. axy = x3 - a. 6. V2(2a-x) =x3. 7. x3 + y3 =3axy. 8. x2y + y2x = a3. A 1STYPTOTES.22 225 9. x-y2 = (a + y)2(b-2 _ y2). 1 0. X2p,2 = a2y2 _ b2X2. 12. (a- y= x2(a ~x). 13. Xy2 =4a2(2ca - x). 14. y2(a — x) =x(b X)2. 15. X2y-=X3 + X +y. 16. cxy2+ a2y-x3 +mx2 +nx +p. 17. X3~+2x2y xg2 -2y3 +4y2~+2xy y- 10. 18S. x3 2 2xy? + Xy2+- xy +2 =0. 19. y(x -y)3 =y(x -y)~2.I 20. x3~+2X2 y -4Xy2 -8y3 -4x +8y=1. 21. (X +y)2(X +2y~+2)= x +9y -2. 22. 3x3~+17x2y +21JXy2-_9y3 —2ax,2- l2axy -18ay2 - 3a 2x +a ay0. 23. rO'= a. 24. rO =a. 25. rsinnO=a. 26. r =a cosecO0+ b. 27. r =2a sin 0tanO0. 28. r sin 20= acos 30. 29. r =a + bcot nO. 30. r~sin nOa. 31. Show that all the asymptotes of the curve r tan nO= a a touch the circle r =-. 32. Show that there is an infinite number of asymptotes of the curve y = (a - x) tan viz, x= -a, x= ~3a, x=+~5a, etc. 3 3. Show that the curve 02 (ar - r2) = b2 has a circular asymptote. 34. Show that there is an in-finite series of parallel asymptotes to the curve r a +6b -sin O P 226 AS YMP TOTES. and show that their distances from the pole are in Harmonical Progression. Find the circular asymptote. 35. Find the asymptotes of the curve y = x-+ a. Find on which side of the oblique asymptote the curve lies in the positive quadrant. Show also that the hyperbola x(y - x)= 2a2 is asymptotic to this cubic curve. 36. Find the asymptotes of the curve y = x2X + a, and find x —ba x - ai on which side the curve approaches these asymptotes. 3 3 37. Show that the curve x= - a has a rectilinear asympay tote y = 0, and a parabolic asymptote y2 == ax. 38. Show that the curve x2y=x4 + x3+x2+x~+ 11 has a parabolic asymptote whose vertex is at the point (- -; a), and whose latus rectum = 1. 39. Show that the curve x2y = x3 + x2 + x + 1 has a hyperbolic asymptote whose eccentricity= 2 /2 +,/2 40. Find the equation of a cubic which has the same asymptotes as the curve xs - 6x2y + 1 lxy2 - 6y3 + x+y + 1= 0, and which touches the axis of y at the origin, and goes through the point (3, 2). 41. Show that the asymptotes of the cubic x2y - xy2 + xy + y2 + x - y = 0 cut the curve again in three points which lie on the line x + y = 0. 42. Find the equation of the conic on which lie the eight points of intersection of the quartic curve xy(x2 - y2) + a2y2 + b22 =- aCb2 with its asymptotes. 43. Show that the four asymptotes of the curve (x2 - y2)(y2 - 4X2) - 6x3 + 5x2y + 3Xy2 - 2y3 - 2 + 3xy - 1 =0 cut the curve again in eight points which lie on a circle. ASYMPTOTES. 227 44. Form the equation of the cubic curve which has x =0 y = 0, X+ y =1 for asymptotes, and cuts itsasymptotes in the a b three points where they intersect the line + = 1, and also passes through the point a, b. 45. Form the equation of a quartic curve which has x = 0, y = 0, y=x, y= - x for asymptotes, which passes through the point a, b, and cuts its asymptotes again in eight points' lying upon the circle x2 + y2 = a2 46. 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 asymptote, and also to touch the axis of y at the origin. 47. Form the equation of a quartic curve with asymptotes = 0, x + y = 0, x - y = 0, the curve being supposed to approach y = 0 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). 48. If the equation of a curve be written (X).- X ) -]( ) + e.-.2(-) +** = and if <,(p/) = 0, i<n(]1) = 0, = n-l(l) = 0, and ~'-1(1) = 0, show that there are two parallel asymptotes equidistant from the origin, whose equations are Y = + 4/d 2n-2(~1) 49. Show that the first approximation to the difference of the ordinates of the curve (k ) + +n- () + n-2)- +. =0 2228 ASYiMPPTOTE/S. and its rectilinear asymptote y = Px + f for a point whose abscissa is x is ~n(IL)Bkni(l)12 - 24jpn(L)I 2(jLL)Sb2i(fL) + 2Pq2(l)[Qf'l(p)12 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. 50. Show that the curve (y - 2x)2(y + x) + (y + 3x)(y - x) +x = 0. has the parabolic asymptote 3y2 - 12xy + 12x2 + 5x =0. 51. Show, by transforming to the point h, k, that the asymptotes of the general curve of the n1h degree (a0, a1,..., ac,,x, y)"' + n(b0, be,..., b1_,3x, y)"-l +... = 0 will all pass through one point if ao0, a, a2, a** I a1, a2, a1,..., a,, =0, and that the co-ordinates of that point are a, b1 - a bo alb0 - a0b, aCa2- a12 a0a2 - a1 [Professor Cayley uses the notation (aa, a..., a,,ix, y)fl for the general binary quantic of the nth degree: a0ex + nalx 'y + n(n -1) a -2y2 +... + anYZ CHAPTER IX. SINGULAR POINTS. 237. Concavity. Convexity. In the treatment of plane curves the terms concavity and convexity with regard to a point are applied with their ordinary signification. Thus, for example, any are 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. 238. 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 limitP Q T /A B Fig. 31. ing position of a 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. 31; so 230 SIVGULAR POINTS. that the immediately neighbouring portions AP, QB must in general lie on the same side of the tangent PT. 239. Point of Inflexion. The kind of point discussed in Art. 238 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, crosses the curve again at a third ultimately coincident point. Such a point can be seen magnified in Fig. 32. - Q R Fig. 32. 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 oj 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. 240. Point of Undulation. Again, there may be a point on the curve for which the PQ R__S Fig. 33. tangent crosses its curve in four ultimately coincident points, P, Q, R, S, as seen magnified in Fig. 33, and the S1NGULAR POINLTS. 231 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 singularities of a. higher order but of similar character may arise. 241. 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= =((x). Let P be the point to be considered, P1 an adjacent point on the curve. Let PN, P1N1 be the ordinates of P Y PX/ O N NT X Fig. 34. and P1, and suppose P1N1 to cut the tangent at P in Q1. Then ON'= x, NP = y = (x). Let ON =x+h, then N1P1= +(x +h) = 0(x) + h' (+(x) +."(x) +....(1) by Taylor's Theorem. Again, the equation of the tangent at P is Y- y = (Px)(X - ). Putting X = x + h we obtain Y= + h'(x) = (x) + he'(x),......(2) which gives the value of VNQ1. 2:32 SINGULAR POINTS. Hence the ordinate of the curve exceeds the ordinate of the tangent by N1P1, - 1-2! " + 3!"'() +.........(3) Now, if h be taken sufficiently small, the sign of the right-hand side will be governed by that of its first term; and this term does not change sign with h because it contains an even power of h, viz., the square. Hence in general, on whichever side of P the point P1 be taken, N1Pj - NV1Q1 will have the same sign-positive if q"(x) be positive, and negative if p"(x) be negative; and therefore the element of the curve at P is convex or concave to the foot of the ordinate of P according as S"'(x) is positive or negative. We have drawn our figure with the portion of the curve considered above the axis of x. If, however, it had been below, the signs of N1P1 and N1AQ1 would both have been negative and we should have had the contrary result. But observing- that +(x) is positive for points above the axis of x, and negative for points below, we may obviously state the unrestricted rule that the elementary portion of the curve y= p(x) in the neighbourhood of the point (x, y) is convex or concave to the d2y foot of the ordinate according as p(x)("(x) or yd- is positive or negative. 242. Points of Inflexion. If (f"(x)= 0 at the point under consideration, we have h3 h7Q / 4h ^P N -1 ~ Q = "'(x) + (x) +., and, as before, the sign of the right-hand side, when h is SINGULA R POINTS. 233 taken sufficiently small, is governed by the sign of its first term. But this now depends on h3, and therefore changes sign with h; that is, the ordinate of the curve Y QI 0 N NI X Fig. 35. is greater than the ordinate of the tangent on one side of P, but less on the other. The tangent now crosses the curve at its point of contact, and the point is of the kind described in Art. 239, and called a point of inflexion. A necessary condition then for a point of inflexion is that '"(x) if not infinite should vanish, and the sign of ("'(x) determines the character of the inflexion; for (assuming the element above the axis of x) if +"'(x) be positive, NlPl- VY1Q1 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 p"'(x) be negative, the change is from convexity to concavity, and this latter is the case represented in the figure. 243. Point of Undulation. Again, if q"'(x)=O at the same point, and +""(x) do not vanish, the first term in the expansion of NlP1 -f NjQ depends on h4, and therefore this expression does not change sign in passing through P. The tangent there 234 SING ULAR POINTS. fore on the whole does not cross its curve at P. The point is of the kind described in Art. 240 and called a point of undulation. 244. Higher Degrees of Singularity. It will now appear that, if by two successive differentiations a result of the form d -= A ( - Ca)2(x - b)2+1 dX2 cl2y be deduced from the equation to the curve, although ddy vanishes both at the points given by x=,a and by x=b, yet it only undergoes a change of sign when it passes through x=b, the index of the factor x-b being odd. Hence at the points given by x=ca there is no ultimate change in the direction of flexure, while at those given by x=b there is a change. The points given by x= a look to the eye like ordinary points on a curve, while those given by x=b 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. 239, which is the case of m == 0. If n = 1, the points given by x = a are points of undulation, such as described in Art. 240. So that for an Inflexional Point the condition d =0, dx2 though necessary, is not sufficient. The complete criterion is that d2 should change sign. If aish, ut deo 2 ds vanish, but do not change sign, the curve at the point under consideration is undulatory. * Dr. Salmon, "Higher Plane Curves," p. 35. Cramer, "Analyse des Lignes Courbes," Geneva. SING ULAR POINT'S. 235, 245. Case when the Tangent is parallel to the y-axis. The test of concavity or convexity has been shown to d2y depend upon the sign of Iy2. In the case, however, of an arc, the tangent to which is parallel to the axis of y, the dy value of 7 and 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 independdC2X ent variable, and then the sign of - will test the condy cavity or convexity to the foot of the ordinate drawn from the point under consideration to the axis of y. Similarly, at a point of inflexion at which the tangent d2x is parallel to the axis of y, dy must change sign. 'y2 And in other cases whenever it is more convenient' to use y instead of x for our independent variable, we are of course at liberty to do so with an interchange of the letters x and y in the formula quoted. P P 0 NL N N2 X Fig 36. 246. The test for conccavity or convexity mnay also be investigated as follows: 236 SING ULAR POINTS. Let P be any point of the curve, co-ordinates x and y. Let the adjacent points on the curve P1 and P2 have coordinates,(x-h, yl) and (x+h, y2) respectively. Let the ordinate of P cut the chord PP2 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 AO, according as NP is < or > NQ, i.e., as y is < or > y-+ 2. 2 ~Now d2=y +h2y dh y+ 2 Yhdx~2 dx2..., dyh2 d2y Y1=Y-dx +2! dx2 "'I so that the criterion depends upon whether I2 d2y y be < or > y+( + d 62..., and proceeding to the limit the curve is convex or cond2y cave to N according as d2- is positive or negative. Ex. 1. Consider the curve y =2 ~/ax. Is it convex or concave to the foot of the ordinate? Here dy -1 N/ a cd2 2 ', Xd2 and d/2 -- a x Hence yd y is negative for all positive values of x (and negative dX2 values 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 +3y2. Hias it a point of inflexion? td2 Here d -6(?J+ 1), so that d2 changes sign as y passes through the value y= -1. Therefore the point (2, - 1) is a point of inflexion on the curve. SILN GULAR POINTS. 237 247. Convexity and Concavity of a Polar Curve. Suppose the equation of a curve to be given in polar co-ordinates, and that it is required to find a test of convexity or concavity towards the pole. \B roP Fig. 37. Let 0 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 (r1, 0-80) and (r2, 0+80). Then the curvi in the immediate neighbourhood of P will be concave or convex to 0, according as AAOP+ ABOPis > or < AAOB when we proceed to the limit. That is, according as rir sin 80 + rr2 sin 60 > or <' rr sin 260, or rlr + rr, > or < 2r1v2 cos 80; i.e., as u2 + I > or < 2u cos 80, where we have written r, = -, etc. Now, by Maclaurin's Theorem, du d2u 802 ul=+dase+ dO 2 2!~+" U= + -d00 d02 2!-. 238 SINGULAR POINTS. and therefore %U + u= 2- + dO2U..02, whence we have concavity or convexity to the pole accordduk S602. ( 2.0 ing as 2u+ 2- +... is > or < 2 1 —+... and proceeding to the limit according as d2k&. + d-2 is > or < 0. 248. Polar Condition for a Point of Inflexion. At a point of inflexion the curve changes from concavity to convexity, and therefore the necessary condition d2U is that -+ do2 should change sign. Ex. Find the point of infiexion on the curve r=a0-. Here au = 01, d2y, 3 therefore a 1 0-2. d62 4 Hence, putting t + u0 to find for what value of 0 a change of sign can occur, we have 0~- 0-i=0, 02 = 1 a= +1 0=-t_ And the positive value only is admissible, giving r=aV2 0=2 5 as the polar co-ordinates of the point of inflexion. MULTIPLE POINTS. 249. Nature of a Multiple Point. A singularity of different nature from those above described occurs on a curve at a point where two branches SI.NGULA R POINTS. 239 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 Fig. 38. curve, one for each branch. 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 A 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 thdegree, each tangent will cut the curve again in n-3 points real or imaginary. In this respect the tangent at such a point resembles the tangent at a point of inflexion, for (Art. 239) the point of contact of a tangent at a point of inflexion counts for three of the n intersections of the line with the curve. 250. Points through which more than one branch of a curve passes are called "mnultiple 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 " nultiple point of the rth 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) 240 SING ULAR POINTS. at a multiple point of the rth 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 in n-r-1 other points real or imaginary. 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 multiple point, and therefore reducing the remaining number of points of intersection to n - - 2. 251. Species of Double Points. Consider the case of a double point. The tangents there may be real, coincident, or imaginary. CASE 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 crunode. Fig. 39. CASE 2. If the tangents be imaginary, there are no real points on the curve in the immediate neighbourhood of the point considered, 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 SIVNG ULA R POINTS. 241 ordinates satisfy the equation to the curve, and is called a " conjugate point" or " acnode." CASE 3. If the tangents at the double point be coincident, the two branches of the curve will touch at the point considered. The point is then in general of the character called a cusp or spinode. 252. Two Species of Cusps. There are two kinds of cusps, as shown in the accompanying figures. TT Q' \\ EA Fig. 40. Fig. 41. (a) In Fig. 40 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). (/3) In Fig. 41 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). 253. 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 Q 242 SINGULAR POINTS. their point of intersection. And in general a multiple point of the rth order may be considered as the result of the combination of v(r- 1) double points, since this is the 2 number of ways of combining the r branches two at a time. 254. 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 + bNx + b2y + C]x2 + cxy + c3y2 +... ++ +klc2x- +... +. +k+ly = 0.............(A) If this be put into polar co-ordinates it becomes a + r(blcos 0 + b2sin 0) + r2(clcos20 + ccos 0 sin 0 + csin20) +... + rn(klccosn0 + k2cosn-10 sin 0 +... + k+lsinn0) = 0... (B) Let 0 be the pole and OA the initial line. Then equation (B) gives the points P1, P2, P,..., in which a radius o A Fig. 42. vector OP1P2..., making a given angle 0 with OA, cuts the curve. The roots of this equation are OP1, OP2, OP3,..... SNG ULAR POINTS. 243 It is clearly of the nth degree, and therefore has in general n roots. These may, however, become imaginary in pairs. I. If a =0, it will be obvious from either the Cartesian equation (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 OP1 = 0. II. In this case, if 0 be so chosen as to make bcos 0 + b2sin 0 =0, a second root of the equation (B) vanishes, and therefore we infer that a straight line making an angle tan1-b-1) 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., bx + b2y =0. 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. 173.) III. If ac=0, bi=0, and b2=0, then in general it is possible to choose 0 so that c cos20 + c2cos 0 sin 0 + csin20 - 0, and then three roots of equation (B) will vanish; that is to say, of the pair of lines whose equation is c12+c2xy+C3y2 = 0 each cuts the cueve 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 244 SING ULAR POINTS. 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., clx2 + c2xy + c3y2, when equated to zero form the equation of the tangents at the origin. The tangent of the angle between these straight lines is given by tan = 2- 4c3 '1 +- C3, If c2>4c c3, the tangents are real and not coincident, and there is a node at the origin. If c22=4cc, the tangents are coincident, and the two branches of the curve touch, and there is in general a cusp at the origin. If c2 <4Cl3, 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 c+c3= 0, the tangents at the origin intersect at right angles. IV. If a = 0, b = 0, b2 =, c1=0, 2 = 0, C = 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 dzx3 + d2x2y + d3xy2 + d4y3 = 0. V. And generally, if the lowest terms of an equation are of the rth degree, the origin is a "mnultiple point of the rth order" on the curve, and the terms of the rth degree equated to zero give the r tangents there. 255. 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. SINGULAR POINTS. 245 Let the straight line -= -- p be drawn through a given point (h, k7) to cut the curve f(x, y)=0. Then x=h+lp, y =k+mp. The use of these equations is obviously equivalent to a double transformation of co-ordinates, the first to parallel axes through h, k, the second to polars. Substituting for x and y in the equation of the curve we obtain f(h + Ip, k + mp) = 0 to find the points P1, P,... in which a radius vector through the point h, k cuts the curve. If this be expanded by the extended form of Taylor's Theorem, the equation becomes +M a c p2 'a + _ (,)+ + )/+!\h+,ac) f+'... which is exactly analogous to equation (B) of Art. 254, and corresponding results follow. I. If f(h, I) = 0, one root of the equation for p vanishes and the point h, kI lies on the curve (which is otherwise obvious). II. In this case, if the ratio 1: m be now so chosen that f - 'af, 0 D'h D '= then another root vanishes, and this relation gives the direction of the tangent, whose equation is therefore af af (x-h) +(y-)f-=o, as found in Art. 169. 246 SINGOUSLAR POINTS. III. But if = 0 and -f 0, as well as f(h, k)=0, then all lines through h, k cut the curve in two contiguous points. But if the ratio 1: m be so chosen that we have in general, as in Art. 254, III., two directions in which a radius vector drawn through (h, 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-mentioned radii vectores meets the curve, two lie on one branch and one on the other. The equation of the two tangents is (x-h + 2(x-) (y-Af + (y-k) a= IV. Further, if 2 = 0, f 0, and =0, in addiDah2 ahak2 tion to af = 0 f, 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 (h, k) is a triple point on the curve. V. And generally the conditions for the existence of a multiple point of the rth order at a given point h, k of the curve are that f(x, y) and all its differential coefficients up to those of the (r-l)th order inclusive should vanish when x = h and y = k; and then the equation of the r tangents at that point will be z _ h+ _ - -l-+...f,~ V+* f(y / f =0. (X - h).V+ r(X - h r-1 - )r ~Xv+~'x-h/ (Y-0)h_~ +. +(y Nor5.o SINGULAR POINTTS. 247 256. 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 k,,~-f a2f,a2f_ (x - h) + 2(- h)(y- )ha + (y - 7)2 = 0, the angle between these tangents is given by 9 1 2f \2 _2f 2f 2 (\? hWck Dh2 'al2 tan 0-= 2f -2 Dh2 Dia,2 and the point h, k is a node or conjugate point according as I(kf \2 is > a \ a 2f 8ts \aWalcQ < wh2' OF' and is in general a cusp if 2f \2 D2f -a2f ahDkac =-h2 'al2' with the preliminary conditions in each case that f(h, )= O,, and =. We say in general a cusp; for it will be seen that in some cases yhen the above conditions hold the curve becomes imaginary 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 necessary. The mode of procedure is indicated below in the method for the investigation of the character of a cusp. It appears that (f -) =xs2 * y2 represents a curve which cuts f(x, y)= 0 in all its cusps; 2n t2f 02 and that + _2f aX2 ay2 248 SINGULAR POINTS. is a, curve which cuts f(x, y)= 0 in all the double points at which the tangents are at right angles. 257. To search for Double Points. The rule therefore to search for double points on a curve f(x, y) =0 is as follows. Find af and f; equate each to,ax 'ay zero and solve. Test whether any of the solutions satisfy the equation of the curve. If so, apply the tests for the character of each of the points denoted, i.e., try whether / 2f \2 b> 32f 32f (-xyl be == x. \xyf -ox" a^y 258. To discriminate tle 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+ by)2 +3+u4+..= 0............. (1) where ax + by = 0 is the tangent at the origin, and u,, ut4,... are homogeneous rational algebraical functions of x and y of the degrees indicated by their respective suffixes. Let P 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. Then p = a. ( 2) Then P.ccx +m................(2) If y 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 P2 we reject SINGULAR POINTS. 249 cubes and all higher powers of P and also all such terms as P2x, P22,... which may arise on substitution. ig 43.-ingle cusp, first species. Fi. -Single cusp, second species. Fig. 43.-Single cusp, first species. Fig. 44.-Single cusp, second species. Fig. 45.-Doublefcusp, first species. Fig. 46. —Double cusp, second species. Fig. 47.-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, 250 SINGULAR POINTS. 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 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 perpendiculars 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 tangent 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 x. 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 extremities 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 /a2 + b2 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 perpendicular itself. Ex. 1. Examine the character of the origin on the curve x4 - 4x2y - 2xy2 + 4y2 = 0. Here the tangent at the origin is y=0. According to the rule put. y = P. The quadratic for P is P2(4 - 2x) - 4Px2 + 4 = 0. SINGULAR POINTS.. 251 The roots of this equation are real or imaginary according as 4x4 is > or < x4(4-2x), i.e., according as x is positive or negative. Hence the cusp is "single" and lies to the right of the axis of y. Moreover the product of the roots is 2 and is positive when x is very small, 4- 2-x 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 quadrant. Ex. 2. Examine the character of the curve x4 -3xy - 3xy2 + 9y2 == at the origin. Here y=O is a tangent at the origin. Put y=P. The quadratic for P is P2(9 - 3x) - 3x2P + 4 = 0. The roots are real or imaginary according as 9x4-4(9-3x)x4 is positive or negative, i.e., as - 27x4 + 12x' is positive or negative. Now, when x is very small, x5 is negligible in comparison with x4, and therefore the above expression is negative for very small positive or negative 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 2m+l y=F(x)~ + (x —h)nyf(x).....................(1) in the neighbourhood of the point x=h, y=F(h), m and n being positive integers. By Taylor's Theorem we may write F(x + ) = F(h) + ax + bx +... and [f(x+ h)] = a + blx..., where a, being [f(h)]2 is necessarily positive. Hence on transforming our origin to the point {h, F(h)} we obtain for the transformed equation 2m + 1 (y- cax- b2-...)2= -~-~ (al~+bl+...).............(2) Examining the form of the curve at the origin, there are obviously coincident tangents if 2m 1 be > 2. n Put y - ax = P, then 2m+ 1 p2 22P(bx2+...)+b2x4_aalx- -... =0. SINGULAR POINTS. 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 cusp extending to the right of the new axis of y. 2m+l Again, the product of the roots = b2x4 - x -... If +1> 4, this product has the same sign as x4 when x is taken sufficiently small, and therefore is positive, giving a cusp of the second species. 2m+ 4- 1 2m+1 If 2+ <4, the term — ax n is the important term in the n product and is negative, x being positive. There is therefore in this case a cusp of thefirst species. We have assumed that the coefficient b or -F"(h) is not zero. If however this coefficient vanish, it is easy to make the corresponding change in the subsequent investigation. Ex. 4. Examine the nature of the double point on the curve (x +y)- 2( - x+ 2)2 0. Here - =3(x+y)2+2 ^/2(y-x+2)=0, } Zx a-3(x+y)2- 2 '2(y-x+2)=o. J These give x + = 0, } and y - x+2=0, ) or x=l, ).J=-l. f Now this point obviously lies upon the curve, and there is therefore a multiple point of some description there. Again, -6(x +y) - 2 /2= -'2 V2 at the point (1, -1), aX2 -6(x +.y) - 2V2 = - 2 /2, go-> 6(x+:/)+ 2 2= 2 2 /2. Hence at this point -2o D2_ ({ 2 )2 and we have a double point at which the tangents are coincident. SINGULAR POINTS. 253 Next, transforming to the point (1, -1) for origin, the equation becomes (x+,y)3 - 2(y- )2= 0. According to the rule we put y-x=P. Then rejecting terms in P3 and P2x we have P2 - 6x2 2 P - 4xs /2 = 0. The roots are real if 18x4 +4 2x3 > 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 = -43 ^/2, and is negative when x is small. This indicates that the cusp is one of the first species. [This curve is obviously only a transformation of the semi-cubical parabola y2= x3.] 259. METHOD II. Another method of discrimination of the species of a cusp depends upon the test for concavity or convexity. Find the two values of d2y (or d2X see Art. 245). If these have opposite signs very near to the cusp, the two branches starting from the cusp are one concave and the other convex to the foot of the ordinate, and the cusp is of the first species. But if the signs be 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., Ex. Discuss the form of the curve y= x + xS at the origin. ld2y -3 1 Here = d -4 * dx"L2 -4 /x' Hence only positive values of x are admissible and the two values of dy, have opposite signs. The origin is therefore a single cusp of the first species. This result is obvious also from the form of the equation to the curve. 25 4 SING ULAR POINTS. 260. Singularities of Transcendental Curves. In addition to the singularities 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." Consider the curve y=a(; (a > 1). When x= -oo, y=l, and as x increases from -oo 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 + o 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. Y o x Fig. 48. Next suppose that the value of y is continuous, but dy that at a certain point d becomes discontinuous, so that two branches of the curve meet at certain angle at the same point and stop there. Such a point is called a " point saillant." SINGULAR POINVTS. 255 261. 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 pointille'es. Ex. In tracing the curve y-x", it is clear that, when x==O, J/=co; and when x =l, y=l1. Also that as x decreases from oo to 1, y also decreases from Go to 1. Between x=1 and x=O0 y is less than 1; and when x-=O, y=l (see Chap. XIII.). There is therefore a continuous branch of the curve, viz., oc PB, above the axis of x. Again, whenever x is a fraction with an even denominator there Y P 0Fig. Fig. 49. are two real values of y, differing only in sign; e.g., (4- ~ 21 whilst, whenever the denominator of x is odd, there is but one real value for y. There is therefore a branch of conjugate points below 256 SINGULAR POINTS. 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 coordinates of any point P on the branch in the first quadrant be (x, y), then ON=x. Take On= -x along the negative portion of the axis of x, then, if p be the corresponding point on the curve, we have pn = ( - x) -, PN = xx, and therefore pn. PNV= (- 1)X, which may be =1, -1, or imaginary, according to the particular value of x. Hence, when the ordinate pn is real, its magnitude is inverse to that of the corresponding ordinate PT. Hence on this curve we 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," p. 275, or a paper by Mr. D. F. Gregory, "Camb. Math. Journal," vol. i., pp. 231, 264. 262. Maclaurin's Theorem with regard to Cubics. We conclude the present chapter with an important theorem with regard to cubic curves, which is due to Maclaurin. If a radius vector OPQ be drawn through a point of inflexion (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 tangent at the point of inflexion as the axis of y, the equation of the cubic must assume the form y3+XU=.=....................... (1) where u is the most general expression of the second and lower degrees, viz., ax2 + 2hxy + by2 + 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 Lr2+Mr+ V = 0, SINGULAR POINTS. 257 where L = sin30 + (a cos20 + 2h sin 0 cos 0 + b sin20) cos 0, M = (2g cos 0 + 2f sin 0) cos 0, N = c os 0. If rl, r2 be the roots of this quadratic, and p the harmonic mean between them, we have 2 1 1 M 2gcos0+2fsin0 P ', ~r, -L c which shows that the Cartesian Equation of the locus of the extremity of the harmonic mean is the straight line gx+fy +c=O. 263. It is obvious from Art. 187 that the equation of the polar conic of the cubic (1) with regard to the origin is x(2gx + 2fy) + 2cx = O, or x(gx+fy + c) = O. 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=O meets the curve, and consequently also that their three points of contact lie in a straight line. 264. 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 R 258 SINGULAR POINTS. point C, which is also a point of inflexion on the cubic. For if B, B, B2 be the three ultimately coincident points on the cubic, which lie in a straight line (B being a point of inflexion), let AB, AB, AB2 cut the curve in C, C0, C2 and let AH, AH1, AH2 be the harmonic means between AB, A C; AB1, A C,; AB,, A respectively, then H, H, H2 lie in a straight line by Maclaurin's Theorem, and B, B, B2 lie in a straight line; therefore by a theorem in conic sections C, CQ, C2 also lie in a straight line, and they are ultimately coincident points. C is therefore a point of inflexion. EXAMPLES. 1. Write down the equations of the tangents at the origin for each of the following curves:(a) y- +c= c cosh - X (/3) = a tan. (7) y2 = X log(l + x). (8) x + y3 = 3axy. 2. Show that the curve y = e" is at every point convex to the foot of the ordinate of that point. 3. Show that for the cubical parabola a2y = (x- b)3 there is a point of inflexion whose abscissa is b. 4. Show that for the semicubical parabola ay2 = X3 the origin is a cusp of the first species. 5. Show that the origin is a cusp of the first species on the curve a(y - x)2= x3. SINGULAR POINTS. 259 6. Show that on the curve (ay - x2)2 = bx3 there is a cusp of the first species at the origin, and a point of 9 infiexion whose abscissa is -b. 64 7. Show that there are points of inflexion at the origin on each of the curves (a) y==cos-. a (P/) y=atan. (7) y= log ( - ). 8. Show that the curves y2 xsin-, y22 x2tan y" = lna a have cusps of the first kind at the origin. 9. Show that at the origin on the curve y2 =bx sin a there is a node or a conjugate point according as a and b have like or unlike signs. 10. Show that there is a point of inflexion on the curve y = e at the point (8, e2). 11. Show that the Trident curve axy + as = x3 has a point of inflexion at the point in which it cuts the axis of x, and show that the tangent at the point of inflexion makes with the axis of x an angle tan-13. 12. Show that the curve b(ay - x2)2 = X5 has a cusp of the second species at the origin. 260 SIN GULAR POINTS. 13. Show that, if n be greater than 2, the curve bV*-4(ay - x2)2= x,n has a cusp at the origin of the first or second species according as n is less or greater than 4. 14. Find the two points of inflexion of the curve y X2 fx- a\X c 9 —a +2 and draw figures showing the characters of the inflexions. 15. Show that every point in which the curve of sines y.. x = sina b cuts the axis of x is a point of inflexion on the curve. 16. Show that the points of inflexion on the cubic a2x x2 + a2 are given by x= 0 and x= a /3. Show that these three points of inflexion lie on the straight line x = 4y. 17. Find by polars the points of inflexion on the curve 2x(,2 + y2) = a(22 + y2). 18. Show that the curve au = " has a point of inflexion where au= {n(1 - )}2. 19. Show that if the origin be a point of inflexion on the curve u + u2 + u +...-= 0 u2 will contain u, for a factor. 20. Show that there is a point of inflexion at the origin on the cubic y = axy + by2 + cx3. 21. Show that there is a point of undulation at the origin on the curve y = ax4 + bx2y2 + cy4. 22. Show that the origin is a triple point on the curve x4 + y4 = ay2, and that there is a cusp of the first species there. SINGULAR POINTS. 261 23. Show that there are two points of inflexion at the origin on each of the curves x2y2.= a2(x2 y2) r2 = a2 cos 20. 24. Show that the abscissae of the points of inflexion on the curve yf =f(x) are roots of the equation n-l - {f'(x)}2=f(x)f"(x). 25. Show that the abscissae of the points of inflexion on the curve y = e-X tan ux are given by 2/z sec2 JUx(u tan jux - A) + X2 tan px = 0. 26. Show that the curve x3 + ax2 + Ca3 - 2 2- x _-a has but one point of inflexion, and that its abscissa is / 3+1 - a/3- 1 27. Find the positions of the points of inflexion on the curve 12y=x4- 16x3+42x2+ 12x+ 1. 28. Prove that the curve y = be (Ga has a point of inflexion given by x=a n-. n 29. Prove that the point (- 2, 2 is a point of inflexion on the curve y = xex. 30. Show that there are two points of inflexion on the cubic X3 + y3 = a3 at the points (a, 0), (0, a) respectively. 262 - SINGULAR POINTS. 31. In the curve x3 + y3 = ax2 show that there is a cusp of the first kind at the origin, and a point of inflexion where x = a. 32. In the curve y2= (x- a)(x- b)(x- c) show that if a = b there is a node, cusp, or conjugate point at x = a according as a is >, =, or < c. Also show that the points of inflexion have for their abscissae x=4 ----. Hence show 3 that the points of inflexion on this curve are real or imaginary according as the curve has a conjugate point or a node. 33. Show that for the curve r =a(1 - cos 0) there is a cusp of the first kind at the origin. 34. Show that the curve.2 cos2 0 a2 cos 20 has a double point at the origin. 35. Show that the curve r = a sin nO has a multiple point at the origin of order n or 2n according as n is odd or even. 36. Show that the curve a02 r = —,1 + '2 has a cusp of the first kind at the/pole. 37. Show that if the cubic xy2 + ey= ax + bx2 + cx + d have a centre, then will b 0 and d= 0 and the centre is at the origin. In this case show also that the origin is a point of inflexion on the curve. 38. Show that there is a conjugate point on the locus x3 + y3 + 3cxy = c' at the point (- c, -c). Trace the curve. SING ULAR POINTS. 263 39. Show that the curve x5 + y5 =- ax2y2 has two cusps of the first species at the origin, and that x + y = a is an asymptote. 40. Show that a cubic curve cannot have more than one double point, and cannot have a triple point. Examine the case of the curve 2(3 + y3) - 3(32 + y2) + 12x = 4 and show that there are apparently two nodes at (1, 1) and at (2, 0) respectively. Explain this result. 41. Show that the curve by2 = x sin"a has a cusp of the first species at the origin and is symmetrical with regard to the axis of x. Show also that it has an infinite series of conjugate points lying at equal distances from each other along the negative portion of the axis of x. 42. Show that the curve ey2 y-x = log, e4x has a node at the point (1, 2). 43. Show that the curve (x2 + y)2 = a(3x2y - y3) has a triple point at the origin, and that the angles between the branches through the origin are equal. 44. Show that the curve (x2 + y2)2 = 4axy(x2 - y2) has a multiple point of the eighth order at the origin, and that the curve consists of eight equal loops. 45. Show that for the Cissoid 2a-x the origin is a cusp of the first species. 264 SINGULAR POINTS. 46. Show that for the Conchoid x2y 2= (a + y)2(b2 y2), if b be > a there is a node at x= 0, y = - a, and if b =- a there is a cusp at the same point. 47. Show that at the point (- 1, - 2) there is a cusp of the first species on the curve x3 + 22 + 2xy - y2 + 5x - 2y = 0. 48. Examine the singularities of the curve 4 - 4ax3 - 2ay3 + 4a2x2 + 3a2y2 - a4= 0. There are nodes at the points (0, a), (a, 0), (2a, a). Find the directions of the tangents at these points. 49. Show that the curve x4 - 2x2y - xy2 - 2x - 2xy + y2 x + 2y + 1 = 0 has a single cusp of the second kind at the point (0, - 1). 50. Examine the character of the curve ay4 - ax2y2 + xI + x4y = O in the immediate neighbourhood of the origin. 51 Show that at each of the four points of intersection of the curve 2 2 2 the curve (ax)' + (by) = (a2 - b2)with the axes there is a cusp of the first species. 52. Show that the origin is a conjugate point on the curve X4 - ax2y + axy2 + a2y = 0. 53. Show that at the origin there isa single cusp of the second species on the curve x4 - 2ax2y - axy2 - a2y2 = 0. 54. Show that the curve Y2= 2x2y + xty + x3 has a single cusp of the first species at the origin. 55. Show that the curve y2= 2x2y + x4y + x4 has a double keratoid cusp at the origin. 56. Show that the curve y2= 2x2y + x4y - 2x4 has a conjugate point at the origin. CHAPTER X. CURVATURE. CURVATURE. 265. 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, intersect R Fig. 50. ing at T and cutting a given fixed straight line LZ in L 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 266 CUR VA TURE. 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. 266. 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 angle of contingence. less. The fraction angle of coningence is called the length of arc average bending or 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 given point the limit of the expression angle of contingence when the limit of the expression when the length of arc length of the arc measured from the given point and therefore also the angle of contingence are indefinitely diminished. That this is a proper measure of the rate of deflection is obvious from the consideration that, for a given length of arc, the deflection is greater or less as the angle of contingence is greater or less, and for a given angle of contingence the deflection is greater or less as the length of the arc is less or greater. 267. Curvature of a Circle. In the case of the circle the curvature is the same at CUR VA TURE. 267 every point and is measured by the RECIPROCAL OF THE RADIUS. R 0Fi. Fig. 51. For let r be the radius, 0 the centre. Then A A are PQ RTQ=POQ= r the angle being supposed measured in circular measure. angle of contingence I Hence -. - - = - length of arc -" and this is true whether the limit be taken or not. Hence the "curvature" of a circle at any point is measured by the reciprocal of the radius. 268. Circle of Curvature. If three contiguous points be taken on a curve, a circle may be drawn to pass through those three points. Let them be P, Q, R. Then, 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 same angle of contingence, viz., the angle RQZ (see Fig. 52). Moreover, the arcs PR of the circle and the curve differ by a small quantity of order higher than their own, and therefore may be considered equal in the limit (see Art. 36). Hence the curvatures of this circle and of the 268 CUR VA TURE. 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 bf the radius being a correct measure of the rate R Fig. 52. 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. 269. Formula for Radius of Curvature. Referring to the figure of Art. 265, let the arc AP measured from some fixed point A on the curve up to P be called s, and A Q, s8-F s; let the angle PLZ=, and QMZ= —V+ d+. Then the angle of contingence RTQ = and the measure of the curvature = LtL=-=-. If there&s ds CUR VA TURE. 269 fore the radius of curvature be called p we have 1 dVI ds -= d' or P= d-.................A) 270. This formula may also be arrived at thus. Let PQ and QR (Fig. 52) be considered equal chords, and therefore when we proceed to the limit the elementary arcs PQ and QR may be considered equal. Call each ds, and A the angle RQZ= 8r. Now the radius of the circum-circle of the triangle PQR is 2 sinPQR' PR = L 2Ss _s o ds Hence p = Lt P Lt sn --- Lt * = ds 2 sin PQR 2 sin ~r r'sin J 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 considered as the point of intersection of the normal at that point with the normal at a contiguous and ultizmately coincident point. 271. 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 ip (Art. 291). For example, that relation for a catenary is known to be s=ctan~; whence we deduce ds at once p= d=csec2/, and the rate of its deflection at any point is measured by 1 cos2/' c p c s2 + 2' 272. Transformations. This formula however must be transformed so as to suit each of the systems of co-ordinates in which it is 270 CUR VA TURE. usual to express the equation of a curve. These transformations we proceed to perform. We have the equations dx. dy cos r=d-, sin = ds. Hence, differentiating each of these with respect to s, d df d2x dcos r d2y -sin ds = ~(82' cos CS = ds2' d2x d2y,1 -~ds2 ds2 whence =...................(B) p dy dx ds ds and by squaring and adding 1 (d2x2 (d2)2( p p= \ -. ~,d:-~/......... These formulae (B) and (c) are only suitable for the case in which both x and y are known functions of s. 273. Cartesian Formula. Explicit Functions. t dy Again, since tan = dx' dx2 we have sec2 - di- d by differentiation with regard to x. Now d ddd pcods f 1 Now dkd ---; dx ds ' dx p cos +; 1 d2y therefore sec3 r. p- = dand sec2f = 1 + tan2 = 1 + (-; therefore......... (D) dx2 CUR VA TURE. 271 This important form of the result is adapted to the evaluation of the radius of curvature when the equation of the curve is given in Cartesian co-ordinates, y being an explicit function of x. 274. 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, y)=0 be the equation to the curve, we have +. dy y 0, 'x 'y dx and 02 +2 o2 0 dy 2 d y\2 +0 d2y. xz2 axay dx Dy2x) y dx2 - Hence, substituting for dy and d in formula (D), dx dx2 I ax I D + 2 y I w 2 a20 +2 _ ia.^J _ z ' x2 axay 0By2 a ( Dy) ) D~( or P 2(pta)2_295a ao 220 Do 2 (E) xA\>ay axaiy x' y ay2\ \x] 275. A curve is frequently defined by giving the two Cartesian co-ordinates x, y in terms of a third variable, 272 CUR VA TURE. e.g., the equation of a cycloid is most conveniently expressed as x=a(O + sin 0), y =(1-cos 0). Formula (D) is very easily modified to meet the requirements of this case. Let x=F(t)l be the equations of y=f(t) the curve. dy dy dt f'(t) Then dx -dx F'(t)' dt d2aydd (dy\ dt and dx2 I~dt *dx) 'dx and d2y dx d2x dy dt2' dt dt2' dt \dt] _f"((t) F'(t)-f(t). F"(t) {F (t)}3 and formula (D) becomes (dx\2 dy2\ \dt ) + dt { [F'(t)]2 + [f(t)]2}1 =d2y dx d2x dy f"(t). F(t)-f'(t). F"(t) (F) dt2 dt dt2 dt Ex. In the above-mentioned case of the cycloid dv d2X = a(l + cos 0), - a sin. dO d-2 y = a Sill 0, ddY a cos 0, dO dO2 and by formula (F) 3 8a cos3 a{( + cos)2+Si}12 _ } 2 0cos0 P cos 0(1 + cos 0) + sin2O 2 cos20 2 2 CUR VA TURE. 273 276. Curvature at the Origin. When the curve passes through the origin the values of dy =and daY( of d(=) and d-2(=q) at the origin may be deduced qx2 by substituting for y the expression px+qx +... (the 2! 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 (1 +p-2)\ once deduced from the formula p = ( + ). [Formula (D)]. Ex. Let the curve be ax + by + a'x2 + 2h'xy + b'y2 +...... = 0. Putting y =px + -X2 +... we have a x+ a' 2 +... '0, + bp + 2h'p +b'p2 +bq 2 therefore a + bp =0, and a' ~~and ~ a' + 2h'p+b'p2+ bq=0, etc., g and =-2-a' +2h'p + b'p2 giving and = - b _7b ' (1+p2) _ 1 (a2 + b2)_ whence p - 1 -b q 2 a'b2-2h'ab+b'a2 This result of course might be deduced at once from formula (E). 277. 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 s 274 CUR VA TURE. 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 y2 _ 32y+ 22 - 3 +y4= 0. Substituting px+- x2 +... for y we have 2! -3p -2 +2 -1 whence p2 -3p + 2=0, p - 1 - 1 = 0, etc., whence p =1 or 2, and q= -2 or 2, and therefore p_ ~PZ) - 2 /2 - 1-414... q -2 53. 5 or - ~5-5 5 =5590.... 2 2 The difference of sign introduced by the q indicates that the two branches passing through the origin bend in opposite directions. B 0 N — 3 _p X A Fig. 53. 278. Newtonian Method. The Newtonian method of finding the curvature of the curve at the origin is instructive and interesting. Suppose the axes taken so that the axis of x is a tangent to CUR VIA TURE. 275 the curve at the point A, and the axis of y, viz., AB, is therefore the normal. Let APB be the circle of curvature, P the point adjacent to and ultimately coincident with A in which the curve and the circle intersect. Then PN2 = A N. INB, PN2 or NrB = AY ABY' Now in the limit NAB = AB = twice the radius of curvature. PnT2 x2 Hence p = Lt Lt -..(........... (G) 2AN 2y. Similarly, if the axis of y be the tangent at the origin, we have p=LtY-. Ex. Find the radius of curvature at the origin for the curve 2x4+3 y4+4xy +x-y2+2 = O. In this case the axis of y is a tangent at the origin, and therefore we shall endeavour to find Lty. 2x Dividing by x 23 + 3/2. + 4xy+y- +2= 0. x x Now, at the origin LtY = 2p, x=0, y= 0, and the equation becomes -2p+2 =0, or p=l. 279. The same method may be applied when the tangent to the curve at the origin does not coincide with one of the axes; but as the method of Art. 276 is very simple we leave the investigation as an exercise to the student. Ex. Establish in the above manner the result of the Example in Art. 276. 276 CUR VA TURE. 280. Formula for Pedal Equations. To find a formula for the radius of curvature adapted to pedal equations. Let 0 be the pole and C the centre c l 0 s ~o~ ~Y Fig. 54. of curvature corresponding to the point P on the curve, P' a contiguous point on the curve ultimately coincident with P, the normals at P and P' intersecting at C (Art. 270). Let OY, the perpendicular on the tangent at P, =p. Then OC2 = r2 + p2 - 2rp cos OPC = r2 + p2 _ 2rp sin p r2+ p2_ 2p9p, 'p since sin -b=-. Again, for the contiguous point P', r becomes r +-r and p becomes p+6p, while p and OC remain the same. Hence OC2 =(r + 6r)2 + p2 - 2p(p + np); therefore by subtraction ( + r)2 _ r2 = 2p81p, or in the limit p = Lt(r-+.2-] =..............() 61 8p 2 6p/ p / CUR VA TURE. 2777 Ex. In the equation p2=Ar2+ B, which represents any epi- or hypo-cycloid [Chap. VII., Ex. 36], we have dr p =Ar, and therefore p XC p. The equiangular spiral, in which p cc r, is included as the case in which B=O. 281. 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. 181 1 - 2 + (di)2 Hence 28/8 or GCUdp I + d2in\ du3 d021 -rcr 1 Now P p and r —, -dp- U, therefore 1dm 1 2L cl+ d= 21\ (m + or ~2).(..................... 282. This may easily be put in the r, 0 form thus:Since U din, 1 dv we have dO I dr a- - r2 C'1 278 C UR VA TURE. and therefore therefore d2u 2 /dr\2 1 d2r. d0r3= \i oJ d-d02' __tr^^W J_ rrq3Xl ~-2 (J2 P= 1f1i 2(dr\2 1 d2r 2^j d r............. (j) 2 2+ dO 2 d2 r2+ 2 -~ d(2 283. Tangential-Polar Form. In Art. 195 it was proved that P= P+........................ (K) giving us a formula for the radius of curvature suitable for p, r equations. Ex. It is known that the general p, i equation of all epi- and hypo-cycloids can be written in the form p= A sin B12 (Chap. VII., Ex. 36). Hence p= A sin B -A B2 sin B2a, and therefore p oc p, thus again proving the result of the Example in Art. 280. 284. 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= GO, pwe get s from(A), we get -!d -0 from (A), CUR VA TUBE. 2 79 d~y0 dX2 = 0 from (D), D2~ /a 2 __ _ =+ ( /x) -o from (E), aaX2 kl aay 3,ax `ay ay2 IaXj U = 0 from idr\2 d22 9.2~ 2 rI - -Ofrom (j), \dO d02 some of which have already been established otherwise. EXAMPLES. 1. Apply formula A to the curves s = aV1, 8= a sin VI, s = a tanfr. 2. Apply formula D to the curves 4ax, y = c cosh 3. Apply formula E to the curve ax + by + a'x2~+ 2h'.,~gb' +9+.. to find the radius of curvature at the origin. 4. Apply formula F to the ellipse x=a cos 0 y=b sin6 5. Apply formula H to the curves p2 =-ar, ap=r2 M+ P ~~~aM 6. Apply formula I to the reciprocal spiral au=6. 285. 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 x; 0 the origin; x, y the co-ordinates of P; X, -Y those of Q; if, the 280 280 ~~CUR VATURE. angle the tangent makes with the axis of x. Draw PN, QM perpendiculars upon the x-axis and PR a perpendi Fig. 55. eular upon QM. Then ~=OM= ON-liP = ON -QP sinqxr =x-p sillr and Ki=MQ=NP~RQ = Y + p COS05 -Now tan \t =dy. dx'~ dy therefore sin i*t = dx V~()2 and Cos qfr = _ 1 + y2 d Y Also CURVTATURE. 281 dy{1~dy/)2 d \ ~Hence d2.(a)~ dX2 INVOLIUTES AND EVOLUTES. 286. 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. To find the locus of the centres of curvature of the parabola X2 Here dy_ d2?' 1 dx 2a' & 2 \'/ +(dy) Hence dx- d2y 4a2 dX2 8?+ 2~ a+, dX2 whence (2ai - 2aC3=27x6 27a,22 64a3 4 Hence the equation of the evolute is 4(y - 2a)3 = 27aX2. 282 CUR VA TURE. 287. Evolute touched by the Normals. Let P1, PP, P3 be contiguous points on a given curve, and let the normals at P1, P, and at P2, P3 intersect at Q1, Q2 respectively. Then in the limit when P2, P3 move along the curve to ultimate coincidence with P1 the limiting positions of Q~, Q2 are the centres of curvature corresponding to the points P,, P2 of the curve. Now Q1 and Q, both lie on the normal at P2, and therefore it P2 Q2 QI Fig. 56. is clear that the normal is a tangent to the locus of such points as Q,, Q2, 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 (Art. 313) 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. 288. There is but one Evolute, but an infinite number of Involutes. Let ABCD... be the original curve on which the successive points A, B, C, D,... are indefinitely close to each other. Let a, b, c,... 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, abcd... is its evolute. And CUR VA TURE. 283 regarding abed... as the original curve, ABCD... is an involute. 0 Fig. 57. 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 of parallel curves. 289. Involutes traced out by the several points of a string unwound from a curve. 284 CUR VA TURE. Since a is the centre of the circle of curvature for the point A (Fig. 57), aA = aB = bB + elementary arc ab (Art. 36). Hence aA - bB = arc ab. Similarly bB-cC= arc be, c -dD = arc cd, etc., fF-gG= arc fg. Hence by addition aA-gG = arcab+arc be+... +arc fg = ar c g. Hence the difference between the radii of curvature at two points of a curve is equal to the length of the corresponding arc of the evolute. Also, if the evolute abc... 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 abed..., one of them coinciding with the original curve ABC.... It is from this property that the names involute and evolute are derived. 290. 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 0 (Fig. 57) be the centre of curvature for the point a of the evolute. The angle +r' between the normals at a, b = the angle between the tangents at a, b = the angle between the tangents at A, B to the original curve = s~. CUR VA TURE. 285 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 c, and p that of the original curve at A, we have, rejecting infinitesimals of order higher than the first, Ss'= are ab = 6p, and therefore p'= 8s' LSp dp d2s p =Lt,-fi =Lti-dq4-d 2 ' 47 ( difr d, 2' s being the arc of the original curve measured from some fixed point up to A, and 4 the angle which the tangent at A makes with some fixed straight line. INTRINSIC EQUATION. 291. 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 (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 =, =f'(x)........................ (1) ~and ds +.2 and =Jl t )j2..[................ (2) If s be determined by integration from (2) and x elimin 286 CUR VA TURE. ated between the result and equation (1), the required relation between s and r will be obtained. Ex. 1. Intrinsic equation of a circle. If g: 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 A A POA = PTX=, and therefore s=ah. 0 A T X Fig. 58. Ex. 2. In the case of the catenary whose equation is y=c cosh' the intrinsic equation is s= c tan V. For tan V d = sinh dx c and I =V /1 + sinh2 = cosh x, and therefore s =c sinh x c the constant of integration being chosen so that x and s vanish together, whence s=c tan '. EXAMPLES. 1. Show that the cycloid x = a( + sin 0)' y=a(l -cos) has for its intrinsic equation s= 4a sin 'p. CUR VATUR E. 287 2. Show that the epi- or hypo-cycloid given by ==(a+b) cos - b cos a+b0 y=(a+b) sin 0-b sin a+b6 has an intrinsic equation of the form s=A sin Be. 292. Intrinsic Equation of the Evolute. Let s=/(-f) 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 0 and P A. 0 T Fig. 59. be the points on the original curve corresponding to the points A, Q on the evolute, p,, p the radii of curvature at 0 and P; V' the angle the tangent QP makes with OA produced, and 4 the angle the tangent PT makes with the tangent at O. Then i '= r, and s'= p- p, = -Po or s' =f'(A ')- o, the intrinsic equation of the evolute. 293. Intrinsic Equation of an Involute. With the same figure, if the curve AQ be given by the equation s' =f('), 288 CUR IrA TURE. ds we have p=s'+ p, p= -7, and = f', whence s =/f{f( ') + po} fj'. 294. Evolutes of Cycloids or Epi- and Hypo-Cycloids. If we apply the result of Art. 292 to the intrinsic equation s=A sin B, we get for the equation of the evolute s'= AB cos B/'- po, or, dropping the dashes, s = AB cos Bz, if s be supposed measured from the point where =2tB' 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.," Vol. VIII, p. 689, and Vol. IX., p. 150.] CONTACT. 295. First, consider the point P at which two curves cut. It is clear that in general each has its own tangent at that point, and that if the curves be of the inth and nth degrees respectively, they will cut in mn -1 other points real or imaginary. Fig. 60. Next, suppose one of these other points (say Q) to CUR VA TURE. 289 move along one of the curves up to coincidence with P. The curves now cut in two ultimately coincident points at P, and therefore have a common tangent. There is then said to be contact of the first order. It will be observed that at such a point the curves do not on the whole cross each other. Again, suppose another of the mnn points of intersection (viz., R) to follow Q along one of the curves to coincidence with P. There are now three contiguous points on each curve common, and therefore the curves Fig. 61. Fig. 61. have two contiguous tangents common, namely, the ultimate position of the chord PQ and the ultimate position of the chord QR. Contact of this kind is said to be of the second order, and the curves on the whole cross each other. Finally, if other points of intersection follow Q and R up to P, so that ultimately k points of intersection coincide at P, there will be l -1 contiguous common tangents at P, and the contact is said to be of the (c- l)th order. And if k be odd and the contact of an even order the curves will cross, but if k be even and the contact therefore of an odd order they will not cross. T 290 CUR VA TURE. 296. Closest Degree of Contact of the Conic Sections with a Curve. The simplest curve which can be drawn so as to pass through two given points is a straight line, do. three do. circle, do. four do. parabola, do. five do. conic. Hence, if the points be contiguous and ultimately coincident points on a given curve, we can have respectively the Straight Line of Closest Contact (or tangent), having contact of the first, order and cutting the curve in two ultimately coincident points, and therefore not in general crossing its curve; the Circle of Closest Contact, having contact of the second order and cutting the curve in three ultimately coincident points, and therefore in general crossing its curve (this is the circle already investigated as the circle of curvature); the Pacrabola of Closest Contact, having contact of the third order and cutting the curve in four ultimately coincident points, and therefore in general not crossing; and the Conic of Closest Contact, having contact of the fourth order and cutting the curve in five ultimately coincident points, and therefore in general crossing. We say in general: for take for instance the " circle of closest contact" at a given point on a conic section. Now, a circle and a conic section intersect in four points CUR VA TURE. 291 real or imaginary, and since in our case three of these are real and coincident, the circle of closest contact cuts the curve again in some one real fourth point. But it may happen as in the case in which the three ultimately coincident points are at an end of one of the axes of the conic that the fourth point is coincident with the other three, in which case the circle of closest contact has a contact of higher order than usual, viz., of the third order, cutting the curve in four ultimately coincident points, and therefore on the whole not crossing the curve. The student should draw for himself figures of the circle of closest contact at various points of a conic section, remembering from Geometrical Conics that the common chord of the circle and conic, and the tangent at the point of contact make equal angles with either axis. The conic which has the closest possible contact is said to osculate its curve at the point of contact and is called the osculating conic. Thus the circle of curvature is called the osculating cir'cle, the parabola of closest contact is called the osculating parabola, and so on. 297. Analytical Conditions for Contact of a given order. We may treat this subject analytically as follows. Let y= (x) yJ= (X) f be the equations of two curves which cut at the point P(x, y). Consider the values of the respective ordinates at the points P1, P2 whose common abscissa is x+h. Let MIN=h, Then NP1 = q(x+h ), NP, = (x + h) 292 CUR VA TURE. and P2 P1 = NPI - NAP2 = (f(x + A) - r(x + h) = [p(x) - ()(x)] + h[p'(x) - V/r(x)] h 2 + - (x )- "()] +.... o M N X Fig. 62. If the expression for P2P] be equated to zero, the roots of the resulting equation for h will determine the points at which the curves cut. If +((x) = -(x), the equation has one root zero and the curves cut at P. If also +'(x)= r'(x) for the same value of x, the equation has two roots zero and the curves cut in two contiguous points at P, and therefore have a common tangent. The contact is now of the first order. If also +"(x) = +"(x) for the same value of x, the equation for h has three roots zero and the curves cut in three ultimately coincident points at P. There are now two contiguous tangents common, and the contact is said to be of the second order; and so on. Similarly for curves given by their polar equations, if r =f(O), r= )(0) be the two equations, there will be n+ 1 equations to be satisfied for the same value of 0 in order CUR VA T UBE. 29 293 that for that value there may he contact of the iith order, viz., f(O) = 0(O), f'(O) = '(O), f"(O) = 5'(O),..., f"(O) p~(O). 298. Osculating Circle. The circle of curvature may now be investigated as the circle which has contact, of the second order with a given curve at a given point. Suppose y=f(x)..................(1) to be the equation of the curve. Let x ~ y) P be the equation of the circle of curvature. By differentiating (2).we have X-x+(-Y)=.....................(3) and differentiating again dy2 yd2y d- + (y -) 2=......................(4)) dy d 2y, Now the x, y, o, d of equations (2), (3), (4) refer to the circle. But, since there is to be contact of the second d1-? d2y order with the curve y =f(x) at the point (x, y), dx and have the -same value as when deduced fromt the eqmation to the emrve, i.e., we may write f'(x) for dy and f"(x) dx d2y 'for From equation (4) I+( l) 1~{ f'(X)}2 Y -Y d~~2 yPx dX2 294 CUR VA TURE. dyx + l\dx} ) f'(x)[1 + {f'(x)}2] whence x -:= - d2 f and by squaring and adding ~1 + {f = l + (x)}2] __ -d2y = ~ f"(x ) dx2 such a sign being given to the radical as will make p positive, i.e., if d2 be positive we must choose the + dX2d2y sign for the numerator, and if -y be negative we must dx2 choose the - sign. The values of x: and y are the same as those found geometrically in Art. 285, viz., dy dx\ y\ 2 x=- d, 2 dx2 8=8+ My \2 dx2 299. Conic having Third Order Contact at a given point. The locus of the centres of all conics having third order contact with a given curve at a given point (i.e., cutting the curve in four ultimately coincident points) is a straight line which passes through the point of contact. CUR VA TURE. 295 Let P be a point on the curve and C the centre of one of the conics having third order contact with the given T Fig. 63. curve at 1'. Let CD be the semiconjugate to CP and CY a perpendicular on the tangent at P. Let CP=r, CD=r', CY=p, and let PC make an angle p with the normal at P. Then we have r2 + r'2 = a2 + b2, and and therefore and for a conic therefore pr, =ab, dclq- + r'dir' =O; CD3 r':3 P= ab =ab; (See Ex. 18, p. 301) dp 3r'2 dr:3' 3 rdr ds a- b ' ds ah ' ds _ 3r d ir sin __ p 'ds cos (' d9r for = cosCPT'= - sin 0, the arcs of the curve and of the conic being measured from the points 0 and 0' up to P, and therefore =-cos 0; ds=3 tan q, I d where dp on l o f the and tan b = I-d' where - is found for one of the conics. *3 d's1 daS I 296 C UR VA TURE. But since the conic and the curve have contact of the dr third order they have the same tangent, the same do, d21, d3r the same -ar, and the same d3 at the point of contact. dO2' dO3 They therefore also have the same p and the same d, for d2r dp d31S p depends on d2- and c- on d3. Hence the value of q found above is the same for all the conics, 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-1Q d) where 4, is founrd from the curve. 300. Osculating Conic. We can now pick out the particular conic which has fourth order contact with the given curve at the given point. Let 0 be the centre of curvature of the point considered and C the required centre of the conic of closest contact. Let P1 be a point on the curve adjacent to the given point P. Join CP, CP1 and draw P1N at right angles to CP. Let OPC=o, OP1C= = + 80, PC= R. Then PEP = POE+, and also = P1 CE+ 0 + 8q, A A whence POE = P1CE + +. Also, neglecting infinitesimals of higher order than the first, PPi= s, CUR VA TUR E. 297 POE 8 P ^ P N 6s cos o and p (,P= -A 3 C = R IC R c 0 P Pt Fig. 64. 8s 8s cos _ Hence + ss c p H or, proceeding to the limit, cos _ 1 dc R p ds' where = tan-1 dp And since the contact is of the fourth order, -. is the same for the curve as for the conic, and may therefore be supposed deiived from the equation of the curve. These equations determine the position of C. 301. 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 298 CUR VA TURE. 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. s2 S3 Assume.x = ( + als + a2 2+ -a33 +-..., s2 S3 y = b+,s+ b2! +!+..., the letters a, a,..., b, b6... denoting constants whose values are to be determined, and s being the length of arc. Then, when s= O, x and y both vanish, and therefore a=b= 0. Again, by Maclaurin's Theorem "I=fct-(6/=("8os 0 '[the suffix zero denoting b =( = (sin r), = the values at the origin] C XY, \ / (sin o 2 0(, —(cosV 1C0 J d+-_(sn _ O 2 \ds2/ ds o p j (= d3X _ (cos sin cZp b3-(O-(- 29 -2 c/ -p2 2 2 C 7y ( ik sin~ cosxclp\ 1d pp etc., s3 whence x = -- +.. s2 S3 dp Y 2p 6 p2 ds CUR VA TURE. 299 EXAMPLES. 1. Prove that in the case of the equiangular spiral whose intrinsic equation is s = a(e" - 1 ), p = mnaeV. 2. For the tractrix s = c log sec f prove that p = c tan b. 3. Show that in the curve y = x + 3x2 - x3 the radius of curvature at the origin = 4714..., and that at the point (1, 3) it is infinite. 4. Show that in the curve y2 - 3xy - 4x2 + x3 -+ 4y + y5 = 0 the radii of curvature at the origin are 85 85 17 and 5 /2. 5. Show that the radii of curvature of the curve y2 2a+X ca-x for the origin = + a /2, and for the point (- a, O) =. 6. Show that the radii of curvature at the origin for the curve X3 + y3 = 3axy 3a are each.. 7. Prove that the chord of curvature parallel to the axis of y for the curve x y = a log sec - a is of constant length. 8. Prove that for the curve s = m(sec`3 - 1), p = 3m tan b sec3/, '1300 CUR VA T URE. and hence that 37mn d d2y = i. dx dX2 Also, that this differential equation is satisfied by the semicubical parabola 27mg2 = 8x3. 9. Prove that for the curve sin~ s =a log eot V\- - + a 2OS27 p = 2a sec3b; and hence that d2y 1 dx2 2 c& and that this differential equation is satisfied by the parabola 2= 4ay. 10. Show that for the curve in which s = aec cp = s(s2 - C2)2.I _ 11. Show that the curve for which s=.,/c8y (the cycloid) has for its intrinsic equation s = 4a sin 1b. Hence prove p 4aV1- 4L. 12. Prove that the curve for which y2 = c2 ~ s2 (the catenary) has for its intrinsic equation s=c tan b. Hence prove p = - the part of the normal intercepted C between the curve and the x-axis. 13. For the parabola y2 = 4ax, prove x = 2a + 3x, Y= -2 12 — a2 9SP SIP being the focal distance of the point of the parabola whose co-ordinates are (x, y). CUR VA TURE. 3()1 14. Show that tile circles of curvature of the parabola y2 = 4axfor the ends of the latus rectum have for their equations x2 + y2 10ax +_ 4ay - 3a2 = 0, and that they cut the curve again in the points (9a, 6Ga). 15. Show that the evolute of the parabola y2=4ax is the semicubical parabola 27ay2= 4(x- 2a)3, and that the length of the evolute from the cusp to the point where it meets the parabola = 2a(3 /3 - 1). x2 2 16. For the ellipse -+ =1, a2 b- a2-b2 9 pro ve x= -- x _ b _ 2 3 - 0y —^ Y Hence show that the equation of the evolute is (ax)i + (by) = (a - 2),and prove that the whole length of the evolute \b a} 17. Show that in a parabola the radius of curvature is twice. the part of the normal intercepted between the curve and the directrix. 18. Prove that in an ellipse, centre C, the radius of curvature at any point P is given by CD3 a2b2 (rr'}) ab = ab where a, b are the semi-axes, r, r' are the focal distances of P, p the perpendicular from the centre on the tangent at-P, and. CD the semi-diameter conjugate to CP. 19. Show that in any conic (l(normal)3 (semi-latus-rectum)2 302 C UR VA TURE 20. Apply the polar formula for radius of curvature to show that the radius of the circle r = acos0 is - 2 21. Show that for the cardioide r = a(l + cos 0) 4a 0. p=-3 cos; i.e., cc ^/r. Also deduce the same result from the pedal equation of the curve, viz., p 2-a. 22. Show that at the points in which the Archimedean spiral r= aO intersects the reciprocal spiral rO = a their curvatures are in the ratio 3: 1. 23. For the equiangular spiral r = ae"o prove that the centre of curvature is at the point where the perpendicular to the radius vector through the pole intersects the normal. 24. Prove that for the curve r a sec 2, r4 p= - 33 25. For any curve prove the formula where tan 4 = d. dr Deduce the ordinary formula in terms of r and 0. 26. Show that the chord of curvature through the pole for the curve p =f(r) is given by chord = 2p- 2f () 27. Show that the chord of curvature through the pole of the crdioie ( 4 cos 0) is cardioide r = a(1 + cos 0) is -r. CUR VA TURE. 303 28. Show that the chord of curvature through the pole of the equiangular spiral r = aen is 2r. 29. Show that the chord of curvature through the pole of the 2r curve m = amcos mO is -_' n + 1 Examine the cases when n 2 - 2, -1, 1-,, 2. 30. Show that the radius of curvature of the curve r= asinnO..... nca at the origin is n 31. Show that for the curve 0V ~ ye _=' we may write p in the form -k (cos- -n 1 where x = k cosm. i - 1 1 —2n 1 ---2m cos m q sin~ 'm Examine the cases rn = 2,, 1. 32. For the rectangular hyperbola xy = k2, r3 prove that p= 2k' r being the central radius vector of the point considered. 33. For the curve r'n = ac" cos mn0, prove that p = = — + i-.. (m + 41." Examine the particular cases of a rectangular hyperbola, leinniscate, parabola, cardioide, straight line, circle. 34. Show that the co-ordinates of the centre of curvature of any curve may be written d2r2 d2r ddy,2 1d2q 1 _\ dy2 dX2 30M CUR VA TURE. 35. If A be the area of the portion of a curve included between the curve, two radii of curvature, and the evolute, prove dA p = 2. as 36. Show that the evolute of an equiangular spiral is an equal equiangular spiral. 37. 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'2 = 2+r2 2pp,...................... (a) '2 =r2 p2...........................() 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', r' between this equation and the equations (a), (3). 38. Show that the curve whose equation is p22 r2 _ a2 is an involute of a circle, and that its intrinsic equation is 2 39. Show that the evolute of the epi- or hypo-cycloid denoted by p2= Ar2+ B is another epi- or hypo-cycloid denoted by 2= Aq,.2 + B( _i AJ 40. Show that the pedal equation of the evolute of the curve rn = a sin m n is obtained by eliminating r between r, 2= m + (2 - 1)r2" (27n 1)2m and P 2='2 M -' 41. Show that the intrinsic equation of the evolute of a parabola is s = 2a(sec3f - 1). CUR VA TURE. 305 42. If x, y be the co-ordinates of a point P of a curve OP passing through the origin 0, then the radius of curvature at 0 = Lt:.2 + y2 x sin a - y 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 4+ y2 = 2a(x + y) at the origin is 2a /2. 43. Show that the curvature at any point of the pedal of an pfr4 + CL2^2) epi- or hypo-cycloid is p(-+ a2p2) where a is the radius of the fixed circle and r and p refer to the pedal curve. [SIDNEY COLL., CAMB.] 44. 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 k2r3 origin is where k2 is the constant of reciprocation. Hence show that the reciprocal of a circle is a conic with the origin as focus. 45. 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 k2p with regard to the origin is k-P 2pp - r2' k2 being the constant of inversion. 46. Show that the parabola whose axis is parallel to the axis of y, and which has the closest possible contact with the curve al 7z-1y = jbn at the point (a, a), has for its equation n(n - 1)2 = 2ay + 2n(n - 2)ax - (n - l)(n - 2)a2. U 306 CUR VA TURE. 47. Show that the locus of the centre of the rectangular hyperbola, having contact of the third order with the conic Ax2+By2=, has for its equation x2 + y2 =(+ ) /Ax2 + By2. 48. Show that the locus of the centres of the rectangular hyperbolae, having contact of the third order with the parabola y2= 4ax, is the equal parabola y2 + 4a(x + 2a) = 0. 49. If the equation to a curve passing through the origin be u, + UI, ++ u3 = 0, where un is a homogeneous function of x, y of n dimensions, show that the general equation to all conics having the same curvature at the origin as the given curve is u12 + + (lx + my)u- = 0. Thence find the circle of curvature. 50. Show that the circle of curvature at the origin for the curve x + y = ax2 + by2 + cx3 is (a + b)(x2 + y2) = 2x + 2y. 51. 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. 52. 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.] 53. 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.] CURVATURE. 307 54. 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 curvature of the curve locus of its extremity is given ~~~~~by, (p2 + C2).by, o2~, p2+ C2- C dp db where p and ~ refer to the corresponding point of the original curve. 55. 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 (p2 + 2 - 2p cos a p2 + c2 - 2pc cos a - csin ad' de where p and ~ refer to the corresponding point of the original curve. 56. If on each tangent to a given curve a length be measured from the point of contact equal to the radius of curvature there, the centre of curvature at any point on the locus of the extremity of the measured length is at the centre of curvature of the corresponding point of the original curve. 57. If. accented letters refer to a point on a curve and unaccented letters to the corresponding point on the involute, prove x = x + P i_ y' s s s Show how, by means of these equations and s' p — l, 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. 308 CUR VAd TURE. 58. Show that the equation of the involute of the catenary y = c cosh C which begins at the point where x = 0, y = c, is the Tractrix x = c cosh-1 C _ /2 - y2. y 59. If a straight line be drawn through the pole perpendicular to the radius vector of a point on the equiangular spiral r = aeOcta 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 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 equiangular spiral. 60. 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 pp 2 \ +ds) [MATH. TRIPOS.] 61. If the tangent and normal to a curve at any point be taken as the axes of x and y respectively, and if s be the distance, measured along the arc, of a point very near to the origin, show that the Cartesian co-ordinates of that point are approximately 6p2 8p3 dsi"' s2 s 4 dp_ 8(1 - +dp + 2.. 2p 6p2 ds 24p ds '... the values of p, dp and d2p being those at the origin. CISd) 2 CUR VA TURE. 309 62. If a line be drawn parallel to the common tangent of a curve and its circle of curvature, and so near to it as to intercept on the curve a small arc of length s measured from the point of contact, of the first order of small quantities, show that the distance between the two points on the same side of the common normal in which the line cuts the curve and the circle of curvature is sp -, i.e., is of the second order of small 6p ds quantities, the values of p and p being those at the point of cis contact; and again, if a line be drawn parallel to the common normal, the distance between the points of intersection with the s3 dp curve and the circle is -- -P and is of the third order of small 6p2 ds quantities. 63. Prove that the circle,/2(2 ++ +2) = 3(x +y) has contact of the third order with the conic 5x2 - 6y + 5y2= 8. 64. Show that for the portion of the curve a5y2 = 7 very near the origin the shape of the evolute is approximately given by 1225x3y2= 16a5. 65. A line is drawn through the origin meeting the cardioide r = (1 - cos 0) 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. 66. If PQ be an arc not containing a point of maximum or minimum curvature, the circles of curvature at P, Q will lie one entirely within the other. 310 C UR VA TURE. 67. If in the plane curve ((x, y) = 0, we have at any point 0, ~=0, ~ 0, prove, that the curvature of one of the ax ay ax 2 branches of the curve which passes through that point is 1 D~ I a2,\.-I 3 Dx3,~Jf:xsy) [CAIUS COLL., CAMB..68. If 0 be the angle between the normal at any point P of a plane curve ~(x, y) = 0, and the line drawn from P to the centre of the chord parallel and indefinitely near to the tangent,at P, prove that cos 0= bp2 ahqq \,./p2 ~q q2 (62 + 1p2 )192 2 (a + b)hp q + (62 +h2)p2} where p=, q= h= and b a~x ' ay) ~" ax xay - Dy2' 69. A curve is such that any two corresponding points of its evolute and an involute are at a constant distance. Prove that the line joining the two points is also constant in direction. CHAPTER XI. ENVELOPES. 302. Families of Curves. If in the equation j(x, y, c)=-0 we give any arbitrary numerical values to the constant c, we obtain a number of equations representing a certain family of curves; and any member of the family may be specified by the particular value assigned to the constant c. The quantity c, which is constant for the same curve but different for different curves, is called the parameter of the family. 303. Envelope. Definition. Let all the members of the family of curves O(x, y, c)= 0 be drawn which correspond to a system of infinitesimally close values of the parameter, supposed arranged in order of magnitude. We shall designate as consecutive curves any two curves which correspond to two consecutive values of c from the list. Then the locus of the ultimate points of intersection of consecutive members of this family of curves is called the ENVELOPE of the family. 304. The Envelope touches each of the Intersecting Members of the Family. It is easy to show that the envelope touches every 312 E1V VELOPES. curve of the system. For, let A, B, C represent three consecutive members of the family. Let P be the point of intersection of A and B, and Q that of B and C. C B/Q.A Fig. 65. Now, by definition, P and Q are points on the envelope. Thus the curve B and the envelope have two contiguous points common, and therefore have ultimately a common tangent, and therefore touch each other. Similarly, the envelope may be shown to touch any other curve of the system. 305. To find the Equation of an Envelope. To find the equation of the envelope of the family of curves of which (x, y, c) =0 is the typical equation., Let ~(x, y, c)= 0, } (A) (x, y, c + c) = 0o, be two consecutive members of the family. Expanding the latter we have 0(x, y, c) + ~c (x, y, c)+...= 0. Hence in the limit, when 6c is infinitesimally small, we obtain -O(X Y, c) = O as the equation of a curve passing through the ultimate point of intersection of the curves (A). If we eliminate c between the equations p(xy, c)= = and a-O(x, y,) = C ENV TTELOPES. 313 we obtain the locus of that point of intersection for all values of the parameter c. That is, we obtain the equations of the envelope of the family of curves of which V(x, y, c) 0 is the type. The polar curves 0(r, 0, c) may be treated in the same manner. Ex. Find the envelope of the syslem of strigyht lines of which,.,,,~ +a, the typ~e, c beinq the para-meter ainzd (a) constant for all lines of the system. Here ~ (PX, y, C)=gy -Cx-% O, C2 Dc a therefore C whence y +i~ 2ax or y'2 = 4ax 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 + a touches the parabola j 2 = 4cx. 306. The Envelope ofAX2 + 2BX + C= () is B2 = A C. If A, B, C he any functions of x and y, and the equation of any curve be AX2+ 2BX~ C= 0, X being an arbitrary parameter, the envelope of all such curves s B2- =. C. For we have to eliminate X between AX2 + 2~BX + C= 0 and 2AX-2B= 0, and the result is clearly B2 =A C. 314 EIV VNELOPES. The result of the example of Art. 305 may be obtained in this way; for the equation y= mx + a may be written m2x - my + a = 0, and therefore the envelope is y2=4ax. 307. Another Mode of Establishing the Rule. The equation A2+ 2BX +C= 0 may be regarded as a quadratic equation to find the values of X for the two particular members of the family which pass through a given point (x, y). Now, if (x, y) be supposed to be a point on the envelope, these members will be coincident. Hence for such values of x, y the quadratic for X must have two equal roots, and the locus of such points is therefore B2= A C. The envelope of the system O(x, y, c) = 0 might be considered in a similar manner. And it is proved in Theory of Equations that if f(c)= 0 is a rational algebraic equation for c, the condition that it should have a pair of equal roots is obtained by eliminating c between the equations f(c)=0, f'(c) = o, a result agreeing with that of Art. 305. EXAMPLES. 1. Show that the envelope of the line x+y=1, where ab=c2. a a b constant, is 4xy = c2. 2. Show that the envelope of the line x + my + = 0, where the parameters 1, m are connected by the quadratic relation a12 + 2m + bm2 + 2g1 + 2fr + c = 0, is the conic A x2 + 2IIy + By2 + 2Gx + 2y + C= (, a, At, g A, B, C, F, G, H being minors of the determinant h, b,.,qf, cl EV VELOPES. 315. 308. Case of Two Parameters. Next, suppose the typical equation of the family of curves to involve two parameters a, / 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. 305, or, as is frequently better from considerations of symmetry, consider one of the parameters capable of independent variation and the other dependent tdpon it. We then proceed as follows. Let +(x, y, a, /)= 0.....................(1) be the typical equation of the curves whose envelope is to be investigated, and f(a, 3))=0.................... (2) the relation connecting a and /3. Then, supposing a the independent parameter, we have a 3 l0 d= 0...................(3) where jj d. (4) where f =+~........................(4) We thus have four equations and three quantities to, eliminate, viz., a, /3, The result of elimination is the Cda equation of the envelope. The parameters a, /3, 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. 309. Indeterminate Multipliers. The equations (3) and (4) may be written +da+~ad3=0 (Art. 139),. +f dd/ =30. Da ao 316 ENV VELOPES. The result of eliminating da, df3 between these equations is aa _Af aa a33 Call each of these ratios X. We then have a-_. - V_...... (5) =., ****** * (6) a^ ~.............. This quantity X is called an " Indeterminate Multiplier." It remains to eliminate a, /3, and X between equations (1), (2), (5), and (6). This method is peculiarly adapted to the case in which (x,, t, y, a,.s0(. y../-a = 0, and f(a, 3) f, (a, /3) - 2 = 0, where qp and fA are homogeneous in a and /3, and of the pth and qth degrees respectively, a1 and a being absolute constants. Multiply equation (5) by a and (6) by /3, and add. Then by Euler's Theorem pal = q\X, so that in such cases X is easily found. Ex. Find the envelope of $+ =1, where a and b are connected a b by the relation a2 b2=C2 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. Here xda + db=, 2 b2 ada +bdb=O, and therefore x a a2 ENVELOPES. 317 Y -Xb. b2 -Multiplying by a and b respectively, and adding, x +. =X(ab2), a b or 1 = Xc2. Hence a3 = C b3 = c2y, and since a2 +b2=c2 we have (c2x)3 + (c2y) = c2, or x3 + = c3. 310. Case of Three Parameters connected by Two Equations. Next, suppose the equation of a curve to contain three parameters connected by two equations. Let the equation of the curve be (x, y, a,, 7) = 0,...............(1) and let fi(a,, y)=O, =............(2) (a,, ) = 0..........(3) be the two connecting equations. Then we have -ia + -d + dy = 0,...............(4) ia rid 'ay dca + Df2dd++ =0dy =................ (6) The result of eliminating da, cd3, dy between these three equations is.. 3 |i ^2 D/f' Dy Da' /3' Dy .318 EN VELOPES. If a, /, y be eliminated between the four equations (1), (2), (3) 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" XA and X2 from the equations + l + =,............... (8) a5+X +Xf-0 o,...............(9) _^^- =~********.(19) 1 la 13 +2a3 +,X +X 0a,...........(10) and it is often advantageous to use these latter equations in place of (4), (5), (6), involving da, df3, dy. The result of eliminating a, 3, y, X, X2 between the six ~equations (1), (2), (3), (8), (9), (10) will then be the equation to the envelope. 311. 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-1 indeterminate multipliers between 2r equations. 312. Converse Problem. Given the Family and the Envelope to find the relation between the Parameters. Suppose we are given the equation of a curve,(x, a, a, )= 0............ (1) containing two parameters. Suppose also the envelope given, viz., F(x, y) 0........................ (2) Required uhe relation between a and 3. Eliminate y between (1) and (2). We obtain an equation of the form f(x, a /3) = 0,....................... (3) EN VEL OPES. 319 giving the abscissa of the point of contact of the curve with its envelope. Since the curve touches its envelope, equation (3) must also be true for a contiguous value of x, viz., x+dx (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(z, a, ) = o,................... (4) f(x+~x, a, b)=0.............. (5) The latter may be expanded in powers of dx, when it becomes f(x, a, b) + -dx+...0,.............. (6) and therefore in the limit 0........................ (7) If, then, x be eliminated between f(l, C, )= o, f(x, a, A) = 0, ax we obtain the relation sought. It will be observed that this is precisely the same process as finding the envelope of 0(x, y, a,/3) =o, considering a, f3 as the current co-ordinates and x, y as paramneters connected by the relation F(x,y)=O. Ex. Given that x-+y-=c3 is the envelope of -+Y=, find the a 0 necessary relation betuween a and b. -dx cy We have — d + c 0 x'~:- y O, dx+; therefore xxa b therefore ^= \a, -)9n 3EN VELOPES. y xbX Hence XX 6 and by addition 1 Xc This gives aC = c bx, b-c,Y anld by squaring and adding a2+b2 C2, the relation required. (See Ex., Art. 309.) 313. Evolutes considered as Envelopes. The evolute of a curve has been defined as the locus of the centre of curvature, and it has been shown (Art. 287) that the centre of curvature is the ultimate point of intersection of two consecutive normals. Hence the evolute is the enzvelope of the nOrmalS to a curve. It is from this point of view that the equation of the evolute of a given curve is in general most easily obtained. x2y Ex. To finzd the evolute of the ellipse - +_=1. The equation of the normal at the point whose eccentric angle is q is ax__ b- =a2-b2......... (1) Cos SillQ We have to find the envelope of this line for different values of the parameter p. Differentiating with regard to p, sinl 9 + bMCOs 95 CYLG ~~y os~in~ o,......:..................... 2 cos2~ sin~cp0(2 sinio Cos345 or + ~~=0. ax Hence sill b COs _ 1.(3) - 9%/b/ \~ax ~J(ax) + (by)l Substituting these values of sinp and cosp in equation (1) we obtain, after reduction, (ax)5+ (b (a)2= (a2 - U ENVELOPES. 321 314. Pedal Curves as Envelopes. It has already been pointed out (Art. 197) that if circles be described on radii vectores of a given curve as diameters they all touch the first positive pedal of the curve with regard to the origin. It is obvious, therefore, that the problem of finding the first positive pedal of a given curve is identical with that of finding the envelope of circles described on the radii vectores as diameters. Again, the first negative pedal is the envelope of a straight line drawn through any point of the curve and at right angles to the radius vector to the point. Ex. 1. Find the first positive pedal of the circle r= 2a cos 0 with regard to the origin. Let d, a be the polar co-ordinates of any point on the circle, then d = 2a cos a. Again, the equation of a circle on the radius vector d for diameter is r= dcos(0-a),................................. (1) or r= 2acos acos(- a)......................... (2) Here a is the parameter. Differentiating with regard to a, - sin a cos(0 - a) + cos a sin(0 - a)= 0, whence sin(0 - 2a) 0, 0 or a=-. (3) or a~ j................. (3) Substituting this value of a in equation (2) r = 2a coS2, or r=a(1 + cos 0), the equation of a cardioide. Ex. 2. Find the equation of the first negative pedal of the cardioide r=a(l +cos 0) with regard to the origin. Here we have to find the envelope of the line x cos a +y sin a = d, where d, a are the polar co-ordinates of any point on the cardioide; i.e., where d= a(1 +cos a). X 322 EN VELOPES. The equation of the line is therefore x cos a +y sin a = a(l + cos a), or (x-a) cos a +y sin a = a, a line which, from its form. is easily seen to be a tangent to (X - a) +y2= a2, or r= 2a cos 0, which is therefore its envelope. EXAMPLES. 1. Find the equation of the curve whose tangent is of the form y = mx + m4, m being independent of x and y. 2. Find the envelope of the curves a2cos 0 b2sin 0 c2 x y c for different values of 0. 3. Find the envelope of the family of trajectories y= tan 0 - 0 being the arbitrary parameter. 4. Find the envelopes of straight lines drawn at right angles to tangents 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. 5. 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. 6. 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. ENVELOPES. 323 7. Find the envelopes of the line - + -1 a b under the following conditions:(1) a+b k, (2) an+bn=k, (3) kcmb = m+n, k being a constant in each case. 8. P is a point which moves along a given straight line. PM, PN are perpendiculars on the co-ordinate axes supposed rectangular. Find the 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. Find the envelope of the line y=mx- 2am - am3 for different values of n; i.e., find the equation of the evolute of the parabola y2= 4ax. 11. Show that the envelope of the family of curves A 3 3B2 + 3C X + D = 0, where X is the arbitrary parameter and A, B, C, D are functions of x and y, is (BC - AD)2 = 4(BD- C2)(A C - B2). 12. Show that the envelope of the family of curves A cos"0 + B sin"0 = C, where 0 is the arbitrary parameter and A, B, C are functions of x and y, is 2 2 2 A2 —z- + B2 — = 02n. 13. Show that the envelope of the lines whose equations are x sec20 + y cosec20 = c is a parabola touching the axes of co-ordinates. 324 ENVELOPES. 14. Find the envelopes of the systems of coaxial ellipses whose semiaxes a and b are connected by the equations (1) a+b=k, (2),/a+ lJb= 1Jk, (3) am + b" = Pkm, (4) abk2, k being a constant in each case. 15. 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 contact are connected by the relations (1) a+ = k, (2) an +Pm= =km (3) a/=3k2 k being a constant in each case. 16. Show that the system of conics obtained by varying X in the equation + 2XX + = - X2 a2 ab b2 have for their envelope the parallelogram whose sides are x = +a, y= +b. 17. Find the envelope of the line which j6ins the feet of the two perpendiculars from any point of a circle upon a given pair of perpendicular diameters. 18. Show that the envelope of straight lines which join the extremities of a pair of conjugate diameters of an ellipse is a similar ellipse. 19. Show that if PM, PNV be perpendiculars from any point P of the curve y = nmx3 upon the axes the envelope of MN is 27y + 4mx3 0. 20. Find the envelope of circles described on the radii vectores of an ellipse drawn from the centre as diameters. ENVELOPES. 325 21. Show that the envelope of a circle whose centre lies on the parabola y2= 4ax and which passes through its vertex is 2ay2 + x(x2 + y2) = 0. 22. Show that the envelope of a circle whose centre lies on the parabola y2= 4ax and whose radius = the abscissa of the centre is made up of the tangent at the vertex and a circle with centre at the focus. 23. If a lamina rotate in its own plane about any fixed point in that plane, show that the directions of motion at any instant of any given curve of points in the lamina have for their envelope the first negative pedal of that curve with regard to the fixed point. Examine the particular cases of a straight line and a circle. 24. Two particles move along parallel straight lines, the one with uniform velocity and the other with the same initial velocity but with uniform acceleration. Show that the line joining them always touches a fixed hyperbola. 25. A series of circles is described having their centres on an equilateral hyperbola and passing through its centre. Show that the locus of their ultimate points of intersection is a lemniscate. 26. Prove that the equation of the normal to the curve + 2 2 x7 + ys = a3 may be written in the form y cos - - x sin < = a cos 24. IHence show that the evolute of the curve is (x + y)i + ( - y) = 2a+. 27. Show that the envelope of the lines x cos ma + y sin ma = ct(cos na)n, where a is the arbitrary parameter, is -n -- n qIm-n = C-2L COS — U 0. m - n 326 ENVEL OPES. 28. If 0 be the pole and P any point of the curve r = a cos mO, and if with 0 for pole and P for vertex a similar curve be described, the envelope of all such curves is r = aCcos. 29. If 0 be the pole and P any point of the curve m = a mcos m0, and if with 0 for pole and P for vertex a curve similar to r7n = an cos nO be described, the envelope of all such curves is m n 9nn mn rm+nz = aCmn COS 0. m+n 30. If 0 be the pole and Y the foot of the perpendicular from 0 on any tangent to the curve rm = - cos m0, and if with 0 for pole and Y for vertex a curve similar to r7 _-= a, cos nO be described, the envelope of all such curves is r" = a" cospO, where p= - mn m + n+mn 31. If a point on the circumference of a given circle be taken as pole, and circles be described on radii vectores of the given circle as diameters, the envelope of these circles is a cardioide. 32. Show that the envelope of all cardioides on radii vectores of the circle r = a cos 0 for axes, and having their cusps at the pole, is ' = at cos 30. 33. Show that the envelope of all cardioides described on radii vectores of the cardioide r = a( + cos 0) for axes, and having their cusps at the pole, is r=(2a) 0os rT = (2a): cos 4. ENVELOPES. 327 34. On radii vectores of r2" = a2n cos 2n0 as axes curves similar to it are described, the curves being all concentric. Show that the envelope of all these is re = c" cos no. 35. Prove that the pedal equation of the envelope of the line x cos 20 + y sin 20 = 2a cos 0 ~is~ 1p2 = (r - a2). 36. Prove that the pedal equation of the envelope of the line x cos mO + y sin mO = a cos nO is m2r2 = (m2 - n2)p2 + n2a2. 37. Two central radii vectores of a circle of radius a rotate from coincidence in a given initial position with uniform angular velocities o and a'. Show that the pedal equation of the envelope of a line joining their extremities is (c + o')2r2 = 4007, 2 + (c - o/)2a2. 38. The envelope of polars with respect to the circle x2 + y2 = 2ax of points which lie on the circle x2 + y2= 2bx is {(a- b)x+a b2=b2{( -a )2 +y2}. 39. 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. 40. 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. 41. Prove that the envelope of the circles obtained by varying the arbitrary parameter a in the equation c(y - a)2 + (cx - a2)2 = (a2 + c2)2 consists of a straight line and a circle. 328 ENVELOPES. 42. 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 parabola which goes through the extremities of the minor axis and whose latus rectum is equal to that of the ellipse. 43. Given the centre and directrices of an ellipse, show that the envelope of the normals at the ends of the latera recta is 27y4 + 256cx3 = 0. 44. 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 limacon. 45. Find the envelope of a parabola of which the directrix and one point are given. 46. Find the condition between c and b that the envelope of the line -+=1 a b may be the curve xPyq = kP+q. 47. S is a fixed point, and with any point P of a curve for centre and with radius PS + k a 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. 48. Rays emanate from a luminous point 0 and are reflected at a plane curve. OY is the perpendicular from 0 on the tangent at any point P, and OY is produced to a point Q, such that YQ = O Y. 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 ENVELOPES. 329 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.] 49. 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 - 4 rolling upon the circumference of a circle of radius. 50. Rays emanate from a point on the circumference of a reflecting circular arc. Show that the caustic after reflection is a cardioide. 51. Show that if rays emanate from the pole of an equiangular spiral and are reflected by the curve the caustic is a similar equiangular spiral. CHAPTER XII. CURVE TRACING. 315. 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. CURVE TRACING. 331 316. Order of Procedure. 1. A glance will suffice to detect symmetry in a curve. 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. If all the powers of both x and y which occur be even, the curve is symmetrical about both y2 V2 axes, as, for instance, in the case of the ellipse -2+ 2 = 1 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 = kc The origin is then said to be a centre of the curve. 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 asymptotes are there? First find those parallel to the co-ordinate axes; next, the oblique ones (Art. 210). These results point out in what directions the curve extends to infinity. Find also on which side of each asymptote the curve lies (Art. 232). 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. 254), 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. 320. 332 CUR VE TRACING. 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. 258). 6. Find what other multiple points the curve has (Art. 257), and ascertain the position and character of each. dy 7. Find y; and for what points it vanishes or becomes dx infinite. These results will indicate the points at which the tangent is parallel or perpendicular to the axis of x. The 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 nth degree in n points real or imaginary, and imaginary intersections occur in pairs. 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 x 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 simplified upon reduction to the polar form. This is especially the case when the origin is a multiple point on the curve. 317. 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 trace to note down the result of each inves CURVE TRACING. 333 tigation 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. 318. We add a few examples to illustrate the points enumerated. Fig. 66. I. To trace the curve y=(x- 1)(x- 2)( - 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=x(x2-1), showing symmetry in opposite quadrants when referred to the new 334 C URYVE TRACING. axes, and that the tangent at the new origin is inclined at an angle -135' to the axis of x. (j) Recurring to the original equation, if Y=O, x==1,2, or 3; If X=O, y= - 6 If X= CI) Y=a0; If X=-% Y=-. When x is > 3 y is positive, < 3 hut > 2 y is negative, x < 2 hut > 1 y is positive, X < 1 y is negative. (y) The curve does not go through the origin, and, although -extending to infinity, it has no rectilineal asymptote. (3) Since y =X3-6X2+1lx-6 we have 3y3V2 12x+llI d~x -which vanishes when x 2+ 1 (e) Also -12-Y=, x- 2), which shows that there is a point of indX2 4lexion at the point where x =2. The shape of the curve is therefore that shown in Fig. 66. Y 0 B-A (a,o 0, 0) Fig. 67. IL. To trace the curve a)2 x-b3 aTa CASE 1. Suppose a > b (Fig 67). -- CURVE TRACING. 335 (a) The curve is symmetrical with regard to the axis of x. (3) While x < b, y is imaginary, and y is real for all values of x from b to w, and the curve meets the axis of x when x= a and when x= b. (y) dy=0 when x =a, and = oo when x=b, so that the curve dX touches the axis x at the point (a, 0), and cuts it at right angles at (b, 0). (a) There is no asymptote; but, when x= oc, y and fy are both 7x co in the limit, the curve ultimately taking the shape of 5 4-2T yY= O aA (a,o) --- Fig. 68. CASE 2. Next consider a < b (Fig. 68). (a') There is in this case also symmetry about the axis of x. (P') The equation to the curve is satisfied by the point (a, 0), but by no other point in its vicinity, for if x be < b, y is imaginary except when x= a. The point (a, 0) is therefore a conjugate point. (y') Moreover d-= c when x=b, and the curve cuts the axis of dv x at right angles at this point. (d') Also, when x= d-= =O; so the curve in departing from dx (b, 0) (the point B in Fig. 68) must bend towards the positive direc 336 CURVE TRACING. tion of the axis of x, and, finally, -y again becomes infinite, showing that there must be a point of inflexion at some point C between B and co. Its exact position is of course given by the equation d2y dx2 The shapes of the curves in the two cases are given in Figs. 67 and 68 respectively. EXAMPLES. 1. Trace the curve y=x2(x-1), showing that its tangent is parallel to the axis of x at the origin and at the point x=-. 2. Trace y=X3, and show that there is a point of inflexion at the origin. This curve is called the cubical parabola. 3. Trace the curve y2=X3 showing that there is a cusp of the first kind at the origin. This curve is called a semicubical parabola. 4. Trace the curve ay2 =(x - a)(x - b)(x- c), where a, b, c are in descending order of magnitude, and examine the cases (1) a b. (2) b=c. (3) a=b=c. III. To trace the curve x3 + ax2 + a3 Y x2 _ a a being positive. a. There is no symmetry about either axis and the curve does not pass through the origin. p. The curve cuts the axis of y at the point (0, - a) and the axis of x at the point given by the real root of x3 + a2 + a3 = 0. (It is clear that two roots of this equation are imaginary, for the sum of the squares of the reciprocals of its roots is negative.) Also, the real root is obviously negative and numerically greater than a. 7. When x is > a, y is positive. When x lies between a and -a, y is negative. CURVE TRA CING. 337 When x is < - a, y is positive until x passes the negative root above referred to, and then is negative afterwards. 8. The asymptotes parallel to the axes are x= +a. To find the oblique asymptote a a3 =xl+a+ I+ + XY ---- X + - 1-2 a2 or y=x +a +.... x Fig. 69. Hence y=x-+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 curve lies above the oblique asymptote. If x be negative, the curve lies below it. Y 338 CURVE TRACING. dy x(3 - 3a2x -4a3) dx~ (-2_ a2)2 which gives d-=0, when x=0 or when x3-3a2x-4a3=0, which clearly has a positive root lying between x=2a and x=3a, and which can be shown to have only this one real root. Also, dy = dx only when x= ~a. p. A point of inflexion lies between x=- 5a and x= - 6a (Ex. 26, Chap. X.). The shape is therefore that given in Fig. 69. IV. To trace the curve y2 +2x3y +X70. a. The curve is not symmetrical about either axis and there are no asymptotes. g. The curve passes through the origin, but cuts neither axis again. 7. There is a cusp at the origin, the equation of the tangent being y=0. Y 0, X Fig. 70. Proceeding according to Art. 258 the quadratic for P is P2+2Px3+ 7-O, an equation whose roots are real if x be very small, positive or negative; for the criterion for real roots is that X6 - x7 should be > 0. CURVE TRACING. 339 This condition is fulfilled until x is > 1, when P or y becomes imaginary. Moreover, the product of the roots=x7 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, while on the negative side it is of the first species. The point is therefore a point of oscul-inflexion (Fig. 47). 8. y2== -3 x+X3 x - x, 48 so that dy= co if x= 1. Also, one value of dq is zerowhen x=. dx dx 49 The shape of the curve is now readily seen to be that shown in Fig. 70. 319. 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. 0, x Fig. 71. V. To trace (x2 +2 - 3ax)2 =4ax?(2a - x). Here y2 = 2ax - x2 + 2 d /ax dJ2ax - x2 + aa =(2ax- x2+ \/7a)2; therefore y= +- /2ax - x' 2 + x ax, or -:7=9 ~Yl 32, 340 CURVE TRACING. where y, and y2 are corresponding ordinates of the circle x2 + y2= 2ax and of the parabola y2= 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+ l)2=(1 -x)5, 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 r=a sec +~a cos 0, the radius vector being the sum or difference of the radii vectores of a straight line and a circle. 320. Newton'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 AxPyq, Bxrys 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) (r, s), which we shall call P and R respectively. Then, since xPyq and x'ys are of the same order of magnitude, xP- and ys-q are also of the same order, s-q and therefore the order of x is that of yp-r CURVE TRACING. 341 Now S — = tan 0, where 0 is the angle which the line PR makes with OX. So that the order of x is that of y-tan, and therefore the order of the term AxPyq is that of yq - tan 0 Now q-p tan 0 = the intercept OA made Yl 117 I R _R B __ _.___ A -_ i 1 - I _____ 0 T 0 X Fig. 72, by the line PR upon OY, so that the order of the terms AxPyq and BxryS is that of yOA] and is measured by the intercept OA. Consider next any other term C'x'ny in the equation. Let its graphical point (in, n) be denoted by M in the figure. Then the order of this term is that of yn - m tan 0 or yOB the line MB being drawn parallel to RP, cutting off the intercept 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. 342 CUR VE TRA CING. 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 X2y3+ 2Xy + 3X4y + 2y2 + y4 = 0, the points A, B, C, D, E represent the 1st, 2nd, etc. terms respectY A B C 0 X Fig. 73. ively, and a glance at the diagram will show that the second and third and the second and fifth j are groups which may be taken together in tracing near the origin, whilst the first and third and the first and fifth are groups which may be taken together in approximating to the form of the curve at an infinite distance from the origin. 321. The above method is a modification of the one adopted in such cases by Newton, and is known as Newton's Parallelogram. A further slight variation on the same method is due to De Gua, and is known as De CURVE TRA CING. 343 Gua's "Analytical Triangle." [De Gua's "Usage de l'Analyse de Descartes," Paris, 1740.] VI. To trace x +5/ - 5a2x,2y =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, whilst at a great distance from the origin the first and y x Fig. 74. second may be taken together. This indicates that at the origin the curve assumes the parabolic forms y2= +-a I/5x, x3 = 5a2y, and that at infinity it approximates to the straight line x+y=-0, which is obviously the only asymptote. 344 CUR VE TRACING. p. Moreover, the equation may be written y-x( 1-5a2 5? Xa2y a2 x when in the limit = -x = a very large quantity. Hence again y = -x is an asymptote, but we gain the additional information that if x be negative and very large the ordinate of the curve is greater than the ordinate of the asymptote. 7y. Since when the signs of x and y are both changed the equation remains of the same form there is symmetry in opposite quadrants. 6. Since dy x(2a2 - X3) dx /4 -a2x2 ' we have dy-0 dx at the points where the curve is intersected by the cubical parabola 2a2y=x3 (which is easily traced), and by the axis of y; and dy3_ dx where the curve is cut by either of the parabolas y2= +ax. The form of the equation is therefore that shown in Fig. 74. EXAMPLES. 1. Trace 5 +?/5 = 5ax2y2, showing that at the origin there are two cusps of the first species, an asymptote x+y=a, two infinite branches below the asymptote, and a loop in the first quadrant. 2. Show that the curve y - a22y2 + x6=0 consists of four equal loops, one in each of the four quadrants and lying entirely within the circle r=a. 322. Polar Equations. Order of Procedure. In tracing a curve from its polar equation it is advisable to follow some such routine as the following:1. If possible form a table of corresponding values of r and 0 which satisfy the equation of the curve. Consider both positive and negative values of 0. CUR VE TRA CING. 345 2. Obtain the value of tan 9, Art. 178. This will indicate the direction of the tangent at any point. The length of the polar subtangent is often useful, Art. 179. 3. Examine whether ahay values of 0 exist which give an infinite value of r. If so, find whether the curve has asymptotes in such directions (Art. 234) and find their equations. 4. Examine whether there be an asymptotic circle (Art. 236). 5. Find the positions of the points of inflexion (Art. 248). 6. It will frequently be obvious from the equation of the curve that the values of r or 0 are confined between certain limits. If such exist they should be ascertained. Kyg., if r = asinno it is clear that r must lie in magnitude between the limits 0 and a, and the curve lie wholly within the circle r= a. 323. Curves of the Classes r = a sin nO, r sin nO = a. VII. To trace r = a sin 50. a. We have the following table of corresponding values of r and: Values of 0 r w 2w 3w % _ — 10 O 10 Cd 10 Cd 10 )5 -4 4 — 4z I — -- Correspond- Pos. Pos. ing Values 0 and and 0 Neg. a Neg. 0 Pos. of r Incr. I Decr. 0 7r 67 ir 7w' 8w Values of 0 etc. 10 i-O 10 o Corresponding Values a Pos. 0 Neg. -a Neg. 0 etc. of r 346 CUR VE TRA CING. /3. r is never greater than a, and there is no asymptote. 7. tan 0-1 tan 50, and therefore vanishes whenever r vanishes and = oo whenever r= +a. The curve therefore consists of a series of similar loops as shown in Fig. 75, all being arranged symmetrically about the origin and lying entirely within a circle whose centre is at the pole and radius a. 3 Fig. 75. 324. Any other curve of the class r = a sin nq may be traced in a similar manner. We annex a figure of the curve r= a sin 60 (Fig. 76). It will be noticed for this class of curves that if n be odd there are n loops, whilst if n be even there are 2n loops. This will be easily seen from. the order of description of the loops, which we have denoted by the numerals 1, 2, 3... in the figures. CURVE TRACING. 347 325. Curves of the class r sin nO = a are inverse to the above species, and their forms are 10 Fig. 76. therefore obvious, going to oo along a radial asymptote whenever the radius of the companion curve r = a sin 0 vanishes, and touching r=asin nO at the extremity of each loop. We give in illustration a tracing of the curves r= a sin 40 ) and r sin 40 = a with the asymptotes of the latter, in one figure (Fig. 77). 326. Class rn = aos nO. The class of curves of which ron = acos nO is the type embraces, as has been previously noticed, several important and well known curves. For instance, 348 CURVE TRACING. we get Bernoulli's lemniscate (n=2), the circle (n=1), the cardioide (n,= ~), the parabola (n - -), the straight line (n =- 1), the rectangular hyperbola (n = - 2). Fig. 77. VIII. To trace r = a2cos 2 (Bernoulli's Lemniscate). a. Negative values of cos 20 give imaginary values of r. Hence the only real portions of the curve lie in the two quadrants 7r 37r 57r bounded by 0= -- and 0= + 4, and by 0=- and 0= 4 4 4 P. 3r=0 when 0= ~+ or 37r or 5 4 4 4' and = ~a when 0=0 or ir. 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 -0 for 0 the curve is obviously symmetrical about the initial line. CUR VE TRA CING. Also, r increases from 0= -7 to 0 and decreases again from 0=0 to 4 and is nowhere infinite or in fact greater than a. The curve therefore consists of two similar loops as shown ill Fig. 78. 0 fX Fig. 78. Other curves of this species may be treated in a similar manner. It will be easily seen that if n be fractional (=P), the curve will have p portions arranged symmetrically about the origin.,, Fig. 79. For example, in the curve 13 3 r26 aTcos 6 5 _.__ _ 3:50 CURVE TRACING. we have the following scheme of values for r and 0: 57 107r 157r 207r 257r 30r 0 0 - etc. r a 0 -a O a 0 -a etc. 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. 79). EXAMPLES. 1. Trace the curves r = a cos 20, r cos 20 = a, r=a cos 30, r=a cos 40. 2. Trace r3=a3cos 30, r3cos 30 = a3, rg=alcos 30, rCcos 0=a, rl = alcos 10, r1cos 30 = a. 3. Trace the curve y2(x2+ a2) =x2(a2 - x2). [T. C. S., 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. aO IX. To trace r1 -1 + 0 a. By giving a set of values to 0 we have the following table: Values of 0 in 3 1 1 1 1 oo 4 3 2 1 - Circular Measure 2 4 4 2 4a 3, 2a a a a Values of r a 0 - - a 5 4 3 2 3 5 3 CUR VE TRA CING. 351 Values of 0 in 3 Circular Measure 4 -1 3 2 -2 -3 Values of r -3a co 5a 4a 3a 2c 3c ~2 4a 3 lOa 9 -- 0 a I3. Since we may write the equation a 1' 1+ when 0 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. 234 shows that rsin (O+1)+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. 80. 6. The points of inflexion (Art. 248) are given by the equation 03+02+2=0, an equation which has one real root which lies between 0 -1 and 0= -2. The curve is therefore that shown in Fig. 80. 352 CURVE TRACING. EXAMPLES. aO2 1. Trace r=021, showing that it lies entirely within the circle er=a, which is an asymptotic circle; also, that there is a cusp of the first species at the origin. aO2 2. Trace r= 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 02=3. EXAMPLES. 1. Show that the curve y2 2a2 + x2 Y C2 _2 -2 x2 consists of two branches each passing through the origin and extending to infinity, and that the whole curve is contained between two asymptotes parallel to the axis of y. 2. Show that the curve y2 x2 - 4a2 2 - a2 has two infinite branches passing through the origin and lying between the asymptotes x= & a, and that there'are in addition two other infinite branches resembling those of the hyperbola X2 - y2 =4C. 3. Show that the curve X3 + y_ = a:: consists of one infinite branch running to the asymptote x + y = 0 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 x3 + y3 = 3axy consists of one infinite branch running to the asymptote CUR VE TRA CIIVG. 353 + + 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) x + y3= a2x. (/) 3 + y= 2a2. (y) ay2= (a2 - x). [R. M. A., Nov., 1883.] 6. Show that the curve ay2 = xy + x3 [R. M. A., July, 1880.] has a cusp of the first species at the origin and an asymptote + y = a cutting th t T he curve at. 7. Trace the curves (a) ay2 - 2xy + x3 = 0. [R. M. A., Nov., 1880.] (/P) y3 + xy + b =0. [R. M. A., Nov., 1881.] a and b both being positive quantities. 8. Trace y2 = 4a2(2a - x). Show that this curve may be constructed thus: take a semicircle APB whose diameter is AB, produce MP the ordinate of P so that MP: MQ = AM: AB, then the given curve is the locus of Q. [This curve is called the Witch and was discussed by Maria Gaetana Agnesi, Professor of Mathematics at Bologna, 1748.] 9. Trace the curve y2(2a- x) = x3. [Cissoid of Diodes.] Show that this curve arises from the following geometrical construction. AB is the diameter of a semicircle APB, BT the tangent at B, APT a straight line through A cutting the semicircle and the straight line in P and T; then, if Q be taken on this line so that A Q = PT, the locus of Q is the Cissoid. z 354 54 CURVKE TRA MCJIV. 10. Trace (x+aY 47 G and show that the oblique asymptote cuts the curve at an angle tan-r8. [R. M. A., Nov., 1882.] I1. Trace 2x(x2 + y2) = a(23X2 + y2) and find by polars the co-ordinates of the points of infiexion. [R. M. A., June, 1883.] 12. Trace y(a2 + x2) = a2x, showing that there are points of contrary fiexure where x =0 or ~ a,J3, that the tangent is parallel to the axis of x where x= + a, and that the axis of x is an asymptote. 13. Trace,2Y2 = a (X - Y sihowing that the curve lies entirely between its asymptotes y = ~ a, and that its tangents at the origin are y = + x. 14. Trace the curve (.2 - a2)(y2 - b2) I '2 15. Trace X4 = a2 W - y2). 1 6. Trace (y2 - at2)2 = X2(x2 - 2a2). 17. Trace axy = xcv - a'. (The Trident.) 18. Trace the, curve X4 - 2MX2y2 + y4 = a4 when m is respectively greater than, equal to, and less than unity, and also when rn is zero. [LONDON, 1880.] 19. Trace Y 2 2 x-a 20. Trace Y2 X2 x2 - C2' 21. Trace x(X + y)2 = a(x - y)2. [. C0. S., 1879.] 22. Trace a3 = y(x - a)2. [OXFORD, 1876.] 4, CUR yE TRACING. 5,355 2 a 23. Trace ( X [H-. C. S., 1881.] \Y-/ Yy+a 24. Find the multiple points on the curve 2(x4 + y4) + 5x2y2 + 4a4 = 6a2(x2 + y2) and the directions of the tangents at those points. [H. C. s., 1881.] Also trace the curve. 25. Trace. the curve X3 + y3 + 3cxy 0, 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)2(x + y)(2x + y) - a=y2. [CAhIn., 1879.] 27. Trace the curve r a(1 + cs 0). (Cardioide.) 28. Trace r a + b cos 0. (The Lima~on of Pascal.) 29. Trace r= a(2 cs 0 + 1). (The Trisectrix.) 30. Trace the following spirals: (a) r = A. (Spiral of Archimedes.) (/ rO = a. (The Hylperbolic or Reciprocal Spiral.) (y) r20 =a. (The Lituus.) (3) r = ae"O. (The Logarithmic or Equiangular Spiral.) Show that in each case. there is an infinite number of convolutions round the pole,,and that r sin 0= a is an asymptote to (/i) and the initial line an asymptote to (4 31. Trace the curves r=acos 50, rcos 50=a, r= a cos-L0. 32. Trace the curves 2 2 2 2 2 O 13 = ai Cos 2, rs = CCUsec01 rl- =aI COS'S What is the relation between them? [CAMB., 1,876.] 356 CURVE TRACAING. 33. Trace the curve 0- a ' - 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. 0 +sin 0 34. Trace r=a -S, - sin 0' 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 =~ asin20 35. Trace rP =:a 2a- (The Cissoid) cos 0 from the polar equation. 36. Trace r=a - a [R. M. A., July, 1880.] 37. Trace r02= tan 0, from 0= 0 to 0 = 2r. [OXFORD, 1876.] 38. Trace r3sin 3(0 - a) = sin 0 - sin a. [CAMB., 1879.] 39. Trace the "curve of sines" x y= b sin - a 40. Trace y = e-~ tan fix. 41. Trace r=- 05 - 1 for positive values of 0. [TRIN. COLL. CAMB., 1873.] 42. Trace r= a [OXFORD.] I - sin 20 43. Trace (x + a)2(y - a) + (y + )2(x - a) = 0. [OXFORD.] C UR VE TR A CING. 3,57 44. Trace 0(x= a(x b (x -a)(x -b)' 45. Trace x ==a(1 - cosO0)} (The companion to the Cycloid.) 4 6. Trace y=c cosh ~.(The Catenary.) 4 7. Trace y = x +cosh r. 48. Trace x 'y2 - a y)2(bl 2 or r = acosecO0 ~b. (The Conchoid of Nicomedes). 4 9. Trace {2+(a + )}{2+(a 7x)2} - examining the cases (1) a< b. (2) a = 6. (Leniniscate of Bernoulli.) (3) a>b. (Cassini's Ovals.) 5 0. Trace Y4 +x ~2Y + 2y3 + x1 -0. [CRAMER.] 5 1. Trace r =a(cosa Cos O-jcos 3a cos 30 iicos 5a cos 5O -.) [MlA TH. TRipos, 1878.] 52. Trace y =6e (The Probability Curve.) 53. Trace the curves (a) y4axy~4 (/3 a3 Y2 -2abX2y Y-x5=0. () y5 + ax4 - b2Xy2 = 0. [CRAMER.] 154. Trace - ax y + by, - 0. [DE GIA.] 5 5. Trace (aL) x5~+y5-=9a,3Xy. (/3 x5 ~I =-xy(ax + b 2y). [FROST.] APPLICATION TO THE EVALUATION OF SINGULAR FORMS AND MAXIMA AND MINIMA VALUES. CHAPTER XIII. UNDETERMINED FORMS. 327. 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," " Undetermined," "Illusory," or " ndeterminate " form. 328. 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. 329. List of Forms occurring. Several cases are to be considered, viz., when, upon substitution of the assigned value of the independent variable, the function reduces to one of the forms 0 Ox o, -, - 00, 00, 00~, or 1I. It is frequently easy to treat these cases b alebraical It is frequently easy to treat these cases by algebraical 362 UNDETERMINED FORMS. or trigonometrical methods without having recourse to the Differential 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. 330. Algebraical Treatment. Suppose the function to take the form 0 when the independent variable x ultimately coincides with its assigned value a. Put x =a + h and expand both numerator and denominator of the function. It will now become apparent that the reason why both numerator and denominator vanish is that some power of h is a common factor of each. This should now be divided out. Finally, put h=O 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 =-, so that when x becomes = oo y becomes =0. The method thus explained will be better understood by examining the mode of solution of the following examples. UNDETERMINVED FORMS. 363 (1x__ bx Ex. 1. Find Lt x b.. x Here numerator and denominator both vanish if x be put equal to 0O We therefore expand 0s and bx by the exponential theorem. Hence a -L {1+ Xlogea + 2 (logea)2 + * -{ b+lOge+ (logb)2+*** = Lto { loge - logxb + -(log12 (-t2) +..e = loga - logb = log.b Ex. 2. Find t - X+ 1,XI- 3x2 + 2' This is of the form 0 if we put x=1. Therefore we put x= 1 +k and expand. We thus obtain L x - -2+x51 = ( + h)7 - 2(1 + )5 + t= 3x 2+2 = t^=l+ h)3- 3(1 +h)j + 2 = Lt (1+7h+21h2+ +...) - 2(1 + 5h + 10h2+...)+ 1 (1 +3h 3h2+ +...)-3(1 +2h+A2)+2 = Lth=o- 3h + h2 +... -3h+... =Lth-o -3+... -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 of h. 364 UNDETERMIN1ED FORMS. Ex. 3. Find Lto0(ta x Since ~tan x 1 sin x X Cos X X we have Lt,0tanxI. x~~~a Hence the form assumed by (- x)Y is 1' when we put =0O. Expand, sin x and cos x in powers of x. This gives Lt( tan xy'L = LtX=0( 1 higher powers ofxX = Lt,0I where I is a series in ascending powers of x whose first term (and therefore whose limit when x = 0)is unity. Hence D~. + I e by Art. 21. Ex. 4. Find Lt= xThis expression is of ~the form 1'. Put 1X and therefore, if x=1, y=0; therefore Limit required = Ltv~0(l -.y')Y e-1 (Art. 21). Ex. 5. Lt~,x(a - 1). This is of the form co, x 0. Put x=i, therefore, if X cx0,y=0, and ay Limit required= Lt..'.-.= ogea (Art. 22). UNDETERMINED FORMS. 365 EXAMPLES. Find the values of the following limits: L oal 1. L.~Oabx- 1' 2 T _ -1 2. Ltx=l- 1 -X-1 3. LtX= ' 4. Lt=) --- —----- 5. Lt =l4 + 3.-2- 5 X3 - x2 - x + 1 6. Lt x5 5-2x3 -4X2 + 9x -4 6. 4_2X3+2;-1 ex - e-x 6. Ltx=ol 8. Lt=o --- — o9. Lt s X - log (1 + ) 0. Ltx=o- 2 10. LtZ=o x - log (1 + ) 2 11. Ltxo - sin x cos x. X3 sin-lx - x 12. Lt=o l —'a;3cosa 1. cosh x - cos x 13. Lt,=o x sin x. sin-4x 14. Ltx=ta --. tan-l.x 1 sin-l - sinh x 15. Lt.=o 5 — x5 X2 x cos3x-log(l + x) - sin-X 16. Lt_=o- - __2 x5 17. t2 si X + tanh X - 3 17. Ltxto ^ eo Si X-sin x x2 18. LtX=o - og(- ) 2 +Xlog (I - X) xQ e 4 -sin 2XZ 19. Lt=o(ai -sin )x3 X7' 20. Lt x0( s 1 25. Lt=0o(covers x) 26. Ltr(cosec )ta. 2sin 25. Lt~o(covers x)0. 26. Zt=7r(cosec x)~2?x. 2 331. APPLICATION OF THE DIFFERENTIAL CALCULUS. John Bernoulli "' was the first to make use of the processes of the Differential Calculus in the determination of % "Acta Eruditorum," 1704. 366 6UNDETERHjIINED FORMS. the true values of functions assuming singular forms. We propose now to discuss each singularity in order. 332. I. FO RIN Consider a curve passing through the origin and defined by the equations y= 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 Y -O NN X Fin. 81. corresponding to the origin, so that (p(a)= 0 and x/,(ca)= 0. Then ultimately we have Lt1 - Lt tan PON"= the value of dy _:d at the origin; ddy and d d ( do-Ix dx -Y'(0 dtt Hence LLtt `aV --- - = a. Z(t) 00) _ _ _ _ 0 and if (k(t) be not of the form - when t takes its assigned 11 V ) zDo, UNDETERMINED FORMS. 367 value a, we therefore obtain "2) ^)_,0'(a)' t 0(t) = ) But, if () be also of undetermined form, we may V'(t) repeat the process and say Lt V'(t) Lt"(t) Ltt=a, Lt, (t) _ etc., proceeding in this manner until we arrive at a fraction such that when the value a is substituted for t its numerator and denominator do not both vanish, and thus obtaining an intelligible result-zero, finite, or infinite. 333. Another Proof of the Method. We may arrive at the same result il another way, thus:Let ( ) take the form - when x approaches and ultimately coincides with the value a. Let x = +h. Then by Taylor's Theorem (x)_ g(a) + h0('(c + h(a ) -_ '(a+ Oh) +(zx) ~ (a) + h/'(ac + O6h) - V/(a + OVh) for p(a) =0 and +(a) =0 by supposition. Hence in the limit when x= a (and therefore h = 0), we have Lt () =Lt, (a + e) '(ca) W-=(*,) ~~ =V-+(a+ 0ah) - ' )' 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. Lt sin -_ 0=o 68 ' Here p(O)-sin 0 - 0, and {(0) =03, which both vanish when 0 vanishes. 368 UNDETERMINED FORMS. 5'(0) =cos 0- 1, and b'(0)=302, and both of these expressions vanish with 0. Differentiating again '"(0) - sin 0, and /'"(0)=60, and still both expressions vanish with 0. We must therefore differentiate again q"'(0)-= -cos 0, and t"'("0)=6, whence 1"'(0)= -1, and m"'(O) 6; sin 0 - 0 1 therefore sin-0= 1 eq - e-0 + MsinO - 40 0 Ex. 2. Lt - +sn0 [Form ] e0 ~ e-0 +2 cos 0- 4 [Form 0] - 0t0;504 0 Lt - e-0 - 2 sin 0 [Form 0 8_LtS e + e0 -2 cos6 [Form 9] =L< ^-F+2^^ rrorm 0I =Lo 6002 e0 - e-0 + 2 sin 0 0For 0=0 1200 0 eeO+e-O+2 cos 0 1 -lt +-0 0=- 120 30 334. The proposition of Art. 332 may also be treated as follows. Let q(a)=0 and 5-(a)=0, and let the pth differential coefficient of q(x) and the qth of qp(x) be the first which do not vanish when x is put equal to a. Then by Taylor's Theorem, putting x = a + h, (x) = (ca) +h'(a) +.. + P+ q P-l(a)+ 7bP(a+Oh) hp = h19(a + Oh). Similarly h7q (x) =- W,q(a + O). UNDETERMINATED FORMS. 369 Hence Ltx=a (x) - LthP-qP(a+ Tq a + O h) 29 1 f(a) t Now, if jp> q, Lt,=ohP-q =. If p <q, Lth,=hP-= -. If p2=q, Lt I(x) j5P(a). so that the limit is 0, (a) or o, according as p is >,P(a)'or 00, according as p is>,or < q. 335. II. FORM 0 x o0. Let p((a)=0 and 4z(a)=oo, so that (x)Vfr(x) takes the form 0 x oc when x approaches and ultimately coincides with the value a. Then Ltx=a(x)fi(x = )Ltx=( 1) 1 1 and since =0 Y(a) o ' the limit may be supposed to take the form 0, and may be treated like Form I. 0 1 Ex. 1. Lt 0 cot = Lt - =Lt 1. =o 0=o0 tan 06 =o sec20 a. a sill - sill - Ex. 2. Ltx=, sira Ltx-, Lt_ a =a. x x 336. III. FORM-. 00 Let p(a)= oo, f(a) = o, so that T() takes the form when x approaches indefinitely near the value a. 00o2 A 2A ,370 UNDETERMINED FORMS. The artifice adopted in this case is to write 1 4( ) r(x),1(x) 1 I I 1 1 Then since ( -- =0, and -=0, we may con*(a) oo 0 p(a) oo sider this as taking the form O, and therefore we may apply the preceding rule. 1 +'(X) Xc&AVx x^a 1 (x) Lt -=a - (X) =Lt= — )(X)]2 p(x) [)(x]) Therefore LLt=tx=a Lt (ArJt. 12). "=" ~(x) "'"v(x) a('(X) Hence, unless Lt =a,() be zero or infnite, we have 1{ L x= _{ X Lt =a(x) or Lt 4 (x) = Lt (x) -a(r(x) '= ()' If, however, Lt (x) be zero, then Lt. (*) + +-(xa)_and therefore, by the former case (the limit being neither zero nor infinite), = Lt ='(x) ~ '(x) 47'(x) UNDETERMINED FORMS. 371 Hence, subtracting unity from each side, Lt jj'= Ltx4,)2. Lt a'(x) - Lt (x) Finally, in the case in which Lt ' Lt+ _'(x) 0 LVx) Y _ (X)_o Z=(Xx) - P~> and therefore by the last case Ltx=a (X) _r. ~~ ). therefore Lt,()-Lt ) () XVfr(x) v2"(XX This result is therefore proved true in all cases. 337. If any function become ivfinite for any finite value of the independent variable, then all its differential coeffcients 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 coefficients 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 x, sec x, cosec x, are really fractional 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 33.7,2 372UNDETERMINED FORMS. log(x - a) when x = a, or for the exponential function bx-a when x = a, b being supp osed greater than unity.-x 338. From the above remarks it will appear that if 0(a) and fr(a)j become infinite so also in general will g' (a) and Vfr'(a). Hence at first sight it would appear that the formula Ltx "(x) is no better than the original form Ltx ak(x) But it generally happens that the limit of X'X the expression when x a, can be more easily evaluated. Ex. 1. Find Lt=-F- which is of the form -. 2= tan 0 0 o Following the rule of differentiating numerator for new numierator, and denominator for new denominator, we may write the above lim.it 1 -7r ==Lt o07W sec2O' 2 which is still of the form on. But it can be written = Lt cos26 which is of the form i2y\ 0' 2 - t - 2 cos 0 sin 0 2 Ex. 2. Evaluate LtX=W-, which, is of the form - ' For further discussion of this point the student is referred to Professor De Morgan's " Duff, and Int. Calculus." UNDETERMINED FORMS. 373 Xi nxn-1 Lt — _ Lto =0 'a_- - S-~X=oo -- '= *.. ex ex n! n' = Lt,=~ = = o0. It is obvious that the same result is true when n is fractional. Ex. 3. Evaluate Ltx=oxm(log x)', In and n being positive. This is of the form 0 x oo, but may be written Lt. oJ I} [Form -] and by putting x-=e —y this expression is reduced to Lty, l z m =0 as in Ex. 2. 339. IV. FORM o - oo. Next, suppose p(a) = oo and /r(a) = oo,so that 5(x)-f(x) takes the form oo - c, when x approaches and ultimately coincides with the value a. Let, = (x)-(x) = (x) ~(). From this method of writing the expression it is obvious that unless Lt. ( =) 1 the limit of Qt becomes Vf(a) x (a quantity which does not vanish); and therefore the limit sought is oo. But if Lt= - -- =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. Lt =o(' - cot x) = L to (1 - x cot x) X VV Sill G - X COS X *:= ~t~six-. o s x (which is of the form - x sin X 01 x sinx__ (which is of the samei Ltx=~sin x + x cos x form still.sin x+ cos. -= Ltx2=o — =0. 2 cos x - x sin x. 374 UNDETERMINED FORMS. 340. V. FORMS 0~, 00~,.I0 Let y =UL, i and v being functions of x; then logey = v logeu. Now logel=0, logoco = oo, loge0= — O; and therefore when the expression uW takes one of the forms 0~, oo 0, 1I, logy takes the undetermined form 0 x oo. The rule is therefore to take the logarithm and proceed as in Art. 335. Ex. 1. Fznd Ltx=o X which takes the ztndetermiledformn 0~. Ltxologe = Ltx x= Lt= o x =Lt=(-) =0, X X21 x X7I whence Ltx=oX=e = 1. Ex. 2. Find Ltx=_r(sin x)talx. This takes the form 1. 2 Ltx=r (sin.) tan =. Ltx, etanxlog sin 2 2 - -T.,y log sin x, cot x and Ltx=7ta n x log sin x= LLt= -- co t e - cot - -cosecYR = Lt2=r( - sin x cos x) = 0, 2 whence required limit =e~ = 1. A slightly different arrangement of the work is exemplified here. 341. The following example is worthy of notice, viz., Ltx=,{l + f(sx)}(), given that 0(a) = O, V(a) = o, Lt=a0(x)fr() ) = n. We can write the above in the form Lty by A. 21,+ C+)hap. I. which is clearly en by Art. 21, Chap. I. It will be observed that many examples take this form, tan x 2 such, for example, as Ltx=o( — on p. 364, and Exs. 20 to 26 on p. 365. UNDETERMINED FORMS. 375 342. dy of doubtful value at a Multiple Point. dx Since a =0 and 'a-=0 at any multiple point on the,ax ay curve z, = 0, it will be apparent that at such a point the value of y as derived from the formula ~daG'a dy ax dx'- -a 3y will be of the undetermined form 0 The rule of Art. 332 may be applied to find the true limiting values of dy 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 inspection and the required values of y' thus found. If the multiple point be not at the origin, the equation of the curve should be transformed to parallel axes through the multiple point and the problem is then solved as before. Ex. Consider the value of -- at the origin for the curve dx x4+ ax y+ bxy2+y4= 0. The tangents at the origin are obviously x=O, y=O, ax+by=0, making with the axis of x angles whose tangents are respectively wi, 0, - which are therefore the required values of. dxo 376 ULNDETERMINVED FORMS. EXAMPLES. Investigate the following limiting forms:1. Lt log( - 2) log cos x 2. Lt 2x3- 32 +1 3. Lt^r -tan 41 - ^/2 sin E 2. /~~-r x5 _ 5x1 + 2' 4. LtI +cos 7rx 4 tan2 r 5. Ltz=ilog (2 - ) cot (x- a). 6. Lt logsinxC~S log sicos 2 22 cot 0 tan-l(m tan 0) - m cos2~ 7. Lt0o=o 2 sin2_ 8. Lt=o0(cos x)cot2. 9. Lt,=(l - x 2)ogl -x). 10. Lt{x=(logr x)1l-^ Axe + Bxe`l + Cxe-2 + 11. Lt= Ax + Bxn-l + Cx"n-2 + ' according as n is >, xax' + bxm~l- cx~-2 +... or < m. 12. Ltx=o-x-, m being positive. 13. Ltx(a+1 ) \1a - oI/ 14. LtA K 0 cos x 2\1 -COSX/ UNDETERMINED FORMS. 15. LtX=o{cot (45~ - )}ot. 1 1 1 1 16. Lt= +... 2x2 - 2e" + 2 cos x3 + sinrzx 17. Ltx=o - x4 18. Lt 1 - 1i+X2 x'-: 1 r T b { ^ f(i.) If a be > 1. 19. Lt..,a sin 1 sax { (ii.) If a be < 1. 20. Lt cosec x - cot x J/Ca2 + Cax' + X2 - / 1 + x 21. LtX =o a x log cos. - Ia a. _,a-x x 1 e a Sill. + log 1- 1- 992 L,a a 2 -/ \a a- a22. Lt=, - a. 23. Lt=o- cot ' ) 24. Lt /a' + ax + x2,/t2 ax + 2 J/F +, - JC - x 25. Ltl~g( + + 2) +]o(1 -+2) sec x - cos x 26. Lt x sin (sin x) - si, 12 26.. tx=o- - 27. Lt ( +)-e _ t —Ox 03178.:378 UNDETERMILYED FORMS. I exII hex2 28. Ltx0o - 30. Lt (x - y)a n~(y -a) x"+~(a -x)ynj" (x-y)('y —a)(a -x) [Put x = a + ht, y = a + k, and expand in powers of h and k, and finally, after reduction, put b =0, k:=- 0.1 3Ltlog x+ log Y x+y- 2 32. Show that generally, if a function of two independent 0 variables take one of the singular forms -,etc., for certain values of the variables, its value is truly indeterminate. 3 3. Given 1) + Y3 + a3 = 3axy, find the values of cywhen x = y = a. d'x 34. Find the values of dyat the origin for the curve dx x 3+ Y3-=3axy. 35. For the curve x2 Y2 = (a2 - y2)(b + y)2 find the values of dyat the point (0, - b). dx 36. For the curve x4 + ax2y = ay' find the valu.es of dYwhen x =0. dxax -1 lbsin x -sin bx it lbV 37. Prove Ltx.o xnS in:; coxcs x) oga UNDETERMIN3ED FORMS. 379 d+ 1z 38. Prove Ltx=o —r z= n- m2, dx'"-1 cos M.y where u = - y and x = sin y. cos y 39. Find Ltod where y - and = cos-(l -x). dx2 sin = [I. C. S., 1S84.J 40. If y= (sin-'x)2, prove that Ct'+ 2. Ltx= dx'= 2 dX^ 41. Prove that Lt^_ -7 is zero or infinite according as n is greater or less than m, a and b being both greater than unity. 42. Prove Lt,(, 2 - ( talla 2 43. Prove Lt^=, J /a- xcot{1/a = -4 -44. Find Ltx=o(cos ax)cose2B. 45. Find Lxsin bx- b~sin ax { (1) If x =0. tan bx - tan ax; (2) If a =b. 46. Find Ltx=l- 1+( 1) (x2- 1 )i - x + 1 AJ - dt + lx - a 47. Find Lta=,. 48. Find Lx=7r(sin x)t"x. 2" 380 UNDETERMIiINED FORMS. 49. Prove that if, when x is infinite, q(x) = co, then will Lt=.,,(x) = Lt{t((x+ 1)- ), (Lt + 1) and also that Lte= _{_(x)_ = Ltx W[TODHUNTER'S DIFF. CALC.] 50. Prove that LtD x ' = e. 50. [TODHUNTER'S DIFF. CALC.] 51. Prove Lt l" + 2" + 34 +... + n 1 m being positive. 52. Prove Lt,^=oharjna + a + h + 2Atlm +... +a + (lt- 1)h1} bm+_ am+l ' m+ are any given qantities. where h==b _ and a, b are any given quantities. in, CHAPTER XIV. MAXIMA AND MINIMA-ONE INDEPENDENT VARIABLE. 343. Elementary Algebraical Methods. Examples frequently occur in elementary 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. For example, the function x2- 4x+9 may be written in the form (x- 2)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 x =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. As a second illustration let us investigate whether any limitation exists to the values of the expression x2-x 1 2+ x for real values of x. 382 MA XIMA AND MINIMA. X 2 — + 1'Putting 2-, x2 + x+1 we have 2(1 _ y) (l + y) +1-y =, an equation whose roots are real only when (1 + y)2 > 4( )2, i.e., when (3y - 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 -1. Its maximum value is therefore 3 and its minimum 3. 344. Method of Projection. Ex. Suppose it be required to determine geometrically the greatest triangle inscribed in a given ellipse. It is obvious from elementary considerations that if the ellipse be projected orthogonally into a circle the greatest triangle inscribed in the given ellipse must project into the greatest triangle inscribed in a circle; and such a triangle is equilateral and the tangent to the circle at each angular point is parallel to the opposite side. This property of parallelism is a projective property, and therefore holds for the greatest triangle inscribed in the given ellipse. Moreover Area of greatest triangle inscribed in the ellipse Area of ellipse Area of equilateral triangle inscribed in a circle Area of the circle _3 /3 47r * Hence the area of the greatest triangle inscribed in an AMA XIMA AND M1INIMA. 383 ellipse whose semiaxes are a, b is./3a 4 EXAMPLES. 1. Show algebraically that the expression x+- cannot lie beX tween 2 and - 2 for real values of x. Illustrate this geometrically by tracing the hyperbola xy - x2= 1. 2 -4x+ +9 2. Prove that, if x be real, mxust lie between 5 and -. r2+4x+9 5 3. Show that, if x be real, +a. x+b cannot lie between the - a x - ),/a+,/b\2 values _( /av b) and -( - ),,a -,/b/ \,a.+,/b/ 4. Show that the triangle of greatest area with given base and vertical angle is isosceles. 5. 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. 6. If A, B be two given points on the same side of a given straight line and P be a point in the line, then A P+ BP will be least when A P and BP are equally inclined to the straight line. 7. Show that the triangle of least perimeter inscribable in a given triangle is the pedal triangle. 8. If A, B, C be the angular points of a triangle and P any other point, then A P + BP + CP will be a miniimum when each of the angles at P is 120~. [AP is a normal to the ellipse with foci B, C and passing through P.] 9. The diagonals of a maximum parallelogram inscribed in an ellipse are conjugate diameters of the ellipse. 10. 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. 384 3MAXIMA AND MIINIMA. THE GENERAL PROBLEM. 345. Suppose x to be any independent variable capable of assuming any real value whatever, and let ((x) be any given function of x. Let the curve y = ((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..I TIS~g" E / Fig. 22. 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 points as A, C, 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. 346. Points of Inflexion. On inspection of Fig. 83 it will be at once obvious that at such points of inflexion as G or H, where the tangent MAXIMA AND MINIMA. 385 is parallel to one of the co-ordinate axes, there is neither a maximum 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. Fig. 83. This point may however be considered as a combination of two such points as A and B in Fig. 82, the ordinate Fig. 84. increasing up to a certain value N1G,. then decreasing through an indefinitely small and negligible interval to N2G,0, and then increasing again as shown in the magnified figure (Fig. 84), the points G1, G, being ultimately coincident. 2B 386 MAXIMA AND MINIMA. 347. We are thus led to the following definition:DEF. If, while the independent variable x increases continuously, a function dependent upon it, say +(x), increases through any finite interval however small until x=a and then decreases, +(a) is said to be a MAXIMUM value of ((x). And if q5(x) decrease to q(a) and then increase, both decrease and increase being through a finite interval, then +(a) is said to be a MINIMUM value of q((x). 348. Properties of Maxima and Minima Values. Criteria. The following statements will now be obvious from the figures 82 and 83:(a) According to the definition given, the term maximum value does not mean the absolutely greatest nor minimum the absolutely least value of the function discussed. Moreover there may be several maxima values and several minima values of the same function, some greater and some less than others, as in the case of the ordinates at A, B, C,... (Fig. 82). (/) Between two equal values of a function at least one maximum or one minimum must lie; for whether the function be increasing or decreasing as it passes the value [MIP1 in Fig. 82] it must, if continuous, respectively decrease or increase again at least once before it attains its original value, and therefore must pass through at least one maximum or minimum value in the interval. (y) For a similar reason it is clear that between two maxima at least one minimum must lie; and between two minima at least one maximum must lie. In other words, maxima and minima values must occur alternately. Thus we have a maximum at A, a minimum at B, a maximum at C, etc. iAXllWA AND MINIMA. 387 (6) In the immediate neighbourhood of a maximum or minimum ordinate two contiguous ordinates are equal, one on each side of the maximum or minimum ordinate; and these may be considered as ultimately coincident with the maximum or minimum ordinate. Moreover as the ordinate is ceasing to increase and beginning to decrease its rate of variation is itself in general an infinitesimal. This is expressed by saying that at a maximum or minimum the function discussed has a stationary value. This principle is of much use in the geometrical treatment of maxima and minima problems. (e) At all points, such as A, B, C, D,,..., at which maxima and minima ordinates occur the tangent is parallel to one or other of the co-ordinate axes. At points like A, B, C, D the value of d vanishes, whilst dy dx at the cuspidal points E, F, d becomes infinite. The positions of maxima and minima ordinates are therefore given by the roots of the equations 0'(x) = () dyx x (5) That dy-=, or cdy=, are not in themselves sufficient conditions for the existence of a maximum or minimum value is clear from observing the points G, H of Fig. 83, at which the tangent is parallel to one of the co-ordinate axes, but at which the ordinate has not a maximum or minimum value. But in passing a maximum value of the ordinate the angle f which the tangent makes with OX changes from acute to obtuse (Fig. 388 MA XIMA A ND MIVNIMA. dy 85), and therefore tan f, or d-, changes fromn positive to negative; while in passing a mnirnimum value f changes from obtuse to acute (Fig. 86), and therefore dchanges dxc from negative to positive. P /fr acute\ 2 obtuse O N X Fig. 85. ___O_______i obtuse Xacute 0 N X Fig. 86. We can therefore make the following rule for the detection and discrimination of maxima and minima values. First find dy and by equating it to zero find for what values of x it vanishes; also observe if any values of x will make it become infinite. Then test for each of these values whether the sign of dy dx changes from + to - oir from - to + as x increases through that value. If the former be the case y has a maximum value for that value of x; but if the latter, a MAXIMAA AND'J IINIMA. 389 minimum. If no change of sign take place the point is a point of inflexion at which the tangent is parallel to one of the co-ordinate axes. 349. Criteria for the discrimination of Maxima and Minima Values. Another Method of Investigation. The same criteria may be deduced at once from the aspect of dy as a rate-measmrer. For is positive or dx dx is positive or negative according as y is an increasing or a decreasing function. Now, if y have a maxirnum value it is ceasing to increase and beginning to decrease, and therefore dy must be changing from positive to negative; and if y have a minimtm value it is ceasing to decrease and beginning to increase, and therefore dy must be changing from negative to positive. Moreover, since a change from positive to negative, or vice versa, can only occur by passing through one of the values zero or infinity, we must search for the maximum and minimum values among those corresponding to the values of x given by 0'(x)= 0 or by 0'(x) = o. dy Further, since d- must be increasing when it changes d2yd from negative to positive, d must then be positive; and ddy ~~~~Ctd2y similarly, when dy changes from positive to negative d2 must be negative, so we arrive at another form of the criterion for maxima and minima values, viz., that there will be a maximum or minimum according as the value 390 MAXIM A ND MINIMfA. of ~whih sd y d2y of x which makes d- zero or infinite gives d2 a negative or a positive sign. EXAMPLES. 1. Find the maximum and minimum values of y where (x - 1)( - 2)2. Here dy_ (x - 2)2+ 2(x - 1)(x - 2) dx =( -2)(3x-4). Putting this expression =O we obtain for the values of x which give possible maxima or minima values 4 x=2 and x=-. 3 To test these: we have, if x be a little less than 2, y = (-)(+)= negative, dx if x be a little greater than 2, d =(+)(+) = positive. d= Hence there is a change of sign, viz., from negative to positive as x passes through the value 2, and therefore = 2 gives y a miniitnum value. Again, if x be a little less than 4, positive and if x be a little greater than 4, d= (-)(+)- negative, dx= showing that there is a change of sign in d-, viz., from positive to dx) negative, and therefore = 4 gives a maximum value for y. Otherwise: =y (x- 2) (3 - 4), dx ely 4 so that when dy is put =0 we obtain x=2 or 3 dx2 so that, when x = 2, - = 2, )? ^~~d2 MAXIMA AND MINIMA. 391 a positive quantity, showing that, when.= 2, y assumes a minimum value, whilst, when x=4 dy- -2, 3' dxik which is negative, showing that, for this value of x, y assumes a maximum value. 2. If d. Y (x-a)2n(x -b)2 +1, dX where n and p are positive integers, show that x=a gives neither maximum nor minimum values of y, but that x=b gives a minimum. It will be clear from this example that neither maxima nor minima values can arise from the vanishing of such factors of -y as have even indices. 3. Show that x 7 +6 has a maximum value when x=4 and a x-10 minimum when = 16. 4. If 4. If -X(X- 1)2( -- 3)3, show that x=O gives a maximum value to y and x=3 gives a minimum. 5. To show that a triangle of maximum area inscribed in any oval curve is such that the tangent at each angular point is parallel to the opposite side. If PQR be a maximum triangle inscribed in the oval, its vertex P lies between the vertices L, M of two equal triangles LQR, MQR inscribed in the oval. Now, the chord LM is parallel to QR and Q Fig. 87. the tangent at P is the limiting position of the chord ~L, which proves the proposition. It follows that, if the oval be an ellipse, the medians of the triangle are diameters of the curve, and therefore the centre of gravity of the triangle is at the centre of the ellipse. 392 MAXIMA AND MINIMA. 6. Show that the sides of a triangle of minimum area circumscribing any oval curve are bisected at the points of contact; and hence that, if the oval be an ellipse, the centre of gravity of such a triangle coincides with the centre of the ellipse. 7. To find the path of a ray of light from a point A in one medium to a point B in another medium supposing the path to be such that the least possible time is occupied in passing from A to B, and that the velocity of propagation of light changes from v to v' on passing the boundary separating the media. [FERMAT'S PROBLEM.] We shall, for simplicity, consider A and B to lie in the plane of the paper, and the separating surface of the media to be cylindrical with its generators perpendicular to the plane of the paper. Let OPP' be the section of the separating surface by the plane of the paper, and let APB, AP'B be two contiguous paths from A to B. Fig. 88. Then, if the times in these two paths be equal, the quickest path lies between them. Let fall perpendiculars P'n, P'n' from P' upon A P and BP, and draw the normal ZPZ' at the point P. Then, since the time in APB=time in AP'B, AP PB A P' BP' Av V' + v,- Pm PJn' or in the limit -= -, v = whence Lt sin nPZ' Pn v sin n'PZ' tPn and therefore, if in the limit the incident ray AP and the refracted ray PB make angles i, i' respectively with the normal at P, we obtain sin i v sin i' v' thus proving Snell's well known law of refraction. MA XIMA AND MINIM.A. 393 350. Analytical Investigation. We now proceed to investigate the conditions for the existence of maxima and minima values from a purely analytical point of view. It appears from the definition given of maxima and minima values that as x increases or decreases from the value a through any small but finite interval h, if ((x) be always less than p(a), then p(a) is a maximum value of +(x); and that if ((x) be always greater than (ca), then +(a) is a minimum value of p(x). We shall assume in the present article that none of the derived functions we find it necessary to employ become infinite or discontinuous for the particular values discussed of the independent variable. We then have by Lagrange's modification of Taylor's Theorem p(x + h) - ~(x)= hf'(x)+ - x-"(x + 01)| h2 j (A) \ ( -h' j........(A) and p(x - h) - (x) = - h'(x) + 2"p'(x- Oh) And when h is made sufficiently small the sign of the right-hand side of each equation, and therefore also of the left-hand side, is ultimately dependent upon that of h('(x), that being the term of lowest degree in h. Hence - (x + h) —p(x) } and 5(x -h) - Q((x) have in general opposite signs. For a maximum or minimum value, however, it has been explained above that these expressions must, when h is taken small enough, have the same sign. It is therefore necessary that 0'(x) should vanish, so that the lowest terms of the right-hand sides of the equations (A) 394 MAXIMA AND MINIMA. should depend upon an even power of h. 0'(x)==0 is therefore an essential condition for the occurrence of a maximum or minimum value. Let the roots of this equation be a, b, c,.... Consider the root x = a. We may now replace equations (A) by the two equations h2 h73.. (B) (a- h) - ((a)=!(a= ) -... (a - 01h) It is obvious now as before that the term -/"(a), being that of lowest degree, governs the sign of the right and therefore also of the left side of each of equations (B); i.e., in general the signs of 9(a +/h) - (a)) and q(a -h) - ~(a) are the same as that of +"(a). Hence if '"(a) be negative p(a+h) and p(a-h) are both < p(a), and therefore +(a) is a maximum value of +p(x); while if +"(a) be positive both ((a+h) and 9(ac-h) are > (c(a), and therefore (p(a) is a minimum value of +(x). But if it should happen that +"(a) vanishes, equations (B) are replaced by,3 4 p(a + h) - (a)= -(a ) = - +,) 4,(a + Wh) p(a -A)- (a)! = - (' a) 4 +()(a0 and therefore when h is sufficiently small 5(a + h) - (p(a) p(a- h) - ((a) J are of opposite signs, and therefore there cannot be a maximum or minimum value of 95(x) when x= a unless MAXIMA AND MINIMA. 395 0"'(a) 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 vanishing should be of an even order, and that if that differential coefficient be negative when x = a a maximum value of +(x) is indicated, but if positive a miniimum. EXAMPLES. 1. Determine for what values of x the function 0(x) = 12x5 - 45x4 + 40x3 + 6 acquires maximum or minimum values. HEere 0'(x) = 60(x4 - 3x3 + 2x2). Putting this =0 we obtain x=0, x=1, x= 2. Again 0'"(x) = 60(4x3 -f 92 + 4x). If x= 1, "'(x) is negative and therefore we have a maximum value; if x=2, '"(x) is positive and therefore this value of x gives a minimum value for 0(x). If x =0, 0(x) vanishes, so we must proceed further. Now "'(x) = 60(l2x2 -]8 + 4), which does not vanish when x = 0, so x =0 gives neither a maximum nor a minimum. 2. Show that x= 0 gives a maximum value, and xs= 1 a minimum, X3 X2 for the function - 3 2 3. Show that x=0 gives a maximum and x=1 a minimum for X5 24 5 4 4. Show that the expression sin30 cos 0 attains a maximum value when = 60~. 5. Illustrate geometrically the statement of Lrt. 350 that in general O(x -+h) - 0(x) and 0(x - h) - ~(x) are of opposite sign. 396 MAXIMA ANAD MINIMA. IMPLICIT FUNCTIONS. 351. In the case in which the quantity y, whose maximum and minimum values are the subject of investigation, appears as an implicit function of x, and cannot readily be expressed explicitly, we may proceed as follows: Let the connecting relation between x and y be (, y)=),...................... (1) then - F 0......................... (2) 'ax ay dx dy But for a maximum or minimum value of, = 0, and therefore < = 0. ax The values of y deduced from the equations 0(x, y)=O _. (3) 0........................ (3) ax therefore include the required maxima and minima. Differentiating equation (2) we have -a, a2, dy + +. ^ aa\ c y q~ + i1 d2 3x2 xy' dx y + dx dx+ 'y dx 2,. (4) dy and, remembering that dy =0, this reduces to 32o -................(5) dxz^ ^' * ** *9 - ) ay Substituting the values of x and y derived from equad2y tions (3) we can test the sign of d2, and thus discriminate between the maxima and minima values. MAXIMA AND MIrNIMA. 397 The case in which this test fails, viz., when 20 = 0 for the values of x and y deduced by equations (3), is complicated owing to the complex nature of the general formulae for dsy and d4y dx3 dX4' Ex. Find the maximum and minimum ordinates of the curve x 3+3= 3axy. Here (x -ay)+ (y2 - ax)d =,..................... (1) and d 0- gives x2=ay. dx Combining this with the equation to the curve we obtain 3 = 2axy; i.e., y = O or y2= 2ax. y=O0 gives x=0, whilst y2=2ax i 4 = 4a and x = ay which presents the additional solution y = a4, x=a V2. Hence the points at which maxima or minima ordinates may exist have for their co-ordinates (0, 0) and (a 3/2, a */4). Now 6 =6x and 3(y2 - a), ox2 ay and therefore at the point x=a3^2, y =aV 4, d2_y x2 2x -2aV2 2 dx22 7 y a 2- 2 a22-a2 -V2 a' and is negative, and therefore at this point y has a maximum value. At the point x=0, y=0, the formulae for!9 and d-s, both become indeterminate, and we have to investigate their true values. 398 MAXIMA AND MINIMA. Differentiating equation (1) we have 2x - 2ady r a,(f ) + (y2 - ax)d-Y = O. 2x -2a dx + 2y( +2 2+ 2( ) + l(6yq- 3 ) + (y2- ax) y0 dx/ dx 2 dX3 And when x and y both vanish these give dYO0 and d2-= 2 do d2 3a' showing that the ordinate y has for this point a minimum value. SEVERAL DEPENDENT VARIABLES. 352. 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 of 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, z,... 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, z, etc., and it is not necessary that this should be immediately done. Suppose, for instance, u= =(x,, y, z) a function such as the one discussed, x the independent variable, y and z dependent variables connected with x and y by the relations F(X, y, ) =0, F2(, y, )=(). du Then, putting — =() for a maximum or minimum, we -IY rr $ dx MAXIMfA A)ND MINVIMA. 399 du __,qo dy, +3 dz have di * * dz > ) have dlb _ + d............(i) dx 'y dx 'Dz dx. ctx ay ' dr a I(3) -a -, d ' a 0,........... (3) dy dz and eliminating -; and d' (Ix dx' 'a vy av OFi 'aF, 3, _0 (4) ax, ay) D9:Dx' Dy' Dz Dx ' Dy' Dz 0,........... 2 2 F 2 ax' dy' Dz an equation in x, y, z which, with u = qp(x, y, z), F1 =0, and F2= 0, will serve to find x, y, z and u. Again, by differentiating equations (1), (2), (3), and eliinaingdy dz d2y d'z eliminating cly dz d "y d2Z we may deduce the value of d2' d - and test its sign for the values of x, y, z found. Ex. A Norman window consists of a rectangle surmounted by a semicircle. Given the perimeter, show that, when the quantity of light admitted is a maximum, the radius of the semicircle must equal the height of the rectangle. [TODHUNTER's DIFF. CALC., p. 214, Ex. 30.] Let y be the height and 2x the breadth of the rectangle, then the area of the window is given by A = rx2 + 2xy, and this is to be a maximum. For the perimeter we have P = 2/ + 2x + rx = constant. 400 MAXIMA A VD MINIMA. Choose x to be the independent variable. Then we have, since A is a maximum, d 0 = 0rx + 2y + 2x dx dx and since P is constant dP- o = 0=2d. +2+Tr. dx dx Eliminating?/Y we have 7r +2yv=.(7r+2), or x =Y =, 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 dd,2' d2p d2y and d2P0 = 2d2 dx~ -" dx" dx2 clx2 (12A therefore -A=+2( - 2 - 7r) = -7r - 4, and is therefore negative. Hence the relation found, viz., x=y, indicates a axinmum value of the area. 353. In the solution of such questions as the foregoing it is frequently unnecessary to employ any test for the discrimination between the maxima and minima, since it is often sufficiently obvious from geometrical considerations which results give the maxima values and which give the minima. 354. 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 dy dzy dz F'(z)f(x), dx - dz ' dx MAXIMA AND MINIMA. 401 the vanishing of either of the factors f(x) or F'(z) dy will give -d=0, and therefore y may have maxima or minima either for solutions of F'(z)= 0 or for such values of x as make f'(x)=O, and which therefore make z a maximum or minimum. Moreover, if z be not capable of assuming all possible values, it may happen that some of the roots of F'(z) =0 are excluded by reason of their not lying within the limits to which z is restricted. 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. If r and r' be conjugate semi-diameters, a and b the semi-axes, and p the perpendicular from the centre on the tangent at the point whose radius vector is r, we have r2 r'2 =a2 + b2 pr' = ab, giving a2 - a2 + b2 - r2 p2 Differentiating with respect to r a2b2 dpo_ p3 dr and putting d o, 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 0, which is 2C 402 MAAXIMA AND MINIMA. susceptible of all values and therefore suitable for an independent variable. We should thus have c2b2 dp dr =r_ p3 do d0' and the vanishing of r- indicates that the maximum and minimum dO values of p are to be sought at the same values of 0 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 a priori 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. 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 P VP', Q VQ' be diametral planes of the planet, perpendicular to the lines EV and S V, and let ZVZ' be the diameter perpendicular to the plane of motion. Draw QN at right angles to PP'. Let c be the planet's radius and x, a, r the lengths of EV, ES, and SV P' Qf E.S Fig. 89. 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 therefore bounded by the line ZQZ'PZ, whose projection upon the plane PZP'Z' is a crescentshaped area bounded by a semicircle and a semi-ellipse, the greatest MAXIMA AND MINIMA. 403 breadth being PN. The area of this crescent is 17C2 - tC. C cosN VQ, and therefore oc 1 - cos N VQ. rpl, 1- * l VO 1+cos EVS The brightness therefore oc 1-cos VQ or +coEV fEV2. EV2 Now cos EVSx 2 - a2 2xqr (X + r)2 - a2 1 2r r2-a2 whence brightness oc 2-a2 or - +-+ X3 x X2 This expression has its maximum and minimum values, (1) when x is a maximum or a minimum, i.e., when x-=a~r; (2) when +4r 3(r2 - a2) 0. (2) when + J x2 x~3 X4 This second relation gives 2 + 4rx +3(r2-a2)=-0, or x = '3a^2+ 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 /3a2 + r2 - 2r > a -r if r < a, and S3a2+r2-2r<a+r and a if r>. 4 For the inferior planets, Venus and Mercury, whose mean distances from the sun are respectively *7a and '39a 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=O, and is obviously a minimum. Hence x= %/3a2+r2 - 2r gives a maximum, and x =a+r a minimum. It is easy to deduce hence that, for the position of maximum brightness, 2 tan E=tan, 404 MAXIMA AND MINIMA. an equation due to Ialley, and 3a cos2E+ 4r cos E- 4a = 0, which determines the angle E. [See GODFRAY'S ASTRONOMY, 2nd Ed., p. 287.] 355. Other Maxima and Minima; Singularities. The accompanying figure (Fig. 90) is intended to illustrate some points with regard to maxima and minima which we have not at present considered. Y K L R / Q N 0 S' T1 VW x,!l/ i, Fig. 90. At S there is an asymptote parallel to the y-axis. The curve y =(x) approaches the asymptote at each side towards the same extremity. Here y= oo and 2y= dy dy= 0o, but dy changes sign in crossing the asymptote, and there is an infinite maximum 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 d- does not change sign; there is dx therefore neither a maximum nor a minimum at T. MAXIMA AND MINIMA. 405 At M there is a "point saillant " giving a discontinuity dy in the value of dx. The ordinate at such a point is a maximum or a minimum. In the case in the figure we have a maximum ordinate. At R the curve has a "point d'arret " and a maximum dy ordinate, though d% does not vanish or become infinite. dy At N there is a cusp, but ~ is neither zero nor infinite. Yet the ordinate at N is the smallest in its immediate neighbourhood, and therefore a minimum. It is to be noticed, however, that in travelling along the branch MN the value of x does not pass through 0 W, and therefore the ordinary theory does not apply. dyi At such points as Q, - =oco 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 OV, but recedes to it from W to V and then increases again. We notice, however, that this dx dx result may be written as d=0, and that -y changes dy dy 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." 406 M4AXIMA AND MINIMA. EXAMPLES. 1. Find the maximum and minimum values of 2x3 - 15x2 + 36x + 6. 2. Show that the expression (x- 2)(x- 3)2 has a maximum value when x= 7, and a minimum value when 3 -x=3. 3. Show algebraically that the greatest value of x(a - x) is a, and illustrate the result geometrically. 4. Show that the expression X3 - 3x2 + 6x + 3 has neither a maximum nor a minimum value. 5. Investigate the maximum and minimum values of the expression 3x5 - 25x3 + 60x. 6. For a certain curve =y - (x- )(.- 2)2( - 3)3( - 4)4; dx discuss the character of the curve at the points x= 1, x= 2, x=3, x=4. 7. Find the positions of the maximum and minimum ordinates of the curve for which dy (x - 2)3(2x - 3)4(3x - 4)5(4x - 5)6. dx 8. Find for what values of x the expression ( - 1)4( + 3)5 has maximum or minimum values. MAXIMfA AND MINIMA. 407 9. Find algebraically the limits between which the expression ax +x must or must not lie for real values of x. Illustrate your result by a sketch of the curve b y = ax + -. x 10. Investigate algebraically the maximum and minimum X* 2-4x + 2 values of the expression -4 2x- 7 for real values of x. Illustrate your answer geometrically. 11. Investigate the maximum and minimum values of the expression 2x3 - 21x2 + 60x + 30. 12. Find the minimum ordinate and the point of inflexion on the curve x3 - axy + b3 = 0. 13. Find the maximum and minimum ordinates of the curve (y- c)2= (x- a)6(x- b). 14. Show that the curve y = xe has a minimum ordinate where x= - 1. 15. Show that the values of x for which exsix has maximum or minimum values may be determined graphically as the abscissae of the points of intersection of the straight line y= -x, with the curve of tangents y = tan x. 16. Show that the expression a + (x - b) + (x - b)4 has a minimum value when x = b. 17. Find the minimum value of a2 b2 siLn2X cos+ 408 MAXIMA AND MIINIMA 18. Show that sinO cosq attains a maximum value when 0 = tan-l,/P 19. Show that the function x sin x - cos x + cos2x continually diminishes as x increases from 0 to. 20. Show that:/e is a maximum value of (1-. 21. Show trigonometrically that the greatest and least values of the expression a sin x + b cos x are /a2 +b2 and - /a2 -b2. 22. Show by trigonometry that the greatest and least values of the function a cos20 + 2h sin 0 cos 0 + b sin2o a - b a- b are respectively -t~ / +h2 23. Find in an elementary manner the maximum and minimum values of the expression (a sin 0 + b cos 0)(a cos 0 + b sin 0). 24. If y=2x-tanlx-log{x+ /1 +x2}, show that y continually increases as x changes from zero to positive infinity. a2 b2 25. If =- +, x y where x y= a, show that z has a minimum value when a2 a + b' a2 and a maximum when x_ a a-b MA XIfAA AND MINIMA. 409 26. Given that +Y=1, a b show that the maximum value of xy is ab and that the mini4 a2b2 mum value of x2 + y2 is a2 +27. 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: 7r. 28. Show that the maximum and minimum values of 2 2 x + y, where ax2 + 2hxy + by2= 1 are given by the roots of the quadratic (a-M~- 1 =2. Hence find the area of the conic denoted by the first equation. 29. PSP', QSQ' are focal chords of a conic intersecting at right angles. Find the positions of the chords when PP' + QQ' has a maximum or minimum value. 30. Divide a given number a into two parts, such that the product of the pth power of one and the qth power of the other shall be as great as possible. 31. 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. 32. 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? 33. What fraction exceeds its pth power by the greatest number possible? 410 MAXIMA AND MINIMA. 34. 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.] 35. The centres of two spheres, radii r, r2, 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. 36. 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.] 37. AC and BD are parallel straight lines, and AD is drawn. Show how to draw a straight line COE, cutting AD and BD in O and E respectively, so that the sum of the triangles EOD, COA may be a minimum. [VIVIANI.] 38. 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. 39. If four straight rods be freely hinged at their extremities the greatest quadrilateral they can form is inscribable in a circle. 40. A tree in the form of a frustum of a cone is n feet long, and its greater and less diameters are a and b feet respectively. Show that the greatest beam of square section that can be cut out of it is ( b) feet long. 3(a - b) 41. 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". 42. The resistance to a steamer's motion in still water varies as the nth power of the velocity. Find the rate at which the MAXIMA AND MINIMA. 411 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. 43. Show that the volume of the greatest cylinder which can be inscribed in a cone of height b and semivertical angle a is. rb3tan2a. 27 44. 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. 45. Show that the chord of a given curve which passes through a given point and cuts off a maximum or minimum area is bisected at the point. 46. Two particles move uniformly along the axes of x and y. with velocities u and v respectively. They are initially at distances a and b respectively from the origin, and the axes are inclined at an angle o. Show that the least distance between (av - 6bu) sin o the particles is ( - bsin (u2 + v2 - 2uv cos o)-' 47. Find the area of the greatest triangle which can be inscribed in a given parabolic segment having for its base the bounding chord of the segment. 48. 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. 49. In a submarine telegraph cable the speed of signalling varies as x21og - 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 1: lJe. MAXIMAL AND MANIMA..412 50. S is the focus of an ellipse of eccentricity e, and E is a fixed point on the major axis, and P is any point on the curve. Show that when PE is a minimum SP SE 51. Find the maximum value of (x - a)2(X -) f (1) when a>b. (2) when a< b. What happens if a=b Illustrate your answers by diagrams,of the curve y = (x - a)2(x - b) in the three different cases. 52. An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expense of lining it with lead will be least if the depth is made half of the width. 53. If two variables x and y are connected by the relation ax2 + by2 = ab, show that the maximum and minimum values of the function x2 + y2 + xy will be the values of u given by the equation 4(u - a)(u - b) = ab. 54. If SP and SQ be two focal distances in an ellipse ineclined to each other at the given angle 2a, find the greatest and least values of the area of the triangle PSQ. 55. 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. 56. Find the point P on the parabola y2 = 4ax 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? 57. Find the area and position of the maximum triangle MAXIMA AND MINVIMA. 413 having a given angle which can be inscribed in a given circle, and prove that the area cannot have a minimum value. 58. From a fixed point A on the circumference of a circle of radius c the perpendicular A Y is let fall on the tangent at P. Prove that the maximum area of the triangle APY is c2 3. 59. 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. 60. 0 is a fixed point without a circle, A one of the extremities of the diameter through 0, OQQ' a chord through 0. Find its position when the area of the triangle QAQ' is a maximum. Does it ever become a minimum 61. Describe the equilateral triangle of greatest area, each of whose sides passes through a given fixed point. 62. 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 of the areas be the least possible the side of the square is double the radius of the circle. 63. Find the least isosceles triangle which can be described about an ellipse with its base parallel to one of the axes, and show that its sides are 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. 64. 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. 65. Prove that a chord of constant inclination to the arc of -414 4MAXIMA AND MINIMA. a closed curve divides the area most unequally when it is a chord of curvature. 66. Show that the normal chord to the parabola y2=4ax Lat. Rect. which cuts off the least arc is normal where y = --—. and ~/3 is inclined to the axis at an angle tan-l 2,13 67. When the product of two perpendicular radii vectores of a curve is a maximum or a minimum, show that they make supplementary angles with the tangents at their extremities. 68. 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~. 69. 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 =P 70. 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. APPENDIX. APPENDIX. GEOMETRICAL PROPERTIES OF THE CYCLOID. 356. DEF. When a circle rolls in a plane along a given straight line, the locus traced out by any point on the circumference of the rolling circle is called a CYCLOID. 357. Description of the Curve. The nature ot 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 Y B D G A C T M X Fig. 91. cycloid to the portion of the curve lying between two consecutive cusps. 2D 418 APPENDIX. Let A, B be 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 C 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 CY the normal, obviously 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 CE. 358. 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. 359. Equations of the Cycloid. Let DQC be the circle described upon DC for diameter and let 0 be its centre. Draw PM, PN perpendicular to CX and CY respectively, the latter cutting the circle DQC in Q. Join DQ, OQ, CQ. Now, since the circle rolls without sliding along the line AB, every point of the circle comes successively into APPEVNDIX. 419 contact with the straight line, so that the length of AD is half of the circumference of the circle, and the portion GA = arc GP = arc DQ. Hence the remainder DG = arc CQ. Now, PQCT, PQDG are parallelograms; whence, if a be the radius of the generating circle and 0 the angle COQ, PQ = DG = arc CQ = a. Hence, if x, y be co-ordinates of P, x = C=NQ + QP = a(0 + sin 0). y = CN=CO-NO=a(1' os0) f. A ) From these equations the Cartesian equation may be at once obtained by eliminating 0; the result being x=a vers-+Y 2ay-y2,.................. (B) but from the form of the result the equation is not so useful as the two equations marked (A). 360. Length of the arc CP. Since x = a( + sin 0) y = a(l - cos 0)' we obtain dx = a(l + cos0 )dO dy = a sin OdO J squaring and adding ds2 = dx2 + dy2 = 2a2(1 + cos 0)d02 = 4a2cos20d02 0 or ds = 2a cos 0dO, 2 and upon integration s = 4a sin.......................... (c) 2 the constant of integration vanishing if s be measured from (, so that s and 0 vanish together. 420 APPENDIX. Again, since chord CQ = 2a sin 0, we have arc CP = 2 chord CQ.............. (D) 90 Further, since y= 2a sin2,8 = 4al2 =, 8ay..... (E) 361. Geometrical Proofs. These results may be established by geometry as follows:Let TPG be any position of the generating circle, G 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 B D G G A C T T' X Fig. 92. 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 APPENDIX. 421 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 CP is double that of the chord TP, and they begin their growth together at C. Hence arc OP = 2 chord TP. 362. Intrinsic Equation. If in Fig. 91 PTX=, we have =; whence the intrinsic equation of the cycloid is s = 4a sin r. 363. Radius of Curvature. The formula of Art. 270 gives p = -lj =4a cos = 4ac cos 2 = 2, i.e., radius of curvature = 2. normal. 364. Evolute. By Art. 292 the intrinsic equation of the evolute of the curve s=f(/) is s=f'(i/). Applying this, we have for the evolute of the above cycloid s = 4a cos ~, which clearly represents an equal cycloid (see Art. 294). 365. Geometrical Proofs. These results may also be established geometrically as follows: — Let AD be half the base and CD the axis of a given cycloid APC. Produce CD to F, making DF equal to CD, and through F draw FE parallel to DA. Through 422 APPENDIX. 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'. F G E B D. A C T X Fig. 93. On GT and G'G as diameters describe circles, the former cutting the cycloid in the tracing point P. Join PT, PG and produce PG to meet the circle GP'G' in P' and join P'G'. 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 AC and therefore its evolute; and P' is the centre of curvature corresponding to the point P on the.original cycloid. 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 A PPENDIX. 423 curve FP'A, will trace out the cycloidal arc APC. Thus a heavy particle may be made to oscillate along a cycloidal arc, by allowing the suspending 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. 366. Area bounded by the Cycloid and its Base. Let PGP', QG'Q' be two contiguous normals. Then G, G' are their middle points, and therefore ultimately the elementary area GPQG' is treble the elementary area P'GG'Q'. Hence, summing all such elements, the area F P GQ C X Fig. 94. APCD is treble the area 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 CXAP. 424 APPENDIX. Now the length of AD= half the circumference of the circle = 7-a. Hence the rectangle AXUD = 7ra.2a = 27ra2 and therefore the semicycloidal areaAPCD = g. 27-c 2= Brra2 and the area bounded by the whole cycloid -and its base =3wc&2, and is therefore three times the carea of the gernerating circle. ANSWERS. ANSWERS TO THE EXAMPLES. CHAPTER I. PAGE 11. 1. (i.) o; (ii.) 1; (iii.) oo. a 2. (i.)~; (ii.)2. 3. oo. 4. +~ a C] 1. Xx + Yy/= c2. 2. XX Y 1. a2 b'2 3. Y y=y(X-x). 5. 1. 6. a. 7. 3a2. b a 8. (i.); i. a U, 9. 2. EIAPTER II. PAGE 30. 4. x(Y-y)=X-x. 5. cos2x(Y-y) =X-x. 6. (l +x2)(Y-y)=X-x. PAGE 32. 2 o t 1 cosX 1 1. sec2x. 2. 2. 3. - 2 x 4. -- 1 + X2' sin2xx ^ /x- 1I 1. 3X2. 2. /a. X.0. 4. ex. PAGE 36. e^/ 5. 2-ev 6. asinxcosx logea.. 7. lalogxlogea. Xn 3x2 8. -. 1 3aX2 9. - tan x. 10. 2 sin. 11. xa(logex+ 1). 428 ANSWERS TO THE EXAMJIPLES. 12. xsin{ cos. logx+ sin}. 14. (Si (log sin x + x cot x). 13. (sin x)x logsinx+xcot }. 17. (0, 0) and (2a, - 4) a2 3- -— b2 18. ( + a2 +,/-+b4 PAGE 48. 1. log sin x + x cot x. - a3 a2- 2x2 '2 2' 1J;C2~1+2' +sinx x 5. a 2 COSx.logeC,a. 3. - ec I - C). 6. eV-" X cot v(sin 3vw)2(log sin w + w cot w). X \ X/ / CHAPTER III. PAGE 65. 21 9. bnxn-lcos (a + bxn). 7r 1. Td~' COS X ~ J115. 2 /x 10. Cos x X0 -x2 2. - z. 2/x 16 1 2z~' 16. 3 b 1 cos x x[1 + (log x)2] c 1 2 sinx 17. rCos 4. 8- 1 os x~. 8 ~2 12. x ez. a4,/x sin Vx 18. logx + 1 5. ex. 6. cos x. 13. pqx-lcosxqsinP'-lxg. 19 exlog(exx). 7. -sin. 14. 2x x 8. bcos(a +bx). ^/1-x 20. cos ex. ex. log x + sn e 21. log cot x 2 tan-le x cosh x sin 2x 22. (x +a)-l(x + b)n-l[(m + n)x + b +na]. x2+2x -2 I -n 27. tanh x. (x + 1)2 2. (2+2) n 28. sech 2x. -1ii sinh x 2 24. I(a+x) n. 26. 29!csh./ n(, 2 /cosh xl^-x 2 cosec 2x 30. - 1/2 log cot - (log cot x)2 cos X 1 1 31i. 32. 33. — _ 1 4 sin~x 1+ xa ANSWERS TO THE EXAMPLES. 429 3 1 x2 - 2xv2 - 2xv + 1 34. 35.S sinm - x cosn2-lx(mn cos2x - n sin2x). 2x2(1 + X)( + X2) 36. (in-lx)m -l( x)n- l(m cos - x-n sin-lx). 37. cos(eVlog x)e2og(xezi ^ (log x)2 sin (elogx) 37. cos(exlog x)exlog xexJ^ Vilogx)' sin (e~k )V log x)3 38. - 42. 2(1-x2) (1-x)'(l x)2 1 + X2 + xI (1-) x )(1 +x) 3x + x' 39. 39.- - '3 43. log(). (1 + X2')' x4 - 2a2x2 + 4a4 2 40. 44. (X2 - a2)(x2 - 4a2) - 2 + 2x- -x2 41. -_ - 2x- x. 45. () X{ 2(1 - x)(l + X + X2) 2 n 46. +ab 2( tan1 —X +-.2 a2+ ^2(tan-) f a a -+x _t. n(log ex -2 +1} 9 _ /b2-a2 b + a cos x 47 cos-lx - x l-l - x. (1 - x2)2 48. asin(a cosec x' N/2- 1 -1 ) 4' 50. 2etal-l1{logsecx +3tanx}. t l+x' f 51. ex acos(btan-lx) 52. xacx(2 + ex logea) 1 + ac'x4 3 x logge sin (log. /a'2 + X2) (a2 + X2) cOS2 (log3 Va2 + X2) 54 2 6 55. 6 56. x log 5 x log x logx... log-l 58.. '+ 1 a + b cos x 59. 1 1 _1-iX 2V /x- x2 - b sin (b tan-x)} 60. /i(1 + 4x) tX2.12 61. - 2_ 2 4 -62. x log ex - 2 log. e 63. x. xx (log x)2 + lo + 1. x+ 64. xe. ex{logx+. 65. ex. xx(log + 1). 66. ex. e. 67. 10. 101ox(loge10)2. 430 ANSWERS TO THE EXAMPLES. 68. (sin X)cOS x( osx - sin x log sin x\ - (cosX)sin ) (Si2x - os x log cos x). sin x \cos x 69. - (cot x)cotx cosec2x log e cot x - (coth x)coth x cosechx log e coth x. 70. acxxsi1 x log( ac e 1 - 2 70. / acxcosx e )+tasn-1(acxinx) Z\2' 1 + X( ) + tcx an-Xxacxxsin 2x"+ x (+ 1 etan~, kx 71. etan 1-e2 tan- 'l 1+ X2 ( sin + cos 1 - sin + cos — x x x x 2 72. 2 '1 + cos)(1 - sin-) 73. /1- x - 2v/ x 4/'x /1 — 2,/Vx + co- + cos-x1 + cos-1x) 74. y 2xeZ sex2log-a + Vx - -sin ex L 1+~2Vax 2 V(I+(1 +Vx)(1 + 2V\x) 75. -ycotx(1 + 2 cosecx log cos x). 76. - { lg cot — + + - x(1 + x)cot-ix' 77. (l+ {log-+l l}+x {x+ l-logx}. 78.b x2 + y2 - ay OY _y ) 78. x2+y2-ay 4. y(x -y) (X2 + y2) sec - bx (x + ) b 85. i' 79. cos x cos 2x cos2yecs2x. x- xy log x' 80. - x+hy 86. Y log y 1 + x log log y hx + by xlog x 1- x log y 81. n- bxn i -1. y{(a+bx)y - b2} 2xL (a+6bx){ (a+6bxl) - a J} 87. x(y- x)(a+ bx) 82. ytanx+logsiny 88. _ax + hy +g log cos x - x cot y hx + by +f 83. x(3 + 2 tan log x + tan2log x). 89. Y 90. 6x(1 + 2) tan x3. etan-l 1 log1e X 1 + -2 log sec2x3. etan-y 2x2 (1 + a2cos2bx)(2 + ax+ a2) [ 2x + a) log cot - osec x(x + ax + a2) '92. - ab sin bx ab tan 94/1- sin ). (c93..acot b2tanx0 94. sin'x(log x+ i — n2b\ I2/ x ANSWERS TO THE EXAMPLES. 431.95 9.. 96.,1 /l+x- /1- - 97.. 98. -. 9. 1. X AA+j2- X 12 -2 - 100. 2 n(1 + X2) tan- x log tan -x + x 2,+3 100. 2 ----.-. --- -,-.C t (1 + x2) tan-lx( Vx cos l/x - 3 sin,/x) 110. e- XZ 8z +5x - (54 + 4x5) sec-lxVz/]. 4z + 2-L2x X2/z-1 I CHAPTER IV. PAGE 82. 1 (-l)nnf a n a a b ' a - b l(x - a)n+l (x-b)n+l ' I 31fn+l. 2 ( l)n! __1_ 321 +1 7 - (X3)+1 (3x-2)n+1 J3 3.n/ 7r)\ 3n Sn3 + nv)\ 4. 2 2 4. e2 2n-f 1 + 2cos(x + }.. 3(a2 + bsin bx + n tan-l ) - (a+ 9b2)s sin(3bx + n, tan-13-b 6. (-1)l-l!-(n + 2)( + 1) 3(n + 1) 4 2(x -l)n+3 (X - 1)12+2 (-I)t+l- (x -2)w+1_ I (-1)n 1 7. __ _ _n _ _ I 2a- It ($- a)+ (+ a)t+l ' 8. ( 1) 2! sin(n + 1)0 sin+l0, where x = a cot 0. aW+2 9. () - 1)n-( - 1)! sin nO sinwO, where x = a cot 0. an 10.( - 1)(n -1 )!sin(n + 1)0 sinn+lO sin(n + 1)b sin'+t-ll * a2L- b2 bn+2 an+2 where x = b cot 0 = a cot 0. ( —),nn! 1 1 2 4a L(x -a)in+ (x +a)n+l- a+ sinsin( + )sinn+ where x = a cot 0. 12. Y a-2( '-3) n)!{sin(n + 1)0sin+'l0-sin(n + 1) sin+:''}, 2where x= acos ( - = cos + wherex- c=os 0 a -cosf0+ 7Y sin 0 6 sin0 6 432 ANS WERS TO THE EXAMPLES. PAGE 89. 1 Y 2(1-3x4) 2. 2. 2 ((+x4 2.. 3. 2= -(+)2 sin (m + )x -( 2 sin ( - n)x. 4. y3=a eax(ax+3), 2 2 If r<n, yr =n(n-1)...(n-r+ 1)xn-r. 5. If r=n, yr=n!. If r>n, /r =0. 18. y,,= an-2ea I a2x2 + 2nax +n (n - 1). 19. y= an-2{a2X2sin ax+ n ) + 2nax sin ax+ n 7) +n(n -1)sin(ax+ n -2 )}. 20.,n (-1)n!f a2 b2 _ 2 (a-b) l(x-a)nl+i (x - b)+1}' 21. y =(-)-lnf(n + 2)(n + 1) n+ + 1 - 1 2(x - )n+3 (x -_)n+2 (x - 1)n+1 (x -2)"n+ 22. = ( - 1)n(n- 2)! sinl-10{sin n-0 - (n - l) cosn sin 0}, where x = cot 0. 23 = -1)n +n! logx+l+ 1 + +. xn- 2 3.... 36. yn = an+2x2eax. CHAPTER V. PAGE 117. 21. 2(x+ + 9- +...). 25. tan-l^/ - tan-x = etc. 22. Double the series n 21. 26 tan- six etc. 26. tan-l - = sm-lx = etc. 23. Treble the series in 21. \/1-x 24. tan -lP- x=tan -P-tan-'x=etc. 27. sec-1 =2sin-x = etc. q +px q 1-2X2 28. sin-l 2x= 2 tan - Ix = etc. 1+ X2 29. cos - x- = 2 cot-lx = - 2 tan-lx = etc. X + X-1 30. sinh-'(3x + 4x) = 3 sinh-lx = etc. 32. x5 34. Expansion of e-kxcos bx. (See question 4.) ANSWERS TO THE EXAMPLES. 433 36. 1 + nx + -2X n(n -12)X3+ n2(n2 - 22)X n(n2 - 12)(12 - 3.2)X +. 2! 3! 4! 5! 37. The relation between three consecutive coefficients is,2(n + l)an+l = 3an + (2n - l)an-i. 38. yr = a 2b - a2 a -6abx. 3. y —+ — ++ X2 +_ XI.... 4 2 4 12 63. mx- (m — 1)(n - 2) 3 + m(m - l)(n - 2)(m - 3)(m - 4).5 3! 5! CHAPTER VI. PAGE 149. 6. (1) Ady 1 + 4xz dz 1+2x dx 1-2z' dx 1- 2z ) dx 1-2z dz 1 + 2x (2) - dy 1 +4xz' dy 1 + 4xz (3) dx 1 - 2z dy 1 4xz dz 1 +2x' dz 1+2x' a3 a3 2a3 a3 2a3 X2y/ xyy 2 y x2y2 xy. PAGE 152. 2. (a) ax hy hx + by ) 4 - 52y 4y3 - 5a2x' ) y tan x + log sin y log cos x - x cot y ) yxlog y + yx-1 - (x + y)x+Ylog e(x + y) xylog x + xyx-l - (x + y)x+ylog e(x + y) Y -lyx+l +. yXYxl0ggy cos- os_ ylogY x. log ~~~(e) - -_____x___- x xy+ lyx-l + x. yxlog x + xcos Ylogx. sin y - log xylog x y ' V _v '3 v ( zv cnxn-1 d6. ^VxJ^ r y y a x x a? -1 -U ' U - I - (n - 1 ____ 'Ax?y ay ' x 18 1 yaz b^?cn X2nX)-n-1 x 3\nz-1 16. dy sin cz c t ()n-l +co dz cosy c- b siny d a ()n- - 2E 434 ANS WERS TO THE EXAMPLES. a2b2 X2 + y2 22. 22 (a2x2 + b2y2) CHAPTER VII. PAGE 163. Ex. 1. Tangents. (1) Xx+ Yy=c2. Y-ysh X-x). (4) Y - y:slnhX(X - x). (2) Yy =2a(X + x). c ( X + Y (5) X(2xy + y+ y2) + 2xy)= 3a3. x y (6) Y-y=cotx(X-x). (7) X(x2 - ay) + Y(y2 - ax) = axy. (8) X{2x(x2 + y2) - a2x} + Y{2y(x2 +y2) + 2y} =a2(x2 -y2). Normals. (1 X=Y. (2) 2 Y =0, etc. xy a y Tangents are Y=- +33 -a 8 8 2. Normals are Y= T 8/3L y + 41a 9 36 For an ellipse, r2 = a2cos2 + b2sin20. 4. For a rectangular hyperbola, r2 = a2cos 20. PAGE 192. 1. (a) Parallel at points of intersection with ax + hy = 0. Perpendicular at points of intersection with hx + by = 0. (P) Parallel where (- -, 3av2^; perpendicular where = 0. { Parallel at (4a 23V4 Perpendicular at (0, 0), (2a, 0).. (a) ax=-by._ 2. (83) ax= + Jb2-ay. 3. ( (y) x=Oand y=0. xy 10. Area= ca4xy. 11. n= -2; n=l. 21 (a) Tangent, xcos0+ysinO0 21.(a) N, a - b - Normal, ax sec 0 - by cosec 0 = a2 - b2. ANSWERS TO THE EXAMPLES. 435 Tangent, x sin - y cos a sin f\ 2 2 2'. (~) 6-+ osin 8 0 a 8 Normal, x cos + ysin a0 cos + 2a sin. ~2 ~~2 2 2 A+fB CosA+B A - BA o (.4 + AB-i ( Tangent, xsin-B-B0- Ycos' 0= A ---sinf-B ' Normal, x cos ---O + Ysin — 0 =(A - B) cos B. 2B 2B 2B 28. u = ecos 0. CHAPTER VIII. PAGE 224.. 2a 6. x - 2a; 12. x= +a. 3 7. x+y+a=O. 13. x=0. 2. x+y=O0. 8. x=0, y=0, x+y=0. 14. x-a. 3. x+y=0. 9. y=0O 15. x=+~l, y=x. 4.- ==O 1 i~-a - 16. ==' 1=~(0. + ~a 5. ic=0. 11. x=a, y=a, x=y. - 2') 17. x+2y=0, x+y=l, x —y= -1. 21. x+y= ~2V2, x+2y+2=0. 18. x=0, x-y=O, x-y+l-0. 22. y=3x-2a, x+3y=+a. 19. y=0, x=y, x-y~+l. 23. 0=0. 20. x-2y=0, x+2y=~2. 24. rsin0=a 25, nr sin(0 - kr) = a sec k7r, where k is any integer. 26. rsin 0 = a. 27. r cos 0 = 2a. 28. 0 r sin 0 a 2 2' 29. r sin (o- krf), where k is any integer. n n' 30. nO = kr, where k is any integer. 34. r =b. 35. x= ~a, y=x. Above. y=X+a, y=-x-a, x=a. 36. In the first quadrant above the first. In the fourth quadrant below (. the second. 40. x- 6x2y + llxy2 - 6y3=x. 42. + 2 = 44. bay(x2- y) (a 2)( + ). ab - b = aa _ a(Cb a y 45. bxy(2 -?/2) = a(a 2 - b2)(X 2f +2- a'2). 436 ANS WERS TO THE EXAMPlPLES. 46. (X2-y2)2= Ax, or r3 a3s c. co. 2 2 -47. 2y2(x2 -y2) = 3aSx. 1. (a) =0. (P) ax= by. (Y) y = x. (0) x=o, y=O0. 14. x = a and x =2a. 17. 0 = sin-ld 27. x=7andx=1. CHAPTER IX. PAGE 258. 38. A straight line and a point. 40. A straight line and a conic. At (0, a), tan t = ~-2 48. At (a, 0), tan, = ~+ /3. At (2a, a), tan p = ~ 2 V3 (Tangents at origin, y =0 and y = ~ x. 50. Single cusp of first species at origin. x + y is a factor. CHAPTER X. PAGE 279. 1. p=a; p=acos\p; p=asec2. 2. p_2(a +x); p_ Y2 1 C a2 (a2sin20 + b2cos20) ab 3 a'2 6. p=a -0 + ) am +mt 1)r1~"1 1. 256y3 + 27x' = 0. a+ b' c4 2. 3+p —* X2 y a2,3 y+ 12g-X2 -2 ' 4. ((1) 4x3+27ay2=0. 1 (2) y2 = 4h(a + h - x). 5. y2 + 4a(x- 2a) =0. CHAPTER XI. PAGE 322. 6. Two straight lines. n n?n 7. (2) xn+1 + yln+l = kn+]. (3) xm - mmnn km+n (m + n)m+t8. A parabola touching the axes. 9. A hyperbola. ANSWERS TO THE EXAMPLES. 437 10. 27ay2 4(x - 2))3 / 2 2 2 (1) x - +y =k. 14. (2) x +y= 2n 2m mn (3) Xm+2 + y+-2 = - k?+2. (4) 2xy= k2. 46. apbq (pb + q)-P+p pPqq 1 1 1 (1) + y = k-, 15. m (2) x274+l + y2nt+l = k2m+1. (3) 16xy =k2 2 2 2 17. x' +y =a. 20.,2 = a2os20 + b2sin2O. 47. A conic. 1. logba. 2. 3 5' m 3. -, n 4. 1 n 5. 4. 6. 4. 7. 2. CHAPTER XIII. PAGE 365. 8. 1. 14. 1. 1 ' 215. 1 15. 1 10. 3 11 2' 16. -- 62 6 113. 18 - 3' 17..12. 1 613. 118. 13. 1. 3' 19. 1. 2' 20. 1. 21. oo. 22. 1. 23. e-. 24. 0. 25. e-. 26. eL. PAGE 376. 1. 2. 2.1 3. 2. 1 2'. 1. a 6. 4. 4m3 7. m — -. 3 8. 1 8 e 9. e. 10. 1. If n >m, oo. 11. fn =lm, A a n<m, 0. 12. 1. 2 13. ea. 14. e2. 15. e2. 16. ala)a3...an. 17. -1. 18. 0. 19. b, 0. 20.. 2 21. -a. 22. 3' 23. 0. 24. /a. 25. 1. 26. 1 18 27. -. 28. -e. 16 29. 0. 30. -( - l)an-2. 2! 31. 1. 33. (1+-V - 3). 34. Ooroo. 35. + b. -,/ a2 - b2 36. 0 or +1. 39. 4 15. 2 44. e 222. 45. i, b- l(b cos bx - sin bx)cos2bx. 46. -3 47.. 48. 1. 2 V/2a 438 ANSWERS TO THE EXAMPLES. CHAPTER XIV. PAGE 406. 1. Maximum Value = 34, Minimum = 33. 5. x= -2, -1, 1, 2, give Maxima and Minima alternately. At x = 1, y = Maximum. 6. x = 3, y = Minimum. At x = 2 and x = 4 there are points of contrary flexure. 4 7. At x = 2, y = Minimum. Atx =, y- Maximum. 8. x= - 7 gives a Maximum, x = 1 gives a Minimum. 9. It cannot lie between +2 V/ab. 11. x = 2 gives a Maximum, x = 5 a Minimum. 12. Minimum Ordinate at x = b A point of inflexion at (- b, 0). Atx=a, y=c. Atx= a6b, y c6a- b a being supposed greater than b. 17. (a +b)2. 23. J Maximum Value = (a + b)2. Minimum Value = -~(a - b)2. A Maximum when the chords coincide with the transverse axis and 29. lat. rectum. A Minimum when the chords are equally inclined to the transverse axis. 30. ap aq p+q' p+q 32. n parts. Continued product = el. 33. 1 "P 4f A Maximum when the segment is a semicircle. A Minimum when the radius is infinite. 35. The distances of the point from the extremities of the line are 2arl 2ar2 ri2 + r'2 /r2 + 22 2 2 36. The point divides the line of centres in the ratio rl3: r2, rl and r. being the radii. 37. AO:AD=1: V2. A NS WERS TO THE EXAMPLES. 439 38. If A be the smallest angle and b, c the adjacent sides, the distance of each end of the fence from A= -2, and the length of the fence A = /2bc sin 2 42. n+a knots an hour. n 47. Half the triangle formed by the chord and the tangents at its extremities, or three fourths of the area of the segment. I( a 2b a> b, Maximum if x 2b 51. a < b, Maximum if x - a. ca = b gives a point of inflexion. /2sin a cos a / 2sin a cos a If cos a be > e, Greatest = si a CO, Least = 1 sincos a (1 - e cosa)2 (1i+ecosa)2 54. If cos a be < e, the above values are both Minima, and there are two Maxima each equal to 2cot a. 55. The tangent at P must be parallel to SQ. If h < 2a, P is at the vertex. 56. If h > 2a, the abscissa of P is h - 2a, and the perpendicular is therefore the normal at P. 57. Maximum area= 4r2sin a cos3a, where i is the radius of the circle and 2a the given angle. 60. sin A OQ = CA C being the centre. 63 The height is three times the semiaxis to which the base is perpendicular. GLASGOW: PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE. January, I89o. A Catalogue OF Educational Books PUBLISHED BY Macmillan & Co. BEDFORD STREET, STRAND, LONDON. b CONTENTS. CLASSICSELEMENTARY CLASSICS.............. CLASSICAL SERIES..................... CLASSICAL LIBRARY, (I) Text, (2) ranslation....... GRAMMAR, COMPOSITION, AND PHILOLOGY...... ANTIQUITIES, ANCIENT HISTORY, AND PHILOSOPHY.. PAGE 3 7 II i6 22 MATHEATHETICSARITHMETIC AND MENSURATION.... ALGEBRA................ EUCLID, AND ELEMENTARY GEOMETRY.. TRIGONOMETRY............. HIGHER MATHEMATICS...... SCIENCENATURAL PHILOSOPHY.......... 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