UNIVERSITY OF CALIFORNIA AT LOS ANGELES J3KJVERSITY _ LIBRARY A TREATISE INTEGRAL CALCULUS FOUNDED ON THE METHOD OF RATES WILLIAM WOOLSEY JOHNSON Professor of Mathematics at the United States Naval Academy A n napolis, Ma ryland FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS LONDON: CHAPMAN & HALL, LIMITED 1907 Copyright, 1907 BY WILLIAM WOOLSEY JOHNSON a hr fcrirntifir $rtEa finhrrl Bntmmnttft anil Company Netn fork Engineering * Mathematical Sciences Library Q A PREFACE. THE present volume is an enlargement and an extension of my Elementary Treatise on the Integral Calculus, of which a revised edition was published in 1898. The enlargement con- sists chiefly in a fuller treatment of formulae of reduction, and of double and triple integrals in connection with their geometrical representation by cylindrical volumes and solids of variable density respectively. The extension, which constitutes nearly half of the volume, consists of Chapter IV on Mean Values and Probabilities, Chapter V on Definite Integrals including the Eulerian Integrals, Fourier's Series, etc., and Chapter VI on Functions of the Complex Variable. The book forms a companion volume to my Treatise on the Differential Calculus, founded on the Method of Rates (John Wiley & Sons, 1904) to which belong the references made in the text. The Integral Calculus may be said to consist of two distinct parts: the first concerns the reduction of integral expressions to previously known functional forms; the second concerns the mode of expressing a required magnitude in the integral form for subsequent reduction when possible. While the fluxional foundation does not affect the treatment of the former (which is essentially an inverse process, depending upon a body of rules established in the Differential Calculus); in the latter, it leads us to regard the required magnitude as a particular value of a iii iv PREFACE. varying magnitude, or fluent, of which the fluxion or rate of growth is to be ascertained. In some of the simpler applications, the differential, or hypo- thetical increment, which, in the treatment of the subject here followed, measures this rate of growth, admits of construction, e.g. in Arts. 103 and 116. In general, however, it is necessary to regard the magnitude in question as the limit of a sum. The identification of this limit with the definite integral is effected in Art. 97 by the aid of the area which has already been shown to represent the integral as originally denned. Accordingly, the integral expression for the fluent magnitude is for the most part found in the usual manner by means of its element. But, while the idea of the limit is of course essential, the rate method pre- sents the advantage that the integral is defined beforehand, so that the student realizes that it is the sum and not the integral which has an approximate character; in other words, the idea of approximation is not carried into the definition of the integral. The direct representation of the fluent integral as the ordinate of a graph, or curve determined directly by the integral expression itself, flows natifrally from the fluxional point of view, and is developed at some length in Arts. 85-91. Somewhat more than the usual amount of space has been devoted to methods of integration and their classification in such manner that the form of the expression to be integrated shall suggest the best method of attack. A careful consideration of this matter has led me to lay particular stress upon trigonometric substitutions. The subject of Mean Values of continuous variables forms one of the most useful and characteristic applications of the Integral Calculus. It seems therefore, next to the strictly geometrical applications, to be entitled to ample treatment, especially as it includes the important subjects of Centres of Gravity and Radii PREFACE. V of Gyration, of which the treatment is both simplified and shortened by reference to the underlying principles of Mean Values. In the examples following Section XXI will be found a con- siderable collection of discontinuous multiple-angle series, and in the text, Arts. 324-327, an interesting illustration of the relation of such series to the ordinary multiple-angle series, and their connection with the occurrence of divergency. To the section on the F -function is appended a logarithmic table of this function, and also a table of the values of s n by means of which many interesting numerical computations may be made. As in the companion volume, the sections are followed by large collections of examples with their answers. Many of the results being properties of well-known curves are rendered acces- sible by references inserted in the index. W. WOOLSEY JOHNSON. AUGUST, 1907. CONTENTS. CHAPTER L ELEMENTARY METHODS OF INTEGRATION. I. PAGB Integrals ............................................ . ...... ,....,.,... x The differential of a curvilinear area ............................ . ........ 3 Definite and indefinite integrals ......................................... 4 Elementary theorems ................ . .................................. 6 Fundamental integrals .......................................... . ....... 7 Examples I ............... . .............................. .......... 10 IL Direct integration .......... . ....................................... . . . c 14 Rational fractions .................................................... , 15 Denominators of the second degree ...................................... . 16 Denominators of degrees higher than the second .......................... . 19 Denominators containing equal roots .................................... . 21 General expression for the numerator of a partial fraction ................. .. 22 Examples II ............................................. . ....... . , 26 III. Trigonometric integrals ............................ ..... ................ 33 Cases in which sin"* cos" e de is directly integrable ....................... 34 The integrals sin"e) ...................................... . . 61 The integration of - _ - . 64 * d*) Transformation to trigonometric forms .................................. . 65 Radicals of the form y(ax y + bx + c ) .................................... 67 The integrals f - - and f - - ^= .............. 68 J y[(x - aX* - /?)] J V[(x - a) (ft -x)} Exampks V ....................................................... 70 VI. Integration by parts ........ ........................................... 77 Geometrical illustration ................................................ 78 Applications .......................................................... 78 Formulae of reduction .................................................. 81 Reduction of J sin"* 6 M and cos" 1 BdB 82 Reduction of f sin 9 cos* 6 dQ 84 CONTENTS. ix I Reduction of I cos m cos n dd. Reduction of algebraic forms 90 General method of deriving a formula of reduction 91 Development of an integral in series 93 Bernoulli's series 96 Taylor's theorem. 95 Examples VI 97 VII. The integral and its limits 105 Condition of continuity 107 Graph of an integral 1 09 Multiple-valued integrals 112 Formulas of reduction for definite integrals 116 Change of independent variable in a definite integral 119 The integral regarded as the limit of a sum 121 Additional formulae of integration 124 Examples VII 125 CHAPTER III. GEOMETRICAL APPLICATIONS DOUBLE AND TRIPLE INTEGRALS. VIII. Areas generated by variable lines having fixed directions 129 Application to the witch 130 Application to the parabola when referred to oblique coordinates 132 The employment of an auxiliary variable 132 Areas generated by rotating variable lines 134 The area of the lemniscate 135 The area of the cissoid 136 A transformation of the polar formula; 136 Application to the folium 137 Examples VIII 140 IX. The volumes of solids of revolution 147 The volume of an ellipsoid 149 Solids of revolution regarded as generated by cylindrical surfaces 156 Examples IX 151 CONTENTS. Double integrals 155 Limits of the double integral 156 The area of integration 159 Change of the order of integration 160 Triple integrals 163 Integration over a known volume 1 65 Representation of a triple integral by a mass of variable density 1 66 Examples X 168 XI. The polar element of area 172 Transformation of a double integral 174 Cylindrical coordinates ... 176 Solids of revolution with polar coordinates 178 Polar coordinates in space 179 Spherical coordinates , 181 Volumes in general 182 Examples XI 185 XII. Rectification of plane curves 189 Change of sign of ds 190 Polar coordinates 190 Rectification of curves of double curvature 101 Rectification of the loxodromic curve 192 Examples XII 193 XIII. Surfaces of solids of revolution 198 Quadrature of surfaces in general i 99 The determination of surfaces by polar coordinates 202 Examples XIII 203 XIV. Areas generated by straight lines moving in planes 206 Applications 207 Sign of the generated area 209 Areas generated by lines whose extremities describe closed circuits 210 Amsler's planimeter 211 Examples XIV 213 CONTENTS. XV. PAGE Approximate expressions for areas and volumes 215 Simpson's rules 217 Cotes' method of approximation 218 Weddle's rule 219 The five-eight rule 219 The comparative accuracy of Simpson's first and second rules 220 The application of these rules to solids 220 Woolley's rule 221 Examples XV 222 CHAPTER IV. MEAN VALUES AND PROBABILITIES. XVI. The average or arithmetical mean 224 The weighted mean 225 The mean of a continuous variable . 225 The mean ordinate. . 226 The mean of equally probable values 227 The mean of a function of two variables 228 The mean of values not equally probable 230 The centre of position of n points 232 The centre of gravity of unequal particles 234 The centre of gravity of a continuous body 235 Average squared distances of points from a plane 237 The moment of inertia and radius of gyration of an area 237 The radius of gyration of a solid 238 Examples XVI 240 XVII. Mean distances from a fixed point 243 Mean distances between two variable points 246 Mean distances connected with a sphere 248 Random parts of a line or number 251 Random division into n parts 252 Mean area of a triangle with random vertices 256 Mean areas found by the method of centroids 258 Examples XVII 263 xii CONTENTS. XVIII. PAGE The measure of probability 266 Probabilities represented by areas '. 268 Local probability 269 The element of probability 273 Curves of probability 276 Mean values under given laws of probability 282 Probabilities involving selected points 284 Selected points upon an area 286 Random lines 287 Probabilities involving variable magnitudes 289 Examples XVIII 292 CHAPTER V. DEFINITE INTEGRALS. XIX. Differentiation of a definite integral 296 Integration under the integral sign 299 Application to the evaluation of definite integrals 301 Employment of double integrals 303 Transformation by change of variable 306 Substitution of a complex value for a constant 310 Examples XIX 311 XX. Infinite values of the function under the integral sign 315 Cauchy's general and principal values 317 Integrals with infinite limits 318 Integrals of certain rational fractions 320 Frullani's integral 325 Integrals obtained by expansion 329 Series in sines and cosines of multiple angles 332 Integrals developed in multiple-angle series 334 Integrals involving the expression A+ B cos x(A* >B 2 ) 336 Examples XX .' 339 CONTENTS. xiii XXI. PAGE Functions expressed in multiple-angle series 341 Fourier's series 342 The series in multiple-sines 345 Developments containing both sines and cosines 347 Discontinuity of the Fourier's sine-series 350 Geometrical illustration 352 Differentiation of multiple-angle series 353 Integral of multiple-angle series 354 Series obtained by transformation 360 Functions with arbitrary discontinuities 361 Formulae involving both sines and cosines. . . , 364 Examples XXI 365 XXII. The Eulerian integrals 372 Gauss's II-function 374 The gamma-function 377 Transformations of the Eulerian integrals 379 Relation between the two Eulerian integrals 381 Reduction of integrals to gamma-functions 382 Reduction of certain multiple integrals 386 The function log F(i +#), and Euler's constant 389 The logarithmic derivative of F(x) 392 Examples XXII 396 Table of log /*() 401 Table oj values oj s n 403 CHAPTER VI. FUNCTIONS OF THE COMPLEX VARIABLE. XXIII. Complex values of the derivative 404 Conformal representation -. 406 Conjugate functions of x and y 407 Two-valued functions 410 Multiple-valued functions 412 Meaning of integration when the variable is complex 414 Integration around a closed contour 416 XIV CONTENTS. PAGE Integration about a pole 417 Integrals of functions with poles 419 Integration about a branch-point 420 Integrals involving radicals 42 1 The modulus of a sum 425 Power series in the complex variable 426 Circle and radius of convergence 428 Taylor's series 429 One-valued function must admit of infinite value 431 Examples XXIII 432 INDEX 437 THE INTEGRAL CALCULUS. CHAPTER I. ELEMENTARY METHODS OF INTEGRATION. I. Integrals. f. IN an important class of problems, the required quanti- ties are magnitudes generated in given intervals of time \vith rates given in terms of the time / ; or else, being assumed to be so generated concurrently with some other independent variable, have rates expressible in terms of this independent variable and its rate. For example, the velocity of a freely falling body is known to be expressed by the equation v=&, (0 in which t is the number of seconds which have elapsed since the instant of rest, and g is a constant which has been deter- mined experimentally. If s denotes the distance of the body 2 ELEMENTARY METHODS OF INTEGRATION. [Art. I. at the time /, from a fixed origin taken on the line of motion, v is the rate of s ; that is, ds _ ~ hence equation (i) is equivalent to ds = gt dt, ........ (2) which expresses the differential of s in terms of / and dt. Now it is obvious that \gf is a function of / having a differential equal to the value of ds in equation (2) ; and, moreover, since two functions which have the same differential (and hence the same rate) can differ only by a constant, the most general expression for s is ' ....... (3) in which C denotes an undetermined constant. 2. A variable thus determined from its rate or differential is called an integral, and is denoted by prefixing to the given differential expression the symbol , which is called the integral sign.* Thus, from equation (2) we have = \gtdt, which therefore expresses that s is a variable whose differential is gtdt ; and we have shown that \gtdt = \gP + C. The constant C is called the constant of integration; its occurrence in equation (3) is explained by the fact that we have not determined the origin from which s is to be measured. * The origin of this symbol, which is a modification of the long s, will be explained hereafter. See Art. 100. I.] THE' DIFFERENTIAL OF A CURVILINEAR AREA. 3 If we take this origin at the point occupied by the body when at rest, we shall have s = o when t o, and therefore from equation (3) C o; whence the equation becomes s \gfi. The Differential of a Curvilinear Area. 3. The area included between a curve, whose equation is given, the axis of x and two ordinates affords an instance of the second case mentioned in the first paragraph of Art. I ; namely, that in which the rate of the generated quantity, al- though not given in terms of t, can be readily expressed by means of the assumed rate of some other independent variable. Let BPD in Fig. I be the curve whose equation is supposed to be given in the form Supposing the variable ordinate o \ \ \ dx PR to move from the position AB to the position CD, the required area ABDC\s the final value of the FIG. i. variable area ABPR, denoted by A, which is generated by the motion of the ordinate. The rate at which the area A is generated can be expressed in terms of the rate of the independent variable x. The required and the assumed rates are denoted, respectively, by -^- and ; and, to express the former in terms of the latter, it Is necessary to express dA in terms of dx. Since x is an independent variable, we may assume dx to be constant ; the rate at which A is gen- erated is then a variable rate, because PR or y is of variable length, while R moves at a constant rate along the axis of x. Now dA is the increment which A would receive in the time A R S C 4 ELEMENTARY METHODS OF INTEGRATION. [Art. J. dt, were the rate of A to become constant (see Diff. Calc., Art. 22). If, now, at the instant when the ordinate passes the position PR in the figure, its length should become constant, the rate of the area would become constant, and the increment which would then be received in the time dt, namely, the rectangle PQSR, represents dA. Since the base RS of this rectangle is dx, we have dA =.ydx- f(x]dx (i) Hence, by the definition given in Art. 2, A is an integral, and is denoted by A = Definite Integrals. 4-. Equation (2) expresses that A is a function of x, whose differential \sf(x)dx ; this function, like that considered in Art. 2, involves an undetermined constant. In fact, the expres- sion f(x]dx is manifestly insufficient to represent precisely the area ABPR, because OA, the initial value of x, is not indi- cated. The indefinite character of this expression is removed by writing this value as a subscript to the integral sign , thus, denoting the initial value by a, we write A ( f( y \f1r (^ " \ J \x) ax i \6) in which the subscript is that value of x for which the integral has the value zero. If we denote the final value of x (OC in the figure) by b, the area ABDC, which is a particular value of A, is denoted by DEFINITE INTEGRALS. writing this value of x at the top of the integral sign, thus, ABDC = \f(x)dx (4) a This last expression is called a definite integral, and a and b are called its limits. In contradistinction, the expression f(x]dx is called an indefinite integral. 5. As an application of the general expressions given in the last two articles, let the given curve be the parabola Equation (2) becomes in this case A = ( Now, since \x* is a function whose differential is x*dx, this equation gives A - x*dx - \x* + C, ..... (i) in which C is undetermined. Now let us suppose the limiting ordinates of the required area to be those corresponding to x i and x 3. The vari- able area of which we require a special value is now represented by I x*dx, which denotes that value of the indefinite integral which vanishes when x = i. If we put x = I in the general expression in equation (i), namely %x* + C, we have -^ + C; hence if we subtract this quantity from the general expression, we shall have an expression which becomes zero when x i. We thus obtain 6 ELEMENTARY METHODS OF INTEGRATION. [Alt. 5. Finally, putting, in this expression for the variable area, x = & we have for the required area 6, It is evident that the definite integral obtained by this process is simply the difference between the values of the indefinite integral at the upper and lower limits. This difference may be expressed by attaching the limits to the symbol ] affixed to the value of the indefinite integral. Thus the process given in the preceding article is written thus, J iv* = The essential part of this process is the determination of the indefinite integral or function whose differential is equal to the given expression. This is called the integration of the given differential expression. Elementary Theorems. 7. A constant factor may be transferred from one side of the integral sign to the other. In other words, if m is a constant and u a function of x, mudx = m udx. Since each member of this equation involves an arbitrary constant, the equation only implies that the two members have the same differential. The differential of an integral is by definition the quantity under the integral sign. Now the second member is the product of a constant by a variable factor ; hence its differential ismdl \udx\, that is, m u dx, which is also the differential of the first member. T.'J ELEMENTAR Y THEOREMS. 7 8. This theorem is useful not only in removing constant factors from under the integral sign, but also in introducing such factors when desired. Thus, given the integral (x n dx\ recollecting that 1 ) = (n + i)x n dx, we introduce the constant factor n + i under the integral sign ; thus, \x n dx - f (n + i)x n dx = x n + z + C. > J n + ij v n + i 9. If a differential expression be separated into parts, its in- tegral is the sum of the integrals of the several parts. That is, if u,v t w,''' are functions of x, \(u + v + w + -}dx = \u dx + \v dx + \w dx + For, since the differential of a sum is the sum of the differ- entials of the several parts, the differential of the second mem- ber is identical with that of the first member, and each member involves an arbitrary constant Thus, for example, | (2 Vx) dx = \2dx \xdx2x \x + C, the last term being integrated by means of the formula deduced in "Art. 8. Fundamental Integrals. 10. The integrals whose values are given below are called the fundamental integrals. The constants of integration are generally omitted for convenience. 8 ELEMENTARY METHODS OF INTEGRATION. [Art. IO. Formula (a) is given in two forms, the first of which is de- rived in Art. 8, while the second is simply the result of putting n = m. It is to be noticed that this formula gives an indeter- minate result when n = i ; but in this case, formula (&) may be employed.* The remaining formulae are derived directly from the for- mulae for differentiation; except that (/'), ('), (I'}, and (in'} * are derived from (/), (/), (/), and (m) by substituting for x. c* { n , _ x n + I [dx _ __ i , . )* -JTTi )x- ~ (m - i) *--' * (A log a = sin0 ............. (d) = - cos 0. ... ........ (e) ft * Applying formula (a) to the definite integral x n dx, we have }a n+ I which takes the form - when n I ; but, evaluating in the usual manner, 3+i_ fl *-M-, ^ + I lo g ^-^ + I logfl-| = log b - log a ; + i J = i i J = i a result identical with that obtained by employing formula (3). f That sign is to be employed which makes the logarithm real. See Diff. Calc., Art. do. I.] FUNDAMENTAL INTEGRALS. f = sec d . = cosec 6 cot# dd = cosec (i) J sin 2 8 } r dx \-7-f -gt = sin" 1 x J V(i - ^) = ~ cos C'. (k) + JT a -cot- 1 -+ C'. . a a C= - C' L -^= i sec- 1 -+ C= - - cosec- 1 - + C . a 2 ) a a a a dx vers . ' 1 dx = vers - . . (m) f f . (m) IO ELEMENTARY METHODS OF INTEGRATION. [Ex. I. Examples I. Find the values of the following integrals : dx i. ' I$- f dx 5- J i x s dx ' X - 3 5. J o 6. f (.r-i)^, J' a_ fT , x . , . ^V~IT 3 7. ( &c)V#, s ^ a^ 4 =,- Jo 3 Jo 3=o,/= ,s = . g 2g 12 ELEMENTARY METHODS OF INTEGRATION. [Ex. L 20. If the velocity of a pendulum is expressed by Tit v = a cos 2T the position corresponding to / o being taken as origin, find an ex- pression for its position s at the time /, and the extreme positive and negative values of s. 2Ta Ttt s = - sin . 7t 2T s = - when / = T, 37-, $r, etc. 21. Find the area included between the axis of x and a branch of the curve y = sin x. 2. 22. Show that the area between the axis of x, the parabola / = and any ordinate is two thirds of the rectangle whose sides are the ordinate and the corresponding abscissa. 23. Find (a) the area included by the axes, the curve and the ordinate corresponding to x = i, and (/?) the whole area be- tween the curve and axes on the left of the axis of y. 24. Find the area between the parabola of the th degree, a n ~ l y = x", and the coordinates of the point (a, a). L] EXAMPLES. 13 25. Show that the area between the axis of x, the rectangular hyperbola the ordinate corresponding to x = i, and any other ordinate is equivalent to the Napierian logarithm of the abscissa of the latter ordinate. For this reason Napierian logarithms are often called hyperbolic logarithms. 26. Find the whole area between the axes, the curve and the ordinate for x = a, m and n being positive. If n > tn, if n 5 m, 27. If the ordinate BR of any point B on the circle be produced so that BR RP = a 1 , prove that the whole area between the locus of P and its asymptotes is double the area of the circle. 28. Find the whole area between the axis of x and the curve 29. Find the area between the axis of x and one branch of the com- panion to the cycloid, the equations of which are x aty y = a (i cos ^>). 14 ELEMENTARY METHODS OF INTEGRATION. [Art. II. II. Direct Integration. II. In any one of the formulae of Art. 10, we may of course substitute for x and dx any function of x and its differential. For instance, if in formula (b) we put x a in place of x, we have f dx - = log (x a) or log (a x), ] x a according as x is greater or less than a. When a given integral is obviously the result of such a sub- stitution in one of the fundamental integrals, or can be made to take this form by the introduction of a constant factor, it is said to be directly integrable. Thus, sinmx dx is directly in- tegrable by formula (e) ; for, if in this formula we put mx for 0, we have sin m x m dx = cos m x , hence sin mx dx = sin m x m dx = cos m x . j m J m So also in J v(a + bx*} x dx , the quantity x dx becomes the differential of the binomial (a + bx*} when we introduce the constant factor 2#, hence this integral can be converted into the result obtained by putting (a + bx*) in place of x in \y xdx, which is a case of formula (a). Thus 2bx dx = r (a 3^ 11.] DIRECT INTEGRATION. 1 5 12. A simple algebraic or trigonometric transformation sometimes suffices to' render an expression directly integrable, or to separate it into directly integrable parts. Thus, since sin x dx is the differential of cos x, we have by formula (&) f f sin x dx tan x dx log cos x . J cos x jtan 2 6 dO = ((sec 2 6- i}dO = tan 6 - B ; by (e) and (a), [sin 3 BdB =\(i- cos 2 0) sin Ode = - cos + j cos 3 d ; by (/) and (a), x = sn ' * - Rational Fractions. 13. When the coefficient of d^ in an integral is a fraction whose terms are rational functions of x, the integral may gen- erally be separated into parts directly integrable. If the de- nominator is of the first degree, we proceed as in the following example. fj _ ^j- _|_ 2 Given the integral - - dx\ } 2.X + I by division, 2 _-__+J _ _ 3. + 15 _ L_ 2X + I 2 4 4 2.T + I* 1 6 ELEMENTARY METHODS OF INTEGRATION. [Art. 13 hence 1 f 3 f 15 f dk J 2X + I 2j * 4] a = ~j ~ 7 + loi 4 J 2.T + I J (2X + i). 3 \ / When the denominator is of higher degree, it is evident that we may, by division, make the integration depend upon that of a fraction in which the degree of the numerator is lower than that of the denominator by at least a unit. We shall consider therefore fractions of this form only. Denominators of the Second Degree. 14. If the denominator is of the second degree, it will (after removing a constant, if necessary) either be the square of an expression of the first degree, or else such a square increased or diminished by a constant. As an example of the first case, let us, take The fraction may be decomposed thus : X + I X I + 2 I 2 (x - I) 2 " (x - i) 2 ~ x - I "*" (x - I) 2 ' hence [ x + i j _( dx ( dx J(F=ip* -J^i 4 'JjF^Ty = log (x- \)- f ff .\_ t 15. The integral II.J RATIONAL FRACTIONS. 1 7 affords an example of the second case, for the denominator may be written in the form x* + 2x + 6 = (x + i) 2 + 5. Decomposing the fraction as in the preceding article, x + 3 x + i 2 (x + i) 2 + 5 ~ (x + i) 2 + 5 + (x + i) 2 + 5 ' whence )x* + 2x + 6' J(*+i) 2 + 5 J(^+i) 2 + 5* The first of the integrals in the second member is directly integrable by formula (<), since the differential of the denom- inator is 2 (x + i) dx, and the second is a case of formula ('). Therefore X + 3 , . , , -. 2 .X+l 2x + o) -\ tan" 1 : . 16. To illustrate the third case, let us take f 2x + i , -3 ^5^ J "* "* ~ ^ in which the denominator is equivalent to (x ) 2 6^, and can therefore be resolved into real factors of the first degree. We can then decompose the fraction into fractions having these factors for denominators. Thus, in the present example, as- sume (0 X* X 6 X 3 ;tr + .2 ' in which yi and B are numerical quantities to be determined. Multiplying by (x 3) (x + 2), 2x + i = A (x + 2) + B(x 3) (2) - f ^ : iLk r /** 1 8 ELEMENTARY METHODS OF INTEGRATION. [Art. 1 6. Since equation (2) is an algebraic identity, we may in it assign any value we choose to x. Putting x = 3, we find 7 = 5^4, whence A = -J, putting x = 2, 3 = 5^, whence ^ = |. Substituting these values in (i), 2* + i 7 3 x*-x-6~ 5(*-3) 5(* + 2)' whence ~ 3) 17. If the denominator, in a case of the kind last considered, is denoted by (x a) (x b}, a and b are evidently the roots of the equation formed by putting this denominator equal to zero. The cases considered in Art. 14 and Art. 15 are respectively those in which the roots of this equation are equal, and those in which the roots are imaginary. When the roots are real and unequal, if the numerator does not contain x, the integral can be reduced to the form f dx and by the method given in the preceding article we find -. r-7 r . = -. log (x a) log (x b) }(x-a)(x b) a b\_ 'J = J-zlog^?, ' (A)* * The formulae of this series are collected together at the end of Chapter II. for convenience of reference. See Art. 101. 11.] DENOMINATORS OF THE SECOND DEGREE. 19 in which, when x < a, log (a x) should be written in place of log (x a). [See note on formula (b), Art. 10.] If b a, this formula becomes f dx \.x-a . ... = log - (A 1 ) ^f> v _L n >. ' J x* a* 20. & x + a Integrals of the special forms given in (A) and (A') may be evaluated by the direct application of these formulas. Thus, given the integral f dx j 2x* + 3^- 2 ' if we place the denominator equal to zero, we have the roots a = , b = 2; whence by formula (A}, _ i I . x $ ~ ' g 5 --i) (X + 2)~2 or, since log (2x i) differs from log (x |) only by a con- stant, we may write f dx i , 2x i 2 5 Denominators of Higher Degree. 18. When the denominator is of a .degree higher than the second, we may in like manner suppose it resolved into factors corresponding to the roots of the equation formed by placing it equal to zero. The fraction (of which we suppose the numerator to be lower in degree than the denominator) may now be decom- posed into partial fractions. If the roots are all real and un- equal, we assume these partial fractions as in Art. 16 ; there being one assumed fraction for each factor. If, however, a pair of imaginary roots occurs, the factor cor- 2O ELEMENTARY METHODS OF INTEGRATION. [Art. 1 8. responding to the pair is of the form (x a) z + ft*, and the partial fraction must be assumed in the form Ax + B (x - of + f? ' for we are only entitled to assume that the numerator of each partial fraction is lower in degree than its denominator (other- wise the given fraction which is the sum of the partial fractions would not have this property). For example, given Assume (^ + \)(x - i) ~ * + i + J^~i' ' CO whence .. I } / ~ T W /4 -* _l_ P\ _1_ I -V-2 I T \ /"" * ~r 3 v* ij ^./i.* -f- Jj) -f- ^^t ( \ )L,. Since in an identical equation the coefficients of the several powers of x must separately vanish, the coefficients of x*, x and x give, for the determination of A, B and C, the three equations From these we obtain A = 2, B = i, C = 2 ; hence, substituting in equation (i), X + 3 2 2X + \ therefore * + 3 f dkr f 2* dx ( dx = 2 log (*i)_ log(^ + i) tan" 1 jr. 19. The method of determining the assumed coefficients illustrated above makes it evident that, the denominator being II.] MULTIPLE ROOTS. 21 of the nth degree, we must assume n of these coefficients because we have to satisfy n equations, derived from the powers of x from x n ~ 1 to x. It is evident that we may take for the denominators of the partial fractions any two or more factors of the given denomi- nator which have no common factor between themselves, pro- vided we assume for each numerator a polynomial of degree just inferior to that of the denominator. But, since our present object is to separate the given fraction into directly integrable parts, when a squared factor such as {x of occurs, instead of assuming the corresponding partial fraction in the form 2 (which would require to be further decomposed as in (x a) Art. 14), we at once assume a pair of fractions of the form A B x - a ' (x a? ' 20. We proceed in like manner when a higher power of a linear factor occurs-. For example, given JrJlflwF+l)^* 5 we assume x + 2 _/4_ C_ D m (x - i)\x + I) ~ (x - if + (x - \J + x - i + x + i 1 whence X + 2 - [^ + B( x - 1) + a^- 1) 2 ](* + o + ^(* - o 3 - (0 Putting x i, we have 3 = 2A .'. A = %. Putting ^= i, we have i = 8Z> .'. D = . The most convenient way to determine the other coefficients is to equate to zero the coefficient of x 3 , and to put x o. We thus obtain 22 ELEMENTARY METHODS OF INTEGRA TION. [Art. 2O. o = C + D, and 2= A B + C D, from which C = -, and B = . Therefore f x + 2 3 f dx i f dx \ [ dx if <&r 1 ___ yV -y - . I , I L _ . I . . _ \ Q / \ W--4- "- ' , O "T" r~ \ I / A- T \O/ ^y- | j \ O I / >* T I" /ill 'V T 1* ^1 -* T ?s I J I ^- 1 1 \*^ ~\ * J ^ J \" * ) *T J \ **' " ' ' A I O J ** " ' " 1 O J ^ 4 lo s _ 4(x - if 4(> - i) 8 x+i 21. The fraction corresponding to a simple factor of the given denominator may be found, independently of the other partial fractions, by means of the expression derived below. Denote the given denominator by (f>(x), and let it contain the simple factor x a, so that (x), ...... (i) in which ty(x) does not vanish when x = a. Let f(x] denote the numerator (it is not necessary here to suppose this to be lower than (f>(x} in degree), and let Q denote the entire part of the quotient. Then we may assume (x] " (x - d)$(x) r x - a ^ ^r/ in which Q and P are in general polynomials. Clearing of fractions, /(*) = Q(* - aW(*> + A'(a) = $(a\ the substitution of which in equation (2) gives another expression for A, namely, A f(a} ~~ /.// r (3) As an example, let us find the value of f * + 2 I i , ^ 5 dx. J X* + 2JT X* 2X The denominator is the product x(x* i)(x + 2), and the first term of the quotient is obviously x ; hence we assume x 5 + 2 A B C D X* + 2X* X* 2.X X X -\- I X \ x + 2' The coefficient of x* in the equation cleared of fractions gives a = 2. Now forming the fraction f(x] _ X 5 + 2 2 I) -dx = 2x + log J X* + 2X 3 X* 2X "2 X 22. We have seen that the decomposition of a given frac- tion into partial fractions presupposes a knowledge of the roots of the equation resulting from equating the given denominator to zero. In the case of the denominator x n i, we can employ the expressions for the imaginary roots involving the circular functions of certain angles. (Diff. Calc., Art. 231). In some simple cases the factors are expressible in ordinary surds. For example, we have 24 ELEMENTARY METHODS OF INTEGRA T1ON. [Art. 22. ^8 _ ! (x _ i)( x _j_ !)( A -2 + !)^2 _ x v/2 + j^ + x y 2 + ^ It is to be noticed that when the fraction is a rational func- tion of some higher power of x, we may simplify the process of decomposition. Thus it is legitimate to assume x 4 A B A-8 y ~A -r ,j*4 I y ' Jt, - i J, - 1 * -f I because, if z = x*, the numerator and factors of the denominator are rational functions of z. The first term could be treated in like manner with respect to x 2 ; but not the second, since the factors of.* 4 + i involve ;tras well as x z , 23. We have seen in the preceding articles that the partial fractions corresponding to the real roots, whether single or multiple, are directly integrable, and also those corresponding to unrepeated imaginary roots. In the following example a case of multiple imaginary roots occurs: given (a -*)(<* + It is readily shown that the partial fraction corresponding t o a _ x i s _ and the remainder is conveniently 2a 3 (a x) found by subtracting this from the given fraction ; thus a -f- x i a 4 -f- 2a * x 2# 2 # 2 x* (a - x)(c? + ^) 2 ~ 2a 3 (a -x) = 2a\a - x)(tf + x*)* ' The numerator of this remainder is found to be divisible by a x, thus verifying the process ; and the given integral re- duces to I i fa 3 + -$a*x + ax 2 + x 3 s lg (a x) -\ 7-2 2T2 ** 20? ' 2a* J (a* + x*)* The integration of the last term may be effected by a trans- formation given later. See Art. 41 and Art. 76. 24. Instead of assuming the partial fractions with undeter- 11.] RATIONAL FRACTIONS. 2$ mined numerators, it is sometimes possible to proceed more expeditiously as in the following examples: Given dx\ (i+**)' putting the numerator in the form i + x* x*, we have dx a - Treating the last integral in like manner, xdx log (i + *) = - + log Again, given putting the numerator in the form (i + xf 2x x*, we have i -, _ (dx 2 + a ' -- dx f dx ~ f dx Hence by equation (A), Art. 17, dx i x 26 ELEMENTARY METHODS OF INTEGRATION. [Ex. IL Examples II. I. log (a x\ }a x* dx i ' xdx i 3' < log (a + . (*(*- Jo f . . ( Jo 7 . II \ ** x- id ~r mx\ 8. \ (a + mxY dx, S 3/0 COt 2-^ sin 2.* 2 I COS* X 10. cos 3 x sin x dx, Jo f cos Q dQ i 11. cosec 9. J sm Q 2 f , sec* 3* 12. sec 3 x tan 7 ~ Jo EXAMPLES. f * I si^.-% /j'y \. J# ax, . tan 3 7T . 4 sec 4 J o n . M 24. sin (a 26) d$ t 13. \a-ax, mloga 14. j ( i + 3 sin 2 jc) 3 15. (i + 3 sin 1 .*)" sin X cos x dx, r*. (^ _ ^) ^ ^T- o 16. -77 sv, y^aa ^ ; J o V(2ax x ) 2 17. Jjcos'**, - 3 - 1 8. j sec 4 6^/6, tane +^tan 3 Q. 1 9. | tan- x dx, \ t^ 2 x + log cos x. 3 -sec x , 4 Jo 4 'sr jg . \ ^ 4. ' a II log 2 cot" e dfe, ~ 2 -\ x x) + a vers , a Q' 29) 28 ELEMENTARY METHODS OF INTEGRATION. [Ex. II. f cos x dx i .,..,. 25 ' \a-b^ X > logO^sm*). f 4 dx t i log 2. ) tan* 26 6 2 7- I ' * * ^Iog2. J tan * ' 4 i I i 2 ^Jjc J2 ; , log (log*) = log 2. Ji * log* ' Ji 28. , ) i X log * 4 + e--^' tan" i x* dx i 3 a I ^ . T > - tan- J *\ 3 x dx i . .** 3 1 - \~77T* ^n" -sin" 1 -^. 1 4/ (tf * )' 2 a dx i 3 2 - I-^T: TI5\ -^T sm ' 5- f , * ,, -T- tan' 1 J 2 + 5* ' I i/io 4* 39 + ^r 4- I , i / \ 2 ,2X 4- _^> ^ + log (^ -^ + i) + -tan . ((2X + ^ dX - J 2 ^ +3 > ^ - * + 2 log (2* + 3). (V . U : 4r<3^, -log (2* + i) . }(2X + I) 3 2 2X + I I 2ax cos a + i i-* cos an a 7i a : tan ' : = : a sm a. a sin a 2a sm a 10 3 ELEMENTARY METHODS OF INTEGRATION. [Ex. IL a sec a a tan a 47 f dx i -r J -* a 2a.r sec a + a? ' 2# tan a .* 48. 2ax sec or + a a ' 2# tan a x a sec + a tan a; ' if 4/2 2JT 2 3 |/2 ^f 7* 12 2X 2 + 3 1/2* *"<: f x*dx 49 - i=3F- 5 1 3-*- i ^ |(ar +a )^ +3 )" 2l g ',2. > J x' - x* - x + i J X I + 2 X + 3 x dx ^[tan-^ + log^^'- 'togfLUL+J^tan-i 6 & A:+ i 3 X + 2 , 9. X + 1 I . X 2 1 dx, -log h - log . 55- ' '-""**' 4 jc i X. ( dx i , (jc + i) s i _, 2^; i 57- T * 7^S~rn T *" ~Zr~ tan ~tf * f oc doc ii i 58. 7 rrTI v, -log(* i) log(# S +l) -j i. J (x - i) (x 8 + i)' 2 4 2(^1) II.] EXAMPLES. dx 59 x + x y + x') ' log x log (i + x} log ( i + x*) tan - ' x. I I 6* ' 6log* + - f x* i 60. L-TT . ax t )X + X* + I ( X* + X I 6l " J * + *- 6 f X*dx I . X 2 62. \- -- 5 - , -log- - + }x x 12' 7 & x + 2 f ^cVjc 3< \ (x* - i)*' 1 , x* x + i -log-jT-r - 2 & X + X + I i x i 4 og ^T7 f 2-v" -to* . 5 A: i . 64- i - ^-dx, ^-tan- 1 --- log } x a 20, a 40 , f 65 * x a . 20, a 40 x + a x dx i x* 2 f dx i f x + b b _ x~~\ 00. ..a -- ^r-j - ; rr , 75 ; - 5 lOg ./ a , - JT + ~ tan '- . J (x + a) (x + b) ' b + a* L V( x + ) ^J dx 7 ' 68 ' f x + i x 09. ; ^rdx, tan- 1 x + log 77 IT jx(i + jr) & V(i +^ s ) f dx i i 7- / 3 . c , tan ~*x -\ -5 , I* V* T ij >^ 3 32 ELEMENTARY METHODS OF INTEGRATION. [Ex. II. dx X i f d* _Ll > Jar (a + &*)' ^ 10g ( ^?r i b a + 73> 74. Find the whole area enclosed by both loops of the curve 75. Find the area enclosed between the asymptote corresponding to x = a, and the curve 76. Find the whole area enclosed by the curve 77. Find the area enclosed by the catenary | X X ~\ the axes and any ordinate. 78. Find the whole area between the witch and its asymptote. See Ex. 23. IIL] TRIGONOMETRIC INTEGRALS. 33 III. Trigonometric Integrals. 25. The transformation, tan 2 sec 2 6 i, suffices to separate all integrals of the form [tan* 040, (i) in which n is an integer, into directly integrable parts. Thus, for example, [tan 5 OdB = [tan 3 6 (sec 2 0- i) dQ tan 4 -1 tan 3 dd. 4 Transforming the last integral in like manner, we have f , - . . tan 4 tan 2 8 f [tan*0^ f sinOdO Since sin 6dO= - ^/(cos 6^, the value of the last integral is, by formula^'), Art. 17, i i - cos . /i - cos . /i log r i and, multiplying both terms of the fraction by I cos 0, we have O i cos 38 ELEMENTARY METHODS OF INTEGRATION. [Art. 31. 31. Since cos = sin (TT + 0), we derive from formula (), f dO ( dO rn 0~\ 3 = -^TI -- -a = log tan - + - . . . (F\ } cos f -- 1 --- ) - f . . (2) J 2(m n) 2 (m + n) When m = n, the first term of the second member of each of these equations takes an indeterminate form. Evaluating this term, we have sin2# , N ..... (3) and cos2^ = g + Sin2 . (4) Using the limits o and n we have, from (i) and (2), when m and n #?r unequal integers, fir ,n sin mQs'm nOdd = \ cos w0 cos nB dd = o ; . . (5) o Jo but, when m and n are ^##/ integers, we have from (3) and (4) = \\os*n8d6 =- ..... (6) Jo 2 34. To integrate 4/(i + cos ^) ^/(9, we use the formula 2 cos 2 \Q i + cos 0. 40 ELEMENTARY METHODS OF INTEGRATION. [Art. 34. whence V(i + cos 6) 4/2 cos 0, in which the positive sign is to be taken, provided the value of is between o and n. Supposing this to be the case, we have [V(i + cos 0) dB = V2 (cos^Odd = 24/2 sin 0. For example, we have the definite integral IT 2 4/(i + cos 6}d8 = 2 4/2 sin - = 2. Jo 4 Integration of 7 5. y a + b cos 35. By means of the formulae. i = cos 2 + sin 2 and cos = cos 2 sin 2 0, we have f dB _ f dB \a + b cos ~ J (a + b) cos 2 B + (a - b} sin 2 10* Multiplying numerator and denominator by sec 2 ^0, this be- comes f sec 2 \6dQ J a + b + (a b} tan 2 ' and, putting for abbreviation tan = y, we have, since sec 2 ^0 the integral becomes 2 f dy ~j I ~o a b a] c jr the value of which is, by formula (A 1 ), Art. 17, i , c + y r-. r lOg . c(b-a) *c-y 42 ELEMENTARY METHODS OF INTEGRATION. [Art. 35. Therefore, in this case, dB i V(b+a}+ V(b-a) tan \B g - - ' + cos 6 V ( 2 - a 2 ) i/( + tf)- V(-) tan 36. If ^ < i, formula () of the preceding article gives f dB 2 .r /i - e. , ."I z, = -77 -- a? tan MA/ -7 -tanl(? . . (i) J i + ^ cos 6 V (i f) \_V i + e - J Putting ='tant0=tant* f ..... (2) and noticing that = o when = o, we may write f _*? _ - _ ^_ J o i +*>cos0~4/(i-^)' Now, if in equation (i) we put ^ for ^ and change the sign of f, we obtain f _ J i hence, by equation (2), f _^_ o i-ecos Equations (3) and (4) are equivalent to dB dj> i+ecos0 (5) ^ dB fi) and 1 = -77 ^ ..... (oj III.] TRIGONOMETRIC INTEGRALS. 43 the product of which gives (i + e cos ff) (i e cos ) = i e* . . . . (7) By means of these relations any expression of the form dB 1 f J(i + e cos where n is a positive integer, may be reduced to an integrable form. For f dB __ f dO __ i m J(i + e cos 6)* ~ Ji + e cos0 (i + e cos d}"-* ' hence, by equations (5) and (7), ~~ e cos By expanding (i e cos ^)*~ I , the last expression is reduced to a series of integrals involving powers of cos ; these may be evaluated by the methods given in this section and Section VI, and the results expressed in terms of 6 by means of equa- tion (2) or of equation (7). Examples III. f 4 tan 3 mx tan mx } yn m a 2. ta.ri'xdx, A~ilg 2 - J o _ /c i ,,,\ 7/- tan \u + oc) /, \ 44 ELEMENTARY METHODS OF INTEGRATION. [Ex. IIL 4. I sin 3 mx dx, yn ' Jo f . , sin 8 sin 5 9 5. sin cos 0^/0, . J 3 5 6. f t/(sin 0) cos 5 4/9. - sin'0 - sin* + sin^" 9. J 3 7 ii 7. 2 cos 4 sin' 04/0, . J 35 f sin 3 4/0 & A J V(cos0)' | cos* - 2 cos 9. 9. -T-J 7 , Multiply by sin 8 + cos" 0. tan cot 9. J sin 6 cos f 3 >. ^dx, Jcos ^ 10. J "\ ~ //jc. See Art. 2%, 4 ii f 4/9 r , 4 (tan 7 - cot' 0) + 2 log tan 9. J sm 3 cos 3 ' fy(sin 0) logtan - + - log cos 0. Jsec0 tan0' L4 2j 24. cos cos 30 ~ ' 10g + a) + V (t - a) tan ($Q- - a' (* + )- V(^- 9 - _ 5 + 3 cos 9' _ 5 4 cos 9 f ' J f " J ( 4i - J ^ [ J(i f- > J (i _ _ 44> (i +*cos9) 3 37' ' ' 2 -taniO , 8 f ' J tan-M3 ^9 J_ i + 4^3 _ log 8 acose-i d* tan - (i + e cose) 2 ' i g + cos9 g sine i+. ( 2 2 + 2 ) it. 50. Find the area of the loop, and also the area between the curve and the asymptote, in the case of the strophoid whose polar equation is r = a (sec tan 6). Solution : Using as an auxiliary variable, we have / \ , sin 2 0~| x = a (i sine) y a\ tan , L cos 6J the upper sign corresponding to the infinite branch, and the lower to the loop. Hence, for the half areas we obtain f* ff f* ff f TT~| + a*\ sin 6 d$ + a* \ sin 8 6 dQ = a*\ i + - and a 8 f sin 9 dB + a* f sin 8 6 whence dx _ dy ~ ydy I, 2y 4- i = y log(2j+ I)"] = 2-log3 6 12 J x 12 Examples IV. IV.] EXAMPLES. 57 2X I !> far 1 ^r + i ' J (2* + i) 2 ' dx f J 2.r + i log (2.* + i) _ 7 ~~ ~~ ~ - log 1 . e* i - log 2 8 ^ + I 2 log ^ ~ l) /^ f 2 + tan 6 6 log (3 cos 6 sin 9) o. I ~ - u9 ~ . . . _,_ J 3 tan 9 2 tan 9 i 9 f dQ J_ I0> Jtan 2 9-i' 4 g tan 9 + i 2 f tan" 9 dQ . tan 9 i 9 ' Jtan 2 9 -i' 4 g tan9+i 2 cos 9 ^9 aft b log (g cos 9 ^ sin e) a cos 9 sin 9 ' a + b* $8 METHODS OF INTEGRATION. [Ex. IV. f COS QdQ - r - T7T' J COS (a + 6) (6 + a) cos a sin a log cos (9 + a). sin (0 + a) (6 + /3) cos (a /3) + sin (a /3) log sin (0 + /?). 15. tan (0 + ct) cos */0, cos + sin a log tan 29 + 2fX + 7t , f a cos Q dQ . . 16. -i - -- - r, cos log (2 cos a) + asm a. J sin (a + 6) * p" cosje ^ fl _i_ 4/2+2 sin flH 6 _ log (3 + 2 ^2) Jo cose ' 4/2 -/2 2 sinfi'Jo 4/2 fsinJ^Q ^ -f Q 18. ^ , log tan -- . J sin Q 4 XT aX f ^* sm 20 //9 = -7- Jo (i + ^r) 4Jo 16 IV.] EXAMPLES. 59 f dx 1 + ' los * + ' 2 3- L.S / , T \i * J-* \^ < - 1 ) f and = *, x f a* we have dx __ f dy Hence, by equation (A') Art. 17, f dx _ i . y a I ^(x 2 + cF) a J x V(x* + # 2 ) 2a y + a 2a V(x* + a?) + a' Rationalizing the denominator of the fraction in this result, we have V(x* + a 2 ) - a _ [ V(x* + a*) - aj V(x* + a*) + a ~ x* Therefore V.] INTEGRALS CONTAINING RADICALS. 63 In a similar manner we may prove that dx i a 51. Integrals of the form ...... (2) are reducible to the form (i) Art. 49, by first putting y = -. X For example ; dx is of the form (2) ; but, putting x - , whence + 6)= rv " ^ VJ> > and dx = -^ y j we obtain f dx f y dy (ax 2 + b)* J (a + The resulting expression is in this case directly integrable. Thus f dx i x . r, . . . \j ) * b V (a + bf) b\f(ax* + b) 64 METHODS OF INTEGRATION. [Art. 52. Integration of 7 ^ ^- . A /I -y* + // i ^/ i .*< ^- u i 52. If we assume a new variable z connected with x by the relation we have, by squaring, 2.zx = cP, (2) and, by differentiating this equation, 2 (z x) dz 2z dx = o ; whence dx dz Z X 2 or by equation (i), dx dz t ^ r ...... (3) 0*) 2' Integrating equation (3), we obtain > (AT) 53. Since the value of x in terms of z, derived from equa- tion (2) of the preceding article, is rational, it is obvious that this transformation may be employed to rationalize any ex- pression which consists of the product of - . , -,. and a rational function of x. For example, let us find the value of V.] TRIGONOMETRIC TRANSFORMATION. 65 which may be written in the form By equation (2) whence Therefore, by equations (3) and (5), f .. , i f( VC^ 3 or) dx = - i , , 4J 2 (dz a* (dz - + - J z 4 ) zr a* - By equations (4) and (5), the first term of the last member is equal to \ x V(x* 2 )- Hence ^)]. . (L) Transformation to Trigonometric Forms. 64. Integrals involving either of the radicals x* or 66 METHODS OF INTEGRATION. [Art. $4. can be transformed into rational trigonometric integrals. The transformation is effected in the first case by putting x a sin 6, whence V(a* X 2 ) = a cos 6 ; in the second case, by putting x = a tan 8, whence V(a z + x*) = a sec 6 ; and in the third case, by putting x = a sec 0, whence ^(x* a?) = a tan 6. 55. As an illustration, let us take the integral putting x = a sin 0, we'have V(#, x 2 ) = a cos 0,dx = a cos B d0- hence f V(<# -x*}dx = a> [ COS Z 0d0 ___ a 2 c? sin cos by formula (C) Art. 29. Replacing by x in the result, //-2 ~2\ -7 , ,,x V (a* ^r) dx = sin - x - H ~ . . . (M) J 2 d 2 Regarding the radical as a positive quantity, the value of may be restricted to the primary value of the symbol % sin- 1 - (see Diff. Calc., Art. 57); that is, as x passes from a to + a, passes from | TT to + n. V.] INTEGRALS CONTAINING RADICALS. 6? Radicals of the Form ^/(a^ + bx + c). 56. When a radical of the form y(a^ + bx + c) occurs in an integral, a simple change of independent variable will cause the radical to assume one of the forms considered in the preceding articles. Thus, if the coefficient of X* is positive, in which, if we put.r+ = v, the radical takes the form 2a + tf 2 ) or \ f (y i tf 2 ), according as ^ac & is positive or negative. If a is negative, the radical can in like manner be reduced to the form y(a? y*) orV( cPy*} ; but the latter will never occur, since it is imaginary for all values of y, and there- fore imaginary for all values of x. For example, by this transformation, the integral p dx J (a* + bx + <:)! can be reduced at once to the form (J\ Art. 51. Thus dx + (4ac ' ^ (a * + bx + c) ' 68 METHODS OF INTEGRATION. [Alt. 5/. 57. When the form of the integral suggests a further change of independent variable, we may at once assume the expression for the new variable in the required form. For example, given the integral V(2ax x*} x dx ; we have \f(2ax x*) = V[a? (x of] hence (see Art. 54), if we put x a = a sin 0, we have V(2ax x 2 ) = a cos 6, x == a (i + sin 0), dx = a cos & dO ; .\V(2ax - x*}xdx = c? fcos 2 0(i + sin 0) dO /T 3 /7^ C* / yj /) /1\ ^- q /J = (6 + sin cos 0) cos s 2 V 3 = sin~ J -\ (x a) V(2ax x*) (2ax 2 a 2 V 3 v Ct , X ~~" d 1 .f o\ r 9 = sin- 1 h 7- V(2ax x*) yzx 2 - ax 2 a o The Integrals dx , r dx 58. An integral of the form -, ^ j r- may by the J V(eur + bx + c] method of Art. 56, be reduced to the form (K), Art. 52, or to the form (/'), Art. 10, according as a is positive or negative. V.] IRRATIONAL INTEGRALS. 69 But, when the integral appears in one of the above forms (the quadratic under the radical sign admitting of linear factors), another mode of transforming is often convenient. Assuming ft > a, we may in the first case, since the differ- ence of the factors is the constant ft a, put x a. (ft a) sec 2 6 \ x - ft = (ft - a) tan 2 j ' whence dx = 2(fi a) sec 2 tan 0dB t and j - *L - _ = 2 \sec0ii0 = 2 log (sec 4- tan 0). J - * x + J i + Vx' 3 . [ /^ , 2 V* + 2 log (l - J T* I r dx 2 /2JC ~ # 7. - r - -TV, -tan" 1 j/- jxV(2ax a) a a , 8. (a f ^ 'liA J 2^ x* 9 7 5 JvA* 315 H 1 1 Q 44 8 V.] EXAMPLES. 12. I - -7 ^ }x M(x* f V(x< + i J 5- J - .6. f(*- + '"' X 3 -f - ^T = + A - 8 5 Ji 10 40 Rationalize the denominator. 2 (x + a)* - 2 (x + 3 ( - *) I j/(jg* + i) i 4 og i/(^ 4 + i) + tan- v a 8 ) a sec- 1 - 7 2 METHODS OF INTEGRATION. [Ex. V. ,x 3 ^-dx. See formulas, (L) and (K\ -xV(x* + *)-- s log [x 20. - dx, a log dx 21. /% n _ /V-y* J*i. I ^^ UvV* t-V dx 23 i r // a , 2\ n ^(^ + a *) log [ 4/(^ a + a 2 ) + j;] -- ^ - - ' * J VCr' + f ) - a ' 4/(^r a + a*) i log X X f /TY T 24. J y(l + g) ^ -w/tf (^). jlog [^ 8 + l/(i -f a; 4 V.] EXAMPLES. 73 25. I V(ax* + b] dx t [a >o] Put V(ax* + b) = z x^a. b i - log [x 4/0 + tf(ax + b}\ + - V d 2 26. [,- }(a dx + x) ^(x* + #} ' 28. -^ 29. dx dx i x g d&c V(-^ a ~ J ) i L 7 7* Q "i 2JC 2 log tan ^ 8 2 74 METHODS OF INTEGRATION. [Ex. V. f dx / 2 \ k-vt**-.)' ( " +l) oo* + 34- __ j ta ,, 3 37 ' 3 8 - 39 ( a // s\ ^ 3 sm " - ,, ax)' ?6 METHODS OF INTEGRATION. [Ex. V. 50. Find the area included by the rectangular hyperbola y = 2ax + x 3 , and the double ordinate of the point for which x = 2a. a*[6 V2 log (3 + 2 ^2)]. 51. Find the area included between the cissoid x (x 9 + y ) = 2ay* and the coordinates of the point (a, a) ; also the whole area between the curve and its asymptote. . a--)-. 52. Find the area of the loop of the strophoid and = o also the area between the curve and its asymptote. z a ( i ) , and 20* 4 For the loop put y = x . 3 -j- , since x is negative between the limits \f \Cl ^~ ^v J a and o. 53. Show that the area of the segment of an ellipse between the x minor axis and any double ordinate is ab sin - l I- xy. a VI. ( INTEGRATION BY PARTS. 77 VI. Integration by Parts. 59. Let u and v be any two functions of x ; then since d (uv) = u dv + v du, uv \u dv + \v du, \u dv = uv \v du ( i ) whence By means of this formula, the integration of an expression of the form udv, in which dv is the differential of a known function v, may be made to depend upon the integration of the expression v du. For example, if ^^cos" 1 -*" and dv = dx, we have , dx hence, by equation (i), f j , f *dx cos' 1 x-dx = ^rcos- 1 ^- + - ] } V(i - x*y in which the new integral is directly integrable ; therefore cos- 1 ^-^ = ^rcos' 1 ^ V(i ^ 2 ). The employment of this formula is called integration by parts. METHODS OF INTEGRATION. [Art. 60. Geometrical Illustration. 60. The formula for integration by parts may be geomet- rically illustrated as follows. Assum- ing rectangular axes, let the curve be constructed in which the abscissa and ordinate of each point are correspond- ing values of v and u, and let this curve cut one of the axes in B. From any point P of this curve draw PR r( and PS, perpendicular to the axes. 1 Now the area PBOR is a value of the indefinite integral \u dv, and in like manner the area PBS is a value of \vdu ; and we have B A o Area PBOR = Rectangle PSOR - Area-fiRS; therefore \u dv = uv \v du. Applications. 61. In general there will be more than one possible method of selecting the factors u and dv. The latter of course in- cludes the factor dx, but it will generally be advisable to in- clude in it any other factors which permit the direct integra- tion of dv. After selecting the factors, it will be found con- venient at once to write the product u-v, separating the factors by a period ; this will serve as a guide in forming the product VI.] INTEGRATION BY PARTS. 79 v du, which is to be written under the integral sign. Thus, let the given integral be J*Mog x dx. Taking x* dx as the value of dv, since we can integrate this expression directly, we have \X*\Q&X dx = log x- x* \x^ J 3 3-1 x = x* log x \x*dx 3 3J x* = -(3 log*- i). (V 62. The form of the new integral may be such that a second application of the formula is required before a directly integrable form is produced. For example, let the given integral be x* cos x dx. In this case we take cos x dx = dv ; so that having x* = u, the new integral will contain a lower power of x : thus \x* cos xdx = ^-sin x 2 LF sin xdx. Making a second application of the formula, we have \x* cos x dx = x* sin x 2\ x(- cos x*) + cos xdx I = ^sin x + 2x cos x 2 sin x. 8O METHODS OF INTEGRATION. [Art. 63. 63. The method of integration by parts is sometimes employed with advantage, even when the new integral is no simpler than the given one ; for, in the process of successive applications of the formula, the original integral may be repro. duced, as in' the following example: e mx sin (nx + a) dx cos (nx + a) m t . = e wx ' - - -\ --- e mx cos (nx + a) dx n } { \ J n e mx cos (nx + a) m sin (nx + a) m z { = -- *- + -e m *- + -- e" !X n n n in which the integral in the second member is identical with the given integral ; hence, transposing and dividing, [ e mx \ v nx sin (nx + a) dx = 2 [m sin (nx + a) n cos (nx + a)]. 64. In some cases it is necessary to employ some other mode of transformation, in connection with the method of parts. For example, given the integral taking dv = sec 2 dd, we have [sec 8 <# = sec 0-tan - fsec0tan 2 0^#. . . (i) VI.] FORMULAE OF REDUCTION. 8 1 If now we apply the method of parts to the new integral, by putting sec 6 tan 6 dd = dv, the original integral will indeed be reproduced in the second member ; but it will disappear from the equation, the result being an identity. If, however, in equation (i), we transform the final integral by means of the equation tan 2 9 = sec 2 \, we have [sec 3 Od6= sec 6 tan B - [sec 8 8 dB + f sec BdB. Transposing, f n in Sin B [ dd j/i 2 sec 8 dO = j-^ + -dO: J cos 2 J cos a hence, by formula (F), Art. 31, f ,/,,/, sin B i , \~n 6~\ sec 8 BdB 5-2 + - log tan - + - . J 2cos 2 # 2 l_4 2j Formula of Reduction. 65. It frequently happens that the new integral introduced by applying the method of parts differs from the given integral only in the values of certain constants. If these constants are expressed algebraically, the formula expressing the first trans- formation is adapted to the successive transformations of the new integrals introduced, and is called a formula of reduction. 82 METHODS OF INTEGRATION. [Art. 6$. For example, applying the method of parts to the integral [x n e**dx, we have t O.X ft t \x n e ax dx x n \x n -*e ax dx. . . . . (l) J a a] in which the new integral is of the same form as the given one, the exponent of x being decreased by unity. Equation (i) is therefore a formula of reduction for this function. Sup- posing n to be a positive integer, we shall finally arrive at the f e ax integral e ajr dx, whose value is . Thus, by successive appli- cation of equation (i) we have Reduction of \sin m BdQand \cos m QdQ. 66. To obtain a formula of reduction, it is sometimes neces- sary to make a further transformation of the equation obtained by the method of parts. Thus, for the integral the method of parts gives VI.] REDUCTION OF TRIGONOMETRIC INTEGRALS. 83 Substituting in the latter integral I sin 2 6 for cos 2 d, [sin Odd = sin- 1 8 cos 8 + (m - i) (sm m ~* 6 d6 (m - i) fsin w dd ; transposing and dividing, we have f . n , Q sin" 2 - 1 6 cos 6 m - i f . . sin w 0^0=i --- 1 -- sm w - 2 0^6>, . . . (i) J mm} a formula of reduction in which the exponent of sin 6 is dimin- ished two units. By successive application of this formula, we have, for example : f B /i JQ sm * # cos # 5 f sin 6 0^0= -- g -- + g-l sin 4 64 sin 5 cos _ 5 sin 3 6 cos 5-3 sin cos 5-3-1 6 6-4 6-4-2 6-4-2 67. By a process similar to that employed in deriving equation (i), or simply by putting 6 = TT 6' in that equa- tion, we find , (2) mm a formula of reduction, when w is positive. 84 METHODS OF INTEGRATION. [Art. 68. 68. It should be noticed that, when m is negative, equation (i) Art. 66 is not a formula of reduction, because the exponent in the new integral is in that case numerically greater than the exponent in the given integral. But, if we now regard the integral in the second member as the given' one, the equation is readily converted into a formula of reduction. Thus, put- ting n for the negative exponent m 2, whence m n + 2, transposing and dividing, equation (i) becomes f dO _ cos n 2 f d0 . . } sin" 6 ~ (n ^sin"- 1 + n i Jsin*- 2 0' Again, putting 6 = ^ n 6' in this equation, we obtain f d6 _ sin 8 n 2 f dO } cos* 6 ~ (n i) cos*- 1 ^ + n i Jcos*- 2 Reduction of \sin m B cos" 9 d0. 69. If we put dv = sin"* 6 cos 6 d0, we have cos"- 1 /? sin"' +'0 in 0cos* 9d9 = m + i 72 j f * m + i J S but, if in the same integral we put dv = cos" sin 0d0 t we have '0 + i i . ... (2) VI.] REDUCTION OF TRIGONOMETRIC INTEGRALS. 85 When m and n are both positive, equation (i) is not a formula of reduction, since in the new integral the exponent of sin 6 is increased, while that of cos B is diminished. We therefore substitute in this integral sin w+2 6 = sin w 6 (i cos 2 #), so that the last term of the equation becomes n ~ l fsin'" B cos"- 2 BdB- n ~ l \ sin" 1 8 cos BdB. m + i J m + i J Hence, by this transformation, the original integral is repro- duced, and equation (i) becomes [~l + - 1 1 f sin'" cos n 8d8 = m + IJJ m + I - I f . , sin* m + i j T>.. ... , n I m + n , Dividing by i ^ = , we have & * m + i w + i f si J ~a *aja sin** 6 cos" 6d8= m + rc~ T [sm^^cos"- 2 (9^, ... (3) + n J a formula of reduction by which the exponent of cos B is diminished two units. B6 METHODS OF INTEGRATION. [Art. 69. By a similar process, from equation (2), or simply by put- ting = % n 6' in equation (3), and interchanging m and n, we obtain f ~n nJa sin**- 1 d COS n+1 B sin** 6 cos" B dd = J m + n m- T# -I- i f sinw _ 2 ^ CQS ede , , J a formula by which the exponent of sin 6 is diminished two units. 70. When n is positive and m negative, equation (i) of the preceding article is itself a formula of reduction, for both exponents are in that case numerically diminished. Putting m in place of m, the equation becomes fcos*# ,._ __ cos"- 1 ^ n i fcos*- 2 # J sin 6 ~ (m ^sin^-'tf m i }sm m - 2 (T ' ' ' '$' Similarly, when m is positive and n negative, equation (2) gives f sin"* ,Q _ sin^-'g m i fsnV"- 2 ~ (w i)cos"- 1 6 n ^- i Jcos"- 2 ^ cos 71. When m and n are both negative, putting m and n In place of m and , equation (3) Art. 69 becomes f_ J sin w dO __ _ B cos" B ~~ (m + n) sin m - I B cos" +1 B + n + i f w 4- J sii s'm m 0cos n +*0 t in which the exponent of cos is numerically increased. We VL] REDUCTION OF TRIGONOMETRIC INTEGRALS, 87 may therefore regard the integral in the second member as the integral to be reduced. Thus, putting n in place of n + 2, we derive J sin d6 0cos*0 (n i) sin**- 1 cos*- 1 m + n 2 f dO , . - i sin-"- 2 V7 ' Putting = TT 0', and interchanging m and , we have dO i . . " ' __ 0~ (mi)s\n. m m + n 2 f ' dO mi 72. The application of the formulae derived in the preced- ing articles to definite integrals will be given in the next sec- tion. When the value of the indefinite integral is required, it should first be ascertained whether the given integral belongs to one of the directly integrable cases mentioned in Arts. 27 and 28. If it does not, the formulae of reduction must be used, and if m and n are integers, we shall finally arrive at a directly integrable form. As an illustration, let us take the integral [ sin 2 cos 4 6 dd. Employing formula (4) Art. 69, by which the exponent of sin 6 is diminished, we have 8 cos' 8M = - + cos- dB. 88 METHODS OF INTEGRATION. [Art. 72. The last integral can be reduced by means of formula (2) Art. 67, which, when m = 4, gives f 4/17/1 cos 3 6 sin 6 3 f _ _ cos 4 6 dO = - + - cos 2 8 dO ; J 4 4 J therefore f o a A a jo sin#cos 5 cos 3 #sin# , sin 6 cos 6 sin 2 cos 4 Odd = -- - -- 1 --- 1 -- - -- \- ., } 6 24 16 16 73. Again, let the given integral be cos J sin 3 By equation (5), Art. 70, we have f cos 6 d dS _ cos 5 6 5 fcos 4 8 dtt J sin 3 6 2 sin 2 6 2 J sin 6 We cannot apply the same formula to the new integral, since the denominator m i vanishes ; but putting n 4 and m i, in equation (3) Art. 69, we have fcos 4 OdS _ cos 3 fl fcos J sin 6 J si 3 sin Q cos 3 >? f dO { . - - + - 5- sin Odd 3 Jsm (? J A ,-. cos 3 , I ^ ,, + log tan 6 + cos 0. 3 2 Hence fcos 6 6dd cos 5 5 cos 8 6 5. i 5 -T-S-7T- == -- ^^Q z -- - lg tan - ~ cos * J sin 8 2 sin 2 6 2 B 2 2 VI.] REDUCTION OF TRIGONOMETRIC INTEGRALS. 89 Reduction of I cos m & cos nO dQ. 74-. Integrating by parts, we have cos"* Ocosn6d&=cos m 6 . -- h sin 0cos m ~ I 0sin 6dB. J n n ] To obtain a formula of reduction we employ the identity sin nB sin d = cos (n i)# cos ## cos 6, so that the last term of the equation becomes Wl f W'l f cos''fl cos ( iW0 -- cos w 6> cosftA/0. n J n] The original integral is thus reproduced, and after transpo- sition and division the equation becomes f cos'"# sin nd m f cos" 1 8cosn6d6= -- 1 -- \ cos*- 1 6 cos (n i}6dd, (i) J m + n m+n] a formula of reduction in which w and n are each diminished one unit. If m is a positive integer, we shall by repeated applications of this formula arrive at a directly integrable form, even when n is not an integer ; and it may be remarked that, since cos ( nff) = cos nd, the sign of n may at any time be changed if convenient. As an example, we have cos j = cos 2 6> sin ^+ cos B sin 0+ sin 75. If m be negative, putting m = m', n = n', equation (I) becomes 90 METHODS OF INTEGRATION. [Art. 75. fcosn'8 sinn'O m' Ccos (n f + i)& } cos' ~ (m 1 + n') cos m 'e + m' + n' } cos'+ 1 ' in which we may regard the integral in the second member as the one to be reduced. Hence, putting n for n' + I and m for m' + I, we derive f j cos nQ sin (n i)0 m + n 2 fcos (n i)6 ^o^e ' ~w-lcos"'-'6>+ m-i } cos- '0 If w is an integer, we can by this formula make the integra- tion depend upon a final integral of the form J - ^^6, which is readily evaluated when n (and therefore />) is an integer, and also sometimes when n is a fraction. For example, 'cos |0 _ sin \B 2$ fcos "" + - dB i[cos|0 2COS 2 4L COS 2} COS sin 5 sin 5 fcos d(j) 2 cos 2 4 cos 4; cos 20 sin 5sin0 5 i + 4/2 sin _ _i_ , i locx 2 cos 2 4 cos 8 |/2 s i 'J sn Formulae of reduction may be deduced in like manner for the integration of cos sin n6dd, and similar expressions. Reduction of Algebraic Forms. 76. The method of reduction may be applied with advan- tage to some algebraic forms. Take for example the form x m dx VI.] REDUCTION OF ALGEBRAIC FORMS. 9! in which m and n are supposed positive. Integrating by parts, taking u = x m ' and xdx we have f x m dx x m ~* m i f x m ~ 2 dx (a + bx*}" ~ 2b(n i)(a a formula of reduction in which m is reduced by two units and n by one unit. If m and n are integers, the expression to be integrated is a rational fraction and includes (after a simple transformation) all cases of multiple imaginary roots. Repeated applications of the formula will reduce n to unity or else reduce m to zero or to unity, the final integral vanishing in the last case. For example, the integral occurring in Art. 23 is thus reduced to + S^x + ax* + x* x* dx ~- - a - { ^ dx } (a 2 + x 2( a * + **) a 2(0* + x*} (a 2 + in which the final integral only requires further transformation ; the result is x The final integral is also of an integrable form when n is half of an odd integer. General Method of Deriving a Formula of Reduction. 77. The method of integration by parts may be said to consist in finding a function, uv, of which the differential con- 92 METHODS OF INTEGRATION. [Art. 77. tains, as one part, the differential to be integrated ; so as to make the integration depend upon that of the other part. If this last is of the same general form as the given differential, we have a formula of reduction. The more general method consists in finding a function, P, whose differential can be expressed as the sum of certain mul- tiples of two or more differential expressions of a given general form. We then have an equation connecting P with two or more integrals of the given general form. Among these we select that of the highest degree as the one to be reduced, and so prepare a formula of reduction for the given integral form. For example, to find a formula of reduction for the form , dx \dx or J.(#+ 2bx -\- ex' 4 where X is a quadratic expression which for convenience is written in the form whence dX 2(b + cx)dx. If we take X~ m for /*, we shall find it impossible to reduce dP\.Q the required form. The same is true if we take xX~ m '; but by a combination of these forms we are able to accomplish our object. Thus, taking X m (in which p and q are constants to be determined), we have qdx m(p + qx]dX _ qdx _ (p + qx)(b + cx}dx dr -- -- - 2W - . X n X X X The first term is of the proposed form, and the second will VI.] DEVELOPMENT OF AN INTEGRAL IN SERIES. 93 be so if/ and q be so taken as to 'make (p + qx](b 4- ex] a mul- tiple of X plus a constant. This will be the case if we put q = c and / = b, which gives - ex] = (b 4- ex} 2 = cX ac -f ^ 2 . TJT U D ^ + CX Hence we have /*=- , . ^JT r^f ac + 2 , and /^ = 2w #;r ^"* ^"" / \dx , , 9x i^r = dl 2w) -f 2m(ac o 2 } , \ / ytn ' \ I v m + i> A. Ji. which is in the required form. Since the last expression is of the higher degree in X, we put m + I = n, and obtain 'dx _ b + ex (2n 3)^ f dx x _ X"~ 2(n- the formula of reduction required. When n is an integer, the final integral, in successive appli- cations of this formula, is reducible to one of the forms (A), p. 18, or ('), p. 9; and when n is half of an odd integer, it dis- appears by virtue of the factor 2n 3, the last integration being equivalent to the use of formula (/), p. 63. Development of an Integral in Series. 78. It is often desirable to express an integral in the form of a series involving powers of the independent variable x, especially in the case of those integrals which cannot be ex- 94 METHODS OF INTEGRATION. [Art. 78. pressed by means of the elementary functions, that is to say, expressions which we now regard as not integrable. The most obvious way to do this is to develop into a series the expression under the integral sign, and then to integrate te* term by term. Given for example \dx. Using the expan- sion of e*, we have e* i x x % x r ~' i - = - + i + -, + -. + ... + r + . . . xx 2 ! 3! r\ Hence . ..... 1 The series integrated is, in this case, convergent for all values of x, from which it readily follows that the result of integration is also a convergent series for all values of x. If we put /w= and f(x) can be developed by Maclaurin's Theorem, we have A*)=. and integrating, j-y \ I // \ _y /~* . r-f _ \ _^/AY" V// / X** 1 / \ *l*) = /(^)^ = ^ +/(o)^ +/ x (o)- ^/''(o)-, + . . ., (i) J 21 3! in which C = F(d). The result is the same as the develop- ment of F(x) by Maclaurin's Theorem, since f(x) = F'(x\ VI.] BERNOULLI'S SERIES. 95 Bernoulli's Series. 79. By successive applications of integration by parts, we have and finally, provided neither /"(#) nor any of its derivatives to the wth inclusive becomes infinite for = o, z = x or any intermediate value of z, \ X f(z)dz = xf(x)-^f'( 3! -M ! ^ ^ ^ I -i t * \ / which is known as Bernoulli's Series. It is not, in the ordi- nary sense, a development in powers of x, because the coeffi- cients themselves contain x\ they are in fact the values of/(;r) and its derivatives at the upper limit, instead of their values at the lower limit, as in equation (i) of the preceding article. The result may in fact be derived from that equation when written in the form f * x* x 5 J\Z)CtZ = J(Q)X -\- j (Ol : -f- J (O) : T (3) Jo 2: 3 For, putting z x y, and denoting f(x y) by (y)dy = 0(o)* + 0'(o)^ + 0"(o)^ + . . ., jo Jo 2: 3 i or, since 'V0' -4 / J0J' - *> I -* C (/Jt-j ** C- 2J*c/ T^ ^C/ Jo f O ( I ^f SCC ^C (IOC) 2 I *^" SCC "" ^* ^~ T \*^ "" ^ / J * P . r^r 1 , /7T \ /ar , \"1? JT-/I to. I 6 sm h 6 d&. 6 cos ( h6)+sm(-+6) = J L4 \4 / \4 /J 4 g METHODS OF INTEGRATION. [Ex. VI, 11. Lcsec a .*d&?, x tan x + log cos x. 12. \xtan. y xdx\ \x (sec 3 x i ) dx , x tan x + log cos x x*. 15. \x* sin j? dx t 2x sin ^ + 2 cos x x* cos x. f * """) * I* 14. I .# sin~ * .#/&:, -^"sin- 1 ^ T si J o . I o * o sm' o = - x s ta.n~ 1 x x* log (i + x 3 ) ft yy. 4. n *% I >* yV A* . ^ i 15- \ X **> 3 66 f 1 i 2 -\- x 1 ~] I 7r2 16. I Ar'sin- 1 ^^, - *' sin - J jc H /j(\ x*} \ =7- Jo 3 9 Jo 6 9 17. &" x CO&x4Xm = ~ I 22 * o Jo 18. U'^^cos^^c, cos e xvin t sin f ^ I T f ^/ ^^ 19. le'^sm xdx\ =-\e-*(i cos2x)ax\, (cos zx 2 sin 2x ;). 10 v ^ r e e if i 20. 4 e e sin6 J cos 4 Q 3 cos 3 9 3 cos 2 cos 01 2 f 2 48 I57T os 4 9 /* = - ^-~ 32 7T f 1 ^0 I [4 *% _ ' J (i + cose) 3 ' 2 J o cos 4 0' 3* -r, sinecosV 3 c o s 6 cos 9 VI.] EXAMPLES. 101 f 40. . 3 c , jP/ ^ = sec 0. %xy(x* i) + ^log [x + y(x* i)]. f ^ 41. J( -jc 2 ) 2 ^, ( dx i |7 4 , A . -r-j jr- 3 , H COSQ^9= J ( + *T Jo f/ a , n\ // 2 ^S\ j S a * , X j ^V'(^ ^ 2 )[ . (^ 2 + ^ 2 ) V(, cos u H -\ cos P H . J 7 L 5 5 5J 2 sin 2 sin f cosjfldfl sinffl (i + sin$0)(i + 53 ' J cos 2 ^ ' cos^ + 6 g (i -sini#)(i - 54. Derive a formula of reduction for J cos** 6 sin nOdti. m _ [cos- sin (*-,)l^fc n J m + n m CE;. I v'cos sin ^QdO, I/cos . cos 3# (cos 6}^. J T 21 ;6. Derive a formula of reduction for ZTTT? J cos m fsin nO dO _ cos ( i)0 m + n .2 f sin ( i}6 d6 J cos'" ^ (w i) cos'"" 1 w i j cos"*" 1 6 dx 57. Derive the formula of reduction for -. ,,.^ by the method of Art. 77, and deduce the value of -. JTT. J (i x) P Jv y vn i t ffv 24(1 -#' .! g IO4 METHODS OF INTEGRATION. [Ex. VI. f x*dx x* x* x 5 J (i - *') ' 10(1 -x'y ~ 16(1 - *') 4 + 32(1 - x'} 3 ___ _ _ 128(1-**)' 256(1-**)" 512 S i -x f x*dx > J ( t + gy * * >-/ , 2 \ s ' / a \a 6(1 + x ) 24(1 + x } x tan" '.a; 16(1 + x*) 4 16 ' 60. Evaluate J -. '- JTJ-, () by the formulae of reduction ; J \I T~ X f (ft} by substituting^ = i + x*. Show that the results are identical. f x*dx x 4 x* i (*) j -^t = T ^ 7 ~^~* ~ 7TT~ ~^\*' )(l + X ) IO(l + X ) 2O(I + X ) 6o(l + X ) (0\ f x*dx _ i i i f -y* //I" "IT I ? /7 -^ /I 1* I ^v w^^. ^i -y ^_ T .< *> -V -4 T ,,& 1 ^Jv 1 4- T *J* I 1 I I tor*" 1 'i \1 ' / 2 \ / Ldll f _ ^ _ 6 3- 7T~ ~Tf> J (^ 2jc cos a + i a ) a .r cos of 2(x cos a) 3 sin 3 a(x* 2X cos a + i) 3 ,3 sin 4 ac(x* 2X cos a + iy I F JC I I . X* + X + I _, 2* + I~| - -5 ; ---- H log : - ^ -- 1-24/3 tan - ; 9\_x* + x + i x i (x 1 ) V$ J VI.] EXAMPLES. 10$ 65- Derive f(a + bz a formula of reduction ; 2 )" (a + bx*} n [(a + bx'Y^ i x m nb t( a + bx*y-* x- 66. Derive r dx (m i)x m ~ a formula of reduction - \/(a + bx*} m fr>r - J x-* J* (m m y(a + bx*}' 2}b f dX \x m y(a + bx*) (m i)ax m ~ l (m i}a\x m - 2 \f(a + bx*}' 67. Develop J dx in the form of a series. Jo x sin x , xx ax = x 5-5 ! 7-7! VII. The Integral and its Limits. 81. Before proceeding to some formulae of integration involving special values of the limits, it will be convenient to resume the consideration of the integral as defined in the first section. In Art. 3 we supposed a variable magnitude of which the values depend upon some independent variable x (but of which the expression as a function of x is as yet unknown) to vary simultaneously with x, while x passes at a uniform rate over a certain range of values. And it is assumed that the rate of the function is then expressed in terms of x and its assumed rate; or, what is the same thing, that the relative rate of the function as compared with that of x is known. Denoting tin's relative rate byf(x], a known function of x, let F(x) denote the function under consideration ; then by the notation introduced F(x}= f(x}dx.. ...... (I) IO6 METHODS OF INTEGRATION. [Art. 8 1 F(x) may now be defined as the function whose derivative isf(x), or whose differential is f(x}dx ; hence, from this point of view, integration is the search for the indefinite integral, or the inverse of the process of finding the derivative of a func- tion, equation (i) implying nothing more than that d\F(x}\=f(x}dx ....... (2) But because this equation was found to be insufficient to fix the values of the function F, we introduced in Art. 4 the nota- tion of the subscript or lower limit ; so that, when we write (3) we fully define the value of F(x) by implying the additional condition that F(a) = o. The integral thus modified is some- times called a corrected integral, because the indefiniteness arising from the unknown constant of integration has been re- moved. It remains " indefinite " in the sense that it is a func- tion of a variable x to which no special value has been assigned. It is to be noticed that, in many applications, the constant is determined (and the integral thus " corrected ") by the condi- tion that some other value (not zero) of F(x) shall correspond to a given value of x. These given simultaneous values of x and F(x] are called the initial values. 82. As explained in Art. 4, the value of x to which corre- sponds the required value of the magnitude is known as the final value of x, and is used as the upper limit of the integral. Denoting it by b, we thus assume that, as the independent variable x passes from a to b, the function F(x) passes by con- tinuous variation from its initial to its final value ; and, when this is the case, we may write F(a\ .... (4) VII.] THE DEFINITE INTEGRAL, 1 07 in which the function F is defined, not necessarily by equation (3), which implies F(d) = O, but by the more general equation (i), or simply by equation (2). We may now define the definite integral in equation (4) as the increment received by a variable whose differential is f(x] dx while x passes from the value a to the value b ; or, as the mag. dx nitude generated at the rate f(x}r while x t with the arbitrary dx . ,. , rate -7, varies jrom a to b. at The condition mentioned above, that F(x) shall vary con- tinuously, requires that, starting from some finite initial value F(a), it shall not become infinite or imaginary for any value of x between a and b. The function F(x) cannot become infinite or imaginary unless its derivative/^) becomes infinite or imagi- nary. Hence the condition will be fulfilled if f(x) remains real and finite for all such values of x. It is, of course, also assumed that there is no ambiguity about the value of f(x) corresponding to any of these values of x. Cases in which such ambiguity might arise will be considered later. 83. Since the rate of x is arbitrary, we may regard dx in the expression for the integral as constant, and in that case it must be regarded as positive or negative according as b is greater or less than a ; in other words, dx must have the sign of b a. When f(x) does not change sign for any value between a and b, we can thus infer the algebraic sign of the definite integral. Thus, if b > a and f(x) is positive for all values between the limits, F(x) is an increasing function of x, and x is itself increasing ; therefore the definite integral denotes a positive increment. For example, (""sin x , dx J x denotes a positive quantity. Moreover, since in this case f(x) IOS METHODS OF INTEGRATION. [Art. 83, is less than unity for all values between the limits, the integral is obviously less than the increment received by x, that is, less than TT. It is evident from these considerations that an interchange of the limits changes the sign of the integral ; thus, which agrees also with equation (4). Again, we infer from the same equation that (*) dx = f(x}dx la. if c is between the limits a and b\ and this is also true when c is outside of these limits, provided the condition mentioned in the preceding article holds for the entire range of values of x implied by the several integrals. 84. Returning now to the corrected integral, equation (3), we see that it is the same thing as J */(*) dx, in which x stands at once for the upper limit and for the in- dependent variable which, in the generation of the integral, is conceived to vary from the initial to the final value. When there is any danger of confusion between these two meanings of x, it is well to use some other letter for the independent variable ; thus [*/(*) ** J a has the same meaning as the expression above ; and in general it must be remembered that an integral is a function of its limits, and not of the variable which appears under the integral sign unless the same letter serves also to represent one of the limits. VII.] THE GRAPH OF AN INTEGRAL. IOQ Graphic Representation of an Integral. 85. A geometrical illustration was given in Art. 3, in which f(x] was taken as the ordinate of a curve, and the integral was in consequence represented by an area. We shall now employ another illustration, in which the ordinate y represents the in- tegral itself, regarded as a function of its upper limit. In other words, we shall consider what is called the grapJi of the func- tion (or graphic representation, employing rectangular coordi- nates), and shall regard this curve as derived from the expression for the integral, and not from the result of any process of integration. Putting we have f| =/(*) = tan 0, ...... (2) where has the same meaning as in Diff. Calc., Art. 38. Thus for every value of x we know the inclination 0of the curve to the axis of x. The notation also implies that when x = a, y o. Starting, then, from the position (a, o), if we imagine the point (x, y) always to move in the proper direc- tion, a direction which changes as x changes, it will trace out a definite curve, so that the function y has a definite value for each value of x. 86. As an example, let us take the function y f sin x ' = -T dx > (0 Jo * a case in which the indefinite integration cannot be performed. Starting from the origin, the describing point must move in the varying direction defined by sin x . . tan = (2) 110 METHODS OF INTEGRATION. [Art. 86. When x O, we have tan I, or = 45 ; the curve therefore starts from the origin at this inclination. But, since equation (2) shows that tan decreases as x increases, the curve as we proceed toward the right lies below the tangent at the origin, as in Fig. 3. The ordinate y continues, how- /i ever, to increase (compare Art. 83) until, at the point A, x FlG - 3- reaches the value 7t, for which = O. As x passes through this value, changes sign ; there- fore^ has reached a maximum value. From x = n to x = 2n tan is negative, hence y decreases ; in this interval sin x in equation (2) goes through numerically the same values as before, but the denominator x is now much larger than before, hence it is plain that y does not decrease to zero. In like manner it is obvious that it will increase again from x = 2n to x = $7t, and so on, reaching alternate maxima and minima, which contin- ually approach an asymptotic value of y corresponding to x =. oo . The form of the curve for positive values of x is there- fore that represented in Fig. 3. For negative values of x the curve has a similar branch in the third quadrant. It is evident that we can in no case have a finite value of the integral when x = oo , unless, as in this case, the quantity under the integral sign approaches zero as a limit when x increases without limit. 87. When the graph or curve of the indefinite integral is drawn, the definite integral between any limits c and d is rep- resented by the increment of y in passing from the abscissa c to the abscissa d, and the condition given in Art. 82 requires that the curve shall be continuous between the points corre- sponding to the limits ; in other words, that these points should belong to the same branch of the curve. In the illustration above, f(x) remains real and finite for all VII.] THE GRAPH OF AN INTEGRAL. Ill values of x ; accordingly, in this case, all values of the limits are possible. But, if f(x] becomes infinite or imaginary for any value of x between the limits, it will usually be found that, although the indefinite integral y F(x) may have a finite value at each limit, the corresponding points will belong to separate branches of the curve, and therefore equation (4) of Art. 82 cannot be used. For example, if f(x) = x~*, which is infinite for x = o, we have for the indefinite integral tdx - I y ' J x x and y is also infinite when x = o. The graph of this integral is an equilateral hyperbola, by means of which we can represent, for example, the definite in- tegral between the limits 2 and i as the differ- ence BR between two values of the ordinate. But we cannot interpret in this case an integral between whose limits the value x = O lies (that is, with one negative and one positive limit), because the corresponding points would lie on dis- connected branches of the curve. It will be noticed that, for the same reason, if we use the notation of the corrected integral we cannot represent the entire curve by a single equation ; thus the two branches as drawn in Fig. 4 have the separate equations r dx ( x dx y = I ^~ and y = \ IF* J -oo X J oo * Lff FIG. 4. 112 METHODS OF INTEGRATION, [Art. 87. the latter expression denoting an essentially negative quan- tity. 88. It may happen, however, that the indefinite integral remains finite for a value of x which makes f(x) infinite. In such a case, if the value of f(x) remains real while x passes through this value, we shall still have a continuous curve and a continuous variation of the ordinate between limits which include this value of x. For example, when /"(;:) = x~%, which is infinite for x = o, and is real for both positive and negative values, we have y fdx }^~~ FIG. 5. The curve which is drawn in Fig. 5 touches the axis of y and is real on both sides of it, form- ing a cusp. Thus the ordinate varies continuously while x passes through zero, and we can write, for instance, in accordance with equation (4), Art. 82, as illustrated in Fig. 5 by the difference of ordinates BR. multiple-valued Integrals. 89. When the indefinite integral is a many-valued function, the condition that it shall vary continuously while x passes from the lower to the upper limit requires the selection of properly corresponding values at the two limits. In illustra- VII.] MUL TIPLE- VA L UED INTEGRALS. tion let us construct the graph of the fundamental integral = sn " * 0) In this case x can only vary between the values i and -+- i. Proceeding from the origin, the curve, Fig. 6,. has the inclination 45, which is gradually in- creased to 90 when x = i, at the point A, for which the value of y is TT. hsx passes from o to i a similar branch is described in the third quadrant, com- pleting the curve drawn in full line, the ordinate of which is in fact the primary value of the indefinite integral sin' 1 ^- (Diff. Calc., Art. 74). . But the complete curve _y=sin~ I ..r, or;r=sin^/, has another branch continuous with this at the point (i, ?r), as represented by the dotted line. In order that the point moving in accordance with the integral expres- sion shall describe this branch, we must suppose that x, after reaching the value unity at the point A, begins to decrease, and that, as dx thus changes sign, the radical |/(i x*}, which then becomes zero, also changes sign, so that the quantity under the integral sign remains positive, and y goes on in- creasing. Since the radical cannot change sign, while varying continuously, except when it passes through the value zero (at which time its two values are equal), it is only when x = i that it can return to its previous values without causing y also to return to the corresponding previous values; that is, with- out making the generating point return upon the path already described. We may thus, by the alternate increase and decrease of x between its extreme values, describe an infinite number of FIG. 6. 1 14 METHODS OF INTEGRA TION. [Art. 89. branches. In this generation of the integral the value of the radical undergoes periodic change of sign but is never am- biguous ; for, in drawing the curve we assumed that in equa- tion (i) the radical had its positive value when x = o and y = o. This assumption determines the value of the radical for every other value of y ; it is, in fact, always equal to cos, y. 90. In the direct application of limits to the integral (i), it is of course sufficient to limit the indefinite integral to its primary value, the radical being positive for all values of x be- tween the limits ; but, when the integral is the result of trans- formation, care may become necessary in the selection of the values at the limits. Consider, for example, the integral f*" dz J _, |/(2 - O' which can be evaluated directly by formula (/') in the form z -i v* ^ n sin" 1 = . But suppose we make the transformation z* = i x, whence dz = , and \/(2 z*) = ^/(i + x}. It is first to be noticed that, because the value of z is neg- ative at the lower limit, we should in the value of dz put dx z = |/(i x}\ thus dz = . :, and the result of trans- w I 1 ^^ ) formation is f 2 dz i f~ : dx z _ i f~ : x _*>-" 2 Jo ~W=~?i The relation z* = i x shows that as z increases from I to o, x increases from o to i, and the corrected integral in the second member (which is sin" 1 x} increases from o to %n. Then as z further increases from o to -j/2, x decreases from I to I ; but the integral continues to increase from \n through VIL] MUL TIPLE- VA L UED INTEGRA LS. values which are not " primary," reaching the final value f TT at the upper limit, so that we have for the value of the second member of the equation ^(f?r o) = f n as before. 91. To illustrate another mode in which an integral may acquire multiple-values, let us consider the graph of the fun- damental integral dx 5- = tan ' x. Proceeding from the origin, at the inclination 45, the curve approaches, as x increases without limit, an asymptote parallel to the axis of x at the distance ;r. The similar branch de- _ scribed for negative values of x completes the full line in Fig. 7, representing the pri- _ mary values of tan" 1 x , which are employed in any direct application of limits to the integral. But the complete curve y = tan" 1 x consists of an unlimited number of branches which are repetitions of this curve at successive vertical intervals each equal to TT, so that each asymptote is approached by two branches. Now, when the integral arises from transformation, it may happen that x passes through infinity, changing sign, and then the ordinate will pass without discontinuity of value from one branch to the next. For example, given the integral FIG. 7. dB J cos' 6 + 9 sin 2 B if we put tan B = x, this becomes _ re secj Jo i +.9 9 tan 8 0' 4/3 3 dx _ i - tan - 3 = - [tan- 1 ( 1/3) tan- 1 o]. Il6 METHODS OF INTEGRATION. [Art. 91. As 6 passes from o to %n, x increases from o to -f co , and tan" 1 ^x increases from O to TT. But as 6 further increases from I-TT to ?r, x changes sign and passes from co to \ 1/3. During this interval tan" 1 ^x still further increases from %-n to f re, which is therefore in this case to be taken as the value of tan" 1 ( 1/3), while we take zero as the value of tan' 1 o. Hence r? do cos* 6 + 9 sin 1 6 9 Formula of Reduction for Definite Integrals. 92. The limits of a definite integral are very often such as to simplify materially the formula of reduction appropriate to it. For example, to reduce x n e-*dx, } O we have by the method of parts \x n e-*dx x n e~ \ Now, supposing n positive, the quantity x n e~ x vanishes when x = o, and also when x = oo. [See Diff. Calc., Art. 159.] Hence, applying the limits o and oo , I x"e~*dx = n\ Jo J By successive application of this formula we have, when is an integer, VII.] FORMULA. OF REDUCTION. 2-1 = n \ I x n e~ x dx = n(n i) ..... Jo 93. From equation (i), Art. 66, supposing m > I, we have V Tt f 2 sin- OdO = ^Hl f 2 sin*-' Odd. J m J If w is an integer, we shall, by successive application of this r JT formula, finally arrive at \ dO = - or * sin 0^0 = i, according J 2 Jo as m is even or odd. Hence ir r f ^ a ja ( m ~ l }( m 3) I ^ / r>v if ? is even, sm w 6 dO = v -- 7^ - r-^ --- , . . (P) J m(m 2) ...... 22 * j -r Jj f 2 . /i jn (m l)(m ).... 2 and if w is odd, sin"* Q dQ =. * -- 7-^ - ^^ J m(m-2) ...... I 94-. From equations (3) and (4) Art. 69, we derive f 2 sin w 6 cos" 0^0= n ~~ l ( 2 sin w 6 cos"- 2 Jo m + n } w w and F sin* cos" v log ^ hence Jo /+*' ,00 11 I nor /r I ay dy I a Jo/ log + ' 2a The Integral' regarded as the Limit of a Sum. 99. We have seen in Art. 3 that, if the curve y =f(x~) be f & constructed, the integral f(x)dx (in which we suppose/^) J a to remain finite and continuous as x varies from a to b] is rep- resented by the area included by the curve, the axis of x and the ordinates corresponding to x = a and x = b. Let CD in Fig. 8 be the curve, and let the base of this area, AB, whose length is b - y )Ax. ... (4) Therefore the difference between the integral and either of the approximate expressions is less than (y n y ) Ax. Now when the number of parts n is indefinitely increased, VII.] THE INTEGRAL AS THE LIMIT OF A SUM. 123 so that Ax is decreased without limit, the limit of (y n j ) Ax is zero ; it follows that I ydx is the limit of 2y Ax for any range of values of x for which y is an increasing function of x. It can be shown in like manner that the same thing is true while y is a decreasing function, and therefore in general [ f(x]dx the limit of^/(*) Ax . . . (5) J a x = a when Ax decreases without limit. 100. The typical term/(;r) Ax is called the element of the sum in equation (5). It will be noticed that the sum will have the same limit whether the x in f (x) Ax be taken as in equation (2) or as in equation (3), or in any other manner, provided it be some value within the interval to which Ax corresponds. The expression f(x]dx is, in like manner, called the element of the integral in equation (5). In many applications of the Integral Calculus, the expres- sion to be integrated is obtained by regarding it as an element. In so doing we are really dealing with the element of the sum ; but as we intend to pass to the limit we may, in accordance with the remark made above, ignore the distinction between any values of x between x and x + Ax. By equation (5), we pass to the limit by simply writing d in place of A, and the integral sign in place of that of summation,* so that in practice it is- customary to form the element of the integral at once by writing d instead of A. * The term integral, and the use of the long s, the initial of the word sum, as the sign of integration, have their origin in this connection between the processes of integration and summation. 124 ADDITIONAL FORMULAE OF INTEGRATION. [Art. IOI. Additional Formula of Integration. 101. The formulae recapitulated below are useful in evalu- ating other integrals. (A) and (A') are demonstrated in Art. 17; (B) and (C) in Art. 29; (D) and (E) in Art. 30; (F) in Art. 31 ; (G) and (G 1 ) in Art. 35 ; (H) and (/) in Art. 50 ; (/) in Art. 51 ; (K) in Art. 52 ; (L) in Art. 53 ; (M) in Art. 55 ; (N) and (O) in Art. 58 ; (P) and (/>') in Art. 93 ; and (0 in Art. 94. dx \ x a t dx \ x a \ dx I # + .2- -a -- 5- == log- - ; -5 -- 2 = log- . . . (A') }x* a? 20. *> x + a \a i x i 2.a * a x (sw?OdO = \(d - sin cos 0) ........... (B) {cos*edd = %(e + sin0cos8) ........... (C) ( df) 2j= log tan ........... .'.(/?) J sin B cos 8 i cos ( dV Fit 0~\ i+sin0 s=logtan - + - =log - - = log (sec + tan 0). (F) JcosO [_4 2J cos^ ^j, - a = ~77~F m tan "' \V ^-TT tan J(7 + b * and w an integer] >. f'sin 4 Jo tznir. 2 ^TQ ? " Jo 2 + COS 9* a j. f 2 sir' "fo Jo sh Jo 3. i a^, 32 16 4. | sm" ficfy, 16 35' Qcos 6^r9, S 12 VII.] EXAMPLES, 127 f ff 4 7. sin 3 cos 4 0 o and z > i. w j f = x (a n J 128 METHODS OF INTEGRATION. [Ex. VII. 16. Deduce from the result of Ex. 15 the value of the integral when m is an integer. -' a-x-x= . .a"*-'. n (n+i) - (n + mi) t a 4 2 18 4/2 JL& 17. (a + x}" (a-xy dx. See Ex. 16. a 8 J-a 454S IT 18. Tsui' 6 (cos 0) 2 ^/9. /'a/ sin a 6 x, and see Ex. 16. Jo w 19. I 4 cos 8 0, Jo and by formula (0 The areas in this case are all generated with the positive sign, since when y is negative dx is also negative. Had the generating point moved about the curve in the opposite direc- tion, the result would have been negative. Area generated by a Rotating Line or Radius Vector. 108. The radius vector of a curve given in polar coordinates is a variable line rotating about a fixed extremity. The angular rate is denoted by -r- and may dt be re- garded as constant, but then the rate at which area is generated by the radius vector OP, Fig. 13, will not be constant, since the length OP is not constant. The differential of this area is the area which would be generated in the time dt, if the rate of the area were con- stant ; FIG. 13. that is to say, if the radius vector were of constant VIII.] AREAS GENERATED BY ROTATING LINES. '35 length. It is therefore the circular sector OPR of which the radius is r and the angle at the centre is d6. Since arc PR = r d0, sector OPR = -r t d8', therefore the expression for the generated area is 0) 109. As an illustration, let us find the area of the right-hand loop of the lemniscate i^cf 1 cos 2,0. FIG. 14. The limits to be employed are those values of 6 which 7t . 7t make r o; that is and -. 4 4 Hence the area of the loop is 2 J_ 4 110. When the radii vectores, r 2 and r v corresponding to the same value of 6 in two curves, have the same sign, the area generated by their difference is the difference of the polar areas generated by r, and r 2 . Hence the expression for this area is A = l -(r? - r?) d8 ....... (2) 136 GEOME TRICA L A P PLICA TIONS. [Art. III. HI. Let us apply this formula to find the whole area between the cissoid ' r-i = 2a (sec 6 cos #), Fig. 15, and its asymptote P Z , whose polar equation is r 2 = 2a sec 6. One half of the required area is generated by the line P\P& while 6 varies from o to 7i. Hence by the formula 2 A = (2 - cos 2 0) dd=~ FIG. 15. Therefore the whole area required is Transformation of the Polar Formulae. 112. In the case of curves given in rectangular coordinates, it is sometimes convenient to regard an area as generated by a radius vector, and to use the transformations deduced below in place of the polar formulae. Put y mx; . (i) now taking the origin as pole and the initial line as the axis of x, we have x r cos 0, y = r sin 6 : . . (2) therefore and = = tan 0, x dm = sec 2 8 - (3) VIII.] TRANSFORMATION OF THE POLAR FORMULAS. 137 From equations (2) and (3), x* dm = r* d&; therefore equation (i) of Art. 108 gives (4) In like manner, equation (2) Art. no becomes 4 ^U -*/)<*" ...... (5) 113. As an illustration, let us take the folium o ...... (i) Putting y = mx, we have X s ( I + nf] ^amx* =o ...... (2) This equation gives three roots or values of x> of which two are always equal zero, and the third is x - whence y - ........ (5) i + nP These are therefore the coordinates of the point P in Fig. 16. Since the values of x and y vanish when m = o, and when m oo , the curve has a loop in the first quadrant. To find 138 GEOMETRICAL APPLICATIONS. [Art. 113. the area of this loop we therefore have, by equation (4) of the preceding article, _ 9^ f 2 J i 21+ 114. The area included between this curve and its asymr> p tote may be found by means of equation (5), Art. 112. The equation of a straight line is of the form y = mx + b, o\ o\ c FIG. 16. an d since this line is parallel to y the value of m for the asymptote must be that which makes x and y in equations (4) and (5) infinite ; that is, m = i ; hence the equation of the asymptote is y + x = b, (6) in which b is to be determined. Since when m = i, the point P of the curve approaches indefinitely near to the asymp- tote, equation (6) must be satisfied by P when m= I. From (4) and (5) we derive y + x = m i + nr i m + m 2 whence, putting m= i, and substituting in equation (6) a = b, the equation of the asymptote AB, Fig. 16, is y + ^ = - a ........ (7) VIII.] TRANSFORMATION OF THE POLAR FORMULAE. 139 Now, as m varies from oo to o, the difference between the radii vectores of the asymptote and curve will generate the areas OBC and ODA, hence the sum of these areas is repre sented by A=-\ (*$-*) dm, 2J-00 in which x z is taken from the equation of the asymptote (7), and x^ from that of the curve. Putting y = mx, in (7), we have i + m ' and the value of x v is given in equation (4). Hence . _ a* , ~ *"" a + nf i T i + m J_ 2 + m 2 i + n _ 2 i m Adding the triangle OCD, whose area is \cP, we have for the whole area required ftf 2 . * This reduction is given to show that the integral is not infinite for the value m = i, which is between the limits. See Art. 82. 140 GEOMETRICAL APPLICATIONS. [Ex. VIII. Examples VIII. i. Find the area included between the curve and the axis of x. 12 2. Find the whole area of the curve a y = * (a* - x*). 4 3. Find the area of a loop of the curve 4. Find the area between the axes and the curve 5. Find the area between the curve and one of its asymptotes. 20*. 6. Find the area between the parabola y* = ^ax and the straight 80 s line y = x. . 3 y. Find the area of the ellipse whose equation is VIII.] EXAMPLES. 141 8. Find the area of the loop of the curve Of = (*-)(*-*)', 8 (b - a)\ tn which c > o and b > a. J- i$Vc 9. Find the area of the loop of the curve 1050* 10. Find the area'included between the axes and the curve 20 ii. If is an integer, prove that the area included between the axes and the curve i s *=^ :' \ i**. zn (2n !)(+ i) 12. If n is an odd integer, prove that the area included between the axes and the curve /-,.\ G)' _ la ^1 7 . 2 (2 2) 2 2 142 GEOMETRICAL APPLICATIONS. [Ex. VII L 13. In the case of the curtate cycloid x = aib b sin ^, y a b cos ip t find the area between the axis of x and the arc below this axis. (20* + P) cos- 1 1 - 3 a V(? - A 14. If b = %a7T, show that the area of the loop of the curtate cycloid is -OH- 15. Find the area of the segment of the hyperbola x a sec ip, y = b tan ^, cut off by the double ordinate whose length is 2b. ab\ 1/2 log tan ^- . 1 6. Find the whole area of the curve r* = a* cos a 9 + t? sin' 9. - (a* + '). 2 17. Find the area of a loop of the curve > o .. . . ab (a b ) _, a r* = a 1 cos b sm* 9. - + 5 1 tan ' - , 22 O 1 8. Find the areas of the large and of each of the small loops of the curve r = a cos 9 cos 29 ; VIII.] EXAMPLES. . 143 and show that the sum of the loops may be expressed by a single integral. no 1 , a' 7ta* a* ?- + - , and --- . 10 6 32 12 19. In the case of the spiral of Archimedes, r = aB, find the area generated by the radius vector of the first whorl and that generated by the difference between the radii vectores of the nth and (n + i)th whorl. 2 3 and Sna'Tr 3 . 3 20. Find the area of a loop of the curve TTO* r = a sin 39. . 12 21. Find the area of the cardioid r = 40 sin 2 9. 6ita*. 22. Find the area of the loop of the curve cos 29 a 1 (4 TT) r = a -- -. - . cos . 2 23. In the case of the hyperbolic spiral, show that the area generated by the radius vector is proportional to the difference between its initial and its final value. 144 GEOMETRICAL APPLICATIONS. [Ex. VIII. 24. Find the area of a loop of the curve ir.cf r = a cos n Q. - 4* N 25. Find the area of a loop of the curve . COS 1 6 ' 2 26. Find the area of a loop of the curve r* sin 6 = a* cos 26. Notice that r /> raz/ and finite from 6 = to 6 = , #;? za and show that the same expression gives the sum of the loops when b < 2a, V1I1.J EXAMPLES. 29. Find separately the areas of the large and small loops of the limafon when b < 2a. If a = cos -I ( \ t large loop = (20" + a ) a + small loop = ( 2 a s + F) (n - a) - 30. Find the area of a loop of the curve 31. Find the area of the loop of the curve 2 cos 261 r = a cos 9 32. Show that the sectorial area between the axis of x, the equi- lateral hyperbola and the radius vector making the angle 9 at the centre is represented by the formula . i i + tan 9 and hence show that - g-SA and y = If A denotes the corresponding area in the case of the circle **+/= i, we have x = cos 2 A, and y = sin 2^4. 146 GEOMETRICAL APPLICATIONS. [Ex. VIII. In accordance with the analogy thus presented, the values of x and y given above are called the hyperbolic cosine and the hyperbolic sine of 2 A. Thus _ = cosh (2 A), - = smh (2 A). 2 ^ 33. Find the area of the loop of the curve o. 35. Find the area between the curve 38. Trace the curve y x za sm . x* and find the area of one loop. . 34. Find the area of the loop of the curve 2 + I and its asymptote^ - a . 36. Find the area of the loop of the curve y 3 + ax* axy o. 37. Find the area of a loop of the curve x* + y* =0*xy. -- Six.] VOLUMES OF GEOMETRIC SOLIDS. IX. Volumes of Geometric Solids. 115. A geometric solid whose volume is required is fre- quently defined in such a way that the area of the plane sec- tion parallel to a fixed plane may be expressed in terms of the perpendicular distance of the section from the fixed plane. When this is the case, the solid is to be regarded as generated by the motion of the plane section, and its differential, when thus considered, is readily expressed. 116. For example, let us consider the solid whose surface is formed by the revolution of the curve APB, Fig. 17, about the axis OX. The plane section per- pendicular to the axis OX is a circle; and if APB be referred to rectangu- lar coordinates, the distance of the section from a parallel plane passing through the origin is x, while the radius of the circle isj. Supposing the centre of the section to move uniformly along the axis, the rate at which the volume is generated is not uniform, but its differential is the vol- ume which would be generated while the centre is describing the distance dx, if the rate were made constant. This differen- tial volume is therefore the cylinder whose altitude is dx, and the radius of whose base isjj>. Hence, if V denote the volume, dV '= rcy*dx. 117. As an illustration, let it be required to find the volume of the paraboloid, whose height is h, and the radius of whose base is b. 148 GEOMETRICAL APPLICATIONS. [Art. 1 1 7. The revolving curve is in this case a parabola, whose equa- tion is of the form and since y = b when x = h, a & 4flh, whence 4^ = -7 ; ft the equation of the parabola is therefore Hence the volume required is ,. J f f* n&h dx = 7r\ xdx= . h Jo 2 118. It can obviously be shown, by the method used in Art. 1 1 6, that whatever be the shape of the section parallel to a fixed plane, the differential of the volume is the product of the area of the generating section and the differential of its motion perpendicular to its plane. If the volume is completely enclosed by a surface whose equation is given in the rectangular coordinates x, y, z, and if we denote the areas of the sections perpendicular to the axes by A x , A y , and A z , we may employ either of the formulas V = \A X dx, V= \A y dy, V = IA Z dz. The equation of the section perpendicular to the axis of x is determined by regarding x as constant in the equation of the surface, and its area A x is of course a function of x. IX.] VOLUMES OF GEOMETRIC SOLIDS. 149 For example, the equation of the surface of an ellipsoid is # / __ L. -L_ _L- _ - T a* + & (x,y)dy, ....... (i) J y\ in which the limits j/, and j 2 may be any functions of x. In the integration, ^ is to be treated like any other quantity independent of y which may be involved in the expression (f>(x, y] ; in other words, as a constant with respect to the variable y. Thus the indefinite integral will contain both x and y ; but since the definite integral is a function not of the independent variable but of the limits, the expression (i) is independent of y, but is generally a function of x. We may therefore denote it by f(x), and write it in place of f(x) in the expression of an integral in which x is the independent vari- able. Thus, putting =/(*)> ...... (2) i y\ \f(x}dx=\ ^(x, y) is finite and continuous while y passes from 7, to /. Further, assuming this to be the case i 5 6 GEOMETRICAL APPLICATIONS. [Art. 122. for all values of x ,from a to b (so that/"(.r) in equation (3) is finite and continuous for all values concerned in the .r-integra- g tion), the double integral is the limit of 2 a f(x) Ax, when Ax is diminished without limit. It readily follows that the double integral is the limit of where both Ay and Ax are diminished without limit.* The typical term fy(x, y] Ay Ax is called the element of the sum, and in like manner (x, y] dy dx is the element of the double integral. As mentioned in Art. 100, in forming the expres- sion for the element, no distinction need be drawn between any values of x and y, between x and x + Ax, y and y -f Ay, re- spectively, because these distinctions disappear at the limit. Limits of the Double Integral. 123. In discussing the limits of the double integral (3), Art. 121, it is convenient to consider first the simpler expression dy dx. FIG. 19. Performing the jj/-integration, this re- duces to the simple integral (2) * Denoting the difference between _/{*) and . y)Ay, of which ifis ihe limit, by e, e is a quantity which vanishes with Ay. Then the difference be- tween ~2 a f(x) Ax and 2*2f^*#(.t, y) Ay Ax is 'S^Ax, a quantity which vanishes with Ay if a and b are finite. Hence, in this case, the double integral which is the limit of the first of these sums is also the limit of the second. But the con- clusion does not follow when the limits are infinite ; in fact, the double integral is not then always independent of the mode in which the limits become infinite. X.] LIMITS OF THE DOUBLE INTEGRAL. 157 In Fig. 19, using rectangular coordinates, let OA = a, OB b, and let CD, EF be the curves y = j, , y = j a ; then (2) is, by Art. 103, the expression for the area CDFE. There- fore the double integral (i) is represented by the area enclosed by the curves y = y l , y = y^ and the straight lines x = a, x = b. I24-. In order that the double integral may represent the area enclosed by a single curve (like the dotted line of Fig. 19) of which the equation is known, the limits jj/, and y^ must be two values of y corresponding to ,the same value of x, and the limits for x must be those values for which y l and y^ are equal. Between the limiting values of x the values of y are real, and beyond them the values of y become imaginary. For example, suppose the curve to be the ellipse 2X 2.xy + y 4x solving for y, y = x+ l ^(-^ + 6x - 5) = x whence This expression is real for all values of x between I and 5 ; hence the entire area is - i)(5 - *)] dx* = 4*. It is evident that we might equally well have used the ex- pression f q f x * ( q A = dxdy = \(x, x,)dy, itlxi JP * It is useful to notice that a definite integral of this form, by a familiar property of the circle, represents the area of the semicircle whose diameter is the difference between the limits. 158 GEOMETRICAL APPLICATIONS. [Art. 124. in which x^ and x^ are obtained as functions of y, by solving the equation of the curve for x, thus x = b(y + 2) i V(~ }' + 8j - 8) ; and the limiting values of y are those obtained by putting the radical equal to zero, namely, p = 4 2 4/2, (x,y\ or 3, remains finite and con- tinuous while the independent variables x and y vary in any way, provided the point (x, y) remains within the area ASBR. This assumption is clearly identical with that made in Art. 122. A definite solid will then be enclosed between the surface (3), l6o GEOMETRICAL APPLICATIONS. [Art. 126. the plane of xy, and the cylindrical surface. We have now to show that this solid will represent the double integral (i). Let SRPQ be a section of this solid by a plane parallel to that of yz, so that in it x has a constant value. The ordinates of the points R and S are the values of y. t and y^ correspond- ing to that particular value of x. The section SRPQ may be regarded as generated by the line z, while y varies from y l to y t \ hence, denoting its area by A x , as in Art. 118, f* 1 A x =\z dy. y\ Therefore the volume is fb fbty t V= A x dx = zdydx, la J ai Ji which is identical with expression (i). Change of the Order of Integration. 127. It is obvious that the order of integration may be reversed as in Art. 124, provided that the integration extends over the same area. Considering the corresponding process of double summation, we may be said, in the first case, to sum up those of the elements z dy dx which have a common value of x into the sum A x dx, which then constitutes the element of the second summation ; while in the second case we first sum up those of the original elements which have a common value of y into the element A y dy, which is afterward summed for all values of y within the given limits. Since, in expression (i), Art. 126, dx is a constant with respect to the /-integration, it may be removed to the left of the corresponding integral sign. In the resulting expression, f 4 V* dx\ la J yi x.j CHANGE OF ORDER IN INTEGRATION. 161 it is to be understood that because the /-integral which is a function of x follows the sign of ^-integration it is " under " it, or subject to it. This notation has the advantage of mak- ing it perfectly clear to which variable each integral sign cor- responds.* 128. The limits of a double integral are sometimes given in the form of a restriction upon the values which the inde- pendent variables x and y can simultaneously assume. Sup- pose, for example, that in the integration neither variable can become negative or exceed a, and that y cannot exceed x. If x and/ are coordinates of a point in Fig. 21, the restriction is equivalent to saying that the point cannot be below the axis of x, to the right of the line x = a or above the line y = x. Therefore the area of integration is the triangle OAB. This may be expressed either by giving to y the limits zero and x, and then giving to x the limits zero and a ; or, reversing the order of integration, by giving to x the limits y and a, and then giving to y the limits zero and a. Thus the area of integration can be represented by either of the expressions dydx> or dx dy. oJo 1 o y 129. As an illustration, let us take the integral U= I I sin' 1 / |/(i x) \/(x y]dydx. oJ o FIG. 21. * We have in the preceding articles supposed the first written integral sign to correspond to the last written differential when the pair of differentials is written last ; but writers are not uniform in this respect. l62 GEOMETRICAL APPLICATIONS. [Art. 1 29. The /-integration, which is here indicated as the first to be performed, namely, the integration of \/(x y) sin" 1 y dy is impracticable. Noticing, however, that the ^-integration in the given expression is of a known or integrable form, we change the order of integration, determining the new limits so as to represent the same area of integration. Since this area is that indicated in Fig. 21, when a = i, we thus obtain (7= sirr 1 /^! ^/[(i x](x y}\dx. Jo J y The value of the ^-integral is ^TT(I /) 2 ; hence U = ~^\ sin-'j/(i J Finally, integrating by parts and then putting y = sin 6, -sn 2 4 o4/(i-/) 2 40 2 4 _4 130. In a case where the first integration can be effected with respect to either variable, it may happen that in one case only are the limits such as to make the second integration possible. Given, for example, f 00 f 3 U\ e- xy dydx, ...... (i) J o J a in which a and/? are both positive. Integration for/ gives u= and for this form the second integration is impracticable. But, integrating expression (i) first with respect to x, we have, X.] TRIPLE INTEGRALS. 163 owing to the special values (zero and infinity) of the limits, the simple form The double integration in this example extends over the infinite strip of area between the parallel lines y = /3, y = a and on the right of the axis of y, yet we have obtained a finite result. It is plain that this would not have been possible but for the fact that the element e~ xy dy dx approaches to the limit zero, when x increases without limit (y remaining between finite limits) ; that is, the element vanishes for all the infinitely distant points which are included in the area of integration. The fact that we have thus obtained the value of the definite simple integral (2), although the corresponding in- definite integral could not be found, will be noticed in con- nection with the methods of evaluating definite integrals. Triple Integrals. 131. An expression of the general form Dy-tf't <}>(x , y , z) dz dy dx (i) y\i*i is called a triple integral. In the first integration the limits #, and z^ are in general functions of x and y. In the next, the limits j, and^ a are in general functions of x; and in the final integration the limits are constants. It is readily seen that the triple integral, like the double in- tegral considered in Art. 122, is the limit (where Ax, Ay, and Az diminish without limit) of the result of a corresponding sum- mation. Accordingly (f> (x,y, z) dz dy dx is called the element of the integral (i). 164 GEOMETRICAL APPLICATIONS. [Art. 131. Consider now the triple integral with the same limits, but having the simpler element dz dy dx ; the ^-integration can here be effected at once, and we have f * pa OS dydx=\ (z^ #,) dy dx. ... (2) JaJjX, Now supposing x, y, and z to be rectangular coordinates in space, the last expression represents the difference between two solids defined as in Art. 126; that is, the triple integral in equation (2) represents the volume included between the surfaces z #,, z = z,, and the cylindrical surface wJwse section in the plane of xy is the contour determined by the limits for y and x. 132. When the volume is completely enclosed by a surface defined by a single equation or relation between x,y, and z, the case is analogous to that of the area discussed in Art. 124; s l and z t will be two values of z corresponding to the same values of x and y, and the limits for x and y must be determined by the area within which the value of z^ z^ is real. This area is the section with the plane of xy of a cylinder which circum- scribes the given volume. In this case it, is plain that we may perform the integrations in any order, provided we properly determine the limits. Considering the corresponding process of summation, we may be said in equation (2) to have summed those of the ultimate elements which have common values of x and y into the linear element (z^ ,) dy dx. If the integration for y is effected next, we combine such of these last elements as have a com- mon value of x into the areal element A x dx which is often called a lamina. Thus each of the formulae in Art. 118 is the result of performing two of the integrations in the general expression for a volume, namely, V= [\\dxdydz. X.] THE VOLUME OF INTEGRATION. 165 In any case where the result of two integrations (that is, the area of a section) is known, we may of course take advantage of this and pass at once to the simple integral, as in Art. 118, where in finding the volume of the ellipsoid we regarded the expression for the area of an ellipse as known. Integration over a Known Volume. 133. In the general expression, (i) of Art. 131, the integra- tion is said to extend over the volume defined by the limits, that is, the volume represented by either member of equation (2). It is sometimes possible, in the case of a triple integral of the more general form, to take advantage of our knowledge of the geometrical solid over which the integration extends. For example, let it be required to evaluate the integral u = for all values of x, y, and z such that does not exceed unity, that is to say, when the integration ex- tends over the volume of the ellipsoid considered in Art. 118. We here perform the integration for y and z first, because we see that the result will be the section A x of this solid. This reduces the integral to the simple form TT J Ttbc t a , 2 y.. I I .. I A-* /I SJ T I V"-/ * .. r -V* lyY'** (_/ - | X, yi y. U,*L - I X, \Cl X \U,X a? }-a = nctbcC 2 - --} = 166 GEOMETRICAL APPLICATIONS. [Art. 134, 134- . A transformation of variables will sometimes enable us to make use of geometrical considerations. For example, let us find the value of x dxdy dz \/(xyz) for all positive values of the variables whose sum does not ex- ceed a ; or, as it may be written, ( a dx r~* dy t a -*- y dz (^/ . t ~ J 4/jo V 2 If we put V ' x = , \/y = rf, ^z = C, we have U = sf f f ddnd, J O O J o and the condition determining the upper limits is & + jf + ? = a. The last integral (regarding &,, rf, and C as a new set of rec- tangular coordinates) represents one octant of a sphere whose radius is 4/0 ; hence U = $na*. Representation of a Triple Integral by a Mass of Variable Density. 135. We may, in any triple integral of the general form J 0(>, y, z) dx dy dz, suppose the function (x,y, z) is to be taken as the average density of the element Ax Ay Az, so that 2 ) The first of these double integrals is the area of the semi- circle mentioned above Hence 2/1 f a "21 f a 1 / !/( 2 - f}dy +- \(d* - Jo 3 J o 6 and finally, U = -^Tt/ia^a 2 -f- Examples X. 1. Find the volume cut from a right circular cylinder whose radius is a, by a plane passing through the centre of the base and making the angle a with the plane of the base. 2 , -a tan a, 3 2. Show that the integral of x dx dy over any area symmetrical to the axis of y vanishes. Interpret the result geometrically, and apply to find the integral of (c +mx + ny)dx dy over the ellipse f / + tfx* = (?b\ nabc. * The case is analogous to that of the double integral when both pairs of limits are constants, that is, when the area of integration is a rectangle. X.] EXAMPLES. 169 3. Show that the volume between the surface z n = a v + jy and any plane parallel to the plane of xy is equal to the circumscrib- ing cylinder divided by n + i. 4. Find the volume enclosed by the surface whose equation is x* y a 2* - + "71 + -4 = I- 4 - . a b c 5 5. A moving straight line, which is always perpendicular to a fixed straight line through which it passes, passes also through the circumference of a circle whose radius is a, in a plane parallel to the fixed straight line and at a distance b from it; find the volume en- closed by the surface generated and the circle. ncfb 2 6. A cylinder cuts the plane of xy in an ellipse whose semi-axes are a and b, and the plane of xz in an ellipse whose semi-axes are a and c, the elements of the cylinder being parallel to the plane of yz ; find the volume of the portion bounded by the semi-ellipses and the surface. 2 -aoc. 3 7. Find the volume enclosed by the surface "...!?- J 1 **"* Ttabc and the plane x = a. --- . 2 8. Find the volume enclosed by the surface Find A z as in Art. 107. 35 9. Find the volume between the coordinate planes and the surface go GEOMETRICAL APPLICATIONS. [Ex. X. 10. Find the volume cut from the paraboloid of revolution y 4 z* = ^ax by the right circular cylinder x* + y = 2ax, whose axis intersects the axis of the paraboloid perpendicularly at the focus, and whose surface passes through the vertex. 3 , i6a' -\ -- . ii. Find the volume cut from a sphere whose radius is a by a right circular cylinder whose radius is b, and whose axis passes through the centre of the sphere. _ / j _ m 12. Prove that the volume generated by the revolution about the axis of y of the area between the equilateral hyperbola and the double ordinate 2y l is equal to the sphere of radius y t . [Ex. 1 1 shows that the circle has the same property.] 13. Change the order of integration in the double integral n-za y (f)(x,y)dx(ty. y t a t x tia tia x (p(x,y)dy dx + (x, y]dy dx. JoJ JrtJo 14. Evaluate the integral taf*a-y dx dy \ o \ y V(x *-xy}' 2 * /2l S ( ^ 2}a ' 15. Integrate xydydx over the area of the circle (x - /O 3 + (y - k)* = a\ ncthk. X.] EXAMPLES. 16. Show that the integral of the element in Ex. 15 over any square circumscribing the given circle is \a^hk. 17. Integrate x*y dy dx over the circle of Ex. 15, and over the circumscribing square with sides parallel to the axes. na*hk(h* + fa 1 ); ^hk(tf + c?}. 1 8. Evaluate i \xyzdxdy dz for all positive values of x,y, and z whose sum is less than a; also for all positive values the sum of whose squares is less than' s . a" a" 720' 48' 19. Evaluate I \xytdocdydz for all positive values subject to the condition 20. Evaluate I A/ dxdydz for all positive values subject to the condition x + y + z < i. \Tt. 21. Find the volume of a cavity just large enough to permit of the complete revolution of a circular disk of radius a, whose centre describes a circle of the same radius, while the plane of the disk is constantly parallel to a fixed plane perpendicular to that of the circle in which its centre moves. ik s (3 7r + 8). 22. An inverted hollow circular cone of radius a and height h is filled with material of which the density varies as the depth below the plane of the base ; find the mass of the material, yu being the density at the vertex. -^-%n ^d'h. 23. A vessel consisting of a hemisphere of radius a and a cylinder having the same base and height h is filled with material having the same law of density as in Ex. 22 ; what is the value of h if half the mass is in the hemisphere ? ( + I ?2 GEOMETRICAL APPLICATIONS. [Art. 137. XL The Polar Element of Area. 137. When polar coordinates are used, if concentric circles be drawn, corresponding to values of the radius vector r hav- ing the common difference Ar, and then straight lines through the pole corresponding to values of the angular coordinate 6 having the uniform difference Ad, any given portion of the plane may be divided, as in Fig. 22, into small areas. The value of any one of these is readily seen to be Ar . rA6, FIG. 22. where r has a value midway between the greatest and least values of r in the area in question. It follows that the result of a double summation of this element between proper limits will give an approximate expression for the given area. Hence the double integral A = {(rdrdO, (i) which (compare Art. 122) is the limit of this expression when Ar and Ad are both diminished without limit, is the exact ex- pression for the given area. 138. The differential expression r dr dO, or polar element of area, is the product of the differentials dr and rdO, which cor- respond to the mutually rectangular dimensions Ar and rABoi the element of summation. These factors are the differentials of the motions of the point (r, 0), respectively, when r alone XL] THE POLAR ELEMENT OF AREA. varies and when alone varies. It is obvious that the ele- ment of area for any system of coordinates can in like manner be shown to be the product of the corresponding differentia] motions, provided only that these motions are at rigJit angles to one another* The element is in such a case said to be ultimately a rectangle. 139. The formula used in section VIII is in fact the result of performing the integration for r in formula (i) above. Thus when the pole is outside of the given area, as in Fig. 22, we have A = { ^rdrdV = $ f (r./ - r*)dO, the formula of Art. HO. The limits for are now the values which make r 2 = r 1 ; that is, in the case of a continuous curve, the values for which the radius vector is tangent to the curve. So also when, as in the example of Art. 109, the pole is on the curve (so that r l = o for all values of (f), the limits for correspond to tangents to the curve. But, when the pole is within the curve, we assume the lower limit zero to avoid nega- tive values of r, and then make 6 vary through a complete revo- lution, that is, from o to 27T. 140. We may, of course, in formula (i) integrate first for 6, the limits being functions of r. Thus A = This corresponds to summation of the elements along a cir- cular arc of radius r, so as to form the element (0 a O^rdr, in which the angular limits are two consecutive values of corre- * In other words, whenever the loci of constant values of one coordinate cut orthogonally the like loci for the other coordinate ; as in the case of the coordi- nates latitude and longitude on a spherical surface, or on a map made on any system in which the representatives of the parallels and meridians cut at right angles. GEOMETRICAL APPLICATIONS. [Art. 140. spending to the same value of r, and, such that the arc lies within the given area. If the pole is outside of the curve, the limits of the final integration will 'be the greatest and least values of r, for each of which # 2 0, will vanish ; but if the pole is within the curve, the arc will at the lower limit become a complete circumference, and the integral will represent the area included between the given curve and this circumference. Transformation of a Double Integral. |4f. We have seen in Art. 126 that a double integral of the general form (i) may, by taking x and y as rectangular coordinates, be repre- sented by the volume of a cylindrical solid whose base is the area in the plane of xy determined by the limits of integration, and whose upper surface is defined by the equation z = . But in the case of the solid of revolution the limits of the 0-in- tegration are O and 2?r, so that the result of this integration is 27i r 2 sin QdrdQ, the element of the double integral derived in Art. 146. 14-9. In the equations of transformation connecting these coordinates with the rectangular system, it is usual to take the axis of z as that about which the angle

\ hence the element of volume is their product, p 2 cos 6 dp d6 dcf>, in which the factor cos 6 occurs, in place of the factor sin 6 which appears in the element derived in the preceding article, because the axis of rotation is now perpendicular to the initial line. In the case of a sphere whose centre is at the pole, all the limits are constant, and we have = f>^r : cos f"--4^ Spherical Coordinates. ISO. If we give to p the constant value a, 6 and become the coordinates latitude and longitude of a point upon a spheri- cal surface, and a relation between 6 and becomes the equa- tion of a line drawn upon the spherical surface. For example, a formula of spherical right triangles gives for the equation of the great circle making the angle A with the equator at the zero of longitude sin 0= cot A tan (i) The product of the differentials of the motions of P corre- sponding to variations in 8 and 0, Fig. 24, that is, a 2 cos 6 dd d(f>, is the element of spherical surface. Hence, for example, to find 1 82 GEOMETRICAL APPLICATIONS. [Art. 150. the triangular area included between the equator = o, the great circle equation (i), and the meridian = a, we have = c? f a f cos Odedtf>= a 2 ["sin o* o o where 6 is the function of (f> defined by equation (i). From that equation sin sin sin " = 4/(cot 2 A -f- sin 2 0) ^/(cosec 2 A cos 2 0)" Therefore r a sin 0<^0 2-1 cos "T S = aM - - . -= arsm' 1 J ^(cosec* 1 ^4 cos* 5 0) cosec ^4 J = a\A sin~'(sin A cos a)].* Volumes in general. 151. We have seen in Art. 123 that the boundary of the area expressed by a double integral may consist in part of lines whose equations contain only one of the variables, namely, that for which the final integration takes place. But, as ex- plained in Art. 125, in the general case, it is necessary to regard the given area as made up of positive or negative parts of the kind just mentioned. This is done by drawing the loci of con- stant values of the final variable through the intersections of the lines forming the boundary, or else tangent to them as in Fig. 19. These parts are then expressed by separate integrals. * If B is the other oblique angle of the right triangle, another trigonometric formula gives cos B = sin A cos ex. ' hence S = a\A + B - \it\, from which it readily follows that the area of any spherical triangle is equal to * X the spherical excess in arcual measure. XL] VOLUMES IN GENERAL. 183 So also we have seen in Art. 126 that the volume expressed by a triple integral may be bounded in part by a surface whose equation contains only two of the variables, namely, those for which the last two integrations take place. But, in the general case, it is necessary to separate the volume into parts, by means of such surfaces passed through the lines of intersections ot the bounding surfaces, or edges of the given solid. 152. Figs. 19 and 2O illustrate this for rectangular coordi- nates, but similar considerations apply to any other system, and enable us to decide whether it is possible to express a given volume by a single integral. For example, let it be required to find the volume common to the solid of revolution produced by the half-cardioid OAB, Fig. 25, revolving about its axis OB, and the sphere whose centre is at the pole and whose radius is OC = c. The volume is the result of integrating the element 2zrr 2 sin 6 dr dB (found as in Art. 146) over the area OA C. For this area of integration, 6 = 7t and r = o give one constant limit for each variable, and the others are determined by the equation of the arc OA of the cardioid r = a(i cos 0), . . (i) and that of the circular arc AC, r = c. ... (2) Since equation (i) contains both variables, while equation (2) contains r only, we can, by integrating first for (and using equation (i) for one of its limits), express the required magni- tude by a single integral ; thus, fVsi sin BdBdr = 211 i*(i + cos B)dr. 1 84 GEOMETRICAL APPLICATIONS. [Art. 152. Substituting the value of cos from equation (i), V= 2 If in this example we integrate first for r, it becomes neces- sary to find a , the value of d for the point of intersection A, and, regarding the area of integration as separated into two parts by the radius vector OA, to form two integrals, in one of which the upper limit for r is taken from equation (i), and the limits for 6 are o* and O l while in the other r is taken from equation (2), and the limits for are O l and TT. 153. As a further illustration of these principles, let us re- sume the consideration of the volume evaluated in Art. 14/5 The volume is completely enclosed by the spherical and cylin- drical surfaces (i) and (2). In the evaluation, after integrating for z, which occurs only in equation (i), the whole volume was represented by a double integral; the limits of the r#-integra- tion were determined by equation (2) regarded as representing an area in the plane # = o.f In like manner, had we integrated first for 6 which occurs only in equation (2), the volume would have been expressed by a single double-integral expression. But suppose we wished to perform first the integration for r of the element r dr dO dz. The indefinite integral is ^r 2 , and the lower limit is zero; but the upper limit is, for some values of z and 0, given by equa *This limit also is in fact determined by the intersection of two sides of the area of integration, namely, that of the curve (i) and r = o the vanishing inner edge of the area. f This area happened to be entirely within the "projection" of the spher; (i) ; that is, the circle r = a, within which only z in equation (i) is real. Had this not been the case, the area of integration would have been only that com- mon to the curve (2) and the projection. XL] VOLUMES IN GENERAL, 1 8$ tion (l), and for others it is given by equation (2). In other words, the extremity of rmoves from the axis of z outward until it reaches either the spherical or the cylindrical surface. Thus the whole field of ^-integration, which extends from z = a to z = a, and from 6 = \n to 6 = ^TT, must be divided into parts corresponding to these values of the upper limit of r. Tak- ing, as we may by symmetry, one-fourth of this field of in- tegration for one-fourth of the volume V, the dividing line corresponds to the intersection of the sphere and cylinder; it is therefore the result of eliminating r between equations (i^ and (2), namely, z = a sin 6. Now, when z < a sin 6, the r of the cylinder is less than that for the sphere, and is therefore to be taken as the upper limit in J* 3 ! 5 but, when ^ > a sin 0, the upper limit must be taken from equation (i). Hence \V = F x + F 2 *, where and sin0 The result will be found to agree with that of Art. 144. Examples XI. i. Find, by integrating rdr d.6 first with respect to 0, the area included by the first whorl of the spiral of Archimedes and the initial line; also, in the same manner, that between the first and second whorls and the initial line. i 71 " 3 ^; 8?rV. * The surface separating F t and Vi is that of a right conoid generated by a line passing through and perpendicular to the axis of 2, and also passing through the intersection of the sphere and cylinder. The field of integration may be conceived of as an area lying upon the surface of the cylinder r = a, The separating surface traces upon the cylinder the line z = a sin 0. 1 86 GEOMETRICAL APPLICATIONS. [Ex. XI. 2. The paraboloid of revolution x* + / = cz is pierced by the right circular cylinder x* + y = ax, whose diameter is #, and whose surface contains the axis of the paraboloid; find the volume between the plane of xy and the sur- faces of the paraboloid and of the .cylinder. 32; 3. Find the volume cut from a sphere whose radius is a by the cylinder whose base is the curve 2 b, about the axis of y. 1 88 GEOMETRICAL APPLICATIONS. [Ex. XI. Transform to polar coordinates. nb(zb* + 30) Tta 4 _, b ~~6~ f 2 ^ - ") C 7i 14. Find the volume, generated by the curve given in the pre- ceding example when revolving about the axis of x. 7ta(2a t + ^} nb* a + <'a* - &') 15. Find the mass of a circular lamina of radius a if the density varies as the distance from a fixed point on the circumference and is /* at the centre. Interpret the integral also as a volume. 9 16. Find the mass of a square lamina of side 20 if the density varies as the distance from the centre and is f* at the middle of a side. 4^ r ^-[|/2 +lo g (|/2 + l)]. o 17. Find the volume between the surface generated by the revolution of the cardioid r = a(i cos B) about the initial line and the plane which touches this surface in circle. no 1 192 1 8. Transform the triple integral element dx dy dz into the polar element p" cos 6 dp d& d, every value of ip corresponds to a definite point of the curve ; and, as /> varies from o to 2?r, the point (x, y) de- scribes the whole curve in the direction ABCD. But it will be noticed that, dtf> remaining positive, the value of ds becomes zero and changes sign when ^ passes through either of the val- ues o, I-TT, TT, or f n. This corresponds geometrically to the fact that the point (x, y) stops and reverses the direction of its mo- tion, forming a stationary point or ctisp at each of the points A T B, C and D, as shown in Fig. 26. Such points are thus indi- cated by a change of sign in ds, and the arcs between them must be considered separately, because the corresponding def- inite integrals have different signs. Polar Coordinates. 157. It is proved in Diff. Calc. that in polar coordinates This is usually expressed in terms of 9. For example, in the case of the cardioid r = a(\ cos 6} = 2a sin 2 0, we have dr = 2a sin # cos %6 d9, whence, by substitution, ds = 2a sin dd. XII.] RECTIFICATION OF CURVES. 19 r The limits for the whole perimeter of the curve are o and 2n, and ds remains positive for the whole interval. Therefore f2ir 0~\ 21r s = 2a\ sin dO = Aa cos - = Sa. Jo 2 2J Rectification of Curves of Double Curvature. 158. Let a denote the length of the arc of a curve of double. curvature ; that is, one which does not lie in a plane, and sup- pose the curve to be referred to rectangular coordinates x, y and z. If at any point of the curve the differentials of the coordinates be drawn in the directions of their respective axes, a rectangular parallelepiped will be formed, whose sides are dx, dy and dz, and whose diagonal is da. Hence da = V(dx* + df + d). The curve is determined by means of two equations connect- ing x, y and z, one of which usually expresses the value of y in terms of x, and the other that of z in terms of x. We can then express da in terms of x and dx. If the given equations contain all the variables, equations of the required form may be obtained by elimination. 159. An equation containing the two variables x and y only is evidently the equation of the projection upon the plane of xy of a curve traced upon the surface determined by the other equation. Let s denote the length of this projection : then, since d& = dx* + dy*, in which d may, if convenient, be expressed in polar coordin- ates ; thus, 192 GEOMETRICAL APPLICATIONS. [Art. l6o. 160. As an illustration, let us use this formula to deter- mine the length of the loxodromic curve from the equation of the sphere, x 2 + = *; and, denoting the latitude of the projected point by , this gives z = a sin ^, r a cos . . . . (3) In order to express dB in terms of , we substitute the value of r in (2)j whence e 6 4. e -*6 2 sec ^ ...... (4) and by differentiation e" - e- = - sec ^ tan ^ -=? ..... (5) w cr Squaring and subtracting, which reduces to I XII.] LI'.NGTH OF THE LOXODROMIC CURVE. 1 93 From equations (3) and (6) dr* = c? sin 2 dz* a 2 cos 2 whence substituting in the value of da (p. 191) da = a V ( i +-s)dfa \ n 2 / Integrating, (3 = a d(b = a ( (3 a). n j a n where a and fi denote the latitudes of the extremities of the arc. Examples XII. i. Find the length of an arc measured from the vertex of the catenary and show that the area between the coordinate axes and any arc is proportional to the arc. X c ( 7 ' s - (e e 2 \ A cs. 2. Find the length of an arc measured from the vertex of the paraooia y* = 4ax. ., n V x + V(x + a) t/(ax + x) -I- aloe . Va > 194 GEOMETRICAL APPLICATIONS. [Ex. XII. 3. Find the length of the curve between the points whose abscissas are a and b. 4. Find the length, measured from the origin, of the curve a -x* y = a log 5 . a + x a log x. a x 5. Given the differential equation of the tractrix, dy__ _ y dx V(&* y*} ' and, assuming (o, a) to be a point of the curve, find the value of s as measured from this point, and also the value of x in terms of y ; that is, find the rectangular equation of the curve. y s = a log-. . a + v(a* y 2 ) ,/ * ^ x = a log 6. Find the length of one branch of the cycloid x = a (ip sin ^), y = a (i cos ^). Sa. j. When the cycloid is referred to its vertex, the equations being x a (i cos ^), y = a ($ + sin ^), prove that s = XII.] EXAMPLES. 195 8. Find the length from the point (a, o) of the curve x = 20, cos y> a cos 21/1, y = 20, sin t/> a sin 2ip. 8a (i cos ^). 9. Show that the curve, x 3 a cos ^ 20, cos* ^, y= 20. sin 3 ^ has cusps at the points given by ^ = o and ^ ?r ; and find the whole length of the curve. 120. 10. Find the length of the arc of the parabola (-)'+($'=' \0 / \ bl between the points where it touches the axes. a 3 + b s (fb* \_V( a * + &*) + a ~\ [V(^* + &*} + ^1 -t- ^ losr I i rt T o\* o ab ii. Show that the curve x = 20, cos 2 B (3 2 cos 4 #), y = 40 sin 6 cos 3 Q o has three cusps, and that the length of each branch is . 12. Find the length of the arc between the points at which the curve x = a cos 9 # cos 20, y = a sin 5 sin 26 2+^2 cuts the axes. - a. 196 GEOMETRICAL APPLICATIONS. [Ex. XII. 13. Show that the curve x = a cos ip (i + sin 3 rf>), y = a sin ^ cos 2 ip is symmetrical to the axes, and find the length of the arcs between the cusps. / i 2 sin- 1 a ( 4/2 + cos- 1 14. Find the length of one branch of the epicycloid a + b . x = (a + ft) cos ip b cos y = (a + b) sin ip b sin , ip. U (a + 6) a 15. Show that the curve x pa sin ?/' 4, y = 30 cos if? + 4.a cos 3 ?/? is symmetrical to the axes, and has double points and cusps : find the lengths of the arcs, O) between the double points, (/?) between a double point and a cusp, and (y) the arc connecting two cusps, and not passing through the double points. (a), a(7t + 31/3); 16. Find the whole length of the curve x 30 sn = a cos' . XII.] EXAMPLES. 197 17. Find the length, measured from the pole, of any arc of the equiangular spiral r = as" 9 , in which n = cot a. r sec a. 1 8. Prove by integration that the arc subtending the angle at the circumference in a circle whose radius is a, is 206. 19. Find the length, measured from the origin, of the curve defined by the equations 20. Find the length, measured from the origin, of the intersection of the surfaces y 4 sin x, z = 21? (2x + sin 2x). (472" + i)x + 20* sin 2X. 21. Find the length, measured from the origin, of the intersection of the cylindrical surfaces (y *Y = 4 and ^>, we have FIG. 28. dz and tan i/} = dz dy> XIII.] AREAS OF SURFACES IN GENERAL. 2OI in which and-r- are partial derivatives derived from the equa- dx ay tion of the surface. If the planes be intersected by a spherical surface whose centre is P, ADE is a spherical triangle right angled at A, whose sides are the complements of ^ and ip. Moreover, if a plane perpendicular to the tangent plane FED be passed through AP, the angle FPG will be y, and the perpendicular from the right angle to the base of the triangle the comple- ment of y. Denoting the angle EAF by 6, the formulae for solving spherical right triangles give tan if} . a tan $ cos 6 - , and sin = - tan y tan y Squaring and adding, tan 2 if} + tan 2 __ tan 2 y tan 2 if} + tan 2 ; whence sec 2 y I + -7- -7- dxj \dyj Substituting in the formula derived in Art. 164 , we have 166. It is sometimes more convenient to employ the polar 2O2 GEOMETRICAL APPLICATIONS. [Art. 1 66. element of the projected area. Thus the formula becomes 5 = sec yr dr dd, where sec y has the same meaning as before. For example, let it be required to find the area of the sur- face of a hemisphere intercepted by a right cylinder having a radius of the hemisphere for one of its diameters. From the equation of the sphere, s?=rf, ....... (i) we derive dz _ x dz _ y dx~ z* dy~ 2 ' whence /f ^ sec 7=4/1+ -r + L (dz\^~\ a (-r) = dxj \dy/ J z ( c [(rdrdB therefore o = a\\ - -, the integration extending over the area of the circle r = a cos d ....... , (2) Since equation (i) is equivalent to = a XIII.] AREAS OF SURFACES IN GENERAL. 203 From (2) the limits for r are r^ = 0, and r 2 = a cos #, hence in which a sin 6 is put for \h& positive quantity V(a* r.?}. The limits for 6 are \n and \n, but since sin 6 is in this case to be regarded as invariable in sign, we must write 71 5 = 2a* [ 2 (i - sin 0) dB = na z - 2a\ Jo If another cylinder be constructed, having the opposite radius of the hemisphere for diameter, the surface removed is 2/rtf 2 4 2 , and the surface which remains is 4*1*, a quantity commensurable with the square of the radius. This problem was proposed in 1692, in the form of an enigma, by Viviani, a Florentine mathematician. Examples XIII. i. Find the surface of the paraboloid whose altitude is a, and the radius of whose base is b. (* + *)*-*]. 2. Prove that the surface generated by the arc of the catenary given in Ex. XII., i, revolving about the axis of x t is equal to n(cx + sy). 3. Find the whole surface of the oblate spheroid produced by the 2O4 GEOMETRICAL APPLICATIONS. [Ex. XIII. revolution of an ellipse about its minor axis, a denoting the major, b the minor semi-axis, and e the excentricity, 27ta + n - log 6 e i e 4. Find the whole surface of the prolate spheroid produced by the revolution of the ellipse about its major axis, using the same notation as in Ex. 3. 2/T^ 2 + 27tab . e 5. Find the surface generated by the cycloid x = a (^ sin ^), y = a (i cos^) revolving about its base. na*. 3 6. Find the surface generated when the cycloid revolves about the tangent at its vertex. 7. Find the surface generated when the cycloid revolves about its axis. _ 4 3 8. Find the surface generated by the revolution of one branch of the tractrix (see Ex. XII., 5) about its asymptote. 2-n a*. X!!I] EXAMPLES. 205 9. Find the surface generated by the revolution about the axis of x of the portion of the curve which is on the left of the axis of y. 7t[\/2 + log (l + V'2)]. 10. Find the surface generated by the revolution about the axis of of the arc between the points for which x = a and x = b in the i _ i _ x hyperbola xy = tf Tlk r i>" L g + $TT, in which $ denotes the inclination of the tangent line. Since the centre of rotation is one extremity of the generating linep, the differential of this area is found by substituting in the general expression p 1 = o and /? 2 =p. Hence when p is expressed in terms of , A = l - 2 expresses the area between an arc of a given curve, its evolute, and the radii of curvature of its extremities, the limits being the values of 3 3 To obtain the whole area between the curve and its evolute, the limits for 8 are o and 2n ; hence the limits for are o and 3?r. Therefore A ! f 317 ^,^ s^r 3 * 2 A = -, f? d

! PR = 2a sin . Substituting in the formula of Art. 168, we have for the area of the cycloid, since PRO varies from o to ?r, A = 6a 2 sin 2 $ d$ = 37^. Jo XIV.] SIGN OF THE GENERATED AREA. 209 Sign of the Generated Area. 173. Let AB be the generating line, and 7 the centre of rotation. The expression, dA = (i) FIG. 29. for the differential of the area, was obtained upon the supposi- tion that A and B were on the same side of C. Then suppos- ing Pz > Pi, and that the line rotates in the positive direction as in figure 29, the differential of the area is positive; and we notice that every point in the area generated is swept over by the line AB y the left hand side as we face in the direction A B preceding. 1 74-. We shall now show that in every case, the formula requires that an area swept over with the left side preceding, shall be considered as positively, generated, and one swept over in the opposite direction as negatively generated. In the first place, if C is between A and B so that P! is negative, as in figure 30, p\ is still positive, and formula (i) still gives the difference between the areas generated by CB and AC. Hence the latter area, which is now generated by a part of the line AB, must be regarded as generated negatively, but the right hand side as we face in the direction AB of this portion of the line is now preceding, which agrees with the rule given in Art. 173. Again, if C is beyond B, the formula gives the difference of the generated areas ; but since p? is numerically greater than p, in this case, dA is negative, and the area generated by AB is the difference of the areas, and is negative by the rule. f IG. 30. 210 GEOMETRICAL APPLICATIONS. [Art. 174- Finally, if the direction of rotation be reversed, d and therefore dA change sign, but the opposite side of each por- tion of the line becomes in this case the preceding side. 175. We may now put the expression for the area in another form. For whatever be the signs of p 2 and p lt the first factor is the length of AB, which we shall denote by /, and the second factor is the distance of the middle point of AB from the centre of rotation, which we shall denote by p m . Hence, putting Pi = /, and + we have = I lp m d. (2) Since p m d$ is the differential of the motion of the middle point in a direction perpendicular to AB, this expression shows that the differential of the area is the product of this differential by the length of the generating line. Areas generated by Lines whose Extremities describe Closed Circuits. 176. Let us now suppose the generating line AB to move from a given position, and to return to the same position, each of the extremities A and B describing a closed curve in the positive direction, as indicated by the arrows in figure 31. It is readily seen that every point which is in the area described by B, and not in that described by A, will be swept over at least once by the line AB, the left side preceding, FIG. 31. and if passed over more than once, there will be XIV.] AREAS GENERATED BY MOVING LINES. 211 an excess of one passage, the left side preceding. Therefore the area within the curve described by B, and not within that described by A, will be generated positively. In like manner the area within the curve described by A, and not within that described by B, will be generated negatively. Furthermore, all points within both or neither of these curves are passed over, if at all, an equal number of times in each direction, so that the area common to the two curves and exterior to both disap- pears from the expression for the area generated by AB. Hence it follows that, regarding a closed area whose perimeter is described in the positive direction as positive, the area generated by a line returning to its original position is the difference of the areas described by its extremities. This theorem is evidently true generally, if areas described in the opposite direction are regarded as negative. Amslers Planimeter. 177. The theorem established in the preceding article may be used to demonstrate the correctness of the method by which an area is measured by means of the Polar Planimeter, invented by Professor Amsler, of Schaffhausen. This instrument consists of two bars, OA and AB, Fig. 32, jointed together at A. The rod OA turns on a fixed pivot at O, while a tracer at B is carried s """""^ v in the positive direction completely around I \ the perimeter of the area to be measured. At ^ some point C of the bar AB a small wheel is c fixed, having its axis parallel to AB, and its circumference resting upon the paper. When B is moved, this wheel has a sliding and a roll- ing motion ; the latter motion is recorded by an attachment by means of which the number FIG. 32. of turns and parts of a turn of the wheel are registered. 212 GEOMETRICAL APPLICATIONS. [Art. 1/8. 178. Let M be the middle point of AB, and let OA=a, AB = b, MC ' = c. Since b is constant, the area described by AB is by equation (2), Art. 175, Krez.AB=b \p m d : substituting in (i) the value of p m d<}>, ) ...... (2) 179. Two cases arise in the use of the instrument. When, as represented in Fig. 32, O is outside the area to be meas- ured, the point A describes no area, and by the theorem of Art. 176, equation (2) represents simply the area described by B. In this case returns to its original value, hence d$ vanishes, and denoting the area to be measured by A, equation (2) becomes A=bs ...... : . . (3) In the second case, when O is within the curve traced by B, the point A describes a circle whose area is no 1 , and the limit- XIV.] AMSLER'S PLANIMETER. 2i 3 ing values of differ by a complete revolution. Hence in this case equation (2) becomes A 7ta z bs 2nbc, or A = bs + 7t (a 2 - 2bc}* ..... (4) In another form of the planimeter the point A moves in a straight line, and the same demonstration shows that the area is always equal to bs. Examples XIV. i. The involute of a circle whose radius is a is drawn, and a tangent is drawn at the opposite end of the diameter which passes through the cusp ; find the area between the tangent and the involute. a*7t (3 + 7T>) 2. Two radii vectores of a closed oval are drawn from a fixed point within, one of which is parallel to the tangent at the extremity of the other ; if the parallelogram be completed, the area of the locus of its vertex is double the area of the given oval. 3. Show that the area of the locus of the middle point of the chord joining the extremities of the radii vectores in Ex. 2, is one half the area of the given oval. *The planimeter is usually so constructed that the positive direction of rotation is with the hands of a watch. The bar b is adjustable, but the distance A C is fixed so that c varies with b. Denoting A C by q, we have c = q \b, and the constant to be added becomes C= it (a 3 zbq + 3 2 ) in which a and q are fixed and b adjusta- ble. In some instruments q is negative. It is to be noticed that in the second case s may be negative ; the area is then the numerical difference between the constant and bs. 214 GEOMETRICAL APPLICATIONS. [Ex. XIV. 4. Prove that the difference of the perimeters of two parallel ovals, whose distance is b, is 27tb, and that the difference of their areas is the product of b and the half sum of their perimeters. 5. From a fixed point on the circumference of a circle whose radius is a a radius vector is drawn, and a distance b is measured from the circumference upon the radius vector produced ; the extremity of b therefore describes a limacon : show that the area generated by b when b > 20, is the area of the limacon diminished by twice the area of the circle, and thence determine the area of the limacon. 71(20" + '). 6. Verify equation (4), Art. 179, when the tracer describes the circle whose radius is a + b. 7. Verify the value of the constant in equation (4), Art. 179, by determining the circle which may be described by the tracer without motion of the wheel. 8. If, in the motion of a crank and connecting rod (the line of motion of the piston passing through the centre of the crank), Amsler's record- ing wheel be attached to the connecting rod at the piston end, deter- mine s geometrically, and verify by means of the area described by the other end of the rod. 9. The length of the crank in Ex. 8 being a, and that of the con- necting rod b, find the area of the locus of a point on the connecting rod at a distance c from the piston end. 10. If a line AB of fixed length move in a plane, returning to its original position without making a complete revolution, denoting the areas of the curves described by its extremities by (A) and (B}, determine the area of the curve described by a point cutting AB in the ratio m : n. n(A) + m(JB) m + XIV.] EXAMPLES. 215 ii. If the line in Ex. 10 return to its original position after making a complete revolution, prove Holditch's Theorem ; namely, that tne area of the curve described by a point at the distance c and c from A and B is c'(A] , 12. Show by means of Ex. n that, if a chord of fixed length move around an oval, and a curve be described by a point at the distances c and c from its ends, the area between the curves will be ncc '. XV. Approximate Expressions for Areas and Volumes. 180. When the equation of a curve is unknown, the area between the curve, the axis of x, and two ordinates may be approximately ex- pressed in terms of the base and a lim- ited number of ordinates, which are sup- posed to have been measured. Let ABCDE be the area to be de- termined ; denote the length of the base by 2h ; and let the ordinates at the ex- tremities and middle point of the base be measured and denoted by y^y* andjy s . Taking the base for the axis of x, and the middle point as origin, let it be assumed that the curve has an equation of the form (i) then the area required is f* BX* Cx* Dx^~\ h A=\ ydx-Ax-\- H +- = J-A 2 3 4 J_ A in which which A and C are unknown. ,. , .(2) 2l6 GEOMETRICAL APPLICATIONS [Art. 1 80. In order to express the area in terms of the measured ordinates, we have from equation (i), jt = A + Bh + Ch* + Dh\ whence we derive y\ 472 and substituting in (2), It will be noticed that this formula gives a perfectly ac curate result when the curve is really a parabolic curve of the third or a lower degree. 181. If the base be divided into three equal intervals, each denoted by h, and the ordinates at the extremities and at the points of division measured, we have, by assuming the same equation, (i) From the equation of the curve, .x > ? 2 /* "4 8 ^7^2 2)ffi >H~ - >2 ~ 2 - ^ ^ T" ^-' a 2 OT .74 ^ 1- * FIG. -u.. 2 + 4 + "8 J XV.] SIMPSON'S RULES. 2 1/ whence }\ + y 2A + - , Ch * 2 From these equations we obtain A 9/3-74 6 and =*- 4 Substituting in equation (i), Simpsons Rules. 182. The formulae derived in Articles iSoand 181, although they were first given by Cotes and Newton, are usually known as Simpson 's Rules, the following extensions of the formulae having been published in 1743, in his Mathematical Disserta- tions. If the whole base be divided into an even number n of parts, each equal to //, and the ordinates at the points of divis- ion be numbered in order from end to end, then by applying the first formula to the areas between the alternate ordinates, we have That is to say, the area is equal to the product of the sum of the extreme ordinates, four times the sum of the even-num- 2 1 8 GEOMETRICAL APPLICATIONS. [Art. 182 bered ordinates, and twice the sum of the remaining odd-num- bered ordinates, multiplied by one third of the common interval. Again, if the base be divided into a number of parts divis- ible by three, we have, by applying the formula derived in Art. 181 to the areas between the ordinates y y^y^y^ and so on, \h ^ = ( Ji + 3J 2 + 373 + 2j 4 + 3j 5 + $y n Cotes Method of Approximation. 183. The method employed in Articles 180 and 181 is known as Cotes Method. It consists in assuming the given curve to be a parabolic curve of the highest order which can be made to pass through the extremities of a series of equi- distant measured ordinates. The equation of the parabolic curve of the th order con- tains n + i unknown constants; hence, in order to eliminate these constants from the expression for an area defined by the curve, it is in general necessary to have n + I equations con- necting them with the measured ordinates. Hence, if n de- note the number of intervals between measured ordinates over which the curve extends, the curve will in general be of the degree.* * If H denotes the whole base, the first factor is always equivalent to H divided by the sum of the coefficients of the ordinates ; for if all the ordinates are made equal, the expression must reduce to Hy. Thus, each of the rules for an approximate area, including those derived by repeated applications, as in Art. 182, may be regarded as giving an expression for the mean ordinate. The coefficients of the ordinates, according to Cotes' method, for all values of n up to n = IO, may be found in Bertrand's Cakul Integral, pages 333 and 334. For example (using detached coefficients for brevity), we have, when n 4, H r A =\.l' 32, I2 > 32, 7J; and when n = 6, TT A g [41, 216, 27, 272, 27, 216, 41], XV.] THE FIVE-EIGHT RULE. 219 I84-. For example, let it be required to determine the area between the ordinates y^ and y z , in terms of the three equi- distant ordinates y^ y and y. 6t the common interval being k. We must assume y - A + Bx + then taking the origin at the foot of y l} ( h , T / Bh A =\y dx h\ A + + from which A, B and C must be eliminated by means of the equations y z = A+Bh + Ch\ y* A - Solving these equations, we obtain A = n, If we make a slight modification in the ratios of these last coefficients by sub- stituting for each the nearest multiple of 42, we have A = - [42, 210, 42, 252, 42, 210, 42], 840 (the denominator remaining unchanged, since the sum of the coefficients is still 840), which reduces to ^ = [1,5, 1,6, 1,5, ij- This result is known as Weddles Rule for six intervals. The value thus given to the mean ordinate is evidently a very close approximation to that resulting from Cotes' method, the difference being 220 GEOMETRICAL APPLICATIONS. [Art. 184. and substituting 1 185. It is, however, to be noticed, that when the ordinates are symmetrically situated with respect to the area, if n is even, the parabolic curve may be assumed of the (n + i)tli degree. For example, in Art. 180, n = 2, but the curve was assumed of the third degree. Inasmuch as A, B, C and D cannot all be expressed in terms of }\, y.^ and y z , we see that a variety of parabolic curves of the third degree can be passed through the extremities of the measured ordinates, but all of these curves have the same area.* Application to Solids. 186. If y denotes the area of the section of a solid perpen- dicular to the axis of x, the volume of the solid is y dx, and * This circumstance indicates a probable advantage in making n an even num- ber when repeated applications of the rules are made. Thus, in the case of six intervals, we can make three applications of Simpson's first rule, giving TT A ~^ t 1 - 4, 2, 4, 2, 4, i] ........ (i> or two of Simpson's second rule, giving A - ^ [i, 3, 3, 2, 3, 3, i] ........ (2> In the first case, we assume the curve to consist of three arcs of the third degree, meeting at the extremities of the ordinates y 3 and/ 5 ; but, since each of these arcs contains an undetermined constant, we can assume them to have common tangents at the points of meeting. We have therefore a smooth, though not a continuous curve. In the second case, we have two arcs of the third degree containing no arbitrary constants, and therefore making an angle at the extremity of _j/ 4 . It is probable, therefore, that the smooth curve of the first case will in most cases form a better approximation than the broken curve of the second case. In confirmation of this conclusion, it will be noticed that the ratios of the coefficients in equation (i) are nearer to those of Cotes' coefficients for n = 6, given in the preceding foot-note, than are those in equation (2). XV.] APPLICATION TO SOLIDS. 221 therefore the approximate rules deduced in the preceding arti- cles apply to solids as well as to areas. Indeed, they may be applied to the approximate computation of any integral, by putting y equal to the coefficient of dx under the integral sign. The areas of the sections may of course be computed by the approximate rules. Woollens Rule. 187. When the base of the solid is rectangular, and the ordinates of the sections necessary to the application of Simp- son's first rule are measured, we may, instead of applying that rule, introduce the ordinates directly into the expression for the area in the following manner. Taking the plane of the base for the plane of xy, and its centre for the origin, let the equation of the upper surface be assumed of the form Let 2,h and 2k be the dimensions of the base, and denote the measured values of z as indicated in Fig. 35 The required volume is |Z th tk V-\ zdydx. } -h ) -k ' This double integral vanishes for every term containing an odd power of x or an odd power of_^: hence 222 GEOMETRICAL APPLICATIONS. [Art. iS/. By substituting the values of x and y in the equation of the surface, we readily obtain (2) , ... (3) . ... (4) From these equations two very simple expressions for the volume may be derived ; for, employing (2) and (4), equation (i) becomes ihk r==j?(a> + &i + 2h + & 9 + cd; . . . . (5) and employing (2) and (3), hk V- fa + (h + 8^ 2 + ^ + c 3 ) ..... (6) o Equation (5) is known as Woolleys Rule; the ordinates employed are those at the middles of the sides and at the centre ; in (6), they are at the corners and at the centre. Examples XV. 1. Apply Simpson's Rule to the sphere, the hemisphere, and the cone, and explain why the results are perfectly accurate. 2. Apply Simpson's Second Rule to the larger segment of a sphere made by a plane bisecting at right angles a radius of the sphere. XV.] EXAMPLES. 223 3. Find by Simpson's Rule the volume of a segment of a sphere, b and c being the radii of the bases, and h the altitude. f W + V> + V). 4. Find by Simpson's Rule the volume of the frustum of a cone, b and c being the radii of the bases, and h the altitude. 5. Compute by Simpson's First and Second Rules the value of dx , the common interval being T ^ in each case. J o I + X The first rule gives 0.69314866; the second rule gives 0.69315046. The correct value is obviously log e 2 = 0.69314718.* 6. Find the volume considered in Art. 187, directly by Simpson's Rule, and show that the result is consistent with equations (5) and (6). 7. Find, by elimination, from equations (5) and (6), Art. 187, a formula which can be used when the centre ordinate is unknown. V = [ 4 (a f +, + ,+ t ) - ( a , + a, + c } + Ol- O * The error in the eighth place of decimals is therefore 148 by the First Rule, and 328 by the Second Rule, the First Rule giving the better result as anticipated in the foot-note of p. 220. Using the same ordinate?, Weddles' Rule (see foot-note, p. 219) gives the extremely accurate result 0.69314722, the error being only 4 in the eighth place. 224 MEAN VALUES AND PROBABILITIES. [Art. 1 88. CHAPTER IV. MEAN VALUES AND PROBABILITIES. XVI. The Average or Arithmetical Mean. 188. A MEAN of several values of a quantity is an interme- diate value such that, when it is substituted for each of the given values in performing a certain operation, the result is unchanged. For example, if the result of the operation is the product of the given values, the mean value found is that known as the geometrical mean. But the usual and most simple mean value is that obtained when the operation is that of summation. This is known as the average or arithmetical mean* Thus, if M denotes the average of n values of y de- noted by y^ , y z , . . . y n , we have (l) Here the aggregate represented by either member of equa- tion (i) is the same whether the n given values of y be taken or the mean value be taken n times. * A mean value of any other kind can usually be defined by the aid of the arithmetical mean. Thus, the geometrical mean is the quantity whose logarithm is the mean of those of the given quantities ; the harmonic mean is that whose reciprocal is the mean of the reciprocals. The mean error in the Theory of Least Squares is the error whose square is the mean of the squares of the errors. XVI.] THE MEAN OF A CONTINUOUS VARIABLE. 22$ 189. When a number p l of the quantities has a common value jj/j, a number p z has a common value y z , and so on, the total number n of formula (V is equal to p l -\- p 2 -f- . . . or 2p, and the formula becomes ^p.M= /^i + A^ 2 +..== ^/J- . (2) The numbers A A> are ca ^ e< ^ the weights of jj/j, jj/ 2 , . . ., and the mean is called the weighted arithmetical mean of the several values. The mean value of a quantity which admits of a continuous series of values (which is the subject of the present Chapter) is a modification of the M of equations (i) and (2), in deriving which, integration takes the place of summation. The Mean of a Continuous Variable. 190. Consider all the values of a variable which varies con- tinuously between certain extreme values, and suppose a large number n of these values to be chosen and their mean taken. Then, supposing the manner of selection to be such that we can pass to the limit when n is indefinitely increased, we shall have a mean value which depends upon all the values of the variable in question, and is, therefore, properly called a mean value of the variable. But the value of this mean obviously depends also upon the method in which the n values were selected. For example, in finding the mean velocity of a point describing a straight line with variable velocity, we shall arrive at a certain result if we take the n values of the velocity at equal intervals of time ; but the result will be different if we select the velocities with which the point passes equally dis- tant points of its path. 191. When the variable in question is a function of some single independent variable, the mean obtained by taking equi- 226 MEAN VALUES AND PROBABILITIES. [Art. 191. distant values of the independent variable is called tlie mean value of the function for the range of values given to the inde- pendent variable. Let y = f(x] be the function, represented, as in Fig. 8, p. 1 2 1 , by the curve CD ; then the figure illustrates the mode in which n values of the function or ordinate are taken in finding the mean value of the function for all values of x between OA = a and OB = b. These values are the y\ }>i > i * y* f Art. 99 erected at the common interval Ax, where n Ax = b a. Now, multiplying equation (i), Art. 188, by Ax, we have n Ax-M = y l Ax -f- y 2 Ax -f . . . ~\-y n Ax, or (b _ a)M = ^yAx. Here M is the mean of the n actual values of y, and the mean value required is found by passing to the limit when n is indefinitely increased. Ji y dx, a which is the area CABD in Fig. 8. Hence the mean value of ./(*) for all values of x between a and b is given by the equation (3) 192. The expression mean ordinate of any portion of a curve is always, unless otherwise stated, held to signify the mean ordinate regarded as a function of the abscissa. Hence it is the height of the rectangle of which the base is the projection of the curve on the axis of x, and the area is equal to that included between the curve, the base and the extreme ordinates. Thus, "the mean ordinate" of a semicircle z 2 , by the \ R O FIG. 36. whose radius is a is found by dividing the area, XVI.] MEAN OF EQUALLY PROBABLE VALUES. 22/ base 2a\ whence M = \na. This is the average value of per- pendiculars erected at equal distances along the diameter. See Fig. 36. 193. On the other hand, if the average value of perpendic- ulars let fall from equidistant points on the arc is required (see Fig. 37), it is necessary to express the perpendicular as a function of the arc or angle sub- / tended at the centre. Denoting this (as measured from one extreme radius) by FlG - 37 6, the perpendicular is a sin 0, and the value of this mean is, by equation (3), e* a\ sin 6 d9 M= 71 The Mean of Equally Probable Values. 1 94-. The expression mean value of a variable quantity selected under given circumstances is often used to designate the mean of all the values which are equally probable under the circumstances. A point is said to be taken at random upon a line of given length when it is equally likely to fall upon any one of any equal segments of the line. Hence the first of the mean values of PR found in the preceding articles may be called the mean value when R is taken at random upon the diameter, and the second is the mean value when P is taken at random upon the semicircumference. So also M in equation (i), Art. 188, is the mean value of y whenj^, y z , . . . y n are equally probable, and are the only possible values of y. In this case, the finite number n is the total number of cases which are possible and equally probable. It will be convenient also in equation (3), Art. 191, to speak 228 MEAN VALUES AND PROBABILITIES. [Art. 194. of b a (which takes the place of n when it is increased in- definitely in passing to the limit) as the total number of cases, and of the definite integral in the second member as the aggre- gate of them's in the total number of cases. The Mean of a Function of Two^ Variables. 195. When the quantity whose mean is required depends "upon two variables admitting of continuous values under cer- tain restrictions, we may represent it by the ^-coordinate of a surface of which the independent variables x and y are the other two rectangular coordinates. The restrictions imposed upon the values of x and y now limit the foot of the ^-coordinate to a certain area in the plane of xy. If these restrictions consist of fixed limits between which x must lie, together with other fixed limits between which y must lie, this area is a rectangle. It may, however, be an area of any other shape; the limiting values of y will then be different for different values of x. In choosing the points for which the values of z are taken in form- ing the mean, the values of y, for a given value of x, are sup- posed to be taken at equal intervals Ay between the limits .corresponding to that particular value of x. In like manner, the values of x chosen are taken at equal intervals Ax, so that the points (x, y] are uniformly distributed over the area. Thus, if the area contains a large number, n, of elementary rectangles of dimensions Ay and Ax, then one value of z is taken corresponding to each element, Ay Ax, of area. 196. Now, putting z for y in equation (i), Art. 188, and multiplying each member by Ay Ax, we have n Ay Ax-M = "2 zAyAx (l) On passing to the limit when Ax and Ay are indefinitely XVI.] MEAN OF A FUNCTION OF TWO VARIABLES. 229 decreased, M becomes the mean value required. The limiting- value of nAyAx is the area mentioned above; and its value is = \\dydx, where the limits of integration are given in the form of restric- tions upon the values of x and y. The limiting value of the second member of equation (i) is the double integral \zdydx, taken with the same limits; or, as expressed in Art. 126, in- tegrated over the area A; and its value, as shown in that article, is the volume of the cylindrical solid whose upper sur- face is in the surface 2 =p 2 ) is x z -f- k y and so on. Hence the mean distance is 2(x -\- nty = ~2x -j- h = ~x -f- h, by equation (i). But this is the distance of the point (x, y) from the line in question. Next let the line be oblique to the axes. It is shown in Analytical Geometry that the perpendicular upon a line from (x, y) is of the form where A, B, and C are certain constants; in other words, p is a linear function of x andj. Hence we have, for the dis- tances of (x l ,y^), (x z >y^, e ^ c - fro m the line A = Ax, + By, + C, /, = Ax, + By, + C, and adding, 2 = A ^x -- B. 2 nC. Dividing by n, the mean distance is, by equations (i) and (2), but this is the distance of the point (x, ~y) from the given oblique line. 234 MEAN VALUES AND PROBABILITIES. [Art. 2O2. The point (J, ~y) whose position thus determines the mean distance of the given points from any straight line in the plane is called their Centre of Position. 203. In like manner, when n points P l , P 2 , . . . P n in space are referred to rectangular planes, their mean distances from these planes are and it can be shown that the mean distance of the points from any plane is the distance of the point (5F, y, ~z) from that plane. The point which thus determines the mean distance from all planes is called the Centre of Position of the points. It is also called, from its occurrence in Statics, the Centre of Gravity of n equal particles situated at the points P lt P 2 , . . . P n . 204-. If a number p l of the equal particles coincide at the point P l , p 2 of them at P 2 , and so on, the centre of gravity of the particles is at a distance from any given plane which is the weighted mean (Art. 1 89) of the distances of the point P l , P 2 , etc. from the plane. Thus, referring to rectangular coordinate planes, its distance x from that of yz is given by the equation Multiplying both sides by the mass of one of the equal particles, the equation becomes = m l x l -j- m ??2 + = ^>mx, (0 where Wj , m 2 , etc. are the masses of unequal particles situated at Pj, P v etc., and 2m is the total mass. The point (!r, ~y, ~z) whose three coordinates are similarly defined is the centre of gravity of unequal particles. XVI.] CENTRE OF GRAVITY OF A CONTINUOUS BODY. 235 The second member of equation, (i), that is, the aggregate of the distances multiplied by the masses, is called in Mechanics the Statical Moment of the total mass 2m with respect to the plane of yz. The Centre of Gravity of a Continuous Body. 205. The property of the centre of gravity given in the preceding articles obviously extends to continuous bodies. That is to say, the position of this point, which is sometimes called the Centroid, is determined by the mean distances of the particles from three given planes; and, when given, it determines the mean distance from any plane. If the body is homogeneous t that is to say, if the same quantity of matter is contained in equal elements of volume, the mean distances are the ordinary arithmetical means. On the other hand, if the body. is not homogeneous, its density at a given point constitutes the weight to be attributed to the element at that point, supposing, as in Art. 135, the geometric elements to be all equal. In any case, the variable density is simply used as a factor of the ele- ment both in finding the total mass 'as in Art. 135) and in finding the statical moment. 206. The centre of gravity of a homogeneous solid is also called the centre of gravity of the volume, and the integral of xdV (the factor of density being omitted) is called the statical moment of the volume. So also the centre of gravity of a thin homogeneous plate of uniform thickness is called the centre of gravity of the area, and the integral of xdA is called the stati- cal moment of the area. Thus, for the circle x* -j- y* = a 2 , the statical moment with respect to the axis of y of the semicircle on its right is [xdA=2 r*yd 2 236 MEAN VALUES AND PROBABILITIES. [_ Art - Dividing by the area %7ra z , we have, for the abscissa of the centre of gravity of the semicircle, x = 207. One or more of the coordinates of the centroid may be obvious from considerations of symmetry. For example, the centroid of a plane area or of a plane curve is in the plane ; that of a straight line is at its middle point ; that of a sphere or of an ellipsoid is at its centre ; that of a hemisphere is on its central radius ; that of a right cone is on its axis. Again, separation of the body into elements whose centroids are known may lead to similar results. For example, an oblique cone can be separated into circular elements of uniform thickness whose centroids are therefore at their centres. These points are all situated on the geometrical axis, therefore the centroid of the cone is also on that axis. In like manner, the centre of gravity of any triangle is upon a medial line. It follows that it is at the intersection of the three medial lines. It is obvious also that, if three equal particles be situated at the vertices of any triangle, their centre of gravity will be at the same point. Hence the distance of the centre of gravity of a triangle from any straight line in the plane is the arith- metical mean of the distances of the vertices. 208. When integration is required, the element of moment employed may be the moment of any convenient element of the mass or volume. For example, in the case of a hemi- sphere of radius a, it is necessary to use integration in finding the distance from the base. We may take for element of vol- ume the hemispherical shell of radius r and thickness dr, of which we have already found in Art. 198 the mean distance from the base to be one-half of the radius. Therefore, since XVI.] SQUARED DISTANCES FROM A PLANE. 237 the volume of the shell is mr* dr, its moment is Integrating, we have for the hemisphere moment r a Tta* = n r s dr = . Jo 4 Dividing by the volume, which is -f TTtf 3 , we find the height of the centre of gravity above the base to be f. This is there- fore the average distance from the base of all the points within the hemisphere. The moment might be found as the result of a simple in- tegration, in this case, also by using the circular element of volume parallel to the base. The method employed above is particularly adapted to a sphere of varying density if the density is a function of the distance from the centre only. Average Squared Distance of Points from a Plane. 209. The mean square of the distances of points in a plane area from a straight line in the area, and that of the distances of points in a volume from a fixed plane, or from a fixed straight line, have important application in Mechanics. In the case of the area A, if we denote the mean squared distance from the axis of y by .r 2 , it is found from the formula = {(x*dydx, where the integration extends over the area A . The second member of this equation is known as the moment of inertia of the area with respect to the axis of y, and is usually denoted by /. TKe distance k, such that k 2 x~, is called the radius of gyration of the area with respect to the same axis ; thus the equation becomes Jf-A = I. 210. As in finding the statical moment, so in finding the moment of inertia, it may be convenient to use some other 238 MEAN VALUES AND PROBABILITIES. [Art. 2IO. element of area in place of dy dx. For example, in finding the moment of inertia, with respect to the axis of y, of the loop of the lemniscate r 2 = < cos 2&, of which the area is found in Art. 109, we use the polar element of area rdr dQ. The dis- tance of the element from the axis is x.= r cos 6. Hence / = 2 [ 4 [V 3 cos 2 BdrdB = - [ 4 cos2 20 cos 2 ), x = a($> sin ip), find the distance of the centre of gravity of the area from the base. 20. Determine the distance from the base of the centre of grav- ity of a hemisphere when the density varies inversely as the distance from the centre. %a. 21. Find the mean squared distance from the base for the par- ticles of a homogeneous hemisphere. oP = ^a 2 . 22. Determine the radius of gyration of a sphere for a diameter by means of the result of Ex. 21, and also directly by the method of Art. 212. k* = \ a \ 23. Determine the radius of gyration of a cylinder with respect to a diameter of the base. J? = \a^ + i// 2 - 24. Find the radius of gyration of the area between a parabola and a double ordinate 2y perpendicular to its axis, with respect to a perpendicular to its plane passing through its vertex. tf = J X * + ^y\ 25. Determine the radius of gyration of a paraboloid about its axis, b denoting the radius of the base. & = $IP. 26. The cardioid r = a (i cos ff) revolves about the initial line; determine the radius of gyration of the solid formed with respect to this line. tf = |f a 2 . XVII.] MEAN DISTANCES FROM A FIXED POINT. 243 XVII. Mean Distances from a Fixed Point. 2I4-. We have seen that the mean value of a variable de- pends not only upon the restrictions imposed in the question upon the values of the variable, but upon the distribution of the admissible values. This distribution is analytically ex- pressed in the selection of the element employed, and the restrictions are expressed by the limits of integration. For example, in finding the mean distance, from a fixed point 0, of the points on a straight line AB of limited length, the variable is OP, Fig. 38, and the distribu- tion of the values (expressed in words by the statement that P falls at random be- tween the limits assigned), is provided for by selecting the element dy, where y is the distance of P from a fixed point of the line. Taking for this point A the foot of the perpendicular OA = a from the point O, the expression for the variable in question is OP = y(d* -f- jj/ 2 ). Thus, if A and B are given as the limiting positions of P, the "whole number of cases " is represented by AB = b, which is the integral of the element dy between the limits o and b. The mean distance is then given by the equation bM = t/0 a +/) dy = -\b ^(&+P}+a* log Jo ^ L_ see formula (Z), p. 125. 215. As a particular case of this result, putting b = a, we have for the mean distance from one corner of a square of a point taken at random upon either of the opposite sides 244 MEAN VALUES AND PROBABILITIES. [Art. 21$. M $a[ |/2 -f- log (I -f- |/2)] = I . 1480, where a is a side of the square. It readily follows that this is the mean distance from the centre of a point on the perimeter of a square of side 2, and therefore 0.574^ is the same mean when the side is a. Again, putting b = a 1/3, so that the angle AOB is 60, we find, for the mean distance from the centre of a point on the perimeter of an equilateral triangle circumscribing the circle whose radius is a, Jf =.[i -fit's toff (V3+*)]i or, if s is a side of the triangle, M= T y[2/3 + log ( 4/3 + 2)] = 0.398*. 216. If/' in Fig. 38 is taken at random within the rect- angle with vertex at O and sides a and b, the mean distance is the value of M in the equation abM ' =. d(x* ~\~ y*}dy dx, . . . . (i) J o J o in which the " number of cases " is represented by the area of integration ab. Here, the result of the first integration would be which gives a troublesome form to the second integration. But, if we replace the upper limit b by mx, where m is con- stant, this becomes -f w>)-{-log[> +|/(i+ m*}-]\** = Kx\ . (2) XVII.] MEAN DISTANCES FROM A FIXED POINT. 245 and the second integration becomes very simple. In doing this, we integrate over a triangle instead of a rectangle. Thus if, in Fig. 38, b = ma, the mean distance from of a point within the triangle OAB is given by -mtPM. 2 f tt r . = J o J o f = J Kx*dx = Ka z (3) 2Ka whence M. = - , where K is the constant defined in equa- Z m tion (2). It is obvious that the value of the integral in equation (i) can thus be obtained by dividing the field of integration into two parts. Again, because the two parts of the rectangle ab are equal, if M 2 is the value of the mean in the second part, we shall have M= \(M^ -f J/ 2 ). 217. The element in the double integral of the area, which here represents the number of cases, is dy dx, which we may denote by d~*A. In the single integral, the element is the result of integrating this for y\ that is dA = y dx taken between certain limits for y. Thus, in integrating over the half square OA C in Fig. 39, it is dA = x dx. Now, if M de- notes the mean value of the variable in question for this element, the result of the first integration of the expression for the aggregate must equal M dA ; hence, M being the final mean, we have 39- M-A (i) M is here a function of the independent variable x, and M dA may be called the aggregate of the new cases introduced when x is changed to x -j- dx. 246 MEAN VALUES AND PROBABILITIES. [Art. 2 1 7. This equation is frequently useful in extending results already found. Thus, in Fig. 39, M is the mean value of OP for a side opposite to O of the square whose side is x; hence by Art. 215, M = 1.148*. Therefore, by equation (i), whence M f X 1.148*= .765*. Mean Distances between Two Variable Points. 218. When each of two points is taken at random upon a fixed line, the distance between them is a function of two variables, since the position of each point is, in that case, de- termined by a single variable. For example, if each of the points lies upon the circumference of the same circle of radius a, it is determined by an angle at the centre measured from some fixed radius OA. The problem is, in this case, to find the mean value of a chord under certain re- strictions. If there are no further restric- tions, so that each end of the chord is \A equally likely to fall at any point of the circumference, it is obvious by symmetry that we may assume one end to be at the fixed point A, and the other, B, to fall at FIG. 40. random upon the semicircumference ABD in Fig. 40. This restricts the number of cases to n, and we have A B = 2a sin i#. Therefore ,_ M= 2a t* . =\ s 7t J is the mean value of a chord when all positions of the extremi- ties are equally probable. XVII.] MEAN DISTANCES BETWEEN POINTS. 247 219. Now suppose that we require the mean value M v of the chord BC which cuts a fixed diameter AD. Then the two variables = A OB and = AOC'in Fig. 40 determine the positions of the extremities, and each is restricted to values between o and TT. The number of cases is now ?r 2 , and the value of BC is 2a sin (0 + 0)- Therefore n f -ie= r sin $(0 + = 4^ cos $(0+0) d - O Jo '0 = r r n* = 40 (cos$0+sin$0)*/0=8tf sin$0 cos0 =160; whence M, = n* 220. When the lines on which the random points fall have a common part (in which case zero will be a possible value of the distance), care must be taken that the expression for the distance does not become negative. For example, if both ex- tremities of the chord B and C' fall at random in the upper semicircumference in Fig. 40, the expression for the chord is BC' = 2a sin (0 0), which has a negative value when < 0. The aggregate formed by integrating over the range o to TT for both variables would obviously be zero in this case. What we require is the mean of the positive value of this ex- pression. This limits the number of cases to i^ 2 , namely those in which TT> > > o; thus, if we integrate first for 0, its limits are o and 0. Denoting the mean in this case by M z , we have for the mean value of the chord which does not cut a given diameter 248 MEAN VALUES AND PROBABILITIES. [Art. 22O. pr ,+ = 2a\ si J o J o = 4a\ (i J o cos Therefore M 2 = ~ . TT The whole number of chords which do not cut the diameter through A, inclusive of those which lie below the diameter, is 7t 2 , the same as that of the chords which do cut the diameter. Therefore the mean value, when all restrictions are removed, is M = \(M^ -\- M^. Accordingly, the values found in this and the preceding article are verified by the value of M found in Art. 218. Mean Distances Connected with a Sphere. 221. In finding the mean distance of the point B taken at random within the volume of the sphere whose radius is a from the point A taken at random upon the surface, we may obviously take A as fixed. Taking this point as the pole in Fig. 41 (which rep- resents a section through A , B and the FIG. 41. centre), the distance is the radius vector of the point B y and the number of cases is the volume, |7r^ 3 , of the sphere. In finding the aggregate of the r's, we may avoid triple integration by taking for dV, as in Art. 146, the volume generated by the element of area rdrdQ when the figure revolves about the initial line. Thus dV = 27rr j sin QdrdS\ XVIL] DISTANCES CONNECTED WITH A SPHERE. 249 then the mean value of r is determined by Oza cos 6 r 3 dr sin dd o TC fa . = ( 2 Jo 2a cos &y sin 8 dd = f / ** J o Therefore J/= #. 222. This result may be extended to the case of two points both taken at random within the sphere by the method of Art. 217, which applies equally well when the total number of cases is not represented by an area. For, suppose N, the total number of cases to be expressed in terms of a single in- dependent variable, say r, so that dN is the number of new cases introduced when r is changed to r -(- dr. Then, if M 9 is the mean value for these new cases, the final mean M will be determined by -Jo-- In the present problem, when both points are taken at ran- dom within the sphere whose radius is r, the whole number of cases is N= (^Trr 5 ) 2 = V-wV, whence dN = J/?r 2 . 6r*dr. The " new cases," of which this is the number, are plainly those in which one of the points is on the surface of the sphere and the other anywhere within; hence, by the preceding article, M n = \r. Therefore : l/7r2.6.f r*dr\ J o whence M = ||#.* *It may be noticed that in this process if N when expressed in terms of the single variable is of the form kz" and M o is of the form cz m , the differentiation of 250 MEAN VALUES AND PROBABILITIES. [Art. 223. Random Parts of a Line or Number. 223. The division of a line (which may be taken to repre- sent a number regarded as a continuous quantity capable of indefinite subdivision) into three parts at random involves the random fall of two points of division. Each* point of division is equally likely to fall at any position on the line. Let AB be the line, and R and 5 the points of division; then, putting AR = x and AS = y, it is to be noticed that x is not one of the three parts unless y > x. If this restriction is made, the parts are x, y x, a y. Without the restrictions, the mean value of x would of course be %a, and the mean value of y x would be zero, since for every case in which y x has a positive value there is a case in which it has an equal negative value. But, when the re- striction is made, the mean value of y x is the same as the mean distance of two points taken at random on a straight line, in finding which it is necessary to exclude negative values, as in Art. 220. 224. That there is no distinction in kind between the three parts thus symbolized is clearly seen if we imagine the line bent into the circumference of a circle upon which three points fall at random. No change is made by taking one of the points as fixed. It follows that the mean values of the three parts are equal. Again, since the sum of the corresponding terms which enter N introduces the factor , and the integration introduces the divisor m-\-n; so that the relation between M and M for the same final value of z is Mo. Thus, in Art. 217, we ha_ M \M Q ; in the present case, we have M = M\ XVII.] RANDOM PARTS OF A LINE OR NUMBER. 25 I the three aggregates in each " case " (or mode of division) is a, the sum of the aggregates is a multiplied by the whole num- ber of cases. Therefore the sum of the means must be a, and each mean is \a. 225. It is sometimes useful in problems involving two in- dependent variables, like the present, to _ >Q represent them by the coordinates of a point, and to consider the area of integra- tion as in Art. 128. Thus, in Fig. 42, let the x of Art. 223 be the abscissa, OR, and y the ordinate of a point P referred to the rectangular axes OA and OB. Then the condition y > x, while each is positive and less than a, restricts P to FlG - 42. the triangle OBC. It is now obvious that the mean value of x is the same as ~x, the abscissa of the centre of gravity of the triangle OBC, which is %a, agreeing with the conclusion in the preceding article. Again, the mean value AT of x* is given by = J whence M = \cfi. This is in fact the square of the radius of gyration (Art. 209) of the area of integration OBC with refer- ence to the axis of y. 226. If we make a real distinction between the parts, for instance, if we let x represent the smallest of the three parts, the area of integration is still further restricted. Thus, if the three parts in Art. 223 are in order of magnitude, a y being the greatest, the conditions are completely expressed by o < .r < j x < a y. Here, the first inequality shows that P is on the right of the line MEAN VALUES AND PROBABILITIES. [Art. 226. x = o, in Fig. 43, or OB. The second shows that P is on the left of the line x =. y x, or y = 2x, which is the line OD, joining O to D the middle point of BC '. The third inequality shows that P is below the line 2y = a -j- x, which is the line CE, where E is the middle point of OB in the figure. Thus P is restricted to the triangle OEF. The intersection F of the two lines is the point ( l .a, \a). The mean value of x is therefore the FIG. 43. abscissa of the centre of gravity of this triangle, namely \a. The mean value of y is the ordinate of the same point, which by Art. 207 is the mean of the ordinates of the vertices, that is \(^a -f- \a] = -^a. Hence we have, for the mean values of the least, middle and greatest parts, la, Atf and J4#. Random Division into n Parts. 227. If a number a is divided into n parts at random, n r of these parts may be taken as independent variables x, y, z etc., subject to the condition that each shall have a positive value and that their sum shall not exceed a ; thus, x -\- y -\- z -f- . . . < a. The number of cases is now represented by the definite inte- gral ~ i x faxy . . .dzdy dXy fa fa x fa J o*-o J o which involves n I integral signs. XVII.] RANDOM DIVISION INTO U PARTS. In like manner, the aggregate of the wth powers of a part (or numerator of the fraction representing the mean value) is the result of integrating the product of the element in this integral by the mth power of any one of the variables. The integral will be found to have the same value, whichever of the variables in the above order of integrations we employ; but it is simplest to employ the variable for which we first integrate. 228. Integrals of this form are readily evaluated by the help of the theorem of Art. 97. Thus, in the case of five parts, we have to evaluate the quadruple integral naxfaxy fa xyz w m dw dz dy dx . . . (i) o J o Jo The value of the first or inner integral is --(,-_ *_,_.,)-+.. Putting ft for a x y, the upper limit, the integral next to be evaluated is By the theorem referred to, this becomes i r0 fi m + 2 (a x -4- T _/-. V m -f i J (m + i\m + 2) (in + i)O + 2)' In like manner, putting y for a x, the result of the next interation is (m +!)(** + 2}]y ~ (m + i)(m + 2](m -f 3)' and finally ~ (m + !)(* + 2)(m -j- 3)(* -J- 4)' 254 MEAN VALUES AND PROBABILITIES. [Art. 22G. 229. This integral is the aggregate of the mth powers of a part; also, putting m = o, it gives the total number of cases, namely ^a*; hence we have, for the mean value of the power of one of five parts selected at random, o+ This gives \a for the mean of a part (agreeing with Art. 224), -j^a* for the mean square of a part, and so on. In the general case of n parts, we have in like manner, for the whole number of cases, N = (n- i)!' and, for the mean mth power of a part, (n - i}\ml M =7 -* - ri *'*. (m -f- n i ) ! 230. The mean value of a product of powers of two or more parts can be readily found by the aid of the equation r\ x\a-xjdx = -. - j f \ J We have now to evaluate the definite integral 7 = j J j *'*l ' <-'0 - 2x)'dx n _ l dx n _ 2 . . . dx lt (2) in which the n i variables are positive quantities subject to the condition 2x < a, and each of the exponents /, q, . . . s is zero or a positive integer. Denoting the limit in the first integration by a (compare Art. 228), the part of a last written * See EK. VII, 16, p. 128. XVII.] RANDOM DIVISION INTO H PARTS. 25$ is a ~2x = a ^_, ; hence the first integral has the form given in equation (i). At the next integration, putting ft for the upper limit; a = ft x n ~n substituting, the integral to be evaluated takes the same form, s being replaced by s -f- r -f- i, and r by a. new exponent ; so that the resulting exponent is of the form s -j- r -f- t -J- 2 . So also at each step, the new value of s, in the application of equation (i), is the sum of the exponents used and the number of preceding integrations. Consider now the numerical factors introduced at the suc- cessive steps. In the denominator, the new factors run from s -j- i to the new value of s ; therefore these factors constitute the series of increasing natural numbers. In the numerator, the new factor introduced is the factorial of the new exponent. Hence we have finally, for the value of /in equation (2), /_ - p. q. . . .s. - _ l+/+ ^ + ...+, (n- i+p + q + ...+s)\ Dividing by the value of TV, Art. 229, we find that, when a is divided at random into n parts, the mean value of the product of the /th power of one part by the ^th power of another and so on, is 231. The mean value of the least part, the next to the least part, and so on to the greatest part, can be found without in- tegration as follows : * Denote the least part by a, the next by a -f- /?, the next by a -f- ft -f- y, and so on. Then a, fi, y, etc., are all positive, and, summing the parts, we have na-\-(n \)ft -f (n 2]y + . . . = a. The terms in the first member may now be regarded as random parts into which a is divided. Therefore, the mean * See Whitworth's "Expectation of Parts" in "Choice and Chance " where a] so an algebraic proof of the value of M in Art. 230 is given. MEAN VALUES AND PROBABILITIES [Art. 231. a a a value of each term is ; whence , . r , etc., are the n n* n(n I ) mean values of ,/?,... The mean value of a -f- ft is obvi- ously the sum of those of a and ft, and so on. Hence we have, for the several mean values, a I n'n'' a /i n \n a I \ n\n ' n I These results will be found to agree with those found by inte- gration, in Art. 226, for the case n = 3. Mean Area of a Triangle with Random Vertices. 232. When the quantity whose mean is required depends upon the position of two variable points, the element or dif- ferential of the " number of cases " is the product of the ele- ments upon which the two points re- spectively fall. For example, let us find the mean area of the triangle APQ, Fig. 44, formed by joining the fixed point A on the circumference and two points P and Q taken at random within ^^ the circle. It is, for this purpose, con- venient to refer the circle to polar co- ordinates, taking A for the pole, and the tangent to the circle for the initial line. The equation of the circle is then r = 2a sin 6. Denoting by r and the polar coordinates of P, and by p and XVIL] TRIANGLES WITH RANDOM VERTICES. fia sin 9 rdrpdpdOdff) = \n*a*>. Q JO Using the same limits, the aggregate of the areas APQ is sin <6 f?a sin 6 Hence mza sin cp Cm sin 9 r* dr p 2 dp sin (0 - ff)dd d$. o Jo (za\ 6 f* f* = -. I sin 3 sin 3 sin (0 B}ded(t>, I<5 Jo Jo 64.0* f n f * M -^j-J sin 3 (sin sin 8 ^ cos 6 cos sin 4 6)d6 d. and 9^ J Since (sin sin 3 V cos cos sin 4 = J sin 5 cos 0(f0 f sin cos 1 sin 3 cos 0), this becomes J/= J (2 sin 8 30 sin 3 cos -|- 3 sin 4 cos 2 -f- 2 sin 8 cos 2 0)^/0. 258 MEAN VALUES AND PROBABILITIES- [Art. 232. * The value of the integral in this expression is 7.5.3.10- r- sin 4 0-1" 3 r" 4sT^- 2 2-l\ ^ +7 sin' 5-O-4-2 2 L 4 Jo 4Jo 6-4-2 2 ' ^8.6.4-22 - 35^ , 3^3j^ ^ , 3^. S^ _ 20 + 9 + 6^ _ 35^ 64 ' 24-22 16"^ 64" 32 32' Therefore J/= - 9 = E . 97T 2 32 36^- 233. From this result we can readily derive, by the method of Art. 222, the mean area of the triangle when all three vertices are taken at random within the circle. For, if r is the distance of the point farthest from the centre, the value of N in terms of r is N = TrV 6 , whence dN =6x?r 5 dr; and the mean, M Q , of the new cases (of which dN is the number) is, by the preceding article, ^. Hence, if M denotes the required mean when the radius is a, whence 6 35 ~ 8 '$67t a Mean Areas Found by the Method of Centroids. 234. The theorems proved below are often useful in find- ing the mean areas of triangles with random vertices. If two vertices, A and B, of a triangle are fixed and the third, C, is taken at random upon a given line or area ("), its area is XVIL] MEAN AREAS BY THE METHOD OFCENTROIDS. 2$Q \AB-p, where/ is the perpendicular from C upon AB. Sup- posing that the line or area (T) lies in a plane containing A and B, and furthermore that the line AB or AB produced does not cut (C}, the /'s will all have the same direction and their mean value will be the perpendicular from G, the centre of gravity of (C}. Hence, with this proviso, the mean area of the triangle ABC is that of the triangle ABG. For example, if A and B are the extremities of a diameter of the circle whose radius is a, and C is taken at random in one o of the semicircles, the mean area of ABC is , see Art. 206. \* If C is taken at random on one of the semicircumferences, the 2tf 2 mean area is - , see Art. 193. By symmetry, these are also the values when C is taken at random on the whole circle and on the whole circumference respectively. 235. Next suppose that B is not fixed, but taken at random on a line or area (Z?), lying in the plane which contains (C) and the point A, and also that the line AB can in no position of B cut the area (7); in other words, that no straight line through A can cut both (B} and (C). Then the aggregate of the- areas in all the cases in which B has a certain fixed position is (C}-ABG, and the aggregate of all values of ABG when B is not fixed is (B}.AFG, where/ 7 is the centre of gravity of (B]. Therefore the aggregate of all values of ABC when neither C nor B is fixed is ()(]&) AFG, and since the total number of cases is (")(/?), the mean value is AFG. For example, let the vertex A of a. triangle be joined to any point D of the base, and let points P and Q be taken at random respectively in the two parts ABD and ADC. Then, denoting by M 2 the mean of the triangles APQ, we have M 2 = AFG, where F and G are the centres of gravity of the two parts. Now, from the known position of the centre of grav- ity of a triangle, the base FG is f of %BC, and the altitude is f 260 MEAN VALUES AND PROBABILITIES. [Art. 235. of that of the triangle ABC. Hence the mean area required is where A is the area of the triangle ABC. 236. Certain extensions of this result may now be made. In the first place, let Mbe the mean area of APQ when P and Q are each taken at random anywhere in ABC. It is plain that for FIG. 45. different triangles Mis proportional to A. Hence putting BD x, DC = y, BC = a, we have, for the means when P and Q both fall in the triangle whose .base is x, or both in the triangle whose base is y, respectively Now putting, for simplicity, x = y, we have for the mean when P and Q fall on the same side of AD, M^ = \M; and for the mean when they fall on opposite sides, as found above, M 2 = \A. But since the areas of the parts are now the same, M \(M V -j- M 2 } ; whence M= \M^ therefore M --A. 237. Next let the triangle have a fixed vertex on a side of the triangle ABC, say at D, Fig. 45. No straight line through D can cut both of the areas ADB and ADC, therefore the con- dition given in Art. 235 is fulfilled, and denoting as before by jM 2 the mean area when one of the other vertices is taken at jrandom in each of the triangles, we now have M 2 = DFG, or Denoting again by M x and M y the mean areas when both of the other vertices are taken at random in ABD and in ADC XVII.] MEAN AREAS BY THE METHOD OF CENTROIDS. 261 respectively, these means have the same values as in Art. 236, because the fixed vertex is still one of the vertices of the tri- angle in which the other two fall at random. Hence , , . a 27 a 27 Now denoting by M D the mean area when the other tw the number of cases is limited to The "event" whose probability we seek is that the distance YX shall exceed a given quantity c, equal, say, to OD in Fig. 46. There are no favorable cases when x <^; but, for each value of x between c and a, favorable cases occur whenever y < x c. Hence the number of favorable cases is r a r x ~ c c a dy dx = (*"" c}dx = J c J o J c (a - Dividing by the whole number, we have for the probability P p = (a ~, c) *. It may be noticed that it is more convenient, in this prob- lem, to find the probability of the distance exceeding c, because 268 MEAN VALUES AND PROBABILITIES. [Art. 241. the number of cases in which it is less than c can not be ex- pressed by a single definite integral. Probabilities Represented by A reas. 242. When two variables occur, it is often useful to rep- resent them as in Art. 225 by the rectangular coordinates of a point which thus falls at random within an area which repre- sents the whole number of cases. The number of favorable cases will then be represented by a portion of this area, and may often be found without integra- tion. Thus, in the example above, the whole number of cases is represented by the half OA C of the square in Fig. 47, constructed on the line OA. Tak- ing OD = c, the line DE parallel to OC is the locus of the equation y =. x c. Hence the favorable area is the area below this line, that is the triangle DAE. In this graphic method the area favorable to the contrary event is exhibited at the same time ; thus ODEC is the area favorable to a distance XY less than c, which, as remarked in the preceding article, is not expressible by a single integral. 24-3. The method is particularly useful when there are con- ditions which still further restrict the favorable area. For example, we have seen in Art. 223 that the points X and Y in Fig. 46 divide the line a into three parts at random. Sup- pose now that we require the probability that no one of the three parts shall exceed c. The three parts are denoted here by y, x y and a x. The condition that x y shall not exceed c cuts off from the whole area OAC, as we have just seen, the triangle ADE. The condition that y shall not exceed c cuts off in like manner the triangle above the horizontal dotted D FIG. 47. XVIIL] PROBABILITIES REPRESENTED BY AREAS. 269 line, and the condition that a x shall not exceed c excludes that to the left of the vertical dotted line. The area excluded is in each case \(a cf\ but when (as represented in Fig. 47) c < \a, these areas overlap, and the remaining area is the small isosceles right triangle whose area is readily seen to be 4(3^ of (which vanishes when c = \a, as should evidently be the case). Thus the probability is - 5 - when c is be- ct / \ rt tween \a and \a\ and it is I 3 ^ when c > \a. Local Probability. 244. Probability questions concerned with the random fall of actual points, lines and other geometrical magnitudes are sometimes called problems of local probability. One of those earliest solved was proposed by Buffon in 1777 as follows: A floor of indefinite extent is ruled with equidistant parallel lines whose common distance is a ; a rod of length c is thrown upon it at random ; what is the chance that it crosses one of the lines? It is plain that one end of the rod may be assumed to fall upon a fixed line perpendicular to the parallel lines. Denot- ing by x its distance from one of the intersections taken as origin, and by 6 the inclination of the rod to this line, as in Fig. 48, all values of x and of 6 are equally probable. But considerations of symmetry show that we need consider only values of x between o and a there will be a value of 6, as shown in the upper part of the figure, such that, for any smaller value of 6, all values of x give favorable cases. The integration for 6 must now be separated into two parts at this point, say 6, = cos" 1 -. The number of favorable cases is now XVIII.] LOCAL PROBABILITY. 2/1 n ,- r dxdd-\- Jej re cos ad l + c(i sin ^) = a cos-' - -j- ^ c which gives The limiting value of this expression when c increases without limit is of course unity. 24-6. When a problem of local probability involves a line drawn from a given point in a random direction in space, it is necessary to assume, as in Art. 199, a spherical surface having the given point for centre, and to regard the line as piercing this surface in a point taken at random upon it. For example, a shot is fired with a given velocity in a random direction from a point on the circumference of a circular field of which the diameter 2 a is equal to the maximum horizontal range. Given the formula R = 2a sin 20 ...... (i) for the horizontal range, or distance at which the shot falls when 6 is the angle of elevation, it is required to determine the chance that the shot may fall within the field. Let O, Fig. 49, be the point of projection, and let the vertical plane through the line of fire make the angle with the diameter OA of the field. Then and are the spherical coordinates of the random point on the surface of a sphere ol arbitrary radius b. The element of surface is d*S = P cos d dO, FIG. 49- and the whole surface, representing all possible cases, is that 2/2 MEAN VALUES AND PROBABILITIES- [Art. 246. of the upper hemisphere 27tb*. Of this area one-half, ?r^ 2 , cor- responding to values of on the second and third quadrants, is unfavorable. A portion of the remaining half is also un- favorable and we proceed to find its value. For a value of between - ^n and ?r, as in the dia- gram, those values of 6 are unfavorable which make R > r, the radius vector of the field, which is r = 2a cos ....... (2) Comparing with equation (i), we have therefore an unfavor- able case if sin 26 > cos 0. Now, sin 2# = cos when and between these limits (which include 45, the elevation for maximum range) sin 20 > cos 0, hence R > r. Therefore the additional unfavorable area is / = 24/2 2 r sin ^0^/0 = 44/2^ 2 cos^0 k 4 (4/2 i)^ 2 . Therefore the total unfavorable area is b\n -\- 4(^2 i)], and the favorable area is b\7t 4(4/2 - i)]. Thus the chance required is -, which is about .24. 2 71 XVIII.] THE ELEMENT OF PROBABILITY. 273 The Element of Probability. 24-7. If the element which, when integrated between differ- ent limits, gives the whole number of cases and also the num- ber of favorable cases, be first divided by the whole number of cases, the result is an element of probability. If this element be integrated with the limits proper to the whole number, it of course gives unity; and if integrated with the limits proper to the favorable number, it gives the probability at once. Thus, when every value of x between o and a is equally probable, the element of probability is dx/a. We cannot speak of this as the probability of a special value of x ; it may however be called the probability of a value between x and x -f- dx. In the case of the random fall of the point P, it is the chance that P falls upon the element dx of the line a. 24-8. Now suppose that a point P falls at random within a circle of radius a, and let r denote its distance from the centre. All values of r between o and a are now possible, but it is plain that they are not equally probable. The chance that r falls between r and r -f- dr is now the chance that P falls upon the elementary annulus 2nrdr; therefore, dividing by the area of the circle, it is 2.rdr Hence the probability in question is proportional, not to dr, but to r dr-* that is, it is of the form krdr. But the value of k depends upon the extreme values between which r is known to fall, and may be determined by the condition that the value * In using polar coordinates, the probability of r for a point falling at random is proportional to r dr when 6 has a fixed value, and that of 6 is d$ when r has a fixed value. Accordingly the probability of the joint occurrence of a particular value of r and a particular value of 6 is proportional to the product r dr dQ. 274 MEAN VALUES AND PROBABILITIES. [Art. 248. of the integral between these limits must be unity. Thus, if values of r between r l and r z only are possible, assuming the form fcrdrand determining k, we find, for the element of prob- ability, In like manner, if a point falls at random within a spherical surface of radius a, its distance from the centre has the ele- mentary probability ^r 2 dr 24-9. When the element of probability of one of the vari- ables involved in a problem is independent of the other vari- ables, and can be written down beforehand, the problem can be made to depend upon the simpler case in which the variable in question has a fixed value. Denote this variable by r, and let dp be its element of probability, which we suppose a known function of r and dr. Let P denote the required probability, and P the probability when the value of r is fixed. We sup- pose P to be first determined; it will of course be a function of r. Then the product P dp will express the probability that the event will happen in connection with a value of r between r and r -f- dr. But this is only one way in which the event may happen, for it may happen in connection with any one of the possible values of r. Hence the entire probability of the event is found by summing up the probabilities of the different ways, that is by integration. Thus /? = taken between the limits given for r. The method is analo- gous to that given for mean values in Art. 222. XV II I.] THE ELEMENT OF PROBABILITY. 275 FIG. 50. 250. As an illustration let us find the probability that the distance between two points A and B taken at random within a sphere shall be less than the radius a. The probability will evidently be unaltered if we assume one of the points A to be taken on a fixed radius CD of the sphere. Let r denote its distance CA from the centre. The value of r is independent of the other variables (which concern the position of the other point, B) and its elementary prob- ability is, as mentioned in Art. 248, 3r 2 dr *=!-. Let us now find P , the probability that AB < a when A has the position given in Fig. 50. Let a spherical surface whose centre is A and radius a be described, then P is the chance that B shall fall in the lens- shaped volume cut from the given sphere by this spherical sur- face. In other words, it is the ratio of the volume of the lens to that of the sphere. The lens is double the segment of the sphere cut off by a plane perpendicular to and bisecting CA at D. Thus its volume is 2/7 Pa (of s?)dx = hr and, dividing by the volume of the sphere, j^rra 3 , we have 1 2 a~r r 3 Therefore dP=P dp = ~ - 5 ^ 3 b and, integrating between o and a, p T _ . a/, i\ T 11 15 TffU f/ ~ ~S~S ??' Thus the odds are 17:15 in favor of a distance greater than a. 276 MEAN VALUES AND PROBABILITIES. [Art. 251. Curves of Probability. 251. The elementary probability of a variable x of which the possible values are not equally probable may be put in the form y dx, where y is a function of x. The curve in which y is the rectangular ordinate corresponding to the abscissa x (or rather that part of it corresponding to possible values of x~] is called the curve of probability for the variable x. Such a curve is a graphic representation of the law of probability of x, as it may be called, that is the mode in which the probability of x varies. The ratio of any two values of y gives the relative probability of the corresponding values of x, although we can- not assign actual values to the probabilities without introducing the element dx. Since the whole probability that x shall fall between given limits is a value of the integral \ydx % the probability that x shall fall between any given limits is rep- resented by the area enclosed between the curve, the axis of x and the ordinates of the limiting values. For this purpose, the total area, of which the base represents the whole range of possible values of x, must of course be taken as unity. 252. In illustration, let us find the law of probability, and construct the probability curve, for the distance from one end of a line AB = a of the nearer of two random points which have fallen upon AB. Let P and Q be the two points which fall at ran- B com; let ^ denote the one nearer FIG. 51. to A , and x the distance AX. The chance that P shall fall upon any given element dx is dx/a. If this happens, the chance that P is the point X is the same XVI 1 1.] CURVES OF PROBABILITY. 277 thing as the chance that Q falls on the segment a x to the right of dx. Thus the chance that P falls on dx, and is the point X, is (a x}dx ~^ ' But there is an equal chance that Q falls upon dx and is the point X. Therefore the whole chance that X falls upon dx, or value of ydx, is in this case 2 (a x]dx -^r-' w hence the probability curve for the nearer to A of two random points is the oblique straight line 2(a x} ****-?-*' which passes through B as represented in Fig. 51. Thus the law of probability, in this example, is a uniform decrease of probability from a maximum at A to zero at B. The ordinate A C corresponding to x = o is 2/a ; this makes the whole area of the triangle ABC unity, as it should be. 253. The probability that x shall fall between any given values is now represented by a part of the area of this triangle. Thus the probability that it shall be less than AE in Fig. 51 is the area of the trapezoid 'AEFC . It is therefore the same thing as the probability that a point falling at random upon the triangle ABC shall fall upon this trapezoid. So also, in gen- eral, the probability curve defines an area such that the variable for which it is constructed may be regarded as the abscissa of a point falling at random upon the area. In particular, if we draw an ordinate which divides the area into two equal parts, the corresponding abscissa is that value which x is just as likely to exceed as to fall short of. This 278 MEAN VALUES AND PROBABILITIES [Art. 253. value is often called the probable value. It is, of course, not generally the most probable value. In this case, in fact, any smaller value has a greater relative probability. 254-. I n like manner, the dotted line in Fig. 5 1 is the prob- ability curve for the distance of the farther from A of the two random points. Its equation is 2x 2x dx y=^ hence p- is the element of probability. That is, the probability of the distance x of the more distant of two points selected at random is proportional to x dx. The probable value, in this case, is a |/^ ; that is to say, it is an even wager that the greater of the distances of the two points shall exceed this value. The alternative event, in this , case, is that both distances shall fall short of a ^/, the probability of which is also equal to . 205. We have seen in Art. 223 that, when two random points fall upon the line a, the distance x of the nearer point from one end is one of three random parts of the line ; hence the expression found in Art. 252 and the line CB in Fig. 51 ex- press the law of probability of one of the parts when a is divided at random into three parts. By inspection of the figure, it is evident that the chance that the part shall be less than \a is |; the chance that it shall exceed %a (its mean value) is -|; and so on. 256, If three points fall at random upon the line AB, and Ji^ Y, Z denote them when selected in order of nearness to A, the probability curves for their distances from A will be found to be _ 3( *? _ 6Q x)x _ 3^ y ~~ ~a 3 ' y ~~ a 3 ' y : : a 3 ' The proof is similar to that in Art. 252. Thus, for the middle XVIII.] CURVES OF PROBABILITY. 279 point: The probability that the points fall in the order P, Q, R, and that Q falls on a given element dx\ is the probability of the joint event that Q falls on dx, R falls on the segment a x to the right of it, and P on the segment x to the left of it. The respective probabilities of these events are dx a a x and a Hence the probability of the compound event is (a x}x dx But this is only one way in which the middle point can fall on dx, for there are six orders in which P, Q and R may fall ; hence the whole probability that X falls upon dx is 6(a x]x dx Taking A as before for the origin, the three curves are the parabolas shown in Fig. 52. The first, CB, for the point nearest to A , is also the proba- ; bility curve for the value of one of four parts into which a is divided at random. It will be noticed that the sum of the three elementary probabilities FIG. 52. is - ; as should be expected, since the sum of the chances a that X, Y and Z, respectively, shall fall upon a given dx is evidently the same as the sum of the chances that P, Q and R shall fall upon dx. 2 SO MEAN VALUES AND PROBABILITIES- [Art. 257. Discontinuous Curves of Probability. 257. Since the ordinates of a probability curve are merely graphic representations of the relative probabilities of the cor- responding values of x, any curve in which the ordinates are proportional to these relative probabilities will serve as the curve of probability. Then, as mentioned in Art. 253, the equally probable cases correspond to the falling at random of a point upon equal elementary areas of the probability curve. Thus the area whose base is a given range of values represents the number of favorable cases; and, to obtain the numerical value of the probability, this must be divided by the whole area which represents the whole number of cases, or, as we may say, the probability unity. 258. The probabilities of the values of the variable x will frequently, as a result of given conditions, follow different laws in different parts of the range of its possible values ; in other words, the probability will be a discontinuous function of x, When the variable is represented by one coordinate of a point which falls at random upon an area defined by limits of integration, this area will at once determine the law of probabil- ity for all values. For example,' in Art. 226 we saw that the restrictions upon x and y, which are necessary to make x represent the least, and a y the greatest, of three random parts into which a is divided, limit the point (x, y) to the triangular area OEF in Fig. 43. Thus the least part is the abscissa of a point falling at random upon this triangle. In- spection of the diagram shows that no value of x greater than \a is possible. If a line parallel to the axis of y be drawn corresponding to any smaller value of x, the segment of it included within the triangle measures the relative probability of that value of x. Thus the diagram shows that the prob- ability of a given value of the least part decreases uniformly XVIIL] FACILITY OF ERRORS OF OBSERVATION. 28 1 from a maximum at the value zero to nothing at the value \a. No discontinuity occurs, in this case, between the extreme possible values. 259. Now consider the greatest part, which was represented by a y. This is the distance of the point (x, y] from the lino BC, The diagram shows that the possible values lie between \ r and a. The relative probability of a given value is measured by the segment within the triangle of a line parallel to the axis of .rat the given distance from BC. The maximum probability occurs, therefore, at the value -^a. From this maximum the probability decreases uniformly to zero at the value a, and it also de- creases uniformly to zero as we pass from \a to \a. Thus the curve of probability, when laid down upon the axis of 4r, is the broken line NPA in Fig. 53. From this figure it is readily shown that the probability _ that the greatest of three parts exceeds * \a is f , the probability that it exceeds # is , and so on. N M 53- Law of Facility of Errors of Observation. 260. In the Theory of Errors of Observation, the prob- ability of the occurrence of an error x is assumed to be propor- tional to e~ ; '' 2 -**, where h is a con- stant depending upon the pre- cision of the observations. The law implies (as indicated by the form of the curve y = ce-' 1 " 1 *'* shown in Fig. 54) that positive and negative errors numerically ^ equal have the same probability, FlG 54- that the maximum probability occurs at x =O, and that it becomes so small for large values 282 MEAN VALUES AND PROBABILITIES- [Art. 260. of x that it is practically unnecessary to" assign any finite limits to the possible errors. Since the number of errors between x and x -)- dx which may be expected in ' ' the long run ' ' is proportional to ce~ h ***dx, the curve is often said to express t/te law of frequency of errors; or the law of facility of the error x. In order that ce~ k * x *dx shall be equal, and not simply pro- portional, to the probability, it is necessary that the whole area of the curve shall be unity, or which can be shown to give c-= . See Art. 280. I/7T Mean Values under Given Laws of Probability. 261. In equation (2), Art. 189, p l , p. 2 etc. express the relative frequencies with which the values z l , z z etc. of a vari- able z occur among the values of which the mean is required. Dividing by 2p, the whole number of cases, the equation becomes ' in which the coefficients of the several values are their prob- abilities. Denoting these by P^, P 2 etc., we have M= P^ + P,Z, +...== ?Pz. ... (I) Thus the mean value under a given law of probability may be defined as the sum of the products of the several values each multiplied by its own probability. In finding the mean of a continuous variable z, the element which takes the place of/, when integration takes the place of summation, expresses the relative frequency or probability of XVIII.] MEAN VALUES UNDER GIVEN LAWS. 283 the various values of z, in the distribution for which M is required. Denoting this element by dp, the formula for the mean is ' dp (2) in which we may regard dp as an element proportional to the probability of z. But, when the actual element of probability dP is employed (so that dP = i when the integration corre- sponds to the whole range of values of z for which M is re- quired), the formula becomes M= \zdP (3) 262. For example, in the theory of errors the mean error, denoted by e, is denned as that whose square is equal to the mean square of an error. Hence, given the law of frequency, dp = e~ / '^' i dx, we have Integrating by parts, we have in which the first term vanishes at each limit. Hence i i e T75 , or e = kl/2' Again, the mean value of the error without regard to sign is, in the same theory, denoted by ?;; hence, using the exact ele- ment of probability, which (see Art. 260) is dP e-*wdx tfn we have i xe-**dx = - ~-e- h f ff = 2 V*J 284 MEAN VALUES AND PROBABILITIES. [Art. 262. It will be noticed that e is the radius of gyration of the area in Fig. 54 about the axis of y, and 77 is the abscissa of the centre of gravity of the area on the right of that axis. Probabilities Involving Selected Points. 263. In any question of probabilities involving a variable x whose values are not equally probable, but follow the same known law throughout its possible values, the element express- ing this law takes the place of the simple element dx. Thus, suppose that, in the problem of Art. 241, Y instead of being taken at random is the farthest from O of three points taken at random, and that X is the farther from of two other points taken at random ; required as before the chance that the dis- tance YX shall exceed c. Assuming that y < x, as in Fig. 46, we proceed as in Art. 241, except that, in accordance with Arts. 254 and 256, dx is replaced by x dx, and dy by y* dy. Thus the whole number of cases is now. f a a 5 y z dy xdx = \ \ x 4 dx = ; J o 15 and, for the number of favorable cases, we have n*-c fa y* dyxdx = f (x cfx dx. * o ' c Putting, for convenience, in this last integral z = x c, it becomes j r*. + cy, = Jo 20 J Hence the probability is P. = (* - c ) 4 (4* + J XVIIL] PROBABILITIES OF SELECTED VALUES. 285 But if y > x, the probability takes a different form. The whole number of cases is now Jy f a a 5 x dx y* dy = k j/ 4 dy = , and the favorable number is fa ey-c 17 J cJ o xdxfdy = / \s hence the probability is P z = --., 5 (6a 2 -j- "$ac -(- 2 ). 264-. If we require the probability that the distance shall exceed c, without distinction of the cases whenj < x and y > x, we must compare the sum of the favorable cases with the total numbers. The sum of the latter is ^a 5 , which is in fact the value of IT J oJ r y* dy x dx. The sum of the favorable cases will be found to reduce to (a Hence the probability that the distance without regard to sign shall exceed c is a-' 265. If, in the problem solved above, we represent x and y by the rectangular coordinates of a point, as in Art. 242, the point (x, y] is restricted to the square OA CB, Fig. 5 5 . The total number of cases represented by the integral of the pre- 286 MEAN VALUES AND PROBABILITIES. [Art. 265. ceding article may now be regarded as a large number of points distributed not uniformly over the area, but in such a ^ G_ G way that the number falling upon any element of surface dy dx is y^xdydx. In other words, the points are distributed with a density proportional to the value of y^x. The point (x, y} must now be considered as one taken at random from this large number of points ; and the probability that it comes from a certain 1'iti- 55- favorable area, determined by the limits of integration, is the ratio of the number of points in the favor- able area to the whole number. Thus, in the present problem, if lines DE and FG parallel to the diagonal OC be drawn cutting off OD and OF each equal to c, the favorable area when y < x is the triangle DAE. Hence the probability P^ found in Art. 263, is the ratio of the number of points in DAE to that in OAC. In like manner, P 2 is the ratio of the number in FBG to that in OBC, and finally P is the ratio of the number in the two triangles to that in the entire square. Selected Points upon an Area. 266. We have seen, in Art. 254, that the selection of the more distant from a fixed point of two points taken at random upon a straight line is equivalent to giving it a probability pro- portional to the distance. Suppose now that two points fall at random upon a circular area of radius a, and that we select the more distant from the centre ; let us find the probability that it falls upon a given element of area. Denoting the random points by P and Q, and by X that which is farther from the center O, the probability that P falls XVIII.J SELECTED POINTS UPON AN AREA. 287 dS upon a given element dS of surface is - y an d the probability 71 Ct that P is X is then the probability that Q falls on the area nearer to O than dS> which bears to the r 2 whole circle the ratio . Hence, doubling CL the product, because Q may fall upon dS and be X, the whole probability that X falls on dS is rta* no FIG. 56. 267. To find the probability that X falls upon any given area traced out upon the circle, we have only to integrate this over the given area. For example, if the given area is the circle r = a cos in the diagram, having for diameter one of the radii of the large circle, we have 4 r- f =- 7ta*) J f* cos e -1 - P=- r*drde = - V cos 4 8 d6 = 4- 16 The odds are therefore 13: 3 that the more distant of the two points shall fall outside of the small circle. Random Lines. 268. A straight line is said to be drawn at random in a plane if all directions are equally probable, while, among the lines having a given direction, all points of intersection with a common * The points Jfare here distributed with a density proportional to r 2 , but the values of r have a probability proportional to r 3 . The result would apply to any area upon which points had this distribution, but it is to be noticed that selection of the more distant from a fixed point of two points upon an area not a circle would not produce this distribution except within a circle whose radius is the least value of r upon the boundary of the given area. 288 MEAN VALUES AND PROBABILITIES. [Art. 268. perpendicular are equally probable. In expressing the whole number of lines which cut a given convex curve, let p be the perpendicular let fall upon the line from some fixed point within the curve, and let be the inclination of this perpendicular to some fixed direction. Then the whole number will be found by the integration of dp d<>. If, in this integration, the limits taken for p are zero and the perpendicular upon a tangent to the curve, must take all values from o to 2n. Thus the number of lines which cut the curve will be r 2 "- f/ r* dp dtp = J o o J where / is the perpendicular upon the tangent to the curve. 269. Now it is shown in Diff. Calc., Art. 348 (and is geo- metrically evident on drawing the figure) that, if r is the part of the tangent intercepted between the point of contact and the foot of the perpendicular, df = ds pd$ t where s is the length of the arc measured from some fixed point. Now when we have completed the whole circuit of the curve, so that T returns to its original value, the integral of dr is zero ; it follows that, denoting by L the whole length of the curve, we have L f 2 = J o Therefore L, the length of the curve, is the measure of the num- ber of lines drawn at random which meet the curve. Thus, if a convex curve of length / is drawn within a convex curve of length L, the chance that a random .line which cuts L shall also cut / is l/L. XVIII.] RANDOM LINES. 289 270. If the given line is not convex, or if it is not closed, the same result holds if L denotes the length of the shortest convex closed line which surrounds the given line like an elas- tic band stretched about it; for it is evident that a straight line which cuts the given line must cut such a band. For example, the number of lines which pass between two given points whose distance is c is measured by ic. Thus, if two points on the circumference of a circle subtend an angle a at the centre, the chance that a random line cutting the circle shall cut both of the arcs into which it is divided by the points is (2 sin \a)/7t. Probabilities involving Variable Magnitudes. 271. When the probability of an event depends upon a vari- able magnitude in such a way that the probabilities correspond- ing to any values of the variable are proportional to the values themselves, the actual probability of the event is the same as that corresponding to the mean value of the variable. For, by hypothesis, the probability that the event will hap- pen when z has the value z^ , z 2 , . . . and M are respectively ^ z 2 M -, , . . . and - ; a a a where a is a constant. Now, dividing equation (i), Art. 261, by a, we have Since P l is the probability of the value s l , the first term of the second member is the chance that z shall have the value z v and that the event shall then happen. But this is only one way in which the event can happen, and the second member is the sum of the probabilities of its happening in all possible ways, that is, the total probability of the event. Hence the equation expresses the proposition to be proved. MEAN VALUES AND PROBABILITIES. [Art. 272. 272. In particular, if a in equation (i) is a fixed area and 2 is a variable area within it, the chance that a point following at random upon a shall fall upon s is M/a, where Mis the mean value of the area z. For example, let A be the area of a given triangle ABC, and z that of the triangle PQR, where P, Q and R are three points taken at random on ABC; then we have found in Art. 238 that the mean value of PQR is -%A . It follows that if three points be taken at random on ABC, the chance that a fourth point taken at random on ABC shall fall within PQR is j 1 ^. Hence, also, if four points are taken at ran- dom, the chance that one of the four shall fall within the tri- angle found by the other three is |-.* On the other hand, the chance that they shall form the vertices of a convex quadri- lateral is f . Conversely, the mean value of an area may sometimes be found by knowing the probability that a point falls upon it. Thus, in the foregoing illustration, the sides of the triangle PQR when produced separate the whole triangle ABC into six other parts beside the triangle PQR. If S is a fourth point taken at random, the chance that P falls within the triangle QRS is the same as the chance that 5 falls within PQR. Hence the mean area of the space in the vertical angle of the angle at P is also -faA. We thus have four areas each of whose mean values is -^A, and it readily follows that each of the other three has the mean value \A. 273. Since the chance that a given line shall be cut by a random line chosen from a given set of random lines is propor- tional to its length, the chance of cutting a variable line is the same as the chance of cutting its mean value. For example, * If P, Q, R and 5" are the points " /'falls within QRS," "^ falls within PJRS," etc. are mutually exclusive events; that is, no two of them can happen at once. The probability that some one of such a set of events shall happen is evidently the sum of their respective probabilities. XVIII.] VARIABLE MAGNITUDES. 2QI a random line is drawn cutting a circle, and two points are taken at random on the circumference ; what is the chance that they lie upon opposite sides of the line ? This is clearly the same thing as the chance that a random line shall cut the chord joining two random points. The mean value of such a chord was found in Art. 218 to be M = -- ; hence (see 71 Art. 270) the chance required is 2M/27ta or . 274. It should be noticed that the question just solved is not the same thing as the chance that two random secants shall cut one another within the circle. The mean value of the por- tion of such a secant intercepted by the circle is obviously the double of the mean ordinate to a given diameter which was found in Art. 192 to be \na. Therefore M= %rra, and the 2M I chance in this case is - = . 2 The mean value of a ' ' chord ' ' in the sense employed above, that is when the line or secant of which it is part (not the extremities of the chord) is taken at random, admits of a simple expression for any convex curve. For, the number of such chords is JJ \dpd=.L\ and, if C is the length of the variable chord, L.M= \\Cdp d$ determines the mean value. Now, in this integral, we may take as limits for / the values of the two perpendiculars cor- responding to the same value of (one of which will be nega- tive if the origin is taken within the curve as in Art. 268), provided varies only between the limits o and n. Then the MEAN VALUES AND PROBABILITIES. [Ex. XVIII. value of the integral of Cdp will be A , the area of the convex curve, independently of the- value of 0. Hence we have TtA This gives, for the chance that two such random chords inter- sect, that is that two random secants intersect within the area, _ 2M _ 2nA p ~T~ ~iy Examples XVIII. i. A floor is ruled with parallel lines at distances 20, and also with another set at distances 2b perpendicular to them ; a rod of length 2C less than either of the distances is thrown upon the floor at random. What is the probability that it crosses a line ? 2. Three points are taken at random on the circumference of a "circle ; what is the chance that no diameter 1 can be drawn having all three points on one side of it ? . 3. Show that the chance that one of three random parts of a is between c, and a c is the same as that for one of two random parts. 4. What is the chance that the middle point of three random points falling on a line shall fall on the middle third of the line ? H- 5. Two points are taken at random in the northern hemisphere; show that the probability that their difference of latitude exceeds a is P = cos a (J TT )sin a. 6. Two points are taken at random upon a semicircle and their ordinates drawn ; find the chance that a point taken at random upon the diameter shall fall between the ordinates. 4 . ~n* 7. Two points are taken at random within a circle of radius a ; find the probability that their distance shall exceed the radius. 3^3 XVIII.] EXAMPLES. 293 8. If Fis the farther from O of two points taken at random upon a line OA of length a, and X the farther of two other points taken at random, what is the probability that YX shall exceed c ? (a - g. Supposing X in the preceding problem to be the point farther from A of the second pair, find the chance that the distance YX shall exceed c : first, when the points are known to fall in the order O YXA ; second, when the order is OXYA\ third, when the order is not known. (a - cY . ( a - c Y( a + <: -)(sa + c} ~ ' ' 3* 10. What is the chance that the distance exceeds c, if A" and Y are the farthest from O respectively of two and of four points ? P = t=A'(r 5 *' - 3 aV + 6ai- + 2, 3 ). 11. Two points are taken at random in the northern hemisphere; find the elementary probability of the smaller of their latitudes. 2(1 sin (/)) cos d(f>. 12. A line a is divided into three parts at random, and that, of intermediate value is taken ; what is its most probable value, and what the "probable value" in the sense explained in Art. 253 ? \a ; .289^. 13. What is the probability that Y, the nearec to the centre of the two points in Art. 267, falls upon the given area ? P T 5 ^. 14. A point is taken at random within a circle whose radius is a, and a line is drawn at random through it ; find the chance that it cuts a concentric circle whose radius is c < a. c 2 sin~ l . p_ _ a + 2f ^(a 1 S) TT no 1 15. If, in Ex. 14, the line is drawn at random, what is the proba- bility ? c a 294 MEAN VALUES AND PROBABILITIES. [Ex. XVIII. 16. If two points are taken at random, one on the surface and the other within a sphere of radius r, find the probability that their dis- tance shall be less than c, supposing c < 2r. c'^r 3*:) i6r 4 17. If P is the probability that the distance between two points taken at random within a sphere shall be less than c, and P that found in Ex. 16, show that d(NP) = P Q dN (compare Art. 222), and thence find the value of P. _ S&r 3 i&r'-f- <:*)_ 6 18. Find the mean distance of the middle of three points taken at random on a line a from the middle of the line. T \ 4 ; 9' 20. Lines of length b and b' fall at random upon a line a ; find the chance that they overlap by an amount less than c, where c is less than either b or b', and a -j- c > b + b'. c(?a zb2b'-\-c) (a -*)(*- *) . 21. A and B are inhabitants of a city which is known to be sit- uated on a river. Assuming, in default of any knowledge, that the river divides the number of inhabitants into two parts at random, if it is known that B lives on the right bank, what is the probability that A lives on that bank? |. 22. If, in the preceding example, m inhabitants are known to live on one side, and n on the other, show that the odds that A lives on the first-mentioned side are m + i : n + i. 23. A line crosses a circle at random; find the chance that a point taken at random within the circle shall be distant from the line by more than the radius a of the circle. 2 3* XVIII.] EXAMPLES. 295 24. Two random chords of a circle, that is chords whose extrem- ities are taken at random, are drawn ; what is the chance that they intersect ? \. 25. Two random lines cut a square; what is the chance that they intersect within the square ? n 8" 26. A line is drawn at random across a circle; what is the chance that two points taken at random within the circle shall lie on oppo- site sides of it ? 128 45*" 27. Two points A and B are taken at random in a triangle; what is the chance that two other points taken at random in the triangle shall fall on opposite sides of AB ? ^. 28. Four points are taken at random within a circle; what is the chance that they form the vertices of a convex quadrilateral ? 35 1271* DEFINITE INTEGRALS. [Art. 275 CHAPTER V. DEFINITE INTEGRALS. XIX. Differentiation of a Definite Integral. 275. THE general symbol for a definite integral of a single independent variable is f f(x)dx] J a where a and b are constants, that is to say, independent of x Denoting its value by u, we have seen in Art. 82 that F(a), ....... (i) where F(x) is such a function that dF(x] dx =/(*), ........ (2) provided F(x} varies continuously while x passes from the value a to the value b. Moreover, this condition will be fulfilled if f(x) is itself one- valued, finite and continuous for the same range of values of x. The independent variable x is used only in defining the integral which, by equation (i), is a function not of x but of the limits. It may be called the current -variable XIX.] DIFFERENTIATION OF A DEFINITE INTEGRAL. 297 when other variables are also under consideration, and it is evi- dently immaterial what symbol is used for the current variable. We have, in Section VII, derived the values of certain definite integrals by means of formulae of reduction which took simple forms by virtue of special values of the limits. We shall in this chapter consider other methcds by which such integrals can be evaluated in cases where, for the most part, the value of the indefinite integral, F(x), cannot be obtained. 276. Regarding the upper limit in r u = f(x)dx Ja as variable, we have from equations (i) and (2) and in like manner for the lower limit = -F'(a)= -/(a) (2) If the limits were functions of some other variable z, we should have (see Diff. Calc., Art. 371) du du db du da f f i.^ f( \^ a 277. Next, writing the integral in the form r* U= udx, (i) n 298 . DEFINITE INTEGRALS. [Art. 277. as in Art. 84, let us suppose that the quantity u under the inte- gral sign is a function of some other variable a beside the cur- rent variable x. Then U is also a function of a, we now have dU whence d dU _du da dx da' Now, since differentiation with respect to independent vari- ables is commutative (Diff. Calc., Art. 381), this gives d dU du dx da da ' Hence, integrating with respect to x, we may write dU f du da in which the constant of integration C has a definite value be- cause we have fixed the lower limit of the integral. To find this value we notice that when x=a in equation (i), U=o inde- pendently of the value of a. Therefore, for this value of x, U is a constant with respect to a, and its derivative assumes the value zero. It "follows that, putting x = a in equation (2), we find C=o. Hence the equation may be written in the form which expresses that an integral can be differentiated with respect to a quantity independent of the current variable and the limits by differentiating the expression under the integral sign. XIX.] DIFFERENTIATION OF A DEFINITE INTEGRAL 299 If the limits were also functions of a, the total derivative with respect to a would also contain the terms given in equa- tion (3) of the preceding article. 278. By means of this theorem, we can derive from the known value of a definite integral the values of a series of other integrals. For example, the first of the fundamental integrals, p. 8, gives, when n> i, 1 (i) i- 1 whence, by taking successive derivatives with respect to n, we find x 11 log x dx = - (logx') 2 dx=- and in general in which n + 1 is positive, and r is a positive integer. Integration under the Integral Sign. 279. Supposing, as in Art. 277, that u is a function of a as .veil as of x, the definite integral U= u dx 300 DEFINITE INTEGRALS. [Art. 279. is a function of a, and Uda may be integrated between any limits log 2 dd. . (i) 'o 2 2J J But it is easily shown that JT f* f 7 log sin d = '2U\ Jo J o hence, substituting in equation (i), we derive * f 2 f 2 7T log 2 w= log sin ^t//?= logcos/9^=- =1.089. ( 2 ) r * n XIX.J TRANSFORMATION BY CHANGE OF VARIABLE. 307 288. Transformation of a double integral may sometimes be used to evaluate a definite integral. For example, let r 00 k=\ e~* z dx; (i) J o then we have also ,00 k = e- v *dy. * o Since each of the expressions under the integral sign is inde- pendent of the variable contained in the other and of its limits, the product of the equations gives k 2 =\ e~* 2 dx- r rr p y f] *\) = I p (x -r y )/7-v rl^} t ' f~>\ c u*y u./v u/y , . \ ) Jo o o Regarding #, ;y and 2 as rectangular coordinates, this double integral represents the volume included between the planes of reference and the surface whose equation is Transforming to the polar coordinates r and 6, where x = r cos 6, y = r sin 6, the same volume is represented by n f- j-30 2 = e~ r? rdrdd* (3) J o ' o * The analytical transformation of the double element dx dy into r dr dO is given in Art. 142. The integral in equation (3) represents the limiting value when the integration extends over a quadrant of a circle, and the radius is then made infinite; moreover this limit is found to be finite. The integral in equa- tion (2) represents the limiting value when the integration extends over a rectangle whose sides are made infinite. It is clear that these limiting values must be equal. 308 DEFINITE INTEGRALS. [Art. 288. and integrating, we have therefore 289. A double integral whose value is k 2 may be formed in a different way, leading to another evaluation of k. Putting x=az in equation (i) of the preceding article, we have f 00 + QQ k=\ e~ a * z *adz=\ e~ a * xZ adx. (i) v ' o * o Again, taking a as the current variable, we may write k=r e -*da J o If the element of this last integral be multiplied by the con- stant k, the value of the integral will be multiplied by k; hence, using the value of k given in equation (i), we find M.rr -*-<* ~LJ O Reversing the order of integration, we have -- = - X 2 +I 4' XIX.] TRANSFORMATION BY CHANGE OF VARIABLE. 309 hence, as before, k = ^7i. Also, by equation (i), k f k -^; aa J ^ 2 hence, comparing with equation (2), dw du -r=2U or =2da. da u Integrating, we have logu=-2a+c or u=Ae~ 2a , ... (3) where A is a constant independent of a. To determine its value; we notice that, when a=o, u becomes the integral whose value 310 DEFINITE INTEGRALS. [Art. 290. is found in Art. 288; therefore, putting a=o in equation (3), we have A = \^K. Hence u=\ e ' **' dx = -^ . Jo 2 Also, by equation (2), I f 3 .Jo X 20 Substitution of a Complex Value for a Constant. 291. If a complex value is given to one of the constants in an integral of known value, the integral becomes a complex quantity, and we may assume that the real and imaginary parts are represented by the real and imaginary parts of the known value. For example, if in the integral f e ax \e ax dx = a we put a=*m+in (where m is positive), and apply the limits o and oo , we have gtf -i* e~ mx (cosnx+isin nx}"}" ^- r- -m+^n }"}" , _J or i m + in nx _j_ i sn m in m 2 + n 2 ' Equating separately the real and :maginary parts, we have the two results, !"* m f 00 m n ~m 2 +n 2 ' SmHX ~m 2 +n 2 ' XIX.] COMPLEX VALUE GIVEN TO A CONSTANT. 311 which we have previously derived by the method of parts, see Art. 281. Again, in the equation Jo aa' Art. 289, put a 2 =ic 2 ; whence a = <:(i/|+Vi)> an d we find f/ 9 9 1 2W I/7T I i/7T (cos c 2 x 2 i sm c 2 x 2 )dx = : = (i i). J C^2 I +1 2C|/2 Hence ,00 / .00 . f 9 9 1 fa J f ' 9 9 J fa co'$>c*x 2 dx= and .0 si J w/v . 2Cf/2 and, if we put y=c 2 x^, Examples XIX. 1. Derive a series of integrals by successive differentiation of the r r 11 \ definite integral e-*doc. \ xe~ ax = . Jo J o & 2. From the fundamental formula (&') (p. 9) derive dx t and thence derive a series of integrals by differentiation with refer- ence to a. f 00 dx it 1.3... (2^3) _ i 312 DEFINITE INTEGRALS. [Ex. XIX. 3. Derive a series of integrals by differentiating the integral used in Ex. 2 with reference to /?. X 2n-2d x - 1.3.5 ... ( 2 M- 3 ) 4. Derive an integral from that employed in Exs. 2 and 3 by differ- entiating twice with respect to /? and once with respect to a. 5. Derive an integral from the result of Ex. II., 67, by differentia- tion. f _ dx_ Jo (x 2 +b 2 ) 6. Derive an integral by a second differentiation with respect to m of equation (i), Art. 281; also, by means of Ex. XII., 18, Diff. Calc., p. in, find the result of r differentiations. f a _ _ . , 2 2 x 2 e mx sin nx dx= J \ r\ f m~] in nxdx= -- sin (r+i)cot~ J . 7. Derive an integral by differentiating equation (i), Art. 281, with respect to n. f" 3 m 2 n 2 I *v*> rVIX r+r\c* 11 -\- - _ _ Jc OCC ' COS HOC ^4 second differentiation gives again the result of Ex. 6. 8. From the definite integral rf\ m e~ mx cosnxdx= m 2 +n 2 derive the result of Ex. 7 by differentiation with respect to m, also an integral by a second differentiation. 2m(m 2 3 2 ) f o ^ J - j * cos nx ax= XIX.] EXAMPLES. The corresponding general integral, which is x r e~ mx cos nx dx= : -r- cos (r+ 1) cot" 1 J L n J (w 2 + n 2 ) be derived directly from Ex. 6 by differentiation with respect to n. 9. Derive an integral by integrating 2 2 = . "J o a + # 20 f 00 F P q~]dx K . p tan" 1 - tan" 1 = log. J L x x _] x 2 q 10. Derive a definite integral by integrating I" 00 n a m% cin -M-V //'y= 3111 .* U.^- J m^ + n* with respect to w. (cos ax cos fct)d#= log 75 s. 1T *> WJL* I /T* Jo-* * '" I <* 11. Derive an integral by integrating r m a 1tlX f*r\G. WV /7^v 1_>J3 ^/ U..V Jo m^+n 2 with respect to w. rg-a* e bx j J2^. W 2 cos nx dx= log 5. J ^ 2 & a^ + n 2 Each of the integrals, for which Exs. 10 and n gwe the differences, of values corresponding to different constants, is separately infinite. The two results together give the more general difference formula e cos ax e ^LJ^ = $ j og j - m 2 + a 2 ' 12. Derive an integral by integration from the result of Ex. II., 67. p(g+b) q(p+bY fi r * a~] dx n tan-'^-tan" 1 ^- 2 , L 2 = -l2 J rx;L x xjx 2 + b 2 2b 2 314 DEFINITE INTEGRALS. [Ex. XIX. 13. Evaluate the integral 4 log (i + tan (f)), using the theorem of Art. 97, J n log 2 f ""^ log x dx 14. 9^ 9-9-. J O \^ "1" ^ J 8 log a tan x * ax TT I5 'J, f . J 16. tan- 1 . _ a ^ 4 +a 4 i6a 2 oo r~ 2 "1 18. log i + ^ hog x dx. Tra(loga-i). Jo L. x ~J 19. f 1 ^ Jo I 20. 21. Prove that x a ~ I cos (b log #)d#= -27rr 2 and ^ ~ x sin (6 log Jo d -rtr" J f u= J 22. Evaluate u= e~ a * x * cos 2rxdx. du l ,/ r _ Integrate -j- o^ ^arfo. u= e 1 T, Ai - f a cos bx , 23. Futtmg M= g2 2 a#, prove by a double integration by XIX.] EXAMPLES. 315 d 2 u parts that b 2 u= -5, which is satisfied by u=Ae ab +Be~ ab . Thence da^ show that, when a and b are positive, cos bx , TZ , -dx=e~ ab . 24. Derive integrals by differentiation and integration of the result of Ex. 23; and thence deduce the equation of Art. 283. p* sin ft* * f sinftag dx= ^ ll _ e - ab] J a 2 +*? d - 2 6 ) oX (a 2 + X 2 r* 2 a* [I . rcos bx dx - 25. Evaluate the integral , 2+x 2\2- ~ s e ~ XX. Infinite Values of the Function under the Integral Sign. 292. We have seen in Art. 82 that when f(x] is a real and finite one-valued function for all values of x between and including the values a and b, the integral f* f(x) dx J a has a real, finite and definite value. In fact, under these circum- stances, the graph of the indefinite integral, that is the curve y= 31 6 DEFINITE INTEGRALS. [Art. 292. (see Art. 85 et seq.} is, for this range of values of x, determined by the fact that it passes through the point (a, o), and that its gradient is given at every point by the equation -==. Now when the value of f(x] is infinite at the limit the integral itself may increase without limit: but not necessarily so, as illus- trated by the graphs in Figs. 5 and 6, pp. 112 and 113, where in each case the kn wn value of the indefinite integral shows that it is finite at the critical points where f(x] is infinite. Thus we can, for example, write f 1 dx x although f(x) is infinite at the upper limit. 293. When the definite integral is regarded as the limit of a sum (see Art. 99) an integral of this kind is generally characterized by the fact that the extreme elements of the sum vanish when we pass to 'the limit and the number of elements becomes infinite. Consider, for example, the integral n \ log tan (j> d(f>, J o in which log tan is infinite at the lower limit. Here the first element of the sum of which the integral is the limit is 4(f> log tan J<. Passing to the limit when J< vanishes, and writing z for J<, this element takrs the indete minate form z log tan z] z=0 , the value of which is found on evaluation to be zero. Thus the given in- tegral is the sum of an infinite number of vanishing elements and admits of a finite value just as in the ordinary case. Com- pare the integral evaluated in Art. 287. XX] CAUCHY'S GENERAL AND PRINCIPAL VALUES. 317 Cauchy s General and Principal Values, 294. In general, if f(x) is infinite only at the upper limit b, the integral may be regarded as the limiting value when e is diminished without limit of rb e /(*) dx a (where, supposing b>a, e is a small positive quantity), and this may have a finite or an infinite value. In like manner, when f(x) is finite for a and b and for all intermediate values except the single value c, for which it is infinite, Cauchy regarded the integral as the limit of fc-iie rb /(#) dx + f(x)dx, J a ' c+ve when e decreases without limit, // and v being positive numbers. If both parts of this expression have finite limits, the integral h;;s a finite value. If both parts become infinite with the same algebraic sign, the integral is infinite; but, if they become infinite with opposite signs, the re ult takes the indeterminate form oo oo , and may have a finite value. In this last ca e, the limiting value r, generally found to depend upon the ratio fjt : v. This was called by Cauchy the general value of the integral; while the special value assumed when n = v he called the principal value of the integral. 295. For example, if a and b stand for positive quantities, [ b dx f(x) in the integral is infinite for the single intermediate J a X re [""'dx f b dx ue b + =log +log J _ x } vf x & a & ve va ue x=o. Here 318 DEFINITE INTEGRALS. [Art. 295, which takes the form +00 when e=o; but, by algebraic reduction, the expression becomes b a log - + log-, which is accordingly the general value of the integral, and putting jj. = v, we have for the principal value log b log a. Integrals with Infinite Limits. 296. An integral with an infinite upper limit is the limit of an integral of the form AI I /f'vA /7'v* /T \ 7 -/W ax > W Ja w r hen x is increased without limit. In order that it may have a finite and definite value, the graph of the indefinite integral, represented by equation (i), must have an asymptote parallel to the axis of x. When this is the case, it is necessary that, if f(x) (which is the gradient or value of tan in the curve) approaches to a definite * limit, that limit shall be zero. 297. On the other hand, /(#) may approach zero as a limit when #=oo, and yet the integral may increase without limit. For example, in the integral 'dx y=' * If (x) does not approach a definite limit, the integral may remain finite as x increases without limit and yet not have a definite limiting value. For example. I* sin x has no definite value when x = o, and sin x dx approaches no definite J o limit as x increases, but its value must lie between o and 2. XX.] INTEGRALS WITH INFINITE LIMITS, 319 f(x}=o when x=, but the integral (whose value is log x) be- comes infinite. The infinite branch of the graph in this case tends to parallelism to the axis of x, but the branch is parabolic, and y increases without 1 mit. When the integral is regarded as a sum, the firs case is analo- gous to an infinite series of terms decreasing without limit and having a finite sum, and this last case is analogou to a series (like that given in Art. 180, Diff. Calc., p. 176) in which the terms decrease without limit, but the sum nevertheless has no limit. 298. When f(x] is real and finite for all real values of x, and approaches zero both for x= and x= oo, the graph of the integral may approach an asymptote at each end. In this case the integral C /(#) dx, has a finite value, as illustrated by Fig. 7, p. 115. Again, when the indefinite integral becomes infinite both for positive and negative values, so that both parts of the definite integral become infinite, the integral may have a finite value, if these parts become infinite with opposite signs. In this case, the integral must be regarded as the limit of (iA J ~ f(x) dx, where /* and v are positive numbers and h increases without limit. This limit, like that considered in Art. 294, is in general dependent upon the ratio // : v, and, by assuming {i=v, we can obtain a "principal value."* * In this case, if the negative branch of the graph were folded over on the axis of y, the principal value would be the limit of the vertical distance between the two branches. 320 DEFINITE INTEGRALS. [Art. 299. Integrals of Certain Rational Fractions. 299. Let us take, for example, the integral corresponding to a pair of imaginary roots cc//? in the decomposition of a rational fraction (Art. 18). It will be convenient to put the quadratic fraction in the form (i) (x-a) 2 +p 2 because this is the sum of the linear partial fractions A+iB A-iB xa+ifl xaijf Thus we have to consider the integral dx. TC ay -tp" The indefinite integral is A log[(*-a) 2 +/? 2 ]+25tan-'^^ The second term is finite at each limit, and the value when the limits are applied is 2B-rt But the first term is infinite at each limit, hence its value is the limit, when h is infinite, of * 2 A(x -a) ~A log 7-: ^50 =2A log . & 22 3 . N2 , ,3, J _,;,(*- a) 2 +/2 2 Thus h general value of the integral is and the principal value is 2 ETC. XX.] INTEGRALS OF CERTAIN RATIONAL FRACTIONS. 321 300. Let us suppose that, in the integral *) -?dx> I (x) is a rational integral function of which x 2n is the highest term, while f(x) is a polynomial of lower degree; also that (x) =o has no r, al roots. The fraction can be decomposed into n quad- ratic fractions corresponding to the n pairs of imaginary roots. Thus f(x) ^ 2 A l (x-a l ')+2Bip l 2A n (x-a n ) + 2 B ni 8 n (x) (x-atf+p? (x-ajt+pj Hence by equation (3) above we have Thus, in general, the integral is infinite at both limits and is indeterminate in value. Let us now further suppose that /"(:*;) is al least two units lower in degree than (x). Then, equating the coefficients of x 2n ~ l in the result of clearing equation (i) of fractions, we have c = 2lA. Therefore, in this case, we have the indefinite integral being now finite at each limit. 301. Let us apply this result to evaluate where m p p in which p is positive, and q must lie between i and p i. This equation in fact includes equation (i), which is therefore not restricted to integral values of m and n. As a particular case, we may put q=o, provided p>i; thus, when p>i, f dx TT it > = 7 cosec ^ (s) XX.] FRULLANI'S INTEGRAL. 325 Frullani's Integral. 303. Suppose that an integral can be put in the form 'dx, x in which c is positive, and that while 'x varies from c to infinity (x) does not become infinite, but retains he same sign (say the positive) and approaches a finite limit not zero; then, although the function under the integral sign approaches zero, the value of the integral will be infinite when x = oo . For, let A be the least value of $(x] for the entire range of values of x, then because every element of the given integral is not less than the corresponding element of the integral in the seccnd member. But, when x increases without limit, the value of the integral in the second member, whi.h is A (log x log c), becomes infinite; hence, a fortiori, the value of the given integral is infinite. 304. But the difference between two integrals of the form considered may be finite. Consider, for example, Frullani's Integral, r$(ax}-$(bx) U = ax, J o ^ in which a and b are positive, and (x] does not become infinite for any positive value of x. We notice, in the first place, that, supposing 0'(o) not to be infinite, zero is admissible as a lower limit, because when x = o the quantity under the in egral sign is found on evaluation to have the finite value (a 6)$'(o). 326 DEFINITE INTEGRALS. [Art. 304. Employing the method of Art 284, we have f f f a f U=\ (j> f (ax)da dx = \ ' (ax}dx da. JoJb Jb Jo Now a or therefore f*V((^)=tan~ r x, 0(oo ) = $n and ^>(o)=o, therefore -dx = ^-\og^. x 2 Compare Ex. XIX., 9. 305. When (x) is a function which, although remaining finite, has no definite limiting value when x =00, equation (i) fails to determine the value of he in egral. Fo example, when <(#)= cos #, cos oo has no definite value. The following mode of investigating the in egral will, however, show that in these cases equaton (i) will hold true if for <(>) we substitute the mean value of <{>(x) over an infinite ange of values of x. For this purpose, we put -i: dz, an integral having a finite value when h is finite. Putting z = ax, we find j>(ax)-^(o) j dx (i) XX] FRULLANl'S INTEGRAL. 327 In like manner, h_ _ rb(bx)-(o) u I Jo dx. X Supposing a>b, this last equation can be written in the form h _ fad>(bx)d>(o) f vi ) vv J dx i J X J h X Equating the values of u in equations (i) and (2), we have Frullani's Integral is the limit when h=cc of the integral in the first member. As to that in the second member, let us first suppose that the mean value of (x) over the infinite range of values of x is zero. Since the greatest value of the factor a under the integral sign is j- (which is its value at the lower limit), a h h Now the mean value of (x) over the range of values to r is a b b a a hence, if the limiting value of M is zero, that of the integral in 328 DEFINITE INTEGRALS. [Art. 305. the second member of the inequality (4) is zero; and, a fortiori, that of the first member is zero. Substituting in equation (3), and then making h infinite, the result is the same as if 0(o) in equation (i), Art. 304, were equal to zero. For example, putting <(#)= cos x, the limiting value of M is zero, and we have fcos ax cos bx b - ax = log . J x 6 a Finally, if the mean value of (x) for an infinite range of values of x is the finite quartity M, we may put in which (x) is a function having zero for its mean value. Sub- stituting in equation (3), and then making h infinite, we derive -, ... (4) where M takes the place of (oo)=o, and also when its mean value is zero, the two parts of Frullani's Integral are not each infinite at the upper limit (as they were under the conditions named in Art. 303). In like manner, if 0(o)=o, they are not infinite at the lower limit. Thus, if both these conditions hold, both zero and infinity are admissible limits f (j)(cix)dx of the single integral - . But, in this case, he second j x member of equation (4) vanishes, and we have I dx I Jo X J X XX] FRULLANI'S INTEGRAL. 329 It follows that an integral of this form has a value independent of a, so long as a i, positive, which has been assumed in our demonstration. This might also have been inferred from the result of substituting z for ax. But even in this last process we cannot infer that the value is the same for positive and negative values of a, because the integral ceases to have a meaning when a by gradual change of value passes through the value zero. An integral of this form may therefore be a discontinuous func- tion of a. For example, we have seen in Art. 283 that fsin ax n K - dx = or Jo x 2 2 According as a is positive or negat ve. Integrals obtained by Expansion. 307. We have seen in Art. 78 that, when .he integrat'on of j, c ( )dx cannot be effected in finite terms, the integration, term by term, of its development in powers of x will give the like de- velopment of he integral. Thus, if we have, by development of the logarithm, f / x x* \ 0(*) = J o ^i +-+j+... )dx; whence, integrating term by term, 2 X 3 3(4 +2+- 2 + ........ (2) 330 DEFINITE INTEGRALS. [Art. 307. This defines (j>(x) by a series which is convergent when x< i. It ie also convergent when x=i, its value (see DifT. Calc., Art. 238) then being \x 2 . Thus ,, . [\ i dx r? d>(l)=\ log- = ...... (?) v ' J 6 i -x x 6 308. In like manner, the development of the quantit under the integral sign in any other form of infinite series may admit of integration term by term. For example, if we transform the integral considered in the preceding article by putting x = iz, we have * and, expanding the factor (i-z)" 1 in a series convergent when z is between o and i, (i -z) = I log z(i +z +z 2 +. . . )(fe. ... (4) J i Each term under the integral sign is of the form z n log z dz, and integrating by parts we have z M log J i z M+I logz z w+1 i zdz= n + i Giving to n the values o, i, 2, etc., and substituting in equa- tion (4), z 2 z 3 n r z 2 z 3 - +-+- XX.] INTEGRALS OBTAINED BY EXPANSION. 331 The coefficient of Iog2 is the series for log (i z), hence the equation may be written <(i - z) = - log z log (i - z) - c/>(z) + <(i ). Putting x for z, we have the relation ... (5) as a property of this function. As a particular result, putting x = %, we find ft I dx_X* (lQg2) 2 I X X 12 2 309. In the case of a definite integral, the result of expansion in a series of integrable form is in general a numerical series, by means of which, if convergent, the value may be computed; while in some cases the sum may be already known. For ex- ample, to find r n oc sin oc doc U= L~< where p is a positive integer. Expanding (i+cos 2 ^)" 1 , we have ft U=\ xsin x(i cos 2t> x+cos 4p x cos 6p x+. . . }dx. J o Integrating the typical term by parts, r #cos 2rH " I :xn' r T f" x cos^x smxdx= + r - cos sr f +I x dx J 2rp + i J 2 p + iJ It ~2rp+i* 332 DEFINITE INTEGRALS [Art. 309. since the final integral vanishes. Hence, giving to r the values o, i, 2, etc., we hav f*xs'mxdx /iii \ J i+cos 2 ^~ \ 2^ + 1^4^ + 1 6/> + i / When p=i, the series is the result of putting x = i in Gregory's series for tan" 1 x; thus, as a particular case, ("X sin x dx / i i i \ n 2 = - i +- + ... = -. Joi+cos 2 * \ 3 5 7 '/ 4 Series in Sines and Cosines of Multiple Angles. 310. When A is numerically greater than B, the expression A +B cos # can be reduced to the form m(i +20. cos x+a 2 ), where a is real and may be taken less than unity. (See Art. 313.) Now the expression i +20 cos x +a 2 admits of conjugate imaginary factors, thus i +20, cos x+a 2 = (i +a cos x+aistn. x)(i +a cos x ai sin x), or, using the exponential notation (Diff. Calc., Art. 222), cos x+a 2 = (i +ae ix )(i The sums and differences of the expansions of like functions of these imaginary factors may, by means of the equations 2 cos x = e ix + e~ ix and 21 sin x = e ix e~ ix , be expressed in terms of the sines and cosines of the multiples of the angle x, thus giving rise to a variety of developments in mul- tiple angles. Among the simplest are the series deduced below. XX.] MULTIPLE- ANGLE SERIES. 333 311. From the expansions = i - ae ix +a 2 e 2ix - i +ae * x we have by addition 2 +20, cosx 121 = 2|i -a cos x+a 2 cos 2X -a 3 cos T.X+. . . .1; i +2a cos whence, subtracting unity to simplify the numerator, i -a - 2 = ! 2d COS# + 2# 2 COS 2X 2 a 3 COS 1.X + . . . Again, the difference of equations (i) and (2) gives, after dividing by 2ai, sin x = sm x a sin 2X+a 2 sm T.X a 3 sin 4^+. . . (4) i +20, cos x+a* These series are convergent when ai, in the integral of equation (i) or in others involving the expression i +20 cos #+a 2 , a development in con- verging series is readily obtained. For, denoting such a value by a', we have i +20! cos x+a' 2 = a' 2 (i +20, cos x+a 2 ), * See Fig. 61, p. 35 3< for the graph of such a function. 336 DEFINITE INTEGRALS. [Art. 313. in which a stands for the reciprocal of a' and is therefore less than unity. In this way, equation (i) becomes dx i F [ sin x sin 2X sin 3^ n J' and equations (2) and (3) become f* dx 7T J i +2d' cosx+a' 2 a' 2 i and p dx i PTT 1 . 72 = ~72 -~ 2Cot ~ la ' > J i+2a cosx+a a i L2 J the values of both integrals being essentially positive. When the more general expression A +B cos x (in which A>B numerically) occurs, we make, as proposed in Art. 310, A +B cos x = m(i +20. cos x+a 2 ) by putting A=m(i+a 2 ), B = . Eliminating m and solving the quadratic for a, we have recip- rocal roots which we may write in the form A- whence also A+ m=- * In these equations, A is regarded as positive (which does not affect the gener- ality of the results), and the radical is by hypothesis real. Thus a denotes that root which is numerically less than unity, and its sign is that of B. XX.] INTEGRALS IN MULTIPLE-ANGLE SERIES. 337 For example, we have, from equation (i), Art. 312, dx i T.X- on substituting the values of m and a above. This integral has already been expressed in finite form, thus, see formula G, p. 41 : r* dx 2 r IA-B "TTTT tan ,A+Bcosx U The inverse tangent in this expression is therefore a quasi- periodic function, and substituting the values of A and B in terms of a and b we find r i a x~\ x r a 2 . a 3 . "| tan" 1 tan = \asmx sin 2X+ sin T>X . . . . [_i+a 2_\ 2 [_ 2 3 J 314. From equation (5), Art. 311, we have by integration f* r~ a 2 a 3 "] Iog(i + 2acos^+a 2 )^ = 2 asin^ ^sin 2^+-^ sin 3^-. . . (i) when ai in the integral of equation (i) belongs to the general case, the corresponding value of m being a 2 ; thus, when x=n, we have f* log( Jo cos x+ax = 2xoga or o, according as a is greater or less than unity. 315. The series in Art. 311 lead to the immediate evalua- tion of definite integrals of a certain form. For example, if f* U= cos rx log(i + 2a cos x +a 2 )dx, J o where a+a3)t/#' 3 8. Show that, when >i, 7T 7T = sec . 1 2H 2H f dx g. Evaluate log(i + 2a cos#+o 2 ) 2 2 when c c or 6 < c. I"" # sin x dx 12. Evaluate ; ; 5 when ci. Jo I cos TC . I 7T , a log ; log . a i a a a i f * cos rx dx IT.. Evaluate - ; 5- where r is a positive integer and J i 20 cos x-\-a i - 14. Show that, when ai, r being J i 20 cos #+ or Tta 1 a positive integer. XX.] EXAMPLES. 341 17. Show that, when a< i, f x sin x dx 7ie~ c J c z +x 2 1 + 20 cosx+a 2 2(1 + ae c )' 18. Derive the expansion a sin x I , a 2 , a 3 , \ dx=7i(a-\ O+-Q+ . . . I \ 2 3 / i + a cos when a< i, and verify the result when a= i. 19. Derive the expansion, when a< i, [* ' I Cfl d^ ( rvlog(i 2acosx+a 2 ')dx=4(a-\ ^-\ +- Jo \ 3 5 .7 and thence deduce 6 log sin 6 dd= , ^ , i > 8 2 v 3 3 s 3 r 20. Evaluate the integral ^-dx, where Jo -""I 1 x ) 7i cot pn. Expand and see Diff. Calc., Ex. XX I II., 20. XXI. Functions expressed in Multiple-angle Series. 316. The infinite series involving cosines of the multiples of an angle, namely, C+Ai cos 0+A 2 cos 26 +A S cos 3#+ . . . , is convergent for all values of 6, provided the absolute values of the coefficients -form a convergent series. The series then 342 DEFINITE INTEGRALS. [Art. 316. expresses an even periodic function of 6. Under similar circum- stances BI sin +B 2 sin 26 +B 3 sin 3^ + ... expresses an uneven periodic function of 6. In either case, the period is 27r, and the values of the function corresponding to values of 6 between o and x determine the values of the function for all values of 6. So also, if we put 6=nx/l, the series express even and uneven periodic functions of x, in which x = l corresponds to $ = TT, so that the values of the function for values of x within the half- period from o to / determine all the values of the function. It is frequently desirable, especially in the physical applica- tions of mathematics, to express a given function f(x) in one of these forms, and it was found by Fourier that, notwithstanding the periodic character of the series, it is possible so to determine the coefficients in either series that the sum of the series shall for a range of values of x from o to I represent any given function f(x). Fourier s Series. 317. Let us now assume that it is possible to put xx TTX TTX + . . . , (l) for all values of x between o and /; * the values of the coefficients can then be determined as follows : * The demonstrations of Poisson and Lagrange establish the possibility of equation (i) in a direct manner. In that of Lagrange, the series is at first assumed to consist of a limited number of terms. The coefficients (n in number) are then so determined that the equation is satisfied for n equidistant values of x, sub- dividing the interval between o and 1. Thus the graph of the series is made to coincide with that of f(x) at n points. Afterward n is made infinite, so that the graphs coincide for an unlimited number of points. XXL] FOURIER'S SERIES. 343 We have seen in Art. 33, that if m and n are positive integers, f* cos md cos nd dd = o or JTT, Jo according as m and n are unequal or equal. It follows that, putting xx/l = d, if equation (i) be multiplied by cos mO d6, and then the integral of each member be taken between the limits o and Tt for 6, that is between o and / for x, every term in the second member will disappear from the result except that in which n = m. Thus we have, when n > o, f*f\C M /7 'V* A V^Uo /* , U/Jv -il ft / 2 ry r TT^C *? r^ / Lu \ n = -r- /(^) cos w -rdx = f( )cosn6dd. /I / 7T I ~ \ 7T / "Jo ' v jo\'"/ (2) Again, multiplying equation (i) by <#? and integrating, r/ whence T r' , f(v\tiv * ^'>^ W"* 13; Using a separate symbol for the current variable in the integrals, the result may be expressed thus: j fl 2 n ~' X ' TtX f ' TT^ /(*) T- f(v)dv +-T- 2 cos w -y /(v) cos -r-rfv, (4) ^ Jo M=I * Jo * which is true for all values of x between o and /. * The absolute term C is in form the same as \A where A is defined by equa- tion (2), but the case n = o sometimes fails to be included in the general evalu- ation of the definite integral. 344 DEFINITE INTEGRALS. [Art. 318. 318. For example, let us put f(x)=x. Substituting in equation (2), we have 2/ 2/ d> cos r Integrating by parts, we find ,M7T -|* fWt <.cos (/> sin d(f> = Jo -J o J o mi i =o or 2, according as n is even or odd. Therefore A n =o when w is even, 4/ and 4= -2~2 wnen w i s dd. Again, equation (3) gives C = \l. Hence, when o < x < I, i THV i +~ 2 COS 3 y +~ 2 COS 5 + . . . (i) This result holds also for the extreme values o and /; as is readily verified by means of the equation III 7T 2 I + ~5 +~o +~5+ . . . =-5-, 3 2 5 2 7 2 8' proved in Diff. Calc., Art. 238. The complete graph of the second member of equation (i) ^incides with the line y = x for the interval between x = o and x = l, Since the cosine-series is an even /^\ function, the portion between x=o \ and x=l is symmetrical to this ' i with respect to the axis of y, and the rest of the graph is a periodic repeti- tion of these parts forming a broken line as represented in Fig. 57. XXL] THE SERIES IN MULTIPLE SINES. 345 The Series in Multiple Sines. 319. In like manner, we may assume TtX KX TCX f(x)=Bi sm -y +B 2 sin 2 -j +B 3 sin 3 -y +. . . (i) Then r since, by Art. 33, r sin mO sin nd dO = o or JTT, J o according as m and n are unequal or equal, multiplying equa- tion (i) by sin mO dO and integrating between o and TT, we find 2 C l 7IX 2 C x flO\ Bn = -r\ f(%) sin nrdx= flhmntidO; , (2) * Jo * ^JoV"/ and the result of substitution in equation (i) may be written 2 M=0 TIX r l -v /(^)=y^sinwy f(v)sw.n-j-dv. ... (3) n= i . The graph of the sine-series thus determined coincides with that of/ I Q sin nO dd = 9 9 1 si T?J o ^r J o Integrating by parts, f we ~~\ nit fn d> sin d = cos +1 Jo Jo Jo whence cos

when./(:v) is neither an even nor an uneven function, .a development in multiple angles applicable from x= I to x = l be required, it will be necessary to separate f(x) into its even and uneven parts, for development, the one in cosines and the other 348 DEFINITE INTEGRALS. [Art. 322. in sines. Denoting these by (x) and (x) respectively, as in Diff. Calc., Art. 216, they are +/(-*)! and ft*) so that/(#) = (j>(x) + 4>(x). Then, because the integral of an even function between the limits / and / is double the integral between o and /, the coeffi- cients for (x) and ^(x) respectively, Arts. 317 and 319, may be written in the form I f l If 1 7TV C=r\ 4>(v)dv t A n =-\ <(V) cos n-rdv. 2l J -I I* } -I l< I f l B n = -j\ I J _; TtV sin nrdv. But, because the integrals between I and / of the uneven functions TtV TlV (b(v) cos nr and (v) sin n-r vanish, the coefficients may also / I be written if* if' nv C= ^l] _ Arfdv, A = - J ^ f(v) cos n-jdv, I f* 7CV Bn=j) /(v) sin n-jdv. Hence =~\ _f(v}dv j H= 00 +7 2 ( cos n b 7tx[ l TtV 7tx{ 1 , 7tV\ r f(v) cos n +sm n f(v) sin n )dv. I J i t I j i I / XXI.J DEVELOPMENT IN SINES AND COSINES 349 323. For example, to obtain a development of e mx true for all values of x between / and /, we have i e mv dv = i t l C = -T\ 2.1} -i fi rw wIO IP 7CU I f" A n = r \ e mv cos n-rdv = -\ e * cos no d6,, I J -I i ^J-n if' xv i r E n = - T \ e mv sin n dv =-\ e* sin nd dd. I J -I I ~J -x To evaluate these integrals, we have, by Art. 63, f e m9 e md cos nd dd = 5 - ~(m cos nd + n sin nd), j m 2 +w 2V r e** e me sin nd dd = z - z(m sin nd n cos n6) ; W 2 +W 2V whence . e md cos e w9 smnOd6=-(-i Y j _ K ' - e Hence, putting ml/it for m, and substituting, /*^J?//>W/ /? ^ W/ \ 4/J, /X>WW S>~" Wll\ . ffil I c t> i f i/n. i e e / T \n_ / _\ 350 DEFINITE INTEGRALS. [Art. 323. Therefore XX XX cos cos 2-7- I / / \ 2 m 2 l 2 m 2 l 2 +x sm 2 sin 2 3 sm The first line is of course the development of cosh mx, and the second that of sinh mx (Biff. Calc., Art. 217). Thus, taking 2 m sinh mx I i * ( i ) M cos w#\ coshwx = - -I z + 2 s ), /r \2W^ j m 2 +n 2 J 2 sinh mx * ( i )" w sin WA; sinh mx = - m 2 +n 2 Discontinuity of the Fourier Sine-series. 324. We have seen in Art. 319 that the Fourier sine-series is in -general a discontinuous function, presenting sudden changes of value as illustrated by the graphs Figs. 58 and 59. The mode in which an ordinary sine-series, which represents a continuous periodic function, takes on this character is well illustrated by equation (7), Art. 311, namely, a sin x a 2 a 3 sm 2X-\ sin *x . . . , d) i +a cos JP 23 XXL] DISCONTINUITY OF THE FOURIER SINE-SERIES. 351 in which ai, the series is divergent and equation (i) ceases to have any meaning. In the intermediate case, when a = i, the series remains con- vergent for all values of x, and becomes, in fact, the Fourier sine-series for $x, when we take l = it in equation (i), Art. 320. Accordingly, the function y now takes the form tan" 1 tan %x, of which one value is \x. But the series always represents the primary value of the tan" 1 (which is x only when x is be- tween -it and TT); thus it becomes a discontinuous function, increasing with x, but dropping the value it suddenly whenever x passes through an odd multiple of it. 325. The development of the same function y in a series convergent when a>i is readily obtained from Art. 311, in which i +ae ix 2iy = log ; --. & i +ae~ tx For this may be written i _ . a i . a in which, if a>i, the second term can be developed as in equa- tion (6). Hence, dividing by 21, we have, when a>i, 352 DEFINITE INTEGRALS. [Art. 325. a sin x /sin x sin 2X sin T.X i -fa cos #_ \ a \ -... . (2) / The series in this equation is convergent, and is an ordinary periodic function, so that y is, in this case, a quasi-periodic func- tion. If we now make a=i in equation (2), the value of y becomes x diminished by the Fourier series for $x, and is again equal to $x when x is between it and TT; but as x increases, this ex- pression for y receives a sudden increase of TT in value when x passes through an odd multiple of TT. Geometrical Illustration. 326. The function y, developed in equations (i) and (2) above, admits of a simple geometrical construction. For, if, in Fig. 60, we take BA =i and AP.= a, and denote the arcual measure of the angle PAC by x, that of the angle PBC will represent y, where a sin x i +acosx As x increases indefinitely, P describes a circle; and, supposing i, the point B falls within the circle, and the value of y is represented by equation (2). The bar BP now makes complete revolutions, and when a is near to unity, swings very rapidly through a half-revolution in the posi- tive direction. Hence, when the limiting case is reached by diminishing a to unity, the value of y increases uniformly at half the rate of x, but receives a sudden increase of TT in value whenever x passes through an odd multiple of n. The graphs of the function /for two recip- rocal values of a, near to unity, are given in Fig. 61. The zigzag lines, bisected respectively by y = o and y = x, are the limiting forms to which the graph approaches as a approaches unity from the one side or the other. FIG. 61. Differentiation of Multiple- angle Series. 328. The differentiation of a sine-series or a cosine-series for f(x) gives a cosine-series or a sine-series for f(x). When the series is an ordinary continuous function, and therefore either 354 DEFINITE INTEGRALS. [Art. 328. an uneven or an even one, this is in accordance with the fact that the derivative of an uneven function is an even one, and vice versa. Moreover, the sine-series has the property of the un- even function that f(o) = o. But when the series is the Fourier 'sine-series for an arbitrary function /(#), we have not generally f(o)=o, although the sine- series as assumed in Art. 316 vanishes with x. This fact imposes discontinuity upon the series, /(o) being not the value of the series when x=o, but the limit to which the value of the series approaches when x approaches to zero from the positive side. 329. The derivative of the Fourier cosine-series for f(x) will in fact give the sine-series for f(x}, true between the same limits. But it is to be noticed that the derivative of the sine-series for f(x) cannot contain the term C, independent of x; therefore it cannot give the cosine-series for f(x) when the latter properly contains this term. The failure of differentiation in these cases is due to the fact that a divergent series is produced. For example, putting l=n in equation (i), Art. 320, # = 2 (sin x \ sin 2X+\ sin yx . . . ), . . . (i) of which the derivative is i =2(cos x cos 2X+CO5 $x + . . . ), which is inadmissible, because the series is divergent. 330. Let us now, in the general theorem, put l = n (which is the same thing as putting x in place of its multiple nx/l, and therefore involves no loss of generality). Then > in the cosine- series for/(#), Art. 317, the coefficient of cos nx is 2 f" A n =-\ f(x} cos nx dx. XXL] DERIVATIVES OF MULTIPLE-ANGLE SERIES. 355 Integrating by parts, this becomes 2 [/(*) sin nx if,,,,. , ~|* A n =-\ \ f (x) srn nx ax \ nl n nj J 2 r i = / f (x) sin nx dx= B n) Tlflj Q . ft where B' n stands for the coefficient of sin nx in the sine-series for f'(x) as found by the method of Art. 319. Thus B' n = nA n ; but this is precisely the coefficient of sin nx in the derivative of the cosine-series forf(x). Thus the derivative series is the same as the result of developing f'(x) in a Fourier sine-series by the general theorem. 331. On the other hand, the coefficient in the sine-series is 2 f* B n = \ f(x) sin nx dx, n J and, integrating by parts, 2 r-/(*) cos # which shows that we shall have v .4J, = .B w , 0^/3; ?w Cfl^e /"(o)=o and f(n)=o. In this case only, therefore, and not in general,* * The expression for B n above shows that the sine-series for f(x) can be sepa- rated into three parts as follows: f(x)=A'\ sin x-\-\Ai sin zx-\- $A 3 sin + [sin x+ i sin 2X-\- $ sin 7T 2 [sin * J sin 2X+ J sin 3^; . . . ]. 356 DEFINITE INTEGRALS. [Art. 331. will the sine-series admit of a convergent derivative. It is to be noticed that this is precisely the case in which the equation of f(x) to the sine-series is true for the limits inclusive, in which case the graph of the series is a continuous curve composed of arcs which meet each other with a common tangent. For example, X(TZ x) is such a function, and the corresponding sine-series (with a graph consisting of connected parabolic arcs) will be found to give a convergent cosine-series for the derivative T: 2X. The result is, in fact, equivalent to equation (i), Art. 318. Integration of Multiple-angle Series. 332. Conversely, if f(x} is a given function of which the development in a cosine-series contains no absolute term, that is, f* if C'=o, we shall have f'(x}dx=o, whence /(/:) =/(o). The o constant of integration can be so taken that the integral f(x) shall vanish when x=o, and we shall then have also/(-)=o. The sum of the series in the third line is, by equation (i), Art. 329, equal to \x, and that in the second line is the result of substituting n x for x in the same equation; hence the last two lines are equivalent to and we have ~ (i) Since C'= I f'(x) dx = ^^- , the derivative of this equation is i f* H?J /' w cos 2x+A 3 cos 3*+ . . . , as given directly by Fourier's method. Thus in equation (i) the part of the ? : ne-series which produces a divergent derivative has been already evaluated and the remainder admits of a convergent derivative. XXL] INTEGRATION OF MULTIPLE-ANGLE SERIES. 357 The sine-series corresponding to f(x) with the constant thus determined will, as shown above, have the property of actually approaching zero when x = o, and also when x = ~. In the case of any given value of f'(x), we obtain a function of the required kind by simply transposing ' C' in the develop- ment in cosines. For example, from equation (i), Art. 318, we have cos 3# + -^ cos 5#+ . . . I; . (i) whence, integrating, 8/. i i \ x z = IsmjcH 5 sin T.X+-^ sin zx+ . . . /, . (2) it\ 3 3 5 3 / xx- 3 5 J the constant of integration being zero, so that the first member vanishes when x = o, and it is found to vanish also when x = n. 333. In integrating a sine-series the direct determination of the constant is not so readily made, but by proceeding to the next integration we can determine both constants by the double condition employed above. For example, integrating the equation obtained above, and multiplying by 3, we have 3?r 24/ i i x 3 x 2 +C = (cos x-\ - A cos ix H -. cos 2 x\ 3 4 S and integrating again, -- 42 (sin ^+-= sin 3^+3= sin 5*+ . . .), ic \ 3 5 5 5 in which, the two conditions that the first member must vanish when x = o and also when x = x give C'=o and C = j7r 3 . 358 DEFINITE INTEGRALS. [Art. 333. Hence the successive integrals are T.7Z , 7T 3 3 _L_ x;2 _j 2 4 24 / ~7T\ i cos^+^a^ i H T cos ^ 5 4 ....), (3) 06 / f . i i \ X*-2XX*+a*X ' = -\ sin x -i ~ sin 33 ^ 3 "+-5 sin 5^; o + ...J. (4) 334. We have seen that the sine-series obtained directly for a given function is in general discontinuous, and represents the function for values between but exclusive of the limiting values of the variable. But the preceding articles show that the expression for the function found by integrating the derivative as a cosine- series gives its true value inclusive of the limits. The value of the sine-series thus found differs from the given function by a certain linear function. As an illustration, if we develop e mx in a sine-series for values of x between o and it (see the formulas of Art. 323), we have B n = I ^'" r sin w# Jjc = 2 2 Ji - ( - 1 } n e m *\ giving the development 2/i+e m * '. ie m * . i+e m * . \ Again, developing e mx in a cosine-series, we have i f * e m * i C=- e m xdx= _ 71 J W7T and ^4 = - e w *cos nxdx= 5-. ^\( i} n e m * i 1. XXL] INTEGRATION OF MULTIPLE- ANGLE SERIES. 359 giving the development e mx = e - I 2m 71 W+I 2 ' i e mn COS2X-\ > COS 3# + ) (2) /i/fiZ I **2 *J / \ / Integrating and multiplying by m, ...), X + I 1 """a""] 2 sin aH r~?~^ ^rsin 2#+ . . .), (3) Tt \W 2 +I 2 2(w 2 +2 2 ) / the constant of integration being determined by either of the conditions that the series vanishes when x=o and when x=n. 335. The coefficient of sin nx in the last series is Tt n(m 2 +n 2 ) ' and subtracting this from B n , the coefficient in equation (i), we have 2[l-(-l) e I 2+m2 . = 2_ rj_/ jNngWKl. 7tn(m* +n z ) mi therefore subtracting equation (3) from equation (i) we find e mn i 2 M = i ( i) M f Wjr . ^ + i= ^ sin nx. 7i 7i n n = i This equation is readily verified when we substitute in the first member the values of x and of unity in the form of sine-series, viz. # = 2(sin x Jsin 2^+^ sin 3^ . . . ), . . . (i) see Art. 329, and i = -[sin x+% sin 3^+^ sin $x+ . . . ], . . . (2) 71 see Art. 336. 360 DEFINITE INTEGRALS. [Art. 335. Thus the linear function in equation (3) forms, as it were, that part of the function e mx which requires for its development in sines a discontinuous series. When the complete graph of the second member of equation (3) is drawn, the linear function represents the straight, line which passes through the extremities of the arc of y = e mx with which the graph coincides between # = o and x = n. Series obtained by Transformation. 336. Denoting by x the angle whose multiples appear in the series (which was denoted by 6 in Art. 316), the limits of applicability of the series derived directly by the methods of Arts. 317 and 319 are x=o and X = TZ; but by transformation of such series, others with different limits may be derived. We notice in the first place, however, that the substitution of TIX for x does not change the limits. Its effect, moreover, is merely to change the sign of alternate terms, namely those containing odd multiples of x in the case of a cosine-series, and those con- taining even multiples in the case of a sine-series. It follows that the sum of the given series and this transformation, and likewise their difference, will contain either odd multiples only or even multiples only of x. For example, equation (i) of the preceding article gives TT # = 2(sin x + % sin 2*+^ sin 3*+ ...),. . . (3) and the sum is 7r = 4(sin x+% sin 3^+^ sin 53:+ . . . ), ... (4) which is equation (2) employed above. Again, the difference of' the series for x and for TI X is sin 4# + & sin 6x + ...),. . (5) which contains only even multiples of x. XXI.] SERIES OBTAINED BY TRANSFORMATION. 361 337. A series -containing even multiples only of x may be transformed by putting (j> = 2x; and then, since the limits for x are o and TT, those for (f> are o and 2x. For example, equation (5) above thus becomes TT < = 2(sin <+i sin 2(f>+^ sin which is identical with equation (3), and shows that that equation is true up to x = 2x. This might also have been inferred from the fact that equation (i) is true between the limits n and TT. If a multiple-angle series whose limits are o and TT be trans- formed by putting \K x for x, the limits for the new variable are \n and + |TT. In general; the transformed series would con- tain both sines and cosines; but, if odd multiples only occur, a sine-series will thus give rise to a cosine-series, and vice versa. For example, equation (4) above becomes 7T =cos x % cos 3#+i cos $x . . . , which is true for values of x between the limits ^?r, exclusive of both limits. Functions with Arbitrary Discontinuities. 338. The restriction of the validity of equations (4), Art. 317, and (3), Art. 319, to values of x between o and I presents itself as a natural consequence of the fact that the definite integrals in the second member depend for their values solely upon the values of the function for values of x between the limits of inte- gration. In fact, it is found that the equations are true when the values of the function for values of x between the limits are defined in 362 DEFINITE INTEGRALS. [Art. 338. L X any manner whatever, provided only there is no ambiguity about the values of the definite integrals. Thus f(x) may be defined by one functional expre-sion, say fi(x), from x = o to x = a, by an- other expression,/^), from x = a to x=b; and so on to x=L It is not even necessary that the values of. fi (a) and f^(a) shall be the same. FlG - 62 - In other words, the graph of y =f(x) may consist, as in Fig. 62, of any disconnected arcs, of which the projections upon the axis of x cover without overlapping the range from x = o to x = l. The values of the definite integrals are now found by separate integrations of the functions f\(x], fz(x) etc., each over its proper range. In particular, that occurring in the value of C, equation (3), Art. 317, namely = lj represents the total area between the axis of x, the arcs, and their ordinates, due regard being paid to algebraic sign. Thus the absolute term is the mean value of the ordinate between *=o and x=l. It is a noteworthy fact, which we here state without proof, that the value of either series corresponding to a value x=a at which discontinuity occurs is $\fi(a)+f 2 (a)}, that is, the arithmetical mean between the two values of the function. 339. As an illustration, let us construct a cosine-series in which fi(x)=x from *=o to x = %l, and/ 2 (^)=o for all values of x between \l and /. We shall now have in equations (3) and (2), Art. 317, C = f/, and TtX 2/ - 6 cos 2/ cos XXL] FUNCTIONS WITH ARBITRARY DISCONTINUITIES. 363 Integrating by parts, * nn nn_ cos d sin d(j> Jo -Jo J o 7t . nx mi = w sin Kos 1. 22 2 The successive values of this integral for n = i, 2, 3, etc., are x 37r 5?r -- 1, -2, --- 1, o, -- 1, -2, etc. 2 22 Hence, substituting in the general equation, we have / / / Ttx i xx i 7r# \ / = 0- +~ I cos T ~7 cos 3 T +7 COS 5 T ""'/ oTrxfr^ *5 ' 2l i TtX I XX I 7T# \ --5\>s y +- 2 cos 3 y +- 2 cos 5 y +. . .) If TtX I TlX I TTiC -- cos 2 --H cos 6 -+-; ,cos 10--+. . It is readily verified, by means of Gregory's series for \n and the series quoted in Art. 318, that x=o and x = l each give to the second member the value zero. Again, if we put x = %l, the second member becomes / // i i \ / and this is the arithmetical mean between /i(|/) = J/ an in accordance with the last paragraph of Art. 338. 364 DEFINITE INTEGRALS. [Art. 340. Formula Involving both Sines and Cosines. 34-0. If, in the half -sum of the formulae of Arts. 317 and 319, the factors sin nxx/l and cos nxx/l be placed under the integral signs, the result may be written I C l I " = f 7 7T f(x)= \f(v')dv+ ^ \ f(v) cos n j(v x) dv. . (i) 2/Jo * n-J l In like manner, the formula of Art. 322 may be written If I n f TC f(x} = j\ f(v)dv + ^ f(v] cos n (v x] dv* . (2) 2/J -i I I J I M = r J * Let Wi=/r/7, U2 = 27t/l, etc., so that u n = mt/l and the series of w's have the constant difference Au = n/l. Equation (2) may then be written f(x) = : f(v) dv-\ 2 cos u n (vx]f(v} dv Au. zlj-i *-*)-* If, in this equation, / increases without limit, so that Au de- creases without limit, the summation with respect to n will be replaced by integration, the limits for u being zero and infinity. r Therefore, if f(x} is a function such that f(v}dv has a finite J 00 value, we shall have f(x} = - cos u(v x}f(v}dv du. 71 J J _oo This is known as Fourier's double-integral theorem. * Poisson's method consists in a direct demonstration of this equation. See Todhunter's Int. Calc., p. 298. XXI.] EXAMPLES. 365 Examples XXI. i. Develop x 2 in a cosine-series, taking /=-; and verify for .Y=O ind for X=TI. 2. Find the sine-series for /(:*;) = i from x=o to x=l, and show that the result is the derivative of equation (i), Art. 318. 4 / . xx i TIX i xx \ i = I sin H sm 3 H sin 5-y+ ) \ O 3 / 3. Develop x(l x} in a series of sines. 8/ 2 / . Ttx i Ttx i nx 1 T+Ii sm 3T + 3a sm 5 + . 4. Expand cos $6 in a series of cosines taking /=-, and show that the result is numerically true for all values of 0. _ 4 / i cos 6 cos 26 cos 3# A 2 i-3 3-5 5-7 5. Expand cos %6 in a series of sines, and show by its graph that the result is true from 6=0 to 6271 exclusive of these limiting values. , 8/sin# 2 sin 26 3 sin ?0 cos i/7= I 1 +- +... x\ i-3 3-5 5-7 6. From Example 4 derive the numerical series i i i_ T , 8 1.3 5.7 9.11 and verify by means of Gregory's series. 7. Show that the result in Example i may be derived by integrating the expression for x in sines, and determine the constant together with the result of the next integration. / J , x \ x a 7t^x= 121 sin x 5 sin 2X+^ sin -ix ... I \ 2 3 3 3 / 366 DEFINITE INTEGRALS. [Ex. XXI. 8. Find the result of two more successive integrations, see Example 7. X 4 2X 2 X 2 + 7T 4 =48( COS X -- COS 2X+~ COS ^ . . . =72o( sin x -- ^sin 2x+~ 5 sin \ = n(l g 2 ^ + - Jo Jo 2 24 (see Art. 321 and Ex. 12); also that JT fT , , . , , , 7T(log 2) 2 7T 3 log cos 9 log sin d) da>= . Jo 2 48 -/T.2 28. Putting 2, Evaluate -. ; ^>, when ai. Jo (1 + 20, cos x+a 2 ) 2 2(1 -a 2 )' 2a 2 (a 2 -i)' 33. By means of Examples 23 and 30 and the trigonometric for- mula cos 2mx=co& 2 mx sin 2 mx, derive the result 7T COt WlTl I I I I 2m 2m 2 i 2 m 2 2 2 m 2 2 34. Prove in like manner TT coth mn _ i i i 2w 2m 2 35. Show that f ** 9 i j ^ lg 2 n represents x from x=o to ac=o:, a from x=a to ^=TT a, and ^ x from #=TT o; to x=n. 372 DEFINITE INTEGRALS. [Art. 341. XXII. The Eulerian Integrals. 341. A definite integral with constant limits, while not a func- tion of the variable whose differential appears under the integral sign, is in general a function of any other algebraic quantity which occurs in its expression; and, if that quantity admits of continuous variation, the integral is a continuous function of it, and admits of differentiation with respect to it. For example, in the integral x n dx, n admits of any value greater than i ; J o therefore the integral is, for the range of values from i to +00, a continuous function of n. In this case, the value of the inte- gral as a function of n and of its derivatives with respect to n are readily expressed by means of the elementary functions. See Art. 278. Certain ones among those definite integrals which are not thus "integrable" in elementary functions have come to be regarded as fundamental ones, and serve to define new functions which are employed in the expression and calculation of other integrals of more complex form. Of these the most important are those known as the First and Second Eulerian Integrals. 342. The first Eulerian Integral is a function of two variables denoted by B(m, n), and hence sometimes called the Beta Func- tion. It is defined by B(m t w)= x m - J o in which each of the exponents must be greater than i ; there- fore m and n are restricted to positive values. By Art. 97, B(n ) m)=B(m,n), so that m and n are interchangeable. The XXIL] THE EULERIAN INTEGRALS. 373 value of the integral, when one or both of the variables, m and n, is an integer, is expressible in elementary functions; it may be found by a formula of reduction, or directly as follows : Let n have any positive value, then f' i f J i x m ~ 1 dx= and x m dx= ; . J m J m + i Subtracting, f 1 ITT \ X m - I (l-X}dx = = ; :. . . . (i) J m m + i m(m + i) In like manner, putting m + 1 in place of m, and subtracting this from equation (i), f x m ~*(-L X} 2 = - = J ; m + i[_m m + 2j Again, putting m + i in place of m, and subtracting, t X m ~ l (lx) 3 dX = - f r-; -J 2-3 / x ... (3) 374 DEFINITE INTEGRALS. [Art. 342. It is evident that, in this manner, we can prove f 1 n\ . m(m + i) . . . (m + ri) ... (4) for all integral values of n. This equation gives the value of B(m, w + i) when n is an -integer. Gauss' JI Function. 34-3. When m is fixed and n increases without limit, B(m, n) approaches zero as its limit; but its ratio to n~ m is found to be finite. This ratio constitutes a function of the single variable m known as the Second Eulerian Integral. The limiting ratio in question may evidently be derived from equation (4) above by making the integer n increase without limit. First transforming the integral by putting x = y/n, the equation becomes i P / y\ n i i y m ~ l i--) dy = n m ] V n / m i +m +m n + m "When n is made infinite, we have, by evaluation of an indeter- minate form, Ml -<-' \ nl J M= oo Hence the equation may be written in the form _ T r * 2 w y m L I-.' ' "> Here the first member has a finite value, while in the second member the factor n m becomes infinite, and the continued prod- uct (in which each factor is a proper fraction) has zero for its XXII.] GAUSS' n FUNCTION. 375 limit when the number of factors is infinite. The integral (in which m must be positive) is the Second Eulerian Integral. 344. Gauss denoted by the symbol 77 the function denned by -.--J ' (I) Thus the equation above may be written f ^ / / / \ ym-i e -y ( ly = / 2 \ J m He also defined II( m) as the result of changing the sign of m in equation (i),* so that P 123 n n H(-*w)H-*~ - f 3- . . (3) I * o _l n = oo In this last expression, the factor n~ m vanishes, and the con- tinued product becomes infinite, when n is infinite. Now multiplying equations (i) and (3), we find f i 2 2 T? n 2 "It -m) = 31 22 2- ~ LI m* 2* m 2 y m 2 r^ ?# 2 J M= , OT (4) * It is to be noticed, however, that the integral in equation (2) does not admit of negative values of m. f The fractions in equations (r) and (3), taken together, form a series running to infinity in both directions. In equation (4), an equal number of factors is taken from each, and then the number is made infinite, the object being to get rid of the power of this number before it is made infinite. This could, however, be done by taking n factors from the series (3) and rn factors from the series (i), r being constant; for (rri) m Xn~ m reduces to r* , and thus remains finite when n is made infinite. Thus the product II(m)II(-jM) may be written in the form mT n n i i i 2 rn ~\ r \ . . . . . . \_n m n i m i mi + m2 + m r;H-wJ n =oo This expression has therefore a value independent of r. Supposing r>i, the expression contains an infinite number of extra factors on the right. It follows that the product of these factors must be r~ m . See the note on p. 234 Diff. Calc., in which the series is the reciprocal of that here considered. 376 DEFINITE INTEGRALS. [Art. 344. Putting W = sin mn This equation expresses a fundamental property of the function II, and shows that H(m) is finite except when m is an integer. 345. If, in equation (i), we put m + i in place of m, we have - hence, comparing with equation (i ), we have ), . . (6) a second fundamental property of the function. Putting w = o in equation (i), we have U(o) = i; whence, by equation (6), we derive successively, 77(i) = i, 77(2) ==2!, and in general when p is a positive integer TL(p)=p\. Thus the func- tion is a generalization of the factorial product; this is in fact directly evident on giving to m in equation (i) an integral value. XXII.] THE GAMMA FUNCTION. 377 The Gamma Function. 346. The definite integral in equation (2), Art. 344, is the Second Eulerian Integral, and according to Legendre's notation is denoted by r(m)\ it is therefore generally called the Gamma Function. Thus ,0 r(m)=\ J *e- x dx\ ...... (i) and, comparing with Gauss' notation, (2) Suostituting m equation (6) of the preceding article, we have the fundamental relation (3) which is, in fact, equivalent to the formula of reduction given in Art. 92. We have therefore also (4) so that a table of the values of the T-function is also a table of values of the ./I-function. Equation (2), together with the general definition of II (m), serves to define F(m) for negative as well as positive values of m. Equation (3) may also be written in the form n Again, by equations (3) and (4), the fundamental property DEFINITE INTEGRALS. [Art. 347. expressed in equation (5), Art. 344, becomes 71 sin (6) tions (3) and (5), I) = ffX etc. 347. Equation (i) gives at once r(i) = i; whence, by suc- cessive applications of equation (3), F(p) = (p i}\, when p is a positive integer. By equation (5), r(o)=oo, and F(ri) is also infinite for all negative integral values of n. Putting m = \ in equation (6), we derive F ^)= |/TT; whence, again using equa- (f) = ii*r, r(|) = |^, etc., T(-|)= -2^, These special values serve to show that the graph of the function y = F(x] has the general form given in Fig. 63, the number of branches for negative values of x being infinite. By equations (3) and (5), the /"-function of any number is readily expressed in terms of a value of F(n), in which n is between i and 2; it is therefore only necessary to tabulate F(ri) for this interval, corresponding to the arc AB in the diagram. For purposes of computation, a table of values of logic r(n) is more useful. Such a table, car- ried to 12 decimal places, was constructed by Legendre.* An abridgement of this table to 5 decimal places is given at the end of this chapter. y= FIG. 63. * Traite des Fonctions Elliptiques et des Integrates Euleriennes, vol. ii, pp 490-499. XXII.] FORMS OF THE EULERIAN INTEGRALS. 379 Transformations of the Eulerian Integrals. 34-8. A variety of definite integral forms of the Gamma func- tion results from changes in the current variable, with correspond- ing changes in the limits. For example, putting z = e~ x , whence x = log z, we find f f I"/ I\ K -' /"= x n - l e-*dx=\ (-logz) M -'dz = (log-) dz* (i) Jo J o J o \ Z / Again, putting x=z 2 , we have f T(w) = 2 e-*z 2n ~ l dz ....... ( 2 ) ^ o This form gives a direct evaluation of r(|); for, by Art. 288, f = 2 e-^0= fa J agreeing with Art. 347. Putting x=ay, where a is any positive quantity, we find r(ri)=a n \ y n - I e-ydy ...... (3) * o 349. If, in the first Eulerian integral, we put x = i/z, we have B(m t n)=[*-<(i-xr-'dx = l (S ~?* n ' dz. . . (i) J o J I Z Again, putting z=y + i in the last member, * The last member is the original form in which Euler and Legendre treated the Gamma function. f It will be found that substituting the reciprocal in this form merely inter- changes m and n. The three forms in equations (i) and (2) are the simplest forms of B(m, n). See also Ex. 24 below. 380 DEFINITE INTEGRALS, [Art. 349. Putting x/a for the current variable in each case, these three forms are rendered homogeneous in x and a; the results are ,n'), .... (3) B(m, n) A useful transformation results from putting x = sin 2 6, whence i # = cos 2 6 and dx = 2 sin 6 cos <#?. We thus find B(m, n}=2itf m - 1 6 cos^-^ddd, ... (5) J o in which m and n may have any positive values. 350. In the-special case where m+n = i / the 5-function in the form (2) has already been integrated; thus, by equation (2), Art. 302, B(m,i-m) = - -- , . . . . (i) sin mn in which o-i) or 7=-/> + i) until they fall within the limits of the table. For example, to compute the integral when p = % and o. They include the general equations given in Exs. XIX. 6 and 8. 356, When a = o, equations (i) and (2) become 386 DEFINITE INTEGRALS. [Art. 356. and f 00 F(m\ . 7t x m ~ ' sin ox dx = r- sin m- t Jo *> m 2 but in this case we must have w< i. Putting m+n = i, these may be written and sn 2 cos * in which, since m cannot be negative, n cannot exceed unity. Since T(i n}= r( . \ / ' with those of Art. 354. Since F(in)= , . : , these last equations are identical ' i (n) sm mi Reduction of Certain Multiple Integrals. 357. The double integral of x 1 ~ l y m ~^dy dx, where x and y have all positive values such that x + yi, i i + " + *~*~ ' ' the development of log F(i + x) may now be written " . . (i) Again, since the sign of x may be changed in the expression for n(x), Art. 344, we have in like manner log r(i -oc) = r* + %SzX 2 + %S*x? + - . . . (2) The sum of the terms containing even powers of x in these series can be at once evaluated. For, by equations (3) and (6), Art. 346, sm therefore the sum of equations (i) and (2) gives . . (3) * Compare Diff. Calc., Art. 230- In Art. 244 the even-numbered S's are shown to be connected with Bernoulli's numbers. f This equation is identical with equation (2), Diff. Calc., Art. 239, of which XXII.] EULER'S CONSTANT. 391 Substituting in equation (i), 7COC log r(i +*)=* log sl --(^+^3^+15^+ ), (4) in v.hich the sign of x may be changed, and the limits of con- vergence are i. 361. The values of S n all exceed the first term, which is unity; hence the series (4) converges slowly. But, if we put S n = i + s n , the part corresponding to the first term takes a known form. Thus, substituting in equation (4), we have in which the sum of the first series is, by DifL Calc., Art. 198, i i I+x * log -- x. Hence & ix 7C3c( 'T ~~" $c ] log r(i + X ) = \ log - - + (i - f)x & (i +x) sin TIX v -[K^ + K^+.-.l (5) and log r(i -*)- j kg ]* (6) the differential [equation (2), Art. 243] gives the values of S 2n in terms of Ber- noulli's numbers. * By means of these series Legendre (having first calculated the values of S n to 1 6 decimal places, together with the value of f) calculated the values of log F(a). Since it was only necessary to cover a range of unity in the values of the argu- ment, only values of * less than | had to be used, and in fact other relations were used to limit the range still further. See, for example, Ex. 4 below. 392 DEFINITE INTEGRALS. [Art. 362. 362. When known values of the F-i unction are substituted in these equations, relations between 7- and the values of s n are found. Thus, putting x = i in equation (5), we have, by evalua- tion of an indeterminate form, r = i-ilog2-|$s 3 + to + }*7+"-] .... (7) Again, by putting x = \ in equation (6), we derive, since (8) l 3-2 2 5-2 4 7.2 A table of the values of s n is given at the end of this chapter, by means of which the value of 7- * to 10 places of decimals will be found to be ^=0.5772156649. The Logarithmic Derivative ofr(jx). 363. Let r'(x) denote the derivative of r(x), and ^>(#) the logarithmic derivative; then, differentiating equation (i), Art. 359 (and putting x in place of i +#), we have * The value of 7- has been calculated by Professor J. C. Adams to 263 places of decimals, Proceedings of the Royal Society oj London, vol. xxvii, p. 94. The method involves the direct summation of a large number of terms of the har- monic series and calculation of the remainder by means of Euler's formula for the summation of series. The same method was used by Euler and Legendre for S 2 , S 3 , etc.; it is in fact the only practicable method for small values of , owing to the slow convergence of the series. See Boole's Finite Differences, ad Edition, p. 93. XXIL] THE LOGARITHMIC DERIVATIVE OF F(x). 393 It follows that if r is a positive integer, This equation may also be derived by successive steps from the logarithmic differentiation of F(x + i)=xF(x). It shows that the sum of a finite number of terms of any harmonic series can be expressed in terms of ^--functions, that is in terms of F- and F'- functions. Putting x = i in equation (i), or x = o in the derivative of equation (i), Art. 360, we find F'(i) =)-. Hence from equa- tion (2) we have, when r is a positive integer, and, since F(r+i)=r\, In particular r'(i)= 7-, and F'(2) = i f, these give the slope of the graph of F(x), p. 378, at the points A and B. Again, when r is very great, we have approximately, from equation (i) and the definition of j, F'(r)/F(r)=logr, which shows that the subtangent of this curve tends to a ratio of equality to the Napier- ian logarithm of the abscissa. 364. An important theorem due to Gauss follows readily from equation (i); for, putting mx in place of x, where m is an integer, it becomes 394 DEFINITE INTEGRALS. [Art. 364. d log r(mx) r, i i i n = log -- ------- mx mx + i mx + n iJ M =oo mdx Multiplying by m, the terms may be written in groups thus: d log F(mx) iii i mlogn - dx x i 2 m i x-\ mm m ii i x+i , i 2 m i m X + 2 I X + 2+ m The number of columns is m and the number of terms in each is n f , where n = mn' and n' is to be made infinite. Now, since m log n=m log n' + m log m, we can assign log n' to each of the m columns, and then sum the column by equation (i). Thus we have d . d d I i \ log r(mx) = m log m + log 1 (x) + log 1 \ x-\ I J/y* O ^ / /7'Y? // "Y" \ YV\ i d / 2 \ d I m i\ + log r (x+- +---+- r iogr (+- dx \ m/ dx \ m / Integrating this equation, / I \ log r(mx)=C + mx log w + log r"(^) + log F(x + ) + \ rft/ / m i\ +iog r (x+- - , X m /' XXII.] THE LOGARITHMIC DERIVATIVE OF r(x). 395 and taking the exponential of each side, (i \ / 2 \ / m i \ x-t - )r(x+- ... r(x+- -), (7) ml \ ml \ m / where A is a constant of integration to be determined. 365. For this purpose, we notice that, putting x = i/m, Now, by equation (6), Art. 346, r( I \ r ( m - I \_ * \m/ V m I rS sin - m and, taking in like manner the products of pairs of factors equi- distant from the ends, sin sin 2 sin (m i) mm m This, by equation (3), Diff. Calc., Art. 234, gives m i ^ . . (5) m Substituting in equation (4), we find DEFINITE INTEGRALS. [Ex. XXII. and this in equation (3) gives Gauss' theorem, = r(mx)(27t) 2 m*-***. . (6) Equation (5) is a special case of this equation. Examples XXII. i. Show that member is the original form in which Elder introduced the integral, p, q, and n being integers. Legendre thus reduced all cases to that in which =i, by introducing fractional values of the arguments. 2. Prove that f 1 * n - I +# m - 1 , f 1 x n - l +x m - 1 . J (I+x r+ dx= Ja ( i +xr + n B(m ' n} ' 3. Prove that 4. From example 3 derive the general property of the T-function Supposing the values of F(x) from x=o to x=$to have been found, those from $ to f can be derived from this formula, and thence by means of equation (6), Art. 346, all values of P(x). XXII.] EXAMPLES. 397 5. Derive the continued product l/ 2 r mn r m 2 -i r w 2 iio"! 1 w2 J L 1 (+i) 2 jL (w+2 6. Show that 7. Prove that B(m, ri)B(m+n, l) = B(m, l)B(m + l, ri). 8. Derive the fundamental property of the /""-function by differen- ,-co tiation of x n ~ l e~ ax dx with respect to a, and also by integration with J o respect to a. Reduce the following integrals to /""-functions : 11 dz 10. ii -TIT ;dx. 3.49 6 J |/(an x) :k f 2 sin 2p l d cos 29 l d dd I3- J (a sin 2 6 + b cos 2 19)^+9' aaf^r(p + qY f i e~ am **d3C. j ma \ m F(-i) 398 DEFINITE INTEGRALS. [Ex. XXII. 16. I x l - l (i-x 2 ) m -*dx. J o f 17. cos(bx n )dx, J o cos . 18. Deduce equation (3), Art. 350, from (6), Art. 346, by using the form of F given in equation (2), Art. 348. 19. Show that, when =o, (/?"-a")r() = log A 20. Find the area in the first quadrant enclosed by the curve x n V* and apply to the cases w=4, n= oo , and n=\. ab Area= w=4 gives A = .g2jab; n=co,A=ab; n=1[, A $ab. 21. Find the volume generated by a loop of the lemniscate r 2 = a 2 sin 26 revolving about the axis of x. i. 233703. 22. Find the length of a loop of the curve ft = a * cos 6. 3-37490- 23. Express - as a definite z-integral by means of equation (2). V x /oo /-/-\C 'V* Art. 289, and thence evaluate dx as a double integral. Com- J o V x pare Art. 354. /n XXII.] EXAMPLES. 399 24. Show that B(m, ri) may be transformed into (a+b) m b n C a y m ~ '(a- y) n ~ x a w + w - 1 J (y + b) m+n y ' where a and b are positive ; whence ra,m-i_)-i _ + n -* M ~ 25- Find the value of the multiple integral . . . x l ~ I y m ~ I z tt ~ l . . . dz dy dx for all positive values of the variables, such that p q r 26. Show that . 3f 3 ^? . I i+x 2(2 + ^)3(3+^) 4(4+^) where 5;=a) w +(i) M + -.., etc. 400 DEFINITE INTEGRALS. [Ex. XXII. 27. Prove that and [ 28. Prove by direct summation of the powers of the reciprocals that 5 2 + 5 3 + 5 4 + 5 5 4 ---- =1, and 5 2 -5 g + 5 4 -5 5 -) ---- = . 29. Derive the numerical series and thence i* i*s+K ---- = log 2+7-- 1. 30. Derive the series l og ^(i-^ 2 ) = 2 ^ ^ ^^ +< m ^ sin TTJC and thence log 2 = 5 2 +i5 4 +^5 6 4 ---- Hence also, from Ex. 29, we may derive 31. Express r'(x) as a definite integral, giving particular results when x = i, x=2 and #=3. /OO |-00 ,-00 e~* logzdz= 7-; ze~ a log 2^2=17-; 2 2 ^~ z log z ^2=3 2^. Jo Jo Jo XXIL] VALUES OF LOG 401 LOG r(n). n o i 2 3 4 s 6 7 8 9 1 .00 3. 9-99975 95 925 900 876 851 826 802 777 I. 01 9-99753 729 704 680 656 632 608 584 560 536 i .02 5 J 3 489 466 442 419 395 37 2 349 326 33 1.03 280 257 234 211 1 88 166 H3 121 098 076 1.04 53 031 009 * 9 8 7 *9 6 5 *943 *92I *899 *8 77 *8 55 i-5 ^.98834 812 791 769 748 727 705 684 663 ^42 1 .06 621 600 579 558 538 5 1 ? 496 476 455 435 1.07 4i5 394 374 354 334 3*4 294 274 254 234 i. 08 215 195 i75 156 137 117 098 079 59 040 1.09 021 002 * 9 8 3 #964 *946 * 9 27 *9o8 #890 *87i *8 5 3 I. 10 9 -97 8 34 816 797 779 761 743 725 707 689 671 1 .11 653 635 618 600 582 565 548 53 5*3 496 I. 12 478 461 444 427 410 393 376 360 343 326 I-I3 310 293 277 260 244 228 211 i95 179 163 I.I4 147 I3 1 US 099 083 068 052 036 O2I 005 !-I5 3.96990 975 959 944 929 914 8 99 884 869 854 1.16 839 824 810 795 780 766 751 737 722 708 i '.17 694 680 666 651 637 623 610 59 6 582 568 1.18 554 54i 527 5M 500 487 473 460 447 434 1.19 421 407 394 381 3 6 9 356 343 33 3i7 35 i .20 292 280 267 255 242 230 218 206 194 181 I . 21 169 i57 146 134 122 no 098 087 075 064 I .22 052 041 029 018 007 *995 * 9 8 4 *973 *g62 * 95 i 1.23 9-95940 929 918 908 897 886 876 865 854 844 1.24 833 823 813 803 792 782 772 762 752 742 1-25 73 2 722 712 73 693 683 674 664 655. 645 I .26 636 627 617 608 599 59 58i. 572 563 554 1.27 545 536 527 5i9 5i 5i 493 484 476 467 1.28 459 45 1 442 434 426 418 410 402 394 386 I . 29 378 37 362 355 347 339 332 324 3i7 39 I. 3 302 295 287 280 273 266 259 2^2 245 238 I-3I 231 224 217 211 204 197 191 184 178 171 1.32 165 159 152 146 140 134 127 121 "S 109 !-33 10^ 098 092 086 080 75 069 063 058 052 i-34 047 042 036 3 I 026 020 OI 5 OIO 005 ooo i-35 9 94995 990 985 981 976 971 966 962 957 953 1.36 948 944 939 935 93 926 922 Ql8 014 910 r-37 90S 901 898 894 890 886 882 878 875 871 1.38 868 864 861 857 854 8^0 847 844 840 837 i-39 834 831 828 825 822 819 816 813 811 8oF 402 DEFINITE INTEGRALS. LOG (Continued). n i 2 3 4 5 6 7 8 9 1 .40 ;. 94805 803 800 797 795 793 790 7 88 785 7^3 1.41 781 779 77 6 774 772 770 768 766 764 7^3 1.42 761 759 757 756 754 752 751 749 748 747 i-43 745 744 743 74i 740 739 738 737 736 735 1-44 734 733 732 73i 73 73 729 728 728 727 I-45 727 726 726 725 725 725 725 724 724 724 1.46 724 724 724 724 724 724 724 725 725 725 1-47 725 726 726 727 727 728 728 729 73 73 1.48 73 1 732 733 733 734 735 736 737 738 740 1-49 74i 742 743 744 746. 747 748 75 75i 753 I . "O 754 756 758 759 761 763 765 767 768 770 1.51 772 774 776 779 78i 783 785 787 790 792 1.52 794 797 799 802 804 807 809 812 815 817 i-53 820 823 826 829 832 835 838 841 844 847 1-54 850 853 856 860 863 866 870 873 877 880 i. 55 884 887 891 895 898 902 906 910 914 917 1.56 921 925 929 933 938 942 946 950 954 959 i-57 9 6 3 967 972 976 981 985' 990 994 999 *oo4 1.58 1 958 013 018 023 027 032 37 042 047 052 J -59 057 062 068 073 078 083 089 094 099 i5 i. 60 no 116 121 127 132 138 144 149 155 161 1.61 167 173 179 185 190 196 203 209 215 221 1.62 227 233 240 246 252 259 265 271 278 285 1.63 291 298 34 3" 3i8 324 33i 338 34$ 352 1.64 359 366 373 380 387 394 401 408 4i5 423 1.65 43 437 445 452 459 467 474 482 489 497 1.66 505 Si 2 520 528 S3 6 543 55i 559 567 575 1.67 583 59i 599 607 615 624 632 640 648 657 1.68 665 673 682 690 699 707 716 724 733 742 1.69 75 759 768 777 785 794 803 812 821 830 1.70 839 848 857 866 876 885 894 93 9i3 922 1.71 93 1 941 950 960 969 979 988 998 *oo8 *oi7 1.72 9.96027 37 047 056 066 076 086 096 106 116 1-7.3 126 136 146 157 167 177 187 198 208 218 1.74 229 239 250 260 271 281 292 302 313 324 i-?^ 335 345 356 367 378 389 400 411 422 433 1.76 444 455 466 477 488 499 Sii 522 533 545 1.7-7 556 567 579 59 602 614 625 637 648 660 i.jS 672 684 695 707 719 73i 743 755 767 779 1.70 791 803 815 827 839 851 864 876 888 901 XXII.] VALUES OF 403 LOG F(n} (Continued). n o i 2 3 4 5 6 7 8 9 1. 80 9.96913 925 93S 950 9 6 3 975 988 *ooo *oi3 *026 1.81 9.97038 5i 064 076 089 IO2 IJ 5 128 141 J 54 1.82 167 1 80 r 93 206 219 232 2 45 259 272 285 1.83 298 312 325 339 352 366 379 393 406 420 1.84 433 447 461 474 488 502 5i6 53 543 557 1.85 S7i 585 599 613 627 641 656 670 684 698 1.86 712 727 74i 755 770 784 798 813 827 842 1.87 856 871 886 900 9i5 93 944 959 974 989 1.88 9 . 98004 018 33 048 063 078 93 1 08 123 139 1.89 154 169 184 199 215 230 245 261 276 291 i .90 37 322 338 353 3 6 9 385 400 416 432 447 1.91 463 479 495 5" 526 542 558 574 59 606 i .92 622 638 654 671 687 703 719 735 752 768 J -93 784 80 1 817 834 850 867 883 900 916 933 1.94 949 966 983 999 *oi6 *033 *5 #067 *o8 3 *IOO 1.95 9.99117 134 151 1 68 185 202 219 237 254 271 i .96 288 35 323 340 357 375 392 409 427 444 1.97 462 479 497 5i5 532 55 567 585 603 621 1.98 638 656 674 692 710 728 746 764 782 800 1.99 818 836 854 872 890 909 927 945 9 6 3 982 VALUES OF S= n s n n s n 2 0.64493 40668 5 14 0.00006 12481 4 3 .20205 69031 6 15 .00003 05882 4 4 .08232 32337 I 16 .00001 52822 6 5 .03692 77551 4 17 .00000 76372 o 6 .01734306198 18 .00000 38172 9 7 .00834927738 19 .00000 19082 I 8 .00407 73562 o 20 .00000 09539 6 9 .00200 83928 3 21 .00000 04769 3 10 .00099 45751 3 22 .00000 02384 5 n .00049 41886 o 23 .00000 01192 2 12 .00024 60865 5 24 .00000 00596 i 13 .00012 27133 5 25 .00000 00298 o N.B. For greater values of w, divide s^ successively by 2. 404 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 366. CHAPTER VI. INTEGRATION OF FUNCTIONS OF THE COMPLEX VARIABLE. XXIII. >v Complex Values of the Derivative. 366. In treating of the functions of a complex variable it is usual to put z = x+iy for the independent variable; then if iv=f(z) it will usually be of the form w = u+iv, in which u and v are real quantities Involving the real quantities x and y. It is explained in Diff. Calc., Art. 223, how a value of z is geometrically represented, in a plane of reference, by the point whose rectangular coordinates are x and y, and also that z may be put in the form re ie , where r and 6 are the polar coordinates of the same point, which we shall call the point z. In this last form r is called the modulus of z and is regarded as its absolute value, while 6 is the argu- ment or angle determining the direction of the unit factor e ie t which is one of the radii of the unit circle. In like manner, we may write w =f(z) = u+iv = pe^ and represent w by a point referred to rectangular axes of u and v, generally taken for convenience in another plane which we may call the w-plane. XXIII.] COMPLEX VALUES OF THE DERIVATIVE. 405 Thus the functional relation w=f(z) establishes a corre- spondence between positions, PI, P 2 , P 3 etc., of the point (x, y) in the -z-plane and positions, Q l} Q 2 , Qs etc., of the point (M, v) in the w-plane. Thus the w-plane becomes, as it were, a map of the z-plane, in which any figure described in the z-plane is represented by a figure which is called its image. The mode in which this mapping takes place constitutes the geometric repre- sentation of the function which gives rise to it, just as in the simpler case of the function of a real variable the curve y =f(x) represents the function. 367. A continuous variation of the independent variable z is now represented by a continuous motion of the point z, which may be along any arbitrary path or track. The point w will then describe a corresponding track in the w-plane, which is the image of the track of z. Let z, represented by P in Fig. 64, u FIG. 64. describe its track in a definite manner. Its motion at any instant involves a definite rate and a definite direction. Accordingly its differential, dz=dx+idy, depends upon two independent arbitrary elements, the values of dx and dy, which measure in fact the resolved velocities of P 406 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 367. in the directions of the two axes. The actual velocity, or abso lute rate of z, is measured by ds, the differential of the arc described. Denoting, as usual, by $ the inclination of the curve to the axis of x, we have dz=dx+idy=dse^ ....... (i) In the last member, the rate and the direction of motion are expressed separately by the values of the independent quan- tities ds and <; ds = y(dx 2 + dy 2 ) being the modulus of dz, and < its argument. 368. In like manner, we have, for the differential of the func- tion, div=du+i dv = ds'e^', ..... (2) ds f in Fig. 64 being the absolute length or modulus of div, and ' its inclination to the axis of u. From equations (i) and (2) we obtain the derivative of the function w = f(z), namely d-w du+idv ds' , = = - = -} J ( J dz dx+idy ds 13J The final member of this equation shows that the derivative is a complex quantity, of which the modulus is the ratio of the rates, and the argument is the difference of the directions of the motions of w and z. Since the derivative has in general a defi- nite value for a given value of z, both this ratio of rates and this difference of direction are independent of the arbitrary path of z. Thus, if two paths of z start from the same point P, Fig. 65, making a given angle, (j>\ 2 = a, their images in the w-plane will make the same angle at Q, that is, $\-4>2=(x- 369. It is an obvious consequence of this preservation of angles in the image that the image of a small area \vill be a simi- lar small area (the ratio of similitude being the ratio of rates, which, as mentioned above, is the modulus of the derivative). XXIII.] CON FORMAL REPRESEN TA TION. 407 Fr this reason, the image is said to constitute a conformal repre- sentation.* Assuming the derivative to be finite and continuous, the scale of representation, or magnification, and the orientation change continuously from point to point. The points for which the FIG. 65. modulus of the derivative is zero or infinite (and its argument therefore indeterminate) are points of discontinuity for the function, and at these points the conformal representation fails. Conjugate Functions of x and y. 370. When the function iv=f(z)=f(x-\-iy} is given in the form u+iv, u and v become known functions of the two real variables x and y. For example, if f(z) = z 2 , y 2 whence '-y 2 , * In like manner, a map of a spherical surface in which angles are preserved is a conformal representation of the surface. The stereographic projection is an example. In that case, the magnification has at every point of the circumference of the primitive circle a value double that at the centre. 408 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 370. In the general equation expressing the derivative, Art. 368, .... du+idv the differentials du and dv are total differentials due to the varia- tion both of x and of y. Thus, by Diff. Calc., Art. 370, du j du . dv . dv , du = dx+dy, dv= dx+ dy, dx dy ' dx dy ' du du in which , etc. are the partial derivatives of the functions dx dy u and 'v, while dx and dy are independent differentials. Substi- tuting in equation (i), du . dv \ (du . dv\ , +* \dx+{+i)dy dx dx > \ d y d y' dx+idy Since f(z) has a value independent of the ratio dy : dx, we have, by putting dy and dx successively equal to zero, ftt \ du , dv _ .du dv J (2 ) + 1 1 + . dx dx dy dy Equating separately the real and imaginary parts of the last equation, we have the following relations between the partial derivatives : du^dv dv du dx dy' dx dy' Again, eliminating v from these two equations, we have d 2 u d 2 u and the same equation may be found for v. XXIII.] CONJUGATE FUNCTIONS OF X AND y. 409 Two functions of x and y which together satisfy equations (2) are called conjugate functions. Each of them necessarily satisfies equation (3). Thus the functions x 2 y 2 and 2xy, derived above from the function z 2 , are conjugate functions and will be found to satisfy these equations.* 371. From the expressions for u and v in terms of x and y we can readily obtain the equation of the image in the i^-plane of a curve in the z-plane whose equation is given. For we have only to eliminate x and y from three given equations. For example, in the case cited above, where u = x 2 y^, v=2xy, we thus find, corresponding to x = a, which is the equation of a parabola with focus at the origin and vertex on the axis of u to the right of the origin. In like manner,. we find corresponding to y = b, 'which is a parabola with focus at the origin and vertex on the axis of u to the left of the origin. If a 1 series of equidistant values be given to a and also to b, we shall have in the z-plane two sets of straight lines parallel to the axes, dividing the z-plane into small squares. The two corre- sponding systems of parabolas in the w-plane cut each other * An expression u + iv in which the functions u and v are taken at random (although a function of z in the sense that it is determined by the position of the point 2) is not, unless u and v satisfy the equations above, an analytical function of z, that is one which can result from algebraic operations performed upon z, that is, upon x+iy as a whole. Analytical functions were called by Cauchy "fonctions monogenes." 410 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 371. orthogonally, in accordance with Art. 368, and divide the w-plane into curvilinear squares. The correspondence of these small squares in the two planes exhibits most clearly the mode in which the function w varies with the independent variable z.* Two-valued Functions. 372. Let w be a two-valued function, and let z be repre- sented by the point P moving along a given track from the initial point z to the final point z\. Then the two corresponding points in the w-plane will in general be distinct points, Q and Q', and will describe tracks, say AB and A'B', which have no common point; that is to say, no point at which Q and Q' arrive simul- taneously. Hence, selecting A as the initial value of w, B will in general be determined without ambiguity as the final value of w. But, if the track of z passes through a point for which the two values of w become equal, the. paths AB and A'B' will have a common point, and there will be an ambiguity in the final value of w, because it will be possible to pass from A to B'. Such a point in the z-plane is called a change-point or branch-point for the given function. f Then, provided it does not pass through a branch-point, the track of z (together with the initial value w ) determines without ambiguity the final value w\. * Figures illustrating in this way many of the elementary functions will be found in Dr. Thomas S. Fiske's Functions of a Complex Variable, p. 13 et seq. (Mathematical Monograph Series, John Wiley & Sons, 1906), to which the reader is referred for an excellent introduction to the Theory of Functions. It will be noticed that, in the case above, illustrating the function z 2 , the whole of the w-plane corresponds to one-half of the 2-plane. On the other hand, only a portion of the w-plane will correspond to a single value of a multiple-valued function. t Two values of w may become simultaneously infinite at a branch-point, in which case the conclusions below may be established by means of the function v/~ l , of which two values become zero. XXIII.] TWO-VALUED FUNCTIONS. 411 373. Moreover, if the track of z be altered gradually, that is by continuous or infinitesimal changes, to any other track between z and Zi, the tracks AB and A'B' in the w-plane will suffer con- tr uous change, but it will be impossible to pass from A to B' unless at some instant these tracks have a common point; that is, unless the track of z passes through a branch-point. It follows that a track which can be altered into a given track without ever passing through a branch-point will determine the same final value of w. Such tracks are said to be reducible, one to the other. This is as much as to say that, if the area enclosed by the two tracks of z does not contain a branch-point, the tracks lead to the same final value of w; so that, if z returns to z after describing the complete contour formed by the two tracks, it will return with the same value of w with which it started. But, if the contour encloses a branch-point, it may return with the other value of w. 374-. Take, for example, the simple function w=\fz. Put- ting z in the form re i0 , we have seen in Diff . Calc., Art. 228, that w admits of the two values re* ie and ^re^ e+ajc) which is equal to yre^ e . As z moves, the argument varies con- tinuously. Now, starting with the initial values z = r e io , iv = ^r e^ e , if z returns to z without encircling the origin, 6 will return to 6 , and w to w ; but, if z encircles the origin in the positive direction, the final value of the argument of z will be d +27t, and the final value of w will be w . If z describes the same contour in the negative direction, the final value will again be w ; but, if it enwraps the origin twice, or any even number of times, w will return to the value w . 375. In like manner, if w^(za).^>(z), a is a branch- point, and a circuit about a described by z will change the sign of w, provided the factor (z) has no branch-point either on or within the circuit described by z. Thus the function a)(z b)\ 412 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 375. is two-valued, and has two branch-points, a and b. If z describes a contour enclosing a, but not enclosing 6, the sign of w will be changed; and so also if the contour encloses b and not a. But, if the contour encloses both a and b, w will- return to the value w when z returns to z , so that the function is virtually one-valued for such a contour. Since the direction in which z moves about either of these branch-points is indifferent, the same thing will be true if the track of z forms a figure-of-eight, of which one loop contains a and the other b. Multiple-valued Functions. 376. The conclusions arrived at in Arts. 372 and 373 apply also to functions having more than two values. Thus, we select an initial value z , to this corresponds a number of values of w . Then, any track (avoiding branch-points) from the initial point to the point Zi leads from any one of the values of w to a defi- nite one of the values of w\. Now, choosing one of the values of w as the initial value, different tracks may lead to different values of w\\ but, if the area between two tracks does not con- tain a branch-point, the values reached are the same. A standard track between z and Z 1} for example the rectilinear one, will establish a correspondence between the values of w and those of w\\ then one value of w and the corresponding value of iv\ are selected as primary values. Now, any track may, without passing through a branch-point, be reduced either to the standard track or to one combining with it one or more loops or contours (from and back to the initial point), each surrounding a single one of the branch-points. The result of describing one of these loops is to pass from one to another of the values of w , and thus, in connection with the standard track from z to z\ } we may pass from the primary value cf w to one of the non- primary values of w\. 377. For example, let the function be w=tyz, for which XXIII.] MULTIPLE-VALUED FUNCTIONS. 413 the origin is the only branch-point. Take unity, represented by the point A in Fig. 66, as the initial value of z, and let B rep- resent the final value of z. Putting z=re io , we take zero for the initial value of 6. Then the value of w resulting from a rectilinear track will be tyre* ie , where 6 must lie between - and 7t in value. Call this the primary value of w, so that the primary value of w at A is unity, and at B it is iv\ = f/rie*^'. The result of making the circuit of the origin in the posi- tive direction is to add 2it to the value of 6, and therefore to multiply w by e&*=io, oie of the r cube roots of unity. (DifL Calc., Art. 230.) Thus, when the track of z is ACB in Fig. 66, the initial value of w being unity, its final value is urw\. Again, the result of making the circuit of the origin in the negative direction is to multiply w by a/*, the other cube root of unity; so that, when the track of z is ADC, the re- sult is uf"w\. This last is also the result of a track enwrapping the origin twice in the positive direction. 378. The function iv = logz presents a branch-point of a different character. We have, on putting FIG. 66. z = re w = log z =Log where Log r is the real logarithm of the modulus. Taking unity (with 6 = 0) for the initial value of z, the initial value of w is zero. Let us denote by the symbol Log z the result when z follows the rectilinear track AB in Fig. 66. Then Log z=Logr+0 , where 60 lies between n and TI. The circuit of the origin in the positive direction adds 271 to 6, and therefore adds 2in to w. Thus, when the track of z is ACB, the final value of w will be z + 2wr, and for the track ADB it is Log z 2r, while the 414 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 378. general value log z=Log r+i0 +2nix is the result of n circuits of the origin combined with the rectilinear track AB. 379. The values selected as primary are said to constitute one branch of the function, but the mode of selection is arbitrary. A given selection corresponds to a cut in the z-plane, or a number of cuts, each starting from a branch-point, and either passing to infinity or terminating in another branch-point. Moreover every branch-point must be one of the extremities cf a cut. Starting from any initial point, it is possible to reach any other point of the plane without crossing a cut, and the values of w so obtained constitute the branch. For example, the selection above for the primary value of the logarithm is equivalent to making a cut along the negative part of the real axis. A Riemanri's Surface consists of a number of leaves identical with the cut plane and joined to one another along the cut, so that when we cross the cut we pass into another branch of the function. In the case of the logarithm, the number of leaves is infinite, and we ascend, as it were, from leaf to leaf as we go round the branch-point (Windungspunkt in German) in one direction. For the function in Art. 375, the cut may be made along the line ab. There are in that case but two leaves, and each is joined to the other along the cut, so that the surface there intersects itself. A multiple-valued function may be regarded as one-valued on its Riemann's Surface; that is to say, to each point of the surface there corresponds a single value of the function. Meaning of Integration when the Variable is Complex. 380. Let us now consider the meaning of an integral \vhen z in the expression, /(z)cte, under the integral sign admits of complex values. Supposing F(z) to be a function such that its derivative F'(z)=f(z), we write F(*)= /(*>&, XXIII.] INTEGRATION OF COMPLEX FUNCTIONS. 415 in which /(z) is sometimes called the integrand, while F(zJ is the indefinite integral. To remove the ir.definiteness due to the constant of integration we write also, as in Art. 82, \f(z)dz=F(z)-F(a), (i) J a in which a is a selected initial value of z, and it is understood that z in the integral varies continuously from the initial value a to the final value denoted by z in the second member. To the latter, we may of course assign a separate symbol, and place it as the upper limit of the integral. The initial value a and the final value z in equation (i) may now be complex quantities, and the track described by the cur- rent variable z, or track of integration, is supposed to be know r n. 381. Let the indefinite integral be represented by a point ]V = F(z) in a new plane; then, as z describes the given track, IF starting from the position F(a) moves in a direction and with a rate which depend at every instant upon the corresponding value of f(z)dz. Provided ; therefore, that, the track of z does not pass through a point for which /(z) is either infinite or ambiguous * in value, the track of W will terminate in a definite point F(b), * It will be noticed that the restriction with regard to intermediate points upon the track of z is the same as that of Art. 82 with regard to the intermediate values in the case of real integrals. But, whereas for real integrals this restriction excluded the interpretation of certain integral expressions, no such exclusion now exists. For example, we have when 2 is a real variable f'dz i but, if the upper limit is negative, the integral is inadmissible, because to reach such a value z must pass through the value zero for which the integrand is in- finite. Whereas, when complex values are admitted, the track of integration need not pass through the origin and the equation holds true. Thus, \1 z= ^, the value of the integral is 2. 4l6 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. '381. and, by equation (i), the integral will have a definite value. Thus the critical points through which the track of z must not pass include the infinity points, or infinities, of the integrand f(z) as well as its branch-points. These critical points are the only positions at which branch- points .of the indefinite integral F(x] can occur. Hence it follows, as in Art. 373, that any two tracks which can be reduced one to the other without passing through a critical point will give the same value to the integral. Integration around a Closed Contour. 382. It follows from the preceding article that, when z describes a closed curve or contour returning to its initial value, the integral must vanish unless the contour encloses one or more critical points. Furthermore, in comparing the results of different tracks of integration between the same limiting points, we may adopt, as in Art. 376, a standard track, and reduce any other track to a combination of this with one or more loops or con- tours, each enclosing a single critical point. For example, let us take the integral of dz/z, for which the origin is an infinity of the integrand. Let /.-!** denote the result of integrating from an initial point c in a contour described in the positive direction "about the origin and back to c. Putting c = re* e , we may take for the contour a circle of radius r and centre at the origin. This is equivalent to putting z=re ie and making r constant while 6 varies from 6 to 6 +2-. We have now dz=ire ie dd ) hence L = i XXIII.] INTEGRATION ABOUT A POLE. 4 ! 7 Thus it appears that, in this case, the value of the contour integral is independent of the initial point c. Taking unity as the initial point, the value of the indefinite integral is log z, and 2in is the difference between consecutive values of this many- valued function. Compare Art. 378. Integration about a Pole. \ 383. If a is an infinity of f(z), we shall assume that there is a positive value of n such that the product (z a) n f(z) has a finite value when z=a. Denoting this product by <(z), we have where (z) is such that 0('(a). This result is also deducible from equation (i) above by taking derivatives with respect to a; and, in like manner, by taking successive derivatives we have the series of equations : and, in general, -'^'W ...... (5) The contours are here supposed to be described in the positive direction, and, if the pole is enwrapped k times, the result is multiplied by k. Again, if a function has two poles a and 6, it is readily seen that the result of a contour encircling both a and b will be Integrals of Functions with Poles. 386. It follows from the preceding articles that the integral of a function with poles will be a function having an unlimited number of values differing by multiples of one or more constants dependent upon the poles. Take, for example, the integral da ',-i)(z+i)' 420 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 386. The poles of the integrand are the points i and i. The contour integral about the first is, by equation (i), 1 < 2r and, in like manner, we find /_ t = --. The initial point is so taken that, for real positive or negative values of z, the integral is the primary value of tan~ J z, as defined in Diff. Calc., Art. 76. The rectilinear track of integration may now be used to define the primary value of the function for all points except those on the imaginary axis beyond the points i and i. This selection of the primary value therefore corre- sponds to cuts of the plane (see Art. 379) made along these por- tions of the axis. It follows that any circuit which crosses a cut from the right side to the left of the axis adds - to the value of the function, and one crossing in the reverse direction sub- tracts TI. In particular, a circuit about both poles restores the value of the function. Art. 382 and the present one illustrate the fact that, when the integrand has a pole, integration gives rise to a function which .admits of an infinite number of values differing by multiples of a, constant. It follows that the inverse of such a function is a periodic function, in which values of the independent variable .differing by multiples of a constant (called the period) correspond to the same value of the function. Thus e* and tan z are periodic functions, the first having the pure imaginary period 2wr and the second the real period n. Integration aboiit a Branch-Point. 387. The value of a contour integral about a branch-point of the integrand, unlike that about a pole, depends upon the posi- tion of the initial point. The reason is that when a circuit of the XXIII.] INTEGRATION ABOUT A BRANCH POINT. 411 branch-point is completed the integrand returns with a new value; thus, referring to Fig. 67, if a were a branch-point, the integrand in the contour cbebc would have a different value when z finally leaves b from that with which it first arrived at 6, hence the two rectilinear parts of the integral would no longer cancel one another. In illustration, let us integrate z n dz in a circle whose centre is the origin, from the initial point c back to c. Except when n is an integer, the origin is a branch-point of the integrand. Put- ting c = re i and z = re id , the contour is described by making r constant while 6 varies from 6 to 6 +2n. Hence the contour integral is fC fSo~^~3it ~n-\- I ~"j0 +2* z n dz = i Y n ~^~ r I c*( n ^~ I ^d0^ gt'i''+i)0 Jfl H + I Jg JO w I/O PO n + i n + i 1 Now - is the indefinite integral, and e 3i( * +l) * is one of n + i the complex values of i" +I , hence the contour integral is (as we should expect) the quantity which must be added to one of the multiple values of the integral in order to produce another. In this example, if n is zero or a positive integer, the contour integral vanishes, the indefinite integral being in that case a one- valued function. When n= i, the contour integral takes the indeterminate form, and is found on evaluation to be 2ix, agreeing with Art. 382, and when n is any other negative integer, the contour integral vanishes in accordance with equation (5), Art. 385. Integrals involving Radicals. 388. When an integral involves an ordinary quadratic radi- cal, every point at which the quantity under the radical sign vanishes is a branch-point. If the branch-point is on the real 422 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 388. axis, and a real initial value is chosen, the value of the contour integral is determined by a real definite integral. Take for example the integral dz -L The critical points are +i and i. The contour from O about A, the first of these points, may be reduced, as in Fig. 68, to the rectilinear track from O to a point as near as we choose to A, followed by a circular track about A, and t e return to O by the rec- tilinear track. The effect FIG. 68. of the circular part is to change the sign of the radical, so that we may write ri3 fig |-o ^ f*-J. ^7(7^) +result of circular track +J ,., ^7(fip). in which the results of the two rectilinear parts are equal. For the circular part, put z=i de t9 , so that the circle is described by keeping d constant and varying 6 from o to 2x. Then dz=ide ie , and the equation becomes in which d may be decreased without limit The second term vanishes with o; hence, putting = 0, we have IA = dz (i) 389. The direction in which the circuit of A is made does not affect the value of I A, but it must be remembered that the XXIII.] INTEGRALS INVOLVING RADICALS. 423 value given in equation (i) implies that the initial point is the origin and that the initial value of the radical is +i. Since the sign of the radical is changed by the circuit, the result of a second circuit would be TT. That is, the result of two circuits about A would be zero, and the integrand would return to its original value. In like manner, we shall find /#= x, assuming as before that the initial value of the radical is +i. But, if the circuit of B is made after the circuit of A (which changes the sign of the radical), the result will be TT, and the whole result of a contour encircling first A and then B, like OCO in Fig. 68, will be 27T. In this contour the integrand is virtually one-valued, see Art. 375, whence it can be shown, exactly as in Art. 383, that the value of the contour integral is independent of the initial point. Hence, for the contour which does not pass through O, as well as for OCO, we may write I AB =2x. The two branch- points are thus together equivalent to a pole.* 390. Let us now, for any point z, define the primary value FI of the integral V as the result of integrating along the recti- linear track Oz in Fig. 69. Consider now the result of inte- grating along the track OCz. This track may be reduced to the contour about A followed by the track Oz. Hence, remem- bering that the sign of the integrand is changed by the contour, the result is nV\. The results obtained in the preceding article also show that a contour encircling A twice before reaching z will produce FI. * Accordingly, if 2 describes a very large circle with centre at the origin, the integral approaches the form R | , for which, by Art. 382, the J V\ z ) J z loop-integral is 2jr. The ambiguous sign is due to the fact that we have not determined which of its two values to give the integrand at any one point of the circuit. In Fig. 68 we obtained + 27: for the circuit OCO, because we assumed the radical to have the value + 1 at O. 424 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 390. One encircling B, like ODz in the figure, produces nV\. Again, one encircling A and then B in either direction produces 27r + Vi. Thus, by means of different tracks, we can reach any of the values included in the formulae 27r + Fi and (2 + i)7r-Fi, where n is an integer. The integral defines the function sin -1 2 for complex values of 2, and we have thus shown that the relations which exist between the multiple values when z is real hold also when z is complex. The fact that a circuit of both branch- points is equivalent to that of a pole gives a periodic character to the inverse function, just as in Art. 386. 391. The use of the rectilinear track to define the primary branch of this function is equivalent to making cuts in those parts of the real axis which are beyond the points A and B. The Riemann surface consists of an infinite number of leaves. If L , LI, L,2, LS etc. denote consecutive leaves, L and LI may be joined along the positive cut so as to form a self -intersecting surface (as in the second illustration used in Art. 379), and L 2 and L 3 are joined in like manner. But, along the negative cut LI and L 2 must be thus joined, and L 3 in like manner with Z 4 , and so on. In Fig. 69, the parts of the several tracks which would lie on different leaves of the surface thus constructed are indicated. The initial point O being on L , we arrive at z by the route FIG. 69. XXIII.] THE MODULUS OF A SUM 425 through C on the leaf L\ ; by the route through E and F, we arrive on the leaf L 2 ', and, by the route through D, we arrive on the leaf L_I. The Modulus of a Sum. 392. The sum of several complex quantities a, b, . . . , I is graphically represented by the straight line OL, where OA, AB, . . . , KL (equal to the moduli of the given quantities) are laid off successively each in its proper direction, forming a broken line beginning at the origin. Therefore the modulus of the sum, which is the length of OL, cannot exceed the sum of the moduli, which is the length of the broken line OBC . . . L. It is in fact equal to this sum only when the arguments of all the terms are the same, so that the broken line becomes a straight line. Applications of this principle in the discussion of infinite series occur in the following articles. We here notice its applica- tion to the value of a definite complex integral. Let where ^ denotes a specified track of integration in the z-plane. While z describes the track L, w describes a track in the w-plane. Regarding the integral as the sum of its elements, this last track is the limiting form of the broken line representing the summation of the elements, hence its length is the limit of the sum of the moduli of the elements. If 5 is the length of arc in the path of z, ds is the modulus of dz, and if /t is the variable modulus of /(z), fj. ds is the modulus of an element; thus // ds is the length JL of the track in the -zf-plane. The modulus of w therefore cannot exceed this integral. Let M be the greatest value of /* for points 426 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 392. on the track L, and let / be the length of that track; it follows a fortiori that the modulus of w cannot exceed ML Power Series in the Complex Variable. 393. A series of complex terms is convergent when the point representing the sum of n terms tends to a fixed limiting position as n increases without limit. Among non-convergent series, we may distinguish between oscillating series for which this point remains at a finite distance (but does not approach a limiting value), and properly divergent series for which the point recedes indefinitely from the origin. A series is necessarily convergent when the moduli of its terms form a convergent series (the terms of which are of course all positive). Such a series is said to be absolutely convergent. For example the series D FIG. 70. is absolutely convergent when z is a complex quan- tity whose modulus is less than unity. In Fig. 70 the sum of a number of terms of this series is constructed. The broken line OABC . . . here consists of links whose lengths form a decreasing geometrical progression, and their inclinations to the axis of reals an arithmetical progression, so that the angles at A, B, C, etc., are all equal. Such a polygon can be inscribed in an equiangular spiral,* hence the pole P of this spiral is the * See Diff. Calc., Art. 324. If successive radii-vectores of an equiangular spiral make equal angles at the pole, they are in geometrical progression, so XXIII.] POWER SERIES IN THE COMPLEX VARIABLE. 427 limiting position of the nth vertex when n increases without limit, and OP represents the sum of the series. 394. When the modulus of z is greater than unity, the nth vertex of the broken line is on an equiangular spiral but recedes from the pole indefinitely. In the intermediate case, when the mcdulus is unity, the polygon is plainly one inscribed in a circle. We can only say of the nth vertex that it must lie somewhere on this circle; therefore the sum of n terms does not tend toward a limit, but oscillates about a mean value. An example of a convergent series which is not absolutely convergent is furnished by the series Z + l2 2 +j2 3 +---+-Z*+-", ft when the modulus of z is unity. The series of moduli is the harmonic series i + \ + J H , which we have seen has an infinite sum (Diff. Calc., Art. 180). But, when z is a complex unit, the series is convergent; the broken line, though of infinite length, is wrapped about a limiting point. So also, when z= i, we have a series of alternately positive and negative terms, and the sum converges to the limit log 2.* that the triangles formed by joining their extremities are similar. Hence the chords are in geometrical progression and they make equal angles with one another. * The peculiarity of the absolutely convergent series is that there is but one limiting value, independent of the order of the terms, whereas in the other case any limiting values or even an infinite value may be reached as the result of a different law of succession of the terms. For example, in the series of real terms of alternating signs considered above, the sum of the terms of either sign is infinite, and it is only when taken alternately that their sum approaches the limit log 2. A series of this kind is said to be semi-convergent. An analogous case in infinite products is considered in Art. 344, see foot-note. 428 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 395. Circle and Radius of Convergence. 395. If the power series A+Bz+Cz 2 +Dz 3 +-- (i) is convergent for a given value Z, it will be convergent for any value of z whose modulus is less than that of Z. Let R be the modulus of Z and r(z)dz q>(a)= , 2^7tJc z a ju (n \ I f ^ dz 0'(a)= r ; T7>, 2^7^:Jcz-a} 2 no critical point of the function (z) being either on or within the contour of integration C. These equations are the starting-point for many investigations in the theory of functions. They enable us, for example, to prove Taylor's Theorem for the complex variable and to determine the radius of convergence as follows: 430 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 398.. 398. Let the contour C be a circle with radius r about the point a as centre, Fig. 71, and let k, the modulus of h, be less than r, so that the point a+h is within the circle. Since the circle is a contour about a+h and there is no critical point of <(z) either on or within its circumference, FIG. 71. Now, n being any positive integer, i i h h" / x u; h " +I z-a-h z-a (z-a} 2 (z-a)** (z-a) n+l (z-a-hy Substituting in equation (i), 0((0)~.+ -", 2 ' * in which f(z) is developed into a series involving positive integral powers of z. Hence, if the origin is not itself a critical point, the function can be developed in powers of z within a circle whose radius R is the least modulus of a critical point. For example, in the case of the function tan -1 z, its integral form, Art. 386, shows that the limits of convergence for the corresponding series are i, because unity is the modulus of each of the poles i and i. A one-valued function which, like s , is infinite only for infinite values of z can be developed into a series convergent for the whole plane. 400. We can now show that a one-valued function cannot fail to become infinite, for at least one value of z, finite or infinite. For, if this were possible, the modulus of z would have some finite maximum value M. Now we have, for auy point a, 432 FUNCTIONS OF THE COMPLEX VARIABLE. [Art. 400. in which, since <(z) has by hypothesis neither pole nor branch- point, the contour C may be taken as a circle with centre a and any radius. Putting, then, z= a +re* e , whence dz= ire t9 dd, we have n\ f 2 * dtz^ire* 6 dd n\ f a) = 2^xJ r n+1 e (tt 2arf*J The modulus of the integral in the last expression is, by Art. 392, less than 2nM; hence the modulus of ( "\a) cannot exceed n\ M/r n . In this expression r can be made as large as we choose, hence '(a) "(a) etc., all vanish, and equation (2), Art. 398, becomes (a+h)=(a); that is to say, (z) reduces to a con- stant and is not a function of z. Since this applies also to the function 1/^(2), it follows that ^(2) must become zero, either when z is infinite or for some finite value of 2. In particular, if (z) is a rational integral function we have thus proved the fundamental theorem of Algebra that (2)=o must have a root. Examples XXIII. 1. Putting w=pe^=z 2 , derive the polar equations of the images of lines parallel to the axes of x and of y. Compare Art. 371. 2O 2 2b 2 P = r > P = r- i+cos ip i cos $ 2. Derive conjugate functions from w=z 3 , and thence the equations of the images of x=a and y=b. u=x 3 T.xy 2 . 2ja 3 v 2 =(a 3 u)( \J S * I \ / \ 3. Derive conjugate functions from w=e e 1 and show that the net-work of lines parallel to the axes in the z-plane is transformed into concentric circles and their radii, while the images of all other straight lines are equiangular spirals. eF cos y, (? sin y. XXIII.] EXAMPLES. 433 4. Show that u=a and v=b give for the inverse of a given function the systems of curves corresponding to parallels to the axes. Thence show that, for the function 2= log w, these systems are x = log secy+C, # = log cosec ;y + C'. Verify that these curves cut orthogonally. 5. Show that, for the function square-root, both systems of curves are rectangular hyperbolas; and, for the cube-root, each consists of cubics having three fixed asymptotes. 6. Show that the polar equations of the images of x=a and y=b in Ex. 2 are p= a 3 sec 3 ^0, p= b 3 cosec 3 ^0. Trace these curves for a=i and b=i and verify their orthogonal intersection at (2, 2). Show also that, at the other two points of intersection, these curves must cut each other at angles of 30. Notice that the image of x= i is also the image of other straight lines in the z-plane. 7. Given w=z n , show that lines parallel to the axes are converted into circles when n= i; into cardioids when n=2\ and into lemnis- cates when n= \. 8. If w= -j/z, show that, when z describes a small circle about a given point a, the two branches of the function w will describe the separate branches of a Cassinian (Diff. Calc., p. 313, Ex. 29); if the circle passes through the origin, the ovals merge into a lemniscate; and if it encloses the origin, the two branches of the function describe halves of a con- tinuous oval. 9. Given w=^/z, assuming the initial values z =i, H> O =I, draw a track of z which will make w the real cube-root of a negative quantity. 10. Putting c=i, b i in the example given in Art. 375, so that w=V(z 2 -i), show that the systems of hyperbolas x 2 y*=A and xy=B 434 FUNCTIONS OF THE COMPLEX VARIABLE. [Ex. XXIII. form an orthogonal net- work, of which the images are u 2 v 2 =A i an( j uv=B, in the w-plane. Show by construction that arcs of the hyperbolas X 2_y2^ X y = ^^ x 2 y 2 =%, xy=& form a contour by which z can pass from the value z =f+^, around the branch-point z=i, back to z . Then, by tracing the images of these hyperbolas, show that iv, starting from the value w =i+fi, will, while z completes the con- tour, pass from iv to w in accordance with Art. 375. 11. If7f=sinz, we have u= sin x cosh y, v= cos x sinh y (Diff. Calc., Art. 220); show that the orthogonal net-work in the to- plane corresponding to lines parallel to the axes in the z-plane consists of confocal ellipses and hyperbolas. Note that for the inverse function, z=sin -1 w, the foci are the branch- points, and the ellipse furnishes an example of the contour about both branch-points equivalent to the circuit of a pole. See Art. 389. 12. Given w=log r, show that lines parallel to the axes in the % w-plane correspond to two systems of circles in the z-plane. If the z-plane be cut by a circular arc joining a and b, the cut plane corre- sponds to a horizontal strip of the w-plane. Put za^pie*^, zb=p 2 e i8 *. 13. Putting a= i and b= i in example 12, show that w2i cot -1 z, w whence z= cot .. Hence show how to construct two circular arcs in 2^ the z-plane whose intersection determines z=cot (c+id). A and B being the points i and i, one passes through the points whereas is cut harmonically in the ratio e~ zd ; the other is a segment on AB containing the angle 2c. 14. Prove the result mentioned in the foot-note on p. 415 by in- tegration along a semicircular track. 1 6. Find the value of - ^~ - dz when the contour C includes XXIII.] EXAMPLES. 435 15. Show geometrically that, when z describes a circle of radius a about the origin, the value of dz/z is ids /a; and thence derive the value of the loop-integral. - ^~ - Jc(a-a)(s- a and b but no critical point of <(z); and thence deduce equation (2), Art. 385. 17. Find the values of the loop-integrals of dz/z 2 +i. 2 1 8. Find the values of the loop-integrals of dz/z 3 i. 19- Given I ~ . ^> find the values of the loop-integrals for the poles in the first and second quadrants, and show that the result of integrating in the positive direction about these two poles is equivalent to the recti- linear integral from oo to + oo . TT(I i) M = 24 / 2 Compare Ex. XX, 2. f dz 20. Given I- sr r, find the result of integration in a contour J(l+Z 3 )|/Z including the three poles but not the branch-point, assuming the argu- ment of z to lie always between o and 271. |/T. 21. Construct the point representing the generating function of the series represented by Fig. 70, and show that it is the pole of the equiangular spiral mentioned in Art. 393. Show also that, when the modulus of z is unity, it is the centre of the circle mentioned in Art. 394. INDEX. [The numbers refer to the pages.] ABSOLUTE convergence, 426, 427 foot- note Absolute value, or modulus, 404 Amsler's planimeter, 211 Analytical functions, 409 foot-note Approximate methods of quadrature, 215 el seq. Area of integration, 159, 161 Areal element, or lamina, 164 Areas described by moving straight lines, 129, 206 et seq. of closed circuits, 210 plane, 3, 129 et seq., 172 polar element of, 172 polar formulae, 134, 135, 136 Argument, 404 Arithmetical mean, or average, 224 BERNOULLI'S series, 95 Beta-function, 372 Branch of a function, 414 Branch-point, 410, 413 Buffon's probability problem, 269 CARDIOID, 143 ex. 21, 183, 187 ex. g, 188 ex. 17, 190, 190, 207, 242 ex. 26 Cassinian, 433 ex. 8 Catenary, 32 ex. 77, 193 ex. i, 203 ex. 2 Cauchy's general and principal values, 317, 3'9 Centre of gravity, 234, 235 of position, 232 Centroids, 235 method of (in mean areas), 258 Change of independent variable, 50, 119, 306 Change of order of integration, 160, 33 Circle of convergence, 428 Cissoid, 76 ex. 51, 136, 150, 151 ex. 3, 242 ex. 17 Closed circuit, area of, 210 Companion to cycloid, 13 ex. *g Complex values of constant in definite integrals, 310 Complex variable, 404 et seq., 426 integration, 414 Conchoid, 154 ex. 18, 187 ex. 9 Conformal representation, 407 Conjugate functions, 407 Constant of integration, 2 Continued products, 374 Continuous variation, 106, 405 Contour, 411, 416 Contour integral, 417, 420, 423 Convergence, 426 et seq. Corrected integral, 106 Cotes' method of approximation, 218 Critical points, 416 Current variable, 296, 415 Curtate cycloid, 142 exs. 13, 14 Curves of probability, 276, 280, 281 Cusp, 190 Cuts in the z-plane, 414, 424 Cycloid, 48 ex. 48, 152 exs. 8, 9, 194 exs. 6, 7, 204 exs. 5, 6, 7, 208, 242 ex. 19 Cylindrical coordinates, 176 element of volume, 150, 239 437 INDEX. DEFINITE integrals, 4, 106, 296 et seq. differentiation of, 298 Definite integrals, integration of, 300 obtained by expansion, 329 el seq. Density, variable, 166, 237, 242 ex. 20 Derivative, complex values of, 406 Development of an integral in series, 94, 329, 330 Differential of an area, 3, 130, 134, 137 of a volume, 147, 150 Direct integration, 14 Discontinuity, 350, 361 Discontinuous functions, 329, 345 Divergent series, 406 Double curvature, curves of, 191 Double integrals, 155 et seq., 174, 303 transformation of, 174, 307 ELEMENT of an integral, 123, 156, 163 of area, 172 of volume, 176, 178, 181 Elementary theorems, 6 Ellipse, 48 ex. 27, 140 ex. 7, 154 exs. 19, 20 Epicycloid, 196 ex. 14 Equiangular spiral, 197 ex. 17, 426 Eulerian integrals, 372 et seq., 381 Euler's constant, 390 FACILITY of errors, law of, 281 Favorable and unfavorable cases, 266 Folium, 137 Fonctions monogene, 409 foot-note Formulae of integration, 8, 124 Formulae of reduction: algebraic forms, 90 et seq., 103 et seq. definite integrals, 116 et seq. indefinite integrals, 81 et seq. trigonometric forms, 82 et seq., 89, 102 Four-cusped hypocycloid, 242 ex. 18 Fourier's series, 342 et seq. double integral theorem, 364 Frullani's integral, 325 Functions of the complex variable, 404 et seq. Fundamental integrals, 8, 124 GAMMA functions, 377 et seq. graph of, 378 table of, 401 r, 39. 392 Gauss' II -function, 374 theorem in /"-functions, 396 Graph of an integral, 99 et seq. of the /"-function, 378 Graph of multiple-angle series, 344 et se 1-> 353 Gyration, radius of, 237 HARMONIC series, 389 Hoi ditch's theorem, 215 ex. n Hyperbola, 76 ex. 50, 142 ex. 15, 170 ex. 12, 205 ex. 10 Hyperbolic (Naperian) logarithms, 13 ex. 25 Hyperbolic sine and cosine, 146 ex. 32 Hyperbolic spiral, 143 ex. 23 Hypocycloid, 242 ex. 18 IMAGE, 405 Increment, 107 Indefinite integral, 5, ic6 Inertia, moment of, 237 Infinite values of the element, 316 Infinite values of the limits, no, 318 Initial value, 4, 106, 410, 415 Instantaneous centre, 206 joot-note Integral, 2, 105, 123 joot-note definite, 4, 53, 107, 296 et seq. double, 155 et seq., 303 indefinite, 5, 106 regarded as limit of a sum, 121 triple, 163, 1 66 Integrand, 415 Integration about a pole, 417 around a branch-point, 420 by expansion, 94, 329 by parts, 77 et seq. by transformation, 33, 50 direct, 14 of forms containing radica's, 59, 421 of rational fractions, 15 et seq. of trigonometric forms, 33 et seq. over an area, 159, 161 over a volume, 165 under the integral sign, 299 Intermediate values, 107, 415 foot-note LEMNISCATE, 135, 179, 186 ex. 7 Limacon, 144 exs. 28, 29, 214 ex. 5 Limit of summation, 121 Limits of an integral, 5, 105 INDEX. 439 LimiiS of application of a Fourier's series, 342 , 360, 362 of a double integral, 156, 161 of a transformed integral, 53 of a triple integral, 164 Linear element, 164 Local probability, 269 et seq. Log r(i+ *), 389 Logarithmic derivative of F(x), 392 Loop, 412, 416 Loop-integral, 418 Loxodromic curve, 192 MAGNIFICATION (in mapping), 407 Mapping, or conformal representation, 45. 47 Mean areas, 256, 258 Mean distances: between variable points, 246, 248 from a fixed point, 243 from a plane, 232 Mean ordinates, 218 foot-note, 226 Mean squared distances: from a plane, 237 from an axis, 238 Mean values, 224 foot-note, 227, 230 Modulus, 404, 425 Moment, statical, 235 of inertia, 237, 238 Multiple -angle series, 332, 334, 341 el seq. differentiation of, 353 integration of, 355 Multiple -valued functions, 412 Multiple-valued integrals, 112, 115 NAPERIAN logarithms, 13 ex. 25 " Number of cases" of a continuous variable, 228, 230 ORDER of integration, 160, 173, 304 Oscillating series, 426 PARABOLA, 12 ex. 22, 132, 140 ex. 6, 153 ex. 13, 193 ex. 2, 195 ex. 10, 241 exs. 14, 16, 242 ex. 24 Parabola of wth degree, 12 ex. 24 Parabo'oid, 147, 203 ex. i, 242 ex. 25 Partial fractions, 15 et seq. formulae for numerator, 22, 23 Parts, integration by, 77 et seq. Periodic functions, 342, 420 Plane areas, 3, 129 et seq., 172 Planimeter, 211 Polar coordinates in space, 179 Polar element of area, 134, 172 Polar element of area, transformed, 136 Pole, 417 Power series in the complex variable, 426 Primary values, 113, 115,351, 412, 420 Principal values of an integral, 317, 319 Probability, 266 curves of, 276, 280, 281 Probable value, 278 QUADRATURE, approximate, 215 comparison of rules, 220 foot-note, 223 ex. 5 and foot-note Quasi-periodic functions, 335, 352, 353 Quick-return motion, 352 RADICALS, integrals involving, 59 et seq. 421 Radius of convergence, 428 Radius of gyration, 237 Random direction, 231 lines, 287 parts, 250 et seq. points, 230, 256 Rational fractions, 15 et seq., 320 Rational integral function, existence of a root, 432 Rectification of curves, 189 et seq. Rectilinear track, 412, 420, 423 Reducible tracks, 411 Riemann's surface, 414, 424 SEMI -CONVERGENT series, 427 Semi-cubical parabola, 241 ex. 15 Series in powers of the complex vari-. able, 426 in sines and cosines of multiple angles, 332, 334, 341 Maclaurin's, 431 . Taylor's, 96, 430 Simpson's rules, 217, 220 foot-note Solids of revolution, 178 Sphere, 170 exs. n, 12, 187 exs. u, 12, 241 ex. 10 Spherical coordinates, 147, 150, 181 440 INDEX. Spheroid, 203 exs. 2, 4 Spiral of Archimedes, 143 ex. 19, 185 ex. i Stationary point, 190 Stereographic projection, 407 fool-note Strophoid, 49 ex. 50, 76 ex. 52 Summation, limit of, 121 Surfaces of revolution, 198 in general, 199 TABLES: /"-functions, 401 I values of 5 n = 5 i, 403 Taylor's theorem, 96, 430 Track of complex variable, 410 of integration, 415 , rectilinear, 412, 423 reduced, 411 Tractrix, 153 ex. 14, 194 ex. 5, 204 ex. 8, 208 Transformation : of definite integrals, 119, 306 Transformation : of double integrals, 174 of triple integrals, 166 Triple integrals, 163 Two-valued functions, 410 ULTIMATE element, 164, 173 Uniform distribution of values of a function, 228, 232 VARIABLE density, 166 Viviani's problem, 203 Volume, integration over given, 165, 167 Volumes, 147 et seq., 159, 168, 178, 182 WEDDLE'S rule, 219 foot-note, 223 foot- note Weighted mean, 225, 234 Witch, 32 ex. 78, 130, 152 ex. 4 Woolley's rule, 221 495$ . 5 Johnson - 3! treatise on the integral calculus founded on the method QK A 000 191 050 4 Engineering & ^Mathematical Sciences Library 7