CORNELL 
 
 UNIVERSITY 
 
 LIBRARY 
 
 GIFT OF 
 
 Alexander Gray 
 Memorial Library 
 
 Electrical Engineering 
 
 ENGINEERING 
 
Cornell University Library 
 QA 308.B99 1888 
 
 Elements of the integral calculus :with 
 
 3 1924 004 779 447 
 
Cornell University 
 Library 
 
 The original of tiiis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924004779447 
 
ELEMENTS 
 
 INTEGRAL CALCULUS. 
 
 KEY TO THE SOLUTION OF DIFFERENTIAL 
 
 EQUATIONS, AND A SHORT TABLE 
 
 OF INTEGRALS. 
 
 WILLIAM ELWOOD BYERLY, Ph.D., 
 
 PBOFESaon OF MATHEMATICS IN HARVARD UNIVERSITY- 
 
 SECOND EDITION, REVISED AND ENLARGED, 
 
 GINN AND COMPANY 
 
 BOSTON NEW YORK • CHICAGO • LONDON 
 ATLANTA ■ DALLAS ■ COLUMBUS • SAN FliANCISCO 
 
c3 OS 
 
 ^ f—l / ■^ -m 
 
 '7 
 
 Entered according to Act of Congress, in the year 1888, by 
 
 avtlltam: elwood bterlt, 
 
 in the Office of the Librarian of CongresB at "Washington. 
 
 ALL RIGHTS 
 
 3(s 
 
 GINN AND COMPANY • PRO- 
 PRIETORS • BOSTON ■ U.S.A. 
 
PREFACE. 
 
 The following volume is a sequel to mj- treatise on the 
 Differential Calculus, and, like that, is written as a text-book. 
 The last chapter, however, a Kej' to the Solution of Differential 
 Equations, ma3' prove of service to working mathematicians. 
 
 I have used freelj' the works of Bertrand, Benjamin Peirce, 
 Todhunter, and Boole ; and 1 am much indebted to Professor 
 J. M. Peirce for criticisms and suggestions. 
 
 I refer constantly to mj- work on the Differential Calculus 
 
 as Volume 1. ; and for the sake of convenience I have added 
 
 Chapter V. of that book, which treats of Integration, as an 
 
 appendix to the present volume. 
 
 W. E. BYERLY. 
 Cambridge, 1881. 
 
PREFACE TO SECOND EDITION. 
 
 In enlarging my Integral Calculus I have used freely 
 Schlomilch's " Compendium der Hoheren Analysis," Cayley's 
 "Elliptic Functions," Meyer's " Bestimmte Integrale," For- 
 syth's " Differential Equations," and Williamson's "Integral 
 Calculus." 
 
 The chapter on Theory of Functions was sketched out and 
 in part written by Professor B. O. Peirce, to whom I am 
 greatly indebted for numerous valuable suggestions touching 
 other portions of the book, and who has kindly allowed me 
 to have his Short Table of Integrals bound in with this volume. 
 
 W. E. BYERLT. 
 Cambbidge, 1888. 
 
ANALYTICAL TABLE OF CONTENTS. 
 
 CHAPTER I. 
 
 SYMBOLS OF OPERATION, 
 irticle. Page. 
 
 1. Functiouai symbols regarded as symbols of operation 1 
 
 2. Compound function ; compound operation 1 
 
 3. Commutative or relatively free operations 1 
 
 4. Distributive or linear operations 2 
 
 5. The compounds of distributive operations are distributive ... 2 
 
 6. Symbolic exponents 2 
 
 7. The law of indices 2 
 
 8. The interpretation of a zero exponent 3 
 
 9. The iuterpretation of a negative exponent 3 
 
 10. When operations are commutative and distributive, the sym- 
 bols which represent them may be combined as if they v^ere 
 
 algebraic quantities 3 
 
 CHAPTER II. 
 
 IMAGIXAWES. 
 
 11. Usual definition of an imaginary. Imaginaries first forced upon 
 
 our attention in connection vrith quadratic equations . . .5 
 
 12. Treatment of imaginaries purely arbitrary aud conventional . . 6 
 
 13. V^ defined as a symbol of operation 6 
 
 14. The rules in accordance with which the symbol V — 1 is used. 
 
 V^ distributive and commutative with symbols of quantity . 7 
 
 15. Interpretation of powers of V^ 7 
 
 16. Imaginary roots of a quadratic 8 
 
 17. Typical form of an imaginary. Equal imaginaries 8 
 
 18. Geometrical representation of an Imaginary. Beals and pure 
 
 imaginaries. An interpretation of the operation V^ ... 8 
 
 19. The sum, the product, and the quotient of two imaginaries, 
 
 a + b V^ and c + d V^, are imaginaries of the typical 
 form 10 
 
VI INTEGRAL CALCULUS. 
 
 Article Page. 
 
 20. Second typical form r (cos ip + V^ sin <i>). Modulus and argu- 
 
 ment. Absolute value of an imaginary. Examples 1" 
 
 21. The modulus of the sum of two imaginaries is never greater 
 
 than the sum of their moduli H 
 
 22. Modulus and argument of the product of imaginaries 12 
 
 23. Modulus and argument of the quotient of two imaginaries . . 13 
 
 24. Modulus and argument of a power of an iiii igiimry ... .13 
 
 25. Modulus and argument of a root of an imaginary. Example . 14 
 
 26. Kelation between the n nth roots of a real or an imaginary . . 14 
 
 27. The imaginary roots of 1 and — 1. Examples 15 
 
 28. Conjugate imaginaries. Examples 17 
 
 29. Transcendental functions of an imaginary variable best defined 
 
 by the aid of series 17 
 
 30. Convergency of a series containing imaginary terms 18 
 
 31. Exponential functions of an imaginary. Definition of e' where 
 
 z is imaginary 19 
 
 32. The Zatoo/ indices holds for imaginary exponentials. Example 20 
 
 33. Logarithmic functions of an imaginary. Definition of log a. 
 
 Log 3 a pcn'otiie function. Example 21 
 
 34. Trigonometric functions of an Imaginary. Definition of sin z 
 
 and cos z. Example 22 
 
 35. Sinz and coss expressed In exponential form. The fundamen- 
 
 tal formulas of Trigonometry hold for imaginaries as well as 
 
 for reals. Examples 22 
 
 36. Differentiation of Functions of Imaginary Variables. The de- 
 
 rivative of a function of an Imaginary is in general indetermi- 
 nate 24 
 
 87. In difi'erentiating. we may treat the V^ like a constant factor. 
 Example. Two forms of the dififerential of the independent 
 
 variable 24 
 
 38. Dififerentiatiou of a power of a. 25 
 
 39. Differentiation of e'. Example 26 
 
 40. Differentiation of log z 26 
 
 41. Differentiation of sin 2 and cos 3 26 
 
 42. Formulas for direct integration (I., Art. 74) hold when x is 
 
 imaginary 27 
 
 43. Hyperbolic Functions 27 
 
 44. Examples. Properties of Hyperbolic Functions 28 
 
 45. Differentiation of Hyperbolic Functions 28 
 
 46. Anti-hyperbolic functions. Examples 28 
 
 47. Anti-hyperbolic ftinctions expressed as logarithms 29 
 
 48. Formulas for the direct integration of some irrational forms. 
 
 Examples 30 
 
TABLE OF CONTENTS. vii 
 
 CHAPTER III. 
 
 GENERAL METHODS OF INTEGKATING. 
 Article. Page. 
 
 49. Integral regarded as the inverse of a differential 32 
 
 50. If /k is any function whatever of x, fx.dx has an integral, and 
 
 but one, except for the presence of an arbitrary constant . . 32 
 
 51. A definite integral contains no arbitrary constant, and is a func- 
 
 tion of the values between which the sum is taken. Exam- 
 ples 33 
 
 52. Definite integral of a discontinuous function 33 
 
 53. Formulas for direct integration 34. 
 
 54. Integration by substitution. Examples 36 
 
 55. Integration by parts. Examples. Miscellaneous examples in 
 
 integration 37 
 
 CHAPTER IV. 
 
 RATIONAL FRACTIONS. 
 
 56. Integration of a rational algebraic polynomial. Rational frac- 
 
 tions, proper and improper 40 
 
 57. Every proper rational fraction can be reduced to a sum of sim- 
 
 pler fractions with constant numerators 40 
 
 58. Determination of the numerators of the partial fractions by 
 
 indirect methods. Examples 42 
 
 59. Direct determination of the numerators of the partial fractions 43 
 
 60. Illustrative examples 45 
 
 61. Case where some of the roots of the denominator are imaginary . 46 
 
 62. Illustrative example 47 
 
 63. Integration of the partial fractions. Examples 49 
 
 CHAPTER V. 
 
 REDUCTION FORMULAS. 
 
 64. Formulas for raising or lowering the exponents in the form 
 
 x'^-'^{a + bx-^)Pdx 52 
 
 65. Consideration of special cases. Examples 54 
 
Viii INTEGRAL CALCULUS. 
 
 CHAPTER VI. 
 
 IRRATIONAL FORMS. 
 Article. '"'^^ 
 
 66. Integration of the form /(.r, V a+bx)dx. Examples ob 
 
 (il. Integration of the form f(x, 'Vc+'\/a + bx)dx. Examples . . 57 
 
 68. Integration of the form f(x, Va + bx + cx^lx 57 
 
 69. Illnstrative example. Examples 59 
 
 70. Integration of the form /fa;, A/^£±i'W Example ... .61 
 
 \ yix+mj 
 
 71. Application of the Beduction Formulas of Chapter V. to irra- 
 
 tional forms. Examples CI 
 
 72. A function rendered irrational through the presence under 
 
 the radical sign of a polynomial of higher degree than the 
 second cannot ordinarily be integrated. Elliptic Integrals . 62 
 
 CHAPTER VII. 
 
 TRANSCENDEKTAL FUNCTIONS. 
 
 73. Use of the method of Integration by Parts. Examples .... 63 
 
 74. Reduction Formulas for sin" X and cos" a:. Examples 64 
 
 75. Integration of {su\-^x)"dx. Examples 65 
 
 76. Use of the method of Integration by Substitution 66 
 
 77. Substitution of 2 =tan- in integrating trigonometric forms . 67 
 
 78. Integration of sin^a; cos"a;. ax. Examples 68 
 
 79. Reduction formulas for j tan" x.tZx and 4—^^^ — Examples. . 69 
 
 J ^tan"x 
 
 CHAPTER VIII. 
 
 DEFINITE INTEGRALS. 
 
 80. Definition of a definite integral as the limit of a sum of 
 
 infinitesimals 7] 
 
 81. Computation of a definite integral as the limit of a sum. Illus- 
 
 trative examples. Examples . . 73 
 
 82. Usual method of obtaining the value of a definite integral. 
 
 Caution concerning multiple- valued functions. Examples . 76 
 
TABLE OF CONTENTS. ix 
 
 Article. Page. 
 
 83. Consideration of the nature of the value of j /x . cfe when fx 
 
 becomes infinite for x = a, or x = 6, or for some value of x 
 between a and 6. Illustrative examples 80 
 
 84. Maxctimurn-Mmimum Theorem. Test that must be satisfied in 
 
 order that I fx .dx may be finite and determinate if fx is 
 
 infinite for some value of x between a and 6. Illustrative 
 examples. Examples 82 
 
 85. Meaning of j fx . dx. Condition that I fx .dx shall be finite 
 
 and determinate 88 
 
 86. Proof that certain important definite integrals of the form 
 
 i 
 
 fx . dx are finite and determinate. Examples 
 
 87. Application of reduction formulas to definite integrals. 
 
 Examples 91 
 
 88. Application of the method of integration by substitution to 
 
 definite integrals. Illustrative examples. Example ... 9.3 
 
 89. Differentiation of a definite integral. Examples 96 
 
 90. Many ingenious methods of finding the values of definite inte- 
 
 grals are valid only in case the integral is finite and de- 
 terminate 100 
 
 91. Integration by development in series. Examples. . . . 100 
 
 IT 
 
 92. Values of j logsinx.iix, I e-'^dx, i dx obtained 
 
 Jo Jo Jo X 
 by Ingenious devices. Examples 102 
 
 93. Differentiation or integration with respect to a quantity which 
 
 is independent of x. Examples 105 
 
 94. Additional illustrative examples. Examples 106 
 
 95. Introduction of imaginary constants 108 
 
 96. The Gamma Function 109 
 
 97. Table giving logr(ra) from « = 1 to « = 2. Definite integrals 
 
 expressed as Gamma Functions Ill 
 
 98. The Beta Function. Formula connecting the Beta Function 
 
 with the Gamma Function. Value of r (J) 113 
 
 99. More definite integrals expressed as Gamma Functions. 
 
 Examples ' 114 
 
INTEGKAl. CAIiCULUS. 
 
 CHAPTER IX. 
 
 LENGTHS OF CURVES, 
 irticle. P»Be- 
 
 100. Formulas for sinr and cos t in terms of the length of the arc 117 
 
 101. The equation of the Catenary obtained. Example . . 117 
 
 102. The equation of the Tractrix. Examples 119 
 
 103. Length of an arc in rectangular coordinates . . . . 121 
 
 104. Length of the arc of the Cycloid. Example . . . . . 122 
 
 105. Another method of rectifying the Cycloid 12.3 
 
 106. Rectification of the Epicycloid. Examples 12.3 
 
 107. Arc of the Ellipse. Auxiliary angle. Example 124 
 
 108. Length of an arc. Polar coordinates . . . .... 125 
 
 109. Equation of the Logarithmic Spiral 125 
 
 110. Rectification of the Logarithmic Spiral. Examples .... 126 
 
 111. Rectification of the Cardioide 126 
 
 112. Involutes. Illustrative example. Example 127 
 
 113. The involute of the Cycloid. Example 129 
 
 114. Intrinsic equation of a curve. Example ... .... 130 
 
 115. Intrinsic equation of the Epicycloid. Example 131 
 
 116. Intrinsic equation of tlie Logarithmic Spiral 132 
 
 117. Method of obtaining the intrinsic equation from the equation 
 
 in rectangular coordinates. Examples 132 
 
 118. Intrinsic equation of an evolute 134 
 
 119. Illustrative examples. Examples 134 
 
 120. The evolute of an Epicycloid. Example . 135 
 
 121. The intrinsic equation of an involute. Illustrative examples 136 
 
 122. Limiting form approached by an involute of an involute . . 137 
 
 123. Method of obtaining the equation in rectangular coordinates 
 
 from the intrinsic equation. Illustrative example .... 138 
 
 124. Rectification of Curves in Space. Examples 139 
 
 CHAPTER X. 
 
 AREAS. 
 
 125. Areas expressed as definite integrals, rectangular coordinates. 
 
 Examples 141 
 
 126. Areas expressed as definite integrals, polar coordinates - 143 
 
 127. Area between the catenary and the axis . 143 
 
 128. Area between the tractrix and the axis. Example .... 143 
 
TABLE OP CONTENTS. xi 
 
 Article. Page. 
 
 129. Area between a curve and its asymptote. Examples . . . 144 
 
 130. Area of circle obtained by the aid of an auxiliary angle. Ex- 
 
 amples 145 
 
 131. Area between two curves (rect. coor.). Examples .... 146 
 
 132. Areas in Polar Coordinates. Examples 147 
 
 133. Problems in areas can often be simplified by transformation 
 
 of coordinates. Examples ... 150 
 
 134. Area between a curve and its evolute. Examples 150 
 
 135. Holditch's Theorem. Examples .... 151 
 
 136. Areas by a double integration (rect. coor.) 153 
 
 137. Illustrative examples. Examples 154 
 
 188. Areas by a double integration (polar coor.). Example . . . 165 
 
 CHAPTER XI. 
 
 AREAS OF SURFACES. 
 
 139. Area, ot a. surface of revolutioji (yect. COOT.). Example . . . 157 
 J40. Illustrative examples. Examples 158 
 
 141. Area of a surface of revolution by transformation of coordi- 
 
 nates. Example 159 
 
 142. Area of a surface o/reBoteHon (polar coor.). Examples . . 161 
 
 143. Area of a cylindrical surface. Examples 161 
 
 144. Area of any surface by a double integration ... ... 164 
 
 145. Illustrative example. Examples 167 
 
 146. Illustrative example requiring transformation to polar coordi- 
 
 nates. Examples 169 
 
 CHAPTER XII. 
 
 VOLUMES. 
 
 147. Volume by a single integration. Example 172 
 
 148. Volume of a conoid. Examples 173 
 
 149. Volume of an ellipsoid. Examples 174 
 
 150. Volume of a solid of revolution. Single integration. Exam- 
 
 ples 175 
 
 151. Yolnme of a, solid of revolution. Double integration. Exam- 
 
 ples 176 
 
 152. Volume of a, solid of revolution. Polar formula. Example . 178 
 163. Volume of any solid. Triple integration. Rectangular coor- 
 dinates. Examples . 179 
 
 154. Volume of any solid. Triple integration. Polar coordinates. 
 
 Examples 183 
 
XU INTEGRAL CALCULUS. 
 
 CHAPTER XIII. 
 
 CENTRES OF GRAVITY. 
 Article. ^age. 
 
 155. Centre of Gravity deflned .... 184 
 
 156. General formulas for the coordinates of the Centre of Gra\ity 
 
 of any mass. Example . 184 
 
 157. Centre of Gravity of a homogeneous body . . . . 186 
 
 158. Centre of Gravity of a. plane area. Examples 186 
 
 159. Centre of Gravity of a homogeneous solid of revolution. Ex- 
 
 amples . . . . . . . . . 189 
 
 160. Centre of Gravity of an arc; of a surface of revolution. Ex- 
 
 amples . . . . . . . 191 
 
 161. Properties of Quldin. Examples . . 192 
 
 CHAPTER XIV. 
 
 LINE, SURFACE, .\ND SPACE INTEGRALS. 
 
 162. Point function. Continuity of a point function 194 
 
 163. Line-integral, surface-integral, and space-integral of a point 
 
 function 194 
 
 164. Value of a line, surface, or space integral independent of the 
 
 position In each element of the point at which the value of 
 the function is taken . . . .... ... . 195 
 
 165. Value of a line, surface, or space integral independent of the 
 
 manner in which the line, surface, or space is broken up 
 into infinitesimal elements . 195 
 
 166. Geometrical representation of a line-integral along a plane 
 
 curve, and a surface-integral over a plane surface .... 197 
 367. Moments of inertia. Examples .... 197 
 
 168. Relation between a surface-integral over a plane surface, and 
 
 a line-integral along the curve bounding the surface. Ex- 
 ample . . . . 200 
 
 169. Illustrative example. Examples . . .... 202 
 
 170. Anotherformof the relation established in Art. 168. Example 203 
 
 171. Relation betweena space-integral taken throughoiit a given 
 
 space and a surface-integral over the surface bounding the 
 space. Example . .203 
 
 172. Illustrative example. Example . _ 205 
 
TABLE OF CONTENTS. xiu 
 
 CHAPTER XV. 
 
 MEAN VALUE AXK PROBABILITY. 
 Article. Page. 
 
 173. References 206 
 
 174. Mean value of a continuously varying quantity. The mean dis- 
 
 tance of all the points of the circumferences of a circle from 
 a fixed point on the circumference. The mean distance of 
 points on the surface of a circle from a fixed point on the 
 circumference. The mean distance of points on the surface 
 of a square from a corner of the square. The mean distance 
 between two points within a given circle ... . . 206 
 
 175. Problems in the application of the Integral Calculus to proba- 
 
 bilities. Random straight lines. Examples 208 
 
 CHAPTER XVI. 
 
 ELLIPTIC INTEGRALS. 
 
 176. Motion of a, simple pendulum. Vibration. Complete revolution 215 
 
 177. The length of an arc of an Ellipse 217 
 
 178. Algebraic forms of the Elliptic Integrals of the first, second, 
 
 and third class. Modulus. Parameter 21T 
 
 179. Trigonometric forms of the Elliptic Integrals. Amplitude. 
 
 Delta. Complementary Modulus ... 218 
 
 180. Landen's Transformation. Reduction formula by which we 
 
 can increase the modulus and diminish the amplitude of an 
 Elliptic Integral of the first class. A method of computing 
 F{k, (p) 219 
 
 181. Reduction formula for diminishing the modulus and increas- 
 
 ing the amplitude of an Elliptic Integral of the first class. 
 A second method of computing J?" (A, f) 222 
 
 182. Actual computation of f(—, -^ and Ef—, -\ .... 22$ 
 
 183. Landen's Transformation. Reduction formula by which we 
 
 can increase the modulus and diminish the amplitude of an 
 Elliptic Integral of the second class. A method of com- 
 puting E(k,<i,) 226 
 
 184. A reduction formula for diminishing the modulus and in- 
 
 creasing the amplitude of an Elliptic Integral of the second 
 class. A second method of computing E{k, <p) 229 
 
XIV INTEGRAL CALCTJLUS. 
 
 Article. _ ^Kt 
 
 185. Actual computation of iV^, -\ and sf^, ^V . • -231 
 
 186. An Elliptic Integral of the first or second class, whose ampli- 
 
 tude is greater than -, can be made to depend upon one 
 whose amplitude is less than -, and upon the correspond- 
 
 ing complete Elliptic Integral 235 
 
 187. Three-place table of Elliptic Integrals of the first class and 
 
 the second class 287 
 
 188. Addition Formulas. Functions defined by the aid of definite 
 
 integrals, logx, sin'^x, tan'^x, F(k,x), E (^k,x). Addition 
 formula for log x . . . . .... .... 239 
 
 189. Addition formulas for sin" Ik and tan" ^ a; .... ... 241 
 
 190. Addition formula for i^(fc,x) ... 243 
 
 191. Analogy between log"i?i, sin ?(, tan ?(, and F"i(&, tt) . . . .245 
 
 192. Tlie Elliptic Functions, snu, cn«, and dna. Their analogy 
 
 with Trigonometric Functions. Formulas connecting the 
 Elliptic Functions of a single quantity 246 
 
 193. Formulas for Elliptic Functions of (m + u) and (u — v). . 248 
 
 194. Formulas for Elliptic Functions of 2 M , . .250 
 
 195. Formulas for Elliptic Functions of - . 250 
 
 196. PenodtoYy of the Elliptic Functions. Real period, 4 A'. . .251 
 197 Elliptic Functions of a pure imaginary. Jacobi's Transforma- 
 tion. Elliptic Functions have an imaginary period, 4 K'\/—i. 
 Table of values of Elliptic Functions having the modulus 
 
 v^ 253 
 
 2 
 
 198. The Elliptic Integral of the second class expressed in terms 
 
 of Elliptic Functions. Addition formula for Elliptic Inte- 
 grals of the second class . . . 256 
 
 199. Application to the rectification of the Lemniscate. Examples. 
 
 -Bisection of the arc of a quadrant of the Lemniscate . . . 258 
 
 200. Rectification of the Ellipse. Examples 261 
 
 201. Use of the Addition Formula in dealing. with Elliptic arcs. 
 
 Fagnani's Point. Examples 262 
 
 202. Rectification of the Hyperbola. Examples . 264 
 
 208. The simple pendulum. Examples 267 
 
 204. Direct integration of irrational Algebraic functions. Examples 275 
 
 205. Transformation of Elliptic Integrals. Examples . . . 281 
 
 206. Integration of Elliptic Functions. Examples . . ... 282 
 
TABLE OP CONTENTS. XV 
 
 CHAPTER XVII. 
 
 INTRODUCTION TO THE THEORY OF FUNCTIONS. 
 Article. Paf;& 
 
 207. Single-valued functions. Multiple-valued functions .... 283 
 
 208. Importance of the grapliical representation of imaginaries. 
 
 Complex quantity 283 
 
 209. When a complex variable is said to vary continuously . . . 284 
 
 210. A continuous function of a complex variable. Critical values 284 
 
 211. Criterion that a function shall have a determinate derivative. 
 
 Monogenic functions ... 285 
 
 212. Any function involving as a whole is a monogenic function 288 
 
 213. Conjugate functions. Their use as solutions of Laplace's 
 
 Equation. Example 288 
 
 214. Preservation of angles 290 
 
 215. If two paths traced by the point representing the variable 
 
 have a common beginning and a common end, and do not 
 enclose a critical point, the corresponding paths traced by 
 the point representing the function and having a common 
 beginning will have a common end 292 
 
 216. Examples where the paths traced by the point representing 
 
 the variable enclose a critical point 294 
 
 217. Critical points at which the derivative of the function is zero 
 
 or infinite are to be avoided. Branch points. Holomorphio 
 functions 295 
 
 218. Definite integral of a function of a complex variable defined. 
 
 Such an integral is generally indeterminate, and depends 
 upon the path by which the point representing the variable 
 passes from the lower limit to the upper limit of the integral 297 
 
 219. If the function is holomorphio, the definite integral is in gen- 
 
 eral determinate 299 
 
 220. The integral around a closed contour, embracing a point at 
 
 which the function is infinite 300 
 
 221. Illustrative examples 301 
 
 222. Convergency of the series obtained by integrating the terms 
 
 of a convergent series where the separate terms are holo- 
 morphio functions 303 
 
 223. Proof of Taylor's and Maclaurin's Theorems for functions of 
 
 complex variables. Circle of convergence 304 
 
 224. Investigation of the convergency of various series which are 
 
 obtained by Taylor's and Maclaurin's Theorems. Examples 307 
 
XVl INTEGKAL CALCULUS. 
 
 CHAPTER XVIII. 
 
 KET TO THE SOLUTION OF DIFFERENTIAL EQUATIONS. 
 Aitiole. P"«e 
 
 225. Description of Key 312 
 
 226. Definition of tlie terms differential equation, order, degree, 
 
 linear, general solution or complete primitive, singular solu- 
 tion, exact differential equation 312 
 
 227. Examples illustrating the use of the Key 314 
 
 228. Simplification of dififerential equations by change of variable . 324 
 
 Key 326 
 
 Examples t'ndei! Key 349 
 
INTEGRAL CALCULUS. 
 
 CHAPTER I. 
 
 SYMBOLS OF OPEBATION. 
 
 It is often convenient to regard a functional symbol as 
 Vidicating an operation to be performed upon the expression 
 which is written after the symbol. From this point of view the 
 symbol is called a symbol of operation, and the expression writ- 
 ten after the symbol is called the subject of the operation. 
 
 Thus the symbol D^ in D^{x^y) indicates that the operation of 
 differentiating with respect to x is to be performed upon the 
 subject {pi?y). 
 
 u 2. If the result of one operation is taken as the subject of a 
 'second, there is formed what is called a compound function. 
 
 Thus logsino; is a compound function, and we may speak of 
 the taking of the logsin as a compound operation. 
 
 3. When two operations are so related that the compound 
 operation, in which the result of performing the first on any 
 subject is taken as the subjeej of the second, leads to the same 
 result as the compound operation, in which the result of per- 
 forming the second on the same subject is taken as the subject 
 of the first, the two operations are commutative or relatively free. 
 
 Or to formulate ; if 
 
 fFu = Ffu, 
 
 the operations indicated by / and F are commutative. 
 
2 INTEGRAL CALCXJLTJS. [Akt. i. 
 
 For example ; the operations of partial differentiation with 
 respect to two independent variables x and y are commutative, 
 for we know that 
 
 D^D,u = DyD^u. (I. Art. 197) . 
 
 The operations of taking the sine and of taking the logarithm 
 are not commutative, for log sin m is not equal to sin log m. 
 
 4. If f(u±v)=fu±fv 
 
 where u and v are any subjects, the operation /is distributive or 
 linear. 
 
 The operation indicated bj- d and the operation indicated by 
 £>! are distributive, for we know that 
 
 d{u ±v) = du± dv, 
 
 and that -D,(m ± v) = D^u ± D^v. 
 
 The operation sin is not distributive, for sin(w + v) is not 
 equal to sin a + sin-y. 
 
 5. The compounds of distributive operations are distributive. 
 Let / and F indicate distributive operations, then fF will be 
 
 distributive ; for 
 
 F{u ±v) = Fu ± Fv, 
 therefore fF{u ±v)= f{Fu ± Fv) = fFu ± fFv. 
 
 6. The repetition of any operation is indicated by writing an 
 exponent, equal to the number of time's the operation is per- 
 formed, after the symbol of the operation. 
 
 Thus log' a; means loglogloga; ; d'u means dddu. 
 
 In the single case of the trigonometric functions a different 
 use of the exponent is sanctioned by custom, and sin^w means 
 (sinu)^ and not sin sinu. 
 
 7. If m and n are whole numbers it is easily proved that 
 /"■/»«=/"■ + "it. 
 
Chap. 1.] SYMBOLS OF OPERATION. 3 
 
 This formula is assumed for all values of m and n, and nega- 
 tive and fractional exponents are interpreted by its aid. It is 
 called the law of indices. 
 
 8. To find what interpretation must be given to a zero ex- 
 ponent, let 
 
 m = in the formula of Art. 7. 
 
 or, denoting/"^ by v, f^v = v. 
 
 That is ; a symbol of operation with the exponent zero has no 
 effect on the subject, and may be regarded as multiplying it by 
 unity. 
 
 9. To interpret a negative exponent, let 
 
 m= —n in the formula of Art. 7. 
 
 /-»/»M=/-» + "M=:/Oj( = M. 
 
 If we call /"m = V, then /"""« = u. 
 
 If n=\ 
 
 we get f-^fu=u, 
 
 and the exponent —1 indicates what we have called the anti- 
 function of /«. (I. Art. 72.) 
 
 The exponent — 1 is used in this sense even with trigonometric 
 functions. 
 
 10. When two operations are commutative and distributive, 
 the symbols which represent them may be combined precisely as 
 if they were algebraic quantities. 
 
 For they obey the laws, 
 
 a{m + w) = am -\- an, 
 
 am = ma, 
 
 on which all the operations of arithmetic and algebra are founded. 
 
4 INTEGRAL CALCULUS. t-A-RT- 10- 
 
 For example ; if the operation {D^ + Dy) is to be performed 
 n times in succession on a subject it, we can expand {D^-\- Dy)" 
 preciselj- as If it were a binominal, and then perform on u the 
 operations indicated by the expanded expression. 
 
 (A + Dy) ^ » = (Z>/ + 3 D^'Dy + 3 D, Z>/ + D/) u 
 
 = DJu + 3 D^WyU + 3 D^Dy'ii + DJ'u. 
 
Chap. II.] IiXAGi:NABIES. 
 
 CHAPTER II. 
 
 IMAGINAEIES. 
 
 11. An imaginary is usually defined in algebra as the indi- 
 cated even root of a negative quantity, and although it is clear 
 that there can be no quantity that raised to an even power will 
 be negative, the assumption is made that an imaginary can be 
 treated like any algebraic quantit3-. 
 
 Imaginaries are first forced upon our notice in connection 
 
 with the subject of quadratic equations. Considering the tj'pical 
 
 quadratic „ , , , „ 
 
 ^ .1;^ + ace + 6 = 0, 
 
 we find that it has two roots, and that these roots possess cer- 
 tain important j^roperties. For example ; their sum is —a and 
 their product is b. We are led to the conclusion that ever}' 
 quadratic has two roots whose sum and whose product are 
 simplj' related to the coefficients of the equation. 
 
 On trial, however, we find that there are quadratics having 
 but one root, and quadratics having no root. 
 
 For example ; if we solve the equation 
 
 a^-2a; + l = 0, 
 
 we find that the only value of x which will satisfy it is unity ; 
 and if we attempt to solve 
 
 a^ — 2a; + 2 = 0, 
 
 we find that there is no value of x which will satisfy the equation. 
 
 As these results are apparentl}' inconsistent with the conclu- 
 sion to which we were led on sol^ang the general equation, we 
 naturallj' endeavor to reconcile them with it. 
 
 The difflcultj' in the case of the equation which has but one 
 
6 INTEGRAL CALCULUS. [Abt. 12. 
 
 root is easil}' overcome by regarding it as having two equal roots. 
 Thus we can say that each of the two roots of the equation 
 
 a^-2a!+l = 
 
 is equal to 1 ; and there is a decided advantage in looking at the 
 question from this point of view, for the roots of this equation 
 will possess the same properties as those of a quadratic having 
 unequal roots. The sum of the roots 1 and 1 is minus the co- 
 eflficient of x in the equation, and their product is the constant 
 term. 
 
 To overcome the difficulty presented by the equation which 
 has no root we are driven to the conception of imaginaries. 
 
 12. An imaginary is not a quantity, and the treatment of 
 imaginaries is purely arbitrary and conventional. We begin by 
 laj-ing down a few arbitrar}' rules for our imaginary expressions 
 to obej', which must not involve any contradiction ; and we 
 must perform all our operations upon imaginaries, and must 
 interpret all our results hy the aid of these rules. 
 
 Since imaginaries occur as roots of equations, they bear a close 
 analogy with ordinary algebraic quantities, and they have to be 
 subjected to the same operations as ordinary quantities ; there- 
 fore our rules ought to be so chosen that the results may be 
 comparable with the results obtained when we are dealing with 
 real quantities. 
 
 13. By adopting the convention that 
 
 V— a^ = a V^^, 
 
 where a is supposed to be real, we can reduce all our imaginary 
 algebraic expressions to forms where V — 1 is the only peculiar 
 sj'mbol. This symbol V —1 we shall define and use as the sym- 
 bol of some operation, at present unknown, the repetition of which 
 has the effect of_changing the sign of the subject of the operation. 
 Thus in a V — 1 the symbol V — 1 indicates that an operation 
 is performed upon a which, if repeated, will change the sign 
 
 of a. That is, 
 
 a(V-l)^=-o. 
 
Chap. II.] IMAGINARIES. 7 
 
 From this point of view it would be more natural to write tiie 
 symbol before instead of after the subject on which it operates, 
 (V — l)c6 instead of aV — 1, and this is sometimes done; but 
 as the usage of mathematicians is overwhelmingly in favor of the 
 second form, we shall emploj' it, merelj' as a matter of con- 
 venience, and remembering that a is the subject and the V — 1 
 the syvibol of operation. 
 
 14. The rules in accordance with which we shall use our new 
 symbol are, first, 
 
 aV^+6V^^ = (a + &)V^l. [1] 
 
 In other words, the operation indicated by V — 1 is to be dis- 
 tributive (Art. 4) ; and second, 
 
 a-\r^l = (■sr^l)a, [2] 
 
 or our sj-mbol is to be commutative with the symbols of quantity 
 (Art. 3). 
 
 These two conventions will enable us to use our symbol in 
 algebraic operations precisely' as if it were a quantitj- (Art. 10) . 
 
 "When no coefficient is written before V — 1 the coefficient 1 
 will be understood, or unity will be regarded as the subject of 
 the operation. 
 
 15. Let us see what interpretation we can get for powers of 
 V — 1 ; that is, for repetitions of the operation indicated by the 
 symbol. 
 
 (^/Zri)''=l (Art. 8), 
 
 (V"=a)'= V^T, 
 
 (V^a)2= -1, by definition (Art. 13), 
 
 (\A^1)3= (V"^)V^n = - V^^, by definition, 
 (V"=^)*=-(V^T)^ =1, 
 
 (V^i)'=]V^n. =v"^, 
 
 (V^-l)e=(V^rT)^ =_1, 
 
 and so on, the values V— 1, —1, — V— 1, 1, occun-ing in 
 cycles of four. 
 
8 INTEGRAL, CALCULUS. [AbT. 16. 
 
 16. The definition we liaxc given for the square root of a 
 
 negative quantit3-, and tlie rules we have adopted concerning its 
 
 use, enable us to remove entirely the diffleultj' felt in dealing 
 
 with a quadratic which does not have real roots. Take the 
 
 equation 
 
 fl^-2a; + 5 = 0. (1) 
 
 Solving by the usual method, we get 
 
 X = 1 ± V^^ ; 
 
 V^^ = 2 V^l, by Art. 13 [1] ; 
 
 hence a;=l + 2V^^ or 1 — 2V^1. 
 
 On substituting these results in turn in the equation (1), per- 
 forming the operations b}' the aid of our conventions (Art. 14 
 [1] and [2]), and interpreting (V — 1)° by Art. 16, we find that 
 they both satisfy the equation, and that they can therefore be 
 regarded as entirely analogous to real roots. We find, too, that 
 their sum is 2 and that their product is 5, and consequently that 
 they bear the same relations to the coefficients of the equation as 
 real roots. 
 
 17. An imaginarj' root of a quadratic can alwaj's be reduced 
 to the form a + b V — 1 where a and b are real, and this is taken 
 as the general type of an imaginary ; and part of our work will 
 be to show that when we subject imaginaries to the ordinary 
 functional operations, all our results are reducible to this typical 
 form. 
 
 If two imaginaries a + bV^T and c + dV^ are equal, 
 a must be equal to c, and b must be equal to d. 
 
 For we have a + 6V — 1 =c + dV^^. 
 
 Tlieref ore a — c =(d — b) V^^ , 
 
 or a real is equal to an imaginary, unless a — c = = d — b. 
 
 Since obviously a real and an imaginary cannot be equal, it 
 follows that a = c and & = d, 
 
Chap. II.] 
 
 IMAGIN ARIES. 
 
 9 
 
 18. We have defined V — 1 as the symbol of an operation 
 whose repetition changes the sign of the subject. 
 
 Several different interpretations of this operation have been 
 suggested, and the following one, in which every imaginary is 
 graphicall}' represented bj' the position of a point in a plane, is 
 commonly adopted, and is found exceedingly useful in suggest- 
 ing and interpreting relations between diflerent imaginaries and 
 between imaginaries and reals. 
 
 In the Calculus of Imaginaries, a + & V — 1 is taken as the 
 general symbol of quantity. If b is equal to zero, a + b V — 1 
 reduces to a, and is real ; if a is equal to zero, a + 6 V — 1 re- 
 duces to b V — 1 , and is called a pure imaginan/. 
 
 a + 6 V — 1 is represented by the position of a point referred 
 to a pair of rectangular axes, as in analytic geometry, a being 
 taken as the abscissa of the 
 point and b as its ordmate. 
 Thus in the figure the position 
 of the point P represents the 
 imaginary a + b V— 1 . 
 
 If 6 = 0, and our quantity is 
 
 real, P will lie on the axis of 
 
 X, which on that account is 
 called the aods of reals; if a=0, 
 and we have a pwre Imaginary, 
 P will lie on the axis of Y, 
 which is called the axis of pure imaginaries. 
 
 It follows from Art. 17 that if two imaginaries are equal, the 
 points representing them wUl coincide. 
 
 Since a and aV^ are represented by points equally distant 
 from the origin, and lying on the axis of reals and the axis of 
 pure imaginaries respectively, we may regard the operation 
 indicated by V^ as causing the point representing the subject 
 of the operation to rotate about the origin through an angle of 
 90°. A repetition of the operation ought to cause the point to 
 rotate 90° further, and it does ; for 
 
 a(V-])2 = -a, 
 and is represented by a point at the same distance from the 
 
10 
 
 INTEGRAL CALCULUS. 
 
 [Abt. 19. 
 
 origin as o, and Ij'ing on the opposite side of the origin ; again 
 repeat the operation, 
 
 and the point has rotated 90° further ; repeat again, 
 
 a(V^l)' = a, 
 
 and the point has rotated through 360°. "We see, then, that if 
 the subject is a real or & pure imaginm~y the effect of performing 
 on it the operation indicated bj' V — 1 is to rotate it about the 
 origin through the angle 90°. We shall see later that even when 
 the subject is neither a real nor a pure imaginarj", the effect of 
 operating on it with V — 1 is still to produce the rotation just 
 described. 
 
 19. The sum, the j)rori«c?, and the quotient of anj- two imagi- 
 naries, a + 6 V — 1 and c + d V — 1, are imaginaries of the typi- 
 cal form. 
 
 a + &V-l+c + dV^ =«+c + (&-f-d)V^I. [1] 
 
 (a + &V^) (c + dV^) =ac. — bd+ {bc + ad)^^. [2] 
 
 g+ftV^ _ (ft+&V^) («— dV^) _ ac+bd+ (bc—ad)^f^ 
 c+dV^~ (c+dV^T) (c-dV^)~ <^ + <^ 
 
 ac + bd be — ad 
 
 1-1. 
 
 (^ + d^ c- + ( 
 All these results are of the form A+B^—1. 
 
 [3] 
 
 20. The graphical representation we have suggested for 
 imaginaries suggests a second typical form for an imaginary. 
 Given the imaginary x + y^ — 1, let the polar coordinates of 
 the point P which represents a + yV — Iber and <^. 
 
 r is called the modulus and i^ the argument of the imaginary. 
 
Chap. II.] 
 
 IMAGINAEIES. 
 
 11 
 
 The figure enables us tx) establish very- 
 simple relations between x, y, r, and </>. 
 
 x = rcosi> 
 y= ?-sin<^ 
 
 ;} 
 
 r = yj3?+ f, 
 <A= tan-i y- . 
 
 ^ X 
 
 x + yy/—l = rcos<l>+ (V — l)?'sin^ 
 = r(cos^+V— l.sin^), 
 
 [1] 
 [2] 
 
 [3] 
 
 where the imaginary is expressed in terms of its modulus and 
 argument. 
 
 The value of r given by our formulas [2] is ambiguous in 
 sign ; and ff> may hare any one of an infinite number of values 
 differing by multiples of ir. In practice we .always take the 
 positive value of ;-, and a value of ^ which will bring the point 
 in question into the right quadrant. In the case of any given 
 imaginary then, r can have but one value, while <j) may have any 
 one of an infinite number of values differing by multiples of 2ir. 
 
 The modulus r is sometimes called the absolute value of the 
 
 imaginary. 
 
 Examples. 
 
 (1) Find the modulus and argument of 1 ; of V — 1 ; of — 4 ; 
 of — 2V^^; of 3+3V^T; of 2+4 V^; and express each of 
 these quantities in the form r (cos (^ + V — 1 . sin ^) . 
 
 (2) Show that every positive real has the argument zero; 
 every negative real the argument ir ; every^ positive pure imagi- 
 nary the argument - ; and every negative pure imaginary the 
 argument — . 
 
 21. If we add two imaginaries, the modulus of the sum is 
 never greater than the sum of the moduli of the given imagi- 
 
12 INTEGRAL CALCULUS. [Akt. tt. 
 
 Thesumof a+6V^^ and c +dV^I is a + c + (&+t^)V-l- 
 The modulus of this sum is V(a + c)' + (6 + d)' ; t he sum of 
 the moduli of a +& V^ and c +d V- 1 is V a^ + 6^ + V?+^^- 
 We wish to show that 
 
 the sign -< meaning '• equal to or less than." 
 
 Now V(a + c)2 + (6 + c?)2 -< Va2 + ^'^ + Vc2+d^ 
 
 if (a + c)^ + {b + ay -< a^ + W + •2V(rt- + 6') (c^ + d^ +0= + d^ 
 
 that is, if «c + 6d -< Va^ c" + a^ cP + V'(?+ V dP ; 
 
 or, squaring, if 
 
 a^c^ + 2 ahcd + S^d^ -< a V + a^rP + h^<? + ft^d^ ; 
 
 or, if -< (((d-k')-". 
 
 This last result is necessarily true, as no real can have a 
 square less than zero ; hence our proposition is established. 
 
 22. The modulus of the product of two imaginaries is the 
 product of the moduli of the given imaginaries, and the ctrgmnent 
 of the product is the sum of the arguments of the imaginaries. 
 
 Let us multiply 
 
 ri(cos<^i+V — l.sin<^i) by ?-2(cos<^2+V — 1. sini^s) ; 
 we get 
 
 ri?-2[cos 01 cos (^2— sin (^i sin <^2+ V— l(sin 0i cos 02+cos <^isin c^j)], 
 cos <^i cos <^2 — sinc^isin02= <-^os{(f>i + <f,^), 
 sin 01 cos 02 + cos 0i sin 02 = sin (0i + 02) 
 by Trigonometry ; hence 
 
 ri (cos 01 + V - 1 . sin 0i) r^ (cos 02 + V^^. sin 02) 
 = r^r^ [cos(0i + 02) + V^. sin(0i + 02)], 
 
Chap. II.] IMAGINARIES. 13 
 
 and our result is in the typical form, r^r^ being the modulus and 
 <^i + 4>2 the argument of the product. 
 
 If each factor has the modulus unity, this theorem enables us 
 to construct ver}^ easily the product of the imaginaries ; it also 
 enables us to show that the interpretation of the operation V — 1 , 
 suggested in Art. 18, is perfectly general. 
 
 Let us operate on anjr imaginary subject, 
 
 ?-(cos^ + V — 1. sin<^), withV— 1, 
 
 that is, with 1 f cos- + V — 1. sin- V 
 
 The modulus r will be unchanged, the argument 4> 'w^iH be in- 
 creased by -, and the effect will be to cause the point repre- 
 senting the given imaginary to rotate about the origin through 
 an angle of 90°. 
 
 23. Since division is the inverse of multiplication, 
 
 ri(cos^i + V — 1. sin^i) -r- r2(cos<^2 + V— 1. sini^g) 
 will be equal to 
 
 -^ [cos (<^i — <^2) + V— 1. sin(</!>i — <^2)], 
 
 '2 
 
 since if we multiplj' this b}' r2(cos(^2+ V — 1. sin<^2)) according 
 to the method established in Art. 22, we must get 
 
 ri(cos(/)i -f- V— 1. sin</)i). 
 
 To divide one imaginary by another, we have then to take the 
 quotient obtained by dividing the modulus of the first by the 
 modulus of the second as our required modulus, and the argu- 
 ment of the first minus the argument of the second as our new 
 argument. 
 
 24. If we are dealing with the product of n equal factors, or, 
 in other words, if we are raising r(cos(j!) + V— 1. sin^) to the 
 
14 INTEGHAL CALCULUS. l^^^'^' ^^■ 
 
 jith power, n being a positive whole number, we shall have, by 
 Art. 22, 
 
 [r(cos<^ + V-^. sin (^)]" = r"(cos« <^ + V^^. Sinn <^). [1] 
 
 If r is unit}-, we have merely to multiply the argument by n, 
 without changing the modulus ; so that in this case increasing 
 the exponent by unity amounts to rotating the point represent- 
 ing the imaginary through an angle equal to <^ without changing 
 its distance from the origin. 
 
 25. Since extracting a root is the inverse of raising to a 
 power, 
 
 V[)-(cos<^ + V^.sin0)] = 7F^cos^ + V^.sin-j; [1] 
 
 for, by Art. 24, 
 
 / <^ , d>\"l" , 
 
 -yrl COS- + V— l.sin— 1 = j-(cos^ + V — l.sin<^). 
 
 Example. 
 
 Show that Art. 24 [1] holds even when n is negative or 
 fractional. 
 
 26. As the modulus of evei-y quantity, positive, negative, 
 real, or imaginary, is positive, it is always possible to find the 
 modulus of anj- required root ; and as this modulus must be real 
 and positive, it can never, in anj- given example, have more than 
 one value. We know from algebra, however, that every equa- 
 tion of the nth degree containing one unknown has n roots, and 
 that consequently every number must have n nth roots. Our 
 formula. Art. 25 [1], appears to give us but one nth root for 
 any given quantity. It must then be incomplete. 
 
 We have seen (Art. 20) that while the modulus of a given 
 imaginary has but one value, its argument is indeterminate and 
 may have any one of an infinite number of values whicli differ by 
 multiples of 27r. If ^o is owe of these values, the full form of 
 
Chap. U.] IMAGINAKIES. 15 
 
 the imaginaiy is not 7-(cos<^ + V^. sin<^o) , as we have hitherto 
 ■written it, but is 
 
 r[eos(^+2m7r) + V^.sin(<^+ 2m7r)], 
 
 where m is zero or any whole number positive or negative. 
 Since angles differing by multiples of 2 tt have the same trigo- 
 nometric functions, it is easily seen that the introduction of the 
 term 2mir into the argument of an imaginarj' will not modify 
 an3- of our results except that of Art. 25, which becomes 
 
 Vr [cos(<^„+ 2m7r) + V— 1- sin(<^„+ 2mTr)] 
 
 = ^r\ cos M + m -^j + V- 1 . sin 
 
 <t>o 2 
 — h m — 
 n n 
 
 ■ [1.1 
 
 Giving m the values 0, 1, 2,3 .... , n—1, n, n-\-l, success- 
 ively, we get 
 
 </>o ^,2jr <^ 27r <^o , o^'^ <^o , / i\27r 
 
 — 5 5 b^ — 1 ^-3 — » — +(n — 1) — , 
 
 n n n n n n n n n 
 
 H27r, h^TT, 
 
 n n n 
 
 as aiguments of our nth root. 
 
 Of these values the first n, that is, all except the last two, 
 coiTespond to different points, and therefore to different roots ; 
 the next to the last gives the same point as the first, and the 
 last the same point as the second, and it is easily seen that if we 
 go on increasing in we shall get no new points. The same thing 
 is true of negative values of m. 
 
 Hence we see that every quantity, reed or imaginary, has n 
 distinct nth roots, all having the same modulus, but with argu- 
 ments differing by multiples of ^^ . 
 
 27. Any positive real differs from unity only in its modulus, 
 and any negative real differs from —1 only in its modulus. All 
 the »tth roots of any number or of its negative may be obtained 
 
16 
 
 INTEGRAL CALCIJLtJS. 
 
 [Art. 27. 
 
 by multiplying the «tli roots of 1 or of — 1 by the real positive 
 nth root of the number. 
 
 Let us consider some of the roots of 1 and of — 1 ; for ex- 
 ample, the cube roots of 1 and of —1. The modulus of 1 
 is 1, and its argument is 0. The modulus of each of the cube 
 
 2,r 4cTr 
 
 roots of 1 is 1 , and their arguments are 0, -— , and -— ; that is, 
 
 3 3 
 
 0°, 120°, and 240°. The roots in question, then, are repre- 
 sented by the points Pi, Ps, Pa, in the figure. Their values are 
 
 l(cosO + V^. sinO), 
 1 (cos 120° + V^. sin 120°), 
 and 1 (cos 240° + V^. sin 240°), 
 
 or 1, -^ + ^V^l, _^_^V-1. 
 
 The modulus of —1 is 1, and its 
 
 argument is tt. The modulus of the 
 
 cube roots of — 1 is 1 , and their arguments are -, - 4- — . 
 
 3 3 3 
 
 J + ^, that is, 60°, 180°, 300°. The roots in question, then, 
 o o 
 
 are represented by the points Pj, Pj, 
 
 P3, in the figure. Their values are 
 
 i + ^^/^, -1, i-4'V=a. 
 
 Examples. 
 
 (1) What are the square roots of 
 1 and - 1 ? the 4th roots ? the 5th 
 roots ? the 6th roots ? 
 
 (2) Find the cube roots of -8 ; the 5th roots of 32. 
 
 (3) Show that an imaginary can have no real nth root- that 
 a positive real has two real nth roots if n is even, one if n is 
 odd ; that a negative real has one real nth root if n is odd none 
 if n is even. ' 
 
Chap. II.] IMAGINAEIES. 17 
 
 28. Imaginaries having equal moduli, and arguments differing 
 only in sign, are called conjugate imaginaries. 
 
 r(cos<^+ V— l.sin<^), and ?-[cos(-<^) + V^.sin(-<^)], 
 or r(cosi^ — V — 1. 8in<^) are conjugate. 
 
 They can be written x + y V — 1 and x — y V^, and we see 
 that the points corresponding to them have the same abscissa, 
 and ordinates which are equal with opposite signs. 
 
 Examples. 
 
 (1) Prove that conjugate imaginaries have a real sum and a 
 real product. 
 
 (2) Prove, by considering in detail the substitution of 
 a + & V— 1 and a — b ^T-i in turn for x in any algebraic poly- 
 nomial in X with real coefficients, that if any algebraic equation 
 with real coefficients has an imaginary root the conjugate of that 
 root is also a root of the equation. 
 
 (3) Prove that if in anj' fraction where the numerator and 
 denominator are rational algebraic poh'noraials in x, we substi- 
 tute a + b V— 1 and u — b V^ in tui-n for x, the results are 
 conjugate. 
 
 Transcendental Functions of Imaginaries. 
 
 29. "We have adopted a definition of an imaginary and laid 
 down rules to govern its use, that enable us to deal with it, in 
 all expressions involving only algebraic operations, precisely as 
 if it were a quantity. If we are going further, and are to sub- 
 ject it to transcendental operations, we must carefully define 
 each function that we are going to use, and establish the rules 
 which the function must obey. 
 
 The principal transcendental functions are e"", loga;, and sino;, 
 and we wish to define and study these when x is replaced by an 
 imaginarj' variable z. 
 
 As our conception and treatment of imaginaries have been 
 entirely algebraic, we naturallj' wish to define our transcendental 
 
18 IHTEGKAL CALCULUS. [Akt. 30. 
 
 functions by the aid of algebraic functions ; and since we know 
 that the transcendental functions of a real variable can be ex- 
 pressed in terms of algebraic functions only by the aid of infinite 
 series, we are led to use such series in defining transcendental 
 functions of an imaginary variable ; but we must first establish 
 a proposition concerning the convergency of a series containing 
 imaginary terms. 
 
 30. If the moduli of the terms of a series containing imaginary 
 terms form a convergent series, the given series is convergent. 
 
 Let Mo + Ml + "2 + + w» + l^e a series containing imagi- 
 
 nar}' terms. 
 
 Let 
 
 Mo = jBo(cos'I>o+ V — l.sin$o)) Mi = i2i(cos*i+V— 1. sin$i), &c., 
 
 and suppose that the series 7i!q + 7?i + i?2 + + -Bn + is 
 
 convergent ; then will the series Mo+ Mi+ M2+ be convergent. 
 
 The series i?o+iJi + is a convergent series composed of 
 
 positive terms ; if then we break up this series into parts in any 
 wa}-, each part will have a definite sum or will approach a defi- 
 nite limit as the number of terms considered is increased in- 
 definitely. 
 
 The series u^ + v^ + U2 + h„ + can be broken up into 
 
 the two series 
 
 i?oC08*o+ KiC0S<l'i-|-i?2C0S*2+ + ^nC0S$„-|- (1) 
 
 and 
 
 V — l(i?osin*o + -Bisin<l>i-)-i?2sin*2 + +^,.sin*„-)- ). (2) 
 
 (1) can be separated into two parts, the first made up only 
 of positive terms, the second only of negative terms, and can 
 therefore be regarded as the difference between two series, each 
 consisting of positive terms. Each term in either series will be 
 
 a term of the modulus series i^o + -Ki + -B2 + multiplied by 
 
 a quantitj' less than one, and the sum of n terms of each series 
 will therefore approach a definite limit, as n increases indefi- 
 nitely. The series (1), then, which is the abscissa of the point 
 representing the given imaginary series, has a finite sum. 
 
Chap. II.] IMAGINARIES. 19 
 
 In the same way it maj- be shown that the coefficient of V^T 
 in (2) has a finite sum, and this is the ordinate of the point 
 representing the given series. The sum of ?i terms of the given 
 series, then, approaches a definite limit as n is increased indefi- 
 nitely, and the series is convergent. 
 
 31. We have seen (I. Art. 133 [2]) that 
 
 O* /y»2 /y»3 /«4 
 
 e' = l + - + — + — + — + rii 
 
 12!3!4! "-^ 
 
 when X is real, and that this series is convergent for all values of a;. 
 Let us define e*, where z = x + y V — 1 , by the series 
 
 12! 3! 4! •- -■ 
 
 This series is convergent, for if z = r (cos (f> + V--1 . sin <^) the 
 series 
 
 l+I + ^ + il + ^ + 
 
 1 2! 3! 4! 
 
 made up of the moduli of the terms of [2] is convergent by 
 I. Art. 133, and therefore the value we have chosen for e' is a 
 determinate finite one. 
 
 Write X + yV— 1 toi-z, and we get 
 
 c-^'-^-l I ^+W~1 , (a!+yV~l)^ , (x+y^~iy ^ ^3^ 
 
 The -terms of this series can be expanded by the Binomial 
 Theorem. Consider aU the resulting terms containing any given 
 power of a;, say xf ; we have 
 
 ^^^ + ^~ + ^ + 3T~+ + n! + ^' 
 or, separating the real terms and the imaginary terms, 
 
 pP 2! 4! 6! ^ 
 
 + ^^ ^^ 3!^5! 7!+ ■'' 
 
20 INTEGRAL CALCULtTS. l-^^"^' ^^■ 
 
 or — (cosy+ V^.siny), by I. Art. 134. 
 
 Giving J) all values from to oo we get 
 
 e. + .vrT^(^^,gy^^—i.sin 2/) (1 + ^ + 1^ + 1^ + 1^+ ) 
 
 = 6^(008^ + V— l.siny), [4] 
 
 which, by the way, is in one of our tj'pical imaginary forms. 
 
 If a;=0, in [4], 
 
 we get e""^"' = C0S2/ + V— l.siny, [5] 
 
 which suggests a new way of writing our tj'pical imaginary ; 
 
 namely, 
 
 9- (cos <^ + V — 1 . sin ^) = re^-^^. 
 
 32. We have seen that 
 
 let us see if all imaginary powers of e obey the law of indices; 
 that is, if the equation 
 
 e"e" = e" + " [1] 
 
 is universally true. 
 
 Let « = a;i + ?/iV— 1 and v = ajj + j/aV — 1, 
 
 then e"= 6^1 + 2/1^^ = e^i(cos2/i + V— l.sin^i), 
 
 e° z=e^2 + y'.~^^= e%(cos2/2 + V^. sin 2/2) , 
 
 e" e" = e^i e^2 [cos (2/1 + 2/2) + V^ . sin (y^ + y,) ] 
 
 = e^i + ^'2 [cos (2/1 + 2/2) + V^ . sin (2/1 + 2/2) ] 
 
 = e^i + 3^2 + (!/i + 2/2) v^^ 
 
 = e" + ", 
 
 and the fundamental property of exponential functions holds for 
 imaginaries as well as for reals. 
 
 Example. 
 Prove that a"a'' = 0" + " when u and v are imaginary. 
 
Chap. II.] IMAGINAKIES. 21 
 
 Logarithmic Functions. 
 
 33. As a logarithm is the inverse of an exponential, we ought 
 to be able to obtain the logarithm of an imaginary' from the 
 formula for 6"+"^-'. "We see readily that 
 
 2 = r (cos<^ + V^. sin<j>) = e'°e'-+<'^^, 
 
 whence log2 = logr + <^ V — 1 ; 
 
 or, more strictly, since 
 
 2= ?'[cos(<^o + 2n7r) + V — l.sin(^i,+ 2mr)], 
 
 Iog2 = logr+(<^o+2n7r) V^ [1] 
 
 where n is anj' integer. 
 
 If z = x + y V— 1, r = VaF+lf, and (^ = tan"'^ ; 
 
 X 
 
 whence logz = ^log(x^ + 2/^) + V— l.tan"^^. [2] 
 
 X 
 
 Each of the expressions for log z is indeterminate, and repre- 
 sents an infinite number of values, differing by multiples of 
 27rV^. 
 
 This indeterminateness in the logarithm might have been ex- 
 pected a priori, for 
 
 rV^l 
 
 = cos 2 TT -f- V — 1 . sin 2 TT = 1 , by Art. 31. 
 
 Hence, adding 2 7rV— 1 to the logarithm of any quantity will 
 have the effect of multiplying the quantity by 1, and therefore 
 will not change its value. 
 
 Example. 
 
 Show that if an expression is imaginary, all its logarithms are 
 imaginarj- ; if it is real and positive, one logarithm is real and 
 the rest imaginary ; if it is real and negative, all are imaginary. 
 
22 INTEGRAL CALCULUS. l^^"^' ^*' 
 
 Trigonometric Functions. 
 
 34. If 2 is real, 
 
 «3 
 
 ^2 gl ^6 
 
 byl. Art. 134. "^ ■ *■ °- , 
 
 If 2 = )-(cos<^+ V— l.sia<^), 
 
 the series of the moduli, 
 
 '■ + 3! + 5!-^y! + ' 
 
 j2 ,A ,.« 
 
 ' + 2! + 4! + 6! + • 
 
 are easilj- seen to be convergent ; therefore if z is imaginary, the 
 series [1] and [2] are convergent. We shall take them as defi- 
 nitions of the sine and cosine of an imaginary. 
 
 Example. 
 
 From the formulas of Art. 31, and from Art. 84 [1] and [2], 
 show that 
 
 e'^'"^ = cos«+ V — l.sinz, 
 
 and e~^^"' = cos« — V — 1. sinz, for all values of «. 
 
 35. From the relations 
 
 e "^^ = cosz— V — l.sinz, 
 
 we get C0S2 = '^-^ , rn 
 
 [2] 
 
 sm a = 
 
 2 V^l 
 for all values of z. 
 
Chap. II.] 
 Let 
 
 cos(a;+yV— 1): 
 
 IMAGIN ARIES. 
 
 23 
 
 x-/^\ — y 
 
 + e-^ 
 
 v^l + » 
 
 2 
 
 _ (c6sg!+V— l.sing!)e~''+(cosa;— V— l.sina!)e' ' 
 2 
 
 by Art. 34, Ex., 
 
 = cosa; — ! — V — l.sina; \3\ 
 
 In the .same waj' it may be shown that 
 
 . , , / — 7-, (cosa;+ V— 1. sina;)e~''— (cosa;— V— l.sina;)* 
 sm(x+y y—l)=- ^ — - — ^ '— 
 
 2 V — 1 
 
 = sma; — ■ h V — 1. cos a; 
 
 2 2 
 
 If z is real in [1] and [2], we have 
 
 cos a; = 
 
 e»^-i -|_ 6-^*^-1 
 
 [4] 
 
 sma; = V — 1. 
 
 If 2 = y V — 1, and is a pure imaginary 
 cos?/ 
 
 V3I=!!±^ 
 
 saiy 
 
 V^ = 
 
 e" — e - 
 2 
 
 'vn; 
 
 [5] 
 [6] 
 
 whence we see that the cosine of a pure imaginary is real, while 
 its sine is imaginary. 
 
 By the aid of, [5] and [6] , [3] and [4] can be written : 
 
 cos (» + .'z V^ ) =cosa;cos.?/V— 1 — sinajsinj/ V— 1, [7] 
 8,in (aj + y V— 1) = sinajcos?/ V— 1 + cosa!sin?/V— 1. [8] 
 
24 INTEGBAL CALCULUS. t-^'^- ^®' 
 
 Examples. 
 
 (1) From [1] and [2] show that 8ia'z+ cos'z = 1. 
 
 (2) Prove that 
 
 cos (u + v) = cos u COS V — sin u sin v, 
 sm(u + v) = sinwcosv + costtsini;, 
 where u and v are imaginary. 
 
 The relations to be proved in examples (1) and (2) are the 
 fundamental formulas of Trigonometry, and they enable us to 
 use trigonometric functions of imaginaries precisely as we use 
 trigonometric functions of reals. 
 
 Differentiation of Functions of Imaginaries. 
 
 36. A function of an imaginary variable, 
 
 z = x + y\r^, 
 
 is, strictly speaking, a function of two independent variables, 
 X and y ; for we can change z by changing either x or y, or both 
 X and y. Its differential will usually' contain da; and dy, and not 
 necessarily dz ; and if we divide its differential bj' dz to get its 
 
 derivative with respect to z, the result will generally contain — , 
 
 dx 
 which will be wholly indeterminate, since x and y are entirely 
 independent in the expression x + y\f^l. It may happen, 
 however, in the case of some simple functions, that dz will appear 
 as a factor in the differential of the function, which in that case 
 will have a single derivative. 
 
 37. In differentiating, the V^l may be treated like a con- 
 stant; for the operation of finding the differential of a function 
 is an algebraic operation, and in all algebraic operations V~i 
 obeys the same laws as any constant. 
 
Chap. II.] 
 
 IMAGINAKIES. 
 
 25 
 
 Example. 
 Prove that d{a?\l^l) = 2 a? V^ . dx ; 
 
 and that dV— 1. sina;= V— 1. cosx.dx. 
 
 We have, by the aid of this principle, 
 if « = a; + 2/V^, 
 
 dz = dx + V — 1. dy; [1] 
 
 if »=r(cos<^ + V— 1. sin^), 
 
 dz= d?'(eoS(^ + V — l.sin(^) + rd(^(— sin</)+V — l.cosi^) 
 = (c?r + rV^.d(/))(coS(^ + V^.sin<^). [2] 
 
 38. Let us now consider the differentiation of «"*, e', log«, 
 sinz, and cos 2. 
 
 Let % = r (cos ^ + V — 1 . sin ^) , 
 
 then 
 z" = r^Ccos m<^ + V^ . sin m<f>) , by Art. 24 [1] ; 
 
 df = mr"-^ dr (cos m^ + V— 1. sin m^) + mr"'d(l> (— sinm<^ 
 + V — 1 . cos m<^), 
 
 dz"= mr'""^[cos (m— 1) ^ + V — l.sin(m— 1) <^] (cos<^ 
 + \/ — 1. sin (t>)dr 
 
 + mr'° [cos (m— 1) <^ + V — l.sin(m— 1)(^] (cos<^ 
 + V— l.sin<^) sf — Ldtj}, 
 
 dz" 
 
 '[cos(m— 1) <^ + V— l.sin(m— 1) </>] (dr 
 
 4- ?-V — 1 .d<j)) (cos<^ + V— 1. sin^), 
 rtz" = mz'^-^dz, [1] by Art. 37 [2] , 
 
 dz 
 
 [2] 
 
 and a power of an imaginary variable has a single derivative. 
 
26 INTBGKAL CALCTJLTTS. l^^"^' ^^ 
 
 39. If z=x + y\/^l, 
 
 e'=e'(cos2/+V^.siny), byArt.31[4], 
 
 de'=e»da;(cos2/+V^. smy) + ^{—siny 
 + -/^.cosy)dy. 
 
 de' = ^{cosy+^/^l.siny) {dx+^T^. dy), 
 de'=e'dz, [1] 
 
 '^^ :e'. [2] 
 
 dz 
 
 Show that 
 
 Example. 
 da' = a' log a.dz. 
 
 40. If « = »-(cos<^ + V^l. sin^), 
 
 logz = logr + <^V^, by Art. 33, 
 
 dlog2; = h V — 1. d<^ = — 2 2-, 
 
 r r 
 
 ,_ (dr +r V— 1 . d<^) (cos (^+ V— 1 . sin <^) 
 
 dlog2: 
 
 ji dz 
 
 dlogz = —, 
 
 z 
 
 dlogg _ 1 
 dz z' 
 
 r(cos</) + V— 1. sin</>) 
 
 [1] 
 [2] 
 
 41. sm« = 
 
 2V^ 
 
 dsm2 = ;!;3==; V— l.dz 
 
 2 V-1 
 
 by Art. 35 [2], 
 
 dz, 
 
 dsinz = cosz.dz. 
 
 by Art. 35 [1], 
 [1] 
 
Chap. II.] IMAGINAEIES. 
 
 
 27 
 
 on'^'^ — '^ ^ 
 
 
 
 dcosz = V — l.d2 = 
 
 2' 
 
 dz, 
 
 y-1 
 
 dcosz = — sinz.dz. 
 
 
 [2] 
 
 42. We see, then, that we get the same formulas for the dif- 
 ferentiation of simple functions of imaginaries as for the dif- 
 ferentiation of the corresponding functions of reals. It follows 
 that our formulas for direct integration (I. Art. 74) hold when x 
 is imaginarj'. 
 
 Hyperbolic Functions. 
 
 43. We have (Art. 35 [5] and [6]) 
 
 , / — T e" + e" 
 cos a; V — 1 — 
 
 2 ' 
 
 and sina;V^ = ^~^"V ^, 
 
 2 
 
 gi I g-X 
 
 where x is real. — — — ■ is called the iiyperbolie cosine of x, 
 
 2 g« _ g -^ 
 and is written cosh a; ; and — 5 — is called the hyperbolic sine 
 
 of X, and is written sinha; ; 
 
 sinha;=?^^^^ = - V^.sina;V^, [1] 
 
 cosha;= — ^ti_ = cosa; V— 1. [2] 
 
 2 
 
 The hyperbolic tangent is defined as the ratio of sinh to cosh ; 
 and the hyperbolic cotangent, secant, and cosecant are the re- 
 ciprocals of the tanh, cosh, and sinh respectively. 
 
 These functions, which are real when x is real, resemble in 
 their properties the ordinary trigonometric functions. 
 
28 INTBGBAL CALCX7LXJS. 
 
 44, For example, 
 
 for 
 and 
 
 cosh" 9; — sinb^a; = 1 ; 
 
 cosh'' a; = ' ■ , 
 
 4 
 
 sinh" a; = - 
 
 2+e- 
 
 [Abt. 44- 
 [1] 
 
 Examples. 
 
 (1) Prove that 1 — tanh^a; = sectfa;. 
 
 (2) Prove that 1 — ctnh'^a; = — csch^a;. 
 
 (3) Prove that sinh(a; + j/)= sinha;cosh2/ + cosha!sinhy. 
 
 (4) Prove that cosh (a; + y) = cosh x cosh y + sinh x sinh «/. 
 
 45. 
 
 ^ — e' 
 
 c? sinh a; = d 
 
 c?sinha;= cosha;.daj. 
 
 e' + e-' 
 
 dx. 
 
 (1) Prove 
 
 Examples. 
 d cosh a; = sin h a;, da;. 
 dJtanha; = sech-a;.da;. 
 dctnha; = — csch"a;.da;. 
 dsecha; = — secha;tanha;.da;. 
 d csch a; = — csch a; ctnh x.dx. 
 
 46. We can deal with anti-hyperbolic functions just as with 
 anti-trigonometric functions. 
 To find dsinh~^a!. 
 
 Let 
 then 
 
 u = sinh~'a;, 
 x = sinhw, 
 dT= cosU u.dn. 
 
Chap, ll.'j IMAGINAEIES. 29 
 
 , dx 
 
 du = , 
 
 coshw 
 
 cosh?{ = Vl + 8inT?M, by Art. 44 [1], 
 
 coshM= Vl +x', 
 
 vr+^ *- -' 
 
 Examples. 
 Prove the formulas 
 
 dx 
 
 t?.cosh"'a; = 
 
 dt&nh~'x = 
 dsech"*a; = 
 
 c?csch"'a; = 
 
 Va^-1 
 dx 
 
 l-ar" 
 dx 
 
 xsll —a? 
 
 dx 
 xVx^ + l 
 
 47. The anti-hyperbolic functions are easily expressed as 
 logarithms. 
 Let M=sinh~'a;, 
 
 then 
 
 X 
 
 = sinhit = 
 
 e"— e-" 
 2 ' 
 
 2x 
 
 — ? 
 
 
 2X6" 
 
 =ze'"-l, 
 
 
 g2»_ 
 
 2a;e"=l, 
 
 
 e»"- 
 
 2xe" + x': 
 a; = ± Vl 
 
 = 1+0?, 
 
 e" - 
 
 i + A 
 
JO INTEGRAL CALCTJLtTS. C-^^' **"^ 
 
 IS e" is necessarily positive, we may reject the negative value in 
 the second member as impossible, and we have 
 
 e" = a; + Vl + a^, 
 
 7/ = log(a! + Vl +!B^), 
 
 5r sinh-' a; = log (a; + Vl + ar") . [1] 
 
 Examples. 
 Prove the formulas 
 
 eosh~'a; = log(a; + Va^ — 1). 
 tanh~'a; = ^log "*" . 
 
 1 —X 
 
 C8ch-a; = log(^i + ^i + l). 
 
 48. One of the advantages arising from the use of hyper- 
 3olic functions is that they bring to light some curious analogies 
 jetween the integrals of certain irrational functions. 
 From I. Art. 71 we obtain the formulas for direct integration. 
 
 = sm ^a;. [1] 
 
 f 
 
 vr 
 
 iZ^ =tan-»a;. [2] 
 
 r — ~ = S9.0.-^X. [3] 
 
 From Art. 46 we obtain the allied formulas : 
 
Chap. 11.] IMAGINAEIES. 31 
 
 
 Examples. 
 Prove the formulas 
 
 Of if If 
 
 (1) sinha; = -+- + -+ 
 
 (2) cosha; = l + |j + Jj + 
 
 (3) sin (x-\-y V — 1) := sin a; cosh y + V — 1 cos a; sinh y 
 
 (4) cos(a; + yV— 1) =cosx coshy— V— 1 sina; sinhy. 
 
 . ._. ^ / , / — Tv sin 2x + V— 1 sinh 2y 
 
 (5) tan(. + W-l) = cosL + cosh2y ' 
 
 (6) sinh (x + y V — 1) = sinh a; cos y + V— 1 cosh x sin y. 
 
 (7) cosh (x-\-y ■\l— 1) = cosh a; cos y + V— 1 sinh x sin ?/, 
 
 .Q. , , , , /-Tx sinh2a;+V— 1 sin 2y 
 
 (8) tanh (rr + w V — 1) = t—^ — ; ^ 
 
 ^ ' V 1 ^ / gQgh 2x + cos 2v 
 
 a!° , X' 
 
 ,6 
 
 (9) tanh-'a; =a;+-+g- + 
 
32 IKTEGKAL CALCULUS. L-^^T. i'J- 
 
 CHAPTER III. 
 
 GBNBKAL METHODS OF INTEGKATING. 
 
 49. We have defined the integral of an}- function of a single 
 variable as the function which has the given function for its 
 derivative (I. Art. 53) ; we have defined a definite integral as 
 the limit of the sum of a set of differentials ; and we have shown 
 that a definite integral is the difference between two values of an 
 ordinary integral (I. Art. 183). 
 
 Now that we have adopted the differential notation in place of 
 the derivative notation, it is better to regard an integral as the 
 inverse of a differential instead of as the inverse of a derivative. 
 Hence the integral of fx.dx will be the function whose differ- 
 ential is fx.dx; and we shall indicate it by i fx.dx. In our old 
 notation we should have indicated precisely the same function bj' 
 I fx ; for if the derivative of a function is fx we know that its 
 differential is fx.dx. 
 
 50. If fx is a continuous function of a;, fx.dx has an integral. 
 For if we construct the curve whose equation is y=zfx, we know 
 that the area included bj- the curve, the axis of X, any fixed 
 ordinate, and the ordinate corresponding to the variable x, has 
 for its differential ydx, or, in other words, fx.dx (I. Art. 51). 
 Such an area always esr'sts, and it is a determinate function of a;, 
 except that, as the position of the initial ordinate is wholly arbi- 
 trary, the expression for the area will contain an arbitrary con- 
 stant. Thus, if Fx is the area in question for some one position 
 of the initial ordinate, we shall have 
 
 I fx.dx = Fx + 0, 
 where O is an arbitrary constant. 
 
 fi 
 
Chap. III.] GENERAL METHODS OF INTEGEATING. 33 
 
 Moreover, Fa; + is a complete expression for j fx.dx ; for if 
 
 two functions of x have the same differential, they have the same 
 derivative with respect to x, and therefore they change at the 
 same rate when x changes (I. Art. 38) ; they can differ, then, 
 at an^- instant only by the difference between their initial values, 
 which is some constant. 
 
 Hence we see that every expression of the form fx.dx has an 
 integral, and, except for the presence of an arbitrary constant, 
 but one integral. 
 
 51. We have shown in I. Art. 183 that a definite integral 
 is the difference between two values of an ordinary integral, and 
 therefore contains no constant. Thus, if Fx + is the integral 
 of fx.dx, 
 
 C fx.dx =Fb — Fa. 
 In the same way we shall have 
 
 fz.da = Fb — Fa; 
 
 £■ 
 
 and we see that a definite integral is a function of the values 
 between which the sum is taken and not of the variable with 
 respect to which we integrate. 
 
 Since 
 
 (""fx.dx = Fa — Fb, 
 fx,dx = — I fx.dx. 
 
 Example. 
 
 s 
 
 fx.dx + j fx.dx = I fx.dx 
 
 52. In what we have said concerning definite integrals we 
 have tacitlj- assumed that the integral is a continuous function 
 between the values between which the sum in question is taken. 
 If it is not, we cannot regard the whole increment of Fx as equal 
 
u 
 
 INTEGRAL CALCULUS. 
 
 [Art. 53. 
 
 to the limit of the sum of the partial infinitesimal increments, 
 and the reasoning of 1. Art. 183 ceases to be valid. 
 Take, for example, j 
 
 j?=j 
 
 a^ 
 
 — 1 
 
 1 
 
 X 
 
 by I. Art. 55 (7) 
 
 and apparently 
 
 ^"*X! 
 
 r^^(-^\ -f-i) =_2. 
 
 dx 1 
 
 ought to be the area between the curve y -■ 
 
 a." ar 
 
 axis of a;, and the ordiuates corresponding to a; = l and a;= — 1, 
 
 which evidently is not — 2 ; and we 
 
 see that the function — is discon- 
 
 ar 
 tinuous between the values a; = — 1 
 and x = l. 
 
 The area in question which the 
 definite integi-al should represent is 
 easily seen to be infinite, for 
 
 
 /■ 
 
 \ 
 
 
 / 
 
 1 
 
 
 \ 
 
 1 
 
 
 -1 
 
 1 
 
 
 J-, x' € J, a? e ' 
 
 and each of these expressions increases without limit as e ap- 
 proaches zero. 
 
 53. Since a definite integral is the , difference between two 
 values of an indefinite integi-al, what we have to find first in any 
 problem is the indefinite integral. This may be found by in- 
 spection if the function to be integrated comes under anj' of the 
 forms we have already obtained by difl'erentiation, and we are 
 then said to integrate directly. Direct integration has been illus- 
 trated, and the most important of the forms which can be in- 
 tegrated directly have been given in I. Chapter V. For the sake 
 of convenience we rewrite these forms, using the differential 
 notation, and adding one or two mw forms from our sections on 
 hyperbolic functions. 
 
Chap. III.] GENERAL METHODS OF INTEGRATING. 35 
 
 ^n+ 1 
 
 x^dx-. 
 
 11+1 
 
 rdx , 
 
 i — = logaj. 
 J X 
 
 Ca''dx = -^. 
 J log a 
 
 i^dx=e'. 
 
 I sinaj.da; = —cos a;. 
 
 I cos a;, da; = sinx. 
 
 I tana;. da; = .— logcosa;. 
 
 j etna;. da; = log sin a;. 
 
 /dx . _. 
 
 = sin '.i;. 
 Vl — ar* 
 
 C—^= = sinh -1 a; = log (a; + VIT^) . 
 
 J VT+^ 
 
 dx 
 
 C~^= = cosh-i a; = log (a; + Va;^ - 1) . 
 
 J V 0^-1 
 
 dx 
 
 = tan ^x. 
 
 / ax 
 l+ar' 
 
 J_^=tanh-a; = ilog^_^ 
 
 dx_ ,„„v,-i ii„„l±^. 
 
 ■ a?' 
 
 dx 
 
 ■yfW^X 
 
 J X 
 
 /dx 
 V2a; — : 
 
36 ITSITEGRAL CALCULUS. C^^''- ^*' 
 
 54. We took up in I. Chap. V. the principal devices used in 
 preparing a function for integration when it cannot be integrated 
 directly. 
 
 The first of these methods, that of integration by substitution, 
 is simplified by the use of the differential notation, because the 
 formula for change of variable (I. Art. 75 [1]), 
 
 I j< = I uD^ becoming | udx= | u — dy, 
 
 reduces to an identity and is no longer needed, and all that is 
 required is a simple substitution. 
 
 — Vl+loga;. 
 
 Let l + loga; = «; then —- = dz, 
 
 and r^ Vl + logx = r«*d2 = 1 2* = I ( 1 + loga;)^ 
 
 When, as in this example, a factor of the quantity to be 
 integrated is equal or i)roportlonal to the differential of some 
 function occurring in the expression, the substitution of a new 
 variable for the function in question will generally simplify the 
 problem. 
 
 (6) Required | — 
 
 dx 
 
 e" + e'" 
 
 Let e' = y\ then e'dx=dy, 
 
 __dx__ _ _f_dx_ _ _dy 
 er + e-^~ e^' + i ~Y + i' 
 
 (c) Required | sec a;, da;. 
 
 seca;=^_ = -22H. 
 cos.r cos^a; 
 
Chap. III.] GENERAL METHODS OP INTEGRATING. 37 
 
 Let »=sina!; then dz = cosx.dx, 
 
 008^ a; = 1 — z^, 
 
 /cosx.dx r dz ,,1+2 1. . .L ro 
 
 fsec a;.da; = ^log L±liE^ = w tan f^ + ^\. 
 ^ 1 — sin a; \4 2/ 
 
 Examples. 
 Prove that (1) fcscx.dx = ilog ^~°"^^ = log tan -• 
 
 Suggestion : Let a; = cos». 
 
 65. The formula for integration by parts (I. Art. 79 [1]) 
 becomes 
 
 I udv = uv — I vdu, [1] 
 
 when we use the differential notation. It is used as in I. Chap. V. 
 (a) For example, let us find | a;"logx.da;. 
 Let M = loga3, and dv = x'^dx; 
 
 then du = — , 
 
 x 
 
 and 
 
 n + l 
 
 /a;"loga!.da;=-^ loga;— \-^—-dx = -^- — -floga; - 
 
 (6) Required I a; sin^ai.da;. 
 
 Let u = sin~^a;, and dv = xdx ; 
 
 then du = ^ 
 
 Vl-a^ 
 
38 
 and 
 
 INTEGRAL CALCULUS. 
 
 X- 
 
 [Art. 55. 
 
 ;i' sin" .r.aa; = — sin 'a; — -A- I - i 
 
 ra;sin-'a;.c?.r = ^sin-'a; + :|^(cos-'a; + Wl — ar'). 
 
 2 
 
 (' xe'fl.c 
 
 Let 
 and 
 then 
 and 
 
 J 
 
 (1 + 
 
 xY 
 
 dv 
 
 u = x^^ 
 dx 
 
 
 
 (1+a;)^' 
 
 
 du 
 
 = (xe 
 
 v = 
 
 1 
 
 1+x' 
 
 + x)da;, 
 
 
 xc' 
 
 - 4- 
 
 re'^cZa;= — 
 
 a;<- , 
 
 + x 
 
 
 Vl-3a;-a;= 
 
 Examples. 
 
 . .i3 + 2a; 
 ; sin ' 
 
 Vl3 
 
 2) I a;tan~'a;.da;= "'"''' tau~'a; — ^a 
 „. r xdx _ 1 , 1 
 
 l-a; 2(l-a;)2 
 
 aida; 
 
 V2 ax — X- 
 5) I V2c 
 
 = — V2 aa; — .'jr' -f 
 
 -la; 
 overs -. 
 
 a 
 
 •A/ • CIJ!/ — ■ 
 
 *'„;„-i X — O, 
 
 ~-, — V2 ax ~ a? + -^ sin~^ 
 
 Suggestion : Throw 2 rtx — *-' into the form a- — (a; — aY. 
 
 6) P +"*:""" da; = log (a; + sin a;). 
 
 J X -\- sin a; 
 
Chap. III.] GENERAL METHODS OF INTEGRATING. 39 
 
 (7) I — ■ aa; = a;tan-- 
 
 J 1 + cosa; 2 
 
 Suggestion : Introduce — in place of x. 
 
 gN r dx 1 
 
 J a!(loga;)" (n — 1) (loga;)"-"' 
 
 9) ri2S_(l2£^c?a;.=:loga;[log(loga;)-l]. 
 
 J X 
 
 Jsin"" ic d'Tj 1 
 '—^ = zt&nz + log cos 2, where 2 = sin~'a;. 
 (1 —arp 
 
 11) r *^ = 4-logtan('^ + ^Y 
 
 Jsmic+cosa; ^2 V"^ °/ 
 
 ..gN r sinigfe _ log (ct + ft cos a;) 
 
 J a + h cos a; 6 
 
 »/ ar + 4a; + 5 
 
 16) r - =i-tan-^r-tan4 
 
 V a^cos^aj + ft^sin^a; a6 \a I 
 
40 INTEGBAL CALCULUS. 
 
 [Art, 
 
 . 56. 
 
 CHAPTER IV. 
 
 EATIOXAL FKACTIONS. 
 
 56. We shall now attempt to consider sj-stematieally the 
 methods of integrating ^•arious functions ; and to this end we 
 shall begin with rational algebraic expressions. Any rational 
 algebraic polynomial can be integrated immediately bj- the aid of 
 the formula 
 
 f 
 
 ^n + 1 
 
 x"dx = -^ 
 
 n + \ 
 
 Take next a rational fraction., that is, a fraction whose nu- 
 merator and denominator are rational algebraic polynomials. 
 ^V rational fraction is proper if its numerator is of lower degree 
 than its denominator ; improper if the degree of the numeratpr 
 is equal to or greater than the degree of the denominator. Since 
 an improper fraction can always be reduced to a polj'nomial 
 plus a proper fraction, by actuallj' dividing the numerator by the 
 denominator, we need onlj' consider the treatment of proper 
 fractions. 
 
 57. Ecery proper rational fraction can be reduced to the sum 
 of a set of simpler fractions each of which has a constant for a 
 numerator and some power of a binomial for its denominator; 
 
 that is, a set of fractions any one of which is of the form — 
 
 ■' (.-B-a)" 
 
 Let our given fraction be ^ . 
 Fx 
 
 If a, 6, c, &c., are the roots of the equation, 
 
 Fx = 0, (1) 
 
 we have, from the Theory of Equations, 
 
 Fx = A{x - a) (x - b) (x - c) (2) 
 
Chap. IV.] RATIONAL FRACTIONS. 41 
 
 The equation (1) may have some equal roots, and then some of 
 the factors in (2) will be repeated. Suppose a occurs p times 
 as a root of (1), 6 occurs q times, c occurs r times, &c., 
 
 then Fx = A{x — ay{x—by{x — cy (3) 
 
 Call A{x — by{x — cy = <^a;; 
 
 then Fx= (x — a)''<j>x, 
 
 fa fa 
 
 /•„. f^ f^~ ~r 4>^ T~ 4>x 
 
 and ^= > - -^^ I '^«^ 
 
 Fx {x — aYfjix {x — ay(f,x {x — ay <j)X 
 /« ^- f^ J. 
 
 -T- tx — -; — (hX 
 
 cjia •' <^a^ 
 
 (x—ay (x — ay^x 
 
 —^ is a new proper fraction, but it can be reduced 
 
 {x — ay<^x ^ ^ 
 
 to a simpler form by dividing numei'ator and denominator by 
 a; — «, which is an exact divisor of the numerator because a is a 
 root of the equation 
 
 K we represent by fx the quotient arising from the division 
 
 otfx — J^ (j>x by x — a, we shall have 
 <t>a 
 
 fa 
 
 fx 4>a , fiX 
 
 Fx (x — ay (x - ay-' ^x' 
 
 where ■^^ is a proper fraction, and may be treated 
 
 {x-ay-^<l>x ' ^ 
 
 precisely as we have treated the original fraction. 
 
 fa 
 
 Hence /i'" - "^"^ i ^^^ 
 
 (x-ay-''(l>x (x-ay-'' {x — ay-^<i>x 
 
 By continuing this process we shall get 
 
 fa fa fa ff-\a 
 
 fx _ <^a >^a <l>a . . <i>a . fpX 
 
 Fx~ {x-ay {x-ay-^ {x-ay-^ x-a 4>x' 
 
42 INTEGRAI. CALCULUS. i-^^"^- ^^■ 
 
 In the same way -^ can be broken up into a set of fractions 
 
 having (a; — 6)', {x — b)'-\ &c., for denominators, plus a frac- 
 tion which can be broken up into fractious having (.i-— c)', 
 
 {x — cY'^ , &c., for denominators; and we shall have, in 
 
 tlie end, 
 
 f^ _ ^^ I ^^ + + ^^+ ^' 
 
 Fx {x-ay (x-a)"-^^ x—a (x-6)« 
 
 ^' ^' + + K, [1] 
 
 where K is the quotient obtained when we divide out the last 
 factor of the denominator, and is consequenth' a constant. More 
 than this, K must be zero, for as (1) is identically true, it must 
 
 fx 
 be true when a; = x ; but when x= co, d. — becomes zero, be- 
 
 Fx 
 
 cause its denominator is of higher degree than its numerator, 
 and each of the fractions in the second member also becomes 
 zero; whence 71 = 0. 
 
 58. Since we now know the form into wliich anj' given rational 
 
 fraction can be thrown, we can determine the numerators by the 
 
 aid of known properties of an identical equation. 
 
 3 J. -y 
 
 Let it be required to break up -^-y-p — into simpler 
 
 fractions. 
 
 By Art. 57, 
 
 3a; -1 A B , O 
 
 {x-\y(x+\) {x~iy^ a;-l^.r + l' 
 
 and we wish to determine ^-1, B, and C. Clearing of fractions, 
 we have 
 
 ?,x-l = A{x+\) + B (.r -1) {x+l) + C{x-\y. (1) 
 
 As this equation is identically true, the coefficients of like 
 powers of x in the two members must be equal ; and we have 
 
 -B + C=0, 
 .-1-L'C=3, 
 A-B + G= -1; 
 
Chap. IV.] RATIONAL FRACTIONS. 43 
 
 whence we find A= 1, 
 
 5=1, 
 
 (7=-l; 
 ^^^ 3a!— 1 1_ 1 _ 1 .^. 
 
 The labor of determining the required constants can often be 
 lessened by simple algebraic devices. 
 
 For example ; since the identical equation we start with is 
 true for all values of x, we have a riglit to substitute for x values 
 that will make terms of the equation disappear. Take equa- 
 tion [1] : 
 
 3.x--l = .-l(a; + l) + 5(a;+l)(a;-l) + C(a;-l)=. [1] 
 
 Leta;=l, 2 = 2 A, 
 
 .4=1, 
 
 then 2x-2 = B(x+l) {x - 1) + 0(0; -1)^ ; 
 
 divide by a;-l, 2 = B {x+l)+C {x- 1). 
 
 Leta;=l, 2=25, 
 
 B=l, 
 
 then — a;+l =C'(a; — 1), 
 
 C=-l. 
 
 Examples. 
 
 (1) Show that when we equate the coefficients of the same 
 powers of x on the two sides of our identical equation, we shall 
 always have equations enough to determine all our required 
 numerators. 
 
 (2) Break up ^^+^^'~^^^ into simpler fractions. 
 '^ ' ^ (x-syix+l) 
 
 59. The partial fractions corresponding to any given factor 
 of the denominator can be determined directly. 
 
fx A 
 
 , fi'^ 
 
 Fx x — a 
 
 ' </.«' 
 
 Fx 
 <bx = , 
 
 
 44 INTEGRAL CALCULUS. l^'^'^- ^^■ 
 
 Let us suppose that the factor in question is of the first degree 
 and occurs but once ; represent it by a; — a. 
 
 (1) 
 
 by Art. 57, where 
 
 d>x = 
 
 x — a 
 
 so that Fx ={x — a) <^. 
 
 Clear (1) of fi'actions. 
 
 fx = A<t>x + {x-a) /i a;. (2) 
 
 As (1) is an identical equation, (2) will be true for any value 
 of X. Let x= a, 
 
 fa = A<lia, 
 
 a result agreeing with Art. 57. 
 
 Hence, to find the numerator of the fraction corresponding to 
 a factor (x — a) of the first degree, we have merely to strike out 
 from the denominator of our original fraction the factor in ques- 
 tion, and then substitute a for x in the result. 
 
 If the factor of the denominator is of the nth degree, there are 
 n partial fractions corresponding to it. Let (x — a)" be the 
 factor in question. 
 
 ^^ ^' + A, A, ,^^,'tl m 
 
 Fx {x-a)"^ {x-a)"-"-^ {x-ay-^^ ^ x-a^ .joi'^ ' 
 
 where Fx = {x—aY •^^• 
 
 Multiply (4) by {x — a)', and represent (x — aY^ by *». 
 
 ' Fx ^ 
 
 4W =A^ + ^12(0; - a) +A^{x - ay + + A^{x - a)-* 
 
 + ^(x-a)». 
 <tix 
 
ClIAl'. IV.] 
 
 RATIONAL FRACTIONS. 
 
 Differentiate successively both members of this identity, and put 
 as = a after differentiation, and we get 
 
 Ai = *a, 
 
 Ai = ^' a, 
 
 2! 
 3! 
 
 A = 
 
 1 
 
 (n-1)! 
 
 -*(»-« a. 
 
 Althougli these results form a complete solution of the prob- 
 lem, and one exceedingly neat in theory, the labor of getting 
 the successive derivatives of *» is so great that it is usually 
 easier in practice to use the methods of Art. 58 when we have to 
 deal with factors of higher degree than the first. So far as the 
 fractions corresponding to factors of the first degree and to the 
 highest powers of factors not of the first degree are concerned, 
 the method of this article can be profitably combined with that 
 of Art. 58. 
 
 60. As an example where the method of the last article 
 applies well, consider 
 
 3a;-l 
 
 . = ^+^ 
 
 C 
 
 a;(a;-2)(a!+l) x x-2 x + l 
 3a;-l 
 
 A = 
 B = 
 
 = 
 
 3a;-l 
 
 _(a;-2)(a; + l) 
 
 " 3a; — 1 1 _5 
 
 _a;(a;-f-l)J,=2 6' 
 
 ' 3x-l ' 
 
 x{x — 2) 
 
 =1 1+^ ^ 
 
 a;(a;-2)(a; + l) 2 x 6 a; - 2 3 ar+1 
 
 [1] 
 
46 INTEGRAL CALCULUS. [^bt- ^^■ 
 
 61. Although the theory expounded in the preceding 
 articles is complete and can be applied without serious diffi- 
 culty to the case where some or all of the roots of F{x)^0 
 [Art. 57, (1)] are imaginary, there is a practical convenience 
 in modifying the method so as to avoid the explicit intro- 
 duction of imaginaries into the process of integrating a 
 rational fraction. 
 
 We know (Art. 28, Ex. 2) that if the denominator of our 
 given fraction contains an imaginary factor (x— a — b V — 1)" 
 it will also contain the conjugate of that factor, namely, 
 (x — a + b V — 1)", and will therefore contain their product 
 [(x — ay -\- J^]"- Moreover, since by Art. 59 the numerator of 
 the partial fraction whose denominator is (x — a + byl — l)*" 
 is the same rational algebraic function of a — & V — 1 that 
 the numerator of the partial fraction whose denominator is 
 (a: — a — b V — 1)'' is of a + b V — 1, these two numerators 
 must be conjugate imaginaries by Art. 28, Ex. 3. Hence, for 
 
 every partial fraction of the form ; — ■ we shall 
 
 {x — a—b-J—iy 
 
 have a second of the form - 
 
 (x — a + b V— l)*" 
 
 Let {x — a-b yj^^y = X+ T V^ 
 
 X and Y being real functions of x ; then 
 
 (x — a-\-b \f^^y = X— Y V^. 
 
 The sum of the two fractions 
 
 A + B V^ A — B V^ 
 
 {x — a—b yl^iy {x — a + b \l^^y 
 
 _ A + B'J^^1 A-B^^^ 2AX+2BY 
 
 X+Y-J-1 X- F V- 1 [(.r - ay + b'y 
 and is a real proper fraction. Hence, 
 
Chap. IV.] RATIONAL FKACTIONS. 47 
 
 /^- ^ /i^ I /»«: 
 
 [(a; - ay + b^f^fyx [(a; - af + b'f "^ [(a; - af + b'f-^ 
 
 + +(x-ay + b^ + ^' W 
 
 every numerator being of lower degree than its denominator. 
 
 fx 
 
 If we take — \^ . ^ and divide numerator and de- 
 
 \_{x a) -r o J 
 
 nominator by (a; — af + b^ we shall get a fraction of the 
 Q + — — 
 
 ^ ' (x aY + b^ 
 
 form pr \ and B will be of the first degree and 
 
 l(x — ay-\-b'J'~'- ° 
 
 therefore of the form L^x -\- M, and we shall have 
 /ix ^ L^x + M^ Q 
 
 l(x - ay + b'f i(x — ay + b'f "•" i(x — ay + b^j- 1 ■ 
 
 By successive repetitions of this process we can reduce 
 
 [(x-ay + l 
 
 to 
 
 Ljx + M^ L^x + M, L„_,x + M„_i 
 
 [(x — ay+b'f'^Kx-ay + b^f-^^ "•" (x-ay + V ' 
 
 Treating all the partial fractions in (1) in this way and 
 adding the results, we shall at last reduce (1) to the form 
 
 fi^ A^x + Bi A^x + Bj 
 
 [(x — ay+b'J'<l>x~[(x — ay + b^f'^[(x — ay+,b^f-'^ 
 
 + + (x~ay + b' +<^x' <-^'* 
 
 and our partial fractions are simple in form and do not involve 
 imaginaries. 
 
 The coefficients in (2) can be found by either of the proc- 
 esses illustrated in Art. 58. 
 
 62. Let us now consider a rather difficult example, where 
 it is worth while to combine all our methods. 
 
48 INTBGKAL CALCULUS. l-^^'^' ^'^^ 
 
 x' + l 
 
 To break up 
 
 x' + l = 
 nary roots. 
 
 (x — l)(x'' + lf 
 x' + 1 = (x + 1) (x' — x + 1) and x' — x + l = has imagi- 
 
 x^ + 1 _ . x^-^-l 
 
 (x-i)(x'+iy~(x-i)(x + iy(x''—x+iy 
 
 A , A ,-82, Cix + A , C'^a + A 
 
 (1) 
 
 " L(^-i)(^'-«=+iyJ.=-i *■ 
 
 Substitute in (1) the values just obtained, clear of fractions 
 and reduce and we have 
 
 — 9 x'' + 2 x" — 6 X* — 8 0!^ + 8 x^ + 6 a; + 7 
 
 = 18 {x' — 1) \_B^ (x^ - X + 1)=' + (Cix + A) (« + 1) 
 + ((7,x + A)(x + l)(x^-x + l)]. 
 
 Divide through by x^ — 1, and we get 
 
 — 9 x' + 2 x^ - 15 x^ — 6 X — r 
 
 = 18 ^ ACa;" — X + 1)2 + (x + 1) [Cix + A 
 ^{C,x-^D,){x^-x^l)-\\. 
 
 Let X = — 1, and we find 
 
 A = -i. 
 
 Substitute this value for A and reduce ; 
 
 — 6 X* — 4 x= — 6 x'' — 12 X — 4 
 
 = 18 (x + 1) [ Cix + i)i + ( C,x + A) (x' - X + 1) ]. 
 Divide by x + 1 and expand and we get 
 [18 C^ + 6] x^ - [18 (C2 - A) + 2] x= 
 
 + [18(C2-A+C0 + 8],T + 18rA + A) + 4 = 0. 
 
Chap. IV.] RATIONAL FRACTIONS. 49 
 
 This equation must hold good whatever the value of x, 
 whence 
 
 18 Cj +6=0, 
 
 18((7,-A) + 2 = 0, 
 
 18(C2-A+Ci) + 8=0, 
 
 18(A+A)+4 = 0, 
 and 
 
 
 
 c. 
 
 = — 
 
 4, 
 
 
 
 A 
 
 = — 
 
 ■f. 
 
 
 
 c. 
 
 = — 
 
 •*, 
 
 
 
 A 
 
 = 0. 
 
 
 Hence, 
 
 
 
 
 
 x^ + l 
 
 1 
 
 1 
 
 1 
 
 
 (k — 1) (a;3 + 1)2 2 « — 1 9 (a; + 1)2 6 a; + l 
 
 _1 a; 1 3a; + 2 
 
 3 ' (a;2 — a; + 1)2 g'cK^-aj + l" 
 
 (2) 
 
 63. Having shown that any rational fraction can be reduced 
 to a sum of fractions which always come under the four forms 
 A A Ax + B Ax + B 
 
 {x — af' x — a (x — ay + h^' \_{x — aY + ¥y 
 
 it remains to show that these forms can be integrated. 
 
 To find (y^„, 
 J (x — a)" 
 
 let s = a; — a, 
 
 then dz = dx, 
 
 1 A 
 
 , r Ad,x . C^z 
 
 and I -. r: = ^ I -:■ 
 
 J (x — af J s" 
 
 (x — af J s" (» — 1) ^""^ 
 
 1 
 
 (w — 1) (x-ay 
 
 To find f-^^, 
 J X — a, 
 
 let « = as — a, 
 
 then d» = dx, 
 
 [1] 
 
50 INTEGRAL CALCULUS. l-'^^'^' ^^- 
 
 and C^^ = AC^^Alog. = Alog(.-a). [2] 
 
 ^ X — a J z 
 
 Turning back to Art. 58 (2), we find 
 
 r {2,x — V)dx _ r dx r dx _ r dx _ _ 1 
 
 J (x — iy(x-[-i)j {x — iy^j x — 1 J x+1 x — 1 
 
 + log(a; - 1) - log(x + 1) = - ^^3 + ^og~j- 
 
 Turning to Art. 60 (1), we have 
 
 / (3x — l)dx _ rdx r dx _ r dx 
 
 x(x — 2)(x + l)~^J x'^Vx — 2 Vx + 1 
 
 = i logx + I log(a3 - 2) - J log(a; + 1). 
 
 To find 
 
 • (Ax + B) dx 
 {x — af + b^' 
 
 Ax + B _ A(x — a) Aa + B 
 
 (x — ay + b^~ {x — «)^ + 6^"^ (a; — af-{-1^' 
 
 If we let z^=(x — ay-\- P, da = 2 (a; — a) dx, and 
 
 /A{x — a)dx A rdz A ^ t r, xo , ,,t 
 
 If we let » ^ aj — a, dz ^ dx, and 
 
 dz 
 
 /g^i^=(-+^/. 
 
 {x — af + b-' "■ ' 'Jz^ + b' 
 
 Aa -\- B ^ ,z Aa + 5 , ,x — a 
 
 = i — taii-i- = — tan-i — r — 
 
 bob b 
 
 Hence CiA^±M^ 
 
 = I log [(X - of + J^] + ^^ tan-^- 
 
 Tofind r/-^- + fIt> - 
 J [(a; — ay + J'']" 
 
 [3] 
 
Chap. IV.] RATIONAL FRACTIONS. 51 
 
 If we let « = (x — o)2 + V^, ds = 2{x~a) dx, and 
 
 / A{x — a)dx A Cdz_ A 
 
 [(x — a)2+62]»~ 2 J ? ~ "" 2 (m - 1) 
 
 — »+i 
 
 2(w — 1) [(x — a)2 + J2]''-i 
 
 If we let s ^ X — a, rfs ^ (fe, and 
 
 r {Aa + £)dx _ r dz 
 
 J i{x-af+¥J ~ *■ * ^ ^U (s^ + *^»' 
 
 /(iqpj^ ''^^ ^^ "^^'^^ *° ^^v^^^ upon /(^qffip-i by *iie 
 
 aid of the reduction formula [6], Art. 64, which for this special 
 form reduces to 
 
 /i 
 
 dz 
 
 _ 1 g . 2>t — 3 r <£g 
 
 "~ 2 (?i — 1) J2 (s^ + by-^ "*■ 2 (re — 1) sO (s2 + 62)«-i ■ L*-l 
 
 Hence f (^^ + S)dx ^ A 1 
 
 xieuuejj ^^^ _ ^^2 ^ j2-]» 2 (to - 1) [(x - af + b"]—^ 
 
 + (-^- + ^) /[(x-$ + .^]~ t^] 
 
 /• rfx 1 X — a 
 
 ^^°^ J [(x - af + b^J ~2{n-l)b^' [(X - af + PJ"^ 
 
 . 2n — 3 r dx 
 
 '^2{n — V)VJ [(x — a)=' + i2]»-i' '-''-I 
 
 /dx 
 ^, _ X2_|_72-|„ to 
 
 /dx 
 g j which has already been found 
 
 to be - tan ^ — j — 
 o 
 
52 INTEGBAL CALCULUS. ^■^^'^- ^^^ 
 
 Turning back to Art. 62 (2), we find that 
 
 r (x' + l)dx _ r dx r dx _ C dx 
 
 J {x — l)(a? + iy~^J x — 1 *J(x + l)^ *Ja3 + l 
 
 / xdx J r(3x + 2}dx 
 (x' — x + lf 'J x^ — x + 1 
 
 = ilog(x-l)+ i-^-^logCa^ + l) 
 
 , a:-2 „ /^, ,2a; — 1 
 
 -i ,._. + l -^TV3tan-'-^- 
 
 — i log (a;= - a; + 1) - 5V V3 tan-^ ?±:zl 
 
 V3 
 
 Examples. 
 
 1^ r oe' — Sx + S , , , x—2 
 
 ^ J (x-l) {x-2) ^x-1 
 
 2) f^^dx = x + ^iog^. 
 
 Ja:^-! ^a:2 + a;+l ^3 VS 
 
 5) r--^ = -i-tan-^^ + -Llog5L±F. 
 ^Ja*-a;* 2a3 a^4a=^a-a! 
 
Chap. IV.] RATIONAL FKACTIONS. 63 
 
 J a^ + x' + l ^ ^x'+x+l 
 
 (10) r«^<i^ - 1 igg ^-a=V2 + l 
 
 -^ [tan-' (a; V2 + 1 ) + tan-^ (a; V2 — 1 ) ] . 
 
 2V2 
 
 (11) r^^_j_iog«^+W2+i _j_^^^_vwi\ 
 
 Ja^+l 4V2 a^-a;V2 + l 2V2 ^-W 
 
54 INTEGRAL CALCULUS. ^^^'^- ®*' 
 
 CHAPTER V. 
 
 REDUCTION FORMULAS. 
 
 64. The method given in the last chapter for the integration 
 of rational fractions is open to the practical objection that it is 
 often exceedingl.y laborious. In man}- cases much of the labor 
 can be saved by making the required integration depend upon 
 the integration of a simpler form. This is usually done by the 
 aid of what is called a reduction formula. 
 
 Let the function to be integrated be of the form x'^~^(a+bafy, 
 where m, n, and p may be positive or negative. If they are in- 
 tegers, the function in question is either an algebraic polynomial 
 or a rational fraction; if they are fractions, the expression is 
 irrational. The formulas we shall obtain will appty to either 
 case. 
 
 Denote a + 6a;" by 2 ; then we want | x'^~^z^dx. 
 
 Let z'' = u 
 
 and a;""-' dx = dv, and integrate by parts. 
 
 du = pz"-^ dz = bnpx"-^ 2f "' dx, 
 
 a;" 
 v = ~, 
 m 
 
 *^ m m J 
 
 This formula makes our integral depend upon the integral of 
 an expression like the given one, except that the exponent of x 
 has been increased while that of z has been decreased. 
 
 We get from [1], by transposition, 
 
 Cx- + ''-'Z^-^dX = ^-r^ r^„_l^ _ 
 
 J bnp bnp./ 
 
Chap. V.] EEDtrCTION FORMULAS. 55 
 
 Change m + n into m and p — 1 into p, whence m is changed 
 into m—n and p into p+1, and we get 
 
 J hn{p + \) hn{p + l)y ' '*"'' L^J 
 
 a formula that lowers the exponent of x while it raises that of ««. 
 
 Since 2 = a + bx", 
 
 z" = z'-\a + bx") , 
 hence 
 
 jar-^z''dx = Cx'^-^z^-^a + bx"") dx = a Caf-'^z^-^dx 
 
 + hC3^+''-'z^-^dx; 
 therefore, by [1], 
 
 ^!!!!_ ^P Caf'+"-'z^-^dx = a Cx'^-'^z^-^dx + b Cx'"+"-^z''-''dx, 
 m m J J J 
 
 Cx-H^-'dx = ^"_ H^ + np) r^»+»-i,,-.^^. 
 J am am J 
 
 Change p into p+1. 
 
 fx--^^dx = ^""'^' - 6(m + r.p + n) r^»+»-Vda,. [3] 
 »/ am am »/ 
 
 Change m into m — w, and transpose. 
 
 Cx-^z^dx = ^"""^"^^ - "^"^-'^l ra;--^.-da.. [4] 
 J 6(m + np) b{m + np)J 
 
 We have seen that 
 
 Cx"-^z''dx = a Caf-'z^-'dx + b Cx'^+''-'^z''-^dx, 
 
 and, from [1], 
 
 6 Cx''+''-'z''-'^dx = ^^ - — faj^-Vda; ; 
 ^ w» npJ 
 
56 INTEGRAL CALCtTLTJS. [A""^- ^^• 
 
 hence 
 
 fx^-^zodx = a fx'"-H'-'dx + ^^ - — Cx''-^z''dx, 
 J J np npJ 
 
 Cx-^'dx^ "'"'"'' + ""^ Cx-'z^-'dx. [5] 
 
 J m + np m + npJ 
 
 Change p into p + 1, and transpose. 
 
 C^-^z^dx= ^-^-"' m + np + n r^™-i,,.>^_ ^g] 
 
 J an{p + l) an{p+l) J '" -" 
 
 Formula [3] enables us to raise, and formula [4] to lowef, the 
 exponent of x bj" n without affecting the exponent of z ; while 
 formula [5] enables us to lower, and formula [6] to raise, the 
 exponent of z by unitj- without affecting the exponent of x. 
 
 Formulas [1] and [3] cannot be used when m = ; 
 
 formulas [2] and [6] cannot be used when p= —\ ; 
 
 formulas [4] and [5] cannot be used when m = —np; 
 for in all these cases infinite values will be brought into the sec- 
 ond member of the formula. 
 
 65. If 71=1, z = a + hx, 
 
 and our last four reduction formulas become 
 
 x^-^z'dx = !^ — -^ ^ ' j x^z^dx. rsi 
 
 am am, J •- -■ 
 
 J b{m+p) b{m+p)J L -I 
 
 faf"-^z'dx = ^^+-S^- Cx^-'z'-^dx. [5] 
 
 •^ m + p m+pJ '- -' 
 
 J a{p+l)^ a{p + l) J"^ ^ *"• LbJ 
 
 Km and p are integers, and to>0 and jj>0, a repeated use 
 of [5] will reduce p to zero, and we shall have to find merely 
 the (x^-^dx. 
 
Chap. V.] REDUCTION FORMULAS. 57 
 
 If m<0 and p>0, [3] will enable us to raise m to 0, and 
 
 then [5] will enable us to lower p to 0, and we shall need 
 
 , rdx 
 only I — 
 
 ^ X 
 
 If m>0 and p<0, [6] will raise p to -1, and [4] will then 
 
 lower 7?i to 1, and we shall need ( — . 
 
 J z 
 
 If m<0 and73<0, [6] will raise p to —1, and [3] will raise 
 
 Jdx 
 — 
 xz 
 
 / 'dx _ r dx _ _ 1 1 a -\-hx 
 xz J x{a + hx) a x 
 
 Hence, when »t = 1, and m and p are integers, our reduction for- 
 mulas alwaj's lead to the. desired result. 
 
 m 
 
 'dx , 
 = logo;, 
 
 X 
 
 Examples. 
 
 m f— ^^ = --loe5^±^ + — ^4--^ 3_ 
 
 ^ ' J a?{a+bx) a' ^ X a*x 2a»ar' 3a=a^ 4aa!*' 
 
 (2) Consider the case where w=2, rewriting the reduction 
 formulas to suit the case, and giving an exhaustive investi- 
 gation. 
 
 (3) f- 
 J (a 
 
 3j CtX X i_ 
 
 (a+bx'y 4:b{a + bx'y- Sab^a + bx') 
 
 -\ tan''a;-vl-- 
 
 8{ab)i \a 
 
68 INTEGRAL CAXiCTJLUS. C^*^' ®® 
 
 CHAPTER VI. 
 
 IRRATIONAL FORMS. 
 
 66. We have seen that algebraic polj-nomials and rational 
 fractions can alwa3-s be integrated. When we come to irrational 
 expressions, however, verj- few forms are integrable, and most 
 of these have to be rationalized bj- ingenious substitutions. 
 
 If an algebraic function is irrational because of the presence 
 of an expression of the first degree under the radical sign, it can 
 be easilj' made rational. 
 
 Let /(cc, "v/a + 6a;) be the function in question. 
 Let z = y/a + bx; 
 
 then z" = a + bx. 
 
 nz"~^dz = bdx, 
 
 nz"-'-dz 
 
 dx = - 
 
 x = - 
 
 b 
 z" — a 
 
 b 
 
 Hence Cfix, ^'^r+bi)dx = ^j'ff^L=-Bi, z\z''-^dz, 
 which is rational and can be treated bj- the methods of Chapter IV, 
 
 Examples. 
 W f^^'^'' = '= + '^V^ + ^iogW^-'^)■ 
 {2) fv(aa; + 6)"da; = '^V(«^- + ^)'"^". 
 •^ a{m-\-n) 
 
 (3) flxA^ix + a) + V(a; + a)^dx 
 
 _ n^{x + ay"+' nay (a; + »)"+' . .. ,, , ,, 
 ~ 2« + l n + 1 +iV{o=+ar. 
 
Chap. VI.] lEEATIONAL FORMS. 59 
 
 67. A case not unlike the last is J7(a;, V7+Va + bx)dx. 
 
 Let z=^/c+ Va+Tx; 
 
 z" = c + Va + bx, 
 («"— c)"' = a + 6a;, 
 
 Hence 
 
 -"/f 
 
 b 
 if{x, Vc + Va + 6a;) dx 
 
 c)'° — a 
 
 Examples. 
 
 {z" — c)"'-^z''-^dz. 
 
 (1) Find C 
 
 (2) Find f- 
 
 xdx 
 
 Vc + Va + bx 
 dx 
 
 Vl+Vl -a; 
 
 68. If the expression under the radical is of a higher degree 
 than the first the function cannot in general be rationalized. 
 The most important exceptional case is where the function to be 
 integrated is irrational bj- reason of containing the square root 
 of a quantitj' of the second degree. 
 
 Required \f(x, Va -\-bx + coi?)dx. 
 
 First Method. Let c be positive ; take out "Vc as a factor, and 
 the radical may be written -VA + Bx + a?. 
 
 Let -y/A +Bx + x' = x + z, 
 
 A+Bx + a? = 3?+ixz+z', 
 
 z'-A 
 
 x = , 
 
 B-2z 
 
60 INTEGEAX, CAXCTTLTJS. ^ART. 68. 
 
 '^- {B-2zy 
 
 z' - Bz +A 
 
 ■^A+Bx + a? = x + z=-- ^_2z 
 
 and the substitution of these values will render the given func- 
 tion rational. 
 
 Second Method. Let c be positive ; take out Vc as a factor, 
 and, as before, the radical, maj' be written V-4 + Bx + 3?. 
 
 Let VA+Bx+^ = yJA + xz ; 
 
 A -k-Bx + y? = A + -l -sjA . xz + 3?7?, 
 
 ^■JA.z-B 
 \ —z' 
 , _ 2 {-^A.z'-Bz + ^IA)dz 
 '^~ (1-2^)'' ' 
 
 ^A+Bx + ar'=^A+xz= ^^-''-^' + ^^ , 
 
 and the substitution of these values will render the given func- 
 tion rational. 
 
 If c is negative the radical can be reduced to the form 
 V^ + Bx — a^, and the method just given will present no 
 difficult^'. 
 
 Third Method. Let c be positive ; the radical will reduce to 
 
 V^ + Bx + a^. Resolve the quantity under the radical into the 
 product of two binomial factors (a; — a) (a; — j8) , a and /3 being 
 the roots of the equation A + Bx + a;^ = 0. 
 
 Let V(a; - a) {x -/3) = (x- a)z ; 
 
 (x-a){x-fi) = (x-ay^, 
 
 x = i^^, 
 ^^^ 2z(^-a)dz 
 
 {i-z'y ' 
 
 ^{x-a){x-l3) = {x - a)z = (^-")^, 
 
 I —z^ 
 
Chap. VI.] IRRATIONAL FORMS. 61 
 
 and the substitution of tliese values will make the given function 
 rational. 
 
 If c is negative the radical will reduce to "si A + Bx — x^, and 
 maybe written V(a — a;) (a; — /3) where ^ and /3 are the roots 
 o( or' — Bx — A = 0, and the method just explained will apply. 
 
 In general, that one of the three methods is preferable which 
 will avoid introducing imaginary constants ; the first, if c > ; 
 
 a a 
 
 the second, if c < and — ; > ; the third, if c < and — < 0. 
 
 ^ -~ c 
 
 If the roots a and j3 are imaginary, and A = -^^ is negative, it 
 will be impossible to avoid imaginaries, for in that case 
 A + Bx — ss' will be negative for all real values of x. 
 
 69. Let us compare the working of the three methods just 
 
 
 
 ■\/2 + 3x + a^ 
 1st. Let VT+Wx+l^ = X + z ; 
 
 r dx ^ r 2(z^-3z + 2)dz 3-2g ^ r 2dz 
 
 •^ V2 + 3a; + ar= -^ iS-2zy 'z'-3z + 2 J 3-22 
 = -Iog(3-22), 
 
 /: 
 
 dx 
 
 V2 + 3a! + a^ 
 
 = log 
 
 = — log(3 + 2x — 2-\/2+3x + a?) 
 
 3 + 2a;-2V2+3a; + a!^ 
 
 _. 3+2a; + 2V2 + 3a; + a^ 
 
 ~^^ 9 + 12a; + 40^ -8 -Ux-AaP 
 
 = log[3+2a; + 2V2 + 3a; + ar']. (1) 
 
 2d. Let ^/2 + 3x + a:? = -s/2+xz; 
 
 r dx __ 2 r(-^2.z'-3z + ^2)dz 1-z' 
 
 J^j2 + 3^+^~-^ (1-^T ^2.z'-3z + y/2 
 
 = 2r-^ = logl+^. (Art. 63) 
 
62 INTEGRAL CALCULITS. l^^'^' ^^■ 
 
 dx _ , a; — V2 + V2 + 3 a; + a' 
 
 f, "^ -log 
 
 V2 + 3a;+a;2 a; + V2 - V2 + 3a; + a;^ 
 
 _, g' + 2a;V2+3a; + a;' + 2 + 3x+g^— 2 
 ~ ^^ x'+2'^2.x+2 — 2-Sx — a? 
 
 _j 3 + 2a; + 2V2 + 3a: + a;^ 
 ~ °^ 2V2-3 
 
 = log(3 + 2a;+2V2+3»+^) - log(2V2-3), 
 or, dropping the constant log (2^/2 — 3), 
 
 r ^ = log(3 + 2 a; + 2 V2 + 3 a; + a^) _ /gx 
 
 •^ V2 + 3a;4-K^ 
 
 3d. Let V2 + 3a; + a;2= V(a;+1) (a; + 2) = (a;+ 1)2 ; 
 
 f '^ ^9 r-"^' 1^=2 c^^=iogi±^. 
 
 J^2 + Sx + af -'(1-22)2 -z ^1-2" 1-2 
 
 r '^ = log ^^+^ = iog^±iiL^E±2 
 
 «^V2 + 3a; + a2 ^ _ j£±2 v«T"l - Vi+2 
 
 \a; + l 
 
 _, a; + l+2V2+3a; + a;^ + a; + 2 
 a; + l— a; — 2 
 
 = log (3 + 2a; + 2 V2 + 3a;+ a;^) + log (- 1), 
 
 or, dropping the imaginary constant log (— 1), 
 
 /dx 
 ;^=====log(3 + 2a; + 2V2T3^+^). (3) 
 
 Examples. 
 
 .1) r ^^^^= ^ lo^ ^^ + '^^-^ JZ^. 
 
 ^ ' J {2 + 3x)-^4:~x' 4V2 V4T2^+V2^^ 
 
Chap. VI.] IREATIONAL FORMS. 63 
 
 70. If the function is irrational through the presence, under 
 the radical sign, of a fraction whose numerator and denominator 
 are of the first degree, it can always be rationalized. 
 
 ^^^^'SK'^^^i^h 
 
 Let z = ^h + \ 
 
 \lx+ m 
 
 ^ _ ax + b 
 Ix + m' 
 
 „ _ 6 — mz" 
 X , 
 
 Iz^ — a 
 ^y. _ ""-{am — &Z)z"-'c?s 
 
 {lz"~aY ' 
 
 and the substitution of these values will make the given function 
 rational. 
 
 Example. 
 
 / dx s/ l— a; _ o 3I /I— aV 
 0.+xy\l+x ^^[l+x)' 
 
 71. If the function to be integrated is of the forma;""\a+6a;")'', 
 m, n, and/i being any numbers positive or negative, and one at 
 least of them being fractional, the reduction formulas of Art. 64 
 will often lead to the desired integral. 
 
 Examples. 
 
 (1) r '^"^'^ =fsin-^a;- ^^^~'"% 3 + 2a;n. 
 ^ ^ J (1-0^)4 8 V -r ; 
 
 dx _ij 1— Vl— a^ Vl— : 
 
 ^ ' J {2ax-a?)i ^ ' \2^ '2 y \2a 
 
64 INTEGRAL CALCULUS. i^^^'^' '^^' 
 
 72. We have said that when an irrational function contains a 
 quantitj' of a higher degree than the second, under the square-root 
 sign, it cannot ordinarily be integrated. It would be more cor- 
 rect to saj' that its integral cannot ordinarilj- be finitely expressed 
 in terms of the functions with which we are familiar. 
 
 The integrals of a large class of such irrational expressions 
 have been specially studied under the name of Elliptic Integrals. 
 They have peculiar properties, and can be expressed in terms of 
 ordinary functions only by the aid of infinite series. 
 
Ch^p. vxi.] transcendental functions. 65 
 
 CHAPTER VII. 
 
 TBANSCBNBENTAL FUNCTIONS. 
 
 73. In dealing with the integration of transcendental functions 
 the method of integration by parts is generallj' the most effective. 
 
 For example. Required j a;(loga;)^cte. 
 
 Let M = (loga;)^ 
 
 dv = x.dx ; 
 
 2 log a;, da; 
 du = > 
 
 3? 
 . = _, 
 
 jxilogxY = ^(1^' - Ja;loga;.c?a; = | [(log xY-\ogx + i]. 
 
 Again. Required Xe'smx.dx. 
 u = sin X, 
 dv = e'dx; 
 du= cos x.dx, 
 v = e', 
 [Vsin x.dx = e" sin a; — I g'cosx.dx, 
 
 Ce cos x.dx = 6== cos a; + j e' sin x.dx ; 
 
 /' . , e'(sina; — cosa;) 
 
 /e'' ( sin a; + cos a;) 
 e'cosa!.da!= -^^ '- 
 
66 INTEGKAL CALCULUS. l^^"^' "*■ 
 
 Examples. 
 
 (1) Cx'-'(iogxydx=^\'(iogxy-^^^^^^ 
 
 , 6 log a; 
 
 (m + lf {m+iy 
 /n\ r^osx.dx xloga; , , /, x 
 
 {3)f€"^{l-e'-).dx = l. 
 •J 2a 
 
 e"V(l-e^'") + sm-ie°- 
 
 74. The method of integration by parts gives us important 
 reduction formulas for transcendental functions. Let us con- 
 sider I sin"a;.da;. 
 
 u — sin""'a!, 
 dv=: sinx.dx; 
 
 du = {n—l)sin'"^x cosx.dx, 
 v = — cosic ; 
 
 j sin"a;.da; = — sin"-*a; cosa; + (n — 1) fsin—^a; cos^x.dx 
 
 = — sin"-'a; cosx + (n — 1) | (sin"-^a; — sin"a;)«fo! ; 
 
 Csm"x.dx = -- sin"-'a; cosx + ^?-ll Csin'-^x.dx. [1] 
 
 Transposing, and changing n into n + 2, we get 
 
 I sin''a;.dx = — — - sin"+'a; cosa; + ^^-±1 rsin"+2a!.cte. r21 
 -^ n + 1 n+lJ ^ -^ 
 
 In like manner we get 
 J cos»a;.da; = ^ sin a; cos""' a; + ^ii^ rcos»-2a;.cte, [3] 
 
 J'eos»a;.da; = -— l_sina;cos-+'a; + ^+iJ*cos»+^a;.dx. [4] 
 
 If n is a positive integer, formulas [1] and [3] will enable us 
 to reduce the exponent of the sine or cosine to one or to zero, 
 
Uhap. VII.] TRANSCENDENTAL FUNCTIONS, 67 
 
 and then we can integrate by inspection. If m is a negative 
 integer, formulas [2] and [4] will enable us to raise the ex- 
 ponent to zero or to minus one. In the latter case we shall need 
 
 /dx (* clx 
 , or I , which have been found in Art. 54 (c) . 
 cos as J since 
 
 Examples. 
 
 1 + ^^- 
 
 (-1) jsin*a;.daj = [sin''a;+-|- 
 
 /n\ r e J sinajcos'a;/ » , 5\ , 5 / . , , 
 
 (2) I cos''a;.da;= i eos''a;-|--] + — (sinKCOSiB+a;). 
 
 /•Q\ r dx cosa; , i,^ . x 
 
 (3) |-r-^ = - „ ■ 2 +^logtan-- 
 J sin' a; 2sin^a; 2 
 
 (4) Obtain the formulas 
 
 /sinh"a;.da;=-sinh""'a;cosha; ^— | sinh"~'a;.da;. 
 n n J 
 
 sinh"a;.da;= — — sinh"+^a;eosha; ^^— I sinh"+^a;.da;. 
 
 n+1 n+lJ 
 
 /cosh"a;.da;=-sinha;cosh" 'a;-)-^^^^^ | eosh""^a!.da;. 
 n n J 
 
 /coah"x.dx= i-sinha;cosb''+'a;-|-^^^i- j cosh"+^a;.da;. 
 n+1 71+lJ 
 
 ,,. r dx __ 1 cosh a; j. cosha;— 1 
 ^ ^ J sinh'a; ~ ^sinh^a; * ^cosha;-t-l' 
 
 75. The (sin"' a;) "da; can be integrated by the aid of a redue 
 tion formula. 
 
 Let 2=sin"^a;; 
 
 then a; = sin 2, 
 
 dx = cosz.dz, 
 
 and J (sin-'a;)"da;= |2"cos2.&. 
 
68 
 
 INTEGRAL CALCULUS. t^BT. 76. 
 
 Let M = »"i 
 
 dv = cosz.dz ; 
 
 v = sinz ; 
 
 Tz" cosz.dz = z"sinz — niz" 'sinz.dz. 
 
 Cz"-^smz.dz can be reduced in the same waj-, and is equal 
 — z"-'cosz + (n — 1) ^"-''cosz.dz ; 
 
 to 
 hence 
 
 fz" cosz.dz = z"sinz + «z"-'cosz — n(n — 1) jz"-'' cosz.dz, [1] 
 
 or j (sin"' a;)" da; = a:(sin-'a;)"+ nVl — a;^(sin-^a;)"-* 
 
 -n(n-l) r(sin-'a;)''-2d«. [2] 
 
 If n is a positive integer, this will enable us to make our re- 
 quired integral depend upon | dx or | sln~'a;.da;, the latter of 
 which forms has been found in (I. Art. 81). 
 
 Examples. 
 
 (1) Obtain a formula for j (vers-^a;)"da;. 
 
 (2) r(sin-'a;)«da; = a;[(sin-ia;)*- 4 . 3 .(sin-'a!)2+4 . 3 . 2 .1] 
 
 + 4 Vl^^sin-'a; [(sin-'a;)^- 3 . 2]. 
 
 76. Integration b}- substitution is sometimes a valuable method 
 in dealing with transcendental forms, and in the case of the trigo- 
 nometric functions often enables us to reduce the given form to 
 an algebraic one. Let it be required to tind | (/sin a;) cos a;. da;. 
 
 Let z=sina;, 
 
 dz = cosx.da; ; 
 
 I (/sin a;) cosa: dx = \ fz.dz. 
 
Chap. VII.] TRANSCENDENTAL FUNCTIONS. 69 
 
 In the same -way we see that 
 j (fcosx) sinx.dx = — Cfz.dz if z=cosx, 
 
 and 
 j [/(sina;, cosa;)]cosa;.da;= Clf(z, Vl— /)]d« if z=smx, 
 
 j lf{cosx, sina;)] sina!.da; = — | [/(a;, Vl — 2^)]dz if «=cosa;, 
 or, more generally, 
 
 //(sina;, cos a;) da; = i f{z, Vl— «^) — — ^ — if »= sin a;, 
 J -s/l—z" 
 
 //(cos a;, sin x)dx = — j /(«, V 1 —2^) — — — if z= cos a;. 
 
 - Since any trigonometric function of x may be expi'essed in 
 terms of sina; and cos a;, the formulas just given enable us to 
 make the integration of any trigonometric function depend ou 
 the integration of an algebraic function, which, however, is 
 frequently complicated by the presence of the radical VT— ?. 
 
 77. A better substitution than that of the last article, when 
 
 the form to be treated does not contain sina; or cosa; as a factor, 
 
 . X 
 
 IS z = tan— 
 
 2 
 
 This gives us ax=- 
 
 1+2^' 
 
 1 +r 
 
 1-2' 
 cos x = - 
 
 1+2^ 
 
 Whence J/(sina;, cosa;)da;= 2j/(^^, f:p|)^- [1] 
 
 /dx 
 ■ ; 
 a 4-0 cos as 
 
70 INTEGRAL CALCULUS. E-^T. 78. 
 
 Here we have 
 
 r dx ^ 2 r ^^ r - — . 
 
 {l+z") 
 
 b-^m 
 
 = _J_ r dz ^ 2 t,^-x( Ja^.\ 
 a - 6 by I. Art. 77, Ex. 1. 
 
 Hence ( = — ^= tan"^ ( -%/ ~ • tan- |, if a > 6. 
 
 J a + 6 cos a; ya^ — f^ \ya + b 2/ 
 
 78. I sin"" a; cos" a;, dec can be readily found by the method of 
 
 Art. 76 if m and n are positive integers, and if either of them 
 is odd. Let n be odd, then 
 
 n-l 
 
 co8"a;=cos"~'a;co8a; = (l — sin^a;) 2 cos a;, 
 
 I sin"a;cos"a;.da;= | 8in'"a!(l — sin ''a;) "a" cos a;, da;. 
 
 Let z = sin a;, 
 
 dz = coBx.dx, 
 
 I sin" a; cos" a;. da; = j z" (1 — g')~i'dz, 
 
 which can be expanded into an algebraic polynomial and inte- 
 grated directly. 
 
 If m and n are positive integers, and are both even, 
 
 I sin^a; cos"a;.da; = | sin^a; (1 — sin^a;)2da;. 
 
 n 
 
 sin'"a;(l — 8in^a;)2 can be expanded and thus integrated by 
 Art. 74 [1]. 
 
 If m or n is negative, and odd, we caa write 
 
 cos"a; = cos"^^a;cosa;, or 8in'"a; = sin^'^aisina;, 
 
 and reduce the function to be integrated to a rational fraction 
 by the substitution of 
 
 z = cos X, or z = sin x. 
 j sin" a; cos" x. da; can also be treated by the aid of reduction 
 formulas easily obtained. 
 
Chap. VII.] TBANSCENDENTAL FUNCTIONS. 71 
 
 79. I tan"a;da! and ( can be handled by the methods 
 
 J J tan"a; 
 
 of Art. 78. but they can be simplified greatly by a reduction 
 formula. 
 We have 
 
 I tan"a;.r2x= | ta,n"~^xtan^x.dx= | tan""^a;(8ec^a;— l)dx 
 
 = I tan""^a!d(tanx)— | tan"~^a;.da;, 
 
 whence ftan" x.dx = ^^""'''" - CtaW-'x.dx ; [1] 
 
 1 C—^^ — r^ec^a; — tan^a ; , _ / ^d(tana;) _ / ^ dx 
 J tan" a; J tan" a; J tan" a; J tan"~^a;' 
 
 whence T-^^ = ^ ^ - f-^- [2] 
 
 J tan" a; (" — 1) tan""' a; J tan"-''a; 
 
 Bin? X cob' X.dx = 
 
 2) i cos'a; Vsinaj.da; = 
 
 „. ^ sin^x.da 
 
 ^ J / 
 
 •^ Vcosa; 
 
 Examples. 
 
 cos'" a; cos' a; 
 "To 8~' 
 
 28in^a; 2sin^a; 
 
 3 7 
 
 dx 2cos^a; 
 
 •2 cos* a;. 
 
 Vcos X ^ 
 
 .. r 9 -4 7 sin a; cos a; /sin^ a; sin^a; IN , a; 
 
 4) J cos^a; sin^a^.da; = ^ {—^ IT ~ sj + Te' 
 
 •5) r ^^ =seca; + logtan|. 
 
 J sin X cos'' a; ■* 
 
 ^^ /* da; cos a; , 3, „,„„a; 
 
 6) ■ " , =seca;-—r-^+-logtan-- 
 
 J sin^ajcos^a; 28in2ai 2 5 
 
 da; 1,1 
 
 7^ r_^ = i-+ 
 
 "^ J tan* a; 4 tan'' a; 
 
 4 tan* a; 2tan^a; 
 
 2 
 
 + log sin X. 
 
72 
 
 INTEGRAL CALCXTLTJS. 
 
 [Art. 79. 
 
 («>/a 
 
 dx 
 
 + 6 cos a; 
 
 ■sIV 
 
 :l0g 
 
 Vft + a + V 6 — a . tail - 
 ■\lb-\-a — V6 — a . tan- 
 
 dx 
 
 /4 + 5tan-^ 
 
 (9) f- 
 
 J 5 + 4sina; 
 
 (10) f ^5 = ilog sin- - log cos^ + ?log(3+2cosa;). 
 
 ^ '^J 3sina; + sin2a! 5° 2 ^25^'' ' 
 
 .r_oosxd^^_ ^5_^nx Atan-'l^itan?^ 
 
 ^ ^J (5 4-4cosa;)'' 9 5 + 4cosa; 27 ^3 V 
 
 (12) f- 
 J a 
 
 15 + 4 cos a; 
 2 
 
 -tan"' 
 
 (a— c) tan- + 6 
 
 t+6sina;+ccosa; Va=— &"— c^ L ^/a'—b'—c' 
 
 (13) Show that the methods described in Arts. 76-79 apply 
 to the Hyperbolic functions. 
 
 (14) f 
 
 dx 
 
 (l^)/« 
 
 a+6cosha; ^/b" — a^ 
 dx 
 
 tau~'| \/ tanh- 1 if 6>a. 
 
 V^6 + « 2j 
 
 -\- b sinh x -\- c cosh x 
 2 
 
 Vc 
 
 tan" 
 
 (c — a) tanh 7^-\-b 
 -^c' — d' — b^ 
 
Chap. VIII.] 
 
 DEFINITE INTEGKALS. 
 
 78 
 
 CHAPTER VIII. 
 
 DEFINITE INTEGKALS. 
 
 80. In I. Art. 183, a definite integral has been defined as the 
 limit of a sum of infinitesimal terms, and has been proved equal 
 to the difference between two values of an ordinary integral. 
 
 We are now ready to put our definition into more precise, 
 and at the same time more general, form. 
 
 If fx is finite, continuous, and single-valued between the 
 values a; = a and a; = 5, and we form the sum 
 
 (aji — a) fa + {x^ — x^fx^ +(»'3- x^fx^A 1- (a!„_i— a!„-2)/a'„_s 
 
 + (6 — a;»-i)/a:n-i, 
 
 where aj, x^^ ^---a^n-i are n— 1 successive values of x lying 
 between a and 6, the limit approached by this sum as n is in- 
 definitely increased, while at the same time each of the increments 
 {x^ — a), (x^ — a^), etc., is made to approach zero, is the definite 
 
 integral of fx from a to &, and will be denoted by | fx.dx. 
 
 If we construct the curve y =fx in rectangular co-ordinates, 
 this definition clearly requires us to break up the projection on 
 the axis of X of the portion of 
 the curve between the points A 
 and B into n intervals, to multi- 
 ply each interval by the ordinate 
 at its beginning, and to take the _ 
 limit of the sum of these products 
 as each interval is indefinitely decreased ; that is, the limit of 
 the sum of the small rectangles in the figure, and this is easily 
 proved to be the area ABA^Bi. 
 
 Now the area ABA^B^, found by the method of I. Chap. V., 
 
74 INTEGRAL CALCULUS. [Akt. 81. 
 
 Therefore C fx.dx = \ ^fx.dx ^ " J/*-'^^ I ,• W 
 
 That is, I fx.dx is the increment produced in j fx.dx by 
 
 changing x from a to 6. 
 
 It is to be noted that the successive increments (aji — a), 
 (ajj — £Di), (Kj — a^), etc., that is, the successive values of dx^ 
 are not necessarily equal ; and also, that if we multiply each 
 interval, not by the ordinate at its beginning, but by an ordinate 
 erected at any point of its length, the limit of our sum wUl be 
 unaltered, {y. I. Arts. 161, 149.) 
 
 81, It is instructive to find a few definite integrals by actu- 
 ally performing the summation suggested in the definition 
 (Art. 80) , and then finding the limit of the sum. 
 
 (a) I x.dx. 
 
 Let us divide the interval from a to 6 into n equal parts, and 
 call each of them dx. 
 
 Then ndx =b — a. 
 
 Our sum is 
 
 S= adx + (a+dx)dx+{a + 2dx)dx-\ \-{a + {n—l)dx)dx 
 
 = nada; + (l + 2 + 3H \-(n — l))dx' 
 
 since ndx = b — a, and the sum of the arithmetical progression 
 1+2 + 3-1 |-(w-l)= "(^~^) . 
 
 2 ^ ^ ^2 2 
 
 Hence S = ^^i^ -i^^l^dx. 
 
 2 2 
 
Chap. VIII.] DEFINITE INTEGRALS. 75 
 
 As we increase n indefinitely, dx approaches zero, and 
 
 Cx dx = ^™'* r &' - »' _ (& - a) dx ~\ ^ 6^ _ a^ 
 X ' da;=0|_ 2 2 J 2 2* 
 
 (6) f ^dx. 
 
 Let dx = 
 
 n 
 
 S = e'dx + e^^dx + e'+^-^dx -| 1- e-'+t—i'^'da; 
 
 = e''da; [1 + e*" + e^'' + 6"'^ H h e'"""'''"] ; 
 
 but 1 +6*" + e^*"+ ••• + e'""''** is a geometrical progression, 
 and its sum is 
 
 gdx _i e*" — 1 
 Hence S = ^"'^ ~ ^ • e« da; = (e» - e") --^-, 
 
 and fe'<^.= (e-e^),S[^]' 
 
 but as dx approaches zero, —r approaches the indeterminate 
 
 form - ; but since the true value of 
 
 Le*— lJa=o Le'J^ ' 
 e'da; = e' — e". 
 
 r 
 
 cos^x.dx. 
 
 («) X 
 
 Let dx = -, and let n be an odd number. 
 n 
 
 Then 
 
 <S = da; + cos'd9;-da;+ cos^ 2 da; ■ da; H |-cos'(n — 2) da;-da; 
 
 + cos' (m — 1) da; • da; 
 
 = da; + cos' da;- da; + cos' 2 da;- da; H 1- cos' (tt — 2 da;) -dx 
 
 + cos' (tt — dx) ■ dx 
 = dx + cos' da;- da; + cos'2 da; -da; -\ cos' 2 da;- da; 
 
 — cos' da; -da;, 
 
76 INTEGliAL CALCULUS. [ART. 81. 
 
 since cos (tt — <#>)= — cos <^. 
 
 Hence the terms cancel in pairs, and we have left 
 S = dx 
 
 and JJcoB'x.dx = ^^\ \^^ = 0. 
 
 id) j\i 
 
 Siv?x.dx. 
 
 Let da; = — , and let n be an odd number. 
 2?t 
 
 S = sm''Q-dx+B\v?dx-dx-\-s\n-'2dx-dx+ \-s,vo?(ji—'2,')dx-dx 
 
 +sin^ (ii — 1 ) da; ■ dx 
 
 = sin''da;-da;4-sin^2da;-da;-| f-sinV-— 2da; ]da;+sinY-— da;Jdar 
 
 = sin''da;.da;+sin^2da;-da;H |-cos^2dx.dx+cos-da;-da;, 
 
 since sinj- — <^j = cos<^. 
 
 n — 1 
 Then ^S = da; + da; + da; • • • = da;, 
 
 since sin^ <^ + cos^ <^ = 1 . 
 
 TT _ da; 
 i Y 
 
 Therefore <S' = ^ — 
 
 J<>2 ^ 
 
 I sin^ a;. da; = — 
 4 
 
 («) 
 
 dx 
 
 'a X 
 
 Here it is best to divide the interval between a and 6 into 
 unequal parts. 
 
 Let the values a;i, x^, Xs ••• x„_^ be such as to form with a and 
 6 a geometrical progression. 
 
 For this purpose take q = -v/-, so that aq" = b. 
 
Chap. VIII.] 
 
 DEFINITE INTEGRALS. 
 
 77 
 
 Then the values in question are ag, aq^, a^ •••aq"'^, and the 
 intervals are a (g — 1) , aq{q — \), aq^ (s — 1) ••• a?""* (9 — 1) » 
 and the sum 
 
 a(g-l) ag(g-l) ag^g - 1) , ag'-'(g-l) 
 
 a "^ ag "^ ' ag^ -r -i- „^„-i 
 
 = «(g-l). 
 
 To prove our division legitimate we have only to show that 
 each of our intervals, a(g— 1), ttg(g— 1) ■■• og"-'(g — 1), 
 approaches the limit zero as n increases indefinitely. Since 
 
 g" = - 
 a 
 
 the limiting value of g as m increases must be 1, as otherwise 
 limit „» would not be finite. 
 
 Therefore I'^'l [_aq\q - i)] = J^f J [ag^Cg - 1)] = 0. 
 
 We have then 
 ■"' dx limit 
 
 limit r 
 
 r^djo limit ^^3^ limit [n(g_l)]= l>'!^it[n(g-l)] 
 
 limit 
 g = l 
 
 l0£- 
 
 logg 
 
 (9-1) 
 
 since 
 
 n log g = log - 
 
 But 
 
 limit 
 9 = 1 
 
 log- 
 a 
 
 logg 
 
 (9-1) 
 
 J 
 
 , 6 limit 
 
 = i°g;;.g=i 
 
 i^1 = log^. 
 loggj a 
 
 For Tf^l =[!" = 
 
 _9jf= 
 Therefore f — = log 6 — log a, 
 
 Ja X 
 
78 INTEGKAL LVViCULUS. L^'^'- «2. 
 
 Examples. 
 
 (1) Prove by the methods of this ai-ticle that 
 
 I a'dx = — 
 
 jb log a 
 
 (2) By the aid of the trigonometric formulas 
 
 cose + cos2^ + cos30H hcos(n — 1)^ 
 
 = ^ sinn5 ctn 1 — cosrt^ , 
 
 sine +siu2e +sin3e -I h sin(?i— l)d 
 
 ■=\\ (1— cosn6)ctn ^sinnd , 
 
 prove that | cosa.da; =sin6— sino, 
 
 and I sincc.do; = cos a — cos 6. 
 
 (3) Show that i sin°a;.Q!a;= 0, 
 
 and that j cos^x.da; = -- 
 
 Jo 2 
 
 3^(1% = , using the method of 
 
 m + 1 
 
 Art. 81 (e). 
 
 82. When the indefinite integral can be found, the definite 
 integral | fx.dx can usually be most easily obtained by em- 
 ploying the foi-mula [1] Art. 80, and this can always be done 
 with safety when fx is finite, continuous, and single-valued 
 between a;=(x and x = h. 
 
 Of course, if the indefinite integral is a multiple-valued func- 
 tion, we must choose the values of the indefinite integral cor- 
 responding to a; = a and x — b, so that they may be ordinates 
 
 of the same branch of the curve y= i fx.dx. 
 
Chap. VIII.] DEFINITE INTEGRALS. 79 
 
 ^- The indefinite integral 
 
 c ax 1 1 + a; 
 
 I = tan~^a; and tan^^a; is a multiple-valued function. 
 
 Jl+a? _ ^ 
 
 Indeed, y = tan~^a; is a curve consisting of an infinite number 
 
 of separate branches so related that ordinates corresponding to 
 
 the same value of x differ by multiples of w. On the branch 
 
 which passes through the origin, when x= — l, y=taxr^x= — ^ ; 
 
 on the same branch, when a; = 1, w = tan~'x = — On the next 
 
 branch above, when x = — 1, y= tan^^a; = — ; and when x=l, 
 
 5 4 ^ 
 
 y = — ■ On any branch, when x=—l,y = tan~^a; = — ~+mr; 
 
 and on the same branch, when x=l, « = - + mr. 
 
 4 
 
 X' dx _bir 37r_7r 
 iTT^~ 4 4 "2' 
 
 or 
 
 or j ""^ „ = ^ + n7r — (— 7-f-Wir) = 
 
 = - + mr — l \-nir ] = -• 
 
 + a^ 4 \ 4:^ J 2 
 
 By ( fx.dx we mean the limit approached by I fx.dx as 6 
 is indefinitely increased. 
 
 Examples. 
 (1) Work the examples of Art. 81 by the method of Art 82, 
 
 
 (4) r% 
 
 Jo a 
 
 4 sin aj.da; 
 
 = V2-1. 
 
 cos' a; 
 
 dx _ 4 /- 
 
 ■\/x + a + ^x 3 
 
 = |Va(V2-l). 
 
 da; _ TT _ 
 + a;2 — 2a' 
 
80 INTEGRAL CALCULUS. [Art. 83 
 
 (5) r _^:§^^I. if a>0, and --if a<0, andOif a = 0. 
 
 e-"dx =- if o>0. 
 
 J ^00 
 
 
 9) f '^'g 
 
 »/o 1 + 2 a; cos <i> + oi? 
 
 e'" smmx.dx =— if a>0. 
 
 a? + m^ 
 
 ""cos ma;. da; =— — '■ — - if a>0. 
 
 a^ + w? 
 
 i_. 
 
 <^ + a;^ 2 sin </> 
 
 dx _ ")^ 
 
 + 2 a; cos <^ + a;^ sin <^ 
 
 83. When fx is finite and single-valued between x=a and 
 a; = &, but has a finite discontinuity at some intermediate value 
 a; = c 
 
 fx.dx = I fx.dx + I fx.dx, 
 
 \ \ \ and therefore I fx.dx can be found by 
 
 ^ " "A ^j.j^ g2 ^jigQ tije indefinite integral 
 
 \ fx.dx can be obtained; but when fx becomes infinite for 
 x=a, or for x=b, or for some intermediate value a; = c, 
 special care must be exercised, and some special investigation 
 is usually required. 
 
 If fx is infinite when x = a and | fx.dx approaches a finite 
 
 limit as e approaches zero, this limit is what we shall mean by 
 
 /a;.da; ; if 1 fx.dx increases indefinitely as e approaches 
 
 zero, we shall say that I fx.dx is infinite ; and if | fx.dx 
 
 neither approaches a finite limit nor increases indeflnitelv as c 
 
Chap. VIII.] DEFINITE INTEGRALS. 81 
 
 approaches zero, we shall say that j fx.dx is indeterminate. 
 
 fx.dx can be safely employed 
 
 in mathematical work. 
 
 If fx is infinite when x = b and I fx.dx approaches a finite 
 
 J<i. ^t 
 
 limit as e approaches zero, that limit is the value of I fx.dx. 
 
 If fx is infinite when a; = c, and each of the expressions 
 
 fx.dx and | fx.dx approaches a finite limit as e approaches 
 
 zero, the sum of these limits is I fx.dx. Should either or 
 
 both of the expressions, 
 
 fx.dx, I fx.dx, 
 
 fail to approach a finite limit as e approaches zero, | fx.dx is 
 either infinite or indeterminate, and cannot be safely used. 
 
 "When the indefinite integral of fx.dx can be obtained there 
 is little difficulty in deciding on the nature of I fx.dx in any 
 
 mJa, 
 
 of the cases just considered, or in getting its value when that 
 value is finite and determinate. 
 For example, 
 
 \ — is infinite, since 
 
 X 
 
 I — = logxand j — ^ = log(l) — log£ = log-, 
 
 »/ X «/e X € 
 
 and increases indefinitely as e approaches zero. 
 
 J°° dx 
 is not finite and determinate, for 
 1 — 3? 
 
 / 
 
 ■^ i-ii«g-;+- 
 
 I —sr l—x 
 
 and increases indefinitely as e approaches zero. 
 
82 INTEGRAL CALCULUS. [Akt. 84. 
 
 (c) \ ""' — is finite and determinate, for 
 Jo Vfl! _ a^ 
 
 ' = sin ' sin^O = sin^ , 
 
 " Va^ — a^ " • " 
 
 and its limiting value as e approaches zero is sin~'(l) or -• 
 
 r is finite and determinate, for 
 
 (1-x)^ 
 
 S 
 
 i 
 
 ^j^,_((i— )'-Ui-"')'. 
 
 '"' ado; 
 
 .3-5 3,§ 3J_3 
 
 /o (l_a;)i 
 and its limiting value as e approaches zero is f — f . 
 
 r" xdx _ 
 
 ■"5^"~Ti"-F 
 
 + |^'+|^S 
 
 and its limiting value as e approaches zero is — f — f , and 
 consequently 
 
 ••2 
 
 r^^_3_3_3_3__ 
 
 84. When, as is sometimes the case, the indefinite integral 
 cannot be obtained and the function to be integrated becomes 
 infinite at or between the limits of integration, we have 
 recourse to a very simple test which is easily obtained by 
 the aid of the following important theorem, known as the 
 Maximum- Minimum Theorem. 
 
 If a given function of x is the product of two functions both 
 finite, continuous, and single-valued, one of which tf}(x) does not 
 change its sign between x = a and x = b, and if M is algebra- 
 
Chap. Vlll.] DEFINITE INTEGRALS. 83 
 
 ioally the greatest and m the least value of the other factor 
 f (x) between x = a and x = b, I f (x) <^ (x) dx lies between 
 
 </>(x)dx and m 1 <^(x)dx. 
 
 To prove this theorem, let us first suppose that ^{x) is 
 positive between a; = « and a; = 6. M — f{x) is positive for 
 the values of x in question, [ilf — /(a;)]<^(x) is positive, and, 
 therefore, 
 
 C\_M—f(x)-\ <l> (x) dx>0 
 
 and M f <ji(x)dx> f f{x)<l>(x)dx. (1) 
 
 f(x) — m is positive for all values of x between x = a. and 
 X := 6, \_f(x) — m] </) (x) is positive, and, therefore, 
 
 f [/(^) — H "^ (*) «^a; > 
 
 /(a;) <j) (x)dx>m I <^ (x) cZaj. (2) 
 
 a t^a 
 
 /(x) <^ (x) (^a; lies between ilf I <^ (a;) dx and to 
 
 X<j} (x) «£x. It is easy to modify this proof to meet the case 
 
 where <p (x) is negative. 
 
 We can briefly formulate the result of the Maximum-Mini- 
 mum Theorem as follows : 
 
 Cf{x)<i>{x)dx=f{i) f\{x)dx, (3) 
 
 where i, is some value of x between a and b. 
 
 Let us apply this theorem to the consideration of j f(x)dx 
 when/(x) becomes infinite for x = a. 
 
84 INTBGKAIi CALCULUS. [Art. 84. 
 
 In order that ™* f/C^) '^^ should be finite and de- 
 
 E==0 [j^a + e J 
 
 terminate it is easily seen to be necessary and sufficient that 
 
 ( I f(x) dx I should be equal to zero. 
 £d=0La = 0VJa+<r ^ ^ /J ^ 
 
 Let us write f(x) in the form . _ fi and let < A < 1. 
 is positive for all values of x greater than a. 
 
 (x — a)' 
 
 does not increase indefinitely as ^ approaches a. 
 
 Call ^ — a ■>;, whence ^ ^ a + ij. Then a sufficient condition 
 that I f(x) dx shall be finite and determinate when /(a) =co 
 is that t^fiob + if) shall not increase indefinitely as f] ap- 
 
 /j-p ^^ Qi\ "F (^Xj\ 
 
 proaches zero, < A; < 1. If we write f(x) = '•> ^ / 
 
 and proceed as above, we can show that a necessary condition 
 that I f{x) dx shall be finite and determinate when/(a) =oo 
 
 */ a 
 
Chap. VIII.] 
 
 DEFINITE INTEGRALS. 
 
 85 
 
 If f(b) = oo our sufficient condition is that Tf'f(b — ij) shall 
 not increase indefinitely asrjzbO, 0<^<1; and iifip) = oo 
 that neither »;'/(c — 7;) nor ri''f{o -\- r/) shall increase indefi- 
 nitely as 17 ==:; 0, < A < 1. 
 
 Let us apply our tests to the examples considered in Art. 83. 
 
 (a) I — = 00 because . - =1. 
 
 ^ ^ Jo X r) =0 \_r]j 
 
 00 I 1 "3 ^^ indeterminate, for 
 
 limit P i? ~| _ limit F 1 ~| _ 
 
 ,=0Ll-(l-.7)d~^=0L2-J-*'. 
 
 and 
 
 limit r 17 "1 _ limit P —1 ~[ _ _ , 
 
 (v + 1) 
 
 J'*" dx 
 , is finite and determinate, for 
 
 » -Ja' — x' 
 
 1 
 
 = Oiii<k<l. 
 
 7J=0 
 
 \ „2*:-l V 
 
 - '? 
 
 T is finite and determinate, for 
 
 (1 — xy 
 
 and 
 
 [p^^^i^], = C-/-'(i + ,)].=.=»«i<*<i. 
 
86 INTEGRAL CALCULUS. [Art. 84. 
 
 Even when, as in the examples just given, the indefinite 
 integral can be obtained, there is a decided advantage iu using 
 the very simple method of this article. For if the application 
 of the test shows that the definite integral in question is infinite 
 or indeterminate, the labor of finding the indefinite integral is 
 saved ; and if the application of the test proves the definite 
 integral finite and determinate, it follows that the indefinite 
 integral does not become infinite for the value of x which 
 makes the given function infinite, and eonsequentl3' when the 
 indefinite integral has been obtained, the method of Art. 82 
 can be used without hesitation. 
 
 As an example, where the indefinite integral cannot be ob- 
 tained, let us consider at some length 
 
 X 
 
 log-) dx. 
 
 If n is positive, (log-j is continuous and single-valued be- 
 tween x=0 and x=l, but becomes infinite when x=0. We 
 must then investigate the limiting value of jj* ( log - 1 as i; 
 approaches zero. 
 
 1/* ( log - ) is indeterminate when i; = 0, but its true value is 
 easily found to be zero if n is positive, whether n is whole or 
 fractional. For positive values of n, j [log-] dx is, theu, 
 finite and determinate. 
 
 If n is negative, call n = — m. 
 
 Then fflog^Xdx^r-J^ 
 
 '<)' 
 
 is continuous and single- valued from x = to x = l, but be- 
 comes infinite when «= 1. 
 
Chap. VIII.] 
 
 DEFINITE IMTEGRALS. 
 
 87 
 
 We must, then, find 
 
 limit 
 
 7, =0 
 
 which proves to be if m < 1 ; if m = 1, 
 
 log 
 
 = 1; 
 
 r,= 
 
 and if m > 1, it is infinite. 
 
 
 is, then, finite and determinate if m < 1, but infinite if m = 1 
 or m > 1 ; and we reach the result that 
 
 XY-i)" 
 
 dx 
 
 is finite and determinate if w > — 1, but infinite if w = — 1 or 
 »i<— 1. 
 
 Examples. 
 
 (1) Prove that 
 
 r loga' ^^ C}Si±.dx, r^logri±5Y 
 Jo 1 — a; " ^ Jo 1 — 3? ^ Jo X \)-— «/' 
 
 are finite and determinate. 
 
 (2) Prove that 
 
 J"" dx C x^^dx 
 . , i , where m and n are integers, and 
 
 . dx, are not finite and determinate. 
 
 1 —a; 
 
 (3) Find for what values of w j (loga!)"da; is finite and 
 determinate 
 
88 INTBGEAL CALCUIiUS. [Art. 85, 
 
 (4) Find for what values of m and n j aj^nog-j dx is 
 
 finite and determinate. 
 
 (5) Show that j a;"-i(l — a;)''"*da! is finite and determinate 
 if m and n are positive. 
 
 (6) Prove that | log sin aj.da; is finite and determinate. 
 
 (7) Show that the following integrals are finite and deter- 
 minate, and obtain their values : 
 
 r 
 
 Jo ■ 
 
 X 
 
 dx 
 
 si a' — a? 2 
 dx 
 
 — = ir. 
 
 "si ax — a^ 
 
 J"^ dx ^ 7r_ 
 I a;Va;2_i 3 
 
 85. It was stated in Art. 82 that by | fx.dx we mean the 
 
 fx.dx as h is indefinitely increased, and, 
 
 " f 
 
 as we have seen, if the indefinite integral \ fx.dx can be found, 
 
 there is no difficulty in investigating the nature of I fx.dx and 
 
 in obtaining its value if it is finite and determinate. There are, 
 
 however, many exceedingly important definite integrals of the 
 
 form I fx.dx whose values are obtained by ingenious devices 
 
 without employing the indefinite integral, and these devices 
 
 are valid only provided that the integral in question is finite and 
 
 determinate, since an infinite value not recognized and treated 
 
Chap. VIII.] 
 
 DEFINITE INTEGRALS. 
 
 89 
 
 as such, or a value absolutely indeterminate, renders inconclu- 
 sive any piece of mathematical reasoning into which it enters. 
 If we construct the curve 
 
 
 'x.dx is the limit- 
 
 ing value approached by the 
 area A BBiAi, as OBi is in- 
 definitely increased ; and in 
 order that this area should 
 
 be finite and determinate, it is clearly necessary and sufficient 
 that the area BCCiBi should approach zero as its limit as 
 first OCi and then OB-^ is indefinitely increased. 
 
 That is, 
 
 limit / limit 
 
 6 = 00 \C =^ 00 
 
 [j;a.*.])=o. 
 
 lt[;Lt(X>H]=» 
 
 86. A sufficient condition that 
 limit 
 b-- 
 
 can be easily obtained by the aid of the Maximum-Minimum 
 Theorem (Art. 84). 
 Let/(a;) be single-valued and continuous. 
 
 We can write /(x) in the form ^. 
 
 '-,k>l; then by (3) 
 
 Art. 84. 
 
 
 mo A 
 
 k—ib" 
 
 ->b<i, 
 
 and 
 
 |^Hj™^('j>(a:)<Zx)] = if ^y(i) does 
 
 not increase indefinitely as i increases indefinitely. 
 
90 INTEGRAL CAIjCULUS. [Art. 86. 
 
 If, then, [a;y(x)]^^„ is not infinite, k > 1, 
 
 J'»oo 
 f(x)dx is finite and determinate. 
 a 
 
 (a) As an example of the use of this test we will prove 
 
 e~^'dx finite and determinate. 
 e^''' is single-valued, finite, and continuous for all values of a. 
 
 x''e~'" , /c > 1, is easily found and profves to be zero. 
 
 6~'''dx is finite and determinate. 
 
 
 
 (b) Let us consider i dx. 
 
 Ja X 
 
 sm ax . , ^ , „ T • n • 
 
 IS equal to a when a; = 0, and is finite, continuous, 
 
 and single-valued for all values of x. 
 Let a be a given constant ; then 
 
 J^" sin ax , r° sin ax , , T" sin ax , 
 rfa!= I dx-¥ I dx, 
 ax Jd X Ja X 
 
 and I dx is finite and determinate. 
 
 «/o X 
 
 By integration by parts. 
 
 /sin ax , cos ax 1 /• cos ax , 
 rfa; — I r-acB, 
 a; ax aJ x^ 
 
 X" sin aa; , cos aa 1 T" cos ax , 
 c(a; = I — ^;—-dx, 
 X aaaJo- xf 
 
 J /*" cos aa; , . - . 
 
 and I — r— rfa; is finite and determinate since 
 
 »/a X-' 
 
 limit ra;*cosaa;"l limit rcos aa;~| „.^, ^, ^„ 
 a; = oo[_ x"" J a; = oo|_ .-k^"*- J 
 
 XOO 
 COS (of) dx is finite and determinate, for cos (a;') is 
 
 finite, continuous, and single-valued for all values of x, hence, 
 
Chap. VUI.] DEFINITE INTEGEALS. 91 
 
 J COS (a;^ dx is finite and determinate ; and 
 
 
 J, ^ , r''2xcos(x^)dx (sina=) , , f" sin (a;^ (Za; 
 a ^ ' Jo. IX ia. Jo. X 
 
 and I ^^-tt — is finite and determinate i 
 
 Ja. x" 
 
 limit rx^smC^n^ l<>fc<2. 
 a; =^ 00 L a! J 
 
 I since 
 
 Examples. 
 
 sinaa; 
 
 (1) Construct the curves y = e-^ ; y = ^^; y = cos{x'). 
 
 (2) Prove that the following integrals are finite and deter- 
 
 dx. 
 
 minate : 
 
 > 
 
 r''e-'^^cos6a;.da;, f e-'^a;'".da;, £ e'^'^.dx, 
 
 (3) Show that f afe-".da; is finite and determinate for all 
 values of n greater than — 1 . 
 
 87. When we have occasion to use a reduction formula in 
 finding the value of a definite integral, it is often worth while 
 to substitute the limits of integration in the general formula 
 before attempting to apply it to the particular problem. 
 
 We can reduce the exponent of x by [4], Art. 64, 
 
92 INTEGRAL CALCULUS. [Art. 87. 
 
 For our example this becomes 
 
 J ^ ' -m+\ -m+lJ ^ ' 
 
 When a; = 0, and also when w = a, ^ '- = 0. 
 
 — m + 1 
 
 Hence 
 
 rar-^(a' - afr^dx = °^'^'"^ ~ ^^ C^-\aF - x'y^dx ; 
 Jo m — 1 Jo 
 
 rV(«=' - x'y^dx =-. a" f'ai'ia' - a?y"^das 
 Jo 6 Jo 
 
 a* C'x'ia'-x'y^dx 
 
 5 3 
 
 '6 ' 4 
 
 6 4 2 Jo ■sla^ — a? 
 
 J'"_^dx__ 
 » Va2-a;2~2 4 6 2 
 
 Therefore '"" ^"'^ - -"^ ^ ^ ""' 
 
 Examples. 
 
 a^'da; 2 4 . 
 
 a". 
 
 2) fVa^-ar^-da; =— • 
 »/o 4 
 
 3) rVVH^^^^da!=i.^*. 
 Jo 4 4 
 
 4) rV(a2 - a^)« . da; = i . A . ^a6. 
 »/o 6 16 
 
 5) r'sin»a!.da! = l-3-5...(w-l) ^ ^^^^ ^ .^ ^^^^ 
 Jo 2.4. 6... n 2 
 
 = — — - — ^^ when n is odd. 
 
 3.5.7...71 
 
Chap. Vni.] DEFINITE INTEGRALS. 93 
 
 (6) Show that I cos"a!.da;= | ^8in"a;.da!. 
 
 ,_, r^ a?''dx ^ 1.3.5. ..(211-1) ^ 
 ^ ' Jo VlT^ 2.4.6. ..2m ' 2' 
 
 Suggestion : let a; = sin A 
 (8\ r ^'"^^^^ ^ 2.4.6 ...2^ 
 J« Vn^ 3.5.7. ..(2»i + l)* 
 
 (9) From Exs. 7 and 8 obtain Wallis's formula 
 
 B-^ 2.2.4.4.6.6.8.8... 
 2 1.3.3.5.5.7.7.9...' 
 
 Suggestion: I > I > I , 
 
 Jo ^/1_-K2 Jo Vl— a^ J» VI^ 
 
 88. When in finding | /a;.da; the method of integration by 
 
 substitution is used, and y=Fx is introduced in place of x, we 
 can regard the new integral as a definite integral, the limits of 
 integration being Fa and Fb, and thus avoid the labor of re- 
 placing y by its value in terms of x in the result of the indefinite 
 integration. 
 
 Let us find f e^Vl — e^ . dx. 
 
 Substitute y=e'". 
 
 dy = ae'^dx. 
 
 Hence C e^Vl - e^ • da; = ^ fVl-?/" . dt;. 
 
 When a; = — 00, 2/ = 0, and when a; = 0, 2/=l' 
 
 Therefore fVvi - e^ .dx = )-C Vl-^' .dy = ~- 
 J~w "■J a ia 
 
94 INTEGRAL CALCULUS. [Art. 88. 
 
 There is one class of cases where special care is needed in 
 using the method just described. It is when y has a maximum 
 or a minimum value between x = a and x=^b, say for x = c, 
 and X is consequently a multiple- valued function of y. 
 
 For suppose y a maximum when a; = c, then as x increases 
 from a to b, y increases to the value Fc, and then decreases 
 to the value Fb, instead of simply increasing or decreasing 
 from Fa to Fb. If ^{y)dy is the result of substituting 
 y for X in fx.dx, <f>y is a multiple-valued function of y, 
 and it will always happen that when y passes through a 
 maximum, we pass from one set of values of x to another, 
 and therefore from one set of values of <^y to another, and 
 in that case it is necessary to express our required integral 
 
 ^y.dy + I 4>y.dy, taking pains to select the correct set of 
 
 Fa. ^ Fc 
 
 values for <j>y in each integral. 
 
 If 2/ is a minimum between x = a and x = b, essentially the 
 same reasoning holds good. 
 
 A couple of examples will make this clearer. 
 
 '2° x.dx 
 
 - 
 
 V2 ax — a? 
 
 Let y=2ax — a?. Then -^ = 2 (a — «) = when x = a. 
 dx 
 
 — f = — 2, and v is a maximum when x=a. 
 da? 
 
 x=a± Va^ — y, 
 dx = q: ^ 
 
 2 Va^ — y 
 
 Since -^ is positive from a; = to x = a, and negative from 
 dx 
 
 = a to x=2a, dx= — ^ and x = a— Va^ — y from 
 
 2 Va^ - 2/ 
 
 a; = to x = a, and dx = ^ , and x= a + '\/a' — y 
 
 2 Va^ — y 
 from x = a to a; = 2a. 
 
 X 
 
CHAr. VIII.] DEFINITE INTEGRALS. 95 
 
 Hence 
 
 J p'" xdx __ r- xdx r ^ xdx 
 
 ■\/2 ax — x' ^1 V2 ax - x^ •/» \l-2ax—a? 
 
 = 1 P g-V^^ , 1 fO g + V^^ 
 
 Va 2/ — 2/'' ^'^'^ ya^y — y^ 
 
 2»^» -^a^y — y^ 2 Jo ^a^y — f 
 
 = f "" "'^^ = Tra. (Ex. 7, Art. 84) 
 
 - ^(^2/ 
 
 c?3/ 
 
 (6) Cy^ 
 
 ./o(si 
 
 rfa; 
 
 sin a; + cos a; )^ 
 
 Let 2/ = sina; + cosa;. -^ = cosa; — sina; = when a; = -- 
 da; 4 
 
 — -| = — sina; — cosa; = — V2 when a; = -. Therefore y has 
 dar _ 4 
 
 a maximum value V2 when a; = -. 
 
 4 
 
 3/= sina; + cosa;= V2 . eos( - — a;], 
 
 a; = - — cos~'-^, c?a; = ± — ^ 
 
 4 V2 V2 - 2/^ 
 
 Since ^ = and ^ < when a; = - , it follows that ^ is 
 ax oar 4 da; 
 
 positive from a; = to a; = -, and negative from a; = - to a; = -■ 
 
 4 4 2 
 
 Hence we have 
 
 IT IT ir 
 
 r^ da; ^ r!___^5___ . r!___^_^ 
 
 Jo (sin a; + cos a;)^ Jo (sin a; + cos a;/ J„(sina; + cosa;)^ 
 
 4 
 
 Jr'v^ d2/ /"^ dy _ „ /^v^ dy 
 1 yN^-y'' J^-^yNi-y" J\ fslY^^ 
 
96 INTEGRAL CALCULUS. [Aet. 89. 
 
 Let -^■=sm6; 
 
 and 
 
 y\/2-y'' 
 
 dx 
 
 Jo (si 
 
 sin a; + cos as/ 
 
 Example. 
 dx 
 
 ;=1. 
 
 J-*" ax 
 T~- — ; V 
 (siiiK + cosa;)' 
 
 89. Differentiation of a definite integral. 
 
 We have seen in Art. 51 that a definite integral is a function 
 
 of the limits of integration, and not of the variable with respect 
 
 to which we integrate ; that is, that | fx.dx is a function of a 
 
 and 6, and not a function of x. Strictly speaking, 1 fx.dx is 
 
 a function of a and &, and of any constants that fx may con- 
 tain, where by constant we mean any quantity that is indepen- 
 dent of X. 
 
 If the limits a and b are variables, they are always indepen- 
 dent of the X with respect to which the integration is performed, 
 which must from tlie nature of the case disappear when the 
 definite integral is formed, as it always may be in theory, from 
 the indefinite integral ; and this assertion holds good even when 
 the same letter which is used for the variable with respect to 
 which the integration is performed appears explicitly in the 
 limits of integration. 
 
 Thus if we write | sina;.cZa;, the x in svax.dx and the x which 
 
 is the upper limit of integration do not represent the same 
 variable, and are entirely unconnected. Indeed, the former x 
 
Chap. VIII.] DEFINITE INTEGRALS. 97 
 
 may be replaced by any other letter without afEeeting the 
 value of the integral. Tor 
 
 dx 
 
 Jsinic. 
 
 
 = I siuz.d. 
 = 1 — cos X. 
 
 Let us now consider the possibility of differentiating a 
 definite integral. 
 
 Required Z>„ j f(x, a)dx, where a is independent of x, 
 
 and a and b do not depend upon a, and D^fix, a) is a finite 
 continuous function of a for all values of x between a 
 and b. 
 
 We have 
 
 limit 
 
 I f(x, a + Aa)dx — I f(x, a) dx 
 
 Aa 
 
 -Da ) f(x,a)dx-= 
 
 ^ limit r p /(a;, a + Aa)-/(a;, g) n 
 
 Aa = |_»/a Aa J 
 
 ^ p/ ^i™i* r /(x, a+Aa)-/(x, a) "| \ ^^ 
 Hence, Z)„ I f(x, a)dx= I \_Daf(x, a)] dx, [1] 
 
 ■/a t/a 
 
 and we find that we have merely to differentiate under the 
 sign of integration. 
 
98 INTEGRAL CALCULUS. [Art. 89. 
 
 If I)^f(x, a) becomes infinite for some value of x between 
 a and b, or if one of the limits of integration is infinite, the 
 proof just given ceases to be conclusive and [1] must not be 
 assumed to hold good. 
 
 The truth of the converse of the proposition formulated 
 in [1] can be easily established by differentiation, and we 
 have 
 
 I I /(^J a.)dx \da 
 =£\^ff(x,a)da^dx, [2] 
 
 or even 
 
 j I /(*) <^) ^"^ ^« 
 
 =X'[J^ /(a;, a) c^] dx, [3] 
 
 if a, b, 0, and d are entirely independent. 
 
 [2] and [3] are of course subject to limitations easily in- 
 ferred from the limitations on [1], stated above. 
 
 If, however, in [3] b is infinite, it can be shown by the aid 
 of the Maximum-Minimum Theorem that a sufficient condition 
 that [3] should hold good is that it shall be possible to find a 
 value of X such that for that value and for all greater values 
 x''f(x, a) shall be less than some fixed value for all values of 
 a between c and d, k being greater than 1. 
 
 If d and b are both infinite there is also the corresponding 
 condition involving x'f(x, a). 
 
 We are now able to state a sufficient condition that [1] 
 shall hold when b is infinite. It is that it shall be possible 
 to find a value of x such that for that value and for all greater 
 values a!*i>a/(x, a) shall be less than some fixed value. 
 
 Suppose now that we are dealing with variable limits of 
 integration. 
 
Chap. VIII.] BBFINITB INTEGRALS. 99 
 
 Let us find first — I fx.dx. 
 
 dzJa. 
 
 definition =/», it follows that =fi. 
 
 dx dz 
 
 Hence 
 
 dz 
 
 Let I fx.dx = Fx, then | fx.dx = Fz —Fa; and since by 
 
 """" dFz 
 
 fx, it follows that =fz. 
 
 dz 
 
 ^fyx.dx = ''(^^-/-) =fz. [4] 
 
 In the same way it may be shown that 
 
 ^Cfx.dx = -fz. [5] 
 
 Let us now take the most complicated case, namely, to find 
 — I /(a;, q.) da;, where a and h are functions of u. 
 
 Let j f{x, a) dx = F{x, a) ; 
 
 then u = C f(x, a) dx = F{b, a) — F{a, a), 
 
 du dF(b,a) dF(a,a) 
 
 and -;- = ; ^ > 
 
 da da da 
 
 but as 6 and a are functions of a, 
 
 'I^V^ = D,F{b,a)f + D.Fib,a), 
 da da 
 
 ,i,d dF^ ^ ^^p^^^ ^) da ^ j^^-p,^^^ ^)^ 
 
 Oa (la. 
 
 by I. Art. 200. 
 
 D,F{b,a)=f{b,a), 
 
 D,F(a,a)=f{a,a). 
 
 da 
 
 1 Cf{x, a) dx = fwix, a)) dx +f{b, a) f -f{a, a) ^- [6] 
 daJf Jo Oa. Oa 
 
 Hence ^ = D. [F{b,a)-F{a, a)]+/(&, a)§^ -/(a, a) ^, 
 da aa aa 
 
 or 
 
 db J., s da 
 
100 INTEGRAL CALCULUS. [Art. 91. 
 
 Examples. 
 
 (1) — flin (x + y)dx=(x+l) sin {xy + y) — Biny. 
 
 dyJd 
 
 (2) — faPdx = -. 
 ^ ' dxJo 3 
 
 (3) ^ J( Vl -COS 4,. d(l> = e* Vl - cose'^. 
 
 90. When the indefinite integral cannot be found, the prob- 
 lem of obtaining the value of the definite integral usually be- 
 comes a more or less difficult mathematical puzzle, which can 
 be solved, if solved at all, only by the exercise of great inge- 
 nuity. Some of the results arrived at, however, are so impor- 
 tant, and some of the devices employed so interesting, that we 
 shall present them briefly here. But we must repeat the warning 
 that most of the methods are valid only in case the definite 
 integral is finite and determinate ; and erroneous results have 
 more than once been obtained and published when a little atten- 
 tion to the precautions described in Articles 83-86 would have 
 prevented the mistake. 
 
 91. Integration by development in series. 
 ^°^-.dx. (v. Art. 84, Ex. 1.) 
 
 X 
 
 ^ =:(l—x)-^=:l+x + a? + a?+—, \{x<l. 
 
 *»' i'S 
 
 1 —X 
 
 J I — 5_.da;= j (loga-f a;loga;-t-a^loga;-| )dx. 
 1 — X i/O 
 
 Cxnogx.dx = - ] . (v. Art. 55 (a).) 
 
 Therefore 
 
 Jo l-o; yv^i' S^ 4" J 6 
 
 (v. Todhunter's Trigonometry, Chap.XXIlI., Ex. 1.) 
 
Chap. VIII. j DEFINITE INTEGEALS. 101 
 
 W JT log ^^^^ da;. (v. Art. 86, Ex. 2.) 
 
 -Si ^-4ar / g-Zi g-Si g-<i 
 
 " " / -X 
 
 Hence 
 
 2 3 4 \^ 2 3 4 
 
 (I. Art. 130.) 
 
 e-^ , e-''" 
 
 5 +-+-]c^a' 
 
 But i+i+i + i^+...=|. 
 
 (v. Todhunter's Trig., Chap. XXIII., Ex. 1.) 
 
 Therefore flog (^^l±i\ da; = ^. 
 
 Examples. 
 ^ ' Jo \+x 12 
 
 (4) r , '^'^ ^-j^+mic^+(hi\\^ 
 
102 INTEGRAL CALCULUS. [Art. 92 
 
 -mi-] '-<"■ 
 
 92. Integration by ingenious devices. 
 
 IT 
 
 (a) j logsinx.da;. (v. Art. 84, Ex. 6.) 
 
 IT 
 
 Let M = j log sinx.dx. 
 
 Jo ° 
 
 Substitute y = - — x. 
 
 IT 
 
 XO /»2 
 
 logcos?/.d?/= I logcosa.da;. 
 
 2 
 
 2m= I (logsina; + logcosa;)da;= j log (sin a cos a;) 
 
 IT 
 
 = — - log (2) + i log sin 2 x,dx 
 = — -log(2) + - I logsina;.dx. 
 
 j log sin aj.dx = j ^log sin x.dx+ C log sin x.dx 
 
 2 
 
 = u+ I log sinaj.cZa;. 
 
 5 
 
 Substitute y = 7r — x. 
 
 dx 
 
Chap. VIII.] DEFINITE INTKGRALS. 103 
 
 and 
 
 IT 
 
 log sin a;. da; = — I logsin2/.d?/= I \o%%va.x.dx = u. 
 
 Hence 2m = — |log(2) + M, 
 
 and 
 
 J '•2 /*2 -_ 
 
 log sin a;, da; = | log cos a;, da; = — - log (2) . [1] 
 
 »/o 2 
 
 (6) pe-'^da;. (v. Art. 86 (a).) 
 
 Let M= I e "^'dx, and let x=ay; 
 
 ae-»''''d2/= j ae-'''«'da;, 
 
 ./o 
 
 Me-«^= r"ae-'i+'*>»^da;, 
 
 But 
 
 1 1 
 
 I 
 
 ae-f'+'^'^'da: 
 
 2 1+a!^ 
 
 2 1 r dx _ir 
 Hence r* = - j^ rT^-4' 
 
 and rV^da; = iV^. [2] 
 
 Jo 2 
 
 (c) r°° 515:^ . da;. (v. Art. 86 (6) .) 
 
 have - = r V"da if a; > 0. (Art. 82, Ex. 6.) 
 
 X Jo 
 
 We 
 
104 
 Hence 
 
 INTBGEAL CALCULUS. 
 
 [Art. 92 
 
 I dx= I I smmx \ e~"da aa; 
 
 Jo X Jo \ Jo J 
 
 ~ ) (1 6~" sin ma;, da J da; 
 
 Therefore 
 
 f"° sin mx 
 
 = j (j e-'-^^sinmcc.dxjda, by[3],Art.89 
 Jo a' 
 
 mda 
 
 .dx= - if m>0 
 a; 2 
 
 (Art. 82, Ex. 7.) 
 
 [3] 
 
 by Art. 82, Ex. 5. 
 
 = — - if m < 
 
 = if ??i = 
 
 Examples. 
 a; log sin a;, da; =— — log(2). 
 
 Suggestion : let a; = tan ft 
 
 3) r e-^-^da 
 
 4) f'^^ 
 
 ' a; 
 
 sin a; cos ma; 
 
 2a 
 
 da; =0 if m < — 1 or to > 1 
 
 = - if m = — 1 or w = 1 
 
 4 
 
 = 1 if -l<m<l. 
 
Chap. VIII.] DEFINITE INTEGRALS. 105 
 
 J"" sin^x It 
 dx = — Suggestion : integrate by parts. 
 ar 2 
 
 93. Differentiation or integration with respect to a quantity 
 which is independent of x. (v. Art. 89.) 
 
 (a) We have Ce"^dx = -- (Art. 82, Ex. 6.) 
 
 Differentiate both members with respect to a, 
 
 J-* 1 /^" 1 
 
 ( — xe-"dx) = or I xe-'^dx^—- 
 ^ a^ Jo ae 
 
 Differentiate again, 
 
 J"" 2 ' 
 
 a? 
 
 Differentiating n times, 
 
 rVe-'"da; = ^. [1] (v. Art. 86, Ex. 3.) 
 
 (6) We have T'e""- dx = \'^. (Art. 92, Ex. 3.) 
 
 ^ ' Jo 2 a!! 
 
 Differentiating n times with respect to a, 
 
 (v. Art. 86, Ex. 2.) 
 (c) We have f e-""da! = -. (Art. 82, Ex. 6.) 
 
 Multiply by dc, and integrate from a to 6, 
 
 r(Ce-d.\dx=r^i. 
 
 Jo \Ja J Ja C 
 
 Hence f '^l^Zl^dx =logl [3] 
 
 Jo X a 
 
106 INTEGRAL CALCULUS. [Akt. 94. 
 
 •Jo a -f- i 
 
 Multiply by da, and integrate from & to o, 
 
 rYrvdaV-= fA- 
 
 Jo \Ji I «/i a + 1 
 
 Hence C^d. ^logffJI} W 
 
 Examples. 
 
 (1) From C-^ = ^ J- obtain 
 
 ^ '' Jo a^ + a 2 Va 
 
 /"" dx ^ T 1.3.5. ..(271-1) _ 1 
 Jo (a? + a)"+i 2 2.4.6... 2?i ' y/^^ 
 
 afdx = obtain 
 
 n + 1 
 
 Jo ^ ^ ' ^ -^ (n + l)^» 
 
 (3) From | e~"coswia;.da; = --— ^^ — - obtain 
 Jo a^ + m^ 
 
 I cos ma;. da; = + log — ■ ; )• 
 
 Jo a; ^ ^V + mV 
 
 J"" m 
 e~" sin iii.x.dx = — obtain 
 a^ + ni' 
 
 sin »ia;.da; = tan tan ' — ■ 
 
 a; mm 
 
 94. Tbe method illustrated in Art. 93 can be applied to 
 much more complicated forms. 
 
 (o) Ce'^-^.dx. (v. Art. 86, Ex. 2.) 
 
Chap. VIII.] DEFINITE INTEGBALS. 107 
 
 then *f = _2r^.e-^-|. 
 
 da Jo ar 
 
 Substitute » = -, 
 
 and ^ = -2 I e^^^d» = -2( e '^(?a! = -2w. 
 
 da i/o Jo 
 
 Hence — = — 2 da. 
 
 u 
 
 Integrating, 
 
 logM = — 2ffl + C, 
 
 and M=Cie""^". 
 
 When a = 0, u= C e-^dx^^-^. (Art. 92 (6) [2].) 
 
 Therefore ' Ci=iV^, 
 
 e-'=-^da;=i— ^. [1] 
 
 £ 
 
 (6) C e-'^'^ coshx.dx. (v. Art. 86, Ex. 2.) 
 
 Let M= I e~''''*cos6a;.daj, 
 
 then *f = - f xe-^^'sinftjB.da;. 
 
 cZ& Jo 
 
 Integrating by parts, 
 
 r a;e-°'"*'sin6a;.da; = — , f e-^'^ coBbx.dx= -—u. 
 Jo 2aVo 2a^ 
 
 Therefore ^ = -A„, 
 
 d6 2a^ 
 
 dw 6 ji 
 
 or — = — -— d&. 
 
 M 2a^ 
 
108 INTEGRAL CALCULUS. LAbT. 96. 
 
 Integrating, we have 
 
 or u=Cie *"='. 
 
 When 6 = 0, u= C e"''^ dx = ^. (Art. 92, Ex. 3.) 
 Jo 2 a 
 
 Hence u= C e-'^'cosbx.dx^^e'^. [2J 
 
 Examples. 
 ^ ' Jo X a 
 
 ^ '' Jo l+!«2 2 
 
 Suggestion: — ^ = 2 f oe-^P-^'da. 
 1 + ar c/o 
 
 95. Introduction of imaginary constants. 
 
 C° COS (a;2) dx. (v. Art. 86 (c).) 
 
 We have r°°e-°°"* do; = — \/^. (Art. 92, Ex. 3.) 
 
 Jo 2 a 
 
 Let a' = c" V^l" = c^ Tcos | + V^ sin |Y 
 
 Then a = cfcos^ + V^8in^^=^(l+ V^), 
 
 ^ (Art. 25.) 
 
 and — = ^ = _L_(1_ VITT). 
 
 2a cV2(l + V^^) 2cV2 
 
Chap. VIII.] DEFINITE INTEGRALS. 109 
 
 Hence X''"'"^'^'" = ^>i ' ^^ -^^^^ 
 
 But e-o^x'-v^x =cos(c»a^)- V^Tsin(c'ar'). 
 
 ([5] Art. 31.) 
 Therefore 
 
 r cos (c='a^)da;- V^ r"sin(c2a^) dx = — y^{l- V^). 
 
 and 
 
 f cos(c2ar')da;= r°°sm(c2a^)da; = — Ji. [1] 
 
 (Art. 17.) 
 Let c = 1 , 
 
 and f cos (a^) dx= C sin (ic^) dx = |-J-. [2] 
 
 If we substitute y = a^iu [2] , we get 
 
 Gamma Functions. 
 
 96. It was shown in Art. 84 that I [ log- ) dx is finite and 
 determinate for all values of n greater than —1, and infinite 
 when n is equal to or less than —1. The substitution of 
 
 y = log- reduces this integral to I y"e''dy, or, what is the 
 
 * ^» Jo 
 
 same thing, to | x"e~''dx; and in Art. 86, Ex. 3, the student 
 
 has been required to show that this integral is finite and deter- 
 minate for all valpes of n greater than — 1 . 
 
 (x"e-''dx = — x"e-'' + n(af'-^e-'dx, 
 
 by integration by parts. 
 
110 INTEGBAL CALCULUS. [Art. 96. 
 
 If n is greater than zero, 
 
 a;"e-"' = when a!=0, 
 
 and a;"e"* is indeterminate when x= oo.. Its true value when 
 »!= 00, obtained by the method of I. Art. 141, is, however, zero. 
 
 Therefore j x"e-'dx = nj x^'^e^'dx [1] 
 
 for all positive values of n. 
 
 If n is an integer, a repeated use of [1] gives 
 
 Jaf e-" dx = n I ( e'^dx; 
 a/0 
 
 but r e-'dx = 1, 
 
 and we have I x"e~'dx = n\ [2] 
 
 provided that n is a positive whole number. 
 
 If n is not a positive integer, but is greater than —1, 
 
 Jafe~'dx is a finite and determinate function of n, and its 
 
 
 value can be computed to any required degree of accuracy by 
 methods which we have not space to consider here. 
 
 I x"~^e~'dx is generally represented by T(n), and has been 
 very carefully studied under the name of the Gamma Function. 
 If n is a positive integer, we have from [2] 
 
 r(»i + l) = n!. [3] 
 
 From [3], r(2) =1. [4] 
 
 Since r(l) = | x''e-'dx= j e~''dx, 
 
 Ja Jo 
 
 r(i) =1. [5] 
 
 We have alwaj-s from [1] 
 
 V{n + \) = nV(n), [6] 
 
 if n is greater than zero. 
 
Chap. VIII.J 
 
 DEFINITE INTEGRALS. 
 
 Ill 
 
 Since j a;"e "dx is infinite when m is equal to or less than 
 — 1, it follows from the definition of T{n) that r(>i) = oo if 
 n is equal to or less jthan zero. It has, however, been found 
 convenient to adopt formula [6] as the definition of V{n) when 
 n is equal to or less than zero, and to restrict the original defi- 
 nition to positive values of n. The result easily deduced is 
 that r (m) is infinite when n is equal to zero or to a negative 
 fnteger, but is finite and determinate for all other values of n. 
 
 97. We may regard the formula 
 
 r(ra + l) = nr(n) 
 
 as a sort of reduction formula; and since each time we apply 
 it we can raise or lower the value of n by unity, we can obtain 
 any required Gamma Function by the aid of a table containing 
 the values of T (n) corresponding to the values of n between 
 any two arbitrarily- chosen consecutive whole numbers. 
 
 Such tables have been computed, and we give one here con- 
 taining the common logarithms of the values of T (n) from 
 m = l to w = 2. The table is carried out to four decimal 
 places, and each logarithm is printed with 'the characteristic 9, 
 which, of course, is ten units too large, the true characteristic 
 being —1. 
 
 10 + logr(«). 
 
 n 
 
 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1.0 
 
 
 9975 
 
 9951 
 
 9928 
 
 9905 
 
 9883 
 
 9862 
 
 9841 
 
 9821 
 
 9802 
 
 1.1 
 
 9.9783 
 
 9765 
 
 9748 
 
 9731 
 
 9715 
 
 9699 
 
 9684 
 
 9669 
 
 9655 
 
 9642 
 
 1.2 
 
 9.9629 
 
 9617 
 
 9605 
 
 9594 
 
 9583 
 
 9573 
 
 9564 
 
 9554 
 
 9546 
 
 9538 
 
 1.3 
 
 9.9530 
 
 9523 
 
 9516 
 
 9510 
 
 9505 
 
 9500 
 
 9495 
 
 9491 
 
 9487 
 
 9483 
 
 1.4 
 
 9.9481 
 
 9478 
 
 9476 
 
 9475 
 
 9473 
 
 9473 
 
 9472 
 
 9473 
 
 9473 
 
 9474 
 
 1.6 
 
 9.9475 
 
 9477 
 
 9479 
 
 9482 
 
 9485 
 
 9488 
 
 9492 
 
 9496 
 
 9501 
 
 9506 
 
 1.6 
 
 9.9511 
 
 9517 
 
 9523 
 
 9529 
 
 9536 
 
 9543 
 
 9550 
 
 9558 
 
 9566 
 
 9575 
 
 1.7 
 
 9.9584 
 
 9593 
 
 9603 
 
 9613 
 
 9623 
 
 9633 
 
 9644 
 
 9656 
 
 9667 
 
 9679 
 
 1,8 
 
 9.9691 
 
 9704 
 
 9717 
 
 9730 
 
 9743 
 
 9757 
 
 9771 
 
 9786 
 
 9800 
 
 9815 
 
 1.9 
 
 9.9831 
 
 9846 
 
 9862 
 
 9878 
 
 9895 
 
 9912 
 
 9929 
 
 9946 
 
 9964 
 
 9982 
 
112 INTEGRAL CALCtTLtTS. [Aet. 97. 
 
 Such a table enables us to compute with Gamma Functions 
 as readily as with Trigonometric Functions, and consequently 
 the problem of obtaining the value of a definite integral is 
 practically solved if the integral in question can be expressed 
 in terms of Gamma Functions. 
 
 For example, let us consider 
 
 (a) r 3f'e~^'"dx. 
 
 Let y = ax; 
 
 then f a;"e-'"da; = -^ f y''e-'dy = —^ C afe-'dx. 
 
 Jo a"+Vo a»+Vo 
 
 Hence C x" e-" dx = ^ '^'^ '^ ^\ [1] 
 
 Jo a"+' •■ -■ 
 
 provided that a is positive and »>— 1. 
 
 (6) r afAoglYdo;. (v. Art. 84, Ex. 4.) 
 
 Let y = — logx. 
 
 then f af"(log- "da;= C y^e-^'^+^^'dy. 
 
 Jo \^ xj Jo 
 
 Hence, by [1], 
 
 if m > — 1 and »i > — 1 . 
 
 (c) r'e-'^da;. 
 
 Let 1/ = a;" ; 
 
 then f e-''dx = iC LL dy = ^k C" ar^e'' dx. 
 
 Jo Jo ^y^ Jo 
 
 Hence f e-'^ dx = ^T {^) . [3] 
 
Chap. VIII.] DEFlKlTE INTEGRALS. 113 
 
 98. C x'"-^{l-x)''-^dx = B{m,n) [1] 
 
 is an exceedingly important integral that can be expressed iu 
 terms of Gamma Functions ; it is known as the Beta Function, 
 or the First Eulerian Integral, V (n) being sometimes called the 
 Second Eulerian Integral. 
 
 In the Beta Function, m and n are positive, and B{m,n) is 
 always finite and determinate. (v. Art. 84, Ex. 5.) 
 
 In r a;"'-i (1 — x)"-^dx let y=l—x, 
 
 and we get 
 
 C x''-'{l —x)"~'dx= f y"-^{l —y)'^-^dy, 
 
 or B {m, n) = B {n, m) . [2] 
 
 In C ar-^n — xY-^dx let x = --^, 
 
 Jo ^ ' 1+y 
 
 and we get 
 
 Jo ^ ' Jo (1 + 2/)"*+" Jo (1 +»)"*+" 
 
 We have seen in [1] Art. 97 (a) that 
 
 r" , r(w + l) 
 
 I x^e-^dx == '■ Ji — • 
 
 Jo a"+' 
 
 Hence r(»i)=j a^aj'-'e-^da;, 
 
 r(m) a"-^ e— = I «"■+»-* af-^ e-''<*+'^> da;, 
 ~ r(m) C°°a"-^e-''da= C nr-'f C a''+"-^ e-"-^*'^ da\ dx, 
 nn.)Tin)=f;x'^-^LSB+^Jx. 
 
114 INTBGBAL CALCULUS. [AiitT99. 
 
 Therefore IMIM = f ^^JlIL-^.cI. ; [4] 
 
 or by [3], 
 
 B(m, n) = Car-' (1 - .r)— dx = ^ ^(w) j-^-j 
 
 ^ ' ' Jo ^ ^ r(m + n) '- -■ 
 
 If n=l— m, then since r(l)=l, 
 
 _^ da;=| -? — dx=T(m)T(l~m). [6] 
 
 (l-a;)" Jo 1 + a; ^ ^ ^ '' i" "^ 
 
 Formula [6] leads to an interesting confirmation of Art. 
 92(&). 
 
 Let m = ^, and we have from [6] 
 
 *■ ^^'-^ Jo (1+x) 
 
 Substitute y = ^Jx, 
 
 A u r" dx „ f " dy 
 
 and we have I ^ = 2 I — ^ = ir. 
 
 Jo (l+a;)^^ Jo 1+2/2 
 
 Hence T{i)=^^; [7] 
 
 and since by Art. 97 (c) 
 
 C e-'^dx =i^^. 
 
 99. By the aid of formulas [4], [5], and [7] of Art. 98 
 a number of important integrals can be obtained. 
 For example, let us consider 
 
 I sin"a!.c?a;, where n is greater than — 1. 
 
 Let y = sina;, 
 
 IT 
 
 and we have | sin"a;.da;= | y" (1 —y^)~idy. 
 
ClIAP. VIII.] DEFINITE INTEGKALS. 115 
 
 Let, now, 2 = ^» 
 
 and 
 
 =iX'.S'-a-.).-.*.iB(i±i,i). 
 
 But 
 
 = V5^ . V ^ ( by [7] Art. 98. 
 
 V2 
 Hence 
 
 CsiW^xAx^^ ) ^ ( . [1] 
 
 J. 2 rr^ + i'i 
 
 'g-^0 
 
 If n is a whole number, this will reduce to the result given 
 in Art. 87, Ex. 5. 
 
 Examples. 
 
 
 4 rf 
 
 .p±I)r(^) 
 
 (2) I sin"a!Cos'"a;.da! = — ^ — -. — '- — ^^^ — r 
 
116 INTEGKAL CALCULUS. CART. 99. 
 
 (3) f'-^- =^ 
 
 ^ ' Jo ^/1 _ 3-. W 
 
 r[i 
 
 '^^(^l) 
 
 \_ n 
 
Chap. IX.] LENGTHS OE CUBVE8. 117 
 
 CHAPTER IX. 
 
 LENGTHS OP CURVES. 
 
 100. If we use rectangular coordinates, we have seen (I. Art. 
 27) that 
 
 *— I' w 
 
 and (I. Arts. 52 and 181) that 
 
 els' = dx' + dy\ [2] 
 
 dy 
 PYom these we get sinr = -=-■> [3] 
 
 dx r,n 
 
 cosT = — -, [41 
 
 ds 
 
 by the aid of a little elementary Trigonometry. 
 
 These formulas are of great importance in dealing with all 
 properties of curves that concern in any way the lengths of arcs. 
 
 "We have already considered the use of [2] in the first volume 
 of the Calculus, and we have worked several examples by its 
 aid in rectification of curves. Before going on to more of the 
 same sort we shall find it worth while to obtain the equations of 
 two very interesting transcendental curves, the catenary and the 
 tractrix. 
 
 The Catenary. 
 
 101. The common catenary is the curve in which a uniform 
 heavy flexible string hangs when its ends are supported. 
 
 As the string is flexible, the only force exerted by one portion 
 of the string on an adjacent portion is a pull along the string, 
 which we shall call the tension of the string, and shall represent 
 by T. T of course has different values at different points of the 
 string, and is some function of the coordinates of the point in 
 question. 
 
118 
 
 INTEGRAL CALCULTJS. 
 
 [Art. 101. 
 
 The tension at any point has to support the weight of the por- 
 tion of the string below the point, and a certain amount of side 
 pull, due to the fact that the string would hang vertically were 
 it not that its ends are forcibly held apart. 
 
 Let the origin be taken at the lowest point of the curve, and 
 
 suppose the string fastened 
 at that point. 
 
 Let s be the arc OP, 
 P being any point of the 
 string. As the string is uni- 
 form, the weight of OP is 
 proportional to its length ; 
 we shall call this weight ms. 
 This weight acts verti- 
 cally downward, and must be balanced by the vertical effect of T, 
 which, by I. Art. 112, is Tsinr. 
 
 Hence Tsinr = ms. (1) 
 
 As there is no external horizontal force acting, the horizontal 
 effect of the tension at one end of any portion of the string must 
 be the same as the horizontal effect at the other end. In other 
 words, rcosr = c (2) 
 
 where c is a constant. Dividing (1) by (2) we get 
 
 s = — tan T, 
 m 
 
 or s = a tanr, (3) 
 
 where a is some constant. From this we want to get an equa- 
 tion in terms of x and y. 
 
 tanr = Vsec^T — 1 = -v — ; — 1 ; 
 
 = dx. Integrate both members. 
 
 hence 
 
 or 
 
 and 
 
 ads 
 
 {o? + - 
 
Chap. IX.] LENGTHS OF CURVES. 119 
 
 alog(s + VcF+?) = x+0; 
 when x=0, s = 0, 
 lience 0= a log a, 
 
 and log(s + V^+?) = ^ + loga, 
 
 s + V^+7 =ae«, 
 
 <i " - 
 s = -(e« — e~«) = atanT by (3). 
 
 Hence a^ = - (ea-e'S), 
 
 and y = ^(ea + e"») + C. 
 
 K we change our axes, taking the origin at a point a units 
 below the lowest point of the curve, y = a when a;=0, and 
 therefore C= 0, and we get, as the equation of the catenary. 
 
 Example. 
 
 Find the curve in which the cables of a suspension-bridge 
 must hang. Ans. A parabola. 
 
 The Tractrix. 
 
 102. If two particles are attached to a string, and rest on a 
 rough horizontal plane, and one, starting with the string stretched, 
 moves in a straight line at right angles with the initial position 
 of the string, dragging the other particle after it, the path of the 
 second particle is called the tractrix. 
 
 Take as the axis of X the path of the first particle, and as 
 the axis of T the initial position of the string, and let a be 
 
120 
 
 INTEGIIAL CALCULUS. 
 
 [Art. 102. 
 
 the length of the string. From the nature of the curve the 
 string is alwa3's a tangent, and we shall have for any point P 
 
 y 
 
 - = — sinr, 
 a 
 
 for T l3ing in the fourth quadrant has a negative sine. 
 
 Y 
 
 [1] 
 
 df 
 
 hence 
 
 and 
 
 ^= ■ 2 ^^ ._ 
 
 a^ ~ ds' da? + dy^ ' 
 
 f{dx' + df) = cedy\ 
 
 y^doi? = {a'-f)dy\ 
 
 {a'-yYidy 
 
 dx= ±- 
 
 y 
 
 is the differential equation of the tractrix. 
 
 On the right-hand half of the curve t is in the fourth quadrant, 
 
 -^ or tanr is negative, and we shall write the equation 
 dx 
 
 dx = — 
 
 {a'-yy^dy 
 
 y 
 
 [2] 
 
 If we allow the radical to be ambiguous in sign we shall get 
 also the curve that would be described if the first particle went 
 to the left instead of to the right. The tractrix curve, generallj- 
 considered, includes these two portions. 
 
 Integrating both members of [2] , and determining the arbi- 
 trary constant, we get 
 
 x = -^W~f + a\o^"L±:^AEI [3] 
 
 y 
 
 as the equation of the tractrix. 
 
chap. ix.] lengths of curves. 121 
 
 Examples. 
 
 (1) Show by Art. 102 (1) that in the tractrix s = alog_ 
 if s is measured from the starting-point. ^ 
 
 (2) Find the evolute of the tractrix. (I. Art. 93.) 
 
 Rectification of Curves. 
 
 103. In finding the length of an arc of a given curve we 
 can regard it as the limit of the sum of the differentials of the 
 arc, and express it by a definite integral. 
 
 We shall have 
 
 
 Of course in using this formula we must express \ldx^ + d^ 
 in terms of x only, or of y only, or of some single variable on 
 which X and y depend, before we can integi-ate. 
 
 For example ; let us find the length of an arc of the circle 
 
 a^ 4- y2 _ (j2_ 
 
 ix.dx + iy.dy^Q, 
 x.dx 
 
 dy = —- 
 
 y 
 
 ,^„, p-J^^qfsin-'B-sin-'^ 
 
 /*» dx tto 
 
 The length of a quadrant = a I , = — ; 
 
 ^ -'o Va^ - 3? 2 
 
 .•. the length of a circumference = 2ira. 
 
122 INTEGBAL CALCULUS. [Art. 104. 
 
 Length of Arc of Cycloid. 
 
 104. For tlie cycloid we have 
 
 x = a$ — a sin I 
 y = a —a cos ( 
 dx = a(l — cosd)dO = ydO, 
 
 6 = vers~^£, 
 
 (I. Art. 99.) 
 
 de = -- 
 
 a 
 1 dy dy 
 
 \ a 0/ 
 
 y y" ^'iay — f 
 
 dX: 
 
 ydy 
 
 V2 ay — y^ 
 
 ds' = d:^ + df=^^^^=^S!^, 
 2ay — y^ 2 a — y 
 
 ds=^/2a.- -^ . 
 
 V2 a — y 
 
 s = V27( P— ^= = 2VFa(V2^"^ - V2a-y0- 
 •^"o v2a — ?/ 
 
 K the arc is measured from the cusp, 2/o = 0, 
 
 s = 4a-2 V2aV2a— 2/i. [1] 
 
 If the arc is measured to the highest point, t/i = 2 a, 
 
 s = ia. 
 The whole arch = 8 a. 
 
 Example. 
 
 Taking the origin at the vertex, and taking the direction down- 
 ward as the positive direction for y, the equations become 
 
 x = u6 + 1 
 y= a- 
 
 Show that s = 2 V 2 ay when the arc is measured from the 
 
 •''^^"^l. (I. Art. 100.) 
 
 ■acosei ^ ' 
 
 '2 ay when 
 summit of the curve. 
 
Chap. IX.] 
 
 LENGTHS OF CURVES. 
 
 123 
 
 and 
 
 105. We can rectify the cycloid without eliminating $. 
 X = a6 — a sin 6 ) 
 y = a —a cos 6 S 
 dx = a{\ — cos6)(W, 
 dy = asmO.dd, 
 da? + df= 2a'de\l - cos^), 
 
 s = a-J2\ {l-cos6)id&, 
 
 J0„ 
 
 
 2sin='^ 
 
 ,^0 
 
 d0=4:a I sin-c?- = 4af cos- — cos 
 A 2 2 I 2 2 
 
 ^1 
 
 If ^0 = and ^1 = 2 TT. we get s = 8a as the whole curve. 
 
 106. Let us find the length of an arch of the eincycloid. 
 x = {a + h) cos (9 - 6 cos^-^^i-^^ 
 
 y— (a -\- h) sin ^ — 6 sin $ 
 
 b 
 
 ,(I.Art.l09[l].) 
 
 dx = 
 
 dy = 
 
 '■ = (a + bydff 
 
 {a + b) sin + (a + b) sin^-!-t-^6i 
 (a + b) cos^ - (o + 6) cos^l±i0 
 
 de, 
 de. 
 
 2- 2 /'cos ^^^61 cose + sin ^^(9 sin 61^ 
 = 2(0, + by-de^fl - COS-61Y 
 
 s = (a + 6)V2 r^/l-cos^^Vde, 
 
 ^^ 4b{n+b) 
 
 cos — Oo— cos - 
 2& 
 
 2& 
 
 To get a comijlete arch we must let ^o = a-nd 61 = — tt. 
 
 a 
 Hence, for a whole arch, 
 
 8 (ba + h) 
 
124 INTEGRAL CALCULUS. [Art. 107. 
 
 Examples. 
 
 (1) Find the length of an arch of a hypocycloid. 
 
 8 6 (a — 6) 
 
 Ans. s = . 
 
 a 
 
 (2) Find the length of an arch of the curve x^ + y^ = a', 
 and show that it agrees with the result of Ex. 1. 
 
 (v. I. Art. 109, Ex.2.) 
 107. Let us attempt to find the length of an arc of the ellipse 
 
 t + l = i. 
 
 We have — — + -2—2- = 0, 
 
 dy= — ^-dx, 
 ay 
 
 ., a'-W , 
 a- — - 
 
 a^y" a' — or a^ — ar 
 
 where e is the eccentricity of the ellipse. 
 
 The length of the elliptic quadrant is 
 
 -r[^ 
 
 dx. [2] 
 
 ■X' 
 
 These integrals cannot be obtained directly, but 
 \a^ — eV]^ 
 can be expanded by the Binomial Theorem, and the fractions 
 formed by dividing the terms of the result by \aj' — x^]' can be 
 integrated separately, and we shall have the required length 
 expressed by a series. 
 
 A more convenient way of dealing with the problem is to use 
 
 an auxiliary angle. Instead of -^ + ^ = 1 we can use the pair 
 
 of equations . , , 
 
 r .\' (I. A.rt. 150), 
 
 2/ = 6cos<^J ^ ' 
 
Chap. IX.] LENGTHS OF CURVES. 125 
 
 dx = a cos (ji.dcj), 
 
 dy = — b sin <j>.d<l>, 
 
 ds' = (a^cos^.^ + &2sin2<^)d0= = [»=> - (a^ - 6'')sm''<^]dc^» 
 
 = a?fl- ^LJI^sia'^^ dijy' = a\l - ^ sin' <l,)d<l>', 
 where e is the eccentricity of the ellipse. 
 
 s = aC \l-e'8in'<j>y^d4> [3] 
 
 = a f '[l-^e=sin2.^-i-ie*sin*</)-i.i-fe«sinV-"]''<^- 
 For the arc of a quadrant we have 
 s, = aC\l-^sm^ct>fd.j>. [4] 
 
 Example. 
 
 (1) Obtain s, as a series from [2], and also from [4], and 
 compare the results with Art. 91, Ex. 5. 
 
 Polar Formulae. 
 108. If we use polar coordinates we have 
 
 ds = Vdr^ + r'd,i>\ (I. Art. 207, Ex. 2.) 
 
 tanc = ^, (I. Art. 207.) 
 dr 
 
 From these we get, by Trigonometry, 
 
 ^^'^^ = 17' As 
 
 109. Let us find the equation of the curve which crosses all 
 
 its radii vectores at the same angle. Here 
 
 rd<i> adr -,, 
 
 tan i = a, a constant, -— ^ = a, = "9? 
 
 ' dr r 
 
126 INTEGRAL CALCULUS. [Abt. 110. 
 
 t+£ - i 
 
 alogr=<^+C, r = e' »=e''e«, 
 
 9- = 6e°. (1) 
 
 where 6 is some constant depending upon tlie position of the origin. 
 This curve is known as the Logarithmic or Equiangular Spiral. 
 
 110. To rectify the Logarithmic Spiral. We have, from 
 109(1), ' . 
 
 d<i> = a — , 
 r 
 
 rd<ji = adr, 
 ds^ = di^ + 'i^d<l>'' = (1 + a')dr' ; 
 
 s= f(l + a-')hdr= (1 +aOK»-i -r„). 
 
 Examples. 
 
 (1) Find the length of an arc of the parabola from its polar 
 
 equation 
 ^ _ m 
 
 1 + cos (^ 
 
 (2) Find the length of an arc of the Spiral of Archimedes 
 
 r = a^. 
 
 111. To rectify the Cardioide. We have 
 
 r = 2a(l-cos<^), (I. Art. 109, Ex. 1), 
 
 dr = 2 a sin tft.dffi, 
 
 d^ = 4:a^sm^(t>.d4>^ + ia'^{l - cos<^)2c?<^2 
 = 8a'd<l>\l-cos<t>), 
 
 s = 2 V2 . a rfl -cos</.)id^=8arcos|2-cos^'1 
 J(p, L 2 2 J 
 
 = 1 6 a for the whole perimeter. 
 
Chap. IX.] 
 
 LENGTHS OF CURVES. 
 
 127 
 
 Involutes. 
 
 112. If we can express the length of the arc of a given curve, 
 measured from a fixed point, in terms of the coordinates of its 
 variable extremitj^, we can find the equation of the involute of 
 the curve. 
 
 We have fpund the equations of the evolute of y =fx in the 
 form 
 
 X' = X — p COSv ' 
 
 y' = y — psmi 
 We have proved that tanv = tanr', 
 
 and that 
 
 
 SUIT 
 
 dp 
 
 ds' 
 dx' 
 
 (I. Art. 91). 
 
 (I. Art. 95), 
 (I. Art. 96) ; 
 
 COSt'=- 
 
 ds'i 
 
 (Art. 100). 
 
 Since tani' = tanT', v = t' or v=180° + t'. 
 As normal and radius of curvature have opposite directions, 
 we shall consider v = 180° + t'. 
 
 Then 
 Hence 
 
 sin V = — sin r ' and cos v = — cos t'. 
 
 , , dx' 
 
 x' = x + p-. 
 
 y' = y +.p 
 
 dy' 
 
 (1) 
 
 (2) 
 
 Since 
 
 ds' 
 dp = ds', 
 P = s' + l (3) 
 
 where I is an arbitrary constant. Since x and y are the coordinates 
 of any point of the involute, it is only necessary to eliminate x', 
 y', and p by combining equations (1) , (2), and (3) with the equa- 
 tion of the evolute. 
 
 As we are supposed to start with the equation of the evolute 
 and work towards the equation of the involute, it will be more 
 natural to accent the letters belonging to the latter curve instead 
 
128 INTEGRAL CALOTTLUS. t^M. 112. 
 
 of those going with the former; and our equations may be 
 
 written 
 
 x = a,' + p'^; y = y'+p<^; p'=s + l. (4) 
 
 ds as 
 
 Since p'=l when s=0, it follows that I is the free portion 
 of the string with which we start. (I. Art. 97.) By varying I 
 we may get different involutes of the same curve. 
 
 To test our method, let us find the involute of the curve 
 
 ■^ 27m ^ ' W 
 
 for which ? = tn. We must first find s. 
 
 2vdv = — (x — mYdx, 
 
 9?)i y 
 
 3m 
 
 1 /^' 1 
 s= I (2a; + m)^da; = (2a; + m)^— m. 
 
 p' = s + m = — ; (2x + m)J, 
 
 3V3m 
 
 , , 2a; + m 
 x = x -\ ■ — , 
 
 ^ , , _^ (2a; + m)(a;-m)^ 
 ^ 27m y 
 
 , a; — m 
 a;' = — : — , 
 
 , 4 , ,2 4a;'2 
 y' = - — (.T - m)2 = , 
 
 92/ 2/ 
 
 a; = 3a;'+m, 
 
 4a;'2 
 
 2/ = r- 
 
 2/ 
 
 Substituting in (5) the values of x and y just obtained, we have 
 y"= 2mx' 
 as the equations of the required involute. 
 
Chap. IX.] LENGTHS OF CURVES. 129 
 
 Example. 
 Find an involute of ay'^ = a?. 
 
 113. An involute of the cycloid is easily found. Take equa- 
 tions I. Art. 100(C). 
 
 X— ad + a sin 6 
 y = —a + aoos6 
 p' = s, 
 
 a 
 
 dx = a(l + cos6)d6 ='2acos-d0, 
 
 6 (9 
 dy = — aB\n6.d6 =— 2asin-cos-d6, 
 
 2 2 
 
 d^ = 'la'dS^il + cose) = 4 a^dO'-cos"-, 
 
 Let 
 
 -H 
 
 cos- do =4asin-5 
 
 G (9 
 
 a5 = a;' + 4asin— eos- = a;' + 2a sinfl, 
 
 2 2 
 
 3/ = 2/' — 4 a sin^- =y' — 2a{\ — cos6), 
 
 ^ a;' = a^ — a sin S ' 
 y' = a — a cos 6 . 
 
 a cycloid with its cusp at the summit of the given cycloid. 
 
 Example. 
 From the equations of a circle 
 
 x = a cos cfi 
 
 y = a sin 4> 
 
 obtain the equations of the involute of the circle. Let 1=0. 
 
 Ans. x'= a(cos<^ + sin 0) ) 
 y':= o(sin <^ — <^ cos 0) J 
 
130 LNTEGEAL CAiCTTLTJS. [Art. 114. 
 
 Intrinsic Equation of a Curve. 
 
 114. An equation connecting the length of the arc, measured 
 from a fixed point of any curve to a variable point, with the 
 angle between the tangent at the fixed point and the tangent at 
 the variable point, is the intrinsic equation of the curve. If the 
 fixed point is the origin and the fixed tangent the axis of X, the 
 variables in the intrinsic equation are s and t. 
 
 "We have already such an equation for the catenary 
 
 s = a tan T, Art. 101(3), [1] 
 
 the origin being the lowest point of the curve. 
 The intrinsic equation of a circle is obviously 
 
 s = ar, [2] 
 
 whatever origin we maj^ take. 
 
 The intrinsic equation of the tractrix is easU}- obtained. We 
 
 have 
 
 2/ = — asiuT, Art. 102 (1), 
 
 and s = a log" ; Art. 102, Ex. 1. 
 
 y 
 
 hence s = a log ( — esc t) 
 
 where t is measured from the axis of X, and s is measOTed from 
 
 the point where the curve crosses the axis of Y. As the curve is 
 
 tangent to the axis of Y, we must replace t by r — 90°, and we 
 
 get 
 
 s = alogseCT [3] 
 
 as the intrinsic equation of the tractrix. 
 
 Example. 
 
 Show that the intrinsic equation of an inverted cycloid, when 
 the vertex is origin, is 
 
 s = 4asinT; (1) 
 
 when the cusp is origin, is 
 
 s = 4a(l— cost). (2) 
 
Chap. IX.] LENGTHS OF CUEVES. 131 
 
 115. To find the intrinsic equation of the epicycloid we can 
 use the results obtained in Art. 106. 
 
 dx=(a+b)(sm^!^0 - sme)de=2(a+b)cos^^±^ esin—e.ae, 
 \ J 26 26 
 
 dy=(a+b)(cose-cos^-t^e)de=2{a+b)sm^^tl^esm~e.de, 
 \ "J ib 2b 
 
 by the formulas of Trigonometrj' 
 
 sina — sinj8 = 2cos^(a + /3) sin|(a — /3), 
 
 cos/S— cosa = 2sin|(a + /3) sin^(a — /8) ; 
 
 . dy . a + 2b„ 
 
 tanr = -^ = tan — ■ 6, 
 
 dx 2b 
 
 , a + 2ba 
 
 hence t = — — - — d ; 
 
 2o 
 
 CL \^ £i J 
 
 therefore s = ^M«±l) A _ cos-^^r") [1] 
 
 a V a+2b J "- -■ 
 
 is the intrinsic equation of the epicj'cloid, with the cusp as origin. 
 If we take the origin at a vertex instead of at a cusp 
 
 a 
 
 ,r(a + 26) ^,. 
 2a ' 
 
 , , 4&(a + 6) . a , 
 
 hence s' = — i— i — '- sin — -t' ; 
 
 a a + 2b 
 
 ib(a + b) . a rm 
 
 or s = — ^ sin T [2] 
 
 a a + 26 ■ ' 
 
 is the intrinsic equation of an epicycloid referred to a vertex. 
 
132 INTEGRAL CALCULUS. [Art. 116. 
 
 Example. 
 Obtain the intrinsic equation of the hypocycloid in the forms 
 
 s = — 5^ i 1-cos —t], (1) 
 
 s= — i ^sin —T. (2) 
 
 a a— 2b 
 
 116. The intrinsic equation of the Logarithmic Spiral is found 
 without difHculty. 
 
 We have r=be', (Art. 109), 
 
 and s=Vl + a-(»-i — ro). (Art. 110). 
 
 If we measure the arc from the point where the spiral crosses 
 the initial line, ?-o = b, and we have 
 
 IB 
 
 s = 6Vl +a\e'-l). 
 
 In polar coordinates t = <^ + 1, and in this case e = tan~' a ; if 
 we measure our angle from the tangent at the beginning of the 
 arc we must subtract e from the value just given, and we have 
 
 s = 6(Vl+a2)(e«-l) ; 
 
 or, more briefly, s = A;(c'" — 1) , k and c being constants. 
 
 117. If we wish to get the intrinsic equation of a curve directly 
 from the equation in rectangular coordinates, the following method 
 will serve : 
 
 Let the axis of X be tangent to the curve at the point we take 
 as origin. 
 
 tanr = ^; (1) 
 
 da; 
 
 and as the equation of the curve enables us to express y in terms 
 of X, (1) will give us x in terms of r, saj- x = Ft ; 
 
Chap. IX.] 
 then 
 
 LENGTHS OF CURVES. 
 
 dx = F'tAt, 
 
 133 
 
 divide by ds ; 
 
 da; 
 
 ™_d- 
 
 ux j7,i ar , . ax 
 
 — = .c'T— , but — - = cost; 
 
 dx 
 
 hence 
 
 ds = secrF'T.dT. 
 
 (2) 
 
 Integrating both members we shall have the required intrinsic 
 equation. 
 
 For example, let us take x^= 2 my, which is tangent to the 
 axis of X at the origin. 
 
 2 xdx = 2 mdy, 
 
 dy . X 
 
 -^ = tanr = — , 
 dx m 
 
 dx = msec'r-dr, 
 
 dx 2 dr 
 
 — = cost= msec^T — , 
 ds ds 
 
 ds =m sec^T.dT, 
 
 (1) 
 
 ;= m j - 
 J c 
 
 dr m 
 
 inr , 1 , fir , t\ 
 —-— + logtani - + - 1 
 
 smr 
 
 COS^r 
 
 cos^T 2 
 s=0 when t = 0; .-. C=0; 
 sinr 
 
 + C, 
 
 TO 
 S = — 
 
 2 
 
 COS't 
 
 + log tan 
 
 (f-i)} 
 
 (2) 
 
 Examples. 
 
 (1) Devise a method when the curve is tangent to the axis 
 of y, and apply it to ?/^ = 2 mx. 
 
 Q 
 
 (2) Obtain the intrinsic equation ofy^ = {x — my. 
 
 (3) Obtain the intrinsic equation of the involute of a circle. 
 (Art. 113, Ex.) 
 
134 
 
 INTEGRAL CALCULUS. 
 
 [Art. 118. 
 
 118. The evolute or the involute of a curve is easily found 
 from its intrinsic equation. 
 
 If the curvature of the given curve decreases as we pass along 
 the curve, p increases, and 
 
 s' = p — p„. (I. Art. 96). 
 
 If the curvatiu'e increases, p decreases, and 
 
 s' = po — p. 
 
 Hence always s' = ±(p — po) ; [1] 
 
 ds 
 
 (I. Arts. 86 and 90). 
 
 We see from the figure that r' = t. 
 
 \_\dTjr=T' XdTjT^oJ 
 
 or, as we shall write it for brevity, 
 
 ds ' 
 
 s = ± 
 
 dr 
 
 [2] 
 
 119. The evolute of the tractrix s = a log sec t is 
 
 dlogsecTl'" , it. i. 
 
 s = a s =atanT, the catenary. 
 
 dr „ 
 
 The evolute of the circle s = ot is 
 
 s = a—- =0, a point. 
 
 dr ^ 
 
Chap. IX.] LENGTHS OF CTJKVES. 135 
 
 The e volute of the cycloid s = 4 a(l — cost) is 
 
 g^4at?(l-C0Sr) 
 
 -:4asinT, 
 dr 
 
 an equal cycloid, with its vertex at the origin. 
 
 Examples. 
 
 (1) Prove that the e volute of the logarithmic spiral is an 
 equal logarithmic spiral. 
 
 (2) Find the evolute of a parabola. 
 
 (3) Find the evolute of the catenary. 
 
 120. The evolute of an epicycloid is a similar epicycloid, with 
 each vertex at a cusp of the given cui-ve. 
 Take the equation 
 
 s 
 For the evolute, 
 
 = iM£±&)A_cos^^A Art.ll5[l]. 
 
 4&(a + 6) 
 
 dfl- 
 
 -cos- 
 a 
 
 a 
 + 2b 
 
 ') 
 
 a 
 
 
 dr 
 
 
 
 Ab(a + b) . a m 
 
 s = — ^^ — ■ — ^sin T. Ill 
 
 a + 2b a+2b '" -* 
 
 The form of [1] is that of an epicycloid referred to a vertex 
 as origin ; let us find a' and &', the radii of the fixed and rolling 
 cixcIgs 
 
 ^^ 4V(a' + &') ,i,^L^,, by Art. 115 [2]; 
 a' a'+2o' 
 
 hence, ma^^l^^H^, 
 
 a' a + 2 
 
 a' a 
 
 a'+2b' a + 2b 
 
136 INTEGRAL CALCULUS. [Art. ]21. 
 
 Solving these equations, we get 
 
 a + 26 
 
 h'=^ 
 
 a + 2h 
 
 a _a 
 
 and the radii of the fixed and rolling circles have the same ratio 
 in the evolute as in the original epicj'cloid ; therefore the two 
 curves are similar. 
 
 Example. 
 
 Show that the evolute of a hj'pocj'cloid is a similar hjTDO- 
 C5'cloid. 
 
 121. We have seen that in involute and evolute t has the same 
 value ; that is, t = r'. 
 
 If s' and t' refer to the evolute, and s and t to the involute^ we 
 
 have found that 
 
 , ds ' 
 s' = — 
 dT 
 
 ds 
 
 or s' = —- —I, I being a constant, 
 
 dr 
 
 the length of the radius of curvature at the origin. 
 
 (s' + l)dT' = ds, 
 
 s= C\s'+l)dT' 
 
 is the equation of the involute. 
 
 The involute of the catenary s=a tanT is, when Z = 0, 
 
 : a I tanr.dr = alogsecr, the tractrix. 
 
Chap. IX.] LENGTHS OF OUEVJES. 137 
 
 The involute of the cj'cloid s = 4a siiir when 1 = is 
 sinr.dT = 4 a (1 — cost), 
 
 an equal cj'cloid referred to its cusp as origin. 
 
 The involute of a cycloid referred to its cusp s =4a(l —cost) 
 when I = is 
 
 s = 4al (1 — cosT)dT = 4a(T — siuT), 
 
 a curve we have not studied. 
 
 The involute of a circle s = ar when Z = is 
 
 a I T.ciT = 
 
 Jo 2 
 
 122. While any given curve has but one e volute, it has an 
 infinite number of involutes, since the equation of the involute 
 
 s'=jr(a + 
 
 dr 
 
 contains an arbitrary constant I ; and the nature of the involute 
 will in general be different for different values of I. 
 
 If we form the involute of a given curve, taking a particular 
 value for /, and form the involute of this involute, taking the same 
 value of I, and so on indefinitely, the curves obtained will con- 
 tinually approach the logarithmic spiral. 
 
 Let ■ s=fT (1) 
 
 be the given curve. 
 
 s =fj(l +fr)dT = h +£^fr-dT 
 is the first involute ; 
 
 .J;Q + Ir +f;fr.ar)dr = Zt + ^-f +£f;fr.d^ 
 d involute ; 
 
 ,=z.+^+g+ +^-L;+rVT« (2) 
 
 2 3! n) Jo 
 
 s 
 is the second involute ; 
 
 is the ?ith involute. 
 
138 INTEGRAL CALCULUS. [Akt. 123. 
 
 By Maclaurin's Theorem, 
 
 fr =fo + rfo + ^f'o + f/'"0 + 
 
 But s = when t = ; hence /o = 0, and 
 
 /t = ^it +:^r^ + 43r^ + , 
 
 2! 3 ! 
 
 2 3! 4! 
 
 Jo -^ 3! 4! 5! ' 
 
 -^ (Ji + l)! (m + 2)!^(n + 3)! ' '^ -* 
 
 as n increases indefinitely all the terms of (3) approach zero 
 (I. Art. 133) , and the limiting form of (2) is 
 
 s = It -\ H 
 
 2! 3! 
 
 s = ?(e^- 1) by I. Art. 133 [2], 
 
 which is a logarithmic spiral. 
 
 123. The equation of a curve in rectangular coordinates is 
 readily obtained from the intrinsic equation. 
 
 Given 
 
 
 s=/r, 
 
 
 we know that 
 
 
 sinT = ^, 
 ds 
 
 and 
 
 
 dx. 
 cos T = — ; 
 ds 
 
 hence 
 
 dx 
 
 = cos rds = cos Tf'r.dT, 
 
 
 dy 
 
 = sin rds = sin Tf'r.dr, 
 
 
 X 
 
 = 1 cos t/V.^t 
 
 
 
 y 
 
 = 1 sinr/V.dT 
 
 
Chap. IX,] LENGTHS OF CURVES, 139 
 
 The elimination of t between these equations will give us the 
 equation of the curve in terms of x and y. Let us apply this 
 method to the catenary. 
 
 s = atauT, 
 
 r'" J 1 fl +sinT 
 
 •■ = a\ secT.aT = aloff-v — ■ , 
 
 Jo ^\l-sinT 
 
 f = a| secTtanT.dT = a(8ecT — 1), 
 
 2z 
 
 _ l + siuT 
 1 — sinr 
 
 i 
 
 
 
 sinr 
 
 2x 
 
 e« —1 
 
 2x 
 
 X 
 X 
 
 e 
 
 a 
 X- 
 
 
 e« +1 
 
 e«-\- 
 
 e 
 
 a 
 
 
 X 
 
 X 
 
 
 
 SCOT 
 
 = i(e' + e' 
 
 «), 
 
 
 
 the equation of the catenai-y referred to its lowest point as origin. 
 
 Curves in Space. 
 
 124. The length of the arc of a curve of double curvature is 
 the limit of the sum of the chords of smaller arcs into which the 
 given arc may be broken up, as the number of these smaller arcs 
 is indefinitely increased. Let (x,y, z), (x + dx, y + Ay, z + Az) 
 be the coordinates of the extremities of anj- one of the small arcs 
 in question; dx,Ay,Az are infinitesimal; Vda;^+A?/^+ Az^ is the 
 length of the chord of the arc. In dealing with the limit of the 
 sum of these chords, any one maj' be. replaced b}' a quantity dif- 
 fering from it by infinitesimals of higher order than the first. 
 Vdas^ + dy^+ dz" is such a value ; 
 
 hence s = i si da? -f d'f + dz^. 
 
140 
 
 INTEGRAL CALCULtTS. 
 
 [Art. 124. 
 
 Let us rectify the helix. 
 
 x = a cos 6 
 
 y=asme -. (1. Art. 214.) 
 
 z=kO 
 dx = — asinO. d6, 
 dy = acosO.dd, 
 dz = kdO, 
 ds' = (a' + ]<?)d6F, 
 
 Examples. 
 
 (1) Find the length of the curve /^y = —,z = —Y "'^V..^ 
 
 Ans. s = x + z-)-l. 
 
 (2) y = 2Vax — x,z = x — ^-^ — Ans. s = x + y — z + l 
 
CllAl'. X.] 
 
 AR£IA.S. 
 
 141 
 
 CHAPTER X. 
 
 AREAS. 
 
 125. We have found and used a formula for the area bounded 
 by a given curve, the axis of X, and a pair of ordinates. 
 
 = I ydx 
 
 "We can readily get this formula as a definite integral 
 area in the figure is the sum of the 
 slices into which it is divided by the 
 ordinates ; if Ax, the base of each 
 slice, is indefinitely decreased, the 
 slice is infinitesimal. The area of 
 any slice differs from yAx by less 
 than AyAx, which is of the second 
 order if Aa; is the principal infini- 
 tesimal. We have then 
 
 The 
 
 Pi 
 
 PyF\ 
 
 Va 
 
 X Ak 
 
 limit *~*i 
 ^ = AiB = 2 y^^ by I. Art. 161. 
 
 Hence 
 
 = j 'ydx 
 
 [1] 
 
 If the curve in question lies above the axis of X, and x^ is 
 less than x^, each ordinate is positive, each Ax is positive, each 
 term of the sum whose limit is required is positive, the sum is 
 positive, and the limit of the sum or the area sought is positive. 
 If, however, the curve lies below the axis of JX", and x<, is less 
 than Ki, each ordinate is negative, each Ax is positive, each 
 term of the sum is negative, the sum is negative, and the limit 
 
142 INTEGRAL CALCULUS. [Akt. 125. 
 
 of the sum or the area sought is negative. If, then, the curve 
 happens to cross the axis between a;o and x^, formula [1] gives 
 us the difference between the portion of the area above the axia 
 of X and the portion below the axis of X, but throws no light 
 upon the magnitudes of the separate portions. Consequently, 
 in any actual geometrical problem it is usually necessary to find 
 the portion of the required area above the axis of X and the 
 portion below the axis of X separately ; and for this purpose 
 it is essential to know at what points the curve crosses the axis, 
 Indeed, if the problem is in the least complicated, it is neces- 
 sary to begin by carefully tracing the given curve from its 
 equation, and then to keep its form and position in mind during 
 the whole process of solution. 
 
 Examples. 
 
 (1) Show that ( ^xdy is the area bounded by a curve, the 
 
 axis of Y, and perpendiculars let fall from the ends of the 
 bounding arc upon the axis of Y. 
 
 (2) If the axes are inclined at the angle <u, show that these 
 formulas become 
 
 A = sin <D I ydx = sin <o I xdy. 
 
 (3) Find the area bounded by the axis of X, the curve 
 x' -{- iy=0, and the ordinate of the point corresponding to the 
 abscissa 4. Ans. 5^. 
 
 (4) Find the area bounded by the axis of X, the curve 
 y = 3?, and the ordinates corresponding to the abscissae —2 
 and 2. Ans. 8. 
 
 (5) Find the area bounded by the axis of X, the axis of F, 
 the curve y = cos x, and the ordinate corresponding to the 
 abscissa Stt. Ans. 6. 
 
Chap. X.] 
 
 AEEAS. 
 
 143 
 
 126. In polar coordinates we can regard the area between two 
 radii vectores and the curve as the limit of the sum of sectors. 
 
 The area in question is the sum 
 of the smaller sectorial areas, any 
 one of which differs from i'r^A<ji by 
 less than the difference between the 
 two circular sectors ^{r + AryA<f> 
 and ^j^A^; that is, by less than 
 
 rArAfj) + i — ' '^ , which is of the 
 
 second order if A<^ is the principal infinitesimal. 
 
 Hence " A= ,^^J^^^.,]. 
 
 127. Let us find the area between the catenary, the axis of 
 X, the axis of Y, and any ordinate. 
 
 
 A= \ ydx= - 1 (e« + e ')dx. 
 
 
 
 „2 I X 
 
 ^=|(e»-e-i), 
 
 
 but 
 
 |(el_e-|) = s, 
 
 by Art. 101 
 
 Hence 
 
 A = as, 
 
 
 and the area in question is the length of the arc multiplied by the 
 distance of the lowest point of the curve from the origin. 
 
 128. Let us find the area between the tractrix and the axis 
 ofZ. 
 
 We have dx = -^ -^a^-f. (Art. 102.) 
 
 y 
 
 A = Cydx — — { dy^a^ — y' 
 
144 INTEGRAL CALCULUS. [ART. 129. 
 
 The area in question is 
 
 A = -jdy Va^- 2/^ = ^ , 
 which is the area of the quadrant of a circle with a as radius. 
 
 Example. 
 
 Give, by the aid of infinitesimals, a geometric proof of the 
 result just obtained for the tractrix. 
 
 129. In the last section we found the area between a curve 
 and its asymptote, and obtained a finite result. Of course this 
 means that, as our second bounding ordinate recedes from the 
 origin, the area in question, instead of increasing indefinitely, 
 approaches a finite limit, which is the area obtained. Whether 
 the area between a curve and its asymptote is finite or infinite 
 will depend upon the nature of the curve. 
 
 Let us find the area between an hyperbola and its asymptote. 
 
 The equation of the hyperbola referred to its asymptotes as 
 
 axes is 2 i 1,2 
 
 cr + V 
 xy = — ^ — 
 
 Let 0) be the angle between the asymptotes ; then 
 
 -A = sin (D I yax = ■ — sm w I — = 00 . 
 
 Jo 4 Jo X 
 
 Take the curve ifx = 4 a^(2 a — x), 
 
 2 , 3 2a — X 
 or y'= ia^ . — ; 
 
 X 
 
 anj- value of x will give two values of y equal with opposite 
 signs ; therefore the axis of x is an axis of symmetry of the 
 curve. 
 
 When x = 2a, y = 0; as x decreases, y increases ; and when 
 x = 0, y= cc . If a; is negative, or greater than 2 a, .?/ is imagi- 
 nary. The shape of the curve is something like that in the 
 
Chap. X.] AREAS. 145 
 
 figure, the axis of y being an asymptote. The area between the 
 curve and the asymptote is then either 
 
 A=i\ ydx or A=2 I xdy ; 
 
 bj' the first formula, 
 
 by the second, 
 
 Examples. 
 
 (1) Find the area between the curve f(v? + a^) = a?x' and its 
 asymptote y = a. Ans. A = 2a . 
 
 (2) Find the area between y^{2a—x) = x^ and its asymptote 
 x=:2a. ^™*- ^ = 3 ira^. 
 
 .,,,., •> fl5^(a + a!) -, 
 
 (3) Find the area bounded by the curve y' = _ — - and 
 
 its asymptote x = a. 
 
 = 2a'fl 
 
 Ans. A = 2a' 1 + 
 
 130. If the coordinates of the points of a curve are ex- 
 pressed in terms of an auxiliary variable, no new difficulty is 
 presented. 
 
 Take the case of the circle 3^ + y' = a\ which may be written 
 
 x = a cos <p 
 y = a sin <j) 
 dy = acoscl>d<j> 
 
 /•27r 
 
 The whole area J. = a" I cos" <l)dcj> = Tra". 
 
146 INTEGKAL CALCULUS. [Akt. 13L 
 
 Examples. 
 
 (1) The whole area of an ellipse , . ^ Ms irdb. 
 
 y = bsm<f) ) 
 
 (2) The area of au arch of the cycloid is Swa^. 
 
 (3) The area of an arch of the companion to the cj'cloid 
 X = aO, y = a(l — cos6) is 2ira^ 
 
 131. If we wish to find the area between two curves, or the 
 area bounded bj' a closed curve, the altitude of our elementary 
 rectangle is the difference between the two values of y, which 
 correspond to a single value of x. If the area between two 
 curves is required, we must find the abscissas of their points of 
 intersection, and the}' will be our limits of integration ; if the 
 whole area bounded by a closed curve is required, we must find 
 the values of x belonging to the points of contact of tangents 
 parallel to the axis of I*". 
 
 Let us find the whole area of the curve 
 
 a*y^ + h''x'^ = d'h''3?, 
 
 or a*y^ = Va?(a^— a?) . 
 
 The curve is symmetrical with reference to the axis of X, and 
 passes through the origin. It consists of two loops whose areas 
 must be found separately. Let us find where the tangents are 
 parallel to the axis of Y. 
 
 y——x V a^ — .'c^, 
 
 dy b a^ ~ "2,3? 
 
 -^ = —.—= = tanT. 
 
 dx a' Va^-a^ 
 
 T = - when tanr = oo, that is, when x = ± a. 
 
 A = i^^Jx-Ja'-x-'.dx + 2|jr^- ^a^-x'.dx=^a-b. 
 
Chap. X.] AKEAS. 147 
 
 Again ; find the whole area of {y — xy = a? — o?, 
 
 y = x± sla^ -— x^, 
 
 ^ =fiy'- y") <i^ = p V^?^^ . dx. 
 
 To find the limits o{ integration, we must see where t = -• 
 2 
 
 dx~ ,n ^ = °° ^"^^^ x=±a. 
 
 A = 2 C-Ja^-y? . dx = •n-a^. 
 
 Examples. 
 
 (1) Find the area of the loop of the curve y^— '^(« + ^) . 
 
 a — x 
 
 Ans. 2a?(\-'^. 
 
 (2) Find the area between the curves y'^ — iax=Q and 
 
 '^-*'^^=°- Ans.''-^. 
 
 3 
 
 (3) Find the whole area of the curve x^ + y^ = a*. Ans. f W. 
 
 (4) Find the area of a loop of o?f = a^{a?-a?). Ans. — • 
 
 5 
 
 (5) Find the whole area of the curve 
 
 22/2 (a^ + a?) -iay^a' -a?) + (a' - x'y = 0. 
 
 Ans. a'TrU-^^X 
 
 132. We have seen that in polar coordinates 
 
 A = ii 7^d<l>. 
 
 Let us try one or two examples. 
 
 (a) To find the whole area of a circle. 
 
 The polar equation is r = a. 
 
 A = il a'd<i> = irol^. 
 
148 INTEGRAL CALCULUS. [Art. 132, 
 
 {b) To And the area of the cardioide j- = 2 a(l — cos <^) . 
 A = i\ -ia^l — cos<l>yd<f> = 2aM (1 - 2cos</> + cos^ d<l>)d<l>, 
 
 (c) To find the area between an arch of' the epicycloid and the 
 circumference of the fixed circle. 
 
 a; = (a + 6) cos ^ — 6 cos ^^-3^ 6 
 
 
 y =z (^a + b)sin — b sin ^-j— 6 
 
 We can get the area bounded by two radii vectores and the 
 arch in question, and subtract the area of the corresponding 
 sector of the fixed circle. 
 
 Changing to polar coordinates, 
 
 x = r cos <^, 
 2/ = r sin <^. 
 We want ^ | r'dif). 
 
 y 
 
 tan(i = -» 
 
 ^ X 
 
 a? 
 but, since x = r cos <l>, sec ^ = - ; 
 
 X 
 
 hence T^^xdy-ydx^ 
 
 US' y? 
 
 and i^d<^ = xdy — ydx ; 
 
 dx = {a + b)(-sme +sm^^^e\de, 
 
 dy = {a+b) fcos 6 - cos 2-±-^ e\ dO. 
 
 xdy - ydx= (a + b) (a + 2b)fl - cos- e\de = r^d^. 
 
 Our limits of integration are obviously and ?^. 
 
 a 
 
Chap. X.] AKEAS. 149 
 
 Hence A = ^(a + b)(a+ 2 6) j <• / 1 - cos-6l ) d0, 
 
 A = ^{a + b){a + -2b), 
 
 is the area of the sector of the epicycloid. Subtract the area of 
 the circular sector rrcib, and we get 
 
 ._b^{3a + 2b) 
 a 
 as the area in question. 
 
 (d) To find the area of a loop of the curve r^ = a" cos 2 <fj. 
 
 For any value of <^ the values of r are equal with opposite 
 signs. Hence the origin is a centre. 
 
 When ^ = 0, r = ±a; as0 increases, r decreases in length 
 
 till ^ = -1 when r = ; as soon as <^ > -, r is imaginary. If ^ 
 
 decreases from 0, r decreases in length until <^= — -, when r = ; 
 
 and when <t>—jj '" is imaginarj'. To get the area of a loop, 
 
 then, we must integrate from <^= to<i = — 
 
 4 4 
 
 A=zi C^r'd<i> = ^a? Pcos 2 <^.d^ = ^■ 
 J_^ J_f 2 
 
 Examples. 
 
 (1) Find the area of a sector of the parabola r = — 
 
 1 + cos (^ 
 
 (2) Find the area of a loop of the curve r^cos ^ = ct^sin 3 <^. 
 
 Ans. log2. 
 
 4 2 ^ 
 
 (3) Find the whole area of the curve r = a (cos 2 ^ + sin 2 ^) . 
 
 Ans. ira'. 
 
 (4) Find the area of a loop of the curve r cos <j> = a cos 2 <^. 
 
 Ans. /'2-^')al 
 
 (5) Find the area between r =a(sec</)+tan<^) and its asymp- 
 tote r cos <^ = 2 a. ^^^^ (-^ ^ ^\ ^,, 
 
 (i + ^)«- 
 
150 
 
 INTEGRAL CALCULCS. 
 
 [Art. 133. 
 
 133. When the equation of a curve is given in rectangular 
 coordinates, we can often simplify the problem of finding its area 
 bj- transforming to polar coordinates. 
 
 For example, let us find the area of 
 
 {x^ + y-y =Aa-3? + ^ Wy^. 
 Transform to polar coordinates. 
 
 r" = 4 i^{a^ cos^ <^ + 6^ sin^ <^) , 
 r = \ (a^ cos- 4> + b^ sin^ ^) , 
 
 Examples. 
 
 (1) Find the area of a loop of the curve (if^ + y^)' = 4 a^a^y'. 
 
 Ans. 
 
 8 
 
 (2) Find the whole area of the curve _-|-i-=_[l--|--i-j. 
 
 (.1 t/ G V 't (J J 
 
 A-ixs. Z^(a2 + 62). 
 'lab 
 
 (3) Find the area of a loop of the curve ^ — 3 axy + S/-' = 0. 
 
 3a^ 
 2 ■ 
 
 Ans. 
 
 134. The area between a curve and its evolute can easilj' be 
 found from the intrinsic equation of the curve. 
 
 It is easily seen that the area 
 bounded by the radii of curvature 
 at two points infinite^ near, by 
 the curve and by the evolute, dif- 
 fers from ^p'dr by an infinitesimal 
 of higher order. The area bounded 
 by two given radii vectores, the 
 curve and the evolute, is then 
 
 
Chap. X.] 
 
 AEEAS. 
 
 151 
 
 P = 
 
 ds 
 
 Hence A = i^C Y— j dr. 
 
 For example, the area between a cycloid and its evolute is 
 
 Let 
 
 = 8 a- j cos^Tdr. 
 
 To = and Tj ^ - ; 
 
 :8 
 
 «!' 
 
 ^cos^TdT = 2-irc^. 
 
 Examples. 
 
 (1) Find the area between a circle and its evolute. 
 
 (2) Find the area between the circle and its involute. 
 
 HoMitch's Tlieorem. 
 
 135. If a line of fixed length move with its ends on any closed 
 carve which is always concave toward it, the area between the 
 
 curve and the locus of a given 
 point of the moving line is 
 eqnal to the area of an el- 
 lipse, of which the segments 
 into which the line is divided 
 by the given point are the 
 semi-axes. 
 
 Let the figure represent 
 the given curve, the locus 
 of P, and the envelope of the 
 moving line. 
 
 Let AP= a and PB = b, 
 and let CB = p. C being the 
 point of contnrt of the mo\ang line with its envelope. Let 
 AB = a + b = c. 
 
152 
 
 INTEGRAL CALCULUS. 
 
 [Art. 135. 
 
 The area between the first curve and the second is the area 
 between the first curve and the envelope, minus the area between 
 the second curve and the envelope. 
 
 Let 6 be the angle which 
 the moving line makes at 
 aay instant with some fixed 
 direction. Let the figure 
 represent two near positions 
 of the moving line ; A6, the 
 angle between these posi- 
 tions, being the principal in- 
 finitesimal. 
 
 PB = p, P'B' = p+Ap. 
 
 The area PBB'P'P differs 
 iVom ^p^dd by an infinitesi- 
 mal of higher order than the first. 
 
 ^p'de is the area of PBMP, and differs from PP'NB by less 
 than the rectangle on PM and PQ, which is of higher order than 
 the first, by I. Art. 153. But PPNB differs from PP'B'B by 
 less than the rectangle on BN and NB', which is of higher order 
 than the first, since NB', which is less than PP'+ Ap, is infini- 
 tesimal and A6 is infinitesimal. 
 
 The area between the first curve and the envelope is then 
 
 i I p'd6; or, since we can take PP'A'A just as well for our 
 elementary area, ^ I {c — pYdd. 
 
 Hence iJ/dd=^C{c-pyde\ 
 
 whence 2 c | pdd = 2c'7r, 
 
 X27r 
 pdd = ttC. 
 
 The area between the second curve and the envelope 
 
 ij^{p-byde. 
 
 (1) 
 
Chap. X.) 
 
 AREAS. 
 
 153 
 
 The area between the first curve and the second is then 
 
 = b Cpde -b^TT 
 
 de 
 
 by (1), 
 
 = Trbc — b^Tr 
 
 = 7rb{a + b)-b'ir, 
 
 A = irctb, (2) 
 
 which is the area of an ellipse of which a and & are semi-axes. 
 
 Q. E. D. 
 
 Examples. 
 
 (1) If a line of fixed length move with its extremities on two 
 lines at right angles with each other, the area of the locus of a 
 given point of the line is that of an ellipse on the segments of 
 the line as semi-axes. 
 
 (2) The result of (1) holds even when the fixed lines are not 
 perpendicular. 
 
 Areas by Double Integration, 
 
 136. If we take x and y as the coordinates of any point P 
 within our area, x and y will be independent variables, and 
 
 we can find the area bounded by two 
 given curves, y = fx and y = Fx, 
 by a double integration. Suppose 
 the area in question divided into 
 slices by lines drawn parallel to the 
 axis of Y, and these slices subdi- 
 A'ided into parallelogi'ams by lines 
 drawn parallel to the axis of X. 
 The area of any one of the small 
 parallelograms is AyA.x. If we 
 keep X constant, and take the sum 
 of these rectangles from y=fx to y = Fx, we shall get a result 
 differing from the area of the corresponding slice by less than 
 
154 
 
 INTEGEAL CALCTJLTJS. 
 
 [Art. 137, 
 
 2 AccAy, which is infinitesimal of the second order if Ax and Ay 
 are of the first order. 
 
 Hence I Ax.dy = Ax \ dy 
 
 Jfx Jfx 
 
 is the area of the slice in question. If now we take the limit of 
 the sum of all these slices, choosing our initial and final values 
 of X, so that we shall include the whole area, we shall get the 
 area required. 
 
 Hence ^= j Y \dy\dx. 
 
 In writing a double integral, the parentheses are usually omit- 
 ted for the sake of conciseness, and this formula -is given as 
 
 dydx, 
 
 the order in which the integrations are to be performed being the 
 same as if the parentheses were actuallj^ written. 
 
 If we begin bj- keeping y constant, and integrating with respect 
 to a;, we shaU get the area of a slice formed by lines parallel to 
 the axis of X, and we shall have to take the limit of the sum of 
 these slices varying y in such a way as to include the whole area 
 desired. In that case we should use the formula 
 
 I dxdy. 
 
 137. For example, let us find the area bounded by the para- 
 bolas 2/^ = 4ax and a^ = 4 ay. 
 
 The parabolas intersect at the origin and at the point (4 a, 4 a). 
 
 (^l 
 
 X4a /■^4ax /'ia /"^4ay 
 
 I dydx, or ^ = I I dxdy ; 
 
 ia ia 
 
 J "^4 ax ™2 
 dy = vTax ; 
 ^ ^ 4a 
 
 4a 
 
 i'')dy"dx=C (\/I^-~')dx = l^aK 
 Jo Jo^^ Jo I 4a i 3 
 
 The second formula gives the same result. 
 
Chap. X.J 
 
 AREAS. 
 
 155 
 
 Examples. 
 
 (1) Find the area of a rectangle by double integration ; of a 
 parallelogram ; of a triangle. 
 
 (2) Find the area between the parabola y^ = ax and the circle 
 ^'='"^-^- Ans. 2(^-?f} 
 
 (3) Find the whole area of the curve {y — mx — cy = a' — 3?. 
 
 Ans. TO?. 
 
 138. If we use polar coordinates we can still find our areas 
 by double integration. 
 
 Let r=/<^ and r = F<l> 
 be two curves. Divide the 
 area between them into 
 slices bj' drawing radii 
 vectores ; then subdivide 
 these slices bj' drawing 
 ares of circles, with the 
 origin as centre. 
 
 Let P, with coordinates 
 r and ^, be any point 
 within the space whose 
 area is sought. The curvilinear rectangle at P has the base rA<^ 
 and the altitude Ar ; its area diifers from rA<^Ar by an infinitesi- 
 mal of higher order than rA^Ar. 
 
 The area of anj' slice as aba'b' is I rA<l>dr, <^ and A^ being 
 ^F<p '^ff 
 
 constant, that is A<j) I rdr. The whole area, the limit of the 
 
 sum of such slices is Jl = I 1 rdrdtji. (1) 
 
 Or we may first sum our rectangles, keeping r unchanged, 
 and we get as the area of efe'f 
 
 ^F-^<t> ^r, ^F-'^'P 
 
 rAr j d<j>, and A= \ j rd^dr. (2) 
 
 JJ-^ 
 
 Jf-l,p 
 
156 
 
 INTEGRAL CALCULUS. 
 
 [Akt. 138. 
 
 It must be kept in mind that r in (1) and (2) is the radius 
 vector of any point within the area sought, and not of a point 
 on the boundary. 
 
 For example, the area between two concentric circles, r = a 
 and r = &, is 
 
 A= C Crd4>dr= f Crdrd4=7r(^a^ -W). 
 
 Again, let us find the area between two 
 tangent circles and a diameter through the 
 point of contact. 
 
 Let a and & be the tw6 radii, 
 
 r = 2acos<^ (1) 
 
 and r = 2 & cos <^ (2) 
 
 are the equations of the two circles. 
 
 A = 
 
 ( rdrd<l> = 2 (a^ - W) j cos^ ^d^ = ^ (a^ - V) . 
 
 «/2 6 COS i/O 2 
 
 If we wish to reverse the order of our integrations we must 
 break our area into two parts by an arc described from the origin 
 as a centre, and with 2 & as a radius ; then we have 
 
 a cos-i^ 
 
 I rd<^dr + | | rd<j)d7- 
 
 J r Jih Jo 
 
 cos-'26 
 nos"^ 
 
 rl cos"^r cos"'-— Idr + j r cos~'--dr 
 
 V 2a 2bJ J2b 2 a 
 
 = -(a'-m. 
 
 2^ ^ 
 
 Example. 
 
 Find the area between the axis of X and two coils of the 
 spiral r=atfi. 
 
Chap. XI.] 
 
 AKEAS OF STJEPACES. 
 
 157 
 
 CHAPTER XL 
 
 AEEAS OP STJRPACBS. 
 
 Surfaces of Revolution. 
 
 139. If a plane curve y=fx revolves about the axis of X, the 
 area of the surface generated is the limit of the sum of the areas 
 generated by the chords of the infinitesimal 
 arcs into which the whole arc may be broken 
 up. Each of these chords will generate the 
 surface of the frustum of a cone of revolution 
 if it revolves completely aronnd the axis ; 
 and the area of the surface of a frustum 
 of a cone of revolution is, by elementary 
 Geometry, one-half the sum of the circum- 
 ferences of the bases multiplied by the slant height. The frustum 
 generated by the chord in the figure will have an area differing 
 by infinitesimals of higher order from 7r{'y + y -\- Ay) As or from 
 2'7iyds. The area generated by any given arc is then 
 
 Ax 
 
 S: 
 
 yds 
 
 [1] 
 
 If the arc revolves through an angle 6 instead of making a 
 complete revolution, the surface generated is 
 
 s=e 
 
 
 [2] 
 
 It must be noted that [1] and [2] will give a positive value 
 for S if the generating curve lies wholly above the axis of X at 
 the start, and a negative value for S if it lies wholly below the 
 axis of X at the start. If the curve happens to cross the axis 
 of X between the points whose ordinates are y,, and y^, [1] and 
 [2] give not the area of the surface generated by the curve in 
 question, but the difference between the areas generated by the 
 
158 
 
 INTEGRAL OALCtJLUS. 
 
 [Akt. 140. 
 
 portion originally above the axis, and the portion originally 
 
 below the axis. 
 
 Example. 
 
 Show that if the arc revolves about the axis of 1 
 
 ', S=2ir I xds. 
 
 140. To find the area of a cylinder of revolution. 
 
 Take the axis of the cylinder as the axis of X Let a be the 
 altitude and 6 the radius of the base of the 
 cylinder. The equation of the revolving 
 
 line is 
 
 2/ = &; 
 
 d2/=0, 
 
 ds = Vda;^ + dy'' = dx ; 
 
 S = 2iri ydx—2Trab, 
 
 or the product of the altitude by the circumference of the base. 
 Again, let us find the surface of a zone. 
 The equation of the generating circle is 
 
 ie2 -|- 1/2 _ (j,2 . 
 
 , adx 
 ds= — ; 
 
 y 
 
 /S = 2 TT I adx = 2 air (a^ — x^) . 
 
 If Wo = — a and a^ = a, (S = 4 a^ir. 
 
 Hence the surface of a zone is the altitude of the zone multi- 
 plied by the circumference of a great circle, and the surface of 
 a sphere is equal to the areas of four great circles. 
 
 Again, take the surface generated by the revolution of a 
 cycloid about its base. 
 
 « = a^ — asin^^ 
 y = a — a cos 6 J ' 
 ds = add V2 (1 - cosO) , by Art. 1 05 ; 
 
 /•2ir 
 
 5==2,rj a'-^2.{\-cosefde = ^^7ra\ 
 
Chap. XI.] AREAS OF SURFACES. 159 
 
 Examples. 
 
 (1) The area of the surface generated by the revolution of 
 the ellipse ^^ a^V^ —^ 
 
 about the axis of X is 2iro6( Vl — e^ + ?^ 
 
 V « 
 
 about the axis of Y is 27ra^ fl + i-IZi! log ^-tl\ 
 
 q2 7j2 
 
 where e^ = 
 
 (2) Find the area of the surface generated by the revolution 
 of the catenary about the axis of X ; about the axis of Y. 
 
 (3) The whole surface generated by the revolution of the 
 tractrix about its asymptote is 4ira^. 
 
 (4) The area generated by the revolution of a cycloid about 
 its vertical axis is 8 7ra^(7r — f ) . 
 
 (5) The area generated by the revolution of a cycloid about 
 the tEingent at its vertex is S^iro'. 
 
 (6) The area generated by the revolution of the curve 
 x^ +yi= a^ about its axis is ^iro'. 
 
 141. If we know the area generated by the revolution of a 
 curve about an}- axis, we can get the area generated by the 
 revolution about anj' parallel axis hy an easy transformation of 
 coordinates. 
 
 Given the surface generated by the arc from Sf, to Sj about 
 
 OX, to find the area generated by 
 the same arc when it revolves 
 -X' about OX'. 
 
 Let S be the surface about OX, 
 and S' about O'X'. 
 "^ We have 
 
 l/o 
 
 S = 27r Cyds, (S'= 2 TT Cy'ds'. 
 
160 INTEGRAL CALCULUS. [^KT. 141, 
 
 B}' Anal. Geom. , x = x\ 
 
 y = yo + y'- 
 
 Hence rlx = dx\ dy = dy', ds = ds\ 
 
 and <S = 2 TT j (j/o + y')ds = 2 tti/oCsj — So) + 2:7 j y''ds, 
 
 Therefore S' = <S - 2 7r?/„(si -s„). [1] 
 
 Si — s„ is the length of the revolving curve ; 2 wyo is the cir- 
 cumference of a circle of which i/o is the radius. Hence the new 
 area is equal to the old area minus the area of a C3-linder whose 
 length is the length of the given arc and whose base is a circle 
 of which the distance between the two lines is radius. 
 
 In using this principle careful attention must be paid to the 
 sign of yo, and it must be noted that the original formula 
 
 ;S = 2 TT j yds will always give a negative value for the area of 
 
 the surface generated, if the revolving arc starts from below the 
 axis ; and hence, that the surface generated 
 by the revolution of any curve about an 
 axis of symmetrj- will come out zero. 
 
 As an example of the use of the princi- 
 ple, let us find the surface of a ring. 
 
 Let a be the distance of the centre of — - 
 the circle from the axis, and b the radius of 
 the circle. Since the area generated bj' the 
 revolution of the circle about a diameter is zero, the required 
 area is 
 
 27r6.2ira = AiT^ab. 
 
 Example. 
 
 Find the area of the ring generated b}' the revolution of a 
 cycloid about any axis parallel to its base. 
 
 Ans. S = Aab.L+'-^^^ 
 
Chap. XI.] AKEAS OF STTEFACES. 161 
 
 142. If we use polar coordinates, 
 S=2TrCyds 
 
 becomes »S = 2 x j j- sin <^.cte. 
 
 where ds = Vdr^~+l^d^. 
 
 For example ; let us find the area of the surface generated bj' 
 the revolution of the upper half of a cardioide about the hori- 
 zontal axis. 
 
 ?•= 2a(l— eos<^) ; 
 
 dr= 2 a sm(f).d<l>, 
 
 ds'=8a^(l — cos <^) d<ji', 
 
 <S' = 27r I 4 VSa^l- cos<^)^sin<^.d<^. 
 
 Examples. 
 
 (1) Find the surface of a sphere from the polar equation. 
 
 (2) Find the surface of a paraboloid of revolution from the 
 polai* equation of the parabola 
 
 m 
 
 1 — cos (^ 
 
 Cylindrical Surfaces. 
 
 143. If a cylindrical surface is generated by a line which is 
 always parallel to the axis of Z, the area of the portion bounded 
 by two positions of the generating line, the plane of XT, and 
 any curve whose projection on the plane of XZ is given, is 
 easily found. 
 
 Let ABCD be the cylindrical area required. 
 
162 
 
 Let 
 
 INTEGEAIi CALCULUS. 
 
 [Art. 143. 
 (1) 
 
 y=fx 
 
 be the equation of AB, the line of intersection of the surface 
 with the plane XY\ and let 
 
 z = Fx (2) 
 
 be the equation of Cj-Di, the projection of CD on the plane 
 
 otXZ. 
 
 If X, y, z are the coordinates of 
 any point P of CD, the required 
 area is evidently the limit of 
 the sum of rectangles, of which 
 
 of pp'pi'pi" differs by an in- 
 
 ^Xi j X flnitesimal of higher order than 
 
 ds from zds, and therefore the 
 
 required area 'S' = | zds. 
 
 x,z are the coordinates of Pi, and 
 satisfy (2), and ds = '\/dx^-\-dy^ 
 where x, y are the coordinates of 
 P' and satisfy (1). 
 
 We have, then, ^= \ z^dx^ + dy^ 
 
 [3] 
 
 For example, let AB be the quadrant of a circle, and let the 
 projection of the required area on the plaue of XZ be the quad- 
 rant of an equal circle, so that the surface required is one-eighth 
 of the surface of a groin. 
 
 Here 
 
 3? + f=a?, 
 
 (4) 
 
 and 
 
 x' + z' = a'; 
 
 (5) 
 
 
 ds-sjda^ + dv^-'^dx- "^"^ . 
 
 
 and 
 
 z=\la'- a?. 
 
 
Chap. XI.] 
 
 AKEAS OP SUEFACES. 
 
 163 
 
 Therefore S 
 
 = Cya"-'. 
 
 adx 
 
 Vq 
 
 • = a ( da; = a^. 
 
 ! Jo 
 
 Again, let us find the area of the curved surface of the 
 portion of a cylinder of revolution lucluded within a spherical 
 surface, whose centre lies on the surface of the cylinder, and 
 whose radius is equal to the diameter of the cylinder. 
 
 If the centre of the sphere is taken as the origin, and a 
 diametral plane of the cylinder as the plane of XZ, the surface 
 required is four times that indicated in the figure. 
 
 The equation of the cylinder is 
 
 x'-ax + y^^O, (6) 
 
 and of the sphere 
 
 iff' + y^ + z'-a'' = 0. (7) 
 
 Subtract (6) from (7), and we get 
 
 n^ + ax-a'^0 (8) 
 
 as the equation of a cylindrical surface 
 
 perpendicular to the plane XZ, and 
 
 passing through all the points of intersection of (6) and (7). 
 
 (8) is, then, the equation of the projection on the plane of XZ 
 
 of the line of intersection of the given spherical surface and 
 
 the given cylindrical surface. 
 
 adx 
 
 From (6) , ds = s/day' + dy' = -—dx = 
 
 ^ 22/ 2^ax 
 
 From (8) , z = Va^ — ax. 
 Hence S = C-Ja^ - 
 
 ax 
 
 ■n? 
 
 a^/a C^/a — x. dx 
 
 adx _ ava r' va — x.t 
 lax — a? 2 Jo y/x^a^ 
 
 2 Vo* 
 
 a-4a C d"' „i. 
 
 ss I — ~ — Ui ) 
 
 2 Jo JrK 
 
 Va 
 
 and the whole area required, 
 
 4*5 = 40". 
 
164 INTEGEAi CALCULUS. [ART. 114. 
 
 Examples. 
 
 (1) Find the area cut from the cylindrical surface whose 
 base in the plane XY is a quadrant of the curve x^ + y^ = a? by 
 the plane x = z. Ans. ^a". 
 
 (2) Find the area of that portion of a cylindrical surface 
 whose base in the plane of XT' is a quadrant of the ellipse 
 
 — + =^ = 1, and whose projection on the plane of XZ is bounded 
 
 by the curve a^z^ = Wa?(a? — x') . Ans. S = — - — — — - ^ - 
 
 3(a + 6) 
 
 (3) Let the base of the cylindrical surface be a tractrix, 
 whose vertex lies at a distance a to the left of the origin, and 
 whose asymptote is the axis of Y, while its projection on the 
 plane of XZ is bounded by the parabola «^ = — 2 mx. 
 
 Ans. S = 2a-J'2ma. 
 
 (4) Let the base of the cylindrical surface be the upper half 
 of a cycloid, having its vertex at the origin and its base parallel 
 to the axis of Y, and at a distance 2 a from the origin, while 
 its projection on the plane of XZ is bounded by the parabola 
 2=2 mx. Ans. S = 4:a -\lam. 
 
 Any Surface. 
 
 144. Let X, y, z be the coordinates of any point P of the sur- 
 face, and x + i^x, y + Ay, z + Aa the coordinates of a second 
 point Q infinitely near the first. Draw planes through P and Q 
 parallel to the planes of XY a.nd YZ. These planes will inter- 
 cept a curved quadrilateral PQ on the surface ; its projection pq, 
 a rectangle, on the plane' of XZ; and a parallelogram p'q' not 
 shown in the figure, on the tangent plane at P, of which pq is 
 the projection. PQ will differ from p'q' by an infinitesimal of 
 higher order, and therefore our required surface will be the limit 
 of the sum of the parallelograms of which p'q' is any one. 
 
Chap. XI.] 
 
 AREAS OF SUEPACES. 
 
 165 
 
 If P is the angle the tangent plane at P makes with XZ, 
 p'q'cosl3=pq or p'q' =pq sec/i =Aa;A« sec/3, and o-, our sur- 
 face required, is equal to 
 the double integral 
 
 o-= I I sec ^dxdz 
 
 taken between limits so 
 chosen as to embrace the 
 whole surface. 
 
 The limit of the sum 
 of the parallelograms, of 
 which p'q' is a type, will 
 be the required surface 
 if the limit of the sum of 
 the rectangles, of which 
 pq is a tj-pe, is the pro- 
 jection of the surface in 
 question on the plane of XZ\ so that the values of x and z 
 
 between which we integrate in o- = j | seo ^dxdz are precisely 
 
 those we should use if we were finding the area of the projection 
 
 of <T by the double integration j | dxdz. (v. Art. 136.) 
 
 The equation of the tangent plane at P is 
 (X - Xo)D^J+ {y - y,)DyJ+ {z - z„)D,J= 0, by I. Art. 217, 
 (x„,yoi^o) standing for the coordinates of the point of contact, 
 and f(x,y,z) = being the equation of the surface. 
 
 The direction cosines of the perpendicular from the origin upon 
 
 the plane are 
 
 cos a = - 
 
 DccJ 
 
 ^{D^Jf+{DyJf + {D,jy 
 by Anal. Geom. of Three Dimensions. 
 
 COSy8 = 
 
 cosy: 
 
166 INTEGRAL CALCULUS. [ART. U4 
 
 Hence, dropping the accents, 
 
 By considering the projections upon the other coordinate planes 
 we shall find 
 
 In each of the formulas the derivatives are partial derivatives. 
 Let us find the area of the portion of the surface of the sphere 
 
 a^ + 2/^ + 2^ = «^ 
 intercepted bj' the three coordinate planes. 
 
 DJ=2x, 
 
 DJ= 2y, 
 
 A/=2«, 
 
 ^/(Djy + {Djy+{Djy=2a. 
 
 -dydz; (1) 
 
 X 
 
 or 
 
 
 a?-1p- 
 
 cr=JJ^-d.d,. (3) 
 
 For, in the second one, which agrees best with the figure, we 
 must take our limits so that the limit of the sum of the projec- 
 tions majr be the quadrant in which the sphere is cut by the 
 
Chap. XI.j 
 
 AKEAS OF STTRFACES. 
 
 167 
 
 plane XZ ; and the equation of this section is obtained by letting 
 y = in the equation of the sphere, and is 
 
 a^ + 2^ = a?, 
 
 z = Va- - 
 
 whence z = va- — a?. 
 
 If we take as our limits in the integral \ -dz zero and Va^— x' 
 
 •^ y 
 
 we shall get the area whose projection is a strip running from 
 the axis of Xto the curve ; then, taking j ( 1 - cte ) da; from to 
 
 a, we shall get the area whose projection is the sum of all these 
 strips, and that is our required surface. 
 
 y= Va^ 
 
 ■a? — z^. 
 
 f 
 
 ^0 Jo ^£(2 
 
 dzdx 
 
 ■a?- 
 
 dz 
 
 if we regard x as constant ; 
 
 ■^a? — 01? 
 
 Jo- 
 
 Va2-a:2 
 
 dz 
 
 " yid' — x' — z' 2 
 
 r-TT, ird? 
 n- = a \ -dx = - — ■ , 
 
 Jo 2 2 
 
 the required area. Formulas (1) and (3) give the same result. 
 
 145. Suppose two cj'linders of revolution drawn tangent to 
 each other, and perpendicular to the plane of a great circle of a 
 sphere, each having the radius of the 
 great circle as a diameter ; required the 
 surface of the sphere not included by 
 the cylinders. 
 
 The surface required is eight times 
 the surface of which the shaded portion 
 of the figure is the projection. 
 
 If we take the plane of the great 
 circle as the plane of X T, 
 
168 
 
 INTEGRAL CALCULUS. 
 
 aP — ax + y^ = 
 is the equation of the cj'linder, and 
 
 a? + y'' + z^ = a' 
 
 [Akt. 145. 
 (1) 
 
 of the sphere. 
 
 We have o" = j 1 - 
 
 From (2) 
 
 (2) 
 
 ^/{D,fr+{D,fr+iD.fy 
 
 DJ 
 
 DJ=2x, 
 
 DJ= 23/, 
 
 DJ=-2z; 
 
 (Djy+{Djr+iI)jy = iaK 
 
 dydx 
 
 dydx. 
 
 Hence . = JJ^ dj/d. = a JJ-^^ 
 
 o?—y^ 
 
 Our limits of integration for y are "si ax — 3? and Va^ — v?; for 
 X. are and a. 
 
 ^„ ii<^-=^dydx 
 
 </d^-x^ 
 
 /^ 
 
 dy 
 
 •^a^ — oa' — y' 
 
 : = sia~ 
 
 y 
 
 Va^ — se' 
 
 IT 
 
 2 
 
 ^ax-3i' 
 
 77 . , a; 
 = sin"'\ 
 
 9 ^n J-m 
 
 a + x 
 
 To find jsin-'-y ^ .dx we must integrate by parts. 
 Jo ya + x 
 
 Let 
 and 
 
 . _i I X 
 
 u = sin '-V , 
 
 ya + x 
 
 dv = dx; 
 
 v = x, 
 du- 
 
 \l-.dx ; 
 
 2 (a + x) yx 
 
 fsin- JI^.*. = . sin- j:^^ _ ^ f^.dx. 
 J \a + x \a+x 2 J a+x 
 
Chap. XI.] AREAS OF SUKFACES. 169 
 
 Let 20 = -^x ; 2 wdw = dx 
 
 Jo \« + : 
 
 .dx 
 
 ■X 
 
 2 4 4 2 ' 
 
 
 So- = 8a^ is the whole surface in question. 
 
 146. Let us find the area of the curved surface of a right 
 cone whose base is the curve x^ -\-y^ = a^, and whose altitude 
 is c. 
 
 If we take the vertex of the cone as the origin of coordinates, 
 and its axis as the axis of Z, the equation of its curved surface 
 is 
 
 Xi + yi=(^j, (1) 
 
 and the projection of the surface on the plane of XFis bounded 
 
 by the curve 
 
 xl + yi= aL (2) 
 
 From (1) we get 
 
 DJ y a? x%y^ " 
 
 where x, y are the coordinates of anj- point within the projec- 
 tion of the base of the cone. 
 
 Since the four faces of the cone are equal, the required 
 surface 
 
 (7 = - f " Cx-iy-'^-^d'xlyi + c\xl + y^y. dydx. (3) 
 
 aj^ Jo 
 
170 INTEGRAL CALCULTJS. [ART. 146. 
 
 Let US substitute v^ = x and w^ = y, whence dx = Sv'dii 
 and dy = Sw'dw, and we have 
 
 cr=— j ivws/a'v^io^ + (f(v' + w'f.dwdv; 
 o- Jo Jo 
 
 or, since in a definite integral it makes no difference what letters 
 we use for the variables, 
 
 CT = ^{ i\y V^^iV+^V+W -dydx. (4) 
 
 The a; and y in (4) , however, must not be confounded with the 
 X and y in (3). 
 
 The integral in (4) is precisely that which we should have to 
 find if we sought the area of a surface of such a nature that its 
 projection ou the plane of X.Y was a quadrant of the circle 
 3? -\-y^ =: a', and the secant of the angle made by the tangent 
 plane at any point {x,y,z) of the surface with the plane of XT 
 was xy Va^ a^y^ +(^{3? + y-f. 
 
 In the latter problem there is nothing to prevent our re- 
 placing X and y in xy '\la?o?y- + c^ (.i- + y-y hy their values in 
 terms of r and <^, the polar coordinates of any point of the 
 projection a? + y^ =z ai, and dividing this projection into polar 
 elements instead of rectangular elements, and then integrating 
 between the limits which we should use if we were finding the 
 
 area of the projection by the formula -4 = j I rd<jidr. 
 
 We have, then, 
 
 36 C^ C^ I 
 
 cr= — I I 9-^sin^cosi^ Va^9'''sin^<^'cos^(^ + c^r* . j"d?-d<^, 
 * Jo J» 
 
 or 
 
 IT 
 
 36 f^ C"^ / 
 
 o- = — I I i'^ sin <ji cos tp ya^ sin^ <f) cos- ^ -|- c° . drd(f>, 
 o-Jo Jo 
 
 TT 
 
 p- = 6 a I sin 1^ cos ^ Va^ sin- ^ cos^ <l>+c^.d<t>. 
 Jo 
 
Chap. XI.] , AREAS OF SUEFACBS. 171 
 
 Substitute M = sin^</), and 
 
 cr=3a| Va^M(l — m)H-c^. dw, 
 <r = f r2 etc + (a^ + 40^) tan-' —I- 
 
 Examples. 
 
 (1) Find the area Included by the cylinders described in 
 Art. 145 by direct integration. 
 
 (2) A square hole is cut through a sphere, the axis of the 
 hole coinciding with a diameter of the sphere ; find the area of 
 the surface removed. 
 
 (3) A cylinder is constructed on a single loop of the curve 
 r = acosm<^, having its generating lines perpendicular to the 
 plane of this curve ; determine the area of the portion of the 
 surface of the sphere ^ + ]f -\- ^ = o? which the cylinder inter- 
 cepts. ^^^_ iiLY-.i 
 
 n V2 
 
 (4) Find the area of the portion of the surface of the cone 
 described in Art. 146 included by the cylinder a? + y^ = W. 
 
 Ans. ?|!r2V^?+3?tan->('^^?Il?)-atan-^1. 
 
 (5) Find the area of the portion of the surface of the sphere 
 a? -\- if -\- z' = 2 ay cut out by one nappe of the cone 
 Aa^ + Bz'^f. ^^^ 4,ra^ 
 
 -J{l+A){\+B) 
 
 (6) Find the area of the portion of the surface of the sphere 
 x^ -\- y'^ J^ z^ = 2 ay lying within the paraboloid y =Ao(? +Bz'. 
 
 Ans. ll^. 
 ■slAB 
 
 (7) The centre of a regular hexagon moves along a diameter 
 of a given circle (radius = a") , the plane of the hexagon being 
 perpendicular to this diameter, and its magnitude varying in 
 such a manner that one of its diagonals always coincides with 
 a chord of the circle ; find the surface generated. 
 
 Ans. o2(2,r + 3V3). 
 
172 INTEGRAL CALCULUS. . ■ [ART. 147. 
 
 CHAPTER XII. 
 
 VOLUMES. 
 
 Single Integration. 
 
 147. If sections of a solid are made bj- parallel planes, and a 
 set of cj'linders drawn, each having for its base one of the sec- 
 tions, and for its altitude the distance between two adjacent 
 cutting planes, the limit of the sum of the volumes of these 
 cylinders, as the distance between the sections is indeflnitelj- 
 decreased, is the volume of the solid. 
 
 We shall take as established bj" Geometry the fact that the 
 volume of a cjlinder or prism is the product of the area of its 
 base bj- its altitude. 
 
 It follows from what has just been said, that if, in a given 
 solid, all of a set of parallel sections are equal, the volume of 
 the solid is its base hy its altitude, no matter how irregular its 
 form. 
 
 Let us find the volume of a pjTamid having h 
 /j\ for the area of its base, and « for its altitude. 
 // u Divide the pjTamid by planes parallel to the 
 
 // A base, and let z be the area of a section at the dis- 
 Ari- — \\ tance a; from the vertex. 
 
 / y ri We know from Geometry that - = - . 
 
 /. / ; \ b a^ 
 
 V \\ Hence z = -^y?. 
 
 Let the distance between two adjacent sections be dx ; then 
 the volume of the C3-linder on % is 
 
 — a^da;, 
 and F, the required volume of the p}Tamid, is 
 
 «U 3 
 
Chap. XII.] VOLUMES. 
 
 173 
 
 Precisely the same reasoning applies to any cone, which will 
 therefore have for its volume one-third the product of its base 
 by its altitude. 
 
 Example. 
 Find the volume of the frustum of a pyramid or of a cone. 
 
 148. If a line move keeping always parallel to a given plane, 
 and touching a plane curve and a straight line parallel to the 
 plane of the curve, the surface generated is called a conoid. 
 Let us find the volume of a conoid when the director line and 
 curve are perpendicular to the given plane. 
 
 Divide the conoid into laminae by 
 planes parallel to the fixed plane. 
 
 Let Ay be the distance between 
 
 two adjacent sections, and let x be 
 
 the length of the line in which any 
 
 Nv section cuts the base of the conoid ; 
 
 let a be the altitude and b the area 
 
 of the base of the figure. Any one of our elementarj^ cylinders 
 
 will have for its volume ^axAy, since the area of its triangular 
 
 base is ^ax, and we have V=^u | xdy, the limits of integra- 
 tion being so taken as to embrace the whole solid. I xdy be- 
 tween the limits in question is the area of the base of the co- 
 noid ; hence its volume. 
 
 Examples. 
 
 (1) Find the volume of a conoid when the director line and 
 curve are not perpendicular to the given plane. 
 
 (2) A woodman fells a tree 2 feet in diameter, cutting half- 
 way through from each side. The lower face of each Cut is 
 horizontal, and the upper face makes an angle of 45° with the 
 horizontal. How much wood does he cut out? 
 
174 INTEGRAL CALCULUS. [Akt. 149. 
 
 149. To find ttie volume of an ellipsoid. 
 
 a" V (? 
 
 Take the cutting planes parallel to the plane oiXY. A sec- 
 tion at the distance % from the origin will have 
 
 for its equation, and — Vc^ — 2^ and - Vc^ — z^ for its semi-axes; 
 c . c 
 
 hence its area will be — — U? — z'') . 
 & 
 
 Anj"^ of the elementary cj'linders will have for its volume 
 ^^(c^— »^)Az, and we shall have for the whole solid 
 
 V= f iraftc. 
 If a, &, and c are equal, the ellipsoid is a sphere, and 
 
 Examples. 
 
 (1) Find the volume included between an hyperboloid of one 
 sheet 
 
 a^ 6^ (? 
 
 and its asymptotic cone 
 
 c? V (? 
 Ans. It is equal to a C3'linder of the same altitude as the 
 solid in question, and having for a base the section made by the 
 plane of XY. 
 
 (2) Find the wiole volume of the solid bounded by the surface 
 
 -, + ^! + -! = !• ' A ^■^o.hc 
 
Chap. XII.] VOLUMES. 175 
 
 (3) Find the volume cut from the surface 
 
 c b 
 
 by a plane parallel to the plane of ( YZ) at a distance a from it. 
 
 Ans. ■7ra'^{bc). 
 
 (4) The centre of a regular hexagon moves along a diameter 
 of a given circle (radius = a), the plane of the hexagon being 
 perpendicular to this diameter, and its magnitude varying in 
 such a manner that one of its diagonals always coincides with 
 a chord of the circle; find the volume generated. 
 
 Ans. 2V3.a'. 
 
 (5) A circle (radius = a) moves with its centre on the cir- 
 cumference of an equal circle, and keeps parallel to a given 
 plane which is perpendicular to the plane of the given circle ; 
 
 find the volume of the solid it wUl generate. 9^3 
 
 Ans. £^(377 + 8). 
 
 o 
 
 Solids of Revolution. Single Integration. 
 
 150. If a soUd is generated by the revolution of a plane curve 
 y = fx about the axis of x, sections made by planes perpendicu- 
 lar to the axis are circles. The area of any such circle is iry^, 
 the volume of the elementary cj-linder is wy^Ax, and 
 
 F = 
 
 
 is the volume of the solid generated. 
 
 For example ; let us find the volume of the solid generated by 
 the revolution of one branch of the tractrix about the axis of X 
 Here we must integrate from a; = to x = 00 . 
 
 V= ir I y^dx. 
 
 We have dx = - ^^—^dy (Art. 102 [2].) 
 
 ill the case of the tractrix : 
 
176 INTEGRAL CALCtfLTTS. [Ani. 15i. 
 
 hence F= - ir Cyla' - f)idy. 
 
 When a; = 0, y = a, and when a; = oo, y = 0. 
 Therefore F= - a- ) y (a^ - 3/^) 4 d?/ = — . 
 
 Examples. 
 
 (1) If the plane curve revolves about the axis of Y, 
 
 F=7r rVdy. 
 
 (2) The volume of a sphere is ^ ttci'. 
 
 (3) The volume of the solid formed bj* the revolution of a 
 cycloid about its base is o-n^a^. 
 
 (4) The curve y^{'2 a — a;) = af' revolves about its asj-mptote ; 
 show that the volume generated is 2 71^ a?. 
 
 (5) The curve x^ + y^ = a^ revolves about the axis of X; 
 show that the volume generated is -^^-^iro?. 
 
 Solids of Revolution. Double Integration. 
 
 151. If we suppose the area of the revolving curve broken up 
 into infinitesimal rectangles as in Art. 137, the element \x^y 
 at anj' point P, whose coordinates are x and y, will generate 
 a ring the volume of which will differ from 2iry£^x\y bj' an 
 amount which will be an infinitesimal of higher order than the 
 second if we regard Aa; and Ay as of the first order. For 
 the ring in question is obviouslj' greater than a prism having 
 the same cross-section AxAy, and having an altitude equal to the 
 inner circumference 2 iry of the ring, and is less than a prism 
 having Aa;A)/ for its base and 2 7r(y + Ay) , the outer circumfer- 
 ence of the ring, for its altitude ; but these two prisms differ by 
 2irAa;(A?/)^ which is of the third order. 
 
Chap. XU.] VOLUMES. 177 
 
 Aa; I 2 irydy, where the upper hmit of integration is the ordi- 
 nate of the point of the curve immediately above P, and must be 
 expressed in terms of x by the aid of the equation of the revolv- 
 ing curve, will give us the elementary cylinder used in Art. 150. 
 
 The whole volume required will be the limit of the sum of 
 these cj-linders ; that is, 
 
 V=2^r Cydydx. [1] 
 
 «/to «-'o 
 
 If the figure revolved is bounded bj- two curves, the required 
 volume can be found by the formula just obtained, if the limits 
 of integration are suitably chosen. • 
 
 Let us consider the following example : 
 
 A paraboloid of revolution has its axis coincident with the 
 diameter of a sphere, and its vertex in the surface of the sphere ; 
 required the volume between the two surfaces. 
 
 Let 2/^ =2 ma; (1) 
 
 be the parabola, and a? + y^ — 2ax = Q (2) 
 
 be the circle, which form the paraboloid and the sphere by their 
 revolution. The abscissas of their points of intersection are 
 and 2 (a — m) . 
 
 We have F= 2 tt | j ydydx, 
 
 and, in performing our first integration, our limits must be the 
 values of y obtained from equations (1) and (2). 
 
 We get V=-ir\\2{a — m)x — x''\dx, 
 
 and here our limits of integration are and 2(a — m). 
 
 Hence F= Jir(a — m)' = ^, 
 
 b 
 
 if h is the altitude of the solid in question. 
 
 Examples. 
 
 (1) A cone of revolution and a paraboloid of revolution have 
 the same vei-tex and the same base ; required the volume be- 
 tween them. j^^ irtrf^ ^j^gj,g ,^ jg ^.jjg altitude of the cone. 
 
178 INTEGRAL CALCULTTS. [Art. 152. 
 
 (2) Find the volume included between a right cone, whose 
 vertical angle is 30°, and a sphere of given radius touching it 
 along a circle. ^^^ ^. 
 
 6 
 
 Solids of .Revolution. Polar Formula. 
 
 152. If we use polar coordinates, and suppose the revolving 
 area broken up, as in Art. 138, into elements of which rd^dr 
 is the one at anj' point P whose coordinates are r and <^, the 
 element rd^dr will generate a ring whose volume will differ 
 from 2 ttt^ sin t^dt^dr hy an infinitesimal of higher order than the 
 second, if we regard cJ0 and 'dr as of the first order ; for it will 
 be less than a prism having for its base rd<^dr, and for its alti- 
 tude 2 7r(?' + d)-) sin(<^ + cZc^), and greater than a prism having 
 the same base and the altitude 2 irr sin <^ ; and these prisms 
 differ b}* an amount which is infinitesimal of higher order than 
 the second. 
 
 We shall have then 
 
 r= 2 TT r fr^ sin <l>drd<j>, [1] 
 
 the limits being so taken as to bring in the whole of the gener- 
 ating area. 
 
 For example ; let us find the volume generated by the r.°.volu- 
 tion of a cardioide about its axis. 
 
 7-= 2a(l — cos<^) 
 is the equation of the cardioide ; 
 
 F= 2 77 r Tr^sin <j,drd^. 
 
 Our first integral must be taken between the limits r = and 
 r= 2a(l— cos <)!)), and is 
 
 — ^(1— cos<^)'sin</)d<^. 
 
 o 
 
 16 Z^"' 
 V= — a^TT I (1— cos<^)^sin<^d^, 
 3 Jo 
 
Chap. XII.] 
 
 VOLXJMES. 
 
 179 
 
 Example. 
 
 A right cone has its vertex on the surface of a sphere, and its 
 axis coincident with the diameter of the sphere passing through 
 that point ; find the volume common to the cone and the sphere. 
 
 Volume of any Solid. Triple Integration. 
 
 153. If we suppose our solid divided into parallelopipeds by 
 planes parallel to the three coordinate planes, the elementary 
 z 
 
 parallelepiped at any point (x,y,z) within the solid will have for 
 its volume AasAj/Az, or, if we regard x, y, and 2 as independent, 
 dxdydz ; and the whole volume 
 
 [1] 
 
 F= j ( I dxdydz, 
 
 the limits being so chosen as to embrace the whole solid. 
 
 The integrations are independent, and may be performed in 
 any order if the limits are suitably chosen. 
 
 As it is important to have a perfectly clear conception of the 
 geometrical interpretation of each step iu the process of finding 
 
180 INTEGRAL CALCULUS. [Akt. 153. 
 
 a volume by a triple integration, we will consider one case in 
 
 detail. 
 
 Let the integrations be performed in the order indicated by 
 
 the formula C C C 
 
 F= I I I dzdydx. 
 
 If the limits are correctly chosen, our first integration gives 
 us the volume of a prism one of whose lateral edges passes 
 through any chosen point P,{x,i/,z) within the solid, is parallel 
 to the axis of Z, and reaches directly across the solid from 
 surface to surface, while the base of the prism is the rectangle 
 dydx ; our second integration gives the volume of a right cylin- 
 der whose base is a plane section of the solid, passes through 
 the point P, and is parallel to the plane YZ, and whose altitude 
 is dx ; and our third integration gives the volume of the whole 
 solid. 
 
 The limits in our first integration are, then, the values of z 
 belonging to the point in the lower bounding surface and the 
 point in the upper bounding surface which have the coordinates 
 X and y ; the limits in the second integration are the values of y 
 belonging to the two points in the perimeter of the projection 
 of the solid in the plane of XT which have the coordinate x ; 
 and the limits in the third integration are the least value and 
 the greatest value of x belonging to points on the perimeter of 
 the projection of the solid on the plane of XT. 
 
 It is easily seen from what has just been said that the limits 
 in the second and third integrations are precisely those we 
 should use if we were finding the area of the projection of the 
 solid by the formula „ -, 
 
 ^ = I I dydx. 
 
 Of course, it is necessary to have a clear idea of the form of 
 the solid whose volume is required. 
 
 For example , let us find the volume of the portion of the 
 ellipsoid „2 a ^ 
 
 cut off by the coordinate planes. 
 
Chap. XII.] VOLUMES. 181 
 
 F = j I j dzdydx, 
 
 and our limits are, for z, and c\ll — ; for y, and 
 
 I x" ^ "" ^ 
 
 6 -til -; and for x, and a. For, starting at any point 
 
 (x,y,z) and integrating on tjie hypothesis that z alone varies, we 
 get a column of our elementary parallelepipeds having dxdy as a 
 base and passing through the point {x,y,z) . To make this col- 
 umn reach from the plane X5^ to the surface, z must increase 
 from the value zero to the value belonging to the point on the 
 surface of the ellipsoid which has the coordinates x and y ; that 
 
 
 is, to the value c-v/l 2~l2" Then, integrating on the hy- 
 pothesis that y alone varies, we shall sum these columns and 
 shall get a slice of the solid passing through {x,y,z) and having 
 the thickness dx. To make xhis slice reach completelj' across 
 the solid, we must let y increase from the value zero to the 
 greatest value it can have in the slice in question ; that is, to the 
 value which is the ordinate of that point of the section of the 
 ellipsoid bj- the plane XF which has the abscissae. The section 
 in question has the equation 
 
 or If 
 
 therefore the required value of ?/ is & \l 1 
 
 a' 
 
 Last, in integrating on the hypothesis that x alone varies, we 
 must choose our limits so as to include all the slices just de- 
 scribed, and must increase x from zero to a. 
 
 jd.=.=c^i-i;-i; 
 
 a? 
 between the limits and c\\\ ^, - 
 
182 INTBGEAL CALCULUS. [Akt. 153, 
 
 ^f^^'i-t^ 
 
 
 ~ 4 I a^ 
 
 ^V'-i 
 
 between the limits and 6 
 
 
 the volume required. 
 
 Examples. 
 
 (1) Find the volume obtained in the present article, perform- 
 ing the integrations in the order indicated by the formula, 
 
 F= I I I dxdydz. 
 
 (2) Find the volume cut off from the surface 
 
 c b 
 
 by a plane parallel to that of TZ, at a distance a from it. 
 
 Ans. 7ra^V(6c). 
 
 (3) Find the volume enclosed by the surfaces, 
 
 a^ + f = cz, off' + y^^ax, z=0. ^^^ Swa* 
 
 32c' 
 
 (4) Obtain the volume bounded b}- the surface 
 
 z = a— Va^ + 2/^ 
 
 2 a' 
 and the planes x = z and x = 0. Ans. -—. 
 
Chap. XII.] VOLUMES. 183 
 
 (5) Find the volume of the conoid bounded by the surface 
 z^ H f- = & and the planes a; = and x = a. Ans. 
 
 154. If we use polar coordinates we can take as our element 
 of volume 
 
 r" sin <f}drd<t>d6, 
 
 an expression easily obtained from the element 2 irr^ sin <l>drd(fi 
 used in Art. 152. 
 
 Then ' F= f f (Vsin<^drd<^dfl, 
 
 where the order of the integrations is usually immaterial if the 
 limits are properly chosen. 
 
 Examples. 
 
 (1) Find the volume of a sphere by polar coordinates. 
 
 (2) Find the whole volume of the solid bounded by 
 
 (a^ + 2/2 ^ z2)3 = 27 a^xyz. 
 Sxtggestion: Transform to polar coordinates. Ans. -a\ 
 
184 
 
 INTEGRAL CALCULUS. 
 
 [Art. 155. 
 
 CHAPTER XIII. 
 
 CENTRES OF GRAVITY. 
 
 155. The moment of a force about an axis perpendicular to its 
 line of direction is the product of the magnitude of the force by 
 the perpendicular distance of its line of direction from the axis, 
 and measures the tendency of the force to produce rotation 
 about the axis. 
 
 The force exerted by gravity on an3' material body is propor- 
 tional to the mass of the bodj-, and may be measured bj- the 
 mass of the bod3'. 
 
 The Centre of Gravity of a bod^- is a point so situated that the 
 force of gravitj- produces no tendency in the bodj- to rotate about 
 anj' axis passing through this point. 
 
 The subject of centres of gravitj- belongs to Mechanics, and 
 we shall accept the definitions and principles just stated as data 
 for mathematical work, without investigating the mechanical 
 grounds on which they rest. 
 
 156. Suppose the points of a body referred to a set of three 
 rectangular axes fixed in the body, aud let x,y,z be the coordi- 
 nates of the centre of gravit}'. Place 
 the body with the axes of X and Z 
 horizontal, and consider the tendency 
 of the particles of the body to produce 
 rotation about an axis through {x,y,z) 
 parallel to OZ, under the influence of 
 gravit}-. Represent the mass of an 
 elementar}- parallelepiped at anj- point 
 (x,y,z) by dm. The force exerted bj' 
 gravitj- on dm is measured b\- dm, and 
 
 its line of dii'ection is vertical. If the mass of dm were concen- 
 trated at P, the moment of the force exei-ted on dm about the 
 
Chap. XIII.] CENTRES OF GRAVITY. 185 
 
 axis through G would be {x — x)dm, and this moment would 
 represent the tendency of dm to rotate about the axis in ques- 
 tion ; the tendency of the whole body to rotate about this axis 
 would be 2 (a; — x)dm. If now we decrease dm indefinitely, the 
 error committed in assuming that the mass of dm is concentrated 
 at P decreases indefinitely, and we shall have as the true expres- 
 sion for the tendenc}' of the whole body to rotate about the axis 
 through (7, J (a — x)dm \ but this must be zero. 
 
 Hence { {x — x)dm = Q, 
 
 I xdm —X I dm = 0, 
 
 /' 
 
 xdm 
 
 I dm 
 
 [1] 
 
 If we place the bodj' so that the axes of Y and X are hori- 
 zontal, the same reasoning will give us 
 
 y = 
 
 Cydn 
 
 [2] 
 
 j dm 
 
 and in like manner we can get 
 
 zdm 
 
 [3] 
 
 7 
 
 Since j dm is the mass of the whole body, if we represent it 
 
 by M we shall have ^ 
 
 I xdm 
 
 X -■ 
 
 M 
 
 i ydm 
 
 y=- ' 
 
 M 
 
 zdm 
 
 M 
 
 f 
 
186 INTBGEAL CALCULUS. [Art. 157. 
 
 Example. 
 
 Show that the efifect of gravity in making a body tend to rotate 
 about any given axis is precisely the same as if the mass of the 
 body were concentrated at its centre of gravity. 
 
 157. The mass of any homogeneous body is the product of 
 its volumL' by its density. If the body is not homogeneous, the 
 densit}- at any point will be a function of the position of that 
 point. Let us represent it by k. Then we ma}- regard dm as 
 equal to kcIv if dv is the element of volume, and we shall have 
 
 f 
 
 f' 
 
 xndv 
 
 [1] 
 
 Kdv 
 
 and corresponding formulas for y and z. 
 
 If the body considered is homogeneous, k is constant, and we 
 shall have 
 
 I xdv I xdv 
 
 I ydv I ydv 
 I zdv I zdv 
 
 '"-T^-'-y- P5 
 
 dv 
 
 [4] 
 
 In any particular problem ^e have only to express dv in 
 terms of the coordinates. 
 
 Plane Area. 
 
 158. If we use rectangular coordinates, and are dealing with 
 a plane area, where the weight is uniformly distributed, we have 
 
 dv = dA = dxdy. (Art. 136). 
 
Chap. XIII.] CENTRES OP GRAVITY. 
 
 Hence, by 157, [2] and [3], 
 
 I I xdxdy 
 
 QC ^= — — 
 
 I I dxdy 
 
 jjydxdy 
 I I dxdy 
 
 If we use polar coordinates, 
 
 dv = dA = rdcl>dr, 
 
 I I r^ cos <^d<j)dr 
 
 187 
 
 [1] 
 
 and 
 
 I j rdcfidr 
 
 I I r" sin cf) d<^dr 
 
 I I rd<j>dr 
 
 y = 
 
 [2] 
 
 For example ; let us find the centre of gravity of the area be- 
 tween the cissoid and its asymptote. From the equation of the 
 cissoid 
 
 r = » 
 
 a — x 
 
 we see that the curve is symmetrical with respect to the axis 
 of X, passes through the origin, and has the line x = a as an 
 asymptote. From the symmetrj' of the area in question, y =0, 
 and we need only find x. 
 
 I I xdydx I xydx 
 
 _ Jo J-v __ Jo 
 
 C C C' ' 
 
 I I dydx I ydx 
 
 Jo J-y Jo 
 
188 
 
 INTEGRAL CALCXTLUS. [ART. 158. 
 
 i)(a — 
 
 aA 
 
 x)^ Jo{a-x)^ by Art. 64 [4]. 
 
 ^^da; I , ^, dx 
 
 o{a-x}i Jo{a-x)^ 
 
 a; = 5- a. 
 
 As an example of the use of the polar formulas [2], let us find 
 the centre of gravity of the cardioide 
 
 r= 2a(l— cos(^). 
 
 Here, from the fact that the axis of X is an axis of symmetry, 
 we know that ^ = 0. 
 
 I jr' cos <i>drd<t> 
 
 rdrdtp 
 
 i r/cos^d<^ ~ fci - COS <t>y COS <l>d^ 
 
 ^0 O ^0 
 
 ijr'd<l> ^a^j {1- cos 4>yd<t> 
 
 X27r 
 (cos^ — 3 cos^^ 4- 3 cos'<^— cos*<^)d<^ = — J^tt ; 
 
 X27r 
 (1 — 2 cos <^ + cos''<t>)d(j) = 3 TT. 
 
 Hence ^ = — Ja. 
 
 Examples. 
 
 1. Show that formulas [1] hold even when we use oblique 
 coordinates. 
 
 2. Find the centre of gravit}- of a segment of a parabola cut 
 off by any chord. 
 
 Ans. x=^a, y = 0. If the axes are the tangent parallel 
 to the chord and the diameter bisecting the chord. 
 
Chap. XIII.] CENTRES OF GRAVITY. 189 
 
 3. Find the centre of gravity of the area bounded by the semi- 
 cubical parabola ay^ = ar' and a double ordinate. Ans. x = -fa;. 
 
 4. Find the centre of gravity of a semi-ellipse, the bisecting 
 line being any diameter. 
 
 Ans. If the bisecting diameter is taken as the axis of Y, and 
 
 — 4a _ 
 the conjugate diameter as the axis of X, x = — , y = 0. 
 
 Sir 
 
 5. Find the centre of gravity of the curve y" = b' 
 
 X 
 
 Ans. X = ^a. 
 
 6. Find the centre of gravity of the cycloid. 
 
 Ans. x = a7r, y=^a. 
 
 7. Find the centre of gravity of the lemniscate r' = a' cos 2 <j). 
 
 Ans. X = - — a. 
 8 
 
 8. Find the centre of gravity of a circular sector. 
 
 Ans. If we take the radius bisecting the sector as the axis 
 
 of X, and represent the angle of the sector by 2 a, « = f 
 
 a 
 
 9. Find the centre of gravity of the segment of an ellipse cut 
 off by a quadrantal chord. Ans. x = %——-, y = | ^_^ - 
 
 10. Find the centre of gravity of a quadrant of the area of the 
 
 curve xi -\~yi = aS. Ans. x — y — ff- 
 
 a 
 
 159. If we are dealing with a homogeneous solid formed by 
 the revolution of a plane curve about the axis of X, we have 
 
 dv=2 wydydx. (Art. 151 [1] ) 
 
 Hence, by Art. 157 [2], 
 
 \ i ydxdy 
 
IQQ nSTTEGEAL CALCULUS. [Akt. 159. 
 
 If we use polar coordinates, 
 
 dv = 27rr2sm<^drd<^. (Art. 152 [1].) 
 
 I j r' sin ^ cos cfidrdtfi 
 
 Hence '^ r r '"^"' 
 
 j J ?*^ sin <f>drd<t} 
 
 For example ; let us find the centre of gravitj- of a hemisphere. 
 The equation of the revolving curve isa^ + y^ = a^ 
 
 C Cxydydx 
 
 -. _ Jo Jo _ ilL — 3 « 
 
 ( I ydydx 
 
 ^M Jo 
 
 <af-ofl ' \a^ * 
 
 If we use polar coordinates the equation of the revolving curve 
 IS r = a. 
 
 a '^ 
 
 J I I r'sinrf) cos<idrf)cZr , . 
 Jo ia_i5„ 
 
 tiere x = ig = f— , = g «• 
 
 I rsai<l)d<f>dr 
 
 Examples. 
 
 1 . Find the centre of gravitj- of the solid formed by the revolu- 
 tion of the sector of a circle about one of its extreme radii. 
 
 Ans. a; = f « cos^^/S, where ji is the angle of the sector. 
 
 2. Find the centre of gravity of the segment of a paraboloid 
 of revolution cut ofTbj- a plane perpendicular to the axis. 
 
 Ans. a = f a, where a; = a is the plane. 
 
 3. Find the centre of gravity of the solid formed by scooping 
 out a cone from a given paraboloid of revolution, the bases of 
 the two volumes being coincident as well as their vertices. 
 
 Ans. The centre of gravity bisects the axis. 
 
Chap. XIII.] CENTRES OF GKAVITY, 191 
 
 4. A cardioide is made to revolve about its axis ; find the 
 centre of gravity of the solid generated. Ans. S = — fa. 
 
 5. Obtain formulas for the centre of gravity of any homo- 
 geneous solid. 
 
 6. Find the centre of gravity of the solid bounded by the 
 surface z' = xy and the five planes a;=0, y=0, z=0, x=a, y=h. 
 
 Ans. x=fa, y = ^b, z = ^a^bi. 
 
 160. If we are dealing with the arc of a plane curve, the 
 formulas of Art. 167 reduce to 
 
 I xds 
 x==^, [1] 
 
 jyds 
 
 [2] 
 
 Examples. 
 
 1. Find the centre of gravity of an arc of a circle, taking the 
 diameter bisecting the arc as the axis of X and the centre as the 
 
 origin. Ans. x = — , where c is the chord of the arc. 
 
 s 
 
 2. Find the centre of gravitj- of the are of the curve jcl-f ?/J=at 
 between two successive cusps. Ans. x = y = ^a. 
 
 3. Find the centre of gra^dty of the arc of a semi-cycloid. 
 
 Ans. i = (7r — |)a, y = — ^a. 
 
 4. Find the centre of gravity of the arc of a catenary cut off 
 
 by any horizontal chord. 
 
 A^g x = v = "^ "*" ^^1 where 2 s is the length of the arc. 
 ' ^ 2s 
 
 5. Obtain formulas for the centre of gravity of a surface of 
 revolution, the weight being uniformly distributed over the 
 surface. 
 
192 INTEGRAL CALCULUS. [ART. 161 
 
 6. Find the centre of gravity- of any zone of a sphere. 
 
 Ans. The centre of gravity bisects the hne joining the centres 
 of the bases of the zone. 
 
 7. A cardioide revolves about its axis ; find the centre of 
 gravit}' of the surface generated. Ans. x = — -'jY "• 
 
 8. Find the centre of gravity of the surface of a hemisphere 
 when the density at each point of the surface varies as its per- 
 pendicular distance from the base of the hemisphere. 
 
 Ans. X = fa. 
 
 9. Find the centre of gravity of a quadrant of a circle, the 
 densit}- at anj- point of which varies as the ntJi power of its 
 distance from the centre. j^^g ^ ~j, — " + '^ 2a 
 
 n + S tt' 
 
 10. Find the centre of gravity of a hemisphere, the density' 
 of which varies as the distance from the centre of the sphere. 
 
 Ans. X = I a. 
 
 Properties of Ouldin. 
 
 161. I. If a plane area revolve about an axis external to 
 itself through anj' assigned angle, the volume of the solid gene- 
 rated will be equal to a prism whose base is the revolving area 
 and whose altitude is the length of the path described by the 
 centre of gravitj- of the area. 
 
 11. If the are of a plane curve revolve about an external axis 
 in its own plane through any assigned angle, the area of the 
 surface generated will be equal to that of a rectangle, one side 
 of which IS the length of the revolving curve, and the other the 
 length of the path described by its centre of gravity. 
 
 First ; let the area in question revolve about the axis of X 
 through an angle ®. The ordinate of the centre of gravity of 
 the area m question is 
 
 J iydxdy 
 
 y='^ ' by Art. 158 [1]. 
 
 I I dxdy 
 
Chap. XIII.] CENTRES OF GRAVITY. 193 
 
 The length of the path described bj- the centre of gravity 
 © j [ydxdy 
 
 y®=^^ (1) 
 
 I I dxdy 
 The volume generated is 
 
 F= ©CCydxdy, by Art. 151. 
 
 Hence V=y®\ i dxdy. 
 
 But I I dxdy is the revolving area, and the first theorem is 
 established. 
 We leave the proof of the second theorem to the student. 
 
 Examples. 
 
 1 . Find the surface and volume of a sphere, regarding it as 
 generated bj- the revolution of a semicircle. 
 
 2. Find the surface and volume of the solid generated by the 
 revolution of a c^'cloid about its base. 
 
 3. Find the volume and the surface of the ring generated by 
 the revolution of a circle about an external axis. 
 
 Ans. y=27rV6, S = A-n^ab, where b is the distance of 
 the centre of the circle from the axis. 
 
 4. Find the volume of the ring generated by the revolution of 
 an ellipse about an external axis. 
 
 Ans. V= 2 TT^ibc, where c is the distance of the centre of the 
 ellipse from the axis. 
 
194 tNTEGEAL CAiCTJLUS. [Akt. 162. 
 
 CHAPTER XIV. 
 
 LINE, ST7KFACE, AND SPACE INTEGRALS. 
 
 162. Any variable which depends for its value soleh- upon 
 the" position of a point, as, for example, any function of the 
 rectangular or polar coordinates of the point, may be called 
 a point- function. 
 
 A point-function is said to be continuous along a gi-i'en line 
 if its value changes continuously as the point, on whose position 
 the function depends for its value, moves along the line ; it is 
 said to be continuous over a given surface if its value changes 
 continuously as the point is made to move at pleasure over the 
 surface ; and it is said to be continuous throughout a given 
 space if its value changes continuously as the point is made to 
 move about at pleasure within the space. 
 
 163. If a given line is divided in any way into infinitesimal 
 elements, and the length of each element is multiplied by the 
 value a given point-function, which is continuous along the line, 
 has at some point within the element, the limit approached by 
 the sum of these products as each element is indefinitelj- de- 
 creased, is called the line integral of the given function along 
 the line in question. 
 
 If a given surface is divided in any way into infinitesimal 
 elements such that the distance between the two most widely 
 separated points within each element is infinitesimal, and the 
 area of each element is multiplied by the value a given point- 
 function, which is continuous over tlie surface, has at some 
 point within the element, the limit approached by the sum of 
 these products as each element is indefinitely decreased, is 
 called the surface integral of the given function over the surface 
 in question. 
 
Chap. XIV.] LINE, STTEPACE, SPACE INTEGRALS. 195 
 
 If a given space is divided in any way into infinitesimal 
 elements such that the distance between the two most widely 
 separated points within each element is infinitesimal, and the 
 volume of each element is multiplied by the value a given point- 
 function, which is continuous throughout the space, has at 
 some point within the element, the limit approached by the 
 sum of these products as each element is indefinitely decreased, 
 is called the space integral of the given function throughout 
 the space in question. 
 
 It is easily seen that the line integral of unity along a given 
 line is the length of the line ; that the surface integral of unity 
 over a given surface is the area of the surface ; and that the 
 space integral of unity throughout a given space is the volume 
 of the space. 
 
 In the chapter on Centres of Gravity we have had numerous 
 simple examples of line, surface, and space integrals. 
 
 164. That the value of a line, surface, or space integral is 
 independent of the position in each element of the point at 
 which the value of the given function is taken can be proved 
 as follows : The distance apart of any two points in the same 
 infinitesimal element is infinitesimal (Art. 163), therefore the 
 values of a continuous function taken at any two points in 
 the same element will differ in general by an infinitesimal ; the 
 products obtained by multiplying these two values by the mag- 
 nitude of the element will, then, differ by an infinitesimal of 
 higher order than that of the element ; therefore, in forming 
 the integral either of these products may be used in place of 
 the other without changing the result. (I. Art. 161.) 
 
 165. The line integral of a function along a given line is 
 absolutely independent of the manner in which the line is 
 broken up into infinitesimal elements, and is equal to the length 
 of the line multiplied by the mean value of the function along 
 the line ; the mean value of the function being defined as fol- 
 lows : Suppose a set of points uniformly distributed along the 
 
196 INTEGKAL CALCULUS. [ART. 165. 
 
 line, that is, so distributed that the number of points in any 
 portion of the line is proportional to the length of the portion ; 
 take the value of the function at each of these points ; divide 
 the sum of these values by the number of the points ; and the 
 limit approached by this quotient as the number of the points 
 is indefinitely increased is the mean value of the given function 
 along the line ; and this mean value is in general finite and 
 determinate. 
 
 To prove our proposition, we have only to consider in detail 
 the method of finding the mean value in question. Let the 
 number of points in a unit of length of the line be k. Then, 
 no matter how the line is broken up into infinitesimal elements, 
 the number of points in each element is k times the length of the 
 element. Since any two values of the function corresponding to 
 points in the same element differ by an infinitesimal, in finding 
 our limit we maj- replace all values corresponding to points in 
 the same element by any one ; hence the sum of the values cor- 
 responding to points in the same element may be replaced by one 
 value multiplied by the number of points taken in that element, 
 that is, this sum may be replaced by k times the product of one 
 value b^' the length of the element ; and the sum of the values 
 corresponding to all the points taken in the line may be replaced 
 by k times the sum of the terms obtained bj- multiplying the 
 length of each element by the value of the function at some 
 point within the element. When we divide this sum by tlie whole 
 number of points considered, that is, by k times the length of 
 the line, the k's cancel out, and the required mean value reduces 
 to the limit of the numerator divided by the length of the line, 
 and the limit of the numerator is the line integral of the func- 
 tion along the line. Therefore the line integral is the mean 
 value of the function multiplied by the length of the line. 
 
 The same proof may be given for a surface integral or for a 
 space integral. The former is the product of the area of the 
 surface by the mean value of the function over the surface ; 
 the latter is the volume of the space multiplied by the mean 
 value of the function throughout the space ; and both are inde- 
 
Chap. XIV.] LINE, SURFACE, SPACE INTEGKALS. 197 
 
 pendent of the way in which the surface or space may be divided 
 into infinitesimal elements. 
 
 166. If the line along which the integral is taken is a plane 
 curve, it is easy to get a geometrical representation of the 
 integral. For, if at every point of the line a perpendicular to 
 the plane of the line is erected whose length is equal to the 
 value of the function at the point, the line integral required 
 clearly represents the area of the cylindrical surface containing 
 the perpendiculars if the values are all of the same sign, and 
 represents the difference of the areas of the portions of the 
 cylindrical surface which lie on opposite sides of the line if the 
 values of the function are not all of the same sign. 
 
 A similar construction shows that a surface integral over a 
 plane surface may be represented by a volume or by the differ- 
 ences of volumes. Consequently, in each case if the function 
 is finite and continuous, the integral is finite and determinate. 
 
 167. As examples of line, surface, and space integrals, we 
 will calculate a few moments of ineHia. 
 
 The moment of inertia of a body about a given axis may be 
 defined as the space integral of the product of the density at 
 any point of the body by the square of the distance of the point 
 from the axis ; the integral being taken throughout the space 
 occupied by the body. 
 
 If the body considered is a material surface or a material 
 line, the integral reduces to a surface integral or to a line 
 integral. 
 
 In the examples taken below the body is supposed to be 
 homogeneous. 
 
 (a) The moment of inertia of a circumference about a given 
 diameter. 
 
 Using polar coordinates and taking the diameter as our axis, 
 
 
 ^"^ sin^ (^ • Tcadtl) = ka^ir 
 = iMa% [1] 
 
198 INTEGRAL CALCUIiTJS. [Art. 167. 
 
 if I is the moment of inertia, and a the radius, k the density, 
 and M the mass of the circumference in question. 
 
 (6) The moment of inertia of the perimeter of a square about 
 an axis passing through the centre of the square and parallel 
 to a side. 
 
 1=2 Cy^My + 2 CaFlcdx 
 
 %J —a %J —a 
 
 = iMa', [2] 
 
 if 2 a is the length of a side. 
 
 (c) The moment of inertia of a circle about a diameter. 
 
 1= \ I r^ svD? <f> .Tcrd^dr = ^k-TTO.* 
 
 Jo Jo 
 
 = iMaK [3] 
 
 (d) The moment of inertia of a square about an axis through 
 the centre of the square and parallel to a side. 
 
 /= P Cy^kdxdy = ^ka* 
 
 = iMa\ [4] 
 
 (e) The moment of inertia of the surface of a sphere about 
 a diameter. 
 
 I a^ sin^ ^ . ka' sin <jid<j>dO = f kira* 
 = iMa\ [5] 
 
 (/) The moment of inertia of the surface of a cube about an 
 axis parallel to an edge and passing through the centre. 
 
 T=4.r r{d' + z')kdxdz-\r'ir C\y^ + z^)kdydz 
 
 = s*ika^ + ^ka* 
 
 = i^Ma\ [6] 
 
Chap. XIV.] LINE, SURFACE, SPACE INTEGRALS. 199 
 
 (g) The moment of inertia of a sphere about a diameter. 
 
 = iMa\ |-7] 
 
 (h) The moment of inertia of a cube about an axis through 
 the centre and parallel to an edge. 
 
 1= C" C" C\y' + z^)kdxdydz = J^&a" 
 
 = iMaK [8] 
 
 Examples. 
 
 Find the moments of inertia of the following bodies : 
 
 (1) Of a straight line about a perpendicular through an 
 extremity ; about a perpendicular through its middle point. 
 
 Ans. iMP; -^^MP- 
 
 (2) Of the circumference of a circle about an axis through 
 its centre perpendicular to its plane. Ans. Ma?. 
 
 (3) Of a circle about an axis through its centre perpendicular 
 to its plane. Ans. ^Ma?. 
 
 (4) Of a rectangle whose sides are 2 a, 26, about an axis 
 through its centre perpendicular to its plane ; about an axis 
 through its centre parallel to the side 2&. 
 
 Ans. iMia' + b'); ^MaK 
 
 (5) Of an ellipse about its major axis ; about its minor axis ; 
 about an axis through the centre perpendicular to the plane of 
 the ellipse. Ans. IMlf; ^Ma^; lM{a? + b'^). 
 
 (6) Of an ellipsoid about the axis a. Ans. \M{W + c'). 
 
 (7) Of a rectangular parallelopiped about an axis through 
 the centre parallel to the edge 2 a. Ans. \M(W + (?). 
 
 (8) Of a segment of a parabola about the principal axis. 
 
 Ans. \MW, where 2& is the breadth of the segment. 
 
200 
 
 INTEGRAL CALCTJLUS. 
 
 [Art. 168. 
 
 168. If Vi, DjU, and D^u are finite, continuous, and single- 
 valued for all points in a given plane surface bounded by a 
 closed curve T, the surface integral o/D^u taken over the surface 
 is equal to the line integral of a cos a taken around the whole 
 bounding curve, where a is the angle made with the axis of .X 
 by the external normal at any point of the boundary. 
 
 This may be formulated thus : 
 
 I I D^ vdxdy = l u cosa . ds. 
 
 [1] 
 
 Let the axes be chosen so that the surface in question lies in 
 the first quadrant, and divide the projection of T on the axis 
 of T into infinitesimal elements of which any one is dy. 
 
 dy 
 
 On each of these elements as a base erect a rectangle ; and 
 since 7" is a closed curve, each of these rectangles will cut it 
 an even number of times. 
 
 Let us call the values of ?t at the points where the lower side 
 of any one of these rectangles cuts Z', u^, u^, Wg, u^, etc., re- 
 spectively ; the angles which this side makes with the exterior 
 normals at these points, aj, aj, as, a^, etc. ; and the elements 
 which the rectangle cuts from T, dsi, dsj, ds^, ds^, etc. 
 
 It is evident that whenever a line parallel to the axis of X 
 cuts into the surface bounded by T, the corresponding value of 
 u. is obtuse and its cosine negative ; that whenever it cuts out, 
 
Chap. XIV.] LINE, SURFACE, SPACE INTEGRALS. 201 
 
 a is acute and its cosine positive ; and that any value of a is 
 the angle which the contour T itself makes at the point in ques- 
 tion with the axis of Y if we suppose the contour traced by a 
 point moving so as to keep the bounded surface always on the 
 left hand. 
 
 We have then approximately, 
 
 dy^= — dSx'C0Sai = dS2' C0Sa2=— dS3-C08a3=dS4'COSa4=-". [2] 
 
 If, now, in I I D^udxdy we perform the integration with 
 respect to x, and introduce the proper limits, we shall have 
 
 j j D^udxdy = j dy{—Ui + U2 — U3 + Ui---); [3] 
 
 and the second member indicates that we are to form a quantity 
 corresponding to that in parenthesis for every rectangle which 
 cuts T, to multiply it by the base of the rectangle, and then to 
 take the limit of the sum of the results as all the bases are 
 indefinitely decreased. 
 
 By [2], 
 
 dy{—Ui + U2 — Ug + Uf-) 
 = Ml cos ai dsi + M2 cos 02 c?S2 + Mg cos as dss + M4 cos 04 ds4 -I ; [4] 
 
 and the limit of the sum of the values any one of which is 
 represented by the second member of [4] is clearly j « cos ads 
 taken around the whole of T. 
 
 Example. 
 
 Prove that under the conditions stated in the last article 
 
 j j DyUda^y = \ u cos/3 . ds, 
 
 where ^ is the angle made with the axis of Y by the exteriot 
 normal. 
 
202 INTEGRAL CALCULUS. [ART. 169. 
 
 169. As an illustration of the last proposition, let us find the 
 centre of gravity of a semicircle. 
 
 We have y = j-C Cydxdy. (1) 
 
 But we may write y = D^{xy). Hence, by Art. 168, 
 y = ^Jj'ydxdy = j^J^y cos ads 
 
 = — ( racos<^a8in<^cos<^ac?<^+ j ce.O.cos^-da; j 
 
 , iror 3 3 TT 
 
 Ic — 
 2 
 
 which agrees with the result of Ex. 8, Art. 158. 
 
 As a second example, we shall find the moment of inertia of 
 a circle about a diameter. 
 "We have 
 
 I=k\ I y^dxdy =Jc | xy^ cos ^ . ds 
 
 = k \ acostf)a'sm'4>cos<l>ad<j> 
 
 sin^<i cos^<bd(b = -ira* = -Ma^, 
 4 4 
 
 which agrees with the result of (c), Art. 167. 
 
 Examples. 
 
 (1) Find the centre of gravity of a semicircle, using the 
 theorem | | DyUdxdy = | ucos/i .ds. 
 
 (2) Find the moment of inertia of a circle about an axis 
 through its centre perpendicular to its plane, using the principles 
 
 j I D^udxdy=\ ucoaa.ds and | i B^udxdy = i ucos/S .ds. 
 
Chap. XIV.] LINE, SURFACE, SPACE INTEGRAX,S. 203 
 
 170. Since, as we have seen in Art. 168, a is the angle which 
 the curve T makes with the axis of Y; if we trace the curve 
 so as to keep the bounded space on our left, it follows that 
 cosa.ds = dy. 
 
 Hence ( j D^udxdy = ( udy; [1] 
 
 and in like manner, 
 
 I I DyUdxdy = — j udx ; [2] 
 
 the first integral in [1] and [2] being taken over the bounded 
 surface, and the second around the bounding curve. 
 
 For example, the moment of inertia of a square about an 
 axis through the centre and parallel to a side is 
 
 I=TcC Cy^dxdy. ((d) Art. 167.) 
 
 and the last integral is to be taken around the perimeter. 
 Hence 
 
 = IMa'. 
 
 Example. 
 
 Work Ex. 8, Art. 167, by the aid of (2). 
 
 171. If U, DjU, DylJ, and D^U are finite, continuows^ 
 single-valued functions throughout the space bounded by a given 
 closed surface T, the space integral of D^U taken throughout the 
 space in question is equal to the surface integral, taken over the 
 bounding surface, of U cosa, where a is the angle made with 
 the axis of X by the exterior normal at any point of the surface. 
 
 This may be formulated thus : 
 
 r r Cd, Udxdydz = Cu eos a . dS. [1] 
 
204 
 
 INTEGRAL CALCULUS. 
 
 [Art. 171. 
 
 The proof is almost identical with that given in Art. 168, 
 except that for elementary rectangle we use elementary prism. 
 We shall merely indicate the steps. 
 
 dydz = — dSi cosoj = dS2 cosoj ^ — dS^ cos 03 ^ ••• 
 C CfD^Udxdydz = JJ^dyd^- U^ + C/2- iJs-] 
 = the limit of the sum of terms of the form 
 
 Ui costti . dSi + Ui cos 02 . dSi + U^ cos 03 . dS^ + ••• 
 = I U COSa. dS. 
 
 Example. 
 Prove that under the conditions of the last article 
 Ci Cd, Udxdydz = Cucos (i . dS, 
 
 and I I I D ^Udxdydz— | U cosy .dS, 
 
 where j8 and y are the angles made with the axes of Y and Z 
 respectively by the exturior normal to the bounding surface. 
 
Chap. XIV.] LINE, STTRFACE, SPACE INTEGRALS. 205 
 
 172. As an illustration, let us find the centre of gravity of a 
 hemisphere. 
 We have 
 
 = ^-TTL I I a^ cos''l> cos 4> 0? sin </> d<i>d6 
 
 IT 
 
 = -rrrz, | ( COs'<isindid<idd 
 2MJii Ja 
 
 which agrees with the result of Art. 159. 
 
 Example. 
 
 Find the moment of inertia of a sphere about a diameter ; of 
 a cube about an axis through the centre parallel to an edge. 
 Make your work depend upon finding the value of a surface 
 ■ integral. 
 
206 INTEGRAL CALCULUS. [Akt. 173. 
 
 CHAPTER XV. 
 
 MEAN VALUE AND PROBABILITY. 
 
 173. The application of the Integral Calculus to questions 
 in Mean Value and Pi-obabilit}- is a matter of decided interest ; 
 but lack of space will prevent our doing more than solving 
 a few problems in illustration of some of the simplest of the 
 methods and devices ordinarily employed. A full and admirable 
 treatment of the subject is given in "Williamson's Integral 
 Calculus" (London: Longmans, Green, & Co.); and numer- 
 ous interesting problems are published with their solutions 
 in "The Mathematical Visitor " and "The Annals of Mathe- 
 matics." 
 
 174. The mean of n quantities is their sum divided by their 
 number. If the number of quantities considered is supposed 
 to increase indefinitely according to some given law, the prob- 
 lem of finding the limiting value approached by their mean 
 usually calls for the Integral Calculus. The mean value of a 
 continuous function of one, two, or three independent variables 
 has been carefully defined in Art. 165, and has been proved to 
 depend upon a line, surface, or space integral. 
 
 (a) Let us find the mean distance of all the points on the 
 circumference of a circle from a given point on the circumfer- 
 ence. 
 
 If we take the given point as origin, the distances whose 
 mean is required are the radii vectores of points uniformly dis- 
 tributed along the circumference of the circle. 
 
 The required mean is, therefore, by Art. 165, equal to 
 
Chap. XV.] MEAN VALUE AND PEOBABILITY. 207 
 
 the quotient obtained by dividing the line integral of r taken 
 around the circumference by the length of the circumference ; 
 that is, 
 
 I rds 
 
 27ra 
 
 The polar equation of the circle is 
 r=2a cos (ft ; 
 ds= 2ad<ji, 
 
 IT 
 
 M=~ r4:a^cos<l>d<l> = —, 
 2iraJji "■ 
 
 the required mean value. 
 
 (6) Let us find the mean distance of points on the surface 
 of a circle from a fixed point on the circumference. 
 
 Here, by Art. 165, the required mean is the surface integral 
 of r taken over the circle, divided by the area of the circle ; 
 that is, 
 
 2a cos (^ 
 
 ira ,/-j»/o Sir 
 
 (c) The problem of finding the mean distance of points on 
 the surface of a square from a corner of the square can be sim- 
 plified slightly by considering merely one of the halves into 
 which the square is divided by a diagonal. 
 
 Here 
 
 M= - I jr. rdrd<t> 
 a^Jo Jo 
 
 = |(^V2+logtan^). 
 
208 IKTEGEAL CALCtTLTJS. [ART. 175. 
 
 (d) As an example of a device often employed, we shall now 
 solve the. problem, To find the mean distance between two points 
 within a given circle. 
 
 If M be the required mean, the sum of the whole number of 
 cases can be represented by {tti^Y^M, r being the radius of the 
 circle ; since for each position of the first point the number of 
 positions of the second point is proportional to the area of the 
 circle, and may be measured by that area ; and as the number 
 of possible positions of the first point may also be measured 
 by the area of the circle, the whole number of cases to be con- 
 sidered is represented by the square of the area ; and the sum 
 of all the distances to be considered must be the product of the 
 mean distance \>y the number. 
 
 Let us see what change wiU be produced in this sum b}' in- 
 creasing 7- by the infinitesimal dr ; that is, let us find d{Trr*M). 
 
 If the first point is anywhere on the annulus 2 in-.dr, which we 
 have just added, its mean distance from the other points of the 
 
 circle is , by (6). 
 
 Therefore, the sum of the new distances to be considered, 
 
 39 J- 
 if the first point is on the annulus, is -^^.tt?-^. S-n-n?)- ; but the 
 
 second point ma}' be on the annulus, instead of the first ; so that 
 to get the sum of all the new cases brought in by increasing 
 r bj- dr, we must double the value just obtained. 
 
 Hence d ( w-r*M) = ifa ^r*d?-, 
 
 Tr'a*M =^P-7r Cr*dr = J^ira', 
 128a 
 
 M = 
 
 457 
 
 175. In solving questions in Probability, we shall assume 
 that the student is familiar with the elements of the theory as 
 given in " Todhunter's Algebra." 
 
 (a) A man starts from the bank of a straight river, and 
 walks till noon in a random du-ection ; he then turns and walks 
 
Chap. XV.] MEAN VALUE AND PROBABILITY. 209 
 
 in another random direction ; what is the probability that he will 
 reach the river by night ? 
 
 Let 6 be the angle his first course makes with the river. If 
 the angle through which he turns at noon is less than -n- — 2 6. 
 he will reach the river by night. For any given value of 6, 
 then, the required probabilitj' is '" ~ - . The probability that 
 6 shall lie between any given value 6„ and O^ + dd is — . 
 
 The chance that his first course shall make an angle with the 
 river between 6q and Oq + dO, and that he shall get back, is 
 
 7r-2g d9 _ {■n--2e)dd 
 
 2ir i^r -TT^ 
 
 As 6 is equally likely to have any value between and — , the 
 required probability, 
 
 •^0 
 
 (7r-2(9)d6l ^. 
 
 (6) A floor is ruled with equidistant straight lines ; a rod, 
 shorter than the distance between the lines, is thrown at ran- 
 dom on the floor ; to find the chance of its falling on one of the 
 lines. 
 
 Let X be the distance of the centre of the rod from the nearest 
 line ; 6 the inclination of the rod to a perpendicular to the paral- 
 lels passing through the centre of the rod ; 2 a the common dis- 
 tance of the parallels ; 2c the length of the rod. 
 
 In order that the rod may cross a line, we must have 
 ccos^ > x; the chance of this for any given value a;„ of x is 
 
 J- cos-i^. 
 
 The probability that x will have the value x^ is — The 
 probability required is 
 
 » = — I cos ^-dx = — 
 
 TrCtv/o C -rra 
 
 This problem may be solved by another method which pos- 
 sesses considerable interest. 
 
I 6xdd = ira. 
 
 4 wo 
 
 XJ TT -^C COS 9 
 ldxde = 2c. 
 
 210 INTEGRAL CALCtTLDS. [ART. 175 
 
 TT 
 
 Since all values of x from to a, and all values of from — — 
 to - are equally probable, the whole number of cases that can 
 arise ma3' be represented b}' 
 
 ■ S' 
 
 The number of favorable cases will be represented by 
 
 ,j7r ^ccosfl 
 -ir, 
 
 2c 
 Hence p = — 
 
 (c) To find the probability that the distance of two stars, 
 taken at random in the northern hemisphere, shall exceed 90°. 
 
 Let a be the latitude of the first star. With the star as a 
 pole, describe an arc of a great circle, dividing the hemisphere 
 into two lunes ; the probabUit}' that the distance of the sec- 
 ond star from the first will exceed 90° is the ratio of the lune 
 not containing the first star to the hemisphere, and is equal 
 
 to ^^3 — ^. The probabilitj' that the latitude of the first star 
 
 77 
 
 wiU be between u. and a + da is the ratio of the area of the 
 zone, whose bounding circles have the latitudes u and a + da 
 respectivelj', to the area of the hemisphere, and is 
 
 2 tto' cos a da , 
 = cos a Ota. 
 
 2 Tra 
 
 Hence p= | " '■^^ ~ "'' cosa da = — 
 
 */0 TT 
 
 (d) A random straight line meets a closed convex curve ; 
 what is the probabilitj' that it will meet a second closed convex 
 curve within the first ? 
 
 If an infinite number of random lines be drawn in a plane, all 
 directions are equallj- probable ; and lines having any given 
 
Chap. XV.] MEAN VALtTB AND PROBABILITY. 211 
 
 direction will be disposed with equal frequency all over the 
 plane. If we determine a line by its distance j) from the origin, 
 and by the angle a which p makes with the axis of X, we can get 
 all the lines to be considered by making p and a vary between 
 suitable limits by equal infinitesimal increments. 
 
 In our problem, the whole number of lines meeting the exter- 
 nal curve can be represented by j \dpda. If the origin is 
 
 within the curve, the limits for p must be zero, and the perpen- 
 dicular distance from the origin to a tangent to the curve ; and 
 for u. must be zero and 2 -it. If we call this number JV, we 
 shall have 
 
 \pda, 
 a 
 
 p being now the perpendicular from the origin to the tangent. 
 
 If we regard the distance from a given point of any closed 
 convex curve along the curve to the point of contact of a tan- 
 gent, and then along the tangent to the foot of the perpendicu- 
 lar let fall upon it from the origin, as a function of the u, used 
 above, its differential is easily seen to be pda. If we sum these 
 differentials from a = to a =• 2 ir, we shall get the perimeter 
 of the given curve. 
 
 J^27r 
 pda = L, 
 
 
 where L is the perimeter of the curve in question. By the same 
 reasoning, we can see that n, the number of the random lines 
 which meet the inner curve, is equal to I, its perimeter. For p, 
 the required probability, we shall have 
 
 I 
 p = -. 
 
 Examples. 
 
 (1) A number n is divided at random into two parts ; find the 
 mean value of their product. ^^^ «^ 
 
 6' 
 
212 INTEGKAL CALCULUS. [Abt. 176. 
 
 (2) Find the mean value of the orcUnates of a semicircle, sup- 
 posing the series of ordinates taken equidistant. Ans. -a. 
 
 4 
 
 (3) Find the mean value of the ordinates of a semicircle, sup- 
 posing the ordinates drawn through equidistant points on the 
 circumference. ^ 2a 
 
 7r 
 
 (4) Find the mean values of the roots of the quadratic 
 
 of ~ax + b = 0, the roots being known to be real, but 6 being 
 
 unknown but positive. < 5 a , a 
 
 _ an -. 
 
 (5) Prove that the mean of the radii vectores of an ellipse, the 
 focus being the origin, is equal to half the minor axis when they 
 are drawn at equal angular intervals, and is equal to half the 
 major axis when they are drawn so that the abscissas of their 
 extremities increase uniformly. 
 
 (6) Suppose a straight line divided at random into three 
 parts ; And the mean value of their product. , a' 
 
 "*■ 60' 
 
 (7) Find the mean square of the distance of a point within a 
 given square (side = 2 a) from the centre of the square. 
 
 Ans. fa^. 
 
 (8) A chord is drawn joining two points taken at random on 
 a circumference ; find the mean area of the less of the two seg- 
 ments into which it divides the Circle. > tto^ a^ 
 
 Ans. — 
 
 4 
 
 (9) Find the mean latitude of all places north of the equator. 
 
 Ans. 32°. 7. 
 
 (10) Find the mean distance of points within a sphere from 
 a given point of the surface. Ans. 4 a. 
 
 (11) Find the mean distance of two points taken at random 
 within a sphere. Ans. f|a. 
 
 (12) Two points are taken at random in a given line a ; find 
 the chance that their distance shall exceed a given value c. 
 
 Ans, 
 
 {^r 
 
Chap. XV. J MEAN VALUE AND PEOBABILITY. 213 
 
 (13) Find the chance that the distance of two points within 
 
 a square shall not exceed a side of the square. Ans. w 
 
 1 
 
 T"- 
 
 (14) A line crosses a circle at random ; find the chances that 
 a point, taken at random within the circle, shall be distant from 
 the line by less than the radius of the circle. , i _ ^ 
 
 (15) A random straight line crosses a circle ; find the chance 
 that two points, taken at random in the circle, shall lie on 
 opposite sides of the line. . 128 
 
 (16) A random straight line is drawn across a square ; find 
 the chance that it intersects two opposite sides. log 2 
 
 ^ IT 
 
 (17) Two arrows are sticking in a circular target; find the 
 chance that their distance apart is greater than the radius. 
 
 Ans. 3V3. 
 
 4,7 
 
 (18) From a point in the circumference of a circular field a 
 projectile is thrown at random with a given velocity which is 
 such that the diameter of the field is equal to the greatest range 
 of the projectile : find the chance of its falling within the field. 
 
 Ans. |-?(V2-1). 
 
 (19) On a table a series of equidistant parallel lines is drawn, 
 and a cube is thrown at random on the table. Supposing that 
 the diagonal of the cube is less than the distance between con- 
 secutive straight lines, find the chance that the cube will rest 
 without covering any part of the lines. 
 
 4a 
 Ans. 1 , where a is the edge of the cube and c the dis- 
 
 tance between consecutive lines. 
 
 (20) A plane area is ruled with equidistant parallel straight 
 lines, the distance between consecutive lines being c. A closed 
 curve, having no singular points, whose greatest diameter is less 
 
214 INTEGKAL CALCTJLTTS. [Akt. 175. 
 
 than c, is thrown down on the area. Find the chance that the 
 curve falls on one of the lines. 
 
 Ans. — , where Z is the perimeter of the curve. 
 
 TTC 
 
 (21) During a heavy rain-storm, a circular pond is formed in 
 a circular field. If a man undertakes to cross the field in the 
 dark, what is the chance that he will walk into the pond ? 
 
Chap. XVI.] ELLIPTIC INTEGEALS. 215 
 
 CHAPTER XVI. 
 
 ELLIPTIC INTEGRALS. 
 
 176. In attempting to solve completely the problem of the 
 motion of a simple pendulum by the methods of I. Chapter 
 VIII. we encounter an integral of great importance which we 
 have not yet considered. The problem is closely analogous to 
 that of the Cycloidal pendulum (I. Art. 119). 
 
 For the sake of simplicity we shall suppose the pendulum 
 bob to start from the lowest point of its circular path with the 
 initial velocity that would be acquired by a particle falling 
 freely in a vacuum through the distance 2/0 i and this by I. Art. 
 114 [1] is V2^o- 
 
 Forming our differential equation of motion as in I. Art. 118, 
 but taking the positive direction of the axis of Y upward, we 
 
 ae ^ ds ^ ' 
 
 ds 
 Multiplying by 2— and integratmg, 
 
 v''=ffj=2g(y,-y). (2) 
 
 If the starting-point is taken as the origin, the equation of 
 the circular path is a^ + y^ — 2ay = 0, whence 
 
 fdsV_^ g' fdyV 
 [dtj 2ay-y\dt)' 
 
 or, determining C, 
 
 and we have 
 
 yj2ay-y' ^t 
 
 ^=y/2g(yo-y). 
 
216 INTEGRAL CALCULUS. [AKT. 176. 
 
 dt = - ^ 
 
 /2g . ^ ivo -y){^ ay -f) 
 
 Integrating, and determining the arbitrary constant, we get 
 
 t = ^ f '^y- (3) 
 
 V29-'° ■^{yo-y){2ay-y') 
 
 as the time required to reach that point of the path which has 
 the ordinate y. 
 
 The substitution of a;^ = ^ reduces (3) to the form 
 
 ^'•'•Nlo-'')('-f;^) 
 
 where the integral is of the form 
 
 dx 
 
 s: 
 
 V(l-a^)(l-fc'a^)' 
 
 (5) 
 
 &^ being positive and less than unity if y^ is less than 2 a. An 
 examination of equation (2) will show that if this is true, the 
 pendulum will oscillate between the two points of the arc which 
 have the ordinate 2/o- 
 
 If ?/o is greater than 2 a, the pendulum will make complete 
 
 revolutions. For this case the substitution of 3?^=-^ in C3') 
 will reduce it to 
 
 t^a^C , -^ (6) 
 
 where the integral is of the form (5) , Ic' being positive and less 
 than unity. 
 
 The time required for the pendulum to reach its greatest 
 height — that is, in the first case, the time of a half- vibration, 
 
 and in the second case, the time of a half -revolution will 
 
 depend upon 
 
 (7) 
 
 X 
 
 » V(l -a^)(l -y^a?) 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 217 
 
 177. The length of an arc of an Ellipse, measured from the 
 extremity of the minor axis, has been found to be (Art. 107) 
 
 - r^=% 
 
 ■0? 
 
 If we replace - by x, (1) becomes 
 
 ~^^^ .dx. (1) 
 
 and the integral is of the form 
 
 s:^ 
 
 ^^ • A^, (3) 
 
 where fc^ is positive and less than unity. 
 
 The length of an Elliptic quadrant depends upon the integral 
 
 U"^. 
 
 ^'^ dx. (4) 
 
 178. It can be shown by an elaborate investigation, for 
 which we have not room, that the integral of any algebraic 
 function, which is irrational tlirough containing under the square 
 root sign an algebraic polynomial of the third or fourth degree, 
 can by suitable transformations be made to depend upon one 
 or more of the three integrals 
 
 F{k,x) = r , ^'^ . [1] 
 
 n (n, k, X) = r — ^==, [3] 
 
 Jo (1+na^) V(l-ar')(l-A;2a^) 
 
 which are known as the Elliptic Integrals of the first, second, 
 and third class respectively. 
 
218 nSTTBGEAL CALCULUS. [Abt. 179. 
 
 fc, which may always be taken positive and less than 1, is 
 called the modulus ; and n, which may be taken real, is called 
 the parameter of the integral. 
 
 K= F(k, 1) = r '^"' [4] 
 
 Jo ^(i-x')(i-k'a?) 
 
 and E=E (&, 1) =£^l^^ . dx, [5] 
 
 are known as the Complete Elliptic Integrals of the first and 
 second classes. 
 
 179. The substitution of a; = sin<^ in the Elliptic Integrals 
 reduces them to the following simpler forms. 
 
 F{ic,<^)=r-=^^= =r^. [1] 
 
 J» Vl— ft^'sin^^ Jo ^<t> 
 
 E (k, <^) = r^Vl - k' sin> . d0 =C^A<t>.d^. [2] 
 
 n in, k, <!>) = f * '^'^ = r ^ 
 
 J» (H-wsin2^)Vl-fc='sin2(^ Jo(l + nsin='^) A<^ 
 
 [3] 
 
 Jo Vl-Ai^sin^.^ Jo A,^ 
 
 E= C y/l-k'sio'^ . d<l> = r*A<^ . d^. [5] 
 
 <^ is called the amplitude of the Elliptic Integral, and 
 A<^= Vl — fc^sin^^ is called the delta of ^, or more simply, 
 delta </), and is regarded as a new trigonometric function : it is 
 always taken with the positive sign, and has an analogy with 
 cos<^. 
 
 For a given value of k, A<^ is easily seen to be a periodic 
 function of ^ having the period tt. It has its maximum value 1 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 219 
 
 when ^ = and when ^ = tt, and its minimum value Vl — fc^ 
 which is usually represented by k' and called the complementary 
 
 modulus, when <f. = -; and A( - + a 1 = A( ^ — aY 
 
 Landen's Transformation. 
 
 180. The approximate numerical value of an Elliptic Integral 
 of the first class, when k and 4> are given, is easily computed 
 by the aid of two valuable reduction formulas due to Landen. 
 
 If in F(k,^)=r--^=^ 
 
 Jo -sjl-k^sm^^ 
 
 we replace </> by ^i, ^i and <^ being connepied by the relation 
 
 . , sin2<f>, ,,. 
 
 tan<i = ; 2:i— , (1) 
 
 A;+cos2<^i ^ ' 
 
 which is easily transformable into either of the following : 
 
 A;sin^= sin(2<^i — <^), (2) 
 
 tan (<^ — <^i) = -^ tan <^i, (3) 
 
 ^ reduces to I 
 
 Vl — k' sin^(^ 1 + «»^» 
 
 C?<^1 
 
 
 {\+k) 
 
 sin^<^ 
 
 which is also an Elliptic Integral of the first class, but has a 
 different modulus and a different amplitude from those of the 
 given integral. 
 
 The steps of the process are as follows : 
 
 From (1) we easily find 
 
 • 2 I sin^2<f>i 
 
 ^ l+A;2 + 2fccos2<^i 
 
 n 7 2 ■ 2 _. • 1 + Zc cos 2 d, 
 
 whence vl — fc^ sin^ ^ = — — 
 
 Vl+&2 + 2fccos2<^i 
 
220 INTEGRAL CALCTTIiUS. [Art. 180. 
 
 Differentiating (1), we get 
 
 2j jj 2(1 +A;cos2<A,) J . 
 secV d<^ = -)--f — -^ d<^j ; 
 
 butfrom (1), ^eeV= (A: + cos2.^0- ' 
 
 hence 2(1 + A. cos 2.^0 
 
 ^ l+fc2 + 2A;cos2<^i ^ 
 
 d<^ ^ 2d(f>i ^ 2d<l>i 
 
 Vl-fc^sin^.^ Vl + fc'+2A;cos2<^i Vl + fc^+ 2A;-4A;sin2^, 
 
 ^ 2 d^i 
 
 ~i+fc /;; 4& ttt' 
 
 (i + kj 
 
 ^j = when <j> = 0, 
 
 hence 
 
 d<h 
 
 r^ d<f> _ 2 r*i 
 
 -'» Vl-A;2sin> 1 + fcJo ~f J]^ 7=' 
 
 -V 1 sin <6i 
 
 \ {1+kf ^^ 
 
 [4] 
 
 and F(k,4>) = j^Fik„ <!.,), 
 
 where fci = -^^, 
 
 1+A; 
 
 and sin(2^i — (^) = ^- sin^. 
 
 fci is less than 1 and greater than A: ; for < 1 reduces 
 
 1 + k 
 
 to 0<(1-Vfc)', which is obviously true, and ^-^ > A 
 reduces to 4>fc(l + fc)^ which is true, since k is less than 1. 
 If <l> is not greater than |, and the smallest value of ^i con- 
 sistent with the relation sin(2<^i — (^) = fc sin<^ is taken, 
 0<<^i<<^. Hence (4) is a reduction formula by which we 
 can raise the modulus and lower the amplitude of our given 
 function. 
 
Chap. XVI.] 
 
 ELLIPTIC INTEGRALS. 
 
 221 
 
 By applying the formula (4) n times, we get 
 
 F{K,<l>n); 
 
 or, since 
 
 :, etc., 
 
 [5] 
 
 1+k l+fci 1+k^ l-l-fc„_i 
 
 —?_ = A 
 
 F(k, <A) = k„ ^ fei-fc.-V-fe„-i j,(fc^^ ^^) ^ 
 
 where 
 
 2 Vifc 
 K = 1 . /"\ and sin(2<^,-^^_i) =A;y_i sin<^^_i. 
 ■I + "'p-i 
 
 If we suppose n in (5) to be indeiinitely increased, we shall 
 
 have [k.] = 1 ; for if we form the series 
 
 {1 -k) + {l -h) + il -k,)+ ... +(1 -k^)+ ..., 
 
 we shall have 
 
 1 2V^ 
 
 1-ft^ l-ft, 1-fc/ l + v^i+fc,' 
 
 which is always less than unitj' ; hence the terms in the series 
 
 must decrease indefinitely as j3 is increased and _ [1— A;„J = 0. 
 
 Since, as we have seen above, <^„ continually diminishes as n 
 increases, but does not reach the value zero, it must have some 
 limiting value $. Hence 
 
 = j sec^di^ = logtan - + — » 
 
 and F(k, <!>) = log tanf"^ + ^l^h-h-k,...^ [G] 
 
 Formulas [5] and [6] lend themselves very readily to numer- 
 ical computation. 
 
222 
 
 INTEGRAL CALCTJLITS. 
 
 [Abt. 181. 
 
 181. Formula [4J, Art. 180, may be used to decrease the 
 modulus and increase the amplitude of a given Elliptic Integral. 
 Interchanging the subscripts, and using (3) Art. 180 instead of 
 (2) Art. 180, we have 
 
 Fik,cl>) = l±^F(k„,l,,), 
 
 where 
 and 
 
 h-. 
 
 1 _ Vl - A;2 
 
 1 + Vl - k^' 
 tan((^i — <^) = Vl — Zs^tani^. , 
 
 [1] 
 
 By repeated application of [1] we get 
 ^{l+k,)...(l- 
 1 - Vl - k\_^ 
 
 F{k, <^) = (1 + fc,)(l +k,)...(l +&„)^!&^, 
 
 where 
 
 k„ 
 
 i + Vi-feVi 
 
 and tan (^y — ^j,_i) = Vl — A;^j,_i tan <^y_i. 
 
 [2] 
 
 limit 
 
 It is easily shown, as in Art. 180, that _ [fc„]=0, and 
 
 limit /^* 
 
 consequently that __ F(k„, <^„) =1 d<j> = ^, where * is the 
 
 n — 00 ^Q 
 
 limiting value approached by <^ as n is increased. 
 
 If ^ = ^. we get from [2], <^i = 7r, <^2 = 27r, ...^„=2»-V; 
 
 hence K=Ffk, ^=^(1+ k,) (1 + k,) (1 + k,) • 
 
 [3] 
 
 Formulas [2] and [3], like formulas [."i] and [6] of Art. 
 180, lend themselves readily to computation. 
 
 With a large modulus, it is generally best to use [5] and [6] 
 of Art. 180 ; with a small modulus, [2] or [3] of the present 
 article will generally work more rapidly. 
 
 We give in the next article the whole work of computing the 
 
 Elliptic Integral ^(-z-, ^) by each of the two methods, and 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 223 
 
 of computing k(——\=f( — , -) by the second method, 
 employing five-place logarithms. 
 
 182. Ff—, -\ Method of Akt. 180. 
 
 Kt^i) 
 
 fc= 0.70712 log&= 9.84949 
 
 l+&= 1.70712 log (1 + fc) = 0.23226 
 
 log ^7^ = 9.92474 
 
 log2 = 0.30103 
 
 colog(l+ A;) = 9.76774 
 
 log X;i= 9.99351 
 
 fti = 0.98518 log&i= 9.99361 
 
 1+^1= 1-98518 log(l+A;i) = 0.29780 
 
 log V^i= 9.99676 
 
 log2 = 0.30103 
 
 colog(l+fci) = 9.70220 
 
 logA;2 = 9.99999 
 
 log &= 9.84949 
 
 log sin- =9.84949 
 
 log sin (2(^1- <^) = 9.69898 
 
 2^1-^ = 30° 0' 3" 
 
 2^1=75° 0' 3" 
 
 ^1 = 37° 30' 2" 
 
 log&i= 9.99351 
 logsin0, = 9.78445 
 
 log sin (2 <t>2 - 0i) = 9.77796 
 
224 
 
 INTEGRAL CALCULUS. 
 
 [Art. 183. 
 
 2<^j-<^i = 36° 51' 3" 
 
 2^2=74° 21' 5" 
 
 ^ = ^2=37° 10' 32" 
 
 ^^ + -=63° 35' 16" 
 4 
 
 log tan f- + i^^ = 0.30393 
 
 log Vfci = 9.99676 
 
 cologVfc =0.07526 
 
 log log tan /'Z: + ^$"1 = 9.48277 
 
 colog/t = 0.36222 
 logJ'(^,^) = 9.91701 
 
 pf^/l, Z\ = 0.82605 
 
 fjL = 0.43429 is the modulus of the common sj'stem of 
 logarithms. 
 
 Vl-fc' = A;' =0.70712 
 
 1-A;' =0.29288 
 
 1+k' = 1.70712 
 
 fci= 0.17157 
 
 1-^1 = 0.82843 
 l+fci = 1.17157 
 
 fei'= 0.98520 
 
 l—fc,'= 0.01480 
 
 l+fci'= 1.98520 
 
 ^2 = 0.00746 
 
 Method of Art. 181. 
 
 iog(l -A;') =9.46669 
 
 colog ( 1 + fc') = 9.76774 
 
 log fci= 9.23443 
 
 log(l-fci) = 
 
 log(l+A;i) = 
 
 logfci'' = 
 
 logfci' = 
 
 : 9.91826 
 : 0.06878 
 : 9.98704 
 
 : 9.99352 
 
 log (1- A;/) = 8.17026 
 
 colog (1 + fc/) = 9.70220 
 
 log fc2= 7.87246 
 
Chap. XVI.] ELLIPTIC INTEGKALS. 225 
 
 1-^2 = 0.99254 log (1-J;2) = 9.99675 
 
 1 + ^2 = 1.00746 log (1 + k^) = 0.00323 
 
 log A;2'^= 9.99998 
 
 lcj= 1 logifcj' = 9.99999 
 
 A;. = 
 
 logfc' = 9.84949 
 log tan ^ = 0.00000 
 
 log tan (<^j — ^) = 9.84949 
 
 ^i-<^ = 35° 16' 53" 
 <^i = 80° 15' 63" 
 
 logfci' = 9.99352 
 logtan(^i = 0.76557 
 
 log tan ((^2 — ^i) = 0.76909 
 
 if>2-<t>i= 80° 7' 17'' 
 .^2=160° 23' 10" 
 
 tan (</)3 — 1^2) =tan^2 
 *=<^3= 2(^2=320° 46' 20" 
 
 i*= 40° 5' 48" 
 23 
 
 = 144348" 
 
 TT = 648000" 
 
 log(|*)" = 
 
 5.15942 
 
 colog7r" = 4.18842 
 log 7r= 0.49715 
 
 logg*) = 
 
 9.84499 
 
226 INTEGRAL CALCULUS. [Akt. 183. 
 
 log (1+A;i) = 0.06878 
 log (1+A;2) = 0.00323 
 
 log('|3<^') = 9.84499 
 
 logpf^, -\ = 9.91700 
 Ff^, -\ = 0.82605 
 
 For Ff~,~\ we have by (3), Art. 181, 
 
 log (1 + h) = 0.06878 
 
 log(l +A;2) = 0.00323 
 
 log TT = 0.49715 
 
 colog 2 = 9.69897 
 
 logPf^, l") = 0.26813 
 
 i.(f, I) ^1.8541 
 
 183. Lande n's Transformation can also be applied to the 
 computation of Elliptic Integrals of the second class, but the 
 task is a more difficult one ; we shall, however, give a brief 
 sketch of the method ; and in so doing we shall applj' it to a 
 more general form 
 
 of which E (&, (fi) is a special case. 
 From Art. 180 we have 
 
 VniFZn^ = l+fecos2.^,_ ^ 
 
 Vl+A;- + 2A;cos2<^i 
 
 cos.^=. - fc + cos2<^, ^ ^ 
 
 Vl+ft- + 2A;cos2^i 
 
 2(l+fecos2.^,) 
 
 [1] 
 
Chap. XVI.] ELLIPTIC INTEGRALS. , 227 
 
 Hence Vl —k^sm^<j> + fc cos ^ = Vl +A;^ + '2fccos2^. 
 
 -'"Vl-fc^sinV 
 
 and 
 
 G{k, ^)-^sin<^ 
 
 a + 
 
 fc^ 
 
 -Vl-fc2sin> ft" 
 
 -^Vl-fc^sin^^ 
 
 d^, 
 
 -r 
 
 yi-Jc'sm'if, k^ 
 Substituting <^i for <j>, this becomes 
 
 ^' — -|(VT^ft2sin2<^ + A;cos«^) 
 
 d<t>. 
 
 0, 
 
 a — ^ COS 2 ^j 
 
 2 r*! 
 
 
 Hence 
 
 \ (1+fc)' ^ 
 
 fC 1 -p A/ 
 
 d^i. 
 
 [2] 
 
 where 
 
 fci = |^, sin(2<^i-<^) = A;sin<^, ai=a-|, A = ^- [3] 
 
 Formulas [2] and [3] enable us to make our given function 
 depend upon one of the same form, but having a greater 
 modulus and 'a less amplitude. A repeated use of [2], together 
 with the j-eductions employed in Art. 180, gives us 
 
228 INTEGRAL CALCULUS. fAKT- 183. 
 
 r-^- vr--^'''^'t■^^''°'^* 
 
 G'nCfcn, <^„), 
 
 [4] 
 
 where 
 
 ^ - 2-^ 
 
 fCfC-^ K^ ' ' ' iCp_\ 
 
 and 
 
 a. = a-ifl+^ 
 
 «, = " — fl i+Tr+TTT"' 'TTT — r~ )' 
 
 Just as in Art. 180 k„ rapidl}' approaches 1 as n is increased ; 
 the limiting value of Qn{\i ^») is then 
 
 limit G„(fc„, </,„) = r*''E!L±A 
 
 ./O CO! 
 
 sin^<^ 
 
 cos<t> 
 
 d<f> 
 
 = (a„ + ^„) log tan ('j + 1^) - )8„ sin <^,. [6] 
 By Art. 180, [5] and [6], 
 
 limit Kyjhhl^ log tanf^ + ^'] = F(&, <^). 
 [4] can thus be written 
 » (A;, ,^) = if (fc, </,) L - f/'n- i + Jl- + ... 
 
 *C \ fCi fC^ rCo 
 
 ^ 2"-' 2" \1 
 
 fcifc2-*„-i A;iA;2fc3...A;„_JJ 
 
 + llsin <^ + A sin <^, + --?L sin <^2 +-i^ sin ,^3 + . 
 
 J. 2»-i . , 2" .1 
 
 + /, , , sin ^„_i sin ^„ . 
 
 [7] 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 229 
 
 If a = 1, and ^ — — ¥, [7] reduces to 
 
 ^ 2"-^ 2" \1 
 
 — k\ sin <|!» + — J- sin <^i + sin <;!>, + •«• 
 
 2 . ^ , 2" 
 — ^ sin <^i -i ^^ , 
 
 2 Vfc 
 where A;^^ /"' , and sin (2^^- ^,^1) = Visin^y-i- [9] 
 
 By Formulas [8] and [9] an Elliptic Integral of the Second 
 Class may be computed without difficulty. 
 
 184. Formula [2], Art. 183, may be used to decrease the 
 modulus and increase the amplitude of an Elliptic Integral. 
 Interchanging the subscripts, we have 
 
 G (k, <f.) = ^^[Gi {\, .^1) - 1 sin <^i1 ; 
 or, since ^=-> (Art. 183 [3]), 
 
 G{k,4>)='^-^ G'i(fc„.^,)-|sin,^,l [1] 
 
 where 
 
 fci = ^^(^^, tan(<^i-<^) = Vr=F tan.^, ai=a+f, A=^. 
 1 + Vl— A^ 2 2 
 
 [2] 
 
230 INTEGRAL CALCULUS. [Abt. 184. 
 
 A repeated use of [1] gives 
 
 '»]. [3] 
 
 , 1+k, l+ki 1+Jc. o • J I 
 
 where ^^^ p hhh-K ^ 
 
 2" 
 
 and a„=a + j-i8A + ^ + ^ + ^^^ + -+ ^'^''"^''-' Y 
 Just as in Art. 181 limit fc„ = 0, therefore limit )8„ = and 
 
 a„ci!(^ = a„<^„. 
 
 By Art. 181, [2], i±^ . i±^ . . . i±A ^„ = 2?- (fc, ^), 
 
 By Art. 180, [5], i±A = 2^i, 1±A2 = 2^, etc. 
 Hence [3] becomes 
 
 If a = 1 and /3 = —!(?, [4] reduces to 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 231 
 
 where 
 
 1 M I 1 |" "'i"'2 I 'h'^i^'S I \\ 
 
 2^ 2 22 "^ 2' VJ 
 
 [^> sin <!>, + ^ sin.^, + ^l^ sin .^3 + -] [5] 
 
 K = , I \ and tan (<^p — ^p_i) = Vl — fc^_i ■ tan <^p_i. 
 
 1+ Vi - r^., 
 
 [6] 
 
 We have seen in Art. 181 that if ^ =- , <^^ = 2''-'^7r. 
 
 Therefore, for a complete Elliptic Integral of the second 
 class we have 
 
 Formulas [.'i] and [7] are admirably adapted to computation. 
 "We give in the next article the work of computing 
 
 E( — , - ) by each of the methods just given, and of com- 
 
 ^ ^ *7V2 ^\ 
 puting E ( — ) o j ^y the second method ; using, as far as 
 
 possible, the values already employed or obtained in Art. 182. 
 185. e(—,'^- Method of Art. 183. 
 
 '■m 
 
 Here, as we have seen in Art. 182, if we carry the work onlj 
 to five decimal places, ^2= 1, and our working formula will be 
 
 E{k,<i>)=^F{k,<i>)\\+k(l-^ 
 
 r 2 2^ n 
 
 — k\ sin<^ H sin<^i — — — -sin ^2 • 
 
 L yfk -Jkk^ J 
 
232 
 
 INTEGRAL CALCULUS. 
 
 [Art. 186. 
 
 log 2 = 
 
 logk-- 
 
 colog fci : 
 
 log(l+.-^^) = 
 
 logJ'i 
 
 V2 ^■ 
 2'4 
 
 lOgfc: 
 
 log sin 4> = 
 
 : 0.30103 
 : 9.84949 
 : 0.00649 
 
 0.15701 
 
 : 9.43391 
 
 : 9.91701 
 9.35092 
 
 : 9.84949 
 : 9.84949 
 
 9.69898 
 
 log2 = 0.30103 
 
 logVA; = 9.92474 
 
 log sin<^i = 9.78445 
 
 0.01022 
 
 log2^ 
 
 logVfc = 
 colog Vfci = 
 log sin <l>2 ■• 
 
 : 0.60206 
 = 9.92474 
 : 0.00324 
 = 9.78122 
 
 0.31126 
 
 = 1.43553 
 = 1.70712 
 
 -(fi 
 
 2fc 
 
 1+k 
 1 + fc -2^ = 0.27159 
 
 21c 
 
 l+fc_±p:) = 0.22435 
 
 %;sin(^ = 0.5 
 
 2fc^ 
 
 sin <^i== 1.0238 
 
 -?^ sin ^j= 2.0477 
 
 Vfcfci 
 
 ■kl sm(j>-\ — - sin <^i — 
 
 2^ 
 
 sin <^2 
 
 V2 V 
 
 E 
 
 / V2 u\ 
 
 = 0.5239 
 = 0.22435 
 = 0.74825 
 
Chap. XVI.] 
 
 ELLIPTIC INTEGRALS. 
 
 233 
 
 El—jjy Method of Akt. 184. 
 fca = 0. Therefore our formula is 
 
 E{k, <!>) =F{k, ,^) Ti _ |Yi + 1 + ^^1 
 
 4- fc (^ sin <^, + ^:l^ sin ^A 
 
 V 2 
 
 logA;i= 9.23443 
 
 log A:2= 7.87246 
 
 colog4 = 9.39794 
 
 6.50483 
 
 22 
 
 
 2 = 0.00032 
 = 0.08578 
 
 1+^ + ^2=1.08610 
 2 2" 
 
 |('l+| + ^^) = 0.271525 
 log 0.728475 = 9.862415 1 -^fl +^ + ^\ = 0.728475 
 
 = 9.91700 
 
 j,/V2_ 
 
 9.779415 i^/^^j (0.728475)= 0.60178 
 
 log A; =9.84949 
 
 log ^=9.61722 
 
 colog2 = 9.69897 
 
 log sin .^1 = 9.99370 
 
 9.15938 
 
 ^sin ,^1 = 0.14434 
 
234 INTEGRAL CALCULUS. [Akt. 185. 
 
 logfc= 9.84949 
 
 log -^=9.61722 
 
 log Vfc2= 8.93623 
 
 colog4 = 9.39794 
 
 log sin <^2= 9-52592 
 
 7.32680 *^^2gin^^^ 0.00212 
 
 2^ 
 
 ]e (^ sin ^1 + ^^ sin <^2 ) = 0. 14646 
 
 (f sin<^.+^sin<^.) 
 
 fC^, -"] (0.728475) = 0.60178 
 
 E 
 
 .sf— ,-\ Method op Art. 184. 
 V 2'2; 
 
 ^ " ^ = 0.74824 
 
 l-^('l+|' + ^^ = 0.728475 log0.728475 = 9.862415 
 
 \ogF(^,'^ = Q.2mn 
 
 ^(^'i) = '-''°' Iog^(^:f,^) = 0.13054 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 235 
 
 f 186/ An Elliptic Integral of the first or second class, whose 
 amplitude is greater than -, can be made to depend upon one 
 whose amplitude is less than -, and upon the corresponding 
 Complete Elliptic Integral. 
 
 We have 
 
 TT TT ir ff 
 
 JoAcj} Jo A<j> ^n^A<^ J^A<f> 
 
 2 2 
 
 In C^ let <^ = TT - li ; 
 
 2 
 
 then d^ = —d\f; and A^ = Vl — A;^ sin^<^ = Vl — A;^ sin^i/^ = Ai/r, 
 
 TT !r 
 
 and we have T d^ _ _ T dt/^ _ _ T d^ _ C'^ — x 
 
 J„^<l> »Ar^>/' •^,r'^^ •^" ^^ 
 
 2 2 2 
 
 Hence ^i.(;^, .)^pg= 2^ [1] 
 
 niT+p 
 
 
 d<t> 
 ^ 
 
 In ( .^ let <^ =p7r + i/- ; then d(#. = dij/, and A(^ = Ai/-, 
 
 A<^ 
 
 and we have ' f^ = r#= fg = 2ir. 
 J A<i> »/ Aii J A<i 
 
 (j)+i)ii 
 
 -,.^<^ -i^"^ S.^*^ 
 
236 INTEGRAL CALCULUS. [Abt. 186. 
 
 MT+p 
 
 The substitution of ;^ f or ^ — nir in \ — gives us 
 
 J A(h J All/ J Ad> 
 
 nir+p p p 
 
 ^^A<t. JA^ JA.^ 
 Therefore i?' (fc, titt + p) = 2 n^+ F (&, p) . [2] 
 
 In like manner it can be proved that 
 
 F{k,nT-p) = 2nK-F{k,p), [3] 
 
 E{le,nw + p)=2nE + E(Jc, p), [4] 
 
 E{k,n7r-p) = 2nE-E(k,p), [5] 
 
 where E = E(k,-\ is the complete Elliptic Integral of the' 
 
 second class. 
 
 A table giving the values of the Elliptic Integrals of the 
 first and second classes for values of the amplitude between 
 
 and - is, then, a complete table. 
 
 Such a table, carried out to ten decimal places, is given by 
 Legendre in his " Traitd des Fonctions Elliptiques.'' "We give 
 in the next article a small three-place table. 
 
 It must be noted that the first column gives -F(0, <^) and 
 
 E {0, <j>), that is, I d<^ = <^ ; and that the last column gives 
 F{1, <j>) and£(l, <l>), that is, log tanj -+ * Jand sin^. 
 
 The complete Elliptic Integrals, 
 
 K=F(k,^ and E=E(k, A 
 are given in the last line of each table. 
 
Chap. XVI.] 
 
 ELLIPTIC INTEGRALS. 
 
 287 
 
 O 
 
 00 00-3 
 
 cnooi 
 
 o 
 
 o wo 
 
 cnooi 
 o 
 
 
 tOtOM 
 
 WOW 
 
 o 
 
 M 
 
 O WO 
 o 
 
 ■e- 
 
 1— ' 
 
 in 
 
 t— ' 
 
 i—i 
 In 
 Oi 
 
 h-i 
 
 In 
 
 CO 
 GO 
 
 I—" 
 On 
 
 1— ' 
 O 
 
 1.309 1.312 1.321 1.336 
 1.396 1.400 1.410 1.427 
 1.484 1.487 1.499, 1.519 
 
 1.047 1.049 • 1.054 1.062 
 1.134 1.137 1.143 1.153 
 1.222 1.224 1.232 1.244 
 
 0.785 0.786 0.789 0.792 
 0.873 0.874 0.877 0.882 
 0.960 0.961 0.965 0.972 
 
 0.524 0.524 0.525 0.526 
 0.611 0.611 0.612 0.614 
 0.698 0.699 0.700 0.703 
 
 0.262 0.262 0.262 0.262 
 0.349 0.349 0.349 0.350 
 0.436 0.436 0.437 0.438 
 
 0.000 0.000 0.000 0.000 
 0.087 0.087 0.087 0.087 
 0.175 0.175 0.175 0.175 
 
 ■io 
 
 S'll 
 
 CD O 
 
 S'll 
 
 H- ' o 
 CO ?:^ 
 
 S'll 
 
 1— > 
 OJ 
 
 1— ' 
 
 IS) 
 
 1.357 1.385 1.426 
 1.452 1.485 1.534 
 1.547 1.585 1.643 
 
 1.074 1.090 1.112 
 1.168 1.187 1.215 
 1.262 1.285 1.320 
 
 0.798 0.804 0.814 
 0.889 0.898 0.911 
 0.981 0.993 1.010 
 
 0.527 0.529 0.532 
 0.617 0.620 0.624 
 0.707 0.712 0.718 
 
 0.262 0.263 0.263 
 0.350 0.351 0.352 
 0.439 0.440 0.441 
 
 0.000 0.000 0.000 
 0.087 0.087 0.087 
 0.175 0.175 0.175 
 
 a II 
 S'll 
 s'll 
 
 W O 
 
 (— * 
 00 
 
 {^ 
 
 4^ 
 
 is 
 
 OJ 
 
 CO 
 CO 
 Oi 
 
 8 
 
 1.488 1.566 ^.703 2.028 
 1.608 1.705 1.885 2.436 
 1.731 1.848 2.077 3.131 
 
 1.142 1.178 1.233 1.317 
 1.254 1.302 1.377 1.506 
 1.370 1.431 1.534 1.735 
 
 0.826 0.839 0.858 0.881 
 0.928 0.947 0.974 1011 
 1.034 1.060 1.099 1.154 
 
 0.536 0.539 0.544 0.549 
 0.630 0.636 0.644 0.653 
 0.727 0.736 0.748 0.763 
 
 0.263 0.264 0.264 0.265 
 0.353 0.354 0.355 0.356 
 0.443 0.445 0.448 0.451 
 
 0.000 0.000 0.000 0.000 
 0.087 0.0S7 0.087 0.087 
 0.175 0.175 0.175 0.175 
 
 DO ?7- 
 
 S'll 
 
 ^l^ O 
 
 2. S" 
 II 
 
 O 00 
 on ?7- 
 
 S'll 
 
 ®p 
 
 otO 
 
238 
 
 INTEGKAL CALCULUS. 
 
 [Art. 187. 
 
 
 . o 
 
 oS 
 II a 
 
 dS 
 d:S 
 
 .11.5 
 
 0.000 0.000 0.000 0.000 
 0.087 0.087 0.087 0.087 
 0.174 0.174 0.174 0.174 
 
 0.260 0.260 0.259 0.259 
 0.346 0.345 0.343 0.342 
 0.430 0.428 0.425 0.423 
 
 0.512 0.509 0.505 0.500 
 0.593 0.588 0.581 0.574 
 0.672 0.664 0.654 0.643 
 
 0.748 0.737 0.723 0.707 
 0.823 0.808 0.789 0.766 
 0.895 0.875 0.850 0.819 
 
 0.965 0.940 0.907 0.866 
 1.033 1.001 0.960 0.906 
 1.099 1.060 1.008 0.940 
 
 1.163 1.117 1.053 0.966 
 1.227 1.172 1.095 0.985 
 1.289 1.225 1.135 0.996 
 
 r-A 
 
 K 
 
 i-H 
 f-H 
 
 00 
 CO 
 
 1—t 
 
 LO 
 
 CO 
 r-5 
 
 Om 
 
 II a 
 
 dg 
 
 II c 
 .« ■» 
 
 ^^ 
 
 0(M 
 
 II .S 
 
 0.000 0.000 0.000 
 0.087 0.087 0.087 
 0.174 0.174 0.174 
 
 0.261 0.261 0.261 
 0.348 0.347 0.347 
 0.434 0.433 0.431 
 
 0.520 0.518 0.515 
 0.605 0.602 0.598 
 0.690 0.685 0.679 
 
 0.773 0.767 0.759 
 0.857 0.848 0.837 
 0.939 0.928 0.914 
 
 1.021 l.OOS 0.989 
 1.103 1.086 1.063 
 1.184 1.163 1.135 
 
 1.264 1.240 1.207 
 1.344 1.316 1.277 
 1.424 1.392 1.347 
 
 .— 1 
 -^ 
 I-H 
 
 NO 
 
 1—5 
 
 s 
 
 to 
 
 II .5 
 
 -^ 'So 
 
 T^ 
 
 O rH 
 II C 
 
 .« '5 
 
 o« 
 
 II .s 
 
 OS. 
 
 II c 
 
 0.000 0.000 0.000 0.000 
 0.087 0.087 0.087 0.087 
 0.175 0.175 0.174 0.174 
 
 0.262 0.262 0.262 0.262 
 0.349 0.349 0.349 0.348 
 0.436 0.436 0.436 0.435 
 
 0.524 0.523 0.523 0.521 
 0.611 0.610 0.609 0.607 
 0.698 0.698 0.696 0.693 
 
 0^ Ti- 00 
 ir^co 0^ 
 
 ddd 
 N c^ 'T! 
 
 OO \0 lO 
 ir^oo ON 
 
 ddd 
 
 CO t^ uo 
 
 1-^00 ON 
 
 ddd 
 
 ly^ PO o 
 00 !>. NO 
 r^ 00 On 
 
 odd 
 
 1.047 1.046 1.041 1.032 
 1.134 1.132 1.126 1.116 
 1.222 1.219 1.212 1.200 
 
 1.309 1.306 1.297 1.283 
 1.396 1.393 1.383 1.367 
 1.484 1.480 1.468 1.450 
 
 PO 
 PO 
 
 to 
 
 1—5 
 
 -^ 
 
 LO 
 LO 
 
 I-H 
 
 to 
 
 p-( 
 
 !>• 
 
 to 
 
 I-H. 
 
 -e- 
 
 o lo o 
 
 
 rH(NN 
 
 O iO o 
 CO CO ^ 
 
 o 
 
 looio 
 
 Tt* U5 »0 
 
 o 
 
 oino 
 
 CD CD t^ 
 
 O 
 
 lo o in 
 
 »> 00 OT 
 
 O 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 239 
 
 Addition Formulas. 
 
 188. The Elliptic Integrals, F(k, x) and E{k, x), may be 
 regarded as new functions of x, defined by the aid of definite 
 integrals ; namely, 
 
 F(k,x)= I , , 
 
 see Art. 178, [1] and [2]. 
 
 We have seen how we may compute their values to any 
 required degree of approximation when k and x are given. 
 It remains to study their properties. 
 
 We are familiar with other and much simpler functions which 
 may be defined as definite integrals, and whose most important 
 properties can be deduced from these definitions. 
 
 J'^^dx 
 — , sin~*a; as 
 la; 
 
 J ^ tan~^a; as ( -, and the theory of these func- 
 
 Vl-CB^ Jo l+af 
 
 tions may be based upon these definitions. For instance, the 
 
 fundamental property of the logarithm is expressed by what is 
 
 called the addition formula, 
 
 loga; + log2/ = log(x2/), 
 
 and the whole theory of logarithms may be based on this 
 property ; and there are addition formulas for the other func 
 tions defined above ; namely, 
 
 sin~'a; + sin"*?/ = sin~'(a;Vl — 2/^ + y Vl — x') , 
 
 tan~'a; + tan~*y = tan"' ( — ^tX )• 
 VI -xyj 
 
240 INTBGKAL CALCULUS. [Abt. 188. 
 
 These three important formulas are usually obtained by more 
 or less elaborate methods involving the theory of the functions 
 which are the inverse or anti-functions of the logx, the sin~'a;, 
 and the tan "'a;, that is, of e", sinx, and tancc; but they may 
 be obtained without difficulty from the definitions of log a, 
 sin^'a;, and tan"' a;, as definite integrals. 
 
 Take first ' 
 
 loga;=J^ 
 
 X 
 
 Let us determine y in terms of x, so that 
 
 logo; + logy = log c, (1) 
 
 where c is a given constant. 
 
 Since logy= ( -^, 
 
 J\ y 
 
 if we differentiate (1), we have 
 
 dx , dy rt 
 
 X y 
 
 or ydx + xdy = 0. (2) 
 
 Integrate (2), and we get 
 
 (ydx+ (xdy= C. (3) 
 
 Simplify the first member of (3) by integration hy parts; 
 
 xy—i xdy -\-xy — i ydx = C, 
 
 or 2xy — i {xdy -f- ydx) = C. 
 
 Reducing by the aid of (3) , 2xy=C, 
 or xy=Ci, (4) 
 
Chap. XVI.] ELLIPTIC INTBGBALS. 241 
 
 where Oi is an undetermined constant. To determine Oj, let 
 a; = 1 in (4), and we have y=Gi when x=\; let a; = 1 in (1) , 
 
 J^'da; 
 — = 0, log 2/= log c, and y = c, when x=\. 
 1 X 
 
 Therefore Ci = c, and xy = c. Consequently y=- \b the 
 
 X 
 
 required value of y, and we have (1) 
 
 logo; + log- = logc. 
 
 X 
 
 We can express this relation more neatly by replacing c by 
 its value xy, and thus we reach our required addition formula 
 
 log x + \ogy = \og{xy). [5] 
 
 189. The addition formula for the sin""' can be deduced in 
 exactly the same way. We wish to determine y so that 
 
 8in~'a; + sin~'2/ = sin~'c. (1) 
 
 We have sin~^a;= | , sin~'y= | — ^ 
 
 Jo -Ji—x' Jo Vl — / 
 
 Differentiate (1). 
 
 -^ + -^==0, (2) 
 
 or s/l —y^ ■ dx+ Vl —x' ■ dy = 0, 
 
 C-yjT^^ ■ dx+ fVl-a^ -dy^C. 
 Integrate by parts, and 
 
 or, reducing by (2) , 
 
 a, Vn^^ + 2/ Vn^ = G. (3) 
 
242 INTEGRAL CALCULTJS. [Aet. 189. 
 
 To determine O, we have from (3) y = C when a; = 0, and 
 
 — _^^^^ = 0, when 
 » Vl— a^ 
 a; = 0. Hence G = c, and x Vl — y" + y^\ — a? = c, and, finally, 
 
 sin"*a: + sin^'j/ = sin""' (a;Vl —y' + y\/l — a;^). [4] 
 
 To get an addition formula for the tan"^, a slight device is 
 required, that of dividing the differential equation correspond- 
 ing to (2) b3- 1 — a^y". 
 
 As before, let 
 
 tan~*a; + tau'^y = tan~*c, (5) 
 
 where tan~'a;= | ■„, 
 
 Jo l+x' 
 
 and tan~'y= I — ^ — 
 
 Jo 1+/ 
 
 dx , dy _f. 
 l + a^ + TT?~ ' 
 
 or il+y')dx + {l+3f)dy = 0. (6) 
 
 Divide by 1 — x^y^ and integrate. 
 
 Integrate by parts. We have 
 
 and 
 
 ^•rr^^= (i-%^y ^<^+^')^'"+'^'y<^+^)'^y^' 
 
 l+y^ 1+a)^ _x + y 
 
 l-x^y^^" \-a?y^ l~xy' 
 
Chap. XVI.] ELLIPIIC INTEGKALS. 243 
 
 Hence 
 
 Therefore, by (6), -^-+1-=C'. (7) 
 
 1-xy 
 
 To determine O, we have from (7) y = C'when x=0, and 
 
 -=0 when 
 
 1 —or 
 x = 0. 
 
 Hence C=c, and "'"^^ =c, 
 
 1— a^ 
 
 and, finally, tan-'a; + tan-»2/ = tan-^f-^-^^Y [8] 
 
 \l-xyj 
 
 190. To get an addition formula for F{k, x), as before 
 let F(7c,x)+F{k,y) = F{k,c), (1) 
 
 where F(k,x)= I - , 
 
 •>'» V(l-a^)(l-fc''ar') 
 
 and F(k, y)= C ^^ —— • ' 
 
 ^"^ - A- ^y =0. (2) 
 
 or 
 
 V(l-/)(l-F2/2).da;+V(l-a^)(l-fc'a^) ■dy = 0. (3) 
 Divide by 1 — l<?3?y^ and integrate. 
 
 J l-lc'x'f J l-k'x'y^ ^ 
 
244 INTEGRAL CALCULUS. [Akt. 190. 
 
 Integrate by parts. We have 
 
 V(l-y^)(l-A;V) _ y .rox^.^, ^ 
 
 • (H-fc2)(l+A:'a^/)] ^ 
 
 ^l(l-f){X-l^f) 
 
 + 2¥xy^J (1 -y^){l -Ify') ■ dx\; 
 
 
 Vci-a^Jll-A'ai^) 
 
 + 2*2x2/7(1 -a^)(l -fc'a^) • dy\- 
 Hence 
 
 l-fc2a^y2 
 
 r d» I (^y "I 
 
 + 2Jc'xy [V(l - 2/0 (1 - fe'2/') • dx 
 
 Reducing, by the aid of (2) and (3) , we have 
 
 l-Jc'x'f ~ ^*' 
 
Chap. XVI.] ELLIPTIC INTEGKALS. 245 
 
 To determine O, from (4) 2/ = C when a; = 0, and from (1) 
 y = c when a; = 0. Therefore C = c^ and we get 
 
 F{k,x)-\.F{k,y) 
 
 = wik W( l-y^)(l-feV)+yV(l-r')(l-fc^r') \ . , 
 
 V ' 1 - y'x'y^ r ^^ 
 
 our required addition formula. 
 
 An addition formula for -E(fc, x) cau be obtained in very 
 much the same way, but the work 13 rather complicated, and 
 it is better to use a method which will be explained later. 
 
 THE ELLIPTIC FUNCOUONS. 
 
 191. We have just seen that there is an analogy between 
 the Elliptic Integral i?'( A;, a;) , and the familiar functions log a;, 
 8in"^a;, and tan"^a;; and we know that the theory of these 
 functions is ultimately connected with that of their inverse 
 functions, log"'w or e'% sinw, and tanw; and, indeed, that the 
 latter are so much simpler than the former that it is customary 
 to regard them as the direct functions, and the logarithm, the 
 anti-sine, and the anti-tangent as the inverse functions. 
 
 For example : the first three addition formulas just obtained 
 are much simpler when we express them in terms of the direct 
 functions, and thej' become 
 
 log"'(»f + v) = log^'w • Xog'^v, 
 or e'"+"' = e" • e', [1] 
 
 sin(M-fi)) =sinM Vl — sin^'y -|-sin^) Vl — sin^w; 
 
 or 8in(M-|-i') = sinwcos'y-l-cositsin'y, [2] 
 
 , , , \ tan M -4- tan v tot 
 
 tan (u + v) = ■ ; [3j 
 
 1 — tanw • tanw 
 
 and in this form they seem to better deserve the name of 
 addition formulas. 
 
246 INTEGRAL CALCULUS. [Art. 192. 
 
 In the same way the addition formula for F(k, x) can be 
 more simply written in terms of the function which we might 
 naturally represent by F^^u (mod. A;) ; and, as we might 
 expect, this function has many interesting and important prop- 
 erties which well deserve investigation. 
 
 Since in most of the work which follows we shall generally 
 employ the same modulus throughout, we shall not take the 
 trouble to write it except in the few cases where its omission 
 might give rise to confusion, and then we shall put (mod. h) 
 after the function, as above with F-^u (mod. k), or we shall 
 write it more briefly as F~ ^ (u, k). 
 
 192. In Arts. 178 and 179 we have adopted two forms of 
 notation for an Elliptic Integral of the first class, F{k, x) and 
 FQc, 4,); 
 
 F{k, x)= r ^^ 
 
 -'" V(l-a;'^)(l-A;V) 
 
 '^o Vl— fc^sin^i^ '^0 ^* 
 
 where x = sin <^, s/l —a^= cos <f>, 
 
 and Vl - A;V = Vl - fc^sin^i^ = A<^. 
 
 If we let u = F{k,x) =F(k,>l>), 
 
 we have in Art. 179 called <^ the amplitude of u, and sin<^, 
 cos<^, and A^ may be called the sine, the cosine, and the delta of 
 the amplitude of u ; and <^, sin^, cos<^, and A<^ may be written 
 amw, sinamw, cosamw, and Aamw, or, more briefly, amw, snw, 
 CUM, and dnw ; and may be read amplitude w, sine amplitude m, 
 cosine amplitude m, and delta amplitude m. Formulating, we 
 have 
 
 u = F(k,x) = F{k,fl>), ) 
 
 <t>= am?<, 
 
 a; = sin^=snM, f- [1] 
 
 Vl —aP = cos <^ = cnw, 
 Vl-fcV=A<^ = dnM, 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 247 
 
 SUM, cntt, diiM, are trigonometric functions of ^, the ampli- 
 tude of u, but they may be regarded as new and somewhat 
 complicated functions of u itself, and from this point of view 
 they are called Elliptio Functions of m. 
 
 am M also is sometimes called an Elliptic Function ; and there 
 are various allied functions that are sometimes included under 
 the general title of Elliptic Functions. We shall, however, 
 restrict the name to snw, cum, and dnw. Thej- have an analogy 
 with trigonometric functions, and have a theory which closely 
 resembles that of trigonometric functions, and which we shall 
 proceed to develop. It must, however, be kept in mind that 
 the independent variable u is not an angle, as in the case 
 of the trigonometric functions. 
 
 Of course, with our notation, u^=F(k, a;) = sn"^(a;, k), or 
 u = F (k, (ji) — am- ' (<^, k). 
 
 The fundamental formulas connecting the Elliptic Functions 
 of a single quantity follow immediately from the definitions 
 [1] , and are 
 
 sn^M + cn^M = l, [2] 
 
 dn'u + ¥sa'u = l, [3] 
 
 = dnw, 
 
 du 
 
 [4] 
 
 du 
 
 [5] 
 
 '^«^"- suu.dnu, 
 du 
 
 [6] 
 
 ddnu_ ;fc2sn„.enM, 
 
 [7] 
 
 du 
 The only one of this set which needs any explanation is [4] 
 
 We have «=J -^, 
 
248 
 hence 
 
 and, finally, 
 
 Since 
 we see that 
 
 INTEGRAL CALCULUS. 
 
 A<f> duu 
 
 [Art. 193, 
 
 daxau 
 du 
 
 = dnu. 
 
 
 r-^d±^ r* d (-<!>) _ r 
 
 Jo Ad) Jo A ( — d)) Jo 
 
 (-</>) 
 
 ani( — It) = — am?i, 
 sn ( — w) = — snM, 
 en ( — m) =cnM, 
 dn(— m) =dnM, 
 
 [8] 
 
 That is, snu is an odd function of u, and cnw and dnw are 
 even functions of u. 
 
 Since 
 we have 
 
 Jo Acj) 
 
 am(0) = 0, ~) 
 sn(0) =0, 
 cn(0) =1, 
 dn(0) =1, 
 
 [9] 
 
 193. Our addition formula for the sine amplitude flows 
 
 immediately from [5], Art. 190. Let u = F{k,x) and 
 v = F(7c,y), and take the sine amplitude of each member of 
 [5], Art. 190; we get 
 
 , , s snw . cniJ . dnw + cnw. sn-y . dn?« 
 sn (^u + v) = — — . 
 
 1 —k^ . szru . sn^v 
 
 If now we replace v by —v, and simplify by [8], Art. 192, 
 we have 
 
 sn ((4 — v) = 
 
 SUM . cnw. . dnv — cnw . snv . dnw 
 
 and the two formulas can be combined if we use the sign ± ; 
 
Cmp. XVI.] ELLIPTIC INTEGRALS. 249 
 
 „ /» J- \ snu.cnv .dav ±ciiu .Bnv .dnu rn 
 
 sniwiv) = — — • ril 
 
 From [1], with the aid of [2] and [3], Art. 192, we can get, 
 after a rather elaborate reduction, the addition formulas for 
 en and dn. 
 
 / , N cnu . CUV ^: snu . SQV . dnu . dnv ra-i 
 
 ^■^ ^" * ^) = 1-A:^.sn^..sn^^ ^^^ 
 
 , , , s dnw . dnv q: 7iP . snu . snv . enw . cnw tot 
 dn (m ± v) = -3^— [3] 
 
 From [1], [2], and [3J a large number of formulas can be 
 
 readily obtained. "We give only those for sn ; there are 
 similar ones for en and dn. 
 
 , , . , , ,, 2snM. cn'w . dnv rA-i 
 
 sn (m + v) + sn (m — -y) =- -• [4] 
 
 , . , , V 2cnM . sni; . dnw rcn 
 
 sn (u + v) — sn (u — v) = j- [5] 
 
 y sn(u + v). sn {u — v) = —• [6 J 
 
 ^ 1 — fc^- sn^M • sn^-y 
 
 1 + sn (m + v) .sn{u — v) = j— [7] 
 
 ^ l — k'.sn'u.sn^v 
 
 l+A;2sn(M + v).sn(M-^) = ■ ^ [8] 
 
 ^ 1—k'. sn^M . sn^v 
 
 r, . / . xnn , / \n (cu W + SUM . dnt;)'' ^a^ 
 
 [l + sn(M+'y)][l + sn(M— y)] = -i; -j- — ^- [9] 
 
 From [2] and [3] comes the useful formula 
 
 cn(M + 'y) = cnu. cni; — snw . snv . da (u + v). [10] 
 
250 INTEGKAIi CALCULUS. [ART. 194. 
 
 194. If in formulas [1], [2], and [3] of Art. 193 we let 
 
 v = u, we get the following formulas for sn2M, en 2m, and 
 
 dn2M : 
 
 „ 2snM.cnM.dnM rn 
 
 sn2M= — — , [IJ 
 
 1 — Ar sn*M 
 
 „ _ cn^M — sn^M . dn^M _ 1 — 2sn^M + fc^sn^M pn-, 
 1— A^sn^M 1— Zc^sn^M 
 
 , n dn^M — fc^. sn^M. cn^M 1 — 2A;^sn^?^ + fc^sn*M ron 
 
 dn 2m = r — = — [3] 
 
 1— A;*'sn*M 1— Arsn*M 
 
 From these come readily 
 
 1 n 2sn^M.dn^M r,-i 
 
 ^-^°^"= l-fe^sn-M ' W 
 
 1 1 o 2cn^M rc-i 
 
 , , „ 2fc^sn^M. cn'^M p„-, 
 
 l-dn2M= ^_^^^^^ , [6] 
 
 ■« I J o 2dn^M rm 
 
 1 + dn 2m = ■• [71 
 
 195. Replacing u by -, and dividing [4] by [7] and [6] 
 by [5], Art. 194, we have 
 
 ^^,M^l-cnM^ 1-dnM 
 
 2 1+dnM fc2(l + cnM) ■- -" 
 
 ^^u _ dnM+ enw _ — fc'^ + FcnM +dnw 
 2 l + duM fc2(l + enM) "' 
 
 ^jj2W _ fe'^ + dnM + fc^cnM _ (cnM + dnw) 
 2 1 + dnM ~ (1 + en m) ' 
 
 [2] 
 [3] 
 
 where V=\ — 'k^, and is the square of the complementary 
 modulus. 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 251 
 
 From [1], [2], and [3], we can get without difficulty the set 
 
 sn^!f= duM-cnw C , 
 
 2 k^ + dnu-Jc'cau '" -^ 
 
 jj^2M ^ k"{l+cnu) p., 
 
 2 fc'^ + dnM-fc^cnw' '■ "^ 
 
 dn2^- ^'-(l+dnw) P -, 
 
 2 ft'^ + dnw-Ai^cnM ■- -" 
 
 Numerous additional formulas can be obtained by the exer- 
 cise of a little ingenuity, but we have given the most useful and 
 important ones, and they form a set as complete as the usual 
 collections of trigonometric formulas. 
 
 Periodicity of the Elliptic Fvnctions. 
 
 196. We have seen (Art. 186, [2]) that 
 
 F(k,mr + p) = 2nK+F(k,p), [1] 
 
 where Kis the complete Elliptic Integral of the first class. 
 
 Let u = F(k, p), and take the amplitude of each member of 
 [1] ; we get 
 
 Sim{u + 2nK) = nTr+&mu; [2] 
 
 or, replacing w by 2n, 
 
 am(u + 4nK) = 2nTr + ainu; [3] 
 
 whence 
 
 sn(M + 4wJS') = SUM, 'I 
 
 ca(u + 4:nK) =enu, I; [4] 
 
 dn(M-f-4n^)= dnw, j 
 
 and sn u, en u, dn u are periodic functions, and have the real 
 period 4:K. dnw actually has the smaller period 2K, as may 
 be seen by taking the delta of both members of [2]. 
 
262 
 
 INTEGRAL CAIiCXJliTJS. 
 
 [Akt. 196. 
 
 Since the amplitude of ^is -, we have 
 
 [5] 
 
 and our addition formulas [1], [2], [3], Art. 193, give us 
 readily 
 
 su {u + K) = 
 
 en (« + £■) =- 
 
 dn(M + ^) = 
 
 enu 
 6nu 
 
 k'snu 
 diiM 
 
 Jc' 
 dnw 
 
 [6] 
 
 sn (« + 2 K) = — sn M, 
 en (m + 2 K) = — cnu, 
 dn(M+2^)= dnit, 
 
 [7] 
 
 sn(M + 3A') = - 
 cn (m + 3 iT) = 
 dn(M + 3A') = 
 
 CUM 
 
 dnw 
 
 k'snu 
 dnu 
 
 k' 
 dnw' 
 
 [8] 
 
 sn(M + 4A') = snM, 
 en (it + 4^") = en M, 
 dn(w + 4Ar) = dnM, 
 
 [9] 
 
 a eonflrmation of [4]. 
 
Chap. XVI.] 
 
 ELLIPTIC INTEGRALS. 
 
 253 
 
 197. It is easy to get formulas for the sn, en, and dn of an 
 imaginary variable, mV— 1, by the aid of a transformation due 
 to Jacobi. 
 
 Let 
 
 Jo A<i 
 
 (1) 
 
 SO that ^ = am'y, sin<^ = sn'«, and cos^ = cn'U. In (1), re- 
 place </> by ij/, <ti and i// being connected by the relation 
 
 whence 
 
 sin<^ = V— 1 . tani/r, 
 cost^= secip, 
 
 A<^ = Vl — 7c^ sm'<t, = Vl + fc^ tan>, 
 and d<^ = V — 1 . seci/f . di/'. 
 
 Since if/ and ^ equal zero together, 
 
 (2) 
 (3) 
 (4) 
 
 Jo A</> Jo 
 
 '-'X* 
 
 # 
 
 Vl— fc'^sin^i/' 
 If now we let u = F(k', </r) , 
 we have v=u V— 1 
 
 seci/'di/f 
 Vr+"FtanV 
 
 = V^1.J'(*',^). 
 
 (5) 
 
 Hence, since i/? = amtt (roodfc'), we have from (2), (3), 
 and (4), 
 
 
 or, as 
 
 v^uy/ — 1, 
 
254 
 
 INTEGKAL CALCTJLUS. 
 
 [Art. 197. 
 
 / — 7 7 N / — 7 sn (u, k') 
 
 en (m V — 1, k) 
 dn (u V— 1, k) 
 
 en (u, k') 
 
 dn (u, k') 
 
 en (u, k') 
 
 [6] 
 
 It is interesting to note that if it is replaced in (6) by 
 mV — 1, tlie formulas reduce to 
 
 sn ( — m) = — snu, 
 en (— w) = cnw, 
 dn (— u) = dnu, 
 
 and are still true. Consequently, in (6) , u may be either a 
 real or a pure imaginary. 
 
 Let 
 
 J- 
 
 dij/ 
 
 •/O , 1 
 
 dl/r 
 
 = ^' 
 
 Ai//(modfc') -'o Vl— /b'^sin^i/' 
 
 Then, by Art. 196, 4 A'' is a period for snM(modA;'), 
 cnu{modk'), and dnit(modA;'). 
 Hence 
 
 sn(MV— l + 4n/r'V— 1) = snwV— 1, 
 
 en (mV— 1 + 4:7iK' ■\/ — 1) =cnit V— 1, 
 
 dn (m V^^+ 4 ?) K' V^^) = dn« V^^ ; 
 
 or, replacing uv — 1 by f, 
 
 sn(v + 4?i/r' V— 1) =sn'y, 
 
 en {v + 4«,7sr' V— 1) = cnw, 
 
 dn (I) + 4 n A'' V^T) = dnv, 
 
 and 4 A''V— 1 is a period for sn, en, and dn. 
 
 We see, then, that our Elliptic Functions, like Trigonometric 
 Functions, have a real period, and, like Exponential Functions, 
 have a pure imaginary period. They are, then, what may be called 
 
 [7] 
 
Chap. XVI.] 
 
 ELLIPTIC INTEGRALS. 
 
 255 
 
 Doubly Periodic Functions, and they are often studied from the 
 point of view of their double periodicity. 
 
 Like Trigonometric Functions, the Elliptic Functions may be 
 developed in series, and from these series their values may be 
 computed, and tables resembling Trigonometric tables may be 
 prepared. 
 
 A partial three-place table is here presented as a sample. It 
 
 V2 
 is complete for Elliptic Functions having the modulus -p- ; that 
 
 is, 0.7. 
 
 MoDULus^^ = 0.7. 
 2 
 
 u 
 
 snu 
 
 cnu 
 
 dnw 
 
 0.00 
 
 0.000 
 
 1.000 
 
 1.000 
 
 0.05 
 
 0.051 
 
 0.999 
 
 0.999 
 
 0.15 
 
 0.150 
 
 0.989 
 
 0.994 
 
 0.25 
 
 0.247 
 
 0.969 
 
 0.985 
 
 0.35 
 
 0.340 
 
 0.940 
 
 0.971 
 
 0.45 
 
 0.429 
 
 0.903 
 
 0.953 
 
 O.SS 
 
 0.512 
 
 0.859 
 
 0.932 
 
 0.65 
 
 0.SS9 
 
 0.808 
 
 0.909 
 
 0.75 
 
 0.659 
 
 0.752 
 
 0.885 
 
 0.85 
 
 0.722 
 
 0.692 
 
 0.860 
 
 0.95 
 
 0.778 
 
 0.628 
 
 0.835 
 
 1.05 
 
 0.827 
 
 0.562 
 
 0.811 
 
 1.15 
 
 0.869 
 
 0.494 
 
 0.789 
 
 1.25 
 
 0.906 
 
 0.424 
 
 0.768 
 
 1.35 
 
 0.935 
 
 0.353 
 
 0.750 
 
 1.45 
 
 0.959 
 
 0.284 
 
 0.735 
 
 1.55 
 
 0.977 
 
 0.213 
 
 0.723 
 
 1.65 
 
 0.990 
 
 0.143 
 
 0.714 
 
 1.75 
 
 0.997 
 
 0.072 
 
 0.709 
 
 K. 1.85 
 
 1.000 
 
 0.000 
 
 0.707 
 
 From this table, by the aid of formulas [4], [6], [7], and 
 
 (8) of Art. 196, sum, cnw, and dnw may be readily obtained 
 
 V2 
 for any value of m if the modulus is — — 
 
256 INTEGRAL CALCtTLUS. [ART. 198, 
 
 As a matter of fact no complete set of tables for the Elliptic 
 Functions has been published, and their values are usually ob- 
 tained indirectly from Legendre's Tables of Elliptic Integrals 
 (v. Arts. 186, 187), unless especial accuracy is required, in 
 which case they must be computed by methods which we have 
 not space to give. 
 
 198. The Elliptic Integral of the second class E (k, <j>) can 
 be expressed iu terms of Elliptic Functions, and for some 
 purposes there is a decided advantage in the new form. 
 
 "We have E{k, ({>)= f A<l> .d<l>. 
 
 Let u = F(k, <j)), then <^ = amw, and E {k, <^) may be written 
 E {k, amw), or, more simply, E (a.mu), if the modulus can be 
 omitted without danger of confusion. 
 
 Xamu 
 dnw . damw; 
 
 or, since by (4), Art. 192, 
 
 da,mu = dnu .du, 
 
 E{a.mu)= j dn^u.du. [1] 
 
 As an example of the usefulness of the form just given in 
 [1], we will employ it in getting an addition formula for 
 Elliptic Integrals of the second class. 
 
 E (amu) + E (a.mv) 
 
 dn^M .du+\ dn^w . dv 
 
 Jo 
 
 = I dn^z .dz+\ du'z . dz 
 
 Jo Jo 
 
 dv?z.dz +i dn^z,dz-\ dn^z.dz 
 
 dn'z .dz-[ da'z . dz. 
 
 Ju 
 
Chap. XVI.] 
 
 ELLIPTIC INTEGBALS. 
 
 257 
 
 Replacing zhy u + z, and remembering that u and v are given 
 constants, 
 
 dn'z .dz=\ dn^ (m + z) dz, 
 
 u Jo 
 
 and 
 
 E {a.mu) + E {&mv) = 
 
 E [am (m + v)] _ CldiV? (u + z) -dn^x] dz. (2) 
 
 dn^ (m + z)—dn^z= [da (m + 2) + dna] [dn (u+z) — dn z] . (3) 
 
 "We can obtain from [3], Art. 193, the following formulas 
 analogous to [4j and [5], Art. 193, 
 
 dn {u + v) + dn (u — v) ■■ 
 
 2dnM . dn-i; 
 1 — Fsn^?t . sv?v' 
 
 (4) 
 
 dn (« + 1;) — dn (it — v) = - 
 
 2fc^snM . sn-y . cnit . env 
 1 — fc^sn^M . sn^v 
 
 (5) 
 
 If in (4) and (5) we let u + v=u + z, and u — v = z, and 
 substitute the results in (3), we get 
 
 dn^(w + ») — dn^2 
 
 [ 
 
 1 — fc^ sn^- sn^ ( - + z 
 2 \2 
 
 and 
 
 C[An^(u+z)-An^z'\dz 
 
 = — 2sn-cn-dn- I ■ 
 2 2 2J 
 
 2sn-cn-dn- 
 
 2fc^sng + 2')cn('^ + z") dng + «)d« 
 
 ['- 
 
 Fsn2-8nY- + 2 
 
 sn''- 
 2 
 
 l-fe^sn^-sn'' 
 
 H 
 
258 
 
 INTEGKAL CALCTJLTJS. 
 
 [Art. 199. 
 
 since - 2k' sn'-sn f'^ + z'\ en f^+z\ dn f'^ + z)dz is the differ- 
 ential of I — A;^sn^-snM - + «V ' 
 2 \2 J 
 
 C\dn^u+z)-dn'z'} 
 
 dz 
 
 „ U U J u 
 
 2sn- en- dn- 
 
 sn 
 
 l_A;2sn^-snf-+'y) l-fc^sn^^ 
 
 fc^ 2 sn - en - dn - 
 2 2 2 
 
 l_^2sn^- 
 
 \2^ y 2 
 
 ^sn - sn'' - 
 2 V2 
 
 -1) j 
 
 = — k^ . snu . sn V . su {u + v) , 
 
 by (1), Art. 194, and [6], Art. 193. 
 Hence by (2), 
 
 E {a.mu) + ^(amt)) = ^ [am (m + "")] + ^sn«. . sn-y . sn(w. + 'y), 
 
 [6] 
 our required addition formula. 
 
 APPLICATIONS. 
 
 Rectification of the Lemniscate. 
 
 199. From the polar equation of the Lemniscate, ?'^=a^cos29, 
 referred to its centre as origin and its axis as axis, we get as 
 the length of the arc, measured from the vertex to any point, 
 P, whose coordinates are r and 0. 
 
 r = al — = a I 
 
 de 
 
 •\/l— 2sin'e' 
 
 [1] 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 269 
 
 and for the arc of the quadrant of the Lemniscate, that is, the 
 arc from vertex to centre, 
 
 'L 
 
 s,= ar—^=. [2] 
 
 '^0 Vl-2sin2e 
 
 These differ from Elliptic Integrals of the first class only in 
 that the coefficient of sin^^ is greater than unity, and they may 
 be reduced to the standard form by a simple device. 
 
 Introduce in [1] 4> i" place of 9, <j> and 6 being connected by 
 the relation sin^<^ = 2 sin^^. 
 
 Then we have -^l—^sia^e = cos ^, 
 
 and d6 = ^ cos.^dt _ . 
 
 2 Vl-isin^c^ 
 
 Hence s =^C -^^ ^^f(^A [3] 
 
 2 -'" Vl-isin^<^ 2 V2 'V 
 
 and 
 
 _ aV2 p d<l> _ aV2 pf^"2 ■A r4-| 
 
 '" 2 Jo Vi_isi„2<^ 2 V 2 ' 2/ 
 
 The auxiliary angle <^ is very easily constructed when the 
 point P of the Lemniscate is given. We have r = aVcos26, 
 and we have seen that Vcos 26 = cos <^ ; hence ?• = a cos <^. If, 
 then, on a as a diameter we describe 
 a semi-circumference, and with the 
 centre of the Lemniscate as a 
 centre, and with a radius equal to r, 
 we describe an arc, and join with 
 the point Q where this arc intersects 
 the semi-circumference, the angle made b y OQ w ith a is equal 
 to 0. For OQ = a cos AOQ and OP=a Vcos 2 B. 
 
260 INTBGEAL CALCULtTS. [Abt. 199. 
 
 Examples. 
 
 d4> 
 
 (1) Find the numerical value of | 
 
 Vl — 4sin2<^ 
 
 Ans. 0.843. 
 
 (2) Eeduce f ^'^ — to an Elliptic Integral of the 
 
 Jo Vl-wsin^^ 
 
 first class, when n > 1. 
 
 ^■ns — ^ ( — — where sin^ ij/=n sin^^. 
 
 • \ ?i 
 
 (3) The half -axis of a Lemniscate is 2. What is the length 
 of the arc of a quadrant? of the arc from the vertex to the 
 point whose polar angle is 30°? Ans. 2.622 ; 1.168. 
 
 In the inverse problem of cutting off an arc of given 
 
 length the Elliptic Functions are of service. As an interesting 
 
 example, let us find the point which bisects the qiiadrantal arc 
 
 of the Lemniscate. 
 
 „ aV2 , E,/'V2 
 Here s= iF( — 
 
 2 ^ V 2 ■ 2 
 
 and we wish to find <^ and then 6. 
 
 Let u = F — ,-1; we need am-- 
 
 V 2 ' 2y 2 
 
 amw = ^, SUM = 1, CUM = 0, and dn?( = — • 
 
 By [1] and [2], Art. 195, 
 
 „ nU 1 — cnw ,M dnit + cnM 
 
 sn''- = , cn- = ■ 
 
 2 l+dnw 2 l+dnw 
 
 Therefore, 
 
 sir - 
 
 2 oti 1 — cnw 1 ,_ 
 
 = tn''- = = — - = Jo 
 
 ^^2^ 2 dnw + cnw V2 ' 
 
 2 'Y 
 
Chap. XVI.] ELLIPTIC INTEGRALS. " 261 
 
 If, then, the required amplitude is <f,, 
 tan^(^ = V2, 
 and tan<^ =V2. 
 
 Since sin'^ = 2 sin^6, we can compute 6 without difficulty, 
 and so get our required point. If, however, a construction will 
 suffice, a very simple one gives the point. 
 
 Erect at ^ a perpendicular 
 whose length is a mean pro- 
 portional between a and o V2. 
 The angle subtended at by g^ 
 this perpendicular is <^, and 
 the corresponding point, P, is 
 found by the method described O 
 on page 255. 
 
 Rectification of the Ellipse. 
 
 200. We have seen in Art. 177 that the length of an arc 
 of an Ellipse measured from the end of the minor axis is 
 
 J" ^ a^ — x^ 
 If we let x = a sin <j}, [1] becomes 
 
 s = a pVl - e^ sin2 ^ .d<i> = aE{e, 4,), [2] 
 
 e, the modulus of the Elliptic Integral, being the eccentricity 
 of the Ellipse. If a; = a, ^ = -, and the length of the Elliptic 
 quadrant is n- 
 
 s, = a r Vl -e^sin^.^ . d<j, = aEfe,-\ [3] 
 
262 
 
 INTEGRAL CALCULUS. 
 
 [Art. 201. 
 
 The length of an arc of the Elliptic quadrant, not measured 
 from the extremity of the minor, axis, can of course be ex- 
 pressed as the difference between two Elliptic Integrals of the 
 second class. 
 
 The amplitude ^, corresponding to a given point P, of the 
 Ellipse, is easily constructed as follows : On the major axis 
 as diameter describe a circumference ; 
 extend the ordinate of P until it meets 
 the circumference, and join the point of 
 intersection with the centre of the ellipse. 
 The angle the joining line makes with 
 the minor axis is seen to be the required 
 amplitude <^. If <^ is given, P may be 
 -^ found by reversing the order of the steps 
 of the construction. 
 
 Examples. 
 
 The equation of an ellipse is 1- ^ = 1, required the length 
 
 16 8 
 
 of the quadrantal arc ; of the arc whose extremities have the 
 abscissas "2 and 2V2. Ans. 5.4 ; 0.944. 
 
 (2) Find the abscissa of the end of the unit arc measured 
 
 from the extremity of the minor axis in the ellipse 1- i- = 1 ; 
 
 16 8 
 of the point which bisects the arc of the quadrant. 
 
 Ans. 0.996 ; 2.57. 
 
 201. By the aid of the addition formula 
 
 E{a.m.u) + E(3,mv)= E [am (ti + ■;;)] +Fsnr( sn«sn(M +'y) 
 
 ([6], Art. 198) 
 
 it is always possible to find an arc of an ellipse differing from 
 the sum of two given arcs by an expression which is algebraic 
 in terms of the abscissas of the extremities of the three arcs. 
 This will be clearer if we modify slightly the form of our addi- 
 tion formula. 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 263 
 
 Let <f>=:a.rau, ij/=a,rav, and o- = am(M + w). 
 
 Then the formula given above becomes 
 
 E(k, ^)+^(fc, ij/) — E{k, (r) + &^sin(^sini/fsincr, [1] 
 
 where 4>, if/, and o- are three angles connected by the relation 
 
 coso- = cost^cosi/r — sini^sini/^Acr, [2] 
 
 by [10], Art. 193. 
 
 If we multiply [1] by a and take A; equal to e, we get 
 
 aE (e, (p) + aE (e, i)/) = aE (e, <r) + —oo^ . x^ . Xg, 
 
 if Xi, X2, and x^ are the abscissas of the points whose amplitudes 
 are <^, ij/, and tr. 
 
 The most interesting case is when o- = -, in which case 
 
 aE(e, a-) is the arc of a quadrant. [2] then reduces to 
 
 = eos^ oosif/ —s\n<l> sini/'Vl— e^, 
 
 or 
 
 - sin (f> sini/f = cos <^ cos ij/, 
 a 
 
 tan <j} tau ij/ -■ 
 
 a 
 
 [3] 
 
 and we get from [1] 
 
 aE{e, <^) — 
 
 aE (e,-\—aE{e, if/) = ae^ sin (/> sin 1^. [4] 
 
 If, then, any point, P, is given, [3] will enable us to get 
 the amplitude of a second point, Q, and ^ 
 thus to find Q, Q and P being so re- 
 lated that the arc BP, minus the arc AQ, 
 shall be equal to a quantity which is 
 proportional to the product of the ab- 
 scissas of P and Q. 
 
264 INTEGRAL CALCULUS. [Abt. 202. 
 
 For the special case where <j> and f are equal we have from 
 
 [3], tan<^ = J« 
 
 and from [4], 
 
 BP -AP=ae'- sin' d>= -^ = a'-b. 
 a + b 
 
 This point, which divides the quadrant into two arcs whose 
 difference is equal to the difference between the semi-axes, has 
 a number of curious properties, and is known as Fagnafii's 
 point. 
 
 Examples. 
 
 (1) Show that the distance of the normal at Fagnani's point, 
 from the centre of the ellipse, is equal to a — 6. 
 
 (2) Show that the angle between the normals at P and Q in 
 the figure is equal to ij/ — <(> ; that the norrnals are equidistant 
 from ; that this distance is BP — AQ. 
 
 Rectification of the Hyperbola. 
 
 202. If the arc of the Hyperbola is measured from the 
 vertex to any given point, P, whose coordinates are x and y, 
 its length is easily found to be 
 
 [1] 
 
 s^j;L-^yy, [2] 
 
 if e is the eccentricity of the Hyperbola. Let 
 
 ae 
 
 
 y = tan ^, 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 265 
 
 aes 
 aeJo 
 
 and [2] becomes 
 
 _ b' C* sed'<l}d<l> 
 
 hence s = ^ f^_^^^^±^^ _ &! T* s ec'0d0 .3, 
 
 if k=-. 
 
 Now 1 ^ 1-A^ 1^ 1 l-fe^sin^-fc'coB'.^ 
 
 and s = ^ . ^—f r*sec2 <l> \<t> d<l> - 1^ f*.^! 
 
 ae 1 — fc^jyo f- t -»- j^ ^^j 
 
 ae 1 — A;''[jl/o J 
 
 If we integrate by parts, 
 
 f secVA<^d<^ = tan<^A<^ + A^ f ?15-^d<^; 
 
 A^ 
 
 -A<6, 
 
 A<^ A</> 
 
 A^ 
 
 but fc'^5i5!^ = J--A<^, 
 
 and A^^'f*?^ ■d<^ = F(A;, .^)-JS;(&, <^) 
 
 »/o Ad 
 
 Hence 
 
 ae ^ '^ ae(l-fc2) 
 
 s=^F{k,i>)-—^±-—[E{k,ct>)-t^n<t>Ai>-]. 
 
 But 1 - A;^= - 
 
266 
 
 12JTBGKAL CALCULUS. 
 
 [Art. 202. 
 
 J2 
 
 therefore s = — F(Jc, (j>)— oeE (k,<p) + ae tan <^ A<^, 
 ae 
 
 or s = —Pf-, <ji\ - aeE (-, <^ + ae tan <^ A<^. [4] 
 
 ae \e J \^ / 
 
 The angle <f> corresponding to a given point P is easily con- 
 
 structed. We have only to erect a perpendicular to the trans- 
 
 from the origin ; that is, 
 
 verse axis at a distance — = - 
 
 b' 
 
 ae ^a' + l 
 
 at a distance from the centre equal to the projection of 6 on 
 the asymptote, and to join the projection of P on this line with 
 the centre. The angle made by the joining line with the trans- 
 verse axis is 4>t for its tangent is clearly ^ 
 
 ■sld? + W 
 
 Examples. 
 
 (1) Find the length of the arc of the hyperbola 2l= i^ 
 
 measured from the vertex to the point whose ordinate is 2. 
 
 Ans. 2.194. 
 
 (2) Show that ae tan <^ A(^ is the distance from the centre to 
 the normal at P. 
 
 (3) Show that the limiting value approached by the difference 
 between the arc and the portion of the asymptote cut off by a 
 perpendicular upon it from P, as P recedes indefinitely from 
 
Chap. XVI.] 
 
 ELLIPTIC INTEGRALS. 
 
 267 
 
 the origin, is aeE I -, 
 \e 2 
 
 This is generally re- 
 
 ae \e 2 J 
 
 ferred to as the difference between the length of the infinite arc 
 of the hyperbola and the length of the asymptote. 
 
 Show that in example (1) this difference is equal to 2.803. 
 
 The Pendulum. 
 
 203. We have seen in Art. 176 that if a pendulum starts 
 from rest at a point of its arc whose distance above the lowest 
 point is 2/o) the time required in rising from the lowest point to 
 a point whose distance above the lowest point is y, is 
 
 where Jc 
 
 , la p d,t> 
 
 = -\l^, and sin<i=-\-^ 
 
 \2a \% 
 
 =^^Fik,4,), 
 
 [1] 
 
 Vo 
 
 In the figure let A be the lowest point of the arc, B the 
 
 highest point reached by the pendulum, and P the point 
 reached at the expiration of the time t. Call AOB a, and 
 
 AOP e. 
 
 Then ^ = 1 
 
 ■cos a, and 
 
 Ni-^« 
 
 1 — cosa) = sin- = k. 
 
 ' 2 
 
 Consequently the modulus of the Elliptic Integral in [1] is 
 
268 
 
 INTEGRAL CALCULUS. 
 
 [Akt. 203. 
 
 the sine of one-fourth the angle through which the pendulum 
 swings. 
 
 1 — cos^, 
 
 1-. 
 a 
 
 and 
 
 and 
 
 \|^ = Vi(l-cos^) =sin-, 
 
 _ -V -^- sm- 
 . , \v >2ffl 2 
 
 \l/„ \y„ 
 
 \2a 
 
 '2/0 
 
 sin- 
 
 and therefore the sine of the amplitude of the Elliptic Integral 
 in [1] is easily computed when the angle through which the 
 pendulum has risen is given. When ^ = a, sin<^=l, and 
 
 <i = -; so that the time of a half -oscillation is \ -i^f sin-, -), 
 ^2' ^g \ 2' 2/ 
 
 a confirmation of [7], Art. 176. The construction indicated 
 
 in the figure gives the angle <^, corresponding to any given arc 
 AP- For 
 
 --^=l-cos^O'Q, 
 i2/o 
 
 and -J- = V^ (1 - cos^O'Q) = sin^^i^ = sin^CQ. 
 
 ','/o 2 
 
 Therefore ACQ = ^. 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 269 
 
 It is very easy to express the angle 6 in terms of t. 
 We have « = -y(^J?'/'sin^, <^^; 
 
 hence t-^M = pfsia-, <^, 
 
 ^=am(.^J^, 
 
 sin<^ : 
 
 and sin-=sin^snU-j2Vmodsin-J; 
 
 then cos ^ = diiftJ3^ fmod sin -Y 
 
 3in e = 2 sin | sn fiyf) dn ftJs\ /'mod sin-Y 
 
 and sin^ 
 
 Examples. 
 
 (1) A pendulum swings through an angle of 180° ; required 
 
 the time of oscillation. /;; 
 
 Ans. 3.708 xp. 
 ^9 
 
 (2) Compare the times required by the pendulum in Ex. (1) 
 to descend through the first 30°, the second 30°, and the third 
 30° of its arc respectively. 
 
 Ans. 1.028 J- ; 0.446 i/-; 0.380 J-. 
 ^9 ^9 ^9 
 
 (3) The time of vibration of a pendulum swinging in an arc 
 of 72° is observed to be 2 seconds ; how long does it take it to 
 fall through an arc of 5°, beginning at a point 20° from the 
 highest point of the arc of swing ? Ans. 0.095 seconds. 
 
270 INTEGKAL CALCULIJS. [Akt. 203. 
 
 (4) A pendulum for which \\- has been determined, and is 
 
 equal to ^, vibrates through an ai-c of 180° ; through what arc 
 does it rise in the first half-second after it has passed its lowest 
 point? in the first ^ of a second? Ans. 69° ; 20° 6'. 
 
 (5) It has been shown in Art. 176 that if yo>2a the 
 pendulum will make complete revolutions, and that the time 
 required to pass from the lowest point to any point whose 
 distance above the lowest point is y, is 
 
 t=a l± f" ^ ^^ =ajAy(fe,^), 
 
 \ 32/o Jo Vl - fc^ sin^ <^ \ gyo 
 
 where A; = \ — and sin<i = -» -^• 
 
 Show that in this case 6= -, and that sin - = sn [ - -v |2^° V 
 
 2' 2 [a^ 2 J 
 
 Note. — In working with a pendulum it is often about as 
 easy to compute F (A-, <^) by developing by the binomial 
 theorem and integrating two or three terms, as to use a table 
 of Elliptic Integi'als. 
 
 We have F{k,4,)= C ^"^ -, 
 
 Jo Vl-it^sin^^ 
 
 (I - 1<^ sm'<t>)'-=l+iT<^ sin' <l> + h^k* sin* <!,+ ..., 
 
 2.4 
 
 and F{k, <^) = r "^^ = .^ + ^ (.^ _ sin.^ cos<^) 
 
 J" Vl-A-sin^<^ ^ ^ vj 
 
 3 9 
 
 ~ 32^^ si>i^<^ cos^ + ~k* {<f, — sin ^ cos<^) -.. 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 271 
 
 Differentiation and Integration. 
 
 204. Eewriting formulas [4], [5], [6], and [7], of Art. 192, 
 we have 
 
 d am a; ^ dn cc dx, [1] 
 
 <i sn a; = en X dn a; dx, [2] 
 
 d en a; = — sn a; dn a; dx, [3] 
 
 <^ dn a; = — ^^ sn a; en a; dx, [4] 
 
 dn X 
 we add dtux^ — 5— dx. r51 
 
 cn-'a; "- -" 
 
 By the usual method of differentiating an inverse function 
 (1, Art. 72) we get readily . 
 
 d sn-i (x, k) = ^'^ , [6] 
 
 d cn~' (x, A) = =, r7"l 
 
 ^ ^ V(l — a;2)(/fc''+A:V) 
 
 d dn-» (a;, A;) = t^a; ^ 
 
 ^ ^ V(l— a;2)(a;2 — A:'^) 
 
 rf tn-' (x, A;) = -■ , ■ [9] 
 
 ^ ^ V(l + x^) (1 + k'V) 
 
 [6], [7], [8], and [9] give at once a very valuable set of formu- 
 las for integration, namely : 
 
 r ^^ = sn-i (x, k) = Fik, sin-^x), [10] 
 
 Jo V(l - x^) (1 - /fc^xO 
 
 f , ^'^ :^Pn-' (x, k)=F(k, COS-'x), [11] 
 
 J- V(l-x=)(/fc'^ + fcV) ^ ^ ^ ' ' 
 
 r\ ^" - dn- (X, A:) = sn- ' f 2^, /A 
 
 Jx V(l— a;^(a;2-A;'2) \ k J 
 
 = ir('A:,sin-'%^V [12] 
 
272 INTEGEAL CALCULUS. [Abt. 204. 
 
 r ^^ = tn-^ (x, k) = F(k, tan- la;). [13] 
 
 J« Va + aj^Xl + ^'V) 
 
 If in [10], [11], [12], and [13] we substitute y = x^ and 
 then change y to x, we get 
 
 C , ^"^ = 2 sn- Y Va, k) =2F(k,sm-^\/x)JU'\ 
 
 Jo -slxil-xXl-k^x) \ J y 
 
 C '^^ = 2 en-' Nx, k)=2F(k,cos-^ Vx) , [15] 
 
 -'«• -Jx(l-x)(k"+k'x) 
 
 r'-_=^== = 2 dn-1 ( Vx, k) 
 -'^ Va;(l — x)(x —k'^ 
 
 ^2F(^k,sin-^^^^^y [16] 
 
 r , ^"^ = 2 tn-' Nx, k) 
 
 Jo Vx(l + x)(l + /c'^x) 
 
 = 2 i*XA;, tan-1 Vx). [17] 
 
 The following formulas are obtained from formulas [10]- 
 [13] by easy substitutions : 
 
 r , '^'^ = -sn-'(^y>-Ytt>&>x>0; [18] 
 
 Jo V(a»-x^)(i2-x=^ a \b aj' 
 
 r , ''"^ =. = -sn-'(-,-\x>a>b>0; [19] 
 
 Jx V(a;'' — a.^)(x^-62) a. \x aJ 
 
 r" rAr 1 _yx b \ 
 
 -'^ VCa'' + x') {b^ — x'') ~ si a" + 62 ^'^ i^S' V«' + ft7' 
 
 a>6>x>0; [20] 
 r* e?x 1 _Vi a \ 
 
 J" ■sj{a? + x^) (x^ — ¥) ~ V^^H^' ^" \^' V^H^V' 
 
 x>6>0; [21] 
 
 f , ^" =^dn-fg, v^!EZY 
 
 -'^ V(a'-x^(x2-J2) a \^a a / 
 
 a>.x>*>0; [22] 
 
 r-=i_=.itn-/-, V^!EEY 231 
 
 •^» V(x2 + a'')(x2 + 6=') '^ Vi a y/ L^ -J 
 
Chap. XVI.] ELLIPTIC INTEGliALS. ' 273 
 
 For example we will take [19]. 
 
 - I f.lifin fid- = J- ■ 
 
 Let y = -j then dx 
 
 X 
 
 y 
 
 ,2 ' 
 
 C' dx ro dy 
 
 -'- '^{x' — a'){x' — ¥')~ -'i V(l — ay)(l — 
 
 b'y^ 
 
 Jo Jn — 
 
 dy 
 
 ^(1-ahj^/ p 
 
 Let now z^=^ay and 
 
 J"^ dy _1 C' 
 
 dz 
 
 Prom [14]-[17] may be derived in like manner 
 
 •^^ V(x—a.)(x — ^)(x — y) Va — y \^X — y ^a — y/ 
 
 a:>a>/3>y; [24] 
 
 •^"^ -J(a — x)(x — I3){x — y) Va-y V^a— /S'^a — y/ 
 
 a>a;>/3; [25] 
 
 r '^ :^-i=sn-fJ^^,JB\ 
 
 -'v V(a — a;)(;8 — a;)(a;— y) Va — y \^/3— y ^a— y/ 
 
 ;8>x>y; [26] 
 
 r , ^^ =-l=cn-f JfaJ°Z^\ 
 
 *^^ V(a— a;)(j8— a!)(y — a;) Va— y \»j8— a; »a — y/ 
 
 y>x; [27] 
 
 Tor example we will take [24]. 
 
 1 rfy 
 
 Let y = ' then dx^ j : 
 
 ^ a; — y f 
 
274 INTEGJBAL CALCULUS. [Akt. 204. 
 
 dx 
 
 s: 
 
 V(x-a)(x-^)(a;-y) 
 1 
 
 ^ 
 
 Vy(l-(a-y)y)(l-(^-7)y) 
 
 Let now z = (a — y)y, then 
 
 1 
 
 V2/(l-(a-y)2/)(l-(^-r)2/) 
 
 ^-v ds 
 
 :sn" 
 
 Va — y 
 
 From [24]-[27] may be obtained 
 
 J"'' dx 
 
 - \l{x — a)(x — 13) (x — y){x — S) 
 
 2 
 
 sn 
 
 ■fJIEr.JtU:') by [14]. 
 
 \ ^ X — y 'a — y/ 
 
 y^a — 8 X — /3 'a — y y8 — 6/ 
 
 V(a-y)(^-S) 
 
 x>a; [28] 
 
 
 - V(a -x){x~ ji) (X — y)(x — S) 
 
 V(a-y)(;8-8) V^«-/8 =» — 8 ^a-y jS-Sy 
 
 a>x>P; [29] 
 
 ^^ V(a — x)(l3 — x) (x — y) (a- — S) 
 
 V(a - y) (;8 - 8) \^/8-y «-x V „ _ .j, ^ _ Sy/ 
 
 |8>a:>y; [30] 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 275 
 
 £ 
 
 dx 
 
 V(a — a;) (/3 — a;) (y — x) (.;■ — 8) 
 2 
 
 : sn" 
 
 \*y — 8 p — x ''a — y p — oj 
 
 V(a-y)(^-8) 
 
 y>a;>S; [31] 
 
 n dx 
 
 -'^ V(a — x) (13 — x) (y — x) (8 — x) 
 
 V(a — y)(;8 — 8) \^a. — h y-x ^a — y /3-Sj 
 
 S>X. [32] 
 
 Formulas [24]-[32] enable us to integrate the reciprocal of 
 the square root of any cubic or biquadratic which has real 
 roots. 
 
 a 
 
 AS an example let us find | ^=z- 
 
 *^o V(2 ax — x") {a" — x^ 
 
 J" 2 dx 
 » V(2 a— x)(a — x)x(x-\- a) 
 
 Jr'a dx 
 
 
 
 V(2 a — x)(a — x)x(x-\-a) 
 dx 
 
 V(2 a — x)(a — x)x (x + a) 
 
27l! ( INTEGRAL CALCULUS. [Aiir. 204. 
 
 Formulas [10]-[32] suggest the appropriate substitution to 
 rationalize any rational function of x and the square root of 
 a cubic or biquadratic having real roots. 
 
 For instance, let us consider I V(a^ — x^) {V — x^) dx. 
 Let y = sn~M '-, - J, [v. formula [18]]. 
 
 Then x = /> sn( y, ~ \, dx ■= b en y dn y dy, a? — a;^=:a"dn^y 
 
 Ir — .>•- ■= W cn^ ;/ ; 
 
 Jo V("' — J-'K^" — •'"')•"'•'■ = «^'| cv^yAn^ydy 
 
 Examples. 
 
 (1) Find r'-.^=- Ans. :^K(moA^\ovl.Zll. 
 
 ^0 Vl — a;* 2 \ 2 ) 
 
 (2) Eationalize I Vl — x^ . dx. 
 
 •Jo 
 
 Ans. 2V2 jT (dn^a- — dn*a;)(&;. /^mod ^Y 
 
 (3) Find C ^ ''"^ . 
 
 *^" V(a2 — S,«) (ia; — x^ 
 
 Am. -sn-/l, -^ or -^Tmod -Y 
 
 (4) Rationalize f\{^E^, dx. 
 
 ^0 ^bx — x' 
 
 Ans. 2a J ^ dn'x.dx,0T2aEf-,'^\ 
 
Chap. XVI.] ELLIPTIC INTEGRALS. 277 
 
 (5) FindX' 
 
 dx 
 
 ■si {a' -\-c' — x") {a? — x') (c' — x") 
 
 Suggestion : let s = — . 
 x^ 
 
 205. If we are integrating the reciprocal of the square root 
 of a cubic or biquadratic having imaginary roots, formulas 
 [24]-[.32] of Art. 204 no longer serve our purpose and we 
 are driven to a more laborious method. 
 
 We need only to consider the biquadratic form as we may 
 regard the cubic as a special case under it. 
 
 dx 
 V(a + 2 6x + cx^) (a + 2 ySa; + yx^) 
 
 Take the form j — , and 
 
 J VCa- ' 
 
 let V = where m and n are at first undetermined. We 
 
 •' x — n 
 
 shall get an integral of the form 
 
 /: 
 
 V(^ + By+ Gf) {A! + B'y + C'f) 
 
 and m and n can then be so chosen that B and B' shall be 
 equal to zero, and the integral can be obtained by one of the 
 formulas [18]-[21] of Art. 204. 
 
 The values of w and n required are easily proved to be the 
 roots of the quadratic equation 
 
 by-O^ by -0/3 ' ^ ^ 
 
 and are always real if the original biquadratic has any 
 imaginary roots. 
 
 dx 
 
 ^0 Vx (1 + x') 
 
 For example let us find J 
 
278 INTEGRAL CALCULUS. [Art. 206. 
 
 Here a = 0, 5 = i, c = 0, a = l, 13 = 0, y = l. Our auxil- 
 iary equation (1) becomes »^ — 1 = and gives 1 and — 1 for 
 
 m and n. Let, then, y = — — r > substitute and reduce and 
 
 x-\-l 
 
 Jdx _ - /'I di£ 
 
 Vx (l + x^)~ ^V-i V(l + f) (1 - tf) 
 
 Jo V(l + x^) (1 — x2) \ ^ / 
 
 = 2 7f fmod ^ J = 3.708. 
 
 Examples. 
 
 Suggestion : let « = x''. 
 
 C2) Eationalize I — =• 
 
 ^ ^ -'» Vx(l+a;=) 
 
 ^r^2-2cn.,-sn>. / ^VgY 
 Jo sn^tj -^ \ 2 y 
 
 (3) Rationalize j Vl + a;* ■ dx. 
 
 r-- l + cn'y 
 ^"""•Jo (1 + cny)^ "'^ 
 
 Jo sn^x \ 2 / 
 
 206. Formulas for integrating snx, cnx, dnx, and their 
 powers positive and negative, are obtained without difficulty. 
 
Chap. XVI.] ELLIPTIC INTEGKALS. 279 
 
 /, 1 C — /i^ sn cc cn a; rfa; \ C dv 
 sna;«a; = — -; I = I — " 
 k J en X kJ ^/,/2 
 
 ■^y^ — k" 
 
 1. 
 
 ^ — T log (y + ylif — k'^ if 2/ = dn x. Hence 
 j snxdx^— -log (daxx + ^dn!' X — k'^ 
 
 j cna;rfz^-cos~'(dna;). [2] 
 
 I dnxdx = am x = sin~' (sn x). [3] 
 
 /c?x Z' sn a; en a- dn a; (Za; T dy 
 sn X J sn^ X en X dn X %/ ^/^n — y\ i\ 'i^y\ 
 
 = _ ^ log r V(l-y)(l-^V) + l _ 1+^1 
 
 if y = sn^ X. 
 Hence 
 
 r*^^ _lo r sn^ "1 r4-| 
 
 J sn X LCD- X + dn X J 
 
 /(Zx 1, r/c' sn X + dn x~| 
 ^=pi°^L — ^^^— J- ^^^ 
 
 /• (^x ]_ . _, r A:'^sn^x — cn^x "1 
 
 1^ , TA' sn X — en x~| 
 
 = — tan-M -; j • [6] 
 
 k LA;'snx + cnxJ '- 
 
 From Art. 198 [1] we get 
 
 J''sn2x(fo; = ^[x--E'(amx,A;)], [7] 
 
280 INTEGRAL CALCULUS. [Art. 206. 
 
 Xi 1 
 
 cn^ xdx=^-^ [_E (am x, k) — k'^x"], [8] 
 
 / dn^ xdx^E (am x, k). [9] 
 
 An important set of reduction formulas by which the integral 
 of any whole power of sn x, en x, or dn x can be made to 
 depend upon the formulas just obtained can be found with- 
 out difficulty. 
 
 We have -;- (sn^+'xcnaidnx) 
 
 dx^ ' 
 
 = (m + 1) sn" cc — (m + 2) (1 + A'') sn" + = a; + (m + 3) k"^ sn"' + * x, 
 whence we get 
 
 (m + 1) I sn" x dx 
 
 = (m + 2) (1 + /«2) r sn™ + ^xdx 
 
 — (m + 3)FJ sn^ + ^xt^x + sn^+'iccnaidna;. [10] 
 
 (m + 1) k'^i CM^'xdx 
 
 = (m + 2) {k'^— k^f cn"' + ^xdx 
 + (m-\-S)k^J cn'"+*a;<ix — cn^ + ^xsnajdna;. [11] 
 (m + 1) k'^j dn'^xdx 
 
 = (m + 2) (1 4- k' 2) J" dn^+^a;^^; 
 
 — (m + 3) J dn'" +*xdx + k" dn"" +'i a; sn a; en a;. [12] 
 
Chap. XVI.] ELLIPTIC INTEGKALS. 281 
 
 Examples. 
 (1) Obtain the following formulas : 
 
 I sn ^xdx = xsn-^x-\-jcosh-^{ 1 
 
 j cn-^xdx = xcn-'x — -Gos-^ ( VA'^ + kV) 
 j dn~^a!(£a;= a;dn~'a; — sin^M )■ 
 
 (2) Find aSj^ [l-^^^ sn'x + ~sn'x']dx. 
 Ans. i ^ r (a'' + 62) ^ ^^ , J^ - (a^ + 2 6^) K Cmod ^ 
 
 (3) Pind^Ji" {&ii'x-6.n''x)dx, Tmod ^Y 
 
 Ans. ^xfrnoi^X or 0.219. 
 
 ,^^ -I7- ^ /•^2-2cnx--sn^ , / , V2\ 
 (^) ^^"Vo ^x '^"' r""^^/ 
 
 Ans. ^/2 + K— 21! (mod. ^\ or 0.567. 
 
 (5) A cannon-ball of radius b is fired horizontally through 
 the middle of a ship's mast (radius a) ; find (a) the volume, 
 and (b) the whole superficial area of the plug required to fill 
 the hole. 
 
 Ans. (a) ^[ (a' + b") E f^, "^^ - (a' + 2 P) K (mod ^ ; 
 (b) 8(a + b) ^aE Q^, |) " (« - *) -ST^mod ^) • 
 
282 INTEGRAL CALCULUS. [Art. 206. 
 
 (6) A cylindrical hole of radius h is bored through a sphere 
 of radius a, and just grazes the centre ; find (a) the area of the 
 inner surface of the hole, (b) the spherical surface removed, 
 and (c) the spherical volume removed. 
 
 Ans. (a) ^,dbE\.-i ^ j; 
 
 (b) 2aV-4a^^Q, j\ 
 
 (7) Find the mean distance of points uniformly distributed 
 along the perimeter of an ellipse from a focus. 
 
 Ans. One half of the major axis. 
 
Chap. XVII.] THEOKY OF FUNCTIONS. 283 
 
 CHAPTER XVII. 
 
 INTRODUCTION TO THE THEORY OF FUNCTIONS. 
 
 207. A function having but a single value for any given 
 value, real or imaginary, of the variable is called a single-valued 
 function. Eational Algebraic Functions, Exponential Func- 
 tions, the direct Trigonometric Functions, and the Elliptic 
 Functions are single-valued. 
 
 A function which has in general two or more values for any 
 given value of the variable is called a multiple-valued function. 
 Irrational Algebraic Functions, Logarithmic Functions, the 
 inverse or anti-Trigonometric Functions, ancl" the Elliptic In- 
 tegrals, are multiple-valued. 
 
 208. In Chapter II. we have explained the customary graph- 
 ical method of representing an imaginary by the position of a 
 point in a plane, the rectangular coordinates of the point being 
 the real term and the real coefficient of the pure imaginary terra 
 of the imaginary in question. 
 
 In the ordinaiy treatment of the Theory of Functions this 
 method of representation is of the greatest service, and enables 
 us to bring the study of functions of imaginary variables within 
 the province of Pure Geometry, and to give it great definiteness 
 and precision. 
 
 For the sake of brevity we shall in future use the symbol i 
 for V — 1 and cis(f> for cos</> + V — 1 sin<^, so that we shall 
 write our typical Imaginary as x-{-yi or as reis<^, instead of 
 using the longer forms x-\-y V — 1, and r (cos ^ -f- V— 1 sin <^) . 
 
 We shall also use the name complex quantity for an imaginary 
 of the typical form when it is necessary to distinguish it from 
 a pure imaginary. 
 
284 
 
 INTEGRAL CALCULUS. 
 
 [Art. 209. 
 
 209. A complex variable z = x + yi is said to vary continu- 
 ously when it varies in such a manner that the path traced by 
 the point (a;,2/) representing it is a continuous line. 
 
 Thus if z changes from the value a to the value /8, so that 
 the point representing it traces any of the four lines in the 
 figure, z varies continuously. 
 
 It will be seen that a variable can pass from the first to the 
 second of two given values, real or imaginary, by any one of 
 an infinite number of different paths without discontinuity if the 
 variable in question is not restricted to real values ; while a real 
 variable can change continuously from one given value to another 
 in but one way, since the point representing it is confined in its 
 motion to the axis of reals. 
 
 210. A single-valued function w oi Sl, complex variable z is 
 called a continuous function if the point representing it traces 
 a continuous path whenever the point representing z traces a 
 continuous path. 
 
 A multiple-valued function of z is continuous if each of the n 
 points representing values corresponding to a value of z traces 
 a continuous path whenever z traces a continuous path. 
 These n paths are in general distinct, but two or more 
 of them will iutersect whenever z passes through a value 
 for which two or more of the n values of w, usually distinct, 
 happen to coincide, Such a value of « is sometimes called a 
 
Chap. XVII.] 
 
 THEORY OF FUNCTIONS. 
 
 285 
 
 critical value, and the consideration of critical values plays an 
 important part in the Theor3- of Functions. 
 
 In studying a multiple-valued function we may confine our 
 attention to any one of its n values, and except for the possible 
 presence of critical points this value may be treated just as we 
 treat a single- valued function. 
 
 In representing graphically the changes produced in a func- 
 tion IV by changing the variable z on which it depends, it is 
 customary to avoid confusion by using separate sets of axes for 
 w and z. 
 
 211. If we use the word function in its widest sense, 
 w=u +vi will be a function of a complex variable z = x + yi, 
 if u and v are any given functions of x and y. For example. 
 
 xi, 6y, cn^ + y^, x — yi, a? — y^ + 2xyi, 
 
 x — y + xi 
 
 Vx^ + y^ + 4c 
 may all be regarded as functions of z. 
 
 We have seen in Chapter II., Arts. 36-42, that with this 
 definition of function the derivative with respect to s of a func- 
 tion w is in general indeterminate ; but that there are various 
 functions of 2, for instance, 2", log«, e", sinz, where the deriva- 
 tive is not indeterminate. We are now ready to investigate 
 more in detail the general question of the existence of a deter- 
 minate derivative of a function of a complex variable. 
 
 Let w = u + vi be a function of « ; u and v, which are real, 
 being functions of x and y. 
 
 Starting with the value 2o = ^o + 2/o* of z and the correspond- 
 ing value w^ = Uo + Vai of w, let us change z by giving to x 
 increment Asb without changing y. 
 
 V 
 
 V 
 
 «0_A.ai *1 
 
 % 
 
 Wo,; 
 
 
 Vi, 
 
 Mo 
 
 U 
 
 Let AjM and A^'W be the corresponding increments of u and 
 V ; and z^ and ■^t'l the new values of z and w. 
 
286 
 
 We have 
 Then 
 
 INTEGRAL CALCULUS. [^KT. 211. 
 
 Zi = Zo + A.X, lOi = Wo 4- A^w + iA^v. 
 
 Wi — Mn _ A^M tA^'i; ^ 
 2i — 2:0 ~ Ax Ace ■ 
 
 and the derivative of w with respect to z under the given cir- 
 cumstances is 
 
 limit 
 A«=0 
 
 Mj— Wo 
 «! — Zo 
 
 — D^M + iD.'U. 
 
 [1] 
 
 K^ 
 
 t«i 
 
 Zo 
 
 % 
 
 SCq 
 
 Wq, 
 
 ./ 
 
 If, however, starting with the same value Zq of z, we change 
 « by giving y the increment Ay without changing x, we have 
 
 Zi = Xo + (2/0 + -^y)* = «o + i^y^ 
 
 Wl = Mo + AsrW + (^0 + AjV) J = Wo + AjM + i\V, 
 
 Wi — Wo _ A„M *Aj,'y 
 
 %- 
 
 iAy iAy ■ 
 
 and 
 
 S'o[lJ^-]-«.-<^. 
 
 limit 
 
 A2 
 
 [2] 
 
 and this is the derivative of w with respect to z when we change 
 y and do not change x. 
 
 Comparing [1] with [2], we see that if we start with a given 
 value of z, and change z in the two different ways just con- 
 sidered, the limits of the ratios of the corresponding changes in 
 w to the changes in z need not be the same. Indeed, the two 
 
 values for — given in [1] and [2] will not be the same unless 
 dz 
 
 w = w + VI is such a function of « = a; + yi that 
 
 D^u=D,v and D^u = — D,v. [3] 
 
Chap. XVII.] 
 
 THEORY OF FUNCTIONS. 
 
 287 
 
 We shall now show that if w is such a function of z that 
 
 Az 
 
 will be the same if we 
 
 equations [3] are satisfied, a ^r, 
 
 start with a given value Zo of 2, no mattter in what manner z 
 may change ; that is, no matter in what direction the point 
 representing z may be supposed to move ; or, in other words. 
 
 no matter what may be the value of 
 
 limit 
 
 Ay 
 Ax 
 
 Aa!=0 
 
 We have in general, since w is a function of the two variables 
 X and y, 
 
 Aw = {D^u + iD^v) Ax + {DyU + iD^v) Ay + e, 
 
 where e is an infinitesimal of higher order than Ax or Ay. 
 
 (I., Art. 198.) 
 
 Az = Ax + iAy. 
 
 Hence — = 
 
 Aw _D^u.Ax+ iDyV . Ay +W^v . Ax -{-D^u . Ay+ c 
 
 Az Ax + iAy 
 
 Ax Ax Ax 
 
 1 + i 
 
 Ay 
 Ax 
 
 and 
 
 limit 
 A»=0 
 
 Aw 
 Az 
 
 „ , ■n limit 
 
 dw 
 dz 
 
 'Af 
 
 Ax 
 
 + ilD^v—iD,u 
 
 limit rAy"T\ 
 •Aa;=0[_Aa;jy 
 
 , , . limit 
 
 ^+'ax= 
 
 mit [Ay-] 
 
 ''=oLaJ W 
 
 value involving J'™ q ^ ' ^^^ therefore dependent upon 
 
 the direction in which z is made to move. 
 If, however, [3] is satisfied, [4] reduces to 
 dw 
 
 dz 
 
 =:D^u + iD^v, 
 
 [5] 
 
 and the derivative of w is independent of ^^ 
 
 limit fAy"! 
 \a!=0[_Aa;J' 
 
288 li^TEGRAL CALCULUS. [Art. 212. 
 
 A function which satisfies equations [3], and which, there- 
 fore, has a derivative whose value depends only upon the value 
 of the independent variable, and not upon the direction in which 
 the point representing the variable is supposed to move, is called 
 by some writers a monoc/enic function, by others a function which 
 has a derivative, 
 
 212. Any function of » which can be formed by performing 
 an analytic operation or series of operations upon s as a 
 whole, without introducing x and y except as they occur in z, 
 is a monogenic function of z. 
 
 For if iv=fz=f{x + yi), 
 
 where /« can be formed by operating upon x as a whole, 
 
 D^'u>=f% and D^w=if'z; 
 
 therefore iD^w = D^w, or iD^(u + vi) — Dy{u-\- vi) ; 
 
 whence D^u = D^v, and D^u = — D^v; 
 
 and [3], Art. 211, is satisfied. Consequently w is monogenic. 
 This accounts for the results of Arts. 38-42. 
 
 If w is a multiple-valued function of z, there may be several 
 
 different values of — , corresponding to the same value of z ; 
 dz 
 
 but if w is monogenic, each of these values depends only upon z, 
 
 and not upon the way in which z is supposed to change. 
 
 In future, unless something is said to the contrary, we shall 
 
 give the name function only to monogenic functions. Thus we 
 
 shall not call such expressions as x — yi, or a? + y^ -^-^xyi, 
 
 functions of z. 
 
 Conjugate Functions. 
 
 213. If u and v are functions of x and y, satisfying equations 
 [3], Art. 211, it is easy to prove that 
 
 « 
 
 DJ'u + D^-'u=0 and DJv + DJ'v = 0. 
 
Chap. XVII.] THEORY OF FtJNCTIONS. 289 
 
 For since D,u = D,v and D^v = -D^u, 
 
 we have D^u = B,D,v and D,^u==- D,D,v, 
 
 D,^v= - D,D,u and Z»/« =D,D,u ; 
 u and V are then solutions of Laplace's equation, 
 
 Z>/F+Zl/F=0. [1] 
 
 Any two functions <J3 and xji ot x and y, such that 
 '!> i^j y) + i^ (^T y) is a monogenic function of x+yi, are 
 called conjugate functions ; and, by what has just been proved, 
 each of a pair of conjugate functions is always a solution of 
 Laplace's Equation [1]. 
 
 Thus x' — y^^ 'ixy; e'cosy, e^siny; ^log (a^+ y^), tan^'^; 
 
 X 
 
 are three pairs of conjugate functions, since af — y^-i-2xyi 
 = (a; + yiy, e' cosy + ie' siny = e*+'"', -J log (x^ + y^) + i tan~^^ 
 
 X 
 
 = log (a; + yi) , and consequently, by Art. 212, are all monogenic. 
 Therelbre each of the six functions at the beginning of this 
 paragraph is a solution of Laplace's Equation [1]. 
 
 It IS clear that we can form pairs of conjugate functions at 
 pleasure by merely forming functions of x + yi and breaking 
 them up into their real parts, and their pure imaginary parts ; 
 that is, throwing them into the typical form u + vi. 
 
 If each of a pair of conjugate functions, <^ and i/*, is written 
 equal to a constant, the equations thus formed will represent a 
 pair of curves which intersect at right angles. For let {x, y) 
 be a point of intersection of the curves ^ = a, ij/=b ; the slopes 
 
 of the two curves at (x, y) axe. respectively — -=~, — — ^ by 
 
 I., Art. 202; and since D^<^=Dy\p and D^ip = — D^4>, the 
 second slope- is minus the reciprocal of the first, and the curves 
 are perpendicular to each other at the point in question. 
 
 Thus x^ — y^ = a, 2xy= b, cut each other orthogonally ; as do 
 
290 LNTEGEAL CALCULTJS. [Akt. 214. 
 
 also ^log(a^ + 2/^) = a, tan~^'^ = 6; or, what amounts to the 
 
 X 
 
 same thing, a^ + «^ = a, ^ = fej. It must be observed, how- 
 
 X 
 
 ever, that x' + y^ and " are not conjugate functions, and that 
 
 X 
 
 in general the converse of our proposition does not hold. 
 
 It maj' be easily proved that if (j> and \f) are conjugate func- 
 tions of X and y, and /and F are any second pair of conjugate 
 functions of x and y, the new pair of functions formed by re- 
 placing X and y in (j> and ij/ bj' / and F respectively wiU be 
 conjugate. 
 
 Thus (e*cos?/)^ — (e^sin?/)^, 2e'cos?/.e''siny, 
 
 or, reducing, e^ cos 2 2/, e^ sin 2 2/, 
 
 are conjugate functions ; 
 
 or, reducing, log (a^ -H 2/^) , tan^M ^^ X 
 
 are conjugate. 
 
 The properties of conjugate functions given in this article 
 are of great importance in many branches of Mathematical 
 Physics. 
 
 Example. 
 
 Show that if x' and y' are conjugate functions of a; and y, 
 X and y are conjugate functions of x' and y' . 
 
 Preservation of Angles. 
 
 214. If 10 is a single-valued monogenic function of z, and 
 the point representing z traces two arcs intersecting at a given 
 angle, the corresponding arcs traced by the point representing 
 w will in general intersect at the same angle. 
 
Chap. XVII.] 
 
 THEORY OP FUNCTIONS. 
 
 291 
 
 For let Zq be the point of intersection of the curves in the z 
 plane, and Wq the corresponding point in the w plane. Let »j be 
 a point on the first curve, and z^ a point on the second ; and let 
 
 Wj and w^ be the corresponding points in the w figure. 
 
 Let Vi, r^, Si, and s^ be the moduli of % — Zq, z^ — »„ Wj — Wq, 
 and Wj — Wo respectively, <^i, ^^, i/rj, and i/zj their arguments ; 
 then, since w is a monogenic function of «, we must have 
 
 r^ 
 
 limit 
 
 limit fil-^i^^' 
 Lricis^i_ 
 
 whence, by Art. 23, 
 
 = limit 
 
 or 
 
 : limit - 
 
 U 
 
 L«2-«0 J' 
 
 »-2 cis <^2 J ' 
 
 limit 
 
 Sl 
 
 cis (l/r, - <^i) 
 
 = limit — cis (i/'g • 
 
 ■<^2)"|; 
 
 and since, when two imagiuaries are equal, their moduli must 
 be equal, and their arguments must be equal, unless the moduli 
 are both zero or both infinite, 
 
 limit (i/'2 — i/fi) = limit (^2 — ^1) ; 
 
 that is, the angle between the arcs in the w figure is equal to 
 the angle between the corresponding arcs in the z figure ; unless 
 
 tdw~] _ 
 dz\z=z^ 
 
 0, or 
 
 dw 
 dz 
 
 If w is a multiple-valued monogenic function of z, and if 
 starting from any point Zq, the point which represents z traces 
 
292 INTEGRAL CALCULUS. [Art. 210, 
 
 out two curves intersecting at an angle a, eacli of the »t points 
 representing the corresponding values of w will trace out a pair 
 of curves intersecting at the angle a ; unless «o is a point at 
 
 which — is zero or infinite. 
 dz 
 
 If, then, w is any monogenic function of 2, and the point 
 representing z is made to trace out any figure however complex, 
 the point representing w will trace out a figure in which all the 
 angles occurring in the z figure are preserved unchanged, except 
 those having their vertices at points representing values of z 
 
 which make — zero or infinite. 
 dz 
 
 This principle leads to the following working rule for trans- 
 forming any given figure into another, in which the angles are 
 preserved unchanged. 
 
 Substitute x' and y' for x and y in the equations of the curves 
 which compose the given figure, x' and 'j being any pair of 
 conjugate functions (Art. 213) of x and y, and the new 
 equations thus obtained will represent a set of curves forming 
 a second figure in which all the angles of the given figure are 
 preserved unchanged, except those having their vertices at 
 points at which Djc' and D^y' are both zero, or at which one of 
 them is infinite. 
 
 For ex-nniple, x — y = a, • (1) 
 
 x+y=b, (2) 
 
 are a pair of perpendicular right lines. Replace xhy a? — y^ 
 and 2/ by 2 xy, and we get 
 
 a? —■2xy — fz=a, (3) 
 
 a?+2xy-y'' = b, (4) 
 
 a pair of hyperbolas that cut orthogonally. 
 
 215. If t« is a single-valued continuous function of z, it is 
 clear that if w„ and Wj are the values corresponding to z„ and «i, 
 
Chap. XVII.] THEORY OF FUNCTIONS. 293 
 
 and the point z moves from «„ to «! by two different paths, the 
 corresponding paths traced by w will begin at Wq and end at Mi, 
 and consequently that if z describes any closed contour, w also 
 will describe a closed contour. 
 
 If «o is a double-valued function of 2, since to each value of 
 z there will correspond two values of w, it is conceivable that 
 if Ml and w-^ are the values of lo corresponding to «i, and z moves 
 from »o to ^i by two different paths, w may in one case move 
 from Wo to Wi, and in the other case from w„ to w/. 
 
 It can be proved, however, that if the two paths traced by z 
 do not enclose a critical point (Art. 210), and w is finite and 
 continuous for the portion of the plane considered, this will 
 not take place, and that the two paths starting from w^ will 
 terminate at the same point Wi. We give a proof for the case 
 where « is a single-valued function of w. 
 
 As z traces the first path, each of the two points repre- 
 senting the two values of w will trace a path, one starting at Wq, 
 and the other at w„\ and unless the z path passes through a 
 critical point, the two w paths will not intersect, but will be 
 entirely separate and distinct, and will lead, one from w^ to %, 
 the other from Wq' to Wi'. 
 
 If, now, the z path be gradually swung into a second position 
 without changing its beginning or its end, since w is a continu- 
 ous function, the two w paths will be gradually swung into new 
 positions ; but, provided that the z path in its changing does not 
 at any time pass through a critical point, the two w paths will 
 at no time intersect, and consequently it will be impossible for 
 the w points to pass over from one path to the other, and there- 
 fore the point which starts at Wq must always come out at Wj, 
 and not at Wi'. 
 
 It follows readily from this reasoning that if z describes a 
 closed contour not embracing a critical point, each of the w 
 points will describe a closed contour, and these contours will 
 not intersect. 
 
 Of course, the proof given above holds for any multiple- 
 valued function. 
 
 In any portion of the plane, then, not containing critical 
 
294 
 
 INTEGEAL CALCULTTS. 
 
 [Akt. 216, 
 
 points the separate values of a multiple-valued function may be 
 separately considered, and may be regarded and treated as 
 single-valued functions. 
 
 216. That in the case of a double-valued function two paths 
 in the z plane, including between them a critical point, but 
 having the same beginning and ttie same end, may lead to 
 different values of the function, is easily shown by an example. 
 
 Let TO = Vx, and let z, starting with the value 1, move to the 
 value — 1 by the semi-circular path in the figure. That one of 
 
 —1 
 
 -fl 
 
 Fig. 1. 
 
 -fl 
 
 the corresponding values of to which starts with -f- 1 will de^ 
 scribe the quadrant shown in the figure, and will reach the 
 
 If, however, z moves from -|- 1 to — 1 by 
 
 point 1 .cis-, or 
 
 Fig. 2. 
 
 -fl 
 
 the semi-circular path in the second figure, the value of w which 
 starts with -f 1 will describe the quadrant shown in the second 
 
 figure, and will reach the value l.cisf — -|, or —i. These 
 
 two paths described by «, then, although beginning at the same 
 point -f- 1 and ending at the same point — 1 , cause that value 
 of the function which begins with -f 1 to reach two different 
 values ; and the two paths in question embrace the point a = 0, 
 which is clearly a point at which the two values of w, ordinarily 
 different, coincide ; that is, a critical point. 
 
Chap. XVII.] 
 
 THEORY OP FUNCTIONS. 
 
 295 
 
 It is easily seen that if z, starting with the value +1, de- 
 scribes a complete circumference about the origin, the value of 
 w which starts from the point + 1 will not describe a closed 
 contour, but will move through a semi-circumference and end 
 with the point l.cisir or —1. Now, by Art. 215 any path 
 
 Fio. 3. 
 
 described by z beginning with + 1 and ending with — 1 and 
 passing above the origin, since it can be deformed into the 
 semi-circumference of Fig. 1 without passing through a critical 
 point, will cause the value of w beginning with + 1 to end with 
 + i ; and any path described by z beginning with + 1 and end- 
 ing with — 1 and passing below the origin, since it can be 
 deformed into the semi-circumference of Fig. 2 without passing 
 through a critical point, will cause the value of w beginning 
 with -|- 1 to end with — i. Therefore any two paths described 
 by z beginning with -fl and ending with —1 will, if they include 
 the critical point z = between them, lead to different values 
 of w, provided that the same value of w is taken at the start. 
 
 217. If w is a double-valued function of z, and z describes a 
 closed contour about a single critical point, this contour may be 
 deformed into a circle about the critical point, and a line lead- 
 ing from the starting point to the circumference 
 of the circle, without affecting the final value of 
 w (Art. 215). Thus, in the figure, the two 
 paths ABODA, AB'C'D'B'A lead from the 
 same initiid to the same final value of lo ; and 
 this is true no matter how small the radius of 
 the circle B'C'D'. 
 
296 
 
 INTEGRAL CALCTJLTTS. 
 
 [Abt. 21 T 
 
 Let z„ be the critical point, and let Wq be the corresponding 
 point in the w figure. As z moves from Zi towards 2o> the points 
 
 representing the corresponding values of w will start at Wi and 
 Ml' and move towards «„, tracing distinct paths. 
 
 If, now, z describes a circumference about Zq, and then 
 returns along its original path to «i, the first value of ,w will 
 either make a complete revolution about t«o and return along 
 the branch (1) to its initial value Mi, or it will describe about 
 
 Wq a path ending with the branch (2) of the w curve, and move 
 along that branch to the value to/. 
 
 In the first case, and in that case only, the value of ?" 
 describes a closed contour when z describes a closed contour, 
 and is practically a single-valued function. 
 
 If Zo is a point at which — is neither zero nor infinite 
 
 dz 
 
 {v. Art. 214), when z describes about Zo a circle of infinitesimal 
 radius, w will make about iVg a complete revolution ; for since 
 if two radii are drawn from Zg, the cur\'es corresponding to them 
 will form at m'o an angle equal to the angle between the radii, 
 when a radius drawn to the moving point which is describing 
 the circle about 2„ revolves through an angle of .360°, the cor- 
 
Chap. XVII.] THEORY OF FUNCTIONS. 297 
 
 responding line joining iv,, with the moving point representing m 
 will revolve through 360°, and we shall have what we have 
 called Case I. 
 
 If, then, we avoid the points at which — is zero or infinite, 
 
 dz 
 we shall avoid all critical points that can vitiate the results 
 obtained by treating our double- valued or multiple-valued func- 
 tions as we treat single-valued functions. 
 
 A critical point of such a character that when z describes a 
 closed contour about it the corresponding path traced by any 
 one of the values of w is not closed, we shall call a branch point. 
 
 When a function is finite, continuous, and single-valued for 
 all values of z lying in a given portion of the z plane, or when 
 if multiple-valued it is finite and continuous, and has no branch 
 points in the portion of the plane in question, it is said to be 
 holomorphic in that portion of the plane. 
 
 Definite Integrals. 
 218. In the case of real variables, | fz.dz was defined in 
 Art. 80 in effect as follows : 
 
 X 
 
 /a . dz = J™'^ [/«o (z,— z,) +fz^ (z, - z,) +fz,{z, - z,) + ■■■ 
 
 +A-l(^-«n-l)], [1] 
 
 where Zi, Z2, Zg, ...z„_i are values of z dividing the interval 
 between Zq and Z into n parts, each of which is made to 
 approach zero as its limit as n is indefinitely increased. 
 
 is the line integral of fz (Art. 163) taken 
 
 along the straight line, joining »o and Z if Zq and Z are repre- 
 sented as in the Calculus of Imaginaries. 
 
 It has been proved that ii fz is finite and continuous between 
 So and Z, this integral depends merely upon the initial and final 
 values of z, and is equal to FZ—Fzg where Fz is the indefinite 
 
 integral I fz.dz. 
 
298 INTEGKAL CALCTJLTJS. [ART. 218. 
 
 If 2 is a complex variable, and passes from z^^ Z along any 
 
 given path, we shall still define the definite integral 1 fz.dz by 
 
 [1] where now z^, %, 23, ...2„_i are points in the [jiven path. 
 
 Two important results follow immediately from this defini- 
 tion : 
 
 r°fz.dz=-r fz.dz, [2] 
 
 1st. That 
 
 if z traverses in each integral the same path connecting Zo and Z. 
 
 2d. That the modulus of | fz. dz is not greater than the 
 
 line-integral of the modulus of fz taken along the given path 
 joining z^ and Z. 
 If we let 
 
 fz — w = u + vi, z = x + yi,u = <j} (x, y) , and v = tj/ (x, y) , 
 then I fz.dz= I {u+ vi) (dx + idy) 
 
 = J<^ («, y) dx + ij ij/ {x, y)dx- jxl/ (x, y) dy + iC<t> (x, y) dy, 
 
 [3] 
 each of the integrals in the last member being the line-integral 
 of a real function of real variables, taken along the given path 
 connecting Zq and Z. 
 
 If the given path is changed, each of the integrals in the 
 last member of [3j will in general change, and the value of 
 
 Jfz . dz will change ; and, since z may pass from z^ to Z by an 
 
 infinite number of different paths, we have no reason to expect 
 
 that I fz.dz will in general be determinate. 
 
 We shall, however, prove that in a large and important class 
 
 of cases | fz.dz is determinate, and depends for its value 
 
 upon Zo and Z, and not at all upon the nature of the path 
 traversed by z in going from z^ to Z. 
 
Chap. XVII.] THEORY OF FUNCTIONS. 299 
 
 219. If fz is holomorphic in a given portion of the plane, 
 
 f'°fz.dz = Q [1] 
 
 if z describes any closed contour lying wholly within that 
 portion of the plane. 
 
 From [o], Art. 215, we have 
 
 ( ''fz.dz= i w.dz= j udx + i | vdx — i vdy + i | udy, [2] 
 
 the integral in each case being the line-integral around the 
 closed contour in question. 
 
 Since iv = fz is holomorphic, m = <^ («, y), and v = ij/ (x, y) , 
 and D^u, DyU, D^v, and D^v are easily seen to be finite, con- 
 tinuous, and single-valued in the portion of the plane considered. 
 Therefore, by Art. 170, 
 
 I udx = I I D^udxdy ; | vdx = ( 1 DyVdxdy ; 
 I vdy = — j j D^vdxdy ; | udy = — j j D^udxdy ; 
 
 the integral in the first member of each equation being taken 
 around tlie contour, and that in the second member being a 
 surface-integral taken over the surface bounded by the contour. 
 We have, then, from [2], 
 
 I '°fz .dz= C C (DyU+ D^v)dxdy+ i C C{DyV—D,u)dxdy, [3] 
 
 but D,u = DyV, and D^u = - D^v from [3] , Art. 211. Therefore, 
 [3] reduces to | °fz .dz = 0. 
 
 From this result we get easily the very important fact that if 
 fz is holomorphic in a given portion of the plane, j fz.dz will 
 
 have the same value for all paths leading from z^ to Z, provided 
 they lie wholly in the given part of the 
 plane. For let z^^aZ and ZgbZ be any 
 two paths not intesecting between Zq 
 and Z. Then z„aZbZo is a closed con- 
 tour, and 
 
300 
 
 INTEGRAL CALCULTTS. 
 
 [Abt. 220. 
 
 but 
 
 J'fz . dz (along ZoaZbZo) 
 = C^fz.dz (aloDg z„aZ) + j °/z.dz (along ^62:0) = 0; 
 
 r°fz . dz (along Zbzo) = - C fz.dz (along ZobZ) 
 
 Jz J'a 
 
 by Art. 218. 
 Therefore, fV^ • f^« (along 2oaZ) = j /z.dx (along Zo6Z). 
 
 If the paths 2oa.Z and z^bZ inter- 
 sect, a third path Zs^cZ may be drawn 
 not intersecting either of them, and 
 by the proof just given 
 
 J fz.dz (along Zoa^)= j fz.dz (along «oC.Z), 
 
 J fz.dz (along z^bZ) = C fz.dz (along ZocZ) ; 
 
 therefore, 
 
 J fz.dz (along «oa.^) = j fz.dz (along z„6Z). 
 
 220. If /z, while in other respects holomorphic in a given 
 
 portion of the plane, becomes infinite for a value T of 2, then 
 
 ifz . dz taken around a closed contour embracing T, while not 
 
 zero, is, however, equal to the integral taken around any other 
 
 closed path surrounding T. 
 
 For let ABCD be any closed eon- 
 tour about T. With T as a centre, 
 and a radius e, describe a circumfer- 
 ence, taking e so small that the cir- 
 cumference lies wholly within^jBCD. 
 Join the two contours by a line AA'. 
 Then ABCDAA'D'C'B'A'A is a 
 closed path within which fz is holo- 
 morphic. 
 
Chap. XVII.] THEORY OF FUNCTIONS. 301 
 
 Therefore, 
 J/z . dz (along ABCDAA'D'C'B'A'A) = 0, 
 
 or Cfz . dz (along ABCDA) + Cfz . dz (along AA') 
 
 + ffz.dz (along A'D'C'B'A') + Cfz . dz (along A' A) = ; 
 
 but 
 
 and 
 
 Cfz . dz (along AA') =- Cfi- dz along (A' A) , 
 
 id 
 Cfz . dz (along A'D'C'B'A') = - Cfz. dz (along A'B'C'D'A') 
 
 Hence 
 
 frz . d2 (along ABCDA) = j/^ • ^2 (along A'B'C'D'A'). 
 
 221. That the integral of a function of z around a closed 
 contour embracing a point at which tlie function is infinite is 
 not necessarily zero is easily shown by an example. 
 
 fz^ , t being a given constant, is single-valued, con- 
 
 z — t 
 
 tinuous, and finite throughout the whole of the plane except at 
 
 the point t, at which becomes infinite, without, however, 
 
 z—t 
 
 ceasing to be single-valued. 
 
 /dz 
 around a circle whose centre is t, and 
 z — t 
 
 whose radius is any arbitrarily chosen value c. If z is on the 
 
 circumference of this circle 
 
 't- ^ 
 
 z — f = £ (cos <^ + I sin <p) 
 
 = £6*' by [5], Art. 31. 
 
 and dz=ue'l"d<j>. ~ 
 
302 INTEGRAL CALCULUS. [Akt. 221. 
 
 Hence f-^ (around abc) = f i!?!l# = 2^'. 
 
 J z — t ^ Jo te*' 
 
 /. 
 
 From what has been proved in Art. 220, it follows that 
 » j^ 
 
 around any closed contour embracing t must also be 
 
 IS ~^ t 
 equal to 2Tri. 
 
 /Fz 
 dz, when Fz is 
 z — t 
 
 supposed to be holomorphic in the portion of the plane con- 
 sidered, and where the integral is to be taken around any closed 
 contour embracing the point « = f . 
 
 Fz 
 
 - — is holomorphic except at the point z = t, where it 
 z—t 
 
 becomes infinite. The required integral is, then, equal to the 
 
 integral around a circumference described from the point t as 
 
 a centre, with any given radius e, that is, by the reasoning just 
 
 r dz 
 used in the case of I , to 
 
 J z — t 
 
 r- F(t + .e*')ue'^<d^ ^ or i T F (t + .e^') d^ ; 
 
 Jo £6*' Jo 
 
 and in this expression e may be taken at pleasure. If now e is 
 made infinitesimal ee*' is infinitesimal, and since Fz is continu- 
 ous F{t + £6*') is equal to i^t -f- j; where rj is some infinitesimal, 
 and F(t + ee*') d<j} is equal to Ft .d<ji -i-r].d<l>. 
 Now, by I. Art. 161, 
 
 {Ft.d<^ + -qd^)= 1 Ft.d^. 
 
 /•2ir ^ail- 
 
 Hence I I F{t + i.(?'-)d<^ =i\ Ft.d<i> = 2-,riFt; 
 
 and we get the important result that i dz^ taken around anj 
 
 contour including the point z = t, is equal to 2 iri . Ft. 
 
 From this we have Ft = — — i^^dz ; 
 
 2 TriJ z — t 
 
Chap. XVII.] THEORY OF FUNCTIONS. 303 
 
 and we see that a holomorphic function is determined every- 
 where inside a closed contour if its value is given at every point 
 of the contour. 
 
 If in the formula Ft = -— f-^ dz m 
 
 'iiriJ z — t ■■ -■ 
 
 we change ttot + Af , we get 
 
 AJ'i^J- Cwz.dzf ^ LVJ_ C Vz.A^.^t . 
 
 'i.-KiJ \z — t—At z—tj iiriJ (z—t){z—t—M) 
 
 whence 
 
 imit r^-\ ^ j_ Tj,, . ^, . limit r 1 n 
 
 '*=OLAfJ 2iTiJ Af=0[ (2 _«)(«_(_ A«) J' 
 
 limit 
 A« 
 
 m = ^CI^^; [2] 
 
 and in like manner we get 
 
 2,rjJ {z-ty' ■■ -■ 
 
 and in general jrwi^^jil C Fz.dz j.^- 
 
 ^ 2W (2-0""" 
 
 each of the integrals in these formulas being taken around a 
 closed contour lying wholly in that portion of the plane in which 
 Fz is holomorphic, and enclosing the point z = t. 
 
 222. The integral of a holomorphic function along any given 
 path is finite and determinate, for, by [3], Art. 218, it iS equal 
 to the sum of four line integrals, each of which is finite and 
 determinate (Art. 166). 
 
 If a series Wq-\-Wi-\- w^ + •", where Wa, w^, w^ •■• are holo- 
 morphic functions of z, is uniformly convergent for all values of 
 s in a certain portion of the plane, the integral of the series 
 along any given path lying in that portion of the plane is the 
 series formed of the integrals of the terms of the given series 
 along the path in question, and the new series is convergent. 
 
304 INTEGRAL CALCULUS. [Akt. 223. 
 
 For, let (S = Wo + Wi + Wa -I !-«'„ + «'„+ 1 -\ 
 
 = Wo + w-i + W2-\ \-w„-\- 2?„, 
 
 where ^n — ■>^n+i + '^n+2+ "■> 
 
 and where by hypothesis ?i may be taken so great that the 
 modulus of i?„ is less than e for all values of « in the portion 
 of the plane in question, e being a positive quantity taken in 
 advance and as small as we please. 
 
 j Sdz = I Wgdz -{- i Wids-\ 1- I w^dz + | B^ds 
 
 for any given value of n. 
 
 By the proposition at the beginning of this article, j Sds 
 along the given path is finite and determinate, as are also 
 
 I Wodz, I Wxdz, etc. 
 
 The modulus of I R^ dz is not greater than the line-integral 
 
 along the given path of the modulus of R^ (v. Art. 218). If, 
 now, n is taken suificiently great, each value of the modulus 
 of -Z?„ will be less than e ; consequently each element of the 
 cylindrical surface representing the line-integral of the 
 
 modulus of JS„ will be less than e (v. Art. 166), and j R^dz 
 
 will be less in absolute value than e multiplied by the length 
 of the path along which the integral is taken. 
 
 Therefore, | Sdz ^= I w^dz -\- I w-^dz -\- i w^dz -\- — ; 
 
 and, since the first member is finite and determinate, the 
 second member is a convergent series. 
 
 Taylor's and Madaurin's Theorems. 
 1 — g" 
 
 identically, if » is a positive integer, even when q is imaginary. 
 
 223. J— ^=l4-j-|-^2_|.j3 4-...j»- 
 
Chap. XVII.] 
 
 THEORY OF PUNOTIONS. 
 
 305 
 
 If the modulus of q is less than 1 
 
 limit 
 
 w=oo 
 
 Hence l+g + g' + g*+- 
 
 [?"] = 0. 
 
 limit 
 
 
 [1] 
 
 even when q is imaginary, provided that the modulus of q is 
 less than 1. 
 
 Suppose, now, that everywhere within and on a certain cir- 
 cumference described with the point z = a as a centre Fz is 
 holomorphic. Let 2= i be any point within this circumference, 
 and a = Z be a point on the circum- 
 ference. Then the modulus ot Z — a 
 is the distance from a to Z, and the 
 modulus of i — a is the distance from 
 a to i; 
 hence mod (J — a) < mod {Z — a), 
 
 and 
 
 mod 
 
 ft-a\ 
 \Z-aJ 
 
 <1. 
 
 Z—t Z—a — {t — a) Z — a ^ 
 
 Hence 
 
 1 r t-a jt-aY (t-aY 1 
 
 ~Z-al'^Z-a'^ (Z-aY {Z-aY J' 
 
 1 ^_l_.__tz^_4.St:z^4.Atz3l!.+ ... m 
 
 Z-t~Z-(t{Z-aY {Z-aY {Z-aY 
 and the second member of [2] is a convergent series. 
 
 Multiply [2] by —,, and the series will still be convergent 
 for each value of z which we have to consider ; we get 
 1 FZ 
 
 2in Z-t 
 
 =i^[S-('-'(i^-«'-°''(i^-} 
 
 [3] 
 
306 INTEGRAL CALCULUS. [Art. 223. 
 
 Integrate now both members of [3] around the circumfer- 
 ence, and we have 
 
 27riJ Z-t -l-KiyJ Z-a ^ 'J {Z-ay 
 
 and, since each of the functions to be integrated is holomorphic 
 on the contour around which the integral is taken, and the 
 second member of [3] is convergent, each integral will be finite 
 and determinate, and the second member of [4] will be con- 
 vergent. 
 
 Substituting in [4] the values obtained in Art. 221, [1], [2]. 
 [3], and [4], we have 
 
 Ft=^Fa + {t-a)Fa+ (^~")V 'a + (* - ") V"a + ... 
 
 n ! 
 If the point z = a is at the origin, a = and [5] becomes 
 
 Ft=Fo-irtFo + ^F"o+~F"'o + .:, [6] 
 
 which is Maclaurin's Theorem. 
 
 That [5] is merely a new form of Taylor's Theorem is easily 
 seen if we let « — a = h, whence t = a+h, and [5] becomes 
 
 F(^a + h)=Fa + hF'a + ^F"a+^F"'a + .... [7] 
 
 [6] can, of course, be written 
 
 Fz = Fo + zF'o + i.^F"o +^F"'o + ..., [8] 
 
 and [5] as 
 
 Fz = Fa + (z - a) F'a + (izl^V"a + (in£lV"a + ... ; 
 
 [9] 
 
 [5] 
 
Chap. XVU.] THEORY OF PUNCTKJNS. 307 
 
 and we get the very important result that if a function of z is 
 holomorpMc within a circle lohose centre is at the origin it may 
 be developed by Maclaurin's Theorem, and the development will 
 hold, that is, tlie series wiU be convergent, for all values of z 
 lying within the circle. 
 
 If a function of z is holomorpliie within a circle described 
 from « =a as a centre it can be developed by Taylor's Theorem 
 into a series arranged according to powers of z — a, and the 
 development will hold for all values of z lying within the circle. 
 
 The question of the convergency of either Taylor's or 
 Maclaurin's Series for the case when z lies on the circum- 
 ference of the circle needs special investigation, and will not 
 be considered here. 
 
 If the function which we wish to develop is single- valued, in 
 drawing our circle of convergence we need avoid onl3- those 
 points at which the function becomes infinite ; but if it is 
 multiple-valued we must avoid also those at which its derivative 
 is zero or infinite (v. Art. 217). 
 
 224. "We are now able to investigate from a new point of 
 view the question of the convergence of the series obtained by 
 Taylor's and Maclaurin's Theorems in I. Chap. IX. 
 
 Let us begin with the Binomial Theorem, 
 
 (a) (ffl-t-/0" = «''-Fwa"-'/t-Hw ^''~^^ a"-''fe^-H-. [1] 
 
 or, following the notation of [9] , Art. 220, 
 
 If n is a positive integer, «" is holomorphic throughout the 
 whole plane, and [2] holds for all values of z and a, and [1] 
 for all values of a and h. 
 
 If m is a negative integer, z" is single-valued, and it is linito 
 and continuous except for 2 = 0, where «" becomes infinite. 
 [2] is, then, convergent for all values of z lying within a circle 
 described with a as a centre and passing through the origin ; 
 
308 INTEGEAL CALCULUS. [Abt. 224. 
 
 that is, for all values of z, such that mod {z — a) < mod a ; and 
 
 consequentl3- [1] holds if modh<moda. 
 
 If n is a fraction, z" is multiple-valued, and our circle of 
 
 dz" 
 convergence must avoid the points at which — becomes zero 
 
 dz 
 
 or infinite ; but as the origin is the only point of this character, 
 the circle of convergence is the same as in the case last con- 
 sidered, and [1] holds for all cases where mod h<, mod a. 
 
 ■When a and h are real our results agree with those obtained 
 in I. Art. 131. 
 
 (6) e' = e'+'" = e'(cosy + 4sin2/) ([4], Art. 31) 
 
 is single-valued and continuous, and becomes infinite only when 
 a; =00. Maclaurin's development for e^ holds, then, for all 
 finite values of z. 
 
 (c) log« = log (r cis <^) = log r + <l>i (Art. 33) 
 
 is finite and continuous throughout the whole plane. It is, 
 
 however, multiple-valued, but its derivative - becomes infinite 
 
 z 
 
 only when 2 = 0, and does not become zero for any finite value 
 
 of 2. log2, then, can be developed into a convergent series, 
 
 arranged according to powers of z — a, for all values of z within 
 
 a circle having the centre a and passing through the origin ; 
 
 that is, for all cases where mod {z — a)< mod a. 
 
 11 z — a = h, we get 
 
 7i 7)2 h^ 7i< 
 log (a + h) = loga -h^ - ±-^ + ^^ - A_ -H ..., [3] 
 
 a 2a^ 3fr 4a^ 
 
 [3] holding for all cases where rnod li < m,od a. 
 If a = 1 and h = z, we get 
 
 log(l+.) = ^^-|Vf-f+-, [4] 
 
 which holds for all values of z where mod z < 1 . 
 
 (d) sin z = sin (a; -f- yj) = sin x . ^""^^ " -f- i cos x ■ ^" ~ ^~l. 
 
Chap. XVII.] THEORY OF FUNCTIONS. 309 
 
 and cosz = cos {x-\-yi) = COS X — t since- -— , 
 
 (v. [3] and [4], Art. 35) 
 are single-valued, and are finite and continuous throughout the 
 plane. Therefore, Maclaurin's developments for sin 2 and cosa 
 hold for all values of z. 
 
 (e) taa2 = , and secz = , are single-valued and 
 
 cosz cosz 
 
 continuous, and become infinite only when cos« = 0; that is, 
 
 when 2 = -. Therefore, Maclaurin's developments for tang and 
 
 secz (I. Art. 138), hold for every value of z whose modulus is 
 
 less than -• 
 2 
 
 (/) etn z = , and csc« = become infinite when 2 = 0, 
 
 sin 2 sin 2 
 
 and cimnot be developed by Maclaurin's Theorem. 
 
 {g) sin"' 2 is finite and continuous throughout the plane; it 
 
 1 
 IS, however, multiple-valued, and its derivative , ^ becomes 
 
 infinite when z = 1, and when z = — 1. Therefore, the develop- 
 ment for sin'^z (I. Art. 135 [2]), holds for any value of z 
 whose modulus is less than 1. 
 
 (A) tan"' 2 is finite and continuous throughout the plane ; it 
 is multiple-valued, and its derivative becomes infinite 
 
 when 2 = i, and when z = — i. Therefore, the development for 
 tan"'z (I. Art. 135 [1]), holds if modz <.modi; that is, if 
 modz < 1. 
 
 Examples. 
 
 (1) Show that the development of 1- log (1 -f z) , given 
 
 1+2 
 
 .in I. Art. 136, Ex. 1, holds if mod2< 1. 
 
 (2) Show that the development of log (1 + e') , given in I. 
 Art. 136, Ex. 2, holds if mod2<ir. 
 
 (3) Obtain the following developments, and find for what 
 real values of x tliey hold good : 
 
310 INTEGRAL CALCULUS. [Abt. 224. 
 
 /SI / , r^r-, — 2x 1 , a; 1 a;' , 1.3 3? 
 
 (6) (e' + e-)" ■ =2»(^l+^ + n(3n-2)|l+. 
 
 (c) e'.cosa; =\-\-x — — — -—■■■ 
 
 /^v .„,. A a^ , 4a;* 31 K^ \ 
 
 (e) eMogCl+x) =a; + |^ + |^ + |^... 
 
 (/) tan^a; = «,. + 4|.' + 6|^... 
 
 (^) (l + 2a;+3a=)~^ =1 -a; + 2w'-^". 
 
 (A) e"*-"'' =\j^x -{■---... 
 
 /■■N . , 2 , , 2' , 2M7 7 , 2«.31 „ , 2'. 691 u 
 
 0) tana;=a;+— a;'+— ar'+^^aj'H — §7~'^+ ^^ , ^ • 
 
 («) a;, etna; = 1- 
 
 3 45 945 4725 93555 
 
 (0 log tanC^+a;^ = logtanJ+2a;+|^af»+||-V+^V ... 
 
 , \ .h,. 1 , a; , a.-^ 3a;* 8a;' 3a;« , 56a;' , 
 ^ ' 1! 2! 4! 5! 6! 7! 
 
 /• \ t.„. 1 , , a^ , Sa--^ , 9a;* , 37ar' , 177a;« 
 ^ ' 2!3!4!5!6! 
 
 (o) (versin-'a;)^ =2(^a; + ^ + ^ ^ + ^4^ -+-Y 
 
 3.2 3.5 3 3.5.7 
 
 , . 1 ^2 2a; a;' 2a;* 2g°_a; 
 
 ^^^ 2-2a; + a;2 4 4 4 4^ 4^ 42'" 
 
 (•o\ 1 + ^ ._._ 3 ^ = /^3_4\ /3_£^ 
 
 ^^'' 6-5a; + a;2 2 - a; 3 - a; U 3/^2= S^^'"' 
 
Chap. XVII.] THEORY OF FUNCTIONS. 311 
 
 Answers. 
 (a) —a<x<a; 
 
 (6) -oD<a;<ooif n>0, -^ < a;<| if n< ; 
 
 (c) — oo<a;<oo; (d) — oo<a;<oo; 
 
 (e) -l<a;<l; (/) -\<^<\-^ 
 
 {g) — iV3<a;<iV3; (h) -l<a;<l; 
 
 (0 -l<a;<l; 0') -|<a'<|; 
 
 (ft) -,r<a;<^; (0 -|<'»<j5 
 
 (m) -oc<a;<oo; (w) -^<^<^'' 
 
 (o) -2<a;<2; (p) -V2<a;<V2: 
 
 (g) -2<a;<2. 
 
312 INTEGRAL CALCULUS. [ART. 22b. 
 
 CHAPTER XVIII. 
 
 KEY TO THE SOLUTION OF DIFFERENTIAL EQUATIONS. 
 
 225. In this chapter an analytical key leads to a set of con- 
 cise, practical rules, embodying most of the ordinary methods 
 employed in solving differential equations ; and tlie attempt has 
 been made to render these rules so explicit that they may be 
 understood and a[)plied by any one who has mastered the Inte- 
 gral Calculus proper. 
 
 The key is based upon "Boole's Differential Equations" 
 (London : IMacmillan & Co.), to which or to " Forsyth's Differ- 
 ential Equations" (London: Macmillan & Co.), we refer the 
 student who wishes to become familiar with the theoretical 
 considerations upon which the working rules are based. 
 
 226. A differential equation is an expressed relation involv- 
 ing derivatives with or without the primitive variables from 
 which they are derived. 
 
 For example : 
 
 (l+a;)2/-f (l-2/)xg = 0, (1) 
 
 x]/--ay = x + l, (2) 
 
 dx 
 
 dx 
 
 x'^ — y + x\la? — if = Q, (3) 
 
 ^m=^.(^,cly 
 
 y+''±Y (4) 
 
 \dx)-'\'^dx, 
 
 da^ dx^^ dx" dx^^~ ' 
 
 (5) 
 
Chap. XVIII.] DIFFERENTIAL EQUATIONS. KEY. 313 
 
 sin^a;— ^ + sina!cosa;-^ — t/ = a; — sina;, (6) 
 «'(l-^rg-2y = 0, (7) 
 
 D,'z-aWJ'z = 0, (9) 
 
 are differential equations. 
 
 The 07-der of a differential equation is the same as that of the 
 derivative of highest order which appears in the equation. 
 
 Equations (1), (2), (3), and (4) are of the first order ; (6), 
 (7), (8), and (9) of the second order; and (5) of the fourth 
 order. 
 
 The degree of a differential equation is the same as the power 
 to which the derivative of highest order in the equation is 
 raised, that derivative being supposed to enter into the equation 
 in a rational form. 
 
 Equations (1), (2), (3), (5), (6), (7), (8), and (9) are all 
 of the first degree ; (4) is of the third degree. 
 
 A differential equation is linear when it would be of the first 
 degree if the dependent variable and all its derivatives were 
 regarded as unknown quantities. 
 
 Equations (2), (5), (6), (7), (8), and (9) are linear. 
 
 The equation not containing differentials or derivatives, and 
 expressing the most general relation between the primitive vari- 
 ables consistent with the given differential equation, is called 
 its general solution or complete primitive. A general solution 
 will always contain arbitrary constants or arbitrary functions. 
 
 The differential equation is formed from the complete primi- 
 tive by direct differentiation, or b}' differentiation and the 
 subsequent elimination of constants or functions between the 
 primitive and the derived equations. 
 
 If it has been formed by differentiation only without sub- 
 sequent elimination or reduction, the differential equation is 
 said to be exact. 
 
314 INTEGRAL CALCULUS. [Akt. 227. 
 
 A singular solution of a differential equation is a relation 
 between the primitive variables which satisfies the differential 
 equation by means of the values which it gives to the deriva- 
 tives, but which cannot be obtained from the complete primitive 
 by giving particular values to the arbitrary constants. 
 
 227. We shall illustrate the use of the key by solving equa- 
 tions (1), (2), (3), (4), (5), (6), (7), (8), and (9) of Art. 226 
 by its aid. 
 
 (1) (l+»)2/+(l-2/)a;^=0, or (l+x)ydx+(l-y)xdy=0. 
 
 Beginning at the beginning of the key, we see that we have a 
 single equation, and hence look under I., p. 326 ; it involves 
 ordinary derivatives: we are then directed to II., p. 326; it 
 contains two variables : we go to III., p. 326 ; it is of the 
 first order, IV., p. 326, and of the first degree, V., p. 326. 
 
 It is reducible to the form 
 
 l±^dx+^^^^dy = 0, 
 X y 
 
 which comes under Xdx + Ydy = 0. 
 
 Hence we turn to (1), p. 330, and there find the specific direc- 
 tions for its solution. Integrating each term separately, we get 
 
 logx + x + logy — y = c, or log{xy) +x — y = c, 
 
 the required primitive equation. 
 
 (2) x^-ay = x + l. 
 
 Beginning again at the beginning of the key, we are directed 
 through I., II., III., IV., to V., p. 326. Looking under V., 
 we see that it will come under either the third or the fourth 
 head. Let us try the fourth; we are referred to (4), p. 330, 
 for specific directions. 
 
 Obeying instructions, the work is as follows : 
 
 ^~-ay=0, 
 dx 
 
Chap. XVIII.] DIFFERENTIAL EQUATIONS. KEY. 315 
 
 xdy- 
 
 - aydx ■■ 
 
 = 0, 
 
 dy_ 
 
 adx 
 
 = 0, 
 
 y 
 
 X 
 
 
 logy — 
 
 a log a;; 
 
 = c, 
 
 
 ,.,x, 
 
 = c; 
 
 i-^' 
 
 
 
 y =(7af, 
 
 
 ^ = aCx^-^ + 
 dx 
 
 x^''(^ 
 dx 
 
 (1) 
 
 Substitute in the given equation, 
 
 aCx' + x'*'^ - aCx' =x + l, 
 dx 
 
 aj'-t-i— -(a;+l) = 0, 
 
 (J I i 1 =: Q\ 
 
 (a— !)«"-' ax" 
 
 Substitute this value for C in (1), and we get 
 
 ^a a— 1 
 the required primitive 
 
 (3) x^-y + x\/¥^^=0. 
 
 (XX 
 
 Beginning at the beginning of the key, we are directed 
 through I., II., III., IV., to v., page 326. Looking under V., 
 we find that our equation does not come under any of the 
 special forms there given. We are consequently driven to 
 obtaining a solution in the form of a series, and for specific 
 instructions we are referred to (13), page 332. Obeying 
 these, our process is the following : 
 
316 INTEGRAL CALCULUS. [Art. 227. 
 
 dy = y _ 7^237^ dyo ^ Lo _ V^^=1^S 
 
 dx X dX(, Xq 
 
 da;^ X da-v a;o 
 
 dx^ X dxo^ Xf, 
 
 da;' a; dxo a'o 
 
 and the general value of y is 
 
 2/ = 2/0 + (a: - X,) (^^^ _ VS?^^^) - ^''~r°^' (yo + ^ V;?"^^ 
 + 4! (2/° + - Va;„^ - 2/0^ ) + 5, — - ^x}-yi 
 
 2/o + ^Va;„'=-2/„^)- 
 
 (a; — a;;,)' 
 6! 
 
 This result can be very greatly simplified by breaking up the 
 series ; we have 
 
 ,,-,, /^1 (a;-Xo)^ (a;-a;o )' (a;-a;o)° \ 
 
 (a:-a;o) A (aj-Xp)^ (aj-ajp)* (a;-a;o)° \ 
 x^ \ 2! 4! 6! ) 
 
Chap. XVIII.] DIFFERENTIAL EQUATIONS. KEY. 317 
 
 _ {x — XoY 
 
 or 
 
 7! ■)' 
 
 y = x[^ cos (a; - x„) - ^'^'-yo" sin (x - Xo)] 
 
 IfCo x„ 'J 
 
 = a;U2cos(a; — a\,)— Jl— ^ . sin (a; — »„) |. 
 ^ is entirely arbitrary ; call it sin a, then 
 
 Xq 
 
 y = a; [sin a cos(a! — aTo) — cosa siu(a; — a!o)] = a;sin[a— (a; — a;o)], 
 y = x sin (c — x), where c is any constant. 
 
 Beginning at the beginning of the key, we are directed 
 through I., II., III., IV., to VI., page 327. Looking under VI. 
 we see that the equation is of the first degree in a; ; we are 
 referred to (17), page 334, for our specific instructions. 
 
 Obeying these, we first replace -^ by p; the equation becomes 
 
 p^ =: y* (y + xp) . 
 Differentiate relatively to y, and we get 
 
 3/^ = 4:f{y + xp) + 2y* + xy* J. 
 Eliminate x, 
 
 dy y V Pj<^y 
 
 or if + y' dp g (2p^ + t/') _o 
 
 p dy y 
 
318 INTEGRAL CALCTTLXTS. [Art. 227, 
 
 Striking out the factor ip' + 1^, we have 
 
 p dy y 
 
 a differential equation of the first order and degree in which 
 the variables are separated, and which therefore can be solved 
 by (1); page 314. 
 
 Its solution is log p — log ?/' = C7, 
 
 or ^=c- 
 
 y 
 
 Eliminating p between this and the given equation, and re- 
 ducing, we have cy{x — c?) = l, as our required solution. 
 
 Beginning at the beginning of the key, we are directed 
 through I., II., III., VII., to (22) (a), page 335, for our 
 specific directions. 
 
 We see at once that y = 1 is a particular solution. 
 
 Obeying directions, we have now to solve 
 
 Let j^ = e"", and we have 
 
 m*— 2m5 + 2m2 — 2m + l = 0, 
 as our auxiliary algebraic equation in m. Its roots are 
 
 1, 1, V=l, -V^T. 
 
 The solution of (2) is then 
 
 y = {A + Bx)e + Ooosx + D sinx, 
 and of (1) is 
 
 y= (.l + -B.-»)e' + Ccosa; + Z>sina; + l. 
 
 (2) 
 
Chap. XVIII.] DIFFEKENTIAL EQUATIONS. KEY. 319 
 
 (b) sin'x—^ + smx(iOSX-f- — y = x — smx. (1) 
 
 (XX (XOj 
 
 Beginning at the beginning of the key, we are difected 
 through I., II., III., VII., to (24), page 337, for our specific 
 instructions. 
 
 Dividing through by sin^a;, the equation becomes 
 
 d^y . ^ dy „ „ ,„. 
 
 -r-|- + ctna;-;2._csc''a!. y = a;csc^a; — escK. (2) 
 
 dar da; ^ ' 
 
 y = etna; is found by inspection to be a solution of 
 
 d^y . J. dy , 
 
 ^r-^ + ctn a;-^ — csc-.t; . w = ; 
 
 da? dx 
 
 (2) can then be solved by (24) (a). 
 Substitute y = zctnx in (2), and it becomes 
 
 ctn X — - + (ctn^a; — 2 csc^a;) — = x cav?x — esc a;, 
 dar dx 
 
 or — - — (tana; + sec a; CSC a;) — = a;seca!csca; — seca;. (3) 
 da? dx 
 
 Referring to (25), page 339, and obeying instructions, we 
 
 dz 
 let «' = — ,and (3) becomes 
 dx 
 
 dz' 
 
 (tana; + sec a; CSC a;) z' = a;seca;csca; — seca;, 
 
 dx ^ ^ 
 
 a linear differential equation of the first order in z'. whose solu- 
 tion by (4) , page 330, is 
 
 z' = A tan X sec a; — a; sec^a; + tan x sec x (log tan log sin a;) ; 
 
 dz 
 but z' = — , whence integrating, we have 
 dx 
 
 z =B + A seca; — x tana; — (1 + seca;) log (1 + cos a;) , 
 
 and 
 
 y = A CSC x + B ctn x — x — (esc x -\- ctn x) log ( 1 + cos x) . 
 
320 INTEGEAL CALCULTTS. [ART. 227. 
 
 Beginning at the beginning of the key, we are directed 
 through I., n., III., VII., to (24), page 337, for our specific 
 instructions. 
 
 Let us try the method of (24) (e), page 339. 
 
 Assume y='%a„x'", and substitute in the given equation; 
 
 we have 
 
 %[m(m — l)o„ar-i — ■2m{m — l)a„a;'" 
 
 + m (m — 1) a„af"+' — 2a,„a:"'] = 0. 
 
 Writing the coeflScient of a;"" in this sum equal to zero, we 
 have 
 
 m(m + l)a„^+i — 2[m(m — l)+l]a„ + (m — l)(m-2)a„_i=0, 
 
 and we wish to choose the simplest set of values that will 
 satisfy this relation. 
 
 Substituting m = 0, m = — 1, m = — 2, etc., in this relation, 
 we find 
 
 a_i=0|,, a_2=a_i, a_3 = a^2, ••■, 
 
 Hence if we take ao = 0, it follows that 
 
 a_i = a_2 = a_3 • • • = 0, 
 
 and no negative powers of x will occur in our particular 
 solution. 
 
 Substituting now m=l, m = 2, m = 3, etc., we have 
 
 ai = aj = Oj = ai = • • • . 
 
 Taking ttj = 1, we get as our required particular solution of the 
 given equation 
 
 y =x + a? + a? + Qd^ -\- ••-. 
 
 This can be written in finite form, since we know that 
 
 l+x + a^ + a^--- = ~ — 
 1 —X 
 
 Hence y = - ^ 
 
 1 -X 
 is a particular solution. 
 
Chap. XVni.] DIFFERENTIAL EQUATIONS. KEY. 321 
 
 Turning now to (24) (a), page 337, we find 
 
 y=-^^ + c' 1+X + 
 
 1 —X 
 
 ^^f^_•^ 
 
 2a; , \ 
 3_- logo^j. 
 
 Beginning at tlie beginning of the Icey, we are directed 
 tiirough I., II., III., VII., to (24), page 337, for our specific 
 instructions. Let us ivy again tlie metliod (24) (e), page 339. 
 
 Assume y-= So„a;'", and substitute in the given equation, 
 
 S[m (m — l)a„a?"-== + a„a;" — 2 a^a!"'-^] = 0. 
 The terms containing oT are 
 
 (m + 2) (m + 1) a^+jsT + a„a!"" - 2a^^3f; 
 writing the sum of the coefHcients equal to zero, we have 
 
 m (m + 3) a„+2 + a„ = 0. (1) 
 
 Letting m = and m = —3, we get ao = and a_^ = ; and all 
 terms of y involving even negative powers of x disappear, as do 
 all terms involving odd negative powers, except the — 1st. 
 
 (2) 
 
 if we take a.i^=\. 
 
 In general 
 From this we 
 
 get 
 
 m 
 
 (m + 3) 
 
 ai =- 
 
 2.5 
 
 
 1 
 " 3! 5' 
 
 We = 
 
 (h 
 
 
 1 
 
 2A.5.7 
 
 ~ 5! 7' 
 
 
 02 
 
 
 1 
 
 "8 — 
 
 2.4.5.6.7.9 
 
 ~ 7! 9' 
 
 «10 = 
 
 02 
 
 
 1 
 
 2.4.5.6.7.8.9.11 
 
 9! 11 
 
 Hence y = — — ——- + — — — - 
 
 3 3!55!7 7!99!11 
 is a particular solution of the given equation. This can be 
 tlirown into finite form without much labor. 
 
;V22 INTEGKAL CALCULUS. [Akt. 227. 
 
 We have «« = ) 1 ••, 
 
 " 3 3!5 5!7 7!9 9!11 
 
 Ajxy) ^^ a^ ai° a;' a^" 
 da; 3! 5! 7! 9!"'' 
 
 , a? , a;" a^ , a^ \ 
 
 = a;a! •••I, 
 
 ' 3!5! 7!9! / 
 
 
 = a; sin a;; 
 
 whence 
 
 a!y= sina; — ajcosa;, 
 
 and 
 
 y = - (sin a; — x cos a;) . 
 
 X 
 
 By going back to (2), and usiug odd values of m, we get 
 another solution of our given equation, namely, 
 
 1 . a; a? a? af 
 
 ~ X 2, 2!44!6 eTs' 
 
 which can be reduced to 
 1 
 
 X 
 
 (cos a; + a; sin a;). 
 
 Hence our complete solution is 
 
 y = -\_A (cos a; + a; sin a;) + B (sin a? — a; cos as)], 
 
 or 
 
 y=A'\ i i + sm (a; — c) L 
 
 if we let — = tanc. 
 A 
 
 (9) D/«-a'i>/« = 0. 
 
 Beginning at the heginninsr of the key, we are directed 
 through I. and IX. to (45), p. 347, for our specific instruc- 
 tions. 
 
Chap. XVIII.] DIFFERENTIAL EQUATIONS. KEY. 323 
 
 Obeying these, our work is as follows : 
 
 dy — adx = 0, (1) 
 
 dy +adx =0, (2) 
 
 dpdy — a?dqdx = 0. (3) 
 
 Combining (1) and (3), we get 
 
 dpdy — adqdy = 0, 
 or . dp — adg = 0. (4) 
 
 (1) gives y — ax = a. 
 (4) gives p — aq = fi. 
 
 (2) and (3) give us, in the same way, 
 
 y + ax = ai, 
 p + aq =/3i ; 
 and our two first integrals are 
 
 p-aq=fi{y-ax), (5) 
 
 p + aq=f2{y + ax), (6) 
 
 /i and f2 denoting arbitrary functions. 
 Determining p and q, from (5) and (6), 
 
 P = iLf2(y + ax) +/, (y - aa;)], 
 
 g = ^ [/2 (2/ + ax) -/i (2/ - aa;)] ; 
 
 d'!^ =ilf2(.y+ax) +fi{y—ax)']dx+—[f2{y+ax)—fi(y—ax)]dy 
 
 _ /2 (y + a^) (dy + a<^'») —fi (y — «^) i^y — adx) 
 2a 
 
 Hence, z = F{y + ax) + Fi(y — ax) , 
 
 whei-e F and F-^ denote arbitrary functions obtained by integral 
 ing/x and/2, which are arbitrary. 
 
324 INTEGRAL CALCULUS. [Akt. 228. 
 
 228. When a differential equation does not come under any 
 of the forms given in the key, a change of dependent or inde- 
 pendent variable, or of both, will often reduce it to one of the 
 standard forms. No general rule can be laid down for such a 
 substitution. It will, however, often suffice to introduce a new 
 letter for the sum, or the difference, or the product, or the 
 quotient of the variables, or for a power of one or of both. 
 Sometimes an ingenious trigonometric substitution is effective, 
 or a change from rectangular to polar coordinates ; that is, the 
 introduction of r cos <^ for x and r sin <^ for y. 
 
 The following examples of such substitutions are instructive. 
 
 (A.) Change of dependent variable. 
 
 (1) (a; + yY-^=: rr, reduces to — ^ — -dz — dx = U. 
 
 dx a^ + z^ 
 
 if we introduce « = a; + y. 
 
 (2) ^ = sin(<^-^), reduces to — d<^ = 0, 
 
 «<^ 1 — sin u) 
 
 \t m=<^ — e. 
 
 (3) {x — y-)dx + 2 xydy = 0, reduces to {x — z)dx + xdz =0, 
 if « = y-. 
 
 (4) x-^-y + x\l3?—y' = 0, reduces to '^^ + dx = 0, 
 
 dx ^-^ _ ^a 
 
 if^ = ^. 
 
 X 
 
 (5) ^ + £ ^ _ „2 0, reduces to ^^ - n^z = 0, 
 dx- X dx dx- 
 
 it z = xy. 
 
 (B.) Change of independent variable. 
 (1) (l-x^)^g + i/ = 0, reduces to 
 
 dx" 
 ' d6- -^"—-"^j0 
 
 (=os.'0^ + smecos6^ + y = 0. if x = sine. 
 
Chap. XVIII.] DIFFERENTIAL EQUATIONS. KEY. 325 
 
 (2) — f + ta.nx-^ + cos'x.y = 0, reduces to — ^ + 2/ = Oj 
 dxr dx dz' 
 
 if 2! = sin a;. 
 
 (C.) Change of both variables. 
 
 (1) ( 1 ^-i\xy = -^<3? — y^ — a^), reduces to 
 
 \ dx^J dx 
 
 v-.z—- — (z-v-a') = 0, if z = afa.ndv = y'. 
 dz^ dz^ ' 
 
 dy 
 dx 
 sm.(ct> — 0)dcl) = dd, if a; = tan 6 and y = ta,n<b. 
 
 (2) (i/-x){l+afy^= (1+2/2)% reduces to 
 dx 
 
 (3) fx^-y'\^ = afl+ ^ {x' + y') K reduces to 
 
 dr _ _ ^ ^ 0^ if a; = 7- cos <^ and 2/ = r sin <^ 
 Vr(l— ar) Va 
 
326 INTEGRAL CALCULUS. 
 
 KEY. 
 
 Single equation I. 326 
 
 System of simultaneous equations VIII. 329 
 
 I. Involving ordinary derivatives II. 326 
 
 Involving partial derivatives IX. 329 
 
 II. Containing two variables III. 326 
 
 Containing three variables and of first degree. 
 
 General form, Pdx + Qdy + Rdz = . . . (36) 343 
 Containing more than three variables and of 
 the first degree. General form, Pdxi + Qdx^ 
 + Bdx^+ — =^0 (37)344 
 
 III. Of first order IV. 326 
 
 Not of first order VII. 328 
 
 rV. Of first degree. General form, Mdx + Ndy = V. 326 
 Not of first degree VI. 327 
 
 V. Of first degree. General form, Mdx + Ndy 
 
 = 0. 
 
 Variables separated or separable ; that is, of 
 
 or reducible to the form Xdx + Ydy = 0, 
 
 where X is a function of x alone, and I^is a 
 
 function of 2/ alone* (1) 330 
 
 M and JV homogeneous functions of x and y of 
 
 the same degree (2) 330 
 
 *0£ course,. X and J" may be constants. 
 
KEY. 327 
 
 Of the form {ax + by+c)dx + {a'x + b'y+c') dy ^°^ 
 
 = ^J (3) 330 
 
 Linear. General form, ^ + X^y = X^, where 
 ax 
 Xi and X2 are functions of a; alone * . . . (4) 330 
 
 Of the form ^ + Xi?/ = X^y, where X, and X^ 
 
 are functions of x alone * (5") 331 
 
 Mdx + Ndy an exact differential. Test, D^M 
 
 = D^N (6; 331 
 
 Mx + Ny = (7) 331 
 
 Mx-Ny = (8) 331 
 
 Of the form i?; {xy) ydx + F^ (xy)xdy = . (9) 331 
 
 —^ — ^ — - — , a function of x alone . . . (10) 331 
 
 —2 — "^^-^ — , a function of 2/ alone . . . (11)332 
 
 ^'d^~ 3^ ' a function of (ajy) (12) 332 
 
 Ny — Mx ^ ^ 
 
 A solution in the form of a series can always be 
 
 obtained (13) 332 
 
 VI. Not of first degree. 
 
 Can be solved as an algebraic equation in p, 
 
 where » stands f or ^ (14) 333 
 
 dx 
 
 Involves only one of the variables and p, 
 
 where » stands for -^ (15) 333 
 
 dx 
 
 Of the first degree in x and y ; that is, of the 
 form xfiP +yf2P=fsP', where p stands for 
 
 ^ (16) 333 
 
 dx ^ ' 
 
 Of the first degree in a; or y (17) 334 
 
 Homogeneous relatively to x and y . , . . (18) 334 
 
 * Of course, X^ and X^ may be conetants. 
 
328 INTEGRAL CALCULUS. 
 
 Page 
 
 Of the form -F(<^, ^) = 0, where ^ and ^ are 
 
 functions of x, y, and — , such that <j>^ a 
 dx 
 
 \f; = h, will lead, on differentiation, to the 
 
 same differential equation of the second 
 
 order (19) 334 
 
 A singular solution will answer (20) 334 
 
 VII. Not of first order. 
 
 Linear, with constant coefficients ; second 
 
 member zero* (21) 335 
 
 Linear, with constant coefficients ; second 
 
 member not zero* (22) 336 
 
 Of the form (a + &a;)»^ + ^(a + 6a;)»-'^l^ 
 
 + ••• + Ly= X, where X is a function of 
 
 X alone f (23) 337 
 
 Linear ; of second order ; coefficients not con- 
 stant. General form, ^+P^+ Qy = B; 
 dar dx 
 
 P, Q, and E being functions of a; ... (24) 337 
 
 Either of the primitive variables wanting . . (25) 339 
 
 d^v 
 Of the form — ^ = X, X being a function of 
 
 dx" ^ 
 
 a; alone f (26) 339 
 
 d^v 
 Of the form — f =Y, Y being a function of 
 da? 
 
 y alone t (27) 340 
 
 Oftheformg=/|^ (28)340 
 
 Oftheform|f=/|^ (29)340 
 
 Homogeneous on the supposition that x and 
 
 * The first member is supposed to contain only those terms involving the dependent 
 variable or its derivatives, 
 t See note, p. 310. 
 
KEY. 
 
 y are of the degree 1, -^ of the degree 0, 
 dx 
 
 dx" 
 
 of the degree — 1 , 
 
 Homogeneous on the suppofeition that x is of 
 
 dy 
 dx 
 
 the degree 1, y of the degree «, ^ of the 
 
 degree n — 1, — f of the degree n — 2, ■•• 
 da? 
 
 Homogeneous relatively to ?/, -^, — ^, • • • 
 
 OjX axi , 
 Containing the first power only of the deriva- 
 tive of the highest order 
 
 Of the form ^ + X^ + y\^= 0, where 
 doir dx \_dx\ 
 
 X is a function of x alone and Y a func- 
 tion of y alone * 
 
 Singular integral will answer 
 
 VIII. Simultaneous equations of the first order . 
 
 Not of the first order 
 
 IX. All the partial derivatives taken with respect 
 to one of the independent variables . . 
 
 Of the first order and Linear 
 
 Of the first order and not Linear . . . . 
 
 Of the second order and containing the deriv- 
 atives of the second order only in the first 
 degree. General form RD^z + SD^DyZ + 
 TDy^z = F, where B, S, T, and V may be 
 functions of x, y, z, D^z, and D^z 
 
 829 
 
 Fige 
 
 (30) 341 
 
 (31) 341 
 
 (32) 341 
 
 (33) 341 
 
 (34) 342 
 
 (35) 342 
 
 (38) 344 
 
 (39) 345 
 
 (40) 346 
 X. 329 
 
 XI. 329 
 
 X. Containing three variables 
 
 Containing more than three variables 
 
 XI. Containing three variables . . . 
 Containing more than three variables 
 
 (45) 347 
 
 (41) 346 
 
 (42) 346 
 
 (43) 346 
 
 (44) 347 
 
 * See note, p. 310. 
 
330 INTEGRAL CALCTTLTTS. 
 
 (1) Of or reducible to the form Xdx + Tdy = 0, where X is 
 a function of x alone and F is a function of y alone. 
 
 Integrate each term separately, and write the sum of 
 their integrals equal to an arbitrary constant. 
 
 (2) M and iV homogeneous functions of x and y of the 
 same degree. 
 
 Introduce in place of y the new variable v defined by 
 the equation y = vx, and the equation thus obtained can 
 be solved by (1). 
 
 Or, multiply the equation through by — — , and its 
 
 first member will become an exact differential, and the 
 solution may be obtained by (6). 
 
 (3) Of the form {ax + by + c)dx + (a'x + b'y + c') dy = 0. 
 If aV— a'b = 0, the equation may be thrown into the 
 
 a' 
 form {ax + by -f- c) do; H — {ax + by + c)dy = 0. If now 
 
 z = ax + by be introduced in place of either x or y, the 
 resulting equation can be solved by (1). 
 
 If ab' — a'b does not equal zero, the equation can be 
 made homogeneous by assuming x = x'— a, y = y'—fi, and 
 determining a and (i so that the constant terms in the new 
 values of M and N shall disappear, and it can then be 
 solved by (2). 
 
 (4) Linear. General form -^+ Xj 2/= Xj, where Xj and 
 
 dx 
 
 X2 are functions of x alone. 
 
 Solve on the supposition that Xj = by (1) ; and from 
 this solution obtain a value for y, involving of course an 
 arbitrarjf constant C. Substitute this value of y in the 
 given equation, regarding C as a variable, and there will 
 result a differential equation, involving and x, whose 
 solution by (1) will express C as a function of x. Sub- 
 stitute this value for C in the expression already obtained 
 for 2/, and the result will be the required solution. 
 
KEY. 331 
 
 (5) Of the form ^ + X^y = X.,y\ where X^ and X^ are 
 
 functions of x alone. 
 
 Divide through by y", and then introduce z = y^-" in 
 place of y, and the equation will become linear and may 
 be solved by (4) . 
 
 (6) Mdx + Ndy an exact differential. Test DyM= D^N. 
 Find I Mdx, regarding y as constant, and add an arbi- 
 trary function of y. Determine this function of y by the 
 fact that the differential of the result just mentioned, taken 
 on the supposition that x is constant, mtist equal Ndy. . 
 Write equal to au arbitrary constant the | Mdx above 
 mentioned plus the function of y just determined. 
 
 (7) Mx + Ny=0. 
 
 Divide the first term of Mdx + Ndy = by Mx, and 
 the second by its equal —Ny, and integrate by (1). 
 
 (8) Mx-Ny = 0. 
 
 Divide the first terra of Mdx + Ndy = by Mx, and 
 the second by its equal Ny, and integrate by (1). 
 
 (9) Of the form /i (xy) ydx 4-/3 (xy) xdy = 0. 
 
 Multiply through by , and the first member 
 
 Mx — Ny 
 
 will become an exact differential. The solution may then 
 
 be found by (6). 
 
 (10) -^ — — - — , a function of x alone. 
 
 / 
 
 •DyM-DxN 
 
 ■dx 
 
 Multiply the equation through by e-^ -sr ' , and 
 the first member will become an exact differential. The 
 solution may then be found by (6). 
 
332 INTEGRAL CALCULUS. 
 
 (11) DxN-DyM^ ^ function of y alone. 
 
 ^ / '-p«-y--Pi'-M " J 
 
 Multiply the equation through by e"' M ' ^, and 
 the first member will become an exact differential. The 
 solution may then be found by (6). 
 
 Multiply the equation through by e-' Ny-Mx ' " where 
 V = xy, and the first member will become an exact differ- 
 ential. The solution may thus be found by (6). 
 
 (13) A solution of Mdx + Ndy = in the form of a series 
 can always be obtained. 
 
 Throw the given equation into the form -^ = , 
 
 dx , N 
 then differentiate, and in the result replace -^ by 
 
 M . . d^v '^^ 
 
 — — , thus obtaining a value of — | in terms of x 
 
 Jy dvf 
 
 and y ; by successive differentiations and substitutions 
 
 get values of — i, — ^, etc. , in terms of x and y. 
 dxr dx* 
 
 If 2/o is the value of y corresponding to any chosen 
 value ajo of x, y can now be developed by Taylor's 
 Theorem. 
 
 We have y=fx =/(a'o + ^ — ^0) 
 
 =/a% + (X - x,)fx, + i^^^/"a;„ + (^o)!/"'a;„ + •.., 
 or 
 
 y-y, + (r-r)^y'' I i^-^oY '^'y^^ i {x-X,y d^y, 
 
 ^ "dx, 2! dx,'^ 3! dx,^^ ' 
 
 where ^, ^, ^, etc., 
 
 dx^ dx^ dx^ 
 
 are obtained by replacing x and y by a;„ and y^ m the 
 
 values of , ,„ ,, 
 
 dy^ cty^ d^ ^^^ 
 
 dx dx?' da?' "' 
 described above. 
 
KEY. 333 
 
 In the general case y^ Is entirely arbitrary, and if the 
 given equation is at all complicated, the solution is apt to 
 bo too complicated to be of much service. If, however, 
 in a special problem the value of y corresponding to some 
 value of X is given, and these values are taken as 2/0 and 
 a!o, the solution will generally be useful. 
 
 (14) Can be solved as an algebraic equation inp, where;) 
 
 stands for — i. 
 
 dx 
 
 Solve as an algebraic equation in p, and, after trans- 
 posing all the terms to the first member, express the first 
 member as the product of factors of the first order and 
 degree. Write each of these factors separately equal to 
 zero, and find its solution in the form V— c = by (V.) . 
 Write the product of- the first members of these solutions 
 equal to zero, using the same arbitrary constant in each. 
 
 (15) Involves only one of the variables and p, where p stands 
 
 for^. 
 dx 
 
 By algebraic solution express the variable as an expli- 
 cit function of p, and then differentiate through relatively 
 to the other variable, regarding p as a new variable and 
 
 dx 1 
 remembering that — = -. There will result a differen- 
 
 dy p 
 tial equation of the first order and degree between the 
 second variable and p which can be solved by (1). 
 Eliminate p between this solution and the given equation, 
 and the resulting equation will be the required solution. 
 
 (16) Of the form xf^p + yf^p =fiV, where p stands for -^. 
 
 Differentiate the equation relatively to one of the vari- 
 ables, regarding p as a new variable, and, with the aid of 
 the given equation, eliminate the other original variable. 
 There will result a linear differential equation of the first 
 
384 lUTKGRAL CALCDLXJS. 
 
 order between p and the remaining variable, which may be 
 
 simplified by striking out any factor not containing -^ or 
 
 5^, and can be solved by (4) . Eliminate p between this 
 
 dy 
 
 solution and the given equation, and the result will be the 
 
 required solution. 
 
 (17) Of the first degree in x or y. 
 
 The equation can sometimes be solved by the method of 
 (16), differentiating relatively to the variable which does 
 not enter to the first degree. 
 
 (18) Homogeneous relatively to a; and y. 
 
 Let y = vx, and solve algebraically relatively to p or v, 
 
 p standing for -^. Tlie result will be of the form p = fo, 
 ^ " dx ^ J ■> 
 
 or ■y = Fp. If 
 
 jy dy J. divx) J, dv , ^ 
 
 dx dx dx 
 
 an equation that can be solved by (1). If 
 
 V = Fp, ^ = Fp, y = xFp, 
 
 .V. 
 
 an equation that can be solved by (16). 
 
 (19) Of the form F{4>, i/') = 0, where 4> and \j/ are functions 
 
 of X, y, and -^, such that <f> = a and i/; = 6 will lead, on 
 dx 
 
 differentiation, to the same differential equations of the 
 second order. 
 
 Eliminate — ^ between d> = a and ip = b, where a and 6 
 
 dx 
 
 are arbitrar}- constants subject to the relation that 
 F{a, 6) = 0, and the result will be the required solution. 
 
 (20) Singular solution will answer. 
 
 dv 
 Let -^=p, and express p as an explicit function of x 
 
 and y. Take J-, regarding x as constant, and see 
 dy 
 
KEY. 885 
 
 whether it can be made infinite by writing equal to zero 
 any expression involving y. If so, and if the equation 
 thus formed will satisfy the given differential equation, it 
 is a singular solution. 
 
 <^) 
 
 Or take x^ , regarding y as constant, and see whether 
 ace 
 
 it can be made infinite by writing equal to zero any ex- 
 pression involving x. If so, and if the equation thus 
 formed is consistent with the given equation, it is a 
 singular solution. 
 
 (21) Linear, with constant coeflBcients. Second member 
 zero. 
 
 Assume 3/ = e™ ; m being constant, substitute in the 
 given equation, and then divide through by e""- There 
 will result an algebraic equation in m. Solve this equa- 
 tion, and the complete value of y will consist of a series 
 of terms characterized as follows : For every distinct 
 real value of m there will be a term Ce"" ; for each pair 
 of imaginarj- values, a + &V — 1, a— 6V— 1, a term 
 Ae"^ cos 6a; + Be" sin bx ; each of the coefficients A, B, and 
 C being an arbitrary constant, if the root or pair of roots 
 occurs but once ; and an algebraic polynomial in x of the 
 (r— l)st degree with arbitrary constant coefficients, if 
 the root or pair of roots occurs r times. 
 
 (22) Linear, with constant coefficients. Second member not 
 zero. 
 
 (a) If a particular solution of the given equation can 
 be obtained by inspection, this value plus the value of y 
 obtained by (21) on the hypothesis that the second mem- 
 ber is zero, will be the complete value of the dependent 
 variable. 
 
336 INTEGRAL CALCULUS. 
 
 (6) If the second member of the given equation can 
 be got rid of by differentiation, or by differentiation and 
 elimination between the given and the derived equations, 
 solve the new differential equation thus obtained, by (21), 
 and determine the superfluous arbitrary constants so that 
 the given equation shall be satisfied. 
 
 In determining these superfluous constants, it will 
 generally save labor to solve the original equation on 
 the hypothesis that its second member is zero, and then 
 to strike out from the preceding solution the terms which 
 are duplicates of the ones in the second solution before 
 proceeding to differentiate, as from the nature of the case 
 they would drop out in the course of the work. 
 
 (c) If the given equation is of the second order, solve 
 on the hypothesis that the second member is zero, 
 by (21), obtain from this solution a simple particular 
 solution by letting one of the arbitrary constants equal 
 zero and the other equal unity, and let y = v be this last 
 solution ; then substitute vz for y in the given equation ; 
 there will result a differential equation of the second order 
 between x and z in which the dependent variable z will be 
 wanting, and which can be completely solved by (25). 
 Substitute the value of z thus obtained in y = vz and 
 there will result the required solution of the given equa- 
 tion. 
 
 (d) Solve, on the hj-pothesis that the second member 
 is zero, and obtain the complete value of y by (21). 
 Denoting the order of the given equation by n, form the 
 
 « — 1 successive derivatives — , . — i . . . i. Then 
 
 dx dx' da;""' 
 
 differentiate y and each of the values just obtained, re- 
 garding the arbitrary constants as new variables, and 
 substitute the resulting values in the given equation ; and 
 by its aid, and that of the n — 1 equations of condition 
 formed by writing each of the derivatives of the second set, 
 
KEY. 337 
 
 except the nth, equal to the derivative of the same order in 
 the first set, determine the arbitrarj' coefficients and sub- 
 stitute their values in the original expression for y. 
 
 (23) Of the form 
 
 <'' + ^''^"S + ^ ("^ + ^*>''" £t + - + ^2/ = ^' 
 
 where X is a function of x alone. 
 
 Assume a + 6a; = e', and change the independent vari- 
 able in the given equation so as to introduce t in place of 
 X. The solution can then be obtained by (22) . 
 
 (24) Linear ; of second order ; coefficients not constants. 
 General form ^ + P^^+Qy = E. 
 
 (a) If a particular solution y = vot the equation 
 
 can be found by inspection or other means, substitute 
 y=:vz in the given equation, which will then reduce to 
 the form 
 
 d'z , fndv , T}\ dZ -r, 
 
 V — - + 12 \-Pv] — = B, 
 
 dar \ dx J ax 
 
 and can be solved by (25j). Substitute the value of a 
 thus found in y = vz, and the result will be the general 
 solution of the given equation. 
 
 (6) The substitution of y = vz in the given equation, 
 where v is given by the auxiliary differential equation 
 
 2— + Pv = 0, 
 dx 
 
338 INTEGRAL CALCtJUTS. 
 
 and can be found by (1), and should be used in the 
 simplest possible form, will lead to a differential equation 
 in z of the form 
 
 % + Iz = R. 
 which is often simpler than the original equation. 
 
 (c) The introduction of z in place of the independent 
 variable x, z being a solution of the auxiliary differential 
 equation 
 
 da^ dx 
 
 the simpler the better, will reduce the given equation to 
 the form 
 
 which is often simpler than the original equation. 
 
 (d) If the first member of the given equation regarded 
 as an operation performed on y can be resolved into the 
 product of two operations, tiie equation can always be 
 solved. The conditions of such a resolution are the 
 following : let the given equation be 
 
 '*d?+^d^ + "^=^' 
 
 where ?i, v^ w, and jK are functions of x ; this can be 
 resolved into 
 
 where p, q, r, and s are functions of a;, if 
 
 Jdr , ^ .. _, .. , .ds 
 
 pr = u, qr+p 1- s ) = ■«, and qs+p — = to ; 
 
 \dx J dx 
 
 and the values of p, q, r, and s can usually be obtained 
 
KEY. 339 
 
 by inspection. We have first to solve p \-q% — B 
 
 (iv ^^ 
 
 by (4), and then to solve r-^ + sy =z by (4). 
 
 (e) A particular solution of the equation 
 
 cZa;'' da; 
 
 can often be obtained b^- assuming that y is of the form 
 2a„a;'", m being an integer, substituting this value for y 
 in the given equation, writing the sum of the coefficients 
 of x" equal to zero, since the equation must be identically 
 true, and thus obtaining a relation between successive 
 coefficients of the assumed series. The simplest set of 
 values consistent with this relation should be substituted 
 in the assumed value of y, which will then be a particular 
 solution of the equation. If this solution can be ex- 
 pressed in finite form, the complete solution of the given 
 equation can be obtained from it by the method described 
 in (24) (a) . If, however, two different particular solu- 
 tions can be found by the method just .described, each 
 of them should be multiplied by an arbitrary constant, and 
 the sum of these products will be the complete solution 
 of the given equation. 
 
 (25) Either of the primitive variables wanting. 
 
 Assume z equal to the derivative of lowest order in the 
 equation, and express the equation in terms of z and its 
 derivatives with respect to the primitive variable actually 
 present, and the order of the resulting equation will be 
 lower than that of the given one. 
 
 (26) Of the form ^ = X. X being a function of x alone. 
 
 Solve by integrating n times successively with regard 
 to X. 
 
 Or solve by (22). 
 
340 INTEGRAL CALCULUS. 
 
 (27) Of the form ^ = F. 1' being a function of ^ alone. 
 
 Multiply by 2 -^ and integrate relatively to x. There 
 will result the equation i-^] ~^ \ ^^y "^ ^' ^^^nce 
 
 -^= (2 j Ydy -{■ C)K an equation that may be solved 
 dx J 
 
 by (1). 
 
 (28) Of the form ^ =/^^- 
 ^ ' dx" -^da;"-! 
 
 Assume 
 
 d""V ,, dz J. , dz rdz , ri 
 
 ^ = z. then — =/« or dx = — . x= I h (7. 
 
 dx"-^ dx fz J fz 
 
 After effecting this integration, express z iu terms of x 
 
 and C Then, since z = — -4, f = F(x, C), an 
 
 da;"-' da;"-' ^ '^ 
 
 equation that may be treated by (26) . 
 Or, since 
 
 S- = z, ^ = I «da; + c = I \- c, since da; = - — 
 
 dx"-' da;»-2 J J /« ' /« 
 
 d"-'y 
 da;"-' 
 
 =X/f+')+-/§(/?+')+- 
 
 Continue this process until y is expressed in terms of 
 z and m — 1, arbitrary constants, and then eliminate z by 
 
 /dz 
 1- C 
 /2 
 
 (29) Of the form ^ = f'^" '^. 
 ^ '^ da;" •'daf-2 
 
 da;"-^ 
 may be solved by (27) 
 
 Let - — -| = z, and the equation becomes — =fi, and 
 da;" dar 
 
KEY. 341 
 
 (30) Homogeneous on the supposition that x and y are of the 
 
 degree 1, -^ of the degree 0, — | of the degree — 1, •••. 
 dx da? 
 
 Assume a;=e*, y = e^z, and by changing the variables 
 
 introduce 6 and z into the equation in the place of x and y. 
 
 Divide through by e' and there will result an equation 
 
 involving only z, — , — -, ■ • • , whose order may be de- 
 ft^ dv 
 
 pressed by (25). 
 
 (31) Homogeneous on the supposition that x is of the degree 
 
 1, w of the degree n, _^ of the degree n — 1, — ^ of the 
 dx daf 
 
 degree n — 2, ••-. 
 
 Assume a; = e', y = e"'«, and by changing the variables 
 introduce 6 and z into the equation in the place of x and y. 
 The resulting equation may be freed from $ by division 
 and treated by (25). 
 
 (32) Homogeneous relatively to 2/, -^, -t4)"'- 
 
 Assume y = e', and substitute in the given equation. 
 Divide through by e* and treat by (25) . 
 
 (33) Containing the first power only of the derivative of the 
 highest order. 
 
 The equation may be exact. 
 
 Call its first member — . If n is the order of the equation, 
 
 represent ^""'^ by o and ^ by ^. Multiply the term 
 da;"-' dx" dx 
 
 containing ^ by dx and integrate it as if p were the only 
 
 dx d"~'^y 
 
 variable, calling the result Ui ; then replacing p by — ^, 
 
342 INTEGRAL CALCULUS. 
 
 find the complete derivative — -', and form tlie expression 
 •^ da; 
 
 ^-^^\ representing it bv ^. If ^ contains the 
 dx dx ' dx dx 
 
 first power only of the highest derivative of y, it may 
 
 itself be an exact derivative, and is to be treated pre- 
 
 dV 
 cisely as the first member of the given equation — has 
 
 dx 
 
 been. Continue this process until a remainder — ^ of 
 
 dx 
 
 the first order occurs. 
 
 Write this equal to zero, and see if the equation thus 
 
 formed is exact, see (6). If so, solve it by (6), 
 
 throwing its solution into the form F„_i = C. A 
 
 complete first integral of the given equation will be 
 
 Ui+ U2+ ■■■ + V„-i = C. The occurrence at any step 
 
 of the process of a remainder — -, containing a higher 
 
 dx 
 
 power than the first of its highest derivative of y, or the 
 
 failure of the resulting equation of the first order above 
 
 described to be exact, shows that the first member of the 
 
 given equation was not an exact derivative, and that this 
 
 method will not apply. 
 
 (34) Of the form ^ + A'^ + T 
 
 = 0, where X is a 
 
 dy 
 dx 
 
 function of x alone and Y a function of y alone. Multiply 
 dy-i-' 
 
 through by 
 
 and the equation will become exact, 
 
 dx 
 and may be solved )\v (33). 
 
 (35) Singular integral will answer. 
 
 Call f p, and — ^ g, and find _5, regarding » and 'y 
 
 dx"~^ dx" dp , 
 
 as the onlj' variables, and see whether — ? can be made 
 
 dp 
 infinite by writing equal to zero any factor containing p 
 
KEY. 343 
 
 If so, eliminate q between this equation and the given 
 equation, and if the result is a solution it will be a singular 
 integral. 
 
 (36) General form, Pdx + Qdy + Rdz = 0. 
 
 If the equation can be reduced to the form Xdx + Ydy 
 + Zdz = 0, where X is a function of x alone, Y a function 
 of y alone, and Z a function of z alone, integrate each 
 term separatelj-, and write the sum of the integrals equal 
 to an arbitrary constant. 
 
 If not, integrate the equation by (V.) on the supposition 
 that one of the variables is constant and its differential 
 zero, writing an arbitrary function of that variable in place 
 of the arbitrary constant in the result. Transpose all the 
 terms to the first member, and then take its complete 
 differential, regarding all the original variables as variable, 
 and write it equal to the first member of the given equa- 
 tion, and from this equation of condition determine the 
 arbitrary function. Substitute for the arbitrar}- function 
 in the first integral its value thus determined, and the 
 result will be the solution required. 
 
 If the equation of condition contains any other varia- 
 bles than the one involved in the arbitrarj- function, thej' 
 must be eliminated bj- the aid of the primitive equation 
 already obtained ; and if this elimination cannot be per- 
 formed, the given equation is not derivable from a single 
 primitive equation, but must have come from two simul- 
 taneous primitive equations. 
 
 In that case, assume anj' arbitrary equation /(a;,2/,z) =0 
 as one primitive, differentiate it, and eliminate between it 
 its derived equation and the given equation, one variable, 
 and its differential. There will result a differential equa- 
 tion containing only two variables, which maj' be solved 
 by (III.), and will lead to the second primitive of the 
 given equation. 
 
344 INTEGRAL CAiCTJLUS. 
 
 (37) General form, Pdxi + Qdx^ + BdXs + = 0. 
 
 If the equation can be reduced to the foi-m XidXi+X^dai, 
 
 + Xgdajj + = 0, where Xi is a function of Xj alone, Xj 
 
 a function of Kj alone, X^ a function of ajs alone, etc., inte- 
 grate each term separately, and write the sum of their 
 integrals equal to an arbitrary constant. 
 
 If not, integrate the equation by (V.), on the supposi- 
 tion that all the variables but two are constant and their 
 differentials zero, writing an arbitrary function of these 
 variables in place of the arbitrary constant in the result. 
 Transpose all the terms to the first member, and then 
 take its complete differential, regarding all of the original 
 variables as variable, and write it equal to the first mem- 
 ber of the given equation, and from this equation of con- 
 dition determine the arbitrary function. Substitute for 
 the arbitrary function in the first integral its value thus 
 determined, and the result will be the solution required. 
 
 If the equation of condition cannot, even with the aid 
 of the primitive equation first obtained, be thrown into a 
 form where the complete differential of the arbitrarj' func- 
 tion IS given equal to an exact differential, the function 
 cannot be determined, and the given equation is not deriv- 
 able from a single primitive equation. 
 
 (38) S3'stem of simultaneous equations of the first order. 
 
 If any of the equations of the set can be integrated 
 separatelj' bj- (II. ) so as to lead to single primitives, the 
 problem can be simplified ; for by the aid of these primi- 
 tives a number of variables equal to the number of solved 
 equations can be eliminated from the remaining equations 
 of the series, and there will be formed a simpler set of 
 simultaneous equations whose primitives, together with the 
 primitives already found, wiU form the primitive system 
 of the given equations. 
 
 There must be n equations connecting n + 1 variables, 
 in order that the system may be determmate. 
 
 Let X, Xi, X2, , x„ be the origmal variables. Choose 
 
KEY. 345 
 
 any two, a and a;,, as the independent and the principal de- 
 pendent variable, and by successive eliminations form the 
 n equations -^=f^{x,x,,x„ ,x:),"^=flx,x^,x^, ,a!„), 
 
 ) up to — ^=/„(a;,a;i,a;2, ,a;„). Differentiate the first 
 
 of these with respect to x n — 1 times, substituting for 
 
 ~, ~-i i~i after each step their values in terms of 
 
 ax ax ax 
 
 the original variables. There will result n equations, 
 which will express each of the n successive derivatives 
 dxi d^x, d^x, d'x, . , „ 
 
 —^ -T-2> -5-55 ' -riri in terms of x, x^, x^, , x„. 
 
 dx dx' da? da;" 
 
 EUminate from these all the variables except x and Xi, 
 obtaining a single equation of the nth order between x 
 and Xi- Solve this \>y (VII.), and so get a value of x^ in 
 terms of x and n arbitrarj' constants. Find by differen- 
 tiating this result values for — J, — i, , h and write 
 
 ^ dx dot? da;»-i 
 
 them equal to the ones alreadj' obtained for them in terms 
 of the original variables. The n—1 equations thus formed, 
 together with the equation expressing x^ in terms of x and 
 arbitrary constants, are the complete primitive system 
 required. 
 
 (39) System of simultaneous equations not of the first order. 
 Regard each derivative of each dependent variable, 
 
 from the first to the next to the highest as a new variable, 
 and the given equations, together with the equations de- 
 fining these new variables, will form a system of simulta- 
 neous equations of the first order which may be solved by 
 (38). Eliminate the new variables representing the 
 various derivatives from the equations of the solution, and 
 the equations obtained will be the complete primitive sys- 
 tem required. 
 
 (40) All the partial derivatives taken with respect to one of 
 the independent variables. 
 
3-16 INTEGRAL CALCULUS. 
 
 Integrate by (II.) as if that one were the only indepen- 
 dent variable, replacing each arbitrary constant by an 
 arbitrary function of the other independent variables. 
 
 (41) Of the first order and linear, containing three variables. 
 General form, PD^z + QD,jZ = R. 
 
 Form the auxiliary system of ordinary differential equa- 
 tions — = ^ = — , and mtegrate by (38) . Express their 
 
 primitives in the form m = a, v = 6, a and b being arbi- 
 trary constants ; and u =fv, where /is an arbitrary func- 
 tion, will be the required solution. 
 
 (42) Of the first order and linear, containing more than three 
 
 variables. General form, PiDx,e + P^Dx^z + = R, 
 
 where x^, ajj, , a;„, are the independent and z the depen- 
 dent variables. 
 
 Form the auxiliary system of ordinary differential equa- 
 tions ^ = ^ = ^ = ffe^ and integrate them by (38) . 
 
 Pi i 2 Pn R 
 
 Express their primitives in the form v-i = a, v^^ b, v^ = c, 
 , and Vi =f{v2,Vs, ,v„), where /is an arbitrary func- 
 tion, will be the required solution. 
 
 (43) Of the first order and not linear, containing three varia- 
 bles, F{x,y,z,p,q)=0, where p = D^z, q = DyZ. 
 
 Express q in terms of x, y, z and p from the given equa- 
 tion, and substitute its value thus obtained in the auxil- 
 
 dx _ , 
 iary system of ordinary differential equations _ ^ — "2/ 
 
 dz _ dp 
 
 ■'V 
 
 Deduce bj- integration from 
 
 q-pD^q D,q+pD,q 
 these equations, by (36), a value of p involving an arbi- 
 trary constant, and substitute it with the corresponding 
 value of q in the equation dz = prta; -|- qdy. Integrate 
 this result by (36), if possible; and if a single primitive 
 equation be obtained, it will be a complete primitive of the 
 given equation. 
 
KEY. 347 
 
 A singular solution may be obtained by finding the 
 partial derivatives D^z and D^z from the given equation, 
 writing them separately equal to zero, and eliminating p 
 and q between them and the given equation. 
 
 (44) Of the first order and not linear, containing more than 
 
 three variables. F{Xi,x,, , x„,z,pi,p^, ,/»„) = 0, where 
 
 Pi = Dx,z, p2 = Dx,z, 
 
 Form the linear partial differential equation %[(DxiF 
 + PiD^F)Dp<^ - Dp.F(Dx.^ +piD^^)-\ = 0, where 4- is 
 
 an unknown function of (Xi, ,x„,p^, 5P»)i and where 
 
 2; means the sum of all the terms of the given form that 
 can be obtained hy giving i successively the values 1,2, 
 3, , n. 
 
 Form, by (42), its auxiliar3' sj'stem of ordinarj- differen- 
 tial equations, and from them get, by (38), n — 1 mte- 
 
 grals, "^i = Oi, .<l>2 = a,, , "I'„_i = a„_i. B3- these equations 
 
 and the given equation express pi, p^, , p„ in terms of 
 
 the original variables, and substitute their values in the 
 
 equation dz =pidx^ +p-2dx2 + +p„dx„. Integrate this 
 
 by (37), and the result will be the requked complete primi- 
 tive. 
 
 (45) Of the second order and containing the derivatives of 
 the second order onlj- in the first degree. General form, 
 RD;'z + SD^D,z -f- TD,^z = V, where B,S,T, and Fmay 
 be functions of x, y, z, D^z, and DyZ. 
 
 Call D^z p and DyZ q. 
 Form first the equation 
 
 Edif - Sd'xdy + Tdx' = 0, [1] 
 
 and resolve it, supposing the first member not a complete 
 square, into two equations of the form 
 
 dy — midx=0, dy — m2dx = 0. [2] 
 
 From the first of these, and from the equation 
 
 Bdpdy + Tdqdx - Vdxdy = , [3 ] 
 
348 INTEGRAL CALCULUS. 
 
 combined if needful witli the equation 
 
 dz = pdx + qdy, 
 
 seek to obtain two integrals m, = a, Vi = p. Proceed- 
 ing in the same way with the second equation of [2], 
 seek two other integrals u^ = aj, iJj = A > then the first in- 
 tegrals of the proposed equation will be 
 
 Ul=flVy, u^=fiVi, [4] 
 
 where /i and^ denote arbitrary functions. 
 
 To deduce the final integral, we must either integrate 
 one of these, or, determining from the two p and q in terms 
 of X, y, and z, substitute those values in the equation 
 
 dz=pdx + qdy, 
 
 which will then become integrable. Its solution will give 
 the final integral sought. 
 
 If the values of m, and m^ are equal, only one first in- 
 tegral will be obtained, and the final solution must be 
 sought b}' its integration. 
 
 When it is not possible so to combine the auxiliary 
 equations as to obtain two auxiliary integrals m = a, i! =/?, 
 no first integral of the proposed equation exists, and this 
 method of solution fails. 
 
KEY. 
 
 349 
 
 Examples. 
 
 (1) since cosy, daj — cosa; sin 2/. %= 0. Ans. cos2/ = ccoso;. 
 
 (2) {x + yy^ = a\ 
 
 ax 
 
 Ans. y — a tan~^ ^ = c. 
 
 ^ns. ctn 
 
 (3) g = sin(<^-«). 
 
 U) x^-y + x-slW^^Q. 
 dx 
 
 (5) {y-x)(l+x')if = {l+y')i. 
 
 r _ qb — I 
 
 = ^ + C. 
 
 ^?is. sin ^" = c — x. 
 
 X 
 
 dx 
 Ans. ctn 
 
 « (-|-»)"=" 
 
 1 + 
 
 4 2 
 'dy 
 
 - (tan~^!/ — tan"' a;) 
 
 = tan~^2/ + c. 
 
 da; 
 
 (a^ + 2/')*. 
 
 ^«s. 2a(a;^ + 2/^) = (a;^ + 2/^)^ —a; cose + 2/ sine. 
 
 (7) [2 V(a;2/) - a;] dy + ydx = 0. 
 
 (8) (a,_2^) + 2a;2/g = 0. 
 
 Ans. y = ce^V|. 
 Ans. XB" = c. 
 
 (9) {2x-y + l)dx + (2y~x-l)dy = 0. 
 
 Ans. x^ — xy +y' + x — y = c. 
 
 ,,„. dw , sin 2 a; 
 
 ^ ^ dx 2 
 
 ^ns. y = sinx — 1 + ee 
 
 (11) (l-ar')^-a;ii/ = aa;y^ ^ns. y= [cV(l -»")-«]" 
 
 (12) xy{l+xf)f^=l. 
 
 1 ::2? 
 
 ^ws. - = 2 — 2/^ + ce 2 
 
 (13) «(«^ + y' + a')? + a'(a^ + 2/'-«')=0. 
 
350 
 
 INTEGRAL CALCULUS. 
 
 (14) xdx + ydy + ^~y'^^ = Q. Ans. ?-+l!.+ taii-»?^ = c. 
 
 x' + y' 2 X 
 
 (15) |-^)— — = 0. Ans. (y—a\osx—c)(y+a\ogx—c) = 0. 
 \dxj a? 
 
 (16) f^\'+2yctnx^ = y'. 
 
 \dxj dx f iX \ f X \ 
 
 ^ ^ Ans. I y siir c]( v cos^ c 1 = 0. 
 
 V 2 A 2 ; 
 
 ^ns. (2y — £C^ — c)[log(a; + y — l) + a; — c] =0. 
 
 (18) ('llj- (a^+ ^+2/')(2Y+ (a^2/+a^2/^+a^) £ -3^2/^=0. 
 
 Ans. ('y-|-cVa; + i-cVlog2/-|-c^ = 0. 
 
 (19)fl-/-2^Vf^Y-2^|/+^:=o. 
 \ ary \dxj x dx a? 
 
 ^„s. (^2, + log^+^^±i^ - cV^/ - log^±^^±F - c) = 0. 
 
 (20) y = 4 + ^_ff" 
 dx dx \ax 
 
 (21) 2/ = 2//'^Y+2a;^. 
 \dxl dx 
 
 Ans. y = cx + c — c". 
 
 lution, y=^ — "*" ' ■ 
 '^ 4 
 
 Ans. y^ = 2cx + c'. 
 
 Singular solution, y =^ — -' ■ 
 4 
 
 (22) 
 
 \dxj 
 
 xy = {of — y^ — aF) 
 
 <^«'=^4:+»-(ij 
 
 dy_ 
 dx 
 
 Ans. y2-car' + -^ = 0. 
 1+c 
 
 ^»is. y^ = 2 ca; + c*. 
 
 ^^^^ '^ST^^^S'^'''^^- ^'*'- «' + «a'.V + a'a^ = 0. 
 
KEY. 
 
 ?,51 
 
 L \axjj ax ^^^^ {b+yy = iax,f{a) + b = 0. 
 
 (26) a^_|=/[-,._,,|J. ^r.s. -^ = 6, /(«) + . = 0. 
 
 (27) ^-4^ + 6^-4^ + 2/=0. 
 ^ '^ djB* da!^^ da^ da; ^ 
 
 ^ns. ?/ = (Co + Cia; + C2K^ + C8a;^)e'. 
 
 (29) tl + 2^+y = e'. 
 
 Ans. y = — + {A-'rBx)casx-\-{C+Dx)amx. 
 
 (30) ^_2^ + 42/ = e''cosa; 
 ^ '^ da,'' da; ^ 
 
 ^ns. 2/ = ^e-^ + e' (b — ^cosx + (g + 
 
 ^_2^ + ^ = a;3. 
 da;* dar* « da;^ 
 
 sin a; 
 
 (31) ^-2^ + -^ 
 ^ ' dx" dar* Z da? 
 
 Ans. y= (^A + Bx)e' + {C + Dx) +12 a? + ^01? + - + — 
 
 (32) ^_4^ + 4w=a;2. 
 
 "'^ "^ ^ws. y = {A + Bx)e^ + l{2a? + i:X + i). 
 
 (33) — 2 + y = cosa;. ^ns. y = ^cosa;+ Ssina; + - sina;. 
 ^ ^ da;^ ^ 2 
 
 (34) ^ + 4v = a;sin2a;. 
 
 ^«s. 2/=M-— jcos2a; + ^5-— jsm2x + -. 
 
 (35) a;»5-a;^ + 22/ = a;loga;. 
 
 daf da; 
 
 Ans. y = -4a; cos (log x) + 5a; sin (log a;) + x log a;. 
 
352 INTBGKAL CALCtTLTTS. 
 
 (36\ a^^_a^^ + 2a;^-22/ = a!» + 3a;. 
 
 da? dx" dx a? / (\aax\^ 
 
 Ans. y = x{A + B\ogx) + C3? + ^-ix{\ + ^^°^J'> \ 
 
 (37) ^ + ? ^ - n^y = 0. Ans. y=- (Ae"' + Be-"'), 
 
 dx- X dx X 
 
 Ans. 2/ = ^cos (8ma!) + Ssin(sina!). 
 
 ^ ' dx" dx " 
 
 (39) (l_a^)2g + 2/=0. 
 
 Ans. y = \l\ —^[A + B 
 
 "-J^)- 
 
 (40) (l+a;^)g-2a;^ + 22/=0. 
 
 "^ ^^ Ans. y = Bx^A{\-o?). 
 
 (41) ^ ^dy_^^_^^^_^_ 
 
 da;^ x-1 dx x-\ ^^^^_ 2/=^e' + 5a;-(l+a;2). 
 
 (42) 0^^ - 2a; (1 + ») ^ + 2 (1 + a;) 2/ = a;'. 
 
 ^ws « = Axf? + Ba; • 
 
 ^2 
 
 (43) sin^a;^— 22/=0. ^ns. 2/ =^4 etna; + 5(1 -a; etna;). 
 
 d'T 
 
 (44) Tl+irr— 2' = «Y- + logs' 
 da;^ XT loe a; ya; ^ 
 
 / /* da; \ 
 ,4ns. 2/ = ^loga; + e'loga; +5( loga; I a;), 
 
 V -J log® / 
 
 (45) ^_2^-«')^ + /'n^-2!^^2/ = e~. 
 dar \ a;y da; \^ x J 
 
 Ans. y = e'"fA + -^^-\ -\ 
 
 ^ ■' Ans. 2/ = a;(^cosaa;+Bsmaa;). 
 
KEY. 
 
 363 
 
 ^^7)^-2bx^+b^a?y=0. 
 
 da? 
 
 dx 
 
 Ans. y = e''{Aco8x\/b + Bsmx'/b). 
 
 (48) ^-Ax'^ + {ioe'-3)y = e''. 
 
 ^"^ '*'* Ans. y = e''{Ae + Be-'-\). 
 
 (49) (\-a?)^-4.x^-{\+a?)y=x. 
 
 doF dx 
 
 Ans. y = 
 
 1-a^ 
 
 {x + A cos a; + B sin a;) . 
 
 (50) ^ L dy I a'+V«'-8 y^o. 
 
 ^ws. y = e'^'(Aa? + - 
 
 (51) ^ + 2nctnna;^ + (m*-n')y = 0. 
 d^Xj^ dec 
 
 Ans. yt=(^Acosmx + Bsmmx)cscnx. 
 
 (52) (9^_l)^+a;J-c^2/ = 0. 
 
 da? dx 
 
 Ans. y = A{x + \la?-\y+B{x--^a?-iy. 
 
 
 ^ns. y = As,m~ + B cos— 
 a; a; 
 
 (54) d!y_3x+_l # 
 ^ -^ da^ a!^ - 1 da; ^ 
 
 6(a:+l) 
 
 (a;-l)(3a! + 5) 
 
 = 0. 
 
 '^ns. 2/ = [^ + Slog((a;-l)'(3ar + 5))]V(a;-l)3(3a! + 5). 
 
 (56) (l-a.)g-.|_c^,= 0. 
 
 ^ns. 2/ = ^e" '■""' ' + Be-' "" '. 
 
 (56) (l+aa!2)^ + aa;^-M22/ = o. 
 
 d^ 
 
 #. 
 
 (67) (a;-l)(a;-2)^-(2a!-3)^ + 22/=0. 
 
 dx' 
 
 Ans. j/ = c(a;-2)2 + c'(a;-2)[(a;-2)log(a!-2)-l]. 
 
364 INTEGRAL CALCULUS. 
 
 (58) (3-a;)^-(9-4a;)^ + (6-3a;)j/ = 0. 
 
 . ., r3x/183 76 ,21, lA 
 
 \ 8 4 4 2 / 
 
 (69) (^a!'-a?)^-8x^-12y = 0. 
 
 c ('a^ + 3ar''> 
 
 ^ns. ■!/ = -+c' '' „ — r-^- 
 
 ^ (o-a;)» (a^-a!^)^ 
 
 <">S+!l+(»--l>-»- 
 
 ^ns. y = — [_A (sin wa; — na; cos 7ix) + £ (cos nx + n« sin na;) ] . 
 (61) ^+1^=0. Ans. y = c\ogx + c'. 
 
 Ans. (? + cxy = c'x. 
 
 \ rta;yaa^ \axj ax pjncl a first integral. 
 
 Ans. a^-J^ + y-f^) +xy=c. 
 ax \axj 
 
 "-^ "^ ^^ Find a first integral. 
 
 Ans. of — f — a;-^ + xy' = c. 
 dar dx 
 
 (65) -^+Jy- + J^ = 0. Ans. {x-a){y-b){z-c)=c. 
 x — a y — b z —c 
 
 (66) {y + z)dx + dy + dz = 0. Ans. e'{y + z)=c. 
 
 (67) ^ + 4a;+f=0, ^ + Sy-x=0. 
 
 dt 4 dt 7, „ 
 
 '2' 
 
 — — ?/ — — 
 
 Ans. x = ce 2— •^, ^/^f^^ + O® ^* 
 
, KEY. 355 
 
 (68) ^+m''x=0, g_^ea, = o. 
 
 Ans. x = Asmmt + B cos mt, x + y = Ct + D. 
 
 (69) D^z=-^. Ans. e~'(x + y + z)=<l,y. 
 
 (70) xzD^z + yzD^z = xy. Ans. z^ = xy + <l>f-\ 
 
 (71) D^z.D^z=l. Ans. z = ax + '^ + b. 
 
 a 
 
 (72) x'DJ'z + 2 xyD^DyZ + yWJ'z = 0. Ans. z = x<j> (t\ ^J^X 
 
 (73) {DyZYD^z -2D,zD„zD,DyZ + {D,zyDyH = 0. 
 
 Ans. y = Xfftz + 1/«. 
 
 (74) D^z.D^D,z-D,z.Diz = Q. Ans. x = ^y + ypz. 
 
APPENDIX. 
 
Chap. V.] 
 
 INTEGKATION. 
 
 6a 
 
 CHAPTER V. 
 
 INTEGKATION. 
 
 74. We are now able to extend materially our list ot formulas 
 for direct integration (Art. 55) , one of which maj' be obtained 
 from each of the derivative formulas in our last chapter. The 
 following set contains the most important of these : — 
 
 D^loga!=- 
 
 X 
 
 D^a" = a'loga 
 
 D^e = e 
 
 DjSina; = cos a; 
 Z>jCOsa;= — sinx 
 Z)jlogsina; = etna; 
 Djlogcosa; = — tana; 
 1 
 
 Z>j,siii~'a; = 
 X>jtan~^a; = 
 
 gives /^- = log a;. 
 
 X 
 
 " f^a'Xoga = a'. 
 
 " Xe"' = e^ 
 
 " Xcosa! = sina;. 
 
 " yj(— sina;) = cosa;. 
 
 ' ' /j ctn X = log sin x. 
 
 " _/^(— tana;) = logcosa;. 
 
 1 .:__1„ 
 
 V(l-a^) 
 1 
 
 " X 
 
 V(i-^) 
 
 D,weis~^x = 
 
 1 
 
 V(2a!-a;^) 
 
 " f, — ^ = tan->a;. 
 
 " A 
 
 -^{2x-a?) 
 
 The second, fifth, and seventh in the second group can be 
 written in the more convenient forms, 
 
66 DIFFBEENTIAIj CAXrCULUS. [Abx. 76. 
 
 /•I «' 
 
 log a ' 
 f^smx= ■— cosa;; 
 /jtanx= —log cos a;. 
 
 75. When the expression to be integrated does not come under 
 any of the forms in the preceding list, it can often he prepared 
 for integration by a suitable change of variable, the new variable, 
 of course, being a function of the old. This method is called 
 integration by substitution, and is based upon a formula easily 
 
 deduced from -D«(-FV) =DyFy. D^y ; 
 
 which gives immediately 
 
 Fy=f,(D,Fy.D^). 
 
 Let u=D,Fy, 
 
 then Fy=f^u, 
 
 and we have f,u =f^(uD,y) ; 
 
 or, interchanging x and y, 
 
 Lu=f,(uD„x). [1] 
 
 For example, required X(" + bx)'^- 
 
 Let z = a + bx, 
 
 and then f(a + hxy =f,z^ =L{z^ ■ D,x) , by [1] ; 
 
 but x=---, 
 
 b 6' 
 
 D^x = l; 
 
 o 
 
 hence /«(« + 6a;)" = i/,«"= -^ ^""^ 
 
 6 6n4-l 
 
Chap. V.] INTEGEATION. 67 
 
 Substituting for z its value, we have 
 
 n + l 
 
 ,.(. + ..)..> fc±M 
 
 Example. 
 
 Find /» . Ans . - log (a + 6a;) . 
 
 a + 6a; o 
 
 76. If /a; represents a function that can be integrated, /(a+6a;) 
 can always be integrated ; for, if 
 
 2 = a + 6a;, 
 
 then D^x = - 
 
 b 
 
 and j:f{a + 6a;) =Uz =fJzD,x = Ifjz. 
 
 Examples. 
 
 Find 
 
 (1) /.sinaa;. -Ans. cosaa;. 
 
 (2) Xcosaa;. Ans. -sinaa;. 
 
 (3) fji&nax. 
 
 (4) y^ctnaa;. 
 
 77. Required/^ 
 
 ■sjia'-oi?) 
 
 '''"'-^'^'Um^ 
 
 Let « = -' 
 
 then a! = a2;, 
 
 • 
 
68 
 
 '-f. 
 
 DIFFERENTIAL CALCULUS. 
 1 1 . . 1 1 r 1 
 
 " IHt 
 
 =1/.-^ ' 
 
 =u- 
 
 =/. 
 
 a-'V(l-«') a 7(1-2") 
 
 = sin '« = sin ' - 
 
 [Art. 78. 
 
 V(i-«^) 
 
 Examples. 
 
 Find 
 
 (i\ r ^ 
 
 Ans. -tan~'-' 
 a a 
 
 ^^ ■'V+a^ 
 
 (=>\ c 1 
 
 x 
 
 Ans. vers"'— 
 
 a 
 
 V(2aa! — or) 
 
 78. Re^yireaL^^^^^^^. 
 
 
 Let z = x + -^{x' + a?)\ 
 
 
 then z — x = yl{3? + a?), 
 
 
 z'-2zx + x' = a? + a\ 
 
 
 2zx = z'-d', 
 
 
 x-^—\ 
 
 
 22 
 
 ^ix'^a') = z-x = z-t_^=t±^. 
 
 D,x = 
 
 !^ + a' 
 2z' 
 
 L 
 
 
 V(a^ + a') V + a^ -^V + a" 
 
 =/. 
 
 2z 2^4- a" 
 
 Find/, 
 
 a^ + a^ 22^ 
 
 V(a^-a') 
 
 -fz- = log2 = log(a; + V^+a^) . 
 
 Example. 
 
 ^Ms . log (a; + Vx* — a') . 
 
Chap, v.] INTEGRATION. 69 
 
 79. When the expression to be integrated can be factored, the 
 required integral can often be obtained by the use of a formula 
 
 deduced from D,{uv) = uD^v + vD,u, 
 
 which gives uv =f,uD,v +f^vD^u 
 
 or LuD^v = uv —f^vD,u. [1] 
 
 This method is called integrating by parts. 
 
 (a) For example, required f^logx. 
 
 log a; can be regarded as the product of log a; by 1. 
 
 Call logx = u and 1 = D^v, 
 
 then D,u = -, 
 
 X 
 
 v = x; 
 and we have 
 
 f^ogx =/,l loga; =f,uD,v = uv -f,vD,u 
 — a; log a; — /«- = ailoga; — x. 
 
 X 
 
 Example. 
 Find/.xloga;. 
 Suggestion : Let loga; = u and x = D^v. 
 
 Ans. -x^f\ogx — -\ 
 
 80 . Required f, sin^ x . 
 
 Let " = sina; and B^v = sina;, 
 
 then -0:rM = cosa;, 
 
 v= — cosa;, 
 
 /.sin^a; = - sin a; cos a; +Xcos^a; ; 
 
70 DIFPBKENTlAIi CAIiCtTLXJS. [Aht. 81. 
 
 but cos'' a; := 1 — sin'o;, 
 
 so f^(X)a^x=f^l—f^sa}?x = x—f^sir?x 
 
 and J'^sa:?x = x — s,va.xcosx—f^s\r?z. 
 
 2f^sa!?x = x — sinoicosa;. 
 /^BW?x=^(x — siniKcosa;). 
 
 Examples. 
 
 (1) Find/icos'a!. Ans. -(a; + sin a; cos a;). 
 
 (2) Xsinajcosa;. Ans. ^^^, 
 
 81. Very often both methods described above are required in 
 the same integration. 
 (a) Eequiredf,sin~^x. 
 
 Let 
 
 sin~*a!s=y. 
 
 then 
 
 a!=smy; 
 
 
 2),aj=cosy, 
 
 
 Xsin-»a; =f,y =f,y cosy 
 
 Let 
 
 u = y and i3,w = cosy ; 
 
 then 
 
 D,u = l, 
 
 
 V = siny, 
 
 and 
 /,ycosy=2/sinr/— /,8in2/=ysin2/+cosy=a;sin-'a;+v(l— a;^). 
 
 Any inverse or anti-function can be integrated by this method 
 if the direct function is integrable. 
 
 (6) Thus, fJ-^x=f,y=f,yDJy^yfy-fJy 
 
 jvhere y =f~^x. 
 
Chap. V.J INTEGEATION. 71 
 
 Examples. 
 
 (1) Findyicos"**. Ans. xcos~^x — ^{l —a?). 
 
 (2) y^tan~*a!. Ans. a;taii~'a; — -log(l + a^). 
 
 (3) f^yers'^x. Ans. (a;— 1) vers~*a; + ^(2a!— a^). 
 
 82. Sometimes an algebraic transformation, either alone or in 
 combination with the preceding vnethods, is useful. 
 
 (a) Required/,- 5. 
 
 _J_^^(_k L_\ 
 
 a?— a^ 2a\x — a x+a" 
 and, by Art. 75 (Ex.), 
 
 r -J— = -1- nos(x - a) - log(a! + a)2= ^ log^^^. 
 •''a^-a^ 2a'- ^^ ^ sv ^ -'J 2a *a; + a 
 
 (6) ^egwired/, J^j^j. 
 
 l/l+^N _ l + Jg ^ 1 . " 
 
 ^|\l-xJ~ ^{i-x') v(i-«^) V(i-«^)' 
 
 r T- = sm~*a;. 
 
 •^ V(i-^) 
 
 /■ can be readily obtained by substituting y = (1 — ar") , 
 
 and is —yj{\ — a?) ; 
 
 hence / /^J-±|) = sin->x- V(l-a^)- 
 
 (c) Required f, -J {a^ — 3?)- 
 
 „2 g'-ar' _ a' ^__ 
 
72 DIFFJEBENTIAL CALCULUS. [Art. 83. 
 
 0^ 
 
 and /.V(««-..) = ay,-^^-/.^^^-^^, 
 
 whence /, V(a' -a?)=a? sin-if _/. ,, f „. , by Art. 77 ; 
 
 but /.V(«'-a^) = xV(a'-a^)+/. f 
 
 V (a — or) 
 
 6^ integration by parts, if we let 
 
 u = ^{a' — a?) and Z>,,« = 1. 
 Adding our two equations, we have 
 
 2/;V(«' - a^) = » VC"' - «") + a»sin-»- ; 
 
 a 
 
 and .:f,-s/(a? - a^) = Va;V^^^ + a^sin-i^Y 
 
 £XA3IFLES. 
 
 Find 
 
 (1) /,V(a^ + a«). 
 
 1 
 
 (2) f,^{a?-a?). 
 
 ^ns. - [a;^ (iB2 - a^) - a21og(a; + ^/a^-a') j . 
 
 83. To find the area of a segment of a circle. 
 Let the equation of the cu'cle be 
 
 and lei the required segment be cut off by the double ordinates 
 through {xo,yo) and {x,y') . Then the required area 
 
 A=2f,y + C. 
 
Chap. V.] 
 
 INTEGBATION. 
 
 73 
 
 From the equation of the circle, 
 
 y = ■\/ {(IT - 3?) , 
 
 hence A = 2/,^ {a' - ar') + C ; 
 
 and therefore, b}- Art. 82 (c) , 
 
 A = a: V («' - x") + a' sirr' ^ + O. 
 As the area is measured from the ordinate 2/0 to the ordinate t/, 
 A= when x = x„: 
 
 therefore 
 
 = a;„V(«--a;„2) + H^sin"' - + C, 
 
 and we have 
 
 C= —Xf, -^(n' — Xo) — a^sin"^— ) 
 
 ^ = a; V(«' - ar') + «'sin-i| - x^-^{a^-xi) - a^sin-^^" 
 If a^= 0, and the segment begins with the axis ofY, 
 
 A = X -^ (a' — x') -\- a^sin"^ — 
 If, at the same time, x = a, the segment becomes a semicircle, and 
 
 A^a-^la" — a') + a'sm '- = - 
 
 The area of the whole circle is jra^. 
 
74 DIFFERENT LAX, CALCULUS. [Akt. 84. 
 
 Examples. 
 (1) Show that, in the case of an ellipse, 
 
 ^ + ^~^' 
 
 the area of a segment beginning with any ordinate ya is 
 
 A = . 
 
 x^{a^ — Q?) + a^sin~' a;|,^/(a^— a;,,^) — a^sin~'_ 
 
 That if the segment begins with the minor axis, 
 
 
 
 A=- x\J{a' — 3?) + a'sin- 
 
 ^x 
 a 
 
 That the area of the whole ellipse is -Kob. 
 
 
 (2) 
 
 The 
 
 area of a segment of the hyperbola 
 
 
 
 
 
 0/ 
 
 — a;oV K^— «0 + a^log(a;o+Va;o^— a^)]. 
 If »ii = a, and the segment begins at the vertex, 
 
 ^ = - [« V(a^ - «') - anog^x+'s/af-a") + a'loga]. 
 a 
 
 84. To find the length of any arc of a circle, the coordinates 
 of its extremities being (a;,,,?/,,) and {x,y) . 
 
 By Art. 52, s=/.VCl+(A2/)']. 
 
 From the equation of the circle, 
 
 x' + y^=a^, 
 
Chap, v.] INTEGEATION. 76 
 
 we have 2x + 2yD,y = 0, 
 
 y 
 
 ir f 
 
 -a ^r 1 _... jX 
 
 s=f,-=af,-jj-±—^ = asm-'^ + C. (Art. 77.) 
 
 When x = Xo, s = 
 
 . *^o 
 
 hence = o sin"^ - +0, 
 
 C'= — asm '-, 
 a' 
 
 and s = « sm~' sin~'— )• 
 
 \ a a) 
 
 If a;o= 0, and the arc is measured from the highest point of the 
 
 X 
 
 circle, s = asin~"'-' 
 
 ' a 
 
 If the arc is a quadrant, x = a, 
 
 s=asin 1(1) = —, 
 
 and the whole circumference = 2-a. 
 
 85. To find the length of an arc of the parabola y^ = 2 mx. 
 We have ^yD^y = 2to ; 
 
 Ay = — ; 
 y 
 
76 
 
 DIFFERENTIAL CALCULUS. 
 
 [Art. 85. 
 
 s=A 
 
 
 -f. 
 
 1 
 
 V('^' + 
 
 f) D,^. 
 
 
 by Art. 73 ; 
 
 lit a lih 
 
 by Art. 82, Ex. 1. 
 If the arc is measured from the vertex, 
 
 s = when 2/ = ; 
 
 0=-^(mnogm) + C, 
 2m 
 
 and 
 
 G-= mlogm, 
 
 '-2|_ iii^ +™^°S ^i J- 
 
 Example. 
 
 Find the length of the are of the curve ^= 271/^ included be- 
 tween the origin and the point whose abscissa is 15. 
 
 Ans. 19. 
 
A SHORT TABLE OF INTEGRALS 
 
 COMPILED BT 
 
 B. O. PEIRCE 
 
 LATE H0LLI3 PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY 
 IN HARVARD UNIVERSITY 
 
 ABRIDGED EDITION 
 
 GINN AND COMPANY 
 
 BOSTON • NEW YORK ■ CHICAGO • LONDON 
 LANTA ■ DALLAS ■ COLUMBUS • SAN FRANCISCO 
 
COPTKIGHT, 1914, BT 
 
 GINN AND COMPANY 
 
 ALL EIGHTS BESEBVED 
 615.11 
 
 gjie gtdemciiiii grent 
 
 GINN AND COMPANY • PRO- 
 PRIETORS • BOSTON • U.S.A. 
 
FUNDAMENTAL EQUATIONS 
 
 I- I a ■f{x)dx — a I f(x)dx; j <J3{y)dx= j ^^dy, where y'=dy/dx 
 
 2. j (u + v)dx= j udx+ j vrfa;, where wand ■ware any functions of a 
 
 3. j udv = uv — j vdu ; ju— dx = uv— jv — dx. 
 
 /a-m+i fdx 
 
 x'^dx = 7 > if m # — 1 ; | — = log x, or log (— x). 
 m -\-l J X D\/ 
 
 J 'J alogS 
 
 6. I smxdx =— coax; I <iosxdx = smx. 
 
 /tan...=-logcos.;/ctn... = logsin.. 
 
 I aea^dx = tana; ; / c 
 
 . I (ioah.xdx = svdh.x; Is 
 
 I tanh xdx = log cosh x ; I ctnh a; = log sinh as. 
 
 /8:7_^^=.Cn-(^W -ictn-(^). 
 
 /' dx ~i. , Jx\ 1 , a + x 
 -z ; = - tanh->( - 1 > or ;r— log 
 a? — a? a \a/ 2 a "a- x 
 
 —^ — 5= ctnh-M-)> or H— log — ; 
 
 ayt — a" a \a/ 2a °x + a 
 
 sedhidx = tana; ; / cse''a;<ia; = — etna;. 
 7. / coshojtia; = sinha;; / sinh a; da; = cosh a;. 
 
FUNDAMENTAL EQUATIONS 
 
 / • = sm~V-)) or — cos-^(-)- 
 
 Vx^ ± a" 
 
 = log (a; + Va;^ ± of). 
 
 dx 1 
 
 : = - COS"' 
 
 -\/x^ - a" a, 
 
 J x-^a'±x'. a "V 
 
 + Va^ ± x^\ 
 
 10. r_^_= 2 ^^^_ r^+S.^^^-2^^^j^_^ f^+fe_ 
 J x^a + bx V— a ^ — a Va ^ « 
 
 In such a case as this, that one of the alternate values of the inte- 
 gral which makes the quantities under the radical signs positive is 
 to be used, and each radical itself is to be considered positive. Of 
 course an arbitrary constant may be added to the value of every 
 integral given in this pamphlet. 
 
 11. e^ = cosa; + i sina;; e~" = cosa; — i sina;. 
 
 12. sinha; = ^(tf— e-'^); cosha; = ^((f+ e-''). 
 
 13. siu xi = i sinh a; ; cos a;i = cosh a;. 
 
 14. sin a; = — t sinh xi ; cos x = cosh xi. 
 
 15. log M = log (cm)— logo. 
 
 16. log x = log {-x) + {2k + 1) iri ; log,a; = (2.3025851) ■ logjos;. 
 
 17. log (x ± yi) = i log (x'* + f)±i tan- ' {y/x). 
 
 For acute angles and some other cases easily to be determined in 
 each instance, 
 
 18. sin-'M = cos-i Vl-M^ = tan-i (m/V1 - v?) = csc"' (1/m). 
 
 19. sin-^M = _ sin-i Vl - w^^ + a constant = J sin-' (2 m^ - 1) + 
 a constant. 
 
 20. tan-^M = — tan-i (1/m) + a constant. 
 
EATIONAL ALGEBRAIC FUNCTIONS 
 
 I. RATIONAL ALGEBRAIC FUNCTIONS 
 
 A. Expressions involving (a + bx) 
 The substitution of y or s for x, where y = xs = a + bx, gives 
 
 21. I (a -h bxydx = j I y'^dy. 
 I 22. i x{a + bxydx = - I 2/'»(y — a)dy. 
 
 23. fx" {a + te)'»rfa; = -^^ f iT (y - a^dy. 
 
 r x'^dx .^ 1 ny-afdy 
 J {a + bxy fe-'+y 3/™ 
 
 r dx 1 r (a! - &)"'+'-^(^g 
 
 j x"(a + bxy~ a^+o-ij s"' 
 
 25 
 
 Whence 
 
 /dx 1 
 
 27 
 
 '■/ 
 
 
 28- J (a + te)" 2 &(* + *»:)' 
 
 29. r_^ = 1 ^« + to - « log (a + te)]. 
 J a + bx r 
 
RATIONAL ALGEBRAIC FUNCTIONS 
 
 r xdx _ IV 1 a 1 
 
 ^^- J (a + bxf ~ 6^ a + bx'^ 2{a + bxf\' 
 
 ^'^- f'^+k " Ki ^* "^ *'"''' -2a(a + bx)+ a"log(a + te)]. 
 
 35. r^ 
 
 J x(a + bxy a(a + bx) a? 
 
 dx \. a-\-bx 
 
 = ^ — log 
 
 a; (a + 6a;) ax 
 
 dx 
 
 1 1 , a. + te 
 
 dx \ b a + hx 
 
 = - — + iiiog— :: — 
 
 a;^ (a + fe) aa; a^ 
 
 37 
 
 B. Expressions involving (a-\-h^') 
 
 / dx 
 
 dx 1 . , X 1 . , a; 
 ; = - tan"' - = - sin"' 
 
 c c 
 
 V?+1 
 
 X -\- c 
 
 nn r dx 1 , c + a; C dx 1 , 
 
 Va + a; V— 6 
 
 + Sa;^ 2 V-a6 Va - a; V^ 
 
 L= tanh-i(a; -y|^) , [a > 0, 6 < 0]. 
 
 v: 
 
 41 
 
 dx 
 
 dx 
 
 {a + bxy ~ 2a(a + bx') "*" 2^J ^1^' 
 
EATIONAL ALGEBEAIC FUNCTIONS 
 
 42. f— ^-__ ^ _J_ a: , 2m-l f dx 
 
 J (a + 6x2)'»+i 2 ma (a + bx'y ■*" 2 ma J (a + Jx^)"' ' 
 
 ■ J (« + to')"'+' 2J (a + fe)"'+i ' L« - « ]• 
 
 45. f-^— =J-W^l-. 
 Ja3(a+fe=') 2a^a + 6a;2 
 
 46 r_^^dx_ _x a r 
 ' J a + bx^~b hj a 
 
 dx 
 + ba?' 
 
 dx 
 
 47. c ^'^ = _ i. _ i r. 
 
 ■ J x\a + bx^) ax a J \ 
 
 a + bx^ 
 48 / ^ "* —X , 1 r dx 
 
 J r^_^dx__ ^ -a; _J_ /• 
 
 ■J (a + bxY-"^ 2mb(a, + bx^y'^27nbj (a • 
 
 - + boi?y 
 
 49 r ^^a' ^ 1 r___^__ _ ^ r t^a; 
 
 ■ J ^^(a + te'^)™*! a J x^(a + te^)"' aj (a + bx^y-^^' 
 
 50 r_^_ * Ri (^ + x )" , /-^ ,2x-A;"l ,„„ ^ 
 
 '"• J ^Tto°=3^[2^°g /fc^-fa+x' +^^'^~^ivrJ' ^^'^ "^- 
 
 El /^ cct^a; 1 ri, k^—kx4-a? rr , ,2x—k'] 
 
 62. r ^ -log^ 
 
 J a;(a + 5a;") an a + bx" 
 
 ' J (a + bx")'"+^ ~ aj (a + baf)'" ~ aj (a + bx-)'"+'^' 
 
 r x'^dx _ 1 r a;"*-" _ a r ^"'-''cZa; 
 J (a + 6a;")^+i ~ 6 J (a + S*")^ *lj (« + ^a:")""^^ ' 
 
 Z' dx _ 1 r__dx b r dx 
 
 ■ J x"'(a + bx")'' +^~ aJ x'"(a + bx")" a J x'"-"(a + te")^+' " 
 
EATIONAL ALGEBRAIC FUNCTIONS 
 
 •! 
 
 ^|S^ 
 
 
 + 
 
 ^a 
 
 s 
 
 + 
 
 + 
 
 ^ 
 
 + 
 
 s 
 
 + 
 
 + 
 
 + 
 
 + 
 
 i-O 
 
 + 
 
 J 
 ^ 
 
 8 
 
 + 
 8 
 
 
 ¥ 
 
 i^ 
 
 
 + 
 
 1-1 
 
 + r^ 
 
 + 
 
 H 1 - 
 
 a* 
 
 
 vl. 
 
 s 
 
 
 § 
 
 
 KS 
 
 
 
 
 CO 
 
 2 
 
 03 
 CO 
 
 B 
 
 Ph 
 X 
 
 o 
 
 n 
 
 eS 
 
 + 
 
 + 
 8 
 
 
 o 
 V 
 
 a 
 
 I 
 8 
 
 > 
 
 + 
 
 8 
 
 bo 
 o 
 
 I—* 
 
 ■^1 
 
 oo 
 
EATIONAL ALGEBRAIC FUNCTIONS 
 
 63 r^^ _ _ bx + 2a b fdx 
 ■J X^ - qX ~qjx- 
 
 g . r xdx _ _ 2a + bx _ b(2n — l) fdx 
 • J Yn + i ~ nqX" ~ nq J X^' 
 
 c,. Cx" J X h , ^ b''-2ao rdx 
 65. J -dx = ---logX + -^^J -. 
 
 66 f— d = (^^-2ac)x + a5 2a rdx 
 ' J X^ ^~ cqX q J x' 
 
 „„ r x"'dx _ a"*-' n — m + 1 b f x'^-^dx 
 
 ■J X"+^~'~ (2n — m + r)cX'''~ 2n-m + l ' ~cj X»+i 
 
 a r x^-'dx 
 
 T'cJ z»+i 
 
 2n — m + '. 
 
 „n rax 1 . a? b rdx 
 
 ^^- J x^X~'2a^ ^^a? ax^\2a^ »/ j ^ ' 
 
 r dx 1 w + m — 1 b r dx 
 
 Jx*»JC»+i~ (m — l)aa;"— iJt" m-1 " aj a;'"-i^''+i 
 
 2w + m — 1 c r dx 
 m — 1 a J a 
 
 -2_yn+l 
 
10 RATIONAL ALGEBRAIC FUNCTIONS 
 
 D. Rational Fkactioks 
 Every proper fraction can be represented by the general form : 
 
 f{x) ^ g,x--^ + y,a:"-^ + y^x-' + • • ■ + y„ 
 F(x) x" + kjX"-^ + k^x"-^ -\ \-k„ 
 
 If a, b, c, etc. are the roots of the equation Fix) = 0, so that 
 F(x) = (x — a)" (x — by (x — ey ■ ■ ■, 
 
 /(«=) _ ^4 A^ , A^ , ^ A 
 
 "^ j?t^\ /^ _ «np ''' c-y _ n^p-l "^ /-.._ /,^p-s ' ■'■ 
 
 £_ 
 
 F{x) (x~ay (a; — a)"-! (x-ay-^ x-a 
 
 ^ ('^ _ /A9 ^ /■-.. _ 7,N«-1 ^ /^ _ A\9-2 1 r- , 
 
 (x-by (x-by-^ (x-by-^ x-b 
 
 ^ (x - cX (a; - c)'-i (» - cy-^ ^ ^ x-c 
 
 + , 
 
 where the numerators of the separate fractions are constants. 
 If a, b, 6, etc. are single roots, then p = q = r=---=zl, and 
 
 f(EL = ^^ + _B_ + -£ , 
 
 F(x) X — a x — b X — c 
 
 where .4 = ^, B = |gl, etc. 
 
 F'(a) F'(b) 
 
 The simpler fractions, into which the original fraction is thus 
 divided, may be integrated by means of the following formulas : 
 
 _- r hdx rhd(mx -\-n) _ h 
 
 ' J (mx + nf J m{mx + ny ~ m{l — l)(mx + n)'-^ ' 
 
 72. I ; — = — log (mx + n). 
 
 J mx + n m ^ ' 
 
 If any of the roots of the equation f{x) = are imaginary, the 
 parts of the integral which arise from conjugate roots can be com- 
 bined, and the integral thus brought into a real form. The following 
 formula, in which i = V— 1, is often useful in combining logarithms 
 of conjugate complex quantities : 
 
 73. log (x ±iji)=i log (x^ + ^)±i tan-i ^ • 
 
IRRATIONAL ALGEBRAIC FUNCTIONS 11 
 
 II. IRRATIONAL ALGEBRAIC FUNCTIONS 
 
 A. Expressions involving ^a-\-lx 
 
 The substitution of a new variable of integration, y = Va + hx, 
 gives 
 
 74. i -Ja + hxdx = ^ V(a + IxJ: 
 
 76. rx-V^T&^c^. = ^(8«'-12«to + 156V)V(^Tto7^ 
 J 105 6° 
 
 77. C:i±±Edx = 2 V^Tto + « f- 
 78./ 
 
 (^a; 
 
 X 's/a + to 
 c^a; 2 Va + te 
 
 Va + bx f> 
 
 r xdx 2(2 a -bx) , ;— 
 
 80. 
 
 81. 
 
 82 
 
 r ar^^ZcB 2(8a''-4ate + 3JV) > t" 
 
 / (Zx _ 1 , / Va + fa — Va \ 
 
 icVoH-fa "V^ \ Va + te + Va/ 
 
 a + 6a; 
 
 /(^x —2 « + 
 . = — ^i^tanh 1 \ 
 X Va + hx Va ^ * 
 
 /dx _ Va + Sx & /* <^a 
 x^VaTITte aa; 2aJ x Va + te 
 
 8*.j(.+..)-i..=f//..*=?fi|f:. 
 
12 IRRATIONAL ALGEBRAIC FUNCTIONS 
 
 x'^dx 2a:"'Va + te 2 ma P x'^'Hx 
 
 86 
 
 /x"'dx _ 2 a:"' V g + bx 2 ma r 
 
 VTTte " {2m + l)b ~ (2m + 1)6 J ^ 
 
 te (2m + 1)6 (2m + 1)6 J V^Ifte 
 
 „„ / (?a; _ V a + 6a; (2 m — 3) 6 
 o7 — 
 
 + 6a; 
 
 88. /■(^i±M!^=6r(a+*.)^<^x+ar(^±Mz:^. 
 
 a; 
 89 ' -^ - ' '^•^ -^ '• ^^ 
 
 / (^a; _ _ Vft + 6a; (2 w — 3) 6 /^ dx 
 
 a;« Va + 6a; ~ (?i-l)ax"-i (2»- 2)a J a;»-i V^ 
 
 n 
 
 /(a + 6x)2(ia; , /*, , ,i:i^ , /• 
 
 !-— L — 2 = 6 / (a + 6a;) 2 dx + a j 
 
 /dx _ 1 r "^^ ^ /" 
 
 x(a + 6a;)2 •>' a; (a + 6a;) 2 J (« 4. Ja;)^ 
 
 B. Expressions involving y/z^ ± a' and Va^ _ ^^ 
 
 90. CVi^^^dx = ^ [a; Va2"±~^ ± a^ iog(x + Vx^"±a^)]-* 
 
 91. C-Ja'-x'dx = iL Va^ - a;^ + a'' sin-i(-U 
 
 92. r^=^ = log(^ + ^^^ 
 J Vx'* ± a^ 
 
 93 
 94 
 95 
 96. 
 
 /dx . _,/x\ /x\ 
 
 — , = sin ^ - I) or — COS"' { - )• 
 
 Va^ _ x" Va/ \a/ 
 
 /dx 1 Ja\ 1 ,/a;\ 
 , = - COS"' I - I J or - sec~' ( - 1 • 
 X Vx^ - a^ « Vx/ a \a/ 
 
 r-^^==-iiog(^+^^si\* 
 
 J X Va^' ± x^ a ^ a; / 
 
IRRATIONAL ALGEBEAIC FUNCTIONS 13 
 
 97 f^x^— a" I a 
 
 \ dx = V ck'' ~ c^ — a cos~> - • 
 
 J X X 
 
 /xdx r-i, 
 
 98 . , ^ 
 
 xdx 
 ■Vx^ ~ a? 
 
 __ r xax /—: 
 
 100 
 
 / X ■yJx' ± a'dx = ^^(a?±ay. 
 
 101. / x Va'= - x^rfa: = -^ ■\/{a' - x^. 
 
 102. f^ix' ± aydx 
 
 = l[x^(x'±ay ± ^ V^^T^ + ^ log(cc + V?±^^)]. 
 
 103. r V(a^ - xydx 
 
 104. f-^L= = -^^^. 
 
 105 
 
 106. 
 107 
 108 
 109 
 
 a^)' a^ Va^ - x^ 
 xdx — 1 
 
 /xt^a; 1 
 V(a'' - xy ~ Va^ - X 
 
 . Cx-</(a^ - x'fdx = - iV(a2 - x^. 
 
 * See note on page 12 
 
14 lEEATIONAL ALGEBEAIC FUNCTIONS 
 
 110. Cx'^x'± a'dx 
 
 111. jxWa'-x'dx 
 
 = _ ^ V(a2 - a;^)« + f (« Va'^ - x' + a^ sin-^ |^ • 
 
 112. r-&= = |v;^r^T^iog(cB+^^^T^).* 
 
 x^dx X i—„ 5 , a'* ■ 1 ^ 
 
 = — - V o.^ — x^ + -^ sin~' - • 
 
 /a; ax 
 
 113. 
 
 ■ X 
 
 /dx Vx^ ± a' 
 
 115. 
 
 /: 
 
 
 116 
 
 1 = ^ + log(x + Vx^ ± a% 
 
 11- , ■^a' — oe' , V a^ — x^ . ^ x 
 
 117. I ; dx= sin-i- 
 
 x' X a 
 
 x^dx — X 
 
 118 
 
 f /^" = ^^^ + log(x + V^^T^).^ 
 
 ..- r x^dx X ■ ix 
 
 119. I _^ ,_ = — ;==; - sm-i -^ 
 
 V(a^ - x^)" VZ^ 
 
 x" a 
 
 C. Expressions involving y/a + hx+c^ 
 
 4c 
 
 Let X = a + bx + cx^ o = 4 ac — S'', and fc = In order 
 
 ^ . q 
 
 to rationalize the function f{x, Va + 6x + ox^) we may put 
 
 Va + bx -\- ca? = ^ ±0 ^A + Bx ± a?, according as c is positive 
 
 or negative, and then substitute for x a new variable «, such that 
 
 • See note on page 12 ) 
 
IRRATIONAL ALG-EBRAIC FUNCTIONS 15 
 
 z =^A + Bx + x^ - X, it c>0 ; 
 
 „_ ^A+Bx-x' -Va .„ ^„ ^ a 
 
 « = ■ . if c < and > ; 
 
 X — c 
 
 _ (x — /? 
 
 * — \\ J wnere a and B are the root^ of the equation 
 
 ya — X ^ ^ 
 
 ^ + £x - a:'' = 0, if c < and -^ < 
 
 a 
 — c 
 
 By rationalization, or by the aid of reduction formulas, may be ob- 
 tained the values of the following integrals : 
 
 1 . , ./ 2cx + b \ .^ 
 P —pSinh-M ^ ^ >ifc>0. 
 
 Vc WAac-by 
 
 121. I —j^= . sin M , =\)ifc<0. 
 
 122. f ^"^ _ 2{2cx + h) 
 
 ■y/x qVx 
 
 r dx _ 2{2cx + b)Vx 2k(n-l) C dx 
 Jx-VJC~ (2n-l)qX" + 2n-l J x-'Vx' 
 
 125. fVxdx = (2^±*)2^ + ;^ f-^. 
 
 / a;<^a; _ "Vx b_ P dx 
 
16 IREATIONAL ALGEBRAIC FUNCTIONS 
 
 /xdx _ 2 {bx + 2 ft) 
 X^X q^X 
 
 xdx "WX b r dx 
 
 r xdx Vx_ b_ r dx 
 
 J X'^/x' (2»-l)c^» 2cJ x'Vx' 
 
 r^dx_(x 3b\,-3i^-_iacrdx 
 
 "'^- J vf - V2-C - 4^; ^^ + ^B^-j ^• 
 
 r x^dx _ (2b^ — 4:ac)x + 2ab 1 f dx 
 'JXVX~ cqVx <^j Vx' 
 
 r x'dx ^ (26^-4fte)x + 2a& 4ac + (2>t-3)&^ r dx 
 ■Jx»Vx {2n-V)cqX''-'^Vx (2n-l)cq Jx-^Vx' 
 
 ... r^dx _ (x^ 5bx SP 2a\ rr^,/3ab 5b^\rdx 
 
 136. f^Vxdx = ^^-:^ fVIdx. 
 J So 2c J 
 
 137. fxX Vxdx = ^^ -^ fx VI dx. 
 J 5c 2gJ 
 
 138 C ^^^dx _ X" Vx b r X"dx 
 
 J VX ~(2n + l)c 2cJ Vx' 
 
 r ^X'dx ^ xX'Vx _ (2n + 3)b f xX'dx 
 J Vx 2(n + l)c 4(w + l)cJ VX 
 a r X^dx 
 
 2(ri + l)cJ Vx' 
 
 J \ 8c ^iSc" 3c) 5c 
 
 139 
 140 
 
lERATIONAL ALGEBEAIC FUNCTIONS 17 
 
 142 r ^^ 1 1 (Vx+-^ h \ ■. 
 
 -' x^X Va \ a; 2Va/ 
 
 143. I — 7= = — — =:sm-M , ) > if » < 0. 
 
 J x^X -NAlTa Xx^b^-iacJ 
 
 144. '^ '^^ 2Vx 
 
 ■/; 
 
 - , if a = a 
 
 ; Vx bx 
 
 145. r_^5_ = Vx 1 r__dx__ _ ^ r__«^a; 
 
 J ccJ^-VZ (2ra-l)aX« aj a;X"-iVx 2aJ x" 
 
 V^ 
 
 146. 
 
 r dx ^ _ jV^ _ _&_ r_^_ 
 
 J ar'Vx aa; 2 a J xVx 
 
 ^^/xdx nr. . b C dx , C dx 
 
 J « ^J ^X J a 
 
 xVx 
 r xHx X" r x^-Hx b r x—^dx 
 
 ' J xVx~ (2n-l)Vx "J xVx 2j Vx ' 
 
 149 f ^^dx _ Vx 5 r_^ , f_^ 
 ' J 0? ~ X 2J X Vx "j Vx 
 
 /x^rfa; _ 1 r x'^-Hx b r x^-^dx a C x'^-'^dx 
 X»Vx~ "J X»-Wx cj x»Vx cj X''Vx 
 
 , gi ' C x'^XHx _ a'"-'X"Vx _ (2w + 2?»-l)& /" «;"■ -^X't^a: 
 
 (m — l)ffl /■ a:'"-''X"<^a; 
 ~ (2n + m)oJ Vx 
 
 152l f-^^— - ^^ 
 
 Ja^^X-V]? (m-l)ax'»-iX» 
 
 (2ro+2wi-3)& r^_dx__ 
 ~ 2a(m — 1) J a;"-iX»Vx 
 
 (2 7i. + m — 2)c /■ c?x 
 (m — l)a J x^-^X'Vx 
 
18 IRRATIONAL ALGEBRAIC FUNCTIONS 
 
 153 f ^"'^^ - _ A-"-^ Vx (2n-l)b r X'^'Ux 
 'Jx'-Vx~ (m-l)x"'-> 2(m-l) J a;"-iVx 
 
 (2n-l)c r X'-'^dx 
 m-1 J X"'~^Vx 
 
 r dx 1 , ,2h + m(a'+b'x) 
 
 154. / 7= = -7= tan-i ^ )— 1- ' ^ > 
 
 J (a' + 6'a;)Vx V- A 2b'^-hX 
 
 or -7= log 3: — !^^: — ^ , 
 
 V A a' + b'x 
 
 wliere m = bb' — 2 a'c and h = aj'^ — a'65' + ca'^. 
 
 If A = 0, the value of the integral is —2 b' 'Vx/[vi(a'+ b'x)"]. 
 
 D. Miscellaneous Algebraic Expressions 
 
 155. / V2 (/./• - a;" dx = \\_{x — a) V2 ax — ^ + a^ ^wr^ix — a)la\. 
 
 ..r. C dx .la — a;\ 
 
 156. / =cos-M • 
 
 J -J 2 ax -J? \ a / 
 
 r dx ^ 2 _i i -b'(a + bx) 
 
 J V^T^ . V^^TV^ V^W^ ^"^ \ b(a' + b'x) ' 
 
 2 
 : tanh~' 
 
 'bb' 
 
 I &'(a + fa) 
 \6(a' + 6'a;)" 
 
 158. J ■Vla + bx){a' + b'x)dx = —^ — ^ V(a + fa)(«'+(i'x) 
 
 -^r , -^" [7c = «*'-«'*]. 
 
 8 M'J Wa + bx- Va' + 6'a; 
 
 159. r l a' + b'x _ -\/ri+bx--</a' + b'x k P dx 
 
 J ya + bx ^ b 2b J Vrt + faVa' + liS 
 
 160. r-Ji±-2(Zx = sin-'x-Vnr^. 
 
IRRATIONAL ALGEBRAIC FUNCTIONS 
 
 19 
 
 161- / yl^^^dx = -\/{x + a)(x + b) + (a-b)\og{^/x + a + Vx + b). 
 
 /dx o • 1 \x — a 
 
 V(a;-a)(a'— a;) >a' — a 
 
 (ya: + q)dx ^ q + a'p r 
 
 {x - a')(x -b')^a + bx + cx^ «'-*'J 
 
 dx 
 
 + bx + cx^ 
 dx 
 
 (x — a') 'Wa + bx + cx'^ 
 
 T + b'p r I 
 
 «'-*'J (x-b')^\ 
 
 a -\-bx ->r CO? 
 
 164. 
 
 J («' 
 
 <^a; 
 
 + b'x)^a + bx + CO? 
 
 - J_ . lo / 2 ^ + "'■ (a' + ^I'a:) -2 6' VA(a + 6a; + ca:' 
 
 tan" 
 
 a' -\-b'x 
 2h + m(a' + b'x) 
 
 •where 
 165 
 
 h \2b' ■y/-h(a + bx + cx^)/ 
 
 m=bb'-2 a'c and h = ab"' - «'66' + ca'\ 
 
 ■ff{^,^ 
 
 a -\-bx 
 
 a' + b'x 
 
 dx 
 
 = n(a'b — ab') \ f-i tzV. -— -,„ 
 
 ^ ^J ■' Xb'z^-b I (b'z"-''^^ 
 
 bf 
 
 where 
 166 
 
 s"(a'+ b'x)= a + bx. 
 
 ■ I f {x, Va + bx + cx^Jdx 
 
 .r./2Va-z-b z^Va — bz+Va^z^Va-bz + ^a 
 
 1-z^ 
 
 (1 - zy 
 
 dz, 
 
 where 
 
 : + Va = Vo. + iz + I 
 
20 TEANSCENDENTAL FUNCTIONS 
 
 III. TRANSCENDENTAL FUNCTIONS 
 
 167. I svaxdx = — cos x. 
 
 168. I sin^xdx = — -J- cos a; sin aj + ^ a; = | a; — i sin 2 a;. 
 
 169. I sin'ajc^a; = — | cos a; (sin^a; + 2). 
 
 .-« r . , sin""'xcosx , n — 1 f . „_„ , 
 
 170. I sm"xdx= 1 I sin" ^xdx. 
 
 J n n J 
 
 171. I cos xdx = sin a;. 
 
 172. j cos'^ajf^a; = ^ sin a; cos a' + |- .'■ = ^ a; + J sin 2 x. 
 
 173. I cos'aic^x = i sin a; (cos'* a: + 2). 
 
 /I w — 1 r 
 cos"x(ia; = -cos"-^a;sina; -) | cos"-^xfZa-. 
 n n J 
 
 r 
 
 175. I sin x cos xdx -^ i- sin^'x. 
 
 176. j sin'^xcos^a-c^x = — •J(Jsin4a- — x). 
 
 /cos^+^x 
 sin X cos^xtix = — p ■ 
 m + 1 
 
 178. I sin^x cos xdx = 
 
 m + 1 
 
 ,„„ r . , cos"'"'^ sin"+^x m — 1 T 
 
 179. I cos^x sin"xax = ; 1 ; — I ' 
 
 J m + n m + nj 
 
 ,„„ r . , sin"-ixcos'"+ix TO — 1 r 
 
 180. I cos'"xsm"xcZx = ; 1 ; — 
 
 J m -\- n m-\-nJ 
 
 roos'"xdx _ cos^+^x m — n + 2 Tec 
 
 'J sin"x (w — l)sin""'x n — 1 J si 
 
 cos"'~''x sin"x 
 
 cos'"xsin"~^x 
 
TEANSCENDENTAL FUNCTIONS 21 
 
 — 1 roos^~^xdx 
 
 jgo f cos"'xdx cos^-^a;- m — 1 T 
 
 J sin"x (m — n) sin''~^x m — n^ 
 
 183. r!E!^=_ p°^"'(i-")Kf~") 
 
 ' J COS"* I ■„/■"" \ 
 
 sin^as 
 
 184 ' '^'" 
 
 sin"'x cos"a; 
 
 1 1 m + re — 2 r tZa; 
 
 n — 1 sin"'~^x • cos""'x n — 1 J sirfx ■ cos'-^x 
 
 1 1 m + n. — 2r rfa; 
 
 m - 
 
 -1 sin^-^x ■ cos""^* m — 1 / sin^'^cc ■ cos"a; 
 
 rt-2 r 
 -1 jsii 
 
 /^ = log tan X. 
 
 sm X cos a; 
 
 ' fti C-.^:^— — ^ *'°^ ^ -L "^ ~ ^ r dx 
 
 ' J s\Ti™x m — 1 sin^'^a; m — IJ sin™"' 
 
 r dx _ 1 sin a; w — 2 r ' dx 
 
 J cos"a; «■ — 1 cos"~^a5 n—1 J eos"~^a; 
 
 187. I tan ajo^x = — log cos x. 
 
 188. I tan'^ajtZx = tana; — x. 
 
 /tan""' 7' Z' 
 tan"a;<Za; = — '■ — r^ — / i&n'—^xdx. 
 n-1 J 
 
 190. I ctn xdx = log sin x. 
 
 ■1. I cin^xdx = — ctn a; — X. 
 
 \ I ctn»xc^x = — ^ '^_..'^ — / ctn"-2xrfx. 
 
 3. I sec x(Zx = log tan ( T + 2 ) ' 
 
 J' 
 
 sec^x cZx = tan x. 
 
22 TRANSCENDENTAL FUNCTIONS 
 
 195. / seo"xdx= I — -—■ 
 J J cos" a; 
 
 196. I CSC xdx = log tan. -J-x. 
 
 197. I csc''a;c?a! =— ctnx. 
 
 198. / oso"xdx = / -. 
 
 J J sin"x 
 
 inn r dx —1 . Jb + aCOSx] ^T^AT 
 
 199. j ; = , = • sin-^ — -— )[a>6>0], 
 
 J a + 6 cos a; ■y/a' — h^ \.a + b cos xj "- 
 
 or • sm-^ --1 j[a>J>0], 
 
 Va^ — b"^ L a + ocosx J "- 
 
 1 i 1 [ ^"^ — &^- sinx "! ^ , ^ AT 
 =^ ■ tan-' — T—, ) [a > 6 > 0], 
 
 2 _ J2 L + a cos X J 
 
 1 , r^ + a cos X + ^J¥^—a? ■ sin x 1 r ^ a 12 ^ an 
 -a^^L a + 6cosx J "- -" 
 
 _, r &^ + g' + g. (6 cos X + c sin as) " 1 
 L V6M-^(a + & cos X + c siaas)J 
 
 °' Vi?^ 
 
 "^- vi 
 
 6 cos X + c sin x 
 
 ■ sm" 
 
 ■\/a'-b^- 
 01- ■ = • log 
 
 [ 
 
 P + c' + a(b cos X + c sin x) + V^" + c^ — a'(b sin x — c cos x) 1 
 V^M-^ (« + (!• cos X + c sin x) J 
 
 201. I X sin xc?x = sin x — x cos x. 
 
 202. / x^sin xrfx = 2 X sin x — (x'^ — 2) cos x. 
 
 203. I x°sin x(^x = (3 x'' — 6) sin x — (x° — 6 x) cos x. 
 204./x™sinx.x=-x™cosx + «./x'»-^cosx.x. 
 
TEANSCENDENTAL FUNCTIONS 23 
 
 205. I X cos xdx = cos x + x sin x. 
 
 206. / x^cosxdx = 2x cos,a; +(a? — 2) sin x. 
 
 207. I x'cos xt^a; = (3 ar" — 6) cos a; + (cc' — 6 a;) sina;. 
 
 208./.™eos... = .»sin.-./.'"-sin.... 
 
 nna Tsinx 1 sina; . 1 rcosx , 
 ^03. / — -— dx = • 7 H / dx. 
 
 J x" m — 1 x"""^ m — 1 J x"""^ 
 
 oin Tcosx , 1 cosx 1 Tsinx , 
 *10. I — — - dx= I dx. 
 
 J x" m — 1 x""-^ rn — lj x"-! 
 
 212. 
 
 X 3-3! 5-5! 7-7! 9-9! 
 
 x' 
 
 /cosx , , a^ X* x' 
 
 6-6! 8-8! 
 
 213. I sin (mx + a) ■ sin (tix + b) dx 
 
 _ sin (mx — »ix + a — 6) sin (mx + tox + a -\- b) 
 2 (m — n) 2 (m + n) 
 
 214. I cos (mx + a) • cos (nx + S) c?x 
 
 sin (mx + wx + » + ^) , sin (mx — wx + <»■ — ^) 
 2 (m + n) 2 (m — re) 
 
 215. I sin (mx + a) • cos (nx + 5)<?x 
 
 cos (wx + nx + a + b) cos (mx — nx + a—b) 
 2(m + n) 2 (m — n) 
 
 ' <i, I sin (mx + a) • sin (mx + &) dx 
 
 X ., . sin (mx + a) ■ cos (mx 4- S) 
 
 = 7^ • cos (6 — a) ^^ ^ ^^ -• 
 
 2 ^ ' 2m 
 
 21'( ^in (mx + a) • cos (mx + S)«?x 
 
 / sin (mx + a) • sin (mx + J) x . ., . 
 ^h = 2m ^-2-«"^(*-«)- 
 
24 TEANSCEKDENTAL FUNCTIONS 
 
 218. / cos (mx + a) ■ cos (mx + h')dx 
 
 X ., . sin (mx + a) cos (mx + V) 
 
 219./sin-... = .sin-^. + Vrr^. 
 
 220. I cos~^xdx = X cos~'a; — Vl — se'. 
 
 221. I tan-^xdx = x tan~*a; — ^ log (1 + a?). 
 
 222. I ctn-^xdx = x etn-ia; + i log (1 + xy 
 
 223. / veisin-^xdx = (a; — 1) versin~*a; + V2 x —-a?. 
 
 224. j(am-^xydx = x (sin-ia;)'' - 2 a; + 2 Vl - a;^ sin-ia;. 
 225. 
 
 . I X ■ sin-^xdx = J [(2x' — 1) sin"*;?; + x Vl — x']. 
 
 226. I a;"siii^a;<Za; = — rT I — := 
 
 J n + 1 » + lJVl- 
 
 227. fx-cos-^xdx = '""^' "°V'^' + — -T r^ 
 
 J «+i »i+ij vn 
 
 
 228. I x''ta.n^xdx = z r I ^; ; 
 
 J n + 1 n + lj 1 + a^ 
 
 229. I log xdx = X log a; ^ a;. 
 
 230. rfl2£^^.= 1 aog.)-.- 
 
 ■J a (log xy ~~ (n-1) (log a:)"-i ' 
 233. fx'"logxdx = x'-+'\^^^^ 1 1. 
 
 J L'^ + i (m + iy\ 
 
 X' 
 
 dx 
 a" 
 
TEANSCENDENTAL FUNCTIONS 25 
 
 234. ((^dx=-^> ' 
 
 xe'^dx = — (ax — 1). 
 
 236. j x'^e'^dx = / x'"~'^e'^dx. 
 
 J « aj 
 
 237 C^—^ 1 e-^ , a C ^ a 
 
 238. Ce^^^P^-l C^-ax. 
 J V____ ^ a «J a; 
 
 239. re-.sin^a,,fo, = £!li2^iHtf225M. 
 
 nAn C „^ 1 e'^(aoospx +psmpx) 
 
 240. I e"^ ■ cospxdx = — ^^ , f ^— ^• 
 
 241. / sinhxrfx = cosh a;; | coshxtix = sinhos. 
 
 242. I tanh xdx = log cosh a; ; I ctnh xdx = log sinh. as. 
 
 243. I sech a; «^a; = 2 tan-^ (e'). 
 
 244. I cschaic^a; = logtanh (^) • 
 
 245. / X sinh xdx = x cosh as — sinh x. 
 
 246. I X cosh xdx = x sinh as — cosh x. 
 
 247. I cosh^ aitZa; = J (sinh a; cosh x + x). 
 
 248. I sinh x cosh a; t^a; = ;^ cosh (2 a;). 
 
 249. I sinh^ xdx = ^ (sinh a; cosh a; — x). 
 
26 MISCELLANEOUS DEFINITE INTEGRALS 
 
 IV. MISCELLANEOUS DEFINITE INTEGRALS 
 
 250. jr"-^ = |, if OO; 0, if a = 0; -|, if a<0. , 
 
 251. I x"-'^6-''dx= I log- dx = T(n). 
 
 T(n + l) = n-T(n),iin>0. r(2) = r(l) = L 
 
 r(n + 1) = nl, if w is an integer. r (J) = Vtt. 
 
 T(n) = n(n- 1). Z (y) = 2>„ [log T (y)]. 
 Z(l)=- 0.577216. 
 
 oeo r^ » I/I N« ij r" ^"''c^a; r(m)r(«) 
 
 J, ^ ' X (i + a^r^" r(m + w) 
 
 IT IT 
 
 253. / sin"a;rfa; = / cos"xcZa; 
 Ja Jo 
 
 1 ■ 3 ■ 5 • • • (>l - 1) TT .. . 
 
 = — 2 — 7 — c ^ , • -^ ' II w IS an even integer ; 
 
 2 . 4 . 6 • • . (»i - 1) ., . 
 
 = --, — 5 — z. — „ ! II re IS an odd integer ; 
 
 l-O'0-7'--re o> 
 
 1 - \2~ 
 = 2 ^'"' — );; \ ^^^ ^^J value of n greater than —1. 
 
 X" sinmxc^a; tt .. ^ tt 
 X ^2' "™>0; 0, ifm = 0; --, ifm<0. 
 
 J f" sin X ■ cos mxdx „ .„ 
 = 0, if m < — 1 or m > 1 ; 
 
 ^1 ifm = -l or m = l; |, if — l<m<l. 
 
 „gg r" sin^a:(^a; _ tt 
 ■Jo a=^ ~2' 
 
 254 
 
 255 
 
MISCELLANEOUS DEFINITE INTEGRALS 27 
 
 '257. jT cos {a?)dx = r"sin {a?)dx = ^ Jf • 
 
 «5o- I sin kx sin mxdx =■ | cos fca; cos ttm tZa; = 0, [A; t^ m]. 
 •^» Jo 
 
 Jr" 2 ;;• 
 sin kx cos mxrfa; = -; ;> if A; — m is odd ; 
 k^ — Tnr 
 
 = 0, if A — m is even. 
 sin^wa;<ia;= I cos'mxdx = '^- 
 
 261 . I sin kx cos Zia; (ia; = 0. 
 
 Jr'' dx IT 
 
 a + ^icosx Va^ - P 
 
 Jf"° QOBmxdx IT 
 
 JT" cosajifo; _ r" sinxc^a; _ yir 
 ~^JT~X ~^r~~N2' 
 
 5865. r , ^" =K 
 
 Jo Vl-yfc^sin^'x 
 
 IT 
 
 266. rVl-Fsin^x.<Za; = ^ 
 
 'rfi /1\%2 /1-3VA:^ / 1-3.5 VA:'' 1 .. ,, 
 
 = 2[^-(2J ^ -(27ij 3 -(2:4:6; 5 — -J'^f ^^ 
 
 267.jrv-..=^v;=^r(i). 
 
 268. rVe-<^.. = 5:%±i)=^. 
 
 <1. 
 
 <1. 
 
28 MISCELLANEOUS DEFINITE INTEGEALS 
 
 „,.« r . ^ , 1 ■ 3 ■ 6 ■ ■ • (2 n - 1) C 
 
 270. / e-^-U.^'^^- 
 
 
 
 Jo 
 
 r 
 
 271. / e"'"cosma;c?x = -;— ;)ifa>0. 
 
 272. / e-'"sinmxrfa; = -;— 5)ifa>0. 
 
 r -— ^ 
 
 273. re-^'^cosbxdx = :'''"' ■ ^ 
 
 274. ' — 2 — ^o.— , 
 
 n^dx=- . 
 
 Ja l-x 6 |7. 
 
 280. jT V l08(l)"<fc = ^l~J^,- [« + 1 > 0, » + 1 > 0]. 
 
 I log sinxdx = / log cos xt^a; = — ^ • log 2. 
 K • log sin xdx= — — log 2. 
 
 281 
 282 
 
TABLES 
 
 29 
 
 
 Natural Logarithms of Numbers between 
 
 1.0 and 9.9 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1. 
 
 0.000 
 
 0.095 
 
 0.182 
 
 0.262 
 
 0.336 
 
 0.405 
 
 0.470 
 
 0.531 
 
 0.588 
 
 0.642 
 
 2. 
 
 0.693 
 
 0.742 
 
 0.788 
 
 0.833 
 
 0.875 
 
 0.916 
 
 0.956 
 
 0.993 
 
 1.030 
 
 1.065 
 
 3. 
 
 1.099 
 
 1.131 
 
 1.163 
 
 1.194 
 
 1.224 
 
 1.253 
 
 1.281 
 
 1.308 
 
 1.335 
 
 1.361 
 
 4. 
 
 1.386 
 
 1.411 
 
 1.435 
 
 1.459 
 
 1.482 
 
 1.504 
 
 1,526 
 
 1.548 
 
 1.569 
 
 1.589 
 
 8. 
 
 1.609 
 
 1.629 
 
 1.649 
 
 1.668 
 
 1.686 
 
 1.705 
 
 1.723 
 
 1.740 
 
 1.758 
 
 1.775 
 
 6. 
 
 1.792 
 
 1.808 
 
 1.825 
 
 1.841 
 
 1.856 
 
 1.872 
 
 1.887 
 
 1.902 
 
 1.917 
 
 1.932 
 
 7. 
 
 1.946 
 
 1.960 
 
 1.974 
 
 1.988 
 
 2.001 
 
 2.015 
 
 2.028 
 
 2.041 
 
 2.054 
 
 2.067 
 
 8. 
 
 2.079 
 
 2.092 
 
 2.104 
 
 2.116 
 
 2.128 
 
 2.140 
 
 2.152 
 
 2.163 
 
 2.175 
 
 2.186 
 
 9. 
 
 2.197 
 
 2.208 
 
 2.219 
 
 2.230 
 
 2.241 
 
 2.251 
 
 2,262 
 
 2.272 
 
 2.282 
 
 2.293 
 
 Natural Logarithms of Whole Numbers from 10 to 109 
 
 N. 
 
 a. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 
 2.303 
 
 2.398 
 
 2.485 
 
 2.565 
 
 2.639 
 
 2.708 
 
 2.778 
 
 2.833 
 
 2.890 
 
 2.944 
 
 2 
 
 2.996 
 
 3.045 
 
 8.091 
 
 3.135 
 
 8.178 
 
 3.219 
 
 3.258 
 
 3,296 
 
 3.332 
 
 8.867 
 
 3 
 
 3.401 
 
 3.434 
 
 3.466 
 
 3.497 
 
 8.526 
 
 3.555 
 
 8.584 
 
 3.611 
 
 8.638 
 
 3.664 
 
 4 
 
 3.689 
 
 8.714 
 
 3.738 
 
 3.761 
 
 3.784 
 
 3.807 
 
 3.829 
 
 3.850 
 
 3.871 
 
 3.892 
 
 5 
 
 3.912 
 
 8.932 
 
 3.951 
 
 3.970 
 
 3.989 
 
 4.007 
 
 4.025 
 
 4.043 
 
 4.060 
 
 4.078 
 
 6 
 
 4.094 
 
 4.111 
 
 4.127 
 
 4.143 
 
 4.159 
 
 4.174 
 
 4.190 
 
 4.205 
 
 4.220 
 
 4.284 
 
 7 
 
 4.248 
 
 4.263 
 
 4,277 
 
 4.290 
 
 4.304 
 
 4.317 
 
 4.331 
 
 4.344 
 
 4.357 
 
 4.369 
 
 8 
 
 4.382 
 
 4.394 
 
 4,407 
 
 4.419 
 
 4.431 
 
 4.448 
 
 4.454 
 
 4.466 
 
 4.477 
 
 4.489 
 
 9 
 
 4.500 
 
 4.511 
 
 4,522 
 
 4.538 
 
 4.543 
 
 4.554 
 
 4.564 
 
 4.575 
 
 4.585 
 
 4.595 
 
 10 
 
 4,605 
 
 4.615 
 
 4.625 
 
 4.635 
 
 4.644 
 
 4.654 
 
 4.668 
 
 4.673 
 
 4.682 
 
 4^91 
 
 Values in Circular Measure of Angles which are given in 
 Degrees and Minutes 
 
 1' 
 
 0.0008 
 
 9' 
 
 0.0026 
 
 3° 
 
 0.0524 
 
 20° 
 
 0.8491 
 
 100° 
 
 1.7453 
 
 2' 
 
 0.0006 
 
 10' 
 
 0.0029 
 
 4° 
 
 0.0698 
 
 30° 
 
 0.5236 
 
 110° 
 
 1.9199 
 
 3' 
 
 0.0009 
 
 20' 
 
 0.0058 
 
 5° 
 
 0.0873 
 
 40° 
 
 0.6981 
 
 120° 
 
 2.0944 
 
 4' 
 
 0.0012 
 
 30' 
 
 0.0087 
 
 6° 
 
 0.1047 
 
 80° 
 
 0.8727 
 
 130° 
 
 2.2689 
 
 6' 
 
 0,0015 
 
 40' 
 
 0,0116 
 
 7° 
 
 0.1222 
 
 60° 
 
 1.0472 
 
 140° 
 
 2.4435 
 
 6' 
 
 0,0017 
 
 50' 
 
 0.0145 
 
 8° 
 
 0.1396 
 
 70° 
 
 1.2217 
 
 160° 
 
 2.6180 
 
 7' 
 
 0.0020 
 
 1' 
 
 0.0175 
 
 9° 
 
 0.1571 
 
 80° 
 
 1.3963 
 
 160° 
 
 2.7925 
 
 8' 
 
 0.0023 
 
 2' 
 
 0.0849 
 
 10° 
 
 0.1745 
 
 90° 
 
 1.5708 
 
 170° 
 
 2.9671 
 
30 
 
 TABLES 
 
 
 
 Natural Trigonometric Functions 
 
 
 
 Angle 
 
 Sin 
 
 Cso 
 
 Tan 
 
 Ctn 
 
 Sec 
 
 Cos 
 
 
 0° 
 
 0.000 
 
 00 
 
 0.000 
 
 00 
 
 1.000 
 
 1.000 
 
 90° 
 
 1 
 
 0.017 
 
 57.30 
 
 0.017 
 
 57.29 
 
 1.000 
 
 1.000 
 
 89 
 
 s 
 
 0.035 
 
 28.65 
 
 0.035 
 
 28.64 
 
 1.001 
 
 0.999 
 
 88 
 
 3 
 
 0.052 
 
 19.11 
 
 0.052 
 
 19.08 
 
 1.001 
 
 0.999 
 
 87 
 
 4 
 
 0.070 
 
 14.34 
 
 0.070 
 
 14.30 
 
 1.002 
 
 0.998 
 
 86 
 
 5° 
 
 0.087 
 
 11.47 
 
 0.087 
 
 11.43 
 
 1.004 
 
 0.996 
 
 85° 
 
 6 
 
 0.105 
 
 9.567 
 
 0.105 
 
 9.514 
 
 1.006 
 
 0.995 
 
 84 
 
 7 
 
 0.122 
 
 8.206 
 
 0.123 
 
 8.144 
 
 1.008 
 
 0.993 
 
 83 
 
 8 
 
 0.139 
 
 7.185 
 
 0.141 
 
 7.115 
 
 1.010 
 
 0.990 
 
 82 
 
 9 
 
 0.156 
 
 6.392 
 
 0.158 
 
 6.314 
 
 1.012 
 
 0.988 
 
 81 
 
 10° 
 
 0.174 
 
 5.759 
 
 0.176 
 
 5.871 
 
 1.015 
 
 0.985 
 
 80° 
 
 11 
 
 0.191 
 
 5.241 
 
 0.194 
 
 5.145 
 
 1.019 
 
 0.982 
 
 79 
 
 12 
 
 0.208 
 
 4.810 
 
 0.213 
 
 4.705 
 
 1.022 
 
 0.978 
 
 78 
 
 13 
 
 0.225 
 
 4.445 
 
 0.231 
 
 4.331 
 
 1.026 
 
 0.974 
 
 77 
 
 14 
 
 0.242 
 
 4.134 
 
 0.249 
 
 4.011 
 
 1.031 
 
 0.970 
 
 76 
 
 15° 
 
 0.259 
 
 3.864 
 
 0.268 
 
 3.732 
 
 .1.035 
 
 0.966 
 
 76° 
 
 16 
 
 0.276 
 
 3.628 
 
 0.287 
 
 3.487 
 
 1.040 
 
 0.961 
 
 74 
 
 17 
 
 0.292 
 
 3.420 
 
 0.806 
 
 3.271 
 
 1.046 
 
 0.956 
 
 73 
 
 18 
 
 0.309 
 
 3.236 
 
 0.325 
 
 3.078 
 
 1.051 
 
 0.951 
 
 72 
 
 19 
 
 0.326 
 
 3.072 
 
 0.344 
 
 2.904 
 
 1.058 
 
 0.946 
 
 71 
 
 20° 
 
 0.342 
 
 2.924 
 
 0.364 
 
 2.747 
 
 1.064 
 
 0.940 
 
 70° 
 
 21 
 
 0.358 
 
 2.790 
 
 0.384 
 
 2.605 
 
 1.071 
 
 0.934 
 
 69 
 
 22 
 
 0.375 
 
 2.669 
 
 0.404 
 
 2.475 
 
 1.079 
 
 0.927 
 
 68 
 
 23 
 
 0.391 
 
 2.559 
 
 0.424 
 
 2.356 
 
 1.086 
 
 0.921 
 
 67 
 
 24 
 
 0.407 
 
 2.459 
 
 0.445 
 
 2.246 
 
 1.095 
 
 0.914 
 
 66 
 
 — 25° 
 
 0.423 
 
 2.366 
 
 0.466 
 
 2.145 
 
 1.103 
 
 0.906 
 
 66° 
 
 26 
 
 0.438 
 
 2.281 
 
 0.488 
 
 2.050 
 
 1.113 
 
 0.899 
 
 64 
 
 27 
 
 0.454 
 
 2.203 
 
 0.510 
 
 1.963 
 
 1.122 
 
 0.891 
 
 63 
 
 28 
 
 0.469 
 
 2.130 
 
 0.532 
 
 1.881 
 
 1.138 
 
 0.883 
 
 62 
 
 29 
 
 0.485 
 
 2.063 
 
 0.554 
 
 1.804 
 
 1.143 
 
 0.875 
 
 61 
 
 30° 
 
 0.500 
 
 2.000 
 
 0.577 
 
 1.732 
 
 1.155 
 
 0.866 
 
 60° 
 
 31 
 
 0.515 
 
 1.942 
 
 0.601 
 
 1.664 
 
 1.167 
 
 0.857 
 
 59 
 
 32 
 
 0.530 
 
 1.887 
 
 0.625 
 
 1.600 
 
 1.179 
 
 0.848 
 
 58 
 
 33 
 
 0.545 
 
 1.836 
 
 0.649 
 
 1.540 
 
 1.192 
 
 0.889 
 
 67 
 
 34 
 
 0.559 
 
 1.788 
 
 0.675 
 
 1.483 
 
 1.206 
 
 0.829 
 
 66 
 
 35° 
 
 0.574 
 
 1.743 
 
 0.700 
 
 1.428 
 
 1.221 
 
 0.819 
 
 68° 
 
 36 
 
 0.588 
 
 1.701 
 
 0.727 
 
 1.376 
 
 1.236 
 
 0.809 
 
 64 
 
 37 
 
 0.602 
 
 1.662 
 
 0.754 
 
 1.327 
 
 1.252 
 
 0.799 * 
 
 . 53. 
 
 38 
 
 0.616 
 
 1.624 
 
 0.781 
 
 1.280 
 
 1.269 
 
 0.788 
 
 52 
 
 39 
 
 0.629 
 
 1.589 
 
 0.810 
 
 1.235 
 
 1.287 
 
 0.777 
 
 61 
 
 40° 
 
 0.643 
 
 1.556 
 
 0.839 
 
 1.192 
 
 1.805 
 
 0.766 
 
 50° 
 
 41 
 
 0.656 
 
 1.524 
 
 0.869 
 
 1.1.50 
 
 1.325 
 
 0.755 1 
 
 49 
 
 42 
 
 0.669 
 
 1.494 
 
 0.900 
 
 1.111 
 
 1.846 
 
 0.743 
 
 48 
 
 43 
 
 0.682 
 
 1.466 
 
 0.9.33 
 
 1.072 
 
 1.367 
 
 0.731 
 
 47 
 
 44 
 
 0.695 
 
 1.440 
 
 0.966 
 
 1.036 
 
 1.390 
 
 0.719 
 
 46 
 
 45° 
 
 0.707 1 
 
 1.414 
 
 1.000 
 
 1.000 
 
 1.414 
 
 0.707 
 
 45° 
 
 
 Cos 
 
 Sec 
 
 Otn 
 
 Tan 
 
 Cso 
 
 Sin 
 
 Angle 
 
TABLES 
 
 31 
 
 Values of the Complete Elliptic Integrals, K and E, for Different 
 Values of the Modulus, k 
 
 K 
 
 Jo Vl- 
 
 dz 
 
 vl — k^siti^z 
 
 E: 
 
 ■/.'vr 
 
 '■sm^z ■ dz. 
 
 sin-lA- 
 
 X 
 
 E 
 
 sin-lA- 
 
 K 
 
 £ 
 
 sm~^lc 
 
 X 
 
 E 
 
 0° 
 
 1.5708 
 
 1.5708 
 
 50° 
 
 1.9356 
 
 1.3055 
 
 81.0° 
 
 3.2553 
 
 1.0338 
 
 1° 
 
 1.5709 
 
 1.5707 
 
 51° 
 
 1.9639 
 
 1.2963 
 
 81.8° 
 
 3.2771 
 
 1.0326 
 
 2° 
 
 1.5713 
 
 1.5703 
 
 52° 
 
 1.9729 
 
 1.2870 
 
 81.4° 
 
 3.2995 
 
 1.0313 
 
 3° 
 
 1.5719 
 
 1.5697 
 
 53° 
 
 1,9927 
 
 1.2776 
 
 81.6° 
 
 3.3223 
 
 1.0302 
 
 4° 
 
 1.5727 
 
 1.5689 
 
 54° 
 
 2.0133 
 
 1.2681 
 
 81.8° 
 
 3.3458 
 
 1.0290 
 
 6° 
 
 1.5738 
 
 1.5678 
 
 55° 
 
 2.0347 
 
 1.2587 
 
 82.0° 
 
 3.3699 
 
 1.0278 
 
 6° 
 
 1.5711 
 
 1.5665 
 
 56° 
 
 2.0571 
 
 1.2492 
 
 82.2° 
 
 3.3946 
 
 1.0267 
 
 7° 
 
 1.5767 
 
 1.5649 
 
 57° 
 
 2.0804 
 
 1.2397 
 
 82.4° 
 
 3.4199 
 
 1.0256 
 
 8° 
 
 1.5785 
 
 1.5632 
 
 58° 
 
 2.1047 
 
 1.2301 
 
 82,6° 
 
 3.4460 
 
 1.0245 
 
 9° 
 
 1.5805 
 
 1.5611 
 
 59° 
 
 2.1300 
 
 1.2206 
 
 82.8° 
 
 3.4728 
 
 1.0234 
 
 10° 
 
 1.5828 
 
 1.5589 
 
 60° 
 
 2.1565 
 
 1.2111 
 
 83.0° 
 
 3.5004 
 
 1.0223 
 
 ll" 
 
 1.5854 
 
 1.5564 
 
 61° 
 
 2.1842 
 
 1.2015 
 
 83,2° 
 
 3.5288 
 
 1.0213 
 
 12° 
 
 1.5882 
 
 1.5537 
 
 62° 
 
 2.2132 
 
 1.1921 
 
 83,4° 
 
 3.5581 
 
 1.0202 
 
 13° 
 
 1.5913 
 
 1.5507 
 
 63° 
 
 2.2435 
 
 1.1826 
 
 83.6° 
 
 3.5884 
 
 1.0192 
 
 14° 
 
 1.5946 
 
 1.5476 
 
 64° 
 
 2.2754 
 
 1.1732 
 
 83.8° 
 
 3.6196 
 
 1.0182 
 
 15° 
 
 1.5981 
 
 1.5442 
 
 65° 
 
 2.3088 
 
 1.1638 
 
 84.0° 
 
 3.6519 
 
 1.0172 
 
 16° 
 
 1.6020 
 
 1.5405 
 
 65.5° 
 
 2.3261 
 
 1.1592 
 
 84.2° 
 
 3.6853 
 
 1.0163 
 
 17° 
 
 1.6061 
 
 1.5367 
 
 66.0° 
 
 2.3439 
 
 1.1546 
 
 84.4° 
 
 3.7198 
 
 1.0153 
 
 18° 
 
 1.6105 
 
 1.5326 
 
 66.5° 
 
 2.3622 
 
 1.1499 
 
 84,6° 
 
 3.7557 
 
 1.0144 
 
 19° 
 
 1.6151 
 
 1.5283 
 
 67.0° 
 
 2.3809 
 
 1.1454 
 
 84,8° 
 
 3.7930 
 
 1.0135 
 
 20° 
 
 1.6200 
 
 1.5238 
 
 67.5° 
 
 2.4001 
 
 1.1408 
 
 85.0° 
 
 3.8317 
 
 1.0127 
 
 21° 
 
 1.6252 
 
 1.5191 
 
 68.0° 
 
 2.4198 
 
 1.1362 
 
 85.2° 
 
 3.8721 
 
 1.0118 
 
 22° 
 
 1.6307 
 
 1,5141 
 
 68.5° 
 
 2.4401 
 
 1.1317 
 
 85.4° 
 
 3.9142 
 
 1.0110 
 
 23° 
 
 1.6365 
 
 1.5090 
 
 69.0° 
 
 2.4610 
 
 1.1273 
 
 86.6° 
 
 3.9583 
 
 1.0102 
 
 24° 
 
 1.6426 
 
 1.5037 
 
 69.5° 
 
 2.4825 
 
 1.1228 
 
 85.8° 
 
 4.0044 
 
 1.0094 
 
 25° 
 
 1.6490 
 
 1.4981 
 
 70.0° 
 
 2.5046 
 
 1.1184 
 
 86.0° 
 
 4.0528 
 
 1.0087 
 
 26° 
 
 1.6557 
 
 1.4924 
 
 70.5° 
 
 2.!J273 
 
 1.1140 
 
 86.2° 
 
 4.1037 
 
 1.0079 
 
 27° 
 
 1.6627 
 
 1.4864 
 
 71.0° 
 
 2.5507 
 
 1.1096 
 
 86.4° 
 
 4.1574 
 
 1.0072 
 
 28° 
 
 1.6701 
 
 1.4803 
 
 71.5° 
 
 2.5749 
 
 1.1053 
 
 86.6° 
 
 4.2142 
 
 1.0065 
 
 29° 
 
 1.6777 
 
 1.4740 
 
 72.0° 
 
 2.5998 
 
 1.1011 
 
 86.8° 
 
 4.2744 
 
 1.0059 
 
 30° 
 
 1.6858 
 
 1.4675 
 
 72.5° 
 
 2.6256 
 
 1.0968 
 
 87.0° 
 
 4.3387 
 
 1.0053 
 
 31° 
 
 1.6941 
 
 1.4608 
 
 73.0° 
 
 2.6521 
 
 1.0927 
 
 87.2° 
 
 4.4073 
 
 1.0047 
 
 32° 
 
 1.7028 
 
 1.4539 
 
 73.5° 
 
 2.6796 
 
 1.0885 
 
 87.4° 
 
 4.4812 
 
 1.0041 
 
 33° 
 
 1.7119 
 
 1.4469 
 
 74.0° 
 
 2.7081 
 
 1.0844 
 
 87.6° 
 
 4.5619 
 
 1.0036 
 
 34° 
 
 1.7214 
 
 1.4397 
 
 74.5° 
 
 2.7375 
 
 1.0804 
 
 87.8° 
 
 4.6477 
 
 1.0031 
 
 35° 
 
 1.7312 
 
 1.4323 
 
 75.0° 
 
 2.7681 
 
 1.0764 
 
 88.0° 
 
 4.7427 
 
 1.0026 
 
 86° 
 
 1.7415 
 
 1.4248 
 
 75.5° 
 
 2.7998 
 
 1.0725 
 
 88.2° 
 
 4.8479 
 
 1.0022 
 
 37° 
 
 1.7522 
 
 1.4171 
 
 76.0° 
 
 2.8327 
 
 1.0686 
 
 88.4° 
 
 4.9654 
 
 1.0017 
 
 38° 
 
 1.7633 
 
 1.4092 . 
 
 76.5° 
 
 2.8669 
 
 1.0648 
 
 88,6° 
 
 5.0988 
 
 1.0014 
 
 39° 
 
 1.7748 
 
 1.4013 
 
 77.0° 
 
 2.9026 
 
 1.0611 
 
 88.8° 
 
 5.2527 
 
 1.0010 
 
 40° 
 
 1.7868 
 
 1.3931 
 
 77.5° 
 
 2.9397 
 
 1.0574 
 
 89.0° 
 
 5.4349 
 
 1.0008 
 
 41° 
 
 1.7992 
 
 1.3849 
 
 78.0° 
 
 2.9786 
 
 1.0538 
 
 89.1° 
 
 5.5402 
 
 1.0006 
 
 42° 
 
 1.8122 
 
 1.3765 
 
 78.5° 
 
 3.0192 
 
 1.0502 
 
 89.2° 
 
 5.6579 
 
 1.0005 
 
 43° 
 
 1.8256 
 
 1.3680 
 
 79.0° 
 
 3.0617 
 
 1.0468 
 
 89.3° 
 
 5.7914 
 
 1.0005 
 
 44° 
 
 1.8396 
 
 1.3594 
 
 79.5° 
 
 3.1064 
 
 1.04.34 
 
 89.4° 
 
 5.9455 
 
 1.0003 
 
 45° 
 
 1.8.541 
 
 1.3506 
 
 80.0° 
 
 3.1534 
 
 1.0401 
 
 89.5° 
 
 6.1278 
 
 1.0002 
 
 46° 
 
 1.8691 
 
 1.3418 
 
 80.3° 
 
 3.1729 
 
 1.0388 
 
 89.6° 
 
 6.3504 
 
 1.0001 
 
 47° 
 
 1.8848 
 
 1.3329 
 
 80.4° 
 
 3.1928 
 
 1.0375 
 
 89.7° 
 
 6.6385 
 
 1.0001 
 
 48° 
 
 1.9011 
 
 1.3238 
 
 80.6° 
 
 3.2132 
 
 1.0363 
 
 89.8° 
 
 7.0440 
 
 1.0000 
 
 49° 
 
 1.9180 
 
 1.3147 
 
 80.8° 
 
 3.2340 
 
 1.0350 
 
 89.9° 
 
 7.7371 
 
 1.0000 
 
32 TABLES 
 
 Common Logarithms of T(n) for Values of n between 1 and 2 
 
 r(n)=r a;»-i-e-^(fo= r logj *»• 
 
 '» 
 
 log,„r{ra) 
 
 n 
 
 iog,„r(«) 
 
 ■lb 
 
 iog,(,r(») 
 
 n 
 
 log,„r(«) 
 
 n 
 
 iog,„r(«) 
 
 1.01 
 
 T.'9975 
 
 1.21 
 
 1.9617 
 
 1.41 
 
 T.9478 
 
 1.61 
 
 1.9517 
 
 1.81 
 
 T.9704 
 
 1.02 
 
 T.9951 
 
 1.22 
 
 1.9605 
 
 1.42 
 
 1.9476 
 
 1.62 
 
 1.9523 
 
 1.82 
 
 1.9717 
 
 1.03 
 
 1.9928 
 
 1.23 
 
 1.9594 
 
 L43 
 
 1.9475 
 
 1.63 
 
 1.9529 
 
 1.83 
 
 1.9730 
 
 1.04 
 
 1.9905 
 
 1.24 
 
 1.9583 
 
 1.44 
 
 1.9473 
 
 1.64 
 
 1.9536 
 
 1.84- 
 
 1.9743 
 
 1.05 
 
 1.9883 
 
 1.25 
 
 1.9573 
 
 1.45 
 
 1.9473 
 
 1.65 
 
 1.9543 
 
 1.85 
 
 1.9757 
 
 1.06 
 
 1.9862 
 
 1.26 
 
 1.9564 
 
 1.46 
 
 T.9472 
 
 1.66 
 
 1.9550 
 
 1.86 
 
 1.9771 
 
 1.07 
 
 1.9841 
 
 1.27 
 
 1.9554 
 
 1.47 
 
 1.9473 
 
 1.67 
 
 1.9558 
 
 1.87 
 
 1.9786 
 
 1.08 
 
 1.9821 
 
 1.28 
 
 1.9546 
 
 1.48 
 
 T.9473 
 
 1.68 
 
 1.9566 
 
 1.88 
 
 1.9800 
 
 1.09 
 
 1.9802 
 
 1.29 
 
 1.9538 
 
 1.49 
 
 1.9474 
 
 1.69 
 
 1.9575 
 
 1.89 
 
 1.9815 
 
 1.10 
 
 1.9783 
 
 1.30 
 
 T.9530 
 
 1.50 
 
 1.9475 
 
 1.70 
 
 1.9584 
 
 1.90 
 
 1.9831 
 
 1.11 
 
 T.9765 
 
 1.31 
 
 1.9523 
 
 1.51 
 
 1.9477 
 
 1.71 
 
 1.9593 
 
 1.91 
 
 1.9846 
 
 1.12 
 
 1.9748 
 
 1.32 
 
 1.9516 
 
 1.52 
 
 1.9479 
 
 1.72 
 
 T.9603 
 
 1.92 
 
 1.9862 
 
 1.13 
 
 r.9731 
 
 1.33 
 
 1.9510 
 
 1.53 
 
 1.9482 
 
 1.73 
 
 1.9613 
 
 1.93 
 
 1.9878 
 
 1.14 
 
 1.9715 
 
 1.34 
 
 1.9505 
 
 1.54 
 
 T.9485 
 
 1.74 
 
 T.9623 
 
 1.94 
 
 1.9895 
 
 1.15 
 
 1.9699 
 
 1.35 
 
 T.9500 
 
 1.55 
 
 1.9488 
 
 1.75 
 
 T.9633 
 
 1.95 
 
 1.9912 
 
 1.16 
 
 1.9684 
 
 1.36 
 
 1.9495 
 
 1.56 
 
 1.94s)2 
 
 1.76 
 
 1.9644 
 
 1.96 
 
 1.9929 
 
 1.17 
 
 1.9669 
 
 1.37 
 
 1.9491 
 
 1.57 
 
 1.9496 
 
 1.77 
 
 1.9656 
 
 1.97 
 
 1.9946 
 
 1.18 
 
 1.9655 
 
 1.38 
 
 1.9487 
 
 1.58 
 
 1.9501 
 
 1.78 
 
 1.9667 
 
 1.98 
 
 1.9964 
 
 1.19 
 
 1.9642 
 
 1.39 
 
 1.9483 
 
 1.59 
 
 1.9506 
 
 1.79 
 
 1.9679 
 
 1.99 
 
 1.9982 
 
 1.20 
 
 1.9629 
 
 1.40 
 
 1.9481 
 
 1.60 
 
 1.9511 
 
 1.80 
 
 1.9691 
 
 2.00 
 
 0.0000 
 
 rr(0+i) 
 Urw.j 
 
 = z.r(z), if z>0; r(2) = r(l) = l;" 
 r(l — a;)] = ir/sin irx, 
 
 r(2)=r(i) = in 
 
 , if 1 > a; > 0. J 
 
 If the values of an analytic function, fix), are given in a table for consecu- 
 tive values of the argument, a;, with the constant interval d, and if h = kd, 
 where k is any desired fraction, 
 
 v/ . i^ r/ V , 7 A , *(*:-!) * , k(k-l)<k-2) . 
 
 /(a + h)=f(a) + ft ■ A, + -^j— ■ \ + -5^ ^ '- ■ A3 + • • ., 
 
 where /(a) is any tabulated value.