btSl" ■ •••.-• :=— ^ ••-,*'. .V,.-. ' I%^\ IN MEMORIAM FLORIAN CAJORI ♦ • AN ELEMENTARY TREATISE ON CURVES, FUNCTIONS, AND FORCES. VOLUME SECOND; CONTAINING CALCULUS OF IMAGINARY QUANTITIES, RESIDUAL CALCULUS, AND INTEGRAL CALCULUS. By benjamin PEIRCE, A. M. Perkins Professor of Astronomy and Mathematics in Harvard University. BOSTON: JAMES MUNROE AND COMPAJMY 1846. Entered according to Act of Congress, in the year 1846, by James Munroe and Company, in tho Clerk's Office of the District Court of the District of Massachusetts BOSTON: PRINTED BY THURSTON, TORRY & CO. 31 Devonshire Street. CONTENTS. BOOK III. CALCULUS OF IMAGINARY QUANTITIES. CHAPTER T. MODULUS AND ARGUMENT ...... CHAPTER n. IMAGINARY INFINITESIMALS CHAPTER m. IMAGINARY ROOTS OF EQUATIONS ..... CHAPTER IV. IMAGINARY EXPONENTIAL AND LOGARITHMIC FUNCTIONS 22 CHAPTER V. IMAGINARY CIRCULAR FUNCTIONS ..... 25 CHAPTER VI. REAL ROOTS OF NUMERICAL EQUATIONS .... 31 BOOK IV. RESIDUAL CALCULUS. CHAPTER L RESIDUATION iV!:?.Of>5^:57 3 12 13 43 IV CONTENTS. CHAPTER II. DEVELOPMENT OF FUNCTIONS, WHICH HAVE INFINITE VALUES 52 BOOK V. INTEGRAL CALCULUS. CHAPTER I. INTEGRATION ........ 63 CHAPTER II. INTEGRATION OF RATIONAL FUNCTIONS . . • . 69 CHAPTER III. INTEGRATION OF IRRATIONAL FUNCTIONS ... 77 CHAPTER IV. INTEGRATION OF LOGARITHMIC FUNCTIONS . . . 101 CHAPTER V. INTEGRATION OF CIRCULAR FUNCTIONS . . . .116 CHAPTER VI. RECTIFICATION OF CURVES ...... 124 CHAPTER VII. QUADRATURE OF SURFACES 166 CHAPTER Vm. THE CURVATURE OF SURFACES 185 CHAPTER IX. THE CUBATURE OF SOLIDS ...... 197 CHAPTER X. INTEGRATION OF LINEAR DIFFERENTIAL EQUATIONS . 207 CONTENTS. T CHAPTER XL INTEGRATION OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER ......... 239 CHAPTER Xn. INTEGRATION OF DIFFERENTIAL EQUATIONS OF THE SEC- OND ORDER ........ 274 CHAPTER Xni. PARTICULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS . 287 CORRECTIONS, Page line 25 12 26 it t( (107) (110) (111) (114) 29 (138) (139) (140) 32 (149) <( (151) 48 7 53 23 54 14 55 (217) 64 (247) 65 (252) 66 2 I 89 (373) 90 (380) 92 1 t< 2 95 14 106 (439) (443) 107 (19 and 20) ) for read n 7C (n-\-fi) 71. 2B B. f.(a-\-i) dcf.(a + i) t I f(a-^i) dcf.(a — i) t I b^—2ab — a2 b^—3ab. finite infinite. 31 33. (^ + a) f.(x + a). /• ^0 f' ^i — /• ^0 • 3 4 m + l-\-np — ns m+l+n/j — ns The fraction is to be multiplied by (1 z^ + — z^.) "' 4 a 4a2 ' s n s pP—i bP—^. bx^ b x^ a a 10 ^10 biomial binomial. I 2A + 1 2A-1. 108 (458) The first and second members are to be divided by ^. 113 (501) 71 + 1 - n — 1. 114 (504) n + 1 n — 1. Vlll CORRECTIONS. Pagt line for read 119 (543) —— + — . 121 5 1 i 125 9 e2z^g-2, e2*_2^e-2*. 126 18 or F' T or F' T. 127 —1 P' p, " 11 Add^ or are supplements of each other (fig. 2). 128 2 and 3 pi p. 129 (574) e e". 134 (612) t V. (618) ± d=^. 136 11 are is. 137 J 3 (635) (630). 138 (643) fl" and y'/ ^^ and i^^ . " (648) f- fl 139 (649) If" if^. " (652) f- jfl 140 (658) (618) (658). " 9 fig. 5 fig. 6. "19 X V- 143 (680) 2e 4 e. " 13 and 15 F' F". 148 2 /;p /;. 149 2 tangent perpendicular. m/«Qo\

6.] MODULUS AND ARGUMENT. Argument of any real quantity. and the value of (1) becomes R (cos. 6 + sin. ^. v' — 1). (8) 4. Corollary. Since two angles, which differ by two right angles, have the same tangent, there are two values of fl less than four right angles, which satisfy (5) ; and of these two values, that one is to be selected which agrees, in the signs of its sine and cosine, with (6) and (7). Any angle, which differs from the value of & thus found by four right angles, or by any multiple of four right angles, may also be taken as a value of <5. Thus, if q^ is this least positive value of 6, the general value of 6 is 5 = ^0 ± 2 n.^, (9) in which n is any integer, and re is the ratio of the circumfer- ence to the diameter. 5. Corollary. When the imaginary part of (1) vanishes, we have .B := 0, sin. ^ = 0; (10) so that ^Q =: 0, a=:d=2w7r, COS. ^ = 1 ; (11) or &o^n, 3 = zfc (2 W-(- 1) TT, COS. 5 =: — 1. (12) and (11) corresponds to the case of a real positive quantity J (12) to that of a real negative quantity. 6. Corollary. When the real term of (1) vanishes, we have ^ = 0, COS. 6 = 0, ^0 = J TT or = J TT, (13) whence fl = zhj^±2n^ = ±(2 7i±})^, sin,5 = ±1. (14) IMAGINARY Q,UANT1T1ES. [b. III. CH. I. Equal imaginary quantities. 7. Theorem. When the quantity represented by (1) vanishes, the real and the imaginary part, and the moduhis, are each equal to zero, while the argument is indeterminate. Proof. For if ^4-jBV — 1 = 0, (15) we have ^ = — JB \/ — 1 ; (16) that is, a real quantity equal to an imaginary one, which is impossible, and (16) cannot be satisfied, unless we have ^ = 0, jB^O; (17) whence, by (4), R — Q, (18) and, by (5), & is indeterminate. 8. Theorem. When hvo imaginary quantities are equals their real and imaginary parts are separately equal, and they have the same moduhis and argument. Proof. For the equation A+B^— l = A' + BW— ^, (19) gives, by transposition, A — A' -[- {B — B') s^ — \ = ^. (20) Hence, by the preceding theorem, « A — A' = (), B — B'=:Oy or, A=:A', B = B; (21) whence, by (4 and 5), R = R', & = &'. (22) <^ 11.] MODULUS AND ARGUMENT. 7 Conjugate quantities. Imaginary product. 9. Two imaginary quantities are conjugate to each other, when they have the same modulus, and when their arguments only differ in being of contrary signs. Thus the conjugate of (8) is R [cos. (— 6) 4- sin. (— a) . ^ _ 1] ; (23) or, by trigonometry, i? (cos. ^ — sin. 5.v/— 1). (24) 10. Corollary. Two imaginary quantities, which are conjugate to each other, differ only in the sign which precedes the imaginary part. Thus A 4- Bs^ — 1 and A —Bs/— 1 are, by (8 and 24), conjugate to each other. 11. Theorem. The modulus of the product of sev- eral imaginary quantities is equal to the product of the moduli of the factors, and the argument of the product is equal to the sum of the arguments of the factors. Proof, a. When there are two factors R (cos. & + sin. a. V — 1) and 7J'(cos. 6'4-sin.5'.\/ — 1), (25) the product is R R' [cos. 6 cos. &' — sin. q sin. S'] + (sin. & COS. 6' 4" ^^^- ^' COS. &) \/ — 1, (26) which, by (26 and 28 of Trig,), becomes R R [cos. (^ + tv) -f sin. {d + 6') . V— 1] ; (27) so that its modulus is the product of the two moduli, and its argument is the sum of the two arguments. 8 IMAGINARY QUANTITIES. [b. III. CH. I. Imaginary power. h. A third factor miglit be multiplied by (27) in the same way, that is, by multiplying its modulus by the new modulus, and adding to its argument the new argument ; and this pro- cess might be extended to any number of factors. 12. Corollary. If the factors are all equal, the pro- duct becomes a power ; whence the modulus of a posi- tive integral power of an imaginary quantity is the same power of its modulus, and the argument of the power is the product of its argument hy the exponent of the power. Thus [/2(cos.6-|-sin.aV— l)]'^=-K"(cos.W5-fsin.w5.V— 1)-(28) 13. Corollary. When jR = 1, (29) (28) becomes (cos. 6 + sin. a. s/ — 1)" i:r (cos. n &-\- sin. n 6. y' — 1). (30) Reversing the sign of ^ (cos.a — sin.a.V — l)'^ = (cos. n a — sm.n^.^/ — 1). (31) 14. Corollary, Half the sum of (30 and 31) is cos.n a= J(cos.a-|-sin.a V— -1 )" + J(cos. 5 — sin.a. /y/— 1 )n. (32) Half the difference of (30 and 31) is sin. w a. y' — 1 = J (cos. a. -|- sin. ^. \f — 1)" — J (cos. d —sin. 5. V — 1)" (33) <§) 18.] MODULUS AND ARGUMENT. 9 Product of two conjugate factors. Imaginary quotient. 15. Corollary. By development, (32) becomes nln — 1 ) _ . ^ cos.w<5 = COS." (3 :j — ^ — - COS.™ — ~(5 sin. 2 d , 7i(n — l)(w — 2)(w — 3) . . ^ _[_ _i_^ — 11 TT^-i cos."-4 & sin.* d — &c. (34) By developing and dividing by \/ — 1, (33) becomes sin. w ^ zz: 71 COS."— ^ ^ sin. ^ n(7^_l) (n — 2) 1 . 2 . 3 cos."-3(3 sin.3(3 -}-&c. (35) 16. Corollary. The reverse of ^^^ 12 is, that the modulus of a positive integral root of an imaginary quantity is the same root of its modulus, and the argu- ment of the root is the quotient of its argument divided by the exponent of the root ; that is, since roots are fractional powers, the rule of § 12 extends to the case of positive fractional powers. 17. Corollary. The product of two conjugate factors is equal to the square of the modulus. For, in this case, (23 and 27) give 6 + ^' = ^ — a=zO, RR'=R2. (36) 18. Corollary. The reverse of ^ 11 is, that the mo- dulus of a quotient is equal to the quotient of the Tnodu- lus of the dividend divided by that of the divisor ^ and the argument of the quotient is equal to the argument of the dividend diminished by that of the divisor. 10 IMAGINARY QUANTITIES. [b. III. CH. I. Imaginary power. Thus R'{cosJ'-}-smJ'.\/—\) RicosJ-{-s'in.6.\/—l) = ^'[cos.(^'— ^) + sin.(5' — ^)V— 1]. (37) 19. Corollauj. When ^' = 0, and R' — \ \ (38) (37) becomes z=-i [cos.(-^) + sin.(-a)V-l], jR(cosJ + sin.5V— 1 ^ or [i2(cos.^+sin.^.V— l)]-^=-R"^[cos.(— ^)+sm.(— 5)V— 1] — i2-i(cos.5 — sin.^V— 1); (39) and raising to the wth power, by means of (30), [jR(cos.^4-sin.^V— l)]~"=^~"[cos.(-n^)+sin.(-w5)V— 1] = K-™(cos.w^— sin.w^^/ — 1); (40) that is, the rule of <§> 12 may he extended to the case of negative powers. 20. Corollary. The rule of§ 12 may, then, be ex- tended, by § 1, to all powers, real or imaginary. 21. Problem. To find the modulus and argumerrt of the sum or difference of several imaginary quantities. Solution. Let the given sum or difference be r(cos.^4-sin.^V— l)i:r'(cos.^'+sin.^'.\/— 1)± &c., (41) and let R be its modulus, and © its argument; we have by (4 and 5) and by (9 and 29 of Trig.) <5> 23.] MODULUS AND ARGUMENT. 11 Imaginary sura or difterence. 122= (r COS. ^±r'cos. a'i &c.)2-j- (r sin. ^ ± r' sin. 6'zh &c.)2 _ ^2 _|. r'2 + &,c. zb 2 rr' COS. (^ — ^') ± &c., (42) r sin. 6 rt r' sin. d' r±z ^c. . ^ ^ tan. =: — ; T— - — J — (43) r cos. 6 i r cos. 5' i &/C. ' 22. Corollary. Since every cosine is less than unity, (42) gives R^ < r2 -|- r'^ -|- &,c. -\-2rr'-\- &c., or i22 < (;. _|_ ,./ ^ &c.)2, *or JR < r +r' + &,c.; (44) that is, the modulus of the sum or difference of several imaginary quantities is less than the sum of their moduli. 23. Corollary. When there are only two terms in (41), (42) becomes 2J2 -_ ^2 _j. y./2 _t- 2 r r' cos. (a — ^') ; (45) and, therefore, R^ > r^ -{- r'^ —2r r', or R >r — r'; (46) that is, the modulus of the sum or difference of two imaginary quantities is greater than the difference of their moduli. 12 IMAGINARY QUANTITIES. [b. III. CH. II. Imaginary infinitesimal. CHAPTER II. IMAGINARY INFINITESIMALS. 24. An imaginary infinitesimal is an imaginary quan- tity, whose modulus is an infinitesimal. The order of an im^aginary infinitesimal is the same with that of its modulus. 25. Corollary. It follows from Chapter II. of the Differential Calculus, and the preceding Chapter, that all the propositions, lohich have hitherto been investi- gated respecting real infinitesimals^ may he extended to imaginary infinitesimals. «§> 27.] ROOTS OF E(iUATIONS. 13 Roots of a binomial equation. (48) CHAPTER III. IMAGINARY ROOTS OF EQ,TTATIONS. 26. Problem. To solve a binomial equation, and re- duce all its imaginarij roots to the form of% 1. Solution. Let the equation be Ax''— M, (47) in which A and 31 are real or imaginary, and a a positive integer. When (47) is divided by A by means of ^ 18, it is reduced to the form X'^ -ZZZ 777, in which m is of the form of ^ 1. Let then m z=z r (cos. 6 -\- sin. 6. \/ — 1), (49) or x« = r (cos. 6 + s'"- ^- V — !)• (50) The ath root of (50) is, by § 16, ^ ^ A x=z/v/r. (cos. {-sin. — .\/ — 1). (51) a a ' ^ ' 27. Scliolium. Since has, by (9), an infinity of values, (51) would at first sight appear to have a like infinity of values. But, by (9), & fl„ 2 7Zrr — = — dt , (52) a a a ^ ' 14 IMAGINARY QUANTITIES. [b. III. CH. III. Number of roots of a binomial equation. whence the values of x are identical, when they correspond to values of <^, for which the difference of the values of n is equal to a, or is some multiple of a. Now, by subtracting from any value of n the greatest multiple of a contained in it, a remain- der is obtained, which is less than a. The number of differ- ent values of x is, therefore, the same with the number of posi- tive integers (zero included) which are less than a ; that is, the number of values of x or the number of roots of equation (48) is just equal to a. 28. Corollary. When m is real and positivCj (11) gives «, / 2n-n: . 2nTv \ X z= \/m i cos. zb sin. — — .V — 1 I > (5^) in which the double sign renders it unnecessary to no- tice those values of n which exceed the half of a. 29. Corollary. The value of n n = 0, (54) reduces (53) to its real positive root a X z=z s/ m, (55) 30. Corollary. When a is even in (53), the value of n nr=.\a, ' (56) 2w^ . gives = ^> V^'y a X =. — \/m. (58) <5> 33.] ROOTS OF Eq,UATIONS. 15 Every equation has a root. 31. Corollary. When in is real and negative, (12) gives «^ / 2/1 + 1 . 2n + l , A ,^^^ zi=V — wifcos. TTitsin. >T.\/ — II, (o9) in which the double sign renders it unnecessary to no- tice those vaUies of n which exceed the half of a. 32. Corollary. When a is odd, the value of n —1 n = %^. (60) 2n + l ' ,^,, gives Tc ^::^ n^ (61 ) X z=: — \/ — m. (62) 33. Theorem. Every equation has at least one real root or one imaginary root of the form (1). Proof. Let all the terms of the equation be transposed to its first member, which reduces it to the form /.x = 0. (63) Let now x^ be any real or imaginary value of x, for which the value of this first member neither vanishes, nor is infinite, and let h be an infinitesimal ; let also dl.f.x^ be the first dif- ferential coefficient o{ f.x^ which does not vanish ; and (533 of Vol. I.) gives f{x^+h) =r/.x, + ^^ 3 ^^ d:.f.x^' (64) 16 IMAGINARY QUANTITIES. [b. III. CH. IIT. Equations whicli have finite roots. Again, let i be «in assumed real infinitesimal, and let h be determined to satisfy the assumed binomial equation r^r — - '^"J.^o = - if^x^. (05) This value of h, being substituted in (G4), gives /.(/, +70 =/.x, - z/:r, = (l-O/.^o ; (66) so that if r is the modulus of/.a-Q, that of/.(.TQ + /O ^^> ^7 §11, (1 — i)r, and therefore less than that o^ f.x^. The least possible modulus of y.a; is then less than r, unless r is zero ; this least modulus must then be zero, and the corresponding value of X is a root of the equation (63). 34. Scholiiun. The preceding argument does not exclude infinity from being the root of the given equa- tion, so that the following is a convenient statement of the above theorem ; Every equation has at least one finite root of the form (1), wheiiy after it is reduced to the form (63), it does not vanish for an infinite value of tlie variable. 35. Corollary. If the first member of (63) is a polynomial of the form xn -]_ a 2-"-i -\- h 3;"-2 + &c., (67) and if x' is a root of the equation, this polynomial must be divisible by x — x' ] and the quotient must be a polynomial of the (/^ — l)st degree, which must be divisible by a similar factor X — x", and so on. <5) 37.] ROOTS OF EQUATIONS. 17 Tlie conjugate of a real function. Hence (67) must he the continued product of n dif- ferent factors of the form {.V — x^); that is, the equa- tion xnj^a x«-i + h 2;"-2 -f &,c. = (68) must have n roots of the form (1), whether a, b, Sf'c. be real or imaginary, 36. A real function is one, which has real values for all real values of ihc variable, and has not imaginary values, unless the variable is imaginary. 37. Theorem. The conjugate of a real function is the same function of the conjugate of the variable ; or, algebraically, if P + QV-1=/.(P + 2V-1), (C9) where /. denotes a real function, then P-Qs/-\^f.{p-qs^-\). (70) Proof. The function, which is the second member of (69), may be developed and arranged according to powers of \/ — 1. Let, then, the aggregate of all the terms which are independ- ent of v' — 1) 'if'd of those which are multiplied by even powers of \/ — 1 be denoted by P ; while the aggregate of all those terms which are multiplied by odd powers of \/ — 1, is denoted by Q'. The value of P is real, and remains un- changed by changing v' — 1 ^^ — \/ — 1, while that of Q' is reversed ; that is, the value of the function is changed from P + Q' to P — Q'. (71) 2* 18 IMAGINARY QUANTITIES. [b. III. CH. III. Every real equation has at least two roots. But the quotient of Q' divided by \/ — 1, containing only even powers of \/ — 1, is a real quantity, which may be de- noted by Q, that is, Q'=zQx/— 1, (72) P+Q' = P+QV — 1; (73) so that by reversing the sign of \/ — 1, (69) is changed to (70). 38. Corollary, When Q = 0, (74) (69 and 70) become -P=/-(i' + 2V-i)=/.(i'-sV-i); (75) that is, every real value of a real function corresponds to two different values of the variable, which are con- jugate to each other. 39. Corollary. When P = 0, (76) (75) becomes 0=/. (p + 5\/-i)=/. (p-?Vi); (77) that is, ivhen the function ^ which is the first meniher of (63), is real^ tlie conjugate of every imaginary root is also a root of the equation. 40. Corollary, If x' is a root of the equation (68), when a, fc, &c. are real, and if x" is the conjugate of x' , x" is also a root of this equation, and the first member is divisible by the product (z — X') (x — x") — x^— {x' + x'') X + x' x". (78) <§. 40.] ROOTS OF EQ,UATIONS. 19 Number of real factors of a real polynomial. If r is the modulus of z' and & its argument, (8, 24, and 36) give %' -\-x" — "ir COS. 6, x' x" = r2 ; (79) whence (78) becomes the real factor x2 — 2rx cos. 6 + r2 ; (80) so that «???/ real polynomial of the form (67) is the con- tinued product of as many real factors of the form x — x' as the equation (68) has real roots, multiplied by the co7i- tinued product of half as many real quadratic factors of the form (80) as (68) has imaginary roots. 41. Examples. 1. Decompose z'^ — h"^ into a continued product of real factors of the tirst and second degree. Solution. The equation a;7 — 67 _ 0, or x^ = b^ , gives in (48) m = b"^ , = 7; whence (53) becomes X z=z b (cos. f n TV :iz sin. f n re . a^ — 1 ) j which becomes, by putting saccessively for Ji all integers less than half of 7, X z=z b, X =1 b (cos. f 7r i sin. f tt . \/ — 1), X z=z b (cos. f- ^ rt sin. f ^ . \/ — 1), X z=: b (cos. f n zh sin. f ^ . \/ — 1 ) ; 20 IMAGINARY Q,UANT1TIES. [b. III. CH. III. Decomposition of a function into real factors. SO that, by (SO), the continued product is X' —b' ={x-^b) {z2—2bx COS. f .T-f&2) (x2_2 6 2;cos. f -f + 62) {x2—2bxcos.^^-\-b2). 2. Decompose x^ -f~ ^* ^^^^ ^ product of real factors of the first and second decree. Solution. The equation x^ -{- b^ =1 0, or a-4 = —. 6*, gives in (48) ?« = — b^, — ?n =z b^, a =z ^; whence (59) becomes x = b (cos. i (2 w -]- 1 ) TT i sin. ^ (2 71 + 1 ) rr . V— 1 ) J which becomes, by putting successively for 7i all integers less than 2, X = b{c0S.irt-^sm.lrc,^—l) = b(h\/2:hiV^'V—l)y a:=6(cos.f ^±sin.f:T.\/— 1) = ^(— W-=FiV-V— 1); so that, by (80), the continued product is 3c4_|_54_(3;2_26a;cos.-i-7r-|-62)(x2_263;cos.j7r + 62) _ (3;2 — ^ 2 . 6 X + 62) (3;2 _|_ ^2 . fe X + 62). 3. Decompose x^ — 6* into a continued product of real factors of the first and second degree, Ans. (x — 6) (x + 6) (x2 + 62). 4. Decompose x^ + 6^ into a continued product of real factors of the first and second degree. Ans. (x+6)(x2— 26xcos.|7i:+62)(x2_26a;cos.f;T + 62). $ 40.] ROOTS OF EQ,UATIONS. 21 Decomposition into real factors. 5. Decompose x^ — h^ into a continued product of real factors of the first and second decree. Arts, {x—b) {x+b) (x2_|_5a;_j_52) (^x^ — bx + b^). 6. Decompose x^ -\- b^ into a continued product of real factors of the first and second deorree. Ans. (22-1-^3.6.2;+ 62) (a;2-)-62) {z2 — ^S.bx-{-b2). 22 IMAGINARY Q,UANTITIES. [b. III. CH. IV. Imaginary power. CHAPTER IV. IMAGINARY EXPONENTIAL AND LOGARITHMIC FUNCTIONS. 42. Problem, To reduce an imaginary power of a real quantity to the form (1). Solution. Let the exponent of the power be ^-[- B a^ — 1, and let R be the modulus and the argument of this power of the real quantity a, that is, let a-^+^v-i — jR(cos. + sin. 0.^—1). (81) The infinitesimal power i of this equation is by (28) «(^+5v-i) — Ri (cos. i + sin. io.^—l). (82) Hence by (418 of Vol. I. and § 22 of Plane Trig.) l-^i{A + B\/—l)\og.a—{l + i\og.R){l + i0^/—l) — l+i{\og.R-{-0\/—l), (83) Hence, by <5> 8, and using e = the base of the Neperian logarithms, (84) log. R — A log. a z= log. a-^, R — a-^ , (85) B log. a z= = log. a^, aB[z= c©; (86) which, substituted in (81), give a^-^B^-i — a-A (cos. B log. a + sin. B log. a . \/—\). (87) <§> 47.] LOGARITHMIC FUNCTIONS. 23 Imaginary logarithm. 43. CuroUarj/. When ^ == 0, (87) becomes a^v/-i — COS. B log. a + sin. B log. a . \/— 1. (88) 44. Corollanj. When a =: e, (87 and 88) become e-^+B^-i — e-^[cos.B + ^m.B,s/—\), (89) gB./-i _ eos. B + sin. B . V— 1. (90) 45. Corollary. Reversing the sign of B, (89 and 90) be- come ^A-B^f-i — e.^(cos. B — sin. S . \/— 1), (91) e-5v-i z= cos. jB. — sin. ^. V— 1- (92) 46. Problem. To reduce the logarithm of an imagi- nary quantity to the form (1). Solution. Let r be the modulus and <3 the argument of the imaginary quantity, and (90) gives r{Q.os.&-\-s\n.&./^—\) = re(^^-^] (93) the logarithm of which is log. [r (cos. (3 -|- sin. 6 . /y/ — 1 )] =: log. r -\- log. c^ -^—"^ =:log. r + 6V— 1. (94) 47. Corollary. By (4, 5, and 94) log.(^+i?V— l)=logV(^2_[_2J2)_|_tan.[-i]^V— 1 = J log. (^2 _j_S2 )_ptan.[-i]^. V— 1 ; (95) 24 IMAGINARY QUANTITIES. [b. III. CII. IV. Number of the logarithms of a number. and as there is an infinity of values of a =1 tan.[-i] — , A every quantity^ real or hnaginary^ has an infinity of logarithms J of w J lich there is never more than one real logaritlim^ and that^ hy ^ 5, only ivhen the quantity is real and positive. 48. Corollary, By § 5, when A is positive, and B — Q, (95) becomes log. A = log. A^'Znn s/— 1, (96) in which log. A of the second member is the real value of this logarithm. 49. Corollary. By § 5, when A is negative and 5 = 0, (95) becomes log. A — log. (_- ,4) ± (2 n + 1) ^ V— 1. (07) 50. Examples. 1. What is the logarithm of J\/2(l +\/— 1)? 2. What is the logarithm of \/3 + \/— 1 ? Ans. log 2 + (^ zh 2 n) 71 ^— 1. <§> 51.] CIRCULAR FUNCTIONS. 25 Sine and cosine of imaginary angles. CHAPTER V. IMAGINARY CIRCULAR FUNCTIONS. 51. Problem. To reduce the sine and cosine of an imaginary angle to the form ( I). Solution, a. Let the angle be B /s/ — 1, which being sub- Btituted for B in (90 and 9^), gives e-B — COS. B V— I + sin. B sf— 1 . V— 1, (98) e^ = COS. 2J^/— 1 — sin .BV— 1 . V— 1- (99) One half of the sum of (98 and 99) is COS. Bs/—\ — l{c^-\- e-B), (100) One half of the difference of (98 and 99), multiplied bj iV — 1, is sin. ^V— 1 = i{e^ — e-^)\/— 1. (101) b. When the angle is A -{- B\/—}, (100 and 101) give Bin,(^-f"^V' — l) = sin.^cos. B/v/ — 1 -f-cos. A s\n.B\/ — 1 = ^sin.4(e^-[-e-^) + icos..4(e« — e-^)>s/— I ; (102) eos.( A-\-B\/ — I ) = cos. A cos. B\/ — I — sin. ^4 sin. By/ — 1 = icos.A(e^ + e-^) — is\iuA(e^—e-^)^^l. (103) 3 26 IMAGINARY QUANTITIES. [b. III. CH. V. The imaginary angle, whose sine exceeds unity. 52. Problem. To reduce the imaginary angle, the absolute value of whose sine is greater than unity, to the form (I). Solution. Let the given sine of the angle be db (1 + «), and let the required angle he A -{- B \/ — 1 ; it is evident from (102) that, when the sine of the angle is real, cos.^(e^ — e-^') = 0; (104) that is, either e^ = c-^, (105) whence e^^ = 1, 2 B = 0, i5 = ; (106) in which case the given angle is real, and the absolute value of its sine cannot exceed unity ; or cos. J = 0, A -=znn^ (107) sin. yl = =b 1, (108) whence, by (102 and 103), (109) sin.(^ + /^V— l) = sin.(7i7r + SV— 1) =:iM'^^ + ^-^) = i(l+«), (110) COS.(^ + i^\/— l)=:C0S.(«7r4-i?V— 1) ^zp^(e^— e-^)x/— 1 _zp^(— .2a— «2)z==Fv/(2a+a2)V— 1. (HI) The sum of (HO), and (111) multiplied by \/ — I, is c^ =: 1 + « ± V (2 « + a^), (112) whence JB = log. [I +fl±\/(2fl + fl2)] = i log. [1 + a + \/(2a + a^)l (113) and the angle is n^ d= log. [I + a+\/(2a + «2)].v'— 1. (114) § 54.] CIRCULAR FUNCTIONS. 27 Imaginary circular functions. 53. Examples. 1. Reduce tang. {A-\- B \/ — 1) to the form (I). 2 sin. 2^ (e2^ — 6-2^)./— 1 ^'^^' ^5_|_e-25 +2 COS. 2 ^ "^ c2^ + e-2^ + 2 cos.24* ^^^^^ 2. Reduce tang. S\/ — 1 to the form (1). 3. Reduce tang, f— ^] B a^/ — 1 to the form (1). Ans. When B is absolutely less than unity, it is ±w^ + J[log.(l+^) — log. (1— J5)].V— 1. (117) When B is positive and greater than 1, it is ±(n+J).^ + Hlog.(S+l)-log.(^-l)].x/-l. (118) When B is negative and less than — 1, it is ± {n+ih+i[\og.-{l+B)-\og.(l-B)W-l' (119) When S := =i= I, it is ^drOD.V— 1. (120) 54. Equations (100 and 101) have suggested a new form of notation of great practical value, and for which tables have been constructed, similar to the common trigonometric tables. It consists in representing — \/ — I . sin. B \/ — 1 and COS. B \/ — 1 by Sin,^ and Cos. ^, which only differ in their initial capital letters from the common trigonometric no- 28 IMAGINARY QUANTITIES. [b. III. CH. T. Potential functions. tatioii ; this notation may also be extended to the other trigo- nometric functions. These new functions are called potential functions. We have, then, fi- -r, , , . ^ # , sin. Bx/ — 1 = i{e^-e-^), (121) Cos.B=icos,Bx/—l =i{eB + e-B)^ (122) -„ „ Sin. 5 inner. Ba^ — 1 55. Corollary. The differentiation of (121 - 123) gives d,. Sin. 5 = J (e^ + e-B) — Cos. B, (124) rf,. Cos. J5 := 4 (e^ — e-^) — Sin. ^, (125) «//ran. jB z= ^7^—, = 7T-V^ = Sec.2J5. (126) ' C0S.2^V— 1 C0S.25 ^ ' 56. Examples. Demonstrate the following equations. 1. C0S.2 B — Sin.2 B = 1. (127) Solution. By (121 and 122) C0S.2 B = 1 (c25 + 2 + C-2^) Sin.2 JB =: ^ (e25 __ 2 _|_ g-sB) Hence Cos.2 ^ _ sin.2 ^ — 1. 2. Sin. {BdtiB') = S\n.B Cos.B' ± Cos. B Sin. B' ( 128) 3. Cos.(JB ± JB') = Cos. B Cos.^'i Sin. B Sin. 5' (129) 4. Sin.(B + JB') + Sin. {B—B') = 2 Sin.£ Cos.^' (130) ^ 54.] CIRCULAR FUNCTIONS. 29 Potential functions. 5. Sin.{B + B') — Sm.{B—B') — 2Cos,BSm,B' (131) 6. Cos.{B-{-B') + Cos.(B—B') = '2Cos.BCos.B' (132) 7. Cos.{B+ B') — Co^B—B') = 2 Sin. ^ Sin.J3' (133) Sin. ^ + Sin.^- _ Tang. Hg + ^0 f.^.. ' Sin. ^ — ^m.B' ~ Tang. ^ (B — B) ^ ' Cos 7? Cos Tl' ^- Co:.B + Co:.B- = Tan.i(iJ+B')Tan.J(B-B') (135) 10. Sin. 2B z=2 Sin. B Cos. 5 (136) 11. Cos. 2 5 = Cos.2 J5 + Sin.25 (137) z= 1 + 2 Sin.2 B = 2 Cos.2 B — 1 12. Sin. ^jB=: x/[|(Cos. 2 5 — 1)] (138) 13. Cos.^B = ^[^{Cos.2B + l)] (139) t4 rn . ^ , / Cos. 2 5 — 1 \ 14. Ta„g.JB = v(c„i:2^+l) (140) 15. Ta„g.(iJ±B')=. 5^^:^-^^^, (141) 1 zh 1 ang jB 1 ang. B' ^ ' 16. Tang. 2B= ^ . (142) 14- 1 ang. 2 5 ^ ' 17. d^.SinS-^}z= {l + z^)-i = -^^±--. (143) Solution. Let 2: == Sin. 5, or jB = Sin.[-i] z Then by (124 and 127) ef,.^.x = Cos.jB = -v/(l + Sin.2 5) = ^(i -j. ^2) 3* 30 IMAGINARY QUANTITIES. [b. III. CH. V. Potential functions. and by (Vol. I. 566) d,. Sin.[-i] X := 4, B = dcB'^ \/(l+^2) 18. rf..C03.[-i]x= (.^-lH:zz ^^J_^ - (144) 19. J,.Tang.[-i]x=-j-^. (145) 20. Sin. X = X + ^^-^ + --^^^^ + &c. (146) 21. Cos. X = 1 + -- + :^^^-^-^ + &c. (147) § 59.] REAL ROOTS. 31 Stern's method of solving numerical equations. CHAPTER VI. REAL ROOTS OF NUMERICAL EQUATIONS. 57. While the imaginary roots of equations are of great subsidiary value in mathematical investigations, and frequently admit of curious and interesting inter- pretations in physical inquiries, real roots are the prima- ry objects of attention, and methods of determining their numerical values are exceedingly important in practice. Ster7i^s method is the simplest which has yet been published, and is of almost universal application. 58. If the values of a given function and of its suc- cessive differential coefficients, as far as the ?ah, are found for a given value a of the variable ; and if the successive signs of these values are placed after each other, the row of signs thus formed is, in this chapter, called the nth row of signs (a), or simply the nth row (a), or the row (a) ; any pair of successive signs in this row is called a permanence^ when the signs are alike, and a variation, when the signs are unlike. 59. Theorem. If a function and its differential co- efficients inferior to the nVa all vanish, but the ?/th does not vanish, for a value a of the variable, the nth. row of signs (a + i)j * being an infinitesimal, consists 32 IMAGINARY QUANTITIES. [b. III. CH. VI. Signs of vanishing functions. — 1 ^ — - wholly of permanences, while the nih row (a — i) con- sists wholly of variations. Proof. It follows from (Vol. I. 533), that if/, x is the giv- en function /• (« + = Y^ — ;;■ ''-■ J"-"- ('^^) the differential coefficients of which, taken relatively to i, are t.f. ia + = 1T2T3 r:Wl)- ''■^- " = "-^^^'^' ^ f(„A.i^- '•-' .. . (n-\)f.(a+i) dl.f. (« + - i.2.3...(„_2)- ''-f-"- i ' &c. &c. &c. (149) that is, all the terms of the series /.(« + /), d..f.{a + i), dl.f.{a + i),&.c. (150) have the same sign. But the reversing of the sign of i in these equations gives nf. (a—i) dc'f- {a—i) = i ctl.f. {a — i) = —^ ^-^^ S&c. (151) that is, the signs of any two successive terms in the series /. (a — 0, d,.f.{a-~i), e/. (« — 0, &c. (J52) are unlike, and the terms are alternately positive and negative. 60. Corolla?^. If, in a series of the successive dif- ferential coefficients of a function terminating with the <5» 61.] REAL ROOTS. 33 Number of real roots between given limits. nth, all vanish except the 7i\h for a vahie a of the vari- able, the signs of this series will in the row of signs (a -}- i) constitute a series of permanences, and in the row (a — i), a series of variations. 61. Theorem, If the first member of the equation f.x = (153) is coiithmons between the values a and b of the variable, a being greater than b. if tJte nnniber of permanences in the nth row of signs (a) exceeds tJie number of per- onaneuces in the ntJi row (6), and if the excess is denot- ed by V, the number of real roots of (153), ivhich are included between a and b, cannot exceed v. Proof. For while the value of x varies from a to b, a change of sign can occur in the row of signs, only when f z, or one of its differential coefficients, or a series of them, pass- es through zero. Now, the case of a single function being included in that of a series, when a series of these functions vanishes, a number of permanences must, by ^^ 59 and 60, be lost, equal to the number of functions. If, then, this series begins with f x, as it must when the variable is equal to a root of the equation, one permanence, at least, nmst be lost; that is, there is a loss of one or more permanences in the row of signs, corresponding to every real root of the equation. If the vanishing series does not begin witli f. r, and con- sists of an even number of functions, the sign of its first func- tion is, by (148-152), the same with that of the function which follows the series, both before and after vanishing. The relation of the first sign of the series to the sign which pre- 34 IMAGINARY (QUANTITIES. [b. III. CH. VI. Number of real roots of an equation. cedes the series is, therefore, unchanged ; and the loss of per- manmrcs is exactly equal to the even number of terms of the vanishing series. If the vanishintr series consists of an odd number of func- tions, the sign of its first function is reversed when it vanishes. If, therefore, it has, before it vanishes, the same sign with the preceding function, another permanence is liere lost, which is to be added to those before noticed. But if it has, before it vanislies, the opposite sign to the preceding function, a new permanence is introduced, when it vanishes, which is to be subtracted from the number of the others. In one case, there- fore, the ivhole number of lost permanences is one greater than the odd number of terms in the vanishing series ; and, in the other case, it is one less than this number. In any case, the nuinber of lost permanences is, at least, as great as the number of real roots of the equation. 62. Corollary. When the loss of permanences does not arise from a real root of the equation, the number of lost permanences is even ; so that if the number of lost perrnanenres is odd, that is, ifv is odd^ the equation must have at least one real root betioeen a and b. 63. Problem. To find all the real roots of an equa- tion. Solution . Reduce the equation to the form (153), simplify it as much as possible; and determine, as nearly as possible by inspection, those limits between which the different real roots must be, if there are any. Find the successive differential coefficients of the first «5> 63.] REAL ROOTS. 35 Stern's" method of finding the real roots. member^ until one is obtained which docs not vanish be- tween two limits a and b, between which there may be real roots. Let this be the nth differoUial coefficient. If^ then., a being greater than b, the nvmber of per- manences in the nth roio of signs (a) is the same ivith that in the row (b), there is no real root between a and b. If the difference between the number of 'permanen- ces is even, the question of a real root between a and b is undecided ; and if this difference is odd, there must be such a root. Let, then, the mth differential coefficient be the hi^'h- cst one, of which the sign is different in tlie row (a) and in the roio {b). The equatiofi f/,.- + i/. xz=0, (154) can then have no real root between a and b, luJtile the equation d^rf.x-^, (155) must have one, which can be found by the process gicen in the sequel of this solution. If c is the rnnt of (155), it may also be a root of (153), which can be discovei^ed by trial. However this may be, the preceding process is to be repeated for the limits a and c -\- i, i being an. infinites- imal, and also for the limits c-\-i, and b, usitig the mth rov) of signs instead of the nth. A continuation of the process '//nist finally lead to a division of the lim- its from a to b, into sets of limits so norroiv, that, be- 36 IMAGINARY QUANTITIES. [b. II I. CH. VI. Stern's method of finding numerical roots. tween each set there con only be one real root of ( 153) and no real root of the equation chfx = 0. (156) Lei a' and h' he a set of these limits, and if they are far apart, substitute for x, in the first member of[ 153), different numbers, the various integers for instance^ between a' and b' , until one is found which does not dif- fer nincfi from the required root, and denote this first approximalio/i to tlie root by x^. Tlien, if the exact root is XQ-\-h, ive have by (Vol. I. 532) /. (..„+/,) =/.r„ + h ch.f. {r„+« A) =0, (157) whence, by neglecting &h, the approximate value of h is obtained, which is h = -j^ (158) and from the new approximation to the root x^ + A, which is thus found, a neio approximation, can be ob- tained ; and so on, to any required degree of accuracy. 64. Corollary. The rate of approximation can be readily determined ; for if two successive values of h are h and A' corresponding to a^ and x^, so that rro z:z To + /i (159) the error of x^ differs from h by a quantity much smaller than A ; and that of x^ is nearly equal to h' . Now suppose A<(Tvr (160) <^ 65]. REAL ROOTS. 37 Rate of approximation. and we have by (158) and by Taylor's theorem f.x'^=f{x,+h):=f,x,+ d,.f,x^,h-\.ldi.f,x,\h^+ &c. (161) c/,/.2:;=rt/,./.Xo+&c. (162) but by (158) f.x, + d,.f.x,.hz^O (163) /.x; = J ./?./. 2:0. A2+&C. (164) whence neglecting W^, &c. If now we find we have, neglecting the signs, /*'<(tVP+* (167) and therefore if one approximation is accurate to s places of decimals, the next will be accurate ^0 2 s + A: places, 65. Corollary. Since the real root is exactly ^==^0 + ^1 we have ^0 = ^ — ^'» (168) whence by (153 and Vol. I. 532) /. x, = f(x — h) =/. X — h,d,,f {x^&h) =z — hd,.f(x^6h), (169) or neglecting & k /. Xo= _ A d,.f. x={Xq^ x) d,.f. X. (170) In the same way for another hypothesis x^, we have /. z; r= _ h' d^.f X = (x; — X) d^.f X. (171) nr 38 IMAGINARY QUANTITIES. [b. III. CH. T. Rule of false or double position. The difference of (170) and (171) is /.7,-/..r; = (zo-r;)c/,./.x (172) and the quotient of (171) by (172), is f'" ^^'.^^I^ (173) which is identical with the famous rule of false, or rule of double positin??., iti arithmetic; and this admir- able rule, the principle of which is obviously at the foundation of all higher mathematics, and pervades all practical science in some form or other, is sufficient for obtaining, with ease and accuracy, the most important numerical results. 66. Examples. 1. Solve the equation xlog/ z— 100 — in which log.' denotes the common tabular logarithms. Solution. The theory of logarithms gives log.' X z=z log.' e . log. X. Hence if f.x=^x log.' x — 100 d,.f.x = log.'z + log.'c The value of d^^'f- ^ is positive between the limits X = 0, and x zn oo and d^.f.x is negative between the limits X = 0, and x =. c~~^ ^66.] REAL ROOTS. 39 Solution of numerical equations. at both which limits /'. x is negative, and the given equation has therefore no real roots between these limits. But d^.f.z is positive between the limits x=:c~' and x = 0, at which limits f, X has opposite signs, and the given equation has, therefore, only one real root, which is between these limits. A very few trials show, then, that the root is not far from 60, for which value f.x — 1, d\.f.x— 1-78 + 0-43:= 2-21 <^,./.xr= -0072, A: = 2 and the rest of the calculation may be arranged as in the fol- lowing form, in the first column of which are placed the suc- cessive values of y.x, in the second those of rf^'/^ ^» sind in the third those of x. 7. 0084860 221 219017 60 57 569612 2. Solve the equation X — cos. X =: 0. Ans. 56-9612. Ans. 0-7391. 3. Solve the equation X — tang, X = 0. Ans. There are an infinity of roots, one being contained between each set of limits n n and (^ + J) tt in which any integer may be substituted for 7i, the value be- tween ^ and J ^ is 4*4934. ^•'•f BOOK IV. RESIDUAL CALCULUS. 4* A BOOK IV. RESIDUAL CALCULUS. CHAPTER I. RESIDUATION. "S^^' 1. For every finite value of x, which satisfies the equation /x = 00 , that is, ~ = (/.x)-i = 0, (174) /• ^ the first term of Taylor's theorem (Vol. I. 442) is infinite, and the development of y. (x ~\- h) by that theorem is impossible. In this case, if i is an infinitesimal, f. (x -\- i) is infinite ; and if we suppose it to be of the ?wth order of infinity, the expression imf,(x + i) (175) is of the order zero, and is usually finite, as in § 26 of the Differential Calculus. The quantity h^f.(x + h) (176) may then be developed, by MacLaurin's Theorem (445, Vol. I.), as a function of A, and the result will be of ihe orm h'"f,(x-\-h) ==A -{- Bh-\-&.c. _j« Qh^-^-{.Rhm-i 4- Sh^^ + r/i'^+i + ifcc. (177) 44 RESIDUAL CALCULUS. [b. IV. CH. I. Residual. To residuate. which, divided by 7i'", gives + Q A-2 4- 7? A-i 4- >S 4- Th + &c. (178) that is, f.{x -\- h) can, evcti for a value of x which satisfies (174), he developed in a series consisting of two pa; ^5, one of luliich S -\- Th ^ 6cc. (179) is, like Taylor^s Theorem, arranged according to posi- tive and ascending powers of h, and the other part R h-^ + Q /i-^ + &c. + B A-(— i) -j- A A-'" (ISO) is arranged according to negative and descending pow- ers of h. 2. The coefficient of h~^, in the development of f.(^x-\-h) by the preceding method, is called the re- sidual of/, x^ and vanishes for all values of x^ except those which satisfy (174). To residuate is to find the residual. 3. Problem. 2 o residuate a given function. Solution. Let f. denote the given function, and let x^ be the value of x which satisfies (174). Since R, which is the residual of ihi.- function by (180), is the coefficient of h™—^ in (177) the development of ]i"^f.{xQ -\- h) by MacLaurin's The- orem ; we have by (445 of Vol. I.), if we regard h as the vari- able, § 4.] RESIDUATION. 45 Method of residuating. provided that after the differentiation we put hz= 0. This vanishing of h may be effected in the general form, bj substituting for h the infinitesimal f, which gives ^ = 1.2.3. ..(m-f)- ^^^^^ 4. Examples. 1. To residuate the function (x — a)~^ (x — b)~'^. Solution, This function becomes infinite of the first order, when x z=i a -\- i ; and infinite of the second order, when x = b -\- i. The residual which corresponds to x =. a, is, then, ' i (i)-i (« — 6 + i)-^ — {a — b)-^ ; and that which corresponds to x :== 6, is — __(6_a)-2. 1 2. To residuate (x—a){x—b) (z— c)3 Ans, The residual for xziz a, is {a — b)—^ (a — c)~^, that for x=b, is (b — a)-'^ (b—c)-^, that for x=c, IS ^7^-7— r~ -• 46 RESIDUAL CALCULUS. [b. IV. CH. I. Residuation. 3. To residuate cosec. 2. Solution. We have cosec. 2 = GO , whenever z=7i;r, 71 being an integer, and the residual of cosec. z is t 1 t cosec. [n jt -\- 1)^=- %$ sin. {m^ -\- i) cos.{nn^-j~i'j 1 COS. n n = ±1. 4. To residuate tang. x. Ans. ± 1. 5. To residuate Cosec. 2. Ans. 1, 6. To residuate (Cosec. 2)2. Ans. 0. 7. To residuate 2~^ cosec. 2. . .. -4/15. When 2 = 0, it is ^ ; % when 2 z= n;rr, it is i {n 7t)~', 8. To residuate x~^ cosec. 2. Ans. When 2 =: 0, it is 0; when 2 = n TT, it is i (wti)"'. 9. To residuate — — for any value r^ of z which 2 — z satisfies the equation /. z = 00 . ^ 4.] RESIDUATION. 47 Method of residuatins. %^SoJution. Let f. {x^ -\- i) be infinite of the ?nth order, and let f . z =r /. z. (z — 3- J™, (183) 80 that f. (Xq -j- i) may be of the zero order, and the required residual is, by (182), d:-'.f.{x^+i).(x — x^ — i)- ' 1.2.3.... (///.— I) ^ dT.T''r{^o+i) [ (x-x^)-'+{x-T^)-H-{.(x^x ^) -H2+&.c. ] 1.2.3 .... (m — 1) __ 1 /d:-' f-K + O 1 . 2 . 3 . . . . »i — I \ x ( 3", +^tSS^' + 'E^?-'H— )(.»•) But it is evident, from M icLaurin's Theorem, that rf-\f. (r. + O./" (185) J .2.3... (m— I) is the coefficient of i'"~' in the development of f.(.r, +{).{" (186) or, dividing by «", that (IvSo) is the coefficient of {"'-"-'^ in the development of f . [x^ -\- i). Hence, by this ilieorem, d^fSJ^^^+i)J^ _ f/— -^ . f . (r^ + _ r/r"-^f.(xj 1.2.3...(m-l)"" 1:2.3.... (m—/i—l) 1.2.3...(77^-/^-l)^ ^ which, substituted in (184), gives, for the required residual, 1.2.3... (m-l)'x—XQ~ 1.2.3... (/«-2)*(x—2q)2~ "T" 1.2 '(x— Jo)"'-2 ' (Z—Xj"'-^ '"(X — Zq)- ^ ^ 48 RESIDUAL CALCULUS. [b. IV. CH. I. Integral residual. The value of f.x^ is found by the equation (183), which by (521 of Vol. I.) gives _ d-{z-x,r _ 1-2 3>- :^^ ,189^ 10. To residuate the preceding example, when z^ -\- ab f.^ = {z—a) (z—l>)^' a^J^ab 1 Ans. When z^^a. the residual is -— . ; (a — by' X — a . . b2—2nb—a2 1 b^-^-nb 1 when z==b, it is — — . -\- (^b—af 'x—b^ b—a ' {x — by' 11. To residuate example 9, when f .z z=. cosec. z. Ans. When 2 = w tt, the residual is ri= X — n 7t 5. The ijiteg?^al residual o( 3. function between cer- tain limits is the sum of all its residuals contained be- tween those limits; and the total residual is the sum of all its residuals. To residuate from one value of a. variable to another is to find the integral residual between these values of the variable ; and to residuate totally is to find the total residual. a. The total residual is indicated by the sign ^, and the ^ 5.] RESIDUATION. 49 Notation. integral residual is denoted by the same sign with letters an- nexed to it, to show the limits of the residuation ; thus t(/-^) (190) is the total residual of /". x ; while ^^■(/•^) (191) is the integral residual ofjT. x from the limit X = Xq to X = Xj. b. The residuation is often limited to those values of the variable, which render one of the terms or factors of the given function infinite, as in Example 9 of the preceding section ; and this is indicated by placing, in double parentheses, the factor which is thus regarded exclusively of the other factors. Thus £•((/• ^))-(/'-^) (192) indicates the residual of (y. x) (/"'. x) with regard to those values of x, which render f. x infinite. In this way £-((/-^)) (193) should be usgd instead of (190) to denote the total residual of f. X. In the same way denotes the simple residual of /.x.(x-r„)^ X — Xq for the value of x, x z=z x^. 5 50 RESIDUAL CALCULUS. [b. IV. CH. I. Notation. 6. The variable in (101) may be itself a function of other variables, as y, z, &c. ; and the residuation may be sought between the limiting values of y y — Vo ^"^ y — yxy and those of z z = Zq and 2 = 2;^, &c. and this may be expressed by the form vy = y,. ^ = ^1, f^-.((/.:r)), (195) or more simply it being conventional in what order the limits are placed. 7. Corollary. The preceding notation gives at once, if x' is a value of x between x^ and x^, ryl l7-i(f-^)) = ei •{{/■^)) + 17 ((/■-))• (19-) 8. Scholium, If x' is a root of the equation /.xzzroo; * (198) the value of the corresponding residual should be equally di- vided between the two terms of the second members of (197), that is, when one of the limits of (191) is a root of (198), one half of the corresponding residual should be included in the expression (191). 9. Corollary. If, in (19G), there are only two variables y and z, and if y is taken to denote the real term of x reduced <5. 10.] RESIDUATION. 61 Residual of differential. to the form (1), and z ilie real factor of the imaginary term, (196) will denote the integral residual for all values of 2", whose real terms are included between y^ and ?/j, and the real factors of whose imaginary terms are included between Zq and z^. 10. Corollary. It is evident from (182) and <5» 5? that the residual is a linear function ; and found, as it is, by differentiation, it must by <§> 52 of B. II. be free relatively to any other linear function^ such as differ- ence, differential^ &c. Thus, if the residuation is taken relatively to .r, we have L.{{d^-.f.{x,z))) = d^-.C{(f.(x,z))), (199) 52 RESIDUAL CALCULUS. [b. IV. CH. II. Development of a function, wJiicli has infinite values. CHAPTER II. DEV^ELOPMENT OF FUNCTIONS, WHICH HAVE INFINITE VALUES. 11. Problem, To develop a funclion which has in- finite values corresponding to finite values of its vari- able^ in a fortn which may be tised for all values of its variable. Solution, het f.x be tlie given function, and let x^ be a value for which it becomes infinite, so that, if i is an infini- tesimal, f{xQ -\-i) is infinite of the mih order. Then if we put f:.xz=fx.{x — x^)'^; (200) we have f . Xq finite, and (200) can be developed according to powers of x — x^. We have, by Taylor's Theorem, d""-^ f X whence, by (200), f r // f r d"^-^ f X 1 •^ (x-a:J-^(x-xJ'«-i^ ^1.2.3...(wz-l) x—x^ d'^ f X d"'+^ f X ^1.2.3... 7»^1.2.3.. .. (m+1) ^ °''~ '^ ^ Now the upper line of the second member of (202) consists of <5> 11.] DEVELOPMENT OF FUNCTIONS. 53 Function, which is always finite, when the variable is so. terms divided by different powers % — Xq, all of which are finite, unless %=x„ (203) in which case they are infinite ; while the lower line is a func- tion of X, which is finite in this case. We will denote the upper line by X^ and the lower line by Y^^ \ and X^ is, by (188), the residual of -^-^^ (204) X — z when z = x^. (205) If, then, we denote by Z^ all the other residuals of (204), when jf.z is infinite; we have, for the total residual of (204), iS^l^-^X^^Z,. (206) But by (202) /. z = Xo + Fo J (207) and therefore /. x — ^. ^^^^-^ = ^o — ^o- (^08) Now Fq and Z^ are both such functions x)f x that they are finite when X = Xq ; (209) that is, the first member of (208) is a function of a;, which is finite for every finite value of t, such as (209), for which f»% is infinite, and if we denote this function by w.x^ we have f-—l^T^ --.■-• (210) Hence the second term of (210) is finite for all finite values of X for which f. x is finite ; and, therefore, -cs x must be Jinite for every Jinite value of x. 5* 54 RESIDUAL CALCULUS. [b. IV. CH. 11. Development of a function, which has infinite values. Hence in the equation /..= £/i4^ + ..., (211) the first term of the second member consists, as in (188), of a combination of terms arranged according to the negative pow- ers of X — Xq, X — a; J, &,c., while or . x is always finite, and can usually be developed according to powers of x by Taylor's Theorem, or by some other simple process. 12. CoroUari/. When the modulus of x is infinite, the first term of the second member of (211) vanishes, and (211) be- comes f.co= cr.x. (212) 13. Corollary. When the first member of (212) is finite for all values of the argument of z, cr. x is always finite. But it has been shown, in ^ 81 of B. III., that the equation -i-rrO, or tjr.x = 00, (213) uf X is always possible, unless cr.z is constant, that is, independent of x; and, therefore, if we put /. O) = P; (214) we have iiif x^F, (215) and in this case f.^^lSSl-llj^F. (216) ^ 16.] DEVELOPMENT OF FUNCTIONS. 5$ Development of a rational fraction. 14. Corollary. When f.x is a rational fraction, zj.x is also a similar rational fraction, because the second term of (210) consists of the sum of such fractions. But cr. x cannot have an entire polynoiniai for its denominator, because such a denominator would vanish for finite values of x, and cj. x would become infinite. Its denominator must then be constant; that is, cr. X must be an integral polynomial. 15. Corollary. If, in the preceding corollary, the degree of the numerator of/", x is greater than that of its denominator, this function is infinite when its variable is infinite; but if the degree of the numerator is equal to that of the denominator, f,x is finite when its variable is infinite ; but if the degree of the numerator is less than that of the denominator, f, x van- ishes when its variable is infinite. For if the function is ^x"+/>x"-i 4- &c. J - — a'x^'+ bx" -'4- 6lc. ^'•^ ' ^ we have /. x = -— — {-- = — (go )"-«' (218) which is infinite, when n > »', finite and = -y i= i^, when n z= n', (219) zero, when n ^ n'. The polynotTiial uj.x is, therefore, reduced to a constant in the second case, and to zero in the third case. 16. Corollary. The easiest way of finding zu . x m the case of § 14, is to reduce the given fraction by division to a mixed expression, consisting of an integral polynomial, and a fraction in which the degree of the numerator is less than that of the 56 RESIDUAL CALCULUS. [b. IV. CH. II. Development of cosecant. denominator. For this last fraction can, by the preceding corollary, furnish no part of the polynomial et. ■x, ^vhich must, therefore, be the same with the polynomial thus obtained by division. 17. Examples. 1, Develop (sin. a)— ^ by the preceding principles. Solution. The general expression for the root of the equa- tion (sin. x)-i = 00 , (220) is x = ±/i-^, (221) in which 7i is any integer at pleasure; and the corresponding value of the residual of (sin. z) X — z is, by Ex. 9, § 4, if we put 1 i.z = — 1 d^ . sin. z cos. z 1 1 cos, lire x^nrc SO that by (21G) (222) 1 1 1 1,1,1 cosec. X = -, = j r— _ H -- sin.'x X x-\-n X — n x-f-'z/v X — 4 TV <5> 17.] DEVELOPMENT OF FUNCTIONS. 57 Development of secants. 2. Develop sec. x by the preceding principles. Anz. sec. 2 = 4 71 1— — -—^ — ., . -7^ + ^^^"^ — v^ — &.c. J (224) 3. Develop {e + c-^)-i = J Sec. 2;. Solution, Let a; z= ?/ + ~ V — 1» (225) and we have, by (89), e^ z=z cv {co%. z -\- A^ — l.sin. %), (226) €—''=: e—y{cos. z — V — 1 . sin. z). (227) Hence the equation e^-|~ ^~" ^= ^^ (228) involves the two (e^ + e~y) cos. z z= 0, (229) (ey__e-y) sin. ;:i =z 0. (230) Hence, cos.^z=0, e'^ — e-y, or e^^—l, y=:0;(231) z =d^(n + i)n, (232) and the root of (228) is ^{n + i)n^-l, (233) 7: If, now, we take f-=^^e---^=^--^'' (~^'') we have, by (90 and 92), 1 2x7—1 and the corresponding residual of X — z {.r, = ± ;r-7—i ■' i~^^) (236) 5S RESIDUAL CALCULUS. [b. IV. Oil. II. Development of a rational fraction. 11 1 ^^ ^ o / 1 - -^/. , 1-, —/— 1=^^ '2V— l^=P(^^+=^)-'V— 1 -^/V— i±(2w+l).T (237) we have, then, -L- = ( ' L__\ 4. Develop (c"" — e— ^)— ^ =z J Cosec. x. 3-5 4- 1 Solution. Since (.i- 1)2 (x + 2) z= x3 3z + 2, (240) re have, by division, *'+! -t- 1 3 1 -2^^+9^-5 (241) (^_,)-2(,+2)-- 1 1 (x_I)2(x + 2)- Now by Ex. 9 of § 4 ^ \\(z-\y^(z^'2)) fx-z 3(x-l)2 I ^^ ^^ . ^24^^^ "T-9(x — 1) 9(z + 2)' ^ "^ <5> 17.] DEVELOPMENT OF FUNCTIONS. 59 Development of rational fractions, whence, by (216), ^^ + ' -.3+3+ 2 {x — \)2 (x-^2) ' ' 3{x— 1) 2 ^ ^^ - ^^ m3^ "^9(x— 1) 9(x + 2)' ^ ^ /*. a; 6. Develop — ^— — ^^ , in which x^., x,, {x—Xo){x—x^){x — X2).... &/C. are all unequal, and the values of x, which rentier f. x infinite, are to be necrlected. Solution. We have at once Zlf — r /l^ 1_ (244) (X-X,){X-X^)... C (((^_3;^)(^_:,J...)) ^_^ V / (^•O — ^^l)('0 — ^2)--- ^—^ f.x 1 {x^—Xq){x^—X2)(x^—x.J... x — x^ 7. Develop (x+l)(x_2) ^"'* 3C^l)"*"3(x— 2)' a Develop ^-^---^^. 2 2 (a;_2)2 3(x— 2) ' 3(x+ 1) BOOK V. INTEGRAL CALCULUS. BOOK V. INTEGRAL CALCULUS. CHAPTER I. INTEGRATION. 1. The iyitegral of a given differential is the func- tion of which It is^th- diifereiitial ; and /Ac integral of a given finite function is the function of which it is the differential coefficient. To integrate is to find the integral. The sign of integra- tion is /. ; thus f.d,x = x, f.d.fx — f.x] f.d,.x = x, f,d,.f.x=fx; (245) f'di.x = x, f:d:.fx=fx,&.c. (246) 2. Corollary. Since we have d,i{x + a)=zd,.fx, (247) for all values of a, it follows that /.c/,./.xr=/.x + fl, (248) that is, the integral of a function may have an arhitra- 64 INTEGRAL CALCULUS. [b. V. CH. I. Increase or decrease of arbitrary constant. ry constant added to it, and in this form the integral is said to be complete, 3. Corollary. Any constant may then be added to, or sub- tracted from the incomplete integral, and the form of the in- tegral may often be changed by this process. 4. Corollary. If the integral contains a term of the form log./ ^, this term may be changed, by the addition of a constant, to the form log./. X + log. a = log. {af. x). (249) 5. Corollary. If the integral contains a term of the form sin.t— ^] X, this term may be changed, by the addition of a constant, to the form sin.C-i] 2 — J^rz: — (Jtt — sin.[-i] x)z=i — cos.[-i] x (250) or it may be changed into coseJ-^] -, or into cosj— i]/\/(l-x~) or into - s'mS~^^\/{l-x^), In the same way, terms of the form cos.t— ^] 2:, tan.[— ^]x, cot.f— ^] z, sect— ^3 x, &,c. may be changed into — sin.[— ^] x, cot.t— 1] X, — tan.-— ^] z, — cosec.t— ^3 2, &:c. or into 1.1 1 1 ' sect— ^-' -, — tan.'-^^ -, tant— ^1-, cos.t— ^]-, &c. Z XXX «J 7.] INTEGRATION. 65 Number of arbitrary constants. Definite integral. or into (251) 6. Corollary. Since every integration introduces an arbitrary constant, the nunib(3r of arbitrary constants in a complete integral must be equal to the number of integrations. 7. Corollary. The difference between the two values of an integral, which correspond to two values of its variable, is called the definite integral from one value to the other value of the variable. Thus if Xq and Xj are the limiting values of the variable, the integral of cl^.f. x from x^ to r^ is, by (248), (/. ^1 + «) - (/. ^0 + «) =/• ^2 -/. ^0 ; (252) and it is written •X. f Kd,.f.x=f.x^. (252) The definite integral is, therefore, independent of the value of the arbifrary constant ; but the places of the arbitrary con- stant and the variable are supplied by regarding one of the limits as arbitrary and the other as variable, thus J ^( ,d,.f.x=f,x^f.x^, (254) which gives, by (248), a — — /. 3-Q, (255) 6* 66 INTEGRAL CALCULUS. [b. V. CH. I. Integrals are linear functions. 8. Corollary. Since /^o.rf../.x=/.x„-/.x., (256) we have, obviously, fi:=-fi:- (257) 9. Corollary. Equation (246) shows that integration may be regarded as negative differentiation, that is, ^"^Z (258) 10. Corollary. It is evident, from B. II. '§^^ 51 and 62j that integrals are linear functions^ which are free relatively to all other linear functions. Thus we have f.af.x — aj.f x. (259) 11. Corollary. Differ eritials, residuals ^ a7id integrals are functions which are relatively free. 12. Corollajy. When a function can be separated into parts connected by the signs + or — , the integral of the ivhole function is the algebraic sum of the partial integrals. This method of integration might naturally be called inte- gration by parts, but the following is a particular case of it, to which this designation has been applied technically. 13. If u and v are functions of a variable, we have (Vol. I. 468) d^.uv z=i udg.v -\- V df.u, (260) <5) 16.] INTEGRATION. 67 Integration by parts. whence ud,,.v ^n d^.uv — vd^.u, (201) and by integration f,ud^.v z= uv — f.vd,.u; (262) and when a given differential coefficient can be sepa- rated into two factors, one of which, d^. v, has a known integral, the integration can often be effected by the aid of (262) ; and the application of this formula is called integration hy parts. 14. Theorem, A definite integral, which is taken be- tween limits differing by a quantity equal to the differ- ential of the variable, is equal to the differential of the integral. Proof. For the equation (252) becomes, when 3-0 = 2;, a; J r= z + c? X, (263) by (Vol. I. 421) fl^'\d,.f.x=:f.{x-\.dx)^f.x^d.f.x. (264) 15. Theorem. \i x^^ x^^ x^^ , , , . x^ are successive values of x, a definite integral from Xq to ^,„ is equal to the algebraic sum of the corresponding definite inte- grals from Xq to x^j from x^ to x^^ 6oc. Proof. We evidently have /.^n— /.2:o = (/.2;i — /.Xq) +(/.Z2 — /.2; J + (/.^3-/.^2) + &'C. (265) 16. Corollary, Hence if Xq^ x^^ x^^ 6cc. differ by dx, 6S INTEGRAL CALCULUS. [b. V. CH. I. Change of variable. the definite integral from x^ to x„ is equal to the al- gebraic sum of all the corresponding differentials from Xq to x,i, taken at intervals equal to dx. 17. Scholuim. Propositions 14 and 16 require that the inte- gral be a continuous function between the limits, and particu- lar caution must be observed to exclude those cases, in which the value of the integral varies from positive to negative, or the reverse, by passing through infinity, so as suddenly to vary from positive to negative infinity, or the reverse. 18. Theorem. If we have the equation f.f.x^F.x (266) and if we substitute for x any function at pleasure, as 25.] RATIONAL FUNCTIONS. 71 Jntegration of rational fractions. 8. Integrate a x^ -{- b x^ -\- - -{- h. Ans. ^ax'^-{-^bx^-\-c log. x -[- hx. 4. Integrate x~^ {x^ -\- x -{- 1)2. A71S. ^x3-|-x2-|-3z4-21og. 2 — ar-i. 5. Integrate 2;(x + x-i)2. v4?i5. :^ x^ + 2;2 -|- log. a;. 23. Corollary. The substitution of y. x for x in (279) and (282), gives by § 18, a (w x')"+^ /.«(c;.x)"^.. we have for w = 3, f'.x^zzil^ so that (291) becomes — -— — -— — : 2(z-{- 1)^ and we have for w z= 2, f.x^z^z 0, for W =: 1, f. X(j =z 0. TJdrdhj. When a;o=r — 1+V— 1, we have for n =r 2, f.xQ = — ^ ; so that by (294 and 295) ^ = — J, 5 = 0, X—x-\-\, Fzn — 1, and (299) becomes ^^^-^_hi__ We have also for n := 1, /,* x^ z= ^, so that by (294 and 296) <5, 28.] RATIONAL FUNCTIONS. 75 Integration of rational fractions. A=0, B^—l, and (301) becomes — ;^ cot.[-i](x + 1). The required integral is, therefore, log.(x-l)_i(.+l)-+^^^5^jdy__jcot.[-.l(. + l). X5 J- 1 2. Integrate (x_l)-^(.T + 2) _ _ X 1 3. Integrate 4. Integrate (x+l)(x_2)- ^ns. f log. (X + 1) + ^ log. {x — 2). (x+ 1) (x — 2)2 ^n5. Jx2+-A__2iog.(x-2) + |log.(x + l). ^ _ n X -4- m 5. Integrate — — ^-rn- n Ans. - log. (x2 — 2 a X -(- «^ + &^) + !iii + i:!tan.t-']?^. (302) _ _ 71 X -4- m 6. Integrate ^^^^. ^„,. "log /x5+*\_-_;!L^tan.[-i]^^. (303) 76 INTEGRAL CALCULUS. [b. V. CH. II. Integration of rational fractions. 7. Integrate a;2 + 1* Ans. tan.Ml x z= cot.M] -. (304) X n X -\- m 8. Integrate {n a-\-ni) X — n {n^ -\-^^^) — ^^ (^ Ans. 2^2(2:2— 2ax+a2^62) ^ na-{-m . n ^ /on-x - A -! tan.[-i] . (305) <5> 30.] IRRATIONAL FUNCTIONS. 77 Integration of irrational functions. CHAPTER III. INTEGRATION OF IRRATIONAL FUNCTIONS. n 29. To integrate f.[x^j^{ax-{- 6)]. Solution, Let y = ^ {ax-\-h)y (306) whence x = ^^ , 4y ^ = -^^ — > (307) and by § 19 = ,.,,(£=5,,). -p. ,308, 30. Examples. n 1. Integrate \/{ax-\-hY'. Solution. Equation (308) becomes, in this case, •^ ^ ' ' «/ a {in-\-n)a n\/{nx -\- Z>)™+»] (w -j- w) a _ _ \/^ -f- 1 3. Intetrrate — -. . ° S^x— 1 ^7i5. X -(- 4\/x + 41og. (\/x — 1). 7* 78 INTEGRAL CALCULUS. [b. V. CH. III. Integration of irrational functions. \/2;4- 1 3. Integrate -^ — . a/x — 1 6 Solution. Let y i=z \/ % and (308) becomes n 4. Integrate x s/ (x -\- a) -\- \/ [x -\- a). 5. Integrate ^^ ^ , , ^ - T, ^ ^ ^ \/ {ax -\- b) Ans. T — j — — . a{?i -\- i) 31. Problem. To integrate f. [.v, ^(ax- -\- bx -\- c)]. 5 52 4 rt c Solution. Let x =^ y — --, m = — - — - — , (309) 2a 4 «2 ^ ' 1)2 — A. a c whence g x^4- 6 a; + c = « ?/2 — a{y^-^m) (310) c/,.yx=l; (311) ^ 31.] IRRATIONAL FUNCTIONS. 79 Integration of irrational functions. and by § 19, f.f.[z, ^/{ax^+bxJrc)]=ff.^j-^^,^/[a{,f-m)]'] (312) There are, then, several cases : First. When a is negative and m negative, the radical s/[a{y2 — m)] (313) is always imaginary, and the integral, being imaginary, admits of no real solution, and may be solved as in either of the other cases which, in this case, become imaginary. Secondly. When a is positive and m negative, y^ must be greater than m, when (313) is real, let, then, z=.s/{y^ — m)—y, (314) whence (y -\- zY ~y^ -\-^y z -\-z^ = y^ —m (315) y = -^-^z . (316) 2z 2z2 ^-3/ = .7^ -J (317) and (312) becomes v^ Thirdly. When ni is positive, let m ,2 ^ -JL-^, (320) 80 INTEGRAL CALCULUS. [b. V. CH. III. Integration of irrational functions. \////.(22-|_rt) whence y =: ^ — Ky^*-) « — z^ 4 as/ m . 2 (a-;:2)2 2 «\/ m. 2 V[«(y^-'»)] =: -r^- (323) and (312) becomes ^ r \/ffl.(22-|-Qr) h 2a\/m.z n \as/m ,z 32. Examples. 1. Integrate — — Trs—, ; • * V(«3;2 + ^z+c) Solution. In this case (312) becomes J' ^[a{y2—n^ (^^^) (319) becomes, by reduction, — f- —, = -f- log- ^ «/ \/a.z \/a = — ;^- log- [V(y^— ^) — y]. or since — m ^{y2 — m)—y = \^{y-—m)-{-y log. s/[y^ — ni)—y = log. (— ?w) — log.[\/(y^ — "0+^] <5> 32.] IRRATIONAL FUNCTIONS. 81 Integration of irrational functions. in which >- ^og- — "* ^^^^y be omitted, as in art. 3. Again (324) becomes which, when a is positive, is _!_ ] cr ^+^^ — J_ io(T V(y+V ^ )+V(y— ^^^) = _!_., og.[V(.^-»)+.]-3-^ W but when a is negative (327) is as in (803) is/— a z \/—a ^ Ww*+3// The form of this last solution may be changed in several ways, which will often be useful ; thus, let ,..tan.[-]l(^^^^^\ (330) \i s/ m — y\ whence tan. ^ z=z \\ —, ■ — I -^Xs/m -\- yf 2 \/ m sec.2 5 =. 1 + tan.2 6 — COS.''^ A = \^m -f- y A^m — y sin. 2 5 z=z tan.2 a . cos.2 a = 2x/wt m-7/2 4r/(r/x2-|_6x-|_c) sin. 2 2^=4 sin. 2 a . cos.2 ^ = = — — m ^ac — 0^ y b ■4-2 ax COS. 2 d = 2 cos.2 6^1 — -^— — Vwt"" V(^--4ac) 82 INTEGRAL CALCULUS. [b. V. CH. III. Integration of irrational functions. and (329) gives J' /s/(ax^+bx-\-c)~ \/—a ^ ^ = —, sin.t— ^J — -— ! ~ — •' V — a v(o — 4^6;) 1 h + 2ax =. cos.L— ■'J ■ 1 . r n ::pb^2ax 2. Integrate - -. Ans. sin.[-i] z. (332) 3. Integrate ^^/_^^,^ . (333) Ans. log. [x + V ( 1+ a;^)]- 4. Integrate ^^J_^y . (334) ^ns. log. [x + V (^^ — 5 )]• 5. Integrate ;;7(X^,^. Ans. — (^x2+ 1)^(1 _x2). . _1_ 1 V(^+6 2:2 )— z ^6— ^q - 1 1^,, V(«+^^")-V« 2/s/« "^ V(« + ^^^)+V« _ 1 sin.[-]ij-^. a/ — a X ^ " V <§> 33.] IRRATTO ^M, FUNCTIONS. 83 Integration of irrational functions. ^ T 1 7. Integrate Ans. log. 8. Integrate — —- r-. Ans. log. ^{l + z^)- (336) -1 X (337) — 1 X ^ '"''^'''^ Wi^) ^'''^ Ans. — sin.t— 1] - or sec.C— ^^x. X 10. Integrate ^(x2+xy Ans. \og.[i + x+^{x^ + x)], n 33. Problem. To integrate f.fx, I.^L+Af.1 (339) L- '^J c ~|~ it X —^ Solution. Let a-\-h X 2'"=7TI-x' (340) whence x — hif—b' (341) , nih c — ah) ?/"— 1 ^'■'■^= \uy^Jk) ~' (342) and, by § 19, the integral of (339) becomes r f V"!—^ ,/"! "" (^ C—ah)j/n-l 84 INTEGRAL CALCULUS. [b. V. CH. III. Integration of irrational functions. 34. Examples. ,. Integrate jl^. Ans. V(l-^)V{l+^)--Iog.[V(l-^) + V{l+^)] - 1 V 3 ta„.[->] 3 V^.Vd-^ _ 2V{i+^)-V(i-»:) 1 3 1 y. 2. Integrate — — ; — — . I- . 3 /I— a:\* T ^- ^j(iT-D 35. Problem. To integrate ^™"V- [/>/(« + ^^")> ^^ ^j"], (344) when m is exactly divisible by n. Solution. Let « + 6 x" r= y' ; (345) whence a:" = ^:— - — , (346) .» = (?l^)==". (347) The differential coefficient of the logarithm of (346), gives ~^^^—=.y—, (348) whence X— irf z - ^y^"' ^y'-«\ n /349X <5> 37.] IRRATIONAL FUNCTIONS. 85 Integration of irrational functions, and, by § 19, the integral of (344) becomes m m 36. Corollary. When q z=z 2 and 2 m is divisible by 71, (350) is integrable by ^ 31. 37. Examples. 5 1. Integrate x^ ^{a -\- b z^). Solution. In this case m =i 4, 7i z= 4, q z=z 5 ; 5 whence ?/ z=: ^(« -|-6 x*) 5V(«+ ^2:^)6 24 6 5 2. Integrate a:^ ^^^j _|_ j 3-2). 3. Integrate 8 86 INTEGRAL CALCULUS. [b. V. CH. III. Integration of irrational functions. a;2 4. Integrate — — — . Ans. — J X V(l — x2) + J sin.M] x, 38. Problem. To integrate x'n-i {a-\-h x'^)! f. (z") (351) when \--2san nte^er. n ^ q ° Solution. Let ax-^ -^ b z=z y% (352) whence x" = — -, (353) 771 The differential coefficient of the logarithm of (353) gives X ■" y'i — 6' whence (356) '=-y \2f — b/ n(y'i — bY ^ ' and, by § 19, the integral of (351) becomes m p § 40.] IRRATIONAL FUNCTIONS. 8T Integration of binomial irrational functions. 1. Integrate 39. Examples. 1 Ans. -(-1 _3^WVM-_if!)!\ 5 2. Integrate ^ ■ ~ L., 8ax8 40. Problem. To integrate z'" (a + 6 x")^ (359) WjAe?i m and n are positive integers^ and p is a posi- tive fraction. Solution. First. Let v z=z x\ (360) whence d,,.v z^ sx^—'^, (361) in which s is to be taken of such a value as may be found most useful \ let, then, ud,.v = x^ {a-\-h x^)P ; (362) whence u =^ - 2"i-s+i (^^ _[_ j ^r^y^ (363) d,.U==r m-s-\-l {a+h X") P + ^^ x^-'+^{a+h x")?-! ; (3G4) 8S INTEGRAL CALCULUS. [b. V. CH. III. Integration of binomial irrational functions, or, since {a-{-bx''y=z {a + bx") (a-{-bx")P-\ (365) (7}i-s-\-l)a-\-(7n-s-\-l-\-n p)hx" , , , „, . ,con\ and, by (262), the integral of (359) is -z^'+Wa 4- bx'')P s and, if s is taken such that m — s + l+wp — 0, that is, s z:^ m -\- \ -\- n p, (368) (367) becomes a;'"+i(rt + 6x")^ + awp/. x'" (a + 6 x" )p-_i) wt -J- 1 -|- w p The value of the required integral is thus made to de- pend upon that of an integral, in which the exponent of the binomial (a + 5.^"') is diminished b^ unity; the value of this new integral may, by the same formula (369), be made to depend upon that of an integral, in which the exponent of the binomial is still farther di- minished ; and so on until the exponent of the binomial is reduced to a fraction less than unity. Secondly. Instead of (360) let, now, v — {a + b x^^y ; (370) in which s is to be taken of any value, which may be found useful ; whence d,.v=znbs x"-i {a + b x^-^, (371 ) I- (378) In one case the integral of (359) is qp 2;7n+i paP-^ b z"H-"+i p{p-l )aP-^ b^ x™+2»+i — m-\-i^ m + 71 4n~" + l.2.(m-\-2n-\-l) "^ ^* [1 p bx" /^f-j-l ?;i-|-7i-f-i a ^1.2.(m + 2w + l) \ a / ^ J ^ ^ and in the other case • _ r ^ a-^_^L^ ! I- &c. (380) = fe? x'lP+'W+l i I . -; h &C. I <§) 41.] IRRATIONAL FUNCTIONS. 91 Integration of binomial irrational functions. 41. Examples. 1. Reduce the integral of x* {a-{-bx^)^ to depend upon one, in which the exponent of a -\- b z^ is less than unity, and the exponent of its factor is less than 3. Solution. By putting in (369) m =4, 717=3, i? = J, it gives and by putting in (376) m = i, 7i=z3, p = i, it gives SO that, by substitution, f.xHa + bx'r={^^^x^-\--°-.—ya + bx^f 2. Develop the integral of x (a-|" ^ •'^")" according to pow- ers of X. Solution. By putting in (379 and 380) m = 1, 71 =1 3, p = i, 92 INTEGRAL CALCULUS. [b. V. CH. III. Integration of binomial irrational functions. =V(6x^)[f+^3-Tv(-^)^-&c.] 4 3. Reduce the integral of X* (a2 — 3-2^3 ^q depend upon one, in which the exponent of a~ — x^ is less than unity, and that of its factor is less than 2. Ans. {^\ x5 — ^2^\ a2 x^ — f^ a^ x) (a^ — x^)~^ 1 4. Develop the integral of («2 — x^)^ according to powers of X. 2 / a;2 2;* \ Ans. a'2\\-^l. — — -^\. — +&L0.^ 2. 5. Develop the integral of x(l -^x^Y according to powers of X. Ans. z2 (^_|-_2_2.3.__i^a^6_|_&c.) or x> (^ + 2a;-3 4-_i_.x-6+&c.) 42. Problem. To integrate (359) for all real values of 771, n, and p. Solution. First. When m is a negative integer and n a positive integer, the substituting of 7n-\-n for m in (376), <5» 41.] IRRATIONAL FUNCTIONS. 93 Integration of binomial irrational functions. freeing from fractions, dividing by a (m + 1), and transposing, give for the required integral X "»+^ (a+b X") P+^—b (m + l+Jip+?i) f. x^+^ {a+b xy .^^.. a(m+l) ^ '(^^^ ) which formulaj since ni is negative, serves to increase the exponent of the factor of the binomial under the sign of integration, until it becomes positive but less than n. Secondly. When p is negative, the substitution of p -|~ 1 for p in (369), gives by reduction for the value of the requir- ed integral which formula serves to increase the exponent of the binomial under the sign of integration, until it becomes positive hut less than unity. Thirdly. When m and n are fractions and n positive, let the common denominator of m and n be /, and let x = y' (383) whence d,yX=zly^-^ (384) and, by § 19, the intregal of (359) becomes / l7f'^+^-^a + b y^^^Y (385) in which the exponents of?/ are integers, so that it may- be integrated by ^ 40, or the preceding part of this sec- tion. 94 INTEGRAL CALCULUS. [b. V. CH. III. Integration of binomial irrational functions. Fourthly. When n is negative, a simple algebraic reduction gives x^ [a + h x")^ = z'»+"P(6 + az-")^ (386) the integration of which may be efTected by «§) 40 or the preceding part of this section. 43. Examples. i_ 1. Reduce the integral of z — 2 (i_|_ a; 3 )-;i to depend upon one, in which the exponent of the binomial is positive and less than unity, and that of its factor is positive and less than 3. Solution. The substitution of 771 = — 2, ?l=r:3, 2?=: — ^, a=:\, 6=1, in (3S1), gives /2-2 (1 4.2;3)-^ = — 2-1 (l+a:3)^-|-y:2(l + x3)-^, the substitution of 77^ zz: 1 , w z= 3, jP = — 4-, a=i 1, b = l, in (382), gives / X (1 + 23)--^=— ia;2 (1 -j-a;3)f _|_oy:2; (i -f x^)?"- Hence / 2-2 (1 -|- z3)-^___ (a;-l_|_ ^ a; 2) (1 _|_ a^3)f _[_2y:a; (l_j.2;3)l _2 2. Reduce the integral of 2-2(1 -^x^) ^ to depend upon one, in which the exponent of the binomial is positive and less than unity, and that of its factor is positive and less than 3. Ans. I (2 — 2-2) (l+xs)^—/ (l + 23)i § 44.] IRRATIONAL FUNCTIONS. 95 Integration of binomial irrational functions. L2 3. Reduce the integral of {l-\-x^) ^ to depend upon one, in which the exponent of the binomial is positive and less than unity. -^ - ■ I Ans. J X (I +x^) ^ (3 + a;5_2;io) + ty: {I + x^)' 2 4 4. Reduce the integral of x-^ {a -{-bx'^y to depend upon one of the same form, but in which the exponents are integral, except that of the binomial. Solution. In (383) we have, for this case, so that (385) gives / ^ {a-\-hx^Y — 15/ ?/24 (^a + h y^^y. 4 2 5. Reduce the integral of x^ (a -\- h x^Y ^^ depend upon one of the same form, but in which the exponents are integral, except that of the biomial. Ans. I5f. y26 (^a + b i/^^y^ ,3. 3 6. Reduce the integral of x'^ (a -\- b 2; ~2)^ to depend up- on one in which the exponent of x in the binomial is positive. 3 A71S. J. {b -{- a x^)^. 44. Problem. To find the value of the definite inte- gral fl x^ {a + b xy (387) in which c = ^--^ (388) and nij n^ and p are positive. 96 INTEGRAL CALCULUS. [b. V. CH. III. Value of binomial definite integral. Soliti.n. Tiie substitution of a = — - 6 c" (389) reduces (387) to ^''/o ^"^ (^'' — c"Y = {— by f^ x"^ (t" — z")?. (390) First. The term of (3G9) a;»«+i {a-\-h x^'Y — JjP i"^+i (x'* — c^'Y (391 ) is zerOj^wheii z = 0, and when x z=z c. (392) Hence (369) gives for the value of (387) ""''P -.Sl,x-{a-\-bxy-\ (393) m-\-\-\-np and, in the same way, by changing p top — 1,|9 — 2, &c., (309) gives &c. The substitution of each successive value, in the preceding one, gives for the value of (387), if pQ is the greatest integer in p, {^ 7l)P0p{p—\) (p— 2) . . . . ( p— po + 1 ) (m-{-l+np)[m-}-l + n{p—l)]....[m+i-{-n{p—pQ-\-\)] X /o x'" (a + 6 x")2'-'o (396) <5> 45.] IRRATIONAL FUNCTIONS. 97 Value of binomial definite integral. Secondly. In the same way, (376) gives /„% z'-ia + bx-y^- 4^":p^]/o- ^•"-" (°+6^")' (397) and for the final value of (387), if h is the greatest integral number of times, which n is contained in tw, Thirdly. The series (379) gives for the value of (387) 45. Corollary. In the particular case, in which m=:0, 71 = 2, p=: — J, (401) we have n. — — h r.^ . r ■ — a/ — b and by (331) a = — 6 c2, c = V — X' (^^^) /V(^^^ = ~V=6^''-^"'V(^ ('^') whence ,7 v(«+ox2) ^ — 5 ' y' J 1 TT (404) 9 98 INTEGRAL CALCULUS. [b. V. CH. III. Value of binomial definite integral. 46. Corollary. It follows from (404), that if b = —g2, cz= ^ (405) s which c = . g 47. Examples. oV(«— ^2a;2)' — 77 -^r-K7, in Solution. The substitution of 77« zzr 4, W =: 2, p z=z — J, A =r 2 in (399) gives, by (406), y'c %^ a2 S.l /*c 1 _3a2 TT V(«— ^' ^•') ~ ^ * T:^JoV(a-g' x2) - 8^ • ■2"- V(«^ a;2) 1.3.5 TrflS ^"^- 274-6-2- . ., ^ -• V(« —^ ) Solution. Equation (399) gives ^« a;3 2a2 r»a a: JoV(«^ — ^^"~~3-./oV(a2 — x^* <5> 47.] IRRATIONAL FUNCTIONS. 99 Value of binomial definite integral. But by (324) whence / . .. ^ :rr = a» and f ,—r, rr == f a^- p. 2;3 2.4 ^W5. — — - a^. o . 5 5. Find the value of the definite integral J^. \/ {a -\- h x^) where c ■=. s/ — -. o Solution. The substitution of m = 0, n — 2, p = iy in (393) gives by (404) /iC fj /*C 1 TT /7 Q vv -r ; 2*^ \/{a+bx2) ^A^—b ^ ' 6. Find the value of the definite integral /*. \/(a2 — x^), Ans. — - — . 7. Find the value of the definite integral f^. i^s^iofi — x2). Ans, I . -J-. 8. Find the value of the definite integral J^. % V («^ — a^^)- Ans. ^a^. 100 INTEGRAL CALCULUS. [b. V. CH. III. Value of binomial definite integral. 9. Find the value of the definite integral f^. x^\/[a^ — x^). Ans. I . ^ a^. 5 10. Find the value of the definite integral f^. (a^ — x^)^. Ans. -^^ n a^. 11. Find the value of the definite integral f^.x^^a^ — x^)^. Ans. ■g-.fV^^^' 12. Find the value of the definite integral f^,x^{a^ — x^y. Ans. f . -i a"^. 13. Find the value of the definite integral /*. x(a^ — x^)'^. A?is. -f a^. O^. (^>c-T^ 49.] LOGARITHMIC FUNCTIONS. 103 Integration of logarithmic functions. SO that the required integral is m 771+1 p 2 2 ~| -r-r I ('og. x)2 ; — - lo2. x-\-- I. Secondly. When ?w = — 1 (417) gives for the required integral ^(log, x)3. 2. Integrate x'" (log. x)^. Ans. When m differs from — 1, it is a;m+l Lrno. .. 3(log.x)^ 3. 2. log. X 3.2.1 -1 1 L^ ^' ^ ^«+3 ^(^/i+l)2 (m+l)3J W -f- and when m z=. — 1 it is I (log. x)*. 3. Integrate f. x. log. x. -471S. When jT. x differs from x~i, it is and when f.x^n- ^ X it is J (log. x)2. 4. Integrate -^— s — '- when w differs frOm — An.. (}^^^. w -|- 1 (418) 104 INTEGRAL CALCULUS. [b. V. CH. TV, Logarithmic definite integrals. 5. Integrate — — - — — -. " (1— 2;)2 X lOff. z 6. Integrate — ^ . Ans. \os.^ x. ^ X log. X ^ 50. Problem. To find the value of the definite in- tegral n.(-\og.x)i (419) in which n is an integer greater than — 2. Solution. First. In this case, (408-410) give /. x=I, F.x^^x, f.f.xz=x', (420) /. (- log. x.Y = X (- log. xf + ^f. (- log. x)^-' (421) in which when x =1 0, or = ] , (422) the first term of the second member vanishes as in example 2 of B. II. § 109, so that the required integral becomes In-i-^og-^-'- (423) By this process, then, the exponent of ( — log. x) is dimin- ished by unity ; and a continued repetition of it gives for the value of (419) i(i-')(i-"") (l-''+^)f:-i-^o,..Mm n in which h is an integer not greater than - -f- 1. «5> 50.] LOGARITHMIC FUNCTIONS. 105 Logarithmic definite integrals. Secondly. When n is even, let h — ^n (425) and (424) gives /i.(-Iog.2;)'^=1.2.3 h. (426) Thirdly. When n is odd, and positive, let h = ln + i, (427) and (424) gives /■.(-log.^)'-*= (A_J)(A_|)...|.J/..(_Iog..)-i. (428) Fourthly. When n = — 1 let -^ = /^(-log.2:)-i (429) The substitution of xz=zay\ (430) in which a is supposed less than unity, so that (-— log. a) is positive, gives ^ — log.a:=: — y^log.a^ d^yXz^^ya^^.hg. a; (431) and when x = 0, y ziz cc , xzzzl, 7/ = 0; (432) K — — 2 /'^ a^\— log.a)*, (433) and K{—\og.a)~iz=i—2f\a^\ (434) But, by taking the integrals relatively to a, we have r\{-]og.a)-i=K, (435) •/ 106 INTEGRAL CALCULUS. [b. V. CH. IV. Logarithmic definite integrals. r\a^^ = -^-a^^-^'=-JL_, (436) Jo J/^+1 3/^ + 1 and, therefore, the integral of (434) with reference to a, is by (304) X2__2 Z**'.— 1— = — 2(tan.[-i]0— tan.[-i]oo) = —2{0 — irr) = ^. (437) or K=n.(^\og.x)-i=f\-^- = ^/rr. (438) 51. Corollary, The substitution of (438) in (428) gives /M>og.ir*=i^2:^'±Dv.. (439) 52. Corollary. The substitution of X = 3/"*+^ or log. X z=z (w -f- 1) log. y, (440) whence d,y. 2; r= (m + 1) 3/"* (441) in (426 and 439) gives ; by dividing, in one case, by (w+1 )^+^ ; and, in the other, by {m -\- 1)^2^ sir (log. ^Y = - ^'I'iyXr ■■ (442) /.,.(,og..)-J = Li:^(^V^. (443) 2^(m + l)*+^ 53. Problem. To integrate F.{ef'-).{f.xYd,.f.x, (444) <§» 55.] LOGARITHMIC FUNCTIONS. 107 Exponential integrals. Solution. Let 1/ z=. e/-% (445) whence d^y. x m (e-^-^ d^.f. x)-^ = (y d^.f. x)-^ (446) Jog. y —f-'^^ (447) and the integral of (444) is, by § 19, f-i'^oS-vY-^, (448) which may be found by § 48. 54. Corollary, When w = (449) (448) gives f.F.(ef-).d,.f.z^f.^. (450) 55. Examples. 1. Integrate e'^^\/(l — e^'^''). In this case if f.x~ ax, dj. xz=La F.y=: ^y^[\^y2) (450) gives /. e«W(l — «^"") = i/V(l — ^2) = i^e«*\/(l— e'''^) + Jsin.[-i]e''^ 2. Integrate e«^. Ans. ie''\ (451) 3. Integrate xe'^^ Ans. ( ^ j e *» *. 108 INTEGRAL CALCULUS. [b. V. CH. IV. Potential integrals. 4. Integrate a*. Ans. log. a 6. Integrate Sin. {kx-\-a). Ans. ^ Cos. {kz-\-a). (452) 6. Integrate Cos. (kx-\- a). Ans. lSin.{kx-\-a). (453) 56. Problem. To integrate f. (Sin. k X, Cos. k x). (454) Solution. Let y := Sin. A; a;, or kx =z Sin.[-i]y ; (455) and by (127 and 143) Co3.kx=V(l+y2), kd,,,.x = ^-^^^^', (456) whence the integral of (454) is, by § 19, /.J/.[y, V(l + y=)].(l + y^ri (457) which can be found by § 31. 57. Examples. 1. Integrate Sin.'" kx . Cos. kx. Solution. In this case, (457) becomes r y^ — ^ — =z ■ — - . (458) ^ T ^7^-7 >. Cos.'^+iZrz 2. Integrate Cos.'^Arx. Sin. A; a;. ^ns. - — , ,, , . ^ (7^4- 1) A: 3. Integrate Tang, k x. Ans. ^ log. Cos. k x. (459) «5> 57.] LOGARITHMIC FUNCTIONS. 109 Potential integrals. 4. Integrate Cotan.kx, Ans. -\ log. Sin. yt z. (4G0) 5. Integrate Sec.kx. Ans. ^ tan. M] Sin. A: a;. (461) 6. Integrate Cosec k x. Ans. .^ lo2. i ~ ^ZT I ^ '* ^ \Cos.kx+l)' or, by (140), ^ log. Tan. ^kx. (462) 7. Integrate e'^^Sin. ^z. (463) Solution. Since by (121 and 122) Sin. ^•2; = J (e^* — e-^'^), Cos. ^z := J (e^x-j-e— Ax) . we have e '^ ^ Sin. ^ a: = J (e('^+^)^ — e^'^-^')^), f.e'^^Sin,kxz=^ — ^ 2{a+k) 2{a—k) \ ^{a2—k2) } ^^ /a Sin. kx — k Cos. k x\ = ' ( ,i^^p )• (464) 8. Integrate e"* Cos. Arz. An^ .aJa^os.kx—kSm.kx\ Ans. e y a2-k2 ^ )• (46o) 9. Integrate c"* Sin. a a;. Ans, i^e'^"^ — J z. (466) 10. Integrate e '' "^ Cos. a z. Ans. ^e'^'"' -\-^x. (467) 11. Integrate e^-^'^^-^'^Cos.kx. Ans. ^€«sin. Az^ (468) 12. Integrate c«^o^-*^ Sin. ^ x. ^ns. ^. e "^ Cos. A: a;^ (469) 10 110 INTEGRAL CALCULUS. [b. V. CH. IV. Potential integrals. 68. Corollary. The differential coefficients of (451, 452, 453, 458, 4C4, 465) with respect to a, A-, or m, give /x^e'^' =rf^.,.ie'^*; (472) / xCos. {kx + a) z=z d,,,. i Cos. (kx + a) (473) X 1 z:^jS'm.(kx-\-a) — — Cos. (A; x + «) f.x^Sin.(kx + a)=idl,.lCos.(kx-{-a) (474) = ('^ + i)^os.(kx+a)^^^Sin.{kx + a), / x2«+iCos.(A;x + a) = dl",+\ i Cos. (k x + a), (475) /. x^" Sin. (A:x + a) z= d^J. i Cos. (A:x + a), (476) / x2«+i Sin.(Z: X + a) = 4.- ^L"+^ i Cos. (A;x + a) z=c/?;,+i.i Sin.(A;x + «), (477) /.x^^Cos. (A;x+a) = c?.M Sin. (kx + a), (478) Sin.™+1 A:x f. Sin.*" A; x . Cos. k x . log. Sin. kx=: c?,.^. , (479) (1 \ Sin.^'+i k X log. Sin. kx j— - I 7—{VT y § 59.] LOGARITHMIC FUNCTIONS. HI Potential integrals. f.Cos.-H.Sin.kx.log.Cos.kx = d„„.^''^ (480) = I loff. Cos. Jcx — I Cos."'+^ k X — Ix— ^"^ \.- aSm.kx-kC os.kx . c-Sin.Arz ,,^^^ /x;e-Sin.^.==^.,.(e^^^^^i!^=|^^ (483) 59. Corollary. Equations (121 and 123) give x" e-^* Sin. « a; = J z" e^ax^. j ^n^ ^^g^^ 2" c'^^Cos. a a; =z J a;" c'-^^ + ^z'' ; (435) so that by (472) (486) / x" c- Sin. a a: r= ^ c/^.^. i- c^- - _L^ a;«+i 2a 2(?i-f-l) 2n+i —2^^ 2(w+l)^ ' /x"e-Cos.ax=ic/,.2..l62- + ^^-l_ (487) 2n+i«c.«.2^« +2(n+l)'^ • 112 INTEGRAL CALCULUS. [b. V. CH. IV. Exponential definite integrals. 60. Corollary. When n— 1 (486 and 4S7) become / 2 c«^ Sin. a z = J J,., . ji e2 '^^ — ^ x2 (488) / X e«» Cos. a I = ^ rf,. . i; c2" + J 12 (489) — A »> 1 /.'-ia* 4 61. Problem. To find the value of the definite inte- gral J».3.ng_ax2^ (490) in which a is positive and n is zero or a positive in- teger. Solution. Let 6-^^ = 2/, c"'=:^, (491) x2 — _ log. 3/ = log. i , (492) 2xd,,,.x = -i, d^^^.x = -^; (493) so that when x r= 0, ?/ = 1, (494) X = OD, y = 0. (495) The value of (490) is, by § 19, 4/^0og.^)^r-'. (496) <§, 64.] LOGARITHMIC FUNCTIONS. 113 Exponential definite integrals. Hence, by (442 and 443), when n is odd, . ^ -a^^ 1.2.3 (^n-i)_ and when n is even TC /.^ „ —az^ 1.3.5 in-X-V) ... I Ac\Q\ /; . x" e = j^^ — ^ — '- V— . (498) 2 (2 a)*" "^ 62. Corollary, When w =: 0, a := 1, (498) gives /«.e-"'=iV-. (499) 63. Corollary. By reversing the sign of x (497 - 499) give, when w is odd, ax2_ 1.2.^...(^w — ^) 2a^ when » is even y_o..,.e = —^^ ^, (500) 2(2a)^" " /_o„.e-^ = ^V-. (502) 64. Corollary. The sums of (497 and 500), of (498 and 501), of (499 and 502), give, when n is odd, /_"«.x"e-^^'=0, (503) 10* 114 INTEGRAL CALCULUS. [b. V. CH. IV. Exponential definite integrals. when n is even, J-co-^ e = — V— , (504) (2«)^" " /-"oo.c-^ =-s/n, (505) G5. Corollary. The substitution of . + ^„ (506) for X in (504) gives by § 18, when n :rz 0, /f„. e-("'+*^+ 4-a)^ ^iL ; (507) which, multiplied by e 4a ^ gives /_"a,.e-(^^' + *^ + ^)=e47 ' ^IL, (508) 66. Corollary. The differential coefficients of (50S), with reference to a and 6, are 67. Examples. 1. Find the value of the definite integral /_"„ . a: e- Ca ^2 + c) gin. A; x. MS. ^^e <5> 67.] LOGARITHMIC FUNCTIONS, 115 Potential definite integrals. 2. Find the value of the definite integral /_"oo . X c- («^^ + '=) Cos. k X. Ans. 0. 3. Find the value of the definite integral /-"oo • {m »2 + n) e- (« ^'^ + ^) Sin. k x, Ans. 0. 4. Find the value of the definite integral /_*«, . (m x2 4- n) e- ^^ *^ + <=) Cos. k x. Ans. (^ + -^+Ae^«~Vf V 4a2~2a ' / « 5. Find the value of the definite integral r a> p — a X J 0**^ i in which a is positive. Ans. ^. (511) 6. Find the value of the definite integral in which a is positivet - 1 . 2 . . . . 7t ^'''' an + i -' (512) 116 INTEGRAL CALCULUS. [b. V. CH. V. Trigonometric integrals. CHAPTER V. INTEGRATION OF CIRCULAR FUNCTIONS. 68. Problem. To integrate f. (sin. k X, COS. k x). (513) Solution, The substitution of xzzryV— ^ <^c.y.a: = \/— 1, (514) gives by (121 and 122) ^v[i.kxz=z\/ — l.Sin. A;y, (515) COS. kx z=. Cos. k y ; (516) and the integral of (513) is y:^_-l./.(v'— 1. Sin.Ajy, Qos.ky), (517) which may be found by § ^Q. 69. Examples. 1. Integrate sin."* kx. cos. kx. Solution. In this case (517) becomes (_ 1 )? ('"+^^ ^'m.^ky. Cos. k y ; whence by (458, 121, 122, and 514) ^ 69.] CIRCULAR FUNCTIONS. 117 Trigonometric integrals. ^ ^ ^ {m+l)k sin.'" + i kx (m+ l)k' 2. Integrate cos.'" k x . sin. k x. (518) COS."* + 1 A: 2: ^"^- 7 r-T-T-- (519) (wi -f 1) A; ^ ' 3. Integrate sin. (kx -\- a). Ans. — i COS. (A; I + a). (520) 4. Integrate cos. (k x -\- a). Ans. ^ sin. (kx -{- a). (521) 5. Integrate tang. A: 2. Ans. — -^ log. cos. A: a;. (522) 6. Integrate cot. A- z. Ans. -^ log. sin. A: 2 . (523) 7. Integrate sec. k x. Ans, ^ log. . ' -. (524) 1. — — Sill* /l 3/ 8. Integrate cosec. k x. Ans. ^ log. tang. J k x. (525) 9. Integrate e°*sin. kx. ^526) Solution. The substitution of (514) in (526), gives by (464), / e'^^sin. A:x= — /6«J/^^-i Sin. ky as/ — 1 Sin. k y — k Cos. k y — _(a2_j_y^-2) _ ^^ , flsin. A:x~A:cos.A:x ^2 _|_ ^2 118 INTEGRAL CALCULUS. [b. V. CH. V. Trigonometric definite integrals 10. Integrate e*** cos. kx. a COS. k X -\- k ^'\n, h X ,rc\a\ Ans. e- ^^ . (528) 11. Integrate c«s'"-'^^ COS. /ex. ^?is. J* c**"'"-*^. (529) 12. Integrate e<'^^^-^^ sm.kx. Ans. —^,e<'^^^-^\ (530) 70. Corollary. The differential coefficients of (518-521, 627, 528), with reference to m, k, and a, give f. sin.'" k X . cos. k x . log. sin. k x /i • 7 1 \sin.»»+iA;x = i log. sin. A; X ttIt TTTZ' (^'^^/ f, COS.'" kx, sin. k x. log. cos. k x = I r-T — log- COS. kx\ r-zrr-r (532) \7?i + l ^ /(m + l)A:' ^ ' f.xcos. {kx-]-a)z=i— dc.k'^ COS. {kx -\- a) (533) 1 a^ = — COS. (^ 2^ + «) + T sin. (A; X + a), (__l)»y:x2'»sin. (ytx + a)=-~ .».,v when m is even, it is (2 w + J) n, (555) 2. Find the value of the definite integral /(2n + i|)7r . ,.^-. . sm. % . cos. X. (556) Ans. 1. 3. Find the value of (548), when m and p are both even. 1.3.5 (m-l)X1.3.5...(p-l) '• 2.4.6 (ro + i)) (2«+J)=^-(557) 4. Find the value of (548), when m is even and p odd. 1.3.5...(« -l)X2.4.6....(j>-l) '• 17375 ('«+^ ' 2.4.6. ..(j) — 1) °' (»»+l)(», + 3)....(»t+^)- (^^^) 5. Find the value of (548) when m is odd, and p even. 2.4 6....(m-l)Xl.3.5....(p-l) 1.3.5 (»»+?) ' 2.4.6.....(m — 1) "■^ (P + l)(i' + 3)..('«+i'r ^ ' 6, Find the value of (548), when m andp are both odd. An. i ^-4-6^-(»-l)X2.4.6. (y 1) 2.4.6 (m +P) or the same with the second answers in (558 and 559), 11 122 INTEGRAL CALCULUS. [b. V. CH. V. Trigonometric definite integrals. 75. Problefu. To find the value of the definite inte- gral . sin.'" a: cos/ x, in which m, n, and p are positive integers. Solution. The reduction may be made in this case, precise- ly as in ^ 73, it being observed that when x=zO, or =2 Tin, y = 0. (561) By this means, the integral, when either m or p is odd, is zero i but, when m and p are both even, it is 135 (,»-!) X1.35...^,,l) ^^^ 2.4.0 {"^-j-p) 76. Examples. 1. Find the values of the definite integrals «/ 2 71 TV , sm.'" X /. cos/" X when m is even. 1.3.5....(m— 1) ^ ,.^„, ^^^S' ^ , ^ — 2nTv. (563) 2 . 4 . 6 . . . . wi ' 4. Find the value of the definite integral .9 . sm. n X . cos. k a:, when h and Jc are integers. A7is. 0. (564) ^ 76.] CIRCULAR FUNCTIONS. 123 Trigonometric definite integrals. 3. Find the values of the definite integrals / . COS. h X . COS. k Xy /2w7r . , . , . Sin. h X, sm. A;x, v.'hen h and k are integers. Ans. It is zero, unless h 3= k, in which case, it is n ^r, (565) 124 INTEGRAL CALCULUS. [b. V. CH. VI. Length of the arc of a curve. CHAPTER VI. RECTIFICATION OF CURVES. 77. Problem. To find the length of an arc of a given curve. Solution. If s denotes the required arc, its length is readily found by ascertaining the value of its differential coefficient, and integrating it. Thus if we adopt the notation recently introduced by some of the most eminent mathematicians, and denote the differen- tial coefficient by the capital letter D, and denote by a small letter annexed to D or /, the corresponding independent vari- able, we have by (570-582 of vol. 1), s =fDs =/. V [I + (^.3/^) =f:c ' sec. T = /, . cosec. r =frV [r' + [D,, rf] =fW[^ + r' (A ^ =zf^ . sec. £ =zf'(p.i- cosec. s. (566) 76. Corollary. The arbitrary constant, which is to be added to complete each of these integrals, corresponds to the indeterminateness of the point at which the measured arc may commence. The condition, by which this point may be determined, will be sufficient to determine the value of the arbitrary constant; or to eliminate it and reduce the result to the form of a definite integral. Thus, if the length of the § 78.] RECTIFICATION OF CURVES. 125 Arc of hyperbola and cycloid. arc is required, which extends from the value of Xq to that of X J, it is evidently represented by the definite integral 5 =/^^ D s. (567) 78. Examples. 1. Find the length of the arc of the curve of which the equation is Solution. In this case, we have D s = J (g^ + e-^) s = J (e^ — e-^), in which the length of the curve vanishes with x = 0. 2. Find the length of the arc of the parabola whose equa- tion is 1/^ z=2 p Xi counted from the vertex. (568) 3. Find the length of the cycloid from equations (130, 13J, of vol. 1), the arc being supposed to commence with x. Ans. 4 i? (1 — cos. i&)=S R sin.^ J &, (569) and the whole length of a branch is 8 R, corresponding to ^ = 2 ^. (570) 11* 126 INTEGRAL CALCULUS. [b. V. CH. VL Arc of hyperbolic and logarithmic spiral. Ans. r sec. « = 4. Find the length of the liyperbolic spiral, the arc being supposed to commence with (p r= (Pq. (5^1) .„.«[v(..L)-v(.+,y+%(rf;^3)]- 5. Find the length of the logarithmic spiral, the arc being supposed to commence with r. The elliptic and hyperbolic arcs possess some peculiar prop- erties, which deserve particular investigation. 79. Theorem. The two tangents^ which are drawn from a given point to a given ellipse or hyperbola, make equal angles with the two lines which are drawn from the same point to the tivo foci. Thus the two tangenti> P T and P T' (figs. 1,2,3), make equal angles with the lines Pi^ and P F' drawn to the foci; that is, the angles F P T and F' P T' are equal. Proof. Each of the two tangents P T and P T' is, by ex- amples 2 and 3 of § 131 of vol. 1, equally inclined to the lines drawn from the foci F T and F'T.oxF'T and F T', so that the angles F Tt = F' TP , and F T' P — F' T' t'. If then the triangles FTP and F' T ' P ^xe turned over, around the sides T P and T' P, which remain stationary, so as to fall into the positions TP S and T' P S\ the points S and >S'' will be in the lines T F' and T' F produced if neces- sary. The triangles P SF' and PS'F are, then, equal; for the sides P S — P F, P F' — PS' ; <§> 81.] RECTIFICATION OF CURVES. 127 Elliptic and hyperbolic arcs. and the side i^' >S^ ^ i^>S^', because each of these two lines is equal to the transverse axis, since in (fig. 1) each is the sum of the two lines F T and F' T, or of the two F T' and F'T'; while in (figs. 2 and 3) each is the difference of the same two lines. The angles S P F' and F P S' are consequently equal. If the angle FP F' is subtracted from each of these angles (fig. 1), or added to each of them (fig. 2), or diminished by each of them (fig. 3) ; the resulting angles SP F and S' P F' (figs. 1 and 3), or the excess of 360° over ,SrPF and S' P F' (fig. 2) are equal. Hence F P T and F' P T', which ar^ the halves of S P F and >S" P F', are equal. 80, Corollary. If an ellipse (fig. 1) or an hyperbola (figs. 2, and 3) be drawn with the points F and F' for foci, and passing through the point P, the tangent to this new curve at the point P will be equally inclined to the two lines P F and PF'; and, therefore, it will also be equally inclined to the two tangents TP and T P. 81. Theorem. If froin any 'point of the ellipse PP' (fig. 1), or of the hyperbola P P' (fig. 2), ivhich has the points F P' for its foci^ tangents are drawn to the ellipse T T' or hyperbola T T' lohich has the same foci, the sum of the tangents P T and P T' exceeds the in- cluded arc T T' by a constant quantity ; that is, by a quantity which is the same, from whatever point of the first ellipse or hyperbola the tangents be drawn. Proof. Let tangents p t" , p t' be drawn from a second point p infinitely near P. The tangent yt" exceeds P T hj the projection of Pp upon P T diminished by the arc Tt", or pt" = P T-\- P P' cos. TP S— Ti", 128 INTEGRAL CALCULUS. [b. V. CH. VI. Elliptic and hyperbolic arcs. In the same way, 2)i'=:PT' — P P' COS. T'PS'+T' t', z= P T' — P P' COS. TPS-}- T' t'-, whence jpt"+2't' = PT-\-P T' + (T' t' — Tt"). But t"t' = T T' -{-{T'f — Tt"); and, therefore, (j)t" + p t') — t" t' = {P T + P T') — T T'. The excess of the sum of the tangents over the included arc does not, then, increase by moving the point P a small distance upon the curve, in which it is situated ; and conse- quently this excess must be a constant quantity. 82. Corollary. Had an hyperbola P Q, (fig. 1), or an ellipse P Q (fig. 2), been drawn, with the foci jP and P', it might easily have been shown in the same way, that the excess of the difference of the tangents P T and P T' over the dif- ference of the arcs Q T and Q T' was constant. But as the point P, in moving along the curve P Q, approaches Q, the tangents and arcs decrease, and finally vanish when P coin- cides with Q. At the point Q, therefore, the excess of the difference of the tangents over the difference of the arcs is nothing, and therefore this excess is nothing for every point of the curve P Q. Hence, if from any 'point P of the hyperbola P Q (fig. l)j or of the ellipse P Q (fig. 2), which has the points F a7id F' for its foci , tangents are drawn to the ellipse T T' (fig. 1), or to the hyperbola T T' (fig. 2)j which has the same foci^ the difference of the tangents P T and P T' is equal to the difference of the arcs q T afid q T'. ^84.] RECTIFICATION OF CURVES. 129 Elliptic arc. 83. Corollary. If the excess of the sum of the tangents P r and P y over the arc TT' is denoted by 2 E, the two preceding theorems give PT+PT=QT-\-QT' + 2E F T' — P T = Q T— Q T; whence PT' = QT' + E PT= QT-\- E. (573) 84. Corollary. Upon the transverse axis A A' (fig. 4) of the ellipse, describe the semicircumference ^li Z.'^', dravy^ the ordinates L T 31 and L' T' M', and join O L, O L', O being the common centre of the ellipse and circle. Let, if O 5 is the semiconjugate axis, if z:^ LOB, tp' z= L' OB, A=OA , B=OB, x = OM , !/ = MT , z = ML , sz=:B T %' = OM', y> = M' T\ %' — M' L', s' =zB T', > (574) 1 ^' we have, by section 163 of vol. 1, and by the triangles LOB, LOB, x-=zA sin. 85.] RECTIFICATION OF CURVES. 131 Elliptic integral of the second order. ^^== — T^ = — i^ — Tr> (-585) COS. LOR COS. h {'P — f/ ) ON= ORcos.RON='^''"-}y +#, (586) COS. ^ {'p '. (594) a, — -j- 1 But the condition that the given ellipse has the same foci with the curve in which P is situated, gives A^—B^ =zA'^^B'^=z——^, (595) a Hl-'hHl-')' (596) ,.^1 ^^-, Ml-«) _ a-h A^~ a (1—6) "~«(1— 6) But, by (593), /«+6— 1\2 4 a (1—6) ,^^^, sin. 2 4 (a — J) , _ {b-a+lY ^ _ . b-a+l . ,503. ^^0 - (a_6+l)2 ' ^^0 -± ^ZZj+r- (^^^) which, substituted in (594), gives COS. ({q = COS. q; COS. (p' i sin. y) sin. ^' ^ . g)^. (599) Arid this equation of condition is the same^ with the con- dition that the poiiit P of fig. 1 is upon an ellipse or hyperbola having the same foci with the given ellipse^ the upper sign of (599) corresponding to the ellipse^ and the lower to the hyperbola. 86. Corollary, If a spherical triangle (fig. 5) be drawn, of which the sides are 9, H>' and q , and the opposite angles ^» ^'i ^0 » ^^® ^^^^ ^y (3^1) °^ Spherical Trigonometry, § 87.] RECTIFICATION OF CURVES. 133 Elliptic integral of the second order. COS. (fin COS. (p COS. m' COS. flo = '-^—. .-^-- '— ; sin. (p sin. (p' (600) whence by (599), COS. 5o = ±: ^ To . (601) Hence C0S.2 ^Q rz: 1 — e^ sin.2 (p^ , (602) esin. (Pq =z \/ (1 — cos.^ &^) =z sin. dy ; (603) and therefore sin. &Q sin. 6 sin, 6' r604^ sin. (pQ sin. (p sin. (p' This equation gives Jif=zj^{\—e^ sin.2 9')=\/(l— sin.2 ^)==Fcos. 5, (605) ^^'^^(l— e2 sin.2 ip')— ^(l— sin.2 &')z=z cos. ^'. (606) The upper signs in (601) and (605) correspond to f the case in which the curve is an ellipse, and the lower to the case in which it is an hyperbola. The signs in (605 and 606) are derived from the consid- r ^""') eration that when y' is zero, (p is equal to (p^ ; but when

' ; (613) whence _l^=V(l+tan.= V')=V(l+5-ta,i.= .) irr\/(l+tan.2 (f — g2 tan.2 (/))=\/(sec.2y — gS tan. 2 ^ =::sec. 9' V (1— e^ sin.2 T) z= sec. 9 ^ . <^, (614) z= ^ sec. (p cos. a. In the same way, 1 J tp' cos. &' (615) COS. '^j' cos. y' cos. 9' which, substituted in (611 and 612), give t =A tan. J (f — 9') ^ g' — =F ^ tan. J (^ — 9)') cos. q (616) <'=:^ tan. I ( 94.] RECTIFICATION OF CURVES. 137 Elliptic integral of the second order. and because the excess of the difference of the tangents over the difference of the arcs counted from Q is constant, t' — t-{-S'—2s^-\-s'=t'o — to + So^2s^, s+s' — s^z=to — t, — t' + t. (635) Hence, by (627 and 628), E (p -\- E cp' — E Vo = e^ sin. (p sin. (p' sin. % , (636) in which (p, sin. g'' sin. y. (643) But since the differentiation of e sin. If =: sin. & gives ^ ^ ^ £Cos^ ^ ecos^ ^ COS. 5 J . (p ^ ' we have i^, ^ f^ ai a — T^ ^^e E'. 6 =/ ^'. Q ■=. t . COS. f/) We have also, by (581), COS.2 q) r= 1 — sin.2 y = 1 ^ + ^ (^ '/')2, (647) which, substituted in (646), gives i — e- /*(? 1 .1 Z*^ 1 __e2 /»,^ 1 1 § 96.] RECTIFICATION OF CURVES. 139 Elliptic integral of first order. Similar equations may be found for E' 6' and E' 6^, all of which, substituted in ((543), give 1—e^r pp J^ , /»r J , /•To J_"j e Ly ^'^ */ ^'f J ^'^J -\ — {E. (p-\-E . (f' — E . yj^ic sin. 9 sin. 9^' sin. " ; (649) whence, by (636), /"^ J_ + /"" _L _ /•"» J_ = 0. (650) 95. Corollary. The integral is the elliptic integral of the first kind, and is denoted by P . (p, that is, Hence, by (650), F . ./> + P. ,/ — F,n = 0, (653) where 9, V' and 9'o are subjected to the same conditions as in (636). 96. Corollary. In the same way in which y is connected with (f' by means of the construction of fig. 1, in which P is upon the ellipse, giving by {Qo^ and 631) the equation F. T =. F. "' + F.(po, &c ; (655) 140 INTEGRAL CALCULUS. [b. V. CH. VL Elliptic integral of first order. whence F.

i/; . <35 sin. V sin.2 go sin. go 1 ^(e.gp) COS." ^ cos. V^ sin. 1 1 1 (666) ^(1 — cos.^r) \/(l-e"sin."^i/;) ^(e.V) J o^i^'f) J ^ ^(e.g') V^ -^(e-V') t/ j{c.yj) J j{e.yj) = F,—F{e.y^); (667) SO that Fcp and -Fv^ are two functions whose sum is the function Fi which is called the complete integral^ and the two functions are called complementary with regard to each other, as well as the angles w andV', upon which they depend. 142 INTEGRAL CALCULUS. [b. V. OH. VI. Transformation of elliptic integrals. 99. Corollary. The three angles V^, v^' and ^q^ which correspond to p = ^7v-.v, (676) The equation (675) gives by putting '" = TT^' (677) ^." cp=^ {e", cp), (678)' F':;^.^= ^^- ^ — , (682) '^ cos. (2 X — V) -|- e COS. V^ ' ^ ' i>;f.V_ 2 2{e+\)j".x ^ ip COS. (2 yir — V^)-f-c COS. v^ l-J-e'^-(-2e cos. 2 x "■ (e + ir(y'.>rr ~(e+l)/'.;^' (683) J J^ J X Dx '^P __ J ■^J // 2 2 p" ( . M^// -^-Jr. ^"- "^ = 4- ^"- ^- (6B4) (e+l) ^"./ c4- 1 a/ « 101. Corollary. If ;r, z', -^'q correspond to V^, V', % , we have the equation i^."z + i^.";?' — J^.";ro = 0, , (685) with the condition that COS. xq = cos. X COS. X' — sin. x sin. / ^" ;ko . (686) 102. Corollary. In the same way in which F' x is ob- tained from F^^ another function F"x^ might be obtained from F' Xf and so on, until a series was obtained in which the values of e, e", e"', &,c. form a series of eccentricities, in which each differs less and less from unity. It may be shown that e" differs from unity less than e does, for (677) gives (689) 1-^ ~(i+v^)'^(i+^)~(i+v^)-^(iW '^' 144 INTEGRAL CALCULUS. [b. V. CH. VI. Multiplication of elliptic integrals. in which the factor of 1 — e is evidently less than unity, and decreases rapidly with the decrease of 1 — e. The value of F. (p may therefore be made to depend upon a value of F= {e„. cp„), in which c„ differs from unity by as small a quantity as we please, and we have (690) 103. Corollary. In the same way F (ecp), by reversing the above process, may be made to depend upon the value of F (cn (fn)) ^^ which e„ is as small as we please. In this case e" may be found from e by reversing the accents in (677) and solving the equation with regard to c", which gives 104. Corollary. If we put in (677) e — tan.2 ^ ?, (692) we have e" = sin. /5. (693) 105. Corollary. We have by (681, 680, 683), COS. 2 X = (694) J -^ — e COS. yj (I c2)cos. ■ ^ 'J xp — e cos. .^' J rl>-\- e COS. Xfj c-f-cos.2;r= rj-^^ ^— — ^^=r(^V^+ccos. v)cos.'^ (695) J" x^ 1+6 <§> 106.] CIRCULAR FUNCTIONS. 145 Reduction of elliptic integrals. /V^ ( J yp-\-e COS. v^)2_ /» ^' 2(^T/;) ^4-^ e cos. ^p J yj-( l-e^) /^ /I -^ /•V 1 — e Z*'/^ « COS. 1// T+c ""^ 2"^ "^^ 14-e = ^-ni-e)F^+'-f^, (696) which may serve to deduce the value o^ E {e . (f) from those of E.(e„cp„), in which c„ is very small, or differs but little from unity. 106. Corollary. Potential functions may be applied to the hyperbola very nearly in the same way in which circular func- tions have been applied to the ellipse. Thus if we put A Cos. (p = X, B Sin. (p = y, (697) X and y are the coordinates of the hyperbola, of which the equation is (i)-(fy->- («^«) The length of the hyperbolic arc is, by putting ^ = v(l+2), (699) s=Bfs/ (l + c^ Sin.s^)). If we let ^ (c (t) = \/ (1 + e^ Sin.'-J q) (700) ar {ecf)= /'^^(eT), (701) »/ we have s ^= B ;j {c(f). ("62) 107. Corollary. The condition that the point of contact in (fig. 2) is upon another hyperbola which has the same foci with 13 146 INTEGRAL CALCULUS. [b. V. CH. VI. Hyperbolic integrals. the given hyperbola, is expressed algebraically by the equa- tion Cos. Cos. cp' — Sin. cp Sin. (f'^ r ^o t (703) and corresponds to the equation 3" (f'+ 3" 111.] RECTIFICATION OF CURVES. 147 Length of hyperbolic arch. we have sin. ^ =: , cot. ^ =z Sin. .(tan. CO ^'(v); (719) which, substituted in (718), gives s = - F'. oi — B e E'o^-\-Be tan. w j' « = V(^^ + ^^)r^^^ — ^'''^ + tan. CO ./'wl. (720) <§) 113.] RECTIFICATION OF CURVES. 149 Arc of curve of double curvature. 112. Examples.* 1. Prove that the tangent let fall from the centre of the hyperbola upon the tangent is-in the notation of § 111. Be tan. to ^'w — ^(^2_|_^2) tan. « ^'<« (721) 2. Prove that if e" and x are so taken that sin. {^x — f") = e' sin. % (723) the length of the arc of the hyperbola is 5=rV(^'+^')[JS''"-2(l+e')^">^+2e'sin./+tan.c. /J. (724) 113. Definition. A curve of double curvature is a curve all the parts of which do not lie in the same plane. 114. Problem. To find the length of an arc of a curve of double curvature. Solution. Since the length of an infinitesimal arc of the curve is equal to the distance apart of the two infinitely near points x,y,z and x-\-dx, y-\-dy, z-\-dz\ or, algebrai- cally, d s —s/ {dx^ -\-dy^ -\- dz^) ', (725) the length of an arc is s =f^(Dx2-{-Dy2+Dz2), (726) *The examples will not hereafter be strictly confined to the subject of the chapter, but will extend to any exercises suggested by the in- vestigations in the chapter. 13* 150 INTEGRAL CALCULUS. [b. V. CH. VI. Helix. 115. Examples. 1. To find the lencrth of an arc of the helix. Solution. The helix is a curve formed by a string wrapped round a cylinder in such a way as to make a constant angle with the side of the cylinder. Hence, if the plane of xt/ is that of the base of the cylinder, if the centre of the base is the oricrin of coordinates, and if Q z= radius of the base of the cylinder "^ ^ (p =: the angle which the projection of radius vec- | tor^on the plane of a?y makes with the ! fjcy^x axis of X, a :=. the angle which the string makes with the side of the cylinder, the equations of the helix are x^=:q cos. 9', y = ? sin. qp, 2 = ^ 115.] RECTIFICATION OF CURVES. ' 151 Rhumb line. Ans. With the notation of the preceding example, the length of the arc is s=z sec «/, . V [l+(^. 0^] = s'sec. «; (730) where s' = the arc of the meridional section, corresponding to the required arc s. The angle ^ may be substituted for z in (730) by means of the equation /Z>. s' — - — • .("3i) 3. To find the length of the arc in Example 2, when the solid is a right cone. Ans. If (? =1 the angle which the side of the cone makes with the axis, Zq = the value of z at the beginning of the arc s =: (z — Zq) sec. « sec. ^ if sin. ^ cot. a = sec. a sec. ^ . e. ("^32) 4. To find the length of the arc in Example 2, when the solid is a sphere j that is, to find the length of an arc of a rhumb line. Ans. If J? = the radius of the sphere "^ 6 = the inclination of the radius vector I to the axis of z, r C'^S) 6q = ink value of ^ at the beginning of the i arc, the length of the arc is s = R{& — &,)sec. a (734) and (f may be substituted for ^ by means of the equation (p = log. tan. J ^ — log. tan. J 6^ . (735) 152 INTEGRAL CALCULUS. [b. V. CH. VI. Shortest line. 5. To find the length of the arc in Example 2, when the solid is an ellipsoid of revolution. ^ Ans. I^ A = the semi-transverse axis, ^ e = the excentricity, \ (736) 6 corresponds to

s usually depends upon many variables, t, X, 1/, &LC., and their differential coefficients, in such a way that, if t is taken for the independent variable, any change in the functions by which a-, y, &,c. depend upon t, gives the equations ds,=Ds,.dt,, Ss^z=:D So . ^t^, (743) d Dsz=X3x+YSi/+&Lc. -\-X'S Dx-\~Y'3Dij-{-&.c. +X" i 2>2 a;_|_ Y" $ Z>2 3,-|-&,c. ; (744) 154 INTEGRAL CALCULUS. [b. V. CH. VI, Method of Variations. which, substituted in (742), give -\- X" 8 D"^ xJ^&LC. +F,Jy+&c.)=0. (745) But, by (262), /. X' d D x=f. X' D 5 x=X' d x—fDX' . $x (746) /. X" d D^xz=f. X" Z/2 8xz=iX" D 8 X—fDX". D $ x —X' D d x-D X". d x+f i>2 X". S X, &LC. (747) The terms in the last members of (746 and 747), which are not under the sign of integration, must, in passing to the defi- nite integrals, be referred to the limits of integration But it must be observed, that the variations in (746 and 747) are taken upon the supposition that the independent variable t does not itself vary, and that only the functions vary, by which Xj 7/, fcc. are connected with it ; whereas the limits of inte* gration may themselves necessarily vary with a change of this function, and therefore t^ and t^ are supposed to vary. If, then, 8' Xq , ^'^Q, iS^c. denote the variations arising from the change of the functions, the values of the complete varia- tions are 3x=zd'x^-\- Dx^.St^, 6lc. (748) whence 8' x^z=z ^ x^ — D x^ J t^, &c. (749) Hence (746 and 747) give f^^^ XdD x—X[ 3'x^—X'o -5' ^o—fl^ DX.dx (750) ^^X'5D^xz=X[ Dd'x^—DX'; d'x^—X'^DS'x^ f — DX; 6'x^+J^^^^ D^X". <5 X, &,c. (751) in which S' is given by (749). § 116.] RECTIFICATION OF CURVES. 155 Method of Variations. These equations, substituted in (745), give +(X;- &c.) D S'. x^—iX'^-SLc) D ^' z^-f &,c. (752) The terms of (752), which are under the sign of integration, express a variation which belongs to each point of the curve independently of all the other points, and which must, there- fore, be equal to zero for each point ; which gives the general equation {X—D X'-\-D^X"—&LQ,) S X+&C. = 0. (753) The variables, t, x, y, &c. may be bound together by some conditions, represented by the equations i = 0, M=:0, (754) in which L, M, may be functions of t, x, y, &,c. The varia- tions of these equations will then give linear equations between ^Xj ^y, &:,c. from which the values of some of the variations iXj <5y, &c. can be determined in terms of the others. These values, substituted in (753), will reduce the number of varia- tions in (753) to the smallest possible number, and those which remain will be wholly independent of each other, and there- fore their coefficients must vanish. The equations, thus ob- tained from making these coefficients equal to zero, will be the required equations of the shortest time. If, in addition to the equations (754), the limits of the curve are subject to peculiar conditions ; these conditions, with those of (754), referred to the limits of the curve, may be combined with the terms of (752), which are not under the sign of in- tegration, and the equations for determining the extreme points / 156 INTEGRAL CALCULUS. [b. V. CH. VI. Maximum or minimum of definite integrals. of the curve may be found by the same method by which the equations of the curve itself are found. 117. Corollary. The preceding process for finding the minimum of (741), may be apphed to finding the maximum or minimum of any definite integral, such as '■• V, (756) by changing in the various formulae D s into F. 118. Corollary. The number of the variations ^x, Sy, &lc. determined by (754), is plainly equal to the number of the equations of (754). The number of the variations left unde- termined, therefore, in (753), and consequently the number of equations obtained from (753), is equal to the number of the variations not determined by (754). The whole number of equations then of the required curve, is equal to the whole number of the variables 3", y, z, &. to (753), may be put equal to zero, and x, ,«, Sfc. may be elim- inated from the result by the usual process. 120. Corollary. If all the variables had been, in the outset, eliminated from Ds (741) or F (75G), which could have been eliminated by means of the equations (754), the remaining ones would have been independent of each other, and would have given, at once, from (753), X—D X -\- Z>2 X" — &LC. = 0, Y— DY' + 2>-^ Y" — &,c. = 0, &.C. } (757) If, moreover, certain of the variables, and among them the independent variable, had been taken so as to be the very functions of the variables which were constant under the ad- ditional conditions at one of the limits, as that of ^q ; we should have for those variables 8t^={), &.C. (758) and there would have been no additional conditions between ^ ^0 » ^Voi ^^-y which in this case would not differ from •5' Xq , 5' yQ , fee. ; so that (752) would give X',—D X;'4-&c.=0, Y',—D F'J-f &c.=0, &,c. (759) Xq— &c. = l"o— ^G- — 0, is subject to two conditions, that is, if it is at the intersection of two given surfaces ; let this line be the axis of Zq , and we have ^Xoz=0, ^yo = 0, (774) 14* 162 INTEGRAL CALCULUS. [b. V. CH. VI. Shortest line between two surfaces or two lines. whence (765) gives fe = 0, (775) or the required line is perpendicular to the given line. Hence The shortest line which can be drawn between two given surfaces, or two given lines, or a line and a sur- face, or a point and a surface, or a point and a line, is the straight line which is perpendicular to the surface or line at the corresponding extremity. 129. Corollary. If the shortest line is required to be drawn upon a surface of revolution, let the axis of z be the axis of revolution, let u be the projection of the radius vector upon the plane of x y, and let (p be the angle which u makes with the axis of oc ; and we have, by taking z for the independent varia- ble, Ds = ^ {u^ D (/'2 ^ Bu^+l), (776) But by the equation of the surface, w is a given function of 2, and, therefore, not subject to variation. Hence 3Ds= .^ ^ . (777) The equation gives, then, D --^ = 0, (778) U s the integral of which is -jf^ = C; (779) J J s in which C is an arbitrary constant, and the independent va- riable may be any variable whatever, because it is only the ratio of two differential coefficients which enters into (779). <5> 130.] RECTIFICATION OF CURVES. 163 Geodetic curve upon the oblate ellipsoid. 130. Examples. 1. To find the shortest line which can be drawn upon the oblate ellipsoid of revolution. Solution. Let A be ihe greater, and B the smaller semi- axis of the generating ellipse, and c the eccentricity ; we have for the equations of the ellipse, as in (575 and 576), X ^1 A sin. ^, y =B cos. & ; (780) and X in this equation is the same with u in (779), and y is the same with z. Hence (776 and 779) give, by taking ^ for the independent variable, and using the notation of elliptic in- tegrals, D s^=A^ sin. 2 & D f^+A^ cos.2 &-\.B2 sin.2 & z=iA2 (sin.2 6 D ,^ ^ z=i COS. « sin. V', (786) ^^2— -i_g2 (l_cos.2 a COS. 2 v^)zz=l — e2(sin.2 a-j-cos.2 a sin.2 1//) =(1— c2 sin.2 u) (l_e'2 sin.2 ^j — (I_e2 sin.2 a) ^' V'2^ (737) 164 INTEGRAL CALCULUS. [b. V. CH. VI. Geodetic curve upon the ellipsoid. Elliptic integral of the third order* V (^- sin. 2 <3— C2) = yl V (sin.2 ^— sin.2 «) z=zA \/ (cos.2 Li — COS. 2 &) ^^ A COS. « sin. V', (788) ^ _ C^ ^ 2)./.^ sin. « V (l-<^^ si n.2 g) . ^^ y/ '^ '^~sin. fi\/(^'sin.2 6-C2j— sm.2 d __ v/(l—e^ sin.2 «) ^> _ V(I-c- sin.2 a) y ,/;2 sin.«(l+col.2asin.2 t/^) sin. " (l-j-cot.2 « g^,^ 2 ^^j ^/ 1/; (789) sin. « r~ 1-f-e^cos. 2 a €^cos.^«~l "~cos.2 « yy/( J -e^ sin.'-^ u) L (i + cot.''' « sTii^'' V') ^' -^ ^' V' J' Hence, if we adopt the notation n(n.erp) =1 f^ , }^ , (790) J (1+/1 sin.2 v^) ^^ ' '^ / and put n = cot. 2 «, (^91) (789) gives ^-^^ ^ (p z= COS.2a^(l fi2 sin.2 a^ (792) [(l+c2 cos.2") (JT(we'V^i)-iz(?ie>o))-e^cos.'«(P' Vi-F'>o)], which is the required equation of the curve. The length of the curve becomes, by substitution in (781), 8= A a/ (1— e2 sin.2 «) {E'H^^—E'^^). (793) The integral (790) is called the elliptic integral of the third order ^ and admits of theorems similar to those of the first and second orders. 2. To find the shortest line upon the prolate ellipsoid. ^ 130.] RECTIFICATION OF CURVES. 165 Elliptic integral of the third order Quadrature. Ans. Let the axes of the ellipsoid be represented by the same letters, as in the preceding example ; and let the equa- tions of the ellipsoid be uzziB COS. &y z:=A sin. &, •\ Cz^B COS. a, sin. ^i=:sin. « sin. V, > C^^^) e'=e sin. "^ nz=. — sin. 2 cc^ J the equation of the required curve is ('95) e^ cos « y=cos.«\/(l-c2)[^(ne>i)-72(nc>o)] + -^-j— ^(Fv^i-i^V^o), and the length of the curve is 5 = A {E'y^i — E'%). (796) 3. Prove that if (pg, (p and 9' satisfy the lovi^er equation (599), and if N=A^[n{n+l){n + e'')] (797) the elliptic integrals of the third order will satisfy the equa- tion n {n e (f') -{- n (^n e (p) — n {n e (fo) ^ ♦ r-n / Ns\n. cp' sm.

^r^r^s ^'"^ IT ^UT ('^') and 1)2 o __ ^ n L _ the area ah c d .^^^. '•" ~ dl ^ dl.dm ' ^ ' But, if a z=i the angle h a c, ^ 5' = an arc of Z Z', > (801) s" = an arc of 3Im ; j we have the arc a b z= d s', the arc a c =: ds", (802) the area ab c dz=i sin. a d s' . d s' and, since m is the only variable in s', and / the only variable m s' D],^o — sin. aD^s' . D^ s", (803) «5> 134.] QUADRATURE OF SURFACES. 167 Area of a curved surface. and the accents may be omitted in (802) without any ambi- guity. Hence c =ifj,, sin. aD^.s, Di.s; (804) in which D^s and DiS may be taken directly from the gen- eral expression for D s, and a is the inclination of two lines drawn through a point, in such a way, that for the one / is constant, and for the other m is constant. 132. Corollary. ^ When the surface is plane, (570) of vol. 1 gives for rectangular coordinates, l>,s=l, />, s=l, (805) and it is obvious that a is a ri^ht angle ; whence «=/,/,.!=/,. a: =/..y, (806) or supplying the place of arbitrary constants by the form of definite integrals, " =fi fi: ' =/:: ^^^-y^) =f'i: (^.-^»)' (^«') in which the values of a-^ Xy 7/0^1, ^^^ determined by the bounding curve. 133. Corollary. When the surface is plane, (574) of vol. 1 gives DcfS=r, DrS=\, (808) and « is a right angle ; whence o =.f^f^ ^r^f^,r^P=^ if^p . r^• (809) or 134. Corollary. When the surface is curved, let Y denote the inclination of the tangent plane to the plane of x y, and, 168 INTEGRAL CALCULUS. [b. V. CH. VII. Area of a curved surface. since the projection of a surface is equal to the product of the surface by the cosine of its inclination to its projection, (806) gives o=fJ^.sec.Y. (811) Hence, by (600) of vol. 1, where V—0 (812) is the equation of the surface, * = fjy ■ V (D. z^ + ^,==' + !)• (813) 135. Corollary. When the surface is developable, it may be supposed to be developed into a plane, and its area found as that of a plane surface; or it must give the same result to refer the surface to axes, drawn upon it in such a way, that they would be straight lines when the surface was developed, and the rectangular coordinates would then be the length of the shortest lines, which would be drawn upon the surface to two of these axes, which would be perpendicular to each other. 136. Corollary. When the surface is one of revolution, the notation of § 129 gives, by § 134, a^f,^f^.u^{D^z'^+\); (814) and if 5 denotes the arc of the generating curve, "" =hfu 'U DuS =fcpf, ,uD,.s =fcff, . u. (815) 137. Corollary. When the surface of revolution is included between four curves, of which two are the intersections with the surface of two planes which are perpendicular to the axis «§> 140.] QUADRATURE OF SURFACES. 169 Quadrature of a surface of revolution. of revolution, and the other two are the intersections with the surface of the planes, which may be called meridian planes, because they include the axis of revolution, and which are in- clined to each other by an angle Vg j (^1^) gives 138. Corollary. If another surface of revolution were gen- erated by the revolution of the arc in the preceding section, about an axis at the distance h from the former axis, and farther from the arc, so that for this new axis we have u' = u + h, (817) (816) gives the value of the corresponding surface o' = ^^fl^^ {u D^^ s + b D,, s) — a^bcp^ (Si—So). (818) 139. Corollary. Had the second axis been upon the oppo- site side of the arc, we should have had u" z= b — u (819) o" = b^,{s,-s,)-o. (820) 140. Corollary. A curve AB A' B' (fig. 8) is said to have a centre when there is such a point that any chord, such as A A\ B B', &c. which passes through it, is bisected by it ; and such a chord is called a diam- 15 170 INTEGRAL CALCULUS. [b. V. CH. VII. Surface of a ring. eter. The surface generated by the revolution of such a curve about an axis C C' which does not intersect the curve, is called mi angular surface^ or, simply, a ring. The notation S =z the perimeter of the generating curve ABD AD' A, o =: the surface which would be generated by the revolution o( D B A D' about the di- ^ (821) ameter D D' parallel to -CC", h z=. the distance of the axis C C' from the cen- tre, gives by (818 and 820) for the whole surface of the ring, =:2b^ S. (822) 141. Problem. To transform the differential coefficient of a surface from one system of variables to another. Solution. Let / and m be the given variables, and let the second member of (803) be denoted by //, that is, Dl^.o^zH (823) If, then, only one of the variables m is to be changed, and t is to be introduced instead of it by means of the equation M = m, (824) in which M is a given function of I and t ; we have D].,.o — D,D,.o^D^D,.o.D,.m — Dl^.G . D,3I=z H D,.M. (825) «J 144.] Q,UADRATURE OF SURFACES. 171 Transformation of differential coefficient. If the Other variable / is also to be changed, and u to be in- troduced instead of it, by means of the equation / = L, (826) in which Z« is a given function of t and u ; we have D] ,o^HD,.M.D^L, (827) 142. Corollary. If J/, in equation (824), instead of be- ing a given function of t and /, were a given function of t and u, u might be eliminated by means of (826). It is more con- venient, however, to eliminate its differential coefficient only from Di . 31, after having determined this differential coef- ficient by means of (826). Thus the differential of (826) relative to t is, by regarding w as a function of t, = D,L + D,L . D,u, (828) whence ^^ , ^ _ g^ , (829) and (824) gives = —j^^ . (830) But 771 is obviously to be substituted for 31 in (827), whence we have by (827 and 830), Dl^ azzzH (D,3I.D^,L—D,,3I.D,L). (831) 143. Corollary, The two preceding articles may be applied to the transformation of any second differential coefficient of two successive variables. 144. Examples. 1. To find the area of the segment of an ellipse included between two parallel lines. 172 INTEGRAL CALCULUS. [b. V. CH. VII. Transformation of differential coefficient. Solutio7i. Let the ellipse be referred to conjugate axes, as in (74) of vol. 1, in which the axis of ?/ is drawn parallel to the given lines ; and (804 and 807) give, since in this case y„ = -y, (832) is the ordinate y of the ellipse, if « is the angle of the axes = 2 f""^ . 7/ sin. «. (833) If, now, we take 5 so that X ^^ A cos. 5, (834) we have y z=. B sin. ^, (835) D^,x = — A sin. ^, (836) a=2sin. « r ^ A B sm.^ &z=zA J5 sin. « /" (i — cos. 2 &) J &i */ ^1 =^jB sin. a [6^—6^—^{s\n. 2 ^q— sin. 2 ^ J] (837) =■ J sin. « (corresponding areaof asegmentof a circle whose radius is yl). 2. To find the area of a sector of an ellipse, when the ver- tex of the sector is at the centre of the ellipse. Solution. In this case (834 and 835) give, when A and B are the semiaxes, r COS. (p =z A cos. Q, B tan. (p zzz — tan. &, A t^Dq. pzzzA B ; whence, by (810), putting zero for r^ , c = iAB(\-i,). (841) r sin. (f z= B sin. 6, (838) ^ B C0S.2 (p T) (p — ^ A C0S.2 a ' (839) (840) <5> 144.] QUADRATURE OF SURFACES. 173 Area of elliptic segment and sector. Corollary. The whole area of the ellipse \s, n A B. (842) 3. To find the area of a sector of an ellipse, when the ver- tex of the sector is at a focus. Solution. If the origin of coordinates is at the focus, (834 and 835) give r cos. (f=iX7^A cos. ^ — A e=A (cos. & — e) (843) r sin. (fz:zi/^zzB sin. & (844) B sin. & ,r.4^. D6,^?. (lz:i££!_!L-^^ (S46) A (cos. <3 — e)2 ^ ^ ' r^ D6cp = AB . (1— e cos. &), (847) whence, by (810), <^=zA B . [6^—6^—e (sin. ^^— sin. dj]. (848) 4. To find the area of the hyperbolic segment included be- tween two parallel lines. A?is. If the hyperbola is referred to conjugate axes as in (90) of vol. ], in which the axis of ?/ is parallel to the given lines, if y is the angle of the axes, and if ^ is taken so that xz= A Cos. ^ y = B Sin. ^, (849) the area is a = J ^JBsin.y (Sin. 2^j— Sin. 2 ^0+2 ^1—2 5 J. (850) 5. To find the area of the hyperbolic sector, the vertex of which is at the centre of the hyperbola. 15* 174 INTEGRAL CALCULUS. [b. V. CH. VII. Area of hyperbolic segment and sector. Ans. With the notation of the preceding example, the area "^^B ("-«„)■ (851) in which A and B are the semi-axes, 6. To find the area of the Ijiyperbolic sector, the vertex of which is at one of the' foci. Ans. With the notation of the preceding example, the area is oz=AB [(^— ^o) — ^ (Si»- ^— Sin. d J]. (852) 7. To find the hyperbolic segment included between an asymptote, the curve, and two straight lines drawn parallel to the other asymptote. Solution. It is convenient, in this case, to take the two asymptotes for the oblique axes, for which « and ^ in (86) of vol. I. must have the values tan. a z= - , tan. ^ = — - ; (853) whence (86) gives for the equation of the hyperbola, referred to its asymptotes, xy:=:zl{A^+B^). (854) The area of the required segment is, then, by (807, 853 and 854), if the axis of y is the asymptote parallel to the given lines, . ^ px. 2AB px. ^ «/ 3 ^ JAi3 1og. ^. (855) <5> 144.] Q,UARDATURE OF SURFACES. 175 Area of parabolic and cycloidal segments. 8. To find the area of the parabolic segment included be- tween two parallel lines. Ans. If the parabola is referred to oblique axes as in (100) of vol. I, of which the axis of 3/ is parallel to the given lines, and if « is the angle of the two axes, the area is ""^l (i/i 2:1— yo 2:0). (856) 9. To find the area of the parabolic sector, of which the vertex is at the focus. Ans. If P is the distance from the vertex to the focus, if the origin is at the focus and the angle y counted from the vertex, the area is — 2P (tan. J ^1— tan. J 144.] QUADRATURE OF SURFACES. 177 Area of a zone of an ellipsoid. Corollary. If a and h are equal, the surfaces are similar, and (869) gives o'=a^a- (870) that is, the areas of shnilar surfaces areproportional to the squares of their dimensions. 17. To find the area of the zone of an oblate ellipsoid of revolution which is included between two planes drawn per- pendicular to the axis of revolution. Solution. Let the notation be that of Example 1, of § 130, and (816) gives, for the area, 7 ^^^^^ Let the angle w be so taken that B Sin. ^== Ac COS. 6 ; (872) and we shall have — Ae sin. 6 Doj.^^B Cos. oj; (873) whence 2 -ft jB2 /: '° . COS.2 e ., 1 = - / ° . (1+CoS. 2 a,) = ^^ [(%--i) +4 (Sin. 2c.^-Sin.2 .J]. (874) 18. To find the area of the zone of a prolate ellipsoid of revolution which is included between two planes drawn per- pendicular to the axis of revolution. 178 INTEGRAL CALCULUS. [b. V. CH. VII. Area of a zone of a hyperboloid. A/is. With the notation of Example I, of § 130, and put- ting COS. oj ^= e cos. &, (875) the area is " = "^^ [(- --'o) + i (sin- V -s'»- 2 %)]• (876) 19. To find the area of the zone of the hyperboloid of revo- lution formed by the revolution of an arc of an hyperbola about the transverse axis, Ans. If the equations of the generating hyperbola are xz=z A Cos. ^ !/ = B Sin. &, (877) and if w is taken so that e Cos. & z=z sec. «», (878) the area is _nABrsm.'^^ sin.o'o , ,_ tang. (45° +^ co )-| ^-^- L^^r^^'o+^'t^^ii:(45^+i^)J-^ 20. To find the area of the zone of the paraboloid of revo- lution, included between two planes, which are perpendicular to the axis of revolution. Ans. If P is the distance from the vertex to the focus, and if & is so taken that y = 2P tan. 5, (880) the area is a = f T P2 (sec.s^i— sec.^dj. (881) 21. To find the area of the zone generated by the revolution of an arc of a parabola about the axis of ?/ of the preceding example. «§> 145.] QUADRATURE OF SURFACES. 179 Area of a zone generated by the arc of a cycloid. Ans. If ^ is taken so that X -{- P = P sec. 6, (882) and if o' is the value of o in (879), the area is P2 e A B (883) 22. To find the area of the zone generated by the revolu- tion of an arc of a cycloid about the axis of x in (130) of vol. 1. The arc is supposed to commence with ^. A?is. With the notation of equations (130 and [31) of vol. 1, the area is G=l6 7tR2 {2_2 COS. i 5 — ^ sin.2 J 6 . cos. J 5). (884) 23. To find the area of the zone generated by the revolu- tion of an arc of a cycloid about the axis of y in (131) of vol. 1. The arc is supposed to commence with 6. Ans. With the notation of the preceding example, the area is a = 16 nR2 (sin. J ^— J & COS. J ^ — ^ sin:3 J 6). (885) 145. Problem. To find the area of the zone gen- erated by the revolution of a given arc of a plajie curve about an axis in the same plane with the arc^ when the areas of the two zones are known which are generated by the revolution of the arc about two axes in the plane, which are perpendicular to each other. Solution. Let the two perpendicular axes be those of x and y, and let the given areas be, by (816), 180 INTEGRAL CALCULUS. [b. V. CH. VIL Greatest or least surface. o'=r 2 TT o"—2 r^'.x. (887) Let the new axis be inclined to the axis of a; by an angle «, and pass at a distance a from the origin, and the required area is 0=1:^2^ f ^ ' {y eos. « — X sin. « — a) — _j_ 2 TT [a' cos. « — o" sin. a — a {s^—s^)], (888) in which that sign is to be adopted which renders the second member positive. 146. Problem. To draw the curve line subject to given cojiditionSj which includes a maximum or mini- mum surface. Solution. This problem, like that of § 116, involves the maximum or minimum of a definite integral, and is therefore solved in a similar way, by the method of variations. There is, in this case, however, a double integral, and the first inte- gral refers evidently not to disconnected points, but to the bounding lines of the surface, so that the determination of these lines may involve the method of variations, even when the general form of the surface is given. The determination of the form of the surface will admit of more lucid dis- cussion in a chapter upon the curvature of surfaces, and the present chapter will be confined to the consideration of the bounding line. The equation of the surface being given, the form of its second differential coefficient is known, and is independent of <§> 148.] QUADRATURE OF SURFACES. 181 Greatest or least surface. the limiting lines, so that an integration can be directly per- formed, and the required integral be reduced to the form (756), and the process of finding the maximum or minimum becomes identical with that of § 1 IG. 147. Corollary. A kind of equntion of condition is often connected with this problem, wholly different from those refer- red to in § 116. Each of the equations (754) is an equation which is satisfied by the coordinates of each point of the re- quired curve, and is thus equivalent to an infinite number of equations. But an equation, of the class here alluded to, is a single equation, involving the coordinates of every point of the curve. An instance of such an equation is the one which expresses that the bounding curve must be of a given length, or that the definite integral (741) must have a given value. All equations of this kind would appear to depend, neces- sarily, upon definite integrals, and they may be introduced into the equation of maximum or minimum for the purpose of elim- ination by the method of § 119. It must be observed, how- ever, that the multipliers ;., //, &c , of these equations arc always constant. For each of these equations does not deter- mine any relation between $r, $y, &c. which is applicable to each point of the curve, but only a particular relation by which one of the variations, as ^x, may be determined for one of the points in terms of the values of the variations for all the points. The corresponding multiplier '', therefore, must have that par- ticular value which shall cause this single value of ^x to disap- pear from the equation ; that is, ;. must be constant. 148. Examples. 1. To find the plane curve which, having a given length, encloses the maximum area. 16 182 INTEGRAL CALCULUS. [b. V. CH. VII. Greatest or least surface. Solution. The function to be a maximum is, by (806), '^ y, (889) and the function (566) is to be constant. Hence if A is the constant multiplier introduced for the purpose of elimination, the equation is, by the reduction of § 121, 1 - ^ D. (g^ ) = 0, (890) or by the notation of § 148 of B. II., and by (577 and 609 of vol. 1, = 1+^1), .COS. „ (891) = DrX -\- A Dr COS. v — sin. r Dys — A sin. ^ (892) Az:z DrS — Q; (893) that is, the curvature is constant, which is the property of no Other curve than the circle ; the required curve is, therefore, a circle ; which has, already, been proved in the Elements of Geometry. 2. To find the plane curve which, being drawn from one given point to another given point, and having a given length, encloses the maximum area between the curve itself, its two extreme radii of curvature and its evolute. Solution. By adopting the notation of the preceding article, the required area may be expressed in the form 1 . e~ ; - (894) that of the arc will be s =y ;:■ . c. (895) § 148.] QUADRATURE OF SURFACES. 183 Greatest or least surface. Equations (576, 577 and 009) of vol. 1, give Di X z= sin. r Di 5 = o sin. ,, (S9G) Dy 7/ Z=Z COS. V Di S = n COS. i . (897) The given differences of the coordinates of the extreme points of the curve are, then, X, — 2, = fl' . Q sin. r, (89S) ft/ ' !/i—!/o——t ' ' • ^ COS. ,. (899) If, therefore, A, B, C are the constant multipliers of (895, 898 and 899), introduced for the purpose of elimination, the equation of the maximum or minimum is o o _|- ^ -|_ J5 sin. r — C COS. r = 0. (900) Let H and « be taken so that B z=zH COS. cc^ C=Hsm.ai (901) and (900) becomes 2q + A +Hsm. (»' — a) = 0; (902) and by putting r' = v — «, (903) 2Q + A + Hsin.r' = 0; (904) which shows that (900) may be reduced to the form (904), from which the term containing cos. ^ disappears, by merely changing the direction of the axis of x. It does not, then, diminish the generality of the solution to put 0=0; (905) by which (900) becomes 2 § + ^ + J5 sin. V = 0. (906) 184 INTEGRAL CALCULUS. [b. V. CH. VII. Greatest or least surface. Tiie curve is easily expressed in rectangular coordinates by the equations 2 = ^ A COS. r -\-l B sin. 2 ' + ^ r, (937) ij — ^A sin. '■ — ^ B COS. 2 »•. (908) Corolla nj. When the extreme points are not fixed, tlie equation (900) becomes '2 c + ^ = ; (909) that is, the curve is a circle. Corollary. When the length of the curve is not given, the equation (906) becomes "Zq + B sin. 1=0; (910) which is, evidently, from example 3 of § 151 of B. II., a cycloid. <5> 149.] CURVATURE OF SURFACES. 185 Curvature of a surface in any direction. CHAPTER VIII. THE CURVATURE OF SURFACES. 149. Problem. To find the curvature of a given surface at any point in any direction. Solution. Let the tangent plane to the surface at any one of its points be taken for the plane of the coordinates x and y, so that the normal may be the axis of z. We have, then, at this point, D,z=0, D,^z = 0; -(911) and if q^ and c^ are the radii of curvature at the point of the intersections of the planes ofocz and 1/ z with the surface, equation (610) of vol. 1 gives -^^Dlz, ^-=I>lz. (912) The radius of curvature (? of a section made in any intermedi- ate direction by a normal plane, which is inclined to the axis of x by the angle «, is derived from the equation — = Bl z, (913) if u denotes the distance of a point of the curve of intersec- tion from the axis of z. But the coordinates of one of these points are xz=iu cos. «, y =^u sin. « ; (914) 16* 186 INTEGRAL CALCULUS. [b. V. CH. VIII. Directions of greatest and least curvature. whence, in general, D^ z = COS. » D^z -j-sin. ^ DyZ, (915) -=zDlz r= COS.- a Biz 4-2 sin. « cos. « B'l_yZ -\- sin.^a Dl z C0S.2 a , sin.~ « , ^ . ^o = 1 1- 2 sin. "COS. oc Dl,y z. (916) 150. Corollary. The radius of curvature (/, in a direction perpendicular to that of ^, is given by the equation 1 sin."« , cos.2« ^ . _. —=: 2 sin. « COS. a D'i y z. (91 / ) 151. Corollary. The sum of (916 and 917) is -+T = -+-; (91S) that is, the sums of the reciprocals of the two radii of curvature of any two perpendicular sections at a given point of a surface is a constant quantity. 152. Corollary. If Q were the maximum radius of curva- ture at the point, o' would obviously be the minimum radius of curvature ; whence The directions of greatest and least curvature of a surface at any point are perpendicular to each other. 153. Corollary. The difference between (916 and 917) is — — i=:cos.2« f - — -^ — 2sin.2«Z>|.. ^, (919) Q' Q \Qy Qz / and in the hypothesis of the preceding corollary, the first mem- <§> 155.] CURVATURE OF SURFACES. 1S7 Motion of point of contact in direction of greatest or least curvature. ber of (919) is a maximum. The differential coefficient of the second member, taken with reference to «, must be equal to zero, that is, , = sin. 2 '^(- — ,7 ) + 2 cos. 2 a D\.yZ. (920) The sum of (919) multiplied by cos. 2 «, and of (920) Uy sin. 2 a, is cos.2«(^--^=:---. (921) \ '' 'I / " Or Hence, from (918), cos.2 « sin.'2 « from which the curvature of the surface can be found in a direction inclined by the angle « to the direction of maximum curvature. 154. Corollary. One half of the difference between (919) multiplied by sin. 2 «, and (920) by cos. 2 «, is D-j,.y z z= — 1 sin. 2 « Q-D- '-' 155. Corollary. For the direction of the maximum or min- imum, « is zero or a right ajigle, and, therefore, for either of these directions, D%.yz = 0- (924) that is, with a small motion of the point of contact in the direction of the greatest or least curvature, the tan- gent plane rotates about a line perpendicular to the direction of the motion of the point. 188 INTEGRAL CALCULUS. [b. V. CH. VIII. Direction of no curvature. 156. Corollary. When « is half a right angle, (921 and 922) give Qy = Qx, (925) l=l=Wi+M. (926) 157. Corollary. When the values of Q and (?' have opposite signs, neither of the corresponding curvatures is strictly a min- imum, but the two curvatures are the greatest curvatures in opposite directions. There are, in this case, two intermediate directions of no curvature, corresponding by (922) to the values of «, tang. «r=±\/(— ^yV (927) The sections of the surface, made in these directionSj have a contact of the second order with the tangent plane, and correspond, in general, to points of contrary flexure. 158. Corollary. In the case of a point of contact for which the greatest and least curvatures are in opposite directions and equal, we have Q^-Q'; (928) whence, by (918), Cz = — (?,; (929) that is, the curvatures iu any two directions, which are perpendicular to each other, are equal and opposite. We have also in this case, by (927), « — zb 45° (930) for the angles, which the directions of no curvature make with the direction of greatest curvature. <§> 160.] CURVATURE OF SURFACES. 1S9 Curvature of a section which is not normal to the surface. 159. Corollary. If the curvature were required of a sec- tion, the plane of which did not include the normal, it might be found by referring the surface to an oblique system of co- ordinates, of which the tangent plane was the plane of xy, the cutting plane that of xz\ the axis of x being the intersec- tion of these two planes, and the axes of y and z' being per- pendicular to that of X. This system might be obtained from the rectangular one, which has the same axes of 2: and y, but in whicli the axis of z is the normal, by putting & = the inclination of the axis of z to that of z' , (931 ) = the complement of the inclination of the given plane to the tangent plane, which gives 2; =r 2' cos. ^, (9i.5'2) D: z = Dlz' cos. <5 ; (9:3:i) or, by putting Q^ =z the radius of curvature of the inclined section -=i cos. 5, (934) (1^ = COS. &. (935) IGO. Corollary. If the axes, in the preceding corollary, were rectangular, that of y being perpendicular to the given plane, and those of x and z situated in any way whatever in that plane, equation (610) of vol. 1 gives L — - — -' - — ^^ . . (936) If we put the anufle of C'^ and z y =z the angle of o and z, ^ v ' ) and observe that the plane of i' and 'jj is perpendicular to that 190 INTEGRAL CALCULUS. [b. V. CH. VIH. Curvature of any point of the surface. of ;:; and z, so that if a sphere were described with the point of contact for the centre, the arcs &, t, y would form upon the surface a right triangle, of which y was the hypothenuse, we have cos. y =: cot. r COS. &. (938) But the comparison of (81 1 and 813) gives sec.y = ^[\+(D,zy- + {D,zf], . (939) and we have, obviously, sec. Tz=V[l+(/>x2)2]; (940) whence 1 1 sec. r Dl . z Q Q^ sec. Y l+(Z>x zf Dl.z 1 . cos. y 1 + {D. zf • V [1 + {Dy ^f+ {D. ^f] ■ (941) 161. Corollary. The curvature of a section of the surface made by a plane which includes the axis of z, and is inclined to the plane of 2; x by the angle £, may be found by the formula in which w = the distance of any point of the section from the axis of z, whence a;=M cos. c, y ^=-u sin. s ; (943) i>„a;=:cos. «, X>„yz=sin. «; (944) D^ 2=cos. t .D^z-\- sin. « Dy z, (945) Dl z=cos.2 a . Dlz+'^ sin. « cos. £ DJ .,2+sin.2 a . I>2 ^ j (946) 1_ _ {Dlz + 2 tan, e . Dl,yZ-\-i^n.^ s . Dl z) cos, y Q "~l+Z>^22_|.2tan.eZ>,2;i>y2;+(l+i>,%2)tan.2e' ^ ^ <§> 163.] CURVATURE OF SURFACES. 191 Curvature of any part of the surface. and since the coordinates x, y, z do not themselves occur in this value of the reciprocal of the radius of curvature, but only their differentials, (947) is applicable to any point of the surface, and to any direction of the curvature, it being ob- served that s is the angle, which the plane, dravi^n through the axis of z and parallel to this direction, makes with the plane of xy. 162. Corollary. When the plane which is parallel to the required direction of curvature is also parallel to the radius of curvature, (601, 598 and 599) of vol. 1 give cos. fi Dy Z /rv40x tan. £ = ■ = — ^ ; (948) COS. « IJ^ z whence the product of the denominator of (947), by Dx z^, becomes D^z^ ^ Dxz^ + 2D. %2 DyZ^ + DyZ^ + D^z^ = {D, z^ + Dyz') (1 +Dzz2 + Dy z2) = {D^z^ -{- Dyz2)sec.2Y; (949) and (947) becomes 1 Drz^ Dlz+2 D, z DyzDl^y^z-^Dyz'Dlz Dx z^+Dy z' cos.3y.(950) 163. Corollary. When the direction of curvature is per- pendicular to that of the preceding article, the plane which is parallel to it is also perpendicular to that of the preceding article ; whence, in this case, Dxz tan. i' -Z cot. « = =::^ — , U z and (917) becomes 19-2 INTEGRAL CALCULUS. [b. V. CH. Till. Sum of any two perpendicular radii of curvature. • 1C4. Corollary. The sum of (950 and 951) is 11^^ [\ +DyZ')Dlz-^D.z D^ zDl,,^z^{ \+D^z^)Dlz ' ^ ' ""'^' (952) which is, by (918), the sum of the reciprocals of the greatest and least radii of curvature at the point x^ y, z ; or it is the sum of any two perpendicular radii of cur- vature. 165. Problem. To find the greatest or least surface which can he drawn under given conditions. Solution. This form of statement embraces that portion of the problem of § 146 which was reserved for this chapter. Since a single equation between the coordinates of each point is sufficient to determine the surface, no such equation can be given ; but there may be particular conditions invohing defi- nite integrals, like those referred to in § J 46. 166. Corollary. When there is no condition what- ever, the required surface is absolutely the least surface of all lohich have the same boundary. In this case, the integral to be a minimum is (Sli or 813), the variation of vrliich gives ff^. COS.., {D,zDJz+DyzD^dz)=i^. (953) But, by integration, fz fy . COS. y Dxz D-^sz =fyfz . cos. y Dj,Z D^Sz =/y . Dz z cos. y 8 z—frfy . D:c (cos. Y Dj^z) s z, (954) (955) ffj . cos.y Dy zDy ^ z—fj. .By z COS./' (J z-fxfy . By (cos. yBy zy z ; <§> 167.] CURVATURE OF SURFACES. 193 Least surface. whence, by regarding only the terms under the double sign of integration, 0=zDj: (cos. y D^ z) + Dy (cos. •/ D,j z) =zcos.y{D%z+D'yz)+Dx z Dz . cos. y-VDy z Dy cos. /. (956) But Dz. COS. y=zDz. {DzZ^ + DyZ'^+X)"^ = — C0S.3 y {DzzDlz + DyzDl.y z), (957) Dy. cos. Y -^ — COS? Y{DzzDl,yZ+DyzDlz) ; (958) which, substituted in (956), give by (952), ^^ (1 +D, %^) Dlz—2 DzzDyz Dl.yZ + jX+D^z-^) Dlz sec.3 y or 5' = — (?; (960) so that this surface is one in which every point is a case of <§) 158 ; that is, i?i which the curvatures, in directions 'perpendicular to each other , are equal and opposite. The plane is the most simple instance of such a sur- face, but there are other examples to an milimited ex- tent. 167. Corollary. The complete determination of these sur- faces must be reserved for a chapter upon the integration of partial differential equations ; but the following ingenious con- struction, proposed by Monge, notwithstanding its obvious want of practical utility, which was acknowledged by its author, is 17 194 INTEGRAL CALCULUS. [b. V. CH. VIII. Construction of minimum surface. sufficient to exhibit the possibility of such a surface, and give some idea of its nature. Let any curve line, of single or double curvature, be drawn at pleasure in space. Produce all its radii of curvature towards the opposite side of the curve from the centres of cur- vature, and to a distance from the curve exactly equal to the corresponding radii of curvature. The given curve line may, then, be assumed as a line of curvature of the required sur- face ; that is, as a line which lies upon the surface and has at each point, the same curvature with the surface in the direction of this line. The produced radii of curvature, will be the radii of curvature of the surface in directions perpendicular to the given curve ; and if the extremities of those produced radii, which are the corresponding centres of curvature, are fixed, and if all the points of the given curve are rotated with the radii about these centres, moving in planes perpendicular to the given line, each element of the given line will describe an element of the required surface. The given line in its new position will acquire a new form and become a new line of cur- vature, from which another elementary zone of the surface may be described by a repetition of the above process. The small arc, through which each point of the curve must move, is not arbitrary, but is limited by the condition that two successive radii must be in the same plane, so as to meet at the centre of curvature. 168. Corollary. If the given curve of the preceding con- struction were a circle, the resulting surface would be a sur- face of revolution about an axis perpendicular to the plane of the circle and passing through its centre. The particular form <5> 168.] CURVATURE OF SURFACES. 195 Minimum surface of revolution. of this surface may be investigated by taking the axis of z for that of revolution, so that if ,, =:^2 + y2^ (961) z will be a function of u^ and will contain no other function of X and 7j. Hence D^zz=zD^z.D^u — ^xD,z, '] ^ ^ ' K (962) which, substituted in (958), give, by dividing by 4 cos.3 y, 2>„ 2 + 2 ^^ B^z"^ ^uDlz= 0. (963) By putting i; = a/ w, (964) we have ^ 1 r^ D^z — —- D,z, Z V B^ z— B ^-1- —~B^z' (965) which, substituted in (963), give B z A- B z^ — [^ +D;z= 0. (966) Hence 1 Biz _ ^ = v+B^z+B^z^-^ \ . Biz B^zBlz the integral of which is, by introducing A as an arbitrary con- stant, log. ^=log. I'+log. B, z— log. V (1+A ^2), (968) 196 INTEGRAL CALCULUS. [b. V. CH. VIII. Minimum surface of revolution. or A D,. z V V{i+D,z2)' (969) Hence _ A and if q is taken so that t;=^Cos. y, (971) (970) gives Dip . z=: D^z . Dip V =1 A Sin. cp D^z = ^Sin. ^ . -r^^— = A, (972) z= Acp; (973) and the equation of the surface is iAieUe-^)- (^'^) § 169.] CUBATURE OF SOLIDS. 197 General expression for the element of volume. CHAPTER IX. THE CUBATURE OF SOLIDS. 169. Problem, To find the measure of the volume of a given solid. Solution. Let the conditions of the bounding line be ex- pressed by an equation between three variables, /, m, and n. Suppose two surfaces drawn infinitely near each other, in such a way that n is constant throughout their extent. If, then, V denotes the required volume, we have €?„ F= the lamina included between these two surfaces. If two other surfaces are drawn infinitely near each other, in such a way that m is constant through their extent, we have djn dn V=z the small solid rod included between these four surfaces. If tw^o more surfaces are drawn infinitely near each other, in such a way that / is constant throughout their extent, we have di djn dn V =: the infinitely small parallelopiped included be- tween these six surfaces. (9T5) If s' denotes an arc of the intersection of two surfaces for which 7n and ?i are constant, s" an arc of the intersection of two surfaces for which / and n are constant, s'" an arc of the intersection of two surfaces for which / and /« are constant; 17* 198 INTEGRAL CALCULUS. [b. V. CH. IX. General expression for the element of volume. and if a! is the inclination of s" to s'" at the point of meeting, a!' that of s' to s'" ^ and a!" that of s' to s" ; and if h is the in- clination of s'" to the surface which includes s' and 5" ; the sides of the small parallelopiped will be c?s', ds" y ds'" ; the face which includes d s' and d s"^= sin a'" d s' d s" the distance of this from the opposite face = sin. h ds'" \ whence di d^ d, F= sin. a'" sin. h ds' d s" ds'". (976) But I is the only variable in 5', m the only one in 5", and n the only one in s'" , whence the accents may be neglected, by di- viding by dl . dm . dn f and (976) gives (977) Dl^,r^ V=D,,D^ . Z>„ V- sin. a'" sin. b D, s . D,,s . D,,s ; in which DiS, D„^ s, and D„ s may be deduced from the gen- eral expression for the differential of an arc in space, by put- ting successively each pair of the quantities /, m and w, equal to zero. The value of V is, then, the third integral of (977). 170. Corollary. If one of the vertices of the parallelopiped is taken for the centre of a sphere, a\ a", a'" will form, by the intersection of the sides of the parallelopiped with the surface of the sphere, a spherical triangle ; in which h will be the dis- tance of a'" from the opposite vertex. Hence, if A' is the angle opposite a', and if M is the ratio of the sines of the sides to the sines of the opposite angles, so that sin. A 31=-. p, 978) sm. a' ^ ' we have sin. h =r sin. a' sin. ^'= iHf sin. a! sin. o!' ; (979) § 173.] CUBATURE OF SOLIDS. 199 Cubature of solids of revolution. and (977) becomes Z>f.„.„.F=^-^sm.a'sin. «"sin. d^.D.s.D^s . D^s. (980) 171. Corollary. If I, m, n are the rectangular coordinates z, y, z, we have by (725), ds^- = cl x^ + chf- + dz^ (981) a' = a" = a'" = ^^, 31 z= i ; (982) and (980) gives Dl.,.^.V=l, (983) V=f.fJ. 1 =U, z -/./. y =fyf. ^. (984) 172. Corollary. If /, m, n are the polar coordinates of § 73 of B. I., the equations (31, 32, 33) of vol. 1 give, by put- ting u :=. r sin. ip '\ y z=i u cos. 6 > (985) z z=z u sin. <5j ) dy'^-\-dz^—d ifi + w2 ^^2 (9S6) = f/r2+r2 rf52_|_^2sin.2^,rf^'2, (987) D^ ^ ^F=r2 sin. c/^, (988) V=fl^^r^^^^n.,=-fl ^ ^.2eos,,=/^^r3sin.,.(9S9) 173. Corollary. If the coordinates are 2:, ?<, 5 of the pre- ceding corollary, (987) gives Dl.u.^V=u (990) 200 INTEGRAL CALCULUS. [b. V. CH. IX. Volume of spliere, ellipsoid. 174. Corollary. If the given solid is one of revolution about the axis z, of which a segment is required formed by two planes perpendicular to the axis of revolution, z may be substituted for X in (991), and the integrals relative to 6 taken from to 2 -^. Hence V—'XnJ^^ ^.u'=nf^.u'^ = 2nr.zu. (992) 175. Examples. 1. To find the volume of the segment of a sphere. Solution. If Jt is the radius of the sphere, and if the axis of z is perpendicular to the bases of the segment, (992) gives = ^RHz,-z,) — in(^zl^zl). (993) Corollary. The solidity of the sphere is fyr/ja^ (994) 2. Given the volume Fof a solid included within any sur- faces whatever, the combination of which, considered as one surface which in general is discontinuous, is represented by the equation i^. (a: . 3/ . z) = 0, (995) to find the volume V of a solid included within the system of surfaces ^.(|>.l) = 0. (996) Ans. V'=ahc V. (997) 3. To find the volume of the segment of an ellipsoid in- cluded between two planes drawn perpendicular to either of the axes of the ellipsoid. «§> 175.] CUBATURE OF SOLIDS. 201 Volume of hyperboloid and paraboloid. Ans. U A, By C are the axes of the ellipsoid, if the planes are drawn perpendicular to the axis of C, and if V is the solidity of the segment of a sphere whose radius is unity, the segment being included between two planes drawn at the dis- tances -^ and -^ from the centre, the required volume is F'= ABCV. (998) 4. To find the volume of the segment of an hyperboloid in- cluded between two planes drawn perpendicular to that axis, for which the sections made by the planes are elliptical. Ans. If Cis the axis perpendicular to the planes, and if ^ and B are the other two axes, the required volume is V=i^^ {zl-zl) ^nAB {z-z,), (999) in which the upper sign corresponds to the hyperboloid of one branch, and the lower sign to the hyperboloid of two branches. 5. To find the volume of the segment of the paraboloid, in- cluded between two planes drawn perpendicular to the axis of z, the equation of the paraboloid being (-.)■+ (!)'= <^- <■■> Ans. ^n ABC{z\ — zl). (2 a) 6. To find the volume of the segment of a solid of revolu- tion included between two planes, drawn perpendicular to the axis of revolution, when the revolving arc is that of a cycloid about the axis of x in (130) of vol. 1. Ans, Vz=i I jR2 jt (sin. 2 fl,— sin. 2 ^^o)— 2 R- ^ (sin. ^,— sin. ^o) + 3i22-(^-^o). (3a) 202 INTEGRAL CALCULUS. [b. V. CH. IX. Solid of least surface. 7. To find the volume of the segment of the solid of revo- lution of § 174, when u = B Cos. J. (4 a) Aus, V=iAB'-n(Sm!^-^'-Shi^-^\-\-iB^7v{z^-z,). (5a) 176. Prohlein. To find the inaximuin or ininimiini volume which can he included by a surface drawn under given conditions. Solution. Since the general expression for the volume is reduced to the form of a double integral, this problem is pre- cisely similar in its solution to that of § 165. 177. Examples. 1. To find the maximum or minimum volume^ which can be included within a surface of a given area. Solution. Since the double integral (984) is to be a maxi- mum, while that of (811) is to be constant. We have, by § 166, if ^ is a constant multiplier, or 1 I 1 ^ + '- = -i; (7 a) that is, the surf ace is one for which the sum of the recip- rocals of the greatest and least radii of curvature at each <5> 177.] CUBATURE OF SOLIDS. 203 Solid of revolution of least surface. point is constant. The general equation of this sur- face has never been obtained, but the sphere and the cylinder are evidently cases of it. 2. To find the solids of revolution which are solutions of the preceding problem. Solution. Let the axis of z be that of revolution, and by putting u = ^(x^+y^), (8 a) (7 a) becomes, by means of (952), Let V be taken so that uD, V (10 a) whence log. V = log. u + log. D,,z — i log. (I + Z), z2) ; ( 11 a) the differential of which is D^v I . Dlz D^z Dl z — =--rTr V ~~ u ' D,z l+D^z^ 1. , mz u '^ Dl{\+D,,z') = ^+7^o4rV7r^; (12a) which, multiplied by (10 a), gives by (9 a), (13 a) The integral of this equation is u-' v = -^-^ + B, (14a) 204 INTEGRAL CALCULUS. [b. V. CH. IX. Solid of revolution of least surface. in which JB is an arbitrary constant. But if r is taken so that D„ z := cot. T, (15 a (10 a and 14 a) give V =: u COS. T, (16 a COS.r=: — -^ + - , (Ha 2 A u V(-^ + cos.3^)=-+-, (18a u =Acos.r-/y{2AB+A''cos.\), (19 a ^1 ^^sin. tCOs. T ^ ' /s/(2AB+A^cos.Ty ^ -A ^ cos ^ "^ ^ ' //(2^S+^^cos.^t) ^ If e is taken so that '= ^{A2 + 2ABY' (^^^ (21 a) gives, by the notation of elliptic integrals, \/(2 A B+A^ cos, 2 r)— a/(^2_j_2 AB) ,^r, (23 a , A^—2AB , 2^^ i>^.=.-^ COS. ^+;7 ^^ V, ^^^, , x + V(Z^^-^)^^- (''' z = — ^ sin. T -|- (yl_2 B)eF'^ + 2eBEr ; (25 a and z may be found in in terms of u, by substituting (19 a) in (25 a). The preceding solution applies strictly to that case only in which A and B have the same sign ; for, when they have op- posite signs, e becomes greater than unity, and when B is also greater than A, e is imaginary ; but these cases are solved without difficulty. <§» 177.] CUBATURE OF SOLIDS. 205 Greatest solid under given conditions. 3. To find the greatest solid of all those for which /./.sec.2y, (26 a) has a given value, 7 being the inclination of the tangent plane of the bounding surface to the given plane of 2?/. Solution. If ^ is the constant multiplier of (26 a), the equa- tion of the maximum is l — A-'Dlz — A-'D;z=0, (27 a) which is easily derived from the equation sec.2 v = l + D, z"^ + /), z\ (28 a) Let V be taken so that z=:lA{x + yYJ^v, (29a) which gives Dlz^^A+Dlv, (30 a) Dl% = iA + Dlv- (31a) which, substituted in (27 a), give Dlv+Dlvz=:0. (32 a) Let now mz^ x-\- y \/ — 1 , w — X — y s/ — 1 ; (33 a) and w^e have DyV ziz {D„,v — D,v)\/\, W34a) Dlv = Dlv\-<^ Dl,„v + Dlv, />> = — Z>> + 2 J9L. V — Dl V ; J which, substituted in (32 a), give r —Dm.D, V. (35 a) >2 m . n 18 206 INTEGRAL CALCULUS. [b. V. CH. IX, Greatest solid under given conditions. Hence D„ i? is a function, whose difTerential coefficient taken relatively to m is constant, and may, therefore, be any function whatever of n, represented by N ; that is, Dr. V = N, (36 a) Hence v =f, N —f.71 + F. nij (37 a) in whichy and JP are any arbitrary functions ; f . n is the function whose differential coefficient is N, and F.ni is the arbitrary quantity which is constant relatively to n ; that is, which does not vary with n, but may be any function whatever of the other variable m, and which is added to complete the integral. By the substitution of (33 a), (37 a) gives v=f.{x + ys/-\)+F[x-y^^\). (38 a) If we put F — f'^^—i,Fi _ (39 a) F^f'-- ^—l,F\ (40 a) in which f and F' are real functions, the value of y becomes (41 a) »=/.(x+V-l)+/.(a:-^-l)+v'-l[-F'(^+'«/-l)--F'(^v-I)], from which the imaginary quantities will wholly disappear. <§> 180.] LINEAR DIFFERENTIAL EQUATIONS. 207 Order and degree of differential equations. CHAPTER X. INTEGRATION OF LINEAR DIFFERENTIAL EQUATIONS. 178. A differential equation is said to be of the same oi'de?' with that of the highest differential coefficient which it involves. The degree of a differential equation is determined in the same way as that of an ordinary equation, except that the independent variables are neglected, and each differential coefficient is counted as a variable. Thus the equation A Dlv + B D"-^ V + &z.c.-\- A'Dlv -\- B' D-' v + &c. -\-E Dl-"". D';; V 4- &c. +ez; -{-n=0. (42a) is of the n order ; but it is only of the first degree, or linear, if the coefficients A, B, &,c. involve the independent variables X, y, &:,c., but do not involve v, &/C. 179. Any equation, which is of a less order than a given differential equation, and satisfies it by the aid of differentiation without the assistance of any other equation, is said to be an integral of the given equation. The integral is said to be complete when it contains the greatest possible number of arbitrary quantities. 180. Problem. To integrate several given equations^ between the variables x, y, z, i^'c, and their differential coefficients taken with respect to the independent varia- 208 INTEGRAL CALCULUS. [b. V. CH. X. Linear diHerential equations with constant coeliicienls. hie tj xolien the given equations are linear^ and contain no term independent of x, y, z, t^'c, and wJien all the coefficients are constant, and the number of equations the same with that of x, y, z, ^*c. Solution. If the following expressions are assumed for the variables x = A c'\ y — Be'' &c., (43 a) in which 5, A, B, &/C., are constant, their differentials give DtX=i As e'\ Dty— Bs e'', &lc. ) D]x=As^e"-, D\y — Bs^e'\^(i. C (44a) &c. &c. 3 If these values are substituted in the given equations, these equations will evidently become divisible bye*'; and the di- vision by this factor will free the equations wholly from variables, and reduce them to equations between s, A^B, &c,, in which A and B will have a linear form. If all of the con- stants A^ B, &LC. but one, as A, are eliminated, the result will be a single equation involving A and s, in which A, however, will be a factor of the whole equation ; so that the division of this equation by A, will lead to a final equation, involving no other unknown quantity but s, and which will serve to deter- mine s. Let the equation for determining 5 be denoted by ^ = 0, (45 a) and each root of it will give corresponding values of A, B, &/C., or rather of their ratios, and thence values of x, y, &>c., which will be integrals of the given equations. 181. Corollary. The number of integrals found by the preceding process, will be the same as that of the different roots of the equation (45 a) ; but all these integrals can be <5» 183.] LINEAR DIFFERENTIAL EQUATIONS. 209 Linear differential equations with constant coefficients. united into one expression. For it is evident that, if x^ , y^ , &c. denote any one of these systems of integrals, x^z Lx,-\-L' x,,-\-^z.y y=^Ly^-[- L' Us' -{• ^C" (46 a) will also be a system of integrals, in which L, i', 6lq,. will be arbitrary ; for the linear form of the given equations will cause the multipliers of L, L\ &,c. to become th esame func- tions of x^ , y^ , &,c., which the whole equations are of x, y, &,c. ; and therefore x,, , y^ , &/C, will satisfy the equations in the same way as they do when they are by themselves ; that is, the aggregate of the terms dependent upon them will be zero. 182. Corollary. If the first member of the equation (45 a) is reduced to the form s" + « s"-^ + &c., (47 a) the expressions will, by the notation of the residual calculus, include all the terms of (46 a), provided that the residuation is performed relatively to s, and that Aj B, C, &lc. assume a new system of values for each root of (-52 a) <§> 186.] LINEAR DIFFERENTIAL EQUATIONS. 211 Integration of linear differential equations with constant coefficients. 186. Corollary. If x^ , ij ^ , (fcc. ; Jo, ?/o , &lc. ; z'J , &/C., represent the values of cT, y, &/C. ; D^ x, D^ y, &/C. ; D]x^ &/C. when ^ vanishes; and if / (53 a) equations (51a and 52 a) give If the number of the equations (54 a) is taken equal to that of the constant «, ,'?, &c., the values of «, (5, &,c., may, by the usual process of elimination, be found in terms of Xq , i/^ , &:,c. The expressions of «, ^, &;c. in terms of x^ , i/^ , &;c. will clearly be linear functions of «, (^, &c. ; so that if these values are substituted in (51 a and 52 a), the expressions of t, y, &/C. will contain x^ , y^ , &c., in the same linear form in which they now contain «, ,^, &/C. The values of «, i^, &lc., in (51 a and 52 a), might, then, have been assumed at once as identical with Xq , ?/q , &c., and the corresponding values of x, ?/, &c. would be J. Zz a;^ + Z; x'o + &c. + iJ/x 1/q + &c. X =z l^ " — . , ^. X — ^ e' &LC,. ; (55 a) s£ 2' - C ((Sj) in which, it may be observed, that the values of Z/j, Mx, &/C. are entirely distinct from those in (51a and 52 a). 212 INTEGRAL CALCULUS. [b. V. CH. X. Residual integral of a rational fraction. 187. Lemma. If JP denotes the value which xf.x acquires when x becomes infinite, we have F=l{(f.x)), (56 a) whenever /. x denotes a rational fraction, of which the degree of the numerator is less than that of the de- nominator. Proof. It follows from (216 and 219), that, in the present case. the product of which by x is .f.. = lp^\ , (58 a) But when x is infinite, (58 a) becomes F=l{{f.z))=t{{f.^)). (59 a) 188. Corollary. When the excess of the degree of the de- nominator of /. x above the numerator is greater than unity, (59 a) becomes 0=t((/-^)). (60 a) 189. Corollary. When the excess of the degree of the de- nominator of y. X above the numerator is exactly unity, and when f ,x is of the value (217), (59 a) becomes a 'a! -,^L{U'A)' (61a) 190. Corollary. Since, when t becomes zero, the values of X, y, &/C. (55 a) are reduced to x^ , y^ , &,c. ; the polyno- <§) 190.] LINEAR DIFFERENTIAL EQUATIONS. 213 Integration of linear differential equations with constant coefficients. mials L^ , 31^ , Ly , L'y , &c. must be of a less degree than the (n — l)th; while Lx , My , &c. must be of the form 5"-i 4- J s"-2 + &c. (G2a) The form of 5 L^ is, therefore, s" + b s"-' -{- &LC. ; (63 a) so that, by (47 a), s i^ — S (64 a) is of a less degree than S. We have, then, by denoting (64 a) But when t vanishes, Dt x is reduced to Jq > and therefore s Lt'x must be of the form (62 a), while L,^, , s Mz , &lc. must be of a smaller degree. We have then, again, by the differen- tiation of (65 a), and a similar train of argument may be continued to the higher differential coefficients. 191. Corollary. If, in the given equations, there are substituted for x, D^ x, D] x, &,c., the quantities contained under the sign of residuation in (55 a, Go a, 6G a, &lc.), those equations must be satisfied. The reverse process, therefore, of substituting for DtX, D^y, &:-c., not 5 z, sy, &,c., but sx — 2;q>S^.— , sy — .yo^-^, -; + &c.+ JH,yo+&'C.)6), 5 ^'^'^ in which |iz &.C. are not proper factors, but express functional operations to be performed. The value of r would be obtained by eliminating x, 1/, &,c. directly from the given equation, in which process D^^D], &c. are to be treated as though they were factors. The values of X, y, &/C. (70 a), will then be obtained by the same process of elimination, from the equations, which are obtained from the given equations, by substituting DtX — a^o ^ ® for DiX, ^ Ay — yo^® forAy, ifcc. > (^la) Djx — (xj + x^ D) re, ) &.C. This is Cauchy's method of integration, and the function ® is called the principal function. <5> 194.] LINEAR DIFFERENTIAL EQ,UATIONS. 215 Linear differential equations with constant coefficients. 193. Corollary. When the equation (45 a) has several equal roots, the corresponding systems of values in (46 a) would seem to coalesce into one. This loss of terms, and therefore of arbitrary constants, is, however, unnecessary ; for if the roots 5, s', s", &c., instead of being equal, differed infi- nitely little from each other, so that s'=s + A, s"=s'+h'z=is+li+h', &c., (72a) in which h, h\ &c. are infinitely small, we shall have, by (416) of vol. 1, upon putting Bz^A'h, B'z=A"h\ C = B'h, &c., (73 a) >-(74a) A"e'"'=A"e'''+B't e'''=A"e''+{A"h-\-B')t e''-hC t^ e'') &c. The new terms, multiplied by t, t'^, &lc., which are thus in- troduced, are just sufficient to replace those which are lost by addition. These very terms are also introduced by the process of residuation, for this process requires, by (182), one or more diflferentiations, whenever the roots are equal, and each dif- ferentiation will have to be applied to e'^ in (55 a). But by (481) of vol. 1, DT e'^zzzt"^ e\ (75 a) whence the differentiation will, evidently, introduce the re- quired terms. 194. Problem. To integrate several linear differen- tial equations between the variables x^ y, 4*^., and their differential coefficients taken relatively to the independent variable t, lohen all the coefficients are constant^ the tei^ms lohich are independent of x, y, cj'c. are given functions of t, and the number of the equations is the same with that of the variables. 216 INTEGRAL CALCULUS-, [b. V. CH. X. Linear differential equations with constant coefficients. Solution. When llie functions of t are reduced to zero, this problem coincides with tlie preceding one ; and if ^, v, &LC. denote the corresponding vahies (70 a) of x, y, &c. obtained by the preceding process ; while X, Y, &c. are par- ticular values of x, ?/, &,c., which satisfy the present problem, the values x = t-\- X, y = r,+ Y, &,c. (76 a) are complete values of x, y, &,c. for £, t], &/C., involve the re- quired number of arbitrary constants. The problem is reduced, then, to obtaining these particular values of x, ?/, &/C. For this purpose, let the subsidiary quantity t be introduced, and let %=^-'^e=t^l, (77 a) so that is the value which assumes when t is changed into t — T. If, then, X, IT, ^-c. are the values, which i, v, &lc. assume, when © is changed to , and when for a-^ , x'q, yo, &c. are substituted ^^ , ^^ , ^^ , &,c., which are functions of Tj and if the integrations in the following formulas are per- formed relatively to t, we may put X = fl,X, Y =fi Y, &c. (78 a) The differentiation of (78 a) relatively to t, involves not only the differentiation of X, Y, &c., under the sign of integration, but also the changes arising from the change in the limits of integration. If then X^, '^t, ^c., are the values which v\, "ST, fcc., assume when t is changed to t, the differentiation of (78 a) gives D,X=X,+f,D,x, •. D,Y=, ¥, +fi D, ¥, &c. 5 (^^^^ <§) 194.] LINEAR DIFFERENTIAL EQUATIONS. 217 Linear differential equations with constant coefficients. If, again, we put X'z=D,x, ^' = D,Y, (80 a) another differentiation of (79 a) gives D',JC=zD, X, + X] +f'o D] X, &c. (81 a) By the substitution of X, F, &/C. for x, y, &c., in the given equations, the terms under the sign of integration must dis- appear, for the terms under this sign in the values of JT, D^ F, &,c. differ from the values of t, v, &c. in nothing but the com- mon factor e~^^, and the writing of the particular forms ^j, STi , &c. for the arbitrary constants a;^ , Xq, &;c. The substitution of t for t reduces t^^^—'^) to unity, and if ^z ? T'x y are the same functions of t, which JTx and ^T^ are of t, we have, by § 190, ___ » x ^ r. + l: t: + &C.+M, 7;+&c. _ _ ^. Zx' r. +s xj r;+&c.+5irf, 7;+&c ._^, p ^^ ^-L ((^)) ^- -I T, = 7;, &c. ^ Hence, by the omission of the parts of (78 a, 79 a and 81 a), which are under the signs of integration, they become jr=0, D, X= n , D] X— %-\-D, . Tx , &c. ^ and the substitution of these values in the given equations, reduces them to linear differential equations in which T^c , T^ , Ty are the variables, and the order of the equations is less by one than that of the given equations. Thus the number of these variables being greater than that of the equations, 19 218 INTEGRAL CALCULUS. [b. V. CH. X. Linear differential equations with constant coefficients. enables us to take certain of them at pleasure. Thus of the quantities Tx , T^ , &c., all but one may be supposed to be zero ; of Ty, T^ , all but one may be zero ; and in the same way with the others. The selection of the quantities T^ , &c., which are to re- main of a finite value, is immediately fixed, by the considera- tion that the resulting equation should be of as low an order as possible. It is generally possible to select those quantities which correspond, respectively, to the highest order of diffe- rential coefficients of a;, i/, &c. ; and with this selection the resulting equations are wholly free from difTerentials, and are solved by simple elimination. In any case, however, it seems possible to make a selection which will avoid the necessity of integration. 195. Corollary. When s is nothing, the values of X, Y must vanish, as well as all their diflferential coefficients of an order inferior to those which correspond to the quantities in the series T^^ Ty, &c., which are retained as finite. Hence the corresponding values of x,7/, D^x^ &c. will be reduced to ^^o > yo> ^0, &c. 196. Examples. 1. To integrate the diflferential equation D'^x + a D"r' X + &c. = C/; (84 a) in which U is a. function of t. Solution. In this case, the value of r becomes at once F =zD'l + a Dr' + &^o. z=f.Dr, (85 a) <5> 196.] LINEAR DIFFERENTIAL EQUATIONS. 219 Linear differential equations witli constant coefficients. Hence S =z s"" + a s""^ + &c. =/. s, (86 a) by taking/", to denote the integral function, which constitutes the second members of (85 a and 86 a). We have also and the equation for determining ^ is, by (71 a), (2>r + aDr' + &c.)^ -[xo(i>r'+a^r'+&c.)+x; (z>^^+az>r'+&c.)+<^c.]/7 0=0 or (88 a) r ^-[a:o(Z)r'+a^r'+&c.)+2;;(J9r2+aZ)r'+&c.)+&c.]f 0=0 ; whence (89 a) If, in the development of the expression the exponents n, n — 1, &c. of x^ , are regarded as expressing the number of accents, and if the term which does not contain Xq is multiplied by x^ the value (89 a) may be expressed in a more simple form ; for we shall have J^t — ^o To obtain the value of X, let Wi be the value which U assumes when t is changed to t, and by omitting the accent of Tx as unnecessary, we have by § 194, if we suppose all the quantities in the series Tx , &;c. to vanish but the (n — l)st, T,=U, K. = m; (92 a) 220 INTEGRAL CALCULUS. [b. V. OH. X. Linear differential equations with constant coefficients. whence, by (78a, 76a and Ola), X=fl,m0^ (93 a) _/-^^-/^o0_j_y..2g0 (94a) a; = — O.-^o 2. To integrate the differential equation Solution. In this case, we have f.D, = D]—{a + b)D', + abD, = D,{D, — a) {D, — b) S= s (s — a)(s — 6) i^^^ =x^[D]-^(a+b) A+a b]+x', [l>,-(a+6)]+x,' ^_ f'Dt-f'^o y x^[s2-(a-^b)s+a b]-{-x'o [s-{a+b)]+x'^ '- A-^o ^^ ({s(s-a){s-b))) __ ct + b , , 1 „ — •*'o """ I ■*'0 t I. ^0 ^'' + rr—ix ^ > a (a — b) b (« — 6) T> ^-y T-C((5(5_^)(5_^,) ))— C52 ((5(s— «) (S— 6) )) _ ct^ c e""'— cjat + 1) ce'' — c(bt + l) ~ 2Vb~^ ^(a—b) 63 (^a—b) and z =: I -f- JT. 3. To integrate the differential equation D', X — S a D]x + S a^ D.x-- a^ X = b c^ <5> 196.] LINEAR DIFFERENTIAL EQ,UATIONS. 221 Linear difierential equations with constant coefficients. Solution. In this case, we have S= (s— a)3 ' ^'^''^'''' =x^(D-a)^+{x',-ax^){D,-a)+{x'--2axl+a^x,) !> Xq (s—aY+{xo—a Xp) (s—a)+{x'^--2 a x'^+a^ x^) 4. To integrate the diflferential equation D\x -\- a^ DtX ■=.}) sin. m t. Solution. In this case, we have /. A = {D\ + a2 D,) = D, [D] + a^) ^-6 ((s(s2+a^))) =a:^ + — sin. at -\-~, (1 — cos. a t) 2:; . ,,2 2;;'. — sin. a^-j a a^ 19* =XQ-f- — Sin. at-\ 2^ sin.2 ^ at 222 INTEGRAL CALCULUS. [b. V. CH. IX. Linear differential equations with constant coefficients. «/c' b sin. m r. e^ ^^~'^^ «- s 6 sin. m t — m h (cos. m f+c"') b mb / COS. m t cos. at\ ma- m^ — a- \ ni^ a^ / * Corollary. When m is equal to a, the value of JiT becomes ,, 6 , ^ btsm.at 26 .... - . , . X=: -;(l-cos. at) — -— ^r- = -^sin.*a^(sin.ia t-hat cos. Aan. a^^ ' 2a2 a^ ^ "^ ^ ' 5. To integrate the differential equation 2>2 3._(^_j_5) j)^ 3._|_^ 6 X = A ^2_^A: e'^'+Z sin. w ^. a — b a^ b^ 2 A (a+6)^ h t^ 2Ae^' 2Ae*' «2 62 ' a6 ' a'^{a—b) b^ (a—b) (m — a) (m — 6) (m-a) (a-b) (m-b) (a-b) l(ab — w^) sin. nt-[-nl (a-\-b) cos. nt "^ (a2_^w-2)(62 + w2) "^ (a2_|.yj2)(a_6J "~ (62-(-;i2) (^_5) • When ?»=:«, the terms multiplied by k become k t e''' k (e°' — e^^) ^ ■^Z:^" (a— 5)2 ' when m'=.by the terms multiplied by k become a — b (a — 6) 2 <^ 196.] LINEAR DIFFERENTIAL EC^UATIONS. 223 Linear differential equations with constant coefficients. 6. To integrate the differential equation D] X — 2 a D, X -{- a2 X = h t^ -\- k e'"' -{- I sin. n t. Ans. X = Xq e"^ -\- {xq — axQ)te''' + "4 [6 + 4ai+a2 t^ + 2 (a t — S) e'^'] k + 7-:; — — 7,r(«2-n2)sin.nM«2+w2)w^c«*+2aw(cos.w^-c«01. When m-=.ay the terms multiplied by A: become i kt^ e««. 7. To integrate the differential equation D"^, X -\- a^ X =:li t^ + k e"^' + I sin. n t. X h Ans. x-=. ~ sin. a t-\-XQ cos. a t-\- — («2 ^2 — 4 siii.2 J « ^) A:(ac'"'-a cos. a^-m sin. «^) l{n^m.at — as,\n.nt) When w = a, the term multiplied by I becomes ■— — - (sm. at — at cos. a t). ^Z a^ ' 8. To integrate the differential equation D\ a: = x. (95 a) Ans, xz=l{x,^xi) (e'+e-* ) +^ (^;+x;")(6'-e-0 + \ (^0— ^0) COS. # + i (2;; — x;") sin. t = i (^0+ ^0 ) Cos. ^ + J (2:0 + aj'o') Sin. t + J (a^o — ^0) COS. # + J (a:^ — x'^') sin. t. (96 a) 224 INTEGRAL CALCULUS. [b. V. CH. X. Linear differential equations with constant coefficients. 9. To integrate the differential equation n\x + x=iO. (97 a) Ans. x = uxo-\- Dl u.Xo + B\ u,x"^^BtU. x'^', (98 a) in which w = Cos. (/\/ J. ^) cos. (\/ J.^). (99 a) 10. To integrate the differential equation Dlx = x, (lb) Ans. x = uxo-\- D^-' w . 2-0 + D^^ u , x^' + (Si-c, (2 b) in which, when n is an odd number, w = -lc* + 2.^.c ^ COS. (^ sin. I I, (3 b) where ^ denotes the sum of all the terms which are obtained by substituting for m all the integers from 1 to J (w — 1) in- clusive. But when n is an even number, which is not divisible by 4, (4 b) u =-| Cos. i-j-2 j^ .Cos. I f COS. l.cos, l^sm. I I, where 2 denotes the sum of all the terms which are obtained by substituting for m all the integers from 1 to J (Jw — 1) in- clusive. When n is divisible by 4, (5 b) 2r^ rt^ / 2m^\ / . 2m7t\-i w=- Cos. ^+cos.^+2.2^.Cos.(+») ^'°•('-^)^^ ("»+»)' (19") we have a; = ^ -j- ^', y = »; -j- j?'. (20 b) «5> 197.] LINEAR DIFFERENTIAL EQUATIONS. 2*27 Linear difFerenlial equations with constant coefficients. TIT, / V • sin.\/(m — 7i) t . ., . When im — n) is negative, —r^ — and cqs.aJ {m-n]t s/ \m — n) ' . , . S'\n.\/(?i — m)t , ^ ,, , , are to be changed to j—^ r^ and Cos. Aj{n-m)t. (21 b) When 7w+nis negative, tt—t- — - ^nd cos. a/ (m +/i) ^ s/ym-f-n) are to be changed to — ' , , -■ and Cos.\/-(m+w)^. (^^ b) V — \m-\-n) \ 1 / \ / 1XTU J 1 si"- V (^i — n) t . ,/ X When 771 and n are equal, '— and cos.vfm— 70< V (?« — w) are to be changed to t and unity. (23 b) rxTu I • sin. \/(m-|-7^)f , „ , x When 77^^-7^ is zero, — ; — — - and cos.v (tw+ti)^ are to be changed to t and unity. (24 b) The changes, which correspond to the case when n is zero, are easily made. 197. Definition. A fluctuating function ^\^ one, which. constantly changes its value by a finite quantity for an infinitely small change in the variable, alternately in- creasing and decreasing without ever being infinite. This singular function is of great use in the integra- tion of equations which involve several independent variables ; there is no name in general use, but the one here adopted was given by Hamilton, and is highly ap- propriate. The expression sin. a x, is an instance of such a function, when a is infinite ; and, in this instance, it is noticeable that the mean value of the function is zero. 228 INTEGRAL CALCULUS. [b. V. CH. X. Linear differential equations with constant coefficients. 198. Theorem. Iff denotes a function of a, which is continuous and finite within the limits a and b, and if N is a fluctuating function of which the mean value corresponding to each fluctuation is zero^ and if the integrations are performed relatively to «, we have fi-N^f^^O. (25 b) Proof. Let the interval between the limits of the integra- tion be divided into portions, each of which is the infinitely small extent necessary for a single fluctuation ; and let the limits of any portion be ? and ? -[~ i' For this portion we may put « = i^ + S (26 b) and the corresponding integral, taken relatively to £, is But by (533) of vol. 1, M, =U + i,/...„ D'f^, (28 b) and, by definition, /5iV^ + , = 0. (29 b) Hence (27 b) becomes = -^^ A ■ N^+. '"• (30 b) But, by integrating by parts, we find f>.N,^^,j:^i3±^fA^ij^ilm., (31b) and the second member of (31 b) is, evidently, an infinitesimal <§> 200.] LINEAR DIFFERENTIAL EQUATIONS. 229 ^ Fluctuating functions. of the (w-f l)st order, and (27 b) is, therefore, an infinitesimal of the same order. The number of all the portions of (25 b) is equal to ^ , and therefore the sum of all the portions (27 b) i is an infinitesimal of the wth order ; that is, this sum is infinitely small, and may be neglected, which gives at once the equa- tion (25 b). 199. Corollary. If we take /■,= /« (32b) and if f is continuous and finite throucrhout its whole extent, (25 b) gives /« • ^a /c = 0- (33 b) 200. Theorem, If the notation of % 198 is adopted, and if x is included between a and 6, loe shall have / l> N^,-^fa = f^f- ILl . . (34 b) Proof. In the identical equation in which i is an infinitesimal, the first and third terms of the second member vanish by § 198, when this equation is substi- tuted in the first member of (34 b). Hence if a = «_2, (36b) we have /. h J^u-zfa ^ ra+i iV,-z fu ^ f-\-i N,f+. J a a — X J « — i « — X *I — i £ ' ^ ' in the third member of which, the integrations are performed 20 230 INTEGRAL CALCULUS. [b. V. CH. X. Fluctuating functions. relatively to «. But f^-^^ differs infinitely little from f^, and, therefore, (37 b) gives r\ I^=il± ^ ff+i. E^ . (38 b) In the same way, when/^ is unity, and x is zero, ' r ^ = r+\ ^ . (39b) J a a. J — I « which, substituted in (38 b), gives (34 b). 201. Corollary. Since we have /,QO (a-2:)v-l p— cc(a-a;)v-l f' g>((x-a)v-l — -^^ ~" («_x)V — I 2 sin. CO (« — x) a — X (40 b) the first member of (40 b) may be substituted for —^ in (34 b), which gives ^ ^ /»' sin. 00 a / . , , \ Si ■ f\ eA(-«)v-i /„ = 2 /. J^ —^ .(41b) 202. Problem. To find the value of /J!!!^. (42 b) Solution. If we put /5«z=«, (43 b) we have /•' sin. /5 a /•' sin. a p' sin. a ^ ... ,. a « J a a t/aa' that is, the first member of (44 b) is independent of the value <5> 202.] lilNEAR DIFFERENTIAL EQ,UATIONS. 231 Fluctuating functions. of 1^, as long as i^ is positive ; so that if yl is the required value of (42 b), we have D^A=zO. (45 b) We have also /•' sin. /? a _ p' ( 1 +a^) sin. (J g a a J a « (I -|- o.^) /•' sin. ^ a /•' «sin. ^a ,.^, . Hence, by putting £=/'^i;i-^, (47b) ''"''"" D,B=f' ""'■/: (48 b) P J a l-|-a2 ^ ' /*i « sin. /? a whence, by (46 b), A = B — DIB. (50 b) This equation may be regarded as a linear differential equa- tion in which ^ is the independent variable, and its integral is B=A + A'e^+ A" e-'^ (51 b) in which A' and A" are arbitrary constants. The values of these arbitrary constants may be determined from the extreme values of Ds B. When (^ is infinite, the value of (48 b) van- ishes by (25 b) ; but (51 b) gives D^ B = A' e^—A" e-^ , (52 b) which will not vanish, when ,^ is infinite, unless A' — 0; (53 b) whence Ds B = — A" e'? . (54 b) 23'2 INTEGRAL CALCULUS. [b. V. CII. X. Fluctuating functions. Again when ^ is zero the value of (48 b) is ^ ; and although (i may never be supposed quite so small as zero, yet when it is an infinitesimal, (48 b) must differ infinitely little fiom tt^ and therefore, by (54 b), 71 = — A", (55 b) whence D^ B =z rv c~^, (56 b) Finally, the comparison of (47 b and 48 b) gives, by {^Q b), B^fiD^B = ^{\-c-l'). (57 b) Hence, by (51 b), A^.-= f ^l^ = r' ?i^^i^. (58b) 203. Corollary. The substitution of (58 b) in (41 b) gives f. = i-„f^-f'>. e^^-'^'-V". (59 b) provided the integral between the limits a and h is performed relatively to «. 204. Corollary. \'i fa is the same as in (33 b), (59 b) gives f.:^iKSLS\ ■ e'-^-"''-V<^ -=hfa',, ^«^-"'^-'/»- (CO b) 205. Corollary. In the same way, we should have L.f= i-LA- e«^-'')^-^ A . ^ , (61 b) f..v = ~2n f? r, • e""-'-*^'-' /.. ,, ■' (62 b) whence, by substitution, <§> 208.] LINEAR DIFFERENTIAL EQUATIONS. 233 Differential equations with constant coefficients. 206. Corollary. In the same way, ={h')%:? .y.i.,., eP(-«)+."(i'-«+.(-.)]v-i/^ . ^ . .^.(64 b) 207. Corollary. The successive differentiation of (60 b) gives mfl = i-nJL ^ ( - 1 ) 1 / e^(-°)^-l /„ ; (65 b) and, in the same way, by the successive differentiation of (64 b), the factor ;. s/-\ is introduced under the signs of inte- gration for each differentiation relatively to x, the factor u\/-l for each differentiation relatively to y, &c. 208. Problem, To find several functions JCt , Yi , (Sfc. of the independent variables t, a:, y, S^c. which sat- isfy given linear differential equations with constant coefficients between various differential coefficients cor^ responding to the different independent variables^ and lohich become given functions X^ , Yq , Sfc. of the va- riable X, y, <^*c., when t becomes zero. Solution. Let Xt , IT^ &c., Xo , ITo , &c. represent the values of JT^, F^ , &c., JTo, Fq, &c. when x, y, &c. are changed into «, ^, &c. ; so that by (64 b) if n denotes the number of the variables x, ?/, &,c., ^'=(i^)V« .'ft &c. A .,„, &c. e['(-«)+"(y-«+&<=-]y-l X, (G6 b) &c. If now R = L (67 b) represents one of the given equations, in which 72 is a linear function of the differential coefficients with constant multipli- 20* 234 INTEGRAL CALCULUS. [b. V. CH. X. Differential equations with constant coefficients. ers, and Z. is a given function of t, x, y, &c. ; and if 3L de- notes the value of L when x, y, &,c. are changed to «, i^, &c. ; and Iv tlie value of R when Xi ^ Y^, &c. are changed to Xm ITr , ^-c., and Z)^ , Z>j^, &c. are changed to ^\/-l, ^>c...„&c. .[^-(-«)+"(i/-^;+&-]v-i (3tl-2L)=0, which is satisfied by putting ^^IL, (69 b) and this equation involves no other differential coefficients than those taken relatively to t. By this substitution, therefore, all the given equations are similarly transformed, and the problem is reduced to the inte- gration of several linear differential equations with constant coefficients, in which there is only one independent variable; and this integration is performed by the method of § 1 79 to "^ 195, The functions to be determined are, in this new form, Xt , "^t ) &c., of which the initial values are ^o > ^o j ^c. 209. Corollary. It may be observed that for a complete solution, the initial values JTq , Yq &lc. of some of the dif- ferential coefficients D^ X^ , D^ Fj , &,c. should also be given functions of x, y, &c. 210. Examples. 1. Integrate the equation Dl X, + D] X, = 0. (70 b) Solution. In this case, the substitution {QQ b) gives <§) 210.] LINEAR DIFFERENTIAL EQUATIONS. 235 Dillerentiiil equations with constant coefficients. whence + fr^ [e^^^-^-^ (l+V-3)e-Kv-l-3)^-^ * - i (i-V-3) ei(-^-i+3) , t\X';^ . . If the values of Xq j X o and X o are written as follows, Xo=/,, X ; = />«/«, X'o = Dlf'S; (71b) we have, by (60 b and Q5 b), /i/I^^o e'(--«v-i e*-(^-l±3).u =/;;+j(^^_3), ; whence we have ^t — ^ {fz-t +/. x+i(l-V-3j£ +/• x-|:i(l+v/-3)t) -i(/:-^ - i ( 1- V-3)/. ;+.kl-V-3). - i( 1+^-3 )/;+i (1+^-3)0 +^(/:'-^i(i+v-3)/.;'+i(i-v-3).-Hi-v-3)/.',+i(i+v-3)e). (72 b) 2.. Integrate the equation a b Dl X, + (« + 6) i>; ., X, + D]X,^ 0. (73 b) 236 INTEGRAL CALCULUS. [b. V. CH. X. Differential equations with constant coefficients. Ajis. With the notation of (71 b), JC,= —laf.-yt - 6/x-at +f'.-U -f'.-at). (74b) a — 6 3. Integrate the equation a'DlX, -f 2aDl ,X, + D',X,=:0. (75b) t Ans. With the notation of (71 b), X, =/_. + ^/U. + at D,./,_a.. (76b) 4. Integrate the equation aD,X,+ D, X, = 6"^^+"^ (77 b) Solution. The value of X^ in this case is pmt-{-nC( p — a tX-J — l + 7i« m -\- a X /s/ — 1 whence, by the notation of (71 b), The value of the definite integral in (79 b) is found from the equation .1- y;2^ e;i(x-«)v-i ew« — g^a; ^ (80 b) which, multiplied by -^ e™'^ and integrated relatively to x gives i r'^ ' ^-i_ — i _ ; (81b) ^n J a.X a m'-\-a i s/ — 1 « m'-\-a n and this equation divided by e'"'"' is, by substituting m for a m', <§> 210.] LINEAR DIFFERENTIAL EQUATIONS. 237 Differential equations with constant coefficients. The results, successively obtained from (82 b) by multiply- ing by e'"' , and again by substituting x — a t for x, reduce the value of the definite integral of (79 b) to f,mt-\-nx pn{z — at) J- , (83 b) m -\- an and the value of X^ is obtained by substituting (83 b) for the definite integral ; so that X,z=/_, +(83b). (84 b) Corollary. When m = — a 7i, (85 b) (83 b) is reduced to t c'^(,^-«". (86 b) 5. Integrate the equation aD,X, + D,X,=:z tx. (87b) Ans. X,-=if,^a «+^— 2 — J a;2 ^ + J « .T #2 _ ^ «o ^3. (88 b) /4 a 6. Integrate the equation Dl X, + D\ X, = (x2 + ^2) ,xt, (S9 b) (90 b) 7. Integrate the equation aD.vX^-^b D,X,-\-D,X^=lc'^^'+^y-^^t, (91 b) 7 ph .V-\-k y r,,7n t „—(h a+k b)n ah -f-o k -j- m ' which, when m = — ah-\-hk (93 b) is reduced to X,=/,._,,, y_j,+/^e''-^ + Ay-(A« + i6;«. (94b) 238 INTEGRAL CALCULUS. [b. V. CH. X. Diirerential equations with constant coefficients. 211. The integration of linear differential equations, in which the coeflicients are not constant, can only be performed in some particular cases, some of which will be found in some of the following chapters. «5>212.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 239 Equations of the first order. CHAPTER XI. INTEGRATION OF DIFFERENTIAL EQ,UATIONS OF THE FIRST ORDER. 212. To integrate a given differential equation of the first order, between two variables x and t. Solution. Let t be the independent variable, and let the value of Z>jX be found from the given equation in the form M D,x=--. {95 b) The integral of this equation must involve an arbitrary con- stant a, from which the value of a can be found in terms of t and X in the form a=A,, (96 b) in which A^ is a function of t and x. The differential of (96 b) gives ^-DxA,.D,x + D,A,, (97 b) DA JJ^XZZZ Hence, by (95 b), ^'^=-i^- (^^'') D^_ll_±M. (90M in which x is wholly arbitrary, and may, therefore, be taken of such a value that ^N-DxA,, (Ic) which gives ^ M— D^ A^; (2c) 240 INTEGRAL CALCULUS. [b. V. CH. XI. Equations of the first order. whence, by the elimination of -4,, Dl,A,^D.(>-N)^D,(xM). (3 c) There is no general process of finding a value of ^ which will satisfy (3 c), and this problem must be solved in each case by the exercise of the ingenuity. When the value of ^ is found, (1 c and 2 c) give a^A,=f,{^M)^f, [7.N), (4c) in which a is the arbitrary constant. 213. Corollary. An arbitrary function of x will be added to the third member of (4 c) to complete the integral, and an arbitrary function of t to the fourth member of (4 c). But these arbitrary functions are at once determined by the con- ditions that the third and fourth members are equal. 214. Corollary. The value of a is usually determined by the condition that x is to have a certain value x^ , when t be- comes T. Ifj then, A^ denotes the value of At when t and x are changed to ^ and x^ , (4 c) gives A,-A^ = 0. (5c) 215. Corollary. It is often the case that the given equation is such that it cannot be reduced to the form (95 b), and in this case the whole process must be Jeft to the skill of the geometer. 216. Corollary. If iW and N are such functions of x and t, that M=3Ia:3l, iV=iY^-iV, (6 c) in which 31x and Nx are functions of x alone, and M^ and Nt are functions of t alone, the value of ^ may be assumed <5»217.] LINEAR DIFFERENTIAL Eq,UAT10NS. 241 Homogeneous equation. For this assumption reduces the two last members of (3 c) to zero. The equations (4 c and 5 c) give 217. Corollary. When M and TV are homogenous functions of the same degree m, the vahie of ^ is ^=i{Nx + Mt)-'. (9 c) Hence 7r^ = Nx-{- 31 1 (10 c) 2>.r ^ = — A2 (iv+ X D.vN+t Dv M) (U c) A ^- = — ^-^ {M -\-x D, N + t D, 31) (12 c) Dx {^- 31) = — -^^ (31 N -\- 31 X Ds N'-N X D,v31 ) (13c) D,(^-N) — — 7.'i(3IN—3ItD^N+NtD,31). (14c) _, , . X But, by puttmg y = — 31, r ■ r . the expression — becomes a function of i/ alone, which may be denoted by 31', whence D.r M'=D, M. D,y= \ D, M'= ^ (15c) X D, 31' = I),^ 31' . D, ij =z — — D^ 31' D, 31 m 31 (16c) and, therefore, X D:cM= — tD,31+m3I; (17 c) and, in the same way, X D:,Nz=z — tD,N+mN, (18c) 21 242 INTEGRAL CALCULUS. [b. V. CH. XI. Infinite number of multipliers. which, substituted in (13 c), give by (14 c) Dx{} M)=—).\MN-Mt D,N+Nt D,M)=DlxN). (19c) ' Hence (3 c) is satisfied, and (4 c) gives _ p M _ ^ N ""-J tNx-YMt-J:cNx+Mt' ^^^^^ 218. Corollary. If h is any function whatever of a, and if Bt is the same function of ^^ , (96 b) gives b = B,. (21 c) It may be shown, precisely as in §212, that if ," is such that uN=^D^.Br, (22 c) u will be a value of ^ capable of satisfying the equation (3 c). If, however, b' is the differential coefficient of b taken rela- tively to a, and if ^^ is the same function of A^ which b' is of a, we have Da;B:= B[D^A,, (23 c) whence (22 c and 1 c) give ^cN—^-B'tN or u — 7.Bl\ (24c) that is, the product of any value of ^ by any function what- ever of A^ is itself another value of i. 219. Corollary. Whenever M and N can be separated into such portions M , 31', 31'", &.C., and N', N", N'", &c., that the equation (95 b) can be integrated when for M and N are substituted 31 and iV', or M" or iV", &c., the inte- gral of the equation itself is often readily obtained. For this purpose, let ;.' and A[ represent the values of/ and A ^ which cor- respond to M' and N', '■" and A'\ those which correspond to <§> 220.] DIFFERENTIAL EQ,UATIONS OF FIRST ORDER. 243 Equations of the first order. M" and N", &c. it is necessary to find functions ^', (f" &c. o^ A't, A'l &;c., which will satisfy the equation ^■' cp'. {A',) = ;." /. (A;) =z x'" y"; (^V') = &c. (25 c) For if the value of each member of (25 c) is denoted by ij we shall have A J/=: /' /. {A\) 31' -f k" ^p" {A';) M" + &c. (26 c) ^ iV= a' .' ^p'. {A[) N'] = Dj; [^.' cp' {A',) J/], &c. . (28 c) and therefore ;. satisfies (3 c). 220. Examples. 1. Integrate the equation {tX'+T)D,xJ^X+xT' = 0, (29 c) in which JC is a given function of x, and JC' its differential coefficient; and T is a given function of t, and T' its diffe- rential coefficient. Solution. In this case, M = X+xT' N= T+tX' D,N= T' + X' = Z>,. 31, and, therefore, (3 c) is satisfied by ;. = 1. Hence the required integral is — Xt + xT; (30 c) 244 INTEGRAL CALCULUS. [b. V. CH. XI. Equations of the first order. or if X and ^ are the values of X and T when t and x are changed to ^ and x^ , Kt-^x^K — Xt^xT. (31c) 2. Integrate the equation {t cos. X -|-sin. €) D,,x -\- sin. x -f- x cos. ^ r= 0. (32 c) Ans. Tsin. x^-j-.x sin. t ^r: i sin. x-j-x sin. ^. (33 c) 3. Integrate the equation x« £' D^x-{- k x'^' ^^'rz: 0. (34 c) Corollary. When « — «' -}- 1 = 0, the answer is when J/ _ 5 _f_ 1 — 0, it is ^a-a'+l _ ^a^a'+l f ____x__+,,og._ = 0; (37 c) and when both these conditions are satisfied, it is X T^ + ^ x^, = 0. (38 c) 4. Integrate the equation t D,x=:x + \/ (x2 + f2). (39 c) Solution. This is a homogeneous equation, and (9 c) gives ^-' = M t+N X—tX—t X+t \/(x2+i2) — t ^(:t2+^2) . hence, by (20 c), the integral is = \og.W(x^ + fi)-%l (40 c) <§> 220.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 245 Equations of the first order. or V(xa + ^2)__3.— ^(3,2_|_r2)_^^^ (4lc) or t^zzz a^-{-2ax, (42 c) in which a is the arbitrary constant. 5. Integrate the equation 6. Integrate the equation D.x^j—— log.-. (45 c) Ans. {;-iY={;^Y. t46c) 7. Integrate the equation (l+^log. ^)A^=:^-(l + /^Iog. j^ (47c) ^"^- (r) =(f) • (''^'^) 8. Integrate the equation (A r x"+i + A' r' x"'+i) -|-(A: ^"»+i a;'* + k' r'+^ z") Z>, X =r 0. (49 c) Solution. This is a case of § 219, and by putting M' =hr x''+\ M" — h' r' x"'+i , ? / \ M' = k «'«+! X", M" z= A;' ^''+1 x"' j ^ ^ *^^ we have ' ' ( f5l c'i Al - log. ^'^ x^ A'/ = log. <''' x^', s 21* 246 INTEGRAL CALCULUS. [b. V. CH. XL Equations of the first order. and if « and a are taken to satisfy the equation which gives ah -\- ?n = a' h' -\- in', « ^ -f w — «' A;' + n' (53 c) _ {m — m') k'— {n — n') li' h< k — h Id I (m — m') k — {n — n') h (54 c) (55 c) h' k — lik' we may put l-\ __ ^aA+m+l 2:«^+n+l ^ (56 c) and the integral of (49 c) becomes 9. Integrate the equation {^ ax t + 2h e-) D^x + ^ ax^ -\-^ h xt = 0. Ans. a x3 t^-\-h fi x^=a x^ r'^+b x| t^. 10. Integrate the equation (3 a x3 ^3_|_2 6 1) D, x+2 a x^ t^+S b xznO. Ans. a (x3 i2_2;3 t2)-[_6 log. -lA — o. 11. Integrate the equation (^hx + kt-\-a) D,x + h'x-\^k' t + a' =z 0. (58c) Solution. Put, in this equation, x = x'-f-/?, t=:t' + »^ ' (59 c) and we have DtXz=. D^x' =z D^, x', (60 c) whence (58 c) gives (61 c) (hx'+kt'+h(i+kcc-{.a) D,.x'-{-h'x'-\-k't'+h' ?+k'a+a'=0; <5> 220.] DIFFERENTIAL EQ,UATIONS OF FIRST ORDER. 247 Riccati's equation. and if « and ^ are taken such that h^-^ka+a — 0, h'(i + k'a+ a'=:0, (62c) (61 c) becomes (h x'-^k t) D,,%'-\- h' X' + k' t — 0, (63 c) which may be integrated like any other homogeneous equation. 12. Integrate the equation n,x-\- Txz=z T', (64 c) in which T and T' are functions of t. Solution. Let i' = ft T, (65 c) whence D,t' —T (66 c) D,x = D,.xD, t'— T D,.x; (67 c) T' and if T" denotes the vakie of — when t' is substituted for t, (64 c) gives D,.x-{-x=:z T", (68 c) which may be integrated by the processes of the preceding chapter, since it is linear, with constant coefficients. The inte- gral is, if t' is the value of t' when t becomes t, x = x^ c-f '-" + e-'' /V- 2'" t" ' = x^ e-'f'^ ^+ e-^7; T' e-f ^. (69 c) 13. Integrate the equation kx a{t+h'f D,x + t-^h~' (f-{- hy ^^^- ^-^r^Tw+ {k'+i)(t+hr • 14. Integrate the equation D,x-{- hx^ = k r, (70 c) which is called Riccati's equation. 248 INTEGRAL CALCULUS. [b. V. CH. XL Riccati's equation. Solution. Let x' and t' be so taken that - + - ht ~ x't and we have t' ^=TT-+T77F' t' = tm+^l (71c) D,x=:Dr X'. D, <'=(;«+3)r+2 D,,x'—{m+'^)-D,.x' (72c) 1 2 D,x' D,x- h t^ X' f X'2 ^2 —— — ^ -^ — (?/i + 3) — -V (73 c) 1 ^^ ^'= Ti, + 17^3+::;^ (74 c) A— m+3^ ^' A a;' Ax'2f^ Hence h li 1 li ?^+4 D„x^'+h' x'^=k't"^\ (79c) which is of the same form with the given equation. Hence if Riccati's equation can be integrated for any value m' of m, it can also be integrated for the value m determined by (78 c) ; and if it can be integrated for the value m, it can also be inte- grated for the value m'. Let i be determined, so that, 4z m z= 2f-i-l ' (80 c) <5) 220.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 249 Riccati's equation. and (78 c) gives '^^---27T3=--2lktT)Tl^ (^^^) so that m' is obtained from m by increasing i by unity. Hence if the equation can be integrated for any value of /, it can also be integrated for the values of i, which are greater or less by unity, and therefore for any value which differs from i by any integer whatever. But when 2 = 0, (82 c) we have m = 0, (83 c) and Riccati's equation becomes Dtx + h%^ = k, (84 c) the integral of which is t-r- r ^ - J- W Wf'+Wll)s/{h-h X^ ) ^ so that Riccati's equation may be integrated whenever i is an integer either positive or negative. When i z=z ± x, (86 c) we have w =: — 2, (87 c) and therefore this case would only be obtained from the pre- ceding, by an infinite succession of substitutions. This case, however, admits of direct integration, for, by the substitution ^=1^ + 7- (^^'^> Riccati's equation becomes in this case ^•2 2>^ X' + x'2 = yt <2, (89 c) which is homogeneous, and its integral is [2a:+^-V(l+4A')][2z^+r+V(l+4X0] _ /r w(i4-4ft) [2x+/+^V(l+4 /c)][2a;^+T-r>/(l+4 k)] " \t ) (90 c) 250 INTEGRAL CALCULUS. [b. V. CH. XI. Equations of the first order. 15. Integrate the equation P = 0, (91 c) in which P is a given function of D^ x. Solution. By solving the equation (91 c) relatively to D^ x, each of its values will be found to be a constant, one of which we may denote by 7n. Hence DtX = m, (92 c) Avhence x — x^ = 7n (t — t) (93 c) and X — X m = j--f=D,x, (94 c) and the second member of (94 c) may therefore be substituted for Dc X in (91 c) ; and if Q represents the value of P arising from this substitution, the integral of (91 c) is Q = 0. (95 c) 16. Integrate the equation D^ x^ = a9. Ans. {x — x^Y=^a^{t — t)2. 17. Integrate the equation \^{\+D,x^)z=za+hD,x. Ans. ^[{x-x^Y+it-^Y]=, X" z= T, (96 c) in which T is a function of t. Ans. x,^x^= /; V T, (97 c) or the equation which is obtained by freeing (97 c) from radi- cals. § 222.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 251 Equations of the first order. 19. Integrate the equation ^ D^ x^ = t. 4. 3 Ans. {x—x^-\-r'') = U < • 20. Integrate the equation P = t, (98 c) in which P is a given function of 2)^ x. Solution. By putting p z=z D^x^ (99 c) (262) gives x—f,D,x—f,p—f,ipB,t — pt—f, tD,p = pt-f,tD,p=pt-f,.P, (Id) and the integral is obtained by eliminating p between (Id) and the equation obtained from (98 c) by changing DiX io p. 21. Integrate the equation D X Ans. It is the equation obtained by eliminating p and ^ between the equations t =i?4" e^ -{-sin p Tr=p^-f-c^T-f-sin. p^ X X^^^p t p T J (p2 p2^ gP_j_gP^_|_COS.^ — COS.p^. 22. Integrate the equation t-\- a D.xzzzb s/ {\ + D, x^), Ans. It is the equation obtained by eliminating p and p^ be- tween the equations t + ap — 6\/(l+p2) r+ap^=h^{l+p^~) x-x^ = pt-p^r-^ia{p^-p^^)-ip^/(l+p^) 252 INTEGRAL CALCULUS. [b. V. CH. XI. Homogeneous equations of first order. 23. Integrate the differential equation of the first degree, which is homogeneous in reference to the variables x and t. Solution, Let y^z-y DtX=p^ (2d) which gives x =z 1/ 1 {^ ^) Dy% = y Dyt + t^D,xDyt=pDyt (4 d) Dyt _ 1 t p—y (5d) ^'S-t=f -^ , (6d) •7 y p y But the substitution of (2d) in the given equation reduces to an equaticm containing only p and y ; hence the integral (6 d) is readily obtained, and the required integral is obtaine'd by eliminating p and y from (3 d, 6 d) and the given equation in the form to which it is reduced by the substitution of (2 d). 24. Integrate the equation Ans. The equation resulting from the elimination of p and p between the equations X z=z p t ~\-n t ^/ {\ -j- p^) ix//i+z!\ _ ( Pr + ^i^+p? ) Y- 25. Integrate the equation xz=zPt+Q, (7d) in which P and Q are functions of D. x. <5> 220.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 253 Equations of the first order. Solution. Let p= D^x, (8 d) and the differential of (7d) gives DpX=2D^x.Dpt=p Dpt = P Dpt + tDpP + DpQ, (9d) or (p^P) Dpt — tDpP^DpQ, (10 d) which is a linear equation of the first order, by taking ^^ as the independent variable. The integral of (10 d) is an equation between t and p from which p can be eliminated by means of the given equation. 26. Integrate the equation ^ * 711'' ' Ans. The integral is found by eliminating p and p be- tween the equations x~{p—i)t-\-e'^^ , ^ nip Jn(p — pj) t—m{p—p^)e ^ =Te ^ ^^' . 27. Integrate the equation x = tD,x + P, (lid) in which P is a function of Z), x. X — x_ Ans. If P denotes the value which P obtains when "^ is substituted for D^x^ the required integral is tx^-xr={^t--r)P^, (12d) 22 254 INTEGRAL CALCULUS. [b. V. CH. XI. Equations of the first order. 28. Integrate the equation x=t DtX + n\/(l + D, x2). or {tx^—x^Yz=in^{t—'tY-\-n^{x^x^Y. 29. Integrate the equation D,xz=i (Af'+Bx')^"^. (13d) a Solution. Let uz=zxt a , a or X^=iUt~b ; 1 _ 1 ffl^ z=i{A+B uy « « S and 1 h D,u ' ""6(^ + ■Bu'')l~a . — au log. t ru b (A+Bu'y' 1 ~« — au in which u^ = x^r- a 30. Integrate the equation t DtX — X -r(. /•T \/(x'+t^W{D,x^+l) L\F.{x^+t^) in which /. and F. are any given functions. (14 d) (15 d) y+i] *(i6d) <5» 220.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 255 Equations of the first order. Solution. Let r and (p be taken so that t :=: r COS. (p, x z=. r sin. g), (17 d) whence ^2^^2 + ^2, tan. <^ = ? (18 d) (19d) (20 d) Dt x=z sin. cp DtV -{- r cos. D,x^ + 1 = 2>, r2 + r2 Z>, 92, and (16 d) becomes or f.idin. (p.Dt(p =r-^ V^r .F.r^, (22 d) and its integral is /;/.tan.,=/;^^: (23d) in which 2: „ „ , „ /r... iv tan. 9)^ = -?, r|=.a:|+T2. (24 d) 31. Integrate the equation Ans. By the notation of the preceding example, r, rV[r2-(/.r^n- 32. Integrate the equation t Dt X — X X 256 INTEGRAL CALCULUS. [B. V. CH. XI. Equations of the first order. Ans. By the notation of example 30, r*cp V [1 — (/-tan. cp)2] _ J 21 . J (p^ f. tan. (f ~~ * r^ ' 33. Integrate the equation tPt ^ — ^ _r-/ M_Y_.-\-h ^{t^—x^)s/{V—D,x^~)~L\F{t^-x^)} A -y I Ans, By putting X r^ z= t^ — x^. Tan. OP = — X y.2 __ t2 — 372 Tan. CO ^r -^ T T '1: j- (26 d) J the integral is /;./.Tan..=/;.^^ (2rd) 34. Integrate the equation \^{x^-t2)s/{D,x^-l) L\F.{x'^^t2)/ ^J '^'^^ ^ Arts. By putting j,2 zz: x^ -\- t^ f p z=z X tf r| = a?| + '^^» Pr^ ^t '^ ' the integral is £rF.r^=fP f.p. (30 d) 221. Problem. To integrate several differential equa- tions between several variables and their differential cO' efficients taken with respect to one of them regarded as the independent variable. (29 d) <5> 222.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 257 Equations of the first order. Solution. By taking the successive differentials of these equations with respect to the independent variable, as many new equations may be obtained as may be necessary to elim- inate from them, combined with the given equations, all the variables but two, of which one is the independent variable, together with their differential coefficients. The resulting equation will be an equation between these two variables, and the successive differential coefficients of one variable taken with respect to the other, which is the independent variable. In most cases, however, the integration can only be obtained by some ingenious device. Examples of this problem will occur under the subsequent problem. 222. Problem. To find a function v of several inde- pendent variables t, x, t/, Sf'c, which satisfies a given differential equation of the first order^ and becomes a given function of the variables x^ y, (J'c, for a given value T of the variable t. Solution. If D' denotes the differential coefficient with reference to the given function of the variables a-, ?/, (S:c., and if s, p, q, &c. denote the differential coefficients D^ U', D . V, 2>y V, &,c., we have ' D' H' =p D'x-\-q D' 1/ + &i.c. (31 d) If, moreover, x, i/, &c. instead of being independent of ^, were assumed to be certain functions of t, we should have D,^ = s+p D,z + q D,ij + &,c., (32 d) the differential coefficient of which, relatively to D' is D D,^=p D' D,x + q D' D,7j-\- &LC. + D's-\-D,x.D'p + D,y.D'q + &LZ. (33d) 22* 258 INTEGRAL CALCULUS. [b. V. CH. XI. Equations ofllie first order. But the differential coefficient of (31 d) relatively to t is, by this assumption, + D,p.D'x + D,q.D'i/-\-&Lc.; (34 d) and the difference between (33 d and 34 d) is = D' s + D, X , D' p + D,7/ . D' q + 6zc. — D,p.D'x-^D,q.D'y-{-&DC. (35 d) If the given differential equation becomes by the substitution of 5, p, q, &,c. for the differential coefficients of v^, 22 = 0, (36 d) its differential coefficient is 0=:Z>^ R . D' ^+D, R . D's+Dp R . D'p+D^ R . D'q+&.c. +/>^. R . D'x^Dy R . D'i/+&LC., (37 d) which becomes, by the substitution of (31 d), 0=D,R.D's+Dj,R.D p-{-D^R.D'q+&LC. +(p D^ R+D^^R)D'x+{q D^ R-\-D,R)D'y+&LC. (38 d) But (36 d) is the only given equation between the quantities t, X, y, &/C., s, p, q, &c., and cannot, therefore, determine more than one of the differential coefficients D' s, &c. in terms of the others ; so that the value of this differential coefficient, determined from (38 d), must be the same with that given by (35 d) ; and, consequently, the product of (35 d) by D^ R must coincide with (38 d). Hence I D,x D, y D,R~ D,R - D^R = &c. jjD^R^D^R- qD^R+D;R-^'''^^^^''^ ^j -•' I ^x ■-■*' \l -^ yj <5> 223.] LINEAR DIFFERENTIAL EQ,UATIONS. 259 Equations of the first order. and (32 d) gives, by the theory of proportions, each of these fractions, equal to s D, R-\-p Up R-\-q A, Ii-{-&^c. ' ^ ' The equations (36 d, 39 d, and 40 d) may then be regarded as several equations of the first order, with one independent variable, and the values of r, y, &c., />, q^ &c., -^ may be de- termined in terms of t and of their values x^, ?/f &c., jt?^, q &LC.J V corresponding to the value t of t. Since for the value t of ^, v^ becomes a given function of X, y, &c., it is evident that -ip, must be the same function of a; , ?/ , &c. ; and also that p^, q^, &;c. must be the differen- tial coefficients of V^^ with reference to x^ , y^, (Sec. If from the integrals of (39 d and 40 d) the quantities Xr , yr ^-c. are all eliminated, and the value of i// obtained, this value is evidently such a function of ^, a;, y, &:,c. that if ^, x, y, &c. are changed to ^, '^^, y^-, ^-c., ^> will become t/; ; but by the simple change of ^ to ^, V^ must become the same function of X-, y, &.C. which V is of x , y^ , &lc. ; that is, the value of ip obtained by this process of elimination satisfies the problem. 223. Examples. 1. Integrate the linear differential equation of the first order J involving any nnmher of independent va- riables. Solution. This equation may be written in the form r 2>, v^ + XD, V^ + Fl>y V + &c. = M, (41 d) in which T, JT, Y, &bc., M, are functions of t^ x, y, &/C. and V^. 260 INTEGRAL CALCULUS. [b. V. CH. XI. Equations of the first order. In this case (36 d) becomes R—Ts-\-Xp+Yg-\-&LC. — 3I=0, (42d) whence D,R=T, DpR — X, &c. (43 d) s D,R+p D^R + &LC.= T s+X p -\- &LC. = 31, (44 d) and (39 d and 40 d) become 1 D,x Dty „ D^-w The fractions in the first line of (45 d) do not involve p, q, &c., and therefore the integrals of the equations in this line give the required value of V^, without resorting at all to the second line of (45 d). Corollary. Whenever, by the combination of the equations in the first line of (45 d), a number of equations is found equal to that of the variables v^, x, y, &c., and admitting of direct integration, such as D, Z7=0, A F=r 0, &,c., ^ (46 d) the required integrals are C/z= ?7^, F=F^,&c. (47 d) But Z7 , F , &/C. are functions of v^ , x^, y^, &lq,., from which, by the elimination of a; , y^ , &C., the value of V^ may be obtained in terms of C/. , F , &c. ; and the required yalue of V' is, consequently, the same function of U, F, &c. 2. Integrate (41 d) when the quantities T, X, Y, &c. are functions respectively of t, x, y, &,c., each function involving but one variable. <§> 223.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 261 Equations of the first order. Solution. In this case, (45 d) gives whence the values of x, y, &c. are determined in terms of t and constants. The substitution of these values in M re- duces it to a function of t^ which may be denoted by iV, , and we have finally V H N, -r-/:^. («d) and the required value of ^P is obtained by substituting in (49 d) the values of x , y &-c., obtained from (48 d). The values of a: , y , &c. may easily be derived from the values of x, i/, &i,c. by chauging ^, t, x , y , &.C. into t, t", a:, y, &c. ; for the values of x, y, &c. belong to one end of the interval t — r, in the same way in which z , y , &c. belong to the other end of the same interval, so that t^ may be considered as the variable, while t is constant. 3. Integrate the equation t D,rp — x D,V^ = 0. (50 d) Ans. .p =f. V(.r^— <-+t2), (51 d) in which y. denotes the function which V^ is of x when t be- comes T. 4. Integrate the equation X D.V' + t D,x = 0. Ans. V'=/(yj, wherey. has the same signification as in (51 d). 262 INTEGRAL CALCULUS. [b. V. CH. XI. Equations of the first order. 6. Integrate the equation' .„. ,= (4)V.(v'> wherey. has the same signification as in (51 d). 6. Integrate tlie equation atD,-^'-\-bxD:,^ — n^. (52 d) ^7zs.^= (^y-/[^(7)""]. (53d) wherey. has the same signification as in (51 d). 7. Integrate the equation at"" D,y^ + b x'' DxV' = n v>^ (54 d) r 1 , n{m-\)/ 1 1 \-i-("«-i) where y. - f V ± -^JtD ( ^^ L_\"| wnere v^^_/. |^^,._^ ^ (A-1) V''-^ r'^-' )j andy. has the same signification as in (51 d). (55 d) -(fc-i) 8. Integrate the equation t DtV^ -}-y^ D^y^ -{- X z=z 0. where x^ =: x cos. log. yj sin. log. — , and f. has the same signification as in (51 d). 9. Integrate the equation t (b+D, '^)—x{a+Dt V^)-\-rp(b D^^p—a Z?, V^)=0. (56 d) «§> 223.] DIFFERENTIAL EQ,UATIONS OF FIRST ORDER. 263 Equations of the first order. Solution. In this case (39(1 and 40 d) give 1 _ D,x _ D,v^ b V — X t — a ^ a x — b t Hence • i>, i/' + 6 Z), z -[- a = and -qjD.^-^-xDiX-^-tzz^O, the integrals of which are ^ — ^^ + b {x—x^) + a (t—r) = V/2_^2 J^x^ — X^-\- t^ — r^ — 0. Hence _ ^2 _ ^2 _[_ ^,2 _ ^.2 and the required integral is obtained by the elimination of X between the equations [2 V^+b{x-x^)+a{t-r)][h{x-Xr)+a{t-r)]z=x^-x2-\-t^-T^, (57d) V^-/- ^r+b {x-x^)+a{t-r)=0, (58 d) where/*, has the same signification as in (51 d). Corollary. The integral of this equation is, by the corollary to the first example, ^ + bx-]-at =v.(^'^ + x^ + t% (59 d) in which 9 . is an arbitrary function to be determined by the condition that f.x + hxJrar^^.lif.xf + x'^ + ^l (60 d) or it may be that V + 6x-|-«^ is a given function y of V/2 _|_ x2 J^ t:2. 10. Integrate the equation at Dt^ -{-b xb^ -\-c y Dyyp+ &C. =z n 1/^. (61 d) 264 INTEGRAL CALCULUS. [b. V. CH. XI. EqtiJitions of the first order. AtlS. V = in which y. is the function which V^ is of x^ y, &c. when t be- comes r. Corollary. When a = 6 = c =r &c. = 1, (63 d) (62 d) becomes so that -ip is, in this case, a homogeneous function of the nth degree, of ^, x, y, &c. , and (61 d) becomes, by the substitution of (63 d), a proposition applicable to such functions. 11. Integrate the equation in which Z^ is a given function of ^, iT a given function of t^, and L a given function of lt-\-h-{-^y-\- ^-c. Solution. Let M=l, + lx + ly + &LC., (66 d) and (39 d and 40 d) give, in this case, L-^al,~ L-^al, L+a l^ ~ ' n ' ^ ' Hence, by the theory of proportions, and since D, M=D, h+D, h . D, x+Dy ly . B, 3/+&C. (68 d) D,M _ B,.{l,-h) _ DM-ly) B,^' ^gj nL^aM — a{l,—h) ^ a{lc-ly) ' ^ ' ^ ^ where n is the number of the^ variables t^ x, y^ &c. <5>223.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 265 Equations of the first order. Whence, since L is a function of iff, (70d) J ]\lnL+am ^ l—L "^ I -I, J yj n* where ^^ =/. (z^ , ?/^ , &c.), (71 d) f having the same signification as in (52 d) ; and the required integral is obtained by the elimination q{ x^, y , &/C. between the equations (70 d and 71 d). 12. Integrate the equation (65 d) when L = m M, and n — h ^j. (72 d) ■ Ans. The equation obtained by eliminating Jf. between 6 a a fM \ m. n-\-a ^ ,M \mn-\-a (M ■" a (73 d) V/--J,/- = ( J/ ) • (74 d) wherey is the function which ■^p becomes of/,, Zy, &c., when ^ becomes t. When mn -\- a = 0, the integral becomes (nl-M Y-^r, , I M(l,-h\ , , 3/(/,-/ ) 13. Integrate the equation (65 d), when L is any given function of the variables ^, z, y, &,c., and V^. Solution. In this case, the first member of (69 d) must be omitted, but the other member€ give the values of all the va- 23 266 INTEGRAL CALCULUS. [b. V. CH. XI, Equations of the first order. riables expressed in terms of any two of them, and, therefore, the value of L expressed in terms of these two, which two may, for instance, be '^ and t, and the equation will be a differential equation of the first order between two variables, and may admit of easy integration. Corollary. This method may be applied in any case, in which all the integrals hut one of a system of diffe- rential equations of the first order have been obtained^ and the final integral will, depend only upon the integra- tion of a differential equation of the first order between two variables. 14. Integrate the equation (65 d), when Z* is a given func- tion oi l^+lx + ly + &c. + l^ , and L-\-al u— ^^K ^ . (77 d) Ans. The equation obtained by eliminating a:^, y^, 6lc. between the equations aJV 7_7 7_7 7 7 T .r T V I- xp T -^T T where V^ has the same signification as in (71 d), and iVrr= r^^ ^- ^, (79 d) J 31 nL+a31 ^ ^ J[f=?, + Z^ + /, + &c. + 7^, (80d) and n is the number of the viriables t, z, y, &c., and V. <§> 223.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 267 Equations of the first order. 15. Integrate the preceding example when l = mt, M—L. (81 d) Ans. The equation, obtained by eliminating L^ between ( — )nm^a = ^ = — , (82 d where i// is the same as in (71 d). When the integral is m » -)- ^ = ^> (83 d) t — ^ nt — L T V^ ~WT L X (84 d) 16. Integrate the equation TJ^X-\- Y+&LC. = 0, (85d) in which T is a function of t and Z)^ V, JIT a function of x and D^ v^ , Y a function of y and Dy ip , &/C. Solution. In this case (39 d and 40 d) become 1 D,x dt y =z &C. D,T D^X D^Y (86 d) _ -P^p ^ Ay _o A ^ ~ AX A ^ "" 5A2^+i^AJR-?A^-+-&,c. which give the equations A^ A^ + A^ Ai'^o A i'.Ay + A ^-A^^O, d6C., (87 d) the integrals of which are, by denoting the values which X^ Y, &LC. have when x, y, 6lc.^p, q, &:,c. become a; , y^ , &:-c. by X^, Y^, &LC., X=:X. Y= F,&c. (88d) 268 INTEGRAL CALCULUS. [b. V. CH. XI. Equations of the first order. Hence, by (85 d), rz= J\. (89 d) These equations give s, p, q, &c. in terms respectively of t, X, y, &/C., which substiluled in (66 d) give, by integration, These equations give r, ?/, &c. in terms of t, which, substi- tuted in (SG d) give where V is the same as in (71 d). The integral of the given equation is, finally, the equation obtained by eliminating x^, y , &c. between (9J d and 91 d). In making this elimina- tion, it is to be observed that Pt = -0. "^r- &'C., (94 d) V; _,^^ i^.i±PA^^^^±^ (,_.). (95 d) «§> 223.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. > 269 Equations of the first order. Hence, if V'^ is used as in (71 d), and if T T the required integral is obtained by eliminating s, p, q, &;c. between the given equation and the equations ^ sV.T + , I>,^X^j^Y^^_^^^ (97 d) 18. Integrate the equation Ans. The equation obtained by eliminating p between the equations V' =/. [X + 2 « p (^ — T)] _ « p5 (^ _ T) and P^f-'Vx + 'Hap [t — r)l where / and/' are used as in (71 d and 95 d). 19. Integrate the equation {DrPf = b{Dj:^y~. (99 d) Ans. ^=f.[x + {t-^)^b], (le) where /is used as in (51 d). 23* 270 INTEGRAL CALCULUS. [b. V. CH. XL Equations of the first order. 20. Integrate the equation /e = 0, (2e) where 7? is a function of T, X, F, &c., which have the same signification as in (85 d). Solution. It may easily be shown that the equations (87 d, 88 d and 89 d) are applicable to this case. Hence the values of s, p, q, &c. may be found in terms respectively of t, x, y, &c., and these values, substituted in (32 d), reduce the suc- cessive terms to functions respectively of f, x, y, &c. ■ The integral of (32 d) gives, therefore, ^.,^=j\s+f^y+f^^^, + 6.o.. (3e) and the value of V', obtained by eliminating^ , q , &lc. be- tween (2 e, 3 e, and 92 d), satisfies (2e). The values of x , 7/ , &c. are finally eliminated by means of the integrals of the upper line of (39 d), which is freed from s,p, q, &c. by means of (88 d and 89 d).* The functions D^R, Dx -K, ^c,, are functions of T, X^ &c., and therefore by (88 d and 89 d) they are constant, so that the integrals of the upper line of (39 d) become (4e) and the required integral is therefore the result of the elimina- tion of a; , y , &/C. between the equations obtained from (3 e * Note. This last process, which is necessary in order that \fj may become a given function of x, y, &c. when t becomes t, is neglected in the ordinary solution of this question given in (Lacroix, Calc. DifF. et Int., 2d ed., Vol. I, p. 572). <5> 223.] DIFFERENTIAL EQUATIONS OF FIRST ORDER. 271 Equations of the first order. and 4 e) by the substitution of (92 d) and of the value of s obtained from (2 e) by changing T, A", Y, &c. to T ^ X ^ F , &c. 21. Integrate the preceding example when T*, X, Y, &;c. are the same as in example 17. Ans. The integral in the equation obtained by the elimina- tion of X , 1/ , ^c. and 5^ , between the equations obtained from V/ = V^^ + ^ (^— ^) + P. (•^— ^r) + Q't ilZ—^r) +^^' (5 e) T T T and ^r — ^> (^®) by the substitution of (92 d). 22. Integrate example 20, when T=.T'D,^, X=X' D,V^,&LC. (8e) where T', X\ Y', &c. are functions, respectively, of t, x, y, &c. Ans. The integral is the equation obtained by the elimina- tion of 5 i '^r i y^i &c. between the equations obtained from r 't (9e) and -R^ == 0, (lie) by the substitution of (92 d). 272 INTEGRAL CALCULUS. [b. V. CH. XI. Equations of the first order. 23. Integrate example 20, when T= T' {D,rp)\ Xz= X'(l>,t/.)'"&c., (12 e) where T', JC', &c. are the same as in the preceding example. Ans. The integral is the equation obtained by the elimina- tion of 5 , X , v/ , &c. between the equations obtained from us-wt;^-'^ v^' 71—1 _ I J ^ = \ r^ — ^- — — &LZ. (14 e) «/ X m p;-^ X; -^ D. R^ ^ ^ s/X' and (He) by the substitution of (92 d). 24. To integrate the equation jR = 0, (15e) when i? is a function of t, x, y, &-c., X?, , V^ , Z>^ V , &c. and (p where g)=z^I>,t/^ + a?X>, V^ + «Sz>c. — V^. (16 e) Solution. If s, ^, q, &c. have the same signification as in §222, (16 e) gives g) z= ^ 5 + ojp -}- &c. — V', (I'^e) and if the differentials are taken, as if 5,^, q, ^-c. were the independent variables of which t, x, &c. are functions, we have D,cp = t-^s D, t-\-p A X+&C.— D, V^ A t—D,^ J9;a;— &c. ==<, (18 e) § 224.] DIFFERENTIAL EQUATIONS OF F I LIST ORDER. 273 Simultaneous equations. and, in the same way, Dp(p = x, X>, (jp = y, &.C. ; (19 e) that is, t, r, ?/, 6lc. are the difFerential coefficients of q) rela- tively to 5, }), q,&LQ>., and, therefore, the integral of {\o e) may he obtained as if cp were the unknown function, s, p, q, Sfc. the independent variables, and t, x, y, <^'c. the respective diffe- rential coefficients. 224. When the required function -^ is dependent upon several variables, there may be several given equa- tions between its differential coefficients, and the solu- tion is possible, provided the number of equations does not exceed the number of variables. In this case of several simultaneous equations^ as many differential co- efficients maybe eliminated as the number of equations exceed unity ; and the resulting equation may be inte- grated by the preceding methods. It is to be observed that, in the integration of this equation, those variables ma}'' be regarded as constant, of which the correspond- ing differential coefficients have been eliminated. The relation of the required function to the variables, which have been thus regarded as constant, is determined by substitution in the given equations of the result of the integration. The limits of this volume do not, how- ever, permit any examples of this process. 274 INTEGItAL CALCULUS. [b. V. CH. XII. E(]uatiuns of ll)e second order. CHAPTER XII. INTEGRATION OF DIFFERENTIAL EQ,UAT10NS OF THE SECOND ORDER. 225. Problem. To find a function ^ of two varia- bles, X arid t, which satisfies a given differential equation of the second order ^ and which becomes a given function of X for a given function r of t, and its first differential coefficient D^ V , taken relatively to t, becomes another give?i function of x for the same value of t. Solution. Let and let the given equation, by the substitution of these values, become R — 0. (21 e) If D' denotes the differential coefficient of a function taken relatively to either of the given functions of x, we have D'-^^pD'x, Dp = iDx, D's = Q.D'x. (22e) Although X is independent of t, it may be assumed to be an arbitrary function of t, and in this hypothesis V^, p, 5, &c. will become functions of t, and will give DtV' = s -{-p D,x, (23 e) D,y^ = Q + ^^D,x (24 e) ^ 225.] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 275 Equations of the second order. The differential coefficients of (24 e) are D'D.s =z D'o-^ D'Q.D,x + o D' D,x, ^ and those of (22 e) give S (26 e) D' DtP = D^ I D' x-\. i D' D,x D' D,s = D,Q D' X -\-Q D' D,x; which, substituted in (25 e), give D'q = D.c.D' X — Df^ X, D',- , (27e) D^ = D,Q . D, X — D, X . D' Q = {D^Q — D/i.D.x) D'x-\-{D,xfD' ^. (28e) If the values of D' M>, D' p, D' s, D'q and D' a (22 e, 27 e, 28 e) are substituted in D' R=0, (29 e) the resulting equation contains the two arbitrary and indepen- dent elements D' x and D' '^ , the coefficients of which, being put equal to zero, give the two equations D^R{D,x)^ — DR.D,x-]-D,R = 0, (30 e) D^R{D,Q—Dic.D,x) -{-D^RD,c + QD,R + iD^R + pD^R + D^R = 0. (31 e) Whenever, from a judicious combination of the equations (21 e, 23 e, 24 e, 30 e and 31 e), three equations can be found capable of integration, the elimination of ? , o^ and ; between the three integrals of these equations and the equation (21 e) will give two equations between p, s, V, x, \ » ^^ , 5^ , p^ , o^ and z . In the two equations thus obtained, D, V and Dx i// may be substituted for s and p, Dz^ xp^ for p^ , Dl '^P^ for t^ , 276 INTEGRAL CALCULUS. [b. V. CH. XII. Equations of the second order. Di- s for e , and two functions of x for -^p and >• corres- ponding to the given functions of a-, which ■^P and .s become when t becomes r. Between tlie two equations thus obtained, X may be eliminated, and tlie resulting equation is a differen- tial equation of tlie first order, and its integral, obtained by the methods of the preceding chapter, is the required inte- gral. ^ This process is precisely similar to that of § 222, and is derived from the same principles. 226. Corollary. The two given functions of x are wholly arbitrary, and may be altogether independent of each other. They involve, therefore, in the general value of V^, two inde- pendent and arbitrary functions of V, x and t, and which may be independent, not merely in reference to the nature of the functional operations themselves, but in regard to the variables, that is, to the cotnbinations of V', x and t, upon which they depend. This variable or combination is represented by x^ in the preceding section. There must, therefore, in general, be two different values of x , each of which will give a different equation of the first order, the integral of either of which leads to the required value of ¥'. But instead of integrating the two equations of this first order independently of each other, it will be found much easier to integrate either of the equations ob- tained from them by the elimination of D^V^ or DtV^. 227. Scholium. There are many cases in which it is ex- pedient to transform the given equation, before applying this process of integration ; and some of them will be considered among the examples. Whenever the three required integrals cannot be obtained, the preceding process is inapplicable, although the given equa- tion may sometimes admit of integration in these cases, by means of analytical artifices. '5>228.] DIFFERENTIAL E(^UATIONS OF SECOND ORDER. 277 Equations of the second order. 228. Examples. 1. Integrate the equation 2>; V^-|_ (a+6) Dl,rp + abDl^p = P, (32 e) in which P is a given function of x and t. Solution. In this case the equation (21 e) is a _(- (a _j- 6) ? + a 6 a = P, (33e) and (29 e) is D'o^ (a + b) D' Q + ab D'^^ = D^ P. D' x. (34 e) Hence, by the substitution of (27 e and 28 e), the coefficient of D' i placed equal to zero, is {D,x)^-'(a + b) D,x-{-abz=0; (35 e) whence DtX =z a or =:b, (36 e) x — x^=z a {t—r) 0Y = b {t — r). (37 e) The first of these two values reduces (23 e and 24 e) to > (38 e) D,s + b D,p = P, (39 e) in which P may by (37 e) be reduced to a function of t^ and denoted by P^•, hence 5 - ^r + ^ {P-Vr) =fi ^r. (40 e) In the same way, if s'^ , p^ and Pi denote the corresponding values when the second equations in (36 e and 37 e) are em- ployed, we find 5-< + a(p-p;)=/;P;. (41 e) 24 278 INTEGRAL CALCULUS. [b. V. CH. XIL Equations of the second order. These two equations become, by the substitution o( f.x for the values of v^ and D^ V^ when t is r, D:rp+b D,rp-/o(x-at+aT)-bD^f{x-at+ar)=Q, (42e) Dt tp+a Z>, V-/o (x-b t+b r)-a D,f{x-b t+h '^)=Q', (43 e) in which Q and Q' are the values which the second members of (40 e and 41 e) acquire by the substitution for x of its value given by (37 e). The combination of these two equa- tions gives {a — b)D, ^ — afo (x — at-\-a '^)-\-bfo {x—b t-\-b r) -abD^f.(x-at+ar)+abD,f{x-bt+br)=aQ-b Q', (44 e) (a—b) D^ V'-L/o (x— « t+a r)—f, (x—b t+b r) +6 D,f {x—a t+ar)—af(x—b t+b r)=Q'—Q^ (45 e) the integral of which is (a-b) y^+f.fo {x-a t+a r) -f^f, (x-b t+b r) +bf{x-at+ar)-af.(x-at+a'^)=f^(a Q-b Q'). (46e) 2. Integrate the preceding example when P = tx. Ans. The equation (46 e) when its second member be- comes i (a-6) {t-rf [i X (<+2 T) -T-V (a+h) {t+3 t) (<-t)]. 3. Integrate the equation t^- DlV^ + 2tx Dl,-^ + x^ Dlrp = P, in which P is a given function of x and t. <5> 228.] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 279 Equations of the second order. Am. v=/. — + (/«• y- + 7 D.f. — ) (<-,)+ Q, t X in which P^ is the value of P when x is changed to — -, Q is the value of J X J r t^ ' X T^ . when is substituted for x . t ^ 4. Integrate the equation n Mj (^^^) in which P is a function of ^±^ Solution. In this case (30 e) becomes by (20 e), p^ {D, 2;)2 + 2 p 5 i>, a: + s2 = 0, 5 whence DtX^z , (48 e) n ' and by (23 e, 24 e and 27 e), A V^ = 0, (49 e) Dtp = Q -, (50 e) p D,s=o-- = --'^+Pp=-D,p + Pp. (51 e) P p p- * P Hence D,.~~P and ll>,.-=l; (52e) p F p ^ ' 280 INTEGRAL CALCULUS. [b. V. CH. XII. Equations of the second order. S and since P is a function of—, the integral of (52 e) may be directly obtained, and gives — in terms of t, and this value substituted in (48 e) gives '' — \ = -fr J* (^^^) whence, by (49 e), ^ = ^r=/-^5 (54 e) and the required integral is obtained by the elimination of x between (53 e and 54 e). 5. Integrate (47 e) when the second member is zero. Ans. x = F.rp— \, ^ (t — r), (55 e) in which JFis the inverse function of f, and f is the differen- tial coefficient of P taken with respect to its variable. 6. Integrate the equation D^y^.Dlv^ — (Dl^ ^f = 0. (56 e) Solution. In this case (21 e) is . a?_^2_o, , (37 e) and (30 e) is s^ {D,xf + 2QD,x + a = 0, whence q'^ {D, xf + 2 Q o D^ x + o^ = o and (23 e and 24 e) give Z)j 5 = 0, S =z Sr Ap=0, j) = p^ Dt^P = s^+p^D^x «§) 228.] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 281 Equations of the second order. and (31 e) gives whence T T and the required integral is the result of the elimination of x^ between the two equations ^-^^=f^{t-r),. (58e) ^ -/. x^ =fo .x^{t- r) +/: x^ {X - x^) , (59 e) in which the accents denote the successive differential coef- ficients of the functions. 7. Integrate the equation when the value of V^ becomes f x . f^ x when t becomes r, and the value of Dt H' becomes f x.f^x—fx.flx, in which f andy^ are the differential coefficients of/* andyo. Ans, ^ = / (z + ^ — t) . /j, (a; —. ^ + T) . 8. Integrate the equation D]^-'DI^ — -D:V^ = 0. (60e) Solution. In this case, the general form of solution is in- applicable without previous transformation. For this purpose, by putting V^' = D^ V, (61 e) 24* 282 INTEGRAL CALCULUS. [b. V. CH. XII. Equations of the second order. the differential coefficient of (60 e) relatively to t is D'ln^' — DlH^'—'^^D.^' + '^^^rp'^O. (62e) The integral of (62 e) may be found by the general process, which gives Axzrril, 2; — x^=±(< — t), (63 e) ■^ 2 5 2 V As = <^±e = rh Ap + -f-— ^, (64 e) D.pzzzQ:^^, (65 e) Z>, V^' = 5 rh i?, (66 e) and the remainder, after subtracting (66 e), divided by t'^ from (64 e) divided by ty is — it the integral of which is The sum of the equations involved in (68 e) is 2D^ f ,(x-t+r)-^f,{x-\-t-r) ^ f;{x+t-r)-f;{x-t+r) ^ T T _^y_/, (.-,+.)+/.(.+,_.) ^ (69e) in which /j x, and f^ x denote the values of V^' and i^^V^' when t is T, and /q x is the differential coefficient of y, 2 relatively ^0 X. <§> 228.] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 283 Equations of the second order. The integral of (69 e) is 2 TV'' -FA^ + t-r)-F,i.-t + .)^ (69 e') in which F^x and F^x are the integrals o{ f^x and y^ x relatively to x. But it follows from (60 e, 61 e and 63 e) that fo^=fi^ (70 e) f,x=f:x + jf,x, (71c) whence (69 e') becomes 2 T 2>, v^ + /o(^ + <— )+/o(^-« + -) and V' is obtained from the integral of (72 e). ^ 9. Integrate the equation PDf ^-\-S Dl^^ -f rDjV'rzrO, (73e) in which P, S, and T are functions of D^ r// , and D yj. Solution. Let

-2A^.-D.V'D^,V+(l+(AV)2)D^Vr=0. (81 e) By the substitution of the preceding example this equation be- comes (l+p2) Dlcp + 2 p s Dl. (p + (I + s^) D', 9=0, (82 e) 4> 228.] DIFFERENTIAL EQUATIONS OF SECOND ORDER. 285 Integration of equation of surface of minimum extent. which cannot be integrated by the direct application of the general process. If^ however, we put cp' = D,cp, (83 e) the differential coefficient of (82 e), relatively to s, is (1 +p^) Dl (p' + 2 ps Dl^ip' + (1 + s2) Dl ^) - 0, (86 e) the integral of which, found by the process of Ex. 27 of §220, is .-S;--^C--(f^')'].'-> in which 5^ and p should be accented when the lower sign is used. Instead of proceeding with the direct process, we may put ,n=P^^^, n=P^, (88e) 5 5^ S S^ T T and (87 e) gives ^ = ws + V (— 1— w^) = 7is— V(— 1— «^), (89 e) (l+s2)m2— 2psm + (l+p2) = | (l+s2)n2 — 2psw+(l+/?2)=0, ) (^^®) Dr,p =zm D^s, D^p = n D^s, (92e) 286 INTEGRAL CALCULUS. [b. V. CH. Xll. Integration of equation of surface of minimum extent. D,, cp' = {m Dp cp' + D, L. s = — [(I +p^)Dlcp' -\.2ps Dip ' + (1 + s2) Dl cp'] l±f (2pDpCp'-{-2s D,cp'){l +5^) 4 V (-l-w^) V (-1-71^) 4 V (— 1 — wi"2} V ( — 1 — ^2) = 0. (94 e) Hence we find by integration, cp' =z F.m — F.m -^ -\-f. n, (95 e) in which f. n is the function which , X. h\ (4 f ) But if ?7i is less than unity, A"*" will be infinitely greater than A", and the equation (4 f ) becomes P' X"^=- 0, (5f) which is impossible, so that in this case (4 f ) cannot be satis- fied, and (97 e) is not a case of (98 e), and is consequently a particular solution. If »» had been unity, (4 f ) would have been reduced to P' X' = D, X', (6 f ) which is easily satisfied. If m were greater than unity, (4 f ) becomes D, X' = 0, X' = constant, (7 f ) so that a particular solution is only indicated by the condition that m is less than unity. The differential of (2 f ) gives X>,, p' =.mP' {x' — xY-\ (8 f ) which, when x' differs infinitely little from x and m is less than unity, gives D.p^-'^i; (9f) that is, DxP is a fraction whose denominator is zero. <5. 232.] PARTICULAR SOLUTIONS. 289 Particular solutions. The diflferentialion of (96 e) relatively to x gives, by substi- tuting J) for D^ X, D^ R . D^p + i>, 72 =: 0, (10 f ) ''■^ = -d;r- (i»o Whence by (9 f ), i>^R=0, (12 f) provided the numerator cannot become infinity, which will be the case when (96 e) is free from radicals and fractions. This equation (12 f) corresponds to the particular solution, and leads to the particular solution by the elimination of p be- tween it and the given equation (96 e). 231. Corollary. A similar method of finding particular so- lutions may be extended to other differential equations. 232. Examples. 1. Find the particular solution of the equation t + xD^x — d,x ^ {x^-\-t^ — a^). (13 f ) Solution. This equation, freed from radicals, becomes whence (12 f) becomes X {t + xp) =ip (x^ + i^ — «-)• The elimination of /; gives for equation (x^ — a^) (x2 4-^2_a2) = 0, of which the factor x2 _!_ ^2 _ ^2 — (14 f) is the particular solution. 25 290 INTEGRAL CALCULUS. [b. V. CH. XlfL Particular solutions. 2. Find the particular solution of the equation x — tD,x-\-P, (15 f) in which P is a given function of Z>, x. Ans. It is the equation obtained by the elimination of 2? between the equation ^^ < (I6f) and t -\- DpP' — 0, ^ ^ ' in which P' is the value of P obtained by the substitution of p for jD^ X. THE END. /•/