268 REPORT— 1897. 



solutions of the system (2'.), which contain none of tlie roots of the 

 equation f{x) ^ 0, or presupposes them in any manner. This more 

 general solution is an extension of Jacobi's method, which was effected by 

 the consideration of mechanical problems (art. 24) and throws new light 

 upon Abel's theorem, of which some fundamental forms are derived through 

 suitable integration. 



Writing F (.k) = [x — x^) {x — x.2) . . . {x — x„), Richelot found as an 

 integral of 2'. 



Const. = F(a)\ ^/{^ O + . . . + -y J\^.) V 



'\{x-x,)Y'{x,) {x-x,) F(.x„)j 



in which/ (a;) = Aq +Aia; + . . . + AgnO;-", and a denotes any arbitrary 

 integer. 



Let a take n — \ different values a^, oj, «„_i, and we have a complete 

 system of n — \ solutions of the equations (2'.) Through suitable integra- 

 tions of (2'.) different forms of Abel's theorem are derived in accordance 

 with Jacobi's suggestions in art. 24. 



(28) Jacobi (Crelle, bd. xxxii. p. 220) found that the n — \ algebraic 

 equations through which the system (S'.) was integrable consist of one 

 equation of the second degree in a, and a.2 (where «, denotes the sum of 

 the quantities a:,, .7'o, . . . x,„ a., the sum of these quantities taken two at 

 a time, rtj the sum taken three at a time, etc.), and of n — 2 equations by 

 means of which a^, . . . a„ are linearly expressed in terms of a^ and a^ ; 

 and further, that between any two of the quantities a there exists a 

 quadratic, and between any three a linear relation (cf. also Weierstrass, 

 • Math. Werke,' bd. i. p. 267). 



Jacobi further showed that if we write 



/{x) = {bx" + hix"-^ + b.^x"-^- + . . . + h„y 

 + {ex" + c,a:"-i + c.^x"-'^ + . . . + c„)2 

 — {ax" + aio;"-' + a^x'"'^ + . . . -f rt„)^ 



the differential equations (2'.) are completely integrated, if for x^, X2, 

 . . . x„ are written the roots of the equation 



x" + aj.r"-' + «2^"-2 + . . . -f- «,, 

 — {bx + 61a;"-' +&2^"-2 + . . . 4. /,_) cos <p 

 + [ex" + c^x"-'^ + c.x"-^ + • • + c„) sin f, 



where 6 denotes a variable angle. 



See also Brioschi (Crelle, bd. Iv. p. 56) ; and Cayley (' Camb. and 

 Dubl. Math. Journ.' vol. iii. 1848, p. 116); 'Math. Papers,' vol. i. 

 p. 366. 



(29) Reduction and Transformation of hypereUiptlc integrals. — We 

 noted Landen's substitutions for the elliptic integrals in art. 8. 



Legendre, in the thirty-second chapter of the 'Traite des fonctions 

 elliptiques,' t. i. p. 254, showed in general how to reduce the integrals 



f Vdx 



where P denotes a rational function in x, to elliptic integrals . 



