274 REPORT— 1897. 



If any integral of the form 



J (2/i<^^i+y2«^^2+ • • • +y^c?a;^) 



is expressible in algebraic and logarithmic functions and elliptic integrals 

 in such a way that 



\(yidxi+y.2dx2+ . . . +i/^dx^)=u + AilogVi + A2\ogV2 + . . . +A„log v„ 



+ «l^l(«l) + «2v/'2(<2)+ . • . +«n'/'n(U. 



where A,, A^ . . . , a^, a.i, . . . are constants, ^^, v^, Vo, . . . i], t,^, . . . 

 are algebraic functions of a;,, x^, . . .and ;//,, \p2, . . . are any elliptic inte- 

 grals of the three kinds with any moduli and parameters, then Abel proved 

 that this integral may be always expressed in the form : 



^\(yidxi + i/.^dx2 + . . . 2/''rfa;^)=r + A'logp' + A"logp" + . . . +A«logp<" 



where o is an integer, a,, a,, . . . a„ are the same as in the preceding ex- 

 pression ; A', A" . . . A"" are constants; ^i, A,(f,), B.^, A2 (tfj) . . . 

 6n, A „(y„) ; r, !>', p", . . . p'"' are rational functions ' of the quantities x^, X2, 



• ■ • aJ^ ; yi> 2/2. ■ ■ ■ 2/m- 



Abel remarks (p. 550) that this theorem is not only fundamental in 

 all that concerns the application of algebraic and logarithmic functions and 

 elliptic integrals to the theory of the integration of algebraic differentials, 

 but it includes all the possible reductions of the integrals of algebraic 

 formulce by the aid of algebraic and logarithmic functions. 



As a corollary is the following theorem : If — ^ , whei'e p is any 



rational function of x, and ^(x,c) denotes ±v^(l— a;^) (l—c'^x'^), is 

 expressible by algebraic and logarithmic functions and elliptic integrals, 

 then we may always suppose that 



+ A. log ?l±iV^ + A, log ?l±i<^+ . . ., 



where all the quantities p, q^, 5-2, .. . q{, ?./, . . . y, 2/,, y-^, . . . are 

 rational functions of x. 



From this may be derived a complete solution of the equation 



dy dx 



where £ is a constant, and whence also the general transformation of 

 elliptic integrals of the first kind. 



(35) Similar problems were discussed by Liouville- (' Memoires des 

 Savants Etrangers,' t. v. pp. 76 and 103) before he had seen the methods 

 used by Abel. Liouville says that the problems proposed by him do not 

 differ in their origin from those enunciated by Lagrange in the ' Th^orie 



-—, when 6' denotes any rational function of x, and Am(«) 



A ;ii(a') 



= ±^/(l-a;')(l-c'„ar^). 



' See Poisson's report on these memoirs in Crelle, bd. xii. p. 342, and the note 



appended by Liouville ; also Jiirgensen (Crelle, bd. xxiii, p. 129). 



