ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 273 

 further write 



then 



d\og<f>{x) _ /^(x) d\og4^(x)^f^(x) 

 dx -f{x) ' dx fix) ' 



(a)\^^-^^-^-4^(x) 

 ' }(x-a)<p(x) 



^'"^ I (a; -a)<p {x) " ' 'J (a - .r). ^P (a) 



da 



is equal to an aggregate of products of the form 



„ Ca"'da 1 1 

 J f (« 



da Cx"dx 



}jwy 



■where ??i and ii are integers, and the quantities C,„ „ are constants. 



By making use of certain theorems from the theory of linear diffe- 

 rential equations, Abel and Jacobi extended the above results. Formula 

 are derived in which the products of the sums of n such integrals remain 

 unchanged when the argument and parameter are interchanged ; and by 

 means of these formultB n"^ integrals may be linearly expressed through n'^ 

 other integrals, in which the parameter and argument have been inter- 

 changed ; and vice versd, these n"^ integrals may in turn be expressed line- 

 arly through the first n"^ integrals. See in this connection Clebsch and 

 Gordan, 'Theorie der Abelschen Tunctionen,' p. 114, where some inter- 

 esting consequences are deduced.' 



(i dx 

 (34) Abel, 'Sur I'integi-ation de la formule differentielle --==, E, et p 



\/ R 



^tant les fonctions entieres ' ('CEuvres,' t. i. p. 104), gave the conditions 



/• die 

 under which the integrals ~7^ niay be expressed through functions of 



the form log ^^, where w and q are integral functions.^ In another 



2y — q-JK 



memoir, published after his death (' CEuvres,' t. ii. p. 87), the general pro- 

 blem is solved, when may an elliptic integral be reduced to algebraic- 

 logarithmic functions ? Weierstrass (' Math. Werke,' bd. i. p. 227) says 

 tliat the general problem of integrating an algebraic difi'erential by means 

 of logarithms, in so far as this is possible, was first proposed by Abel, 

 who had arrived at very important results, as is seen from a letter written 

 to Legendre (' ffiuvres,' t. ii. p. 271), and it is very probable that just these 

 investigations led him to his celebrated theorem. 



Abel (' CEuvres,' t. i. p. 549) derived the more general theorem relative 

 to the form which one must give to the integral of any algebraic function 

 when it is possible to express this integral by means of algebraic and loga- 

 rithmic functions and elliptic integrals : 



^^ 2/i) 2/2) • • • ?/^ he algebraic functions of x,, x^, . . . x^, and let 

 the x's be connected by any number of algebraic equations. 



' (y. also Weierstrass, WerJte, bd. i. p. 113, where the source of the proper'y of 

 interchange of parameter and argument is revealed ; Frobenius (Crelle, bd. Ixxiii. 

 p. 93). 



^ Cf. papers by Tschebyscheffi (Liovv. Journ. 2nde serie, t. ii. p. 1); Pick (^Sitz- 

 vngsb. der kaiserl. Akad. der Wmenschaften in Wien, 1882, p. 643); Plana (Crelie, 

 bd. xxxvi. p. 1). 



1897. T 



