ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 263 



Abel's theorem, analogous properties exist for all such transcendents, in 

 which the function X is any rational integral function of x. For, taking 

 the next simplest case, let X be of the fifth or sixth degree in o-, and write : 



xdx 



and further write 



or 



JoVA JoVA 



Jo n/^ 



IX 



n(a;)=A'I)(a;) + Ai*i(a;), 



where A and A, denote constants. 



Then from Abel's theorem it follows that of the innumerable solutions 

 of the equation 



(1) n{x^) + U(x2) + U{x:,)=U{a) + U{b), 

 there is one algebraic solution ; that is, a and b may be algebraically 

 determined in terms of x^, x.2, x^ from the two equations that are derived 

 from (1) : 



(b{x^) + <t>(x.,) + ^x.i)=(i>(a) + <i{b), 

 * 1 (^1 ) + * 1 (-'^2) + * 1 (a^a) = '1' 1 («) + <!> 1 (6)- 



(22) In general, if we write : 



f (A + Aia; + Aoa;''+ • • • +J^m-i^"'~'')(^X -.T\M 



J 7x 



where X=/(x) is a rational integral function of the 2ntth or 2m — 1th degree 

 in X, it follows from Abel's theorem that, if m values Xi, x^, . . . x^ of the 

 variable x be given, through these m quantities it is possible to determine 

 (art. 18) in an algebraic manner m— 1 quantities a,, a.^, . ■ . «m-i) which 

 satisfy the transcendental relation : 



n(a;i) + n(^2)+ • • • +n(a:J=n(a,) + n(a2)+ . . . +n(«,„_i) ; 

 and Abel further showed that the quantities a,, «2, . . . cr„,_i are the roots 

 of an algebraic equation of the m — lth degree, and that each of the 

 coefficients in this algebraic equation may be rationally expressed in terms 

 of the quantities : x,, x^, . . . a;,,,, and 



V X7) n/ >^. • • • v^. where X;^=/(a;,) ,x=i, 2 . . . m)- 



It also follows from Abel's theorem that, when any number whatever 

 of values of x are given, tlie sum of the transcendents Tl(x) which 

 belong to given values of x maybe expressed through m—l transcendents 

 n(x') when the m - 1 values of x in these transcendents are algebraically 

 determined from the given values. 



(23) We consider next the case where the sum of four transcendents 

 are expressed as the sum of two, and where the arguments of these two 

 transcendents depend algebraically upon the arguments of the first four. 

 As above, we write : 



r_^ = ${.)andr4^=$,(4 



