ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 277 



by connecting two integrals n and v with the variables Xi and x.2 in the 

 following equations : 



Jo VX Jo V A 



. (2).= r^^^+p±=n,(x-o+n,(a.,). 



J \/ X J s/ X 



In these two equations, when ii is given, the upper limits x^ and X2 

 are not yet determined, there being two of them ; so that we may regard 

 71 and V as independent of each other. This we cannot do when we 

 consider the integrals (I.) and (II.) separately ; for if x was determinate 

 for a given value of « in (I.), then it would be determined in (II.), and 

 therefore (II.) would be determined. 



When u and v are given, the totality of the upper limits (that is, of 

 Xi and x.2) is known ; but these quantities may be permuted, so that 

 x^+X2 and a;,..r2 are definitely determined when n and v are given, and 

 maybe expressed as the roots of a quadratic equation Aa;^ + B.r + C=0, 



where a;i+a;2=— T^=^(",i'),anda;i.a;2=-x-=4'('*>^)- -A-, B, and C are 



functions of u and v, which have definite finite values for all finite values, 

 real or imaginary, of the two arguments u and v ; and the functions 

 ^{ii,v) and yp{u,v) have with reference to the arguments 20 and v four 

 simultaneous independent periods. 



Let the values of x^ and X2, determined from the quadratic equation 

 above, be Xl=A(?^,^•), a'2^/\i(it,'y). 



In these functions it is seen that when one of the arguments goes to 

 infinity, the other becomes indeterminate, and when one of the arguments 

 changes by a constant quantity the other argument is also changed, so 

 that both arguments undergo an alteration at the same time, and the 

 period of one argument is determined by the period of the other ; this is 

 the characteristic property of the periodicity. 



(38) The functions X{n,v) and \]{u,v) are analogous to the elliptic and 

 trigonometric functions, and may be algebraically expressed in terms of 

 functions that contain only one variable.^ 



For let Xi" and X2° be the values of Xi and X2 when we put w=0, and 

 a;/*" and .t,"" the values of these variables when u=0. 



Then from equation (1) and (2) above 



Tl(x^o) + Uix2')=^i ; n,(.T,°) + n,(.r2'')=0, 

 n(a;/'«) + n(x2*'")=0 ; n,(.7j/°') + II,(a;2«")=v. 



Hence 



nCr,") + n(«-/) + u{xn + n(.r2"")=«, 

 n,(.x-;) + n,(x2°) + n, (..-.«») + " i(-'^V°')=^-- 



Owing to Abel's theorem, the two quantities a;, and ajj may be alge- 

 braically expressed as functions of a;,°, a--^", a;/"', and 0:2"" in such a way 

 that 



n(.r,°) + n(x2«) + nCx/o') + n(.T2«') = n(a:,) + U{x2)=n, 



i]j(rc,°) + n,(x-2°) + ii,(.Tr) + "i(^n="iC'^i) + ni(^-2)=^ 



• See Jacobi (Crelle, bd. xxx. p. 183 ; WerJte, bd. ii. p. 85). 



