276 REPORT — 1897. 



Making use of Richelot's transformation, Jacobi wrote the general 

 integral under consideration in the form : 



(A ) r ia + Px)dx 



^ J Va;(l-a;)(l-K-2a;)(l-\2x)(l-^2jc)" 



He distinguished the values of this integral within the six intervals : 



K 



(1) -co ... 0, (2) ... I, (3) 1 . . . 



W^r--^. (=)l^----->. (6)^.-..~. 



When the upper limit rein formula (A.) is considered as a function of u and 

 written x=:\(u), this function behaves like a periodic function within 

 each of these intervals, and therefore seems to have six periods, of which 

 four are independent. Further, this function remains unchanged when u 

 assumes any real or imaginary value, or better expressed, of the values 

 which u may take, there are always those which differ from any real or 

 imaginary quantity by less than any assignable quantity, however small. 



Jacobi found in this a troublesome paradox, which, however, he had 

 already in a measure overcome by means of Abels theorem (see following 

 article). 



Jacobi next proved that if a given function of two variables is an 

 one- valued function of these variables, it is impossible for this function to 

 have more than four independent periods.' 



(37) The inverse functions. — Corresponding to the function .T = sn u 

 ^art. 9), if we try to introduce into analysis a transcendent a; ^\(w), where 



Jos/X 



X being a rational integral function of the fifth or sixth degree in x, then 

 there is no analogy between this function and the elliptic function a;=sn ?*, 

 since such a function, as seen above, has for every value of u not only 

 many values, but is wholly indeterminate,- if for the definition of the 

 integral we consider only the limits and not the paths which the variable 

 describes from one limit to the other. Hence, when we consider the 

 integral (I.) by itself, its inversion does not give useful results. 



The close connection between the integral n(a;) and the integral 



(II.) n,(a:)=r^J 



Jo V X 



was seen in art. 23. 



Jacobi conceived the very fortunate idea of inverting these integrals 



' The more general theorem that a one-valued fimctum of n variaUes cannot 

 have more than 2n indepnidfiit periods was proved much later by Eiemann (Crelle, 

 bd. Ixxi. p. 197). See also Weierstrass {Munatsh. der Akad. der Wiss. zu Berlin, 1876, 

 p. 680; Functionenlehre, p. 166). 



« Cf. Jacobi (Crelle, bd. ix. p. 394 ; Werke, bd. ii. pp. 7 and 516). 



