APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 83 



in common which vary with the arbitrary parameters. The sum of 

 the values of the Abelian integral v (a?, y), in which each point of 

 intersection is sucessively taken as the upper limit. 



^ = v (a?„ y,) +...+ •?; (a?^, yfj)=^ f R (a^, y) dx (4) 



is a function of the parameters a. It is proposed to find the analytic 

 form of this function. 



The quantities a?,, x^, ...,«„ are roots of a certain equation. 



A. (.-», a,, a„ . . . , aj =0 (5) 



of degree fi whose coefficients are rational with respect to the para- 

 meters a. If the curves (1) and (3) have no particular position with 

 respect to the axes it is always admissible to assume the correspond- 

 ing value of y in the form. 



y = ^{«^,^i,(^2,-",<^r)' (6) 



Designate now by S Y the total differential of a function V with re- 

 spect to the parameters a^, a^, . . , , a^, then 



S S -K (x„ y,)8x, + ... + U {x^, y^) Bx^. (7) 



Differentiating relation (6), one can calculate in succession Sx„ 8x^^ 

 . . . , Sxfjb and substitute in the expression (7). Replacing the y's by 

 their values yjr, one has for the coefficient of Sa, a rational function 

 of a?j, ajj, . . . , a?„ and of the a's. It is moreover symmetrical with re- 

 spect to the a^'s and consequently the coefficient of Sr, is a rational 

 function of the «'s. Similar relations hold for the other coefficients. 

 Hence 



S S = P, («„ ...,«,) 8a, + P^ {a„ ...,<?,) 8a, + .. . 

 + P, {a„ ...,«,) 8a^, 



where the P's are rational functions of a,, a^, . , . , a^, and 



S = JP. 8a, + F, K+ . . .H-P, S,/, (8) 



The integration cannot introduce any other transcendental functions 



