( 452 ) 



On the ouutrary by crossing' -/->'/, the integral rwi increases by 



— Ai + T^'n "hi, 

 and as 



(/ lot/ (rw^ — /)'/,! 4" r'u '^/'l 'i ^ll) ^^ '^ '''.'/ '^ ('''"l ' '^'ll) — " -^ ' '''/il '^ '''"l' 

 we finally get 



2m J iT .' h 



Consequently the fiuietion {rwi ; t\{) admits r zeros on Ï", so 

 that any quotient of the squares of two thetas must be a uniform 

 function of position on the undissected surface T with r double 

 zeros and »• double poles. Hence as soon as one of the integrals 

 of the first kind W is reducible, there exist four adjoint curves 

 Zv'i, i^o, -^^3 iii^<^^ -^H, belonging to a pencil, which each separately, 

 letting alone any possible common points of intersection with the 

 fundamental curve ƒ, touch — or at least intersect in two coinci- 

 dent points — the latter r times. The three quotients Ri : R^, 

 R^ : li^j i?3 : R^, being quotients of the squares of two thetas save 

 as to some constant factor, may be taken equal to^jW^ — h,P^^ — ^2? 

 pW — fy, whence 



2 



P ^V=^„[/ Ri A'2 Rs Ri. 



The function p' W being however likewise uniform on T it must be 

 possible to replace the product of the four functions R by the square of 

 a rational function J'\ otherwise said : through the 4r points of contact 

 of the curves R there can be made to pass an adjoint curve -f, 

 the order of which is the double of the order of the curves R, 

 which touches the fundamental curve / in the common points of 

 intersection v/itli the curves R and which for the rest intersects ƒ 

 only in the double points. The elliptic integral itself is now given 

 by the equation 



dRi dRi 



Ri — ^ —R^^J^ 



da: dv 



d W = ^^- dx, 



V 4 A\ A'a ^3 R^ 



