APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 117 



a>l)>G>d. (12) 



In our case we always have J)>x>c, so that according to a well-known 

 formula' 



Je V {x-a){x—h){x-c){x—d) V{a-c){h—d) ^{f>-G){x-d) 



with the modulus K?=k=z) (A.- -i. 



[a — G){b — d) 



Putting this integral, as in formula (5), equal to w, we have: — 



\{J)-c){x-d) \ 2 /' ^ ' 



Via-G){h-d) 

 or, putting -i O L . u='2, 



{x-d){b-c)-^''-^- ^^^^ 



From this 



c(h—d)—d (b—c) sn^s " .._. 



x= — ^^ (lb) 



{^b-d)-{l-e) sn's ^ ' 



For ^=0, x=c={r — ey. The corresponding value of y is easily 

 found as 



y=re+ =P- (17) 



/• — e 



This value of y belongs to the argument v=h, since v — ti=h', hence 

 the constant h is determined by 



/v+ 



'•-* A_^_(j_,),„=(i£fc=2lJi=i!).«) 



Designating the real half -period of sn .z by 2K, we have: — 



(18) 



(>) See Greenhill. Elliptic Functions, pp. 53-55. 



