THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 285 



Next putting 



and with 



.!-/>_, <J* 1+c* 



2 1 - c +" 21 -c 4 



(1 -c 4 ) 2 



6a 3^2 _|_ fe 4 a _ ft -2 



(a 2 + fr~) " 2?>(l a) 



_ _ (1 2ni) a _ pot 



m,(\ m) j^ 1 



s) ( 3a s + a s + (14. 3a) 6 2 } 

 2a(l -?, 2 )( 2 + ft 2 ) 



8 positive, v = oj, + /w 3 , 8 negative, v =/w v 

 p . m 2 a 1 _ S 



MD "a ~ 2 



a m <r c 2 



so that when normalised hy M = ^/Co, D becomes c 2 and R becomes <r" +l . 

 In the sequel it is convenient to put 



_ a - b dn/K' _ o , . 



a+h o dn(l-/)K' 



p = l-2 W = 6 -_^ (19), 



--^W^^^^^T" (20); 



so that 



^ (a - by 



- = 





a - (24) 



a (l _6 z )(a s + 6) 



(1 + a) ( 2 - &') (a + 6 2 ) (25) 



8 



