THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 291 



V + ric 4- a + c = (10), 



- 1 a 4. 



A == (1 + )2 -- 4a ;i - (1 - a) (1 + 3 + 4a s ) (12) ; 



so that, with 



a = a(). = J"^ (13), 



then 



c = (w ^ w) (14) 



(KIEPKRT, ' Math. Ann.,' 32, p. 108). 



Introducing these values into (35), 41, 



,/l x \ 2 - 128^(1 + a- 2a 2 - 3 ) , 1M 



1 U " (1 + ^ (1 - > ! [7 (1 ~ 2 ) + (2 - a + a 2 ) /A] 



to be compared with the expression for , in fj. = 14, L.M.S., 27, p. 440, putting 



r = [ - n , 1 + 2c = l (16). 



2'f 't 



Writing q or j/ for p u , 



_ a s 4. (_ 2 4. rr + rt +) ^ + ( L + a - -) 6 + - ab [ > . . 



(1 + ft) (a - b~)(<r - //) 

 and after reduction 



. > / . \ / . ,, (18) 



or 



,yJ a(t+ 1)7 + fit. = 0, t = n (11 <) (19), 



to be interpreted as an elliptic function relation. 

 Rationalising again 



(1 _|_ fA- ,/' 2 (l + 3) (/ y + 2a a (l + a ) ( / 2 



+ 2a- (1 - a) r/ - 2 (1 - a 2 ) = (20), 



or arranged in powers of , 



a 4 4- 2 ((/ 1) a 3 + ('/'' i ^ f / 3 "I" 2</ 2 -f 2^ 1) a 2 



a 94_ r/ ( 9 _l)a4-? 2 = v/Q ( 22 )' 



Q = 4tf 3 + (<? - I) 2 ( 23 )> 



2 P 2 



