THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 241 



in (2), 11, makes the ellipsotomic relation y u = equivalent to 



a-c z 2ac - a c I = O (2), 



an addition equation of the elliptic functions a, and c of the argument 



C=l + 4c(l+c)* (3), 



a and c differing in phase by one-fifth of a period ; and the five division values of the 

 arguments 2 r r, v = -nr&) 3 , are derivable from each other when considered as elliptic 

 functions of / + i rw > r 0, 1, 2, 3, 4. 



The connexion with KLEIN'S parameter T is made through 



K=-llr, K'' = 4K~(K -- 11) + (10K + 11)'-== 121r'- (4), 



and the quintic transformation 



TT 1 + 4c 



* . c = ,+ 



-- ; r=(^+ 1)* (^ +!)' c ^x 



TT/' _ (2 + 8c + 12<~ + 9c 3 - c* - 3c s - c' 1 )- C 



c(l+e) 

 and then 



H=K 3 - 110HK - 121 (H + K) + 1331 = O (6), 



an elliptic-function addition equation (L.M.S., 25, p. 244; 27, p. 428). 



If analogy is to help us in passing from /x = 1 L to 19, the ellipsotomic equation 

 y, 9 =r should he reducible to the relation 



H 3 K 3 - 152HK -- 3G1 (H + K) = O (7), 



where, in terms of FBTCK'S T and T' (' Math. Ann.,' 40), 



K = - 19r, K'- = 4K 3 + (8K + 19)- = 361^ (8), 



with the addition of the cubic transformations 



__ a 3 _ 5a 2 _|_ 2a + 1 w , _ (a 3 - 2a - 2) 2 {4a 3 + (2a + I) 2 } / y x 



-il " } -Li V / > 



a 2 a 



and K, K' the same functions of 6. 



This combination of (7) and (9) leads to an equation of the 12 th degree between 



a = a(u) and 6 = a I u -f : - ) , functions of the elliptic argument 



\ y / 



AL. (10). 



i/rii\')i * 



VOL. CCIII. A. 2 I 



