THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 293 



45. ,1 = 44, r = K+fK'i, f= 2r + l . 



tM 



Here the relation to be reduced becomes 



D n __ e [(e - 2)p 2 + e] [(e 2 - e - 1)^ + e] 

 - 



" N,, - D,, - ,> [>* - e 8 ] [> - 6 ;! + c 2 - e] 



The algebraical work has been carried out by Mr. 11. H. MACMAHON, and he 

 reduces the relation to a quintic in b [ , 



A + A,M + A.&* + A,//- + A^ 10 + A 5 fc 20 = (2), 



A = a 1 " (1 - 2a 5a~ + 2" :! + 4a' + a 5 ), 



A 1 = a 7 ( 1 + 2 + 15 +8 - 10 - 10a 5 ), 



Aj = a (i (-15-27+14 +45 +0 -11+0 + 3 + a s ), 



A, = :! (G + 22 + 5 40 - 23 + 20 + 5 - 8 - 3a s ), 



A t = 1 5a - 5a- + <) + J3 5 -- 10 + 6 + 3a s , 



A, = - a 6 (3). 



As before, putting 



the relation can be halved in degree, and becomes 



C'Vf 1 ' + c- (c + 1) (2f + 2r + L) ft 3 + (-' + 3r 4 + Gr ! + 8f- + 4c + 1) a- 



- c* (2t-- + 5c + 3) /t - (t- + 1 )' = (5), 



a quartic in a of similar structure to the one in /x = 22, 33 (3), and capable of 

 resolution in a similar manner into 



[ 2c V + (< + 1 ) (2c- + 2r + 1 ) a - (< + I )]- = ( ! [(< - - 1 ) a + (c + 1 )]' (fi), 



C = 4c(c+l) 8 + l (7). 



Put 



2cV + (c+l)_(2 C l+2c+lla-( C +l) = 2 ^_ 2c _ 1 ( } 



( c _ ! a + (c+ 1) 

 and then 



if 



c = c(u), = 



and 



c 2 2 + (c 3 + 3c+l) + c + l 



(o - l)~a~+ (o"+l) 



a relation which requires interpretation, connecting c, x, and a, elliptic functions of it. 



