108 Professor Gayley, On the elliptic-function [May 23, 



we have 



8 = 2B($A 2 + B 2 ) m 3 ( I + en uf (S A 2 +B 2 ) 



X ~ (A+Bf ' ~{2rsnudnw+(l + cnu) 2 [ 3. 

 We find 



3^1 2 + B 2 = 12r 2 en 2 u da 2 u + (1 + en «) 4 , 



= (1 +cn m) (12r 2 (1 - en w) (k' s + k 2 en 2 m) + (1 + en u) 3 }, 



where the term in { } is in fact a perfect cube 



= [1 + en u + r 2 (1 — en u)] 3 . 



(The last mentioned expression is in fact 



= (1 + en uf + r 2 (1 - en u) [3 (1 + en u) 2 + 3r a (1 + en u) (1 - en u) 



+ r 4 (l-cnw) 8 ], 

 where the second term is 



= J 2r 2 (1 - en u) ft (1 + en 2 u) + ±>- 2 (1 - en 2 »)], 

 that is, it is 



= 12r (1 - en u) (k' 2 + k 2 en 2 u)) : 

 we have consequently 



1 _ ,3 = w 3 (1 + en uf {1 + r 2 + (1 - r 2 ) en w} 8 

 {2r sn u dn w + (1 + en uf] 3 



or extracting the cube root y, = Jl — x 3 , has its foregoing value : 

 and the differential expressions are then verified. 



Suppose y= 1, we have 



(m — 1) (1 + en uf + mr 2 (1 - en 2 u) = 2r sn u dn i*, 

 that is 



(m - l) 2 (1 + en uf + 2m (m - 1) r 2 (1 + en u) 2 (1 - en u) 



+ 3m 2 (1 + en u) (1 - en uf = r 2 (1 - en m) {4 - 4Ar (1 - en 2 u)} 



or observing that the right-hand side is 



= r- (1 - en u) {(1 + en uf + (1 - en w) 2 + r 2 (1 + en w) (1 - en u)) 

 and multiplying by ±r 2 , the equation becomes 



= |(m - l) 2 ?' 2 (1 + en uf + (2m 2 - 2m + 1) (1 + en uf (1 - en u) 



+ (m 2 - 1) r 2 (1 + en u) (1 - en uf - (1 - en uf ; 

 viz. this is 



= {lr (> 2 - 1) (1 4- en u) - (1 - en a)} 8 , 



