244 
f 594 
594. 
ON A DIFFERENTIAL EQUATION IN THE THEORY OF ELLIPTIC 
FUNCTIONS. 
[From the Messenger of Mathematics, vol. iv. (1875), pp. 69, 70.] 
The following equation presented itself to me in connexion with the cubic trans 
formation : 
Writing as usual k = w 4 , I was aware that a solution was 
where u, v are connected by the modular equation 
w 4 — v* + 2 uv (1 — wV) = 0 ; 
but it was no easy matter to verify that the differential equation was satisfied. After 
a different solution, it occurred to me to obtain the relation between (Q, u)\ or, what 
is the same thing, (Q, k), viz. eliminating v, we find 
or say 
whence also