290 
[45 
45, 
ON THE THEORY OF ELLIPTIC FUNCTIONS. 
[From the Cambridge and Dublin Mathematical Journal, vol. n. (1847), pp. 256—266.] 
Adopting the notation of the Fund. Ffova, except that for shortness sn u, cn u, 
dn u are written instead of sinam u, cosam u, V am u‘, let the functions © (u), H (it) 
be defined by the equations 
@ w 0 * u ' i ( 1_ i° du io du sn % J ^ 
7r (15? - 2iw) 
Hu= — ie %(u + iK) (2), 
it is required from these equations to express sn u in terms of the functions H (u), 
©(w). To accomplish this we have 
d 2 . Id 2 l id V 
y—„ log sn 'll = T „ sn u — —— -y- sn u) 
du sn u du 2 sn 2 u \du J 
= — (1 + k 2 ) + 2Id sn 2 u — j g Jr w — (1 + k 2 ) + k 2 sn 2 wj 
= k 2 sn 2 u — ; 
sn 3 w 
whence also 
d 2 
log sn u — k 2 sn 2 u — k 2 sn 2 (u + iK'). 
If for a moment 
^r,u = du sn 2 u, ‘ X I / '„ U — du I du sn 2 u, 
Jo J Q J o 
then log sn« = k 2 y\r t/ u — k 2 (u + iK') + Au + B;