294 
ON THE THEORY OF ELLIPTIC FUNCTIONS. 
[45 
cl cl 2 cl 
Substituting these values of ^ %u, ^ ®u and ^ ©m in the equation (4) in the place 
of the corresponding differential coefficients of X, all the terms vanish, or the equation 
is satisfied by 2 = © (u), and similarly it would be satisfied by X = H (u). 
Assume now 
7tK 1 
K 
ITU 
2 K'’ 
then observing the equation 
d_JE 
dk K '' 
we have 
K 2 kk 
1 r 2 (KIy - KK - K'E) = - 77 
2K 2 kk' 2 ’ 
dX 
7r dX 
d 2 X 
7T 2 d 2 X 
du 
2 K dv 
du 2 
4 K 2 dv 2 ’ 
dX 
v (v* 
E\ dX 
7T 2 dX 
dk 
kk' 2 { 
K) dv 
2 K 2 kk' 2 dw ’ 
•( 5 )> 
whence, substituting in the equation (4), this becomes 
^-4,^=0 
dv 3 dco 
which is of course satisfied as before by X = © (a), or X — H (u), an equation demon 
strated in a different manner (by means of expansions) by Jacobi in the Memoirs 
referred to. 
Consider next the equation 
(EX 
du 2 
— 2 nu ( k'' 2 
E\ dX 
K du 
di, 
+ 2nkk' 2 ^ = 0 
dk 
■(6). 
(n being any positive integer number). Then, by assuming 
7tK' 
CO = n 
K ’ 
111TU 
~~K’ 
we should be led as before to the equation (5). Hence, considering %u or Hu as 
K' 
functions of u and ^ , the equation (6) is satisfied by assuming for X a corresponding 
nK' 
function of nu and . Let \ be the modulus corresponding to a transformation of the 
w th order; then A, A' being the complete functions corresponding to this modulus, 
A / K' 
A = n K ’ S ° ^at equation (6) will be satisfied by assuming X = © / (nu) or 
X = H, (nu), where ©,, H, correspond to the new modulus