45] 
ON THE THEORY OF ELLIPTIC FUNCTIONS. 
295 
Assume now in the equation (6), 
Hence, substituting, 
cP 
E\ d 
d ~ 2 (® n u . z) - 2nu (k' 2 - K j du (<ò n u. z) + 2nkk' 2 (KV)- w ^ [(Kk')~% n u.z] = 0 ; 
but 
<**)*a M ®”» • *] - a (e ”“ •- y ~w ®““ • *• 
d®u 
or effecting the differentiation, and eliminating by means of the equation obtained 
from (4) by writing 2 = ©», 
d 
(.Kkj^ dk [{Kk'Y^-v . ¿\ 
= 
2kk' 2 ©m [ d 
Substituting in (6) and reducing, 
(h _ nz id 2 ©w ^ / ^'2 A’X d ©m] w — 1 y 
K) du \ + 2kk' 2 
E 
K 
d 2 z 
did 
+ 2 n 
jL^©w_ / ^ 
©n du V K 
~ + 2 nkk' 2 — 
du dk 
, „ (n _ ix if 1 _ JL 
( _© 2 ?ì V du ) %u du 2 
+ ' 1 ~ K ) < z = 
i.e. 
But 
whence 
dPz 
dip 
+ 2 n 
~rf log %u f E 
du “r K 
+ n (n — 1) 
~ +2 nkk’ 2 % 
du dk 
d 2 log ©w E 
dii 2 
K 
z = 0. 
du 2 
K 
d 2 ^ 
du 2 
dz 
dz 
+ 2nk 2 (f 0 du cn 2 -a) + 2nkk' 2 -j^+n(n—l)k 2 sn 2 u.z = 0 (7) ; 
which is therefore satisfied by 
^ i2Kk'\ * (n-1) ©,?im 
V 7T / 
© ? fi< 
£ = 
[2Kk'\ * (n_1) 
V 7T 
© n it