1882.] an elliptic function solution. 225 



the constant is |A", so that we may put 



aiso ^- 63 =Hr^y 



sn 2 u dn 2 u 



= 3V3 



(1 + cnw) 4 ' 



and therefore 



1 + x _ 4 sn u dn u 

 l^x~ y (1 + cnw)* 



and also ^= 2#3 i**^ 



1 — y v (1 + en v 



where u + v = f K. 



2 ' 



Since 



snt/vdnlA" 



(1 + cnfAT) 2 2^3' 

 1 + x sn m dn u /l + en t A\ 2 



, „ 1+x sn m dn m /l+cnfAy 



therefore ^ = — ^-y-^ -y— — - — J , 



1 — x sn|A dnfA \ 1 +cn u J 



1 + y _ sn v dn v /l + en § A~\ 2 

 r^y-snfirdnfiJTV 1 + en v J ' 



and denoting sn u, en w, dn u by Sj , c x , d 1 ; sn v, en v, dn v by 

 s a , c 2 , d 2 ; snfiT, cnfAT, dnfA" by s 3 , c 3 , c£ 3 respectively; 



