PARTICULARLY THOSE OF TWO VARIABLES. 
817 
By the values of X, Y we have 
±-i /_A_ + A + A + A) 
2 \ dX~ dX» dXj 
A =i /A__A. A + _A\ 
clx% 2 \(^X 1 dX 3 ' dX% dXJ 
A-iLA + i- + L + d \ 
dy~^\ dY~dY~dY s ^dYj 
hence 
d 3 _ & & & _ <& d 3 # d 3 
dx l dyfi r dx % dy 3 dx s d,y s dxfiy^ dX l dY l dX 3 dY 3 dXfiY s dX/lY^ 
(99). 
By means of (97) and the corresponding theorem for U0 ViP (y), an expression is 
obtained for containing 16 terms; substitute this in (98) and transpose 
the operator by (99), and then, by (94), express each term as the product of four d>’s, 
and there will result the theorem already given in (23). 
On the differential equations satisfied by ®\ (^ P p ) x > V \■ 
30. From the theory of elliptic functions it is known that if 
K= 
A= 
d6 
o\/l — id sin 3 6 
% d6 
I q\/1 — A 3 sin 3 6 
K, A, h, A are definite functions of p, q ; viz.: 
^2Ul+-2j>+2y+2p 9 + . . 
/ y/^=l+2 2 '+2 2 *+2 ? 9 + . . . 
/- + 2p* + 2p“ 4- . , . 
K I + 2p + 2p 4 4 2p 9 + . . . 
yy 2^-f 2<p + 2<p 5 + . . . 
V _ l + 22 + 2 ? 4 + 2 2 9 + . . . 
Also K, k are given each as a function of the other by the respective differential 
equations 
5 m 2 
