PARTICULARLY THOSE OE TWO VARIABLES. 
819 
and 
clf__ 
dx 
=K ; 
~d d s Q s dK d s 0 
die dx s K dx dx s 
Hence 
®A+v ! ^=o 
K d* (to" 1 "" rf/c 
and therefore also <E>, the sum of the terms xp, satisfies the differential equation 
d*cf> 
dx 3 + 
. . . (102) 
or restoring the usual coefficient 
in the second term 
d 2 4> 
- 2x ( k2 k)« fc +2 “^ =0 - ■ • • 
... (103) 
Similarly <I> satisfies 
#4) 
dy z 
. . . (104), 
or 
, 2U'^A # , ou „# A 
—r - ^r=° • • • 
A dX J dy dX 
. . (105); 
and from (94) it at once follows that 
d , 2KA d 2 4> „ 
r -—I-— ————0 
dr t r 2 dxdy 
32. All these equations can be deduced from the general definition of <h, viz.:— 
"s (_1 
(10G). 
171 — — 00 n= — CO 
The equation (106) is obviously satisfied. Consider the general term in <1> to obtain 
(102); it is a multiple of 
( 2m +A a 
u=p 4 ewe 
the coefficient being independent of x, k. How 
