748 
PEOFESSOE CAYLEY ON PEEPOTENTIALS. 
Also 
dV 
e* q+1 ^- e = —Qi +1 e 2q+2 U— 
Jl 
_*y y-y 
7 2 + A j \j 
V 1 dV 
\P + 6 “ da • 
+ 
l (W dV\ 
+ d y dy A db , 
116. To integrate the equation for Y we assume 
V=0<p, 
where 0 is a function of 6 only, and <p a function of a, /3 ... y (without 6), such that 
V<P=/£<P, 
x being a function of 0 only. Assuming that this is possible, the remaining equation to 
be satisfied is obviously 
Y 2 © . „ d © 
40 
^ + 2 ^{22+ 2 +o(- ?5 4s...+ / A J )}+*©- 0 - 
Solutions of the form in question are 
<P=1 , *=0, 
C _ o <7 _ o. 
:a , *=- 
p+t 
<p=(3 , *= „ 
V+r” a 2 +a 
o _ a0 ;r - ■- 1 i-grr-2 5 ®_l 
V * / 2 + 0./ + 0 V 2 +S ' | ^ ^ 2 + 3'" A 2 + 0)’ 
-u 1 /_ 2y— 2— 6 l • 
1 ^ / 2 +* /+ e ) 
and it can be shown next that there is a solution of the form 
<p=|(Aa 2 +B/3 2 . . . + Cy 2 )+D. 
117. In fact, assuming that this satisfies V<p— ;s<p=0, we must have identically 
+^Ti{-pr s *‘-F- ■■■■ -]&+} 
+ 4* + #{ _ jP P'”— 5 ,2 + 1 } 
Q ) 
-_A_| 
p+n 
— s— 2<7— 1 + 
* V + 0 ^A 2 + 0 
+ ^«{“ s_22_1 + 7 ^ 9 -- + FT«} 
/i 2 + 0 
/ 2 + 9 1 g 2 
+*{l(A a 2 +Bj3 2 ... + Cy 2 ) + D [; 
