607] 
A MEMOIR ON PREPOTENTIALS. 
401 
and therefore 
viz. multiplying by this is 
or 
80 
dH 
dd 
dz 
+ 2 PHJ z + 46H = 0 ; 
Jd + ^0 PH2z=0 ’ 
*5 »<**> + h p -°‘ 
viz. substituting for P its value, this is 
1 d 
H 2 z 
Hence, integrating, 
W (H2z) + ¥e( 2q + 2 + pTe + fT~e + ”- + ¥^re)-°- 
H 2 z = 
C6-1-' 
and 
0 = GH 
V/ 2 + O.g 2 + 0 ... h-+d’ 
6~ q ~ l dd 
■ x H 2 \!f 2 + 0. g 2 + 6... A 2 + 0 
G an arbitrary constant, 
, x arbitrary, 
where the constants of integration are C, \; or, what is the same thing, taking T 
the same function of t that H is of 0 (viz. T is what </> becomes on writing therein 
V/s + i 
\V+1 
VA 2 -f t 
V/ 2 + g 2 + ... + A 2 * V/ 2 + <7 2 + ... + A 2 ’ V/ 2 + $r 2 + ... + A 2 ’ 
in place of a, /3,.., y respectively), then 
¿-9- 1 
© = — GH 
i: 
T 2 *If 2 + t.g 2 + t... A 2 + i ’ 
where % may be taken = qc : we thus have 
t~ q ~ 2 dt 
Recollecting that 
+1. g 2 + t... A 2 +1 
a 2 A 2 c 2 e 2 
“ + ^+0 + •'‘ + W+~d + 0 ’ 
so that for 0 = 00 we have a 2 + b 2 + ... + c 2 4- e 2 = 0, the assumption % = 00 comes to 
making R vanish for infinite values of (a, b,.., c, e). 
125. We have to find the value of p corresponding to the foregoing value of V; 
viz. W being the value of V, on writing therein (x, y,.. ,z) in place of (a, b,.., c), 
then (theorem A) 
P = ~ 
F + q) 
2(r*)T( ? + l) 
_ dW\ 
e 2q+l ) . 
do ) 0 
C. IX. 
51