700 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
39. This may be accomplished most easily by means of a particular case of the last- 
mentioned theorem ; viz. writing W — 1, we have 
J^S+J<fe...^cZwVU=0, 
or the required value is = — j ^ over the surface of the last-mentioned sphere. 
We have, if for a moment r 2 —x 2 . . .-\-z 2 +w 2 , 
dU /x d 
z d w d\ _ / x d 
z d w d\ 
t dJJ 
dU 
da \rdx" 
' r dz' r dw) ’ l r dx' 
‘ r dz' r dw i 
V' dr ’ 
dr 9 
that is, — 1, ^ ; and hence 
da r s R s 
where JcZS is the whole surface of the sphere x 2 . . . +z 2 + w 3 =It 2 , viz. it is =R S into 
the surface of the unit-sphere^ 2 . . . -\-z 2 -\-w 2 = 1. This spherical surface, say 
c d2 i9 
JUZ is — r i( s+1 )» 
4 (ny 
(s-i)T%{ s -iy 
J du 4(ri) s+i 
-j- dS— px(/_-i) , an d consequently 
f dx . . . dzdwV 
J r(is-i) 
40. Treating in like manner the case where W at a point indefinitely near the point 
(a, . . . c, e) within the surface becomes 
l 
— {{x-af . . . + ( 0 - c) 2 + («,— e) 2 }^’ 
and writing T to denote the same function of (a, . . . c, e) that IT is of (x ... z, w), we 
have, instead of the foregoing, the more general theorem 
Jw ^ <ZS+ Jfe . . dz dw WVU-ifij^A V 
=Jxi^<JS+j’&...*fcuvw-i|pA ) T , 
where in the two solid integrals respectively we exclude from consideration the space 
in the immediate neighbourhood of the two critical points (a ... c, e) and (a . . . c, e) 
respectively. 
Suppose that W is always finite within the surface, and that U is finite except at 
the point (a ... c, e ), and moreover that U, W are such that VTJ=0, VW=0, then 
the equation becomes 
f„ju JO 4(rw +l T7 . r TT <iw 70 
