PROFESSOR CAYLEY ON PREPOTENTIALS. 
701 
In particular this equation holds good if U is = x y. + 
41. Imagine now on the surface S a distribution gdS producing at a point (a ' . . . c', e') 
within the surface a potential V', and at a point (a " . . . c", e") without the surface a 
potential Y"; where, by what precedes, Y" is in general not the same function of 
(a " . . . c", e ") that V' is of (a! ...</, e'). 
It is further assumed that at a point (a ... c, e) on the surface we have Y'=Y" : 
that V', or any of its derived functions, are not infinite for any point (a' .. . d, d) 
within the surface : 
that V", or any of its derived functions, are not infinite for any point ( a " . . . c", e") 
without the surface : 
and that Y"=0 for any point at infinity. 
Consider Y' as a given function of (a. . . c, e) ; and take W' the same function of 
(x . . . z, w). Then if, as before, 
U= 
then 
{(a— a?) 2 . . .+ (c — z) 2 + (e — w) 2 }* s 
(£•■+£+£)»-». 
Similarly, considering Y" as a given function of (a ... c, e) and take W" the same 
function of (#-. . z, e). Then, by considering the space outside the surface S, or say 
between this surface and infinity, and observing that U does not become infinite for any 
point in this space, we have 
I 
dW" 
dx" 
dJJ 
rd8=)W"~dS; 
and adding these two equations, we have 
C TT /dW , dW"\ 7C1 C /„ 7 dU , _ x „, dU\ 7 _ 4 (TiY + 1 tt , 
J U (- -M+Wr) *=J ( W ^+ W "&®) dS -T&k) V ' 
But in this equation the functions W' and W" each of them belong to a point 
(x . . . z, w) on the surface, and we have at the surface W'=W", =W suppose; the 
term on the right-hand side thus is dS, which vanishes in virtue of 
d\ J dU A . 
d^^ds 1 T = v ’ ana the e( l ua tion thus becomes 
J 
(dW dW" 
\ dx' ‘ dx" 
dS 
that is, the point (a . . . c, e) being interior, we have 
-T($s—$)/dW dW"\ 
4 ( r i) s+1 \dx ,Jr ~dx ]l ) 
dS 
{ (a - xf. . . + (c - zf + (e - wyy s ~ * * 
5 A 2 
