607] 
A MEMOIR ON PREPOTENTIALS. 
345 
0. IX. 
44 
where, in the two solid integrals, we exclude from consideration the space in the imme 
diate 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 V £7= 0, V IF = 0 ; then 
the equation becomes 
In particular, this equation holds good if U is 
1 
{(a — x) 2 + ... + (e — ' 
41. Imagine now on the surface S a distribution pdS 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 V"; where, by what precedes, V" is in general not the same 
function of (a",.., c", e") that V' is of (a',.., c', e'). 
It is further assumed that at a point (a,.., c, e) on the surface we have V' = V": 
that V', or any of its derived functions, are not infinite for any point e) 
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 V" = 0 for any point at infinity. 
Consider V' as a given function of (a,.., c, e); and take W’ the same function 
of {oc,.., z, w). Then if, as before, 
{(a — xf + ... + (c — z) 1 + (e — w) 2 p~s ’ 
we have 
Similarly, considering V" as a given function of (a,.., c, e), take W" the same 
function of (x,.., 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 
adding these two equations, we have 
4(r|)« +1 F , 
r (*«-*) *