The collected mathematical papers of Arthur Cayley.

607]

A MEMOIR ON PREPOTENTIALS.

345

where, in the two solid integrals, 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 V U= 0, V W=0; then
the equation becomes
W d dS- 24 (- VT= UdW dS.
rs — J de
In particular, this equation holds good if U is
1
a +... + (e- w)2} -
41. Imagine now on the surface S a distribution p dS 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 (a',.., c', 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 (x,..,z,w). Then if, as before,
1
U=
{(a - x2+... + ( - Z)2 + (e -w)2}S-'
we have
(d+ d-2 d 2d 2
(lTV' [ u- 4 U 4 (rI+= 1 v'.
u7 ~,, ~ = W - r (- -re)
/ dW' /w' dU 40' 2 84-1
U dS U dS d (s-).
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
aU drt dS = W "dUdS;
d811 d81;
adding these two equations, we have
d W' dw"\ dS dU U, (Jr )s+l '
U + d 17)dS i a-, + Wd dS - F \/ \
C. IX. 44~~~~~~~~~~~~~

C. IX.

44