607] 
A MEMOIR ON PREPOTENTIALS. 
337 
The potential (assuming that the matter which gives rise to it lies wholly within 
a finite closed surface) must vanish for points at an infinite distance: or, more 
accurately, it must for indefinitely large values of a 2 +... + c 2 + e 2 be of the form, 
Constant -r- (a 2 -t- ... + c 2 + e^ s ~K It may be a discontinuous function; for instance, 
outside a given closed surface it may be one function, and inside the same surface a 
different function of the coordinates (a,.., c, e); viz. this may happen in consequence 
of an abrupt change of the density of the attracting matter on the one and the 
other side of the given closed surface, but not in any other manner; and, happening* 
in this manner, then V' and V" being the values for points within and without the 
surface respectively, it has been seen to be necessary that, at the surface, not only 
dV' dV" dV' dV" dV' dV" 
V = V", but also 
Subject to these conditions as 
da da de de ’ de de 
to value at infinity and continuity, V may be any function whatever of the coordinates 
(a,.., c, e); and then taking W, the same function of (#,.., z, w), the foregoing 
equation determines p, viz. determines it to be =0 for those parts of space which 
do not belong to the material space, and to have its proper value as a function of 
(x,.., z, w) for the remaining or material space. 
The Prepotential-Plane Theorem A. Art. Nos. 29 to 36. 
29. We have seen that, if there exists on the plane w = 0 a distribution of 
matter producing at the point (a,.., c, e) a given prepotential V—viz. V is to be 
regarded as a given function of (a,.., c, e)—, then the distribution or density p is 
given by a determinate formula; but it was remarked that the prepotential V cannot 
be a function assumed at pleasure: it must be a function satisfying certain conditions. 
One of these is the condition of continuity; the function V and all its derived 
functions must vary continuously as we pass, without traversing the material plane, 
from any given point to any other given point. But it is sufficient to attend to 
points on one side of the plane, say the upperside, or that for which e is positive; 
and since any such point is accessible from any other such point by a path which 
does not meet the plane, it is sufficient to say that the function V must vary 
continuously for a passage by such path from any such point to any such point; 
the function V must therefore be one and the same function (and that a continuous 
one in value) for all values of the coordinates (a,.., c) and positive values of the 
coordinate e. 
If, moreover, we assume that the distribution which corresponds to the given 
potential V is a distribution of a finite mass 
over a finite portion of the 
plane w = 0, viz. over a portion or area such that the distance of a point within the 
area from a fixed point, or say from the origin (a,.., c) = (0, .., 0), is always finite; 
this being so, we have the further condition that the prepotential V must, for in 
definitely large values of all or any of the coordinates (a, .., c, e), reduce itself to the 
form 
:/ 
p dx ... dz ) -r- (a- + ... + c 2 4- e 2 )^ s ' +9 . 
C. IX. 
43