PROFESSOR CAYLEY ON PREPOTENTIALS. 
693 
viz. this equation determines § as a function, in general a discontinuous function, of 
(x ... z, w) such that the corresponding integral 
may be the given function of the coordinates (a ... c, e). The equation is, in fact, the 
distribution-theorem D. 
28. It is to be observed that the given function of (a ... c, e) must satisfy certain 
conditions as to value at infinity and continuity, but it is not (as in the distribution- 
theorems A, B, and C it is) required to satisfy a partial differential equation ; the 
function, except as regards the conditions as to value at infinity and continuity, is abso- 
lutely arbitrary. 
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 . . . -j -c 2 -\-e 2 be of the form, Constant -f- by 
(a 2 . . . -\-c 2 -\-e 2 ) is ~ i . 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', V" being 
the values for points within and without the surface respectively, it has been seen to be 
dV' dY" dY' dY" dY' dY" 
necessary that, at the surface, not only V'=Y", but also ~de~~de' 
Subject to these conditions as 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 
(x ... z, w ), the foregoing equation determines g>, 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 Prejpotential 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 that the distribution or density g> 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 Y 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 acces- 
sible from any other such point by a path which does not meet the plane, it is suffi- 
cient 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 Y must therefore be one and the same 
4 z 2 
