676 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
then changing the signification of g, so as to denote by it the product g dv, the expression 
for the element of mass becomes § dS, which is the formula in the case of the superficial 
distribution. 
The space or surface over which the distribution extends may be spoken of as the 
material space or surface; so that the density g is not =0 for any finite portion of the 
material space or surface ; and if the distribution be such that the density becomes =0 
for any point or locus of the material space or surface, then such point or locus, consi- 
dered as an infinitesimal portion of space or surface, may be excluded from and regarded 
as not belonging to the material space or surface. It is allowable, and frequently con- 
venient, to regard q as a discontinuous function, having its proper value within the 
material space or surface, and having its value =0 beyond these limits; and this being 
so, the integrations may be regarded as extending as far as we please beyond the material 
space or surface (but so always as to include the whole of the material space or surface) — 
for instance, in the case of a spatial distribution, over the whole (s-f-l)dimensional 
space ; and in the case of a superficial distribution, over the whole of the s-dimensional 
surface of which the material surface is a part. 
In all cases of surface-integrals it is, unless the contrary is expressly stated, assumed 
that the attracted point does not lie on the material surface ; to make it do so is, in 
fact, a particular supposition. As to solid integrals, the cases where the attracted point 
is not, and is, in the material space may be regarded as cases of coordinate generality ; 
or we may regard the latter one as the general case, deducing the former one from it 
by supposing the density at the attracted point to become =0. 
The present memoir has chiefly reference to three principal cases, which I call 
A, C, D, and a special case, B, included both under A and C : viz. these are : — 
A. The prepotential-plane case; q general, but the surface is here the plane w= 0, 
C qdx ... dz 
J {{a-x)*... + {c-z)* + e*} is+9 ' 
B. The potential-plane case; q = — and the surface the plane w— 0, so that the 
integral is 
qdx ... dz 
Si 
(a — x ) 2 . . .+ (c— ,s) 2 + e 2 } 2S 2 
C. The potential-surface case; q=—^, the surface arbitrary, so that the integral is 
q dS 
J {(a— x) 2 . . . + (<?— zY+ {e—wf} hs ~ k 
D. The potential-solid case; q=—\, and the integral is 
