A MEMOIR ON PREPOTENTIALS. 
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plane w = 0, the superficial element dS is = dx... dz. The cases of a less-than-s-dimen- 
sional volume are in the present memoir considered only incidentally. It is scarcely 
necessary to remark that the notion of density is dependent on the dimensionality of 
the element of volume dnt : in passing from a spatial distribution, pdx... dz dw, to a 
superficial distribution, p dS, we alter the signification of p. In fact, if, in order to 
connect the two, we imagine the spatial distribution as made over an indefinitely thin 
layer or stratum bounded by the surface, so that at any element dS of the surface 
the normal thickness is dv, where dv is a function of the coordinates (x, .., z, w) of the 
element dS, the spatial element is = dv dS, and the element of mass pdx...dz dw is 
= pdv dS; and then changing the signification of p, so as to denote by it the product 
p dv, the expression for the element of mass becomes p 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 p 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, 
considered 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 convenient, to regard p 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 + 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, so that the integral is 
B. The potential-plane case; q = — and the surface the plane w = 0, so that 
the integral is