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XXII. A Memoir on Prepotentials. By Professor Cayley, F.B.S. 
Eeceived April 8, — Eead June 10, 1875. 
The present Memoir relates to multiple integrals expressed in terms of the (s+1) ulti- 
mately disappearing variables (x . . z, w), and the same number of parameters (a . . c, e ), 
and being of the form 
C § dzr 
J {{a-x )*. . + (c-z?+ 
where g and dzs depend only on the variables (x . . z, w ). Such an integral, in regard 
to the index \s-\-q_i is said to be “ prepotential,” and in the particular case q=—^ to 
be £; potential.” 
I use throughout the language of hyper-tridimensional geometry : (x . . z, w) and 
(a . . c, e) are regarded as coordinates of points in (s-j-l)dimensional space, the former 
of them determining the position of an element qdm of attracting matter, the latter 
being the attracted point ; viz. we have a mass of matter = ^ distributed in such 
manner that, dvr being the element of (s-f-1)- or lower-dimensional volume at the point 
{oc . . z, w), the corresponding density is g, a given function of (x . . z, w), and that the 
element of mass gdvr exerts on the attracted point (a . . c, e) a force inversely propor- 
tional to the (s+2g'4'l)th power of the distance {(a— x) 2 . .-\-(c — z) 2 -\-{e — w) 2 \ i . The 
integration is extended so as to include the whole attracting mass J qdvr ; and the integral 
is then said to represent the Prepotential of the mass in regard to the point (a . . c, e). 
In the particular case s— 2, q = — the force is as the inverse square of the distance, 
and the integral represents the Potential in the ordinary sense of the word. 
The element of volume dvr is usually either the element of solid (spatial or (s-j-1)- 
dimensional) volume dx . . dz dw, or else the element of superficial (s-dimensional) 
volume dS. In particular, when the surface (s-dimensional locus) is the (s-dimensional) 
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 d zu : in passing from a spatial distribution, qdx . . .dz dw, to a super- 
ficial distribution, § dS, we alter the signification of g. 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 q dx ... dz dw is =% dv dS; and 
mdccclxxv. 4 x 
