336 
A MEMOIR ON PREPOTENTIALS. 
[607 
up of an indefinitely small sphere, radius e and density p, which includes within it 
the attracted point, and of a remaining portion external to the attracted point. 
portion VF=0; hence, as regards the whole attracting mass, V V has the first- 
mentioned value, that is, we have 
where p is the same function of the coordinates (a,.. , c, e) that p is of (x,.., z, w); 
viz. the potential of an attracting mass distributed not on a surface, but over a 
portion of space, does not satisfy the potential equation 
but it satisfies the foregoing equation, which only agrees with the potential equation 
in regard to a point (a,.., c, e) outside the material space, and for which, therefore, 
p is =0. 
The equation may be written 
or, considering V as a given function of (a,.., c, e), in general a discontinuous 
function but subject to certain conditions as afterwards mentioned, and taking W the 
same function of («,.., z, w) that V is of (a,.., c, e), then we have 
viz. this equation determines p 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 absolutely arbitrary.