PEOFESSOE CAYLEY ON PEEPOTENTIALS. 
681 
As to this, remark that V is not an arbitrary function of (a . . . c, e) : non constat 
that there is any distribution of matter, and still less that there is any distribution of 
matter on the surface, which will produce at the point (a . , .c, e), that is at every point 
whatever, a prepotential the value of which shall be a function assumed at pleasure of 
the coordinates (a ... c, e). But suppose that V, the given function of (a ... c, e), is 
such that there does exist a corresponding distribution of matter on the surface (viz. 
that V satisfies the conditions, whatever they are, required in order that this may be the 
case), then the foregoing formula determines the distribution, viz. it gives the expression 
of that is, the density at any point of the surface. 
6. The theorem may be presented in a somewhat different form ; regarding the pre- 
potential as a function of the normal distance a, its derived function in regard to a is 
2g (Tl)T g 
g 2 2 +i s’ r(± s +g)’ 
that is 
and we thus have 
___j_ 2(rJ r )T( g +i) . 
8 2j+ 1 f P(^S + g) ’ 
dw_ _J_ 2(r^r(g+i) 
ds 8*«+ r(i* + g) ’ I s — u h 
or, what is the same thing, 
§=■ 
r(is+g) 
2(r^r(g+i) 
da . 
d W 
where, however, W being given as a function of (x . . . z, w), the notation requires 
explanation. Taking cos «... cos y to be the inclinations of the normal at N, in the 
direction NP in which the distance a is measured, to the positive parts of the axes of 
(x . . . z ), viz. these cosines denote the values of 
dS dS 
dx ' ’ ' dz ’ 
each taken with the same sign + or — , and divided by the square root of the sum of 
the squares of the last-mentioned quantities, then the meaning is 
dW dW , dW 
~di=S co S“---+^ c °sy. 
7. The surface S may be the plane w= 0, viz. we have then the prepotential-plane 
integral 
J{(«- 
qdx . . .dz 
x)^..+{c-z)2 + e*} is - ,I, 
. . (A) 
where e (like s) is positive. In afterwards writing e=0, we mean by 0 the limit of an 
indefinitely small positive quantity. 
The foregoing distribution-formulse then become 
e=wf (eS,W) -’ W 
