607] 
A MEMOIR ON PREPOTENTIALS. 
343 
36. It is assumed in the proof that 2q + 1 is positive or zero; viz. q is positive, 
or if negative then — q ^ 2 > the limiting case q= — \ is included. 
It is to be remarked that, by what precedes, if q be positive (but excluding the 
case q = 0), the density p is given by the equivalent more simple formula 
_ T + q) 
P (0)Tg 
(eWW),. 
The foregoing proof is substantially that given in Green’s memoir on the Attraction 
of Ellipsoids; it will be observed that the proof only imposes upon V the condition 
of vanishing at infinity, without obliging it to assume for large values of (a,.., c, e) 
the form 
M 
{a 2 +... + ^ + 
The Potential-surface Theorem C. Art. Nos. 37 to 42. 
d 2 d? 
dx 2 1 dz 2 ^ de 2 
3? 3? 3? 
37. In the case q = — \, writing here V = + ...+ , we have, precisely 
as in the general case, 
JW-J^dS+J dx... dzde FV77 = j U^dS + f dx...dzde tTV W; 
and if the functions U, W satisfy the equations V U — 0, V W = 0, then (subject to the 
exception presently referred to) the second terms on the two sides respectively each of 
them vanish. 
But, instead of taking the surface to be the surface positive infinity together with 
the plane e = 0, we now leave it an arbitrary closed surface, and for greater symmetry 
of notation write w in place of e; and we suppose that the functions U and W, or one 
of them, may become infinite at points within the closed surface; then, on this last 
account, the second terms do not in every case vanish. 
38. Suppose, for instance, that U at a point indefinitely near the point (a,.., c, e) 
within the surface becomes 
1 
}(« — a) 2 + ... + \z — c) 2 + (w — e) 2 p~* ’ 
then if V be the value of W at the point (a,.., c, e), we have 
j" dx ... dz dw WV U = V j dx ...dz dw V U; 
and since V U= 0, except at the point in question, the integral may be taken over any 
portion of space surrounding this point, for instance, over the space included within the