PROFESSOR CAYLEY ON PREPOTENTIALS. 
699 
36. It is assumed in the proof that 2^+1 is positive or zero; viz. q is positive, or if 
negative then — q >* ; the limiting case q=—\ is included. 
It is to be remarked that by what precedes, if q be positive (but excluding the case 
^=0) the density § is given by the equivalent more simple formula 
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 
foim +“ +e «r*' 
The Potential-surface Theorem C. — Art. Nos. 37 to 42. 
37. In the case q= — writing here V we h ave precisely, as in the 
general case, 
jV ^ dS+jjdx ... dzde WVU=ju ^ dS+^dx ...dzde UVW ; 
and if the functions U, W satisfy the equations VU=0, VW=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 ; 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 . 
{{oc— a ) 2 . . . + [z— c ) 2 + (w— e) 2 } is_ " ’ 
then if V be the value of W at the point (a... c , e), we have 
J dx . . . dz dw W V U = ‘ V J dx . . . dz dw V U ; 
and since VU = 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 
sphere, radius R, having the point (a . . . c, e) for its centre ; or taking the origin at this 
point, we have to find ^dx . ..dz dw VU, where 
U= 
\x z . . . + 
and the integration extends over the space within the sphere x 2 . . .+5 2 +w 2 =R 2 . 
MDCCCLXXV. 5 A 
