680 
PEOFESSOE CAYLEY ON PEEPOTENTIALS. 
density at every point thereof being =g. It is proper to remark that the neglected 
terms are of the orders 
JLjYi-Y 3 /_^\ 2?+2 
’ VeJ ’ 
(ri)T g 
so that the complete value multiplied by a 2q is equal to the constant g + ^ + terms 
s g \ 2 q / S \ 2 5'"^ 2 
of the orders ( ’(e) ’ & c * 
4. Let us now consider the prepotential of the remaining portion of the surface ; 
every part thereof is at a distance from P exceeding, in fact far exceeding, R ; so that 
imagining the whole mass jg dS to be collected at the distance R, the prepotential of 
the remaining portion of the surface is less than 
jgdS ' 
E s+2? ’ 
viz. we have thus, in the case where the mass J g dS is finite, a superior limit to the 
prepotential of the remaining portion of the surface. This will be indefinitely small in 
comparison with the prepotential of the disk, provided only is indefinitely small 
compared with R s+2? , that is s indefinitely small in comparison with R 1+5 ?. The proof 
assumes that the mass J* § cZS is finite ; but considering the very rough manner in which 
the limit was obtained, it can scarcely be doubted that, if not universally, at least 
for very general laws of distribution, even when jg d S is infinite, the same thing is true ; 
viz. that by taking s sufficiently small in regard to R, we can make the prepotential of 
the remaining portion of the surface vanish in comparison with that of the disk. But 
without entering into the question I assume that the prepotential of the remaining 
portion does thus vanish ; the prepotential of the whole surface in regard to the inde- 
finitely near point P is thus equal to the prepotential of the disk; viz. its value is 
_ 1 (T$yTq 
(±s+ q y 
which, observe, is infinite for a point P on the surface. 
5. Considering the prepotential V of an arbitrary point (a ... c, e) as a given function 
of (a . . . c, e) the coordinates of this point, and taking (x ... z, w) for the coordinates 
of the point N, which is, in fact, an arbitrary point on the surface, then the value of V 
at the point P indefinitely near to N will be =W, if W denote the same function of 
(x . . . Zf w) that Y is of (a ... c, e). The result just obtained is therefore 
or, what is the same thing, 
vv -^r(i S + ? )’ y- u J’ 
_r(|s+?) , 
e-lrijTs 0 • ■ 
