362 
A MEMOIR ON PREPOTENTIALS. 
[607 
which, in the same case q = — ?, is 
where the ¿-integral is 
and the value of V is therefore 
(Fiym /«+1 
r (*« + *) a- 1 
Recurring to the case of the internal point; then, writing V = ^ + - + as + 
and observing that V (/e 2 ) = 4 (|s + ■£), we have 
(in particular, for ordinary space s + 1 = 3, or the value is —j—, = — 47r, which is 
68. The integrals referred to as the ring-integral and the disk-integral arise also 
from the following integrals in two-dimensional space, viz. these are 
y*~ x dS 
y s ~ x dx dy 
{(¿c — /e) 2 -l- y 2 ]^+s ’ 
in the first of which dS denotes an element of arc of the circle x- + y' 2 = the 
integration being extended over the whole circumference, and in the second the 
integration extends over the circle x 2 + y 2 =/ 2 ; y*- 1 is written for shortness instead of 
(y 2 ) i(s “i ) , viz. this is considered as always positive, whether y is positive or negative; 
it is moreover assumed that s — 1 is zero or positive. 
Writing in the first integral x=fcos0, y =/sin 6, the value is 
viz. this represents the prepotential of the circumference of the circle, density varying 
as (sin 6) s ~ l , in regard to a point x = k, y = 0 in the plane of the circle; and similarly 
the second integral represents the prepotential of the circular disk, density of the 
element at the point (x, y) — y s ~ 1 ) in regard to the same point x — K,y — 0; it being 
in each case assumed that the prepotential of an element of mass p dvr at a point 
at distance d is = • 
dS+2q ■