607] 
A MEMOIR ON PREPOTENTIALS. 
361 
Transforming so that the coordinates of the attracted point are 0, .., 0, k, the 
integral is 
_ f dx ... dzdw 
J {x 2 + ... + z 2 + (k — w) 2 p+9’ 
where the equation is still x 2 + ... + z 2 + w 2 =f 2 . Writing here x = r%,.., z = r£, where 
£ 2 + ... + £ 2 = 1, we have dx ... dz = r 8-1 dr dX, where dX is an element of surface of the 
s-dimensional unit-sphere f 2 + ... + £ 2 = 1; the integral is therefore 
■/ 
r 8-1 dr dX dw 
{r 2 + (k - w) 2 }* s +3 
r 8-1 dr dw 
=I dS f 
[r 2 + (k- w) 2 }^ s+q ’ 
where, as regards r and w, the integration extends over the circle r 2 + w 2 = f 2 . The 
2 (T 1 ) 8 
value of the first factor (see Annex I.) is = 1, writing y and x in place of 
1 
2 (IV-) 8 
r and w respectively, the integral is = -p,— multiplied by 
I 
f y°- : 
J 1 (x-K 
2/ 8-1 dx dy 
{(# — K) 2 + y 2 \t s+ V 
over the circle x 2 + y 2 =f 2 ; viz. this last expression ^without the factor 
disk-integral discussed in the present Annex. 
67. We find, for the value in regard to an internal point k < f 
(n) s+1 
2(rj> 
r(W 
* 
is the 
V: 
x/ S+1 [ (t +/ 2 - fC 2 ) h - q t-*- q (t + p)-^+q-l dt, 
— Q) Jo 
r G8+q)rQ-qY 
which, in the particular case q = — \, is 
= rE) f + ' /" +/■ - **) (« +/ S )" i * _s dt; 
viz. the integral in t is here 
-f, K‘ ■+ rw dt • (*£*- pTi)■ 
T7- (nr 1 
i s +v' 
It may be added that, in regard to an external point tc> f the value is 
or we have 
V — 
r (i® + q) r (i 
r/ 8+1 and [ (t+f 2 - K 2 p q t-*- q (t +/ 2 )-i 8 +9~i dt, 
-q) 
C. IX. 
46