360 A MEMOIR ON PREPOTENTIALS. [607 
the equation of the surface being x 2 + ... + z 2 + w 2 —f 2 ; there are the two cases of an 
internal point k <f, and an external point k >f (a 2 + ... + c 2 4- e 2 = k 2 as before). 
where the equation of the surface is still x 2 + ... + z 2 + w 2 =/ 2 . Writing x =f%,.., z =/£, 
fs 
w — fco, where f 2 + ... 4- £ 2 + ®* = B we have dS ———- , or the integral is 
■w 
d%... 
<w (/ 2 - 2*/w + /e 2 )^ 
Assume l~ = px,..,Z=pz, where xr + ... + z 2 = 1; then p 2 + co 2 = 1. Moreover, d%...d%. 
= p s ~ 1 dpd'Z, where dX is the element of surface of the s-dimensional unit-sphere 
x~ 
4- .,. -f z 2 = 1 ; or for p, substituting its value Vl — co 2 , we have dp = 
— coda) 
vr 
and 
thence d% ... d£= - (1 — co dco dt. The integral as regards p is from p = —1 to 
+ 1, or as regards co from 1 to —1; whence reversing the sign, the integral will be 
from co = — 1 to + 1; and the required integral is thus 
(1 - <u 2 )* s — 1 dco 
í 1 (1 -co^-'dcodZ = rs[ dl f 1 _0L 
j J _! (/ 2 - 2fc/o) + K 2 )* S+q ’ j J ~ J _! (/ 2 - 
2/cfco + K 2 )i s+q ’ 
r _ 2 (T 1 )* 
where d2 is the surface of the s-dimensional unit-sphere (see Annex I.), = ■ i. - ; 
J i j s 
and for greater convenience transforming the second factor by writing therein co — cos 6, 
(TAV 
the required integral is = multiplied by 
r (*«) 
W 
sin s_1 6 dd 
o (/ 2 — 2«/’cos 6 + K 2 )$ s+q ’ 
which last expression—including the factor 2f s , but without the factor —is the 
ring-integral discussed in the present Annex. It may be remarked that the value can 
be at once obtained in the particular case s = 2, which belongs to tridimensional space: 
viz. we then have 
7= 2*-/^' sinl)M 
_ 2-77/> 
2/cfq 
o (f 2 — 2/c/cos 6 + K 2 ) q+l 
(f 2 — 2k/ cos 6 + K 2 )~ q 
= ^((/-0-^-(/+*)-*»). 
which agrees with a result given, Mécanique Céleste, Book xir. Chap. n. 
66. Consider next the prepotential of the uniform solid {s + l)-dimensional sphere, 
dx... dzdw 
V = 
{(a — x) 2 + ...+(c — z) 2 + (e — w) 2 \^ s+q ’