PROFESSOR CAYLEY ON PREPOTENTIALS. 
715 
dl . . . dl ) = — (1— a 2 ) is ~ l u dco d%. 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 oj— - 1 
to -f 1 ; and the required integral is thus 
C 1 (l-q^lfrrfS ( 1 —W-'d* 
—J J _ , (f - 2 k/0J + K 2 ) * s+ 2’ ■«/ J (/ 2 - 2*> + K 2 ) **+2’ 
2 (ri)* 
where J d% is the surface of the s-dimensional unit-sphere (see Annex I.), == ^ 1 $ > 
and for greater convenience transforming the second factor by writing therein co= cos 6, 
(f 1 )® 
the required integral is — into 
n/s p sin 5-1 $ dd 
J 0 (/ 2 -2x/cos 0 + X 2 )i s+ «’ 
which last expression ^including the factor 2 f% but without the factor 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 
Y 2 tt/' 2 r sinJdd 
~ J J 0 (/ 2 -2x/cosfl + x 2 )2 +1 
= i^(/*-2*/cos«+*V 
=i I 
which agrees with a result given, ‘ Mecanique Celeste,’ Book XII. Chap. II. 
66. Consider next the prepotential of the uniform solid (s + l)dimensional sphere, 
f dx ... dzdw 
V ~ J {{a-w) 2 ...+ (c-zf +{e-w ) 2 } ii+2 ’ 
equation of surface x 2 . . . -]-z 2 -\-w 2 =f 2 , and the two cases of an internal point x<f, 
and an external point z>f (a 2 . . . -\-c 2 -j-e 2 =z 2 as before). 
Transforming so that the coordinates of the attracted point are 0 . . . 0, «, the integral 
J{x 2 ... +z 2 + (x-wfl is+q ’ 
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 s ~ 1 drdX, where d% is an element of surface of the 
.s-dimensional unit-sphere | 2 . . . -f- £ 2 =1; the integral is therefore 
4 
1 dr d1*dw 
{r 2 + (: 
J J{>- s +(*-i 
w) 2 \ is+ * 
drdw 
w) 2 ^ s+i ’ 
where, as regards r and w, the integration extends over the circle r 2 -\-w 2 =f 2 . The value 
mdccclxxv. 5 c 
