PROFESSOR CAYLEY ON PREPOTENTIALS. 
707 
extended to all positive values of such that | . . .+£+<"< 1 ; and we obtain 
this by a known theorem, viz. 
(JUp+i 
Volume of (s+l)dimensional sphere =/' s+l + 
Writing x=fB , . . . z=f%, w=fco, we obtain dS=f s d%, where is the element of 
surface of the unit-sphere £ 2 . . . + £ 2 + <y 2 =l ; we have element of volume d%. . .d% da 
—V s dr d%, where r is to be taken from 0 to 1, and thence 
that is, 
J d \ . . . d% dr X dt= dZ, 
§dZ=(s+l)§dt... dtd», = 2 (^+ 1 )^ 
2(ri) s 
J 2 (ri ) s+1 
^S= surface of (s -f- ljdimensional sphere =/ s 
52. Writing s— 1 for s, we have 
(W 
Volume of (s— ljdimensional sphere=/ s p/i g ^_-f 
Surface of do. =f s 1 "p^ , 
which forms are sometimes convenient. 
Writing in the first forms s+l = 3, or in the second forms s=3, we find in ordinary 
space 
Volume of sphere=/ 3 =f 3 — — — =z—f~, 
r J r(f) J f.i.VTr 3 
and 
Surface of sphere=/ 2 —f 2, =4 nf 2 , 
as they should be. 
T s ~^dv 
Annex II. The Integral 1 ^ 8+g8 ^ s+q . — Nos. 53 to 63. 
53. The integral in question (which occurs ante, No. 2) may also be considered as arising 
from a prepotential integral in tridimensional space ; the prepotential of an element of 
mass dm is taken to be=J^- 2 , where d is the distance of the element from the attracted 
point P. Hence if the element of mass be an element of the plane z — 0, coordinates 
(x, y ), being the density, and if the attracted point be situate in the axis of z at a 
distance e from the origin, the prepotential is 
xt f pdxdy 
V J (a? 2 +y 2 +e 2 )j* + 2' 
For convenience it is assumed throughout that e is positive. 
MDCCCLXXV. 5 B 
