706 
PROFESSOR CAYLEY ON PEEPOTENTIALS. 
over the surface of the (s+l)coordinal sphere x 2 . . . -\-z 2 -\-w 2 =f 2 , 
2 (r i ) s+i / s i 
— r(J s + l) for exterior point z>f 
and 
2{TV S+ H° 1 
— ^(4 + 4 f^~ x ^ or i n ^ er i° r P°i n t *</, 
where %, 2 =a 2 .. .-\-c 2 -\-e 2 . Observe that for the interior point the potential is a mere 
constant multiple of f. 
The same Annex VI. contains the case of the s-coordinal cylinder# 2 . . . -\-z 2 =f 2 , which 
is peculiar in that the cylinder is not a finite closed surface, but the theorem C is found 
to extend to it. 
49. As regards theorem D, we might in like manner obtain potentials relating to the 
(s-f l)coordinal sphere x 2 . . . -\-z 2 -\-w 2 =f 2 and ellipsoid^ . . . +p+-p=l; but I confine 
myself to the case of the sphere (see Annex VII.). We here assume values "V 7 and V" 
belonging to an internal and an external point respectively, and thence obtain a 
value g, or distribution over the whole (s+l)dimensional space, which density is found 
to be =0 for points outside the sphere. The result obtained is 
v C dx...dzdw 
Ji («-*) 2 • • - + (c-z)*+ (e-w) 2 }^ 
over (s+l)coordinal sphere # 2 . . .-\-z 2 -\-w 2 =f 2 , 
(T-s') s+ i f s+l 
== j'{ls + -?-) x 7 - 1 ^ 0r ex ^ er ^ or P°i n t *>f 
= TJfs > +f { + \)f 2 — ( 2 5 — f° r interior point z<f, 
where 7?=d ? . . .-J -c 2 -\-e 2 . 
The remaining Annexes VIII. and IX. have no immediate reference to the theorems 
A, B, C, D, which are the principal objects of the memoir. The subjects to which they 
relate will be seen from the headings and introductory paragraphs. 
Annex I. Surface and Volume of Sphere x 2 . . . j rz 2j r vf=f 2 . — Nos. 51 & 52. 
51. We require in (s-f-l)dimensional space, J dx ... dz dw, the volume of the sphere 
x 2 . . .-\-z 2 -\-w 2 =f 2 , and J cZS, the surface of the same sphere. 
Writing x=f\Z%. . . z=f\/%, w=f*/ co, we have 
dx ...dz dw = -Ti f s+1 d\... d% du, 
with the limiting condition § . . . ; but in order to take account as well of the 
negative as the positive values of x ... z, w, we must multiply by 2 S+1 . The value is 
therefore 
=/* +i jr*... ••<*?*». 
