PBOPESSOB CAYLEY ON PEEPOTENTIALS. 
757 
which at the surface is =- 
(s— 1)M 
f* ' 
Hence 
(g-i)r(jg-D.M r(jg+j).M 
4(r^) s+i / s 
2(r ±y + 'f 
2(r *)•+*/' 
(viz. g is constant). 
130. Writing for convenience M= ^ r ^ +I y a constant which may be put =1), 
also a 2 . . . + c 2 -f e 2 =% 2 , we have and consequently 
8/dS 

(e— wy\ 
2 li s ~i 
= 2( r(ir+i)' / ^ f ° r exteri ° r p° int *>/» 
2(rh !+1 / s s/ i , . , . . , - 
= 2 x for interior point « </. 
i (,2 s + 2 ) / 
By making a ... c, e all indefinitely large we find 
P/<®= 
2(ri) s+i / s s/ 
r(^+i) 5 
viz. the expression on the right-hand side is here the mass of the shell thickness hf. 
Taking s= 3 we have the ordinary formulae for the Potential of a uniform spherical 
shell. 
131. Suppose s=3, but let the surface be the infinite cylinder x 2 -\-y 2 —f 2 . Take here 
V'=Mlo g^o 2 +b 2 , Y"=M log/, 
each satisfying the potential equation ^ 2 + = 0 ’ hut Y', instead of vanishing, is 
infinite at infinity, and the conditions of the theorem are not satisfied ; the Potential of 
the cylinder is in fact infinite. But the failure is a mere consequence of the special 
value of s, viz. this is such that s— 2, instead of being positive, is =0. Keverting 
to the general case of (s-j-1) dimensional space, let the surface be the infinite cylinder 
x 2 . . . -\-z 2 =f 2 ; and assume 
V ~ («^..+ C 2 P~ i); Y " = J^ ( a constant )> 
these satisfy the potential equation ; viz. as regards V', we have 
■£+£) v -°> 
that 
jf-+h)r=o. 
dc 2 
Y' is not infinite at any point outside the cylinder, and it vanishes at infinity, except 
indeed when only the coordinate e is infinite, and its form at infinity is not 
=M -p(a 2 .. . +c 2 +e 2 )^" 1 >. 
V" is not infinite for any point within the cylinder ; and at the surface we have V'= V". 
5 h 2 
