CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
335 
The potentials at a point outside the shell and in its substance may be written 
respectively 
P=2D' 
bV f+1 
U,,e 4 tt 1 
and 
P=SDW(S j .(®)T,_ 1 08) -T^S, _ 1 (^))e- 
wherein, as before, x=\r, /3=Xb, 
Let us write 
H,=(S ) X. 1 (^)-m- 1 (^)W.(66) 
where T,,..^/?) and S n -\(fi) are to be treated as simply certain constant multipliers ; H„ 
will satisfy all the formulae of reduction which we have found to hold separately for S* 
and T„; and we have further these particular values 
H„(/3)=l, H,, +1 (a) = 0 by equation (62) 
and by the equation (B g ), H„ +1 (/3) = — ^j^H„(/3) = — ~' n +1 
Ab 
(67) 
Now H„(x) satisfies the equation 
q j + (x 3 _ , A+i)' )H=0 , 
dr r dr \ r ' 
and if Eh correspond to any other value of X, viz. : we have also 
^+ 2 f + A J _Aqrv=o, 
dr r dr \ r 
whence 
i3/ Tr dW TT ,d H\ 1 >> 
But, by equations (B 4 ), 
i -~ = »H,+XrH. +1 
dll' 
=nHi,+\VH'„ 
dr 
+ l 
and, therefore, 
( 68 ) 
