CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 329 
are to undergo no sudden change in passing across the surface of the sphere, and that 
P must be finite at the centre. 
The equation in P (inside the sphere), is satisfied by 
P= AU„S„(\r)e -V2 -4ir~.(55) 
where U /t is any surface harmonic of degree n, and S u (Xr) is the solution of the equation 
d?Q 2 dQ 
dr r dr + 
—T~) 
Q=o 
(B) 
which does not become infinite at the centre. In fact the two solutions of this 
equation are, as I have elsewhere shown 
s A'->=(x,y,(lA 
T.(Xr)=(\r)-(L£( 
n 
n 
Sill 
\r 
cos Xr j 
\r J 
(BJ 
of which the former vanishes at the centre, except when n= 0, and the latter vanishes 
at oo ; it is the former of these expressions which is chosen. 
Outside the sphere the value of P is 
P = BU*.r* _1 e" 4, 'S.(5(f) 
and the above boundary conditions give 
A„S,(Xa) = B.a-- 1 , A,~ = -B,(«+ l)a— 
or 
B=a" +1 .A ;i S(Xa) 
a^+(#+i)S.=0.(57) 
This equation is also equivalent to the following somewhat simpler one (see equation 
B 5 , § 10), 
S„_ 1 (Xa) = 0. .(57,) 
The values of X being thus determined, we may proceed as in the former problem 
to find the coefficients due to any initial circumstances, and thence the state of the 
