MOTIONS IN A SPHERICAL CONDUCTOR. 
529 
Outside: 
(41). 
F= 
G= 
H= 
dQ n d 
dx dx 
dQ /t . dQ_ u _j 
dy ' dy 
dCl, j dD, n ,_-j 
dz dz 
(42). 
Heie (f>,n n—\i ^u> 
indicated. 
These formulae give 
Inside: 
fl„, n_, ; _ l5 are solid harmonics of the algebraical degrees 
Outside : 
a— 0, 5=0, c=0 
(44). 
The sort of reciprocal relation between the formulae (27) and (34) on the one hand, 
and (40) and (43) on the other, is very remarkable. 
The continuity of F, G-, H at the surface of the sphere implies two relations which 
we shall not require ; whilst that of a, b, c involves 
\Jj /t (TR).oj n =0 .(45). 
This result follows also from (26), since 
xu-\-yv-\-zw=— ^(a:v 3 F+yv 3 G+2;v 2 H) 
■jp 
= 4“•^(w+l)'/'«(^’)-<u„.(46). 
4. From this point we must discuss separately the cases of free and forced motion, 
respectively. First let us take that of free motion. We assume that (no matter 
how) electric currents have been started in the sphere and then left to themselves. 
First Type. The equations (24) must now hold not merely in the space immediately 
