MOTIONS IN A SPHERICAL CONDUCTOR. 
533 
decay of any system of currents arbitrarily given in the sphere. The determination 
of the harmonics oj u in terms of the initial circumstances, although interesting 
mathematically, would occupy too much space to be given in full here. It may suffice 
to remark that if u, v, tv be any three functions satisfying the solenoidal condition (8), 
and if £ —du'ldy — dv/dz , &c., &c., then the values of u, v, tv are completely determinate 
throughout any spherical region having its centre at the origin when we know the 
values of xu-\-yv-\-zw and of x^-fyy-fz'Q throughout that region. This is most readily 
seen from hydrodynamical considerations. The problem then resolves itself into the 
identification of the given initial values of these expressions with those which result 
from our formulae, viz., 
and 
7v 3 
xu-\-yv-\-zw=TZ —.n.n-\- 1 .xf/ u (kr)a)„ . 
ft 2 
%€+yy+z£= —tt — n.n + l.xj/„(kr)xH 
(56a), 
(56b). 
The summations here embrace all integral values of n and all admissible values of 1c. 
O 
In (56a) these are given by t//„(FR) = 0, and in (56b) by ^_ 1 (^R) = 0. The identifica¬ 
tion can be effected by known methods. 
5. Let us next proceed to consider the currents induced in the sphere by operations 
outside it; and for simplicity let us suppose that the changes in the field are periodic 
and follow the simple harmonic law. The value of X is now prescribed, viz., it = 27 rip, 
where p is the frequency. Hence, by (20), 
1$ = — 8 Trip / p, 
and 
1 
t-H 
T 
.(57), 
provided 
(f—^p/p . 
.(58). 
j 
| 
Since all our formulae involve only even powers of q there is no loss of generality 
in taking q always positive. 
From (45) we see that co n = 0, so that we have to deal exclusively with solutions of 
tke first type. The complete solution of the problem is then given by the equations 
(27) and (32) in which the values of X_«_ T in terms of X„ are to be obtained from 
the surface-conditions (37) and (38), viz., we have 
X 
x " 4-iW x ” 
f 1 
R, 2,i+1 
2*2 n Fl 
X 
n 
(59), 
(60). 
MDCCCLXXXIIJ. 
3 Z 
