MOTIONS IN A SPHERICAL CONDUCTOR. 
537 
In fact, if we write 
where the lower limit is taken at such a depth that the currents there are insensible, 
we readily find that the currents are approximately equivalent to an infinitely thin 
spherical current sheet of radius It, the components of the current at any point of the 
sheet being given by 
u — 
2n+l f d 
4ttK \ J dz 
2n +1 
47tR 
w — 
2n +1 
47tR 
(73).* 
6. The foregoing methods can be readily adapted to the case of a shell bounded by 
two concentric spherical surfaces. The most interesting case is when the shell is 
infinitely thin. The free motions of the second type then decay with infinite rapidity, 
and there are no forced motions of this type. Hence we have practically to deal only 
with solutions of the first type. The theory of these has been given by Professor 
Niven, but for the sake of completeness it is here discussed from the point of view of 
the present paper. 
Let u, v, iv be the components of the total current at any point of the shell, and 
let p'=pfS, where S is the thickness of the shell. Then if all our functions vary as 
e M we shall have 
pu'——\ F, pv = —\ G, p'w '=—\H ... . . (74). 
In the hollow space inside the shell 
F =(^- a 
G=(s -—x 
dx d: 
dyj 
d 
Xn 
X" r- 
(75),t 
H: 
( d d\ 
= x -- v — 
dy J dxJ 
whilst (32) hold for the external space. The functions F, G, H must vary continually 
as we cross the shell, so that 
.( 76 ), 
at the surface 
* The conclusions of this section have an obvious bearing on the results obtained by Professor D. E. 
Hughes in his experiments with the Induction Balance (Proc. Roy. Soc., May 15, 1879). 
t It is here assumed that the inducing system, if any, is situate in the space external to the shell. 
