CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
313 
(a.) When both the substances are good conductors, we may take K=K'=0 ; the 
equations then become 
(c- 0 ?+c|-d|=» 
dt ' d N 
e'=0 
(?) 
( b .) When the first substance is a conductor and the second a non-conductor (air), 
we shall have K = 0, C'= 0 ; the equations are then, electro-magnetic measurements 
being still employed, 
/ Iv' d\d£ r d±_V d m dT 
V 47 rdt) dt~*~ dN krrdt dN 
4W=-K'C(C-~| 
/d\Jr 
V*n‘ 
c W \ 
'dN ) 
(8) 
But K' is infinitely small compared to C; and therefore, if we write (d) for the 
electrostatic measure of e, we shall have 
dt "W/N 
= 0 
47r(e / ) 
dip' d\]r 
dN~dN 
1 
l 
r 
I 
J 
( 9 ) 
We may derive an important corollary from these results. 
If $ is always =0 at the surface of the conductors (as will be the case in the 
following problems), we shall have 
( 10 ) 
But from equations (5), remembering that e=0, we derive 
vV=0, Where .00 
within the conductor, therefore, ip is everywhere zero; and since xp' always satisfies the 
equation v~xp' = 0 outside the surface and is zero at every point of it, it follows that i p' 
is also everywhere zero, and there is no free electricity either within or upon the 
conductor. 
Let us now collect the results of the above discussion, so far as it relates to a 
conductor disturbed by the introduction or variation of a magnetic system in a space 
surrounded by air ; putting K = 0, and writing a for —, 
