CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
321 
Let x^Po) represent the law of decay of a system which was initially represented by 
P 0 ; then, combining equations 26 and 27, we see that the complete value of P at any 
time for the system is given by 
To complete the solution, therefore, we have only to determine the nature of x T - 
Case of an infinite plate of finite thickness. 
§ 5. Let us now take up the case of an infinitely extended plane plate of thickness 
2b, and suppose the origin somewhere midway between the two faces of the plate so 
that its faces are determined by 
The scalar and vector potentials of external magnets or currents may be denoted by 
o F 0 C IT -o 
ds‘‘ L dy\ ' dx' 11(1 
. . . (2!)) 
The equations of the currents in the plate will have for them type 
(Til- 
d(F + F 0 ) dF 
dx 
dt 
(30) 
and we shall prove that all the conditions of the problem may be satisfied by taking 
dP „ d P TT 
*=°- f =-f g =*/ h=0 
d<f> 
U =~ly’ V= ~d%’ W=0 
. (31) 
47r 
<h= — v 2 P 
J 
The equations of electromotive force are now reduced to the single characteristic 
equation 
rc rl 
.( 32 ) 
iU sp =l< p +L) 
Since there is no free electricity anywhere present, the currents in the plate are 
closed currents, and therefore F, G, H and their differential coefficients are all con¬ 
tinuous in passing across the boundary of the conductor. All these conditions will be 
satisfied by having 
