CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
31L 
The total electric current u, v, iv is connected with the magnetic force by equations 
of the form, 
4 , See. 
dy dz 
( 2 ) 
and satisfies the equation of continuity 
If we put 
dx'dy dz 
U=p + 
df 
dt* ' ' 
df dg dh 
dx'dy ' dz ’ 
dp dq dr de 
= 0 
( 3 ) 
expressing that the loss of electricity by conduction through the faces of an element 
is equal to the loss of free electricity in the element, a result which may be taken as 
self evident. 
If we put, with Maxwell, y>.= CP,/= — ‘P, &c., this equation may be written 
If e 0 be the initial value of e, 
, < K de n 
' £+ Sr m~°- 
= e„e _4 "G 
showing that any initial electrification will rapidly disappear in a conductor for which 
k is small compared to C. When the substance does not conduct we shall have 
e=e (J ; so that if we suppose air and other non-conductors initially uncharged, they 
cannot acquire any charge. 
The equations which determine the vector potential or electro-magnetic momentum 
in terms of the current are 
F — [-w/a 7 dydz!, &c. f 
where r=\/(x-x')' 2 -{-(y—y')~ J r (z—z')' 2 , and the integrations are to be extended over 
all space where there are currents, whether these be currents due to conduction, or 
time-variations of the electric displacement. F, G, H are thus the potentials of 
distributions of imaginary matter of finite density ; and, therefore, in crossing the 
surface which divides two substances in the field, we shall have 
