CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
315 
where <y 1; au, w 3 are the angular velocities of the element at x, y, z. We cannot, 
therefore, look upon i// as the potential of free electricity; in fact it is easy to see that 
there will be no free electricity inside it, just as when it was at rest. If the conductor 
be symmetrical about an axis, and revolve about it sufficiently long for the currents to 
become steady, the total current will become identical with the conduction current, and 
there will be no flow across the surface. We may then take at every point of the 
surface 
lu-\-mv-\-nw— 0. 
The triple integrals in the expressions for F, G, H are then to be taken only 
dF 
throughout the conductor, and we shall have as before F and , &c., continuous in 
& dN 
passing across the surface; and outside, F, G, H satisfy the equations 
V 2 F=0, v 2 G=0, v 2 H=0. 
The special problems which are solved in the present paper depend for their solution 
on certain properties of the vector potential; and it will therefore not be out of place 
to devote a little space to their preliminary discussion. We shall thereby gain a 
clearer insight into the subsequent analysis. 
The vector potential. 
§ 3. (A.) We shall first examine the nature of the vector potential inside a space 
due to magnets or currents outside that space. It is connected with the magnetic 
potential fl by the equations. 
(a.) dH dG_ d/2^ 
dy clz dx 
dF 
dll 
cUl 
1 ** 
1 ^ 
1 
in 
II 
dy 
dG 
dF 
dn 
dx 
1 
I# 
1 
~ dz 
(13) 
and it is clear that, if we can find one set of values F, G, H satisfying the equations 
when fl is supposed given, the complete values will be 
F +S> 
g+A h+ 
dy 
(l X 
dz ’ 
where y is an arbitrary function of x, y, z. It is therefore necessary for our purpose to 
find one solution only. 
MDCCCLXXXI. 2 T 
