I 
THE ELECTRIC AND LUMINIFEROUS MEDIUM. 757 
tinuity. The irrotational motion in the surrounding medium, which is thus continuous 
with the vortex sheet, and therefore determined by it, represents the magnetic field 
of the current flowing in the wire. On the other hand, in the illustration of this 
section, the motion in the medium is not irrotational, for it represents the field deter¬ 
mined by the displacement currents in the medium and the conduction current in the 
wire, taken together.] 
52. To return to our condenser illustration ; it does not follow from the superficial 
character of the current djdt {f g, h) that the velocity-vector djdt (^, 17 , Q is also 
very small throughout the field except at the very surface of the wire. We have in 
fact (/, g, h) = curl (i, rj, Q, therefore 
d d d \ /d^ dr} d^ 
'I-- [Tv ■ Tf • 
SO that, the compression d^jdx + drjldy -f dijdz being null, djdt 17 , Q are the 
potentials of certain ideal mass-distributions close to the surface of the wire ; therefore 
they are of sensible magnitude throughout the surrounding field. 
It appears from the surface character of the disturbance of the electric displacement 
{/> 9) which is thus introduced for current-systems flowing in complete circuits, 
that if we transform the kinetic-energy function 
in which it is convenient to take the density to be unity, so that it shall be expressed 
in terms of the current d/dt (/, g, h), at the same time treating the rotational 
displacement of the medium as continuous, we shall have practically reduced it to a 
surface integral along the wire. To effect this, let (F, G, H) be the potentials, 
throughout the region, of ideal mass-distributions of densities d/dt {f, g, h ): so that 
(F, G, H) = I (/', g\ h'). 
where r is the distance from the element of volume 
c/t to the point considered ; then 
