756 
MR. J. LARMOR ON A DYNAMICAL THEORY OF 
ments of the flow which are situated round the periphery of the section, much as the 
energy of a vortex sheet is introduced in the theory of discontinuous fluid motion ; 
and its variation will now lead to electro-dynamic equations of continuous electric 
flow in the ordinary manner. There is no diflicuity in extending this view to cases 
in which the breach of circuital character of the displacement-current djclt { f g, li) 
may have to be made up by an ideal distribution of flow throughout the volume, that 
is, by a volume instead of a surface distribution of electric currents, as in an actual 
conductor of finite resistance. 
[51. (Added June 14.) The velocity of a fluid is derivable in hydrodynamics, by 
kinematic formulse, from the vorticity of its flow, provided we suppose the vorticity 
to include the proper vortex sheets spread over the surfaces of discontinuity of flow, 
if such exist; in the same way the magnetic force is derivable as above from the 
displacement-current, provided this current includes the proper current-sheets over 
the surfaces of the conductors or other surfaces of discontinuity of the magnetic field. 
Let us consider an isolated uncharged conductor, and imagine an electric charge 
imparted to it. This charge is measured by the integral of the electric displacement 
[f, g, li) taken over any closed surface surrounding the conductor. Now if this 
rotational displacement were produced by continuous motion in the surrounding 
medium, its surface integral over any open sheet would be equal to the line integral 
of the linear displacement of the medium taken round the edge of the sheet. In a 
closed sheet the surface-integral would therefore be null; thus a charge cannot be 
imparted to a conductor without some discontinuous motion, or slip, or breach of 
rotational elasticity, in the medium surrounding it. If we imagine the charge to be 
imparted, by means of a wire, the integral of electric displacement over any open 
surface surrounding the conductor and terminated by the wire is equal to the line- 
integral of the linear displacement of the medium round the edge of this surface 
where it abuts on the wire. If the wire is thin, this line integral is therefore the 
same at all sections of it, and thus involves a constant circulatory displacement of the 
medium around it. If the wire is a perfect conductor, there is no elasticity and there¬ 
fore no rotational displacement of the aether inside its surface ; thus there is slip in 
the medium at the surface of the wire; and if we desire to retain the formulce of con¬ 
tinuous analysis, we must contemplate a very rapid transition by means of a vortex 
sheet at the surface, in place of this discontinuity. This vortex sheet is in the present 
example continuous with rotational motion in the outside medium ; the tubes of 
changing vorticity, i.e. of electric current, are completed and rendered circuital by 
displacement currents in the surrounding dielectric. But in the case of the con¬ 
denser-circuit above considered, the alteration of the density of the vortical lines 
between a pair of plates, which is produced by sepai’ating them, involves a trans¬ 
lational circulatory movement around the edge of the condenser and throughout the 
medium outside, which is almost entirely of irrotational type, except at the surface ol 
the conducting wire where a vortex sheet has to be located in order to avoid discon- 
