758 
MR. J. LARMOR ON A DYNAMICAL THEORY OF 
on integrating by parts. The medium is supposed here to be mathematically con¬ 
tinuous as above, thus avoiding separate consideration of the conducting channels,— 
though its structure may change with very great rapidity in crossing ceidain 
interfaces ; and it is taken to extend through all space, so that the surface-integral 
terms may be omitted, no active parts of the system being supposed to be at an 
infinite distance. Thus 
_ \ cm cll^ , , , 
~ Stt J J r \ dt dt dt dt ^ dt dt) ’ 
which is the form required, expressed as a double integral throughout space. 
For a network of complete circuits carrying currents tj, . . . we may express this 
formula more simply as 
47rT = \ q® 11 “^ (iS] + . . . + tpa + • • • , 
where e is the angle between the directions of the two elements of arc; which 
is Neumann’s well-known form of the mechanical energy of a system of linear 
currents. The currents are here simply mathematical terms for such flows of electric 
displacement along each wire as would be required to make the displacement 
throughout the field perfectly circuital, if the effective elasticity were continuous in 
accordance with the explanation above. 
53. Now if two wire cii’cuits carry steady currents, generated from condensers in 
this manner, and are displaced relatively to each other with velocities not considerable 
compared with the velocity of propagation of electromotive disturbances, the electric 
energy of the medium is thereby altered. There is supposed to be no viscous resist¬ 
ance in the system, and no sensible amount of radiation ; therefore the energy that is 
lost by the medium must be transferred to the matter. This transfer is accomplished 
by the mechanical work that is required to be done to alter the configuration of the 
wires against the action of electrodynamic forces operating between them ; for these 
mechanical changes have usually a purely statical aspect compared with the extremely 
rapid electric disturbances. The expression T, with its sign changed, is thus the 
potential energy of mechanical electrodynamic forces acting between the material 
conductors which carry the currents. 
Furthermore, as above observed, the electro-kinetic energy and the electrodynamic 
forces at which we have arrived are expressed in terms of the total current flowing 
across any section of the wire supposed thin, and do not involve the distribution of 
the current round the contour of the section to the neighbourhood of which it is 
confined, nor the area or form of the section itself. It therefore does not concern us 
whether the wire is a perfect conductor or not ; the previous argument from the 
circuital character of the rotation {f, g, h) shows that the total current is still the 
