702 
J. LARMOR ON A DYNAMICAL THEORY OF 
displacement increases T by ST ; this increase must come from some source; as there is 
supposed to be no dissipation it must come ultimately from the energy of the material 
system. During the displacement the electromotive system is at each moment 
sensibly in an equilibrium condition, so that there is practically no interaction between 
the kinetic energies of the electromotive and the material systems such as would arise 
from mixed terms in the energy-function involving both their velocities,—a fact verified 
experimentally by Maxwell. Thus somehow by means of unknown connecting 
actions, the displacement alters the mechanical energy of the system by an amount 
— 8T, and of this, considered as potential energy, the mechanical forces are the result. 
The mechanical force acting to increase the co-ordinate is therefore d'Y: d(b^. In 
fact, instead of considering the material system to be represented by the co-ordinates 
. . . which enter into the electro-kinetic energy, we must consider it to be an 
independent system linked on to the electro-kinetic system by an unknown mechanism, 
which however is of a statical character, so that energy passes over from the electro- 
kinetic system to the other one as mere statical work, without any complication arising 
from the effects of mixed kinetic reactions. In the discussion in Maxwell’s “Treatise,” 
§ 570, this idea of action and reaction between two interlocked systems, the electro¬ 
motive one and the mechanical one, has in the end to be introduced to obtain the 
proper sign for the mechanical foi’ce. The energy T is electro-kinetic solely; no 
energy of the material system is included in it. 
58. This deduction of the electrostatic and the electrodynamic mechanical forcive 
may now be re-stated in a compact form, which is also noteworthy from the circum¬ 
stance that it embodies perhaps the simplest method of treatment of the energy- 
function in all such cases. Let us consider the dynamical system undei' discussion to 
be the purely electric one, that is, to consist of the dielectric medium only, so that 
it has boundaries just inside the surfaces of the conductors, which are supposed to be 
perfectly inelastic. The energy function T + W remains as above stated, for all the 
energy is located in the dielectric; the electro-kinetic part T arises from motion of the 
medium, and the electrostatic part W from its rotational strain. But in the equation 
of Least Action we must also take account of tractions which may be exerted by the 
matter of the conductors on the boundary of this dielecti'ic system. If Sa’c?S denote 
the work done on the dielectric by these tractions extended over the element JS of 
the surface, the equation of Action will be 
S|(T — W) dt — I I hiv = 0, 
the time of ])assage from initial to final position being unvaried. When the dis¬ 
turbances considered are, as usually taken, too slow to generate sensible waves in the 
dielectric, and even when this restriction is not imposed, it equally follows that the 
* IMaxwell, ‘Treatise,’ Part IV., “ Electromagnetism,” Chap. VI. The apparatus was constructed as 
early as 1861. 
