812 
im. J. LARMOR OX A DTA^AMICAL THEORY OF 
which is equal to 47r/3a.(eu)% if the nucleus which bounds internally the strained 
medium is s}>herical and of radius a. The potential energy of elastic strain in the 
medium is, on the same supposition, by the ordinary electrostatic formula, -§ {eVyja^ 
where V is the velocity of electric propagation. We assume that the nucleus of the 
electron has no other intrinsic inertia of its own, and no other potential energy of its 
own ; under these circumstances its potential and kinetic energies will be of the 
same order of magnitude only when its velocity is comparable with that of radiation. 
In that case the present formulae are not ap^dicable, except merely to indicate the 
orders of magnitude; but we can conclude that, in a steady molecular configuration 
of electrons, where there must be an increase of kinetic energy equal to the potential 
energy which has run down in their approach, the velocities of the constituent 
electrons must be comparable with that of radiation, just as the above estimate from 
inaOTetic data simgested. 
Suppose there are two electric systems in the field producing velocities (w, v, w) 
and [n', v', w') respectively. The kinetic energy is now 
\\{{u + u'f + (v + v'f + {'10 + w'f} dr, 
of which the part that involves their mutual action is + vv' + luw') dr. 
If the velocity {u, v, w) belongs to an electron {e, v) as above, the mutual part 
of the kinetic energy is v' d/dz + w' djdy) dr, or on integration by parts 
— evl{v'n — lo'm) dS — ev\{dw'{dy — do':dz) dr, of which the former part is 
null when the external boundary is very distant. Thus the mutual electro-kinetic 
energy is — ev\r~^ df'jdt dr, where f is the component parallel to v of the electric 
displacement belonging to the other system. 
If the other system is also an electron [e , v) the total electro-kinetic energy is 
T = i L {evf + 4 L' {evf -f M . cu . e'v', 
where L, L' are as determined above, having the values Stt, 3«, 87r/3a' when the 
nuclei are spherictd, wlnle M = r~^ cos {ds . ds) + d^rjds ds, in wliich ds, ds are in 
the directions of v, v, and r is the distance between the monads.^ The potential 
eneroy is 
W = I A (cV)2 + 1 A' {e\f + B . eV . c'V, 
where A and A' are as determined above, being the reciprocals of the radii when the 
nuclei are spherical, and B = The equations of motion of the two electrons may 
medium wliicli is due to an electron; for the electron is part of the oi’igiual constitution of the medinm, 
and we cannot imagine it to be removed altog’ether. It may, however, be moved on into a new position, 
and we can then determine, as above, the displacement in the medium produced by this change of its 
locality. 
* The calculation of M is given concisely by H. Lamb, ‘ Proc. Lond. Math. Soc.,’ June 1883, p. 407; 
the result is given aho by Heaviside, ‘ Electrical Papers,’ vol. 2, p. 501. 
