position and motion of the charges, satisfy the Maxwellian equations
The theory of Lorentz is based on these four aethereal equations of Maxwell, together with the equation which determines the ponderomotive force on a charged particle; this, which we shall now derive, is the contribution furnished by Clausius' theory. The Lagrangian equations of motion of the electron e are
,
and two similar equations, where L denotes the total kinetic potential due to all causes, electric and mechanical. The ponderomotive force exerted on the electron by the electromagnetic field has for its x-component
,
or
;
which, since
,
reduces to
,
or ,
so that the force in question is
.
This was Lorentz' expression for the ponderomotive force on an