Riemann, and Clausius, and from Lorentz' own earlier work, lies in the conception which is entertained of the propagation of influence from one electron to another. In the older writings, the electrons were assumed to be capable of acting on each other at a distance, with forces depending on their charges, mutual distances, and velocities; in the present memoir, on the other hand, the electrons were supposed to interact not directly with each other, but with the medium in which they were embedded. To this medium were ascribed the properties characteristic of the aether in Maxwell's theory.
The only respect in which Lorentz' medium differed from Maxwell's was in regard to the effects of the motion of bodies. Impressed by the success of Fresnel's beautiful theory of the propagation of light in moving transparent substances,[1] Lorentz designed his equations so as to accord with that theory, and showed that this might be done by drawing a distinction between matter and aether, and assuming that a moving ponderable body cannot communicate its motion to the aether which surrounds it, or even to the aether which is entangled in its own particles; so that no part of the aether can be in motion relative to any other part. Such an aether is simply space endowed with certain dynamical properties.
The general plan of Lorentz' investigation was to reduce all the complicated cases of electromagnetic action to one simple and fundamental case, in which the field contains only free aether with solitary electrons dispersed in it; the theory which he adopted in this fundamental case was a combination of Clausius' theory of electricity with Maxwell's theory of the aether.
Suppose that e(x, y, z) and e′(x′, y′, z′) are two electrons. In the theory of Clausius,[2] the kinetic potential of their mutual action is
;
so when any number of electrons are present, the part of the
