Hendrik Lorentz

“ Parallel to OX we have the same electric force in S and S0, but in a direction perpendicular to OX the electric force in S will be times the electric force in S0.
By means of this result every electrostatic problem for a moving system may be reduced to a similar problem for a system at rest; only the dimensions in the direction of translation must be slightly different in the two systems. If, e.g., we wish to determine in what way innumerable ions will distribute themselves over a moving conductor C, we have to solve the same problem for a conductor C0, having no translation. ”

“ It is easy to show that if the dimensions of C0 and C differ from each other in the way that has been indicated, the electric force in one case will be perpendicular to the surface of C, as soon as, in the other case, the force is normal to the surface of C0. ”

“ For this purpose we consider a system of ponderable bodies, the ions in which are capable of vibrating about determinate positions of equilibrium. If the system be traversed by waves of light, there will be oscillations of the ions, accompanied by electric vibrations in the aether. For convenience of treatment we shall suppose that, in the absence of lightwaves, there is no motion at all ”

“ According to these views an electric current in a conductor is to be considered as a progressive motion of the ions, and a dielectric polarization in a non-conductor as a displacement of the ions from their positions of equilibrium. The ions were supposed to be perfectly permeable to the aether, so that they can move while the aether remains at rest. ”

“ We have already seen that, in the states of equilibrium, the electric forces parallel to OX, OY, OZ, existing in S differ from the corresponding forces in S0 by the factors
and . From (Ve) it appears that the same factors come into play when we consider the part of the electric forces that is due to the vibrations. If, now, we suppose that the molecular forces are modified in quite the same way in consequence of the translation, we may apply the just mentioned factors to the components of the total force acting on an ion. ”

“ In former investigations I have assumed that, in all electrical and optical phenomena, taking place in ponderable matter, we have to do with small charged particles or ions, having determinate positions of equilibrium in dielectrics, but free to move in conductors except in so far as there is a resistance, depending on their velocities. ”

“ Always provided that we start from a vibratory motion in S0 that can really exist, we have now arrived at a motion in S, that is possible in so far as it satisfies the electromagnetic equations. The last stage of our reasoning will be to attend to the molecular forces. In S0 we imagine again, around one of the ions, the same small space, we have considered in § 7 and to which the molecular forces acting on the ion are confined; in the other system we shall now conceive the corresponding small space, i.e. ”

“ Thus, some of the dependent variables in our equations (Id) — (IVd) will be represented in S0 and S by the same functions of x', y', z', t', whence we conclude that the equations will be satisfied by values of , which are likewise in both cases the same functions of x', y', z', t'. By what has been said at the beginning of this §, not only , but also the total electric force will be the same in S0 and S, always provided that corresponding ions at corresponding times (i. e. for equal values of t') be considered. ”