Page:Electromagnetic phenomena.djvu/11

 Hence, if we alter the relative position of the centres of the electrons in Σ by applying the deformation (kl, l, l), and if, in the points thus obtained, we place the centres of electrons that remain at rest, we shall get a system, identical to the imaginary system Σ' , of which we have spoken in § 6. The forces in this system and those in Σ will bear to each other the relations expressed by (21).

In the second place I shall suppose that the forces between uncharged particles, as well as those between such particles and electrons, are influenced by a translation in quite the same way as the electric forces in an electrostatic system. In other terms, whatever be the nature of the particles composing a ponderable body, so long as they do not move relatively to each other, we shall have between the forces acting in a system (Σ' ) without, and the same system (Σ) with a translation, the relation specified in (21), if, as regards the relative position of the particles, Σ'  is got from Σ by the deformation (kl, l, l), or Σ from Σ'  by the deformation $$\left(\frac{1}{kl},\ \frac{1}{l},\ \frac{1}{l}\right)$$.

We see by this that, as soon as the resulting force is 0 for a particle in Σ' , the same must be true for the corresponding particle in Σ. Consequently, if, neglecting the effects of molecular motion, we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted be its neighbours, and if we take for granted that there is but one configuration of equilibrium, we may draw the conclusion that the system Σ' , if the velocity w is imparted to it, will of itself change into the system Σ. In other terms, the translation will produce the deformation $$\left(\frac{1}{kl},\ \frac{1}{l},\ \frac{1}{l}\right)$$.

The case of molecular motion will be considered in § 12.

It will easily be seen that the hypothesis that has formerly been made in connexion with MICHELSON'S experiment, is implied in what has now been said. However, the present hypothesis is more general because the only limitation imposed on the motion is that its velocity be smaller than that of light.

§ 9. We are now in a position to calculate the electromagnetic momentum of a single electron. For simplicity's sake I shall suppose the charge e to be uniformly distributed over the surface, so long as the electron remains at rest. Then, a distribution of the same kind will exist in the system Σ'  with which we are concerned in the last integral of (22). Hence