Page:Electromagnetic phenomena.djvu/10



passing at the same time to x', y', z', t'  as independent variables. The final result is

$$\varphi'=\frac{w}{4\pi c^{2}r'}\frac{\partial\left[\mathfrak{p}_{x}^{'}\right]}{\partial t'}-\frac{1}{4\pi}\left\{ \frac{\partial}{\partial x'}\frac{\left[\mathfrak{p}_{x}^{'}\right]}{r'}+\frac{\partial}{\partial y'}\frac{\left[\mathfrak{p}_{y}^{'}\right]}{r'}+\frac{\partial}{\partial z'}\frac{\left[\mathfrak{p}_{z}^{'}\right]}{r'}\right\}$$.

As to the formula (18) for the vector potential, its transformation is less complicate, because it contains the infinitely small vector $$\mathfrak{u}'$$. Having regard to (8), (24), (26) and (5), I find

$$\mathfrak{a}'=\frac{1}{4\pi cr'}\frac{\partial\left[\mathfrak{p^{'}}\right]}{\partial t'}$$.

The field produced by the polarized particle is now wholly determined. The formula (13) leads to

and the vector $$\mathfrak{h}'$$ is given by (14). We may further use the equations (20), instead of the original formulae (10), if we wish to consider the forces exerted by the polarized particle on a similar one placed at some distance. Indeed, in the second particle, as well as in the first, the velocities $$\mathfrak{u}$$ may be held to be infinitely small.

It is to be remarked that the formulae for a system without translation are implied in what precedes. For such a system the quantities with accents become identical to the corresponding ones without accents; also k=1 and l=1. The components of (27) are at the same time those of the electric force which is exerted by one polarized particle on another.

§ 8. Thus far we have only used the fundamental equations without any new assumptions. I shall now suppose that the electrons, which I take to be spheres of radius R in the state of rest, have their dimensions changed by the effect of a translation, the dimensions in the direction of motion becoming kl times and those in perpendicular direction l times smaller.

In this deformation, which may be represented by $$\left(\frac{1}{kl},\ \frac{1}{l},\ \frac{1}{l}\right)$$, each element of volume is understood to preserve its charge.

Our assumption amounts to saying that in an electrostatic system Σ, moving with a velocity w, all electrons are flattened ellipsoids with their smaller axes in the direction of motion. If now, in order to apply the theorem of §6, we subject the system to the deformation (kl, l, l), we shall have again spherical electrons of radius R.