Page:BatemanElectrodynamical.djvu/41

These are equivalent to the relations

${\begin{array}{rl}B_{x}=&\mu H_{x},\\B_{y}+\mu E_{z}=&\mu \left[H_{y}+uD_{z}\right],\\B_{z}-uE_{y}=&\mu \left[H_{z}-uD_{y}\right],\\E_{x}=&\epsilon ^{-1}D_{x},\\E_{y}-uB_{z}=&\epsilon ^{-1}\left[D_{y}-uH_{z}\right],\\E_{z}+uB_{y}=&\epsilon ^{-1}\left[D_{y}+uH_{y}\right],\end{array}}$

obtained by Minkowski, Einstein, and Laub.

It is also easy to verify that the constitutive relations (7) take the form

${\frac {s_{x}-u\rho }{\sqrt {1-u^{2}}}}=\sigma E_{x},\ s_{y}=\sigma {\frac {E_{y}-uB_{z}}{\sqrt {1-u^{2}}}},\ s_{z}=\sigma {\frac {E_{z}-uB_{y}}{\sqrt {1-u^{2}}}},$

given by Minkowski, Einstein, and Laub.

We have thus shown that the scheme of constitutive relations indicated by the invariance of a quadratic form agrees with the known scheme of relations in particular cases, and is invariant for any relevant space time transformation which is biuniform in a certain domain.

These relations are not the most general possible, and so the configuration and state of motion under consideration is of a special type, but the relations are sufficiently general for most ordinary purposes.

[Note added October 8th, 1909.]

Spherical wave transformations are not the only ones which can be used to transform problems occurring in the theory of electrons, for there are large classes of transformations which can be applied to particular problems, but cannot be applied to an arbitrary problem. The equations of transformation in this case involve the magnitudes of the electric and magnetic forces occurring in the particular problem.

Let us suppose that the electrodynamical field in a particular problem is of such a nature that the components $\left(E_{x},E_{y},E_{z}\right),\ \left(H_{x},H_{y},H_{z}\right),$ of the electric and magnetic forces are connected by the relations

$E_{x}H_{x}+E_{y}H_{y}+E_{z}H_{z}=0,$ $E_{x}^{2}+E_{y}^{2}+E_{z}^{2}=H_{x}^{2}+H_{y}^{2}+H_{z}^{2}$

and let $\left(s_{x},s_{y},s_{z}\right)$ be the components of Poynting's vector.