Page:BatemanElectrodynamical.djvu/40

 With these constitutive relations the equations of electrodynamics are invariant for a very large class of space time transformations. The relations simplify considerably when the quadratic form is of a simple type.

The vanishing of the quadratic form may be supposed to represent the condition that two neighbouring particles are in a position to act on one another.

We shall now verify that in a particular case the constitutive relations obtained in this way agree with the ones obtained by Minkowski, Einstein, and Laub.

Starting with a system at rest, we assume that the invariant quadratic form is

This gives

and if we take

the relations (2) take the form

Now make the Lorentzian transformation

the quadratic form then becomes

We again have $$\Delta=-\epsilon\mu$$, and the relations (2) take the form

$$\begin{array}{l} B_{x}=\mu H_{x},\\ \\B_{y}=\frac{\epsilon\mu-u^{2}}{\epsilon\left(1-u^{2}\right)}H_{y}+u\frac{\epsilon\mu-1}{\epsilon\left(1-u^{2}\right)}D_{z},\\ \\B_{z}=\frac{\epsilon\mu-u^{2}}{\epsilon\left(1-u^{2}\right)}H_{z}-u\frac{\epsilon\mu-1}{\epsilon\left(1-u^{2}\right)}D_{y},\\ \\E_{x}=\frac{1}{\epsilon}D_{x},\\ \\E_{y}=\frac{1-u^{2}\epsilon\mu}{\epsilon\left(1-u^{2}\right)}D_{y}+u\frac{\epsilon\mu-1}{\epsilon\left(1-u^{2}\right)}H_{z},\\ \\E_{z}=\frac{1-u^{2}\epsilon\mu}{\epsilon\left(1-u^{2}\right)}D_{z}-u\frac{\epsilon\mu-1}{\epsilon\left(1-u^{2}\right)}H_{y}.\end{array}$$