Page:CunninghamExtension.djvu/18

94 These equations expanded become the second and third of Lorentz' equations for ponderable bodies.

If, now, at a point (xyzt) the velocity u vanishes, and we assume that consequently $$d=\epsilon e,\ b=\mu h$$, we at once obtain from (7'), (10'), (11') equations identical with (10), (11), (12), § 2.

Thus, if we assume those equations together with Lorentz' fundamental equations, to be the correct scheme of the electrodynamics of moving bodies, we arrive at the conclusion that that scheme is invariant, not only under the extension of the transformation of Lorentz and Einstein given by Mirimanoff, but also under the transformation obtained by inversion in the four dimensional space here developed, and therefore also under a transformation obtained by combining any number of such operations.

8. In a paper in the Annalen der Physik, starting from the equation

$$\frac{1}{c}\left(\frac{\partial D}{\partial t}+S\right)=\mathsf{curl}\ H,$$

Einstein and Laub deduce that the conditions to be satisfied at a surface of discontinuity in the material media are that the following quantities must be continuous in crossing the surface:

$$D_{n},\ \left\{ H+\frac{1}{c}[DW]\right\} _{\overline{n}},\ B_{n},\ \left\{ E-\frac{1}{c}[BW]\right\} _{\overline{n}},$$

n denoting the direction of the normal, and $$\overline{n}$$ of a tangent line to the surface. S, as in Minkowski's paper, denotes the sum of the conduction and convection currents.

If we apply the extended transformation to these conditions, we find that they are not conserved. But if we take the Lorentz equation which is obtained from that above by the substitution of the vector for

$$Q=H-\frac{1}{c}[PW]$$

the conditions are that

$$D_{n},\ \left\{ Q+\frac{1}{c}[DW]\right\} _{\overline{n}},\ B_{n},\ \left\{ E-\frac{1}{c}[BW]\right\} _{\overline{n}},$$