# Page:CunninghamExtension.djvu/5

1909.]
81
The principle of relativity in electrodynamics.

If a denote a volume which is moving with velocity ${\displaystyle \left(w_{x},w_{y},w_{z}\right)}$, the corresponding volume A is given by

 ${\displaystyle A=a/\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right).}$ (5)

In applying the transformation to the consideration of the number of free electrons per unit volume, it must be remembered that they are moving with different velocities, and that this velocity affects the relative volumes which they occupy in the two systems of coordinates.

But the magnitude of the charge of each electron and the number of them are unchanged. If, therefore, we confine our attention to a group of electrons all having the same component of velocity ${\displaystyle w'_{x}}$ in the one system and ${\displaystyle W'_{x}}$ in the other, the volumes occupied, δa, δA, are connected by the relation (5).

Thus this group contributes to the density of the free electricity as measured in the two systems different amounts, ${\displaystyle \delta \rho =\sum e/\delta a}$, ${\displaystyle \delta {\mathsf {P}}=\sum e/\delta A}$. Using (5), we have

${\displaystyle \delta {\mathsf {P}}={\frac {\sum e}{\delta A}}={\frac {\beta \sum e}{\delta a}}\left(1-{\frac {vw'_{x}}{c^{2}}}\right)}$

Let ${\displaystyle w_{x}}$ be the mass velocity of the moving medium as distinguished from ${\displaystyle w'_{x}}$ the velocity of the individual electron.

Then

${\displaystyle {\begin{array}{ll}\delta {\mathsf {P}}&={\frac {\beta \sum e}{\delta a}}\left\{\left(1-{\frac {vw_{x}}{c^{2}}}\right)-{\frac {v}{c^{2}}}\left(w'_{x}-w_{x}\right)\right\}.\\\\&=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta \rho -{\frac {\beta v}{c^{2}}}{\frac {\sum e\left(w'_{x}-w_{x}\right)}{\delta a}}\\\\&=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta \rho -{\frac {\beta v}{c^{2}}}\delta j_{x},\end{array}}}$

where ${\displaystyle \delta j_{x}}$ is the contribution of the same group of electrons to the conduction current through the medium.

This last relation is true for each particular value of ${\displaystyle w'_{x}}$, so that on summing for all electrons, we have

 ${\displaystyle {\mathsf {P}}=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\rho -{\frac {\beta v}{c^{2}}}j_{x},}$ (6)

an equation identical with Minkowski's and Mirimanoff's.

Similarly the contributions of the same group of electrons to the