1909.]

88

The principle of relativity in electrodynamics.

Using (4) and (5) this becomes

$\left.{\begin{array}{cll}&\delta P_{x}&={\frac {\sum ex}{\delta a}}=\delta p_{x},\dots \\\\{\mathsf {Similarly,}}&\delta P_{x}&={\frac {\sum eY}{\delta A}}\\\\&&={\frac {\sum e}{\delta a}}\left\{\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)y+{\frac {\beta vw_{y}x}{c^{2}}}\right\}\\\\&&=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta p_{y}+{\frac {\beta vw_{y}}{c^{2}}}\delta p_{x},\dots \\\\{\mathsf {and}}&\delta P_{z}&=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta p_{z}+{\frac {\beta vw_{z}}{c^{2}}}\delta p_{x},\dots \end{array}}\right\}.$ |
(a) |

Since the polarization electrons have no velocity relative to the body, they contribute nothing to the magnetization in either system.

Consider next the *magnetization electrons*, that is, systems of electrons spinning round centres fixed in the body and having a magnetic moment; possibly also an electric moment in a moving system.

If these have an electric moment, consider the contribution to the polarization

$\delta P_{X}={\frac {\sum eX}{\delta A}}={\frac {\sum e}{\delta a}}\beta \left(1-{\frac {vw'_{x}}{c^{2}}}\right)X={\frac {\sum ex}{\delta a}},$
where we consider in the first place only those electrons having a definite value of $w'_{x}$ and afterwards effect a summation.

Similarly $\delta P_{Y}$, *i.e.*, $\sum eY/\delta A$, becomes, after reduction,

$\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta p_{y}+{\frac {\beta vw_{y}}{c^{2}}}\delta p_{x}+{\frac {\beta v}{c}}\delta m_{z},$
and likewise

$\delta P_{z}=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta p_{z}+{\frac {\beta vw_{z}}{c^{2}}}\delta p_{x}-{\frac {\beta v}{c}}\delta m_{y}.$
In the same manner for the same group of electrons

$\delta M_{X}={\frac {1}{2c}}\sum e\left\{Y\left(W'_{Z}-W_{Z}\right)-Z\left(W'_{Y}-W_{Y}\right)\right\}/\delta A,$
which, on reduction, becomes

$\delta m_{x}-\left(\beta vW_{Y}\delta m_{y}+\beta vW_{Z}\delta m_{z}\right)/c^{2},$