# Page:CunninghamExtension.djvu/7

1909.]
88
The principle of relativity in electrodynamics.

Using (4) and (5) this becomes

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

${\displaystyle \delta P_{X}={\frac {\sum eX}{\delta A}}={\frac {\sum e}{\delta a}}\beta \left(1-{\frac {vw'_{x}}{c^{2}}}\right)X={\frac {\sum ex}{\delta a}},}$

where we consider in the first place only those electrons having a definite value of ${\displaystyle w'_{x}}$ and afterwards effect a summation.

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

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

and likewise

${\displaystyle \delta P_{z}=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta p_{z}+{\frac {\beta vw_{z}}{c^{2}}}\delta p_{x}-{\frac {\beta v}{c}}\delta m_{y}.}$

In the same manner for the same group of electrons

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

which, on reduction, becomes

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