Page:SahaElectrodynamics.djvu/29

(Ions, electrons), then these equations form the electromagnetic basis of Lorentz's electrodynamics and optics for moving bodies.

If these equations which hold in the system K, are transformed to the system k with the aid of the transformation-equations given in § 3 and § 6, then we obtain the equations :—

${\frac {1}{c}}\left(\rho 'u_{\xi }+{\frac {\partial X'}{\partial \tau }}\right)={\frac {\partial N'}{\partial \eta }}-{\frac {\partial M'}{\partial \zeta }},\ {\frac {\partial L'}{\partial \tau }}={\frac {\partial Y'}{\partial \zeta }}-{\frac {\partial Z'}{\partial \eta }}$, ${\frac {1}{c}}\left(\rho 'u_{\eta }+{\frac {\partial Y'}{\partial \tau }}\right)={\frac {\partial L'}{\partial \zeta }}-{\frac {\partial N'}{\partial \xi }},\ {\frac {\partial M'}{\partial \tau }}={\frac {\partial Z'}{\partial \xi }}-{\frac {\partial X'}{\partial \zeta }}$, ${\frac {1}{c}}\left(\rho 'u_{\zeta }+{\frac {\partial Z'}{\partial \tau }}\right)={\frac {\partial M'}{\partial \xi }}-{\frac {\partial L'}{\partial \eta }},\ {\frac {\partial N'}{\partial \tau }}={\frac {\partial X'}{\partial \eta }}-{\frac {\partial Y'}{\partial \xi }}$,

where

${\frac {u_{x}-v}{1-{\frac {u_{x}v}{c}}}}=u_{\xi }$, ${\frac {u_{y}}{\beta \left(1-{\frac {vu_{x}}{c^{2}}}\right)}}=u_{\eta },\ \rho '={\frac {\partial X'}{\partial \xi }}+{\frac {\partial Y'}{\partial \eta }}+{\frac {\partial Z'}{\partial \xi }}$ $=\beta \left(1-{\frac {vu_{x}}{c^{2}}}\right)\rho$, ${\frac {u_{z}}{\beta \left(1-{\frac {vu_{x}}{c^{2}}}\right)}}=u_{\zeta }$.

Since the vector $(u_{\xi },u_{\eta },u_{\zeta })$ is nothing but the velocity of the electrical mass measured in the system k, as can be easily seen from the addition-theorem of velocities in § 4—so it is hereby shown, that by taking