Page:CunninghamExtension.djvu/16

The velocity of a moving point

$\left.{\begin{array}{l}W_{R}=-{\frac {w_{r}-v}{1-{\frac {vw_{r}}{c^{2}}}}}\\\\W_{N}={\frac {w_{n}}{\beta \left(1-{\frac {vw_{r}}{c^{2}}}\right)}}\end{array}}\right\},$

(2')

where $W_{N}$ is the velocity in any direction perpendicular to R.

The relative velocity of two moving points

$\left.{\begin{array}{l}W'_{R}-W_{R}=-{\frac {w'_{r}-w_{r}}{\beta ^{2}\left(1-{\frac {vw_{r}}{c^{2}}}\right)\left(1-{\frac {vw'_{r}}{c^{2}}}\right)}}\\\\W'_{N}-W_{N}={\frac {w'_{n}-w_{n}}{\beta \left(1-{\frac {vw_{r}}{c^{2}}}\right)}}-{\frac {w'_{n}v\left(w'_{r}-w_{r}\right)}{\beta c^{2}\left(1-{\frac {vw_{r}}{c^{2}}}\right)\left(1-{\frac {vw'_{r}}{c^{2}}}\right)}}\end{array}}\right\}.$

(4')

The distance between two points (r, t), (r', t') at the time T corresponding to (r, t),

$\left.{\begin{array}{l}\lambda \delta R=-{\frac {\delta r}{\beta \left(1-vw'_{r}/c^{2}\right)}}\\\\\lambda \delta N=\delta n+{\frac {vw'_{n}\delta r}{c^{2}\left(1-vw'_{r}/c^{2}\right)}}\end{array}}\right\}.$

(3')

The element of volume

$\lambda ^{3}\delta A=-{\frac {\delta a}{\beta \left(1-vw_{r}/c^{2}\right)}}.$

(5')

The density of free electricity

$-\lambda ^{-3}\rho =\beta \left(1-vw_{r}/c^{2}\right)\rho -\beta vj_{r}/c^{2}.$

(6')

The current density

$\left.{\begin{array}{rl}\lambda ^{-3}J_{R}=&\beta \left(1+vW_{R}/c^{2}\right)j_{r}\\-\lambda ^{-3}J_{N}=&j_{n}-\beta vW_{N}j_{r}/c^{2}\end{array}}\right\}.$

(7')

The polarization

$\left.{\begin{array}{rl}\lambda ^{-2}P_{R}=&p_{r}\\\lambda ^{-2}P_{\theta }=&-\beta \left(1-vw_{r}/c^{2}\right)p_{\theta }-\beta vw_{\theta }p_{r}/c^{2}-\beta vm_{\phi }/c\\\lambda ^{-2}P_{\phi }=&-\beta \left(1-vw_{r}/c^{2}\right)p_{\phi }-\beta vw_{\phi }p_{r}/c^{2}+\beta vm_{\theta }/c\end{array}}\right\}.$

(8')