Page:CunninghamExtension.djvu/16

92 The velocity of a moving point

{{MathForm2|(2')|$$\left.\begin{array}{l} W_{R}=-\frac{w_{r}-v}{1-\frac{vw_{r}}{c^{2}}}\\ \\W_{N}=\frac{w_{n}}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)}\end{array}\right\} ,$$}}

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

The relative velocity of two moving points

{{MathForm2|(4')|$$\left.\begin{array}{l} W'_{R}-W_{R}=-\frac{w'_{r}-w_{r}}{\beta^{2}\left(1-\frac{vw_{r}}{c^{2}}\right)\left(1-\frac{vw'_{r}}{c^{2}}\right)}\\ \\W'_{N}-W_{N}=\frac{w'_{n}-w_{n}}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)}-\frac{w'_{n}v\left(w'_{r}-w_{r}\right)}{\beta c^{2}\left(1-\frac{vw_{r}}{c^{2}}\right)\left(1-\frac{vw'_{r}}{c^{2}}\right)}\end{array}\right\} .$$}}

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

{{MathForm2|(3')|$$\left.\begin{array}{l} \lambda\delta R=-\frac{\delta r}{\beta\left(1-vw'_{r}/c^{2}\right)}\\ \\\lambda\delta N=\delta n+\frac{vw'_{n}\delta r}{c^{2}\left(1-vw'_{r}/c^{2}\right)}\end{array}\right\}.$$}}

The element of volume

The density of free electricity

The current density

{{MathForm2|(7')|$$\left.\begin{array}{rl} \lambda^{-3}J_{R}= & \beta\left(1+vW_{R}/c^{2}\right)j_{r}\\ -\lambda^{-3}J_{N}= & j_{n}-\beta vW_{N}j_{r}/c^{2}\end{array}\right\} .$$}}

The polarization

{{MathForm2|(8')|$$\left.\begin{array}{rl} \lambda^{-2}P_{R}= & p_{r}\\ \lambda^{-2}P_{\theta}= & -\beta\left(1-vw_{r}/c^{2}\right)p_{\theta}-\beta vw_{\theta}p_{r}/c^{2}-\beta vm_{\phi}/c\\ \lambda^{-2}P_{\phi}= & -\beta\left(1-vw_{r}/c^{2}\right)p_{\phi}-\beta vw_{\phi}p_{r}/c^{2}+\beta vm_{\theta}/c\end{array}\right\} .$$}}