Page:CunninghamExtension.djvu/19

shall be continuous, provided there is no true electricity on the surface of separation.

There is no difficulty in proving that these conditions are conserved under Mirimanoff's extended transformation.

It will be proved now that they are also conserved under the transformation by inversion, and therefore by the most general transformation of the whole group which leaves the fundamental equations invariant.

From the equations (18'), we obtain if we write

$H'=H+{\frac {1}{c}}\left[EW\right]=Q+{\frac {1}{c}}\left[DW\right],$

$\left.{\begin{array}{l}H'_{R}=-{\frac {\lambda ^{2}}{1-{\frac {vw_{r}}{c^{2}}}}}\left\{h'_{r}-{\frac {v}{c^{2}}}\left(w_{r}h'_{r}+w_{\theta }h'_{\theta }+w_{\phi }h'_{\phi }\right)\right\}\\\\H'_{\theta }={\frac {\lambda ^{2}}{\beta \left(1-{\frac {vw_{r}}{c^{2}}}\right)}}h'_{\theta }\\\\H'_{\phi }={\frac {\lambda ^{2}}{\beta \left(1-{\frac {vw_{r}}{c^{2}}}\right)}}h'_{\phi }\end{array}}\right\}.$

(14')

If we consider an element of length $(\delta s,\delta S)$ of which the components are $(\delta r,\delta n),(\delta R,\delta N)$ in the two systems respectively, $\delta n,\delta N$ being the components perpendicular to r, R respectively, we have equations analogous to (4),

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

Hence

${\begin{array}{ll}H'_{R}\delta R+H'_{N}\delta N&={\frac {\lambda }{\beta \left(1-{\frac {vw_{r}}{c^{2}}}\right)}}\left\{{\frac {h'_{r}\delta _{r}}{1-{\frac {vw_{r}}{c^{2}}}}}+h'_{n}\delta _{n}-{\frac {v\left(w_{r}h'_{r}+w_{n}h'_{n}\right)}{c^{2}\left(1-{\frac {vw_{r}}{c^{2}}}\right)}}+{\frac {vw_{n}h_{n}}{c^{2}\left(1-{\frac {vw_{r}}{c^{2}}}\right)}}\right\}\\\\&={\frac {\lambda }{\beta \left(1-{\frac {vw_{r}}{c^{2}}}\right)}}\left\{h'_{r}\delta _{r}+h'_{r}\delta _{n}\right\},\end{array}}$