Page:CunninghamExtension.djvu/14

Hence

$LR^{2}\sin \theta \ dR\ d\theta \ d\phi \ dT=lr^{2}\sin \theta \ dr\ d\theta \ d\phi \ dt;$

and therefore for a closed system, if we integrate through the whole volumes and corresponding times,

$\int \int L\ dV\ dT=\int \int l\ dv\ dt.$

Thus, if the electrodynamic equations are based on the principle of least action, the action being the time integral of the difference of the magnetic and electric energies of the system, we may say that the action is invariant under the transformation.

The other invariant is expressed in the equation

$E_{R}H_{R}+E_{\theta }H_{\theta }+E_{\phi }H_{\phi }=-\lambda ^{4}\left(e_{r}h_{r}+e_{\theta }h_{\theta }+e_{\phi }h_{\phi }\right).$

Thus, if the electric and magnetic vectors are at right angles at any point in the one system, they are also at right angles at the corresponding point and instant in the other.

Consider now the conditions which hold at a reflecting surface which is at rest in the (r, t) system, viz.,

${\frac {e_{r}}{l}}={\frac {e_{\theta }}{m}}={\frac {e_{\phi }}{n}},$ $lh_{r}+mh_{\theta }+nh_{\phi }=0,$

where (l, m, n) are the direction cosines of the normal to the surface referred to the directions of $(r,\theta ,\phi )$ respectively.

The mechanical force per unit charge moving with the surface in the transformed system is

$F=\left(E_{r},\ E_{\theta }-{\frac {v}{c}}H_{\phi },\ E_{\phi }+{\frac {v}{c}}H_{\theta }\right)=\lambda ^{2}\left(e_{r},\ -{\frac {e_{\theta }}{\beta }},\ -{\frac {e_{\phi }}{\beta }}\right).$

Hence

${\frac {F_{r}}{l}}={\frac {F_{\theta }}{-m/\beta }}={\frac {F_{\phi }}{-n/\beta }}.$

But if we consider corresponding areas ds, dS and their projections, we obtain

${\frac {l\ ds}{L\ dS}}={\frac {r^{2}}{R^{2}}}=\lambda ^{2},$ ${\frac {m\ ds}{M\ dS}}={\frac {n\ ds}{N\ dS}}={\frac {r\ \delta r}{R\ \delta R}},$

δr being an element of length at rest.