1909.]

98

The principle of relativity in electrodynamics.

The magnetization

$\left.{\begin{array}{rl}-\lambda ^{-2}M_{R}=&m_{r}-\beta v\left(W_{\theta }m_{\theta }+W_{\phi }m_{\phi }\right)/c^{2}\\\lambda ^{-2}M_{N}=&\beta \left(1-vW_{R}/c^{2}\right)m_{n}\end{array}}\right\}.$ |
(9') |

7. The question now arises as to whether in a *ponderable* body the transformations above given between (*E, H*) and (*e, h*) stand as they are, or whether they should be really between (*E, B*) and (*e, b*). Examination shews that relativity is fully maintained if the latter is adopted in accordance with Mirimanoff's paper on the Lorentz-Einstein transformation. We know, changing (*H, h*) into (*B, b*) that the transformation obtained leaves invariant the equations

${\begin{array}{rl}-{\frac {1}{c}}{\frac {\partial b}{\partial t}}=&{\mathsf {curl}}\ e\\\\0=&{\mathsf {div}}\ b.\end{array}}$
The transformation being now

$\left.{\begin{array}{ll}E_{R}=\lambda ^{2}e_{r},&B_{R}=-\lambda ^{2}b_{r}\\E_{\theta }=\lambda ^{2}\beta \left(-e_{\theta }+vb_{\phi }/c\right),&B_{\theta }=\lambda ^{2}\beta \left(b_{\theta }+ve_{\phi }/c\right)\\E_{\phi }=\lambda ^{2}\beta \left(-e_{\phi }-vb_{\theta }/c\right),&B_{\phi }=\lambda ^{2}\beta \left(b_{\phi }-ve_{\theta }/c\right)\end{array}}\right\},$ |
(10') |

with the use of (8'), (9'), the following equations are deduced:

$\left.{\begin{array}{ll}D_{R}=\lambda ^{2}d_{r},&Q_{R}=-\lambda ^{2}q_{r}\\D_{\theta }=\lambda ^{2}\beta \left(-d_{\theta }+vq_{\phi }/c\right),&Q_{\theta }=\lambda ^{2}\beta \left(q_{\theta }+vd_{\phi }/c\right)\\D_{\phi }=\lambda ^{2}\beta \left(d_{\phi }-vq_{\theta }/c\right),&Q_{\phi }=\lambda ^{2}\beta \left(q_{\phi }-vd_{\theta }/c\right)\end{array}}\right\},$ |
(11') |

where the new quantities are defined by

${\begin{array}{ll}D=E+P,&d=e+p,\\B=M+H,&b=m+h,\\\\Q=H-{\frac {1}{c}}[PU],&q=h-{\frac {1}{c}}[pu].\end{array}}$
These together with (6'), (7') will, by comparison with the transformation in its original form, leave unchanged the equations

${\begin{array}{rl}{\frac {1}{c}}\left({\frac {\partial d}{\partial t}}+\rho u\right)=&{\mathsf {curl\ }}q,\\\\\rho =&{\mathsf {div}}\ d.\end{array}}$