# Page:CunninghamPrinciple.djvu/2

539
Electromagnetic Mass of a Moving Electron.

of light. The present paper reconsiders Abraham's discussion and comes to the conclusion that the objection is not valid. The discussion was suggested by the fact that it has been proved that Maxwell's equations represent equally well the sequence of electromagnetic phenomena relative to a set of axes moving relative to the aether, as relative to a set of axes fixed in the aether. More explicitly this is stated as follows : —

If there are two sets of rectangular axes (A, A') coinciding at a certain instant, of which A' is moving relative to A with velocity v in the direction of the axis of x, which is conceived as at rest, and if x, y, z, t be space and time variables associated with A, and x', y' z', t' similar variables associated with A', then the equations

${\displaystyle {\frac {1}{c}}{\frac {\partial E}{\partial t}}=curl\ H,\quad {\frac {1}{c}}{\frac {\partial H}{\partial t}}=-curl\ E}$,

transform identically into the equations

${\displaystyle {\frac {1}{c}}{\frac {\partial E'}{\partial t'}}=curl\ H',\quad {\frac {1}{c}}{\frac {\partial H'}{\partial t'}}=-curl\ E'}$,

the accented and unaccented magnitudes being connected by the relations

${\displaystyle {\begin{array}{ll}x'=\beta (x-vt],\\y'=y,\\z'=z,&\beta =\left(1-v^{2}/c^{2}\right)^{-{\frac {1}{2}}}\\t'=\beta \left(t-{\frac {vx}{c^{2}}}\right),\end{array}}}$

${\displaystyle {\begin{array}{l}E'=\beta \left({\frac {E_{x}}{\beta }},\ E_{y}-vH_{z},\ E_{z}+vH_{y}\right),\\\\H'=\beta \left({\frac {H_{x}}{\beta }},\ H_{y}-vE_{z},\ H_{z}+vE_{y}\right)\end{array}}}$

Further, if ${\displaystyle \rho ={\tfrac {1}{4\pi }}div\ E}$, and ${\displaystyle \rho '={\tfrac {1}{4\pi }}div\ E'}$, the volume integrals taken through corresponding regions ${\displaystyle \tau ,\ \tau ',\ \int _{\tau }\rho d\tau }$ and ${\displaystyle \int _{\tau '}\rho 'd\tau '\!}$ are identically equal, giving an exact correspondence as regards distribution of electric charge.

Thus the above transformation renders the electromagnetic equations of a system independent of a uniform translation of the whole system through the aether.[1]

1. The transformation in question is given by Einstein in a paper in the Annalen der Physik, xvii. (q. v.). It is in substance the same as that given by Larmor in 'Aether and Matter,' chap, xi., though the correlation is only proved to hold as far as the second power of v/c. Prof. Larmor tells me he has known for some time that it was exact. Vide also Lorentz, Amsterdam Proceedings, 1903-4.