Page:Zur Theorie der Strahlung in bewegten Körpern. Berichtigung.djvu/3

Therefore, the amount $(2\varkappa -1)$ stems from the total radiating energy in the moving cavity:

$h\epsilon _{0}(1-\beta ^{2})^{-2}\qquad (=h\epsilon _{0}(\varkappa +\tau ))$

thus

$h\epsilon _{0}\left({\frac {\beta ^{2}}{1-\beta ^{2}}}+{\frac {1}{2\beta }}\log {\frac {1+\beta }{1-\beta }}\right)=\varkappa 'h\epsilon _{0}$

from the heat supply of the walls, while the amount

$h\epsilon _{0}((1-\beta ^{2})^{-2}-\varkappa ')=h\epsilon _{0}\left({\frac {1-\beta ^{2}+\beta ^{4}}{(1-\beta ^{2})^{2}}}-{\frac {1}{2\beta }}\log {\frac {1+\beta }{1-\beta }}\right)=\tau 'h\epsilon _{0}$

is gained from the work.

This result is now in full agreement with the one of Abraham. Because the work which was spent to bring the system to a certain velocity, can be calculated from momentum by

$\int w\ dt\ {\frac {dG}{dt}}={\overset {\beta }{\underset {0}{\int }}}wd\beta {\frac {dG}{d\beta }};$

if one inserts for $G$ its value, then the integration indeed provides the value $h\epsilon _{0}\tau '$.

If one neglects magnitudes beginning with order $\beta ^{4}$, then it is

${\begin{array}{lr}\varkappa '=&1+{\frac {4}{3}}\beta ^{2},\\\\\tau '=&{\frac {2}{3}}\beta ^{2}.\end{array}}.$

(These values are related to quasi-stationary, reversible velocity changes; twice of the work must be spent at sudden accelerations of the system. In the latter case, one obtains ${\tfrac {8}{3}}{\tfrac {\epsilon _{0}}{c^{2}}}$ for the apparent mass; the relevant calculation executed by me in an earlier work^[1], is free of the mentioned calculation error. Though the concept of an apparent mass is probably to be confined to quasi-stationary motions.)

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The following thermodynamic considerations remain unchanged in principle; one only has to understand by $\varkappa$ the

1. F. Hasenöhrl, Sitzungsber. d. k. Akad. d. Wissensch. zu Wien, IIa. 113. p. 1039. 1904.