Page:AbrahamMinkowski1.djvu/20

From that it follows

(49c)

${\begin{cases}2{\mathfrak {E'{\dot {P}}}}&=2(\epsilon -1)\left\{{\mathfrak {E}}'_{x}{\mathfrak {\dot {E}}}'_{x}+k^{-2}{\mathfrak {E}}'_{y}{\mathfrak {\dot {E}}}'_{y}+k^{-2}{\mathfrak {E}}'_{z}{\mathfrak {\dot {E}}}'_{z}\right\}\\\\&+4{\mathfrak {\dot {q}}}_{x}|{\mathfrak {q}}|k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}\right\}\\\\&-2{\frac {{\mathfrak {\dot {q}}}_{y}}{|{\mathfrak {q}}|}}\left\{{\mathfrak {E}}'_{x}{\mathfrak {P}}_{y}-{\mathfrak {E}}'_{y}{\mathfrak {P}}_{x}\right\}-2{\frac {{\mathfrak {\dot {q}}}_{z}}{|{\mathfrak {q}}|}}\left\{{\mathfrak {E}}'_{x}{\mathfrak {P}}_{z}-{\mathfrak {E}}'_{z}{\mathfrak {P}}_{x}\right\},\end{cases}}$

while (49a) gives

(49d)

${\frac {d}{dt}}({\mathfrak {E'P}})=2(\epsilon -1)\left\{{\mathfrak {E}}'_{x}{\mathfrak {\dot {E}}}'_{x}+k^{-2}{\mathfrak {E}}'_{y}{\mathfrak {\dot {E}}}'_{y}+k^{-2}{\mathfrak {E}}'_{z}{\mathfrak {\dot {E}}}'_{z}\right\}+2{\mathfrak {\dot {q}}}_{x}|{\mathfrak {q}}|k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}\right\}$.

Since one now has according to (48c)

${\begin{array}{c}{\mathfrak {E}}'_{x}{\mathfrak {P}}_{y}-{\mathfrak {E}}'_{y}{\mathfrak {P}}_{x}=|{\mathfrak {q}}|^{2}{\mathfrak {E}}'_{x}{\mathfrak {P}}_{y},\\{\mathfrak {E}}'_{x}{\mathfrak {P}}_{z}-{\mathfrak {E}}'_{z}{\mathfrak {P}}_{x}=|{\mathfrak {q}}|^{2}{\mathfrak {E}}'_{x}{\mathfrak {P}}_{z},\end{array}}$

thus it is given

(49c)

${\begin{cases}{\mathfrak {E'{\dot {P}}-{\dot {P}}E'}}&=2{\mathfrak {\dot {q}}}_{x}|{\mathfrak {q}}|k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}\right\}\\&-2{\mathfrak {\dot {q}}}_{y}|{\mathfrak {q}}|{\mathfrak {E}}'_{x}{\mathfrak {P}}_{y}-2{\mathfrak {\dot {q}}}_{z}|{\mathfrak {q}}|{\mathfrak {E}}'_{x}{\mathfrak {P}}_{z}.\end{cases}}$

The insertion of this expression and the corresponding magnetic term into (31), gives (instead of value (32) of momentum density) the corrected value

(50)

${\begin{cases}c{\mathfrak {g}}&={\mathfrak {[EH]+\left[E'[qP]\right]+\left[H'[qM]\right]}}\\&+{\mathfrak {q}}|{\mathfrak {q}}|^{2}k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}+{\mathfrak {E}}'_{y}{\mathfrak {M}}_{y}+{\mathfrak {H}}'_{z}{\mathfrak {M}}_{z}\right\}\end{cases}}$

That relation (18) is satisfied, can easily be verified.

If the value (50) of $c{\mathfrak {g}}$ is inserted in the general formula (19) for the energy density, then it follows instead of (33)

(51)

${\begin{cases}\psi &={\frac {1}{2}}{\mathfrak {E}}^{2}+{\frac {1}{2}}{\mathfrak {H}}^{2}\\&+{\frac {1}{2}}{\mathfrak {E'P}}+{\frac {1}{2}}{\mathfrak {H'M}}+|{\mathfrak {q}}|^{2}k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}+{\mathfrak {E}}'_{y}{\mathfrak {M}}_{y}+{\mathfrak {H}}'_{z}{\mathfrak {M}}_{z}\right\}\end{cases}}$

One also obtains, because of (20), the corrected formula for the energy current

(52)

${\begin{cases}{\frac {\mathfrak {S}}{c}}&={\mathfrak {[EH]+\left[E'[qP]\right]+\left[H'[qM]\right]}}\\&+{\mathfrak {q}}|{\mathfrak {q}}|^{2}k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}+{\mathfrak {E}}'_{y}{\mathfrak {M}}_{y}+{\mathfrak {H}}'_{z}{\mathfrak {M}}_{z}\right\}\end{cases}}$

From (50) and (52) one can see, that also in Lorentz's theory (when modified in the given way) the relation between the energy current and momentum density exists:

(53)	${\frac {\mathfrak {S}}{c}}=c{\mathfrak {g}},$

which we already encountered in Minkowski's theory.

This result was to be expected; after the equations connecting ${\mathfrak {D}}$ and ${\mathfrak {B}}$ with ${\mathfrak {E'}}$ and ${\mathfrak {H'}}$, are brought into agreement, no essential difference exists any more between both theories from the standpoint of our system.