# 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.