Page:AbrahamMinkowski1.djvu/10

which represent $\psi$ and ${\mathfrak {g}}$ by electromagnetic vectors, and finding them is our next goal. Accordingly, we can separate the terms in (17), which only contain derivatives with respect to time, from those ones, into which the divergence of ${\mathfrak {w}}$ is inserted as a factor; thus the equations are given

(17a)

${\dot {\varphi }}+{\mathfrak {g{\dot {w}}}}={\mathfrak {E'{\dot {D}}+H'{\dot {B}}}}$

(17b)

$\varphi ={\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}$

The elimination of $\varphi$ gives:

(18)	$2{\mathfrak {g{\dot {w}}}}={\mathfrak {E'{\dot {D}}-D{\dot {E}}+H'{\dot {B}}-B{\dot {H}}'}}$

This relation will serve us to determine the components of momentum density, after the right-hand side is expressed as a linear function of the acceleration components, based on the characteristic relations (of this theory) between the electromagnetic vectors.

For the components of ${\mathfrak {g}}$ perpendicular to ${\mathfrak {w}}$, the condition is given from (Vb) and (11)

(18a)

$[{\mathfrak {wg}}]=[{\mathfrak {DE}}']+[{\mathfrak {BH}}']$

This has to be satisfied in any case, since otherwise the system would exhibit an inner contradiction.

From (16) and (17b), the energy density is determined

(19)	$\psi ={\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}+{\mathfrak {wg}}$

According to (Va), the sum of relative normal stresses amounts to

$X'_{x}+Y'_{y}+Z'_{z}=-\left\{{\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}\right\}$

consequently it follows in accordance with (10)

$X{}_{x}+Y{}_{y}+Z{}_{z}=-\left\{{\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}+{\mathfrak {wg}}\right\}$

so that the remarkable relation exists

(19a)

$X{}_{x}+Y{}_{y}+Z{}_{z}+\psi =0$

If one inserts value (19) of $\psi$ as well as expressions (Va) for the relative stresses into (12), then one obtains for the energy current

(20)	${\mathfrak {S}}=c[{\mathfrak {E'H'}}]+{\mathfrak {w}}\{{\mathfrak {E'D}}+{\mathfrak {H'B}}\}-{\mathfrak {D(wE')-B(wH')+w(wg)}}$

an expression, which because of the known calculation rules, passes into

${\frac {\mathfrak {S}}{c}}{\mathfrak {=[E'H']+\left[E'[qD]\right]+\left[H'[qB]\right]+q}}({\mathfrak {q}}c{\mathfrak {g}}),$

when it is put for brevity's sake

${\mathfrak {q=}}{\frac {\mathfrak {w}}{c}}$

Instead of it, it can also be written

(21)	${\frac {\mathfrak {w}}{c}}{\mathfrak {=\left[E'-[qB],\ H'+[qD]\right]-q(qW)}}$