Page:AbrahamMinkowski1.djvu/15

 From the standpoint of the system used by us, the task also arises to derive the momentum density from relation (18). It follows from (36)

$\begin{array}{l} \mathfrak{E'\dot{D}-D\dot{E}'=\dot{q}[E'H]+q[E'\dot{H}]+q[\dot{E}'H]},\\ \mathfrak{H'\dot{B}-B\dot{H}'=\dot{q}[EH']+q[\dot{E}H']+q[E\dot{H}]}.\end{array}$

Thus the right-hand side of (18) becomes

{{MathForm1|(38)|$$\left\{ \begin{array}{c} \mathfrak{E'\mathfrak{\dot{D}}-D\dot{E}'+H'\dot{B}-B\dot{H}'=\dot{q}\left\{ [E'H]+[EH']\right\} }\\ +\mathfrak{q\left\{ [E'\dot{H}]+[\dot{E}H']-[\dot{E}'H]-[E\dot{H'}]\right\} }\end{array}\right.$$}}

We express, on the basis of (37), $$\mathfrak{EH}$$ and $$\mathfrak{\dot{E}\dot{H}}$$ by the vectors arising in the main equations, and we find

By inserting (38a,b) into (38), we obtain

However, now it follows from (36)

thus the second and third row of the right-hand side of (28c) assume the values

$\begin{array}{l} 2\left\{ \mathfrak{(\dot{q}D)(qE')-(qD)(\dot{q}E')}\right\} =2\left(\mathfrak{[\dot{q}q][DE']}\right),\\ 2\left\{ \mathfrak{(\dot{q}B)(qH')-(qB)(\dot{q}H')}\right\} =2\left(\mathfrak{[\dot{q}q][BH']}\right).\end{array}$

If it indeed holds, as required by (18a)

then the second and third row are providing together:

$2\left(\mathfrak{[\dot{q}q][\mathfrak{q}c\mathfrak{g}]}\right)=2\mathfrak{[\dot{q}q][\mathfrak{q}c\mathfrak{g}]}-\mathfrak{q}^{2}(\mathfrak{q}2c\mathfrak{g})$

Therefore it eventually follows from (18)

The comparison with (20) gives the important relation