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

(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

(38a)

${\mathfrak {[E'H]+[EH']=}}2{\mathfrak {[E'H']+q(E'D)-D(qE')+q(H'B)-B(qH')}}$

(38b)

${\begin{cases}{\mathfrak {\qquad [E'{\dot {H}}]+[{\dot {E}}H']-[{\dot {E}}'H]-[E{\dot {H'}}]}}\\={\mathfrak {{\dot {q}}(E'D)-D({\dot {q}}E')+{\dot {q}}(H'B)-B({\dot {q}}H')}}\\+{\mathfrak {q\{E'{\dot {D}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}'}}\}\\-\left\{{\mathfrak {{\dot {D}}(qE')-D(q{\dot {E}}')+{\dot {B}}(qH')-B(q{\dot {H}}')}}\right\}.\end{cases}}$

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

(38c)

${\begin{cases}\qquad {\mathfrak {E'{\dot {D}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}'}}\\=2{\mathfrak {\dot {q}}}\left\{{\mathfrak {[E'H']+q(E'D)+q(H'B)-D(qE')-B(qH')}}\right\}\\+{\mathfrak {({\dot {q}}D)(qE')-(qD)({\dot {q}}E')-(q{\dot {D}})(qE')+(qD)(q{\dot {E}}')}}\\+{\mathfrak {({\dot {q}}B)(qH')-(qB)({\dot {q}}H')-(q{\dot {B}})(qH')+(qB)(q{\dot {H}}')}}\\+{\mathfrak {q}}^{2}\{{\mathfrak {E'{\dot {D}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}'}}\}.\end{cases}}$

However, now it follows from (36)

${\begin{array}{l}{\mathfrak {-(q{\dot {D}})(qE')-(qD)(q{\dot {E}}')=({\dot {q}}D)(qE')+(qD)({\dot {q}}E')}}\\{\mathfrak {-(q{\dot {B}})(qH')-(qB)(q{\dot {H}}')=({\dot {q}}B)(qH')+(qB)({\dot {q}}H')}}\end{array}}$

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)

(39)	$[{\mathfrak {q}}c{\mathfrak {g}}]={\mathfrak {[DE']+[BH']}},$

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)

(39a)

$c{\mathfrak {g=[E'H']+q(E'D)+q(H'B)-D(qE')-B(qH')+q(q}}c{\mathfrak {g)}}$

The comparison with (20) gives the important relation

(40)	${\mathfrak {g=}}{\frac {\mathfrak {S}}{c^{2}}}$