# Page:AbrahamMinkowski1.djvu/12

From them it follows, that when ${\displaystyle {\dot {\epsilon }}}$ and ${\displaystyle {\dot {\mu }}}$ are set equal to zero:

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

Now, since relation (18) requires

${\displaystyle 2{\mathfrak {\dot {q}}}c{\mathfrak {g=E'{\dot {D}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}}}'}$

one thus places Cohn's theory in our system, by setting

 (27) ${\displaystyle c{\mathfrak {g=[E'H']=}}{\frac {{\mathfrak {S}}'}{c}}}$

In Cohn's electrodynamics, the momentum density has to be set equal to the relative ray divided by ${\displaystyle c^{2}}$.

That relation (18a) is satisfied by (26) and (27), can easily be confirmed by considering, that the identity

${\displaystyle {\mathfrak {\left[q[E'H']\right]=\left[E'[qH']\right]-\left[H'[qE']\right]}}}$

exists. From (19) it follows now for the electromagnetic energy density.

 (28) ${\displaystyle \psi ={\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}+{\mathfrak {q[E'H']}}}$,

an expression, which according to (26) can also be written

 (28a) ${\displaystyle \psi ={\frac {1}{2}}\epsilon {\mathfrak {E}}'^{2}+{\frac {1}{2}}\mu {\mathfrak {H}}'^{2}+2{\mathfrak {q[E'H']}}}$;

it is in agreement with E. Cohns approach.

 (29) ${\displaystyle {\begin{cases}{\mathfrak {D}}=\epsilon {\mathfrak {E}}'-[{\mathfrak {qH}}],\\{\mathfrak {B}}=\mu {\mathfrak {H}}'+[{\mathfrak {qE}}];\end{cases}}}$
 (30) ${\displaystyle {\begin{cases}{\mathfrak {E'}}={\mathfrak {E}}+[{\mathfrak {qH}}],\\{\mathfrak {H'}}={\mathfrak {H}}-[{\mathfrak {qE}}].\end{cases}}}$
Besides four vectors contained in the main equations, two new vectors ${\displaystyle {\mathfrak {E,H}}}$ occur here. This circumstance makes Lorentz's theory more complicated than Cohn's one. The latter directly connects the components of ${\displaystyle {\mathfrak {D,B}}}$ with those of ${\displaystyle {\mathfrak {E',H'}}}$ by equations, which are linear in the velocity components; at this one, however, the connecting equations (§ 10, eq. 37b) given by elimination of ${\displaystyle {\mathfrak {EH}}}$, are not linear in the velocity components any more.