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which connect ${\mathfrak {DB}}$ and ${\mathfrak {E'H'}}$ with each other, have no such illustration. There is also no reason, when one takes the standpoint of the relativity principle, to speak about the aether and its electromagnetic properties. This principle only considers matter in its motion relative to the observer, and the electromagnetic processes in this matter.

However, for our system of electrodynamics of moving bodies, the vectors ${\mathfrak {E}}$ and ${\mathfrak {H}}$ are of lower importance than the vectors ${\mathfrak {DBE'H'}}$. When we, under elimination of ${\mathfrak {E}}$ and ${\mathfrak {H}}$, directly connect those four vectors with each other, then the relatedness of the theories of Minkowski and Lorentz will become clear.

A) Theory of Minkowski.

From equations (36) and (37) of § 9 it follows

(45)	${\begin{cases}{\mathfrak {D+\left[q[qD]\right]}}=\epsilon {\mathfrak {E'-[qH']}},\\{\mathfrak {B+\left[q[qB]\right]}}=\mu {\mathfrak {H'+[qE']}}.\end{cases}}$

If we let the $x$-axis coincide with the direction of ${\mathfrak {q}}$, then it is given for the components taken in this direction

(45a)

${\begin{cases}{\mathfrak {D}}_{x}=&\epsilon {\mathfrak {E}}'_{x},\\{\mathfrak {B}}_{x}=&\mu {\mathfrak {H}}'_{x}.\end{cases}}$

On the other hand, for the components perpendicular to the direction of motion, it is given

(45b)

${\begin{cases}k^{2}{\mathfrak {D}}_{y}=&\epsilon {\mathfrak {E}}'_{y}-[{\mathfrak {qH}}']_{y},\\k^{2}{\mathfrak {B}}_{y}=&\mu {\mathfrak {H}}'_{y}+[{\mathfrak {qE}}']_{y}.\end{cases}}$

B) Theory of Lorentz.

From equations (30) of § 8 it follows

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

Thus the components of ${\mathfrak {E}}$ and ${\mathfrak {H}}$ are parallel and perpendicular to the direction of velocity

(46a)

${\begin{cases}{\mathfrak {E}}_{x}={\mathfrak {E}}'_{x},&k^{2}{\mathfrak {E}}_{y}={\mathfrak {E}}'_{y}-[{\mathfrak {qH}}']_{y};\\{\mathfrak {H}}_{x}={\mathfrak {H}}'_{x},&k^{2}{\mathfrak {H}}_{y}={\mathfrak {H}}'_{y}+[{\mathfrak {qE}}']_{y}.\end{cases}}$

While in Minkowski's theory, $\epsilon$ and $\mu$ in isotropic bodies are independent from the direction, it is allowed in Lorentz's theory, that different values of $\epsilon$ and $\mu$ are possible for excitations parallel and perpendicular to ${\mathfrak {q}}$. Consequently one obtains from (29) and (46a) for the longitudinal components of ${\mathfrak {D}}$ and ${\mathfrak {B}}$

(47a)

${\begin{cases}{\mathfrak {D}}_{x}=&\epsilon _{x}{\mathfrak {E}}'_{x},\\{\mathfrak {B}}_{x}=&\mu _{x}{\mathfrak {H}}'_{x}.\end{cases}}$