Page:Das Relativitätsprinzip und seine Anwendung.djvu/12

Furthermore, the following auxiliary vectors are useful:

${\begin{array}{cc}{\mathfrak {H}}_{1}={\mathfrak {H}}-{\frac {1}{c}}[{\mathfrak {w}}\cdot {\mathfrak {D}}],&{\mathfrak {B}}_{1}={\mathfrak {B}}-{\frac {1}{c}}[{\mathfrak {w}}\cdot {\mathfrak {E}}]{,}\\{\mathfrak {E}}_{1}={\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {w}}\cdot {\mathfrak {B}}],&{\mathfrak {D}}_{1}={\mathfrak {D}}+{\frac {1}{c}}[{\mathfrak {w}}\cdot {\mathfrak {H}}].\end{array}}$

The given field equations now are still to be supplemented by stating the relations, which exist between the vectors ${\mathfrak {E}},{\mathfrak {H}}$ and ${\mathfrak {D}},{\mathfrak {B}}$. One can derive these relations in two ways.

The first phenomenological method follows that procedure: One considers an arbitrarily moving point of matter, and introduces a reference system in which it is at rest; then, in case the volume element surrounding the point is isotropic in the rest system, e.g. the equations valid for resting systems hold between ${\mathfrak {E}}$ and ${\mathfrak {D}}$:

${\mathfrak {D}}=\varepsilon {\mathfrak {E}}{,}$

or also

${\mathfrak {D}}_{1}=\varepsilon {\mathfrak {E}}_{1}{,}$

because the auxiliary vectors ${\mathfrak {D}}_{1}$, ${\mathfrak {E}}_{1}$ are identical with ${\mathfrak {D}}$, ${\mathfrak {E}}$ for ${\mathfrak {w}}=0$. However, ${\mathfrak {D}}_{1}$ and ${\mathfrak {E}}_{1}$ are transformed in the same way, and from that it follows, that the equation

${\mathfrak {D}}_{1}=\varepsilon {\mathfrak {E}}_{1}$

remains valid in the initial reference system as well. Accordingly it is

${\mathfrak {B}}_{1}=\mu {\mathfrak {H}}_{1}.$

As regards the conduction current, we only remark that it depends on ${\mathfrak {E}}_{1}$.

The second method is based upon the mechanics of the electrons. In the same way, as (for resting bodies) equation ${\mathfrak {D}}=\varepsilon {\mathfrak {E}}$ proves to be the consequence of the assumption of quasi-elastic forces, which draw back the electron in their rest states, one will obtain the equation ${\mathfrak {D}}_{1}=\epsilon {\mathfrak {E}}_{1}$ at moving bodies, when one ascribes to those quasi-elastic forces those properties, which are required by the relativity principle. The latter will be satisfied, when one uses the expression of the generalized attraction law for these forces, where $R$ must be taken proportional to $r$.

The similar is valid for the explanation of the conduction resistance. A satisfying electron-theoretical explanation of the magnetic properties of the bodies is not present for the time being.

At last, the importance of the previous equations shall be shown at three remarkable cases.

The first remark is based on the equation

$\varrho '_{l}=a\varrho _{l}-{\frac {b}{c}}{\mathfrak {C}}_{z}$

According to it, $\varrho '_{l}$ can vanish without the need of $\varrho _{l}=0$,