Page:Grundgleichungen (Minkowski).djvu/23

are the same as those of ${\displaystyle {\mathfrak {e}}+[{\mathfrak {wm}}]}$ and ${\displaystyle {\mathfrak {m}}-[{\mathfrak {we}}]}$, multiplied by ${\displaystyle {\frac {1}{\sqrt {1-{\mathfrak {w}}^{2}}}}}$. On the other hand ${\displaystyle {\mathfrak {E}}'}$ and ${\displaystyle {\mathfrak {M}}'}$ shall stand to ${\displaystyle {\mathfrak {E}}+[{\mathfrak {wM}}]}$, and ${\displaystyle {\mathfrak {M}}-[{\mathfrak {wE}}]}$ in the same relation us ${\displaystyle {\mathfrak {e}}'}$ and ${\displaystyle {\mathfrak {m}}'}$ to ${\displaystyle {\mathfrak {e}}+[{\mathfrak {wm}}]}$ and ${\displaystyle {\mathfrak {m}}+[{\mathfrak {we}}]}$. From the relation ${\displaystyle {\mathfrak {e}}'=\epsilon {\mathfrak {E}}'}$, the following equations follow

 (C) ${\displaystyle {\mathfrak {e}}+[{\mathfrak {wm}}]=\epsilon ({\mathfrak {E}}+[{\mathfrak {wM}}])}$.

and from the relation ${\displaystyle {\mathfrak {M}}'=\mu {\mathfrak {m}}'}$ we have

 (D) ${\displaystyle {\mathfrak {M}}-[{\mathfrak {mE}}]=\mu ({\mathfrak {m}}-[{\mathfrak {we}}])}$

For the components in the directions perpendicular to ${\displaystyle {\mathfrak {w}}}$, and to each other, the equations are to be multiplied by ${\displaystyle {\sqrt {1-{\mathfrak {w}}^{2}}}}$.

Then the following equations follow from the transfermation equations (12), 10), (11) in § 4, when we replace ${\displaystyle q,\ {\mathfrak {r_{v},\ r_{\bar {v}}}},t,{\mathfrak {r'_{v},\ r'_{\bar {v}}}},t'}$ by ${\displaystyle \left|{\mathfrak {w}}\right|,\ {\mathfrak {s_{w},s_{\bar {w}}}},\varrho ,{\mathfrak {s'_{w},s'_{\bar {w}}}},\varrho '}$.

${\displaystyle \varrho '={\frac {-\left|{\mathfrak {w}}\right|{\mathfrak {s_{w}}}+\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}},\ s'_{w}={\frac {s_{w}-\left|{\mathfrak {w}}\right|\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}},\ {\mathfrak {s'_{\bar {w}}}}={\mathfrak {s}}_{\bar {w}}}$,
 (E) ${\displaystyle {\begin{array}{c}{\frac {{\mathfrak {s_{w}}}-\left|{\mathfrak {w}}\right|\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}}=\sigma ({\mathfrak {E}}+[{\mathfrak {wM}}])_{\mathfrak {w}},\\\\{\mathfrak {s_{\bar {w}}}}={\frac {\sigma ({\mathfrak {E}}+[{\mathfrak {wM}}])_{\mathfrak {\bar {w}}}}{\sqrt {1-{\mathfrak {w}}^{2}}}}\end{array}}}$

In consideration of the manner in which ${\displaystyle \sigma }$ enters into these relations, it will be convenient to call the vector ${\displaystyle {\mathfrak {s}}-\varrho {\mathfrak {w}}}$ with the components ${\displaystyle {\mathfrak {s_{w}}}-\varrho {\mathfrak {\left|w\right|}}}$ in the direction of ${\displaystyle {\mathfrak {w}}}$ and ${\displaystyle {\mathfrak {s_{\bar {w}}}}}$ in the directions ${\displaystyle {\mathfrak {w}}}$ perpendicular to ${\displaystyle {\mathfrak {\bar {w}}}}$ the Convection current. This last vanishes for ${\displaystyle \sigma =0}$.

We remark that for ${\displaystyle \epsilon =1,\ \mu =1}$ the equations ${\displaystyle {\mathfrak {e'=E',\ m'=M'}}}$ immediately lead to the equations ${\displaystyle {\mathfrak {e=E,\ m=M}}}$ by means of a reciprocal Lorentz-transformation with ${\displaystyle -{\mathfrak {w}}}$ as vector; and for ${\displaystyle \sigma =0}$, the equation ${\displaystyle {\mathfrak {s}}'=0}$ leads to ${\displaystyle {\mathfrak {s}}=\varrho {\mathfrak {w}}}$, that the "fundamental equations of Æther" discussed in § 2 becomes in fact the limitting case of the equations obtained here with ${\displaystyle \epsilon =1,\ \mu =1,\ \sigma =0}$.