Page:Grundgleichungen (Minkowski).djvu/23

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

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

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

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

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

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

$\varrho '={\frac {-\left|{\mathfrak {w}}\right|{\mathfrak {s_{w}}}+\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}},\ s'_{w}={\frac {s_{w}-\left|{\mathfrak {w}}\right|\varrho }{\sqrt {1-{\mathfrak {w}}^{2}}}},\ {\mathfrak {s'_{\bar {w}}}}={\mathfrak {s}}_{\bar {w}}$,

(E)

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

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

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