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

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 $\epsilon = 1,\ \mu = 1,\ \sigma = 0$.