Page:Grundgleichungen (Minkowski).djvu/23

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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.