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

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

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 $$\left|\mathfrak{w}\right|,\ \mathfrak{s_{w},s_{\bar{w}}},\varrho,\mathfrak{s'_{w},s'_{\bar{w}}},\varrho'$$.

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