Page:Grundgleichungen (Minkowski).djvu/25

 Now by putting $$\mathfrak{H}=\mathfrak{B}$$, the differential equation (29) is transformed into the same form as eq. (1) here when $$\mathfrak{m}-[\mathfrak{we}]=\mathfrak{M}-[\mathfrak{wE}]$$. Therefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of for moving non-magnetised bodies at last agree with the relativity postulate.

If we make use of (30) for non-magnetic bodies, and put accordingly $$\mathfrak{H}=\mathfrak{B}+[\mathfrak{w},\ \mathfrak{D}-\mathfrak{E}]$$, then in consequence of (C) in §8,

$$(\epsilon-1)(\mathfrak{E}+(\mathfrak{wB}])=\mathfrak{D}-\mathfrak{E}+(\mathfrak{w}[\mathfrak{w},\ \mathfrak{D}-\mathfrak{E}])$$,

i.e. for the direction of $$\mathfrak{w}$$

$$(\epsilon-1)(\mathfrak{E}+(\mathfrak{wB}])_{\mathfrak{w}}=(\mathfrak{D}-\mathfrak{E})_{\mathfrak{w}}$$,

and for a perpendicular direction $$\mathfrak{\bar{w}}$$,

$$(\epsilon-1)(\mathfrak{E}+(\mathfrak{wB}])_{\mathfrak{\bar{w}}}=(1-\mathfrak{w}^{2})(\mathfrak{D}-\mathfrak{E})_{\mathfrak{\bar{w}}}$$,

i.e. it coincides with 's assumption, if we neglect $$\mathfrak{w}^2$$ in comparison to 1.

Also to the same order of approximation, 's form for $$\mathfrak{F}$$ corresponds to the conditions imposed by the relativity principle [comp. (E) § 8] — that the components of $$\mathfrak{F_{w}}$$, $$\mathfrak{F_{\bar{w}}}$$ are equal to the components of $$\sigma(\mathfrak{E}+(\mathfrak{wB}])$$ multiplied by $$\sqrt{1-\mathfrak{w}^{2}}$$ or $$\frac{1}{\sqrt{1-\mathfrak{w}^{2}}}$$ respectively.

§ 10. Fundamental Equations of E. Cohn.
E. assumes the following fundamental equations

where E, M are the electric and magnetic field intensities (forces), $$\mathfrak{E,M}$$ are the electric and magnetic polarisation (induction).