# Page:Grundgleichungen (Minkowski).djvu/25

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Now by putting ${\displaystyle {\mathfrak {H}}={\mathfrak {B}}}$, the differential equation (29) is transformed into the same form as eq. (1) here when ${\displaystyle {\mathfrak {m}}-[{\mathfrak {we}}]={\mathfrak {M}}-[{\mathfrak {wE}}]}$. Therefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of Lorentz 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 ${\displaystyle {\mathfrak {H}}={\mathfrak {B}}+[{\mathfrak {w}},\ {\mathfrak {D}}-{\mathfrak {E}}]}$, then in consequence of (C) in §8,

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

,

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

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

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

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

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

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

### § 10. Fundamental Equations of E. Cohn.

E. Cohn[1] assumes the following fundamental equations

 (31) ${\displaystyle {\begin{array}{c}curl\ (M+[{\mathfrak {wE}}])={\frac {\partial {\mathfrak {E}}}{\partial t}}+{\mathfrak {w}}\ div\ {\mathfrak {E}}+{\mathfrak {F}},\\\\-curl\ ({\mathfrak {E}}-[{\mathfrak {wM}}])={\frac {\partial {\mathfrak {M}}}{\partial t}}+{\mathfrak {w}}\ div\ {\mathfrak {M}},\end{array}}}$
 (32) ${\displaystyle {\mathfrak {F}}=\sigma E,\ {\mathfrak {E}}=\epsilon E-[{\mathfrak {w}}M],\ {\mathfrak {M}}=\mu M+[{\mathfrak {w}}E]}$,

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

1. Gött. Nachr. 1901, w. 74 (also in Ann. d. Phys. 7 (4), 1902, p. 29).