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 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 ${\mathfrak {H}}={\mathfrak {B}}+[{\mathfrak {w}},\ {\mathfrak {D}}-{\mathfrak {E}}]$, 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 Lorentz's assumption, if we neglect ${\mathfrak {w}}^{2}$ in comparison to 1.

Also to the same order of approximation, Lorentz'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. Cohn^[1] assumes the following fundamental equations

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

(32)	${\mathfrak {F}}=\sigma E,\ {\mathfrak {E}}=\epsilon E-[{\mathfrak {w}}M],\ {\mathfrak {M}}=\mu M+[{\mathfrak {w}}E]$,

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

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