Page:Grundgleichungen (Minkowski).djvu/39

while the relation

(69)	$lor\ ({\overline {lor\ F^{*}}})=0$

signifies that of the four equations in (B), only three represent independent conditions.

I shall now collect the results.

Let w denote the space-time vector of the first kind

${\frac {\mathfrak {w}}{\sqrt {1-{\mathfrak {w}}^{2}}}},{\frac {i}{\sqrt {1-{\mathfrak {w}}^{2}}}}$

(${\mathfrak {w}}$ = velocity of matter),

F the space-time vector of the second kind ${\mathfrak {M}},\ -i{\mathfrak {E}}$ (${\mathfrak {M}}$ = magnetic induction, ${\mathfrak {E}}$ = Electric force),

f the space-time vector of the second kind ${\mathfrak {m}},\ -i{\mathfrak {e}}$ (${\mathfrak {m}}$ = magnetic force,)

${\mathfrak {e}}$ = Electric Induction.

s the space-time vector of the first kind ${\mathfrak {s}},i\varrho$ ($\varrho$ = electrical space-density,)

${\mathfrak {s}}-\varrho {\mathfrak {w}}$ = conductivity current,

$\epsilon$ = dielectric constant,

$\mu$ = magnetic permeability,

$\sigma$ = conductivity.

then the fundamental equations for electromagnetic processes in moving bodies are

{A}

$lorf=-s$

{B}	$lorF^{*}=0$

{C}	$wf=\epsilon wF$

{D}	$wF^{*}=\mu wf^{*}$

{E}	$s+(w{\bar {s}})w=-\sigma wF$.

$w{\bar {w}}=-1$, and wF, wf, wF*, wf*, $s+(w{\bar {s}})w$ which are space-time vectors of the first kind are all normal to w, and for the system {B}, we have

$lor\ ({\overline {lor\ F^{*}}})=0$

Bearing in mind this last relation, we see that we have as many independent equations at our disposal as are necessary for determining the processes when proper fundamental data are given, where the motion of matter, thus the vector ${\mathfrak {w}}$ as a function of x, y, z, t are given.