# Page:Grundgleichungen (Minkowski).djvu/34

 (54) ${\displaystyle \Psi _{4}=i({\mathfrak {w}}_{x}\Psi _{1}+{\mathfrak {w}}_{y}\Psi _{2}+{\mathfrak {w}}_{z}\Psi _{3})}$

The vector ${\displaystyle \Psi }$ is perpendicular to w; we can call it the Magnetic rest-force.

Relations analogous to these hold among the quantities ${\displaystyle iwF^{*},{\mathfrak {M,E,w}}}$ and Relation (D) can be replaced by the formula

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

We can use the relations (C) and (D) to calculate F and f from ${\displaystyle \Phi }$ and ${\displaystyle \Psi }$, we have

${\displaystyle wF=-\Phi ,\ wF^{*}=-i\mu \Psi ,\ wf=-\epsilon \Phi ,\ wf^{*}=-i\Psi }$

and applying the relation (45) and (46), we have

 (55) ${\displaystyle F=[w,\Phi ]+i\mu [w,\Psi ]^{*}}$,
 (56) ${\displaystyle f=\epsilon [w,\Phi ]+i[w,\Psi ]^{*}}$,

i.e.

 ${\displaystyle F_{12}=(w_{1}\Phi _{2}-w_{2}\Phi _{1})+i\mu (w_{3}\Psi _{4}-w_{4}\Psi _{3})}$, etc. ${\displaystyle f_{12}=\epsilon (w_{1}\Phi _{2}-w_{2}\Phi _{1})+i(w_{3}\Psi _{4}-w_{4}\Psi _{3})}$, etc.

Let us now consider the space-time vector of the second kind ${\displaystyle [\Phi \Psi ]}$, with the components

 ${\displaystyle \Phi _{2}\Psi _{3}-\Phi _{3}\Psi _{2},\ \Phi _{3}\Psi _{1}-\Phi _{1}\Psi _{3},\ \Phi _{1}\Psi _{2}-\Phi _{2}\Psi _{1}}$, ${\displaystyle \Phi _{1}\Psi _{4}-\Phi _{4}\Psi _{1},\ \Phi _{2}\Psi _{4}-\Phi _{4}\Psi _{2},\ \Phi _{3}\Psi _{4}-\Phi _{4}\Psi _{3}}$,

Then the corresponding space-time vector of the first kind

${\displaystyle w[\Phi ,\Psi ]=-(w{\overline {\Psi }})\Phi +w({\overline {\Phi }})\Psi }$

vanishes identically owing to equations 49) and 53).

Let us now take the vector of the 1st kind

 (57) ${\displaystyle |\Omega =iw[\Phi ,\ \Psi ]^{*}}$

with the components

${\displaystyle \Omega _{1}=-i\left|{\begin{array}{ccc}w_{2},&w_{3},&w_{4}\\\Phi _{2},&\Phi _{3},&\Phi _{4}\\\Psi _{2},&\Psi _{3},&\Psi _{4}\end{array}}\right|}$, etc.

Then by applying rule (45), we have