Page:AbrahamMinkowski2.djvu/5

and from $V_{II}^{4}[-i{\mathfrak {H}},\ -{\mathfrak {D}}\}$, when combined according to scheme (6), we obtain $V_{I}^{4}$, which is called the "magnetic rest force":

(8)	${\mathfrak {R}}^{m}={\frac {{\mathfrak {H}}-[{\mathfrak {qD}}]}{\sqrt {1-{\mathfrak {q}}^{2}}}},\ U^{m}={\frac {i({\mathfrak {qH}})}{\sqrt {1-{\mathfrak {q}}^{2}}}},$

The vectors "electric rest force and magnetic rest force" are connected with $V^{3}$, which determine the ponderomotive forces on electric and magnetic poles in motion, and which I wrote down in the first paper as ${\mathfrak {E}}'$ and ${\mathfrak {H}}'$:

(9)	${\mathfrak {E}}'={\mathfrak {E}}+[{\mathfrak {qB}}],\ {\mathfrak {H}}'={\mathfrak {H}}-[{\mathfrak {qD}}]$

Evidently we have:

(9a)	${\mathfrak {R}}^{e}=k^{-1}{\mathfrak {E}}',\ U^{e}=ik^{-1}({\mathfrak {qE}}'),$

(9b)	${\mathfrak {R}}^{m}=k^{-1}{\mathfrak {H}}',\ U^{m}=ik^{-1}({\mathfrak {qH}}'),$

setting

(9c)	$k={\sqrt {1-{\mathfrak {q}}^{2}}}$

From the two $V_{I}^{4}$

$\left\{{\mathfrak {R}}^{e},\ U^{e}\right\}$ and $\left\{{\mathfrak {R}}^{m},\ U^{m}\right\}$

which are composed according to formula (1a), we obtain the $V_{II}^{4}$:

${\begin{array}{l}{\mathfrak {a}}=k^{-2}[{\mathfrak {E}}'{\mathfrak {H}}']\\{\mathfrak {b}}=ik^{-2}\left\{{\mathfrak {E}}'({\mathfrak {qH}}')-{\mathfrak {H}}'({\mathfrak {qE}}')\right\}=ik^{-2}\left[{\mathfrak {q}}[{\mathfrak {E}}'{\mathfrak {H}}']\right]\end{array}}$

By insertion of $V^{3}$:

(10a	${\mathfrak {f}}'=[{\mathfrak {E}}'{\mathfrak {H}}']$

the last $V_{II}^{4}$ can be written:

(11)	${\mathfrak {a}}=k^{-2}{\mathfrak {f}}',\ {\mathfrak {b}}=ik^{-2}[{\mathfrak {qf}}']$

By multiplying the $V^{3}{\mathfrak {f}}'$ by the speed of light ($c$), we obtain the "relative radius" vector of my first paper [l. c.), equation IV].

Finally we combine $V_{I}^{4}$ which is represented by (11), with the $V_{I}^{4}$-"velocity" according to scheme (6). $V_{I}^{4}$ is calculated as follows:

${\begin{array}{l}{\mathfrak {R}}=ik^{-3}\left\{{\mathfrak {f}}'+\left[{\mathfrak {q}}[{\mathfrak {qf}}']\right]\right\}\\U=-k^{-3}({\mathfrak {qf}}').\end{array}}$

When multiplied by ($-i$), we arrive at the $V_{I}^{4}$-"rest radius" of Minkowski:

(12)	$\left\{{\begin{array}{l}{\mathfrak {R}}=k^{-1}{\mathfrak {f}}'+k^{-3}{\mathfrak {q}}[{\mathfrak {qf}}']\\U=ik^{-3}({\mathfrak {qf}}').\end{array}}\right.$

§ 3. Four-dimensional tensors .

A "Four-dimensional tensor" ($T^{4}$) is a set of ten variables, which are transformed into the orthogonal Lorentz transformations, such as we transform the squares and products of the coordinates $x,y,z,u$:

$x^{2},\ y^{2},\ z^{2},\ yz,\ zx,\ xy;\qquad xu,\ yu,\ zu;\qquad u^{2}.$