Page:LorentzGravitation1916.djvu/12

§ 12. In what precedes we were concerned with the volumes of parallelepipeds expressed in natural units. When we have introduced coordinates $x_{1},\dots x_{4}$ we may also express these volumes in the "$x$-units" corresponding to the coordinates chosen.

Let us consider e.g. the three-dimensional extension $x_{4}=const.$, which cuts the conjugate indicatrix in the ellipsoid

$g_{11}x_{1}^{2}+g_{22}x_{2}^{2}+g_{33}x_{3}^{2}+2g_{12}x_{1}x_{2}+2g_{23}x_{2}x_{3}+2g_{31}x_{3}x_{1}=-\epsilon ^{2}$

If we agree that in $x$-measure spaces in this extension will be represented by positive numbers and that a parallelepiped with the positive edges $dx_{1},dx_{2},dx_{3}$ will have the volume $dx_{1}\ dx_{2}\ dx_{3}$, we find for that of the parallelepiped on three conjugate radius-vectors

${\frac {\epsilon ^{3}}{\sqrt {-G_{44}}}}$

where it has been taken into consideration that $G_{44}$ is negative.

The volume of the same parallelepiped being expressed in natural measure by — $-i\epsilon ^{3}$ (§ 8), we have to multiply by

$l_{123}=-i{\sqrt {-G_{44}}}\,$

(8)

if we want to pass from the expression in $x$-measure to that in natural measure.

For the extension $\left(x_{2},x_{3},x_{4}\right)$, i.e. $x_{1}=0$ the corresponding factor is

$l_{234}=-{\sqrt {G_{11}}}$

(9)

§ 13. In the theory of electromagnetic phenomena we are concerned in the first place with the electric charge and the convection current. So far as these quantities belong to a definite element $d\Omega$ of the field-figure they may be combined into

$\mathrm {q} d\Omega$

where $\mathrm {q}$ is a vector which we may call the current vector. When it is resolved into four components having the directions of the axes, the first three components determine the convection current, while the fourth component gives the density of the electric charge.

As to the electric and the magnetic force, these two taken together can be represented at each point of the field-figure by two rotations

$\mathrm {R} _{e}$ and $\mathrm {R} _{h}$

in definite, mutually conjugate two-dimensional extensions. These quantities are closely connected with the current vector, for after having introduced coordinates $x_{1},\dots x_{4}$ we have for each closed surface $\sigma$ the vector equation