# Page:LorentzGravitation1916.djvu/21

$1^{*},2^{*},3^{*},4^{*}$. Their components and the magnitudes of different extensions can now be expressed in $\xi$-nits in the same way as formerly in $x$-units. So the volume of a three-dimensional parallelepiped with the positive edges $d\xi_{1},d\xi_{2},d\xi_{3}$ is represented by the product $d\xi_{1}d\xi_{2}d\xi_{3}$.

Solving $x_{1},\dots x_{4}$ from (19) we obtain expressions of the form

 $\left.\begin{array}{c} x_{1}=\gamma_{11}\xi_{1}+\gamma_{21}\xi_{2}+\dots+\gamma_{41}\xi_{4}\\ \cdots\cdots\cdots\cdots\cdots\cdots\\ \cdots\cdots\cdots\cdots\cdots\cdots\\ x_{4}=\gamma_{14}\xi_{1}+\gamma_{24}\xi_{2}+\dots+\gamma_{44}\xi_{4}\\ \gamma_{ba}=\gamma_{ab} \end{array}\right\}$ (20)

If we use the coordinates $\xi$ the coefficients $\gamma_{ab}$ play the same part as the coefficients $g_{ab}$ when the coordinates $x$ are used. According to (18) and (20) we have namely

$F=\sum(a)\xi_{a}x_{a}=\sum(ab)\gamma_{ab}\xi_{a}\xi_{b}$

so that the equation of the indicatrix may be written

$\sum(ab)\gamma_{ba}\xi_{a}\xi_{b}=\epsilon^{2}$

§ 24. Let the rotations $\mathrm{R}_{e}$ and $\mathrm{R}_{h}$ of which we spoke in § 13 be defined by the vectors $\mathrm{A^{I},A^{II}}$ and $\mathrm{A^{III},A^{IV}}$ respectively, the resultants of the vectors $\mathrm{A_{1^{*}}^{I},\dots A_{4^{*}}^{I}}$, etc. in the directions $1^{*},\dots4^{*}$. Then, according to the properties of the vector product that were discussed in § 11,

$\begin{array}{ll} \left[\mathrm{R}_{e}\cdot\mathrm{N}\right] & =\left[\mathrm{\left(A_{1^{*}}^{I}+\dots+A_{4^{*}}^{I}\right)\cdot\left(A_{1^{*}}^{II}+\dots+A_{4^{*}}^{II}\right)\cdot N}\right]\\ & =\sum(\overline{ab})\left\{ \left[\mathrm{A}_{a^{*}}^{I},\ \mathrm{A}_{b^{*}}^{II}\cdot\mathrm{N}\right]-\left[\mathrm{A}_{a^{*}}^{II},\ \mathrm{A}_{b^{*}}^{I}\cdot\mathrm{N}\right]\right\} \end{array}$

where the stroke over $ab$ indicates that each combination of two different numbers $a, b$ contributes one term to the sum. For the vector product $\left[\mathrm{R}_{h}\cdot\mathrm{N}\right]$ we have a similar equation. Now two or more rotations in one and the same plane, e.g. in the plane $a^{*}b^{*}$, may be replaced by one rotation, which can be represented by means of two vectors with arbitrarily chosen directions in that plane, e.g. the directions $a^{*}$ and $b^{*}$. We may therefore introduce two vectors $\mathrm{B}_{a^{*}}$ and $\mathrm{B}_{b^{*}}$ directed along $a^{*}$ and $b^{*}$ resp., so that

 $\left[\mathrm{B}_{a^{*}}\cdot\mathrm{B}_{b^{*}}\right]=\left[\mathrm{A}_{a^{*}}^{I}\cdot\mathrm{A}_{b^{*}}^{II}\right]-\left[\mathrm{A}_{a^{*}}^{II}\cdot\mathrm{A}_{b^{*}}^{I}\right]+\left[\mathrm{A}_{a^{*}}^{III}\cdot\mathrm{A}_{b^{*}}^{IV}\right]-\left[\mathrm{A}_{a^{*}}^{IV}\cdot\mathrm{A}_{b^{*}}^{III}\right]$ (21)

Then we must substitute in (10)

 $\left[\mathrm{R}_{e}\cdot\mathrm{N}\right]+\left[\mathrm{R}_{h}\cdot\mathrm{N}\right]=\sum(\overline{ab})\left[\mathrm{B}_{a^{*}}\cdot\mathrm{B}_{b^{*}}\cdot\mathrm{N}\right]$ (22)

Here it must be remarked that the magnitude and the sense of one of the vectors $\mathrm{B}$ may be chosen arbitrarily; when this has been done, the other vector is perfectly determined.