# Page:LorentzGravitation1916.djvu/21

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

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

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

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

${\displaystyle F=\sum (a)\xi _{a}x_{a}=\sum (ab)\gamma _{ab}\xi _{a}\xi _{b}}$

so that the equation of the indicatrix may be written

${\displaystyle \sum (ab)\gamma _{ba}\xi _{a}\xi _{b}=\epsilon ^{2}}$

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

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

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

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

Then we must substitute in (10)

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

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