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

{{MathForm2|(20)|$$\left.\begin{array}{c} x_{1}=\gamma_{11}\xi_{1}+\gamma_{21}\xi_{2}+\dots+\gamma_{41}\xi_{4}\\ \cdots\cdots\cdots\cdots\cdots\cdots\\ \cdots\cdots\cdots\cdots\cdots\cdots\\ x_{4}=\gamma_{14}\xi_{1}+\gamma_{24}\xi_{2}+\dots+\gamma_{44}\xi_{4}\\ \gamma_{ba}=\gamma_{ab} \end{array}\right\} $$}}

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,

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

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

Then we must substitute in (10)

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.