Page:LorentzGravitation1916.djvu/21

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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.