$1^{*},2^{*},3^{*},4^{*}$. 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 $1^{*},\dots 4^{*}$. 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.