. Their components and the magnitudes of different extensions can now be expressed in -nits in the same way as formerly in -units. So the volume of a three-dimensional parallelepiped with the positive edges is represented by the product .

Solving from (19) we obtain expressions of the form

(20) |

If we use the coordinates the coefficients play the same part as the coefficients when the coordinates are used. According to (18) and (20) we have namely

so that the equation of the indicatrix may be written

§ 24. Let the rotations and of which we spoke in § 13 be defined by the vectors and respectively, the resultants of the vectors , etc. in the directions . Then, according to the properties of the vector product that were discussed in § 11,

where the stroke over indicates that each combination of two different numbers contributes one term to the sum. For the vector product we have a similar equation. Now two or more rotations in one and the same plane, e.g. in the plane , 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 and . We may therefore introduce two vectors and directed along and resp., so that

(21) |

Then we must substitute in (10)

(22) |

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