In the following calculations the vector has one of the directions . As this is also the case with the vectors and , the vector product occurring in (22) can easily be expressed in -units. After that we may pass to natural units and finally, as is necessary for the substitution in (10), to -units.

In order to pass from -units to natural units we have to multiply a vector in the direction by a certain coefficient , and a part of the extension by a coefficient . These coefficients correspond to (§ 10) and (§ 12). The factors e.g. can be expressed by means of the minors of the determinant of the quantities . If this is worked out and if the equations

are taken into consideration, we obtain the following corollary, which we shall soon use:

Let and also be the numbers 1, 2, 3, 4 in any order, being not the same as , then we have, if none of the two numbers and is 4,

(23) |

and if one of the two is 4

(24) |

§ 25. We shall now suppose (comp. § 24) that in -units the vector has the value +1, and we shall write for the value that must then be given to . If the -components of the vectors etc. are denoted by etc., we find from (21)

(25) |

This formula involves that

(26) |

It may be remarked that is the value that must be given to the vector if is taken to be 1.

The quantities may be said to represent the rotations .

At the end of our calculations we shall introduce instead of the quantities t defined by

(27) |

In the first of these equations are supposed to be the numbers 1, 2, 3, 4, in an order obtained from 1, 2, 3, 4 by an even number of permutations.