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,
and if one of the two is 4
§ 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)
This formula involves that
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
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.