Page:LorentzGravitation1916.djvu/22

 In the following calculations the vector $$\mathrm{N}$$ has one of the directions $$1^{*},\dots4^{*}$$. As this is also the case with the vectors $$\mathrm{B}_{a^{*}}$$ and $$\mathrm{B}_{b^{*}}$$, the vector product occurring in (22) can easily be expressed in $$\xi$$-units. After that we may pass to natural units and finally, as is necessary for the substitution in (10), to $$x$$-units.

In order to pass from $$\xi$$-units to natural units we have to multiply a vector in the direction $$a^{*}$$ by a certain coefficient $$\lambda_{a}$$, and a part of the extension $$a^{*},b^{*},c^{*}$$ by a coefficient $$\lambda_{abc}$$. These coefficients correspond to $$l_{a}$$ (§ 10) and $$l_{abc}$$ (§ 12). The factors $$\lambda_{abc}$$ e.g. can be expressed by means of the minors $$\Gamma_{ab}$$ of the determinant $$\gamma$$ of the quantities $$\gamma_{ab}$$. If this is worked out and if the equations

$\gamma_{ab}=\frac{G_{ab}}{g},\ g_{ab}=\frac{\Gamma_{ab}}{\gamma},\ g\gamma=1 $|undefined

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

Let $$a, b, c, d$$ and also $$a', b', c', d'$$ be the numbers 1, 2, 3, 4 in any order, $$a'$$ being not the same as $$a$$, then we have, if none of the two numbers $$\alpha$$ and $$\alpha'$$ is 4,

and if one of the two is 4

§ 25. We shall now suppose (comp. § 24) that in $$\xi$$-units the vector $$\mathrm{B}_{a^{*}}$$ has the value +1, and we shall write $$\chi_{ab}$$ for the value that must then be given to $$\mathrm{B}_{b^{*}}$$. If the $$\xi$$-components of the vectors $$\mathrm{A^{I}}$$ etc. are denoted by $$\Xi_{1}^{I},\dots\Xi_{4}^{I}$$ etc., we find from (21)

This formula involves that

It may be remarked that $$\chi_{ba}$$ is the value that must be given to the vector $$\mathrm{B}_{a^{*}}$$ if $$\mathrm{B}_{b^{*}}$$ is taken to be 1.

The quantities $$\chi_{ab}$$ may be said to represent the rotations $$\left[\mathrm{B}_{a^{*}}\cdot\mathrm{B}_{b^{*}}\right]$$.

At the end of our calculations we shall introduce instead of $$\chi_{ab}$$ the quantities t$$\psi_{ab}$$ defined by

In the first of these equations $$a, b, a', b'$$ 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.