# Page:LorentzGravitation1916.djvu/22

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In the following calculations the vector ${\displaystyle \mathrm {N} }$ has one of the directions ${\displaystyle 1^{*},\dots 4^{*}}$. As this is also the case with the vectors ${\displaystyle \mathrm {B} _{a^{*}}}$ and ${\displaystyle \mathrm {B} _{b^{*}}}$, the vector product occurring in (22) can easily be expressed in ${\displaystyle \xi }$-units. After that we may pass to natural units and finally, as is necessary for the substitution in (10), to ${\displaystyle x}$-units.

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

${\displaystyle \gamma _{ab}={\frac {G_{ab}}{g}},\ g_{ab}={\frac {\Gamma _{ab}}{\gamma }},\ g\gamma =1}$

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

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

 ${\displaystyle {\frac {l_{bcd}\lambda b'c'd'}{l_{a'}\lambda _{a}}}=-1}$ (23)

and if one of the two is 4

 ${\displaystyle {\frac {l_{bcd}\lambda b'c'd'}{l_{a'}\lambda _{a}}}=1}$ (24)

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

 ${\displaystyle \chi _{ab}=\left(\Xi _{a}^{I}\Xi _{b}^{II}-\Xi _{a}^{II}\Xi _{b}^{I}\right)+\left(\Xi _{a}^{III}\Xi _{b}^{IV}-\Xi _{a}^{IV}\Xi _{b}^{III}\right)}$ (25)

This formula involves that

 ${\displaystyle \chi _{ba}=-\chi _{ab}}$ (26)

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

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

At the end of our calculations we shall introduce instead of ${\displaystyle \chi _{ab}}$ the quantities t${\displaystyle \psi _{ab}}$ defined by

 ${\displaystyle \psi _{ab}=\chi _{a'b'}(a\mp b),\ \psi _{aa}=0}$ (27)

In the first of these equations ${\displaystyle a,b,a',b'}$ 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.