Page:LorentzGravitation1916.djvu/24

${\begin{array}{c}{\frac {l_{ab4}\lambda _{bc4}}{l_{a}\lambda _{c}}}\chi _{b4}=\chi _{4b}=\psi _{ac},\\\\{\frac {l_{ab4}\lambda _{ac4}}{l_{b}\lambda _{c}}}\chi _{4a}=\chi _{a4}=\psi _{bc},\\\\{\frac {l_{ab4}\lambda _{abc}}{l_{4}\lambda _{c}}}\chi _{ba}=\chi _{ba}=\psi _{4c}.\end{array}}$

Taking also into consideration the opposite side $\left(dx_{a},dx_{b},dx_{4}\right)$ we find for $X_{a},X_{b},X_{4}$ the contributions

${\frac {\partial \psi _{ac}}{\partial x_{c}}}dW,\ {\frac {\partial \psi _{bc}}{\partial x_{c}}}dW,\ {\frac {\partial \psi _{4c}}{\partial x_{c}}}dW.$

This may be applied to each of the three pairs of sides not yet mentioned under $a$; we have only to take for $c$ successively 1, 2, 3.

Summing up what has been said in this § we may say: the components of the vector on the left hand side of (10) are

$X_{a}=\sum (b){\frac {\partial \psi _{ab}}{\partial x_{b}}}dW$

§ 27. For the components of the vector occurring on the right hand side of (10) we may write

$i\mathrm {q} _{a}d\Omega$

if $\mathrm {q} _{a}$ is the component of the vector $\mathrm {q}$ in the direction $x_{a}$ expressed in $x$-units, while $d\Omega$ represents the magnitude of the element $\left(dx_{1},\dots dx_{4}\right)$ in natural units. This magnitude is

$-i{\sqrt {-g}}dW$

so that by putting

${\sqrt {-g}}\mathrm {q} _{a}=w_{a}$

(28)

we find for equation (10)

$\sum (b){\frac {\partial \psi _{ab}}{\partial x_{b}}}=w_{a}$

(29)

The four relations contained in this equation have the same form as those expressed by formula (25) in my paper of last year^[1]. We shall now show that the two sets of equations correspond in all respects. For this purpose it will be shown that the transformation formulae formerly deduced for $w_{a}$ and $\psi _{ac}$ follow from the way in which these quantities have been now defined. The notations from the former paper will again be used and we shall suppose the transformation determinant $p$ to be positive.

1. Zittingsverslag Akad. Amsterdam, 23 (1915), p. 1073; translated in Proceedings Amsterdam, 19 (1910), p. 751. Further on this last paper will be cited by l. c.