# 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.