Page:LorentzGravitation1916.djvu/24

 $\begin{array}{c} \frac{l_{ab4}\lambda_{bc4}}{l_{a}\lambda_{c}}\chi_{b4}=\chi_{4b}=\psi_{ac},\\ \\ \frac{l_{ab4}\lambda_{ac4}}{l_{b}\lambda_{c}}\chi_{4a}=\chi_{a4}=\psi_{bc},\\ \\ \frac{l_{ab4}\lambda_{abc}}{l_{4}\lambda_{c}}\chi_{ba}=\chi_{ba}=\psi_{4c}. \end{array}$|undefined

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

$\frac{\partial\psi_{ac}}{\partial x_{c}}dW,\ \frac{\partial\psi_{bc}}{\partial x_{c}}dW,\ \frac{\partial\psi_{4c}}{\partial x_{c}}dW.$|undefined

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

$X_{a}=\sum(b)\frac{\partial\psi_{ab}}{\partial x_{b}}dW$|undefined

§ 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

we find for equation (10)

The four relations contained in this equation have the same form as those expressed by formula (25) in my paper of last year. 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.