Page:LorentzGravitation1916.djvu/24

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\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.