Page:LorentzGravitation1916.djvu/39

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obtained by differentiating the factor $\sqrt{-g}$ only and the other part by keeping this factor constant.

For the calculation of the first of these parts we can use the relation

$\frac{\partial\log\left(\sqrt{-g}\right)}{\partial g^{ac}}=-\frac{1}{2}g_{ac}$

and for the second part we find

$\frac{1}{2}\sqrt{-g}\sum(pq)g^{pq}\bar{\psi}_{ap}\bar{\psi}_{cq}$

If (32) 1915 is used (67) and (68) finally become

$\begin{array}{c} \sum(q)\psi_{cq}\bar{\psi}_{cq}+\sum\limits _{a\ne c}(a)\psi_{ac}^{*}\psi_{a'c'}=2\mathrm{L}\\ \\ \sum(q)\bar{\psi}_{cq}\psi_{bq}+\sum\limits _{a\ne c}(a)\psi_{ab}^{*}\psi_{a'c'}=0 \end{array}$

These equations are really fulfilled. This is evident from $\psi_{aa}=0$, $\bar{\psi}_{aa}=0$, $\psi_{ba}=-\psi_{ab}$ and $\bar{\psi}_{ba}=-\bar{\psi}_{ab}$, besides, the meaning of $\psi_{ab}^{*}$ (§ 11, 1915) and equation (35) 1915 must be taken into consideration.

§ 43. In nearly the same way we can treat the gravitation field of a system of incoherent material points; here the quantities $w_{a}$ and $u_{a}$ (§§ 4 and 5, 1915) play a similar part as $\psi_{ab}$ and $\bar{\psi}_{ab}$ in what precedes. To consider a more general case we can suppose "molecular forces" to act between the material points (which we assume to be equal to each other); in such a way that in ordinary mechanics we should ascribe to the system a potential energy depending on the density only. Conforming to this we shall add to the Lagrangian function $\mathrm{L}$ (§ 4, 1915) a term which is some function of the density of the matter at the point $P$ of the field-figure, such as that density is when by a transformation the matter at that point has been brought to rest. This can also be expressed as follows. Let $d\sigma$ be an infinitely small three-dimensional extension expressed in natural units, which at the point $P$ is perpendicular to the world-line passing through that point, and $\bar{\varrho}d\sigma$ the number of points where $d\sigma$ intersects world-lines. The contribution of an element of the field-figure to the principal function will then be found by multiplying the magnitude of that element expressed in natural units by a function of $\bar{\varrho}$. Further calculation teaches us that the term to be added to $\mathrm{L}$ must have the form

$\sqrt{-g}\varphi\left(\frac{P}{\sqrt{-g}}\right)$

(71)