# Page:LorentzGravitation1916.djvu/6

so far as it belongs to the space ${\displaystyle \Omega }$ enclosed by that surface. Then the quantity ${\displaystyle H_{1}}$ is the sum, taken with the negative sign, of the lengths of all world-lines of material points so far as they lie within ${\displaystyle \Omega }$, each length multiplied by a constant ${\displaystyle m}$, characteristic of the point in question and to be called its mass.[1]

It must be remarked that the elements of the world-lines of material points intersect the corresponding indicatrices themselves. The lengths of these lines are therefore real positive quantities.

A deformation of the field-figure leaves ${\displaystyle H_{1}}$ unchanged.

§ 7. We shall now pass on to the part of the principal function belonging to the gravitation field. The mathematical expression for this part was communicated to me by Einstein in our correspondence. It is also to be found in Hilbert's paper in which it is remarked that the quantity in question may be regarded as the measure of the curvature of the four-dimensional extension to which (1) relates. Here we have to speak only of the interpretation of this quantity. To find this the following geometrical considerations may be used.

Let ${\displaystyle PQ}$ and ${\displaystyle PR}$ be two line-elements starting from a point ${\displaystyle P}$ of the field-figure, ${\displaystyle QR}$ the line-element joining the extremities ${\displaystyle Q}$ and ${\displaystyle R}$. If then the lengths of these elements in natural measure are

${\displaystyle PQ=ds',\ PR=ds'',\ QR=ds}$

we define the angle ${\displaystyle (s',s'')}$ between ${\displaystyle PQ}$ and ${\displaystyle PR}$ by the well known trigonometric formula

 ${\displaystyle {\begin{array}{c}ds^{2}=ds'^{2}+ds''^{2}-2ds'ds''\cos(s',s'')\\\\\cos(s',s'')={\frac {ds'^{2}+ds''^{2}-ds^{2}}{2ds'ds''}}\end{array}}}$ (4)

from which one can derive

 ${\displaystyle \cos(s',s'')=\sum (ab)g_{ab}{\frac {dx'_{a}}{ds'}}{\frac {dx''_{b}}{ds''}}}$ (5)

By means of this formula we are able to determine the angle between any two intersecting lines. Of course the two other angles of the triangle ${\displaystyle PQR}$ can be calculated in the same way.

Now two cases must be distinguished.

a. The plane of the triangle ${\displaystyle PQR}$ cuts the conjugate indicatrix, but not the indicatrix itself. Then the three sides have positive imaginary values. Moreover each of them proves to be smaller than

1. This agrees with the value of the Lagrangian function, which is to be found e.g. in my paper on "Hamilton's principle in Einstein's theory of gravitation." These Proceedings 19 (1916). p. 751.