Page:LorentzGravitation1916.djvu/8

conjugate hyperbolae. In both cases we derive our unit from the area of a parallelogram described on conjugate radius-vectors.

A three-dimensional extension cuts the conjugate indicatrix in an ellipsoid, or the indicatrix and its conjugate in two conjugate hyperboloids. Now our unit will be derived from the volume of a parallelepiped described on three conjugate radius-vectors.

In a similar way the magnitude of four-dimensional extensions will be determined by comparison with a parallelepiped the edges of which are four conjugate radius-vectors of the indicatrix and the conjugate indicatrix.

It must here be kept in mind that, according to well known theorems, the area of the parallelogram and the volume of the parallelepipeds in question are independent of the special choice of the conjugate radius-vectors.

We shall further specify the units in such a way (comp. § 5) that the numerical magnitude of a parallelogram or a parallelepiped described on conjugate radius-vectors is found by multiplying the numbers by which the edges are expressed in natural measure.

From what has been said it follows that the area of the parallelogram described on two line-elements is given by the product of the lengths of these elements and the sine of the enclosed angle. Similarly the area of an infinitely small triangle is determined by half the product of two sides and the sine of the angle between them.

We need hardly add that the numerical value of any two-, three- or four-dimensional domain expressed in natural measure is not changed by a deformation of the field-figure.

§ 9. Let, at any point $P$ of the field-figure, 1, 2, 3, 4 be four arbitrarily chosen conjugate radius-vectors of the indicatrix. Two of these determine an infinitely small part $V$ of a two-dimensional extension. We may prolong this part to finite distances from $P$ by drawing from this point geodetic lines whose initial directions lie in the plane $V$. In this way we obtain six two-dimensional extensions (1,2), (2,3), (3,1), (1,4), (2,4) and (3,4). Let us now consider in one of these e. g. ($a,b$) an infinitesimal triangle near the point $P$, the sides of which are geodetic lines (viz. geodetic lines in ($a,b$)). If in calculating the angles of this triangle we go to quantities of the second order with respect to the sides and to the distances from $P$, the sum $s$ of the angles proves to have no longer the value $\pi$ (comp. § 7). The "excess" $e=s-\pi$ is proportional to the area $\Delta$ of the triangle, independently of the length of the sides, of their ratios and of the position of the triangle in the extension ($a,b$). For the three extensions