Page:LorentzGravitation1916.djvu/5

they always remain hyperboloids of the said kind. The gravitation field will now be determined by these indicatrices, which we can imagine to have been constructed in the field-figure without the introduction of coordinates. When we have occasion to use these latter, we shall so choose them that the "axes" $x_{1},x_{2},x_{3}$ intersect the conjugate indicatrix constructed around their starting point, while the indicatrix itself is intersected by the axis $x_{4}$. This involves that the coefficients $g_{11},g_{22},g_{33}$ are negative and that $g_{44}$ is positive.

§ 5. The indicatrices will give us the units in which we shall express the length of lines in the field-figure and the magnitude of two-, three or four-dimensional extensions. When we use these units we shall say that the quantities in question are expressed in natural measure.

In the case of a line-element $PQ$ the unit might simply be the radius-vector in the direction $PQ$ of the indicatrix or the conjugate indicatrix described about $P$. It is however desirable to distinguish the two cases that $PQ$ intersects the indicatrix itself or the conjugate indicatrix. In the latter case we shall ascribe an imaginary length to the line-element^[1]. Besides, by taking as unit not the radius-vector itself but a length proportional to it, the numerical value of a line-element may be made to be independent of the choice of the quantity $\epsilon$.

These considerations lead us to define the length that will be ascribed to line-elements by the assumption that each radius-vector of the indicatrix has in natural measure the length $\epsilon$, while each radius-vector of the conjugate indicatrix has the length $i\epsilon$.^[2]

It will now be clear that the length of an arbitrary line in the field-figure can be found by integration, each of its elements being measured by means of the indicatrix or the conjugate indicatrix belonging to the position of the element. In virtue of our definitions a deformation of the field-figure will not change the length of lines expressed in natural measure and a geodetic line will remain a geodetic line.

§ 6. We are now in a position to indicate the first part $H_{1}$ of the principal function (§ 1). Let $\sigma$ be a closed surface in the field-figure and let us confine ourselves to the principal function

1. This corresponds to the negative value which (1) gives for $ds^{2}$.
2. For a radius-vector on the asymptotic cone we may take either of these values; this makes no difference, as the numerical value of a line-element in the direction of such a radius-vector becomes 0 in both cases.