1345 
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 in- 
troduction of coordinates. When we have occasion to use these 
latter, we shall so choose them that the “axes” «,, v,,., intersect 
the conjugate indicatrix constructed around their starting point, 
while the indicatrix itself is intersected by the axis z,. This involves 
that the coefficients g,,,9,,, Jo, ave negative and that g,, 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’). 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 «. 
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 ¢, while each 
radius-vector of the conjugate indicatrix has the length ve. *) 
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 geodetie line will remain a 
geodetic line. 
§ 6. We are now in a position to indicate the first part /7, of 
the principal function (§ 1). Let o be a closed surface in the 
field-figure and let us confine ourselves to the principal func- 
1) This corresponds to the negative value which (1) gives for ds?. 
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 O in both cases. 
