1348 
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 hyper- 
boloids. 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 insuch 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 paral- 
lelogram 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 con- 
sider in one of these e. g. (a, 6) an infinitesimal triangle near the point P, 
the sides of which are geodetie lines (viz. geodetic lines zn (a, 6)). 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 a (comp. $ 7). 
The ‘‘excess” ¢—=s — is proportional to the area A 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 exten- 
