Page:LorentzGravitation1916.djvu/7

the sum of the others, from which one finds that the angles have real values and that their sum is $\pi$.

b. The plane PQR cuts both the indicatrix and the conjugate indicatrix. In this case different positions of the triangle are still possible. We can however confine ourselves to triangles the three sides of which are real. These are really possible, for in the plane of a hyperbola we can draw triangles the sides of which are parallel to radius-vectors drawn from the centre to points of the curve (and not of the conjugate hyperbola).

By a closer consideration of the triangles now in question it is found however that by the choice of our "natural" units one side is necessarily longer than the sum of the other two. Formula (4) then shows that the cosines of the angles are real quantities, greater than 1 in absolute value, two of them being positive, and the third negative. We must therefore ascribe to the angles imaginary or complex values. If for $p>+1$ we put

$\arccos p=i\log \left(p+{\sqrt {p^{2}-1}}\right)$

and

$\arccos(-p)=\pi -\arccos p$

we find for the three angles expressions of the form

$i\alpha ,\ i\beta$ and $\pi -i(\alpha +\beta )$

so that the sum is again $\pi$.

From the cosine calculated by (4) or (5) the sine can be derived by means of the formula

$\sin \varphi ={\sqrt {1-\cos ^{2}\varphi }}$

where for the case $\cos ^{2}\varphi >1$ we can confine ourselves to the value

$\sin \varphi =i{\sqrt {\cos ^{2}\varphi -1}}$

with the positive sign.

It deserves special notice that two conjugate radius-vectors of the indicatrix and the conjugate indicatrix are perpendicular to each other and that a deformation of the field-figure does not change the angle between two intersecting lines determined according to our definitions.

§ 8. Before proceeding further we must now indicate the natural units (§ 5) for two-, three-, or four-dimensional extensions in the field-figure. Like the unit of length, these are defined for each point separately, so that the numerical value of a finite extension is found by dividing it into infinitely small parts.

A two-dimensional extension cuts the conjugate indicatrix in an ellipse, or the indicatrix itself and the conjugate indicatrix in two