Page:VaricakRel1912.djvu/14

The infinitesimal form of the L-E transformation is

(36)	$Uf\equiv -l{\frac {\partial f}{\partial x}}-x{\frac {\partial f}{\partial l}}$

The invariants of first kind are

(37)	$\omega (l,x)=l^{2}-x^{2}-y^{2}\coth ^{2}b=0$

because $U(\omega )=0$. These are the distance lines Y = b.^[1]

The invariants of second kind are the perpendiculars to the X-axis

(38)	$\omega (l,x)={\frac {l}{x}}-\coth u=0$

because it is

$U(\omega )=-1+\omega ^{2}=F(\omega )$

In Lobachevskian right angled coordinates the equation of that perpendicular is X = u.

In space we obtain the distance lines with the X-axis as their center line, by the intersection lines of two distance areas

$y-d_{1}=0,\ z-d_{2}=0$

The center areas are the coordinate areas XY and XZ.

The L - E transformation (34), to which also equations z' = z have to be added, can be interpreted as a translation along the intersection line of these two equidistant areas. The orbital curve of a point of a rigid body during a translation along the X-axis is the distance line to the X-axis. The transverse dimensions of the body remain unchanged during this displacement.

If we take the parameter c = ∞, then the Lobachevskian space goes over to the euclidean space; the limiting arcs x, y, z become straight lines; Weierstrass coordinates are transformed into ordinary Cartesian coordinates, those distance lines become parallels to the X-axis; the transformation (34) or (26) goes over into (25).

6. Transformation of the time parameter.

From two observers moving uniformly but at different velocities on parallel paths, each of them can claim with the same justification, that they are at rest in respect to each other. Expressed in terms of geometry, this means that we can transform any point of the plane into a state of rest by relating it to a new coordinate system. From the related point, let fall the perpendicular upon the X-axis and use this perpendicular as the new ordinate axis. If we take, for example, the perpendicular through M, then XY is the

1. H. Liebmann, Nichteuklidische Geometrie, 172