Page:VaricakRel1912.djvu/12

 thus

$$l^{2}-x^{2}-y^{2}=\operatorname{ch}^{2}Z$$

and eventually

This relation is valid between the coordinates of every single point. It is known which role this invariant plays in 's four-dimensional interpretation of relativity theory.

5. The Lorentz-Einstein transformation.

The transformation

represents the translation along the X-axis in euclidean space. The transformation

similarly can be interpreted as a translation along the X-axis in an space.

If we remain in the plane then we can say: The Lorentz-Einstein transformation defines a motion along the distance line with the X-axis as its center line.

This distance line Y = b is the location of the points having a constant distance b from the X-axis. The length of its arc between two points M and $$M'$$ is (Fig. 6).

The displacement by the distance s along that equidistant line is defined by the equations

For the passage from $$M'$$ to $$M''$$ we have

$$X=X'-\frac{s'}{\operatorname{ch}\, b},\ Y=Y'.$$

or