then we obtain from the first formula, the principle of Doppler:
The parallel angle corresponds to the length , and so we have in this case
It is also
Higher powers of u can be neglected for small values, and we can replace by . The preceding formula goes over into the expression of Doppler's principle of ordinary mechanics:
Note, that in the primed reference frame .
The ratio of the frequencies and in the formula can be represented as the ratio of two limiting circular arcs between two common axes.
The expression of aberration is transformed into
The aberration equation is thus
A light ray T coming from an infinitely distant light source, is striking the x-axis at point M under the acute angle φ. We lay off the line toward the increasing abscissa, and from N apply the Lobachevskian parallel to T. This parallel encloses the angle with the x-axis.
If , also , hence , and the angle goes over into its supplement .
The formulas of relativity theory are very simplified in this interpretation. For example, for a moving electron of mass μ we have
A body in uniform translational motion in the direction of the increasing x-coordinate, has according to relativity theory the kinetic energy
where μ denotes its mass in the ordinary sense.[3] We can write this in a simpler way
Instead of we can write , where p is the area of a Saccheri isosceles quadrilateral with two right angles. Its three sides enclosing two right angles, have a length of u.
Here it was presupposed that this body is not subjected to external forces. However, if external forces act on that body, which are in equilibrium with each other, i.e. they don't accelerate the body, then according to the investigations of Einstein[4] its kinetic energy is (oddly enough) increased by
According to our definition this expression goes over into
We take a Lobachevskian right angled triangle. If the legs a