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and b are the lines that correspond to the parallel angles ${\tfrac {1}{2}}\pi -\Pi (u)$ and $\Pi (u)$, and A and B are its acute angles, then we simply have

$\Delta E=-\operatorname {tg} \,B\sum \left(\xi K_{\xi }\right)$

From these few examples we can see, what advantage (even by mathematical evaluation) the non-euclidean interpretation of the relativity formulas could give to us. We have excellent tables for hyperbolic functions, which were published by the Simithsonian Institution in 1909.

The analogies that exist between relativity theory and Lobachevskian geometry are in any case interesting. The formulas of recent mechanics for $c=\infty$ are reduced to the formulas of Newtonian mechanics. Similarly also the Lobachevskian geometry, if we take the so called radius of curvature as infinite, goes over into the euclidean geometry. For ordinary velocities, the results calculated according to the relativity formulas, practically do not differ from these calculated according to the ordinary mechanical expressions. Also for distances of ordinary lengths, the calculations according to the Lobachevskian geometry do not differ from the euclidean calculations. In relativity theory there exists an absolute speed, in Lobachevskian geometry there exists an absolute length.

In relativity theory all bodies in motion are subjected to a certain deformation. In Poincaré's interpretation of Lobachevskian geometry, we can take the line element $d\sigma ={\tfrac {ds}{y}}$ which cannot be moved without deformation.

(Received January 19, 1910.)