Page:VaricakRel1912.djvu/8

 for the composition of velocities. Now, recently an interesting paper was published by, in which he arrived in an odd way at results previously published by me already. On page 9 he namely says: "If v be the absolute velocity of the particle with respect to the system, then the inverse hyperbolic tangent of v will be spoken of as the rapidity. Thus if w be the rapidity,

As w increases from 0 to &infin;, v increases from 0 to 1. For small values of w, practically, velocity is equal to rapidity, but we shall see latter that, for large values, it is the rapidity and not the velocity which follows the additive law." Then on p. 29: "Thus instead of a Euclidean triangle of velocities, we get a Lobatschefskij triangle of rapidities. ''For small rapidities, however, we may identify rapidity and velocity, and the Lobatschefskij triangle may be treated as an Euclidian one. It is also seen that rapidities in the same straight line are additive.''"

The difference in the ways, by which one arrives at the same results, strengthen the confidence in it.

3. The addition of velocities is not commutative. In an geometry no parallelograms exist; the resultant of two velocities cannot be represented by a diagonal of a parallelogram. As a consequence the components are noncommutative. Because of simplicity we take two velocities under an angle $$\alpha=\tfrac{\pi}{2}$$. From formula (5) we obtain

from which we can easily derive

or

In figure 3 we have