Page:VaricakRel1912.djvu/8

for the composition of velocities.^[1] Now, recently an interesting paper was published by Robb,^[2] 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,

$v=\tanh {\text{ }}w$

As w increases from 0 to ∞, 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 Lobachevskian 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

(13)	$\operatorname {ch} \,u=\operatorname {ch} \,u_{1}\operatorname {ch} \,u_{2},$

from which we can easily derive

$\operatorname {th} ^{2}u=\operatorname {th} ^{2}u_{1}+\operatorname {th} ^{2}u_{2}-\operatorname {th} ^{2}u_{1}\operatorname {th} ^{2}u_{2}$

or

(14)	$v={\sqrt {v_{1}^{2}+v_{2}^{2}-\left({\frac {v_{1}v_{2}}{c}}\right)^{2}}}$

In figure 3 we have

$OA=u_{1},\ AB=u_{2},\ OB=u.$

1. G. Herglotz, Über den vom Standpunkt des Relativitätsprinzips aus als "starr" zu bezeichnenden Körper. Ann. d. Phys. 32, 404, 1910
2. Alfred A. Robb, Optical geometry of motion; a new view of the theory of relativity, Cambridge 1911. His preface is dated 13. May 1911. My relevant investigations were published in Physikalische Zeitschrift of February 1st and April 1st 1910. Also my mentioned Serbian treatise was completed by the end of the year 1910.