Page:VaricakRel1912.djvu/4

infinitely great velocity. It cannot be reached by an accumulation of velocities less than the speed of light; it also cannot be changed by addition or subtraction of a velocity less than the speed of light. Now, by a suitable definition we can simply make that the speed of light is represented by an infinite quantity. As unit length we take $c=3\cdot 10^{10}$, i.e. the light-path in a second, and then put

(1)	${\frac {v}{c}}=\operatorname {th} \,{\frac {U}{c}}=\operatorname {th} \,u$

To velocity v we relate the length U of measure unit u according to the relation

(2)	$u=\operatorname {th} ^{-1}{\frac {v}{c}}$

Following the English style of writing this denotes the inverse function of the hyperbolic tangent. Now we want to investigate whether this definition is not in sharp contrast to the ordinary illustration of velocities. The distances proportional to the relevant velocities are used in ordinary mechanics as representing the velocities of uniform motions. Formula (2) leads to the same result at the limits of our ordinary experience. Only at velocities nearly comparable to the velocity of light, a notable difference occurs that quickly leads to infinite distortion.

We have put

(3)	${\frac {U}{c}}=u={\frac {v}{c}}+{\frac {1}{3}}\left({\frac {v}{c}}\right)^{3}+{\frac {1}{5}}\left({\frac {v}{c}}\right)^{5}+\dots$

If we take v = 1 km/sec at first, then

$u={\frac {1}{3\cdot 10^{5}}}+{\frac {1}{3^{4}\cdot 10^{15}}}+{\frac {1}{5\cdot 3^{5}\cdot 10^{25}}}+\dots$

If we neglect everything after the first term on the right-hand side, then we commit an error that not even exerts an influence upon the 10th decimal. So by our definition, a length of 1 km is representing a velocity of 1 km/sec.

If we take the velocity of 100 km/sec which is in any case an extremely high velocity in ordinary mechanics, then it is given

${\frac {U}{c}}={\frac {1}{3\cdot 10^{3}}}+{\frac {1}{3\cdot 3^{3}\cdot 10^{9}}}+{\frac {1}{5\cdot 3^{5}\cdot 10^{15}}}+\dots$