Page:VaricakRel1910a.djvu/2

 If the velocities $$v_1$$ and $$v_2$$ enclose the angle &alpha;, and

$$\frac{v_{1}}{c}=\operatorname{th}\,\ u_{1},\ \frac{v_{2}}{c}=\operatorname{th}\, u_{2},$$

then lay off the line $$OA=u_{1}$$ from point O into the direction of $$v_1$$, and apply the line $$AC=u_{2}$$ under the angle &alpha;. The resultant corresponds to the line $$OC=u$$. In the an triangle OAC the relation is given

$$\operatorname{ch}\, u=\operatorname{ch}\, u_{1}\ \operatorname{ch}\, u_{2}+\operatorname{sh}\, u_{1}\operatorname{sh}\, u_{2}\cos\ \alpha$$

If we denote herein

$$\operatorname{ch}\, u_{i}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}\, u_{i}=\frac{\frac{v_{i}}{c}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},$$

then we obtain after some simple transformations the general ian addition theorem for velocities. In the case $$\alpha=\tfrac{\pi}{2}$$, we have

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

or

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

respectively

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

That this addition is not commutative can be easily seen from the first figure of that, however, we now have to interpret as a figure in the an plane. Additionally we have to put

In hyperbolic geometry the angle sum in any triangle is smaller than two right angles. Hence

$$\alpha_{1}+\alpha_{2}<\frac{\pi}{2},$$

thus OD does not coincide with the direction of OC. For the direction difference $$\delta=\sphericalangle COD$$ we find

It is also

$$\operatorname{tg}\,\frac{\delta}{2}=\operatorname{th}\,\frac{u_{1}}{2}\operatorname{th}\,\frac{u_{2}}{2}$$

If $$v_1$$ and $$v_2$$ are not in the xy-plane, but arbitrarily in space, then we obtain six terminal points, while above we only had points C and D.

In addition, I want to show by some examples, how 's formulas of can be interpreted as real in the an geometry.

Equations (3) in § 5 of the mentioned paper of define (in respect to the stationary system S) the velocity components $$u_{x},u_{y},u_{z}$$ of a point uniformly moving in relation to S'. If $$u_{z'}=0$$, then

$$\frac{u_{y}}{u_{x}}=\frac{u_{y'}}{u_{x'}+v}\cdot\sqrt{1-\left(\frac{v}{c}\right)^{2}}$$

If we take $$c=\infty$$, then the straight line upon which that point is moving, encloses the angle &lambda; with the x-axis. However, if c remains finite and equal to the propagation velocity of light in empty space, then we find the direction coefficients of that straight line as:

$$\operatorname{tg}\,\lambda'=\frac{\operatorname{tg}\,\lambda}{\operatorname{ch}\, u}$$

If u is the hypotenuse and $$\tfrac{\pi}{2}-\lambda$$ is an acute angle in the right angled an triangle, then $$\lambda'$$ is the second acute angle. It will be the smaller, the greater the translation velocity of S'. For v = c we have $$\lambda'=0$$.

We define s as the extension of a stationary electron in the direction of the x-axis. If it is set into motion with velocity v in the same direction, then its contracted extension is

$$s'=\frac{s}{\operatorname{ch}\, u}$$

Upon the distance line y = u having the x-axis as its center line and u as its parameter, and beginning from its intersection point U with the ordinate axis, we measure the length UM = s. The abscissa of M is $$s'$$.



In the same way the dilation of a clock in uniform motion relative to a reference frame can be interpreted.

If we further put

$$\varphi=\Pi\left(u_{1}\right),\ \varphi'=\Pi\left(u'_{1}\right),\ \frac{v}{c}=\operatorname{th}\, u,$$