Page:VaricakRel1910c.djvu/1

 The Reflection of Light at Moving Mirrors.

By.

In the following I would like to give a non-Euclidean interpretation of 's formulas for the reflection of light at moving mirrors. The ray of light incident at a reflecting coordinate plane $$\xi=0$$ shall be defined by the quantities $$A$$, $$\cos\varphi$$, $$\nu$$. These quantities are related to a stationary coordinate system. Mirror $$\xi=0$$ shall move with velocity $$v$$ in the direction of the positive abscissa axis of the stationary system. For the direction cosine of the reflected ray, one thus has the formula according to :

If one puts herein

then it becomes

$\operatorname{th}\ u_{1}=\frac{\operatorname{th}\ u_{1}''+\operatorname{th}\ u}{1+\operatorname{th}\ u_{1}\operatorname{th}\ u}$

or

Now it is furthermore

and therefore

's formula for $$\cos\varphi'$$ was already transformed by me as the aberration equation (in my first report ), into the form:

and thus one has:

and

Here, $$u_1$$ means the perpendicular belonging to the parallel angle $$\varphi$$. Equation (8) replaces 's formula

$\cos\varphi'''=\frac{\left(1+\left(\frac{v}{c}\right)^{2}\right)\cos\varphi-2\frac{v}{c}}{1-2\frac{v}{c}\cos\varphi+\left(\frac{v}{c}\right)^{2}}$|undefined

From Fig. 1, the construction of the reflected ray is easily to be seen by formula (8). In the construction it is advantageous, to use angle $$\psi$$ being supplementary to $$\psi'''$$



It is $$\psi=\Pi\left(u_{1}-2u\right)$$. For $$u_{1}-2u$$ one has $$\psi=\tfrac{\pi}{2}$$. One ordinarily also considers $$\psi$$ as the reflection angle.

However, we can arrive at equation (8) in a still shorter way. Namely, the reflection angle at the moving mirror can be determined in the same way as in the stationary one, by means of construction on the basis of ' principle. I only mention the relevant explanations of and , undertaken by them in the course of investigating the  experiment.

assumes $$v$$ as being positive if the mirror is approaching the incident rays. In his formula (1) we thus have to assume $$v$$ as being negative, to bring it into accordance with our definitions. Then it reads in our notation

According to the relation that holds between the parallel angle and the corresponding perpendicular, we can write

$e^{-u_{1}'''}=\frac{1+\operatorname{th}\ u}{1-\operatorname{th}\ u}e^{-u_{1}}$|undefined

or

It's known that one has to take $$u_{1}$$ as being negative for angle $$\varphi$$, since it is supplementary to $$\psi$$. In which relation the magnitude of angle $$\psi$$ is with respect to angle $$\varphi$$, depends on the direction of motion of the mirror relative to the light source. In the case considered, angle $$\psi$$ is larger than $$\varphi$$, since it is related to the smaller perpendicular as parallel angle.