Page:VaricakRel1912.djvu/22

 If we divide this formula by (47), then we obtain

$$\frac{\sin\varphi'}{\cos\varphi'}=\frac{\sin\varphi}{\cos\varphi-\frac{v}{c}}$$

from which we easily find in agreement with ordinary theory

However, in this case we have to take $$\varphi'_{0}$$ instead of $$\varphi$$ and $$\tfrac{v}{c}$$ instead of u.

Here, we also want to give another expression for aberration, that was geometrically interpreted by in two ways. From (56) it follows

$$e^{-u'_{1}}=e^{u}\cdot e^{-u_{1}}$$

according to the relation existing between the perpendiculars and the corresponding parallel angle, this can be written in the form

10. The reflection of light at a moving mirror.

Let the coordinate plane $$X'=0$$ be a perfect mirror. The light ray incident at the reflecting coordinate plane at point $$O'$$, is defined by the angle $$\varphi$$ and the frequency &nu;. These magnitudes are related to the stationary system. The mirror $$X'=0$$ is moving with velocity u in the direction of the positive abscissa axis of the stationary reference frame.

Instead of the parallel angle $$\varphi$$ we want to consider the corresponding line $$u_1$$. In the primed system the corresponding length according to (56) is

$$u'_{1}=u_{1}-u$$

To the parallel angle of the reflected ray, we have the corresponding length

In the primed system $$S'$$ the incident ray encloses the angle $$\varphi'$$ with the $$X'$$-axes; but after the reflection the angle is $$\varphi''=\pi-\varphi'$$. According to the definition of the parallel angles for negative perpendiculars, the angle $$\varphi''$$ corresponds to the perpendicular $$-u'_{1}$$ when the supplementary angle $$\varphi'$$ corresponds to the perpendicular $$u'_{1}$$. If we consider the aberration equation (56), then we can see that for the observer resting in O,