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

(59)	$\sin \left(\varphi '-\varphi \right)={\frac {v}{c}}\sin \varphi '$

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 Plummer in two ways.^[1] 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

(60)	$\operatorname {tg} \,{\frac {\varphi '}{2}}={\sqrt {\frac {c+v}{c-b}}}\operatorname {tg} \,{\frac {\varphi }{2}}$

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 ν. 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

(61)	$u''_{1}=-u'_{1}=u-u_{1}$

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,

1. H. C. Plummer, On the Theory of Aberration and the Principle of Relativity. Monthly Notices of the Royal Astronomical Society, XX, 1910, 259