Page:VaricakRel1910c.djvu/2

For the ratio of amplitudes and frequencies, Einstein gives the following equation:

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

(11)

Due to relation (2) we can write it in the form

${\frac {A'''}{A''}}={\frac {\nu '''}{\nu ''}}=\left(1+\operatorname {th} \ u\ \operatorname {th} \ u_{1}'\right)\operatorname {ch} \ u$

or also

${\frac {A'''}{A''}}={\frac {\nu '''}{\nu ''}}={\frac {\operatorname {ch} \left(u+u_{1}''\right)}{\operatorname {ch} \ u_{1}''}}$

(12)

If equation (7) is considered, then it becomes

${\frac {A'''}{A''}}={\frac {\nu '''}{\nu ''}}={\frac {\operatorname {ch} \left(2u+u_{1}\right)}{\operatorname {ch} \left(u+u_{1}\right)}}={\frac {\operatorname {ch} \left(u_{1}-2u\right)}{\operatorname {ch} \left(u_{1}-u\right)}}$

(13)

However, for the reflected ray it is

$A''=A',\ \nu ''=\nu '\,;$

(14)

according to formula (28) on p. 292 of this journal, one has

${\frac {A'}{A}}={\frac {\nu '}{\nu }}={\frac {\operatorname {ch} \left(u_{1}-u\right)}{\operatorname {ch} \ u_{1}}}$

(15)

and thus it becomes

${\frac {A'''}{A}}={\frac {\nu '''}{\nu }}={\frac {\operatorname {ch} \left(u_{1}-2u\right)}{\operatorname {ch} \ u_{1}}}$

(16)

The relations of the amplitudes and frequencies of the incident and reflected light, can be represented by the relation of the arcs of two distance lines between shared normals. Equation (16) replaces Einstein's equations

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

(17)

Fig. 2

We have graphically represented formulas (15) and (16) in Fig. 2. It is easily seen, that one obtains $\nu '''(A''')$ by reflection of $\nu (A)$ upon $\nu '(A')$. In this way, also angle $\varphi '''$ can be determined by reflection of the incident ray at the aberrated ray. Formula (15) for Doppler's principle and formula (16) for the amplitude and frequency of the reflected light are of the same form; as well as aberration equation (6) and formula (10) for the reflection angle.

We denoted this velocity by $v$, which is represented by distance $u$ (for $c=1$), and by $v'$ we want to denote that velocity corresponding to the double distance $2a$. Then it follows from the previously mentioned equations, that the same light ray appears to an observer moving with velocity $v'$, as of the same constitution as it would appear for a resting observer after the reflection at a mirror moving with velocity $v$. In both cases the motion must be of the same direction.

Also the procedure by Bateman^[1] is in connection with this result, who derived the laws of reflection at moving mirrors on the basis of the presupposition: the image of an object shall emerge by the space-time transformation

${\begin{array}{l}x'=-x\ \operatorname {ch} \,2u+l\ \operatorname {sh} \,2u\\l'=l\ \operatorname {ch} \,2u-x\ \operatorname {sh} \,2u\end{array}}$

(18)

He writes this in another form.

For a light ray which is incident perpendicularly, we have $\varphi =0$, thus $u_{1}=\infty$, and formula (16) goes over into

$\nu '''=\nu e^{-2u}\,.$

(19)

The relation of frequencies and amplitudes can in this case be represented as the relation of two coaxial limiting arcs.

Agram, May 14, 1910

(Received May 23, 1910.)

1. H. Bateman, The reflection of light at an ideal plane mirror moving with a uniform velocity of translation. Phil. Mag. 18, 892, 1909.