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Therefore we can obtain the laws for the reflection of light upon a moving mirror, by replacing u by 2u in formulas (34). However, as the image is located at the object's opposite side of the plane $X'=0$, we have to take $-x'$ instead of $x'$, i.e., we have to subject the light vector to the transformation

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

Now, it follows from (1)

$\operatorname {ch} \,2u={\frac {c^{2}+v^{2}}{c^{2}-v^{2}}},\ \operatorname {sh} \,2u={\frac {2vc}{c^{2}-v^{2}}}$

and the preceding equations go over to

(71)	${\begin{array}{l}x'=-{\frac {c^{2}+v^{2}}{c^{2}-v^{2}}}\cdot x+{\frac {2vc^{2}}{c^{2}-v^{2}}}\cdot t\\\\t'={\frac {c^{2}+v^{2}}{c^{2}-v^{2}}}t-{\frac {2v}{c^{2}-v^{2}}}\cdot x.\end{array}}$

H. Bateman^[1] has derived the laws of reflection on a moving mirror on the basis of the presupposition: the image of an object is caused by that space-time transformation (71).

The reflection angle at the moving mirror can be defined in the same way as at a stationary mirror, by means of construction according to the principle of Huyghens. I only mention the related statements by W. M. Hicks^[2] and E. Kohl^[3], performed by them with respect to the Michelson-Morley experiment. From our figure 14 we see, that $\psi =\Pi \left(u_{1}-2u\right)$ or $u_{2}=u_{1}-2u$, when we denote by $u_{2}$ the perpendicular corresponding to angle ψ. From this it follows $e^{-u_{2}}=\left(e^{u}\right)^{2}e^{-u_{1}}$, or

(72)	$\operatorname {tg} \,{\frac {\psi }{2}}={\frac {c+v}{c-b}}\operatorname {tg} \,{\frac {\varphi }{2}}$

This is the formula of Hicks. However, he assumes v to be positive when the ray moves towards the incident rays. In his formula (1) we have to take v as negative, to bring them into accordance to our definition.^[4] In the same way we have to alter his figure.

1. H. Bateman, The reflexion of light at an ideal plane mirror moving with a uniform velocity of translation. Phil. Mag. 18, 892, 1909
2. Phil. Mag. 3, 1902, 15
3. Ann. d. Phys. 28, 1909, 262
4. See also Laue, Das Relativitätsprinzip, 93