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system is as high as in the unprimed. Additionally, light is propagating in all valid reference frames in all directions in an isotropic way. For the moving observer the infinitely thin light wave emerging from the instantaneous signal, propagates as a spherical wave, which is also the case for the stationary observer. The equation for the sphere or the circle will be transformed by (34) into the very same equation. In Weierstrass coordinates we can write the equations of the circle in the form $l=\operatorname {ch} \,r$ or^[1]

$x^{2}+y^{2}=l^{2}\operatorname {th} ^{2}r,$

that is

(40)	$x^{2}+y^{2}=\varrho ^{2}$

if we put $\varrho =\operatorname {sh} \,r$. In a point M(x,y) we lay the tangent at this circle, and from center O we let fall a limiting arc that is normal to the tangent; then $\varrho$ is the length of this limiting arc. If we write (40) in the form

$x^{2}+y^{2}=l^{2}-1$

then because of (43) it goes over into

$x'^{2}+y'^{2}=l'^{2}-1$

At this place I want to remind the words of Minkowski: "It cannot be a great problem for a mathematician, who is accustomed to considerations on multi-dimensional manifolds and also on the expressions of the so called non-euclidean geometry, to adapt the notion of time to the application of the Lorentz-transformation."^[2] In connection with the things that were mentioned by Klein in his lecture "Geometrische Grundlagen der Lorentzgruppe", those primed words obtain a deeper meaning, and it would be very important to know, whether Minkowski has brought some of "his related inner considerations"^[3] to paper?

7. Application of the transformation equations on some problems of optics.

The light vector of a plane light-wave propagating in vacuum, which is related to system S, shall be proportional to^[4]

$F=\sin \nu \left(t-{\frac {x\ \cos \varphi +y\ \cos \psi }{c}}\right)$

1. H. Liebmann, Nichteuklidische Geometrie, 1905, 172
2. H. Minkowski, Zwei Abhandlungen über die Grundgleichungen der Elektrodynamik, 1910, 19.
3. Jahresbericht der Deutschen Mathematiker-Vereinigung, XIX, 1910, 299.
4. A. Einstein, Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen. Jahrb. d. Radioaktivität und Elektronik, 4, 1907, 424