Page:VaricakRel1912.djvu/21

If $\varphi ={\tfrac {\pi }{2}}$, also $u_{1}=0$, hence $u'_{1}=-u_{1}$, and the angle $\varphi '$ goes over into its supplement $\varphi '_{1}$. The diverted light ray is RJ.

The construction of the required parallel is executed in Fig. 13 as follows.^[1] Put $MN=u=\operatorname {th} ^{-1}{\tfrac {v}{c}}$. From N we let fall the perpendicular NP upon T, then we construct the perpendicular NR upon NP in N, then we lay off the line MS equal to the absolute unit distance of our Lobachevskian space upon T, and from S we let fall the perpendicular SR upon NR. If we additionally make NU = PS, then NU will be parallel to T in the Lobachevskian sense, and this parallel encloses with the X-axis the angle $\varphi '$.

Since point U lies always between S and R, it can easily be seen from the figure that in relativity theory $\varphi '$ is smaller than $\varphi '_{0}$ of ordinary mechanics.

In Einstein's formula for aberration, we denote the cosines on the left-hand side and in the numerator of the fraction by corresponding sines, then we square and obtain in this way after some transformations

(57)	$\sin \varphi '={\frac {\sin \varphi \cdot {\sqrt {1-{\frac {v}{c}}}}}{1-{\frac {v^{2}}{c^{2}}}\cos \varphi }}$

During the motion of earth in its orbit relative to the fixed stars as reference frame we have ${\tfrac {v}{c}}=0,0001$. For such small velocities we can neglect ${\tfrac {v^{2}}{c^{2}}}$, and then we approximately obtain

(58)	$\sin \varphi '={\frac {\sin \varphi }{1-{\frac {v}{c}}\cos \varphi }}$

1. Lobatschefskij-Engel. Zwei geometrische Abhandlungen, 256