Page:VaricakRel1910b.djvu/7

 Put $$MN=u=\operatorname{th}\,^{-1}\frac{v}{c}$$. From N we let fall the perpendicular NP upon MT, then we construct the perpendicular NR upon NP in N, then we lay off the line MS = 1 upon MT, and from S we let fall the perpendicular SR upon NR. If we additionally make NU = PS, then NU will be parallel to MT in the an sense, and this parallel encloses with the X-axis the angle $$\varphi'$$.

As point U always lies between S and R, it can be easily seen from the figure that in relativity theory $$\varphi'$$ is smaller than $$\varphi'_0$$ of ordinary mechanics.

In '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 get in this way after some transformations

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

Formula (34) and formula (32), in which we will take $$\varphi'_0$$ instead of $$\varphi_0$$, give

from which we can, in accordance with the ordinary theory, easily find the formula

By comparison of formulas (33) and (34) we find

However, if we take in Fig. 3 $$OM' = u$$, then we can put

If would even be better, if Fig. 1 of my first treatise is used. Then we can take

7. Light pressure
The following remarks are related to 's formulas located in § 8 of his first paper on the relativity principle. We can easily see that

and this can be interpreted, as earlier $$\tfrac{\nu'}{\nu}$$, by using Fig. 6.

For light pressure, gave the formula

We transform them at first into the form

$$P=2\frac{A^{2}}{8\pi}\left(\operatorname{th}\,\ u_{1}-\operatorname{th}\,\ u_{2}\right)^{2}\cdot\operatorname{ch}\,^{2}u$$

from which we easily obtain

If we take a an right angled triangle with hypotenuse $$u_{1}-u$$ and an acute angle $$\alpha=\Pi\left(u_{1}\right)$$, then we obtain for the side adjacent to angle &alpha;, the expression

hence


 * , February 12, 1910

(Received February 18, 1910)