Page:Ueber die Ablenkung eines Lichtstrals von seiner geradlinigen Bewegung.djvu/8

 specified. It would describe a parabola when 4g = v², an ellipse when 4g were greater than v², and a circle when 2g = v². Since we don't know any celestial body whose mass is so great that it can generate such an acceleration at its surface, then the light ray always describes a hyperbola in our known world.

Now, it only remains to investigate, to what extend the light ray will be deflected from its straight line; or how great is the perturbation angle (which is the way I want to call it).

Since the figure of the trajectory is now specified, we can consider the light ray again as arriving. And because I at first want to specify only the maximum of the perturbation angle, I assume that the light ray comes from an infinitely great distance. — The maximum must take place in this case, because the attracting body longer acts on the light ray when it comes from a greater than from a smaller distance. — If the light ray comes from an infinite distance, then its initial direction is that of the asymptote BR (Fig. 3.) of the hyperbola, because in an infinitely great distant the asymptote falls into the tangent. Yet the light ray comes into the eye of the observer in the direction DA, thus ADB will be the perturbation angle. If we call this angle $$\omega$$, then we have, since the triangle ABD at A is right-angled:

$$\operatorname{tang}\ \omega=\frac{AB} {AD}$$.

However, it is known from the nature of the hyperbola, that AB is the semi-major axis, and AD the semi-lateral axis. Thus this magnitudes must also be specified. When a is the semi-major axis, and b the semi-lateral axis, then the parameter is:

$$p=\frac {2b^2} {a}$$.

If we substitute this value into the general equation of hyperbola