Page:Ueber die Ablenkung eines Lichtstrals von seiner geradlinigen Bewegung.djvu/7



From this finite equation between r and $$\phi$$, the curved line can be specified. To achieve this more conveniently, we again want to reduce the equation to coordinates. Let (Fig. 3) AP = x and MP = y, then we have:

$$\begin{array}{ll} & x=1-r\cos \phi\\ & y=r\sin \phi\\ \mathrm{and} & r=\sqrt{\left(1-x\right)^{2}+y^{2}}\end{array}$$

If we substitute this into equation (VIII), then we find:

$$y^2=\frac {v^2(v^2- 4g)} {4g^2}[1-x]^2- \frac {v^2(v^2- 2g)} {2g^2}[1-x]+ \frac {v^2} {4g^2}$$,

and if we properly develop everything,

Since this equation is of second degree, then the curved line is a conic section, that can be studied more closely now.

If p is the parameter and a the semi-major axis, then (if we calculate the abscissa with its start at the vertex) the general equation for all conic sections is:

$$y^2=px+ \frac {p} {2a} x^2$$.

This equation contains the properties of the parabola, when the coefficient of x² is zero; that of the ellipse when it is negative; and that of the hyperbola when it is positive. The latter is evidently the case in our equation (IX). Since for all our known celestial bodies 4g is smaller than v², then the coefficient of x² must be positive.


 * If thus a light ray passes a celestial body, then it will be forced by the attraction of the body to describe a hyperbola whose concave side is directed against the attracting body, instead of progressing in a straight direction.

The conditions, under which the light ray would describe another conic section, can now easily be