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

$r+\left[{\frac {v^{2}-2g}{2g}}\right]r\ \cos \phi ={\frac {v^{2}}{2g}}$.

(VIII)

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,

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

(IX)

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