. | (VIII) |
From this finite equation between r and , 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:
If we substitute this into equation (VIII), then we find:
and if we properly develop everything,
(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:
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