Page:Theory of the motion of the heavenly bodies moving about the sun in conic sections- a translation of Gauss's "Theoria motus." With an appendix (IA theoryofmotionof00gaus).pdf/32

 ellipse to call the value of the expression $\frac{p}{1-e e},$ even here where it becomes negative, the semi-axis major of the hyperbola, then this quantity indicates the distance of the point just mentioned from the perihelion, and at the same time the position opposite to that which occurs in the ellipse. In the same way $\frac{e p}{1-e e},$ that is, the distance from the focus to the middle point between these two points (the centre of the hyperbola), here obtains a negative value on account of its opposite direction.

5.

We call the angle $v$ the true anomaly of the moving body, which, in the parabola is confined within the limits $-180^{\circ}$  and $+180^{\circ},$  in the hyperbola between $-(180^{\circ}-\psi)$  and $+(180^{\circ}-\psi),$  but which in the ellipse runs through the whole circle in periods constantly renewed. Hitherto, the greater number of astronomers have been accustomed to count the true anomaly in the ellipse not from the perihelion but from the aphelion, contrary to the analogy of the parabola and hyperbola, where, as the aphelion is wanting, it is necessary to begin from the perihelion: we have the less hesitation in restoring the analogy among all classes of conic sections, that the most recent French astronomers have by their example led the way.

It is frequently expedient to change a little the form of the expression $r=\frac{p}{1+e \cos v};$ the following forms will be especially observed:

$$\begin{aligned} & r=\frac{p}{1+e-2 e \sin ^{2} \frac{1}{2} v}=\frac{p}{1-e+2 e \cos ^{2} \frac{1}{2} v} \\ & r=\frac{p}{(1+e) \cos ^{2} \frac{1}{2} v+(1-e) \sin ^{2} \frac{1}{2} v}. \end{aligned}$$

Accordingly, we have in the parabola

$$r=\frac{p}{2 \cos ^{2} \frac{1}{2} v};$$

in the hyperbola the following expression is particularly convenient,

$$r=\frac{p \cos \psi}{2 \cos \frac{1}{2}(v+\psi) \cos \frac{1}{2}(v-\psi)}.$$