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/31

 axis; hence the semi-axis major, called also the mean distance, $=\frac{p}{1-e e};$ the distance of the middle point of the axis (the centre of the ellipse) from the focus will be $\frac{e p}{1-e e}=e a,$  denoting by $a$  the semi-axis major.

IV. On the other hand, the aphelion in its proper sense is wanting in the parabola, but $r$ is increased indefinitely as $v$  approaches $+180^{\circ},$  or $-180^{\circ}.$  For $v= \pm 180^{\circ}$  the value of $r$  becomes infinite, which shows that the curve is not cut by the line of apsides at a point opposite the perihelion. Wherefore, we cannot, with strict propriety of language, speak of the major axis or of the centre of the curve; but by an extension of the formulas found in the ellipse, according to the established usage of analysis, an infinite value is assigned to the major axis, and the centre of the curve is placed at an infinite distance from the focus.

V. In the hyperbola, lastly, $v$ is confined within still narrower limits, in fact between $v=-\left(180^{\circ}-\psi\right),$  and $v=+\left(180^{\circ}-\psi\right),$  denoting by $\psi$  the angle of which the cosine $=\frac{1}{e}.$  For whilst $v$  approaches these limits, $r$  increases to infinity; if, in fact, one of these two limits should be taken for $v,$  the value of $r$  would result infinite, which shows that the hyperbola is not cut at all by a right line inclined to the line of apsides above or below by an angle $180^{\circ}-\psi.$  For the values thus excluded, that is to say, from $180^{\circ}-\psi$  to $180^{\circ}+\psi,$  our formula assigns to $r$  a negative value. The right line inclined by such an angle to the line of apsides does not indeed cut the hyperbola, but if produced reversely, meets the other branch of the hyperbola, which, as is known, is wholly separated from the first branch and is convex towards that focus, in which the sun is situated. But in our investigation, which, as we have already said, rests upon the assumption that $r$ is taken positive, we shall pay no regard to that other branch of the hyperbola in which no heavenly body could move, except one on which the sun should, according to the same laws, exert not an attractive but a repulsive force. Accordingly, the aphelion does not exist, properly speaking, in the hyperbola also; that point of the reverse branch which lies in the line of apsides, and which corresponds to the values $v=180^{\circ},$ $r=-\frac{p}{e-1},$  might be considered as analogous to the aphelion. If now, we choose after the manner of the