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/29

 the sun, in order that these phenomena may be continually produced. In this way it is found that the action of the sun upon the bodies moving about it is exerted just as if an attractive force, the intensity of which is reciprocally proportional to the square of the distance, should urge the bodies towards the centre of the sun. If now, on the other hand, we set out with the assumption of such an attractive force, the phenomena are deduced from it as necessary consequences. It is sufficient here merely to have recited these laws, the connection of which with the principle of gravitation it will be the less necessary to dwell upon in this place, since several authors subsequently to the eminent have treated this subject, and among them the illustrious, in that most perfect work the Mécanique Céleste, in such a manner as to leave nothing further to be desired.

3.

Inquiries into the motions of the heavenly bodies, so far as they take place in conic sections, by no means demand a complete theory of this class of curves; but a single general equation rather, on which all others can be based, will answer our purpose. And it appears to be particularly advantageous to select that one to which, while investigating the curve described according to the law of attraction, we are conducted as a characteristic equation. If we determine any place of a body in its orbit by the distances $x, y,$ from two right lines drawn in the plane of the orbit intersecting each other at right angles in the centre of the sun, that is, in one of the foci of the curve, and further, if we denote the distance of the body from the sun by $r$  (always positive), we shall have between $r,$  $x,$  $y,$  the linear equation $r+\alpha x+\beta y=\gamma,$  in which $\alpha,$  $\beta, \gamma$  represent constant quantities, $\gamma$  being from the nature of the case always positive. By changing the position of the right lines to which $x,$ $y,$  are referred, this position being essentially arbitrary, provided only the lines continue to intersect each other at right angles, the form of the equation and also the value of $\gamma$  will not be changed, but the values of $\alpha$  and $\beta$  will vary, and it is plain that the position may be so determined that $\beta$  shall become $=0,$  and $\alpha,$  at least, not negative. In this way by putting for $\alpha,$ $\gamma,$  respectively $e,$  $p,$  our equation takes the form $r+e x=p.$  The right line to