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/30

 which the distances $y$ are referred in this case, is called the line of apsides, $p$  is the semi-parameter, $e$  the eccentricity; finally the conic section is distinguished by the name of ellipse, parabola, or hyperbola, according as $e$  is less than unity, equal to unity, or greater than unity.

It is readily perceived that the position of the line of apsides would be fully determined by the conditions mentioned, with the exception of the single case where both $\alpha$ and $\beta$  were $=0;$  in which case $r$  is always $=p,$  whatever the right lines to which $x,$  $y,$  are referred. Accordingly, since we have $e=0,$ the curve (which will be a circle) is according to our definition to be assigned to the class of ellipses, but it has this peculiarity, that the position of the apsides remains wholly arbitrary, if indeed we choose to extend that idea to such a case.

4.

Instead of the distance $x$ let us introduce the angle $v,$  contained between the line of apsides and a straight line drawn from the sun to the place of the body (the radius vector), and this angle may commence at that part of the line of apsides at which the distances $x$  are positive, and may be supposed to increase in the direction of the motion of the body. In this way we have $x=r \cos v,$ and thus our formula becomes $r=\frac{p}{1+e \cos v},$  from which immediately result the following conclusions :-

I. For $v=0,$ the value of the radius vector $r$  becomes a minimum, that is, $=\frac{p}{1+e}:$  this point is called the perihelion.

II. For opposite values of $v,$ there are corresponding equal values of $r;$  consequently the line of apsides divides the conic section into two equal parts.

III. In the ellipse, $v$ increases continuously from $v=0,$  until it attains its maximum value, $\frac{p}{1-e},$  in aphelion, corresponding to $v=180^{\circ};$  after aphelion, it decreases in the same manner as it had increased, until it reaches the perihelion, corresponding to $v=360^{\circ}.$  That portion of the line of apsides terminated at one extremity by the perihelion and at the other by the aphelion is called the major