6 RELATIONS PERTAINING SIMPLY [BOOK I. 



ellipse to call the value of the expression ^~ , 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 ep -, 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 and -(-180, in the hyperbola 

 between (180 - - 1/&amp;gt;) and -)- (180 y&amp;gt; ), 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 



: the following forms will be especially observed : 

 1 -|- e cos v J 



r P _ P 



1 -)- e 2e sin 2 ^v 1 e-\-2e cos 2 ^ v 



- P 



Accordingly, we have in the parabola 



-_ P . 

 ~2cos 2 lt&amp;gt; 



in the hyperbola the following expression is particularly convenient, 



