SECT. 1.] TO POSITION IN THE ORBIT. 5 



axis ; hence the semi-axis major, called also the mean distance, =. ^ ; the dis 

 tance of the middle point of the axis (the centre of the ellipse) from the focus will 

 be e j^ =ea, 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, or 180. For 

 v = + 180 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 = (180 if), and v = -{-(180 if), denoting by if the angle of 



which the cosine =-. 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 if. For 

 the values thus excluded, that is to say, from 180 if to 180 -(-if, 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 sepa 

 rated 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 hyper 

 bola also ; that point of the reverse branch which lies in the line of apsides, 



and which corresponds to the values z&amp;gt; = 180, r== j~i&amp;gt; might be consid 

 ered as analogous to the aphelion. If now, we choose after the manner of the 



