ELLIPTIC AND PARABOLIC ORBITS. 



Fig. i. 



d is 5 a, and the aphelion distance d' is s a', the mean distance a 

 being s c, or half the major axis. 



The curvature of the ellipse continually increases from the mean 

 distance to perihelion, and constantly decreases from perihelion 

 to the mean distance, being 

 equal at equal angular dis- 

 tances from perihelion as seen 

 from the sun. 



It is evident that if a body 

 move in a very eccentric 

 ellipse, such as that repre- 

 sented in fig. 1, whose plane 

 coincides exactly or nearly 

 with the common plane of the 

 planetary orbits, it may in- 

 tersect the orbits of several 

 or all of the planets, as it is 

 represented to do in the figure, 

 although its mean distance 

 from the sun maybe less than 

 the mean distance of several 

 of those which it thus inter- 

 sects. The aphelion distance 

 of such a body may, there- 

 fore, greatly exceed that of 

 any planet ; while its mean 

 distance may be less than 

 that of the more distant 

 planets. 



6. The form of a parabolic 

 orbit having the same peri- 

 helion distance as the elliptic 

 orbit is represented at a p p, 

 in fig. 1. This orbit consists 

 of two indefinite branches, 

 similar in form, which unite 

 at perihelion a. Departing 

 from this point on opposite 

 sides of the axis a a', their curvature regularly and rapidly 

 decreases, being equal at equal distances from perihelion. The 

 two branches have a constant tendency to assume the direction 

 and form of two straight lines parallel to the axis a a'. To 

 actual parallelism, and still less to convergence, these branches, 

 however, never attain, and consequently they can never reunite. 

 They extend, in fine, like parallel straight lines, to an unlimited 



149 



