» 
ON THE ORBITS OF THE ASTEROIDS. 143 
Judging from its brilliancy, Atalanta is the smallest known asteroid. Its brilliancy 
is only about -4y as great as would be that of Vesta at the same distance. Its mean 
distance is 2.748, — considerably greater than the average. Supposing that by the resist- 
ance of a medium it was brought from the farthest limit of the zone of the asteroids 
to the position in which we find it, it would have been caused to approach the sun by 
the amount .407. The brilliancy of different asteroids at equal distances being pro- 
portional to the squares of their diameters, while the effect of the resisting medium 
is inversely proportional to their diameters, the effect of the medium on Vesta would 
be only 41, as great as on Atalanta; the mean distance of the latter would, therefore, 
have been diminished by 0.034; and this we may regard as the extreme limit of the 
possible change in the mean distance of Vesta from this cause. 
Hygea is smaller than Vesta, and Themis smaller than either Vesta or Flora. Hence, 
if these asteroids have been affected by a medium, the former positions of their orbits 
were more unfavorable for a common point of intersection than their present ones; 
hence our conclusions respecting the possibility of a common point of intersection are 
not invalidated by our not having taken into account the action of this possible cause. 
Moreover, one effect of the medium would be to increase the eccentricities of all the 
asteroids, and for this reason the former forms of the orbits were less favorable to in- 
tersection if this cause has acted. 
§ 4. 
Of the Relations between the Masses of the Asteroids and certain Elements of their Orbits. 
On any probable hypotheses that we can make respecting the cause of an explosion 
of a planet, the smaller fragments ought, on the whole, to be thrown off with a greater 
velocity than the larger ones. Moreover, when, as in the case of the asteroids, each 
fragment is very small compared with the original mass, it seems at least highly prob- 
able that the velocities of those thrown in any one direction would be nearly the same, 
on the whole, as the velocities of those thrown in a direction at right angles to that of 
the first. 
Thus we have two probable tests of Olbers's hypothesis. To apply them, we shall 
first deduce certain relations between the velocity with which a fragment would be 
thrown, and the elements of the orbit in which it would afterwards move. 
For this purpose, take, for the axis of X, a line passing through the sun and the 
position of the planet at the time of the explosion, let the axis of Y be in the plane of 
the orbit of the planet, and that of Z perpendicular to it. Represent by a, the dis- 
tance of the planet from the sun, and by ¿ and ¢ the velocities of the projected fragments ; 
