148 ON THE ORBITS OF THE ASTEROIDS. 
less velocity than the others. To the smallness of y is alone due the fact, that the 
eccentricities are not as many times larger than the inclinations as would be required 
on the hypothesis of explosion. Now this smallness of y may be accounted for on 
Olbers's hypothesis, if we reflect that all the asteroids for which aj is greater than 
.120 would be thrown without the limit of the zone in which they are now found; 
and that owing to their consequent frequent approach to Jupiter when in their 
aphelion, or to Mars when in their perihelion, their orbits might be entirely deranged. 
We may now determine whether there exists between the masses of the asteroids, 
and the velocities with which, on Olbers's hypothesis, they were thrown, any relation 
in virtue of which the smaller asteroids were thrown with greater velocity than the 
larger ones, or vice versa. This question would be solved with most theoretical rigor 
as follows: — If n asteroids be numbered in the order of their magnitude, and also in 
the order of their velocity of projection, if m, represents the number of any asteroid in 
the order of magnitude, and v, in the order of velocity of projection, then will the 
condition of no relation between these two classes of numbers be, when » is large, 
A zm, Y. 
n 4-1 
B being a small quantity of the order of magnitude of the chance errors of distribu- : 
tion, or of »*. But a more simple method will give a result practically quite as good. 
If we take the forty-eight asteroids of which both the magnitudes and velocities are 
given, we find that the average velocity of projection of that half of which the mag- 
nitude is greatest is .209, or about 2.49 miles per second ; and of that half of which the 
magnitude is least, .217, or about 2.59 miles per second. This difference is much less 
than that which might result from the chance inequalities; hence no relation like that 
sought for exists between the masses of the asteroids, and the velocities with which they 
were projected, if Olbers's hypothesis be true. 
The velocities 7 are, some positive, others negative, which indicates that the frag- 
ments must have been projected both backward and forward with respect to the direc- 
tion of the planet's motion. The signs of § and £ it is not possible to determine, since 
the eccentricity and inclination of a fragment would have the same value, whether these 
quantities were positive or negative. - 
In finding the values of £, no allowance could be made for the latitude of the planet 
at the time of the explosion. This must have been very small relative to the invariable 
plane, else the inferior limits of the inclinations of none of the asteroids could have been 
nearly zero. | | 
cw WM 
