142 ON THE ORBITS OF THE ASTEROIDS. 
§ 3. 
Have the Orbits of the Asteroids ever been materially affected by a resisting Medium ? 
It is highly probable that all the asteroids are of nearly the same density ; it is at 
least highly improbable that there exists any relation between the density and the 
magnitude, by virtue of which the smaller asteroids are more dense than the larger 
ones. If, then, these bodies are retarded by the action of a resisting medium, we may 
expect to see its effect more manifest on the small ones than on the large ones; the 
resistance being proportional to the superficial area, while the inertia is proportional 
to the mass, and probably to the volume. If, then, the large and small asteroids were 
originally arranged indiscriminately with respect to their distance from the sun, the 
effect of a resisting medium would be manifested in a tendency among the smaller 
asteroids to be nearer the sun than the larger ones. If we represent the mass of an 
asteroid by m, and the number of asteroids by n, the condition of indiscriminate 
arrangement would be 
Zma Za 
= SES: 
23m n 
in which 8 is a quantity of the order of magnitude of the chance errors of distribu- 
tion; which diminishes inversely as the square root of n ; and therefore vanishes when 
n is infinite. A tendency in the smaller asteroids to be near the sun will be manifested 
ma 
=m 
a tendency in the larger asteroids to be near the sun will be manifested by the same 
by the quantity = z being greater than any probable value of B, and vice versa, 
expression being negative, and greater than any probable value of $. 
Unfortunately, however, if we apply this method to the actual case now in question, 
we shall fall into error from a cause which seems unavoidable. In fact, an asteroid 
near the sun will be more easily discovered than a more distant one of the same mag- ` 
nitude, owing to its greater brilliancy ; and this circumstance is of itself sufficient to 
cause a tendency in the smaller known asteroids to be near the sun, though no such 
tendency should exist in the whole group, known and unknown. We could eliminate 
the effects of this cause, provided that we knew the general law which connects any 
assumed magnitude with the number of asteroids of that magnitude. But such a law - 
can be derived only from observation, and the discussion of the observations will be 2 
subject to the same difficulty with the application of the test. Still, we may make 
some deductions respecting the effect of a resisting medium by considering its different 
effects on bodies of different magnitudes. : 
