42 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
It appears, therefore, that the results from the two hypotheses difPer but little for 
some distance outside the region of collisions, and either line may be taken as near 
enough to the correct result. We see then that the effect of annulling collisions and 
allowing each body to describe an orbit is that the density at first falls off more 
rapidly than if the medium were in convective equilibrium, and that further away the 
density falls off less rapidly. At more remote distances the density would be found 
to tend to vary as the inverse square of this distance. Thus, the formulae would make 
the mass of the system infinite. In other words, the existence of meteorites with 
nearly parabolic and hyperbolic orbits necessitates an infinite number, if the loss to 
the system is constantly made good by the supply. 
The subject of this section is considered further, from a physical point of view, in 
the Summary at the end. 
14. On the Kinetic Theory ivhere the Meteorites are of all sizes. 
In an actual swarm of meteorites all sizes occur, for, even if this were not the 
case initially, inequality of size would soon arise through fractures. Hence, it becomes 
of interest to examine the kinetic theory on the hypothesis that the colliding bodies 
are of all possible sizes, grouped about some mean value according to some law of 
frequency. 
If there be two sets elastic spheres in such numbers that there are respectively 
A and B in unit volume, and if the mean squares of the velocities of the two are cA 
and respectively, and if a and h are the radii of the spheres of the two sets, then it 
is proved that the number of collisions between them per unit time and volume is 
+ [f;r 
We shall now change the notation, and for a and h write and Sg, and for a and /3 
write and u^. 
Then, if S be the density of the spheres, their masses are and ^Trhsf 
The condition for the permanence of condition is that the spheres of all masses 
shall have the same mean kinetic energy. Hence, we refer the mass to a mean sphere 
of radius 5 , and the velocity to a square of velocity Fh 
T’hen 
Thus, our formula may be written 
+ U 
* ‘ The Kinetic Theory of Gases,’ by H. W. Watsox, p. II. 
