I. On the Application of the Kinetic Theory to Dense Oases. 
By S. H. Burbury, F.B.S. 
Received January 12,—Read February 7, 1895. 
The motion of a great number of elastic spheres, when their aggregate volume 
does not bear an evanescent ratio to the containing space, has received little attention 
from writers on the kinetic theory. In what respect, beyond the shortening of the 
mean free path, will it differ from that of the rare medium usually discussed? I 
think that the answer to this question is that there exists in all systems, dense or 
rare, a tendency for the spheres to move together in masses or streams, and so to 
diminish the mean pressure per unit of area, and the number of collisions per unit of 
volume and time. And this tendency has an appreciable influence on the form of 
the motion as soon as the ratio of the aggregate volume of the spheres to the con¬ 
taining space becomes appreciable. 
If a part of the system, say n spheres, be at any instant contained in a volume V, 
they have energy, IV, of the motion of their common centre of gravity. And they 
have energy, T,., of relative motion. As the spheres increase in diameter, the ratio 
T r /T, will be found to diminish on average. But the number of collisions per unit 
of volume and time, given T, or T, -f T y , depends on TV, and therefore diminishes by 
the diminution of T,.. 
1. Let M be the mass, c the diameter of a sphere, p the number of spheres per 
unit of volume, p the pressure per unit of surface. Also let us now denote by pT r 
the energy of the motion of the p spheres relative to their common centre of gravity, 
so that T r is now the mean value per sphere of this energy. 
We have then, as is well known, 
P = */>T r + .(1), 
in which R is the repulsive force, r the distance between a pair of spheres, and the 
summation includes all pairs in unit of volume. We must first evaluate 5ARr on 
the assumption that no forces act except during collisions. 
2. Let q be the relative velocity of two spheres. Let 6 be the angle made with q 
by the line of centres at collision, if a collision takes place. The angle 6 may have 
any value from zero to \tt. As the effect of collision the velocity, \cq cos 6, in the 
line of centres is reversed for each sphere. We may assume this reversal to be effected 
MDCCCXCVI.—A. B 27.2.96. 
