OF THE KINETIC THEORY TO DENSE GASES. 
7 
and time is proportional to p, and, therefore, increases in the ratio I : 1 «> when 
the spheres, from being material points, acquire diameter c. We now see that this 
is true for each separate class of the vertical momenta. 
9. Corresponding to N spheres with vertical velocity u at the plane s = 0 at the 
beginning of dt, we have at the end of dt N spheres, the same or substituted, whose 
average height is (1 + k) ds or (1 + k) u dt. But their loss of kinetic energy due to 
the ascent is on average M fds for each sphere. It follows that the average loss due 
to an ascent ds is, allowing for substitutions 
M/efc 
’ 1 + K ' 
10. Let then the number per unit of volume of spheres whose energy of vertical 
velocity is ^ u 2 . . . \ (id + du 2 ) be at the base, where s = 0, p 0 e -,lQ did in which 
Q = M (I + k) id (A). Then the number per unit of volume which at height s 
have 
M 
f* 
1 + Kj 
for energy of their vertical velocity is (remembering (5)) 
Po e~ m '° dtde-w-"*, 
that is p 0 e /<Q dvd (B). The two classes A and B are equally numerous per unit of 
volume, and since, allowing for collisions and substitutions, the loss or gain of energy 
due to the force in passing from the base to ds or vice versa is Mjds /1 + k, either 
class can by ascending or descending (the proper number of substitutions taking- 
place) replace the other. And the assumed law of distribution of velocities is not 
disturbed by the force f as spheres pass up and down the column. Now, make 
k = 0 in the above reasoning, and we find it is exactly the reasoning from which, in 
the ordinary case of a rare medium, we conclude that T is constant throughout the 
column. The same reasoning leads in the general case to the conclusion that 
(1 + k) T is constant throughout. 
11. The above results are obtained on the hypothesis that no account need be 
taken of stream motion among our spheres. If, however, there be such stream 
motion, we have to suppose that the N spheres crossing the plane s = 0 were 
members of a large group having a vertical velocity U of their common centre of 
gravity, and that the N spheres have vertical velocity u relative to this common 
centre of gravity. Then in time dt the N spheres will have risen on average the 
distance ds by virtue of their relative velocity u, where it = ds/dt, and a distance 
U dt by virtue of the common velocity U. Their loss of kinetic energy by the 
action of the force jf in this ascent is 
M/U dt + M/’eZs/l -b k, 
or if U dt = ds 
M f (ds + ds /1 -j- k). 
