of Edinburgh, Session 1885-86. 397 
where the last term in the value of y is due to 2v sets of impacts 
between Qs and their Rs, after the Qs have collided in pairs. 
To get a notion of the nature of this result, suppose as before 
w = p, v = v. Also let Q = 3R, P = 4R, then we find by the above 
methods that ultimately x is nearly the double of y. 
If we had unit coefficient of restitution between Q and R, and 
were to assume Boltzmann’s result here, we should have x to y as 
1:2; because the Qs consist of two separate parts, each of which 
(having the same number of degrees of freedom) would ultimately 
have the same energy as a P. 
12. Without more formidable mathematical processes we cannot 
well push these investigations further : — but enough has been done to 
show on what bases Clerk-Maxwell’s Theorem really rests ; and, at 
the same time, to show that even were Boltzmann’s extension of it 
rigorously proved, it need not prevent us from accepting the kinetic 
theory, which has furnished such simple and complete explanations 
of many puzzling phenomena. It is not at all likely that the 
particles of any gas (be it even mercury vapour) behave as if their 
coefficient of restitution were exactly unity. They would probably 
require, in order to do so, a steady supply of heat ; perhaps other 
things of which we have as yet no knowledge. Among these un- 
known conditions may be mentioned, so far as the specific heat 
question is concerned, the nature of the impacts between the particles 
and the walls of the containing vessel. The law of these impacts, 
and the mode in which the energy thus received is distributed 
among the degrees of freedom of a particle, may differ widely 
from those which regulate the impact of particle on particle. 
( b ) On the Length of the Mean Path among Equal Spheres. 
The following investigation has been made as elementary as 
possible. It will be seen that it leads to a result somewhat different 
from that usually accepted. The source of the discrepancy is 
pointed out. 
Let 8 be the diameter of a sphere. It protects a circular area tts 2 
in any plane through its centre ; in the sense that another sphere, 
of the same diameter, moving perpendicularly to that plane, will 
necessarily collide with the first if the line of motion of its centre 
pass within the circle 
