of Edinburgh, Session 1885 - 86 . 
387 
1. Prop. VI. of Clerk-Maxwell’s well-known earliest investigation 
of this subject (Phil. Mag. 1860, I. 25) deals with two systems of 
colliding spheres (those in each system being equal to one another). 
Their coefficient of restitution is unity, and they rebound without 
loss of energy from the walls of the containing vessel. His state- 
ment is that, after many collisions , the average kinetic energy of a 
sphere is (ultimately) the same in each of the colliding systems. 
Particular stress must be laid on the words I have put in italics, 
because they form the basis of the whole theory. Without colli- 
sions there could be no law of any kind ; the arrangement would 
be, and would continue to be, an absolutely haphazard one about 
whose character we could not possibly reason. And only after many 
collisions, among great numbers of spheres, can there be any approach 
to a statistical finality of arrangement. 
The theorem is undoubtedly true, provided the number of spheres 
in each system be extremely large, while those of one system are 
not extremely numerous in comparison with those of the other • 
but the proof given by Maxwell has more than one very objection- 
able feature. 
2. The chief of these is the assumption (for it is nowhere justified) 
that the transference of energy from system to system can be 
calculated without taking account of the mode of its distribution 
among the particles of either system. This assumption enables 
Maxwell to reduce the question to the treatment of the consequences 
of a single impact between a P dnd a Q (these letters represent the 
mass of a sphere of either system) ; each having the average energy 
of the system to which it belongs, and being thus regarded as typical 
of its system. It is typical of all the impacts between a P and a Q 
(here called simultaneous ) which take place in the average time 
which elapses between any two successive impacts of a particular 
P on some Q or other. 
The elegance of the investigation is farther enhanced by the addi- 
tional assumption that, in obtaining the results of this typical impact, 
the original directions of motion of the P and Q may be taken as at 
right angles to one another. 
The basis for this assumption is, apparently, a previous proposi- 
tion, which shows that the mean square relative velocity of a P and 
a Q is the sum of the mean square speeds of the Ps and the Qs. 
