of Edinburgh, Session 1885-86, 
389 
energy in the two systems would be instantaneous if the masses of 
the Ps and Qs were equal ! 
Sir W. Thomson has suggested that part of Clerk-Maxwell’s MSS. 
must have been, by mistake, omitted in sending to press. But I do 
not think that this idea is confirmed by a careful examination of 
the text. 
Thus it appears that the objections to Maxwell’s proof depend, 
in the main, upon his having commenced too soon to simplify by 
means of averages. It does not appear that his method, when the 
objectionable assumptions are put aside, can be applied in any simple 
manner (see Proc. R.S.E., Dec. 15, 1884), But the investigation 
may be conducted very simply, as follows, by a method which shows 
clearly, at every step, what assumptions are made and how they are 
to be justified. 
3. When two impinging spheres are of equal mass, their velocities 
in the direction of the line joining their centres at impact are simply 
interchanged. Hence the impact of a P on a P alters (in general) 
the distribution of kinetic energy in the system, but does not alter 
its average value per particle. These results are, of course, obvious ; 
but they show how it comes about that one particle among a very 
great number of equal ones (which originally had equal speeds, let 
us say) may attain any speed, however great ; while others may be 
brought (for a brief period) to rest. This has always, in my expe- 
rience, formed a serious difficulty to beginners. But, as will be 
seen, it is a necessary characteristic of statistical uniformity. 
To take a simple case, this will occur whenever a special particle 
always impinges on others, so that its own direction of motion is 
perpendicular to, and that of the other along, the line of centres at 
each impact. For it thus gets at each impact the whole energy of 
the two, and it might go on doing so till it had reduced all the 
others to rest. Ho doubt the acquirement of large speeds is common 
enough, but only for a few at a time even among a very large group 
of particles. It will presently be shown that there is a special 
distribution of relative position and velocity among the particles ; 
towards which there is, on the whole, an approximation, though not 
necessarily a continuous one. In this special distribution, all speeds 
occur, but the number of particles which have either high or low 
speeds (as compared with the mean square speed) is a very small 
