388 
Proceedings of the Eoyal Society 
This proposition is true, under the conditions assumed, and can be 
proved as below,* by a process much easier for the beginner than 
that of Maxwell, 
But if this assumption were justifiable, a farther application of 
the same principle should bring out that P and Q (being treated as 
typical particles after, as well as before, colliding) also separate 
after impact with motions in directions at right angles to one 
another. For, in the cases here considered, an instantaneous 
reversal of each velocity would make the whole system retrace its 
motion, f Maxwell’s formulae, however, show that in general the 
directions of motion after the typical impact are not at right 
angles to one another. • It is clear that this objection is fatal to the 
method. 
But there is yet a farther objection. In interpreting the result 
obtained in virtue of the assumptions already described, Maxwell 
exaggerates the rate of equalisation of the average energy in each 
system by treating the question as if every P impinged on a Q 
in the “ simultaneous ” sense above spoken of ; and thus ignoring 
the almost incomparably more numerous particles of each system 
which, during the very short period contemplated, were either free 
from impacts or impinged on fellow particles. 
So that, finally, he arrives at the very startling conclusion that 
the difference of the average kinetic energies of a particle from each 
system is reduced at every group of simultaneous impacts in the 
ratio (P - Q) 2 /(P + Q) 2 . Thus the equalisation of average kinetic 
* The mean value, of the square of the distance of any point on a sphere 
from an internal or external point A, is the sum of the squares of the radius of 
the sphere and of the distance of A from the centre. Divide the spherical 
surface into pairs of elements by double cones, of very small angle, whose 
vertices are at the centre. For each pair of these the theorem is obviously 
true. Hence if the speeds of two points be p and q, their mean square relative 
speed is p 2 + <f. From this the above statement follows at once ; provided 
that all directions are equally likely for each amount of speed. 
f Here we meet with a quasi-metaphysical difficulty, which must be men- 
tioned in passing. For, it may be said, since there is perfect reversibility, the 
mere instantaneous reversal of a state which is approaching finality will give 
a state whose tendency is to depart from finality, i.e ., to get back to the exact 
reverse of its original condition. True, and most important, but not fatal to 
the conclusion ; unless an infinite time has elapsed since the start. For, when 
the reversal has brought the system back to the same configuration as at start- 
ing, but with velocities reversed, it is a new departure : — which will lead 
towards, but never to, its own state of finality. 
