392 
Proceedings of the Royal Society 
excess of energy of the impinging Ps over that of the corresponding 
Qs. Similarly the average value of P (u' 2 - id) may he looked on as 
two-thirds of the increase of energy of the impinging Ps, and so 
forth. But the assumptions above enable us to say that, because 
there are many “ simultaneous ” collisions, the average value of Tu 2 
for the impinging Ps is the average value for all the Ps. &c. Then 
our equation shows that, with the above assumptions, the impinging 
Ps lose energy on the whole if, and only if, their average energy 
is greater than that of the Qs they impinge om But such gains 
or losses of energy distribute themselves through the systems of Ps 
and Qs separately; so that, on the whole, there is transference of 
energy from the Ps to the Qs so long, and only so long, as the 
average energy of a P is greater than that of a Q. Thus there is an 
approach (persistent in the long run, hut not in general continuous) 
to equality between the average energy of a P and a Q; and, with 
these assumptions, Maxwell’s proposition is undoubtedly true, 
7. We may, in passing, make an approximation (of a very rough 
kind) to the rate at which this equalisation goes on, as follows i — ■ 
Let w be the whole number of Ps , 
R 5S QS j 
v the number of impacts between a P and a Q, in 
r the average interval which elapses, for each one P, 
between impacts on Qs. Let Vp 2 /2 he the average energy of a P, 
Q^ 2 /2 that of a Q. Then our equations give (omitting the numerical 
factor J) 
= - A Qs 2 ) • 
Writing x for Pp 2 , y for Q<? 2 , and N forPQv/(P + Q) 2 , this becomes 
= -TS(x-y) = -py, 
whence 
where 
Pp 2 -x — A- pBg " * 
Q q 2 = y = A + zf&%~ et 
e = W- 
7 + p 
zvp 
8. It is foreign to my present purpose to enter into the calculation 
of the values of v and r, which depend on the diameters of a P and 
a Q, and the average distance between the centres of any two 
