of Edinburgh, Session 1885-86, 
391 
to alter it For then the uniformity of distribution of the spheres 
in space, and the symmetry of distribution of velocity among them, 
enable us (by the principle of averages) to dispense with the only 
limitation above imposed, viz., the parallelism of the lines of centres 
in the collisions considered. 
5. When a P impinges on a Q, let u and v be their velocity-com- 
ponents (measured towards the same parts) in the line of centres at 
impact. Let these be changed by the impact to v! and v' respect- 
ively. Then the ordinary text-book result is 
P(«' - u) = - -*’)=- Q(l>' - V) . 
From this we deduce immediately 
P(»' 2 - « 2 ) = - pr^( p “ 2 - V = ( p - Q)*0 = - Q(«' 2 - «** 
This shows the amount of energy transferred between the P and 
the Q at one impact, 
6. To obtain an average from this we begin by assuming that 
the Ps and Qs are thoroughly mixed, and are separately in the 
“ special ” condition of § 4. Of course, this implies that there is a 
very large number of particles of each kind. We also assume, what 
will probably on consideration be granted, that the mutual actions 
of the Ps alone, and of the Qs alone, still tend to preserve in each 
system this “ special ” state ; ot to restore it if it should be disturbed 
by the action of some Ps upon Qs. This is based partly on the 
uniform mixing of the Ps and Qs, partly upon the small percentage 
of each system which is involved in any a simultaneous ” collisions. 
If this be granted, it is clear that we may assume that for a great 
number of simultaneous (§ 2, above) impacts of Ps on Qs, the average 
value of uv is at least approximately zero.* The reader must bear in 
mind that u and v are velocity-components parallel to the line of 
centres, which may have any direction. But we also see that we may 
now look on the average value of P u 2 - Qz; 2 as being two-thirds of the 
* There is no inconsistency between the two expressions above, viz. , ‘ ‘ great 
number of simultaneous impacts,” and “ small percentage of each system 
which is involved in any simultaneous collisions.” For we must remember 
that the whole number of particles is very great; and even a ‘‘small per- 
centage ” of a very great number may itself be “ a great number.” 
