396 
Proceedings of the Poyal Society 
In the present example, ISP vanishes when e — l, in which case we 
have the equations of § 7, For them we had X 2 = l, which is its 
least (numerical) value., Hence when e<\ we have ultimately 
x>yi 
that is, the average energy per Q tends to smaller value than that 
per P. 
11. To work out the consequences of such equations as those of 
§ 9, in the most general form, would lead to details too complex 
for the ordinary reader. We will therefore take another special 
case, in order to point out how the results are modified by other 
simple assumptions of the kind spoken of there. 
Let there be three systems of particles such that Ps and Qs are 
as before, but each Q has an P, which cannot impinge on anything 
but its special Q. Thus we may suppose each Q in the former 
arrangement (§7) to be made hollow, and to have an P (free) put 
inside it. 
We are not prepared, so far as we have gone^ to treat this 
question if the P’s, like the Ps and Qs, have unit coefficient of 
restitution. For, § 8 above, the P’s do not impinge on one another. 
But if we suppose the coefficient of restitution of Q and R to be 
less than unity, and the interior of a Q so nearly equal to the P 
inside, that between every two collisions of the Q with an external 
particle there is time for the R inside it to be reduced to relative 
rest (or what may be treated as such) we may approximate to the 
ultimate state of things. 
The equations in § 6 still hold good for each impact of a P on a Q. 
But, immediately after the impact, the Q impinges on its R ; with the 
result that, before the Q suffers another collision, o' is reduced to v'\ 
where (Q + R)F' = QF + R?;, so that instead of the. equations in 
§ 7, we have now 
4PQ„ r Q 
(P + Q) 2 ) Q + R 2 ' 
4PQv f y Qa PQy ) 
py P + Q | Q + R + (P + Q)(Q + R) + (P + Q)(Q + R) 2 J 
4QR + 2R 2 , 
(Q'+ R) 2 yv 
