of Edinburgh, Session 1885-86. 
395 
We have omitted terms in uc from each of the right-hand sides j 
taking for granted that, in this case also, we may treat their mean 
value as nil if the number of colliding pairs is sufficiently great, 
and if the equalising process goes on in each system. These equa- 
tions, with the proper alterations, apply to the internal impacts in the 
systems of Ps and Qs separately. The values of e may be different 
for a P and P and a Q and Q ; but from them the value for a P and 
Q can be calculated. [Hodgkinson, B . A. Report, 1834.] 
10. In particular there is a specially interesting case when e—l 
for a P and P, and for a P and Q ; but e < 1 for a Q and Q. Here 
it is easy to see that the equations of § 7 are modified to 
rarX = — 17 — y) 
py— TS{x-y)-Wy 
wfiere N ? depends upon e and upon the frequency of impacts 
among the Qs. 
[This, and all the equations which correspond to questions of the 
kind above proposed, are of the type 
x— -ax + cg/ \ 
y = cps — by ) 5 
ab - c x c 2 > 0 ; 
and the solutions are always of the form 
x + \y = 
x - \ 2 y = 
where the values of A are given by 
c 2 \ 2 -(a - b)\ - e x = 0 . 
We have also c 2 \ a — fx , so that the equation for y, is 
fx 2 + (a + b)/x + ab - c x c 2 - 0 . 
The root /x 2 , which corresponds to the negative value of is 
(numerically) greater than /x 1 ; so that the value of x - A . 2 y dies away 
faster than that of x + A,-^. W e may suppose the whole energy to be 
constantly recruited through the Ps, so that ultimately x - A 2 y = 0. 
This gives the final ratio of the energies of particles, one from each 
system.] 
