22 
Proceedings of the Royal Society 
but PGQ becomes P'GQ' where GP' = GP, GQ' = GQ, and (by a 
previous proposition) all directions of P Q (in space) are equally 
likely. Let the speed OP' be called / ; OQ', q; and let 
< OGP' = 0 . 
Then QC« r , + ^ 2g2 + 2P ^g cos a ; 
Hence, taking mean values, with regard to a, to 0, and to the plane 
through OG in which 0 is measured (which, of course, presupposes 
an immense number of impacts), we have 
which is Clerk-Maxwell’s result. The above investigation, however, 
shows the somewhat complex process of averages by which it is 
obtained. 
The Corollary, in which theproposition is extended to a number 
of systems of different sets of particles, follows in the same way ; 
P 
0 
(P + Q ) 2 
also P/2 _ q ^'2 = p(OG 2 + GP 2 - 20G . GP cos 6) 
- Q(OG 2 + GQ 2 + 20G . GQ cos 6) 
= /p , m 2 (Py + QV + 2PQ pq. cos a - PQ(^ 2 + q 2 - cos a)) 
(P + QP 7 
