of Edinburgh, Session 1885-86. 
537 
3. On the Partition of Energy among Groups of Colliding 
Spheres. By Prof. Tait. 
(Abstract.) 
The second, only, of two short papers which I communicated to 
the Society a month ago, on points connected with the Kinetic 
Theory of gases, took account of the greater probability of collision 
between two particles as their relative speed is larger. I propose 
now to calculate the effect of this consideration on the results of my 
first paper. 
The question resolves itself into finding the mean value of the 
expression 
Pu 2 - Qv 2 - (P - Q)uv (1) 
of that paper, in terms of the mean square speeds in the two systems ; 
because, only when this is zero, does kinetic equilibrium set in. 
We are here concerned only with collisions between particles 
belonging one to each system. 
In my Note on the Mean Free Path (ante, p. 397) it is shown that 
the fraction of a group of particles, having speed v perpendicular to 
a layer of thickness Sr, which collide, at angles from /3 to /3 + S/3, 
with particles in that layer whose speed is v v is proportional to 
Sr 
— Jr 2 + ^ - 2vv x cos /3. sin /3d/3 . 
Hence, in a given time, the number of such collisions is propor- 
tional to 
Jv 2 + v\ - i lvv 1 cos j3. sin /3d/3 = v sin /3d/3, suppose. 
Let Y, Y 1 be the projections of v, v x on the unit sphere, C that of 
the line of centres at collision. Then YY 1 = /?. Let also YC = a, 
CYY 1 = ^>. Finally, let n, ?i v represent the proportionate numbers 
of particles in the separate systems which have speeds v to v + Sv, 
v Y to v 1 + Sv 15 respectively. Thus 
n = 
Jit . d 
2 v 2 dv , 
where, for simplicity, p is put for 1/a 2 and q for 1//3 2 in the 
