MR. S. H. BURBURY ON THE COLLISION OF ELASTIC BODIES. 411 
And, therefore, since E and R are arbitrary, pf, &c., are not altered by collision 
at all, that is, the Maxwell-Boltzmann distribution, given existing, is not altered 
by collisions. _ 
The above proof also shows that p 3 — p\ 2 is zero without the factor R, that is, the 
average value for all systems is zero, as well as for all collisions ; and in proving that 
p 2 = p' 2 f it does not matter whether we introduce the factor R or not. 
9. We will give certain examples of the functions S x . . . S„_! R. 
I. Elastic spheres of masses m and M respectively. Here a colliding pair, which 
corresponds to a system in the general treatment, has six degrees of freedom, there 
should, therefore, be five linear functions of the velocity unaltered by collision. They 
are x, y, X, Y, the tangential components of velocity at the instant of collision, and 
mu + MU = (M + m) V = mu’ + MU', 
where u, U are the normal components. 
We have from the last equation 
m (w-O + M(U-U') = 0, 
and by the equation of energy 
m (u* - u z ) + M (U 2 - U' 2 ) = 0, 
whence 
u + u = U +U r , 
or 
u - U = - (u - U') = R. 
II. The system consists of a sphere of mass m colliding with a spheroid of mass M. 
It is assumed that the spheroid will acquire no rotation about its axis of figure, but 
may have rotation about any other principal axis. It has then five degrees of 
freedom, and the system of sphere and spheroid has eight. 
We require, then, seven linear functions of the velocities to be invariable. 
Let 0 be the centre of the spheroid, OZ in the plane of the paper, its axis of 
figure, P the point of collision, PN normal at P, ON perpendicular to PN, and 
ON = c, A the moment of inertia of the spheroid round an axis through O perpen¬ 
dicular to OZ. Then our seven constants are 
3 g 2 
