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X. On the Collision of Elastic Bodies. 
By S. H. Biirbury, F.R.S. 
Received October 24,—Read November 19, 1891. 
Revised June 25, 1892. 
In a paper read before the Society on June 11, Sir William Thomson expressed a 
doubt as to the general truth of the Maxwell-Boltzmann doctrine concerning the 
distribution of energy among a great number of mutually acting bodies, and suggested 
that certain test cases should be investigated. The test that he proposed on that 
occasion was a number of hollow elastic spheres, each of mass M, and each containing 
a smaller elastic sphere of mass m, free to move within a larger one. This pair he 
calls a doublet. This case is within the general proof of the doctrine given below. 
It is, however, I think amenable to a simpler treatment, which has been applied to 
the case of elastic spheres external to one another. 
1. Every doublet has a centre of inertia of the sphere M and its imprisoned m. 
Let V he the velocity of that centre of inertia, R the relative velocity of M and m. 
If V and R be given in magnitude, R given in direction, V may have any direction, 
and in Maxwell’s distribution, for given direction of R, all directions of Y are 
equally probable. Conversely, if, whatever be the values of Y and R, for given 
direction of R all directions of Y are equally probable, Maxwell’s law prevails. 
Now consider a very great number of doublets, all having their relative velocity and 
the velocity of centre of inertia within limits R, R + dR, and Y, Y -f dN. Consider 
them before and after collisions between M and m. Nothing is changed by collision 
except the direction of R, and that change of direction is independent of the direction 
of V. Therefore after collision for given direction of R all directions of Y are equally 
probable, and therefore Maxwell’s distribution prevails after as well as before col¬ 
lision, and is therefore not affected by collisions. 
To proceed to more general cases. 
2. The characteristic property of collisions of conventional elastic bodies is that 
with continuous variation of the coordinates, and without variation of the kinetic 
energy, the velocities at a certain instant change discontinuously. The general 
treatment adapted to systems of this kind is as follows :—Let there be a system 
defined by n coordinates p 1 . . . p n , the corresponding velocities being p x . . . p n , and 
the generalised components of momentum q l . . . q n . At a certain instant a collision, 
i.e., discontinuity of p Y . . . p' n occurs. After the collision, let the velocities and 
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