1891.] On the Collision of Elastic Bodies. 175 



III. On the Collision of Elastic Bodies." By S. H. BURBURY, 

 F.R.S. Received October 24, 1891. 



(Abstract.) 



1. In this paper I discuss, firstly, a case suggested by Sir W. 

 Thomson on June 11 as a test of the truth of the Maxwell-Boltzmann 

 doctrine concerning the distribution of energy. Sir W. Thomson, 

 supposes a number of hollow elastic spheres, each containing a smaller 

 sphere free to move within it. This pair he calls a doublet. 



If V be the velocity of the centre of inertia of a doublet, R the 

 relative velocity of the two spheres, then, under the distribution in 

 question, for given direction of R, all directions of Y are equally prob- 

 able, and the converse is also true. If a collision occurs, the change 

 of direction of R due to it is independent of the direction of Y, as well 

 as of the magnitudes of Y and R. Therefore, after collisions, as well 

 as before, for given direction of R all directions of Y are equally 

 probable. Whence it follows that the distribution of velocities is 

 unaffected by collisions. This appears to me to be sound as well for 

 internal as for external collisions. 



2. The characteristic of collisions of conventional elastic bodies is 

 the discontinuous change in the velocities without alteration of the 

 kinetic energy. If that occurs for any material system of n degrees of 

 freedom, there are n 1 independent linear functions of the velocities 

 v l . . . .v n which remain unaltered, call them Si .... S_!, and one, R, 

 which is unaltered in magnitude but reversed in sign. 



3. The kinetic energy cannot contain any of the products RS, but 



must be of the form 2E = \R a +/(Si S M _,), where /(^ S_0 



is a quadratic function of these quantities. 



4. If after collision the velocities v\ . . . . v' n were all reversed in 

 sign, R and Si .... S M _! would be reversed in sign. The system 

 would retrace its course, undergoing collision, changing v into v. 

 R would be positive before and negative after collision. S! . . . . S M _! 

 would be throughout negative, i.e., of opposite signs to the signs they 

 had in the first case. 



5. To define a collision, we assume that a certain function ty of the 

 coordinates and constants cannot become positive, and when ty = 0, 

 d^-ldt being positive, dty/dt changes sign discontinuously, and a colli- 

 sion occurs. It follows that dty/dt is equal or proportional to R. 



6. What has been proved for a system holds equally for a pair of 

 systems, having coordinates p l . . . .p r for the one, and p r +i . . . . p n 

 for the other, if ty be a function of pi . . . .p n , which cannot become 

 positive. 



7. All those systems for which, at a given instant, "^ lies between 



