502 PROFESSOR BURNSIDE ON THE PARTITION OF ENERGY IN THE 



about the A-, B-, and C-axes at the centre of inertia lie between a^ and 

 o) 1 + dco l , o) 2 and w 2 + </&> 2 , and w 3 and w 3 + <iw 3 respectively is 



•J- 



fC-lfCnrCt) 



7T 



[This assumption is made by analogy from the form for the speeds, and can 

 only be justified by results.] 



(iii.) That for any sphere all directions of the velocity of the centre of 

 inertia with regard to the principal axes are equally likely. 



(iv.) That the distance c between the centre of figure and centre of inertia 

 of a sphere is very small compared with the radius. 



The last assumption is made for the following reason :— The " opacity," or 

 power of intercepting impinging particles, of a layer of such spheres as are being 

 considered will depend both on the linear and angular velocities of the spheres, 

 and the probability of a collision between different spheres will no longer be 

 proportional to their relative speed, but to some function of their linear and 

 angular velocities, which even if it could be expressed analytically would almost 

 certainly be of a most intractable form. If, however, the distance c is assumed 

 to be very small in comparison with the radius, the probabilities of a collision 

 between different spheres, and the mean free path, will be sensibly independent 

 of the angular velocities, and hence the same as for a system of homogeneous 

 spheres, while there will still be an interchange between the energies of trans- 

 lation and rotation at each collision. 



Let u, u' be the velocities of the centre of inertia of a sphere in the line of 

 centres before and after an impact : — 



cd 1 , o> 2 , &) 3 , oi\, &/ 2 , o/ 3 the angular velocities about the principal axes at the 

 centre of inertia before and after an impact : — 



/, m, n the direction-cosines of the line of centres with respect to the principal 

 axes at an impact : — 



and let large and small letters be used to distinguish the values of these 

 quantities for the two impinging spheres. 



Write also for brevity, 



N/3-My = P, n{3-my=p, 

 Ly — Net = Q , ly — na = q , 



Ma-L/3 = R, ma-l/3=r. 



The spheres are said to be elastic in the sense that the energy of a pair of 

 colliding spheres is unaltered by the impact ; and Maxwell shows, in the 

 paper already referred to, that in the impact of two such spheres the relative 

 velocity of the points of contact in the direction of the line of centres is simply 

 reversed after impact. 



