OX THE KINETIC THEORY OF GASES. 223 



Otherwise we may express it thus : If there be N 

 spheres within S, the chance that their common centre of 

 gravity shall have velocity U . . . U + d\J in.r is in the rare 

 medium proportional to f _N/iU VU, because in the rare 

 medium the velocities are all independent. If now the sur- 

 face S were an elastic boundary, the N spheres would by 

 their collisions with it and with one another in a very 

 short time acquire what Tait calls "mass motion" with 

 velocity U in x, their relative velocities being those of a 

 system of elastic spheres in the special state, see (5) above. 

 But remove the elastic boundary and they will in the rare 

 medium escape out of S, and mix with the surrounding 

 spheres, in much less time than it takes them to subside 

 into mass motion. 



But in the dense medium the process of diffusion or 

 escape out of S is slower, and the process of acquiring mass 

 motion more rapid. So, as you increase the density, you 

 increase the chance that for neighbouring spheres the 

 velocities have a common ancestor or many such, and there- 

 fore the chance that they are coincident in direction. The 

 spheres will develop a tendency to move together in masses, 

 to form "streams" in fact. 



38. In such a medium the chance that the N spheres 

 within S shall have respectively^- velocities ?t 1 . . . u 2 . . . 

 u N is not, as it would be on the ordinary hypothesis, repre- 

 sented by E -A(« 1 2 +« 2 2 + • • • +«V but will be of the form 



£ — «j "j 2 etc. /a^+b ufl + a^\ + etc.,\ 



in which the co-efficient b r% will vanish if the spheres to 

 which ?/,. and u s relate are not very near each other. And it 

 will be found that the energy of the motion of the common 

 centre of gravity of the group, i.e., the energy of stream 

 motion, bears a larger proportion to the whole energy of the 

 group than it would do if the chances were all independent. 



39. To work out this problem in a mathematical form 

 would be very difficult. I can do no more within the 

 limits of this article than indicate what, as appears to me, 

 the general character of the motion will be. I hope that the 

 long delay in the appearance of the concluding parts of 

 Professor Tait's work is due in part at least to the fact 



