478 Mr. S. H. Burbury on certain 



We bave here an irreversible process. We might take 

 n' — n for characteristic function, but for symmetry let 



H = n log n — n + n' log n r — n\ 



and let "fin or ~dn' denote the change of n or n f due to one 

 drawing and return. Then "d?i={q—p)r and 



C)H = log n ~&n + log n f ~Qn r 



— ~&n log -j, because d (n -f n') — 0, 

 n'+(l — 2k)n , w 



n 



= 7^102 



n-\-n n 



And this is negative or zero. H is the characteristic function 

 of the process. The directing condition is that each drawing 

 is at random, and independent of the past history of the 

 system, that is independent of all former drawings, except as 

 they have altered n and n f . 



Equal Elastic Spheres. 



8. A great number of elastic spheres, each of unit mass and 

 diameter a, are at an initial instant set in motion within a 

 field of no force S bounded by elastic walls. The initial 

 motion is formed as follows : — (1) One person assigns com- 

 ponent velocities u, v, w to each sphere according to any 

 law subject to the condition that 1u = Xv=*Liy = and 

 £(w 2 + v 2 + w 2 ) = a given constant. (2) Another person, in 

 complete ignorance of the velocities so assigned, scatters the 

 spheres at haphazard throughout S. And they start from 

 the initial positions so assigned by (2) with the velocities 

 assigned to them respectively by (1). 



The condition of independence of Art. 6 is evidently satisfied 

 in this initial state. 



J). Jjetfdudvdw be the number of spheres having in the 

 initial distribution (1) component velocities between the limits 



u . , . u -J- du } 



v . . . v +dv, 

 w . . . w + dw. 



These we will call /spheres, or spheres of the class/. And 

 let'FdUdVdW be the number having in the initial distri- 



