supposed Irreversible Processes. 477 



theoretical purpose it is sufficient to deal with the test-case 

 of elastic spheres, and for that case the condition may be 

 expressed as follows : — The chance that two spheres approach- 

 ing collision shall have velocities within assigned limits is 

 independent of their relative position, and of the positions 

 and velocities of all other spheres, and also independent of 

 the past history of the system, except as this has altered the 

 distribution of the velocities inter se. 



This condition of independence is, as I propose to show^ 

 the directing condition required to make the diminution of H 

 in the test-case an irreversible process. I propose also to 

 consider as far as possible whether this directing condition 

 can be assumed to exist in fact. 



7. Let us first take au algebraic example. There are in a 

 bag n white and n! black balls. I draw at random r balls, r 

 being small compared with n or n, and they happen to be p 

 white and q black, with p + q = r. I then return to the bag, 

 not jo white and q black which I drew, but q white andjo black, 

 so that as the result of the drawing and return, n has become 

 n + q— p, and n' has become n' -\-p — q; n+n' is unchanged, 

 but n—n ! has become rc — ?i / -\-2(q— p). Now, on average, 

 the most probable values of p and q are 



n n f 



so the effect of the drawing and return is on averaoe to dimi- 

 nish the difference n^ n' . And if the process be repeated 

 indefinitely, n^n' will, on the whole, diminish. But at any 

 drawing the actual values of p and q will generally differ 

 from their mean values. We may express this by saying 



that the actual value of p is k ,r, where k is a positive 



quantity, not greater than— , whose mean value is unity 



and which is more likely to have a value nearer to unity than 

 one more remote. Then 



n' + (l — k)n , , N n'+(l — 2k)n 



q — r~p= i-^-r. and (q—p)= —LSI — *"> % 



To give a concrete example, let r=100. Let P be the 

 chance that jo — (7 = 2, and V the chance that q—p = 2. Then 

 approximately 



p/p'=g), 



which is very great if njri is great, but approaches unity 

 as njn f approaches unity. 



