Theory of the Dissipation of Energy. 297 



Hence the probability of the number of molecules in A being 

 exactly ?', and in B exactly n—i, irrespectively of individuals, 

 is a fraction having for denominator (a + b) n , and for numera- 

 tor the term involving a i b n ~ i in the expansion of this binomial ; 

 that is to say, it is — 



n{n— l). ..{n— i + 1) ( a Y7 b \ n ~* 



1.2 ... 



If we call this T { , we have 





n—i a 

 I+lb 



Hence T { is the greatest term, if i is the smallest integer 

 which makes 



n — i b 



i + 1 a' 



this is say, if i is the smallest integer which exceeds 



a b 



a + b a + b' 



Hence if a and b are commensurable, the greatest term is that 

 for which 



a 

 a + b 



To apply these results to the cases considered in the pre- 

 ceding article, put in the first place 



n=2 xlO 12 , 



this being the number of particles of oxygen ; and let i=n. 

 Thus, for the probability that all the particles of oxygen shall 

 be in A, we find 



a V* 10 ' 2 



/ a \ 2X 

 \a + b) 



Similarly, for the probability that all the particles of nitrogen 

 are in the space B, we find 



ixio 12 



\a + b) 



Hence the probability that all the oxygen is in A and all the 

 nitrogen in B is 



/ a \ 2xl ° 12 / b \ 8X1 ° 12 

 \a + b) X \a + b) 



