608 Mr. J. H. Jeans on the 



second, and so on. The number of ways in which this can 

 be done is 



..... (id 



|fl! |«_2 |«g • • • |On 



The systems o£ class A are therefore comprised in this 

 number of elements, each element being of the same size as 

 element c. Hence we see that the systems of class A form 

 a fraction 6 a of the whole number of systems, where 



*.=-Ji— ( 12 ) 



8=1 



This again is true whether we consider all possible systems 

 or only those systems having a given energy. 



§ 20. We can evaluate a when N is large compared with n, 

 and there are (in the limit) a very great number of molecules 

 in each of the n cells. We have, when a s is very great, the 

 approximate formula used by Boltzmann* 



a„= \Z27Ttf, 



(?)' 



or taking the logarithm of each side 



log [^=4 (log 2?r+ 1) + (a, + i)(log a,-l) ; 



and when a s is very great this assumes the limiting form 



log \as=a s log a n 



an equation in which the difference between the two sides is 

 infinite, but is vanishingiy small in comparison with either 

 side. Hence from equation (12), 



log a = log jN — N log n—% log \a 8 



=N (log N— log n) — %a log a s . 



Now N=^« s , and we may write ~N/n = a , where a is the 

 mean number of molecules in each cell. Making these substi- 

 tutions, our equation becomes 



log a = ta s log fl -2«,log a= —ta 8 log (^J. 



The right-hand member of this equation is, in general, an 



* Vorlesungen, i. p. 41. 



