38-40] The Method of General Dynamics 39 



Let us, for the present, suppose that the radius of the molecules is vanish- 

 ingly small, so that those parts of the generalised space excluded by the 

 conditions of 33 may be neglected. Then the representative points of 

 systems which are such that the centre of the molecule A lies within a single 

 specified cell, i.e., is restricted to a range &> out of the whole volume H of 

 the containing vessel will clearly occupy a fraction o>/t of the whole of the 

 generalised space. Since nw =H, this may be written n~\ If two molecules 

 A and B both lie within specified cells, the representative points occupy a 

 fraction n~- of the whole, and so on. Thus if each of the N molecules lies 

 within a specified cell, the representative points will occupy a fraction n~ N of 

 the whole of the generalised space. 



Now the number of different ways in which the N molecules can be 

 assigned to the n different cells, so that the system shall belong to class A, 

 defined as above, is 



N\ 

 ojaalo,! ... a n \ 



where, since the total number of molecules is N, we have of course 



a 1 + a. 2 + a 3 + ... +a n = N ........................ (59). 



Hence it follows that the representative points of systems of class A will 

 occupy a fraction 



V^ - -.*>- .............................. (60), 



chiOjIOti ... a n \ 



of the whole of the generalised space. 



40. We have already supposed N to be great ; we now suppose that 

 a 1} a. 2> a 3 ... a n separately are very great. This enables us to express (60) in 

 a simpler form, by using the well-known theorem of Stirling, 



........................... (61). 



p = ao 



If we take logarithms of both sides, this becomes 



p .................. (62). 



Denoting expression (60) by 6 a , we have 



s=n 



log O a = log N ! 2 log a s \ N log n, 

 *=i 



and in the limit, when a l} a 2 ... a n , N are all infinite, this may, in virtue of 

 equation (62), be replaced by 



log a = \ log 27r + (N+ 1) log N-N 



- i { log 2?r + (a, + J) log a, - a s ] - N log n. 



s=l 



