52 Dr. L. Natanson on the 



mutual interchange between these, being now limited to the 

 interior of the first element, will not lead to new and distinct 

 molecular configurations. Since to the N 2 molecules in the 

 second element, to the N 3 molecules in the third, &c, the same 

 remark applies, the arrangement (1) cannot be realizable 

 but in 



N! 

 Ni!N 2 !...NJ. (3) 



distinct manners ; and if we agree to put ! = 1 the expres- 

 sion just given will not be altered in the case in which some 

 of the "N. are equal to zero. To find the value of the pro- 

 bability Q we have to divide (3) by the sum of all terms of 

 the form (3) which are consistent with the condition 



N 1 +N s +...+N,=N (4) 



Accordingly, we obtain 



W- « N N.!N 2 !...]N T ! w 



From this it follows at once that the smallest value of Q, say Q', 

 is assumed in the case in which of the numbers ~N 1} N 2 , . . ., N n 

 all are equal to zero, except one, which is therefore equal to 

 N ; and that the greatest value of Q (or Q" say) corresponds 

 to the case 



N 1= N 2 =...=N n =? (6) 



The corresponding probabilities are 



QT =4 and Q"= r/ SU» ; • • • ( 7 ) 



a- 



hence 



Q' 



[(?)']" 



or, approximately, N and ~N/n being very large, 

 1 



(*) 



,H+ 



s a/(2«-N)"->, 



a very small fraction for any large value of N. In words : that 

 all the molecules should be accumulated in one element is the 

 most improbable, and that they should be evenly distributed 



