55 



On the other hand, (he chance Wm for (he realisalion of that 

 case, in whicli tlie different types are found represented amongst 

 the collective tables in proportion to their probability, will contain 

 a large permutation-factor, and consequently — with a suffi- 

 ciently large number of tables the ratio Wu/W^n may reach any 

 degree of smallness. It makes a great difference, therefore, — and 

 of course not only to the calculation of the maximum — whether 

 we take the tables collectively as an object of higher order in the 

 calculation of combinations or whether we determine the probability 

 for each table separately and calculate that of the whole as product 

 of the separate probabilities. 



III. Suppose that the number of tables and holes for each table are 

 not yet given, but only the total number of hollows in all the tables 

 together, and that it was left to our choice to divide them amongst 

 the tables, then an opinion as to what was the most probable 

 division would be even more arbitrary. 



IV. It is obvious, that the above considerations may be applied 

 to the gas, taking into consideration, where necessary, additional 

 conditions. 



If we inti'oduce the restriction that in the parts only we attend 

 to all the possible permutations, in defining the most probable 

 division, and that in the system as a whole we do not take into 

 consideration any further permutations between these parts, only then 

 does the probability for the state of the whole appear as the pi'oduct 

 of the probability of the states of the parts. 



If on the other hand the total system is regarded as a new object 

 for combinations, an object of a higher order, the probability of the 

 distribution of a special state in the whole is not equal to the 

 product of the probabilities of the parts corresponding to this state. 

 The latter must be corrected by a certain permutation-factor, the 

 magnitude of which is dependent upon the number of the parts, 

 that is either upon the fineness of the division to be chosen at will, 

 or — with a permanently fixed fineness of division — upon the 

 magnitude of the total system. 



The question arises: with which hy IT should the entropy be 

 identified? 



Only when the said permutation-factor is neglected can it be said 

 that the tending of the parts towards the maximum of their entropy 

 brings with it a striving towards a , maximum of the entropy of 

 the whole. 



