THE NEW STATISTICAL MECHANICS 371 



"theory of probability." I cannot do better than repeat, with slight changes 

 in wording, the process of solution given by the Mayers in their book. 

 "A distribution is characterized by the number of balls in each of the 

 numbered baskets, since the balls are indistinguishable. Consider the 

 arrangement in a line of the symbols Zi, Z2, • • • Zc, fli, 02, * • • Oiv, as for 

 instance, 



Zi Oi Gi Zs as flg Z5 S4 29 fle • • • 



Such an arrangement could be used to define an assignment of .V numbered 

 balls, the o's, to C numbered baskets, the 2's, by adopting the convention 

 that the balls to the right of each numbered z belong to the basket of that 

 number. For instance, the above corresponds to balls 2 and 4 in basket 1, 

 balls 8 and 9 in basket 8, no balls in baskets 5 or 4, and ball 6 in basket 9. 

 One must observe the convention that the row starts with a z, and we shall 

 consider only arrangements of the symbols which start with zi. However, 

 (C — 1) ! such arrangements of the symbols correspond to one arrangement of 

 numbered balls in the same baskets, since permutations of the (C — 1) groups 

 of each Zi with its following a's correspond to the same arrangement of 

 numbered balls in the baskets. In addition all TV^! permutations of the a's 

 correspond to the same distribution of un-numbered and indistinguishable 

 balls in the baskets. In all, each distribution of the indistinguishable balls 

 among the numbered baskets corresponds to TV! (C — 1) ! arrangements of the 

 symbols, and the {N -\- C — 1) symbols (after the first) may be arranged in 

 (.V + C - 1)! different ways." 



Thus we come to the formula for IF tot, the total number of ways in which 

 all possible distributions of the cells of a region among the populations can be 

 realized; it is, 



_ (iV,+C- 1)! 

 ^^^°* " NMC- 1)! ^ ^ 



Dropping the "ones" for the amply sufificient reason that they are insigni- 

 ficant by comparison with TVy and C, and taking the logarithm with use of 

 the super-Stirling approximation, we find: 



In TFtot = (A^• + C) In (.Vy + C) - .V,- In Nj - C\nC (23) 



which with a little regrouping of terms is found to be the very same ex- 

 pression appearing in (15) for In Tl^nax. 



2 J. E. and M. G. Mayer, "Statistical Mechanics" (John Wiley & Sons, 1940); p. 438. 

 Reprinted by permission. 



2 For the historian of science it is interesting to note that the formuhi (22) was used 

 by Planck in his earliest derivation of the black-body radiation law. His un-numbered 

 balls were quanta of energy, his baskets were linear oscillators, and his k In W was the 

 entropy of the system of C oscillators sharing N j quanta among themselves. Cf. Natur- 

 wissenschaften, April 2, 1943. 



