390 BELL SYSTEM TECHNICAL JOURNAL 



by experiment years before it was explained, and a great puzzle it was. 

 There, the old statistics and the new (in the limit of extreme rarefaction) 

 led to the same result. Here the old statistics was impotent, and the new 

 had to be invented. 



Reverting to the identification of B with 1/kT: this may be proved in 

 the following way^. Refer back to equations (74) and (71), and for ease 

 of operation write Xj for (e^'^' — 1)~^ We then have: 



Nj = Cxy (78) 



S = k\nW = ki: {C(l + Xj) In [C(l + Xj)] - Cxj In [Cx^] - Cln C| (79) 



t/ = 2 ALEj = C S EjXj (80) 



Differentiate .5 with respect to B and do the like with t/, and divide the 

 former derivative by the latter, so as to get the derivative dS/dU. It 

 will be found that this is equal to kB\ and since by definition of absolute 

 temperature it is also equal to T~^, the identification is made. . 



The Bose-Einstein and the Fermi-Dirac Statistics 



Hitherto in this article, except for one protective allusion, I have spoken 

 as if the new statistics were one and indivisible. There are, however, two 

 branches of it, known respectively as the Bose-Einstein statistics and the 

 Fermi-Dirac statistics. It is the former of which I have treated throughout 

 this essay. The point at which the latter branches off is to be found on 

 page 365, where I introduced the game of balls and baskets, the balls stand- 

 ing for cells and the baskets for populations. On reaching this point the 

 game is to be played with the supplemental assumption that there are only 

 two baskets, those numbered and 1. That is to say: a cell may either be 

 empty or may contain a single atom, but never more than one. 



I leave to the student the task of revising equations (7) to (16) accord- 

 ingly, but I take it upon myself to point out how easily the problem can be 

 solved by the second method — that of pages 370-71, the method involving 

 the counting of all the different ways in which un-numbered atoms can be 

 distributed among numbered cells. In the Bose-Einstein case the funda- 

 mental formula is (22), which is not very easy to derive. In the Fermi-Dirac 

 case we proceed by playing anew the game of balls and baskets. There are 

 but the two baskets, one being set out to receive the balls corresponding to 

 the empty cells and the other for the cells containing one atom each — the 

 "filled cells," we may call them. There being in thejth region N, atoms 

 and C cells all together, the first basket is destined to contain (C — Nj) 

 balls and the second to contain N j. The question is then: in how many 



' I am indebted for this proof, as well as for much other assistance in the preparation 

 of the article, to Dr. L. A. MacColl, 



