﻿300 Simplified Deduction of Planch's For inula. 



however, that Planck obtains the totally different formula 

 (Nx-l + pi . gSl-l + P)! M 



(Ni-lJlP! ' (N 2 -l)!P! ' ' ' K7J 

 (which only corresponds approximately with (/3) for very 

 large values of P) and a corresponding law of dependence 

 of the entropy on N. This can be simply explained as 

 follows : Planck does not deal with really mutually free 

 quanta e; the resolution of the multiples of e in separate 

 elements e. which is essential in his method, and the intro- 

 duction of these separate elements have to be taken cum 

 grano salis ; it is simply a formal device entirely analogous 

 to our permutation of the elements e or O. The real object 

 which is counted remains the number of all the different 

 distributions of N resonators over the energy-grades 0, e, 

 2e . . . with a given total energy Pe. If, for instance, P = 3, 

 and N = 2, Einstein has to distinguish 2 3 = 8 ways in which 

 the three (similar) light-quanta A, B, C can be distributed 

 over the space-cells 1, 2. 



A. B. 0. 



I. 



1 



1 



1 



II. 



1 



1 



2 



III. ' 



1 



2 



1 



IV. 



1 



2 



2 



V. i 



2 



1 



1 



VI. 



2 



1 



2 



VII. 



2 



2 



1 



VIII. 



2 



2 



2 



Planck, on the other hand, must count the three cases 

 II., III., and V. as a single one, for all three express that 

 resonator Rj is at the grade 2e, R 2 at e ; similarly, he has to 

 reckon the cases IV., VI., and VII. as one ; Rj has here e 

 and R 2 2e. Adding the two remaining cases I. (R 2 contains 

 3e, R 2 Oe) and II. (Ri has Oe, R 2 3e), one actually obtains 

 (N-l + P)! _ (2-1 + 3)! 

 (N-1)!P! "" (2-1)! 3! " 4 



different distributions of the resonators R b R 2 over the 

 energy-grades. 



