﻿298 Profs. P. Ehrenfest and H. K. Onnes : Simplified 



energy-grade 4e (R r contains the energy 4e), H 2 the grade 

 2e } R 3 the grade Oe (contains no energy), R 4 the grade e. 

 Our symbol will, read from left to right, indicate the energy 

 o£ R 1? R 2 > R35 R±-> i n the distribution chosen, and particularly 

 express that the total energy is 7e. For this case the 

 symbol will be : — 



JJ>i^i^iOi-I-iOOi-iI 



or also more simply : — 



|eeeeO-OOl 



With general yalues of N and P the symbol will contain 

 P times the sign e and (N — 1) times the sign O *. The 

 question now is, how many different symbols for the dis- 

 tribution may be formed in the manner indicated aboye 

 from the given number of e and O ? The answer is 



( N-l+P)! m 



P!(N-1)! ^ } 



Proof : first considering the (N — 1 + P) elements e . . . e, 

 O . . . O as so many distinguishable entities, they may be 



arranged in 



(N-l + P)! (2) 



different manners between the ends | j|. Next note, that 

 each time 



(N-1)!P! (3) 



of the combinations thus obtained give the same symbol for 

 the distribution (and give the same energy-grade to each 

 resonator), viz. all those combinations which are formed 

 from each other by the permutation of the P elements e| or 

 the (N — 1) elements O. The number of the different symbols 

 for the distribution and that of the distributions themselves 

 required is thus obtained by dividing (2) by (3) q. e. d. 



* We were led to the introduction of the (N — 1) partitions between 

 the N resonators in trying to iind an explanation of the form (N — 1) ! 

 in the denominator of (A). Planck proves that the number of dis- 

 tributions must be equal to the number of all " combinations with 

 repetitions of N elements of class P," and for the proof, that this number 

 is given by the expression (A), he refers to the train of reasoning- 

 followed in treatises on combinations for this particular case. In these 

 treatises the expression (A) is arrived at by the aid of the device of 

 " transition from n to ra+l>" and this method taken as a whole does 

 not give an insight into the origin of the final expression. 



f See Appendix. 



