871 



is the followino-: resonator /?, lias reached the energv-grafle 4f (7?j 

 contains the energy 4f), R.^ the grade 2?, R^ the grade Op (contains 

 no energy), R^ the grade e. Our s^niibol will, read from left to right, 

 indicate the energy of R.^, R.^, R^. /?, in the distribution chosen, and 

 pai'ticnlarly express, that (he total energy is 7f. For this case the 

 symbol will be: 



or also more simply : 



f f f f 



0"00^I 



With general values of .V and P the symbol will contain /^ times 

 the sign g and (^V— 1) times the sign O ^). The question now is, 

 how many dijjerent symbols for the distribution may be formed in 

 the manner indicated above from the given number of e and O? 

 The answer is 



(iV-l + P)/ 



^— ^ (1) 



Proof: first considering the {X — 1 -f- ^) elements e . . . f , O... O 

 as so many distinguishable entities, they may be arranged in 



(.V - 1 + 7^) / (2) 



different manners between the ends T[]T- Next note, that each time 



(A' ^)!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 8 ^) or the {N — 1) elements 0. The 

 number of the dijf'erent symbols for the distribution and that of the 



^) We were led to tlie introduction of the (N — \) parlilioiis between the N 

 resonators, in trying to find an exph\nalion of the form {N — 1)/ in the denomi- 

 nator of {A) (compare note 1 on page 872). Planck proves, that the number of 

 distributions must be equal to the number of al! 'combinations with repetitions 

 of N elements of class P" and f.-r the proof, Ibat this number is given by the 

 expression (.4), he refers to the train of reasoniug followed in treatises on com- 

 binations for this parlicular case. In these treatises the expression {A) is ari ived 

 at by the aid of the device of "transition from n to n -\- 1", and this method taken 

 as a whole does not give an insight into the origin of the final expression. 



2) See appendix. 



