﻿Deduction of Planck's .Formula. 299 



APPENDIX, 



\es 



The contrast betiveen Planch's hypothesis of the energy-gradi 

 and Einstein s hypothesis of energy-quanta. 



The permutation of the elements e is a purely formal 

 device, just as the permutation of the elements O is. More 

 than once the analogous, equally formal device used by 

 Planck, viz. distribution of P energy-elements over N reson- 

 ators, has by a misunderstanding been given a physical 

 interpretation, which is absolutely in conflict with Planck's 

 radiation-formula and would lead to Wien's radiation- 

 formula. 



As a matter of fact, Planck's energy-elements were in 

 that case almost entirely identified with Einstein's light- 

 quanta, and accordingly it was said that the difference 

 between Planck and Einstein consists herein, that the latter 

 assumes the existence of mutually independent energy- 

 quanta also in empty space, the former only in the interior 

 of matter, in the resonators. The confusion which underlies 

 this view has been more than once pointed out*. Einstein 

 really considers P similar quanta, existing independently of 

 each other. He discusses, for instance, the case that they 

 distribute themselves irreversibly from a space of N t cm. 3 

 over a larger space of N 2 cm. 3 , and he finds, using Boltz- 

 mann's entropy-formula, S = &log W, that this produces 

 a gain of entropy f : 



S — S =Hog(^V , («) 



i. e. the same increase as in the analogous irreversible distri- 

 bution of P similar independent gas-molecules, for the number 

 of ways in which P quanta may be distributed, first over 

 N 1? then over N 2 cells in space, are to each other in the ratio 



Nf : Nf 08) 



If with Planck the object were to distribute P mutually 

 independent elements e over N resonators, in passing from 

 N x to N 2 resonators the number of possible distributions 

 would in this case also increase in the ratio (/3) and corre- 

 spondingly the entropy according to equation (a). We know, 



* P. Ehrenfest, Ann. d. Phys. vol. xxxvi. p. 91 (1911) : G. Krutkcnv, 

 Phijsik. Zschr. vol. xv. pp. 133, 3G3 (1914). 



t A. Einstein. Ann. d. PJn/s. vol. xvii. p. 132 (1905). 



