872 



distributions themselves required is tlms obtained bv dividing- (2) by 

 (3) q. e. d.^). 



APPENDIX. 



Tke contrast between PiiANCK'.s' lujpotlie.^'is of the ener(/>/-(/f(((/es (ind 

 EiNSTEiis'.v Jijipothexis of ener()i/-fjuanta. 



The permutation of the elements e is a purely lormal tlevice, ju.st as the per- 

 mutation of the elements is. More than once the analogous, equally formal device 

 used by Planck, viz. distribution of P energ-y-elemenls over N resonators, has by 

 a misunderstanding been given a physical interpretation, wliich is absolutely in 

 conflict with Planck's r'adialion-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 Elnstein's light-quanta and accordingly it was said, thai the difference 

 between Planck and Einstkin 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^ cm'^ over a larger space of 

 Nz cmS and he finds using Boltzman's entropy-formula: S = klog W, that this 

 produces a gain of entropy 3) ; 



S — S^=klogi-~-\ (a) 



1) It may be added, that the problem of the distribution of N resonators over 

 the energy grades corresponds to the following: On a rod, whose length is a mul- 

 tiple Pd of a given length ••, notches have been cut at distances s, 2t, etc. from 

 one of the ends. At each of the notches, and only there, the rod may be broken, 

 the separate pieces may subsequently be joined together in arbitrary numbers and 

 in arbitrary order, the rods thus obtained not being distinguishable from each other 

 otherwise than by a possible difference in length. The question is, in how many 

 different manners (comp. Appendix) the rod may be divided and the pieces distri-- 

 buted over a given number of boxes, to be distinguished horn each other as the 

 1st 2i\d, .... A^th, when no box may contain more than on*; rod. If the boxes, 

 which may be thought of as rectangular, are placed side by side in one line, they 

 form together as it were an oblong drawer with {N — 1) partitions, formed of two 

 walls each, (comp. the above symbol in its first form, from which the second 

 form was derived by abstracting from the fact, that each multiple of t forms one 

 whole each time), and these double partitions may be imagined to be mutually 

 exchanged, the boxes themselves remaining where they are. The possibility of this 

 exchange is indicated by the form of the symbol chosen. 



As a further example corresponding to the symbol we may take a thread on 

 which between P beads of the same kind, (A^— 1) beads of a different kind are 

 strung, which divide the beads of the first kind in a 1st, ^nd . . . iVth group. 



2j P. Ehrenfest, Ann. d. Phys. 36, 91, 1911, G. Krutkow, Physik. Zschr. 15, 

 133, 363, 1914. 



») A. Einstein, Ann. d. Phys. 17, 13^, 1905, 



