873 



1. e. the same increase as in the analogous irreversible distribution of P similar, 

 indopendent gas-molecules, for the number of ways in which P quanta may be 

 dislrihuted fust oven' iVj, llien over iV., cells in space, are to each other in the ratio 



iV> : A^.^ 0^) 



II' with Pr-ANCK the object were to distribute P mutually independent elements 

 £ over N resonators, in passing from N^ to N.;, resonators the number of possible 

 distributions would in this case also increase in the ratio ( ■) and correspondingly 

 the entropy accord. ng to equ .tion (a). We know, however, that Planck obtains 

 the totally dilYerent formula 



(.V,-l+P)/ (iV,-l+P)/ 

 (N^-l)! P! ' {N^-iy P! •••••• u) 



(which only coincides approximately with (/?) 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 t , 

 the resolution of the multiples of e into separate elements f, which is essential in 

 his method, and the introduction of these separate elements have to be taken 

 "cum grauo salis" ; it is simply a formal device entirely analogous to our permut- 

 ation of the elements r or 0. The real object which is counted remains the number 

 of all the different distributions of N resonators over the energy-grades 0, e, 2 , ... 

 with a given total energy P.. If for instance P = 3, and N—2, Einstein has to 

 distinguish 2' = 8 ways in which the three (similar) light-quanta A, B, C can be 

 distributed over the space-cells 1, 2. 



PlaNci^ on the other hand must count the three cdses It, III and V as a single 

 one, for all three express that resonator R^ is at the grade 2 , i?^ at g; similarly 

 he has to reckon the cases IV, VI and VII as one ; R^ has here s and R^ 2t-. 

 Adding the two remaining cases 1 (R^ contains 3e, R^ Os) and II {Ri has Og, R^ He) 

 one actually obtains 



{N—\)!P! (2 — 1)/ 3/ ~ 



different distributions of the resonators R^, R^ over the energy-grades. 



We may summarize the above as follows ; Einstein's hypothesis leads necessarily 

 to formula (y.) for the entropy and thus necessarily to Wien's radiation-formula, 

 not Planck's Planck's formal device (distribution of P energy-elements e over 

 N resonators) cannot he interpreted in the sense of Einstein's light-ciuania. 



(December 24, 1914). 



