﻿of Radiation and Line Spectra. 797 



so that we have 



p_cc <t— co r=co 



2 2 2 •••./?-..• =i. • • • (4) 



p = (7 = r = 



For the sake of brevity we shall say that Np ff r . . . systems 

 are on the locus {par . . .), N p Vr' . . . on the locus (/oW . . .), 

 and so on. We have for the energy o£ the whole collection 

 of systems the following expression: — 



p = co o-=oo r = oo 



E = N X % % •• • E„ ffr .../p ffr ... . (5) 



p=0 o = r=0 



where E^r ... is the energy of a system on the locus 

 (par . . .). If P is the number of ways in which N systems 

 can be distributed, so that N^r • • • lie on the locus 

 (par . . .), .NpVr'.. on the locus (p'cr'r' . . .), and so on, we 

 have 



N! 

 P = (N pffr ..,)! (N,,vr'...)! ... ' * ' (6) 



We shall call P (after Planck) the i: thermodynamic 

 probability " of the distribution in question, and identify the 

 quantity 



* = *logP (7) 



with the entropy of the assemblage of systems ; the quantity 

 k is the entropy constant. We may assume N^r . . . , 

 NpVr' . . . , and a fortiori N to be individually very lar^e 

 numbers, and therefore, by Stirling's theorem, 



( N v-..y N K pvr'...\ _ 



\ por . . . J\ pa T ... J 



This last equation, together with (3) and (7), leads to the 

 following expression for the entropy of the collection of 

 systems : — 



^=-Mlii.../ pvr ...log/o ffr ... . . (8) 



t v 



The condition for statistical equilibrium is expressed by 

 where the variation is subject to the total energy being 



