﻿840 Messrs. C. Gr, Darwin and R. H. Fowler on 



of our former paper the number of weighted complexions 

 will be 



M! ao a, CIO, 



^UJT..P°* >1 -"F' * ' ' ' (91 > 



and we must have 



S r «r=-M, 2rOre r + € = E. . . . (9'2) 



The probability that the mechanical system is in dQ, is 

 measured by the total number of complexions for which 

 it is there, and so (9'1) must be summed over all values of 

 the as consistent with (9*2). Now if & is the solution of 



V-e = M.$~logf, .... (9-3) 



this sum is 



[/(■&)] M * d£l 



* B { 2 ^(*ip° gf y 



h n ' 



by virtue of § 6 of our former paper. Here 3 is, strictly 

 speaking, the exact temperature of the bath at the moment 

 under consideration, and so will be liable to fluctuation 

 according to the value of e ; but, by virtue of the assumption 

 that the bath is very large, e wil) practically always be 

 insignificant in the solution of (9 # 3), and so 3 may be taken 

 as a constant. Then the probability that the mechanical 

 system is in the cell dD. is proportional to $ e dQ, and all the 

 other factors are constant and may be omitted in taking 

 averages. Using (7*5) we thus obtain Gibbs' expression 

 for the density-in-phase of the " canonical ensemble/'' 

 namely 



This leads to the impossibility of perpetual motion and 

 all his work on the laws of thermodynamics. 



§ 10. The Deduction of the Elementary States from 

 Thermodynamic Data. 



An interesting result follows from the inversion of the 

 argument of § 8. Suppose that we have an assembly of 

 unknown constitution in which the temperature is sole 

 variable. Then a knowledge of the specific heat determines 

 the characteristic function, and thence the partition function. 

 If this can be expanded in terms of •&, we can determine 

 the energies and weights of the elementary states ; but the 

 matter is complicated by the fact that we cannot tell in 



