92 REPORT— 1894. 



accounts, as, personally, I think they make the matter clearer. They are 

 not Boltzmann's. 



In his first paper ' Boltzmann starts by taking a finite number n 

 of molecules, and supposing that the kinetic energy of each molecule must 

 have one or other of a discrete series of values «, 2f, 3f, . . . ^je. Taking 

 the total energy T of the system as equal to \f, he investigates the 

 probability that it should be divided between the molecules in a given 

 manner, each value of the energy being a priori equally probable for a 

 given molecule. If wq, w,, w^, . . . w^, be the numbers of molecules having 

 energies 0, e, 2e, . . . j)e, the number of permutations of molecules satis- 

 fying this distribution or ' complexion ' is 



V=-r-^r 1 • . • • (42) 



subject to the conditions 



WQ-f- W.J + . . . +ii)p ^n .... (43) 

 wi + 2w2+ • • • +pi^p=^ .... (44) 



The cl jMsteriori most probable distribution is that for which the number 

 of permutations is greatest. Taking M to be the logarithm of the denomi- 

 nator of ^, or 



M=log(wo!) + log(<.»i 0-f- . . . . (45) 



we haA'e, therefore to make M a minimum subject to the conditions (43) 

 (44). To simplifiy the calculation, when w is very great, w ! may be 

 replaced by its approximate value 



v/ 



(2-)(")" (46) 



Passing to the case in which the energy is capable of continuous 

 variation, if/(.x) dx denote the number of molecules with energy between 

 X and x + dx, we have to put wq=i/ (0), w,=£/(f), . . . w^=itf {pt), and 

 to make t=-dx in the limit, so that the problem reduces to finding the 

 minimum of 



fco 



M'= /{x)\ogf(x)dx .... (47) 

 Jo 

 subject to the conditions 



n= f{x)dx (48) 



Jo 



T={'"xf(x)dx (49) 



Jo 



Avhere M' differs from M by a constant, and is what Boltzmann calls the 

 ' measure of permutability.' 

 The solution is 



/ (x) dx^Ce-'"^ dx (50) 



This, therefore, is the a j^osteriori most probable distribution of the 

 energy among the molecules on the hypothesis that the a p)riori pro- 



' ' Ueber die Beziehung zwischen dem zweiten Hauptsatz der mechanischen 

 Wiirmetheorie und der Warscheinlichkeitsrechnung respective den Satzen iiber 

 das Warmegleichgewicht,' Sitzlcr. der k. Wiener Akad., Ixsvi. (ii.), Oct. 1878. 



