94 REPORT — 189-i. 



mann's minimum function with the entropy is estabHshed more briefly by 

 Burbury in his recent paper, to be discussed shortly.' 



54. The particular assumption as to the law of a jjriori probability 

 precludes the above investigations from furnishing a complete proof of 

 the Boltzmann-Maxwell Law. In a subsequent paper ^ Boltzmann has 

 removed this restriction, and has considered the a posteriori probabilities 

 corresponding to any assumed law of a jmori probability. In other 

 words, we start with a large number (N) of molecules having a given 

 distribution of energy, and from them a smaller number {n) are selected, 

 and their mean energy is found to have a certain value which may be 

 either the same or diiferent from that of the original N. It is required to 

 find the most probable law of distribution in the n selected molecules, or, 

 generally, the probability of any given distribution. 



Boltzmann first considers the case where the original molecules follow 

 the Boltzmann-Maxwell Law for two dimensional space, and points out the 

 necessary modifications for space of three dimensions. In the general 

 case, supposing /j, /a, • • . fp to denote the a priori probabilities of a 

 molecule having energies f, 2f, . . . pe, the a j)osteriori probability of a 

 combination in which the numbers of molecules having these energies are 

 ojq, ojj, . . . Wp respectively is proportional to O, where 



"=/o- /."■■• • . f? ^ \ , . . . (56) 



iiIq! ta<| ! 0)2' 

 where as before 



The approximate expression for w ! now gives 



log Q.■=■^^^o^ logy) — ^(ij, log w; + constant 



and Boltzmann finds the following results. 



If the mean energy of the selected n molecules is equal to the mean 

 energy of the original N, the most probable distribution of energy in the 

 latter is identical with the distribution in the former. 



If, however, the mean energy of the smaller number is unequal to that 

 of the larger, the most probable distribution is that given by the form 



<..=^/,e-'"- (57) 



Boltzmann's investigation was probahly an attempt to arrive at the 

 Boltzmann-Maxwell distribution as the ultimate result of a immber of 

 successive processes such as the above, independently of the initial dis- 

 tribution. This has recently been actually accomplished by Burbury by 

 the application of a different method as follows : — 



55. Burbury ^ bases his investigation on a generalisation of the theory 

 of Least Squares, which asserts that if we regard the variations of a series 

 of n quantities a;,, ajj) • • • ^n ^s being each the result of an infinite number 

 N of independent simultaneous increments divided each by a/N, then the 



' ' On the Law cf Distribution of Energy,' Plnl. Mag., January 1894. 



• ' Weitere Bemerkungen iiber einige Probleme der mechanischen Warmetheorie,' 

 Sitz^). der It. Wietier Akad., Ixxviii. (ii.), June 1878. The second part of the paper 

 deals with the equilibrium of a gas under gravity, and is less interesting. 



^ Pliil. Marj., January 1894. 



