﻿82i Messrs. C. G. Darwin and R, H. Fowler 



on 



average properties o£ the assembly, which are, on any 

 statistical theory, those which experiment determines, 

 instead o£ the most probable values, as is usually done. 

 It can also be carried out simply and rigorously without 

 the use of Stirling's theorem, and thus provides satisfactory 

 proofs of all the usual partition laws, including Maxwell's 

 Distribution Law. 



In this discussion the partition laws were all obtainable 

 without any reference to thermodynamical ideas, in par- 

 ticular without any mention of entropy. This we claim 

 as an advantage. But a great deal of work has been 

 done on partition laws, in which the idea of entropy has 

 played a leading part ; so that, for this if for no other 

 reason, it is fitting to examine its position in our pre- 

 sentation of statistical theory. But the power of our 

 method on the statistical side invites a somewhat more 

 general review of the fundamental connexion between 

 classical thermodynamics and statistical mechanics both 

 of classical dynamics and the quantum theory. In the 

 former work we were content with purely statistical results, 

 and identified the temperature scale simply by the perfect 

 gas laws ; here we attempt a more strictly logical deve- 

 lopment, and prove the laws of thermodynamics for 

 assemblies composed of systems of a fairly general type, 

 and, by linking on to Gibbs' work, also for general systems 

 which obey the laws of classical mechanics. 



After summarizing our previous results in § 2, we pass 

 in § 3 to a comparison between the empirical temperature 

 in thermodynamics and the parameter which acts as 

 temperature in our previous work. In §§4, 5, 6, we 

 make a critical study of the usual presentation of entropy 

 in statistical theory. This is ordinarily introduced by 

 means of Boltzmann's Hypothesis, which relates it to 

 probability, and, though no objection can be made to much 

 of the work based on this hypothesis, it appears to us 

 that the development is often marred by somewhat loose 

 reasoning. Though much that we here say is general 

 and not at all dependent on our special methods of 

 treatment, yet it has been far easier to examine the validity 

 of the arguments on account of the way in which it is 

 possible to combine assemblies together at will. In con- 

 sequence of this discussion we are led, in § 7, to a pre- 

 sentation of entropy which is very closely related to that 

 of classical thermodynamics, which frees it from the com- 

 binatory complications with which it is normally associated 

 and brings it back to direct dependence on the partition 



