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VI. On the Conditions necessary for Eqidpartition of Energy. 

 (Note on Mr. Jeans's Paper^thil. Mag. November 1902.) 

 j/ By S. H. Burbuby, F.P.S* 



MR. JEANS'S conclusions are in substantial agreement 

 with mine, as explained in this Magazine for December 

 1900, so far as regards the nature of the required conditions. 

 On the question whether these conditions exist in any natural 

 system, there is perhaps difference of opinion. 



Mr. Jeans deals, as I also dealt, with two supposed proofs 

 of the law: — (1) that given by Lord Eayleigh interpreting 

 Maxwell in this Magazine for January 1900; (2) the proof 

 given by Boltzmann in the Vorlesungen ilber Gas Theorie. 

 As both profess to prove the truth of the law, the personal 

 authority in favour of it is at first sight overwhelming. If, 

 however, we find on examination that the two proofs are 

 inconsistent with each other, the authority for the law is 

 not the sum, but the difference of two very great magnitudes. 



Eayleigh supposes a natural system defined by coordinates 

 and momenta q 1 . . . g n , p 1 , . ,p n , which is supposed to move 

 under its own internal forces unaffected by any other body, 

 and therefore to have constant total energy E. With that 

 constant energy it passes in cycle through the phases fa . . . <£,.. 

 Mr. Jeans prefers to conceive the system as a " point " in a 

 space of 6n dimensions. Next we suppose a great number of 

 systems similar in constitution to the first, and each passing 

 in cycle through the same phases fa . . . <j) r unaffected by any 

 body external to itself. In that ensemble of systems, all 

 having the same energy E, and passing in cycle through the 

 same series of phases, the number of systems in any one of 

 the phases fa. . . <$> r at any instant is equal to the number at 

 the same instant in any other of the phases (p { . . . fa. This 

 is what Willard Gibbs proves in ' Principles of Statistical 

 Mechanics,' Chapter I. 



If fa . . . fa. include all possible phases in which the system 

 can be with energy E, the law of equipartition follows. 

 But there are two objections to this result, one namely 

 that according to Mr. Jeans, with whom I agree, no such 

 system exists or can exist in nature, and the other that the 

 motion assumed is cyclic and reversible ; whereas Boltzmann 

 appears to me to prove (if only his fundamental assumption be 

 admitted) that the motion in which energy is equally par- 

 titioned is irreversible. The two great authorities are thus 

 inconsistent with each other. If the phases fa . . . fa. through 



* Communicated by the Author. 



