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XXXVIII. Remarks on Maxwell's Investigation respecting 

 Boltzmann's Theorem. By Lord Ravleigh, Sec. U.S.* 



rpHE investigation in question, which was published by 

 JL Maxwell in the 12th volume of the Cambridge Philo- 

 sophical Transactions f only a short time before his death, has 

 been the subject of some adverse criticism at the hands of Sir 

 W. Thomson % and of Mr. Bryan §. The question is indeed a 

 very difficult one ; and I do not pretend to feel complete 

 confidence in the correctness of the view now to be put 

 forward. Nevertheless, it seems desirable that at the present 

 stage of the discussion some reply to the above-mentioned 

 criticisms should be hazarded, if only in order to keep the 

 question open. 



The argument to which most exception has been taken is 

 that by which Maxwell (Reprint, p. 722) seeks to prove that 

 the mean kinetic energy corresponding to every variable is 

 the same. In the course of it, the expression T for the kinetic 

 energy is supposed to be reduced to a sum of squares of the 

 component momenta, an assumption which Mr. Bryan charac- 

 terizes as fallacious. But here it seems to be overlooked that 

 Maxwell is limiting his attention to systems in a given con- 

 figuration, and that no dynamics is founded upon the reduced 

 expression for T. The reduction can be effected in an infinite 

 number of ways. We may imagine the configuration in 

 question rendered one of stable equilibrium by the introduc- 

 tion of suitable forces proportional to displacements. The 

 principal modes of isochronous vibration thus resulting will 

 serve the required purpose. I do not see the applicability to 

 this argument of the warning quoted from Routes 'Rigid 

 Dynamics/ Perhaps the objection is felt that the conclusion 

 cannot be true in the absence of a complete specification of 

 the variables. This is a point that may require further 

 examination. I admit that the argument seems to imply 

 that the conclusion possesses something of an invariantic 

 character. 



The nature of the question may be illustrated by an example 

 approximately realized in the billiard-table, viz. the path of an 

 elastic particle moving in a plane without loss of energy and 



* Communicated by the Author. - 



t ' Reprint,' vol. ii. p. 713. 



% Proc. Roy. Soc. June 1891. 



§ " On the Present State of our Knowledge of Thermodynamics," Brit. 

 Assoc. Report, 1891. I am indebted to the author for an advance copy 

 of this valuable report. 



