[ 98 ] 



VI. The Law of Partition of Kinetic Energy. 

 By Lord Rayleigh, F.R.S* 



THE law of equal partition, enunciated first by Waterston 

 for the case of point molecules of varying mass, and the 

 associated Boltzmann- Maxwell doctrine respecting steady dis- 

 tributions have been the subject of much difference of opinion. 

 Indeed, it would hardly be too much to say that no two writers 

 are fully agreed. The discussion has turned mainly upon 

 Maxwell's paper of 1879 f, to which objections J have been 

 taken by Lord Kelvin and Prof. Bryan, and in a minor degree 

 by Prof. Boltzmann and myself. Lord Kelvin's objections 

 are the most fundamental. He writes § : " But, conceding 

 Maxwell's fundamental assumption, 1 do not see in the mathe- 

 matical workings of his paper any proof of his conclusion 

 ' that the average kinetic energy corresponding to any one of 

 the variables is the same for every one of the variables of the 

 system/ Indeed, as a general proposition its meaning is not 

 explained, and it seems to me inexplicable. The reduction of 

 the kinetic energy to a sum of squares leaves the several parts 

 of the whole with no correspondence to any defined or definable 

 set of independent variables." 



In a short note || written soon afterwards I pointed 

 out some considerations which appeared to me to justify 

 Maxwell's argument, and I suggested the substitution of 

 Hamilton's principal function for the one employed by 

 Maxwell If. The views that I then expressed still commend 

 themselves to me ; and I think that it may be worth while to 

 develop them a little further, and to illustrate Maxwell's 

 argument by applying it to a particular case where the 

 simplicity of the circumstances and the familiarity of the 

 notation may help to fix our ideas. 



But in the mean time it may be well to consider Lord 

 Kelvin's " Decisive Test-case disproving the Maxwell-Boltz- 

 mann Doctrine regarding Distribution of Kinetic Energy "** ? 

 which appeared shortly after the publication of my note. The 

 following is the substance of the argument : — 



" Let the system consist of three bodies, A, B, C, all 

 movable only in one straight line, KHL: 



* Communicated by the Author. 



t i Collected Scientific Papers/ vol. ii. p. 713. 



X I am speaking here of objections to the dynamical and statistical 

 reasoning of the paper. Difficulties in the way of reconciling the results 

 with a kinetic theory of matter are another question. 



§ Proc. Rov. Soc. vol. 1. p. 85 (1891). 



|| Phil. Mag. Apr. 1892, p. 356. 



•[[ See also Dr. Watson's ' Kinetic Theory of Gases,' 2nd edit. 1893. 

 ** Phil. Mag. May 1892, p. 466. 



