1900.] on the Dynamical Theory of Heat and Light. 371 



Maxwell * gave a demonstration extending it to the generalised 

 Lagrangian co-ordinates of any system whatever, with a finite or 

 infinitely great number of degrees of freedom. The words in which 

 he enunciated his supposed theorem are as follows : 



" The only assumption which is necessary for the direct proof is 

 "that the system, if left to itself in its actual state of motion, will, 

 " soouer or later, pass [infinitely nearly f] through every phase which is 

 " consistent with the equation of energy " (p. 714) and, again (p. 716). 



" It appears from the theorem, that in the ultimate state of the 

 " system the average J kinetic energy of two portions of the system must 

 " be in the ratio of the number of degrees of freedom of those portions. 



" This, therefore, must be the condition of the equality of tem- 

 " perature of the two portions of the system." 



I have never seen validity in the demonstration |] on which Max- 

 well founds this statement, and it has always seemed to me exceed- 

 ingly improbable that it can be true. If true, it would be very 

 wonderful, and most interesting in pure mathematical dynamics. 

 Having been published by Boltzmanu and Maxwell it would be worthy 

 of most serious attention, even without consideration of its bearing on 

 thermo-dynamics. But, when we consider its bearing on thermo- 

 dynamics, and in its first and most obvious application we find 

 it destructive of the kinetic theory of gases, of which Maxwell was 

 one of the chief founders, we cannot see it otherwise than as a cloud 

 on the dynamical theory of heat and light. 



§ 10. For the kinetic theory of gases, let each molecule be a cluster 

 of Boscovich atoms. This includes every possibility (" dynamical," 

 or " electrical," or " physical," or " chemical ") regarding the nature 

 and qualities of a molecule and of all its parts. The mutual forces 

 between the constituent atoms must be such that the cluster is in 

 stable equilibrium if given at rest ; which means, that if started from 



* ' On Boltzmann's Theorem on the Average Distribution of Energy in a 

 System of Material Points.' Maxwell's Collected Papers, vol. ii. pp. 713-741, 

 and Canib. Phil. Trans., May 6, 1878. 



t I have inserted these two words as certainly belonging to Maxwell's 

 meaning. — K. 



J The average here meant is a time-avernge through a sufficiently long time. 



|j The mode of proof followed by Maxwell, and ics connection with antecedent 

 considerations of his own and of Boltzmann, imply, as included in the general 

 theorem, that the average kinetic energy of any one of three rectangular com- 

 ponents of the motion of the centre of inertia of an isolated system, acted upon 

 only by mutual forces between it* parts is equal to the average kinetic energy of 

 each generalised component of motion relatively to the centre of inertia. Con- 

 sider, for example, as " parts of the system " two particles of masses m and m' free 

 to move only in a fixed straight line, and connected to one another by a massless 

 spring. The Boltzmann-Maxwell doctrine asserts that the average kinetic energy 

 of the motion of the inertial centre is equal to the average kinetic energy of the 

 motion relative to the inertial centre. This is included in the wording of 

 Maxwell's statement in the text if, but not unless, m = m'. See footnote on 

 § 7 of my paper ' On some Test-Cases for the Boltzmann-Maxwell Doctrine 

 regarding Distribution of Energy.' Proc. Roy. teoc., June 11, 1891. 



