10 Lord Kelvin on the 



diminishing distance, according to airy law for distances less 

 than the sum of the radii, subject only to the condition that 

 it would be infinite before the distance became zero. In fact 

 the impact, oblique or direct, between two Boscovich atoms 

 thus defined, has the same result after the collision is com- 

 pleted (that is to say, when their spheres of action get outside 

 one another) as collision between two conventional elastic 

 spheres, imagined to have radii dependent on the lines and 

 velocities of approach before collision (the greater the relative 

 velocity the smaller the effective radii) ; and the only as- 

 sumption essentially involved in those demonstrations is, that 

 the radius of each sphere is very small in comparison with 

 the average length of free path. 



§ 17. But if the particles are Boscovich atoms, having 

 centre of inertia not coinciding with centre of force ; or quasi 

 Boscovich atoms, of non-spherical figure ; or (a more accept- 

 able supposition) if each particle is a cluster of two or more 

 Boscovich atoms : rotations and changes of rotation would 

 result from collisions. TTaterston's and Olausius' leading' 

 principle, quoted in § 13 above, must now be taken into 

 account, and Tait's demonstration is no longer applicable. 

 Waterston and Clausius, in respect to rotation, both wisely 

 abstained from saying more than that the average kinetic 

 energy of rotation bears a constant ratio to the average 

 kinetic energy of translation. With magnificent boldness 

 Boltzmann and Maxwell declared that the ratio is equality ; 

 Boltzmann having found what seemed to him a demonstra- 

 tion of this remarkable proposition, and Maxwell having 

 accepted the supposed demonstration as valid. 



§ 18. Boltzmann went further* and extended the theorem 

 of equality of mean kinetic energies to any system of a finite 

 number of material points (Boscovich atoms) acting on one 

 another, according to any law of force, and moving freely 

 among one another ; and finally, Maxwell | gave a demon- 

 stration extending it to the generalized Lagrangian co-ordi- 

 nates 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 

 u proof is that the system, if left to itself in its actual state of 



* " Studien iiber das Gleiehgewicht der lebendigen Kraft zwischen 

 bewegten materiellen Punkten." Sitzb. K, Akad. Wien, October 8 r 

 1868. 



j " 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 Camb. Phil. Trans., May 6, 1878. 



