80 Sir W. Thomson. On the Maxwell-Boltzmann [June 11, 



Akad. Wien,' October 8, 1868), enunciated a large extension of this 

 theorem, and Maxwell a still wider generalisation in his paper " On 

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

 System of Material Points " (' Cambridge Phil. Soc. Trans.,' May 6, 

 1878, republished in vol. 2 of Maxwell's 'Scientific Papers,' pp. 

 713_l741) ? to the following effect (p. 716): 



"In the ultimate state of the system, the average kinetic energy of 

 two given portions of the system must be in the ratio of the number 

 of degrees of freedom of those portions." 



Much disbelief and doubt has been felt as to the complete truth, or 

 the extent of cases for which there is truth, of this proposition. 



3. For a test case, differing as little as possible from Maxwell's 

 original case of solid elastic spheres, consider a hollow spherical 

 shell and a solid sphere globule we shall call it for brevity within 

 the shell. I must first digress to remark that what has hitherto by 

 Maxwell and Clausius and others before and after them been called 

 for brevity an " elastic sphere," is not an elastic solid, capable of 

 rotation and of elastic deformation ; and therefore capable of an 

 infinite number of modes of steady vibration, into which, of finer and 

 finer degrees of nodal sub-division and shorter and shorter periods, 

 all translational energy would, if the Boltzmann-Maxwell generalised 

 proposition were true, be ultimately transformed by collisions. 

 The " smooth elastic spheres " are really Boscovich point-atoms, 

 with their translational inertia, and with, for law of force, zero force 

 at every distance between two points exceeding the sum of the radii 

 of the two balls, and infinite repulsion at exactly this distance. We 

 may use Boscovich similarly for the hollow shell with globule in its 

 interior, and so do away with all question as to vibrations due to 

 elasticity of material, whether of the shell or of the globule. Let us 

 simply suppose the mutual action between the shell and the globule 

 to be nothing except at an instant of collision, and then to be such 

 that their relative component velocity along the radius through the 

 point of contact is reversed by the collision, while the motion of their 

 centre of inertia remains unchanged. 



4. For brevity, we shall call the shell and interior globule of 3, a 

 double molecule, or sometimes, for more brevity, a doublet. The 

 " smooth elastic sphere" of 3 will be called simply an atom, or a 

 single atom ; and the radius or diameter or surface of the atom will 

 mean the radius or diameter or surface of the corresponding sphere. 

 (This explanation is necessary to avoid an ambiguity which might 

 occur with reference to the common expression " sphere of action " of 

 a Boscovich atom.) 



5. Consider now a vast number of atoms and doublets, enclosed in 

 a perfectly rigid fixed surface, having the property of reversing the 

 normal component velocity of approach of any atom or shell or doublet 



