[From the Cambridge Philosophical Society's Transactions, Vol. xii.] 



XCIV. On Boltzmann's Theorem on the average distribution of energy in a system 



of material points. 



LUDWIG BOLTZMANN, in his "Studien liber das Gleichgewiclit der 

 lebendigen Kraft zwisclien bewegten materiellen Punkten " [Sitzb. d. k. Akad. 

 Wien, Bd. LVIII., 8 Oct. 1868], has devoted his third section to the general 

 solution of the problem of the equilibrium of kinetic energy among a finite 

 number of material points. His method of treatment is ingenious, and, as far 

 as I can see, satisfactory, but I think that a problem of such primary importance 

 in molecular science ought to be scrutinized and examined on every side, so 

 that as many persons as possible may be enabled to follow the demonstration, 

 and to know on what assumptions it rests. This is more especially necessary 

 when the assumptions relate to the degree of irregularity to be expected in 

 the motion of a system whose motion is not completely known. 



Mr H. W. Watson, in his Treatise on the Kinetic Theory of Gases*, has 

 developed with great clearness the steps of the investigation of the distribution 

 of energy among a set of particles which are supposed to act on each other 

 only at very small distances. The particles may be acted on by external forces 

 such as gravity, but it is expressly stipulated that the time during which a 

 particle is encountering other particles is very small compared with the tune 

 during which there is no sensible action between it and other particles ; and 

 also that the time during which a particle is simultaneously within the distance 

 of molecular action of more than one other particle may be neglected. 



Now this method of treating the question, however necessary it may be 

 in the subsequent investigation of the processes of diffusion, &c. in gases, is 

 inapplicable to the theory of the equilibrium of temperature in liquids and 



* Clarendon Press Series, 1876. 



VOL. ii. 90 



