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VIII. On the Second Law of Thermodynamics in connexion 

 with the Kinetic Theory of Gases. By S. H. Bukbuky*. 



IN a memoir by Boltzmannf is contained an analytical 

 proof of the second law of thermodynamics, founded on 

 results obtained in the earlier part of the same memoir. The 

 writer's treatment of this part of the subject is not characte- 

 rized by his usual lucidity, and has given much trouble to some 

 mathematicians. It is worth while to show that the second 

 law can be deduced by a simple process from Boltzmann's 

 results. 



A great number, X, of material particles are moving in a 

 closed spherical vessel. Forces of mutual attraction or repul- 

 sion act between each pair of particles and between the par- 

 ticles and the Avails of the vessel. To simplify the case, we 

 will suppose that these are the only forces which act on the 

 particles. 



Boltzmann has shown that tne mean kinetic energy is the 

 same for every particle. Let T be this mean kinetic energy, and 



let T= nj. The mean kinetic enegy of the system is then -^. 



Let the positions of the particles at any instant be defined 

 by reference to three rectangular axes passing through the 

 centre of the vessel, the coordinates of the several particles 

 being x 1 ,y 1> z 1 ,...a K ,y K9 z k . 



Let v be the volume of the vessel. 



The pressure exerted by the system of moving particles on 

 any point on the surface of the sphere may vary from instant 

 to instant according to the positions which the particles hap- 

 pen to have. But if the motion be stationary, this pressure 

 must have on the average of any very long time a certain 

 mean value P per unit area, which in the absence of im- 

 pressed forces will be uniform over the whole surface. 



Then P is the force per unit area which must be exerted on 

 the surface of the sphere from without in order to maintain 

 equilibrium. 



Let x De the ergal for any given positions of the particles 

 of all the forces acting on the particles — that is, of all the forces 

 which act in any manner on the system, except the external 

 force P. 



All those positions of the particles for which the coordinates 

 of the first particle are between #, and x^ + dx^ 



z 1 and z l + dz 1 , 



* Communicated by the Author. 



t Sitzungsberichte der Wiener Akad. vol. lxiii. 



