ON OUR KNOWLEDGE OF THERMODYNAMICS. 75 



and therefore 



average value of m'* = average value of ^;^ . . -(IS) 



in accordance with Maxwell's Law. The same thing would also be true 

 if, the equations of the orbits of the particles being 



(j,(x, 2/)= constant, 



/ were a function of <l> as well as of E, or indeed of any integral of the 

 equations of motion other than E, involving co-ordinates only and 7iot 

 velocities. 



Lord Kelvin's ' Decisive Test Case .' 



26. The test case by which Lord Kelvin claims to have ' effectually 

 disposed of ' Maxwell's law of partition ' really confirms all that has 

 been said about the law in this Report. It shows the impossibility of 

 drawing general conclusions as to the distribution of energy in a single 

 system from the possible law of permanent distribution in a large number 

 of systems. In other words, it tells us once more that the mean value of 

 any portion of the energy obtained by integrating with respect to the 

 multiple differential of the co-ordinates and momenta is not necessarily 

 equal to the mean value obtained by integrating with respect to the time 

 for a single system. 



This test case has been criticised in a general sort of way by Mr. E. P. 

 Culverwell,^ and the following investigation will, I think, accord with his 



views : — 



27. The equal masses A and C are supposed to be separated by a 

 ' simple vibrator ' B with which they can collide, and Lord Kelvin assumes 

 that in the course of a large number of collisions this vibrator will equalise 



K p A II B C L 



J y cs ^ ^3 & 1 



and keep equal the average kinetic energies with which A and C rebound 

 from B. C is reflected by a fixed wall at L ; but A, in addition to being 

 stopped by a fixed reflecting wall at K, is acted on by a repulsive force 

 from K while it lies within a certain space, KH. Part of the energy with 

 which A left B then becomes potential while the energy of C always 

 remains kinetic, and Lord Kelvin infers that the average kinetic energy 

 of A is less than that of C. 



Now it is not obvious that the vibrator at B will actually always 

 equalise the average kinetic energies of rebound of A and C. If KB is 

 much greater than BL, C will collide with B much more frequently than 

 A, and I should be inclined to think that without investigation no definite 

 relation could be assumed between the average energies of rebound of 

 A and C. But this is quite irrelevant to the point. Accordingly, let us 

 assume the vibrator to be possessed of the property in question, so that 

 the average energies of A and C are equal. Let x, x' be the co-ordinates 

 of A and C, u, u' their velocities, x the potential energy of A, so that x 

 vanishes when A is outside the region KH. Take each particle of unit 

 mass. 



Then the law of distribution as stated and proved in this Report asserts 



• Phil. Mag., May 1892, p. 466 ; Nature, May 5, 1892. p. 21. 

 » Nature, May 26, 1892, p. 76. 



