On the Law of Partition of Kinetic Energy. 99 



" B being a simple vibrator controlled by a spring so stiff 

 that when, at any time, it has very nearly the whole energy 

 of the system, its extreme excursions on each side o£ its 

 position of equilibrium are small : 



" C and A, equal masses : 



" C, unacted upon by force except when it strikes L, a fixed 

 barrier, and when it strikes or is struck by B : 



"A, unacted on by force except when it strikes or is struck 

 by B, and when it is at less than a certain distance, HK, from 

 a fixed repellent barrier, K, repelling with a force, F, varying 

 according to any law, or constant, when A is between K 

 and H, but becoming infinitely great when (if at any time) 

 A reaches K, and goes infinitesimally beyond it. 



" Suppose now A, B, G to be all moving to and fro. 

 The collisions between B and the equal bodies A and on 

 its two sides must equalize, and keep equal, the average 

 kinetic energy of A, immediately before and after these 

 collisions, to the average kinetic energy of C. Hence, when 

 the times of A being in the space between H and K are 

 included in the average, the average of the sum of the potential 

 and kinetic energies of A is equal to the average kinetic 

 energy of C. But the potential energy of A at every point 

 in the space HK is positive, because, according to our suppo- 

 sition, the velocity of A is diminished during every time of 

 its motion from H towards K, and increased to the same value 

 again during motion from K to H. Hence, the average 

 kinetic energy of A is less than the average kinetic energy 

 of C ! " 



The apparent disproof of the law of partition of energy in 

 this simple problem seems to have shaken the faith even of 

 such experts as Dr. Watson and Mr. Burbury * M. Poincare, 

 however, considering a special case of Lord Kelvin's pro- 

 blem f, arrives at a conclusion in harmony with Maxwell's 

 law. Prof. Bryan J considers that the test-case "shows the im- 

 possibility 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." It is indeed 

 true that Maxwell's theorem relates in the first instance to a 

 large number of systems; but, as I shall show more fully 

 later, the extension to the time-average for a single system 

 requires only the application o£ Maxwell'' s assumption that all 

 phases, i. e. all states, defined both in respect to configuration 

 and velocity, which are consistent with the energy condition 



* Nature, vol. xlvi. p. 100 (1892). 



t Revue yener ale cles Sciences, July 1894. 



| " Report on Thermodynamics," Part II. § 26. Brit. Ass. Rep. 1894. 



H2 



