﻿of Radiation in any Material System. 537 



It is well known that dV 2a is a differential invariant. 



At any time we start a number of systems proportional to 

 dV 2 ,i and having their coordinates in the 2n'ple element 

 dV2n* This distribution is invariant and in each system tbe 

 state o£ temperature equilibrium is reached. I have shown 

 (Phil. Mag., Jan. 1911, p. 18) that we need only assume 

 that a finite value of H involves finite momenta. Tben 

 by averaging over all these systems having the same total 

 energy H, it can be shown that any degree o£ freedom 

 has the same kinetic energy as any other. This does not 

 indeed prove that in any one body at a constant 

 temperature there is equipartition o£ the kinetic energy. 

 For conceivably the same total energy // may be distri- 

 buted in different ways between kinetic and potential and 

 result in different temperatures. But it is a result im- 

 possible to reconcile with the fact expressed in such a 

 formula for the complete radiation as M. Planck's. For 

 there it appears that the smaller wave-length has the 

 lesser energy at all temperatures whatever. No process 

 of averaging over bodies in temperature equilibrium can 

 assign the same energy to the two wave-lengths. 



The only escape from this reasoning is to suppose that 

 there are certain values of the coordinates which cannot 

 occur in nature. But the principle on which these are 

 excluded must be dynamical. For example, a vortex theory 

 of matter would allow only such distributions as have 

 throughout the same circulation in corresponding circuits. 

 Unless some such ground for ruling out particular initial 

 conditions can be given, dynamics is confessed to be unequal 

 to the task of bringing about temperature equilibrium. But 

 so far as I am able to see, constant circulation involves merely 

 a reduction in the number of the degrees of freedom. Thus 

 in the case of a mass of liquid moving irrotationallv it 

 makes the motion of any point in the interi®r determinate 

 when the motion of the surface is given. The degrees of 

 freedom of a gravitating spherical volume of water for 

 example, become an enumerable infinity of normal vibrations 

 for small disturbances from the spherical shape. 



Where an atomic theory of matter is retained, it seems 

 impossible to see how any configuration can be excluded, and 

 here dynamics leads to the paradoxes of equipartition. 

 Where matter is continuous it may be pleaded that the 

 contradiction is not certain. But to say that the distribution 

 of energy depends upon the dynamics of a continuous fluid 

 and that no light is thrown upon it by anything that 

 happens in a system having only a finito number of degrees 



