Systems, and Planck's Theory of Radiation. 919 



Hence we have E 3 = »iR,T/s. Thus in any part of the 

 energy which is expressible as a homogeneous and necessarily 

 positive function of the co-ordinates, the average energy of 

 any m co-ordinates is proportional to m and to T ; but this is 

 exactly the theorem of equipartition of energy. 



Experimental knowledge of wave-motion seems to place it 

 beyond question that the energies of waves of different 

 frequencies must be represented by different sets of co- 

 ordinates, and that each energy must be necessarily positive. 

 If this is granted, the necessity for equipartition of energy 

 between the different waves follows, and in a state of maxi- 

 mum entropy the total radiant energy must always be 

 proportional to the temperature. 



This establishes the main proposition of the present paper. 

 It may in addition be of interest to examine in detail the 

 form assumed by the general argument when it is applied 

 to the special problem under discussion. 



Special Investigation of Wave-motion. 



8. The system to which we shall now confine our attention 

 will consist of a volume of aether in which a very small 

 amount of matter is embedded, the function of the matter 

 being solely to make possible the transfer of energy in the 

 aether between vibrations of different wave-lengths. Let 

 there be supposed to be n vibrations in the aether, and let 

 the co-ordinates of the 5th vibration be Q g and R . Let the 

 number of additional degrees of freedom introduced by the 

 presence of the matter be m, and let a typical one of these 

 co-ordinates be S r . 



We shall suppose that 2m, the number of co-ordinates 

 associated with the matter, is infinitesimal in comparison 

 with 2n, the number associated with the aether. It will also 

 be assumed that m is so small that, for all configurations 

 which are of importance, the energy residing in the matter 

 is negligible in comparison with that residing in the aether. 



For this system equation (1) assumes the form 



?(&♦&--!.&•••« 



In this equation the number of terms on the right is small 

 compared with that on the left. If all matter were entirely 

 absent the terms on the right would vanish altogether, and 

 since the waves of different periods would then become 

 independent dynamical systems, the terms on the left would 



