and the Partition of Energy in Continuous Media. 245 



Thus £ 1? £ 2 > ••• are distributed according to the law of trial 

 and error, and the mean values o£ their squares are given by 



The energy of the trains of waves being, as has been seen 

 in equation (26), 



jfew+tf +■».+«." 



it now follows that the average value of each term is £RT. 



20. Thus we have seen that in the normal state the random 

 motions and positions of the molecules result in certain 

 departures from uniformity, both of density and of momen- 

 tum. We have seen how these departures from uniformity 

 can be regarded as due to regular trains of waves, and have 

 effected their analysis into such trains of waves. 



In any general motion of the medium, the kinetic and 

 potential energies T V are given by 



2T=« 1 1 9 +* 2 2 + ... 



where </>i, <£ 2 -.- measure the amplitudes of different trains 

 of waves. We have found that in the normal state the 

 average value of each of the terms 



is the same, namely qRT, the value given by the theorem 

 of equipartition of energy. Incidentally, we have j also 

 verified that the values of d> u (j> 2 , ... <£i, ••• are ranged round 

 their mean values according to the law of trial and error, as 

 they ought to be. 



To put the matter in another way, we have found that the 

 law of distribution 



A e-hmW+vf+wS+u?*...)^ ^ ^ dfi2 ^ dyi d ^ dj , 2 



is identical with, and may be transformed into, the law of 

 distribution 



A , ,-***&+&+ ... +a^»+/w+ -)4 i 4 2 ... #1 .... 



The former law regards the energy as that of a system of 

 Phil. Mag. S. 6. Vol. 17. No. 98. Feb. 1909. S 



