﻿Partition of Energy and Newtonian Mechanics. 389 

 theory, E x is known to be equal to ^mJiO. Thus we have 



BS _ Cl-Jc 



Wi~M ; 



or i£ we use a 1 £=R then we see that 



as as i 



which is the general result expressed by the second law of 

 thermodynamics. 



The present considerations do not therefore affect the truth 

 of the second law of thermodynamics, but this does not 

 appear to justify us in the conclusion that the theorem of 

 equi-partition is also true, as the following analysis shows. 



Suppose that any other part of the energy, say E 2 \ can 

 also be expressed in the same form as E x ', viz. 



Eo' = s/r 



the summation now extending to m 2 terms. The value of 

 V 2 can then be calculated in the same way as Y. lf and now 

 we find 



so that 



as 



BE 2 



1 



" e~ 



E 2 = 



a 2 m 2 k 



OL\ '2 



a 3 

 «i 



??i 2 R 



E 2 



a resultant which is not consistent with the theorem of the 

 equi-partition of energy since the average energy in this 



particular type of coordinate is now only — * -=-, and not 



-n— as in the case of the perfect gas, which forms part of the 



same system. 



It would thus appear that if there is any reason to suppose 

 that the various coordinates, and in particular the coordinates 

 of the Fourier series, are to be differentiated from one another 

 on the lines suggested above, then equi-partition of energy is 

 hardly to be expected ; but any such differentiation in type, 

 although not actually contained in our usual stock of 

 dynamical ideas, is at least as consistent with these ideas as 

 the usual assumption made regarding this point, so that 

 there is every reason for adopting it as a useful, if arbitrary, 

 additional hypothesis to replace the one already in use. 



Applications in Radiation, — In attempting to apply the 

 statistical principles of the preceding paragraph to the 



