658 Mr. G. H. Livens on the 



The average amount of energy in the coordinate of the 

 first type is thus 



and for those of the other types it is 



HKP— 1 



■ = 1,2, ...r) 



In the limit when e is very small the average amount in the 

 first type of coordinate is correctly JcT ; it will be ot s kT * in 

 the other types unless <x s is very small, in which case the 

 amount per coordinate is very small but of the form 



l kT -l 



wherein T . e 



rj s = ut — . 



e=0 *s 



This is the modified form of the general theorem of the 

 partition of energy in the steady thermal state, which must 

 replace the theorem quoted by Jeans and Rayleigh, when 

 the application is to problems of the nature of those sketched 

 above. Equi-partition no longer exists, and however ideal 

 the system, could never possibly be attained, not however 

 for any peculiar physical reason, but just simply because of 

 the mathematical (and physical) impossibility of such a 

 state. 



Jeans contends f that, since equi-partition of energy does 

 not exist among all the coordinates, including those specifying 

 the oscillations, a steady state of thermal equilibrium is never 

 reached, a doctrine which, as he admits, would entirely re- 

 move the foundations of the application of thermodynamic 

 principles to this subject. It must, however, be noticed that 

 equi-partition is a test of thermal equilibrium only when all 

 possible types of coordinate are equally probable. If, as 

 suggested above, the various types of coordinates ultimately 



* This confirms our general interpretation of the extent or limitation 

 of a coordinate imposed by the analytical conditions. 



t See his papers " Temperature radiation and the partition of energy 

 in continuous media," Phil. Mag. [6] xvii. p. 219 (1909) ; " The motion 

 of electrons in solids," ibid. [6] xvii. p, 773 (1909), & xviii. p. 209 (1909). 



