﻿466 Messrs. 0. G. Darwin and R. H. Fowler on 



§ 7. The Meaning of '&. 



Returning to the subject of § 5, we see that the partition 

 of energy between the A's and the B's depends on a para- 

 meter which can be determined in terms of" the energ}- by 

 means of the equation (5*8). It is fairly evident that 3 is 

 connected with the temperature, though it requires more 

 general considerations to prove this properly. But if we 

 assume this connexion and are content to replace the thermo- 

 dynamic definition of absolute temperature by one based on 

 the law of equipartition of energy for systems obeying the 

 laws of classical mechanics, then we can at once identify 

 the meaning of $. 



For let us suppose that the B's are vibrators of very low 

 frequency. They will then obey the classical laws, and the 

 average energy of each will be kT. But 



Tit v — — £T 



^Js-i-i.-iogCi/*)-*- 1 ' 



which shows that 



*=«-^ T (7-1) 



Substituting this in (5*9), we obtain the well-known form 



Me 



E A = 



e e/kT 



Observe that while rj is tending to zero there need be no 

 difficulty about the condition that rj and e are to be com- 

 mensurable. We shall later return to this question of 

 temperature and establish it for much more general types o£ 



system. 



§ 8. Other mean values. 



Exactly the same methods can be applied to evaluate any 

 other mean value besides E A . For example : — 



p - _ v jvn ni 



Ua r-A,» a r^j ai !. ... 6 ! b, !....' 



_ (M-l)l N! 



- M ^WUi'!.... MM-.:..:' * ( } 



summed over all zero and positive values such that 



X r a r ' = M-l, 2A=N, % r r*a r ' + 2,sf,b=$-re. 

 Applying the multinomial theorem and reducing the 



