﻿456 Messrs. C. G. Darwin and R. H. Fowler on 



v r may be. In all cases we have the result that (a r — a r ) 2 is 

 less than a~ r , and therefore that the average departure of 

 a r from ~a r is of order (o^)*. We can also interpret this fact 

 by saying that departures of a r from li r which are much 

 greater than (a7-)* will be relatively rare ; as M is large and 

 (or)i small compared with oT r , this is precisely equivalent to 

 saying that the possession of the average value of a r is a 

 normal property of the assembly in the sense used by Jeans *« 

 We have thus a simple and complete proof that uniform 

 density is a normal property of this assembly. 



§ 4. The Distribution of Energy among a Set of 

 Planck Vibrators. 



Another case where the treatment can be almost entirely 

 algebraic is that of the partition of energy among a set of 

 Planck vibrators which all have the same frequency. Let e 

 be the unit of energy so that every vibrator can have any 

 multiple of e. As we saw in § 2, the weight attached to every 

 state is to be taken as unity. 



Let there be M vibrators and let there be Pe of energy 

 (P is an integer) to be partitioned among them. To specify 

 a statistical state, let a be the number of vibrators with no 

 energy, a x with e, a 2 with 2e, etc. Then we have 



a + a 1 + a 2 + a 3 + ... = 'M., .... (4'1) 



a 1 + 2a 2 + 3a 3 +... = P, .... (4'2) 



and anv set of a's which satisfy these equations corresponds 

 to a possible state of the assembly. By the principles of § 2 

 each of the complexions will have unit weight. Now count 

 up the number of complexions corresponding to the speci- 

 fication. By considering the various permutations of the 

 vibrators, it is seen to be 



M! 



a \ ail a 2 l 



(4-3) 



We must next find C the total number of all possible 

 complexions. Let X a denote summation over all possible 

 values of the a's which satisfy (4*1) and (4*2). Then 



a ! aj : a 2 l ... 

 Consider the infinite series 



(i+s + si +5 3 + ; >)M - 



* Jeans, ' Dynamical Theory of Gases/ passim. 



