﻿the Partition of Energy, 455 



which conform to the specification is 



M! 



a x ! a 2 ! . . . a m I ' 



and each of these must be weighted with a factor 



u l u % • • • ' »i 



The total number of all the weighted complexions is 

 ai ! a 2 ! . . . 



= (, 1+! . 2+ ...) M = VM, 



by the multinominal theorem. This could have been deduced 

 at once by working direct with probabilities v r /Y instead of 

 weights v y , but the argument has been given in detail to 

 illustrate the method for more complicated cases. 



We next find the average value of a r . This is given by 



M' 

 Qa r = % a a r , vfivf* ... . 



Ui 1 «2 ' • • • 



To sum this expression we only have to cancel a r with the 

 first factor in a r ! in the denominator, and then it is seen to 

 be equal to 



Mv r (v l + v 2 + ...yi-\ 



and so, as is implicit in our assumptions, 



a, = Mt V /V (3'3) 



But we can now go further and find the range over which 

 a r will be likely to fluctuate. This is estimated by averaging 

 the square of the difference of a r from its mean value. We 

 shall throughout this paper describe such a mean square 

 departure as the fluctuation of the corresponding quantity. 

 Thus the fluctuation of a r is (ar — a r ) 2 . Now 



(a r — a r ) 2 = a r (a r — 1) + a r — 2a ~d r -\-~a r 2 , 



and averaging the separate terms by the multinominal 

 theorem, we have 



(a r a r ) — y 2 + y V ' V + \ V / 



= ^(l-v) = -( 1 -M> • • • ^ 



This result represents the fluctuation however large or small 



