﻿the Partition of Energy. 467 



expression to a complex integral, we have 



which, by virtue of the value of C from (5' 71) and the 

 argument of § 6, at once yields 



a;=Mr(i-y), 



=M*-«/**(l-e-*/* T ) J . . . . (8-3) 



which is the formula of § 4 over again, the presence of the 

 B's being immaterial. 



When we come to evaluate fluctuations the matter is a 

 little more complicated, because the leading terms cut out, 

 and so the second term of the asymptotic expansion will in 

 general play a part. For example, consider the fluctuations 

 of a_ : 



( a r- a r) 2 = a X a r- 1 ) + ^-<X) 2 ' 



By arguments exactly similar to those above, we have 



C«„( 



a.-l^MCM-D^j^Tw 



(1_/)N- 2 (1_^)*" 



and so by (6-2) 



«>,- 1) =M(M-ip 2 " £ (l-y) 2 -U+ 0(1/E) }. 



Thus the fluctuation is 



^_M3 2r6 (l-3 e ) 2 + 0(M 2 /E). 



This is sufficient to show that the possession of ~a~ r (8'3) is a 

 normal property of the assembly. 



The complete calculation of the O-term is rather com- 

 plicated ; the result is given at the end of this section. But 

 a great simplification arises if we suppose that there are 

 many more B's than A's, while E is so adjusted that $ the 

 temperature is unchanged. In this case the term 0(M 2 /E) 

 becomes small and may be neglected. We shall describe 

 this case by saying that A is in a bath of temperature S. 

 Then, provided this is so, we have 



(^EJ=a r Q--aJ-M.) (8-4) 



A much more important quantity is the fluctuation of E A . 

 This is found by evaluating E A 2 . Now, just as CE A was 

 given by operating with zd/dz on the first factor in 



2R2 



