ENSEMBLE OF SYSTEMS. 83 



Now we have identically 



A l Ai = (Ai 2T) e ) + (2T1 1 -4)> 



where A l ~A^ e denotes the excess of the force (tending to 

 increase a^ exerted by any system above the average of such 

 forces for systems of the same energy. Accordingly, 



But the average value of (A l A^\f) (A^\ e A^) for systems 

 of the ensemble which have the same energy is zero, since for 

 such systems the second factor is constant. Therefore the 

 average for the whole ensemble is zero, and 



Atf. (248) 



In the same way it may be shown that 



(A, - A l ) (e-e) = (^ - AJ (e - e). (249) 



It is evident that in ensembles in which the anomalies of 

 energy e e may be regarded as insensible the same will be 

 true of the quantities represented by A^\ f A^ 



The properties of quantities of the form A^\ will be 

 farther considered in Chapter X, which will be devoted to 

 ensembles of constant energy. 



It may not be without interest to consider some general 

 formulae relating to averages in a canonical ensemble, which 

 embrace many of the results which have been given in this 

 chapter. 



Let u be any function of the internal and external coordi- 

 nates with the momenta and modulus. We have by definition 



**-.>, V: .fc! 



u-J...Jue e d^.^dq, (250) 



phases 



If we differentiate with respect to , we have 

 du f a r/du u u e 



d = J J (35-3 <#--^i 



phases 



