ALL SYSTEMS HAVE THE SAME ENERGY. 117 



(e q ) vary in the different systems, subject of course to the con- 

 dition 



p -f e q = e = constant. (373) 



Our first inquiries will relate to the division of energy into 

 these two parts, and to the average values of functions of e p 

 and e q . 



We shall use the notation y\ 6 to denote an average value in 

 a microcanonical ensemble of energy e. An average value 

 in a canonical ensemble of modulus (D, which has hitherto 

 been denoted by M, we shall in this chapter denote by '^@, to 

 distinguish more clearly the two kinds of averages. 



The extension-in-phase within any limits which can be given 

 in terms of e p and e q may be expressed in the notations of the 

 preceding chapter by the double integral 



*dV p dVq 



taken within those limits. If an ensemble of systems is dis- 

 tributed within those limits with a uniform density-in-phase, 

 the average value in the ensemble of any function (u) of the 

 kinetic and potential energies will be expressed by the quotient 



of integrals 



/ r 



udVpdVq 



dVpdVq 



Since d V p = e^ p de p , and de p = de when e q is constant, the 

 expression may be written 



To get the average value of u in an ensemble distributed 

 microcanonically with the energy 6, we must make the in- 

 tegrations cover the extension-in-phase between the energies 

 e and e + de. This gives 



