ALL SYSTEMS HAVE THE SAME ENERGY. 123 



and s 5l =^\ = ^ (395) 



de | rfej J0 rfe 2 |e 



We have compared certain functions of the energy of the 

 whole system with average values of similar functions of 

 the kinetic energy of the whole system, and with average 

 values of similar functions of the whole energy of a part of 

 the system. We may also compare the same functions with 

 average values of the kinetic energy of a part of the system. 



We shall express the total, kinetic, and potential energies of 

 the whole system by e, e p , and e g , and the kinetic energies of the 

 parts by e^, and e 2p . These kinetic energies are necessarily sep- 

 arate : we need not make any supposition concerning potential 

 energies. The extension-in-phase within any limits which can 

 be expressed in terms of e g , e^, e zp may be represented in the 

 notations of Chapter VIII by the triple integral 



taken within those limits. And if an ensemble of systems is 

 distributed with a uniform density within those limits, the 

 average value of any function of e q , e^, e^ will be expressed 

 by the quotient 



fffue^ded V Zp d V q 



or 



To get the average value of u for a microcanonical distribu- 

 tion, we must make the limits e and e + de. The denominator 

 in this case becomes e^ de, and we have 



C 2p =C Cq 



(396) 



