ALL SYSTEMS HAVE THE SAME ENERGY. 121 



which shows that d$ dlog V approaches the value unity 

 when n is very great. 



If a system consists of two parts, having separate energies, 

 we may obtain equations similar in form to the preceding, 

 which relate to the system as thus divided.* We shall 

 distinguish quantities rekting to the parts by letters with 

 suffixes, the same letters without suffixes relating to the 

 whole system. The extension-in-phase of the whole system 

 within any given limits of the energies may be represented by 

 the double integral 



taken within those limits, as appears at once from the defini- 

 tions of Chapter VIII. In an ensemble distributed with 

 uniform density within those limits, and zero density outside, 

 the average value of any function of e 1 and e a is given by the 

 quotient 



which may also be written f 



If we make the limits of integration e and e + de, we get the 



* If this condition is rigorously fulfilled, the parts will have no influence 

 on each other, and the ensemble formed by distributing the whole micro- 

 canonically is too arbitrary a conception to have a real interest. The prin- 

 cipal interest of the equations which we shall obtain will be in cases in 

 which the condition is approximately fulfilled. But for the purposes of a 

 theoretical discussion, it is of course convenient to make such a condition 

 absolute. Compare Chapter IV, pp. 35 ff., where a similar condition is con- 

 sidered in connection with canonical ensembles. 



t Where the analytical transformations are identical in form with those 

 on the preceding pages, it does not appear necessary to give all the steps 

 with the same detail. 



