116 A PERMANENT DISTRIBUTION IN WHICH 



this process fails to define a limiting distribution in any such 

 distinct sense as for other values of the energy. The difficulty 

 is not in the process, but in the nature of the case, being 

 entirely analogous to that which we meet when we try to find 

 a canonical distribution in cases when ^ becomes infinite. 

 We have not regarded such cases as affording true examples 

 of the canonical distribution, and we shall not regard the cases 

 in which e^ is infinite as affording true examples of the micro- 

 canonical distribution. We shall in fact find as we go on that 

 in such cases our most important formulae become illusory. 



The use of formulae relating to a canonical ensemble which 

 contain e^de instead of dp l . . . dq n , as in the preceding chapters, 

 amounts to the consideration of the ensemble as divided into 

 an infinity of microcanonical elements; 



From a certain point of view, the microcanonical distribution 

 may seem more simple than the canonical, and it has perhaps 

 been more studied, and been regarded as more closely related 

 to the fundamental notions of thermodynamics. To this last 

 point we shall return in a subsequent chapter. It is sufficient 

 here to remark that analytically the canonical distribution is 

 much more manageable than the microcanonical. 



We may sometimes avoid difficulties which the microcanon- 

 ical distribution presents by regarding it as the result of the 

 following process, which involves conceptions less simple but 

 more amenable to analytical treatment. We may suppose an 

 ensemble distributed with a density proportional to 



where &> and e 1 are constants, and then diminish indefinitely 

 the value of the constant &>. Here the density is nowhere 

 zero until we come to the limit, but at the limit it is zero for 

 all energies except e'. We thus avoid the analytical compli- 

 cation of discontinuities in the value of the density, which 

 require the use of integrals with inconvenient limits. 



In a microcanonical ensemble of systems the energy (e) is 

 constant, but the kinetic energy (e^) and the potential energy 



