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 ${\displaystyle \psi }$ 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 ${\displaystyle e^{\phi }}$ is infinite as affording true examples of the microcanonical 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 ${\displaystyle e^{\phi }\,d\epsilon }$ instead of ${\displaystyle dp_{1}\ldots dq_{n}}$, 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 microcanonical 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

${\displaystyle e^{-{\frac {(\epsilon -\epsilon ')^{2}}{\omega ^{2}}}},}$

where ${\displaystyle \omega }$ and ${\displaystyle \epsilon '}$ are constants, and then diminish indefinitely the value of the constant ${\displaystyle \omega }$. Here the density is nowhere zero until we come to the limit, but at the limit it is zero for all energies except ${\displaystyle \epsilon '}$. We thus avoid the analytical complication 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 (${\displaystyle \epsilon }$) is constant, but the kinetic energy (${\displaystyle \epsilon _{p}}$) and the potential energy