function of ${\displaystyle \epsilon }$, which becomes infinite with ${\displaystyle \epsilon }$, and vanishes for the smallest possible value of ${\displaystyle \epsilon }$, or for ${\displaystyle \epsilon =-\infty }$, if the energy may be diminished without limit.

Let us also set

${\displaystyle \phi =\log {\frac {dV}{d\epsilon }}.}$

(266)

The extension in phase between any two limits of energy, ${\displaystyle \epsilon '}$ and ${\displaystyle \epsilon ''}$, will be represented by the integral

${\displaystyle \int _{\epsilon '}^{\epsilon ''}e^{\phi }\,d\epsilon .}$

(267)

And in general, we may substitute ${\displaystyle e^{\phi }\,d\epsilon }$ for ${\displaystyle dp_{1}\ldots dq_{n}}$ in a ${\displaystyle 2n}$-fold integral, reducing it to a simple integral, whenever the limits can be expressed by the energy alone, and the other factor under the integral sign is a function of the energy alone, or with quantities which are constant in the integration.

In particular we observe that the probability that the energy of an unspecified system of a canonical ensemble lies between the limits ${\displaystyle \epsilon '}$ and ${\displaystyle \epsilon ''}$ will be represented by the integral^[1]

${\displaystyle \int _{\epsilon '}^{\epsilon ''}e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ,}$

(268)

and that the average value in the ensemble of any quantity which only varies with the energy is given by the equation^[2]

${\displaystyle {\overline {u}}=\int _{V=0}^{\epsilon =\infty }ue^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon ,}$

(269)

where we may regard the constant ${\displaystyle \psi }$ as determined by the equation^[3]

${\displaystyle e^{-{\frac {\psi }{\Theta }}}=\int _{V=0}^{\epsilon =\infty }e^{-{\frac {\epsilon }{\Theta }}+\phi }\,d\epsilon ,}$

(270)

In regard to the lower limit in these integrals, it will be observed that ${\displaystyle V=0}$ is equivalent to the condition that the value of ${\displaystyle \epsilon }$ is the least possible.

1. ↑ Compare equation (93).
2. ↑ Compare equation (108).
3. ↑ Compare equation (92).