Page:Elementary Principles in Statistical Mechanics (1902).djvu/112

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

Let us also set

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

(266)

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

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

(267)

And in general, we may substitute $e^{\phi }\,d\epsilon$ for $dp_{1}\ldots dq_{n}$ in a $2n$-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 $\epsilon '$ and $\epsilon ''$ will be represented by the integral^[1]

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

(268)

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

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

(269)

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

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

(270)

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

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