Page:Elementary Principles in Statistical Mechanics (1902).djvu/111

CHAPTER VIII.

ON CERTAIN IMPORTANT FUNCTIONS OF THE ENERGIES OF A SYSTEM.

In order to consider more particularly the distribution of a canonical ensemble in energy, and for other purposes, it will be convenient to use the following definitions and notations.

Let us denote by $V$ the extension-in-phase below a certain limit of energy which we shall call $\epsilon$. That is, let

$V=\int \ldots \int dp_{1}\ldots dq_{n},$

(265)

the integration being extended (with constant values of the external coördinates) over all phases for which the energy is less than the limit $\epsilon$. We shall suppose that the value of this integral is not infinite, except for an infinite value of the limiting energy. This will not exclude any kind of system to which the canonical distribution is applicable. For if

$\int \ldots \int e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}$

taken without limits has a finite value,^[1] the less value represented by

$e^{-{\frac {\epsilon }{\Theta }}}\int \ldots \int dp_{1}\ldots dq_{n}$

taken below a limiting value of $\epsilon$, and with the $\epsilon$ before the integral sign representing that limiting value, will also be finite. Therefore the value of $V$, which differs only by a constant factor, will also be finite, for finite $\epsilon$. It is a function of $\epsilon$ and the external coördinates, a continuous increasing

1. This is a necessary condition of the canonical distribution. See Chapter IV, p. 35.