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

88 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 The extension in phase between any two limits of energy, $$\epsilon'$$ and $$\epsilon''$$, will be represented by the integral  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 and that the average value in the ensemble of any quantity which only varies with the energy is given by the equation  where we may regard the constant $$\psi$$ as determined by the equation  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.