Page:Elementary Principles in Statistical Mechanics (1902).djvu/58

linear function of the energy, we shall say that the ensemble is canonically distributed, and shall call the divisor of the energy ($\Theta$) the modulus of distribution.

The fractional part of an ensemble canonically distributed which lies within any given limits of phase is therefore represented by the multiple integral

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

(93)

taken within those limits. We may express the same thing by saying that the multiple integral expresses the probability that an unspecified system of the ensemble (i. e., one of which we only know that it belongs to the ensemble) falls within the given limits.

Since the value of a multiple integral of the form (23) (which we have called an extension-in-phase) bounded by any given phases is independent of the system of coördinates by which it is evaluated, the same must be true of the multiple integral in (92), as appears at once if we divide up this integral into parts so small that the exponential factor may be regarded as constant in each. The value of $\psi$ is therefore independent of the system of coördinates employed.

It is evident that $\psi$ might be defined as the energy for which the coefficient of probability of phase has the value unity. Since however this coefficient has the dimensions of the inverse $n$th power of the product of energy and time,^[1] the energy represented by $\psi$ is not independent of the units of energy and time. But when these units have been chosen, the definition of $\psi$ will involve the same arbitrary constant as $\epsilon$, so that, while in any given case the numerical values of $\psi$ or $\epsilon$ will be entirely indefinite until the zero of energy has also been fixed for the system considered, the difference $\psi -\epsilon$ will represent a perfectly definite amount of energy, which is entirely independent of the zero of energy which we may choose to adopt.

1. See Chapter I, p. 19.