Page:Elementary Principles in Statistical Mechanics (1902).djvu/34

efficient to indicate that they are to be regarded as constant in the differentiation.

We may give to this principle a slightly different expression. Let us call the value of the integral

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

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taken within any limits the extension-in-phase within those limits.

When the phases bounding an extension-in-phase vary in the course of time according to the dynamical laws of a system subject to forces which are functions of the coördinates either alone or with the time, the value of the extension-in-phase thus bounded remains constant. In this form the principle may be called the principle of conservation of extension-in-phase. In some respects this may be regarded as the most simple statement of the principle, since it contains no explicit reference to an ensemble of systems.

Since any extension-in-phase may be divided into infinitesimal portions, it is only necessary to prove the principle for an infinitely small extension. The number of systems of an ensemble which fall within the extension will be represented by the integral

$\int \ldots \int D\,dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}.$

If the extension is infinitely small, we may regard $D$ as constant in the extension and write

$D\int \ldots \int dp_{1}\ldots dp_{n}\,dq_{1}\ldots dq_{n}$

for the number of systems. The value of this expression must be constant in time, since no systems are supposed to be created or destroyed, and none can pass the limits, because the motion of the limits is identical with that of the systems. But we have seen that $D$ is constant in time, and therefore the integral

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