Page:Elementary Principles in Statistical Mechanics (1902).djvu/161

Rh But since and   This proves (448), and shows that the sign $$=$$ will hold only when  for all phases, i. e., only when the distribution in phase of the original ensembles are all identical.

Theorem IX. A uniform distribution of a given number of systems within given limits of phase gives a less average index of probability of phase than any other distribution.

Let $$\eta$$ be the constant index of the uniform distribution, and $$\eta+\Delta\eta$$ the index of some other distribution. Since the number of systems within the given limits is the same in the two distributions we have where the integrations, like those which follow, are to be taken within the given limits. The proposition to be proved may be written or, since $$\eta$$ is constant,  In (451) also we may cancel the constant factor $$e^\eta$$, and multiply by the constant factor $$(\eta+1)$$. This gives The subtraction of this equation will not alter the inequality to be proved, which may therefore be written