Page:Elementary Principles in Statistical Mechanics (1902).djvu/42

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

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where the limits in the two cases are formed by corresponding phases. When the integrations cover infinitely small variations of the momenta and coördinates, we may regard $P'$ and $P''$ as constant in the integrations and write

$P'\int \ldots \int \,dp_{1}'\ldots dq_{n}''=P'\int \ldots \int \,dp_{1}''\ldots dq_{n}''.$

Now the principle of the conservation of extension-in-phase, which has been proved (viz., in the second demonstration given above) independently of any reference to an ensemble of systems, requires that the values of the multiple integrals in this equation shall be equal. This gives

$P''=P'.$

With reference to an important class of cases this principle may be enunciated as follows.

When the differential equations of motion are exactly known, but the constants of the integral equations imperfectly determined, the coefficient of probability of any phase at any time is equal to the coefficient of probability of the corresponding phase at any other time. By corresponding phases are meant those which are calculated for different times from the same values of the arbitrary constants of the integral equations.

Since the sum of the probabilities of all possible cases is necessarily unity, it is evident that we must have

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P\,dp_{1}\ldots dq_{n}=1,$

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where the integration extends over all phases. This is indeed only a different form of the equation

$N=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}D\,dp_{1}\ldots dq_{n},$

which we may regard as defining $N$.