Page:Elementary Principles in Statistical Mechanics (1902).djvu/158

where the integrations cover all phases of the first system, the integral (431) will reduce to the form

$\int \ldots \int e^{\eta _{1}}\,dp_{1}\ldots dp_{m}\,dq_{1}\ldots dq_{m},$

(434)

when the limits can be expressed in terms of the coördinates and momenta of the first part of the system. The same integral will reduce to

$\int \ldots \int e^{\eta _{2}}\,dp_{m+1}\ldots dp_{n}\,dq_{m+1}\ldots dq_{n},$

(435)

when the limits can be expressed in terms of the coördinates and momenta of the second part of the system. It is evident that $\eta _{1}$ and $\eta _{2}$ are the indices of probability for the two parts of the system taken separately.

The main proposition to be proved may be written

${\begin{aligned}\int \ldots \int \eta _{1}e^{\eta _{1}}\,dp_{1}\ldots dq_{m}+&\int \ldots \int \eta _{2}e^{\eta _{2}}\,dp_{m+1}\ldots dq_{n}\leq \\&\qquad \int \ldots \int \eta e^{\eta }\,dp_{1}\ldots dq_{n},\end{aligned}}$

(436)

where the first integral is to be taken over all phases of the first part of the system, the second integral over all phases of the second part of the system, and the last integral over all phases of the whole system. Now we have

$\int \ldots \int e^{\eta }\,dp_{1}\ldots dq_{n}=1,$

(437)

$\int \ldots \int e^{\eta _{1}}\,dp_{1}\ldots dq_{m}=1,$

(438)

and

$\int \ldots \int e^{\eta _{2}}\,dp_{m+1}\ldots dq_{n}=1,$

(439)

where the limits cover in each case all the phases to which the variables relate. The two last equations, which are in themselves evident, may be derived by partial integration from the first.