Theorem V. If an ensemble of systems is so distributed in phase that the index of probability is a linear function of ${\displaystyle F_{1}}$, ${\displaystyle F_{2}}$, etc., (these letters denoting functions of the phase,) the average value of the index is less than for any other distribution in which the functions ${\displaystyle F_{1}}$, ${\displaystyle F_{2}}$, etc. have the same average values.

Theorem VI. The average value in an ensemble of systems of ${\displaystyle \eta +F}$ (where ${\displaystyle \eta }$ denotes as usual the index of probability and ${\displaystyle F}$ any function of the phase) is less when the ensemble is so distributed that ${\displaystyle \eta +F}$ is constant than for any other distribution whatever.

Theorem VII. If a system which in its different phases constitutes an ensemble consists of two parts, and we consider the average index of probability for the whole system, and also the average indices for each of the parts taken separately, the sum of the average indices for the parts will be either less than the average index for the whole system, or equal to it, but cannot be greater. The limiting case of equality occurs when the distribution in phase of each part is independent of that of the other, and only in this case.

Let the coördinates and momenta of the whole system be ${\displaystyle q_{1}\ldots q_{n},p_{1},\ldots p_{n}}$, of which ${\displaystyle q_{1}\ldots q_{m},p_{1},\ldots p_{m}}$ relate to one part of the system, and ${\displaystyle q_{m+1}\ldots q_{n},p_{m+1},\ldots p_{n}}$ to the other. If the index of probability for the whole system is denoted by ${\displaystyle \eta }$, the probability that the phase of an unspecified system lies within any given limits is expressed by the integral

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

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taken for those limits. If we set

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

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where the integrations cover all phases of the second system, and

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

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