of the probability-coefficients of the original ensembles, the average index of probability of the resulting ensemble cannot be greater than the same linear function of the average indices of the original ensembles. It can be equal to it only when the original ensembles are similarly distributed in phase.

Let ${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, etc. be the probability-coefficients of the original ensembles, and ${\displaystyle P}$ that of the ensemble formed by combining them; and let ${\displaystyle N_{1}}$, ${\displaystyle N_{2}}$, etc. be the numbers of systems in the original ensembles. It is evident that we shall have

${\displaystyle P=c_{1}P_{1}+c_{2}P_{2}+\mathrm {etc.} =\Sigma (c_{1}P_{1}),}$

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where

${\displaystyle c_{1}={\frac {N_{1}}{\Sigma N_{1}}},\qquad c_{2}={\frac {N_{2}}{\Sigma N_{1}}},\qquad \mathrm {etc.} }$

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The main proposition to be proved is that

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P\log P\,dp_{1}\ldots dq_{n}\leq \Sigma {\Bigg [}c_{1}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}P_{1}\log P_{1}\,dp_{1}\ldots dq_{n}{\Bigg ]}}$

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or

${\displaystyle \mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}[\Sigma (c_{1}P_{1}\log P_{1})-P\log P]\,dp_{1}\ldots dq_{n}\geq 0.}$

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If we set

${\displaystyle Q_{1}=P_{1}\log P_{1}-P_{1}\log P-P_{1}+P}$

${\displaystyle Q_{1}}$ will be positive, except when it vanishes for ${\displaystyle P_{1}=P}$. To prove this, we may regard ${\displaystyle P_{1}}$ and ${\displaystyle P}$ as any positive quantities. Then

${\displaystyle {\bigg (}{\frac {dQ_{1}}{dP_{1}}}{\bigg )}_{P}=\log P_{1}-\log P,}$

${\displaystyle {\bigg (}{\frac {d^{2}Q_{1}}{dP_{1}{}^{2}}}{\bigg )}_{P}={\frac {1}{P_{1}}}.}$

Since ${\displaystyle Q_{1}}$ and ${\displaystyle dQ_{1}/dP_{1}}$ vanish for ${\displaystyle P_{1}=P}$, and the second differential coefficient is always positive, ${\displaystyle Q_{1}}$ must be positive except when ${\displaystyle P_{1}=P}$. Therefore, if ${\displaystyle Q_{2}}$, etc. have similar definitions,

${\displaystyle \Sigma (c_{1}Q_{1})\geq 0.}$

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