136
MAXIMUM AND MINIMUM PROPERTIES.
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 , , etc. be the probability-coefficients of the original ensembles, and that of the ensemble formed by combining them; and let , , etc. be the numbers of systems in the original ensembles. It is evident that we shall have
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(445)
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where
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(446)
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The main proposition to be proved is that
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or
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If we set
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will be positive, except when it vanishes for
. To prove this, we may regard
and
as any positive quantities. Then
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Since
and
vanish for
, and the second differential coefficient is always positive,
must be positive except when
. Therefore, if
, etc. have similar definitions,
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(449)
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