THERMODYNAMIC ANALOGIES.
177
To make the problem definite, let us consider a system consisting of the original system together with another having the coördinates , , etc., and forces , etc., tending to increase those coördinates. These are in addition to the forces , , etc., exerted by the original system, and are derived from a force-function () by the equations
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For the energy of the whole system we may write
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and for the extension-in-phase of the whole system within any limits
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or
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or again
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since
, when
,
,
,
, etc., are constant. If the limits are expressed by
and
,
and
,
and
, etc., the integral reduces to
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The values of
,
,
,
, etc., which make this expression a maximum for constant values of the energy of the whole system and of the differentials
,
,
, etc., are what may be called the most probable values of
,
, etc., in an ensemble in which the whole system is distributed microcanonically. To determine these values we have
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when
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That is,
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