THE CANONICAL DISTRIBUTION.
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where denotes the value of for the modulus . Since the last member of this formula vanishes for , the less value represented by the first member must also vanish for the same value of . Therefore the second member of (359), which differs only by a constant factor, vanishes at the upper limit. The case of the lower limit remains to be considered. Now
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The second member of this formula evidently vanishes for the value of
, which gives
, whether this be finite or negative infinity. Therefore, the second member of (359) vanishes at the lower limit also, and we have
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or
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(362)
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This equation, which is subject to no restriction in regard to the value of
, suggests a connection or analogy between the function of the energy of a system which is represented by
and the notion of temperature in thermodynamics. We shall return to this subject in Chapter XIV.
If , the second member of (359) may easily be shown to vanish for any of the following values of viz.: , , , , where denotes any positive number. It will also vanish, when , for , and when for . When the second member of (359) vanishes, and , we may write
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(363)
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We thus obtain the following equations:
If ,
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(364)
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