OF THE ENERGIES OF A SYSTEM.
91
the integration being extended, with constant values of the coördinates, both internal and external, over all values of the momenta for which the kinetic energy is less than the limit . will evidently be a continuous increasing function of which vanishes and becomes infinite with . Let us set
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(277)
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The extension-in-velocity between any two limits of kinetic energy
and
may be represented by the integral
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(278)
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And in general, we may substitute
for
or
in an
-fold integral in which the coördinates are constant, reducing it to a simple integral, when the limits are expressed by the kinetic energy, and the other factor under the integral sign is a function of the kinetic energy, either alone or with quantities which are constant in the integration.
It is easy to express and in terms of . Since is function of the coördinates alone, we have by definition
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(279)
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the limits of the integral being given by
. That is, if
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(280)
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the limits of the integral for
, are given by the equation
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(281)
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and the limits of the integral for
, are given by the equation
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(282)
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But since
represents a quadratic function, this equation may be written
|
(283)
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