Page:Elementary Principles in Statistical Mechanics (1902).djvu/48

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CONSERVATION OF EXTENSION-IN-PHASE

R=\mathop {\int \ldots \int } ^{F=k}Ce^{-F}\,dp_{1}\ldots dq_{n},

(59)

U_{1}=\mathop {\int \ldots \int } ^{F=1}dp_{1}\ldots dq_{n}.

(60)

But since

F

is a homogeneous quadratic function of the differences

p_{1}-P_{1},p_{2}-P_{2},\ldots q_{n}-Q_{n},

we have identically

{\begin{aligned}&\mathop {\int \ldots \int } ^{F=k}d(p_{1}-P_{1})\ldots d(q_{n}-Q_{n})\\&=\mathop {\int \ldots \int } ^{kF=k}k^{n}\,d(p_{1}-P_{1})\ldots d(q_{n}-Q_{n})\\&=k^{n}\mathop {\int \ldots \int } ^{F=1}d(p_{1}-P_{1})\ldots d(q_{n}-Q_{n}).\end{aligned}}

That is

U=k^{n}U_{1},

(61)

whence

dU=U_{1}nk^{n-1}\,dk.

(62)

But if

k

varies, equations (58) and (59) give

dU=\mathop {\int \ldots \int } _{F=k}^{F=k+dk}dp_{1}\ldots dq_{n}

(63)

dR=\mathop {\int \ldots \int } _{F=k}^{F=k+dk}Ce^{-F}\,dp_{1}\ldots dq_{n}

(64)

Since the factor $Ce^{-F}$ has the constant value $Ce^{-k}$ in the last multiple integral, we have

dR=Ce^{-k}dU=CU_{1}ne^{-k}k^{n-1}\,dk,

(65)

R=-CU_{1}n!e^{-k}\left(1+k+{\frac {k^{2}}{2}}+\ldots +{\frac {k^{n-1}}{(n-1)!}}\right)+\mathrm {const.}

(66)

We may determine the constant of integration by the condition that

R

vanishes with

k

. This gives