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

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

(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