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

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

(59)

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

(60)

But since ${\displaystyle F}$ is a homogeneous quadratic function of the differences

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

we have identically

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

That is

${\displaystyle U=k^{n}U_{1},}$

(61)

whence

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

(62)

But if ${\displaystyle k}$ varies, equations (58) and (59) give

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

(63)

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

(64)

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

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

(65)

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

(66)

We may determine the constant of integration by the condition that ${\displaystyle R}$ vanishes with ${\displaystyle k}$. This gives