Page:Elementary Principles in Statistical Mechanics (1902).djvu/99

posing that there were means of measuring these quantities separately) had each separately uniform values.^[1] Exceptions might occur when for particular values of the modulus the differential coefficient $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ takes a very large value. To human observation the effect would be, that in ensembles in which $\Theta$ and ${\overline {\epsilon }}_{p}$ had certain critical values, ${\overline {\epsilon }}_{q}$ would be indeterminate within certain limits, viz., the values which would correspond to values of $\Theta$ and $\epsilon _{p}$ slightly less and slightly greater than the critical values. Such indeterminateness corresponds precisely to what we observe in experiments on the bodies which nature presents to us.^[2]

To obtain general formulae for the average values of powers of the energies, we may proceed as follows. If $h$ is any positive whole number, we have identically

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon ^{h}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}=\Theta ^{2}{\frac {d}{d\Theta }}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon ^{h-1}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n},$

(214)

i. e., by (108),

${\overline {\epsilon ^{h}}}e^{-{\frac {\psi }{\Theta }}}=\Theta ^{2}{\frac {d}{d\Theta }}{\bigg (}{\overline {\epsilon ^{h-1}}}e^{-{\frac {\psi }{\Theta }}}{\bigg )}.$

(215)

Hence

${\overline {\epsilon ^{h}}}e^{-{\frac {\psi }{\Theta }}}={\bigg (}\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h}e^{-{\frac {\psi }{\Theta }}},$

(216)

and

${\overline {\epsilon ^{h}}}=e^{\frac {\psi }{\Theta }}{\bigg (}\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h}e^{-{\frac {\psi }{\Theta }}}.$

(217)

1. This implies that the kinetic and potential energies of individual systems would each separately have values sensibly constant in time.
2. As an example, we may take a system consisting of a fluid in a cylinder under a weighted piston, with a vacuum between the piston and the top of the cylinder, which is closed. The weighted piston is to be regarded as a part of the system. (This is formally necessary in order to satisfy the condition of the invariability of the external coördinates.) It is evident that at a certain temperature, viz., when the pressure of saturated vapor balances the weight of the piston, there is an indeterminateness in the values of the potential and total energies as functions of the temperature.