Page:Elementary Principles in Statistical Mechanics (1902).djvu/228

${\frac {\psi _{\rm {gen}}-\epsilon }{\Theta }}=\eta _{\rm {gen}},$

(549)

which corresponds to the equation

${\frac {\psi -\epsilon }{\Theta }}=\eta ,$

(550)

we have

$\psi _{\rm {gen}}=\psi +\Theta \log {\nu _{1}}!\ldots {\nu _{h}}!,$

and

$\mathrm {H} -\eta _{\rm {gen}}={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi _{\rm {gen}}}{\Theta }}.$

(551)

This will have a maximum when^[1]

${\frac {d\psi _{\rm {gen}}}{d\nu _{1}}}=\mu _{1},\qquad {\frac {d\psi _{\rm {gen}}}{d\nu _{2}}}=\mu _{2},\qquad \mathrm {etc.}$

(552)

Distinguishing values corresponding to this maximum by accents, we have approximately, when $\nu _{1},\ldots \nu _{h}$ are of the same order of magnitude as the numbers of molecules in ordinary bodies,

${\begin{aligned}\mathrm {H} -\eta _{\rm {gen}}&={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi _{\rm {gen}}}{\Theta }}\\&={\frac {\Omega +\mu _{1}\nu _{1}'\ldots +\mu _{h}\nu _{h}'-\psi _{\rm {gen}}'}{\Theta }}\\&\!\!\!\!\!\!\!\!\!\!\!\!\!\!{}-{\bigg (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}^{2}}}{\bigg )}'{\frac {(\Delta \nu _{1})^{2}}{2\Theta }}-{\bigg (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}\,d\nu _{2}}}{\bigg )}'{\frac {\Delta \nu _{1}\Delta \nu _{2}}{\Theta }}\ldots -{\bigg (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}^{2}}}{\bigg )}'{\frac {(\Delta \nu _{h})^{2}}{2\Theta }}\end{aligned}},$

(553)

$e^{\mathrm {H} -\eta _{\rm {gen}}}=e^{C}e^{-{\big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}^{2}}}{\big )}'{\frac {(\Delta \nu _{1})^{2}}{2\Theta }}-{\big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}\,d\nu _{2}}}{\big )}'{\frac {\Delta \nu _{1}\Delta \nu _{2}}{\Theta }}\ldots -{\big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}^{2}}}{\big )}'{\frac {(\Delta \nu _{h})^{2}}{2\Theta }}},$

(554)

where

$C={\frac {\Omega +\mu _{1}\nu _{1}'\ldots +\mu _{h}\nu _{h}'-\psi _{\rm {gen}}'}{\Theta }},$

(555)

and

$\Delta \nu _{1}=\nu _{1}-\nu _{1}',\quad \Delta \nu _{2}=\nu _{2}-\nu _{2}',\quad \mathrm {etc.}$

(556)

This is the probability of the system ($\nu _{1}\ldots \nu _{h}$). The probabilty that the values of $\nu _{1},\ldots \nu _{h}$ lie within given limits is given by the multiple integral

1. Strictly speaking, $\psi _{\rm {gen}}$ is not determined as function of $\nu _{1},\ldots \nu _{h}$, except for integral values of these variables. Yet we may suppose it to be determined as a continuous function by any suitable process of interpolation.