Page:Elementary Principles in Statistical Mechanics (1902).djvu/229

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

(557)

This shows that the distribution of the grand ensemble with respect to the values of $\nu _{1},\ldots \nu _{h}$ follows the "law of errors" when $\nu _{1}',\ldots \nu _{h}'$ are very great. The value of this integral for the limits $\pm \infty$ should be unity. This gives

$e^{C}{\frac {(2\pi \Theta )^{\frac {h}{2}}}{D^{\frac {h}{2}}}}=1,$

(558)

or

$C={\tfrac {1}{2}}\log D-{\frac {h}{2}}\log(2\pi \Theta ),$

(559)

where

$D={\begin{vmatrix}{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}^{2}}}{\Big )}'&{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}\,d\nu _{2}}}{\Big )}'&\ldots &{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}\,d\nu _{h}}}{\Big )}'\\{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{2}\,d\nu _{1}}}{\Big )}'&{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{2}^{2}}}{\Big )}'&\ldots &{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{2}\,d\nu _{h}}}{\Big )}'\\\ldots &\ldots &\ldots &\ldots \\\ldots &\ldots &\ldots &\ldots \\{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}\,d\nu _{1}}}{\Big )}'&{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}\,d\nu _{2}}}{\Big )}'&\ldots &{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}^{2}}}{\Big )}'\end{vmatrix}}$

(560)

that is,

$D={\begin{vmatrix}{\big (}{\frac {d\mu _{1}}{d\nu _{1}}}{\big )}'&{\big (}{\frac {d\mu _{1}}{d\nu _{2}}}{\big )}'&\ldots &{\big (}{\frac {d\mu _{1}}{d\nu _{h}}}{\big )}'&\\{\big (}{\frac {d\mu _{2}}{d\nu _{1}}}{\big )}'&{\big (}{\frac {d\mu _{2}}{d\nu _{2}}}{\big )}'&\ldots &{\big (}{\frac {d\mu _{2}}{d\nu _{h}}}{\big )}'&\\\ldots &\ldots &\ldots &\ldots \\\ldots &\ldots &\ldots &\ldots \\{\big (}{\frac {d\mu _{h}}{d\nu _{1}}}{\big )}'&{\big (}{\frac {d\mu _{h}}{d\nu _{2}}}{\big )}'&\ldots &{\big (}{\frac {d\mu _{h}}{d\nu _{h}}}{\big )}'&\end{vmatrix}}$

(561)

Now, by (553), we have for the first approximation

${\overline {\mathrm {H} }}-{\overline {\eta }}_{\rm {gen}}=C={\tfrac {1}{2}}\log D-{\frac {h}{2}}\log(2\pi \Theta ),$

(562)

and if we divide by the constant $K$,^[1] to reduce these quantities to the usual unit of entropy,

${\frac {{\overline {\mathrm {H} }}-{\overline {\eta }}_{\rm {gen}}}{K}}={\frac {\log D-h\log(2\pi \Theta )}{2K}}$

(563)

1. See page 184-186.