Page:Elementary Principles in Statistical Mechanics (1902).djvu/136

${\overline {(e^{\phi }-{\overline {e^{\phi }}}){\bigg (}{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {e^{\phi }{\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {e^{\phi }}}{\Theta }}=-{\overline {e^{\phi }{\frac {d\phi }{d\epsilon }}}},$

(365)

or

$2{\overline {e^{\phi }{\frac {d\phi }{d\epsilon }}}}={\overline {e^{\phi }}}{\overline {\frac {d\phi }{d\epsilon }}}={\frac {\overline {e^{\phi }}}{\Theta }},$

(366)

${\overline {(\epsilon -{\overline {\epsilon }}){\bigg (}{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {\epsilon {\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {\epsilon }}{\Theta }}=-1,$

[1](367)

${\overline {(\epsilon ^{m}-{\overline {\epsilon ^{m}}}){\bigg (}{\frac {d\phi }{d\epsilon }}-{\frac {1}{\Theta }}{\bigg )}}}={\overline {\epsilon ^{m}{\frac {d\phi }{d\epsilon }}}}-{\frac {\overline {\epsilon ^{m}}}{\Theta }}=-m{\overline {\epsilon ^{m-1}}}.$

(368)

If $n>4$,

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

[2](369)

If $n>2h$,

${\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}{\frac {d\phi }{d\epsilon }}}}-{\frac {1}{\Theta }}{\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}}}={\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}{\frac {d\phi }{d\epsilon }}}}-{\overline {e^{-\phi }{\frac {d^{h+1}V}{d\epsilon ^{h+1}}}}}$

or

${\overline {e^{-\phi }{\frac {d^{h+1}V}{d\epsilon ^{h+1}}}}}={\frac {1}{\Theta }}{\overline {e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}}}$

(370)

whence

${\overline {e^{-\phi }{\frac {d^{h+1}V}{d\epsilon ^{h+1}}}}}={\frac {1}{\Theta ^{h}}}.$

(371)

Giving $h$ the values 1, 2, 3, etc., we have

${\overline {\frac {d\phi }{d\epsilon }}}={\frac {1}{\Theta }},\quad \mathrm {if~} n>2,$

${\overline {\frac {d^{2}\phi }{d\epsilon ^{2}}}}+{\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}^{2}}}={\frac {1}{\Theta ^{2}}}\quad \mathrm {if~} n>4,$

as already obtained. Also

${\overline {\frac {d^{3}\phi }{d\epsilon ^{3}}}}+3{\overline {{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\frac {d\phi }{d\epsilon }}}}+{\overline {{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}^{3}}}={\frac {1}{\Theta ^{3}}}\quad \mathrm {if~} n>6.$

(372)

1. This equation may also be obtained from equations (252) and (321). Compare also equation (349) which was derived by an approximative method.
2. Compare equation (350), obtained by an approximative method.