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

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112

THE FUNCTION $\phi$ AND

{\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)

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