Page:Elementary Principles in Statistical Mechanics (1902).djvu/143

From the last equation, with (298), we get

${\overline {e^{-\phi _{p}}V_{p}}}|_{\epsilon }=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }V_{p}\,dV_{q}=e^{-\phi }V.$

(375)

But by equations (288) and (289)

$e^{-\phi _{p}}V_{p}={\frac {2}{n}}\epsilon _{p}.$

(376)

Therefore

$e^{-\phi }V={\overline {e^{-\phi _{p}}V_{p}}}|_{\epsilon }={\frac {2}{n}}{\overline {\epsilon _{p}}}|_{\epsilon }.$

(377)

Again, with the aid of equation (301), we get

${\overline {\frac {d\phi _{p}}{d\epsilon _{p}}}}{\bigg |}_{\epsilon }=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }{\frac {d\phi _{p}}{d\epsilon _{p}}}e^{\phi _{p}}\,dV_{q}={\frac {d\phi }{d\epsilon }},$

(378)

if $n>2$. Therefore, by (289),

${\frac {d\phi }{d\epsilon }}={\overline {\frac {d\phi _{p}}{d\epsilon _{p}}}}{\bigg |}_{\epsilon }={\bigg (}{\frac {n}{2}}-1{\bigg )}{\overline {\epsilon _{p}^{-1}}}|_{\epsilon },\quad \mathrm {if} \quad n>2.$

(379)

These results are interesting on account of the relations of the functions $e^{-\phi }V$ and ${\frac {d\phi }{d\epsilon }}$ to the notion of temperature in thermodynamics,—a subject to which we shall return hereafter. They are particular cases of a general relation easily deduced from equations (306), (374), (288) and (289). We have

${\frac {d^{h}V}{d\epsilon ^{h}}}=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }{\frac {d^{h}V}{d\epsilon _{p}^{h}}}\,dV_{q},\quad \mathrm {if} \quad h<{\tfrac {1}{2}}n+1.$

The equation may be written

$e^{-\phi }{\frac {d^{h}V}{d\epsilon ^{h}}}=e^{-\phi }\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{-\phi _{p}}{\frac {d^{h}V}{d\epsilon _{p}^{h}}}e^{\phi _{p}}\,dV_{q}.$

material points with this particular value of the energy as nearly as possible in statistical equilibrium, or if we ask: What is the probability that a point taken at random from an ensemble in statistical equilibrium with this value of the energy will be found in any specified part of the circle?