Page:Elementary Principles in Statistical Mechanics (1902).djvu/120

etc., when ${\displaystyle V_{q}}$ is a continuous function of ${\displaystyle \epsilon _{q}}$ commencing with the value ${\displaystyle V_{q}=0}$, or when we choose to attribute to ${\displaystyle V_{q}}$ a fictitious continuity commencing with the value zero, as described on page 90.

If we substitute in these equations the values of ${\displaystyle V_{p}}$ and ${\displaystyle e^{\phi _{p}}}$ which we have found, we get

${\displaystyle V={\frac {(2\pi )^{\tfrac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1)}}\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }(\epsilon -\epsilon _{q})^{\tfrac {n}{2}}dV_{q},}$

(304)

${\displaystyle e^{\phi }={\frac {(2\pi )^{\tfrac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n)}}\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }(\epsilon -\epsilon _{q})^{{\tfrac {n}{2}}-1}dV_{q},}$

(305)

where ${\displaystyle e^{\phi _{q}}d\epsilon _{q}}$ may be substituted for ${\displaystyle dV_{q}}$ in the cases above described. If, therefore, ${\displaystyle n}$ is known, and ${\displaystyle V_{q}}$ as function of ${\displaystyle \epsilon _{q}}$, ${\displaystyle V}$ and ${\displaystyle e^{\phi }}$ may be found by quadratures.

It appears from these equations that ${\displaystyle V}$ is always a continuous increasing function of ${\displaystyle \epsilon }$, commencing with the value ${\displaystyle V=0}$, even when this is not the case with respect to ${\displaystyle V_{q}}$ and ${\displaystyle \epsilon _{q}}$. The same is true of ${\displaystyle e^{\phi }}$, when ${\displaystyle n>2}$, or when ${\displaystyle n=2}$ if ${\displaystyle V_{q}}$ increases continuously with ${\displaystyle \epsilon _{q}}$ from the value ${\displaystyle V_{q}=0}$.

The last equation may be derived from the preceding by differentiation with respect to ${\displaystyle \epsilon }$. Successive differentiations give, if ${\displaystyle h<{\tfrac {1}{2}}n+1}$,

${\displaystyle {\frac {d^{h}V}{d\epsilon ^{h}}}=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }{\frac {d^{h}V_{p}}{d\epsilon _{p}^{h}}}\,dV_{q}={\frac {(2\pi )^{\tfrac {n}{2}}}{\Gamma ({\tfrac {1}{2}}n+1-h)}}\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }(\epsilon -\epsilon _{q})^{{\tfrac {n}{2}}-h}dV_{q}.}$

(306)

${\displaystyle d^{h}V/d\epsilon ^{h}}$ is therefore positive if ${\displaystyle h<{\tfrac {1}{2}}n+1}$. It is an increasing function of ${\displaystyle \epsilon }$, if ${\displaystyle h<{\tfrac {1}{2}}n}$. If ${\displaystyle \epsilon }$ is not capable of being diminished without limit, ${\displaystyle d^{h}V/d\epsilon ^{h}}$ vanishes for the least possible value of ${\displaystyle \epsilon }$, if ${\displaystyle h<{\tfrac {1}{2}}n}$. If ${\displaystyle n}$ is even,

${\displaystyle {\frac {d^{\tfrac {n}{2}}V}{d\epsilon ^{\tfrac {n}{2}}}}=(2\pi )^{\tfrac {n}{2}}(V_{q})_{\epsilon _{q}=\epsilon }}$

(307)