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

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

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

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

(304)

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

(305)

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

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

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

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

(306)

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

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

(307)