Page:Elementary Principles in Statistical Mechanics (1902).djvu/128

as practically equivalent to the values relating to the most common energy

${\bigg (}{\frac {d\phi }{da_{1}}}{\bigg )}_{0},\quad {\bigg (}{\frac {d\phi }{da_{2}}}{\bigg )}_{0},\quad \mathrm {etc.}$

In this case also $d{\overline {\epsilon }}$ is practically equivalent to $d\epsilon _{0}$. We have therefore, for very large values of $n$,

$-d{\overline {\eta }}=d\phi _{0}$

(337)

approximately. That is, except for an additive constant, $-{\overline {\eta }}$ may be regarded as practically equivalent to $\phi _{0}$, when the number of degrees of freedom of the system is very great. It is not meant by this that the variable part of ${\overline {\eta }}+\phi _{0}$ is numerically of a lower order of magnitude than unity. For when $n$ is very great, $-{\overline {\eta }}$ and $\phi$ are very great, and we can only conclude that the variable part of ${\overline {\eta }}+\phi _{0}$ is insignificant compared with the variable part of ${\overline {\eta }}$ or of $\phi _{0}$, taken separately.

Now we have already noticed a certain correspondence between the quantities $\Theta$ and ${\overline {\eta }}$ and those which in thermodynamics are called temperature and entropy. The property just demonstrated, with those expressed by equation (336), therefore suggests that the quantities $\phi$ and $d\epsilon /d\phi$ may also correspond to the thermodynamic notions of entropy and temperature. We leave the discussion of this point to a subsequent chapter, and only mention it here to justify the somewhat detailed investigation of the relations of these quantities.

We may get a clearer view of the limiting form of the relations when the number of degrees of freedom is indefinitely increased, if we expand the function $\phi$ in a series arranged according to ascending powers of $\epsilon -\epsilon _{0}$. This expansion may be written

$\phi =\phi _{0}+{\bigg (}{\frac {d\phi }{d\epsilon }}{\bigg )}_{0}(\epsilon -\epsilon _{0})+{\bigg (}{\frac {d^{2}\phi }{d\epsilon ^{2}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{2}}{2}}+{\bigg (}{\frac {d^{3}\phi }{d\epsilon ^{3}}}{\bigg )}_{0}{\frac {(\epsilon -\epsilon _{0})^{3}}{3!}}+\mathrm {etc.}$

(338)

Adding the identical equation