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

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

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

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

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

(337)

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

Now we have already noticed a certain correspondence between the quantities ${\displaystyle \Theta }$ and ${\displaystyle {\overline {\eta }}}$ 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 ${\displaystyle \phi }$ and ${\displaystyle d\epsilon /d\phi }$ 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 ${\displaystyle \phi }$ in a series arranged according to ascending powers of ${\displaystyle \epsilon -\epsilon _{0}}$. This expansion may be written

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

(338)

Adding the identical equation