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

104 as practically equivalent to the values relating to the most common energy In this case also $$d\overline\epsilon$$ is practically equivalent to $$d\epsilon_0$$. We have therefore, for very large values of $$n$$, 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 Adding the identical equation