Page:Elementary Principles in Statistical Mechanics (1902).djvu/132

108 falls within any given limits of energy ($$\epsilon'$$ and $$\epsilon$$) is represented by If we expand $$\eta$$ and $$\phi$$ in ascending powers of $$\epsilon - \epsilon_0$$, without going beyond the squares, the probability that the energy falls within the given limits takes the form of the 'law of errors'—  This gives  and  We shall have a close approximation in general when the quantities equated in (355) are very small, i. e.'', when  is very great. Now when $$n$$ is very great, $$-d^2\phi/d\epsilon^2$$ is of the same order of magnitude, and the condition that (356) shall be very great does not restrict very much the nature of the function $$\eta$$.

We may obtain other properties pertaining to average values in a canonical ensemble by the method used for the average of $$d\phi/d\epsilon$$. Let $$u$$ be any function of the energy, either alone or with $$\Theta$$ and the external coördinates. The average value of $$u$$ in the ensemble is determined by the equation