138.

The thermodynamic state of the system of oscillators is fixed by the fact that the values of the distribution densities $w_{1}$ , $w_{2}$ , $w_{3}, \dots$ of the oscillators among the separate region elements are given. Within a region element the distribution of the oscillators is according to the law of elemental chaos ([sect:122.] Sec. 122), i.e., it is approximately uniform.

These data suffice for calculating the entropy $S$ as well as the energy $E$ of the system in the given state, the former quantity directly from [eqn:(173)] (173), the latter by the aid of [eqn:(205)] (205). It must be kept in mind in the calculation that, since the energy varies appreciably within a region element, the energy $E_{n}$ of all those oscillators which lie in the $n$ th region element is to be found by an integration. Then the whole energy $E$ of the system is: $E = E_{1} + E_{2} + \dots E_{n} + \dots . \Label [e q n] (215) \upshape (215)$ $E_{n}$ may be calculated with the help of the law that within every region element the oscillators are uniformly distributed. If the $n$ th region element contains, all told, $N_{n}$ oscillators, there are per unit area $\frac{N_{n}}{h}$ oscillators and hence $\frac{N_{n}}{h} d f \cdot d ψ$ per element of area. Hence we have: $E_{n} = \frac{N_{n}}{h} \iint U d f d ψ .$ In performing the integration, instead of $f$ and $ψ$ we take $C$ and $ϕ$ , as new variables, and since according to [eqn:(211)] (211), $f = C \cos ϕ ψ = 2 π ν L C \sin ϕ \Label [e q n] (216) \upshape (216)$ we get: $E_{n} = 2 π ν L \frac{N_{n}}{h} \iint U C d C d ϕ$ to be integrated with respect to $ϕ$ from $0$ to $2 π$ and with respect to $C$ from $C_{n - 1}$ to $C_{n}$ . If we substitute from [eqn:(205)] (205), [eqn:(209)] (209) and [eqn:(216)] (216) $U = \frac{1}{2} K C^{2}, \Label [e q n] (217) \upshape (217)$ we obtain by integration $E_{n} = \frac{π^{2}}{2} ν L K \frac{N_{n}}{h} (C_{n}^{4} - C_{n - 1}^{4})$ and from [eqn:(214)] (214) and [eqn:(208)] (208): $E_{n} = N_{n}(n - \tfrac{1}{2})h\nu = Nw_{n}(n - \tfrac{1}{2})h\nu,

\Label[eqn]{(218)}\tag*{\upshape (218)}$ plus 0.75em minus 0.25em that is, the mean energy of an oscillator in the $n$ th region element is $(n - \frac{1}{2}) h ν$ . This is exactly the arithmetic mean of the energies $(n - 1) h ν$ and $n h ν$ which correspond to the two ellipses $C = C_{n - 1}$ and $C = C_{n}$ bounding the region, as may be seen from [eqn:(217)] (217), if the values of $C_{n - 1}$ and $C_{n}$ are therein substituted from [eqn:(214)] (214).

The total energy $E$ is, according to [eqn:(215)] (215), $E = N h ν \sum_{n = 1}^{n = \infty} (n - \frac{1}{2}) w_{n} . \Label [e q n] (219) \upshape (219)$

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