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: $E=E_1+E_2+\dots E_n+\dots .\mathrm{\backslash Label}[eqn](215)\text{upshape (215)}$ $E_n$ 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 ${\displaystyle \frac{N_n}{h}}$ oscillators and hence ${\displaystyle \frac{N_n}{h}}df·d\psi$ per element of area. Hence we have: $E_n=\frac{N_n}{h}\iint Udfd\psi .$ In performing the integration, instead of $f$ and $\psi$ we take $C$ and $\varphi$, as new variables, and since according to [eqn:(211)] (211), $f=C\cos \varphi \psi =2\pi \nu LC\sin \varphi \mathrm{\backslash Label}[eqn](216)\text{upshape (216)}$ we get: $E_n=2\pi \nu L\frac{N_n}{h}\iint UCdCd\varphi$ to be integrated with respect to $\varphi$ from $0$ to $2\pi$ 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) $U=\frac{1}{2}K{C}^2,\mathrm{\backslash Label}[eqn](217)\text{upshape (217)}$ we obtain by integration $E_n=\frac{{\pi }^2}{2}\nu LK\frac{N_n}{h}(C_n^4-C_{n-1}^4)$ 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 $(n-\frac{1}{2})h\nu$. This is exactly the arithmetic mean of the energies $(n-1)h\nu$ and $nh\nu$ 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), $E=Nh\nu {\sum }_{n=1}^{n=\infty }(n-\frac{1}{2})w_n.\mathrm{\backslash Label}[eqn](219)\text{upshape (219)}$