155.

In the preceding chapter we have made ourselves familiar with all the details of a system of oscillators exposed to uniform radiation. We may now develop the idea put forth at the end of [sect:144.] Sec. 144. That is to say, we may identify the stationary state of the oscillators just found with the state of maximum entropy of the system of oscillators which was derived directly from the hypothesis of quanta in the preceding part, and we may then equate the temperature of the radiation to the temperature of the oscillators. It is, in fact, possible to obtain perfect agreement of the two states by a suitable coordination of their corresponding quantities.

According to [sect:139.] Sec. 139, the "distribution density" $w$ of the oscillators in the state of statistical equilibrium changes abruptly from one region element to another, while, according to [sect:138.] Sec. 138, the distribution within a single region element is uniform. The region elements of the state plane $(f\psi )$ are bounded by concentric similar and similarly situated ellipses which correspond to those values of the energy $U$ of an oscillator which are integral multiples of $h\nu$. We have found exactly the same thing for the stationary state of the oscillators when they are exposed to uniform radiation, and the distribution density $w_n$ in the $n$th region element may be found from [eqn:(270)] (270), if we remember that the $n$th region element contains the energies between $(n-1)h\nu$ and $nh\nu$. Hence: $w_n=\frac{(p𝖨{)}^{n-1}}{(1+p𝖨{)}^n}=\frac{1}{p𝖨}{\left(\frac{p𝖨}{1+p𝖨}\right)}^n.\mathrm{\backslash Label}[eqn](271)\text{upshape (271)}$ This is in perfect agreement with the previous value [eqn:(220)] (220) of $w_n$ if we put $\alpha =\frac{1}{p𝖨}\text{and}\gamma =\frac{p𝖨}{1+p𝖨},$

and each of these two equations leads, according to [eqn:(221)] (221), to the following relation between the intensity of the exciting vibration $𝖨$ and the total energy $E$ of the $N$ oscillators: $p𝖨=\frac{E}{Nh\nu }-\frac{1}{2}.\mathrm{\backslash Label}[eqn](272)\text{upshape (272)}$