151.

Let us now calculate the most important properties of the state of statistical equilibrium thus produced. Of the $N$ oscillators situated in the field of radiation the number of those whose energy at the time $t$ lies in the interval between $U=nϵ+\rho$ and $U+dU=nϵ+\rho +d\rho$ may be represented by $N{R}_{n,\rho }d\rho ,\mathrm{\backslash Label}[eqn](253)\text{upshape (253)}$ where $R$ depends in a definite way on the integer $n$ and the quantity $\rho$ which varies continuously between $0$ and $ϵ$.

After a time $dt={\displaystyle \frac{d\rho }{a}}$ all the oscillators will have their energy increased by $d\rho$ and hence they will all now lie outside of the energy interval considered.

On the other hand, during the same time $dt$, all oscillators whose energy at the time $t$ was between $nϵ+\rho -d\rho$ and $nϵ+\rho$ will have entered that interval. The number of all these oscillators is, according to the notation used above, $N{R}_{n,\rho -d\rho }d\rho .\mathrm{\backslash Label}[eqn](254)\text{upshape (254)}$

Hence this expression gives the number of oscillators which are at the time $t+dt$ in the interval considered.

Now, since we assume our system to be in a state of statistical equilibrium, the distribution of energy is independent of the time and hence the expressions [eqn:(253)] (253) and [eqn:(254)] (254) are equal, i.e., $R_{n,\rho -d\rho }=R_{n,\rho }=R_n.\mathrm{\backslash Label}[eqn](255)\text{upshape (255)}$ Thus $R_n$ does not depend on $\rho$.

This consideration must, however, be modified for the special case in which $\rho =0$. For, in that case, of the oscillators, $N=R_{n-1}d\rho$ in number, whose energy at the time $t$ was between $nϵ$ and $nϵ-d\rho$, during the time $dt={\displaystyle \frac{d\rho }{a}}$ some enter into the energy interval (from $U=nϵ$ to $U+dU=nϵ+d\rho$) considered; but all of them do not necessarily enter, for an oscillator may possibly emit all its energy on passing through the value $U=nϵ$. If the probability that emission takes place be denoted by $\eta$ ($<1$) the number of oscillators which pass through the critical value without emitting will be $N{R}_{n-1}(1-\eta )d\rho ,\mathrm{\backslash Label}[eqn](256)\text{upshape (256)}$ and by equating [eqn:(256)] (256) and [eqn:(253)] (253) it follows that $R_n=R_{n-1}(1-\eta ),$ and hence, by successive reduction, $R_n=R_0(1-\eta {)}^n.\mathrm{\backslash Label}[eqn](257)\text{upshape (257)}$

To calculate $R_0$ we repeat the above process for the special case when $n=0$ and $\rho =0$. In this case the energy interval in question extends from $U=0$ to $dU=d\rho$. Into this interval enter in the time $dt={\displaystyle \frac{d\rho }{a}}$ all the oscillators which perform an emission during this time, namely, those whose energy at the time $t$ was between $ϵ-d\rho$ and $ϵ$, $2ϵ-d\rho$ and $2ϵ$, $3ϵ-d\rho$ and $3ϵ\dots$. The numbers of these oscillators are respectively $N{R}_{0}d\rho ,N{R}_{1}d\rho ,N{R}_{2}d\rho ,$ hence their sum multiplied by $\eta$ gives the desired number of emitting oscillators, namely, $N\eta (R_0+R_1+R_2+\dots )d\rho ,\mathrm{\backslash Label}[eqn](258)\text{upshape (258)}$ and this number is equal to that of the oscillators in the energy interval between $0$ and $d\rho$ at the time $t+dt$, which is $N{R}_{0}d\rho$. Hence it follows that $R_0=\eta (R_0+R_1+R_2+\dots ).\mathrm{\backslash Label}[eqn](259)\text{upshape (259)}$

Now, according to [eqn:(253)] (253), the whole number of all the oscillators is obtained by integrating with respect to $\rho$ from $0$ to $ϵ$, and summing up with respect to $n$ from $0$ to $\infty$. Thus $N=N{\sum }_{n=0}^{n=\infty }{\int }_0^{ϵ}{R}_{n,\rho }d\rho =N\sum R_nϵ\mathrm{\backslash Label}[eqn](260)\text{upshape (260)}$ and $\sum R_n=\frac{1}{ϵ}.\mathrm{\backslash Label}[eqn](261)\text{upshape (261)}$ Hence we get from [eqn:(257)] (257) and [eqn:(259)] (259) $R_0=\frac{\eta }{ϵ},R_n=\frac{\eta }{ϵ}(1-\eta {)}^n.\mathrm{\backslash Label}[eqn](262)\text{upshape (262)}$