169.

We shall now calculate the number of the possible systems of values $𝖺$, $𝖻$, $𝖼$, which correspond to the vibrations within a certain small range of the spectrum, say between the frequencies $\nu$ and $\nu +d\nu$. According to [eqn:(306)] (306), these systems of values satisfy the inequalities ${\left(\frac{2l\nu }{c}\right)}^2<{𝖺}^2+{𝖻}^2+{𝖼}^2<{\left(\frac{2l(\nu +d\nu )}{c}\right)}^2,\mathrm{\backslash Label}[eqn](309)\text{upshape (309)}$ where not only ${\displaystyle \frac{2l\nu }{c}}$ but also ${\displaystyle \frac{2ld\nu }{c}}$ is to be thought of as a large number. If we now represent every system of values of $𝖺$, $𝖻$, $𝖼$ graphically by a point, taking $𝖺$, $𝖻$, $𝖼$ as coordinates in an orthogonal coordinate system, the points thus obtained occupy one octant of the space of infinite extent, and condition [eqn:(309)] (309) is

equivalent to requiring that the distance of any one of these points from the origin of the coordinates shall lie between ${\displaystyle \frac{2l\nu }{c}}$ and ${\displaystyle \frac{2l(\nu +d\nu )}{c}}$. Hence the required number is equal to the number of points which lie between the two spherical surface-octants corresponding to the radii ${\displaystyle \frac{2l\nu }{c}}$ and ${\displaystyle \frac{2l(\nu +d\nu )}{c}}$. Now since to every point there corresponds a cube of volume $1$ and vice versa, that number is simply equal to the space between the two spheres mentioned, and hence equal to $\frac{1}{8}4\pi {\left(\frac{2l\nu }{c}\right)}^2\frac{2ld\nu }{c},$ and the number of the independent variables of state is four times as large or $\frac{16\pi l^3{\nu }^{2}d\nu }{c^3}.$

Since, moreover, the partial energy ${\displaystyle \frac{\overline{L\mathrm{\backslash Strut}}}{3}}$ corresponds on the average to every independent variable of state in the state of equilibrium, the total energy falling in the interval from $\nu$ to $\nu +d\nu$ becomes $\frac{16\pi l^3{\nu }^{2}d\nu }{3c^3}\overline{L\mathrm{\backslash Strut}}.$ Since the volume of the cavity is $l^3$, this gives for the space density of the energy of frequency $\nu$ $𝗎d\nu =\frac{16\pi {\nu }^{2}d\nu }{3c^3}\overline{L\mathrm{\backslash Strut}},$ and, by substitution of the value of $\overline{L\mathrm{\backslash Strut}}={\displaystyle \frac{L}{N}}$ from [eqn:(200)] (200), $𝗎=\frac{8\pi {\nu }^{2}kT}{c^3},\mathrm{\backslash Label}[eqn](310)\text{upshape (310)}$ which is in perfect agreement with Rayleigh's formula [eqn:(285)] (285).

If the law of the equipartition of energy held true in all

cases, Rayleigh's law of radiation would, in consequence, hold for all wave lengths and temperatures. But since this possibility is excluded by the measurements at hand, the only possible conclusion is that the law of the equipartition of energy and, with it, the system of Hamilton's equations of motion does not possess the general importance attributed to it in classical dynamics. Therein lies the strongest proof of the necessity of a fundamental modification of the latter.

VIrreversible Radiation Processes