71.

Though the manner in which the volume density $u$ and the specific intensity $K$ of black radiation depend on the temperature is determined by the Stefan-Boltzmann law, this law is of comparatively little use in finding the volume density ${𝗎}_{\nu }$ corresponding to a definite frequency $\nu$, and the specific intensity of radiation ${𝖪}_{\nu }$ of monochromatic radiation, which are related to each other by equation [eqn:(24)] (24) and to $u$ and $K$ by equations [eqn:(22)] (22) and [eqn:(12)] (12). There remains as one of the principal problems of the theory of heat radiation the problem of determining the quantities ${𝗎}_{\nu }$ and ${𝖪}_{\nu }$ for black radiation in a vacuum and hence, according to [eqn:(42)] (42), in any medium whatever, as functions of $\nu$ and $T$, or, in other words, to find the distribution of energy in the normal spectrum for any arbitrary temperature. An essential step in the solution of this problem is contained in the so-called "displacement law" stated by W. Wien,W. Wien, Sitzungsberichte d. Akad. d. Wissensch. Berlin, Febr. 9, 1893, p. 55. Wiedemann's Annal., 52, p. 132, 1894. See also among others M. Thiesen, Verhandl. d. Deutsch. phys. Gesellsch., 2, p. 65, 1900. H. A. Lorentz, Akad. d. Wissensch. Amsterdam, May 18, 1901, p. 607. M. Abraham, Annal. d. Physik 14, p. 236, 1904. the importance of which lies in the fact that it reduces the functions ${𝗎}_{\nu }$ and ${𝖪}_{\nu }$ of the two arguments $\nu$ and $T$ to a function of a single argument.

The starting point of Wien's displacement law is the following theorem. If the black radiation contained in a perfectly evacuated cavity with absolutely reflecting walls is compressed or expanded adiabatically and infinitely slowly, as described above

in [sect:68.] Sec. 68, the radiation always retains the character of black radiation, even without the presence of a carbon particle. Hence the process takes place in an absolute vacuum just as was calculated in [sect:68.] Sec. 68 and the introduction, as a precaution, of a carbon particle is shown to be superfluous. But this is true only in this special case, not at all in the case described in [sect:70.] Sec. 70.

The truth of the proposition stated may be shown as follows:

Let the completely evacuated hollow cylinder, which is at the start filled with black radiation, be compressed adiabatically and infinitely slowly to a finite fraction of the original volume. If, now, the compression being completed, the radiation were no longer black, there would be no stable thermodynamic equilibrium of the radiation ([sect:51.] Sec. 51). It would then be possible to produce a finite change at constant volume and constant total energy of radiation, namely, the change to the absolutely stable state of radiation, which would cause a finite increase of entropy. This change could be brought about by the introduction of a carbon particle, containing a negligible amount of heat as compared with the energy of radiation. This change, of course, refers only to the spectral density of radiation ${𝗎}_{\nu }$, whereas the total density of energy $u$ remains constant. After this has been accomplished, we could, leaving the carbon particle in the space, allow the hollow cylinder to return adiabatically and infinitely slowly to its original volume and then remove the carbon particle. The system will then have passed through a cycle without any external changes remaining. For heat has been neither added nor removed, and the mechanical work done on compression has been regained on expansion, because the latter, like the radiation pressure, depends only on the total density $u$ of the energy of radiation, not on its spectral distribution. Therefore, according to the first principle of thermodynamics, the total energy of radiation is at the end just the same as at the beginning, and hence also the temperature