On an Improvement of Wien's Equation for the Spectrum.
by (Max) (Planck), 1900
(\Huge{The}) interesting result of long wavelength spectral energy measurements which were communicated by Mr. Kurlbaum at today's meeting, and which were obtained by him and Mr. Rubens, confirm the statement by Mr. Lummer and Mr. Pringsheim, which was based on their observations that Wien's energy distribution law is not as generally valid, as many supposed up to now, but that this law at most has the character of a limiting case, the simple from of which was due only to a restriction to short wave lengths and low temperatures ^{1}. Since I myself even in this Society have expressed the opinion that Wien's law must be necessarily true, I may perhaps be permitted to explain briefly the relationship between the electromagnetic theory developed by me and the experimental data.
The energy distribution law is according to this theory determined as soon as the entropy $S$ of a linear resonator which interacts with the radiation is known as function of the vibrational energy $U$. I have, however, already in my last paper on this subject ^{2} stated that the law of increase of by itself not yet sufficient to determine this function completely; my view that Wien's law would be of general validity, was brought about rather by special considerations, namely by the evaluation of an infinitesimal increase of the entropy of a system of $n$ identical resonators in a stationary radiation field by two different methods which led to the equation
\begin{equation*} dU_n \cdot \Delta U_n \cdot f(U_n) = ndU \cdot \Delta U \cdot f(U), \end{equation*}where
\begin{equation*} U_n = nU \qquad \mbox{and} \qquad f(U) =  \frac {3}{5} ~\frac{d^2S}{dU^{2}}. \end{equation*}From this equation Wien's law follows in the form
\begin{equation*} \frac{d^2S}{dU^2} = \frac{\mbox{const}}{U}. \end{equation*}The expression on the righthand side of this functional equation is certainly the abovementioned change in entropy since $n$ identical processes occur independently, the entropy changes of which must simply add up. However, I consider the possibility, even if it would not be easily understandable and in any case would be difficult to prove, that the expression on the lefthand side would not have the general meaning which I attributed to it earlier, in other words: that the values of $U_n$, $dU_n$ and $\Delta U_n$ are not by themselves sufficient to determine the change of entropy under consideration, but that $U$ itself must also be known for this. Following this suggestion I have finally started to construct completely arbitrary expressions for the entropy which although they are more complicated than Wien's expression still seem to satisfy just as completely all requirements of the thermodynamic and electromagnetic theory.
I was especially attracted by one of the expressions thus constructed which is nearly as simple as Wien's expression and which deserves to be investigated since Wien's expression is not sufficient to cover all observations. We get this expression by putting ^{3}
\begin{equation*} \frac{d^2S}{dU^2} = \frac{\alpha}{U(\beta + U)}. \end{equation*}It is by far simplest of all expressions which lead to $S$ as a logarithmic function of $U$which is suggested from probability considerations  and which moreover reduces to Wien's expression for small values of $U$. Using the relation
\begin{equation*} \frac{dS}{dU} = \frac{1}{T} \end{equation*}and Wien's "displacement" law ^{4} one gets a radiation formula with two constants:
\begin{equation*} E = \frac{C\lambda^{5}}{e^{c/ \lambda T}  1}, \end{equation*}which, as far as I can see at the moment, fits the observational data, published up to now, as satisfactory as the best equations put forward for the spectrum, namely those of Thiesen ^{5} ^{6} LummerJahnke ^{7}, and LummerPringsheim ^{8}. $($This was demonstrated by some numerical examples.$)$ I should therefore be permitted to draw your attention to this new formula which I consider to be the simplest possible, apart from Wien's expression, from the point of view of the electromagnetic theory of radiation.
FOOTNOTES

Mr. Paschen has written to me that he has also recently found appreciable deviations from Wien's law.

I use the second derivative of $S$ with respect to $U$ since this quantity has a simple physical meaning.

The expression of Wien's displacement law is simply
\begin{equation*} S = f(U/\nu), \end{equation*}where $\nu$ is the frequency of the resonator, as I shall show elsewhere..

One can see there that Mr. Thiesen had put forward his formula before Mr. Lummer and Mr. Pringsheim had extended their measurements to longer wave lengths. I emphasize this point as I have made a statement to the contrary [M. Planck, Ann. Physik 1, 719 $($1900$)$;] before this paper was published.

O. Lummer and E. Jahnke, Ann. Physik Lpz. 3, 288 $($1900$)$.

O. Lummer and E. Pringsheim, Verh. Dtsch. Phys. Ges. Berlin 2, 174 $($1900$)$.
References
 Planck, M. Über eine Verbesserung der Wien'schen SpectralGleichung, Verhandlungen der Deutschen Physikalischen Gesellschaft, 1900, Band 2, Seite 202204.
 Planck, M.; D. ter Haar $($translator$)$. On an Improvement of Wien's Equation for the Spectrum, The Old Quantum Theory, Pergamon Press, 1967, p. 79.
Notes
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