It is shown that, for any distribution law in which Boltzmann's law holds for the various quantum states, $\frac{dlogW}{\mathrm{dT}}=\frac{({\overline{\epsilon }}_W-\overline{\epsilon })}{k{T}^2}$, where $W$ is the fraction of the molecules in certain specified quantum states, ${\overline{\epsilon }}_W$ the average energy of the molecules in these states, $\overline{\epsilon }$ the average energy of all the molecules, $T$ the absolute temperature, and $k$ the gas constant. The distribution law will appear to be continuous if not viewed too closely, even though motions of the molecules are quantized. In what follows we consider the continuous outline and neglect the fine structure. If $W_{\epsilon }d\epsilon$ is the fraction of molecules whose energy lies between $\epsilon$ and $\epsilon +d\epsilon$, we have the general rule $\frac{{\partial }^{2}log{W}_{\epsilon }}{\partial T\partial \epsilon }=\frac{1}{k{T}^2}$. Proceeding along these lines we can find in a new and simple way the distribution law over a range of energies if the average energy of the molecules (or the energy at which $W_{\epsilon }$ is a maximum) is given over a range of temperatures. At a given temperature we can compare the actual distribution law with a classical one which makes $W_{\epsilon }$ have a maximum at the same energy, and we find a limit beyond which the actual distribution law cannot depart from this particular classical law, provided the molecule is made up of a group of rotators and harmonic oscillators, and is sufficiently complex. An example is considered, which is of interest in the theory of the decomposition of azomethane.

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