Theoretical derivation of expressions for fluctuations in energy, pressure, and density.—The fluctuation of a quantity, such as the energy, is defined as the ratio of the average deviation from the mean energy to the mean energy. The following expressions are deduced from a new statistical theory for ordinary temperature and pressure: Fluctuation in energy at constant $\mathrm{volume}=\sqrt{\left(\frac{2}{\gamma \mathrm{nf}}\right)}$ and at constant $\mathrm{temperature}=\frac{2}{f\sqrt{n}}$; Fluctuation in pressure or density at constant $\mathrm{temperature}=\frac{1}{\sqrt{n}}$; where $n$ is the number of molecules, $f$ the number of degrees of freedom of each, and $\gamma$ the ratio of total energy to kinetic energy. At the critical condition the first of these becomes ${\left(\frac{12}{\gamma \mathrm{nf}}\right)}^{\frac{1}{4}}$ and the last ${\left(\frac{6}{n}\right)}^{\frac{1}{4}}$. The expressions obtained by Smoluchowski, Hauer, Fürth and Laue are shown to be special cases of the above. Relations are obtained between the various types of fluctuations for the case of an ideal gas.

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