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By Rudolf Clausius.

Publication information: cir.nii.ac.jp/crid/1362544420729366912

Published in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.

Online text: commons.princeton.edu/josephhenry/wp-content/uploads/sites/71/2021/01/1870-Virial-Thm-Classius-The-London-Edinburgh-and-Dublin-Philosophical-Magazine-and-Journal-of-Science.pdf

Cited passages: On a Mechanical Theorem applicable to Heat by R. Clausius (1870), pp. 123-127; p. 122 footnote dates the lecture to 13 June 1870.

Rudolf Clausius established a relation between average kinetic energy and the forces acting on a bounded mechanical system. The virial theorem follows by differentiating the sum of position-momentum products and averaging when the boundary term remains bounded. For inverse-square gravity it gives $2⟨K⟩+⟨U⟩=0$ under the appropriate equilibrium assumptions. The result connects microscopic motion with binding energy and supplies a bridge between mechanics, gases and gravitating systems.^[ref]