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Who first discovered the Virial Theorem? Who first wrote it down? When? Where?

My guess: a 19th century thermodynamicist

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According to the OED, Clausius coined the German word "virial" (from vīs force, strength):

a. In Clausius' kinetic theorem of gases: (see quots.). virial theorem, the theorem that for a steady-state system of particles obeying an inverse square law of force, the time-average of the kinetic energy equals the time-average of the virial; or equivalently, that the potential energy is twice the total energy and the kinetic energy is the negative of the total energy.

See:

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