Bodies" published in 1772. Karl Jacobi's generalization of the identity to N bodies and to the present form of Laplace's identity closely resembles the classical virial theorem. However, the interpretations leading to the development of the equations were very different, since at the time of development, statistical dynamics had not yet unified the separate studies of thermodynamics and classical dynamics. The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell, Lord Rayleigh, Henri Poincaré, Subrahmanyan Chandrasekhar, Enrico Fermi, Paul Ledoux, Richard Bader and Eugene Parker. Fritz Zwicky was the first to use the virial theorem to deduce the existence of unseen matter, which is now called dark matter. Richard Bader showed the charge distribution of a total system can be partitioned into its kinetic and potential energies that obey the virial theorem. As another example of its many applications, the virial theorem has been used to derive the Chandrasekhar limit for the stability of white dwarf stars. Illustrative special case Consider particles with equal mass , acted upon by mutually attractive forces. Suppose the particles are at diametrically opposite points of a circular orbit with radius . The velocities are and , which are normal to forces and . The respective magnitudes are fixed at and . The average kinetic energy of the system is Taking center of mass as the origin, the particles have positions and with fixed magnitude . The attractive forces act in opposite directions as positions, so . Applying the centripetal force formula results in: as required. Note: If the origin is displaced then we'd obtain the same result. This is because the dot product of the displacement with equal and opposite forces , results in net cancellation. Statement and derivation Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step. For a collection of point particles, the scalar moment of inertia about the origin is defined by the equation where and represent the mass and position of the th particle. is the position vector magnitude. The scalar is defined by the equation where is the momentum vector of the th particle. Assuming that the masses are constant, is one-half the time derivative of this moment of inertia In turn, the time derivative of can be written where is the mass of the th particle, is the net force