Since the i and j both go over the full range, 1 to N, we could interchange the i and j on the right-hand side of equation (13.20) without really changing anything. If we rewrite the right-hand side, interchanging i and j, and then adding it to

The term on the right-hand side is the sum of the gravitational potential energies of each pair of particles. Note that each pair appears twice in the sum, since the energy of the pair is independent of which particle in the pair we count first. For example, for particle 1 and 2, both the quantities Gm2/|r12| and Gm2/|r21| appear. This means that the double sum on the right-hand side of equation (13.26) gives us twice the gravitational potential energy, but there is also a factor of one-half in front, so the right-hand side is equal to the gravitational potential energy U. We can therefore rewrite equation (13.17) as


If we take the time average of these quantities over a sufficiently long time, the left-hand side

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