PROOF OF BERTRAND'S THEOREM 605 reference orbit,then the force laws are restricted to the discrete set given by Eq. (A-8)with rational values of B.In order for the orbits to remain closed for larger deviations from circularity,no matter what the initial conditions of the reference orbit,only two of these rational values are permitted:B=1 and B=2.Since we know that these attractive force laws do in fact give closed orbits for all E and I leading to bounded motion,they must be the only force laws leading to closed orbits for all bounded motion.PROOF OF BERTRAND'S THEOREM 605 reference orbit, then the force laws are restricted to the discrete set given by Eq. (A-8) with rational values of P. In order for the orbits to remain closed for larger deviations from circularity, no matter what the initial conditions of the reference orbit, only two of these rational values are permitted: = 1 and IPI = 2. Since we know that these attractive force laws do in fact give closed orbits for all E and 1 leading to bounded motion, they must be the only force laws leading to closed orbits for all bounded motion