The sum rules for elementary discontinuities of inelastic scattering amplitudes, derived by Veneziano, are shown to be inconsistent with the definition of the discontinuities. New sum rules are derived. Each of the sum rules is equivalent to the basic discontinuity formula.The author would like to thank Professor B.E.Y. Svensson for encouragement and stimulating discussions on the subject and for suggestions concerning the manuscript. He is also grateful for the hospitality extended to him in Lund.