This paper exploits and extends results of Edmonds, Cunningham, Cruse and McDiarmid on matroid intersections. Letr 1 andr 2 be rank functions of two matroids defined on the same setE. For everyS ⊂E, letr 12(S) be the largest cardinality of a subset ofS independent in both matroids, 0≦k≦r 12(E)−1. It is shown that, ifc is nonnegative and integral, there is ay: 2 E →Z + which maximizes\(\sum\limits_S {(k - r_{12} (E - S))y(S)} \) and\(\sum\limits_S {(k + 1 - r_{12} (E - S))y(S)} \), subject toy≧0, ∀j∈E,\(\sum\limits_{S \mathrel\backepsilon j} {y(S) \leqq c_j } \).