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Some bones have slender marrow cavities and relatively thick walls. Others have wide cavities and thinner walls. Such differences are described by a quantity K, the ratio of the internal to the external diameter. A theoretical analysis shows that the optimum value of K, which allows the mass of a bone to be minimized, depends on whether the bone is selected principally for yield or fatigue strength, for ultimate strength, for impact strength or for stiffness. It also depends on whether the cavity is filled with marrow or with gas. The values of K found in the limb bones of mammals, birds and a few reptiles are surveyed, and compared to the theoretical optima.