I t is a generally appreciated deduction from St. Venant's solution of the flexure problem that a beam in which the material is disposed at a distance from the neutral axis is superior to the solid section in economy of material. St. Venant's solution, however, suggests th at this advantage increases without limit as the thickness of the material is reduced and the distance from the neutral axis is increased. I t has, of course, been generally realised th at this conclusion is not supported by ordinary engineering practice, and recent experience in the use of high tensile steels and problems of aircraft structure have emphasised the desirability of a further examination of the flexure problem.St. Venant's solutions are obtained when the equations of equilibrium of an isotropic elastic solid are made linear by the neglect of terms of higher orders than the first: and by Kirchhoff's theorem of determinancy these solutions may then be considered unique and stable. To attack problems of stability it is necessary, as is shown by R. V. Southwell* in his * General Theory of Elastic Stability,' to include some of the second order effects. I t is, in fact, only when these become considerable that Kirchhoff's theorem fails and instability becomes possible. By this general treatment various classes of instability are obtained or indicated, but the only ones susceptible to analysis or of practical interest (on account of the " elastic limit " which is a feature of all practical materials) are those in which at the moment of instability the strains are still small. Bryantf has shown that this will only occur, as in the case of thin rods and shells, when one dimension of the body is small compared with others.In a similar way we may expect that when one dimension of the cross section of the body is small compared with others it will be necessary to include second order terms in problems purely of flexural equilibrium. This is really evident in the case of the flexure problem from St. Venant's explicit description of the stress at any element of the cross section as a function of the initial position of the element. The longitudinal stress, for example, is prescribed as directly proportional to the initial distance of the element from the line of centroids or

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