The Theory of Structures is for the most part concerned with the calculation of stresses in a given structure under given external conditions of loading and temperature. The real problem of structural design, however, in aeronautics at any rate, is to find that structure, which will equilibrate the external loads, without failure or undue deformation, under such conditions of temperature, as may be appropriate, and which at the same time will have the least possible weight. The solution of this general design is obviously very difficult and cannot be resolved at the present time. However, on the basis of certain classical theorems due to Maxwell and Michell and using methods and suggestions derived from these theorems by H.L. Cox, one can make certain progress, and in addition point the way to profitable lines of research. The present paper reviews the classical results and their current application, develops the mathematical theory for the two-dimensional case and derives a number of special solutions. It is hoped that its publication will encourage research in this very important field.