Page:American Journal of Mathematics Vol. 2 (1879).pdf/19



It is not intended in this discussion to give the exact theory of flexure for all materials and shapes of pieces subjected to bending, nor indeed for any one kind of material. The present state of knowledge regarding the internal molecular action developed in any piece of elastic material by the action of external forces, is not such as to enable one to treat any problem of this kind with mathematical rigor if the piece be of finite dimensions. The illustrious Lamé, however, has remarked that the exact solutions of all problems in natural science are usually obtained by successive approximations, and such has certainly been the case in regard to the theory of flexure.

If the following investigation shall be found to constitute even a short step in the direction of the correct theory, the object of the writer will have been accomplished.