This work is consequent on a forthcoming joint paper by A. C. Stevenson and the present writer on the use of the torsion function in the solution of flexure problems. The treatment is general, but special attention is paid to doubly-connected cross-sections which can be mapped conformally on a circular annulus. The eccentric circles cross-section (4) is solved in order to illustrate the power of the method, and results are obtained for the cross-section bounded by confocal limacons. This is of interest, since it has been customary to regard these problems as largely determined by the cross-section of the cylinder; but these results show that the terms involving the elastic constants are now no longer negligible. The method is capable of giving useful information to aeronautical engineers, and the writer hopes to present in a later communication a treatment of an aerofoil-like profile.