The torsional stiffness of a thin-walled tube of given perimeter and thickness varies as the square of the enclosed area. The shape giving maximum torsional stiffness is therefore a circle, this being the curve with maximum area for given perimeter. We consider here the nature of the corresponding curve for maximum bending stiffness. The quantity to be made a maximum is now, not the area within the curve, but the second moment for the whole curve about the bending axis. This will give the greatest “moment of inertia” for a tube of uniform thickness. The theory is given in the Appendix and typical forms are shown in Fig. 3. A finite length of base has to be assumed; otherwise the curve reduces to a single straight line normal to the axis.