It is known from both theoretical and experimental investigations that St. Venant's assumption on the constancy of the shape of the cross section of girders in pure bending does not hold true in case of thin-walled sections. The greater flexibility than calculated according to ordinary bending theory of initially curved tubes, as experimentally found by Professor Bantlin, was perfectly explained by Professor von Kármán in 1911 on the assumption of a flattening of the section.

In 1927 Brazier with the aid of the variational method determined exactly that the shape of an originally circular thin-walled bent cylinder corresponding to the least potential energy is quasi elliptical and that the cross section of the cylinder, therefore, must flatten, even if the centre line of the cylinder was originally straight. In consequence of the flattening St. Venant's linear law for the curvature loses its validity and the curvature increases more rapidly than the bending moment. For a certain value of the curvature the bending moment is a maximum, and after this value was reached the curvature increases even if the applied moment remains unchanged or decreases, fulfilling thereby the criterion of instability. This instability occurs when the rate of flattening, i.e., the maximum radial displacement of any point of the circumference of the tube divided by the original radius of the tube, will equal 2/9.

In recent years much attention has been devoted to what Professor Southwell has appropriately called relaxation (1) methods in the solution of structural problems. These methods are essentially physical applications of iteration processes for solving simultaneous equations. In this paper it is not intended to discuss the advantages or otherwise of physical relaxation as compared with analytical iteration. The object is to examine the mathematical basis of the relaxation methods and to investigate the conditions of their convergency.

Observations on the wing-beats of pigeons, based on cinematograph films taken by the author; with conclusions regarding the characteristics of the wing and its lift in the various phases of the beat.

In case of normal monocoque aeroplane fuselages relatively strong rings are spaced at distances equal to diameter of the fuselage, failure of the structure always occurs by instability of the most stressed stringer. The buckling stress, however, is mostly very low, especially, when the number of stringers is great and consequently the cross section of one stringer is small. The value of the stress at failure may be raised by spacing the rings nearer and reducing thereby the free length and, consequently, the slenderness ratio of the stringers. In this case the rings should be constructed of lighter sections in order to balance the increment of weight of the whole structure due to the greater number of rings.

By decreasing gradually the interval between rings a critical distance is reached when the most stressed stringer does not buckle between adjacent rings, but forces one or more supporting rings to deform simultaneously. This form of instability will in the following be termed local buckling of the monocoque structure.