Thin-walled tubes, in. in diameter, of three hard aluminium alloys and of mild steel have been tested in fatigue under three systems of alternating stresses while subjected to biaxial mean tensions imposed by means of internal pressure. In fatigue under direct or bending stresses the hoop tension in the walls of the aluminium alloy tubes did not seriously reduce the fatigue endurance, but it did markedly affect the mode and rate of crack propagation; cracks initially transverse to the tube axis tended to develop very rapidly in the axial direction. This tendency was present under both fluid and gas pressure, and under gas pressure the cracks propagated so fast that the test piece was often blown to pieces before the gas pressure fell by leakage through the cracks. The gradual taper in wall thickness along the fillets joining the test section to the enlarged ends offered no barrier to propagation of the axial cracks and the whole test piece, including its enlarged ends, was often shattered. Propagation of the axial cracks was preventible by sufficiently reducing the fillet radius, or by a ring glued on. Under alternating torsion, both endurance and mode of failure were affected by internal gas pressure. The initial fatigue crack, either circumferential or axial, often extended over a length comparable with the diameter of the tube, except under low ranges of shear stress when the crack length was sometimes very short. At each end the cracks forked in a characteristic manner and under moderate gas pressure the portions of tHe wall between the prongs of the fork were blown outwards. Under high pressure explosive failure and fragmentation often occurred. Mild steel under alternating torsion with internal gas pressure exhibited the same modes of failure, and two or more fatigue cracks were often formed simultaneously. Under high hoop tension, cracks propagated rapidly and one test piece, after two million cycles endurance, failed by exploding. However, no mild steel test piece was fragmented. A tentative explanation is offered of the reason why, in tubes of small diameter, rather short fatigue cracks may be expected to lead to fast fracture under the static loading. Attention is drawn to the inference that the initial fatigue crack itself must develop very quickly to considerable length.