Several analytical methods are outlined to deal with the title problem when the vessel is free of external constraints. In essence these correspond to the Haigh equation, which provides a nonlinear relationship between the induced circumferential stress and the applied internal pressure. To make use of these analyses a survey of the cross sectional profile is required. The approach using an internal swinging arm has been found most useful. The extent to which an improved Haigh equation is valid is examined with reference to the pneumatic loading of a twin saddle supported horizontal vessel which contains imperfections and restraints at the supports.