The paper reports on a theoretical and an experimental investigation carried out on a thin-walled hemi-ellipsoidal prolate dome in air and also under external water pressure. The investigation found that there was good correlation between experiment and theory. The theoretical investigation was carried out using the finite element analysis to model both the structure and the fluid. The theoretical investigation used two different programs, one of which was the giant computer program ANSYS and the other was an in-house program developed by Ross. For the shell structure, the ANSYS program used 2 different doubly curved thin-walled shell elements, while the in-house program used a simpler axisymmetric thin-walled shell element. This axisymmetric element allowed a sinusoidal variation of the displacements in the circumferential direction, thus, decreasing preparation and computational time. Agreement between the 3 different finite elements was found to be good. This was found particularly encouraging for the in-house software, as it only took a few hours to set up the computer model, and a few seconds to analyse the vessel, whereas the ANSYS software took several weeks to set up the computer model, and several minutes to analyse the shell dome. The ANSYS software, however, did have the advantage in producing excellent graphical displays in both the pre-processing and post-processing modes.