The boundary-value problem of deformation of a rotationally symmetric shell is stated in terms of a new system of first-order ordinary differential equations which can be derived for any consistent linear bending theory of shells. The dependent variables contained in this system of equations are those quantities which appear in the natural boundary conditions on a rotationally symmetric edge of a shell of revolution. A numerical method of solution which combines the advantages of both the direct integration and the finite-difference approach is developed for the analysis of rotationally symmetric shells. This method eliminates the loss of accuracy encountered in the usual application of the direct integration approach to the analysis of shells. For the purpose of illustration, stresses and displacements of a pressurized torus are calculated and detailed numerical results are presented.