Approximate elasticity relations are derived for the axially symmetric deformation of a thin shell of revolution made of a non-linearly elastic material using the three-dimensional equations of the theory of elasticity. The deformations are assumed to be of the order of a small parameter which is proportional to the square root of the dimensionless thickness of the shell. Terms of the second order of smallness with respect to the deformations are retained in the elasticity relations, as a result of which the equations obtained have an error of the order of the dimensionless thickness of the shell, which is customary in the linear theory of shells. The Kirchhoff-Love hypotheses are satisfied only in the first approximation. The axial compression of a shell, assuming that one of the extreme parallels can freely slide along a plane of support, which is perpendicular to the axis of revolution, is considered as an example. A formula is obtained for the limiting load, which physically and geometrically takes account of non-linear effects in the first approximation.