An analytical study is made of non-equilibrium effects on hypersonic, inviscid flow over slender, axisymmetric bodies. Also, two-dimensional results are obtained for the purpose of comparison. The rate process under consideration is that of molecular vibration of the gas. The exact problem is solved by successive approximations based on a double-expansion scheme involving two small parameters: one represents the fact that the bodies considered are slender; the other represents the facts that the vibrational internal energy is small in comparison to the total enthalpy. The exact differential equations and boundary conditions are simplified to the hypersonic-small-disturbance-approximation form. The unknown quantities in this approximate problem are expanded into series of the small parameter, (γ − 1)/(γ + 1), which is ⅙ for a diatomic gas. In this formulation it is found that the classical hypersonic similitude can be extended by slight modifications to cover the added consideration of vibrational non-equilibrium. The modifications introduced are the normalization of all lengths by the characteristic relaxation length of the gas and the addition of a new dimensionless parameter, which is a measure of the excitation level of the vibrational internal energy in the flow field. Explicit, uniformly valid solutions are obtained for the specific problems of flow over a slender cone and of that over a thin wedge. The successive approximations are carried as far as necessary to show the non-equilibrium effect, which differs in order of magnitude for the various flow quantities. One interesting feature of the solutions is the non-monotonic behaviour in the relaxation of the surface pressure of both the cone and the wedge, in contrast to intuitive expectation. The result for a 20° cone in a free stream of oxygen at 300° K and a Mach number of 15 is displayed and compared with the numerical solution of the exact problem using the method of characteristics.