The equations of conservation of momentum, energy and mass together with the equations of state are examined for free convection from a vertical paraboloid. A transformation due to Saville & Churchill is applied to the first- and second-order boundary-layer equations, which are then solved using series about the stagnation point, using asymptotic series far up the body and in between by a method due to Merk. The second-order outer inviscid flow is given in terms of infinite integrals as a solution of Laplace's equation in paraboloidal co-ordinates.Eight second-order effects are distinguished, depending on longitudinal and transverse curvatures, the displacement flow, heat flux into the boundary layer and the variation of density, viscosity, thermometric conductivity and the coefficient of expansion with temperature. Expressions for the skin friction, heat-transfer coefficient and various flux thicknesses are obtained and a comparison of the second-order effects is made.