Prandtl's boundary-layer theory is embedded as the first step in a systematic scheme of successive approximations for finding an asymptotic solution for viscous flow at large Reynolds number. The technique of inner and outer expansions is used to treat this singular-perturbation problem. Only analytic semi-infinite bodies free of separation are considered. The second approximation is analysed in detail for steady laminar flow past plane or axisymmetric solid bodies. Attention is restricted to low speeds and small temperature changes, so that the velocity field is that for an incompressible fluid, the temperature field being calculated subsequently. The additive effects are distinguished of longitudinal curvature, transverse curvature, external vorticity, external stagnation enthalpy gradient, and displacement speed. The effect of changing co-ordinates is examined, and the behaviour of the boundary-layer solution far downstream discussed. Application to specific problems will be made in subsequent papers.