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An investigation is carried out into the structure of the laminar boundary layer originating from the forward stagnation point of a prolate spheroid at incidence in a uniform stream, assuming that the external velocity distribution is given by attached potential theory. The principal new results of the study are: A new transformation of the body co-ordinates is devised which facilitates the computation of the solution near the nose. Two variations of the standard box method of solving the equations are devised to enable solutions to be computed in regions of cross-flow reversal. They are referred to as the zigzag box and the characteristic box. Whereas in two-dimensional flows the effect of the boundary layer approaching separation on the external flow may be represented by a blowing velocity, in the present study we find that this is only true near the windward line of symmetry. Near the leeward line of symmetry the blowing velocity must be replaced by a suction velocity even though the boundary layer is being significantly thickened. The boundary layer over the whole of the spheroid cannot be computed in an integration from the forward stagnation point. The accessible region is largely bounded by the separation line, if α ≤ 6°, and develops a wedge-like shape whose apex is named the accessibility ok, pointing towards the nose of the spheroid. On the windward side of this line the solution develops a singularity; on the leeward side the situation is less clear but it is also believed to occur there. For α ≥ 15° the accessible region on the leeward side of the ok is largely determined by the external streamline through the ok.