In an attempt to throw light on the complex flow pattern of the North Brazil Current near its point of separation from the coast, the flow in the neighborhood of boundary- and internal stagnation points ("critical points") has been analyzed. The underlying hypothesis is that fluid masses of widely different potential vorticity come in contact near such points. In a one-and-a-half layer idealization of inertial, frictionless flow the key control parameter is the ratio of potential vorticities of the converging fluids. This determines the angle of flow after separation, and, for given buoyancy and depth scale, the volume of each kind of fluid transported past the stagnation point. A quasi-geostrophic calculation also gives a realistic picture of the streamline field near a stagnation point. Using the analytical results, a critical point analysis has been carried out on the observed pressure field of the separating North Brazil Current. The results support the idea of direct leakage along the coast into the Guiana Current. They also suggest a second, indirect route of water transport from the North Brazil Current to the North Equatorial Current, via the interior of the cyclonic gyre between the North Equatorial Counter Current and the North Equatorial Current.