A theoretical analysis is made of the flow of an incompressible non-Newtonian viscous liquid in a curved pipe, looking for differences in observable characteristics from the corresponding case of Newtonian flow. Such a motion is of interest to the experimentalist because the flow could be readily attained and controlled in practice, the most easily measurable quantities being the axial pressure gradient and the volume rate of flow. It is assumed, for the purpose of mathematical analysis, that the curvature of the pipe is small, more precisely that the radius of the circle in which the central line of the pipe is coiled is large in comparison with the radius of the cross-section. A solution is developed by successive approximations, the first approximation corresponding to the flow of a Newtonian viscous liquid as given by Dean (1). The streamlines in the plane of symmetry and the projection of the streamlines on a normal section are compared with those of a Newtonian liquid.