Summary In accordance with experiments on a horizontal rectangular channel supplied steadily from a reservoir through a contraction, the boundary layer on the bottom is taken as laminar and two-dimensional, and the main stream above it is supposed at every cross-section to possess uniform velocity. Simple alternative shapes are assumed for the boundary layer, as in well-known aeronautical theory. With the aid of the continuity and momentum equations, convergent numerical results are obtained; and it is found that, as the boundary-layer thickens downstream, the free surface falls very slightly and the velocity of the main stream increases. The energy equation is also examined and provides a check on the previous work. The calculated boundary-layer thickness is not inconsistent with measurements made in a channel 14 in. wide and 6 ft long.