1—Although many investigations have been made on the wave resistance of submerged bodies, no case has been solved completely in the sense of taking fully into account the condition of zero normal velocity at the surface of the body. The simplest case is that of the two-dimensional motion produced by a long circular cylinder, with its axis horizontal and perpendicular to the stream, submerged at a certain depth below the upper free surface. This problem was propounded many years ago by Kelvin, and it was solved later, as regards a first approximation, by Lamb; in that solution the cylinder was replaced by a doublet, and the effect of the disturbance at the surface of the cylinder was neglected. Applying the method of images, I examined a second approximation, and also by the same method obtained a first approximation for the vertical force on the cylinder. Although the problem is not in itself of practical importance, it seems of sufficient interest to obtain a more complete analytical solution, and this is given in the present paper. The solution contains an infinite series, whose coefficients are given by an infinite set of linear equations; expansions are given for the coefficients in terms of a certain parameter, and corresponding expressions obtained for both the wave resistance and the vertical force. Numerical calculations have been made from these for various velocities and for different ratios of the radius of the cylinder to the depth of its axis. These confirm the general impression that the first approximation is a good one over a considerable range. The effect of the complete expressions appears in an increase in the wave resistance at lower velocities and a slight decrease at high velocities; this may be described as due largely to a shifting of the maximum of resistance towards the lower velocities, an effect which might have been anticipated.