## Extract

1. The horizontal force on a circular cylinder immersed in a stream is familiar as an example of wave resistance. The following note supplies a similar calculation for the resultant vertical force. The problem was suggested in a consideration of the forces on a floating body in motion, the horizontal and vertical forces and the turning moment; but the case of a partially immersed body presents great difficulties. It seemed, however, of sufficient interest to compare the resultant horizontal and vertical forces for a simple case of complete immersion for which the calculations can be carried out. The horizontal force, or wave resistance, has usually been obtained indirectly from considerations of energy, but a different method is adopted here for both components of force and the turning moment. In a former paper the method of successive images was applied to the problem of the circular cylinder, taking images alternately in the surface of the cylinder and in the free surface of the stream. Using these results to the required stage of approximation, the complete force on the cylinder is now obtained as the resultant of forces between the sources and sinks within the cylinder and those external to it. The same method can be applied to any submerged body for which the image sytems are known, and the resultant force and couple calculated in the same manner. The proposition used in this method is that for a body in a fluid, the motion of which is due to given sources and sinks, the resultant force and couple on the body are the same as if the sources and their images attract in pairs according to a simple law of force, inverse distance for the two-dimensional case and inverse square of the distance for point sources. This fairly obvious proposition follows directly from a contour integration in the two-dimensional case; and, in view of the application, the extension is given in 2 when the flow is due to a distribution of doublets. In 3 the horizontal and vertical force on a circular cylinder are obtained by this method, the former agreeing with the usual expression for the wave resistance. The different variation of the two components with velocity is of interest, and the expressions are graphed on the same scale. The additional vertical force due to velocity changes direction at a certain speed, and is clearly associated more with the surface elevation immediately over the centre of the cylinder. In 4 reference is made to the couple on the cylinder. This should, of course, be zero for a complete solution; it is verified that the method used here gives zero moment up to the stage of approximation in terms of the ratio of the radius of the cylinder to the depth of its centre.

## Footnotes

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- Received November 28, 1928.

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