THE study of the vibration of a straight elastic bar is a very old one. The vibrations may be classified as lateral (or transverse) torsional and longitudinal. The first two are the more important ones and also the more difficult. We shall consider the lateral and torsional vibrations of a bar clamped at one end and free at the other, which may rotate like the blades of an airscrew. Both cases, rotation or none, are important problems of civil engineering, marine and aeronautical engineering and the theory of elasticity.