Conical sleeves, known as collets, have long been used for clamping sliding bodies to cylindrical columns in any desired position. Quantitative expressions for their properties are presented. Of particular interest is the ability of a collet to hold a shaft stationary against axial forces in either direction, and yet be able to release its grip with a minimum of additional force. This treatment relates the firmness of grip, the ease of release and the interface stresses to the cone angle, the coefficients of friction and the applied axial forces. The central part played by hysteresis is revealed. Experiments employing a test model give good support to the theoretical conclusions. They show that, up to the interface pressures reached, about 250 MPa, the friction between lubricated hardened steel and mild steel surfaces is independent of pressure. The strong dependence of friction on surface roughness is demonstrated. The possible adaptation of the collet test equipment to the measurement of friction at high interface pressures is touched upon. It could be a complementary alternative to the more flexible pin-on-disc method. With much larger working areas, it should have the advantages of better defined areas of contact, reduced ploughing effects and less leakage of lubricant.