Abstract: In improved equipment for lapping internal surfaces, use is made of a composite lap, the rod of which is held by the tool spindle and inserted with a marginal degree of clearance into a cone that carries the abrasive shell, and a chucking mechanism located beneath and coaxial with the spindle and lap; errors are avoided, particularly with smallbore work, by embodying the lap rod in three rigidly associated sections, namely, a polygonal tip, insertable and fastened with a transverse pin in a matching polygonal socket offered by the cone, a flexible coupling section, and a shank uppermost that is screwed into or otherwise clamped to the spindle. Work to be lapped is held in a chuck suspended from at least three cables threaded through relative peripheral mountings afforded by the chuck itself.