One of the rails in a contact rail, suitably the inner rail (12) is provided over its length with a linearly increasing electric resistance that can be acquired as a measurable value at any point along the rail. A preferred embodiment has for this purpose a metallic conductor (11) helically wound around an isolating body (10). This inner rail (12) is held in an outer rail (13) with an upper U-shaped cross-section, and is separated therefrom by an isolating layer (14). The helically wound metallic conductor (11) has a considerably higher resistance compared to a massive rail, suitable for applying a measurement method based on the measuring bridge principle for locating a drop wire that has fallen on the contact rail in case of thread breakage. The distance between the fallen drop wire and one end of the rail can be determined from the relationship between the resistance of the whole inner rail (12) and the resistance of the section of inner rail extending from one end of the rail to the fallen drop wire, this resistance relationship being determined by the electric measuring bridge.