A unitary assembly supporting an electrode wire guide member and a machining fluid nozzle providing a jet of fluid coaxially disposed with the wire, in a travelling wire EDM apparatus. The unitary assembly is mounted on a plate connected through a universal swivel joint to the lower slide of a cross-slide table. Lateral displacements of the plate cause the unitary assembly to orient itself under the action of a pair of fingers mounted on the plate co-operating each with a ramp mounted on the cross-slide table such as the unitary assembly constantly remains coaxial with the electrode wire.