A system for driving first and second accessory machines of a double-bodied turbine engine with an LP shaft and an HP shaft is disclosed. The system includes a first power train disposed between the shaft of the HP rotor and the first machines, which mechanically drives the first machines, and a first coupling means and a second power train between the shaft of the LP rotor and the second machines arranged so as to drive the said second machines. The system also includes a second coupling means between the first power train and the second power train allowing the second accessory machines to be driven by the shaft of the HP rotor, particularly when the engine operates at high speeds. The system may include a gearbox with at least two speed ratios and the first coupling means may be arranged to selectively couple one or other of the two ratios or else to decouple the mechanical transmission from the shaft of the LP rotor.