Precessing accretion discs have long been suggested as explanations for the long periods observed in a variety of X-ray binaries, most notably Her X-1/HZ Her. We show that an instability of the discs response to the radiation reaction force from the illumination by the central source can cause the disc to tilt out of the orbital plane and precess in something like the required manner. The rate of precession and disc tilt obtained for realistic values of system parameters compare favourably with the known body of data on X-ray binaries with long periods. We explore other possible types of behaviour than steadily precessing discs that might be observable in systems with somewhat different parameters. At high luminosities, the inner disc tilts through more than 90 degrees, i.e. it rotates counter to the usual direction, which may explain the torque reversals in systems such as 4U 1626-67.
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