Prior work had shown that smooth eye movements depend both on the motion of the target on the retina and on the subject's expectations about future target motion (Kowler and Steinman, 1979a,b). Effects of expectation cannot be eliminated by making target motions unpredictable (Kowler and Steinman, 1981). The experiment reported here shows that effects of expectations on smooth eye movement depend in a lawful way on the history of prior target motions. Anticipatory smooth eye movements (involuntary drifts in the direction of future target motion) were measured while subjects fixated a stationary target that was expected to step in an unpredictable direction (right or left). Anticipatory smooth eye movement velocity depended on the sequence of steps in prior trials, e.g. velocity was faster to the right when the prior steps were to the right. The influence of prior steps diminished the further back into the past the step occurred. Sequential dependencies were also observed for the saccades used to track the target steps. Anticipatory smooth eye movement velocity was predicted by a two-state Markov model developed by Falmagne et al. (1975) for similar sequential dependencies observed in a manual reaction-time task (button-pressing). The model uses the prior sequence of target motions to predict the subject's expectation, and assumes that the expectation determines anticipatory smooth eye movement velocity. The fit of the model to the data was good which shows that taking expectations into account is both necessary and feasible. Taking expectations into account, quantitatively, allows accurate predictions about smooth eye movement velocity when target motions are unpredictable.