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Previous models of visual search have hypothesized either a random search or a repeated systematic search strategy. Although both models reproduce well the cumulative search time distribution, F(t), neither fully accords with eye movement data. A new model is proposed in which search is intended to be systematic but suffers from imperfect memory. Systematic search is then a special case in which the memory is perfect, and random search a special case in which the memory is totally lacking. The model was derived for single and multiple occurrences of a single fault (or target) type. Where the model could be proved to be insoluble, a simulation model was used. Simulation results were compared with the previous calculations of Morawski, Drury, and Karwan (1980) and were shown to give identical results for pure random and pure systematic search. As the parameters of the memory model were varied, a family of curves between these extremes was produced.