When a target has an apriori existence in an area A and a fraction Q of the area is searched, there are well-known expressions for the detection probability when the target is stationary. In this paper the detection probability is worked out for a more important case when the target is in motion. It must be assumed, however, that the target manages to remain in the area in which it has apriori existence by permitting suitable changes in its direction of motion. The detection probability will depend on the ratio of the speeds of the target and the searcher in a complex way. The computation should involve a computer programme but analytical expressions can be approximately derived for I#