wild goose chase (n.)

"pursuit of anything in ignorance of the direction it will take," hence "a foolish enterprise," 1592, first attested in "Romeo and Juliet," where it evidently is a figurative use of an earlier (but unrecorded) literal sense in reference to a kind of follow-the-leader steeplechase. Wild goose (as opposed to a domesticated one) is attested in late Old English (wilde gos).