ABSTRACT In an attempt to understand the mechanism governing separation the possibility of there being further solutions to the Falkner-Skan equation f ′′′ + ff ″ + β(1 − f ′2 ) = 0, in addition to those found by Hartree, is investigated. It is found that when β < − 0·1988 there are no solutions satisfying | f ′| ⩽ 1 for all η, and that if 0>β> − 0·1988 there are two acceptable solutions, one with f ″(0) < 0. The new ones are computed to three places of decimals for various values of β and tabulated. In addition, it is shown that if − 0·5 < β < 0 there is a family of solutions corresponding to boundary layers bounded on one side by free streamlines. These are also computed and graphs of f ′ (0) and δ 1 are displayed. The impact of these solutions on the theory of separation is discussed.