Similar solutions of an equation governing the flow in the plane of symmetry of a corner boundary layer with favourable pressure gradient are extended to a “critical” value of the pressure gradient for which no solution could be found previously. It is shown that the failure to achieve this result in the past was due to the “singular” nature of this solution rather than to its non-existence as one was tempted to suspect. The existence and uniqueness of this crucial solution are demonstrated and the solution itself is obtained to a high degree of numerical accuracy. Criticism of the theory on which this corner boundary-layer equation is based is discussed in an Appendix.