To solve a mathematical problem involving a small parameter, it is customary to expand the solution in powers of that parameter. In singular cases the resultant linearized problem may be insoluble, and in some such cases it is appropriate to expand the solution in powers of the square-root of the small parameter. These cases are associated with bifurcation of the solution. The method is illustrated by applying it to the Falkner-Skan equation and to a problem in hydrodynamic instability. In particular, Hartree's conjecture, that near separation the skin friction vanes like the square-root of the appropriate parameter of the Falkner-Skan equation, is substantiated.