SUMMARY The manner in which the variable x, measuring distance from the leading edge, first enters into the solution of the title problem is elucidated. The structure of the corresponding eigenfunctions is different in each of four regions of the boundary layer and the corresponding contribution to the velocity is weaker than any finite power of — x + Ut, where t measures time and the variable x first enters at x = Ut. A comparison between the predicted variations of displacement thickness and of the skin friction with the computed values obtained by Dennis (2) is favourable. IN part I of this paper Stewartson (6) considered the development of the unsteady boundary layer on a semi-infinite flat plate immersed in an infinite viscous fluid. The plate is given, impulsively, a velocity U at time t = 0 in a direction parallel to its length and thereafter moves in the same direction with the same velocity. It was found that the structure of the boundary layer at a fixed distance x from the leading edge is different, according as T (= Utjx) Jjl. If T 1, x enters the structure, which approaches the steady Blasius form asr->-oo. Physically, the significance of T = 1 is due to the wave-like character of the governing equation when regarded as a function of x and t. This means that the disturbance caused by the presence of the leading edge travels through the boundary layer with the maximum local velocity at any station of x, i.e. the mainstream velocity. Hence, the effect of the leading edge is felt when Ut > x, i.e. T > 1. Two interesting questions which naturally arise relate to the mathematical mechanism by which the variable x enters the solution at T = 1+ and the variable t disappears as T->-OO. An answer to the second question was indicated in the paper but very little progress was made with the first question. Since then the problem has aroused considerable interest and of the subsequent contributions we mention especially the first full numerical solution by Hall (3), the numerical solution of the similarity form of the governing equation by Dennis (2) and the completion of the mathematical description of the solution as T -> oo by Watson. This study has just been