Importance of Tollmien's counter example

Efforts to construct a general theoretical basis containing the essential f eatures of Tollmien's counter example to the sufficiency of Rayleigh's theo rem on point of inflexion have resulted in the determination of a pair of u pper bounds of the rate of growth of arbitrary unstable disturbances; where as, the necessary condition of the existence of these upper bounds have pro vided access to a sufficient condition of stability in its simplest form in the equilibrium of homogeneous incompressible inviscid parallel shear flow s that are not known as yet and go beyond the works of Rayleigh [1], Tollmi en [2], Friedrichs [3], Fjortoft [4], Holland [5], Howard [6, 7], Hickernel l [8], and Banerjee et al. [9]. An alternative proof of the result that a w ide class of such flows could be made stable by bringing the boundaries suf ficiently dose, although the flow has a point of inflexion inside the domai n of flow with the Fjortoft's criterion satisfied, which is derived by Draz in and Howard [10] from variational formulation of the problem follows as a n outcome of the expressions of these upper bounds. The counter example has played the role of a forerunner for much of the development that followed in its wake after 1935, and the present succession of papers is especially undertaken to investigate the trail left behind by the counter example and, it is hoped, to arrive at a necessary and sufficient condition of stabilit y in its simplest form, which is still missing in the literature on the sub ject.