Introduction. While considerable progress has been made in recent years in various stability problems of hydrodynamics (see C. C. Lin [1] and the bibliography contained in the reference) and in the discussion of fully developed turbulence (see G. K. Batchelor [2]), the problems associated with transition from laminar to turbulent motion are in a less satisfactory state. To be sure the phenomenological theory of H. Emmons [3, 4] shows considerable promise, but it replaces explicit dependence on the hydrodynamical equations by general stochastic and probabilistic considerations. Unfortunately, the usual laminar stability theory of the Orr-Sommerfeld equation seems incapable of reasonable extension to the nonlinear regime. In addition, even the linear stability analysis is very difficult and requires delicate mathematical consideration. In contrast to this, the techniques used by E. Hopf [5] to establish the existence of a weak solution for all time of the Navier-Stokes equations in a bounded region are in principle simple and constructive, and it therefore seems reasonable to investigate their practicality for, first, the investigation of the stability problems of hydrodynamics and, second, any insight they might be capable of providing for the transistion problem. In essence, Hopf's method consists of proving the existence of a complete orthonormal set of functions of the space variables which have divergence zero and which vanish outside sets of compact support. A solution to the Navier-Stokes equations is then sought as an orthogonal series expansion with unknown coefficients of time. This, when substituted into the Navier-Stokes equations and sampled by all members of the orthogonal set, leads to an infinite set of ordinary nonlinear differential equations in the infinite number of unknown time dependent coefficients. The existence of a solution of this system is then established by a process of successive approximations involving a square truncation of the system. The idea of the weak solution and Hopf's treatment of the boundary conditions closely parallel those of the direct methods of the calculus of variation. Certainly, if for a given problem, a sufficiently judicious choice of a complete orthonormal set of functions in the space variables could be made so that the main remaining features of the Navier-Stokes equations were to be contained in a relatively few non-