This problem was discussed and formulated for machine computation by John von Neumann and others. His own original draft of a discussion of the differential and difference equations is given in Appendix I, and an iteration scheme for solving systems of linear equations is given in Appendix II. In the main body of this paper we shall outline the derivation of the equations employed by the computer and refer to these appendices for detailed discussions concerning them where necessary. Some of von Neumann's difference equations were modified in the course of the work. The reasons for these modifications and their nature will be enlarged upon in the course of the discussion. 2. The Equations of Motion and Boundary Conditions. We denote by x and y the Cartesian abscissa and ordinate of a point in a fixed coordinate system in a vertical plane oriented as in Fig. 1; that is, x and y are Eulerian coordinates. The velocity of the fluid at this point at time t will be said to have x and y components u(x, y, t) and v(x, y, t), respectively. The density of the fluid will be denoted by p, the pressure by p, and the acceleration of gravity by g. The system of equations describing the motion of an incompressible fluid subject to the force of gravity in the vertical direction is then