Problems in fluid mechanics of essentially irrotational flow are often so complex that numerical methods must be used in the irresolution. A simple, yet generally useful method is the relaxation process, which is based on a network of values of the required function determined by means of the finite-difference theory. The numerical process is simple, and techniques are available for satisfying various types of boundary conditions. The computations are useful in solving problems of efflux and see page and in analyzing flow through transitions or around submerged bodies. Examples of these various types have been solved, and the results are presented as illustrations of the process.