Quasi-geostrophic equations are derived, and the limitation of the approximations is discussed on the basis of a scale analysis. It is found that friction at the sea bottom is an important dynamical factor which determines an intrinsic time scale of importance to the accuracy of the equations. Analytical solutions are derived, giving the current resulting from specified external forces, and a comparison is made to results reported elsewhere. A special solution is shown to represent a trapped wave resembling a Kelvin wave, but with a reduction in phase speed caused by the implicit approximations. The implications of this solution are discussed in the concluding part of the paper, together with some remarks on a numerical model based on the quasi-geostrophic equations.