The problem of axisymmetric deformations of an elastic membrane has been formulated in terms of three first-order nonlinear ordinary differential equations. In this paper, the application of these equations is made to the problem of a circular flat membrane indented by a smooth sphere. The membrane is then deformed into an axisymmetric surface. The deformed membrane is divided into two regions: a region of contact with the sphere, and a region which is free of external load except at the boundaries. The common boundary of the two regions is not known a priori. The nonlinear membrane equations are applied specifically to each region and continuity of stress and deformation are required at the common boundary. Numerical solutions are obtained for the membrane of Mooney model material. Applications are pointed out in the discussion.