Mandelstam's program for constructing the scattering amplitude from its analytic properties and unitarity is analysed in the case of nonrelativistic scattering by a cutoff potential or by a hard sphere. The asymptotic behavior of the scattering amplitude in the momentum transfer plane is obtained, leading to a double dispersion representation for the amplitude. The usefulness of this representation is limited by an essential singularity at infinity in the momentum transfer plane. An infinite system of dispersion relations, connecting each partial wave with all succeeding ones, is derived from the dispersion relation for fixed momentum transfer. The partial-wave amplitudes must be constructed from this system together with the unitarity condition. Possible ambiguities in the solution of this problem are investigated. It is shown that ambiguities in the exact solution affecting only a finite number of partial waves (Castillejo, Dalitz and Dyson ambiguities) do not exist. They would arise, however, in approximate solutions and it would be very hard, in practice, to eliminate them from the exact solution. The ambiguities can be formulated in terms of the positions of the poles of the S-matrix. A series of sum rules which must be fulfilled by the poles is derived. The solution of the system is investigated in the particular case of scattering by a hard sphere. In this case, if one assumes that the exact solution is known for angular momenta larger than some (arbitrarily given) valae, each partial-wave dispersion relation for smaller values of the angular momentum can be exactly solved, and it follows from the sum rules that the solution is unique.