The experimental data of Foley et al. on elastic proton-proton scattering have been analyzed in a phenomenological manner using the impact-parameter representation for the scattering amplitude in the form given by Blankenbecler and Goldberger. The detailed shape and energy dependence of the elastic differential cross section, the small ratio of the elastic-scattering cross section to the total cross section, and the constant total cross section can all be explained with remarkable ease using a smooth weight function with reasonable properties. Two cases were considered. In the first, the scattering amplitude was assumed to be purely imaginary. In order to account for the observed narrowing of the elastic diffraction peak, considered as a function of the square of the invariant momentum transfer t it is then necessary that the radius of the interaction region increase, and its opacity decrease, with increasing energy. The necessary energy dependence cannot be determined uniquely, but is consistent with that expected on the basis of semiclassical considerations. However, some unexpected difficulties are encountered at the lower momenta, and evidence is adduced for the existence of a significant real part of the scattering amplitude for moderate values of |t|. The consequences of a nonzero real amplitude were investigated in a second model. It was found that the shrinking of the diffraction peak can be explained using real and imaginary amplitudes with fixed t dependence, with the magnitude of the former decreasing relative to that of the latter with the energy dependence expected on the basis of potential-scattering considerations. It is therefore not possible to conclude from the observed shrinking of the diffraction peak that a Regge pole mechanism is operative, or even that the effective radius of the absorptive region is increasing, without first establishing experimentally the behavior of the real part of the scattering amplitude for t<0. The observation that |Ref(s, 0)Imf(s, 0)|2<<1, as established using the optical theorem, is insufficient for this purpose. The π--p elastic scattering data of Foley et al. have also been analyzed.