The Hill model predicts that the transpolar potential (ΦPC) saturates for strong solar wind electric fields (ESW) and saturates at higher values for larger solar wind ram pressures (PSW) and/or lower ionospheric Pedersen conductances (ΣP). Three months of Defense Meteorological Satellite Program (DMSP) measurements of polar cap potentials have been compared with predictions of the Hill model. Data from representative days that span a wide range of interplanetary and solar-cycle conditions were selected for presentation. In the comparison we augmented the Hill model by adding a constant term to the magnetospheric potential and expressing ΣP in terms of the 10.7-cm solar radio flux (F10.7). Temporal variations in observed ΦPC are in good agreement with the predictions of the augmented Hill model. Comparison of two events with significantly different average PSW but similar F10.7 and ESW found that DMSP observed higher potentials when PSW was larger. DMSP also observed higher potentials when ΣP was lower, when comparing two events with significantly different F10.7 (solar minimum versus solar maximum) but similar PSW and ESW. Both comparisons agree with the predictions of the Hill model.