The partial-wave equation dσdΩ=|[12ik]l=0L(2l+1)(1- al)Pl(θ)|2 has been used to fit most of the recent π+p differential cross-section measurements above 1 GeV/c. The al were determined by the method of weighted least squares, with the further requirement that they be real and they satisfy either constraints of the form 1>=1-al>=0 (which allows the scattering to be interpreted as purely absorptive) or the more relaxed constraints 2>=1-al>=0. This equation with the requirements does not allow the scattering amplitude to have a spin-flip part or a real part, but for one set of data further terms were added to allow these additional parts of the scattering amplitude. For each differential cross section at the various energies, a set of al values was determined which in almost all cases fit the measured cross sections quite well. These sets of al parameters have two properties in common. First, all al except a0 satisfy 1>=1-al>=0. The a0 parameters (s-wave amplitudes) required 1-a0>=1 except for the higher energies where 1>=1-a0>=0 was obtained. Second, graphs of 1-al versus l (one graph for each different cross-section measurement) show that 1-al decreases rather smoothly with increasing l and that the curve is either roughly linear or concave upward. No striking variations in the al parameters are observed when the energy is close to one of the π+p total cross section resonances. The al parameters are interpreted using 1-al as a measure of the absorption of the lth partial wave by inelastic processes. Differential cross section measurements of π-+p at 2.01 GeV/c and of π++p at 2.02 GeV/c, previously published only in graphical form, are given in the Appendix.