The main content of this article is a phenomenological treatment of the energy dependence of proton-proton scattering in the energy region 0≤$E_{\mathrm{lab}\mathrm{\le }800}$ MeV, using previously developed two-variable expansions of scattering amplitudes. Instead of turning directly to the experimental results, we make use of the recent Geneva-Saclay phase-shift analysis to calculate the proton-proton partial-wave amplitudes $A_{\mathrm{lsl}\mathcal{’}s\mathcal{’}}^j$ (together with their error bars). These amplitudes are expanded in terms of O(4) transformation matrices in a manner implied by the two-variable expansions. The expansion coefficients, i.e., the O(4) amplitudes, are then obtained by a ${\chi }^2$ minimization, treating the phase-shift values of $A_{\mathrm{lsl}\mathcal{’}s\mathcal{’}}^j$ as data. We find that the O(4) expansions provide a good fit with a reasonable number of parameters, that the fit is quite stable with respect to the variation of the number of free parameters allowed, and that the expansions have a good threshold behavior. The results are sufficiently encouraging to warrant a future reanalysis of all nucleon-nucleon elastic-scattering data, using the O(4) expansions, and to extend the analysis to higher energies than those that have been treated by phase-shift analysis.

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