We combine new data on fo(S*) production in J/+ and D, decays with earlier information on central production and elastic PP, KK processes to make a fresh examination of the f o ( S * ) resonance. The key feature of our amplitude analysis is its strict enforcement of unitarity. This allows the good energy resolution of the new J/+-&T~P(KE) data to play its full role in delineating the fo(S* ) resonance structure that experiment demands. This enables us to distinguish alternative resonance mechanisms that have been proposed: we conclude that fo(S* ) is most probably not a KE molecule, nor an amalgam of two resonances, but a conventional Breit-Wigner-like structure. In this preferred description, the fo(S* ) has rather a narrow width (To-52 MeV) and comparable couplings to av and KZ. Possible spectroscopic interpretations are considered. PACS number(s): 14.40.Cs, 13.20.Fc, 13.20.Gd, 13.25. + m D. MORGAN AND M. R. PENNINGTON 48 ture on the continued scattering amplitudes. This sheet structure arises because the functional form of the scattering amplitude, in fact, depends upon the c.m. momentum of the opening channel.The paradigm case is the simple Breit-Wigner resonance with For narrow resonances remote from relevant thresholds the various partial widths r, can be well approximated by constants; in general, unitarity requires energydependent partial widths-for S-wave channels typically of the form r, =k, y, , with k, the corresponding channel c.m. momentum and the reduced width y, roughly constant. Since k, and -k, correspond to the same energy, yet give different values for the continued scattering amplitude, Eq. (1.1), we need to distinguish these. It is this specification of the signs of the momenta that defines the sheet structure of the energy plane. In the present discussion we are mainly concerned with two channels, n-n-and KC and we label their corresponding c.m. channel mo-