
Recent angular momentum analyses of the three-pion system produced in the reaction ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}p$ have been criticized because the three-pion states used in the analysis do not satisfy unitarity. In order to answer this criticism we develop a set of unitarity equations which can be solved to yield a set of three-pion states which explicitly satisfy both three-particle unitarity and two-particle unitarity (Watson's theorem). A numerical method for solving the equations is given. When the unitary states are used to reanalyze the data for the reaction ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}p$ we find some changes in the detailed numerical fits but no change in the general conclusions. In particular, the phase of the amplitude for the ${A}_{1}$ state varies only slightly in the ${A}_{1}$ mass region, in contrast with the behavior expected for a resonant state.
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