Quantization Conditions for Linear and Nonlinear Trajectories

It is shown that the relation between linear trajectories of opposite normality found by Ademollo, Veneziano, and Weinberg also holds for nonlinear trajectories which occur in a generalization of the Veneziano form. It is also shown that half-integer spacing of trajectories of opposite normality which are connected by pion emission is possible only if the lowest spin particle on the upper trajectory is missing. Experimentally, there is a distinct absence of such particles (e.g., low-lying ${\mathrm{½}}^{-}$ baryons and baryon resonances).