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the isospin-i and -f states. Each fit involved five arbitrary parameters and gave a good x 2 . Since the cross-section errors are mostly of a systematic nature, it is difficult to ascribe an exact meaning to the errors in the parameters derived from the least-squares fit. However, our opinion is that allowing a range of uncertainty of ±2 standard deviations in these parameters is reasonable.If fit I is used to evaluate the asymptotic contributions to the pion-nucleon forward dispersion relations, good agreement is obtained with recent experimental values 2 of the real part of the pion-nucleon scattering amplitude. Fit III also gives good agreement within 1 standard deviation. However, because of the slow convergence of the 77 + and IT" total cross section obtained from fit II, a good fit to the real part is obtained only by going to the limits of the results of the two experiments. Even if we take the most convergent fit to the difference (fit I) and estimate where the Pomeranchuk theorem will be valid to ~0.1%, we obtain E Since the recent success of the Adler-Weisberger relation 1 there has been considerable interest in attempts to saturate current algebra and superconvergence sum rules with a set of single-particle intermediate states. 1 " 5 The superconvergence relations reflect a pure Regge asymptotic behavior which is a multiparticle effect and it is very difficult to see how one could obtain a solution for a range of t with only single-particle states. 4 > 5 But with current-algebra sum rules things are different; here we are dealing with a perturbative amplitude whose high-energy behavior is dominated by a fixed pole in the j plane 4 ' 8 : A(s,t).-+F(t)/sS -oo + superconverging Regge terms, (1) and it is quite possible that a saturation with single-particle intermediate States could generate rather a lot of the structure of the fixedpole term. Of course we need an infinite number of particles if F(t) is to have any singularities in the finite t plane.On the one hand, we could be very optimistic and seek an exact single-particle representation of current algebra in the P z -*<*> frame, 2 but the only solution found so far is the degen-erate-mass case where vector current conservation can be satisfied locally and the completeness properties of the functions describing the particles plays a very prominent role. 3 ' 4 Faced with the lack of nontrivial solutions, Gell-Mann and Zachariasen have tried to find any general restrictions on the possible mass spectra of any solution (assuming one exists), and have failed to do so up to order 1/m 2 . 2 On the other hand, whether one believes in the existence of exact solutions or not, it is clearly interesting to treat the approximation of saturating the sum rules with arbitrarily narrow resonances.In this Letter we demonstrate that the saturation of current-algebra sum rules by singleparticle intermediate states indeed implies a strong limitation on the mass spectrum: m(j) cannot grow asymptotically faster than j, the spin of the particle.In order to avoid as much co...

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