Forward dispersion relations, indexed by continuous parameters and 6J0 , are written for the class of functions B+ ( c o ) = F± (cu) / { ( oj ±co,) ' (oj ± cuo)' 3 , oijS<i , o< cu«<oj, , i defined in terms of the K N forward scattering amplitude F+= A+ + i D+ . The magnitude of the dispersion integral involving F over the 'unphysicalf region of energies below the K N threshold is sensitively controlled by yß and Ct)0 . Thus, this class of dispersion relations is well suited to test the

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... ation of a model amplitude into the unphysical region. It is found that none of the models tested -a constant scattering length analysis (6 parameters), a zero-range K-matrix analysis (9 parameters), and an effective-range K-matrix analysis (A4 parameters)adequately represents F when extrapolated to unphysical energies, but that the first two of these models are superior to the last. The data currently available on D_ is seriously in doubt; with substantially improved D_ data, the methods described in this work would almost certainly provide a valuable test of the extrapolations to unphysical energies of the models for F . STATEMENT