Pion-Pion Interactions in the StatesT=0andT=1

New dispersion relations are derived for the $s$-wave pion-nucleon scattering amplitudes. These relations are specifically chosen to facilitate the task of separating the two-pion exchange term from the other effects contributing to low-energy pion-nucleon scattering. In these equations the contribution from the unknown short-range terms is markedly suppressed, a greater emphasis is placed on the experimentally better established very low energy pion-nucleon data, in particular on the scattering lengths, and in the contribution of the two-pion exchange term the lower energies are more heavily stressed. The terms due to the two-pion exchange, which we isolate, are very clearly recognized by their characteristic energy dependences.