The scattering of ${\ensuremath{\pi}}^{+}$ and ${\ensuremath{\pi}}^{\ensuremath{-}}$ mesons by protons and the production of ${\ensuremath{\pi}}^{+}$ mesons by gamma-rays incident on protons are studied with the quantized theory of pseudoscalar mesons with pseudoscalar coupling. The Dyson transformation is applied to the Hamiltonian to yield a new representation in which the velocity-independent aspects of the nucleon dynamics are more readily identified. A large "core" term appears in the meson-nucleon interaction in this representation and is incorporated into the zero-order Hamiltonian. The remaining interaction terms are treated perturbation-wise or according to the Heitler damping procedure. It is found that even after separating out the large core term, this method of approach fails to give agreement between theory and experiment. It does, however, improve on the results of the straight perturbation approximation and brings the theory in closer accord with the observations. It is suggested that a direct test of the role of the "core" term is possible in the process of double meson production by gamma-rays incident on protons.