  Starting from hyperbolic dispersion relations for the invariant amplitudes of
pion-nucleon scattering together with crossing symmetry and unitarity, one can
derive a closed system of integral equations for the partial waves of both the
s-channel (pi N --> pi N) and the t-channel (pi pi --> Nbar N) reaction, called
Roy-Steiner equations. After giving a brief overview of the Roy-Steiner system
for pi N scattering, we demonstrate that the solution of the t-channel
subsystem, which represents the first step in solving the full system, can be
achieved by means of Muskhelishvili-Omn\`es techniques. In particular, we
present results for the P-waves featuring in the dispersive analysis of the
electromagnetic form factors of the nucleon.
