Trace formula and singularities of the $S$ matrix for a system of three one-dimensional particles. Third group integral

A new expression is obtained for the third group integral in terms of two- and three-particle $S$ matrices. For Bose and Fermi statisttes, the result is essentially simpler than for Boltzmann statistics. The singularities of the three-particle $S$ matrix are described compactly. The particles are assumed to be one-dimensional.