A detailed description of the system of peripheral nerve-cells upon the gut and diverticulum of amphioxus (Branchiostoma) is given; it is shown experimentally by means of degeneration experiments that these cells are connected with the central nervous system by their own axons, which run in the dorsal-root nerves. The form and connexion of the cells are described, special attention is paid to the problems of the multinucleate cells in the plexus, and to the occurrence of possible asynaptic connexion between neighbouring nerve-cells. No sheath-cells have been observed upon the peripheral nerve-fibres, either within the atrial plexus or upon the dorsal-root nerve bundles; earlier misinterpretations of the nuclei of the cells of the epineurium around the dorsal nerve bundles are discussed. The origin of the atrial system in ontogeny is discussed; it is suggested that it arises in an analogous manner to the enteric plexuses of vertebrates, by outgrowth from the central nervous system. The part that this system of nerve-cells plays in the life of the animal is not known. Finally, the relation of this system of cells to that found upon the guts of other groups of animals is discussed, and it is concluded that the system is not homologous with the enteric systems of nerve-cells in the vertebrates.