This is a sequel to a paper of 3 years ago, which studied the orbits of `comets' near a `sun' regarded as a point source of gravitation according to general relativity. That paper expressed the forms of the orbits in terms of elliptic functions, but its method was not so well adapted to a study of the time in those orbits. In the first half of the present work these orbits and their associated times are described in a simple form, the results being expressed in terms of integrals of elementary functions, which can be easily worked out either by quadratures or by approximation. One result of the earlier paper was the proof that no orbit can have perihelion inside r = 3m, and in the later part of the present work a method is proposed in order to study this region, since no comet can return from it. It is supposed that flashes are emitted both from a distant observatory and from a comet, each signalling the ticks of his clock according to the time it is keeping. These are observed by the other and compared with the time on its own clock. The method serves to describe occurrences between r = 3m and the `barrier' at r = 2m, and it points to some unexpected results in the matter of the comet passing the barrier, which call for explanation.