The orbits of free particles in the neighbourhood of a massive nebular nucleus are calculated on the kinematic theory of gravitation. They are shown to be Keplerian orbits on the dynamic time-scale, but open equiangular spirals on the absolute or kinematic time-scale, and it is shown how they would be so recognized by an observer at the nucleus. The suggestion of E. W. Brown, that nebular arms are the envelopes of actual orbits, is examined in detail, and shown to be untenable. It is shown in fact that the actual case must be the reverse, namely that the orbits cannot have an envelope. The condition for this is found, and shown to give the present positions of the particles forming a nebular arm. This locus satisfies the functional equation for a nebular arm, and corresponds to ejection from a fixed point near the centre of the nebula, in accordance with Jeans's ideas and with a conclusion of Lindblad's. The theoretical locus unwinds for a finite number of convolutions in the sense of the orbital motion, and then doubles back, unwinding thereafter for the same number of convolutions in the retrograde sense. It is accordingly suggested that the spiral structure consists of two parts, one interior to a critical distance, the other exterior. It is further suggested that “barred” spirals are systems for which ν0 , the present number of complete convolutions, is small, and for which much of the material lies near the critical distance. Comparison is made with observation.