The present note continues the discussion, begun in the first paper with the above title, of classes of spaces H which are extensions of L2 (0, ∞) and whose elements, which we call ‘sequence-functions’, exhibit some of the properties of distributions. The previous paper defined the spaces, and described how they can be used to extend the domain of definition of Waston transforms. Further related applications to transform theory are described in [2] and [3]. In this paper I pursue the analogy between these sequence- functions and other types of generalized functions further by discussing their local behaviour, the existence of ordinary and convolution-type products, and of derivatives and integrals.