The presence of the cosmical constant introduces a fundamental scale and prevents there being any simple self-symmetry. Henriksen, Emslie and Wesson (HEW), who studied spherically-symmetric models with a positive cosmical constant, have, however, demonstrated the possible existence of a self-similarity of the second kind and identified the similarity variable. They obtained interesting analytic solutions which are homogeneous in density, but not in pressure. We have extended this work and investigated the general behaviour of these cosmological models which possess a self-similarity of the second kind and in which the requirement of homogeneity is relaxed.