the measured quantities and the vacancy density is. It would be necessary to directly measure, say, the pressure due to the vacancies [@Steve], and then compare this with (\[pee\]). In such an experiment it would be crucial to take the hard-core effects into account. Work is in progress to calculate the coefficient $b_2$ for the hcp lattice; however, the results are not expected to differ much from those given for the simpler cubic lattice here.

The results for the two-dimensional system might have relevance for low-density $^4$He films on textured substrates [@Rehr]. There, the presence of adsorption sites localizes the Helium particles on the points of a two-dimensional lattice, defined by the substrate. Thus the tight-binding approach used in this paper becomes applicable. The adsorption sites cannot hold more than one particle, so that the hard-core repulsion is also present.

The authors thank J. Goodkind for making available some of his experimental data, and for useful discussions; useful discussions with A.F. Andreev and S. Steel are also acknowledged. Part of this research was supported by the Natural Sciences and Engineering Research Council of Canada, and the ‘Stichting voor Fundamenteel Onderzoek der Materie’ (FOM), which is financially supported by the ‘Stichting 