Solid helium can apparently grow on certain surfaces while on others it is inhibited and solidification takes place away from the adsorbent walls. In this paper we present experimental data on helium inside grafoil, on which surface s&d He-4 appears to be able to grow, and also inside the pores of Vywr glass in which evidence suggests that the solid phase forms in the centre of the pores. We present data on both the specific heat and the pressure-temperature relation for these two situations. Melting takes place at pressures below those of the bulk melting curve in the case of grafoil, and above in the case of Vycor. An analysis is made of the measured heat capacities below the melting anomalies, and their temperature dependence indicates that inside both grafoil and Vycor a possible explanation ls that the main excitations present are rotons wlth an energy gap of about 6K to 7K. This would indicate the presence of a liquid phase down to our lowest temperatures, and this observation is backed up by the fact that the entropy under our latent heat anomalies is less than the value expected for bulk helium of the same mass.