We describe measurements of the specific heat of parahydrogen in Vycor
porous glass, using two different methods, between 6K and 15K. Result
s from the first method are very similar to those from other laborator
ies, showing broad peaks at solidification and melting at temperatures
well below the bulk, and decreasing smoothly at lower temperatures. T
he second set were carried out very slowly and we believe are much clo
ser to thermodynamic equilibrium. They show new phenomena not resolved
in the previous measurements, namely three distinct peaks in the free
zing anomaly, and at lower temperatures a small cusp-like feature whic
h is reproducible in four separate experiments over a period of severa
l weeks. Below the anomaly the specific heat follows a roton-type func
tion, but these measurements cannot determine whether superfluid is pr
esent.