Superfluidity Without Superflow in Unsaturated Helium Films

It is shown experimentally that the superfluid fraction $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$ is continuous and finite at the point at which superflow vanishes in unsaturated helium films. It follows that there is a region of superfluidity without superflow. In addition it is shown that the behavior of the partial molar entropy may account for the disappearance of superflow without requiring that $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$ vanish.