Phys. Chem. Liq. 2, 213 (1971). 12 C. A. Tracy and B 0 M. McCoy, Phys. Rev. B JL2, 368 (1975).We report the first observation of persistent currents in unsaturated superfluid 4 He films on a large-diameter glass substrate in an open geometry. Although such currents have been observed in filter material 1 and Vycor, 2 previous attempts to observe these currents in similar open geometries have not been successful. 3 ' 4 This led to the suggestion that high-quantum-number circulation states may be very unstable. 4 Recent experiments 5,6 in saturated films have shown this suggestion to be incorrect for those films. We present here the first measurements of the stability of persistent currents in unsaturated superfluid films as a function of film thickness at constant temperature and observe a strong thickness dependence for films thinner than about 12 atomic layers. Currently it is believed that thermally activated dissipation is responsible for the decay of persistent currents near critical velocity. 7 The observed behavior of our decays places strong restrictions on models used in this theory in the case of films. The decay of the flow velocity becomes immeasurably small for films thicker than 12 atomic layers. For these thicker films we find the circulation stable with respect to translation and to changes in the film thickness through condensation and evaporation.The persistent currents are produced on the outside of a cylindrical Pyrex glass ring with posts 13 A. . attached to it schematically shown in Fig. 1(a). To produce the thin films investigated here the entire apparatus is enclosed in a copper cell where the pressure is maintained at a value below the saturated vapor pressure at the chosen operating temperature of 1.45 K. The film thick-GLASS .SUBSTRATE HEATERS ^^" ^\ Nef^5 ^^^^ HEATER S H f^^^L^C^AL. STRIPS V S2 ~4 , Is 1 " 0 -^A 1 V S 3 V S1 FIG. 1. (a) Pyrex glass ring substrate of average perimeter l t +l 2 =21.15± 0.2 cm. The long-to shortflow-path ratio l 2 /l t is 15.63 with height of 0.73 cm and thickness of 0 o 24 cm. The aluminum strips are on the outer side and directly opposite the two posts. (b) Schematic of the flow paths. We report the first measurements of the stability of persistent currents in unsaturated 4 He films in an open geometry. Persistent currents are generated without rotation and exhibit several features which place strong restrictions on models of thermally activated dissipation in thin films. 1484