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Third-sound techniques have been used to measure the thickness of an unsaturated superfluid 4He film as a function of flow velocity. Under the conditions of persistent flow the film thins according to the predictions of Kontorovich. In situations in which a heater is used to continuously drive a film current the measured thinning is in disagreement with the predictions of Kontorovich. A possible explanation for these results is presented. We conclude that the film thinning problem presents no fundamental discrepancy with the Bernoulli equation.