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Data for the temperature difference produced by the thermal counterflo~of HeII in long glass tubes of rectangular cross section are given and compared to earlier data from circular tubes. Both geometries display a region of mutual friction described by the Vinen theory. In the circular tubes this region terminates at a critical heat current above which an additional source of dissipation is present. This critical heat current and dissipation are not observed in any of the three rectangular tubes studied. The significance of this observation to the nature of the flow in the circular tubes is discussed.

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