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Measurements have been made of the relative superfluid density psp/ p and the critical mass currentIsc for the onset of dissipation in the superfluid flow of pure 4He and 3He/4He mixtures in the circular cylindrical pores of Nuclepore near the h transition, using a double-ended Helmholtz resonator. Pore diameters of O. 03, O. 1, and 0.4 tzm were studied using 3He mole fractions of O. O, ~ 0.2, and 0.4. For each pore diameter dp and mole fraction x our results for psp/ p were • 0 675 fitted relattvely well by the form C(t-to1) " and those for Isc by the form I~co(t-to~) g, where t=-[Tx -T]/Tx. For each combination of dp and x the reduced onset temperatures to1 and toc are in rough agreement with each other but somewhat larger than values found by others for Nuclepore. For the smaller two pore diameters, these onset temperatures are consistent with the unithat near Tx, h p~b~/m4kBT should be a constant versality hypothesis 2 e independent of T and x. Here h is the reduced Planck constant, P~b is the bulk superfluid density, ~ is the coherence length, m4 is the 4He atom mass, kB is Boltzmann ' s constant, and T is the absolute temperature. At each value of x the amplitudes C agree well with the corresponding amplitude for Psb/P, and g equals 1.17 + 0.04 for all cases. Measurements of the velocity of second sound at x = 0. 000, 0.190, 0.401, and 0.450 are also reported.

A numerical evaluation of the average superfluid density of 4He in a cylindrical pore near T a predicted by the Ginzburg-Pitaevskii-Mamaladze theory has been carried out as a function of pore radius and temperature. The results are presented in graphical form and in terms of an accurately fitting closed algebraic form.The only detailed prediction for the behavior of the superfluid density ps of 4He in a cylindrical pore of which we know is provided by the free-energyexpansion theory of Ginzburg and Pitaevskii 1 as modified by Mamaladze.2'$ Although frequent reference has been made in the literature to the temperature To at which ps disappears in the pore and its increasing depression below the bulk lambda temperature Ta as the pore radius is decreased, we are not aware of the publication of any calculation of the explicit dependence of as on T below To for comparison with experiment. § To fill this gap we have calculated the average value 0,p of p, over the cross-sectional area of the pore as a function of temperature and pore radius predicted by the Ginzburg-Pitaevskii-Mamaladze (GPM) theory.In the GPM theory p~(r) is given locally by malO(r)l 2, where m4 is the mass of the 4He atom, and ~O(r) is an order parameter which satisfies the differential equation -(h2/2m4) V24, -ag, +/31¢]2~ = 0(1) subject to the boundary condition that ~ should vanish at the walls of the

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