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Total scattering cross sections have been measured for 0.3 -31 eV positrons colliding with helium atoms and for 0.25 -24 eV positrons colliding with neon atoms using a beam-transmission technique. These measurements have resulted in the first direct observations of Ramsauer-Townsend effects for positrons colliding wih helium and neon, and also provide clear indications of the onset of positronium formation in each of these gases. As an overall check of the reliability of the experimental approach for measuring total scattering cross sections for positrons, electron-atom scattering cross sections have been measured in the identical apparatus, using the same target gases, and the same technique as was used for the positron measurements.