The transmission of positrons through neon gas has been studied in the elastic region below the positronium formation threshold at energies from 4 eV to 14 eV, and the same experiment was carried out for argon from 4 eV to 8 eV. By repeating the measurements using three values of magnetic field along the scattering chamber, in the direction of transmission, some sensitivity to individual phase shifts is obtained, and a phase shift analysis has been carried out. The l = 0, 1, and 2 shifts are each parametrized as a two term power expansion in the wave number, while the forms of δ3, δ4 are assumed from theory. The results of the analysis are compared with published theories and indicate the need for more accurate calculations, particularly for argon. The total cross sections deduced for neon are in disagreement with the smaller values measured by Canter, Coleman, Griffith, and Heyland, but the cross sections for argon are in agreement with theirs.