Tables that have been used as a reference for nearly 50 years for the intensity and polarization of reflected and transmitted light in Rayleigh scattering atmospheres have been found to be inaccurate, even to four decimal places. We convert the integral equations describing the X and Y functions into a pair of coupled integro-differential equations that can be efficiently solved numerically. Special care has been taken in evaluating Cauchy principal value integrals and their derivatives that appear in the solution of the Rayleigh scattering problem. The new approach gives results accurate to eight decimal places for the entire range of tabulation (optical thicknesses 0.02-1.0, surface reflectances 0-0.8, solar and viewing zenith angles 0°-88.85°, and relative azimuth angles 0°-180°), including the most difficult case of direct transmission in the direction of the sun. Revised tables have been created and stored electronically for easy reference by the planetary science and astrophysics community.
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