The value of numerical integration in obtaining approximate solutions of an equation of transfer, and the different methods at our disposal, are discussed. It is shown that although the Newton-Cotes method, used by Kourganoff, is better than the Gauss method, used by Chandrasekhar, both are inferior to a new method, the double-Gauss, discovered by the author. The errors in the approximate values of the source-function and the limb-darkening in all three methods are tabulated for various approximations, and illustrated by graphs.
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J. B. Sykes (1951) studied this question.