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Gauss' method of mechanical quadrature has the advantage over most methods of numerical integration in that it requires about half the number of ordinate computations. This is desirable when such computations are very laborious, or when the observations necessary to determine the average value of a continuously varying physical quantity are very costly. Gauss' classical result 2 states that, for the range (-1, +1), the "best" accuracy with n ordinates is obtained by choosing the corresponding abscissae at the zeros xi, • • • , x n of the Legendre polynomials P n (x). With each Xi is associated a constant ai such that (1) I f(x)dx ~ ai/Oi) + a 2 f(x 2) + • • • + a n f(x n). The accompanying table computed by the Mathematical Tables Project gives the roots Xi for each P n (x) up to n = 16, and the corresponding weight coefficients a*, to 15 decimal places. The first such table, computed by Gauss gave 16 places up to n-l. z More recently work was done by Nyström, 4 who gave 7 decimals up to w = 10, but for the interval (-1/2, +1/2). B. de F. Bayly has given the roots and coefficients of Pu(x) to 13 places. 5 The Gaussian quadrature formula for evaluating an integral with arbitrary limits (p, q) is given by

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