The chief use of this table of the zeros and weight factors of the Hermite polynomials is in the calculation of integrals over the interval [— °°, °° ], when the integrand is either the product of e~* and a polynomial, or may be closely approximated by e~ times a polynomial. The zeros and weight factors, xi and ai, respectively, together with the auxiliary quantities /3i = ai exp [{xf)], which are useful in computation, are all tabulated here for the first twenty Hermite polynomials. The zeros x'?' and 0' are given to 13 or more decimals, and the weight factors a/' to 13 significant figures. Although other shorter tables have appeared, this present table will enable one to cope with problems requiring much higher degree and accuracy, both iii problems involving direct quadratures and those arising in the numerical solution of linear integral equations for the range [— , co ]. Thus the use of this table in any direct quadrature can give exact accuracy as far as the 39th degree (the only inexactitude arising from the use of rounded values of a;'' and ai').