In an earlier study of the uses of moments and cumulants in the specification of statistical distributions, the authors developed explicit asymptotic expansions, expressing any desired percentile point of such distributions in terms of known cumulants. The general formulae are now presented as far as the sixth adjustment, based on the eighth cumulant, and also numerical tables showing the coefficients of all terms for ten chosen levels of significance over the range 0.5 to 0.0005 (single tail), together with the first five Ilermite polynomials and tables for the common tests of significance, χ2, t and z. at the same levels.