Mr. G. F. Hardy has contributed a valuable paper on this subject (J.I.A., xxiv, p. 95) and (p. 103) has done good service by introducing a theorem by Gauss for the accurate integration of algebraic functions limited to any stated degree, and by showing how the same may be applied to the approximate integration of other functions. Again, Mr. George King (J.I.A., p. 276) has made good use of one of the best of Mr, Hardy's formulas in his Numerical Calculation of the Values of Complex Benefits, practically elucidating the important advantages gained by its adoption. I have read these papers with much interest and have thereby been induced to investigate the matter still further, from my own point of view, in its especial relation to assurance calculations, and with results that may be considered worthy of communication.