When a series ov quantitys proceed by a regular law, ther is no difficulty in interpolating between each ajacent two ov them any number ov terms with any desired degree ov accuracy. The methods to be adopted for this purpos hav been described in varios original and reprinted papers containd in the Jurnl ov the Institute ov Actuarys : (see, for instance, Mr. Woolhouse's paper On Interpolation, Summation, etc., xi, 61 , 301, and xii, 136; Mr. Gray's paper On the Construction ov Tabls by the Method ov Differences, xiii, 61, 149, 293, and xiv, 307; and the translations ov M. Maurice's Essay on Interpolation, xiv, 1; and ov Briggs's explanation ov his Method ov Quintisection, xiv, 73). We somtimes, however, hav to interpolate between quantitys which do not accuratly folio any law. For instance, we may hav calculated premiums for every quinquennial age, and wish to obtain the premiums for the intermediat ages by interpolation. In this case, it wil somtimes happen that the ordinary formulas ov interpolation do not giv satisfactory results, unles we take a very large number ov differences, and then the amount ov labor is offen more than the result is worth. In my paper On the Value ov Anuitys payabl half-yearly, quarterly, etc., I indicated briefly a method ov interpolation which I tho't miht be employd with advantage in such cases (xiii, 322), and it is now my intention to work out the formulas resulting from that method, and giv a practical ilustration ov their use.