ponding positions in the two compartments of the formation just given. The joint-life annuities also, with which we have to do, occupy positions corresponding to the others in the formation on pp. 1, li. Thus, to verify the rows opposite 92: 92'92 92'93 92'94 92'95 92'96 92'97 476990 534078 599527 670603 742292 800332 476990 422532 360285 292917 292917 225212 170541 *953980 956610 959812 963520 967504 970873 The annuity values are : *5800 *4898 *3798 *2524 '1157 '0000 and Orchard's Table gives for the corresponding assurances : *95398 95661 *95982 96352 96751 97087 This will no doubt be considered a sufficiently close correspondence between the two sets of values. The two formations will, however, stand a yet severer test. If we determine the assurances from the annuities by the formula, Axy=I—(IV)(1 +Ax·y), (log (1+ a.) being found in the annuity formation,) we obtain, 0.198668 0.173120 0.139815 0.097745 0.047538 0.000000 2-464284 2-464284 2-464284 2-464284 2.464284 2-464284 2-662952 2-637404 2-604099 2-562029 2-511822 2-464284 *046021 '043391 *040188 *953979. 956609 *959812 *036478 032495 *029126 *963522 *967505 *970874 and we have now an all but perfect agreement in the sixth place. This method of verification will be available when the work is undertaken by a single computer; but it may also be occasionally had recourse to when the work is done in dupli cate. The final arrangement of the results will be analogous to that of the annuity values, pp. 141, &c.; and they will occupy the entire quadrangular space. The values in which y will the values of a be put at the top and those of y at the side, the values of the function which occupy the rows in the first compartment of the formation will here take their places, also in rows, above the diagonal, while those that occupy the rows in the second compartment, will take their places in columns below the diagonal. This will be understood by reference to the following specimen of the final arrangement : Values of Assurance of a Unit, on (x) against (y). (x at the top, y at the side.) 55402 60719 66551 72674 5189157425 63567 70154 76670 89 90 2 2 47699 53408 59953 67060 74229 80033 95 29292 34070 40365 48351 57730 66163 95 The specimen forms the concluding portion of the last six columns. It is composed of the values formed in the examples on pp. lxxx to lxxxiii. Enough has now been done, in regard to the function under consideration, to enable any one, otherwise qualified for the task, easily to complete the three per-cent. formation, or to undertake the like at either of the other rates of three-anda-half and four per-cent. It does not seem too much to expect that among the younger members of the profession, who hope yet to occupy higher positions in it, there are to be found some who, recognising the claim their profession has upon them, as so well set forth in the passage from Bacon which forms the motto of the Journal, are willing to devote a portion of their spare * "I hold every man a debtor to his profession, from the which as men of course do seek to receive countenance and profit, so ought they of duty to endeavour themselves by way of mends to be a help and ornament thereunto." time to the extension of the deductions from the tables which have, at the cost of so much time, labour, and expense, been prepared and put forth by the Institute. Such a course would be remunerative in various ways:-First, it would beget the satisfaction arising from the discharge of a duty which, by hypothesis, they feel to be incumbent upon them. Secondly, it would lighten and facilitate their labours when they come to occupy the positions to which they aspire, by increasing the stock of material required for the proper and satisfactory conduct of their business; and, thirdly, it would operate to their more immediate advantage by imparting to them a dexterity in manipulation and a confidence in the use of tables not easily acquired by other means. The present Introduction has grown to what it is feared will be considered by many, an undue length. The object mainly kept in view in its preparation has been, to show with what facility, by proper arrangement, and with the aid of the materials contained in the present volume, the tables required for actuarial use may be constructed: it is hoped that it will be found to contain little which does not bear more or less closely on that object. P. G. EXPLANATION OF THE ADJUSTMENT OF THE TABLES. BY W. S. B. WOOLHOUSE. THE Experience Committee having intimated a desire that I should adjust the graduation of the Tables of Mortality for Healthy Lives, the task undertaken by me is completed, and, for the information of the Committee, it only remains to accompany the resulting Tables by an explanation of the details of what has been done. The subject being one of very great practical importance, inasmuch as the adjusted tables are designed to form the basis of an extensive superstructure of other tables, I have given to it a careful and independent consideration, and, after testing various schemes, I have at length succeeded in devising a method that may be regarded as efficient and satisfactory. As suggested in my paper "On the Construction of Tables of Mortality," the number-living at each age is the most manageable element for final adjustment; at ages beyond the limiting age of the Table it is at once conveniently, as well as accurately, put down as zero, a practical facility that cannot be over-estimated; and it has this essential advan * Journal, vol. xiii., p. 95. tage, that precisely the same aggregate tabular mortality or decrement must necessarily be retained between all points of actual coincidence, in whatever way the intermediate numbers may be modified. As a consequence of this principle the number of such coincidences with experience in the curve of the number-living may be regarded as one test of close adjustment and substantial exhibition of the actual mortality. The method I have adopted may be briefly stated, and the rationale of the process and its accurate adaptation to what is chiefly required will be at once apparent. The data, for the reasons already stated, are the numbers-living at successive years of age as deduced, without any adjustment, from the original facts. If we begin at the first age in the Table and extract the numbers-living at quinquennial intervals, that is, according to the usual notation, 40, 15, 20, 25, .... we can by the formula for interpolation determine all the intermediate values at the other ages, and so obtain a complete series of values that shall be continuous. Geometrically speaking, we shall thus pass a continuous curve-line through the indicated quinquennial points. Against the adoption of such curve-line as the basis of the final Table there is manifestly this tangible objection, that the numbers at the ages 10, 15, 20, 25, .... are made use of exclusively, and that the original numbers between these ages are wholly ignored as data. This rather material objection, which is inherent in other methods of adjustment, is entirely removed by varying the epoch of the adopted quinquennial data, that is, by taking the five distinct series hereunder stated, viz. : then by separately interpolating the intermediate values for each of these series, and by finally taking the arithmetical average, or mean value, of the five completed sets of results, the series of adjusted values is obtained. Reverting again to a graphical illustration, all the points of the original data are thus occupied by five distinct curves, assimilating to the experience and to one another, and forming in combination a sort of network; and at every age the resulting ordinate of the adjusted curve is |