y Formation of ary, and Dary x-y=0 x-y=1 x-y=2 15 298112 19.8661 294938 19 7214 291732 19:5764 984760 1447 984713 319429 2804 316383 313338 14 325239 2646 322286 304200 20 1465 301132 200047 297918 19.8572 984627 1418 984693 1475 984646 1482 319239 17 II ΙΟ 330576 2386 327907 314914 20 6497 312367 20.5290 983691 1207 983983 1364 984159 1475 319143 20 8518_317006 207494 314422 20 6264 982896 1024 983294 1230 983586 1391 1561 337442 14 294664 19.7090 291489 19:5654 288522 19°4322 316098 61 13 313053 85 1332 983809 1218 310294 21 300642 199822 297343 19.8309 294124 19-6845 321810 318668 41 315622 23 308289 20 3371304943 20*181c 1567 329051 325906 80 IO 315943 20 6987 312964 20 5572 309728 20:4046 The column headed 15, p. 141, is copied out, (the value of a15-15 being taken from p. 140,) and differenced. On the right a15-10 is formed as shewn, and the successive terms a511, &c., are formed by the continuous addition of two series of differences, of which the first is Aa,, formed on p. lxii, and the second is Ary, formed in the column adjoining on the left. 3. To form Aryary. This difference comes into use only in the construction of the initial terms of the several columns in the main formation; and in these r=y. The formula consequently for this case becomes, and the series to be differenced will be that on p. 140, headed ax.x The formation of Arrar was, like that of Aary superposed upon the joint-life computations; and on reference to the specimen on p. lxiv, the values of the former of these will be found in the first column, (in which r=y,) under those of the latter which fall in the same column, but necessarily in reverse order. The following is an example of the formation of the initial terms of the several columns: Of the two series of differences here employed the first is 2Aa, the double of the series formed on p. lxii, and the second is Arrar, whose formation has just been described. The principal formation, like others that have preceded, occupies a triangular space, having 87 columns and 87 rows. The initial portion of the first six columns is here given. The values of r are at the side and those of y at the top. The initial terms were first inserted, and next the verification series. The series Aa, (8469, 8189, &c.) was then entered in column 10; and the successive terms of it were carried out horizontally, as shewn. The series Aar, were then inserted, that which in the specimen (pp. lxiv, lxv,) occupies the last row, in the first column; that which occupies the next to the last row, in the second column; and so on. The numbers opposite the several rows, in fact, indicate the columns in which the differences in those rows are to be entered, respectively. The final additions were then proceeded with, without interruption. The method employed in the construction of this table is in description somewhat tedious; but it will not be considered so in application, by any one accustomed to the construction and verification of tables, when it is mentioned, that the time. occupied in the construction of a complete table did not exceed 46 hours. Of this time 24 hours were required for the formation and insertion in their places of the initial and the verification series, and the remaining 44 sufficed for the formation and transference of the differences, and the final additions. It may not be out of place to point out that while, as in the case of a joint-life annuity and its corresponding assurance, we may pass from a last-survivor annuity to a last-survivor assurance either by the usual formula Ay=I—(1—v)(I+Ary), or by the use of Orchard's Table, a like analogy does not hold in regard to the mode of passing from a whole life to a deferred last-survivor annuity. The component annuities here, a+α-ar must be dealt with separately. 3. Of the Construction of Survivorship Assurance Tables. The present volume contains no Survivorship Assurance Tables. A complete set of such tables would form a most desirable addition to those that are here given. The Council have no immediate intention of undertaking their construction, what they have done, in connexion with the HM table at least, being sufficient to admit of the exact treatment of all cases in which not more than two lives are involved. The field is therefore open for such members of the Institute, or others, who, recognising the additional facility in the treatment of the more complex cases that would be conferred by the possession of a set of Survivorship Assurance Tables, have the time and the inclination to undertake their construction. While by so doing the benefit they would confer on their profession would be great, that which they would derive personally from the exercise would be by no means small. To facilitate this construction, by whomsoever it may be undertaken, the tables on pp. 236 and 237 have been formed. We proceed to explain the methods employed in the formation of these, and shall then shew the manner in which they are to be used in the construction of Survivorship Assurance |