■T:\\-yy:<('K-\->r.-:>i ! nsUin Actuaries'' Life Tables- TABLES DBiDUCED FROM THE MORTALITY EXPERIENCE OP -LIFE. ASSURANCE G0MPANIB8. WITH AN APPENDIX CONTAINING A COMPLETE SYSTEM OF NOTATION FOR LIFE CONTINGENCIES. LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Clcua INSTITUTE OF ACTUARIES' LIFE TABLES. TABLES DBDUOED FnOM THE MORTALITY EXPERIENCE LIFE ASSUEANCE COMPANIES, AS COLLEOTBD ASD AEBANOED BY THE INSTITUTE OF ACTUARIES i' OF GREAT BRITAIN AND IRELAND: WITH AS INTRODUCTION EXPLANATORY OF THE CONSTRUCTION AND APPLICATION OF THE TABLES, AND AS APPENDIX CONTAINING A COMPLETE SYSTEM OF NOTATION FOR LIFE CONTINGENCIES. Publtihtd by tlu Authority, and under the Superintendence, of the Imtitute. LONDON : CHARLES & EDWIN LAYTON. 1872. H& ^3 PREFACE In the Preface to "The MortaHty Experience of Life Assurance Companies, collected by the Institute of Actuaries," published in May, 1869, it was stated, "The preparation of monetary and other Tables for official purposes is a work involving so much time, labour, and expense, that the Council of the Institute of Actuaries have decided not further to delay the publication of the original facts, on which they will have to be computed." The liberality of several Life Assurance Companies promptly relieved the Council from all anxiety in regard to the expense attending the completion of so extensive a work. Their further contributions, in answer to a circular issued in June, 1869, amounted to £60^, making the total Fund collected £11^2. $5. A List of the contributing Offices is appended to this Preface. As a preliminary to the determination of the classes of monetary Tables to be computed, it was considered important — I. That the Tables of Mortality deduced directly from the Observations should be graduated in such a manner as to preserve, to the utmost possible degree consistent with their practical use, all the original features of the Observations. 2. Tliat further investigations should be made into the question of the effect of selection, with the view of deciding whether distinct monetary Tables should be given, applicable to assured lives from which that effect has in a great measure passed away. The graduation of the three Tables, HM, H^, and Hmf was kindly undertaken by Mr. Woolhouse, whose great skill and experience suggested to him a new method, combining in a probably unprecedented degree adaptation for all prac- tical purposes with close adherence to the original facts. As this method cannot fail to be of great interest, the Tables, as adjusted by Mr. Woolliouse, together with his explanation of the process of adjustment, are printed at the end of the Introduction. In order to examine more closely into the question of the effect of selection, a Table of Mortality was deduced from the Observations H^, excluding the years of Assurance O to 4; this was similarly graduated by Mr. Woolhouse. It is designated by the symbol Hm:(5). The particular period adopted as most suitable generally for exclusion was deter- mined by the marked difference, in the classified original data, between the first five years of assurance and the subsequent periods. It was then decided, after a careful consideration of the whole subject, that Commutation Tables for Single Lives, and Values of Annuities to 4 places of decimals, at 3, 3^, 4, 4^, 5, and 6 pcr-cent, should be deduced from the graduated H^: and H^ Tables. It was also decided that similar Tables, at 3, 3^, and 4 pcr-cent, should be deduced from the H-m:(-^) Table. These were considered sufficient to afford to Life Assurance Companies a basis for approximate valuations of their position and prospects independent of the influence of recent selection. The Table H^f i^^s not been made a basis of calculation in the present volume. The Tables for Two Lives comprise an entire set of Joint Life Annuities for every combination of ages from (lo, lo) upwards, at 3, 3^, and 4 pcr-cent, as being the rates generally sufficient for official calculations. The values of Annuities on the last survivor of two lives for the same combinations of age are also given at the same rates of interest, and by a simple arrangement fill up, with a new and valuable set of Tables, what would otherwise have been a great waste of space in the Joint Life Columns. The laborious and responsible task of superintending the computation and printing of all these Tables was undertaken in the most disinterested manner by Mr. Peter Gray, whose name will be accepted as a guarantee of the accuracy and consequent value of the completed results. To the Tables themselves Mr. Gray has written an Introduction, fully explaining the methods of formation and various modes of their application. This renders it unneces- sary for the Council to do more than draw attention to his clear and able exposition: but they desire, at the same time, to record their sense of the great obligations which all interested in the present work are under to Mr. Gray for his invaluable assistance in its production. Other Tables are required to complete the Canon, such as Single and Annual Premiums for Survivorships and Endow- ments and the values of Temporary and Deferred Annuities, but it appears to the Council that it would not be advisable to delay any longer the publication of the present volume, which, with these explanations, they now submit to the Profession. The Council have thought it desirable to append to this volume a comprehensive Scheme of Notation, settled by a Special Committee of Actuaries, after long and careful con- sideration of the subject, aided by valuable suggestions and VI co-operation from various quarters. The Council believe that this notation will be found extremely well adapted to express all the benefits that occur in practice, and they strongly recommend it for general adoption, in the expectation that its use will contribute materially to advance the study of life contingencies by promoting exactness of thought and expression. LIST OF CONTRIBUTING OFFICES. British Equitable Assurance Company. Briton Medical and General Life Association. Church of England Life Assurance Institution. Clergv Mutual Assurance Society. Clerical, Medical, and General Life Assurance Society, Commercial Union Assurance Company. Crown Life Assurance Company. Eagle Insurance Company. English and Scottish Law Life Assurance Association, Equity and Law Life Assurance Society. Friends' Provident Institution. General Reversionary and Investment Company. Guardian Fire and Life Assurance Company. Hand-in-Hand Fire and Life Insurance Society. Imperial Life Insurance Company. Law Union Fire and Life Insurance Company. Legal and General Life Assurance Society. Liverpool, London, and Globe Insurance Company. London and Provincial Law Assurance Society. London Assurance Corporation. Metropolitan Life Assurance Society. National Life Assurance Society, National Union Assurance Company. North British and Mercantile Insurance Company. Northern Assurance Company. Pelican Life Assurance Company. Prudential Assurance Company. Oucen Insurance Company. Royal Farmers' Insurance Company. Scottish Imperial Insurance Company. Sovereign Life Assurance Company. Sun Life Assurance Society. Universal Life Assurance Society. West of England Fire and Life Insurance Company. Whittington Life Assurance Company. Vll CONTENTS. Preface ....... Introduction ...... Explanation of the Adjustment of the Tables Page iii ix . Ixxxvii TABLES. /. — One Life. %* Under each of the three Mortality Tables, HM^ H^, and IIM:(5) are given : — First, a Table of Elementary Values, with Results deduced from it involving the rate of mortality only ; and, Secondly, the following Tables involving the rates of both mortality and interest, for each rate of interest em- ployed, namely : — 1. Commutation Table ; 2. Logarithms of the same ; 3. Results deduced from the Commutation Table. The rates of interest will be apparent from the following enumeration : — HM:_ Elementary Table, and Results Three Per-Cent. Three and a Half Four Four and a Half Five Six I 2 7 '5 23 31 39 +7 Vlll HF- Page. • 55 El.KMENTARY TaBLE , and Results 56 Three Per-Cent 61 Three and a Half ,, 69 Four ,, 77 Four and a Half >» 85 Five >> 93 Six >> lOI HM(5):_ 109 Elementary Table , AND Results . 1 10 Three Per-Cent i^5 Three and a Half >> 123 Four » 131 //. — Two Lives. Of Two-Life Tables the values given, deduced from HM only, are those of Annuities on foint Lives and on Last Survivors. As these, for the same rate of interest, occupy between them a single quadrangular space, a reference for each rate of interest is sufficient. H^ : — Three Per-Cent Three and a Half „ Four „ Auxiliary Tables for the Formation of Survivorship As- surances Constants System of Notation 139 171 203 • 235 • 238 • 239 INTEODUCTION. Three distinct Mortality Tables are, in the present work, made bases of computation. Thc\- are designated respectively by the suggestive s}-mbols, HM, H^ and HM(o\ The first two, HM^ and H^, have been deduced, by a highly scientific process of graduation, devised and applied by Mr. Woolhouse,* from the two, similarly designated, on pages 273 to 276 of the Mortality Experience of Life Assurance Companies, collected by the Institute of Actuaries, ^whWshQd'm 1869. The original tables commence at age o, with a radix of 10,000. The new tables, on the other hand, commence with a radix of ioo,ooc> at age 10, the numbers observed upon, between ages o and 10, being considered too small to afford trustworthy results; and therefore the numbers-living, in corresponding tables, do not admit of being directly compared. It will be shown hereafter how closely the results of the graduated tables are assimilated to those of the tables from which they have been respectively deduced. As regards H^^^*, the third table which is here made a basis of computation, there is no table in the Experience volume to which it holds a relation corresponding to that held by the two H^: and H^ to the tables similarly designated in the volume referred to. It has been formed from the same data as H^, when modified by the exclusion from them of the experience of the first five years of assurance. Comparison of the results of this table with the corresponding results of HM serve to show the effect of recency of selection. The arrangement that has been adopted, in regard to the Single-Life Tables, has been to group together, under each fundamental table, all the tables deduced from it. It is » Much attention has been given by Mr. Woolhouse to the subject of the graduation of tables. A full description of his method, as applied to the Table H^, follows this Introduction. In the application of this method every fact in the table operated upon has its due weight accorded to it ; and the effect is, that in the resulting table, while asperities are softened down, ever)' well-pronounced characteristic in the original is faithfully reproduced. h believed that this arrangement will be found to be more con- ducive to facility in the use of the tables than any other that could be suggested. The Two-Life Tables, which have been deduced from HM, follow the Single-Life Tables. L Of the Single-Life Tables. The Single-Life Tables, deduced from the several funda- mental tables, may be classified as follows: — 1. An Elementary Table. This contains the Mortality Table and such deductions from it as are requisite for the construction of the succeeding tables, and of any others that may be proposed. 2. Results involving the rate of mortality only. These are/^, q^ and r^. 3. A Commutation Table. 4. Logarithms and Cologarithms of the principal columns of the Commutation Table. 5. Results deduced from the Commutation Table. These are ^j, A^ and cr^. The monetary tables, comprising 3, 4 and 5 of the above enumeration, arc given at various rates of interest. For each of the tables H^ and H^ they are given at the following six rates, viz., 3, 3^, 4, \\, 5 and 6 per cent.; and for HM(5) they are given at the three rates 3, 3^ and 4 per cent. The tables as above classified will be now more particularly described ; and attention will be directed to any specialties in their form, or in the methods employed in their construction, that may seem to deserve or to demand notice. The illustrative references will be solely or chiefly to the table Hm: and the deductions from it, with interest, where this element enters, at 3 per cent. I. TJie Elementary Tabic. The functions here tabulated for each age are 4 and d^y with their logarithms and cologarithms; and also the logarithms and cologarithms of /^, the logarithmic functions being in all cases accompanied by their differences. It has not been usual, for what reason is not apparent, to tabulate log dj. ; nor have the differences been heretofore tabulated. These find their uses, as will hereafter appear, in the construction of tables. The bases of the logarithmic portion of the Elementary Table are log 4 and log^^.. The manner in which the remaining columns are deduced from the columns containing these functions is sufficiently obvious ; and it is therefore necessary to explain only the methods employed for their verification. Since, ?/j. being any function of .r, it appears that if in any column of differences we insert at intervals the proper values of w^, they will in each case be the sum of all the terms which precede them. Thus, referring to p. 2, if in the blank spaces in the column headed log/^. (which contains the differences of log 4) we insert log/io, log As, log /20, &c., the column will be verified by continuous addition. But the requisite additions can very well be made without the actual transference of the terms from the adjoin- ing column. All the columns of differences were checked in this way. Again, since the sum of a logarithm and its complement is o, so the sum of any number of logarithms and their com- plements is alsoo. The complementary columns were verified therefore by adding together, as they stand, the corresponding groups of five in the two columns. 2. Results involving the Rate of Mortality only. These results, as respects H^", are on p. 6. They consist of A. Qx ^"<^ ^x, foi" each age. The first two are complements of each other to unity ; and they might have been formed by taking out the numbers corresponding to log A in the Elementary Table, and subtracting the results from unity. But it was preferred to employ for this purpose a method which brings into use certain of the tabulated functions ; and as it is typical of other operations to follow, it will be described at some length. h 2 Since q,.— -,-, and consequently, logr/r^log^^/j. — lo^'4 ==log^^4-colog4; therefore A log (7^;= A log ( UNIVERSITY ) Log rt'io is set down on the first line, and on the fourth, seventh, and every third succeeding hnc the differences of logc/j in order, commencing with Alogc/jo; colog/io is set down on the second line, and the differences of colog4, in other words the successive terms of colog/r, commencing with colog/io, on the fifth, eighth, Sec, that is, on every third line as before. The third, sixth, ninth, &c., lines thus left vacant, are intended to receive the results of the final additions, which results are the logarithms of the values sought. Before proceeding to the final additions the terms set down should be added continuously in groups ; and the successive sums, which should be previously formed from the Elementary Table, and inserted in their places, will serve, as they are reached in order, to verify also the final additions. In most, or all, of the logarithmic series with which we have to do, the increase (or decrease) in the successive terms is so gradual that it is not necessary, in computation, to write the index of more than the first term. We have sufficient intimation of a change in the index by that which takes place at the same time in the mantissa of the logarithm. It is thus unnecessary to set down the indexes of the differences, and they are not inserted in the tables. It is frequently desirable to form a series in reverse instead of direct order; and the Elementary Table furnishes the means of doing so. Thus: — Since log^^+i = log qx+ ^ log rt'^ + colog/^, therefore, log ^x= log ^x+i — -^ log o'^— colog p^ =log ^^+1 + A colog d^ + log/^, (Acolog(^.5 being the negative difference of log^.r. marked — A in the Elementary Table) ; and the series may be formed by continuous addition, as before. The following is the construction of the last six terms ; and the results of this and the preceding formation may be compared with the complete table on p. 6. XIV Log f/97 0-954243 !?x Pr (1-7.) Col. /97 1-045757 97 0-000000 G47817 2G4047 roooooo -000000 96 -9118G4 332438 5598G2 •816327 ■^W^n 95 •8041G4 208517 G92583 •637036 •362964 94 •7052G4 147020 7GG578 •507300 •^92700 93 •C18862 114799 812035 •415778 •584222 92 •545G9G ■35 ^3H •648686 Logr/92 2-404834 Col. ^92 3-140862 92 T-545696 It is hardly necessary to remark that, the column q^. shewing the annual decrement on a iniit of life at each age, it shews also, by the proper disposition of the decimal point, the annual mortality per hundred, per thousand, &c. Thus, we see that at age 52 the mortality is if per cent., or \'j\ per thousand, and so on. The next result is r^., the Average Duration of Life at each age. The column has been formed b}- the aid of the well-known formula, in which Cj. denotes the Curtate Duration at age x, and from which the Complete Duration, r^, is deduced by addition of "5 to each term. The formation of the first few terms of the scries (the last few as the table is arranged) is here subjoined. The initial term, ^,j; being o, is logr,H; = log/'>j,;. The operation is con- ducted by means of the table Log (i -^-x) in Gray's Tables and Forninhc, and verification is obtained from point to point by the use of the formula e,.= < j+l + '92 l-^?> Log 936 Col. 723 92 2-971276 3-140862 0-112138 •9296 1-2044 I "495 7 1-7946 Comparison of the series thus formed with the corres- ponding series formed from the same table, H^ previous to graduation, Expcrioicc, pp. 281, 282, gives, for ages 10 to 90, the following result: — In 20 cases the corresponding terms of the two series are identical ; in 26 they differ by ±'0i ; in 11 they differ by ±'02 ; in 12 by ±'03 ; in 2 by ±"04; in i by — •05 ; in 3 by ±'o6; in 3 by ±'08; in i by +"io; in i by — •15 ; and in I by — 'iS. Or it may be put thus : — The sum of the positive devia- tions is "81, and that of the negative deviations is roi ; and we have XVI 20 cases of coincidence ; 34 with an average deviation of + '024 ; and 27 „ „ ,. -037. From which it appears that the graduated table may be unlicsitatingly accepted as a faitliful representation, but deprived of its angularities, of that from which it is deduced. A similar comparison to the foregoing, instituted between the Average Durations deduced from the original and the graduated H^ tables, for the same ages, 10 to 90, gives a somewhat less favourable result. The former is to be found in the Experience volume, on pp. 281, 282, and the latter on p. 60 of the present work. Here wc have : — 3 cases in which the corresponding terms are identical; 15 in which they differ by ±'0i ; 14 in which they differ by ±02; 14 by ±"03; 5 by ±-04; 5 by ±-05 ; 3 by ±-o6 ; 2 by —-07 ; i by +-o8 ; 3 by — '09; 2 by — "lO ; 3 by ±-ii ; 3 by ±-i2 ; i by —-13 ; i by +-14; i by 4- '16 ; I by +-i8; i by —-19; i by 4- -24 ; i by —-27; and i by + *43- The totals are, of positive deviation 2-48, and of negative 2'I2 ; and we have finally : — 3 cases of coincidence ; 38 with an average deviation of + "065 ; 40 „ „ „ -"OSS- The deviations are here, as already remarked, confined within less narrow limits than in the case of the HM table. The cause of this is, that the original H^ table contains more asperities than the original H^: ; and this again arises, no doubt, from the comparative smallness of the number of lives entered in the observations on which it is founded. And it is to be borne in mind that the deviations (which legitimately arise in the process of graduation) are in the direction of softening down the asi)eritics by which, like other results of original observations, the present table is characterized. The graduated table may therefore be confident!}- accepted as an exponent of the law of mortalit}- likely to prevail amongst a class similar to that subjected to observation. XVI 1 The tables H^ and HM(5) are formed, as has been already stated, from the same series of observations. In the case of the former table the lives are brought under observation at the date of assurance, that is, at the period of their selection, as healthy lives ; in the case of the latter table they are not brought under observation till they have been assured five* years. Hence, if recency of selection has any effect in modifying the rate of mortality in a community, this ought to become apparent on comparison of the results of the two tables formed as above described. And accordingly we find, on referring to the Average Durations, pp. 6 and 1 14, that those on the latter page, derived from the table HM(.5)^ are throughout less than those on the former, derived from the table H^:. The difi"erence, which at the outset is 1-83 years, remains nearly stationary for 12 terms, after which it gradually decreases ; but it does not entirely disappear till the close of the series. The conclusion then is, that recency of selection exercises a by no means unimportant influence on the rate of mortality ; and the measure of its effect in the present case is the series of differences between corresponding terms in the two sets of Average Durations now under consideration. 3. The Coimmitation Table. We now come to the tables involving the rate of interest as well as the rate of mortality. The principal and funda- mental of these is the Commutation Table. It is necessary to say, that in the Commutation Tables here given, Davies's relation of the several columns to each other has been adopted ; so that we always have ^.= D,' and so on. And hence the formulae for their application are those of Professor De Morgan's paper in the Compmiion to the Almanac for 1840, reprinted in the Journal of the Institute, vol. xii., p. 328. This form has been adopted after fully • More exactly four and a half years, on the average. XVI 11 considering and duly weighing the arguments that have of late years been adduced for a departure from it, and which need not here be recapitulated. The methods employed in the construction of tlic funda- mental columns will now be described. And first as to Column D. We have log D^. = log 4 + log z'^, and log D^+, = log 4+, + log v^-^^ ; therefore A log D^ = log/^ + log t' = \ogvp^. Hence, since logDj.4.i = logDj,-j- AlogD;r=logDj. + log7/';r. it appears that the column log Dj. may be formed in direct order by the continuous addition to log Djo of the terms of the series logt/^. But it is more convenient, b)- facilitating the formation of Column N, to form log D^. in reverse order. From the foregoing we have log D^ = log D^+, - log vp^ Hence, commencing with log D97, the column is formed by the continuous addition of colog z; + colog/;r. A specimen of the operation follows : — 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 Logt* 716278 729115 741952 754789 767G26 7804G4 + 793301 806138 818975 831813 + 844650 857487 870324 883161 895999 + 90883G Log t"^'' 91 96 95 94 9.3 92 LogDx 2-754789 0-954243 1-709032 012837 735953 -457822 012838 440138 Dx and N^ •910798 + 012837 307417 -231052 012837 233422 •477311 012837 187965 -678113 96 •5^17 2-8696 3'38 1.3 8-1433 11-5246 17-0236 28-5482 30'oi3i 58-5613 92 95 94 93 XIX 58-5613 92 -678113 47'6'5')^ SI ^^^^ ^' 91 -853829+ 71-4215 012837 177-6383 00 142337 II ^ ^ y 90 -009003 102-095 012837 126794 279733 89 89 -148634 140-810 012837 420- <43 88 118786 ^ •280257 190-659 012837 67F^ 114203 8; -407297 255-445 012838 866^ 86 107713 86 -527848+ 337*^^9 012837 1203-816 85 102311 85 -642996 439'538 1643-354 84 Logt;" 2-908836 „ /gs 3-734160 85 2-642996 The first line is occupied by log 7^^^ and the fourth, seventh, tenth, &c., by colog v ; the second line is occupied by log /gr, and the fifth, eighth, eleventh, &c., by the successive terms of colog/a; in reverse order. Before proceeding to the additions it is necessary to introduce certain corrections. The true value of colog 7; is -012837,2247 . . . . , and the neglect of the figures beyond the sixth would give rise to constantly increasing error, which, at the close in log D,o, would amount to no less than 20. To counteract this a periodical correction is made, and the points where it is to be in- troduced are easily determined by reference to the series log if, a portion of which, in reverse order to correspond with the order in which the table is being formed, is given in c2 XX ihc margin.* Commencing with logc'" (which is the term that enters into log D^;) and attending to the last figures only, we see that the differences are 7, 8, 7, 7, 7, 8, 7, 7, 7, 7, 8, ... . Let the terms on which the 8's fall be marked with +, meaning that those terms have been increased by unity. The marked terms here are 95, 91, 86, &c., and the terms in logD^ corres- ponding to those ages will thereupon have to be similarly increased, since log D^^^log Z^+logz/"'. The + marks are therefore placed against the proper ages in the working column, and it is then seen that what is requisite is to alter the value of cologt^ next preceding each marked line from ■012837 to 012838. This correction being madet the addition is proceeded with, and the successive sums are the terms of the series log D^., true to the nearest unit in the last place. It is obvious that the frequent writing of cologt* may easily be avoided. It should be written at the bottom of a card ; and this being held over the sum last formed, the) loga- rithm can be included in the next summation. This method will be exemplified in the formation of log C^.. The numbers corresponding to log Dj. were taken out by one computer and carefully examined by another. Further verification was obtained when Columns N and M were formed, as will be shewn hereafter. The formation of a portion of Column N is shown ; but it is unnecessary to exemplify that of Column S. The method employed for the formation of Column C will now be described. We have logC^ = log:v+i + log^^. Hence, A log C,=logt^ + A log^/^. • A six-figure table of lo^ v', which shall be true in the last place, cannot be formed from a seven-figure table, since we are left in doubt as to whether or not a terminal 5 in the seventh place indicates an increase in the sixth. And in especial it cannot be formed from Mr. David Jones's values of this function, these being very incorrect in the last place, particularly the 4 per-cent. column. An excellent set of tables of logo/*, extending to ten places, and therefore suited for every purpose, is given in Mr. William Thomas' Thomson's yictuarial Tables, to which work, Mr. Thomson informs his readers, it was contributed by Mr. Filipowslti. -| The corrected figures are printed in hc.uiir type. Consequently, for the formation of log C^. in direct order, we have log C^+i = log C^ + A log C^ = log C;r + log ^' + A log ^j, ; and for the formation in reverse order, which is the more convenient, we have log Car = log Cj,+i + colog t^ + A colog dj. ; in which A colog ^j. is the function occupying the column marked —A in the Elementary Table. The following is an example of the formation: — Log Gx Ci and M^ Log V98 2-741952 J> ^97 0-954243 97 1-696195 647817 •4968 97 96 •356849 332438 i'2743 2^77ii 96 95 -702124 208518 5-^365 7-8076 9S 94 -923479 + 147020 8-3845 16-1921 94 93 -083336 12-1154 114799 28-3075 93 92 •210972 16-2544 112362 44^5619 92 91 •336171 2r6856 93465 66-2475 91 90 •442473 + 83945 27^6995 93'947o 90 89 -539255 34'6i43 94270 128-5613 89 88 -646362 44-2957 99304 172-8570 88 87 •758503 57-3460 85411 230-2030 71-9036 87 86 •856751 82553 302-io66 86 8^ -952141 + 89-5656 39r6722 85 « • * • Log C?85 3-056142 t,96 2-895999 85 r952141 XXII The formation here is effected in the manner just referred to. The only addends written down arc the terms of the series A colog (^^^ (in reverse order), and colog v is included in each summation (after the first) by means of a card. In consequence of the constant addend, colog f, not being set down, the corrections rendered necessary by its curtailment to six places must in this formation be applied to the terms of A colog ^^. Their places are determined at once by reference to the formation of log D^.. In the present formation the marked terms are those corresponding to the ages one year younger than those to which they correspond in the last- mentioned formation. The powers of v which there enter into Dj., enter here into Cj:_i. Hence, applying the marks, and increasing the addend immediately preceding each by a unit in the last place, continuous addition gives the terms of log Cx true to the nearest figure. The numbers being taken out and examined, the formation of Column M is proceeded with, and a specimen of the opera- tion is exhibited. We can now procure verification of all the preceding work. Since M^='t'N^_, — N^, we have Mgj = •:'N84 — Ngs ; and the formula is applied as follows : — 103)1643354(1595490 fNg^. 613 1203816 Nfls- 983 391674 Mas. 504 92 This process is easier than the logarithmic one, when, as here, division by i -l-z is used instead of multiplication by^'. It also, usually, gives us one or two figures more in tlie result than can be obtained by the use of logarithms. A similar relation holds between the Columns S and R. Thus, And by the application of these formulae the whole of the Commutation Tables in the present volume have been verified, from point to point. XXlll 4. Logarithms and Cologariihms of the Principal Columns of the Commutatio7i Table. This table, which contains also the differences of the loga- rithms enumerated, occupies the four pages next following each of the Commutation Tables. The foundations of the table are the columns containing log D^;, log N^^ and log M^. ; and the manner in which the remaining columns are derived from these is sufficiently obvnous to preclude the necessity for description. The columns were severally verified in the same way as the corresponding columns of the Elementary Table. 5. Results derived from the Commutation Table. The results here tabulated are ax, A^. and cr^, the values of annuities, and the single and annual premiums for assurance of a unit. They were formed as follows : — 1. ^x=j3;; log «^ = log N^ + colog D;, . Hence. A log ^x= ^ log N^. + A colog D^, = A log Na- + colog vpx ; and \ogax+\ = \oga^+ !l\ogax = log ax-\- -^ log Nj. + colog vPx. 2. Similarly: — A-Mf and log A;j = log M^ + colog Dj,. Alog A^=A log jM^ 4- colog c^^. log A^+i = log A, + A log A^ = log Aa; + A log M, + colog vPx ■ 3. And finally: — _ M. and log -sr^rr log Ma- + colog Na._,. A log -s^ar^ A log Mj. + A colog N^_, . log -zar^+i = log TiT;, + A log -57^ = log -sij. + A log M^r + A colog N,._i . XXIV The following are examples of the three operations :- LogA^ 4-298988 5-128372 13 14 '5 16 16 Loga, "x 6-254516 5-128372 1-382888 982268 14971 24-1484 -380127 982136 14573 23-9953 •376836 981984 14281 23-8142 •373101 981818 14105 23-6103 -369024 981641 14039 23'3897 •364704 981454 14086 23-1581 -360244 22-9215 T-427360 992208 _14971 •434539 993771 14573 •442883 994922 14281 6-145817 5-214427 -452086 995637 14105 -461828 995961 14039 1-360244 -471828 995896 14086 -481810 4-267383 5-214427 1-481810 A. LogflTj: 4-298988 7-727862 267523 2-026850 992208 17622 271981 -036680 993771 17732 277257 -048183 994922 17864 283195 -060969 995637 18016 289620 -074622 995961 18182 296366 -088765 995896 18359 •303256 -103020 ♦ « 4-267383 7-835637 2-103020 -010638 -010881 -01 1 173 -01 1507 -011875 •012268 -012677 The initial values are log Nio + cologD,o,logM,o + cologD,o, and logMio + cologNs, the addends being in accordance with the preceding formulze. Neither colog N9 nor its difference is in the table. They arc easily formed, thus: — 1796867 74409 Ng 187I276 log 6-272138 col. 7-727862 A 017622 The verifications at the bottom are log Nis + colog D,6, log M 16 + colog D ,6, and log Mi^ + colog N,5. 6. 0)1 the Extent of t/ie Tables. The limits prescribed at the outset for the extent of the tabulated values, as sufficient for practical purposes, were as follows : — For the logarithms, six places, throughout ; for D^. and Mx of the Commutation Tables, also six places ; for the remaining columns, N^, S^ and R^, seven places ; for the assurances, six places ; and for the annuities, four decimal places. The Elementary Tables were formed by the aid of six-figure logarithms ; but for the Commutation Tables and the deductions from them, seven-figure logarithms were used, the whole, logarithms and numbers, being subsequently cut down to the prescribed limits. It may fairly be presumed therefore that the tables thus formed are true in their last figures. The object in view, in the employment of seven figures, at the cost of considerable enhancement of labour, was the attainment of a greater degree of exactitude than was to be had by the use of six figures. It is well enough known that a result formed by means of logarithms may generally be depended on, subject to an error of a unit or so, to the same number of places as are contained in the logarithms used in its formation ; but it is not so well known what degree of exactitude may be looked for in the results of such a series of mixed operations as is required in the formation, for example, of a commutation table. With a view to throw a little light upon this point, six- figure logarithms only have been employed in the portions of the commutation table given as examples ; and comparison of these portions with those corresponding to them in the printed tables, reveals the existence of a few discrepancies of a unit in the last place. But, to obtain a wider field for com- parison. Column D, commenced in the example, p. xviii, has been completed on the same basis, Column N has been formed from it, and from those columns, still using the six-figure tables, a table of annuities has been formed. The following small table shows the result of a careful comparison of the columns formed as just described with the d XXVI corresponding columns in ihc tables, ihese latter being assumed to be correct : — ElTOlS. Logl)^ D. Nx 1 r/j. Tutals. o I ,3 60 28 48 39 I 50 3 I 77 10 235 1 10 4 I 1 88 88 87 87 1 350 This means that in log D^; there are 60 values correct, and 28 with an error of ± i ; in D^, 48 correct, 39 with an error of ±1, and one with an error of ±2, and so on ; the state of the annuity column being, 77 correct, and 10 with an error of ± I ; that is, in the fourth decimal place, tJirce only being usually tabulated. These deviations from strict accuracy will not be considered very ;^9i •9871G3 •837122 15 •985911 91 •824285 This scries, in whatever way it is to be used, whether in the formation of Column D, or in the construction of a table of annuities, requires correction in consequence of the curtail- ment of the value of log t', which enters every term. The true value of log?' is -987162,775 . . . ., the \alucs used being •987163 ; and the correction is the abatement of a unit in the sixth place at the proper points, which may be determined, as already shewn, by reference to a six-figure table of log v'^. The requisite corrections are introduced in the example. XXIX Problem 2. To form a table of loaded assurance premiums. The formation of u^. has been already exemplified. The present formation differs from that referred to in no other respect than that here the logarithm oi {i plus its loading) is included in the initial term. Thus, to form -arj. with a loading of i/-^ per-cent. : — Log 1-175 0-070038 „ M,o 4-298988 Col. N9 7-727862 10 2-09G888 -012499 992208 17622 •106718 -012785 993771 17732 -118221 -013129 994922 17864 13 -131007 '013521 « • • Log 1-175 0-070038 „ M,3 4-279889 Col. N12 7-781080 13 2-131007 The premiums here formed may be compared with those for the same ages on p. 14. Problem 3. To form a table of the values of endowments payable at a specified age. The age at which the endowment becomes payable being X + n, the value of the benefit is or, in logarithms, log D^+,, + colog D^. And the difference of this expression, in which .r + ;/ is constant, is A colog D^, or colog vpn- Two examples follow, in which .'.• + ;/ takes the values 21 and 60, respectively. XX \ Values of Endoivuioits. Payable at 21. Tajablc at GO. LogDij 4-710910 Log Doo 3-999G31 Col. D,o 5-128372 Col L^io 5^128372 10 T-839312 14971 •690736 10 1 12.^00:5 11971 •J34277 1 1 •8.^.4283 14573 -714962 1 1 •142974 14573 •138987 12 •8G885G 14281 739360 12 •157547 14281 •143730 J3 •883137 14105 •764077 13 •171828 14105 •H8535 H •897242 1403!) .789300 14 •185!»33 14039 •153438 I J •911281 14086 •815232 i.^ •199972 1408G •158479 i6 •9253G7 •842106 16 •214058 • • • * • • Log Da, 4^710940 Log Dgo 3^999631 Col. D„ 5-214427 1 Col. Dig 5^214427 •164704 i6 r925367 1 1 16 r2 14058 Problem 4. To form a tabic of the assurances equivalent to a present value of a unit. The \-alue of this assurance, when the age is ,v, is, the logarithm of which is, logD.r+ cologMj.. And the difference of this expression is, logu/x+Acolog M^. The required table may be consequently constructed as follows : — Assurances ivJiosc value is a Unit. LogDio 4-871628 Col. M,o 5-701012 10 0-572G-iO 37380 985029 7792 •565461 y6';6'] 985427 G229 •557117 3 •6068 985719 5078 13 -547914 i-S?>^'^ 985895 43G3 14 -538172 3'4528 985961 4039 15 •528172 - 3-3742 * » • LogD,5 4-799659 Col. M,5 5^728513 i.i 0-528172 The use of such a table as that of which the formation is here exempHfied is to facihtate the conversion of a cash bonus into a reversionary bonus. Example. A poHcy on (15) has assigned to it a present bonus of 42*375. ]-'ind the equivalent reversionary bonus. 42-3/5X 3-3742 =142-982. This table could be \'cry readily formed from the Assurances on p. 14 by means of Colonel Oakes's Table of Reciprocals. The method here employed possesses the advantage of giving also the logarithms of the values formed. Problem 5. To form a complete table of the values of deferred and temporary annuities. By a complete table is to be understood a table, that shall comprise, in regard to the classes of annuities specified, all the cases that can present themselves in the use of the mortality table employed. XXXMI The present ai;e, (or, preferably, tlie ai^e to wliicli the value to be formed has reference,) being as usual ,r, denote that at which the deferred annuity is to be entered upon, and the corresponding temporary annuity to cease, by y. Then, the value of tiie deferred annuity being of which tiic logarithm is, logN,, + cologD^, we shall have to form the values of this expression for all the combinations of .r andj', in which x docs not exceed j. This can be very commodiously done in a series of paralel collumns. Either of the quantities ,*■ andj', as maybe arranged, will vary in the columns ; and the other will vary in passing from column to column, that is in the rows. If, as is most conve- nient, it is chosen to commence with the younger ages, the work will assume one or other of the following forms : — The first, if in the columns ,v be made to vnr}', and the second, if y be made to vary: — 96 The forms are theoretically equally eligible. The first, however, is to be preferred, becau.sc in it the addends will consist of fewer significant figures than in the second : A colog D^ contains generally only five significant figures, while A log N„ contains six throughout. The following is a specimen of the formation, in which the values of y are at the top and those of x at the side. xxxni Formatioti of Deferred and Temporary Annuities. lO 11 12 10 382888 24-1484 24-1484 365156 14971 24-1484 23-1823 0-9661 23 '9953 23 '9953 347292 14971 362263 24-1484 22-2480 1-9004 23 ■9953 23-0284 1 1 380127 14573 0-9669 12 376836 23-8142 23-8142 Forvtation — (continued). xo 329276 14971 11 344247 14573 12 358820 14281 J3 373101 14 T5 13 24-1484 2 [-3440 2-8044 23-9953 22-0926 1-9027 23-8142 22-8465 0-967.: 23-6103 23-6103 311094 14971 326065 14573 340638 14281 354919 14105 369024 I 4 24-1484 20-4688 3-6796 23-99^3 292735 14971 21-1868 307706 2-8085 14573 23-8142 21-9098 322279 1-9044 14281 23-6103 22-6422 0-9681 336560 14105 23*3897 23-3897 350665 14039 364704 15 I 24-1484 19-6216 4-5268 20-3098 23-8142 21-0029 '9'^S?> 23-3897 22-4215 0-9682 23-1581 23-1582 The first six columns are here given complete. The initial terms are the successive values of log N^ -h colog D^., when_y is made to vary ; and they are formed as follows, the addends being the terms of Alog Nj, : — LogNio 6-254516 Col. D,o 5 128372 ID 1-382888 982268 •365156 982136 -347292 981984 13 -329276 981818 14 -311094 981641 15 -292735 * * LogN,5 6^64363 Col. D,o 5128372 ir; 1-292735 In the columns, x being the variable, the addends are, as already intimated, the terms of A colog D^, or colog vp^, com- mencing in each column with colog vp^^, and continued till the next value of x at the side is the same as the value of ^ at the top. The addends having been written down, it is well before proceeding to the additions to form series for verifica- tions of the several columns at suitable intervals. These will be formed in the same way as the series of initial terms. The following is an example for x-=. 12 : — LogN,2 6-218920 Col. D12 5-157916 12 1-376836 981984 13 -358820 981818 14 -340638 981641 15 « -322279 * Log N,5 Col. D,2 6-164363 5-157916 I'J 1-322279 The verification values being inserted in their places the addition is proceeded with. A final verification is had at the XXXV close, the last sum in each column being the logarithm of the whole-life annuity on the age at the top and also at the side. The manner of forming these logarithms has been already shown (p. xxiv). To pass now to the columns of numbers. It will be observed that in each column the lines next above the values of the deferred annuities are occupied by terms of the series of whole-life annuities (from p. 14), commencing, in each column, with the annuity corresponding to the age opposite, viz., 10, and ending with that corresponding to the age at the top. Subtraction now gives the values of the complementary annuities ; i. e., the temporary annuities which, together with the deferred annuities that form the subtrahends, are equal to the corresponding whole-life annuities that form the minuends. The two sets of results may be arranged in a table of the following form : — Values of Deferred and Temporary Annuities. 10 II 12 13 14 15 10 I 2 3 4 16 * 23-1823 22-2480 23*0284 21-3440 22-0926 22-8465 20-4688 21-1868 21-9098 22*6422 19-6216 20-3098 21-0029 21-7050 22-4215 10 I 2 3 4 16 * •9661 1-9004 2-8044 3-6796 4-5268 -9669 1-9027 2-8085 •9677 1-9044 2-8113 -9681 1-9053 -9682 * The horizontal lines separate the deferred from the tempo- rary annuities, the former occupying the portion of the table above the lines, and the latter the portion below.* For the deferred annuities, the present ages are at the side, and the ages at which the annuities are respectively to * The spaces occupied by the horizontal lines might be legitimately occupied by the whole-life annuities. But these are already given elsewhere, and the lines more effectually serve the the purpose of separators. XXXVI be entered upon are at the top. Thus, the value of an annuity on (12), to be entered upon at 15, is 21-0029, and so on. For tlie temporary annuities, the present ages are at the top, and the ages at which the annuities are respectively to cease, are at the side. Thus, the value of an annuity on (12) to last three years, and consequently to cease at 15, is 2-8113. These two annuities being complementary, their sum, 23-8142, is the value of the whole-life annuity at 12. Inspec- tion will show how the values formed in the example arrange themselves in the table. The values in the two portions of the table are related as follows: — Take 12,12, for example; the values that follow vertically from this point, are respectively complementary to those that follow horizontally. Thus, the sums of the pairs, 22-8465 -t- -9677, 2 1 -9098 -f- 1 -9044, 2roo29-|-2-8ii3, are each equal to 23-8142, which is the value of the whole- life annuity at 12. Deferred annuities in which the period of deferment is the same, say n years, are to be found ranging diagonally down- wards, commencing opposite x, and under x-\-n. Thus the series of annuities deferred three years, (;^ = 3) are 21-3440, 2ri868, 21-0029, &c. In like manner temporary annuities, whose duration is the same, n years, are also found, in their own compartment of the table, ranging diagonally downwards, commencing under X and opposite x^-n. Thus, the annuities to last three years are 28044, 28085, 2-8113, &c. And the annuities in these two sets are complementary, each to each ; since the initial value of x is the same in both sets, and also the value of n. Finally, it may be mentioned that the foregoing table, if completed, would occupy exactly the same space as one of the Two-Life Tables, nanicl\- 30 pages. XXXVll It is obvious that such a table as that just described would be of much utility for the formation of temporary premiums, which are frequently required in the case of endowments and other benefits. Problem 6. To construct a table of the values of policies. The value of a policy of assurance is the excess of the value of the sum assured over that of the premium. The sum assured being a unit, and the premium Pj,, where y is the age at which the policy was effected ; then, the present age being ,r, the value of the policy is, M,-P,N,_i *V I = = D.-(l-^)N._i-P,N._i = .-(._.^P,)i^^ Putting for -jy- its value, i +«^, and denoting {\—v-{-Y^ by Qy, this expression may be conveniently written, I— Q/i+^x) (I) To construct the required table then we shall have first to form the values of Oj,(i +«x) for every combination of x and y in which x is not less than y, and then to subtract the indi- vidual results from unity. The formation here requisite, is entirely analogous to that in Problem 5. In both it is similar series of the products of two factors that have to be formed. There, however, the logarithms of the factors were given, while here their determination is a preliminary part of the process. The premium Yy may be either the pure premium, that is the premium at the age y given by the table used in the valua- tion ; or it may be the office premium, in its integrity, or any how modified. There are thus two cases, which will be dealt with separately. * The usual symbol for the value of a policy is Vi-jn, where *• is the age at which the assurance was effected, and n the number of years since elapsed. This is at variance with the convention by which x represents the present age. For this reason, and to fecilitate the investigation, the symbol is here modified as in the text. xxxvni Case I. Let P^ be the pure premium at agc>'. Then 1 =:ZJ :=■ — = D,-(i -t;)N,.. (i-v) N _T\ Substituting this in O^, (i) becomes, I+^y (2); and the preliminary operation here will be the formation of log(H-«^), which will give us also colog (i +a,j), since y takes no values different from those taken by x. A specimen of this preliminary formation follows : A Colog(l+ff,) A *Log Ng Col. Dio xo Log(l+ffx) G-272138 5-128372 1-400510 982378 14971 997349 2-599490 13 14 15 16 -3U7859 982268 14573 -394700 98213G 14281 •391117 981984 14105 -38720G 981818 14039 •3830G3 981 G41 1408G •378790 996841 •602I4I 996417 •605300 996089 •608883 995857 -612794 995727 •61693; * •62 1210 • 002651 003159 003583 00391 1 004143 004273 LogN,5 G-1G4363 CoLDir, 5-214427 16 1-378790 * See page xxiv. XXXIX After the examples already given, the above does not seem to require explanation. The addends are A log N^^.i + A colog D^. Observing that A colog (i+a^) contains fewer significant figures than A log (i +«a,), j, in the construction, is made to vary in the columns ; x, therefore, will vary in the rows, and the initial terms will be formed as follows : — Log(l+«io) 1400510 Col. 2-599490 0-000000 997349 13 H •997349 996841 •994190 996417 •990607 996089 •986696 995857 •982553 Log(l+a,5) 1^-383063 Col. (1-foio) 2-599490 15 T-9S2553 A specimen of the formation, consisting of the first six columns, is now given, in which the values of x are at the top and those of j/ at the side. Formation of the Values of Policies. 10 II 12 10 000000 I^OOOOO -00000 997349 2651 000000 •99391 •00609 I'OOOOO •00000 994190 2651 996841 3159 000000 •98671 •01329 •99275 •00725 I^OOOOO •00000 xl lO 990607 2651 TI 993258 3159 12 996417 3583 13 000000 14 15 13 Formation — (continued). •97860 •02140 •98460 •01540 •99178 •00822 I -00000 •00000 986G96 2651 989347 3159 992506 3583 996089 3911 000000 •96983 •03017 •97577 •02423 •98289 •OI7II •99103 •00897 fOOOOO ■00000 15 982553 2651 985204 3159 988363 3583 991946 3911 995857 4143 000000 •96062 •03938 •96650 •03350 •97356 •02644 •98163 •01837 •99051 •00949 I'OOOOO •00000 The logarithmic operation may be verified from point to point as in last problem. The following is the verification of the row in which _;'= 12. Log(l + «„) 1-394700 Col. 12 13 ^4 15 2-605300 OmOOOOOO 996417 •996417 996089 •992506 995857 •988363 Col. (l + a,2) 2-605300 Log(H-ai5) r383063 15 r-988363 A final verification is had at the close of the work, the last sum in each column being 'oooooo, which is what log (i +^x/^ + colog (i ■\-a^ becomes when x=y. When the numbers corresponding to the several logarithmic scries have been taken out, and their complements to unity- formed, these last, which are the policy values required, may be abstracted and arranged in a table in either of the follow- ing forms : — xli Values of Policies. — Form No. i. y 10 1 1 12 13 H 15 10 •00000 •00609 •01,329 •02140 •0,3017 •0,3938 10 II •00000 •00725 •01540 •0242,3 •03350 II 12 •00000 •00822 •01711 •02644 12 13 •00000 •00897 •01837 ^3 H •00000 •00949 14 ^5 •00000 I ft Values of Polici 'es. — Form No. 2. X 10 10 11 12 13 14 15 •00000 10 II •00609 •00000 II 12 •01329 •00725 •00000 12 ^3 •02140 •01540 •00822 •00000 ^3 14 •03017 •02423 •01711 •00897 •00000 14 15 •03938 '^^oV^"^ •02644 •01837 •00949 •00000 15 « * * * * » « t, In the first form the present ages are at the top, and the ages at which the policies were severally effected are at the side ; in the second form the ages are interchanged, the present ages being at the side, and the others at the top. In both cases, the difference between the two ages to which a specified value corresponds, is the number of years that the policy whose value is in question has been in force. The second form seems, on the whole, the preferable one. But whichever is chosen, a large portion of the columns will be left blank. The blanks might be filled by the values of some other function, say the logarithms of the policy values ; and the two sets of functions might be rendered more distinguish- able by being written in differently coloured ink, or, if printed, by being set up in different type, as in the Two-Life Tables of the present volume. The table, if completed as suggested, would occupy thirty pages. xlii Case 2. Let Py be tlie office prLiniuin, or any modification of it. It was found in the last Case that Oy, or i — t' f P^ reduced to (i +(Jy)~\ so that tlic expression for the policy value assumed the form 1 +aj- In the present Case Oy admits of no such simplification; and therefore the expression here to be dealt with is in which Qy=i~'t' + Py, and P^has a given value. The nega- tive portion of the expression is, however, of the same form as the corresponding portion of the expression in the first Case, and the mode and form of its construction will be analogous. The preliminary formation here will be, since log(i+rtj) is already formed, that of log Q,,, which, in practical applica- tions, will be a very easy matter, since P„ for all ages will be given. For the present purpose, however, it is necessary to form Pj,. Premising that we assume for this the pure premium loaded to the extent of ten per-cent,* the formation can be easily conjoined with that of log Oy, as follows : — Col. I'lO TO LogPy U-0U.393 4-29S988 T-727862 2^068243 992208 17G22 Py and Qy Log Qy •029126 •040828 2-610958 •078073 993771 17732 •089576 •994922 17864 •OII969 •041095 •OI229I •04I4I7 •613789 •617179 002831 003390 003831 13 •1023G2 • Ten pcr-ccnt. will be considered but a small loading for an office premium. It scr%c3 our pre:*nt purpose, however, as will be seen presently, better than a heavier one. xliii 13 •102362 995637 18016 •012658 -041784 14 •116015 -013062 995961 •042188 18182 15 •130158 995896 18359 •013495 -042621 16 •U4413 ■013945 * * •043071 * Log I'lo 0-041393 » Mic 4-267383 Col. N,g 7-835637 16 M 444 13 •621010 004179 •625189 004435 •629624 004561 •634185 13 14 15 The above needs little explanation. Log Py = log (rioMj,) + colog N_„_i, is first formed, the addends being A log M^ and A colog N^_i. The numbers being taken out are each increased by I —v = -02gi26, thus forming 0„. The logarithms of Oj^ and their differences complete this preliminary formation. The following shows the construction of the first six columns of the final formation, the values of x being as before at the top, and those of j at the side. Format ion of the ]^ a lues of Policies. 10 ■100510 610958 01U68 1^02676 — '02676 397859 I 610958 I 008817 I I -020.^ I 2831 ,— '020'^ I 011648 1 •027 18 394700 610958 005658 2831 008489 3390 011879 I •013 1 1 — •0I3II I •01973 — •01973 1 •02773 " •02773 xliv roriiintioit — continued). 13 391117 610958 lO 002075 2831 II 004906 3390 13 00829G 3831 IJ 012127 14 16 1 387206 G10958 1-00479 — -00479 998 164 2831 I -01 1 36 — -011,36 000995 3390 1-01929 — -01929 004385 3831 1-02832 - -028,32 008216 4179 012395 •995 7 « -00422 1-00229 — -00229 1 -01 01 ■J — -01015 1-01910 — -019 10 1-02895 — -02895 383063 610958 994021 2831 996852 3390 000242 3831 004073 4179 008252 4435 012687 •98'533 -01367 •99278 -00722 1-00056 — -00036 1*00942 — -00942 1*01918 — -01918 I -02964 — -02964 The only point in respect of which the logarithmic part of this formation differs from that in Case i is, that here the addends arc the differences of log Oy. The operation may be verified horizontally at any point, as in Case i ; but we have not here the final verification that we have there. The last sum in each column is no longer o : it is necessary, therefore, to form a series consisting of the terminal values of the several columns in order. This may be done as follows : — Log[Q„(l+0] ^og(l-f-«io) 1-400510 Q,o 2-610958 10 0-011468 997349 2831 II •011648 996841 3390 12 -011879 996417 3831 13 •012127 996089 4179 14 -012395 xlv J4 012395 995857 4435 IS -012687 * « Log(l+f/,5) 1 -383003 Qi5 2-629G24 15 0-012687 Both ;r and J' here vary: the addends, therefore, are A log (i+aj.) and AlogQ^; and the terms thus formed should be inserted in their places at the outset. It will be observed that, with the exception of three, the policy values here formed are all negative ; and a very slight addition to the loading would have rendered the three excep- tions negative also. With so small a loading as ten per-cent. a policy effected at 10 has no positive value till 14, when it is worth just one penny. Yet there are not wanting those who maintain that every policy, whatever may be its duration, has, or ought to have, a surrender value ; and that, consequently, the abandonment of a policy, under whatever circumstances, is a source of profit to the office. II. Of the Two-Life Tables. The Two-Life Tables in the present volume comprise, first a complete set of the values of annuities on the joint duration of two lives, at each of the rates, three, three-and-a-half and four per-cent. ; and, secondly, a similar set of annuity values, and at the same rates, on the last survivor of two lives. These, as has been already intimated, have been deduced from the H^: table. The two classes of tables will be treated in order ; and, as before, the illustrations will have reference to the rate of three per-cent. I. 0/f/^c Joint-Life Tables. The tables were constructed by aid of the well-known formula, xlvi which is very simpl)- dcducctl as follows : — The annuity on {.v.j) consists of two portions, that iiavinjj reference to the first year, and that having reference to all the years after the first, of the possible joint duration of (x) and 0'). The value of the first portion is 7/^.^ and that of the second portion is iy^;r.i/^x+i';/+i ; whence the total \aluc is, or, in logarithms, Iog^^.j,=log7/;,.y + log (i +^rj+i.„+i). For the application of this formula wc want the values of \ogz'/>j;.y for the requisite combinations of x and j'. These are, if for convenience we make x the older age, the combinations in which each value of ;r is connected in succession with all the values of J' that do not exceed it. The formation of the values of this function was, therefore, the first step ; and the following is a type of the arrangement adopted : — X-IJ 96 95 94 93 o 96-96 95 "95 94'94 93-93 I 96-95 95 '94 94-93 93-92 2 96-94 95-9.5 94-92 93-91 • * « « « » ♦ * « « 84 96-12 95'^^ 94-10 85 96-11 95-10 86 96-10 The values of ,t- arc at the top, and those o^ x—y at the side. The former proceed from 96 to 10, and the latter from o to 86. Hence x is constant in the columns, and x—y is constant in the rows. It is the fulfilment of this latter condition that has been mainly kept in view in devising the arrangement. It is the series in which x—y is constant that are wanted in the final construction ; and they are most conveniently abstracted when formed in rows. Tlie following is a portion o^ the actual formation : — xlvii Fonnatiou of Logrp^ o 96 515256 295815 811071 132721 943792 73995 017787 45457 063244 250S7 088331 20541 108872 95 106887 132721 239608 73995 313603 45457 359060 25087 384147 20541 404688 15543 420231 94 372329 73995 446324 45457 491781 25087 516868 20541 537409 15543 552952 8008 560960 93 520319 45457 565776 25087 590863 20541 611404 15543 626947 8008 634955 4583 639538 92 611232 25087 91 661408 20541 681949 15543 636319 20541 656860 15543 672403 8008 697492 8008 705500 4583 710083 6490 680411 4583 684994 6490 691484 * 716573 5402 721975 80 81 82 83 84 85 86 249797 163 249960 48 250008 999933 249941 999824 249765 999708 249473 999603 249076 545776 48_ 545824 999933 545757 999824 545581 999708 545289 999603 544892 678545 999933 678478 299824 678302 999708 678010 999603 677613 752473 999824 752297 999708 752005 999603 751608 797753 999708 822550 999603 797461 999603 797064 822153 The construction was effected as follows. The function to be formed is log vp^.y^ which is equal to \ogv + \ogp^ + \ogpy\ and, connecting the constant, logu, with log/^., it will be written thus, \ogvp^ + \ogp,j. Now if in this expression both x and y vary, as in the rows, wc have, xh'iii Axvlog»'/r-y = AlogT/^+Alog/,; . . . (i.) if )' alone vary, as in tlic columns, \vc have, Avlogx'/^.,, = Alog/„; (2.) and if .r alone vary, as in ascending diagonally, we have, AjogrA.,, = Alogt/^ (3-) From (i) we get, for construction of the initial and verifica- tion scries, in the rows, proceeding from left to right, log i/;r-;/=log ty^j:+i.,+i + Acolog v/>^+ A colog/^; from (2), for the principal formation in the columns, log yA-y=log vp^.,j+i + A colog/,,; and from (3), for the construction of the series of terminal values, log ^A-y = log t^A+l-y + A Colog/^. The following arc examples of the application of the first and third formulae: — Initial X — Log tyjgs 96-96 95 V5 94-94 93 '93 92-9: 91-91 Log 17591 91-91 Values. y=o r251209 1 -204047 2^515256 295816 295815 riOG887 132721 132721 -372329 73995 73995 •520319 454:56 45457 •01 1232 25O89 25087 -001 408 « r82428G 1-837122 roe 1408 Verifications. X- Log vpM V 2^90 96*90 95-89 94-88 93-87 92-86 Log r/?9, 92-85 ■I/=6 1-251209 r-8570G3 ri08872 295816 15543 •420231 132721 8008 •5G09G0 73995 4583 •039538 45456 0490 •091484 25O89 5402 91-85 -721975 1/824280 1-897089 r-721975 Term In Log vpoG 96-10 95-10 94-10 93-10 92'IO Logt^Jg, 9i'io x\ Values. r251209 1-997807 r24907e 295816 -544892 132721 •077013 73995 •751008 45456 •797004 25089 91-10 •822153 1-824280 1 -997867 r822153 xlix As appears by the formulae the differences are here used in reverse order. Theoretically Acologiy^^ is the same thing as A colog/^, since v is constant. Practically, however, in conse- quence of the unavoidable inaccuracy in log v, these functions will occasionally differ by a unit (sometimes two) ; and there- fore, since A colog vp., is not tabulated, in using A colog/^ in its stead, the last figure must be corrected, by reference to the column colog vp„ pp. lo, 12, before proceeding to the additions.* The corrected figures are here printed in different type. The main formation illustrates the application of the second of the foregoing formulae, and the whole of the results just obtained will be found in their places in the portion of it given on p. xlvii. It will be observed that it is the same series, cologpy, (in reverse order,) that forms the addends in all the columns ; and consequently, when it has been inserted in the first column, it was an easy matter to transfer it thence to the other columns in succession. It may be mentioned that after the initial and two or three verification series had been inserted, the entire formation of this function did not occupy more than twelve hoiirs. A specimen of the annuity construction occupies the fol- lowing two pages, and to it we now refer. The arrangement is here obvious from the indications at the tops of the columns and at the side. In the columns x—y is constant, which is necessitated by the requirements of the working formula, and y, the younger of the two ages, is constant in the rows. The result of the arrangement is that the values which arc to occupy the rows in the table in its final form are here formed also in rows. * See pnge xxviii. 1 Fonnaiion of (7^.„. y 96 93 9f 93 92 91 x-y = (l 515256 -0.528 x-y=\ X — ^ 1 106887 01398'J 8 1 120884 -1.321 811071 -0647 372329 053873 9 239608 027233 64 426211 -2668 266845 -1849 943792 -0879! 520319 102711 2 446324 073660 < 313603 036509' 64 623032 -4198 519991 -3.3 1 1 350176 -2240 611232 152215 9 505776 124196^ 23' 491781 087754 14 763456 -5800 689995 -4898 579549 •3798 661408 198647 21 636319. 173089, 31i 59(»863j 139801, 860076 -7246 8O9439; -6448 730678 •5379 • • • • • * y 15 ^-y 249960 = 81 •1778 The same. •'■—.'/ = 83 i i 1 545824, 071067 9 14 616900 ■4139 250008 •1778 678478 150420 545757 071082 1 u 828898 752297 223812 39 _;6744 616840 •4138 249941 -1778! 67S302, i:)()391 12' 545581 071067! 6 1 13 976148 -9466 1-2210 828705 •6741 616654 ■4137 797461 289244 24 086729 822153 346540 16 16S7()!J 752005 223772 2 678010 150332 16 11 975779 ■9458 '■-'94 828358 •^1?>^ JO '■4747 797064 289050 39 os(ii:).s 751608 223611 23 '••75242 ' 1 ■9446s The terming portions of these six (double) columns i« given hcreatccr, pp. ixiv and Ixv. y Formation X— y=3 I ofa^-y. — (conli X— 7/=4 nued.) =5 9.3 017787 •1042 359060 043032 8 92 402100 -2524 063244 •1157 516868 097745 384147 047534 4 Qi 614613; -4117 431685 •2702 088331, -1226 » * * » « • • ! y X-XJ = 84 The same. X — 1/ = 86 12 249765 •'777 545289 071037 10 II 616336 •41.34 2494731 -1776 677613 150244 11 544892 070991 11 10 827868 •6728 615894 •4129 249076 •1775 The initial terms are formed as follows. In the equation if .r is 96, (the next to the oldest tabular age,) aj-^y^/^i vanishes, and '.vc get whence log^9c-v=logyAG-j/. From this it appears that the initial terms of the several columns arc the terms of the series log v/y^.,, which occupy the first column, (that headed 96,) in the formation on page xlvii. The initial terms being then inserted in their places, each is followed, in its own column, by the succeeding terms of the series, {x—f constant,) of which it forms the leading term. The logarithmic operation now commences ; and the first column, (x—j'=o,) is referred to for illustration. As shewn by the working formula, the chief part of this operation con- sists in the formation of log(i +rt:j..„) from lo^ax-,,, which is known. This of course can be done by aid of the common logarithm tables, used first inversely and next directly. But it can be done much more easily by the use of Tabic I. in Gray's Tables and Fonnuhs; which table, giving the values of log(l+.r) corresponding to successive values of log.r, enables us, by a single direct entry, to pass at once from the given to the required logarithm. The operation was conducted, there- fore, by means of this table. The method of proceeding is so simple as hardly to need explanation. The table is entered with the initial term 2*515256, which is the logarithm of a^^i^.,^^, and the result •013989 + 8, (8 being the pro-parts,) is set down as shewn. Addition then gives the annuity value on the next younger com- bination, (95'95)- Here, observing that 120884 is less than 515256, we know that an increase of a unit has taken place in the index. The next entry accordingly is made with I •120884, and an operation similar to that just described gives liii log%,.9,, and so on. Having always sufficient warning of an increase in the index by the decrease that at the same point takes place in the mantissa, it is found to be unnecessary to set down the indices, whereby some writing, as well as space, is saved. The operations in the remaining columns being in all respects similar to those in the first it is not necessary further to refer to them. From and after column 4, {x—y='^,) the index of the initial term is i, and in none of them does a change take place till a considerable number of terms has been formed. When the logarithmic part of the operation has been com- pleted there remains the taking out of the numbers. It has been usual heretofore to tabulate annuity values to three decimal places only. It is, however, now well enough under- stood that such annuity values " are not sufficiently exact for survivorship questions, the results of which depend on the differences of nearly equal annuities"* ; and therefore it was determined that the annuity values in the present volume should be given to four decimal places. The additional labour involved in this determination is very small indeed. So long as the values do not exceed 10 the numbers can be obtained to the required extent by inspection, by the use of the common seven-figure tables, or Bremikcr's six-figure table. For values from 10 to 20 we can now also obtain them by inspection by the use of Mr. Sang's recently published seven-figure table. In this table the numbers extend to 2(X),ooo ; and therefore in the portion from 100,000 to 200,000 the argument consists of six places. It is consequently in only the comparatively small number of cases in which the value of the joint-life annuity exceeds 20, that interpolation in taking out the numbers is necessary; and, as only one figure has in those cases to be interpolated for, if Brcmiker's * De Morgan, Q/mpanion to The yilmanac, 1 842, p. 7 ; and Journal of the Imtitute^ vol. xiii. p. 136. liv tabic be used (ov the purpose tlic oper.iti /'g4 /as P. 150 l+«G5-37 8-7779 logO-943391 238 \-v „ 2-464284 ■^65-37 •25567 •74433 „ 1-407675 p. 150 5 '^64-37 «64-37 «65-36 8-0791 8-41934 „ 0-907363 col. 0-017915 log 0-925278 150 3 7^7999 „ 0-892089 col. 0-003974 '^65-3G 7-87160 log 0-896063 8-41934 P36 ^05-37 7-87160 ■54774 •74433 Sum, Diff., 1-29207 •19659 A' A 1 -64603 = •09830= =half sum. -A.C5-37 •74433 Iviii The difference between '''''■" and "'"■'''' is here added to and subtracted from Ar,.v37 '• ^^K li^^^ sum and the half difference arc the complementary survivorship assurances, being together equal to the joint-Hfe assurance. Here too Orchard's Tables would afford material help. By a single entry in them we should find at once Ar,.v37 = .74433- Notwithstanding this aid, the operation is sufficiently tedious to render the possession of complete tables of survivorship assurances exceedingly desirable. The true values of the above benefits, as formed, first, by a continuous process which will be hereafter described, and secondly, by the use of the six-figure logarithms which arise in the annuity operation, are 646010 and •098324. Those here formed arc consequently aftcctedby errors of 4- 2 and —2 in the fifth place, respectively. These errors, which arise in the use of the tabulated four-decimal annuities, may or may not be con- sidered of importance. Generally speaking, if three decimals only are used in the annuities, the errors will be of ten times the magnitude of those that arise when four are used. Finally, it may be mentioned, that the value of this benefit, deferred ;/ years, will be formed, as in other cases, by multi- plying the value for the entire duration of (■X'.}') by the factor, V .,./>,.„ or j^- .^-. This is overlooked by Milne, (p. 184,) and Jones, (p. 176,) the expressions given by both of whom involve in their numerical application just three times the amount of work that is really necessary. 2. Of tJie Last-Siii-vivor Tables. These tables occupy the lower portion of the quadrangular space of which the upper portion is occupied, as already described, by the joint-life annuities. Those of them in which the rate of interest is three per-ccnt. are on pages 140 to 170 ; and from these the illustrations now to be given, of the methods employed in the construction of the Last-Survivor Tables, will be taken. lix The function under consideration is, and the values of it have to be formed for all the different combinations of x andj'. It is symmetrical with respect to the variables x and y : its value is not affected by the inter- change of these; and therefore it suffices to form it for the cases in which each value of one of the variables, say x, is combined with all the values of the other, y, in succession, that do not exceed it. It would be an exceedingly onerous task, and withal very unsatisfactory in its results, to form the required values directly, by the addition and subtraction of the single and joint-life annuities. But here the continuous method of con- struction by differences, which has been already found so effectual and satisfactory, does not fail us ; and the tables liave been constructed by this method, at the cost of a com- paratively small amount of labour, and with all needful assurance of accuracy. The following is a type of the arrangement adopted in the construction. The main object in view here, as in preceding instances, is to present in columns or in rows the results which in the final arrangement are to occupy similar positions. X ro II 12 13 14 15 lO IO"IO 1 1 r no ii'ii 12 I2'I0 I2-II 12-12 13 i3'io 13-11 13-12 13-13 H 14* 10 14- 1 I 14-12 i4'i3 14-14 15 15-10 15-II 15-12 1^-13 15-14 ^S'K^ • « • » * • * Call {x) the older of the two lives in each combination, and (j') the younger. Then we have here the successive values of X at the side, and those of y at the top ; and it will be obser\'ed that x is constant in the rows, and ;' in the columns. Ix Now Ilcncc, if .r alone vary, as in the columns, we have, and if J' alone var)-, as in the rows, we have. From the first of these we get, as the formula for the general construction in the columns, ^7JTTT/=^rr/ + A^.j— A.^.r.j,; (i-) and from the second, for the construction of verification values in the rows, a^:ff-\=(r7ij+ ^<^y—-^,j(^x-p'y (2-) Also, if X and y both vary, we have, and this gives as the formula for the construction of series of terms dcscendinrr diagonally, rt7|T:^ = «T/+ A^''j:+ Arr„ — Aj.,,rt'j,.y . . . (3.) Formula (3) comes into use only in the formation of the series of initial terms. In seeking to apply these formula: we are met by a difficulty at the outset : while the differences arc all essentially negative, those that occur in each formula are affected with unlike signs. This would seem to imply that in each case, to pass from one value to the next, both an addition and a subtraction would be necessary. But the diflficulty, such as it is, is easily surmounted. We use the differences which, in their normal form, would give rise to arithmetical subtraction, in their complementary form ; and thus the operation becomes one of uniform addition. Ixi The arithmetical equivalcnf^ of a negative number is formed in the same way as the mixed form of a negative logarithm. Thus the arithmetical equivalent of— 2576 is 77424. In the first form the whole of the number is negative, while in the second the last four figures are positive, and the prefix i only is negative. The two forms being equal in value they may be used indiscriminately in arithmetical operations. Thus, the algebraical sum of 4865923 and —2576 will be determined by either of the following operations : — 4865923 48'55923 -2576 17424 4863347 4863347 But the arithmetical equivalent is not restricted to the above form. We may write it 77424, "i 97424, '^9974^4, and so on. We have thus the power of bringing up the positive portion of an arithmetical equivalent to any number of places we please, which enables us often to dispense with the necessity of writing the negative prefix at all, or of otherwise particularly attending to it. Thus, to form the aforesaid sum : — 4865923 9997424 4863347 We simply neglect the last carriage, which in such cases is always a unit. It must not be supposed that when, as in the case before us, it is necessary to use a series of differences in their mixed form, wc require first to construct tliem in their normal form, and thence to pass to the other. They arc just as easily con- structed in the one form as in the other ; and therefore they are to be constructed in that form only in which they are required for use. * This is the term applied by the late Peter Nicholson to th;^^se complementary numbers. He appends a brief tract on Arithmetical Equivale.Tts to the second edition of his Essay en Involution and E-voliition, published in 1820, Ixii Inspection of the forcgoint^ formul.x will shew that it is the differences of the sini^ie-life annuities that have to be used in their niixctl form in order that the operation shall be one of addition throutjhout. The following shews the mode of constructing the differences of the series ax, (p. 14), in both forms : — ax A A 10 24-1484 — 1531 8469 II 2.3 •995.3 — 1811 8189 12 23-8142 -2039 7961 1.3 23-6103 —2206 7794 H 23-3897 -2316 7684 ^5 23-1581 * * * * Since A«x=^j-+i~^j-' ^"^ \iQ.XQ. rt'^.^i is less than a^, the first of the two columns marked A, which contains the differences in their normal form, is constructed by subtracting, arithmetically, the several lines from those above them ; and the results arc consequently negative. The differences in their mixed form, on the other hand, are constructed by subtracting, arithmetically, each line from that below it. These occupy here the second of the two columns marked A. The negative prefixes, however, being omitted, as not required for our purpose, the portions exhibited are essentially positive. This column was completed for the purpose of the present construction ; and it will be well now to shew how the differences of the other series, a^.y, were formed. In- spection of the formul-c shews that these, to admit of their combination with the others by arithmetical addition, require to be exhibited in their normal negative form. The differences of a^.,, involved in the formula: are, as indi- cated by their symbols, deduced from different successions of the terms of that function. Thus, the difference involved in formula (i), symbolized by A^^..,,, implying that in the primary function;irtakes the increment unity in passing from term to term, while J' remains constant, is formed from tlie rows in the table, pp. 140 to 170. That in formula (2), which is symbolized by Ixiii ^yUx.y, is formed from the columns, in which, as we know, y varies, while x remains constant. And that in formula (3), symbolized by Aj,.,/?j;.„, is formed from series of terms des- cending diagonall)', in which order both x and y increase by a unit. I. To form Aj^rt,..^. It would no doubt have been quite possible to form all the series of differences wanted from the copy of the table written out for the printer. To have done so, however, in the case of the present series, would obviously have been a most irksome task. Every row has to be differenced, the subtractions have to be performed sideways, and there is no space whereon to set down the results. The formation was therefore superposed upon the original com- putations. There the rows, although in reverse order, are the same as in the printed table ; and there is ample space for setting down the differences as formed, so as to permit revision in case of a discrepancy shewing itself The method employed will be clearly understood from the following specimen. It consists of the terminal portion of the six (double) columns of the formation of ax-y, the initial portion of which is given on pp. 1, li. The subtractions being made sideways, the differences are seen here ranged each under its minuend. 2. To form L^^x-y. This difference, as stated, is formed from the columns. It comes into use for the formation of series for verification ; and as only four or five of these are required, perhaps the best method of proceeding is to copy out the proper columns and difference them in the usual way. A specimen of the main formation will be given pre- sently. The formation of the series requisite for its verifica- tion is here shewn. Ixiv Formation of (7^.,,, a>id A^x- y X-Xj '=0 x—y=\ j--y = 2 1.1 298112 981760 19-8661 1447 294938 197214 291732 19-5764 984550 1406 984713 1450 319429 2804 316383, 313338 11 36 30 14 304200 9'84"G27 20-1465 301132 200047 297918 19-8572 1418 984693, 1475 984646, 1482 325239 2646 322286, 319239 309860 20-4111 30! 17! 13 307009, 20-2773 303902 20-1327! 984275 1338 9844511 1446 984517, 1505' 330576 2386 327907J 324953 63 9| 2i 12 314914 20-6497 3123671 20-5290 309472 20-3926' 983691 1207 983983 1 364 984159, 1475 335439 2021 332959 330194' 13 64| 69! II 319143 20-8518 317006] 20-7494 314422 20-6264 982896 1024 983294 1230 983586, 1391 339446 1561 337442 334962 41 6 21! 10 322383 21-0079 320742 20-9287 318569,20-8243 792 1044 1 ^^-^^ Ver ifications of a—. 1" ^ «lo 23-1581 p. 14 10 20-4046— 675 );10 24-1484 „ ., II '3371— 9^0 47 '3065 12 -2451 — 1124 "lo-lO 20-4046 ,, 141 13 -1327 — 128c 15-10 26-9019 14 ■0047— 1 3 86 8469 15 19 •866 1 675 15-11 •8163 S1S9 920 15-12 •7272 7961 1 1 24 i5*U •6357 7794 1280 I5"i4 •5431 7684 1386 "^s^^ •4501 Ixv Formation of a^.y and Aaj;.y (continued). y 15 H 13 X — 3 28S603 19-4358 x — ^ 285680 983876 307537 76 /=4 19-3055 X — I 282998 /=5 19-1 86C 984272 310389 3 ^303 1 189 983459 304970 93 1148 294664 197090 291489 19-5654 28852- 19-4322 984483 316098 61 1436 984205 1332 313053 85 983809 310294 21 1218 300642 i9'9822 297343{ 19-8309 294124 19-6845 984470 321810 40 1513 984307 1464 318668' 41 984029 315622 23 1361 19-937; 1493 306320 20-2451 303016] 20-0917 299674 984225 327240 19 1534 984178 324096 15 1540 984015 320858 7( 311484 20-4873 308289 20-3371 304943 20-l8lC 1567 983762 332101 80 1502 983828 329051 85 1561 983781 325906 41 315943 20-6987 1415 312964 20-5572 1526 309728 20-404^ 1586 10 The column headed 15, p. 141, is copied out, (the value of «i5.i,5 being taken from p. 140,) and differenced. On the right ^isTo is formed as shewn, and the successive terms rtisTf, S:c., are formed by the continuous addition of two series of differences, of which the first is Any, formed on p. Ixii, and the second is Aya^-.y, formed in the column adjoining on the left. 3. To form Aj^.ya^f.y. This difference comes into use only in the construction of the initial terms of the several columns in the main formation ; and in these x=f. The formula conse- quently for this case becomes, ^x+TTFTi = ^-r-x + 2Aa^-A^.j^^.^ ; and the series to be differenced will be that on p. 140, headed aj;.x- The formation of A^.a:^?x-x> was, like that of Aj-a^-.y, superposed upon the joint-life computations ; and on reference to the specimen on p. Ixiv, the values of the former of these will i Ixvi be found in the first column, fin whicli .v = y,) under those of the latter which f.dl in th.e same column, but necessarily in reverse order. The following; is an example of the formation of the initial terms of the several columns : — « Initial Terms. n,o 24-1484 p. 14 »io 24-1484 ,, ,, 48-2968 »10-10 21 "0079 ,,140 io"io 27-2889 6938 I 01 xi-ii -^ 6378 «15 23-1581 Verification. '15 23"i58 2021 „„ 23-1581 12-12 26-9787 46-3162 5922 „i.vi5 i9'866i 2386 lyi^ 26-4501 13-13 -8095 5588 2646 14-14 -6329 .r368 2804 1 5" ^5 '45° i Of the two series of differences here employed the first is 2Aa the double of the series formed on p. Ixii, and the second is A -x^x-x. ^vhose 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. Ixvii lO 10 27-2889 8469 792 Format II 27-1388 8189 1024 ion ofa-z 12 26-9787 7961 1207 13 26-8095 7794 1338 ax-y. 14 26*6329 7684 I4I8 II •2150 8189 1044 12 •1383 7961 1256 •0601 7961 1230 13 •0600 7794 1415 26-9792 7794 1391 •8955 7794 13*54 14 26-9809 7684 j 1526 •8977 7684 1502 •8113 7684 1475 -7227 7684 1446 15 •9019 • •8163 » •7272 * * •5431 * 15 2 6-450 1 The values of x are at the side and those of y at the top. The initial terms were first inserted, and next the verification series. The series ^ax, (8469, 8189, &c.) was then entered in column 10 ; and the successive terms of it were carried out horizontally, as shewn. The series Axa^-y were then inserted, that which in the specimen (pp. Ixiv, Ixv,) occupies the last rozif, in the first column ; that which occupies the next to the last rozv, in the second column ; and so on. The numbers opposite the several roivs, 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 tliis lime 2^ hours were required for tlic for- mation and insertion in their places of the initial and the verification series, and the remaining 44 sufficed for the for- mation 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 Ax-j,= i-(i-J')(i+^rp), 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 de- ferred last-survivor annuity. The component annuities here, ax + (Tiy—aj..y, 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 H^: 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 per- sonally 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 ha\c been formed. We proceed to explain the methods employed in the forma- tion of these, and shall then shew the manner in which they arc to be used in the construction of Survivorship Assurance Tables. Ixix The functions tabulated are log (/^~' — i) and log|(/^~' + i). We have, log(A"'-i) = log-^y-^ = log^=log4 + colog/, 6 '-J+i- . •. A log (/^-' - 1) = A log ^^ + colog/,+1 ; and hence, log (/;;\ - 1 ) = log (A"' - 1 ) + A log 4^ + colog/^+i. Again, logi(A-+i)=log/4^^ = logi + log(4 + 4+0 + colog4+.. Hence, ^ log KA"' + I ) = a log (4 + 4+,) + colog/^+, ; •••logi(AVi + i) = logKA~'+0 + ^log(4 + 4+i) + cologA+,. By these formulae the series were constructed, a prelimi- nary in the case of the second of them being the formation of log(4 + 4+i) with its differences, a very simple matter. The following is an example : — Formation of log ^{px~^ + !)• Log "5 T-698970 „ (^10 + ^0 5-299965 Col. Ill 5-002133 10 0-001068 998065 1736 11 -000869 998409 1444 12 -000722 998644 1268 Formation of log (p_c~^—l Log 6^10 2-690196 Col. /„ 5002133 10 3-692329 908595 1736 II •602660 918405 1444 12 •'522509 942196 1268 13 •465973 # Jc u •000634 A survivorship assurance on (x) against (j) is an assurance payable at the end of the year in which the combination (xj/) is dissolved, provided the dissolution is caused by the death of (x). When the sum assured is a unit the value of the assurance is denoted by A;:^. The value of this assurance may be conceived to be com- posed of two portions, one having reference to the first year, and the other to all the years after the first. Now it is shewn, (Gray's Tables ami FoiDiuhu, pat;e j},,) that the probability of {x) dying before {y) in the first year is Ki-A)(i+A); hence the value of the first portion of tlic assurance is, Mi-A)(i+A); and_thc value of the second portion is obviously, in which A ^\^_„^^ denotes the value of a similar assurance on (.r+i) against (j'+i). Hence, = ^^A-.v[i(A~'-l)(A-'+ + AjTTT^TTi ; or, in logarithms, logA^- =logt;A.,+ log[i(A-'-i)(A"'+ + A,-TTiHn]- This is'thc working formula, by means of which, commenc- ing with a combination in which x and y, having between them a specified difference, one or other, (or both,) is the greatest tabular age, we may form in succession, the values correspond- ing to all the combinations in which the variables have the same difference. For this purpose, and for the formation of a complete table, we require, first, log vp^.,j for all the different combinations of x and y. These logarithms have been already formed for use in the construction oi a^^-m and would of course be immediately available. And we want, secondly, to be used as will be presently 'shewn, logKA~'-0(A"'+ 0. not for the different combinations merely, but for all the combinations of the two variables x and y. The cause of this duplication of the number of logarithms to be formed— for it amounts to a duplication— is that this function is not, like the other, symmet- rical with respect to x and y. The simplest arrangement for the formation of log l(^_^-i_i)(/>,-i+ i) would be in a quadrangular space, consist- ing of 87 columns and the same number of rows, having the Ixxi successive values of, say, x, at the top, and those oi y at the side. But this arrangement is objectionable inasmuch as the series in which x—y3.n^y—x are constant, (which are the series more particularly required,) would find their places in it in diagonal lines ; and thus a difficulty would be interposed to the taking of them out in order. The formation will there- fore be effected in two compartments, of both of which the following is a type : — 96 95 94 9?> 92 91 86 96'io 8.5 96-11 95-10 84 96-12 95'ii 94-10 83 96-13 95-12 94-11 93-10 82 96-14 95-13 94-12 93-11 92-10 81 96-15 95"i4 94"i3 93-12 92-11 91-10 * * * * * * * In the first compartment ;ir>jj'; and, the successive values of X being at the top, those of x—y are at the side. In the second compartment xy. In descending diagonally x varies in reverse order. Hence the initial terms will be formed by the con- tinuous addition of the negative differences of log(/x~'— 1)> in reverse order. In the rows both x and y vary, in reverse order. Hence verification series will be formed by the continuous addition of the two series of negative differences in the table, in reverse order. In the columns ^y varies, in direct order. Hence the prin- cipal formation will consist in the continuous addition of the terms of the series A log i(A~^ + 0- 2. xJ' Inituil Terms. Verification S eries. x-t/ = 81 x-y = Log W-1) 0-647817 0-001068 0-617817 0-000625 0-647817 0-5OS155 96-10 0-648885 596485 96-15 0-648442 596485 96-96 1-155972 596485 9^10 -245370 999976 765437 94-10 768379 •013749 839603 9y^4 •244903 768379 33 9S'95 •517894 768379 906767 93-10 •853352 881377 94'i3 -013315 839603 94-94 •193040 839603 92-10 •734729 88 951849 91-10 * 924397 •6"59lT6 * 93-12 •853006 881377 147 93-93 •984492 881377 971865 92-11 •734530 924397 199 92-92 '837734 924397 984957 9rio •659126 91-91 * •747088 * x y. 94 013749 999801 013550 999853 013403 999912 013315 93 853352 999801 853153 999853 853006 92 734729 999801 734530 91 659126 ■; 736847 321276 080689 915726 794506 715246 15043 12056 8966 4566 2597 3657 4 751890 333332 089655 920292 797103 718903 28135 15043 12056 8966 4566 2597 ^ 780025 348375 101711 929258 801669 721500 48151 28135 15043 12056 8966 4566 2 828176 376510 116754 941314 810635 726066 93233 48151 28135 15043 12056 8966 I 921409 424661 144889 956357 822691 735032 1 234563 93233 48151 28135 15043 12056 ! 155972 517894 193040 x,., + log[i (/,-'- i)(/^-i + I) + A,^:^,]. which, for th.e moment, we Vvill write, log rt = log <^ + log {c + d). Here log b, log rand log^/* are known ; and to obtain \oga it is requisite first to form log (^+^0- This of course can be done b}' the use of the common tables, but tlie operation would be a tedious and laborious one. Fortunately this labour need not be incurred ; the end in view can be attained with great fncility by the aid of the • It will be prcicntly slicwn that when either *• or _y is the oldest tabular age, log A '-p— , that is log*/, is known j and fiomthis value wc descend to those corres- ponding to the younger ages, in :uccc9sion. Ixxv table employed in the formation of a^.^. The table in question, as already mentioned,* gives log ( i + ;ir; corresponding to suc- cessive values of logx; its characteristic equation being, T [log x] = log ( I + x), in which T is a functional symbol denoting the tabular result corresponding to the appended argument. In this equation write ^ for a", and we get, T [log ^] or T [log d- log c] = \og(i+'^^) = \og('-+~) = \og(c + d)-\ogc. ■'■ logfr + ^;=:logr4-T[log«^— log^]. It thus appears that log(^ + ^) is formed by adding to logr the tabular result corresponding to log<'/— logr. We have, therefore, log(7=:log/-' + logr + T [logd— logc] ; and, changing the order of the first two terms, the work is vet}' commodiously arranged as follows : — log^ Sum log^ log^ T[log^-logf] log ^— log r log a Log d is supposed to have been just formed ; and, log c and log 1^ having been previously inserted in their places, log ^ is sub- tracted from log d, and the tabular result answering to the difference is written under log d. Addition of the three lines then gives log a, which forms the log d, of the next operation. The type just given is here repeated, with the values of a, h, c and d restored ; and the arithmetical operation by which we pass from log A^^y;^, to logA^;; is clearly exhibited. The numbers corresponding to the logarithms successively formed are not to be taken out till the close of the operation. * p. lii. Ixxvi D, log A,-p7yn A,^,.,^-, A B C T[D,-A] D,-A H A + B + C =iogA;, A' The a rran element adopted in the formation of A^^ is similar to tliat employed in the construction of a^-^. It is typified as follows : — I 2 3 4 ' 6 97 97-97 96 96-96 97-96 95 95-95 9<^'95 1 97*95 94 94-94 95*94 1 96'94 97-94 93 93-9.3 94-93 95-93 96-9.3 97-93 92 92-92 93-92 94-92 95-92 9692 97-92 91 91-91 92-91 93-91 94-91 95-91 96-91 97-91 • • • « « » • » Here for x>j>, x—j, (constant in the columns,) is at the top, and J, (constant in the rows.) is at the side ; and for ,v<^',y — .v is at the top, and x at the side. This arrange- ment affords the greatest facility for the transfer of the values formed to the tables arranged for use ; since in these the values that are here found in the rows, will take their places in either columns or rows. The first thing now to be done is to form the initial values for the several columns. The expression in its first form is, Ab = '^ [K I -A) ( I +A) +A-V A-:^r,v,]. Case I. x=}'. Let .rand jj/ each equal 97. Then Z^. A- and />x'y vanish, and the expression reduces to, whence, logAj4:i^. = log-5 + log^'. U^41VeRSlTY ) Case 2. x>j'. Let .i'—97 ; then/j; and /^.y vanish, and we have, whence, logA4, = logyA + logi(//"'+i). Case 3. xy Logf/v. 1-251209 „ ^( ;;,,-' + !) 0-50815O 97-96 1-759364 295816 Initial Terms. X j' the addends are, A colog vpy and A colog ^ (/y~' + i) ; and {ox xy), 15597- i^ subtracted from the initial term, which is log AJj-j;. and the diftcrencc 530161 is set down opposite the subtrahend. It is easily ascertained that the index of this logarithm is 2. Entering the table then with 2-530161, the result, 014477, is set down as shewn ; and summation of the three lines gives for log A^, •685705. This forms the initial term for another similar operation, the result of which is 684407 = log AJj^^. And so by a succession of such operations the column is completed, and all the values of log A^,, in which x=y are formed. The operations in the other columns also, are entirely analogous to that described, the values formed in each being those in which the differences between ,r and j' are as at the top, respectively. The same remarks apply to the second compartment, in Ixxix which .r s(»31 327543 2 •017457 702228] •50.376; 733404 * •5^^257 Ixxxi Formation of ^:~ (continued). y- -■^=3 y- -:r=4 y- -^=5 1 1 1 ! t 1 1 1 1 1 1 94 860105 724612 780025 017787 049306 9 1-080080 93 847127 703278 885949 •769040 M34059 348375 359060 119019 12 •498752 751890 063244 055434 7 92 826466 ■670603 870575 •742292 903270, •8ooqq2 101711 516868 184838 19 803436 •724755 333332, 384147! 128564 Hi •537243 736847" 088331) 059446, 3' 1-166423 1 1 9A •635669 846054] •701 '543 8846271 .':66toc! 929258 611404 242626 34 •874178 089665 537409 196054 32 •756389 321276 -563351 404688 1 135402 I 14 ' 90 783322 •607186 823160 •665516 861380 •726742 801669 672403 291927 26 •981653 920292 626947 255144 30 •902868 080689 552952 205040 35 •780691 89 * 766025 08347s » 802413 ■634473 • • ^38716 •689788 97' y- G8G133 r=o x< y— V X= I V- x — 2 \ 96 •4849.';9 1 597996, -396274 | 1 752457 2-845539 811071 029377 35 95 •48,^'; 12 {592940 -391688:490297, -,3092411 286273 1-306667 520836 2-969461 239608 943792 0801151 038664, 11, 41; 94 •481550; 606007 -403562503333 .318664I 032643 -573364 12587G 1-377457 446324 313603' 138103: 092879, 18'j 1 Hi 93 •479.323 617088 i -414084 532369 ; -751219 914020 t 491781 1 150830 i 15 •340698 865869 565776 194208 7 •618349 92 -476990 625860! •4235325566461 -360285! 762131 -863729 790266 -766380 636319' 590863' 238205 199714! 12! 29 Qi -474886 636667 -433179,5808721 -380954 67767^ 68194C 28096( 64063'5 [ -958993 692717, -^^^lob )i , 6568601 ) 248674; ) 241 90 ■472777 ^: '4.3 7^54 y -033408 ! : 598275 6T9282 69749i2 290608 4G 1 •.396';29 60722( 71803] 31805] •978993 89 • • -470697 • 643311 • 5 -439858 m 607428 •404975 1 • Ixxxiii X 1 x=3 y— r=5 i i Q4 391397 -246262' 360439 017787 044294 6 422526 1-030958 0^ •26456J 304995 •201834! 007253 359060 100417 15 •415273 241816 063244 047523 9 •292917 352592 1-063179 Q2 466745 •225212I 231828 •170541 838417 516868 153788 8 •628328 931650 384147 101600 9 •420942: 166213 1088331 047783 2 1-065615 Qi 509081 •322910 417406 •26146c •648402 1 302329 •200599 720852 611404 207916 11 •788229 769003 537409 159880 1 [862236 404688 105654 20 •440093 go 540183 •346883 466293 •292613 372598 •2358,30 634325 672403 256479 26 •905858 662460 626947 213917 13 •803833 710611 552952 164083 27 •661987 89 • 563233 * •36579 ♦ 1 [| 503337 1 i • •31866^ « •| 427673 i • 1 ■267715 * Ixx: XIV ponding positions in the two compartments of the fornKition just given. Tlic joint-life annuities also, with which we have 10 do, occupy positions corresponding to the others in the for- mation on pp. 1, li. Thus, to verify the rows opposite 92 : — 92*92 9293 92'94 9295 92*96 476990 534078 599527 670603 742292 476990 422532 360285 292917 225212 9297 800332 170541 •953980 -956610 *9598i2 -963520 -967504 *97o873 The annuity values are : — •5800 •4>'"98 -3798 -2524 -1 157 -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 cor- respondence between the two sets of values. The two forma- tions will, however, stand a yet severer test. If we deter- mine the assurances from the annuities by the formula, A;,.y=i-(i-z'){i+a^.,j), (log (i +ajc.y)helng found in the annuity formation,) we obtain, 0-198668 0-173120 0-139815 0-097745 0-047538 0-000000 2-464284 2-662952 2-464284 2-637404 2-464284 2-604099 2-464284 2"-562029 2.464284 f-511822 2-464284 2'-464284 •04602 1 •953979 •043391 •956609 •040188 -959813 •036478 -963522 •032495 •96750.5 •029126 •970874 and we have now an all but perfect agreement in the sixtli 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 v. ill be analogous to that of the annuity values, pp. 141, S:c. ; and they will occupy the entire quadrangular space. The values in which .v=j' will take their places in the diagonal of the quadrilateral ; and, if Ixxxv the values of x be put at the top and those of j at the side, the values of the function which occupy the rows in the first compartment of the formation will here take their places, a/so 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 (:r) against {y). (x at tlie top, y at the side.) y 92 93 • 94 1 95 » 1 * 96 * 97 « y * • * • # # * * • i 89 •58348 •63447: •68979 89 '90 I ■ 3 1 4 1 •55402 •51891 •47699 •42253 •36029 •60719' •57425 •53408 •47932 •41408 •66551-72674 •63567 "70154 •59953 -67060 •54715 ■6^2>^s •48155^ -56184 ■76670 •74229 •70328 •64965 •80033 •76904 -72461 90 I ! 2 3 4 95 6 7 •29292 •22521 •17054 •34070 •26456 •20183 40365 -48351 •31866 -39169 •24626 ^30924 •57730 •48496 •39627 •57460 •48544 95 6 7 The specimen forms the concluding portion of the last six columns. It is composed of the values formed in the examples on pp. Ixxx to Ixxxiii. 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-ccnt. formation, or to undertake the like at either of the other rates of three-and- a-half and four per-cent. It docs 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, recog- nising 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 even- man a debtor to his profession, from the which as men of coune do seek to receive countenance and profit, so ought they of duty to endeavour themselves byway of mends to be a help and ornament thereunto." Ixxxvi 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 feci to be incumbent upon them. Secondly, it would lighten and facilitate their labours when they come to occupy the positions to wliich 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 witli what facility, by proper arrangement, and with the aid of the materials contained in the pn^scnt volume, the tables required for actuarial use may be constructed : it is hoped that it will be found to contain little which docs not bear more or less closely on that object. P. G. EXPLANATION 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- * Journaly vol. xlii., p. 95. Ixxxviii tagc, that precisely the same aggregaic 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, arc 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, /,o, /jj, /jg, /jj, .... we can by the formula for interpolation determine all the intermediate values at the other ages, and so obtain a complete scries 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 lo, 15, 20, 25, .... arc 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 scries hereunder stated, viz. : — 'I0» M5) '20> '25) • • • • Al» A6> k\y ^■Idy • • • • '12» '17 » '22 » '27» • • • • M3> '18 > VJS* '28> • • • • 'U» '19 J '24» 'iO 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 the arithmetical mean oi the five corresponding ordinatcs, and, in other Ixxxix words, the five curves arc, as it were, mutually drawn in towards a central course. That such central curve must exhibit a correct average of the original observations without giving undue weight to any of them, it is unnecessary to explain, and it will be at once perceived that every element of the data is equally employed in its determination. It is not requisite, however, to compute these five curves separately, a labour which would be unnecessarilv circuitous. For the purpose of actual calculation wc proceed mathematically to reduce the preceding system of operations to a direct process. For any given age let / denote an interpolated value of the number living, and let A, which denotes the original number tor an age x. years older, be the nearest quinquennial point of the corresponding curve. Then the series of values from which / is found are /j-j, /-, 4-)-5, &c., and by interpolating with central values and stopping after sccond-difier- ences, we shall have = lz-h'\ 'Z ~ IzArb- 50 25 50 For the several values of /as deduced from the five respective curves we must make ;: separately equal to — 2,— r, o, i, 2. The five values of /are therefore /=--i2/_7+-84/_, + -28/+3 /=_-o^/_, + -9$/_i + -i2/+, /= o + / + o /=-f ■j2/_4 + -95/+i--o8/+,-, /= + -28/_3+-84/+2-'I2/+7 Hence if we put 7l=/-l+/+I. 72 = ^-2 + ^+2' 73 = ^-3 + ^+3'! 74=/-4 + /+4, 7g=/-g + /+g. 77=^-7 + 47) /=7i — 73.^=72 — 73. -^' = 70— 73. ^^ = 77 — 74 • • 0^) and if (/) denote the required average number-living at the given age, we shall have by adding together the five values of /, m (") xc 5(0=^+ •96y, + •84y2H--28y3+ -I 2y, — -oSyo- -I 2y7 =/+y, + y,-'04((y,-y3) + 4Cy.^-y3)+2(yo-y3) + 3(77-7.1)} = /+7i + 7.-04(/+4^+2^' + j^) (7) By means of these last formulx, (ct), (/j), {-/), the required adjusted values of the numbers-living are readily computed. Example. In the Table H'', Healthy Lives — Male, it is required to find the adjusted value of /for age 25. The data are taken from " The Mortality Experience," Tabic H^, page 273, and the calculation in cxtenso is annexed: — (25) I 9297 yi 18610 yi 18619 4 6526 Con-'' + 4-60 5)4C530-60 (/) 93061 1 3 4 G 7 /_i 9361 /+i 9249 1-1 9434 /+; 9185 /_3 9493 ;+.i 912.5 ;_4 9560 /+4 9054 /_« 9684 ;+r. 8913 /_; 9743 U; 8848 yi 18610 73 10618 y; 18619 ya 18618 y:, 18618 y4 18614 X - -04 yc 18597 ya 18618 y7 18591 74 18614 •Ih - 42 3/t - 69 -111 - 4 / -« 4^ +4 -4 'J +1 h -21 = +4-60 k -23 Corr" -115 In filling in the data the computer will bear in mind that a chasm precedes the last two columns, and that l_^ and /^^ arc there passed over unheeded. This is further indicated by the numerals placed at the top of the respective columns. It \vill be further observed that the final result is here put down and retained to an extra place of figures. Since the exp'-rsion (y) is linear with respect to the several values of /, it is evident that precisely the same formula; may be applied to adjust the yearly decrements, and these being much smaller numbers the calculation of the table may be thus considerably abbreviated and expedited, the numbers-living being then deduced by the successive subtraction of the adjusted decrements. The adjustment of the decre- ments undoubtedly oftcrs the greatest possible facility. As an example, taking the same Tabic as before, the following is the calculation of the adjusted decrement {d) at age 25; — XCl (25) d 48 n 128 y.j 133 309 Corr" - 0-52 5)308-48 {J) 617 93061 92444 1 2 3 4 G 7 (/_i 64 d+i 64 rf-2 73 (f+2 60 rf_3 59 6f+3 71 . 67 67 d.6 68 rf+o 65 5 45 77919 950 4-8916434 9-9946724 -9878079 •012 192 I 6 76969 996 4-8863158 9-994 343 5 •9870597 •0129403 7 15913, 1 04 1 4-8806593 9-9940080 -9862978 •013 702 2 8 74932 1082 4-8746673 9-993 683 2 -985 560 2 -0144398 9 n 850 1 1 24 4-8683505 9-993 339 2 -984 7800 •015 220 50 72726 1 160 4-8616897 9-9930170 •984 049 7 •0159503 I 71566 1193 4-854 706 7 9-9926994 -983 330 I •0166699 2 70373 1235 4-847406 I 9-9923107 •9S2 450 7 •0175493 3 69138 1286 4-8397168 9-9918459 •9813995 •018 600 15 4 678^2 1339 4-8315627 9-9913438 •980 265 9 -019734 I xcv Healthy Lives — Male. — Adjusted Tahle. Age. X No. Living. I. Decre- ment. Log l^. T nn. , Pro^- Sur- 1-Oe/^.f vivinglYcar. = A log/^ p^ Prob. Dying in 1 Year. 1 55 66 SI 3 1399 4*8229065 9-990 767 9 •9789665 .021 033 5I 6 65 ^H 1462 4-8136744 9-990 1377 •977 547 1 -0224529 7 63652 1527 4-803 812 I 9-9894543 -976 010 2 •023 989 8| 8 62 125 ^59^ 4-7932664 9-9887258 -9743742 •025 625 8 9 60533 1667 4-781 992 2 9-9878723 -9724613 -0275387 6o 58866 1747 4-7698645 9-986916 I -9703224 -029 677 6 I :57 1^9 1830 4-7567806 9-985 858 I -967961 6 •0320384 2 55289 ^9^5 47426387 9-984691 I -965 363 8 -0346362 3 53 374 2001 4-7273298 9-983 405 I -962 509 8 •0374902 4 51 373 2076 4-7107349 9-982 085 6 -9595897 -0404103 65 49297 2141 4-692 8205 9-9807x65 •9565694 •043 430 6 6 47156 2196 4-673 537 9-9792893 -9534312 •046 568 8, 7 44960 2243 4-652 8263 9-977 7744 -950 I II 2 •049 888 8 8 42717 2274 4-630600 7 9-9762427 -9467659 •053 234 1 9 40443 2319 4-606 843 4 9-974 355 1 .942 660 •0573400 70 38124 2371 4-581 1985 9-973 114 •9378082 •062 191 8 I 35 753 2433 4-5533125 9-969 392 5 -93x9498 •068 050 2 2 333^0 2497 4-522 705 9-966 1699 -925 060 •0749400 3 30823 2554 4-488 8749 9-9624355 -9x7 X398 -0828602 4 28 269 2578 4-4513104 9-9584706 •908 804 7 -091 1953 75 25691 2527 4-409 781 9-9550326 -901 6387 •09836x3 6 23 164 2464 4-3648x36 9-951 1567 -893 628 I •10637x9 7 20 700 2374 4-3159703 9-9470974 -8853140 •I 14 6860 8 18326 2258 4-263 067 7 9-942 894 I •876787 r •X23 2X2 9 9 16068 2138 4-205 961 8 9-9379893 -8669405 •1330595 80 13930 201 15 4-143951 1 9-932 1429 -855 348 I ■1446519 I ^^9^5 1883 4-0760940 9-925 293 5 •841 963 9 •158 036 I 2 10032 1719 4-001 387 5 9-9183703 •8286483 •1713517 3 8313 1545 3-9197578 9-910 702 6 •8x41465 •1858535 4 6768 ^346 3-8304604 9-903 699 I •80X 1229 •X98877 1 85 5422 1138 3-7341595 9-897 690 •790 1143 •209 885 7 6 4284 941 3-6318495 9-892 2S69 -7803455 •2196545 7 3 343 773 3-5241364 9-885 796 7 -768 7706 •231 2294 8 2570 615 3-409933 1 9-88X 2137 -760 7004 •2392996 9 ' 955 495 3-291 1468 9-873 206 I •746803 X •253 1969 90 I 460 408 3-1643529 9-8576628 -7205479 •2794521 I I 0^2 329 3-022015 7 9-837 X22 6 -687 2623 •3127377 2 723 254 2-859 138 3 9-8x20345 -648 686 •3513140 3 469 ^95 2-671 172 8 9-7665778 -584221 7 •4157783 4 274 139 2-4377506 1 9-692 583 2 -492 700 8 -5072992 95 ^35 86 2-1303338 9-559 862 3 1 -362 962 9 •637037 1 6 49 40 1-690 196 I 9-264 046 4 •1836735 -8x63265 7 9 9 0-954 242 5 00 •000 000 I -000 0000 8 CO HF. Healthy Lives. — Fcmak. — Adjusted Table. Age X No. j Dccre- Living. menL /x 1 005 1029 8 73976 1067 9 72909 1102 50 71807 ii33 I 70674 1167 2 69507 1204 3 68 303 1 25 I 4 67 052 1304 Log/j 5-000000 o 4*998 076 2 4-996297 I 4-994 607 5 4-992951 O 4-991 2748 4*989 516 6 4-987617 1 4*985 516 3 4-9831525 4-980521 5 4-977691 6 4"974 732 7 4-971 7164 4-9686884 4'965 657 9 4-962 606 1 4'959 5i3 6 4*956 3173 4-9530103 4*949 594 9 4-9460787 4-942 494 2 4-938 844 8 4*935 I H i 4*931 295 1 4*9273704 4*923301 o 4-919078 1 4*9147133 4-910 229 4 4-905 644 8 4-9009785 4-896 2120 4-891 2865 4-886 163 5 4-8807793 4-8750902 4-869 090 8 4-862 781 I 4-8561668 4*849 259 7 4-842 028 5 4-8344398 4-826411 7 9-9980762 9-998 220 9 9-9983104 9*998 343 5 9*9983238 9-998 241 8 9-998 1005 9*997 899 2 9-9976362 99973690 9-9971701 9-997041 I 9-996 983 7 9*996972 o 9-996 969 5 9-996948 2 9-9969075 9*996 803 7 9-996 693 o 9-996 5846 9-996483 8 9*9964155 9-9963506 9-9962693 9-996 181 o 9*9960753 9*995 930 6 9*995 777 ^ 9*995 635 2 9*9955161 9*9954154 9*995 333 7 9*995 233 5 9*9950745 9*9948770 9*9946158 9-9943109 9-994 000 6 9*993 690 3 9*993 385 7 9-993 092 9 9-992 768 8 9-992 41 1 3 9-991 971 9 9-9914708 Prob. Siir- Prob. Dying viving 1 Year, in 1 Year. Px ! qx -995 5800 -9959119 -996 1170 -996 193 o •996 148 o ■995 959 6 ■9956359 •9951744 -9945720 -993 960 2 •993 505 1 -993 2100 -9930788 •9930519 •993 046 3 •9929976 •992 904 6 •992 667 3 -9924141 -992 166 7 •9919364 •991 7803 •991 632 2 -9914465 •9912450 •991 0037 •990 673 8 •9903235 ■990 000 o -989 728 6 •989 499 1 •989313 1 -9890846 •9887228 •988 273 1 •9876790 •986 98 1 8 -986 280 9 *985 5764 •9848853 -984221 6 -983 487 6 -982 678 o •981 6846 -9805524 •004420 o •004088 1 -003 883 o -003 807 o •003 852 o •0040404 •004 364 I •004825 6 -005 428 o -006 039 8 •006494 9 •006 790 o •006921 2 •006948 I -006 953 7 -007 002 4 -007 095 4 ■007332 7 *oo7 585 9 *oo7 833 3 •008 063 6 -008 219 7 -0083678 •^08 <,<,i 5 •008 755 o •0089963 •009 326 2 •009 676 5 •01 0000 o -010 271 4 •0105009 •0106869 -0109154 -01 1 277 2 •oil 7269 •012 321 o •013 0142 •013 7191 •014423 6 •015 1147 -0157784 •0165124 -017 322 o -018 315 4 -0194476 XCIX Healthy Lives,- 'Male and Female. -Adjusted Table. Age X No. Living. decre- ment. Log Ij: Log p^. ^ =Alog I J, Prob. Sur- iving 1 Year. Prob. Dying in 1 Year. 9s 55 65748 1358 4-8178825 9-990 935 9 '979 345 4 *o2o654 6 6 64 390 1414 4-8088184 9-9903567 -978 040 I ■02 T 959 9 7 62 976 1471 4799 175 I 9-9897353 -9766419 -023 358 I 8 61 5°5 153 1 4-7889104 9-989 052 6 -975 107 7 -0248923 9 59 974 1601 4-7779630 9-988 249 ■973 305 I *026 694 9 6o 58373 1677 4-766 2120 9-9873404 *97i 271 •028 729 I 56 696 1760 4753 55^ 4 9-9863046 •9689572 *o3i 042 8 2 54936 1849 47398570 9-985 "31 2 -966 342 7 ■033 657 3 3 53 087 1936 4-724988 2 9-983 865 9 '9^3 53^6 *o36 468 4 4 51 151 2014 4-7088541 9-9825545 -960 6264 -0393736 65 49 137 2080 4-691 408 6 9-981 215 6 -9576694 •0423306 6 47057 2138 4*672 6242 9-9798059 •9545657 •045 434 3 7 44919 2186 4-652 430 I 9-9783333 -951 3346 *o48 66^ 4 8 42733 2224 4-630 7(53 4 9-976 788 I •947 955 9 ■052 044 I 9 40509 2268 4-6075515 9-9749777 ■9440124 •055 987 6 70 38 241 2331 4-5825292 9-972 6862 -939 044 5 '060 955 5 I 35 9^° 2401 4-5552154 9*969 946 I '933 1384 *o66 86i6 2 33 5°9 2469 4-5251615 9*966 760 2 -9263183 •0736817 ,3 3 1 040 2531 4-4919217 9*963 060 3 *9i8 460 •081 5400 4 28509 2567 4-454 982 9*9590215 -9099583 *o9o 041 7 75 25942 2542 4-4140035 9-9552124 -902 012 2 •097 987 8 6 23 400 2476 4-3692159 9*951 428 8 -8941881 •105 811 9 7 20 924 2369 4-3206447 9*9478163 •8867807 •1132193 8 18555 2247 4*268461 9-943 939 7 -878 9006 •121 0994 9 16308 2110 4-212 400 7 9*9398265 -8706156 •1293844 80 14 198 1969 4*152 227 2 9-935 ^'^3 7 •861 3185 •138 681 5 I 12 229 1823 4*0873909 9*929 892 9 •850928 I •149 071 9 2 10406 1672 4-0172838 9*9239294 •8393234 •1606766 3 8734 1522 3-941 2132 9-916 842 5 -8257385 •174261 5 4 7 212 1360 3-8580557 9*909 248 6 -8114254 •1885746 85 5852 1186 3-7673043 9*901 6404 -7973342 •202 665 8 6 4666 1014 3-6689447 9-8935861 *782 683 2 •2173168 7 3652 849 3*562 530 8 9*885 092 3 -7675247 •232 475 3 8 2803 689 3-447623 I 9*8774819 -754 191 9 •245 808 I 9 2 114 548 3*325 105 9*8696868 -7407758 •2592242 90 1566 435 3*1947918 9*8586708 •722 222 2 •2777778 I I 131 33^ 3-053 462 6 9*846 904 5 *702 9i7 7 •297 082 3 2 795 247 2*900 367 I 9-8384135 *689 308 I •310 691 9 3 548 181 2*738 7806 9-8258855 *669 708 •3302920 4 367 131 2^^6.^666 I 9-808 245 9 •6430518 •3569482 95 236 86 2*372 912 9*803 1793 •635 593 2 •3644068 6 150 56 2*176091 3 9*7970366 -626 666 7 '373 333 3 7 94 44 1*973 1279 9*725 842 I •5319^9 •468 085 I 8 50 33 1*698 9700 9-5314789 -340 000 *66o 000 9 17 17 1*2304489 — CO *ooo 000 i*ooo 000 100 \ 1 — 00 TABLES. ONE LIFE. I. H". HEALTHY MALE LIVES. 2 Klcincnturtf Vdhws. X lO /x Log/^ I^og/'^ A Colug /^ X 100 000 5 '000 000 T-997 867 000 397 yooo 000 10 I 99.110 4-997 867 •998 264 000 292 •002 133 I 2 99 H3 •996 131 •998 55^ 000 1 76 •003 869 2 .3 98784 •994 687 •998 732 000 067 •005313 3 4 98496 •993 4^9 ■998 799 999 952 •006581 4 '5 98224 •992 218 •998 75 1 999837 •007 782 15 6 97942 •990 969 •998 588 999722 •009 03 1 6 7 97624 •989557 •998 3 1 999 604 •010443 7 8 97245 •987 867 •997 914 999 584 •OI2 133 8 9 96779 •985 781 •997 498 999 744 •014 219 9 20 96 223 •983 279 •997 242 999 828 •016 721 20 I 9S 614 •980 521 •997 070 999 947 •019479 I 2 94971 ■97759^ •997017 000 036 •022 409 2 .3 94 32 1 •974 608 •997 053 000 054 •025 392 3 4 93683 •971 66 1 •997 107 000 004 •028 339 4 25 93 061 •968 768 •997 III 999976 •031 232 25 6 92444 •965 879 •997 087 999 904 •034 121 6 7 91 826 ■962 966 •996991 999 883 ■037034 7 8 91 192 •959 957 •996874 999 886 •040 043 8 9 90538 •956831 •996 760 999 873 •043 169 9 30 89865 •953 591 •996 6^3 999914 •046 409 30 I 89 171 •950 224 ■996 547 999919 •049 776 I 2 8846^ •946771 ■996 466 999921 •053 229 2 3 87748 •943 237 •996 387 999 904 •056 763 3 4 87 021 •939 624 •996 291 999 882 •060376 4 35 8628J ■93s 9^ S •996 173 999 853 •064 085 35 6 85524 •932 088 •996 026 999 844 •067 912 6 7 84745 •928 114 •995 870 999 862 •071 886 7 8 83943 •923 984 •995 732 999 867 •076 016 8 9 83 122 ■919 716 •995 599 999 902 •080 284 9 40 82 284 •915 315 •995 501 999921 •08468^ 40 I 81 436 •910 816 •995 422 999891 •089 184 I 2 80582 •906 238 •995313 999 828 •093 762 2 3 79717 •901 5S^ •995 141 999 8 1 •098 449 3 4 78830 •896 692 •994951 999 722 •103308 A 45 77919 •891 643 •994 (^13 999 670 •108357 45 6 76 969 •886 316 •994 343 999 ^^S •113 684 6 7 75 973 •880 6. •;9 •994 008 999 ^n •119 341 7 8 74 932 •874667 •993 683 999657 '^^5 333 8 9 73850 •868 350 •993 340 999677 •131 650 9 50 72 726 •861 690 •993017 999 682 •138 310 50 I 71566 ■854 707 •992 699 999 612 •145 293 I 2 70373 •847 406 •992 311 999 S3 5 •152594 2 3 69138 •839717 •991 846 999 498 •160 283 3 4 67852 •831563 •991 344 999 423 •168437 4 3 HM. Elementary Values. X Colog;?^ A d. Log f/. A — A X 10 lO 0"002 133 999 603 490 2^690 196 908 595 091 40^ I •001 736 999 708 397 •598 79^ 918405 081 595 I 2 •001 444 999 824 329 •517 196 942 196 057 804 2 .3 •001 268 999 933 288 •459 3'r- 975 177 024 823 3 4 •001 201 000 048 272 ■434569 o[5 680 984320 4 15 •001 249 000 163 282 •450 249 052 178 947 822 15 6 •001 412 000 278 318 •502427 076 212 923 788 6 >7 1 •001 690 000 396 379 •578 639 089 747 910253 7 8 •002 086 000 416 466 •668386 076 689 923 311 8 9 •002 502 000 256 SS^ •745 075 °39 542 960 458 9 20 •002 758 000 1/2 609 •784617 023 594 976406 20 I •002 9 30 000 053 643 •808 211 004 702 995 298 I 2 •002 983 999 964 650 •812 913 991 908 008 092 2 3 •002 947 999 946 638 •804821 988 969 on 031 3 4 •002 '6^^ 999 996 622 ■793 790 996 495 003 505 4 25 •002 889 000 024 617 •790 285 000 703 999 ^91 25 6 •002 913 000 096 618 •790 988 on lOI 988 899 6 7 •003 009 000 117 634 •802 089 013489 986 51 1 7 8 •003 126 000 114 654 •815578 012437 987 S^3 8 9 •003 240 000 127 673 •8280x5 OJ3 344 986 656 9 30 •003 367 000 086 694 •841 359 007 446 99^ 554 30 1 ■003 453 000 081 706 •848 805 006 714 993 286 I 2 •003 534 000 079 7^7 '^S5 S^9 006 015 993 985 2 3 •003 613 000 096 727 •861534 007 698 992 302 3 4 •003 709 000 X18 740 •869 232 009 864 990 136 4 35 •003 827 000 147 757 •879 096 012 441 987 559 35 6 •003 974 000 1^6 779 •89 i 537 012637 987 3(^3 6 7 •004 130 000 138 802 •904174 010 169 989831 7 8 •004 268 000 133 821 ■914 343 008 90 [ 991 099 8 9 •004 401 000 098 838 •923 244 005 152 994 848 9 40 ■004 499 000 079 848 •928396 003 062 996938 40 I •004578 000 109 854 •93^458 ^05 558 994 442 I 2 •004 687 000 172 865 •937016 010 908 989 092 2 3 •004 8j-9 000 190 887 •947 924 on 594 988 406 3 4 •005 049 000 278 911 '9595^^ 018 206 981 794 4 45 •005 327 000 330 950 •977 724 020 S35 979465 45 6 •005657 000 3^^ 996 •998 259 019 192 980 808 6 7 •005 992 000 325 I 041 3-017451 016 776 983 224 7 8 •006317 000343 1082 •034227 oj'5 539 983 461 8 9 •006 660 000 323 I 124 •050 766 013 692 986308 9 5° •006 983 000 318 I 160 •064 458 012 182 987 818 50 I •007301 000 388 I 193 •076 640 015027 984973 I 2 •007 689 000 465 ^^35 •091 667 017574 982426 2 3 •008 154 000 502 1 286 •109 24 J 017540 982 460 3 4 •008656 000577 1339 •126 781 019037 980 963 4 HM. Elementary Vulues — (( jontinucd. ) X /. L..g/, •^%'/'x A Cul..-/, X S^ '^>'^>5i3 4-822907 7-990 767 999 37' 5' J 77 093 :>S () 65 114 •813674 •990 138 999316 •186 326 6 7 63 6112 •803 812 •989 454 999272 •196 188 7 8 62 12'; •793 266 -988 726 999 '47 •206 734 8 9 60 533 •781 992 •987 873 999 043 •218008 9 6o 58 866 •769 86 i; •986 916 998 942 •230 135 60 I 57 "9 •756 78'i •985 858 998 833 -243 219 I 2 55 289 •742 639 -984 691 99S714 -257 361 2 .3 SiiU •727 330 •983 405 998 680 •272 670 3 4 51373 •710735 -982 085 998632 •289 265 4 6.S 49297 •692 820 •980717 998572 •307 180 ^^S 6 47 J5'5 '^n 537 •979 289 998 486 •326 463 6 7 44960 •652 826 ■977 775 998 467 •347 '74 7 8 42 717 •630601 •976 242 998 1 13 •369 399 8 9 40 443 •606 843 •974355 997 759 •393 '-51 9 7° 38 124 •581 198 •972 114 997 279 •418802 70 I is m '5^3 3^^ •9'59 393 996777 •446 688 I 2 :^^ 320 •522 705 •966 170 996 265 •477 295 2 3 30 823 •488 875 •9^2 435 996 036 •5" '25 3 4 28 269 •451 310 •95847^ 996 562 •548 690 4 75 25 <^'9i •409 78 1 •955 033 996 1 23 •'590 219 75 6 23 164 •364814 •951 ^^6 995 942 •635 '86 6 7 20 700 ■315970 •947 098 995 796 •684 030 7 8 J8326 •263 068 •942 894 995 095 •736932 8 9 16068 •205 962 •937989 994 154 •794038 9 8o 13 930 •H3 95i •932 143 993 '51 •856 049 80 I 119^5 •076 094 •925 294 993 076 •923 906 I 2 10 032 •001 388 •918370 992332 •998 612 2 3 8313 3 -9 '9 758 •910 702 992 998 4-080 242 3 4 6768 •830 460 •903 700 993 989 -169 540 4 8.5 5422 •734 160 •897 689 994 598 •26'5 840 85 6 4284 •63 1 849 •892 287 993 5 JO •368 151 6 7 3 343 •524 i3<5 •885 797 995417 •475 864 7 8 2570 •409 933 •881 214 991 992 •590 067 8 9 ^955 •291 147 •873 206 984457 •708 853 9 90 I 460 •164353 •857 (^(^3 979 459 •835 647 90 1 I 052 •022 016 •837 122 974913 •977 984 1 2 723 2-859 138 •812 035 954 543 3-140 862 2 3 469 •671 173 •766578 926 005 •328827 3 4 274 •437 75 i •692 583 867 279 •562 249 4 95 ^^S •130334 •559 862 704 185 -869 666 95 6 49 r69o 196 •264047 2-309 804 6 7 9 0-954 243 '•045757 7 Elementary Values — (continued.) X Colog;;^ A Cl:r Log d^. A —A X 55 0-009 ^33 000 629 I 399 3-145 818 019 129 980871 55 6 •009 862 000 684 1 462 •164947 018 892 981 108 6 7 •010 546 000 728 1527 •183839 018 104 981 896 7 8 •oil 274 000 853 1592 •201 943 019993 980 007 8 9 ■012 127 000957 1667 •221 936 020357 979 643 9 6o •013 084 001 058 I 747 •242 293 020 158 979 842 60 I •014 142 001 167 1830 •262451 019 718 980 282 1 2 •015309 001 286 I 915 •282 169 019 078 980 922 2 3 •016595 001 320 2 001 •301 247 or5 980 984 020 3 4 •017915 00 1 368 2 076 •317 227 013 390 986610 4 65 •019 283 001 428 2 141 •330617 on 015 988 985 65 6 •020 711 001 514 2 1^6 •341 632 009 197 990 803 6 7 •022 225 001 1533 2243 •350 829 005 96 1 994 039 7 8 •023 758 001 887 2274 •35'^ 790 008 511 991 489 8 9 •025 645 002 241 2319 •365301 009 6;^ 1 990 369 9 70 •027 886 002 721 2371 •374932 on 210 988 790 70 I •030 607 003 223 2433 •386 142 on 277 988 723 1 3 ■033 830 003 735 2497 •397 419 009 802 990 198 2 3 ■037.05 003 964 2554 •407 221 004 062 995 938 3 4 •041 529 003 438 2578 •411 283 991 322 008 678 4 75 •044 967 003877 2527 •402 605 989 036 010 964 75 6 •048 844 004 058 2 464 .391 641 983 840 016 160 6 7 •052 902 004 204 2374 •375481 978 243 021 757 7 8 •057 106 004 905 2258 ■353 724 976 284 023 716 8 9 •062 oil 005 846 2138 •330008 974267 025 733 9 80 •067 857 006 849 2015 •304275 970575 029425 80 I •074706 006 924 1883 •274850 960 426 039 574 I 2 •081 630 007 668 I 719 •235 276 953 652 046 348 2 3 •089 298 007 002 1545 •188928 940117 059 883 3 4 •096 300 006 on 1346 •129045 927097 072 903 4 85 •102311 005 402 I 138 •056 142 917448 082 552 85 6 •107 713 006 490 941 2-973 590 914589 085 411 6 7 •114 203 004 583 773 •888 179 900 696 099 304 7 8 •118786 008 008 615 •788875 905 730 094 270 8 9 •126794 015 543 495 •694 605 916055 083 945 9 90 •142337 020 ■'541 408 •610 660 906536 093 464 90 I ■162878 025 087 329 •517 196 887 638 112 362 1 2 •187965 045457 254 •404 834 885 201 114799 2 3 •233 422 073 995 195 •290 035 852 9S0 147 020 3 4 •307417 132 721 139 •143 015 791 483 208517 4 95 •440 138 ^95 815 86 I '934 498 667 562 332438 95 6 •735 953 40 •602 060 352 183 647817 6 7 9 0-954 243 / HM. ProhahiJities of Lhiug over, aiul of DijiiKj in, a Year, nwl the Average Duration of Life, at earli Aijc X lO ;'r °> X ;'r '.r •995 100 •004 900 50^291 55 -978967 •021 033 16-962 I •996010 •003 990 49^536 6 ■977547 •022 453 16-316 2 •996 08 1 •00.3,319 4S^733 7 -976 010 •023 990 15-679 ,3 ■997 085 •002 911; 47 ■893 8 ■974374 -025 626 15-052 4 •997 2.38 •002 762 47^032 9 -972461 •027539 14-435 i.^ •997 129 •002 871 46-161 60 -970322 -029678 13-830 6 ■99^ 1S?> •00,3 247 45-292 I -967 962 •032038 13-237 7 •996 1 18 •00,3 882 44^438 2 •965 364 -034636 12-659 8 •995 208 •004 792 43-609 3 -962 510 ■037490 12-095 9 •994 ^-5S ■005 745 42^8i7 4 65 6 -959590 -040 4 JO 11-547 20 •99.3671 •006.329 42-061 •956 569 •043 43 1 11-012 I ■993 275 •006 725 41-326 -953 43 1 -046 569 10-489 2 ■993 156 •006 844 40-603 7 ■950 1 1 1 -049 889 9-977 3 •993 2.3'5 •006 764 39"879 8 -946 766 ■053 234 9-475 4 ■993361 •006 6.39 39'H7 9 -942 660 ■057340 8-980 25 •993370 •006 6,30 38-405 70 •937808 -062 192 8-495 6 ■993315 •006 685 37-658 I -931950 -068 050 8-026 7 ■993 096 •006904 36^908 2 •925 060 -074940 7-575 8 •992 828 •007 172 36^i62 3 •917 140 -082 860 7-148 9 •992 567 •007 433 35 '4 1 9 4 •908 805 -091 195 6-749 30 •992277 •007723 34-681 75 •901 639 -098361 6-376 I •992 08.3 •007917 33-946 6 •893628 -106372 6-017 2 ■991 895 •008 105 33^213 7 -885314 -1 14 686 5^674 3 ■991 715 •008 285 32-481 8 •876787 •123 213 5^344 4 •991496 •008504 3i^748 9 •866941 •133059 5-025 35 •991 226 •008 774 31-016 80 •855 348 -144652 4-719 6 •990891 •009 109 30-286 I •841 964 •158036 4-433 7 ■990 s:/> •009464 29-';6o 2 •828648 •171352 4-171 8 ■990 220 •009 780 28-838 3 •814 147 ■'85853 3 '930 9 •989918 •010082 28^118 4 •801 123 •198877 3-713 40 •989 694 •010306 27-399 85 •790 115 •209885 3-511 I •9895^3 •010487 26-679 6 •780345 •219655 3"3io 2 •989 266 •010734 25'956 7 •768770 •231 230 3-101 3 •988873 •01 1 J27 25-233 8 ■760 700 •239300 2-884 4 •988444 -^^^S5(^ 24-511 9 •746804 -253 196 2-634 45 •987 808 •OI2 192 23-792 90 •720548 •279452 2-357 6 •987 060 •012 940 23-079 I •687263 •312737 2-077 7 •986 298 •013 702 2 2^375 2 •648 686 •351314 1-795 8 •985 560 •014440 21-679 3 •584222 •415778 1-496 9 •984 780 •015 220 20-989 4 -492 700 -507300 1-204 .50 •984 050 ■015950 20-306 95 •362 964 •637036 •930 I ■983 330 •016670 19-627 6 -183673 -816327 -684 2 •982451 •017549 18-951 7 -000 000 I -000 000 -500 3 •981 400 •018600 18-281 4 •980 266 •019734 17-618 THREE PER-CENT. 3 n:i;cKNT. Cominutatum Tahlc. W Nx S. M. lO I 2 3 4 6 7 8 9 20 I 2 3 4 6 30 I 2 3 4 35 6 7 8 40 I 2 3 4 45 6 50 I 2 3 4 74 409*4 71 888-1 ^9 5'.';'9 67 267*1 65117-5 6;i 046" 2 61 034'2 59 064-I 57 121*2 55I9I7 53 27<5-3 5^397*2 495647 477917 46083-8 44446-5 42 865*8 41339-1 39857-9 38419-5 37023*2 35667*3 34354-2 33083*3 31853*6 30662*8 29 ^08-6 28388*1 27300-5 26246*1 25224-7 24237*6 23 284*9 22 364-0 21 471-1 20 604-8 19 760*8 18936*9 18133-5 17351-1 16589*3 15 849-2 15 131-1 14432*6 i^ 7-;i-6 1 796 867 1 724979 3 655 463 I 588 196 1523079 I 460 03 2 1 398 998 I 339 934 I 282 813 I 227 621 I 174345 I 122947 I 073 383 I 025 591 979505-3 935 058-8 892 193-0 850 854-0 810996*1 772576*6 135 553-4 699886*1 632448-6 600595-0 569932*2 540423*6 484735-0 458 489*0 433 264*2 409026-6 385741-7 363377-6 341 906-6 321 301-8 301 541-0 282 604-1 264470*7 247 1 19*6 230 530*3 2 14681*0 199549-9 185 J 17-3 171 365-8 36413 646 34616779 32 891 800 31 236337 29648 141 28 125062 26 665 030 25 266 032 23 926098 22 643 285 21 415 664 20 241 319 19 1 18 37 1 18044988 17019397 16 039 892 15 104833 1 4 2 1 2 640 13361786 12550790 II 778 214 1 1 042 660 10342774 9677242 9 044 794 8444199 7 874266 7 333 843 6821807 6337072 5878583 5445319 5 036292 4650551 4287 173 3 945 267 3 623 965 3322424 3 039820 2775349 2 528 229 2 297 699 2 083 018 I 883 468 T 698 ^'^ I 9906*2 9552-2 9273-8 9049-7 8 859-3 8684*7 8 ';o9.o 8316*6 8 094-0 7828-2 7 520*4 7 193-0 6857-4 6528-1 6 214-2 5917-2 5631-1 5 352-9 5 075-8 4798*2 4521-0 4243-4 3 969-2 3 698-9 3432-8 3 169-8 2 908-6 2 647-6 2386-8 2 127*6 1870*7 1 6i8*3 I 371-5 1 128*8 0887*3 o 646*3 0402*4 o 154*2 9 902*27 9 648-05 9391-66 9 134-76 8878-25 8 620-44 8359-81 756181' 736275' 716723' 697 449' 678399' 659 540' 640855' 622346' 604 030' 585936' 568 107' 550587' 533 394' 5^^K^3y qooooS 483 794- 467877- 452 246* 436893* / 5 5 6 421 817*9 407 019' 392 498' 378255' 364.286' 350587' 337 154' i>23 984' 31 1 076' 298428' 286041 273914*1 262 043*4 250425*1 239053-6 227924-7 217037-5 206391*1 195988-7 185834-5 175932-2 166284*2 156892*5 147 757-8 138879-5 130 259- 1 I 2 3 4 15 6 20 1 2 3 4 25 6 7 8 30 I 2 3 4 35 6 40 I 2 3 4 45 6 7 8 50 1 2 3 4 3 PER-CEKT. Commutation Table — (continued). X D. N. s. M, K X 5S 13 087-6 158278-3 1526985 8 096-34 121 899-2 55 6 124,59-1 145 839* J 1 368 707 7 829-08 113 802-9 6 7 1 1 80^-7 ^34033-4 1 222868 7 557-92 105973-8 7 8 II i86-8 122846-6 I088S34 7 282-95 98415-91 8 9 10582-7 112 263-9 965987-7 7 004-63 91 132-95 9 6o 9991-51 102 272-4 853 723-8 6721-69 84 128-32 60 I 9412-61 92859-77 751451-4 6433-80 77406-64 I 2 8845-67 84014-10 65859 '-6 6 141-02 70972-84 2 3 8290-57 75 723-52 574577-5 5 843-56 64831-82 3 4 7 747-34 67976-19 498 854-0 5541-80 58988-26 4 65 7217-74 60 758-45 430877-8 5237-85 53446-26 65 6 6703-17 54o55'28 370119-4 4933-50 48208-62 6 7 6 204-86 47 850-42 3 16 064- 1 4630-44 43 275'ii 7 8 5723-60 42 126-82 268 213-7 4329-90 38644-67 8 9 5 261-08 36865-74 226086-9 4034-08 34314-77 9 70 4814-96 32050-78 189 22I-I 3741-20 30280-69 70 I 4383-99 27 666-79 157 170-3 3 450-47 26539-49 I 2 3 966-66 23 7oo'i3 129503-5 3 160-83 23 089-01 2 3 3 562-52 20 137-61 105 803-4 2872-23 19928-18 3 4 3 172-17 16965-44 85 665-80 2 5^5-63 17 o55'96 4 75 2798-91 14 i66-s3 68 700-36 2304-77 14470-32 75 6 2 450-10 II 716-43 5-^ 533-83 2037-49 12165-55 6 /' 2 125-71 9590717 42 817-40 I 784-46 10 128-06 7 8 I 827-11 7 763-609 33 226-68 1547-77 8343-609 8 9 1555-32 6208-285 25463-07 I 329-20 6 795-842 9 80 1309-10 4899-184 19254-79 I 128-28 5 466-642 80 I I 087-12 3 812-061 14355-61 944-429 '4 338- 365 I 2 888-659 2 923-402 10543-54 777-628 3393-937 2 3 714-938 2 208-464 7 620-14 629-790 2 616-309 3 4 5^5'^^^ I 643-354 5411-68 500-787 19S6-518 4 85 439'537 I 203-817 3768-32 391-672 1485-732 85 6 337-I70 866-647 2 564-5 i 302-107 I 094-060 6 7 255*445 61 I-202 I 697-86 230-203 791-953 7 8 190-659 420-543 I 086-66 172-857 561-750 8 9 140-810 279733 666-11 128-561 388-893 9 90 102-095 177-638 386-38 93-9470 260-33 1 90 I 71-4214 106-217 208-74 66-247 5 166-384 I 2 47 '655 6 58-561 102-53 44-561 9 100-137 2 3 30-013 I 28-548 43-97 28-3074 55-575 3 4 17-0236 1^-525 i5'42 16-192 I 27-268 4 95 8-1432 3-381 3-89 7-8076 11-076 95 6 2-8696 •512 •51 2.771 1 3-268 6 7 •511 7 *ooo •00 -4968 ■497 7 10 HM. 3 i'i:i; (HXT. Logdrithms nml C<>-/"i/(ir/t/tni.'< '_^l)j., Nj., am/ M,.; wi't/i tin ir Diffi rcncfti. X Log D^ (Logr;;^) " ' A (Cologi-/jj) LogN, A X lO 4-871628 1^985 0295* 1 28 372 0014971 6-254516 982 268 10 I •856657 •985427 -143 343 •014573 -236784 982 136 I 2 •842 084 •985719 •157916 •014 281 -218920 981984 2 3 •827803 •985 895 -172 197 •014 105 ■200904 981818 3 4 •813698 •985961 •186302 •014039 •182 722 981 641 4 15 •799659 •985914 ^200 34 1 •014086 •164363 981454 15 6 •785573 •985751 ^2 14 427 •014 249 •145817 98 I 266 6 7 •771324 •985 473 •228676 •014527 •127083 981 080 7 8 •756797 •985077 •243 203 •014923 -108 163 980901 8 9 •741874 •984 660 •258 126 •015340 -089 064 980732 9 20 •726534 •9S4406 •273 466 •015594 -069 796 980 564 20 I •710940 •984 232 •289060 •015 768 -050 360 980 395 1 2 •695 172 •984 180 •304828 •01 15 820 ■030755 980 219 2 3 •679352 •984215 •320 648 ■015 785 •010 974 980033 3 4 ■663 567 •984270 'li^Ali •015730 5-991007 979832 4 25 •647837 •984 274 •352 163 •015 726 •970839 979620 25 6 •632 III •984 250 •367 889 •015 750 •950 459 979396 6 7 •616361 •984 153 •383 639 •015847 ■929 855 979 164 7 8 •600 ^14 •984037 •399 486 •015963 -909019 978923 8 9 •584 55 J •983 923 •415449 •016077 •887942 978672 9 30 ■568474 •983 796 •431526 •016 204 •866614 978413 30 I ■552 270 •983710 •447 730 •016 290 •845027 978 142 1 2 •535980 •983 6291 ^464 020 •016371 •823 169 977 856 2 3 •519609 •983550 ^480 391 -016450 •801 02 <; 977 557 3 4 ■'5°3 159 •983 453 •496841 •016547 •778582 977241 4 35 •486612 •983 ^?,(> •513388 •016664 •755 823 976911 2,^ 6 •469 948 •983 189 •530052 •016 811 •732 734 976566 6 7 •453 137 •983 033 •546863 -016967 •709300 976 204 7 8 •436170 •982 894 •S^^-i?, 830 •017 106 •685 504 975 825 8 9 •419064 •982 762 •580936 •017238 •661329975424 9 40 •401 826 •982 664 •598174 -017336 •636753974999 40 1 •384490 •982585 •615 510 •017415 -611 752974545 I 2 •367 075 •982 475 •632925 •017525 -586297974061 2 3 •349 550 •982 304 •6';o45o •017 696 -560358973549 3 4 •331854 •982 114 •668 146 -017886 •533907973006 4 45 •313968 •981836 •686032 -018 164 •506913972433 45 6 •295 804 •981 506 •704 196 -018494 •479346971833 6 7 •277310 •98ri7i •722 690 •018829 •451 179971 199 7 8 •258481 •980 846 •741 519 -019154 •422378I970529 8 9 •239327 •980502 •760673 019498 -392 907 969 821 9 50 •219 829 •980 179 •780 171 •019821 -362 728 969 066 50 I •200008 •979862 •799992 •020 138 •331 794968258 I 2 •179870 •979474 •820 130 •020 526 •300052967395 2 3 •159344 •979 009 •840 6^6 -020991 •267447966477 3 4 •^^.^^.-i^ -978 506! -861 647 -021494 •233924965497 4 11 3 PERCENT. Logarithms and Co-logarithms ofDj^., N^., and M^.; with their Differences. X Colog N^ A LogM^ A Colog M^ A X lO 7745484 017732 4^298988 992 208 y7oioi2 007 792 10 I •763216 017 864 •291 196 993771 •708 804 006229 1 2 •781080 018 016 •284967 994922 ■715033 005 078 2 3 •799096 018 182 •279 889 995^^1 •720 111 004363 •3 4 •817278 018359 •275526 995961 •724474 004 039 4 15 •835637 018546 •271487 995 896 •728513 004 104 ^5 6 •854 ^83 018734 •267383 995 462 •732617 004538 6 7 •872917 018920 •262 845 994689 •737 ^S5 005 3 1 1 7 8 •891837 019 099 •257534 993 574 •742 466 006426 8 9 •910936 019268 •251 108 992436 •748 892 007 564 9 20 •930 204 019436 •243 544 991 808 •756456 008 192 20 I •949 640 019 605 •235 352 991440 •764648 008 560 1 2 •969 245 019 781 226 792 991431 •773 208 008 569 2 3 •989 026 019967 •218 223 991 674 •781777 008 326 3 4 6-008993 020 168 •209 897 991 969 •790 103 008 031 4 ^5 •029 161 020 380 •201 866 992 123 •798134 007 877 25 6 •049541 020 604 •193989 992 200 •806011 007 800 6 7 •070 145 020 836 •186189 992 090 •8138x1 007 910 7 8 •090981 021 077 •178279 991931 •821 721 008 069 8 9 •112 058 021 328 •170 210 99^ 785 •829 790 008 215 9 30 •133386 021587 •161995 991 618 •83800^ 008382 30 I •154973 021858 •153613 991 559 •846387 008441 1 2 •176831 022 144 •H5 ^72 991 513 •8^828 008487 2 3 •198975 022 443 ■136 6Ss 991 480 •863315 008 520 3 4 •221 418 022 759 •128 165 991413 •87^835 008587 4 iS •244177 023 089 •119578 991301 •880422 008 699 35 6 •267 266 023 434 •110 879 991 130 ■889121 008 870 6 7 •290 700 023 796 •102 009 990 950 •897991 009 050 7 8 •314496 024175 •092 959 990 815 •907 041 009 185 8 9 •338671 024576 •083 774 990 702 •916226 009 298 9 40 •3^i^ 247 025 001 •074476 990 666 •925524 009 334 40 I •388 248 025 455 •065 142 990 676 •934858 009324 I 2 •413 703 025 939 •055818 990632 •944182 009368 2 3 •439 642 026451 •046 450 990 468 •9S3 550 009532 3 4 •466 093 026 994 •036918 990 283 •963 082 009717 4 45 •493 087 027567 •027 201 989935 •972 799 010 065 45 6 •520654 028 167 •017 136 989509 •982 864 010491 6 7 •548821 028801 •006 645 989 090 ■993 3^S 010910 7 8 •577622 029471 3^995 735 988 705 4*004265 01 1 295 8 9 •607 093 030179 •984 440 988 302 •015560 oil 698 9 50 •637272 030 934 •972 742 987 955 •027 258 1 2 045 50 1 •668 206 03 1 742 •960 697 987 630 •039 303 012370 I 2 •699 948 032605 •948327 987 203 •051673 012797 2 3 •n'^SS3 033 523 •935530 986 666 •064470 013334 3 4 •766076 034 503 •922 196 986 093 •077 804 013907 4 \2 3 i'i;i:cKNT. Ijogarithms ami Co-loynrlthms of l)j^y Nj., aiiJ M^.; with (heir Differences. X Log IV A (Logr/jj.) Colog Dj. A (Colog tpj.) Log N, A X 5a 4*1 16 859 '•977931 5-883 HI 0^022 069 5^199 42 1 964453 55 6 •094 790 ■977 300 •905 2 1 •022 700 •163874963339 6 7 •072 090 •976617 •927910 -023 383 •127 213962 150 7 8 •048 707 •975 889 •951 293 •024 1 1 1 •089363960 877 8 9 •024596 •975 °35 •975404 •024 965 •050 240 959518 9 6o 3-999 <53 1 •974079 4^ooo 369 •025921 •009 758 958 070 60 I •973 7'o •973021 •026 290 •026979 4^967 828 956524 ] 2 •94'5 73i •97 J 854 •053 269 •028 146 •924352 954879 2 3 •918585 •970568 •081 415 •029432 •879231 953 126 3 4 •889153 •969 248 •no 847 •030752 •832357 95x250 4 ^3 •8S8401 •967 879 •141599 •032 121 •783 607 949231 65 6 •826 280 •966452 •173 720 •033 548 •732838 947 048 6 7 •792 7.32 •964937 •207 268 -035 063 •679886 944 673 7 8 •757669 •963 406 •242331 •036 594 •624 559 942 064 8 9 •721075 •961518 •278925 •038482 •566623 939216 9 7° •682 593 ■959277 -317407 •040 723 •505 839 936 120 70 I •641 870 •956 555 ■358 130 •043 445 •441959 932792 I 2 •598 425 •953 33.^ -40 X 575 •046 667 •374751 929257 2 .3 ■55^ 758 •949 598 •448 242 •050402 •304008 925 557 3 4 •5^^35(> •945 633 -498 644 -054367 •229565 921 699 4 75 •446989 •942 196 -553 01 1 -057 804 •151 264 917 531 75 6 ■389185 •938319 •610815 •061 681 •068 795 913056 6 7 •327 504 •934 260 •672496 •065 740 3 •981 851 90S 213 7 8 •261 764 •930057 •738236 -069 943 •890 064 902 908 8 9 •191 821 •925 152 •808179 •074 848 •792972 897 152 9 So •J1697.5 •919306 •883 027 •080 694 •690 124 891 036 80 I •036279 •912456 •963721 •087 544 •^81 160 884729 I 2 -■948 735 ■905 53 ry 5-051 265 •094467 •465 889 878201 2 3 •854 268 •897 866 •H5 732 •102 134 •344 090 871 641 3 4 •752 '34 •890861 •247 866 •109 139 •215 73 X 864 829 4 8.5 •642 995 •884853 •357 005 -115 147 •080 560 857282 85 6 •527848 •879 450 •472 152 •120550 2-937 842 848343 6 7 •407 298 •872959 •592 702 • r 2 7 04 1 •786 185 837635 7 8 •280 257 •868377 •719743 •J31 623 •623 810 822 933 8 9 •148634 •860369 •851366 •139631 ■446 743 802 793 9 9° •009003 •844 82 r, •990997 •^55^75 •249536 776657 90 I 1-853828 •824 286 2^146 172 •175 7 '4 •026 193 741 417 I 2 •678 114 •799 ^97 •321886 •200803 i'767 610 687967 2 3 •477 31 ' 153 741 •^22 689 •246 259 '455 577 606 045 3 4 •231052 •679 746 •■768 948 •320254 •061 622 467 46*2 4 95 0*910 798 •547 025 7'o89 202 ■452 975 0^529 084 179948 95 6 •457 823 •251 209 •542 177 •748791 1^709032 6 7 I • 709 032 0*290968 7 13 3 PER-CENT. Logarithms and Co-hgarithms of Dj., N_j., ancl'M^; with their Differences. X CologN^ A Log M,. A CologM^ A X 55 6-800579 °35 547 3-908 289 985422 4^091 711 014578 55 6 ■836126 036661 •893 711 984691 -106 289 015309 6 7 •872 787 037850 -878402 983 906 •121598 016 094 7 8 •910 637 1039123 -862308 983077 •137692 016923 8 9 •949 760 040 482 •845385 982 093 •154615 017907 9 6o •990 242 041 930 •827478 980990 -172522 019 010 60 I 5-032 172 043 476 •808 46S 979772 -191 532 020 228 I 2 •075648 045 121 •788 240 978438 -2JI 760 021 562 2 3 -120 769 046 874 •766678 976973 •233322 023 027 3 4 -167643 048750 •743651 975502 •256349 024 498 4 65 •216393 1050769 •719^53 974003 -280847 025 997 65 6 -267 162 052952 •693 156 972 466 -306844 027534 6 7 •320 114 1055327 •665 622 970 856 •334378 029 144 7 8 ■375441 057 936 -636478 969 267 •363522 030733 8 9 ■433 377 060 784 ■605 745 967 266 •394255 032 734 9 70 •494 161 063 880 •573 on 964 868 -426989 035 132 70 I •558 041 067 208 •537879 961 922 -462 121 038078 1 2 -625 249 070 743 -499 801 958418 -500 199 041582 2 3 •695 992 074443 •458219 954 348 •541 781 045 652 3 4 ■770435 078301 •412567 950 061 •587 433 049 939 4 75 -848 736 082469 -362 628 946 467 •637372 053 533 75 6 -93 T 205 086944 -309095 942411 -690905 057589 6 7 4-018 149 091 787 •251 506 938 199 •748494 061 801 7 8 -109936 097 092 -189705 933 885 -810295 066 115 8 9 -207 028 102848 ■123590 928 826 •876410 071 174 9 80 -309876 108964 -052 416 922753 •947584 077247 80 I -418840 115271 2-975 169 915603 3-024831 084397 1 2 •534 1 JI 121 799 -890772 908 424 -109 228 091576 2 3 '^55 9^° 128359 •799 196 900457 •200 804 099 543 3 4 •784269 ^35 171 -699 6S3 893 270 •300347 106730 4 85 •919440 142 718 •592 923 887237 •407 077 112 763 85 6 3-062 158 ^51657 •480 r6o 8819^-1 •519840 118 049 6 7 -213 815 162375 •362 III 875 576 •637889 124^24 7 8 -376190 177067 •237687 871423 •762313 128577 8 9 •553 257 197207 • 1 09 1 1 863773 •890 890 136227 9 90 -750464 223 343 1-972883 848 286 2-027 1 17 151 7H 90 I -973 807 258583 •821 169 827 794 -178831 172 206 1 • 3 2-232390 312033 -648 963 802938 ■35^^37 197 062 2 3 ■544423 393 955 ■451 901 757403 •548 099 242 597 3 4 •938 378; 532 538 ■209304 683213 -790 696 316787 4 95 f-470916 820052 0-892517 550 13S 1-107483 449 862 95 6 0-290968 _^442 6^5 253 540 •557345 746 460 6 7 1-696 195 0-303 805 7 14 3 PKKCKNT. J'dlw's of Annuities^ and Sinfjle and Annual Premiums for Assurance of a Unit. X lO "x A. ■07^ T Ox A. ur^r 24-1484 -267 523 •010638 sa 12-093 8 -618628 -047 246 I ^3-99.5 3 •271 981 -010881 6 11-7242 -629392 -049 464 2 23-814 2 -277257 •Oil 173 7 1 1-353 3 -640 195 -051 824 .3 2.3-610 3 •283 19.) -01 1 507 8 10-981 3 -6';i 029 •054337 4 23-3897 -289620 -oil 875 9 10-6083 -661 895 •057019 15 23-1581 -296366 -012268 60 10-2359 -672 740 •059 874 6 22-921 ^ •303 256 •012 677 I 9-865 5 -683 530 -062 908 7 22-6861 -310 114 -013 093 2 9-497 8 -694 240 -066 1 3 2 8 22-457 7 •3i<5 7^^5 •013504 3 9"i33 7 •704844 •069 555 9 22-242 9 •323024 -013 898 4 8-7741 -715316 •073 ^85 20 22-042 ^ -328859 -014 272 65 8-4179 -725691 •077054 I 21-8484 ■33'\S^3 -014641 6 8-064 I •735 996 -081 199 2 21-656 2 -340 1 10 -015 012 7 7-7118 •746259 -085661 .3 21-4596 •345836 -015398 8 7-3602 •756499 -090488 4 21-2539 •351827 -015810 9 7^0073 •766779 •095 761 25 21-0379 -358 120 -016 250 70 6-656 5 •776995 -101 482 6 20-813 6 -364651 -016 717 1 6-3109 -787 062 -107 656 7 20-582 3 •371389 -017 208 2 5-9748 -796849 -114 246 8 20-347 2 •378237 -017 718 3 5-6526 •806 234 -121 190 9 20-109 ° ■385175 -018 247 4 5-348 2 -815 100 -128398 30 19-8674 -392 212 -018795 75 5-061 4 •823 453 -^35851 I 19-6227 •399340 •019364 6 4-782 -831592 -143824 2 19-3726 -406622 -019959 7 4-5118 •839 463 -152304 .3 19-1 169 -414072 -020 ^'6^ 8 4-249 I •847 113 -161382 4 18-8549 ■42 1 703 -021 239 9 3-991 6 -854613 -171 209 35 18-587 1 •429 503 -021928 80 3-7424 -861872 •181 737 6 18-314 1 •437452 -022 649 I 3-'5o66 -868 741 -192773 7 18-037 •445525 -023 403 2 3-289 7 -875 058 -203 992 8 17-755'^ •453721 -02419: 3 3-0890 -880902 -215431 9 17-4689 -462 072 •025019 4 2-908 ■886174 -226758 40 17-1762 •470597 -02 !; 891 85 2-7388 -891 102 •238337 I 16-8757 •479 349 -026816 6 2-5704 -896 009 •250957 2 i&^66 2 -488 364 -027 801 7 2-392 7 -901 184 •265 625 3 16-2483 -497622 -028851 8 2-205 7 ■906629 -282815 4 15-924 1 -507 066 -029961 9 1-9866 -913 012 •305 703 4.5 I5-593 5 ■516693 -031 138 90 1-7399 •920 196 •335 846 6 15-2596 -526420 •032376 I 1-4872 -927 558 •372935 7 14-923 4 .536211 •033 674 2 1-2288 -935 082 •419537 8 H'584 7 •54'5o77 •035 039 3 -9512 -943 169 •483 383 9 14-2423 ■ss^' 049 •036481 4 -6770 -951 157 •567 187 50 13-8963 ■(,66 127 •038 005 95 -415 2 -958 780 •677477 I ^3-54.'; 2 •576354 •039625 6 •1783 -965 679 -819543 2 13-188 1 •586755 •041356 7 -000 •970874 •970874 3 12-8264 -597291 •043 199 4 12-461 5 -607 916 -045 160 THREE AND A HALF PER-CENT. ir. 3^. I'KRCENT. Cvmmutation Table, S, ^^x H. lO I 2 15 6 7 8 20 I 2 3 4 6 7 30 I 2 3 4 35 6 7 40 I 2 3 4 45 6 50 I 2 3 4 70 891*9 68 i;39"o ^'i.";9'-3 6.3 162-9 60 849'o 58 629-0 564S57 54 396-5 52352-9 "JO 340-2 48 358-4 46427-4 44555-7 42 754-4 41 029-1 37794-6 36272-4 34803-8 33385-7 32 017-0 30695-4 29422-6 28 197-2 27017-9 25 882-3 24 787-7 23 731-3 22 711-8 21 729-1 20 782-7 19872-9 18999-6 18 160-0 17350-7 16 ^70*2 15814-7 15082-1 14372-4 13 685-9 13 021-8 12380-8 11762-7 10587-3 I 488 195 I 422 604 I 359 44 1 I 298592 1239 963 I 183479 1 1 29 083 1076730 I 026389 978031-1 931 603-7 887048-0 844293-7 803 264-5 763 886-0 726091-5 689819-1 655oi5"2 621 629-5 589612-6 558917-2 529494-6 501297-5 474279-5 448397-2 423 609-6 399878-3 377 ^^^'5 334654-7 314 781-7 295 782-2 277 622-2 260 271-5 243 701-3 227886-7 2 1 2 804-5 198 432-1 184 7462 171 724-4 159343-5 147 580-8 136415-3 125 828-1 29 720 786 28 1 64 43 2 26676 237 25 253 633 23 894 192 22 595 600 21355638 20 172 159 19 043 076 1 7 966 346 16939957 15 961 926 15 030 322 14 143 274 13 298981 12495716 II 731 830 ^ I 005 739 10 3 15 919 9 660 904 9039275 8 449 662 7 890 745 7361 250 6 859 9S3 ^3^5^n 5937276 5 5^3^^^! 5 113 788 4736622 4381 185 4046530 3731748 3'\35 9^>^ 3 158344 2 898072 2654371 2 426484 2 213 680 2015248 I 830501 1658777 I 499 434 ^35^^53 I 215437 <; 864-2 5528-6 5 265-9 5 055*5 4877-6 4 7i5"3 4552-6 4 375*4 4171-4 3 929-0 3 649-6 3353-9 3 052-2 2757-6 2478-1 2 214-9 I 962-7 I 718-6 I 476-6 I 235-4 0995*7 o 756-8 0522-0 o 291-6 0065-9 9843-87 9 624-46 9406-32 9189-33 8974-71 8 763-05 8556-11 8354-76 8157-71 7 962-48 7 768-74 T 513-55 7375-82 7 176-X5 6975*64 6774-38 6573-70 6374-29 6174-85 5974*19 567 167-5 55^ 30 y 3 535 774-6 520508-7 505 453-2 490575*6 475 860-3 461 307-7 446932-3 432 760-9 4 1 8 83 1 -9 405 182-3 391 828-5 378776-3 366018-7 353 540-5 34^ 325-^ 329362-9 3 1 7 644-3 306167-7 294932-3 283 936-6 273 179-8 262 657-8 252366-3 242 300-4 232456-6 222 832-1 213425-8 204 236-4 191; 261-7 186498-7 177942-6 T69 587-8 i6i 430-1 153467-6 145 698 9 138 125-3 130 749*5 123573-3 T16597-7 109823-3 103 249-6 96875-33 90 700*49 10 I 2 3 4 15 6 7 8 20 1 2 3 4 25 6 30 1 2 3 4 35 6 7 8 40 1 2 3 4 45 6 7 8 50 1 2 3 4 17 HM. 3 1 PERCENT. Commutation Table (continued) . X D. N. s. M, R. X 55 10027-4 115 800-7 I 089 609 5772-32 84 726-30 55 6 9 484'5 1 106316-2 973808-7 5 568-54 78953-98 6 7 8958-02 97358-14 867 492-6 5 362-79 73 385-4-1 7 8 8447-46 88910-68 770134-4 5 ^55-'^ 68022-65 8 9 795^-65 80958-04 681 223.7 4 946-00 62 867-49 9 6o 7472-11 73 485-93 600 265-7 4 734-40 57921-49 60 I 7005.18 66480-75 526779-8 4520-15 53 187-08 I 2 655^-45 59 929-30 460 299-0 4303-30 48 666-94 2 3 6 110-66 53818-65 400369-7 4 084-06 44363-63 3 4 5682-67 48 135-97 34655^'^ 3862-72 40279-57 4 65 5 268-63 42 867-34 298415-1 3 640-84 36416-85 65 6 4869-38 37997-96 255 547-8 3419-76 32776-01 6 7 4485-63 33 5^^-34- 217549-8 3 200-67 29356-25 7 8 4117-72 29394-61 184037-5 2 984-45 26 155-58 8 9 3 766-69 25 627-93 154642-9 2 772-66 23 171-13 9 70 3 430'63 22 197-30 129 014-9 2 563-99 20 398-46 70 I 3 108-48 19088-82 106 817-6 2357-84 17 834-48 I 2 2 798-98 16 289-84 87 728-81 2 153-46 15476-63 2 3 2501-67 13 788-18 7 J 438-97 I 950-80 13323-17 3 4 2 216-79 ii 571-39 57 650-80 I 750-52 11372-37 4 75 I 946-50 9624-883 46079-41 I 555-20 9621-842 75 6 I 695-69 7929-191 36 454' 53 1370-21 8 066*643 6 7 I 464-08 6465-116 28525-34 I 195-94 6 696-430 7 8 I 252-34 5 212-781 22 060-22 1033-71 5500-491 8 9 I 060-90 4151-881 16847-44 884-622 4466-784 9 80 888-634 3 263-247 12695-56 748-233 3 582-162 80 I 734-388 2 528-859 9432-314 624-037 2 833-929 I 2 597-419 1931-440 6903-455 511-902 2 209-892 2 3 478-309 J 453-131 4972-015 412-995 I 697-990 3 4 376-245 I 076-886 3518-884 327-106 I 284-995 4 85 291-226 785-660 2441-998 254-809 957-8895 85 6 222-321 5^3-3395 1 656-338 '^95-75^ 703-080 2 6 7 167-620 395-7^94 I 092-998 148-570 507-3279 7 8 124-504 271-2157 697-279 III-I22 358-758 8 9 91-5073 179-7084 426-063 82-335 7 247-636 I 9 90 66*027 113-681 4 ^46-355 59-949 9 165*3003 90 I 45-966 8 67-7146 132-673 42-1225 i 05-350 5 I 2 30-5229 37-1917 64-959 28-233 I 63-228 00 2 3 19-130 2 i8'o6i 5 27-767 17-8725 34-99495 3 4 10-7984 7-263 I 9-706 10-187 6 17-12241 4 95 5'i40 5 2-122 6 2-443 4-8948 6-93483 95 6 1-802 7 •3199 -320 1-7309 2-040 00 6 7 -3199 -0000 •000 -309 i -309 09 7 18 3i I'KR-CENT. Logaritlnns and Co-!o(/^ 974 230 8 9 •337 042 •980 659 •662 958 •019341 •550 763|973 834 9 40 •317701 ■980561 •682 299 •019439 •524 597 973 413 40 I •298262 ■980481 •701 738 •019519 -498 010 972 962 I 2 ■278743 •980373 •721257 -019627 -470972 972 482 2 3 •259 116 •980 200 •740884 -019 800 •443 454 971973 3 4 •239316 ■980012 •760684 -019988 •415427 971431 4 43 •219328 •979732 •780672 -020 268 •386858970861 45 6 •199 060 ■979403 ■800940 -020597 •357719970262 6 7 •178463 •979 06b •821537 -020932 •327981969631 7 8 •^57531 •978 742 •842 469 -021 258 •297 612968964 8 9 •136273 ■978399 •863 727 -021 601 •266576968256 9 .';o ■114672 ■978077 •885328 -021 923 •234832 967 502 50 I •092 749 ■977 759 •907 251 -022 241 •202334 966 696 I 2 •070 ^oS ■97737c •929492 -022 630 •169 03c 965 833 2 3 ■04787^ •976906 ■952 122 •023 094 ■134863 964915 3 4 •024784 ■976403 ■975 2 iC •023 597 -099778^963933 4 19 3i PER-CENT. Logarithms and Co-logarifhins of Dg, N^., and M^; with their Diferenccs. X Colog N^ A LogM, A Colog M^ A X 10 lO 7-807 892 019448 4-200420 990713 r799 580 009 287 I •827340 019576 •191 133 992590 -808867 007 410 I 2 •846916 019724 •183723 993 973 -816277 006027 rt 3 •866640 019887 •177696 994837 •822304 005 163 3 4 •886527 020 064 •172533 995 235 •827467 004 765 4 ^S •906591 020 248 •167768 995 ^74 •832232 004826 15 6 •926839 020435 •162942 994 679 •837 058 005321 6 7 •947 274 020 619 •157 621 993 792 •842379 006 208 7 8 •967 893 020 795 •151413 992 507 •848587 007 493 8 9 •988688 020 959 •143920 991 199 •856080 008 801 9 20 6^oo9 647 021 122 •135 119 990 488 •864881 009 512 20 I •030 769 021 284 •125607 990077 •874393 009923 1 2 •052053 021453 •115 684 990 084 -884316 009916 2 3 •073 506 021635 •105768 990382 •894232 009618 3 4 •095 141 021 830 •096 150 990 742 •903 850 009 258 4 25 •116971 022 038 •086892 990937 •913 108 009 063 25 6 •139 009 022 256 •077829 991 046 •922 171 008 954 6 7 •161 265 022484 •068875 990 939 •931 125 009 061 7 8 •183749 022 719 •059814 990777 •940 186 009223 8 9 •206468 022965 •05059^ 990 6t, I •949 409 009369 9 30 ■229433 023 219 •041 222 990 460 •958778 009 540 30 I •252652 023 486 •031 682 990415 •968318 009585 1 2 •276138 023 766 •022 097 990385 ■977903 009 61^ 2 3 •299904 024 062 •012 482 990369 -987518 009 6^ I 3 4 •323966 024371 •002 851 990315 -997 149 009685 4 35 •348337 024697 3^993 166 990 2 1 1 4-006 834 009 789 3S 6 •373034 025 038 •983377 990 043 -016623 009 957 6 7 •398072 025 395 •973420 989 864 •026580 010 136 7 8 •423 467 025770 •963 284 989 736 •036 716 010264 8 9 •449237 026 166 •953 020 989 6^6 •046980 010 364 9 40 •475 403 026587 •942 6^6 989621 •057344 010379 40 I •501990 027038 •932277 989657 •067 723 010343 I 2 •529028 027 518 •921 934 989 634 •078066 010366 2 3 •556546 028 027 •91 1 568 989480 -088432 010520 3 4 •584573 028 569 •901 048 989 303 •098952 010 697 4 45 •613 142 029 139 •890351 988948 •109649 on o';2 45 6 •642 281 029738 •879299 988 '512 •120 701 011488 6 7 •672 019 030369 •867811 988081 •132 189 on 919 7 8 •702388 031 036 •855 892 987 692 •144 108 012308 8 9 •733424 03 1 744 •843 584 987 286 •156416 012 714 9 50 •765 168 032498 •830 870 986 940 •169 130 013 060 50 I •797 666 033 304 •817810 986622 •182 190 013378 I 2 •830970 034167 •804432 986 194 •195568 013 806 2 3 •865137 035 085 •790 626 985 (^:^?> •209374 014347 3 4 •900 222 036 067 -776279 985 072 •223721 014928 4 '20 3! PKR-CENT. Li^garlthvifi ami (^o-U'fjnrltlnns of 1)^., Nj., and Mj.; icith tluir Diffcrtncc's. X LogD^ A (Logf/j^) Colog D^ 1 A (Colog ryv) LogN, A X ss 4*ooi 187 T^975 828 ^■998 813 0^024 172 5-063 7 1 1 962 888 55 6 r977oi.'; •975 197 4-022 985 -024 803 ■026599961 773 6 7 •952 212 ■974 5 '4 •047 788 •025 486 4^988 372J960 582 7 8 •926 726 ■973 786 •073 274 •026 214 ■948954959306 8 9 •900 5 I 2 •972932 •099488 -027 068 •908 260 957 944 9 6o •873 444 •971975 •126556 •028025- •866204 956492 60 I •845 4 '9 •970918 ■154 581 •029082 •822696954943 I 2 •816.337 •969751 •183663 •030 249 ■777639953294 2 3 •786088 •968 465 •213 912 •031535 •730933951537 3 4 •754553 •967 H5 ■245 447 •032855 -682470I949657 4 65 •721 698 •965 776 •278302 •034224 •632 1 27J947 633 65 6 •687474 •9^4 349 •312 526 ■035 65 1 •579 7601945 445 6 7 •651823 •962 834 ■348177 •037 166 •525 205 943 063 7 8 •614657 •961 302 •385343 •038698 •468 268:940 446 8 9 •575959 ■959415 •424041 •040 585 •408 714 937586 9 70 •535374 •957 ^74 •464626 •042 826 •346300934479 70 1 •492 548 •954452 •507452 •045 548 •280779931 138 1 2 •447 000 •951 229 'SS'i 000 •048771 •211 917927 590 2 3 •398 229 •947 496 •601 771 •052504 •139507923879 3 4 •345 725 ■943 530 •654275 •056470 •063 386920009 4 75 •289 255 •940 092 ■710745 •059908 3^983 395915 834 75 6 •229347 •936 216 •770653 •063 784 -899 229^91 1 347 6 7 •^65563 ■932 157 •834437 •067 843 •810576I906493 7 8 •097 720 ■927954 •902 280 •072 046 •717 o69'9oi 176 8 9 •025 674 ■923 049 •974326 •076951 •618245 895 405 9 80 2 •948 723 •917203 ro5i277 •082 797 •5^3 ^5° 889275 80 I •865 926 •910353 ■J 34 074 •089647 •402 925 882 956 T 2 •776279 ■903 430 •223 721 •096570 •285881876424 2 3 •679709 •895 762 •320 291 •104238 •162 305869 865 3 4 •575 47 i •888 759 •424529 •III 241 •032 170 863 065 4 85 •464 230 •882 749 •535770 •117251 2-895 235 855 535 85 6 •346979 ■877 347 ■6'i3 02 1 •122653 •750770 846617 6 7 •224326 •870856 ■775 674 •129 144 ■597 387 835 928 7 8 •095 182 •866 274 •904818 •133726 ■433 3^5 821 253 8 9 r96i 456 •858 265 2-038544 ■HI 735 •254568801 121 9 90 •819 721 •842 723 •180 279 ■157277 •055689774993 90 I •662 444 •822 182 -337 55^ •177818 1-830682 739 763 I 2 •484 626 •797094 ■5^5 374 •202906 ■570445 686 306 2 3 •281 720 ■751638 •718280 •248362 ■256751 604368 3 4 •033358 ■677 643 •966 642 ■322357 0-861 119 ■{6575^ 4 95 0711 001 ■544922 1^288999 ■455 078 •326870 178 159 95 6 ■255923 •249 106 ■744077 ■750894 1^505 029 6 7 I •505029 0^494971 7 21 3^ PERCENT. LogarWims and Co-logarithms of D^, their Diferences. N. f?M, X CologN,. A Log ]\I^. A Coiog :\L. A X 55 < 5-936289 037 112 r76i35i 984391 4^238 649 015 609 55 6 •973401 038 227 •745 742 983 649 •254258 016 35 1 6 7 5-011 628 039418 •729391 982851 -270609 017 149 7 8 -051 046 040 694 -712 242 982 012 •287 758 017988 8 9 -091 740 042 056 -694 254 981 01 1 •305 746 018989 9 6o •133796 043 508 •675 ^65 979888 •324735 020 112 60 I •177304 045 057 ■655 '53 978 649 •344847 021351 I 2 •222 ;^6i 046 706 ■633 802 977290 •366 198 022 710 2 3 •269 067 048 463 •611 092 975801 -388908 024 199 3 4 •3^7530 050343 •586893 974309 •413 107 025 691 4 65 •367873 052367 •561 202 972 794 -438798 027 206 65 6 •420 240 054555 '533 996 97 1 245 -466 004 028 755 6 7 •474 795 056937 •505 24 i 969 624 •494 759 030376 7 8 •531 732 059 554 •474 865 968032 ■5^5 '35 031 968 8 9 •591 286 062 414 •442 897 966019 ■557 103 033981 9 70 ■653 700 065 521 •408916 963 599 •591 084 036401 70 I •719 221 068 862 •372515 960 623 •627485 039377 1 2 •788083 072 410 •333 138 957075 •666 862 042 925 2 3 •860 493 076 r2i •290 213 952 955 •709787 047 045 3 4 •936614 079991 •243 168 948618 •756832 051382 4 75 4^oi6 6o5 084 166 •191 786 945 002 •808 214 054 998 75 6 •100 771 088 6S3 •136788 940921 •863 212 059 079 6 7 •189424 093 507 •077 709 936689 -922 291 063 3 1 1 7 8 •282931 098 824 •014398 932360 •985 602 067 640 8 9 •381755 104595 2^946 758 927279 ro53 242 072721 9 80 •486350 110725 •874037 921 173 •125963 078827 80 I •597 °75 117044 •795210 913977 •204 790 086 023 1 2 •714119 123576 •709187 906 758 •290 813 093 242 2 3 •837 695 130 ^35 •615 945 898 743 •384055 loi 257 3 4 •967 830 136935 •514688 891528 •485312 108472 4 85 rJ04 765 144 465 •406216 885491 •593 784 114509 85 6 •249230 1533^3 •291707 880 224 -708 293 119776 6 7 •402613 164072 •171931 873 869 -828069 126 131 7 8 •566685 178747 •045 800 869 788 -954 200 130 212 8 9 •745432 198879 1-915588 862 200 2-084412 137 800 9 90 •944 3 1 1 225 007 •777 788 846726 •222 212 '53 274 90 I 2^i693i8 260237 •624514 826 244 •375486 173 756 1 2 •429 55S 313694 •450 758 801 428 •549 242 198572 2 3 •743 249 395632 -252 186 755 885 •7478X4 244 1 15 3 4 1-138 881 534249 -008 071 681 667 •991929 3'^333 4 95 ■6y3 130 821 841 0-689738 548537 1-310262 451463 95 6 0-494971 •238275 251813 •761725 748 187 6 7 1^490 088 0^509912 7 22 HM. 3^ PERCENT. Vdlucs of Annui(H\i, and Stiujle and Atinual Prcmi'ians for Assunmn' of a Unit. X «x A. tt-j. X Ox A. tPj lO 21-9539 •223781 •009 749 '^^> H-5485 'sir^f^^fi ■045 875 I 21-8342 •227 830 •009 978 6 I r209 5 •';87 120 •048087 2 21-6889 •232 743 •010 258 7 10^8683 -598 658 •050 442 ,3 2I-j;22 8 •238360 •010583 8 ' 0-525 I -610 261 •052 950 4 21-341 2 ■244 500 •0 J 0944 9 lo^iSoo -62 J 932 •055 629 15 21-1493 •250990 •on 332 60 9-834 7 -633 610 •058480 6 20-952 6 -257 '^43 •on 736 I 9^490 2 -645 258 •061 510 7 2o-7.';6 6 •264271 -012 147 2 9-1475 -656 848 •064 730 8 20-566 8 •270690 •012 551 3 8-8073 -668350 •068 148 9 20*389 I •276698 -012 936 4 8^4707 •679736 ■071 773 20 20-2246 •282 259 •013 299 65 8-1363 •691 042 ■075637 I 20-065 ^ •287 629 ■013654 6 7-8034 •702299 ■079 775 2 19-908 7 •292941 -OI40IO 7 7-471 I -713539 •084 233 3 ^9747 5 ■298392 ■014382 8 7T386 -724783 •089 055 4 19-5779 •304129 •014 780 9 6-803 8 -736 102 •094326 ^.'i 19-3986 •310 194 -015 207 70 6-4703 •747 380 •100047 6 19-2 1 1 5 •3 16 519 -015 660 I 6-1409 -758521 •106 222 7 19-017 7 •323072 •016 139 2 5-8199 -769375 • 1 1 2 8 1 3 8 18-8202 -329751 -016637 3 S'S^^^^ •779801 ■I 19 756 9 18-6196 '33^535 •017 153 4 5-2199 •789666 ■126958 30 i8-4i'56 ■343 433 -0x7688 75 4-944 7 •798971 ■134400 I 18-2085 •350437 -018244 6 4-6761 •8o8o-;6 •142362 2 17-9962 •357616 •018826 7 4-415 8 •816856 •150827 3 177783 •364986 -019437 8 4-1624 •825 425 •159 890 4 17-5542 •372562 •020 080 9 3'9'^3^ •833 842 •169703 35 17-3245 -380332 •020 755 80 3-6722 •842 003 •180 215 6 17-0895 •388277 •021 464 I 3-4435 ■849737 •191 232 7 16-8503 •396368 •022 205 2 3-233 •8';680 •202 424 8 16-6066 •404 606 ■022 980 3 3-038 I ■863 448 -213828 9 16-3576 •413 026 -023 795 4 2-8622 •869 395 -225 104 40 16-102 6 •421 652 -024 654 85 2-697 8 ■874955 -236617 I 15-8397 -43054^ ■025 567 6 2-533 9 •880496 •249 156 2 ^5-5^79 •439 ns •026 541 7 2-3608 •886349 -263731 3 15-2876 •449213 -027 ';8o 8 2-1784 •892519 •280810 4 15-000 7 -458915 -028681 9 1-9639 ■899773 ■303 580 45 14-7072 •468838 -029 849 90 1-721 7 .907 960 -333 595 6 14-4098 •478894 ■031 077 I 1-473 1 •916368 ■370531 7 14-1097 •489 044 •032 366 2 I -21 8 5 •924979 •416942 8 13-8064 •499 300 -033 722 3 -9441 ■934 256 -480 f,',3 9 13-4990 ■509 '^95 -035 154 4 -6726 •943 438 ■564 053 50 13-1874 •520 232 •036669 95 •4129 •952 219 ■673 936 I 12-870 2 •530 959 •038281 6 •^775 •960 178 •81 ^464 2 12-5465 •541 906 •040 003 7 •966184 •966184 3 12-217 6 •\S3 029 •04 J 840 4 11-8849 •564 280 ■043 794 FOUR PEE-CENT. 24 4 I'KK-CENT. Commiitdtum Tahlc. lO I 2 3 4 6 7 20 I 2 3 4 25 6 30 I 2 3 4 35 6 40 I 2 3 4 45 6 50 I 2 3 4 l>. (^1 556-4 64639-8 61 905-7 59327-1 568790 54 540-3 52 292*0 50 117-6 48 002-9 45 935*4 43914-9 4] 958-6 40073-5 38 268-5 36547-8 34908-7 33 Hr^ 31846-8 30410-5 29 03 1- 1 27707-1 264357 25217-7 24 05 1 - 2 22934-6 21 864-9 20 839-5 19 855-5 18911-1 18 005-9 17 138-9 16309-8 15518-1 14 761-1 •4035-4 i3 339"6 12 670-2 12 025-2 1 1 404-3 10 807-3 10 233-5 9682-92 9 155-30 8 648-68 8 161-36 N. ' 356 297 1 291 658 I 229 752 I 170425 1 1 13 546 I 059 005 I 006 713 956595*8 908 "593-0 862 657-6 818742-6 776784-0 736710-5 698442-0 661 894-2 626985-5 593641-9 561 795 'I 531384-7 ^°'-35y5 474 646-5 448 210-8 422993-2 398941-9 376007-4 354 142-4 333302-9 313447-4 294536-3 276530-4 2.';9 39i'5 243 081-7 227563-6 212 802-5 198 767-2 185427-5 172757-4 160 732-2 149327-9 138520-6 128 287-2 118 604-2 109448-9 100 800-3 92638-91 S. 2^434671 23078374 21 786 716 20 ^y;^6 964 19 386 540 18 272 994 17 213 988 16 207 275 15 250 679 14342 086 13 479 429 12 660686 II 883 902 II 147 192 10448 750 9786855 9159870 8 566 228 8 004 433 7 473 048 6970695 6 496 048 6047 838 5 624 844 5 225 902 4 849 895 4 495 753 4 162 450 3 849 002 3 554 466 3 277936 3 018544 2775463 2 547 899 2335096 2 136329 I 950 902 I 778 144 I 617 412 I 468 084 1329564 I 201 276 I 082 672 973 223-3 872423-0 M. 1 2 7928 12474-5 12 226-6 12 029-0 II 862-6 1 1 71 v6 II 561 -I 1(397-8 II 210-7 10989-5 io735'8 10468-5 10 197-2 9933-49 '9 684-60 945i'27 9228-73 9 014-40 8802-97 8593-27 8385-77 8 180-02 7978-77 7 782-25 7 590-64 7403-12 7218-66 7036-14 6 855-46 6 6"7'7-62 6503-07 6333-24 6168-78 6 008-61 5850-68 5694-72 5 538-34 5 380-69 5 222-25 5 063-91 490575 4748-80 4593-59 4 439" 1 1 4 284-42 K X 429295-1 416502-2 404027-7 391 801-2 379772-2 10 I 2 3 4 367 909-6 356 198-0 344636-9 333239-^ 322 028-4 15 6 7 8 9 311 038-8 300303-1 289834-5 279637-3 269703-8 20 I 2 3 4 260019-2 250567-9 241 339-2 232324-8 223 521-8 25 6 7 8 9 214928-6 206 "542-8 198362-8 190 384-0 182601-8 30 I 2 3 4 175 01 1- 1 167 608-0 160389-3 ^^3 3^y^'' 146 497-7 35 6 7 8 9 139 820-1 133 3 1 7-0 126983-8 120 815-0 1 14 806-4 40 I 2 3 ^08955-7 103 261-0 97 722-69 92 342-01 87 119-76 45 6 7 8 9 82055-85 77150-10 72401-30 67 807-70 6;^ 368-60 50 I 2 3 4 2.') 4 PER-CENT. Commutation Table — (continued). X D. N. s. M, R. X 55 7 ^92 "59 84946-32 779784-1 4129-56 59084-17 55 6 7 24i*i5 77 705-17 694837-8 3 973-98 54954-61 6 7 6806-31 70898-86 617 132-6 3817-65 50980-63 7 8 6387-53 64511-33 546 233-8 3 66o-6^ 47 162-98 8 9 5 984'46 58526-87 481 722-4 3 503-26 43 502-34 9 6o 5 595-83 52931-04 423 195-6 3 344-79 39 999-08 60 I 5 220-92 47710-12 370264-5 3 185-11 36654-29 I 2 4859-28 42 850-85 322554-4 3 024-27 33469-18 2 3 4510-55 38340-30 279703-6 2 862-44 30444-91 3 4 4174-47 34 165-83 241 363-3 2 699-84 27582-47 4 65 3851-71 30314-12 207 197-4 2537-64 24882-62 65 6 3542-72 26771-40 176883-3 2-376-79 22344-99 6 7 3 247-83 23523-58 I ^0 1 1 1-9 2 218-16 19968-20 7 8 2967-11 20556-47 126588-3 2 062-36 177^0-04 8 9 2 701-11 17855-35 106 031-9 I 910-48 15687-68 9 70 2448-30 15407-05 88176-53 1761-56 13 777-20 70 I 2 207-73 13 199-32 72 769-47 I 615-15 12015-65 I 2 1978-36 II 220-97 59570-15 I 470-69 10400-50 2 3 I 759-71 9461-258 48349-18 1328-13 8 929-809 3 4 155^-83 7909-431 38887-92 I 187-93 7 601-675 4 75 1356-07 6 533-3^^ 30978-49 I 051-86 6413-742 75 6 I 175-66 5377-711 24425-13 923-602 5361-886 6 7 I 010.19 4367-521 19047-42 803.355 4438-284 7 8 859-938 iS^rs^i 14679-90 691-957 3 634-928 8 9 724-983 2 782-600 II 172-31 590-076 2942-972 9 80 604-344 2 178-256 8389-713 497-321 2352-895 80 I 497-043 1 681-213 6211-457 413-264 I 855-575 I 2 402-396 1 278-817 4530-244 337-734 I 442-3 1 1 2 3 320-620 958-1973 3251-427 271-435 I 104-577 3 4 250-992 707-2053 2 293-230 214-138 833-143 4 85 J 93-342 s^y^^?>:, T 586-024 i66"i42 619-004 85 6 146-887 366-9769 I 072-161 127-123 452-863 6 7 110-214 256-763 1 705-184 96-099 3 325-740 7 8 81-4703 175-2928 448-421 71-5948 229-641 8 9 59-5908 115-7020 273-128 52-8488 158-046J 9 90 42-791 72-911 157-426 38-3409 105-197190 I 29-647 I 43-2639 84-515 26-8428 6&Ss6 I 2 19-591 7 23-6722 41-251 17-9277 40-013 2 3 12-220 11-4522 17-579 1 1-309 6 22-086; 3 4 6-8646 4-5876 6-124 6-424 2 io-776| 4 95 3-252 I ^■Z3SS ^-53f> 3-075 7 4-35 2J 95 6 1-1350 -2005 -201 1-0836 1-276 7 -200 5 -000 -000 •1927 •^93 7 2<; 4 I'KK-CENT. Lognrithms niul Cu-li>i/aritliin.s ry'Dj.,. X^., and Mj.; icith thvir Dijf'i renrts. X LogD, A (Logf/;^) Colog D^ A (Cologf/;^) LogNj. A X lO 4-829667 7-980 833 l'^1^2>?,2, 0^019 167 6-132355 978 792 10 I •810500 -981 231 •189500 •018 769 •III 147978671 I 2 79' 73' •981522 •208 269 •018478 •089818978525 2 3 •773253 •981 699 •226747 •018301 ■068343978365 3 4 ■754952 •981 766 •245 048 •018 234 •046 708978 190 4 '5 •736718 •981 718 •263 282 •018282 -024898978 008 '5 6 •7'8436 •98' 554 •281 564 •018446 •002 9061977 823 6 1 •699 990 •981277 •300010 •018723 5-980729977640 7 8 ■681 267 •980881 •318733 •019 119 •958369977470 8 9 •662 148 •980 464 •337852 •019536 •935 839977 308 9 20 •642 612 •980 209 •357388 •019791 •913 '47977 ^S?> 20 I •622 821 •98c 037 •377 179 •019963 •890300976997 I 2 •602 8 ■;8 •979984 •397 142 •020016 •867 297 976833 2 3 •582 842 •980019 ■417 '58 •019981 ■844 '30 976 659 3 4 •562 861 •980073 •437 ^?>9 •019927 •820789 976 469 4 ^5 •542 934 •980078 •4^7 066 •019922 -797258976267 25 6 •523012 •980 054 -476 988 •019946 ■773 525'976o53 6 7 •503 066 ■979957 •496 934 -020 043 ■749578975831 7 8 •483 023 •979841 •516977 -020 159 -725409975601 8 9 •462 864 •979726 •537^36 -020 274 •701 010975 360 9 3° •442 590 •979600 •557 4'° •020400 •676370975 I'2 30 J •422 190 •9795'5 •577810 •020485 •65' 482,974 851 I 2 •401 705 •979432 ■598 295 •020568 •626333974577 2 3 ■381 137 ■979354 •618863 •020646 -600910974286 3 4 •360491 •979^57 ■639 5°9 •020743 •575^96973982 4 35 •339 748 •979 '4° -660 252 •020860 •549178973661 iS 6 •318888 •978993 •681 112 •021 007 •522839973326 6 7 •297881 •978837 -702 119 •021 163 •496165972974 7 8 •276718 •978698 •723282 •021 302 •469139972604 8 9 •255 4"5 •978566 ■744584 •021434 •441743972213 9 40 •233982 •978468 •766018 •021532 •413 956 971796 40 I •212450 •978388 •787550 •021 612 •385 752 97' 35' 1 2 •190838 •978279 •809 162 •021 721 •357 'O3 970874 2 3 •169 117 •978 108 -830883 •021 892 •327977 970368 3 4 •147225 •977918 ■852 775 •022 082 •298345 969 829 4 45 •125 143 •977639 •874857 •022361 •268 174 969 263 45 6 •102 782 •977310 •897218 •022690 ■237 437968666 6 7 •080092 •976975 •919908 •023 025 •206 103968038 7 8 •057067 -976650 •942 933 •023 350 •174 141 967 374 8 9 •033 7' 7 -976306 •966283 •023 694 -141 515966668 9 50 •010023 •975 983 •989977 •024017 •108 183I965917 50 I 3-986006 -975667 4-013994 ■024333 -074 100J965 112 I 2 •961 673 •975277 •038327 •024723 -039 212964 250 2 3 •936950 •974812 •063 050 •025 188 •003462,963331 3 4 •91 1 762 ^974 31 1 •088 238 •025 689 4-966793962352 4 27 4 PERCENT. Logarithms and Co-logarithms ofDj., N^., a7id M^.; with their Differences. X Colog N^ A LogM, A Colog M^ A X lO 7-867 645 021 208 4^106966 989 058 y893 034 010942 10 I •888 853 021 329 •096 024 991 280 •903 976 008 720 I 2 •910 182 021475 •087304 992 924 •912 696 007 076 2 3 ■93^657 021 6^^ •080228 993 954 •919772 006 046 3 4 •953 292 021 810 •074 J82 994 435 -925818 005 S^!i 4 15 ■975 102 021 992 -068617 994381 ■93-^ 3^ i 005 619 15 6 _-997 094 022 177 -062 998 993 823 -937 002 006 177 6 7 6-019271 022 360 •056821 992813 •943 179 007 187 7 8 -041631 022530 -049 634 991 345 -950366 008 6ss 8 9 -064 161 022 692 -040979 989855 -959021 010145 9 20 •086853 022 847 -030 834 989052 -969 166 010948 20 I -109 700 023 003 -019886 988596 -980 114 1 1 404 1 2 •132703 023 167 •008482 988 620 -991 518 oil 380 2 3 •155870 023 341 3^997 102 988980 4-002 898 oil 020 3 4 •179 211 023531 •986082 989 408 -013918 010592 4 25 •202 742 023 733 •975 490 989 652 •024510 010348 25 6 •226475 023 947 •965 142 989 795 •034 858 010205 6 7 •250422 024 169 •954937 989 692 •045 063 010308 7 8 •27459^ 024399 •944 629 989529 •°5S31^ 010471 8 9 •298990 024 640 •934 158 989385 •065 842 010615 9 3° •323630 024888 •923 543 989 2T2 •076457 010788 30 I ■348518 025 149 •912 755 989 181 •087 245 010819 1 2 •373667 025423 •901 936 989 169 •098 064 010 831 2 3 -399 090 0257x4 •891 105 989 174 •108895 010826 3 4 •424804 026018 •880 279 989 136 •119 721 1 864 4 ?,S •450822 026339 •869415 089 042 •130585 010958 3S 6 •477 161 026674 •858457 988878 •HI 543 oil 122 6 7 ■S°?-> ^''5 027 026 •847 33S 988 702 •152 665 Oil 298 7 8 -530861 027396 •836037 988 585 -163963 OII415 8 9 •558257 027 787 -824622 1 988 497 •175378 011503 9 40 •586044 028 204 -813 119 988 507 •186 881 01 1 493 40 I -614 248 028 649 •801 626 1 988 573 •198374 oil 427 1 2 -642 897 029 126 -790199 988575 •209 801 oil 425 2 3 •672023 029632 •7787741988432 •221 226 oil 568 3 4 •701655 030 171 -767 206 988 266 •232 794 oii 734 4 45 •731826 030737 •755472 987 907 •244528 012 093 45 6 •762563 031334 •743379 987 459 -256621 012541 6 7 •793 897 031 962 -730838 987 020 -269 162 012980 7 8 •825 859 032 626 -717858 986628 -282 142 013372 8 9 •858485 033 332 •704486 986 219 •295 5H 013 781 9 50 •891 817 034 083 -690 70^ 985 879 •309 295 014 121 50 I •925 900 034 888 -676584 985 569 •323416 014431 1 2 •960 788 035 750 •662 153 985 142 •337 847 014858 2 3 •996538 036 669 •647 295 984597 •352 705 015403 3 4 5^033 207 037 648 •631 892 984012 ' ^368 108 015988 4 '>M 4 TKIi CKNT. IjOgarithiiis and Cu-Iot/(irt'thms (»/'Dj., N^., and M^.; ivit/i their Differences. T Log IV A (Log'7'x) Colog D^ A (Colog r/jj.) Log N.r A X 55 3-88607.5 ^■•973 734 4-113927 0^026 266 4-929 145 961 305 s$ 6 •859 807 •973 105 •140 193 ■026 895 •890 450 960 189 6 7 •832912 •972421 •167088 -027 579 •850639958997 7 8 ■S05333 •971692 •194667 •028308 •809636957719 8 9 ■777 oa.-; ■970839 •222975 •029 161 •767355 956355 9 6o •747864 •969883 •252 136 •030 1 17 ■723710 954901 60 I ■7'7 747 •968 825 •282 253 •031 175 •678611 953 349 I 2 •686572 ■967 657 •313428 -032 343 ■6;^ 1 960 95 1 696 2 3 •654229 ■966372 •345771 ■033 628 •583656949936 3 4 •620601 ■965 052 •379399 •034948 -533, 592 948 053 4 (^:^ 'S^S(^S3 •963 684 •414347 •036316 •481645 946 026 65 6 •549337 .962 256 •450 663 •037 744 •427671 943832 6 7 •51^593 •960741 •488407 -039 259 •37^503 941 445 7 8 •472334 •959 209 ■527 666 •040791 •312948 938821 8 9 •431543 •957322 •568457 •042 678 •251 769935951 9 7° •388865 •955 080 •611 135 •044920 •187720932832 70 1 •343 945 •952 360 •656 055 •047 640 •120552929478 I 2 •296305 •949 136 •703 695 ■050 864 •050030925919 2 .3 •245441 •945 402 •754559 -054 598 3-975949922196 3 4 •190843 ■941 438 •809157 ■058 562 •898 145 918319 4 75 ■132281 •937999 ■867719 ■062 001 •816464 9H 134 75 6 •070 280 •934^23 ■929720 -065877 •730598909637 6 7 ■004405 ■930 064 ■995 597 •069936 •640 235 904773 7 8 2^934467 ■925 86 J 3-065 S33 •074^39 ■545 008 899 443 8 9 •860328 •920956 •139672 -079044 •444451 893 658 9 8o •781 284 •915 110 •218716 ■084 890 •338109887514 80 I ■696 394 ■908 260 •303 606 ■091 740 •225623881 186 1 2 ■604 654 •901 337 •395 346 ■098 663 •106 809874646 2 3 •505 99 ' ■893 669 •494 009 ■106331 2-981 455j868o9i 3 + •399 660 •886 666 •600 340 •I 13 334 -849546 861 302 4 85 •286326 •880 656 •7^3674 •I 19 344 •710848 85379' 85 6 •166982 •875 254 •833018 •124746 •564 ^39 844894 6 7 •042 236 •868 763 •957 764 •131 237 •409533 834 23 1 7 8 I •91 0999 •864 181 2^089 001 •135 819 -243 764 819577 8 9 •775180 ■856172 •224820 •143828 •063 341 799452 9 90 •631352 ■840 630 •368 648 *^';9 37o 1-862793 773332 90 I •471982 •820 089 •528018 •179911 -636125 738 114 I 2 ■292071 •795001 ■707929 •204999 •374239 684 650 2 3 •087 072 •749545 ■912928 •250 455 -058889 602 694 3 4 0-836 61 7 •675550 r- 1 63 383 •324450 o-66i 583 464 045 4 95 •512 167 •542829 •487 833 •457 171 •125 628 176381 95 6 ■054 996 ■247013 •945 004 •752987 7^302 009 6 7 • ■302 009 . 0^697991 7 29 4 PERCENT. Logarithms and Co-logarithns ofY>j.^ N_^., and M_p; with their Differences. X Colog N^. A Log M^ A CulogM, A X 55 5'°7o855 038695 3-615904 983322 4^3 84 096 016678 55 6 •109550 039 81 1 •599 226 982570 •400774 017430 6 7 •149 361 04 J 003 •581 796 981 762 •418 204 018238 7 8 •190364 042 281 •563 558 980914 •436442 019 086 8 9 •232645 043 645 ■544472 979897 •455528 020 103 9 6o •276290 045 099 •524369 978755 •475631 021 345 60 1 •321389 046 6^ I -503 124 9/7 497 -496 876 022 ^03 I 2 •368 040 048304 •U.80621 976 115 •5^9379 023 885 2 3 •416344 050 064 -456736 974602 •543 264 025398 3 4 •466 408 051947 •431338 973 092 -568662 026908 4 65 •518355 053 974 •404430 971 561 •595570 028439 65 6 •572329 056 i68 -37599^ 970 001 -624 009 029 999 6 7 •628497 058555 ■345 992 968372 -6'54oo8 031 628 7 8 •687 052 061 179 •314364 966779 ■68^636 033 221 8 9 •748231 064 049 •281 143 964 754 •718857 035 246 9 70 •812 2S0 067 168 •245 897 962315 -754 103 037685 70 I •879448 070 52-2 •208 212 959309 •791788 040 69 1 I 2 •949970 074081 •167 521 955 72 J •832479 044 279 2 3 4^024051 077804 •123 242 95^550 •876758 048 450 3 4 •101855 081 681 •074792 947 165 •925 208 052 835 4 75 •183536 085 866 •021957 943 528 •978043 056472 75 6 •269402 090 363 2-965485 939 423 3-034515 060577 6 7 •359 765 095227 •904908 935 n^ •095 092 064829 7 8 •454992 100557 •840079 930 829 •159 92 1 069 171 8 9 '555 549 106342 •770908 925729 •229092 074271 9 80 •661 891 112 486 •696637 919590 •303363 080410 80 I •774377 X18814 •616 227 912348 •383 773 087 652 I 2 •893 19 r i 25 354 •528575 905 090 •471425 094910 2 3 3-018545 131909 •433 665 897 029 '5^^335 102971 3 4 •150454 138698 •330694 889784 •669306 no 216 4 85 •289 152 146 209 •220478 883 745 -779522 1 16 255 85 6 •435361 155 106 •104 223 878497 •895777 121 1503 6 7 •590 467 165 769 i^982 720 872 162 2^017 280 127838 7 8 •756236 180423 •854 882 868 154 •145 1 18 131 846 8 9 ■936 6 j9 200 548 •723 036 860627 •276964 139373 9 90 2-137207 226 668 -583 f^63 845 ^'^5 •416337 154 835 90 I •363875 261 886 •428828 824696 •571 172 175304 1 2 •625 761 3^5 35° •253 524 799921 -746476 200 079 2 3 •941 III 397 306 •053 445 754371 •946555 245 629 3 4 1-3384^7 535 955 0^807816 680 125 1-192 184 319875 4 95 -874372 823 619 •487 941 546942 -512059 453 058 95 6 0-697991 •034883 250 092 -965 117 749 908 6 7 1-284975 0-715025 7 :\o 4 PKK-CENT. ]'ulues of Annuities^ and Singh and Annual Premiums for Assurance of a Unit. X lO Ox Ax •Wt X (>x Ax ■BTj. 20-076 j; •189365 •008 985 5!^ 11-042 6 -536 823 -044577 I i(;>S2 4 •192 985 i -009 197 6 10-731 1 -548 805 -046 782 2 19-864 y •197 503 -009466 1 10-416 7 •560 899 -049 130 3 19728.3 •202 757 -009 782 8 10-0996 '57^ 093 •051632 4 ^9*577 5 -208559 -010135 9 9-7798 -585 392 -054305 15 19-4169 -214733 •010517 60 9-459 -597 730 -057 150 6 19-2518 -221 086 •OI09I7 I 9-1383 •610067 •060 175 7 19-087 -227422 •oil 322 2 8-8184 •622371 •063 388 8 18-9279 ■^3^ 543 •Oil 719 3 8-';oo I -634610 •066 800 9 18-7798 -239239 •012 095 4 8-1845 -646751 •070418 20 18-6438 •244468 •012445 C^S 7^8703 •658 834 •074274 I 18-5131 •249497 •012 786 6 7^5567 •670 894 •07840s 2 18-3840 -254463 •013 127 7 7-2429 -682 966 -082 855 3 18-251 1 -259574 •013484 8 6-928 1 -695 073 •087 672 4 18-1104 •264985 -013866 9 6-6104 -707 294 •092938 25 17-9607 •270742 -014279 70 6^293 •719502 -098657 6 17-8038 •276777 •014719 I 5-978 7 •731589 •104832 7 17-640 6 -283 055 -015 185 2 5-6719 -743 390 •11 1 422 8 17-4737 •289472 •015 669 3 5-3766 -754 746 -118362 9 i7'303 9 •296 002 •016 171 4 5-0969 -765506 •125558 30 17-1309 -302 658 •016693 75 4-8326 -775668 •132988 I 16-9548 -309432,-017234 6 4-5742 -785606 ■140935 2 16-7737 •316396 •017 801 7 4-3235 -795^51 •149386 3 16-587 2 -323570 •018398 8 4-0789 -804659 -158432 4 16-3948 •330969 -or9027 9 3-8382 -813917 •168 229 i^ 16-1968 •338584,-019689 80 3-6043 •822 910 •178725 6 15-993 8 -346393-020384 1 3-3824 -83 1 445 -189 722 7 15-7864 ■354368 -021 110 2 3-1780 ■839307 •200887 8 15-5747 -362 509! -021 871 3 2-9886 ■846 593 •212254 9 15-3577 -370857 j -022672 4 2-8176 ■853 167 •223 480 40 15-1347 •379434! •023517 85 2-657 8 ■859316 •234927 I 14-9040 -38S308, -024416 6 2^498 4 •865 448 •247 386 2 14-6644 -397 522 j -025 377 7 2-329 7 ■871 935 •261 867 3 14-4165 •407058 ^026 404 8 2-1516 •878784 -278836 4 14-161 9 •416852 -027494 9 1-9416 -886862 •301 489 45 13-9005 -426903 -028 6.';o 90 1-7039 -896004 •331376 6 13-6350 •437 116 •029 868 I 1-459 3 -905412 •368 159 7 13-3663 -447 45 1 •031 146 1-2083 •915 066 •414380 8 13-0940 ■457921,-032490 3 -9372 -925 493 •477 756 9 12-8173 -468 564 •033 911 4 •6683 -935835 •560954 50 12-5360 -479383 -035415 95 •4106 -945 745 •670437 I 12-2488 -490 43 1 -037017 6 •1766 ■954 746 •811438 2 11-9547 •501 742 •038 730 7 ■961 539 •961 539 3 11-6550 '5^?> 270 -040 559 4 11-3509 -524964 -042504 FOUE AND A HALF PER-CENT. .^2 4.1 i'KR-CKNT. Coin m utation Table, D, Nx S. Mx Hx lO I 2 1.1 6 7 8 20 I 2 3 4 6 30 I 2 3 4 35 6 7 40 I 2 4.1 6 / 8 9 50 I 2 3 4 64.392-8 61 317-9 .iy44r4 .1.1 74 1 "o 53 185-2 .1° 754'3 48429-3 46 19.3-4 44 052-6 41 9.34-5 .39898-2 3 7 9.5B-4 36060-6 34 27 1 "5 32573-9 30964-2 29 4.U-4 27 978-6 26 588-9 25261-5 23 994-0 22 783-4 21 629-7 20530-5 9483-7 8486-1 7 534-8 6626-9 5 760-3 4934'2 4 147-0 3 398-3 2 686-9 2 010-2 i3'55-i 0750-1 o 161-7 9598-29 9 059- 1 1 8543-82 8051-47 7581-86 7 134-43 6707-39 6 299- 1 7 188642 127324 068881 013 140 959 954- 909 200' 860 771- 814577" 770545' 728 610' 688 712' 650 773' 614 713' 580441' 547867' 516903' 487469' 459 490' 432 901 407 640 383 646 360 862 339^33 318 702 299 219 280 733 263 198 246571 230 810 2U 876-8 201 729- 188331' 175644' 163634' 152 269' 141 519' 131357' 12 1 759' 1 12 700 104 156 96 104' 88 t;23' 81 388 74681 68382 20 229 255 1 9 040 6 1 3 17 913 289 16844408 1 5 83 1 268 14 87 1 3^3 13 962 1 13 13 loi 342 12 286 764 n 516 219 10 787 609 10098 896 9448 122 8 833 409 8252967 705 099 188 196 700 726 241 236 808334 400 694 017047 656185 316951 998 249 699030 418 297 K'in 098 908527 677716 461839 260 no 071 778 896 133 732499 580 230 438710 307353 185594 072 893 968737' 872 632' 784 109' 702 720' 628039' 10434-33 10 132-39 9898-29 9712-65 9557-H 9416-59 9277-15 9 126-68 8 955-07 8753-15 8 522-61 8 280-96 8036-81 7 800-64 7578-80 7371-84 7 175-39 6987-09 6802-23 6619-76 6440-07 6262-75 6 090-13 5922-38 5 759-60 5 601-05 5 445-85 5 293-01 5 142-43 4994-93 4850-85 4711-33 4576-88 4446-56 4318-68 4 192-99 4067-57 3941-74 3815-88 3 690-70 3 566-27 3443-37 3 322-43 3 202-61 3 083-23 327960-1 317525-8 307 393-4 297495-1 287 782-5 278 2215-4 268808-8 259531-6 250 404-9 241 449-9 232696-7 224 174T 215893-2 207 856-, 3 200055-7 92476-9 85 1 05 -I 77929-7 70942-6 64 140-4 57520-6 51 080-5 44817-8 38727-6 32 805-3 27045-7 21 444-6 15998-8 10705-8 05 5^3-3 00 568-4 95)17-54 91 oo6-2i 86429-35 81 982-77 77664-09 73471-10 69403-53 65461-79 61645-91 57955-20 54388-94 50 945-56 47623-14 44420-52 10 1 2 3 4 6 30 I 3S 6 7 8 9 40 I 2 3 4 45 6 7 50 1 2 3 4 33 4 1 PERCENT. Commutation Table — (continued). X D. N. s. M. Ex X 55 5 908-95 62 473-10 559657-2 2964*27 41337-29 55 6 SS^S'SI s^9^rs?> 497 184*1 2 845*34 38373-02 6 7 5178-26 51 759-28 440 246*5 2 726*40 35527-69 7 8 14 836-39 46922-89 388487-3 2 607-52 32 801*29 8 9 4509-53 42413-36 341 564*4 2488-92 30 193-76 9 60 4 196-50 38216-86 299151*0 2 370-09 27 704-84 60 I 3 896-61 34320-25 260934*2 2 250-91 25 334-75 1 2 3 609-35 30710-90 226613*9 2 131-44 23083-85 2 3 3334-29 27 376-61 195903-0 2 011-81 20952-40 3 4 3071-09 24305-52 168 526*4 I 892*19 18940-59 4 65 2 82o-o8 21485-44 144 220*9 I 773-43 17 048-40 65 6 2581-44 18904-00 122 735*4 I 6^6-2^ 15 274-97 6 7 2355-24 16548-77 T 03 831-4 I 541-19 13618-74 7 8 2 141-38 14407-39 87 282-66 1 428*75 12077-55 8 9 I 940-08 12467-31 72875-28 1 3ig-66 10648-80 9 70 I 750*08 10717-23 60407*97 I 213*21 9329*138 70 I I 5'7o-57 9 146-663 49690-74 1 109*06 8115*927 1 2 I 400-66 7 746-005 40 544-08 I 006-78 7 006-870 2 3 I 239-90 6506*108 32 798-07 906-337 6000*088 3 4 r 088- 19 5417-918 26 291*96 808-023 5093-751 4 75 946-366 4471-552 20 874-04 713-059 4285*728 75 6 816-536 3 655-016 16402*49 623*981 3572*669 6 7 698-258 2956-758 12747-48 540-865 2 948-688 7 8 S9^-SS^ 2 365-200 9790-717 464-233 2407-823 8 9 496-335 1 868-865 7425-5^7 394-484 1943-590 9 80 411-764 1457-102 5 SS^'^S'^ 331-286 1549*106 80 1 337-035 1 120-067 4 099-550 274*289 t 217*820 I 2 271-551 848-516 2979-482 223*319 943 -53 J 2 3 2 15-33 J 633-186 2 130-966 178*792 720-213 3 4 167-761 465-424 1497-781 140-495 541*421 4 85 128-610 336-814 1032*357 108-568 400-926 85 6 97-2408 239-573 695-543 82-7368 292*358 6 7 72-613 8 166-960 455-969 62-297 2 209-622 7 8 53-4194 113-540 289-010 46-2298 H7-324 8 9 38-8863 74-654 175-470 33-9970 101*095 9 90 27-7899 46-864 100-816 24-575 I 67-097 90 I 19-161 7 27*702 53-952 17-1436 42*522 I 2 I2-602 1 5*100 26-250 11*409 1 25-379 2 3 7-822 7 7-278 ii'i 50 7*1724 13-970 3 4 4-3734 2-904 3-872 4*0600 6*797 4 95 2*062 -842 •968 1*9369 2-737 95 6 •716 2 •126 *I26 -6799 *8oo 6 7 -1259 •000 *ooo *1205 -121 7 :i\ 4! I'KIK'KXT. IjOgarhlims ^S 030 760 •660570 987 454 •339430 1 2 546 2 3 ■786125 031 262 •648024 987327 •351976 012 673 3 4 •817387 031 797 •^?,s?>s^ 987173 •364 649 012 827 4 45 •849 184 032361 •622 s^^. 986 81 1 •377476 013 189 45 6 •881 545 032953 •609335 986353 •390 665 013647 6 7 •914498 ,033577 •595 688 985 907 •404312 014093 7 8 •948075 034239 •581595 985514 •418405 014486 8 9 -982314 034940 -567 109 985 J 05 •432 891 014895 9 50 5'oi7 254 035 690 •552214 984770 •447 786 015 230 50 I -052944 036492 •536984 984472 -463 016 015528 1 2 -089436 037353 •521456 984 049 •478544 015951 2 3 •126 789 038 269 •505 505 983501 •494 495 016499 3 4 •165058 039 249 •489006 982 912 •510994 017088 4 :\r, 4:1 l'i:ii <'l^^'■l'• Z/oya^/V////^s• and Co-lo(jarithms tifDj., Nj, and Mj ; icit/i their Differvnvrs. X Log D^ A (Log rp_r) Colog \)j. A (Colog vp^) LogN^ A X 5.? 377' 511 7-971651 4^228489 0-028 349 4^795693 959 706 55 6 •743 '^^2 -971022 •2.^6838 -028978 •755399958589 6 7 •714184 •970 338 •2135 816 •029 662 •713988,957397 7 8 •684522 -969 609 •315478 •030391 •671385 9'fi 1 '8 8 9 •654131 -968 756 •345 869 -03 1 244 •627 503 954752 9 6o •622887 •967 800 •377113 -032 200 ■582 255 953 295 60 I •590 68; -966 742 •409313 •033 258 ■S3S 550 951743 I 2 •557429 ■965575 •442571 •034425 •487 293 95008; 2 3 •523 004 •964288 •476996 •035712 •437380 948325 3 4 •487 292 -962970 •512 708 •037030 •385705946439 4 65 •450 262 -961 600 •549738 •038 400 •332 144 944410 (^5 6 •411 862 -960173 •588 138 •039827 •276554 942 212 6 7 •372035 -958658 •627965 •041 342 •218 766939819 7 8 •330 69.3 •957 126 •669307 •042 874 •158 585^937 188 8 9 •287819 •955 239 •712 181 -044761 •095773934310 9 70 •243 058 •952 998 •756942 -047 002 •030083931 180 70 I •196 0156 -950 276 •803 944 ■049 724 3-961 263927 815 I 2 •146332 •947 054 •853 668 •052 946 -889078924243 2 3 •093 386 •943319 •906614 -056681 -813 321 920511 3 4 •036 70 j •939 354 •963 295 -060 646 •733832916626 4 75 2-976059 •935917 3-023 941 •064 083 ■650458912431 75 6 •911 976 -932 040 -088024 -067 960 •562 889907 927 6 7 •844016 -927981 •155984 •072 019 •470 816903 052 7 8 •771997 •923 778 •228003 •076 222 •373868897 710 8 9 •695775 •918873 •304225 •081 127 ■271578891912 9 80 •614648 •913027 •385 352 ■086973 •163490^885754 80 I •527675 -906177 •472325 -093 823 •049 244 879416 I 2 •433 852 -899254 •■566148 •100 746 2-928 660 8;28;i 2 '3 '333 106 -891 586 -666 894 -108414 •801531 866318 3 4 •224692 •884583 •775308 ■I 15 41 7 •667 849 859541 4 85 •109275 •878574 •890 725 -121 426 •527 390 852 048 85 6 1-987849 •873 170 2-012 1 151 •126 830 •379438 843 173 6 7 •861 019 -866681 •138981 •133319 •222 6i 1 832538 7 8 •727700 -862 097 •272 300 •137903 •055 149 817903 8 9 •589 797 •854 090 •410 203 •145910 1-873052 797 786 9 90 •443 887 •838 546 •SS^^^3 •161 454 •670 838 771676 90 I •282 433 •81800; •717567 •181993 •442514 736468 I 2 • 1 00 440 •792918 •899 560 •207 082 •178982 682 998 2 3 0-893 358 •747461 1-106642 •252539 0861 980 601 027 3 4 •640 819 •673 467 •359181 •326533 •463 007 462 345 4 95 •314286 •540 746 -685714 •459254 1-925 352 174610 95 6 ^•855o32 •24493c 0-14496J •755070 -099 962 6 7 •099 962 •900 038 7 37 4| PER-CENT. Logarithms and Co-logarithms of D^., N^, and M^.; ivith their Differences. X CologN^ A Log :m^. A Colog ]\L A X 55 5-204307 040 294 3^471918 982 216 +-528 082 017784 55 6 •244601 041 41 1 1 ■454134 981456 •545 866 018544 6 7 •286012 042 605 •435590 980638 -564410 019362 7 8 •328613 043 882 •416 228 979784 •583 772 020 216 8 9 •372497 045 248 •396012 978752 -603 988 021 248 9 6o •417745 046 705 •374764 977 594 -625 236 022406 60 I •464450 048257 •352358 976316 -647 642 023 684 I 2 •51270710499:3 .328674 974913 •671326 025 087 2 3 •562 620 ! 051 675 •303 587 973 378 •696413 026622 3 4 •614295 053561 •276965 971850 •723 035 028 150 4 65 •667 856 055 590 •248815 970305 •751 185 029 695 65 6 •723 446 057788 •219 120 968 736 •780880 031 264 6 7 •781 234 060 181 •187856 967 100 •812 144 032 900 7 8 •841 4^5 062 812 •154956 965 508 •845 044 034492 8 9 •904227 065 690 •120464 963 472 •879536 036528 9 7° •969917 068820 •083 936 961 018 •916 064 038982 70 I 4'o38 737 072185 •044 954 957982 '9SS 046 042 018 1 2 •110922 "^IS 151 •002 936 954 354 •997 064 045 646 2 3 •186679 079489 2^957 290 950 134 3-042710 049 866 3 4 •266 168 083374 •907 424 945 701 -092 576 054 299 4 75 •349 542 087 569 •853 125 942 047 -146875 057953 75 6 •437 m 092 073 •795 172 937917 -204828 062 083 6 7 •529 184 096 948 •733 089 933 647 -266911 066 3 S3 7 8 •626 132 102 290 •666 736 929 294 '333 264 070 706 8 9 •728422 108088 •596030 924173 •403 970 075827 9 8o •836510 114 246 •520 203 918 005 •479 797 081 995 80 I •950 1$^ 120 584 •438 208 910717 -561 792 089 283 1 2 3^o7i340 127 129 •348925 903 422 •651075 096578 2 3 •198469 133682 •252347 895314 •747 ^53 104686 3 4 •332 151 140459 •147 661 888 040 •852339 1 1 1 960 4 85 •472 610 147952 •035701 881 998 •964 299 118 002 85 6 •620562 156827 1-917699 876770 2-082 301 123230 6 7 •777389 167 462 •794469 870453 •205531 129547 7 8 •944851 182 097 -664922 866519 •33S 078 133481 8 9 2"i26 948 202 214 •531441 859 054 •468 559 140946 9 90 •329162 228324 •390 495 843 607 -609 ^0^ ^5^^393 90 I •557486 263532 -234102 823 148 •765 898 176852 1 2 •821 018 317002 •057 250 798417 •942 750 201 583 2 3 T^i38o2o 398973 0-855 667 752 858 i'i44 333 247 142 3 4 •536993 531^55 •608525 678584 •391475 321416 4 95 0-074648 825390 •287 109 545 342 -712 891 454658 95 6 •900 038 T- 83 2 451 248395 0-167549 75^605 6 7 -080 846 ■919154 1 ;{s HM. 4] I'KH-CENT. Values of AnnuUlea, and Singh ami Annual Premiums for Assurnncc of a Unit. X «x A. w^ X Ox A. IBj. lO 18-4592 -162 042 -008327 55 10-5726 •501 658 •043 349 I 18-3849 •165243 -008524 6 10-2858 •514010 ■045 545 2 18-2892 -169366 -008 780 7 9^995 5 -526509 •047 884 3 18-1758 -174246 -009 0S7 8 9-702 -539 146 •050378 4 18-0493 -179695 ■009433 9 9^405 3 -551926 •053 043 15 17-9137 -^^5 533 -009 809 60 9-1068 •564777 •055881 6 177738 -191561 •010 204 1 8-807 7 -577658 •058 898 7 17-6341 '^91 515 •010 603 2 8-5087 -590 534 •062 105 8 17-4994 -203374 -010994 3 8-2106 -603370 •065 508 9 17-3750 -208734 •oil 360 4 79H3 •616 130 •069 1 1 7 20 17-2617 -213 609 -on 697 (^5 7-6187 •628859 •072964 I 17-^534 -218 274 -012 024 6 7^3230 -641 591 •077 086 2 17-0467 -222 870 -012350 7 7-0264 -654367 •08 1 527 3 16-9366 -227613 -012 690 8 6-728 I -667 211 •086336 4 16-8192 -232 66s -013 057 9 6-426 2 -680 212 •091 596 25 16-6936 -238 076 -013456 70 6-1238 -693 23 J •097311 6 16-561 2 -243 776 -013 881 I 5-8238 -706151 •103484 7 16-422 9 •249 730 •0143.33 2 5-5303 •718792 •110071 8 16-2813 -255830 -014804 3 5^2473 -730978 •117007 9 16-1368 -262 050 -015292 4 4-978 8 •742538 •124195 30 15-9893 -268404 -015 798 75 4^7250 •753470 •131611 I 15-8388 -274882 -016 324 6 4-4762 -764181 •139545 2 15-683 7 -281 ^64 -016877 7 4^2345 •774592 •147979 3 15-5234 •288467 -017458 8 3'998 3 •784764 •157007 4 ^S'iSlS -295612 -oj8 072 9 3-7<^5 3 •794794 •166787 3.5 15-1862 -302988 -018 719 80 r,. ^^0 - .0 .•).:) / •804554 •177 266 6 15-0100 -310573 -019399 I 3-3^33 -813829 •188243 7 14-8297 -318340 -020 1 10 2 3'i247 -822381 •199380 8 14-6450 -326 290 -020 856 3 2-9405 -830312 •210 711 9 14-455 2 -334463 -021 641 4 27743 -837469 •221886 40 14-2596 -342 890 -022470 85 2-6189 •844 163 -233 266 I 14-0564 -351638 -023 355 6 2-463 7 •850 844 -245 645 2 13-8446 -360 758 -024302 7 2^2993 •857925 •260 034 3 13-6246 -370232 -025316 8 2-1254 -865411 •276892 4 13-3979 •379993 -026392 9 1-9198 -874 266 -299427 45 13-1645 •390 044 -027537 90 1-6864 -884318 •329188 6 12-9267 •400 284 -028742 I 1^445 7 -894681 -365817 7 12-6855 •410671 -030008 2 1-198 2 •905337 •411 847 8 12-4405 •421 221 -031340 3 •930 3 -916874 •474991 9 12-1908 •431973 -032 748 4 -6640 •928337 -557 885 50 11-9363 -442 934 -034 240 95 •4084 •939 340 •666 964 I 11-6756 -454159 •035 829 6 •1758 •949 333 •807418 2 11-4079 •465 690 •037532 7 •956938 -956938 3 11-1342 -477476 •039350 4 10-8557 ,-489466 •041 285 FIYE PEE-CENT. 40 5 I'Ki; CF^NT. Commutation Table. X D. N. Sx M, I J., .r lO 61391-3 I 047 140-2 16859782 8 604-11 252 897-6 10 I 58 181-4 988 958-8 I <; 8 1 2 642 8317-61 244293-5 I 2 Sa 1898 933 7<^J9-o 14823683 8 096-55 235 975-9 2 3 52 387-3 881381-7 13 889914 7922-07 227879-3 3 4 49 7472 83 1 ^4-5 13008533 7 776-61 219957-3 4 15 47 247'4 784387-1 12 176898 7 645-78 212 180-6 15 6 44 868-4 739 5 J 8-8 II 392 511 7 5'6-S9 204534-9 6 7 42593-0 696925-7 10652 992 7377-85 197 018-3 1 7 8 40407-3 656518-4 9 956 066 7 220-36 189640-4 1 8 9 38298-7 618 219-7 9 299 548 7 035-95 182 420-1 9 20 _^6 265-4 581954-3 8681328 6826-40 175384-1 20 I 34 3i9"9 547 634-4 8099374 6607-81 168557-7 I 2 32465-8 SK'i 168-5 7551740 6388-00 161 949-9 2 .3 30 708-2 484 460-3 7036571 6176-38 155 561-9 3 4 29048-1 455412-2 6552 III 5978-55 H9 385-5 4 25 27481-2 427 93 1- 1 6 096 699 5 794-87 143 407-0 25 6 25999-0 401 932-1 5 668 768 5621-35 137612-1 6 7 24595-4 377 336-6 ^526683^ 5 455-82 13 1 990-8 7 8 23 262-5 354074-1 4 889 499 5 294-09 126534-9 8 9 21995-9 332 078-2 4 535 425 5 135-20 121 240-9 9 30 20792-7 3^1285-5 4203347 4979-48 116 105-7 30 I 19649-7 291 635-8 3 892 061 4826-55 III 126-2 I 2 18 ^6';-8 273070-0 3 600 425 4678-39 106 299-6 2 3 '7 538-4 ^5r)S3i'6 3 327 355 4535-08 loi 621-2 3 4 16564-9 238966-8 3071824 4 396-69 97 086-14 4 35 15 641-9 223 324-8 2 832 857 4 262-54 92 689-45 35 6 14766-4 208558-5 2609532 4 131-84 88426-91 6 7 i3 935'i 194623-4 2400973 4003-74 84 295-07 7 8 13 145-9 181477-5 2 2063 150 3878-14 80291-33 8 9 12397-5 169080-0 2024873 3 755-69 76413-19 9 40 II 68S-1 157391-9 I 855 793 3 636-66 72657-50 40 I II oi6-8 146375-1 I 698401 3521-94 69020-84 I 2 10382-2 135993-0 I 552 026 3411-91 65 498-90 2 3 9781-63 I262II-3 I 416033 3305-77 62 086-99 3 4 9212-18 116 999-2 I 289821 3 202-12 58 781-22 4 45 8672-11 108327-0 I 172 822 3 100-72 55579-10 45 6 8 158-46 100 i68-6 I 064 495 3 000-03 52478-38 6 7 7 669-41 92499-17 964326-5 2 899-48 49478-35 7 8 7204-12 85 295-05 871827-3 2 799-40 46578-87 8 9 6761-99 78533-06 786532-3 2 700-33 43 779-47 9 50 6341-98 72 191-08 707 999-2 2 602-31 41079-14 50 I 5 943-64 66247-44 635 8o8-2 2 505-97 38476-83 I 2 5 5^'^^-^5 60681-19 569560-7 2 41 i-6i 35970-86 2 3 5 208-16 55473-04 508879-5 2318-58 33 559'^5 3 4 4867-89 50605-15 453 406-5 2 226-31 31 240-68 4 41 6 PERCENT. Commutation Tahle — (continued). 6 9 6o J 2 3 4 6 7 8 9 7° I 2 75 6 8o I 2 3 4 85 6 7 8 9 90 I 2 3 4 95 6 7 D.. 4 544' 4^37' 3 944 3666 3402' 3 151' 2 912' 2684' 2468' 2 262' 2 067' I 883' I 710' 1547' ^395- 1252' I 119' 993' 875' 764' 66v 568' 483' 407' 340' 28r 228' i83' 144' 1 12 85' 64 47' 35' 25' i8' 12' 8' 5' 2' 59 ^5 77 80 70 42 28 74 34 66 84 84 58 85 67 99 1 1 289 097 368 582 103 498 664 414 065 961 '597 893 346 7^74 5°^ 5 936 6 0974 4272 0849 4105 123 I 0184 7923 •310 2 •4529 •0792 o6o'j6 823-41 878-63 21J-84 809-14 76 745*43 060-69 592-36 329-69 261-86 378-02 667-44 ii9'59 723-926 470-936 351-827 358-538 483-441 719-074 057-492 489-389 005-891 598-227 257-813 976-748 I 747787 5 564-1907 419-298 2 306-95 1 8 221-2344 156-7329 108-7962 73-6989 48-271 6 30-1867 17-7762 9"653 1 4'634 7 1-8424 •5322 •0792 •000 o M, 402 801-3 35^ 740'8 3i49^7'4 277038-7 242 826-9 212 017-8 184360-1 159 614-6 137553-9 117 961-6 1 00 63 1 -9 85370-02 71 992-00 60324-56 50 204-97 41 481-04 34010-11 27658-28 22 299-7^ 17 816-30 14097 1 1 039 855°' 6544 4946' 3688 2 7II' 1963' 1 399' 980' 673' 452' 295' 1 86' 112' 64' 34' i6' 7' 2- ■23 74 ■347 ■455 •228 ■415 ■66j •879 •688 •3901 ■4383 ■2040 •471 I 6749 •9760 •7044 ■5177 741 5 0884 4538 611 4 0792 000 o 134-83 2 043-79 1953-18 I 863-05 i 773-56 I 684*32 I 595-25 I 506-39 1417-83 I 329-69 I 242-61 I i57X)8 t 073*53 992-256 913-781 837-565 763-350 690-821 619-928 550-871 484-483 422-508 364-955 312-145 264-308 221-169 182-449 147-988 118-026 92-3799 71-100 7 53-9^6;^ 40-473 2 29-916 6 21-917 8 15-7862 10-9730 7-2766 4-5588 2-5716 1-222 5 -4276 -0755 29 014-36 26879-54 24835-75 22 882-57 21 0J9-51 19 245-95 17561-63 15966-38 14459-99 13042-17 II 712-48 10469-86 9312-781 8 239-250 7 246-994 6333'2^3 5 495-648 4732-298 4041-477 3421-549 2870-678 2386-195 I 963-687 1598-732 I 286-587 1 022-279 801-109 4 618-660^ 470-672 7 352-6464 260-266 5 189-1659 ^35' ^99 3 94-7262 64-809 6 42-891 8 27-J055 16-132 5 8-855 9 4-297 1 1-7255 •5030 -0755 55 6 9 60 I 2 3 4 65 6 I 2 3 4 75 6 90 I 2 3 4 95 6 42 6 I'Ki; CKNT. LoqtiritJnns und Co-loijarithms of 1)^., N,. , and M^; icitli tluir Differences. X LogD^ A {Lo^vps) Colog D^ A (Colog vpj.) LogN, A X lO 4788 107 T976677 5-211893 0-023323 6-020 005 975173 10 I •764784 •977 075 •235 216 •022925 5^995 178 975061 I 2 .741 8.:;9 -9T1 3^1 •258 141 •022 6;^^ -970239 974925 2 3 •719 226 -911 S-^^ •280774 •022458 •945 164 974 769 3 4 •696 768 •977610 •303232 •022 390 •9-9 933 974 598 4 ^5 •674378 •977 562 •325 622 •022438 •894531 974418 15 6 •651940 •977399 •348 060 •022 601 •868949974238 6 7 •6293.39 •977 121 •370661 •022 879 -843 187 974060 7 8 •606 460 •976724 •393 540 -023276 ■817247973896 8 9 •583 184 •976309 •416816 •023 691 •791 143973746 9 20 •559493 •976053 •440507 •023 947 -764889973 602 20 1 'S3S 546 •975880 •464 454 •024 120 -738491973458 I 2 •511 426 •975 829 •488574 •024 171 •711 949973309 2 3 •487 ^55 •975 863 •512745 •024137 •685258973 147 3 4 •463 118 •975917 •536882 •024083 •658 405 972 969 4 25 •439035 •975922 •560 965 •024078 •631374 972779 25 6 •414957 •915 898 •585 043 •024 102 •604153 972576 6 7 •390 855 •975 801 •609 145 -024 199 •576729 972365 7 8 •366656 •975 685 '^33 344 •024315 •549094972146 8 9 ■342341 •975571 •657659 •024429 •521 240971 919 9 30 •317912 •975 443 •682 088 •024557 •493 159 97 1 682 30 I -^93 355 •975 359 •706645 •024641 •464841 971433 1 2 •268 714 •975276 •731286 •024724 •436274 971 171 2 3 •243 990 •975 198 •756010 •024802 •407 445 970 893 3 4 •219 188 •975 102 •780812 •024898 •378338 970 599 4 35 ■194 290 •974 983 •805710 •025017 •348937 970291 3S 6 •169273 •974837 •830727 •025 163 -319228 969967 6 7 •144 110 •974681 •855890 •025319 -289 195 969628 7 8 •118 791 •974542 •881 209 •025 458 •258 823 969 269 8 9 •093 333 •974410 •906 667 •025590 -228092 968 890 9 40 •067 743 •974312 •932257 •025688 •196982 968 485 40 I •042 055 •974232 •957945 •025 768 •165467 968 050 1 2 •016 287 •974124 ■983 713 •025 876 •133 517 967581 2 3 3^990 41 1 •973951 4-009 589 •026049 •101 098 967 085 3 4 •964362 ■973 763 •035638 •026237 •068183 966554 4 45 •938125 •973 483 •06 1 875 •026 517 •034737 965 995 45 6 •911 608 •973 154 •088392 •026 846 •000732 965406 6 7 •884 762 •972 819 •115 238 •027 181 4^966 138 964 786 7 8 •857581 •972494 •142 419 •027 506 -930924 964 129 8 9 •830075 •972 150 •169925 •027 850 ■895 053 963 43 1 9 50 •802 225 •971827 •197 775 •028 173 -858484 962685 50 I •774052 •971511 •225948 •028489 •821 169 961 885 1 2 •745 5^3 •971 121 •254437 •028879 ■783 054 961 028 2 3 •716684 •970657 •283316 -029343 -744082 960 113 3 4 •687341 •970 154 •312659 -029846 •704 195 959 134 4 43 6 PERCENT. Logarithms and Co-logarithms of D_^,, N^, and M^.; loith their Differences. X Colog N,. A Log INI J. A Colog M,. A X 10 lO 7-979995 024827 3^934 706 985 293 4^065 294 014707 I 6"oo4 822 024939 •919999 988301 ■080 001 on 699 I 2 •029 761 025 075 •908 300 990539 •091 700 009461 2 3 ■054 836 025231 •898 839 991 952 •loi 161 008 048 3 4 •080067 025 402 •890 791 992 63 1 •109 209 007 369 4 15 •105469 025582 •883 422 992 599 •116578 007 401 15 6 •131 051 025 762 •876021 991909 •123979 008 091 6 7 •156813 025 940 •867 930 990 629 •132070 009371 7 8 ■182753 026 104 •858559 988 764 •141441 on 236 8 9 •208857 026 254 •847323 986 869 •152677 013 131 9 20 •235 III 026398 •834 192 985 865 ■i6s 808 014135 20 I •261 509 026542 •820 057 985 308 •179943 014692 I 1 •288051 026 691 •805 Z6^ 985 369 •194635 014631 2 3 •3H742 026 853 19° nA 985862 •209 266 014 138 3 4 •341 595 02/031 ■776 59(> 986 448 •223 404 013552 4 25 •368626 027 221 •763 044 986797 •236956 013 203 25 6 '39^ 847 027424 •749 841 987 019 •250 159 012981 6 7 •423271 027635 •736860 986931 •263 140 013 069 7 8 •450 906 027 854 •723791 986767 •276 209 013 233 8 9 •478 760 028081 •710558 986626 •289442 013374 9 30 •506841 028318 •6971841986453 ^302 816 013547 30 I 'Sis 159 028567 •683637 986459 •316363 013 541 1 2 •^6^ 726 028 829 •670096 986489 •329904 0135^^ 2 3 •S9'^SSS 029 107 •656585 986541 •343 415 013459 3 4 •621 661 029 40 J •643 126 986542 •356874 013458 4 35 •651 063 029 709 •629 668 986475 •370332 013525 35 6 •681 772 030033 •616 143 986323 •383857 013677 6 7 *7 to 805 030372 •6024661986 158 •397 534 013 842 7 8 •741 177 030731 •588 624 ! 986 066 •411376 013934 8 9 •771908 03 1 1 1 ■574690 1986013 •425310 013987 9 40 •803018 0315^5 •560 703 986079 •439 297 013 921 40 I •834533 031950 •546 782 986 216 •453 218 013784 I 2 ■866 483 032419 •532998:986275 •467 002 013725 2 3 •898902 032915 •519273 986164 •480727 013 836 3 4 •931817 033 446 •505437 986026 •494 563 013974 ^ 45 •965 263 034 005 •491 463 985 662 •508537 014338 45 6 •999 268 034 594 •477 125 985 195 •522875 014805 6 7 ro33 862 035214 •462 320 984 745 •537 680 015255 7 8 •069076 035871 •447 065 984351 •552 935 015 649 8 9 •104947 036569 •431416 983 943 •568584 016057 9 50 •141516 037315 •415359 983617 •584641 016383 50 I •178831 038 115 •398976 983331 •601 024 016669 1 2 •216946 038972 •382307 982 914 •617693 017 086 2 3 •255918 039887 •365221 982 366 •634779 017634 3 4 •295 805 040 866 •347 587 981775 •652413 018 225 4 44 HM. 5 PERCENT. LogarUhnis ami (Jo-l>{jarit]nnf< of 1)^., Nj , ami M,; with their Dijfircncrs. X LogD^ A (Logr;;^) Colog Dj. A (Colog ry^^r) LogNx A X 3-657495 ^•969579 4^342 5°5 0*030421 4^663329 958091 55 6 •627074 •968 948 •372926 •031 052 •621 420 956 974 6 7 • '196 022 •968 265 •403 978 •031 735 •578394 955 783 7 8 •564287 ■967 537 •435713 •032463 •534177 954503 8 9 •53^824 •966 683 •468176 ■0333^7 •488 680 953 136 9 6o •498507 •965726 •501 493 •034274 •441 816 951679 60 I ■464 2,5,5 •964 669 ■535 767 •035 33 f •393 495 9 "JO 124 I 2 •428902 •963 502 •571098 •036 498 •343619 948 468 2 3 •392 404 •962 216 •607 596 •037784 •292 087 946 704 3 4 •354620 •960 896 •645 380 •039 104 •238791 944816 4 65 ■3^5r>^6 •959527 •684 484 •040 473 •183607 942 785 65 6 ■275043 •958 100 •724957 •041 900 •126 392 940 584 6 7 •233 143 •956 5^5 •766857 •043415 •066976938 187 7 8 •189728 ■955 054 •810 272 •044 946 •005 163935549 8 9 •144782 •953 166 •855218 •046 834 3^940712932663 9 7° •097 948 •950 924 •902 052 •049 076 •873375929524 70 I •048872 •948 204 •951 128 •051 796 •802 899926 147 I 2 2-997 076 •944980 3'002 924 •055 020 •729046 922 ';66 2 3 •942 056 •941 246 •057944 ■058 754 •651 612 918823 3 4 •883 302 •937 282 •116698 •062 718 •570435 914930 4 75 •820 584 •933 843 •I794I6 •066 157 •485 365 910 728 75 6 •754427 •929967 •245573 •070033 •396 093 906 214 6 7 •684394 •925908 •315606 •074092 •302307 901331 7 8 •610 302 •921 705 •389698 •078 295 •203 638 895978 8 9 •532007 •916 800 •467 993 •083 200 •099 6 1 6 890 167 9 8o •448 807 •910954 '55^ 193 •089 046 2-989 783 883 995 80 I •359 761 •904 104 •640 239 •095 896 •873778 877648 T 2 •263 86.5 •897 181 •736135 •102 819 •751426 871097 2 3 -i6i 046 •889513 •838954 •110487 •622 523 864547 3 4 •050 559 •882510 •949441 •117490 •487 070 857 783 4 85 1^933 069 •876501 ^•o6693i •123499 •344 853 850307 85 6 •809 570 •871097 •190430 •128903 •195 160 841 454 6 7 •680667 •864 608 •319333 •135392 •036614 830847 7 8 •545 275 •860024 •454 725 •139976 r867 461 816 231 8 9 •405 299 •852 017 •594701 •H7983 •683 692 796124 9 90 •257316 •836474 •742 684 •163526 •479 816 770023 90 I •093 790 •815933 •906 210 •184067 •249839 734827 I 2 0^909 723 •790845 7^090 277 ■209155 0-984 666 681351 2 3 •700568 •745 389 •299432 •254611 ■666 017 599364 3 4 •445957 •671393 •554043 •328 607 •265381 460 653 4 95 '^n 35° •538673 •882 650 •461 327 i^726o34 172847 95 6 r656o23 •242 858 0^343 9*77 •757 142 2^898 881 6 7 2^898 88i rioi 1 19 7 45 6 PER-CENT. Logarithms and Co-logarithms ofDj., N^,, and M^, ; with their Differences. X Colog N^ A Log M.r A Colog M^ A X 55 5'33(S67i 041 909 3-329362 981 074 4-670 638 018926 55 6 •378580 043 026 •310436 980307 •689564 019693 6 7 •421 606 044217 •290 743 979 483 •709257 020517 7 8 •465 823 045 497 •270 226 978 621 ■729774 021379 8 9 •511 320 046 864 •248 847 977578 -75^ ^53 022422 9 6o •558 1S4 048 321 •22642^ 976403 •773 575 023 597 60 I •606505 049876 •202 828 975 108 •797172 024892 I 2 •656381 051 532 •177936 973 687 •822 064 026313 2 3 ■707913 053 296 •15] 623 972 128 •848377 027 872 3 4 •761 209 055 184 •J23751 970585 •876 249 029415 4 65 •816393 057 215 ■094336 969 028 •905 664 030972 65 6 •873 608 059416 -063 364 967451 ■936636 .032549 6 7 •933 024 061 813 •030815 965 809 •969 185 034 191 7 8 •994837 064451 2-996624 964 218 3-003376 035 782 8 9 4-059 288 067337 •960 842 962 176 •039 158 037824 9 70 •126625 070476 •923 018 959 706 •076982 040 294 70 I •197 lOI 073 853 •882 724 956 641 •117 276 043 359 I 2 •270954 077434 -839365 952976 •160635 047 024 2 3 •348388 081 177 •792341 948 709 •207 659 051291 3 4 ■429 .565 085 070 •741 050 944229 •258950 055 771 4 75 •5^-^ 63s 089 272 •685279 940556 •314721 059444 75 6 •603907 093 786 •625 835 936405 •374^65 063 595 6 7 •697 693 098 669 •562 240 932 117 •437 760 067 883 7 8 •796362 104 022 •494357 927753 •505 643 072247 8 9 •900384 109833 •422 no 922 615 •577 890 077385 9 80 3^oio 217 1 16 005 ■344725 916416 '655^75 083 584 80 I •126222 122 352 •261 141 909 085 •738859 090915 I 2 •248574 128903 •170 226 901 753 •829774 098 247 2 3 ■377477 ^35 453 •071979 893 598 -928 021 106402 3 4 •512930 142217 1-965 577 886 297 2-034423 113 703 4 85 '655 147 149 693 •851874 880 250 •148 126 ^19750 85 6 •804 840 ^58546 •732124 875 043 •267876 124957 6 7 •963 386 169 ^53 •607 167 868 746 •392 833 ^31254 7 8 2-132539 183 769 •4759^3 864 883 •524087 ^35 ^^7 8 9 •316308 203 876 -340 796 857483 •659 204 142517 9 90 •520 184 229977 •198279 842 048 •801 721 157952 90 I •750 j6i 265 173 •040327 82 1 603 •959 673 178397 I 2 1-015 334 318649 ©•86i 930 796917 1-138070 203 083 2 3 -333983 400 636 •658847 75^35^ •341 ^53 248 649 3 4 •734619 539 347 •410 198 677 053 •589 802 322947 4 95 0^273966 827 153 •087251 543 768 •912749 456232 95 6 i^ioi 119 r63i 019 246672 0^368 98 1 753328 6 7 ^•877 69i ri22 309 7 40 HM. 5 PER-CENT. ]'alues of Annuities, and Sinrjlc and Annual Preniiutns Jor Assuranrr of a Unit. X Ox A. IBj. X f'x A. t^x 10 i7-oj;6 8 •140 152 •007 762 55 10-1352 ■469751 •042 186 I 16-9978 •142 960 •007 943 6 9-8707 •482 350 •044372 2 16-919 2 •146704 •008 187 7 9-602 2 ■495 132 -046 701 .3 16-8243 •151 221 •008484 8 9-3302 •508088 -049185 4 16-7 t; 2 •156323 •008 823 9 9-054 3 •521 223 -051 841 15 i6-6oi 7 •161824 •009 194 60 8-7763 ■534464 -054 670 6 16-482 -J67525 -009583 I 8-4969 ■547 766 •057 678 7 16-3624 •173217 •009977 2 8217 I -561 092 -060875 8 16-2475 •178690 -010360 3 7-9375 ■574406 -064 269 9 16-142 •183 7^3 -010717 4 7-6590 •587667 •067 868 20 16-047 X •188234 -Oil 042 65 7-3806 -600924 •071 704 I 15-9568 •192536 •oil 355 6 7-1015 •61421 ^ •075815 2 15-8680 •196 761 -oil C:)C)r^ 7 6-8208 -627583 -080 246 3 15-7762 -201 131 -on 989 8 6-5378 -641 055 •085 045 4 ^^6/7 9 •205 816 -012341 9 6-250 7 -654727 •090 298 25 15-5718 •210 867 •012724 70 5 ■962 5 -668453 -096 008 6 15-4595 •216 214 ■013 136 I 5-675 8 -682 IO-; -102 176 7 15-341 7 -221 822 ■013574 2 5-394 7 -695 488 -108759 8 15-2208 -227 580 •014030 3 5-1234 -70841 1 • 1 1 5 690 9 15-0973 •233462 •014503 4 4-865 6 -720688 -122868 30 14-9709 •239482 •014995 75 4-6215 -732310 -130270 I 14-841 8 -245 630 •015505 6 4-3819 ■743717 -138 188 2 14-708 2 •251989 -016042 7 4-1487 ■754824 -146604 3 14-5698 •258580 -016608 8 3-9205 ■765 693 ■155 614 4 14-426 I •265423 -017 206 9 3-6950 -77(^43^ -165376 3.'; 14-2773 •272508 -017837 80 3-4752 •786897 ■175836 6 14-1239 •279814 •018501 I 3-2660 ■796857 •186 792 7 13-9664 •287313 -019 197 2 3-0730 •806048 •197901 8 13-8049 •295007 -019926 3 2-893 9 -814578 •209 196 9 ^2>-(y?>'^?, •302940 -020 695 4 2-7322 -822 277 •220320 40 13-4660 •311 142 -02 'i 509 85 2-5810 -829478 •23 1 6;^'') I 13-2865 •319688 •022377 6 2-4299 -836671 ■243 934 2 13-0987 •328632 -023309 7 2-269 6 ■844 305 ■258 230 3 12-902 9 •337957 -024308 8 2-099 8 •8 ■52 389 •274979 4 12-7005 -347 S9^ -025371 9 1-8984 •861 980 -297396 4.'; 12-4914 •2>^1 SS"^ •026 502 90 1-669 2 •872897 •327030 6 12-2779 •367 720 -027 694 1 1^4324 •884173 -3<^3 506 7 12-060 8 -378058 •028946 2 1-1883 ■895 793 •409 347 8 11-8398 •388583 -030 264 3 •923 5 •908 404 -472 260 9 11-6139 ■399339 -03 I 659 4 -6598 •920961 •554856 50 11-3831 •410331 •033 136 95 •406 I •93,3041 ■^^63 544 I II-U5 9 •421 622 ■034713 6 -1749 •944051 -803 498 2 10-901 6 •433 25^ •036 403 7 •952381 -952381 3 10-651 2 •445 '82 -038 209 4 10-395 7 ■457347 ■040 133 SIX PER-CENT. •IM 6 I'KKC'KNT. Commutation 7a/ile. X Hx Nx Sx Mx Ux X lO I 2 3 4 55 8 59-5 52 420-6 49 256" I 463*13-8 43 564-9 824 105-6 771 685-0 722 428-9 676 1 15" I 632550-2 » I 933 387 1 1 109 282 10337597 9615 168 8 939 053 6031-27 5773-14 5575-84 5421-59 5294-21 1 154662-1 148630-8 142857-7 137 281-9 131 860-3 10 I 2 3 4 15 6 7 8 9 40 985-4 38554-5 36254-1 34069-2 31986-7 591 564-7 ^r^_:^ 010-2 516756-2 482 687-0 450 700-3 8306503 7 714 938 7 161 928 6 64 5 172 6162485 5 180-72 5069-71 4951-61 4818-83 4664-81 126566-0 12 1 385-3 116315-6 I I I 364-0 106545-2 15 6 7 8 9 20 I 2 .3 4 30002-8 28 125-4 26 354-9 24693-0 23137-7 420 697-5 392572-1 ^66 217-2 341524-2 318386-5 5 711 784 5 291 087 4898515 4532298 4190774 4491-45 4312-31 4 133-87 3963-7^ 3806-13 loi 880-4 97388-9 93 076-60 88942-73 84979-02 20 I 2 3 4 25 6 7 8 9 21 683-1 20320-r 19 041-8 J 7 839-9 16 709-4 296703-4 276383-3 257341-5 239501-6 222 792-2 3872387 3575684 3 299 300 3 041 959 2802457 3 661-21 3 525-59 3 397-43 3 273-40 3152-70 81 172-89 77511-68 73 986-09 70588-66 67315-26 25 6 7 8 9 30 I 2 3 4 15646-4 14 646-8 13 708-3 12 827-6 12 OOI-2 207 145-8 192 499-0 178790-7 165 963-2 153962-0 2579665 2372519 2 180020 2 001 230 I 835 266 3 o?,S'S3 2921-53 2 812-13 2707-32 2 607-06 64 162-55 61 127-03 58 205-49 55 39y3^ 52 686-04 30 I 2 3 4 35 6 7 8 9 II 225-6 10497-3 9812-89 9 169-84 8566-18 142 736-4 132 239- r 122426-2 113 256-4 104 690-2 I 681 304 1538568 1406329 I 283 903 I 170646 2510-78 2417-86 2 327-66 2 240-05 2 155-44 50078-98 47 568-20 45 150-34 42822-68 40582-63 3S 6 7 8 9 40 I 3 4 7 999-83 7 469-23 6972-55 6507-27 6 070-63 96690-35 89 221-12 82 248-57 75741-30 69 670-68 I 065 956 969 265-8 880044-7 797796-2 722054-9 2073-97 I 996-19 1 922-30 I 8151-69 I 783-38 38427-18 34357-02 32434-72 30583-03 40 I 2 3 4 45 6 7 8 9 5 660-82 5275-29 4912-29 4570-73 4249-75 64 009-86 58734-57 ^r^ 822-28 49251-55 45 001-81 652 384-2 588374-3 529 639-8 475 817-5 426565-9 ] 717-20 I 652-09 I 587-69 1524-19 I 461-92 28 799-65 27 082-45 25 430-36 23 842-68 22318-49 45 6 7 8 9 50 I 2 3 4 3948-18 3 665-28 3400-17 3 151-42 2917-74 41 o^y6^ 33988-17 30836-76 27 919-02 381564-1 340510-5 303 I 2 2- 1 269 134-0 238297-2 I 400-90 1341-49 I 283-85 1227-56 I 172-26 20 856-56 19455-66 18 114-17 16830-32 15602-76 50 1 2 3 4 49 6 PER-CENT. Commutation Table — (continued). 6 7 8 9 6o I 2 3 4 65 6 7 S 70 I 2 3 4 75 6 9 80 3 4 85 6 90 I 2 3 4 95 6 D, 2 698"26 491-99 2 298- 1. "J 2 1 1 6-05 I 945- 1 i I 784-48 16.33-51 I 491-68 I 358-50 1 116-71 1007-74 906-428 812-460 72J-669 645*339 570-947 501-976 438-073 379-033 324-968 276-419 233-034 194-630 160-990 131-669 106-248 84-393 o 65-973 7 50-672 o 38-296 7 28-5460 21-014 15-241 I 10-9377 7-7059 5'238 3-3962 2-07 I-I45 5 •5324 •1823 •03 1 6 N. 220-76 728-77 430-62 314-57 369-45 584-97 95 1 '46 45978 101-28 867-718 751-011 743-268 83'6-S40 024-380 298-710 653-37^ 082-424 580-448 142-375 763-343 438-374 161-955 928-922 734-291 573-30^ 441-633 335*386 250-993 185-019 134*347 96-050 3 67*5043 46-489 4 31-2483 20-310 6 12-604 7 7 '366 s 3-9703 1-891 9 ■7464 •2139 •03 1 6 •000 o Sx 210 ^85 162 141 123 07 92 79 68 58 49 41 34 29 23 19 16 12 10 378-2 157-4 428-7 998-0 683-5 314-0 729 05 777*59 317-81 216-54 348-82 597-81 854-54 017-70 993-32 694-61 041-24 958-82 378-37 235*992 472-649 034-275 872-320 943-398 209-107 635*805 194-172 858-786 607-794 422-775 288-428 192-378 124-874 78-384 47-136 26-825 14-220 6-854 2-884 -99 •246 •032 -000 M, 117-94 064-40 011-61 959-601 908-445 857-911 807-950 758*577 709-835 661-788 614-761 569-007 524*734 482-073 441-270 402-016 364' ^53 327*499 292-010 257*766 225-157 195-002 167-263 142-050 119-426 99-2175 81-2495 65-408 9 51-7666 40-199 2 30-692 23-109 2 17-1938 12-609 6 9-1689 6-55^3 4-5247 2-9793 1*853 7 1-0384 •4902 -170 2 -029 8 14430 13 312 12 248 11 236 10 276 50 •16 55 95 9 368-502 8510-591 642 065 2.^0 6944 6334 5 572' 4 957' 4388 3863 3381 2940 2538 2 174 1846 1554 1297 T 071 876 709 567 448- 349* 267- 202- 150- no- 79* 56- 39' 26' II' 6' 3' I 442 681 ^75 941 868 598 582 ■429 ■931 920 ■^55 998 9964 7338 6841 2579 ■0405 7910 ■382 I 6155 ■4163 ■7242 •615 o ■421 1 ■8115 •642 6 •0863 5616 •582 3 •7286 •690 2 •200 o -029 8 55 6 9 60 I 2 3 4 (^5 6 70 I 2 3 4 75 6 80 I 2 3 4 85 6 7 8 90 I 2 3 4 95 6 .)(» 6 1'i:k-cknt. Li'tgarithms mid Co-lixjarithms f>fDj., X^, (inJ M,.; in't/t thiiii' Differences. X LogD^ A (Log vp^) Colog Dj. A (Colog rp^) LogN^ A X lO 4.746941 1^972 561 y253 059 0-027439 5'9'5 9^^ 971457 10 I •719 .';o2 •972958 •280498 •027042 •887440971355 I 2 •692 460 •973 250 •307 540 •026 750 •858795971 226 2 .3 •665 7 1 •973427 •334 290 ■026 573 •830021 971 074 3 4 •639^37 •973 493 •360 863 •026507 •801 095970907 4 15 •612 630 •973 445 •387370 ■026555 •772002970731 15 6 •586075 •973 282 •413 925 •026 718 •742733970553 6 7 •559357 •973 005 •440 643 •026995 •713286^970380 7 8 •532362 •972 608 •467638 •027 392 •683 666j97o 222 8 9 •504970 •972 192 •495 030 •027 808 •653888970082 9 20 •477 162 •971936 •522838 •028064 •623 970 969 949 20 I •449 098 •971 764 •550902 •028 236 •593 919969820 I 2 •420862 •971 712 •579138 •028 288 •563 739 969 683 2 3 •392574 •971 746 •607426 •028 254 •533422969533 3 4 •364320 •971 801 •63^ 680 •028 199 •502955969368 4 25 •336 121 •971805 •663 879 •028 195 •472323969189 25 6 •307926 •971 781 •692 074 •028 219 ■441 512968998 6 7 •279707 •971686 •720293 •028314 •410 510968 798 7 8 •251393 •971568 •748607 •028432 •379308968592 8 9 •222 961 •97M54 •777039 •028 546 •347900I968376 9 30 •194415 •97^327 •805585 •028673 •316276 968 153 30 J •165742 •971 242 •834 258 •028758 •284429 967 916 I 2 •136984 •971 160 •863016 ■028 840 -252345 967 667 2 3 •108 144 •971 081 •891 856 •028 919 •220 012 967 402 3 4 •079225 •970 985 •920775 •029015 •187414 967 121 4 2>S •050 210 •970867 •949 790 •029 133 -'5-^535 966 825 35 6 •021 077 •970 720 •978923 •029 280 •121 360 966514 6 7 3-991 797 •970565 4-008 203 •029435 •087874 966188 7 8 •962 362 •970425 •037638 •029575 •054 062 965 844 8 9 •932787 •970294 •067 213 •029 706 •019906 965477 9 40 •903 081 •970 195 •096919 •029 805 4^985383 965 085 40 I •873276 •970 116 •126 724 •029884 •950468 964 660 I 2 •843 392 •970007 •I •56 608 •029993 •915 128 964 205 2 3 •813399 •969 834 •186 601 •030 166 •879333 963717 3 4 •783 233 •969 647 •216 767 •030353 -843 050 963 197 4 45 •752880 ■969 366 •247 J 20 •030 634 •806 247 962 647 45 6 •722 246 •969 038 •277754 •030962 •768894 962 068 6 7 •691 284 •968 702 •308716 •031 29S •730962 961458 7 8 •6^9986 •968377 •340014 •031623 •692 42c 960810 8 9 •628 363 •968033 •371637 •031967 •653 23c 960 122 9 50 •596396 •967712 •403 604 •032 288 ■613352 959384 50 I •564 108 •967 393 •435 892 •032 607 •572 736 958592 I 2 •531 501 •967 005 •468 499 •032995 •531328 957741 2 3 •498 506 •966 540 •501 494 -033 460 •489 069 956 83 1 3 4 •465 046 ■966 03 J •534 954 -033 962 •445 90c 955 858 4 51 6 PER-CENT. Logarithms and Co-logaritlims^ ofldxi N^., and M^; with tlieir Differences. X CologN^ A LogM, A Colog M,. A X lO 6-084017 028 543 3^780408 981 004 ? 2 19 592 018996 10 I •112 560 028 645 •761 412 984 899 •238588 015 lOI 1 2 •141 205 028774 •746311 987816 •253 689 012 184 2 3 •169979 028926 •734127 989674 •265 873 010 326 3 4 •198905 029 093 -723801 990589 •276199 009 41 1 4 15 •227998 029 269 -714390 990593 •285610 009407 15 6 •257267 029 447 -704983 989 764 •295017 010 236 6 7 •286714 029 620 •694747 988 195 •305 253 on 805 7 8 •316334 029778 -682 942 985 892 •317058 014 108 8 9 •346112 029 918 -668 834 983 553 •331 166 016447 9 20 •376030 030051 -652387 982323 •347613 017677 20 I •406081 030 180 •634710 981 647 ■3^5 29° 018353 I 2 •436261 030317 ■6163^7 981745 •383 643 018 255 2 3 •466578 030467 -598 102 982382 •401 898 017 618 3 4 •497 045 030632 •580484 983 141 •419516 016859 4 ^S •527677 030 811 •563 625 983 606 •436375 016394 25 6 •558 488 031 002 •547231 983920 •452 769 016080 6 7 •589 490 031 202 •53^ ^5^ 983 849 •468 849 016 151 7 8 •620 692 031 408 •515000 983 683 •485 000 OI63I7 8 9 •652 100 031 624 •498 683 9^3 55^ •50^317 016449 9 30 •683 724 031847 -482 234 983377 •517766 016623 30 I •7^5571 032 084 •465 61 1 983 425 •534389 016575 I 2 •747 ^55 032333 •449 036 983 503 •550964 016497 2 3 •779988 032 598 •432539 983 612 •567461 016388 3 4 •812586 032879 •416 151 983 658 •583 849 016342 4 35 ■845 465 033 175 •399809 983 623 •600 191 016377 35 6 ■878640 033 486 •3^3 432 983 488 •616568 016512 6 7 ■912 126 033812 •366920 983338 •633 080 016662 7 8 •945 938 034 1^6 •350258 983 279 •649 742 016 721 8 9 •980 094 034523 '333 537 983 266 •666 ^63 016734 9 40 "5-014617 034915 •316803 983400 •683 197 016600 40 I "•049532 035 340 •300203 983 618 •699 797 016382 I 2 •084872 035 795 •283821 983 747 •716179 016253 2 3 •120667 036283 •267568 983 676 •732432 016324 3 4 •156950 036 803 •251 244 983 576 •748 756 0:6424 4 45 •193 753 037353 •234 820 983213 •765 180 016787 45 6 •231 io6 037932 •218033 982 732 •781967 017 268 6 7 •269038 038542 •200 765 982 274 •799235 017 726 7 8 •307 580 039 J90 -183039 981 886 •816961 018 114 8 9 •346 770 039878 -164925 981483 •835 °75 018517 9 5° •386648 040 616 -146408 981 181 •853 592 018819 50 I •427 264 041 408 •127589 980826 -872 41 1 019174 1 2 •468672 042 259 •108415 980 627 •891 585 019373 2 3 •5 TO 93 I 043 169 •089 042 979982 •910958 020018 3 4 •554 100 044 142 [ ^069 024 979 394 •930 976 020 606 4 r.2 6 I'Kli-CKNT. Logaridnns (uiJ C'o-h>i/(trit/n/is o/'Dj, N^, ami ^^^'j with their Dijj'crcncai. X Log IV A (Logr/;^) Colog D^ A (Cologtyjj.) Log N., A X ss 5-431 084" r-965 462 +■568916 0-034538 4-401 758 ^54818 ss 6 •39'5 .H''' •964832 •603 454 •035 168 •356576 953 706 6 7 •361378 •964 148 •638622 •035852 •310 282 952515 7 8 •3255^6 •963 420 674474 •036580 •262 797 951 237 8 9 •288946 •962 567 •71 J 054 •037 433 •214034 949872 9 6o •2515^3 •961 610 ■748487 •038 390 • 1 6^ 906 948413 60 I •213 123 •960552 •786877 •039448 • 1 1 2 3 1 9 946857 I 2 •173675 •959385 ■826325 •040615 •059176 945 200 2 3 •133 060 958 100 •866940 •041 900 •004376 943 436 3 4 •091 160 •956779 •908 840 •043221 3^947812 941 546 4 65 •047 939 ■955 41 1 ■952 061 •044 589 •889358 939512 65 6 •003350 ■953 983 •996 650 ■046017 ■828 870 937308 6 7 2-957333 •952469 3-042 667 •047 5i 1 •766178 934 905 7 8 •909 802 •950937 090 198 •049063 •701 083 932 255 8 9 •860 739 •949 049 •139 261 •050951 '^33 33^ 929356 9 70 •S09788 •946 808 190 212 •053 192 •562 694 926 198 70 1 •756 596 .944087 •243 404 •055913 ■488892 922 803 1 2 •700683 •940 864 •299317 •059 1^6 •411695 919 201 2 .3 •641547 •937 129 •358 453 •062 871 •330896 915441 3 4 •578676 •933 165 •421324 •066835 •246337 911535 4 75 •511 841 •929727 •488159 •070273 •157872 907317 75 6 •441 568 925851 558432 •074149 065 189 902 790 6 7 •367419 •921 791 •632581 •078 209 2-967979 897 889 7 8 •289 210 917588 •710790 •082412 •865868 892515 8 9 •206 798 •912684 •793 202 •087316 •758383 886679 9 80 •J 19 482 •906837 •880518 •093 ^(^3 645 062 880483 80 I •026 319 •899 988 •973681 •100 012 •525 545 874 116 1 2 1-926307 •893 064 2-073 693 •106936 •399661 867 ^:,s 2 3 •819371 •885 397 •180629 •114603 •267 216 861 012 3 4 •704768 •878393 •295 232 •121 607 •128228 854271 4 8.^ •583 ^6i •872384 •416839 •127 616 1^982499 846832 85 6 •455 545 •866 98 1 •544 455 •133019 •829331 838023 6 7 •322526 •860491 •677474 •139509 •667354 827472 7 8 • 1 83 1 7 •855 908 -816983 •144092 •494826 812897 8 9 •038925 •847 900 961 075 •152 100 •307 723 792 810 9 90 0-886 825 •832357 I-XI3 175 •167 643 ■100533 766728 90 I •719 182 81J 817 280818 -188183 0867 261 73 1 558 I 2 •530 999 •786728 •469 001 •213272 •598819 678072 2 3 ■317 727 •741 272 •682273 •258 728 •276891 596056 3 4 •058999 •667278 •941 001 •332722 1872 947 457 287 4 95 r726 277 'S3A5S^ 0-273 723 •465 444 •330 234 1 69 340 95 6 •260833 238741 •739167 •761 259 •499574 6 7 2 "499 574 ■500426 7 53 6 PER-CENT. Logarithms and Co-hrjari'thms of D^., N^, and M^.; with their Differences. X CologN^ A Log M^ A CologM^ A z 55 5-598 242 045 182 3-048418 978 686 4-951582 021314 55 6 •64,5 424 046 294 •027 104 977910 -972896 022090 6 7 •689718 047 485 •005014 977077 -994986 022923 7 8 •737 20,3 048 763 2-982 091 976 208 3-017909 023 792 8 9 •785 966 050 128 -958 299 975 143 -041 701 024857 9 6o •836094 051587 •933 442 973 942 •066558 026058 60 I •887681 053 143 •907 384 972 616 -092 616 027 384 I 2 •940 824 054 800 -880 000 971 158 •120 000 028 842 2 3 ■995 624 056564 •851 158 969561 •148842 030439 3 4 4^052 188 058 454 •820 7191967 987 •179 281 032013 4 65 •110642 060 488 •788706 966 41 1 •211 294 033 589 65 6 •171 130 062 692 •755 "7 964 822 •244883 035178 6 7 •233 822 065 095 •719939 963 174 •280061 036 826 7 8 •298917 067 745 •683113 961 592 •316887 038408 8 9 •^66 662 070 644 •644 705 959 53^ •355 ^95 040 462 •9 70 •437306 073 802 •604 243 957 041 '395 757 042 959 70 I •511 108 077 197 •561 284 953 926 •438716 046 074 1 2 •588305 080 799 •515210 950 188 •484 790 049 8 1 2 2 3 ■669 104 084559 •465 398 945 828 •534602 054172 3 4 •753663 088 465 •411 226 941 259 •588774 058741 4 75 •842 128 092 683 •352485 937 553 ■6'^7 5'5 062 447 75 6 •934811 097 210 •290 038 933361 •709962 066 639 6 7 3-032 021 102 III •223 399 929041 -776601 070959 7 8 •134 132 T07485 •152440 924 660 •847 560 075 340 8 9 -241 617 113 321 •077 100 919488 •922 900 080512 9 80 •354938 119517 ^996588 913 233 2^003 412 086767 80 I ■474455 125884 •909 821 90:5 816 •090 179 094 1 84 I 2 -600339 132445 •815637 898413 •184363 101587 2 3 •732784 138988 -714050 890 167 •285950 109833 3 4 •871772 145 729 -604217 882810 •395 783 117 190 4 85 2^017 501 153 168 -487 027 876758 •512973 123 242 85 6 •170669 161977 •363 785 871 588 •636215 128412 6 7 •332646 172528 •235373 865 330 •764627 134670 7 8 •505 174 187 103 -100703 861 614 -899 297 138386 8 9 •692 277 207 190 0-962317 854340 1-037683 145 660 9 90 •899 467 233272 -816657 838935 •183343 161 06^ 90 I 1-^32739 268442 ■655 59^ 818518 •344 408 181482 1 2 •401 181 321928 -474 no 793 920 •525 890 206080 2 3 •723 109 403944 -268030 748 346 •731970 251654 3 4 0-127053 5427^3 -016376 674001 •983 624 325 999 4 95 •669 766 830 660 T-690377 540619 0-309 623 459381 95 6 r-500426 •230996 243 272 -769 004 756 728 6 7 2-474268 ^•525732 7 .M HM. 6 PERCENT. Values of Annuities, and Singh and Annual Premiums for AsKuranri- of a Unit. X "x A. 'Wj. X Oj- A. iPj. lO '47585 •108 01 1 -006854 55 9-347 •414318 •040 042 I 147210 - 1 1 131 -007 005 6 9-1207 •427 128 •042 203 2 14-6668 •I 13 201 -007 226 7 8-890 •440 186 •044508 3 14-5986 -117 062 -007 505 8 8-655 1 •453 487 •046 969 4 14-5J9 7 •121525 •007 830 9 8-4157 -467 038 •049 602 15 14-433 5 •126404 -008 190 60 8-1732 •480 761 -052409 6 14-343 6 -131495 -008570 1 7-9286 •494 608 -055 396 7 14-253 7 •136581 •008 954 2 7-6825 •508 539 -058571 8 14-1679 •141443 •009325 3 7-435 6 -522513 •061 941 9 14-090 2 -145836 •009 664 4 7-1887 •536487 '°^5 5'5 20 14-021 9 •149701 •009 965 65 6-941 •550512 •069326 I 13-9579 •iS^i^'S •010 250 6 6-6915 •564 635 -073411 2 138956 -156854 -010530 7 6-439 4 •578903 -077 816 3 13-8308 •160520 -010 823 8 6-1842 •593 350 •082 591 4 13-7605 -164499 •oil 145 9 5-923 8 ■608087 -087 826 ^5 13-6836 •168 851 •on 499 70 5-661 2 •622953 -093 520 6 13-601 5 •173502 •on 883 I 5-398 8 •637804 -099676 7 13-5146 •178420 •012 292 2 5-1406 -6';2 42o -106 247 8 13-4250 -J85488 •012 720 3 4-890 4 -666578 -113 i63 9 iyi33 3 -188678 •013 164 4 4-652 2 -680 063 -120318 3° 13-2392 •194008 •013625 75 4-426 2 -692 857 •127 687 I J3-I428 - 1 99 466 •014 104 6 4-203 6 -705456 •135571 2 13-042 5 -205 141 •014 608 7 3-9862 -717762 •143949 3 12-938 -211 055 •015 142 8 3-7728 •729844 •152919 4 12-8289 •217233 •015 709 9 3-561 I •741825 -162 641 35 12-7152 •223 66s •016308 80 3-354 I -753 540 -173063 6 12-597 5 •230332 •016939 I 3-1566 •764719 -183975 7 12-476 I •237204 -017 602 2 2-9741 -775052 -195 026 8 12-3510 •244285 •018 297 3 2-8044 •784655 •206 247 9 12-221 3 •251 622 •019032 4 2-6513 -793322 •217271 40 12-0866 •259 252 •019 81 T 85 2-508 I •801431 •228454 I 11-9452 •267 256 •020 645 6 2-3648 -809 542 •240595 2 1 1 -796 I •275 695 •021545 7 2-212 2 -818177 •254 708 3 11-6395 -284557 •022513 8 2-0503 -827344 •271237 4 11-4767 -293 772 •023 546 9 1-8569 -838286 •293421 45 11-3075 •303 348 •024647 90 1-6357 •850 809 •322 800 6 11-1339 ■3^3 n 5 •025 810 I 1 -406 3 -863 794 •358971 7 10-9567 •323207 •027 032 2 1 - 1.69 -877 225 •404435 8 10-7754 •333 467 •028319 3 •9103 ■891 873 •466 886 9 10-5893 •344 003 -029 683 4 -6516 -906519 •548 889 50 10-398 J •354 823 •031 130 95 •401 8 -920 66i ■6^6 794 I I0-200 7 •366 000 •032677 6 -1733 -933 605 '795 725 2 9-996 -377497 -034 330 7 •0000 -943 396 •943 396 3 9-7850 •389526 •036 117 ^ 9-568 7 -401 770 •038 015 ONE LIFE. II. HEALTHY FEMALE LIVES. Elementary Values. X /x Log Ij: hogpj, A Colog Ix X lO 1 00 000 5'ooo 000 7-998 634 999 533 yoooooo 10 I 99 686 4-998 634 •998 167 999 595 •001 366 I 2 99 266 •996801 •997 762 999676 •003 199 2 3 9« 75'^^ •994 563 •997438 999 757 •005437 3 4 98 1 75 •992 001 •997 195 999 825 •007 999 4 '5 97 543 •989 196 •997 020 999 907 •010 804 15 6 96876 •986 2 ] 6 •996927 999992 •013 784 6 7 96 193 •983 «43 •996919 000 075 •01*6857 7 8 95513 •980 062 •996994 000 088 •019 938 8 9 94854 •977056 -997 082 999921 •022944 9 20 94219 •974 138 •997 003 999820 •02"^ 862 20 I 93571 •971 141 •996823 999722 •028 859 1 2 92889 •967 964 •996545 999 607 •032 036 2 .3 9^ ^S^ •964 509 •996 152 999552 •035491 3 4 91340 •960661 •995 704 999 574 •039339 4 ^5 90441 '9S^3^5 •995 278 999 752 •043 (^35 25 6 89463 •951643 •995 030 999 808 •048357 6 7 88445 •946 673 •994838 999913 •053327 7 8 87400 ■941 511 •99475^ 000028 •058489 8 9 86350 •936262 •994779 000 044 •063 738 9 30 85318 ■931 041 •994 823 000 062 •068 959 30 I 84307 •925 864 •994 885 000061 •074 136 I 2 83320 •920 749 •994946 999 962 •079251 2 3 82356 •915695 •994908 999972 •084305 3 4 8 J 396 •910603 •994 880 999 982 •089397 4 35 80442 •905 483 •994 862 999939 •094517 3^ 6 79496 •900 345 •994 801 999 93 7 •099 ^S^ 6 7 78550 •895 h6 •994 738 999 935 •104854 7 8 77604 •889884 •994673 999 935 -no 116 8 9 76658 •884557 •994 608 999 908 •1^5 443 9 40 75712 •879 165 •994516 999912 •120835 40 I 74762 •873 681 •994428 999916 •1263 19 I 2 73 809 •868 109 •994 344 999908 •131 891 2 3 72854 •862 453 •994 252 999 897 ■137 547 3 4 71 896 •856 705 •994 149 999 896 •143 295 4 45 70934 •8^0854 •994 045 999 937 •149 146 45 6 69 968 •844 899 •993982 999 946 •155^01 6 7 6900!^ •838881 •993 928 999 947 •161 119 7 8 68 047 •832809 •993 875 999 93 1 •167 191 8 9 67094 •826684 •993 806 999 872 •173 316 9 50 66 144 ■820490 •993 678 999777 •179510 50 I 65188 •814 168 •993 455 999 708 -185832 I 2 64213 •807 623 •993 ^^3 999 ^S3 -192377 2 3 6^ 210 •800 786 •992 816 999 S^^(^ •199 214 3 4 62173 •793 602 •992382 999 611 •206398 4 57 HP. Elementary Values. X Colog;;^ A d. Logf/^ A —A X lO o-ooi 366 000 467 3^4 2^496930 126319 873681 10 I •001 8,33 000 405 420 •623 249 084321 915679 I 2 •002 238 000324 5^0 ■70757° 056 606 943 394 2 3 •002 c^62 000 243 58 i •764176 036541 9^^ 459 3 4 •002 805 000175 632 •800 717 023 409 97659^ 4 15 •002980 000 093 667 •824 126 010 291; 989 705 ■15 6 •003073 000008 683 •834421 998 088 001 912 6 7 •003 081 999 9^$ 680 •832509 986376 013624 7 8 •003 006 999912 659 •818885 983 889 016 III 8 9 •002 918 000079 ^iS •802 774 008 801 991 199 9 20 •002 997 000 180 648 •811 575 022 209 977791 20 I ■003 177 000 278 682 •833 784 03 2 894 967 106 I 2 ■003 455 000393 736 •866678 043413 956587 2 3 •003 848 000448 813 •910 091 043 669 956331 3 4 •004 296 000426 899 •953 760 03*5579 963421 "^ ^5 •004 722 000 248 978 •990339 017 409 982 !;9i 25 6 •004970 000 192 IOI8 3-007 748 on 368 988632 6 7 •005 162 000 087 1045 •019 116 002 073 997927 7 8 •005 249 999972 1050 •021 189 992491 007 509 8 9 •005 221 999 9S^ 1032 •0J3 680 991 071 008929 9 30 •005177 999938 lOII •004751 989566 010434 30 I •005 1 15 999 939 987 2-9943x7 989 760 010 240 I 2 •005 054 000038 964 •984077 998 194 001 806 2 3 •005 092 000028 960 ■982 271 991 277 002 723 3 4 •00 j 120 000018 954 •979548 99*5 343 003657 4 35 •003 138 000 06 r 946 •975891 000 000 000 000 35 6 •00 5 199 000 063 946 •975891 000 000 000 000 6 7 •005 262 00006^ 946 •975891 000 000 000 000 7 8 •005327 000 0615 946 •975891 000 000 000 000 8 9 •005 392 000 092 946 •975891 001 833 998 167 9 40 •005 484 000 088 950 •977724 001 369 998 63 1 40 I •005 572 1 000 084 953 •979093 000910 999 090 I 2 •005 6j6 000 092 955 •980 003 001 363 998637 2 3 •005 748 000 103 958 •981366 001 809 998 191 3 4 •005 851 000 104 962 •983 175 001 802 998 198 4 45 •005 955 000 063 966 •984977 998 649 001 351 45 6 •006 018 \ 000 054 9'53 •983 626 997 740 002 260 6 7 •006072 000 053 958 •981 366 997 727 002 273 7 8 •006 125 000 069 953 •979 093 998 63 1 001 369 8 9 •006 194 000 128 950 •977 724 002 734 997 266 9 50 •006322 000 223 956 •980458 008 547 991453 50 I •006 545 000 292 975 •989 005 012 296 987704 I 2 •006 837 000347 1003 3^001 301 014478 985522 2 3 •007 184 000434 1037 •015 779 018 047 981953 3 4 •007618 000389 1081 •033 826 013838 986 162 4 58 Elementary I alucs — (continued). X /, Log/x hogps A Colog If X ss 61 092 4-785984 T99I 993 999 644 y2i4oi6 55 6 59 97<5 111 911 •991637 999 639 •222 023 6 7 5««32 •769614 •991 276 999621 •230386 7 8 ^57662 •760 890 •990 897 999 530 •239 no 8 9 56466 •75' 787 •990427 999165 •248213 9 6o 55^35 •742214 •989 592 999 026 •257 786 60 I 53927 •731 806 •988618 998 843 •268 194 I 2 52 532 •720424 •987 461 998697 •279576 2 .3 5^037 •707 883 •986 i-jS 998 590 •292 1 15 3 4 49436 •694 043 •984 748 998 709 •305 957 4 65 47 730 •678791 •983 457 998 734 •321 209 65 6 45 946 •662 248 •982 191 998538 •337 752 6 7 44 100 •644 439 •980 729 998372 •355 561 7 8 42 186 •625 168 •979 101 998 170 •374832 8 9 40 204 •604 269 •977271 997 865 •395731 9 /O 38154 •581540 •975 136 997 092 •418 460 70 I 36 03 1 •556676 -972228 996 64 1 ■443324 1 2 33 799 •528904 -968 869 996 295 •471 096 2 .3 31461 •497 773 •965 164 995 898 •502 227 3 4 29036 •462937 •961 062 995 657 '531 063 4 75 26 546 •423 999 •956719 995 567 •576001 75 6 24028 •380 718 -952 286 997219 •619 282 6 7 21 528 •3^i 004 •949 505 997412 •666 996 7 8 19165 •282509 •946917 997 891 •717491 8 9 16960 •229426 •944 808 998 793 •770574 9 8o 14936 •174234 •943 601 999 III •825 766 80 I 13 117 •117 835 •942712 995 959 •882 165 I 2 II 496 •060 547 •938671 993 162 •939 453 2 3 9982 3-999218 ■93 J 833 990 997 4-000 782 3 4 8532 ■931051 -922 830 987988 -068 949 4 85 7 143 •853 881 •910818 985 ^33 •146 119 85 6 5817 •764699 •896451 98609'; •235301 6 7 4583 •661 150 •882546 987 728 •338850 7 8 3 497 •543 696 •870274 991528 •456 304 8 9 2594 •4^3970 •861 802 998 5 1 2 •586030 9 90 1887 •275 772 •860314 003 600 •724228 90 I 1368 •136086 •863914 02^1 447 •863914 I 2 I 000 •000 000 •885 361 028 619 3-000000 2 3 768 2-885361 -913980 996 796 -114639 3 4 630 •799341 -910776 987633 -200659 4 95 5^3 •7101x7 •898 409 973072 •289883 95 6 406 -608 526 •871481 949542 •391474 6 7 302 -480007 •821 023 877947 •5^9993 7 8 200 •301 030 •698 970 •698970 8 9 too •000 000 2-000 000 9 59 Elementary Values — (continued) . X 55 Cologj9^ A (L- Logd^ A -A X o'ooS 007 000 3^6 1116 3^047 664 010 762 989238 55 6 •008 363 000 361 1144 •058426 009 760 990 240 6 7 •008 724 000379 1170 •068 186 009 545 990455 7 8 •009 103 000 470 1196 •077731 012527 987473 8 9 •009573 000 835 1231 •090 258 026350 973 650 9 6o •010408 000 974 1308 •116608 027 966 972034 60 I •on 382 001 157 1395 •144574 030 067 969933 1 2 •012539 001 303 1495 •174641 029 750 970 250 2 3 •013 842 ooj 410 1601 •204391 027588 972 412 3 4 •015252 001 291 1706 •231979 019416 980584 4 65 •016543 001 266 1784 •251395 014837 985 163 65 6 •017 809 00 r 462 1846 •266232 015710 984 290 6 7 •019 271 001 628 1914 •281 942 015 162 984838 7 8 •020 899 001 830 1982 •297 104 014650 985350 8 9 •022 729 002 135 2050 •3^1754 015196 984 804 9 7° •024 864 002908 2123 •326950 021 744 978256 70 I •027772 003 359 2232 •348 694 020 151 979849 I 2 •031 131 003 705 2338 •368845 015867 984 ^33 2 3 •034836 004 102 2425 •384712 on 487 988513 3 4 •038938 004343 2490 •396 199 004857 995 143 4 75 •043 281 004433 2518 •401 056 996 884 003 116 75 6 •047714 002 781 2500 •397 940 975524 024476 6 7 •050 495 002 588 2363 •373464 969 945 030 055 7 8 •053 083 002 109 2205 •343 409 962 802 037 ^98 8 9 •055 192 001 207 2024 •306211 953 622 046378 9 8o •056399 000 889 1819 '259 833 949950 050 050 80 1 •057 288 004041 1621 •209783 970343 029657 I 2 •061 329 006 838 1514 •180 126 981 242 018758 2 3 •068 167 009 003 1450 •161368 981334 018666 3 4 •077 170 012 012 1389 •142 702 979 842 020 158 4 8^ •089 182 014367 1326 •122544 968771 031 229 85 6 •^03 549 013905 1234 •091315 944515 055 485 6 7 •117 454 012 272 1086 •035 830 919858 080 142 7 8 •129 726 008472 903 2^955688 893 73 J 106 269 8 9 •138 198 001 488 707 •849419 865 748 134252 9 90 •139686 996 400 5^9 •715 ^67 850681 1493^9 90 I •136086 978553 368 ■565 848 799 640 200 360 1 2 •I 14 639 971 381 232 •365488 77439T 225 609 2 3 •086020 003 204 138 •139879 928307 071693 3 4 •089 224 012367 Ji7 •068186 961 198 038 802 4 95 •101591 026928 107 •029384 987 649 012351 95 6 •128519 050458 104 •017033 991567 008 433 6 7 •178977 122053 102 •008 600 991 400 008 600 7 8 •301030 100 •000 000 000 000 000 000 8 9 100 •000 000 9 60 Probabilities of Living ovcr^ and of Dying in, a Year, and the Avcraxje Duration of Lift' , at each Age. X '" (!-/'.) '■' X P^ 9r (I -P.) h lO •996 8^)0 •003 140 48-195 SS •981 732 •018268 I8I92 I ■995 787 •004213 47^34<> 6 •980926 •019074 17-521 2 •994862 -005 138 [ 46-1544 7 •980 1 13 •019887 i6-8=;2 3 ■994 1 1 7 •005 883 1 45^782 8 •979 258 •020 742 1 61 84 4 ■993 562 •006438 45"05o 9 •978199 •021 801 ^5-516 I.") •993 >62 •006838 44-338 60 •976319 •023 681 14^85 1 6 •992 9j;o '007 050 43-640 I ■974132 •025 868 14-199 7 ■992931 •007 069 ! 42946 2 ■97 « 541 •028459 13-563 8 •993 100 •006900 42-249 3 •968631 •03 1 369 12-945 9 •993 305 •006 695 41-539 4 •965491 •034509 12-349 20 ■993 122 •006878 40-815 65 •962 623 -037377 n-772 I •992 711 •007 289 40-094 6 •959822 •040 178 1 1-210 2 ■992 080 -007 920 39^385 7 ■956 599 •043401 10-658 3 •991 178 •008822 38-696 8 ■953018 •046982 lo-j 19 4 •990 158 •009 842 38036 9 •949010 -050990 9^593 25 •989186 •010814 37^409 70 -944357 ■055 643 9-082 6 •988621 ■on 379 36-812 I ■938053 •061 947 8^587 7 •988 181 •on 815 1 36-230 2 •930826 •069174 8-I2I 8 •987 986 •OJ2 014 35-657 3 922 921 •077079 7-688 9 •988 049 •on 951 35^085 4 ■914245 -085 755 7-288 30 •988 150 -oi I 8;;o 34-503 75 ■905 146 -094 854 6-925 I •988293 •on 707 i 33^91 1 6 ■895 955 •104045 6-598 2 •988430 •on 570 33'307 7 •890 236 •109 764 6^306 3 •988 343 •on 657 32691 8 •884 946 •115 054 6^022 4 •988 280 •01 1 720 32^070 9 •880660 -U9340 5'74o 35 •988 240 •on 760 3r445 80 •878214 •121 786 5-450 6 •988 100 •on 900 30-813 I •876420 •123 580 5-^37 7 ■987957 •012043 3o^J78 2 •868302 •J31 698 4- 790 8 •987 810 } -012 190 29^540 3 ■854 739 •145 261 4^441 9 •987 639 •012 341 28-898 4 •837201 •162 799 4' III 40 ■987 453 •012547 28^253 85 ■814363 •185637 3-81.3 I ■987 ^S3 •C12 747 27^606 6 •787864 -212 136 3-568 2 •987 061 •012939 ' 26^956 7 •763038 •236962 3-394 3 •986 8,50 •013 150 26^302 8 -741 779 •258 221 3-293 4 •986 620 •0 J 3 380 25^"646 9 •727448 -272552 3^266 45 •986382 -013 618 24-987 90 •724960 -275040 3-302 6 •986237 •013 763 24-325 I -730994 -269 006 3-365 7 •986 117 -013 883 1 23-658 2 -768000 •232 000 3-4J9 8 •985 995 014 005 1 22^984 3 -820 312 •179688 3-301 9 •985841 •014 159 22^303 4 •814286 -185 714 2914 50 ■985 547 •014453 2r6i6 95 ■791423 •208577 2-465 I ■985 043 •014957 20926 6 •743 843 ■256157 1-983 2 ■984380 •015 620 20^236 7 •662 252 -337748 1-493 3 ■983 594 •016406 J 9^549 8 •500 000 •500 000 I'OOO 4 •982613 •017387 i8^867 9 •000 000 i-ooo 000 •500 THREE PER-CENT. r.2 HF 3 rrjicKNT. Commutation TahJi . X lO D. N. s. M. K X 74 409'4 I 729035- 34562827- 2X 88X-9 744232-8 10 I 7^oi5'3 X 657 020- 32833 792- 2X 655-0 722350-9 I 2 69623-2 J 587 397- 31 X76772- 2X 360-5 700695-9 2 3 67 248*0 1520149- 29589375- 2X 0x3-2 679335-4 3 4 64905-2 I 455 244- 28069 226* 20629-1 658322-3 4 1.5 62 609- 1 I 392 634- 266x3 982- 20 223-4 637693-2 15 6 60 369-9 I 332 265- 25221 348- 19807-7 6x7469-8 6 7 58198-4 I 274 o66- 23889083- 19394-5 597 662-x 7 8 56103-8 I 217 962- 22 6x5 0x7- X 8 995-1 578267-5 8 9 54093-9 I 163 868- 21397055" 186x9-3 559272-4 9 20 52 i66-8 I III 702- 20 233 186- x8 267-7 540653-2 20 I 50 299-0 I o6x 403- 19 121 484- x7 9x9-4 522 385-5 I 2 :f8478-i I 012 925- x8 060 082- 17563-4 504466-1 2 3 46693-2 966 23 r -4 17047 157- 17 190-5 486902-7 3 4 44 933 '2 921 298-2 x6 080 926* 16 790-6 469 712-2 4 25 43 195' I 878 103-1 15 159628- 16361-2 452921-6 25 6 41 483-5 8366x9-6 14 281 524- 15907-7 436560-4 6 7 39817-0 796802-6 13444905- 15449-4 420652-7 7 8 38 200-5 758602-1 12 648 102- 14992-7 405 203-3 8 9 36642-3 721959-8 XI 889500- 14547-i 3902x0-7 9 30 3S i49'9 686 809-9 XI 167540- 14 121-9 3is^^y^ 30 I 33721-7 653 088-2 XO48073X- 13 717-5 361 541-6 1 2 i^3S^3 620 731-9 9 827 642- 13 334-3 347824-1 2 3 31050-4 589 68 X -6 9 206910- X2970-8 334489-8 3 4 29 794-6 559887-0 8 617 229- X2 6x9-4 321519-0 4 35 28587-8 531299-2 8 057 342- 12 280-4 308 899-6 35 6 27428-7 503870-5 7526043- 1 1 954-0 2966x9-3 6 7 26312-9 477557-6 7 022 172- 11637-1 284665-3 7 8 25 238-9 452318-7 65446x5- 11329-4 273 028-2 8 9 24205-0 428 1 13-7 6 092 296- 11 030-7 26X 698-8 9 40 23 2I0-0 404903-7 5 664 182- 10740-7 250668-x 40 I 22 251-3 382652-4 5 259 2 79" 10 458-0 239927-4 I 2 21 327-8 361324-6 4876626- 10 182-6 229469-5 2 3 20438-7 340 886-0 4515302- 9914-65 2x9286-9 3 4 19582-4 321303-5 4174416- 9653-72 209372-3 4 45 18757-7 302 545-8 3853 112- 9 399-33 X 99 7x8-5 45 6 179^3-3 284582-5 3 550 ^6(^- 9151-33 X903I9-2 6 7 17 200-1 267382-4 3 265 984- 8911-29 x8x 167-9 7 8 16467-3 2509x5-1 2 998601- 8 679-45 X72 256-6 8 9 15 763-8 235 ^S^'3 2 747686- 8455-55 163577-J 9 50 15087-9 220063-4 2512535- 8 238-85 155 121-6 50 I 14436-7 205 626-7 2 292472- 8o27-r3 146882-7 1 2 l^i 806-6 19X 820-1 2086845- 7817-49 138855-6 2 3 13 195- X 178 624-9 I 895 025- 7 608-11 131 038-1 3 4 12 600-6 x66 024-3 X 716400- 7397-94 123430-0 4 63 3 PERCENT. Commutation Table. X D^ N, s. M, Rx- X 5a 12 020"9 154003-4 1550376- 7 185-24 116032-1 55 6 1^457-6 H2 545-9 1396372- 6972-04 108846-8 6 7 109117 131 634-2 I 253 826- 6 759-86 loi 874-8 7 8 10383-2 121 251-0 I 122 192- 6549-18 95114-93 8 9 9871-67 ^^^379' 3 I 000941- 6340-09 88565-75 9 6o 9375'2i 102 004-1 889561-8 6131-15 82 225-66 60 I 8 886-60 93 117-50 787557-7 5915-60 76094-51 1 2 8404-58 84712-92 694440-2 5 692-42 70 178-91 2 3 7927-57 76785-35 609 727-3 5 460-20 64486-49 3 4 7455-^3 69330-12 532942-0 5218-76 59026-29 4 65 6988-31 62 341-82 463 61 1-8 4968-98 53 807-53 65 6 6531-17 55^^°'(^5 401 270-0 4715-39 48838-55 6 7 6086-18 49724-47 345 459-4 4460-62 44 1 33 • 1 6 7 8 5 652-45 44072-02 295 734-9 4204-17 39663-54 8 9 5 229-99 38842-03 251 662-9 3 946-34 35-^5^-37 9 7° 4818-75 34023-28 212 820-9 3 687-43 31512-04 70 I 4418-08 29605-20 178797-6 3427-11 27 824-61 1 2 4023-68 25581-52 149 193-4 3 161-39 24397-50 2 3 3 6^6-26 21945-26 123 610-9 2 891-17 21 236-11 3 4 3258-23 18687-03 101 66y6 2 619-05 18344-94 4 75 2 892-06 15 794-97 82978-57 2347-78 ^5 725-89 75 6 2541-49 13 253-48 67 183-61 2 081-44 13378-11 6 7 2310-74 11043-74 53930-13 1824-71 11 296-67 7 8 I 910-76 9131-980 42 887-40 1589-12 9471-955 8 9 I 641-67 7490-313 33 755-42 I 375'69 7882-832 9 8o I 403-64 6086-671 26265-10 1 185-48 6507-145 80 I I 196-79 4889-877 20 178-43 1019-51 5321-668 I 2 T 018-34 3871-533 15 288-55 875-920 4302-156 2 3 858-476 3 013-058 11 417-02 745-713 3426-236 3 4 713-400 2300-658 8403-964 624-641 2 680-523 4 85 579-051 I 721-607 6103-307 512-041 2 055-882 85 6 457-823 I 263-784 4381-700 407-679 I 543-841 6 7 350-196 913-588 3 117-916 3^3-387 1 136-161 7 8 259-430 654-158 2204-329 232-820 822-774 8 9 186-835 467-323 1550-171 167-781 589-954 9 90 131-954 335-370 I 082-848 118-342 422-173 90 I 93-8750 242-495 747-478 83-1069 303-830 I 2 65-9136 176-581 504-984 58-850 7 220-723 2 3 49-1473 127-434 328-403 44-004 1 161-873 3 4 39-1419 88-293 200-969 35-4302 117-869 4 95 30-944 3 57-347 112-677 28-3727 82-438 95 6 23-7767 33-57^ 55-330 22-1064 54-066 6 7 17-1710 16-400 21-759 16-193 2 31-959 7 8 11-0403 5-359 5-359 10-562 7 15-766 8 9 5-359 4 -000 •000 5-2033 5-203 9 M 3 PKKCKNT. Logarithms and Co-Iofjnrit/ims of D^., N^, anJ M^; ivith tliclr J)iJ}'vrenci'S. X LogD, A cdog n. A (Cologr/J^) LogN, A X lO 1 +•871 62811-985 797 5*»28 372 o^o 1 4 203 6^ 23 7 804 98 1 524 10 I ■8.=; 7 42."; ■985329 •142575 •014671 •219 328981 358 I 2 •842 754 •984 926 •157246 •015074 •2CO 686981 200 2 3 •827680 •984 600 •172320 •015 400 •181 886981 050 3 4 •812 280 ■984358 •187 720 •015 642 •162936980901 4 15 •7966.38 •984 183 •203 362 •015 817 •H3 837 980 754 15 6 •780821 •984 090 •219 179 •015910 •I 24 59 1 980 601 6 7 •76491 1 •984082 •235 089 •01 5 918 •105 192 980442 7 8 •748993 ■984 ^SS •251 007 •0^5845 •085 634980 270 8 9 •733 h8 •984 246 •266852 •015 754 •065 904980084 9 20 •717394 •984 1 66 •282 606 •015834 •045 988979 892 20 I •701 360 •98398^ •298440 •016 015 •025 880979 697 I 2 -^^s "545 •983 708 ■314455 •016 292 ■005577979504 2 3 •669 25,3 ■983315 ■330 747 •016685 5-985081979319 3 4 •652 568 •982 867 ■347432 •017 133 •964400979 146 4 25 •63.'; 435 •982441 ■3645*^5 •017559 •943 54'^ 9 78 982 25 6 •617876 •982 192 •382 124 •017808 -922528978823 6 7 •600 068 •982 001 ■399932 •017999 •901 351 978 663 7 8 •582 069 •981 914 •417 931 •018086 •880014978499 8 9 ■'s^:^ 983 ■981 94 J •436017 •018 059 •8585^3978324 9 30 •545 9H •981986 •454076 •018 or4 •836837978135 30 I •527910 •982048 •472 090 •0179^2 -814972977932 I 2 •5099-8 •982 109 •490 042 •017891 ^792904977 714 2 3 •492 067 •982 071 ■507 933 •017929 •770618977482 3 4 •474 138 •982 042 •525862 •017958 •748 100977239 4 35 •456 180 •982 025 •543 820 •017975 •725339976980 3.^' 6 •438 205 •981 964 ■5^51 795 •018 036 •702319976707 6 7 •420 169 ■981 901 ■579831 •018 099 •679026976419 7 8 •402 070 ■981 836 ■597 9 >o •018 164 ■655445,976 114 8 9 •383906 •981770 •616094 •018 230 •63^559.975793 9 40 •3*55 ^1^^ •981 679 ■634324 •018 321 ■607 352 975 452 40 I •347 ^ss •981 591 •652 645 •018409 •582 804975 094 I 2 •328946 •981 507 •67 1 054 •018493 •557 898974 711 2 3 •3 J 0453 •981414 .689547 •018 ';86 •532609974306 3 4 •291 867 •981 312 ■708133 -018 (388 •506915 973 876 4 45 ■273 ^79 •981 208 •726 821 •018 792 •480 791 973417 45 6 ■254387 •981 144 •745 '^13 •018856 •454208972925 6 7 ■235531 •981 091 •764469 •018 909 •427 133972394 7 8 •216 622 •981038 ■783378 •018 962 ■399527971 821 8 9 •197 660 •980 969 •802 340 •019031 •371 348071 200 9 50 •178 629 ■980840 •821 371 •019 160 ■342548970532 50 I •159469 •9806)8 •840531 •019382 •313 080969 814 I 2 •140087 •980326 ■859913 •019674 -282 894969 048 2 3 •120413 ■979979 •879587 •020 021 -251 942*968 230 3 ' •100392 •979 545 •899 608 •020 455 •220 X 72^967 358 4 65 3 PER-CENT. Logarithms and Co-locjarithms ofT>^, N^, and 31,.; tvith their Differences. X CologN^ A LogM, A Colog Mj,. A X lO 7762 196 018476 4-340 085 995 474 5 '659 915 004526 10 1 •780672 018 642 •i:iS 559 994 05 1 •664441 005 949 I 2 •799314 018 800 •329 610 992 882 •670390 007 118 2 3 •818 114 018 950 •322492 991987 •677 508 008 013 3 4 •837064 019 099 •314479 991375 •685521 008 625 4 15 •856163 019 246 •305 854 990981 •694 146 009 019 15 6 •875 409 019399 •296835 990 844 •703 ^65 009 156 6 7 •894808 0J9 558 •287679 990 962 •712321 009 038 7 8 •914366 019730 •278 641 991322 ■721359 008678 8 9 •934 096 019916 •269963 991 721 ■730037 008 279 9 20 •954012 020 108 •261684 991 639 •738316 C08 361 20 1 •974 120 020 303 -^:)3 3^?> 991 286 •746677 008 714 1 2 J994423 020496 •244 609 990 680 •nS?>9^ 009320 2 3 6-OI49I9 020 681 •235289 989776 •764711 010 224 3 4 •035 600 020 854 •225 065 988 750 ■774935 on 250 4 25 ■056454 021 018 •213815 987 792 •786 185 012 208 25 6 •077472 021 177 •201 607 987305 ■798393 012 695 6 7 •098 649 021337 •I889I2 986967 •811088 013 033 7 8 •I 19 986 021 501 ■J75879 986897 •824 121 013 103 8 9 •141 487 021 676 •162 776 987 118 •837 224 012 882 9 30 •163 163 021 865 •149894 987 382 •850 106 012618 30 I •185028 022 068 •137276 987 693 •862 724 012307 1 2 •207 096 022 286 •124969 987998 •875031 012 002 2 3 •229 382 022 518 •112 967 988072 •887 033 on 928 3 4 ■251 900 022 761 •101 039 988172 •898961 oil 828 4 35 •274661 023 020 •089 211 988301 •910 789 on 699 3^ 6 •297 681 023 293 •077512 988 332 •922488 011668 6 7 •320974 023 01 •065 844 988 :^6^ ■934 156 011637 7 8 ■344555 023 886 •054 207 988 396 ■945 793 1 1 604 8 9 •368441 024 207 •042 603 98 8 430 •957397 on 570 9 40 •392 648 024548 •031033 988414 •968 967 on 586 40 I •417 196 024906 •019447 988411 •980 SS3 011589 I 2 •442 T02 025 289 •007 858 988 420 •992 142 on 580 2 3 •467391 025 694 3-996278 988417 4-003 722 on 583 3 4 ■493 085 026 124 •984 695 988 402 •015 305 011598 ■ 4 45 •519209 026583 •973 097 988 387 •026903 on 61^ 45 6 •545 792 027 075 •961 484 988457 •038516 on 543 6 7 •572867 027 606 •949941 988551 -050 059 011449 7 8 •600473 028 179 •938492 9886^0 •06 1 1508 011350 8 9 •628652 028 800 •927 142 988 724 •072858 on 276 9 50 '^'•>1 452 029468 •915866 988 694 •084 134 on 306 50 I •686920 030 186 •904 560 988 507 •095 440 011493 I 2 •717 106 030952 •893 067 988 210 •106933 on 790 2 3 •748058 031770 •881 277 987 834 •118723 012 166 3 4 •779828 03 2 642 •869 111 987330 •130889 012 670 4 <;<; 3 I'HIiCKNT. Lofjarithms ami C(j-lo,/(iritfiin.s of' D^., Nj., and^ij-j w?//// their Differences. X Log I), A (Logr/*^) Colog D, A [Colog vp^) Log N> A X 55 4*079 9.37 'i^'979 "5'''; "^•92006.3 0^020 844 5-187 530966425] ^^ 6 •059 09.5 •978799: •940907 •021 201 '^!,^9:^^ 965 414 6 7 •0.57 8y2 •9784.59 •962 108 •021 561 -1 19 369 964 3 1 6 7 8 •016 3 5 1 ■978060^ ■98.3 669 •021 940 •08,} 685 963 120 8 9 3 994 .391 •977590 4005 609 •022 410 •046 805 961 813 9 6o •971 981 91^ 755 •028019 •02,3 245 •008618 960413 60 1 •948 7.56 •975 780 •051 264 •024 220 4^969 03 1 958919 1 2 •924516 ■974624 •075 484 •025 .^76 •927950 957 328 2 .3 •899140 •973.321 •100 860 •026679 •885278 955 644 3 4 •872 461 •971911 •1275.39 •028 089 •840922 953857 4 65 •844.372 •970619 •155628 •0293S1 •794779 951 938 ^^ 6 •814991 •9*59 354 •185 009 •030 646 •746717 949 853 6 7 •784.345 ■967 892 •2156.15 •032 108 •696570 947 593 7 8 •7522.37 ■966 264 •247 763 •033 736 •644 163 945 '39 8 9 •718501 •9<54 433 •281499 •035567 •589302 942 474 9 70 •682 934 •962 299 •317066 •037701 '^?>^ 11^ 939 592 70 1 •'545 2.3.3 •9,^9.391 •354 767 •040 609 -471368 936 558 I 2 ■604624 •956 0,3 1 •395376 •043 969 •407 926 933415 2 .3 •560 655 •952327 •439 345 •047 673 •.341 34' 930 199 3 4 •512982 •948 225 •487018 •051775 -271540 926979 4 75 •461 207 •943 882 •538 793 •056 1 18 .198519 923 811 n 6 •405 089 •939 448 •594 91 1 •060 552 •122330 920747 6 7 -.344 5.3 7 •936 668 ^(^:^'^ 463 •063332 •043 077 917488 7 8 •281 205 •9.34 080 •718795 •065 920 3^960565 913 935 8 9 •215285 •931 971 •784715 •068 029 •874500 909 880 9 80 •147256 •930 763 •852 744 •069237 •784380 904918 80 I •078 019 ■929875 •921 981 •070 125 •689 298 898 585 I 2 •007 894 ■925 834 •992 106 •074 166 •587 883 891 124 2 .3 2-9.3.3 728 •918996 ^■o66 272 •081 004 •479007 88284-; 3 4 •852724 •909 99,3 •147 276 •090007 •361 852 874082 4 85 •762717 •897981 •2.3728.3 •102 019 •235 934 865 739 85 6 •660 698 ■88,361.3 •339302 •116387 •10 1 673 8.^9077 6 7 •544 .3 1 ' •S69 709 •455 689 •130 291 2^9607501854933 7 8 •414020 •8,w 437 •585980 •142563 •815683 853 934 8 9 •271457 •848 965 •728543 •151 035 •669 617 855907 9 90 •120422 •847477 •879578 •152523 •525524 859178 90 I 1^967899 •8'; I 076 2^0.32 lOI •148924 •384 702,862 242 1 2 •818975 •872524 •181 025 •127476 •246 944 858 340 2 .3 •691499 •901 142 •.308 501 •098 858 •105 284 840636 3 4 •592 641 •897 940 ■407 359 •102 060 I •94592c 812594 4 95 •490581 •885571 ■509419 •114429 •758 5 M 767447 95 6 -.3 76 '152 •858644 •62,3 848 ■141 356 •525961 688875 6 7 •2,34 79^ ! -808 186 •76'; 20^ •191 814 •21483^ 5H279 7 8 ■042982 \ ^^86 1.3.3 •957018 •313867 0-729 iij 8 9 0-729 J 15 ' r27o 88j 9 67 3 PER-CENT. Logarithms and Co-hgarithms o/Dj., Nj., and M^; with their Differences. X Coiog N^ A LogM, A Colog U^ A X 55 6*8 1 2 470 033 575 3-856441 986919 4^143 559 013 081 55 6 •846 045 034586 •843 360 986578 •156640 013 422 6 7 •880631 035 684 •829938 986 249 •170 062 013751 7 8 •9163x5 036880 •816 187 985 908 •183 813 014092 8 9 ■953 195 038 187 •802 095 985447 -197905 014553 9 6o •991 382 039587 •787542 984457 •212458 015543 60 I 5-030969 041 081 •771999 983 298 •228001 016 702 I 3 ■072 050 042 672 '755^91 981 912 -244 703 018088 2 3 •114 722 044356 •737209 980358 •262 791 019642 3 4 •159078 046 143 -'t^l 5^7 978700 ■282433 02T 300 4 65 •205 221 048 062 •696 267 977 250 -303 733 022 750 65 6 -^53 ^^3 050 147 ■^13 $11 975879 •326483 024 121 6 7 ■303 430 052407 •649 396 974284 •350 604 025716 7 8 -2SS^31 054861 •623 680 972514 •376320 027 486 8 9 •410698 057 S^^ ■Sg6 194 970530 •403 806 029470 9 70 •468 224 060 408 •566724 968 204 •433 276 031 796 70 I •528632 063 442 •534928 964951 -465 072 035 049 I 2 '59^- 074 066 585 •499879 961 194 •500 121 038 806 2 3 ■6sS6s9 069 801 •461 073 957071 •538927 042929 3 4 •728460 073 021 •418 144 95^5^3 •581 856 047487 4 IS •801 481 076 189 •370657 947 708 -629 343 052 292 75 6 •877670 079 ^53 •318365 942 830 •681 63s 057170 6 7 •956923 082 512 •261 195 939 963 -738805 060 037 7 8 4-039 435 086065 •20I 158 937362 •798842 062 638 8 9 •125500 090 120 •138520 935 373 •861 480 064627 9 80 •215 620 095 082 -073 893 934 499 •926 107 065501 80 I •310702 101415 •008392 934072 •991 608 065 928 I 2 •412 117 108876 2-942464 930 108 ro57 S3(^ 069 892 2 3 •520993 ^^1 ^55 •872572 923 059 •127 428 076941 3 4 •638 148 125 918 •795631 913674 •204369 086326 4 85 •764066 134 261 •709305 90; 014 •29069^ 098 986 85 6 •898327 140923 •610319 885 762 -389681 114238 6 / 3'039 25o J 45 067 •496081 870940 ■503919 I 29 060 7 8 •184317 146066 -367021 857 723 -632979 142277 8 9 •330383 144093 •224744 848 396 •775 ^S^ 151604 9 90 •474476 140 822 •073 140 846 497 -926860 153 503 90 I •615298 137758 1-919637 850 I r 15 2-080363 149 88 T 1 2 •753056 141 660 ■769752 873 742 -230 248 126258 2 3 •894 716 159364 •643 494 905 880 -356506 094 120 3 4 2^054080 187 406 •549 374 903527 -450626 096 473 4 95 •241 486 ^3^SS3 •452 901 891 618 ■547 099 108382 95 6 •474 039 311 125 •344519 864815 •655481 135 185 6 7 •785 164 485721 •209334 814440 •790666 185560 7 8 i^27o885 •023 774 692 504 •976226 307 496 8 9 0-716 278 7-283722 9 08 3 rKU-CKNT. Valuer of Annuitua, and Single ttnd Annual Premiums for Assurance of a Unit. X "x A. W^ X «x A. ^x lO 2y2]6 8 -294074 -012 133 55 12-81 1 3 -597729 -043278 I 23-0093 -300 700 -012 524 6 12-441 2 -608 5091 -045 272 2 227998 -306801 -012 891 7 12-063 6 -619507 j -047 422 3 22*605 I •3 J 2 473 -013237 8 1 1-677 6 -630749 049753 4 22-421 •317833 •013570 9 11-282 7 -642251 -052289 15 22-24.3 3 -323 010 -013897 60 108802 -^53 975 •055 048 6 22-o68 4 -328 106 ■014223 £ 10-4784 •665 677 •057 994 7 21-891 ^ '33^ 249 ■014558 2 10-0794 •677300 -061 132 8 21-709 I -338570 •014909 3 9-685 9 -688 761 1 •064455 9 21-5157 •344 203 -015 287 4 92995 -700014! •067 966 20 21-310 <; •350179 -015 696 65 8-9209 -711 042 •071 671 I 21-101 9 -356257 -016 1 19 6 8-545 3 -721 982 •075 638 2 20-8945 -362 296 -016547 7 8-170 I •732911 •079924 3 20-693 2 •368 159 -016971 8 7-7970 -743 778 •084549 4 20-503 7 -373678 -017377 9 7-4268 -754559 •089 543 25 20-3288 •378774 •017759 70 7-0606 -765 225 •094 934 6 20-1675 •383470 -018 1 16 I 6-7009 •775701 -100 728 7 20-01 1 6 •388 on •018466 2 6^357 7 -785697 -106785 8 19-858 4 •392473 •018816 3 6-035 I -795093 -113 018 9 19-702 9 •397 003 -019 176 4 5135 3 -803 825 •119345 30 J 9*539 5 •401 763 -019 561 75 5-461 5 -811 801 -125637 I 19-367 -406 787 •019973 6 5-2148 •81898^ •131 779 2 19-1843 -412 108 •020417 7 4-995 -825387 -137678 3 18-991 I -417 734 -020 896 8 4-7792 -831672 •143907 4 J8-791 6 -423 546 -021 400 9 4-562 6 •837982 •150645 3.5 18-5849 •429 567 •021 934 80 4-33^3 ■844 573 •158268 6 18-3702 -435 820 •022499 I 4-08^8 •851 870 -167499 7 18-1492 •442257 •023 095 2 3-801 8 -860 142 -179 129 8 17-9215 -448887 •023 724 3 3-5098 -868 648 -192 614 9 17-687 -455719 -024387 4 3-2294 -876813 -207311 40 1 7 "445 2 -462 761 -02 i; 088 85 2-973 2 -884277 •222 563 I 17-1969 -469 994 •025 828 6 2-7604 •890473 -236802 2 16-941 5 -477432 -02661 1 7 2-608 8 -894 889 •247 975 3 16-6785 •485 093 -027 440 8 2^521 5 -897432 -254842 4 16-407 7 -492979 -028320 9 2-5013 -898021 •256485 45 16-1292 -501 092 •029 254 90 2-5416 -896848 •253 235 6 15-8424 -509 444 -030 248 1 2-611 -894826 -247 807 7 i5'545 4 -5 '8 095 -031 314 2 2-6790 -892 846 -242 689 8 15-2372 -527072 -032461 3 2-592 9 -895 353 -249 201 9 14-9172 -536 392 •033 699 4 2-255 7 -905 174 -278029 50 14-585 4 -546 056 -035 036 95 1-8532 -916896 •321352 I 14-243 3 •556021 •036477 6 1-4119 -929 750 -385482 2 13-8933 •566 213 •038018 7 -955 I -943 056 -482362 3 13-537 2 •576 586 '-039 663 8 •485 4 •956 735 -644076 4 13-1759 -587 IIO: -041 416 9 -000 -970874 -970874 THREE AND A HALF PER-CENT. 70 31 PEK-C'ENT. Com m utation Table. 1>. N. Mx lO I 2 .3 4 6 7 8 9 20 I 2 3 4 6 3° I 3 4 35 6 7 8 9 40 I 2 3 4 45 6 7 8 9 I 2 3 4 70891-9 68279-5 65 692-6 6,3 i4<;-o 60650-7 58 222-5 ^^$ 869-0 53 599" I 5' 420-5 49338-9 47 35''3 45 435 "4 43 578-9 41 771-6 40003-0 38 269-8 34936-9 31 841-4 30397-0 29021-0 27711-4 26464-5 25271-5 24130-7 23 040-5 21996-5 20996-7 20039-4 19 122-8 18244-3 17 402-6 16596-6 15824-5 15084-8 14376-2 13698-9 13 051-9 12433-9 "843-3 11277-4 10733-1 10 208-2 9701-14 I 499 667 I 43 I 387 I 3^^ '^95 I 302 550 I 241 899 I 183677 I 127 808 I 074 208 I 022 788 973 449' J 926097-9 880 662- <; 837083-6 795312-0 755 309-0 717039-1 680463-3 645 526-4 612 169-8 580328-4 54993^-5 520910-4 493 1 99' I 466 734-6 441463-1 417332-3 394291-8 372295-3 351 298-6 331 259-2 312 136-5 293 892-2 276489-6 259893-0 244 068-5 228983-8 214607-6 200 908-7 T 87 856-9 175423-0 163 579-7 152302-3 141 569-2 131 361-0 121 659-9 28 199634 26699967 25 268 580 23 902 885 22 600335 21358436 20174759 19046952 17972743 ^6949955 15 976 50*5 15 050408 14 169 746 13 332 662 ^^S313S^ II 782 041 II 065 002 JO 384539 9739012 9126843 8546514 7996583 7475672 6982473 6515739 6074 276 S^^^9A3 5 262 6<, I 4890356 4539058 4 207 798 3 895 662 3601 770 3 325 280 3 065 387 2 821 319 2 592 33S ^311 V-1 2 176818 I 988962 I 813 539 I 649 959 I 497 657 1356087 I 224726 7781-2 7566-1 7288-2 6962-1 6603-1 6 225-9 5841-3 5460-7 5 094-6 4751*8 4432-7 4 118-0 3798-1 3 464-5 3 108-4 2 728-0 2328-1 r 926-0 1527-2 I 140-0 0772-3 0424-3 o 096-0 9 786-27 9488-21 9 202-03 8927-85 8 662-94 8 406-99 8 15970 7 920-76 7 688-93 7 464-24 7 246-68 7035-82 6831-24 6632-76 6441-59 6257-84 6 081*23 59ii"i3 5 74574 5582-77 5 420-79 5 258-98 563 83 7- J 546 055-9 528489-8 511 201-6 494 239'6 477636-4 461 410-5 445 569-2 430 108-6 415 014-0 400 262-2 385 829-5 371 711-4 357913-4 344 448-9 33^ 340-5 318612-5 306 284-4 294358-4 282 831-2 271 691-2 260918-9 250 494-5 240398-5 230612-2 221 124-0 21 1 922-0 202 994-1 194331-2 185924-2 177764-5 169 843-7 162 154-8 154690-6 147443-9 140 408-1 133576-8 126944-1 120 502-5 1 14 244-6 108 163-4 102 252-3 96 506-54 90923-76 85502-97 10 1 2 3 4 15 6 7 8 20 1 2 3 4 25 6 30 1 2 3 4 35 6 7 8 40 I 2 3 4 45 6 7 8 50 I 2 3 4 71 3J PERCENT. Commutation Table. 55 6 7 8 9 6o I 2 3 4 65 6 9 I 2 3 4 75 6 7 8 9 80 X 2 3 4 85 6 7 8 9 90 I 2 3 4 95 6 7 8 9 D, 9 2IO'I2 8 736-11 8279-68 7 840-60 7418-33 7 01 1-22 6613-71 6224-75 5843-10 5468-41 5 101-16 4 744'44 4399-82 4 066-54 3 744'43 3 433'33 3 132-65 2 839-22 2553-45 2 276-94 2 011-28 I 758-94 I 522-64 1309-67 X 119-79 952-810 808*474 684-602 574-339 474-309 383-664 30X-876 229-794 169-412 X2 [-417 85-3376 59-7743 42-217 O 31-3263 24-8283 19-533 7 14-9366 10-7348 6-8687 3-3182 N, r [2 449-8 Jto3 713-7 95433-97 ^7593-37 80 175-04 73 163-82 66 550-1 X 60325-36 54482-26 490x3-85 43912-70 39 168-26 34 768-44 30 70X-90 26957-48 23524-15 20391-50 X7 552-28 H 998-83 X2 721-90 xo 710-62 8 95 X -67 6 7429-0371 6 1x9-368; 4999-5741 4046-764 3 238-290, 2553-688, X 979-348 I 505-039 ^ J2X-375 8x9-499 589-7051 420-292! 298-876 213-538 -^53-764 m"547 8o"220 55'39^ 35-858 20-922 X0-187 3-318 -000 X 103 067- 990616-8 886903-x 79X469-X 703875-8 623 700-7 55° 536-9 483 986-8 423 66x-4 369x79-2 320 165-3 276 252-6 237084-4 2023x5-9 17X 614-0 144656-5 X2X 132-4 xoo 740-9 83 X 88-62 68 X89-79 55 467-89 44757-28 35 805-60 28376-56 22 257-'i9 17 257-62 13 2XO-86 9972-568 74x8-880 5 439-532 3 934-493 2 8x3-1x7 1 993-6x8 I 403-914! 983-62x1 684-746, 47X-208 317-444! 205-898; 125-6671 70-285 34-427' ^3- 5° 5 3-318 -000 M, 5 096-0 X 4933*46 4772-46 4 613 "3 7 4 456*24 4 299-98 4139-57 3974-27 3 803-1 X 3 626-ox 3 443 -68 3 259-47 3 075-29 2 890-79 2 706-20 2521-72 2337-14 2 149-65 X 959-89 1 76913 I 58x-o7 I 39&ys X 219-92 I 058-45 9x2-859 783-742 67 X -62 7 575-094 487-983 407-375 332-769 ^63'955 202*082 X 49-47 X X07-204 75-2307 5^' 5 53 2 37-0173 27-5542 22-1x5 6 X 7-660 5 13-7240 10-0273 6-524 2 3-2060 E. 80 243-99 75 147-98 70 2x4-52 65442-07 60828-70 56372-46 52 072-48 47932-91 43 958-65 40 155-54 36 5^9' 53 08^-84 29826-38 26 75X-09 23 860-30 21 X54-X0 18632-38 x6 295-23 14 145-58 X2 185-69 XO 4x5-96 8 834-888 7 43 8- X 43 6 218-2x9 5 159-774 4246-9x5 3463-172 2791-545 2 2X6-45X I 728-468 ] 321-094 988-325 724-369 522-288 372-817 26^613 X90-382 137-829 X 00*8 1 2 73-258 51-142 33-482 19-758 9-730 3-206 55 6 9 60 X 2 3 4 65 6 70 X 2 3 4 75 6 90 X 2 3 4 95 6 72 HF. 3.7 I'KR-CKNT. Loynrithms aiiil Co-luyarithms of 1)„ X^, end .M^ ; n-ilh their Differences. ro I 2 3 4 I.? 6 7 8 9 20 J 2 3 4 ^) 6 7 8 9 30 I 2 3 4 3.5 6 7 8 40 I 2 3 4 45 6 7 8 9 50 I 2 LogD, 4 3 ■8.50. 597 •834 290 •817 516 •8003.39 •782836 •765 091 747 171 •729 1 58 •711 136 •693 189 •657394 •639277 •620882 •602 093 •582 857 'S^i 194 •543 284 •523 182 •502992 •482 830 •462 713 •442 658 •422 664 •402631 •382571 '362 493 •322 151 •301 884 •28x551 •261 J27 •240615 •220018 •199329 •'78539 ■157643 •136684 •115 672 •094607 ■073 473 •052 210 •030725' •008947 •986823 ,r ^ . CologDJ^,'^ J LogN, 983 ^9^ 5 983 2 261 982 823 982497 982 255| 982080 981987! 981978; 982 053! 982 143J 082 062 981 883 981 605 981 211 980 764 980337 980 090 979 89S 979810 979838 979883 979945 980 006 979967 979 940 979922 979860 979 798 919 nz 979667 9795761 979488; 979403; 9793ii 979 2lo| 979 X04 979041 9789881 978935 978866 978737 978515 978222 977876 9774424 149 403 0^016 3o7|6-i 75 995 ■'65710, -0167741 '^55 757 182484 -017177 -135 354 •I 14 794 •094 080 •1996611 •21/ 164I •2349091 •252 829! •270 842 •288864 •306 81 J •324668 •342 6o6| •360 7 23 1 •379118 •397907 •4^7143 •436806 •456716 •476818 •497 008 •517 170 •537287 '."^51 342 •577 336 '597 369 •617 429 •637507 •657 647 •677 849 •698 116 •718449 •738873 •759385 •779982 •800671 •821 461 •842357 •863316 •884328 •905 393 •926527 •947 790 •969275 •991 °r)3 •013 177 •017503 •0^7745 •017 920 •018 013 •018022 •o . 7 947 •017857 •017938 '018 1J7 •018395 •018789 •019 236 •019663 '019910 •020 102 ■020 190 •020 162 •020 117 •020 055 •019994 •020033 •020060 •020078 •020 140 •020 202 •020 267 -020333 •020424 •020512 •020597 •020 689 •020 790 •020896 •020959 •021 012 •02 1 065 •021 134 •021 263 •021 485 •02J 778 •022 124 •022558 •073 233 I ^052 235 •031 089 •009 786 5-9883x3 •966657 •94481c •922 769 •900538 •878 125 •855 543 •832805 •809914 •786872 •763 674 •740309 •716763 •693 022 •669 070 •644 89^ •620482 •1595 81& •570888 •545676974492 •520x68074177 •494 345:973 843 •468 1 88|9 73 491 •441679973 116 •4x4795972717 •387512^972293 •359805I971840 •33X 645971 354 •302 999970 828 •273827970260 •244 0871969642 •213729968977 •182 706968 263 •I ijo 969 967 498 •I'l 8 467:966680 •0S5 i47'965 8i2| 979762 979 597 979440 979 292 979147 979002 978854 978 697 978527 978344 97815 977 959 977 769 977587 977418 977262 977 109 976958 976802 976635 976454 976259 976 048 975 824 975588 975336 975 070 974 788 10 I 2 3 4 15 6 20 I 2 3 4 25 6 7 30 I 2 3 4 35 6 7 8 9 40 r 2 3 4 45 6 7 8 9 50 1 2 3 4 73 3;V PER-CENT. LogaritJnns and Co-hgartfhms of D^, N^., and M^; zoilJi their Differences. lO I 2 15 6 20 I 2 3 4 6 7 8 30 1 2 3 4 35 6 7 8 40 I 2 3 4 45 6 7 8 Colo.ir N. •824005 •844 24.3 •864646 •885 206 •905914 •926767 ■947 7(^5 •968 91 1 •990 214 6^oi I 687 '^33 343 •055 190 •077231 •099 462 •12x875 •144457 •167 195 •190086 •213 128 •236326 •259691 •283 237 •306978 •330930 •379518 •404 182 •429 112 •454324 •479832 •505 ^'55 •531 812 •558321 •585205 •612488 •640 195 •668355 •697 007 •726173 '7559^3 020 338 020403 020 560 020 708 020853 020998 021 146 021 303 021 473 021 6^^6 021 S47 022 041 022 23 1 022413 022 582 022 738 022 891 023 042 023 198 023365 023 546 023 741 023 953 024 176 024413 024 664 024930 025 212 025 508 025 823 026 157 026 509 026 884 j 027 383 I 027 7071 028 160 028 646 029 172 029 740 030358 lio- :\L 4-249961 •244676 ■237749 •229479 •220 191 ■2 10 209 •199789 •189 229 •178821 •168845 •^59347 ■^9 774 •139 818 •129 189 ■J 17 550 •104 760 •090898 •786271 03 I 023 •817294 031737 •849031 032502 ■881533 033320 ■9 r 4 853 034188 •076 496 •061 724 •046 885 •032 310 •018 047 •004 151 3'99o6i7 •977 184 •963 S84 ■950 747 '93 7 ^^^ •924641 •911 674 •898 767 •885 866 •872985 •860 139 ■847315 •834 ^00 •821 "695 •808 993 •796424 •783991 •771670 •759346 •746850 •734063 •720903 994 7^5 993 073 991 730 990712 990 018 989580 989 440 989592 990 0:24 990 502 990427 990 044 989371 988361 987 210 986 138 985 598 985 228 985 161 985 425 985737 986 104 986466 986 567 1 986 700 '986863 986919 986975 987 033 987 093 987 099 987 119 987 154 987176 987 i85 987 195 987298 987431 987 5*^7 987679 987676 987504; 987213] 986 839 986329 CologM, 5750 039 •755324 •762 251 ■770521 ■779809 •789791 •800 311 •810 771 •821 179 •83 J ^55 ■840653 •850 226 •860 183 •870811 •883450 •895 240 •909 103 •923 504 •938276 '953 115 •967 690 •981953 ■995 849 4-009383 •022 816 •036 1 16 1 •049 253 ' •063 334 •075359 •088326 •10 1 333 ■J14134 •137015 •139 861 I •152685 •165 500 •178305 • 191 007 ■20,5 576 •216009 •228330 •240654 •253 150 •265937 •279098 A 005 285 006927 008 270 009288 009 982 010420 010 560 010 408 009 976 009 498 009573 009956 ojo 629 on 639 012 790 I 013 862 014 403 014772 014839 014575 014 263 013 896 013 534 013433 013 300 3/ 013 013 081 013 035 OI3 967 013 907 013 901 012881 012 846 012 824 012 8 1 5 012 805 012 702 O 1 2 569 012433 012 321 012 324 ')0 012496 X 012 787 2 013 r6i 3 013671 4 10 I 2 3 4 ^5 6 7 8 20 I 2 3 4 25 6 7 8 30 I 2 3 4 35 6 7 8 40 1 3 4 45 6 7 8 74 3i J'Ell-CENT. Lojarithms and Co-hffariihms ofl^^, Nj., and Mj, ; with their JJifferaiccs — (coTitinued). J* Log D, A (Log vp^) 1 CologD, A (Cologtyjj.) LogN, A X 55 3-964 265 '^■911 053 ^•o35 735 0^022947 5^050 959 964877 ss 6 •941 318 -976696 -o';8 6H2i •023 304 •015836 963 867 6 7 •918014 •976335 •081986! •023 665 4^979 703 962 768 7 8 •894 349 •975957 •105 65 1 1 •024043 •942471 961568 8 9 •870 joO •975487 •I 29 6941 •024513 1 •904039 960 257 9 6o •«45 793 •974652 •154207 •025 348 •864 296 958853 60 I •820445 •973 677 -^19S:i'> •026323 ■823 149 957351 1 2 •794122 •972521 •205878 •027479 •780500 955755 2 3 •766643 ■971 218 •233 337 •028 782 •736255 954064 3 4 •737861 •969 808 •262 139 •030 192 •690319 952271 4 65 •707 669 -968516 •292331 •03 1 484 •642 590 950 344 65 6 •676185 •967 250 •323815 •032 750 •592 934 948 251 6 7 •643 43 !i •965790 ■3.5<'-> :/^s •034210 •541 '85 945980 7 8 •60922'; •964 160 •390775 •035 840 •487 ^^5 943514 8 9 •573385 •962 2,3 1 •426615 •037669 •430679 940835 9 70 •5357^6 •960 196 •464284 •039 804 •371514 937 935 70 I •495912 •957 287 •504 0S8 •042713 •309449 934885 I ! 2 •4.i3 199 •953 928 •546801 •046072 •244334 931 724 2 3 •407 127 •950 224 •592 873 •049776 •176058 92S494 3 4 '351 ?>S^ •946122 •642 649 •053 878 •104552 925262 4 75 ■303 473 •941 778 •696527 •058 222 •029814 922 090 13 6 •245251 ! ^937346 •754749 •062 654 3^951904 919029 6 7 •182597 •934565 •817403 •065 435 ■870933 915774 7 8 •117 162 •931976 •8S2 838 •068 024 •786707 912 226 8 9 •C49 138 •929 868 •950862 •070132 ■698933 908175 9 80 2-979006 •928660 3-020994 •071340 •607 108 903 208 80 I •907 666 •927772 •092334 •072 228 •510 316 8068^2 I 2 •835438 •923731 •164562 •076 269 •407 168 889354 2 3 •759 169 •916892 •240 831 •083 108 -296522 881 026 3 4 •676061 •907 890 •323939 •092 no ■177548 872 203 4 85 •583951 •895 878 •416049 •104 122 ■049751 863 798 85 6 •479829 •881 511 •520 171 •118489 2-913 549 857086 6 / •36 1 340 •867 605 •638 660 •132395 •770635 852917 7 8 1 -228945 •855 334 •771055 •144666 •623 552 851938 8 9 -084279 •846861 •915 721 '^53 K>9 •475 490 853985 9 90 1-93 1 140 •845374 1-068 860 •154626 •329475 857379 90 I ■776514 •848974 •223 486 •151 026 -186854 860603 I 2 •625488 •870421 ■374512 •129579 •047457 856 828 2 3 •495 909 •899039 •504091 •100 961 1-904 285 839 162 3 4 •394948 •895 836 •605 052 •104 164 •743 447 811 143 4 95 •290 784 •883 468 •709 216 •I 16 532 •554 590 766007 93 6 •174252 •856541 •825 748 •143439 •320597 687447 6 7 •030 793 •806 083 •969 207 •193917 ■008 044 5 1 2 86 J 7 8 0836876 •684029 r-j63 124 •315971 0-520 905 8 9 •520905 •479 095 9 75 HP. 3^ PER-CENT. Logarithms and Co-logarithms ofD^, N^, and M^.; wkh their Differences — (continued). X CologNx A LogM, A CologM^ A X 55 6-949 041 035 123 3^707231 985 920 4^292 769 014080 55 6 •984 164 036 133 •693 151 985591 •306 849 014409 6 7 yo20 297 037232 •678 742 985276 •321258 014724 7 8 •057529 03S432 •664018 984 950 •335982 015 050 8 9 •095 961 039 743 •648 968 984 499 •351032 015 501 9 6o •■^35704 041 147 '^33 467 983 488 •366533 016512 60 I •176831 042 649 •6i69SS 983 302 •383 045 017 698 I 2 •219500 044 245 '599^57 980882 -400 743 019 118 2 3 •263 745 045 936 •580 139 979290 -419861 020 710 3 4 •309681 047 729 ■559 429 977 594 •440571 022 406 4 65 •357410 049 6^6 •537023 976123 •462977 023 877 65 6 •407 066 051 749 •5^3 146 974 740 •486 854 025 260 6 7 •458815 054020 •487 886 973 131 •512 114 026 869 7 8 •512835 056486 •461 017 971342 •538983 028658 8 9 •569321 059 ^^5 •432359 969339 •567 641 o;o 661 9 70 •628486 062 065 •401 698 966988 •598302 033012 70 1 •690551 065115 •368 686 963 682 •631314 036318 I 2 •755666 068 276 •332368 959 864 •667632 040 136 2 3 •823 942 071506 •292 232 955 675 •707 768 044325 3 4 •895 448 074738 ■247907 951045 •752093 048955 4 75 •970 186 077910 •198952 946 165 •801 048 053 835 75 6 4^048 096 080971 •145 117 941 216 -854883 058784 6 7 •129067 084226 •086 333 938335 •913 667 061 665 7 8 •213293 087 774 •024668 935 736 •975332 064 264 8 9 •301 067 091 825 2-960404 933 769 3^039596 066231 9 80 •392 892 096 792 •894173 932955 •105827 067 045 80 I •489 684 103 148 •827 128 932611 •172 872 067389 I 2 •592832 1 10 646 •759 739 928 665 -240 261 071335 2 3 •703478 118974 •688 404 921590 •3 1 1 596 078410 3 4 •822452 127797 •609 994 912 149 •390 006 087851 4 85 •950 249 136202 •522 143 899388 •477 857 100612 85 6 3*086451 142 914 •421531 883996 •578469 X 16 004 6 7 •229365 147083 •305527 869 029 •694 473 130971 7 8 ■376448 148 062 •174556 855655 -825 444 144345 8 9 •524510 146015 •030 2x1 846 1 8^ •969789 153816 9 90 •670525 142 621 1-876395 844 204 2-123605 155 796 90 I •813 146 139397 •720599 847 806 •279401 152 194 I 2 •952543 143 172 •568 405 871782 •431595 1282x8 2 3 2-095715 160838 •440187 904511 •559813 095 489 3 4 •^S^S53 188857 •344 698 902 306 •655302 097 694 4 95 •445410 233 993 •247 004 890478 •752 996 X09522 95 6 •679403 312553 •137482 863 701 •862 518 136299 6 7 •991 956 487 139 •001 183 813347 •998817 186653 7 8 1 "479095 0-814530 691435 1-185 470 308565 8 9 •505 965 •494 035 9 7r, 3 J PERCENT. Values of Annuities, nnd Hincjlc and Annual Premiums for Assurance of a Unit. X "x A. ■CTj. X (^j Ax •cr^ 10 2i'i54.3 -250821 -01 I 322 ^^S 12-2094 '^53 306 •041 887 I 20-96} 6 •2'';7 268 •Oil 713 6 11-871 8 -564 720 -043 873 2 20-789 2 •263 168 -012 078 7 11-5263 -576406 -046 016 .3 20-627 9 -268621 -012 420 8 11-171 8 -588394 -048341 4 20-476 2 •273750 -012747 9 10-807 7 -600 706 -050874 »5 20-330 2 -278688 -013 065 60 10-4353 •613300 -053632 6 20-186 7 -283 543 •013383 I 10.0624 -625907 •056579 7 20-041 5 •288450 -013709 2 9-691 2 •638462 -059718 8 19-8907 -293 S^^ -014052 3 9-3242 -650872 -063 043 y 19-7299 -298990 -014423 4 8-963 I -663 084 -066554 20 19-558 -304800 -014826 ^S 8-6084 -675079 -070 259 I 19-3828 -310728 •015245 6 8-2556 •687 008 -074 226 2 19-2084 -316623 -015668 7 7-902 2 -698958 -078515 3 '9"o39 5 •i'^'2'3-iS -016085 8 7-5499 •710873 -083 144 4 18-8813 •327685 -016482 9 7-1994 -722 727 •088 144 ^5 18-7364 '33^5^5, -OI685I 70 6-8517 •734483 -093 545 6 18-6042 -337057 -017 193 I 6-509 3 -746060 -099351 7 18-4769 ■341359 •017526 2 6-182 I •757128 -105419 8 18-3523 •345 575 -017857 3 5-8740 -767547 -III 660 9 18-2256 -349 859 -018 198 4 5-5873 -777241 •117991 30 18-091 7 -354388 -018 562 75 5-3^S3 -786 102 •124280 I 17-9494 -359 199 -018 956 6 5-0892 -794083 -130408 1 2 17797 7 -364328 -019382 7 4-879 1 -801 191 •136279 3 17-6362 -369789 -019 842 8 4-6725 -808178 •142474 4 17-4688 •375451 -020329 9 4-464 7 •815 202 -149175 35 17-2946 •381 341 -020844 80 4-247 2 •822559 •156762 6 17-1130 •387485 -021393 1 4-005 4 -830 734 •16^^66 7 16-9252 •393 833 -021 971 2 3^730 2 -840042 '^11 59^ 8 16-731 I -400 396 -022 582 3 3^4463 -849 642 -191 089 9 16-5304 -407 183 -023 227 4 3^i73 I -858880 -205812 40 16-3228 •414 206 •023 911 85 2-922 8 -867 345 -22 1 103 I 16-1087 •421444 -024633 6 2-7147 -874383 •235 385 2 15-8878 -428914 -025 398 7 2-566 2 -879403 -246592 ,3 1 5 '659 4 -436637 •026 210 8 2-4809 -882 289 -253467 4 i5'423 5 -444617 •027 072 9 2-461 6 -882 942 -255070 4.'; 15-1798 •452857 •027 989 90 2-502 3 -881566 -251712 6 14-9280 -461 372 -028 966 1 2-5724 -879 194 -246 107 7 14-6661 -470 228 •030016 2 2-642 2 -876833 -240 742 8 14-393 I ■479460 •031 148 3 2-560 8 -879587 -247019 9 14-1085 -489 085 -032372 4 2-231 -890 739 -275685 50 13-812 -499 i 1 1 -033 696 95 1-8357 -904 106 -318828 I 13-505 1 -509 490 -035 125 6 1-4007 -918817 -382 729 2 13-1900 -520 146 -036 656 7 -9490 -934 093 -479276 3 12-868 2 '!^3 1 025 -038291 8 •483 1 •949 847 -640451 4 12-5408 -542 099 .040 035 9 •000 ■966184 -966184 FOUR PER-CENT. 7s 4 rKK-CENT. Comnnifdfion TnhJc. lO I 2 3 4 6 7 8 9 20 I 2 3 4 6 7 30 I 2 3 4 35 6 7 40 I 2 3 4 45 6 7 8 9 I 2 3 4 I>x f^'? 55'^>'4 64 754" I 62 00 r 2 59 3io'3 56693-6 54 l62"2 51722-9 49382-9 47 H7"9 45021-7 43 ooo*3 41 062-1 :^9 i95'o 37388-9 35^331 33 9^y9 32 268-4 30 674-2 29 145-9 27688-3 26305-1 249937 23 75 1 'o 22 573'3 21 452-1 20 385"3 93707 8 404-0 7483-1 6605-7 5770-0 4973-2 4 213-8 3 490-3 2 800-8 2 143-8 I 517-7 0922-3 o 35*^-4 9818-61 9307-30 8819-98 8353-90 7907-13 7478-28 N. I 308 776 I 244021 1 182 020 I 122 710 I 066016 I on 854 960 13 1-2 910748-3 863 600-4 818578-7 775578-4 734 5 J '5-3 695321-3 657932-4 622 298-7 588372-7 55^ 104-4 525430-2 496 284-2 468 596-0 442 290-9 417297-2 393 546-2 370972-9 349520-8 329 i35'S 309 764-8 291 360-8 273877-8 257272-1 241 502-1 226528-9 212 315-1 198 824-9 1 86 024- 1 173880-3 162 362-6 151 440-3 141 083-9 131 265-3 121 958-0 113 138-0 104 784-1 96876-96 89 398-69 S, 23 182 517 21 873741 20 629 720 19447 700 18324990 17258973 16 247 1 19 15286988 14376 240 ^3512639 12 694061 II 918482 II 1 83 966 I o 488 645 9830712 9 208414 8 620041 8063937 7538507 7 042 222 6573626 6 13^335 5714038 5320492 4949519 4599998 4 270863 3961 098 3669737 3 395 860 3 138 588 2 897 085 2670557 2458 241 2259417 2 073 392 I 899 512 I 737 "150 I 585 709 I 444 625 I 313 360 I 191 402 I 078 264 973 480-2 876603-2 M. 4 620-6 4416-6 4 154-3 3 848-0 3512-5 3 161-5 2 805-4 2 454-8 2 ] 19-1 I 806-3 ^5^6-s I 232-2 o 944-4 o 645-8 0328-6 9991-38 9638-62 9 285-^6 8937-08 8 600-40 8 282-21 7982-49 7 701-14 7436-91 7 183-90 6942-14 6 7 1 1 -6;;^ 6 489-99 6276-87 6071-94 5 874-90 5 684-64 5501-11 5324-28 5153-71 4989-02 4 830-00 4677-57 4531-77 4392-31 4 258-63 4 129-28 4002-44 3 876-97 3752-24 K. 43 I 7609 417 140-3 402 723-7 388569-4 374721-5 361 209-0 348 047-5 335 242-0 322 787-2 310668-1 298861-8 287 345-3 276 1 13- 1 265 168-7 254522-9 244 194-3 234 202-9 224564-3 215 278-7 206341-7 ^97741-3 1 89 459- 1 iSi 476-6 ^73 775-4 159 154-6 152 212-5 H5 500-9 139 010-9 132734-0 126662-1 120 787-2 115102-5 109 601-4 104277-1 99123-41 94 134-40 89 304-40 84626-82 80 095-05 75702-74 71 444-11 67314-83 63312-39 59435-42 10 1 2 3 4 ^5 6 7 8 9 20 1 2 3 4 25 6 35 6 40 1 2 3 4 45 6 50 I 2 3 4 79 4 PERCENT. Commutntion Table — (continued). X Dx Nx s. M, R. .r 5S 7 065-63 82333-06 787 204-5 3 627-22 ^5683-18 5S 6 566977 75 663-30 704871-5 3503-11 ^2 055-96 6 7 5 290"9i 69372-39 529 208-2 3380-78 48552-85 7 8 5 928-65 <^3 443-74 559835-8 3 260-48 45 172-07 8 9 5582-39 57861-35 496392-0 3 142-24 41 911-59 9 6o 5 250-66 52 610-69 438530-7 3025-22 38 769-35 60 I 4929-16 47681-53 385 920-0 2905-67 35 744-12 I 2 4.616-97 43 064-56 338238-5 2 783-06 32838-45 2 3 + 3i3'o5 38751-51 295 173-9 2656-72 30055-39 3 4 4017-07 34 734-44 256422-4 2526-63 27 398-67 4 66 3729-27 31 005-16 221 688-0 2393-33 24872-04 65 6 3451-81 ^7553-35 190682-8 2 259-3 1 22478-70 6 7 3 ^85-70 24367-65 163 129-5 2 i.2^-g6 20 219-40 7 8 2930-23 21437-42 138 761-8 1993-01 18093-44 8 9 2685-15 [8752-27 117324-4 I 860-64 16 100-43 9 7° 2 450-23 16302-05 98572-11 I 728-99 14239-80 70 I 2 224-89 14077-15 82 270-07 t 597-89 12 510-81 I 2 2 006-80 12 070-36 68 192-92 1465-37 10912-92 2 3 I 796-13 10 274-22 ^6 122-56 1331-89 9447-551 3 4 I 593-93 8 680-290 45 848-34 I 198-77 8115-662 4 75 I 401-20 7 279-095 37 168-05 I 067-34 6916-893 75 6 I2I9-5X 6 059-589 29 888-96 939-541 5 849-555 6 7 I 050-60 5008-991 23829-37 817-537 4910-014 7 8 899-308 4 109-684 18820-38 706-654 4092-477 8 9 765-230 3 344-454 14 710-69 607-165 3 385-823 9 8o 647-988 2 696-465 11 366-24 5^355 2778-658 80 I 547-185 2149-281 8669-773 443-475 2 259-302 I 2 461-119 1688-162 6520-493 378-454 1815-828 2 3 384-991 1303-171 4832-331 320-062 1437-373 3 4 316-410 986-761 3529-161 266-288 I J17-312 4 85 254-7 1 1 732-050 2 542-400 216-758 851-024 85 6 199-449 532-601 I 810-350 171-293 634-265 6 7 15^-095 381-506 I 277-749 130-610 462-972 7 8 110-857 270-650 896-243 96-1834 332-362 8 9 79-0684 191-581 6^5'593 68-6588 236-179 9 90 56\^°5 9 136-275 434-012 47-9374 167-520 90 I 38-552 5 97-723 297-737 33-311 I 119-583 I 2 27-0977 70-625 200-014 ^3-339 1 86-271 2 3 20-010 1 50-615 129-389 17-2943 62-932 3 4 15-783'^ > 34-831 78-774 13-8369 45-638 4 95 12-358 J 22-473 43-943 ii-oi8 ^ 31-801 95 6 9-404^ 13-069 21-47C 8-539 9 20-783 6 7 6-726 J 6-342 8-402 6-2236 12-243 7 8 4-283 ] 2-059 2-05$ 4-039 2 6-019 8 9 2-059: •00c •00c 1-980 c I -98c 9 80 4 PER-CENT. Logarithms and Co-Io^/ariflims of 1)^., N^., and M^.; with thi ir Dij)< renews. X LogD^ (Log vp^) Colog Dj. A (Colog r;j.r) LogN, A X lO 4-829667 T*98i 601 y 1 70 333 ©•oi8 399 6T1686'; 977 963 JO I •811 268 •981 132 •188732 •018868 •094 828 977 797 I 2 •792400 •980 730 -207 600 •019 270 •072 625 977 643 2 3 ■773 130 •980404 •226870 •019 ';96 •050 268977 496 3 4 •753 534 •980 J 62 • 246 466 •019 838 •027764 977 354 4 '5 •733 f^¥^ •979987 •266 304 •020 013 •005 118 977213 15 6 •71368.3 •979 894 •286317 •020 106 5*982 33 1 977067 6 7 •693577 •979885 •306423 •020 1 15 •959 398 976915 7 8 •67.3462 •979960 •326538 •020 040 ■936313 976 747 8 9 ■6^1 422 •980 050 •346578 •019950 •913 060 976566 9 20 •^?>^ 472 •979969 •366528 •020 03 1 •889 626 976375 20 I •61.3441 •979 790 •386559 •020 2]0 •866001 976185 1 2 •593 23 1 •979 5^2 •406 769 •020488 •842186 975 995 2 3 •572743 •979 118 •427257 •020882 *8i8i8i 975818 3 4 •551861 •978671 •448 139 •021 329 •793 999 975 654 4 ^5 'S?>°S^''- •978 245 •469 468 •02 1 755 •769653 975 503 25 6 •508777 •977996 •491 223 •022 004 •745 ^S^ 975 359 6 7 •486773 •977805 •513227 •022 [95 1^0 ^ir, 975216 7 8 •464578 ■977718 •535422 •022 282 •695 73 1 975 068 8 9 •442 296 •977745 •557 704 •022 255 •670799 974909 9 .30 •420041 •977 789 •579959 •022 211 •645 708 974738 30 I •397 830 •977852 •602 170 •022 148 •620446 974550 I 2 •.375682 •977913 •624318 •022 087 '594 996 974346 2 3 '?>S?> 595 •977875 •646405 .022 125 •569342 974 131 3 4 •331470 •977846 •dd^si^ •022 154 •543 473 973902 4 35 •309316 •977829 •690 684 •022 171 •'^n?>is 9n(>5l 3^ 6 •287 145 •977768 •7J2 8.';5 •022 232 •491 032 973 399 6 7 •264913 •977 704 •735 087 •022 296 •464 43 1 973 i26 7 8 •242 617 •977640 •1^12>'^?> •022 360 •437557 972836 8 9 •220 257 •977574 119 U?, •022 426 •410393 972528 9 40 •197 831 •977 483 •802 169 •022 517 •382921 972 203 40 I •175314 •977395 •824686 •022 605 •^^S 124 971857 I 2 •152709 •9773J1 •847291 •022 689 •326981 971490 2 3 •130 020 •977218 -869 980 •022 782 •298471 971 098 3 4 •107238 •977 ^16 •892 762 •022 884 •269569 970681 4 45 •084354 •977012 •915646 •022988 •240 250 970236 45 6 •061 366 •976948 •938 634 ■023 052 •210486 969 755 6 7 •038314 -976895 •961686 •023 105 •180 241 969 236 7 8 •015 209 •976841 •984791 •023 159 •149477 968 673 8 9 3-992 050 •976774 4-007 950 •023 226 •118 150 968 060 9 .50 •968 824 •976643 •031 176 •023357 •086210 967399 50 I •945 467 •976422 •054 s:i3 •023578 •053 609 966686 I 2 -921 889 •976 130 -078 III •023 870 •020 295 965 926 2 3 •89S019 •975 783 •loi 98; •024 217 4-986221 965 no 3 4 -873802 •975349 •126 198 •024651 ■9 5-^33^ 964 243 4 81 4 PERCENT. Logarithms and Co-logarithms of T)^, N^., and Islj.) with their Differences. X Colog N.,. A LogM, A Colog M,, A X \o 7-883 135 022 037 4^164964 993 899 5'835 036 006 10 1 \o I •905172 022 203 •158863 992 025 •841 137 007975 I 2 ■9'^lilS 022357 •150888 990 499 •849 112 009 501 2 3 •949732 022 504 •141 387 989348 •858613 1 65 2 3 4 ■972236 022 646 •130735 988572 •869 265 01 1 428 4 1.5 _-994 882 022 787 •119307 988087 •880 693 01 1 913 15 6 6-017 669 022 933 •107394 987 943 •892 606 012057 6 7 •040 602 023 085 •095337 988 134 •904 ^^^?> on 866 7 8 •063 687 023 253 •083471 988 644 •916 529 on 356 8 9 •086940 023 434 •072 115 989 207 •927885 010793 9 20 •110374 023 625 •061 322 989 142 •938678 010858 20 I •133999 023 815 •050 464 988728 •949 S?>^ on 272 I 2 ••157814 024005 •039 192 987 985 •960 808 012 015 2 3 •181 819 024 182 •027 177 986 865 •972 823 013135 3 4 •206001 024346 •014 042 985 583 •985958 014417 4 25 •230347 024497 3^999625 984390 4"ooo375 015 610 25 6 •254844 024 641 •984015 983 793 •015985 016207 6 7 •279485 024 784 •967 808 983 388 •032 192 016612 7 8 •304269 024932 •951 196 983323 •048 804 016677 8 9 •329201 025 091 ■9?>\S''9 983 627 •065 481 016373 9 30 •354 292 025 262 •918 146 983 992 •081 854 016008 30 I •379554 025 450 •902 138 984417 •097 862 015583 I 2 •405 004 025 654 •886555 984 838 •113 445 015 162 2 3 ■■430658 025 869 •871393 984 967 •128 607 015033 3 4 •456527 026 098 ■856360 985 134 •143 640 014866 4 35 •482 62^5 026343 •841 494 985334 •158506 014666 ?>S 6 •508 968 026 601 •826828 985416 •173 172 014584 6 7 •s?>s 569 026874 •812 244 985 499 •187756 014501 7 8 •5'52 443 027 164 •797 743 985 585 •202 257 014415 8 9 •589607 027472 •783328 985 673 •216672 014327 9 40 •617079 027797 •769001 985 702 •230999 014 298 40 I •644 876 028 143 •754 703 985 748 •245 297 014252 I 2 •673 019 028510 •740451 985810 •259 549 014 190 2 3 •701529 028902 •726 261 985 859 •273 739 014 141 3 4 •730431 029319 •712 120 985 895 •287880 014105 4 45 •759 750 029 764 •698015 985932 •301985 014068 45 6 •789514 030 245 •683 947 986074 •316053 013 926 6 7 •819759 030 764 •670 021 986 247 •329979 013753 7 8 •850523 031327 •656268 986425 •343 732 013575 8 9 •881 850 031940 •642 693 986577 •357307 013423 9 50 •913790 032 601 •629 270 986 605 •370730 013395 50 1 •946391 033 314 •615875 986450 •384125 013550 I 2 ■979705 034074 •602 325 986 168 •397 675 013 832 2 3 5^013 779 034890 •588493 985 798 •411507 014 202 3 4 •048 669 035 757 •574291 985 282 •425 709 014 718 4 S2 4 PEH-CENT. Logarithms ami Co-lofjarithins of Dj., N^., and M,^; j/^iV/e Mt/V X'///'" renas — (continued). Log Dj. L, V 6 7 8 9 6o I 2 3 4 6 70 I 2 3 4 75 6 9 80 I 2 3 4 6 7 8 9 90 I 2 3 4 95 6 7 8 3"849 '5'' •824 1 1 1 ■7987131 ■772 9./^! 746 820J •720 214 •692773! •^'34 7«5l •60J 910^ •571 624 •538047 •503 205 •466901 •428 969 •389 206 •347 309 •302 504 ■254339 •202 470 •146499 •086 184 •021437 2'953 908 •883 792 •811 567 •738 134 •663813 •585 450 ■500 250 •406 04 •299832 •179249 •044 762 I • 898 003 ■74277' •5860:52 •432933 •301 261 •198 207 "091 950 0-97332 •82777.^ •631 763 •313699 Colog IV •974960 •974 602 •974 243 •973864 •973 394 •972 559 •971584 •970 42S •969 125 •967714 •966 423 •965 158 •963 696 •962 068 •960237 •958 J 03 •955 195 •948 131 ■944 029 ■939 ^^85 ■935 253 ■932471 A (Colog f/jj.) 4^1 50 849 ■175889 •201 287 •227 044 •253 180 •279 786 •307 227 •3?,':, 643 ■365215 •396 090 •428376 •461 953 ■496 79. '^33 099 •57^031 •610 794 •652 691 ■697 496 ■745661 ■797530 ■853501 •913 816 ■978563 929 884 3'046 092 •927 775 •926567 •925 679 •921637 •914 800 •905 797 ■893 785 •879417 •865513 •853 241 •844 768 •843 28J •846881 •868328 •896946 ■893 743 •881375 •854448 •803 990 •68x936 •116 208 •188433 •261 866 •336187 •414550 •499 750 •593 953 •700 168 •820 751 •955 238 2^JOI 997 •257 229 •413948 •567 067 •698 739 •801 793 •908 050 7-026 675 •172 227 ■368237 •686 301 Log Nj -025 040 •025 398 •025 757 •026 136 •026606 •027441 •028 416 •029572 ■030 875 •032 286 ■033577 •034 842 •036304 ■037932 •039 763 •041 897 •044 805 •048 165 •051 869 ■055971 •060315 •064 747 •067 529 •070 1 16 •072 225 •073433 •074321 •078363 •085 200 •094 203 •106 215 •120583 ■134487 •146759 ■'55232 •156719 '^53 119 •131 672 ■103054 •106257 •118 625 ■145552 •196010 •3 1 8 064 4^915 574963 31 1 •878885I962302 •841 187,961 202 •802 3S9960 000 7623891958^85 957 276 955 770 954 ^69 •721074 •678 350 •634 120 •588 289 •540 760 952471 950674 948 740 946 640 944 359 941 881 939188 936273 933 205 930029 926785 923 543 920366 917307 ■6997501914058 •613 808910517 ■524325906470 ■430 795 901 498 •332293895 121 •227 414887 587 •1 15 001 879 21 1 2^994 212 864541 •491 434 •440 174 •386814 •33^ 173 ■273 054 •212 242 ■148515 •081 720 •oil 749 3^938 534 862 077 782443 •726402 •581 502 •432 407 •282 o^S3 ■134417 1^989996 •848 960 ■704275 •541 965 •351659 •116229 0-802 250 •313699 870329 861 861 855 100 850905 849 946 852 064 855 579 858 964 855315 837690 809 694 764570 686 02 1 511 449 55 6 7 8 9 60 3 4 6 70 1 2 3 4 75 6 7 8 9 80 T 2 3 4 85 6 90 I 2 3 4 9S 6 7 8 83 4 PERCENT. Logarithms and Co-logarithms of Dj., N^., and M^; wkh their Differences — (continued). X CologN^ A LogMj, A CologM^ A X 55 5-084426 036689 3-559573 984881 4-440427 015 119 55 6 •121 115 03 7 698 •544454 984 5^3 •455 546 015437 6 7 •158813 038 798 •529017 984265 •470983 015735 7 8 •197 611 040 000 •513282 983958 •486718 016042 8 9 •237 611 0413^5 •497 240 983 5^8 •502 760 016482 9 6o •278926 042 724 •480 758 982488 •519242 017512 60 I •321 650 044 230 •463 246 981277 •536754 018723 I 2 •36^ 880 045 83 1 •444523 979823 •555477 020 177 2 3 •411 711 047 529 •424346 978 196 •575 654 021 804 3 4 •459 240 049326 •402 542 976461 •597458 023 539 4 65 •508 ^^66 051 260 •379003 974972 •620 997 025 028 65 6 '559 '826 053 360 '353 975 973 579 -646 025 026421 6 7 •613 186 °55 641 •327554 97^955 -672 446 028 045 7 8 •668827 058 119 •299 509 970 152 •700491 029 848 8 9 •726946 060812 •269 661 968 130 •730339 031 870 9 70 787 758 063 727 •237791 965 75*^ •762 209 034 244 70 I •851485 066 795 •203 547 962400 •796453 037 600 I 2 •918 280 069971 •165947 958521 •834 053 041 479 2 3 •988 251 073215 •124468 954 268 •875532 045 732 3 4 4-061 466 076457 •078736 949 5^^ •921 264 050 434 4 75 •137923 079 634 •028 302 944614 •971 698 055386 75 6 •217557 082 693 2-972916 939591 3-027084 060 409 6 7 •300 250 085 942 •912507 936700 -087493 063 300 7 8 •386 192 089 483 •849 207 934 100 •150793 065 900 8 9 •475675 093 530 •783307 932 158 -216693 067 842 9 80 •569 205 098 502 •715465 931404 •284535 068 '596 80 I •667 707 104879 •646 869 93 i H4 '353 131 068 856 I 2 •772586 112 413 •578013 927 221 •421 987 072779 2 3 •884999 120 789 •505 234 920 118 •494 766 079 882 3 4 3^005 788 129 67 J •425 352 910 623 •574648 089377 4 85 '^35^^59 138139 '335975 897 765 •664 025 ro2 235 85 6 •273598 144900 '^33 740 882237 •766 260 117 763 6 7 •418498 149 095 •115977 867 123 •884 023 132877 7 8 •5'57 593 150 054 1^983 100 853 59<^ 2-016900 146404 8 9 •717647 147936 •836 6c)6 843 978 •163304 156022 9 90 •865583 144 421 •680674 841 915 •319326 158 085 90 I 2'OIO 004 141 036 •522589 845 496 •477 41 1 154504 I 2 •151 040 144685 -368085 869817 •63 1 915 130 183 2 3 •295 725 162 310 •237902 903 ^37 •762 098 096 863 3 4 •458 035 190306 •141039 901 080 •858961 098 920 4 95 •648341 235430 •042 119 889 336 •957881 no 664 95 6 •883771 313979 0-931455 862588 I -068 545 137 412 6 7 I •197750 488551 •794043 812 253 •205 957 187747 7 8 •686301 ■606 296 690370 -393 704 309630 8 9 •296 666 -703 334 9 •^i 4 I'KKCKNT. I'tiliu'^ of Aniiuiticji, and Single and Jnnnal Premiums fur Assurance of a Unit. X "j A. ^j X "i A. ifff lO i9'373 1 •216420 •010623 ss 11-652 6 •5^3361 ■040573 I 19-2115 -222 636 •01 1 015 6 11-3442 -525 222 •042 548 2 19-0645 -228 290 •on 378 7 11-0274 •537 408 •044 682 3 18-9294 '^33 484 •on 716 8 10-701 2 •549954 •047 000 4 18-803 I •238342 •012 036 9 10-3650 -562 885 •049528 15 18-6819 •243 003 •0 1 2 346 60 10-019 8 -576161 •052 284 6 18-563 •247577 •012 655 1 9'673 4 -589486 •055 230 7 18-442 6 •252 209 -012 972 2 9^327 5 -602 790 •058 368 8 18-3168 ■257045 -013307 3 8-984 7 -615973 •061 692 9 18-1819 •262 236 -OI3671 4 8-646 7 •628973 •065 201 20 18-0366 •267 824 -014069 65 8-3140 •641 770 •068 904 I i7-8'87 9 •273541 •014482 6 7-982 3 •654517 •072 869 2 17-7400 •279 229 -014900 7 7-649 1 •667 343 •077 158 3 J 7 '597° •284731 -015 311 8 7-3160 •680155 •081 789 4 17-4638 •289855 -015699 9 6-983 7 •692 935 •086 794 25 17-3429 •294506 -016056 70 6-6S3 3 •705643 •092 201 6 i7'233 7 •298 702 -016382 I 6-327 I -718188 •098018 7 17-1294 •302 716 -016698 2 6-0147 •730202 •104095 8 17-0276 -306632 -017 009 3 5-7202 •741 531 •110344 9 16-9240 •310615 -017330 4 5-445 8 -752083 •116677 30 16-8139 •314852 -017675 75 5^^94 9 -761 734 •122961 I 16-696 I •319380 -018048 6 4-968 9 -770427 •129074 2 16-5696 •324 244 •018455 7 4-7678 -778163 •134916 3 16-434 I •329456 •018897 8 4-5698 -785776 •141077 4 16-293 I •334881 -019365 9 4-3705 •793442 •147740 3.5 16-1458 •340 547 -019 862 80 4-1613 -801 489 •155289 6 15*991 4 -346 484 -020 392 1 3-9279 ■810466 •164465 7 I5'83i4 •352 640 -020951 2 3-661 -820 730 -176084 8 ^S'(^^S3 •359026 -021 543 3 3-3849 ■831348 -189592 9 1 5 '4930 •3^S 654 -022 170 4 3-1186 •841 592 •204339 40 i5'3Ho •372537 -022 835 85 2-8740 •8 150 998 •219666 I 15-1290 •379655 -023 539 6 2-6704 -858832 •233991 2 14-9373 •387027 -024 284 7 2-5249 -864425 •245231 3 1473B4 -394 676 -025077 8 2-4414 -867637 •252115 4 14-532 2 •402 608 -025921 9 2-4230 -868347 •253681 4.'; 14-3185 •410 829 -026819 90 2-4640 -866768 •2^0 220 6 14-096 8 •419354 •027778 1 2-534 8 -864046 •244440 7 13-865 3 •428 260 -028 809 2 2-6063 -861 296 •238830 8 13-622 9 •437582 -029925 3 2-5294 •8642^15 •244874 9 13-3690 •447 345 •031 i33 4 2-2068 -876662 •273378 50 13-103 5 •457 558 •032443 95 1-818 <; •891 597 •316340 I 12-827 5 •468 174 •033 858 6 1-389 6 -908 091 •380 01 1 2 12-543 I •479110 •035377 7 •9429 •925272 -476 226 3 12-2519 •490313 -037 000 8 •4808 •943 048 •636 863 4 "■9545 •501 752 •038732 9 -000 •961 539 •961 539 FOUR AND A HALF PEE-CENT. 80 4] J'KK-CENT. Cum mutation Table. X lO I 2 3 4 n. Nx s. M, K. X 64392-8 61 4264 58533-6 s^ 725-2 53011-8 I 148675 I 087 248 I 028 715 972989-5 9'997/-7 19 196 516 18047 842 16960594 159.31 879 14958889 12 155-4 1 1 961-9 •I 714-3 1 1 426-5 1 1 1 1 2-8 334 185-9 322030-4 310068-5 298 354-2 286927-8 10 I 2 3 4 6 7 8 9 50 402-5 47 902-2 45516-3 43 248-3 41 100-4 869575-2 821 673-0 776 156-7 732908-4 691 8080 14 038912 ^3 169336 12347663 'I 571 507 10838598 10 786-2 10456-4 10 133-2 9 825-30 9539-76 27981^-0 265028-8 254572-4 244439-2 234613-9 15 6 7 8 9 20 I 2 3 4 39 067-2 37127-8 35 270-0 33 483-8 31759-2 652 740-8 6i<) 613-0 580342-9 546859-1 515099-9 10 146 790 9 494 050 8878437 8 298 094 7751234 9 276-46 9019-34 8 760-39 8492-96 8210-28 225074-1 215797-7 206778-3 198 017-9 189525-0 20 I 2 3 4 ^5 6 7 8 9 30 092-5 28485-2 26 948-4 25483-3 24092-9 485 007-4 456522-2 429573-8 404 090-5 379997-5 7 236 135 6751 127 6 294 605 5 865 03 1 5460941 7911-15 7 599-76 7 289-58 6984-89 6691-92 181 314-7 173403-5 16^; 803-8 158514-2 151529-3 25 6 7 8 9 3° I 2 3 4 22 779-9 21 540-6 20371-7 19 268-9 18 224-2 357217-6 335677-0 315305-2 296036-3 277 812-1 5 080 943 4 723 726 4388049 4072743 3776707 6416-38 6158-07 5916-74 5 691-20 5476-26 144837-4 138421-0 132 263-0 126346-2 120 655-0 30 I 2 3 4 35 6 7 8 9 17 235-1 16 298-9 ^5 41 1-5 14570-2 13772-8 260 577-0 244 278-1 228866-7 214 296-5 200523-7 3 498 895 3238318 2 994 040 2765 ^13 2550877 5 271-86 5077-90 4892-30 4714-68 4544-72 115 178-8 109906-9 104 829-0 99936-71 95 222-03 3S 6 7 8 9 40 I 2 3 4 13017-1 12 300-2 1 1 620-5 10976-2 10365-4 187 506-6 175 206-4 163585-9 152 609-7 142 244-2 ^-3<>oo,^^ 2 162 846 I 987 640 1 824054 1671445 4382-08 4225-78 4075-74 3931-86 3 793-74 90677-31 86 295-24 82 069-46 77993-72 74061-87 40 I 2 3 4 45 6 7 8 9 9786-37 9237-41 8717-96 8 226-73 7762-21 132457-8 123 220-4 114502-5 106 275-8 98513-54 I 529 200 1 396 743 I 273522 I 159020 1 052 744 3 661 -02 3 533-48 3411-82 3 296-00 3 185-74 70 268-13 66607-12 63 o-]y63 59 661-82 45 6 7 8 9 50 I 2 3 4 7322-78 6906-16 6509-92 6 132-29 5771-94 91 190-76 84 284-60 77774-67 71642-39 65870-44 954230-3 863 039-5 778 754-9 700980-3 629337-9 3 080-57 2979-29 2 880-44 2783-14 2 686-87 c^^ 180-07 50099-51 47 120-22 44239-77 41 456-64 50 I 2 3 4 87 4^ PER-CENT. Commutation Table — (continued). X 55 D. N. S. M, Rx X 5 427'3'5 50443-09 ^ 563467-4 2590-83 ^ 38769*77 55 6 509877 x'; 344*32 , 503 024-4 1495-96 , 36178-94 6 7 + 786-14 50558-18 . 1-47 680-0 1402-89 33682-98 7 8 + 488-95 4.6069-23 , 397 121-9 2 311-80 31 280-10 8 9 + 206-55 +1 862-68 351052-6 2 222-71 28968-29 9 6o 3937*65 37925*03 309 189-9 2 134*95 26 745*59 60 I 3678-85 34246-18 271264-9 2045-72 24610-64 I 2 3429*37 30816-81 237018-7 I 954*65 22 564-92 2 3 3 188-30 27 628-52 206 201-9 I 861-26 20 610-27 3 4 2955*29 24673-22 178573*4 I 765*55 18 749-01 4 65 2730-44 21 942-78 153 900-2 I 667-96 16983 -46 65 6 25x5-20 19427-58 131-957*4 I 570-30 15315*50 6 7 2310-19 i-j 117-40 112 529-8 1 473*59 13 745*21 7 8 2 114-76 15 002-64 95412-42 J 377*64 12 27 1-62 8 9 1928-61 13074-02 80 409-78 1282-57 10893-97 9 7° 1751*46 11322-57 67335*76 I 188-46 9611-403 70 J 1582-78 9739*789 56013-19 I 095-20 8422-941 I 2 I 420-79 8318-996 46273-40 I 001-38 7327*739 2 3 I 26^-56 7 053*434 37954*41 907*327 6 ^2&2,(i2 3 4 I 117-72 5935*719 30900-97 813-979 5419*035 4 75 977-861 4957*858 24965*25 722-256 4605-056 75 6 846-992 4 IIO-866 20007-39 633*496 3 882-800 6 7 726-188 3384*677 15 896*53 549-166 3 249*303 7 8 618-640 2 766-037 12511-85 472-889 2 700-138 8 9 523-889 2 242-148 9745*815 404*777 2 227-249 9 So 441-500 I 800-648 7 503*667 344-949 I 822-472 80 I 371*035 1429-613 5703-019 293-495 1477*524 I 2 311-180 I 118-433 4273*406 249-617 I 184-028 2 3 258-563 859-871 3 154*973 210-400 934*411 3 4 211-486 648-384 2 295*102 174*458 724-01 1 4 85 169-432 478-952 1646-718 141-511 549*552 85 6 132-038 3A^'9'^S I 167-766 111-413 408-041 ' 6 7 99-548 247*367 820-851 84-609 1 296-628 7 8 72-6879 174-679 573*485 62-035 8 212-019 8 9 5^*5965 123-082 398-806 44-0744 149-983 9 90 35*9175 87-165 275-724 30-6173 105*909 90 I 24*9175 62-247 188-559 2 1 • 1 64 75*291 I 2 17-4302 44-817 126-312 14*749 7 54*127 2 3 12-8095 32-007 81-495 10-8800 39*378 3 4 10-055^ 21-952 49-488 8-6773 28-498 4 95 7*835^ ) i\-iit 27*53^ 6-890 3 19-820 95 6 5*9342 8-182 13-420 ST-^3 12-930 6 7 4-224 c > 3-95^ 5*235 3*871 7 7-604 ■ 7 8 2-6765 ) 1-281 1-281 2-506 j 3*732 8 9 1-280^ i -ooc •ccc 1-225} V22t S 9 88 4,j PERCENT. Logarithma and Co-^o'/ariflims of Dj., Nj., and M^.; tcitli tilt ir ])ij}'i fences. X LogD^ A (Log vp^) Colog Dx A (Colog r/)j.) LogN, A X lO 4-808837 1-979518 5-191 163 0^020482 6-060 197 976 132 10 I •788.355 •979050 •21 1 645 •020950 -036 329975 966 I 2 •767 405 •978647 •232 595 •021 i53 •012295975813 2 .3 •746052 •978321 •253 948 •021 679 5-988 108 975 669 3 4 724373 •978079 •275627 •021 921 •963777 975 530 4 1 5 •702452 •977904 •297 548 •022 096 •939307 975 392 15 6 •680 3 r/i •977 81 r •319644 •022 189 -914699 975251 6 7 •658 167 •977 802 •341 833 •022 198 -889950 975 100 7 8 •635 969 •977877 •364031 •022 123 -865 o-jo 974936 8 9 •613846 •977967 •386 154 •022 033 •839986 974 155 9 20 •591 813 •977886 -408 187 •022 1 14 -814 741 974 567 20 I •569 699 •977 707 •430301 •022 293 -789308974377 1 2 •547 406 •977429 •452594 •022 571 •763 685 974 190 2 .3 •524835 •977035 •475 165 •022 965 •737875974017 3 4 •501 870 •976588 •498 130 •023 412 -711 892973 856 4 ^5 •478458 •976 162 •521542 •023838 •685748973714 25 6 •454 620 •975 9H •545 380 •024086 -659462973576 6 7 •430534 •975721 -569 466 •024279 •633038973441 / 8 •406 255 •975 ^35 •593 745 •024365 -606479973302 8 9 •381 890 •975 662 -618 liO •024338 •579781 973 152 9 ,30 '357 ^^^ •975 707 -642 448 •024 293 •552933 972989 30 I '333 259 •975 769 -666 741 •02423X -525922 972 809 1 2 •309028 •975 830 -690972 •024170 •498731 972 614 2 3 •284858 •975 791 •7^5 H2 •024209 •471345 972406 3 4 •260 649 '915 764 ■139 35^ •024236 •443 751 972 185 4 35 •236413 '915 746 •763 58; •024254 •415936 971949 35 6 •212 159 •975 685 -787841 -024315 •387885 971698 6 7 •187844 •975621 -812 156 -024379 •359583 971432 7 8 •163465 •975557 ■^3^5?>5 -024 443 '3?>'^ 015 971 151 8 9 ■139 022 •97549^ -860978 •024 509 •302 166 970851 9 40 •I '4 513 •975400 -885487 •024600 •273017 970533 40 I •089913 •975312 -910087 •024688 •243 550 970 196 I 2 •065 225 ■975228 •934775 •024772 •2x3746 969836 2 .3 ■040 453 '975^35 '959 547 •024 865 -183582 969 453 3 4 •015588 '915 033 -984412 •024967 '^53^35 969 043 4 45 3*990 621 •974929 4-009379 •025 071 •J22 078 968 605 45 6 ■9'55 550 •974865 •034 450 •025 135 •090 683 968 132 6 7 •940415 •974812 •059 585 ■025 188 -058815 967 619 7 8 •9J5227 ■974 759 •084 113 -025241 -026434 967 062 8 9 •889986 •974690 -110014 -025310 4-993 496 966455 9 50 •864676 •974561 •135324 •025 439 '95995-^ 965 797 50 I •839237 •974 339 -160763 -025 661 -925748965090 I 2 •813576 •974046 -186424 •025 954 -890838964332 2 3 •787 622 •973 700 •212378 -026300 ■855 170963 521 3 4 •761 322 •973 266 •238678 -026734 •818 691 962 656 4 89 4i PERCENT. Logarithms and Co-lugarithms of D^, Nj., and ^l^\ vnth their Differ ences. X 10 Colog N^. A Log M, A Colog M^. A X 7*939803 0338684-084770 993 03 1 5-915230 006969 10 T •963671 024034 •077 801 990914 •922 199 009 086 1 2 J987 705 024 187 •068715 989 198 •931 285 010 802 2 1 3 6'oi I 892 024331 •057913 987 909 •942087 012091 3 ! 4 •036223 024470 •045 822 987 046 •954178 012954 4 1 15 •060 693 024608 •032868 986 5 r4 •967 132 013 486 15 6 •085301 024749 •019382 986365 •980618 013 60,^ 6 7 •no 050 024900 ■005 747 986 599 ,•994253 013401 7 8 •134950 025 064 3^992346 987 191 4-007 654 ! 012 809 8 9 •160 014 025 245 ■979537 987 845 -020463 012 155 9 20 •185259 025433 •967382 987793 •032 618 012 207 20 I •210 692 025623 -95S^15 987348 •044825 012 652 1 1 2 ■^S^i'^S 025 810 •942523 99>6c^o^6 •057477 013 464 1 2 1 3 •262 125 025 983 •929 059 985 299 •070941 014 701 3 4 •288108 026 144 •914358 983 882 •085 642 016 118 4 25 •314252 026 286 •898 240 982 560 •loi 760 017 440 25 6 '340538 026424 ■880 800 I981 902 •I 19 200 018 098 6 7 •366962 026 559 •862 702 981 458 •137298 018 542 7 8 •393 5'^'^ 026 698 •844 160 981 391 •^^5840 018 609 8 9 •420 219 026 848 •825 ss I 981739 •174449 018 261 9 30 •447 067 027 on . ^807 290 982154 •192 710 017 846 30 I •474078 027 191 •789444 982 639 •210556 017361 I 2 •501 269 027 386 •772083 983 T20 •227917 016 880 1 2 1 3 ■^s^'^^ss 027594 •755 203 983281 •244797 016 719 3 4 •5S^ H9 027815 •738484 983 480 •261 516 016 520 4 35 •584064 028 051 •721964 983 720 •278 036 016280 3!^ 6 •612 115 028302 •705684 983 829 •294316 016 171 6 7 •640417 028568 •689513 983 940 •310487 016060 7 8 •668 985 028 849 ■673 453 984 054 •326547 015946 8 9 •697 834 029 149 •657507 984173 •342 493 015827 9 40 •726983 029467 •641 680 984227 •358320 015773 40 I •756450 029 804 •625907 984299 •374093 015 701 1 2 •7862';4 030 164 •610 206 984393 •389 794 015 608 2 3 •8164^18 030547 •594 598 984 469 •405 402 015 531 3 4 ■846965 030957 •579067 984535 ■420933 015465 4 45 •877922 031395 'S^3 602 984601 •436398 015 399 45 6 •909317 03 1 868 •548 203 984 783 •451 797 015217 6 7 •941 185 032381 •532986 985 001 •467 014 014999 7 8 •973 5'56 032938 •517987 985224 •482 013 014776 8 9 5-006504 033 545 •503 211 985420 •496 789 014580 9 5° •040049 034203 •488 60, 1 985482 •511 369 014518 50 1 •074252 034910 •474 I J3 985 346 •525887 014654 I 2 •109 162 035 668 •459 459 985 076 •540541 014924 2 3 •144830 036479 •444535 984711 ■sss 465 015 289 3 4 •181309 037344 •429246 984193 ■570754 015 807 4 00 41 rKKcKNT. Logarithms and Cu-logurithms of D,, X^, and .M_r ; »///' their Difjerences. LoK D, /T .IS 6 7 8 9 6o I 2 ^.5 6 7 8 7."; 6 7 8 9 8o I 2 .3 4 85 6 7 8 90 I 2 3 4 9.-; 6 7 8 r7.34.s88 ■707 4^S •679985 •65 2 H.S •623 926 •.595^.37 •.S65713 ■.S3.5 2i4| •503:) 59; •470601, ■436 233 •400 .S 7 3 '3^3 647 •321; 261 •285245 •243 400 •199420 •1.S2531 •102 283 •048331 2-990277 •927 880 •861 049] •791438 •719239 •644 93 1 •569 4 1. S •493 on •412 566 •325 282 •228996 •120 698 1-9980331 •861 462J •712 6201 •.S5.S 306 •396 504 •241 301 •107546 •002 409' 0*894 070 773 3<^^ 625 727 427 634 107487 " (Uolog//;^.) " •972877 •972520 •972 i6o ■971 781 •971311 •970476 •969 i;©! •9^'8 34.S •967 042 •965632 •964340 ■963 074 •961 614 •959984 •958 ^55 •956020 •953 1 1 ^ •949752 4-265 4120 •292535 •320015 •347 855 •376074 •404763' •434287 •464 786| •496441 •529399 '^^?> 767: •599427: ••'^36 353; •674 739; •7^4 755: •7156600 •800580' 847 469 •946 048 I ^897 7 1 7 •941 946 '937603 "933 169 "930389 •927801 •925 692 •924484 •923 596 •9 '9 555 •91 ■16 •903 7 H •891 702 •877335 •863429 •851 10 •842 686 -841 198 •844 797 •866 24 5 •894 863 •891 661 •879292 •852365 -801 907 •679 853 951669 3-009 723 -072 120 •^38 95^ •208 562 •280 761 . I •355069' •430585 •506989, •587 4341 -674718; •77J004| _-879 302| 2-001 967 •I38538I •287380 ■444 694 •603 496, •758699' •892 454 •997591 [-1015 930 •226638 •374273 •572366 -892513 027 123 4-781347 961 726 027480 -743073 960719 027840 -703 792 '959 619 028219 -663 411 '958416 028689 -621 827 957 099 •029524 -578926 955686 •030499 '534612 954 176 •031655 -488788 952570 •032958 -441358 950868 •034368 -392226 949066 •035 660 -341 292 ] 947 127 •036926 -288419 945019 •038386 -233438 942 730 •040016 -176 168 940241 •041 845 -116409 937536 043 980 -053 945 ! 934 605 931 521 928330 925072 921 821 •046 889 '3-988 550 -050248 -920071 •053 952 I •018 054 -848401 'J" ^.)t : '"73 473 •062397 -695 294 1918 639 •066831 •6139331915584 •069 61 1 I -529517 I 912 341 •072199 -441 858 908 806 •074308 -350 664 '904 765 •075516 -255429 899789 •076404 -155 218 893 392 •080445 -0486101885823 •087284 2-934433 877400 •096286 -811 833 868459 •108298 •6802921859930 •122665 -540222 853 119 •136 571 '393 341 1848 899 •148842 -242 240 847 955 •157314 '090 195 850 146 •158802 1-940341 '853 779 •^S5 203 "794 120 857323 '^^^l^S '651 443 853804 •105 137 -505247 j 836 217 -108339 -341 464 808 246 •120708 -149 710 763 136 •147 635 0^912 846 684600 •198093 -597446 510041 •320147 -107487 55 6 7 8 9 60 I 2 3 4 6 70 I 2 3 4 75 6 7 8 9 80 i 2 3 4 85 6 7 8 9 90 I 2 3 4 95 6 7 8 ( U^4IyERS!TY j Logarithms and Co-logarithms o/'D^., N^,, and M^,; with their Differences. X Colog N^ A Log Mj. • A Colog M^ A X S5 5*2 1 8 65,3 038 274 3^4i3 439 983 798 4-586561 016 202 55 6 •236927 039281 •397237 983 497 •602 763 01-6503 6 7 •296208 040381 •380 734 983 217 -619 266 016783 7 8 •336589 041 584 •36395^ 982931 •636049 1 7 069 8 9 •378173 042 901 •346 8S2 982 506 •653118 017494 9 6o •421074 044314 •329388 981458 •670612 018542 60 I •465 388 045 824 •310846 980 223 •689 154 019777 I 2 •511 212 047 430 •291 069 978 738 •708931 021 262 2 3 •558 642 049132 •269 807 977073 •730 193 022 927 3 4 •607774 050934 •246880 975305 '753 120 024 695 4 65 •658 708 052873 •222185 973 796 ■777815 026204 65 6 •711 581 054981 •195 981 972396 •804019 027 604 6 7 •766 562 057 270 •168377 970 760 •831623 029 240 7 8 •823832 059 759 •f 39 137 968 943 •860 863 031057 8 9 •88359^ 062 464 •108080 866 905 •891 920 033 095 9 70 ■946 OjJ 065 395 ■074985 964 509 •925015 035491 70 I 4^oii45o 06S479 •039494 961 103 •960 506 038897 I 2 •079929 071 670 •000597 957 J 67 ■999 403 042 833 2 3 ■151 599 074928 2 '95 7 764 952849 5-042 236 047 15 J 3 4 •226527 078179 •910 613 948078 •089387 051922 4 75 •304 706 081 361 •858691 943 053 •141309 056947 75 6 •386067 084416 •801 744 937 959 •198 256 062 041 6 7 •470483 087 659 •739703 935 °S'^ •260 297 064944 7 8 '55^ J 42 091 194 ■674 759 932457 •325241 067 543 8 9 •649336 095 ^35 •607 216 930538 •392 784 069 462 9 80 •744571 100 211 •537754 929 847 •462 246 070153 80 I •844782 106608 •467 601 929674 •532399 070326 I 2 •951 390 114 177 •397 275 925771 •602 725 074 229 2 3 3'o65 567 122 600 •323046 918646 •676954 08 1 354 3 4 •188 167 ^31541 •241 692 909 099 •758308 090901 4 85 •319708 140 070 •150 791 896 r45 •849 209 103 855 85 6 ■459 778 J46881 •046936 880481 •953 064 ii95'9 6 7 •606 659 151 iOI £•927417 86!^ 225 2-072583 134 775 7 8 •757 760 152045 •792642 85"^ 545 •207358 H8455 8 9 •909 805 149 854 •644187 841 780 •355^^3 158 220 9 90 2'o59 659 146 221 •485 967 839630 •5H033 160370 90 I •205880 142677 ' 3^=^597 843 185 •674403 ^56^'^5 1 2 •348557 146 196 •168782 867 847 •831218 ^32153 2 3 •494 753 163 783 •036629 901 757. ■963371 098 243 3 4 ■63S536 191 754 0-938386 899 85 1 i^o6i 614 100 149 4 95 •850 290 236864 •838237 888 192 •161 763 in 808 95 6 •087 154 315400 •726429 86x473 ■273 57 J f 38 527 6 7 •402 554 489959 •587 902 811 162 •412 098 188838 7 8 •892513 •399064 689307 •600936 310693 8 9 •088371 •91; 629 9 \)2 4:\ IKKCKNT. Vahte^s of AnnuitH's^ and tixngh'. and Ainmal Premiums far Assurance of a Unit. X "s A. tTj. J fs A. «r^ lO .7-8386 -.88770 -010020 55 ..■1367 -477365 •039332 I 177000 •194736 •0.04.4 6 1 0854 5 -489522 -041 294 2 • 7-5748 •200 .29 •010774 7 1 0^563 5 •502 052 •043417 .3 1 7-460 j; -205 05. •0. I 108 8 10^262 8 •5H999 •045 726 4 17-3542 -209 628 -Oil 421 9 995 » 8 -528392 -048 247 i.-; i7-2<;2 6 -214002 -01 1 724 60 9^6314 •542 189 •050 999 6 17-153 I •218286 -012 025 1 9"3o8 9 -556 075 -053941 7 .7-0523 •222 628 -012332 2 8^986 2 •569974 -057 076 8 16-9465 -227184 -012 659 3 8^66 <; 6 ■583 778 -060398 9 .6832 I •232 109 -013 016 4 8^348 8 -597419 -063 903 20 16-708 I -237449 -013409 ^\^^ 8-0364 •610875 •067 602 1 1 16-5809 ■242927 •013818 6 7-724 i ■624323 -071563 2 16-4543 -248380 ■014230 7 7-4095 -637867 •075851 3 16-3320 •253 643 •014634 8 7-0943 '(^5 1 444 •080482 4 16-2189 -258517 -015014 9 67790 ■66^ 020 ■085 489 25 .6-1172 -262 895 ■015358 70 6^464 7 -678555 •090 903 6 16-0266 -266 796 -015 669 1 6-1536 ■691950 ■096 727 7 15-9406 -270501 -015968 2 5-8552 -704801 -102 813 8 15-857 I -274097 •016 260 3 5-573 4 -71.6937 -109067 9 15-772 I -277 755 -016561 4 5-3106 -728252 -115402 ,30 15-681 3 -281 668 -016885 75 5-070 I -738608 •121 679 I 15-5834 •285881 ■017239 6 4-853 5 -747937 •127776 2 J 5-47 7 <5 •290439 ■017 626 7 4-6609 -756230 -133589 3 15-3634 -295 35^ ■018 050 8 4-47 1 2 -764400 -1397^5 4 15-2441 -300493 ■018499 9 4-2798 -772639 -146339 35 .5-1190 -305 880 ■018976 80 4-0785 -781 310 -153847 6 14-9874 -3 1 1 548 ■019487 I 3-853 -791 017 •162 994 7 14-8504 •3 1 7 446 •020 028 2 3-5942 •802 165 -174605 8 14-7079 -323585 ■020600 3 3-325 6 •813 731 •188121 9 H-5594 -329978 •021 208 4 3065 8 -824916 ■202 889 40 14-404 7 •336641 •021 853 85 2^8268 -835 209 ■218 252 I 14-2442 -343 553 ■022537 6 2^6274 •843 797 ■232 6:8 2 14-07^73 ■35° 736 •023 263 7 2-4849 •849933 •243 891 3 13-903 7 •358216 •024035 8 2-403 J •853 454 •250 785 4 137229 '3^^5 999 •024859 9 2-385 5 •854214 •252317 45 13-5349 -374094 •025 738 90 2^426 8 •852434 -248 755 6 ^y339 3 -3825.9 ■026676 1 2^^98 1 •849 3^3 ■242 804 7 13-134 I ■39^ 355 ■027 689 2 2-571 2 -846215 •236953 8 12-9.83 •400 645. ■028 785 3 2-4986 ■849 342 -242 765 9 12-691 4 •4104. > ■029976 4 2-1830 •862933 -271 106 50 12-4530 -420 683 ■031 271 95 1-801 5 •879360 -313886 I 12-2043 •431396 ■032671 6 1-3787 ■897 567 -377328 2 11-9471 •442 469 •034175 7 -9370 -916591 -473213 3 11-6828 ■453 850 ■035 785 8 -4785 ■936334 -633314 ' 1 1-412 2 ■465 504 •037504 9 -000 ■956938 ■956938 FIVE PER-CEi\T. d\ 6 rER-CENT. Comnuitativn Table. X 1^. Nx Sx • Mx Rx X lO I 2 .3 4 61.391-3 58 284-3 55 2 75-0 52372-4 49 5^S-^ I 013415- 955 '30"8 899855-8 847 483-4 797 898-3 16006 094- 14992679- H 037 549* ^3 137693- 12 290 209- 10 2IO-1 10026-5 9 792*60 9522-14 9 228-69 261 430-3 251 220-2 241 193*7 231401*1 221 879-0 10 1 2 3 4 15 6 7 8 9 46919-9 44380-0 41 968-7 39687-6 37 536-9 750978-5 706 598-5 6646298 624942-1 587 405*2 11 492 311- 10 741 333' 10034734- 9370 104- 8745 '62- 8924-69 8 619-13 8 321-14 8038-59 7 777-80 2 1 2 650*3 203 725-6 195 106-5 186785-3 178746*7 6 7 8 9 20 I 2 3 4 31 754-1 30002-4 28 321-6 551 895-1 518308-5 486 554-4 456552-0 428230-4 8157757- 7 605 862- 7 087 554- 6 600 999- 6144447- 7 538-47 7 305-88 7072-74 6833-12 6581*03 170968*9 163430-5 156124-6 149051-8 142 218-7 20 1 2 3 4 25 6 7 8 9 26707-5 25 i6o-6 23 689-8 22 295-2 20978-4 40T 522-9 376362-3 352672-4 330377'2 309398-8 5716217- 5314694- 4938332- 4 585 659' 4255282- <^3'^S-SS 6 040- •JO 5 767-83 5501-26 5246-17 135637-7 129322-1 123 281-6 117513-8 1 12 012-5 25 6 7 8 9 30 I 2 3 4 19740-7 18577-8 17 486-1 16460-7 15494-1 289658-2 271 080-3 253 594-3 237 ^33'^ 221639-5 3945883- 3656225- 3 385 145" 3 131 550- 2894417- 5007-38 4 784-60 4577-46 4384-78 4 202*04 106 766-4 101 759*0 96974-39 92 396-93 88012*15 30 1 2 3 4 35 6 7 8 9 14583-4 13725-6 12916-4 12 153*2 11433-4 207 056-1 ^93 330-5 180414-1 168 260-9 156827-5 2672777- 2465721- 2272391- 2091977- 1923716- 4029-09 3 865-76 3 710-20 3 562-05 3 420-96 83 810-10 79 781-01 75 915-25 72 205-05 68 642-99 35 6 / 8 9 40 I 2 3 4 10754-6 10 113-9 9 509"52 8 939*5 1 8401-86 146073-0 135 959" I 126449-5 117 510-0 109 1082 r 766888- 1 620 815- 1 484 856- I 358407- I 240897- 3 286-59 3 158-07 3 035-28 2 918-10 2 806-15 6^ 222-03 61 935-45 58777-38 .55742*10 52 824-00 40 I 2 3 4 45 6 7 8 9 7894-71 7416-38 6966-00 6542-18 6143-39 loi 213-5 93 79 7 "'08 86831-08 80 288-90 74i45'5i I 131 788- 1030575- 936778-0 849 946-9 769 658-0 2 699-08 2 596-69 2 499-47 2407-37 2 320-11 50017-85 47318-77 44 722-08 42 222-61 39815*24 45 6 7 8 9 50 I 2 3 4 5 768-00 5 4 13 '94 5079-01 4 761-60 4 460-46 68377-50 62 963-56 57884-55 53 122-95 48 662-49 ^95 S^^' 5 627 135-0 564171-4 506 286-9 453 163-9 2237-27 2 157-87 2 080*75 2 005-19 I 930-80 37495-13 35257-86 S3 100-00 31019*25 29014-05 50 I 2 3 4 95 5 PERCENT. Commutation Tahh — (continued). X D. N. s. M, R. X 55 6 7 8 9 + 174-20 3 902-80 3 646-06 3 403'38 3 i74'o8 +4488-29 +0585-49 36939-43 33 536-05 30361-97 404501-4 360 013-1 319427-6 282488-2 248952-1 1 856-94 1784-31 I 713-42 1 644-36 1577-13 27 083-26 25 226-32 23442-01 21 728-59 20 084-23 6 7 8 9 6o I 2 3 4 2957-04 2 749"53 2550-87 2 360-26 2177-35 27 404-93 24 655-40 22 104-53 19 744-28 17566-93 218590-2 191 185-2 166529-8 144425-3 124681-0 1511-23 1 444-54 1 376-80 I 307-66 1237-15 18 507-10 16995-88 ^5 55^' 3^ 14174-54 12866-88 60 1 3 4 65 6 7 8 9 2 002-I1 ^ 835-50 1 677-86 1528-61 1387-42 15564-82 13 729-32 12 051-46 10 522-86 9135-435 107 114-1 91 549-29 77819-96 65 768-50 55 245-65 1 165-59 I 094-32 1 024-08 954-729 886-331 11629-73 10464-15 9369-831 8 345-749 7391-020 ^5 6 7 8 9 70 I 2 3 4 I 253-98 I 127-81 1007-57 893-210 785-107 7881-459 6 753-648 5 746-080 4852-870 4067-763 46 110-21 38 228-75 31475-10 25 729-02 20 876-15 818-955 752-503 685-966 619-587 554-018 6 504-689 5685-734 4933-231 4 247-266 3627-678 70 1 2 3 4 75 6 7 8 9 683-600 589-293 502-838 426-328 359-312 3384-164 2794-871 2 292-034 I 865-706 I 506-394 16808-39 13424-23 10 629-36 8337-322 6471-616 489-896 428-142 369-748 317-183 270-468 3 073-660 2583-764 2 1 5^-622 I 785-874 1 468-691 75 6 7 8 9 80 I 2 3 4 301-363 252-059 210-390 173-983 141-628 1 205-03 1 952-973 742-583 568-600 426-972 4965-222 3 760-191 2 807-218 2 064-635 1 496-035 229-630 194-676 165-010 138-622 114-552 r 198-222 968-592 773-916 608-906 470-284 80 I 2 3 4 85 6 7 8 9 I T2-925 87-5829 '55-7175 47-7570 33-7382 314-047 226-464 160-747 112-990 79-251 1 069-063 755-016 528-552 367-806 254-816 92-593 1 72-6283 54-933 5 40-102 15 28-3578 355-732 263-139 190-511 135-577 95-475 85 6 7 8 9 90 I 2 3 4 23-3741 16-1384 11-2353 8-2178 6-420 2 55-877 39-739 28-503 20-286 13-866 i75\''/\'^ 119-688 79-949 5 1 -445 31-160 19-600 2 13-4776 9-343 6-86o 5 5-4542 67-117 47-517 34-039 24-696 17-836 90 I 2 3 4 95 6 7 8 9 4-9789 3-7528 2-6586 1-6768 •7985 8-887 5-134 2-475 -798 17-294 8-408 3-274 •798 4-3186 3-3296 2-414 I 1-5589 -7604 12-382 8-063 4-733 2-319 -760 95 6 7 8 9 90 6 PERCENT. IjogaritJiTJis and Co-lcxjnrithma of Dj., N_r, a/ul ^l^.] with their Difference.f. X Log 1), A (Logr/>x) Colog D^ A (Colog Vp:r] LogN. A X lO 4788 107 ■^•9 7 7 445 5-2 1 1 893 0^022 i^t^r^ 6^005 787 974276 10 I l^K^ 552 •976977 •234448 •023 023 5^980 063 974 1 10 I 2 742529 •976574 •257471 •023 426 •954 1 73 973 958 2 3 719 10.3 •976248 •280897 •023 752 •928 13 I 973817 3 4 •69535' •976006 ■304649 •023 994 •901948^973680 4 15 •671357 •975 830 •328 643 •024 170 •875628 973 545 15 6 •647 187 •975 738 •352813 •024 262 •849 173 973 407 6 7 •622 925 •975 730 •377075 •024 270 •822 580(973 260 7 8 •598 65.S •975 804 •401345 •024 196 •795840 973 098 8 9 ■574459 ■975 894 •425541 •024 106 •768938 972919 9 20 •550 3S^ •975813 •449647 •024 187 •741 857 972731 20 I •526 166 •975 634 •473 834 •024366 •714588 972 544 1 2 •501 800 •975 356 •498 200 •024 644 •687 132 972358 2 ,3 •477 156 •974962 •522844 •025 038 •659 490 972 188 3 4 •452118 •974 5 ^^ •547882 •025485 •63 1 678 972032 4 ^5 •426 63.3 •974089 •573 367 •025 911 •603710 971 896 25 6 •400 722 •973 840 •599 278 •026 160 •575606 971 766 6 7 •374562 •973 649 •625438 •026351 •547372 971638 7 8 •348 211 •973 562 •651 789 •026438 •519010 971509 8 9 ■321773 •973 589 •678 227 •O264II •490519 971367 9 30 •295362 '913 633 •704638 •026367 •461 886 971 212 30 I •268995 •973 697 •731005 •026 303 •433 098 971041 1 2 •242 692 •973 756 •757308 •026 244 •404 139 970 854 2 3 •216448 •973719 •783552 •026 281 •374993 970654 3 4 •190 167 •973 690 •809833 •026310 •345 647 970441 4 zs •^63857 •973 674 •836 143 •026326 •316088 970 212 35 6 '^37 S3'i^ •973 61 1 •862 469 •026389 •2S6300 969970 6 7 •I u 142 ■973 549 •888858 •026451 •256270 969713 7 8 •084691 •973 484 ■9^5309 •026516 •225 983 969 439 8 9 -05^^15 •973418 •941 825 •026582 •195422 969 148 9 40 •03 1 593 •973327 •968407 •026673 •164570 968838 40 I •004920 •973 239 •995 080 •026 761 •133408 968 509 1 2 3-978 159 •973 155 4"02i 841 •026 845 •101 917 968 158 2 3 ■95^14 •973 062 •048 686 •026938 •070075 967 782 3 4 •924376 •972960 •075624 •027 040 •037857 967381 4 4j ■897 3?,^ •972 856 •102 664 •027 144 •005 238 966951 45 6 •870 192 •972 792 •129808 •027 208 4-972 189 966 486 6 7 •842 984 ■972 739 •1^7 016 •027 261 •938 675 9^5 981 7 8 •815723 •972 685 •184277 •027315 •904 656 965 429 8 9 •788408 •972 618 •211592 •027382 •870085 964 828 9 50 •761 026 •972487 •238974 •027513 •834913 964 1 •j6 50 I 133 5^3 •972 266 •266487 •027734 •799089963474] 1 2 •705779 •971974 •294 221 •028 026 •762 563 P62719 2 3 ■677753 •971627 •322247 •028373 •725 282 ■)6i 912 3 4 •649 380 •971 193 •350620 •028807 ^687 194 p6i 052 4 97 5 PERCENT. Logarithms and Co-logarithms of D^., Nj., and M^.; with their Differences. X Colog Nj. A LogM, A Colog M^ A X lO 7-994 2 [3 025 724 4^009 028 992 120 5-990972 007 880 10 I 6-019 9,37 025 890 •001 148 989 750 ■998 852 010 250 I 2 •045 827 026 042 3-990 898 987 837 J4"oo9 102 012 163 2 3 -071 869 026 183 •978 735 986405 •02 1 265 1 013 595 3 4 •098 052 026320 ■g6$ 140 985 453 -034860 014547 4 15 •124,372 026 455 •950 S93 984871 -049407 015 129 15 6 ■150827 026593 •935 464 984719 -064536 015 281 6 7 •177420 026 740 •920 183 984997 •079817 015003 7 8 •204 160 026902 •905 180 985677 -094 820 014323 8 9 •231 062 027 081 •890857 986426 -109 143 013574 9 20 •258 143 027 269 •877283 986389 •122 717 1 013 611 20 1 •285412 027456 -863 672 985 9 1 6 •136328 ' 014084 1 2 •312868 027 642 -849 588 985031 •1150412 014969 2 3 •340510 027 812 •834619 983 675 •165381 016325 3 -1 •368322 027 968 •818294 982 118 •181 706 017882 4 25 •396 290 028 104 •800412 980661 •199588 019339 25 6 •424 394 028 234 •781073 979940 •218927 020 060 6 7 •452 628 028 362 •761 013 979 449 •238987 020 551 7 8 •480 990 028491 •740462 979380 •259538 020 620 8 9 •509481 028633 •719842 979 769 •280 158 020 231 9 30 •538 114 028788 •699 61 1 980 235 •300389 019 765 30 I •566902 028959 -679846 980 779 •320 154 019 221 I 2 •595861 029 146 -660 625 981323 '339 31 5 018 677 2 3 •625007 029 346 •641 94'8 981513 •358052 018487 3 4 •654 353 029 S59 •623 461 981746 •37'5539 018254 4 35 •683 912 029788 •605 207 982028 •394 793 017972 35 6 -713700 030 030 •587 235 982 163 •412765 0x7837 6 7 •743 730 030287 •569 398 982 303 •430 602 017697 7 8 •774017 030561 -S5'>- 701 982 447 •448 299 017553 8 9 •804578 030852 •534 148 982597 •465 852 017403 9 40 •835 430 031 162 •5^'5 745 982 677 •483 255 017323 40 I •866 592 031491 •499422 982777 •500578 017223 I 2 ■898 083 03 1 842 •482 199 982 901 •517801 017 099 2 3 •929925 032 218 •465 100 983 oil •534900 016989 3 4 •962 143 032 619 •448 1 1 1 983 105 •551 889 016895 4 45 •994762 033 049 •431216 983 204 •568 784 016796 45 6 5-027 811 033 5 H •414420 983 428 •585580 016572 6 7 •061325 03 4 1 9 -397 848 983 ^595 •602 1 52 016305 7 8 •095 344 034571 •381543 98,5 965 •618457 016035 8 9 •129915 035^72 •3^5 508 984 209 •634 492 0x5791 9 50 •165087 035 824 •349717 984308 •650 283 015 692 50 I -20091 1 036526 •334025 984 195 ■665 975 015 805 I 2 •237437 037 281 •318 220 983936 •681 780 016064 2 3 •274718 038088 •302 156 983581 -697 844 016419 3 4 •312806 038948 •285737 983 060 -714263 016940 4 Its HF 5 riCK-CKNT. Logarithms and Cn-h>i/an'tfiins c/Dj:, Nj., and M^; u't't/t llieir JJijd'cicnns. LogD, 55 6 7 8 9 6o I 2 3 4 65 6 70 X 2 3 4 75 6 7 8 9 80 I 2 3 4 85 6 7 8 90 I 2 3 4 95 6 7 8 A 3-620573 •59' 377 •^i 824 'S'i^ 9'o •ijoi 618 ■4708.57 •439 259 •406 687 •372 9;. 9 ■337928 •301 487 ■263 754 •224756 •184296 •142 208 •098 2S9 •052 236 •003 274 2-950 954 •894929 •834802 ■77033 •701 428 •629 744 ■555 47 J •479 090 •401 501 •323 024 •240 jO( •151 150 ■052 790 1^942 419 •817681 •679037 •528 122 ■368735 •207 860 ■050 583 0914756 •807 546 •697 134 •574353 •424 645 •224479 1-902259 C"'«6^'(Cologf/v) •970804 ■970447 •970 086 •969 708 •969 239 •968 402 •967 428 •966 272 •964969I ■963 559J •962 267 •961 002 •959 540 •957912 •956081 ■953 947 ■951038 •947 680 ■943 975 3 ■9398731 •935529 •93 1 097 •928 316 •925 727 •923619 •922 411 •921523 ■917 482 •9 1 o 644 •901 640 4-379 427:o^o29J96 •408623 "029553 •438 176 ^029 914 •468 090 "030 292 •498382 -030761 •529143 -031598 ■560741 -032572 '593 -i^ 3 "033728 •627041 -035031 •662072 ^036441 •889629 •87 ij 262 •861 356 •849 085 •840613 •839125 •842 725 •864 171 •892 790 •889 58S •877219 •850 292 •799834 •677780 •69851 •736246 ■775244 •815704 •857 792 •901 71 1 •947 764 •996 726 049 046 •J05071 •16!; 198 •229669 •298572 •370256 •444529 •5209ro •598 499 •676976 ■7. '9 494 •848 850 LogN, -947 210 2-057 581 -182319 -320963 -47187 -631 265 •792 140 ■949415 1*085 244 •192454 •302 866 -425 647 •575355 •775521 0-097 741 ■0.37 733 •038998 •040 460 •042 088 •043919 •046 053 •048 962 •052320 •056 025 •060 127 •064471 •068903 •071 684 •074273 •076381 •077589 •078477 •082518 •089 356 •098 360 •110371 •124738 •138644 •150915 •^59387 •160875 •157275 •135829 •107 210 •110412 •122 781 •J49708 •200 [66 •322 220 4^648 246 960 1 25 •608371 959 119 •567490958 022 •5255121956818 •482330^955499 •437829I954083 •391 912J952 569 •344481950960 •295441949255 •244 696 947 448 •192 144945505 •13764994339' •081 040941 094 •022 134 3-960 729 •896607 ■8295.38 •759372 ■685 999 •609 356 ■529451 -446362 •360 221 •270843 ■177939 •080998 i'979o8o ■870 745 ■75480 •630400 ■496 99 •354999 •206 142 •0530.38 1-899006 •747 235 •599215 •454898 ■307 189 •141936 938 59: 935878 932931 929834 926 627 923 357 920 095 916911 913859 910 622 907 096 903 059 898 082 891665 884 062 875 593 866 595 858004 851 143 846 896 841968 848229 851980 855683 852 291 834747 806 799 0^948 735 •710440 •393621 /•902 259 761 705 683 1 508638 55 6 7 8 9 60 I 2 3 4 65 6 I 2 3 4 75 6 80 I 2 3 4 85 6 90 I 2 3 4 95 6 99 5 PER-CENT. Logarithms and Co-hgarithms of Dj;, N^-, and M^jj with tlieir Differences. X Colog N^ A Log M^ A Colog ]M,. 1 A X 5^ 'S'iS'^ 754 039 875 3-268797 982 674 4^731 203 017326 55 6 •391 629 040881 •251471 982392 •748529 017 608 6 7 •4325^0 041978 •233 863 982 134 •766137 017866 7 8 •474488 043 182 •215997 981870 •784003 0x8 130 8 9 .•5^7670 044501 •197867 981463 •802 133 018537 9 6o "•562 171 045917 •179330 980 399 •820 670 019601 60 I •608 088 047 43 I ■159729 979 141 -840 271 020 859 I 2 •655519 049 040 -138870 977 625 •861 130 022375 2 3 •704559 050 745 •116495 975 926 •883 505 024074 3 4 15530^. 052552 •092421 974123 ■907 579 025877 4 65 •807 856 054 495 -066 544 972 599 •933 456 027 401 65 6 •862351 056 609 •039143 971 192 •960857 028808 6 7 •918960 058906 •010335 969 545 •989 66s 030455 7 8 •977 866 06 1 405 2-979 880 967716 yo2o 120 032 284 8 9 4-039271 064 122 ■947 596 965 664 •052 404 034336 9 7° •J 03 393 067 069 •913 260 963 248 •086 740 036752 70 I •170463 070 166 •876 508 959 794 -123492 040 206 I 2 •240 628 073373 •836302 955 801 •163 698 044 199 2 3 •3 14 00 J 076643 •792 103 951421 •207 897 048579 3 4 •390 644 079905 •743 524 946580 •256476 053 420 4 75 •470549 083 089 •69c 104 941 484 •309 896 058516 75 6 •553 638 086 141 •631588 936318 •3684x2 063 682 6 7 '^39 779 089378 •567 906 933 404 •432094 066 596 7 8 •729157 092 904 •501310 930 807 •498 690 069 193 8 9 •822061 096941 •432 117 928912 •567 883 07x088 9 8o •919002 loi 918 •361 029 928284 •638971 07X 7x6 80 I 3-020920 108335 •289313 928 197 •710 687 07 1 803 X 2 '•129255 115938 •217510 924321 •782490 075 679 2 3 •245 193 124407 •141 831 917172 •858 X69 082 828 3 4 •369 600 J 33 405 ■059 003 907575 •940997 092 425 4 85 • 503 005 141 996 1-966 578 894 528 2-033422 105472 85 6 •645001 148857 -861 106 878731 •138894 X2X 269 6 i •793 858 153 104 •739837 863 334 •260 163 136666 7 8 •946 962 154032 -603 171 849501 •396 829 150499 8 9 2' 100 994 151771 •452672 839589 •547328 X604XI 9 90 •252 765 148 020 ■292 261 837350 •707739 162 650 90 X •400 785 144317 •129611 840874 •870389 159 126 X 2 •545 J 02 147 709 0-970485 865871 1-029 5 15 134129 2 3 •692 81 I 165253 ■^36 35(^ 900374 •163644 099626 3 4 •858 064 193 201 •736730 898618 •263270 loi 382 4 9^ 1-051 265 238 295 •635 348 887 046 •364652 1 12 954 95 6 •289560 316819 •522394 860 358 •477 606 X 39 642 6 7 •606379 491 362 -382 752 810 072 •617 248 X 89 928 7 8 0-097 74X •192 824 688 246 •807 176 311754 8 9 i^88i 070 o^i 18930 1 9 1(MI 5 rEH-CENT. Values of Anuuitua, and Siiujlc ami Annual I'reniunns for Assnrana' of a Unit. X «, A, ICj X «j A., fflTj, lO i<5-.^o7 5 •166 31 1 •009 499 ss 10-6579 •444861 •038 159 1 16-.587 4 •172 027 •009 894 6 10^3990 •457 188 ■040 108 2 i6-279 6 -177 161 •010 253 7 io^i3i 4 •469 936 •042 217 3 16181 9 •181 816 •010582 8 9^853 7 •483 155 •044515 4 16-091 5 •186 119 •010890 9 9^565 6 •496877 •047 028 15 i6'ooi; 6 -190 212 •01 1 185 60 9^267 7 •511 062 •049 774 6 i;)"92'i 5 • 1 94 2 1 2 •011477 I 8^967 1 •525375 •052711 7 15-8363 •198 270 •01 1 776 2 8-66 fj 5 •539738 •055 842 8 157465 •202547 •012 095 3 8-365 3 •554033 •059 158 9 15-6487 •207 204 -012 446 4 8-o68o •568 189 •062 658 20 1 5 '54 1 9 •212 290 •012834 ^K^ 7^7742 •582 180 •066351 1 1 5 '43 20 •217524 •013 238 6 7^479 9 •596 196 •070307 2 15-3226 •222735 -013 646 7 7-1826 ■610350 •074591 3 15-2172 -227 753 •014044 8 6-8840 ■624574 •079 221 4 15-120.3 •232368 •014415 9 6-5845 ■638834 •084 229 25 15-0341 •236471 •014748 70 6-28^2 '^S3 087 •089 646 6 14-9584 •240077 •015044 1 5-9883 ■667 225 •095478 7 14-887 I -243473 •015325 2 5^7029 ■680813 •loi ';7o 8 14-8183 •246 746 •015599 3 5 '433 I •693 664 •107 828 9 14-7484 •250074 •015879 4 5-1812 •705659 •114 163 30 14-673 2 -253 658 •016 184 75 4^950 5 •716643 ■120434 I 14-591 6 •257543 •016 518 6 4^742 8 •726535 •126513 2 14-502 7 •261 778 •016886 7 4-558 2 •735324 ■132295 3 14-406 I •266379 •017 291 8 4-3762 •743989 •138385 4 i4'304 7 •271 202 •017 720 9 4^1924 •752741 •144969 3."; 14-198 I •276280 •018 179 80 3^9986 •761972 •152437 6 14-085 4 •281 646 •018 670 I 3-7808 -772345 •161 553 7 13-9678 -287247 019 191 2 3^5296 •784307 •173 i5i 8 13-8450 •293 096 •019744 3 3-2681 •796755 •186675 9 13-7166 •299 208 •020331 4 3-0147 •808822 ■201 463 40 13-5824 -305599 •020957 85 2-781 •819952 ■216860 I 13-4428 •312250 •02I 620 6 2-585 7 •829252 ■231 266 2 13-297 I -319 183 •022325 7 2-446 •835 904 •242571 3 13-^450 •326428 •023077 8 2^365 9 •839719 •249476 4 12-986 2 ■33399^ ■023880 9 2-349 ■840524 •250977 45 128204 •341 885 •024 738 90 2-3906 •838545 •247318 6 12-6473 -350129 •02"; 60 1 2-462 4 •835 125 •241 200 7 12-465 •358810 •026648 2 2-5370 -831574 •235 110 8 12-2725 -367977 •027 725 3 2-468 5 -834834 •240 690 9 12-069 2 -377659 •028897 4 2-1597 ■849540 •268869 50 11-8546 •387875 •030174 95 i-7849 ■867388 •3 1 1 460 I 1 1-6299 -398 576 ■031 558 6 r368o ■887 238 •374678 2 11-3968 •409676 -033 047 7 •931 I ■908 045 ■470232 3 ^'^"^':)^S •421 1 18 •034641 8 •4762 ■929 706 ■629 800 4 10-909 8 -432 8^9 •036346 9 -000 ■952381 ■952381 SIX PER-CENT. 102 6 PER-CENT. Commutation Tnhle. lO I 2 3 4 6 20 I 2 3 4 6 7 8 30 I 2 3 4 35 6 7 8 40 I 2 3 4 45 6 50 I 2 3 4 1'. N. 55 839'5 525^3*3 49332-2 46 3007 43 422-9 40701-3 38 i34'9 357227 33 462-4 3^3S^'5 377-9 524*4 777"2 125-4 559-0 072-6 664-9 340-7 098-1 936-5 854-7 847-8 91 i-i 039-3 225-5 465-9 757-40 Air3l 900-03 360-89 857-10 386-50 947-05 536-64 ^S3'3^ 795-45 461-75 150-76 860-97 590-85 338-63 102-55 881-21 (>13-S3 799851-0 747337-7 698 005-5 651 704-8 608 281-9 567 580-7 529445-8 493 723-1 460 260-7 428910-3 399532-3 372007-9 346230-7 322 105-3 299 546-3 278473-7 258 8o8-8 240468-2 223 370-1 207433-6 192578-9 178731-1 165 820-0 ^S3 780-7 142555-2 132089-3 122331-9 113236-4 104759-0 96858-96 89498-08 82 640-98 76 254-48 70307-43 64770-79 59617-43 54821-97 50360-23 46 209-47 42348-50 38 757-65 35419-02 32316-47 29435-26 26761-73 S. 1 1 345 587 10545736 9798399 9 100393 8 448 688 7840407 7 272 8a6 6743380 6249657 5789396 5 360486 4 960 954 4588946 4242 715 3 920 610 3 621 063 3 342 590 3 083 781 2843313 2619943 2 412 509 2 219930 2 041 199 1875379 1721598 1579043 1 446 954 I 324622 I 211 385 I 106 626 J 009 768 920 269-4 837628-5 761 374-0 691 066*6 6262958 5666783 511 856-4 461 496-1 415286-7 372938-2 334180-5 298761-5 266445-0 237 009-8 M, 7404-17 7238-76 7 030-03 6 790-92 6 533-94 6 270-23 6007-67 5 754-03 5515-80 5 297-99 5 099-99 4909-38 4 720-12 4527-44 4326-64 4 r 17-18 3 902-20 3 69 1 - 1 o 3 486-67 3292-88 3 113-20 2947-14 2 794-20 2 6,3-28 2 520-88 2 396-76 2 280-65 2 171-11 2067-77 970-28 878-30 791-17 708-71 630-75 556-98 487-09 420-88 358-62 300-18 245-34 193-76 144-80 097-69 05 1 -98 007-38 165 052 157 648 150 409 143379 136588 130054 123784' 117776' I 12 022' 106506' 101 2o8' 96 108 91 199 86479' 81951' 77625' 73 507' 69 605' 65914' 62427' S9 134- 56021- S3 074- 50 280- 47627' 45 io6' 42 709' 40428' 38257' 2,6 190' 34219- 32341- 30550' 28841- 27 210" ^$(^53- 24 166' 22 745' 21 387' 20087' 18841' 17647' 16503' 15405' T^\3S3- X •\ 10 -0 I •2 2 2 3 3 4 3 15 1 6 4 7 4 8 6 9 6 20 60 1 22 2 10 3 66 4 02 2^ 84 6 64 7 54 8 87 9 98 30 78 1 64 2 44 3 16 4 28 3S ■'2 6 88 7 77 8 00 9 13 40 42 1 25 2 54 3 79 4 81 45 72 6 84 7 22 8 04 9 70 50 94 I 14 2 44 3 47 4 103 6 PEPt-CENT. Commutation Table — (continued) . X D, N. s. M. R. X 55 2478-35 24283-39 210 248-1 963-530 13 346-08 6 2 295-35 21 988-04 185964-7 920-819 12382-55 6 7 2 124-12 19863-91 163 976-6 879-515 11461-73 7 8 I 964*04 17899-88 144 112-7 839-664 10582-22 8 9 1 814-43 16085-44 126 212-8 801-233 97 425-54 9 6o J 674-41 14411-03 1 10 127-4 763-916 8941-322 60 I I 542-23 12868-80 95716-37 726-509 8 177-406 I 2 1417-30 i I 451-51 82847-56 688-872 7 450-897 2 3 I 299-02 10 152-49 71396-06 650-821 6 762-025 3 4 I 187-05 8965-442 61 243-57 612-378 6 1 1 1-204 4 65 I o8i-2i 7884-231 52278-13 573-732 5 498-826 65 6 981-885 6902-346 44393-89 535-608 4925-094 6 7 889-090 6013-257 37491-55 498-391 4389-486 7 8 802-360 5 210-896 31478-29 461-987 3891-095 8 9 721-381 4489-516 26267-39 426-424 3429-108 9 7° 645-847 3 843-669 21 777-88 391-723 3 002-684 70 I 575-387 3 268-283 17934-21 357-821 2 610-961 I 2 509-192 2759-091 H 66^-9 3 324-195 2253-140 2 3 447-141 2311-950 1 1 906-84 290-966 I 928-945 3 4 389-316 I 922-634 9594-885 258-451 1637-979 4 75 335-783 1586-851 7672-251 226-955 1379-528 75 6 286-729 I 300-122 6085-400 196-507 1 152-573 6 7 242-355 I 057-767 4785-278 168-763 955-666 7 8 203-541 854-226 3727-512 143-667 786-902 8 9 169-927 684-299 2873-285 121-575 643-235 9 8o 141-177 543-122 2 188-986 102-443 521-661 80 I 116-966 426-156 I 645-864 86-223 2 419-217 I 2 96-708 7 329-447 I 219-708 72-5867 332-994 2 3 79-2192 250-228 890-261 60-5713 260-407 3 4 63-8790 186-349 640-033 49-7152 199-836 4 85 50-452 4 135-897 453-684 39-9044 150- 121 85 6 38-761 97-136 317-787 31-0687 110-217 6 7 28-809 7 68-326 220-651 23-3115 79-148 7 8 20-738 6 47-587 152-325 16-871 I 55-836 8 9 14-5127 33-075 104-738 11-819 I 38-965 9 90 9-959 7 23-115 71-663 8-0875 27-146 90 I 6-81 1 6 16-303 48-548 5'5°3 3 19-059 1 2 4-697 4 1 1-606 32-245 3-7746 hr555 2 3 3-4034 8-203 20-639 2-7465 9-781 3 4 2-633 8 5-569 .12-436 2-1695 7-034 4 95 2-0233 3*545 6-867 1-708 I 4-865 95 6 1-5107 2-035 3-322 1-3100 3-157 6 7 I -060 I •975 1-287 •9449 1-847 7 8 -6623 -312 •312 -607 I •902 8 9 •3124 ! -000 1 •000 •2947 •295 9 101 HF. 6 PEll-CENT. Logarithms ami Co-Iot/arif/nns of D^., N^., and M^. ; in'fh their J)i^erenres. X LogD^ A (Logtyv) CologD^ A (Colog rpj.) LogN^ A X lO 4746941 1-973329 5-253 059 ©•026671 5-903009970508 10 I •720 270 •972 860 •279730 •027 140 •873517970342 I o •693 1.30 •972457 •306870 •027543 •843859970192 2 3 •66.^587 •972 132 ■334413 •027 868 •814051^970054 3 4 •637719 -971889 •362 281 •028 II 1 •7841051969923 4 !.■; •609608 "971 714 •390 392 •028286 •754028969794 15 6 •581 .322 -971 622 •418678 •028378 •723 8221969 661 6 7 •552944 '971 613 •447 056 •028387 •693 483 969521 7 8 •524557 ■971 ^^87 •475 443 •028313 •663 004 969363 8 9 •496244 -971 777 'S''2> 1S<^ •028 223 •632367 969 185 9 20 •468021 ^971697 ■531979 •028 303 •601 552 969 000 20 I •4397^8 -9715^7 •560 282 •028483 •570552 968814 I 2 •411 2.35 •97; 240 •588 7^55 •028 760 -539366 968 63 2 2 3 •382475 •970 845 •617525 •029 155 •507998968466 3 4 ■?>S2>^^^^ '91^^99 •646 680 •029601 •476464 968320 4 25 •323 719 ^969 972 •676281 •030028 -444 784 968 195 25 6 •293691 •969724 •706 309 •030 276 •412979 968 079 6 7 •263415 -9^^9532 •736585 •030468 •381 058 967 967 7 8 •232947 -9^>9445 •767053 ■030555 -349025 967 854 8 9 •202 392 -9<59 473 •797 608 ■030527 •316879 967 730 9 30 •171 865 -9^9517 •828 i^S •030483 •284609 967 59^ 30 I •141 382 •969 580 •8-^8 618 •030420 •252 200 967437 I 2 •110962 •969 640 ■889038 •030360 •219637 967 265 2 3 •080602 •969 602 •919398 •030 398 •186902 967 081 3 4 •050 204 -9^9574 •949 796 •030426 -153983 966 885 4 35 •019778 -9^9 Sj^^ •980 222 •030444 •120 868 966 672 3S 6 3-989334 -9^^9 495 4-010666 ■030505 •087 540 966 446 6 7 -958829 -969432 •041 171 •030 568 •053 986 966 205 7 8 •928 261 ■969 368 •071739 •030632 •020 191 965 949 8 9 •897629 •969 30J •102371 •030 699 4-986 140:965 674 9 40 •866930 •969 210 •133070 -030 790 •951 814965382 40 I •836 140 •969 123 •163860 •030877 •917 196 965 069 I 2 •805 263 •969 038 •194737 •030962 •882 265 964 736 2 3 •774301 •968 946 •225 699 -03 1 054 •847 001 964378 3 4 •743 247 •968 844 •256753 •031 156 •811 379 963 994 4 45 •712 091 •968 739 •287 909 •031 261 •nsM3 963 582 45 6 •680 830 •968 675 •319 170 •031325 -738955 963 ^33 6 7 •fH9 ^°!^ •968 62 3 •350495 •031377 •702088 962 643 7 8 •618 128 •968 568 •381872 •031432 ■664 73 1 962 107 8 9 •586696 •968501 •413 304 •03 1 499 •626838 961 520 9 50 •^^S ^91 •968372 -444 803 •03 I 628 -588358 960 879 50 I •523 569' ^968 149 -476 43 I -03 1 85 1 -549237 960 187 I 2 •491718 -967857 •508 282 -032 H3 •509424 959 444 2 3 •459575 -9^7510 -540425 •032490 •468 868 958 646 3 4 •427085, ^967 077 •572915 •032923 -427514 957 795 4 105 6 PERCENT. Logaritlnns and (Jo-logarithms of Dj., N^., and ]\I^; icith their Differences. X lO Colog N.r I A i LogM^r A Colog M^ A X &oi)6()gi 029492 3-869 476 1 990 188 ^•i3o 524 009 812 10 I •12648.3 1 029658 •859 664 '987 293 •140336 012 707 I 2 •156 141 029808 •846957 984972 • J 53 043 015 028 2 3 •185949 029946 •831929 9S3 246 -168 071 016754 3 4 ■215895 030077 -815 J 75 982109 -184825 017 891 4 15 •245972 030206 •797 284 981 422 •202 716 018578 15 6 •276178 0,30339 •778 706 981 266 •221 294 018734 6 7 ■306517 030479 •759 972 '981 636 •240 028 018 364 7 8 •3369961 030 637 •741 608 982 503 •258392 0x7497 8 9 •S^l^^i 030815 •724111 983 459 •275889 016541 9 20 •398 448 03 1 000 •707570 983457 -292430 016543 20 I •429 448 ,031 186 •691 027 982926 ■308973 017074 I 2 •460 634 031368 •673 953 981 900 •326047 018 100 2 3 •492 002 031534 •^SS ^53 980 298 ■344 147 019 702 3 4 '^^3 536 03 1 680 •636 151 978 449 •363 849 021551 4 25 ■555216 i 03 1 805 •614 600 976710 •385400 023 290 25 6 •587 021 031 921 •5913^0 975 846 •408 690 024154 6 7 •618942 032033 ■5'^7 i5<5 975 255 •432844 024 745 7 8 ■650975 032 146 ■542 411 975 ^66 ■457 589 024834 8 9 •683 121 032270 ■5^7577 975 630 •482 423 024370 9 30 •715391 032409 ■493 207 976 194 •506 793 023 806 30 I •747 800 032 ^^63 •469 40 1 976856 ■530599 023 144 I 2 •780363 032735 •446 257 977525 '553 743 022475 2 3 •813098 032919 ■423 782 977770 •576 218 022 230 3 4 •846017 033 115 •401 552 978073 •598 448 021 927 4 35 •879 132 033328 ■379625 978 433 •620375 021567 35 6 •912 460 033 554 ■358058 978 623 •641 942 021377 6 7 •946014 033 795 •336681 978821 •663319 021 179 ■ 7 8 ■979 809 034051 •315502 979 025 •684 498 020975 8 9 5-013 860 034326 •294527 979239 ■705473 020 761 9 40 •048186 034618 •273766 979371 •726234 020 629 40 I •082 804 034 93 1 •253 137 979531 ■746863 020 469 I 2 •1^7 735 035 264 •232 668 979720 •767332 020 280 2 3 •152999 035 622 •212388 979895 ■787 612 020 105 3 4 •188 621 036 006 •192 283 980 054 -807 717 019946 4 45 •224627 036418 •^72337 980 221 •827 663 019779 45 6 •261 045 036867 •^^2558 980 539 -847 442 019461 6 7 •297912 037357 '^33 097 980907 •866 903 019 093 7 8 ■335 269 037 893 •I 14 004 i 981 284 •885 996 018 716 8 9 '373 ^62 038480 ■095 288 981 631 •904712 018 369 9 50 •411 642 039 121 •076919 981 812 •923081 018 188 50 I •450 763 039813 •058731 981 750 •941 269 01 8 250 I 2 ■490 576 040 ss^ •040481 981 525 •959519 018475 2 3 •531 132 041 354 •022 006 1 981 189 ■977994 018811 3 4 ■572486 042 205 •003 195 980670 •996 805 019330 4 lor. 6 PERCENT. Ijogarithms and C(>-/i>pj.) LogN, A X .15 ?\394 i'^J2 [■966687 4-605 838 0-033313 4^385 309 956878 55 6 •360 849 •966 330 •639 151 -033 670 •342 187 955 878 6 7 •327 179 ■965 97 I ■672821 ■034 029 •298065 954 785 7 8 •293 1 50 •9^.1391 •706850 •034 409 •252 850 953 583 8 9 ■2.S8741 •965 122 •741 259 •034878 •206433 952262 9 6o •223 863 •964 286 •77^5 «37 ■035 7 H •158695 950 843 60 I •188 149 •9^^5 3 1 ' ■8 1 1 85 1 •036 689 •109 ';38 949325 I 2 • 1 5 1 460 ■962 156 ■848 ';4o •037 844 •058 863 947 710 2 .3 • I X 3 6 1 6 ■960 852 •886384 ■039 148 •006 ■573 945 999 3 4 •074468 ■9.19 442 •925532 -040558 3-952 57=^ 944 187 4 -^.^ •033910 •9.18 I. SI •966 090 -041 849 •896 759 942238 ^5 6 2^992 061 ■956 885 roo7 939 •043 115 •838997 940113 6 7 •948 946 ■9.1.1 424 •051054 •044576 •779 iio 937802 1 8 •904370 •9.13 794 •095 630 -046 206 •716912 935 288 8 9 •858 164 •9.1i9'^.1 ■141 836 •048 035 •652 200 932546 9 70 •810 129 •949 83 1 ■189871 -050 169 •584 746 929574 70 I •759960 ■946 92 J •240040 •053 079 •514320 926 446 I 2 •706881 •943 .163 ■293 119 •056437 •440 766 923 2 [2 2 3 •650444 •939859 •349 S'/^ •060 141 •363 978 919919 3 4 ■.190 303 •935 75^J ■409 ^97 •064 244 •283 897 916639 4 7.1 •526 059 •931 413 •473941 •068587 •200 1536 913448 75 6 •4.17472 •926980 •542528 -073 020 •I 13 984 910406 6 7 •3844.12 ■924 199 •615548 •07'; 80J •024390 907 183 7 8 •308651 •921 61 1 •691349 •078389 2^931 573 903 673 8 9 •230 262 •919503 •769738 -080497 •835 246 899 65 I 9 80 ■H9 76,1 •918 294 •850235 -081 706 •734897 894672 80 I •068 059 ■917407 •931 941 •082 593 ■629 r;69|888 2i7 T 2 I-98'; 466 •9U3'^5 2-014534 •086 635 •517786 880550 2 3 •898 831 •906527 •10 1 169 •093 473 •398336 871991 3 4 •80.13.18 ■897 524 •194642 -102 476 •270327 862 882 4 8.5 ■702882 •885513 •297 118 -114487 •133 209 854 170 85 6 •.18839.1 ■871 144 ■411 605 •128856 1^987379 847 207 6 7 •4.19 .139 ■857 240 •540 461 -142 760 •834 586 842 906 7 8 •316779 •844 9^9 •683 221 •155031 •677492 842 004 8 9 •161 748 ■836496 •838252 •163504 •519496 844 399 9 90 0-998 244 •83 1 008 T-ooi 756 •164992 '3^3 895 848 384 90 I ■833 252 •838 608 •166748 •161 392 •212 279 852403 I 2 ■671 860 •860056 •328 140 •139944 •064682 849 268 2 3 •.13191^^ •888 67 5 •468 084 •1J1327 0-913950 83 1 806 3 4 •420 589 ■885471 •579 41 1 -114529 •745 756 803913 4 9.1 ■306060 •873 103 ■693 940 •126 897 •549 669 758 849 95 6 •179 163 •846175 ■820837 •1.13825 •308 ';i8 680 ^^6 6 7 •025 338 •795 7' 7 ■974662 •204283 1-988874 505 845 7 8 1-82I 055 •673 664 0-178945 •326336 •494719 8 9 ■494719 •505 281 9 107 HI". 6 PER-CENT. Logarithms ami Co-logarithms ofT)-,., N^, and M^; with their Differences — (continued). X Colog Nj- A LogM^ A Colog M^ A X 55 ^•614691 043 122 2^983 865 980 309 V016135 019691 55 6 '•657813 044 122 •964174 980070 •035 826 019930 6 7 ■70^935 045215 •944 244 979 862 •055 756 020 138 7 8 •747 150 046417 -924 106 979 ^53 •075 894 020347 8 9 ■793567 047738 ■903 759 979 286 •096241 020714 9 6o •841305 049 157 •883 045 978 196 •I 16 955 021 804 60 I •890462 050 675 •861 241 976898 •138759 023 102 I 2 •941 137 052 290 •838 139 975322 •161 861 024678 2 3 •993427 05^ 001 •813 461 973 559 •J 86 539 026441 3 4 4-047 428 055813 •787 020 971 689 •212 980 028311 4 65 •103 241 057 762 •758 709 970 138 •241 291 029862 65 6 •16 J 003 059887 •728847 968 723 •271 ^53 031277 6 7 •220 890 062 198 •697 570 967 060 •302430 032940 7 8 •283088 064 7 1 2 •664 630 965 212 •335-370 034788 8 9 •347 800 067 454 •629 842 963 137 •370158 036 863 9 70 •415254 070426 •592979 960686 •407 021 039314 70 1 •485 680 073 554 '553 (^(^5 957 141 •446335 042 859 I 2 ■559 234 076788 •510 806 953 036 •489 194 046964 2 3 •636022 080 081 •463 842 948537 •53(^ 158 051463 3 4 •7 16 103 083361 •412379 943 5^^ •587621 056439 4 75 •799464 086552 -355 940 938322 •644 060 061 678 75 6 •886016 089 594 •294 262 933016 •70^738 066984 6 7 •975610 092 817 •227 27S 930079 •772722 069921 7 8 3-068427 096327 •157357 927486 •842 643 072514 8 9 •164754 100349 •084 843 925 641 •915^57 074359 9 80 •265 103 105328 •010484 925 140 •989516 074 860 80 I •3 70 43 J III 783 1^935 624 925 233 2-064376 074767 I 2 •482214 1 19 450 •8608^7 921 410 •139 H3 078 ^go 2 3 •601 664 128 009 •782267 914 222 •217 733 085778 3 4 •729673 137 118 •696489 904 53 i •3035^1 095 469 4 85 •866791 14^830 •601 020 891 303 -398 980 108 697 85 6 2-OI2 621 152 793 •492 323 875 247 •507 677 124753 6 7 •1654x4 157 094 •367 570 859 573 •632430 140427 7 8 -322 508 157996 ■227 143 845 440 •772857 154560 8 9 •480 504 155 601 •072583 835 23 J •927417 164 769 9 90 ■636 105 151 616 0^907 814 83280^ T'o92 186 167 195 90 I -787721 147 597 •740 619 836252 •259381 163748 I 2 •935318 150 732 •576871 86 1 905 •423 129 138095 2 3 1-086050 168 194 •438 776 897 592 •561 224 102 408 3 4 -254 244 196087 •336368 896 143 ■'663 632 103857 4 95 •45° 33 1 241 151 •232 511 884 748 •767489 1 15 252 95 6 •691 482 319644 •117 259 8';8 128 •882741 h'i 872 6 7 o-oi I 126 494 ^55 1^975 387 807 894 0^024613 192 106 7 8 •^05 281 •783281 686133 •216 719 313867 8 9 " •469414 •530 586 9 (IS 6 I'KIi CKNT. I alue^ of Annuities^ mid Si/u/lc nml Annual Pmninins ftr Asintniiice of o Unit. X fl^ K Wg X "x A. ^s lO 14324 » ■132597 -008 653 ^f, 9^7982 -388 780 036 004 I 14-2,31 4 •137846 -009 050 6 9^579 4 •401 167 -037920 2 14149 I •142504 •009407 7 9^3516 -414061 -040 000 3 '4-07.S5 •146 670 •009 729 8 91 13 8 -427520 -042 271 4 14-0083 -150472 -010026 9 88653 -441 589 -044 762 »5 13-945 ■154055 -010308 60 8-6o6 6 -456 229! -047 491 6 X3-883 5 '^Sl S^l •010 ^8^ I 8-344 3 -471078 •050413 7 13-821 •161075 •010868 2 8-0798 -486047. •053 530 8 137546 •164836 01 1 172 3 7^8155 -501 009 •056833 9 13-681 I -168992 ■oil 511 4 7^5527 •515884 •060318 20 13-5997 ■^73599 01 I 891 ^S 7^2920 -530639 •063 994 I ^3-S'S^ •i 78 365 •012288 6 7*0297 -545 489 -067 934 2 13-431 7 -183 112 •012688 7 6-7634 - 5^(^563 -072 206 3 ^yi5^3 -187663 •013076 8 6'494 5 •575785 -076828 4 13-2783 -191 792 •013432 9 6-2235 -591 123 •081 833 25 13-2150 ■195380 ■013 745 70 5-95 M ■606527 •087 253 6 I3-I6I ■198435 •014013 I 5^6802 ■621 879 •093 093 7 I3-III 2 •201 253 •014262 2 5^41 8 6 ■636 685 -099 194 8 13-064 I -203 922 •014500 3 yi705 -6 ■JO 726 •105457 9 13-0163 -206626 •014742 4 4-9385 ■663 859 -III 789 30 12-964 I ■209577 •015 008 75 4-7258 -675898 - 1 1 8 044 I 12-9068 -212 823 •015304 6 4-5343 •686737 -124087 2 12-8433 -216419 -015634 7 4-364 5 -696 347 •129806 3 12-7732 -220384 -016001 8 4^196 8 -705 840 -135 821 4 12-6993 •224568 •016393 9 4-0270 •715452 -142321 35 12-6209 •229006 •016813 80 3-847 I •725637 •149 706 6 12-5373 ■233 735 017 266 1 3-6434 •737 ^6s •158755 7 12-4496 •238700 ■017748 2 3-4066 •750571 •170329 8 J 2 -35 7 5 •243916 •018 261 3 3-1587 •764603 •183857 9 12-2606 •249401 •018808 4 2-9172 -778271 -198679 40 12-1586 ■255 174 •019392 85 2-693 6 -790931 -214138 I 12-0519 •261 214 020 014 6 2-5060 -801 547 -228620 2 11-9400 •267 5^0 ■020676 7 2-3716 -809 154 -239989 3 11-822 2 •274 212 021 386 8 2-2946 -813512 -246921 4 11-6986 •281 214 •022 145 9 2-279 •814395 -248365 4.? 11-5687 •288567 •022 959 90 2-3209 -812 027 ■244522 6 11-432 I -296298 ■023 833 I 2-393 5 -807918 -238081 7 n-287 1 ■304503 •024 782 2 2-4707 •803 545 -231522 8 11-1328 •313239 •025818 3 2^410 I -806976 -236643 9 10-9684 -322 546 026950 4 2-1143 •823 720 -264496 50 I o- 793 5 -332446 •028189 9a 1-7523 -844 210 •306729 I 10-6089 -342 896 ■029537 6 1-3470 •867 154 -369479 2 10-416 I •3S3 804 -030992 7 •9195 -891 352 -464375 •3 10-2163 ■365 116 ■032552 8 -471 7 -916697 -622 884 4 10-0099 ■376799 •034224 ^ •000 •943 397 -943 397 III. H:M(f'). HEALTHY MALE LIVES, OMITTING THE FIRST FIVE YEARS OF ASSURANCE. no HM(.1). Ehynentary Values. X lO /. Log/, I-Og/Jx A Colog fj. X 1 0000 4-000 000 ''•998 259 000 256 4"ooo 000 10 I 9960 3-998259 •998515 000 171 •001 741 I 2 9926 •996774 •998 686 000 039 •003 226 2 3 9896 •995 4*^0 •998 725 999 953 •004 540 3 4 9867 •994 '85 •998678 999907 •005815 4 ^5 9837 •992 863 •998 585 999 773 •007 137 15 6 9805 •991448 •998358 999726 •008552 6 7 9768 •989 806 •998 084 999 542 •010 194 1 8 9725 •987 890 •997 626 999 446 •012 1 10 8 9 9672 •985516 •997072 999 296 •014484 9 20 9607 •982 588 ■996368 999418 •017412 20 I 9527 •978956 •995 786 999726 •021 044 I 2 9 435 •974 742 •995512 999 8 1 2 •025 258 2 .3 9338 •970 254 •995324 999 949 •029 746 3 4 9238 •9^55578 •995 273 000 141 •034422 4 25 9138 •960 85 1 •995414 000 193 •039 149 25 6 9042 •956 265 •995 607 000 053 •043 735 6 7 8951 •951872 •995 660 000 105 •048 128 7 8 8862 ■947532 ■995 7^55 000 108 •052 468 8 9 8776 •943 297 ■995 873 000 X I I •056 703 9 3° 8693 •939170 -995 984 000 01 5 •060 830 30 I 8613 ■935 154 •995 999 999 962 •064 846 J 2 8534 •931 ^53 •995 9*51 000014 •068 847 2 3 8455 •927 114 •995975 999 909 •072886 3 4 8377 •923 089 •995 884 999751 •076911 4 35 8298 •918973 •995 ^35 999 848 •081 027 3S 6 821c; •914 608 •995 483 999 844 •085 392 6 7 8130 •910 091 •995327 999841 •089909 7 8 8043 •905 418 •995 168 999 945 •094582 8 9 7 954 •900586 •995113 999 944 •099414 9 40 7865 •895 699 •995057 000 000 •104 301 40 1 7 776 •S90 756 -995 °57 999 886 •109244 I 2 7688 •88^813 •994 943 999 883 •114 187 2 3 7 599 •880 7 -ji^ •994826 999 820 •119 244 3 4 7509 •875 582 •994 64*5 999 69^ •124418 4 45 7417 •870228 •994342 999 685 •129772 45 6 7321 •864570 •994027 999 ^73 •135430 6 7 7 221 •858597 •993 700 999 597 •141403 7 8 7 ^n •852 297 •993 297 999580 •147 703 8 9 7 008 •845 594 •992877 999 625 •154406 9 50 6894 •838471 •992 502 999 608 •161 529 50 I 6776 •830973 •992 1 10 999 '^54 •169027 I 2 66^4 •823083 •991764 999501 •176917 2 3 6529 •814847 •991 265 999 544 •185 1,^3 3 4 6399 ■806 1 12 •990 809 999 447 •193888 ' Ill HM(5). Elementary Values. X Colog/j^ A d. Log d^ A — A X lO o-ooi 741 999 744 40 i-6o2 060 929419 070581 10 I •001 48J 999 829 34 '53 1 479 945 642 054358 I 2 •001 314 999961 30 •477 121 985277 014723 2 •001 275 000 047 29 •462 398 014723 985 277 3 4 •00 X 322 000 093 30 •477 121 028 029 971971 4 ij •001 41 j 000 227 32 ■505 150 063 052 936948 15 6 •001 642 000 274 37 •568 202 065 266 934 734 6 7 •001 916 000458 43 •633 468 090 808 909 192 7 8 •002374 000554 53 •724276 088637 911 363 8 9 •002 928 000 704 65 •812913 090177 909 823 9 20 •003632 000582 80 •903 090 060 698 939302 20 I •004214 000 274 92 •963 788 022 984 977016 I 2 •004488 000 188 97 •986 772 013 228 986772 2 3 •004676 000 05 1 100 2-000 000 000 000 000 000 3 4 •004727 999 ^59 100 •000 000 982 271 017729 4 25 •004586 999 S°7 96 r982 271 976770 023 230 25 6 ■004393 999 947 91 •959 041 990349 009 651 6 7 •004340 999 ^95 89 •949 390 985 108 014892 7 8 •004 235 999 S92 86 ■934498 984580 015 420 8 9 •004 127 999 889 83 •919 078 984012 015988 9 30 •004 016 999 985 80 •903 090 994 537 005 463 30 I •004 001 000 038 79 •897 627 000 000 000 000 I 2 •004039 999 986 79 •897 627 994 468 005532 2 3 •004025 000 091 78 •892 095 005532 994 468 3 4 •004 116 000 249 79 •897 627 021451 978549 4 35 •004365 000 152 83 •919 07S 010341 989 659 35 6 •004517 000 156 85 •929419 010 100 989900 6 7 •004673 000 159 87 •939519 009 871 990 129 7 8 •004 832 000 055 89 ■949390 000 000 000 000 8 9 •004887 000 056 89 •949 390 000 000 000 000 9 40 •004 943 000 000 89 •949 390 995 093 004 907 40 I •004 943 000 1 14 88 •944 483 004907 995 093 I 2 ■005 057 000 1 17 89 ■949390 004 853 995 H7 2 3 •005 174 000 180 90 •954 243 009 545 990 455 3 4 ■005 354 000304 92 •963 788 018483 981 517 4 45 •005 658 000315 96 •982 271 017729 982 271 45 6 ■005973 000 327 100 2^000 000 017033 982967 6 7 •006300 000 403 104 •017033 020 393 979607 7 8 •006 703 000420 109 •037426 019479 980521 8 9 •007 123 000375 114 •056 905 014977 985 023 9 50 •007 498 000392 118 •071 882 014478 985522 50 I •007 890 000 346 122 •086360 010550 989 450 I 2 •008 236 000 499 125 •096910 017033 982 967 2 3 •008 735 000456 130 ■I 13 943 013 162 986 838 3 4 •009 191 000553 134 •127 105 015910 984 090 4 112 Elementartf Values — (continued) . X /x Log/^ Log/v A Colog Ir X 5S 6 26'; 3796921 7-990 2 ';6 999 4'3 4-203 079 55 6 6 126 ■787 '77 •9S9 669 999303 •2 12 823 6 7 5982 •776846 •988972 999329 •223 154 7 8 S^i^ •76/; 818 -988301 999 204 •234 182 8 9 5677 •754119 •987 j°5 998981 •245881 9 6o 5516 ■741624 •986486 998979 •258376 60 I 5 347 •728 1 10 •985 4^5 998 799 •271 890 I 2 5171 '^3 515 •984 264 998 686 •286425 2 3 4987 •697 839 •982 950 998 547 •302 161 3 4 4 795 •680 789 •981497 998 682 •319 21 I 4 6.^ 4 595 •662 286 -980 179 998532 •337 7H 65 6 4390 •642 465 -9787x1 998575 •357 53S 6 7 4 180 •621 176 •977 286 998 399 •378824 7 8 3967 •598 462 •975685 998 3 1 2 -401 538 8 9 3751 ■'574 H7 •973 997 997818 •425 853 9 /o 3 533 •548 144 •971 815 997 340 •451856 70 J 3 311 •519959 •969 K^5 996 728 •480041 I 2 3084 •489 114 •g6^ 883 996 2-9 •510886 2 3 2851 •454997 -962 142 995 834 ■545 003 3 4 2 61,3 •417 139 ■957 97*5 996510 •582861 4 75 2372 •375 115 ■954 486 995 808 •624 885 75 6 2 136 •329601 •950 294 996 41 1 •670399 6 7 1905 •279895 •946 705 995487 •720 105 7 8 1685 •226 600 -942 192 994867 ■773400 8 9 M75 •168 792 •937059 994516 •831 208 9 8o I 276 •10.^851 ■93^ 51 5 993 368 •894 H9 80 I I 090 •037426 •924943 992 930 •962574 r 2 917 2-962 369 •9^7873 992 873 3^037 63 1 2 3 759 •880 242 •910 746 992 871 •119758 3 4 6:8 •790988 •903617 993 955 •209 012 4 85 495 •694 60'; •897572 994551 •305 395 85 6 391 •592 177 -892 123 992 793 •407 823 6 7 305 •484 300 -884916 993 841 •515700 7 8 234 •369 216 •878757 990541 •630784 8 9 ^77 •247973 •869 298 986559 •752027 9 90 131 •117271 •855857 977 '95 •882 729 90 I 94 1-973 128 •833 052 973552 2-026872 I 2 64 •806 t8o -806 604 942 340 •193 820 2 3 41 •612784 •748944 930721 •387 216 3 4 23 •361 728 •679 66 i, 881 002 •638 272 4 95 1 1 ■041 393 •560 667 •958 607 95 6 4 o-6o2 060 1^397 940 6 113 HM(5). Elementary Values — (continued). X Co\o%Px A d. Log d. A — A X 5S 0-009 744 000587 139 2^143 015 015347 984 ^53 55 6 •010331 000 697 144 •158362 017729 982 271 6 7 •on 028 000 671 150 •176091 014 241 985 759 7 8 •on 699 000 796 ^5S •190332 016494 983 506 8 9 •012495 001 019 161 •206826 021 061 978 939 9 6o ■013514 001 021 169 •227887 017 626 982374 60 I •014535 001 201 176 ■245513 019305 980 6gs 1 2 •015736 001314 184 •2648x8 018483 981 517 2 3 ■017 050 001 453 192 •283301 017729 982 271 3 4 •018503 001 318 200 •301 030 010724 989 276 4 65 •019 82 I 001 468 205 ■311 754 010465 989 53 S 65 6 ■021 289 001 425 210 •322 219 006 161 993 839 6 7 ■022 714 001 601 213 ■328380 006074 993 926 7 8 ■024315 001688 216 •334454 004002 995 998 8 9 •026003 002 182 218 ■338456 007 897 992 103 9 70 •028185 002 660 222 •346 3 S3 009 673 990327 70 I •030 845 003 272 227 •356026 011330 988670 1 2 •034 1 17 003 74 T ^33 •367 35^ 009 221 990779 2 3 •037858 004 166 238 ■31^511 005 440 994560 3 4 •042 024 003 490 241 •382017 990 895 009 105 4 75 •045514 004 192 236 •372912 990 700 009 300 15 6 ■049 706 003 S^9 231 •363612 978811 021 189 6 •7 / ■053 295 004513 220 •342423 979 79*5 020 204 1 8 •057 808 005 133 210 •322 219 976634 023 ^66 8 9 •062941 005 484 199 •298853 970 660 029340 9 80 •068 425 006632 186 •269513 9''^8 533 031467 80 I ■075057 007 070 ^13 •238 046 960 611 039389 1 2 •082 127 007 127 .^58 •198657 950562 049438 2 3 •089 254 007 129 141 •149219 940 686 059314 3 4 ■096383 006 045 123 •089905 927 128 072 872 4 85 •102 428 005 449 104 •017033 917465 082535 85 6 •107877 007 207 86 1-934498 916 760 083 240 6 7 •115 084 006 159 71 •851 258 9046x7 095 383 7 8 •121 243 009 459 51 •755875 906 883 093117 8 9 •130 702 013441 46 •662 758 905 444 094 55^ 9 90 •144 H3 022 805 31 •568 202 908919 091 08 X 90 I •166948 026 448 30 •477121 884607 1 15 393 I 2 •193396 057 660 23 •361 728 893 545 106455 2 3 •251 056 069 279 18 •255273 823 908 176092 3 4 •3^0 33S 1 18 998 12 •079 181 765917 234 083 4 95 •439 2,33 7 •845 098 756962 243 038 95 6 4 •602 060 6 114 HM(5). Probabilities of Living over, and of Dying in, a Year, and the Average Duration ff Life, at each Age. X ;'x '> X Px 9r i^-Pr) t'x lO •996 000 •004 000 48^463 ^^ •977813 •022 187 16^767 I •996 586 •003414 47^^>5^J 6 •976494 •023 506 i6^i36 2 •996 978 •003 022 46^817 7 •974925 •025 075 ^5-512 .3 •997 069 •002 931 45-958 8 •973 423 ■026577 14^898 4 •996 960 •003 040 45-091 9 •971 640 •028 360 14-291 ^5 •996 747 •003 253 44-227 60 -969362 -030638 13-694 6 •996 226 •003774 43^370 I •967 084 -032 916 i3-ni 7 •995 598 •004402 42^532 2 -964417 -035 583 12-540 8 ■994 550 •005 450 4r7i8 3 -961 500 -038500 n-984 9 •993 280 •006 720 40-944 4 -958 290 -041 710 n-444 20 •991673 •008327 40-218 ^K^ •955386 •044614 10-921 I ■990 343 •009657 39^551 6 -952 164 •047 836 10-407 2 •989719 •010 281 38-932 7 -949 043 ■050957 9^905 3 •989 291 •010 709 38-331 8 -945551 -054 449 9-410 4 ■989 ^ 73 •010825 37^741 9 -941 882 -058 n8 8-923 25 •989 494 •010506 37-148 70 -937 164 •062 836 8-443 6 •989936 •010064 36-537 I -931 441 •068 559 7-975 7 •990057 •009 943 35^904 2 •924449 -075551 7-526 8 •990 296 •009 704 35-259 3 -916520 •083 480 7-100 9 •990 542 •009458 34-600 4 •907 769 •092 231 6-701 30 •990 797 •009 203 33-926 75 -900 506 -099 494 6-331 I •990 828 •009 172 33-236 6 •891854 •108 146 5-975 2 •990 743 •009257 32-539 7 •884514 •115486 5-639 3 •990775 •009 225 31-838 8 -87537^ •124629 5-310 4 •990 569 •009 43 1 31-130 9 •865 085 •^349^5 4-995 35 •989998 •010002 30^422 80 •854232 •145768 4-696 6 •989 ^^53 •010347 29-724 I •841 284 -158 716 4-412 7 •989299 •010 701 29^029 2 •827 699 -172 301 4-150 8 •988 935 •on 065 28^338 3 •814 230 ■185 770 3-910 9 •988 81 1 •oil 189 27-650 4 •800970 -199030 3-688 40 •988 684 •on 316 26-957 85 •789899 -210 lOI 3-480 I •988683 •011317 26^260 6 •780051 •219949 3-272 2 •988 424 •on 576 ^S'SSS 7 •767213 •232 787 3-054 3 •988 156 •01 1 844 24-848 8 •756410 •243 590 2-829 4 •987 748 •012 252 24-140 9 •740 I 12 •259888 2-579 45 •987057 •012943 23-433 90 -717557 •282 443 2-309 6 •986341 •013659 22-734 I •680 851 •3»9 '49 2-021 7 •985 598 •014402 22-042 2 •640 625 -359375 1-734 8 •984685 •015 315 2i-357 3 •560974 •439 026 1-427 9 •983 733 •016 267 20-681 4 •478 262 •521 738 1-152 50 •982 884 •017 116 20^015 95 •363637 •636363 •864 I •981995 •018 005 19-354 6 •000 000 I-OOOOOO •500 2 •981 214 •018786 18^700 3 •980 089 •OI99I I 18^049 4 •979059 ■020941 17-405 h:m(5). THREE PER-CENT. iir. 3 PER-CENT. Commutation Tdhlc. X 1>. N. Sx 1 iJ. X lO I 2 3 4 7440-941 \-J<,i22'2 7 i95'.32 168 126-9 6961-90 161 i6i;-o 6738-69 i54 42<^i"3 ^p~y^^ 147903-0 3487413-4 3 312 091-2 3 '43964-3 2982799-4 2828373-1 2 117-74 2 088-84 2 065-00 2044-57 2 025-40 75864-78 73 747-04 71 658- 19 69593-19 67 548-62 10 I 2 3 4 6 7 8 9 6 314-00 6 1 1 0- 1 ■; 5 909-So 5 7i2"4i 5515-81 141 589-0 135478-9 129 569-1 123 856-7 1 1 8 340-9 2 680470-1 2538881-0 2 403 402-2 2273833-1 2 149976-4 2 006-14 1 986-20 1 963-81 1 938-56 1 908-33 63517-09 61 530-89 59567-07 57628-52 15 6 7 8 9 20 I 2 3 4 .5319-16 5121-2.3 4924-06 4731-49 4544-48 113 021-7 107900-5 102 976-4 98 244-92 93 700-44 2 031 62, J 6 1 918613-9 1 810713-4 1 i^mT"^ 1 609492-1 1 872-34 1 829-34 1 781-32 1 732-17 1 682-98 55720-19 S3 847-85 52 018-51 50237-18 48505-01 20 1 2 3 4 25 6 7 8 9 4 3'54-36 4 192-73 4029-64 3873-37 3 724-06 89336-08 85 143-35 81 113-71 77240-33 7351^-27 1515791-6 1426455*5 1 260 198-5 1 182958-2 1 635-22 1590-71 1549-74 1 510-84 1 474-35 46822-03 45 186-81 43596-10 42 046-36 40535-52 25 6 7 8 9 30 I 2 3 4 3581-40 3 445-09 3314-07 3187-76 3 066-36 69 934-87 66489-78 63 175-71 59987-95 56921-60 1 109441-9 1039507-0 973017-2 909841-5 849853-6 1440-15 1 408-15 1 377-47 1 347-69 1319-14 39061-18 37 621-03 36212-88 34835-40 33487-72 30 I 2 3 4 35 6 7 8 9 294897 2 834-44 2723-41 2615-79 2511-50 S3 972-63 51 138-18 48414-77 45 798-98 43287-47 792932-0 738959-3 687821-2 639 406-4 593 607-4 1 291-06 1 262-42 1 233-95 1 205-65 1 177-55 32 168-58 30877-52 29 615-10 28381-15 27 175-49 3S 6 7 8 9 40 I 2 3 4 2 41 1-07 2314-36 2 221-52 2 131-85 2 045-24 40876-41 38562-05 36 340-53 34208-69 32 163-45 550319-9 509443-5 470 881-5 434541-0 400332-3 1 150-27 1 123-78 1 098-35 1 073-38 1 048-87 25 997-94 24847-67 23 723-89 22 625-54 21552-15 40 1 2 3 4 45 6 7 8 9 1961-34 I 879-57 I 799-90 I 722-31 I 646-53 30 202-IO 28322-54 26522-64 24800-33 23 153-80 368 1688 337966-7 309 644-2 283 1 21-5 258 321-2 1 024-54 999-895 974-969 949-801 924-192 20503-28 19478-74 18478-8'; 17503-88 16554-08 45 6 7 8 9 50 I 2 3 4 1572-57 I 500-63 I 43070 I 362-93 I 296-89 21^81-23 20080-60 1 8 649-90 17 286-97 1599008 235 '67-4 213 586-2 193 505-6 174855-7 157568-7 898-188 872-055 845-823 819-730 793-383 15629-88 14731-70 13 859-64 13013-82 12 194-09 50 1 2 3 4 117 HM(5). 3 PEE-CEXT. Commutation Table — (continued). D. N, ■s. M. S5 6 7 8 9 6o I 2 3 4 6 7 8 70 I 2 3 4 75 6 9 80 I 2 3 4 85 6 90 I 2 3 4 95 6 23275 i7o'29 109-49 050-17 992-482 936-248 881-129 827-307 774"630 723-113 672-769 624-033 576-876 531-534 487-954 446-209 405-991 367-142 329-5^9 293-214 258-418 225-929 195-627 167-995 142775 119-915 99*45 1 5 81-230 I 65-2758 51-6014 40-1274 30-7734 23-305 7 17-3596 12-7485 9-160 J, 6-381 8 4-218 2-623 8 1-429 o -663 5 •2343 14757-34 13587-05 12477-56 11427-39 10434-91 9498-661 8617-532 7 790-225 7015-596 6 292-483 5619-714 4995-680 4418-805 3 887-271 3399-317 2953-109 2 547-118 2179-976 1850-457 I 557-243 I 298-825 I 072-896 877-269 2 709-2742 566-4995 466-584 7 347-1332 265-903 I 200-627 3 149-025 9 108-898 5 78-125 54-8194 37-4598 24-7113 15-5507 9- 1 69 o 4-9505 2-3268 •8978 •2343 •000 o 141 578-6 126 821-3 113 234-2 100756-7 89329-28 78894-37 69395-71 60778-18 52987-95 45972-35 39679-87 34060-16 29 064-48 24 645-67 20 758-40 17359-08 14405-98 II 858-86 9678-88 7 828-425 6 271-1 4972-356 3 899-460 3 022-191 2312-917 1746-417 I 299-833 952-699 686-796 486-169 337-143 228-245 150-120 95-300 57-840 33-129 17-578 8-409 3-459 1-.J32 •234 -000 767-016 740-462 713-754 686-743 659-645 632-318 604-469 576-311 547-730 518775 489-493 460-352 431-370 402-831 374-732 347-199 319-978 292-954 266-024 239-317 213-061 188-099 164-378 142-443 122-116 103-415 86-444 2 71-1194 57-5310 45-7579 35-7869 27-601 6 21-0302 15-7629 ^^■657 5 8-440 8 5-9288 3-9514 2-4796 i'36i 2 •6374 -2274 1 1 400- 10633- 9893' 9179' 8492' 7833- 7 200' 6 596' 6019' 5 472' 4953' 4463' 4003' 3 572' 3 169 71 69 •228 ■474 731 085 767 •298 •987 ■257 •482 •989 •637 ■267 ■436 2 794-704 2447-504 2 127-526 I 834-572 I 568-548 1329' I ii6' 928' 763' 621' 499' 395' 309' 238' i'8o' 134" 99' 71' 50' 34' 23' 14- 8- 4' 2- •231 169 •070 ■693 •249 ■^33 718 •274 ■^55 ■624 ■866 •079 ■477 ■447 ■684 027 ■586 657 706 226 -865 •227 55 6 7 8 9 60 I 2 3 4 65 6 7 70 I 2 3 4 75 6 9 80 I 2 3 4 85 6 7 90 I 2 3 4 95 6 118 3 PKR CENT. Logarithms nml Cn-fot/arit/inis of I)^., Nj., anS 980 027 9 20 •725843 •983532 •274157 •016468 •053 162 979861 20 I •709375 ■982948 ■290 625 •017052 •033 023 879715 1 2 •692323 •982 6715 •307 677 ■017325 ■012 738 ,979572 2 3 •674998 •982 487 •325002 •017513 4^992310 979432 3 4 •657 485 ■982436 •342515 •017564 •971 742 979285 4 ^5 •639921 ■982 576 ■360079 •017424 •951027 979124 25 6 •622497 •982 770 •377503 •017230 •930151 978943 6 7 •605 267 •982822 •394 733 •017 178 •909 094 978750 7 8 •588 089 •982928 •41 1 911 -017 072 •887 844 978 540 8 9 •571017 •983 036 •428 983 -016964 •866384 978310 9 30 •554 053 •983 147 •445 947 -016853 •844 694 978061 30 1 •537200 •983 161 •462 800 •016839 •822 -/t^s 977 795 I 2 •520361 •983 124 •479639 ■016876 •800 550 977514 2 3 •503 485 ■983 138 •496515 ■016862 ■778064 977213 3 4 •486623 •983 048 •S^?>^11 ■016952 •755277 976897 4 35 •469 67 1 •982 797 ■530329 ■017203 •732174 976571 ?>S 6 •452468 •982 645 •547532 •017355 •708 745 976233 6 7 •435113 •982 491 •564887 •017509 •684978 975878 7 8 •417 604 •982330 •582396 ■017 670 •660 856 975506 8 9 ■399 934 •982 276 •600 066 •017724 •62,6 "^^i 975 111 9 40 •382 210 •982 220 •617790 •017 780 •611473 974687 40 I •364430 •982 220 -^2>SS1^ •017 780 •586 160 974231 I 2 •3466^0 ■982 106 -^S?> 350 ■017894 •560391 973 745 2 3 ■328756 ■981988 •671 244 ■018 012 •^?,\^z^ 973227 3 4 •310744 ■981 809 •689 256 ■018 191 •507 363 972674 4 45 •292553 •981505 •707447 ■018495 •480037 972095 45 6 •274058 ■981 190 ■725942 •018810 •452 132 971485 6 7 •255 248 •980 862 •744752 •019 138 ■423617 970841 7 8 •236 1 10 ■980 460 •763 890 •019540 ■394458 970 164 8 9 •216570 ■980 040 •783430 •019960 •364622 969 454 9 50 •196610 •979 665 •803 390 •020335 •334076 968 701 50 I •176275 •979272 •823 725 •020 728 ■302 777 967 900 T 2 •155547 •978927 •844453 •021 073 •270677 967 042 2 3 •134474 ■978428 ■865 526 •021572 •237719 966 132 3 4 •112 902 •977972 -887098 •022 028 •203851 965 157 4 119 3 PER-CENT. Logarithms and Co-logarithms of D^,, N^., and M^.; with their Differences. X Colog N,. A LogM, A Colog M^ A X lO &']^6i62, 018200 3^325 873 994033 4-674127 005 967 10 I •774363 018366 •319906 995 014 •680 094 004986 1 2 •792 729 018550 •314920 995 682 •685080 004 318 2 3 •8n 279 018744 •310 602 995 908 •689398 004 092 3 4 •830023 018947 •306510 995 852 •693 490 004 148 4 15 •848970 019 158 •302 362 995 661 •697638 004339 15 6 •868128 019370 •298023 995077 •701977 004 923 6 7 •887 498 019582 •293 100 994378 •706900 005 622 7 8 •907080 019785 •287478 993 176 •712522 006 824 8 9 •926 865 019973 •280654 991 n-^ -719346 008 269 9 20 •946838 020 139 •272385 989 909 •727615 010 091 20 I ■966977 020 285 •262 294 988 449 •737706 0^1551 I 2 •987 262 020428 •250 743 987 849 •749257 012 151 2 3 5-007 690 020568 •238592 987487 -761 408 012513 3 4 •028258 020715 •226079 987 498 •773921 012 502 4 25 •048973 020876 •2T3 577 988013 •786423 on 987 25 6 •069 849 021057 •201590 988 669 •798410 on 331 6 7 •090906 021 250 •190259 988959 •809 741 II 04 1 7 8 •112 156 021 460 •179 218 989381 •820782 010 619 8 9 •133 616 021 690 •168599 989 809 •831401 010 191 9 30 •155306 021939 •158408 990 241 -841 592 009 759 30 I •177245 022 205 •148649 990434 •851 351 009 566 I 2 •199450 022486 •139083 990 506 •860917 009 494 2 3 •221936 022 787 •129589 990 701 •870411 009 299 3 4 ■244723 023 103 •120 290 990657 -879710 009 343 4 ^5 •267826 023 429 •110947 990 258 •889053 009 742 3S 6 •291 255 023 767 •loi 205 990 093 •898 795 009907 6 7 •315 022 024 122 •091 298 989925 •908 702 010075 7 8 ■339 J 44 024 494 •081 223 989 758 •918777 010 242 8 9 •363638 024 889 •070981 989818 -929019 010 182 9 40 •38S527 025313 •060 799 989 882 •939201 010 118 40 I •413840 025 769 •050681 990 061 •949319 009 939 1 2 ■439 '^09 026 255 •040 742 990013 •959 258 009 987 2 3 ■465 864 026 773 •030 755 989967 •969 245 010 033 3 4 •4.92637 027 326 •020 722 989 808 •979278 010 192 4 45 •5 19 963 027905 •010 530 989 425 •989470 010575 45 6 •547868 028 515 2-999 955 989 036 3-000045 010964 6 7 •576383 029 159 •988991 988 642 •oil 009 011358 7 8 •605 542 029 836 ■977633 988 129 •022 367 on 871 8 9 •635378 030 546 ■965 762 987 605 •034238 012395 9 50 '66^ 924 031 299 ■953 367 987177 •046 633 012 823 50 1 •697 223 032 100 •940 544 986736 •059 456 013 264 I 2 ■729323 032 958 •927 280 986391 -072 720 013 609 2 3 •762 281 033 868 •913 671 985812 •086329 014188 3 4 •796 149 034 843 •899 483 985321 •100517 014679 4 120 3 PERCENT. Logarithms and Co-Io'/(iritIinis o/'D^.) N^., oikI }>l^', with their IJiffhrcnn.s. X LogD, A (Logr/;^) Colog D.,. A (Colog vpj. ) Log N.f A X 55 .3-090874 T-977418 ■+•909 126 0^022 582 4-169005 1964117 55 6 ■068 292 •976833 •931 708 •023 167 -133 12 r 963 005 6 7 •045 1 25 -976 134 •954875 •023 866 •096 1301961 817 ■ 7 8 •02 1 259 •975464I •978741 ■024536 •0.57947 960542 8 9 2-996 72.3 -9746683-003277 •025332 •018489 959 173 9 6o •97139' -973 649 -028 609 -026351 3'977 662 957721 60 I '945 040 -972627 -054 960 •027373 •935 383 956 167 I 2 •917667 •971427 •082 333 ■028573 ■891550 954 5 M 2 3 -889 094 •970 112 •I 10906 -029888 •846 065 952757 3 4 •859206 •968 660 -140794 -031340 •798822 950 892 4 65 •827866 •967 342 •172 134 -032 658 •749 7 14; 948 881 65 6 •795 208 •965 874 -204 792 -034 126 •698595 946 7 JO 6 7 -761 082 •964 449 •238918 ■°35 55' •6453051944340 7 8 •72."; 53 1 •962 848 •274469 •°37 152 •589 645 94'" 747 8 9 -688379 •961 159 •311 621 -038841 •531392 938 888 9 70 •649538 •958978 •350462 -041 022 •470 280 935 769 70 I •608516 •956318 •391484 -043 682 •406 049 932403 I 2 •564834 •953 046 •435 166 -046 954 •338452 928827 2 3 •517880 •949305 •482 120 •050 695 •267279 925077 3 4 •467 185 •945 138 •532815 -054 862 •192356 921 195 4 75 •412323 •941 649 •587677 •058351 '^^355^ 917007 75 6 '353 97^ •937457 •646028 •062 543 •030558 912575 6 7 -291 429 •933 867 •708571 •066 133 2^943 ^33 907 681 7 8 •225 296 '9^9355 •774704 -070 645 •850814 902386 8 9 •J5465J •924 222 •845 349 •075 778 ■753 200 896 704 9 80 •078873 •918738 •921 127 -081 262 •649 904 890 592 80 I i"997 61 1 •912 [06 2^002 389 -087894 •540 496 884227 I 2 •909 7 1 7 ■905 ^35 •090 283 -094965 •424 723 877667 2 3 •814752 •897 910 •185248 -102 090 •302390 870872 3 4 •712662 •890779 •287338 -109 221 •173 262 S63 760 4 85 •603441 •884735 •396559 -115 265 -037022 855 769 85 6 •488 176 •879 285 •511 824 -120 715 1-892 791 846 144 6 7 -367461 •872 079 •632539 -127 921 •738935 834631 7 8 -239540 •86'; 920 •760460 -134080 '573 566 819 329 8 9 -105 460 •856461 •894540 •^3 539 -392 895 798 856 9 90 0-961 921 •843 020 7-038079 •156980 -191751 770571 90 I •804941 •820 214 •J 95 059 •179786 0-962 322 732328 I 2 •625 15.' •793 767 •3 74 84;-; •206233 •694 650 672 103 2 3 •418922 •736107 •581078 •263 893 '366 753 '186416 3 4 -155029 •666827 •844 97 J, ■333 ^73 1-953 169 416 51 7 4 95 I -82 1 8';6 •547 830 0^1 78 144 -452 170 •36g6S6 1 95 6 •369 686 •630314 6 121 3 PER-CENT. Logarithms and Co-logarithms ofD^, N^,, and Mj.; with their Differences — (con tinued) . X CologN^ A LogM^ A Colog M^ A 55 55 5-8,3o 992 035883 2-884804 984699 yii5i96 015301 6 •866875 036995 -869 503 '984 045 •130497 1 015 955 1 6 7 ■903870^038183 •853 548 983 246 •146452 016 754 7 8 •942053 039458 •836794 982517 •163 206 017483 8 9 •981 511 040827 -819311 981 625 -180689 018375 9 6o 4-022338 042279' -800936 980438 -199 064 019562 60 I •064617 043833 -781374 979283 -218626 1 020 717 I 2 •108450 045 485 -760657 977910 ■239 343 022 090 2 3 '^S3 93S ' 047 243 1 738 567 , 976 412 •261 433 ' 023 588 3 4 -201 178 0491081 -714979 974767 •285 021 025 233 4 65 •250 286 051 119 •689 746 973 344 •310254 026 6^6 65 6 •301405 053290 •663 090 971 760 •336910 028 240 6 7 •354 695 °5^ 660 ■634850 970273 '365 150 029727 7 8 ■4 J 0355 058 253 -605 123 ;968 ^98 •394877 03 1 402 8 9 •468 608 061 112 -573 721 966 858 •426 279 033 H2 9 70 •529 720 064 231 •540579 964541 ■459421 035 459 70 I '593 95^^ 067597 •505 120 961 680 •494880 038320 I 2 •661 548 071 173 •466 800 958 121 •533 200 041 879 2 3 •732721 074923 -424921 954053 •575079 045 947 3 4 •807 644 078805 •378974 949531 •621 026 050469 + 75 •886449 082 993 •328505 945 882 ■671495 054 118 75 6 -969442 087 425 •274387 941 455 •725613 058 545 6 7 3-056867 092319 -215842 937801 •784158 062 199 7 8 •149 186 097614 ■K'>3 6^3 933 130 •846357 066870 8 9 •246 800 103 296 •086773 927 810 •913227 072 190 9 80 •350096 109408 •014583 922 153 •985417 077847 80 I ■459 504 1 15 773 1-936736 915252 2-063 264 084 748 1 2 •575277 122333 •851988 907914 •148 012 092 086 2 3 •697 610 129 128 -759902 900 564 •240098 099436 3 4 -826738 136240 •660 466 893 258 •339 534 106 742 4 85 -962 978 144 23 1 '553 724 887210 •446 276 112 790 85 6 2-107 209 ^53 856 ■440 934 881 909 '559 066 11 8 09 1 6 7 •261 065 165369 -322843 874 794 •677 ^57 125 206 7 8 •426434 180671 •197637 868967 •802 363 i3'i 033 8 9 -607 105 201 144 -066 604 859 779 -933 396 140 221 9 90 -808 249 229429 0-926383 846 ';86 1-073 617 153 4H 90 I 7-037678 267 672 ' •772969 823 784 -227031 176210 I 2 •305350 327897 •596753 797621 •403 247 202379 2 3 '633 247 413584 •394374 739 554 -601; 626 260446 ' 3 4 0-046 83 1 583483 •133928 670467 •866072 329533 4 95 -630314 1 T^8o4 395 552454 0^195605 447 546 95 6 i ■SS*^ 849 •643 151 i 6 122 3 PERCENT. Values of AnnuitHS, and Single and Annual Pi'cmvnm for Assurance of a Lnit. X flx Ax Vj: X Ox A, «r. lO i3\/ji9 • 284607 01 I 587 55 II-971 I - 622 200 - 047 968 I 23-366 2 290 306 01 I 914 6 I r6io 632719 050176 2 23-1496 296614 012 282 7 1 1-246 2 643314 052 532 .3 22-9163 303 407 012 686 8 10-881 5 653 936 055 038 4 22-673 2 310489 013 116 9 10-5140 664 643 057 725 15 22-4247 3 1 7 730 -013564 60 10-1454 675375 060 596 6 22-172 8 -325 066 -014028 1 9-780 1 -686016 -063637 7 21-924 5 -332 298 -014495 2 9-4164 -696 610 •066877 8 21-682 I •339358 -014962 3 9*056 7 -707 087 -070310 9 21-4549 •345 975 -015 408 4 8-701 9 -717420 -073 946 20 21-248 I •351999 -01 5 822 65 8^353 I ■727579 -077790 I 21-069 2 -357206 -016 186 6 8-005 5 ■737704 -081 917 2 20-912 9 •361 759 -016 509 7 7^6599 ■747770 -086349 3 20-764 I -366095 -016821 8 7^3133 •757865 -091 163 4 20-6185 •370334 -017 130 9 6-966 5 -767 966 -096 400 25 20-469 4 -374676 -017452 70 6-6182 -778 no -102 138 6 20-307 4 •379396 •017 806 I 6-2738 -788 142 •108353 7 20-129 2 •384585 -018 202 2 5*937 7 •797932 -II5014 8 19-941 4 -390058 -018626 3 5'6i5 6 -807312 •I2203I 9 19-7409 •395 896 -019 088 4 5'3 10 9 -816186 -129329 30 19*527 3 •402 1 19 -019590 75 15-026 I •824484 -136820 I 19-2999 -408 742 •020 135 6 4-748 8 •832559 -144823 2 19-0629 •415645 -020 717 7 4-484 4 ■840 259 -153209 3 18-8182 -422770 -021 332 8 4-222 •847 904 •162372 4 18-5633 •430 197 •021 990 9 3-967 8 •855307 •172171 35 18-302 2 -43 7 800 -022681 80 3^7242 •862 402 •182 551 6 18041 7 •445 387 -023 390 I 3^4905 -869210 •193567 7 177773 •453 091 •024 130 2 3^2735 •875 530 •204877 8 i 7 -508 7 -460913 •024903 3 3^073 5 •881354 -216361 9 17-2356 •468 864 •025711 4 2-8880 -886 756 •228074 40 16-953 6 -477078 -026573 85 2-7138 -891 832 •240 139 I 16-662 1 •485 569 •027492 6 2^538 7 -896928 -253 462 2 16-3584 -494416 •028483 7 2-3522 -902 364 -269 186 3 16-046 5 -1503 1500 •029537 8 2-1579 •908024 ■287 543 4 15-726^ -512836 -030661 9 ^•938 4 -914416 -311 199 45 i5*398 7 •522 369 •03 1 854 90 1-697 6 -921430 •341577 6 15-0686 '5:i^ 983 •033 107 1 1^436 8 -929026 •381 257 7 147356 -541 680 •034424 2 ^••73 5 •936 694 •430 954 8 H-399 5 •551 471 •035811 3 -886 8 •945 044 -500869 9 14-062 2 -561 296 •037265 4 -6283 •952575 •585 026 50 13723 5 •571 159 -038 792 95 ■35:i I -960 592 •709947 I i3'38m -581 12^ •040 40J 6 -000 c -970874 -970874 2 13-035^ ) -591 199 •042 121 3 12-683/ -601 44*^ > •043954 4 12-329 (' ) -611 76c ) ^045 89' HM(5). THEEE AND A HALF PER-CENT. ILM 3! I'KK-CENT. Committatum Tdlth'. X D. N. Sx M, K X lO I 2 3 4 7 089*19 6822-06 6,<;68-86 6327-54 6 095-65 152 122-0 145 300-0 138 731-1 132 403-6 126307-9 2851473-6 2699351-6 2554051-6 2415320-5 2 282 917-0 1 705-23 1677-84 1636-15 1 618-24 57 400-61 ss 695-38 54017-54 52 362-21 50 726-06 10 I 2 3 4 6 7 8 9 5871-61 5 654-60 5442-77 5 235*56 5 030-95 120436-3 114781-7 109338-9 104 103-4 99072-42 2 156 609-1 2 036 172-8 1921391-1 1 8i2 052-1 1 707 948-8 I 600-33 1581-88 I 561-26 1 538-11 1510-54 49 107-82 47 507-49 45925-61 44364-35 42 826-24 15 6 7 8 9 20 I 2 3 4 4828-15 4626-03 4426-44 4232-78 4045-85 94244-27 89618-24 85 191-80 80 959-02 76913-17 1 608 876-4 1425013-9 1 339822-1 1 258 863-0 1477-88 1 439-03 I 395-87 1351-90 1 308-10 41315-70 39837-82 38 398-79 37002-93 3S^S^'^3 20 I 2 3 4 25 6 7 8 9 3866-72 3696-71 3 535-76 3 382-22 3236-13 73 046-45 69349-74 65813-99 62431-77 59 195-63 1 181 949-9 1 108903-4 1039553-7 973 739-7 911307-9 I 265-79 1 226-54 1 190-59 1 156-63 1 124-91 34342-92 33077-13 31850-59 30 660-00 29503-37 25 6 7 8 9 30 I 2 3 4 3097-X3 2 964-86 2838-32 2716-95 2 6oo-86 56098-51 S3 ^3y^s 50 295-33 47578-38 44977-52 852 112-3 791 013-8 742 8S0-1 692584-8 645 006-4 I 095-34 1 067-81 I 041-53 I 016-14 991-927 28378-46 27 283-11 26 215-31 25173-78 24157-63 30 I 2 3 4 35 6 7 8 9 2489-21 2 380-98 2 276-66 2 176-13 2 079-28 42488-31 40107-33 37830-68 '^s 654-55 33575-27 600028-9 557540-6 517433-3 479602-6 443 948-0 968-229 944-173 920-370 896-831 873-566 23 165-71 22 197-48 21 2^y^l 20332-93 19436-10 35 6 7 8 9 40 I 2 3 4 I 986-48 I 897-59 1812-67 1731-10 X 652-75 31588-79 29 691-20 27 878-53 26 147-44 24 494-69 410372-8 378784-0 349092-8 321 214-2 295 066-8 851-087 829-368 808-619 788-345 768-535 18562-54 17 711-45 16882-08 16073-46 15285-12 40 1 2 3 4 45 6 7 8 9 ^';77•29 1 504-23 1433-51 1 365-09 1 298-72 22917-39 21 413-16 19979-65 18614-57 ^7315-84 270572-1 247654-7 1 226241-5 206 261-9 187647-3 748-971 729-246 709-394 689-446 669-246 14516-58 13767-61 13038-37 1232897 1 1 629'S2, 45 6 7 8 9 50 I 2 3 4 1 1 234-39 1 172-24 1 112-20 1 054-41 998-466 16081-45 14909-21 13 797-01 12 742-60 II 744-13 1 70 33 1-5 154 250-0 139340-8 125543-8 1 ji 801*2 648-834 628-420 608-028 587-841 567-557 TO 970-28 10321-45 9693-026 9 084-998 8497-157 50 1 2 3 4 125 3^ PERCENT. Commutation Table — (continued). M, 55 6 9 6o I 2 3 4 65 6 70 I 2 3 4 75 6 9 80 I 2 3 4 85 6 90 I 2 3 4 95 6 944-500 892-313 841-873 793-007 745"827 700-170 655-766 612-735 57o'949 530"403 491-092 453"3i<5 4i7"035 382-400 317-921 287-869 259-065 23 1 '394 204-905 179717 156-363 134737 ii5"^47 97"387' 81-399 67-182 54-608 43-671 ?>\-?>ss 26-587 20-291 15-292 11-336 8-284 5'924 3 4' J 07 3 2-701 9 1-672 4 •9064 -4189 -1472 10 799-634 9907-321 9 065-448 8 272-441 7526-614 6826-444 6 170-678 5557-943 4 986-994 4456-590 3 965*499 3 512-182 3095-147 2712-746 2 045-474 I 757-605 I 498-540 I 267-146 I 062-240 882-5238 726-1605 591-4232 476-276 2 378-8884 297-4887 230-3059! 175-6974! 132-0264' 97-6707; 71-083 3 1 50-7922 35 '499 3 24-163 I 15-8783 9-954 o 5-8467 3-1448 1-4724 -5660 •1472 -0000 lor 057-09 90 257-45 80350-13 71284-68 63 012-24 ss 485-63 48659-19 42488-51 36930-57 31943-57 27486-98 23521-48 20009-30 16914-15 14 201-41 11838-01 9 792-539 8 034-934 5 269-249 4207-008 3 324-485 2598-324 2 006-901 I 530-625 I 151-736 854-248 633-942 448-244 316-218 218-547 147-464 96-672 61-173 37-009 21-131 iri77 5-330 2-186 •713 ■147 •000 547*355 527-108 506-843 486-446 466-083 445-646 424-920 404-065 382-999 361-761 340-386 319-217 298-266 277-734 257-616 237-999 218-698 199-629 180-719 162-055 143-796 1 10-181 95'i47 2 81-2818 68-587 57-122 46-8204 37-7295 29-891 I 23-2845 17-8874 10-1357 7-467 7 5-3874 3110 1 2-504 2 1-5660 -8566 •3997 -142 2 7929 7382- 6855- 6348- 5861- 5 395- 4 950- 4525- 4 121- 3738- 3376" 3 035" 2 716- 2418' 2 140" '600 245 •137 294 •848 765 •118 •198 ■133 ■134 •373 987 770 504 770 154 o 154 64 426-456 226-827 046-108 884-053 3 740-257 8 613-7383 503-557 I 408-409 9 327-128 1 258-541 I 201-418 3 ^54*597 9 116-8684 86-977 3 63-692 8 45-805 5 32-2302 22-0945 14-6268 9-2394! 5-4687 2-9646 1-3985 ■5419 -142 2 5S 6 9 60 I 2 3 4 65 6 70 1 2 3 4 75 6 9 80 1 2 3 4 85 6 7 8 9 90 1 2 3 4 95 6 120 3 J I'EIU'ENT. Ij^ 793 -016 791 I 6^io^ 6 •759715 •106918 7 18-6138 'i^Ci 730 -017 168 2 5^7844 •770575 •113580 8 i8-45"8 8 •341 972 •017574 3 5^476 2 -781 000 •120597 9 18-292 1 •347610 •018018 4 5-1841 •790877 •127 890 3° J 8* 1 13 I '353 664 -018504 75 4-910 6 -800 122 •135370 I 17*921 2 •360 154 -019034 6 4-6441 -809137 •143361 2 17-7201 •366 953 -019 602 7 4*3895 •817 749 •151 731 3 i7\';ii7 -374002 -020 204 8 4-1362 -826310 •j6o 879 4 i7"293 3 •381 384 •020 848 9 3-8905 -834620 •170 661 3.? 17-069 -388971 •021527 80 3-6547 -842 596 •181 022 6 16-8449 •39'5 549 -022 222 I 3-4280 -850 260 •192 017 7 1 6*6 16 7 -404 264 -022 948 2 3"2/74 •857382 .203 297 8 16-3844 -412 122 -023 706 3 3-023 2 •863 948 •214742 9 i6'i47 6 •42c 130 ■024501 4 2842 9 •870046 •226402 40 15-9019 -428 439 ■025 349 85 2-6736 •875 772* •238 399 I 15-6468 -437 064 -026 255 6 2-503 2 •881 '5361 -251 639 2 i5'379 8 -446 09,3 -027 234 7 2-3213 •887688 •267271 3 15-1046 •455 402 -028278 8 2-X3I5 •894104 -285518 4 14-8206 -465 004 -029392 9 1-9166 -901372 -309053 1 4.=; I4-.529 5 -474 846 •030577 90 1-6802 •909 3641 '339 292 6 H'235 3 •484 797 -031 821 1 I "42 3 5 -918048-378813 13-9376 -494 866 •033 129 2 1-1639 •926 824' -428306 8 13-636 2 •505 057 •034 507 3 -880 5 -936410 -497970 9 i3'333 '')^5 3io •035 953 4 •6244 -945 068 -581 782 50 13-027 8 •525 '^30 •03 7 47 i 95 •3S^3 •954303 • 706 190 1 12-7186 •536085 •039077 6 -000 •966 184 ^966 184 2 12-405 I •546 688 -040 782 3 12-085 I •557 509 -042 606 4 11-762 2 •568 429 -044 540 HM(5). FOUR PER-CENT. i;{2 HM(.'.). 4 i'Eli-CKNT. Cinnmutation Tnhle JO I 2 3 4 6 7 8 20 I 2 ,3 4 w N. .3° I 2 3 4 3 J 40 I 2 3 4 6 7 8 .SO I 2 3 4 6 7.';.';'^>4 6 469- a,} 6 J997'5 5 943-i8 5462-14 5 234"97 5014-63 4800-53 4 590'74 4 3 84-5 1 4180-77 3 98'"i5 3 788-67 3603-94 3427-82 3 261-35 3 104-36 2 955*28 2 814-04 2 680-21 ^553-41 2432-68 23'7'47 2 207-78 2 J02-84 2 001-74 I 904-83 J 811-97 I 723-00 I 638-19 1 5.';7"36 I 480-52 I 407-09 1 269-78 1 205-14 1 142-96 1 083-17 1 025-56 970-073 916-797 865-664 816-732 769-683 M. If. 32 786-0 26316-1 20 1 16*4 H4 i73"i [08 47 5" 2 103 013-0 97 778-06 92 7''>3"43 87 962-90 83372-16 78987-65 74 806-88 70825-73 67 037-06 63 433" 1 1 60005-29 ^^i 743'94 :)?> <539\'i9 50 684-3 1 47870-27 45 190-06 42 6;^6-6e, 40 203-96 37886-50 35 678- 7 i 33 575-88 31 574' 1 4 29 669- 31 27 85 7 '34 26134-34 24496-14 22 938-78 21 458-27 20051-17 18714-22 17444-44 16 239-31 '5 096'35 14013-18 12 987-62 '2017-55 1 1 100-75 10 235-09 9 4>8-35: 8648-67: 2 348401 2 2 1 5 6 1 •■; 2 089 298 I 969 182 1 855 009 I 74*^534 I 643 521 I 545 743 1452979 I 365 016 I 281 644 I 202 657 J 127 850 I 057 024 989 986-9 926553-8 866 548-5 809 804-5 756 164-9 705 480-6 657 610-4 612 420*3 569 7837 529579-7 491 693-2 456014-5 422 438-6 390 864-4 361 195- 1 307 203-5 282 707-3 259 768-5 238310-3 218 259-1 199544-9 182 100-4 I6586I-I 150 764-8 136751-6 123 764-0 1 1 1 746-4 100 645-7 90410-61 80992-26 388-66 43851-53 362-67 42462-87 341-44 41 100-20 323-42 39758-76 306-67 38435-34 290-02 37 128-67 272-93 35838-65 253-94 34565-72 •232-71 33311-79 I 207-55 32 079-08 1 177-89 30871-53 1 142-78 29693-64 1 103-96 28 550-86 1 064-61 27 446-89 r 025-59 26 382-29 988-082 25 356-69 953-456 24 368-6I 921-896 23415-16 892-216 22493-26 864-640 21 601-04 839-050 20 736-40 815-333 792-813 771-160 750-603 730-583 710-359 690-443 670-844 651-564 633-027 615-202 598-255 581-775 550-001 534-198 518-370 502-541 486-590 470-549 454-584 438-712 423-075 407-438 19897-35 19082-02 18 289-21 17518-05 16767-44 16 036-86 15 326-50 14636-06 13965-22 12 680-62 12 065-42 11467-17 10885-39 10 3 19-64 9 769-640 9 235-442 8717-073 8214-531 7727-941 7257-392 6 802-809 6364-097 5 941-022 10 1 2 3 4 15 6 7 8 20 1 2 3 4 25 6 7 8 9 30 I 2 3 4 6 40 I 2 3 4 45 6 7 8 50 1 2 3 4 133 HM(5), 4 PERCENT. Commutatioi TahJe — (continued). X D. Nx s. M, R. X y:) 724-582 7 924-089 72343*59 391-941 5S3yS^3 55 6 681-256 7 242-833 64419-50 376*483 5 HI -643 6 7 639-656 6603-178 57 176-67 361-085 4765-160 7 8 599'63 I 6003-547 50573*49 345-662 4404-075 8 9 561-244 5 442*303 44569*94 330-339 4058-413 9 6o 524-353 4917*950 39 127-64 315*034 3728-074 60 I 488-738 4429-211 34 209-69 299*587 3 413*040 I 2 454"472 3 974*739 29 780-48 284-118 3 113*454 2 3 421-443 3 553*296 25 805-74 268-569 2 829-336 3 4 389-632 3 163-663 22252-44 252-967 2560-767 4 65 359-020 2 804-644 19088-78 237*340 2 307-800 65 6 329-810 2474*833 16284-13 221-939 2 070-460 6 7 301*955 2 172-878 13 809-30 206-769 I 848-520 7 8 275"547 I 897-332 1 1 636-42 191-974 1641-751 8 9 ^S^'S^i 1 646-809 9 739*092 177*548 I 449*777 9 70 226-887 I 419-922 8092-283 163-548 I 272-228 70 I 204-452 1 215-470 6672-361 149-840 I 108-680 I 2 183-111 1 032-359 5 456*892 136-362 958*840 2 3 162-766 869-592 8 4424*533 123-060 822-478 3 4 143*441 726-152 I 3 554*940 109-995 699-418 4 75 125-203 600-949 2 2828-788 97*2739 589-424 75 6 108-4 10 492*5396 2227-839 85-296 1 492*150 6 7 92-966 8 399*5728 I 735*299 74-023 406-853 7 8 79-067 8 320-5050 1335*726 63-6996 332-831 8 9 66- SS I 6 ^i)3-9S2S 1 015-221 54-2244 269-131 9 80 55*358 4 198*595 I 761-268 45'59i 214-906 80 I 45*470 ^ 153*1250 562-673 37-8318 i6g-^i6 I 2 36-7820 116-3429 409-548 30-8926 131*484 2 3 29*2735 87-0694 293-205 24-798 8 100-591 3 4 22-9186 64-1509 206-135 19-5698 • 75*7923 4 85 17-651 1 46*499 8 141-984 15*1837 '56-222 6 85 6 13*4063 33*093 4 95*485 11-6179 41-0388 6 7 10-0554 23-038 62-391 8-7826 29-4209 7 8 7-41792 15-620 I 39'3!:i3 6*5318 20-6384 8 9 5*395 ^8 10-2249 ^3133 4*7944 14-1065 9 90 3*83947 6*385 5 13*508 3*446 2 9-312 1 90 1 2-64907 3*7364 7-123 2*403 5 5*8659 I 2 1*73425 2-002 I 3*386 1*5905 3*462 4 2 3 1-06828 *933 9 1*384 •9913 1-8719 3 4 •57623 *357 6 *45o •5403 -8806 4 95 -26499 '092 7 •093 •251 2 *340 3 95 6 ■092 65 -ooo •000 -089 1 -089 I 6 134 4 PEHCKNT LogariOans and Co-lj.) CologD^ A (Cologty),) Log Nx A X lO 3-829667 7-981 226 4^^70333 0-018 774 5^123 152 978307 10 I •81089,3 •981 481 -189 107 •018 •;i9 •101459 978144 I 2 •79^374 •981 652 •207 626 -018348 •079 603 977961 2 3 •774026 •981 692 •225974 •018308 •057564 977 766 3 4 •755 7i« •981 645 •244 282 •018355 •035 330 977562 4 15 •737363 •981551 •262637 •018449 •012 892 977 349 15 6 •718914 -981325 •281086 •018 675 4^990 241 977 136 6 7 •7002,39 -98 1 05 1 •299 761 •018949 •967377 976923 7 8 •681 290 •980 593 •318 710 •019407 ■944 300 976721 8 9 •661 883 -980 03 S •338 1 1 7 •019962 •921 021 976538 9 20 •641 921 •979 335 •358 079 •020 665 ■897 559 976383 20 I •621 256 •978752 •378 744 •021 248 •873 942 976249 1 2 •600 008 -978479 •399992 •021 521 •850 191 976124 2 3 •578487 -978 291 •4215^3 •021 709 •826315 976001 3 4 •556778 •978 240 •443222 -021 760 •802 316 975 874 4 ^5 •535 oj8 -978380 •464982 •021 620 •778190 975 729 25 6 •5^3398 •978 51 ?> •486602 •021 427 •753919 975 567 6 7 •491 971 •978627 •508029 •021373 •729486 975 388 7 8 •470598 •978732 •529402 •021 268 •704874 975 192 8 9 •449330 •978 840 •550 670 •021 160 •680 066 974977 9 30 •428 170 •978951 •571830 021 049 ■^ss 043 974 740 30 r •407 121 -978965 •592 879 •02 X 035 •629 783 974486 I 2 •386086 •978927 •613 914 •021073 •604 269 974216 2 3 •365013 •978942 •634987 •021 058 •578485 973924 3 4 ■343 955 ■978 852 •656045 -021 148 -55^ 409 973618 4 3.5 •322 807 •978600 •677 193 -021 400 •526027 973 305 35 6 •301 407 -978450 •698 593 -021 550 •499332 972975 6 7 •279857 -978294 ■720143 -021 706 •472307 972633 7 8 •258 151 •978 134 -741 849 •021866 •444 940 972272 8 9 •236285 -978080 •1^3 1^5 .021 920 ■417212 971886 9 40 •214365 -978024 •785635 -021 976 •389098 971472 40 I •192389 •978024 -807 611 -021 976 •360570 971025 I 2 •170413 •977910 -829 ^87 -022 090 •331595 970545 2 3 •148323 ■977792 -851677 -022 208 -302 140 970032 3 4 •126115 •977613 •873 885 •022 387 •272 172 969 485 4 45 •103 728 •977309 -896 272 •022 691 •241 657 968910 45 6 •081037 •976993 •918963 •023 007 •210 567 968 305 6 7 •058 030 •976667 •941970 •023 333 •178872 967 665 7 8 •034 697 •976 264 •965 303 •023 736 •146537 966993 8 9 •010961 •975 843 •989 039 •024157 •113 530 966 286 9 50 2-986804 •975469 3-013 196 •024531 .079816 965 536 50 I •962 273 -975 076 •037 727 •024924 ■045352 964 740 I 2 •937349 •974731 -062 6^1 •025 269 •010 092 963 883 2 3 •912 080 -974232 -087 920 .025 768 3^973 975 962974 3 4 •886312 ■973 775 -113688 •026 225 •936949 962 000 4 135 4 PERCENT. Logarithms and Co-hgarithms of Dxi N^, and M^.; with their Differences. X CoiogN^ A Log Mj; A Colog Mx A X lo 6-8768481 021693 3^142595 991797 4-857405 008 203 10 I •898541 021856 •134392 993 178 ^865 608 006822 1 2 •920397 022039 ^127 570 994128] -872430 005 872 2 3 •942 4.36 022 234 •121 698 9944691 -878302:005531 3 4 •964670 022 438 •116 167 994428; -883833,005572 4 15 •987 108 022 651 •I 10 595 994210 -889405 005790 15 6 5-009 759 022 864 •104805 993470! -895 195 006530 6 7 •032623 023077 •098 275 992 586! -901725:007414 / 8 •055 700 023 279 •090 861 991045 '909 139 '008955 8 9 •078979 023462 •081 906 989 198 1 -918094 010 802 9 20 •102 441 023617 •071 104 986859 -928896 013 141 20 I •126058 023751 •057 963 984991 1 -942037 015 009 I 2 • 149 809 023 876 •042 954 984235! -957046 015765 2 3 •173685 023 999 •027 189 983 786 -972811 016 214 3 4 •197 684 024 126 •010975 983818 -989 025 016 182 4 25 •221 810 024 271 2^994 793 984508 3^-005 207 015492 25 6 •246081 024433 •979301 985379 -020 699 014 621 6 7 •270514 024612 -964 680 985 790 •035320 014 210 7 8 •295 126 024808 •950470 986365 -049530 013635 8 9 •319934 025 023 •936835 986953 •063 165 013 047 9 30 •344957 025 260 •923 788 987 547 •076212 012453 30 I •370217 025 5H •911 335 987 836 -088 66^ 012 164 1 2 •395731 025 784 •899171 987 974 •100 829 012 026 2 3 •4215^5 026076 •887 145 988 265 •112855 on 735 3 4 •447591 026 382 •875410 988 260 •124590 oil 740 4 35 •473973 026 695 •863 670 987 808 •136330 012 192 35 6 •500 668 027 025 •851478 987 650 -148 ^22 012 350 6 7 •527 693 027367 •839 128 987 493 •160872 012507 7 8 ■555 060 027 728 •826621 987336 -173379 1 2 664 8 9 •582 788 028 114 •813957 987 465 •186043 012535 9 40 •610902 028528 •801 422 987 596 •198578 1 2 404 40 I •639 430 028975 •789018 987 869 •210982 012 131 1 2 •668 405 029 455 •776887 987 868 •223 113 012 132 2 3 •697 860 029968 •764755 987870 •235 245 012 130 3 4 •727828 030515 •752625 987738 •247375 012 262 4 45 •758343 031 090 •740363 987339 •259637 012 66 i 45 6 •789433 031695 •727702 986938 •272 298 013 062 6 7 •821 128 032 335 •714640 986532 •28^ 360 013 468 7 8 •853 463 033 007 •701 172 985991 •298 828 014009 8 9 •886470 033714 •687 163 985 442 •312837 014558 9 50 •920 184 034 464 •672 605 985 009 •327395 014991 50 1 •954 648 035 260 •657614 984 565 •342386 015435 I 2 •989 908 036117 •642 179 984 239 •357821 015761 2 3 4^026025 037 026 •626418 983644 •373582 016356 3 4 •063 051 038 000 •610 062 983 158 •389938 016 842 4 130 4 rEK-CKNT. Logarithms nud C(>-lo3 954 •033 289 3 -8742 •927916 •495 104 9 12-6639 •474462 -034724 4 •620 7 -937668 •578573 50 12-3883 •485 067 -036 231 95 ■3497 -948088 •702470 [ 12-108 2 -495 839 -037827 6 •000 -961538 -961538 2 11-8234 •506 792 •039521 3 ''■5317 •518 010 •041336 4 11-2367 ■529359 •043 260 TAVO LIVES. THREE PER-CENT. IKi HM. 3 ri:i{ ("KNi". Values of Annuities on Two Joint LinSj tnvl on the Last Survivor of Tu'ti Lives. Ltjua/ Ao:)r) 70 •2226 •0904 6 •3398 •2S7-i I 3 9453 8 -07 05 7 • lOI I •0(;.35 2 •6788 •2708 8 i6"86oo 23S3-U 3 •4279 7-8773 9 •6176 •G004 4 ■^975 •4989 30 •3734 •3r,u 75 2-9876 •1352 I •1281 -1173 6 •7852 G-7788 2 15-8782 22-8(;70 1 ■5924 -4312 3 •6229 •OlOU 8 •4068 •0914 4 •3615 •3483 9 •2247 5-7585 3.^ •09 -jo •0792 80 •0488 •4300 6 14-8242 21-8010 I 1-S844 •1288 7 •5509 -5231 2 •7379 4-8415 8 •2752 •23(;o 3 •6069 -5711 9 ■9954 20-9424 4 •4970 ■3190 40 1.3-710,3 -0421 8S •4025 •0751 I •4173 1 -3341 6 ■?,'^?>9 3-8209 2 •I 142 i •01S2 7 •2224 •5030 3 128024 19-(;942 8 •1,305 •2809 4 ■4848 -3G34 9 •0122 2-9010 4.^ •1619 •0251 90 0-8693 ■G105 6 1 1 '83 79 18GS13 I •7246 •2498 7 •5148 •3320 2 •15800 1-8776 8 •1921 17-9773 3 •4198 •4820 9 10-8681 •G1G5 4 -2668 •0872 50 •.5428 -2408 95 • 1 3 2 1 0-0983 I -2140 1G-87G4 6 •0328 •3238 2 98802 •49(;0 3 •5434 •1094 4 •2058 15-7172 141 HM. 3 PERCENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. TO II 13 13 14 15 lO 20-9287 20-8243 20-6987 20^5572 20-4046 10 I 27-2150 •7494 -6264 •4873 •3371 I 2 •1383 27-0601 •5290 •3926 •2451 2 3 •0600 26-9792 26-8955 •2773 •1327 3 4 26-9809 -8977 -8113 26-7227 •0047 , 4 15 -9019 •8163 •7272 •6357 26-5431 15 6 -8239 •7358 •6440 •5496 -4540 26^3582 6 7 •7476 -6571 •5626 •4655 -3668 •2678 7 8 -6733 -5806 •4835 •3835 •2820 •1800 8 9 -6021 -5070 •4076 •3048 -2004 •0955 9 20 •5334 •4361 •3342 •2290 •1218 •0140 20 I •■4666 -3672 •2629 •1551 •0453 25^9347 I 2 •4011 -2995 •1929 •0825 25-9700 •8568 2 3 •3362 -2325 •1235 •0107 •8955 •7794 3 4 •2715 •1656 •0543 25-9389 •8211 •7022 4 ^5 •2069 •0991 25-9854 •8674 •7469 •6252 25 6 •1427 •0328 -9169 •7964 •6731 •5485 6 7 •0791 25-9672 •8490 •7260 •6000 •4726 7 8 •0166 •9025 •7821 •6567 •5281 •3978 8 9 25-9548 •8390 •7162 •5884 •4571 •3242 9 30 -8941 -7763 •6516 •5211 •3874 •2517 30 X -8343 -7147 •5878 •4552 •3187 •1804 I 2 -7752 •6539 •5248 •3898 •2511 •1099 2 3 •7171 -5940 •4629 •3256 •1843 •0407 3 4 •6597 -5349 •4018 •2623 •1184 24^9722 4 35 •6030 •4765 •3415 •1997 •0535 •9047 35 6 •5471 •4191 -2821 •1382 24-9896 •8383 6 7 •4923 •3627 -2239 •0779 •9270 •7731 7 8 •4384 •3073 •1666 •0186 •8654 •7091 8 9 •3854 •2528 •1103 24-9602 •8048 •6462 9 40 •3333 •1992 -0550 -9029 •7452 •5842 40 I •2819 •1465 -0005 •8464 •6866 •5232 I 2 •2314 •0946 24-9470 •7908 •6288 •4632 2 3 •1815 •0434 •8942 •7362 •5720 •4040 3 4 •1328 24-9934 •8426 •6826 •5164 •3461 4 4? •0849 •9444 •7919 •6301 •4618 .2893 45 6 •0385 •8967 •7427 •5791 •4087 •2341 6 7 24-9932 •8502 •6947 •5293 •3569 •1802 7 8 •9492 •8051 •6482 •4811 •3067 •1279 8 9 •9064 •7612 •6029 •4340 •2578 •0769 9 5° •8647 •7187 •5589 •3883 •2102 -0273 5° I •8243 •6771 •5161 •3439 •1639 23-9790 I 2 •7850 •6369 •4744 •3007 •1189 •9320 2 3 •7469 •5979 •4342 •2588 •0752 •8864 3 4 •7100 •5602 •3952 •2183 •0329 •8422 4 [156] Age of elder Life:- j Joint Life Annuitj-, at Top. ( Last Surviror Annuity, at Side. 142 3 I'KU-CEXT. Values of AiviiKtus on Two Juint Lives, (ih4 •3649 •2299 •0877 9 30 •GOoG 23-4830 •2486 •I 176 ij-9796 30 I -4933 -3GG8 23-2415 •0010 •8670 I 2 •3823 -2517 •1224 22-9943 •7483 2 •272G •1G41 •1382 •0258 -0047 22-8884 -872G -7520 22-7412 •6164 3 4 22-4819 3 4 35 •0571 22-9148 •7735 •6330 -4930 •3541 35 6 22-9518 •8057 •G(;oi •5156 •3713 •2280 6 7 •8487 •6986 •5491 -4002 •2518 •1040 7 8 •7473 •5934 •4398 -2870 •1341 21-9820 8 9 •6478 •4902 •3327 -1756 -0186 •8619 9 40 •5502 •3887 •2274 -06G3 21-9050 •7440 40 I •4541 •2891 •1239 21-9588 •7933 •6279 I 2 •3599 •1912 •0222 -8531 •6835 •5137 2 3 •3G74 •0951 21-9223 -7494 •5756 •4016 3 4 •1770 •0013 •8249 •6481 •4704 •2921 4 4'; •0887 21-9095 •7296 •5491 •3674 •1851 45 "6 -0030 -820G •6372 ■4531 •2677 •0811 6 7 21-9197 •7342 •5475 •3599 •1707 20-9804 7 8 •8391 •6506 •4607 •2697 -0769 •8828 8 9 •7608 •5693 •3764 •1821 20-9859 •7881 9 50 •6849 •4906 •2947 -0973 •8977 •6963 50 I •6113 •4142 •2155 •0150 -8123 -6075 I 2 •5397 •3401 •1386 20-9353 •7295 -5213 2 3 •4705 •2684 •0G43 •8582 •G494 -4381 3 4 •4036 •1991 20-9924 •7838 •5721 •3578 4 [l.il.J , , ,j , .f t Joint Life Annuitv, at 1 op. Aj;o of elder Life: — w . c- \ ' •• . ^j ( hast Survivor Anniiitv, at >iae. 145 3 PER-CENT. Values of Annuities on Tico Joint Lives, and on the Last Survivor of Two Lives. lO 34 35 36 37 38 j 39 17*3436 i7'^325 16-9154 16-6931 16-4656 j 16-2319 10 I •31.53 •1059 -8903 •6696 •4436 1 •2114 I 2 •2673 •0598 •8462 •6273 •4032 j •1728 2 3 •2029 J6-9977 •7862 •5694 •3473 ' •I 190 3 4 •1262 •9233 •7142 •4997 •2799 •0538 4 15 •0408 •8405 •6339 •4220 •2046 15-9808 15 6 i6'95i2 •7536 •5497 •3402 •1254 ' •9042 6 7 •8618 •6668 •4656 •2589 •0466 I •8279 7 8 ■7771 •5849 •3863 •1822 15-9725 •7563 8 9 ■7021 •5126 •3167 •II ^0 ' •9079 •6942 9 20 •6378 •4510 •2576 ■0587 •8539 •6426 20 I •5782 •3941 •2034 •0069 •8048 •5959 I 2 •5199 •3386 •1507 15-9569 •7573 •55^^ 2 3 •4580 •2797 •0947 •9038 •7070 '5°33 3 4 •3888 •2136 •0317 -8438 •6500 •4492 4 25 •3109 •1390 15-9604 •7758 •5852 •3875 25 6 •2258 ■0575 •8823 •7011 •5138 •3195 6 7 •1341 15-9694 •7979 •6202 •4365 '2457 7 8 •0380 •8772 •7095 •5355 '3555 •1683 8 9 15-9381 •-7813 •6174 '4474 •2712 •0877 9 30 •8339 •6810 •5214 '3553 •1832 •0036 30 I •7256 •5768 •4212 •2595 •0913 i4'9i6o I 2 •6111 •4667 •3154 •1578 14-9941 -8229 2 3 •4899 •3499 -2030 •0499 -8905 •7239 3 4 •2259 •0835 14-9351 •7802 •6181 4 35 22-2ir3l 1 4-95 7 J •8133 •6633 •5059 35 6 •0855 21-9441 •6851 •5398 •3874 6 7 21-9568 -8108 21-6660 •4105 •2630 7 8 •8303 •6794 •5299 21^3821 •1327 8 9 •7057 •5501 •3956 •2429 21-0918 9 40 •5830 •4228 •2635 •1057 20-9495 20^7949 40 I •4625 •2974 •1332 20-9704 •8090 •6492 I 2 •3437 •1740 •0048 -8370 -6704 •5052 2 3 •2271 -0526 20-8787 •7056 •5339 •3632 3 4 •1132 20-9341 •7552 •5772 •4001 •2242 4 4^ •0017 •8181 •6344 •4514 •2693 -0879 45 6 20-8937 •7054 -5171 •3293 •1419 19-9553 6 7 •7886 •5962 -4032 •2105 •0182 •8263 7 8 •6871 •4902 •2929 ! -0955 19-8983 •7013 8 9 •5884 •3875 •1857 19-9838 •7818 •5798 9 "JO •4930 •2880 •0821 •8756 •6690 •4620 5° •4004 •1916 19-9814 •7708 •5594 •3477 I 2 •3107 •0981 -8841 i -6691 - -4535 •2369 2 3 •2241 ■0079 •7900 •5710 •3510 •1300 3 4 •1406 19-9209 •6993 •4764 •2523 •0268 4 [1()0] ^ge of eUlor Lifc:- \ Joint Life Annuity, at Top. ( Last Survivor Annuity, at Side. 146 3 rER-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor i>f Tiri) LiviS. 40 41 42 43 44 45 lO '5 "99 1 3 15-7422 15*4832 152152 H-9397 14-6570 10 I •97-3 •7243 •4669 •2002 •9260 •6444 I 2 ■9 5.-; 4 ■6894 •4334 •1683 •8957 •6158 2 .3 •8856 •6396 •5857 •1224 •8518 ■5737 3 4 •8207 •5788 •3271 •0660 '7974 •5214 4 15 •7501 •5106 •2611 •0024 •7361 •4623 15 6 •6758 •4387 •1917 14-93^3 •6713 ■3999 6 7 •6021 •3^>73 •1227 •8688 •6071 •3380 7 8 ■5329 •3008 •0584 •8068 •5475 •2806 8 9 •4732 •2433 •0034 ■7540 •4969 •2321 9 20 •4240 •1964 14-9587 •7"5 •4564 •1936 20 I ■3797 •1544 •9189 •6739 •4210 •1602 I 2 ■3372 •I 144 •8812 •6384 •3876 •1290 2 3 •2922 •0719 •8412 •6008 •3523 •0959 3 4 •2409 -0234 •7953 •5575 •3116 •0576 4 25 •1822 14-9676 •7426 •5075 •2643 •0130 25 6 •II74 •9060 •6840 •4521 •21 17 13-9632 6 7 •0471 •8390 •6203 •3916 •J 544 •9089 7 8 14-9732 •7688 'SS'>S •3281 •0943 •8520 8 9 •8965 •6956 •4840 •2622 •0318 •7930 9 30 •8162 •6192 •4114 •1934 13-9666 ■7313 30 I •7.326 •539^5 •3358 •1216 •8987 •6671 I 2 •6438 ■4.)3o •2553 •0453 •8263 •5987 2 3 •5491 •3647 •1694 13-9636 •7489 •5253 3 4 •4481 •2681 ■0774 •8761 •6658 -4467 4 33 •3403 •i'554 13-9793 •7828 •5771 •3625 3S 6 •2268 •0566 •8755 •6837 •4830 -2732 6 7 ■1075 1 3 "9423 •7662 •5797 •3839 •i79r 7 8 13-9823 •8223 •'5514 •4700 •2796 •0798 8 9 •8502 •6954 •5610 •5299 •3540 •2306 •1688 12-9745 •8622 9 40 •4010 •0510 40 I 20-4909 •2629 •0981 12-9243 -7413 I 2 •3414 20-1790 12-9553 •7874 -6104 2 3 •1939 -0259 19-8592 •6405 •4696 3 4 -0493 19-8755 •7029 195319 -3202 4 45 19-907") •7279 •5493 •3722 191974 45 6 -7093 •5841 -3995 •2102 •0352 18-8565 6 7 •6350 •4440 -2530 •0041 18-8708 -0917 7 8 -504") -3079 -1110 18-9161 •7225 •5309 8 9 -3777 -1750 18-9735 -7719 •5721 •3740 9 50 •2548 -0472 -8393 •0318 •4257 •2213 50 I •1354 18-9225 -7090 -4950 •2833 •0724 I 2 -0198 -8010 •5825 -3632 •1449 17-9277 2 3 18-9079 -0847 •4002 -2352 •0108 •7874 3 4 -8002 •5718 •3421 •1115 17-8813 •0517 4 [161] Age of elder Life:- j -Joint Life Annuity, at Top^ ° { Last Survivor Annuity, at Side. 147 3 PER-CENT. Values of Annuities on Two Joint Lives, arid on the Last Survivor of Tico Lives. 46 47 48 49 50 51 to H"3'595 14-0786 137839 1 3 '4843 13-1800 12-8693 10 I •3582 •068.$ •7749 •4764 •1729 -8634 I 2 •3311 •0429 ■75°7 ■4536 •I516 -8433 2 3 •2908 •0044 •7139 •4186 •I183 •8116 3 4 •2406 i3^9562 •6677 •3742 •0758 •7710 4 15 •1836 •9013 •6149 -3^-3^ ■0271 •7243 15 6 •1234 •8433 •S'>9° •2698 12-9754 •6747 6 7 •0638 •7858 •5037 •2165 •9243 •6256 7 8 •0087 •7328 •4528 •1677 •8773 •5806 8 9 13-9622 •6885 •4104 •1271 •8386 •5436 9 20 •9257 •6538 •3776 •0960 •8092 •5158 20' I •8942 •6241 •3497 •0698 •7847 •4927 I 2 •8649 •59^7 •3241 •0460 •7625 •4723 2 3 •8340 •5678 •2971 •0209 •7391 •4506 3 4 •7980 •5341 •265.:; i2^99r3 •7^15 •4248 4 25 •7559 •4945 •2282 •9.03 •6787 •3940 25 6 •7090 •4302 •1865 •9170 •6417 '3593 6 7 •6576 ■4016 •1407 •8738 •6010 •321 1 7 8 •6038 •3509 •0928 •8287 •5586 •281 1 8 9 •5480 •2982 •043 1 •7820 •5147 •2400 9 30 •4898 •2433 12-9914 •7333 •4690 •1971 30 I •4292 •1862 •9377 •6829 •4217 •1,529 I 2 •3645 •1253 •8S04 •6290 •3712 •1055 2 3 ■2954 •0599 •8188 •571 1 •3169 •0546 3 4 •2208 I2-9897 IS'-S •5088 •2582 11-9997 4 35 •1413 •9143 •6816 •4419 •1954 •9407 35 6 •0566 •8343 •60.59 •3707 •1283 •8779 6 7 12-9673 •7499 •5262 •2955 •0577 •8114 7 8 •8733 •6608 •4420 •2161 11-9829 •7414 8 9 '773^ •5660 •35^3 •1314 •9032 •6664 9 40 ■666s •4646 •2564 •040S •8177 •5860 40 I •5512 ■3S5^ •1525 11-9424 •7248 •4984 I 2 •4263 •2360 •0393 •8350 •6232 •4024 2 3 ■2917 •1076 ii^9i69 •7187 •5128 •2979 3 4 •1485 11-9707 •7863 •5943 •3947 •i860 4 45 1 1-9966 •8252 •6473 •4618 •2685 •0663 45 6 •6730 •5016 •3227 •I 36 I 10-9403 6 7 18-5100 •3501 •1779 10-9979 •8089 7 8 -3427 18^1580 •0266 •8535 •6714 8 9 •1792 17^9878 17^8004 •7020 •5268 9 ^0 -0198 •8218 •6275 17-4366 •3748 50 I i7-8tur) •6597 •4585 •2607 17 0667 I 2 •7132 •5017 •2936 •0887 16^8876 16-6899 2 3 •o()r)3 •3482 •1333 16-9214 •7130 •5080 3 4 -4243 •1995 16-9779 •7590 •5434 •3310 4 .. \ Joi It Life Ann jity, at Top. [IC ■2] A Te of eldsr I "fc :-!.,. t ^iirviv.-.r r knnnitv at S \'\Ab. 148 3 TKHrEXT. Values of Annuities on Two Juint Livrs^ ami on the Last Survivor of Two Lives. 52 53 54 55 56 57 lO ^^•55^5 12-2279 I r 8999 11-5676 11-2321 10-8940 10 I •54^>5 •2238 •8966 •56.52 •2305 •8931 I 2 •.1279 •2064 •880 <; '55°3 •2168 -8805 2 3 •4977 •1779 •8535 •5249 •J929 •8581 3 4 '4589 •1409 •8183 •4914 -1611 -8280 4 K'i •4142 •0981 •7774 •4524 •1239 -7926 15 6 •.3666 '0525 •7338 •4106 •0841 •7545 6 / •319) •0073 •6906 •3692 ■044.5 -7169 / 8 •2764 ii^966i •65 1 1 •3316 •0085 -6825 8 9 •241 1 ■9325 •6191 -3012 10-9797 '655^ 9 20 •2148 •9077 •59.58 -2791 ■9590 •6357 20 1 •1933 •8876 •5770 -2617 ■9427 •6206 I 2 •1743 •8700 •5607 •2467 •9290 •6079 2 3 •1543 •8.514 •5437 -2310 ■9145 •5948 3 4 •1303 •8292 •5230 •2 1 19 •8969 •5785 4 25 •101 1; •8023 •4980 •1886 •8753 •5584 25 6 ■0689 •7718 •469.5 •1620 ■850-1 ■5353 6 7 •03,30 •7382 •4379 •1325 •8229 -5096 7 8 11-9956 •7031 •40.51 •J019 •7943 -4829 8 9 ■9570 •6670 •3714 •0704 ■7650 •4556 9 30 •9169 •6295 ■3365 •0378 ■7347 •4275 30 I ■8755 •5909 •3004 •0043 ■7036 •3987 J 2 •8.312 •5496 •2620 10-9686 •6704 •3679 2 3 ■7837 '5^5^ •2206 •9301 •6347 •3348 3 4 '73^3 •4572 •1758 -8884 •59.59 •2989 4 35 •6771 •4056 •1277 •8436 •5542 •2602 35 6 •6181 ■3500 •0763 ■79.57 •5098 •2190 6 7 •5560 •2924 •0221 •7453 •4629 •1755 7 8 •4902 •23 10 10-9648 •6920 •4134 -1297 8 9 •4201 •1653 •9036 •6349 ■3605 •0806 9 40 ■3445 •0947 •837.5 •5734 •3033 -0276 40 I •2622 •0174 •7654 •5060 -2405 99692 I 2 •1718 io^9324 •6856 •43 H •I 70S •9043 2 3 •0732 •8395 •5983 ■349.5 ■0943 -8328 3 4 10-9673 •7397 ■5043 •2613 •0116 •7556 4 45 •8539 -Cly^S •4033 •1663 9-9225 -6721 45 6 ■7345 '5 '97 •2968 •0660 -8283 •5839 6 7 •6098 •4016 •^8,54 9^961 1 •7297 •4914 7 8 •4792 •2778 •0683 •8.508 -6259 •3942 8 9 •3417 •1473 9"9448 •7342 •5162 -2910 9 50 •1968 •0097 ■8144 -6109 ■3999 -1817 50 I •0434 9^8636 •7080 •6757 •5276 •4795 •27.58 -1428 -0648 8-9392 •8054 I 2 iG-3or.r) '3390 -1898 2 3 •3707 ■00 1 3 3 4 •1220 15 •9172 •0328 8^852'i -6641 4 [163] Age of elder Lifc:- \ Foint I^ife Annuity, at Top. ( Last Survivor Annuity, at Side. 149 Vah 3 PERCENT. of Annuities on Tico Joint Lives, and on the LoM Survivor of Two Lives. 33 59 60 1 61 62 63 10 io"5534 IO"2I04 9-8667 9*5234 9-1815 8-8418 10 I '553:^ •2109 •8678 '5^5^ •1837 -8445 I 2 •5417 •2004 •8583 ■5^66 •1761 •8379 2 3 •5207 •1806 •8399 '4995 •1603 •8231 3 4 •4922 •1538 •8145 •4756 •1379 •8022 4 15 •4586 ■I2I9 •7844 •4470 •1109 •7767 15 6 •4224 •0874 •75^7 •4160 •0815 •7489 6 7 •3864 •0532 •7191 •3851 •0522 •7211 7 8 •3538 •0221 •6896 •3571 •0256 •6959 8 9 •3277 9"9975 •6662 ■335° •0047 •6762 9 20 •3094 •9804 •6503 •3200 8-9907 •6631 20 I ■'-955 •9674 •6382 •3090 •9805 •6538 I 2 •2839 ■9569 •6286 •3003 •9727 •6467 2 3 •2719 •9460 •6187 •2913 •9645 •6394 3 4 •2570 •9322 •6061 •2798 •9541 •6299 4 25 •2383 •9150 •5902 •2651 •9406 •6174 25 6 •2168 •8950 •57^7 •2480 •9247 •6028 6 7 •1929 •8727 ■5510 •2288 •9070 •5864 7 8 •1680 •8497 •5296 •2089 •8885 •5694 8 9 •1427 •8262 •5078 •1888 •8699 ■5522 9 30 •1166 •8020 •4855 •1682 •8509 •5347 30 I •0900 •7774 •4628 •1472 •8317 •5^70 I 2 •0616 •7511 •4386 •1249 •8112 •4982 2 3 •0308 •7228 •4125 •1009 •7891 •4780 3 4 9-9976 •6921 •3841 •0748 •765 T •4559 4 35 •9618 •6590 •3536 •0467 •7392 -4322 35 6 •9236 •6237 •3211 •0167 •7117 •4069 6 7 •8834 •5866 •2868 8-9853 •6828 •3804 7 8 •8410 •5475 •2508 •9522 •6525 •3527 8 9 •7956 '5°56 •2123 •9168 •6200 •3230 9 40 ■7465 •4603 '^7°5 •8784 •5849 •2909 40 I •6925 •4102 •1244 •8359 •5459 •2551 I 2 •6321 •3544 •0726 •7882 •5018 •2147 2 3 ■5656 •2925 •0153 •735^ •4529 •1695 3 4 '4935 •2254 8-9531 •6775 •3996 •1203 4 45 •4156 •1528 •8855 •6148 •3416 ■0667 45 6 ■333^ •0760 •8140 . -5484 •2801 •0099 6 7 •2467 8-9953 •7390 •4788 •2156 7-9503 7 8 •1556 •9103 •6599 •4054 -1476 •8874 8 9 •0591 •8201 ■5758 •3272 •075 i -8204 9 50 8-9565 •7242 •4864 •2439 7-9978 •7489 50 I •8467 •6213 •3902 -1543 •9H5 •6716 I 2 •7284 ■5^03 •2862 •0572 •8240 •5876 2 3 •6022 •3915 •1748 7-9529 •7267 •4970 3 4 •4686 •2656 ■0565 •8421 1 •6230 •4003 4 [164] A?e of elder Life: Joint Life Annuity, at Top. Last Survivor Annuity, at Side. 150 3 PKH-CKNT. Values of Annuities on Two Joint Lircs^ and on the Last Survivor of Tiro Lives. 64 65 66 67 68 69 lO 8'i;o<;2 81707 7-8373 7*504 1 7-1706 6-8347 TO I •5084 •1742 •8412 •5084 •1752 •8394 I 2 •3026 •1693 •8370 •5049 •1723 •8372 a 3 •4890 •1568 •8257 •4946 •1630 •8289 3 4 •4694 •1386 •8087 ■4789 •h'85 •8155 4 15 •4455 •1160 •7876 •4592 •1302 •7985 15 6 •4192 •0913 •7642 •4373 •1096 •7793 6 7 •3928 •0664 •7408 •4 15 1 •0889 •7599 7 8 •3690 •0438 •7195 •3950 •0699 •7421 8 9 •3504 •0263 •7030 •379'5 •0554 •7285 9 20 •3383 •0149 •6924 '3^91 •0463 •7200 20 I •3296 •0071 •68^2 •3^3i •0403 •7146 I 2 •3^33 •0013 •6801 •3586 •0363 •7110 2 3 •3^67 7 ■99,)'^ •6749 •3541 •0323 •7076 3 4 •3081 ■9878 •6680 •3478 •0267 •7026 4 ^5 •2967 •9774 •6585 •3391 •0188 •6955 ^5 6 •2832 •9649 •6471 •3287 •0093 •6868 6 7 •2680 •9509 •6341 •3168 69984 •6768 / 8 ■^5^3 •93<^H •6208 •3045 ■9872 •6666 8 9 •2365 •9218 •6074 •2922 •9759 •6563 9 30 •2204 •9071 •5938 •2799 •9646 •6460 30 I •2042 •8922 •5803 •2675 •9534 •6359 I 2 •1870 •8765 •5659 •2545 •9415 •6251 2 3 •1684 •8595 ■5504 •2403 •9287 •6135 3 4 •1482 •8410 •S33S •2249 •9H7 •6008 4 35 •1264 •8211 •5154 •2084 •8996 ■ •5871 35 6 •1033 ■7999 •4960 •1908 •8836 •5726 6 7 •0791 "1119 ■4759 •1725 •8671 •5576 7 8 •0538 •7548 •4549 'K'iSS -8499 •5421 8 9 •0266 •7301 •4325 •1332 •8315 •5^5^ 9 40 7 "997 3 •7034 •4083 •I 112 •8117 •5°78 40 I •9646 •^735 •38x1 •0866 •7894 •4878 I 2 •9275 •6396 '3S^'^ •0584 •7639 •4646 2 3 •8860 •6016 '3^ S3 •0266 •7349 •4383 3 4 •8407 •5599 •2772 6-9917 •703 i •4094 4 45 •79 '3 •5H5 •2354 •9535 •6682 •3777 45 6 •7389 •4662 •1912 •9130 •63 1 2 •3440 6 7 •6840 •4157 •1448 •8705 •5925 •3088 7 8 •6260 •3624 •0959 •82£;8 'ssn •2717 8 9 •5642 •3055 •0437 •7780 •5082 •2321 9 50 •4981 •2447 6-9878 •7270 ■4615 •1897 5° I •4267 •1788 •9272 •6714 •4108 •1435 I 2 ■3487 •1067 •8609 •6104 •3549 •0926 2 3 •2646 •0289 •7890 •5443 •2943 •0371 3 4 •1747 6-9455 •7119 •4732 •2290 59774 4 [16.5] , r ij T -r \ -Toint Life Annuity, at Top. Age of elder Life:- j ^a^t Survivor Annuity, at Side. 151 3 PER-OENT. Values of Anma'ties on Tico Joint Lives, and on the Last Survivor of Two Li res. 70 71 72 73 74 75 lO 6-4997 6-1687 5-8462 5-5362 5-2429 4-966 1 10 I ■5047 •1738 ^•8513 ■5414 •2481 -97^3 I 2 •5030 •1727 ■8507 ■54^2 •2483 •9718 2 3 ■4956 •1661 •8449 ■536^ •2438 •9680 3 4 •4^34 ■1550 •8348 •5269 •2356 •9605 4 ^5 •4676 •1405 •8214 •5147 •2243 •9502 15 6 •4498 •1239 •8060 ■5004 •21 1 1 •9380 6 7 •4316 •1069 •7902 -4857 •1974 ■9253 7 8 •4150 •0914 •7757 -4722 •1848 '9^35 8 9 •4023 •°795 •7647 •4619 •^752 ■9045 9 20 •3944 •0723 •7580 •4557 •1694 •8992 20 I •3894 •0677 •7539 •4520 •j66i •8962 I 2 •3864 •0651 '75^5 •4500 •1644 -8947 2 3 •3834 •06215 •7494 •4481 •1628 -8934 3 4 •3790 •0586 •7460 ■4452 •1602 •8912 4 25 •3726 •0529 •7408 •4406 •I '')6i -8875 25 6 •3647 •0457 ■7344 ■4347 •1508 •8827 6 7 '3556 •0374 •7268 -4278 •1446 •8770 7 8 •3462 •0289 •7190 •4208 •1382 •8712 8 9 •3369 •0204 •7113 •4138 •1318 •8654 9 30 •3275 •0119 •7036 •4068 •1255 -8598 30 I •3183 •0036 •6962 •4001 •1194 -8543 I 2 •3086 5-9948 •6882 •3930 •I 130 -8485 2 3 •2980 -9853 •6797 •3852 •1061 •8422 3 4 ■2866 '975° •6703 •3768 •0985 -8354 4 35 •2742 ■9638 •6602 •3677 •0903 •8279 35 6 •2611 •9520 •6496 •3581 •0816 •8201 6 7 •2476 •9398 ■6386 •3483 •0728 •8122 7 8 •^337 ■9273 •6275 •3383 •0638 •8042 8 9 •2189 •9141 ■6136 •3277 ■0544 -7958 9 40 ■2029 •8998 •6029 •3163 •0443 •7868 40 I •1848 •8836 •5884 •3034 •0327 -7765 I 2 •1640 •8648 •5715 •2883 •0192 •7643 2 3 •1402 •8434 •5522 •2709 ■0035 •7502 3 4 •1140 •8197 •5308 •2516 4-9861 •7346 4 45 •08 5 1 •7935 •5072 •2302 •9668 •7171 45 6 •0546 •7658 •4821 •2076 -9463 •6986 6 7 •0226 •7370 .4560 •1840 •9251 -6795 7 8 5-989 i •7066 •4286 •J 593 •9028 -6594 8 9 ■9532 •6742 •3994 •J330 •8791 •6381 9 50 •9148 •6395 •3681 •1048 •8537 •6152 50 I •8728 •6015 •3338 ■0739 •8259 •5902 I 2 •8265 '5595 •2958 •0396 -7949 •5622 2 3 •7760 '5^ 36 ■2542 ■0019 •7608 -5314 3 4 •7215 •4640 •2092 4'96i I •7^39 •4980 4 [16G] Age of elder Life: Joint Life Annuity, at Top. Last Survivor Annuity, at Side. 152 3 PKlt^CEN'I'. Vahifs of Annuities on Two Joint Lii'es, and on tlm Lost Survivor of Two fjins. 76 77 78 79 80 81 lO 4-69.^8 4'4340 4-1791 3^9287 3-68.59 3^4558 10 i 7010 •439' •1840 •933.5 •6906 -4604 J 2 •7018 •4401 •18,53 ■9349 •6920 •4619 2 3 •698.S •4374 •1830 •9330 •6905 ■4607 3 4 •6918 •4314 •^777 •9283 -6864 •4571 4 15 •6824 •4228 •1699 •9213 -6801 •4514 ^5 6 •6712 •4125 •1604 •9126 •6722 •4442 6 7 •6.594 •4015 •i.';o3 •9033 •6637 •4364 7 8 •6484 •3913 •J 409 •8946 '^^!\6 -4290 8 9 •6400 •3836 •J33(^ •8879 •6494 -4232 9 20 '<^35^ •3790 •1294 •8840 •6458 -4199 20 I •632,3 •3 7 '^5 •1271 •8819 •6439 -4182 I 2 •6,5 1 1 •3754 •1262 ■8811 •6433 ■4177 2 3 ■6300 •3746 •1256 •8807 •6429 •4174 3 4 •6281 •3729 •1242 •879.5 •6419 -4166 4 25 •6248 •3700 •1216 •8772 •6400 •4149 25 6 •6205 •3662 •1182 •8741 •6372 •4124 6 7 •6154 '3('ii5 •I 139 •8703 •6338 •4094 7 8 •6101 -3S<^7 •1096 •8664 •6302 •4062 8 9 •6049 ■3520 •1053 •8626 •6268 •403 J 9 30 •5997 •3473 •ion •8588 •6234 -4000 30 I •5948 ■3429 •0972 •8.5.53 •6203 •3972 I 2 •5896 •3383 •0930 •8516 •6170 •3943 2 3 •5840 •3332 •088.5 •847.5 •6134 -391 1 3 4 •5778 •3277 •0835 •8431 •6094 •3875 4 35 •57^1 •3216 •0781 •8382 •605 1 •3836 3$ 6 •5641 '3K'i3 •0724 •8331 •6005 •3796 6 7 •5.'; 70 •3089 •0667 •8280 •5960 •375^ 7 8 •5498 •3026 •0610 •8250 •5915 •3716 8 9 •5423 •2959 •0552 •8178 •5869 •3(^75 9 40 •5344 •2889 •04S9 •8123 •5821 •3633 40 I •5252 •2807 •0417 •8060 •5765 •3584 I 2 •5143 •2710 ■033 1 •7983 ■5697 ■3524 2 3 '5017 ■2597 •0230 •7893 '5^n •34.53 3 4 •4876 •2470 •0116 •7792 •5527 •3372 4 45 •4718 •2329 3^9989 '7^77 •5425 •3281 45 6 ■4552 •2179 •98.H •75.56 '53^6 •3184 6 7 ■4.3 79 •2024 •9715 •7432 ■5205 •308.5 7 8 •4199 •1862 '957^ •7303 •5091 -2983 8 9 •4007 •1690 •9417 •7167 •4969 •2875 9 50 •3802 ■^507 •9253 •7021 •4839 •2760 50 I •3577 ■130.1 •9073 •6860 •4697 •2633 I 2 •3326 •1079 •8870 •6679 •4535 ■2490 2 3 •3048 •0829 •8646 •6478 •4356 •2330 3 4 •2746 '°557 •8402 ■6260 •4161 •2156 4 [167] Age of elder Life:- \ Joint Life Annuity, at Top ° I Last Survivor Annuity, at Side. 153 3 PER-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. 82 83 84 85 86 87 lO 3'244i 3-0480 2 8711 27056 2-5407 2-3665 10 I •2484 •0522 875^ •7095 •5444 •3700 I 2 •2501 •0540 8769 •7114 •5463 •3719 2 3 •2492 •0533 8765 •7112 •5463 •3721 3 4 •2460 ■0505 8741 •7091 •5447 •3708 4 15 •2409 •0459 8700 •7055 •5414 •3681 ^5 6 •2344 •0400 8646 •7006 '537° •3642 6 7 •2272 •0334 8585 •69^0 '53^9 ■3596 7 8 •2203 •0270 8525 •6895 •5269 ■3550 8 9 •2150 •0220 8480 •6852 •5229 •35^3 9 20 •2119 •0192 8454 •6828 •5206 ■ "3493 20 I •2103 •0178 8440 •6816 ■5195 •3483 1 2 •2099 •0174 8438 •6813 ■5194 •3481 2 3 •2098 •0174 8438 •6814 '5^95 •3483 3 4 •2091 •0169 8433 •6811 '5^93 •3482 4 25 •2076 •0155 8422 •6801 •5185 ■3475 25 6 •2054 •0136 8405 •6786 •5171 •3464 6 7 •2026 •our 8382 •6766 '5^53 •3448 7 8 •1998 •0085 8359 ■6745 '5-^35 •34.32 8 9 •1970 •0060 8336 •6725 •51^7 •3416 9 30 ••1943 •0036 8315 •6705 •5099 •3401 30 I •1918 •0013 8295 •6687 •5084 ;3387 I 2 •1891 2-9990 8274 •6669 •jo68 "3373 2 3 •1863 •9964 82^1 •6649 '5050 •3358 3 4 •1831 ■9936 8225 •6626 •5029 •3340 4 35 •1796 •9905 8197 •6600 ■5007 •3320 35 6 •1760 •9872 8168 ■6574 •4983 •3300 6 7 •1724 •9840 8139 •6548 •4961 •3280 7 8 •1688 •9809 8ri2 •6524 ■4939 •3261 8 9 •1653 •9777 8084 •6500 •4919 •3243 9 40 ■1616 •9745 8o.';6 •6476 •4898 •3226 40 I •1573 •9707 8023 •6447 •4873 •3205 I 2 •1520 •9661 7982 •641 1 •4842 ■3179 2 3 •14.0 •9604 7932 •6366 •4803 •3145 3 4 •1384 ■9540 7874 '63 15 •4758 •3106 4 45 •1303 •9467 7809 •6257 •4706 •3061 45 6 •1216 •9389 7739 •6193 •4651 '3013 6 7 •1128 •9310 7668 •6132 ■4595 ■2964 7 8 •1036 •9229 759^ •6067 ■4539 •2913 8 9 •0940 •9H3 7520 ■6000 •4479 •2864 9 50 •0838 •9052 •7439 •59^9 •4417 •2810 50 1 •0726 •8953 7351 •585 T •4349 •2732 I 2 •0598 •8839 •7250 , "5761 •4270 •2684 2 3 ■0455 •8712 •7137 •'y66l •4182 •2607 3 4 •0300 ■8573 •7013 I -555 r •4086 •2524 4 [168] Age of elder Lite: Joint Life Annuity, at Top. Last Survivor Aniiuitv. at Side. 1.54 HM. 3 I'KK-CKNT. Values of Annultits on Two Joint Licrs^ (ind on the Last Survivor of Tico Liceti. 88 1 89 90 1 91 92 1 93 10 2-1829 i9^>73 1-7242 1^4747 1-2194 0-9446 1 10 I •1862 •9702 1 •7267 •4768 •2210 ■9458 , I 2 •1881 •9720 •7283 •4781 ■2221 •9466 2 3 •188^ ■9725 •7288 •4787 •2227 1 •9470 3 4 •i875 •9719 •7285 , •4786 , •2226 1 •9470 4 15 •J852 •9701 •7272 •4776 •2220 ' •9467 '5 6 •1819 •9^^ 7 3 •7250 •4760 ' •2209 •9460 6 7 •1778 1 ■9^^39 •7222 •4739 •2193 •9449 7 8 •^737 •9603 •719:^ •4714 •2175 •9436 8 9 •1704 j •9574 •7167 •4694 •2159 ' •9425 9 20 •168.^ ' ■9.)57 •7153 •4682 ■2149 I ■9417 20 I •I 676 , •9.149 ■714.1 •4675 •2144 •9413 I •1675 •9.14« •714.1 •4675 •2143 1 •9413 2 3 •1677 ■9.15^ ■7147 •4677 •2i45 ' •9414 3 4 •1677 -955' •7148 ■4678 •2146 ■9415 4 25 •1672 ■9.147 •714.1 •4676 •2145 •9414 25 6 •1662 •9,139 •7139 •4672 •2142 •9413 6 1 7 •J 648 •9528 •7130 •4665 •2137 •9409 7 8 •1634 •9516 ■7121 ■46.58 •2131 •9405 8 9 •1620 •9505 711X •4650 •2126 •9402 9 30 •1607 •9494 •7102 •4643 •2121 •9398 30 I ■1596 •9484 •709.1 •4638 • 2 1 16 ■9395 I 2 •1.04 •947.1 •7088 •4632 •2112 ■9392 2 3 •i.vi ■9464 •7079 •4626 •2108 •9390 3 4 •^556 •9452 •7069 •4618 •2103 •9386 4 3.1 •1539 •9438 •70.18 •4610 •2096 •9382 35 6 •1521 •9422 •7046 •4600 •2089 •9377 6 7 ■^.104 •9408 •7034 •4591 •2082 •9372 7 8 •1488 •939.1 •7023 •4.182 •2076 •9367 8 9 •^473 •9382 •7014 •4575 •2070 •9363 9 40 •H59 •9371 •7005 •4.168 •2065 •9360 40 I •1442 •93.18 •699.1 •4.561 •2060 •9357 I 2 •1420 •9340 •6981 •45.11 •2054 •9353 2 3 •1392 •9317 •6964 •4538 •2044 •9347 3 4 ■^359 •9291 •6943 •4522 •2033 •9339 4 45 •1320 •9258 •6917 •4502 •2018 •9329 45 6 •1279 •9224 •6889 •4481 •2002 ■9319 6 7 •J237 •9189 •6861 ■4459 •1986 •9308 7 8 •^195 ■9M4 •6834 •4438 •1971 •9297 8 9 •I 151 •9119 •6803 •4416 '^955 •9286 9 50 ■ 1 106 •9082 •6776 •4394 •1939 ■9275 50 I •io-i7 •9041 •674.1 •4370 •1921 •9264 I 2 •0999 •8994 •6707 •4342 •I 90 1 •9250 2 3 •093.5 •8941 •666s •4309 •1878 •9234 3 4 •0863 •8883 •6619 •4274 •1852 1 •9217 4 [169] Ape of elder Life:- \ J^'i'it^ifo. Annuity, at Top •• ■■ '^ / I-aet Survivor Aiinuitv. at Sio 155 3 PER-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. 94 95 96 10 1 lO : O 6728 0-4129 1 J 7 75 I , 6735 ■4134 1 X776 I 2 6741 •4137 ^777 2 3 6744 •4138 1778 3 4 6744 •4139 1778 4 15 6743 •4t38 1778 15 6 6739 •4136 1777 6 7 6732 •4133 1776 7 8 6724 •4129 ^75 8 9 6716 •4125 ; 1773 9 20 6712 •4122 ' 1772 20 1 6709 •4120 J771 I 2 1 • 6708 •4120 177^ 2 3 • 6709 •4120 ^771 3 4 6710 •4121 1771 4 25 ■ 6710 •41 2 1 1771 25 6 6709 •4120 1771 6 7 6707 •4119 1771 7 : 8 6704 •4118 1770 8 j 9 6702 •4117 i 1770 9 30 6699 •41 15 1769 30 , I 6698 •41 14 J 769 I 1 2 6696 •4113 1769 2 1 3 6694 •4^13 1768 3 4 6692 •41 12 1768 4 35 6690 •4110 1768 35 6 6687 •4109 1767 6 7 6683 •4107 1766 7 8 6681 ■4105 1766 8 9 6678 ■4104 n65 9 40 6676 •4103 1765 40 I 6674 •4102 1765 I 2 6672 •4101 1764 2 3 6668 •4099 1763 3 4 1 6664 •4097 1763 4 1 45 6658 •4094 1762 45 6 6651 •4090 1760 6 7 6644 •4086 ^759 7 8 6638 •4083 ^758 8 9 6631 •4079 ^75^ 9 50 6624 •4076 ^755 50 I 6617 •4072 1754 1 2 6609 •4068 1752 2 3 6599 •4063 1750 3 4 •6589 ■4057 •1748 4 [170] Age of elder Life :- Joint Life Annuity, at Top. Last Survivor Annuitv. at Side. i:*»; 3 IM-Pi-rENT. Values flf .Iniiiii'lirs on Tvo Joint Lives, and on the Last Survivor iif Two Lives. 10 II 12 13 • H 15 55 .1.^ 240740 24-5239 24-3577 241 792 23-9921 237995 1 ^' •fit or. •48!lO •3210 •1410 •9528 •7584 6 7 •(;()77 ' •4555 •2870 •1055 •9150 •7188 7 8 •r.7(;3 •4233 •2538 •0709 ■8788 •6808 8 9 ■r)4(i;? •3927 •2221 •0380 •8442 •6445 9 6o •r)i7fi •3034 •1918 -O003 -8111 •0090 60 I •4!)<)r) •3357 •1031 23-9703 •7796 •5700 I 2 •4047 •3094 •1359 i •9478 •7496 •5450 2 3 •4403 •2845 •1100 1 •9209 •7212 •5151 3 4 •4173 •2610 •0857 •8954 •6944 •4867 4 Ci', •3950 •2390 •0028 •8714 •0090 •4000 65 '6 •3752 •2182 •0413 •8487 •0451 •4346 6 7 •3501 •1987 •0211 ■8275 •0220 •4107 7 8 •3380 •1803 •0021 ■8075 •6014 •3881 8 9 •3210 •1632 23^9843 •7887 •5815 •3669 9 70 •3052 •1471 •9077 •7712 ■5628 •3470 70 I •2900 •1324 •9524 •7551 ■5456 •3285 I 2 •2770 •1188 •9383 •7402 ■5297 •3115 2 3 •2648 •1005 •9250 •7208 •5154 •2960 3 4 ■2537 •0954 •9141 ■7147 •5023 •2820 4 75 •2437 •0854 •9038 •7037 •4906 •2093 75 6 •2340 •0703 •8944 ■0938 •4799 •2577 6 7 •2262 •0080 •8859 •0847 •4701 •2471 7 8 •2184 •0604 •8780 •0704 ■4611 •2373 8 9 •2113 •0534 •8709 •6689 •4530 •2284 9 80 •2049 •0471 •8046 •6622 •4457 •2204 80 I •1992 •0415 •8589 •6562 •4392 •2133 T 2 •1940 •0300 •8538 •6508 •4334 •2009 2 3 •1894 •0321 •8492 •6460 •4282 •2012 .3 4 •1853 ■0282 •8453 •0418 •4236 •1961 4 85 •1810 •0240 •8410 ■0379 ■4194 •1914 85 6 •1781 •0213 •8383 ■0344 •4154 •1871 6 7 •1740 •0180 •8350 ■0309 •4116 •1827 7 8 •1712 •0148 •8318 •0275 ■4079 •1786 8 9 •1077 •0117 •8288 •6244 •4044 •1746 9 90 •1041 ■0085 •8258 •0214 •4011 •1708 90 I •1009 •0057 •8233 •0188 ■3983 •1077 I 2 •1578 •00:M •8209 •0104 ■3959 •1649 2 3 •1550 •0007 •8188 •0145 ■3939 •1620 3 4 •1520 23-9988 •8171 •6129 ■3923 •1008 4 9.5 •1507 •9971 •8157 •6117 ■3910 •1595 9S 6 •1492 [ -9900 •8148 •6108 •3902 •1580 6 [141] Ape of elder I.ife:- .loint Life Annuity, at Top. Last Survivor Annuity, at Side. lo7 3 PERCENT. Values of Annukies on Tico Joint Lives^ and on the Last Survivor of Two Lives. 16 17 18 19 20 21 23-6047 23 4107 23 2199 2 3 '0355 22-8572 22 6805 ■5^ 6 •5010 3658 1734 22^9874 •8077 6299 "6 7 ■5203 3225 1285 •9411 •7601 5811 7 8 ■4804 2810 0852 •8905 •7144 5342 8 9 •4424 2412 0439 •8537 •6704 4893 9 6o •4057 2029 0040 •8120 •6281 4461 60 I •3710 1665 22 9661 •7734 •5880 4049 I 2 •3378 1317 9299 •7360 ■5496 3657 2 .3 •3063 0987 8955 •7004 ■5131 3283 3 4 •2704 0674 8628 •6666 •4783 2929 4 ^.■) •2481 0376 8318 •6345 •4455 2592 6^^ 6 •2214 0094 8023 •6040 •4142 2273 6 7 •1900 22 9828 7745 •5751 •3846 1971 7 8 •1721 9574 7480 •5477 •3564 1083 8 9 •1495 9335 7229 •5217 •3298 1411 9 7° •1282 9110 6992 •4971 •3046 1155 70 I •1085 8901 6772 •4743 •2811 0910 I 2 •0903 8707 6568 •4530 •2593 0093 2 3 •0737 8530 6381 •4336 •2394 0490 3 4 •0580 8309 6211 •4159 •2213 0305 4 IS •0449 8222 6050 •3998 •2047 0136 75 6 •0323 8087 5913 •3849 •1894 21 9981 6 7 •0208 7964 5782 •3711 •1753 9837 7 8 •0102 7849 5659 ■3584 •1622 9704 8 9 •0005 7744 5547 •3466 •1501 9581 9 8o 22^9917 7648 5445 •3359 •1391 9469 80 I •9839 7563 5353 •3263 •1292 9308 I 2 •9768 7486 5271 •3176 •1203 9278 2 3 •9705 7417 5197 •3099 •1123 9196 3 4 ■9649 7356 5132 •3029 •1051 9124 4 8.^ •9597 7299 5070 •2965 •0985 9056 85 6 •9549 7246 5012 •2904 •0923 8993 6 7 •9500 7192 4954 •2843 •0859 8928 7 8 •9453 7140 4897 •2782 ■0797 8865 8 9 •9408 7088 4840 •2721 •0734 8801 9 90 •9364 7038 4784 •2661 •0671 8738 90 I •9327 6994 4735 •2607 •0615 8681 I 2 •9294 6956 4690 •2558 •0564 8628 2 3 •9267 6924 4653 •2510 •0520 8583 3 4 •9246 6899 4623 •2483 •0483 8545 4 95 •9231 6880 4000 •2456 •0455 8516 95 6 •9221 0868 4585 •2439 •0436 8496 6 [14-2J Ago. of cider Lifc:- \ Joint Lifo Annuity, at Top. ( Last Survivor Annuity, at Side. 158 HM. 3 THR-CENT. Values of Ainuiiti'i's on Tiro Johit Lti'c., •0891 •8276 •5622 ■2932 •0223 ■7497 2 3 •0190 •7543 ■4852 ■2125 16-9375 ■6605 3 4 17 •9530 •6852 •4128 •1364 ■8575 ■5763 4 6', •8907 •6201 ■3445 ■0646 ■7821 •4969 65 6 •8320 •5587 ■2802 16^9971 ■7110 •4222 6 / •7768 •5009 ■2196 •9335 ■6442 •3518 7 8 •7247 •4465 •1625 •8736 ■5812 •2855 8 9 •6757 •3952 •1089 ■8173 ■5220 •2231 9 7° •6298 •3474 •0587 ■7646 ■4666 •1649 70 1 •5873 ■3030 •0123 •7158 •4153 •1109 I 2 •5481 •2621 16-9695 •6709 •3681 •0611 2 3 •5125 •2249 •9305 •6300 •3251 •0159 3 4 •4801 •1912 ■8952 •5930 •2862 15-9749 4 75 •4508 •1606 ■8633 ■5595 •2509 •9378 75 6 •4238 •1325 ■8339 ■5286 •2185 •9037 6 7 •3991 •1068 ■8070 ■5004 •1889 ■8724 7 8 •3764 •0831 ■7822 ■4744 ■1616 ■8437 8 9 •3555 •0613 ■7595 ■4506 ■1365 •8174 9 8o •3365 •0416 ■7389 ■4290 •1138 •7934 80 I •3195 •0239 ■7204 ■4096 •0935 •7720 I 2 •3013 •0081 ■7039 ■3924 ■0754 •7529 2 3 •2907 16-9940 ■6891 •3769 ■0591 •7358 3 4 •2786 •9814 ■6760 ■3631 ■0447 ■7206 4 85 •2674 •9698 '6639 ■3505 ■0314 ■7066 85 6 •2568 •9588 ■6524 ■3384 •0187 •6933 6 7 •2463 •9479 ■6410 ■3265 •0062 •6801 7 8 •2360 •9372 ■6299 ■3148 15-9939 ■6672 8 9 •2257 •9265 ■6188 ■3032 •9816 ■6543 9 90 •2156 •9161 ■6080 ■2918 •9697 ■6417 90 I •2066 •9068 •5983 •2817 •9591 ■6305 I 2 •1985 •8985 ■5896 •2727 •9496 ■6205 2 3 •1914 •8912 ■5821 •2648 •9414 ■6118 3 4 •1856 •8853 ■5760 •2585 •9347 ■6047 4 95 •1811 •8807 ■5713 •2536 ■9296 ■5993 95 6 •1780 •8775 ■5681 •2503 ■9261 ■5956 6 [14 6] A ge of elder I ■ r 1 Joil It Life Annu t Survivor A ity, at Top. nnuitv. nt S de. 162 HM. 3 ri:i;-(i;N'r. Values of Aiiiii/itiis <>n Jiro Joint Ju'resj a/td on tlie Last Survivor of Two Lircs. 46 1 47 48 49 jO .51 55 17-2874 170561 16^ 8277 166019 163792 16 1595 55 6 ^\-^:^'^ 16^9179 6830 •4503 •2206 15 9936 6 7 •0290 •7853 5438 •3046 •0679 8337 7 8 101)078 •6580 4104 -1645 159211 6798 8 9 •7919 •5364 2827 •0305 •7804 5322 9 6o •G815 •4203 1607 15^9024 •6458 3909 60 I •57G7 •3101 0448 •7806 •5179 2564 I 2 •4773 •2056 15 9349 •6650 •3963 1285 2 .3 ■3834 •1068 8310 •5556 •2811 0073 3 4 •2948 •0135 7328 •4522 •1723 U 8926 4 65 •2113 15-9256 6402 •3547 •0695 7843 65 6 •1325 •8427 5529 •2627 14-9726 6821 6 7 •0584 •7647 4707 •1761 -8811 5856 7 8 15-988G •6911 3932 •0943 -7950 4946 8 9 •9229 •6219 3203 •0175 -7139 4090 9 70 •8615 •5573 2521 14-9456 •6380 3289 70 I •8047 -4973 1890 -8790 -5677 2546 I 2 •7523 •4422 1309 -8177 -5030 1862 2 3 •7046 •3920 0780 -7619 •4441 1239 3 4 •6615 •3465 0301 -7114 •3908 0675 4 75 •6224 •3053 14 9867 -6656 •3425 0164 75 6 •5864 •2675 9468 -6236 •2981 13 9695 6 7 •5535 •2328 -9103 -5851 -2574 9265 7 8 •5233 •2010 -8767 •5497 -2201 -8870 8 9 •4956 •1718 •8460 •5172 •1858 8508 9 80 •4704 •1453 •8180 •4878 •1548 -8179 80 I •4478 •1215 •7930 •4614 •1269 -7885 I 2 •4277 •1003 7708 •4380 •1022 7623 2 3 •4097 •0814 •7508 •4170 •0801 1 -7389 3 4 •3937 •0646 •7331 •3983 •0604 •7181 4 85 •3789 •0490 •7168 •3811 •0422 -6989 85 6 •3649 -0343 7012 •3648 •0250 -6807 6 7 •3510 -0197 1 •6859 •3486 •0080 j 6627 7 8 •3374 •0054 •6709 •3329 139914 6452 8 9 •3238 U-9911 •6559 •3170 •9747 6277 9 90 •3106 •9772 6412 •3017 •9586 6106 90 I •2987 •9647 i •6281 ■2879 -9441 -5954 I 2 •2882 •9536 -6164 ■2756 -9312 5819 2 3 •2789 •9438 6062 •2649 -9200 1 5700 3 4 •2715 •9360 •5979 •2562 -9109 1 5605 4 95 •2658 •9300 5916 •2496 -9039 ' 5532 95 6 •2619 •9258 5872 •2450 •8991 1 5481 ' tl47J Age of elder Life:- I Joint Life Annuity, at Top ■" ° ( Last ?umvor Annuity, at . Side. 163 3 PER-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Txoo Lives. 52 53 54 55 56 57 1 55 15-9429 15-7304 15-5225 8-6949 8-5149 55 6 •7695 -5493 •3336 15-1231 •3584 6 7 •6022 -3743 •1507 14-9322 14-7191 7 1 8 •4410 •2055 14-9742 -7477 •5265 14-3113 8 , 9 •2861 •0432 -8042 •5698 •3405 •1171 9 6o •1378 14^8875 -6409 •3987 •1615 13-9299 60 I 14-9964 •7390 -4849 •2351 13-9901 •7505 1 I 2 •8619 •5975 -3363 •0789 -8262 •5787 2 3 -7342 •4631 -1949 13-9303 -6700 •4147 3 1 4 •6135 •3359 -0609 •7892 •5216 •2588 4 65 •4993 •2154 13-9339 •6555 •3808 •1106 65 6 •3913 •1015 -8137 -5287 •2471 12^9698 6 1 •2895 13-9939 •7001 •4087 •1205 •8363 7 8 •1934 -8923 •5927 -2953 •0007 •7097 8 9 •1028 •7966 •4914 •1881 12-8874 •5900 9 70 •0181 •7069 •3965 •0876 •7810 •4775 70 I 13-9395 -6237 •3084 12-9943 •6821 ■3728 I 1 2 ■8671 •5470 -2271 -9081 ■5908 •2760 2 3 •8011 •4771 •1530 -8295 •5074 •1876 3 4 •7414 •4138 •0858 •7582 •4318 ■1073 4 75 •6873 •3564 •0249 •6936 •3631 ■0344 15 6 •6375 •3036 12-9689 •6340 •2999 11-9672 6 7 •5920 •2553 -9176 •5795 •2419 ■9055 7 8 •5502 -2109 •8704 •5293 •1885 -8488 8 9 •5118 •1702 •8271 •4833 •1396 ■7966 9 80 •4770 •1332 •7878 •4415 •0951 ■7492 80 i I •4457 •1000 •7525 •4040 •0551 -7067 1 2 •4180 ■0706 •7212 •3706 •0196 -6688 2 3 •3932 -0442 •6932 •3408 11-9878 •6349 3 4 •3711 -0207 •6682 •3142 •9594 •6047 4 85 •3508 12-9991 •6452 •2897 ■9333 •5768 85 6 •3315 -9786 -6233 •2663 -9084 •5502 6 7 •3124 •9584 •6018 •2434 •8839 •5240 7 i 8 •2939 •9386 •5809 •2210 •8600 •4985 8 9 •2753 •9189 •5598 •1986 •8360 •4728 9 90 •2573 •8998 •5395 •1769 •8129 •4481 90 I •2411 •8827 •5213 •1575 •7922 •4260 I 3 •2268 •8674 •5051 •1403 •7738 •4063 2 3 •2143 •8542 •4910 •1253 •7578 •3891 3 4 •2042 •8435 •4796 •1131 •7448 •3753 4 95 •1965 •8353 •4710 •1039 •7350 •3648 95 6 •1912 •8297 •4650 •0975 •7282 •3576 6 [US] Asje of elder Life: Joint Life Annuity, at Top. Last Survivor Annuity, at Side. 1G4 HM. 3 PKHCENT. Values of Annuities on Tiru Joint Lives^ and on the Last Survivor t>f Tu-o Lirt'S. 1 58 59 60 61 62 63 5a 8-3274 8-1323 7-9310 7-7242 7-5127 7-2972 55 6 •1790 7^9920 •7986 •599'5 •3958 •1879 6 7 •023.3 •«44.-; ■^593 •4683 -2724 •0723 7 8 •6898 •5129 •3301 •1423 6-9 -,o I 8 9 13-8908 •3590 •1847 •0031 •8210 9 6o •704.'^ 13-4852 •0328 &8616 •6857 60 I •r>l()7 -2891 130080 •7125 '5449 I 2 •33G8 -1010 12-8721 120508 ■3989 2 3 •1G4D 12-9210 •G839 •4543 122326 3 4 •0012 •7494 •5041 •2602 •0362 11 •SMS 4 65 12-84.34 •5858 -3325 •0805 ir8483 •6182 65 6 •G972 -4299 -1088 11-9148 •0084 •4302 6 7 •5505 -2818 -0129 -7511 •4907 •2503 7 8 ••4230 -1409 11-8047 -5951 •3328 •0784 8 9 •2965 -0074 -7238 -4468 •1768 10^9145 9 70 •177G 11-8817 -5911 -3068 •0293 •7593 70 I •0GG7 -7045 -4071 -1759 10-8912 •6138 I 2 11-9G42 -0559 -3523 -0545 •7630 ■4786 2 3 •8705 -5500 -2471 10-9431 •6453 •3543 3 4 •7853 •4662 •1513 -8417 •5380 •2409 4 75 •7079 -3840 -oe4i -7493 •4401 •1374 75 6 •G3G4 -3081 10-9835 -0037 •3495 •0414 6 7 •5708 -2384 -9093 -5850 •2059 9^9528 7 8 •5105 -1741 •8410 -5123 •1888 •8710 8 9 •4550 ■1150 -7780 -4454 •1170 •7954 9 80 •4044 -0012 -7207 -3843 •0520 •7204 80 I •3591 •0128 -0091 •3293 9-9941 •6641 I 2 •3187 10-9098 -6232 •2804 •9420 •0087 2 3 •282G -9312 -5821 •2305 •8952 •5589 3 4 •2503 -8967 •5453 •1973 -8534 •5143 4 85 •2205 -8649 •5113 -1011 -8148 •4732 85 6 •1921 -8345 •4789 -1204 -7778 •4338 6 7 •1041 -804G •4408 -0922 -7413 •3948 7 8 •13G8 -7753 •4155 -0587 -7054 ■3505 8 9 •1093 -7459 -3839 •0248 -6692 •3178 9 90 •0828 -7174 -3534 99920 •6341 •2802 90 I •0591 -0919 •3200 •9626 •6024 •2403 I 2 •0380 -0093 -301G •9364 •5743 •2101 2 3 •0197 -0495 -2803 •9134 •5495 •1895 3 4 •0048 -0330 -2032 •8949 •5296 •1080 4 95 10-9930 -0215 -2501 •8809 •5144 •1517 95 6 •9858 -0132 -2412 •8712 •5040 •1404 6 [149] Age of elder Lifc:- Joint Life Annuity, at Top. Last Survivor Annuity, at Side. 165 3 PER-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. 64 65 66 67 68 69 55 7 0787 6-8562 6- 6292 6 3969 ^•^587 5"9i3o 55 6 6 97^J7 •7613 5412 3K'^5 •0837 •8441 6 7 8686 •6606 4476 2288 •0038 •7706 7 8 7542 •5538 3482 1366 5"9J85 •6921 8 9 6330 •4404 2425 0383 •8276 •6082 9 6o 5059 •3213 T312 5 9348 •7314 •5194 60 I 3734 •1969 0148 8262 •6306 •4260 I 2 2357 ■0674 5 8935 7129 •5252 •3283 2 3 C933 5'933-i- 7676 6377 5952 ■4155 •2265 3 4 — ■7952 4735 •3019 •1210 4 6=; 11 39G8 5028 3470 •1836 •0109 65 6 2005 1 A. 7(10 2147 •0597 4-8954 6 lUvivZ — 7 012i •7827 10 ^a^ 4^9292 •7734 7 OK) i Ji — 8 10 8321 •5945 3646 10 1428 •6441 8 9 G604 •4143 1760 9 9457 9^7234 9 70 4974 •2432 9 9968 7581 •5272 9^3040 70 I 3443 •0824 8280 5811 •3418 •1101 I 2019 9^9326 6705 4157 •1683 8^9283 2 3 0708 •7945 5252 2629 •0078 •7597 3 4 9 9511 •6682 3921 1228 8^8603 •6047 4 75 8417 •5527 2703 8 9943 •7250 •4623 75 6 7402 •4454 1569 8746 •5987 •3290 6 7 64G4 •3462 0520 7637 •4815 •2052 7 8 5597 •2544 8 9548 6608 •3726 •0900 8 9 4796 •1694 8647 5654 •2715 7^9828 9 80 4062 •0916 7823 4779 •1786 •8842 80 I 3401 •0214 7077 3988 •0946 •7949 I 2 2812 8^9588 6412 3280 •0194 •7150 2 3 2282 •9025 5812 2643 7^9516 •6428 3 4 1808 •8520 5275 2072 •8908 •5779 4 85 1369 •8053 4778 1541 •8342 •5175 85 6 0949 •7605 4300 1031 •7797 •4593 6 7 0534 •7162 3828 0526 •7257 •4013 7 8 0126 •6728 3364 0030 •6724 •3442 8 9 B 9713 •6287 2893 7 9526 •6184 •2860 9 90 9312 •5859 2436 9037 •5659 •2293 90 I 8949 •5472 2023 8596 •5185 •1781 I 2 8626 •5127 1655 8202 •4762 •1324 2 3 8341 •4822 •1329 7854 •4389 •0922 3 4 8110 •4575 •1065 7572 •4087 •0596 4 95 7935 •4387 •0864 7357 •3857 •0348 95 6 7813 •4256 0724 7207 •3697 •0175 6 [150] Age of elder Life:- Joint Life Annuity, at Top. Last Survivor Annuity, at Sido. 16G HM. 3 i-KR-CEXT. Values of Annuities on Tn-o Juint Lives, and on the Last Survivor (//' Tivc ) Lives. 70 71 72 73 74 75 5.1 5-6627 5 "4 1 04 5" '605 4^9169 4-6838 4^4616 SS 6 ■5997 •3530 •1082 •8694 •6406 ■422.5 6 7 'S.^^i •2914 •0521 -8183 •5942 •3803 7 8 •4602 •2255 4-9919 •7634 ■5442 •3348 8 9 •3831 •1547 •gi-ji •7043 •4903 •2857 9 6o •3013 •0797 •8584 -6414 -4328 •2332 60 I •2152 -0005 •7858 •5750 •3720 •1776 I 2 -•i2-;o 4-9175 •7096 '5<^S I •3080 •I 191 2 3 •0309 •8308 •6299 •4320 •2410 ■0577 3 4 4'9332 •7407 •5470 'iSS9 •1712 3-9938 4 65 •8312 -646+ •4601 -2760 •0979 -9266 65 6 •7^38 ■5470 -3684 •J915 •0202 •8.552 6 7 •6102 -4416 •2709 -10x5 3-9372 •7789 7 8 '4895 •3293 •1667 -0050 •8481 •6966 8 9 •3598 -2081 •0538 3-9002 •7508 •6064 9 7° •0796 3-9338 •7882 •6466 •5095 70 I 8-8878 •8079 -<^;o5 ■yM •4069 I 2 •6075 8-4778 ■5494 •4232 •3009 2 3 •5209 -2930 8-0780 •3091 •1940 3 4 -3581 -1225 7-8998 7-6917 •0893 4 75 •2084 7-9654 •7353 •5200 7-3203 7 5 6 •0G80 -8178 •5807 •3584 -1519 6-9602 (5 7 7^9373 -6802 •4361 •2070 6-9940 •7959 7 8 •8156 -5517 •3009 -0652 -8457 •6415 8 9 •7021 -4318 •1744 6-9323 -7066 •4962 9 8o •5975 •3211 •0575 -8092 -5775 -3614 80 I •5027 •2204 6-9510 -6969 -4596 -2381 I 2 •4177 -1302 -8554 -5961 -3536 -1270 2 3 •3409 ■0485 •7689 •5046 -2574 -0261 3 4 •2717 6-9750 •6908 •4220 -1704 5-9349 4 85 •2073 -9064 •6179 •3449 -0890 -8494 85 6 •1450 -8399 •5471 -2698 ■0097 -7659 6 7 •0829 •7734 •4762 •1944 5^9299 -6817 7 8 •0215 •7075 •4056 •1191 -8500 -5972 8 9 6-9587 -6399 •3330 •0413 -7672 •5094 9 90 •8975 -5737 •2615 5-9646 -6852 •4222 90 I •8420 -5134 •1963 •8943 •6098 •3419 I 2 •7924 -4595 •1377 ■8308 •5417 •2692 2 3 -7487 -4117 •0855 •7741 -4805 •2037 3 4 •7133 -3730 •0431 •7278 -4303 •1498 4 9S -6863 -3435 •0107 •6922 -3916 -1082 95 6 -6676 •3230 5^9881 •6673 •3644 -0789 6 r,-ii A f ij I T i Joint Life Annuitv, at J op. [loll Age of elder Luc: — {, .^ . •'. /... "• ■" " ( Last Survivor Annuitv, at bi Side. 167 3 PER-CENT. Values of Aimukies un Two Joint Lives ^ and on the Last Survivor of Two Lives. 76 77 78 79 80 81 5S 4-2418 4 0261 3 8136 3-6021 3-3947 3-1964 55 6 •2063 3 9941 7848 -5762 -37^"; •1757 6 7 •1681 959^ 7536 ■5483 •3465 •1532 7 8 •1269 9223 7^99 •5^79 •3193 •1288 8 9 •0822 8817 6833 •4849 •2895 •1021 9 6o •0344 8384 6440 ■4495 •2576 •0734 60 I 3-9838 7923 6025 •4117 •2236 •0428 I 2 •9303 7437 5581 •3718 •1876 •0103 2 3 •8743 6927 5118 •3299 -1497 2^9762 3 4 •8159 6395 4635 •2861 •II03 •9406 4 65 •7545 5835 4126 •2401 •0687 •903 1 65 6 •6892 5239 3584 •1910 •0242 -8630 6 7 •6192 4599 3001 •1380 2-9763 •8196 7 8 •5435 ro9os 2367 -0803 •9240 ■7722 8 9 •4603 3T39 1664 -0161 •8655 •7x90 9 70 •3705 2310 0900 2-9460 •8014 •6604 70 I •2751 1425 0083 •8707 -7322 •597 Jt I 2 •1761 0505 2 9230 •7920 •6597 -5304 2 3 •0762 2 9574 8365 •7119 -5858 •4623 3 4 2-9783 8660 75^6 •6332 '5^3^ -3952 4 75 6 •8832 7773 6857 6690 5837 •5568 •4424 -3690 -3299 -2620 75 6 ■4775 7 n-nosi — 4966 •3964 •2938 •1923 7 U UUO i 8 •4474 6 li ,1 •3127 •2160 •1200 8 _04:0 — 9 •2961 1070 5 9280 •1339 ■0435 9 80 •15.")4 5 •0266 9604 8261 7755 6357 5^6001 •4547 1-9639 80 5^2851 I I 2 5-9105 7049 5094 •3233 •1486 4-9874 2 3 ■8049 5946 3945 •2036 •0241 -8585 3 4 •7093 4948 2903 -0951 4^9113 -7415 4 85 •6197 4010 1923 4-9929 •8051 -6314 8^ 6 •5319 3089 0959 •8923 •7002 -5225 6 7 •4432 2156 4 9979 •7896 •5929 -4108 7 8 •3539 1214 8988 •6854 •4836 -2967 8 9 •2608 0229 7946 •5753 •3676 •1749 9 90 •1683 4 9246 6904 •4647 •2504 •0513 90 I •0828 8337 5936 -3616 •1408 3^9351 I 2 -0051 7509 5052 -2671 •0398 •8276 2 3 4-9351 6761 4252 •1813 3-9477 •7289 3 4 -8773 6142 3590 •1101 •8709 •6462 4 95 •8326 5662 3076 •0547 •8109 •5813 95 6 -8009 5322 2710 •0153 -7682 •5348 6 n cm A f ij T f \ Joint Life Annuity, at Top, [152] Ace of elder Life : — <, .c • a •. /o-j •- ^ ° ( Last burvivor Annuity, at Side. 168 3 PERCENT. Values of A/i)iuilies on Tiro Joint Livis, invl mi ihc Last Survivor of Two Lives. 82 83 84 85 86 87 ' 55 3-0129 2-8420 2-6876 2-5429 2-3979 2-2431 SS 6 2-9943 •8254 -6728 •5297 •3862 •2330 6 7 •9742 -8074 -6566 'S'^:-> -3735 •2220 1 8 •9.) 23 •7877 -6390 -4996 -359'5 •2099 8 9 •9282 -7661 •6196 -4822 -3442 •J 964 9 <5o •9024 -7428 •5986 •4^34 -3274 •1818 60 I •8748 -7180 •5762 •44.32 -309.-; •1660 I a •8455 -6916 •5.) 2 4 •4218 •2904 •1492 2 3 •8147 -6638 ■5274 •3993 -2703 •1316 3 4 •7826 •6349 •50JI3 •37'^o -2496 •1134 4 65 •7488 -6044 •4739 'iS^^ -2278 •0944 65 6 •7126 •5719 •4446 '?>^$'^ •2045 •0740 6 7 •(>ns •s^f^s •4126 -2965 •1791 •0519 7 8 •6305 -4976 ■3774 •2648 •1509 •0272 8 9 ■5S20 •4535 •3374 •2286 •1184 i-9987 9 /O •5285 -4046 •2928 •1880 •0819 •9663 70 I •4704 •3514 •2439 •1433 -0414 -9.302 I 2 •4091 -2949 •1920 •0957 I -998 1 -8913 2 3 •3462 -2370 •1386 -0461 -95.32 -8509 3 4 •2843 •1798 •0858 1-9980 -9089 -81 10 4 75 •2241 •1243 •0343 -9508 -8659 -7724 75 6 •1612 •0661 1-9807 -901 1 -8205 •7315 6 7 •0966 •0062 -9250 -8496 •7733 •6889 7 8 •0294 1-9436 -8668 •7956 •7236 •6439 8 9 1*9580 •8770 •8045 ■7375 •6697 ■5947 9 8o •8835 •8073 •7391 -6761 •6126 •5422 80 I •8089 •7371 •673^ -6140 •5545 •4885 I 2 •6704 -6102 •5548 •4990 ■4370 2 3 4-7083 •ss^?> •4983 •4460 ■3877 3 4 -5875 4-4467 •4482 -3991 -3443 4 85 •4737 -3295 4-1986 'iS^S •3051 85 6 •3611 -2134 •0793 39527 •2662 6 7 •2454 ■0940 39564 ■8264 3-6969 7 8 •1268 3-9713 ■8300 •6963 -5630 3-4245 8 9 3-9998 -8393 -6936 -5557 -4179 -2743 9 GO -8702 -7042 •5536 -4111 -2685 -1194 90 1 -7477 -5761 -4205 •2734 -1261 2-9715 I 2 -6340 -4505 •2958 -1441 2-9923 -8325 2 3 ■5289 -3453 -1793 -0231 -8668 -7021 3 4 -4403 •2511 -0802 2-9197 -7593 -5902 4 9^ ■3705 •1763 -0011 -8368 -6729 -5002 9') (5 •3202 •1221 2-9434 •7762 -6095 •4339 6 [1.53] , r 11 T •/■ 1 Joint Life Annuity, at Top. Affc of elder Life: — 1 r . o • * ' •» /o-j '^ { Last Survivor Annuity, at Side. 169 3 PEE-CENT. Values of Annuities on Txoo Joint Lives, and on the Last Survivor of Tico Lives. 88 89 90 91 92 93 i 55 2-0785 1-8818 1-6568 1-4235 1-1823 1 0-9197 55 6 •0699 •8748 •6512 •4192 •1792 -9176 6 7 •0605 •8671 •<545 r •4145 •1758 •9^)4 7 8 •0502 •8586 •6384 •4094 •1721 -9128 8 9 •0387 •8490 -6308 •4036 •1678 -9100 9 6o •0261 •8386 •6224 '397^ •1631 •9068 60 I •0125 •8273 -6134 •3901 -'579 •9033 I 2 1-9981 -8152 •6036 •3826 ■J523 •8995 2 3 -9829 •8025 '5934 •3746 •1464 •8954 3 4 -9672 •7894 •5828 •3664 •-1403 •8912 4 65 •9508 7758 '57^9 ■3579 •1340 •8869 65 6 ■9334 •7614 •5604 •3490 •1274 •8824 6 7 •9145 •7458 •5480 •3394 •1204 ■S776 7 8 •8935 •7284 •5342 •3289 •1128 •8725 8 9 -8688 •7079 ■5^79 •3164 •1037 •8663 9 7° -8407 •6844 •4989 •3017 •0929 •8590 70 I -8091 •6576 •4771 •2847 •0802 •8504 I 2 •7749 •6284 •4532 •2657 •0659 ■8405 2 3 •7392 •5979 •4279 •2455 •0506 -8297 3 4 •7039 •5676 •4029 •2256 '0353 -8189 4 75 •6699 •5386 •3791 •2067 •0210 -8089 75 6 •6338 •5078 ■3536 •1864 •0057 -7981 6 7 •5961 ■4755 •3271 '^6^3 0-9897 •7869 7 8 •5560 •441 1 •2986 •J427 -9727 •7751 8 9 '5^ ^9 •4029 •2668 •1172 -9533 '76j5 9 8o •4645 •3614 •2319 •0888 •9314 ■7459 80 I •4156 •3183 •1952 •0587 -9078 •7289 I 2 •3686 •2765 •1594 •0292 -8845 •7120 n 3 •3234 •2363 •1247 •0001 -8613 •6949 3 4 •2837 •2010 •0943 o'9747 -8410 •6799 4 85 -2482 ■1697 •0676 •9526 •8235 •66C9 85 6 •213 1 •I 39 1 •0418 •9315 -8069 •6548 6 7 •1739 -1050 •0132 •9084 •7890 •6418 7 8 •0676 0-9824 -8840 ■7709 •6296 8 9 3-1247 •9356 -8461 •7417 •6090 9 90 2-9632 2-7909 •7908 •6979 •5770 90 I •8089 •6277 2-4363 •6448 •5379 I 2 -G636 -4737 -2708 2-0712 •4898 2 3 -5273 •3288 •1141 1-9005 1-6902 3 4 -4107 •2045 1-9791 -7525 ■5260 r2971 4 95 -3171 -1048 •8701 -6322 -3916 •1424 95 6 -2483 •0317 •7897 -5429 •2914 ■0253 6 [154] , r ij T T I Joint Life Anniutv, at lop. Age 01 elder Lite: — < , ^ r. • . • '.. /f-j o ( Last Siuvivor Annuity, at Side. 170 3 rKK-CKNT. Values vf Aniuiitii's on Tico Joint Livis, and on the Last Survivor of Two Lives. 1 1 '' 95 96 i 55 0-6577 ©•405 r 1746 sa 6 •6564 •4044 1743 6 7 ■<555o •4037 1740 7 8 •6535 ■4029 1738 8 9 •6517 •4020 1734 9 6o •6497 •4010 1730 60 1 •6476 •3998 1726 I 2 •6452 •3986 [721 2 3 •6427 ■3972 1716 3 4 •6401 •3958 1711 4 65 •6374 •3944 ; 1706 65 6 •6.346 •3929 1700 6 7 •6316 •3913 1694 7 8 •6285 •3897 r688 8 9 •6247 •3877 1 1681 9 70 •6202 •3854 1672 70 I •6149 •3826 1662 I 2 •6087 •3793 1650 2 3 •6018 •3756 1636 3 4 •5949 •37^8 1621 4 75 •5886 •3684 1608 75 6 •5817 •3646 i 1594 6 7 •5746 •3608 i 1579 7 8 •5^71 '3!i67 1564 8 9 •5585 •3521 1546 9 80 •5485 •3467 i '5^5 80 I '5374 •3405 1501 I 2 •5264 •3344 1478 2 3 •5149 •3279 : 1452 3 4 •5048 •3221 1429 4 85 •4961 •3172 1409 85 6 •4881 •3127 1392 6 7 ■4795 •3077 ^371 7 8 •4720 •3038 1357 8 9 •459^ •2970 ^33^ 9 90 •4378 •2850 1285 90 I •41 1 7 •2702 1226 I 2 •3798 •2524 "57 2 3 ■33^^ •2240 •1849 1042 0879 3 4 4 95 0-9073 0647 95 6 •7674 0-5288 - 6 [155] Age of elder Life:- .Toint Life Annuity, Live.-*. Equal Ages. X fx.x ^"x—ax.x 1 "x.x ^(Ix—Os.x 10 19-3289 24-0789 5S 8-5546 14-5424 I •2029 •405:) 6 -2385 •1805 2 •0.346 ■•^^'^■l 7 79231 13-8135 3 18-8.522 -2i:U 8 -608.^ ' -4417 4 -6054 -077U 9 -2944 : -0656 15 •■i^3:i 23-03r)l 60 6-9834 12-6860 6 •I 158 -7894 I •6767 •3037 7 17-8722 -G410 2 -3754 11-9196 8 -6423 •4913 3 -0805 -5341 9 •4360 •3422 4 5-7931 •1483 20 ■2554 -1938 65 '51^1 10-7611 I -0876 -0440 6 •2342 -3726 2 16-9259 22-8915 7 4-9595 9-9827 3 •7606 -7344 8 •6863 -5909 4 •5843 -5715 9 •4111 •1965 25 '3949 -4023 70 •1378 8-8028 6 •i960 -2270 I 3-8695 -4123 7 15-9893 -0461 2 •611 1 -0287 8 •7798 21-8G06 3 •3676 7-6556 9 •5689 •6703 4 •J437 •2961 30 •3561 •4751 /J 2-9395 6-9499 I •1419 -2751 6 •7424 -6098 2 14-9231 -0G93 7 '5543 -2773 3 •6988 20-8578 8 •3730 5-9518 4 •4685 -6399 9 '1 949 -6321 35 •2329 -41 Gl 80 •0225 •3219 6 13 '9930 -18G0 I 1-8612 -0258 7 •7502 19-9504 2 -7174 4-7486 8 ■5047 •7085 3 •5886 •4876 9 •2549 i ^4003 4 •4806 •2438 40 12-9996 1 ^2056 85 •3877 •0079 I ■73^3 18-9431 6 •3007 3-7671 2 -4629 -G729 7 •2107 ■5109 3 -1805 -3947 8 . ^1202 -2366 4 11-8921 •1093 9 •0036 2-9242 45 •5979 17-81G5 90 0-8625 -5809 6 •3019 •5177 I -7^93 -2269 7 -G062 -2132 2 ■57'52 1-8G08 8 10-7102 16-9026 3 ■4173 -4709 9 -4122 -5858 4 •2654 -0798 50 •J123 -2625 95 ••1314 0-6944 I 9-8082 15-9322 6 -0326 -3224 2 -4986 -5944 3 •i8j4 -2498 4 R'^\-c7 ' ll-S'.i'.ll 173 3 J PER-CENT. Values of Annuities on Two Joint Lives, and on tlie Last Survivor of Tico Lives. JO II 12 13 14 15 lO 19-2650 19-1783 I9^0722 18-9516 18-8209 10 I 24-5231 •1178 •0136 •8949 •7661 I 3 •4G45 24-4053 1 18-9323 •8157 •6890 2 3 •4045 •3434 24-2794 ■7177 •5932 3 4 •3435 •2805 •2144 24^1463 •4833 4 15 •2823 •2174 •1492 •0789 24-0072 15 6 •221G •1549 •0847 •0120 23-9379 23-8635 6 7 •1G20 •0934 •0212 23^9463 -8699 -7930 / 8 •1039 •0336 23-9593 •8823 -8036 -7242 8 9 •0481 23-9759 •8997 •8206 •7397 -6580 9 20 23^9943 •9204 •8422 •7611 •6780 -5941 20 I •9418 •8662 •7862 •7030 •6178 •5317 I 2 •8901 •8129 -7310 •6458 •5585 •4702 2 3 •8387 •7598 -6761 •5890 •4995 •4089 3 4 •7874 •7067 •6211 •5319 •4404 •3476 4 25 •7357 •6536 •5661 •4748 •3811 •2862 25 6 •6842 •6003 •5111 •4178 •3218 •2246 6 7 •6331 •5474 •4562 •3611 •2629 •1633 7 8 •5823 •4953 •4022 •3049 •2047 •1028 8 9 •5321 •4435 •3489 •2495 •1471 •0430 9 30 •4826 •3924 -2959 •1949 •0902 22-9838 30 I •4337 •3420 •2438 •1406 •0342 -9255 I 2 •3852 •2921 •1922 •0871 22^9784 •8678 2 3 •3372 •2426 •1410 •0342 •9234 •8105 3 4 •2895 •1935 •0903 22-9815 •8688 •7537 4 35 •2424 •1451 •0403 •9296 •8149 •6976 3S 6 •1957 •0970 22-9907 •8782 •7614 •6421 6 7 •1498 •0499 •9418 •8277 •7089 •5875 7 8 •1044 •0032 •8936 •7776 •6570 •5335 8 1 9 •0597 22^9571 •8461 •7284 •6059 •4804 9 1 40 •0155 •9118 •7992 ■6798 •5554 •4279 40 I 22^9718 •8669 •7529 •6317 •5055 •3759 I 2 •9285 •8226 •7071 •5843 •4561 •3247 2 3 •8859 •7787 •6619 •5374 -4074 •2739 3 4 •8439 •7356 •6173 •4913 -3594 •2240 4 45 •8025 •6933 •5737 •4458 •3123 •1749 45 6 •7621 •6518 •5309 •4016 •2661 •1269 6 7 •7227 •6114 •4892 •3583 •2212 •0800 1 8 ■6813 •5720 •4485 •3160 •1772 -0343 8 9 •6467 •5336 •4087 •2749 •1343 21-9895 9 50 •6101 •4961 •3701 •2346 •0924 •9459 5° I •5744 •4596 •3323 •1955 •0516 •9033 I 2 •5397 •4239 •2955 •1572 •0117 •8616 2 3 •5058 •3892 •2597 •1200 21-9728 •8210 3 4 •4731 •3557 •2250 ■0839 -9352 •7816 4 [188] Age of elder Life :- j f °'^V^^''". ^"T^>'' .^^ ^op. -■ ° I Last Survivor Annuity, at S Side, 174 3i PKK-C'ENT. Valuer of Annui'tus on Two Joint Lhrs, and on the Last Survivor (if Tiro ] jives. 16 ^7 18 1 ^9 20 21 1 to 18-6849 '8-.H85 18-4168 18-2949 18-1842 18-0779 ,0 I •6,) 19 •4974 •3'574 •2474 ' -1384 1 -0338 I 2 *55^>S •4H3 •2964 •1783 , 'O/^S ' i7'9'^85 1 2 3 •4'5.H •3331 •2073 •0913 17-9863 i -8836 3 4 •3559 -2279 •1044 1 7^9906 i 8878 •7892 4 15 •2384 -1129 17-9919 •8804 •7798 •6834 15 6 17-9927 •8742 •7651 •6668 •5726 6 7 23-7105 •7560 •6494 •5534 •4613 7 8 •64o2 23-5674 •5379 , "4443 •3547 8 9 -5766 -4963 23-4180 , '3445 '^sn 9 20 •5104 •4278 •3471 ' 23-2692 •1703 20 I •4458 •3609 •2779 •1976 23^1201 I 2 •3820 •2949 •2095 •1270 •0472 22-9089 2 3 •3185 •2290 •1414 ■0500 22^9744 •8939 3 4 •2548 •1630 •0730 22^9859 •9014 •8185 4 ^5 •1910 •0967 •0044 •9148 •8280 •7427 25 6 •1272 •0300 22-9357 ■8437 •7540 •0008 6 7 •0635 22-9040 •8073 •7728 ■6813 •5912 7 8 •0006 •8992 •7990 •7028 ■0088 •5163 8 9 22-9384 •8346 •7326 •6335 •5372 •4421 9 30 •8770 •7708 •6664 •5649 •4663 •3689 30 I •8163 •7078 •6010 •4972 •3962 •2965 I 2 •7563 •6453 •5363 •4300 •3268 •2247 2 3 •6969 •5837 •4722 •3638 •2581 •1538 3 4 •6377 •5224 •4086 ■2978 •1901 •0833 4 35 •5795 •4618 •3459 •2328 •1228 •0139 SS 6 •5218 •4019 •2836 •1685 •0563 2r9451 6 7 •4650 •3429 •2225 •1051 2r9909 •8776 7 8 •4089 •2846 •1621 •0426 •9262 •8109 8 9 •3536 •2272 •1025 21-9810 •8626 •7452 9 40 •2991 •1706 •0438 •9202 •7998 •6804 40 I •2451 •1145 21-9856 •8600 •7377 •6163 I 2 •1917 •0590 •9281 ■8005 •0704 •5530 2 3 •1389 •0041 •8712 ■7418 •6157 •4906 3 4 •0870 21^9502 •8152 •6839 •5561 •4291 4 45 •0359 •8971 •7602 •6270 •4974 •3688 45 6 21^9859 •8451 •7063 •5713 •4401 •3097 6 7 •9372 •7944 •6538 •5170 •3841 •2522 7 8 •8894 •7449 •6024 •4640 •3295 •1960 8 9 •8429 •6963 •5522 •4121 •2761 •1411 9 50 •7974 •6491 •5030 •3614 •2240 j •0876 5° I •7529 •6029 •4551 •3118 •1730 1 •0353 I 2 •7095 •5576 •4082 •2034 •1231 j 20-9841 i 2 3 •6671 ■5135 •3624 •2101 ■0740 ! ■9342 I 3 4 ' •0260 ■1700 •3180 •1002 •0274 •SS59 4 ' [189] Age of I'lder Life: Joint Life Aiiiuiity, at Top. Last Survivor Annuitv, at Side. 175 3^ PERCENT. Values of Annuities on Tico Joint Lives ^ and on the Last Survivor of 2wo Lives. 22 23 24 25 26 27 lO X7-9725 17-8627 1 77444 17-6168 17-4812 17-3385 10 I •9.300 •8219 ■7054 •5792 ■4454 -3045 I 2 •8666 •7603 •6457 •52H •3893 -3504 2 3 ■7857 •6813 •5688 •4466 ■3165 •1794 3 4 •6914 •5892 •4787 •3587 •2309 -0960 4 15 •5878 •4879 •379^ •2617 •1362 •0037 15 6 •4793 •3816 ■2757 •1602 •0369 16-9068 6 7 •3704 •2751 •J715 •0585 16-9375 •8097 7 8 •2660 •1729 •0717 i6^96io •8426 •7172 8 9 •1708 •0800 16-9811 •8729 7569 •6340 9 20 •0861 16-9977 •901 1 79.52 •6815 •5610 20 I •0056 -9194 -8420 •8252 •7217 •6492 •6105 •4923 I 2 7502 •5404 -4247 2 3 22 -8112 •6712 '5727 •4665 '3533 3 4 •7364 22-6542 •4883 •3847 ■2743 4 25 •6581 -5734 22-4882 •2940 •1864 25 6 •5798 -4925 -4047 22-3161 -0012 6 . •5017 ■4243 •4119 •3320 -3213 •2387 •2299 •1444 22-1380 •0497 ^ / 8 21-9549 / 8 9 •3477 •2528 •1569 •0598 21-9621 -8643 9 30 •2719 •1744 •0758 21-9760 -8754 •7746 30 I •1972 •0971 21-9958 •8932 •7897 •6858 I 2 •1231 •0205 •9165 •8110 •7045 •5976 2 3 •0497 21-9447 •8380 •7297 •6203 •5103 3 4 2V9769 •8694 •7601 •6490 •5367 -4236 4 35 •9051 •7951 •6833 •5694 •4542 -3380 35 6 •8341 •7217 •6072 •4906 •3725 •2534 6 7 •7642 •6495 •5325 •4132 •2922 •1700 7 8 •6953 •5782 •4588 •3369 •2129 •0879 8 9 •6276 •5082 •3863 •2617 •1351 •0069 9 40 •5607 •4392 •3149 •1877 •0584 20-9274 40 I •4946 •3710 •2444 •1147 20-9826 •8488 I 2 •4294 •3036 -1748 -0427 •9080 •7712 2 3 •3650 -2372 •1061 20-9716 •8344 •6949 3 4 •3017 •1719 •0387 -9018 •7621 •6200 4 45 •2396 •1079 20-9726 •8335 •6912 •5465 45 6 •1788 -0453 •9080 •7667 •6221 •4748 6 7 •1196 20-9844 •8452 •7016 •5548 •4051 7 8 •0619 •9249 •7839 •6384 •4892 •3372 8 9 •0055 •8669 •7241 •5766 •4254 •2709 9 50 20^9504 •8104 •6658 •5165 •3632 •2066 50 I •8968 •7552 -6090 •4578 •3025 -1438 I 2 •8443 •7013 -5535 -4006 -2434 •0827 2 3 •7932 •6489 -4996 -3450 -1859 •0232 3 4 •7436 -5980 -4473 -2910 •1302 19-9656 4 [190] Age of elder Life:- j Joint Life Annuity, at Top ° ( Last sunivor Annuity, at bide. 176 3 1 PER-CENT. Values of Aiinuitli a on Tiro Joint Licrs^ mvl on the Jjiiat Survivor of Two Liv(S. 28 29 30 31 32 33 10 171918 17-0414 16^8869 16^7287 16^5649 1 6^3950 10 X •J.591 -0103 •8.174 ■7007 •5383 •3699 r 2 •1069 J 6-9596 •8086 '(^$?.(^ •4929 •3262 2 3 •0.381 -8929 •7435 'S9°l ■4319 •2669 3 4 1 6-9.=; 6 7 •8r37 •6666 -5^S5 ■3590 •J961 4 15 •8667 •7259 •581 1 •4323 •2777 •1171 15 6 •7722 •6338 •4912 •3448 •1925 •0340 6 7 •6776 •5416 ■4014 •2573 '^°n ^5-95 12 7 8 •5874 •4538 •3160 •^743 •0267 •8729 8 9 •5065 '37S^ •2398 ■J 004 ^5'9553 •8036 9 ao •4360 •3070 •J739 ■0369 •8940 7-448 20 1 •3697 •2433 •I 135 ^1-9778 •8373 •6903 I 2 •3046 •1806 •0524 ■9200 •7818 '^373 2 3 •2357 •1^43 15-9887 ■8589 •7232 •5811 3 4 •1594 -0^06 •9^11 •7906 •6576 •5182 4 25 •0744 ^5'9:M •8382 7139 •5838 -4472 25 6 15-9820 -8690 •7517 •6303 '5^3^ •3695 6 7 ' -8830 7730 •6587 ■.5404 •4163 •2857 7 8 -6728 •5617 ■44'55 •3254 -1980 8 9 21-7670 •4609 •3489 •23 1 1 -1070 9 30 •6741 21-5743 •2473 •1328 -0120 30 I -5822 -4792 2r3768 •0307 ^49^33 I 2 -4910 •3847 •2790 2M740 -8091 2 3 •4005 •2909 •1819 ■0735 20^9654 3 4 •3106 ■1978 •0854 20^9735 •8619 20-7507 4 3') •2218 -1057 20^9900 ■8746 •7594 -6446 35 6 •1340 -0146 •8954 •7769 -6578 •5392 6 7 •0476 20-9250 •8024 •6800 •5576 •4353 7 8 20-9623 -8365 •7106 •5847 •4586 •3326 8 9 •8784 -7494 •6201 •4909 •3611 •2313 9 40 •7957 -6635 •5310 •3983 •2r.5i •1315 40 I •7141 ■5788 •4430 •3069 •1701 ■0328 I 2 •6337 -4953 •3562 ■2167 ■0764 19^9354 2 3 •5543 -4129 •2707 ■1278 19-9840 ■8393 3 4 •4766 •3322 •1868 ■0407 •8934 •7451 4 4') •4004 •2531 •1046 19^9553 •8046 •6527 45 6 -3260 •1759 •0245 ■8720 •7180 •5627 6 7 -2537 •1008 19-9466 ■7911 •6339 •4751 7 8 -1833 •0278 •8707 •7123 •5520 •3899 8 9 •1146 19-9566 •7969 ■6356 •4723 •3070 9 50 •0479 ■8874 •7251 •5611 •3949 •2265 50 I 19-9829 •8200 ■6552 ■4886 •3195 •1481 I 2 -9196 ■7544 ■5871 •4180 •2462 •0719 2 3 •8580 ■6907 ■5211 ■3494 ■1750 18^9979 3 4 •71JS4 •6290 ■4572 •2832 •IOC, 2 •9264 4 [lyij . ,, , .,, , Joint Life Annuity, at Top. Age of elder Lite:- j j .^^^ ^^^^^.■^,^^. Annuity, at Side. 177 3i PERCENT. Values of' Annuities on Two Joint Live^; and on the Last Survicor of Two Lives. 34 35 36 37 38 39 10 lO 1&21S6 16 0360 ^5*8477 i5'6544 15-4561 15-2518 I •1949 0136 •8267 •6346 •4376 •2347 I 2 •1528 15- 9731 ■7877 •5974 •4019 •2004 2 3 •0955 9177 •7341 •5454 •3518 •I 520 3 4 •0266 8508 •6693 •4826 •2908 •0929 4 15 15-9498 7762 •5967 •4121 •2224 •0265 15 6 •8691 6976 ■5203 •3379 •1503 14-9566 6 7 •7884 1 • 6193 •4442 •2640 •0786 •8870 7 8 •7124; • 5454 ■3727 •1946 •0113 •8219 8 9 ■6455 4808 •3101 •1343 14-9531 •7657 9 20 •5887 • 4263 •2578 •0840 •9050 •7196 20 I •5367 ' • 3764 •2102 •0385 •8615 •6782 I 2 •4860 3281 •1641 14-9948 •8200 •6387 2 3 •4323 2769 '^^53 •9483 ■7759 •5969 3 4 •3720 2191 •0602 •8957 ■7257 ■5492 4 25 •3038 • 1537 H-9975 •8357 •6683 ■4945 25 6 •2290 0818 ■9285 •7696 •6052 •4340 6 7 •1483 0042 •8538 •6980 •5364 •3684 7 8 •0638 14 9229 ■7757 •6229 •4645 •2994 8 9 14-9760 8384 •6945 ■5449 -3897 •2278 9 30 ■8844 750^ •6097 •4635 •3116 -KS3i 30 I •7892 1 6584 •52li •3788 •2304 •0752 I 2 •688 ■■; 5613 •4279 •2889 •1442 13-9927 2 3 •5818 4582 3487 •3286 •1933 •0523 •9046 •8104 3 4 •222Q •0915 ^3'95'^3 4 35 20 5300 — •I 109 13-9834 -8503 •7104 35 6 4208 20 3031 •8695 •7404 •6048 6 7 3130 1914 20-0703 ■6253 •4939 7 8 20G5 0808 19-9557 19-8316 '3775 8 9 1014 19 9717 •8423 -7140 19-5867 9 40 19 9975 8639 •7304 •5978 •4660 19-3353 40 I 89i9 7572 -6196 •4827 •3464 •2111 I 2 7937 6518 •5099 •3688 -2280 •0881 2 3 6938 5479 •4018 •2562 -1110 18^9665 3 4 5957 4458 •2956 •1456 18-9959 •8467 4 45 4996 ' 3458 •1914 •0371 •8829 •7290 45 6 4058 2481 •0897 18^9312 •7725 •6140 6 7 3U7 1531 18-9907 •8280 •6650 •5019 7 8 2259 0607 •8944 •7276 ■5603 -3928 8 9 1397 18 9708 •8006 •6298 •4583 •2864 9 50 0558 [ 8834 •7095 •5348 •3592 •1829 50 I 18 9742 7984 •6209 •4423 •2627 •0822 I 2 S949 7158 •5347 -3525 •1689 17^9842 2 3 8179 6356 -4512 -2653 •0780 •8892 3 4 •7436 5582 •3704 -1812 17^9901 •7974 4 [19-2] Age of elder Life : - ! Joint Life Annuity, at Top. Lubt Survivor Annuity, at Side. 2 A 178 85 PERCENT. Values of Atmuitics an Tu'd Joint Jji'vis^ and an (he Last Surcivm' of Two Jji've^. 40 4' 42 43 1 44 1 45 10 IJ-04I0 14-8218 14-5933 ' i4'3556 14*1 107 138586 10 I •02-,o •8070 •5795 1 ■343 1 •0993 •8481 I 2 14-9923 '7757 -5497 •3146 •0723 •8224 2 .3 ■94.16 •7308 •5064 •2730 •0322 •7842 3 4 •8884 •6754 •4530 •2214 139825 •7361 4 1.1 •8240 •6131 •3925 •1630 •9260 •6816 15 6 •7.1^1 •5472 •3288 1 •1013 -8663 •6239 6 7 •6886 •4818 •26.55 ' •0401 •8071 •5667 7 8 •62 -,6 -4209 •2066 13-9832 •7523 •5138 8 9 •5 7"^ 5 •3688 'Kl^j ■9349 •7059 •4693 9 20 ■5274 •5266 •I 161 -8965 •6692 •4344 20 I ■4880 •2892 •0807 -8628 •6374 •4042 I 2 •4506 •2538 •0472 •8313 •6077 -3763 2 3 •4109 •2162 -0118 •7979 •5763 -3468 3 4 '3656 •J732 13-9710 •7594 '5399 -3125 4 ^5 -3^35 •1236 •9238 •7146 •4975 •2723 25 6 •2.557 •0686 •8714 •6647 •4501 •2275 6 7 •J929 •0086 •8144 •6104 •3984 •1784 7 8 •1271 13-9458 •7.544 •5535 -3443 •J270 8 9 •0587 •8803 •6922 •4943 •2881 -0737 9 30 13-9872 •8123 •6273 -4325 •2295 •0182 30 I •9128 •7413 ■5597 •3683 •1685 12^9604 I 2 •8337 •6658 •4877 •2998 •1035 •8988 2 3 •7494 •5852 •4108 •2266 -0339 •8328 3 4 •6593 -4990 •3284 •1480 12-9592 •7618 4 35 ■5632 -4070 •2406 -0642 •8794 •6859 35 6 •4617 •3096 •1475 12-9753 •7946 •6053 6 7 '355^ •2073 •0494 -8817 -7054 •5204 7 8 -2432 •0999 12-9465 •7832 •61 14 ■4309 8 9 •1249 12^9862 •8655 •8374 •6787 •5676 •5116 •3358 9 40 40 •7215 •4053 •2343 I 19-07G8 •5970 •4482 •2909 , -1249 I 2 18-9-490 18-8106 •3190 •1669 •0062 2 3 -8226 •6791 18^5365 •0336 ; 11-8782 3 4 •6980 •5495 •4017 18-2547 •7422 4 45 •5755 •4220 •2689 •1166 17^9657 45 6 •4556 •2972 •1389 17-9812 •8248 ! 17^6700 6 7 •3388 •1754 •0120 •8488 •6869 1 -5264 7 8 •2249 -0566 17 •8880 7195 •5521 •3858 8 9 •1139 17-9407 •7670 •5932 •4202 •2483 9 5° •0059 •8279 •6491 •4700 •2915 •1139 50 I 17-9007 •7180 •5342 •3498 •1659 16-9826 I 2 •7983 •6109 •4223 •2327 •0433 •8544 2 3 •099 1 •5071 -3136 •1190 169242 •7297 3 4 •6031 •4068 1 -2085 1 , -0088 •8089 1 •6088 4 [ly i] A', c of elder L ., Join 'f«=- Last t Life "Annu Survivor A ity, at Top. nnuity, nt Si de. 179 3 J PER-CEXT. Values of Anmntieft on Tico Joint Lives^ and on the Last Survivor of Two Lives. 46 47 48 49 50 51 lO 13-6016 1 3 •3409 13-0760 12-8062 12-5312 12-2497 JO I •5922 •33^-5 -0686 -7996 •5255 •2448 I 2 •5^578 •3094 -0468 •7792 -5062 -2268 2 3 •5310 -2742 -0132 -7469 •4756 •T975 3 4 •4849 .2297 12-9704 •7059 •4362 •1598 4 i5 •4322 •1790 -9214 •6^-88 -3908 •1162 15 6 •3765 •1251 -8696 •6087 -3426 ■0699 6 7 •3213 •0719 -8181 •5593 -2949 •0239 7 8 •2703 •0227 -7708 ■5^3(^ -2512 11-9819 8 9 •2276 12-9818 •7315 -4760 -2151 •9475 9 ; 20 •1943 •9502 •7015 •4475 -1880 •9218 20 1 1 •1659 '9^33 •6762 •4237 -1656 •9007 I 2 '^397 •8988 •^532 -4022 •1457 •8821 2 3 •1120 •8728 -6290 •3796 •1245 •8625 3 4 •0797 •8424 •6004 •3528 •0995 •8391 4 25 •0417 -8067 -5666 •3210 -0695 -8110 25 6 12-9992 -7664 •5287 -2851 •0357 -7792 6 7 •9527 -7223 -4869 •2458 11-9985 •7441 7 8 •9040 -6762 •4433 •2046 •9597 ■7075 8 9 ■8535 •6285 •3982 •1620 •9196 -6698 9 30 •8009 •5787 •35^3 •1177 -8779 -6306 30 I •7463 •5271 •3026 •0719 -8348 •5901 I 2 •6880 •4720 •2506 •0229 •7887 •5469 2 1 3 •6254 •4129 •1948 11-9703 •7392 •5004 3 1 4 ■55S2 •3492 •1347 •9^35 -6858 •4502 4 135 •4862 •2811 •0702 •8527 -6285 •3963 35 ! 6 •4096 •2085 -0015 -7879 •5674 •3388 6 7 •3289 -1320 1 1 -929 1 •7195 •5029 •2782 7 8 •2439 •05^3 •8527 •6473 •4348 •2141 8 9 '^534 11-9654 •7712 •5702 •3621 •J456 9 40 •0568 •8735 -6841 •4877 •2841 •0721 40 I 11-9523 •7740 •5895 •3980 •1992 10-9919 ' ; 2 •8388 •6656 •4863 •2999 •1062 •9039 2 j 3 •7162 •548.1 •3745 •J934 •0050 •SoSo 3 4 ■5857 •4235 •2550 •0795 10-8966 •7050 4 45 •4470 •2905 -127S 10-9579 •7807 •5948 45 6 •1511 10-9942 •8302 •6588 -4788 6 7 17-3684 •8552 •6971 •5317 •357'^ 7 ' 8 -2220 17-0609 •5.581 •3988 •2309 8 9 -0786 16^9116 16-7473 -2591 •0975 9 50 16-9384 •7654 •5950 16-4273 -- - - 9'957o 50 r •8012 -62-23 •4457 •2717 16-1006 I 'J, •6672 •4822 •2995 •1192 15-9416 15-7666 2 3 •o307 •3457 •1569 15-9703 •7862 •6045 3 4 •4101 •2132 •0183 •8254 •6349 •4465 4 ; [194] . r u T -f 1 Jomt Life Annnitv, at Top. Asi of elder Life: — i , . a > _ i - •. . c- 1 * ' Last Sunivor Anmiitv, at bide. 180 SJ PKK-f'ENT. Va?u€Jt of Anmn'fii's on Tiro Joinf Liv,s^ ami on the LoM Survivor of liro Ijir^s. 52 s:^ 54 5S 56 57 to 1 1 -960 7 11-6657 ii-3'557 iro6ii 10-7528 10-4410 10 I •9368 -6626 •3634 •0595 •7518 •4408 I 2 •9399 -6468 •3488 -0460 •7393 •4293 2 3 •91 2 1 -6204 •3238 -0224 -7171 •4084 3 4 •8760 •5860 -2909 10-9910 •6873 •3801 4 »5 •8.342 •5459 •2^26 •9544 -6-23 •3468 15 6 •7896 •503 1 -2115 '9^S^ -6147 •3109 6 7 •7455 •4607 •1709 -8761 •5774 •2752 7 8 •7051 -4220 •^337 -8406 •5434 •2428 8 9 •6722 -3906 •1038 -8120 ':^^^'^ -2168 9 20 •6480 -.3676 -0821 •7916 -4969 •1987 20 I •6282 •3492 •0648 •7754 •4819 -1847 I 2 •6109 •?,i^^ -0500 •7617 •4^9> •1731 2 .3 •5927 •3162 •0344 ■7474 -456 1 -1610 3 4 •5709 •2959 '"^^55 ■7299 •4399 •1460 4 ^.-J '5445 -2712 10-9925 •7083 •4198 -1274 25 6 •5146 •2432 -9662 -6837 -3968 •1058 6 7 •4815 -2121 •9370 -6564 -3712 •0819 7 8 •447^ •1798 -9067 -6280 •3447 •0571 8 9 •41 1 7 •^465 •8755 •5989 •3175 -0318 9 .3° •3750 *II2I •8433 •5688 •2894 •0056 30 I •3370 •0767 -8102 •5379 •2607 9-9790 I 2 •2965 -0388 •7749 •5049 •2300 •9505 2 3 •2529 10-9980 •7368 •4695 -1969 -9197 3 4 •2058 ■9539 '^^955 •4310 -1612 -8864 4 35 -^^5^ •9065 -6512 •3896 •1226 •8507 :i$ 6 •:oi3 •8559 -6040 •3455 •081^ -8 1 24 6 7 •0443 •8026 •5540 •2990 -0382 •7721 7 8 10-9842 ■7462 •5014 -2498 9"9925 •7297 8 , 9 •9199 •6860 •4451 •1972 •9435 •6843 9 40 •8508 •621 I •3844 •1406 -8907 -6352 40 J •7753 'S:)°^ •3178 •0784 •8327 •58 1 1 I 1 ^ •6921 ■4719 •2443 •0094 •7681 •5209 2 3 •6014 •3862 •1637 9'9337 •6972 •4545 3 4 •5039 •2941 -0767 •8519 •6204 •3826 4 45 •3993 •^951 9*9833 •7638 '^?>1(^ •3049 45 6 •2891 -0907 •8846 •6708 •4500 •2228 6 7 •1740 9-9816 -78x4 ■5734 •3584 -1366 7 8 •0534 -8671 -6730 -4710 -26[9 -0461 8 9 9-926.3 •74'^3 •S5^5 •3627 •^598 8-9500 9 50 •7923 •6188 •4374 •2481 •0516 •8480 50 I •6501 •4833 •3086 -1259 8-9359 •7389 I 2 •3386 •1707 8-9949 -8118 •6215 2 3 15 -4205 -0246 •8558 -6796 •4963 3 4 •2G07 15-0779 •7091 •5401 -3639 4 [i9.M Arc of elder Life:- .loint Life Annuity, at Top. Last Survivor Annuitv. at 8tde. 181 3J PERCENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. lO 58 59 60 61 62 63 10-1263 9-8083 9-4891 9-1694 8-8^-02 8-5324 10 I •1266 •8093 ■4905 -17J4 •8527 ■s?>s?> I 2 •I 162 -7998 •4819 •1637 •8459 •5293 2 .3 •0965 •7814 •4647 ■1477 -8309 •s-^ss 3 4 •0697 •7561 •4409 •1251 •8098 •4956 4 15 •0380 •7258 •4122 •0980 •7841 -47H 15 6 •00,57 •6932 -381 r -0685 ■7561 •4449 6 7 9-9697 •6607 •3502 -0390 -7282 -4183 7 8 •9.5 S 7 •Cr->i?, •3221 -0124 •7027 •3943 8 9 •9141 •6079 •3000 8-9914 -6829 •3754 9 20 -8969 •3918 •2849 •9772 -6696 •3630 20 I -8840 •5797 •2737 -9669 •6601 ■3542 I 2 ■8733 •5700 •2648 •9588 •6528 •3476 2 .3 •8623 •5599 •2557 •9505 •6453 •3409 3 4 •8485 ■5472 •2440 •9398 -^^ss •3320 4 25 •8311 '32-^^ •2291 •9261 •6229 •3203 25 6 •8112 -s^^s •2119 -9100 •6080 •3066 6 7 -7888 •4917 •1924 •8920 -5912 •2910 7 8 •7656 -4701 •1724 •8733 ■5740 -2750 8 9 •7420 •4482 •15 19 •8544 •5564 •2388 9 30 •7178 •4256 •13 1 1 •8351 •5385 •2423 30 I •6930 •4027 •1099 •8155 •5205 •2237 I 2 -666.^ •3783 •0873 •7947 ■5013 •2080 2 ,3 •6381 ':^s^9 ■0629 •7722 -4806 •1890 3 4 -6072 T-?>?> ■03^5 •7478 •4581 •1683 4 .35 •5739 •2924 •0080 ■7215 •4339 -1460 r 6 •5385 •2596 8-9776 •6934 •4080 •1222 6 7 •iJOI I •22'5I •9457 •6640 •3810 -0974 7 8 •4618 •1887 -9121 •(^r-,^ •3526 •0714 8 9 •4196 ■^497 •8762 •6001 •3222 -0435 9 40 •3742 •X076 •8373 •5<543 •2893 •0134 40 I ■r-:^9 -061 1 •7943 -3246 ■2528 7-9798 I 2 -2678 ■0090 •7460 ■4799 •21 l6 •9419 2 .3 -2059 8-9513 -6924 •4302 •1636 •8994 3 -^ •1387 •8887 -6341 •376^ -^^^ss •8531 4 45 •0660 •8208 •5709 •3173 •0610 •8026 45 6 8-9891 •7490 •5039 •255^ -003 1 •749^ 6 / •9084 -6736 ■4337 -1897 7 '9425 -6929 7 8 •8234 •5941 '?>59^^ •1208 •8785 •^^?,Ti 8 9 •7333 •5097 -2808 •0475 -8103 •5706 9 50 •6376 •4201 •1970 7*9693 •7378 ■5033 5° I ■5349 T-ri •J 068 •88p -6394 •4304 I 2 •4242 •2196 -0092 •7938 •5741 •?>S'^^ 2 .3 •3059 •1082 7 '9045 -^9 SI -4824 •2635 3 4 •1806 7-9899 •7932 •5912 •3846 •1742 4 [196] Age of elder Life : \ Joint Life Annuity, at Top. / Last Survivor Annuity, at Side, 182 HK. 3J PER-CENT. Values of Anmndea on Tico Joint Lives, and on the Last Survifnr of Tiro Lives. 64 65 66 67 68 69 lO 8-2168 7-9025 7^5885 7-2739 6-9582 6-6393 10 i •2202 •9062 •59^5 •2782 •9628 •6441 I 2 •2149 •9017 ■5887 •2751 ■9603 -6422 2 3 •2022 -8900 •5780 ■26 H ■9515 •6343 3 4 •1836 •8726 •5619 •2.504 •9377 -6216 4 15 •1608 •8512 •.5417 •2316 •9201 •60152 15 6 •^35 7 •827.5 •5194 •2105 -9004 •5868 6 7 •I 101; ■8036 -4968 •1893 -8803 -5680 7 8 •0876 -7820 -4764 -1699 -8621 ■.'5509 8 9 •0699 -7652 ■4606 •^55^ -8482 •5378 9 20 ■0.583 •7.544 •4504 •i45<^ •8394 -5296 20 I •0502 •7469 •4436 •J 394 •8337 ■5244 1 2 ■0442 •7416 -4388 '^35^ -8299 -5211 2 3 •0382 •7362 "4341 •1309 •8262 •5179 3 4 •0301 •7289 •4275 •1250 •8209 '5^3^ 4 25 •0194 -7191 •4185 -1168 •8135 ■5065 ^5 6 •0067 •7074 •4077 -1069 -8044 -4982 6 7 7-9923 -6940 •3954 -09^6 -7940 -4886 7 8 ■9774 •6803 -3828 •0839 •7833 •4788 8 9 •9625 -6665 •3701 •0723 -7726 •4690 9 30 •9473 •6525 •3572 -060:; -7618 ■4592 30 I -9320 -6386 •3444 •0488 •7512 ■4495 I 2 -9158 •6237 •3308 •0364 •7399 ■4393 2 3 •8983 -6077 •3162 -0230 •7277 -4282 3 4 •8793 •5902 •3002 -0084 •7144 -4 16 1 4 35 •8588 •.5714 •2830 6-9927 -;ooi ■403 1 35 6 •8370 •.55 H •2647 -9760 -6849 ■3893 6 7 -8142 •5306 •2457 •9587 -6692 ■375^ 7 8 •7904 -r,o89 •2259 •9407 -6529 •3604 8 9 •7650 •485'^ -2048 -9216 ''"'355 ■3447 9 40 ■7374 •4605 •1819 -9008 •6168 -3278 40 I •7066 •4324 •I 562 •8775 ■.5957 •3087 I 2 •6717 -4004 -1270 -8.^08 •57 H -2S67 2 3 -6326 •3644 •0940 -8206 ■5439 -2617 3 4 •5899 •3251 •0579 •7876 •5136 -2342 4 45 '.5433 •2821 •0184 •75^3 -4805 •2039 45 6 •4938 •2365 6-9764 •7128 •4453 •17J9 6 7 •4420 •1887 •9325 •6726 -4085 ■1383 7 8 •3872 •1383 -8862 -6301 ■3697 •1030 8 9 •3289 •0845 -8367 •5848 •3283 ■0654 9 50 -2666 ■0270 •78.39 '53(^3 -2840 •0250 50 I •1991 69647 •7264 •4836 ■2357 5-9810 I 2 •1255 •8965 •663S •4257 •1826 •9325 2 3 ■0459 •8227 •5953 -3628 -1248 •8795 3 4 6-9608 •7437 •5220 •2952 -0626 -8225 4 [107] Age of cider Life: \ Joint Life Annuity, at Top. i Lxst Survivor Annuity, at Side. ina 3i PERCENT. ]'ahies of Annuities on Tiro Joint Lives, and on the Last Surcivor (if Two Lives. 70 71 72 13 74 75 lO 6\52o6 6*0049 S-6g66 5-3998 5-1185 4-8528 10 I ■3255 •0100 •7017 •4050 -1237 •8580 I 2 •3241 •0091 •7012 •4049 •1240 -8586 2 3 •317^ •0028 •6957 •4001 •1198 -8549 3 4 •3054 5-9921 •6860 •39^3 •iriS -8478 4 15 •290.3 •9782 ■6732 -3794 •1009 -8378 ^5 6 •2/3^ •9622 •6583 ■3657 •0882 •8260 6 7 ■2556 •9458 •6430 '35^'^ •0749 •8136 7 8 •2395 •9308 •6290 •3383 •0627 •8022 8 9 •2272 •9 '93 •6183 •3283 •0533 •7934 9 20 •2197 •9123 •6118 -3223 •0478 •7883 20 I •2149 •9080 •6079 •3188 •0446 •7854 I 2 •2120 '9°^:^ •6057 •3169 •0429 •7839 2 3 •2092 •9031 •6037 -3151 •0415 •7827 3 4 •2051 •8994 •6004 •3123 •0390 •7806 4 25 •1990 •8939 -39^S •3079 -03S'' •7771 25 6 '^9K^ •8871 •5893 •3023 •0300 •7725 6 7 •1827 •8791 •5821 •2957 •0240 •7670 7 8 •1738 •8709 •5746 •2889 •0178 •7614 8 9 •1648 •8628 ■5672 •2822 •0117 •7558 9 30 •^559 •8547 •5599 -2755 •0056 •7503 30 I •1471 •8467 •5527 •2690 4-9998 •7451 I 2 •1378 •8384 •5451 •2622 •9936 -7395 2 3 •1278 •8293 •5369 -2548 ■9869 •7335 3 4 •1169 •8194 •5279 •2467 ■9796 •7269 4 35 •1050 •8087 •5x83 •2380 •9717 •7197 3S 6 •0926 •7974 •5081 •2288 ■9^34 •7122 6 7 •0797 •7858 ■4976 •2194 -9549 •7046 7 8 •0664 1139 •4870 •2098 •9464 •6969 8 9 •05 H •7613 ■4756 •1997 •9373 •6888 9 40 •0372 lATi •4635 •1888 •9277 •6802 40 I •0200 •7322 •4497 ■1765 •9166 •6703 I 2 •0001 •7143 '4335 •1620 •9036 •6586 2 3 5-9775 •6938 •415^ •1453 •8885 -6451 3 4 •9525 •6712 •3946 •1268 •8718 •6300 4 45 •9249 •6462 •3719 •1063 •8532 •6132 45 6 •8957 •6197 •3479 •0845 •8335 •5954 6 7 •8653 •5921 •3229 •0619 •8131 •5769 7 8 •8332 •5631 •2967 •0382 •7918 -5576 8 9 •7990 '53^^ •2687 ■0130 •7690 -5371 9 50 •7624 •4989 •2387 4-9860 •7446 •5152 5° 1 •7224 •4627 •2060 •9564 •7179 •49 1 1 I 2 •6782 ■4225 •1696 •9234 •6881 •4641 2 3 •6299 •3786 •1297 •8873 •^5S3 •4345 3 4 ■5778 To'i^ •0865 -8481 •6x98 •4023 4 [198] Age of elder Life : Joint Life Annuity, at Top. Last Survivor Annuity, at Side. 1«4 HM. 3\ I'I:K CKNT. Values of .limitid's <>n Tiro Jnint Livis^ iiwl on the La.st Suroivor of Two Lives. 76 77 1 78 79 80 81 lO 4-5929 4-3406 4-0946 3 8524 3-6173 3-3941 10 I •5980 •3457 •099:) 8572 -6219 -3986 I 2 •5988 •3467 -J 008 8586 -6234 •4002 2 3 •5958 •3441 •0986 8*568 -6220 •3990 3 4 •5893 •3384 ■0935 8523 •6180 •3955 4 15 •5802 '3301 •0859 8455 -6118 •3900 ^1 6 •5<593 •3201 ■0768 8.371 -6042 •3830 6 7 •5578 ■3094 -0669 8281 •5959 •3754 7 8 •5472 •2995 ■0577 8195 -5880 -3681 8 9 •5390 •2919 -0506 1 8129 •5819 -3624 9 20 •5342 •2875 •0465 8092 •5784 •3592 20 I '53^5 •2850 •0443 : 8071 •5766 •3576 I 2 •5303 •2840 •0434 , 8064 •5759 •3570 2 3 •5294 •2832 •0428 , 8059 -5756 •3568 3 ' 4 •5275 •2816 •0415 ; 8048 ■5747 •3560 4 25 •5244 •2789 -0390 1 8026 •5728 •3544 25 6 •5203 •2751 •0357 7997 ■5701 •3520 6 7 •5^53 •2706 •0316 7960 -5668 •3490 7 8 •5101 •2659 •0274 1 7922 •5^i33 •3459 8 ; 9 •505 i ■2614 •0232 , 7884 -5600 •3429 9 130 •5001 •2569 -0192 7848 •5567 •3399 30 1 I ■4954 •2526 •0154 1 7814 -5536 •3372 I : 2 •4904 '2481 •0113 1 7778 -5504 ■3344 2 3 •48.50 •2433 -0070 7739 •5470 •33^2 3 4 •4790 •2379 •0021 7695 •5431 -3278 4 ' 35 •4726 •2321 3-9969 7648 •5389 •3240 35 1 6 •4658 •2260 •9914 7599 ■5344 •3200 6 7 •4589 •2198 •9859 755'' ■5300 -3 161 7 8 •4521 •2137 •9804 7501 •5-57 •3123 8 9 •4449 •2073 ■9748 7451 •5213 -3084 9 40 •4372 •2005 -9687 7398 -5166 ■3043 40 I ■4284 ■J927 -9618 7337 -5112 •2995 I 2 '4179 ■1833 '9535 7263 •5047 •2937 2 3 •4057 •1724 •9437 7^75 -4969 -2868 3 4 •3921 ■1601 •9327 7077 -4881 -2790 4 45 •37<59 •1464 -9204 ; 6966 -4782 -2701 45 6 •3609 •1319 •9073 6849 •4677 -2607 6 7 •3442 •I 1 70 •8939 6729 •4569 •2510 7 8 •3269 •1014 •8799 6604 •4^58 •2411 8 9 •3084 •0848 -8651 6471 •4340 -2306 9 50 •2887 •0671 ■8493 6330 -4214 -2194 50 I ■2670 •0477 -8318 6175 -4076 -2071 I 2 •2427 •0258 -8122 6000 -3920 -1932 2 3 •2159 •0017 •7906 5805 -3746 -1777 3 4 •1868 3-9755 -7670 5594 •3557 -1608 4 [Hl!l] . r ij T f \ .Toint Life Annuity, at Top. Aue of elder Life: — { i . <- • * " •• /cj "^ / Last Survivor Annuitv, at Side. 185 3^- PER-CENT. Values of Annuities on Two Joint Lives^ and on the Last Survivor of Two Lives. 82 10 I 2 3 4 6 7 20 I 2 3 4 6 7 8 30 I 2 3 4 35 6 7 8 9 40 I 2 3 4 45 6 83 3-1886 •1928 •J945 •1936 •1906 •1856 •1792 ■1722 ■1654 •1602 ■^572 •^557 •1.553 •^552 •^545 •1531 •1510 •1482 ■1454 •1427 •1401 •1376 •^3S^ '1323 •1292 •1258 •1223 •1187 '^^53 •1119 •1083 •1041 •0990 •0928 •0858 •0778 •0694 •0608 2-9980 3-0022 •0039 •0033 •0006 2-9961 •9903 •9838 •9776 ■9727 -9700 •9686 •9682 •9682 -9677 •9664 ■9645 •9621 -9596 •957^ •9547 -9526 '95°3 -9478 ■9450 ■9420 •9388 '935^ -9326 -9296 -9264 -9227 •9182 •9127 -9064 ■8993 -89x7 -8840 84 » -0519 •8761 9 •0426 -8678 50 -0327 -8590 1 -0217 •8492 2 ■0093 -8382 3 2-9954 -8258 4 -9803 •8123 2-8261 -S30I •8319 •8315 -8292 •8251 •8198 -8139 •8081 •8035 -8010 •7997 •7994 •7994 •7990 •7979 •7963 •7941 •7918 -7896 •7874 •7855 •7835 ■7812 ■7787 •7760 •7731 •7703 •7676 -7650 -7622 •7590 •7550 ■75°i ■7445 •7381 •7313 •7244 ■7173 •7099 -7021 •6934 •6836 •6725 -6604 85 2-6652 -6691 •6709 -6708 -6688 -6652 -6604 •6549 ■6495 •6453 -6429 -6418 •6415 -6416 -6413 -6404 -6389 ■6369 •6349 -6329 ■6310 -6292 -6274 •6254 -6232 -6207 -6181 -6156 •6133 -6109 •6085 -6057 -6022 •5979 '59^9 •5872 -5811 •5749 -5686 -5620 '555' •5475 •5387 -5289 •5182 86 2-5048 •5085 •5104 ■5104 -5088 '5°56 'SO '3 •49^3 •49^3 •4874 -4852 -4841 -4840 -4841 ■4839 •4831 -4818 ■4800 •4782 -4764 •4747 •4732 -4716 •4699 •4679 ■4657 •4634 •461 1 •4591 •4570 •455° •4526 •4495 ■4457 •4413 •4363 •4309 •4254 -4198 •4140 •4080 •4013 •3936 •3850 ■3755 2-3350 •3385 •3404 •3407 ■3394 -3367 '33^9 -3284 •3238 -3202 •3182 -3172 •3171 ■3173 •3172 ■3165 ■3 154 •3138 -3122 -3092 ■3079 ■3065 ■3050 •3032 ■3013 •2993 •2973 •2955 •2937 -2920 -2900 -2874 •2841 •2803 •2759 ■2712 -2664 •2616 -^5^5 •2455 -2389 -2314 -2232 10 I 2 3 4 15 6 7 20 I 2 3 4 25 6 7 30 J 2 3 4 35 6 9 40 45 6 5° I 2 3 4 [-200] Age of elder Life , \ Joint Life Aumiity, at Top. Last Survivor Anmiitv, at Si 186 3,V PKR-e'ENT. Values of Aniiitidi^ un Ttco >h>iiit Livis^ and uii the Last Survivor of 2 wo LlVrs. 88 89 1 90 ! 1 9' 1 1 92 93 lO 1 ^•1559 ; 1-9449 17062 I '4608 1-2092 09376 10 I •I 592 •947« •7087 •4628 •2108 •9387 I 2 ■ 1 6 1 •949,'; •7102 •4642 •2118 •939.5 2 3 •1614 •9.^00 •7108 •4648 •2124 , •9400 3 4 •1605 •9494 •7104 •4646 -2123 •9400 4 i.^ •1,582 •9476 •7092 •4<>37 •2118 •9397 15 6 •1550 •9449 •7070 •4621 -210(5 •9390 6 7 •15 10 •9416 •7043 •4600 •2091 •9379 i 8 •1469 •9380 •7013 •4.>76 •2072 ■9366 8 9 •1437 •9351 •6989 -4555 •2056 •9355 9 20 •1418 •9335 •6974 •4543 •2047 •9348 20 I •1409 ■9327 •6967 •4.537 •2042 •9344 I 2 •1408 •9326 •6966 •4537 •2041 •9343 2 3 •1410 •9328 •6969 ■4539 •2043 •9344 3 4 •1410 •9329 •6970 •4540 -2044 •9345 4 25 •J 405 •9325 •6967 •4.538 •2043 •9345 25 6 •^395 ■9317 •6961 •4534 •2040 •9343 6 7 •1382 •9306 •6952 •4527 •2035 •9340 7 8 •1368 •9294 •6943 •4519 •2029 ■9336 8 9 ■'355 •9283 •6933 •4512 •2024 •9332 9 30 •1342 •9272 •6925 •4505 •2019 •9328 30 I '^33' •9263 •6917 •4500 •2015 •9326 I 2 •1319 •9254 •6910 •4494 -201 1 •9323 2 3 •1306 •9243 •6902 •4488 •2006 •9320 3 4 •1292 •9231 •6892 •448 J •2001 •9317 4 35 •1273 •9217 •6881 •4472 •J 99.5 -9312 35 6 •I2J7 •9202 •6869 •44^3 -1988 •9307 6 7 •1240 •9188 •6857 •4453 •1981 •9303 7 8 •1224 •9175 •6847 •4445 •J974 •9298 8 9 •1210 •9163 •6837 •4438 •1969 -9294 9 40 '^m •915 1 •6828 •443 1 -1964 ■9291 40 I •1179 •9139 •6819 •4424 •1959 -9288 I 2 •^157 •9121 •6805 •4414 ••953 -9284 2 3 •1130 •9099 •678'8 •4401 •^943 -9278 3 4 •1098 •9072 •6767 •4386 •1932 •9270 4 45 •1059 •9041 •6742 •4366 •1918 •9261 45 6 •1019 •9007 •6714 •4345 •1902 -9250 6 7 •0978 •8973 •6687 •4324 •1886 ■9239 7 8 •0937 •8939 •6660 •4302 •1871 •9228 8 9 •0894 •8903 •6632 •4281 •1815 •9217 9 5° •0850 •8867 •6603 •4259 •1839 •9207 50 1 •0801 •8827 •6572 •423.'; •1822 •9195 I 2 •0744 •8781 '6535 •4208 •1802 •9182 2 3 •0681 •8729 •6494 •4176 •1778 •9x66 3 4 •o6ii •8671 •6448 •4141 •1753 •9149 4 [-201] A„o ^t ^\A^r. t f^. S Joint Life Annuity, at Top. Ace of elder Lite: — \ , . r. . , • '-. . c-j " ( Last Sumvor Annuity, at !?iQc. 187 3i PER-CEKT. Values of Annuities on Tico Joint Lives, and on the Last Survivor of Two Lives. [•20-2] lO 94 95 96 10 0-6684 o'4io7 o"i766 I •6692 •4111 •1768 I 2 •6697 •41 14 1 •1769 2 3 •6700 •4116 1 •1769 3 4 •6701 •41 \6 •1770 4 15 ■6699 •4^15 •1770 15 6 •6695 •4113 •1769 6 7 •6689 •4110 •1768 7 8 •6681 •4106 •1766 8 9 •6673 •4102 •1764 9 20 ' •6668 •4099 •1763 1 20 I 1 •6666 •4097 •176,3 I 2 •666^ •4097 •1763 2 3 ■6666 •4097 •1763 3 4 •6667 •4098 •T763 4 25 ■6666 •4098 •1763 25 6 ■6666 •4097 ■1763 6 7 ■6663 •4096 •1762 7 8 ■6661 ■4095 •1762 8 9 •6659 •4094 •1761 9 30 •66s6 •4092 ■1761 30 I •6655 •4092 •1761 I 2 •^^53 •4091 •1760 2 3 •6651 •4090 •1760 3 4 '66/Sf^ •4089 •1760 4 35 •6647 •4087 •1759 35 6 •6644 •4086 •1759 6 7 •6640 •4084 •1758 7 8 •6638 •408.3 •J 757 8 9 •66?,s •4081 •^757 9 40 '6633 •4080 ■1756 40 I ■6631 •4079 •175^ I 2 •6629 •4078 •1756 2 3 •6625 •4076 •^755 3 4 •6621 •4074 •1754 4 45 •6615 •4071 •1753 45 6 •6608 •4067 •1752 6 7 •6602 •4064 •^750 7 8 •6595 •4060 •1749 8 9 •6588 •4057 •1748 9 50 •6582 •4053 •1746 5° I ■6575 •4050 ■1745 1 2 ■6567 •4045 •1744 2 3 •6557 •4040 •1742 3 4 •6546 •4035 1 -1740 4 Ac e of e dcr Life: — ( Joint Life 1 Last Sllr^ Annuity, a ivor Annuit t Top. V, at Si 188 3^ PERCENT. Values of Annuities on Tvo Joint Lives^ and on tlir Last Survivor of Tico Lives. 10 II 12 13 H 15 ^:^ 5') 22-4413 22-3232 22-1914 220489 2r8987 217434 6 •4106 •2919 -1591 •0152 ■8634 •7065 6 7 •3812 •2617 ■1279 2r9827 ■8294 •6708 7 8 •3527 •2327 1 -0978 •9514 •7966 •6364 8 9 •3256 •2049 -0691 •9214 •7651 •6035 9 6o •2995 •1784 ■0417 •8928 •7350 •5718 60 I •2747 •1530 ■0154 ■8653 •7063 •5415 I 2 •2512 •1290 21 9905 ■8394 ■(i7si) ■5127 2 3 •2288 •1062 ■9669 ■8146 ■6529 ■4852 3 4 •2078 •0847 -9447 ■7913 ■6283 •4592 4 65 •1877 •0643 -9235 •7691 •6049 •4344 6'i 6 •1688 •0451 -9036 •7482 , ^5827 •4110 6 7 •1511 •0271 ■8849 •7285 •5619 •3888 7 8 •1343 •0100 ■8672 •7099 •5421 •3678 8 9 •1184 21-9939 ■8505 •6923 •5234 •3479 9 70 •1036 •9790 ■8351 •6760 1 ^5061 •3293 70 I ■0899 •9651 ■8207 ■6609 •4900 •3120 I 2 •0772 •9524 •8076 ■6470 •4751 •2960 2 3 •0657 •9408 ■7956 •6343 •4615 •2815 J 4 •0553 •9304 ■7848 •6229 •4493 •2683 4 75 •0458 •9209 -7750 •6126 •4381 •2562 75 6 •0371 •9123 -7662 •6031 •4280 •2452 6 7 •0291 •9043 ■7580 •5945 •4186 ■2350 7 8 •0217 ■8971 ■7505 •5866 •4101 ■2258 8 9 •0150 ■8905 •7438 •5795 ■4024 ■2173 9 80 •0088 ■8845 •7377 •5730 ' -3954 ■2097 80 I •0033 ■8791 •7322 •5673 ■3892 ■2028 I 2 21-9983 •8744 •7274 •5622 ■3836 ■1967 2 3 •9940 ■8701 •7231 ■5576 •3787 ■1913 3 4 •9900 •8663 ■7192 ■5535 ■3742 ■1864 4 85 •9865 •8629 ■7158 •5498 •3702 ■1819 85 6 •9830 •8596 ■7124 ■5463 ■3663 •1776 6 7 •9797 •8565 •7093 ■5429 ■3626 ■1734 7 8 •9764 •8534 ■7063 ■5398 ■3591 ■1695 8 9 •9729 -8503 ■7033 ■:)367 ■3557 ■1656 9 90 •9694 -8472 ■7004 ■5337 •3525 •1618 90 I •9662 -8445 i ^6978 i 5311 ■3497 ■1587 I 2 •9632 -8419 ■6956 ■5289 ■3474 ■1560 i 2 3 •9604 -8396 ■6935 ■5269 ■3453 ■1537 ' 3 4 •9581 -8376 ■6918 ' -5254 1 3437 •1520 1 4 95 •9561 -8360 •6904 ■5241 i -3425 •1507 95 6 •9548 -8349 •6895 ■5234 ■3417 i •1498 6 [173] Age of elder Life: — Joint Life Annuity, at Top. Last Survivor Annuitv, at Side. 189 3 J PEK-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. ' i6 n 18 19 20 21 55 21-5860 21-4290 2r2747 21 1256 20^9815 20-8389 55 6 •5474 •3887 •2329 0823 ■9372 •7934 6 7 •5100 •3497 •1923 0386 •8942 •7494 7 8 •4740 •3120 •1532 20 9981 •8528 •7069 8 9 1 •4394 •2759 •1155 9602 •8128 •6661 9 6o ' •4062 •2411 •0794 9238 •7744 •6268 60 I •3743 •2078 •0446 8879 •7376 •5891 I 2 •3440 ■1759 •0116 8537 •7025 •5532 2 3 •3150 •1456 20-9798 8210 •6689 •5189 3 4 •2876 •1168 •9499 7899 •6370 •4863 4 65 •2614 •0893 •9211 7602 •6065 •4552 65 6 •2366 •0632 •8938 7319 •5776 •4256 6 : 7 •2132 •0384 •8680 7051 •5501 •3975 7 8 •1908 •0149 •8433 6795 •5238 •3707 8 9 •1696 20^9924 •8197 6551 •4988 •3452 9 70 •1498 •9713 •7976 6322 •4752 •3212 7° I •1313 •9517 •7769 6107 •4532 •2987 I . 2 •1142 •9335 •7577 5907 •4327 •2778 2 3 •0985 •9168 •7401 5724 •4139 •2586 3 4 •0843 •9016 •7240 5557 •3967 ■2411 4 75 •0713 •8877 •7093 5404 •3810 •2251 75 6 •0594 •8749 •6957 5262 •3665 •2104 6 7 •0483 •8630 •6831 5130 •3529 •1966 7 8 •0382 •8521 •6715 5009 •3405 •1839 8 9 •0290 •8420 -6608 4897 •3289 •1722 9 ; 80 •0206 •8329 •6510 4794 •3184 •1G14 80 1 I •0131 •8247 ■6422 4702 •3089 •1517 I ! 2 •0064 •8174 •6344 4619 •3004 •1431 2 3 •0004 •8109 •6273 4545 •2927 •1353 3 4 20^9950 •8049 •6209 4478 •2858 •1283 4 85 •9900 •7995 •6151 4416 •2795 •1218 85 6 ■9852 •7942 -6094 4356 •2733 •1156 6 7 •9805 -7890 -6038 4297 •2672 ■1094 7 8 •9760 •7840 •5983 4238 •2612 •1033 8 9 •9716 •7789 •5927 4179 •2550 •0970 9 90 •9673 •7740 •5872 4119 •2489 •0908 90 I •9636 •7697 •5823 •4067 •2434 •0852 I 2 •9605 •7660 •5781 •4020 •2384 •0801 2 3 •9577 ■7628 •5743 •3977 •2339 •0755 3 4 ■9557 •7603 •5713 •3944 •2304 •0718 4 I95 •9542 •7585 •5691 •3918 •2276 •0690 95 6 ■9532 •7573 •5677 •3902 •2258 •0670 6 [174] Age of elder Life: — ( Joint Life Annuit\-, at Top. ( Last Siuvivor Annuity, at Side. 190 3J PERCENT. Values of Annuities on Two Joint Livrs^ and on the Last Survivor of Two Lives. 55 22 ^3 24 25 26 27 20G955 20548G 203905 ! 20^2388 200763 ' 19 9098 55 6 •0489 •5009 •3475 •1883 •0242 •8560 6 7 •no:?'.) •4548 1 •3002 •1395 199740 •8041 7 8 •;■)(;(».■) •4103 •2545 ■09 20 •9254 •7540 8 9 •r.lsT •3070 1 •2107 i 0474 •8790 •7000 9 6o •47SG •3265 •1080 i ^0042 ■8343 •0600 60 1 •4401 •2872 •1283 19^9027 ■7917 •6159 I 2 •4034 •2497 •0899 •9232 •7510 •5740 2 . 3 •3(384 •2139 •0532 ' ^8856 •7122 1 -5340 3 1 -^ •3:3r)2 •1800 •0185 •8499 •6755 ' ^4961 4 ' 6=; •3034 •1476 19^9853 •8158 •6404 •4600 65 6 ■2733 •1168 •9538 •7835 •6072 •4257 6 7 •244(3 •0877 •9240 •7529 •5757 •3932 7 8 •2174 •0599 •8950 •7237 •5457 •3623 8 9 •1914 •0334 •8085 •6959 •5171 •3329 9 7° •1070 •0080 •8431 •6099 •4903 •3053 70 I •1441 19-9853 •8194 •6456 •4053 •2795 I 2 •1229 •9(337 •7974 •6230 •4421 •2555 2 3 •1034 •9440 •7772 •6023 •4208 •2336 3 4 ■0857 •9259 •7588 •5834 •4014 •2136 4 7.^ •0095 •9095 •7420 •5662 •3837 •1954 75 6 •0545 •8942 •7205 •5503 •3073 •1785 6 7 •0405 •8801 •7121 •5355 •3522 •1629 7 8 •0277 •8071 •6988 •5220 •3382 •1485 8 9 •0158 •8551 •6866 •5095 •3253 •1352 9 So •0050 •8441 •0754 •4980 •3136 •1231 80 I 19^9952 •8342 •0054 •4877 •3030 •1122 I 2 •9804 •8253 •0504 •4785 •2935 •1025 2 3 •!)780 •8174 •6483 •4703 •2851 •0937 3 4 •9715 •8103 •6411 •4629 •2774 •0858 4 85 •9050 •8037 •6344 •4560 •2704 •0786 85 6 •9580 •7973 •6279 •4494 •2030 •0716 6 7 •9524 •7910 •0215 •4429 •2569 •0647 7 8 •9403 •7849 •0153 •4305 •2504 •0579 8 9 •9400 •7786 •0089 •4300 1 •2437 •0510 9 90 •9338 ■7723 •0020 •4230 •2371 •0442 90 I •9281 ■7007 •5970 •4179 •2312 •0381 I 2 •9231 ■7017 •5920 •4128 : •2260 •0327 3 •9185 ■7572 •5875 •4082 1 •2213 •0278 3 4 •9148 ■7535 •5838 •4046 i 1 •2175 1 •0240 4 9.5 •9119 •7507 •5810 •4017 ; •2147 •0210 95 6 ■9099 •7487 •5791 •3998 i •2127, •0190 6 [175] Age of elder Life:- \ J^i^t ^ife Annuity, at Top "• ■" * I Lnst Survivor Annuity, at Side. 191 Sh PEll-CENT. Valued of Annidtits on Tico Joint Lives, and on the Last Survivor of Two Lives. 28 29 30 31 [ 32 33 55 19-7407 19-5692 19-3953 19 2191 19 0398 18 8573 55 6 ■G850 •5116 •3357 j 1573 18 9757 7909 6 7 •6314 •4561 •2783 0978 9140 7269 7 8 •5797 •4027 •2229 0406 8548 6653 8 9 •5301 •3514 •1700 18 9858; 7979 6064 9 \ 6o •4825 •3024 •1192 9333 7436 5501 60 1 •4371 •2554 •0707 8832 • 6917 4963 I 2 •3937 •2107 •0246 8355 6424 4452 2 3 •3525 •1681 18-9806 7901 5955 3966 3 4 •3135 •1278 -9390 7472 5511 3507 4 65 •2762 •0894 -8994 7062 5088 3069 65 6 •2408 •0529 •8618 6675 4688 2655 6 7 •2074 •0184 •8262 6308 4309 2264 7 8 •1755 18-9856 •7924 5959 3949 ' 1892 8 9 •1452 •9544 •7602 5628 3607 1539 9 70 •1167 •9251 •7300 5317 3287 1208 70 I •0902 •8977 •7018 5027 2987 : 0899 I 2 •0655 •8723 •6756 4757 2710 1 0613 2 3 •0429 •8490 •6517 4511 2456 i 0351 3 4 •0223 •8278 •6299 4286 2225 0113 4 75 •0035 •8085 •6100 4081 2014 i 17 9895 75 6 18^9862 •7906 •5916 3892 1819 9694 6 7 •9701 •7740 •5745 3717 1639 9508 7 8 •9552 •7588 •5588 3555 1473 9337 8 9 •9415 •7447 •5443 3406 1319 i 9179 9 80 •9291 •7318 •5311 3271 1180 ; 9035 80 I •9178 •7202 •5192 3148 1053 8906 I 2 •9078 •7099 •5085 3039 0941 8790 2 3 •8987 •7006 •4990 2940 0840 : 8686 3 4 •8906 •6922 •4904 2852 0749 8593 4 85 •8831 •6845 •4824 2771 0666 ': 8507 85 6 •8759 •6771 •4748 2692 0585 '. 8423 6 7 ■8688 •6697 •4672 2614 0505 8341 7 8 •8618 •6625 •4598 2538 0427 8261 8 9 •8547 •6552 •4523 2461 0347 8179 9 90 •8476 •6480 •4448 2385 0269 8098 90 I •8414 •6415 •4382 2316 0199 8026 I 2 •8358 •6357 •4322 2255 0136 7962 2 3 •8307 •6305 -4269 2200 0080 7904 3 4 •8267 •6263 •4226 2156 0035 1 7858 4 95 •8236 •6231 •4193 2122 0000 7822 95 6 •8215 •6210 •4170 2099 17 9977 , 7798 6 [176] Age of elder Life : Joint Life Annuitj', at Top. Last Surrivor Annuity, at Side. 192 3 J PERCENT. Vafucs of A/i unifies on Two Joint Livs, (Did on the JAist Sio'vivor of Two Livc.'i. 34 35 36 37 38 39 as 18 6717 18 4834 18-2925 18^0998 17-9053 17-7089 55 6 6025 1 4114 •2175 -0216 -8236 -6236 6 7 5361 ! 3421 •1454 17-9465 -7452 -5416 7 8 4721 ' 2757 •0761 -8743 •6699 •4631 8 9 4109 ; 2121 ■0099 ■8052 -5979 •3879 9 6o 3524 1512 17-9466 ■7393 •5292 •3161 60 I 2966 0932 •8863 ■6765 •4637 •2477 I 2 2436 0381 •8290 •6168 •4015 •1829 2 3 1932 17 9858 •7746 ■5602 •3425 •1214 3 4 1456 9364 •7232 ■5068 ■2869 •0633 4 6S 1003 8894 •6744 ■4560 ■2340 •0083 65 6 0574 8449 -6282 ■4080 •1841 16^9562 6 7 01G9 8029 ■5846 ■3627 •1370 •9071 7 1 8 17 9784 7630 •5432 ■3197 •0923 •8607 8 1 9 9419 7252 •5040 -2790 •0500 •8167 9 ;o 9076 6898 •4672 ■2409 •0105 •7755 70 1 8757 6567 •4330 -2054 169736 •7372 I 2 8462 6261 •4013 •1726 •9395 •7019 2 3 8191 5981 •3723 •1425 •9084 •6695 3 4 7945 5727 •3460 ■1153 •8801 •6402 4 75 7720 5495 •3220 •0904 •8544 •6135 75 6 7513 5280 -2998 •0675 •8306 •5888 6 7 7321 5082 •2793 ■0463 ■8087 •5661 •7 8 7145 4900 •2605 ■0268 ■7886 •5452 8 , 9 6982 4732 •2431 ■0088 •7700 ■5260 9 : 8o 6833 4578 •2273 169925 •7531 •5085 80 I 6699 4440 -2130 ■9777 •7378 •4927 1 I 2 6580 4317 •2002 ■9646 -7243 •4787 2 3 6473 4206 •1888 ■9528 •7121 •4661 3 -^ 6377 4107 •1786 •9422 •7012 •4548 4 1 85 6288 4016 •1692 •9325 •6911 •4445 85 6 6202 3927 -1600 •9231 •6814 •4345 6 7 6118 3840 -1510 •9138 •6719 •4247 7' 8 6034 3754 •1422 •9047 •6626 •4150 8 9 5950 3667 •1332 •8954 •6530 •4052 9 ' 90 5867 3581 •1243 •8863 •6436 •3956 90 I 5792 3504 •1163 •8781 •6352 •3869 I 2 5726 3435 •1092 •8707 •6277 •3792 2 3 5666 3374 -1029 •8641 •6209 •3723 3 4 5619 3324 •0977 •8589 •6154 •3667 4 , 95 5582 3287 •0938 •8548 •6112 •3624 95 i 6 5557 3261 •0911 •8520 •6084 -3594 6 i 1 [177J Age of elder Life : — Joint Life Annuity, at Top. Last Survivor Annuitj-, at Side. 193 3 J PER-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. 1 1 40 41 42 43 44 45 55 IT-olOo 17 3098 17-1070 16-9024 16 6973 16-4919 55 6 •42U 2165 •0093 •7999 5898 •3791 6 7 •3357 1269 16-9153 •7014 4864 •2706 7 8 •2535 0409 •8252 •6068 3871 •1663 8 9 •1750 16 9586 •7389 •5163 2920 •0664 9 6o •1000 8801 •6566 •4299 2013 15-9710 60 I •0285 8053 •5782 •3476 1148 •8801 I 2 16-9608 7344 •5038 •2695 0327 •7937 2 3 •8965 6672 •4333 •1955 15 9549 •7119 3 4 •8359 6038 •3669 •1257 8815 •6346 4 65 •7784 5436 •3038 •0595 8119 •5614 65 6 •7241 4869 •2443 15-9970 7462 •4922 6 7 •6729 4333 •1882 -9381 6842 •4270 7 8 •6244 3826 •1351 -8823 6257 •3653 8 9 •5786 3348 •0850 -8297 5703 -3071 9 7° •5357 2900 -0381 •7804 5185 •2526 70 I •4958 2484 15-9945 •7347 4704 •2019 I 3 •4590 2099 •9543 •6924 4260 •1552 2 3 •425 i 1748 •9175 •6539 3855 •1125 3 4 •3948 1430 •8842 •6190 3488 •0739 4 75 •3671 1141 •8540 •5872 3154 •0387 75 6 •3415 0874 •8261 •5580 2847 •0064 6 7 •3179 0628 •8004 •5310 2564 14-9766 7 8 •2963 0403 •7768 •5063 2304 •9492 8 9 •2763 0195 •7551 •4836 2065 •9241 9 So •2582 0007 •7354 •4629 1848 •9012 80 I •2418 15 9837 •7177 •4443 1652 •8806 I 2 •2273 1 9686 •7019 •4278 1479 •8624 2 3 •2143 9551 •6878 •4130 1324 •8460 3 4 •2026 9429 •6751 •3997 1184 •8313 4 85 •1919 9318 •6635 •3875 1056 •8178 85 6 •1815 9210 •6523 •3758 0933 •8048 6 7 •1714 9105 •6413 •3643 0812 •7921 7 8 •1615 9002 •6306 •3530 0693 •7797 8 9 •1514 8897 •6197 •3416 0574 •7670 9 90 •1415 8795 -6091 •3305 0457 •7547 90 I •1326 8704 •5996 •3206 0352 •7437 I 2 •1247 8623 -5911 •3118 0260 •7339 2 3 •1176 8550 •5836 •3039 0178 •7252 3 4 •1119 8492 •5776 •2977 0112 •7183 4 95 •1075 8447 •5730 •2929 0062 •7130 95 6 •1045 8416 •5698 •2896 0028 •7094 6 [17.1] Age of elder Life:— I Joint Life Annuity, at Top. Last Survivor Annuity, at Side. 2 c 194 S\ PEK-CENT. Values of Annutlics on Two Joint Livcs^ and on the Last Survivor of Two Livi's. 1 46 , 1 47 48 49 50 51 55 16-2875 IG 0848 15 8839 15 6848 15^4878 152928 55 6 •1G93 15 9608 •7540 5487 •3453 •1438 6 7 •0553 8414 ■6286 4173 •2077 149996 / 8 15-9458 72G4 •5081 2908 •0749 •8604 8 9 •8408 6161 •3923 1693 149473 •7265 9 6o •740G 5107 •2815 0529 •8251 •5981 60 I •G449 4102 •1758 14 9417 •7083 •4753 I 2 •5542 3147 ■0754 8360 •5971 •3583 2 3 •4680 2241 14^9800 7357 •4914 •2471 3 4 •3867 1384 •8899 6408 •3915 •1418 4 65 •3096 0573 •8044 5508 •2967 •0418 65 6 •23G8 14 980G •7236 4657 •2069 139472 6 7 •1G81 9082 •6474 3853 •1222 •8577 7 8 •1031 8398 •5753 3093 •0420 •7731 8 9 •0417 7752 •5072 2374 139662 •6930 9 7° 14-9844 7147 •4435 1703 •8953 •G181 70 I •9310 6585 •3842 1078 •8294 •5484 1 2 •8818 6067 •3296 0502 •7686 •4841 2 .3 •8369 5594 •2798 13 9976 •7130 •4254 3 4 •7962 5165 •2345 9499 •6627 •3722 4 7.5 •7591 4775 •1935 9066 •6169 •3238 75 6 •7250 i 4416 •1556 8667 •5748 •2793 6 7 •6937 j 4085 •1208 8300 •5361 •2383 7 8 •6649 3782 •0889 7963 •5005 •2008 8 9 •6384 ' 3503 •0595 7654 •4679 •1662 9 8o •6143 3250 •0328 7372 ■4382 •1348 80 I •5926 1 3022 •0088 7119 •4115 ■1066 I 2 •5734 2819 13-9875 6894 •3877 •0815 2 3 •5562 2638 •9684 6693 •3665 •0591 3 4 •5407 2475 •9513 i 6513 •3475 •0390 4 85 •5265 2326 •9356 6348 •3301 •0205 85 6 •5128 2182 •9205 1 6189 •3133 •0028 6 7 •4994 2041 •9056 6033 •2969 12^9855 7 8 •4863 1903 •8911 5880 •2808 •9685 8 9 •4730 1763 •8764 5726 •2646 •9514 9 9° •4601 1627 ■8621 5575 •2488 •9347 90 I •4484 1504 •8493 5440 •2346 •9198 I 2 ■4381 ' 1396 •8378 5320 •2220 •9065 2 3 ■4289 1299 •8277 5214 •2108 •8948 3 4 •4216 1221 •8195 5128 •2018 •8853 4 95 •4160 ' 1162 •8133 5062 •1950 •8781 95 6 ■4121 1122 •8090 1 5017 •1903 •8732 6 [170] . r ij T T \ Joint Life Annuity, at fop. Arc of elder Life: — ', . c \ ' ■. . c-j ° ( Last Survivor Annuity, at iMde. 195 3J PER-CENT. Values of Annuities on Two Joint Lives, and on the LoM Survivor of Tiro Lives. b'^ 53 54 SS 5^ 57 ''{ 15-1001 14-9103 14-7243 •5543 14-3651 ' •1928 8-3929 8*2240 5S 6 6 \ i4-i'4i:i •7475 14-0007 •0771 7 ; -7933 8 -CAll •5896 •3893 1 7 •4368 ■2294 1 •0256 ; 13-8261 13-6314 8 9 1 -5069 •2894 •0750 13-8640 ■6571 •4548 9 6o i -3720 •1478 13-9264 •7083 ■4940 •2843 60 I 1 -2429 •0121 -7839 •5587 ■3372 •1201 I 2 j -1199 13-8827 •6478 •4157 ■1871 12-9626 2 .3 ' -0027 •7594 •5180 •2792 ' ■0436 •8120 3 4 13-8917 •6424 •3948 •1494 ; 12-9071 •6684 4 1 65 -7863 •5312 •2775 •0258 ■7769 •5314 65 6 -6804 •4257 •1663 12-9084 '■ ■6531 •4010 6 7 •5919 •3259 •0608 •7971 j ■5356 •2770 7 8 •5025 •2314 12^9609 •6914 ! ■4239 •1591 8 9 •4178 •1419 •8662 •5912 1 ■3180 ■0471 9 70 •3386 •0580 •7774 •4972 j •2185 11^9418 70 I •2649 12-9799 •6947 •4096 1 •1256 •8435 I 2 •1968 -9078 •6183 •3286 •0398 •7525 2 3 •1347 •8419 •5484 •2545 11-9612 •6691 3 4 •0783 -7822 •4850 •1871 •8897 ■5932 4 75 •0271 •7278 •4273 •1259 -8247 ■5241 75 6 12-9799 •6778 ■3742 •0694 •7647 •4603 6 7 •9305 •6317 •3252 •0175 •7094 •4016 7 8 •8967 •5894 •2803 11-9697 •6586 •3476 8 9 •8600 •5506 •2390 •9258 •6119 •2978 9 80 •8267 •5152 •2014 •8858 •5693 •2524 80 I •7968 •4834 •1676 •8498 •5310 •2116 I 2 •7702 •4552 •1376 •8178 •4969 •1753 2 3 •7464 •4299 •1107 •7892 •4664 •1428 3 4 •7251 •4073 •0867 •7636 •4391 •1136 4 85 •7056 •3865 •0645 •7400 •4139 •0868 85 6 •6868 •3665 •0433 •7174 •3897 •0610 6 7 •6684 •3470 •0225 •6952 •3661 •0357 7 8 •6505 •3279 ■0022 •6735 i ^3429 •0110 8 9 •6323 •3086 11^9817 •6516 1 -3196 10-9860 9 90 •6147 •2899 •9618 •6304 1 ^2969 •9618 90 I •5988 •2731 •9439 ■6114 i ^2766 •9400 I 2 •5848 •2583 •9281 ■5945 1 ^2586 •9208 2 3 •5724 •2451 •9141 •5796 i ^242 7 •9038 3 4 •5624 •2345 •9029 •5676 ' ^2299 •8901 4 95 6 ■5549 •2265 •8943 •5585 •2202 •8797 95 •5496 •2209 •8884 •5523 1 ^2135 •8726 6 [180] Age of elder Life : — Joint Life Annuity, t\t Top. Last Survivor Anniuty, at Side. 196 3\ I'KK-CKN'i'. I'alues of Annuities on 7\co Joint Lives^ and on the Last Survivor of Tiro Lives. 55 58 i 59 1 60 61 62 63 8-0480 7-864.5 ' 7-6749 7^48oo 7-2803 7-0766 55 6 7-9085 -73M ■55°^ •3625 •1699 6-9732 6 7 •7620 •5935 , •4187 •2,584 •0532 •8636 ; 7 8 •4476 •2S05 •1077 6-9300 -7478 8 9 13-2575 '^35° 6-9700 -7999 •6252 9 Co •0793 12^8797 •8261 -6637 •4968 60 I 1-2-9070 •7002 ' 12-4988 •5221 •3629 I 2 •7420 •5270 •3185 12^1156 •2240 2 3 •58-16 •3021 •1452 ir9340 ir7308 3 4 •1338 •2039 11^9794 •7011 •5496 ir3451 4 6^ •2898 •0525 •8206 •5947 •3755 •1632 65 '6 •1524 11^9080 •0088 •4354 •2080 10-9886 6 7 •0217 •7703 •5239 •2831 •0487 •8211 7 8 11-8973 •0391 •3857 •1377 10^8959 •6607 8 9 •7789 •5142 •2539 10-9988 •7497 •5071 9 7° •0070 •3964 •1295 •8070 •6115 •3616 70 I •5035 -2803 •0131 •7440 •4810 •2248 I 2 •4071 -1841 10-9049 •3303 •3609 •0974 2 3 •3780 •090 1 •8050 •5251 •2498 9-9800 3 4 •2981 •0050 •7151 •4292 •1482 •8727 4 75 •2248 10-9271 •6325 •3416 •0555 •7746 75 6 •1570 •8551 •5559 •2004 9-9694 •6834 6 7 -0945 •7886 •4853 •1854 •8897 •5990 7- 8 -0370 •7274 •4201 •1101 •8162 •5210 8 9 10-9840 •0709 •3000 •0522 •7482 •4487 9 8o •9357 •0194 •3051 9^9937 •6860 •3826 80 I •8921 •5730 •2557 •9410 •6299 •3229 I 2 •8535 •5317 •2116 •8940 •5799 •2697 2 3 •8188 •4947 •1721 •8519 •5350 •2218 3 4 •7870 •4015 •1367 •8141 •4947 •1789 4 85 •7590 •4309 •1040 •7792 •4575 •1393 85 6 •7314 •4014 •0725 •7450 •4216 •1011 6 7 •7043 •3725 •0415 •7125 •3863 •0633 7 8 •0779 •3441 •0112 •6800 •3515 •0263 8 9 •0511 •3154 9^9804 •6470 •3162 8^9885 9 90 •6252 •2870 •9505 •6149 •2818 •9517 90 I •0019 •2625 •9236 •5860 •2508 •9185 I 2 •5813 •2404 •8997 •5603 •2232 •8889 2 3 ■5031 •2208 •8787 •5370 •1987 •8020 3 4 •5484 •2051 •8017 •5194 •1790 -8413 4 95 •5373 •1931 •8489 •5055 •1640 •8252 95 6 •5297 •1819 •8 100 •4959 •1537 •8140 6 [1111] 4 f 1 1 T T 1 Joint Life Annuitv, at fop. Al'c of elder Luc: — i t . o • * ■.. „. c'j„ '^ ' Last Survivor Annuitv, at bide. 197 3 1 PEE-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. 64 65 66 67 68 69 55 6 8698 6 6590 6-4435 6-2225 5'9957 5*7611 55 6 7731 5689 •3598 •H,)0 •9242 *6953 6 7 6706 4732 •2707 •0624 •8478 •6250 7 8 5620 3716 •1761 ■9745 •7664 '5500 8 9 4468 2638 •0754 5^88o8 •6795 •4696 9 6o 3260 1504 5*9693 •7819 •5876 •3846 60 I 1998 0318 •8582 -6782 •4911 •2952 I 2 0686 5 9083 •7433 -5699 •3902 •2016 2 3 5 9329 7804 •6221 '4573 •2852 •1040 3 4 "~ 6484 •4979 -3408 •J 764 •0028 4 6^ 10 9586 — •3689 -2197 •0630 4-8971 65 6 7762 10 5708 ■0929 4*9442 •7862 6 7 6010 3877 10-1816 •8189 •6689 7 8 4329 2119 9-9978 9-7908 '— ■5446 8 9 2717 0430 -8210 •6060 9-3978 9 7° 1188 9 8824 -6527 -4297 •2134 9-0038 70 'i 9 9748 7311 •4938 •2630 •0388 8-8210 I 2 840G 5898 •3453 -1070 8-8750 -6493 2 3 7167 4593 -2079 8-9625 -7231 •4897 3 4 6034 3398 -0819 -8297 -5834 -3428 4 75 4997 2303 8-9663 -7078 -4550 -2075 75 (5 4032 1282 -8585 •5940 -3348 •0807 6 7 3139 0337 -7585 •4882 -2230 7^9625 7 8 2312 8 9461 -6658 •3900 •1190 •8525 8 9 1546 8649 •5797 •2988 -0223 •7500 9 8o 0844 7904 •5007 •2150 7-9334 •6555 80 I 0209 7230 •4291 •1390 -8527 •5697 I 2 8 9643 6629 •3652 •0711 -7805 •4929 2 3 9134 6088 •3076 •0099 -7153 •4236 3 4 8677 5601 •2559 7-9547 -6566 •3610 4 85 8255 5-151 •2080 •9037 -6021 •3028 85 6 7848 4718 •1617 •8543 -5494 •2464 6 7 7-146 4289 •1160 -8054 •4970 •1903 7 8 7051 3868 •0710 -7572 •4454 •1348 8 9 66-48 3438 -0251 -7082 •3927 •0781 9 90 6255 3019 7-9804 -6603 •3414 •0227 90 I 5900 2G40 -9399 -6170 •2949 6-9724 I 2 5583 2301 -9037 -5783 •2534 •9276 2 3 5301 2000 •8716 -5440 •2166 •8879 3 4 5073 1756 •8455 -5161 •1867 •8557 4 95 4900 1570 -8256 •4948 •1639 •8311 95 6 4779 1440 -8117 -4800 •1481 •8140 6 [182] . f ij r T ( Joint Life Annuity, at Top. Ase of elder Life: — < , . ^ . a " •. . ^-j ° I Last Survivor Annuity, at bide. HM. 31 J'EK-CENT. Valuer of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. 70 7' 72 73 74 75 55 55 5-5216 5-2798 5-0398 4*8056 4-5813 4-3673 6 •461.3 •2248 4-9896 -7599 -5397 •3295 6 7 •3968 -1657 -9357 ■7108 -4950 •2889 7 8 •3278 •1025 -8779 •6.581 •4469 •2450 8 9 ■2539 •0346 •8158 •6012 •3949 •J976 9 6o •J 755 4^9625 •7497 •5407 •3395 •1469 60 I •0929 •8865 •6798 •4767 •2809 •0933 I 2 •006.3 •8068 -606^ -4093 •2192 •0367 2 3 4'9i6o -7234 •5298 -3389 ■1545 3-9774 3 4 •8222 •6368 •4500 •2656 •0872 -9157 4 65 •7242 ■5461 •3664 •1886 •0164 -8507 65 6 •6210 -4505 •2780 •J071 3-9414 •7818 6 7 ■5^^7 -3490 •1840 -0202 •8613 •7080 7 8 •3955 •2407 •0835 3-9271 •7751 •6283 8 9 •2703 •1237 3-9744 •8257 •6809 ■5410 9 70 3-9995 8583 •7173 •5799 -4470 70 I 8-6117 •7363 -6031 ■4732 -3474 I 2 •4319 8-2245 •4856 •3630 -2443 2 3 •2640 •0494 7^8459 •2522 -•1403 3 4 •1103 7-8876 •6768 7-4793 •0,584 4 75 7-9680 •7382 •5203 -3100 7^1262 75 6 •8344 •59?7 •3730 -1019 6-9655 0-7830 6 7 •7097 •4663 •2349 •0173 •8146 •6259 7 8 •5933 •3435 •1056 6-8816 •6727 -4780 8 9 •4847 •2286 6^9844 •7542 •5393 •3388 9 80 •3845 •1225 •8722 •6361 •4154 •2092 80 I •2933 •0258 •7699 •5282 •3021 •0906 I 2 •2116 6-9390 •6780 •4311 •2000 5-9837 2 3 •1378 •8605 •5947 •3431 •1073 -8865 3 4 -0711 •7896 -5194 •2635 •0235 •7985 4 85 •0090 •7234 •4491 -1890 5-9449 •7160 85 6 6-9487 •6590 •3806 •1103 •8081 •6351 6 7 •8886 •5946 -3119 •0432 •7907 •5534 7 8 •8290 •5306 •2433 5-9701 •7131 •4714 8 9 •7678 •4647 •1725 •8943 •0324 •3857 9 90 -7079 -3990 •1020 •8192 •5522 •3005 90 I -6534 •3408 •0386 •7502 ■4783 •2218 I 2 •6048 •2879 5-9810 -6879 •4114 •1503 2 3 •5616 •2408 -9296 •6320 •3510 •0857 3 4 •5266 •2025 •8877 •5862 •3014 •0324 4 95 ■4999 •1733 •8556 -5510 •2031 4-9013 95 6 -4814 •1530 •8332 -5203 •2301 •9622 6 [183] Age of elder Life: Joint Life Annuitj', at Top. Last Survivor Annuity, at Side. 190 3J PER-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Tioo Lives. 76 77 78 79 80 81 55 55 4-1552 3-9468 3-7412 3-5362 3-3349 3-1422 6 •1209 ■9159 -7^33 •5111 •3124 •1220 6 7 •0841 -8825 •6831 •4840 •2881 -I002 7 8 •0442 •8464 -6505 •4546 •2616 -0765 8 9 •0010 •8072 •6150 •4226 •2328 -°5°5 9 6o 3-9549 •7652 ■5770 •3882 •2018 -0225 60 I -9059 •7206 •5365 ■35^5 •1687 2-9927 I 2 •8542 •6736 -4937 •3128 '^337 •96 1 1 2 3 •8000 •6241 •4487 •2721 •0969 •9279 3 4 -743'5 •5726 •4019 •2296 •0585 -8933 4 65 •6842 •5 '84 -35^6 •1849 •0181 -8568 65 6 •6210 •4607 •3000 •1372 2-9749 •8178 6 7 -553^ -3987 -2435 •0858 •9283 •7756 7 8 -4799 -3314 •1820 •0298 -8774 •7294 8 9 •3992 -2571 -J137 2-9673 •8205 •6776 9 7° •3120 •1764 -0394 •8991 -7580 •620^ 70 I •2193 •0904 2-9598 •8258 •6906 •5586 I 2 •1230 •0008 •8767 •7490 •6199 ■4935 2 3 •0258 2^9IOI •7924 •6709 -5477 •4269 3 4 2-9305 •8211 •7096 -5941 -4767 -3613 4 *- r' -8378 -7346 •6291 ■5194 -4077 -2976 75 6 -6453 -5458 •4420 •3360 -2312 6 7 6-446G •4607 •3629 •2625 •163 1 7 8 •2927 6^1175 •2810 •1863 •0923 8 9 •1476 5-9664 5^7949 •1059 •0173 9 8o •0123 -8255 •6483 5^4798 ^'9393 80 I 5-8884 •6962 •5136 •3397 5^1764 I 2 •77GG •5795 •3920 -2131 •0448 4-8894 2 .3 •6749 •4733 •2812 •0976 4-9248 •7650 3 4 •5827 •3769 •1806 4^9928 •8158 •6519 4 85 •49G1 •2863 •0859 •8941 •7131 •5454 85 6 •4111 •1971 4^9926 •7966 •6115 •4399 6 7 •3250 •1066 •8975 •6970 •5074 •3316 7 8 •2384 •0152 •8012 •5957 •4012 •2206 8 9 •1476 4-9191 •6996 •4883 •2880 •1018 9 90 •0570 •8229 •5976 •3801 •1733 3^9808 90 I 4^9732 •7337 •5026 •2789 •0657 •8668 I 2 •8969 •6524 •4159 •1862 3^9G65 •7612 2 3 •8278 •5785 •3369 •1015 •8756 •6637 3 4 •7707 •5174 •2715 •0311 •7997 •5820 4 9t •7264 •4699 •2205 3-9762 •7403 •5177 95 6 •6950 •4362 •1843 •9371 •6979 •4716 6 [184j Aire of elder Life: — Joint Life Annuity, ;it Top. Last Survivor Annuity, at Side. 200 3^ rKll-CENT. Values of Annuities on Tao Joint LivcSj mid on the Last Survivor of Tico Lives. 82 83 84 85 86 87 5S 55 29637 2-7974 2-647 1 2-5063 2-3650 2.2141 6 •945<5 •7812 •6326 •4934 ■3537 •2042 6 7 •9260 •7636 •6169 •4793 -3412 •1934 7 8 •9046 •7444 •5997 ■4^39 •3276 -1816 8 9 •8813 •7234 •5807 •4469 -3125 -1683 9 Co •8561 •7007 -^602 •428.5 -2961 •1540 60 I •8292 •6764 •5383 •4088 •2785 •J 38.5 I 2 •8006 •6i;o6 •5150 •3878 •2598 -1220 2 3 7706 •6236 •4906 •3658 -2401 •1048 3 4 •7394 •5954 •4652 •3430 •2198 •0869 4 65 •7064 ■s^:^(^ •4384 ■3190 •1984 •0682 65 6 •6712 •5339 •4097 •2932 •J75'5 •0482 6 7 •6330 •4993 •3786 •2652 '^5^1 •0265 7 8 •59^1 •4614 •3442 •2343 •1231 •0024 8 9 •5439 •4183 •3050 •1988 •0913 I "9743 9 70 ■4917 •3706 •2614 '^59^ ■^:^s^ '9425 70 I •4349 ■3185 •2135 •^^^2, •0158 •9071 I 2 •3749 •2633 •1627 •0686 I '9 73 2 •8688 3 'i^5S •2066 •1103 •0204 •9292 •8292 3 4 •2529 '^S^l •0586 1-9728 •8857 •7900 4 75 •1940 •0963 •0084 •9265 •8435 •7521 IS 6 '^3^5 •0393 I '9556 •8778 •7989 •7119 6 7 •0693 I •9806 -9011 •8273 •7526 •6700 7 8 •0034 •9193 •8440 •7743 •7037 •6257 8 9 1-9334 •8540 •7829 -7172 •6508 •5773 9 80 •8604 •7855 •7186 .6569 •5946 '5^5^ 80 I •7871 •7166 •65 1 1 •6.^38 •5959 •5376 •5375 •4828 •4727 I 2 '59^9 '4220 3 4-6200 •5330 -4820 •4306 •3734 3 4 •5033 4^3673 -4326 •3844 •3306 4 85 •3932 •2539 41274 •3426 -2921 85 6 •2841 •1414 •0117 3-8891 •2537 6 7 •1718 •0255 3-8924 •7665 3-6410 7 8 •0565 3-9062 •7694 •6400 -5107 33762 8 9 3-9325 •7774 •6363 •5028 •3691 •2296 9 90 •8056 •6451 •4992 •3612 •2228 -0779 90 I •6855 •5194 •3686 •2261 •0830 2-9328 I 2 •5737 •4018 •2460 •0990 29515 •7962 2 3 •4700 •2921 •1311 2-9795 •8276 -6674 3 4 •3825 •1990 •0331 •8773 •7214 •5569 4 95 •3133 •1249 2-9548 •7952 -6358 •4677 95 6 •2634 •0711 •8975 •7351 •5729 •4019 6 [185] Age of elder Life:- J Joint Life Annuity, 3] PERCENT. I rt/MC5 of Annuities on Tico Joint Lives^ anti on t/ie Last Snrvicor of Tiro Lives. 94 9S 96 S!i 5.1 0-6535 0^4029 0-1737 6 •6322 •4022 •1735 6 7 •6508 •4015 •1732 7 8 •649.3 •4007 •1729 8 9 •6475 •3998 •1726 9 6o •6456 •.3987 •1722 60 I •6434 •3976 •1718 I 2 •6411 ■3964 •1713 2 3 •6386 •3950 •1708 3 4 •6.360 •3936 •1703 4 65 '^3:^3 •3922 •1698 65 6 •630.^ •3907 •1692 6 7 •6276 •3892 •1686 7 8 •6245 •.3876 •1680 8 9 •6207 ■3856 -1673 9 70 •616,3 •3833 •1664 70 I •61 10 ■3805 -^654 I 2 •6048 ■3772 •1642 2 3 •5980 •3735 •1628 3 4 •591 1 •3697 •1613 4 75 •5849 •3663 •1600 75 6 •5780 •3626 •1586 6 7 •5710 •3588 •1571 7 8 ■S^3^ •3548 •i5.';6 8 9 '^S5° •3502 '^Si9 9 80 •5451 •3448 •I5I8 80 I •5341 •3387 •1494 I 2 ■5231 •3326 •I47I 2 3 ■51^7 •3261 -M45 3 4 •.)0^7 •3203 •1422 4 85 •493 T •2,^55 •1402 85 6 •485^ •3110 -1385 6 7 ■4765 •3060 ■J364 7 8 •4691 •3021 -•350 8 9 •4563 •2954 -^325 9 90 •4352 •2835 •1279 90 I ■4093 •2688 •1220 J •3776 •25 1 1 •1151 2 3 •3293 •2228 •1037 3 4 •1839 •0874 4 95 0-901G •0644 95 6 •7627 0^5200 6 no-T > r 11 1 -t Lloint Life Annuitv, at Top. rin,] Aec of elder Life: — {, .„ . a •• » t:-j ' ■■ ° j Last burvivor Annuitv, at bide. TWO LIVES. FOUR PER-CENT. 2<4 HM. 4 rKKCKNT. Values of Annuities on Two Joint Lives, and on the Last Survival- of Two Lives. Equal Ages. X Os.x 2(/., — a J . ., 2874 T 1 -Cx.J lO 1 7-8656 22- 55 i 8-2598 13-8254 I • 7*^3 7 L'dll 6 7^9''>33 4989 2 •6225 1(»73 / •6666 1008 3 •4497 (KKi'J 8 •3700 12 8292 4 •2539 21 'JU 11 9 •0733 1803 i.^ •043.5 7903 60 6^7787 1393 6 16-827^ 0701 I •4877 11 78S9 7 •6148 551)2 -) •2013 4355 8 •4144 44U ] 5^9204 0798 9 •23.58 3238 4 -6462 10 7228 20 •0809 2067 65 •3771 3635 I ^5*9379 0883 6 •I 1 15 0019 2 •8006 20 •<.I074 7 4-8479 '• 0379 3 ■6599 •8423 8 •5852 2710 4 •5088 -7 120 9 -3200 8 9008 2.1 •345.1 575'J 70 ■0560 5300 6 •^731 -43-15 I 3-7962 1012 7 14-9931 -2881 2 •5457 ' 7981 8 •8104 ; -1370 .; •309- 4440 9 •6262 ID ■9810 4 •0915 1023 30 •4399 •8219 75 2-8928 'J 7724 I •2522 -6574 6 -7007 4477 •0.598 -4870 7 •5^r- 1298 3 13-8622 3122 8 -3401 8177 4 •658.5 -1311 9 -J 65 8 -5100 35 •4496 18 •9440 80 1-9969 2117 6 -^3(^3 -7513 I •8386 i 9202 7 •0200 •5528 2 •6973 0587 8 12-8007 -3487 ,) •5707 4065 9 •5770 -1384 4 •4645 1707 40 •3479 1^ •9215 85 •3732 3 9424 I •1107 -0973 6 -2876 7092 2 11-8635 4053 7 •I99I 4003 3 •6073 2257 8 •JIOI 1931 4 •3447 1 16 9791 9 0-9952 2 8880 45 •0760 7250 90 •8.557 ! 5521 6 J 0-8052 4048 J •7141 2045 7 •5339 1987 2 •5724 1 8442 8 •2618 15 9262 ,•) •4148 4590 9 9*9873 6473 4 •2639 ! 0727 50 •7103 3017 95 •1308 ' 0904 I •4288 oosy 6 -0324 3208 2 •1412 1-1 7082 3 8-8495 40(15 4 '5551 1401 205 HM. 4 PER-CENT. Values of Annuities on Tico Joint Lives, and on the Last Survivor of Tico Lives. 10 II 12 13 14 15 lO I 3 4 8140 1550 1072 0584 177413 •6923 17-6511 •6035 '5353 2r9549 i7'5476 n •50^5 •4348 '35°9 4348 3901 3^5' 2429 1478 10 I 2 3 4 17 22-2449 — •2001 22 •1537 •10G4 22-0579 •0076 15 •0586 0092 2r9567 ■9023 01 •QKif; ii ZL O-iUO 6 ■0111 21 9603 •9062 ■8500 ■7925 i 21 7341 6 7 21 •9642 9119 •8563 ■7983 ■7389 j 6787 7 8 •9188 8649 •8077 •7480 ■6868 6246 8 9 •8747 8197 •7608 •6994 -6365 5725 9 20 ■8324 7758 •7156 •6526 •5879 5221 20 I •7909 7332 •6714 •6068 •5404 4728 1 2 •7501 6909 •6278 •5615 •4935 4242 2 3 •7093 6487 •5841 •5163 ■4465 3755 3 t '^ •6681 6064 •5401 •4707 ■3993 j 3264 4 ^S •6267 5635 •4959 •4249 ■3516 j 2770 25 6 •5852 1 5208 •4516 •3790 ■3039 2274 6 7 •5437 4781 •4074 •3330 •2564 { 1779 7 8 •5024 4354 •3634 •2874 •2088 1286 8 9 •4615 3933 •3197 •2422 •1619 1 0798 9 30 ■4211 i 3516 •2765 •1974 -1155 ' 0315 30 I •3808 3102 •2338 •1530 -0693 20 9836 I 2 •3408 2689 •1912 ■1089 -0234 9359 2 3 •3011 2280 •1488 •0650 20-9778 8884 3 4 •2614 1873 •1067 •0213 -9325 8413 4 35 •2220 1466 •0648 20^9778 -8873 7945 35 6 •1830 1066 ■0233 •9349 -8427 7480 6 7 •1443 0668 20-9823 •8922 •7985 7020 7 8 •1060 i 0275 •9416 •8502 ■7546 6565 8 9 •0682 : 20 9884 •9014 •8084 ■7114 6114 9 40 •0306 i 9499 •8615 •7671 ■6685 5668 40 I 20^9933 1 9116 •8220 ■7261 ■6258 5225 I 2 •9562 8735 •7826 •6854 •5834 4784 2 3 •9194 8359 •7437 •6450 •5416 4347 3 4 •8832 7986 •7055 •6052 ■5002 3917 4 45 •8474 7619 ■6674 •5659 •4592 3490 45 6 •8123 7260 •6303 •5273 ■4102 3073 6 7 •7779 1 6907 •5939 •4896 ■3798 2664 7 8 •7442 6562 •5583 ■4526 •3414 2262 8 9 ■7113 6224 •5234 •4164 •3036 1869 9 5° •6790 5894 •4893 •3810 -2668 1484 50 I •6475 5571 •4559 •3463 -2306 1107 I 2 •6165 5255 •4233 ■3124 -1952 0737 2 3 •5864 4946 •3914 ■2792 •1606 0375 3 4 •5571 1 i 4645 •3603 ■2469 •1269 0023 4 [•220] Age of elder Life:- | Joint Life Annuity, at Top " { Last burvivor Annuity, at Side. 20G 4 PER-CEN'I'. ]'(ih(fs of Anuiiitic^- on Tiro Joint Livfs^ and on ilic Last Survivor of Two Livts. 16 n 18 J9 20 21 to 17-3172 J 7" 1993 i7*o8 169816 16-8879 167987 lO I •2739 •1575 •0454 •9425 •8504 •762.3 I 2 •210^ •0956 16-9851 •88.39 •7931 •7066 n 3 •1301 •0170 -9082 •8087 •7195 -6346 3 4 •0.368 16-9256 •8186 •7208 •6.334 •5502 4 15 16-9,346 •8252 •7202 •6242 ■5386 •4572 15 6 •7202 •6i7J •5-30 •4392 ■359'5 6 i 7 21-618G •5^37 •4215 ■3395 •2617 7 . 8 •5626 2r5012 •3242 •2440 •1680 8 9 •5086 •4453 2r3835 '^575 •0833 9 20 •4564 •3913 •3277 21-2661 •0086 20 I •4053 •3384 •2730 •2096 2M483 I 2 •3549 •2861 •2189 •1537 •0906 21-0287 2 3 •3043 •2337 •1646 •0977 •0328 20-9690 3 4 •2535 •1809 •1099 •0411 20^9744 •9088 4 25 •2021 •1277 •0547 20-9840 •9154 •8479 25 6 •1506 •0742 20-9994 -9267 •8561 •7867 7 •09l»2 -0209 -9440 -8694 •7969 •7255 7 1 8 •0480 20-9677 •8889 •8123 •7379 •6645 8 9 20-9973 •9151 •8343 •7558 •6794 •6040 9 30 -9470 •8629 •7801 •6997 ■6214 •5441 30 I -8972 •8111 •7264 •6441 •5639 •4847 I 2 •8477 •7597 •6730 •5887 •5067 •4255 2 3 -7984 -7085 •6199 •5337 •4498 •3667 3 4 •7493 •6575 -5671 •4790 •3932 •3083 4 35 •7006 •6068 •5146 •4247 •3371 •2502 3S 6 •6524 •5568 •4625 •3708 •2814 •1928 6 7 -G045 •5071 •4110 •3174 •2263 •1359 7 8 •5572 -4579 •3601 •2647 •1717 •0795 8 9 •5104 •4093 •3096 •2125 •1179 •0239 9 j 40 .4639 -3610 •2596 •1607 •0644 19^9688 40 I •4179 -3131 •2099 •1094 •0114 •9142 X 2 •3719 •2655 •1605 •0583 19-9587 •8598 2 3 •3265 •2182 •1116 •0077 •9066 •8061 3 1 4 •2817 •1717 •0632 19-9577 •8551 •7531 4 45 •2374 -1255 •0155 -9083 •8041 •7007 45 6 •1939 -0804 19-9686 •8599 •7543 •6493 6 7 •1513 •0361 •9227 •8124 •7054 -5990 7 8 •1094 19-9926 •8775 •7658 •6574 -5498 8 9 •0685 •9500 •8334 •7202 •6105 -5015 9 50 •0284 -9082 •7901 •6755 •5645 •4543 50 I 19-9890 -8673 •7475 •6316 •5193 •4079 I 2 •9504 •8270 •7059 •5885 •4751 •3625 2 3 •9126 •7877 •6650 •5464 •4317 •3181 3 4 •8758 •7493 •6251 •5052 •3894 •2747 4 [•2-21] Age of elder Life: — [ Joint Life Annuity, at Top, ( Last Survivor Annuity, at Side. ■20; 4 PERCENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. 22 23 24 25 26 27 lO 16-7104 i&6iS^ i6^^i88 16-4105 16-2951 16-1734 10 I •675.5 •5848 •4864 , •3796 •2654 •1449 I 2 ■6211 ■5319 •4352 •3297 •2171 ■0981 2 3 •550'^ •4631 •3680 •2641 ■^53^ ■0359 3 1 + •4680 •3821 ■2886 •1866 •0774 i5^96i7 4 1 '^ ■3767 •2925 •2009 •1006 K^'9933 •8796 15 1 6 •2809 •1986 ■1087 •0104 •9050 •7932 6 i 7 •1849 •1044 ■0165 I5^9200 ■8166 ■7067 7 1 8 •0930 •0144 15-9284 •8339 •7323 ■6245 8 9 •oioi 15*9332 ■8491 ■7565 •6569 -55'° 9 20 15 "93 72 •8621 •7798 •6891 •5915 •4875 20 ' I •8684 •7952 •7147 •6259 •5302 ■4282 I ' 2 •7293 •6508 •5640 •4702 •3702 2 3 20-90o8 •5834 •4986 ■4069 ■3089 3 4 •8436 20^7781 ■4262 ■3366 ■2408 4 ^5 •7807 •7132 20-6449 •2583 •1647 25 ' 6 •7176 •6480 ■5776 20^5062 ■0820 6 1 7 •6544 •5828 •5102 •4366 20^3624 7 8 -.3914 •5177 •4429 •3671 •2905 20-2137 8 9 •5290 •4531 •3762 •2980 •2192 -1398 9 : 30 •4670 •3892 •3100 •2295 •1482 -0665 30 ' j I •4055 •3256 •2443 •1615 •0778 19-9935 I 2 •3445 •2624 •1789 •0938 •0076 ■9208 2 , ' 3 •2837 •1996 •1139 •0264 19-9379 •8485 3 4 •2233 -1371 •0493 19-9595 -8685 ■7766 4 ! -''^ •1633 -0751 1 9^9851 -8930 -7995 ■7051 3'i \ 6 •1039 •0137 •9215 -8272 -7313 •6343 ^ i 7 •0452 19^9529 •8586 -7621 -6638 •5643 1 7 ! 8 19-9871 •8930 •7965 ■6977 -5971 •4950 8 ' 9 -9296 ■8337 ■7352 ■6341 •5311 ■4266 9 40 •8728 •7749 ■6745 ■5713 -4659 ■3590 40 I •8165 •7167 ■6143 -5090 -4014 ■2920 I 2 •7604 •6590 •5546 -4472 -3374 •2256 2 3 •7050 •6019 •4957 ■3861 ■2741 ■1599 3 4 •6505 •5455 •4375 •3260 -2118 •0953 4 45 •5965 ■4900 •3801 •2667 ■1504 ■0316 4') 6 •5138 •4357 •3241 ■2086 ■0903 18^9694 6 7 •4920 ■3825 •2692 •1520 •0315 ■9086 7 8 ■4414 ■3303 •2154 •0965 18-9742 ■8490 8 9 -3918 ■2794 •1629 •0422 •9181 ■7910 9 50 •3433 ■2295 •1115 18^9892 •8633 ■7342 50 I •2958 ■1806 ■0612 •9373 •8096 ■6787 I 2 •2491 ■1328 •0120 •8865 -7571 ■6244 2 3 •2036 ■0860 18-9639 ■8370 ■7059 •5714 3 4 •1592 ■0404 •9171 1 •7886 -6561 •5199 4 [•2-2-2] Age of elder Life :- \ Joint Life Annuity, at Top ° ( Last Survivor Annuity, at bn Side, 208 HM. 4 PERCENT. Value.'i of A tin It it i'S on Tuxi Joint Livis, aw? on the Last Snrvicor of Two Li VIS. 28 29 30 lO 16-0478 1 5*9 '89 X5786.3 I •0207 •89.30 •7617 2 '5-9752 •849' •7193 .3 •914(5 •7900 •6618 4 •8424 •7195 •5929 i.l •7620 •6410 •51^5.3 6 •6775 •5584 •4357 7 •5930 •4758 '.'iS.'i^^ 8 •5127 •3975 •2787 9 •4412 •3279 •21 10 20 ■379<5 •2683 'i533 1 •^^^■■3 •2130 •0999 2 •266:i •1589 •0479 .3 •2071 •1019 i4^9928 1 4 •1412 •0.381 '93^3 25 •0^73 1^-066 •862 1 6 14-9870 •8885 •7865 7 •9006 •8047 •7050 8 •7171 •6200 „ 20-0605 •5318 9 19-9846 •9091 19-9030 •8249 .30 I 19-7410 2 •8338 •7469 -6G03 3 •7589 •6693 -5799 4 •6843 •5920 •4998 35 •6102 -5151 •4200 6 •53G7 •4390 •3411 7 •4640 -3635 •2G28 8 •3922 -2889 •1854 9 •3212 -2153 -1089 40 •2510 •1424 -0332 I •1814 •0701 18-9582 2 •1125 18-9985 -8838 3 •0444 •9278 •8104 4 18-9773 •8583 •7381 1 45 •9113 •7896 •6669 6 •8467 •7227 •5973 7 •7836 •6572 •5294 8 •7219 •5933 •4630 9 •6617 •5308 •3982 50 •6029 •4698 •3349 I •5455 •4102 •2731 2 •4892 •3520 •2127 3 •4344 •2952 •1540 4 •3810 •2400 •0967 j 31 32 I 33 i5"65o5 •6270 ■5859 •5301 j •4'53o j •.3881 I ■3094 i •2307 'K1^3 •0905 I i5"5094 •4872 •4474 •393 1 •3278 •2547 •1778 •xoio -0286 i4"9'548 •0.347 "9108 14-9832 "8613 •9333 '8132, •8803 7624 i •8209 7052 ] •7540 -6406 •6808 ■.)'599 •6019 I •4935 ! ■5^94 '4136 ; •4338 I -3307 j •3447 1 "2443 : . *i545 19-5740 ! i -4907 19-4014 •4077 ^3154 •3250 ■2431 •1619 •0816 -0021 18-9235 •8456 -7684 -6920 -6169 •5429 •4707 •4001 -3312 •2639 -1982 •1341 •0714 •0105 17-9512 •2298 -1448 -0605 18-9772 -8946 -8130 •7320 •6518 -5724 -4943 •4174 •3423 •2690 •1972 •1274 •0591 17-9925 -9275 -8642 •8027 J 5^3626 ■•3416 '3033 •2505 •1869 •I 157 •0406 i4-9'557 ■8952 •8333 •7812 7336 •687.5 •6387 •5837 •5215 ■4531 •3793 •3020 •2218 •1382 '°.'i^3 ^3-9j9.'i 19-2232 -1345 -04G4 18-9589 -8724 -7867 •7018 -6177 -5342 -4518 -3706 -2905 •2124 •1361 •0615 17-9888 -9179 •8486 -7810 •7152 -6513 10 I 2 3 4 15 6 7 8 20 I 2 3 4 30 I 2 3 4 3.^ 6 7 8 40 I 2 3 4 45 6 7 8 9 50 I 2 3 4 [•.'•J.-Jj Aee of flder Life: J Joint Life Annuity, at Top. I Last Survivor Annuity, at Side. 209 4 TER-CENT. Values of Annuities on Tico Joint Lives, and on the Last Survivor of Two Lives. id 34 35 36 37 38 39 10 15-2099 15-0513 14-8873 147186 H*5452 i4'366o I ! •J 899 •0326 • ^8696 •7020 •5 29-3 ■3517 I 2 •1530 1 4-9959 •8354 -6C90 •4980 •3212 2 3 •1018 ■9173 •7872 •6225 •4528 •2776 3 4 •0398 •8870 •7286 •5654 •3976 •2238 4 15 14-9704 -8192 •6627 •S°''?> ■335'^ •1632 15 6 •8973 •7480 •5932 •4337 •2693 •0991 6 7 ■8243 •6770 •5240 ' 3(^(^3 •2038 •0354 7 8 ■755^ •6joi ■4.-92 •303 ? ■1425 13-9760 8 9 •^956 -55^9 •4028 •2488 •0898 -9250 9 20 ■6454 •5035 '35^'^- •2039 •0468 •8836 20 1 •5996 ■4597 •3 141 •1636 •0083 •8469 I 2 •5555 ■4175 ■2739 •1252 13-9716 •8121 2 3 •5088 •3728 •2312 •0846 •9328 •7751 3 4 ■4559 •3221 •1827 •0382 ■8886 •7329 4 25 •3960 •2645 •1273 13-9850 •8377 •6843 25 6 •3301 •2011 •0663 ■9264 •7814 •6304 6 7 •2s88 •1323 •coo I •8627 •7203 •5717 7 8 •1842 •0603 13-9308 •7961 ■6^62 •5102 8 9 •1067 i3-985^ •8587 •7268 •5897 •4463 9 30 •0259 ■9077 •7836 •6545 •5202 •3797 30 I 13-9419 -8266 ■7055 •5793 •4479 •3104 I 2 •8j3i •7407 •6227 •4996 •3712 •2368 3 3 •7588 •6495 •5346 •4 '47 ■2895 •1582 3 4 •5524 •4408 •3241 •2022 ■0742 4 3.S 19-0392 •3412 •2279 •1094 12-9848 3^ 6 18-9478 18-8494 •I 264 •01 14 ■8903 6 7 •8571 •7553 18-6538 1 2^9085 •7911 7 8 •7073 -6G21 -5571 18 -4526 •6870 8 9 ■6783 •5697 •4612 •3530 18^2454 9 40 •5901 •4780 •3660 •2542 •1428 18-0319 40 1 •5026 •3871 •2715 •1559 •0407 17-9259 I 2 •4159 •2968 •1776 -0583 17-9392 •8204 2 3 1 -3300 •2075 -0846 17-9616 •8386 •7158 3 4 •2455 •1195 17-9931 -8662 •7393 •8125 4 4i •1623 ■0329 •9028 •7722 ■6414 -5105 45 6 •0809 17-9482 ■8145 -6802 ■5456 •4106 6 7 -0014 •8654 ■7-283 •5903 •4519 •3129 7 8 17-9238 •7845 •6440 •5025 •3603 •2174 8 9 •8482 •7057 •5618 •4167 •2709 •1241 9 5° •7743 •6287 •4816 •3331 •1835 -0330 50 I ■7022 •5536 •4033 •2515 •0984 10-9440 J 2 •6318 •4803 •3268 -1718 •0152 •8572 2 3 •5634 •4090 •25-25 -0942 16-9343 •7727 3 4 •4969 •3308 •1804 •0190 •8558 •6907 4 [224 J Ajie of dd.T Life: — Joint Life Annuity, at T»p. Last Survivor Annuity, at SiJc. 2 V. 210 HM. 4 PER-CENT. ]'^aluc3 of A)) nit it Its on Tiro Joint Livts^ and on the Last Survivor of 2'iro Lives. 40 41 42 43 44 45 13-1296 ! 10 lO 14-1806 13-9872 137847 i3"573'5 i.r355^ I •1672 -9748 7733 •5^30 •3457 -1210 I 2 •1.581 -9469 7467 •5377 •32^3 -0980 2 ■ 3 •0959 -9062 7073 •4998 -2850 -0629 3 4 •0437 •8557 •^>5^5 •4524 •2392 -0188 4 K'i 13-9848 -7984 -6029 •3987 •1871 12-9684 15 6 -9226 •7379 •5443 -3418 •J320 •9149 6 7 -8607 -6779 •4859 •2853 •0772 •8620 7 8 -8030 •6220 •43 i 8 -2328 -0266 •8129 8 9 753S •5744 •3859 -1886 1 2*9840 -7720 9 20 -7141 •5364 ■3495 ■^53 7 -9306 -7402 20 \ I •6790 •5029 •3177 •1235 -9219 -7129 I ' 2 '6459 ■4715 -28S0 -^955 •8954 -6880 2 i 3 •6109 •4384 •2.565 -0657 •8675 -6616 3 4 •5706 •4001 -2202 -0312 •8348 -6308 4 25 •5241 •3557 •1779 12-991 1 -7966 •5945 25 6 -4726 •3064 •1308 -9462 7539 •5539 6 7 •416.3 -2526 •0794 -8972 -7072 •5095 7 ! 8 •3574 -1963 •0256 •8458 •6383 -4629 8 9 •2962 •13/8 12-9698 -7926 -6073 -4148 9 : 30 •2324 -0767 •91 15 7370 •5547 •3645 30 '\ I -1660 -0132 •8508 •6793 -4998 •3124 I ; 2 ■0954 ^2 '945 7 -7863 •6178 •4413 •2368 2 ! 3 -0201 ■8735 7174 •5519 •3785 -1972 3 4 12-9394 -7962 •6433 •4813 -3112 •1330 4 35 •8535 •7137 •5644 •4058 -2392 •0644 35 ; 6 •7625 -6263 -4806 •3257 -1626 11-9913 6 , 7 -6669 •5545 •3925 -2413 -0821 •9H7 7 ! 8 -5666 •4380 -2999 -1526 11-9973 •8338 8 1 9 •4605 ■335^ •2017 -0384 11-9581 -9071 -8110 7477 '^557 9 40 -2272 -0972 40 J 17-8115 11-9849 •8501 7073 -5565 I 2 -7019 17-5835 7331 •5948 •4485 2 3 •5931 -4704 17-3478 •4736 •3320 3 4 -4856 -3586 -2315 17-1048 •2079 4 45 •3795 -2480 •1164 16-9850 1685-15 45 6 •2753 -1396 ■0033 -8673 •7319 16-5974 6 7 -1734 •0333 16^8926 -7517 •6115 •4720 7 8 •0738 16^9293 •7840 •6385 •4934 •3488 8 9 1G97G4 -8277 •6778 -5276 -3776 -2280 9 50 •8813 -7283 -5740 -4190 -2641 -1095 50 I •7884 -6311 -4724 -3128 •1530 15-9933 I 2 -6976 -5363 -3732 -2089 •0444 -8795 2 3 -6093 -4439 -2765 -1077 15 9383 -7686 3 4 -5236 •3542 -1826 •0093 •8352 -6605 4 1 [225] Age of elder Life: Joint Life Annuity, at Top. Last t?urvivor Annuity, at Side. 211 HM, 4 PERCENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Tioo Lives. 46 47 48 49 1 50 51 10 lO 12-8992 12-6649 12-4263 12-1825 "i 1-933^ 11-6778 I •8914: -6380 •4202 •1773 -9290 •6741 I 2 •8696 •6373 •4006 •1588 -9116 •6578 2 3 •8360 •6050 •3697 •12921 •8833 •6308 3 4 •7933 ' •5640 •3301 •0912; -8467 •5957 4 15 •7446 •5168 •2847 ■0473 -8045 '5SS° 15 6 •6929 •4668 •2364 •0006 •7594 •5116 6 7 •6416 •4172 •18H4 ^^•9543 •7148 •468.5 7 8 •5943 •3715 •J 444 •9118 •6738 •4292 8 9 •5549 ; •3337 •1080 •8769 •6403 •3970 9 20 ■5245 1 •3047 •0804 •8506 ' •6153 •3133 20 I •4988 ' •2804 •0573 •8289 •5948 •3540 I 2 •4752 •2583 •0366 •8095 •5767 •3370 2 3 •4504 - •2349 •0148 •7890 •5576 ■3^93 3 4 •4213 •2073 11-9890 •7648 •.5349 •2980 4 2.=; •38711 •17.1° -9582 •7358 ■.507.5 •2722 25 6 •348J •1386 -9236 •7030 ; •4765 •2430 6 7 •3062 •0983 •88^6 -6669 ' •4424 •2107 7 8 ■2620 •0564 ■8458 -6293 •4068 •1770 8 9 •2X63 -0130 -8046 •5904 •3701 •1425 9 30 •1686 11-9678 -7619 'SS°° •3320 •1066 30 I •IT9I -9210 •7176 -5082 •2926 •0695 I 2 •0664 -8710 •6705 •4636 •2506 •0300 2 3 •0098 •8174 -6197 •4157 •2053 10-9874 3 4 11-9489 ■7597 •5650 •3^39 •1565 •9414 4 35 •8836 .6977 •5063 •3084 •X041 •8920 3S 6 •8143 -6318 •4438 •2493 •0482 •8393 6 7 •7412 ■5624 •3779 -1870 10-9893 ■7837 7 8 •6641 -4891 •3084 -121 1 •9272 •7251 8 9 •0821 -4111 •2343 •0509 •8607 •6625 9 40 •4944 •3276 'K^A9 10-9756 •7894 •S9S^ 40 I •3994 •2370 ■0687 •8936 •711 7 •5217 I 2 •2961 -1381 10-9744 -8039 •6264 •4408 2 3 •1842 -03 1 1 •8720 •7062 •533.5 •3S^S 3 4 •0650 10-9167 •7625 •6016 •4338 •^511 4 45 10-9381. •7948 •64.57 1 •4898 •3270 •1560 45 6 -6670 •5230 •3723 •2X47 •0489 6 7 16-3343 •3952 •2498 •0976 9*93 7 1 7 8 -2060 16 0651 •1218 9"975o •8200 8 9 •0800 15-9338 15-7895 •8460 -6966 ! 9 5° 15-9563 -8047 •6550 15-5073 •5666 50 I •8349 -6780 -5228 , -3695 15^2182 I 2 -7159 •5537 -3930 -2340 •0768 14-9216 2 3 -5997 •4321 •2659 -1012 U-9382 -7770 3 4 -4866 1 -3136 •1419 ' 14-9716 •8028 •6355 4 r,-,^, . c ij I r \ Joint Life Annuity, at Top. [•2-2G] Age of elder Life:- j ^^st Survivor Annuity, at Side. •2\'2 4 I'Ki'.-CHNT. ]\ihtcs of Anuuiti'rs on Tico Joint L>vrSj and on the Last Sitrvwor of Tico Lives. iO 52 11-4147 Si 54 10-8703 S5 10-3065 57 111451 'o'59"5 100186 10 I •4116 •1428 -8688 •589.-; -3061 •0189 I 2 •39^53 •1285 •8555 •5772 -2948 -0083 2 3 •3706 • 1 04 1 •8323 'SS'^:^ -2741 9-9890 3 4 •3370 •0719 -8015 1 -15260 -2462 •9625 4 15 •2979 •0344 •7655 •49 '5 •2132 •9309 15 6 •2361 10 9942 -7269 1 •4544 •'777 -8969 6 7 •2147 •9543 •6886 •4177 -1424 -8632 7 8 •1767 •9179 •6537 •3842 -1 103 •8324 8 9 '1 460 •8884 •^255 •3573 -0847 -8080 9 20 •1234 •8671 •<'^o53 •3381 -0666 -7909 20 I •1053 •8';oo •5893 •3232 -0326 '1119 1 2 •089') •8354 •5757 •3107 •0410 -7672 2 .3 •0730 •S20: -5616 •2975 -0290 -7562 3 4 •053 » •8of5 "5442 -28i4 -0140 •7423 4 25 •0289 •7787 -C230 -261 15 9'9954 •7249 25 6 •0014 •7529 -:i'986 •23S7 -9740 •7050 6 7 10-9709 •7242 -4716 •2133 ■9303 -6827 7 8 •9392 •<^943 •443^ -1870 ■9255 •6595 8 9 •9066 •6637 •4148 -1600 -9003 •6359 9 30 •8729 •6319 •3851 •1322 -8742 -6:17 30 I •8381 •5993 '3.') 45 -J 036 -8476 -3868 I 2 •8009 '5^45 -3219 •0731 -8192 -3604 2 3 •7609 •5270 -2868 -0404 -7886 •5319 3 4 •7177 •^864 -24S8 -0048 •7554 -3009 4 35 •6712 •4428 -2079 9-9'^^.' •7197 •4677 iS 6 •6217 •39-'>3 •^^H3 •9258 -6816 -4322 6 7 •5^^93 •3472 -1183 •8828 •(5415 •3948 7 8 •5142 •2954 -o':98 •8374 •5993 'i:)^S 8 9 '4552 •2400 •0179 -7890 •5540 •3134 9 40 •3918 •1S04 9-9620 -7366 •5052 -2679 40 I •3224 •I 151 •9007 •6791 •4514 •2177 I 2 •2459 •0429 •8327 ■6153 ■3915 1 "1617 2 3 •1623 9-9638 •7581 •5451 •3256 -0999 3 4 •0722 -8786 •6776 -4692 •2542 •0330 4 45 9-9757 -7869 ■59^9 •3873 •1772 8-9606 45 6 •8738 -6903 ■4993 -3009 •0957 -8839 6 7 •7673 -3892 -4036 •2103 -0103 -8036 7 8 •6557 •4831 •3030 -1152 8-920 <; •7191 8 9 '5380 •3711 -1966 •0144 •8254 -6294 9 50 "4^39 -2^28 -0841 8-9078 -7244 ■5343 50 I •2S19 -1268 89642 • 79:^9 -6163 •4323 I 2 8-9922 -8358 -67.17 -5006 •3224 2 3 14-G175 -6994 •541^ •37<^9 •2032 3 4 •-1G98 U-3065 -4044 -2462 •0810 4 ro.TT-i > r 1 1 T f \ Joint Life AnnuitA-, at Top. [227J Age of eMer Life :- j ^^^ ^^^.^.^^ ^^^^.^^^ J^.^^ av5/TY i ^ J 4 PER-CENT. Values of Annuities on Two Joint Lives, and on the LoM Survivor of Two Lives. 58 :^9 60 61 62 63 ,o 97372 9-4321 9*^55° 8-8369 8-5385 8-2409 10 I •7280 •4335 -1369 -8392 •5413 -2440 I 2 •7186 •4248 •1291 •8322 •5350 •23S5 2 i 3 •7002 •4077 •1130 -8172 -521 1 -2256 3 ; '^ •6750 •3838 •0904 •7959 -5011 •2068 4 15 •6450 ■3SS'- •0633 •7702 ■4767 •1837 15 ; 6 •6125 •3243 •0338 -7421 •4500 •'584 6 i ^ •5802 •2934 -0043 •7140 •4233 •i330 7 I « i •5508 •2653 8-9776 -6885 •3990 •1099 8 9 i •5^75 •2432 '95^5 •6685 •3801 -0920 9 . 20 •5^14 •2280 'gA,22 -^SS'' ■3^1S •0801 20 I 1 •4993 •2167 -9318 •6454 •3585 •0719 I 2 ; •4895 •2077 -9236 •6379 •3517 •0657 3 3 ! •4793 •1985 '9^S^ •6303 •3448 •0594 3 4 •4666 •1867 •9043 -6203 •3357 -0512 4 25 •4504 •1717 ■8904 •6075 •3238 •0402 25 6 ■4317 ■1543 •8742 •5924 •3098 •0272 6 7 •4109 •1347 •8559 •5754 -2940 •012 5 7 8 •3892 •I 145 •8370 •5579 '^111 7-9973 8 1 9 •3672 •0940 -8180 •5400 '2612 -9821 9 30 •3445 •0730 •7984 -5220 •2443 •g66s 30 1 •3215 •0516 •7786 -5036 •2274 •9509 I 2 •2969 •0288 ■7575 •4841 •2094 •9343 2 3 •2704 -0041 •7347 -4630 •1900 -9164 3 4 •2417 8-9775 •7099 •4401 •1688 -8969 4 35 •2107 ■9487 •6833 •4155 •1461 •8759 35 6 'nil •9:8£ •6549 •3892 •1218 •8535 6 7 •1430 •8859 •6251 ■3617 -0964 -8302 7 8 •1065 -8^20 •5938 •3328 •0698 •8057 8 9 •0673 -8ij8 •5603 •3019 •0413 •7796 9 40 •0251 •7766 •5241 •2685 •0106 •7513 40 I 8-9783 •7333 •4839 •2.313 7-9763 •7198 I 2 •9261 •6846 •4.588 •'895 -9376 •C841 2 3 ■8683 -6307 -3886 •1429 -8944 -6441 3 4 •8056 •5721 ■ -3340 •092 X •8473 -6ooj 4 45 •7378 •5086 •2747 •0369 •7960 •5529 45 6 •6659 •4414 •2118 7-9783 •7415 •5024 6 7 •5905 •3708 •1460 -9169 •6845 •4494 7 8 •5111 •2965 -0766 •8.-;22 •6243 '39?/^ 8 9 •4269 ■2 '75 •0027 •7833 •5602 •3341 9 50 •3373 •1334 7-9240 •7099 •4918 '2706 5° I •2412 •043 1 -8393 •6307 •4179 •2019 I 2 •'374 7-9454 ■7475 •5447 •3376 •1269 2 3 •026^ •8407 ■6490 •4522 •2510 -0461 3 4 j 7-9088 7295 •5441 '353^ •^585 6-9596 4 [•3-23] , p 1, T •/• \ Joint Life Annuity, at Top. Age of elder Life:- j^ast Sumvor Annuity, nt Side. 214 4 I'KR-C'ENT. Values of Annni'fns' on 'J'mt Joint Lh'cs^ and on the Last S^trvivor of Tn'o Lifvs. 64 65 66 67 68 69 10 lO 7 '9447 7-6489 7-3528 7-0554 6-7563 6-4533 I •9482 •6^28 •3570 ■0599 •7610 •4582 1 2 ■9434 •6487 '^S^i^ •0571 •7587 •4565 2 .3 •9314 •-^377 •3435 •0479 'n°s •4490 3 4 •9138 •6212 •3281 •0337 1313 •4369 4 15 •8920 •6007 •3088 -0156 -7204 -4212 15 6 •8681 •5780 •2874 ^'9954 •7015 •4035 6 7 •8439 ':^S5^ •2658 •9750 , -6822 : ^3854 / 8 •8220 •5343 •2461 •9564 -6646 •3689 8 9 •8050 •5183 •2309 •9421 •65 ' 2 •3562 9 20 •7939 •5079 •2212 •9330 •6427 •3484 20 I •786.3 •..•;oo9 •2147 -9271 •6373 •3434 I 2 7807 •4959 •2103 •923 1 •6338 •3403 2 .3 •7751 •4909 •2058 •9191 •6303 '3313 3 4 •7676 •4840 •1997 -9136 •62.13 •3328 4 25 •7575 •4748 •191 1 •9058 ■6183 •3264 25 6 •7455 •4636 •1809 •8964 -6096 •3185 6 7 •7318 •4510 •1692 -88^6 •5997 •3094 7 8 •7178 •4380 •1572 •8745 •5895 •3000 8 9 10^6 •4249 •H5I -8634 "5793 -2907 9 30 •6893 •41 17 •1330 ■8^22 •5690 -2813 30 I •6749 •3985 •120S •8412 •5589 -2721 I 2 •6596 •3844 •1080 •8294 •5482 -2623 2 3 •6431 •3693 •0941 •8167 -5366 •2518 3 4 •6252 •3528 •0789 -8028 '5^39 •2402 4 35 •6059 •3350 •0626 •7879 '!'^-^o^ -2278 3^ 6 •5853 •3162 •0453 -772t •4959 -2x47 6 7 •5638 •2965 •0274 ■7557 •4809 -2011 7 8 •5414 •2760 •0086 ■7387 •4655 -1872 8 9 •5175 ■2541 6-9887 •7205 -4490 •1723 9 40 •4916 •2304 -9671 -7009 •4312 '^5^3 40 I •4626 •2039 -9429 -67S8 -4112 -1381 I 2 •4297 •J736 -9152 ■653.'; •38S2 •1172 2 3 •3928 •1397 -8839 -6249 •3620 •0934 3 4 'iS^S •1025 -8497 •5934 -3332 -0672 4 45 •3084 •0618 -8122 ■5590 •3016 •0383 45 6 •2617 •0186 •7724 •5224 -2682 •0077 6 7 •2126 <5-9733 -7308 -4842 -2331 S'97:y1 7 8 •i6xo •9256 -6868 •4439 -1962 •9421 8 9 •1059 •8748 -6400 -4009 1 -I. .169 •9062 9 50 •0470 •8204 •5899 •3.148 •1147 •8677 50 I 6-9833 •7613 •5354 •3047 -0688 •8258 I 2 •9136 •6968 ■4757 •2497 -0182 ■7795 2 3 •8383 •6268 •4109 •1898 5*963 1 •7290 3 4 •7576 •5517 •34 '3 -1255 -9038 -6745 4 [22i>] A"c of elder Life : Joint Life Annuity, at Top. Lnst Survivor Annuity, at Side. 215 4 PERCENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. 70 7^ / ■" 73 74 IS lO 6-1498 5-8484 !^'S':>ZS 5-2691 4-9993 4-7440 10 I 1 •1548 •8535 •5586 •2743 5-0044 •749^ I 2 •1536 -8528 •5583 •2743 -0048 •7498 2 3 •1469 -8469 'ss:^ ^ •2698 •0008 -7464 3 4 •1358 -8367 •5438 •2614 4-9932 7395 4 15 • 1 2 1 3 -8232 •5314 -2499 •9827 7298 15 6 •1047 -8078 •5171 •2366 ■9704 •7184 6 7 •0877 •7920 •5023 -2228 ■9575 -7064 7 8 •0722 ■7774 -4887 -2101 ■9456 ■^^953 8 9 •0603 •7663 •4783 -2004 ■93^ S •6868 9 20 •0531 7596 -4720 -1946 •93 1 ^ -6817 20 I • -0485 '7554 -4683 •191 1 -9280 -6789 I 2 •0458 •7530 •4662 •1893 •9264 •6775 2 3 •0432 •7508 •4643 -1877 •9251 •6764 3 4 •0392 •7473 -4612 -1851 -9227 •6744 4 25 •0334 -7421 •4565 ■1808 •9190 -6710 25 6 •0263 •7356 -4506 •^755 •9141 •6666 6 7 •0179 •7279 •443 <5 -1691 -9083 -6613 7 8 •0093 -7201 •4365 •1626 •9023 •6559 8 9 •0008 ■7^23 •4294 -1561 •8964 -^5° 5 9 3° 5-9922 ■7045 -4223 •1497 •8906 •6452 30 I •9839 •6969 "4^54 ■1435 -8850 -6402 I 2 ■9750 •6889 •4082 -1369 •8790 -6348 2 3 •9654 •6802 •4003 •1298 •8726 •6290 3 4 •9550 •6707 •3917 •1220 -8656 •6226 4 ZS •9437 •6605 •3825 •1137 -8580 '6157 3S 6 -9318 •6497 •3727 •T048 •S^oo -6085 6 7 •9195 -6386 •3627 -0938 •8419 -6012 7 8 -9068 -6273 '35^S -0866 •8336 •5938 8 9 ■8935 •6152 •3417 •0770 -8250 •5860 9 40 •8790 •6023 ■3301 -0666 •8157 '5777 40 I -8627 •5875 •3169 •0547 •8051 •5683 I 2 •8437 '5724 •3014 •0409 •7926 '557° 2 3 •8221 •5508 -2837 -0248 •7782 •5440 3 4 •7982 •5292 •2641 •0071 •7621 •5294 4 45 •7719 •5053 •2424 4-9874 7442 •5132 45 6 •7440 •4799 •2193 •9665 7253 -4961 6 7 •7149 •4535 •1954 •9448 705^ •4783 7 8 -6843 •42.57 •J 703 '9221 -6851 '4597 8 9 •65x7 •3961 •1435 •8978 -6632 •4400 9 5° -6167 •3644 •1148 •8719 •6398 •4189 50 1 •5786 •3297 •0834 •8435 -6141 '3957 I 2 '5i^i •2913 •0485 •8119 •5855 '3697 2 3 •4902 •2493 •0103 7772 •5540 •3412 3 4 •4403 •2038 4-9689 739'5 •5198 -3102 4 [230] . , ,, . -e \ Joint Life Annuitv, at Top. Age of elder Life:- j ^^ s^.j^.^^ ^^^-^j^^^ J^^^^ lM(i 4 PERCENT. ]'aliir.'i of Aimuitics on Tiro Joint Lives, and on the Last Surn'cor of Tico Lives. 76 77 78 79 80 81 lO 4M939 4'25o7 4-0131 37788 3-5509 3-3344 10 I •4989 •2557 •0180 •7836 'S^SS •3388 I 2 •4998 •2568 •0193 •7850 ■5570 '34-34 2 3 •4969 '2543 •0172 •7833 •5557 •3393 3 4 •4907 •2488 •0123 •7789 •55 '8 •3359 4 15 •4819 •2408 •0050 •7723 -5459 •3305 ^5 6 •4713 •2310 3-9961 •7642 •5384 •3237 6 7 •4602 •2207 •9865 •7553 •5303 '3^^3 7 8 •4498 •21 10 •9775 •7470 •5226 •3091 8 9 •4418 •2036 •9706 •7406 •5166 •3036 9 20 •4372 •1993 •9666 •7369 '^^3^ '3^^^ 20 I •434^ •1969 •9644 •7349 •5^H •2988 I 2 •4334 •1959 •^6^6 •7342 •5108 •2983 2 3 •4325 •1952 •9630 •7338 •510/; •2981 3 4 •4308 ■^93 7 •9617 •7327 -5096 •2974 4 25 •4278 •1910 •9593 •7306 •5078 •2958 25 6 •4238 •1874 •9561 •7277 •5052 •2935 6 •7 / '4^9° ■J830 •9521 •7241 •50 J 9 ■2905 7 8 •4140 •1785 •9481 •7204 •4986 •2875 8 9 •4091 •1741 •9441 •7168 -4953 •2846 9 30 ■4043 •1698 ■9401 •7133 •4922 •2817 30 I •3998 •1656 •9364 •7100 •4892 •2791 J 2 •3949 •1613 ■93^5 •7065 •4861 •2763 2 3 •3897 •1566 •9283 •7027 •4827 •2733 3 4 •3839 ■i5H •9236 •6985 •4789 •2699 4 35 •3777 •1458 •9186 •6939 *4749 •2662 55 6 •3712 •1399 •9133 •6892 •47o'5 •2624 6 7 •3646 •1339 •9079 •6844 •4663 •2586 7 8 •3579 •1280 •9026 •6797 •4621 •2548 8 9 •3510 •1218 •8971 •6748 •4578 •2510 9 40 •3437 •I 153 •8913 •6697 •4533 ■2471 40 I '3^5^ •1078 •8847 •6638 -4481 •2425 I 2 '3^5^ •0988 •8766 -6s66 -4417 •2368 2 3 '3134 •0882 •8672 •6482 ■4342 •2301 3 4 •3002 •0764 •8565 •6386 •4256 •2225 4 45 •2856 •0631 •8446 •6279 •4160 •2139 45 6 •2701 •0491 •8320 ■616.5 •4058 •2047 6 7 •2540 •0346 •8190 •6049 •3954 •1953 7 8 •2373 ■0196 •80.H •5928 •3845 •1857 8 9 •2195 •0036 •7911 •5799 •3731 •1755 9 50 •2005 3-9865 •7758 'S(^^3 ■3609 •1647 50 1 •1796 ■9677 •7590 '^S^^ •3475 •1527 I 2 •1562 •9466 •7400 •5343 •3324 •1392 2 3 •1303 •9233 •7190 •5154 -3^5^ ■1241 3 4 •1022 •8979 •C962 •4949 •2971 •J077 4 [•231] Age of elder Life:- Joint Life Annuity, at Top. Last Survivor Anntiitv, at S.de. 217 4 PER-CENT. Values of Annuities on Two Joint Lives, and on the Last Survivor ' of Two Lives. lO 82 83 84 85 86 87 10 3-1347 2-9495 2-7823 2-6259 2-4698 2-3044 I •1389 ■9536 •7863 •6297 ■4735 •3078 I 2 •1406 •9553 •7880 •6316 •4753 •3097 2 3 •1398 •9547 •7877 •6314 •4754 •3099 3 4 •1368 •9521 •7854 •6295 •4738 •3087 4 15 •1320 •9478 •7815 •6260 •4707 •3061 15 6 •I2tj8 •9421 •7763 •6213 •4665 •3023 6 7 •I 189 •935^ •7705 •6159 •4616 •2979 7 8 •I 123 •9296 •7648 •6106 •4567 ■2934 8 9 •1071 •9249 •7604 •6065 •4528 •2899 9 20 •1042 •9222 •7579 •6042 •4507 •2879 20 1 ■1027 •9208 •7566 •6030 •4496 •2869 I 2 •1023 •9205 •7563 •6028 •4494 •2868 2 3 •1022 •9204 •7564 •6029 •4496 •2870 3 4 •J016 •9199 •7560 •6026 •4494 •2869 4 25 •1002 •9187 •7549 •6017 •4486 •2862 25 6 •0981 •9x69 '7533 •6002 •4473 •28^1 6 7 -°955 -9145 •75^^ •5983 •445^ •2836 7 8 •0928 •9120 •7489 •5963 •4438 •2820 8 9 •0901 •9096 •7467 '5943 •4421 •2805 9 30 •0875 •9073 •7446 '59^5 •4404 •2790 30 I •0852 •9052 •7427 •5908 •4389 •2777 X 2 •0827 •9030 •7408 •5890 -4374 •2764 2 3 •0800 •9005 •7386 •5871 •435^ •2749 3 4 •0770 •8978 •7361 •5849 •4337 •2732 4 35 •0737 •8949 -7335 •5825 •4316 •2713 35 6 •0702 •89x7 •7307 •5799 •4293 -2693 6 7 •0668 •8887 •7279 •5775 •4271 •2674 7 8 •063J •8857 ■7^53 •575 1 •4251 •2656 8 9 •0601 •8828 •7227 •5729 •4231 •2639 9 40 •0566 •8/97 •7200 •5705 •4211 •2622 40 I •0526 .8762 •7169 •5678 •4187 •2602 I 2 •0476 •8717 •7130 -5644 •4157 •2577 2 3 •0416 •8664 •7082 •5601 •4120 •2545 3 4 •0348 •8603 •7027 '5553 •4077 •2508 4 45 •0270 •8533 ■696'; •5497 ■4027 •2464 45 6 •0188 •8459 •6898 •5437 •3974 •24x8 6 7 •0104 •8384 ■6831 •5377 •3921 •2371 7 8 •0018 •8307 '6762 '53 15 •3867 •2323 8 9 2-9927 •8226 •6689 '5^5^ •3810 •2274 9 50 •9830 •8140 •661.3 •5^83 •5io8 '375° •2223 50 I •9724 •8045 •6529 •3685 •2166 X 2 •9603 •7937 •6433 '5°^3 •3610 •2I0X 2 3 •9468 •7817 '^325 •4927 '35^5 •2028 3 '^ •93 2 J •7685 •6207 •4822 •3433 •^947 4 [232] Age of elder Life : — Joint Life Annuity, at Top. Last Survivor Annuity, at Side. 218 4 PKllCHNT. I'alncs of Annuities on Two Jinnt Lives, and on the Last Survivor of Two Lives. 88 89 90 91 92 93 lO 2-1295 1-9229 1-6885 1-4471 1. 1 990 0-9307 10 I •1327 •9257 •6910 -4491 •2006 -9318 I 2 •1345 •9274 •6925 •4505 -2017 -9326 2 3 '^35° •9280 •693 1 •4510 •2022 •9330 3 4 •1340 •9274 •6928 •4509 •2022 •9331 4 iS "1319 •9256 •6915 •4^00 •2016 -9328 15 6 •1286 •9230 •6894 •4484 •2005 -9321 6 7 •1247 •9197 -6867 •4463 •1990 -9310 7 8 •1207 •9162 •6858 •4440 •1972 -9297 8 9 •^175 •9133 •6813 •4419 -1936 •9286 9 20 •1157 .9117 •6799 •4408 •1946 -9279 20 I •1148 •9109 •6792 -4401 -1941 '9275 I 2 •1147 •9108 •6791 -4401 -1940 •9274 2 3 •1149 •91 10 •6794 •4403 •1942 •9275 3 4 •I 149 •91 1 1 ■6795 -4404 •1943 -9276 4 ^5 •1144 •9107 -6792 •4402 -1942 -9276 25 6 '^^35 •9100 •6786 •4398 ■1940 -9274 6 7 ■J 121 •9089 •6777 •439^ ■1935 -9271 7 8 •I 108 •9077 •6768 •4384 -1929 -9267 8 9 •1094 •9066 •6759 •4377 •1924 -9263 9 30 •1082 •9055 •^750 ■4370 -1919 -9260 30 I •107I •9046 •674.3 •4364 -1914 •9257 I 2 •1059 •9°37 -6736 •4359 -191 1 •9254 2 3 •1047 •9027 •6738 ■4353 •1906 -9252 3 4 •1032 •9015 -6718 •4346 •1901 •9248 4 35 •1016 •9001 •6707 •4337 -189,- -9244 35 6 •0999 •8986 -6695 •4328 -1888 •9239 6 7 •0982 •8972 •6684 •4319 -j88i •9234 7 8 •0967 •«959 -6673 •43^0 •1875 -9230 8 9 •0952 •8948 •6664 •4303 ■1869 -9226 9 40 •09.38 ■8937 -^^K'^'y -4297 -186.-; -9223 40 i •0922 •8924 -6646 ■4290 •i860 -9220 I 2 •0901 • "8907 ■66.U •4280 •i8,-;3 -9216 2 3 •0874 •888^ .6616 •4267 -1844 •9209 3 4 •0842 •8859 •6595 •4252 •1833 -9202 4 45 •0805 ■8828 •6570 •4232 -i8r9 •9193 45 6 '<^7^^ •8794 '6543 -421 1 -1803 •9182 6 7 •0724 •8761 •6516 -4190 -1787 •9171 7 8 •0684 •8727 •6489 •4169 •1772 •9161 8 9 •0642 •8692 •6461 -4148 •1756 -9130 9 50 •0598 •8657 •6433 •4127 •1740 •9139 50 I '°5S^ •8618 •6403 ■4103 •1724 •9128 I 2 •0495 •8572 -6367 •4076 •1704 ■91 '5 2 3 •04.33 •8';2i -6326 ■4045 •1681 -9099 3 4 •0364 •8464 -6281 •4010 -1656 -9082 4 rrt,,, A r ij T f ( Joint Life Annuity, at Top. [2331 Ace of elder Life : — ]j . ^ a -^ ^ o- ^■10 I Last Survivor Annuity, at Si Side. 219 4 PERCENT. Values of Annuities on Tico Joint Lives, and on the Last Survivor of Two Lives. 94 95 1 .« 1 JO 0-6642 0-4084 °'nS7 10 I •6649 -4088 •^759 I 2 •6654 -4091 •1760 2 3 ■6657 •4093 -1761 3 4 •6658 •4093 -1761 4 15 ■6636 •4093 •1761 15 6 •6652 -4091 •1760 6 7 •6646 -40S8 '^759 7 8 •6638 -4083 •1758 8 9 •6630 •4079 •1756 9 20 •6626 -4076 •^755 20 I •6623 •4075 •1754 I 2 •6622 -4074 •1754 2 3 •6623 •4075 ■^754 3 4 •6624 ■4075 •^754 4 25 •6624 •4075 •1754 25 6 •6623 •4075 •1754 6 7 •6621 -4074 •^754 7 8 •6618 -4072 •1753 8 9 •6616 -4071 •1753 9 30 •6614 -4070 •1753 30 I •6612 -4069 ■J752 I 2 •6610 -4068 '^75^ 2 3 •6609 •4067 ■•1752 3 4 •6607 -4066 -^5' 4 35 •6604 -4065 •1751 35 6 •6601 -4063 •1750 6 7 •6598 -4062 •^749 7 8 ■6595 -4060 •J 749 8 9 ■6593 •4059 -1748 9 40 •6591 -4058 -1748 40 I •6589 •4057 •1748 I 2 •6587 •4056 •1747 2 3 •6583 •4054 -1746 3 4 •6579 -4052 •1746 4 45 •6573 -4049 •1745 45 6 ■6';66 •4045 •1743 6 7 '^559 -4041 -1742 7 8 •6553 -4038 •1741 8 9 •6546 •4034 •1739 9 50 •6539 ■403 1 ■nss 50 I •6533 -4027 •1737 1 2 ■^5^5 •4023 •1735 2 3 •6515 •4018 '^33 3 4 •^505 •40 13 •^73 1 4 [•234] A „^ r.r „t 1^^ T ;f , 5 Joint Life Annuity, at Top. Ase ol elder Lite: — { ■, ^ r, . a • r.--, ( Last burvivor Annuitv, at Side. 220 4 ri:i;cKNT. I'dhics of AniniiliLS on 'J)r,> Joint Lii\s, auil on the Lost Snrviror of Two Juices. 10 II 13 13 14 ij 'i'; 20 f) 280 20-4.355 20-3.303 20^2 150 200941 19-9680 sa '6 •/)011 •4074 •3012 •1853 •0624 •9348 6 7 •4746 •3802 •2731 •1560 •0317 •9027 7 8 •4 189 •3540 •2459 •1277 •0021 •8715 8 y •4242 •3287 •2199 •1004 19-9735 •8415 9 r.o •4005 •3045 •1948 •0743 -9461 •8126 60 I •.'57 7 'J •2815 •1710 •0494 -0199 •78.50 I 2 •nr.r.4 •2595 •1483 •0256 •8948 •7586 2 ,> ■•^■^:^7 •2385 •12G5 •0028 •8708 •7333 3 4 •310.3 •2187 •lOGO 199814 •8482 •7094 4 6^ •2979 •1999 •0865 •9609 •8266 •6865 65 6 •2804 •1821 •0680 •9415 •8061 •6648 6 7 •2010 •1G54 •0507 •9233 •7867 •6442 7 8 •2483 •1-195 •0343 •9059 •7683 •6246 8 9 •233G •134G •0188 •8897 •7510 •6061 9 /O •2107 •120G •0043 •8744 •7347 •5886 70 I •20G8 •107G 19-9908 •8601 •7195 •5724 I 2 •1949 •0957 •9785 •8471 •7056 -5574 2 ' .7 •1S40 •0847 •9672 •8351 •0927 -5436 3 4 •17U •0749 •9570 •8244 •6812 -5311 4 7.1 •1G51 •0G59 •9477 •8145 •6706 •5197 /.'; 6 •1.^)68 •0577 •9393 •8056 •6610 •5092 6 7 •1493 •0502 •9316 •7975 •6522 •4996 7 8 •1423 •04.33 •9245 •7900 •6441 •4908 8 9 •1359 •0370 •9181 •7832 •6368 •4828 9 So •1299 •0312 •9122 •77G9 •6300 •4753 So 1 •124.5 •02G0 •90G9 •7714 •6240 •4088 I 2 •1198 •0215 •9023 •7665 •6187 •4629 2 3 •iir.G •0174 •8982 •7622 •6140 •4577 3 4 •1118 •0137 •8945 •7582 •6097 •4530 4 8^ •1084 •0105 •8911 •7547 •6058 •4487 85 6 •1051 •0073 •8880 •7513 •6021 •4446 6 7 •1018 •0043 •8849 •7481 •.••085 •4405 7 8 •09 8 G •0013 •8220 •7449 •5951 •4366 8 9 •0952 19-9983 •8791 •7419 •5917 •4329 9 90 •0919 •9953 •8763 •7.391 •5886 •4293 90 I •OSS 7 •9926 •8737 •7366 •5859 •4262 I 2 •0858 •9901 •8715 •7344 •5836 •4236 2 .3 •0830 •9878 •8695 •7325 •5816 •4213 3 4 •080G •9858 •8678 •7309 •5800 •4196 4 9.) •0787 •9842 •8664 •7296 •5788 •4182 93 6 •0774 •9831 •8655 •7288 •5780 •4174 6 [205] . f ij T T \ .Niint Life Annnitv, at Top. Age of elder Life:- j j ^^^^ ^^^^.:^.^^ Annuity, at Side. 221 4 PER-CENT. Values of Annuities on Two Joint Lives, and on (lie Last Survivor of Two Lives. 16 ^7 18 ^9 20 21 55 19-8i00 19-7119 19-5863 19^4651 19-3483 19-2325 5") 6 •8052 •6757 •5487 •4262 •3083 •1916 6 7 •7716 •6405 •5122 •3885 •2696 •1519 7 1 8 •7389 •6064 •4767 •3519 •2320 •1134 8 i 9 •7073 •5734 •4424 •3164 •1956 •0762 9 1 i 6o •G770 •5417 •4093 •2823 •1606 •0403 60 I •C480 •5113 •3777 •2496 •1270 •0060 I ' 2 •G202 •4821 •3473 •2181 •0947 18-9730 2 3 •5935 •4541 •3181 •1879 •0638 •9413 3 4 •5G82 •4276 •2904 •1593 •0344 -9113 4 65 •5441 •4022 •2639 •1318 •0062 •8825 (5^ 6 •5211 •3779 •2385 •1056 18^9793 •8551 6 7 •4993 •3549 •2144 •0806 •9537 •8289 / 8 •4784 •3329 •1914 •0567 •9292 •8039 8 9 •4587 •3120 •1694 •0340 •9058 •7801 9 70 •4401 •2923 •1487 •0125 •8837 •7576 70 I I •4227 •2737 •1292 18-9922 •8629 •7364 2 •4006 •2566 •1111 •9734 •8437 •7167 2 .3 •3918 •2408 -0944 •9560 •8258 •6986 3 4 •3783 •2264 •0792 •9402 •8096 •6820 4 i 75 •3660 •2132 •0652 •9256 •7947 •6668 75 6 •3547 •2010 •0523 •9122 •7808 •6527 6 7 •3443 •1898 •0404 •8997 •7680 •6397 7 8 •3346 •1794 -0293 •8881 •7561 •8276 8 9 •3258 •1699 •0191 •8774 •7451 •6164 9 80 •3177 •1610 •0096 •8675 •7349 •6060 80 I •3105 •1531 -0012 •8586 •7257 •5967 I 2 •3040 •1461 18-9936 •8507 •7176 •5884 2 3 •2983 •1398 •9869 •8435 •7102 •5809 3 4 •2931 •1341 •9807 •8370 •7035 •5741 4 85 •2883 •1289 •9751 •8311 ■6974 •5679 85 ! 6 •2837 •1238 •9696 •8254 •6915 •5619 6 1 7 •2792 •1188 •9642 •8196 •6856 •5559 / i 8 •2748 •1139 •9588 •8139 •6797 •5499 8 9 •2704 •1089 •9533 •8081 •6737 •5438 9 90 •2663 •1042 •9480 •8024 •6678 •5378 90 I •2627 •1000 •9432 •7972 •6623 •5323 I 2 •2596 •0963 •9390 •7925 •6575 •5273 2 3 •2569 •0932 •9354 •7884 •6531 •5228 3 4 •2549 •0907 •9324 •7851 •6495 •5191 4 95 •2533 •0888 •9302 •7825 •6468 ■5162 95 6 •2524 •0877 •9287 ■7808 ■6449 •5143 6 [-20C] A<'e of elder Life: — j Joint Life Annuity, at Top. ( L:i3t Survivor Anuuitj-, at Side. 222 4 PERCENT. Values of Ainvntlcs on Two Joint Lives^ nnd on the Last Survivor of Two Lives. 22 -3 24 2.5 26 27 .55 19 1159 18-9902 18^8710 18^7418 18 0077 18 4699 5.5 6 0741 •9532 •8275 •0904 5009 4214 6 7 033") •9110 •781S •0525 5155 3740 7 8 18 91) U •8714 •7434 •0099 4717 3293 8 9 9561 •8324 •7035 •5688 4293 2857 9 6o 9194 •7:t50 •0051 •5293 3880 2437 60 I 8844 •75111 •0L'8i ■4915 3497 2035 I 2 8507 •7247 •5931 •4553 3124 1050 2 .3 8184 •0918 •5593 •4-200 2707 1282 .3 4 7878 •0005 •5273 •3877 2428 0933 4 ^K^ 7584 •0305 •4907 •3502 2105 0599 ^S 6 7304 •0020 •4074 •3203 1790 0281 6 7 7038 •5749 •4397 •2978 1503 17 9979 7 8 0783 •5489 •4132 •2705 1223 9090 8 9 0541 •5242 •3880 •2447 0957 9410 9 70 0312 •5009 •3042 •2203 0705 9157 70 I 0097 •4790 •3418 •1973 0409 8914 I 5897 •4587 •3211 •1701 0251 8089 2 3 5713 •4400 •3019 •1505 0049 8481 3 4 5545 ■4229 •2840 •1380 17 9800 8292 4 75 5391 •4073 •2080 •1223 9098 8119 IS 6 5248 •3928 •2538 -1071 9542 7958 6 7 5110 •3794 •2402 -0932 9399 7811 7 8 4993 •3070 •2270 •0803 9200 7074 8 9 4880 •3555 •2159 •0083 9143 7547 9 So 4775 •3449 •2051 •0572 9029 7430 80 I 4iiSl •3354 •1954 •0473 8927 7325 I 2 4 5 'J 7 •3209 •1808 •0385 8837 7231 2 .3 ' 4521 •3193 •1791 •0300 8755 7147 3 4 4453 •3123 •1720 •0234 8081 7071 4 8.^ 4390 •3000 •1050 •0108 8014 7001 85 6 : 4330 •2999 •1594 •0105 8549 0934 6 7 1 4209 •2938 •1532 •0042 8484 0807 7 8 4209 •2878 •1471 17-9979 8419 6801 8 9 ' 4148 •2817 •1409 •9910 8354 6733 9 90 4088 •2750 •1348 -9854 8291 6608 90 I 4032 •2701 •1293 -9798 8233 0008 I 2 3983 •2052 •1244 •9748 8181 0554 2 .3 3938 •2008 •1200 •9703 8130 0507 3 4 3901 •2571 •1103 •9000 8098 6408 4 1 9.5 3872 •2542 •1135 •9038 8009 6438 95 6 3852 •2523 •1110 •9019 8050 6418 6 [207] A lie of elder Life: — j Joint Life Annuity, .it Top. I Last Survivor Annuity, at Side. 223 4 PER-CENT. Values of Annuities on Tico Joint Lives, and on the Last Survivor of Tioo Lives. 28 29 30 3^ 32 33 55 55 18 3293 18 1865 18-0413 17^8938 17 7432 17 5894 6 2793 1 1347 17-9878 •8383 6856 5297 6 7 2309 0847 -9359 •7847 6300 4720 7 8 1841 0363 •8860 •7329 5764 4164 8 9 1390 17 9897 -8377 •6830 5247 3629 9 6o 0957 9449 •7915 •6352 4752 3115 60 I 0541 9022 -7472 •5895 4279 2625 I 2 0144 8611 •7050 •5458 3827 2156 2 3 17 9765 8219 -6645 •5040 3395 1709 3 4 9404 7848 •6261 •4644 2986 1286 4 65 9060 7493 •5895 •4266 2596 0882 65 6 8732 7155 •5546 •3907 2224 0498 6 7 8421 6834 •5216 •3565 1872 0134 7 8 8123 6527 •4900 •3240 1536 16 9787 8 9 7841 6236 •4600 •2931 1218 9458 9 70 7574 5961 •4317 •2639 0917 9148 70 I 7323 5703 •4051 •2366 0635 8857 I 2 7091 5464 •3805 •2113 0374 8588 2 3 6877 5244 •3578 •1879 0134 8340 3 4 6683 5044 •3372 •1667 16 9916 8115 4 75 6504 4860 •3183 •1472 9715 7908 75 6 6339 4690 •3008 •1292 9530 7717 6 7 6187 4533 •2846 •1127 9359 7541 7 8 6045 4387 •2697 •0973 9201 7378 8 9 5915 4253 •2558 •0830 9054 7227 9 80 5794 4129 •2430 •0699 8919 7088 80 I 5686 . 4017 •2316 ■0581 8798 6963 I 2 5589 3918 •2214 •0476 8690 6852 2 3 5503 3829 •2122 •0382 8593 6753 3 4 5424 3748 •2039 •0297 8505 6662 4 85 5352 3674 •1962 •0218 8425 6579 85 6 5283 3602 •1889 •0143 8347 6500 6 7 5214 3531 •1816 •0068 8270 6420 7 8 5145 3461 •1743 16^9993 8194 6341 8 9 5076 3389 •1670 •9918 8116 6261 9 90 5008 3319 •1598 •9844 8040 6183 90 I 4946 3255 •1532 •9777 7971 6112 I 2 4891 3198 •1473 •9717 7909 6049 2 3 4842 3148 •1421 •9663 7855 5992 3 4 4802 3106 •1378 •9619 7810 5946 4 95 •4771 3074 •1345 •9585 7775 5911 95 6 4750 3052 •1322 •9562 7751 5886 6 [208j . r ij T •/• ( Joint Life Annuity, at Top. Age of elder Life:- j ^ast Survivor Annuity, at Side. 224 4 ii:r-cent. Vafiies q/Jinuiitirs o/i Ttro ffniiit Livrs^ awl on (lie Last Survivor of Ttro Lives. 34 35 i i 36 37 38 1 i 39 55 17-432G 17 2729 17 1106 16-9462 16 7799 16 6113 55 6 •3705 2082 0433 •8760 7065 5348 6 7 •3106 1458 16 9783 •8083 6359 4610 1 8 •2r)27 0857 1 9157 •7430 5678 3900 8 9 •1971 0279 8555 •6803 5025 3217 9 6o •1439 IG 9725 7979 •6203 4399 2564 60 I •0930 9196 7429 •5630 3802 1941 1 2 •0414 8691 6904 •5084 3233 1348 3 lG-9980 8210 1 6404 •4563 2691 0782 3 4 •9541 7754 1 5930 •4071 2178 0247 4 65 •9123 7321 1 5479 •3602 1690 15 9739 65 6 •872G 6909 5052 •3157 1228 9257 6 7 •8349 6518 4646 •2736 0789 8801 7 8 •7990 6146 4260 •2336 0373 8368 8 9 •7G50 5794 3895 •1957 15 9979 7958 9 7° •7328 5461 i 3550 •1599 9609 7572 70 X •7028 5150 3228 •1265 9261 7212 J 2 •6750 4862 1 2930 •0956 8941 6879 3 •6494 4597 ! 2656 •0672 8647 6573 3 4 •6261 4357 2407 •0414 8380 6296 4 75 •6048 4137! 2179 •0178 8135 6043 75 6 •5851 3933 ; 1968 15-9960 7910 5809 6 7 •5669 3745 1 1774 •9760 7702 5594 7 8 •5501 3571 ! 1594 •9574 7510 5395 8 9 •5345 3411 14^28 ■9402 7332 5211 9 8o •5202 32G2 1276 •9244 7169 5042 80 I •5073 3130 1138 •9102 7023 4891 I 2 •4958 3011 ' 1016 •8976 6892 4756 2 3 •4856 2905 1 0907 •8863 6776 4635 3 4 •4763 2809 j 0807 •8761 6670 4526 4 85 •4677 2721 0717 •8667 6574 4426 85 6 •4595 2636 0629 •8577 6480 4330 6 7 •4513 2552 0542 •8487 6388 4235 7 8 •4432 2468 0455 •8398 6296 4141 8 9 •4349 2383 0368 •8308 6204 4045 9 90 •4269 2300 0282 •8219 6113 3952 90 I •4195 2224 0203 •8138 6030 3867 I 2 •4130 2156 0133 •8066 5955 3791 2 3 •4072 2096 0071 •8002 5889 3723 3 4 •4024 2047 0020 •7949 5835 3667 4 95 •3988 2009 15 9981 •7908 5793 3624 95 6 •3963 1983 9954 •7881 5764 1 3595 6 [209] 4 r ij T •/• i Joint Life Annuity, at Top. Age of elder Life:- j ^ast Survivor Annuity, at Side. 225 4 PER-CENT. Values of Annuities on Tim Joint Lives, and on the Last Survivor of Two Lives. 40 4i 42 43 44 45 55 16 4407 16-2675 16^0917 15-9140 15-7353 15^5558 55 6 3606 •1837 •0040 •8220 •6388 •4544 6 / 2835 •1030 15^9194 •7333 ■5456 •3566 7 8 2092 •0253 •8379 •6478 •4559 •2623 8 9 1379 15-9505 •7596 •5656 •3696 •1717 9 6o 0696 •8791 •6846 •4869 •2869 -0848 60 I 0045 •8110 •6132 •4119 •2081 ■0019 I 2 15 9425 •7461 •5452 •3405 •1330 14-9229 2 3 8835 •6843 •4804 •2725 •0615 •8477 3 4 8276 •6259 •4192 •2082 14^9939 •7766 4 65 7746 •5704 •3611 •1471 -9297 •7090 65 6 7243 •5178 •3059 •0893 •8689 •6450 6 7 6767 •4681 •2538 •0345 •8114 •5844 7 8 6316 •4209 •2043 14^9826 •7568 •5270 8 9 5888 •3763 •1576 •9335 •7051 •4726 9 70 5487 •3313 •1137 •8874 •6567 •4216 70 I 5111 •2952 •0727 •8444 •6114 •3739 I 2 4765 •2590 •0349 •8047 ■5697 •3300 2 3 4417 •2259 •0001 •7683 •5314 •2897 3 4 4159 ■1958 14^9687 •7352 •4967 •2532 4 75 3896 •1683 •9400 •7051 •4651 •2199 75 6 3652 •1430 •9135 •6773 -4359 •1891 6 7 3429 •1197 •8891 •6518 -4090 •1609 7 8 3223 •0982 •8667 •6282 •3843 •1348 8 9 3032 •0784 •8460 •6065 •3615 •1108 9 80 2857 •0602 •8270 •5866 •3406 •0888 80 I 2700 •0439 •8100 •5688 •3218 •0690 I 2 2561 •0294 •7948 •5529 •3051 •0515 2 3 2436 •0164 •7813 •5387 •2902 •0358 3 4 2323 •0047 •7690 •5259 •2768 •0216 4 85 2220 14^9940 •7578 •5142 •2644 •0086 85 6 2120 •9837 •7471 •5029 •2526 13^9962 6 7 2022 •9735 •7364 •4917 •2408 •9838 7 8 1925 •9634 •7259 •4807 •2293 •9716 8 9 1826 •9532 •7153 •4696 •2176 •9593 9 90 1731 •9433 •7050 •4588 •2063 •9474 90 I 1643 •9343 •6957 •4491 •1960 •9366 I 2 1565 •9263 •6874 •4404 •1869 •9269 2 3 1496 •9192 •6800 •4328 • •I 789 •9184 3 4 1439 •9134 •6740 •4265 •1723 •9115 4 95 1395 •9089 •6694 •4217 •1673 •9062 95 i5 1365 •9058 •6663 •4185 •1639 •9026 6 r.TiAi ^ f u I r t Joint Life Annuity, at Top. [-210] Age of elder Life :- j ^^^^ ^^^^.-^.^^ Annuity, at Side. 2 G 226 4 rEU-CKNT. Vahus of Annuities on Tiro Joint Livis^aml on the List Survivor of Two Lircs. 46 47 48 49 50 5^ 51 15-3707 15^1980 15-0214 14-8455 14-6708 14-4975 SS 6 •2704 •0871 14-9040 •7230 •5427 •3634 6 7 •1678 14-9794 •7910 •6046 •4184 •2332 7 8 •oc;s7 •8754 •0825 •4900 •2983 •1072 8 9 14U731 •7753 •5773 •3796 •1824 13-9855 9 6o •8822 •0793 •4704 •2736 •0710 -8085 60 I •7i)r)0 •5877 •3801 •1723 13-9644 •7564 I 2 •7119 •5002 •2881 •0755 •8626 •64;»3 2 3 •0327 •4170 •2005 13-9833 •7655 •5470 3 4 •5578 •3382 •1175 •8959 •0735 •4500 4 '5'; •4807 •2033 •0387 •8128 •5859 •3578 65 "6 •4193 •1922 13-9039 •7340 •5028 •2701 6 7 •3555 •1250 •8930 •6593 •4241 •1S70 7 8 •2949 •0013 •8259 •5885 •3494 •1081 8 9 •2377 •0010 •7623 •5215 •2787 •0334 9 7° •1840 13-9444 •7027 •4586 •2123 12-9632 70 I •1338 •8915 •0470 •3999 •1503 -8978 I 2 •0870 •8428 •5950 •3457 •0931 •8373 2 3 •0451 •7981 •5485 •2961 -0407 •7819 3 4 •0060 •7570 •5058 •2510 12-9931 •7316 4 75 13-9715 •7200 •4009 •2099 •9497 •0857 75 6 •9391 •0805 •4309 •1720 •9097 •0434 6 7 •9094 •0552 •3979 •1372 •8730 •0046 7 8 •8819 •0202 •3075 •1051 •83'Jl •5687 8 9 •85G7 •5990 •3394 •0756 •8079 •5358 9 So •8335 •5752 •3138 •0485 -7794 ■5056 80 I •8127 •5534 -2907 •0242 -7537 •4785 I 2 •7942 •5339 •2702 •0026 -7310 -4544 2 3 •7777 •5105 -2519 12-9833 -7106 -432 'J 3 4 •7028 •5008 •2354 •9660 •0923 -4135 4 «S •7491 •4804 •2203 •9500 •0755 •3958 85 6 •730O •4720 •2057 •9347 •G5D4 •3787 6 7 •7229 •4589 •1914 •9196 •6434 •3619 7 8 •7101 •4455 •1772 •9047 •6278 •3453 8 9 •0972 •4318 •1029 •8897 •6119 •3286 9 9° •0840 •4180 •1490 •8751 -5966 •3124 90 I •0732 •4000 •1304 -8618 •5826 •2978 I 2 •0030 •3959 •1251 •8500 -5703 •2847 2 3 •0540 •3804 •1151 •8395 -5593 •2732 3 4 •0467 •3787 •1070 -8310 •5504 •2638 4 95 •GUI -372S •1008 •8245 •5435 •2567 95 6 •0373 •3687 •0905 •8200 •5388 •2517 6 I'^llJ Age of elder Life:- Jdiiit LitV Aiiiiiiity, .'il 'I'lip. Last Survivor Auuuity, at Side. 227 4 PERCENT. Values of Annuities on Two Joint Lives, and on the Last Survivor of Two Lives. 52 S3 54 55 56 57 55 U-32r)6 14-1560 13-9891 B^ioBi 7*9496 55 6 •1852 •0092 •8358 13-6656 •81 15 6 7 •0490 13-8665 •6866 •5097 13-3363 7 8 13-91G9 -7281 •5417 •3581 •1779 13-0015 8 9 •7891 •5941 •4012 •2110 •0240 12-8406 9 6o •66G2 •4650 •2658 •0689 12-8751 •6849 60 I •5483 •3411 •1356 12-9323 •7318 •5347 I 2 •4355 •2224 •0108 •8011 •5941 •3902 2 3 •3279 •1090 12-8914 •6754 •4620 •2515 3 4 •2256 •0012 •7778 •5558 •3360 •1190 4 65 •1282 12-8985 •6695 •4416 •2157 11-9923 65 6 •0357 •8008 •5663 •3328 •1009 •8713 6 7 12-9479 •7081 •4683 •2292 11^9916 •7560 7 8 •8646 •6200 •3752 •1307 •8875 •6460 8 9 •7856 •5364 •2868 •0372 •7886 •5414 9 70 •7114 •4578 •2036 11-9491 •6953 •4427 70 I •6421 •3844 •1258 •8667 ■6080 •3502 I 2 •5781 -3166 •0539 •7905 •5272 •2645 2 3 •5194 •2544 11-9879 •7205 •4529 •1857 3 4 •4661 •1979 •9280 •6568 •3853 •1140 4 75 •4176 •1464 •8733 •5988 •3237 •0485 75 6 •3727 •0989 •8229 •5452 •2667 10^9879 6 7 •3316 •0552 •7765 •4958 •2142 •9321 7 8 •2936 •0149 •7336 •4503 •1658 •8806 8 9 •2586 ir9778 •6942 •4083 •1211 •8330 9 80 •2266 •9438 •6581 •3699 •0803 •7895 80 I •1979 •9133 •6256 •3354 •0435 •7503 I 2 •1724 -8862 •5968 •3047 •0108 •7155 2 3 •1496 •8619 •5710 •2772 10-9815 •6842 3 4 •1290 •8401 •5478 •2525 •9551 •6561 4 85 •1102 •8201 •5265 •2298 •9309 •6303 85 6 •0921 •8009 •5060 •2080 •9076 •6054 6 7 •0743 •7819 •4859 •1865 •8847 •5809 7 8 •0568 •7633 •4661 •1654 •8622 •5568 8 9 •0391 •7445 •4461 •1441 •8394 •5324 9 90 •0219 •7263 •4267 •1234 •8173 •5089 90 X •0064 •7098 •4092 •1047 •7974 •4875 I 2 ir9926 •6952 •3936 •0881 •7797 •4686 2 3 •9804 •6823 •3799 ■0735 •7641 •4519 3 4 •9705 •6718 •3687 -0616 -7514 •4384 4 95 •9630 •6638 •3602 -0525 •7417 •4280 95 6 •9578 •6583 •3544 -0463 •7351 •4209 6 [212] A f Irl T -f • \ •^°''*' ^''"'^ Annuity, at Top. ° - j Last Survivor Annuity, at Side. 228 4 rKUCKNT. II/Zma-? of Annuities on Tiro Joint Linus, uml on t/ie Last Survivor (if Tn-o Lives. 5« .■;9 60 61 62 1 ^'3 5.1 77841 7-6114 7*4327 7^2486 7-0599 6-8673 5^ 6 •6. 128 •4869 •3150 •^37'^ 6-95.';4 •7692 6 7 •5148 *35,';9 •1908 •0203 •8449 -66.1.5 7 8 •2181 •0601 6-8966 •72S2 •5554 8 9 12GG13 6-9224 •7661 •6047 •4390 9 60 •4985 12 -3164 •6296 •4755 •3169 60 I •3413 •1520 11-9677 •3409 •^895 I 2 •18S •5369 ^^(^1 •1785 •0061 •8403 4 65 •6213 •4496 2761 •1043 3-9378 •7775 65 6 •5220 '?>yis 1909 ■0257 •86,H •7J09 6 7 •4168 •2.-;97 1002 3'94i8 •7879 •<5394 7 8 •3048 '^S:^^ 0032 •8518 •7045 •5623 8 9 •J 840 •0423 3 8978 7854 •6487 '(i^2>?> •4777 •3865 9 7° 3-9222 •5154 70 I 8-3495 6672 ■5379 •41 18 •2897 I 2 •1795 7.()Q01 •4238 •3048 •1894 2 I Jo.)4: — 3 •0-209 -8174 7 6247 •1970 •0883 3 4 7-8745 •6638 4640 7-2765 2-9891 4 75 •7391 •5216 3151 -1209 6-9404 7'J 6 -6118 •3877 1746 6-9739 -7871 6-6130 6 7 •4929 -2624 0428 -8359 •6429 -4629 7 8 •3817 •1449 6 9191 -7060 -5071 -3213 8 9 •2777 •0319 8030 -5839 -3791 -1877 9 8o •ISU 6-9328 6951 -4703 -2599 -0630 80 I -0938 •8399 5967 •3665 -1509 5-9489 I 2 -0152 -7564 5082 -2730 ■0526 -8459 2 3 G-9441 •6808 4280 -1882 5-9633 •7523 3 4 -8798 •6124 3554 -1114 -8823 ■6673 4 85 -8199 •5485 2875 -0395 -8065 •5876 85 6 •7617 •4863 2213 5-fl692 -7323 -5094 6 7 •7031 •4239 1547 •8984 -6572 •4302 7 8 -6454 •3616 0880 -8273 ■5817 •3504 8 9 -5857 •2974 0189 •7533 ■5030 •2668 9 90 -5273 •2341 5 9507 •6800 -4246 •1835 90 I -4739 •1762 8879 •6123 •3521 •1063 I 2 •4261 •1241 8313 -5511 -2863 •0360 2 3 •3835 -0777 7806 -4900 -2269 4-9724 3 4 •3489 -0399 7392 •4506 -1778 -9197 4 95 •3224 -0109 7074 •4157 •1398 •8789 95 6 •3040 5-9907 6851 •3912 •1130 •8500 6 roiRT A r ij T ■/• < Joint Life Annuity, at Top. 1^15] Age of elder Life: — K-, .c • a •» .c ° ( Last biir\ivor Annuity, at S Side. 231 4 PER-CENT. Values of Annuities on Two Joint Lives ^ and on the Last Survivor of Tuoo Lives. 55 76 77 78 79 80 81 55 4-0716 3-8703 3-6712 3-4725 3-2770 3^0896 6 •0386 •8404 •6442 •4482 -2551 •0700 6 7 •0030 -8081 -6150 •4219 -2315 •0488 7 8 3-9^45 •7732 •5834 •3933 •2058 ■0257 8 9 •9227 -735^ •5490 •3622 •1778 •0004 9 6o •8781 •6945 •5121 •3288 •1476 2-9732 60 I •8.507 •6514 •4728 •2933 •^K^5 ■9442 1 2 •7807 •6057 •4313 -2556 •0814 -9134 2 ,3 •728,3 •5579 •3877 •2161 •0456 •8811 3 4 •6737 ■5080 •3423 -■1749 •0084 -8474 4 65 •6162 ■4555 •2945 •1314 2-9690 •8118 65 6 •5550 •3995 ■2435 -0851 •9271 -7739 6 7 •4894 •3394 •1886 -0352 •8818 •7328 7 8 •4 1 S3 •2742 •1289 2-9807 •8322 •6879 8 9 •3400 •2020 •0625 •9200 -7768 -6373 9 70 •2554 •1236 2-9902 •8535 -7159 •5816 70 I •1652 •0398 •9127 •7820 •6502 •5212 I 2 •o/~i5 2-9526 •8317 •7071 •581 1 ■4576 2 3 2-9769 •8(542 *7495 •6309 •5106 -3925 3 4 -8840 '7775 •6687 -5560 -4413 •3284 4 75 •7938 •6932 •5902 •4831 -3739 •2661 75 6 •6061 •5089 -4075 •3037 •2012 6 7 6-2916 •4258 •3301 •2319 -1345 7 8 •1442 5-9766 •2501 •1574 •0652 8 9 •0049 •8316 5^6670 •0786 I •991 7 9 80 5-8748 •6959 •5258 5^3639 •9152 80 T •7554 •6477 •5714 •4589 •3961 •2788 •2289 •1067 5-0715 4-9445 1 2 4^7946 I 2 3 •5496 •3564 •1719 4-9953 •8286 •6744 3 4 •4606 •2633 •0747 -8940 •7233 •5651 4 85 •3770 •1758 4-9833 •7986 •6240 •4622 85 6 •2948 •0896 •8930 •7043 •5256 •3600 6 7 •2113 •0018 •8008 •6076 •4246 •2548 7 8 •1270 4-9128 •7070 •5090 •3212 •1468 8 9 •0384 •8191 •6079 •4043 •2108 •0308 9 90 4^9499 •7251 •5082 •2984 ■0986 39125 90 I •8677 •6376 •4150 •1992 3-9931 •8006 I 2 •7927 •5576 •3297 •1080 •8956 •6967 2 3 •7246 •4848 •2519 •0246 •8059 •6007 3 4 •6681 •4244 •1872 3-9549 •7309 •5199 4 95 •6242 •3773 •1366 •9005 •6720 •4561 95 6 •5930 •3437 •1006 •8617 •6298 •4103 6 [216] Age of elder Life: Joint Life Annuity, at Top. Last Survivor Annuity at Side. 232 4 rilKCKNT. VnhiiS of Annuities on Ttro rlaint Livrs^ and on the Ldsf Surriror t>f Tw ■9090 •7540 •5997 •4359 95 6 •2082 0214 •8527 •0949 •5372 •3705 6 [•J17J , r 1 . I T \ Joint Life Annuity, at Top. Age of elder Life:- j ^ast Survivor Annuity, at Side. 233 4 PER-CENT. Valv£s of Annuities on Two Joint Lives ^ and on the Last Survivor Of Tii:o Lives. 88 89 90 91 92 93 $5 2-0288 1-8401 i^623i 1-3972 1-1628 0-9063 SS 6 •0205 •8333 -6177 •3930 ■1597 -9042 6 7 •on "5 -8259 -6117 •3885 •1564 -9020 7 8 •0016 -8176 -60^2 •3835 •1527 •8995 8 9 1-9904 -8084 •5978 •3778 •i486 •8967 9 6o •9783 •7983 •5897 •3715 •1440 -8936 60 I •9652 •7S73 •5809 •3647 •1389 •8901 I 2 •9512 •7756 'S1H •3573 ■1334 •8864 2 3 •9365 •7633 •56^4 •3495 •1276 -S824 3 4 -9213 -7506 '5S^^ •3415 •1217 -8783 4 65 -9<^SS •7374 •5406 •3332 '^^SS -8740 65 6 -8887 •7234 •5294 •3245 •I 09 1 -8696 6 7 •8705 •7083 •5173 •3152 -1022 •8649 7 8 -8502 •6915 •5039 •3050 -0947 ■8599 8 9 -8264 -6717 •4881 •2927 •0858 •8538 9 70 •7992 •6489 •4696 •2784 •0752 -8467 70 I •7687 •6229 •4485 -2618 •0629 -8382 I 2 •13SS •5946 •4251 •2433 •0489 -8285 2 3 •7009 •5649 •4005 •2236 •0338 •8178 3 4 -6668 '53S5 •3762 •2041 -0189 •8072 4 75 •6338 •5074 •3530 •1856 •0049 •7974 IS 6 •5988 •4774 •3282 •1658 0-9898 •7868 6 7 •5623 •4460 •3023 •1452 •9742 •7759 7 8 '5^?>S •4126 •2746 •1232 •9575 •7642 8 9 -4808 •3755 ■2437 •0983 •9385 •7508 9 8o ■4347 ■3351 •2096 •0705 -9170 •7356 80 I •3872 •2932 •1738 ■041 1 •8940 •7189 I 2 •3415 •^5^5 •1389 •0121 -8711 -7022 2 3 •^91^ -2132 •1049 0-9837 •8484 •6854 3 4 •2589 •1788 -0752 •9589 •8284 •6706 4 85 •2243 •1483 -0491 •9372 •8112 •6579 85 6 •1903 •1185 -0240 •9165 •7949 -6460 6 7 •1522 •0854 0-9961 •8939 ■7774 •6332 7 8 •0491 -9661 •8702 •7596 •6212 8 9 3-0441 •9205 •8331 •7311 •6010 9 90 2-8894 2-7250 -7790 •6881 •5695 90 I •7407 -5678 2-3842 •6360 ■5310 I 2 •6003 -4188 •2241 2-0316 •4837 2 3 •4676 •2778 •0716 1-8655 1-6618 3 4 •3536 •1563 1-9396 -7207 -5012 1-2781 4 95 •2617 •0584 •8325 -6026 -3691 -1262 95 6 -1938 1-9863 -7532 -5145 -2703 -0106 6 roioT A f ij f T i Joint Life Annuity, at Top [218] Age of elder Life : — {, .^ ■ * " •• 1 *■ ■* ° ( La«t Survivor Annuity, at Side. 2 H 234 HM. 4 rKK-("ENT. Values of Annnitos on Tiro Joint Licvn^ and on the Last Burviwr of Tiro Lives. 94 9.S 96 5.? 0-6493 0-4007 0-1729 5S 6 •6480 •4000 -1726 6 7 •6466 •3993 -1724 7 8 •<'h.';i •3985 -1721 8 9 •6434 •3976 •1718 9 6o •6415 •39^3 •1714 60 1 •6393 •39.';4 •1710 I 2 •6370 •3942 ■1705 2 3 ■6345 •3929 -1700 3 4 •6319 ■3915 •1695 4 65 •6293 •3901 -1689 6.5 6 •6265 •3886 -1684 6 7 •623(5 •3870 -1678 7 8 •6205 •3854 -1672 8 9 •6168 •3835 -1665 9 70 •6124 -3812 •1656 70 1 •6071 •3784 •1646 I 2 •6010 •3751 -1634 2 .3 •5943 •3715 -1620 3 4- •5874 •3677 -1605 4 7.^ •5812 •3643 -1592 7.5 6 •5744 •3606 •1578 6 7 •.5674 •3568 ■1.564 7 8 •5600 •3529 •1.549 8 9 •5516 •3483 '^53^ 9 80 •5417 •3429 •1511 80 I •5308 •3369 -1487 I 2 •5199 •3308 •1464 2 3 •5086 •3243 ■1438 3 4 •4986 -3186 'H^S 4 85 •4901 •3138 •139.5 8.5 6 •4822 •3093 •1378 6 7 •4737 •3044 •J358 7 8 •4663 •3005 ■'344 8 9 ■ "4536 -2938 •1319 9 90 •4326 ■2820 •'273 90 I •4069 -2673 •1214 I 3 •3754 -2498 •I 146 2 3 ■3274 •2216 -J 830 -1032 -0870 3 4 4 95 0-8959 -0641 9.5 6 1 -7579 0-5231 6 ,,,,., . f ij T 'i- \ Joint Life Annuitv, at Ton [1'19J Age of eldei Lile:- j ^ast Survivor Annuity, at i Side. "S^. AUXILIAKY TABLES. 286 Auxiliary Series, for the formation of Survivorship Assurance Tables. X Log A — A Log A —A X 10 3-692329 910 331 089669 0-00 1 068 999 801 000 J 99 10 I •602660 919849 080 151 -000 869 999853 000 J 47 I 2 •522509 943464 o':^6<,o,(i -000 722 999912 000 088 2 3 •4*55 973 976378 023622 •000 634 9999671000033 3 4 •442351 016929 983 071 •000601 000 024 999976 4 15 •459 280 053590 946410 •000625 000081 999919 15 6 •512870 077902 922098 •000 706 coo 140 '999 860 6 7 •590772 091 833 908 167 •000 846 000 199 '999 801 7 8 •682 605 079 191 920 809 •001 045 000 208 999 792 8 9 •761 796 042300 957700 •001 253 000 128 999872 9 20 •804 096 026524 973476 •001 381 000087 9999n 20 1 •830620 007 685 1 992 3 15 •001 468 000027 999 973 I 2 •838305 994 855 005 145 •001 495 999981 000 019 2 3 •833 160 991 862 008 138 ■001 476 999 973 000 027 3 4 •825022 999384 000 616 •001449 999998 000 002 4 25 •824406 003 616 996384 •001 447 000012 999 988 25 6 •828022 014 no 985 890 •001 459 000 048 999 952 6 7 •842 132 016615 983385 •001507 000 059 999941 7 8 '^58 747 015 677 984323 •001 566 000057 999 943 8 9 •874424 016 711 983 289 •001 623 000 063 999 937 9 30 •891 135 010 899 989 lOI •001 686 000 044 999956 30 I •902 034 010 248 989 752 •001 730 000041 999 959 I 2 •912 282 009 628 990372 •001 771 000 040 999960 2 3 •921 910 on 407 988593 •001 811 000 048 999952 3 4 •933317 013 691 986309 •001 859 000 059 999941 4 35 •947 008 016415 983 585 -001 918 000 074 999926 ZS 6 •963 423 016 767 983 233 -001 992 000 078 999922 6 7 •980 190 014437 985 563 -002 070 000 069 999 93 1 7 8 •994627 013302 986 698 -002 139 000 068 999932 8 9 2-007929 009651 990349 •002 207 000 049 999951 9 40 •017580 007 640 992360 •002 256 000 040 999 960 40 I •025 220 010245 989 755 •002 296 000 054 999 946 I 2 '0354^5 015767 984 233 -002350 000 086 999914 2 3 •051 232 1 6 643 983357 -002 436 000 096 999904 3 4 •067 875 023533 976467 -002532 000 139 999861 4 45 •091 408 026 192 973 808 •002 671 000 167 999 833 45 6 •117 600 025184 974816 •002838 000 169 999 83 1 6 1 •142784 023 093 976907 -003 007 000 164 999 8.36 7 8 •165877 023 J 99 976801 -003 171 000 172 999 828 8 9 •189076 020675 979325 •003 343 000 162 999 838 9 50 •209751 019483 980517 -003 505 000 161 999 839 50 I •229 234 022 716 977284 •003 666 000 195 999 805 I 2 •251950 025 728 974272 -003 86j 000 235 999 765 2 3 •277678 026 196 973 804 •004 096 000 253 999 747 3 4 •303874 028 270 1 971 730 -004349 000 292 999 708 4 237 Auxiliary Series^ for the formation of Survivorship Assurance Tables — (continued). X Log (i>x-'-l) A — A Log A — A X 65 2-332 144 028991 971 009 0*004 641 000318 999 682 55 6 •361 135 029438 970562 •004959 000347 999653 6 7 •390573 029378 970 622 •005306-000368 999632 7 8 •419951 032 120 967 880 •0056741000431 999569 8 9 •452071 033441 9(^^559 •006105 000486 999514 9 6o •485512 034300 965 700 •0065911000537 999463 60 I •519812 035027 964973 •007128 000594 999406 I 2 •554839 035673 964327 •007722 0006551999345 2 3 •590512 033 895 966 105 •008377 000673 999327 3 4 •624407 032673 967327 •009 050 000699 999301 4 65 •6^7080 031726 968274 •009 749 000730 999270 65 6 •688806 031 422 968578 •010479 000 776 j 999 224 6 7 •720 228 029719 970 281 •on 255 000 786 999 214 7 8 •749947 034156 965 844 •012 041 000971 999029 8 9 •784103 037517 962483 •013 012 001 155 998845 9 70 •821 620 041 817 958183 •014 167 001 4061998 594 70 I •863437 045 107 954 893 •015573 001 671 998 329 I 2 •908 544 047367 952633 •017244 001 945 998 055 2 3 •955 91 1 045591 954409 •019 189 002 072 997928 3 4 I'ooi 502 036289 963 711 •021 261 001 804 998 196 4 75 •037791 037880 962 120 •033 065 002043 997957 75 6 •075671 036 742 963 258 •025 108 002 148 997 852 6 7 •II24I3 035349 964651 •027 256 002235 997765 7 8 •147762 038295 961 705 •029491 002 620 997 380 8 9 •186057 042 124 957876 •032 III 003 142 996 858 9 80 •228 181 045 281 954719 •035 253 003 704 996 296 80 I •273462 042 056 957 944 •038957 003 773 996 227 I 2 •3155^8 042 950 957 050 •042 730 004 210 99S 790 2 3 •358468 036417 963 583 •046 940 003 874 996 126 3 4 •394885 029408 970592 •050814 003347 996 6S3 4 85 •424 293 025 161 974839 •054161 003 027 996973 85 6 ■449 454 028 792 971 208 •057188 003 657 99^ 343 6 7 •478 246 019482 980518 •060 845 002597 997403 7 8 •497 728 032524 967476 •063 442 004 ^66 995 434 8 9 •530252 058392 941 608 •068 008 008966 991 034 9 90 •588644 0694:4 930586 •076974 012 056 987 944 90 I •658058 ^ 075 603 924397 •089 030 015 043 984957 I 2 •733661 118 623 881377 •104073 028 12S 971865 2 3 •852 284 160397 839 603 •132 208 048 151 951849 3 4 0-012681 231 621 768379 •180359 093 233 906 767 4 95 •244302 403515 596485 •273592 234563 765437 95 6 •647817 •508 155 6 238 Constants. Numbers. Logarithms. r 1-r V 1-r 3 pcr-cent. 3.^ .. 4 4i » 5 6 •970874 •966184 •9'5^538 •959938 ■952381 •94339^5 •029126 •033816 •038462 •043062 •047619 •056604 T-9871G3 •98.-)000 •982907 •980884 •978811 •974094 2-404284 •529128 •585027 •G3409G •077781 •752845 END OF TABLES. NOTATION. NOTATION. ' = interest on £1 (or any other monetary unit, sny on 1,) for ono year, f = r := tlic \.ihif of 1 Lojij; (I +/)= thc/nrce of discount ; = the nominal yearly rati- of interest when (interest heing convcrtihlf ni'iintiitly) tlie t-flective nite of interest is i. f'" = the effective rate of interest when the nominal rate of interest i is convertible m times a year. ,•(!) ^ ]3(, jIq half-yearly. »*'■•' = Do. do. quarterly. t = Do. do. momently. 1^ = the nnmher of persons living at the age x according to any mortality table. N.B. Althouirh the subscrii)t form of notation is used hei-e ami throuiirhout this scheme, it will be seen that the notation can equallv well be adopted in its essential features by those who mav jirefer tlie functional form, /(x), «tc. '/j = ^.1 ~\= ^J'^ numV^er of persons living at the age .r-f- 1. "'j- = 'j-+n= t, „ x+n. -1/,. = /,_,= „ ,, j._]. 't — 't-ii — „ „ X — )l. (Ij: = /r — ''j+i= the number of persons dying bctwcm the ages X and :r4- 1. ^(I.r = 'lr+\ '.. + 1 '.r+2 "'/. = '' T+ii^^'jT+n 'j-+;i + I -v/. = '/,_,=/,._,-/. -'v/. = 'lr-i, = lr-,—ls-u+\ hi. = 'j'* ;/ 'xiiz = Uyh '/xv = '-r+l-'y+l ^^xyz = 'x+\Lj+\-lz+\ -Vr, = h-\-^y-\ ~^hi,z = lx-\-h-\-^:-\ fl.rv = 'xij 'xy'=-'x-'ii — '.r f 1 • 'i/f 1 241 V — f] — "7 "+1/ — / 7 7 7 -H^ — .7 — -«/ -«+i7 — 7 7 7 7 "ri/ — ".r-?i'i/— n — 'x'j 'xy — 'x-n-^ij—n — '•x-n+l-'y-n+i n-mj — 7 n-niff ,7 ''X — ''.r+n-OT "x — '■''x+n-m- N.B. Wlien there is iio risk of laisunderstanding, the subscript X, ry, itc. may be omitted. Thus in lieu of a,=.)^-+r2'i?+z;3!i-+ '.T 'x 'x we may write a—v- 4-^2 - +t)<' - -{- jj.x=^ — T ' ~r ^x= J- loge lx=ii^Q force of mortality at the age x. We have approximately bi (omega) = the limiting age, which none of the lives actually attain. „jt;_j. = the probability of a person whose age is x living n years. ;j_j, = the probability of a person whose age is x living 07ie year. n2yxy] = t'le probability of the joint existence of two in years, Pay] persons whose ages are x and y continuing], 1 year. nPxyi} = the probability of the joint existence of ("« years, Pxyz] three lives x, y, z, continuing (l year. ni xy rxy = the probability of the existence of the last jn years, survivor of x and y continuing \ 1 year. ^^— = the probability of one at least of x, y, and z, being alive at the end of n years. nV;^,? = the probability of two at least of x, y, and 2, being alive at the end of n yeai's. "^■^1= the probabOitv of a; dying within-^" qx] ' (1 year. nixy\ = t^e pi'obability of the joint existence of a: (n years, qj.y ) and y failing within \ 1 year. |„y— 1= the probability of the last survivor of a: fn years, qj^-^j ) and y dying within i 1 year. ^1= the probability of x dying before y within] qry) 11 year. )l I 242 ^'nQx = the prol-ability of x 2- .= the values of intercepted annuities depending on the statuses denoted by the suffixes. \ n — 'i' n.V^xyt n't^xync.. .. J ISr.B. If the above annuities are payable in advance, it is under- stood that a temporary or intercepted annuity-due for t years is one in respect of -nhich t years' payments are to be made, thus, t^j; =1+ t-\ax, In fact, whatever the status on which the annuity depends, denotiiig it by X, we have tax=l+ t-\ax: or simply, \t^=^ + ,t-i(t- "We also have ri^jcy = 71-iCi.ryi or simply „!a=„_i'a. nfi'x = 71-1 tO^x or simply ■n-\m- IST.B. If any of the above annuities is payable by half-yearly instalments, this is denoted by appending '" ; thus Ux, ^'xi n\Clx} n^x, \n ^x=i-rax, »x — -+ax- 246 Also that wo have Aa particulai* cases, wc have Obs. The period for which an annuity Ls deferred need not be an exact number of years ; hut if tlio first payment is made after a time r, wliich may be either integi-al or fractional, we shall have ^[a^ = the value of an annuity on x, of which a payment is made fii-st at the time r, and thenceforward at intervals of a year. ^ja* = the value of an anniuty on x, payable half-yearly, of which a paj-ment (=i) is made fix*st at the time r, and thenceforward at intervals of half a year. ^ a?) .^ \ = the values of similar annuities payable respectively by quai-terly and in-\y instalments, the first at the time r. N.B. Wo still have 7 Sj =T- \\cit:, T**jr — T- Jiflx, la'"' 1 Li-) These annuities may be temporary I Q I n'^) ; o") Obs. Any of the preceding annuities may be coinplcie, i.e. con- tinued up to the day of the death of the life or lives, or of the failure of the status, on which it depends. This is denoted by placing a small circle above the a or a. Thus : — Ux, &x, OS) ttx} a?. n Ox) tiUxy 10(7) nCtx) n\tUx) Uryy &C. Obs. If the annuity is supposed to be payable momently, it Is called a continuous annuity, and is denoted by (7. Thus, 247 a±> djcy, &c. n\iflx) n\tdxy) N,B. In this case it makes no difierence wlietlier the anniiity is payable in advance oi* not. Obs. Since a term of n years may be considered as a special kind of life, such that the probability of living over each of the 71 years is certainty, and the probability of dying at the end of the n years is also certainty, denoting such a term by n| , we have f-=: the value of an annuity duiing the joint existence of a life X and a term of n years certain, i. e., = a temporary annuity for n years on the life x, aiH, Ua^ n'x Similarly a=\ — n\ {xyz n\xy av\ I = the value of an annuity duiing the existence of the life X, and the term of n years certain, ■whichever may endure the longer, i.e., ^= the value of an annuity for n years certain, and so much longer as x may Kve. = the value of an annuity for n years certain, and so much longer as either x or y may live. = the value of an annuity for n years certain, and so much longer as x and y shall both live. ■=■ the value of an annuity to continue for n years certain, and as much longer as x, y, 2 . . . . shall all live. = the value of an annuity to continue until the failure of two of the three statuses represented by the lives x and y and the term of n years certain; i.e., an annuity to cease on the death of both X and y within n yeai-s, or at the cxpii*ation of « years if either xov y shall have died ■v\'ithin that term, or on the failvire of the joint existence of X and y after that term. Reversionary Anmiities. = the value of an annuity on the life of x after the death of y, 248 aJjy = do. oil tlic joint lives rif .r ami y aftor the death of z. Oj'i^ = do. ou the life of tlio last survivor of x and y after the death of z. ayJ'r = (Ic- '^H the life of x after the failure of the joint existence of y and z, aj^lx = do. on the life of .r after the death of both y and z. rt„lj^ = do. to commence on the death of y, and to continue jiayable to the end of n yeai".s from the present time or to the death of X, ■whichever shall first happen.* a ^ = do. do. do. whichever shall II ml last happen. a,;^|j = do. on the life of x, to commence on the death of y or at the expu'ation of ?j years, whichever may first hapi^en. a^i = do. on the life of x, to commence on the death of y or at the expii-ation of n years, whichever shall last hapjien. Obs. These annuities may be deferred, temporary, or intercepted. Thus m(^!/[x> [ndyxi mCti\x' N.B. In explanation of the pi-inciple of tliis notation, it is to be observed that the letters or numbei"s placed at the right-hand lower comer of the principal symbol for an annuity or assui^ance, refer exclusively to lives; while letters or numbers placed at the left-hand lower corner refer exclusively to terms of years : also, that a letter or number placed on the left-hand side of the upright stroke, 1 , denotes a life or a term of years, as the case may be, for which the benefit is deferred. Obs. Deferred and intei'cepted annuities may be considered as a special kind of reversionary annuities, if we consider a term of years as a special kind of life, as explained above. Thus Also, n'£ax=«^!M-'?x; * This benefit must not be confounded with the annuity to commence on the death of _y, and to continue payable for wjcars if .v shall live so long; for in that case the term of n y.-ars does not begin to run until the death of y. 249 that is, ;in annuity on the life of x, deferred n yeai'S and then to continue t yeai'S, is equal to an aiuiuity for so much of the joint existence of the life x and the term of n + ^ years reckoned from the present time, as remains after the termination of n years. N.B. In the above reversionary annuities, it is assumed that the first pajTnent of the annuity is made at the end of the year in which the first life (or status) fails; and that the last payment is made at the end of the year preceding that in which the life (or status) during which the annuity is to be enjoyed, fails. If, however, the annuity is to run from the day of the failure of the first life (or status) to the day of the failure of the second life (or status) ; as, for example, from the day of the death of y up to the day of the death of x, the first payment being made one year after the death of y ; the annuity is by analogy called complete, and is denoted by ay\xi Cilxy, ayz\xi ^^^ If the first pajTnent falls due on the day of the failure of the first life (or status), or the annuity is payable in advance, being still sup- posed to be continued to the day of the failure of the second status, it is denoted by k,}x, ^z\xy, ^z\xi &C' If these annuities are payable half-yearly, quarterly, &c., this Avill be denoted in the same way as for annuities on single lives. Thus, ay £ = the value of an annuity on the life of x, of which a payment, i, is first made six months after the death of y, and thenceforward half- yearly, a proportionate part of the half-year's annuity being payable on the death of x. the value of an annuity on the life of x, of which a payment, \, is first made on the day of the death of y, and thenceforward half- yearly, a proportionate part of the annuity being payable on the death of x. Contingent Annuities. aj^i^ = the value of an annuity on the life of x after the death of y, provided he die before z. Oy,!_j = the value of an annuity on the life of x after the death of y, provided he die after z. a^j ^ = the value of an annuity on x after the death of y. provided he die within n yeai-s. 2 k 250 dxyz rys Cxyxlxyz — Survivorship Atumitics. the value of au annuity to commence on the death of the first of x and y, and continue to the death of the second, the value of an annuity to commence on the death of the first of the three lives x, y, and z, and continue to the death of the last, the value of an annuity to commence on the death of two of the three lives x, y, and z, and continue to the death of the last. the value of an annuity to commence on the death of the first of the three lives x, y, and z, and continue so long as the other two are jointly in existence. Annuities Certain. a;^ = the value of an annuity certain for n years. Ooo = a perpetuity. Aclm = 8' pei-petuity deferred n years. „|a<:^'^"| = a temporary annuity certain of t years deferred or a;;i|r^J « years. A7iniiities on Successive Lives. N.B. T\^len a letter or number is placed within brackets, ( ), the status denoted by it is to be nominated at the expiration of the status denoted by the letter or number immediately preceding. flw^j = the value of an annuity on the successive lives of X and y ; y being nominated at the death of x. a— =r = the value of an annuity on the life of x, and for n years longer. a ^., - = the value of an annuity on the successive lives of 3-, y, 2- a—=r = the value of an annuity for the joint lives of x and y, and for n years longer if y shall live so long. the value of an annuity on the life of y, who is to be nominated on the death of x. the value of an annuity for n years certain, com- mencing at the death of x. 251 a^^\g) = the value of au annuity on the life of z, who is to be nominated on the failure of the successive lives of X and y. <^x\(Jt\)y = the value of an annuity to commence on the death of X, and continue for ?? years if y shall live so long. a ;-=]— = the value of an annuity to commence on the death of X, and continue for 7i years certain and as much longer as y may live. Obs. Any of the preceding annuities may be continuous, in which case « is used instead of a ; or they may be complete, in which case a is used. The last five annuities, when complete, run, like ordinary reversionary annuities, from the day of the failm-e of the status, or life, for which the annuity is defen-ed, to the day of the failure of the status, or Hfe, for which it is enjoyed. Asstirances. A3. = the present value of an assurance of 1, payable at the end of the year in which x shall die ; or, simply, an assurance on x. N.B. An assurance is always understood to be for 1 unless otherwise stated. Obs. An assurance may, just like an annuity, depend on the failure of two or more joint lives, or on the failvire of the last survivor of two or more lives. It may also be deferred, temporary, or inter- cepted. Thus we have assurances Immediate. Deferred. Temporary. Intercepted. Ax n\-^x [n-^x «!'A, ■^xy ■^xi/s xyi A-2 xyt A— .tl-j-w _ ft ^{xy)z — the value of an assurance payable on the death of z or on the death of the last survivor of x and y whichever may first happen. the value of an assurance payable on the failure of the joint lives x and y or on the death of r which- ever shall last happen. -'A, = A,_, n -m A A 252 'A„, = A ■r+l-l/+l Similarly for the other assurances. A'r^ _ A, ^ A, ^ the values of assurances payable at the end of that ?H-part of a year in -which the life or status shall fail. = the values of continuous assurances, or assurances payable at the instant "when the life or status shall fail. or A,;;! In^ry) or Axyii]. U-^xy'^ or A;^.^^ t^n-t-^x' or t'A^rnl ■°-Xj/ X\r- u or Endoivment Assuyatices. the value of an assurance on the joint duration of the life x and the teiin of 7i years certain ; the value of an endowment assurance on x, payable on his attaining the age x+7i, or at the end of the year in which he shall pre\'iously die. the value of an endo^^^nent assui^ance payable in 11 years if both x and y are then alive, or at the end of the year in wliich either x or y shall previously die. the value of an endowment assiu'ance payable in 71 years if either x or y is then alive, or at the end of the year in which the survivor of them shall previously die. the value of an endowment assurance deferred t years, that is, an assurance payable if x shall die after t years but before w years, or if he shall attain the age x-\-7i. Contingent Assurances. the value of a contingent assurance on x against y, J. e. payable at the end of the year in which x shall die, if he leave y sxirvivuig. the value of an assurance payable at the end of the year in wliich x shall die provided he die after y. 253 Obs. These assurances may be deferred, temporary, or inter- cepted : — Deferred. Temporary. Intercepted. n\-^xyy U-^ry> n\t^xy «l-^yU> Iji-^lar* n*t-^\x They may also be modified in other respects as ordinary assur- ances. Thus ^ ly> A. l.^^ or Ay(^, denote assurances payable at the end of that m-part of a year in which X shall die first and second respectively of the two lives. And ^i,' ^L, ov A,!„ denote continuous contingent assui-ances payable at the instant when X shall die first and second respectively of the two lives. Ai -^ = the value of an assurance on x provided he die before y or Avitliin i years after him. A-^l = the value of an assurance on x provided he die more than i years after the death of y. = the value of an assurance on y who is to be nomi- nated on the death of x. = the value of an assurance to be payable n years after the death of x. or -^xliiH)) We have A^ = the value of 1 payable at the end of 71 years pro\dded X be then alive ; = the value of an endowment on x payable in n = nEj years; A.^^ = the value of 1 payable on the death of x provided that take place before the end of ?i years ; = the value of a temporary assurance for n years on x; = In Ax . Just a.sA]r,j + Ajj=Ar,j, we have Ai.;;]+A^=Ax;;],— an equation expressing that an endowment assurance is the sum of a temporary assui'ance and an endowment. 264 A'-3 A" Survivorship yissurances involving Three Lives. first ■ := the vftluo of an assurance on x- provided he die the value of an assur- ance on X pro\'ided that he die second third of the three lives. either 1st or 2ndl ,, 1st ,, 3rd I „ 2nd,, 3rd J A-xy, = the value of an assurance on the joint lives of x, y, s, provided x die firet. Hence A' - '■■^xyz • also A g .yj, — Aj^la Ai^,\ ■^xyt 'first, -= the value of an assurance on the last sur-i second, vivor iM'o%'ided that x die j third. Hence :Aj-i/»— A., yt\x • ■^xyt \ »= the value of an assurance payable on the second death provided that x die first, second, third. Hence A 2 A 2 A- A (fil'St, second, thii'd. .j,^3 = the value of an assurance on the joint lives of x, y, ' r, provided either x or y die first. Hence ^Ty : a -^Tijz- ry:t ^xy:t ay.t Aj^..A = the value of an assui-ance payable on | ^^ " ' the death of x or y provided either-^ 1st „ 3rd, of them die [ond „ 3rd. ■ 1' = the value of an assurance on the last survivor of x and y provided that-i second - z die I thii'd I of the [ three. 255 Hence Aj^,,=A^; 1 1 A— — A — • ■^^xy.z — -^xi/z ) 2 2 A5^.^=A— 2. 3 3 = the value of an assui'ance on the ; last survivor of x and y pro- J vided that z die I ^ the value of an assurance on X provided that he die 1st or 3rd of the 2nd „ 3rd three 2nd • 3rd 2nd or 3rd and that s -die firet of the three. x{yz) ^•■nth =z the value of an [1st or 2nd \ and provided that in assurance on a;i , „ , r the latter case z die 1st „ 3rd I ^ , , provided he die v ) first of the three. IIP 1 1 ( before y, = the value of an assurance on the last survivor of x and y provided that z die! after y. ^ the value of an assurance on x provided that either X or y die first of the three. = the value of an assiu'ance on x provided he die after y, but mthin i years after his death. = the value of an assurance on x provided he die after y, hut more than t years after his death. H ■nr^ or P^ = '^xy or Pxy 'zy2 OV i^xyz 'Bjyy or PjJ] _ la^y or r^y ^ ^\n^x = P|rtAxy'| PlnAjT/aJ PnA, P A* tyi) Annual Premiums. annual premium for an assurance on x. do. do. on joint lives of J ' ', [x, y, and z. [x and y, do. do. on last survivor of -^ {x, y, and z. do. do. on X provided he die before y. do. for a temporary assui'ance for n years on x. J 1 • • 1- of* imtl y, do. do. on joint lives oi ■ {x, y, and s. J 1 1 , . o f-^ and y, do. do. on last survivor of -^ , [x, y, and z. do. do. on X provided he die before y. 256 N.B. If desired, the letter A can bo dispensed with in the case of tempor.iry assurances, by regarding a temporary assiu'ance as a special kind of contingent assurance, as explained above. Thus, P^ =, annual premium for Ai^i] ; or for a temporaiy assur- sxnco for « yeai-s on x. Px;T] = annual premium for Aj^ , or for an endowment on X payable in « yeara Similarly, P_j,1] will denote the annual premium for Aj„"l , that is to say, for an endo^\Tncnt assurance on the life of x, payable in n years. Thus, v= annual premiums for endowment assurances pay- able in 11 ycai*s, depending on the life or lives denoted by the suffixes. If the premixim is payable half-yearly or quarterly, we denote the half-yearly premium by P, and the quarterly premium by P. 2 i The symbols P'-', P'^', P'""), will, by analogy with a'-\ m^xtr,f