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° . asa Ca i ee
INSTITUTE OF ACTUARIES LIFE TABLES.
TABLES
DEDUCED FROM THE
MORTALITY EXPERIENCE
oF
LIFE ASSURANCE COMPANIES,
48 COLLECTED AND ARRANGED BY THE
INSTITUTE OF ACTUARIES
OF GREAT BRITAIN AND IRELAND:
WITH 4X INTRODUCTION EXPLANATORY OF THE CONSTRUCTION AND APPLICATION OF THE TABLES,
AND AN
' APPENDIX CONTAINING A COMPLETE SYSTEM OF NOTATION
FOR LIFE CONTINGENCIES.
Published by the Authority, and under the Superintendence, of the Institute.
a
LONDON: CHARLES & EDWIN LAYTON.
1872.
) YO 4-0. oO. Ft. 2 —U
TPac/ase 3-3- as WNR
PREFACE.
In the Preface to “The Mortality 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 £605, making the total Fund collected £1142. 5s. 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—
1. 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.
V\Eetee ep
iV
2. That 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, H™, HF, and H™F, 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. Woolhouse, 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 © to 4; this was similarly graduated by Mr. Woolhouse. It is designated by the symbol H™(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, 34, 4, 44, 5, and 6 per-cent, should be deduced from the graduated H™ and HF Tables. It was also decided that similar Tables, at 3, 34, and 4 per-cent, should be deduced from the H™(5) 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 has not been made a basis of calculation in the present volume.
" N
Vv
The Tables for Two Lives comprise an entire set of Joint Life Annuities for every combination of ages from (10, I0) upwards, at 3, 34, and 4 per-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. Clergy 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. Queen 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.
——
Vil
CONTENTS.
Page
PREFACE iii
InrRopucrion . ee . 1x
EXxrLaNnaTiow OF THE ADJUSTMENT OF THE TABLES . . » Lexxvil
TABLES. L.—One Life. *.* Under each of the three Mortality Tables, HM, HF, and H{M(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 :—
H*®;— 1 ELEMENTARY TABLE, AND ResuLts 2 THREE Per-Cenrt. 7 Turee anv a Hair » 15 Four ” 23 Four anp A Har ” ‘ 31 Five » 39 SIx
Vili
HF:— . ; ; ; . . ae ELEMENTARY TABLE, AND Resutts. ; . ; . 56 THREE Per-Cent . . ; , . 61 ‘THREE AND A Har » . . . . . 69 Four ” . . . . - 97 Four anp a Har 9 . . ; . . 85 Five - ; ; ; ; - 93 Six - . . . . . IO] HM(5):— . . ; ; ; , , oo, . 109 ELEMENTARY TABLE, AND RESULTS. . . . . ITO THREE Per-Cent . , ; , . 115 THREE AND A Har » . ; . . . 123 Four » ; . ; . . 131
IT.—Two Lives.
*,* Of Two-Life Tables the values given, deduced from H™ only, are those of Annuities on Joint 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™ :—Turze Per-Cent ras 21°
Turee AND A Hatr » . . . . . I71
Four ” . . . . . 203 Auxiuiary TABLES FOR THE FoRMATION OF SURVIVORSHIP As-
SURANCES . . . . . . . - 235
ConsTANTS . . . . . ; . . . . 238
System or Notation ; . . . . . ; . 239
id
INTRODUCTION,
THREE distinct Mortality Tables are, in the present work, made bases cf computation. They are designated respectively by the suggestive symbols, H™, H¥, and H™(), The first two, H™ and HF, 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, published in 1869. The original tables commence at age 0, with a radix of 10,000. The new tables, on the other hand, commence with a radix of 100,000 at age Io, the numbers observed upon, between ages 0 and I0, 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™M(5), 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 HF, 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 H{™ 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 HM, 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, every well-pronounced characteristic in the original is faithfully
reproduced. 6
x
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 H™, follow the Single-Life Tables.
I. 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 ~,, Jz and éy.
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 a,, A, and g,.
The monetary tables, comprising 3, 4 and 5 of the above enumeration, are given at various rates of interest. For each of the tables H™ and HF they are given at the following six rates, viz., 3, 34, 4, 44, 5 and 6 per cent.; and for H™(5) they are given at the three rates 3, 34 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 H™ and the deductions from it, with interest, where this element enters, at 3 per cent.
1. The Elementary Table.
The functions here tabulated for each age are /, and @,, 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
Xi
usual, for what reason is not apparent, to tabulate log d@,; 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/, and logd,. 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, #, being any function of z,
Uzpt Ate t+ Atri t . . . tHAUr a1 = Ugia,
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 logs, (which contains the differences of log /,) we insert log dp, log ¢,5, log fy, &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 Mortalsty only.
These results, as respects H™, are on p.6. They consist of ~,, g- and ¢,, for 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 f, 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. 7
Xil
Since 9.= a, and consequently, log g,= log a,—log /, | =log d,+colog Z, ; therefore A log 7,== A log d, + A colog /, =A log d,+colog f,, (since A colog /,=colog 7,). Now log 7.4,=log 9,+ A log ¢,.
Hence, by substitution,
log Jz41=log 7. + Alog d, + colog pz. That is, we shall pass from log ¢, to log ¢,,, by adding to the former the differences of its components ; and the logarithms of the series g, will consequently be formed by continuous addition.
The formation of the first few terms of the series is here exhibited.
Lo Ps
on (l=gs) Logdyy 2°690196 *Col. 449 5-000000
10 3°690196 ‘004900 = ‘995100 908595 2133
rr 600924 "003990 "996010 918405 1736
12 ‘321065 003319 "996681 942196 1444
13 °464705 "002915 997085 975177 | 1268
14 °441150 "002762 "997238 15680 1201
15 °458031 7002871 "997129
d @ * # Verification. Logd,; 2°450249 Col. bs 5-007782 . 1s 63458031 * Col. is adopted in the illustrative examples as an abbreviation for Co/og, in order to range better with Log.
Xill
Log dj) is set down on the first line, and on the fourth, seventh, and every third succeeding line the differences of log d, in order, commencing with Alogd,,; colog 4 is. set down on the second line, and the differences of colog/,, in other words the successive terms of colog~,, commencing with colog 1), on the fifth, eighth, &c., 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 g.4:=log g,+ A log d,+ colog Z,, therefore, logg,=logg,.,—A log d,—colog p, =log 7211+ A colog d, + log p,,
(Acolog ad, being the negative difference of logd,, 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 7s Gz (19,) Log dpy 0-954243 Col. yy 1-045757
97. 9:000000 __t-e00000 "000000 647817 264047
96 =. 911864 "816327 "183673 332438 559862
95 °804164 "637036 "362964 208517 692583
94 705264 507300 ‘492700 147020 766578
93 ‘618862 "415778 *584222 114799 812035
g2 545696 "351314 "648686 @ @ # # Log dy, 2'404834 Col. fog 3140862 92 1545696 It is hardly necessary to remark that, the column g¢, shewing the annual decrement on a unit 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 1? per cent., or 174 per thousand, and so on. The next result is ¢,, the Average Duration of Life at each age. The column has been formed by the aid of the well-known formula,
€g=P,(1 + Cri); in which e, denotes the Curtate Duration at age x, and from which the Complete Duration, ¢,, is deduced by addition of *5 to each term.
The formation of the first few terms of the series (the last few as the table is arranged) is here subjoined. The initial term, 7 being 0, is log ég=logfo,. The operation is con- ducted by means of the table Log (1+) in Gray’s Zables and Formula, and verification is obtained from point to point by the use of the formula, frei losat 2s os
(= +.
£
xV
Log ez es Cr
97 “0000 "5000
96 1:264047 1837 ‘6837 559862 073225 __f
95 633094 "4296 "9296 692583 155195 28
94 847806 "70.44 1°2044 766578 231564 ___#
93 998144 9957 _1°4957 812035 300081 22
92 112138 1°2946 1°7946
# # # # bogtlogt .. 47 _ 936 log 723
Log936 =. 2971276 Col.723 3140862 92 0-112138
Comparison of the series thus formed with the corres- ponding series formed from the same table, H™, previous to graduation, Experience, pp. 281, 282, gives, for ages 10 to Qo, the following result:—In 20 cases the corresponding terms of the two series are identical ; in 26 they differ by +or; in 11 they differ by +02; in 12 by +'03; in 2 by +°04; in 1 by —°05; in 3 by +06; in 3 by +'08; in 1 by +°10; in 1 by —'I5; and in 1 by —'18.
Or it may be put thus:—The sum of the positive devia- tions is ‘81, and that of the negative deviations is 1°01; 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 unhesitatingly accepted as a faithful 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 HF 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 we have :—3 cases in which the corresponding terms are identical; 15 in which they differ by +'o1; 14 in which they differ by +:02; 14 by +°03; 5 by 4°04; 5 by +°05; 3 by +°06; 2 by —‘07; 1 by +08; 3 by —‘09; 2 by —10; 3 by H°11; 3 by 4°12; 1 by —*13; 1 by +°14; 1 by +°16; 1 by +°18; 1 by —*19; I by +'24; 1 by —°27; and 1 by +°43-
The totals are, of positive deviation 2°48, and of negative 2°12; and we have finally :-—
’ 3 cases of coincidence ;
38 with an average deviation of +065 ;
40 ” ” ” —'053. The deviations are here, as already remarked, confined within less narrow limits than in the case of the H™ table. The cause of this is, that the original HF 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 asperities by which, like other results of original observations, the present table is characterized. The graduated table may therefore be confidently accepted as an exponent of the law of mortality likely to prevail amongst a class similar to that subjected to observation.
xvii
The tables H™ and H™(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 114, that those on the latter page, derived from the table H™(5), are throughout less than those on the former, derived from the table H™. The difference, 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 Commutation 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
and soon. And hence the formule for their application are those of Professor De Morgan’s paper in the Companton to the Almanac for 1840, reprinted in the Fournal of the Institute, vol. xii. p. 328. This form has been adopted after fully
® More exactly four and a half years, on the average.
XVili
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 the funda- mental columns will now be described. And first as to Column D.
We have log D,=log /, + log v*, and log D,,,=log /,,, + log v**' ; therefore 4 log D,=log f, + log v =log up,.
Hence, since log D,,,=log D,+ Alog D,=log D, + log uf,, it appears that the column log D, may be formed in direct order by the continuous addition to log D,) of the terms of the series log vp,. But it is more convenient, by facilitating the formation of Column N, to form log D, in reverse order. From the foregoing we have
log D,=log D,,, —log uf, =log D,4, + colog up,. Hence, commencing with log D»;, the column is formed by the continuous addition of colog v+colog Z,. A specimen of the operation follows :-—
Log v* Log Ds D, and N, 100 62°716278 Loge 2-754789 99 °729115 yy Ig7 §=0°954243 98 741952 ‘709032 . 97 “754789 s7 012837 su7 9 96 767626 735953 33 a aesol 96 “457822 2°8696 012838 paRia 93 ‘806138 3°3813 95 92 818975 440138 91 8318134 95 9107984 81433 90 844650 012837 ak Bo 857487 307417 "524094 88 = 870324 94 ‘231052 17°0236 87 = 883161 012837 38° <aBa 85 Spee 93 ‘477311 30°OI31 . 012837 TRcGla 187965 5850139 92 678113
eee ee eee 7
xix
58°5613 92 678113 47°6555 012838 Toe a1 tA 162878 106°2168 ro g1 °853829 + 71°4215 012837 yo>-aRo 149337 *77'0383— 90 go 009003 102°095 012837 : 126794 77973389 8g -°148634 140°810 012837 , 118786 #7054388 88 ‘280257 190°659 012837 Sr i-a0a » 114903 611°203 87 8; -407297 255°445 012838 6 107713, 84788 86 ‘527848+ 337°169 012837 3202316 1oagi1 70 BHO BS 85 °642996 439°538 1643°354 84 1d # ® a Log 8 2-908836 ‘a5 3°734160 - «Bg 2642996
The first line is occupied by log v”, and the fourth, seventh, tenth, &c., by cologv; the second line is occupied by log 4, and the fifth, eighth, eleventh, &c., by the successive terms of colog g, in reverse order. Before proceeding to the additions it is necessary to introduce certain corrections. The true value of colog uv 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 Dj, 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 v*, a portion of which, in reverse order to correspond with
the order in which the table is being formed, is given in c2
xX
the margin.* Commencing with logv® (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 log D, corres- ponding to those ages will thereupon have to be similarly increased, since log D,=log/.+logv*. 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 cology next preceding each marked line from 012837 to ‘012838. This correction being madef 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 cology 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 D, 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
log C,=log v,.1+ log d,.
Hence, A log C,=log v+ A log d,.
® A six-figure table of log 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 ¢ 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 log ©, extending to ten places, and therefore suited for every purpose, is given in Mr. William Thomas Thomson's Actuarial Tables, to which work, Mr. Thomson informs his readers, it was contributed by Mr. Filipowski.
+ The corrected figures are printed in heavier type.
Xxi Consequently, for the formation of log C, in direct order, we have log C,,,=log C,+ A log C, =log C,+ log v+ A log a, ; and for the formation in reverse order, which is the more convenient, we have . log Cp=log C,,,+ colog v + A colog d, ; in which Acologd, is the function occupying the column marked —A in the Elementary Table. The following is an example of the formation :—
Log C, C, and M, Log v 3741952 » Gg, 0°954248 9 1:696195 . F GATBIT 490891 96 °356849 . 332438 ~ 243 ——__ 27711 96 208518 78076 gs 94 °'923479 + 8°3845 147020 16°1931 94 93 083336 I2°II 54 114799 383075 8 92 :210972 16°3.544 112362 44°5619 2 9: °836171 21°6856 __93465 66'°2475 g! 90 06 442.473 + 27°6995 889% 93°9470 go 89 539255 34°O143 94270 128'5613 89 88 646362 44°2957 __99304 172'8570 88 87 °758508 57°3460 85411 Ggo'a030 87 86 856751 71°9036 82558 302°1066 86 85 9521414 89°5656 391°6722 85 ® d # & Log ds 3-056142 vw 2-895999
85 1952141
XXil
The formation here is effected in the manner just referred to. The only addends written down are the terms of the series Acologd, (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 v, not being set down, the corrections rendered necessary by its curtailment to six places must in this formation be applied to the terms of Acologa,. 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 wv which there enter into’ D,, enter here into C,.,. 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 C, 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,=vN,,—N:, we have My=vNg—Nes; and the formula is applied as follows :-—
103)1043354( 1595490 oNg.
13 1203816 85°
983 301674
565 391074 Mg; 5°4 92
This process is easier than the logarithmic one, when, as here, division by 1 +4 is used instead of multiplication by v. It also, usually, gives us one or two figures more in the result than can be obtained by the use of logarithms.
A similar relation holds between the Columns S and R. Thus,
. R,= VS2-1— Sx , And by the application of these formule the whole of the Commutation Tables in the present volume have been verified, from point to point.
Xxili
4. Logarithms and Cologarithms of the Principal Columns of the Commutation 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 obvious 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 @,, A, and a, the values of annuities, and the single and annual premiums for assurance of a unit.
They were formed as follows :—
I. ()
.*. log a4,=log N,+ colog D,. Hence, A log a,=A log N,+ Acolog D,, =A log N,+colog vf, ; and log @,,,=log a,+ A log a, =loga,+ A log N,+colog uf,. 2. Similarly :—
and log A,=log M,+colog D,. A log A,=A log M,+colog z,. log Ay, =log Ag+ A log A; =log A, + A log M,+colog v,. 3. And finally :-—
_ M, ve New’ and log #,=log M,+colog Nz).
A log #,= A log M,+ Acoloz N,_). log #,,,=log #w,+ A log w, =log#,+ A log M,+ Acolog N,_;.
xxiv
The following are examples of the three operations :—
Log az
6°254516 5-128372
10 | 1-382888 982268 14971
rr] °380127
982136
14573
12| ‘376836
981984
14281
13} ‘373101
981818
14105
14| ‘369024
981641
14039
15] °364704
981454
14086
16] °360244 e ®
6°145817
5°214427
16 | 1°360244
az
Log A,
4298988 §-128372
24°1484 | 1:427360
23°9953
23°8142
23°6103
23°3897
23°31 581
23°9315 ®
992208 14971
434539 993771 14578
"442883 994922 14281
452086
995637 14105
461828 995961 14039
471828 995896
14086 481810
4267383 5-214427
1-481810
A;
"267523
"271981
"277257
283195
F
"289620 296366
"303256 #
Log w,
4-298988 7727862 2-026850
992208 17622
036680
993771 17732
048183 994922 17864
060969 995637 18016
074622 995961 18182
088765
995896 18359
"103020
4°267383 7-836637 2°103020
Wr
"010638 "010881 “OLII73 "OL1507 "011875 "012268
"013677 #
The initial values are log Ny» + colog Dj, log Mj) + colog Dio, and log M,y)+colog No, the addends being in accordance with the preceding formule. Neither colog N, nor its difference is
in the table.
Nio Dio
Ny
A
They are easily formed, thus:—
1796867
74409 1871276 log 6'272138 col. 7°727862 017623
The verifications at the bottom are log N¢+colog Dg,
log Migt+ colog Die, and log M,. + colog Nis-
XXV
6. On the Extent of the 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 M, 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 a
XXvi
corresponding columns in the tables, these latter being assumed to be correct :—
N. a, | Totals,
48 50 77 235 39 33 10 IIo T 3 e¢@ 4 @s6 I @e I
This means that in log D, there are 60 values correct, and 28 with an error of +1; 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 +1; that is, in the fourth decimal place, three only being usually tabulated.
These deviations from strict accuracy will not be considered very great. In order to determine the cost by which the greater exactitude of the tables as printed has been purchased, an experiment was instituted. Three gentlemen, well prac. tised in the use of both the seven and the six-figure tables, kindly lent their aid. Each of them made a certain numbef of direct entries, first in the seven and next in the six-figure tables. A like number of inverse entries was then made, first in the seven and next in the six-figure tables as before ; the times occupied being carefully noted in each case. Com- bination of the results gave the following ratios of the times occupied in the two cases :—
Direct entries . . . . . . . I: 663 Inverse _,, rr Orie) €;
In both cases the ratio is greater than that of 3 : 2, showing that in each there is a saving in time (to say nothing of the saving in mental effort) of at least one-third, by the use of the six-figure tables, the saving in the case of the inverse operation being the greater of the two.
The practical inference fron. what precedes is that, where
XXVii
such a degree of exactitude as that which was found to have been attained by the use of six-figure logarithms in the con- struction of the Commutation Table is judged sufficient for the purpose in hand, it is nothing else than an improvident waste of time and labour to use the seven-figure tables. Had the present investigation been made previous to the construction of the Commutation Tables, and its result known, in all likeli- hood the employment of seven figures here would not have been thought of.*
7. Miscellaneous Formations.
It will be proper now to give, in the form of Problems, some examples of more general formations, those already given having reference to tabulated functions. The differences of the principal logarithmic functions are given in these tables for the first time; and some further justification of this departure from previous usage than has yet been afforded, will perhaps be looked for.
PROBLEM 1. To construct the series log vf,.
This series presents itself as the series of differences of log D,. It is frequently wanted however when log D, has not been formed, as for the construction of a table of annuities,
* There is at present no English six-figure table, properly so called, available ; which is the reason why the late Professor De Morgan, as he informs us,f on occasions when he suspected five figures to be insufficient, immediately had recourse to a seven-figure table, We have abundance of tables in which the logarithms extend to six places; but as the argu-
“ments extend to no more than four, they are more troublesome to use than the seven-figure
tables, with a figure fewer in the results. The differences in them range from 434 to 43, and for want of room there are no proportional parts, while there are two figures to be proportioned for. A proper six-figure table, on the other hand, has, like our seven-figure tables, five-figure arguments, and consequently gives five-figure results by inspection; the differences range from 44 to 4, affording ample space for the exhibition of the requisite proportional parts; while, moreover, there is never more than one figure to be proportioned for.
It is hoped that ere long an English Table, with various improvements for facilitating its use, will be available. In the meantime a very good German table, possessing the above-described qualifications, and whose title follows, may be had at a very moderate cost :—Logarithmisch-trigonometrische Tafeln, mit sechs Decimalstellen. Vom Dr. C. Bremiker, Berlin, 1869.
+ Penny Cyclopedia, vol. xxiii, p. 499; and English Cyclopedia, Division, Arts and - Sciences, vol. vii., Col. 1005.
XXVIII
for instance, and therefore the method of its formation will be here shown.
Since log vp,=log uv + log 2,, therefore, A log vp,= A log p,. ~ Hence, log vf,,,=logvp~,+Alogp~,; . . . , (1) and, log 7,=log vp,,, + Acolog p, . . . (2)
Of the two formula deduced (1) is for the direct, and (2) for the reversc formation. The latter is the more convenient for two reasons: first, the series is usually wanted in reverse order ; and, secondly, the differences which form the addends in this order, have many fewer significant figures than those that form the addends in the other. An example of each formation is given :—
Log tp, Logv ‘987163 Logv ‘987163 1 Pro ‘997867 » Pog °264047 to °985030 96 251210 397 295815 11 °985426 95 °947024 292 132721 12 °985719 94 “679746 176 73995 13 °985895 93 °753741 67 45457 14 *985962 92 799198 999952 25087 15 ‘985913 g1 *824284 ° ° ° ° Logvy -987163 Logy -987163 — » P15 "998751 » Poi 837122 15 °985914 gr °824285
This series, 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 zv, which enters every term. The true value of logy is ‘987162,775 ...., the values 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 refererice to a six-figure table of log v’*. The requisite corrections are introduced in the example.
XXi1xX
PROBLEM 2. To form a table of loaded assurance premiums,
The formation of w, has been already exemplified. The present formation differs from that referred to in no other respect than that here the logarithm of (1 plus its loading) is included in the initial term.
Thus, to form #, with a loading of 174 per-cent. :—
Log 1175 0:070038 rT) Mio 4°298988 Col.Ny 7°727862
10 §62°096888 "012499 992208 17622
1r *106718 012785 993771 17732
12 °118221 "013129 994922 17864
13 *131007 "013521 * * ° Log 1°r75 0-070038 » Mis 4:279889 Col.Nj, _7°781080 13 2°131007 The premiums here formed may be compared with those for the same ages on p. I4. 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 z+n, the value of the benefit is Desa.
’ #
or, in logarithms, log Dz.,+colog D,. . And the difference of this expression, in which «+ is constant, is A colog D,, or colog vf,,. Two examples follow, in which «+ takes the values 21 and 60, respectively.
Values of Endowments.
Payable at 21, Payable at 60.
LogD,, 4:710940 | Log De 3-999631 Col. Dig 5°128372 | Col. Dig 5°128372 10 61-839312 690736 | ro 61:128003 = +1342 14971 ? ! 14971 4070 11 *854283 714962 rr = 142974 "13898 14573 14573 9 7 12 °868856 739360 12 °157547 "143730 14361 | ca, 13. 883137 764057 13 ‘171828 —_-148535 _ 14105 _ 14105 14 °897242 789300 14 185933 "153438 14039 14039 15 911281 815232 15 ‘199972 "158479 14086 14086 16 925367 842106 16 °214058 e ° e ! # e ® LogD,, 4°710940 | LogDg 3-999631 Col. D,, 5°214427 | Col Dig 5214427 +1647 04 16 1:925367 16 1:214058
PROBLEM 4. To form a table of the assurances equivalent to a present value of a unit.
The value of this assurance, when the age is 2, is,
Dz. M,’ the logarithm of which is, log D,+ colog M,.
And the difference of this expression is, log up, + A colog M,.
The required table may be consequently constructed as follows :—
XXX!
Assnrances whose value 1s a Untt. Log Dip 4871628 Col. Myo 5°701012
10 6 0:572640 3°7380 985029 7792
11 ‘365461 3°6767 985427 6229
12 °557117 3°6068 985719 5078
13 °547914 3°5311 985895 4363
14 ‘538172 3°4528 985961 4039
15 ‘628172 3°3742 # 2 Log Dis 4:799659 Col. M)s 5°728513 15 0:528172
The use of such a table as that of which the formation is here exemplified is to facilitate the conversion of a cash bonus into a reversionary bonus.
Example. A policy on (15) has assigned to it a present bonus of 42°375. Find the equivalent reversionary bonus.
42°375 X 3°3742= 142°982.
This table could be very readily formed from the Assurances on p. 14 by means of Colonel Oakes’s Zable 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.
dr, XXXili
The present age, (or, preferably, the age to which the value to be formed has reference,) being as usual 2, 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 the deferred annuity being
N, D, of which the logarithm is, log N,-+colog D,,
we shall have to form the values of this expression for all the combinations of « and y, in which x does not exceed y._ This can be very commodiously done in a series of paralel collumns. Either of the quantities « and y, as may be 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 zx be made to vary, and the second, if y be made to vary :—
10 96 10 96 10 10 10 10
96 96 9 96
The forms are theoretically cqually eligible. The first, however, is to be preferred, because 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 z at the side.
XXXill
Formation of Deferred and Temporary Annusties.
Io
It
[2
24°1484 24°1484 24°1484 10 (382888 | 24°1484 | 365156 | 23°1823 | 347292 | 22-2480 14971 | 0-966: | 14971} 1:9Q004 23°9953 23°9953 KI 380127 | 23°9953 | 362263 | 23°0284 14573 | 09669 23°8142 r2 376836 | 23°8142 Formation—(continued).
13 14 15 24°1484 24°1484 24°1484 10 | 329276 | 21°3440 | 311094 | 20°4688 | 292735 | 19°6216 14971 | 2°8044] 14971} 3-6796] 14971] 4°5268 ———_—— | 23°9953 23°9953 | —_— | 23°9953 11 | 344247 | 22°0926 | 326065 | 21°1868 | 307706 | 20°3098 14573 | 179027 14573 | 2°8085 | 14573) 36856 23°8142 23°8142]_ ss | 23°81 42 12 | 358820 | 22°8465 | 340638 | 21°9098 | 322279 | 21:0029 —___ | 23°6103 23°6103 23°6103 13 | 373101 | 23°6103 | 354919 | 22°6422 | 336560 | 21°7050 14105 | o9681] 14105} 1:9053 23°3897 23°3897 14 369024 | 23°3897 | 350665 | 22°4215 14039 0°9682 23°1581 15 364704 | 23°1582
The first six columns are here given complete. The initial terms are the successive values of log N,+colog D,, when y is made to vary; and they are formed as follows, the addends being the terms of Alog N, :—
XXXIV
Log Nio 6°254516 Col. Dig 5°128372 10 1°382888
982268 rr - °365156
982136 12 347292
981984 13 329276
981818 14 311094
981641 15 292735 *% *%
Log Nis 6164363 Col. Dio 5-128372 15 1292736
In the columns, x being the variable, the addends are, as already intimated, the terms of A colog D,, or colog uf,, com- mencing in each column with colog vf,>, and continued till the . next value of x at the side is the same as the value of y 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 :—
Log Nj, 6-218920 Col. Dig 5°157916
12 1:376836 981984
13 358820 981818 14 340638 981641
rs ‘322279 #
#
Log Ni, 6164363 Col. Dp 5:157916
1s 1322279
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 ; 2. ¢, 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 Deferved.and Temporary Annutties.
20°4688 | 19°6216
21°1868 |20°3098 | 1 21°9098 |21'0029 | 2 22°6422 |21°7050 | 3 22°4215 | 4
°9682 |————| 16
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 the 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, 238142, 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°8405 + °9677,
21°9098 + 1°9044,
21'0029 + 2°8113, 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 2 years, are to be found ranging diagonally down- wards, commencing opposite #, and under ++. Thus the series of annuities deferred three years, (#=3) are 21°3440, 21°1868, 21'0029, &c.
In like manner temporary annuities, whose duration is the same, # years, are also found, in their own compartment of the table, ranging diagonally downwards, commencing under x and opposite +2. Thus, the annuities to last ‘hree years are 2°8044, 2°8085, 2°8113, &c. -And the annuities in these two sets are complementary, each to each; since the initial value of z is the same in both sets, and also the value of x.
Finally, it may be mentioned that the foregoing table, if completed, would occupy exactly the same space as one of the Two-Life Tables, namely 30 pages.
XXXVI
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 P,, where y is the age at which the policy was effected ; then, the present age being +, the value of the policy is,
M,—P,N,_1 *V I += = Do =D,—(1—v)N,_1—P,N,1
\N =1—(1—v+ P,)—=". | ” Dy
Putting for ae its value, 1+4,, and denoting (1—v+ P,) by
Q, this expression may be conveniently written,
I—O,(l4+as) ee ee (1)
To construct the required table then we shall have first to form the values of Q,(1+4,) for every combination of + and y in which # 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 P, 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 V,|,, where x is the age at which the assurance was effected, and 2 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 facilitate the investigation, the symbol is here modified as in the text,
XXXVI1i
Case I. Let P, be the pure premium at age y. Then M P= ®) =H
=D,—(1—v)Ny1
=r _ (1-1)
N,-1 awe! Oy “Tha, (1~—v). Substituting this in Q,, (1) becomes,
and the preliminary operation here will be the formation of log (1+4,), which will give us also colog(1+a,), since y takes no values different from those taken by z.
A specimen of this preliminary formation follows :
Log(1-++-a,) A Colog(1-+-a,) A *LogN, 6-272138 Col. Dig 5°128372
10 1:400510 997349 ©=—-2°599490 = 002651 982378
14971
11 «397859 = 996841 602 141 003159 982268 14573
12 ‘394700 996417 "605300 003583 982136 14281
13 °391117 996089 60888 3 003911 981984 ; 14105
14° °387206 995857 612794 004143 981818 14039
15 *383063 995727 616937 004273 981641 14086
16 °378790 "621210 Ps Ps
LogNis 6°164363 Col.D,, 5:214427 16 1-378790
* See page xxiv.
XXXI1X
After the examples already given, the above does not seem The addends are A log N,1+
to require explanation. A colog D,.
Observing that A colog (1+.a,) contains fewer significant figures than A log (1+4,;), y, in the construction, is made to vary in the columns; z, therefore, will vary in the rows, and
the initial terms will be formed as follows :—
Log (1+ a4) 1-400510
Col. i, 2°599490 10)6©=—- 9000000 997349
II 997349 996841
12 994190 996417
13 990607 996089
14 "986696 995857
ts °982553 * Sd
Log (1-+-;5) 1:383063 Col. (1-449) 2°599490 15 1982553
A specimen of the formation, consisting of the first six columns, is now given, in which the values of # are at the top
and those of y at the side.
Formation of the Values of Poltctes.
10 II 10 | 000000 | 1:00000 } 997349 | -99391 "00000 2651 | -oo609 it 000000 | 100000 . "00000
7
994190 2651
996841 3159
000000
12
98671 "01329 "99275 00725 I°00000 "900000
xl
Formation—(continued).
13 14 15
10 | 990607 | -97860 | 986696 | -96983 | 982553 | 96062 2651 | ‘oz140 2651 | *03017 2651 | °03938 tr | 993258 | -98460 | 989347 | :97577 | 985204 | “96650 3159 | ‘o1540 3159 | °02423 3159 | °03350 12 | 996417 | ‘99178 | 992506 | ‘98289 | 988363 | °97356 3583 | °00822 3583 | ‘o1711 3583 | °02644
13 | 000000 | rro0000 | 996089 | ‘99103 | 991946 | “98163 ' ‘00000 3911 | *00897 3911 | °01837 14 000000 | 1'00000 | 995857 | “99051 "00000 4143 | ‘00949 15 000000 | r°00000 “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 y=12.
Log (1+4,,) 1:394700
Col. ,, 2°605300 12 0:000000 996417 13. 996417 996089 14 "992506 995857 15 ‘988363
# #
Col. (14,3) 2°605300 Log (1+4,5) 1-383063 15 1-988363 A final verification is had at the close of the work, the last sum in each column being ‘ooo000, which is what log (1+ a4,)+colog (1+4,) becomes when +=y.
When the numbers correspondihg to the several logarithmic series 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. 1.
II 12 13
"00609 | °01329 | ‘02140 "00000 | *00725 | ‘01540
00000 | °00822 "00000
Values of Policies-—Form No. 2.
2 <0) 1J 12 13 14 15
Io} *00000 Xo) Ir] *00609 | *00000 II 12] *01329 | *00725 | ‘oo000 12 13] *02140 | ‘01540 | °00822 | ‘00000 13 I4| “03017 | °02423 | ‘oI1711 | ‘00897 | ‘o0000 14 15| °03938 | °03350 | °02644 | 01837 | ‘00949 | ‘00000 | 15 * * * * * * * ®
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. ft
xlii
Case 2. Let P, be the office premium, or any modification of it.
It was found in the last Case that Q,, or 1—v + P,, reduced to (1+4,)"', so that the expression for the policy value assumed the form
I+, ~ T+ay
In the present Case Q, admits of no such simplification ; and therefore the expression here to be dealt with is
I —-Q, (I + &z); in which Q,=1—v-+ P,, 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 (1+ ,) 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 P,. 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 Q,, as follows :-—
Log P, P,and Q, Log Q, A Log i'1o = 0°041393 =-——_——_ » My 4:298988 |-029126| Col. Ng 7°727862 . 10 2068243 -o11yoo. | 992208 -os0828 2°610958 002831 10 17622 rr °078073 = ‘or1969 993771 041095 6138789 003390 411 17732 12 ‘089576 -o12291 994922 = -o41417 ‘617179 = =003831 12 17864 13 ‘102362
- © Ten per-cent. will be considered but a small loading for an office premium. It serves our present purpose, however, as will be seen presently, better than 2 heavier one.
xiii 13 °102362 = -012658 995637 041784 621010 18016 ™ ‘116015 -013062 995961 :042188 °625189 18182 . 15 130158 °013495 . 995896 042621 629624 18359 16 °144413 ‘013945 043071 634185 * * * e Log i'1o §=0°041393 » Mig 4267383 Col. Nig 7:835637 16 2-144413
004179
004435.
004561
13
14
‘The above needs little explanation. Log P,=log (1°10M,) + colog N,_;, is first formed, the addends being A log M, and A colog N,_;.. The numbers being taken out are each increased by 1—v='029126, thus forming Q,. The logarithms of Q, and
their differences complete this preliminary formation.
The following shows the construction of the first six columns of the final formation, the values of + being as before
at the top, and those of y at the side.
Formation of the Values of Poheies.
10 II 400510 397859 610958 610958 ro | 011468 | = 102676] 008817 | 102051 — ‘02676, 2831 |— ‘02051 011648 | 1°02718 iT — 02718
12 394700 610958 005658 | 1°01311 2831 |— ‘°o131I 008489 ; = 1°01973 3390 |= °01973 011879 | 1°02773
— °02773
xliv
Formation—(continued).
13 14 15
391117 387206 383063
610958 610958 610958 10 | 002075 1'00479| 998164 "99578 994021 "98633 2831 |— ‘o0479] 2831 00422) 2831 "01367 rr | 004906 | 1:01136] 000995 | 100229] 996852 "99278 3390 |— 01136] 3390 |— ‘90229} 3390 00722 12 | 008296 T°01929 004385 I‘O1OI5 000242 100056 3831 |— -o1929] 3831 |— ‘orors} 3831 |— *00056 13 | 012127 | 102832) 008216 | rorgro} 004073 | roog42 — 02832] 4179 |— ‘org10] 4179 |— ‘oog42 14 012395 | r:02895] 908252 | 101918 — 02895 4435 |— 01918 16 012687 | 1°02964 — '02964
The only point in respect of which the logarithmic part of this formation differs from that in Case 1 is, that here the addends are the differences of log Q,. The operation may be verified horizontally at any point, as in Case 1; but we have not here the final verification that we have there. The last sum in each column is no longer 0: 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 [QAI i tae) Log (1 Law) * ‘400510 » Qio 2610958 10 0:011468 997349 2831 II 011648 996841 3390 12 011879 996417 3831 13 012127 996089 4179
Ig 012395
xiv
14 012395 995857 4435
15 012687
Log (1 +15) 1 383063 >» Qay-«—«2°629624 15 0-012687
Both and y here vary: the addends, therefore, are A log (1+4,) 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.
1. Of the Foint-Life Tables.
The tables were constructed by aid of the well-known formula,
xvi
Agy=Upey (I+ Gory) 5 which is very simply deduced as follows :—
The annuity on (#y) consists of two portions, that having reference to the first year, and that having reference to all the years after the first, of the possible joint duration of (~) and (y).
The value of the first portion is vf,, and that of the second portion is upsy 2ei1-y41; Whence the total value is,
Azy=Uey (1+ Goprytt) 3 or, in logarithms, log @,.,=log upzy + log (1 + @e41-y41):
For the application of this formula we want the values of log up, for the requisite combinations of # and y. These are, if for convenience we make + the older age, the combinations _ in which each value of x is connected in succession with all the values of y 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 :—
eye |e eee] oe |
The values of z are at the top, and those of +—y at the side. The former proceed from 96 to 10, and the latter from 0 to 86. Hence + is constant in the columns, and +—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 7—y is constant that are wanted in the final construction ; and they are most conveniently abstracted when formed in rows.
The following is a portion of the actual formation :—
zZ—¥! 96 95 94 93 92 gI © | 515256 | 106887 | 372329 | 520319 | 611232 | 661408 295815 | 132721 | 73995 | 45457 | 25087 | 20541 1 | 811071 | 239608 | 446324 | 565776 | 636319 | 681949 132721 | 73995 | 45457 | 25087 | 20541 | 15543 2 | 943792 | 313603 | 491781 | 590863 | 656860 | 697492 _73995 | 45457 | 25087 | 20541 | 15543 | 8008 3 | 017787 | 359060 | 516868 | 611404 | 672403 | 705500 46457 | 25087 | 20541 | 15543 / 8008 | 4583 4 | 063244 | 384147 | 537409 | 626947 | 680411 | 710083 25087 | 20541 | 15543 8008 4583 6490 5 | 088331 | 404688 | 552952 | 634955 | 684994 | 716573 20541 | 16543 | 8008 | 4583 | 6490 | 5402 6 | 108872 | 420231 | 560960 | 639538 | 691484 | 721975 s o s o eo 2 rT] e 2 # # # eo 2 80 | 249797 | 545776 | 678545 | 752473 | 797753 | 822550 163 | 48 | 999933 | 999824 | 999708 | 999603 8: | 949960 | 545824 | 678478 | 752297 | 797461 | 822153 48 | 999933 | 299824 | 999708 | 999603 82 | 250008 | 545757 | 678302 | 752005 | 797064 999933 | 999824 | 999708 | 999603 83 | 249941 | 545581 | 678010 | 751608 999824 | 999708 | 999603 84 | 249765 | 545289 | 677613 999708 | 999603 8s | 249473 | 544892 999603 86 | 249076
The construction was effected as follows.
Formation of Log psy.
xl vii
be formed is log uf,.,, which is equal to log v+log p,+log p, ;
The function to
and, connecting the constant, logy, with logs,, it will be written thus, log uf, + log p,.
Now if in this expression both z and y vary, as in the rows, we have,
xi viii
Az, log up.y=A log up, + A log f,;. - - (1.) if y alone vary, as in the columns, we have, A, log vpey=A log~,;.-.-+. ~~. (2) and if # alone vary, as in ascending diagonally, we have, A, log up,y=A log up, . » « (3-) From (1) we get, for construction of the initial and verifica- tion series, in the rows, proceeding from left to right, log upg y==log upssi»41 + Acolog up, + A colog J,; from (2), for the principal formation in the columns, log up.-y=log vpeyi1 + 4 colog J,; and from (3), for the construction of the series of terminal values,
log wp ey = log wp g41-4+ A colog p,. The following are examples of the application of the first and third formule :—
Initial Values. Verifications. f—y=0 2—y=6 Terminal Values. Log tpyg 1:251209 | Logepy, 1-251209 | Logepes 1:251209 » Pos 1°264047 | 4, poo 1°857663 | » Pio 1:997867 96°96 2°515256 96°90 1°108872 96°10 ©1:249076 295816 295816 295816 295815 15543 gs10 544899 95°95 1:106887 95°89 420231 132721 132721 132721 ——_— 132721 8008 | 94°10 ah oe 94°94 ‘372329 94°88 °560960 —— 73995 73995 93°10 °751608 73995 4583 45456 93°93 °520319 93°87. °639538 g2'10 §=—. °797064 45456 45456 | 25089 45457 6490 orto °822153 92°92 = °611232 92°86 °691484 _ _—=—s 25089 25089 | Log eps, 1824286 25087 5402 Pio 1997867 91-91 661408 91°85 721975 or10 ©1-822153 Log yp, 1:824286 | Log upp, 1-824286 » Po 1°837122 | ,, pes 1°897689 g1or 1661408 92°85 1721975
xlix
As appears by the formule the differences are here used in reverse order. Theoretically A colog vf, is the same thing as A colog f,, 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 Acolog vf, is not tabulated, in using A colog f, in its stead, the last figure must be corrected, by reference to the column colog vf,, pp. 10, 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 formule, and the whole of the results just obtained will be found in their places in the portion of it given on p. x\vii.
It will be observed that it is the same series, colog g,, (ir feverse 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 of thred verification series had been inserted, the entire formation of this function did not occupy more than twelve hours.
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 are to occupy the rows in the table in its final form are here formed also in rows.
* See page xxvili: £
]
* Formation of az-y. y | #-y=0 96 | 915256 +0328 106887 | 013989 8
372309 239608!
053873. 027233 9 64 94 [426211 +2668] 266845, 520319 446324 102711: 073660: 2: 7 93 | 623032 -4198/519991 350176 611232 565776” 152215. 124196 9 23: 92 | 763456 -5800) 689995 —*4898/ 579549 661408 636319:
198647 173089 21 31,0 14 gr | 860076 +7246 809439 "6448) 730678! +5379 * e ! ® e * | ® ® The same. y 2—y=81 2—y==82 15 | 249960, | 545824, | 071067! | a
14 |616900, -4139) 250008: +17
545757
~
150420) ___9 13 828898 O74 mote 752297, 1678302; 223812 150391 39 12 12 976148) _ 9466 828705) °67 797461! 752005 239244 223772 24 2 TI 080729! TaaT0 975779] 9458 828358 822153) 797064 346540 289050 16 39)
086153 s-asg
|
* The termin‘! portioas of these six (double) columns is given hereafter, pp. Ixiv and Ixv.
li
Formation of a,,—(continued.) y | #—y=3 a—y=4 2—y=5 |
eS |
"1157
129702 088331) __*1226 z—y==86
ES | SS ES NE ND | RS
inne! lane
rr |616336| +4134] 249473) +177 677613 544899 150244 070991 1} 11
lii The initial terms are formed as follows, In the equation
Agay=OP gry (I+ Aes ryt),
if z is 96, (the next to the oldest tabular age,) @24)-54) vanishes, and we get Agg-y = UP o6+y 5 whence log @og-y= log Upgg-ye
From this it appears that the initial terms of the several columns are the terms of the series log uf,., 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, (7—y constant,) of which it forms the leading term.
The logarithmic operation now commences ; and the first column, (x—y=0,) is referred to for illustration. As shewn by the working formula, the chief part of this operation con- sists in the formation of log (1+4,.,) from loga,.,, 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 Table I. in Gray’s Tables and Formule, which table, giving the values of log (1 +.) corresponding to successive values of log z, enables us, by a single direct entry, to pass at once from the given to the required logarithm. The opération 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 dogo, 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 1°120884, and an operation similar to that just described gives
litt
log ag4.9,, and soon. 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, ig 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=3,) the index of the initial term is 1, 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 Bremiker’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 200,000; 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 Bremiker’s
* De Morgan, Companion to The Almanac, 1842, p. 7; and Sournal of the Institute, vol, xiii, p. 136.
liv table be used for the purpose the operation is a very easy
one.
There will thus be small excuse for the restriction of annuity tables that may hereafter be published to three decimal places.
The following diagram will aid in the comprehension of the final arrangement of the results :—
JOINT~LIFE ANNUITIES
telat eC tea eens -)
a ESaaununmwvenwweeeeve es a ree
40 396
There are two sets of results to be provided for, the joint- life annuities and the last-survivor annuities. They are of the same extent, each requiring for its proper display a triangular space. These are combined, therefore, as shewn in the diagram, the joint-life annuities occupying the upper triangle and the last-survivor annuities the lower. To distinguish the two—although the distinction is hardly needed—the two sets of annuities are printed in different type; and they are separated from each other in the several columns by horizontal lines. The pages not being of sufficient depth to receive the whole series of ages from 10 to 96, the quadrangular space is divided horizontally into two, of which the first, (of the three- per-cent table,) occupies pp. 141 to 155, and the second pp. 156 to 170. To facilitate the following out of any of the
lv
columns numbers are placed at the bottom of the pages, which in the first part, pp. 141 to 155, refer to the pages where the remaining portions of the columns thus indicated are to be found; and in the second part, pp. 156 to 170, the refer- ences are to the pages where the preceding portions of the several columns are to be found.
The directions at the bottom of the several pages of the table, will be found, after a little practice, quite sufficient to enable the annuity, whether joint-life or last-survivor, on any specified combination of ages, to be readily found.
There being no room in this arrangement for the annuities on the combinations in which the ages are equal, these are cisposed by themselves in a separate table on page 140.
To ensure accuracy in the results, in the absence of columns D and N for two lives, the annuity construction was conducted in duplicate, by independent computers. Two sets of books, properly ruled, were prepared, in each of which the terms of the preliminary series, log vf,.,, were entered, and carefully compared. The operation then proceeded simultaneously in both sets, comparison being made from point to point, and discrepancies, revealing the existence of error, traced to their sources and removed, till the whole was completed.*
The numbers were afterwards taken out in both sets of books, and carefully compared throughout; and it is hoped that, as a result of the care exercised, no error of any conse- quence will be found to have escaped detection.
It may be of use to mention that the two sets of annuity values formed on pp. | and li, will be found in their places in the printed table, those formed in the first specimen on pp. 169 and 170, and those formed in the second on pp. 154 and 155.
* In point of fact, owing to the care exercised in this part of the work, and the simpli- city of the operation, very few errors were committed. In one set of the books—the first that comes to hani—the errors committed in the logarithmic process appear to have been just five, being on an average one error in 765 operations, It may be noted also, that the com- puter in this case was a gentleman who, although an expert arithmetician, had no previous practice in the use of mathematical tables.
lvi
And it will be observed that, as already intimated, the values which occupy the rows in the formations on pp. | and li, take their places also in rows in the printed table.
It will be well, before leaving the subject of the Joint-Life Tables, to give a few examples of their applications.
Example 1. Find the value of an annuity on (48°36) to be entered upon in I5 years. (48°36) + 15 =(63°51) The formula is, nlFary = U" nf yt eta ytn which in the present case becomes, 101@4o-26=0"*.15P49-362¢0-01- In applying the formula we use the equality,
De 4 15 63 “51 v -16248-36 = Dus e L e
P. 149 Ge3-51 7 ‘0716 log 0°884886
12 Des 8918585 fe) Ds se ee col: 5: 741519 2 bs) se ee log 4854707 ” lg .... col.5-067912
15.448:36 2°9350 log 0°467609 nstas-ae § 4°7300 The required deferred annuity is thus 2°9350; and the corresponding temporary annuity, formed by subtraction, is 4°7366. Example 2. Find the value of an assurance of a unit on (37°30). The formula is, Agy=1—(I—v) (1+azy); which in the present case becomes, A379 = I —(1—v) (1 + ag7-30)- P. 145) 1+437-39 16°3553 log 1213659 238 Il—v .... 4, 2°464284 "456368 4, 1°677943 Agra9 °523632
lvii
The logarithm of the divisor for the annual premium, (payable till the first death,) is in the process; and hence the premium may be readily determined.
It will be unnecessary to have recourse to the foregoing pro- cess, when Orchard’s Tables are at hand. These give at once "52304.
Example 3. Required the value of a survivorship assur- ance of a unit on (65) against (37).
The formula here is, (Milne, p. 184,)
AlL_=2(A,, 4+ 224), zy $( ay t po-! Py-1 ’
and in the present case it becomes, a Age = 4(Ass. a7 > a)
P. 150 1+4¢5-37 8°7779 Mee 943391
238 1l—o wee 2°464284
; 25567 5 " T407675 Agys; 74433
P. 150 Geg-g7 8°0791 ~=—,,_ 0°907363
5 Pe ++ o010°017915 “ne 8:41934 log 0925278 64 EEE eee 150 Ge5-36 7°7999 — , 0°892089 3 pa .... ool, 0:003974 65-36 787160 log 0-896063 Pre Se Ee 264-37 8° I Pes 41934 65-36 487160 Pe 34774 Acs-a7 "74433 Sum, 1°29207 Diff, "19659 Ae 64603 =half sum. Tord 09830= ,, diff.
A6s-87 74433
viii
he difference between “92! and “86 is here added to and
Pos pws
subtracted from Ags.3;: the half sum and the half difference are the complementary survivorship assurances, being together equal to the joint-life assurance.
Here too Orchard’s Tables would afford material help. By a single entry in them we should find at once Ags.37=.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 are consequently affected by errors of +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 (%.y) by the factor,
n Dan “yen | U" Poy OF D. a.
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 the Last-Survivor 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-cent. 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 + andy. 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 have 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.
12 13 Id 15
12°12
13°12] 13°13
I4sI2] 14°13 | 14°r4 15°12] 15°13 | 15°14] 15°15
Call (z) the older of the two lives in each combination, and (y) 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 observed that x is constant in the rows, and y in the columns.
lx
Now
Hence, if z alone vary, as in the columns, we have,
and if y alone vary, as in the rows, we have,
byz5=Aa,— Ayazy
From the first of these we get, as the formula for the general construction in the columns,
Oey y= Gay t Nag —Agdgys 6. es (1.)
and from the second, for the construction of verification values in the rows,
Asyti= Gay t Ady—Ayaey, ...- (2.) Also, if # and y both vary, we have, Ap y4z,= 4a, + Ady— Age, 5
and this gives as the formula for the construction of series of terms descendine diagonally,
Qsy 1 y+l = Gay + AGz+ Aay—Apyaey . ~~ (3.)
Formula (3) comes into use only in the formation of the series of initial terms.
In seeking to apply these formulze we are met by a difficulty at the outset: while the differences are 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 difficulty, 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.
xi
The arithmetical equivalent*® 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 17424. In the first form the whole of the number is negative, while in the second the last four figures are positive, and the prefix 1 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 4865923 —2576 17424 4863347 4863347
But the arithmetical equivalent is not restricted to the above form. We may write it 17424, 197424, 1997424, and soon. 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, we require first to construct them in their normal form, and thence to pass to the other. They are 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 these complementary numbers,
He appends a brief tract on Arithmetical Equivalents to the second edition of his Essay on involution and Evolution, published in 1820.
Lxii
Inspection of the foregoing formulz will shew that it is the differences of the single-life annuities that have to be used in their mixed form in order that the operation shall be one of addition throughout. The following shews the mode of constructing the differences of the series a,, (p. 14), in both forms :—
a, A A se) 24°1484 — 1531 8469 II 23°9953 — 811 8189 12 23°8142 — 2039 7961 13 23°6103 —_ 2206 7794 14 23°3897 — 2316 7684 15 23°1581 * * * *
Since Aa,=@,4:—@,, and here a,,, 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 are 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, were formed. In- spection of the formulz 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 @,., involved in the formule are, as indi- cated by their symbols, deduced from different successions of the terms of that function. Thus, the difference involved in formula (1), symbolized by A,2,.,, implying that in the primary functionz takes the increment unity in passing from term to term, while y remains constant, is formed from the rows in the table, pp. 140to 170. That in formula (2), which is symbolized by
Lxiti
4,4, is formed from the columns, in which, as we know, y varies, while x remains constant. And that in formula (3), symbolized by A,.,4,.,, is formed from series of terms des- cending diagonally, in which order both x and y increase by a unit.
1. To form A,a,, 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 @,,, the initial portion of which is given on pp. I, li.
The subtractions being made sideways, the differences are seen here ranged each under its minuend.
2. To form Aya,,. 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.
) fe)
10 II 12 13 14 15
lxiv
Formation of agy, and Maz,
982896 339446 41
Verifications of a;,. 5 A 215 20°4046— 675 7910 "3371 920 "245I—1 Be 215-10 °1327 1250 , °0047—= 1386 15°10 19°8661 IS‘11 15°12 15°13 15°14 15°15
23°1581 p. 14 24°1484 47°3065 20°4046 26'9019
3?
» I4I
Ixv:
Formation of ayy and ha,., (continued).
ee fe | es 9 a, |
983828 329051 85
— eee | Oe, § eee | eee
312964) 20°54 572 152
The column headed 15, p. 141, is copied out, (the value of @15.15 being taken from p. 140,) and differenced. On the right @isio is formed as shewn, and the successive terms ajsn, &c., are formed by the continuous addition of two series of differences, of which, the first is Aa, formed on p. Ixii, and the second is A,@,.,, formed in the column adjoining on the left.
3. To form A,.,@,... This difference comes into use only in the construction of the initial terms of the several columns in the main formation; and in these xs=y. The formula conse- quently for this case becomes,
and the series to be differenced will be that on p. 140, headed
Qe The formation of A,.,4@,,, was, like that of A,@,,,
superposed upon the joint-life computations ; and on reference
to the specimen on p. Ixiv, the values of the former of these will t
lxvi
be found in the first column, (in which x=y,) under those of the latter which fall in the same column, but necessarily in reverse order.
The following is an example of the formation of the initial terms of the several columns :—
Initial Terms. Qo «=: 2.41484 pp. 14 9910 24°1484 99 «99 482968 . 91010 21°007Q ,,140 IO°I0 27°2889
6938
1561 ae rrr °1388 Verification.
6378 Qj, 23°158%
20231 9999 23°1581 12°12 26°9787 46°3162
5922 99 15°15 19°8661
2386 15°15 26°4501 13°13 “8005
5588
2646 I4'l4 6329
5368
2804
1515 “4501
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,.-@,,, Whose formation has just been described.
The principal formation, like others that have preceded, occupies a triangular space, having 87 columns and 87 rows. The initial portion of the first six columns is here given.
ixvii
Formation of a5,==05+ ay — Gzg-y.
14 15 26'9787 7961 1207 26°6329 7084 1418
"5431 | 26°4501
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 Aa,, (8469, 8189, &c.) was then entered in column 10; and the successive terms of it were carried out horizontally, as shewn. The series A,a,-y were then inserted, that which in the specimen (pp. L xiv, lxv,) occupies the last vow, in the first column; that which occupies the next to the last row, in the second column; andsoon. The © numbers opposite the several vows, 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
lxviii
464 hours. Of this time 24 hours were required for the 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 _
Asy=I—(1—2)(1 + aay), 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, @,+@y—a,,, 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 have been formed. We proceed to explain the methods employed in the forma- tion of these, and shall then shew the manner in which they are to be used in the construction of Survivorship Assurance Tables.
Ixix The functions tabulated are log (f,-!— 1)and log$(p,7' + 1). We have,
l.— aoe L545.
log( ~,°!'—1)=log = = =log 2 =log d,+ colog /;,,. batt
*, Alog (~.'— = =A log d, + colog ps4, ; and hence, log (p23, — 1) =log (p,7'— 1) + A log dz + colog 5,1. Again,
log (2 '+ 1)=log i=" pees
=log $+ log (4, + la41) + colog 4,41.
Hence, A log (p21 + 1) =A log (4,+ 4a41) + colog p41 ;
* logd (Pepi + 1) =logh( pe" + 1) + Alog Ue +441) + colog p41.
By these formule the series were constructed, a prelimi- nary in the case of the second of them being the formation of log (/,+441) with its differences, a very simple matter. The following is an example :—
Formation of log ( pz~'—1). Formation of log 4( pr +1).
Log dy 2°690196 Log 5 1698970 Col. 2, 5:002133 » (hot) 5°299965 10 3692329 Col. Ai 5-002133 908595 10 0001068 1736 998065 iI -602660 __ 1736 918405 rr 000869 1444 998409 12 522609 _ 1444 942196 12 000722 1268 998644 13. °465973 __ 1268 * * 13 ‘000634
@ #
A survivorship assurance on (x) against (y) is an assurance payable at the end of the year in which the combination (xy) is dissolved, provided the dissolution is caused by the death of (x). When the sum assured is a unit the value of the assurance is denoted by Aj;
The value of this assurance may be conceived to be com- posed of two portions, one having reference to the first year,
; xx
and the other to all the years after the first. Now it is shewn, (Gray's Tables and Formula, page 73,) that the probability of ; (x) dying before (y) in the first year is
4(1—Pe)(1+fy) ;
hence the value of the first portion of the assurance is,
40(1—22)(1 +f) ; and the value of the second portion is obviously,
i UoyAz- l-y+ 9
in which A=j>7; denotes the value of a similar assurance on (7+ 1) against (y+1). Hence, Az, =8(1 —pz)(1+f,) —peyAsignl =Upeylt( he '—I)( 2+) + Asari ; or, in logarithms, . log AZ, = log Ur, + log [4$(2.7'—1)(4,71 +1) + Asami:
This is the 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 uf,., for all the different combinations of # and y. These logarithms have been already formed for use in the construction of a,.,, and would of course be immediately available. And we want, secondly, to be used as will be presently shewn, log 4(,-'—1)(f,'+ 1), not for the different combinations merely, but for a// 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 3(p2-'—1)(f,7'+ 1) would be in a quadrangular space, consist- ing of 87 columns and the same number of rows, having the
Ixxi
successive values of, say, z, at the top, and those of y at the side. But this arrangement is objectionable inasmuch as the series in which «—y and y—# 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 :—
In the first compartment +>y; and, the successive values of x being at the top, those of +—y are at the side. In the second compartment +<y; hence, the values of y being at the top, those of y—-+ are at the side.
1. *>y. 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 (~,~'—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 3(f,-'+ 1).
2. x<y. Between this case and the last there is simply an interchange of the functions in x and y. The modification this causes in the working formule is easily seen; and it will receive illustration in the examples now to be given.
It is necessary first to form the initial and verification series. The following are specimens of these formations :—
Initial Terms.
i +1) 0-001068
Leg, (rai 1) 0-647817 9 4
96°10 0°648885
596485
"10 «6 °2.45370 9s 768379 94°10 °013749 839603
93°10 © 853352 881377
92°10 6734729 924397
giro §=—°659126
* * Initial Terms.
Log 4(pal+-1) 0-508155
?
(pio! 1) 3°692329
g6"10 9-200484
95°10 94°10 93°10 92°10
gI°1o *
765437
‘965921
906767
Ixxii
a>y
z—y=81
Log (px!
» 4(pia!+1) 0-000625 9615 0-648442
95°14 94°13 93°12 92°11
QgI°1o
<Y
596485 999976 ‘244903 768379 33 013315 839603 88 "853006 881377 147 "734530 924397 199 "659126
y—2=81 ‘| Log 4(p.g'+1) 0°508155 Log 4(pxy'+1) 0:508155 » (py'—1) 0647817 96°96 1:155972
» (prt—1) 3°459280
96°15 3967435
95°14
94°13
93°12
Q2°11
gI‘ro
765437 983071 716943 906767 023622 "646332 951849 056536 "654717 971865 080151 706733 984957 089669
781359
Verification Series. z—y=0 —1) 0°647817 Log (pg!—1) 0:647817
» $(pa't+1) 0508155 96°96 1:155972
95°95 94°94 93°93 92°92
gr°g! *
Verification Series. ¥ -——7==0
95°95 94°94 93°93 92°92
Org! *
596485 765437
*617894
768379 906767
"193040
839603 951849
"984492
881377
765437 596485
"517894
906767 168379
‘193040
951849 839603
"984492
971865
IXxiti
The following are specimens of the formation of the principal series :—
|
36
85 84 83 82
81
82
81 |
Formation of Log} (pe'—1 (py! +1).
96 648885 999801
95
r>y.
94
93
Q2
oY — ——— el en ed wwe fo See ee
648686 999853
ee eee
648539 999912
245370 999801
eee
245171 999853
013749 999801
———ee— ff fee fee fo Oe
648451 999967
245024 999912
013550 999853
853352 999801
one =f nee ef = Oo eee
| 648442
736847
999967 244903
‘321276
013403 999912
013315
080689 8966
853153 999853
853006
915726 4566
734729 999801
734530
5
794506
Cane) oak | Sesame § oe Se
333332 15043
659126 |
715246 | 3657
a, a ee ee ee
780025 48151
828176 93233
921409 234563
155972
g6 | 200484 910331
110815 919849
030664 943464
974128 976378
950506 16929
967435
348375 28135
376510 48151
424661
95
a)
965921 910331
876252 919849
796101 943464
739565 976378
715943
&
101711 15043
—_—
116754 28135
144889 48151
193040
t<y 94
929258 12056
941314 15043
956357 28135
984492
93
801669 8966
810635 12056
822691 15043
837734
g2
721500 | 4566 |
726066 | 8966 |
735032 12056 |
747088 |
gI
eee OR eee
eee § eee fee 0 eee
872688 910331
783019 919849
702868 943464
' 646332
824537 910331
—_—_ ——- —
734868 919849
ee ee
654717
796402 910331
106733
781359
k
Ixxiv_
96 95 94 93 92 QI 5 | 166213 | 862236 | 710611 | 629936 | 582319 | 538484 75603 | 69414 | 58392 | 32524 | 19482 | 28792
pe eee ee Oe ae
4 | 241816 | 931650 | 769003 | 662460 | 601801 | 567276 118623 75603 69414 58392 32524 19482
3 | 360439 | 007253 | 838417 | 720852 | 634325 | 586758 160397 | 118623 75603 "| 69414 58392 32524
ne re EE eel
2 | 520836 | 125876 | 914020 | 790266 | 692717 | 619282 231621 | 160397 | 118623 75603 69414 | 58392
1 | 752457 | 286273 | 032643 | 865869 | 762131 | 677674 403515 |.231621 | 160397 | 118623 75603 69414
pS NE minnie erm neem ce, TI TO el
o | 155972 | 517894 | 193040 | 984492 | 837734 | 747088
It may be mentioned in regard to the preceding formations, that when the initial and verification series have been inserted in their places, the whole of the subsequent work need not, and will not, occupy a fairly expert arithmetician more than twenty-four hours.
Also, since the element of interest is not involved in these series, when they have been once formed, they can be used in connexion with any rate of interest that may be proposed.
Having now got the auxiliary series, we are prepared for the construction of the required tables of Az. .
The formula is,
log AZ,=log up, + log[$ (.°'—1)( py! + 1) + Anza, which, for the moment, we will write, . log a=log 6+ log (¢+ 2).
Here log 4, log c and log d* are known; and to obtain
log a@ it is requisite first to form log (¢+¢@).
This of course can be done by the use of the common tables, but the operation would be a tedious and laborious one. Fortunately this labour need not be incurred ; the end in view can be attained with great facility by the aid of the
* It will be presently shewn that when either x or y is the oldest tabular age, log Asisty that is log d, is known; and from tius value we descend to those corres-
ponding to the younger ages, in succession.
lxxv
table employed in the formation of @,,. The table in question, as already mentioned,* gives log (1+ 2) corresponding to suc- cessive values of log; its characteristic equation being,
T [log z]=log (1+), in which T is a functional symbol denoting the tabular result corresponding to the appended argument. a
In this equation write = for z, and we get,
T [log ss or T [log d—log ¢] _ a c+, =log (1+ ¢ )=log(——) =log (c+ d) —logc.
. log (c+d@)=log c+ T [log d—log c]. It thus appears that log(c+d) is formed by adding to log c the tabular result corresponding to log d—log c. We have, therefore, log a=log 6+ log c+ T [log d—log e] ; and, changing the order of the first two terms, the work is very commodiously arranged as follows :—
: log d
loge log d—log c log 4 T [logd—log c]
| | | | | Sum | log a
Log d is supposed to have been just formed; and, log ¢ and log 5 having been previously inserted in their places, loge is sub- tracted from log d, and the tabular result answering to the difference is written under log 6. 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, 6, c and d restored; and the arithmetical operation by which we pass from log A>, to log Ab is clearly exhibited. The numbers corresponding to the logarithms successively
formed are not to be taken out till the close of the operation.
* P. li.
lexvi
1 i log Asisz | Aisne
| Ds A | log4(p.—1)(f,14+1) | DA B ” Pay C pf
An
The arrangement adopted in the formation of Aj, is similar to that employed i in the construction of @,,. It is typified as follows :—
4. § 6
| | | | { 97°93 96°92 | 97°92 95°91! 96°91 | 97°91 ¢ | e ¢
Here for +>y, «—y, (constant in the columns,) is at the top, and y, (constant in the rows,) is at the side; and for +<y, y—+ 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,
A =0[R(1 —h.) (1 +4) + hey Aszizail: Case 1. x=y. Let x and y each equal 97. Then £, fA, and f,., vanish, and the expression reduces to,
A sy = to ; whence, log Aga, =log *5 + log v.
Ixxvil
Case 2. +>y. Let r=97; then 2, and f, vanish, and we
have,
A=! +h) = Upy (py +1);
whence,
log Aj, =log up, + log 4( py" + 1). Case 3. x<y. If y here equal 97, p, and f,, vanish, and
we get,
Ajg=4tr(t ps) =4Uhs(P2'—1) ;
from which,
log Ai; =log ‘5 + log up. + log (ps7
For Case 1 we have,
log*5 1-698970 » 2 1987163 1686133
For Cases 2 and 3 the initial terms may be formed in series,
as follows :—
Initial Terma.
a>Y Log tpye 1:251209 » (pos! +1) 0°508155 97°96 1-759364 295816 765437 97°95 ‘820617 132721 906767 97°94 860105 73995 951849 97°93 885949 45456 971865 97°92 °903270
9791 913316
® e
z<y Log °5 x Upge ”? (pog7! — 1)
96°97
95°97
94°97
93°97
92°97
91°97 e
1:698970 1:251209
0:647817
1597996
295816 596485 490297 132721 768379 391397 73995 839603 304995
Ixxviii
These series being, almost necessarily, formed in reverse order, it is the complementary differences of their component functions which constitute the addends. Thus for +>y the addends are, A colog uf, and A colog 4 (f,-'+ 1); and for r<y they are, Acolog vf, and A colog (g,-'—1). For Acolog #, and A colog uf,, (the series being identical,) we use A colog fy, as on p. xliv, subjected to the corrections there specified ; and the differences of colog }(f,-'+ 1) and colog (g,-'—1) are the negative differences of log 3(¢,-'+1) and log(g,-'—1). They are of course used in reverse order.
The specimens of the main formation will be found on pp. lxxx, Ixxxi, lIxxxii and Ixxxiii.
The initial terms are first inserted in their places; and then, in accordance with the type on p. Ixxvi, the terms of the two auxiliary-series, which are taken, the first from the formation on p. Ixxiii, and the second from that on p. xxiv.
The working then commences, and is carried on in each column, till all the columns in succession, in both compartments, are brought down to the same point, say y=85, and +=85, when measures will be taken for verification, as will be presently explained. Referring to the column r—y=0, (2 >), 155972 is subtracted from the initial term, which is log Aig, and the difference 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 Aim, 685705. This forms the initial term for another similar operation, the result of which is ‘684407=log Ag. And so by a succession of such operations the column is completed, and‘ all the values of log A; in which e=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 2 and y are as at the top, respectively.
The same remarks apply to the second compartment, in
lxxix
which z<y. The first column here would be identical with the corresponding column in the first compartment, and therefore it is unnecessary to insert the figures.
It is, as has been said, the remaindets of the subtractions that occur in the several operations, that form the tabular arguments. These regularly increase in magnitude, and it is therefore sufficient formally to determine the index of the first of them, since, in succeeding remainders an increase (of a unit) in the index is always pointed out by a decrease in the mantissa. Examples are seen in the first three columns of both compart- ments, where, after the first step, the index increases from 2 to 1, the increase in each case being accompanied by a decrease in the mantissa.
id
For the verification of the foregoing construction, it is re- commended that the work be performed in duplicate by independent computers. It ought to be carried on in sections, each embracing twelve or fifteen of the ages at the side, and ending in both sects of computations with the same row. Comparison of this row in the two sets will show if error has been committed; and if so it must be sought out and removed. Each section must be finished and corrected ere commencing the next.
But if, as will usually be the case, the annuities on the joint-lives have been previously formed, verification can be otherwise obtained. For we can, by Orchard’s Table, pass from the annuities to the corresponding assurances ; and we can also, by addition of the proper values in the table being formed, find the values of the same assurances. This follows from the equation,
Azy=A5+ A=;
zy?
that is, the sum of the two survivorship assurances in which . the ages are interchanged, is equal to the assurance on the joint-lives. And the values in which the ages are thus interchanged are very readily found, as they occupy corres-
97
96
95 | 68
9+ | 682641
93 |
g2
Ixxx
Formation of At,
&>y
685705| 484959] 759864) -574598
738103:
837734! -842895 611232 565776 | 229604 205454
39 17!
661408 636319 268083 256078 16 6!
681949 291145 30
{ -577298| 820617
: 77604 "534078 822691! 90491
661633
01 376510 '1-436169 313603 meer
547140] 794940! 623651 956357) °7817461116754' -678188
491781) 169245' 29 777809! "599527 941314) -836495 590863 226888
676589] -47488¢| 715004 518912 750i 7574254 735039| 980062 810635} 948469
656860 275996 32
697492 309817 29
733404] °541257
94
93
Q2
gi
go
89
Ixxxi
Formation of Az, (continued). 9 >y
y—T=3 ¥—xr=4
|
/
y—2=5
870575| *7422092| 903270! -800332
101711| -724755| 333332| -537243| 736847| 1-166423
384147 128564
611404 537409 404688 242626 196054 135402 34 32 14
ee ee |) en eee add
783322| -607186| 823160) -665516| 861380
801669) 9816531 920292] -9028681 080689
672403 626947 552952 291927 255144 205040 26 30 35
—ee | ee ee | ff er | nen
766026) -58347¢] 802413) 634473] 838716
* * * * *
r<Y x ! y~r=1 | Y—z--2 97 ee 96 484959 |597996 “39674 752457. 2-845539, 811071 029377 35, 95 592940; ~ "391688 490297, 309241 286273) 1 306667, 520836: 9-969461 239608 943792! 080115 038664 11, 41 94 606007 +403 562, 603333) .318664 032643) -673364' 125876, 1377457 446324 313603) 188109 092879, , ll 93 "4.79323 STORE 908 "414084! 539369! -340606 "340098 865869, 751219 914020, -618349 565776 491781 194208 150830 7 15 92 °4$76990 | 625860! -422532) 556646 _*360285 762131| 863729, 790266| -766380 590863 199714 29 91 | "474886 '433179| 580872) -380954 °958993)| 692717, -888155 681949 656860 280966 24867 4 45 24 go | 492777 640634 "437 154) 59827 598275 "396529 607226} -033408 619282) -978993 718031 697492 318051 290608 4 ; 46 89 "470697 | 643312) °43¢858) 607428) -404975
Ixxxiii
—
y—2=4 y—ar=5
i
er ee me
——— Ee ee eee | eee | eee |
SS
O4 391397 _'2.46262) 360439, 1-030958!
oO
93 | 422526) -264567, 304995; *201834 415273) 241816) 1063179) 063244 047523 es SS "292917| 352592) :225212| 231828 628328) 931650 384147 101600 9 322910) 417406) - °200500 788229; 769003) - 440093 537409 159880 1 340883) 466293 "235830 905858) 662460) - 661987 626947
*170541 1065615
256479 213917 26 _j|__18 see 8, |563233| -365791| 805337| -318607| 427673) 267715
e * @ e ® e
‘Ixxxiv
ponding positions in the two compartments of the formation just given. The joint-life annuities also, with which we have to do, occupy positions corresponding to the others in the for- mation on pp. |, li.
Thus, to verify the rows opposite 92 :—
92°92 92°93. 92°04) (92°95 s«Q92'96.—s-«9)2'97 476990 534078 599527 670603 742292 800332 476990 422532 360285 292917 225212 170541
— -eee
"953980 ‘956610 ‘959812 "963520 967504 °970873 The annuity values are :-— "5800 «6. 4898S 37908 )=— 2524 = II57 = 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—v)(I + az), (log (1 +4,y) being found in the annuity formation,) we obtain,
0:198668 0°173120 0°139815 0-097745 0-047538 0-000000 9464284 2:464284 9-464284 9464284 2.464284 9-464284
SES ES eR es
2662952 2°637404 2:604099 2:562029 2-511822 2-464284 "046021 7043391 ‘040188 . 036478 = "032.495 "029126 953979 . 956609 ‘959812 963522 = 967505 = -9 70874 and we have now an all but perfect agreement in the sixth place. This method of verification will be available when the work is undertaken by a single computer; but it may also be occasionally had recourse to when the work is done in dupli-
cate.
The final arrangement of the results will be analogous to that of the annuity values, pp. 141, &c.; and they will occupy the entire quadrangularspace. The values in which x=y will take their places in the diagonal of the quadrilateral; and, if
Ixxxv
the values of # be put at the top and those of y at the side, the values of the function which occupy the rows in the first compartment of the formation will here take their places, a/so i rows, above the diagonal, while those that occupy the rows in the second compartment, will take their places 7% 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 (z) against (y). (z at the top, y at the side.)
a * % % 2
#
+ * | * * | |
| 89 | °58248| 63447 ors | 89 | 90 | °55402| "60719 66551) °72674) go | 1 | °51891 57425) 63567, ‘79154 °76670 T | | 2 | *47699] °53408, 59953. 67060] 74229] ‘80033 2
| 3 | 42253) 47932) “54715, 62365, °70328| 76904) 3 | 4 | 36029] *4.1408) °48155 56184 64965| "72461, 4° 95 | °29292| -34070} 40365! -48351, 57730, 66163] 95 | 6 | '22521} 26456] °31866) -39160) -48496) °57460 |
7 | *£7054) '20183] 24626) -30924! *39627] 48544) 7
The specimen forms the concluding portion of the last six columns. It is composed of the values formed in the examples on pp. [xxx 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-cent. formation, or to undertake the like at either of the other rates of three-and- a-half and four per-cent.
It does not seem too much to expect that among the younger members of the profession, who hope yet to occupy higher positions in it, there are to be found some who, 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 Fournal,* are willing to devote a portion of their spare
* ¢¢] hold every man a debtor to his profession, from the which as men of course do seek
to receive countenance and profit, so ought they of duty to endeavour themselves by way of mends to be a help and ornament thereunto.”
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 feel to be incumbent upon them. Secondly, it would lighten and facilitate their labours when they come to occupy the positions to which they aspire, by increasing the stock of material required for the proper and satisfactory conduct of their business; and, thirdly, it would eperate to their more immediate advantage by imparting to them a dexterity in manipulation and a confidence in the use of tables not easily acquired by other means.
The present Introduction has grown to what it is feared will be considered by many, an undue length. The object mainly kept in view in its preparation has been, to show with what facility, by proper arrangement, and with the aid of the materials contained in the present volume, the tables required for actuarial use may be constructed: it is hoped that it will be found to contain little which does not bear more or less closely on that object.
P, G.
EXPLANATION
OF THE
ADJUSTMENT OF THE TABLES.
BY
W. 8S. B. WOOLHOUSE.
Tue 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 carcful 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-
* Fournal, vol. xiii., p. 95.
Ixxxvili
tage, that precisely the same aggregate tabular mortality or decrement must necessarily be retained between all points of actual coincidence, in whatever way the intermediate numbers may be modified. As a consequence of this principle the number of such coincidences with experience in the curve of the number-living may be regarded as one test of close adjustment and substantial exhibition of the actual mortality. The method I have adopted may be briefly stated, and the rationale of the process and its accurate adaptation to what is chiefly required will be at once apparent.
The data, for the reasons already stated, are the numbers-living at successive years of age as deduced, without any adjustment, from the original facts. If we begin at the first age in the Table and extract the numbers-living at quinquennial intervals, that is, according to the usual notation, /,9, /15» /es 445, .... we can by the formula for interpolation determine all the intermediate values at the other ages, and so obtain a complete series of values that shall be continuous. Geometrically speaking, we shall thus pass a continuous curve-line through the indicated quinquennial points. Against the adoption of such curve-line as the basis of the final Table there is manifestly this tangible objection, that the numbers at the ages 10, 15, 20, 25, .... are made use of exclusively, and that the original numbers between these ages are wholly ignored as data. This rather material objection, which is inherent in other methods of adjustment, is entirely removed by varying the epoch of the adopted quinguennial data, that is, by taking the five distinct series hereunder stated, viz. :—
lio» Ass hi, Lys, eves Ar» hie» fas he, cee 4a» Ar has fers cee hs, Ags hs» he, cere Nias hos fea» Mop, ++
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 of the five corresponding ordinates, and, in other
Ixxx1X
words, the five curves are, 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 unduc 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 unnecessarily circuitous, For the purpose of actual calculation we 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 /,, which denotes the original number for an age z ycars older, be the nearest quinquennial point of the corresponding curve. Then the series of values from which / is found are /,_,, 4, 445, &c., and by interpolating with central values and stopping after second-differ- ences, we shall have
ie =a (na
SH), 4 ot, XS)
For the several values of /as deduced from the five respective curves we must make z separately equal to —2,—1, 0,1, 2. The five values of / are therefore
== —°121_7+ °84/_5+°28/,3 d= — *08/_ 6+ °96_1 + 12044 f= o+ 2+ Oo f= +1 2l_a+ "96/41—"08/,, d= + 2873+ °84/,9—*1 2/47 Hence if we put N=lLitles, ye=lat lye, ete . ) YeHLatla, ve=lot lee, y= tr fen 8=1—- Ys h=yom ys A= + MP) and if (/) denote the required average number-living at the given age, wc,
shall have by adding together the five values of /, m
xc
S(Q=l+'96y + B4yo+'28 73+ 12 y4—"O8yg— "I 2yz =l+ yi + y2—'04{ (v1 — y3) + 4(va— 73) + 2(¥e— 3) + 3(77 4) }
=l+ yit72—04(/+ 4g + 24+ 3h) .
(y)
By means of these last formula, (a), ((3), (y), 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,” Table H™, page 273, and the calculation sz extenso is annexed :—
1 2 3 4 | G | 7 a a t 9297 | 7, 9861 | ts 9434 | 1-3 9493 | 24 9860 | /¢ 9684 | 1.7 9743 yi 18610 | 241 9249 | 242 9185 | Uys 9125 | Lys 9054 | lye 8913 | l47 8848 y2 18619 | 1 18610 | 318619 | ys 18618 | y4 18614 | yo 18597 | y7 18591 | 46526 |_ys 18618 | ys 18618 | 24 - 42 -ys.18618 | 7, 18614 Cor®+460| f -8| g +1| 3k - 69 h -21/k -23 5)46530°60 | 4g +4 -l () 93061 <4] ot. - 4 -115 | x - 04 | = 44:60 | Cor
In filling in the data the computer will bear in mind that a chasm precedes the last two columns, and that /_, and /,, are there passed over unheeded. This is further indicated by the numerals placed at the top of the respective columns. It will be further observed that the fina! result is here put down and retained to an extra place of figures.
Since the expression (y) is linear with respect to the several values of /, it is evident that precisely the same formule 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 offers the greatest possible facility.
As an example, taking the same Table as before, the following is the calculation of the adjusted-decrement (¢@) at age 25:—
(25) d 48 d_; 64 yi 128 | dy, 64 183 [y RE
309 | y3 130
5)308-48) 4g + 12 (d) 617 +10 93061 |Dage25 92444 [() ,, 26
The resulting adjusted decrement 617 being here subtracted from 93061, the adjusted number-living at age 25, we obtain 92444 for the adjusted number-living at age 26.
In the actual construction of the Tables I have first made independent computations of the adjusted numbers-living for every fifth year of age, and afterwards calculated the values throughout for every age by adjusting the decrements, thence deducing by successive sub- traction the numbers-living, and making use of the former calculations at each quinquennial stage, as a periodical check on the accuracy of the work. The final results, on being differenced to second-differences, are generally found to be remarkable for their orderly progression, though at exceptional places there may yet exist some slight traces of irregu- larity, but they are quite isolated, and so minute as to be readily amended by inspection.* ‘The simplicity and efficiency of the manipulation will be duly appreciated by those who may hereafter have occasion to put it in practice.
Another matter incidental to the completion of the Table it will be requisite to explain. As the data for each separate calculation must extend over an interval of seven years preceding and following the age, the formule will obviously not apply to the first seven years of the Table, and the numbers for those years, viz., ages 10 to 16, will therefore be wanting. To effect a continuous junction at age 17, I have considered it most expedient to supply the required numbers by means of constant third-differences. At age 10, the radix of the table is /g=z100000. If A,, Ag, Ag be the differences immediately following age 17, and 1==7, we shall have
* Those who are not practically familiar with progressions of differences, and the dis- turbances caused by isolated errors, need only have recourse to the elementary Rule given by me in a former paper, Fournal, vol. xii, page 140.
XCli
yo 4yy—2A, + —— Het Da, — at et) wet 2) A; hy 7h-+28A4— 84A3
from which
When the series of numbers is put down in a retrograde order, the differences that are of an odd order will change sign. In such case therefore we shall have to begin with 4,, and apply the three orders of differences after having reversed the signs of the first and third. The third-difference should be calculated to one or perhaps two additional places of figures, and then the continued summation of the differences will sufficiently check the accuracy of the computation.
The first calculated numbers of the H™ Table, beginning at age 17, with the accompanying differences, are
Age17 = 97624
18 97245 _
“19 = 96779
Here we have A, =—379, Ag=— 87, and
—379 466 °7
_ _ 87 , 379 2376 _
3 +2 84 — 25°70 Hence, changing the signs of the odd orders, the three commencing differences are -+379°0, —87°0, +25°7, and the retrograde calcula-
tion is as follows:—
+25°7 | =As = 87°
I §? ‘ | +379°0 mee
7 | 97624 319-7 1°3
16 | 97941°7 go: | 356
15 | 98223°8 272'2 | ~ 99
14 | 984960 2880 «| +15'8
13 98784'0 +41°5
12 | ggtizs | 3795 | 467-2 11 | ggsr02z | ,38°7 | +929 10 | 99999°8 | T4°9 The column of second-differences is formed by the repeated addition of
the constant third-difference placed above it, and the other columns are hence obtained by continued addition, or the inverse operation to that of differencing.
VClil
In conclusion it may be observed that in the method of adjustment here laid down and adopted, every individual element of the data supplied by the experience has its proper and legitimate influence in determining the several results, and that there does not exist anything of an arbitrary nature in the process, * No extraneous condition or restriction is placed upon the quantities, which are freely permitted to manifest and assert their own law. In fact, if the original points, taken in groups, were to range in curves of the third order, they would in such case not be subject to any alteration whatever by the operation of adjustment. Thus the process, unlike othcr methods of adjustment, does not in any way interfere with the organic relations which exist amongst the true values but operates exclusively upon the incidental imperfections, whether in excess or defect, that is to say, its efficiency is wholly directed towards the neutralization of the small positive and negative portions of the data which constitute the errors of observation. Hence, also, if there should be any particular phases in the absolute law of mortality or any special peculiarities at certain periods of life, the same will be brought out with greater clearness and significance in the final Table, after the casual irregu- larities have been eliminated.
The adjustments have been separately made on the H™, H¥ and H™? Tables; and also on an H™*) Table, excluding the first five years of assurance, this last Table being designed for the general purposes of valuations.
“ Oo
=
ed
we
pb © &> WH PHONO CONF ON RW BH OW ONT AND & BH OLD ONT OM BD OM} DT ON bw BD
as
tun
I00 000
99 510 99 113 98 784 98 496 98 224 97 942 97 624 97 245 96779 96 223 95 O14 94971 94 321 93 683 93 061 92 444 g1 826 QI 192 90 538 89 865 89 171 88 465 87 748 87 021 86 281 85 524 84745 83 943 83 122 82 284 81 436 80 582 719737 78 830 77919 76 969 75973 74932 73 850 72 726 71 566 7° 373 69 138 67 852
Detre-
ment. dy
1339
XCIV
H*®., _ Healthy La Lives.—Male. —Adjusted Table.
Log /, |
5°000 000 0
4°997 866 7 4°996 1306 4°994 686 6 4°993 418 6 4°992 217 6 4°990 969 0 4989 556 6 4°987 867 3 4°985 781 1 4°983 2789 4°980 521 5 4977 591 9 4°974 608 4 4'971 6608
4°968 7677
4°965 8787
4°962 9657 +9599567 4°956 8309 4°953 590 6 4°950 223 6 4°946 775 5 4'943 237 2 4°939 624 1 4°935 9152 4°932 088 o 4°928 1141
4°923 984 5
4°919 7160
4°915 315 4 4°910 8164 4°906 2380 4°9OI 5509 4°896 691 5 4°891 643 4 4°886 315 8 4°880 659 3 4°874 667 3 4°868 350 5 4861 689 7 4°854 706 7 4°847 4061 4°839 7168 4°831 5627
Log pz =A logl,
9°997 8667 9°998 263 9 9°998 5560 9°998 7320 9°998 7990 9°998 7514 9°998 587 6 9°998 3107 9°997 9138 9°997 497 8 9°997 242 6 9°997 059 5 9°997 017 4 9°997 0524 9°997 1069 9°9907 Irz1o 9°997 087 0 9°996 9910 9°996 874 2 9°996 7597 9°996 633 0 9°996 5479 9°996 465 7 9°996 386 9 9°996 2911 9'996 1728 9°996 026 1 9°995 8704 9°995 7315 9°995 599 4 9°995 5010
9°995 4216 |"
9°995 3129
9°995 1406 ]°
9°994 9519 9994 672 4 9°994 343 5 9°994 008 o
9°993 683 2 |° 9°993 339 21° 9°993 0170 |° 9°992 699 4 | °
9°992 3107 9°991 845 9 9°991 343 8
ee « id e e e e
Prob. Sur-
Prob. Dying
viving 1 Year.) in 1 Year.
Pz
"995 1000 "996 O10 5 996 680 6 "997 084 5 "997 238 5 "997 1290 "996 7532 996 1178 995 208 o 9942550 "993 6710 993 2750 993 1558 9903 2359 "993 3606 "993 359 9 "9935 3149 "993 095 6 "992 828 3 "992 5667 "992 277 3 "992 082 6 "991 895 1 "OOI 7149 "991 496 3 "QOI 226 3 "990 891 4 "990 536 3 "990 2196 "989 918 4 "989 694 2
"981 399 5 "980 265 9
Yr
004 900 0 "003 989 5 "003 319 4 "002 915 5 "002 761 5/ "002 8710 "003 2468 ‘003 882 2 "004 7920 005 7450 006 329 0 "006 7250 "006 844 2 "006 764 1 006 639 4 "006 6301 006 685 1 "006 904 4 ‘oo7 1717 "007 433 3 "007 7227 "0079174 "008 1049 "008 285 1 008 503 7 "008 773 7 "009 108 6 "009 463 7 "009 780 4 "010 081 6 "O10 3058 "010 486 8 "O10 734.4 ‘OIL 1269 "OIl 556 5 "O12 192 1 "012 940 3 "013 7022 "O14 439 8 "015 2200 7 |°015 950 3 "016 669 g
“O17 549 3 ‘018 600 5
po 7341
on oe
No. | Decre- Living. | ment.
z£ &
Age. z
| I 66 513} 1399 65 114] 1462 63 652| 1527 62 125] 1592 60 533 | 1667 58 866| 1747 57 119] 1830 55 289] 1915 53 374) 2001 51 373 | 2076 49 297) 2141 47 156| 2196 44 960 | 2243 42717 | 2274 , 49 443 | 2319 38 124| 2371 35 753 | 2433 33 320] 2407 30 823} 2554 28 269 | 2578 25 691 | 2527 (23 164] 2464 20 700 | 237-4 18 326| 2258 | 16 068 2138 | 13.930] 2015 11915] 1883 10 032| 1719 1545 1346 1138 941 773 615 495 408 329 254 195 139 135] 86 40
i WO Cr Qh
>.)
© WAT be wn 8
ha |
~“
co OM OAM &® HK OW Or An Ff & BH | O
co
‘oO
Oo ., Bn~T CaGnr & DN m
xCV
H*™,
Log pz =A logl,
Log J,
4°822 906 5 4°813 6744 4°803 812 1 4°793 266 4 4°781 992 2 4°709 864 5 4°756 780 6 4°742 638 7 4°727 3298 4°710 7349 4°692 820 5 4°673 537 ©] 9979 289 3 4°652 826 3 | 9°977 7744 4°630 600 7 | 9'976 242 7 4°606 843 4|
4°581 198 5 4°553 312 5 4°522 705 0 4488 8749 4°451 310 4| 4+°409 781 0| 9°955 032 6 4°304 8136 9°951 1567 4°315.970 3| 9°947 097 4 4°203 067 7 | 9°942 894 1 4°205 961 8| 9°937 989 3 4°43 951 1 | 9932 1429 4076 O94 0 | 9°925 293 5 4001 387 5 | 9°918 370 3 3°919 757 81 g'g10 702 6 37830 460 4 | 9°903 699 1 3°734 159 5 | 9°897 6900 3°631 849 5 | 9°892 2869 3°524 136 4| 9°885 796 7 3°409 933 1! 9881 213 7 3°291 146 8 | 9°873 2061 3°104 352 9 | 9857 662 8 3°022 015 7| 9" 837 1226 2°859 138 3: 'Q° 812 034 5 2°671 1728 9766 5778 2°437 750 6' 9692 583 2 2°130 333 8 9'55y 862 3 1'6G0 196 1 | 9°264 046 4 0°9542425| —o
aw 3
9°99° 137 7 9°989 454 3 9°988 725 8 9°987 872 3 9°986 9161 9°985 8581 9°984 691 1 9°983 405 1 9°982 085 6 9°980 716 5
9°990 7679
id ¢ ’ Ne ee aa ame ed
Healthy Inves —Male.— Adjusted Table,
Prob. Sur-
viving 1 Year.
Px , "978 966 5 "977 5471 "9760102 "974 3742 "972 461 3 "970 322 4 "967 961 6 "965 363 8 "962 509 8
"959 5897 "956 569 4 "953 4312 "950 111 2 "946 7659 942 660 0 937 808 2 "931 949 8 "925 0600 "917 1398 "908 804 7 "901 6387 "893 628 1 885 3140 876 787 « "866 O40 5 855 3481 ‘B41 963 9 "828 648 3 814.1465 "801 1229 "799 T14 3 "780 345 5 | 768 7706 "760700 4 *746 803 1 "129 5479 687 262 3 648 686 0: "584.2217
"492 700 BY "362 9629 | ‘183 673 5
*000 000 O
Prob. Dying |
in 1 Year. |
Vr
021 033 5 "022 4529 "023 989 8 "025 625 8 027 538 7| "029 677 6! "032 038 4' "034 6362 "037 490 2, "040 410 3' "043 430 6, "046 568 8 "049 888 8 053 234.1 "057 340 0: 062 191 8 068 o50 2! "074. 940 0; ‘082 860 2: O9T 195 3 ‘098 361 3] "106 3719) "114 6860, "123 212 9 "133 059 5; "144 651 QI 158 036 1' "171 3517)
1858535
"198 877 | 209 885 7 i "219 654 5: "2312294 "239 299 6 253 1969 "279 4521 "312 73771 381 3140) "415778 3,
"597 299 2
"037 037 1 816 326 5 17000 000 o
a
XCvi
HF. Healthy Inves.—Female.—<Adjusted Table. °
Prob. Sur- | Prob. Dying
>. oOo G9 i] (+) - © DI An Rw WH ONO OXF ANP WHOM) DWYIOANADO HMO
hw bP HOO Or Anh BHO
wr
65 188 64213 63 210 62 173
a a
1003 1037 108!
4°998 634 2 4°996 800 5 4°994 563 5 4°992 000 9 4°989 1961 4°986 216 2 4°983 143 5 4°980 062 5 4°977 0556 4°974 138 5 4°97 1 141 3 4°967 964 3 4°964 509 5 4°960 6610 4°956 305 4 4°951 043 5 4°946 673 3 4°O41 SIT 4 4°936 262 3 4°931 0407 4°925 863 6 4°920 749 3 4°915 695 2 4°910 603 I
6| 4°905 4829
4°900 345 3 4°895 1462 4°889 884 1 4°884 557 5 4°879 1647 4873 680 9 4°868 ro09 3 4°862 453 4 4°856 704.7 4°850 854 5 4°844 899 5 4°838 880 6 4°832 809 0 4°826 683 7 4'820 490 5
5] 4°814 167 7
4°807 623 0 4°800 785 8 4°793 601 8
9°998 634 2 9°998 166 3 9°997 763 0 9°997 4374 9°997 195 2 9°997 0201
9°995 7944 9°995 278 1 9°995 029 8 9°994 8381 9°994 7599 9°994 7784 9°994 822 9 9°994 885 7 99949459 9°994 907 9 9°994 879 8 9'994 862 4 9°994 800 9 9°994 7379 9°994 973 4 9°994 607 2 9°994 5162 9°994 428 4 9°994 3441 9°994 251 3 9°994 149 8 9°994 045 0 9°993 981 1 9°993 928 4 9'993 8747 9°993 8068 9'°993 577 2 9°993 455 3 9°993 1628 9°992 8160
9°992 382 5
"996 8600 "995 786 8 "994 862 3 "994.116 8 "993 562 5 "993 1620 "992 949 8 "992 93°99 "993 100 4° "993 395 5|° "993 1224 "992 7114)" "992 076 6]° "99117771" "999 1577)" "989 186 31° 988 6210 "988 1847] ° "987 986 3 "988 048 6|° 988 1502 988 292 8|- "988 430 2 988 343 3 988 279 5|° "988 2400 988 1000 "987 9567 |° "987 809 9 "987 959 5 "987 452 5 "987 2529 "987 061 2 "986 8504 "986 619 6° 986 3817]: 986 2366|° °986 1169]° "985 995 0 "985 840 8 |° "985 5407 "985 043 3 984 380 1 983 5944 |" "982 6130
Pz
eee
"014.005 0
viving 1 Year.| in | Year. .
| qe | |
"003 £400! "004 213 2 "005 1377 "005 883 2 "006 437 5 "006 838 0 "007 050 2 "007 069 I
006 899 6 006 694 5
7006 877 6!
007 288 6 007 923 4 008 8223 009 842 3 0108137)
"O11 3790!
o51 815 3|
"012 013 7,
OST OSI 4’
‘o11 849 8
O11 707 2
"OIL 569 8 "O11 6567
O11 720 5
‘O11 7600 ‘O11. 9000)
O12 043 3 | O12 1901 O12 340 5 O12 547 5 0127471 0129388) 013 149 6 013 380 4, 013 618 3 013 763 4| 013 883 1 |
O14 159 2!
O14 453 3 "014.9567 "O15 6199
016 4056
‘017 3870}
|
xcvii
HF. Healthy [nves.—Female.— Adjusted Table.
N Decre- Prob. Sur- | Prob. Dyi Age. Living. ment Log I, Log pz viving 1 Year| in] Year. F L, d. =A log /, Pz Qn 55 {61 092) 1116} 4°785 984 3 | 9°991 993 2 | "981 732.5] °018 267 5 6 159976) 1144) 4°777 977 5 | 9991 636 x | 980925 7| *019 074 3 7 | 58 832| 1170| 4°769 613 6! Q'991 2761 |"Q80 1129] *o19 8871 8 | 57 662) 1196 | 4°760 889 7 | 9°990 897 3 |'979 2584] -020 7416 9 | 56 466) 1331 | 4°751 787 0| 9990 427 4 | "978 199 3| *021 8007 60 | 55 235| 1308 | 4°742 214 4| 9°989 591 9 | °976 319 4| 023 6806 I | 53927 | 1395 |4°731 806 3 | 9°988 617 6 |°974.131 7] *025 868 3 2 | 52 532] 1495 | 4°720 423 9| 9'987 461 2 | "971 5412] 028 4588 3 | 51 037) 1601 | 4°707 885 1 | 9°986 158 2 | "968 6306] -031 36904 4 | 49 436| 1706 | 4°694 043 3 | 9°984 741 8 | 965 4907} 034 509 3 65 |47 739] 1784 | 4°678 791 4 | 9°983 456 3 |"9626231| °037 3769 6 | 45 946| 1846 | 4°662 247 7 | 9°982 1909 | "959 822 4) °040177 6 7 |44 100] 1914 | 4°644 438 6 | 9°980 729 7 |°956 5986) °043 401 4 8 | 42 186] 1982 | 4°625 168 3 | 9°979 101 0 | "953 0176] *046982 4 Q | 40 204] 2050 | 4°604 269 3 | 9°977 2708 | "949 0100] *o50Q900 70 138. 154| 2123 |4°581 540 1 | 9°975 1362 | "944357 1| °055 6429 I | 36.031 | 2232 | 4°556 676 3 | 9°972 227 6 | "938.053 3] °0619467 2 |33 799 | 2338 | 4°528 903 9 | 9°968 868 6 | 9308264! ‘069 173 6 3 | 31 461 | 2425 | 4°497 772 5| 9°905 164 3 |"9229204| °077079 6 4 | 29.036 | 2490 | 4°462 9368 | 9°961 062 3 | 914.2444) 085 755 6 75, | 26 546 | 2518) 47423 9091/1 9°956 718 5 | "905 1458} “0948542 6 |24 028 | 2500] 4°380 717 6] 9°952 2861 |°8959547| “104045 3 7 |21 528 | 2363 | 4°333 003 7 | 9°949 505 I |°890 2360) ‘109 7640 8 | 19 165 | 2205 | 4°282 508 8} 9°946 917 0 | 884.946 5] "115053 5 9 | 16960] 2024 | 4°229 425 8| 9°944 808 5 | 880 6604) “119 339 6 80 | 14.936] 1819 | 4°174 234 3 | 9°943 600 2 |°878 213 7| 121 7863
13 117| 1621 | 4°117 834 5 | 9°942 712 3 |°8764199| °123 5801 11 496| 1514] 4060 546 8 | 9°938 670 8 | 868 302 0| +131 6980 9 982 | 1450! 3°999 217 6| 9°931 833 2 | "854.738 5| "145261 5 8 532} 1389 | 3°931 0508 | 9°922 829 9 | 8372011] "1627989 7 143] 1326 | 3°853 8807 | 9°910818 4 | 814 363 8! +185 6362 1234 | 3°764 699 1 | 9°896 450 8 | -787 863 2| °212 1368 4 583 | 1086 | 3°661 1499 | 9°882 545 7 | °763 037 3) °236962 7 3.497| 903 | 3°543 695 6| 9°8702744 |°741. 7786) -2582214 2594] 707 |3°413 9700] 9°861 Bol g | "727 448 0) °272 5520) 1887] §519|3°275 7719 | 9860 314 2 |°724. 9603] °2750397
CO 0'O C™ Qurr bh GD B mm O tn Cc bone ~
ve)
1 | 1368) 368] 3°136 0861 | 9°863 913 9 |°7309942] °269005 8 2 | 1000| 2321] 3°000 0000; 9'885 361 2 |°768 0000! ‘232 0000 3} 768) 138} 2°885 361 2| 9'913 979 3 | 820 312 5| *179 687 5 4 | 630] 11712°799 340 5| 9°910 776 9 |°814 285 7| °186 714 3 95 513] 107/2°710 117 4| 9°898 408 6 |°791 423 0] *208 5770 6 406| 104] 2°608 5260| 9°871 480 9 |°743 842 4| °2561576 7 302] 102] 2°480 0069| 9°821 023 1 |'6622516| 3377484! 8 200] 100] 2°301 0300] 9°6 989700 |*5000000| .5000000 9 100} 100] 2°000 0000 —o "000 000 0} I'000 000 0 | 100
Oo — |
xeviii H™, Healthy Lives.—Male and Female.— Adjusted Table.
No. Living. le
Decre- ment,
4,
Log ls
Log Ps =Alog ly
Prob. Sur- | 1 lriving | Year, Ps
- w & » » “ 6 Rew So wr ah ae on Bo Or hae wn So Or anne wn SO OV Ana nnd
100 000 99558 99451 98 766 98 390 g8orr 97 615 97 189 96720 96 195 95 614 94993 94.348 93 695 93 044 92 397 91750 91099 90.431 89 745 89 042 88 324. 87 598 86 865 86 122
85 368;
84.600 83 811 83.000 82170 81 326 80472 79 612 78 743 77855 76942
5°000 000 0 4°998 0762 4996297 1 +994 607 5 49929510 49912748 4989 5166 4°987 6171 4985 5163 4°983 152 5 47980 521 5 4977 6916 49747327 49717164 4°968 688 4 4965 9579 4°962 606 1 +959 513 6 4°956 3173 4953 010 3 4949 5949 49460787 4°942 4942 47938 844 8 4935 1141 49312951 4927 3704 4°923 3010 49190781 49147133 49102294 4°905 6448 4°900 978 5 4/896 2120 4°891 286 5 4°886 163 5 4880 779 3 4°875 0902 4869 090 8 4/862 781 1 4856 166 8 4849 2597 4°842 028 5 4834 439 8 4826 4117
9°998 0762 9°998 2209 9°998 3104 9°998 343 5 97998 3238 97998 2418 9°998 100 5 9°997 899 2 9°997 6362 9°997 3690 9°997 1701 9°997 0411 9°996 983 7 9°996 9720 9°996 969 5 9996 9482 9°996 907 5 9°996 803 7 9°996 693 0 9°996 584.6 9°996 483 8 97996 415 5 97996 3506 97996 269 3 9°996 1810 9°996 075 3 9'995 9306 99957774 9°995 6352 9995 5164 99954154 9°995 3337 9°995 233 5 9°995 0745 9°994 8770 9°994 6158 9°994 3109 9°994 000 6 9°993 690 3 9°993 3857 9°993 0929 9°992 768 8 9°992 4153 9'991 9719 9°991 4708
"995 580 0| "004 4200 "995 911 9| 004 088 1 "996 117 0] 003 883 0 "996 193 0| 003 807 0 "996 148 0] 003 8520 "995.959 6| 004 040.4 “995 635 9] "004 364 1 "995 174 4| "004 825 6 “994 572 0] :005 4280 "993 960 2] 006 039 8 "993 505 1]°006 4949 "993 210 0| 006 7900 "993 078 8|-006 921 2 "993 051.9] ‘006948 t "993 046 3 |°006953 7 "992.997 ©| 007 002 4 "992.904 6] *007 095 4 "992 667 3 | 007 3327 "992.414 1|'007 585 9 "992 166 7 | "007 833 3 "991 936 4|*008 063 6 "991 780 3 |°008 219 7 "991 632 2| "008 367 8 “991 446 5|"v08 553 5 "991 245 0| 7008 7550 "991 003 7 |"008 996 3 "990 673 8| ‘009 3262 "990 323 5|*009 676 5 *990 000 00100000 "989 7286 ‘oro 271 4 "989 499 1 | 010 5009 +989 313 1|"0106869 +989 0846 |‘o10915 4 +988 722 8)o1t 2772 +988 273 1|"011 7269 "987 679 0|*012 3210 "986 985 8] 013 0142 *986 2809] °013 7191 "985 576.4 |°014 4236 "984 885 3] 015 1147 +984 221 6) 015 7784 "983 487 6|°016 5124 "982 678 0} ‘017 3220 +981 684 6] ‘018 315 4 “980 552 4|°019 4475
in in
Spo wn OO GAT QO
~y
\O Ve) foe) co ~J OO On An P ® BH et OLO CONT Qin & © DH HOM} CNT Ant © OH ONO COX OA
o ie)
Age Living. | ment.
No. _ | Decre-
1358 1414 1471 1531 160r 1677 1760 1849 1936 2014 2080 2138 2186 2224 2268 2331 2401 2469 2531 2567
33
XC1X
Hr
Log /,
4°817 882 5
4°808 818 4 4°799 175 1 4°788 910 4 4°777 963 0 4°766 2120 4°753 552 4 4°739 8570 4724 988 2 4°708 854 1 4°691 408 6 4°672 624 2 4°052 4301 4°630 763 4 4°607 551 5 4°582 529 2 4°555 215 4 4.525 161 5 4°491 9217 4°454 982 0 4°414 003 5 4°369 2159 4°320644 7 4°268 461 0 4'212 4007 4°152 227 2 4°087 3909 4°017 283 8 3°941 2132 3°858 055 7 3°797 304 3 3°668 944 7 3°562 5308 3°447 023 1, 3°325 1050 3°194 7918 3°053 462 6 2°900 367 I 2°738 780 6 2°564 666 1 2°372 9120 2°176 og! 3 1°973 1279 1°698 9700
17| 1°230 4489
au= OO
9°990 9359 |" 9°990 3567 |° 9°989 7353 1° 9°989 052 6 |" 9°988 249 0 |° 9°987 3404 1° 9°986 3046 |° 9°985 1312 ]° 9°983 865 9 |- 9°982 554 5 |" 9°981 2156 ]° 9°979 805 9 |° 9°978 333 3]° 9°976 788 1 |° 9°9749777|° 9°972 686 2 |° 9°969 9461 |° 9°966 7602 |: 9°963 060 3 | ° 9°959 021 5 |° 9°955 212 4 |° 9°951 4288 |: 9°947 816 3 |° 9°943 9397 |° 9°939 826 5 |° 9°935 1637 |° 9°929 8929 |° 9°923 929 4 |° 9°916 842 5 |° 9°909 2486 |-
9°901 6404 9°893 586 1 9°885 092 3 9°877 4819 9°869 686 8 9°858 6708 9°846 904 5 9°838 413 5 9°825 885 5 9°808 2459 9°803 179 3 9°797 0366 9°725 842 1 9°531 4789
md °°)
Prob. Sur-
Healthy Lrves,—Male and Female.— Adjusted Table.
Prob. Dying
viving 1 Year| in 1 Year.
"797 3342 *782 683 2 "797 5247 “7541919 ‘7407758 "722 2222 "792 9177 689 308 1 "669 708 o 643.051 8 635 3932 "626 666 7 "5319149 *340 000 0 "000 000 0
fz
0206546
"027 9599 "023 3581 "024 892 3 "026 694 9 ‘028 7290 "031 042 § "033 657 3 "036 468 4 "039 373 6 "042 3306 "045 434 3 "048 665 4 "052 O44 1 eas 060 9 "066 861 7 °073 681 7 "O81 5400 "090 O41 7 "097 987 8 "105 8119 "113.219 3 "121 099 4 "129 384 4 "138 681 5 "149 0719 "160 6766 "174261 5 "188 5746 "202 665 8 "217 3168 "232 475 3 "245 808 x "259 2242 "2777778 "297 082 3 "310 6919 °330 2920 "356.948 2 "364 406 8 "373 333 3 *468 085 1 *660 000 0 T°000 000 0
TABLES.
ONE LIFE.
—EEe ee CO —————— ———
H™,
HEALTHY MALE LIVES.
Lan ime
we
wo tS) WO WOr~3 An POH HO OHO WT AN PW! vw WO /}O W~ 7 Aw Pw WD HO
9
Sas >> > 0 OO On~T AM RO DY | O
G Db
2 H™
Elementary Values.
Log t,
5°000 000
4°997 867
"996 131 "994 687 "993 419 "992 218 "990 969 "989 557 "987 867 "985 781 "983 279 "980 521 ‘977 595 "974 608 971 661
"968 768 "965 879 "962 966 "959 957 "956 831
953 591 "950 224 "946 771 "943 237 "939 624
"935 915 "932 088 "928 14 "923 984 "91g 716 "915 315 "910 816 "906 238 "QOL 551 "896 692 "891 643 886 316 "880 659 "874 667 868 350 "861 690 "854 707 "847 406 839 717 "831 563
1997 867 "998 264 "998 556 "998 732 "998 799
"998 751 ‘998 588 "998 310 "997 914 "997 498
997 242 "997 070 "997 017 "997 053 "997 107
"997 {II "997 087 "996 991 "996 874 °996 760
"996 633 "996 547 "996 466 "996 387 "996 291 "996 173 "996 026 "995 870 "995 732 "995 599
"995 501 "995 422 "995 313 "995 T41 _ 994951
"994 673 "094 343 "994 008 "993 683 "993 340°
"993 O17 "992 699 "992 311 "991 846 "991 344
000 397 000 292
000 176 000 067
999 952
999 837 999 722 999 604 999 584 999 744
999 828 999 947
000 036 000 054 000 004
999 976 999 904 999 883 999 886 999 873
999 914 999 919 999 921 999 904 999 882
999 853 999 844 999 862 999 867 999 902
999 921 999 891 999 828 999 810 999 722
999 670 999 665 999 675 999 657 999 677
999 682 999 612 999 535 999 498 999 423
Colog #,
5°000 000 002 133 "003 869 "005 313 "006 581
"007 782
"009 031
"O10 443 "O12 133 "014.219 7016 721 "019 479 "022 409 "025 392 028 339 "031 232 "034 121 "037 934 "040 043 "043 169
‘046 409 "049 776 "053 229 "056 763 "060 376
"064 085 "067 O12 "071 886 "076 016 "080 284
"084. 685 "089 184 093 762 "098 449 "103 308
"108 357 "113 684 "1IQ 341 "125 333 "131 650
"138 310
"145 293
"152 594
"160 283 "168 437
Zz
a an
ww
wo w WO COr~ 7 Qn PW OW HO WO DWr~ At HON HO LO Cr Qn PO WB HO
©o
bwenad
tin hObYH HO HO CON AN
0°002 133
on pow nS
in POH mM O OO DOr ANH
"O01 736 "OOl 444 "001 268 "OOl 201
"OO! 249 "OO! 412 "001 690 "002 086 "002 502
"002 758 "002 930 "002 983 "002 947 "002 893
“002 889 "002 O13 "003 009 "003 126 "003.240
"003 367 "003 453 "003 534 "003 613 "003 709 003 827 003 974 "004 130 "004 268 "O04 401
"004 499 ‘004 578 "004 687 "004 859 "005 049
"005 327 "005 657 "005 9g2 "006 317 "006 660
006 983 "007 301 "007 689 "008 154 "008 656
Elementary Values.
999 933 000 048
000 163 000 278 000 396 000 416 000 256
000 172 000 053 999 964 999 946 999 996
000 024 000 096 000 II7 000 114 000 127
000 086 000 081 000 079 000 096 000 118
000 147 000 156 000 138 000 133 000 098
000 079 000 109 000 172 000 190 000 278
000 330 000 335 O00 325
000 3.43 000 323 000 318 000 388 000 465 000 502
000 577
H™,
397 329 288 272
282 318 379 466 556 609 643 650 638 622
617 618 634 654 673 694. 706 717 727 740 757 779
802 821 838 848 854 865 887 gil
95° 996 I O41 xr 082 1 124
1 160
I 193 I 235 1 286
I 339
Log 4,
490|,2°690 196
"598 791 "517 196 "459 392 "434 569
"450 249 502 427 "578 639 *668 386 "745 975 "784 617 “808 211 "812 913 "804 821 "793 79°
"790 285 “7y0 988 "802 089 "815 578 "828 015
"841 359 "848 805 "855 519 "86 534 "869 232 879 096 "B91 537 "904 174 "O14 343 "923 24y
"928 396 "931 458 "937 016 "947 924 "959 518
"O77 724 "998 259 3°017 451 "034227 "050 766
"064 4.58 "076 640 "og! 667 "109 241 "126 781
A
908 595 918 405 942 196 975177 O15 680
052178 076 212 089 747 076 689 039 542
023 594 004 702 991 908 988 969 996 495 000 703 OII 10] 013 489 O12 437 O13 344
007 446 006 714 006 O15 007 698 009 864
O12 441 O12 637 O10 169 008 gol 005 152
003 062
005 558
O10 go8
OIT 594 018 206
020 535 O19 192 016 776 016 539 013 692 o12 182 O15 027 017 $74 017 540 O19 037
O9I 405 081 595 057 804 024 823 984 320
947 822 923 788 O10 253 923 311 960 458
976 406 995 298 008 092 O11 031
003 505
999 297 988 899 986 511 987 563 986 656
992 554 993 286 993 985 992 302 ggo 136
987 559 987 363 989 831 99T 999 994 848
996 938 994 442 g89 092 988 406 981 794 979 465 980 808 983 224 983 461 986 308 987 818 984 973 982 426 982 460 980 963
bh |
we
©» ©) © cer An POH HO OO ONAN PO HHO LO Or Qn HO DW =» O
aX)
aN && DH O
>
—& OO Be O WO COM AN
=
FS pw wn
ee] fo ¢) ~J ~Y WO CO~NrAMH HW He GO LW GW TAN HON MO WO OH” A
& & DY
‘Oo “IAN
66 513 O65 114 63 652 62 125 60 533
58 866 57 119 55 289 53 374 51 373
49 297 47 156 44 960 42717 40 443
38 124 35 753 33 320 30 823 28 269
25 691 23 164 20 700 18 326 16 068
13 930 IIQIS 10 032 8 313 6 768
5 422 4 284 3 343 2 57° 1955 1 460
Log f;
4°822 907
"813 674 "803 812 "793 266 "785 992 ~769 865 "756 781 "742 639 "727 33° 710 735 692 820 ‘673 537 652 826 "630 Gor 606 843 581 198 "553 312 "522 705 "488 875 "451 310 "409 781 "304 814 "315 970° "263 068 "205 962
"143 951
"076 094 "001 388
3°919 758
"830 460
"734 160 631 849 524 136 "499 933 "2O1 147 "164 353 "022 016
2°859 138
"671173 437751 "130 334
1°690 196 0°954 243
4 HM,
Log Pr
1990 767 "990 138 "989 454 "988 726 "987 873 986 916 "985 858 984 6g! "983 405 982 085
‘980 717 "979 289 °977 775 "976 242 "974 355
"972 114 ‘969 393 °966 170 "962 435 "958 471
"955 933 "951 156 "947 098 "942 894 "937 989
"932 143 "925 2904 °918 370 "910 702 "903 700 "897 689 "892 287 "885 797 "881 214 873 206 "857 663 837 122 "812 035 "766 578 "692 583 "559 862 264 047
Elementary Values—(continued.)
A
999 371
999 316 999 272 999 147 999 943
998 942 998 833 998 714 998 680 998 632
998 572 998 486 998 467 998 113 997 759
997 279 996 777 996 265 996 036 996 562
996 123 995 942 995 796 995 995 994 154 993 151 993 076 992 332
‘992 998
993 989
994 598 993 510 995 417 991 992 984 457 979 459 974913
954 543 926 005
867 279 704 185
Colog 2,
5°77 993 "186 326 "196 188 "206 734 °218 008
"230135 "243 219 "257 361 *272 670 289 265 *307 180 "326 463 "347 174 "309 399 "393 157 "418 802 *446 688 "477 295 "511 125 "548 690 "590 219 635 186 "684 030 "736 932 "794 038
"856 049 "923 906 998 612 4°080 242 "169 540 "265 840 "368 151 "475 864 *590 067 708 853
835 647 _"977 984 3°140 862
328 827
"562 249
"869 666 2°309 804 1045 757
qn WO co~nr Qin
A A © WH O
“wy OO cor~7 Cin PO HD O LW Or~ 3 Quan
Ve) 0o (o2) ~} OO CON AM PW WH HO
Pwpr oO
\O “IAN
in - Fre vn8 oor ot | &
~r
6 | 8 9 ° I 2 3 4 5 6 7 8 9
co oe] Bo wow akd P&H 1 O
wae f O DN HH
Colog pz
0°009 233 "009 862 "O10 546 "O11 274 "O12 127
"013 084 "O14 142 "O15 399 "016 595 017 O15 "019 283 "020 7II "022 225 "023 758 "02.5 645 "027 886 "030 607 "033 830 037 565 O41 529 "044 967 "048 844 "052 902 "057 106 -°062 O11 "067 857 "074 706 "081 630 089 298 "096 300 "102 31r "107 713 "114 203 "118 786 "126 794 "142 337 "162 878 "187 965 "233 422 "397 417 "440 138 735 953
5 H™
Elementary Values—(continued.)
A
000 629 000 684 000 728 000 853
000 957
oor 058 oor 167 oor 286 OOL 320 oor 368
oor 428 OO! 514 OO! 533 oo1 887 002 241
002 721 003 223 003 735 003 964 003 438 003 877 004 058 004 204 004 905 005 846 006 849 006 924 007 668 007 002 006 or!
005 402 006 490 004 583 008 008 O15 543
020 541 025 087 045 457 073 995 132721
295 815
ad, | Logd,
I 399 I 462 I 527 I 592 1 667
1747 1 830 Tgt5 2 Oo: 2 076
"104 947 "183 839 "201 943 "221 936
"242 293 "262 451 282 169 "303 247 317 227
*330 617 "341 632 "350 829 "356 790 "305 501 "374 932 *386 142 "397 419 "407 221 "411 283
214! 2 196 2243 2274 2 319
2 371 2 433 2 407 2 554 2578 2 527 2 464 2 374 2258 2 138
"402 605 391 641 "375 481 353 724 "330 008
"304 275 274 850 "235 276 "188 928 "129 045 "056 142 2°973 590 888 179 788 875 "694 605 "610 660 "517 196 "404 834 "290 035 "143 O15 86 1°934 498 40! ‘602 060
9} 9°954 243
2015 1 883 1719 1 545 I 346 1 138 941 773 615 495 408 329 254 195 139
3°145 818
A
O19 129
018 892 018 104 O19 993 020 357
020158 org 718 019 078 o15 980 O13 399 O11 O15 009 197 005 96: 008 511 009 631
OII 210 O1l 277 009 802 004 062 Q9I 322
989 036 983 840 978 243 976 284 974 267 97° 575 960 426 953 652 940 117 927 097 917 448 914 589 goo 696 995 73° 916055 906 536 887 638 885 201 852 980 791 483 667 562 352 183
—A
980 871
981 108 981 896 g80 007 979 643 979 842 980 282 980 922 984 020 986 610
988 985 990 803 994 939 991 489 99° 369
988 790 988 723 ggo 198 995 938 008 678 O10 964 ©16 160 O25 757 023 716 025 733
029 425 039 574 046 348 059 883 072 903 082 552 085 411 099 304 094 270 083 945 093 4064 112 362 114 799 147 020
208 517
332 438 647 817
in OO Gs~7 An
WN .@)
\O ioe) Cc ~T ~T! ~A 0 OO G7 Qn POH ew O WO Wr QAKr HW HD CO CO CGO~ AN ADO D &
Pw bP
Le) ~s CAG
Probabilities of Living over, and >
ration of
|
=
wo
Oo
Go
fe) I 2 3 4 5 6 7 8 9 ) I 2 3 4 25 6 7 8 9 ° I 2 3 4 5 6 7 8 9
iw, ro] © wx Oth pwn nd
~~ &W WY
Average
"995 100 "996 O10 "996 681 997 085 997 238
"997 129 "996 753 "996 118 "995 208 "994 255
"993 671 "993 275 "993 156 "993 236 "993 361
"993 370 "993 315 "993 096 992 828 "992 567
"992 277 "992 083 "991 895 "OOT 715 "991 496 "991 226 "990 8g! "990 536 "990 220 "989 918
"989 694 "989 513 "989 266 "988 873 "988 444 "987 808 "987 060 "986 298 "985 560 "984 780
"984 050 "983 330 "982 451 °981 400 ‘980 266
"002 871 "003 247 "003 882 "004 792 "005 745 "006 329 7006 725 "006 844 "006 764 "006 639
“006 630 "006 68 5 "006 go4 “007 172 "007 433
"007 723 "007 917 “008 105 "008 285 "008 504
"008 774 "009 10g "009 464 "009 780 "010 082
"O10 306 "010 487 "O10 734 ‘OII 127
“OII 556
"O12 192 "012 940 "O13 702 "014.440 "O15 220
"015.950 "016670
"O17 549 "018 600
"O19 734
Oe ws om
~r &Oowvwm O OW CT A
~
co © on Ar SPO Hm O YO ON Qn
1° 2)
‘Oo
he) “saat PO WP HO
g
fe,
"004 YOO | 50° "003 990 "003 319 "002 O15 "002 762
"979 322 "967 962 "995 364 "962 510 "959 59°
"956 569 "953 431 "Q5O 1 "946 766 "942 660
"937 808 "931 95° "92,5 060 "917 140 908 805
"QOT 639 "893 628 "885 314 876 787 "866 O41
"855 348 "841 964 "828 648 "814 147 801 123
"790 II5 "780 345 °768 770 "760 700 "746 804 "720 548 687 263 648 686 "584 222 °492 700
"362 964 "183 673
ing in, a Year, and the oe each A ge
Tz (1—pz)
"021 033 "022 453 "023 9QO "025 626 027 539
"029 678 "032 038 "034 636 "037 49° "040 410
"043 431
"062 192 "068 050 "074940 "082 860 "091 195 "098 361 "106 372 "114 686 "123 213 "133 959 "144652 "158 036 "171 352 "185 853 198 877 "209 885 "219 655 "231230 "239 300 "253 196
°279 452 "312737 "351314 415778 "507 300 637 036 °816 327 "000 000 [5°000 000
H™,
THREE PER-CENT.
Lm! bout
wo
©o
&o we WO Cn~T Qin BPW HD Mm CO CO DOr? AN HMO OHO WO ONAN HPO HD mm O
pwend
pa
Or —&woObK eM CO OO Cr Qn
74 400'4 71 888'1 69 515°9 67 267°! 65 117°5 63 046°2 61 034'2 59 064°! 57 121°2
55 1907
53 276°3 54 3972 49 504°7 477907 46 085°8 44 446°5 42 865°8 41 339°1 39 857°9 38 419°5 37 02 3°2
35 667°3 34354°2 33 083°3 31 853°6
30 662°8 29 508°6 28 388°1 27 300°5 26 246°1
25 224°7 24 237°6 23 284'9 22 364°0 21 471°!
20 604°8 Ig 760°8 18 936°9 18 133°5 17 35Ul 16 589°3 15 849°2 15 131°! 14 4326 137516
8 H™.,
3 PER-CENT. Commutation Table.
I 796 867 1724979 1655 403 I 588 196 I §23 079 I 460 032 I 398 998 T 339 934 ¥ 282 813 1 227621
1174345 1 122947 1073 383 1025 59!
979 5§95°3 935 058'8 892 193°0 850 854'0 810996"! 772 576°6
735 553°4. 699 886°1 665 531°9 632 448°6 600 595°0
569 932°2 540 423°6 512 035°5 484 735°0 458 489°0 433 264°2 409 026°6 385 741°7 363 377°6 341 906°6 321 301°8 301 541°0 282 604°! 264 470°7 247 119°6 230 530°3 214 681'0 199 549°9 185 117°3 171 3658
36 413 646 34 616779 32 891 800 31 236 337 29 648 141 28 125 062 26 665 030 25 266 032 23 926 098 22 643 285 21 415 664 20 241 319 19 118 371 18 044 988 17 O19 397 16 039 892 15 104 833 14 212 640 13 361 786 12 550790 11778214 11 042 660 10 342 774
9 677 242
9 044 794
8 444 199
7 874 266
7 333 843
6 821 807
6 337072
5 878 583 5 445 319 5 036 292 4.650 551 4287 173
3945 267 3 623 965 3 322 424 3 039 820 2775 349 2 528 229 2 297 699 2 083 018 1 883 468 T 698 351
19 906°2
19 552°2 19 273°8 19 049°7 18 859°3 18 684°7 18 509.0 18 316°6 18 o94'0 17 828°2
17 520°4 17 193°0 16 857°4 16 528°1 16 214°2
159172 15 6311 15 352°9 15075'8 14 798°2 14 521°0 14 243°4 13, 969°2 13 698'9 13 432°8 13 169'8 12 908°6 12 647°6 12 386°8 12 127°6
11 870°7 II 618°3 II 371°5 rr £28°8 10 887°3
10 646°3
10 402°4
10 154°2 9 902°27 9 64805 9 391°66 9 134°76 8 878:25 8 620°44 8 359°81
756 181°7
736 275°5 716 723°3 697 449°6 678 399°8 659 54°°5 640 855°8 622 346°7 604 030°1 585936": 568 107°9 55° 587°5 533 394°5 516 537°0 500 008°9
483 794'7 467 877°5 452 246'5 436 8936 421 817°9
407 0196 392 498°7 378 255°3 364 286° 350 587°2
337 154°5 523 984°7 311 0761 298 428°4 286 0o41°6
273 914°1 262 043°4 250 425°1 239 053°6 227 924°7 217 037°5 206 391°! 195 988°7 185 834°5 175 932'2 166 284'2 156 892°5 147 757°8 138 879°5 130259'1
a. o> & ee S = = 0 OO ONAN PO YHOO HO CONAN HW BK EO LO Gr? An Pw WY =O
OK pwen
iw Hove O OO CONTA
CO —— anni <
z D,
55|13 087°6 6|12 439°! 7111 805°7 8i1x 186°8 g|10 582°7 999F51 9 412°61 8 845°67 8 290°57 7 147°34
7 217°74 6 703°17 6 204°86 5 723°60 5 261°08
4 814°96 4 383°99 3 966°66 3 562°52 3.172°17 279891
2450°10
DP pow D
~
“I
9 HM™,
3 PER-CENT. Commutation Table—(continued),.
Nz
158 278-2
145 839°! 134933°4 122 846°6 112 263°9
102 272°4 92 859°77 84 014'10 75 723°52 67 976°19 60 758°45 54055728 47 850°42 42 126°82 36 865°74 32050°78 27 666°79 23 700°Y3 20 137°61 16 965°44 14 166°53 II 716°43
Sz
I 526.985 1 368 707 1 222 868 1 088 834
965 987°7
853 723°8 751 451°4 658 591°6 574 577°5 498 8540
430 877°8 37° T19*4 316 064°1 268 213°7 226 086'9
18Q 221°% 157 170°3 129 503°5 105 803-4
M;
8 096°34 7 829°08 7 557°92 7 282°95 7 004°63 6 721°69 6 433°80 6 141°02 5 843° 56 5 541°80
5 237°85 4933°5° 4 630°44 4 329°90 4 034°08 374120 3 45°°47 3 160°83 2 872°23
85 665°80|2 585°63 68 700°36|2 304°77 54 533°83)2 037°49
|121 899°2
OO COr~7 Ain WH WO WO Dr T1 OA
OO OO Cr? Qn huwpne
ve) °
\©O “ran # © b
2125°71 1827°11 I 555°32 I 309°10 1 087°12 888°659 714°938 _ 565111
439°537 337°170 255°445 190°659 140°810
102'095
714214 47°655 6 30°013 I 17°023 6 8°143 2 2869 6 “517
9 590°717
7 763°609)| .
6 208°285
4 899°184 3 812°061 2 923°402 2 208°464 I 643°3 5¢ I 203°817 866°647 611°202 420°543 279°733 177°638 106°217 58°561 28°548 11°525 3°381 512
“000
42 817-40|1 784°46 33 220°68)1 547°77 25 463°07|I 329°20 19 254°79|1 128°28
14 355°61 10 543° 54 7 620°14 5 411°68 3 768°32 2 504°51 1 697°86 1 086°66 666'11 386°38 208°74
944°429 777-628 629°790 500°787 391°672 302°107 2.30°203 172°857 128°561
93°947 0 66°247 5 44°5619 2B°307 4 16°192 I 7°807 6 2.9711 "4968
R, . |2
wn © con~r Aw
113 802°9
105 973°8 98 41591 OT 132°95 84 128°32 77 406°64 7° 972°84 64 831°82 58 988°26
53 440°26 48 208°62 4327511 38 644°67 34 314°77 30 280°69 26 539°49 23 089'01 19 928°18 17 055°96
14 470°32 12 165°55 10 128°06 8 343°609] 8 6 795°842} 9 5 466°642/ 80 4 338°365) 1 3 393°937| 2 2616°3009| 3 1986°518] 4 1 485°732|85 1.094'060|] 6
791°953| 7 561°750] 8
388°893) 9 260°331}g0 166°384] 1 I00°I37} 2 55°575} 3 27'268) 4 11'076195 3°268] 6
497} 7
Cc
~r OS pwn
wy WIA FW RHO OO DOXA
10
Ho®™, 3 PER-CENT. Logarithms and Co-logarithms of D,, N,, and M,; with their Differences.
A A (Log vp;)| 1° P= \ Cologep,
te ee Pe eel
4°871 628/1°985 029 5°128 372/0°014 97116'254 5161982 268
Ln]
"856 657
we ww »p Sy me BS ema &ObwN me O WO CH~3 An PHD = O [OO DW~ 1 Qn HON - O
PO