► VITAL STATISTICS. BY f E. B. ELLIOTT, | '■’S'-. OF BOSTON. From the Proceedings of the American Association for the Advancement of Science. * 50 A. MATHEMATICS AND PHYSICS. u VITAL STATISTICS. A. Tables of Prussian Mortality, interpolated for Annual Intervals of Age; accompanied with Formulce and Process for Construction. B. Discussion of Certain Methods for converting Ratios of Deaths to Population, within given Intervals of A ge, into the Logarithm of the Probability that one living at the Earlier Age will attain the Later ; with Illustrations from English and Prussian Data. C. Process for deducing accurate Average Duration of Life, present Value of Life-Annuities, and other useful Tables involving Life- Contingencies, from Returns of Population and Deaths, without the Intervention of a General Interpolation. The mortality and accompanying tables, to which the attention of the Association is called, comprise portions of a series of tables that have been and are being prepared, for the New England Mutual Life Insurance^ Company of Boston, from official returns of the British, Swedish, Prussian, and Belgian governments, and from such reliable American statistics as are obtainable. In several of the United States of America the decennial enumer¬ ation of the numbers and ages of the living effected for the General Government have been quite accurate and reliable, while the only official mortality returns (viz. those ordered in connection with the last census, 1850) are inaccurate and deficient. In Massachusetts, since its Registration Act of 1849, certain districts have furnished valuable and satisfactory information respecting the numbers and ages of the dying ; but from the published abstracts it has been impossible to separate imperfect from reliable data. In the yet unpublished abstracts of the returns for 1855 an improvement is being effected, under the direction of the present Secretary of State, which, although augmenting somewhat the expense, will afford fit material for the construction of a Life-Table that shall satisfactorily represent the rates of mortality prevailing among the inhabitants of the larger part of the Commonwealth. The leading paper (A) presents a new Life-Table, complete for annual intervals of age, and calculated from over a million (1,197,407) V\A I. ^ r ' -‘ " ?’ uV^ ne-nnajti c *5 MATHEMATICS. 51 _ A * > of observations regarding the ages of the dying, in a population of fifteen millions (14,928,501), and in a community where observations a— on vital statistics, for many years, are believed to have been made with care and accuracy. It adds one to the very limited list of National Life-Tables. The remaining papers (B and C) are devoted to the discussion of certain methods for converting rates of mortality for different inter¬ vals of age into probability of living ; and to the presentation of abridged methods for calculating, at certain ages, accurate tables of - practical value, involving life-contingencies, accompanied with simple rules for determining any required value intermediate. A. Tables of Prussian Mortality, interpolated for Annual Inter¬ vals of Age ; accompanied with Formulae and Process for Con¬ struction. The data from which the following tables have been calculated were obtained from documents sent by Mr. Hoffman of Berlin to the English Ministry of Foreign Affairs, and published in the Sixth Annual Report of the Registrar-General in England. Population of Prussia, Civil and Military (exclusive of Neuf- chatel).* At the end of the year 1834, 13,509,927. 46 44 1837, 14,098,125. 44 44 1840, 14,928,501. The documents above mentioned give no statistics of immigration or emigration. The increase of population during the three years 1838, ’39, ’40, was 830,376. The excess of births over deaths during the same three years was 486,937. Leaving 343,439, which is 41.36 per cent of the total increase of population, unaccounted for by excess of births over deaths. * “ The population of Neufchatel, not included in the above, was 59,448 in 1837, 52,223 in 1825” ^ 2 . £$4 2 - 52 A. MATHEMATICS AND PHYSICS, Population of Prussia at the End of the Year 1840, classed accord¬ ing to Age and Sex. Ages. Males. Females. Males. Females. Ages. 0- 5 1,134,413 1,114,871 ) 5- 7 370,740 336,429 ( > 2,603,699 2,550,022 0-14 7-14 1,098,546 1,068,722 \ 14-16 344,179 331,039 14-16 16-20 586,059 20-25 692,704 25-32 777,183 -3,238,434 3,253,643 16-45 32-39 646,122 39-45 536,366 45-60 816,726 881,280 45-60 60 and upwards, 445,544 463,935 . 60 and upwards. All Ages, 7,448,582 7,479,919 Assuming the distribution of the (3,253,643) females for the several intervals between the ages 16 and 45 to be proportioned to the distri¬ bution of (3,238,434) the corresponding number of males, we have Ages. Females. 16-20 588,812 20-25 695,957 25-32 780,833 32-39 649,156 39-45 538,885 Total, 16-45 3,253,643 Hence the following Numbers and Ages of the Population of Prussia at the End of the Year 1840. Ages. Persons. 0- 5 5- 7 7-14 14-16 16-20 20 - 25 25-32 32-39 39-45 45-60 60 and upwards, 2,249.284 737,169 2,167,268 675,218 1,174,871 1,388,661 - 1,558,016 1,295.278 1,075,251 1,698,006 909,479 All Ages, 1 14,928,501 We wish to distribute the population from ages 25 to 45, from 45 to MATHEMATICS. 53 60, and from 60 upwards in quinquennial or decennial periods, to corre¬ spond with the ages of the dying as presented in the mortality returns. We first determine the quinquennial distribution between ages 25 and 45. Let P x/y represent the population between ages x and y, or the numbers living under age y, less the numbers living under age x. P16/20 = 1,174,871 P16/25 = 2,563,532 P16/32 = 4,121,548 P16/39 = 5,416,826 P16/45 = 6,492,077 P16/60 = 8,190,083 Let a = 20 , b = 25, c Let a = 25, b = 32, c Let a — 32, b = 39, c Let a = 39, b = 45, c Assume P 16/ , = P l6/a + P 16/6 4“ P 16/c 32 ; then will Pi 6 / 3 o = gq # (then will P 16/30 — ’ 1 and P 16/35 = then will P 16/35 = and P /40 = 60 ; then will P 16/40 = x—b. x - c a-b . a-c x — a . x-c b-a . b-c x-a . x-b c-a . c-b : 3,722,366. 3,703,211, 4,708,839. 4,682,050, : 5,598,277. : 5,611,751. } } i Taking the arithmetical mean of the above duplicate results, we have P l6l30 = 3,712,789 Hence P 25 , 30 = 1,149,257 P 16/35 = 4,695,445 P 30/35 = 982,656 ■P.6,40 = 5,605,014 Pas/40 = 909,569 P 40/15 = 887,063 which results cannot vary materially from the actual distribution. The following Table gives the distribution of the population of Prussia between ages 45 and 60 (1,698,006) ; and of the population from age 60 upwards, according to the corresponding proportional dis¬ tribution of the numbers of the population of the Northwestern Division of England (the Eighth of the eleven Districts into which England and Wales are divided in the Reports of the Registrar-General). Ages. Distribution of Prussian Population over age 45. 45-55 1,257,322 55-60 440,684 60-65 353,657 65-75 398,925 75-85 137,188 85 - 95 18,638 95 and over, 1,026 1 1 45 and over, 2,607,485 5 * 54 A. MATHEMATICS AND PHYSICS. Numbers and Ages of the Population of the Northwestern Di¬ vision (Eng.) in 1841, according to which the above Distribution was made. — (9th Rep. Reg.-Gen.) Ages. Northwestern Division (Eng.), 1841. Numbers and Ages of the Living above Age 45. 45-55 151,064 55 - 60 52,947 60 - 65 39,656 65 - 75 44,732 75-85 15,383 85-95 2,095 95 and over, 115 1 45 and over, 305,992 The numbers living above age 15 in the Northwestern Division (England) were grouped, with reference to age, only in decennial classes. By assuming the algebraic equation, 55 lx P 55 55/45 x —55 . x — 65 . * — 75 45 —55.45 —65.45 — 75 x —45.07 — 55.0: — 75 + i si65 ■ 65 — 45 . 65 — 55.65 — 75 + P: 07 — 45. 07 — 55. 07— 65 55/75 75 — 45 .75 — 55.75 — 65 a close approximation to the probable number of persons living be¬ tween ages 55 and 60 (52,947) and between ages 60 and 65 (39,656) resulted. P 55/x represents the number of persons reported living under age 07, less the number living under age 55. We remark that P 55/45 is essentially negative. The population of the Division, as returned for the night of June 6-7, 1841, was two millions (2,098,820), being one eighth of the entire population of England and Wales (15,914,148) at that date. The counties of Cheshire and Lancashire constitute this Division. The latter county includes the densely populous and unhealthy district^ of Liverpool . /fyl The ratios of deaths to population for the intervals from age 45 to 60, and from age 60 upwards, more closely approximated the corre¬ sponding ratios for Prussia, than did those of any other large commu¬ nity concerning which reliable population and mortality statistics were to be obtained. MATHEMATICS. 55 Table comparing Ratios of the Annual Number of Deaths to the Numbers living in certain Communities from Age 45 to 60, and from Age 60 to extreme Old Age. Ages 45 - 60. Ages 60 and upwards. Prussia. Deaths, 1839, ’40, ’41 ,1 .024^? * 0 .089- Population, 1840, J Northwestern Division [England). Deaths, seven years, 1838 - 44, \ Population, middle, 1841, 1 .023 .079+ Sweden. Deaths, twenty years, 1821 -40, ) Population, mean of 1820,’30,’40, J .021 .079— Belgium. Deaths, nine years, 1842-50, ) Population, October 15, 1856, J ’ .020 .073 Enqland and Wales. 1841, . .019 .069 A comparison of the distribution of the numbers of the living in Prussia in these intervals of age according to that of the North¬ western Division (.Eng.), with a distribution of the same numbers according to the mean of the corresponding distribution of equal numbers of the populations of England in 1841 and of Belgium in 1846, would give the following results. Distribution of the Population of Prussia according to I Ages. The Mean of Equal Numbers The Northwestern Division in England and Belgium. (England). 45-55 1,285,567 1,257,322 55-60 412,439 440,684 45-60 1,698,006 1,698,006 60-65 330,425 353,657 65-75 398,646 398,925 75-85 155,077 137,188 85 and over, 25,331 19,709 60 and over, 909,479 909,479 45 and over, 2,607,4^85 v 2,607,485 The distribution according to the English and Belgian facts would give larger numbers after about age 75, in the resulting Life-Table. The distribution according to that of the Northwestern Division was adopted as the best representation of the probable corresponding distribution of the population of Prussia, within the intervals of age above mentioned. Hence the following Table. 56 A. MATHEMATICS AND PHYSICS, Deaths, Population, Mortality, and Logarithms of the Probability of Living in Prussia. The Numbers of the Living between ages 45 and 60, and from 60 to extreme old age are distributed according to corresponding proportional distributions of the numbers of the population of the Counties of Cheshire and Lancashire (Northwestern Division ), in England , in 1841. Deaths. Aggregate Numbers and Ages of the Dying during the Three Years 1839, ’40, ’41 Population. Numbers and Ages of the Living at the End of the Year 1840. Ratios of the Average Annual Numbers of the Dying during the Three Years 1839, ’40. ’41, to the Numbers of the Living com¬ puted with reference to the Middle of 1840. Mortality. Logarithms of Probabil¬ ities of Surviving each Interval. Duplicate Values, each deduced from two consecutive Ratios in the Column of Mortality. Values derived from Comparison of the Duplicates. Ages. x > y- D 0, V ®0,x P —P 0/y X 0/x D ^P o/y ^Po/x M X Px 0-1 1-3 3-5 5-7 7-14 14-20 20-25 25-30 30-35 35-40 40-45 45-55 55-60 60-65 65-75 75-85 85 and upwards Total, 310,527 162,356 62,734 33,272 27,156 34,585 36,849 31,594 35,579 38,094 78,503 46,704 58,576 107,653 61,697 15,572 2,249,284 737,169 2,167,268 1,850,089 1,388,661 1,149,257 982.656 909,569 887,063 1,257,322 440,684 353.657 398,925 137,188 19,709 .0802238 .0152056 .0077790 .0062978 .0089397 .0096939 .0108317 .0131780 .0144675 .0210345 .0357042 .0557995 .0909134 .1515098 .2661784 1,197,407 14,928,501 — .013106 — .013201 — .023480 — .023628 — .016399 — .016432 — .019433 — .019418 — .021057 — .021059 — .023533 — .023542 — .028646 — .028630 — .031434 — .031464 — .092155 — .092527 — .077947 — .078021 — .122543 — .121891 — .424020 — .408584 — .716433 — .730677 — .013155 — .023557 — .016416 — .019425 — .021058 — .023537 — .028637 — .031449 — .092322 — .077981 — .122189 — .415608 — .722021 14928501 — Population of Prussia, as returned for the end of the year 1840. 14770727 — Population of Prussia, estimated for the middle of the year 1840, from the numbers returned as living at the end of 1834,1837, and 1840. MATHEMATICS. 57 It will be observed that the values derived from comparison of the duplicate logarithms, and which have been adopted in constructing the Interpolated and other Tables, are not in all cases arithmetical means. The difference is of little moment, but there is no sufficient reason for preferring the former. Logarithms of the Probability of Surviving, Computed from the Returns of the Numbers of the Living under Age 5; and of the Numbers of the Dging annually under 1 Year of Age, over 1 and under 3, over 3 and under 5. Ages. ^Px,y: 0-1 — .082920 1-3 — .051670 3-5 — .022522 The successive addition of the logarithms of the probabilities of surviving the consecutive intervals of age to 5.001688, the logarithm assumed for the proportional numbers born alive, gives the following Table of the Logarithms of the Proportions, and the Proportions of Persons born alive and surviving certain Ages in Prussia, ac¬ cording TO THE CALCULATED LAW OF MORTALITY. Deaths , 1839, ’40, ’41. Population computed with reference to middle of 1840. Distribution of Population above Age 45, Northwestern Division {Eng.). 1 Survivors. Age. Logarithms. Persons. X L L X X 1 0 5.001688 100,389 1 4.918768 82,941 3 4.867098 73,637 5 4.844576 69,916 14 4.807864 64,249 25 4.772023 59,159 35 4.727428 53,386 45 4.667342 46,488 55 4.575020 37,585 65 4.374850 23,706 75 3.959242 9,104.2 85 3.237221 1,726.7 j 95 1.982879 96.1 105 1.803755 .636 58 A. MATHEMATICS AND PHYSICS. The values opposite ages 95 and 105 were computed from the logarithms of the numbers surviving at ages 65, 75 , and 85, by the exponential formula, X L x = $ x = $ 65 + ($ 75 -$ 65 ) Jt 5~ -65_ j 5 in which 0 75 — 65 (or q'°) = $85-$75 $75 - $ 65 * These values were adopted as bases for the construction of the accompanying Life-Table interpolated for annual intervals of age; and also for computing by abridged methods certain practical life- contingency tables. Before presenting this table and these methods, we will state some of the principles which underlie, and indicate the process by which ratios of the numbers of the dying to the numbers of the living, during the several intervals of age, have been converted into logarithms of the probabilities that one living at the earlier age will attain the later. Whenever, in any community, the intensity of mortality at each age, or the ratio of the numbers momentarily dying during each minute interval of age to the numbers then living within the same interval, has been constant for a period of time equal to the difference between the specified age and the extreme of old age, an invariable law of mortality is said to prevail in that community. The law of human mortality is seldom strictly invariable. It fluc¬ tuates within certain limits , not only with different communities and localities, but in the same community during successive periods, and in the same localities. The habits, occupations, and social condition of the members of the community remaining unchanged, the larger their numbers the narrower these limits. It is within the province of the vital statistician to determine, not merely an average of the rates of mortality prevailing in a community, but also the sensible limits within which the rates fluctuate. Our present inquiries have reference to the determination of a law of mortality which shall satisfactorily represent the average of the rates prevailing among the inhabitants of a populous state, with fixed geographical boundaries ; and in which the numbers of the inhab¬ itants vary with births and with deaths, with immigration and with emigration. MATHEMATICS. 59 If, in a large community, varying with births, deaths, and migra¬ tions, but in which the numbers of the population have not been subject to sudden and irregular change, the number of the dying during a given year or period of time between ages not very remote be divided by the number of the living between the same ages at the middle of that period of time, the quotient resulting from the division has generally been assumed closely to approximate the quo¬ tient that would have resulted had the numbers of the population within the limits of these ages been stationary; that is, assuming an invariable law of mortality, had the numbers of the population for each minute interval of age within the limits of these ages re¬ mained constant during that period, and unaffected by either immi¬ gration or emigration. The. errors involved in this assumption are of small moment com¬ pared with probable errors of observation, and vanish when the inter¬ vals of age are taken exceedingly minute, and where the excess or deficiency of the deaths in the former half of the period of time, with reference to half the deaths of the entire period, is exactly counter¬ balanced by a corresponding deficiency or excess of the deaths in the latter half of that period of time. We adopt this hypothesis, and assume that each of the ratios in the column headed Mortality is identical with that which would have resulted had the population of Prussia within the limits of the ages been stationary for a period of years equal to the specified interval ; and we also assume the accuracy of the Prussian mortality and pop¬ ulation returns. From these ratios we now proceed to determine duplicate logarithms of the probability that one surviving the earlier age in each interval will attain the later. Let P 0lx = the number living under age a?, in a stationary popula¬ tion, in which the same law of mortality prevails as in Prussia. D 0/x = the number of annual deaths under age x in the station¬ ary population. Z 0 = the number born alive each moment of time , in the stationary population. l x — the number surviving x years, out of (/ 0 ) the momentary number of births. 60 A. MATHEMATICS AND PHYSICS. j) alb — j — the probability that one surviving the earlier age («) will attain the later (b). Kit — h — 4 — the number dying in x years out of (7 0 ) the mo¬ mentary number of births. In a stationary population and therefore, n — 0/x Vq,x ~ dx' d Po/x - ^ x‘l =f (l 0 - K.) = ~f\.- But M alb , the ratio of the average annual number of deaths in Prus¬ sia between ages a and b to the number of the living between these ages, computed with reference to the middle of the period in which the deaths occur, equals ^ a/b _ •*-alb -^ 0 /b -t-0/a Assume d 0/a . = Qx Rx 2 , Q and R being unknown, and inde¬ pendent of the variable x. Then Q x -f- R x 2 A/, = dx and Hence M a/b ^which Palb _ Pq ib — Po/a\ Qb — a -f- Rb 2 — a 2 “ Kb-Pj ~ 7 7 _ lob — a — * 4—71 -o Q —j— R b —|— u l 0 Q b-\- R b 2 -|- a b -|- a 2 Mr, Q —j— jR c -j- b and MATHEMATICS. 61 So also, \p alb (which and ^ . Z 0 - ^0/6 X T = X TT- \ _ — Q& — Rb 2 ) ~ X l Q — Qa—Ra^ *Pb,c = * Z 0 — Qc — Rc 2 l 0 —Qb—Rb 2 Given M a/5 and M i/C , required X^ tt/6 and X j? 6/c . First determine values for Q and E ; the values of X p alb and \p bJC are then readily found. and in which n _ y'M a/b - p'M b/c ^ y'P — P'y * °’ v y M a/b — /3 M ble , R = y/3'- fiy 1 - ? “ ; 0 = i + ft 1 — b -[- a -(- (b*+ba + a*)M alb . 3 _ (c + &)Jf„. , 7 2 - 7 y = c + J+ (£± 5 i+^ 3 / The reduction may be simplified by letting Z 0 = 1, and by the use of addition and subtraction logarithms. In like manner, from M b/C and M c/d obtain \p b/c and \p c/d ; and so on for all intervals of age which the returns give. We thus obtain duplicate values for the logarithms of the proba¬ bilities of surviving all the intervals specified except two. For the first and the last interval we have but single values. We may, without material error, adopt for the true probability the mean of these dupli¬ cate results. It will be observed that the conversion, in each of these cases, is made for the entire interval, not, as is more frequent, for the middle 6 62 A. MATHEMATICS AND PHYSICS. year of the interval. We are thus enabled, without the intervention of a general interpolation, to compute directly the number surviving at certain ages in the resulting life-table, out of a specified number born alive. Usually the conversion is from a single ratio, based upon the assumption of a uniform distribution of deaths throughout the interval. By the present method, however, the conversion is effected, taking into account the actual or variable distribution of deaths, from three consecutive ratios, one preceding and another following the interval. A comparison of the relative accuracy and simplicity of several methods for effecting the conversion will be given on a following page. We now proceed to indicate methods for obtaining 'probabilities of surviving from birth to ages one, three , and five. We have the average annual number of deaths in Prussia under the ages of one, three, and five (jD 0/1 , D 0/3 , D 0/5 ), for the period of the three years 1839, ’40, ’41 ; and the population under the age of five (P 0/5 ) at the end of the middle year of the period (end of 1840); also the ratio © of the annual increase in the number of births deduced from the numbers registered for each of the six years 1836-41. The average annual number of deaths for the three years 1839, ’40, ’41 we shall consider identical with the number of deaths for the year 1840. From the following, it would appear that the accurate number of those born alive cannot be obtained directly from official reports, be¬ cause of probable deficiencies in registration. If the numbers of the living and of the dying at the earlier ages have been accurately observed and returned, if the numbers at these ages have been but little affected by immigration and emigration, and if the ratio of annual increase in the number of births can be obtained, a close approximation to the actual number of those born alive may be computed. Let Z 0 be the number of those momentarily born alive in Prussia at the time for which the census was taken (end of 1840). the ratio of the annual 1 /births 1839, ’40, ’41\*__ , V — (births 1836, ’37, ’38/ increase in the number of births estimated from those registered for each of the six years 1336-41. MATHEMATICS. 63 X - = .0066586, the logarithm of this ratio. v Let So,* he the number that died before attaining the age of x years (according to the prevailing law of mortality) out of (/ 0 ) the number born alive in Prussia during the moment of time (end of 1840) that the enumeration of the living is supposed to have been made. v x d 8 0/3 . will express the number of those aged x years that died in Prussia during the supposed moment of enumeration. / v x d 8 0/x S' v x = Dq/x , 0 */ 0 the annual number of deaths in Prussia under the age of x years, for the year ending with the census, i. e. for the year 1840. /* x /% x v x I v~ x I v x d8 0/x J 0 J o represents the total number that died in Prussia during the x years preceding the time of the enumeration of the living, out of the num¬ bers born alive within that period. This expression obviously equals jy v x represents the number born alive during the x years preceding the time of the enumeration. The numbers born alive within this period of x years, Jess the num¬ bers dying within the period out of the numbers born alive, obviously represent the numbers of the living at the end of the period under the age of x years ; immigration and emigration among those under age x being considered null. the numbers born alive during (1840) the year immediately preceding the time of enumeration. 64 A. MATHEMATICS AND PHYSICS. Hence l 0 (= z»/V) = |p„,.+ O/x >/> V> * /o fov*y 0 *v* , r r , »* — 1 in which V is the Napierian logarithm of v. Let a? = 5 ; then will L 0 = To simplify, let 75 , r v 5 7 Doudx) V — 1 ^ Vo ^^=1 ' U 5 - 1 Do,- d x v 5 V D 0/x d, v — 1 v Then L 0 — \Po/jt'\-D'o, x dx\ v 5 The returns give D m = 103,509 D 0i3 = 157,628 D 0i5 = 178,539 average annual deaths under ages one, three, and five. P 0/5 = 2,249,284 population under age five at the end of the year 1840. From these, and from .0066586 X the logarithm of the ratio of annual increase among registered births, we find B' m = 98,100, D' 0l3 = 154,043, D' 0/S = 179,912. Assume Do, x = D'o, o [=0] -[- a? 0 -|- a?. a? — 1 0 2 -(- a?. a? — 1 . a? — 3 0 3 -f- x . x —1 . a? — 3 . x — 5 R = a?^ (a: 2 —— 4 a? 2 -)-3a?) 0 3 -[-(a? 4 —9a? 3 -(-23a? 2 —15a?) R. MATHEMATICS. 65 _x' i 20.-3 (3*-16)*+18 2 r+—S - ■ 6 +-6- 6 . [(12a — 135)o; +460] o: — 450 ) +-30- R \ dJK, d x = 0 + (2a— l)d 2 + (3a 2 — 8a -f 3) 0 3 + (4a 3 — 27a 2 + 46a — 15) R. = 20 2 -|- (6 a — 8) 6 s -f (12a 2 — 54a + 46) R. yd X) 8 , 8 2 , and 8 3 are the divided differences of the values V 0l0 (= 0), D 0/n D 0/3 , and D 0/5 ; and R is indeterminate. D'oio — D'o/3 - A d 0 000 98,100 154,043 179,912 98,100 98,100. 55,943 27,971.5 25,869 12,934.5 A 0 8* — 70,128.5 —23,376.17 — 15,037.0 — 3,759.25 Ad 2 d 3 19,616.92 3,923.38 We observe that the divided differences of the first order are posi¬ tive, and that they diminish as the age advances. Required for R a value such that the first differential coefficients of the function assumed for D' 0/x be positive. It would also be de¬ sirable, if possible, that the second differential coefficients, from birth to at least age five, be negative. The latter is not possible for the entire period, with our present values for Don ? D'o# ? and Do/ 5 , if we assume but one arbitrary value (R). Our object, however, is sufficiently attained by taking, for R, a value such that for ages three and five the above conditions shall be observed. That the first differential coefficients be positive for ages three and five, it is requisite that R < 396.6 > — 920.1; that the second differential coefficients be negative for the same ages, it is requisite that 6 66 A. MATHEMATICS AND PHYSICS. R > — 939.8 < — 520.6; from which it appears that R should be negative, and that its value be between — 920.1 and — 520.6 Let R — — 700. We now have * $ = 98,100 6 2 = — 23,376.17 6 3 = 3,923.38 R = — 700. / Hence m j t i . , 25 . 175 „ . 325 3 125 _ D', lx d x (which = — 6 + — e - 12 R ) ~ 657 ^95. L 0 — |p 0 ,5 P(J/X d Z ! ^5 in which X V ~ *■ = 1.3142433, i>5— 1 and P 0I5 = 2,249,284 ; therefore, v — 1 ?— 1 ’ L 0 = 599,418. By the above process the probable number born alive during the year 1840 is found to have been 599,418 instead of 562,394, the average of the numbers registered as born alive during each of the three years 1839, ’40, ’41 ; thereby indicating an annual deficiency in the registration of 37,024, or about 6.2 per cent of the probable number born. In the above we have supposed the numbers of the dying and of the living at early ages accurately returned. If either be represented less than truth, the resulting correction would give still larger the probable number of births. Correction for deaths that escape registration, if any, would tend to reduce the probabilities of living. v _j i Having found L 0 (which equals — y — . the annual number of births for the year 1840, we next seek values, corresponding to intervals of age 0-1, 1-3, and 3-5, for D" 0/x (which equals v — 1 (t, d x the annual number of deaths in a stationary popula- MATHEMATICS. 67 tion in which L 0 is the annual number of births ; or the number that must die in x years, according to the law of mortality prevailing in Prussia, out of £ 0 > born alive. v 5 V v x D'o/x —— ~ ^ -Doix v 5 V r x „ d\, x v — 1 — - iv x v — 1 J o dx V v 5 V , — -. / V x d v — 1 J o T)u 0 IX • V - 1 v x d D"o,x = ^y- * d i vV D 'o /*)• But d ( v x . D' 0/x ) = v x . d D'o/x -|- D'o/x • d v x — v x (d D' 0/x -f- V D' 0 , x d x ); v — 1 .. dD\ lx = {dD' 0/x + VD' 0/X dx) v 5 V Integrating, W^+V-/. D '°’* dx - V , V, D' o/i, jD'o/ 3 ? an( ^ -D'o/s are already known; also 0, 0 2 , and 0 3 , and I? in the expression y *x , X^ ( . , 2*-3 (3*- 16)*+ 18 +- g -* . [(12 x — 135) a? + 460] x — 450 _ 30 Substituting for x values 1, 3, and 5, we have J' 1 ^D ! 0!X dx — 55,899, D'o/x d x = 323,335, D' 0lx dx = 657,995. D ,y 0/ i = 104,184, D" m = 159,735, D\ 5 = 181,955. Therefore, 68 A. MATHEMATICS AND PHYSICS. X^o/x’ (the logarithm of the probability that one born alive will sur¬ vive x years) — X-——^ = X ——- ^ 0/ x . Iq ho Therefore, \p M = T.9170804 = X .82619, Xp 0/3 = 1.8654097 = X .73352, \p 0/5 = 1.8428880 = X.69645. Hence of 100,000 born alive there will attain the age of one year 82,619, three years 73,352, five years 69,645 ; or of 100,389 born alive there will attain the age of one year 82,941, three years 73,637, five years 69,916. The latter results are those adopted in the accompanying inter¬ polated and other tables. These tables, as first constructed, repre¬ sented the probability of surviving five years from birth to be .69916, computed by a process less rigorous and satisfactory than the one just described. By assuming the same number surviving at age five (69,916) as in the original table, modification of the values for ages greater than five becomes unnecessary. The logarithms of the numbers surviving certain ages out of 100,389 born alive may be continued for ages greater than five, by successively adding to 4.8445759 (the logarithm of the number surviving age five), the logarithms that have previously been determined for the proba¬ bilities of surviving the consecutive intervals. The table will then be ready, either for a general interpolation of the numbers surviving each anniversary of birth, or for obtaining, by abridged methods, the accurate average duration of life, life annuities, annual premiums, single premiums, and other practical tables involv¬ ing life contingencies, for certain ages, without the intervention of a general interpolation. Simple rules may also be added for computing from these periodical results any specified values intermediate. The following is a brief method for finding approximate values for the probabilities of surviving the intervals from birth to ages one, three, and five, on the supposition of a probable deficiency in the registered number of births, and that the ratio between the numbers registered and the true numbers is constant. MATHEMATICS. 69 The same interpretation of symbols is observed as in the last demonstration. We already have U = 7. f + = P °' 5 +/o , Jo / «* 0 ■D'o/« d * f'oV* ’ and A>/* = fl V * v *• When the interval (0-a;) is not large, xv 2 is a close approximation to the value of j v x d x ; hence the following approximate relations. L > = 7i'*=\ p "+S! **»■**] 5 4r D' ou = ^ • % v x 1 ^ ^0/x d Do/a? ax v x Let us first seek an approximate value for L 0 . It is obvious that J* q D' 0/x d X = f D'o/x d X +/ D'o/x d x +/ D o ix d x. Assuming each term, in the right-hand member, to be the integral of the general term of an equidijferent progression, we have y^D ' 0( ,dx = 5 — 3 JP ' C ' 5 + f*D’ IIU dx = 3^1 />„„<**= 1=0^!. (* D'o/x dx = D'o, 5 -j- 2 D'q/3 § D'on • */ o Therefore, Since _ v 5 D 0/x V ° ,x ~ vl * l 5 "’ D 1 0/l , D 1 0/3 , and D^g equal respectively 98101, 154044, and 179913. 70 A. MATHEMATICS AND PHYSICS. Therefore, But L.= f S D' aix dx = 635,152. J 0 P 0/5 1 .Xo D'o/xdx 5v 2 2,249,284 -f 635,152 = 2,884,436 5 v 2 = 594,851. 594,851, the computed number of births for the year 1840 by this approximate method, is less by about three fourths of one per cent than 599,418, the corresponding number of births computed by the previous method. Having found an approximate value for * • r* (° r x »>’ we next wish approximate values for rf . ^ (or r x ^), dx J o v x J corresponding to intervals of age 0-1, 0-3, and 0-5. When the interval b - a is small, 6 d B 0lx , , D 0lb - D / nearly equals b + a ? , 2 Da l V 0/a JSa,b # b+a * Hence the following approximations : vi V _ Aw _ 104 306. dx v * dx D, = 55,804. == = 22,234. da; v 4 = 104,306. dx vi^= 160,110. dx v i^ = 182,344. dx * This approximation was adopted by Dr. Farr in constructing his Austrian Life-Table. — Rep. Reg. Gen. MATHEMATICS. 71 X- 1 1.9162709 = .82463. X- 1 1.8638226 = .73084. §0/5 -L'o - ~n \ p m (which = - ) = X- l T.8410233 = .69346. '■"0 Then of 100,000 bom alive there will be living at ages (1) 82,456, (3) 73,084, (5) 69,346 ; or assuming the number living at age five to be 69,916, the same as in accompanying tables, then out of 100,821 born alive there will be living at ages (1) 83,143, (3) 73,684, (5) 69,916. This table, joined with the interpolated for ages greater than five, gives for average future duration of life 36.51 years from birth, instead of 36.66, according to the values previously obtained, and adopted in the interpolated and other tables. If we substitute 562,394, the average annual number returned as born alive in Prussia for a period of time (1839, ’40, ’41) of which the year 1840 was the middle, for 594,851, the approximate number just computed, we shall find \p m = 1.9109082 = X .81453, X p 0l3 = 1.8544920 = X .71531, Xp 0l5 = 1.8298000 = X .67577 ; and out of 103,461 bom alive 84,272 will survive one year, 74,006 “ three years, 69,916 u five years, and the average future duration of life from birth will appear to have been 35.61 years. A general interpolation of the logarithms of the proportions surviv¬ ing each anniversary of birth intermediate the specified ages, gives the following. Hence Pm (which = • Pm (which = L 0 dx L 0 ' T §0/3 -^0 d x L 0 72 A. MATHEMATICS AND PHYSICS, Prussian Life-Table, calculated from the ages of those dying during THE THREE YEARS 1839, ’40, ’41 ; AND FROM THE AGES OF THE LIVING COMPUTED WITH REFERENCE TO THE MIDDLE OF THE YEAR 1840. Logarithms Differences between Consecutive Logarithms of the Probability of Living. . Persons Average Future Duration (or Expec¬ tation) of Life. of the Numbers Born, and Living at each Age. of the Proba¬ bility, at each Age, of Living One Year. Born, and Living at each Age. Dying during each Year of Age. Ages. IL lL x\\ — lL x k Px—*Px+l r L —L x *+1 # IP X *Px -4*Px JU D x Hj X 0 5.001688 .917080 — 1. — 51797 100,389 17,448 36.66 1 4.918768 .968877 — 10576 82,941 5,736 2 4.887645 .979453 — 7597 77,205 3,568 3 4.867098 .987050 — 3378 73,637 2,163 4 4.854148 .990428 — 1854 71,474 1,558 5 4.844576 .992282 — 1678 69,916 1,232 47.06 6 4.836858 .993960 — 1241 68,684 948 7 4.830818 .995201 — 886 67,736 745 8 4.826019 .996087 — 601 66,991 601 9 4.822106 .996688 — 381 66,390 504 10 4.818794 .997069 — 210 65,886 443 44.81 11 4.815863 .997279 — 84 65,443 409 12 4.813142 .997363 4 65,034 393 i 13 4.810505 .997359 63 64,641 392 14 4.807864 .997296 99 64,249 399 15 4.805160 .997197 120 63,850 411 41.17 16 4.802357 .997077 121 63,439 425 17 4.799434 .996956 123 63,014 440 18 4.796390 .996833 114 62,574 455 19 4.793223 .996719 107 62,119 468 20 4.789942 .996612 101 61,651 479 37.54 21 4.786554 .996511 95 61,172 489 22 4.783065 .996416 95 60,683 499 23 4.779481 .996321 100 60,184 508 24 4.775802 .996221 106 59,676 517 25 4.772023 .996115 118 59,159 527 34.02 26 4,768138 .995997 121 58,632 538 27 4.764135 .995876 125 58,094 549 28 4.760011 .995751 * 128 57,545 560 29 4.755762 .995623 133' 56,985 571 30 4.751385 .995490 137 56,414 583 30.55 31 4.746875 .995353 140 55,831 594 32 4.742228 .995213 144 55,237 606 33 4.737441 .995069 151 54,631 617 34 4.732510 .994918 153 54,014 628 35 4.727428 .994765 158 53,386 640 27.14 36 4.722193 .994607 159 52,746 651 37 4.716800 .994448 161 52.095 661 38 4.711248 .994287 165 51,434 673 39 4.705535 .994122 171 50,761 682 40 4.699657 .993951 185 50,079 693 23.76 41 4.693608 .993766 204 49,386 703 42 4.687374 .993562 230 48,683 717 43 4.680936 .993332 258 47,966 731 44 4.674268 .993074 291 47,235 747 45 4.667342 .992783 323 46,488 766 20.40 46 4.660125 .992460 352 45,722 787 47 4.652585 .992108 391 44,935 809 48 4.644693 .991717 438 44,126 834 49 4.636410 .991279 498 43,292 860 50 4.627689 .990781 —1. 570 42,432 892 17.11 MATHEMATICS, 73 Ages. Logarithms Differences between Consecutive Logarithms of the Probability of Living. Persons Average Future Duration (or Expec¬ tation) of Life. Of the Numbers Born, and Living at each Age. Of the Proba¬ bility, at each Age, of Living One Year. Born, and Living at each Age. Dying during each Year of Age. lL x lL x+\ — lL x *Ps—*P x + 1 L X ^x+\ E X I ~ Jk P x D X 51 4.618470 .990211 — 1. 651 41,540 925 --— 1 52 4.608681 .989560 745 40,615 965 53 4.598241 .988815 851 39,650 1,008 54 4.587056 .987964 970 38,642 1,057 55 4.575020 .986994 1101 37,585 1,108 13.98 56 4.562014 .985893 1307 36.477 1,166 57 4.547907 .984586 1491 35,311 1,231 58 4.532493 .983095 1657 34,080 1,302 59 4.515588 .981438 1803 32,778 1,371 60 4.497026 .979635 1934 31,407 1,439 11.22 ! 61 4.476661 .977701 2042 29,968 1,500 1 62 4.454362 .975659 2137 28,468 1,551 63 4.430021 .973522 2215 26,917 1,592 64 4.403543 .971307 2275 25,325 1,619 65 4.374850 .969032 2323 23,706 1,632 9.03 66 4.343882 .966709 2330 22,074 1,629 67 4.310591 .964379 2337 20,445 1,610 68 4.274970 .962042 2345 18,835 1,576 69 4.237012 .959697 2351 17,259 1,530 70 J 4.196709 .957346 2360 15,729 1,471 7.36 71 4.154055 .954986 2367 14,258 1,404 72 4.109041 .952619 2414 12,854 1,328 73 j 4.061660 .950205 2828 11,526 1,249 74 4.011865 .947377 2988 10,277 1,173 75 3.959242 .944389 3157 9,104 1,094 5.97 76 ! 3.903631 .941232 3338 8,010 1,014 77 i 3.844863 .937894 3527 6,996 932 78 j 3.782757 .934367 3726 6,064 851 79 3.717124 .930641 3939 5,213 769 80 3.647765 .926702 4162 4,444 690 1 4.80 81 3.574467 .922540 4399 3,754 613 82 3.497007 .918141 4648 3,141 540 83 3.415148 .913493 4913 2,601 470 84 3.328641 .908580 5191 2,131 404 85 3.237221 .903389 5485 1,727 345 3.82 86 3.140610 .897904 5799 1,382 289 87 3.038514 .892105 6126 1,093 241 88 2.930619 .885979 6475 852 196 89 2.816598 .879504 6842 656 159 90 2.696102 .872662 7231 497 127 3.02 91 2.568764 .865431 7641 370 98 92 2.434195 .857790 8076 272 76 93 2.291985 .849714 8534 196 57 94 2.141699 .841180 9019 139 43 95 1.982879 .832161 9530 96 31 96 1.815040 .822631 10072 65 22 97 1.637671 .812559 10644 43 15 98 1.450230 .801915 . 11248 28 10 99 1 252145 .790667 11887 18 7 100 1.042812 .778780 12562 11 4.4 101 0.821592 .766218 13275 6.6 2.7 102 0.587810 .752943 14029 3.9 1.7 103 0.340753 .738914 14826 2.2 1.0 104 0.079667 .724088 — 1. 1.2 .6 105 1.803755 1 .6 .6 7 74 A. MATHEMATICS AND PHYSICS. The leading features in the interpolated Life-Table for Prussia are two. 1st. Strict conformity at certain points to values calculated from actual data. 2d. Regularity in the graduation. It will be observed that the logarithms of the proportions born alive and surviving ages 1, 3, 5, 14, 25, 35, 45, 55, 65, 75, and 85, as calculated from actual data, are identical with those in the interpolated table. More frequent coincidence would fail, for certain intervals of age, to secure the desired regularity. From these values we find, by inspection, that the logarithms of the reciprocals of the probabilities of surviving equal consecutive in¬ tervals of age diminish from birth, until they attain a minimum between ages 5 and 25 (near age 14), then gradually increase for subsequent intervals. In effecting the interpolation, we sought to arrive only at results that, coinciding at the ages above specified with those derived from the actual data, should represent the logarithms of the reciprocals of the probabilities of surviving consecutive annual intervals of age as diminishing from birth to a minimum at some point between ages 5 and 25, then gradually increasing for subsequent intervals of age ; and that the differences between these logarithms should also advance without manifest irregularity, increasing from, at latest, age 25 to extreme old age. Two distinct functions of interpolation were employed. 1st. The exponential. For X L x , write

a + (06 — 0a) 1 r~ u — i in which 0 a , 0 6 , 0 C are known values of the function 0*, correspond¬ ing to ages a , &, and c. q is to be determined. If the terms be equidistant , that is, if c — b — b — a , q — and 0c ~~ 0 6 06 - 0 / 0* — 0a + (06 — 0 a ) 2 (0c — 06) — (06 — 0a) . (q x ~ a - 1). If the terms be not equidistant, the determining of q will involve the solution of quadratic or higher equations. MATHEMATICS. 75 2d. The algebraic. x - in which *.T + B r , , C, , D, + Qn,; rL. = # — a . x — b . x — c . x — d and n. , B x = n* r _ n , — b' Lx X — D x = - C X - u. d ’ Q may be zero, or an arbitrary constant real and finite, or a real function involving only integral powers of the variable, and which cannot cause the term (Qn x ) to become infinite or indeterminate for any value of the variable within the limits assigned for interpolation, or between those corresponding to the extreme given values of the function. A x ( obviously becomes unity , and terms indepen- When x = ’ A a ( dent of this factor vanish. h T> _L a u t& B„ n r x CC U « ’ c: r J u u u ’ Da Another convenient function for interpolation when three terms are given, but which was not employed in framing the present table, is the general parabolic. in which — tya -(“ ($6 - $«) ^-> . c~ Qa b — c — a A -- The exponential involves but three known values of the function. The number of known values that may enter into the interpolation by the algebraic is unlimited; but, without care, the resulting series will often be quite eccentric. Given, logarithms of the proportions born alive, and surviving ages 76 A. MATHEMATICS AND PHYSICS. 1, 3, 5, 14, 25, 35, 45, 55, 65, 75, and 85; required (X L x ) the logarithms of the proportions surviving each intermediate anniversary of birth. By the exponential formula values between ages 25 and 381 . . , , „ f 25, 35. and 45. 35 d 48 ! were res P ectlvel y interpolated from j ’ ’ . known values of the function for ^ 35, 45, and 55. 45 and 58 | ao . eg | 45, 55, and 65. 55 and 68 J [ 55, 65, and 75. Values from 73 to age 105, inclusive, were interpolated from known values of the function for ages 65, 75, and 85. By the above it appears that duplicate values were obtained at ages 36 and 37, 46 and 47, 56 and 57. The results deduced from these interpolations conform strictly to the conditions imposed, except near the joining points of the several series, where appear irregularities in the first and second orders of differences. These manifest irregularities were then corrected for the several in¬ tervals, by adding to the result at each of the several ages the corre¬ sponding value of A,, derived from the simple algebraic function A, = x — a . x — b . x — c . x — h g a . g b . g — c.g h + a . x — b . x — c . x — g a . h — b . h — c . h — g In applying the correction to ages between 35 and 48, a, Z>, c, g , and h equalled respectively 35, 45, 55, 36, and 37. A :6 and A 37 were the differences between the duplicate values for ages 36 and 37. The difference was positive when the one of the duplicate values first obtained was the greater. In correcting between ages 45 and 58, a , Z>, c, g , and li equal re¬ spectively 45, 55, 65, 46, and 47, and A 46 and A 17 were the differences respectively between the X L 46 and X L„ just corrected, and the corre¬ sponding results derived by the exponential formula from values for ages 45, 55, and 65. By a similar process the correction was made for values between ages 55 and 68. A slight irregularity still existing in the second differences (the first differences from the logarithms of the probability of living) near MATHEMATICS. 77 the joining of the series about age 36, another correction was made to the values between ages 37 and 45, viz. : x — 35 . x — 3 6.x — 37 . x — 45 . x — 46 . x — 47 = 34 —35.34 — 36.34 — 37.34 —45.34—46.34 — 47' The values for A 34 , being the difference between the first of the duplicate \L 3i and the corrected second of the duplicates. A x is additive, if the first of the duplicate values for X L 3i is the greater. The values between ages 67 and 73, inclusive, were computed from the known values at ages 65, 66, 67, and 73, by assuming the third order of differences constant. X L 73 = X L 65 -]— 8 A — |- 28 A 2 -j— 56 A 3 . A and A 2 were derived from the original value for X L 65 , and from the corrected values for X L 66 and XL 67 . a 3 was then readily found, and consequently the values required between ages 67 and 73. A modification of the method here indicated might have been applied with advantage to the correction of irregularities near the points of junction in other parts of the table. From the given values of X L x for ages 3, 5, 14, 25, 35, together with the values of X L x for ages 26 and 27, computed as above, the unknown values between ages 5 and 26 were interpolated by the algebraic formula X L x = 0, = $3 + $5 -gj + $14 ~f" $25 + $26 jjr + , F* , , G. $27 -ft- T - $35 ~FT' -^27 ^35 The forms of the functions A , B, C, &c. have been previously given. From XL m XL 3 , and X L 5 values were deduced by the exponential formula for X L 2 (= 4.887645) and X ] b 4 (= 4.854456). By the same formula, from X L 3 , X L 5 , and the computed value for \L 7 was deduced a duplicate value for x£ 4 (= 4.853532). From comparison of the duplicate values for \L 4 , giving to the former double weight, we obtain 4.854148. We remark that the desired regularity in the graduation, for the greater part of the table, was attained by making identical three or more consecutive values of adjoining series. It will be observed that the interpolated results represent mortality 1 * 78 A. MATHEMATICS AND PHYSICS. diminishing from birth, until attaining a minimum about age 12, then increasing gradually to age 105, the assumed terminating age of the table. Also, that the values in the column of differences headed — A \p x gradually increase through the greater part of the entire table, diminishing, however, between ages 17 and 22. A curve, to which the intervals of age and corresponding intensities of mortality are co-ordinates, will be concave downwards through the space where these differences diminish, if elsewhere concave upwards. The attain¬ ment of regularity at joining points in the order of differences next higher, was deemed unimportant. For the accuracy with which much of the arithmetical computation has been performed, in the preparation of this and certain other tables following, credit is due to Mr. Howard D. Marshall, of Boston.* Life-Tables, advancing, by regular gradations, from birth to extreme old age, and conforming strictly at convenient intervals to values de¬ rived from original data, are uncommon. The graduation of the older tables was very imperfect. The Car¬ lisle gives the annual rate of mortality at age 20 greater than at 23; at 31, greater than at 34; at 46 greater than at 51; at 88 greater than at 89 ; and at 91 the same as at 101. Mr. Milne’s excellent table for Sweden and Finland, (1801-5,) though less faulty, is still irregular; so also those of De Parcieux, Kersseboom, Finlaison, and others. The valuable and elaborate English Life-Tables prepared by Dr. Farr, and published in the Reports of the Registrar-General (Eng¬ land), and also the one prepared by a committee of eminent actuaries to represent a law of mortality according to the combined experience of Insurance Companies, as published by Mr. Jenkin Jones, vary the results derived from actual data, to conform to assumed laws. The graduation of the Actuaries’ Table is unexceptionable ; that of the tables of Dr. Farr nearly so. The important tables presented by Mr. E. J. Farren, in his instruc¬ tive treatise entitled, “ Life Contingency Tables, Part I.,” begin with age 21, and conform strictly at decennial points to values derived from actual data. The function of interpolation adopted by him was the Calculus of Finite Differences, so far as possible ; assuming, how¬ ever, the intensity of mortality to advance by a constant ratio, when, either from paucity of data or other sufficient cause, the Calculus of * Mr. Marshall has deceased since this paper was prepared, and in press. MATHEMATICS. 79 Finite Differences was inapplicable. The results attained are entire¬ ly regular. Many writers on this subject have felt it desirable that some simple generic law be discovered, which, by suitable changes in the constants, will approximate the specific laws of human mortality indi¬ cated by known tables. Among the more philosophical conceptions is the one of Mr. Gompertz (Philosophical Transactions, 1825), that for the greater part of life man momentarily loses 44 equal proportions of his remaining power to oppose destruction ”; and consequently, that the intensity of mortality increases with advancing age by a con¬ stant ratio. Mr. Edmonds would have u the force of mortality at all ages ” 44 expressible by the terms of three geometric series, so con¬ nected that the last term of one series is the first of the succeeding series.” Dr. Farr recognized the principle in framing his English Tables for 1841; treating 44 the two series of numbers representing the mortality from 15 to 55, and from 55 to 95, as geometrical pro¬ gressions. The ratios were derived from a comparison of the increase in the mortality at 15-20, 25-30, 35-40, &c.; and the increase at 20-25, 30-35, 40-45, &c.; and the first terms were derived from these ratios, and the sums of the series which they formed.” Mr. Orchard’s method, as described by Mr. Gray in the Assurance Magazine, (London,) for July, 1856, was the adoption of 44 two con¬ secutive series, having constant second differences,” to represent the proportions living from age 20 to 80 and from 80 to 96, the ter¬ minating age of his table. He wished 44 to find a simple algebraical relation which should passably well represent some of our best tables.” The advantage claimed for a table so constituted 44 is, that it admits, by the application of simple analytical processes, of the independent formation of any of the values which ordinarily require the aid of a formidable array of the results of previous computation.” The same paper gives a single algebraical function of the second degree pro¬ posed by Mr. Babbage, which is said to represent, nearly, the Swedish Table of Mortality. Other methods have been proposed by mathematicians of estab¬ lished reputation.* * A valuable contribution to this department of the science of Vital Statistics was read before the American Association, at its late meeting, by President McCay of South Carolina. 80 A. MATHEMATICS AND PHYSICS. 15. Discussion of Certain Methods for converting Katio of Deaths to Population, within given Intervals of Age, into Logarithms of the Probability that one living at the Earlier Age will attain the Later. With Illustrations from English and Prussian data. In the paper immediately preceding, a method has been indicated for the conversion of mortality into probability of living from a com¬ parison of three consecutive ratios, one preceding and another follow¬ ing the specified interval. In the present paper the results so derived will be compared with others obtained from a single ratio. Let m (identical with M in the preceding paper) represent the rate of annual mortality for any interval of age, or the ratio (t) of the number annually dying to the number living in the community within that interval of age. If the population be stationary , m a/b (which equals f the rate of -t-alb annual mortality for the interval between ages a and b, will equal L x dx f b a L;dx' If also the deaths be supposed uniformly distributed throughout the interval of age, i. e. — dL x , constant, the numerator (/:--•) will represent the sum of a series of constants, and the denominator L x dx ^ the sum of a series of values progressing by a common difference ; hence the value of the fraction will be independent of the extent of the interval , and will vary only with the mean age. If for —, the mean age, we substitute 2 , and assume for k any arbitrary value, mjzr k ,7+z will be constant for all values of the arbitrary , provided 2 k does not exceed ( b — a) the limits of age within which the uniformity of distribution was assumed. It will follow that y—% the value of the probability that one living ■^z + k at the earlier age, 2 — k , will attain the later, 2 -f- k, expressed in terms of the known annual rate of mortality (ffi a/i ), and of the arbi¬ trary ( k ), is 1 — k m 0/b l+k m alb MATHEMATICS. 81 Hence the probability of surviving the entire period ( b — a) is _ b — a 1 - 2 — ma ' b , , b-a 5 H-2~ m a,b and the probability of surviving the middle year of the period is 1 — i ™ alb 1 + 2- m a/b Again, if the population be stationary , m xl x ~ + dx . dx (for which put m dx • dx), the intensity of mortality at age x, or the rate of momen¬ tary mortality at that age, will equal — dL x L z = — d\ = — X l x + dx - ^ Px/x + d x ? (for which put — X p d3 ), the Napierian logarithm, with the algebraic sign changed, of the probability of surviving a moment of time from age x. Hence j* m dx d x, the integral within the limits of the ages a and b of the intensity of mortality, will equal —X p all the Napierian log¬ arithm, with its sign changed, of the probability that one living at the earlier age (a) will attain the later (b). “ A rate of mortality ” “ derived from the integration — d L ” has been happily styled the “ integral rate of mortality.” * Assuming deaths uniformly distributed , m d z becomes equal to m a/6 ; that is, the rate of annual mortality at the mean age equals the rate of annual mortality for the entire interval; consequently — l)Pdz = m a ib • d % ; that is, the Napierian logarithm, with its sign changed, of the proba¬ bility of surviving a moment of time at the middle of the specified interval, equals the rate of annual mortality for the interval, multiplied by the differential of the mean age. * Life Contingency Tables, Part I., by E. J. Earren. In the same connection is stated the important proposition, that “ whatever progression prevails among the integral rates of mortality at different ages, the same progression will be found to prevail among the logarithms of the probabilities of living, and vice versa." 82 A. MATHEMATICS AND PHYSICS. The intensity of mortality at age z (when deaths are uniformly dis¬ tributed) being the middle term (~dL z V L z ^ of a series of reciprocals of an arithmetical progression, is less by a small proportion than the average value of the terms constituting the series ; hence ( m a/b ) the rate of annual mortality for the interval of age b — a is somewhat less (in the case of such uniform distribution) than (/' the integral rate of mortality for the interval, or than its equivalent (—X 'Path ), the Napierian logarithm, with the sign changed, of the probability that one living at the earlier age (a) will attain the later age (b). To convert the Napierian to the common logarithm, we multiply by p (= .4342945), the modulus of the common system. PRUSSIA. 1839, ’40, ’41. Table comparing Logarithms of Probabilities of Surviving, comput¬ ed by different Methods. Ratio of Deaths to Population. Common Logarithm, with changed Sign, of the Probability that one liv¬ ing at the Earlier Age in each Interval will attain the Later. Mortality. Integral. Approximate. Ages m — Xp each from three consecutive Ratios. b — a 1 —m —— x . 2 b — a \ + m — (b a) X m 1 2 — ( b — a) ( u . m a, b. A B c D E 0- 5 .0802238 .157112* .176598 .174297 .174204 5- 7 .0152056 .013155 .013208 .013208 .013208 7-14 .0077790 .023557 .023655 .023649 .023649 14-20 .0062978 .016416 .016413 .016411 .016411 20-25 .0089397 .019425 .019416 .019412 .019412 25 - 30 .0096939 .021058 .021054 .021050 .021050 30-35 .0108317 .023537 .023527 .023521 .023521 35-40 .0131780 .028637 .028626 .028616 .028616 40-45 .0144675 .031449 .031430 .031416 .031416 45-55 .0210345 .092322 .091691 .091355 .091352 55-60 .0357042 .077981 .077738 .077539 .077531 60 - 65 .0557995 .122189 .121962 .121199 .121167 65 - 75 .0909134 .415608 .425992 .395105 .394832 75-85 85 and i upw’ds \ .1515098 .2661784 .722021 .860283 .659260 1.162896 .657999 1.155998 * This value was calculated by a process described in the preceding paper, from population under age 5 ; from deaths for the intervals of age 0 - 1, 1-3, and 3 — 5 ; and from the rate of annual increase of births estimated from registered births for the six years 1836-41. MATHEMATICS, 83 ENGLAND AND WALES. Table comparing Logarithims op Probabilities of Surviving, com¬ puted BY DIFFERENT METHODS. Deaths {Seven Years) 1838-44. Population computed to Middle o/’184l. Ninth Rep. Reg.-Gen., pp. 176, 177. Ratio of .Deaths to Population. Common Logarithm, with changed Sign, of the Probability that one living at the Earlier Age in each Interval will attain the Later. Mortality. Integral. Approximate. - Xp m Duplicate Values, Mean of , b —a l-m^T i-T — {b—a)j.i . m Ages a, b. each from two con¬ secutive Ratios. the Duplicate Values. . b ~ a ' + “-2- -{b-a)X^ m 2 A B c D E 0- 1 .1792379 \ .077265 .073073* .078052 .078052 .077842 1 - 2 .0654971 .028133 .028379 .028256 .028455 .028455 .028445 2- 3 .0351076 .015206 .015235 .015220 .015249 .015249 .015247 3- 4 .0250056 .010850 .010854 .010852 .010860 .010860 .010860 4- 5 .0184203 .007995 .007998 .007997 .008000 .008000 .008000 5-10 .0091272 .019686 .019789 .019738 .019823 .019820 .019819 10-15 .0052572 .011397 .011425 .011411 .011417 .011416 .011416 15-25 .0081967 .035723 .035643 .035683 .035618 .035598 .035598 25-35 .0098929 .043033 .043045 .043039 .042999 .042966 .042964 35-45 .0124582 .054239 .054261 .054250 .054176 .054105 .054105 45-55 .0165886 .072341 .072636 .072489 .072209 .072044 .072043 55-65 .0295429 .130216 j .130452 .130334 .129249 .128313 .128303 65-75 .0622301 ! .283777 j .278012 .280895 .279528 .270349 .270262 75-85 .1374474 .718169 ! .649789 .683979 .731961 .597869 .596926 85-95 .2842092 1.127822 1.242716 1.234305 9o —(- .4146003 1 1.827064 1.800586 * This value was derived from the registered births for the eleven years 1839 - 49. and from the registered deaths under one year of age for the ten years 1840-49. 84 A. MATHEMATICS AND PHYSICS. In each of the preceding Tables, column A gives rates of annual mortality for the several specified intervals of age, or ratios of the average numbers annually dying in the community within the speci¬ fied intervals to the numbers living within the same intervals, estimated with y^ference to the middle of the year or period in which the deaths occurred. Column B with changed signs gives the common logarithms of the probabilities of surviving the specified intervals, each computed from three consecutive ratios in the column of mortality by a process de¬ scribed in the preceding paper. These values, which we designate integral values, may be assumed without appreciable error to represent truly the results demanded by actual data, and with them may be com¬ pared approximate values obtained by simpler processes. The approximate values in the columns C, D, and E were each derived from single ratios in A. The values in C were each obtained by first multiplying ( m ) the annual rate of mortality by (t) half the number of years in the interval of age; then finding the logarithm, with changed sign, of the quotient of unity less this product divided by unity plus this product. The values in D were each found by multiplying the number of years in the respective interval by the logarithm, with changed sign, of the quotient of unity less half the rate of mortality divided by unity plus half the rate. The values in E were each found by multiplying the mortality by .4342945 (/*), the modulus of the common system of logarithms, and by (b — a) the number of years in the respective interval of age. Whenever the decrements in the Life-Table resulting from the origi¬ nal data are constant, the corresponding result in C represents the logarithm, with changed sign, of the probability of surviving the entire interval; that in D represents the product of (b — a) the number of years in the interval, multiplied by the logarithm, with changed sign, of the probability of surviving the middle year of the interval; and that in E the product of fc“) the number of equal moments in the in¬ terval, multiplied by the logarithm, with changed sign, of the proba¬ bility of surviving a moment of time at the middle of the interval. Whenever the decrements in the Life-Table are increasing , the above MATHEMATICS. 85 results are each less than the respective logarithm ; and when de¬ creasing , greater. The results in E should in all cases be somewhat less than those in D, although generally the approximation is so close that values in E may without appreciable error be substituted for those in ^.J) * The results in D are likewise less than corresponding ones in C. The results in C are less than the integral values in B , whenever the decrements between the proportions surviving at equidistant ages in the Life-Table, derived from actual data, form a series increasing with the age ; they are equal to them, when the series is uniform, and greater than truth when the series diminishes. By reference to the Prussian Table interpolated for annual intervals, we observe that the decrements diminish from birth to age 13; increase thence to age 65 ; and again diminish to the age terminating the Table. The process for deducing values in D is identical with that adopted by Dr. Farr,* in briefly calculating approximate Life-Tables. After determining from values so obtained the proportions of the living at certain ages, he assumed that the proportions within the several inter¬ vals were series in arithmetical progression.f It is not unusual, in framing Life-Tables from population and mortality statistics, to let i -j JL 2 interval, then, assuming some law of relation, to determine values for intermediate ages. Results so deduced will commonly represent the probability of living for a large part of life somewhat greater than truth demands. aal the probability C. Process for deducing accurate Average Duration of Life. Present Value of Life-Annuities, and other useful Tables in¬ volving Life-Contingencies, from Returns of Population and Deaths, without the Intervention of a General Interpolation. The logarithms of the proportions surviving at certain ages (X L x ) are obtained, by successively adding to the logarithm of a number as- * To this distinguished writer the science of vital statistics is largely indebted for valuable, extensive, and varied contributions, t Fifth Report Reg.-Gen. Eng., p. 362. 8 86 A. MATHEMATICS AND PHYSICS. sumed living at birth, or other specified age, the logarithms of the probabilities of surviving subsequent intervals. Processes for accu¬ rately and for approximately determining the logarithms of the prob¬ abilities of surviving have been indicated in the previous papers. Average future duration (or expectation) of life (E x ) expressed in years for any age (a?) may be obtained by multiplying by the differ¬ ential of the age (d x) the integral of the proportions surviving within (/; 3 4 the limits of the given age and of the greatest tabular age and dividing the product by the proportions living at the given age. That is, E„ = ixj?L, in which 105 is assumed the greatest tabular age. A close approximation to this value may be found by dividing l x = l x -\~l x+ 1 + .. . . L l05 ) the sum of the proportions liv¬ ing at the given age and at each subsequent anniversary by ( L x ) the proportions living at the given age, and from the quotient deducting the half of unity ; that is, E x = ■'£^106 j Zj.r — i , nearly. The latter is the more common process. The formula expressing the value of a life-annuity, or the present value of one dollar payable at the end of each year during the re¬ mainder of the life of the annuitant after attaining a given age, is E x+ i v -j- L x+2 r 2 -j- . ... -I x = c — b 2 c — b C + B — l + « d — j b + c d — a [ ) a — b -j- c j ) _o r * d — * b -h c -l ft d-« \ 3 1 a — d 2 )"!• C + B — D + A 12 Then dx! b X = c — b { q _ |_ £ _j_ B — D A 12 and C — B 2 +s;*-V>+*+ The three-point exponential formula X = a -f- £ y* ? a, /3, and y being unknown, and independent of the variable x, may take the form MATHEMATICS. 89 X=A + (B-A) q ~ 1 in which «—■— 1 ’ C — A q l ~° — 1 — B — A ’ whence may be determined the value of q. B — A + s * = dx! X ii± 1 1 B — A -\- b — a \A fl y C-B + c-b)A B—A 6-a__ l B — A 2 1 ~ q — 1 Q is the Napierian logarithm of q. When the terms are equidistant, i. e. c — b = b — a, _ (C - B\ h ~ -Kb —a) ’ a . fjL B ’ B and B — A B — A q b ~ a — 1 C — B—B — A • fi is (.4342945) the modulus of the common system of logarithms. When the terms are not equidistant, the application of the expo¬ nential function involves the resolution of equations of higher than the first degree. The three-point parabolic formula X = a + j8 (# — a) y may become X=A + (B-A)(^j; in which A . c — « +1 B — A . b — a b — a 8 90 A. MATHEMATICS AND PHYSICS. Then will *b x f x — b — a A + B — A and ? + i d x X=c — bA + — a . C — A b — a . B q -\- 1 q + 1 Th e finite integral of (a? — a) q advances in the form of a series , no application of which has been made in the illustrations which follow. Of the functions above enumerated, the algebraic will commonly prove the most simple in practice, but will not in all cases satisfy the conditions required. When assumed to express the law of relation between certain known values of the function L x or L x v x , a portion of the resulting series of numbers between the known values may increase with advancing age, rather than diminish. The values between B and C, derived from the algebraic formula assigning a law of relation between the four known values A , B , C, and D, lie between corresponding duplicate values derived from two algebraic formulae, one a function of the three known values A, B , and C , and the other of the three values B , C, and D. When the relations of the known values to each other are such that the- series resulting from each of the latter formulae diminish continuously with advancing age from A to C and from B to D respectively, then that portion between B and C of the single algebraic series con¬ necting the four values A , B , C, and D must diminish continuously. The series of values resulting from the algebraic formula assigning a law of relation between the three known values A, i?, and C will continuously diminish from A to C only when , . . , , A — C are each positive and greater than _; B and « b — a or, if the three terms be equidistant, only when -^ the value of the ratio of the first differences is between and 3, and the differences themselves nega¬ tive. Similar relations obviously obtain when the three known values are B , C, and D. Applying this test to the Prussian Life-Table, we first find that the algebraic function assigning a law of relation between the three known values does not completely satisfy the conditions for the proportions surviving ages 0, 1, 3; 3, 5, 14; 75, 85, 95; and 85, 95, 105; that MATHEMATICS. 91 is, for the extremes of the table, the values there rapidly diminishing; and also for ages 3, 5, and 14, where there is a great disparity in the length of the intervals of age. It will hereafter appear that eccen¬ tricities at the older ages may be disregarded in constructing tables of future duration of life, and of life-annuities, without materially affecting the correctness of the results for earlier ages. TABLE I. Average Future Duration op Life in Prussia. Algebraic Integration. 1 ^ Proportions Born, and Living at Specified Ages in Prussia, calculated, by the Integral Method, from three Consecutive Ratios of Deaths to Population. Sum of ( L x ) the Propor¬ tions Living at the given Age and at all subse¬ quent Ages specified. | 1 Aggregate Number of Future Years that ( L x ) the Proportions surviving Specified Ages will live. Average Future 1 Duration of 1 Life. £*+£*+10 + £*+20 + •••• 10$ , 8 * + 8 * + n "2) , ^* + ^*+10 ^*—10 ^-20 \ £* ! < + ■" 12 So L x Ages. L x dx r l J X x E x 5 69,916 5 69,916 364,916 !l4 64,249 15 63,748 295,000 2,626,778 41.21 ! 25 59,159 25 59,159 231,252 2,012,408 34.02 ! 35 53,386 35 53,386 172,093 1,448,721 27.14 1 45 46,488 45 46,488 118,707 948,047 20.39 | 55 37,585 55 37,585 72,219 524,773 13.96 65 23,706 65 23,706 34,634 215,941 9.11 1 75 9,104.2 75 9,104.2 10,927.6 54,597 6.00 : 85 1,726.7 85 1,726.7 1,823.4 5,847 3.39 95 96.1 96.7 — 232 105 .64 .64 In the preceding table the first of the columns headed L x gives the proportions surviving certain ages according to the law of mortality prevailing in Prussia; the probability of surviving each of the several intervals being calculated from three consecutive ratios of deaths to populations. In the second of the columns headed L x the values for ages 95 and 105 were computed from the logarithms of the propor¬ tions surviving ages 65, 75, and 85, by the exponential formula which expresses the value of the required logarithms in terms of the age, and of the three given logarithms. The number surviving age 15 (63,748) was obtained by assuming an algebraic law of relation for the proportions surviving the four ages 5, 14, 25, and 35. 92 A. MATHEMATICS AND PHYSICS. The next column ( S x ) gives the sum of the proportions surviving the given age and all subsequent specified ages. z * 105 The values in column headed d x J - L x give the aggregate number of future years of life that the proportions surviving given ages will enjoy, according to the prevailing law of mortality, and were each computed from four equidistant values in the preceding column ( S x ), by means of the formula ^ •*105 „ ( L x = - }S.+ &+, + Sx + s x 12 The values thus obtained, divided by the corresponding proportions living, give the average future duration of life. We have already called attention to the unsatisfactory nature of the values resulting from the use of the algebraic formula when the given numbers rapidly diminish, as in the Life-Table after about age 75. Table II. will compare corresponding results obtained by different 1 formulae and processes. In the third column of Table II. the integrations were effected by the exponential formula when the three given values involved in the equation were equidistant; when not equidistant, the parabolic formula was adopted. The parabolic and the exponential formulae each afford results that constantly diminish with advancing age. The process of integration by the algebraic formula involving four known values is the simplest, and between ages 15 to 75 is entirely satisfactory; from 5 to 15, and from 75 upwards, the values afforded are not so reliable ; and from 95 to 105, duration of life is represented as negative. In Table IV. we observe that the first three columns of average future duration of life present results, for the larger part of life, almost identical. Values by the algebraic formula slightly exceed those of the following column, calculated by a combination of parabolic, expo¬ nential, and algebraic formulae. The excess at specified ages from 15 to 55 inclusive is only the one-hundredth part (.01) of a year, or about four days. MATHEMATICS. 93 TABLE II. Comparison op Temporary Aggregate Future Duration of Life, CALCULATED BY DIFFERENT METHODS, FROM THE PROPORTIONS SURVIVING ACCORDING TO THE PRUSSIAN LiFE-TABLE. Ages. Proportions Born alive, and Surviving certain Ages. The Aggregate Number of Years of Life which the Proportions Surviving at the Commencement of certain Intervals of Age will enjoy during each Interval. -^x+n Ar ^x + n + 4: C ix r + " 2 X L x J X * + ?Tl x 2 Parabolic and Exponential Algebraic By Annual Equidifferent Method. Duplicates. Mean. Formula. Interpolation. 0 1 100,389 82,941 87,827* 155,560* 155,176 91,665 1’55,494 91,665 ! 156,578 3 73,637 142,992 142,359* 142,792 143,251 143,553 5 69,916 664,403* j 666,802 l f 665,602 656,124 661,937 668,320 ; 15 63,748 613,405 | 615,410 1 614,407 614,370 616,016 614,535 25 59,159 563,826 1 563,581 l f 563,653 563,687 563,655 562,725 35 53,386 500,393 ] 500,836 j I 500,614 500,674 500,322 499,370 45 46,488 422,257 ] 423,648 j 422,952 423,274 422,990 420,365 55 | 37,585 311,573 j 306,973 j 1 309,273 308,830 311,376 306,455 65 23,706 164,599 } 155,807 j 160,203 161,341 159,662 164,050 75 9,104 49,992 ] 45,211 j 1 47,601 48,750 47,769 54,155 85 1,727 7,137 j 5,688 j 6,412 6,081 6,368 9,115 ; 95 96 285 j 209 j 247 — 194 228 485 105 1 115 0 1 A comparison of the results in the third column of Table IV. with those arrived at by a general interpolation and direct summation of the proportions surviving each anniversary of birth, exhibits a differ- * These four values were calculated by the parabolic formula; the other values in the column by the exponential. 94 A. MATHEMATICS AND PHYSICS. ence at specified ages from birth to age 85 inclusive, that in but one case (at age 65) exceeds three one-hundredth parts (.03) of one year, or about eleven days. The former results are deemed in every respect as satisfactory as the latter. We observe that results by the equidifferent method compared with approved results, from birth to age 45 inclusive, are. usually about one tenth of a year in excess ; and that for ages above 45 the excess is much greater. TABLE III. Euture Duration of Live in Prussia. The Temporary Future Duration of Life for the Proportions Surviving, was computed hy the Parabolic Formula from Birth to Age 1 ; Exponential, from 1 to 3 and 3 to 5 ; Mean of Parabolic and Exponential, from 5 to 15 ; Algebraic, from 15 to 75 ; and Mean of Exponential Duplicates, from 75 to 105. • Ages. Temporary Future Duration of Life. Future Duration of Life. Average Future Duration of Life. X d *f: +nL * dx r 05 l J X X dx r m L x 0 87,827 3,678,033 36.64 1 155,176 3,590,206 43.29 3 142,992 3,435,030 46.66 5 665,602 3,292,038 47.09 15 614,370 2,626,436 41.20 25 563,687 2,012,066 34.01 35 500,674 1,448,379 27.13 45 423,274 947,705 20.39 55 308,830 524,431 13.95 65 161,341 215,601 9.09 75 47,601 54,260 5.96 85 6,412 6,659 3.85 95 247 247 2.57 MATHEMATICS, 95 TABLE IV. Comparison of Average Future Duration of Life. Computed, by different Processes, from the Prussian Life-Table. Ages. By the Algebraic Formula. Parabolic, 0 — 1; Exponential, 1-3 and 3 - 5 ; Mean of Parabolic and Exponential, 5-15; Algebraic, 15 - 75; Mean of Exponential Duplicates, 75 -105. By Annual Interpolation. Assuming L x within each Interval to advance by an Equidifferent Progression. 0 36.64 36.66 36.77 1 43.29 43.27 43.40 3 46.66 46.63 46.76 5 47.09 47.06 47.19 15 41.21 41.20 41.17 41.28 25 34.02 34.01 34.02 34.09 35 27.14 27.13 27.14 27.24 45 20.39 20 39 20.40 20.53 55 13.96 13.95 13.98 14.21 65 9.11 9.09 9.03 9.61 75 6.00 5.96 5.97 7.00 85 3.39 3.85 3.82 5.55 95 2.57 2.37 5.05 TABLE V. Value, at certain Ages, of One Dollar to be paid at the End of each Year during the Remainder of Life, according to the Prus¬ sian Life-Table, with Process for determining. Interest of Money , Four per Cent. Algebraic Finite Integration. I Ages. L (-L)* L 'x + ^ + 10 + — \S f + £' , in -f 2 * * *+ 10 100 * •§T L’ x * Vl.04/ £'* + 20 “H • • • • £'* 1 X £'* — ^ + ^ ,05 L< 2 1 x x Life-Annuity, 4 per Cent. 1' 5 57,466 143,287 i 15 35.397 85,821 666,675 18.33 j 25 22,192 50,424 384,260 16.82 j 35 13,529 28,232 208,804 14.93 i 45 7,959 14,703 103,448 12.50 55 4,347 6,744 43,186 9.44 65 1,852 2,397 13,115 6.58 75 481 545 2,306 4.29 85 61.6 63.9 133.6 1.67 95 2.3 2.3 — 13.8 105 .01 .01 ff.-lO S'x—so¬ il — S' x -}- S'z-fio 96 A. MATHEMATICS AND PHYSICS. In Table V. we observe that the ratios of the first differences of the values ( L x second column, for ages 65 and over, are not within the required limits, J and 3, and, consequently, that the values of the annuity resulting from integration by the algebraic for¬ mulae are to an extent unsatisfactory. Had the integration of S' x for the older ages been effected by the exponential formula, the following would have resulted. 1 . 1 ! Ages. -^ + ST&\ sr si. Duplicates. Mean. - 1 55 j 43,539.1 ) ( 42,671.2 J 43,105 9.42 65 J 13,417.5 ) | 12,711.6 J 13,065 \ 6.55 75 j 2,525.4 ) \ 2,281.3 \ 2,403 4.50 85 1 ( 233.3 \ ] 188.2 J 210.8 * 2.92 In Table VI. the values in the third column corresponding to inter¬ vals between ages 65 and 95 are arithmetical means of dupli¬ cate values resulting from finite integration of L' x by the exponential formula. The value opposite age 95 is a single' value similarly ob¬ tained. Duplicate Values and Mean. Ages. Duplicates. Mean. 65 j 10,990.8 i 1 10,356.0 ■ | 10,673 75 f 2,312.6 i 1 2,076.1 i [ 2,194 85 f 229.4 ( 1 183.0 * Single Value. ■ 206.2 95 6.3 6.3 MATHEMATICS. 97 In the same column (the third) the value of the summation from 5 to 15 (455,694) is the arithmetical mean of a result (456,371) derived by integration from the parabolic formula involving values in the preceding column for ages 3, 5, and 15, and of another (455,017) obtained by finite integration from the exponential formula involving values for ages 5, 15, and 25. The remaining values in that column were obtained by the finite integration of a series of distinct algebraic formulae, each involving four given values of L x When the terms in the second column ( L' x ) are equidistant, and the intervals unity, 1 — , C, c, and D, d, corresponding known values of X , x ; required values intermediate between B and C. If the given terms be equidistant, and n—d — c — c — b = b — a, the algebraic formula will give the following : — TABLE VII. Special Formulae for Interpolation, involving Four known Equi¬ distant Values of the Function. Algebraic. Ages. X * + Tff n 1 Tff (9 B+ C) + 3 Sffffff (3 . 9 B + C- -19 A- -11 D) + T 2 0 n 1 Tff (8 5 + 2 C) + 8 Tffffff ( 8 B + 2 c- - 6 A- -4 D) + Iff 71 1 Iff (7 5 + 3 C) + affffff (3 .IB + 3 C- - 17^4 — - 13 D) + Tff W 1 Tff (6 5 + 4 C) + 4 Tffffff (3 .65 + 4 C- - 16 A - - 14 D) + Iff n 1 ? ( B+ C) + Tff ( B + C- - A- - D) + 1 6 ff n 1 1 ff (4 5 + 6 C) + 4 Tffffff (3 .4 B + 6 C- -UA- -16D) + Tff 71 1 TO (3 5 + 7 C) + tffffff (3 .3 B + 7 C- - 13 A- -17 D) + Tff 71 1 Tff (2 5 + 8 C) + 8 Tffffff ( 2 B + 8 C- - 4A- - 6 D) » + Tff n Tff ( 5 + 9 C) + affffff (3 . B + 9 c- -11 A- -19 D) Example. — Given, unaugmented annual premiums, from Table VI., corresponding to ages 15, 25, 35, and 45 ; required the premium for age 28. 100 A. MATHEMATICS AND PHYSICS. The difference between ages 25 and 28 is of (n) the interval of age from 25 to 35 ; that is 28 is b “h t 3 o n - Then -^28 — tV B ~h ^ C) -|- (3.7 B -f- 3 C — 17 A — 13 D), in which A, B , C, and D equal respectively 1.33, 1.77, 2.43, and 3.57. T V(7JB + 3C) = 1.968 3 (7 B + 3 C) == 59.04 11A+13B = 69.02; .-. V 28 = 1.93. Required a value corresponding to age 40, from data in the column headed- - -j- JJ x . The formula is £ X» = i (5 + C) + T V (5 + C - A - D), and the given values are for ages 25, 35, 45, and 55. B + C = 312,080, A -f D = 427,275 ; X i0 = 148,840. Table VII. may take the symmetrical form of TABLE VIII. Special Formula: for Interpolation, involving Four known Equi¬ distant Values op the Function. Algebraic. Ages. X X b "h io n A (9 C) 9x1(3.95+ C— 19 A— 11 D) 4(85 + 2 0) 8x2 (3.85 + 20— 18 4 — 12 5) 4 (7 -B + 3 C) 7x3(3.75 + 30— 17 A— 13 5) 4(65 + 4C) 6x4(3.65 + 4 0— 16 A — 14 5) is n 4(55 + 5 0) + ^ 5x5(3.5 5 + 5 0— 15 4— 15 D)j A» 5 * 5 (4£ + 6C) 4x6 (3.45 + 60 — 14 — 16 5) t 7 o n 4(3 5 + 7 0) 3x7 (3.35 + 70 — 13 A —17 5) 4(25 + 8 0) 2x8 (3.25 + 80— 12 j! — 18 5) ^+I0 W 4 ( 5 + 9 C) 1x9(3. 5 + 9C — 11 A —19 5) v / ASTRONOMY. 101 And generally, when the four terms A, B, C , and D are equidistant, * = c-I > — n +x ~ic) + 6(c -^ X jc — x. x — & [3 . (c — x B -\-x — b C) — d —x A — x — a D~\\. The writer is not aware that any previous attempt has been made to pass by direct and summary processes from the immediate results of actual observations to solutions of monetary and other practical questions involving life contingencies, as accurate and reliable as those obtained by the intervention of a formidable interpolation. In view of the large and rapidly accumulating mass of population and mortality statistics, such processes seem to be demanded. The approximate methods heretofore published have already been adverted to ; allusion has also been made to certain formulae adopted in the construction of theoretical Life-Tables, which afford facilities for the independent formation of required monetary and other values. Note. — The average future duration of life for ages 15, 25,35,45, and 55, deduced from values in column C, on page 82, are from .1 to .2 of a year less , and those derived from values in column E are from .2 to .3 of a year greater , than corresponding durations obtained by more accurate methods. Arithmetical means of these results exceed the true values by about .05 of a year. By giving greater compara¬ tive weight to values deduced from C, closer approximations will ensue.