LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class GENERAL ^9^ < SYSTEM AND TABLES OP Life Insurance. A TREATISE DEVELOPED FROM THE EXPERIENCE AND RECORDS OF THIRTY AMERICAN LIFE OFFICES, UNDER THE DIRECTION OF A COMMITTEE OF ACTUARIES. BY LEVI W. MEECH, ACTUARY IN CHARGE. REVISED EDITION OF 1898, WITH AMERICAN LIFE TABLES, INCLUDINa VALUATION OF ANNUITIES OR PENSIONS, REVERSIONS AND DOWER. THE SPECTATOR COMPANY, NEW YORK. wy\^ rx^\\i^^^ 100.000 98,000 90.000 85.000 AMERICAN LIFE TABLE CMALES) FROM COMBINATIONS OF THE CENSUS. f SCALE OF AGES Copyright, 1881, by Levi W. Meech. NEPAL S>i'^<^^^ PREFACE. THE origin of the present collection of statistics, dates from the re-organization of the Chamber of Life Insurance in 1873. Among the leading objects, it was provided that, "there shall be a standing Committee on Mortality Experience, appointed by the Executive Committee, immediately after the Annual Election of the Chamber, in each year, and consisting of five members, to whom shall be referred the work of collecting the statistics of Mortality Experience, classifying, arranging, and tabulating them for practical use, with power to employ skilled assistance." In entering upon the duties assigned them, this Committee, after some prelimi- nary consultation, decided to invite a more full expression of opinions in a general meeting of Actuaries, which was held in New York, Nov. 19, 1874. Among those present and participating in the deliberations, besides the Committee, were S. Co Chandler Jr., D. Parks Fackler, C. M. Hibbard, Sheppard Homans, Jas. Weir Mason. Prof. C. F. McCay, Emory McClintock, George W. Phillips, William D. Whiting, and Charlton T. Lewis, Secretary of the Chamber. Besides these, a considerable number communicated their views by letter. An enlarged plan of operations having been generally called for, the Committee, in concurrence, subsequently issued the following Circular, under date of March 16, 1875 ; copies of which were addressed to all Life Insurance Companies in the United States : Sir: The Chamber of Life Insurance has, through its Executive Committee, appointed the under- signed a Committee upon Mortality Experience, with instructions to procure from every American Company willing to furnish it, their experience in full up to a recent period. In accordance therewith, we respectfully and most earnestly request you to favor us with the data of your company to Decem- ber 31, 1874, as indicated below. While the amount of labor to be performed both by the companies and the Committee mil be considerable, the importance of this work, if well done, to American companies, cannot be over estimated. Tables made from such experience can justly be regarded as of higher authority, and will probably more nearly approximate the true average of mortality among insured American lives than any now in use. And should similar examinations be made hereafter, as we trust may be the case, in which the results of the work now in hand may be merged, by such periodic examinations, and the aggregation of experience, we shall be able measurably to trace any general improvement in longevity, the effect of causes operating specially within the period under consideration, the effect of public sanitary measures, and probably many other points of interest incident to the development and improvement of a new and growing country. We hope that the present undertaking is but the beginning of a systematic and profound study of the mortality among insured American lives. In settling the details of this work, we have had in view certain important classifications, leaving many minor though interesting ones to be wrought out by the companies themselves ; and the saving of all but indispensable labor to the companies contributing their data. The points agreed upon by this Committee for consideration are : First, The general mortality among insured lives ; Second, The mortality by sexes ; Third, The mortality by localities, taking the State as a basis, except in certain States where the County is to be designated ; 115492 4 PREFACE. Fourth, The mortality by years of insurance, in order to sbow the eflfect and duration of medical selection ; Fifth, The ratio of financial loss to sums exposed to risk ; Sixth, A classification of the causes of death, in general, by locality, and such other special relations as may be deemed advisable hereafter. For these purposes the Committee have adopted the enclosed card, as enabling the companies to furnish the necessary data vfith the least possible labor. The instructions upon another page will enable a clerk to make correct and rapid use of it. Companies which have already arranged their data for their own examination in a manner to supply the Committee what is called for by the card, need only furnish their cards or sheets already prepared or their classifications if these correspond to those above stated, and any such material will be returned after use ; and, if desired, all cards will be returned. No tabulation or exhibit will be made in any manner of the separate mortality of any company, and the data will be held as confiden- tially given. It is our desire that the data may be given with all reasonable speed, and in view of the great interest and importance attaching to the work, we trust all will make this matter one of special care and attention, and employ in it sufficiently skilled labor to insure neat and accurate execution. The cards should be carefully verified t)efore being sent to the Committee, which will have no means of correcting errors made at the offices of the companies. The results of this labor will be reported to the Chamber at the earliest practicable moment, and by it published so as to be accessible to all desiring it. We believe it to be the general conviction among American and foreign Insurance officials and actuaries that the material now capable of being supplied here will give results of peculiar value and importance, and that American companies owe it to the business at large and to their own interests that these should be collated and properly classified ; and that it would be a cause of regret should any Company fail to join in this work, which is not undertaken for the benefit of the Chamber alone, or of the companies belonging to it, but for that of all companies doing business in this country. We promise our best endeavors to make the character of the work worthy of its importance. Please indicate at once to the Chairman at Hartford, Conn., the number of cards you may need, and the probable time at which they can be completed. Respectfully, Jacob L. Gkeene, Chairman. Edwin W. Bryant, Robert A. Granniss, i Levi W. Meech, Howell W. St. John. The way had been opened for this investigation, by the London Institute of Actuaries, which had collected the records of twenty Life Offices in England and Scotland. But the strongest incentive, lay in the recent remarkable increase of Life Insurance in the United States, creating a demand for a more systematic exposition of insured experience. The following summary from the current Massachusetts and New York State Reports, comprising the greater part of the Companies in the Union, may serve to illustrate the magnitude of the business at this epoch. In a general statement, the fifty thousand policies of 1859 had, in four years or in 1863, increased to a hundred thousand. Then the rate of increase was temporarily doubled, and within two years more, at the end of the year 1865, the number exceeded two hundred thousand. In another two years, or in 1867, it exceeded four hundred thousand. And in 1872, the number of policies had risen to eight hundred thousand. By this time, the annual discontinuances began to balance the diminished number of entrants ; and at the epoch of the experience, Dec. 31, 1874, the total number exposed to risk had been nearly stationary for five or six years, with about eight hundred thousand policies in force. The preceding Circular, was accompanied by the adopted form of policy cards, and instructions, which are given hereafter. In the final result, twenty- three Companies returned the cards, filled out from their office records ; and seven gave the results of their own investigations, brought down to the beginning of the year 1874. The latter returns were accompanied by collateral explanations and results sufficient for verifying PREFACE. 5 and resolving them into the original elements, in uniformity with the card reductions. In this manner the entire returns were united ; and it was found that, of all the valid experience, recorded upon the office registers in the United States, so favorable had the enterprise been regarded, fully three-fourths had been contributed for this collection. The following are the names of the Companies whose combined experience is given in the succeeding pages, with the year of organization and aggregate number of policies: COSIPANIES. OBGANIZATION. 1. Aetna Life, Connecticut, . . . . . 1850 2. American Mutual, Connecticut, . . 1848 3. Brooklyn, New York, 1864 4. Charter Oak, Connecticut, .... 1850 5. Connecticut General, Connecticut, . 1865 6. Connecticut Mutual, Connecticut, . 1846 7. Continental, New York, 1866 8. Covenant, Missouri, 1853 9. Equitable, Iowa, ' . . . 1867 10. Equitable, New York, 1859 11. Germania, New York, I860 12. Globe, New York, ........ 1864 Companies. OBOANizATtoN. 16. Massachusetts Mutual, Massachusetts, 1851 17. Metropolitan, New York, .... 1867 18. Mutual Life, New York, 1843 19. Mutual Benefit, New Jersey, . . . 1845 20. National of United States, Illinois, 1868 21. New England Mutual, Massachusetts, 1844 22. Northwestern, Wisconsin, .... 1858 23. Penn Mutual, Pennsylvania, .... 1847 24. St. Louis Mutual, Missouri, .... 1858 25. Travelers, Connecticut, 1866 26. Union Central. Ohio, ...... 1867 27. Union Mutual, Maine, 1849 13. John Hancock, Massachusetts, , . . 1862 1 28. United States, New York, .... 1850 14. Life Association, Missouri, .... 1868 | 29. Washington, New York 1860 15. Massachusetts Hospital, Mass., . . 1823 i 30. Western New York. ...... 1868 Summary in 1874. Net Number op Policies or Lives. Amount Insured. Existing or not terminated, . . 549,418 $1,367,217,000 Discontinued, 431,568 1,159,867,000 Died, 46,5 43 130,2 24,000 Whole Number, . . . . 1,027,529 $2,657, 3 08.000 Exposed tc Risk one Year, . 4,504,797 $11,839,005,500 These figures represent the net numbers after more than a hundred thousand secondary and other policies have been omitted or transferred according to the rules of reduction. Thus the total Experience of the Thirty Life Offices has a net basis of more than a million of policies, or lives, insured for more than twenty-six hundred millions of dollars. The arrangement of so extensive a serjes of tables has been guided, as far as possi- ble, by the best established precedents, and, mere innovation has been carefully avoided. In several particulars, however, the system of Life Tables as described in that great repository, the Journal of the Institute of Actuaries, was found to be still in a state of progress. Different courses were indicated for determining the proper resultant between the effects of the medical selection and the counter selection of the insured, such as the omission of the first three or five years experience. Also shorter methods were demand- ed for changing the standard life table to higher or lower grades of mortality. And another desideratum was the condensation within a convenient compass of the vcflunii- nous tables for term policies, endowments, and joint lives, to give monthly as well as annual values. It has been much easier to describe, than to supply what is needed. That an advance might be made to meet these difficulties, the actuary in charge has given them all the consideration which other duties have permitted ; Avith what success the examination of the following pages prepared by him, must determine. Among the improvements to meet the necessities of the work, may be named, the principles of transformation of Office data, the system of final series, the life table graded for different 6 PREFACE. climates, the proof of the logarithmic law of mortality, the general yaluation tables, the arrangement of joint lives and survivorships, with new investigations described hereafter. The first reductions of the Mortality Experience, were commenced in November, 1875 ; but the principal part of the returns arrived in the following summer, when the number of clerks engaged in their reduction, was augmented from three to ten. Other returns came in at later dates ; and the last was received at the close of the year 4877. With an adequate clerical force, the reductions were pressed forward as fast as the returns arrived ; and in March 1878, the first fruits appeared in the issue of copies of the Climatic Tables for examination. It may here be mentioned, that after the preliminary arrangements had been initiated and were fairly in progress, two of the Committee withdrew from membership though not from interest in the work, and their places were supplied in succession by the election of Emory McClintock and George W. Phillips. The former actuary had especially advocated the importance of the returns of losses and amounts insured ; while the latter actuary, by elaborate office reductions, had prepared the way for the present climatic statistics. For the position of chairman of the committee, Levi W. Meech was chosen, who had at an early stage been designated to take charge of the returns, and to make the proper reductions at the expense of the Chamber. And in May, 1878, after the dissolution of the Chamber, John M. Taylor, Secretary of the Connecticut Mutual Life Office, accepted the position of Treasurer. In the medical department, the classification of diseases given by the cards, required the services of a physician long conversant with office examinations, and the current applications from different parts of the United States. Such an examiner Avas found in J. C. Jackson, M.D., of Hartford, by whom the numerous returns of diseases were faith- fully examined during several months, till the medical statistics were classified under the most favorable conditions of accuracy and authority. It should also be mentioned that the returns from two Western Life Companies, were generously furnished by the late William E. Harvey, Actuary, after his own reductions. And in this connection, for maintaining the highest attainable accuracy, thanks and commendation are due to our clerical force, especially to M. C. Spring, S. E. Warner, and S. M. Hayward, assist- ants. From the amateur collection courteously furnished by Mr. J. Downes, of Wash- ington, D. C, we have copied the accurate table of anti-logarithms, reduced from the eleven place values of Dodson, compared with Vega ; and have added proportional parts for the middle of each division, computed with three extra decimals. To give confidence in employing the vital statistics, it may be proper to state, that all the original returns had been copied under responsible direction, from regular Office records, alike free from objection. The subsequent reductions by the Committee, have been verified by two independent operations ; and a further check was obtained by comparison of the parallel columns of Lives and Amounts. The work has been nearly six years in progress, under a single direction of large experience; and with favorable conditions, has thus resulted in an authentic standard for reference. Levi W. Meech, in Charge. ' EDWijf W. Bryant, Emory McClintock, George W. Phillips, Howell W. St, John. CONTENTS. ELEMENTARY OBSERVATIONS AND TABLES. DIVISION FIRST. Mortality and Loss Experience. SECTION PA«B I. Instbuctions and Sample Cakds 13 II. Adjustments op Cards. Dtjkation 31 Method for Joint Lives, Transfers, Original Death Claims 32 III. Classifications op Data 33 IV. Calendar Years and Policy Years 34 V. Notation and Annual Equations - 35 VI. Transformation op Data 36 VII. Exposed to Risk , 38 VIII. Comparative Results. Ratios op Mortality and of Loss to 1874 39 Influence of Medical Examinations and the Amounts at Risk 30 Insured Females 34 IX. Continuation of the Experience op^ 1874 to Final Series 35 X. Notes on Graduation. Short Method for the Expectation of Life 38 Graduation by Formula and by Divided Differences 40 Loss relative to Term and other Species of Insurances 43 DIVISION SECOND. Climatic Tables. I. Computation op Probable Loss, with Tabular Factors 43 Change of Probable Loss from the standard of one Life Table to another 46 II. Classification op Counties. Miscellaneous Statistics 46 Climate Extra 48 III. The Life Table Graded for Different Climates 49 Climatic Formulas and Special Ratios for Tables 50 CONTENTS. DIVISION THIRD. Medical Statistics. SECTION FAeS I. Desckiption of the Tables 52 II. Statistics of Consumption Compabed 54 Sanitary Survey of the United States 56 Table of the Relation of Height and Weight 57 III. The Germ Theory op Malarial Diseases 57 Table of Sickness and Recovery in Adult Life 58 DIVISION FOURTH. Time and Monetary Tables. I. The System of Interest and Annuities 59 II. Time and Interest Tables for Days 60 Copy Multiplication 61 III. Compound Interest and Discount Tables. For Twelve Rates of Interest continued to 100 Years 64 Connection of the Tables 65 IV". To FIND THE Rate of Interest of an Annuity 41, 65 V. To FIND THE Rate of Interest on the Price of Funded Stocks or Bonds 67 Table of the Auxiliary Factor 68 VI, Interest Compounded oftener than Once a Year 69 Correction of the Present Value of Half- Yearly and Quarterly Annuities 70 Illustrations of Compound Interest 71 F^PtT SEOOISTD. LIFE ANNUITIES* AND INSURANCES. I. First Principles op Probability for Reference 233 Mathematical and Moral Expectation 235 II. The Life Table. Law op Mortality Established 237 Construction of Life Tables from the Final Series of Tables III and IV 237 Formula and Constants of the Life Table for Males 237 Fluctuating periods of health or Climacterics 238 Expressions for the relative Force and the Rate of Mortality 239 III. Present Value op Life Annuities 240 Temporary and Deferred Annuities. Joint Lives ... 241 Simpson's Rule. Half- Yearly and Quarterly Annuities 242 Annuities payable to the day of Death 242 IV. Expectation of Life and Probable Life 242 The Curtate and the Even Expectation 243 Probable Life in Europe. 244 CONTENTS. 9 SECTION PAGE V. Commutation Tables 244 Methods of Construction 246 VI, Single and Annual Premiums of Insurance - 248 1. Formulas for Single Premium 249 2. Annual Premium 249 3. Five, Ten, or Limited Annual Premiums 250 4. Temporary Insurances 250 5. Pure Endowments ... 251 6. Endowment Insurances, 251 7. Deferred Insurances 252 8. Return Premiums 252 9. Increasing and Decreasing Scales of Premium 253 10. Increasing and Decreasing Insurances 254 11. Price of Annuities secured by Life Insurance 355 12. Insurance on Female Life 256 13. Insurance on Joint Lives 256 Method for Three or Four Joint Lives 258 Equalization of Annuities. Tables for Simpson's Rule 259 14. Survivorship Insurances. Makeham's General Solution 260 15. Insurances payable at Death. 16. Semi- Annual and Quarterly Premiums 264 VII. Reserve or Valuation of Policies 264 Months and Days. Reserve on Paid-up Policies 265 Valuation of Annual Premium Policies. Prospective Formulas 265 Valuation of Gross Premiums 266 Valuation of Annual Premium Policies. Retrospective Formulas 267 Elementary Formula of Valuation 268 Application to Return Premium and other Species of Policy 268 General Valuation Table. Adjustment for Months 269 Valuation Table for Joint Lives by Single Entry 270 Reserve on Survivorship Insurances, new process 271 Short Method for the equivalent Sum Insured by Paid-up Policy 271 Non-Forfeiture. Annual Valuation of an Insurance Office 272 Reference to Methods for the Distribution of Surplus 273 Illustrations of the Contribution Formula 274 Tontine Dividends 295 VIII. Life Annuities. Survivorshlp and Reversionary Annuities. Successive Lives, 275 Present Value of Single Life Annuities of different Species „ . . 275 Reserve or Valuation of Annuity Policies 276 Annuity extended to an extra payment ; also to the day of Death 277 Present Value of an Annuity on the longest Life or last Survivor 277 Annuity on the last n Survivors of m Lives 278 Reversionary Annuities. Verification 279 Partition of Annuities. Conditional Periods 281 Survivorship Problems. Successive Lives 283 IX. Derivative Formulas. The Continuous Method 285 Annual Derivation. Tables of Addition and Cubtraction Logarithms 283 Summation Formula applied to Annuities and Instalments 288 Change from Ordinary to Continuous Annuities and Premiums 290 Applications of the Differential and Integral Calculus 291 Total Probability of Survivorship between Two Lives 294 Remarkable Simplification of the Survivorship Problems 295 X. Insurance on Lives according to the Order of Survivorship 296 Notation 302 Interpolation. CoNSTANTf 304 Tumbles iint p^^rt p^irst. DIVISION FIRST. Mortality and Loss Experience. TABLE PAGE I. Observations on Insured Male Life 77-123 Summary (A). According to Ages of Exposure 74- Summary (B). According to Years of Insurance 76 II. Observations on Insured Female Life 127-158 Summary (A). According to Ages of Exposure 124 Summary (B). According to Years of Insurance 126 III. Male Life (A). Mortality and Death. Claims per cent, by Ages 159 Male Life (B). Mortality and Discontinued per cent, by Durations jl61 Male Life (C). Logarithm of Multiplier for Final Series. 162 Average Sum Insured 159-161 IV. Female Life (A). Mortality and Death Claims per cent, by Ages 164 Female Life (B). Mortality" and Discontinued per cent, by Durations 166 Female Life (C). Logarithm of Multiplier for Final Series 166 Average Sum Insured 164-166 Mean Percentage of Mortality by the Year of Insurance 30 Male Life. Deaths and Claims per cent, to 1874 31 Female Life. Death and Claims per cent, to 1874 32 Comparison from several Tables of Mortality 34 Comparisons of Experience to 1874 with Final Scries 36-38 Short Method for Expectation of Life 39 Loss relative to Species and to Amount of Policy 42 DIVISION SECOND. Climatic Tables. V. Probable Loss and Actual Loss by States and Territories 167 VI. Order and Variability of Climatic Ratios in thirty-five States 168 VII. Probable Loss and Actual Loss by Counties in twelve States 169-179 VIII. Climatic Commutation Table. Interest 4 per cent 180 Multiplier to give the Probable Loss 44 Classification of Counties 46 Age of Mean Rate of Probable Loss 47 Comparison of Northampton and Carlisle Annuities 49 DIVISION THIRD. Medical Statistics. IX. General Table of Diseases and Deaths with Variability 182-185 X. (A) Males. (B) Females. Diseases and Durations of Policy at Death 186 XL (A), (B). Diseases and Ages at Death 188 XII. (A), (B). Diseases and Deaths by States and Territories 190 Xm. (A), (B). Ratio of Deaths to Exposed, by Duration of Policy 192 XIV. (A), (B). Ratio of Deaths to Living by Ages of Exposure 194 XV. (A), (B). Percentage Distribution of Disease by Duration of Policy. 196 XVI. (B), (A). Percentage Distribution of Diseases by Ages of Exposure 197 Note.— The Formola of Computation is generally given ou the first Page of each Table. TABLES. .11 TABLE PAGE XVII. (A), (B). Climatic Distribution of Diseases among 100 Deaths. Results of the Census 199 (C). Diseases and Deaths among 10,000 Exposed in each Group of States 201 A Sanitary Survey of the United States 56 Table of Relative Height and Weight 57 Table of Sickness and Recovery in adult Life 58 DIVISION FOURTH. Time and Monetary Tables. XVIII, Calendar and Decimal Parts of a Year 202 XIX. Simple Interest for Days at 5 per cent. Omitted 204-207 Copy Multiplication 62 XX. Amount of $1 at the End of any Number of Years 208-211 XXI. Present Value of $1 due at the End of any Number of Years 212-215 XXII, Amount of $1 per annum at the End of any Number of Years 216-219 XXIII. Present Value of $1 per annum for any Number of Years 220-223 XXIV. Annuity which $1 will purchase for any Number of Years 224-227 XXV. Logarithm of the Present Value of $1 due at the End of any Number of Years. . . 228-231 Table of the Auxiliary Factor. . . ^ 68 Illustrations of Compound Interest 71 TABLES IlSr F.^I^T SEOOISTID. THIRTY AMERICAN OFFICES' EXPERIENCE. XXVL Male Life Table. Expectation of Life and Probable Life *3 XXVII. Common Logarithms of the preceding Life Table *4 XXVm. Life Annuities. Present Value (a) *6 XXIX. Animal Premium to insure $1000 for Life *8 XXX. Single Premium to insure $1000 for Life *10 XXXI. Reversion or Life Insurance equivalent to Single Premium of $1 *12 XXXn. Life Annuity which $1000 will purchase *13 XXXIII. Commutation Table and Logarithms, 3 per cent *14 XXXIV. Commutation Table and Logarithms, 3^ per cent *18 XXXV. Commutation Table and Logarithms, 4 per cent *22 XXXVI. Commutation Table and Logarithms, 4^ per cent *26 XXXVII. Commutation Table and Logarithms, 5 per cent *30 XXXVIII. Commutation Table and Logarithms, 6 per cent *34 XXXIX. Commutation Table and Logarithms, 7 per cent *38 XL. Commutation Table, 8 per cent ''42 XLI. Commutation Table, 9 per cent *44 XLII. Commutation Table, 10 per cent , *46 XLIII. Common Logarithms for Special Formulas *48 .^L.JV. Life and Temporary Annuities, or Log (1 + «"~*), 4 per cent *50 XLV. Life and Temporary Annuities, or Log (1 + a"~^), 4^ per cent *61 ^VI. Life and Temporary Annuities, or Log (1 + «""■'), 5 per cent *72 12 TABLES. TABLE TAGE XLVII, Life and Temporary Annuities, or Log (1 + a^ ), 6 per cent *83 XLVIIL Conversion Table from Annuities to Annual Premiums *94 XLIX. Conversion Table from Annuities to Single Premiums *99 ju. Single Premium for Life or Reserve on 1000 by Months *104 LL Logarithm of Single Premium by Months on $1 Insured *108 LIL Single Premium or Reserve for Term Insurance of 1000 *112 LIII. Annual Premium for Term Insurance with Valuation Factor *118 LIV. Single Premium or Reserve for Endowment Insurance *123 LV. Continued Annual Premium for Endowment Insurance of 1000 payable at Death or Age 30, 35, .... , 75. Also X (P-n;) *135 LVI. Five Annual Premium for Endowment Insurance as above *138 LVII. Ten Annual Premium for Endowment Insurance as above *141 LVIII. Continued Annual Premium for Endowment Insurance of 1000, payable at Death, or at the end of 5, 10, ,40 Years *144 LIX. Five Annual Premium for Endowment Insurance as above *146 LX. Ten Annual Premium for Endowment Insurance as above *148 LXI. Limited Annual Premiums for Life Policies *150 LXII. Net Value or Reserve on Ten Payment Life Policies *151 LXIII. General Valuation Table for Annual Premium Policies *154 LXIV. Value of / and /' for every Year and Month of Age *200 JOINT LIVES MALE. LXV. Correction for reducing Two Lives or Three to the case of Two Equal Ages *204 LXVI. D and N Columns for Two Equal Ages, 3 per cent *206 LXVII. D and N Columns for Two Equal Ages, 3^ per cent *207 LXVIII. D and N Columns for Two Equal Ages, 4 per cent ^ *208 LXIX. D and N Columns for Two Equal Ages, 4^ per cent »209 LXX. D and N Columns for Two Equal Ages, 5 per cent ^ *210 LXXL D and N Columns for Two Equal Ages, 6 per cent *211 LXXII. Annuitieis on the Joint Continuance of Two Lives . . .Page 301. New Table. . . . *212 LXXIII. Annual Premium for Insurance on Two Joint Lives *213 LXXIV. Single Premium or Reserve on Two Joint Lives *214 LXXV. Ten Payment Premiums with Valuation Factors *215 LXX VI. General Valuation of Annual Premium Policies on Joint Lives *216 LXXVII. Female Life Table. Expectation of Life *224 LXXVIII. Female Life. D, N, a, and Correction of Age. 4 per cent *226 MISCELLANEOUS. LXXIX. Comparison of different Life Tables *228 LXXX. Expectation of Life by various Tables *230 Probable Life in Europe Text, 244 LXXXI. Percentage of Mortality by different Life Tables *232 LXXXII. Anti- Logarithms, or Numbers to Logarithms *234 TABLES. 12a PAGE Amkricajst Life Tables. From Combinations of the Census. — Introduction *255-*259 The Mathematical Law of Mortality Tontines. Method of Least Squp-res, *254 *257 Tables of D and N for Males and for Females. Interest 3, 4, 5 and 6 per cent *360-*263 Present Value of Life Annuities. Interest 3, 4, 5, 6 and 7 per cent *264 Single Premiums and Reversions *266, *267 Present Value of Joint Life Annuities, Parts I and II *268, *269 Present Value of the Right of Dower *370 NOTES AND ADDITIONS. The additions made to the present Revised Edition will be found on the four pages inserted after page 12, and those which follow page *254, comprising American Life Tables and their mode of analysis. Some slight corrections have been made, and the separate paging of the Insurance Tables denoted by stars, after page 298 of the original treatise, is still retained. Or Ages in the Tables. — To facilitate a correct and ready reference to tabular values, the letter x everywhere denotes the Office Age at Entry, in entire years. After Entry, x-\-n denotes the Age ai the begitming of each Policy Year. And h denotes the Months ov fraction of a year elapsed since the last regular payment of Annual Pre- mium. Thus, a; + w always denotes "the Office Age last birth-day" \n integer years, and x4-n-\-h denotes the exact age of the insured at any date, in years and months. Instead of n years, the proper side column can also be entered with one less than the number of premiums paid. The above system is applied uniformly in all the Tables, except the Life and Tem- porary Annuities, Tables XLIV to XLVII. In these four Tables, pages *50-*93, the Age of Entry x at the head of the columns is the same as in the other Tables. But the n " Payments " at the side denote the number of Policy years at next birth-day, or the number of annual premiums already paid, in the case of life policies, or w =: w + 1. It may also be noted that •' 4 Per Cent." or any other rate given at the head of a column or Table, signifies the rate of Interest employed in its construction. Reductions of Mortality Experience. — Commencing with the original obser- vations on Lives and Amounts, the relation which has been conventionally adopted between Calendar and Policy Years, and the mode of averaging fractional parts of Years, are described with "sample cards" on pages 17, 23, and 24. Page 24 also sketches the extended classifications given in Table I. by double entry, that is, "Age of Entry" above, and "Years of Insurance" at the side, shown on pages 77-123. These double-entry elements are next classified in groups to become the single- entry — Summary (A), page 74. Here on the line of age 25, for example, is given the sum of Death Claims 1,002, and of Exposed to Risk 152^867*5. How these results were found for each separate age is shown by the grouping on page 38. Observing that 25 scattering were omitted from this illustration, but included in Summary (A), all the other elements in the central Columns under (1874) can be readily traced to theiy original places on pages 88, 87, etc. The next reduction gives the Mortality Per Cent. (1874), page 159, third column. On the line of age 25, for example, the Tabular per Cent, is 0.657. This, like th© 123 NOTES AND ADDITIONS. other values, is derived from two adjacent ages of Summary (A), as follows : — Taking the sum of death claims opposite ages 25 and 26, or 2,231 as numerator and the sum of the exposed 339,592-5 as denominator, the value of the fraction is 0-657, as before quoted. One more simple process, described on page 237, would change these per- centages to the relative numbers in Table XXVI, which completes the whole of the former course of reduction from the original registry of Policies to the Life Tables. The Method of Final Series. — By this extension of the former method, the experience of the Existing was carried forward, so that the general experience was determined in the final Table with superior accuracy. To aid in the explanations of the text, the specimen columns of page 38 may here be translated into general formulas : ^ Let D denote the Death Claims, E the Exposed to Eisk, and 1 the relative weight for the first combination of ages 0-25. Let Di, Ei, ii\, v denote the like numbers for the next combination 1-24, and so on ; where v denotes the present value of $1 dis- counted for 1 year. By the Former Method of reduction : Death Claims per Cent. _ (D + Dx-f-D, + D3 + D 4+. -QlOO - E + E, + E, + E3-rE.-h... By the Method of Final Serieg : Death Claims per Cent. _ (D + D i Wiz; + 'D^iv^v^+ ^z W^i^-\- . . . ) 100 ~ E +"EiWiV + E2W2V2 + E3?t'3t73 + . : . The weights w were at first computed by direct solution, as described on page 37. A parallel Example may be found in De Morgan!s Essay on Probabilities, page 137. It may further be noted that as w increases from year to year, the discount factors V, v^, 't^. . .continually diminish. On page 36, the question may arise why are two ages of entry united ? and why is the process of Final' Series stopped with the 27th year? In reply, the derivation of the fundamental Formula for jp" on page 25, pre-supposes the Observations to be suffi- ciently numerous to change uniformly (or approximately so) from the beginning to the end of each year, as illustrated by the trapezoid. After the 27th year, the Claims were so few and irregular that the Formula for p", thus derived conditionally, could not be applied; and so 46,413— 215— 2,361, that is, 43,837 Existing still remain at the 28th year. The real errors of their omission from the Preceding Formula of Final Series are deemed insensible. For, were the 43,837 distributed to older ages and then dis- counted by 1^, v^. . .the whole present values would be less than 13,000. Whereas in the former Method of constructing Life Tables, the future experience of a much greater number, or 102,484 Existing, would be left undetermined and omitted. The factors v, 1?, ifi. . .introduced on page 37, may be further illustrated. A careful inspection of Commutation columns and formulas for premiums discloses the fact that the Commutation factors v, v^, 1^^ . . after the manner of Weights, are effective only as they commence with the "age of entry;" thus practically attenuating the more distant data, they make the Value of Premiums and Annuities depend chiefly on the ages nearest to the "age of entry." Therefore, it appeared important to recog- nize this inevitable attenuation, before as well as after the construction of the Life Table, by discountijig the original data to Present Value at the respective ages of entry. A majority of the Committee held that an Experience Table based on the Amounts would represent the actual business of a Life Office more closely than one based upon N0TE8 AND ADDITIONS. 12c Lives. And the different sums insured found on pages 159, 160 sustained this view; while the method of Lives virtually assumes the sum insured to be uniform at every age. In separate offices, 'however, the ratios of Death Claims by ages, although more correct, were more fluctuating, than the ratios from Lives. This difficulty was obviated by resorting to the larger numbers of the Thirty Offices united, which give a very regular curve from the Amounts, as shown in the fundamental Table XXVI. The inquiry has arisen, what is the relation of Table XXVI to the Actuaries jjM(5) Table which omitted the experience of the first 4.5 years, and to the earlier plan of a series of separate Life Tables for every Age of Entry? Without attempting a full answer, we note from page 30 that the mean duration of the Thirty Offices Policies was 4.36 years, and for equal ages of exposure the average rate of mortality was attained in 2.5 years after Entry. The corresponding numbers for the Twenty British Offices were 9.12 years and 4.5 years. Hence the inference that the latter terms 2.5 and 4.5 being half the former nearly, are not fixed periods, but vary with the mean age of Policies in each Office. Therefore returning to the earlier plan, so perfect in theory, suppose one Life Table constructed from the Amounts under Age-at-Entry 37, another for 38, and so for every other Age of Entry, — taking not the usual 10,000, but the entrant amounts, as 94,029, etc., for their initial numbers. This secures to the and D Commutation elements their proper weights in one general Life Table which might be constructed from all of them. Were the distribution of insured ages very dissimilar in different Life Offices, the many single Tables might be required ; but the researches of King, Chandler, and other actuaries prove the ages to be so nearly similar, that one general Life Table has appeared sufficient. The method of Final Series might ha^■e been derived as an improvement on this earlier plan. Climate Extra, Page 48. — In the case of extra premium on a life long resident in India, returning after many years to reside permanently in Europe, Mr. Makeham proposed to determine the future reduced premium P, by the following equation of Eeserves : l_l±^^ = l-(l-t;) (l + «^+«)-P(l+«x-H.). i.-\-a X Here the Eeserve on the left by the Bengal or* other foreign Table is equated to the Reserve by the home Office Table ; whence is determined P = l±^tlt (1 — v). (l + a'a,)(l + aa; + n) ^ ^ Besides this method, the two letters of Mr. Makeham in the 14th volume of the Assurance Magazine, also contain the first extension of the commutation principle or factor fc^ to climatic mortality, here revived as described in Section III. Pages 62, 63. — The Copy Multiplication Table is now entirely correct. In the first edition were two errata, the one requiring the first 9X in column 100 to be changed to 8X ; the other in column opposite 70 at the side, requiring the first 4556 to be changed to 4456. Page 270, line 38. — The Valuation of Ordinary Joint Life Policies may be effected with great facility, as shown by the Example at the foot of Page 270. Since in these Policies F = n, the general formula becomes V — f'" — V" h' nd NOTES AND ADDITIONS. Price of Stocks and Bonds, Page 67. — To find the Price (not the interest) at which a specified Bond should he bought or sold, to yield an assigned rate of Interest on the purchase. In the notation of the text, i — i' can be regarded as the temporary annuity, which p — 1 will purchase with i' as the rate of interest, till maturity. Example. — Required the equivalent Price p to yield 4 per cent, interest, which should be paid for a 6 per cent. Bond, that will mature in 18 years. Answer, p = $1.25318 on each II of par value. Formula, p = l-^{i—i') V'„. Table XXIII, Page 220, .... Valuation of Ordinary Endowments, Page 266, lines 2 and 32. — Both of these Formulas, like the general Table LXIII, have the usual simple cases. When w = 0, before the first premium was paid, Va; = 0. Also comparison with page 251 proves that, one year before maturity, when n^m — 1, Va;+n = v — Pa;. And at maturity, when n ■= m, Yx+m = 1> the sum insured. To interpolate for Months, besides the use of Table LXIII, we may substitute the preceding value of V into line 7, page 270 ; and thus, \in =: n-\-l, 1 I 'y^^~^ J '~J\ , m-l ! + «« Example. — Required the 4 per cent. Reserve V on each $1 insured, at the end of 7^ years, for an Endowment Insurance by annual premium, beginning at the age x = 26 years, to mature in m =: 15 years, solution." (1) By the General Table LXIII: Page *144. x = 26, m = 15, 1.581343 = A (P— tt). '' *163. x,n + h = 7 yrs. 6 m. 70.79 0.986978 = A (B/). " *245. • .,.+ 370. 10 ...2.568321 V = 440.89 = Reserve on $1000. (2) By the above Formula : Page *201. x + n = Sd,h = 6"',f' — 0.98077, T. 989888 = A/. " *53, x-^n = S4:,m—n = 7, 0.785055 " *52. X = 26, 7n = 15, 1.042648 sub. '' *248. n=zn-{-l, /— 0.53988 ... 1. 732295 V = 0.44089 = Reserve on $1. Commutation Formulas, Page 293, 237. — Instead of integration, we may develop T)x in series, and take the sum of the geometrical progressions between x and the end a> of the life-table, by the common rule. If Xh = 5.028254: (1 u w^ ?/ ) N^ = -C.-^rTc{vsY -j ^— -^+ — _^- J 2 (^,^2_-I) + 1.2.3 {vsq^-1) -•'•]' With interest 4 per cent., A {vs) = T.9802201 ; Xu = 4.837557 + 0.041279507a; ; Na; = — 121,828 -tk{vs)^ {22.46015 + 19. 70403W- 3.214919^2 + 0.6155736w3— 0.1048335^4 + 0.0155241 w5_ . _ |, PART FIRST. ELEMENTARY OBSERVATIONS AND TABLES. The Elementary Observations and Tables of Part First are classed into four Divisions, as follows : First. Mortality and Loss Experience, - Second. Climatic Tables, ------ Third. Medical Statistics, Fourth. Time and Monetary Tables - - Tables I-IV. V-VIII. IX-XVI. XVII-XXV. DIVISION FIRST. ORTALITY AND LOSS EXPERIENCE. The received method of reduction presupposes the new business of different calendar years to be superimposed or brought together, as if the entrant insurances were all effected in one calendar year. Thus the entrants at the age of thirty, for example, are all classified together, as entering in the same initial year. From this origin, the annual deaths and discontinuances are traced forward and noted year by year, through the whole course of insured experience. Such are the elementary observations given in Tables I-IV, for male, and for female life. Let us first glance at the methods from the beginning. SECTION I. Instructions and Cards. The following Instructions of the Committee in March, 1875, were published and attached to folio card-boards, for constant reference in filling out the blank cards, from the Office records. INSTRUCTIONS. Let each figure and word be plainly written with a pointed pencil of medium hardness. The first blank, in the left hand upper corner, is for the con- venience of the Committee in separating the cards, and will contain their private mark for the several companies, and need not be regarded by you, — nor the second, which contains the policy-num- ber, — inserted consecutively by the printer's numbering machine. Private mark and number of fotictf. 14 MORTALITY AND LOSS EXPERIENCE. Mesidenee. Amount and Kind. Tears of JEntry and Exit. The third blank, "residence," is to be filled with the name of the State only, except that in the following States the County is also to be inserted, viz.: Virginia, North Carolina, South Caro- lina, Georgia, Florida, Alabama. Tennessee, Arkansas, Mississippi, Louisiana and Texas. In case the residence be in any foreign country, draw a line through the word " State " and write the name of the country in the same space. The series of blanks relating to the amount, kind, and duration of the policy are to be treated as follows : In the top space, under " Amount," write the amount insured, thousands at the left hand, and hundreds at the right hand of the light perpendicular line, using no ciphers, where even hundreds or thousands are entered. On the same horizontal line under " Kind," indicate the kind of policy by L for Life, E for Endowment, and T for Term Policies. On the same horizontal line, under " Cal. Year of Entry," enter the calendar year in which the policy was issued. If the policy be still in force, make a dash after the -figures " 18" in the same horizontal line, under the " Cal. Year of Exit." If it has ceased to be in force for any reason except death, enter upon that line and under the '• Cal. Year of Exit," the calendar year in which it ceased, and make a dash in the blank " Cause of Death." If it ceased by death, make the same entries, but carefully write D after the year of exit. If any changes have occurred in the amount or kind, the original number, age and date being retained, other entries must be made as follows : The original amount, kind and year of entry having been entered as above, the calendar year in which the change was made is to be taken as the year of exit of the original policy, and entered in the top line under that head. Then an entry of the changed amount or kind, as the case may be, is to be made under the proper head on the second line, and the date of the change is to be taken as the calendar year of entry of the changed policy, which will of course correspond with the calendar year of exit of the original policy. xiKit by Death under The termination by death or otherwise of the second or changed CHANGED Policies, policy is to be treated in the same manner as has been pointed out for the original Should the changed poHcy have been still further changed, the proper entries are to be made upon the third line, under the several heads, in the same manner as required for the changes from the original. If more than two changes have taken place, the subsequent ones may be entered in the bottom blank for memoranda. Exit by Death. Changes in Amount and Kind. Second Changes, Third and subse- quent Changes, MORTALITY AND LOSS EXPERIENCE. 15 Wlien changed Pol- icy is in force- Revival of lapsed Policies, Revival or change triTHO UT new med- ical examination. Policies lapsed and good for ''PRO RA- TA " amounts. Revival of " PRO RATA " Policies. Aye of Entry. "Rated Age^^ " True Age." Sew, Cause of Death. "Xot Taken" Pol's. " JRe-insur," Pol's. Joint Life Policies, If the policy in which change has been made, be in force, make a dash after the figures " 18 " under the " Oal. Year of Exit," in the horizontal line occupied by that policy. If a policy has lapsed and has subsequently been revived under the same number, date, age, etc., and the ;"evival has taken place in the calendar year in which the lapse occurred no entry of these facts need be made. If the revival has taken place in a later calendar year than that in which the lapse occurred, then the calendar year of lapse is to be entered as the year of exit of the original policy, and the revival is to be treated as a change, and the proper entries are to be made upon the second line, the calendar year of revival being taken as the year of entry of the changed policy, and write the letter K. in the memorandum blank at the bottom. If revival or change has been made by a policy under a new number, without a new medical examination, let the card for the new policy contain in the memorandum blank the number of the original policy also, with an written before it. In case policies after lapse remain good by their own terms for a fixed or ^^ pro rata^^ amount of the original sum insured, the year of lapse is to be taken as the year of exit of the original, and also as the year of entry of the changed or ^'pro rata" amount. In case such policies, after continuing for a time for the '^ pro rata " amount, are revived under the same number, date, etc., such revival is to be treated as a second change and entered accordingly. The blank " Age at Entry," is to be filled with the office age, or that upon which the premium is written. When for any special reason, the age has been rated higher or lower than the true age, the true age at nearest birthday should be stated in the memorandum blank with the words " true age " before it. The blank following "Age of Entry," upon the same line, may be disregarded, being for the use of the Committee only. The blank " Sex " is to be marked with a dash for a male life, and with a capital F for a female life. In the blank " Cause of Death " write the cause stated upon the Company's register. The following special cases must be carefully noted : Omit all policies " not taken " or not actually put in force. Omit all policies of re-insurance, or for the benefit of other companies. In case of a "joint life policy" mahe a card in full for each life, using a separate card for the second, but writing in the number. 16 MORTALITY AND LOSS EXPERIENCE. J'oitU lAfe Policies. Ziimited premium Policies. Mark each card in the memorandum blank with a capital J, and place them both in an envelope also marked J. It may sometimes happen that a " joint life policy " is discon- tinued as to one of the lives, and so adjusted as to continue upon the other life singly. The card representing the life upon which the insurance is dis- contiyiued will show the amount, kind, calendar year of entry and calendar year of exit, in the usual manner of a terminated policy. The card representing the life upon which the insurance is continued will show the change in the same manner as other changes are indicated, with the addition of the letter S in the memorandum blank, signifying " Single Life." No special designation need be made of policies with a limited number of premium payments, whether completed or otherwise. Policies terminated "When a policy has terminated for any cause other than death, by causes other than j-j^g causc need uot be Stated, but be careful to make a dash in the Death, blank " Cause of Death." SAMPLE CARDS. A. \ RESIDENCE. State. Ci>4%tz-c^'e. County. Amount. Kind. Cal. Year of Entry. Cal. Year of Exit. /^ --^ 18^¥ 18 — 18 18 18 18 Age at Entry. SO /O Sex. 1 Cause of Death. ..... AA. \ RESIDENCE. State. O^^zWte. County. Amount. Kind. Cal. Year of Entry. Cal. Year of Exit. /O .?' 18^-^ 18 18 18 18 18 Kge: at Entry. SO /O Sex. (^ Cause of Death. 1 MORTALITY AND LOSS EXPERIENCE. 17 B. \ RESIDENCE. State. ^l^u-aM^t-a. County. ^i{?e'^^€.co . Amount. Kind. Cal. Year of Entry. Cal. Year of Exit. / S # 18<^^ 18^^ S ^ IS^J* 18 18 18 Age at Entry. sr Sex. Cause of Death. . o. \ RESIDENCE. State. C/iMad-l. County. Amount. Kind. Cal. Year of Entry. Cal. Year of Exit. S s^ 18^^ 18<^-^ S 18^^ \%e^ / S 18^^ 18 — Age at Entry. SS Sex. Cause of Death. X). \ RESIDENCE. State. C/!utdd. County. C/J a-Ct-vtZ't. Amount. Kind. Cal. Year of Entry. Cal. Year of Exit. ^0 18^^ 187^ jf^ 18 18 1 18 18 Age at Entry. ^i» Sex. Cause of Death. Cyy,^-n-a'i.c/ C/e^e^. E. \ RESIDENCE. State. J^iQ, 18 18 18 18 Age at Entry. JS' Sex. Cause of Death. H. -/-/JS^S \ RESIDENCE. State. Cy!/(Qit-cAe-ry.€i'n, County. Amount. Kind. Cal. Year of Entry. Cal. Year 1 of Exit. ] S S ^ "^18^^ 18 18 18 18 18 Age at Entry. M Sex. Cause of Death. / HH. ^Y Cause of Death. N. \ RESIDENCE. State. lan of a census of living and deaths only, without the sums insured, has been supplemented by parallel columns of the Amounts, treating dollars as lives. The present statistics, therefore, are two-fold ; and have required more than double the usual labor in their preparation. The cards, on reception, were first examined from beginning to end, to select out all the secondary insurances, such as the preceding sample card N, marked under- neath. This particular entry — 19,263 simply refers from the secondary to the original insurance noted on card No. 19,263. The secondary or " 0-cards " so called, were next copied on the middle line of the originals, which were sought out for this pur- pose, before the regular order of the cards by number was broken up. The 0-cards, being about one-tenth of the whole, together with Reversionary Dividends, were laid aside as supernumerary. 23 MORTALITY AND LOSS EXPERIENCE. After this and other preparations, presently described, the next step was to note the Duration on each card of single insurance, which was found by subtracting the first "Cal. Year of Entry" from the last "Cal. Year of Exit," or (in case of Non- terminated or Existing policies) from the Epoch 1874, when the experience was inven- toried. Thus on the preceding cards A, and AA, the difference of 1864 and 1874, that is, Duration 10, is noted opposite the age. We may observe that the sum of the age and duration, or 30 and 10 on the card A, gives the advanced age of the insured, 40 years, in 1874. A more full account of the connection of Age and Duration will be given hereafter. With respect to cards having two or more successive insurances, the Durations are noted, after preparation, in the following manner. To refer to a few instances : — Where the amounts are alike as on the preceding card E, the entries may be regarded as two parts of one experience, or $5,000 exposed to risk through a duration of 12 years, like a single policy. Where the amounts are different, as shown on the preceding card F, the above rules require its resolution into three new separately written cards, all commencing with the medical examination of 1860; that is, (1) a Death card of the last amount $800 for the whole Duration 13 years ; (2) a Discontinued card of the difference 2,500-800, or $1,700 from 1860 to 1868 or Duration 8 years ; and (3) another Discontinued card of the difference 3,000-2,500, or $500 from 1860 to 1866 or Duration 6 years. All the other similar cards B, 0, M, can be superseded by single cards on the same principle. Indeed, experience seems to indicate that separate cards are more convenient in practice, than the three ruled lines of the present arrangement. Thus the card B is to be resolved (1) into an Existing card of the last amount $3,000 for Duration 14 years, and (2) a Discontinued card of the difference 7,500-3,000 or $4,500 for 3 years. On cards 0, M, the lower amount always subtracted from the next preceding amount over it, will give a negative difference. The corresponding cards of this species are to be reserved for future subtraction from positive results of the same age and duration. Thus, card C is to be superseded by (1) an Existing card of the last amount $7,500 from 1855 to 1874 or 19 years ; (2) a Discontinued card for the negative difference — $4,500 from 1855 to 1869 or Duration 14 years; and (3) another Discon- tinued card for the negative difference — $2,500 from 1855 to 1864 or Duration 9 years. These cards (2), (3) are to be reserved for future subtraction as above mentioned. These reductions will be best understood by clerks who are familiar with algebraic operations. Heretofore, the cards of Joint Lives have usually been reduced as single lives. But the death of the first of a couple, is occasionally followed by the death of the second in the same year; and such second deaths, although noted on the single life plan, escape the record Of joint lives. To correct for this omission, let q, q' or 1 —p, 1 —p denote the probabilities that two persons insured for %a each will die within one year. As single livQS, the Expectation of Loss in one year will be a{q-{-q'). As joint lives, the Expectation of Loss will be a{l—pp'), or a—a{l—q){l—q'),th2it is, a{q + q'—qq')'y which may evidently take the form The last expression of Probable Loss may also represent the Actual Loss on Joint Lives. Hence if each Actual Loss is divided by the six months probability (1— i^') of the survivor, the results will correspond to the experience of single lives. Since the average value of \q' is about \ oi 1 per cent, we may for every 200 Death cards of Joint Lives, select and change into a Death card one Discontinued card of the same MORTALITY AND LOSS EXPERIENCE. 23 sex, and of similar age, duration and amount. After this simple correction, first devised in the present investigation, the pairs of cards of joint lives can be separated, and the whole series united with, the other cards for single lives. Another practical artifice was employed to change a few thousand policies from the })lan of next birthday to the American custom of nearest birthday. Instead of dating back each card of the whole series half a year, each alternate card was dated back a whole year ; since all the policies were for ordinary amounts, and opposite errors would compensate each other. Another large collection of data with ages according to lad birthday, was dated forward on the same convenient principle. There were also several thousand Transferred Policies, or policies which on the winding-up of the Company that issued them, had been transferred to another Com- pany. The transfer was accompanied with the original ages or years of Entry of the Existing policies, but the corresponding Discontinuances and Deaths up to the time of transfer, were not stated. Thus, suppose ]00 or more persons to have entered the Cadmus Life Company at the age of 30 years. Five years after, the Cadmus is closed up, and the survivors at the age of 35, are transferred to a second Company. The valid experiences commencing at age 35, or five years after the medical examination, should enter Tables I and II, not on the first line, but on the fifth line below, at Duration 5. This object can be accomplished by writing two cards for each transfer; the card (1) beginning with the medical examination at the age of 30, and extending forward into the returns of the second Company, as usual ; and the card (2) beginning at the same age of 30, to be discontinued after a Duration of 5 years. The latter is to be reserved for future subtraction from positive Discontinuances of the same age and duration. In this way, the first five years exposure of the card (1) will finally be can- celled by the card (2), as was proposed. In relation to Endowment Insurances, composed of two parts ; that is, of a Temporary Insurance, and a Pure Endowment for the same amount; since the maturity of the Endowments was not specified on our cards, their effect on the whole experience is that of Temporary Insurance only. Death Claims reduced by compromise or by litigation have been restored to the original amount, as the uniform standard of this investigation. In the case of a single Company where the amount actually paid, had been given in lieu of the normal Death Claims, a small correction of about 2| per cent, was applied for compromised and litigated claims, taken together. Having thus represented the whole series by single cards, each commencing with a medical examination, and having noted on each of them its proper Duration, we apply the advantages of the card system as follows. SECTION III. Classifications of Data. The next operation was to separate the cards of Male Life from those of Female Life. Each of these groups was then divided into the three classes of Existing, Discontinued, and Died. Each of these classes was then sub-classified according to the Durations of Policy, 0, 1, 2, 3, 4, And these were next sub-divided into smaller packages of cards corresponding to the Age of Entry. The counting of cards in these packages, gave the Existing, Discontinued, and Died, of Lives in Table I and II. And the addition of the corresponding Amounts, by the same packages, gave the parallel columns of 24 MORTALITY AND LOSS EXPERIENCE. ExistiDg, Discontinued and Death Claims. The latter were first added in full, and then changed to the nearest TJwusands of Dollars, omitting the last three residual figures. For every Age of Entry, the sum of the Existing, Discontinued, and Died, expresses the total Number of Entrants at the head of the columns. For example, at the age of 27 years, on Page 90; 36,221 had entered, and were insured for the initial amount of $87,030,000. Of these policies, 18,328 were still in force, or existing at the epoch in 1874; also 16,790 had been discontinued; and 1,103 had died, as noted at the foot of the columns. Of the Deaths, the largest number in any one year, and more than one-third of the Discontinuances had occurred in the second year, shown opposite Duration 1. This feature is common to all the Companies. Tlie first ten or fifteen years Duration comprise the greatest part of the business; and the whole is practically limited to thirty years ; although a few cases are of older date. SECTION IV. Calendar Years and Policy Years. For greater convenience, as before mentioned, the new business of different cal- endar years is superimposed, for this investigation, as if all the insured entered the Company uniformly during one initial year. On the average they enter at the middle of this year ; and so the average are exposed for only the latter half of the first calendar year. The Exits as well as the Entrances of subsequent years, are likewise assumed to occur uniformly during each separate calendar year. Average Policy Years thus begin and end loith the middle of Calendar Years ; and the average birthday is taken at the middle of the initial year. The ratio of mortality for the first year has been a subject of discussion in the Journal of the Institute ; but, as there stated, " it is difficult to devise a better plan ; " and m practice, the regular formula for subsequent years has been generally applied to the first year, as will be presently exhibited. Throughout the Tables I and 11, the first column headed " Years of Insurance," 0, 1, 2, 3, etc. expresses the Durations found by subtracting the Year of Entry on the cards from the Year of Exit or termination. For future reference, the more definite scale, — ^, ^ — 1^, \\ — 2^, etc. is also given in the right hand columns of Summary A and B, at the beginning of Tables I and II. As before described, assuming the entrances to occur uniformly during the twelve months of the initial calendar year, the average date of insurance or entry is taken at the middle of the year, when the " Years of Insurance " begin, as follows. The title " Years of Insurance " is synony- mous with " Durations" and with "Policy Years." Calendar Teabs. Years of Insurance. Existing, Discontinued, Died. i Beginning ] Middle ( End f Middle ] Middle ( End r Middle ] Middle (Etc 1 1 H 2 2 3i Insurances commence. Average date of Entrance. First Year's " Existing " record closes. Average date of Deaths and Discontinued. Survivors enter on second Policy Year. Second Year's "Existing" record closes. Average date of Deaths and Discontinued. Survivors enter on third Policy Year. Third Year's " Existing " record closes. Beginning Middle End Beginning Middle End Beginning Etc MORTALITY AND LOSS EXPERIENCE. 25 SECTION V. Notation and Annual Equations. Let N = number of admissions during any year of Age. J) _ a it Deaths during any year of Age. r = " " Discontinuances during any year of Age. e = " " survivors entering on any year of Age. R = " " Existing at the end of the year. Ni, Di, r,, e„ Ri = the like numbers for the next higher year of age, etc. These are here adopted from the short method of W. S. B. Woolhouse in the Journal of the Institute, Vol. 13, page 10. M = e+^N— ^(r + D) = number exposed at the middle of the year, if the changes occur uniformly. M can also denote the middle altitude of a trapezoid, whose base is unity or one year, and whose area represents the total exposure of the year. Assuming half the annual decrement to occur before, and the other half to occur after, the middle of the year, we have, p = ^ — jjT = the proportion that survive one year's exposure to Death. M— ir p' = ^ — J- = the proportion that survive one year's exposure to Discontinuance. f> M— ^(r+D) e, + R ,, ^' ..x. t - _, p = ^ .u ,j){ = = the proportion that survive one years exposure to ^^ '^ both Death and Discontinuance. q = rr= — -r=- = 1 — p = proportion that die in one year. j3 = M+|D=£+^N—^r= Exposed to Risk [of Death]. The last denominator M + ^D or i3 has been named the " Exposed to Risk," although it differs from the mean exposure M shown above. q = -p^ r, for the initial year of Entry or Duration 0. T T 1 — p' = ^ — -- =: -p^ — T^pr = proportion that discontinue in one year. ^ M + ^r i3 + |r— ^D ^ ^ ^ r-i-J) r + D 1 — »" = ST? — TT =Frr = 73 — r- = proportion of both deaths and discontinuances ''^ ' r m one year. For Duration 0, Co = N= Sum of (R + D+r), j3o = ^(N— ro), 1, e, = N-(R„+Do + ro), /3, = e,-^r„ 2, e, = e,-(R, + D, + r,), jS, = e,-^r,, " " 3, E, = E,-(R, + D, + r,), Pz = e,-lrs, etc. etc. etc. By eliminating e, and by transposition, or substitution, 3, = 2/3o-Ro-Do-K /3, = /3.-R,-D,-ir,-ir„ Pa = /J2— Rq— Dj— ^rj— |r3, etc. etc. 26 MORTALITY AND LOSS EXPERIENCE. etc. etc. M, = 2Mo-E.-i(r, + D,), M, = M,-E,-i(D,+n)-i(D,+r,), M3 = M2-E,-i(D, + r,)-i(D3 + r3). etc. etc. A method for carrying forward the Existing to their proportional terminations in the columns of Discontinued and Died, will be described hereafter in Section IX. SECTION VI. Transformation of Data. The foregoing equations have been particularly serviceable in transforming the results, which had been wrought out by different Actuaries and Companies, to a uniform system. A brief memorandum of some of the operations may be useful for future reference. The Experience of one, designated as Company A, gave the values of N, /J, and D for Lives at every age, with but an abstract of the values of r. After the latter had been interpolated for all ages, the Existing were then found from the above equations of R. Having now obtained the elementary data for Lives, the omitted Amounts were next supplied by proportions wrought out for the Existing, Discontinued, and Died, separately at each age. Thus, taking the first two terms from the general experience of the other similar Companies, we have the simple statement, as R of the Lives is to R of the Amounts, so is R of the Lives in Company A, to the corresponding Amount at the same age. The returns from another, which may be designated as Company B, gave the Deaths and Exposed to Risk by Policy years, accompanied by the following explanation. " The classification is by Calendar Years of Issue and Policy Years after Issue. In regard to fractions of years, a policy for $10,000 lapsed or discontinued at the end of 3^ years stands pro rata in the fourth year as $5,000, etc. If a policy was terminated by Death, it was entered as at risk for the whole year, and entered as a death for the whole amount." In order to change to the forms of the card system, the data P, D in thousands of dollars were first added as follows. Here P denotes the Amount Exposed to risk in one Policy year, and D the corresponding Death Claims. Age OF Entbt. Values of P. Ist Pouct Yeab. Age OF Entbt. P. 2d Pouct Yeab. 1869 '70 '71 •72 '73 '74A P. 1869 '70 '71 '72 '73A P. 7 8 1 14 8 14 IS 16 17 18 19 5 5 6 S 1 1 8 8 3 6 1 1 1 7 7 6 12 18 18 14 18 16 17 18 19 8 8 1 1 8 8 3 6 1 MORTALITY AND LOSS EXPERIENCE. 27 For any one Age of Entry, as 18 years, let D\, P„ denote the Deaths and the Exposure of the 1st Policy Year; D'a, Pa, the like quantities for the 2d Policy Year; and go on. Let us assume the entrants to begin, on an average, in the middle of the initial calendar year. Then if the deaths occur uniformly in each year, and Do, D,, D.J, . . . denote the Deaths on the card system, we have Do = iD\; Di = iD\ + iD'a; D, = iD', + ^J)',; etc. In like manner if Eo, Ej, etc. denote the Existing on the card system, at the end of Calendar years or the middle of Policy years, and Deaths were included as above described, we shall have by the middle ordinate of the trapezoid denoting the exposure of the pohcy year, restricted to the last columns denoted by '74A, '73 A, '72A, . . . E«=Px-iD\; E, = P,-p'a; Ea = P3-iD'3; etc. Lastly if ro, Ti, r^, — denote the Discontinued on the card system, Tq only is to be estimated, being about one-half, or rather 0.45 of rj. It is evident that discontin- uances and lapses will occur mostly at the end of policy years, that is, at the middle of calendar years, on the average. We also note that by adding and subtracting |D'„ the middle ordinate Pi — ^D', becomes P,, at the beginning, and P,— D', at the end of the policy year. Consequently Pa is the value at the beginning of the next year; and the Existing Eo were withdrawn (at the end of the calendar year) six moiiths previous, Taking the difference, we have ri = P,— D',— Eo— Pa; r^ = Pa— D'a— E,— P3 ; etc. After transformation of the Amounts given by Company B, in accordance with these three series of equations, investigated for the purpose, the corresponding normal number of Lives or Policies was then obtained conversely from the principle of pro portions employed for Company A. The records of a third Company, denoted as Company C, gave the Deaths and Exposed for both Lives and Amounts, according to Calendar years of exposure, with correction for fractional parts of the year. The age nearest to Jan. 1st had been taken as the Age of Entry. After tlie first Policy on any life, the subsequent policies or surrenders, had been regarded merely as increase or decrease of the first Policy. By aid of further statistics, the stated Loss actually paid, was corrected to the Loss as insured. The lives were assumed to enter the Company at the middle of the initial calendar year, on an average, and in this sense, at their real ages. The results annually wrought out by the Office, had been given in the form of summation tables of the quantities N, M, D, E, which, besides determining the final ratios of mortality, were sufficiently extensive for other purposes. In order to unite with the returns of other Companies, as here presented, it became necessary to resolve the results into their original or annual elements. A single example will illustrate the special process devised for this object. Years of Insurance. Probpble Deaths Ages. 34 35 36 37 38 1 2 8 4 6 136.33 258.86 2 3 0.55 212.97 198.11 179.33 Probable Deaths. 2,283.48 53.56 55.37 57.83 60.04 61.33 28 MORTALITY AND LOSS EXPERIENCE. The horizontal and vertical sums were thus given, and it was required to fill out the series of squares, with such compatible numbers as should make up these sums. Instead of resorting to equations of condition for five year periods, the squares were first filled out approximately, from the Experience of another similar Company. Their vertical sums, when compared with the true sums, gave a multiplier for each column, such that the vertical sums of products agreed with the true sums, shown at the foot of the columns. Next, the same products were added horizontally ; their sums when compared with the actual sums in the left hand column, gave a new set of multipliers, such that the horizontal sums of the new products agreed with those stated in the left- hand column respectively. The same products were next added vertically, to get new multipliers and products as first described, and so on. It proved to be a converging process, which terminated when the last multipliers became virtually 1 ; and the joint conditions of horizontal and of vertical summation were satisfactorily fulfilled. The resulting Probable Losses when divided by the same tabular probabilities previously used as multipliers to determine the Probable Loss, gave the corresponding values of M as required. After this leading determination, different courses were suggested. Approximate values of D could readily be found by applying the known ratios of mortality to the values of M ; and approximate values of E and r could then be derived from the foregoing Equations for M (page 26), by assuming the ratio of E to r to be the same as in the general experience of other similar Companies, at the same age and duration. Each equation would then have only one unknown quantity. And these approximate values could lastly be made to accord with the given sums by slight percentage corrections. But the work being already performed, the present sketch sufficiently indicates the methods of interpolation. The new converging process, which opened the way as above described, could be applied more extensively, as where the sums of the columns had been taken in the two directions vertically and diagonally. SECTION VII. Exposed to Risk. After the original data had been reduced to the three %rms of Existing, Discon- tinued, and Died, shown in Tables I and II, the next process was to fill out the columns of Exposed to Eisk [of Death]. This peculiar term has already been defined under Notation, by the quantnty i3, or M + ^D. To explain the application, — in Table I Male Life, Age at Entry 33 years, for example, the Number of Entrants 40,498 at the head of the first columns, was found by adding the first three sums at the foot of the columns. And the Entrant Amount Insured $104,474,000 was found in like manner. Then by the preceding equations /or i3 (page ^5), half the number of entrants less half of 1,872 the discontinuances of the year 0, gives 19,313 years of exposure in the year 0. And the number of entrants N diminished by the sum of Existing, Discontinued, and Died of the Year 0, gave the surviving entrants ei, on the year 1. And f, diminished by half the discontinued 6,484 of the year 1, gave (i, or 32,927 Exposed to risk during the year 1. After the first year, the process is uniform, as follows. MORTALITY AND LOSS EXPERIENCE. 29 FROM TABLE I. AGE AT ENTEY 33 YEARS. Year of Insukance. Entrants less Existing, Discontinued, and Died. Surviving Entrants. Subtract ^ Disc'd. Exposed TO Risk. 1 2 3 40,498-(2,349 + 1,872 + 108) 36,169-(2,181 + 6,484 + 219) 27,285-(l,776 + 2,845 + 218) 1(40,498) 36,169 27,285 22,446 1(1.872) 1(6,484) 1(2,845) K1.856) 19,313 32,927 25,862| 21,518 In the columns of Amounts, the Exposed to Kisk [of Death] are found in the same manner. Thence, the ratios of mortality, and of discontinuance, or of both combined, can be determined by the preceding formulas. The collective data will be found unusually complete for these, as well as other inquiries. The preceding principles require that, after passing from Deaths to the Ratios of mortality, or to the life table, the current Ages on the plan of " nearest birthday " are to be diminished by \ year, as if changing the Ages from the middle to the beginning of the year of the ratio. Thus 29^ current is change^ to 29 years. The English custom of age "next birthday" would require a similar, final change of one year, reducing the current age 30 to 29, for example, as noted in the first volume of the Institute Experience, page 18. SECTION VIII. Comparative Results. Ratios of Mortality and of Loss to the Year 1874. The collection of the Thirty Life Offices represents an aggregate of about 1,177,000 original policies. But under the rules of reduction, before described, these were adjusted to the net number stated below, comprising above a million of regular observations. It will be seen that the number of female entrants was relatively small ; the total propor- tion being twenty-two males to one female insured. The ages of entry group themselves above and below the mean age of thirty-five, the half of " three-score years and ten." Numbers. Males. Females. Total. Number of Entrants Whole number of Deaths 982,734 44,485 35.23 36.27 44,795 2,058 34.46 34.87 1,027,529 46,543 35.20 36.23 Average Age of Entry for Lives, years " " " " Amounts, years ' Proportions : Existing. Discon- tinued. Died. Total. Of 982,734 Male Entrants 53.7 49.7 53.5 41.8 4.5 4 (^ 100.0 100.0 100.0 " 44,795 Female Entrants " 1,027,529 Male and Female Entrants 42.0 4. 5 30 MORTALITY AND LOSS EXPERIENCE. Dividing now the corrected number " Exposed to Eisk " by the whole number of Entrants, we find at the epoch of the Experience in 1874, the Mean Duration of Policies was 4.36 years. Adding this to the Average Age of Entry 35.23 years, we find that the Lives Insured had attained the Average Age of 39.6 years at the epoch. In the older Experience of the Twenty British Offices, the Mean Duration of Policies at the epoch in 1863 was 9.12 years; the average Age of Entry was 35.3 years; and the average age attained by all the lives insured, was 44.4 years. Resuming the Experience of the American Offices, we have next the following results. MEAN PERCENTAGE OF MORTALITY ACCORDING TO THE YEAR OF INSURANCE. ALL AGES. TO 1874. Ykar OP Insurance. Male Life. Female Life. Year OF Insurance. Deaths Per cent. Claims Per cent. Deaths Per cent. Claims Per cent. 1 2 3 4 5 6 7 8 9 Total, 0.63 0.81 0.92 1.00 1.09 1.13 1.17 1.22 1.20 1.23 1.03 0.63 0.85 1.00 1.06 1.19 1.23 1.27 1.34 1.27 1.25 1.10 0.86 1.08 1.06 1.14 1.16 1.28 1.40 1.33 1.06 1.73 1.16 0.93 1.12 1.09 1.27 1.32 1.43 1.54 1.35 1.16 1.54 1.23 1 2 3 4 6 6 7 8 9 Total. The preceding abstract, from Tables III and IV (B), shows how many die out of 100 Exposed to one year's mortality, the average being above one per cent. In the " Year of Insurance " 0, as before explained, the observations extend, on the average, over half a year from the day of entry, although the percentage is the annual rate. The "Year of Insurance" 1 extends from ^ a year to 1^ years from entry; and so on. The principal effect of the medical selection is more correctly exhibited on Pages 31-33, for Equal Ages of Exposure ; the average rate being generally attained within 2^ years. At the same time, the counter selections of the insured are proceeding by changes of amount, so that the final claims are generally about four per cent, in excess of what would be the loss for equal Policies or amounts insured. — Compare p. 192. The computation of the following summary for Male Life may be thus illustrated. For "Years of Insurance" 0, for example, we add the Deaths and Exposed in Table I for Ages of Exposure 30, 31, 32, 33, 34, 35 Years, which give 89, 90, 95, 108, 99, 94, or a total of 575 Deaths, and in like manner 115,458.5 Exposed to Kisk. These Deaths divided by the Exposed give 0.498 per cent, as stated in the Table on next page. The six middle Ages current, strictly represent the period 29^-35^; the central five years of which are 30-35 years; the fraction being excluded only to facilitate future comparisons. From the true middle age 32^, dropping ^ year leaves the middle initial age 32, as before specified. It will be seen that the ages bordering on the five-year periods, as 25, 30, etc. are added twice. The ratios of Death Claims were found in the same manner, subject in case of future graduation to a smaU correction shown in Section X, 3. MORTALITY AND LOSS EXPERIENCE. 31 For the STimmary of Female Life, a parallel method was employed ; the data being first added in regular five-year periods, such as 30, 31, 32, 33, 34; then 35, 36,.37, 38, 39, etc., or 29^344; 344-39-|; etc. To advance these periods |- year, that is, to 30-35 ; 35-40, etc., one-tenth of the increase from one quinquennial to the next, was added to the former ratio. Thus for the periods (0) 24|-2f)|^, 29^34|^, the ratios of Deaths per cent, are 27-^3956 and 36-^4258 or .6825 and .8455. One-tenth of their diflference .0163 added to the former gives .6988 or .699 as stated in the abstract on next page under 25-30 at the head of the column. As a method of adjustment, there appears little or no ground of preference between this and the former operation for Male Life, with a fair measure of regularity in the data. MALE LIFE. DEATHS AND CLAIMS PER CENT. TO 1874. Ages of Exposure. OF Insurance. 15-20 20- 25 • 25- 30 30-35 1 Deaths CLAms Deaths Claims Deaths Claims Deaths Claims PER CENT. PER CENT. PER cent. per cent. PER CENT. PER CENT. PER CENT. PER CENT. .735 .597 .571 .446 .481 .442 .498 • .485 1 .673 .625 .677 .620 .607 .603 .617 .618 2 .802 .746 .782 .846 .691 .735 .691 .764 3 .634 .544 .685 .626 .777 .737 .744 .758 4 .590 .776 .740 .719 .811 • .868 .833 .854 5 .400 .421 .899 1.038 .796 .885 .827 .914 6 1.388 .305 1.223 .650 .879 .905 .801 .882 7 2.116 1.227 1.003 1.209 .860 .879 .837 .827 8 .306 .548 .812 .897 .799 .856 9 1.022 .337 .561 .459 .801 .841 4 .709 .636 .672 .620 .647 .645 .666 .683 5-9 .725 .432 .967 .910 .823 .879 .817 .879 10-14 1.890 1.572 1.295 .850 1.037 1.194 15-19 .863 .386 1.243 1.634 20-24 2.778 1.105 Total, .709 .627 .689 .634 .671 .673 .705 .782 35-40 40-45 45- -50 50- 55 .520 .528 .663 .666 .840 .802 1.186 1.354 1 .709 .831 .845 .839 1.034 1.031 1.332 1.309 2 .785 .871 .931 .942 1.126 1.107 1.442 1.495 3 .852 .884 .958 .950 1.096 1.118 1.403 1.525 4 .920 1.000 1.019 1.108 1.184 1.252 1.469 1.512 5 .865 .956 1.045 1.124 1.185 1.206 1.392 1.424 6 .914 .963 1.028 1.071 1.145 1.241 1.557 1.489 7 .946 1.003 1.007 1.038 1.127 1.291 1.506 1.611 8 .789 .843 1.013 1.020 1.099 1.102 1.525 1.517 9 .889 .918 .900 .961 1.036 1.014 1.568 1.398 0-^ .758 .829 .892 .905 1.069 1.074 1.379 1.442 5—9 .883 .948 1.013 1.061 1.133 1.194 1.497 1.488 10-14 .933 .936 1.007 1.015 1.185 1.309 1.423 1.467 15-19 .929 1.006 .903 1.031 1.134 1.212 1.379 1.453 20-24 2.201 2.010 1.135 1.566 1.019 1.067 1.460 1.567 25-29 3.150 6.349 1.393 2.534 1.838 2.653 Total, .802 .866 .938 .963 1.102 1.138 1.426 1.469 33 MORTALITY AND LOSS EXPERIENCE. MALE LIFE. DEATHS AND CLAIMS PER CENT. TO 1874. Ages op ] Exposure. Years OF 55- 60 60-65 65-70 70-75 TVRTTR A TTPK Deaths Claims Deaths Claims Deaths Claims Deaths Claims FEB CENT. PEB CENT. PER CENT. PER CENT. PER CENT. PER CENT. PER CENT. PER CENT. 1.550 1.739 2.425 2.429 3.200 5.010 4.138 3.003 1 1.851 1.806 2.721 2.703 3.142 2.713 4.902 5.066 2 1.887 2.071 2.644 2.423 3.460 3.821 3.663 4.484 3 2.021 2.113 2.867 2.886 3.610 4.169 3.338 2.193 4 2.158 2.018 2.913 3.164 3.689 3.899 3.464 3.136 5 2.077 2.117 3.008 2.820 3.811 3.797 6.452 4.626 6 1.992 2.121 2.761 3.402 3.706 4.341 6.656 6.849 7 2.079 2.121 2.903 3.153 4.631 4.285 4.928 4.937 8 2.121 2.179 2.585 2.727 3.819 3.339 5.405 6.438 9 2.074 2.060 2.724 3.074 3.537 4.140 7.355 7.901 0^ 1.926 1.968 2.g56 2.767 3.482 3.809 3.726 3.437 5—9 2.064 2.122. 2.820 3.043 3.921 3.998 6.109 6.027 10-14 1.824 1.920 2.619 2.583 3.794 3.819 6.014 5.531 15-19 1.875 2.037 2.681 3.005 4.211 4.624 5.037 4.934 20-24 1.790 2.053 2.775 3.214 4.523 5.494 6.131 6.396 25-29 1.727 1.708 2.871 2.940 3.727 4.098 6.061 6.145 TotaJ, 1.942 2.017 2.756 2.899 3.934 4.236 5.631 ' 5.550 0—4 75-80 80-85 85- -90 All Ages. 5.096 5.150 14.286 9.524 .881 .937 5—9 8.430 8.066 3.636 1.550 1.179 1.267 10-14 9.707 8.808 9.677 9.160 1.171 1.433 15-19 7.942 7.119 11.834 13.598 4.445 5.333 1.679 1.798 20-24 8.234 6.943 12.012 9.825 21.239 25.478 2.339 2.582 25-29 10.836 9.848 11.282 9.773 20.779 18.065 2.943 3.173 Total, 8.483 7.724 11.332 10.342 16.667 17.757 1.028 1.096 FEMALE LIFE. DEATHS AND CLAIMS PER CENT. TO 1874. Ages of Exposure. Years 4 OP 15- 20 20-25 25- -30 30-35 Instirance. Deaths Claims Deaths Claims Deaths Claims Deaths Claims PEB cent. PEB CENT. PEB CENT. PER CENT. PER CENT. PER CENT. PEB CENT. PER CENT. .808 .784 1.062 1.042 .699 .621 .840 1.104 1 .585 .463 .952 .944 1.056 1.004 1.076 1.148 2 .600 .396 1.049 1.031 1.019 1.091 .946 .814 3 1.415 2.307 1.122 1.383 1.023 1.113 1.139 1.292 4 .993 .526 1.047 1.561 1.018 1.085 5 .380 .455 1.175 1.054 1.010 .882 6 1.894 .924 1.065 1.231 1.372 1.491 7 1.020 1.193 1.184 .959 1.023 .713 8 1.916 1.674 2.083 6.180 1.250 1.206 9 . . . •• 2.071 1.388 0—4 .714 .707 1.024 1.017 .971 1.015 1.009 1.080 5—9 .991 .764 1.180 1.419 1.199 1.081 10-14 .949 2.110 1.013 .948 15-19 S.310 3.124 Total, 1 .676 .671 1.108 .999 .997 1.065 1.052 1.084 MORTALITY AND LOSS EXPERIENCE. 33 FEMALE LIFE. DEATHS AND CLAIMS PER CENT. TO 1874 Years 1 Ages of Exposure. 1 OP 35- 40 40-45 45- 50 50- 55 Insxjbance. Deaths Claims Deaths Claims Deaths Claims Deaths Claims PER CENT. PER CENT. PER CENT. PER CENT. PER CENT. PER CENT. PER cent. PER CENT. .795 .946 .838 .955 .975 1.013 1.193 .842 1 1.157 1.322 1.101 1.007 .899 .763 1.064 1.545 2 .917 1.080 1.009 1.027 .806 .724 1.496 1.570 3 .916 .785 1.108 1.159 1.239 1.399 1.179 1.470 4 .789 .780 1.145 1.197 1.124 1.563 1.550 1.524 5 1.086 1.145 1.090 1.390 1.393 1.533 1.804 1.943 6 1.264 1.383 1.097 1.027 1.287 1.656 1.601 1.627 7 .937 1.078 .909 1.118 1.497 1.332 1.427 1.470 8 1.509 1.531 .410 .479 .351 .233 .954 .976 9 .913 .868 1.456 1.646 1.257 .963 1.941 1.481 0—4 .946 1.030 1.051 1.064 .997 1.052 1.295 1.441 5-9 1.142 l;22l 1.000 1.147 1.239 1.312 1.568 1.594 10-14 .843 .839 1.294 1.435 1.027 .994 1.189 1.153 15-19 2.440 3.128 1.031 1.252 .830 .766 1.299 1.509 20-24 .940 .678 1.116 .986 .826 .590 25-29 2.564 1.786 Total, 1.002 1.082 1.049 1.104 1.059 1.103 1.341 i.432 55- -60 60-65 65-70 70-75 1.388 1.932 28.571 18.182 1 1.492 1.964 1.957 1.541 4.032 2.899 2 1.862 2.149 3.434 3.301 4.496 6.344 6.250 5.000 3 1.351 1.962 2.125 2.438 4.445 8.375 4 1.898 1.743 3.173 4.035 5.489 8.595 3.279 i.482 5 2.067 2.521 3.479 5.128 3.509 4.301 6 1.936 2.411 1.651 2.953 5.368 4.989 10.127 14.595 7 2.867 2.842 3.042 3.603 6.017 0.049 2.899 5.042 8 1.145 0.878 2.406 1.989 3.429 1.250 9 2.035 1.242 2.160 2.316 7.429 8.154 17.168 10.222 0—4 1.601 1.950 2.418 2.590 4.211 6.038 4.407 8.100 5—9 2.057 2.215 2.650 3.585 4.799 4.722 5.458 6.171 10-14 1.621 1.819 1.847 2.100 4.533 4.643 5.262 4.905 15-19 2.372 1.877 1.207 2.561 2.367 2.124 3.112 6.316 20-24 2.251 2.093 3.070 2.758 3.292 4.466 5.377 7.292 25-29 » 1.695 1.064 Total, 1.831 2.015 2.385 2.822 4.039 4.937 4.783 5.617 0-4 75 -80 80-85 85- -90 All J ^.ges. 14.909 14.107 40.000 28.571 1.060 1.135 5—9 8.207 8.523 23.000 16.250 50.000 50.000 1.329 1.423 10-14 25.352 17.391 1.373 1.439 15-19 6.203 8.516 9.091 24.000 1.545 1.876 20-24 21.536 23.482 31.666 35.952 16.667 16.667 2.179 2.270 25-29 0.717 0.431 Total, 14.587 14.428 20.128 22.013 16.282 21.363 1.158 1.234 34 MORTALITY AND LOSS EXPERIENCE. Such are some of the effects in the aggregate. But a more exact impression will be given by glancing down the columns of these tables, and noting the incessant fluctuations of the rates. For instance, in the periods 50-55, 65-70 the claims per cent, of the first year have exceeded those of the second year for Male Life ; and in Female Life, the same feature prevails as often as its opposite, for all ages. It should be noted that in each vertical column, the percentages refer to different groups of persons insured. The percentages for the same group proceed forward and downward in a diagonal direction, for five-year periods. Thus for the same class of females entering at the Ages 25-30, the successive Claims per cent, in five-year periods are given as 1.015, 1.081, 0.839, 1.252, 0.986, It may be proper to observe that in the national census, the mortality of females, in the great mass, is less than that of the male population. In life insurance statistics, the same feature would undoubtedly prevail, if a fair representation of the female class in the community were insured. But in practice, the class that actually insure have so many unhealthy Uvea, especially in the period of maternity, that the total claims per cent, of females (1.23) have very considerably exceeded the rate (1.10) of insured males. The principal causes of this difference will be developed hereafter in the Medical Statistics. Table XIV (B) indicates especially the risk incident to the birth 6f the first child, which is well known to be far greater than at any subsequent delivery, with compensations in the future conditions of life. Compare also results from the Census of Scotland in the Journal of the Institute of Actuaries, Vol. 22, page 233. COMPARISON OF DEATHS PER CENT. FROM SEVERAL TABLES OF MORTALITY. 30 Amekican Offices. 20 British Offices. American Life, 1858. Carlisle Table. Mai.es. Females. DlFF. Males. Fexales. DrPF. M. AND F. M. AND F. 15^-20 0.71 0.68 + .03 0.47 ' 0.76 -.29 1.31 0.68 20-25 0.69 1.10 -.41 0.69 0.85 -.16 1.07 0.70 25-30 0.67 1.00 -.33 0.69 1.18 -.49 0.78 0.82 30-35 0.71 1.05 -.34 0.82 1.13 -.31 1.04 1.01 35-40 0.80 1.00 -.20 0.95 1.21 -.26 0.98 1.09 40-45 0.94 1.05 -.11 1.07 1.28 -.21 0.83 1.41 45-50 1.10 1.06 + .04 1.36 1.39 -.03 1.13 1.44 50-55 1.43 1.34 + .09 1.74 1.57 + .17 1.55 1.52 55-60 1.94 1.83 + .11 2.40 2.02 + .38 1.87 2.20 (50-65 2.76 2.39 + .37 3.48 2.86 + .62 4.34 3.68 65-70 3.93 4.04 -.11 5.02 4.37 + .65 4.15 4.45 70-75 5.63 4.78 + .85 7.33 6.84 + .49 5.19 6.97 75-80 8.48 14.59 -6.11 11.00 10.66 + .34 8.48 10.54 80-85 11.33 20.13 -8.80 16.52 12.51 + 4.01 .... 13.86 85-90 16.67 16.28 + .39 22.35 22.82 -.47 .... 19.93 90-95 .... .... 32.73 22.68 + 10.05 28.61 Deaths, 44,485 2,058 20,521 3,3a5 750 1,840 Entrants, 982,734 44,795 130,243 16,604 19,725 .... This Table represents the percentages from original observations simply. Gradu- ation would require a small correction described in Section X, 3. In the columns of the Thirty Ofiices, under the age of forty-five, the mortality of females will be seen to exceed that of males by at least one-third part, for a considerable extent ; from the age of forty-five to sixty-five, the order is reversed, and the mortality of males exceeds that MORTALITY AND LOSS EXPERIENCE. 35 of females by a much smaller difference relatively. Above sixty-five, the female experience is small Id numbers, and fluctuating in value. Yet in the larger experience of British females from sixty-five* to eighty, the excess of male mortality is still contin- ued, though reversed afterward. The last column but one, is derived from the published Eeport of the Experience of the Mutual Life Insurance Company of New York to Feb. 1, 1858, pages 10, 11. From its middle ages, between thirty and sixty, Mr. Sheppard Homans, the Company's Actuary, constructed the common American Experience Table of 1858, with some modifications. Passing now to the last colifmn, we note that all the percentages of the Carlisle Table exceed those of male life in the Thirty Offices Experience. And the percentages of the Twenty British Offices, male as well as female, with one or two exceptions at the beginning, show a similar excess over that of American insured males. SECTION IX. Continuation of the Experience of 1874 to Final Series. The record of a Life Office noted upon the card returns will now he regarded as an inventory of the state of its business up to that particular time ; from which the average for a longer series of years remains to be determined. The applications of the Life Table to the current business of Insurance, whether to the surrender or change of policies, or to reserves, all presuppose that such Life- Table is based on terminated experience. In the analogous problem of the theory of probabilities, where the game is stopped prematurely, it is proved that the stakes should be divided among the players, in proportion to their probabilities of winning. Although the game stops, the determination of the several portions evidently depends on a knowledge of the chances, or law of the game when played through. In the present case, the returns of 1874 comprise 549,418 Existing policies, of which the record closes before their future experience is determined. The column of Existing in Tables I and II now represents only first experiences, cut into dissimilar portions by the end of the 1st, the 2d, and other years after the medical examination, according to the year chosen for making out the cards or returns. Had the canvass of the office registers been deferred fifteen or twenty years, the most of the policies or lives would evidently have passed to their termination, and would have been returned as Discontinued or Died. And this mature experience, including the present returns, would have furnished the genuine elements for constructing the Life Table. But not to wait for such supplementary experience by natural termination, we have found that it can be approximately supplied by carrying forward tlie Existing to their proportional terminations in the cohimns of Discontinued and Died ; and the same for Amounts Insured and Claims. The present experience down to 1874 will give the requisite multipliers, and the products so found, may be conveniently termed Final Series. Let us here investigate the weights or multipliers for this purpose. On page 25 is a formula assigning the probability p" of surviving one year's exposure to both death and discontinuance. Hence from an.y age of Entry in Tables I and II we can determine the surviving entrants, N, N;j,", Njo,";?/', .... of Final Series denoted by N, e'l, e'a, .... as they would be, if the present Existing, all proceeded to their termina- tions in Discontinuance or Death. And the annual differences, N— e,', e/— e^', etc. are 36 MORTALITY AND LOSS EXPERIENCE. evidently multipliers of the corresponding sums of discontinued and died on the same line of Table I. Hence their respective ratios will give the required weights or multi- pliers (w) of Tables III and IV. Thus in the accompanying illustration, the first columns of Existing, Discon- tinued and Claims are simply the sums of the corresponding Amounts from Table I in Thousands of Dollars under Ages of Entry 36 and 37, (mean age ^^). The product of these (except Existing), by the Multiplier opposite in the last column, gives the Discontinued Amounts and Claims, (which now include the Existing carried forward), under Final Series. AMOUNTS FROM TABLE I. AGES AT ENTRY. 36 AND 37 TEARS. Years OF Insur- Experience ENDING 1874. Final Series. Existing Discon- Entrants Entrants DlSCON- Death Exposed to Multiplier ance. (R). tinued. Claims. (^). {e). TINTJED. Claims. Risk. {w). 11,763 7,443 457 196,462 196,462 7,443 457 94,509.5 1.0000 1 10,058 37,361 1,743 176,799 188,563 39,181 1,859 173,971.5 1.0665 2 8,433 15,564 1,350 137,637 157,523 17,813 1,431 148,615.5 1.1445 3 7,693 • 10,655 959 113,390 138,378 13,109 1,180 131,733.5 1.3304 4 8,705 7,753 916 93,084 123,989 10,337 1,330 118,835.5 1.3330 5 9,533 5,665 923 75,710 113,442 8,414 1,369 108,335 1.4852 6 9,639 3,433 665 59,590 103,659 5,914 1,146 99,703 1.7228 7 8,991 3,617 470 45,863 95,599 5,455 980 92,871.5 2.0845 8 7,915 1,508 366 -33,785 89,164 3,980 966 87,174 2.6393 9 5.659 707 217 23,996 84,318 3,481 762 82,977.5 3.5098 10 3,659 496 179 17,413 80,975 3,307 833 79,831.5 4.6505 11 3,437 330 334 13,079 77.836 1,964 1,333 76,854 5.9515 12 1,107 191 149 10,098 74,539 1,410 1,100 73,834 7.3819 13 853 167 113 8,651 72,029 1,391 941 71,333.5 8.3364 14 937 143 134 7,519 69,697 1,316 1,149 69,039 9.3700 15 730 78 78 6,316 67,232 830 830 66,817 10.645 16 596 57 98 5,430 65.572 688 1,184 65,328 13.076 17 339 44 77 4,679 63,700 599 1,050 63,400 13.615 18 341 37 59 4,219 62,051 544 868 61,779 14.709 19 390 40 55 3,782 60,639 641 881 60,318.5 16.035 20 373 45 49 3,397 59,117 807 879 58,713.5 17.931 21 196 16 53 3,930 57,431 313. 1,037 57,374.5 19.603 22 198 31 74 3,665 56,081 653 1,557 55,755 31.044 23 364 17 56 3,363 53,873 388 1,378 53,678 33.808 24 347 6 69 2,025 53,306 155 1,783 53,138.5 35.781 25 477 3 48 1,603 50,368 94 1,504 50,331 31.363 26 377 30 30 1,075 48,670 903 1,354 48,318.5 45.377 27 345 3 33 648 46,413 315 3,361 46,305.5 71.618 28 160 11 267 .... 29 35 1 96 30 56 1 60 32 1 38 2 103,484 84,439 9,549 The first column of Entrants (e) is readily computed by the formula of page 25, preparatory to obtaining the values of p" by the formula there given. 'P w = V „ _ R + fii r = wr. D' = wD. Here e^ denotes the entrants on the line or space next below that of the existing R and entrants e. Multiplying N or 196,462 in this example by the first value of p" gives MORTALITY AND LOSS EXPERIENCED 188,562, that is, the second value of e' ; and this by the next value of p" gives the next value of e' ; and so on. The entrants e' are here computed through 27 years, omitting the rest on account of the smallness and irregularity of the last data. Now the first diflference of column e' or 7,900 divided by the sum of the first Discontinued and Claims 7,443 + 457 gives 1, the first multiplier in column {w). The second difference of e' or 31,040 divided by 27,361 + 1,743 gives 1.0665 the second multiplier in column {w), and so on. Another Solution. Since Entrants (e) always commence where Existing (E) end, at the beginning of any one year, we have first the proportion ; Entrants : Discon- tinued in the following twelve month : : Existing : proportional Discontinued. The sum of the second and fourth terms gives the Discontinued of Final Series, on the same line of the Table. And the Deaths of Final Series are found in like manner. Adding the third term to the Existing at the beginning of the next year, and then subtracting the two-fourth terms of Discontinued and Deaths just found, will give the entire surviving Existing from previous years, which will be the tliird term of the proportions for the next year, and so on. Exposed to Risk. After applying either solution, the Discontinued and Deaths so found, will give the Exposed to Risk, in Final Series through the usual formula on page 25, by making the Existing (E) to be zero. Or in a different way, the Exposed to Eisk may be found by multiplying the Exposed to Eisk in the first series of 1874, by the same multiplier (w) that is used to change the Discontinued and Died on the same line to Final Series, from Tables I or II. This implies that the percentages of Death and Discontimiance, but not their absolute numbers, are virtually alike on the same line of the series of 1S74, and of Final Series. This agreement is shown in the following examples, computed with five-place logarithms, from Amounts Insured. ELEMENTARY CLAIMS PER CENT. Years Ages ob Entry Ages of Entry Years Ages of Entry Ages op Entry OF Insub- 36 AND 37. 56 AND 57. OF Insur- 36 AND 37. 56 AND 57. ANCE. (1874). Final Series. (1874). FiNAI, Series, ance. (1874). Final Series. (1874). Final Series. 0.484 0.484 1.352 1.352 13 1.319 1.319 7.816 7.817 1 1.069 1.069 1.852 1.850 14 1.665 1.664 4.847 4.850 2 0.963 0.963 2.206 2.206 15 1.243 1.243 3.170 3.175 3 0.896 0.896 2.436 2.436 16 1.814 1.815 4.248 4.249 4 1.037 1.027 3.125 3.123 17 1.653 1.656 11.007 11.012 5 1.265 1.265 2.761 2.760 18 1.405 1.405 0.899 0.899 6 1.149 1.149 4.243 4.240 19 1.462 1.461 9.848 9.852 7 1.055 1.055 3.716 3.718 20 1.496 1.497 10.687 10.674 8 1.108 1.108 3.200 3.198 21 1.814 1.811 5.066 5.072 9 0.918 0.918 3.237 3.235 22 3.793 2.793 19.954 19.914 10 1.043 1.042 4.200 4.301 23 3.380 2.381 7.233 7.339 11 1.735 1.734 5.478 5.488 24 3.413 3.420 4.396 4.412 12 1.490 1.490 5.071 5.071 25 3.997 2.995 13.123 13.115 Weights. The foregoing system of multipliers (w) will be seen to coincide with the method of weights, used in the process of average. It is evident that the weights (w) have the same relation to a life table designed for questions of population, without interest, — that discounted weights {w. v"), where n denotes the " Years of Insurance," have to a life table designed to give the well known Commutation Columns, for different rates of interest. 38 MORTALITY AND LOSS EXPERIENCE. Accordingly, discounted weights (w. v"), assuming v to be 1-^1.04, have been preferred in Tables III and IV, for determining the Claims of Final Series. In con§equence of large irregularities in the statistics of Female Life, only the mean weights, (w) and {to. v") were tabulated by their logarithms in Table IV (0), Deferring further explanations, and the construction of life tables to Part II, we subjoin an example of the preparatory operation for a single Age of Exposure of Male Life; the weights of Table III being changed from (w) to {tv. v"). AGGREGATE CLAIMS PER CENT. EXAMPLE. AGE OF EXPOSURE 24^ TO 25^. Tears op Insurance. Age of Entry. LOG'M Weight. (1874). Final Series. CT.ArMS. Exposed. Ct.atms. Exposed. 1 2 3 4 5 6 7 8 9 10 11 12 13 25 24 23 22 21 20 19 18 17 16 15 14 13 12 0.0000 0.0142 0.0318 0.0510 0.0908 0.1348 0.1938 0.3009 0.4288 0.5670 0.6271 0.6781 0.6875 0.6559 168.0 305.0 257.0 111.0 68.0 52.0 18.0 14.0 6.0 1.0 1.0 1.0 36,205.0 50,185.0 30,560.5 17,851.0 9,781.0 4,374.5 2,079.5 951.0 448.0 227.5 93.0 67.0 8.0 11.5 168.0 315.2 276.5 124.8 83.8 70.9 28.1 28.0 16.1 3.7 4.2 4.8 36,205.0 51,856.0 32,885.0 20,077.0 12,056.0 5,966.2 3,249.4 1,901.5 1,202.5 839.7 394.1 319.3 38.9 52.1 Sum 1,002.0 152,842.5 1,124.1 167,042.7 Claims p er cent 0. 656 0.673 1 SECTION X. Notes on Grraduation. Short Method for the Expectation of Life. G-radnation by Formula and by Divided Differ- ences. Loss relative to Term and other species of Insurances. In further preparation for the construction of life tables in Part II, and other applications, the following notes of investigation are here placed on record. To correct for minor irregularities in the data, the observations may be grouped in five or ten-year periods. And from these five or ten-year sums, the annual values are again derived in a more regular and continuous series, by the process of graduation. This may be done in several ways. 1. Tlie Graphic Method. On a base line representing years, or one of the two dimensions, set off distances corresponding to the given periods of age. Ea<;h of these distances is made the base of a rectangle, whose altitnde is the average obtained by dividing the respective group of observations by its number of years. Having plotted this series of rectangles, whose upper contour forms a broken line, draw with a free MORTALITY AND LOSS EXPERIENCE. 39 hand, a mean curve to supersede the broken line. This curved line must be drawn as little curved as the conditions admit of, and never change its direction abruptly. It will generally cut off a triangular space from one corner of each rectangle, and by compensation take from without, an equal space above the next corner of the rectangle. A preliminary curve is often drawn in pencil, and then successively adjusted by measuring, till the spaces exterior and interior to each rectangle are made equal. Finally, the ordinates of the curve measured year by year, give the graduated series required. The living and the deaths for the Carlisle Table were graduated separately by this method, as described in Milne on Annuities and Insurances, Vol. I, page 100, (London, 1815). In respect to the vital statistics of the city of Carlisle and other places, he observes, it is very desirable that the exact numbers be given monthly for the first three months after birth; quarterly for the remaining three quarters of the first year of age ; and after that, separately for the 2d, 3d, 4th and 5th years of age. And then intervals of five years to the extremity of life may do very well. And if the number of inhabitants be very considerable, intervals of ten years may be employed after the age of twenty or thirty. 2. Short Method. When life tables have once been formed by the graphic or other correct method, they may furnish corrections to shorten future applications. From the three life tables constructed by Milne, Vol. II, pages 404, 405, 564, 534, 5G6, we have derived for future reference, the followii;g method for passing directly from the original data, to an outline of the Expectations of Life. And the same process might easily be adapted to Life Annuities. At the age of 80 years, the Expectation of Life is generally 5 years, more or less ; and the deviations of the Expectations, and the Logarithms of the probabilities of life, from their mean values, appear very nearly proportional to each other, in different Tables. On this clue, are founded the equations below, which give a close approximation to the Relative and to the True Expectation at the age of 80. From the Relative Expectation at 80, we derive the Relative values at 70, 60, 50 .... 5 years, to which a small correction is to be added to give the true Expectations.- EXAMPLE FROM THE ORIGINAL DATA OF THE CARLISLE TABLE. Pebiod n. Living L. Deaths D. -L-JD Age. Expectation. DiFPEBENCK. Cob. Calc'd. True. Cabl. S. M. S. F. 0- 5 9,868.5 812 1.816225 38.72 5- 10 8,703 89 1.977795 5 51.26 51.25 -.01 -.02 + .02 -.04 10- 15 6,853.5 84 1.989230 10 48.85 48.82 -.03 + .01 + .01 -.01 15- 20 6,47l 44 1.985235 15 45.02 45.00 -.02 CO + .02 20- 30 12,730.5 96 1.96725 20 41.49 41.46 -.03 00 + .0;^ : 30- 40 8,406 89 1.95402 30 34.35 34.34 -.01 -.01 + .01 40- 50 8,223 118 1.93770 40 27.63 27.61 -.02 00 + .03 50- 60 5,638.5 103 1.92067 50 21.13 21.11 -.02 -.01 + .02 ikO 60- 70 4,194 173 1.82083 60 14.37 14.34 -.03 + .01 + .03 ^c 70- 80 1,831.3 152 1.63936 70 9.22 9.18 -.04 + .05 00 iVC 80- 90 558 98 1.23529 80 5.46 5.51 + .05 + .01 -.05 c 90-100 94.5 28 2.70366 90 3.28 .... 100-105 18 4 100 .... 2.28 .... 73,593 1,840 Xp\ 40 MORTALITY AND LOSS EXPERIENCE. The third column shows the Deaths in the city of Carlisle during nine years ; and the second, the corresponding Population exposed to risk, as adjusted from two cen- suses. The third column is derived from these by the formula at the head of the column, where L and D correspond to the period n, that is, 5 or 10 years, on the same Hne. It expresses the common logarithm of an approximate probability of surviving the period, as noted at the foot of the column. Let s denote the sum of the Logarithms of the three given probabilities from the age of 50 to 80. In the present example s is L38086 or —0.61914. Also let e'„ e^, denote the Eelative and the True Expectation at the age 80, as defined by the following equations : e'so = 10.844 + 10.3155, e^ — 7.833 4-3.833s. The multipliers of C to give the correction. Cor. are shown in the lower part of the last column of the accompanying Table. From e' at 80, the decennial values of e are derived one from another successively down to the age of 20, by the relative formula, e'_,o = 5+^'V^ + 5). e^ = e', + Cor. C ^ ^go — ^'so- Under 20 years of age, the formula for five-year intervals will be eU = 2.5+/(e'. + 2.5). The probability y or p^ is obtained at once f fom the fourth column of the Table, corresponding to the given interval of age. The first equations applied to the Carlisle data give e^ = 4.46 and ego = 5.46 ; whence their difference C = 1.00. And the cor- rections to be added to e' are known from the last column of the Table. From the age of 80 down to 5, the Expectations calculated by this simple process, differ but slightly from the standard results of the graphic method, as shown in the last column but three. And the next two columns headed S. M. and S. F. derived from the data and Tables of Swedish Males and Swedish Females found in Milne's treatise, show even smaller residuals. 3. Correction. A plausible but erroneous assumption is occasionally made, that when the sum of the Deaths from 60 to 65 years of age, for example, is divided by the sum of the Exposed to Risk at the same ages, the resulting ratio of mortality will be that of the middle age, 62-63. In respect to the values of /, so found in the life table, a correction will be required, which is additive up to about the age of 72, and sub- tractive after that age. Mr. Woolhouse finds the required correction to be the central second difference plus one-fifth of the fourth difference (of the five values of I in the period), to be applied with the sign just described. In the same communication will be found Wool house's Method of Final Adjustment, after a previous graduation. {Journal, Vol. XXI, p. 58, and Vol. XIII, p. 98.) 4- Graduation hy Formula. Among various formulas presented for choice, let us here employ a modified geometric progression, with three constants to be determined from the data. For example, the observations of three five-year periods of age A, B, C are given, to be resolved into the series for single years. Let u^ denote the number for the age x ; then for annual and for five-year periods : w, = a + 5(f, Wo = a4-J> Mi = a + 5cS u^:=a-\-hc^, . . ., K — ha + hs, B = 5a + Jsc*, C = 5« + *sc"*; MORTALITY AND LOSS EXPERIENCE. 41 s = l + c+c' + c'+c* = ^— ^, c' = g3^, ^.-R ^-^ (B-A)(g-1). Wg = « + 5c^ Mj = a + Jc', Wg •= « + ^c^- Thus from A, B, C may be found u^, w„ iig, the three middle terms of B. From B, 0, D may be similarly found the three middle terms of C, and so on. Then u^, Wio the two omitted terms between these triplets can be supplied by divided differences, as presently described. And the six other initial or final terms of the whole series, can be regularly computed by the local process above. In case the sum of any five computed terms differ from the original given sum, one-fifth of the excess can be apportioned to each term. Also instead of five-year periods, the same solution applied to ten-year periods will give five sums of two terms each, which can afterwards be separated into single terms, by applying series. And instead of interpolating Deaths D, and Living L, separately, it may be preferable to take the combinations L — ^D, and L + ^D, for greater regularity. Conversely, it will sometimes be required to find x corresponding to a value of u or w,, intermediate to Uq, Ui, u^, noted above. In this case, logarithms can be applied to either of the results of elimination below, involving x. After solution, if the suffixes of u, instead of 0, 1, 2, are changed to n, n-\-h, n-\-2h, for instance, then x should be changed to n+xh. > , = ^^^11^', c' = l + '^^^^{c-l). Ui—Uq U\ — Uq w.-Wo~^ 1.2 y^-^)+ 7x7273 (<^-l)+-... 5. Interpolation hy Divided Differences. It has just been shown in 3. how far the divided sum or average of five terms differs from the middle term. But to sub- stitute the divided difference of the extremes at the middle point, for the middle difference, is much more accurate, since it differs from the truth by only half the central third difference. Although often insensible, yet provision will be made for correcting even this slight residual, whenever it appears. Embarrassment is sometimes experienced in bringing together portions of the life table interpolated separately, so that the differences at the junctions shall be free from break or abrupt changes. The new artifice here suggested, is that of substituting divided differences at the middle points. For one of the simpler applications, let us first suppose that two terms d, e, interpolated by different formulas are to be reconstructed so as to present a regular continuity. Let the series be ..\hc{d,e)fgh.. Zy = (/_.)_ (/=^ +5^ + ^^]. d-=c^yV—^, e=d+y+^—^, /=e+y+_^. Thus, at first canceling the terms d, e, which include the junction, we find a slight correction y, such that the three divided differences each corrected by y, shall make up the proper quantity / — c. Then d, e are computed ; also /, to prove the operation, by coinciding with the previous value of/, as above indicated. 42 MORTALITY AND LOSS EXPERIENCE. For the reconstruction of four terms /, g, h, i by five differences, let the adjacent terms he ab c d e {f g, h i) j Tclm n 5y=U-e)— j—a h—h l—c m—-d n—e j—a jy . , 1c — h , , l—c , f=e+y+'^, g=f+y + -Q-, h=g+y + -;^,eU;. The same principle might evidently be extended to the omission and re-computation of second or third differences; so that it offers facilities of unusual precision and importance. 6. Loss Experience on different Species of Policies. An inspection of the ratios below, may give the most correct impression. The experience of Term Policies is fluc- tuating, and appears generally dependent on particular circumstances in connection with other classes of Insurance. Besides ten Companies classified in regard to Species, the statistics of six Companies in the general collection, were classified in respect to the Amount of Policy. A difference of opinion exists in respect to the value of such minor classifications, which might be indefinitely extended to Paid-up Life, Ten Payment Life, Paid-up Endowment Policies, and others. LOSS ON DIFFERENT SPECIES OF POLICIES. Number op Companies. LrPK Policies. Endowment Insubancbs. Term Policies. Pbobable L. Actual L. 2 Probable L. Actual L. 6 Probable L. Actual L. d 3 2 2 3 3,661,757 7,076,232 3,611,112 2,997,823 $ 3,799,225 7,068,483 3,766,759 . 3,614,288 100: 104 99 104 120 645,702 1,119,234 302,716 399,790 $ ' 533,013 859,350 264,537 391,172 100: 83 77 87 98 169,204 408,101 55,419 77,285 $ 165.500 448,530 84,067 42,900 lOO: 98 no 152 56 105 10 17,346,924 18,248,761 105 2,467,442 2,048,072 83 710,009 740,997 LOSS RELATIVE TO THE AMOUNT OF POLICY. Spbcibs or POLICIBS. Under $4,000. $4,000 to $10,000. Over ^10,000. Probable L. Actual L. g Probable L. Actual L. © 1 100: 106 73 90 106 Probable L. Actual L. 6 ■< OS life Policies . . . Endowment Ins. Term Policies. . 7 Companies . . $ 1,994.890 258,228 88.717 3,827,440 $ 1,915,905 222,699 101,967 3,672,003 100: 96 86 115 96 $ 2,718,160 261,060 82,126 4,393,274 $ 2,874,100 190,000 74,000 4,679,100 $ 33,440 1,575 4,750 119,157 $ 35,000 20,000 85,000 100: 105 71 CLIMATIC TABLES. 43 DIVISION SECOND. CLIMATIC TABLES. Of thirty life insurance Companies included in the general collection, twenty- seven have returned the locality of the insured, by States or Counties. The results are given in Tables V, VI and VII. In Table VI, the Order and Variability of Ratios is shown in detail, according to twelve Companies whose mean ratios are less than 94, six Companies whose mean ratios were between 94 and 100, and nine Companies whose mean ratios were greater than 100. The last six columns exhibit the Probable and the Actual Loss in Thousands of Dollars ; that is, 870 there signifies $870,000, and so on. For any State or group of adjacent States, the variability may be estimated by comparing the mean with the associated values horizontally. SECTION I. Computations of Probable Loss. For future applications, as well as to illustrate the method of computing the columns of Probable Loss, we here insert the two accompanying Tables of multipliers (A), (B), to three decimals, based on the Thirty Offices Experience. A policy of 155,000 entered at the Age of 37, for example, has the Duration or Years of Insurance 10\, and is in force or Existing at the epoch of Office investigation. The tabular prob- ability of dying in this interval is .101, which multiphed by 5,000 gives $505, the total Probable Loss since the day of Entry. Had the same policy been Discontinued or Died, the other Table (B) would have given the multiplier .095, and the product of .095 by 5,000 or $475 would be the Probable Loss for ten years. The result differs from the former for Existing, by half a year's exposure, according to the general system of reduction described in Division I, page 24. Accordingly, the values (A) are simply the means of the adjacent values (B). Thus the first Table gives Multipliers for tlie sum of Probable Losses for |, 1|, 2^, Z\, years Existing, and the second Table for 0, 1, 2, 3, ... . years till Death or Discontinuance. A modification to give the annual values separately, has been practiced by several Life Offices, that compute the Probable Loss at the end of every business year. For this object, Table XXVI will give the annual multiplier q for every age attained, denoted by x-\-n. A policy of $6,000 exposed for the half or third of a year, is accounted as $3,000 or $2,000 respectively exposed for the whole year; and so for other fractional parts. A joint-life policy terminates at the first death. To shorten the labor, sometimes, the tabular multipliers represent only the two nearest decimals ; and a compensation of positive and negative errors is assumed in the final sum. 44 CLIMATIC TABLES, (A). THIRTY OFFICES EXPERIENCE. Multiplier for Existing to give the whole Probable Loss, Dura- Age of Entry z. DUBA- tion TIOK 10-14 15-19 30-34 25-29 30-34 35-39 40^4 45^9 50-54 55-59 60-64 65- 89 "■''^• 0^ .003 .003 .003 .003 .004 .004 .005 .006 .008 .011 .016 .023 OJ IJ .010 .010 .011 .011 .012 .013 .015 .019 .024 .033 .047 .069 1^ 2| .017 .017 .017 .018 .020 .033 .036 .032 .041 .*056 .079 .116 2i 3^ .033 .034 .025 .026 .037 .031 .037 .045 .059 .080 .114 .16 5 3i ^ .029 .030 .033 .033 .035 .040 .047 .059 .077 .105 .149 .21 6 4i 5* .036 .037 .038 .040 .044 .050 .059 .074 .096 .132 .186 .2C 7 - 5i 6i .043 .044 .045 .048 .052 .059 .070 .089 .117 .160 .325 .31 9 6i 7i .049 .050 .053 .055 .061 .069 .083 .104 .138 .189 .264 .3'; 2 7i 8i .055 .057 .059 .063 .069 .079 .096 .121 .159 .219 .305 .4S -5 8i 91 .063 .064 .066 .071' .079 .090 .109 .138 .183 .350 .347 .4'3 7 9^ 10^ .068 .070 .074 .079 .088 .101 .122 .156 .207 .283 .389 .52 9 10^ IH .074 .077 .081 .087 .097 .113 .137 .175 .232 .317 .433 .56 so 11 J 12i .081 .084 .088 .095 .106 .124 .152 .194 .259 .352 .477 .6? 12^ 13i .087 .091 .096 .103 .116 .136 .167 .215 .286 .388 .521 .e-; 7 13i i4 .094 .098 .103 .112 .126 .149 .184 .237 .315 .425 .565 .72 2 m 15^ .100 .104 • .110 .121 .137 .162 .201 .259 .345 .462 .608 .76 .5 15^ 16J .107 .111 .118 .130 .148 .175 .218 .283 .376 .500 .650 .8C >4 16i ' 17^ .113 .118 .136 .139 .159 .189 .337 .308 .408 .539 .692 .88 (8 17i 1 is! .119 .136 .134 .148 .170 .205 .356 .333 .440 .578 .731 .8-; '0 18i m .137 .133 .143 .158 .183 .220 .277 .360 .474 .616 .769 .8J >8 19^ 20^ .133 .140 .151 .168 .195 .236 .298 .388 .509 .654 .804 .92 Jl 20i 21i .139 .148 .159 .178 .208 .253 .321 .417 .544 .692 .837 .9^ tl 2U 22 i .147 .155 .169 .189 .221 .271 .344 .446 .579 .728 .866 M »7 22^ 23 i .154 .163 .178 .200 .336 .289 .369 .477 .614 .763 .892 .9'- ro 23* m .161 .171 .187 .212 .351 .309 .394 .509 .650 .796 .915 M ^1 24i 25i .168 .1-79 .196 .224 .366 .330 .420 .541 .685 .827 .934 .96 58 25i 26| .175 .187 .307 .237 .383 .351 .447 .574 .719 .856 .951 .9{ )3 26i 27i .183 .196 .317 .350 .300 .374 .476 .607 .753 .882 .964 .9{ )6 27i 28^ .190 .205 .328 .264 .318 .397 .505 .640 .784 .905 .975 .9t )8 28i 29i .198 .214 .339 .278 .337 .431 .535 .673 .814 .925 .984 .9J )9 291 30^ .305 .223 .351 .293 .356 .446 .565 .705 .843 .942 .989 .9$ )9 30i 31i .313 .233 .363 .309 .377 .473 .596 .737 .868 .957 .994 M )9 31i 32i .333 .343. .376 .336 .398 .499 .627 .768 .893 .969 .997 32i 331 .331 .254 .389 .343 .431 .537 .659 .798 .913 .978 .999 . 33i 34J .240 .265 .303 .361 .444 .555 .690 .826 .932 .986 .999 . 34i 35^ .349 .276 .318 .379 .468 .585 .721 .853 .947 .991 .999 . 35J 36^ .358 .287 .333 .399 .493 .614 .751 .877 .960 .995 .999 . 36i 37i .367 .300 .349 .420 .519 .644 .781 .898 .971 .997 21* 38i .378 .313 .366 .441 .546 .674 .809 .919 .980 .999 . 38i 39^ .288 .326 .383 .463 .573 .704 .835 .936 .987 .999 . 39J 40^ .298 .340 .401 .487 .601 .733 .860 .951 .992 .999 . 40i 41i .310 .355 .430 .513 .630 .763 .883 .963 .995 .999 • -*!* 42^ .322 .370 .440 .537 .658 .790 .904 .973 .997 !2* 43i .335 .386 .461 .562 .687 .817 .923 .982 .999 . 43i 44^ .349 .403 .483 .588 .716 .842 .940 .988 . 44J 10-14 15-19 30-24 25-29 30-34 35-39 40^4 45^9 50-54 55-59 60-64 65- 69 Fo RMULA Multi jlier = -H ^_±\ CLIMATIC TABLES. 45 THIRTY OFFICES' EXPERIENCE. (B). Multiplier for Discontinued and Died to give the whole Probable Loss. DCBA- Age of Entry X. DUBA- TION n. _ TION 10-14 15-19 30-34 35-39 30-34 35-39 40^4 45-49 50-54 55-59 60-64 65-6 9 "• .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .OOC > 1 .006 .007 .007 .007 .008 .008 .010 .013 .016 .022 .031 Mt • 1 2 .013 .013 .014 .014 .016 .017 .030 .035 .033 .044 .063 .09S i 2 3 .030 .020 .031 .033 .023 .036 .031 .038 .050 .068 .096 .14( ) 3 4 .036 .037 .038 .039 .031 .035 .043 .053 .068 .092 .131 .19( ) 4 5 .033 .033 .035 .036 ,039 .045 .053 .066 .086 .118 .167 .34] ^ 5 6 .039 .040 .041 .044 .048 .054 .064 .081 .106 .146 .305 .39( i 6 7 .045 .047 .049 .051 .056 .064 .076 .096 .137 .174 .344 .34{ ) 7 8 .053 .053 .056 .059 .065 .074 .089 .113 .148 .203 .384 .39^ ^ 8 9 .058 .060 .063 .067 .074 .084 .103 .139 .171 .234 .336 .45] L 9 10 .065 .067 .070 .074 .083 .095 .115 .147 .194 .266 .368 .mi 5 10 11 .071 .073 .077 .083 .093 .106 .139 .165 .219 .299 .411 .55^ » 11 12 .077 .080 .084 .091 .101 .118 .144 .184 .245 .334 .455 .60f 5 12 13 .084 .087 .093 .099 .111 .130 .159 .304 .272 .369 .499 .654 I 13 14 .090 .094 .099 .107 .131 .143 .175 .335 .300 .406 .543 .70( ) 14 15 .097 .101 .106 .116 .131 .155 .193 .248 .329 .443 .586 .744 t 15 16 .103 .107 .114 .135- .143 .168 .309 .271 .360 .481 .639 .78f i 16 17 .110 .114 .133 .134 .153 .183 .337 .395 .391 .519 .671 .825 5 17 18 .116 .133 .130 .143 .164 .197 .346 .330 .434 .558 .713 .85{ > 18 19 .133 .139 .138 .153 .176 .313 .366 .346 .457 .597 .751 .88f 5 19 2b .130 .136 .147 .163 .188 .328 .387 .374 .491 .685 .787 .91( ) 20 21 .136 .144 .155 '.173 .301 .344 .309 .403 .526 .673 .831 .93S J 21 22 .143 .151 .164 .183 .314 .363 .333 .431 .561 .710 .853 .941 ) 22 23 .150 .159 .173 .194 .338 .380 .356 .461 .596 .746 .879 .964 t 23 24 .157 .167 .183 .306 .343 .399 .381 .493 .632 .780 .904 .97f i 24 25 .164 .175 .191 .318 .358 .319 .407 .524 .667 .813 .935 .98^ } 25 26 .171 .183 .301 .330 .374 .340 .433 .557 .702 .843 .943 .99] L 26 27 .179 .191 .313 .343 .391 .363 .461 .590 .735 .869 .958 .99{ > 27 28 .186 .300 .333 .356 .308 .385 .490 .623 .768 .894 .970 .99'- ^ 28 29 ,194 .309 .333 .371 .337 .408 .519 .656 .799 .915 .980 .99J ) 29 30 .301 .318 .345 .385 .346 .433 .550 .689 .828 .934 .987 .99J ) 30 31 .309 .338 .357 .301 .366 .459 .580 .721 .856 .950 .993 .991 ) 31 32 .317 .338 .369 .317 .387 .485 .611 .753 .880 .963 .995 .991 ) 32 33 .336 .348 .383 .334 .409 .513 .643 .783 .903 .974 .998 33 34 .335 .359 .396 .351 .433 .541 .674 .812 .923 .982 .999 34 35 .344 .370 .310 .370 .456 .570 .705 .840 .940 .989 .999 35 36 .353 .381 .335 .389 .480 .599 .736 .865 .954 .993 .999 36 37 .363 .393 .340 .409 .506 .639 .766 .888 .966 .996 .999 37 38 .373 .306 .357 .430 .533 .659 .795 .909 .976 .998 38 39 .383 .319 .374 .453 .559 .689 .833 .928 .984 .999 39 40 .393 .333 .393 .475 .587 .718 .848 .944 .990 .999 40 41 .304 .347 .410 .499 .615 .748 .873 .957 .994 .999 41 42 .316 .363 .430 .524 .644 .776 .894 .968 .996 .999 42 43 .338 .378 .450 .549 .673 .804 .914 .978 .998 43 44 .341 .395 .471 .575 .701 .830 .932 .985 .999 .... 44 10-14 15-19 30-24 35-39 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-6 9 % F OmSXTLi V, Mult iplier = -'i Ht 46 CLIMATIC TABLES. The climatic Tables being already in use, are retained in the original form, based on multipliers from the common American Table of 1858. Parallel computations of the total, lead to the following precepts : To change the Probable Loss Am. 1858, in Tables V, VI, or VII, to the standard of— The Thirty American OflBces Table ; « Twenty British " « H**- " Seventeen British " " Deduct ^th part. Add 5^ per cent. Add 8f per cent. The corresponding factors are 0.9667, 1.0519, 1.0837, and their logarithms 1.98530, 0.02198, 0.03492. The columns of Actual Loss of course remain unchanged. By these precepts, a present Probable Loss of $1,204,521 to be represented on the basis of the Thirty Offices Table, is changed to $1,164;370. And the tabular ratio of 100 : 134 becomes 96f ; 134, or 100 : 139 nearly. SECTION II. • Classification of Counties. Misoellaneons Statistics. Preparatory to the classification of Counties in respect to Mortality, the simple division shown on the common Geological maps of the United States may be noted. First, the lowland or Alluvial Counties commencing with the eastern shores of Maryland and Virginia, and following the line of tide-water on the Atlantic and Gulf coasts to the borders of Mexico, and up the valley of the Mississippi to the mouth of the Ohio river. These alluvial Counties are denoted in Table VII by one dot annexed. Between these and the line of 400 feet elevation, are the Middle Counties, of which the tertiary portion are here denoted by three dots, and the few cretaceous by two dots. The remaining or Upland Counties above the, line of 400 feet elevation, according to Prof. Guyot, are here indicated by the absence of dots. The Counties of Indiana admit of separate consideration. Also the present object may be found to require a modifica- tion of these divisions. CLASSIFICATION OP COUNTIES AiiLuviAii Count's. Middle Counties. Upland Counties. Total Counties. States Pbob. Loss. Act. Loss. Prob. Loss. Act. Loss. Pkob. Loss. Act. Loss. Pbob. Loss. Act. Loss. Alabama . . 396,236 544,453 175,977 194,580 523,007 729,014 1,204,521 1,618,048 Arkansas . . 98,449 221,995 125,202 196,084 46,238 80,040 295,413 508,119 Florida 133,347 224,596 28,758 32,550 . . • • .... 181,635 303,852 Georgia 2,117 7,500 524,221 603,540 355,738 257,860 950,653 911,536 Louisiana. . 1,252,180 2,194,255 23,901 30,660 .... . . • . 1,363,520 2,400,915 Mississippi 301,087 552,260 416,636 585,344 13,491 15,660 795,476 1,308,632 N. Carolina 225,053 353,555 390,924 443,633 178,804 201,721 843,815 1,069,409 S. Carolina. 374,487 446,433 192,307 183,198 179,009 227,004 753,060 864,635 Tennesee. . 708,436 1,488.636 181,460 209,380 467,280 544,055 1,383,648 2,262,121 Texas 214,043 511,467 332,901 572,267 184,655 264,238 778,713 1,358,972 Virginia. . . 3,567 13,009 385,727 349,100 141,204 202,750 646,188 673,350 Total 3,709,002 6,558,155 2,778,004 3,398,331 2,089,436 2,522,342 9,196,642 13,279,589 Ratio (Am. 1858), 100:177 100 : 122 100 : 121 100 : 144 By 30 Offices Table, 100 : 1 83 100 : 127 100:125 100 : 149 By 20 Offices Table, HM., 100 : 168 100 : 116 100 : 115 100 : 137 By 1 7 Offices Table, 100 : 1 63 100 : 113 100 : 111 100 : 133 CLIMATIC TABLES. 47 A singular feature is presented in the close agreement of the climatic Katios for Middle and Upland Counties, indicating that these may be united ; or that hereafter, two (AoE&e^ Alluvial and Upland,\i\\\ generally be sufficient. A wide contrast is shown between their characteristic Ratios, 100 : 183 and 100 : 126, by the Thirty Offices Table. It will also be noted that the Upland Counties of Georgia at the southern termination of the Alleghany mountains are exceptionally healthy in the ratio of 100 : 72. In the State of Indiana, the experience of eleven smaller Companies gave a Probable Loss of $563,770 and an Actual Loss of $743,866; Ratio 100 : 132. By the accession of other Companies, the total ratio was materially altered, as shown at the end of Table VIL The less healthy Counties are reported to lie in the north-eastern part of the State, and along the Wabash and Ohio rivers. The following additional data pertaining to the Southern States, are arranged from the statements of particular Companies, more especially to show in the. last column the Age corresponding to the mean Rate of Probable Loss divided by the number Exposed to Risk. The results of the 27 Companies in Table V, and the 3 other Companies, for the whole United States, are subjoined. AGE OP MEAN RATE OP PROBABLE LOSS. COXFANIBS. Deaths OB Loss. Exposed TO Risk. Pbobable Deaths OK Loss. Rate OP Prob- able. Age OI" Rate. Table employed. Pbob. : Act. As 100: .^tna Life (III) 1,366.3 677 243.3 459.8 139 74,270 37,940 17,594 20,937 5,966 966.90 442.96 203.95 249.31 79.94 1.3019 1.1675 1.1592 1.1910 1.3400 46.27 46.26 46.07 4456 46.60 *17 Offices Am. 1858 << It 20 Offices 141 153 119 184 174 St. Louis Mutual Covenant " Continental, N. Y Scottish Amicable 27 Companies, U. S 3 « " 192,049.0 35,961 9,327,400 2,909,037 104,531.5 37,580.7 1.1209 1.2919 45.11 48.69 Am. 1858 .98 85 It may here be noted that the Experience of the ^Etna, Covenant and Continental, also of the 27 Companies and 3 Companies, is stated in thousands of dollars. Three- fourths of the business of the Covenant of St. Louis was located in the State of Missouri. The Experience of the Scottish Amicable here represents only the portion located in the British colonies of the West Indies ; where the climatic ratio by the 20 Offices Table, 100 : 174 so far exceeds the like ratio 100 : 168 before found for Alluvial counties in the Southern States. With the omission of the Continental, N. Y., whose business appears somewhat exceptional, the Age of the mean Rate of Probable Loss relative to exposure, differs little from 46.3 years, in the Southern or tropical experience of the other four Companies. With regard to the comparative Mortality of insured Females, the Southern experience of the St. Louis Mutual, comprising 628 deaths of Males, and 48 of Females, gave the ratio of Probable to Actual Loss (Am. 1858) for Males 1.51, and for Females 1.70; thus indicating greater mortality on the part of Females. In relation to the Mortality iy Policy years, the total experience of the St. Louis 48 CLIMATIC TABLES. Mutual, comprising 1486 deaths, North and South, gave the following ratios of Probable Loss, denoted by 1, to Actual Loss (Am. 1858) : ST. LOUIS MUTUAL. Policy Years. Deaths Ratios 1st. 3d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 10th. Total. 360 330 232 180 146 87 73 40 20 7.. 1486 1.13 1.85 1.55 1.26 1.68 1.21 1.52 1.81 1.34 1.36.. 1.33 SCOTTISH AMICABLE IN THE WEST INDIES. Deaths 9 1.59 10 0.97 13 1.47 12 1.59 14 2.12 6 1.03 After 6th year, 75 (Am. 1858), 1.91 139 1.65 Ratios The mean rate 1.33 of the former Company is reached in the second year; that of the latter, nearly in the first year, but the ratios are irregular from paucity of data. For information upon other portions of the globe, reference may be made especially to the following articles in the Journal of the Institute of Actuaries : On the rates of extra premium for foreign travelling and residence. H. Bailt, Esq. Actuary of the London Assurance Corporation. 77-94, 1869. By Arthur Vol 15, pp. By On the Additional Premium required for Residence in Foreign Climates, the Honorary President, James Meikle, Felloiv, etc. Vol. 19, p. 268, 1876. On the Death Rate among Assured Lives in the West Indies, heing the Experience of the Scottish Amicable Life Assurance Society during Tliirty Years, 184.6-76. By John Stott, late Manager of the Society, Fellow, etc. Vol. 21, p. 153, 1878. In proceeding from the temperate to the torrid zone, it is generally admitted that, " other things being equal, — height, soil, and river drainage, — the mortality will increase as we go South ; and that heat stimulates the vital functions, so that maturity is reached earlier, and decay commences earlier." Climate Bxtra. The net extra to meet the additional risk, is roughly found by dividing the excess of the Actual above the Probable Loss, by the Exposure. Thus among the examples given by the distinguished actuary and manager T, B. Sprague, in the Journal of the Institute, Vol. 19, page 297, the years of Life exposed in India Avere 229, Actual Deaths 11, Probable Deaths, H« 3.499. And (11— 3. 499)^229=. 03276 ; or the Climate Extra is 3.276 per cent, of the sum insured. By a more extensive method, Mr. A. H. Baily now President of the Institute, -in his memoir above quoted, first constructed a life table from the foreign experience, and thence a scale of annual premiums ; the difference between this and the home scale, gave the climate extras for the several ages of exposure. Besides the ordinary extra, which averaged one per cent, in the West Indies, the difference between unacclimated and acclimated lives led the Scottish Amicable to charge an additional extra of one per cent, of the sum insured, during the first three years of foreign residence. On returning to reside in Great Britain, with ordinary health, all extras are omitted. CLIMATIC TAB LBS. 49 . SECTION III. The Life Table graded for different Climates. Let us now consider the application of Table VI to permanent life insurances for different grades of mortality. In a State, for example, where the ratio of Probable to Actual Loss by Table VI corrected to the Thirty Offices Table, has been 100 : 139 in the mean Experience of 27 Companies ; and the ratios were 100 : 112, 100 : 153, and 100 : 161 in the three component groups of Companies, the most experienced actuaries and managers could best determine the single resultant or probable climatic ratio for Office premiums. Such examination would be influenced by the probable extent of new business, the medical and counter selections, prevailing causes of death, — and south of the latitude of 36° 30', also extending north of it, the alluvial Counties distinguished in Table VII. For convenience, the net ratio thus conventionally adopted may be denoted by 100 : lOOp, that is 1 : p. The Thirty Offices collection defines the experience of insured mortality along a wide belt of the temperate zone, the middle line centering nearly on the latitude of the city of New York. The resulting standard. Table XXVI has the advantage of unusually large data, critical accuracy and improved construction. To construct other life Tables, from the small numbers of more distant climates. North or South, would involve great labor and complexity. The difficulties are marvellously reduced by the discovery of a double gradation. That is. Table XXVI being graduated in the direction of the scale of ages, is virtually graded in another direction also, by its commutation columns, which represent life tables for so many different climates, or climatic ratios. And the values for intermediate ratios, can be found by simple interpolation. This improvement was first suggested by the entire agreement of Table XXVI with the well-known law of Gompertz, modified by Makeham, observing that in previous comparisons of the annual rate or (log p), change of the first term (log s) of Make- ham's formula, corresponded with changes of mortality due to local or climatic causes; the other term or power of {q) being nearly constant for a long series of years or ages. Again, from a different source, but on the same principle, the experience of the Scottish Amicable in the West Indies, was stated by the late manager and actuary to be represented by the Northampton table better than any other of the current life tables. Yet our examination indicates, that by a simple change of the rate of interest, the Carlisle Table might be substituted, as shown by the following comparison of life annuities ; and the same might be extended to life premiums, with a simple modification presently described. LIFE ANNUITIES. Age, Years Northampton 4 per cent. Carlisle 5 per cent. .... Northampton 5 per cent, Carlisle 6 per cent 25 35 45' 55 65 15.44 14.04 12.28 10.20 7.76 15.30 14.13 12.65 10.35 7.77 75 4.fi6 4.99 13.57 13.46 12.50 12.57 11.11 11.43 9.38 9.52 7.28 7.28 4.74 4.76 50 . CLIMATIC TABLES. In the first publication of the climatic Tables, it was provisionally assumed that, above the age of twenty or some lower limit, the different grades of mortality [q) are proportional to those of the standard Table. The preceding comparison of annuities as well as other reasons, afterwards suggested to the writer the more convenient and exact hypothesis, to which Mr. McClintock appears to have arrived in a different way, that the prohahilities of survival {p) for differetit grades of mortality, are proportional to those of the standard Life Table. In another expression, the rates of survival and of interest may be commuted. That is, so far as relates to life annuities, an increase of the rate of mortality, may be represented by an increase in the rate of interest; and conversely. For illustration, let us distinguish the Northampton values by an accent, and write the well-known formulas below ; then since annuities are assumed to be equal: Northampton Jf. p>er cent. Carlisle 5 per cent. Relation, a'x = v'p'xO- + «'«+i)- «x = vpx{l + ax+i). v'p'x = vpx. -~, ,1 JJ x+n I^aj+n Consequently -------- — - — = — - — . 1 + ax-^^-j^^-l+ax. D'.-D,-d/d,- -X = lOOo] ^^ - (!-.') [, nx = lOOo] ^^ - (l-v) [. Ordinary Life, Tr'ar = 7ra; + 1000(v'— v). Reserve = l _1±^ = 1 _1 +^£±!L. 1 + ax l + «ar m *•!.'«« Na;+l — -N^ai+a+l Temporary Annuity, a^ = a^ = ^r . ^ * \Jx Premium of Term Insurance, P' = «;'- ^%i^=^!^±^ii, P' = P + 1000(t;'-2;). - JN X — ^ x+n Hence it appears that with very simple modifications, the Northampton values at 4 per cent, can all be derived virtually from the 5 per cent. Carlisle values. In like manner, the values corresponding to different grades of climatic mortality, can be conveniently derived from the columns of the Thirty Offices Table. The age of the mean ratio of mortality has been determined on page 47 to be about 46 years. For tlie present purpose, this determination will supply the place of another element, the mean ratio of probable loss to the number exposed to risk, in the climatic district. In a given locality, let the probable loss as computed by the Thirty Offices Table, be to the actual loss as 1 : p. That is, let p denote the climatic ratio ; and let an accent distinguish the values of v, q, p for the locality. Then by the Relation preceding, and by page 47, v'p'46 = vpu, or v'{l—q't6) = vp^^. ? « = PQw Elimmatmg g «, p = --7- — • Let the value of v' depend constantly on 4 per cent, interest ; that of pt^, q^^ on the Thirty Offices Table. Then if v depends on 4, 4^ or 5 per cent, interest, the values of p thus found will be 1, L405, 1.805 respectively; which may be termed Special Climatic Ratios. CLIMATIC TABLES. 61 For the First Special Ratio 1.405, we have p'x = y~c^P^' -^^'^^ ^^^®» ^^ might construct a special life table, and 4 per cent, commutation columns, which would give the net life premium on 1000, at the age of 30 years for example, 19.50. But the same can be found much more easily by the preceding formula, -n'x = {"^Xf 4^ per cent.) + 1000(v'— v); where -^-——^ = 0.00460066, to be multiplied by 1000. And so for other similar formulas. 1.04 For the Second Special Ratio 1.805, we have p'x = -:r-^px. Also tt'x = (ttx, 5 per cent.) + 1000(v'— v) ; where Y7iA~T7\'K = 0.00915751 to be mul- tiplied by 1000. Therefore for the D, N, S columns according to 4 per cent, interest and the Climatic Ratio 1,405, we have in Table VIII substituted the 4^ per cent, columns of the Thirty OflBces standard. And for the D, N, S columns corresponding to 4 per cent, and the Climatic Eatio 1.805, Table VIII gives the equivalent 5 per cent, columns of the standard. But it is needful to observe, that this substitution could not be extended to the C, M, R columns, without modification. Nor is it necessary, since all the formulas can be expressed in terms of D, N, S, v' and v. In conclusion, the Climatic Ratios given in Tables V, VI and VII, being first changed to the basis of the Thirty Offices Table, the corresponding 4 per cent, premiums and annuities may be found from those of the Special Ratios in Table VIII by proportion or interpolation. And the preceding formulas may be extended to other species of Premiums and Reserves. The relation v'p' = vp also occurs in the treatise on Policy Life Lines, by James Meikle, Edinburgh, 1871. 52 MEDICAL STATISTICS, DIYISION THIED. MEDICAL STATISTICS. Statistics of Diseases or Causes of Death have been returned by twenty-seven of the thirty Companies included in the general collection. These twenty-seven are the same which gave the Climatic Statistics, with one exception. The total number of Deaths from All Causes was 37,624 ; of which 35,442 were Deaths of Males, and 2,182 of Females; also there were 8,919 Deaths additional in the three Companies not specifying diseases. Under experienced medical advice and direction, the results are given in the most approved forms for reference in Tables IX-XVII. The classification of diseases accords nearly with the last plan of Dr. Farr, as recently applied in the Medical Report of the Mutual Life of New York. An examination of medical treatises, including special works on Medical Examina- tion for Life Insurance, like Brinton or Allen, will suggest many inquiries, to which the present Tables will often give an instructive answer. "What diseases increase, and what diminish as age advances? What diseases or causes of death are beyond the cognizance, and what are most easily recognized by medical examination, and at what ages ? What is the distribution of diseases in different parts of the United States ? On these and kindred topics, the statistics give valuable information, and will suggest, in lieu of extended commentary, a ready reference to the Tables themselves. In examining these records, we are reminded of the more numerous unrecorded cases, in which, the innate vital force has reverted from morbid or diseased conditions into the old channels of health. The struggle to rise from a diseased condition, may be short and decisive, terminating in death or in convalescence, or it may be longer prolonged with varying fortunes. The contending forces often appear so equally balanced, that a very little will turn the scale. The struggle between the tendency to health and the tendency to disease, acute or chronic, may last from a day to months, and even years. Thus medical statistics illustrate the nature of the mysterious vital force, by its final developments under varied conditions of age, duration, sex and climate. SECTION I. Description of the Tables. The first of the series. Table IX, is a general Table giving firstly the Number of Deaths of insured males, and similarly of females, by 160 Diseases, in seven classes, commencing with zymotic. The next three columns give the results for two, for five, and for twenty Companies arranged to include about one-third of the whole, or nearly 12,541 in each group. By glancing across the three columns horizontally, the Variability by each disease is ordinarily indicated. For example, the deaths by Typhoid Fever, at the head pf the list, appear to have been quite uniformly distributed. MEDICAL STATISTICS. 63 In the last column, the 37,624 deaths are represented on the percentage scale of 100.00 distributed proportionally to the number of deaths by each disease. The prin- cipal or leading diseases in order, are Typhoid Fever 5.99 per cent, other Fevers 4.35, Consumption 18.31, Apoplexy 4.70, Paralysis and Disease of Brain 4.34, Heart Disease 3.61, Pneumonia or Inflammation of the Lungs 7.68, Accidents and Injuries 7.21 per cent, of all. These and other results may be deemed as correct as can possibly be obtained under existing circumstances. Some allowance is to be made for the inexact- ness of medical science, with gradual changes and transposition of medical terms. Also when one disease is accounted as the producing cause of another, the last or fatal disease is the one noted in these statistics, from Office registers. Table X, XI, XII contain only the absolute numbers of Deaths, which constitute one of the two terms of ratio or comparison in the Tables following, with separate columns for males (A), and Females (B). Tables XIII and XV are a further development of Table X with respect to Duration of Insurance. After the seven general classes, the statistics are subjoined for thirty-nine leading Diseases. The corresponding Exposure of the living is given at the foot of the columns in Table XIII. For example, in the first column, 2,445 Deaths divided by 356,210 years Exposed gives 0.00686, that is, 0.686 Deaths per cent. or 68.6 in 10,000 Exposed to Eisk, as shown at the head of the column. It will be noticed that during the first two or three years of Male Life, the deaths from Zymotic diseases, and from Accidents and Injuries, have generally exceeded the average, thus showing this part of the mortality to be uncontrolled by the Medical examination. And the same may be remarked in Female Life, including Childbirth and Puerperal Diseases. On the other hand the efficient diagnoses of Constitutional and Heart diseases, and, in general, of " All Causes," is very evident from the inspection. Tables XIV and XVI give, according to the Ages of Life or Exposure, in extension from Table XI, the ratio of deaths to the Living, and the ratio to total Deaths. The Exposed to Risk by Ages are inserted at the foot of Table XIV ; and the number of diseases is further reduced in Tables XV (B) and XVI (B) for female life. All these results by Ages of Life, will be found not less valuable than those from the difierent point of view of the Durations of insurance. Finally, Tables XII, and XVII (A), (B), (C), indicate the Climatic Distribution of Diseases in the United States. These figures relate to Actual Deaths, while the Climatic Tables V, VI, and VII are based on Amounts Insured; the one class may supplement the other. In comparing the deaths of 703 Foreign Males, Table XVII (A), with ail equal number of the native-born insured, the greater exemption of foreigners from Zymotic and Constitutional Diseases is counterbalanced by an excess of deaths from Apoplexy (chiefly of Germans in one Company), from Abscess, Hemorrhage etc. of Lungs, and Accidents and Injuries. About two-thirds of this experience, however, occurred in British America, and the remaining numbers are too small for final conclusions. Again, comparing insured Male Deaths with those of the TJ. S. Census of 1870, shown in Table XVII, for equal totals, the proportions dying from three of the seven classes, that is, from Circulatory, Respiratory and Miscellaneous Diseases, are nearly equal. And the decreased mortality of insured lives from Constitutional Diseases is counterbalanced by a marked excess of deaths from Nervous and Digestive Diseases, and a small excess from Zymotic Diseases. In respect to Female life the difierences of distribution evidently follow a different order. Representing the total deaths by 100, in each of the seven groups of States, ve have in Table XVII (B) the relative proportions of insured males dying from thirtj- 54 MEDICAL STATISTICS. seven principal diseases. It is the comparison of a part with the whole, that is, deaths with total deaths, since the living or Exposed to Risk are liere omitted. But the last Table, XVII (C) gives a sanitary survey of the United States, according to the deaths and diseases among 10,000 living insured in each group of States. To arrive at the living, the following process was employed. First, as verified in the mean of two of the largest Life Companies, the ratio of Probable to Actual Claims in Table V was assumed to be the same as the like ratio for Deaths ; and the Age of mean Probable Deaths, to be 46 years, as already indicated, page 47. That is, in any group of States, Probable Deaths Probable Claims Probable Deaths Actual Deaths Actual Deaths Exposed Actual Claims ' Exjyosed Actual Claims Tabular Deaths 946 ', or = q4iX Probable Claims 10,000 Living ' Here the second equation is divided by the first, to give the third. By changing the middle member to a decimal, then multiplying both numerator and 1 the denom- inator by 10,000 we obtain the last member, as shown at the head of Table (C). The value of q or 0.011562 corresponds to the Actual and Probable Claims found in Table V. The rest of the values in Table C were then readily found by multiplying the head of the column into the proportional parts of Table XVII (B). By this technical process, an additional and very important Table is obtained. SECTION II. Statistics of Consiimption Compared. Sanitary Survey of the United States. To illustrate the relation of the statistics to any specified disease, let us take for a first example. Consumption, the most fatal of all. Firstly in Table IX, the deaths 2,283, 2,231, and 2,372 in three groups, prove the deaths from Consumption to have occurred at a nearly uniform rate of frequency in all the Life Insurance Companies. Again, the 18.31 per cent, of total deaths shown in the last column, when increased, as it probably should be, by one-thirteenth of the deaths by Pneumonia (chronic cases), and by deaths registered from Abscess of Lungs, Hemorrhage of Lungs and Disease of Lungs, becomes 20.67 per cent., or more than one-fifth of the whole mortality. Table XIII confirmed by XV, shows the effect of medical selection exhausted and the average mortality of consumptive cases reached in the third year of Duration of the Insurance. Table XIV confirmed by XVI, shows the largest mortahty by con- sumption to occur between the Ages of 20 and 30 years. But after this period, the proportion of consumptive deaths of Males, bears a two-fold aspect. That is, when compared with the number of living exposed (Table XIV), it slowly decreases to a minimum rate between 50 and 60 years, and then gradually rises again, with a heavy mortality all the while, in every period of age. FKOM TABLE XIV (A). Ratio of Consumptive Deaths, to 10,000 Living at each Age. Afire Under 20 12.0 2a-30 22.7 30-40 19.6 40-60 17.4 50-60 15.8 60-70 17.6 70-80 19.i All Ages. 18.6 Ratio MEDICAL STATISTICS. 65 But when compared with the total contemporary deaths (Table XVI), the propor- tion from consumption continually decreases, as the complementary proportion from other causes, increases above the age of thirty years. For Female Life, with more extensive data, the general statement would doubtless, be similar to that before given for males. FEOM TABLE XVI (A). Ratio of Consumptive Deaths, to 100 total Deaths at each Age. Affe 20-30 80.8 80-40 25.8 40-50 17.8 50-60 10.8 60-70 6.5 70-80 3.5 All Ages. 18.3 1 Ratio Passing now to the Climatic Distribution in Table XVII (A) we observe that North of latitude 36° 30', the deaths from consumption are 18.7 per cent, of the total Deaths for male life, and 19.3 for females. \ While soutli of 36° 30', the percentages are 12.3 and 9.0 respectively. The apparent deficiency from consumption in the latter, proceeds only from comparison with a larger total, including an excess from malarial or other diseases. Compared with the United States Census of 1870, the percentage of consumptive deaths 18.3 in a total of 100 deaths of insured, is less than the result erf the Census, which is represented by 26.2 for males, and by 29.7 for females. Thus the rejection of lives uninsurable from incipient consumption is manifest. Table XVII (B) exhibits in more full detail, the percentage of insured males dying from consumption regarded as a part of a total of 100 deaths in each group of States. In the composition of 100 deaths, the consumptive element will be seen to decrease in the following order of groups, iii, i, iv, ii, v, vii, vi : as the numbers 21, 20, 18, 17, 16, 15, 12. Thus in proceeding towards the tropical regions, the ratio of deaths by consumption to those from other diseases, gradually diminishes. But further statis- tics in the next Table (C) will prove such change of ratio to proceed, not from absolute decrease of consumptive deaths, but entirely from increased mortality by other causes. The practical question, what part of the United States is most favorable to con- sumptive invalids, can now be satisfactorily answered, so far as average yearly results are concerned, from Table XVII (C). Ratio of Consumptive Deaths., to 10,000 Living. Group of States Ill 22.3 V 21.5 VI 21.0 I 20.8 IV 18.5 li 16.9 VII 16.9 Ratio Group III comprises New Jersey and Pennsylvania ; V, Delaware, Maryland, Dist. of Columbia, Virginia, Kentucky and Missouri ; VI, States south of lat. 36° 30' ; I, New England and New York; IV, Ohio, Indiana, lUinois, Iowa and Kansas; II, Michigan, Wisconsin, Minnesota and Nebraska ; VII, Washington, Oregon, Cali- fornia, Utah, Dacotah and New Mexico. By an obvious generalization, the first four groups, comprising the A tlantic and Gulf States, from Maine to Florida and from Florida to the borders of Mexico, have very nearly the same rate of consumptive deaths, or 21 annual deaths to 10,000 living. The Western States show a decrease of the consumptive rate to \%\ in group IV; while the groups II, and VII, or the Northwestern and Pacific States agree in the 56 MEDICAL STATISTICS. A SANITARY SURVEY OP THE UNITED STATES. The bold figures refer to the seven groups of States in the margin. The respective annexed figures are arranged In ascending order, to show the Proportional Deaths by each disease, among 10,000 Insured Males, living in each group of States. From Table XVII (C). Diseased. AH Causes. Zymotic Diseases Constitutional Diseases. Nervous Diseases Circulatory Diseases Respiratory Di seases .... Digestive Diseases Miscellanous Diseases. . . Typhoid and Typhus I Fever J Malarial Fever Erysipelas Dysentery Diarrhoea Cholera Alcoholism Other Zymotic Diseases. Dropsy Cancer Consumption Other Constitutional } Diseases f Apoplexy Congestion of Brain Paralysis, Softening, J etc. of Brain J Epilepsy and Convul- 1 sions ( Other Nervous Diseases. Diseases of Heart Other Circulatory Dis- ) eases J Pneumonia Congestion of Lungs Bronchitis and Pleurisy. Abscess, Hemorrhage, ) etc. of Lungs J Other Respiratory Dis- 1 eases ( Diseases of Stomach Diseases df Bowels Peritonitis Diseases of Liver Other Digestive Dis- I eases... ~. f Diabetes...; Diseases of Kidneys Other Urinary Diseases. Abscess, Hemorrhage, I Old Age ( Debility, Exhaustion, (^ Prostration ) Accidents and Injuries.. Suicides Unknown Causes Least Mortality. 2 97.7 4 104.5 1 105.3 3 107.1 15.9 21.4 11.5 4.4 12.1 8.6 16.0 1.6 1.1 16.9 1.0 16.6 22.1 14.4 4.6 13.3 9.1 16.6 1.8 1.5 16.9 1.0 17.6 23.0 15.4 6.6 14.6 10.0 17.3 1.9 1.6 18.5 1.4 18.5 26.3 15.4 6.7 14.6 11.2 17.4 1.9 1.9 20.8 1.4 Greatest Mortality. 7 112.2 5 130.5 6 170.5 18.7 26.4 18.4 6.8 16.6 11.2 18.9 6.9 3.3 1.3 1.7 1.0 2.0 .4 3.7 2.2 2.1 21.0 1.4 27.5 27.3 20.1 7.3 18.1 11.9 22.9 2.2 2.1 21.5 1.7 10.8 2.2 2.2 3.4 .9 .6 3.1 1.1 1.2 1.3 12.8 2.1 1.8 48.4 27.9 22.3 9.2 21.5 22.0 23.2 8.5 11.8 2.2 5.8 3.6 3.5 .6 18.2 2.6 2.2 22.2 2.0 12.6 2.2 3.4 2.4 1.4 3.5 1.2 1.3 24 13.3 3.3 2.5 Mean 118.3 2.3.3 24.9 16.8 6.5 15.8 12.0 18.9 6.6 3.8 1.4 2.3 1.3 1.7 .4 5.9 2.0 1.8 19.7 1.4 5.8 2.4 6.8 .6 1.1 6.0 .6 9.2 1.7 2.1 2.1 .8 2.3 2.7 .7 .3.6 2.6 .5 2.4 .9 1.0 1.2 9.6 1.8 i.5 Groups. 1. New England. New York. 2. Northwest. Michigan. Wisconsin. Minnesota. Nebraska. New Jersey. Pennsylvania. 4. Ohio. Indiana. Illinois. Iowa. Kansas. 5. Delaware. Maryland. Dist Columbia. Virginia. Kentucky. Missouri 6. South of 36° 30'. North Carolina. South Carolina. Tennessee. Gteorgia. Florida. Alabama. Mississippi. Arkansas. Louisiana. Texas. Pacific, etc. Washington. Oregon. California. Utah. Dakotah. New Mexico. MEDICAL STATISTICS. 57 more favorable rate of 17 consumptive deaths to 10,000 living. In establishing these important conclusions, the statistics entirely concur without discordance. Invalids experiencing this or any other disease will be interested to note the effect of climatic conditions shown in the accompanying Sanitary Survey of the United States. Relation of Height and "Weight. For convenient reference, the following Standard Table is introduced, showing the average or most healthful Weight relative to a given Height or stature. The medium circumference around the chest is added, although of less importance. "As a rule, twenty per cent, or one-fifth is almost the maximum variation from this within the limits of health." The Table is especially applicable to the medical examination of males between 30 and 60 years of age, relative to consumption. Height. Weight. 5 feet 1 inch 5' « 2 (( 5 '' 3 « 5 " 4 (( 5 " 5 a 5 « 6 (( 5 " 7 (I 5 " 8 li 5 " 9 u 5 « 10 li 5 " 11 « 6 « a Should weigh 120 lbs. a (I 125 « a a 130 " (( i( 135 « a li 140 « <( 11 143 « a 11 145 « (( li 148 « (( 11 155 " ti 11 160 « ti 11 165 " u ii 170 " Medium Chest. 34.06 inches. 35.13 11 35.70 11 36.26 (I 36.83 11 37.50 « 38.16 (( 38.53 11 39.10 11 39.66 11 40.23 u 40.80 li SECTION III. Tlie G-erm Theory of Malarial Diseases. Table of Sickness and Recovery in Adult Life. The statistics of Fever in Table IX comprise nine or ten kindred varieties, all of which are included in the general class of Zymotic diseases, of which the family of fevers constitutes about three-fifths. Malarial Fever is used to designate yellow fever, remittent fever, and intermittent fever or fever and ague ; to which some would add one or two other varieties, with a part of the cases of fever unspecified. Table XVII (C) will show that Zymotic diseases including fevers, have been the most prevalent in the groups of States, V and VI. And the more extended Table VII will suggest the separation of Alluvial counties, as a class, in which the greatest mortality has occurred. It may also be observed, that a large portion of the insurances in group V are located in the chief cities, including Baltimore, Washington and St. Louis. The space already given to medical Statistics, will here forbid a more extended discussion of fevers and other diseases. Yet the growing importance of the germ theory of miasm, and its favorable acceptance among physicians, may claim a passing notice. Our knowledge of the phenomena of fevers, is announced to be especially pro- moted by discoveries with the microscope. In examining the organs and secretions of persons who have died of fever, the membrane lining the stomach was almost 55 MEDICAL STATISTICS. invariably covered with a multitude of microscopic plants. These often covered the whole intestinal tract ; some were found on the surface of the lungs, and some were detected in the blood. On living patients, these parasites have been found in the substance, which is formed at an advanced stage of the fever, at the corners of the mouth and eyes. In the secretions of persons in full health they cannot be detected. Water taken up and placed in the rays of the sun till stagnant, became coated with a green superficial film, on which the microscope showed plants closely resembling those found in the human body. Some of these parasites showed cell-articulations; others were hollow and others showed nuclei and spores. A writer on the causes of Ague remarks that of these "ague plants" is formed the greyish film wherever black damp earth is turned up and exposed to the sun. Thence issue innumerable atomic seeds or spores which rise into the air, carrying pestilence with them. The danger from their growth is greatest in a hot dry season following a wet one. By day, the atmosphere is shown by the microscope to be free from those organ- isms. After sunset, the spores of the "ague plants" rise with the evening dews, recalling " the pestilence that walketh in darkness." In different parts of the world, these cryptogamic spores rise in the night mists, to definite heights. In the United States, they seldom rise more than thirty-five to sixty-five feet above the low levels ; in England, not more than from fifteen to thirty feet. But the spores of the "ague plants" having risen and become entangled in the mists, may all be carried by the wind, far perhaps from the place of germination. "Around the entire continent of Africa, except where the headlands jut out into the sea, there extends a belt of low malarious country from twenty to sixty miles wide. Within this belt it is unsafe for an unacclimated person to spend a night, although, as the malaria does not extend higher than about twelve feet from the ground, it is believed that it greatly conduces to health to sleep in the second or third story. Travelers visiting Africa should at once go to the higher lands, not allowing themselves to sleep a single night on the coast." According to this theory, the therapeutical effect of cinchona bark and its alka- loids, in malarial fevers, is due to a process of withering or destroying the fast growth of the poisonous fungoids in the human system. Recently, Pasteur has extended inocula- tion with "cultivated " germs of disease, to protect life against the more malignant tvpes. TABLE OF SICKNESS, RECOVERY AND MORTALITY.* At the BEGmNiNG of the Yb. or AoB. In the ensuing Twelve Months. Pbecise AeE. 4 Weeks of Sickness. AVEBAGE. Numbeb NUMBEB NuitBEB Casks of Cases of Died. ^ Living. Healthy. Sick. Sickness. Kecovebt. 25 1000 981.3 18.7 222.4 214.2 7.0 .969 (1872.) 0.78 30 1000 980.5 19.5 206.6 197.1 8.5 .998 0.86 35 1000 980.7 19.2 206.6 197.5 8.5 1.024 1.01 40 1000 977.7 22.3 209.4 198.0 11.4 1.160 1.17 45 1000 975.3 24.7 212.4 199.8 12.0 1.304 1.48 50 1000 968.8 31.2 227.1 211.6 14.8 1.644 1.98 65 1000 962.2 37.8 244.4 224.6 19.8 1.973 2.72 60 1000 950.4 49.6 2651 233.8 27.6 2.684 4.03 65 1000 927.6 72.4 282.6 242.9 34.4 3.913 630 70 1000 885.4 114.6 303.7 236.1 55.1 6.306 10.08 75 1000 811.8 188.2 307.8 231.8 78.2 9.760 15.07 80 1000 723.6 276.4 298.6 183.6 135.4 13.664 20 71 * Abstract of 800,000 observations in Friendly Societies, from Assurance Magazine, Vol. 16, p. 413. To show Healthy Life, cases of chronic sickness and the returns from sailors, miners and colliers were left out by Mr. Finlaison. Only the last column, from Mr. Batcliffe's collection in (1873), iuclodes AU Classes. TIME AND MONETARY TABLES. 59 DIVISION FOURTH, TIME AND MONETARY TABLES. Fob Interest and Annuiiies Certain. SECTION I. The System of Interest and Annuities. By general custom, interest is understood to increase uniformly, by equal incre- ments of interest in equal times, during twelve months or the period of conversion, and is then added to the principal, to form a new principal. When the interest is added at the end of each successive year, it is said to be compounded annually ; when the interest is added to the principal at the end of every six months, it is said to be compounded semi-annually, and so on. When the rate of interest is 6 per cent, for example, the increase on $1 principal is $0,005 monthly, which at the end of twelve months or a year, amounts to $1.06. Or omitting the character $ or £ the yearly interest on 1 unit of money is in this case 0.06. Let i denote .04, .05, .06 or any other rate of interest on 1 for one year; then will 1 + 1 denote the amount at the end of a year. By proportion, 1 : l + ^ : : new principal \-\-i\ (l + i)*, the amount at the end of two years. In like manner (1 + i)^ will be the amount of 1 at the end of three years. And in general, (1 + iY expresses the amount of 1 at the end of n years, with compound interest. The 'present value of 1 payable at the end of a year, is such a sum v as with its interest, will exactly amount to 1. By proportion 1 : 1 + t : : v : 1 ; whence V = - — : = (l + i)-i. Thus defined,. V is the present value of 1 receivable at the end of a year; and v" is the present value of 1 payable at the end of n years. Annuities. On this principle, the whole amount of 1 per annum, payable at the end of each year, with compound interest, will at the end of any number n of years, evidently be the sum of the geometric series, (l+^•)«-l4.(l + ^);^2+ .... +(i4.f) + i = (l+t)»- l ^ ^ In like manner, the whole present value of 1 per annum, payable and discounted at the end of each year, will be, for n years, \ — V I It may be proper to observe, that the sum of the preceding series has long since been demonstrated without geometrical progression. Thus at the rate of interest i, 60-61 TIME AND MONETARY TABLES. the annual income from the so-called perpetuity - will evidently be 1, to continue for- ever. Deducting from this, the present value of tlje perpetuity deferred n years, that is ^ . — , we have the expression equal to V above, or the present value of n annual payments of 1. And multiplying this whole present value by (1 + t)", we reproduce the above amount A for the end of n years. In practice, these formulas are seldom computed ; since time and labor are saved by recourse to the standard Monetary Tables of present values and amounts, calculated for various rates of interest. And ge^ierally, $1 and its tabular result, or the latter alone when multiplied hy the proposed principal or number of dollars, gives the value required, in practical business. — For the Value of an Increasing Annuity, see page 376. Tabic XVIII is a Time Table gi^■ing in triple columns the day of the month and the day of the year, also the corresponding decimal of a year of 365 days. The star (*) designates Monday, or one day in the week (as shown in the last column), from which any adjacent day of the week is readily noted. Also from any given date, the interval to the end of the year is found by subtracting the day of the year from 365, in com- mon years, or the corresponding decimal of a year from 1; to be increased for any extension into the following year. SECTION II. Interest Tables for Days. Required the interest on $4500 from May 27 to Sept. 8 at 6 per cent. The difference of the days of the year in the Time Table, 147 and 251, gives the interval 104 days ; which guides in the same Table, to the fraction of a year 0.28493. The interest for a whole year is 4500 x.06; that is 270; and the product of 270 by the preceding fraction gives $76.93, the interest required. But the next Table is designed to give the interest by a more convenient process. Copy Multiplication Table. In this Table, from which the Interest Table origi- nated, the carrying figure is added to the next product figure, as usual, and the sum given by inspection. It presents within a convenient compass, a generalization of the common process, and was suggested to the writer, by the early design of "Napier's rods or bones," by the first Table of Crelle, the Tables of Laundy, and others. For large numbers, or "contracted multiplication," it appears specially adapted, as well as for " proof," or correction of the common method, and various otlier objects. The bold types or figures of the Table, taken singly, represent every possible multiplicand, however extended. These are repeated immediately over their product figures, which may be copied rapidly in any order, as from left to right. In a few places, the construction of the Table required 10 to be tabulated as a single figure, denoted by X or z, and so written. But after the usual addition, this character will disappear in the final product. Also by practice, an easy habit may be acquired of reading off two figures at a time. Thus on reading 94 (nine, four) of the multiplicand, look first under 9 only, to read off its product figure 7, passing immediately under 4 to find the other figure 1, and copy 71. The same routine must be strictly followed in every example, as follows : 1. Prefix a small cipher to the multiplier to distinguish it, and to be used as its first figure. On account of "carrying figures" in the construction of the Table, all single ciphers, when effective, are to be treated like significant figures. TIME AND MONETARY TABLES. MULTIPLICATION. 100 200 300 400 ( 00 01 23 45 67 89 00 00 00 00 00 01 23 45 67 89 01 23 45 67 89 0123 45 67 89 03 46 80 24 68 01 23 45 67 89 03 69 35 81 47 01 23 45 67 89 04 83 60 48 36 00 12 13 15 17 01 23 45 67 89 00 00 00 00 01 01 23 45 67 89 00 00 00 00 11 0123 45 67 89 00 00 00 01 11 01 23 45 67 89 00 00 00 11 n 01 23 45 67 89 01 23 45 67 8x 01 23 45 67 89 01 33 45 67 9x 0123 45 67 89 01 23 45 68 9x 0123 45 67 89 01 33 45 78 9x 01 23 45 67 89 02 46 80 24 69 01 23 45 67 89 02 46 80 24 79 01 23 45 67 89 03 46 80 25 79 01 23 45 67 89 03 46 80 35 79 01 23 45 67 89 03 69 25 81 48 01 23 45 67 89 03 69 35 81 58 01 23 45 67 89 03 69 25 82 58 01 23 45 67 89 03 69 25 92 58 01 23 45 67 89 04 83 60 48 27 01 23 45 67 89 04 82 60 48 37 0123 45 67 89 04 82 60 49 37 01 23 45 67 89 04 82 60 59 37 12 13 15 17 20 23 25 29 012345 67 89 00 00 01 11 11 01 23 45 67 89 00 00 01 11 12 01 23 45 67 89 00 00 11 11 22 0123 45 67 89 00 00 11 13 22 0123 45 67 89 01 23 46 78 90 01 23 45 67 89 01 33 46 78 91 01 23 45 67 89 01 33 56 78 xl 01 23 45 67 89 01 23 56 79 xl 01 23 45 67 89 03 46 81 35 79 01 23 45 67 89 03 46 81 35 7x 01 23 45 67 89 02 46 91 35 8x 01 23 45 67 89 02 46 91 36 8x 0123 45 67 89 03 69 26 93 58 01 23 45 67 89 03 69 26 92 59 0123 45 67 89 03 69 36 92 69 0123 45 67 89 03 69 36 93 69 01 23 45 67 89 04 83 61 59 37 01 23 45 67 89 04 82 61 59 38 0123 45 67 89 04 82 71 59 48 01 23 45 67 89 04 82 71 5x 48 20 23 25 29 30 34 38 01 23 45 67 89 00 00 11 12 22 01 23 45 67 89 00 01 11 32 33 01 23 45 67 89 00 01 11 33 33 01 23 45 67 89 01 23 56 79 01 01 23 45 67 89 01 34 56 89 03 01 23 45 67 89 01 24 56 89 12 01 23 45 67 89 02 46 91 36 80 01 23 45 67 89 02 47 91 46 81 01 23 45 67 89 02 47 91 46 91 01 23 45 67 89 03 69 36 93 69 01 23 45 67 89 03 6x 36 x3 6x 0123 45 67 89 03 6x 36 x3 7x 01 23 45 67 89 04 82 71 50 48 01 23 45 67 89 04 83 71 60 49 01 23 45 67 89 04 83 71 60 59 30 34 38 40 43 45 01 23 45 67 89 00 01 13 33 33 01 23 45 67 89 00 01 13 33 33 01 23 45 67 89 00 01 13 33 34 01 23 45 67 89 01 24 57 89 12 01 23 45 67 89 01 24 57 8x 12 0123 45 67 89 01 24 57 8x 13 0123 45 6789 02 47 92 46 91 01 23 45 67 89 02 47 92 47 91 01 23 45 67 89 03 47 93 47 92 0123 45 6789 03 60 37 03 70 01 23 45 67 89 03 60 37 04 70 01 23 45 67 89 03 60 37 04 71 01 23 45 67 89 04 83 72 60 59 01 23 45 67 89 04 83 73 61 59 01 23 45 67 89 04 83 73 61 5x 40 43 "45 50 56 58 01 23 45 67 89 00 11 33 33 44 01 23 45 67 89 00 11 33 33 45 01 23 45 67 89 00 11 33 34 45 01 23 45 67 89 01 34 67 90 23 01 23 45 67 89 01 34 67 90 34 01 23 45 67 89 01 34 67 91 34 0123 45 67 89 02 57 02 57 02 01 23 45 67 89 03 57 03 57 03 01 23 45 67 89 03 57 03 58 03 01 23 45 67 89 03 70 47 14 81 0123 45 67 89 03 70 47 14 82 01 23 45 67 89 03 70 47 15 82 01 23 45 67 89 04 93 83 71 60 01 23 45 67 89 04 93 83 71 61 01 23 45 67 89 04 93 83 73 61 50 56 58 60 63 67 0123 45 6789 00 11 33 34 45 01 23 45 67 89 00 11 33 34 55 01 23 45 67 89 00 12 23 44 56 01 23 45 67 89 01 34 68 91 34 01 23 45 67 89 01 34 68 91 34 01 23 45 67 89 01 85 68 xl 35 01 23 45 67 89 03 57 03 58 03 01 23 45 67 89 03 57 03 58 13 01 23 45 67 89 03 58 03 68 14 01 23 45 67 89 03 70 48 15 82 0123 45 67 89 03 70 48 15 92 01 23 45 67 89 03 71 48 25 93 01 23 45 67 89 04 93 83 73 61 01 23 45 67 89 04 93 83 72 71 01 23 45 67 89 04 94 83 82 72 60 63 67 70 72 75 78 01 23 45 67 89 00 12 23 44 56 01 23 45 67 89 00 13 33 45 56 01 23 45 67 89 00 13 33 45 66 01 23 45 67 89 00 13 33 45 67 01 23 45 67 89 01 35 68 01 35 01 23 45 67 89 01 35 68 03 35 01 23 45 67 89 01 35 78 02 45 01 23 45 67 89 01 35 78 02 46 01 23 45 67 89 03 58 03 68 14 01 23 45 67 89 02 58 03 69 14 01 23 45 67 89 02 58 13 69 24 0123 45 67 89 03 58 13 69 35 01 23 45 67 89 03 71 48 25 93 01 23 45 67 89 03 71 48 26 93 0123 45 67 89 03 71 58 26 x3 0123 45 67 89 03 71 58 26 x4 01 23 45 67 89 04 94 83 82 72 01 23 45 67 89 04 94 83 83 72 01 23 45 67 89 04 94 93 83 82 01 23 45 67 89 04 94 93 83 83 70 72 75 78 80 84 86 88 89 01 23 45 67 89 00 13 34 45 67 01 23 45 67 89 00 12 34 55 67 01 23 45 67 89 00 12 34 56 67 01 23 45 67 89 00 12 34 56 77 01 23 45 67 89 00 12 34 56 78 0123 45 6789 01 35 79 02 46 01 23 45 67 89 01 35 79 12 46 01 23 45 67 89 01 35 79 13 46 0123 45 67 89 01 35 79 13 56 0123 45 67 89 01 35 79 13 57 01 23 45 67 89 03 58 14 69 35 01 23 45 67 89 03 58 14 79 25 0123 45 67 89 02 58 14 7x 35 01 23 45 67 89 03 58 14 7x 35 0123 45 67 89 03 58 14 7x 36 01 23 45 67 89 03 71 59 36 04 0123 45 67 89 03 71 59 36 04 0123 45 6789 03 71 59 37 04 01 23 45 67 89 03 71 59 37 14 01 23 45 67 89 03 71 59 37 15 01 23 45 67 89 04 94 94 83 83 01 23 45 67 89 04 94 94 93 83 01 23 45 67 89 04 94 94 94 83 01 23 45 67 89 04 94 94 94 93 01 23 45 67 89 04 94 94 94 94 80 84 86 88 89 90 01 23 45 67 89 00 13 34 56 78 01 23 45 67 89 01 35 79 13 57 01 23 45 67 89 02 58 14 70 36 01 23 45 67 89 03 71 59 37 15 01 23 45 67 89 04 94 94 94 94 90 100 200 300 400 For intermediate nombers in the side colamn, use the upper or less. Thus, for 85, 86, 87, or 88, use the line of 85. TIME AND MONETARY TABLES. 63 MULTIPLICATION. 500 600 700 800 900 00 01 23 45 67 89 05 05 05 05 05 01 23 45 67 89 06 28 40 62 84 01 23 45 67 89 07 41 85 39 63 01 23 45 67 89 08 64 30 86 43 01 23 45 67 89 09 87 65 43 21 00 12 13 15 17 01 23 45 67 89 05 05 05 05 06 01 23 45 67 89 05 05 05 05 16 01 23 45 67 89 05 05 05 06 16 0123 45 67 89 05 05 05 16 16 01 23 45 67 89 06 38 40 62 85 01 23 45 67 89 06 28 40 62 95 01 23 45 67 89 06 28 40 63 95 01 23 45 67 89 06 28 40 73 95 01 23 45 67 89 07 41 85 39 64 01 23 45 67 89 07 41 85 39 74 01 23 45 67 89 07 41 85 2x 74 01 23 45 67 89 07 41 85 3x 74 01 23 45 67 89 08 64 20 86 43 01 23 45 67 89 08 64 30 86 53 01 23 45 67 89 08 64 20 87 53 0123 45 67 89 08 64 20 97 53 0123 45 67 89 09 87 65 43 22 01 23 45 67 89 09 87 65 43 32 01 23 45 67 89 09 87 65 44 32 01 23 45 67 89 09 87 65 54 33 12 13 15 17 20 23 25 29 012345 67 89 05 05 06 16 16 01 23 45 67 89 05 05 06 16 17 01 23 45 67 89 05 05 16 16 27 01 23 45 67 89 05 05 16 17 27 01 23 45 67 89 06 28 41 73 95 01 23 45 67 89 06 38 41 73 96 01 23 45 67 89 06 38 51 73 x6 01 23 45 67 89 06 28 51 74 x6 01 23 45 67 89 07 41 86 30 74 01 23 45 67 89 07 41 86 30 75 01 23 45 67 89 07 41 96 30 85 01 23 45 67 89 07 41 96 31 85 01 23 45 67 89 08 64 21 97 53 01 23 45 67 89 08 64 21 97 54 01 23 45 67 89 08 64 31 97 64 01 23 45 67 89 08 64 31 98 64 01 23 45 67 89 09 87 66 54 33 01 23 45 67 89 09 87 66 54 33 01 23 45 67 89 09 87 76 54 43 01 23 45 67 89 09 87 76 55 43 20 23 25 29 30 34 38 01 23 45 67 89 05 05 16 17 27 01 23 45 67 89 05 06 16 27 28 01 23 45 67 89 05 06 16 27 38 01 23 45 67 89 06 38 51 74 06 01 23 45 67 89 06 39 51 84 07 01 23 45 67 89 06 39 51 84 17 01 23 45 67 89 07 41 96 31 85 01 23 45 67 89 07 42 96 41 86 01 23 45 67 89 07 42 96 41 96 01 23 45 67 89 08 64 31 98 64 01 23 45 67 89 08 65 31 x8 65 01 23 45 67 89 08 65 31 x8 75 01 23 45 67 89 09 87 76 55 43 01 23 45 67 89 09 88 76 65 44 01 23 45 67 89 09 88 76 65 54 30 34 38 40 43 45 01 23 45 67 89 05 06 17 27 38 01 23 45 67 89 05 06 17 38 38 01 23 45 67 89 05 06 17 28 39 01 23 45 67 89 06 39 52 84 17 01 23 45 67 89 06 29 52 85 17 01 23 45 67 89 06 29 52 85 18 01 23 45 67 89 07 42 97 41 96 01 23 45 67 89 07 42 97 42 96 01 23 45 67 89 07 42 97 42 97 01 23 45 67 89 08 65 32 08 75 01 23 45 67 89 08 65 32 09 75 01 23 45 67 89 08 65 82 09 76 01 23 45 67 89 09 88 77 65 54 01 23 45 67 89 09 88 77 66 54 01 23 45 67 89 09 88 77 66 55 40 43 45 50 56 58 60 63 67 01 23 45 67 89 05 16 27 38 49 01 23 45 67 89 05 16 37 38 4x 01 23 45 67 89 05 16 27 39 4x 01 23 45 67 89 06 39 62 95 28 01 23 45 67 89 06 39 62 95 29 01 23 45 67 89 06 39 62 96 29 01 23 45 67 89 07 52 07 52 07 01 23 45 67 89 07 52 07 52 08 01 23 45 67 89 07 52 07 53 08 0123 45 67 89 08 75 42 19 86 0123 45 67 89 08 75 42 19 87 01 23 45 67 89 08 .75 43 Ix 87 01 23 45 67 89 09 98 87 76 65 01 23 45 67 89 09 98 87 76 66 01 23 45 67 89 09 98 87 77 66 50 i 56 58 01 23 45 67 89 05 16 28 39 40 01 23 45 67 89 05 16 28 39 50 01 23 45 67 89 05 17 28 49 51 012345 67 89 06 39 63 96 29 01 23 45 67 89 06 39 63 96 39 01 23 45 67 89 06 3x 63 x6 3x 01 23 45 67 89 07 53 08 53 08 01 23 45 67 89 07 53 08 53 18 01 23 45 67 89 07 53 08 63 19 01 23 45 67 89 08 75 43 10 87 0123 45 67 89 08 75 43 10 97 01 23 45 67 89 08 76 43 30 98 01 23 45 67 89 09 98 88 77 66 0123 45 67 89 09 98 88 77 76 01 23 45 67 89 09 99 88 87 77 60 63 67 70 72 75 78 01 23 45 67 89 05 17 28 49 51 01 23 45 67 89 05 17 28 4x 51 01 23 45 67 89 05 17 38 4x 61 01 23 45 67 89 05 17 38 4x 62 01 23 45 67 89 06 30 63 06 30 01 23 45 67 89 06 30 63 07 30 01 23 45 67 89 06 30 73 07 40 01 23 45 67 89 06 30 73 07 41 01 23 45 67 89 07 53 08 63 19 01 23 45 67 89 07 53 08 64 19 01 23 45 67 89 07 53 18 64 39 0123 45 67 89 07 53 18 64 3x 01 23 45 67 89 08 76 43 30 98 01 23 45 67 89 08 76 43 31 98 0123 45 67 89 08 76 53 31 x8 0123 45 67 89 08 76 53 31 x9 01 23 45 67 89 09 99 88 87 77 01 23 45 67 89 09 99 88 88 77 01 23 45 67 89 09 99 98 88 87 01 23 45 67 89 09 99 98 88 88 70 72 75 78 80 84 88 88 89 01 23 45 67 89 05 17 39 40 62 01 23 45 67 89 05 17 39 50 63 01 23 45 67 89 05 17 39 51 62 0123 45 67 80 05 17 39 51 72 01 23 45 67 89 05 17 39 51 73 01 23 45 67 89 06 30 74 07 41 01 23 45 67 89 06 30 74 17 41 0123 45 67 89 06 30 74 18 41 0123 45 67 89 06 30 74 18 51 0123 45 67 89 06 30 74 18 53 01 23 45 67 89 07 53 19 64 30 01 23 45 67 89 07 53 19 74 20 01 23 46 67 89 07 53 19 75 30 01 23 45 67 89 07 53 19 75 30 01 23 45 67 89 07 53 19 75 31 0123 45 67 89 08 76 54 31 09 01 23 45 67 89 08 76 54 31 09 0123 45 67 89 08 76 54 32 09 01 23 45 67 89 08 76 54 32 19 01 23 45 67 89 08 76 54 32 Ix 01 23 45 67 89 09 99 99 88 88 01 23 45 67 89 09 99 99 98 88 01 23 45 67 89 09 99 99 99 88 01 23 45 67 89 09 99 99 99 98 01 23 45 67 89 09 99 99 99 99 80 84 86 88 89 90 01 23 45 67 89 05 17 39 51 73 01 23 45 67 89 06 30 74 18 53 01 23 45 67 89 07 53 19 75 31 01 23 45 67 89 08 76 54 32 10 01 23 45 67 89 09 99 99 99 99 90 500 600 700 800 900 • 64 TIME AND MONETARY TABLES. 2. The rule of arithmetic requires " the multiplicand to be successively multiplied by each figure of the multiplier." So here the left-hand pointer is to be located in the Table once for each single figure of the multiplier, by finding such single figure at the head of the column, and passing down to its two right-hand companion figures, found at the side ; such three figures may be termed a triplet. At each location of a triplet, the single figures of the multiplicand are all to be read, and the product figures under them copied ; each line beginning one place to the right. 3. When the right-hand figure of the triplet, noted above, locates between two given figures of the side column in the Table, always use the upper, or less of the two. And when such two nght-hand figures of a triplet are one or both lacking on the right hand of the multiplier, let ciphers be assumed in their place. Example. Multiply 34986 by 753. To the multiplier 752 prefix a cipher making "752, which shows there will be four locations of triplets. First "locate " the left-hand pointer in column 0, under the line 75. Here finding each figure of the multiplicand, as 34, etc. read off and copy the figures underneath, as 23 etc. from left to right, with the right hand. Common Process. 34986 By the Table. 34986 Locations 752 752 in Table. 69972 23664 - - - 075 174930 20705 - - - 752 244902 , 50611 - - - 52» , 68862 - - - 2o« Product, 26309472 Product, 26309472 Again, locate the left-hand finger or pointer for the triplet 752, in the column 7, under line 52, or rather line 50 which is the next " upper," and beginning to copy 20 etc. one place to the right, proceed as before. Again, locate in column 5, under line 20, etc. Again and lastly, locate in column 2, under line 00, and copy as before. In adding up to obtain the total product, the sum of each vertical column is the same as when found by the common method. The separately written triplets "75, 752, etc. noted on the right of the example, for illustration, are to be omitted in practice. SECTION III. CompoTind Interest and Discount Tables. For Twelve Kates of Interest and for 100 Years. Table XX. The Amount of $1 at the end of any number of years; or (1 -ft)'*. Table XXI. The Present Value of $1 due at the end of any number of years ; that is v^ or (1 -f i)—^. Table XXII. The Amount of $1 per annum, at the end of any number of years ; or or The annuity of II is payable at the end of each year. Table XXIII. The Present Value of II per annum, for any number of years ; or TIME AND MONETARY TABLES. «5 Table XXIV. The Annuity which $1 will purchase for any number of years ; i 1—(1 + *■)-« It is the Annual Payment or Annual Sinking Fund to pay a present debt of $1 with its interest, in any number of years. Table XXV. Common LogaritJim of the Present Value of $1 due at the end of any number of years; a(14-i)-», or Xv^. As in Milne's treatise, a denotes the common logarithm of the quantity, before which it is placed. The above named Monetary Tables are derived from the original series published by John Smart, London, 1726. A portion of Smart's Tables, for half years depend on y/l + i as the interest factor for six months. But mercantile usage has decided in favor of (1 + as the yearly factor, and (1+^i) as the factor for the odd six months ; hence the half-yearly columns in his Tables are not now in use. A part of Smart's Tables, those tinder 7 per cent., we have here collated with new values computed with four extra decimals, or with the accurate Tables of Chisholm. Connection of the Tables. It will be observed that for a given value of n and *, Tables XX and XXI are each the reciprocal of the other. Tables XXIII and XXIV, are another couple haviug the same reciprocal property. Again, the amounts in Table XX, are the annual differences of Table XXII ; and the present values in Table XXI, are the differences of Table XXIII, as will be seen by inspection. Amount of Premiums paid. For illustration, the annual premium of $50 has just been paid at the beginning of the seventh year. Eequired the amount at that date, of the seven premiums, at 4 per cent. By Table XXII the amount of $1 annual premium is given at $7.89829; the product of this by 50 is $394.91, the amount required. Had it been a single premium of $50 paid six years ago, the different Table XX would give 50 x 1.2653 = $63.27, the amount at compound interest 4 per cent. Deferred Annuity. If the annuity of $1 for n years is to be deferred t years and then commence, the present value is evidently shown in the expression, ,. ., , 1— (l + «)-" ,^ (l-f«)- — z -„ given in Tahle XXIV. 1— v" TABLE OF THE AUXILIARY FACTOR (i-0- Pbicb on 100. 1 2 3 4 5 6 7 8 9 5 1.0000 0.9608 0.9231 0.8868 0.8519 0.8182 0.7857 0.7544 0.7241 0.6949 6 0.6667 .6393 .6129 .5873 .5625 .5385 .5152 .4925 .4706 .4493 7 0.4386 .4085 .3889 .8699 .3514 .3333 .3156 .2987 .2821 .2658 8 0.2500 .2346 .2195 .2048 .1905 .1765 .1628 .1494 .1364 .1236 9 0.1111 .0989 .0870 .0753 .0638 .0526 .0417 .0309 .0204 .0101 10 -.0000 -.0099 -.0196 -.0291 -.0385 -.0476 -.0566 -.0654 -.0741 -.0826 11 -.0909 -.0991 -.1071 -.1150 -.1228 -.1304 -.1379 -.1453 -.1525 -.1597 12 -.1667 -.1736 -.1803 -.1870 -.1935 -.2000 -.2063 -.2126 -.2187 -.2248 13 -.2308 -.2366 -.2424 -.2481 -.2537 -.2593 -.2647 -.2701 -.2754 -.2806 14 -.2857 -.2908 -.2958 -.3007 -.8056 -.3103 -.3151 -.3197 -.3243 -.3289 The new expression for i' will be sufficient for most purposes. By successive sub- stitution, however, in the right-hand member of the original formula preceding, the approximation <;an be carried to any extent of accuracy. In the Example stated above, the price on 100 is 105, or j» = 1.05. Also n' = 20 — 0.6 X .0476 X 20 = 19.4 years ; whence the required rate of interest i' = .06— .048 X .088 = .06— .0042 = .0558. Here the factor —.048 or rather .0476 is found in the above Table, corresponding with the price 105; and the other factor V or .088 corresponds to n' or 19.4 years and 6 per cent, in Table XXIV. Provided the Bond is continued to maturity, the approxi- mate interest realized on the purchase will be 5.58 per cent. Example 2. Required the rate of interest to be realized when the price is 75 on the 100, or p=:0. 75, the rate of interest i being 7 per cent., and the Bond to be redeemed in 40 years. Here ^i' = 40 + 0.6 x. 333 x 40 = 48 years; then i'=.07 + .333 x .0728 = .0942 ; that is, 9.42 per cent, on the purchase price. Example 3. When n = 30 years, i = .06,p = 0.5. Then w' = 30 + . 6 x 1 x 30 = 48 years, i' = .06 + 1 x. 0639 = .1239. The result 12.39 per cent, is correct to four- places, as verified. It is worthy of remark that in these three arbitrary examples, having periods of 20, 40 and 30 years, the approximate formula, i' = i-^p, should give results so near the truth as 5.71, 9.33 and 12.00; the true rates being 5.58, 9.42, and 12.39 per cent respectively. TIME AND MONETARY TABLES. 69 SECTION VI. Interest Compounded Oftener than Once a Year.. When interest is compounded at m equal intervals in a year, the amount at the end of the first of them is [ 1 H — j ; and this raised to the with power, gives the amount at the end of m terms, or one year ; thus : H) When interest is compounded momently or at the end of every instant, the number of intervals w in a year becomes very great or infinite, which evidently causes m to vanish in the right-hand member, leaving at the year's end, i \m 1 1 Here e is the number whose Napierian logarithm is 1; that is e = 2.7182818285. Thus on the supposition of compounding momently, the amount of $1 principal at the end of n years, will be £'«. In the common formulas of compound interest, the interest for one interval may take the place of annual interest, when periods of conversion are substituted for the number of years. And if there are m equal periods in one year, there will evidently be nin equal periods in n years. Writing m equal to 1, 2, 4, or infinity, let Y, H, Q, M denote the Amount of $1 with its compound interest, at the end of n years, accord- ing as the interest is convertible into principal, Yearly, Half-yearly, Quarterly, or Momently. 4n Then Y = (l + *)~ Q=\^+4 H^^l+I)"", M=ein. The Present Value of 1 due n years hence, with interest compounded m times in the year, is y \ — »i« 1+- m) In the case of Annuities payable half-yearly, quarterly, or generally m times in the year, let us ^rs/?^ assume the interest to be compounded in the same way, or m times in the year, and that the annuity or yearly payment of 1 is divided into m payments of each, at equal intervals. Then by substitution as shown by Dr. Price: m Amount of Annuity for n years = — . — Present Value of Annuity for n years = - 1 + :^) -ij- ml 1 + ^n When payable yearly, m — l; half-yearly, wi = 2 ; quarterly, w = 4. 70 TIME AND MONETARY TABLES. Secondly and more generally, let interest be convertible m times in a year, and the annuity be payable m! times, each payment being — ,. In this case, the discount factor m —m to give the present value for one year, is I IH — j ; or for ^ years, or a fraction, the / J \—mt factor is IH — 1 . Now dividing one year by the number of its payments of annuity, we have —, = t, the time or interval from one payment to another. Hence the present value of the pi'n payments in n years, will be the sum of the geometric progression ; since 77i't = 1 : w 1_\ mj \ m) \ m) J I \ —mt j i \ — nm — mt j j \ — mn . .1+- - 1+- 1 1- 1+- 1 \ m \ m 1 \ m m' , L . i\-^^ m' L . *\'^ 1- 1+- 1 + - -1 \ mj \ mj When m and m' are equal, tbis manifestly coincides with the last formula on page 69. In the denominator, developing by the binomial theorem, since —;, = t, mj m m m' }nJ2 \7n' m)\m' in)l.2.d i+{~^]l+{^.-^M^.--.]~+---~\ Substituting the series in the above denominator, and dividing. iii-^ Present Value = R+l(i_l\-(i,_i,).l-,-^+ . ..1 x fl-fl ^^Y'^']. \_i 2\m m j \m^ my 12 J L V ^/ J In the case of annual interest, or w = 1, we have, Present Value = (-+ ^ ~ ].[1— (l-ht)-^], nearly, 1— (l + t)-« m'—l , ^r , m'—l When the annuity is half-yearly, or m' = 2, the last term becomes 0.25; which agrees with the usual approximate correction for Life Annuities. More accurately, when i = .04, the third term of the series above, when united to the second, gives 0.24755. This correction from Annuities Certain, strikingly agrees with those com- puted by Mr. Sprague from Life Annuities at ages where the initial rate of mortality plus that of interest equals 4 per cent. Compare Journal of the Institute, Vol. 13, pp. 212, 308. Finally for illustration, the annexed Table from David Jones on Annuities will indicate the effect of compounding oftener than once a year to increase the yearly rate of interest. In long periods of years, the difference becomes more material: TIME AND MONETARY TABLES. n AMOUNT OF $1 IN ONE YEAR, AND ITS LOGARITHM. AlfNUAL BATE OP Ikterest. Pat- able. Amount in ONE Yeae. LOGABITHH. Annual BATE OP Intebest. Pay- able. Amount in ONE Ybab. Logarithm. 2 Y 1.020000 .00860 01718 5 Y 1.050000 .0^118 92991 per cent. H 1.020100 .00864 27476 per cent. H 1.050625 .02144 77808 Q 1.020150 .00866 42470 Q 1.050946 .02158 0:275 M 1.020201 .00868 58896 M 1.051271 .0217147241 2| Y 1.025000 .01072 38654 6 Y 1.060000 .02530 58653 per cent. H 1.025156 .01079 00638 per cent. H 1.060900 .02567 44494 Q 1.025235 .01082 35735 Q 1.061364 .02586 41690 M 1.025315 .01085 73620 M 1.061837 .02605 76689 3 Y 1.030000 .01283 72247 7 Y 1.070000 .0293837777 per cent. H 1.030225 .01293 20845 per cent. H 1.071225 .02988 06996 Q 1.030339 .01298 02193 Q 1.071859 .03013 76716 M 1.030454 .01302 88345 M 1.072508 .03040 06137 3i Y 1.035000 .0149403498 8 Y 1.080000 .03342 37555 per cent. H 1.035306 .01506 88358 per cent. H 1.081600 .03406 66786 Q 1.035462 .01513 41909 Q 1.082432 .03440 06870 M 1.035630 .01520 03069 M 1.083287 .0347435586 4 Y 1.040000 .0170333393 9 Y 1.090000 .03742 64979 per cent. H 1.040400 .01720 03435 per cent. H 1.092025 .03823 25809 Q 1.040604 .01728 54951 Q 1.093083 .03865 32667 M 1.040811 .0173717793 M 1.094175 .03908 65034 Note. — The period required for a given principal to double itself at compound interest, ia defined by the well-known equation for n years, (1-Ht> = 2, n=, — Ti—K = — ^+0.35 nearly. Also (l-f-) =2. Thus, dividing 70 by the rate of interest as an integer, will give a close approximation, easily remembered. In the last formula making m = l, 2, 4, and infinite as before, we find the following more exact series. Doubling Period in Years and Decimals. Interest compounded yearly: Rate per cent. 2 3 4 Doubling Period. 35.004 23.450 17.673 Interest compounded semi-annually: Rate per cent. 2 3 4 Doubling Period. 34.830 23.278 17.502 Interest compounded quarterly: Rate per cent. 2 3 4 Doubling Period. 34.743 23.191 17.415 Interest compounded momently: Ratp, per cent. 2 3 4 DouDllng Period. 34.657 23.105 17.329 5 14.207 5 14.036 5 13.946 5 13.863 6 11.896 6 11.725 6 11.639 6 11.552 7 10.245 7 10.075 7 9.989 7 9.902 8 9.006 8.837 8 8.751 8 8.665 TABLES IN PART FIRST, MORTALITY AND LOSS EXPERIENCE, Tables I-IV. Pages 74-166 CLIMATIC TABLES, - - " V-VIH. ** 167-181 MEDICAL STATISTICS, " IX-XVI. " 182-198 TIME AND MONETARY TABLES, - - - « XYII-XXV. '' 199-231 74 TABLE I. MALE LIFE. SUMMAKY (A). OBSERVATIONS CLASSIFIED ACCORDING TO THE AGE OF EXIT OR EXPOSURE. Age of Whole Number < jp Ektbants, 982,734. 2,568,856 Thousand Dollabs Insubed. Age at Exit or Exposure. lib Exit. Existing. Discon- tinued. Died. Exposed to Risk. Amounts Existing. Amounts Discontln'd. Death Claims. Exposed to Risk. 1 3.5 1 6.5 0- i ^ 1 9 1 34 18 2 G6 i- H 2 8 7 69 19 9 144.5 H-n 3 6 8 86 13 26 186 2i-3i •4 5 8 99 11 15 211.5 H- 4^ 5 10 12 107 19 25 230 4i- H 6 8 9 105.5 16 19 230.5 H-6i 7 10 10 2 110 20 21 6 241 6i- n- 8 14 15 108.5 31 ' 24 237 7i- 8i- 9 7 11 1 107.5 23 25 1 228.5 8^- 9i 10 20 14 139 49 20 289.5 9i-10i 11 20 28 174 42 43 361.5 10^11% 12 32 35 239 58 59 490.5 lli-12^ 13 34 46 2 361.5 76 77 3 741.5 12i-13^ U • 74 97 3 732.5 129 199 3 1,499 13i-Ui 15 96 212 3 1,405 226 388 4 2,863 14i-15i 16 224 394 11 2,375.5 411 764 17 4,632.5 15^-16^ 17 353 615 24 3,693 667 1,132 36 7,134 16i-17^ 18 639 972 32 5,736.5 1,188 1,817 55 11,035.5 17l-18^ 19 940 1,782 68 8,857.5 1,666 3,261 118 16,774.5 18i-19^ 20 1,536 2,845 116 13,773.5 2,783 5,308 200 26,120 19^-20^ 21 2,383 4,323 158 21,040.5 4,376 8,210 266 41,378.5 20^21i 22 3,246 6,335 200 30,596 6,297 12,672 333 62,732 21^-22^ 23 4,452 8,488 312 42,003 8,711 17,543 609 89,221.5 22^-23^ 2U 6,044 10,333 326 54,367:5 12,056 22,187 707 119,483 23^24i 25 7,309 11,753 468 67,546.5 14,780 26,663 1,002 152,867.5 24^-25} 26 8,879 13,346 530 80,629 18,739 30,876 1,229 186,725 25^-261 27 10,221 13,891 639 92,781 21,831 33,441 1,452 218,231.5 26^27i 28 11,799 14,782 758 106,513 25,843 35,733 1,734 250,743.5 27i-28\ 29 12,876 15,751 784 118,901 28,841 38,245 1,925 281,450 28^-29^ 30 17,480 16,082 819 129,868 32,578 40,227 2,093 312,842.5 29i-30^ 31 15,092 16,571 961 136,354 34,273 42,664 2,410 339,736 SOiSli 32 15,991 16,130 979 143,956.5 37,781 42,288 2,444 361,484 31I-32I 33 16,618 16,148 1,057 151,141 38,901 42,554 2,788 381,961 32^-33^ 34. 17,784' '16,351 1,193 157,277.5 42,133 43,599 3,213 400,764.5 33^-34^ 35 17,712 16,286 1,196 161,673.5 43,697 43,855 3,234 414,914.5 34i-35^ 36 18,444 16,477 1,218 165,280.5 46,172 44,175 3,380 427,076 35^-361 37 18,806 15,497 1,299 166,330 47,066 43,243 4,028 432,046 36i-37^ 38 18,850 14,888 1,403 166,100 45,893 41,408 3,865 432,402 37^-38^ 39 19,317 14,325 1,398 164,729.5 47,916 40,246 3,931 432,874 38l-39i 40 19,351 13,675 1,401 162,619 48,845 38,935 3,823 430,914.5 39^-40^ 41 19,683 13,234 1,430 158,904.5 49,273 37,094 4,091 425,161.5 401^4H ^ 18,465 12,134 1,401 152,323 46,579 35,982 4,023 413,268 4H-42i 4-5 17,165 11,438 1,456 146,097 44,676 33,604 3,941 401,060 42i-43i U 17,305 10,671 1,446 139,838 44,846 31,433 4,055 387,327 43i-m 45 16,605 9,847 1,244 133,033.5 43„646 29,657 3,453 372,103 44h-45^ 46 16,518 9,235 1,339 126,308.5 43,846 27,922 4,012 356,838.5 45^4H 47 15,230 8,397 1,325 117,740 41,080 25,782 3,827 336,969 46i-47i 48 13,811 7,536 1,280 109,560.5 37,684 24,249 3,905 317,001 47i-48i 49 13,191 6,891 1,157 102,105 36,291 21,180 3,565 298,249.5 \48i-49i TABLE 1. '3'0 MALE LIFE. SUMMARY (A). OBSERVATIONS CLASSIFIED ACCORDING TO THE AGE OF EXIT OR EXPOSURE. Age Wholb Number < )p Entbants , 982,734. 2,568,856 Thocsand Dollabs Imscbed. "I Age at n.t Exit or at Exit. 50 Existing. Difecon- tinued. Died. ^ Exposed to Risk. Amounts Existing. Amounts Discontin'd. C Death laimB. Exposed to Risk. Exposure. 12,511 6,243 1,192 1 34,951.5 34,761 19,532 3 ,554 280,485 49^-50^ 51 11,365 5,780 1,187 i ^7,291 32,004 18,619 3 ,606 260,386 50^1^ 52 10,643 4,935 1,108 ' ^9,336.5 30,397 16,366 3 ,487 238,306.5 5'H-52i 53 9,686 4,413 1,032 ' a, 744 27,339 14,529 3 ,227 216,122 524,-53'^ 54 9,480 3,849 988 ( 34,574.5 27,483 13,619 2 ,975 195,270 53^54h 55 8,335 3,294 982 I )7,089.5 25,161 11,550 3 ,183 172,957.5 54^-55^ 56 7,582 2,824 876 { )0,184 22,555 9,552 2 ,744 151,660 55%-56\ 57 6,519 2,360 813 ^ t3,729 19,400 8,194 2 ,718 132,555 56^^-57^ 58 5,612 1,922 756 { 58,164.5 16,246 6,837 2 ,313 115,575 57i>-58\ 59 4,985 1,640 721 c 53,284.5 14,851 5,756 2 ,300 101,140.5 58^-59i 6'0 4,178 1,348 733 ^ J8,892 12,630 4,891 2 ,102 87,621 59^-60^ 61 3,637 1,060 649 ^ J4,959.5 11,274 3,784 2 ,130 75,419.5 60^-61^- 62 3,313 798 531 ^ Jl,189 9,528 2,915 1 ,624 63,379.5 61^-62^^ 63 2,849 605 509 ] L7,725 8,712 2,275 1 ,686 53,175 62^-63^ 64 2,333 499 457 ] L4,678.5 6,748 1,907 1 ,336 43,325 - 63^-641 65 2,056 407 416 ] .2,116.5 6,023 1,258 1 ,517 35,712 64i-6H 66 1,634 311 368 9,732 4,797 1,204 1 ,080 28,233 65^-661 67 1,395 246 271 7,716.5 3,867 798 815 22,082.5 66i-67i 68 1,071 150 259 6,042.5 3,176 601 792 17,188.5 674j-68i 69 837 130 224 4,694 2,610 431 600 13,025.5 68^-69^ 70 586 95 189 3,595.5 1,602 334 528 9,625 69^-70^ 71 469 68 134 2,783.5 1,289 211 353 7,322 70^-71^ 72 365 52 136 2,148 1,043 134 348 5,576.5 71^-72^ 73 300 31 93 1,627 749 77 236 4,131.5 72^-73^ n 216 24 80 1,221.5 612 73 176 3,104.5 73^-74i 75 146 20 60 914.5 426 37 143 2,382 74i-75i 76 94 7 53 701.5 305 11 129 1,800 75i-76i- 77 77 10 54 547.5 244 20 96 1,353 76^-77^ 78 51 8 47 410 135 16 102 998.5 77i-78^- 79 39 7 29 306.5 86 14 86 749 78i-79i 80 39 6 21 232 109 13 50 563.5 79^-80^ 81 28 3 18 168.5 77 7 36 395.5 80^-81^ 82 16 7 21 119 46 15 38 273 81^-82^j 83 12 7 79 43 20 182 82^-831 84 3 1 7 59.0 11 1 13 118.5 83^-84^ 85 12 2 6 48 29 4 11 92 84^-85^ 86 3 5 29 10 11 50 85^-86^ 87 1 4 21 3 5 29 86^-87^ 88 4 2 16.5 5 2 21.5 87^-88^ 89 1 3 11 1 7 15 88^-89^ 90 1 2 6.5 1 2 6.5 89^-90^ 91 4 4 90^—91^ 92 1 1 4 1 1 4 9^-92^ 93 1 2 1 2 92^-93^ 94 1 1 93^-94i 95 1 1 1 1 94^-95^ 527,157 411,092 44,485 4,3 27,086.0 1,328,404 1,114,487 12^ ),965 11,493,967.5 % TABLE I. MALE LIFE. SUMMAKT (B). OBSERVATIONS CLASSIFIED ACCORDING TO YEARS OF INSURANCE. 09 o3 Whole Numbeb OF Entbants, 982,734. 2,568,856 Thousand Dollars Insubbd. Duration of Insmance. Existing. Discon- tinued. Died. EzDosed to Risk. Amounts Existing. Amomits Discontin'd. Death Claims. Exposed to Kisk. 60,190 46,913 2,943 467,910.5 153,950 99,417 7,787 1,234,719.5 0- i 1 51,189 158,722 6,428 793,327 132,847 378,590 18,131 2,118,407 i- H 2 49,307 67,850 5,733 622,424 113,202 208,799 16,668 1,673,734.5 n- H 3 45,000 44,573 5,114 511,172.5 105,929 140,830 14,531 1,369,050 2i-3i 4 48,697 30,418 4,623 423,563 117,147 98,064 13,409 1,129,143 3f-4i 5 48,237 21,452 3,911 344,308 126,106 69,899 11,213 914,605.5 4i- 5i 6 48,663 14,043 3,205 274,412.5 127,146 44,140 9,166 720,267 5i- 6i 7 45,160 11,563 2,569 209,741.5 117,454 30,606 7,311 546,582 6i- n 8 38,200 5,352 1,837 153,555 100,312 15,703 5,045 398,662.5 7i- 8i 9 28,037 3,231 1,344 109,226.5 69,922 8,937 3,525 280,985.5 8^9i 10 18,210 2,146 982 77,157 43,800 6,006 2,619 200,067 ^h-m 11 10,977 1,359 751 56,212.5 26,698 3,688 2,153 148,801 lOi-lH 12 5,120 830 647 43,390 13,301 2,349 1,805 116,931.5 lli-12^ 13 3,779 586 519 36,915 10,656 1,730 1,472 99,786 12i-13i 14 4,123 574 469 32,037 11,331 1,624 1,290 85,981 13i-14| 15 3,110 304 399 27,006 8,506 843 1,080 72,126.5 14^15^ 16 2,254 261 401 23,214.5 6,229 734 1,201 61,752 15i-16i 17 1,916 180 349 20,339 4,706 528 987 53,691 16i-17i 18 1,707 157 298 17,905.5 4,452 390 951 47,539 17i-18i 19 1,377 124, 303 15,760 3,914 325 758 41,778.5 18i-19i 20 1,148 110 270 13,963 3,114 337 758 36,775.5 19i-20i 21 950 95 269 12,442.5 2,499 261 771 32,604.5 20i-21i 22 1,019 77 274 11,137.5 2,529 230 837 29,089 211-22^ 23 1,541 61 261 9,775.5 3,790 184 768 25,516 22^23| 24 2,140 35 218 7,925.5 4,977 82 605 20,825 23i-24| 25 1,946 24 155 5,538 4,679 53 490 15,175.5 244-251 26 1,420 28 105 3,411 4,094 85 329 9,937.5 25|-26| 27 930 10 54 1,867 2,838 23 161 5,460.5 26|-27i 28 467 6 31 875 1,391 9 83 2,445.5 27^28^ 29 189 6 10 371.5 453 17 15 958.5 28i-29i 30 154 1 8 169.5 432 3 31 480.5 29^30i 31 7 16 30f-3H 32 1 7 1 16 31h32i 33 1 6 5 15 32^33^ 34 2 1 4 1 2 9.5 33^34^ 35 2 7 34^-354 36 2 7 36i-36i 37 2 7 36|-37i 38 1 2 - 2 7 37i-38i 39 1 1 5 5 38^39^ 527,157 411,092 44,485 4,327,086.0 1,328,404 1,114,487 125,965 11,493,967.5 TABLE I. OBSERVATIONS ON MALE LIFE. 77 Excepting the first Aggregates, the Amounts Insured are stated in THOUSAND DOLLABS in the columns $, where 1 denotes $1000, etc. Age at Entry, Years. Age at Entry, 2 Years. ii 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 No. OF Entbants, 7. &« 3.5 6 5.5 4.5 4 4 2 2 2 2 2 1 1 39.5 $13,000 Insubed. Si Q QU ^3 $ 6.5 12 11-5 9.5 91.5 Age at Entry, 1 Year. bo < 5 6 7 8 9 10 11 12 No. OF Entrants, 57. $110,000 Insubed. 9 1 28 18 2 54 4 6 44 9 8 86 5 37 11 73 3 3 30.5 8 5 59.5 1 4 24 2 7 45.5 4 21 8 40 2 17 3 32 4 2 14 10 3 27.5 1 8.5 2 15 2 8 3 14 2 6 4 11 1 4 1 7 2 3 1 5 6 1 1 .5 1 .5 39 18 253.5 ;82 1 28 479.0 1 2 3 -4 5 6 7 8 9 10 11 12 13 u 15 16 17 18 19 20 21 22 3g 1 2 3 4 5 6 7 8 9 10 11 12 13 No. OP Entrants, 39, 5"" 15 19.5 32 27.5 24.5 19 16 9.5 6 5 1 1 1 1 1 164.0 $94,000 Insured. 611 "C >J '55 » T o a Is $ $ 10 47 22 73 1 6 59 9 2 54 2 4 42 10 3 1 36.5 8 3 22.5 2 13 8 11 3 3 3 3 3 3 53 40 1 373.0 Age at Entry, 3 Years. Age at Entry, 4 Years. 4 5 6 7 8 9 10 11 12 13 U 15 No. OF Entrants, 26. 162,000 Insured. 1 1 12.5 2 1 30.5 1 2 23 4 57 2 1 5 19.5 1 13 48.5 3 3 14.5 8 37 4 1 12.5 3 31.5 6 2 2 11 5 3 28.5 6 1 1 7.5 3 5 19.5 7 1 6 2 14 8 3 5 7 12 9 1 2 3 5 1 1 2 2 11 15 114.5 25 37 285.5 3 5 6 7 8 9 10 11 12 13 No. OF E ntrants, 28. $56,00G Insured. 1 14 2 28 2 26 3 52.5 1 2 24 2 4 49 2 3 20.5 4 5 42.5 1 16.5 2 35 78 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 4 Years. Age at Entry, 7 Years. «! No. OP Entrants, 28. $56,000 Insured. 4J No. OF Entrants, 24. $53,000 Insured. •a M 1 .2 "P. 1.1 ■q 1 hS 1 II 5'" Si ^2 ? be < •5 ' 5 § i • . II T % $ % $ $ $ ■ 1 5 1 16 2 34 9 1 1 11.5 1 2 1 26 7 6 1 3 13.5 3 5 29.5 10 1 1 4 20 2 7 46.5 <§ 7 1 1 10.5 2 1 23.5 11 2 1 1 1 16.5 12 2 1 40 9 8 2 9 5 21 12 3 1 3 12.5 1 5 23.5 10 9 3 7 7 16 13 4 3 8.5 4 18 11 10 3 4 4 9 u 5 2 7 5 16 12 11 1 2 3 5 15 6 1 5 3 11 13 12 1 2 16 7 1 4 2 8 U 13 1 1 2 36 20 — 2 349.0 17 8 9 1 1 1 3 1.5 1 3 31 2 6 4 15 16 16 12 165.0 10 13 1 89.5 21 1 199.0 Age at Entry, 5 Years.. 1 Age at Entry, 8 Years. No. OF Entrants, 18. $43,000 Insured. 1 _ 2 9 5 21.5 5 1 1 1 1 15.5 2 2 37 6 2 1 10.5 2 1 17.5 8 2 1 2 1 13 1 7 5 30.5 7 1 2 3 1 17.5 2 7 29.5 9 3 1 9.5 2 20 8 2 3 12.5 3 22.5 10 4 1 1 8.5 2 3 17.5 9 3 4 1 2 1 10 7.5 1 4 2 19 15 11 12 5 2 7 4 14 10 6 1 5 2 10 11 5 7 14 13 7 1 3.5 1 7.5 12 6 1 6.5 1 13.5 U 8 1 3 3 7 13 7 6 13 15 9 1 2 2 4 U 8 9 1 2 6 5 2 4 13 11 16 17 10 1 2 lb 11 1 2 16 10 3 7 18 12 1 1 2 2 17 11 3 7 19 11 6 1 79.0 23 15 5 175.0 12 13 14 3 3 3 7 7 7 20 21 22 Age at Entry, 6 Years. 15 16 1 2.5 2 1 6.5 6 23 No. OF Entrants, 20. $40,000 Insured. 17 1 2 2 6 25 18 4 26 1 1 9.5 2 1 19.5 6 19 4 27 1 1 17.5 2 86 7 2 2 8 15.5 4 3 88.5 s 20 4 28 3 4 10 6 25 9 21 4 29 4 1 1 7.5 5 2 21 10 22 4 30 5 6 7 2 1 6 6 4 5 3 15 15 • 10 11 12 13 23 1 1 4 4 31 10,12 116.0 17 19 245.5 8 2 3 4 7 U 9 1 1 3 3 15 10 10 80.0 26 14 185.0 TABLE I. OBSERVATIONS ON MALE LIFE. 79 Age at Entry, 9 Years. Age at Entry, 11 Years. No. OP Entbants, 80. $58,000 Insured. •a a P bo < ■si No. OP Entrants, 78. $148,000 Insured. coed si 'h .22-5 1 -« . 5 1 h2 to a .2 H ".2 5a Si no 1 1^- aw eg §1 ^ 5" 5 If 1 T §1 T $ T ~$ $ T $ 15 29 9 7 4 37 10 8 70 11 1 1 3 28.5 1 4 56 10 1 2 22 56 3 30 115 12 2 7 22.5 5 50.5 11 2 3 8 39 1 6 13 90.5 13 3 1 19 2 48 12 3 3 4 30 i 6 11 72.5 U 4 1 1 17.5 1 2 45 13 4 2 5 22.5 13 9 56.5 15 5 1 1 15.5 1 1 42.5 n 5 1 1 17.5 2 3 37.5 16 6 1 14 2 41 15 6 1 2 15 3 5 31.5 17 7 1 1 12.5 5 1 38.5 16 7 3 13 8 26 18 8 3 1 10.5 5 3 31.5 17 8 2 10 4 18 19 9 1 1 6.5 2 1 2-4.5 18 9 1 7.5 1 13.5 20 10 1 5 1 22 19 10 1 6.5 4 11 21 11 2 4 18 21 20 11 6 9 22 12 1 2 1 3 21 12 6 9 23 13 1 2 22 13 6 9 H 14 1 2 23 14 1 6 2 9 25 15 1 1 1 2 2 H 15 16 17 18 19 20 21 1 1 1 4.5 4 4 3 2 2 2 2 1 2' 1 6.5 6 6 5 3 3 3 3 26 27 28 29 30 31 32 33 15 15 175.5 41 17 458.5 Age at Entry , 10 Years. No. OP EM:rant8, 74. $157,000 Insured. 6 1; 36.5 19 1 78 10 AX 22 1 5 11 61.5 16 21 126.5 11 23 2 3 Slf 2 3 6 48 7 19 90.5 12 24 2 3 35 3 4 4 4 6 3 39 30.5 3 2 15 10 66.5 51 13 U 25 26 1 2 1 2 3 1 36 37 5 6 7 3 C 2 2^ 3! 24 19.5 13 6 13 4 1 3 43.5 35.5 21 15 16 17 27 JL 1 1 1 38 29 '49 309.5 63|85 624.5 8 9 1 11 11 10 8.5 8 7 6 1 17 17 16 13 11 10.5 10 18 19 20 21 22 23 24 Age at Entry, 12 Years. 10 11 12 13 14 1 1 -L 1 4 No. OP Entrants, 155. 1 ' 1 $296,000 Insured. 12 17 5 75 24 7 144.5 2 1 1 2 3 16 23 12 1 121.5 100 9 20 30 30 1 250 210 13 U 3 6 12 72 10 16 167 15 15 6 10 25 4 3 7 56.5 5 15 \ 141.5 16 16 17 18 19 2 6 4 4 4 2 10 8 8 8 26 27 28 29 5 7 4 48 6 7 125.5 17 6 14 1 38.5 19 3 114.5 18 7 8 9 2 1 23.5 14 29 37 1 93.5 64 19 20 20 4 8 30 9 2 12 3 27 21 21 22 23 24 2 2 4 4 2 2 2 6 8 8 6 6 31 32 33 10 11 12 13 1 1 1 1 1 9.5 8 1 7 : 6.5 5 2 3 5 1 21.5 14 12 11.5 22 23 H 25 41 33 373.5 82 75 687.0 14 1 4.5 2! 7 26 80 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 12 Years. Age at Entry, 14 Years. No. OP Entrants, 155. $396,000 INSTJBED. hr 13 . i fcb -a . 1 O =! go 5 «2 c $ O 3 o S ft" ■SB Expose to Risk $ $ 15 1 4 1 6 16 3 5 17 1 2.5 1 4.5 18 2 4 19 1 2 a 4 20 1 1 21 1 1 22 1 1 1 1 83 69 3 613.0 175 118 3 1,430.0 27 28 29 SO SI S2 S3 SJ, Age at Entry, 13 Years. No. OP Entbants, 235. $458,000 INSUBED. 14 8 1 113.5 34 17 2 220.5 1 16 40 2 192 24 82 1 364 2 8 18 145 21 35 280.5 3 11 16 120 32 38 223 4 16 16 93 20 22 161 5 13 10 1 64 12 24 4 118 6 11 6 1 42 18 16 1 82 7 5 1 2 26.5 10 5 3 52.5 8 4 2 1 18 8 4 1 35 9 2 1 11.5 4 1 23.5 10 2 2 8 2 4 17 11 1 5 5 13 12 4 8 13 1 4 1 8 14 3 7 15 3 7 16 3 7 17 1 3 3 7 18 1 2 1 4 19 1 3 20 1 3 21 1 3 22 1 3 23 1 1 3 3 105 120 10 865.5 192 248 18 1,653.0 13 u 15 16 17 18 19 20 21 22 23 25 26 27 28 29 30 31 No. OP Entrants, 722. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 28 26 38 40 45 50 38 34 27 18 8 4 2 1 3 1^ 375 36 117 60 44 26 18 9 10 4 1 327 20 a« 343 598.5 482 390 314 246 181.5 131 90 60.5 39 28 23 19 18 14 14 13 13 13 12 10 8 8 4 2 2 1 1 1 3,079.5 $1,491,000 Insured. $ 64 52 62 75 75 83 51 76 41 28 25 3 4 1 11 3 9 14 9 695 .2 -T- i C.5 $ 64 234 142 125 68 64 39 20 12 4 777 19 1^ h2 $ 713.5 1,244 1,002 802.5 630 487 351.5 270 178 129 95.5 67 63 58 57 '45 45 44 44 44 40 37 27 27 13 4 4 3 3 3 6,531.0 60 u 15 16 17 18 19 20 21 U 26 26 27 28 29 30 31 32 33 S4 35 36 37 38 39 US TABLE I. 81 OBSERVATIONS ON MALE LIFE. Age at Entry, 15 Years. :i NUMBEB OF Entbants, 1,086. $2,088,000 Ihsubed. Age »2S at Discon- Exposed to Bisls. Amounts Amounts Death Exposed to Risk. Exit. >'^ Existing. tinued. Died. Existing. Discontinued. Claims. i $ $ $ 46 58 1 514 112 93 2 997.5 IS 1 44 207 4 877.5 81 386 7 1,688 16 2 50 88 7 682 82 221 6 1,296.5 17 3 46 59 4 551.5 66 155 6 1,020.5 18 4 67 35 5 454.5 99 101 16 820.5 19 5 62 29 1 350.5 94 82 1 614 20 6 54 26 2 260 92 30 2 463 21 7 41 17 3 182.5 71 22 7 343 22 8 32 1 129.5 70 9 249.5 23 9 30 4 .95 77 4 173 2^ 10 12 a 1 62.5 10 2 1 93 25 11 13 3 1 47.5 13 2 1 80 26 12 2 32 2 65 27 13 3 1 29.5 7 4 61 28 14 2 2 25 6 2 51 29 15 1 22 2 44 30 16 2 21 2 42 31 17 1 1 19 4 1 40 32 18 1 17 2 35 33 19 16 33 3J^ 20 . 16 33 35 21 1 16 3 33 36 22 1 15 1 30 37 23 4 1 14 11 1 29 38 24 3 9 11 17 39 25 5 6 5 6 UO 26 1 1 U 27 1 1 1 1 J^ 524 531 31 4,466.5 924 1,113 51 8,359.5 Age at Entry, 16 Years. N [THBEB OF ] ENTRANTe , 1,659. $3,063,000 iNStTRBD. 16 118 100 5 779.5 206 174 6 1,444.5 1 93 318 7 1,277 189 524 12 2,415 17 2 95 139 7 948.5 193 287 17 1,808.5 18 3 61 93 5 730.5 112 237 11 1,336.5 19 4 83 -66 3 585 114 142 4 1,024 20 5 78 51 2 440.5 125 110 1 780 21 6 75 23 2 323.5 114 61 2 568.5 22 7 60 15 1 227.5 87 26 2 409 23 8 44 7 155.5 68 9 302.5 n 9 32 2 1 107 74 5 1 227.5 25 82 TABLE I. OBSERVATIONS ON MALE LIFE. |; Age at Entry, 16 Years. 1 of NnXBER OP Entrants, 1,659. $3,063,000 iNStJRED. Age ' Is at Dii»con- Exposed to Risk. Amounts ' Amounts ' Death Exposed to Risk. Exit. ■ >"& Existing. tiuued. ; Died. Existing. IDiscontin'ed. Claims. $ $ $ $ 10 18 4 1 .71 30 s 1 146 26 11 7 1 49,5 • 17 1 110.5 21 12 5 3 40.5 19 9 88.5 28 13 8 1 34 7 1 65 29 14 1 1 3; 24.5 1 2 2 56 30 15 ' 2 20: 3 52 ■31 16 3 1 17.5 12 ■2. : 48 32 17 3 i I 14 3 35 33 18 2 ; 11: 7 32 SJf 19 , ' 9! ; 25 ■ 35 20 [ 3 1 a 6 1 : 25 36 21 ! ^ 1 1 19 37 22 i; 6; i X 19 38 23 1 » 5; 5 1 : 18 39 1 24 2 i 1 4^ 11 j ; i 13 Jfi 25 2 i 1 2i 1 2 i 1 ! 2 \U1 ■ 796 824 39; . 5,897.5 1,405 1,597; ^ ll]070.0 \ Age at Ell try, 17 1 1 fears. ! N nUBBR' OF ENTBAKTf 5, 2,455. $4,692,000 INSUBED. i i 17 137 — ; — : — t 14^ f 9- 1,156.5 275 225' 17 2,'233.5 1 134 445 10-: 1,9445 I 250 813 12 3,768.5 18 2 126 23^ 10? 1,459 ' 220 554 16 2. 823 ■19 3 '95 142 9 1,133 168 314 13 2, 153 20 i 4 , 121 88 10: 916; 5 1 184 194 9 1, 718 21 5 124 70 71 709' 217 157 I'B 1, 349.5 22 6 134 39 6 523.5 ' 212 98 8 989 23 7 94 17 3 355.5 228 30 7 705 2U ; 8 61 8 1 246 120 14 6 448 25 i 9 64 4 1 178 119 6 1 312 26 10 30 2 111 49 4 189 27 11 11 2 2 78 20 5 1 133.5 .28 12 5- 2 63 12 2 109 29 13 . 8 3 55.5 13 3 94.5 30 14 4 2 46 8 4 80 \31 15 6 1 40 5 4 ., 68 '>32 16 33 59 ,33 17 4 1 32.5 3 1 I ■ 58.5 34 18 1 28 1 f ' i 55 35 19 3 1: 21^ 9 i i 54 \36 '20 4 1 23 4 4 44 37 21 2 18 7 36 •38 22 16 ■ ; 29 39 23 2 1 16 2 2 29 Jfi 24 6 13 12 25 Jfl TABLE I. 83 OBSERVATIONS ON MALE LIFE. Age at Entry, 17 Years. !i Number op Entrant 3, 2,455. $4,692,000 Insured. Age B at Discon- Exposed to Amounts Amounts Death Exposed to Risk. Exit. 1 >"& Existing. tinued. Died. Rigli. Existing. $ Discontin'cd. Claims. $ $ :$ 25 3 7 4 13 r^ 26 1 4 5 9 43 ■27 1 2.5 2 3 u 28 3 2 2 2 45 1,182 1,197 76 9,237.0 2,149 2,418 125 17,590.0 Age at Entry, 18 Years. N UMBER OF Entrants , 3,973. $7,466,000 INSUKED. 18 288 291 9 1,841 563 466 14 3,500 1 225 832 21 2,969 386 1,413 36 5,716.5 19 2 181 329 21 2,142.^5 341 813 42 4,181.5 20 3 175 232 9 1,660 263 463 14 3,160.5 21 4 178 139 7 1,290.5 292 303. 16 2,500.5 22 : 5 174 113 11 979.5 322 253; 14 1,914.5 23 6 198 47 5 714.5 362 lor 7 i;398.5 24 ! 7 123 30 5 473 279 50' 14 951 25 8 111 20 1 320 194 32 2 617 26 9 66 1 2 197.5 129 6 6 402 27 10 39 6 2 126 ;74 11' iB 258.5 28 11 16 4 80 ^5 Hi ! 165.5 29 12 3 1 61.5 ; 5 3! r 123.5 SO 13 5 58 9 1 i 117 31 ;14 9 1 1 52.5 :21 2' n 107 32 15 7 42 10 84 33 16 3 2 35 2 7 74 34 17 5 1 30 4 4 65 35 18 1 24 1 57 36 19 5 23 10 56 37 20 2 18 3 46 38 21 1 16 6 43 39 22 2 15 3 37 40 23 2 1 13 6 ; 4 34 Al 24 3 10 8 24 42 25 2 7 3 16 43 26 2 1 5 3 4 13 U 27 1 2 ■ 4 6 45 28 1 1 . 2 2 46 1,827 2,046 100 13,20^.5 3,336 3,933 197 25,670.5 84 TABLE I OBSERVATIONS ON MALE LIFE. Age at Entry, 19 Years. H NUMBEB OP Entrant s, 6,365. $11,576,000 INSUBED. Age §§ at ^3 Existing. Discon- tinued. Died. Exposed to Risk. Amounts Existing. Amounts Discontin'ed. Death Claims. Exposed to ffisk. Tlxit. $ $ $ $ 387 559 25 2,903 713 875 37 5,350.5 19 1 363 1,451 39 4,668.5 625 2,497 78 8,702.5 20 2 294 526 26 3,278 521 1,024 65 6,239 21 3 277 353 13 2,518.5 435 825 24 4,728.5 22 4 273 191 15 1,956.5 419 456 34 3,629 23 5 298 121 10 1,512.5 478 • 256 39 2,820 n 6 285 80 20 1,104 498 191 18 2,079.5 25 7 219 48 11 735 353 91 30 1,422.5 26 8 146 16 2 473 304 39 4 974.5 27 9 102 8 1 313 218 18 1 638 28 10 48 7 2 202.5 94 24 3 398 29 11 34 5 1 146.5 65 10 2 284 30 12 10 3 1 107.5 24 7 3 208.5 31 13 11 1 2 94.5 26 4 1 176 32 14 6 4 1 79 12 5 3 144.5 33 15 8 1 69.5 14 2 126 S4 18 7 1 3 60.5 9 5 8 108.5 35 17 5 50 5 89 36 18 4 45 8 84 37 19 6 1 41 5 4 76 38 20 6 1 2 33.5 8 4 5 65 39 21 5 1 25 7 3 50 UO 22 5 1 1 18.5 4 1 5 39.5 Jfl 23 6 12 20 30 42 24 2 6 4 10 43 25 2 4 2 6 U 26 2 2 4 4 Jf5 2,811 3,377 177 20,459.0 4,875 6,334 367 38,483.0 Age at En try, 20 1 fears. Ni JMBEB OF 1 Sntkants , 10,110. 119,352,000 Insubsd. 20 704 817 38 4,646.5 1,324 1,415 58 8,918.5 1 562 2,297 59 7,402.5 1,085 4,232 112 14,339 21 2 464 790 43 5,238 800 1,769 64 10,141.5 22 3 438 589 31 4,041.5 678 1,359 51 7,713.5 23 4 450 368 18 3,094 721 954 45 5,828 U 5 420 191 26 2,346.5 713 421 52 4,374.5 25 6 414 131 18 1,739.5 697 291 33 3,253.5 26 7 361 81 8 1,201.5 671 141 19 2,307.5 27 8 245 27 7 778.5 455 58 21 1,518 28 9 164 22 2 502 297 39 4 993.5 29 1 TABLE I 85 OBSERVATIONS ON MALE LIFE. Age at Entry, 20 Years. ^i Number op Entbants , 10,110. $19,252,000 Insured. Age £ 2 " at ►2S Discon- Exposed to Risk. Amounts Amounts Death Exposed to Exit. >"& Existing. tinued. Died. Existing. Discontin'ed. Claims. Risk. $ $ % $ 10 77 11 3 319.5 161 22 3 662 SO 11 36 6 5 231 66 21 13 476.5 31 12 25 6 3 184 52 12 11 381 32 13 9 1 3 152.5 12 6 3 309 33 14 16 2 139 43 4 289 3U 15 27 . 122 45 244 35 16 14 1 94.5 31 4 197 36 17 15 80 20 164 37 18 • 2 3 63.5 2 6 141 38 19 10 1 59.5 17 1 135.5 39 20 6 49 13 118 Jfi 21 5 2 42 15 3 103.5 Jfl 22 5 1 35.5 7 1 86.5 ^ 23 10 30 31 79 kS 24 4 2 1 19 11 2 4 47 u 25 6 13 8 31 Jt5 26 4 7 16 23 ^6 27 2 3 . 4 7 Ul 28 1 3 Jt8 29 1 1 3 3 Jfi 4,496 5,349 265 32,636.5 7,998 10,761 493 62,887.5 Age at Ee itry, 21 1 fears. Nr ruBEB or I Intrants 14,896. 130,749,000 Insured. 21 1,058 1,094 49 6,901 2,018 2,125 62 14,312 1 794 3,382 68 11,004 1,628 6,571 115 23,258.5 22 2 672 1,183 70 7,859.5 1,428 2,948 184 16,756 23 3 662 852 42 6,100 1,194 1,994 89 12,673 2^ 4 697 495 40 4,722.5 1,170 1,224 68 9,781 25 5 622 318 24 3,579 1,254 833 40 7,514.5 26 6 638 176 25 2,686 1,179 428 66 5,590 27 7 503 133 26 1,868.5 1,086 252 54 4,005 28 8 360 48 13 1,249 777 113 31 2,682.5 29 9 265 21 5 841.5 546 61 7 1,787.5 30 10 108 18 2 552 236 44 4 1,182 31 11 79 8 2 429 154 22 8 909 32 12 28 4 6 342 66 11 12 730.5 33 13 40 8 2 302 95 26 1 634 SJf 14 33 3 3 254.5 56 6 8 522 35 15 28 2 2 216 62 4 3 453 36 16 35 2 1 184 78 2 1 385 37 17 26 2 1 146 41 6 4 302 38 18 15 1 1 117.5 29 3 2 252.5 39 19 23 1 2 100.5 62 1 5 219.5 UO 86 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 21 Years. 20 21 ^2 23 24 25 26 27 28 29 Nfmbbb op Entbants, 14,896. Existing. 13 4 10 6 21 6 4 2 1 1. 6,754 Discon- tinued. DlQd. 7,754 388 Exposed to Risk. 75 61 56 45 38 17 10 5 3 1.5 49,766,0 $30,749,000 Insured. Amounts j Amounts Death Existing. Discontin'ed. Claims. 15 8 23 14 44 14 9 8 3 3 13,300 16,678 771 Exposed to Rislc. $ 152 133 124 100 85 41 26.5 17 9 4.5 104,641.5 Age at Exit. 41 4S U 45 46 47 48 49 50 Age at Entry, 22 Years. 1 2 \Z |4 is 6 ;7 !9 io ii i2 13 14 15 16 17 18 19 20 21 22 23 24 is 26 27 28 29 io NuMBEB cJf Entrants, 19,955. 1,263 1,554 57 1,019 4,603 114 913 1,627 75' 812 1,057 58. 895 594 48 827 437 44 869 244 32 700 192 24 515 73 16 332 28 12 218 31 8 132 16 6 41 12 4 54 6 7 47 9 56 5 3 31 3 3 40 1 2 40 2 3 13 1 1; 17 2 10 1; 7 2 28 1 23 1 li 16 ( 6' ! 1' 4 ! i 2 i i 1 ! 3 1 ■ i 8,934 10,501 520 9,200.5 14,779.5 10,531.5 8,201.5 6, .506 5,047.5 3,836 2,717 1,860.5 1,279 905.5 . 656 504 450 381.5 327.5 264.5 -228.5 185 140.5 125 107 95 i 86; 5 57j5 33; 17i lOi 6' 4 68,546.0 $42,124,000 Insured. 2,690 2,004 1,745 1,452 1,749 1,684 1,830 1,364 1,240 786 532 282 94 130 • 96 113 73 74 90 23 40 19 16 76 56 32 17 12 6 3 18,333 2,945 89 9,035 193 4,058 158 2,712 111 1,482 128 1,020 96 640 72 376 55 177 31 80 28 72 19 30 14 33 9 12 22 15 7 3 5 2 4 1 11 5 1 5 11 3 7 4 3 2 5 22,740 1,051 19,589.5 31,882.5 2S,139 17,851 14,191 11,063 8,453 6,043 4,347.5 2,948 2,058 1,456 1,128.5 1,003 837.5 730.5 608.5 529 446.5 345.5 311.5 266 240.5 219 139.5 80 :48 26 14 8 150,008.0 23 H 25 26 27 28 29 30 31 32 33 S4 35 37 38 39 40 ^^ 42 43 44 45 Jfi 47 48 49 50 51 52 TABLE I. 87 • OBSERVATIONS ON MALE LIFE. i Age at Entry, 23 Years. NUMBEB OP ] Entbants , 24,574. $54,330,000 INSUEED. Age at Exit. Discon- Exposed to Risli. Amounts Amounts Death Exposed to Bisk. ,^^ Existing. tinued. Died. Existing. Discontin'ed Claims. i $ ■ $ $ 1,629 1,748 62 11,413 3,459 3,349 122 25,490.5 23 ; I 1,379 5,393 113 18,438.5 2,926 10,977 253 41,911.5 H 2 1,105 1,897 96 13,301.5 2,135 5,367 257 30,560.5 26 3 1,025 1,319 85 10, 491 5 2,008 3,475 176 23,747.5 26 ! 4 ■ 1,195 792 55 8,327 2,348 2,172 149 18^740 27 5 1,093 497 52 6,432.5 2,283 1,305 136 14.504.5 28 : 6 968 325 42 4,876.5 2,001 816 112 li;025 ■29 7 829 243 17 3,582.5 1,756 495 30 8,256.5 30 : 8 689 83 29 2,573.5 1,587 206 66 6-120 \81 9 450 49 16 1,789.5 1,008 117 53 4[305:5 '32, 10 280 34 14 1,282 683 103 37 3,134.5 [33 11 150 31 12 955.5 362 81 50 2,322.5 •Si 12 113 ] L3 1^ 771.5 254 29 44 1,855.5 \35 il3 62 9 i 634.5 153 14 7 1,536 36 il4 79 7 f 56(j).5 244 11 f 1;363.5 \37 15 55 8 2 470 122 It 8 1,096.5 38 16 54 ;7 1 405.5 144 11 :957.5 \39 17 48 ■ ;3 t 347 91 ^ 3S i805.5 .40 118 36 3 4 289.5 59 4 3 ;675 M il9 I 33 3 t 246.5 78 ^ 1603 \Jt2 20 34 3 209 101 7 517 J^3 '21 14 1 3 171.5 30 5 9 406.5 U .22 21 2 4 153 32 3 16 363.5 ¥ 23 33 1 126.5 99 4 312 46 24 33 2 93 85 211 47 25 '■ 28 . 1 58 52 5 119 \4S |26 10 29 18 . 62 49 27 10 19 31 . 44 50 ,28 7 9 11 13 51 29 2. 2 2 2 52 11,464 12,467 643 88,059.5 24,162 28,572 1,596 201,061.0 * 1 Age ati En try, 24 1 rears. j 1,967 UBEB OF E NTB&NTS 28,504. $64,563,00( ) Insubed. 1 n 1,895 57 13,304.5 4,230 3,793 103 30,385 1 1,546. 5,871 135 21,649.5 3,321 12,504 305' 50,185 ■25 2 1,388 2,225 119 15,920.5 3,113 6,330 296 37,142 26 3 1,225 1,400 108 12,601 2,323 3,768 225 28,684 27 4 1,344 890 74 10,123 2,502 2,456 172 23,024 28 5 1,278 599 65 7,966.5 2,645 1,608 150 18,318 29 i 6 1,246 369 42 6,133.5 2,686 981 103 14,228.5 SO 1 7 1,012 308 27 4,507 2,167 580 59 10,659 31 1 8 856 124 15 3,252 2,111 292 44 7,997 ,32 ! 9 580 63 12 2,287.5 1,325 140 33 5,626 S3 88 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 24 Years. It NXT3CBEB OF Entkants i, 28,504. $64,563,000 IHSUBBD. Age at Exit. Existing. Discon- tinaed. Died. Exposed to Risk. Amounts Existing. Amounts Discontin'ed Deatti Claims. Exposed to Risk. $ $ $ $ 10 394 57 13 1,635.5 1,006 119 41 4,138.5 Si 11 192 28 10 1,186 495 84 16 2,990 35 12 108 29 11 955.5 304 75 29 2,399.5 36 13 89 12 8 816 233 33 29 2,012.5 37 14 93 10 13 708 211 10 33 1,729 38 15 80 10 5 592 187 23 15 1,468.5 39 16 63 6 5 499 191 20 10 1,245 1^0 17 58 5 2 425.5 126 14 7 1,027 ^1 18 47 2 1 362 132 5 3 884.5 42 19 46 1 3 312.5 129 4 9 745 JfS 20 27 2 2 262 64 2 10 604 U 21 20 1 3 231.5 45 2 10 528 ^ 22 25 1 207.5 54 3 470.5 J^6 23 31 2 4 181 72 7 7 * 411.5 47 24 38 3 145 76 5 329 Jf8 25 36 1 104 94 2 248 49 26 31 2 66 69 8 148 50 27 17 1 34 40 - 5 75 51 28 6 1 16 14 4 30 52 29 2 9 5 12 53 30 7 7 7 7 54 13,852 13,912 740 106,494.0 29,977 32,861 1,725 247,751.0 Age at En try, 25 1 fears. Ni nCBSB OF I Cntbakts 32,680 $76,582,000 INSUBED. 25 2,209 2,120 84 15,280 4,998 4,172 168 36,205 1 1,840 6,496 137 25,019 4,184 14,301 338 60,093.5 26 2 1,585 2,396 136 18,596 3,388 6,966 299 44,938 27 3 1,513 1,606 127 14,874 3,182 4,307 276 35,614.5 28 4 1,575 964 110 11,949 3,479 2,834 293 28,586 29 5 1,562 688 66 9,438 3,422 1,927 218 22,433.5 30 6 1,427 427 56 7,252.5 3,066 1,209 157 17,225.5 31 7 1,261 341 .<7 5,385.5 2,900 777 116 13,009.5 32 8 972 149 34 3,832.5 2,240 380 88 9,415 33 9 618 82 20 2,711 1,747 197 49 6,798.5 34 10 443 49 18 2,007.5 1,034 124 56 4,842 35 11 239 32 11 1,506 605 66 32 3,657 36 12 137 28 14 1,226 348 53 33 2,960.5 37 13 116 22 13 1,050 321 35 27 2,535.5 38 14 116 19 8 900.5 304 45 33 2,147.5 39 15 102 15 9 759.5 281 30 23 1,773 40 16 77 7 9 637.5 189 23 24 1,442.5 41 17 70 4 3 546 167 17 8 1,209.5 42 18 46 4 10 469 67 14 29 1,019 43 19 49 3 4 409.5 112 7 13 912.5 U TABLE I. 89 OBSERVATIONS ON MALE LIFE. Age at Entry, 25 Years. taa NUMBBB OP ] Entrants , 32,680. $76,582,000 Insured. Age at Exit. Discon- Exposed to Bisk. Amounts Amounts Deatli Exposed to Risli. >*5 Existing. tinued. Died. Existing. Discontin'ed Claims. $ $ $ $ 20 26 1 2 354.5 79 5 7 781.5 45 21 31 1 3 325.5 77 1 5 692.5 Jf6 22 27 1 3 290.5 61 8 4 606 Jf7 23 44 1 3 259.5 142 1 4 536.5 Jf8 24 46 1 1 211.5 90 4 1 388 W 25 66 4 164 72 16 295 50 26 47 4 94 95 17 207 61 27 22 1 43 58 5 95 62 28 15 20 19 32 53 29 2 5 4 13 6U 30 3 3 9 9 55 16,286 15,457 937 125,619.5 36,740 37,503 2,339 300,473.5 Age at Ee itry, 26 " fears. Ni TUBER OI- 1 Sntbants , 34,138. $80,336,000 Insured. m 2,265 2,182 82 15,978 5,018 4,022 182 38,107 1 1,864 6,541 175 26,338.5 4,258 14,556 394 63,736 27 2 1,592 2,560 142 19,749 3,498 7,161 348 48,225.5 28 3 1,530 1,647 111 15,911.5 3,244 4,517 277 38,540.5 29 4 1,561 1,111 115 12,891.5 3,538 2,985 280 31,268.5 SO 5 1,506 776 94 10,272 3,391 2,194 216 24,861 31 6 1,451 488 70 8,040 3,213 1,353 169 19,480.5 32 7 1,304 396 61 6,077 3,074 931 174 14,956.5 33 8 1,045 181 37 4,423.5 2,523 462 115 11,012 SU 9 750 100 32 3,201 1,813 234 78 8,026 35 10 498 66 17 2,336 1,213 194 36 5,921 36 11 318 41 12 1,767.5 755 102 32 4,524 37 12 126 29 16 1,402.5 327 74 29 3,649 38 13 124 22 11 1,235 329 57 25 3,227.5 39 14 170 20 12 1,079 426 63 29 2,813.5 40 15 130 10 2 882 359 18 5 2,318 41 16 72 9 8 740.5 184 31 23 1,929.5 42 17 90 6 3 653 208 13 9 1,700.5 43 18 51 3 3 555.5 126 8 10 1,473 44 19 65 2 2 499 209 7 2 1,329.5 45 20 44 4 2 429 111 14 4 1,108 46 21 31 2 5 380 69 3 17 984.5 47 22 32 2 6 342 75 3 17 895.5 48 23 50 2 4 302 128 10 4 797 49 24 65 3 4 245.5 179 5 12 657.5 60 90 TABLE I OBSERVATIONS ON MALE. LIFE. 1 Age at Entry, 26 Years. i NcMBER OF Entrants 34,1.38. $80,236,000 Insured. Age at Discon- Exposed to Bisk. Amounts Amounts Death Exposed to Risk. >*A Existing. tinued. Died. Existing. $ Discontin'ed. Claims. $ $ $ 25 63 3 175 135 18 464 51 26 44 1 1 108.5 123 1 5 310.5 52 27 29 1 1 62.5 76 4 4 180 53 28 17 1 32 46 5 98 ' H 29 9 14 24 • 47 55 30 5 i ; 5 23 t i 23 56 16,901 16,205 1,032 136,127.5 38,695 39,022 2,519 332,664.0 i Age at Entry, 27 Years. « Nt rHBER OF I ! Intrants , 36,221. 1 $87,030,000 Insured. 27 2,274 2,050 81 17,085.5 ' 5,478 4,344 189 41,343 1 2,019 6,753 201 28,439.5 4,678 15,314 479 69,362 28 2 1,660 2,877 150 21,404.5 1 3,687 8,413 367 52,341.5 29 3 1,689 1,672 128 17,320 i 3,604 4,741 306 41.710.5 30 4 1,703 1,168 126 14,083 ; 3,634 3,360 307 33,750 31 5 1,651 776 90 . 11,282 3,578 2,224 247 27,017 32 6 1,658 473 63 8,916.5 3,781 1,270 188 21,445 33 7 1,413 431 53 6,743.5 ' 3,225 984 118 16,349 3U 8 1,213 201 36 4,961.5 2,883 489 94 12,269.5 35 9 873 120 32 3,552 ; 2,133 300 77 8,898 36 10 559 77 15 2,548.5 ! 1,343 222 50 6,427 37 11 347 45 22 1,913.5 814 118 63 4,864 38 12 150 35 19 1,504.5 371 81 43 3,887.5 39 13 140 33 15 1,301.5 381 86 31 3,390 40 14 161 23 12 1,118.5 410 74 31 2,898 Ul 15 99 11 8 928.5 239 20 22 2,410 J^ 16 78 11 7 810.5 196 24 17 2,127 US 17 74 4 7 718 154 5 24 1,899.5 u 18 63 11 5 629.5 179 28 8 1,705 Jf5 19 55 \ 3 3 554.5 1 158 11 13 1,498.5 46 20 42 5 1 492.5 132 17 2 1,313.5 47 21 41 3 5 445.5 100 5 10 1,168.5 48 22 41 3 4 396.5 1 104 6 10 1,053 49 23 47 3 4 348.5 111 15 11 928.5 50- 24 79 1 2 295.5 174 4 7 797 51 25 71 1 4 213.5 188 3 9 612.5 52 26 58 3 138 164 2 414 53 27 36 4 77 118 17 248 54 28 16 1 37 38 1 113 55 29 9 20 24 74 56 30 9 2 11 44 6 50 57 18,328 16,790 1,103 148,290.5 42,123 42,158 2,749 362,364.0 TABLE I. 91 OBSERVATIONS ON MALE LIFE. Age at Entry, 28 Years, i Number of Enthanm , 41,636. $93,734,000 Insured Age at Exit. Discon- Exposed to Amounts Amounts Death Exposed to >*A Existing. tinued. Died. Risk. Existing. Discontin'ed. Claims. Risli. $ $ $ $ 2,457 2,052 91 19,792 5,979 4,193 165 44,770.5 28 1 2,078 7,044 178 33,514 5,178 16,133 415 75,330.5 29 2 4,905 2,917 153 26,277.5 3,940 8,513 423 57,414.5 SO 3 1,696 1,797 143 18,862.5 3,631 5,175 318 46,207.5 31 : 4 1,834 1,210 132 15,520 3,972 3,598 258 37,872 32 i 5 1,705 815 113 12,541.5 3,695 2,382 285 30,652 33 \ 6 1,691 527 97 10,052.5 3,701 1,458 228 24,752 3J^ \ 7 1,565 478 80 7,762 3,759 1,281 187 19,453.5 35 8 1,330 210 50 5,773 3,299 557 128. 14,588.5 36 9 983 129 43 4,223.5 2,392 339 114 10,713.5 37 10 685 78 31 3,094 1,670 212 69 7,932 38 11 407 45 19 2,316.5 986 156 42 6,009 39 12 223 37 21 1,849.5 565 117 49 4,844.5 40 13 164 30 12 1,572 459 89 34 4,127.5 U 14 182 26 14 1,368 488 65 38 3,557.5 42 15 143 is' 12 1,151.5 381 33 23 2,982.5 43 16 98 16 10 981 239 34 38 2,545 U 17 100 7 7 861.5 219 34 20 2,234 4^ 18 102 7 8 747.5 260 18 26 1,969 46 19 65 5 7 631.5 197 4 20 1,672 47 20 48 4 8 555 120 13 26 1,446.5 48 21 40 6 3 494 84 15 5 1,286.5 49 22 45 3 7 446.5 118 7 21 1,186.5 60 23 54 1 6 392.5 146 4 10 1,042 51 24 94 4 4 330 243 7 15 880.5 52 25 74 2 5 229 167 12 20 613 53 26 63 3 1 147.5 187 1 6 419.5 54 27 38 2 82 119 . 9 226 55 28 24 1 42 66 f 1 98 56 29 11 17 8 31 57 30 5 6 18 23 58 31 5 59 32 5 60 33 5 61 34 5 62 35 5 63 36 5 64 37 5 65 38 5 66 39 1 -5 5 67 22,909 17,468 1,259 171,642.0 46,286 44,450 2,998 406,925.0 92 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 29 Years J* 5 NlTMBBB OP ] 3NTHANT8 , 38,787. $96,811,000 Iksubed. Age at Discon- Exposed to Risk. Amounts Amounts Death Exposed to Risk. >'A Existing. tinaed. Died. Existing. Discontin'ed. Claims. ri ft 1 ti $ S $ $ 2,483 2,017 86 18,385 6,013 3,356 217 46,727.5 29 1 2,119 6,783 181 30,809.5 5,078 15,605 470 79,422.5 SO 2 1,829 2,959 147 23,638.5 4,258 8,724 388 61,710 31 3 1,557 1,842 124 19,262 3,940 5,801 326 49,801.5 32 4 1,821 1,228 111 16,046 4,189 3,614 306 40,828 33 5 1,770 884 133 13,058 4,154 3,082 360 32,985 Si 6 1,765 592 79 10,417 4,321 1,848 241 26,006 35 7 1,631 476 59 8,039 4,005 1,151 141 19,944.5 36 8 1,399 204 53 6,009 3,398 599 121 14,923.5 37 9 1,081 135 34 4,387.5 2,574 343 69 10,933.5 38 10 734 88 22 3,161 1,779 239 56 7,999.5 39 11 448 55 16 2,333.5 1,133 146 35 5,972 40 12 193 40 17 1,822 444 103 37 4,679.5 U 13 158 22 13 1,581 398 43 35 4,125.5 J^ 14 184 36 16 1,381 459 86 30 3,628 JfS 15 130 12 7 1,157 356 33 14 3,079.5 u 16 94 10 9 1,009 229 28 47 2,679 45 17 91 12 8 895 247 23 14 2,377.5 46 18 78 6 6 787 188 5 9 2,102.5 47 19 59 6 6 697 210 18 14 1,894 48 20 50 9 7 624.5 127 14 21 1,654 49 21 44 6 4 560 111 17 14 1,490.5 50 22 44 5 5 506.5 127 17 28 1,348.5 51 23 71 2 8 454 163 6 14 1,182 52 24 90 1 8 373.5 246 4 13 1,000 53 25 109 3 1 273.5 268 4 4 737 54 26 57 1 2 161.5 162 3 8 461.5 55 27 58 2 101 167 1 289.5 56 28 17 1 42 50 3 122 57 29 15 1 23.5 41 4 67 58 30 8 8 24 24 59 20,187 17,437 1,163 168,003.0 48,859 44,917 3,035 430,195.0 Age at Ee itry, 30 1 fears. Ni nCBEB OF ] Sntramts , 42,300. $105,978,000 Insubbd. 30 2,663 2,184 89 20,058 6,517 4,702 218 50,638 1 2,279 7,141 226 33,793.5 5,266 17,016 593 86,033 31 2 1,952 2,998 176 26,219 4,622 9,145 466 67,093.5 32 3 1,855 1,937 163 21,623.5 4,052 5,665 437 54,600.5 33 4 2,125 1,317 150 17,978.5 4,528 4,158 398 45,200 34 5 1,931 995 92 14,547.5 4,774 2,974 271 36,708 35 6 1,862 650 119 11,702 4,748 1,926 293 29,213 36 7 1,802 616 80 9,138 4,358 1,347 241 22,535.5 37 8 1,648 228 45 6,884 3,834 672 136 16,927 38 9 1,264 155 45 4,999.5 2,896 371 131 12,435.5 39 TABLE I. OBSERVATIONS ON MALE LIFE. 98 Age at Entry, 30 Years. it Number op ] Entrants , 42,300. 1105,978,000 Insured. Age at 'Rrit Discon- Exposed to Risk. Amounts Amounts Death Exposed to Risk. >'A ExiBting. tinued. Died. Existing. Discontin'ed Claims. $ $ $ $ 10 805 118 33 3,554 1,868 293 90 9,076.5 40 11 547 60 21 2,627 1,284 150 52 6,897 41 12 215 41 19 2,008.5 544 122 40 5,425 42 13 179 25 14 1,741.5 460 69 43 4,745.5 43 14 182 33 20 1,519.5 486 90 50 4,163 44 15 160 21 12 1,290.5 468 66 37 3,549 45 16 102 10 12 1,103 302 36 52 2,993 46 17 95 12 9 978 263 40 35 2,601 47 18 69 8 7 864 175 19 10 2,273.5 48 19 72 5 7 781.5 174 12 19 2,073 49 20 65 4 4 698 163 22 13 1,863 50 21 49 2 4 626 139 4 / 9 1,674 51 22 57 2 5 571 138 7 23 1,520.5 52 23 98 4 8 506 243 6 23 1,353 53 24 116 2 6 397 286 11 12 1,078.5 54 25 93 1 6 273.5 241 4 23 773 55 26 73 3 174 219 10 507 56 27 45 1 1 97.5 152 4 1 276 57 28 29 2 50 67 2 120 58 29 13 20 33 52 59 30 7 7 19 19 60 22,452 18,472 1,376 186,831.0 53,319 48,933 3,726 474,416.5 Age at Ee itry, 31 ^ fears. Ni TKBEB OF I Sntbants , 39,923. $100,042,000 Insured. 31 * 2,450 1,857 90 19,033 6,132 4,048 253 47,997 1 2,197 6,386 196 32,333 5,549 14,806 476 82,206 32 2 1,870 2,747 175 25,373.5 4,235 8,270 517 64,643 33 3 1,872 1,851 162 21,029.5 4,198 5,522 447 52,995 3^. 4 1,813 1,276 149 17,432 4,300 3,858 438 43,660 35 5 1,867 917 125 14,373.5 4,657 2,842 348 35,572 36 6 1,862 622 105 11,612 4,499 1,818 225 28,237 37 7 1,625 532 84 9,068 3,908 1,331 216 21,938.5 38 8 1,578 206 52 6,990 3,696 646 150 16,826 39 9 1,229 132 36 5,191 2,938 338 114 12,488 40 10 1,142 103 28 3,808.5 2,040 273 81 9,130.5 41 11 479 63 24 2,555.5 1,139 155 58 6,795.5 42 12 228 38 21 2,002 591 104 51 5,469 43 13 152 24 26 1,722 540 67 77 4,741.5 44 14 207 27 15 1,518.5 613 75 35 4,053.5 45 15 150 18 15 1,274 374 39 40 3,348.5 46 16 119 18 17 1,091 334 40 50 2,895 47 17 99 8 15 942 234 29 49 2,476.5 48 18 88 7 8 820.5 222 14 22 2,172 49 19 63 8 10 717 165 14 25 1,914 50 94 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 31 Years. li Ndmbeb of Entrants , 39,9-23. ! $100,042,000 Insured. Age S3 1 at Existing. Discon- tinued. Died. Exposed to Risk. Amounts Existing. Amounts Discontin'ed. Death Claims. Exposed to Bisk. Exit. i $ $ $ $ 20 66 4 4 638 i 199 8 14 1,713 51 21 44 2 11 565 1 124 4 26 1,494 52 22 42 5 10 506.5 111 9 27 1,337.5 53 23 85 4 8 450 205 11 32 1,189.5 5J, 24 95 9 355 230 26 947 55 25 82 1 2 250.5 201 3 10 689.5 56 26 80 2 166 214 5 477 57 27 43 3 84 122 5 258 58 28 . 20 38 95 131 59 29 11 18 26 36 60 30 7 7 10 10 61 21,665 16,856 1.402 181,964.5 51,901 44,324 3,817 457,841.0 Age at Entry, 32 Years. Nt 7XBBR OF I 5NTRANT8 , 40,069. j $101,772,000 Insured. 32 2,428 1,868 95 19,110.5 ■ 6,080 4,061 244 48,855.5 1 2,037 6,409 187 32,493.5 5,163 15,640 459 83,567 33 2 1,916 2,813 189 25,658.5 4,583 8,484 543 65,883 SJ^ 3 1,805 1,776 168 21,259 4,000 5,326 434 53,852 35 4 1,943 1,316 139 17,740 4,488 3,985 368 44,762.5 36 5 1,856 905 118 14,547.5 4,742 2,556 288 36,636 37 6 1,860 589 125 11,826.5 4,435 1,742 339 29,457 38 7 1,841 486 100 9,304 4,332 1,273 297 23,175.5 39 8 1,582 230 61 7,005 4,084 644 156 17,588 Jfi 9 1,288 . 161 48 5,166.5 2,997 410 133 12,821 U 10 833 91 30 3,704.5 2,055 224 87 9,374 Jt2 11 504 87 23 2,752.5 1,104 159 77 7,040.5 JfS 12 261 38 21 2,163 626 115 56 5,722.5 U 13 187 24 19 1,850 456 72 57 4,947 45 14 197 28 16 1,618 494 81 36 4,357.5 i6 15 145 13 14 1,384.5 389 33 37 3,770.5 ^7 16 105 9 16 1,214.5 262 7 44 3,324.5 J,8 17 90 12 18 1,083 i 237 43 32 2,993.5 Jfi 18 82 10 11 964 252 31 33 2,687.5 50 19 61 9 8 861.5 159 24 27 2,375 51 20 93 3 9 786.5 313 21 36 2,166.5 52 21 47 6 12 680 127 11 30 1,801.5 53 22 59 3 10 . 616.5 162 8 20 1,635 5U 23 65 1 1,3 545.5 130 4 39 1,447 55 24 117 2 3 466 310 6 6 1,273 56 25 129 1 3 344.5 299 1 10 953.5 57 26 99 2 4 211 279 10 14 639 58 27 62 1 106.5 176 3 339.5 59 28 25 1 54 84 4 162 60 29 11 28 23 74 61 30 17 17 51 51 62 21,735 16,893 1,461 185,562.0 52,892 44,974 3,906 473,732.0 TABLE I. OBSERVATIONS ON MALE LIFE. ,05 Age at Entry, 33 Years. eta 1 2 3 4 5 6 7. 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Number of Entbants, 40,498. „ . . . Discou- T»j„j Exposed to Existing. tinned. Died. ^^g^ 2,349 2,181 1,776 1,843 2,000 1,866 1,958 1,920 1,676 1,282 756 524 240 161 187 130 86 80 109 79 57 63 41 77 117 127 83 38 34 4 21,848 1,872 6,484 2,845 1,856 1,292 900 577 525 232 142 108 63 32 30 31 9 12 10 6 4 4 4 5 2 6 1 1 17,052 108 219 218 173 161 132 97 95 73 53 51 26 19 25 15 21 13 17 7 14 9 7 14 10 4 6 5 4 2 1,598 19,313 32,927 25,862.5 21,518 17,928 14,671 11,934.5 9,328.5 6,935 4,999 3,539 2,646.5 2,049 1,759 1,542.5 1,320.5 1,159 1,049 944.5 824 727 657 582.5 524 433 309 175.5 86.5 44 «104,474,000 INSXTRBD. 185,800.0 Amonnts Existing. 6,073 5,438 4,580 4,327 4,821 4,652 4,900 4,549 4,057 3,117 1,793 1,262 522 433 543 359 223 171 280 216 136 151 99 180 269 286 223 126 100 13 20 Amounts Discontinued I 53,919 $ 4,107 15,517 8,877 5,817 3,966 2,815 1,824^ 1,252 668 385 261 157 84 82 91 48 49 22 6 16 11 19 21 11 11 2 4 Deatli Claims. $ 232 584 604 511 502 385 264 223 197 131 118 102 49 75 55 69 35 39 41 30 26 30 33 43 10 14 13 15 2 Exposed to Bislc. 46,123 I 4,432 50,183.5 86,303.5 68,084.5 55,553.5 45,824 37,110.5 29,754 23,052 17,320 12,539.5 8,968.5 6,848.5 5,364 4,710 4,115.5 3,448 2,971.5 2,678 2,454 2,122 1,862.5 1,685.5 1,484.5' 1,336.5 1,102.5 818 517 278 135 33 20 478,677.5 Age at Exit. 33 34 35 36 37 38 39 J,0 U 42 43 U 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 Age at Entry, 34 Years. NuMBBB OF Entrants, 2,491 1,669 99 2,026 6,097 192 1,828 2,895 163 1,813 1,791 174 1,864 1,258 169 1,976 836 139 1,977 574 109 1,975 476 85 1,599 249 62 1,237 144 49 18,977.5 32,316.5 25,602.5 21,268.5 17,757 14,677 11,857 9,246 6,823.5 4,966 $102,664,000 Insubbd. 6,401 5,508 4,435 4,285 4,395 4,849 5,160 4,937 4,025 2,967 3,469 262 14,753 497 8,303 494 5,707 481 3,899 446 2,729 402 1,696 305 1,224 223 683 155 422 118 i 49,597.5 85,155.5 67,622.5 55,688.5 46,119.5 37,964.5 30,501 23,576 17,462.5 12,730 34 35 36 37 38 39 40 41 42 43 TABLE I. OBSERVATIONS ON MALE LIFE. Age at EnU7, 34 Years. 9 NUMBEB OP Bntbants , 39,624. • 1103,664,000 Inscbed. Age at Exit. Existing. Discon- tinued. Died. Exposed to Risk. Amounts Existing. Amounts Discontin'ed. Death Claims. Exposed to Risk. $ $ $ s 10 789 98 45 3,559 1,972 290 105 9,289 U 11 458 74 25 2,639 1,079 213 76 6,960.5 45 12 277 36 29 2,101 722 134 107 5,632 46 13 160 27 21 1,763.5 430 88 55 4,692 47 14 169 27 12 1,555.5 425 67 37 4,129.5 48 15 161 19 16 1,351.5 423 39 49 3,614.5 49 16 111 16 13 1,157 298 40 40 3,103 60 17 89 11 14 1,019; 5 208 14 48 2,738 51 18 96 10 8 906 263 37 27 2,456.5 52 19 61 7 14 793.5 183 25 42 2,135.5 53 20 49 8 9 711 124 6 21 1,895 54 21 52 6 12 646 147 14 30 1,740 55 22 59 7 6 575.5 152 25 17 1,543.5 56 23 80 5 11 504.5 216 13 37 1,355.5 57 24 119 8 411 287 29 1,096 58 25 86 1 3 283.5 245 1 10 779.5 69 26 87 2 9 193 229 4 14 522 60 27 45 2 96 153 4 277 61 28 29 2 49 84 6 120 62 29 10 18 15 31 63 30 8' 8 16 16 64 21,781 16,343 1,500 183,832.5 54,633 43,895 4,136 480,543.5 Age at En try, 35 ^ fears. Ni FHBEB OP I Sntbants 39,759. $103,782,000 iNaCBEB. 36 2,192 1,826 94 18,966.5 5,722 3,955 231 49,913.5 1 2,023 6,257 209 32,518.5 5,292 15,063 633 86,342.5 36 2 1,738 2,673 203 25,821.5 1 4,423 8,570 597 68,601 37 3 1,833 1,761 190 21,663.5 ' 4,324 5,703 519 56,444.5 38 4 1,993 1,252 187 18,134 4,887 4,157 517 46,671.5 39 5 1,967 891 153 14,882.5 5,120 2,684 426 37,847 40 6 2,034 605 126 12,014.5 4,923 1,760 311 30,079 41 7 1,923 520 87 9,292 4,358 1,325 255 23,302.5 42 8 1,577 233 67 6,905.5 3,905 662 178 17,696 43 9 1,254 155 48 5,067.5 3,079 440 120 13,062 U 10 835 97 47 3,639.5 1,899 262 119 9,512 45 11 500 59 27 2,679.5 1,142 184 80 7,271 46 12 249 37 28 2,104.5 724 105 73 5,904.5 47 13 189 23 26 1,797.5 496 94 71 5,008 48 14 183 33 12 1,554.5 509 114- 31 4,337 49 15 139 16 19 1,335 351 54 56 3,713 60 16 115 11 15 1,163.5 332 25 40 3,266.5 61 17 79 8 13 1,024 235 14 32 2,875 62 18 69 9 13 923.5 151 14 33 2,594 53 19 72 6 12 834 245 16 43 2,395 54 TABLE I. OBSERVATIONS ON MALE LIFE. 97' * Age at Entry, 35 Years. If NlTHBEB OF EKTBAltTS, 39,759. $103,782,000 Insubed. Age at Exit. Discon- Exposed to Risk. Amounts Amounts Death Exposed to Risk. >'A Existing. tinued. Died. Existing. Discontin'ed. Claims. $ $ $ $ 20 54 6 17 744 146 9 51 2,094.5 55- 21 50 9 10 665.5 156 24 31 1,881 56 22 32 5 12 598.5 70 19 31 1,672.5 67 23 91 3 10 550.5 195 7 35 1,558.5 68 24 93 1 7 447.5 213 2 20 1,324 69 25 120 1 7 346.5 293 2 17 1,089 60 26 101 4 4 217 423 5 15 775.5 61 27 63 1 110 178 1 335 62 28 28 2 46 90 1 156 63 29 11 1 15.5 39 4 63 64 30 2 2 4 7 15 22 66 21,609 16,502 1,648 186,066.5 53,927 45,273 4,582 487,806.5 Age at En try, 36 1 fears. Nt rUBEB 07 £ Intbasts 38,034. $102,433,000 Insubed. 36 2,233 1,632 103 18,201 6,295 3,859 279 49,287 1 1,973 5,731 231 31,200.5 5,509 14,327 1,120 84,836.5 37 2 1,734 2,564 206 24,849 4,343 8,256 656 66,916 38 3 1,757 1,675 198 20,789.5 4,279 5,440 505 55,069 39 4 1,850 1,176 174 17,409 4,348 4,151 454 45,489.5 JfO 5 1,885 841 148 14,376.5 4,849 2,799 496 37,212.5 U 6 1,844 575 110 11,635.5 4,985 1,745 346 29,595.5 Jf2 7 1,845 507 97 9,140.5 4,806 1,279 206 22,752.5 Jf3 8 1,588 260 89 6,815 4,034 792 214 16,705 U 9 1,234 146 39 4,935 2,867 345 110 11,888.5 46 10 888 78 31 3,550 1,754 204 97 8,637 46 11 556 58 36 2,563 1,315 170 115 6,599 47 12 206 40 18 1,922 514 84 56 5,042 48 13 165 29 19 1,663.5 423 99 54 . 4,380.5 49 14 188 28 17 1,451 496 68 66 3,820 60 15 147 11 12 1,226.5 401 49 28 3,199.5 61 16 94 11 19 1,056.5 236 24 65 2,734 62 17 65 5 14 935.5 138 16 35 2,413 63 18 78 7 8 850.5 185 30 17 2,217 64 19 64 9 12 756.5 217 18 31 1,991 66 20 44 6 7 673 . 109 23 32 1,722.5 66 21 51 4 11 617 135 10 27 1,565 67 22 38 4 10 551 106 10 41 1,393 68 23 62 3 9 499.5 129 9 30 1,236.5 69 24 107 1 10 426.5 156 3 27 1,071.5 60 25 109 1 12 308.5 279 3 33 885.5 61 26 69 2 4 186 198 20 5 562 62 27 56 1 4 111.5 181 3 19 347.5 63 28 23 2 51 89 6 146 64 29 13 26 1 13 51 66 98 TABLE I. OBSERVATIONS ON MALE LIFE. i 1 Age at Entry, 36 Years. '' Kj d NUMBEK OP ENTBANTf 5, 38,034. $102,433,000 INSUBBD. Age at Eiit Discon- Exposed to Risk. Amounts Amounts Death Exposed to 94 H Existing. tinued. Died. Existing. Discontin'cd. Claims. $ $ 1 $ -30 11 , 13 35 38 66 31 2 3 67 32 1 2 1 3 68 33 2 69 34 2 70 35 2 71 36 2 72 37 2 73 38 1 2 2 U 20,977 15,405 1,652 178,799.5 53,424 43,836 5,173 469,821.5 Age at Entry, 37 Years. Number of I ilTBANTS , 35,363. $94,029,000 INSXJKED. 37 2,074 1,462 73 16,950.5 5,468 3,584 178 45,222.5 1 1,731 5,294 222 29,107 4,549 13,034 623 78,282 38 2 1,624 2,432 188 23,291 4,090 7,308 594 62,939 39 3 1,572 1,506 171 19,510 3,413 5,215 454 51,993.5 40 4 1,753 1,160 162 16,434 4,357 3,602 462 43,718 U 5 1,782 814 139 13,532 4,684 2,866 426 35,665 42 6 1,690 536 113 10,936 4,644 1,688 319 28,278 JfS 7 1,687 472 96 8,629 4,185 1,338 264 21,802 U 8 1,538 258 63 6,481 3,881 716 152 16,326 45 9 1,212 129 48 4,686.5 2,792 362 107 11,754 46 10 785 92 35 3,316 1,905 292 82 8,528 47 11 465 66 32 2,417 1,112 160 109 6,315 48 12 221 37 28 1,868.5 593 107 93 4,960.5 49 13 165 28 23 1,587 429 68 59 4,187 50 14 168 25 23 1,372.5 441 74 58 3,628 51 15 117 . 12 18 1,163 329 29 50 3,077.5 52 16 116 15 14 1,014.5 360 33 33 2,667.5 53 17 87 5 13 874.5 201 28 42 2,244 54 18 64 6 16 769 156 7 42 1,983.5 55 19 60 6 8 683 173 22 24 1,771 56 20 57 4 7 610 ! 164 22 17 1,552 57 21 28 2 8 543 61 6 26 1,357 58 22 39 6 9 503 92 21 33 1,256.5 59 23 60 3 11 1 450.5 135 8 26 1,117 60 24 93 2 10 377 191 3 42 950.5 ^1 25 82 5 273 ■ 198 15 716 62 26 61 6 186 ' 179 25 503 63 27 58 4 119 164 14 299 64 28 28 1 57 71 5 121 65 29 14 1 28 1 22 1 45 66 30 12 1 13 i 21 1 22 67 19,443 14,372 1,548 167,781.5 49,060 40,593 4,376 443,281.0 TABLE I. OBSERVATIONS ON MALE LIFE. 99 1 Age at Entry, 38 Years. 1 Number op Entbants , 34,7r2. $93,522,000 INSUBED. Age 1 at Exit Discon- r Exposed to Risk. 1 ! Amounts Amounts Death Exposed to Risk. Existing. tinued. Died. i Existing. Discontin'ed. Claims. $ $ $ $ 2,110 1,427 92 16,672.5 5,262 3,130 240 45,196 S8 1 1,746 5,169 224 28,558.5 4,751 12,952 591 78,414 39 2 1,650 2,301 186 22,853.5 4,304 7,316 515 62,938 ¥> 3 1,545 1,608 176 19,063 1 3,879 5,226 468 51,848 41 4 1,816 1,120 138 15,978 1 4,492 3,724 468 43,026 J^ 5 1,542 747 141 13,090.5 4,422 2,400 334 35,004 J,B 6 1,796 509 125 10,779.5 4,860 1,657 308 28,219.5 U 7 1,743 432 84 8,388 4,499 1,188 242 21,629 J^o 8 1,591 212 64 6,239 4,062- 582 170 16,003 m 9 1,158 143 47 4,406.5 2,829 362 103 11,299 Jf7 10 711 91 38 3,084.5 1,625 243 96 8,064.5 Jt8 11 457 58 31 2,261 1,080 174 91 6,135 Jf9 12 231 34 25 1,727 537 112 53 4,821 50 13 149 24 15 1,442 447 60 45 4,145 51 14 195 23 12 1,254.5 610 64 49 3,591 52 15 101 13 14 1,029.5 295 37 34 2,881.5 . 53 16 88 13 14 901.5 238 34 35 2,517 5k 17 64 7 13 789.5 163 24 43 2,215 55 18 69 7 12 705.5 187 15 39 1,989.5 56 19 47 2 11 620 120 10 29 1,751 57 20 38 4 9 559 108 11 20 1,591.5 58 21 35 3 6 508.5 115 16 24 1,450 59 22 33 3 11 464.5 88 3 31 1,301.5 60 23 45 2 9 418 113 11 39 1,175.5 61 24 102 1 10 362.5 234 4 25 1,016 62 25 85 10 250 237 37 755 63 26 60 1 4 154.5 182 10 7 476 6U 27 48 1 90 167 i 1 282 65 28 22 2 41 74 1 114 66 29 6 1 1 16.5 18 3 3 37.5 67 30 ^8 19,291 1 9 12 3 15 68 13,955 1,526 162,717.5 50,010 39,368 4,144 439,901.0 Age at Ed try, 39 1 fears. Nt rHBEB OF I Entrants 32,206. $88,592,000 Insured. 39 1,839 1,307 79 15,449.5 4,986 2,922 258 42,835 1 1,636 4,705 201 26,628.5 4,765 11,866 588 74,493 Jfi 2 1,550 2,030 195 21,424 1 4,133 6,324 579 60,045 Jfl 3 1,521 1,417 190 17,955.5 3,629 4,767 518 49,787.5 1^ 4 1,547 1,029 172 15,021.5 4,034 3,385 510 41,564.5 43 5 1,624 753 146 12,411.5 ! 4,488 2,360 410 34,148 U 6 1,633 493 108 10,018.5 : 4,605 1,596 273 27,272 45 7 1,649 412 75 7,825 : 4,507 1,173 302 21,009.5 46 8 1,513 197 53 5,796.5 3,773 591 139 15,318.5 47 9 1,042 130 47 4,067 2,645 384 118 10,920 48^ 100 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 39 Years. »2S Number op Entrants , 32,206. $88,592,000 iNsmiED. 1 Age at Fixit Discon- Exposed to Risk. Amounts Amounts Death Exposed to Bisk. ^►3 Existinff. tinued. Died. Existing. Discontin'ed. Claims. $ $ $ $ 10 698 80 31 2,873 1,759 182 110 7,874 J^o 11 403 65 37 2,071.5 996 155 108 5,836.5 50 12 191 26 29 1,586 523 85 89 4,612.5 51 13 139 19 17 1,343.5 433 72 51 3,922 52 14 148 19 15 1,168.5 393 80 46 3,362 53 15 112 10 14 991 356 17 31 2,874.5 5^ 16 102 7 14 856.5 275 22 50 2,468 55 17 63 4 9 735 192 7 20 2,128.5 56 18 70 4 15 659 171 27 55 1,899.5 57 19 53 4 11 570 141 6 28 1,657 58 20 39 5 8 501.5 93 10 34 1,480 59 21 32 3 12 450.5 91 9 46 1,343.5 60 22 34 2 9 404 166 5 22 1,199.5 61 23 48 2 5 359 107 5 16 1,006.5 62 24 66 9 305 145 30 881 63 25 75 2 5 229 160 1 22 705.5 6U 26 68 6 148 227 24 523 65 27 30 3 74 123 13 272 66 28 21 1 3 40.5 61 3 11 134.5 67 29 4 3 16 12 4 61 68 30 9 9 45 45 69 17,959 12,726 1,521 151,988.0 48,033 36,054 4,505 421,679.0 Age at En try, 40 ^ fears. Ni JHBEB OP 1 Sntbants . 33,003. $90,364,000 Insured. Jfi 1,883 1,346 90 15,828.5 5,034 3,001 271 43,681.5 1 1,712 4,760 222 27,304 4,910 11,946 668 76,085 Jil 2 1,534 2,114 204 21,933 3,276 7,046 564 61,011 J^ 3 1,449 1,447 193 18,414.5 3,332 4,951 594 51,172.5 JfP 4 1,639 1,045 173 15,526.5 3,850 3,490 560 43,026 U 5 1,758 735 121 12,824.5 5,273 2,560 330 35,591 Jf5 6 1,950 478 109 10,339 5,219 1,534 307 27,941 Jf6 7 1,736 386 101 7,848 4,520 1,082 289 21,107 Jf7 8 1,387 208 55 5,714 3,808 650 170 15,432 U8 9 1,043 138 45 4,099 2,702 398 135 10,930 49 10 713 85 38 2,899.5 1,811 216 132 7,786 50 11 436 53 30 2,079.5 1,103 136 86 5,667 51 12 191 25 22 1,574.5 491 72 66 4,374 52 13 120 21 12 1,338.5 390 49 44 3,756.5 53 14 160 23 26 1,184.5 486 71 83 3,262.5 5U 15 122 9 13 982.-5 361 32 36 2,642 55 16 84 8 21 839 259 32 78 2,213 56 17 66 7 ■ 17 726.5 155 35 46 1,842.5 57 18 54 2 15 639 126 1 40 1,623.5 58 19 37 5 12 566.5 88 13 26 1,450.5 59 TABLE I. OBSERVATIONS ON MALE LIFE. 101 Age at Entry, 40 Years. 11 NUMBEB OP ] Entrants , 33,003. $90,364,000 Insubbd. Age at Exit. Discon- Exposed to Risk. Amounts Amounts Death Exposed to >'A Existing. tinued. Died. Existing. Discontin''ed. Claims. Eisk. $ $ $ 20 37 5 11 512.5 110 13 23 1,323.5 60 21 41 4 14 460 93 17 55 1,175.5 61 22 43 2 13 402 96 4 34 1,017 62 23 54 2 10 344 119 3 32 •883.5 63 24 65 6 279 149 20 731 6J, 25 89 9 208 208 33 562 65 26 45 1 2 109.5 138 3 4 319.5 66 27 30 62 89 176 67 28 19 1 32 48 1 87 68 29 9 12 35 38 69 30 3 3 3 3 70 18,509 12,909 1,585 155,085.0 48,282 37,355 4,727 426,910.5 Age at Er itry, 41 ^ fears. Nt THBKB OF I Sntbants , 27,981. $79,495,000 Insubbd. U 1,711 1,076 84 13,452.5 4,784 2,382 257 38,556.5 1 1,399 3,943 202 23,138.5 4,102 10,345 577 66,899.5 42 2 1,275 1,824 178 18,654 3,111 6,085 519 54,005.5 43 3 1,366 1,251 159 15,663.5 3,262 4,196 417 45,235 U 4 1,386 872 133 13,077 3,561 3,106 369 37,905 45 5 1,494 631 123 10,806.5 4,454 2,164 355 31,340 46 6 1,519 434 101 8,657 4,552 1,769 356 24,564.5 47 7 1,412 360 76 6,640 3,937 1,012 216 18,266 48 8 1,298 171 68 4,886.5 3,599 508 162 13,353 49 9 827 98 39 3,386 2,139 306 96 9,185 50 10 602 68 27 2,437 1,594 222 60 6,686 51 11 369 32 23 1,758 937 83 62 4,879.5 52 12 179 23 18 1,338.5 490 86 74 3,796 53 13 122 19 10 1,120.5 373 64 23 3,157 54 14 132 17 14 970,5 427 72 36 2,693 55 15 92 12 11 810 261 48 32 2,170 56 16 64 6 13 698 182 33 48 1,836.5 57 ar 38 5 9 615.5 106 22 24 1,579 58 18 50 6 9 563 152 19 22 1,428.5 59 19 47 4 13 499 130 4 25 1,243 60 20 31 2 10 436 68 4 27 1,084 61 21 19 3 9 392.5 56 2 19 986 62 22 39 1 12 362.5 87 4 30 908 63 23 35 2 9 310- 79 5 27 786.5 64 24 71 1 10 264.5 108 3 35 676.5 65 25 63 6 183 152 23 532 66 26 37 1 4 113.5 100 3 17 355.5 67 27 34 3 72 113 7 237 68 28 17 1 35 70 4 117 69 29 7 1 17 22 1 43 70 103 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 41 YesLTs. ■a c3 NlTHBBR OF Entbants, 27,981. $79,495,000 Insured. Age at "Exit. Discon- j Exposed to 1 Risk. Amounts Amounts Deatli Exposed to Bislc. ^5 Existing. tinued. Died. Existing. Discontin'ed Claims. $ $ $ $ 30 6 1 9 18 1 20 71 31 2 1 72 32 2 1 73 33 2 1 7Jf 34 2 1 1 .5 75 15,741 10,864 1,376 131,373.5 43,026 32,548 3,921 374,527.5 Age at Entry, 42 Years. Ni JMBEB OP ] Entrants , 36.450. j $76,522,000 Insured. 1 42 1,538 965 92 12,742.5 1 4,399 2,396 263 37,063 1 1,295 3,703 191 22,003.5 3,950 9,795 511 64,566.5 A3 2 1,265 1,712 173 17,810 3,388 5,643 510 52,386.5 U 3 1,217 1,141 122 14,945.5 3,123 4,110 339 43,612 J,5 4 1,372 823 141 12,624.5 3,637 3,009 462 36,590.5 Jf6 5 1,310 608 137 10,396 3,910 2,133 389 29,920.5 Jf7 6 1,413 389 89 8,450.5 4,058 1,398 279 23,856 48 7 1,452 345 58 6,581.5 1 4,150 1,030 167 18,305 49 8 1,205 176 70 4,811 i 3,391 506 195 13,220 50 9 929 90 59 3,403 2,405 293 147 9,234.5 51 10 553 71 26 2,334.5 1,394 205 59 6,433.5 52 11 350 47 25 1,696.5 903 168 59 4,794 58 12 156 25 20 1,285.5 465 58 52 3,719 54 13 120 20 15 1,087 352 62 47 3,142 55 14 113 14 13 935 285 43 40 2,690.5 56 15 63 8 12 798 177 18 43 2,335 57 16 57 5 12 716.5 175 26 36 2,093 58 17 55 8 11 641 155' 27 44 1,855.5 59 18 49 2 10 570 148 3 46 1,641.5 60 19 39 3 9 508.5 121 8 20 1,442 61 20 18 4 12 457 53 20 31 1,287 62 21 25 4 14 423 83 19 27 1,183.5 63 22 29 3 10 380.5 , 51 13 36 1,057.5 64 23 57 2 15 339 159 4 58 962 65 24 81 3 13 264.5 226 5 46 740.5 66 25 68 5 169 189 11 466 67 26 44 3 96 102 15 266 68 27 29 2 2 48 101 2 10 148 69 28 11 1 16 27 5 36 70 29 i 4 4 4 4 71 14,917 10,173 1,360 126,537.5 41,581 30,994 3,947 365,051.0 TABLE I. OBSERVATIONS ON MALE LIFE. 103 1 Age at Entry, 43 Years. ji Number op Entbants 24,402. $69,852,000 IKSUBKD. Age at Exit. Discon- Exposed to Bisk. Amounts Amounts Deatli Exposed to Risk. >'S Existing. tinued. Died. Existing. Discontinued. Claims. 1 $ $ $ 1,384 937 90 11,733.5 3,938 3,170 235 33,841 4S 1 1,340 3,382 187 20,300 3,731 8,783 548 59,128 u 2 1,344 1,529 172 16,317.5 3,334 5,404 497 47,756 ^ 3 1,205 1,030 166 13,622 3,171 3,511 503 39,567.5 46 4 1,344 723 146 11,374.5 3,378 2,691 434 32,792.5 47 5 1,266 518 136 9,364 3,589 1,880 434 26,705 It8 6 1,259 392 85 7,507 3,444 1,304 293 21,090 49 7 1,329 281 92 5,826.5 3,639 858 289 1'6,373 50 8 1,012 152 60 4,189 3,835 533 155 11,663.5 51 9 806 88 49 3,997 3,157 364 107 8,280 52 10 466 49 32 2,073.5 1,196 170 91 5,799 53 11 359 32 23 1,535 976 135 77 4,364.5 54 12 134 26 19 1,124 385 68 43 3,215 55 13 105 14 15 951 333 58 48 2^724 56 14 108 8 16 820 352 38 48 2,395 57 15 85 6 13 689 335 13 39 1,870 58 16 43 7 10 584.5 131 30 22 1,585 59 17 41 6 8 525 138 15 27 1,419.5 60 18 40 5 470.5 115 16 20 1,249 61 19 23 3 15 419.5 49 8 43 1,102 62 20 34 6 8 377 74 15 22 998.5 63 21 23 3 15 340.5 48 8 48 891 64 22 30 8 301 73 35 791 65 23 50 1 13 263.5 138 3 35 681.5 66 24 62 1 8 198.5 149 1 26 506.5 67 25 31 3 5 137 77 4 17 329 68 26 33 1 3 89.5 85 6 8 230 69 27 24 2 53 69 3 134 70 28 13 2 3 36 40 1 9 61.5 71 29 -'.2' 1 1 8.5 6 3 1 10.5 72 30 5 5 2 2 73 13,790 9,205 1,407 114,310.5 37,737 37,968 4,147 327,354.0 Age at En try, 44 1 fears. Ni THBBB OF ] Sntbants 22,431. $64,953,000 Inscbsd.- 4U 1,374 830 73 10,800.5 3,956 1,911 187 31,521 1 1,076 3,072 163 18,618 2,999 7,803 432 54,998 Jt5 2 1,083 1,477 186 15,104.5 2,921 4,943 563 45,194,5 46 3 1,135 987 150 13,603.5 2,793 3,583 445 37,448 47 4 1,202 679 119 10,485.5 3,155 3,531 404 31,153.5 48 5 1,124 535 103 8,557.5 3,291 1,676 305 25,491 49 6 1,256 354 102 6,886 1 3,823 1,350 294 20,432 50 7 , 1,097 284 72 5,309 1 3,409 811 207 15,285.5 51 8 933 132 59 3,833 3,765 415 186 11,056.5 52 9 691 78 47 3.735 1.887 371 130 7,762.5 53 104 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 44 Years. 1 i !i NUMBEB OF EnTB&NTS 22,431. $64,9.53,000 Ihsdbbd. Age at S33 £3 . Discon- Exposed to Bisk. Amounts Amounts Death Exposed to Exit >*A Existing. tinued. Died. Existing. Discontin'ed. Claims. Risk. $ $ f $ 10 447 56 33 1,930 1,198 179 90 5,520.5 H 11 266 34 28 1,405 686 93 88 4,096.5 55 12 141 28 17 1,080 438 73 52* 3,239.5 56 13 106 10 16 903 320 35 58 2,695.5 57 14 101 8 17 772 292 19 46 2,290.5 58 15 77 4 13 648 218 32 . 43 1,927 59 16 51 7 20 552.5 153 19 56 1,640.5 60 17 33 4 13 476 93 4 54 1,420 61 18 28 4 13 426 76 8 33 1,267 62 19 30 2 10 382 84 24 30 1,142 63 20 20 2 9 340 55 7 49 1,012.5 6Jf 21 26 1 14 309.5 91 3 51 903.5 65 22 15 13 269 37 92 760 66 23 61 3 16 239.5 159 7 42 627.5 67 24 50 11 161 117 25 423 68 25 38 5 100 98 19 281 69 26 25 1 2 56.5 58 3 10 162.5 70 27 19 29 64 93 71 28 7 10 16 29 72 29 1 3 3 13 73 30 2 2 10 ' 10 7^ 12,515 ■ 8,592 1,324 104,925.5 35,264 25,698 3,991 309,896.5 Age at En try, 45 1 fears. * Nt THBBB OF I Sntbants 21,980. $63,491,000 iKSTTBXt). 45 1,241 859 69 10,560.5 3,441 1,947 222 30,772 1 1,137 3,043 180 18,289.5 3,319 8,169 537 53,796.5 46 2 1,076 1,361 170 14,770.5 2,724 4,349 455 43,681.5 47 3 1,055 906 158 12,391 3,034 3,277 450 36,689.5 48 4 1,217 657 131 10,396.5 3,136 2,223 418 3(r,455.5 49 5 1,135 482 96 8,479 3,329 1,624 286 24,978 50 6 1,193 339 104 6,837.5 3,635 1,145 279 19,978.5 51 7 1,101 274 79 5,234 3,312 897 246 15,043.5 52 8 949 113 55 3,860.5 2,732 406 169 10,834 53 9 745 95 42 2,752.5 2,054 330 91 7,565 54 10 490 56 30 1,890 1,266 176 73 5,167 55 11 289 38 16 1,323 773 104 37 3,688 56 12 112 17 13 990.5 336 44 43 2,804 57 13 90 6 20 854 250 35 52 2,385.5 58 14 81 14 23 734 252 35 59 2,048.5 59 15 58 6 20 620 164 14 46 1,713 60 16 38 6 11 536 120 17 38 1,487.5 61 17 45 3 10 482.5 99 10 30 1,316 62 18 44 5 12 423.5 120 6 34 1,179 63 19 15 4 7 363 39 11 13 1,016.5 64 TABLE I. OBSERVATIONS ON MALE LIFE. 105 Age at Entry, 45 Years. 'A Existing. tinued. Died. Existing. Discontin'ed Claims. $ $ $ $ 20 19 3 11 337.5 68 4 23 957 65 21 22 4 14 304 54 11 35 858.5 66 22 31 1 11 265.5 68 8 38 760 67 23 32 2 6 222 86 6 50 647 68 21 52 13 183 146 32 508 69 25 42 8 118 104 15 330 70 26 21 2 68 75 13 211 71 27 25 2 45 71 4 123 72 28 11 1 1 17.5 33 3 3 46.5 73 29 2 5 6 9 7A 30 3 3 3 3 75 12,371 8,295 1,314 103,356.5 34,849 24,851 3,791 301.052.0 Age at Er Ltry, 46 ^ fears. Nt TMBEB OF I Entrants 19,350. $57,757,000 Insubbd. J,6 1,135 711 6V 9,319.5 3,449 1,638 155 28,059.5 1 959 2,613 156 16,130.5 3,146 6,791 481 49,119.5 Jf7 2 900 1,221 137 13,098.5 1 2,507 4,299 486 39,947.5 ^ 3 937 816 122 11,043 2,607 2,843 421 33,383.5 ^9 4 1,002 573 126 9,289.5 2,825 2,202 385 27,833 50 5 1,053 417 107 7,666.5 3,102 1,554 327 22,745 51 6 1,093 282 89 6,157 3,301 1,082 256 17,998 52 7 1,002 259 78 4,704.5 3,054 807 247 13,496.5 53 8 891 125 54 3,432.5 2,547 432 141 9,576 51, 9 711 76 41 2,387 1,837 245 108 6,549.5 55 10 408 51 26 1,571.5 1,106 134 79 4,415 56 11 262 24 11 1,100 638 59 37 3,133.5 57 12 123 18 16 806 352 49 70 2,404.5 58 13 67 10 11 653 210 60 63 1,928 59 14 69 8 17 566 236 22 45 1,614 60 15 62 2 5 475 187 6 22 1,319 61 16 28 5 6 404.5 73 21 30 1,096.5 62 17 43 2 11 367 128 10 12 978 63 18 15 1 15 311.5 62 4 40 831 6A 19 19 3 11 279.5 41 6 22 724 65 20 20 1 12 247.5 46 8 35 654 66 21 14 3 8 213.5 50 10 24 564 67 22 30 1 9 189.5 75 3 31 483.5 68 23 27 2 6 149 57 11 15 370.5 69 24 30 2 3 114 68 4 8 291 70 25 26 5 80 58 21 213 71 26 20 3 49 61 7 134 72 27 16 2 26 27 2 66 73 28 4 1 8 21 3 37 74 29 2 3 10 13 75 30 1 1 3 3 76 , . 10,969 7,226 1,155 90,843.0 31,884 22,300 3,573 269,980.5 106 TABLE I. OBSERVATIONS OX MALE LIFE. r 1 i Age at Entry, 47 Years. ; -si NUJfitBB OF EnTBANT! i, 16,859. 151,924,000 Ihsurbd. Age {SS at Exit. Existing. Discon- tinued. i Died. 1 Exposed to Risk. [ Amonnts Existing. Amoants Discontin'ed j Death Claims. Exposed to Risk. $ $ $ $ 955 634 61 8,112.5 2,704 1,528 178 25,198 47 1 788 2,246 173 14,086 2,394 6,252 546 44,388 4-8 2 781 1,082 133 11,461 2,185 3,860 369 36,392 49 3 853 703 96 9,654.5 2,231 2,608 317 30,604 50 4 863 479 120 8,114.5 2,450 1,903 358 25,800.5 51 5 953 368 66 6,708 " 2,829 1,409 214 21,336.5 52 6 1,019 252 77 5,379 3,137 950 210 17,114 53 7 938 206 65 4,054 2,936 644 210 12,970 H 8 765 118 53 2,889 3,341 391 150 9,306.5 55 9 572 64 43 1,980 1,528 160 94 5,540 56 10 341 35 22 1,315.5 937 ' 102 87 3,787 57 11 187 27 14 921.5 454 60 39 2,682 58 12 107 9 17 702.5 275 53 54 2,132.5 59 13 56 8 7 570 i 168 25 18 1,764.5 60 14 45 8 10 499 119 21 32 1,555.5 61 15 41 1 8 439.5 136 2 17 1,393 62 16 37 3 13 388.5 113 10 55 1,234 63 17 23 3 11 335.5 47 17 20 052.5 64 18 32 3 8 298.5 123 7 119 973.5 65 19 18 2 12 256 116 4 35 726 66 20 13 2 7 224 25 8 23 569 67 21 14 1 9 202.5 48 3 20 515.5 68 22 13 1 7 178.5 i 37 4 15 444. 69 23 20 2 6 157 i 53 4 17 388 70 24 31 6 130 73 14 316 71 25 27 1 4 92.5 69 3 17 227*5 72 26 28 2 61 1 66 3 140 73 27 18 4 31 43 7 71 74 28 2 2 9 7 7 21 76 29 2 1 5 2 1 7 76 30 2 2 4 4 77 9,544 6,258 1,057 79,257.5 28,650 20,028 3,246 248,653.0 Age at En try, 48 1 rears. Nt rXBBB OF I Intrants 15,825. $47,985,000 Insubbo. 48 905 582 72 7,621.5 2,555 1,775 175 23,105 1 785 1,986 152 13,273 2,466 5,366 535 40,797 49 2 810 987 ! 121 10,849.5 2,426 3,274 358 33,476 50 3 758 720 112 9,065 1,888 2,844 362 27,633 51 4 849 431 108 7,619.5 2,342 1,729 373 23,096.5 52 5 838 381 95' 6,256.5 2,634 1,499 313 18,767.5 53 6 920 251 80 5,007.5 2,795 879 202 14,631.5 54 7 854 163 48 3,800.5 i 2,576 540 154 10,925 55 8 801 93 58 2,770.5 2,412 297 203 7,776.5 56 9 1 540 49 31 1,840.5 1,442 149 80 4,938.5 67 TABLE I. 107 OBSERVATIONS ON MALE LIFE. Age at Entry, 48 Yfears. "1 NUMBEB OF ] EKTBANTe , 15,825. $47,985,000 iHsxiBED. Age I Js at Discon- T\Jnjl Exposed to Risk. Amounts . Amounts Death Exposed to ETit 1 ^& Existing. tinued. Uiea. Existing. Discontin'ed. Claims. Rislc. $ 1 $ $ 10 278 31 22 1,229.5 646 115 68 3,284.5 58 11 227 19 13 904.5 575 68 40 2,479 59 12 83 13 16 648.5 208 37 56 1,811.5 60 13 60 10 10 538 157 31 27 1,513.5 61 14 62 9 8 458.5 173 23 31 1,302.5 62 15 50 3 11 382.5 184 8 40 1,083 63 \ 16 22 3 8 318.5 62 11 23 849.5 6J^ i 17 19 8 287 63 19 759 65 \ 18 23 3 9 258.5 65 4 20 675 66 19 18 3 11 223.5 48 5 31 585.5 67 20 9 3 11 191.5 27 8 33 500 68 21 11 1 8 169.5 33 5 18 433.5 69 22 17 1 7 149.5 47 3 23 378.5 70 23 26 1 5 124.5 51 2 8 306 ' 71 24 37 11 93 81 35 246 ■ 72 25 13 3 45 28 9 130 73 26 10 4 29 29 11 93 7A 27 6 15 14 53 75 28 4 9 21 39 76 29 2 5 11 18 77 30 2 3 2 1 78 31 1 5 79 32 1 5 80 33 1 1 5 5 81 9,039 5,743 1,043 74,190.0 26,061 18,672 3,252 221,708.5 Age at En try, 49 1 fears. Nt rHBER OP I Sntrants 13,873. 143,119,000 Insured. Jf9 795 456 57 6,708.5 2,405 1,090 181 21,014.5 1 686 1,776 128 11,677 2,155 4,937 356 36,974.5 50 2 680 841 125 9,554.5 2,071 2,887 383 30,551.5 51 3 725 614 118 8,022 1,973 2,204 383 25,552 52 4 758 394 90 6,675 1,907 1,594 277 21,297 63 5 854 291 96 5,484.5 2,594 1,993 289 17,319.5 H 6 826 219 86 4,279.5 2,545 851 295 13,014.5 55 7 774 148 63 3,184 2,350 481 173 9,508.5 56 8 588 82 38 2,232 1,874 296 120 6,597 57 9 411 54 38 1,538 1,163 150 120 4,380 58 10 261 30 32 1,047 715 72 81 2,986 59 11 146 18 17 730 419 53 54 2,127.5 60 12 71 7 10 554.5 206 35 36 1,610.5 61 13 45 8 10 466 134 29 26 1,336.5 62 14 51 4 10 405 166 9 25 1,157.5 63 108 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 49 Years. ^1 Number of Entrants , 13,87.3. 143,119,000 Insured. Age at Discon- Exposed to Rislc. Amounts Amounts Deatti Exposed to Risk. FrTit Existing. tinued. Died. Existing. Discontin'ed. Claims. $ s $ $ 15 39 2 11 341 107 10 39 957 6i 16 32 18 290 114 55 806 66 17 10 1 16 239.5 30 1 40 636.5 66 18 21 2 6 212 67 3 15 564.5 67 19 10 2 11 183 25 3 22 479.5 68 20 17 1 9 160.5 53 11 17 425.5 69 21 9 1 8 133.5 18 1 34 349.5 70 22 14 12 116 30 28 297 71 23 9 6 90 19 12 239 72 24 17 6 75 43 28 208 73 25 15 2 4 51 28 1 9 136.5 n 26 13 3 31 50 6 99 75 27 4 1 15 10 4 43 76 28 5 10 14 29 77 29 2 5 4 15 78 30 1 1 1 2.5 3 3 5 9.5 79 7,889 4,954 1,030 64,512.5 23,292 16,714 3,113 200,721.5 Age at En try, 50 ' JTears. Ni TMBKB OP I Entrants , 13,650. $41,776,000 Insured. 50 857 491 85 6,579.5 2,445 1,110 280 20,333 1 667 1,796 163 11,319 1,949 4,860 581 35,511 51 2 652 836 147 9,173 1,893 3,108 487 28,997 52 3 673 575 132 7,668.5 1,875 1,986 467 24,070 53 4 831 382 104 6,385 2,266 1,447 340 20,011.5 5^ 5 823 252 93 5,133 2,707 1,032 298 16,166 55 6 768 177 62 4,002.5 2,393 615 209 12,337.5 56 7 784 150 72 3,009 2,541 558 287 9,149 57 8 575 66 44 2,045 1,725 254 131 5,915 58 9 364 36 29 1,375 1,020 100 86 3,882 59 10 236 38 26 945 603 188 59 2,632 60 11 120 14 17 657 320 43 54 1,854.5 61 12 63 5 16 510.5 142 29 57. 1,444.5 62 13 52 8 11 425 176 19 19 1,221.5 63 14 53 6 9' 355 153 23 25 1,005.5 H 15 34 4 12 288 119 12 28 810 65 16 19 3 6 238.5 56 16 11 649 66 17 18 2 6 211 53 5 22 571.5 67 18 15 10 185.5 33 1 55 493.5 68 19 11 4 159.5 26 4 13 403 69 20 16 10 143.5 49 6 27 359 70 21 6 3 116.5 17 3 7 278.5 71 22 13 7 106.5 30 1 19 252.5 72 23 10 3 86 21 8 203 73 24 19 5 73 42 8 174 7Jt TABLE I. 109 OBSERVATIONS ON MALE LIFE. Age at Entry, 50 Years. •si n d >2S NUMBKR OF Entbantb , 13,650. $41,776,000 Insubkd. Age at Discon- Exposed to Rislv. Amounts Amounts Deatli Exposed to Bislf. Exit. >"& Existing. tinued. Died. Existing. Discontin'ed. Claims. $ $ $ $ 25 17 4 49 46 10 124 75 26 7 2 3 27 15 2 7 67 76 27 8 1 16 18 1 44 77 28 3 1 7 8 2 25 78 29 3 15 79 30 3 3 15 15 80 7,717 4,848 1,085 61,295.0 22,756 15,422 3,598 189,014.0 Age at Entry, 51 Years. Ni JMBEB OP ] Sntrants , 10,458. $32,807,000 Insukbd. 61 619 406 56 5,026 1,703 1,067 177 15,870 1 546 1,381 135 8,686.5 1,565 3,899 445 27,910.5 52 2 556 607 95 7,011.5 1,642 2,373 362 22,764.5 5S 3 513 423 90 5,845.5 1,510 1,641 359 18,753.5 5U 4 626 317 83 4,872.5 1,847 1,266 285 15,431 55 5 607 225 83 3,892.5 2,035 918 260. 12,207 56 6 591 155 61 3,012.5 1,924 654 215 9,176 57 7 557 127 41 2,219.5 1,683 426 132 6,547 68 8 398 44 49 1,536 1,149 140 141 4,449 59 9 254 31 29 1,051.5 837 159 95 3,009.5 60 10 171 12 25 747 422 33 55 1,981. "5 61 11 91 9 22 540.5 234 31 63 1,472.5 62 12 58 5 7 420.5 162 18 12 1,151 68 13 35 5 11 350.5 126 18 29 959 6Jf 14 47 3 6 300.5 134 6 16 792 65 15 31 1 8 245.5 109 8 18 635 66 16 26 1 10 205.5 70 3 17 502.5 67 17 22 3 3 167.5 67 8 7 410 68 18 17 2 7 140 64 7 9 328.5 69 19 7 1 4 114.5 16 2 8 251 70 20 3 1 5 102.5 5 2 9 ^25 71 21 8 1 4 93.5 6 1 10 209.5 72 22 19 6 81 22 11 193 73 23 12 9 56 28 23 160 74 24 14 2 35 33 1 109 75 25 5 4 19 48 9 75 76 26 1 10 . 5 18 77 27 2 2 9 4 1 13 78 28 3 5 6 8 79 29 2 2 2 2 80 5,841 3,760 857 46,799.0 17,458 12,580 2,769 145,613.5 110 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 53 Years. NUMBBB OF Entrant 3, 9,452. $29,239,000 Insured. Age at Discon- T\fy^.l Exposed to Risk. Amounts Amounts Death Exposed to BLli Exit. >*& Eristing. tinued. JJiecl. Existing. Discontin'ed. Claims. Risk. $ $ $ 566 299 42 4,576.5 1,745 694 162 14,272.5 52 1 476 1,222 104 7,934 1,372 3,329 286 24,973.5 53 2 529 551 113 6,467.5 1,507 1,959 325 20,671.5 5Jf 3 497 354 92 5,373 1,375 1,531 303 17,094.5 55 4 566 263 83 4,475.5 1,719 1,025 246 14,138.5 66 5 561 178 62 3,606 1,736 730 199 11,296 57 6 546 100 65 2,844 1,747 405 223 8,793.5 58 7 503 90 75 2,138 1,741 272 181 6,485 69 8 444 44 33 1,493 1,336 208 91 4,323 60 9 266 19 21 984.5 833 62 67 2,761 61 10 166 11 21 682.5 464 45 58 1,807.5 62 11 111 8 18 486 303 17 47 1,254.5 63 12 44 5 10 350.5 75 14 25 889 64 13 28 1 5 293.5 75 6 18 779 65 14 25 2 6 259 48 21 17 672.5 66 15 31 3 7 225.5 72 9 26 592.5 67 16 25 1 7 185.5 92 3 19 488.5 68 17 13 2 6 152 46 5 10 373.5 69 18 2 1 10 131.5 6 2 22 314 70 19 9 . 1 4 118.5 25 2 4 284 71 20 6 8 105 16 16 254 72 21 10 1 3 90.5 14 3 5 220.5 73 22 5 1 4 76.5 11 2 9 199 n 23 14 1 7 66.5 39 1 20 177.5 75 24 13 3 45 28 8 118 76 25 10 1 6 28.5 23 3 18 80.5 77 26 3 4 12 7 16 38 78 27 3 5 10 15 79 28 2 5 80 29 2 5 81 30 1 2 3 5 82 31 1 2 83 32 1 2 8U 33 1 2 85 34 . 1 1 2 2 86 5,473 3,159 820 43,215.5 16,468 10,348 2,423 133,389.5 Age at Ei itry, 53 ^ fears. N UMBER OF ] Entrants , 8,213. 125,055,000 Insured. 53 537 301 32 3,956 1,306 579 130 12,238 1 477 1,061 74 6,812.5 1,336 3,145 218 21,467.5 5If 2 412 493 99 5,484.5 i 1,070 1,963 384 17,359.5 55 3 415 306 93 4,574 1,187 1,222 294 14,313 56 4 432 219 89 3,803.5 1,276 835 273 11,803.5 57 TABLE I. Ill OBSERVATIONS ON MALE LIFE. Age at Entry, 53 Years. ill Number of Entrants, 8,213. $25,055,000 Insured. ' Age 11 ►•5 at Existing. Discon- Died. Exposed to Amounts Amounts Death Exposed to Exit. tinued. Risk. Existing. Discontin'ed Claims. Risk. 1 $ $ $ $ 5 457 173 63 3,086.5 1,465 714 172 9,480 58 6 530 101 47 2,429.5 1,555 399 122 7,286.5 59 7 393 82 36 1,761 1,353 255 . 89 5,282.5 60 . 8 358 41 39 1,270.5 1,135 137 120 3,644.5 61 9 226 26 20 840 645 74 53 2,284 62 10 117 10 14 576 303 40 26 1,529 63 11 92 7 16 436.5 172 37 46 1,161.5 64 12 38 2 14 324 98 6 44 922 65 13 28 4 10- 269 110 14 24 770 66 14 32 1 5 228.5 97 3 11 627.5 67 15 26 1 10 190.5 85 3 24 516.5 68 16 13 2 7 153 53 6 22 403 69 17 9 3 7 130.5 20 3 12 323.5 70 18 15 3 113 45 8 290 71 19 9 2 6 94 34 2 15 236 72 20 5 1 5 77.5 7 4 16 184 73 21 4 6 67 2 24 159 n 22 5 2 57 17 4 133 75 23 4 4 50 5 7 112 76 24 16 6 42 47 3 100 77 25 5 1 20 16 2 50 78 26 4 2 14 7 10 32 79 27 8 8 15 15 80 4,667 2,836 710 36,868.5 13,461 9,441 2,153 112,723.0 - Age at En tv^ 54 1 rears. N aMBER OF Entrants , 7,146. i 122,521,000 Insured. H 495 241 50 3,452.5 i 1,657 467 160 11,027 1 356 898 88 5,911 1,040 2,624 283 18,925 55 2 403 412 87 4,812 i 1,243 1,541 308 15,519.5 56 3 361 272 81 3,980 : 964 1,194 302 12,601 57 4 370 197 83 3,303.5 1 1,097 789 206 10,343.5 58 6 440 141 64 2,681.5 1,447 549 263 8,371.5 59 6 386 94 50 2,060 i 1,155 380 141 6,197 60 7 351 61 51 1,546.5 ! 1,237 201 161 4,610.5 61 8 343 28 20 1,100 971 71 79 3,076.5 62 9 203 15 25 715.5 540 56 79 1,963 63 10 117 11 12 474.5 285 49 24 1,291.5 6Jf 11 51 6 12 337 158 22 26 947 65 12 20 1 12 270.5 65 1 18 751.5 66 13 25 3 4 236.5 75 6 11 665 67 14 25 3 11 205 69 4 15 574 68 113 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 54 Years. NCHBEB OF Entbante , 7,146. $22,521,000 Insured. Age |h at £S Discon- , Exposed to Risk. Amounts Amounts Deatii Exposed to Exit. >"& Existing. tinued. Died. Existing, Discontin'ed. Claims. Risk. " $ $ $ $ 15 24 3 11 166.5 74 17 25 479.5 69 16 8 1 5 129.5 35 1 9 371.5 70 17 10 1 5 115.5 34 1 23 326.5 71 18 3 2 3 99 11 2 7 268 72 19 5 6 92 25 14 249 73 20 8 1 5 80.5 21 . 3 6 208.5 u 21 3 5 67 8 14 180 75 22 6 6 59 13 13 158 76 23 13 1 1 46.5 29 2 1 131 77 24 7 2 32 15 5 100 78 25 9 2 1 22 25 4 2 78 79 26 4 1 3 10.5 12 3 13 47.5 80 27 1 3 3 21 .81 28 1 2 8 18 82 29 1 4,048 1 10 10 83 2,394 704 32,012.0 12,323 7,987 2,211 99,510.0 Age at Er itry, 55 "5 fears. N UMBER OF Entrants i, 5,963. $18,980,000 Insttbkd. 55 384 193 49 2,885 1,190 478 214 9,230.5 1 319 770 76 4,952 1,018 2,255 241 15,929.5 56 2 285 348 74 3,998 906 1,264 231 12,911 57 3 325 227 65 3,351.5 1,009 1,079 198 10,602.5 58 4 308 172 59 2,762 926 721 160 8,495.5 59 5 349 111 64 2,253.5 ^ 1,114 443 170 6,827.5 60 6 343 61 43 1,754.5 ' 1,172 308 120 5,168 61 7 333 44 31 1,316 958 150 128 3,647 62 8 249 26 23 917 674 98 59 2,437 68 9 193 15 20 624.5 540 47 59 1,631.5 6^ 10 108 10 7 399 278 23 29 997.5 65 11 75 2 7 278 175 6 14 676 66 12 33 4 5 193 80 15 15 476.5 67 13 13 2 5 152 49 4 16 372 68 14 19 2 4 132 43 5 13 302.5 69 15 10 1 5 107.5 * 25 2 9 243 70 16 10 4 7 90 20 4 21 206 71 17 7 7 71 21 24 163 72 18 4 1 1 56.5 11 1 1 117.5 73 19 5 4 51 8 6 105 U 20 8 3 42 24 3 91 75 21 2 31 5 64 76 22 1 29 1 59 77 23 2 2 28 4 2 68 78 24 9 4 24 15 9 52 79 TABLE I. • 113 OBSERVATIONS ON MALE LIFE. Age at Entry, 55 Years. !i Number op Entrants, 5,963. $18,939,000 iNstJBED. Age ei0 at a> CO Discon- Exposed to Risk. Amounts Amounts Death Exposed to Risk. Exit. >'A Existing. tinued. Died. Existing. Discontin'ed Claims. $ $ $ $ 25 1 11 1 28 80 26 7 10 19 27 81 27 2 1 3 6 2 8 82 3,403 1,993 567 26,522.0 10,291 6,903 1,745 80,926.0 Age at Entry, 56 Years. N TTMBEB 07 ENTBANTf i, 4,978. $16,256,000 Insubed. 56 335 155 38 2,411.5 1,056 409 117 7,923.5 1 220 634 77 4,1.33 686 1,837 288 13,755.5 57 2 231 284 68 3,377 726 999 244 11,363.5 58 3 230 194 64 2,839 829 808 266 9,490 59 4 243 140 69 2,378 857 538 201 7,722 60 5 299 86 59 1,953 1,008 450 160 6,170 61 6 328 48 40 1,528 942 180 160 4,687 62 7 285 43 32 1,114.5 929 127 113 3,431.5 63 8 212 18 22 767 710 77 67 2,287.5 6Jf 9 159 8 14 520 443 20 50 1,462 65 10 74 8 13 339 217 23 41 947.5 66 11 56 2 8 247 130 9 40 673.5 67 12 23 5 11 179.5 76 8 28 495 68 13 20 2 10 142 72 4 25 385 69 14 19 2 8 110 50 4 24 284 70 15 5 1 82 19 3 208 71 16 3 7 76 15 14 186 72 17 6 1 6 65.5 9 1 27 156.5 73 18 4 1 1 52.5 8 1 1 119.5 7^ 19 2 5 47 2 13 110 75 20 3 3 40 11 7 95 76 21 2 1 4 33.5 5 2 7 76 77 22 3 3 27 8 8 63 78 23 3 1 21 9 1 47 79 24 8 2 17 20 2 37 80 25 1 1 7 3 1 15 81 26 2 1 5 7 1 11 82 27 1 2 2 3 83 28 1 1 8J, 29 1 1 i 1 1 85 2,778 1,632 568 22,516.0 8,850 5,497 1,909 72,206.5 114 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 57 Years. i Number of Entrants, 4,212. $13,878,000 Insured. Age at Discou- Exposed to Amounts Amounts Death Exposed to Risk. £zit. ^.2 Existinor. tinacd. Died. Risk. Existing. Discontined. Claims. % f $ $ 290 126 28 2,043 1,042 296 82 6,791 57 1 199 464 66 3,536 579 1,406 184 11,755 58 2 275 244 54 2,917 795 871 224 9,853.5 59 3 200 139 60 2,396.5 655 mb 162 8,081.5 60 4 238 119 57 2,007.5 758 456 250 6,719 61 5 297 53 47 1,626.5 923 265 158 5,350.5 62 6 2G4 42 35 1,235 974 243 209 4,015.5 63 7 230 38 37 896 801 127 113 2,647.5 6^ 8 171 12 22 604 548 36 59 1,652 65 9 122 10 13 400 328- 32 30 1,011 66 10 63 11 7 254.5 144 27 25 623.5 67 11 42 3 6 177.5 60 6 21 438 68 12 13 2 8 127 44 2 15 353 69 13 11 8 105 33 28 293 70 14 10 1 1 85.5 28 1 1 231.5 71 15 5 1 5 73.5 23 1 10 201.5 72 16 9 2 2 62 40 2 1 167 73 17 6 2 50 18 4 125 U 18 3 1 42 9 1 103 75 19 2 4 38 3 7 93 76 20 2 6 32 9 12 83 77 21 3 24 8 62 78 22 1 4 20.5 3 15 52.5 79 23 1 16 5 36 80 24 8 1 15 13 1 31 81 25 1 1 5.5 3 3 15.5 82 26 2 1 4 7 3 11 83 27 1 2,466 1 1 1 8Jf 1,269 477 18,794.5 7,843 4,412 1,623 60,797.0 Age at Eb try, 58 1 fears. N ITUBEB OF ENTRANTa , 3,605. $11,429,000 Insured. 58 243 100 16 1,752.5 715 220 77 5,604.5 1 212 375 62 3,058.5 609 1,171 166 9,831.5 59 2 183 167 54 2,513.5 486 675 165 8,133.5 60 3 171 117 59 2,134.5 501 546 225 6,872 61 4 242 72 42 1,810 716 336 125 5,705 62 5 294 58 43 1,461 1 920 219 143 4,586.5 63 6 228 43 36 1,073.5 ^81 173 118 3,327.5 H 7 197 31 28 772.5 668 99 90 2,392.5 65 8 156 7 20 528.5 517 28 44 1,571 66 9 100 8 10 345 299 19 38 986.5 67 TABLE I. US- OBSERVATIONS ON MALE LIFE. i Age at Entry, 58 Years. A H NUMBBB OP Entbants, 3,605. 1 111,429,000 Insubed. Age ►2s 1 at 1 Discon- 1 Exposed to Kisli. Amounts Amounts Deatli Exposed to Risk. Exit. >'A Existing. tinued. 1 Died. Existing. Discontin'ed. Claims. 1 $ $ $ 10 65 4 12 229 206 10 37 635 68 11 36 1 5 149.5 99 8 12 383 69 12 9 3 8 106.5 34 6 17 265 70 13 10 1 6 87.5 34 1 19 210.5 71 14 13 1 3 70.5 42 1 5 156.5 72 15 3 2 54 3 5 109 73 16 6 3 49 6 6 101 7A 17 3 3 40 7 7 89 75 18 2 2 34 5 4 75 76 19 2 5 30 6 7 66 77 20 4 23 10 53 78 21 1 1 19 4 1 43 79 22 1 3 17 3 3 38 80 23 2 2 13 11 4 32 81 24 4 2 9 6 4 17 82 25 1 3 1 7 83 26 2 6 8J, 27 1 2 3 6 85 28 1 3 86 29 1 1 3 3 87 2,186 988 431 16,389.5 6,585 3,512 1,332 51,308.5 Age at En try, 59 ^ fears. N tIMBKB OF Entbants , 2,933. $9,413,000 Insubed. 59 190 110 20 1,411.5 581 252 73 4,580.5 1 154 330 76 2,448 504 1,007 250 8,003.5 60 2 144 133 56 1,986.5 476 498 162 6,497 61 3 151 94 42 1,673 492 400 117 5,410 62 4 180 65 38 1,400.5 611 330 145 4,436 63 5 197 41 37 1,129.5 644 175 88 3,427.5 64- 6 176 29 31 860.5 589 91 135 2,562.5 65 7 169 17 33 630.5 528 64 60 1,761 66 8 137 7 19 416.5 361 19 38 1,131.5 67 9 59 3 10 255.5 195 14 38 716 68 10 38 1 13 184.5 120 6 29 473 69 11 27 1 14 132.5 63 3 25 319.5 70 12 8 1 6 90.5 34 1 17 229.5 71 13 10 1 7 75.5 32 3 16 176.5 72 14 13 1 2 57.5 41 3 7 125.5 73 15 5 1 1 , 41.5 8 1 1 75.5 74 16 1 35 6 66 75 17 1 1 3 33.5 2 1 4 59.5 76 18 29 53 77 19 1 5 29 3 4 53 78 116 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 59 Years. NUMBBB OP Entrants, 2,93a $9,413,000 Insured. Age U at £S Discon- Exposed to Amonnts Amounts Death Exposed to Risk. Exit. >"& Existing. tinued. Died. Risk. Existing. Discontin'ed. Claims. $ $ $ $ 20 1 1 22.5 1 1 45.5 79 21 1 1 2 20.5 7 2 5 43 80 22 2 2 17 5 2 30 81 23 2 4 13 8 4 23 82 24 5 1 7 8 1 11 83 25 1 2 8h 26 1 1 2 2 85 1,672 838 423 13,002.5 5,320 2,871 1,222 40,313.0 Age at Ee Ltry, 60 1 fears. N CniBEB OF Entrants , 2,682. $8,497,000 Insuked. 60 179 81 32 1,300.5 619 165 103 4,166 1 147 282 55 2,249 456 747 191 7,236.5 61 2 144 147 55 1,832.5 485 516 147 5,958 62 3 152 82 50 1,519 446 345 187 4,895.5 63 4 132 53 49 1,249.5 386 271 138 3,954.5 61, 5 165 50 34 1,017 535 167 115 3,211.5 65 6 187 27 27 779.5 562 144 99 2,406 66 7 139 18 20 543 409 55 59 1,645.5 67 8 117 7 10 371.5 402 55 25 1,122.5 68 9 74 6 10 238 202 18 23 659 69 10 39 4 3 149 94 23 7 413.5 70 11 26 1 6 104.5 61 4 11 299 71 12 8 1 5 71.5 26 4 18 223 72 13 3 2 5 57 10 17 16 168.5 73 14 5 1 2 47.5 15 3 3 132.5 7^ 15 6 2 1 39 12 6 4 110 75 16 2 2 31 7 2 91 76 17 1 3 27 5 11 82 77 18 2 23 5 66 78 19 21 61 79 20 1 1 21 5 5 61 80 21 3 19 8 51 81 22 1 4 16 1 3 43 82 23 1 1 11 12 7 39 83 24 1 9 1 20 8Jf 25 4 1 1 7.5 11 1 1 18.5 85 26 1 2 3 6 86 27 1 3 87 28 1 3 88 29 1 1 3 3 89 1,536 765 381 11,758.5 4,769 2,541 1,187 37,149.0 TABLE I. 117 OBSERVATIONS ON MALE LIFE. Age at Entry, 61 years. •si Number OF Entrants, 1,683. $5,239,000 Insured. Age at Exit. Existing. Discon- tinued. Died. Exposed to Risk. Amounts Existing. Amounts Discontin'ed. Deatli Claims. Exposed to Rislc. $ $ $ $ 90 53 20 815 219 117 77 2,561 61 1 74 187 24 1,426.5 207 591 80 4,530.5 62 2 73 68 29 1,201 277 257 89 3,819.5 63 3 121 54 32 1,038 291 266 90 ' 3,192 6J, 4 128 46 27 835 342 228 125 2,564 65 5 109 35 23 639.5 250 171 77 1,897.5 66 6 118 8 15 486 401 30 40 1,470 67 7 101 15 20 341.5 306 28 67 1,000 68 8 61 5 10 210.5 180 9 35 608.5 69 9 30 3 10 135.5 72 11 37 383.5 70 10 18 1 6 93.5 53 1 12 . 268.5 71 11 11 2 4 68 58 9 12 198.5 72 12 4 2 1 51 4 6 5 121 73 13 4 1 2 44.5 11 2 1 108 U 14 4 2 38 8 8 95 75 15 1 2 32 1 5 79 76 16 1 2 29 4 1 73 77 17 2 1 5 25.5 5 3 12 66.5 78 18 2 18 12 48 79 19 1 16 3 36 80 20 1 15 3 33 81 21 1 1 1 13.5 2 2 3 29 82 22 1 11 3 23 83 23 1 1 10 4 ' 1 20 8^ 24 3 8 4 15 85 25 2 1 5 6 2 11 86 26 1 2 2 3 87 27 1 1 88 28 1 1 89 29 1 1 1 1 90 957 482 244 7,611.5 2,705 1,731 803 23,257.0 Age at En try, 62 1 fears. Ni JMBER OF Entrants , 1,206. $4,188,000 Insured. 62 89 27 20 589.5 341 71 38 2,058.5 1 52 118 33 1,011 213 333 115 3,571.5 63 2 55 42 21 846 185 181 71 2,986.5 6J, 3 64 48 29 725 157 161 114 2,559.5 65 4 95 25 27 595.5 305 186 79 2,115 66 5 62 16 18 453 239 155 33 1,560.5 67 6 89 6 11 362 330 19 28 1,201.5 68 7 78 13 19 252.5 329 33 55 817.5 69 8 46 4 6 147 131 8 16 413 70 9 28 1 --- 6 92.5 76 14 12 255 71 118 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 62 Years. !i Number op Entrants, 1,306. $4,188,000 Insured. Age !3b at Existing. Discon- tinued. Died. Exposed to Risli. Amounts Existing. Amounts Discontin'ed. Death Claims. Exposed to Risli. Exit. $ $ $ $ 10 5 2 58 20 3 160 72 11 8 1 4 50.5 23 3 6 135.5 73 12 3 1 1 37.5 5 1 5 104.5 u 13 • 1 2 33 6 6 94 75 14 1 1 30 3 1 82 76 15 3 " 1 4 27.5 9 2' 9 77 77 16 1 3 21 10 7 58 78 17 1 17 1 41 79 18 1 1 16 5 1 40 80 19 1 2 14 2 3 34 81 20 1 11 4 29 82 21 1 10 3 25 88 22 1 1 9 6 3 22 8J^ 23 4 7 9 13 85 24 3 4 86 25 3 4 87 26 1 1 3 2 1 4 88 27 1 1 1 1 89 687 303 216 5,425.5 2,406 1,167 615 18,466.5 Age at Entry, 63 Years. > fUMBER OF Entrant 8, 957. $2,897,000 Insured. 63 52 24 9 466.5 159 62 24 1,417.5 1 47 116 23 814 171 316 60 2,494 6J^ 2 56 52 24 660 192 143 79 2,033.5 65 3 37 37 18 535.5 92 107 74 1,637.5 66 4 64 32 . 15 446 181 106 42 1,365 67 5 62 15 12 343.5 161 144 48 1,017 68 6 54 10 14 257 172 30 43 721 69 7 47 6 6 181 166 17 16 482.5 70 8 46 2 5 124 99 8 13 288 ■71 9 15 2 6 71 31 5 14 169.5 72 10 9 1 3 48.5 34 1 5 121.5 73 11 5 1 4 35.5 6 3 6 80.5 U 12 3 1 4 25.5 10 2 2 66 75 13 4 1 1 17.5 22 1 5 52.5 76 14 1 -2 12 1 2 25 77 15 1 9 1 22 78 16 8 21 79 17 1 8 5 21 80 18 1 1 7 6 3 16 81 19 . 1 4.5 3 5.5 82 TABLE I 119 OBSERVATIONS ON MALE LIFE. Age at Entry, 63 Years. Number op Entrants, 957. $2,897,000 Insured. Age ^u at «m , , Discon- , Exposed to Risk. Amounts Amounts Death Exposed to Exit. >'A Existing. tinued. Died. Existing. Discontin'ed. Claims. Kisk. $ $ $ $ 20 1 4 1 4 83 21 3 3 84 22 3 3 85 23 1 3 1 3 86 24 2 2 87 25 2 2 2 2 88 506 301 150 4,091.0 1,506 948 443 12,073.5 Age at Entry, 64 Years. 1^ lUMBBB OF Entrant 8, 770. 12,381,000 Insured. 64 56 18 6 376 179 32 19 1,174.5 1 35 75 23 652.5 101 189 58 2,056.5 65 2 34 39 13 537.5 105 127 51 1,739.5 66 3 39 33 12 454.5 114 93 49 1,473.5 67 4 50 7 12 383.5 179 64 39 1,232 68 5 63 13 14 311.5 159 99 42 932.5 69 6 50 10 12 223 134 25 47 669.5 70 7 27 4 11 154 102 29 41 461.5 71 8 41 2 10 113 140 3 34 302:5 72 9 16 3 3 59.5 37 7 5 123.5 73 10 11 2 1 38 20 3 1 76.5 74 11 1 25 4 54 75 12 1 2 24 2 6 50 76 13 2 3 20 1 7 41.5 77 14 2 2 16 2 4 34 78 15 1 12 2 28 79 16 2 1 11 1 3 26 80 17 8 22 81 18 1 3 • 7.5 3 13 20.5 82 19 1 4 3 6 83 20 2 3 2 3 84 21 1 1 85 22 1. 1 86 23 1 1 1 1 87 427 209 134 3,436.5 1,275 675 431 10,530.0 120 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 65 Years. Age at Entry, 66 Years. ^a" No. OF Entrants, 592. $1,780,000 Insubed. ■i-> •a « bo < No.< 3F Entrants, 801. ■$858,000 Insured. •a bo < a>c3 a 11 3*" i s 1^ ^2 1 S W §1 5*^ H2 1 1 1 -a C o 1 5*" 5 73 . ion c 1 $ S 2 T 5i S3 $ on ^2 $ $ $ $ 46 17 6 287.5 53 22 36 852 65 15 3 7 5 12 81 1 20 66 11 490 37 192 32 1,519 66 16 1 4 2 7 82 2 24 42 12 405 71 110 43 1,299 67 17 3 5 83 3 27 17 15 339.5 77 79 41 1,090.5 68 18 3 5 8J, 4 34 11 10 283.5 97 43 27 911.5 69 19 1 2.5 3 3.5 85 5 44 14 13 227 127 123 39 704.5 70 20 2 2 86 6 31 8 10 159 95 39 27 457.5 71 21 1 2 1 2' 87 7 45 6 5 111 111 31 9 300.5 72 22 88 8 21 5 58 56 23 165 73 23 89 9 12 1 4 31.5 33 9 12 81.5 U 24 90 10 3 15 12 32 75 25 91 11 12 13 14 1 1 1 2 12 12 11 9 3 1 2 2 20 20 19 14 76 77 78 79 26 1 130 1 92 94 77 1,363.5 397 259 202 4,215.5 15 1 7 1 12 80 Age at Entry, 67 Years. 16 17 18 19 1 1 5.5 4 4 4 .1 2 10.5 8 8 8 81 '82 83 8J, No. OF Entrj INTS, 229. $597,000 Insured. 67 11 8 8 110.5 22 14 45 291.5 20 1 1 4 3 1 8 85 1 8 22 8 191 12 64 25 484 68 21 1 2 3 4 86 2 4 13 7 157.5 11 22 25 404 69 22 1 1 87 3 8 11 3 134.5 17 44 9 335 70 23 1 1 88 4 18 8 3 114 47 12 9 281 71 24 1 1 89 5 11 6 4 86 25 15 7 211.5 72 25 1 1 775 649 1 1 90 6 7 8 18 9 9 3 2 3 4 2 66.5 42 28 5 66 13 39 12 5 2 4 6 4 169.5 94 74 33 73 74 75 76 309 183 100 2,485.5 302 7,548.0 A Ke . it Entry, 66 Year s. 9 6 18 10 11 2 1 1 j- 12 9 7 21 13 77 78 No. OP E ■ntba NTS, 301. $858,000 In 5UBED. 12 8 12 7Q 9 11 5 145 37 25 11 416.5 66 13 1 8 12 80 1 23 29 8 261.5 5 69 14 750.5 67 14 7 11 81 2 11 20 11 206 29 58 29 668 68 15 1 7 11 82 3 7 17 8 165.5 42 50 32 556 69 16 1 6 10 83 4 10 3 8 140.5 44 4 25 455 70 17 1 5 8 8J, 5 21 6 9 118 64 28 16 370 71 18 4 7 85, 6 12 2 5 84 47 8 18 272 72 19 ^ 4 4 7 66 7 13 3 2 64.5 36 2 4 202 73 20 1 3 3 87 8 13 1 2 47.5 32 11 4 155.5 n 21 1 2 2 88.' 9 6 1 2 31.5 20 1 9 113.5 75 22 1 1 89 10 2 8 23 20 7 84 76 23 1 1 90 11 2 3 18 20 3 57 77 24 1 1 91 12 13 3 3 13 10 6 16 34 28 78 79 25 1 1 1 1 92 14 7 12 80 105 76 48 1,027.5 275 182 140 2,498.5 TABLE I. 121 OBSERVATIONS ON MALE LIFE. Age at Entry, 68 Years. Age at Entry, 70 Years. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 2 3 4 5 6 7 8 9 10 11 12 13 No. OF Entrants, 151. si's O 13 13 1^* 6 5 1 73 3 16 2 131 1 6 5 115 5 7 102.5 6 6 1 91 16 3 8 79.5 11 1 5 53.5 4 2 3 36 8 3 28 3 1 2 16.5 1 1 2 10.5 1 6.5 1 6 1 1 4.5 1 3 2 2 2 2 2 2 1 2 ' 1 .5 64 51 36 771.0 3,000 Insdbed. 17 5 14 19 50 22 15 36 16 202 83 93 h: $ 188 357 320.5 287.5 262 233.5 165 120.5 94 42 20.5 10 9 7.5 4 3 3 3 3 3 3 3 .5 2,142.5 Age at Entry, 69 Years. 68 69 70 71 72 73 7U 75 76 77 78 79 80 81 82 83 8J, 85 86 87 88 89 90 No, OF Entrants, 92. $284,000 Insured. 1 46 1 132 3 6 5 88 8 24 12 251 1 7 1 73.5 1 34 1 202 4 65.5 9 14 6 176 6 1 2 53.5 17 3 3 152.5 9 3 2 43.5 36 14 3 124 8 2 5 30 19 2 10 77 6 1 16 15 4 47 1 9 10 28 2 7 4 16 2 1 6 2 1 14 2 3 1 10 11 1 1 1 1 1 42 26 24 443.0 127 95 42 1,232.5 69 70 71 72 73 75 76 77 78 79 80 81 82 Cm <0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 No. OF Entrants, 58. 23 22 13 25.5 43.5 34.5 29.5 24 19.5 15 9 3 3 3 3 2.5 1 1 1 1 1 233.0 $120,000 Insured. o a 5'^ 5s W2 $ % 1 $ 4 8 3 56 1 11 5 99.5 1 9 4 83.5 10 5 1 71.5 15 58 7 1 1 42.5 10 6 34 2 18 6 16 1 1 10 8 8 8 8 1 5 7.5 2 2 2 2 2 2 55 35 30 538.5 Age at Entry, 71 Years. Age at Entry, 72- Years. No. OF Entrants, 84. $59,000 Insurbd. 1 1 11.5 2 1 1 29 3 1 20.5 7 1 1 54.5 1 15.5 3 1 45.5 1 13.5 3 4 40.5 1 1 11.5 5 1 4 34.5 bo < 70 71 72 73 7Jt 75 76 77 78 79 80 81 82 83 81f 85 86 87 88 89 No. OF Entrants, 31. $79,000 Insured. 1 15.5 2 39.5 1 5 2 27.5 1 8 9 73 4 2 2 21 1 3 8 57.5 3 1 14 11 1 47 1 1 10 1 1 35 1 8 1 33 2 1 6 6 5 29 3 1 3.5 20 1 20.5 14 10 7 105.5 37 18 24 334.5 71 72 73 u 75 76 77 78 72 73 71^ 75 76 132 TABLE I. OBSERVATIONS ON MALE LIFE. Age at Entry, 72 Years Age at Entry, 75 Years. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 11 No. OP Entrants, 24. $59,000 Insubed. to a a Q 1.3 5^ ■t •o . §.2 h2 si a 1 1.1 T si ax w2 1 5 1 1 8.5 2 2 24 6 1 1 6.5 2 5 20 7 1 1 5 1 5 14 8 1 1 3 2 1 8 9 1 5 10 1 5 11 1 5 12 1 5 13 1 "9" 7 8, 1 100.5 5 25 11 5 23 295.0 Age at Entry, 73 Years. Age at Entry, 74 Years. No. OF Entrants, 11. 1 1 5 1 8.5 1 7.5 7 3 1 7 3 3 1 1 2.5 1 6 4 1 50.5 $22,000 Insubed. 10.5 19.5 18.5 18 18 • 8 5.5 1 1, 1 1 1 1 1 113.0 77 78 79 80 81 83 8Jf 85 No. OF ENTBANT8, 19. $44,000 Insured. 1 1 9.5 3 1 22 2 16 9 35^ 2 3 13.5 2 8 27 1 1 9.5 1 ^ 1 18.5 1 8 1 15 2 7 1 14 2 5 2 13 2 2 6 8 1 1 5 5 4 8 7 71.5 10 28 6 158.0 73 u 75 76 77 lO 79 80 81 u 75 76 77 78 79 80 81 82 83 8^ 85 86 87 Is (D ^ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 No. OF Entrants, 11. §1 .• u a % 1 2 $19,000 Insured. 1? W2 5.5 11 10 9 < 6.5 2 1 1 1 1 1 1 1 1 1 1 1 1 1 65.0 .2I 1-3 14 $ ' $ 2 w2 $ 9.5 19 17 16.5 15 14.5 4 1 1 1 1 1 1 1 1 1 1 1 1 1 109.5 Age at Entry, 76 Years. Age at Entry, 77 Years. to < 75 76 77 78 79 80 81 \ 82 \ 83 \ Sh 85 86 87 88 89 90 91 92 93 9A 95 No. OF Entrants, 2. $a,000 Insured. 1 1.5 2 3 2 3 \ 2 3 1 2 1 2 3 1 1 .5 1 .5 1 1 10.5 2 1 15.0 76 77 78 79 80 81 82 No. OF Entrants, 1. $2,0X) Insured. .5 1 1 2 1 1 2 2 2 1 2.5 5.0 77 78 79 TABLE I. 123 OBSERVATIONS ON MALE LIFE. Age at Entry, 78 Years. Age at Entry, 82 Years. "1 ^1 No. OF Entrants, 4. 15,000 Insured. is" No. OP Entrants, 1. $1,000 Insured. a 1 Died. Exposed to Risk. be B a) i.5 «5 14 is si g hi 1.1 5*" i 1^- w2 bi) % 1 1.1 5'^ 51 §•5 as 1 2 3 4 1 1 1 1 2 3.5 1.5 1 .5 $ $ 1 1 2 $ 1 $ 2.5 4.5 2.5 2 1 7§ f^2 M 1 , •! •o 6b . -^ "» • 2S M -a . bb 1 . _! 13 . &U .S tPo' w^ a =^ »; s-^ 03 .a o'li^ ^x c a% ui ^M 03 1 1 1|.| i H fi 1 ft ©•2 h2 o - 5" q5 M bo < {Si 1 (5" 5 h2 ig'-s OS if MS « bo < $ $ $ $ $ % 1 $ 2.5 3 1 1 1 1 1 •4 1 1 4.5 1 5.5 2 1 1 1.5 1 1.5 5 2 4 5 3 2 1 1 6 3 4 5 Jf 3 1 1 7 4 1 4 2 5 5 4 1 1 8 5 1 2.5 1 2.5 6 5 1 .5 1 .5 9 6 7 8 1 2 2 2 1 2 2 2 V 8 9 3 6.0 3 6.0 9 10 1 1 1 1 1' 1 10 11 Age at Entry, 5 Years. — — No. OF Entrants, 1. $1,000 Insubed. 3 2 29.5 4 2 34.0 ■ .5 .5 5 Age at Entry, 2 Years. 1 2 1 1 .5 1 1 .5 6 7 No. OF Ektbants, 2. $2,000 Inscbbd. 2 ■ 1 2.0 1 2.0 1 1 Age at Entry, 6 Years. 1 2 3 4 i 1.5 1 1 1 1 1.5 1 1 1 8 5 6 No. OF Entrants, 1. $1,000 Insured. 6 .5 .5 5 1 .5 1 .5 7 1 2 3 4 7 2 6.0 2 6.0 8 9 JO Age at Entry, 3 Years. 5 B 11 1 No. OF Entrants, 8. $3,000 Insured. 3 5 7 8 9 1 .5 1 .5 13 U 15 1.5 3 1.5 3 2.5 2 3 1 1 2.5 2 1 1 1 ^ 9.0 1 9.0 2 6 4 5 1 1 .5 1 1 .5 7 Age at Entry, 7 Years. 8 No. OF Entrants, 4. ' $4,000 Insured. i 2 10.5 1 2 10.5 1 7 2 2 1 1 3.5 1 3.5 8 2 3 3 9 3 1 2.5 1 2.5 10 4 2 i 2 11 128 TABLE II. OBSERVATIONS ON FEMALE LIFE. Age at Entry, 7 Years. Age at Entry, 9 Years. No. OP Entrants, 4. $4,000 Insured. •a be •«5 >^5 No. op Entrants, 7. $21,000 Insured. *a bio .2 '53 §1 5= ^M j 1 h 1 1^ ^2 bi c 'p. a a) 1.1 5*" -a 5 1^ ■la w2 'x $ 11 1 'tf • po w2 $ $ $ $ $ $ 5 1 2 1 2 i^ 10 2 19 6 1 1 i5 11 \ 2 20 7 1 .5 1 .5 n 12 2 21 1 3 16.5 1 3 16.5 13 14 15 16 2 2 2 2 22 23 25 Age at Entry, 8 Years. 17 18 1 2 14 2 2 26 27 No. OP Entrants, 5. $5,000 Insured. 8 4 3 32.0 7 84.0 2.5 1 2.5 1 2 3 4 1 2 5 4 3 2 1 2 5 4 3 3 9 10 11 Age at Entry, 10 Years. No. OP Entrants, 18- $22,000 Insurbd. 5 6 1 2 1 2 IS U 15 1 1 2 1 9 17 2 3 1 11 20.5 10 11 7 2 1 13.5 1 15.5 12 8 9 1 1 16 17 3 4 1 1 13 12.5 1 1 15 14.5 13 10 11 12 13 14 / 18 19 20 21 22 5 6 7 8 9 1 2 3 11 10 7.5 6 6 1 3 2 13 11.5 9 8 8 15 16 17 18 19 15 16 1 1 23 10 11 12 1 1 5 4.5 3 1 1 7 6.5 . 5 20 21 9.9, 3 2 29.5 3 2 29.5 13 14 3 3 5 5 23 U . Age at Entry, 9 Years. 15 16 17 1 2.5 2 2 1 4.5 4 4 25 26 97 No. OP Entrants, 7. $21,000 Insured. 9. 18 2 2 4 4 28 1 1 3 5 1 10 6 11 1 132.5 9 12 1 171.0 1 2 3 4 5 1 1 1 1 4.5 3 3 3 2.5 5 2 1 5 14.5 9 9 9 6.5 10 11 12 IS U Age at Entry, 11 Years. No. OP Entrants, 25. $40,000 Insured. 11 2 1 12 6 1 19.5 6 1 2 15 1 1 1 21.5 5 1 32.5 12 7 1 2 16 2 1 7 16.5 5 5 24.5 13 8 1 2 17 3 2 3 1 10.5 2 4 2 15 U 9 1 2 18 4 1 6 1 9 15 TABLE II. 139 OBSERVATIONS ON FEMALE LIFE. Age at Entry, 11 Years. Age at Entry, 12 Years. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Years of Insurance. No. OF Entrants, 25. $40,000 Insured. be a to 9. ^ 5'^ i 1^- P h2 .S c« 1 J "S^- $ $ $ $ 5 5 8 6 1 5 2 8 7 4 6 8 4 6 9 4 6 10 4 6 11 2 3 4 4 12 2 2 13 2 2 14 1 1.5 1 1.5 15 1 16 1 17 1 18 1 19 1 20 1 21 1 1 1 9 15 1 108.0 22 16 2 157.0 Age at Entry, 12 Years. 16 17 18 19 20 21 H 25 26 27 28 29 30 31 32 No. OF Entrants, 27. $62,000 Insured. 1 13.5 10 31 2 4 24 10 3 50.5 2 3 1 18.5 3 10 1 34 1 2 13 1 5 22.5 1 11 1 19 3 10 3 18 1 1 6.5 1 1 14.5 2 4 3 3 3 3 3 6 10 7 7 7 7 7 1 3 2 2 2 2 2 2 5 7 • 2 2 2 2 2 2 13 U 15 16 17 18 19 20 21 2^ 25 26 27 28 29 30 31 > 130 TABLE II. OBSERVATIONS ON FEMALE LIFE. Age at Entry, •14 Years. Age at Entry, 15 Years. 0)5 4 1 7 2 2 3 4 4 2 5 2 6 3 7 3 8 1 9 2 10 11 1 12 13 1 14 1 No. OF Entrants, i 33 51 lii ^2 41.5 70.5 51.5 41.5 31.5 26 20.5 14.5 9 6.5 4 4 2.5 2 1 326.5 $130,000 Insuebd. bb 1 ■2a W2 $ $ $ $ 8 3 63.5 13 13 112.5 3 17 1 84.5 2 9 1 67.5 2 5 57.5 2 11 47.5 7 2 39 4 2 30 2 2 24 15 2 20 4 1 4 1 2.5 1 2 1 67 1 61 2 559.5 Age at Entry, 15 Years. U 15 16 17 18 19 20 21 22 2S 25 26 27 28 No. OP Entrants, 136. $308,000 Insured. 6 5 65.5 13 4 102 1 6 33 108.5 6 33 174.5 2 8 11 80.5 16 18 143 3 1 6 2 64 1 13 c i 111.5 4 5 11 52.5 11 19 87.5 5 5 4 40 6 4 65 6 7 2 2 32 12 2 2 56 7 5 5 19.5 5 10 36 8 5 12 6 26 9 3 7 5 20 10 1 3.5 1 i 1 14.5 11 3 1 14 12 3 14 13 3 14 14 3 14 15 1 3 5 14 16 2 9 17 2 9 18 2 9 19 2 9 20 2 9 21 2 9 22 2 9 23 2 9 24 1 2 5 9 15 16 17 18 19 20 21 22 23 2J^ 25 26 27 28 29 SO 31 3J, 35 36 37 38 39 No. OF Entrants, 136. 54 i».5 78 520.0 $808,000 Insured. 95 104 9 ■5B h2_ $ 4 4 995.0 Age at Entry, 16 Years. Age at Entry, 17 Years. No. OF Entrants, 209. 16 6 •9 4 12 51 11 13 6 98.5 155.5 116.5 95.5 81 $.336,000 Insured. 160 273.5 219.5 175 142.5 bo < UO No. OF Entrants, 164. $229,000 Insured. 9 4 80 11 6 111.5 1 6 41 1 130.5 7 50 1 187 2 11 13 1 96.5 13 18 1 145 3 1 6 75 1 11 116.5 4 6 8 67 12 19 100.5 5 5 4 55 4 8 75 6 13 3 46.5 15 7 63.5; 7 5 2 31 13 2 44 1 8 9 1 25 7 1 30 1 9 ' 5 2 14 1 2 21 1 1 10 2 8 8 19 ' 11 1 6 5 11 12 5 6 13 5 6 14 3 5 2 6 15 2 4 16 2 4 17 2 4 18 2 4 19 1 2 1 4 20 3 21 3 22 3 23 3 24 1 3 3 77 83 4 664.5 102 123 4 977.0 16 17 18 19 20 21 2J, 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 17 18 19 20 21 TABLE II. 131 OBSERVATIONS ON FEMALE LIFE. Age at Entry, 17 Years. «a* No. OF Entrants, 209. $.3-36,000 a it 5'^ 13 S Exposed to Kisk. so c 5" $ $ 5 8 9 64.5 14 18 6 17 5 49.5 33 7 7 9 3 1 29 15 1 8 6 2 17 9 2 9 5 10 12 10 1 4.5 1 11 4 12 4 13 4 14 4 15 4 16 4 17 4 18 1 4 19 3 20 3 21 3 22 1 3 1 23 1 2 1 24 1 90 1 1 112 7 768.5 142 178 OS 16 113 86.5 49.5 31 21 9 3 3 3 2 1 1,359.5 Age at Entry, 18 Years. 22 23 21 25 2G 2 ( 28 29 30 31 32 33 3J^ 35 .36 37 38 39 40 41 No. OP Entrants, 372 $651,000 Insured. 16 31 3 170.5 28 32 4 309.5 1 11 109 1 267.5 11 230 1 472 2 21 40 181 27 72 309 3 12 26 3 127 18 57 7 217.5 4 8 12 1 93 16 16 1 156 5 11 11 1 72.5 12 23 2 119.5 6 16 5 2 52.5 21 6 1 91 7 8 4 30 25 4 64 8 1 2 19 1 8 33 9 4 1 16.5 5 1 27.5 10 2 12 4 22 11 2 1 10 3 5 18 12 7 10 13 7 10 14 1 7 1 10 15 1 6 1 9 16 5 8 17 5 8 • 18 5 8 19 5 8 18 19 20 21 22 23 25 26 ?7 28 29 30 31 32 33 34 35 36 37 Age at Entry, 18 Years. *,g Existing. tinued. Died. Existing. Discontin'cd. Claims. Risk. $ $ $ $ 56 75 7 533 88 116 16 935 2Z 1 45 278 8 864 79 482 . 23 1,525 23 2 60 121 9 611.5 94 262 14 1,051 n 3 60 61 4 451.5 68 145 5 739.5 25 4 44 39 6 337.5 74 60 21 564 26 5 34 25 2 255.5 56 57 5 410.5 27 6 35 11 3 201.5 40 14 7 314 28 7 33 16 1 150 50 44 1 238 29 8 29 19 1 98.5 46 29 2 150.5 30 9 11 2 1 58 11 6 1 85 31 10 14 2 45 21 4 70 32 11 5 1 28.5 10 1 44.5 33 12 3 23 2 34 3J, 13 1 19.5 3 30.5 35 14 1 19 5 2^9 36 15 1 17.5 1 23.5 37 16 17 23 38 17 2 2 17 2 2 23 39 18 13 19 J,0 19 2 13 1 . 19 Jfl 20 11 18 ^ 21 11 18 JfS 22 2 11 3 18 U 23 ■ .2 9 3 15 U5 24 4 7 8 12 46 25 2 3 2 4 47 26 1 1 2 2 Jf8 445 650 46 3,826.5 665 1,220 101 6,415.0 Age at Entry, 23 1 fears. N [TUBER OF ] Bntrantb , 1,264. $2,268,000 Insured. 23 68 81 9 591.5 114 130 22 1,069 1 59 300 7 956 92 536 11 1,734 2J, 2 82 104 8 688 114 221 .11 1,252.5 25 3 44 73 7 509.5 60 176 16 929 26 4 57- 37 5 403.5 84 87 15 721.5 27 ■ 5 51 29 5 308.5 66 60 9 549 28 6 52 14 3 231 80 27 5 430.5 29 7 55 18 160 106 18 323 SO 8 30 2 95 67 4 206 31 9 10 2 63 26 4 135 32 10 7 52 11 107 33 11 6 1 1 44.5 13 2 1 95 SA 12 3 2 36 5 1 79.5 35 13 32 74 36 14 9 1 31.5 8 2 73 87 134 TABLE II. OBSERVATIONS ON FEMALE LIFE. Age at Entry, 23 Years. zi . sfMBEB OF ] Bntbants , 1,264 $2,268,000 Insueed. Age §i at £S . . Difscon- Exposed to Risk. Amounts Amounts Death Exposed to Risk. Exit. 1 ^M Existing. tinned. Died. Existing. $ Discontin'ed. Claims. $ $ $ 15 2 2 29 9 6 64 38 16 25 49 39 17 1 24.5 1 48.5 JfO 18 2 24 16 48 U 19 2 1 22 2 3 • 32 ^ 20 3 19 3 27 JfS 21 2 16 2 24 U 22 2 1 14 3 1 22 Jf5 23 3 11 7 18 Jf6 24 4 8 4 11 A7 25 3 4 3 7 It8 26 1 4 U9 27 1 550 1 4 4 50 665 49 4,400.5 899 1,269 100 8,136.5 Age at Entry, 24 "5 fears. N (THBEB OF Entbant 5, 1,399. $2,603,000 Insttbkd. H 64 77 6 661 102 132 5 1,235.5 1 57 328 11 1,088 108 567 17 2,080.5 25 2 88 130 10 791 133 240 25 1,552 26 3 61 84 7 586 100 191 17 1,178.5 27 4 67 51 5 450.5 109 156 13 888 28 5 59 34 3 336 114 96 6 640 29 6 54 11 5 251.5 106 25 10 459.5 30 7 42 14 2 180 63 26 3 318 31 8 34 8 1 125 75 15 1 231.5 32 9 22 7 2 82.5 33 14 2 141 33 10 5 1 1 54.5 8 1 •2 98.5 Si 11 8 1 47.5 11 1 87.5 35 12 39 76 36 13 1 39 10 76 37 14 3 1 37.5 8 2 65 38 15 1 34 5 56 39 16 1 33 1 51 JfO 17 1 32 . 5 50 Jfl 18 1 30.5 1 44.5 J^ 19 30 44 43 20 30 • 44 U 21 2 30 5 44 45 22 1 28 1 39 46 23 4 2 27 5 2 38 Jf7 24 6 21 13 31 48 TABLE II. 135 OBSERVATIONS ON FEMALE LIFE. Age at Entry, 24 Years. ^1 Number of Entrant 3, 1,399. $2,603,000 Insured. Age 11 at Exit Discon- Exposed to Amounts Amounts Death Exposed to task. tHU tinued. Risk. Existing. Discontin'ed Claims. $ $ $ $ 25 8 1 15 9 1 18 49 26 5 6 7 8 50 27 1 1 1 1 51 595 748 56 5,086.5 1,032 1,467 104 9,596.0 Age at Entry, 25 Years. N CHBER OF Entrants i, 1,547. . $3,119,000 Insubed. 25 66 103 3 722 134 161 3 1,479 1 68 347 15 1,201.5 94 697 22 - 2,472.5* 26 2 79 126 9 882 107 337 24 1,839.5 27 3 49 93 6 684.5 71 275 12 1,402.5 28 4 73 58 4 554 93 206 10 1,079 29 5 81 40 3 429 136 94 3 826 30 6 62 , 18 4 31G 142 45 8 617.5 31 7 49 19 2 231.5 126 37 3 426.5 32 8 45 8 1 167 77 19 2 269.5 33 9 23 4 4 115 33 4 4 179 SJf 10 15 1 85.5 23 3 138.5 35 11 7 2 69 15 2 113 36 12 1 2 1 60 1 1 1 96.5 37 13 2 2 56 3 3 92.5 38 14 1 1 53 2 1 88 39 15 2 1 1 50.5 5 4 1 83 1,0 16 2 2 47 2 " 4 75 41 17 3 43 3 2 69 42 18 2 1 38.5 2 1 63.5 43 19 1 36 2 61 44 20 1 35 2 59 45 21 3 34 3 57 46 22 1 31 2 54 47 23 6 2 30 9 5 52 48 24 1 10 22 16 38 49 25 7 12 15 22 50 26 i 1 5 5 7 7 51 663 825 59 6,010.0 1,125 1,889 105 11,760.0 136 TABLE II. OBSERVATIONS ON FEMALE LIFE. Age at Entry, 26 Years. •si NUMBEB OF Entrants, 1,664. $3,058,000 Insured. Age ►2S at Discon- Exprtsed to Risk. Amounts Amounts Death Exposed to Risk. T-rit >"& Existing. 1 tinued. 1 Died. Existing. Discontin'ed Claims. ri ft 1 1, 1 $ $ $ S 81 119 5 772.5 132 231 6 1,410.5 £6 1 77 345 17 1,286.5 129 649 35 2,358.5 27 2 82 136 8 952 128 329 16 1,705.5 S8 3 71 , 94 6 747 91 244 10 1,275 29 4 73 66 6 590 117 165 10 969.5 SO 5 73 37 3 459.5 92 76 2 722 SI 6 76 18 6 356 112 40 10 570 S2 7 60 23 3 253.5 113 42 4 407 S3 8 57 4 5 177 94 6 8 266 SU 9 30 7 1 109.5 42 10 2 156 S5 10 p 1^ 4 1 73 17 5 2 104.5 36 11 ' 7 1 57.5 11 1 82.5 37 12 3 48.5 8 67 38 13 1 47 1 63 39 14 2 46 3 62 Jfi 15 2 44 6 59 Jfl 16 2 42 1 53 J^ 17 2 40 2 52 43 18 1 37.5 1 1 49.5 U 19 36 48 45 20 3 36 5 48 46 21 4 1 33 6 1 43 47 22 3 28 3 36 48 23 • 3 25 8 33 1 49 24 9 1 22 11 1 25 60 25 8 12 8 13 51 26 4 4 5 5 52 742 858 64 6,335.0 1,137 1,807 108 10,083.5 Age at Eu try, 27 ^ fears. Ni 7HBEB OP ] Sntbants , 1,624. 13,233,000 Insured. 27 79 75 9 774.5 113 118 16 1,557.5 1 69 335 12 1,293.5 138 627 20 2,672.5 28 2 105 148 9 971 176 390 18 2,006 29 3 74 108 6 729 123 288 10 1,473 SO 4 82 60 5 565 140 173 13 1,109.5 31 5 85 28 5 434 137 99 12 820.5 32 6 56 18 6 321 88 41 15 601.5 33 7 59 17 4 241.5 93 42 3 457 34 8 51 8 4 166 105 18 6 331 35 9 31 4 105 51 5 208.5 36 TABLE II. 137 OBSERVATIONS ON FEMALE LIFE. Age at Entry, 27 Years. i Number of Entrants, 1,624. $3,233,000 Insured. Age u at 1 £S . Discon- , Exposed to Amounts Amounts Death Exposed to Risk. Exit. >"& Existing. tinued. Died. Risli. Existing. Discontin'ed. Claims $ $ $ $ 10 10 3 1 70.5 26 8 2 151 37 11 9 1 57.5 17 2 118 38 12 4 1 47.5 10 1 99.5 39 13 1 1 1 42.5 1 1 1 88.5 40 14 1 40 2 86 41 15 39 84 42 16 1 1 38.5 1 2 83.5 43 17 1 37 3 81 44 18 1 1 36 1 1 78 45 19 2 1 33.5 3 1 75.5 46 20 3 2 30 8 4 70 47 21 26 60 48 22 3 26 6 60 49 23 6 23 9 54 50 24 6 17 14 45 51 25 8 11 23 31 52 26 3 3 8 8 12,510.0 63 749 811 64 6,178.5 1,295 1,819 119 Age at Entry, 28 1 rears. N CTMBEB OF ] Entrants , 1,852. $3,462,000 Insubbd. 28 70 102 4 875 Ill 168 7 1,647 1 75 382 12 1,485 112 678 29 2,837 29 2 112 158 8 1,128 167 422 8 2,146 30 3 93 91 10 883.5 187 158 21 1,681 31 4 100 67 8 701.5 143 180 13 1,304 32 5 88 36 8 542 127 77 13 1,019.5 33 6 107 15 2 420.5 185 36 3 823 34 7 89 15 4 296.5 197 42 4 596 35 8 57 6 2 193 116 24 3 362 36 9 37 6 3 128 64 18 4 222 37 10 18 4 1 83 27 8 2 141 38 11 8 1 1 61.5 18 2 1 107 39 12 4 1 51.5 14 1 86.5 40 13 2 47 3 72 41 14 2 45 1 3 69 42 15 2 1 43 3 2 66 43 16 2 1 40 6 1 61 44 17 2 37 7 54 45 18 1 1 34.5 1 1 46.5 46 19 2 1 32.5 2 5 43.5 47 138 TABLE II. OBSERVATIONS ON FEMALE LIFE. !i Age at Entry, 28 Years. NXTMBBB OF Entrants, 1,852. $3,462,000 Insured. Age at Discon- Exposed to Risk. Amounts Amounts Deatli Exposed to Exit. >*S Existing. tinued. Died. Existing. Discontin'ed. Claims. Risli. $ $ $ $ 20 2 1 30 2 1 38 48 21 4 27 4 35 49 22 3 23 5 31 50 23 7 20 10 26 51 24 2 1 13 2 1 16 52 25 8 10 10 13 53 26 1 2 1 3 H 27 1 2 55 28 1 1 2 2 56 897 886 69 7,255.0 1,526 1,820 116 13,549.0 Age at Ee itry, 29 1 fears. N [7MBEB OF Entrants , 1,713. $3,520,000 Insured. 29 78 89 6 812 150 139 12 1,690.5 1 63 323 18 1,378.5 107 599 54 2,919.5 30 2 79 133 16 1,069.5 130 354 31 2,282 31 Q 81 104 10 856 125 309 25 1,789.5 32 4 93 57 8 684.5 146 208 18 1,381 33 5 89 47 5 531.5 152 124 8 1,051 SI, 6 90 23 5 402.5 186 60 8 799 35 7 79 27 3 282.5 160 60 8 545 36 8 55 8 3 183 94 21 6 336.5 37 9 38 3 1 119.5 77 9 2 221.5 38 10 15 6 3 76 24 10 7 133 39 11 5 3 1 53.5 12 5 2 94.5 40 12 1 46 2 78 41 13 1 1 3 44.5 2 1 13 75.5 42 14 3 40 9 60 43 15 3 1 1 36.5 2 1 3 50.5 u 16 3 32 11 45 45 17 1 1 28.5 1 1 33.5 46 18 1 1 27 1 2 32 47 19 25 29 Jt8 20 2 25 3 29 49 21 1 23 1 26 50 22 2 22 3 25 51 23 5 20 6 22 52 24 8 15 9 16 53 25 4 7 4 7 54 26 3 3 3 3 55 803 826 84 6,843.5 1,420 1,901 199 13,774.5 TABLE II. 139 OBSERVATIONS ON FEMALE LIFE. Age at Entry, 30 Years. ;i Number or Entrants, 1,944. 13,837,000 iNsirBBD. Age II at Discon- Exposed to Amounts Amounts Deatli Exposed to Rislv. Exit. Existing. tinued. Died. Risk. Existing. Discontin'ed. Claims. t $ $ $ 90 79 8 932.5 149 150 16 1,843.5 30 1 76 356 18 1,589 116 659 51 3,192.5 31 2 111 146 9 1,244 174 354 14 2,519 32 3 93 102 18 1,000 144 299 . 38 2,004.5 33 4 103 82 8 797 156 232 17 1,557 3^ 5 125 50 6 620 207 174 9 1,181 35 6 87 23 11 452.5 166 61 34 847.5 36 7 110 21 332.5 176 67 583.5 37 8 67 15 2 204.5 109 31 3 358.5 38 9 43 2 127 76 6 228 39 10 14 . 1 1 82.5 22 1 1 148.5 40 11 11 3 67 31 3 125 41 12 2 2 1 52 6 6 5 88 42 13 1 1 48 1 1 74 43 14 1 1 46 5 1 72 44 15 2 44 3 36 45 16 1 1 41.5 1 3 61.5 46 17 1 40 1 59 47 18 1 n 1 38 1 3 1 56.5 48 19 35 2 53 49 20 1 . 33 1 51 50 21 1 32 1 50 51 22 1 2 1 30 2 4 1 47 52 23 10 1 27 13 1 42 53 24 8 16 16 28 54 25 4 • 8 3 12 55 26 4 4 9 9 56 968 884 92 7,943.0 1,589 2,050 198 15,357.5 Age at Ed try, 31 1 fears. Ni JMBEB OF ] Entrants , 1,674. $3,563,000 Insdtred. 31 75 64 4 805 116 108 18 1,727.5 1 93 313 13 1,374.5 158 628 15 3,007 32 2 107 125 10 1,049.5 135 368 17 2,336 33 3 85 98 6 821 141 306 25 1,847 34 4 91 75 5 643.5 148 232 8 1,412 35 5 79 39 6 490.5 . 134 225 11 1,027.5 36 6 85 18 3 377 215 46 4 747 37 7 80 18 5 271 144 27 12 . 491.5 38 8 53 6 4 174 77 13 10 315.5 39 9 29 5 1 111.5 54 9 1 217.5 40 140 TAHLE II. OBSERVATIONS ON FEMALE LIFE. Age at Entry, 31 Years. ^1 n ei is 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 NuHBEB OF Entrants, 1,674. Existing. 15 11 3 1 4 2 2. 2 3 1 2 5 4 833 Discon- tinued. Died. 778 63 Exposed to Risli. 77 55.5 39.5 35 33.5 28 25 23 20 20 20 18 16.5 15.5 13 6 2 6,565.0 $3,563,000 INSCBED. Amounts Existing. 24 19 7 1 14 5 4 11 2 5 13 4 2 1,441 A.mount8 Discontin'ed. 12 7 3 1,992 Death Claims. 130 Exposed to Risk. $ 152 117.5 88.5 80 78 61.5 55 51 42 42 42 31 28.5 27 21 6 2 14,053.5 Age at Exit. ¥i U ^6 50 51 52 53 H 55 56 57 Age at Entry, 32 Years. Number of Entrants, 1,763. 81 60 8 1 65 383 14 2 87 139 10 3 78 83 10 4 101 62 5 5 84 34 8 6 103 35 4 7 87 27 2 8 53 6 1 9 30 7 1 10 19 11 14 3 12 1 1 1- 13 1 14 2 3 1 15 4 1 16 2 1 17 1 18 1 19 2 1 851.5 1,422.5 1,082.5 ' 874.5 714 560 433. 295. 190 129. 95 74.5 58.5 56 53.5 48.5 44 40.5 40 39 $3,671,000 Insured. 139 122 193 120 158 162 183 174 106 60 22 27 2 4 16 1 5 100 755 392 256 150 145 85 50 18 15 3 2 5 1 17 23 21 19 7 22 6 6 1 5 1,785.5 3,037.5 -2,319 1,781 1,439 1,126.5 827.5 571 357 233.5 161 137.5 108 104 98.5 85 65 64 63 32 33 3Jf 35 36 37 38 39 Jfi Jf2 U Jfi Jf7 Jt8 50 51 TABLE II. 141 OBSERVATIONS ON FEMALE LIFE. Age at Entry, 32 Years. !i NUMBEB OF Entrants, 1,763. $3,671,000 Insubbd. Age at Exit. Discon- Exposed to Risk. Amounts Amounts Deatli Exposed to Bisk. ^'A Existing. tinued. Died. Existing. Discontin'ed Claims. $ $ $ $ 20 2 1 35.5 3 1 56.5 S2 21 2 33 3 53 53 22 7 31 8 50 5J, 23 6 24 11 42 55 24 8 18 15 31 56 25 6 10 10 16 57 26 4 4 6 6 58 849 846 68 7,258.5 1,554 1,980 137 14,707.5 Age at Entry, 33 Years. N UMBBR OF Entrant 5, 1,785. 13,471,000 Insured. 33 101 56 9 864.5 180 88 24 1,691.5 1 69 324 14 1,457 ; 112 626 26 2,866 SU 2 100 129 11 1,147.5 163 310 23 2,260 35 3 87 100 10 922 158 268 12 1,785 36 4 107 65 5 742.5 186 179 15 1,391.5 37 5 103 41 6 577.5 153 145 7 1,028.5 38 6 92 24 4 436 159 62 5 765 39 7 94 22 3 317 f 162 50 3 545 Jfi 8 71 7 205.5 97 10 350 U 9 28 4 2 129 53 7 1 244.5 J^ 10 15 3 1 95.5 30 17 1 178.5 ^ 11 8 3 76.5 16 12 133 U 12 3 4 2 65 5 11 2 105.5 Ji5 13 1 58 1 93 U6 14 2 3 2 55.5 5 3 3 90.5 A7 15 6 1 49.5 8 2 80 1^8 16 5 43 10 71 Jfi 17 3 1 38 6 1 61 50 18 2 2 1 33 7 2 1- 53 51 19 3 29 3 44 52 20 26 41 53 i 21 1 1 25.5 1 1 40.5 H 22 1 1 23.5 1 3 37.5 55 23 4 22 6 35 56 24 7 2 18 9 2 29 57 25 6 1 9 14 1 18 58 26 2 2 3 3 59 921 790 74 7,467.5 1,548 1,796 127 14,040.5 ... i.. 142 TABLE II. OBSERViVTJOXS ON FEMALE LIFE. Age at Entry, 34 Years. •si Number of Ektrants, 1,688. $3,343,000 Insured. Age ii at Exit. Discon- Exposed to llisk. Amounts Amounts ' Death Exposed to >'A Existing. tinued. Died. Existing. Discontin'ed. Claims. Risk. $ i $ $ 102 79 7 804.5 181 144 22 1,600.5 3i 1 61 308 18 1,346 100 605 46 2,695.5 35 2 109 124 11 1,051 201 313 27 2,090.5 36 3 84 96 3 821 130 282 9 1,565 37 4 85 62 7 655 130 179 17 1,195.5 38 5 73 48 4 508 86 102 10 908 39 6 91 24 3 395 175 98 5 712 Jfi 7 76 20 3 279 94 71 10 447.5 U 8 62 11 1 184.5 84 38 2 289 J^ 9 32 3 4 114.5 44 5 10 181.5 43 10 14 4 75 19 4 123 u 11 6 1 1 58.5 9 1 1 101.5 43 12 4 2 50 4 4 89 46 13 3 45 3 83 47 14 1 42 2 80 48 15 3 ' 1 40.5 8 4 76 49 16 1 37 1 66 50 17 36 65 51 18 36 65 52 19 2 1 36 6 1 65 53 20 3 33 4 58 54 21 3 30 4 54 55 22 2 27 6 50 56 23 9 ^ 25 14 44 57 24 7 16 15 30 58 25 5 9 6 15 59 26 2 4 2 9 60 27 2 2 7 7 61 842 783 63 6,760.5 1,335 1,850 160 12,765.5 Age at Ee itry, 35 1 fears. N [JMBBB OF ] Entrants , 1,705. $3,622,000 Insured. 35 82 77 7 814 158 123 19 1,749.5 1 69 270 16 1,404 124 569 45 3,037.5 36 2 94 121 11 1,123.5 188 375 20 2,396.5 37 3 102 94 9 911 156 275 13 1,863.5 38 4 100 67 4 719.5 125 163 4 1,475.5 39 5 91 36 3 564 142 132 5 1,199 40 6 117 28 7 438 188 115 14 928.5 41 7 94 17 3 291.5 182 69 10 634.5 4^ 8 53 6 1 183 108 58 1 379 43 9 34 10 1 121 41 38 2 222 44 10 18 3 1 79.5 35 4 1 158 45 11 • 7 ' 1 59 12 1 120 46 12 2 51 2 107 47 13 4 1 2 48.5 20 3 5 103.5 48 14 3 1 41.5 2 2 76 ^j . T^ABLE II. OBSERVATIONS ON FEMALE LIFE. 143 Age at Entry, 35 Years. Number of ENTBiNTS, 1,705. $3,622,000 Insured. Age II at Existing. Discon- tinued. Died. Exposed to Rislc. Amounts Existing. Amount? Discontin'ed. Deatli Claims. Exposed to Risk. Exit. $ $ $ $ 15 2 1 38 5 1 73 50 16 1 35 5 67 51 17 1 2 1 33 1 5 2 59.5 52 18. 1 1 29.5 5 1 53.5 53 19 3 1 1 27.5 4 2 3 47 H 20 24 3 39 55 i 21 23 36 56 22 2 23 4 36 57 23 8 21 9 32 58 24 6 13 7 23 59- 25 4 7 10 16 60 26 2 3 3 6 61 27 1. 1 3 3 62 899 735 71 7,137.0 1,534 1,934 154 14,941.0 Age at En Ltry, 36 ^ fears. N UMBER OP Entrant! i, 1,551. $.3,002,000 Insured. 36 63 63 3 744 93 127 9 1,437.5 1 70 281 13 1,281.5 133 505 29 2,520.5 37 2 80 110 8 1,003 121 262 31 1,975 38 3 88 78 7 821 186 217 11 1,583.5 39 4 111 47 8 663.5 181 115 8 1,220.5 AO 5 88 38 3 502 123 86 3 931 Jfl 6 78 25 5 379.5 110 47 8 738.5 J^ 7 77 22 1 273 166 50 2 572 AS 8 66 8 1 180 169 19 5 369.5 AA 9 26 7 1 105.5 44 13 1 179.5 A5 10 17 3 1 73.5 24 5 1 125.5 A6 11 6 4 52 19 9 93.5 A7 12 4 44 8 70 A8 13 2 1 39.5 2 1 61.5 AO 14 3 1 36.5 7 1 58.5 50 15 33 51 51 16 33 51 52 17 2 33 3 51 53 18 1 31 1 48 5A 19 1 30 1 47 55 20 1 29 1 46 56 21 6 1 28 10 1 45 57 22 5 21 5 34 58 23 5 16 9 29 59 24 4 11 9 20 60 25 6 7 10 11 61 26 1 1 1 ! 1 • 62 810 688 53 6,471.5 ! 1,435 1,457 110 1 12,370.5 1 144 TABLE II. OBSERVATIONS ON FEMALE LIFE. Age at Entry, 37 Years. •si 1 1 Number op Entbants, 1,548. $3,045,000 Insured. Age at Exit. j Existing. Discon- tinued. Died. Exposed to Risk. Amounts Existing. 1 Amounts Discontin'ed Deatli . Claims. Exposed to Rislc. $ 1 $ $ $ 73 64 8 742 147 ' 86 16 1,479.5 37 1 81 270 15 1,268 146 500 35 2,546 38 2 91 103 7 985.5 143 223 15 2,003.5 39 3 84 92 6 790 143 280 12 1,594 40 4 100 51 3 628.5 167 123 6 1,237.5 41 5 79 32 7 484 156 94 14 956 Jt2 6 91 18 1 373 134 41 2 718.5 4S 7 94 12 2 266 164 39 5 542.5 U 8 47 7 1 160.5 140 24 1 342 45 9 29 1 4 108.5 41 10 6 184 46 10 12 3 73.5 19 4 130 47 11 7 2 59 8 1 108.5 48 12 2 1 1 50.5 5 1 5 99.5 49 13 2 1 46.5 4 1 88.5 60 14 2 44 3 84 51 15 1 2 41 1 6 78 52 16 2 1 2 38.5 8 1 . 7 73.5 53 17 3 1 33.5 4 5 55.5 54 18 1 30 1 49 55 19 . 3 27.5 3 46.5 56 20 1 26 4 45 57 21 25 41 58 22 5 25 6 41 59 23 5 20 ■ 7 35 60 24 7 1 15 12 2 28 61 25 4 7 7 14 62 26 3 3 7 . 7 63 826 664 58 6,371.0 1,477 1,442 126 12,627.5 Age at Eu try, 38 ^ ["ears. Ni TMBER OF I jNTBANTS 1,508. $3,086,000 Insttbed. 38 69 66 5 721 148 110 13 1,488 1 65 240 14 1,248 150 464 26 2,583 39 2 82 119 15 989.5 181 316 27 2,017 40 3 77 76 10 795 134 237 23 1,532.5 41 4 88 67 5 636.5 153 155 7 1,179.5 ^ 5 107 32 6 494 159 89 14 897.5 43 6 97 8 5 361 159 24 7 668 4-^ 7 65 17 5 246.5 123 45 11 467.5 45 8 45 6 165 82 13 304.5 46 ■ 9 t._... 43 4 115 68 8 212 47 TABLE II. 145 OBSEKVATIONS ON FEMALE LIFE. " Age at Entry, 38 Years. NUHBEB OF Entbants , 1,5()8. $3,086,000 Insured. Age II at Exit. Existing. Discon- tinued. Died. Exposed to Rislt. Amounts Existing. Amounts Discontin'ed. Deatli Claims. Exposed to Rislc. $ $ $ 10 10 2 1 '69 23 12 1 134 48 11 8 2 56 16 2 103 49 12 1 47 1 86 50 13 2 46 6 85 51 14 5 1 44 14 1 79 52 15 38 64 53 16 3 1 37.5 5 3 62.5 54 17 1 34 1 56 55 18 1 5 32.5 1 4 54.5 56 19 1 27 1 50 57 20 3 5 26 4 10 49 58 21 1 1 1 17.5 5 5 1 32.5 59 22 5 15 9 24 60 23 10 15 61 24 3 10 4 15 '62 25 5 7 7 11 *6S 26 2 2 4 4 64 1 787 642 79 6,290.0 1,456 1,484 146 12,274.0 Age at Entry, 39 Years. Ni CTMBBR OP ] Entrants , 1,330. $2,782,000 Insured. 39 81 45 G 642.5 148 80 14 1,351 1 51 209 10 1,093.5 102 414 14 2,333 40 2 101 105 4 875.5 209 290 5 1,865 41 3 72 89 8 673.5 130 237 12 1,387.5 42 4 65 53 10 522.5 73 185 25 1,034.5 4S 5 77 29 7 406.5 133 78 30 805 U 6 76 20 1 298 120 48 1 579 45 7 60 10 4 206 116 29 7 419.5 46 8 41 5 134.5 99 16 274 47 9 21 3 89.5 37 3 165.5 48 10 1<> 67 34 127 49 11 8 1 51 10 1 93 50 12 3 2 1 41 6 6 2 79 51 13 36 68 52 14 1 1 36 1 1 68 53 15 34 66 54 16 3 34 15 66 , 55 17 2 30 4 49 56 18 2 3 29 2 3 47 57 19 1 2 23 5 7 38.5 58 146 TABLE II. OBSERVATIONS ON FEMALE LIFE. Age at Entry, 39 Years. H Number op Entrants, 1,330. $2,782,000 Insured. Age is at s£S . Discon- Died. Exposed to Amounts Amounts Death Exposed to lUsk. Exit. >'A Existing. tinued. Risk. Existing. Discontinued Claims. J $ $ $ 20 1 21 1 30 59 21 2 1 20 3 1 29 60 22 17 25 61 23 3 2 17 ! 4 2 25 62 24 5 12 6 19 63 25 4 7 5 13 H 26 1 3 4 8 65 27 2 2 4 4 66 697 574 59 5,422.0 1,267 1,397 118 11,068.5 Age at Entry, 40 ^ fears. tf N UMBER OP Entrant i, 1,444. $2,888,000 Insured. Jfi 73 G3 5 690.5 129 108 8 1,390 1 67 242 17' 1,182 119 527 33 2,379.5 Al 2 94 110 5 922 174 243 19 1,842.5 J^ 3 62 77 10 729.5 88 204 20 1,426 43 4 •83 65 9 586.5 164 184 21 1,124 U 5 84 22 7 451 133 64 17 815 4S 6 87 18 5 340 160 34 10 616 46 '7 81 16 3 231 147 33 4 412.5 47 8 45 4 137 87 9 240.5 48 9 25 2 1 89 37 6 1 146 49 10 IG 2 01 20 3 103.5 50 11 7 ■ 2 43 14 3 80.5 51 12 1 2 34 1 7 61.5 52f 13 2 1 31.5 3 5 54.5 63 14 2 1 28.5 10 3 47.5 54 15 1 26 2 36 55 16 25 34 56 17 1 1 1 24.5 1 2 2 33 57 18 22 29 68 19 1 22 2 ^9 59 20 21 27 60 21 2 21 1 27 61 22 1 1 1 18.5 1 1 1 25.5 62. 23 4 16 7 23 63 24 7 1 11.5 9 2 15 64 25 4 4 5 5 65 747 630 67 5,768.0 1,311 1,438 139 11,023.0 TABLE II. 147 OBSERVATIONS ON FEMALE LIFE. Age at Entry, 41 Years. •Si Number of Ektbants, 1,077. $2,277,000 Instjbed. Age ii at Discon- Exposed to Bisk. Amounts Amount^ Death Exposed to Risk. Exit. >"& Existing. tinaed. Died. Existing. Discontin'ed. Claims. $ $ $ $ 50 35 3 521 84 67 8 1,105 41 1 43 182 10 898 97 374 16 1,931 42 2 58 .74 10 717 122 192 20 1,535 43 3 53 62 5 581 107 168 14^ 1,213 U 4 74 43 4 470.5 139 117 8 949.5 45 5 68 26 5 358 111 61 12 713.5 46 6 63 12 4 266 116 37 9 541.5 47 7 64 19 2 183.5 ,138 60 3 368 48 8 46 3 1 106.5 78 12 1 191 49 9 15 2 57 21 5 103.5 50 10 , 8 6 38 11 25 67 51 11 2 27 4 44 52 12 2 1 25 2 1 40 58 13 1 22 1 37 54. 14 3 21 8 36 55 15 18 28 56 16 1 •18 5 28 57 17 2 1 17 3 2 23 58 18 1 1 14 1 1 18 59 19 12 16 60 20 1 12 1 16 61 21 11 15 62 22 1 11 1 15 63 23 1 1 1 9.5 1 1 1 13.5 64 24 4 7 5 11 65 25 1 3 3 6 66 26 2 2 3 3 67 564 465 48 4,426.0 1,062 1,119 96 9,067.5 Age at Entry, 42 Years. Ni [TMBBE OP ] Entkakts , 1,101. $3,242,000 Insubed. 42 64 42 4 529.5 124 71 9 1,085.5 1 39 176 10 903 56 348 27 1,864 43 2 65 74 10 729 119 174 . 23 1,520 ^-^ 3 69 55 8 589.5 92 162 29 1,210 45 4 64 35 4 467.5 137 81 9 967.5 46 5 69 22 7 371 134 67 14 747.5 47 6 65 25 5 271.5 ! 131 58 7 537 48 7 54 11 2 183.5 t 100 23 2 358.5 49 8 43 5 119.5 !: 86 10 240 50 9 17 2 1 73 1 42 3 1 147.5 51 148 TABLE II. OBSERVATIONS ON FEMALE LIFE. Age at Entry, 42 Years. it Number op Entrants, 1,101. $2,342,000 Insured. Age gg , at k £S . Discon- Exposed to Risk. Amounts Amounts Death Exposed to Risk. Exit. >"& Existing. tinued. Died. Existing. Discontin'ed. Claims. $ $ $ $ 10 11 2 53 17 4 101 52 11 9 41 26 82 53 12 2 1 32 4 1 56 5A 13 1 29 1 51 55 14 1 1 27.5 2 2 49 56 15 2 26 ^ 7 46 57 16 1 2 23 2 {> 36 58 17 2 21 2 31 59 18 19 29 60 19 1 19 2 29 61 20 18 27 62 21 1 1 18 1 2 27 63 22 16 24 eu 23 1 16 1 24 65 24 6 15 8 23 66 25 4 . 9 9 15 67 26 5 5 6 6 68 596 452 53 4,624.5 1,109 1,009 124 9,333.5 Age at Entry, 43 1 fears. J Dumber of Entbani s, 981. $2,065,000 Insured. 43 56 37 5 472 79 49 13 1,008 1 38 144 8 811 92 338 17 1,755 U 2 65 60 7 663 114 179 19 1,387.5 45 3 52 49 6 536.5 82 149 12 1,090.5 46 4 69 31 5 438.5 140 68 14 888 47 5 62 21 3 338.5 153 52 2 674 48 6 64 12 3 257 106 17 19 484.5 49 7 66 16 3 176 123 32 4 335 50 8 40 3 97.5 77 4 190 51 9 12 3 54.5 15 9 106.5 52 10 12 3 3 39.5 25 3 9 85.5 53 11 2 2 1 22 4 4 1 48 54 12 2 18 6 ' 41 55 13 16 35 56 14 1 1 15.5 5 3 33.5 57- 15 14 27 58 16 1 14 1 27 69 17 1 13 3 26 . 60 18 12 23 61 19 1 12 2 23 62 TABLE II. 149 OBSERVATIONS ON FEMALE LIFE Age at Entry, 43 Years. -4 Number op Entrants, 981. $2,065,000 Insured. Age ii at Discon- Exposed to Risk. Amounts Amounts Death Exposed to Exit. >"& Existing. tinued. Died. Existing. Discontin'ed. Claims. Risk. $ $ $ 20 2 11 2 21 63 21 1 1 9 , 1 1 19 64 22 1 7 5 17 65 23 1 6 1 12 66 24 1 5 5 11 67 25 3 4 5 6 68 26 1 1 1 1 69 552 382 47 4,063.5 1,041 907 117 8,375.0 Age at Entry, 44 Years. J {UUBER OF Entbani 8, 926. $1,871,000 Insured. U 67 22 5 452 125 37 14 917 1 28 136 10 764 48 296 15 1,547 J-5 2 55 54 5 631 101 125 5 1,273.5 l6 3 45 .47 5 520.5 62 139 8 1,035.5 47 4 67 32 6 431 1 103 78 19 857 48 5 66 13 3 335.5 140 50 9 671 Jf9 6 63 19 5 250.5 1 155 48 13 473 50 7 49 6 2 170 73 14 2 274 51 8 42 5 2 113.5 64 11 3 186.5 52 9 23 1 2 66.5 43 2 3 113 53 10 9 3 1 39.5 18 3 2 64.5' 54 11 8 1 2 27.5 13 1 5 42.5 55 12 1 1 17 1 2 24 56 13 15 21 57 14 15 21 58 15 1 15 1 21 59 16 2 14 2 20 60 17 1 12 5 18 61 18 1 10.5 1 12.5 62 19 10 12 63 20 1 1 10 1 3 12 H . 21 8 8 65 ' 22 1 7.5 1 7.5 66 23 1 7 1 7 67 24 3 6 3 6 68 25 1 3 1 3 69 26 1 2 1 1 2 70 27 1 1 \ 1 1 71 532 341 53 3,954.5 959 806 106 7,650.5 150 TABLE II. OBSERVATIONS ON FEMALE LIFE. Age at Entry, 45 Years. Age at Entry, ' 46 Years. No. OP Entrants, 465. 1 1905,000 Insured. 1 <«§■ No. OF Entrants, 433. $902,000 Insured. •^ o2 M °d w 1 tl i t» 1 •< t-l tit c it •6 5 h2 ! u a \ 1 i $ 1~ si Si 14 bo < '"I si kt Is 1 P ti c pa $ p $ p5 T m2 $ $ 30 21 1 222 37 33 1 436 51 15 1 11 1 13 57 1 27 66 3 380 38 136 9 766 52 16 2 1 10 1 1 12 68 2 34 23 7 305.5 52 66 11 618 53 17 1 7 2 10 69 3 25 18 2 244 43 38 5 503 5J, 18 6 8 70 4 42 16 1 200 68 47 1 412.5 55 19 1 6 1 8 71 5 33 10 4 144 65 26 12 307 56 20 5 7 72 6 26 7 1 98.5 60 14 1 210 57 21 1 5 1 7 73 7 19 7 3 64.5 35 20 4 132 58 22 2 4 3 6 u 8 15 1 39 37 1 83 59 23 1 2 1 3 75 9 10 11 6 7' 1 22.5 16 8 18 10 1 44.5 26 13.5 60 61 62 24 1 273 . 1 2 2 76 138 22 1,827.5 531 321 50 3,580.5 2 5 12 13 14 1 7 6 6 1 11 10 10 63 6Jf 65 Age at Entry, 53 Years. ' 1 No. ar Entrants, 3SJ0. $633,000 Insured. 15 6 10 66 16 6 10 67 15 16 2 152 14 35 2 299 53 17 6 10 68 1 26 45 3 264.5 61 108 18 528 H 18 1 6 1 10 69 2 17 22 2 202 41 51 6 369.5 55 19 5 9 70 3 21 11 2 166.5 27 20 3 287 56 20 5 9 71 4 21 9 1 133.5 34 21 1 236.5 57 21 1 6 2 9 72 5 15 8 3 103 24 31 6 175.5 58 22 4 7 73 6 25 2 2 80 33 2 2 129 59 23 1 4 1 7 7Jf 7 12 2 2 51 '28 6 2 90 60 24 1 3 3 6 75 8 16 2 35 l23 1 56.5 61 25 1 2 1 3 76 9 4 1 17.5 14 1 32.5 62 26 1 268 171 1 2 470 2 77 10 11 12 13 14 15 4 1 1 1 1 1 13 8 7 6 5 5 - 6 2 1 1 1 1 1 18 11 9 8 7 7 63 H 65 66 67 68 261,816.0 386 49 3,674.5 Age at Entry, 52 Yea rs. No. OF Entb ANT8, 433. $902,000 iNi JURED. 16 4 6 69 31 6 1 213.5 68 8 1 447 52 17 4 *6 70 1 34 53 2 368.5 80 101 6 774.5 53 18 1 3.5 2 5 71 2 26 22 1 295 50 60 3 608 5J, 19 3 4 72 3 37 16 3 249 59 54 6 498 55 20 3 4 73 4 34 18 7 192 641 53 14 379.5 56 21 3 4 74 5 25 8 3 138 60 i 17 8 266.5 57 22 1 3 1 4 75 6 26 5 2 103.5 41 9 6 185.5 '58 23 1 2 1 3 76 7 8 22 11 6 3 70 43.5 48 9 13 9 129.5 72.5 59 60 24 1 1 2 2 77 181 119 20 1,275.5! 312 278 43 2,301.5 9 10 13 3 1 31 17 25 3 55 61 1 7 *J 27 62 11 2 14 6 20 63 12 12 14 6Jf 13 12 14 65 14 1 11.5 1 13.5 66 ' TABLE II. 153 OBSERVATIONS ON FEMALE LIFE. Age at Entry, 54 Years. Age at Entry, 55 Years. F Entbants, 297. $638,000 Insubed. ' j 14 12 2 142.5 38 26 3 306 1 18 46 2 246 38 81 4 530.5 2 17 21 4 192.5 22 57 21 419.5 3 20 13 3 154.5 46 -2'Z 22 337 4 14 8 3 121 30 16 5 250 5 12 4 2 98 26 12 1 4 201 6 26 3 80.5 42 10 10 160 7 11 1 51.5 25 5 2 100.5 8 12 1 1 38.5 20 1 70.5 9 4 2 24 5 1 48.5 10 4 2 17 11 8 38 11 1 10.5 1 21.5 12 1 9 1 19 13 7 17 14 6 16 55 56 57 58 50 60 61 62 63 64 65 66 67 68 69 .06416 .06304 .06108 .05870 2 3 .11133 .11282 .10211 1 .10598 1 .09944 .09678 .09379 .09200 3 4 .15424 .15897 .14333 .14795 .14100 .13500 .13075 .12865 4 5 0.21081 0.22007 0.20526 0.20511 0.19508 0.18536 0.17896 0.17856 5 6 .29008 .30419 .28311 .27934 .26141 .24597 .24303 .24077 6 7 .42017 .41418 .38348 .37144 .34663 .32425 .32288 .31871 7 8 .56518 .56338 .49498 .48051 .44877 .42394 .41637 .41552 8 9 .72039 .72029 .63628 .00940 .56752 .53940 .52961 .53480 9 10 0.90595 0.88140 0.76332 0.73569 0.08621 •0.65580 0.64871 0.65832 10 11 1.03608 .98935 .88288 .85079 .78971 .76447 .75669 .76347 11 12 .13428 1.06705 .06695 .93098 .87108 .85081 .84310 .84680 12 13 .18797 .11745 1.02247 .98875 .91419 .90053 .89185 .89522 13 l'' .24499 .17096 .07471 1.04638 .90450 .94898 .94126 .93814 14 15 1.29687 1.22848 1.14839 1.10994 1.03495 1.01171 1.00502 0.99013 15 16 .34188 .30143 .20883 .17915 .09429 .06871 .06272 1.04408 16 17 .38040 .39219 .27400 .24687 .13776 .11044 .11365 .08012 17 18 .40808 .45412 .33330 .30876 1 .18436 .15738 .15951 .11317 18 19 .44173 .49038 .39645 .35785 .22871 .20863 .20061 .10153 19 20 1.44609 1.42734 1.28963 1.26198 1.23978 1.20003 20 21 ■ .5x5848 .47540 .33599 .29878 .28672 .25242 21 22 .56209 .52180 .37204 .33082 .32485 .28933 22 23 .53941 .57132 .41461 .37611 .36530 .32815 23 24 .77465 .68447 .48049 .44200 .45187 .38124 24 34,35 36,37 38,39 40,41 1 42, 43 44,45 46,47 48,49 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 .02736 .02797 .02612 .02682 .02637 .02667 .02592 .02522 1 2 .05863 .05861 .05685 .05787 .05685 .05506 .05485 .05414 2 3 .09011 .09003 .08992 .08450 .08871 .08557 .08437 .18785 3 4 .12711 .12451 .12542 .11724 .12499 .12342 .12055 .12281 4 5 0.17563 0.17178 0.17432 0.16137 0.17407 0.17291 0.16813 0.16904 5 6 .24009 .23023 .23728 .23311 .23927 .23703 .23415 .24216 6 7 .32266 .31900 .32222 .32714 .32248 .33015 .32644 .34079 7 8 .42283 .42148 .43019 .43705 .43852 .44456 .44387 .46653 8 9 .53758 .54528 .55948 .57120 .56882 .57540 .61384 .63870 9 10 0.65714 0.66750 0.68024 0.69490 0.70469 0.70987 0.76135 0.77771 10 11 .76034 .77463 .^9305 .81533 .81133 .82799 .89096 .88884 1 11 12 .83(509 .86817 .87803 .91130 .91464 .92110 .98369 .99857 12 13 .89591 .92046 .93184 } .96849 .97285 .98229 1.05109 1.05690 13 1 14 .94054 .96708 .98309 i 1.02026 1 i.02834 1.03559 .09971 .10516 14 ^ 1 TABLE III. 163 MALE LIFE (C.) LOGARITHM OF THE MULTIPLIER (tv) TO CARRY FORWARD THE AMOUNTS EXISTING TO THEIR PROPORTIONAL TERMINATION INCLUSIVE IN THE FOLLOWING COLUMNS OF TABLE I, THAT IS, TO FINAL SERIES. AGES OP ENTRY. 15 34,35 36,37 38,39 40,41 42,43 44,46 46,47 48,49 15 0.99334 1.02716 1.05255 1.09514 1.08940 1.09572 1.15306 1.17165 16 1.04190 .08194 .10557 .15674 .13412 .14549 .20908 .24176 16 17 .08799 .13401 .15389 20898 • .17147 .18669 .24690 .29332 17 18 .12433 .16757 .19163 .24450 .21191 .21942 .28698 .32466 18 19 .16214 .20506 .23491 .28703 1 .25451 .25687 .33896 .37522 19 20 1.20632 1.25361 1.27022 1.32455 1.28544 1.28311 1.39109 1.40787 20 21 .23740 .29230 .30026 .35877 .31108 .31230 .41859 .44997 21 22 .27625 .32312 .33458 .39109 .34072 .35177 .46206 .48151 22 23 .30803 .35808 .38224 .43614 .37225 .38598 .52132 .53881 23 24 .37613 .41130 .42984 .49325 .46483 .48744 .59343 .60108 24 25 1.47904 1.49640 1.53001 1.58489 1.63230 1.64291 1.71368 1.76675 25 26 .62925 .65588 .67472 .76986 .81786 .83406 .87907 .87790 26 27 .94425 .85502 .91484 .96749 2.03817 2.03844 2.16395 2.13867 27 28 2.28653 2.21526 2.24928 2.26639 .47507 .47829 .50773 .26995 28 50,51 52,53 54, 55 1 56, 57 58,59 60,61 62,63 64,65 0.00000 0.00000 0.00000 O.OOOCO 0.00000 0.00000 0.00000 0.00000 1 .02578 .02589 .03195 .03235 .02877 .02832 .03272 .02596 1 2 .05292 .05435 .06092 .05647 .05943 .05577 .06376 .04455 2 3 .08601 .08721 .09767 .09116 .09101 .09351 .10001 .07247 3 4 .12422 .12676 .13934 .13230 . .13035 .13840 .12885 .10867 4 5 0.18118 0.18340 0.19199 0.18767 0.19541 0.19452 0.20007 0.17234 5 6 .26561 .26048 .27789 .27283 .29568 .27301 .28121 .26813 6 7 .36828 .36581 .38372 .38968 .40959 .40603 .42073 .37843 7 8 .51439 .50572 .52739 .54590 .56757 .55388 .65071 .54104 8 9 .66341 .67694 .68911 .72297 .74676 .74438 .83539 .82438 9 10 0.80739 0.83452 0.85481 0.89414 0.90593 0.88890 0.97509 1.03826 10 11 .92290 .95311 .98318 1.01557 1.07042 1.00005 1.07081 .19434 11 12 1.00623 1.05377 1.08678 .10290 .19273 .12747 .13848 .19434 12 13 .06246 .09947 .14336 .17348 .26270 .17079 .18068 .20835 13 14 .12941 .15676 .20055 .25364 .35345 .20889 .28163 .23468 14 15 1.20760 1.20986 1.26238- 1.32928 1. 51441 1.25803 1.29866 1.25587 15 16 .28547 .27998 .33082 .37798 .54211 .29116 .35042 .25587 16 17 .33760 .35163 .37702 .45536 .57575 .32190 .41536 .27011 17 18 .39651 .40659 .43473 .50500 .60526 .35842 .42304 .27011 18 29 .45631 .44707 .46092 .53992 .62313 .35842 .52626 .27011 19 20 1.48627 1.50172 1.50610 1.55195 1.66091 1.35842 1.55182 1.27011 20 21 .53167 .52717 .57961 .61034 j .66091 .38443 .60981 .47423 21, 22 .55351 .54756 .60488 . .65609 1 .72603 .44570 .65903 .47423 22' 23 .61165 .58759 .63576 .69605 j .78501 .45300 .79733 .47423 23 24 .68100 .66743 .72109 .75003 i 2.00995 .61650 .79733 24 164 TABLE IT. FEMALE LIFE (A). MORTALITY AND DEATH CLAIMS PER CENT. OF THE LIVING OR EXPOSED, BY EXPERIENCE ENDING (1874), AND BY FINAL SERIES. ALSO AVERAGE SUM INSURED (1874). 1 1 Mortality, Per Cent. Death Claims. Per Cent. Death Claims. Per Cent. Final Se- Average Sum Insured (1874) .S o3 bb « ^ to 00 -^ (1874.) (1874.) ries. 1 Existing. Discontin- Died. Exposed. SI < fl $ ued. < « $ $ 9 10 1.639 1.111 2,800 1,143 1,000 1,475 10 11 1.047 .671 3,571 1,143 1,000 1,560 11 12 4,125 769 1,663 12 13 1.072 1.003 2,500 1,385 1,500 1,603 13 14 .668 .654 1,933 1,372 1,500 1,531 14 15 .213 .141 1,444 1,197 1,000 1,514 15 16 .448 .290 1,207 1,270 1,000 1,546 16 17 1.071 1.099 .754 1,333 1,249 1,700 1,657 17 18 .822 .869 (.671) 1,485 1,733 1,818 1,720 18 19 .519 .575 [.757] 1,552 1,797 1,900 1,715 19 20 1.007 .906 1,554 1,679 1,556 1,729 20 21 1.011 .899 1,522 1,783 1,556 1,749 21 22 .965 1.136 1.013 1,608 1,854 2,068 1,758 22 23 1.109 1.104 (.999) 1,55.5 1,867 1,758 1,767 23 24 .990 .720 [.850] 1,545 1,858 1,323 1,821 24 25 1.007 .929 1,579 1,920 1,737 1,883 25 26 1.139 1.291 1,563 2,016 2,165 1,910 26 27 1.046 1.134 1.160 1,582 2,108 2,081 1,918 27 28 .904 1.026 (1.065) 1,581 2,164 2.181 1,923 28 29 .919 1.069 [1.182] 1,633 2,176 2,252 1,937 29 30 1.009 1.131 sl,696 2,126 2,200 1,964 30 31 1.0^8 1.059 1^697 2,186 2,055 1,996 31 32 1.119 1.046 1.108 1,643 2,324 1,882 2.013 32 33 1.126 1.136 (1.084) 1,660 2,352 2,027 2,009 33 34 1.069 1.102 [1.133] 1,749 2,359 2,070 2,007 34 35 1.0(i8 1.160 1,785 2,425 2,176 2,003 35 36 .958 1.092 1,840 2,452 2,252 1,977 36 37 .951 1.079 1.127 1,793 2,338. 2,208 1,945 37 38 .977 1.040 (1.082) 1,712 2,213 , 2,071 1,946 38 39 .944 .815 [1.207] 1.773 2,302 1,697 1,968 39 40 .970 .857 1,748 2,495 1,740 1,968 40 41 1.012 1.067 1,723 2,417 2,076 1,969 41 42 1.095 1.232 1.163 1,700 2,366 2,225 1,978 42 43 1.157 1.383 (1.104) [1.285] 1,749 2,482 2,375 1,987 43 44 1.172 1.300 1,853 2,507 2,296 1,978 44 45 1.085 1.088 1,796 2,358 1,960 1,955 45 46 .981 . .946 1,857 2,335 1,884 1,953 46 47 1.011 .989 ' 1.049 1,935 2,388 1,904 1,946 47 48 .995 1.094 i (1.103) [1.391] 1,868 2,261 2,130- 1,936 48 49 1.040 1.192 1.837 , 2,207 2,227 1,943 49 Note. The percentages by Mortality and by Death Claims were derived from the sum or mean of data at two adjacent Ages in Table II (A). The quinquennial percentages (Final Series) depend on w . t" from Table IV (C) applied to Death Claims and Exposed to Riijk in Table II. Under these and TABLE TV. 165 FEMALE LIFE (A). MORTALITY AND DEATH CLAIMS PER CENT. OF THE LIVING OR EXPOSED, BY EXPERIENCE ENDING (1874j, AND BY FINAL SERIES. ALSO AVERAGE SUM INSURED (1874). Mortality. Death 1 Cl&iins. Death Claims, Average Sum Insured (1874). ^ « " hn Per Cent. Per Cent. Per Cent. Final Se- bo .5 < a (1874.) ! (1874.) ries. Existing. Discontin- ued. Died. Exposed. a> bo < a < $ $ f 50 1.171 1.252 1,825 2,346 2,078 1,945 50 51 1.251 1.287 1,759 2,504 . 1,987 1,932 51 52 1.500 1.480 1.310 1,770 2,483 1,902 1,928 52 63 1.451 1.633 (1.432) [1.567] 1,838 2,556 2,169 1,927 53 54 1.180 1.443 1,846 2,654 2,333 1,908 54 55 1.577 1.610 • 1,802 2,517 1,933 1,894 55 56 2.081 2.245 1,800 2,329 2,058 1,908 56 57 2.169 2.719 1.959 1,750 2,282 2,429 1,938 57 58 1.854 2.002 (2.015) [2.015] ! 1,796 2,172 2,128 1,971 58 59 1.721 1.757 1,942 2,194 2,026 1,985 59 60 1.773 2.306 1,893 2,581 2,543 1,955 60 61 1.866 2.277 1,898 2,741 2,344 1,921 61 62 2.266 2.569 2.687 1,792 2,454 2,151 1,898 62 63 2.880 3.131 (2.822) 1,712 2,032 2,083 1,916 63 64 3.868 4.023 [2.863] 1,914 1,769 2,049 1,970 64 65 3.348 3.421 2,130 1,889 2,033 1,990 65 66 3.237 4.340 2,078 2,089 2,625 1,958 66 67 4.024 6.409 4.357 1,750 2,039 3,000 1,884 67 68 5.134 6.496 (4.937) 1,529 2,042 ^333 1,844 68 69 5.674 5.961 [4.370] 1,607 2,077 2,000 1,904 69 70 3.697 4.664 1,515 1,400 2,500 1,981 70 71 3.563 5.495 1,750 2,143 3,125 2,027 71 72 4.167 5.989 5.250 1,773 2,167 2,875 2,000 72 73 4.459 4.831 (5.617) [6.844] 1,864 2,000 2,143 1,978 73 74 8.907 8.696 2,071 2,571 1,909 1,956 74 75 10.369 11.475 1,300 2,250 2,100 1,887 75 76 9.032 8.108 1,333 1,000 1,714 1,910 76 77 16.000 13.492 14.776 1,600 .... 1,700 2,016 77 78 19.149 16.915 (12.232) [10.662] 2,250 1,500 1,889 3,138 78 79 15.385 3.889 3,250 1,500 2,000 2,215 , 79 80 25.000 20.408 3,500 1,667 2,042 80 81 40.000 32.836 .... 1,571 1,914 81 82 18.182 30.435 .... 3,500 2,091 82 83 11.111 37.500 (22.01) [12.51] ! 1,000 6,000 1,778 83 84 1,000 ■1,286 84 85 16.667 25.000 * • • • 2,000 1,333 85 86 20.000 33.333 .... 2,000 1,200 86 87 .... 1 1,000 .... 1,000 87 88 .... .... (21.43) [22.82] 1,000 .... 1,000 88 89 AU Ages. 50.000 50.000 1,000 1,000 89 1.1581 1.2344 1 i 1,744 2,216 2,070 1,942 enclosed in parentheses are the quinquennial values from the preceding column or from Death Claims (1874). And the values in brackets are the quinquennial ratios from the Female Experience of the Twenty British Offices. 166 TABLE lY FEMALE LIFE (B) ANNUAL MORTALITY AND DISCONTINUED PER CENT. OF THE EX- POSED ACCORDING TO YEARS OF INSURANCE. ALSO, AVERAGE SUM INSURED. «>- 2 Mortality Discontinued D. and Dis. Average Sum Insured. <» « 1 Per Cent . (1874.) Per Cent . (1874.) Per Cent. (1874.) (1874.) •0 oS B7 Lives. By Am'ts. By Lives. By Am'ts. By Am'ts. Existing. Discon- tinued. Died. ■ Exposed. jS s .861 .934 9.146 7.716 8.579 1,790 $ 1,665 2,158 1,989 1 1.081 1.120 20.576 19.700 20.610 1,782 1,914 2.082 2,010 1 2 1.062 1.088 11.549 14.037 14.978 1,708 2,476 2,061 2,010 2 3 1.139 1.274 10.633 14.148 15.248 1,622 2,672 2,206 1,973 3 4 1.156 • 1.321 8.781 12.083 13.250 1,688 2,701 2,206 1,930 4 5 1.280 1.434 7.210 10.566 11.853 1,685 2,821 2,121 1,893 5 6 1.401 1.541 5.561 7.282 8.713 ' 1,776 2,476 2,063 1,875 6 7 1.331 1.353 6.955 8.636 9.875 1 1,830 2,335- 1,895 1,864 7 B 1.064 1.159 4.504 6.270 7.358 1,854 , 2,578 , 2,000 ! 1,836 8 U 1.726 1.541 4.226 4.923 6.390 j 1,718 2,097 i 1,600 1,792 9 10 1.382 1.359 3.892 4.896 6.190 1 1,685 2,286 1,778 1,808 10 11 1.494 1.598 3.365 3.316 ' 4.861 1,952 1,800 1,955 1,828 11 12 1.329 1.197 2.397 2.516 3.684 j 1,869 1,897 1,625 1,804 12 13 1.538 2.110 1.269 1.362 3.443 i 2,419 1,928 2,471 1,801 13 14 1.056 .921 1.628 1.512 2.419 2,463 1,647 1,545 1,772 14 15 1.143 1.796 1.247 1.440 3.210 1,978 ' 2,000 2,727 1,736 15 16 1.672 2.742 .672 .922 3.639 2,649 2,333 2,800 1,708 16 17 1.314 1.455 1.195 1.527 2.960 1,906 2,100 1,818 1,642 17 18 2.171 1.646 1.538 1.100 2.728 2,077 1,167 1,235 1,630 18 19 1.509 1.605 1.373 1.689 3.266 1,815 2,000 1,727 1,624 19 20 1.901 2.813 .589 .734 1 3.526 1,756 2,000 2,385 1,611 20 1 21 2.875 2.833 .646 1.325 1 4.121 1,511 3,250 1,556 1,579 21 22 .897 .568 1.253 1.358 1.919 1,633 1,714 1,000 1,578 22 23 2.459 1.949 .414 .393 1 2.335 1,434 1,500 1,250 1,577 23 24 3.121 3.293 .288 .352 3.633 1,633 , 2,000 1,727 1,637 24 25 1.031 .631 .631 1,615 1,000 1,634 25 26 1,593 1,686 26 27 2,500 2,182 27 28 1,500 1,333 28 29 1,000 29 30 1,000 1,000 1,942 30 1.158 1.234 10.954 12.412 \ 13.499 1,744 2,216 2,070 TABLE lY. FEMALE LIFE (C.) LOGARITHM OF THE AVERAGE FACTORS (tcl (WV^), TO CARRY FORWARD THE AMOUNTS EXISTING IN TABLE II TO THEIR PROPORTIONAL TER- MINATIONS INCLUSIVE IN THE FOLLOWING CJOLUMNS, i. e. TO FINAL SERIES. Years of • Insurance. n. Log w. hog(w.V^) Years. n. 10 Log w. hog{tV.V"') Years. n. Log w. Log(w.'y") 0.00000 0.00000 0.86339 0.69306 20 1.21692 0.87625 1 1 .02133 .00430 11 .95711 .76974 21 .24727 .88957 z .04603 .01196 12 1.03036 .82596 22 .27938 .90465 3 .08533 .03423 13 .05439 .83296 23 .33135 .93958 4 .12601 .05788 14 .07156 .83309 24 .44621 1.03741 5 0.18963 0.10446 15 1.10460 0.84910 25 1.69101 1.26518 6 .27044 .16824 16 .12902 .85649 7 .38586 .26663 17 .15871 .86914 1 V — 1.04 8 .54308 .40681 18 .17888 .87228 9 .72467 .57137 19 .19810 .87447 TABLE Y. PROBABLE LOSS AND ACTUAL LOSS BY STATES AND TERRITORIES. 167 states, &c. Alabama . . Arizona . . Arkansas . California. Colorado. . Connecticut Dakota Delaware District of Columbia. Florida Georgia Idaho Illinois Indiana Indian Territory. Iowa Kansas. Kentucky ...... Louisiana Maine Maryland Massachusetts. Michigan. .. . Minnesota. . . . Mississippi Missouri , Montana Nebraska Nevada New Hampshire. New Jersey New Mexico New York. North Carolina . Ohio Oregon Pennsylvania . . Rhode Island . . South Carolina. Tennessee Texas Utah Vermont Virginia Washington . . . West Virginia. . Wisconsin Wyoming Unknown , British America . Other Foreign . . Total $104,531,520 Probable Loss. $1,204,521 2,230 295,413 2,489,469 150,821 3,604,783 5,412 253,963 580,298 181,635 950,653 23,532 7,320,753 2,465,769 10,007 1,659,034 454,910 2,169,340 1,363,520 1,826,915 2,573,670 11,977,364 2,371,856 735,380 795,476 4,336,484 21,824 223,736 157,256 1,264,848 2,666,988 17,778 22,271,323 843,815 5,689,260 164,380 10,393,537 1,058,688 753,060 1,383,648 778,713 14,328 723,750 646,188 23,270 314,438 3,009,760 4,022 37,337 1,425,150 841,215 Actual Loss. $1,618,048 5,000 508,119 3,414,340 131,000 3,369,951 5,000 224,526 660,006 303,852 911,536 15,743 6,365,344 2,590,728 5,000 1,255,512 . 391,470 2,237,220 2,400,915 1,511,492 3,311,815 10,309,004 2,119,010 785,907 1,308,632 4,810,720 5,000 136,300 177,905 1,123,093 2,515,832 29,500 21,059,726 1,069,409 5,292,302 141,000 9,578,003 945,152 864,635 2,262,121 1,358,972 39,500 578,092 673,350 6,000 255,755 2,316,213 1,000 67,818 1,033,812 948,616 $102,048,996 Satio. 100:134 224 172 97 87 94 92 88 114 167 96 67 87 105 50 76 86 103 176 83 129 86 89 107 164 111 23 61 113 89 94 166 95 127 93 86 92 •89 115 163 175 276 80 104 81 77 182 73 113 98 Note. Tables V, VI and VII refer to the common American Table of 1858. To change to the Thirty Offices Table, subtract J^tt part from the Probable Loss. Or add ^ih to the last term of the Ratio. 168 TABLE YI. ORDER AND VARIABILITY OF RATIOS OF PROBABLE TO ACTUAL LOSS IN THIRTY-FIVE STATES. States and District. Iowa Wisconsin Vermont West Virginia. Maine EatioB. Number of Companies. 27 12 6 9 Oregon . Kansas Massachusetts. Illinois Delaware Rhode Island Michigan New Hampshire. Pennsylvania . . . Ohio Connecticut. New Jersey. . New York. . Georgia .... California . Kentucky . Virginia . . Indiana . . . Minnesota . Missouri. . District of Columbia. South Carolina...... North Carolina Maryland Alabama 100 : 76 77 80 81 83 86 86 86 87 .88 89 89 89 92 93 94 94 95 96 97 103 104 105 107 111 114 115 127 129 134 Tennessee . Mississippi. Florida Arkansas. . Texas Louisiana. Average . 62 74 78 58 93 78 102 68 80 81 77 96 81 82 83 73 78 90 89 96 106 86 104 114 96 95 123 103 137 108 163 156 164 121 1671120 94 88 81 100 172 175 176 98 230 170 103 112 88 86 109 98 84 92 95 97 104 111 98 118 105 81 100 99 94 121 124 71 105 131 148 123 43 357 180 192 119 218 90 96 80 81 82 93 83 50 92 95 98 104 84 93 91 97 102 104 105 98 96 123 130 116 92 122 141 113 163 119 156 179 203 219 117 177 211 111 Thousand Dollars Loss. 12 Frob. L. Act. L, 870 2,122 305 108 512 72 218 3,654 2,776 43 538 1,559 236 63 475 74 149 2,934 2,259 1 316 242 686 657 377 307 3,030 1 2,472 2,107 1,746 1,234 1,185 9,071 353 959 701 309 1,045 469 1,835 280 499 306 847 531 498 310 120 141 256 525 1,074 905 921 8,126 314 920 743 . 265 1,088 537 1,768 266 615 315 1,161 571 777 376 145 324 434 625 970 Frob. L. Act. L 611 530 288 23 1,113 40 124 7,244 2,823 80 687 1,203 742 3,018 2,229 1,683 598 7,690 117 880 736 97 855 144 906 173 91 214 746 70 176 33 18 i 6 ! 35 I 43 999 575 468 234 23 870 42 139 6,346 2,427 87 572 1,012 683 2,872 2,169 9 Frob. L. Act. L 177 358 131 184 202 52 113 1,080 1,722 132 143 289 107 170 167 26 103 1,029 1,680 137 156 131 484 450 146 i 134 4,346 14,234 1,354 377 1,751 687 .663 885 7,515, 5,510 137 481 923 651 595 97 846 136 1,101 214 65 225 979 104 218 14 63 10 67 93 954 733 241 565 122 1,595 714 932 5,419 460 571 900 312 657 113 1,942 127 179 163 ' 185 324 529 981 , 1,171 603 943 710 1,268 453 918 44 149 488 796 749 97 174 857 1,683 838 TABLE YII. 169 PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OP ALABAMA. Counties. Frob. Loss. Actual Loss. Counties. Prob. Loss. Actual Loss. Autauga . . $3,403 • 5,220 Lowndes . . 18,785 17,720 Baker . . 219 .... Macon . . 18,814 25,820 Baldwin . 1,952 Madison 26,163 44,600 Barbour . . . 22,093 10,220 Marengo . . 25,991 21,000 Bibb .. 1,882 5,220 Marion 535 Blount 198 . • • • Mobile . 394,284 544,453 Bullock . . 11,868 18,440 Monroe . . . 3,782 5,220 Butler . . . 12,833 20,880 Montgomery . . 94,225 171,574 Calhoun 7,739 15,420 Morgan 2,763 1,000 Chambers 16,168 20,880 Perry . . 63,414 63,260 Cherokee 528 Pickens . . 19,328 39,000 Choctaw . . . 4,220 .... Pike . . . 2,091 Clarke ... 985 .... Kaudolph 1,236 ! Clay 1,169 .... Russell . . 6,529 30,440 Coffee . . . 552 Shelby 4,831 6,500 Colbert 3,029 10,440 St. Clair 2,685 5,220 Conecuh . . . 1,831 5,000 Sumpter . . 14,185 10,000 Coosa 1,291 > • • • Talladega 18,576 19,100 Covington . . . 19 .... Tallapoosa ' 3,169 .... Crenshaw . . . 2,049 * * . • Tuscaloosa 18,138 5,000 Dale ... 1,389 - • . Walker 411 5,000 Dallas . . 114,320 151,700 Washington . . . 1,581 10,220 Elmore 7,586 2,500 Wilcox . . 28,311 33,000 Escambia . . . 227 Unknown 6,085 15,000 Etowah Fayette 2,596 597 Additional Total 103,216 134,941 1,204,521 1,618,048 Franklin 3,575 .... Ratio 100 :134 Greene . . Hale .. 37,285 7,901 79,140 6,000 Henry . . . 2,895 ! Alluvial. 396,236 544,453 Jackson 1,065 .... Ratio 100 :137 Jefferson 6,371 10,440 Middle . . 175,977 194,580 Lauderdale 4,616 10,440 Ratio 100 :111 Lawrence 11,310 5,220 Upland 523,007 729,074 Lee 22,109 27,600 Ratio 100 :139 Limestone 7,523 5,220 STATE OP ARKANSAS. Arkansas . $3,270 15,660 Crittenden . 5,901 16,630 Ashley . 995 5,000 Cross . 837 .... Benton 411 Dallas . . . 3,457 10,000 Bradley . . . 1,656 Desha . 3,625 13,220 Calhoun . . . 1,169 .... Drew . 5,876 15,660 Carroll 95 Franklin 197 5,000 Chicot . 4,588 5,220 Fulton 334 2,500 Clarke 2,390 Grant' ... 496 .... Columbia . . . 2,071 5,220 Greene . . . 1,890 5,220 Conway 2,178 5,000 Hempstead . . 4,046 10,220 Craighead . . . 199 5,220 Hot Springs . . . 1,439 .... Crawford 6,868 10,440 Independence 4,718 5,000 170 TABLE YII. PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OP ARKANSAS.— Continued. Counties. Frob. Loss. Actual Loss, Counties, Prob Loss, Actual Loss. : Izard 131 Scott 322 Jackson . . . 9,044 19,440 Sebastian 7,581 15,220 Jefferson . 25,782 42,050 Sevier . . 104 .... Johnson 1,556 1,000 Sharpe 142 .... Lafayette . . . 1,148 St. Francis. 2,677 10,220 Lawrence 599 .... Union . . . 9,463 Little River . . . 1,484 10,000 Washington 1,071 .... Madison 103 .... White 3,322 5,220 Marion 65 .... Woodruff . . . 9,075 8,000 Mississippi . 2,849 Yell 1,361 6,000 Monroe . . . 3,041 5,424 Unknown 15,615 4,000 Montgomery Ouachita Perry , Phillips . Pike 19 8,744 15,660 Additional Total 9,909 6,000 295,413 508,119 42,049 570 98,335 3,000 Ratio 100 : 172 Poinsett . . . 149 Alluvial . 98,449 221,995 Polk 7 Ratio 100 :226 Pope 482 i,'66o Middle ... 125,202 196,084 Prairie . . . 3,829 7,220 Ratio 100 :157 Pulaski . . . 71,442 110,120 Upland 46,238 80,040 Randolph 2,724 5,000 Ratio 100 -.173 Saline 248 ... S .TATE OP PLORIDA. Alachua . 6,419 42,500 Orange . 350 Baker . 404 ' ' ' * Polk . 12 .... Bradford . 282 Putnam . 3,385 1,000 Clay . 566 5,'66o Santa Rosa . 4,455 .... Columbia . 86 3,000 St. Johns . 2,527 .... Dade. 369 .... Sumpter . 62 .... Duval. 31,126 66,111 Suwannee . 1,201 Escambia . 23,796 29,000 Volusia . 17 1,000 Franklin . 7,689 7,000 Wakulla . 57 Gadsden . 5,465 15,000 Walton . 164 Hamilton . 753 Unknown 380 11,000 Hernando . 380 . • . . Additional 18,781 35,706 Hillsborough . 564 Jackson . . . 5,960 25,000 Total 181,635 303,852 Jefferson . 9,911 10,000 Ratio 100 : 167 Leon (Tallahassee). Levy . 23,461 90 33,435 Madison . . . 22,798 7,550 Alluvial . 133,347 224,596 Marion . 1,942 Ratio 100 : 169 Monroe(KeyWest). 4,971 2,000 Middle . . . 28,758 32,550 Nassau. 3,212 9,550 Ratio 100 :113- TABLE YII. 171 PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OF GEORGIA. Counties. Prob. Loss. Actual Loss. Counties. Prob. Loss. Actual Loss. Appling . . . 1263 Jackson 1,058 .... Baker . . . 1.028 Jasper 916 .... Baldwin 9,221 Jefferson . . . 9,169 Bartow 3,524 14,500 Johnson . . , 4,477 11,250 Berrien . . . 99 . •> r . Jones 404 .... Bibb 63,130 78,380 Laurens . . . 945 • • • • Brooks . . . 2,003 .... Lee . . . 4,811 10,220 Brvan . . . 52 .... Liberty . . . 1,045 .... Bullock ... 95 Lincoln 2,185 .... Burke . . . 5,033 7,666 Lowndes . . . 5,637 2,500 Butts 13 Lumpkin 297 .... Calhoun . . . 99 Macon . . . 4,294 5,220 Camden . 1,876 2^566 Marion . . 2,005 10,220 Campbell 2,913 .... Mcintosh . . . 429 • • . • Carroll 1,174 • • • • Merriweather 343 . . - . Case 285 Miller . . . 180 • • . • Catoosa 47 .... Mitchel . . . 1,718 Charlton . 241 5,000 Monroe 7,202 5,220 Chatham . . . 200,436 259,895 Montgomery . . . 544 .... Chattahoochee . ; 185 .... Morgan 7,512 10,440 Chattooga 496 .... Murray 808 .... Cherokee 231 .... Muscogee . . 47,445 59,597 Clarke 26,408 15,220 Newton 3,407 .... Clay . . . 1,449 Oglethorpe 4,318 .... Clinch ... 279 Pierce . . . 883 7,500 Cobb 17,259 10^666 Pike 1,757 5,000 Coffee . . . 73 .... Polk 492 6,000 Columbia 7,171 2,000 Pulaski . . . 7,502 11,440 Coweta 9,870 10,440 Putnam 9,462 Crawford 142 Quitman . . . 633 • • • . Dade 269 .... Eandolph . . . 6,901 Decatur . . . 4,064 17,220 Eichmond . . . 107,140 83,160 De Kalb 1,868 Schley . . 959 .... Dooly . . . 492 Scriven . . . 483 .... Dougherty . . . 16,371 28,308 Spalding 16,084 . ■ . . Early . . . 3,137 Stewart . . 4,885 Effingham . . . 427 Sumpter . . . 20,040 21,240 Elbert 1,305 Talbot . . 5,004 5,000 Floyd 13,686 Tatnall . . . • 475 .... Forsyth 703 Taylor . . 1,815 .... Fulton 101,439 59', 666 Telfair . . . 1,273 .... Glynn . . . 3,361 5,000 Terrell . . . 10,814 5,000 Gordon 664 Thomas . . . 10,459 5,330 Greene 1,963 3^666 Troup 9,145 5,220 Gwinnett 198 .... Twiggs . . . 2,882 10,000 Hall 98 Union 76 • • • . Hancock 4,262 13,666 Upson . . 578 • • • . Harris . . 558 .... Walker 82 • • • • Hart 18 Walton 3,753 • • •". Heard 793 Ware . . . 707 • • • • Henry 15 .... Warren 3,506 . • . . Houston . . . 12,837 17,940 Washington . . . 3,451 15,000 Irwin . . . 424 Wayne. . . . 347 173 TABLE YII. PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OF GEORGIA.— Continued. Counties. Prob. Loss. Actual Loss. Counties. Froh. Loss. Aotual Loss, Webster ... * $394 Alluvial . 2,117 7,500 White 760 .... Ratio 100 :354 Whitfield 5,686 10,440 -Middle . . . 524,221 603,540 Wilcox . . . 83 1,000 Ratio 100 : 115 Wilkes 7,320 10,000 Upland 355,738 257,860 Wilkinson . . . 983 Ratio 100 : 72 Worth . . . 66 i Unknown 6,691 8,500 Additional 61,886 38,636 Total 950,653 911,536 Ratio 100 : 96 STATE OP LOUISIANA. r Parishes. Parishes. Ascension . $937 • • ■ • Plaquemines . 2,577 7,000 Assumption . 2,088 .... Point Coupee . 19,361 27,380 Avoyelles . 19,799 20,880 Rapides . 9,458 7,000 Bienville . . . 2,968 5,220 Richland . 2,320 5,220 Bossier . 4,734 5,220 St. Charles . 472 .... Caddo , 50,677 134,090 St. Helena . . . 2,070 .... Calcasieu . 505 St. James . 1,040 10,000 Caldwell . 1,297 3,000 St. Landry . 10,677 10,440 Carroll . 12,683 15,440 St. Martin . 13,457 ^ 5,220 Catahoula . . . 3,347 10,220 St. Mary . 6,668 11,440 Claiborne . . . 5,832 St. Tammany . 502 Concordia . 4,094 Tangipahoa . 848 i,6oo De Soto- . . . 4,138 15,220 Tensas . 14,095 20,880 E. Baton Eouge . 14,447 18,440 Terrebonne . 3,703 30,000 E. Feliciana . 21,616 61,100 Union . . . 2,413 .... Franklin . 4,861 14,220 Vermilion . . 468 5,220 Grant . ■ 2,747 .... Washington . . . 2,631 . . . .• Iberia . 1,091 ... W. Baton Rouge . 3,181 10,220 Iberville . 5,275 15,660 W. Feliciana . 10,751 32,440 Jackson . . . • 487 Winn . . . 15 Jefferson . 16,427 li,300 Unknown 20,208 23,000 Lafayette . 6,290 .... Additional 67,231 153,000 Lafourche . Livingston . Madison . 7,505 583 3,618 26,262 26,'220 Total Ratio 1,363,520 •100 2,400,915 : 176 Morehouse . • 8,024 8,350 Alluvial . 1,252,180 2,194,255 Natchitoches . 4,096 10,000 Ratio 100 : 175 Orleans . 949,144 1,593,793 Middle . . . 23,901 30,660 Ouachita . 10,064 52,820 Ratio 100 :128 TABLE YII. 173 PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OF MISSISSIPPI. Counties. Frob. Loss. Actual Loss, Counties. Pr ob, Loss. Actual Loss. Adams . 33,747 43,220 Montgomery . . . 49 5,220 Alcorn . . 3,239 Neshoba . . 105 .... Amite . . . 21,673 5,000 Newton . . . 910 .... Attala . . . 1,546 10,000 1 Noxubee . . 23,170 15,255 Bolivar . 16,677 93,520 Oktibbeha . . 12,837 5,179 Calhoun . . . 1,933 Panola . 17,765 26,100 Carroll . . . 14,214 15,440 Perrv . . . 99 5,220 Chickasaw . . 5,995 10,440 Pike" . . . 10,667 51,440 Choctaw . . . 5,584 12,720 Pontotoc . . 2,006 . • • . Claiborne . 7,535 5,000 Prentiss . . 29 • • • • Clark . . . 12,427 17,220 Eankin . . . 3,947 11,500 Coahoma . 12,571 27,440 !' Scott ... | 942 \ Copiah . . . 18,284 19,930 Simpson . . . 132 .... Covington . . . 2,500 : Smith ... 2,075 .... De Soto . 22,695 43,760 Sunflower. 3,488 10,220 Franklin . . . 3,183 ] Tallahatchie 13,442 10,440 Grenada . . . 22,809 39,540 1 Tippah . . 2,151 .... Hancock . 794 3,000 Tishemingo . . 1,927 Harrison . 5,734 12,440 Tunica . 7,191 10,220 Hinds . . . 41,040 90,660 Union . . 1,141 5,220 Holmes . . 8,614 23,440 Warren(Vicksb'g). 95,136 151,200 Issaquena . 5,243 5,000 Washington - 19,770 51,760 Itawamba . . 1,438 .... Wayne . . . 194 .... Jackson . 4,850 .... Wilkinson . 8,559 20,000 Jasper . . . 2,305 Winston . . 2,545 2,000 Jefferson . 9,951 30,660 Yalabusha . . . 22,858 26,100 Kemper . . 3,034 7,000 Yazoo . 29,381 18,720 Lafayette . . . 14,357 15,660 Unknown 21,650 51,000 Lauderdale . . 12,515 18,000 Additional 42,622 104,368 LaWrence . . . Leake . . . Lee . . 6,739 877 3,151 6,564 10,000 5,000 5,220 Total ^ Eatio 95,476 100 1,308,632 : 164 Lincoln . . . Alluvial . a 01,087 552,260 Lowndes . . . 50,669 82,670 Ratio 100 : 184 Madison . . . 25,873 41,220 Middle ... 4 -16,626 585,344 Marion . . , Eatio 100 : 140 Marshall . . . 34,520 16,*550 Upland 13,491 15,660 Monroe . . 6,308 15,220 Eatio 100 : 116 STATi 3 OP NORTH CAROLINA. ' Alamance 4,556 5,000 Camden . 878 Alexander 8 . • • • Carteret . 7,652 18,500 Anson 5,714 1,000 Caswell 1,488 15,000 Beaufort . 7,920 2,000 Catawba 1,841 Bertie . 15,815 24,000 Chatham 6,455 8,000 Bladen . . , 10,342 12,000 Cherokee 57 ... Brunswick . 66 .... Chowan . 4,866 16,000 Buncombe 653 .... Cleaveland 3,450 7,500 Burke 458 7,'o6o Columbus . . . 2,068 2,000 Cabarrus 3,494 2,000 ' Craven . 41,103 82,620 Caldwell 2,299 1,000 1 Cumberland . . . 23,112 17,000 174 TABLE YII. PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OP NORTH CAROLINA.— Continued. Counties. Prob. Loss. ■ Actual Loss. Counties. Prob. Loss. Actual Loss. Currituck . $286 Pasquotank . 8,794 15,000 Davidson 3,280 Perquimans . 4,307 4,000 Davie 1,215 Person 110 .... Duplin . . . 6,115 .... Pitt . . . 13,556 22,000 Edgecombe . . . 48,389 11,000 Polk 123 .... Forsyth 1,508 2,000 Randolph 1,653 .... Franklin . . . 8,369 6,000 i Richmond ... | 4,481 ... Gaston 3,246 20,000 Robeson . . . 6,924 6,500 Gates . 4,909 16,500 Rockingham 2,100 .... Granville 14,830 . • • Rowan 18,079 15,000 Greene . . . 13,266 13,000 Rutherford 1,719 .... Guilford 7,260 7,000 Sampson ... 7,421 2,000 Halifax . . . 28,183 55,000 Stanley 123 .... Harnett . . . 3,368 ... * Stokes 180 Haywood 114 .... Surry 285 .... Henderson 178 .... Tyrrell . 1,526 2,000 Hertford . . . 3,576 5,000 Union 761 .... Hyde . 726 .... Wake . . . 128,886 131,633 Iredell 5,252 5,000 Warren 11,350 13,000 Jackson 1,045 .... Washington . 5,868 1,085 Johnston . . . 12,848 20,000 Watauga 38 .... Jones . 1,593 7,000 Wayne . . . 22,752 33,000 Lenoir . . . 22,149 62,000 Wilkes 354 .... Lincoln 4,430 . . . . ' Wilson . . . 21,075 36,500 Macon 285 Yadkin 855 5,000 Madison 237 Unknown 16,053 33,500 Martin . McDowell Mecklenburg Mitchell 13,783 3,872 47,667 45 20,'o6o 2,000 66,720 Additional Total Ratio 32,981 38,000 ] 843,815 100 1,069,409 :127 Montgomery 101 More 2,104 .... Alluvial . 225,053 353,555 Nash . . . 2,740 5,000 Ratio 100 :157 New Hanover . 103,265 144,850 Middle . . . 390,924 442,633 Northampton . . . 1,304 3,000 Ratio 100 :113 Onslow . 1,696 Upland 178,804 201,721 Orange 13,932 19,501 Ratio 100 :113 STATE OP SOUTH CAROLINA. Abbeville 26,231 23,940 Edgefield 7,285 Anderson 14,784 33,820 Fairfield 13,682 28,390 Barnwell . . . 9,449 2,000 Georgetown . 7,177 .... Beaufort . 8,462 15,000 Greenville 15,789 17,914 Charleston . 347,589 417,440 Horry . Kersnaw . . . 5,360 10,340 Chester 21,728 31,000 9,373 • • • • Chesterfield . . . 4,883 5,000 Lancaster 4,257 2,500 Clarendon . . . 4,263 1,500 Laurens 17,988 38,720 Colleton . 5,899 3,658 Lexington . . . 1,190 . . . Darlington . . . 5,804 7,000 Marion . .• 16,865 10,000 TABLE YII. 175 PBOBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OF SOUTH CAROLINA— Continued. Counties. Prob. Loss. Actual Loss. Counties. Prob. Loss. Actual Loss. Marlborough . . . $5,412 Unknown 5,542 9,000 Newberry 26,516 10,720 Additional 1,715 1 Oconee Orangeburgh . . . Pickens 7 20,718 628 16,253 Total Eatio 753,060 100 864,635 :115 Eichland (Cohim- bia) . . . 90,086 120,440 Alluvial . 374,487 446,438 Spartanburg 7,446 15,000 Ratio 100 :119 Sumter . . . 16,034 15,000 Middle . . . 192,307 182,193 Union 6,707 15,000 Ratio - 100 :95 Williamsburg . . . 8,230 5,000 Upland 179,009 227,004 York 15,961 10,000 Ratio 100 :127 - STATE OF TENNESSEE. Anderson 1,365 Henderson . . 1,683 Bedford 16,423 31,660 Henry . . . 4,250 10,220 Benton . . 614 Hickman 2,329 • • • • Bledsoe 545 Humphreys 710 5,000 Blount 1,263 • • . ■ Jefferson 1,337 • • • • Bradley 8,422 2,000 Johnson 99 • . . • Campbell 1,585 Knox 27,321 28,440 Cannon 99 Lake . 616 .... Carroll 2,692 .... Lauderdale . 16,212 15,660 Carter 1,022 .... Lawrence 314 .... Cheatham 130 Lincoln 18 Claiborne 397 .... Macon 160 Cocke 149 5,000 Madison . . . 44,568 38,320 Coffee 552 2,000 Marion 2,555 2,000 Crocket 99 ■ • • • Marshall 4,325 Cumberland 802 Maury 59,708 68,441 Davidson (Nash- McMinn 3,602 1,000 ville) 147,911 208,386 McNairy 1,632 .... Decatur . . 560 Meigs 992 .... De Kalb 397 2,000 Monroe 4,599 3,0*68 Dickson 665 .... Montgomery 54,768 48,640 Dyer . 9,871 10,440 Obion . . . 9,517 20,880 Fayette . . . 40,039 60,600 Perry 1,041 • • • • Franklin 7,482 2,500 Polk 3,238 .... Gibson . . . 12,263 20,880 Rhea 518 . • . • Giles 22,868 15,660 Roane 1,953 .... Grainger 2,629 .... Robertson 3,454 10,440 Greene 2,675 2,000 Rutherford 26,064 73,880 Grundy 294 .... Scott 4,215 .... Hamilton 13,934 5,000 Sequatchie 49 .... Hancock 496 • * • • Sevier 397 .... Hardeman . . . 8,174 5,320 Shelbv(Memphis . 652,127 1,439,536 Hardin . . 3,369 .... Smith 1,187 • • . • Hawkins 496 Stewart 2,534 • • • • Haywood . . . 49,859 47,040 Sullivan 1,381 176 TABLE Yll. PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OP TENNESSEE. -Continued. Counties. Prob. Loss. Actual Loss. Counties. Prol). Loss. Actual Loss. Sumner Tipton . Union $5,652 29 610 1,190 11,220 23,000 Additional Total Eatio 14,405 5,000 1,383,648 100 2,262,121 :163 Warren 146 Washington 458 .... Wayne 843 5,220 x\lluvial . 708,436 1,488,636 Weakley . . . 6,564 6,220 Ratio 100 :210 Wiiite 99 .... Middle . . . 181,460 209,380 Williamson 3,683 5,000 Ratio 100 :115 Wilson 9,317 5,550 Upland 467,280 544,055 Unknown 12,067 15,000 Ratio 100 :116 STATE OF TEXAS. Anderson 3,419 2,500 Freestone 5,956 20,880 Angelina . . . 642 Frio . . . 846 .... Atascosa . . . 1,522 Galveston . 158,697 389,487 Austin ... 11,964 26,880 Gillespie 595 .... Bastrop . . . 3,163 Goliad . . . 3,523 .... Bee ... 351 Gonzales . . . 12,321 20,677 Bell 1,720 10,000 Grayson 9,657 22,440 Bexar 35,418 32,220 Grimes . . . 10,750 5,220 Blanco 528 .... Gaudalupe . . .' • 5,869 5,220 Bosque . . . 2,625 Hardeman 95 .... Bowie 5,458 10,440 Hardin . 290 5,000 Brazoria . 14,540 41,220 Harris . . . 69,767 177,280 Brazos . . . 24,633 25,660 Harrison . . . 17,623 26,100 Burleson . . . 2,321 10,000 Hays 2,712 .... Burnet 2,412 - > • • Henderson 347 .... Caldwell . . 4,412 10,000 Hill . . . 2,650 .... Calhoun . 8,331 . 15,000 Hood 5 .... Cameron . 2,210 5,000 Hopkins 644 .... Chambers . 458 • • . . Houston . . . 4,699 8,000 Cherokee . . . 690 Jack 563 • • • • Coleman 76 > • « • Jackson . 49 • • • • Collin . . . 1,864 .... Jasper . 1,552 2,000 Colorado . . . 8,446 13,720 Jefferson . 3,331 16,660 Comal 3,514 1,000 Johnson 1,700 .... Comanche 480 Karnes . . . 4,726 Cook 758 . . • . Kaufman 1,227 5,660 Coryell 899 ... Kendall 517 .... Dallas . . . 5,105 10,220 Kerr 47 .... Davis 1,007 5,000 Kinney 960 Denton 1,719 Lamar 4,201 10,440 De Witt . . . 4,042 16,060 Lampasas 248 .... Ellis . . . 6,164 5,220 Lavacca , . , 14,609 5,220 El Paso 190 .... Leon . . . 1,991 .... Falls 3,912 20,440 Liberty . 2,987 6,220 Fannin . . . 5,821 10,440 Limestone 13,644 10,220 Fayette . . . 10,816 21,390 Live Oak . . 972 5,220 Fort Bend . 3,696 20,660 Llano 1,266 TABLE YII 177 PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OP TEXAS.— Continued. Counties. Prob. Loss. Actual Loss. Counties. Prol). Loss. Actual Loss. Madison . . . 1396 Titus 2,789 Marion 28,156 85,498 Ti'avis 13,656 7,500 Matagorda . ' 6,716 5,000 Trinity . . . 397 McLennan ... 34,387 46,880 Tyler... 47 Medina 532 3,000 Upshur 1,898 Milam . . . 4,625 6,330 Uvalde 1,759 .... Montgomery . . . 3,417 5,000 Van Zandt 190 .... Nacogdoches . . . 902 .... Victoria . 5,609 5,220 Navarro 14,073 5,220 Walker . . . 3,203 7,500 Newton . 7 Washington . . . • 30,767 47,100 Nueces . 1,986 * . • * Wharton . 144 Orange . 1,661 Williamson 1,113 3,66o Palo Pinto 787 Wilson . . . 930 Panola . . . 1,633 Wise 8 i,66o Parker 3,316 5,230 Wood 418- .... Polk . . . 168 .... Unknown 36,794 5,000 Red River Refugio . Robertson . . . 5,887 1,779 33,096 66,100 Additional Total Ratio 31,065 6,000 778,713 100 1,358,973 :175 Rusk . . . 3,239 3,000 Sabine . . . 967 Alluvial . ■ 314,043 511,467 San Saba 376 Ratio 100 :339 Shelby . . . 3,013 .... Middle . . . 333,901 573,267 Smith 1,301 Ratio 100 :172 Starr 396 Upland 184,655 264,238 Tarrant 3,661 5,330 Ratio 100 :143 S-] [•ATE OP VIRGINIA. Accomack . 3,163 13,000 Elizabeth City 3,087 Albemarle 8,819 15,000 Essex . . . 3,388 i6,'66o Alexandria 37,319 35,500 Fairfax 7,147 9,000 Amelia . . . 100 5,000 Fauquier 2,647 Amherst 1,350 Fluvanna 123 5,660 Appomattox 47 Franklin 411 . * • > Augusta 10,080 18,500 Frederick 897 15,750 Bedford 541 Giles 228 - • « • Botetourt 385 . • • ■ Gloucester . . . 2,356 2,500 Brunswick . . . 1,397 13,000 i Goochland . . . 139 .... Buckingham 449 ■ Greenville . . . 1,436 .... Campbell 38,090 10,000 Halifax 1,181 .... Caroline . . . 1,276 1 Hanover . . . 3,342 .... Charles Citj . . . 884 .... Henrico . . . 149,712 127,500 Charlotte 3,117 31,500 Henry 7 .... Chesterfield . . . 8,259 13,500 Highland 48 ' Clark 1,036 .... Isle of Wight . . . 260 .... Craig 6 .... James City . . . 617 .... Culpepper 2,026 10,000 King and Queen... 48 .... Cumberland 115 King George . . . 665 .... Dinwiddle . . . 134,043 84,'50'0 j King William . . . 847 178 TABLE YII. PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OP VIRGINIA.— Continued. Counties. Prob. Loss. Actaal Loss. Counties. Prob. Loss. Actual Loss. Lancaster . . . ■ $279 2,000 Roanoke 3,972 5,000 Lee 11 .... Rockbridge 1,662 5,000 Loudon 12,313 3o,*o6o Rockingham 636 Louisa 48 .... Shenandoah 271 .... Lunenburg . . . 230 .... Smyth 815 1,000 Madison 584 Southampton . . . 1,285 Matthews . . . 595 • • • . Spottsylvania . . . 25,838 18,100 Mecklenburg . . . 1,171 2,000 Stafford 693 . . . - Middlesex . . . 1,075 .... Surry . . . 33 .... Montgomery 503 .... Sussex . . . 798 5,000 Nansemoiid. . . 2,932 .... Warren 261 5,000 Nelson 296 Warwick . . . 30 New Kent . . . 749 2,'o'6o Washington 1,347 9,000 Norfolk . . . 42,309 59,000 AVestmoreland . . . 551 .... Northampton . 38 .... : Wythe 363 j Northumberland- 813 .... i York ... 1,950 i,'66o Nottoway . . . 47 Unknown 11,378 27,000 Orange 1 Page Pittsylvania Powhatan ... Prince Edward 1,359 190 5,075 1« 564 3,000 Additional Total Ratio Alluvial . 104,312 81,500 673,350 1 :104 646,188 100 3,567 13,000 Prince George . . . 922 5,000 Ratio 100 :364 Princess Anne 1,095 2,500 Middle . . . 385,727 349,100 Prince William . 366 1,000 Ratio 100 :90 Rappahannock 91 2,000 Upland 141,204 202,750 Richmond . . . 7,543 Ratio 100 :144 STATE OF INDIANA. Adams $2,636 2,500 Floyd 68,366 60,713 Allen 83,143 84,500 Fountain 6,716 7,000 Bartholomew 24,058 29,500 Franklin 4,579 7,000 Benton 293 .... Fulton 5,691 8,100 Blackford 310 750 Gibson 17,638 21,000 Boone 9,993 14,500 Grant 6,392 3,000 Brown 1,078 5,000 Green 3,805 2,000 Carroll 9,535 9,000 Hamilton 17,261 21,249 Cass 21,080 20,000 : Hancock 8,126 2,500 Clarke 35,387 40,000 Harrison 3,741 .... Clay 12,967 9,000 Hendricks 14,113 11,000 Chnton 1,217 .... Henry 13,671 19,390 Crawford 6,709 16,900 Howard 5,267 5,000 Daviess 11,227 16,000 Huntington 10,303 6,343 Dearborn 35,840 24,500 Jackson 14,297 13,000 Decatur 18,782 14,000 Jasper 679 .... DeKalb 6,249 2,000 Jay 668 .... Delaware 21,134 45,220 Jefferson 69,401 68,000 Du Bois 3,402 11,000 Jennings 2,003 .... Elkhart 8,649 1,100 Johnson 21,018 7,300 Fayette 17,726 4,200 Knox 7,807 3,000 TABLE YII, 179 PROBABLE LOSS AND ACTUAL LOSS BY COUNTIES IN TWELVE STATES. STATE OF INDIANA.— Continued. Counties. Prob. Loss. Actual Loss. Counties. Prob. Loss. ■ Actual Loss. Kosciusko $10,464 4,500 Scott 124 1,000 La Grange 1,879 4,000 Shelby 20,163 37,000 Lake 1,392 Spencer 31,383 43,894 La Porte 54,045 45,760 Starke 360 .... Lawrence 7,266 348 Steuben 1,757 .... Madison 48,863 65,088 St. Joseph 50,975 60,350 Marion 220,246 214,050 Sullivan 5,157 4,000 Marshall 15,564 18,200 Switzerland 10,845 9,200 Martin 425 2,000 Tippecanoe 40,246 42,080 Miami 16,601 10,500 Tipton 6,181 2,800 Monroe 10,969 4,000 Union 2,481 .... Montgomery 20,974 26,100 Vanderburgh 54,739 75,642 Morgan 14,592 8,500 Vermillion 2,107 .... Newton 2,064 .... Vigo 57,339 72,500 Noble 6,564 5,000 Wabash 18,655 12,325 Ohio 1,662 . • . • Warren 2,758 2,000 Orange 7,355 4,165 Warrick 8,856 1,000 Owen 5,134 2,000 Washington 5,378 9,500 Parke 5,799 12,400 Wayne Wells 69,563 56,500 Perry 7,791 5,500 1,632 .... Pike 302 White 745 .... Porter 27,004 21,800 Whitley 1,888 5,000 Posey 33,638 50,100 Unknown 7,077 8,900 Pulaski Putnam Randolph 27 49,785 15,568 34,*ldo 13,900 Additional Total Ratio 836,541 975,321 2,465,769 100 2,590,728 :105 Ripley 957 2,000 Rush 8,932 2,500 180 TABLE YIII. THIRTY OFFICES' EXPERIENCE. COMMUTATION TABLE FOR THE SPECIAL CLIMATIC RATIOS 1.405 AND 1.805. 4 PER CENT. Arvp. 1.405 1.805 .400 1.405 1.805 .400 1.405 1.805 .400 A ftp. AvTCi* Da,. Dx. Diff. N.. 1,089,675 Diff. TTa.. TTfl). Diff. iLVJXii lo- 10 64,393 61,391 3,002 1,232,048 142,373 13.803 17.878 4.075 11 61,221 58,089 3,132 1,167,655 1,028,284 139,371 13.969 18.030 4.061 ll 12 58,204 54,963 3,241 1,106,434 970,195 136,239 14.143 18.190 4.047 12 13 55,334 52,005 3,329 1,048,231 915,231 133,000 14.327 18.360 4<033 13 14 52,605 49,205 3,400 992,897 863,226 129,671 14.520 18.539 4.019 14 15 50,009 46,554 3,455 940,291 814,022 126,269 14.723 18.728 4.005 15 16 47,540 44,045 3,495 890,282 767,468 122,814 14.937 18.928 3.991 16 17 45,192 41,670 3,522 842,742 723,423 119.319 15.164 19.139 3.975 17 18 42,959 39,422 3,537 797,550 681,753 115,797 15.402 19.362 3.960 18 19 40,834 37,294 3,540 754,591 642,332 112,259 15.652 19.598 3.94^ 19 20 38,813 35,279 3,534 i 713,757 605,038 108,719 15.917 19.847 3.930 20 21 36,891 33,372 3,519 674,944 569,759 105,185 16.196 20.111 3.915 21 22 35,061 31,566 3,495 638,054 536,387 101,667 16.489 20.389 3.900 22 1 23 33,322 29,857 3,465 602,992 504,820 98,172 16.799 20.683 3.884 23 24 31,666 28,239 3,427 569,670 474,963 94,707 17.126 20.993 3.867 24 25 30,092 26,707 3,385 538,004 446,724 91,280 17.470 21.322 3.852 25 26 28,593 25,256 3,337 507,912 420,018 87,894 17.834 21.669 3.835 26 27 27,167 23,882 3,285 479,319 394,762 84,557 18.217 22.036 3.819 27 28 25,810 22,581 3,229 ! 452,152 370,880 81,272 18.622 22.425 3.803 28 29 24,519 21,350 3,169 426,342 348,298 78,044 19.049 22.835 3.786 29 30 23,290 20,183 3,107 401,822 326,949 74,873 19.500 23.269 3.769 30 31 22,120 19,078 3,042 378,532 306,766 71,766 19.976 23.729 3.753 31 32 21,007 18,031 2,976 356,412 287,688 68,724 20.479 24.215 3.736 32 33 19,947 17,040 2,907 335,405 269,657 65,748 21.010 24.730 3.720 33 34 18,938 16,101 2,837 315,458 252,617 62,841 21.571 25.274 3.703 34 35 17,977 15,211 2,766 296,520 236,516 60,004 22.164 25.851 3.687 35 36 17,062 14,368 2,694 278,543 221,305 57,238 22.791 26.461 3.670 36 37 16,190 13,569 2,621 261,482 206,937 54,545 23.454 27.108 3.654 37 38 15,359 12,812 2,547 245,292 193,369 51,923 24.155 27.793 3.638 38 39 14,568 12,094 2,474 229,933 180,557 49,376 24.898 28.519 3.621 39 40 13,815 11,413 2,402 215,364 168,463 46,901 25.683 29.288 3.605 40 41 13,096 10,768 2,328 201,550 157,050 44,500 26.514 30.104 3.590 41 42 12,411 10,156 2,255 188,454 146,282 42,172 27.396 30.969 3.573 42 43 11,758 9,576.12,182 | 176,043 136,125 39,918 28.328 31.886 3.. 558 43 44 11,135 9,025.7|2,109 164,285 126,549 37,736 29.317 32.860 3.543 44 45 10,541 8,503. 4'2,038 153,150 117,523 35,627 30.366 33.893 3.527 45 46 9,974.0 8,007.71,966.2 142,609 109,020 33,589 31.478 34.991 3.513 46 47 9,432.9 7,537.3 1,895.0 132,635 101,012 31,623 32.658 36.156 .3.498 47 48 8,916.3 7,090.6 1,825.7 123,202 93,474.9 29,727 33.910 37.394 3.484 48 49 8,423.0 6,660.41,756.0 114,286 86,384.3 27,902 35.240 38.710 3.470 49 50 7,951.9 6,263.5 1,688.4 105.863 79,717.9 26,145 36.653 40.109 3.456 50 51 7,501.6 5,880.71,620.9 97,910.9 73,454.4 24,457 38.154 41.597 3.443 51 52 7,071.2 5,516.9 1,554.3 90,409.3 67,573.8 22,836 ; 39.751 43.181 3.430 52 53 6,659.7 5,171.11,488.6 83,338.1 62,056.7 21,281 41.450 44.867 3.417 53 54 1 6,266.1 4,842.3 1,423.8 76,678.5 56,885.8 19,793 43.258 46.663 3.405 54 D,4^p.c. D,5p.c.; N, 4^ p.c. N, 5 p. c. Insurance 1000 TABLE YTII. 181 THIRTY OFFICES' EXPERIENCE. COMMUTATION TABLE FOR THE SPECIAL CLIMATIC RATIOS 1.405 AND 1.805. 4 PER CENT. 1.405 1.805 .400 1.405 1.805 .400 1.405 1.805 .400 AQE. 55 D^. Da:. Diff. N«. N.. DiflF. TTfl).' T^x' DifE. AGE. 55 5,889.8 4,529.8 1,360.0 70,412.4 52,043.4 18,369.0 45.184 48.577 3.393 56 5,529.4 4,232.4 1,297.0 64,522.7 47,513.7 17,009.0 47.235 50.616 3.381 56 57 5,184.5 3,949.6 1,234.9 58,993.3 43,281.3 15,712.0 i 49.422 52.791 3.369 57 58 4,854.8 3,680.4 1,173.9 53,808.8 39,331.7 14,477.1 51.752 55.111 3.359 58 59 4,538.1 3,424.3 1,113.8 48,954.5 35,651.4 13,303.1 54.239 57.588 3.349 59 60 4,235.4 3,180.7 1,054.7 44,416.4 32,227.1 12,189.3 56.896 60.233 3.337 60 61 3,945.5 2,948.8 996.7 40,181.0 29,046.4 11,134.6 159.732 63.060 3.328 61 62 3,667.9 2,728.3 939.6 36,235.5 26,097.6 10,137.9 i 62.763 66.081 3.318 62 63 3,402.2 2,518.6 883.6 32,567.6 23,369.3 9,198.3 66.004 69.313 3.309 63 64 3,147.9 2,319.3 828.6 29,165.4 20,850.1 8,315.3 69.472 72.770 3.298 64 65 2,904.7 2,129.9 774.8 26,017.5 18,531.4 7,486.1 73.182 76.472 3.290 65 66 2,672.2 1,950.1 722.1 23,112.8 16,401.5 6,711.3 77.154 80.434 3.280 66 67 2,450.3 1,779.6 670.7 20,440.6 14,451.5 5,989.1 81.411 84.682 3.271 67 68 2,238.6 1,618.1 620.5 17,990.3 12,671.9 5,318.4 85.971 89.232 3.261 68 69 2,037.1 1,465.5 571.6 15,751.7 11,053.8 4,697.9 90.866 94.116 3.250 69 70 1,845.6 1,321.4 524.2 13,714.6 9,588.3 4,126.3 96.111 99.351 3.240 70 71 1,664.1 1,185.8 478.3 11,869.0 8,266.9 3,602.1 101.74 104.97 3.23 71 72 1,492.5 1,058.4 434.1 10,204.9 7,081.1 3,123.8 107.79 111.00 3.21 72 73 1,330.8 939.23 391.6 8,712.4 6,022.8 2,689.6 114.28 117.49 3.21 73 74 1,179.0 828.18 350.8 7,381.7 5,083.5 2,298.2 121.26 124.45 3.19 74 75 1,037.2 725.09 312.1 6,202.62 5,160.42 4,255.34 1,947.3 128.76 131.93 3.17 75 76 905.40 629.93 275.5 3,530.25 1,635.2 136.82 139.97 3.15 76 77 783.67 542.64 241.0 4,260.02 2,900.32 1,359.7 145.50 148.63 3.13 , 77 78 671.93 463.05 208.9 3,476.35 2,357.68 1,118.7 154.82 157.94 3.12 78 79 570.22 391.09 179.2 2,804.41 1,894.63 909.8 164.87 167.96 3.09 79 80 478.45 326.59 151.9 2,234.19 1,503.54 730.65 175.69 178.75 3.06 80 81 396.47 269.33 127.1 1,755.74 1,176.95 578.79 187.35 190.38 3.03 81 82 324.06 219.10 105.0 1,359.27 907.62 451.65 199.95 202.94 2.99 82 83 260.89 175.55 85.34 1,035.21 688.52 346.69 213.55 216.51 2.96 83 84 206.60 138.36 68.24 774.32 512.97 261.35 228.36 231.26 2.90 84 85 160.63 107.06 53.57 567.713 374.607 193.11 244.48 247.33 2.85 85 86 122.41 81.199 41.21 407.081 267.547 139.53 262.25 265.03 2.78 86 87 91.272 60.254 31.02 284.668 186.348 98.320 282.16 284.88 2.72 87 88 66.390 43.619 22.77 193.395 126.094 67.301 304.82 307.46 2.64 88 89 47.042 30.760 16.28 127.006 82.475 44.531 331.93 334.50 2.57 89 90 31.844 20.723 11.12 79.9643 51.715 28.249 359.77 362.26 2.49 90 91 20.473 13.260 7.213 48.1201 30.992 17.128 387.00 389.39 2.39 91 92 12.567 8.1007 4.466 27.6470 17.732 9.9150 416.10 418.37 2.27 92 93 7.3390 4.7081 2.631 15.0799 9.6315 5.4484 448.21 450.36 2.15 93 94 4.0382 2.5783 1.460 7.7408 4.9234 2.8174 483.21 485.21 2.00 94 95 2.0773 1.3199 .7574 3.70262 2.3451 1.3575 522.56 524.38 1.82 95 96 .99391 .62854 .3654 1.62536 1.0252 .6002 573.04 574.65 1.61 96 97 .41960 .26409 .1555 .63145 .39664 .2348 626.04 627.37 1.33 97 98 .16062 .10061 .0600 .21185 .13255 .0793 719.70 720.58 0.88 98 99 .05123 , .03194 .0193 .05123 .03194 .0193 961.54 961.54 0.00 99 D,4|p.c. D,5 p.c. 1 N,4|p.c. N, 5 p. c. Insurance 1000 182 TABLE IX. GENERAL TABLE OP DISEASES AND DEATHS IN TWENTY-SEVEN LIFE INSURANCE COMPANIES FROM THEIR ORGANIZATION TO THE YEAR 1874. The columns headed (2), (5,) (30) record the Deaths in so many Companies. Number of Deaths. Variability. M and F. Per Cent. DISEASES. of Total. 1 Hales. Females. Total. 1 (2) (5) (20) All Causes. 35,442 2,182 37,624 12,*845 12,456 12,323 100.00 Zymotic Diseases 6,356 303 6,659 ! 2,285 2,172 2,202 17.70 ; >* Constitutional Diseases 8,175 548 8,723 i 2,898 2,868 2,957 23.19 ! P 1 < Nervous Diseases 5,106 193 5,299 1 1,977 1,715 1,607 14.08 1 ^ Circulatory Diseases 1,986 106 2,092 1 785 668 639 5.56 P ■Respiratory Diseases 4,771 291 5,062 1,682 1,641 1,739 13.45 CQ Digestive Diseases 3,344 273 3,617 1 1,251 1,229 1,137 9.61 Miscellaneous Diseases' 5,704 468 6,172 j 1,967 2,163 2,042 16.42 Typhoid Fever 2,147 107 2,254 i 744 775 735 5.99 Typhus Fever 159 11 170 i 82 48 40 .45 1 Cerebro-spinal P'ever 23 1 24 ' 23 1 .06 1 Yellow Fever 252 6 258 ! 57 88 113 .69 Remittent Fever 412 25 437 i 156 122 159 1.16 Intermittent Fever 159 7 166 , 38 56 72 .44 Congestive Fever 213 13 226 94 66 66 .60 Typlio-malarial Fever 46 4 50 17 21 12 .13 Fever 255 12 267 66 132 69 • .71 ' Small Pox 298 7 305 74 101 130 .81 Measles 13 2 15 4 5 6 .04 1 Scarlet Fever 38 2 40 19 15 6 .11 i Diphtheria and ) p Malignant sore throat ) 127 6 133 36 44 53 .35 .o Erysipelas 374 10 384 161 128 95 1.02 1 , Pyaemia P»| Carbuncle 70 4 74 21 27 26 .20 62 1 63 28 16 19 .17 Influenza 12 2 14 4 7 3 .04 Dysentery 587 35 622 254 167 201 1.65 Diarrhoea 328 22 350 129 96 125 .93 Cholera 431 15 446 155 136 155 1.19 Cholera-morbus 195 8 203 63 79 61 .54 Goitre 4 4 2 2 01 Malignant pustule 11 11 2 , 4 5 .03 Glanders 1 1 1 ^ ; .00 Purpura haemorrhagica 21 21 3 5 13 I .06 Alcoholism 117 1 118 50 31 37 1 .31 Other Zymotic Diseases 1 2 3 2 20 1 .00 Anaemia 59 12 71 33 18 .19 '^ Cancer 621 44 665 241 216 208 1.77 g Dropsy 622 56 678 205 237 236 1.80 ■| Gout 23 23 13 4 6 .06 •2 Rheumatism 169 11 180 52 75 53 .48 1 Gangrene 51 51 18 13 20 .14 V Tubercular meningitis 10 1 11 3 3 5 .03 Lumbar abscess 11 11 6 3 2 .03 TABLE IX. 183 GENERAL TABLE OF DISEASES AND DEATHS IN TWENTY-SEVEN LIFE INSURANCE COMPANIES FROM THEIR ORGANIZATION TO THE YEAR 1874. The columns beaded (3), (5), (20) record the Deaths in so many Companies. 1 Number of Deaths. Variability. M and F. Per Cent. DISEASES. of Total. 1 1 Hales. Females Total. (2) (5) (20) All Causes. 35,442 2,182 37,624 31 12,845 12,456 12,323 100.00 i 1 Scrofula 25 6 15 10 6 1 .08 •1 Tabes Mesenterica 88 5 93 31 36 26 .25 5 Morbus Coxje 12 12 5 3 4 .03 ' 3 Consumption 6,474 412 6,886 2,283 2,231 2,372 18.31 ,S Other Constitutional 10 1 11 6 4 1 .03 Apoplexy 1,705 61 1,766 653 599 514 4.70 Congestion of brain 655 14 669 234 239 196 1.78 Softening of brain 399 9 408 i 177 93 138 1.09 Paralysis 841 32 873 360 250 263 2.32 Disease of l)rain 721 37 758 239 251 268 2.02 Convuls'ns and epilepsy 130 8 138 52 50 36 .37 Insanity 140 6 146 63 41 42 .39 Anxiety 2 2 2 .01 Fright- 1 1 1 .00 . En cephalitis 277 10 287 109 101 77 .76 1 Cerebro-spinal sclerosis 1 1 ! 1 .00 1 Cerebral embolism 1 1 1 .00 ^ Anffimia of brain 8 8 4 2 2 .02 Effusion on brain 48 3 51 10 23 18 .14 Neuralgia 17 1 18 5 8 5 .05 Progressive muscular ) atrophy j 3 3 2 1 .01 Tetanus 47 4 51 10 24 17 .14 Inflam't'n of spinal cord 18 18 10 5 3 .05 Disease of spinal cord 41 2 43 .21 10 12 .11 Congest'n of spinal cord 3 3 2 1 .01 — • Other Nervous Diseases 48 6 54 21 18 15 .14 Disease of heart 1,297 62 1,359 534 418 407 3.61 Peri- and endo-carditis ^ 104 9 113 29 38 46 .30 Hypertrophy of heart 100 4 104 28 47 29 .28 Valvular disease of heart 98 6 104 14 55 35 X .28 Fatty degen't'n of heart 42 3 45 21 15 9 .12 Dropsy of heart 56 6 62 27 13 22 .16 Rheumatism of heart 66 6 72 30 17 25 .19 i" Atrophy of heart 4 4 4 .01 1 Paralysis of heart 27 1 28 3 12 13 .07 1 Abscess of heart 5 5 1 4 .01 -^ Angina pectoris 79 4 83 46 17 20 .22 Aneurism of aorta 51 1 52 25 14 13 .14 Rupture of aorta 16 1 17 6 7 4 .05 Embolus of pulmo- ) nary artery | 5 1 6 1 3 2 .02 Phlebitis 18 18 4 9 5 .05 Other Circulatory 18 2 20 12 3 5 .05 184 TABLE IX. GENERAL TABLE OP DISEASES AND DEATHS. Number of Deaths. Variability. M. andF. 1 Per Cent. DISEASES. Males. Females. Total. (2) (5) (20) j of Total. 1 All Causes. 35,442 2,182 37,624 12,845 12,456 12,323 1 100.00 Epistaxis 8 8 2 3 3 .02 Disease of Larynx 112 3 115 j 30 44 41 .31 Bronchitis 437 21 458 1 144 144 170 1.21 Pleurisy 172 7 179 75 64 40 .48 Congestion of Lungs 568 29 597 187 198 212 1.59 i Pneumonia 2,713 176 2,889 990 899 1,000 7.68 1 Abscess of Lungs 78 6 84 25 36 23 .22 Hemorrhage of Lungs 283 7 290 93 90 107 .77 OS Disease of Lungs 264 31 295 92 111 92 .78 Emphysema, Asthma 63 8 71 21 25 25 .19 Pulmonary Apoplexy 34 1 35 10 16 9 .09 Gangrene of Lungs ' 13 13 4 3 6 .03 CEdema of Lungs 16 16 1 7 8 .04 Other Kespiratory 10 2 12 8 1 3 .03 Inflammat'n of Stomach 319 39 358 121 116 121 .95 Ulceration of Stomach 75 6 81 34 29 18 .22 Disease of Stomach 150 5 155 58 57 40 .41 Hemorrhage of Stomach 57 1 58 18 20 20 .15 Congestion of Stomach 23 2 25 7 9 9 .07 Tumor of Stomach 4 4 3 1 .01 Inflammation of Bowels 425 45 470 189 130 151 1.25 Ulceration of Bowels 67 6 73 39 15 19 .19 Hemorrhage of Bowels 84 2 86 33 24 29 .23 Congestion of Bowels 22 5 27- 8 14 5 .07 Disease of Bowels 100 10 110 47 32 31 .29 Obstruction of Bowels 36 1 37 9 12 16 .10 Perforation of Bowels 6 1 7 3 2 2 .02 Peritonitis 246 41 287 101 102 84 .76 Gastro-euteritis 165 17 182 41 74 67 .48 > Disease of Stomach \ and Bowels ) 127 7 134 31 41 62 .36 ♦3 s Hemorrhage 9 9 4 3 2 .02 to Strangulated Hernia 43 12 55 f8 17 20 .15 p Colic, Tympanites and ) Constipation f 69 3 72 22 28 22 .19 Dyspepsia 43 3 46 13 24 9 .12 Gangrene of Tongue 3 3 1 2 .01 Stricture of (Esophagus 11 11 1 8 2 .03 Fistula in Ano 12 12 3 5 4 .03 Disease of Spleen 22 22 14 4 4 .06 Leucoeythaemia 7 1 8 3 5 .02 Ascites 41 8 49 15 20 14 .13 Abdominal Tumor 35 3 38 11 20 7 .10 Undefined Diseases, ) Abdominal Organs ) 11 11 ' 7 4 ,03 Jaundice 75 2 77 37 21 19 .20 Inflammation of Liver 268 15 283 80 93 110 .75 Cirrhosis of Liver 104 4 108 38 33 37 .29 Abscess of Liver 79 7 86 28 32 26 .23 TABLE IX. 185 GENERAL TABLE OF DISEASES AND DEATHS. Number of Deaths. Variability. M. and F. | 1 Per Cent. | of Total. 1 DISEASES. Males. Females. Totol. (2) (5) (20) All Causes. 35,442 2,182 37,624 12,845 12,456 12,323 100.00 ! Diseases of Liver 448 19 467 153 180 134 1.24 Congestion of Liver 63 3 66 29 22 15 .18 Hypertrophy of Liver 36 2 38 11 14 13 .10 o Acute Yellow Atrophy [ of Liver ) 9 9 3 3 3 .02 s 60 Fatty Degeneration of ) Liver J 15 1 16 4 6 6 .04 S Biliary Calculus 15 15 6 6 3 .04 Obstruction of Hepa-) tic Duct ) 4 4 1 3 .01 Kupture of Gall Bladder 2 2 1 1 .01 Other Digestive 14 2 16 6 181 9 i .04 Bright's Disease 550 17 567 184 202 L51 Inflammation of Kidneys 60 1 61 17 24 20 .16 Abscess of Kidneys 12 12 3 5 4 .03 Tumor of Kidney 1 1 1 . , . . Disease of Kidneys 255 9 264 116 81 67 .70 Diabetes 158 3 161 78 • 44 39 .43 Addison's Disease 12 12 3 5 4 .03 Inflammation of Bladder 74 2 76 '24 16 36 .20 Disease of Bladder 54 1 55 24 16 15 .15 Hemorrhage of Bladder 5 5 1 2 2 .01 Eupture of Bladder ' 1 1 1 . . Urinary Calculi 20 1 21 4 ii 6 .06 Gravel 13 13 8 5 .03 3 Disease of Prostate ) Gland f 39 39 16 13 10 .10 1 Stricture of Urethra 5 5 4 1 .01 a Gangrene of Scrotum 1 1 1 , , , , S 09 Other Urinary 66 5 71 16 25 30 .19 » Childbirth and Puer- ) peral Diseases ) Diseases of Breast ) and Uterus [ •' 197 197 61 73 63 .52 110 110 29 53 28 .29 Debility, Exhaustion [ and Prostration \ 377 28 405 145 136 124 1.08 Abscess 117 5 122 43 38 41 .32 Hemorrhage 66 9 75 18 33 24 .20 Tumors 55 3 58 21 23 14 .15 Inflammation of Joints 15 15 8 2 5 .04 Old Age 87 12 99 63 18 18 .26 Accidents and Injuries 2,678 34 2,712 806 1,019 887 7.21* Suicides 475 7 482 150 180 152 1.28 ! Causes unknown or ) ill defined \ 508 24 532 125 161 246 1.42 *Deaths from Accidents Only were 2,058 or 3.48 per cent of 59,008 total deaths to the year 1890. 186 TABLE X. (A.) DISEASES AND DURATIONS OF POLICY AT DEATH OF 35,442 INSURED MALES. r~ Duration or Years of Insurance. Total. DISEASES. 0-i i-H H-2|2|-4i H-n 9|-19i 19i-29^ [ All Causes. 2,445 842 5,161 4,651 7,803 10,062 4,089 1,197 35,442 6,356 1 — 1 Zymotic Disaases 1,223 943 1,297 1,424 509 118 1^' Constitutional Diseases 226 978 1,124 2,091 2,635 937 182 8,175 iS Nervous Diseases 297 620 591 1,078 1,525 745 247 5,107 !» Circulatory Diseases 75 190 199 411 641 330 140 1,986 '8 p Respiratory Diseases 312 761 626 1,031 1,322 553 165 4,770 CQ Digestive Diseases 216 493 461 717 949 402 105 3,344 Miscellaneous Diseases 477 896 707 1,178 468 1,566 613 240 5,704 Typhoid, Typhus Fever 316 473 357 492 160 40 2,306 Malarial Fever 108 163 123 176 186 62 13 831 « Erysipelas 22 41 54 79 113 53 12 374 1 Dysentery 92 105 75 124 124 52 15 587 1 Diarrhcea 27 49 52 66 74 48 12 328 Cholera 102 97 67 75 67 21 2 431 Alcoholism 8 24 13 29 26 17 . . 117 Other Zymotic Diseases 167 271 202 66 280 342 96 24 1,382 611 ^ 1 Dropsy 25 74 147 180 96 23 tJ Cancer 15 55 51 155 216 95 44 632 ■i Consumption 155 790 947 1,691 2,113 689 89 6,474 Other Constitutional 31 59 60 208 98 126 _ 57 250 26 458 Apoplexy 115 210 357 468 96 1,705 i Congestion of Brain 64 102 85 147 179 63 15 655 Paralysis, Softening [ etc., of Brain \ 101 253 234 452 724 357 112 2,236 ^ Epilepsy, Convulsions 4 15 19 35 37 17 3 130 Other Nervous 13 66 40 173 45 87 117 58 21 381 !>• Diseases of Heart 189 386 597 313 131 1,855 ■H Other Circulatory 9 17 10 25 572 44 17 9 131 . Pneumonia 188 447 355 750 310 89 2,711 g Congestion of Lungs 44 92 81 107 149 70 25 568 g j Bronchitis and Pleurisy 21 87 71 154 170 76 31 610 Abscess, Hemorrhage^ etc., of Lungs ^ 46 102 89 159 200 75 12 683 Other Respiratory 13 33 30 39 53 22 8 198 Diseases of Stomach 36 84 91 139 160 95 23 628 _> Diseases of Bowels 62 118 99 163 194 85 18 739 Peritonitis 18 51 37 48 66 18 8 246 Diseases of Liver 43 134 151 232 326 122 29 1,038 Other Digestive 57 4 106 83 135 203 82 27 693 Diabetes 17 11 33 60 19 14 158 Diseases of Kidneys 31 85 100 178 302 140 50 886 Other Urinary 8 24 26 55 85 56 28 282 Childbirth and Puer- 1 per^il Diseases ' s • • • ■ * • * * e Diseases of Breast and ) Uterus ' PI • • ' • S Abscess, Hemorrhage, [ Old Age C 11 45 30 60 78 49 66 339 IM Debility, Exhaustion, } Prostration, etc. f 8 40 35 75 120 61 38 378 Accidents and Injuries 334 526 384 573 639 194 28 2,678 Suicides 37 86 68 97 139 37 11 475 Unknown Causes 44 73 53 107 143 57 5 508; Note. The Total, 3.'5,442, includes 26 more unknown and 8 above 29| Years. TABLE X 187 (B.) DISEASES AND DURATIONS OF POLICY AT DEATH OF 3,182 INSURED FEMALES. Duration or fears of Insurance. J DISEASES. 0-i i-li H-2i 2i-4i 4i-9i 9f-19i 19i-29i Total. All Causes. 197 403 341 467 513 190 70 2,182 "^ Zymotic Diseases 48 62 48 61 57 23 4 303 ftH Constitutional Diseases 19 86 101 118 148 65 12 549 Nervous Diseases 16 30 31 47 35 23 12 194 d Circulatory Diseases 9 11 12 24 32 9 9 106 » g Respiratory Diseases 23 62 45 - 65 66 20 12 293 Digestive Diseases 28 53 37 56 67 19 9 269 Miscellaneous Diseases 54 99 67 96 108 31 12 468 Typhoid, Typhus Fever 16 24 21 26 22 7 2 118 Malarial Fever 8 12 11 7 6 5 49 d Erysipelas 1 1 3 4 1 10 1 Dysentery 7 1 5 6 12 3 1 35 Diarrhoea 4 7 3 4 3 1 22 N Cholera Alchoholism 4 2 3 1 4 2 •• •• 15 1 Other Zymotic Diseases 9 2 15 3 11 8 6 i 53 . Dropsy 8 7 8 14 6 45 ^ Cancer 2 4 9 18 8 11 3 55 1 Consumption 14 68 78 85 119 41 8 413 Other Constitutional 1 6 . 7 7 7 7 1 36 61 Apoplexy 7 10 8 16 13 5 2 09 Congestion of Brain 2 2 4 2 3 2 1 16 1 Paralysis, Softening i etc., of Brain 6 13 15 19 14 11 8 86 ^ Epilepsy, Convulsions 1 4 3 2 , , 10 Other Nervous i 4 4 6 2 3 1 21 >. Diseases of Heart 7 11 10 23 32 9 9 101 u Other Circulatory 2 2 1 5 Pneumonia 19 37 24 43 37 12 5 177 ^ Congestion of Lungs 5 5 4 9 3 2 28 +> Bronchitis and Pleurisy i 5 5 10 3 2 2 28 Abscess, Hemorrhage) etc., of Lungs ) 2 15 8 6 13 1 1 46 Other Respiratory 1 3' 2 4 2 2 14 Diseases of Stomach 6 12 6 11 14 3 2 54 > Diseases of Bowels 7 19 10 9 17 6 2 70 9) Peritonitis 5 7 9 13 4 3 41 p Diseases of Liver 2 8 6 13 15 4 48 Other Digestive 8 7 6 10 17 3 6 2 56 Diabetes , , .. 3 Diseases of Kidneys 3 4 4 4 9 2 1 27 Other Urinary 3 2 3 1 9 3 Childbirth and Puer- ) peral Diseases ) 30 51 25 45 40 6 197 s 03 Diseases of Breast and } Uterus f 6 23 20 25 28 6 2 110 :3 Abscess, Hemorrhage ) Old Age j 2 3 6 5 3 6 3 28 JM Debility, Exhaustion, ) Prostration, etc. ) 2 3 4 3 9 6 2 29 Accidents and Injuries 5 7 4 7 8 1 2 34 Suicides 2 1 1 2 1 . . 7 Unknown Causes 4 4 3 3 4 4 1 24 NoTB. The Total, 3,182, includes 1 more unknown. 188 TABLE XI. (A.) DISEASES AND AGES AT DEATH OF 35,443 INSURED MALES. Ages at Death. DISEASES. All Causes. 91-19^ 133 19i-29i 3,476 29f39i 39i-49| 49i-59| |59H9i 69i-79| 79i-89| Total. 9,321 10,840 7,576 3,357 647 92 35,44^ Zymotic 54 903 1,866 1,943 1,137 390 58 5 6,356 pH* Constitutional 22 1,117 2,691 2,441 1,342 484 76 2 8,175 M Nervous 9 '238 997 1,576 1,410 733 135 8 5,107 » Circulatory 3 61 328 537 582 389 81 5 1,986 » g Respiratory 11 370 1,168 1,524 1,117 484 88 9 4,770 Digestive 12 216 758 1,134 811 361 45 7 3,344 Miscellaneous 22 571 1,513 1,685 1,177 516 164 56 5,704 Typhoid,Typhus 28 472 702 620 361 116 7 2,306 Malarial Fever 4 72 252 286 154 58 5 831 ^ 'Erysipelas 27 86 132 85 37 6 1 374 ■^ Dysentery 7 62 165 160 136 44 12 1 587 a Diarrhoea 3 27 77 98 74 37 10 2 328 N Cholera 5 55 134 141 73 21 2 431 1 Alcoholism 6 41 55 14 1 , . 117 ^ Other Zymotic 7 182 409 451 240 76 16 1 1,382 . Dropsy 19 103 191 183 99 15 1 611 ^ Cancer 8 72 186 239 107 20 632 Consumption 21 1,052 2,406 1,933 819 219 23 i 6,474 Other Const'l 1 38 110 131 101 59 18 458 Apoplexy 1 42 238 500 598 279 46 1 1,705 IB CongestionBrai n 3 53 190 231 130 43 5 655 § ^ Paralysis, Soft- 1 ening Brain 1 4 109 431 663 572 355 79 6 2,236 ^ Epilepsy and { Convulsions ) 10 48 48 25 13 2 130 Other Nervous 1 24 90 134 85 43, 3 1 381 -?;' Diseasesof Heart 3 59 307 . 488 553 362 78 5 1,855 ■B Other Circulat'y 2 21 49 29 27 3 131 Pneumonia 6 183 594 882 689 305 49 3 2,711 >> i^ Congest'n Lungs 2 41 132 189 130 57 16 1 568 1 Bronchitis and [ Pleurisy ) 2 42 152 202 132 62 15 4 610 Abscess, Hem- ) or'age Lungs ) 1 90 240 192 112 42 6 •• 683 Other Respirat'y 14 50 59 54 18 2 1 798 . Dis. of Stomach 3 24 124 226 159 75 16 1 628 > Dis. of Bowels 2 72 219 206 159 73 o 3 739 Peritonitis 2 35 73 77 42 15 1 1 246 bo P Dis. of Liver 2 37 209 410 , 266 101 11 2 1,038 Other Digestive 3 48 133 215 185 97 12 693 Diabetes 5 32 58 26 33 4 o . 158 Dis. of Kidneys 4 59 160 291 239 107 25 1 886 Other Urinary 11 31 53 78 82 25 2 282 Childbirth.Pu- } » erperal Dis. f • • ' • * • g Dis. Breast ) § and Uterus ) • * "3 Abscess, Hem- ) S or'age,01dAgef 1 16 77 77 55 37 . 52 25 339 1^ Debility, Ex- ) haustion, etc. \ 21 60 88 102 79 25 2 378 Accid'ts, Injuries 16 365 894 820 449 119 15 ^,678 Suicides 1 49 126 173 106 18 2 475 1 UnknownCauses 45 133 125 122 41 16 26 508 TABLE XI. 189 (B.) DISEASES AND AGES AT DEATH OF 2,183 INSURED FEMALES. Ages at Death. ' DISEASES. 9f 19i 19f 29i 29^-39139^491 49f59i 59^69^ 69|-79| 79^89^ Total. AH Causes. 18 360 698 563 317 164 51 11 2,182 Zymotic 6 48 98 75 51 17 6 2 303 tM Constitutional 5 99 195 158 70 20 2 549 pS Nervous 26 45 44 40 29 9 2 194 < Circulatory 5 22 34 23 18 5 106 » Respiratory 36 78 78 48 34 12 1 293 ^ Digestive 3 33 89 73 47 24 5 269 Miscellaneous 4 113 171 101 39 22 12 6 468 Typhoid, Typhus 2 25 33 32 21 5 118 Malarial Fever 1 2 15 15 9 5 i i 49 d Erysipelas 1 2 1 5 1 , , 10 o Dysentery 4 11 7 5 5 3 1 35 n Diarrlicea i 4 14 2 1 . . 22 N Cholera Alcoholism 1 4 5 4 1 1 •• •• • • 15 1 Other Zymotic 1 8 18 14 8 1 2 53 Dropsy 2 12 15 15 4 . , 45 ^ Cancer 1 5 24 16 4 2 55 s o Consumption 3 91 165 105 38 10 , . 413 o Other Const'l 2 5 13 14 1 2 36 Apoplexy 4 10 14 19 9 3 1 61 n CongestionBrain 3 8 2 2 2 16 1 Paralysis, Soft- ( ening Brain i 12 21 22 15 12 6 1 86 ^ Epilepsy and [ 1 1 3 2 1 10 Convulsions ) Other Nervous 6 5 3 2 5 21 >. Diseasesof Heart 5 21 31 22 17 5 101 b [Other Circulat'7 1 3 1 5 Pneumonia 18 45 53 26 27 7 177 >> S Congest'n Lungs 4 8 6 4 1 3 1 28 4^ Bronchitis and } Pleurisy ) 4 9 6 5 4 1 28 OB Abscess, Hem- ) or'age Lungs ) 8 13 11 8 2 •• 46 Other Respirat'y 2 3 2 5 1 14 Dis. of Stomach 6 19 11 5 7 5 54 •-< Dis. of Bowels 1 9 27 15 14 5 . , 70 1 Peritonitis 1 9 17 7 5 ' 2 . . 41 bo Dis. of Liver , , 3 16 20 9 3 48 Other Digestive 1 6 10 20 14 7 56 Diabetes , , 2 1 , , 3 Dis. of Kidneys 5 9 8 4 1 27 Other Urinary 1 3 4 1 9 Childbirth.Pu- / 3 66 94 34 197 1^ erperal Dis. j" o Dis. Breast ) and Uterus ) 1 22 34 33 15 4 1 110 1 Abscess, Hem- } or'age.OldAgef 1 7 5 •• 5 5 4 28 u Debility, Ex- | haustion, etc. f 4 7 4 3 8 1 2 29 Accid'ts, Injuries 6 13 6 6 3 1 . . 34 Suicides , , 3 2 2 7 UnknownCauses •• 5 4 7 4 1 3 •• 24 190 TABLE XII. (A) MALE LIFE. DISEASES AND NUMBER OF DEATHS BY STATES AND TERRITORIES. 1. _6 o B o 1 B i i i . §1 >> 3 O t 1 O 1 O .2 o • o 1 ■ft u a> 1 >, 1 .1 00 C i CD <0 < 1 p 1 o i 1 (5 O P 5 ^ Alabama 10 50 26 7 10 23 12 20 22 40 5 26 3 6 260 Arkansas 21 6 3 2 8 4 11 1 10 11 5 82 California 38 78 121 38 47 84 67 63 38 65 20 92 24 30 805 Colorado 1 1 3 1 2 1 1 2 2 2 6 1 1 24 Connecticut 179 165 314 96 67 148 131 108 71 122 42 103 19 83 1648 Delaware 4 11 10 4 3 3 2 7 2 11 3 60 Dist. of Col'bia 7 23 50 7 9 21 9 19 13 17 4 8 4 12 203 Florida 1 10 10 4 6 3 3 2 7 2 48 Georgia 8 45 20 *6 i2 20 7 16 14 25 2 14 1 7 197 Idaho 1 1 4 6 Illinois 151 270 398 106 92 233 105 235 134 261 46 206 25 76 2338 Indiana 63 135 181 37 24 92 38 126 68 98 11 83 12 28 996 Iowa 33 65 97 28 19 73 36 46 38 62 11 39 9 22 578 Kansas 11 21 19 7 1 14 3 13 5 13 4 22 4 4 141 Kentucky 19 73 93 24 33 49 26 53 33 49 8 57 19 21 557 Louisiana 9 112 54 12 18 44 13 15 14 50 10 31 2 14 398 Maine 74 114 158 47 18 79 34 48 50 64 12 73 12 33 816 Maryland 54 104 176 57 59 113 66 82 57 91 22 49 6 43 979 Massachusetts 297 392 857 229 182 379 244 245 207 275 136 359 51 255 4108 Michigan 69 105 162 41 40 77 51 91 55 95 13 88 21 37 945 Minnesota 43 20 70 9 21 29 20 30 25 30 9 40 2 8 356 Mississippi 7 54 26 9 13 14 9 20 10 24 2 19 1 12 220 Missouri 75 277 178 49 66 122 49 101 71 120 16 99 17 62 1302 Mon.Neb.ISTev. 9 13 11 5 2 14 2 10 6 14 1 16 1 4 108 New Hamps're 58 60 110 22 33 65 23 40 40 56 13 41 5 34 600 New Jersey 54 97 249 44 49 97 74 73 46 95 32 52 11 56 1029 New Mexico 1 3 1 , , 1 6 New York 449 643 143i 395 424 768 474 572 473 66i 290 422 95 346 7443 N. Carolina 7 39 25 7 6 14 8 17 11 29 3 9 6 181 Ohio 140 232 408 91 111 203 91 166 160 233 38 185 33 95 2186 Oregon 2 3 8 3 3 1 1 3 3 3 5 1 36 Pennsylvania 199 264 575 167 135 291 195 174 157 326 85 174 35 170 2947 Rhode Island 15 38 80 24 26 42 29 30 11 35 10 16 3 18 377 S. Carolina 3 13 14 6 10 10 4 9 7 17 3 2 3 101 Tennessee 8 160 63 8 24 22 16 27 16 46 4 27 6 14 441 Texas 7 51 20 10 9 18 6 20 13 34 1 21 5 10 225 Utah 1 3 1 1 2 1 9 Vermont 25 33 58 8 13 24 17 23 15 29 1 21 2 12 281 Virginia 15 33 34 5 13 17 17 14 9 17 3 20 1 7 205 Washington 1 1 1 1 2 , . 6 West Virginia 5 6 13 i 10 10 2 < 5 15 5 i 4 84 Wisconsin 121 142 227 71 46 106 51 114 86 145 17 139 33 64 1362 Unknown 1 5 2 1 4 1 1 1 4 1 24 45 Brit'h America 29 37 73 9 23 45 24 35 45 39 10 46 5 19 439 Other Foreign 6 33 41 11 25 21 17 16 17 22 2 33 3 17 264 Total 2307 4049 6472 1705 1703j3403 1987 2711 2056 3345 887 2678474 1665 35442 TABLE XII, 191 (B) FEMALE LIFE. DISEASES AND NUMBER OF DEATHS BY STATES AND TEBBITOEIES o a. o o a o a o ft a § 3 1-3 5 o 1 < 3 O 1 1 O o 5 a o B g o g 1 o O B o to CO 1 5 6>> o 5 -d a a a) 1 2 e 2 i" (5 1 5 1 o i i 1 1 ^ Alabama 2 3 3 3 2 1 3 3 1 21 Arkansas 1 1 1 3 California 2 4 2 2 2 1 1 5 6 2 - , 33 Colorado , , Connecticut 14 16 36 10 6 19 7 7 10 20 2 9 5 13 174 Delaware 1 2 1 1 2 1 8 Distof Col'bia 2 3 1 2 2 , . 1 1 12 Florida 1 1 Georgia 7 2 1 1 3 . . i; .. . . 2 20 Idaho . . Illinois 11 15 35 7 2 7 9 7 10 23 19 8 16 169 Indiana 11 12 9 2 4 3 4 20 5 13 10 5 5 103 Iowa 5 5 11 2 1 3 4 3 4 8 1 5 3 55 Kansas 1 3 1 2 2 , . 1 1 2 13 Kentucky 6 3 1 4 3 3 1 4 3 28 Louisiana . . 2 1 1 i i 1 2 2 11 Maine 1 5 1 1 . . 2 1 2 . . 13 Maryland 2 4 16 5 4 7 4 4 2 11 4 3 3 3 72 Massachusetts 7 15 52 17 4 12 12 14 8 23 3 29 8 16 220 Michigan 4 8 19 11 3 4 4 7 5 8 1 6 7 6 93 Minnesota 1 1 1 ^ , 3 3 6 15 Mississippi 3 3 1 1 1 1 2 3 , . 1 i i 18 Missouri 4 6 6 4 2 2 1 7 3 10 8 2 3 58 Mon.Neb.Nev. ^ , , , 1 1 1 1 4 New Hamps're 5 15 5 1 3 2 5 3 9 1 6 1 2 58 New Jersey- 5 14 3 5 3 4 4 5 5 2 5 2 5 62 New Mexico . , New York 17 33 68 31 7 28 24 33 27 45 8 21 24 19 385 N. Carolina 5 2 3 2 1 1 3 1 1 19 Ohio 6 14 31 10 7 11 5 21 12 16 1 18 9 10 171 Oregon , , 1 1 Pennsylvania 8 7 30 9 4 11 10 8 4 23 9 9 8 140 Rhode Island 1 6 2 3 1 2 4 1 1 21 S. Carolina 1 1 1 1 4 2 1 11 Tennessee 1 3 3 1 2 1 3 4 , , 1 19 Texas 1 , , 1 1 1 1 1 3 1 10 Utah , , ^ , , , . , Vermont 1 1 9 2 1 1 2 1 3 3 1 2 27 Virginia 2 . . ^ , 1 1 1 1 6 Washington . . West Virginia i 2 1 i 5 Wisconsin 7 3 is 4 1 4 1 5 1 8 2 9 5 3 66 Unknown 1 1 li 3 Brit'h America 2 , , 6 1 i 3 1 4 3 3 2! 26 Other Foreign 1 3 •• 1 • • 2 1 8 Total 118 184 409 138 63 132 105 176 116 273 27 198 110 133 2182 192 TABLE XIII. MALE LIFE (A). RATIO OF DEATHS BY EACH DISEASE TO 10,000 LIVING EXPOSED TO RISK, AN EQUAL NUMBER 10,000 IN EACH STATED PERIOD OF INSURANCE. Duration or Tears of Insurance. DISEASES. 0-h fH If2i 2Hi .4f9i H-^H 19f29| r Total. ActTD'th All Causes. 68.6 80.4 93.1 102.5 114.5 140.9 213.5 102.0 35,442 Zymotic 23.7 19.0 18.8 17.1 16.3 17.5 21.1 18.3 6,356 >i Constitutional 6.3 15.3 22.5 27.5 30.0 32.3 32.4 23.5 8,175 2§ Nervous 8.3 9.6 11.8 14.1 17.2 25.6 43.8 14.7 5,107 B Circulatory 2.1 3.0 4.0 5.4 7.3 11.3 25.0 5.7 1,986 P Respiratory 8.9 11.9 12.6 13.6 15.0 19.1 29.4 13.7 4,770 *" i Digestive 6.0 7.6 9.2 9.3 10.7 13.8 18.7 9.6 3,344 jMiscellaneous 13.3 14.0 14.2 15.5 18.0 21.3 1 43.1 16.5 5,704 Typhoid,Typhus 1 8.9 7.4 1 7.2 6.2 5.7 5.6 1 7-2 6.6 2,306 Malarial Fever 3.0 2.5 ! 2.4 2.3 2.1 2.1 i 2.3 2.4 831 .s Erysipelas .6 .6 1.1 1.0 1.3 1.8 1 2.1 1.1 374 1 Dysentery 2.6 1.6 1.5 1.6 1.4 1.8 2.7 1.7 587 Diarrhoea .8 .8 1.0 .9 .8 1.6 2.1 1.0 328 Cholera 2.8 1.5 1.3 1.0 .8 .7 .4 1.2 431 Alcoholism .2 .4 .3 .4 .3 .6 .3 117 Other Zymotic 4.8 4.2 4.0 3.7 3.9 3.3 4.3 4.0 1,382 . ^Dropsy .7 1.1 1.3 1.9 2.0 3.3 4.1 1.8 611 *j Cancer .4 .9 1.0 2.0 2.5 3.3 7.8 1.8 632 o Consumption 4.3 12.4 19.0 22.3 24.1 23.7 15.9 18.6 6,474 Other Const'l .9 .9 1.2 1.3 1.4 2.0 4.6 1.3 458 Apoplexy 3.2 3.3 4.1 4.7 5.3 8.6 17.0 4.9 1,705 CongestionBrain 1.8 1.6 1.7 1.9 2.0 2.2 2.7 1.9 655 1 Paralysis, Soft-) ening Brain '' 2.8 3.9 4.7 5.9 8.2 12.2 19.9 6.4 2,236 £ Epilepsy and } Convulsions ) .1 .2 .4 .5 .4 .6 .5 .4 130 Other Nervous .4 .6 .9 1.1 1.3 2.0 3.7 1.1 381 ^ Diseasesof Heart 1.8 2.7 3.8 5.1 6.8 10.7 23.4 5.3 1,855 o lOther Circulat'y .3 .3 .2 7.2 .3 .5 .6 ' 1.6 .4 131 Pneumonia 5.4 7.0 7.6 8.6 10.7 15.9 7.8 2,7U u iCongest'n Lungs 1.2 1.4 1.6 1.4 1.6 2.4 : 4.4 1.6 568 % Diseasesof Heart 3.1 2.9 3.4 5.4 7.7 7.2 25.9 5.2 100 U Other Circulat'y .9 .7 .3 5 Pneumonia 8.4 9.7 8.1 10.2 9.0 9.5 14.4 9.3 176 >> Congest'n Lungs . , 1.3 1.7 .9 2.2 2.4 5.8 1.5 30 u Pi Bronchitis and ) Pleurisy ) •,• 1.3 1.7 2.4 .7 1.6 5.8 1.6 31 (0 Abscess, Hem- ) or'age Lungs ) .9 3.9 2.7 1.4 3.1 2.2 . 42 Other Respirat'y 1.0 .5 1.0 l.,6 5.8 .7 13 Dis. of Stomach 3.6 3.1 2.0 2.6 3.4 2.4 5.8 2.7 52 •5 Dis. of Bowels 3.1 5.0 3.4 2.1 4.1 4.8 5.8 3.6 69 Peritonitis 2.2 1.8 3.0 3.1 1.0 8.6 2.1 41 bo P Dis. of Liver .9 2.1 2.0 3.1 3.6 3.2 2.8 53 Other Digestive 3.5 1.8 2.0 2.4 4.1 4.8 5.8 3.1 58 Diabetes .7 .2 3 Dis. of Kidneys 1.3 1.0 i.3 .9 2.2 i.6 . . 1.5 27 Other Urinary .8 .4 .7 .5 9 OQ Childbirth,Pu- } erperal Dis. ) 13.2 13.3 8.5 10.7 9.7 4.8 •• 10.4 198 b a Dis. Breast ) and Uterus ) 2.6 6.0 6.8 5.9 6.8 4.8 5.8 5.8 110 Abscess, Hem- ) or'age.OldAgey .9 .8 2.0 1.2 .7 4.8 8.6 1.5 28 3 Debility, Ex- ) haustion, etc. j .9 .8 1.4 .7 2.2 4.8 5.8 1.6 31 Accid'ts, Injuries 2.2 1.8 1.4 1.7 1.9 5.8 1.7 32 Suicides .9 .5 .3 7 {UnknownCausea 1.8 1.0 1.0 .7 1.0 3.2 1.2 23 Actual Deaths 197 403 341 467 513 190 70 2,182 2,182 Exposed to Risk 22,610 38,142 29,431 43,032 41,124 12,529 3,458 189.340 189,340 194 TABLE XIT. MALE LIFE (A). RATIO OF DEATHS BY EACH DISEASE TO 10,000 LIVING EXPOSED TO RISK, AN EQUAL NUMBER 10,000 IN EACH PERIOD OF AGE. 75.7 Ages of Expoi nire. DISEASES. 19i-29^ 29f39| 39f 49i 49|-59i 59f 69^ 69^-79^ 79^-891 1389. 7 AllAges il02.0 All Causes. 74.8 76.0 97.5 146.0 270.4 538.0 Zymotic 30.7 19.5 15.2 17.5 21.9 31.4 48.2 75.5 18.3 >i Constitutional 12.6 24.0 22.0 22.0 25.9 39.0 63.2 30.2 ! 23.5 P4 Nervous 5.1 5.1 8.1 14.2 27.2 59.0 112.3 120.9 14.7 Circulatory 1.7 1.3 2.7 4.8 11.2 31.3 67.4 75.5 5.7 ?^ Respiratory 6.3 8.0 9.5 13.7 21.5 39.0 73.2 135.9 13.7 03 Digestive 6.8 4.7 6.2 10.2 15.6 29.1 37.4 105.7 9.6 Miscellaneous 12.5 12.3 12.3 15.1 22.7 41.6 136.4 845.8 16.5 Typhoi(l,Typlius 15.9 10.2 5.7 5.6 7.0 9.3 5.8 .. 6.6 Malarial Fever 2.3 1.6 2.1 2.6 3.0 4.7 4.2 2.4 u Erysipelas .6 .7 1.2 1.6 3.0 5.0 15.1 1.1 i Dysentery 4.0 1.3 1.3 1.4 2.6 3.5 10.0 15.1 1.7 i Diarrhoea 1.7 .6 .6 .9 1.4 3.0 8.3 30.2 1.0 N Cholera 2.8 1.2 1.1 1.3 1.4 1.7 1.7 1.2 Alcoholism .1 .3 .5 .3 .1 . . .3 Other Zymotic 4.0 3.9 3.3 4.1 4.6 6.1 13.3 15.1 4.0 1.8 Dropsy , ^ .4 .8 1.7 3.5 8.0 12.5 15.1 4A Cancer . , .2 .6 1.7 4.6 8.6 16.6 1.8 g Consumption 12.0 22.7 19.6 17.4 15.8 17.6 19.1 15.1 18.6 1 Other Const'l .6 .8 .9 1.2 2.0 4.8 15.0 1.3 Apoplexy .6 .9 1.9 4.5 11.5 22.5 38.3 15.1 ' 4.9 to CongestionBrain 1.7 1.1 1.5 2.1 2.5 3.5 4.2 1.9 > Paralysis, Soft-) ening Brain J 2.3 2.3 3.5 6.0 11.0 28.6 65.7 90.6 6.4 »z; Epilepsy and Convulsions .2 .4 .4 .5 1.1 1.7 .4 Other Nervous .6 .5 .7 1.2 1.6 3.5 2.5 15.1 1.1 y jDiseasesof Heart \ Dis. of Bowels 1.1 1.0 1.8 1.9 3.1 5.9 4.2 45.3 2.1 S Peritonitis 1.1 .8 .6 .7 .8 1.2 .7 bo Dis. of Liver 1.1 .8 1.7 3.7 5.1 8.1 9.1 30.2 3.0 Other Digestive 1.7 1.0 1.1 1.9 3.6 7.8 10.0 2.0 Diabetes , , .1 .3 .5 .5 2.6 3.3 0.5 Dis. of Kidneys 2.3 1.3 1.3 2.6 4.6 8.6 20.8 15.1 2.5 i Other Urinary .2 .3 .5 1.5 6.6 20.8 30.2 .8 Childbirth, Pu-/ erperal Dis. j ^ e Dis. Breast ) and Uterus ) • • • • • • • ' * • Abscess, Hem- ) or'age,01dAge ) .6 .3 .6 .7 1.1 3.0 43.2 377.6 1.0 i4 Debility, Ex- { haustion, etc. \ .5 .5 .8 2.0 6.4 20.8 30.2 1.1 Accid'ts, Injuries 9.1 7.9 7.3 7.4 8.7 9.6 12.5 7.7 Suicides .6 1.1 1.0 1.6 2.0 1.5 1.7 1.4 UnknownCauses 1.0 1.1 1.1 2.4 3.3 13.3 392.7 1.5 Actual Deaths 133 3,476 9,321 10,840 7,576 3,357 647 92 ; 55,442 Exposed to Risk 17,576 464,444 1,226,542 1,113,191 518.747 134,148 12,025 663 3.476,335 TABLE XIY. 195 FEMALE LIFE (B). RATIO OF DEATHS BY EACH DISEASE TO 10,000 LIVING EXPOSED TO RISK, AN EQUAL NUMBER 10,000 IN EACH PERIOD OF AGE. Ages of Ezposare. DISEASES. 19f29i|29f39i 39H9i 1 49f 59i 59f69i 69i-79^ All Ages. I Act'l Deaths All Causes. 104.7 101.4 106.5 i 134.7 262.7 630.2 114.9 2,182 Zymotic 14.1 14.2 14.2 i 21.6 27.2 74.2 16.1 1 302 JM Constitutional 28.7 28.4 29.9 29.8 32.0 24.6 28.8 547 Nervous 7.6 6.5 8.3 17.1 46.5 111.3 10.2 195 s Circulatory 1.4 3.3 6.5 9.4 28.8 61.8 5.5 105 M p Respiratory 10.5 11.3 14.7 20.3 54.4 148.3 15.3 292 oa Digestive 9.5 12.9 13.8 19.9 38.5 61.8 14.3 273 Miscellaneous 32.9 24.8 ' 19.1 16.6 35.3 148.2 24.7 468 Typhoid,Typhu8 7.3 4.7 6.0 8.9 8.0 6.3 118 Malarial Fever .6 2.2 2.7 3.9 8.0 12.4 2.6 49 u Erysipelas .3 .3 .2 2.1 1.6 .5 10 1 Dyse ntery 1.2 1.6 1.4 2.1 8.0 37.1 1.9 35 Diarrhoea 1.3 2.0 .4 .4 1.2 22 N Cholera Alcoholism 1.2 .8 .8 .4 .4 .8 .1 15 1 Other Zymotic 2.3 2.6 2.7 3.4 i.6 24.7 2.7 52 Dropsy .6 1.7 2.8 6.4 6.4 2.6 49 la Cancer .3 .8 4.6 6.8 6.4 24.6 2.7 51 i Consumption 26.4 24.0 19.8 16.2 16.0 , . 21.5 ' 409 1 C9 Other Const'l 1.4 1.9 2.7 .4 3.2 2.0 38 Apoplexy 1.2 1.4 2.6 8.0 14.5 37.1 3.3 63 B CongestionBrain .9 1.2 .4 .9 3.2 .9 16 O t Paralysis, Soft-) ening Brain J 3.5 3.0 4.1 6.4 19.2 74.2 4.5 86 ^ Epilepsy and ) Convulsions ) .3 .2 .6 .9 1.6 .4 9 Other Nervous 1.7 .7 .6 .9 8.0 1.1 21 •^ Diseasesof Heart 1.4 3.1 5.9 9.4 27.2 61.8 5.2 1 00 3 Other Circulat'y .2 .6 1.6 .3 5 . Pneumonia 5.2 6.5 10.0 11.0 43.2 86.6 9.3 1 76 o Congest'n Lungs 1.2 1.2 1.1 1.7 1.6 37.1 1.5 30 (8 >4 Bronchitis and ) Pleurisy ■ j ].2 1.3 1.1 2.1 6.4 12.3 1.6 31 ■a Abscess, Hem- } or'age Lungs f 2.3 1.9 2.1 3.4 3.2 •• 2.2 . 42 Other Respirat'y .6 .4 .4 2.1 12.3 .7 13 ^- Dis. of Stomach 1.7 2.7 2.1 2.1 11.2 61.8 2.7 52 .5 Dis. of Bowels 2.6 3.9 2.8 5.9 8.0 3.6 69 S Peritonitis 2.6 2.5 1.4 2.1 3.3 , , 2.1 41 2 |Di8. of Liver .9 2.3 3.8 3.9 4.8 , , 2.8 53 lOther Digestive 1.7 1.5 3.7 5.9 11.2 3.1 58 Diabetes .4 .4 .2 3 Dis. of Kidneys i.4 1.3 1.6 1.7 12.3 1.5 27 Other Urinary .3 .4 1.7 1.6 .5 9 3 Childbirth, Pu- i erperal Dis. j 19.3 13.6 6.3 10.4 1 98 S Dis. Breast ) § and Uterus f 6.4 5.0 6.2 6.4 6.4 12.3 5.8 1 10 g Abscess, Hem- ) .2 'or'age,01dAge ' .3 1.0 .9 8.1 61.8 1.5 ?,8 ^ Debility, Ex- } haustion, etc. ) 1.2 1.0 .8 1.3 12.8 12.4 1.6 31 Accid'ts, Injuries 1.7 1.9 1.1 2.5 4.8 12.3 1.7 32 Suicides .9 .4 .9 .3 7 1 UnknownCauses 1.4 6 1.4 1.7 1.6 37.1 1.2 23 Actual Deaths 360 698 563 317 164 51 2,182 2,1 32 Exposed to Risk 34,427 68,769 52,783 23,536 6,245 809 189,340 189,3 40 196 TABLE XY. (A.) MALE LIFE. PERCENTAGE DISTRIBUTION OF DISEASES AMONG 100 DEATHS, THE SAME NUMBER 100 IN EACH PERIOD OF INSURANCE. '~ 1 Dtiration or Years of Insurance. DISEASES. 3-i ^-li H-2i 2^-4i H-9i H-19i I9i-29i Total. ' All Causes. 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Zymotic Diseases 34.5 23.6 20.3 16.7 14.1 12.5 9.9 17.9 r>H Constitutional Diseases 9.2 18.9 24.2 26.8 26.1 23.0 15.2 23.1 < Nervous Diseases 12.1 12.1 12.7 13.8 15.2 18.1 20.6 14.4 Circulatory Diseases 3.1 3.6 4.3 5.2 6.3 8.1 11.7 5.6 to Respiratory Diseases 12.8 14.8 13.3 13.2 13.1 13.5 13.^ 13.5. s Digestive Diseases 8.8 9.6 10.0 9.2 9.5 9.8 8.8 9.5 Miscellaneous Diseases 19.5 17.4 15.2 15.1 6.0 15.7 15.0 20.0 16.0 Typhoid, Typhus Fever 13.0 9.2 7.7 4.9 3.9 3.3 6.5 Malarial Fever 4.4 3.1 2.7 2.3 1.8 1.5 1.1 2.3 « Erysipelas .9 .8 1.2 1.0 1.1 1.3 1.0 1.1 1 Dysentery 3.8 2.0 1.6 1.6 1.2 1.3 1.3 1.7 t Diarrhoea 1.1 .9 1.1 .8 .7 1.2 1.0 .9 N Cholera 4.2 1.9 1.4 1.0 .7 .5 .2 1.2 Alcoholism .3 .5 .3 .4 .3 .4 , , .3 Other Zymotic Diseases 6.8 5.2 4.3 3.6 3.4 2.4 2.0 3.9 Dropsy 1.0 1.4 1.4 1.9 1.8 2.4 1.9 1.7 '^ Cancer .6 1.1 1.1 2.0 2.1 2.3 3.7 1.8 1 Consumption 6.3 15.3 20.4 21.6 21.0 16.9 7.4 18.3 Other Constitutional 1.3 1.1 1.3 4.5 1.3 1.2 1.4 6.1 2.2 1.3 Apoplexy 4.7 4.1 4.6 4.6 8.0 4.8 ^ CongestiQn of Brain 2.6 2.0 1.8 1.9 1.8 1.5 1.3 1.8 1 Paralysis, Softening \ etc., of Brain ) 4.1 4.9 5.0 5.8 7.2 8.7 9.3 6.3 ^ Epilepsy, Convulsions .2 .3 .4 .4 .4 .4 .2 .4 Other Nervous .5 .8 1.0 1.1 1.2 1.4 1.8 1.1 >. Diseases of Heart 2.7 3.3 4.1 4.9 5.9 7.7 10.9 5.2 S Other Circulatory .4 .3 .2 .3 7.3 .4 .4 .8 .4 Pneumonia 7.7 8.7 7.6 7.4 7.6 7.4 7.7 ^ Congestion of Lungs 1.8 1.8 1.7 1.4 1.5 1.7 2.1 1.6 Bronchitis and Pleurisy .9 1.7 1-5 2.0 1.7 1.9 2.6 1.7 "ft Abscess, Hemorrhage 5 etc., of Lungs 1.9 2.0 . 1.9 2.0 2.0 1.8 1.0 1.9 Other Respiratory .5 .6 1.6 .6 .5 1.8 .5 1.6 .5 .7 .6 - Diseases of Stomach 1.5 2.0 2.3 1.9 1.8 , .5 Diseases of Bowels 2.5 2.3 2.1 2.1 1.9 2.1 1.5 2.1 1 • Peritonitis .7 1.0 .8 .6 .7 .4 .7 .7 1 g Diseases of Liver 1.8 2.6 3.3 3.0 3.3 . 3.0 2.4 2.9 Other Digestive 2.3 2.1 .3 1.8 1.7 2.0 2.0 2.3 2.0 Diabetes .2 .2 .4 .6 .5 1.2 .4 Diseases of Kidnevs 1.3 1.6 2.1 2.3 3.0 3.4 4.2 2.5 Other Urinary .3 .5 .6 .7 .9 1.4 2.6 .8 Childbirth and Puer- / peral Diseases \ 3 • % Diseases of Breast and Uterus S Abscess, Hemorrhage, Old Age .4 .9 .6 .8 .8 1.2 5.5 1.0 "^ Debility, Exhaustion, ] Prostration, etc. f .3 .8 .8 1.0 1.2 1.5 3.2 1.1 x\ccidents and Injuries 13.7 10.2 8.3 7.3 6.4 4.7 2.3 7.5 Suicides 1.5 1.7 1.5 1.2 1.4 .9 .9 1.3 Unknown Causes 1.8 1.4 1.1 1.4 1.4 1.4 .4 1.4 \ Actual Deaths 2,445 5,161 4,651 7,803 10,062 4,089 1,197 35,442 TABLE XY. 197 (B). FEMALE LIFE. DURATIONS. PERCENTAGE DISTRIBUTION OF DISEASES AMONG 100 DEATHS, THE SAME NUMBER 100 IN EACH STATED PERIOD OF INSURANCE. All Causes. Duration or Years of Insurance. 0-i Hi H-H H-H 4f 9i 1 9f 19i 19f29i Total, 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Typhoid and Typhus 8.1 6.0 6.2 5.6 4.3 3.7 2.9 5.4 Other Zymotic Diseases 16.3 9.4 7.9 7.5 6.8 8.4 2.8 8.5 Consumption 7.1 16.9 22.9 18.2 23.2 21.6 11.4 18.9 Other Constitutional 2.5 4.4 6.7 7.1 5.6 12.6 5.7 6.3 Apoplexy 3.6 2.5 2.3 3.4 2.5 2.6 2.9 2.8 Other Nervous Diseases 4.5 4.9 6.8 6.6 4.3 9.5 14.3 6.1 Diseases of Heart 3.6 2.7 2.9 4.9 6.2 4.7 12.9 4.6 Pneumonia 9.6 9.2 7.0 9.2 7.2 6.3 7.1 8.1 Other Respiratory 2.1 6.2 6.2 4.7 5.7 4.2 10.0 5.3 Digestive System 14.2 13.2 10.9 12.0 13.1 10.0 12.8 12.3 Childbirth and Puer- { peral Diseases ) 15.2 12.7 7.3 9.6 7.8 3.2 9.0 Diseases of Breast and ) Uterus f 3.1 5.7 5.9 5.4 5.5 3.2 2.9 5.1 All other Diseases 10.1 6.2 7.0 5.8 7.8 10.0 14.3 7.6 TABLE XTL (B.) FEMALE LIFE. AGES. PERCENTAGE DISTRIBUTION OF DISEASES AMONG 100 DEATHS, THE SAME NUMBER 100 IN EACH STATED PERIOD OF AGE, All Causes. Ages of Death. 19^-291 29f39| 39H9i 100.0 49f59i 59f69^ 69^-79i Total. 100.0 Act'lD'ths. 100.0 100.0 100.0 100.0 100.0 2,182 Typhoid and Typhus 6.9 4.7 5.7 6.6 3.1 5.4 118 Other Zymotic Diseases 6.4 9.3 7.6 9.5 7.3 11.8 8.4 184 Consumption 25.3 23.6 18.7 12.0 6.1 18.8 409 Other Constitutional 2.2 4.3 9.4 10.1 6.1 4.0 6.3 138 Apoplexy 1.1 1.4 2.5 6.0 5.5 5.9 2.9 63 Other Nervous Diseases 6.1 5.1 5.3 6.6 12.2 11.7 6.1 132 Diseases of Heart 1.4 3.0 5.5 6.9 10.4 9.8 4.6 100 Pneumonia 5.0 6.5 9.4 8.2 16.5 13.7 8.0 176 Other Respiratory 5.0 4.7 4.5 7.0 4.2 9.8 5.3 116 Digestive System 9.2 12.7 13.0 14.8 14.6 9.8 12.5 273 Childbirth and Puer- [ peral Diseases \ 18.3 13.5 6.0 9.1 198 Diseases of Breast and ) Uterus ij 6.1 4.9 5.9 4.7 2.4 2.0 5.0 110 All other Diseases 7.0 6.3 6.5 7.6 11.6 21.5 7.6 165 198 TABLE XYI. (A.) MALE LIFE. PERCENTAGE DISTRIBUTION OF DISEASES AMONG 100 DEATHS, THE SAME NUMBER 100 IN EACH PERIOD OF INSURANCE. DTRF.ASES. Ages of Exposure. 1 i9hm 29H9i 39H9^ 49f 59i 59^-691 69f79i Total. All Causes. 100,0 100.0 100.0 100.0 100.0 100.0 100.0 Zymotic Diseases 26.0 20.0 17.9 15.1 11.6 9.0 17.8 ^ Constitutional Diseases 32.1 28.9 22.5 17.7 14.4 11.7 23.1 P^ Nervous Diseases S Circulatory Diseases 6.8 10.7 14.5 18.6 21.9 20.9 14.5 1.7 3.5 5.0 7.7 11.6 12.5 5.6 ^ Respiratory Diseases 10.7 12.5 14.0 14.8 14.4 13.6 13.4 «o Digestive Diseases 6.3 8.1 10.5 10.7 10.8 6.9 9.5 Miscellaneous Diseases 16.4 16.3 15.6 15.4 15.3 25.4 16.1 Typhoid, Typhus Fever 13.5 7.5 5.7 4.8 3.5 1.1 6.5 Malarial Fever 2.1 2.7 2.6 2.0 1.7 .8 2.3 « Erysipelas .8 .9 1.2 1.1 1.1 .9 1.1 o Dysentery 1.8 1.8 1.5 1.8 1.3 1.9 1.6 P. Diarrhoea .8 .8 . .9 1.0 1.1 1.5 .9 •^ Cholera 1.6 1.4 1.3 1.0 .6 .3 1.2 Alcoholism .2 .5 .5 .2 , , .3 Other Zymotic Diseases 5.2 4.4 4.2 3.2 2.3 2.5 3.9 ^ Dropsy .5 1.1 1.8 2.4 2.9 2.3 1.8 *J Cancer .2 .8 1.7 3.2 3.2 3.1 1.7 o Consumption 30.3 25.8 17.8 10.8 6.5 3.5 18.3 Other Constitutional 1.1 1.2 1.2 1.3 1.8 2.8 1.3 Apoplexy 1.2 2.6 4.6 7.9 8.3 7.1 4.8 g Conprestion of Brain 1.5 2.0 2.1 1.7 1.3 .8 1.9 ^ Paralysis, Softening { ^ etc., of Brain J 3.1 4.6 6.1 ■ 7.6 10.6 12.2 6.3 ^ Epilepsy, Convulsions .3 .5 .5 .3 .4 .3 .4 Other Nervous .7 1.0 1.2 1.1 1.3 .5 1.1 ^ Diseases of Heart 1.6 3.3 4.5 7.3 10.8 12.0 5.2 "o Other Circulatory .1 .2 .5 .4 .8 .5 .4 . Pneumonia 5.3 6.4 8.1 9.1 9.1 7.6- 7.6 g Congestion of Lungs 1.2 1.4 1.7 1.7 1.7 2.5 1.6 oa Bronchitis and Pleurisy 'g» Abscess, Hemorrhage ) |S etc., of Lungs f 1.2 1.6 1.9 1.8 1.8 2.3 1.7 2.6 2.6 1.8 1.5 1.3 .9 , 1.9 Other Respiratory .4 ,5 .5 .7 .5 .3 .6 • Diseases of Stomach .7 1.3 2.1 2.1 2.2 2.5 1.8 .5 Diseases of Bowels 2.1 2.4 1.9 2.1 2.2 .8 2.1 S Peritonitis 1.0 .8 .7 .6 .5 .7 g Diseases of Liver 1.1 2.2 3.8 3.5 3.0 1.7 2.9 Other Digestive 1.4 1.4 2.0 2.4 2.9 1.9 2.0 Diabetes .1 .4 .5 .3 1.0 .6 .5 Diseases of Kidneys 1.7 1.7 2.7 3.2 3.2 3.9 2.5 Other Urinary .3 .3 .5 1.0 2.4 3.9 .8 Ciiildbirth and Puer-/ g peral Diseases J § Diseases of Breast and g Uterus ■ •3 Abscess, Hemorrhage, S Old Age .5 .8 .7 .7 1.1 8.0 1.0 S Debility, Exhaustion, / Prostration, etc*. f .6 .7 .8 1.3 2.4 3.9 1.1 Accidents and Injuries 10.5 9.6 7.6 5.9 3.5 2.3 7.5 Suicides 1.4 ' 1.4 1.6 1.4 .5 .3 1.3 Unknown Causes 1.3 1.4 1.2 1.6 1.2 2.5 1.4 Actual Deaths 3,476 9,321 ' 10,840 7,576 3,357 647 35,442 TABLE XYII. 199 (A). CLIMATIC DISTRIBUTION OP DEATHS AND DISEASES AMONG 100 DEATHS OP INSURED MALES AND PEMALES. Ages, : 20-75. Besidence at Death. North of ae-'SO' South 36" 30'. j Foreign Male. XJnited States. TJ. S. Census 1870. SEX. Male. Fern. Male. Fem. Male. 100.0 Fem. ioo7o 14.0 Male. 100.0 Fem. Too.o^ Deaths from all Causes. 100.0 100.0 100.0 100.0 100.0 Zymotic Diseases 17.3 13.3 28.4 22.5 15.0 17.8 : 15.9 14.5 {H Constitutional Diseases 23.6 25.6 15.4 16.5 19.1 23.1 24.9 i 32.1 37.4 Nervous Diseases 14.5 8.9 13.1 11.3 16.1 14.5 8.9 , 9.5 8.1 s Circulatory Diseases 5.8 5.0 4.0 3.8 5.8 5.6 4.8 ! 5.9 5.9 >i Respiratory Diseases 13.4 13.4 12.6 12.8 16.1 13.4 13.4 13.3 11.4 ^ Diseases of Heart 5.4 4.7 3.9 3.8 5.2 4.6 I 5.9 5.9 •H Other Circulatory .4 7.7 .3 .1 1.3 7.3 .4 .2 Pneumonia 7.9 7.4 9.8 7.6 8.1 ilO.2 8.3 s Congestion of Lungs 1.6 1.5 1.3 . , 1.6 1.6 1.4 1 Bronchitis and Pleurisy 1.7 1.4 2.0 1.5 2.1 1.7 1.4 1.1 1.0 Abscess, Hemorrhage } etc., of Lungs f 1.9 2.0 1.1 .8 4.3 1.9 1.9 1 2.0 2.1 Other Respiratory .5 .6 2.4 .8 .7 .8 1.6 .6 1.8 .6 Diseases of Stomach 1.7 3.0 1.5 2.4 1,0 1.1 > Diseases of Bowels 2.0 3.0 3.4 5.3 1.7 2.1 3.2 2.6 2.4 1 Peritonitis .7 1.9 .3 .8 1.0 .7 1.9 .2 .5 r? Diseases of Liver 2.9 2.2 2.8 4.5 2.8 2.9 2.4 2.0 1.7 Other Digestive 1.9 2.7 3.4 2.2 1.6 .3 2.0 2.7 .7 .9 Diabetes .5 .2 .1 .5 .1 .4 .1 Diseases of Kidneys 2.6 1.3 1.4 1.7 2.5 1.2 1.9 .8 Other Urinary .8 .4 • .7 .3 .8 .4 .4 .1 i Childbirth and Puer- peral Diseases 9.0 ' 9.0 9.1 6.5 Diseases of Breast and _ Uterus 1 5.0 5.3 5.0 2.4 9 Abscess, Hemorrhage, | Old Age ) Debility, Exhaustion, } Prostration, etc. f 1.0 1.1 1.4 1.4 .6 .7 1.5 1.0 .7 1.0 1.1 1.3 1.4 [ 2.7 2.7 i Accidents and Injuries 7.5 1.5 7.8 1.5 11.2 7.5 1.5 9.1 1.4 Suicides 1.4 .4 .8 1.1 1.3 .3 .9 .3 Unknow^n Causes 1.3 82,544 1.0 2,012 1.5 1.5 2.9 1 703 1.4 1.1 1.4 1.8 91.898 ■ ■- — 1 Actual Deaths 2.153 133 35,442 2.182 104,531 ^00 TABLE XVII. (B). PROPORTIONAL DEATHS AND DISEASES TO 100 TOTAL DEATHS OP INSURED MALES IN EACH GROUP OF STATES Group of States. L* XL III. IV. V. VI. VII. Mean Group. Groups. All Causes. 100.0 100.0 100.0 100.0 100.0 100.0 100.0 Zymotic 16.7 19.0 15.5 17.9 21.1 28.4 14.2 19.0 I * § Constitutional 25.0 21.9 26.0 22.0 20.9 15.4 19.8 21.6 Nervous 14.7 11.8 14.4 13.8 15.4 13.1 16.4 14.2 New England. Circulatory 6.3 4.5 6.8 4.4 5.1 4.0 8.2 5.6 New York. a Respiratory 13.6 14.9 11.3 15.9 13.9 12.6 13.0 13.5 ^ Digestive 8.2 10.2 10.5 10.7 9.1 12.9 8.1 10.0 II. Miscellaneous 16.5 17.7 15.5 15.3 14.5 13.6 20.3 16.2 Northwest. Michigan. Typhoid.Typlius 7.2 8.8 6.4 6.4 5.3 2.8 4.6 5.9 Malarial Fever 1.5 2.4 1.7 3.2 2.9 6.9 1.7 2.9 Wisconsin. u Erysipelas 1.0 1.3 .9 1.2 1.0 .9 2.0 1.2 Minnesota. o Dysentery 1.6 1.4 .8 1.4 2.8 3.4 1.3 1.8 Nebraska. I Diarrhoea .9 .7 .9 .7 1.2 2.1 .2 1.0 in. N Cholera .8 .8 1.0 1.9 2.7 1.4 .8 1.3 Alcoholism .3 .3 .3 .4 .4 .2 .5 .3 New Jersey. Other Zymotic 3.4 3.3 3.5 2.7 4.8 10.7 3.1 4.5 Pennsylvania. IV . j Dropsy 1.7 1.7 2.4 1.8 1.7 1.3 1.7 1.8 *> Cancer 2.0 1.6 2.0 1.5 1.7 .7 1.7 1,6 13 o Consumption 19.7 17.3 20.7 17.7 16.4 12.3 15.1 17.0 Ohio. U Other Const'] 1.6 1.3 .9 1.0 1.1 1.1 1.3 1.2 4.9 Indiana. Illinois. Apoplexy 5.0 4.0 4.6 4.0 5.5 4.8 6.1 1 Congestions rain Paralysis, Soft- i^ ening Brain f 1.4 6.9 1.5 5.3 2.0 6.1 2.4 5.9 2.2 6.0 2.9 4.4 1.5 6.5 . 2.0 5.9 Iowa. Kansas. V. Delaware. Maryland. Dist. Columbia. in Epilepsy and I Convulsions J Other Nervous .3 1.1 .3 .7 .6 1.1 .4 1.1 .4 1.3 .4 .6 3.9 1.3 1.0 .5 1.0 >» Diseasesof Heart 5.8. 4.2 6.6 4.2 4.8 6.6 5.2 a w Other Circulat'y .5 .3 .2 .2 .3 .1 1.6 .5 Virginia. Kentucky. Pneumonia 7.0 8.7 6.2 9.4 8.3 7.4 8.0 7.9 ^ Congest'n Lungs 1.7 1.7 1.1 1.6 1.7 1.3 .8 1.4 Missouri. o Bronchitis and | " Pleurisy ) 1.7 2.0 1.4 1.9 1.4 2.0 2.0 1.8 VI. « Abscess, Hem- { • or'age Lungs \ Other Respirat'y 1.8 .4 2.1 ^.4 1.9 .7 2.3 .7 1.8 .7 1.1 .8 1.6 .6 •1.8 .6 South of 36° 30'. North Carolina, , Dis. of Stomach 1.5 2.4 1.9 1.8 1.6 3.0 1.2 1.9 Tennessee. > Dis. of Bo web 1.8 2.2 2.3 2.4 1.9 3.4 1.4 2.2 Georgia. a e Peritonitis .7 .8 .9 1 .8 .5 .3 .3 .6 Florida. bo P Dis. of Liver 2.6 2.5 3.5 3.5 3.0 2.8 3.7 3.1 Alabama. Other Digestive 1.6. 2.3 1.9 2.2 2.1 3.4 1.5 2.1 Mississippi. Arkansas. Diabetes .5 .5 .5 .4 .6 .1 .5 .4 Dis. of Kidneys 3.3 1.4 2.9 1.8 1.6 1.4 2.3 2.1 , Louisiana. Other Urinary .9 .8 l.O .5 .7' .7 .6 .7 Texas. Childbirth.Pu- ) VII. " erperal Dis. f • • ' " • * • * S Dis. Breast and Uterus Pacific, etc. g • • • * 5 o m Abscess, Hem- [ or'age.OldAgej" 1.1 .8 1.0 1.0 1.0 .6 .3 .8 Washington. Oregon. » Debility, Ex- haustion, etc. 1.0 1.0 2.2 .7 1.0 .7 1.1 1.1 California. Utah. Accid'ts, Injuries 6.8 10.1 5.7 8.6 7.1 7.8 11.4 8.2 Dakota. Suicides 1.2 2.1 1.2 1.3 1.4 .8 2.9 1.6 New Mexico. UnknownCauses i 1.7 1.0 1.0 1.0 1.1 1.5 1.2 1.2 Actual Deaths 15,273 j2,7ie 3,976|6,239 |3,306 j2,153 863 .. TABLE XYII. 201 (O- PROPORTIONAL DEATHS AND DISEASES TO 10,000 MALES LIVING IN EACH GROUP OP STATES. Group of States. I.* 11. HI. IV. V. VI. VII. Mean Group. Groups. All Causes. 105.3 97.7 107.1 104.5 130.5 170.5 113.2 Zymotic 17.6 18.5 16.6 18.7 27.5 48.4 15.9 23.3 L* i Constitutional 26.4 21.4 27.9 33.0 37.3 26.3 22.1 34.9 Nervous 15.4 11.5 15.4 14.4 30.1 22.3 18.4 16.8 New England. < Circulatory 6.6 4.4 7.3 4.6 6.7 6.8 9.2 6.5 New York. s Respiratory 13.3 14.6 12.1 16.6 18.1 21.5 14.6 15.8 II. ^ Digestive 8.6 10.0 11.2 11.2 11.9 22.0 9.1 12.0 Miscellaneous 17.4 17.3 16.6 16.0 18.9 23.2 22.9 18.9 Northwest. Michigan. Typhoid,Typhus 7.6 8.5 6.8 6.7 6.9 4»-8 5.2 6.6 Malarial Fever 1.7 2.3 1.8 3.3 3.8 11.8 1.9 3.8 Wisconsin. o Erysipelas 1.0 1.3 1.0 1.3 1.3 1.5 2.2 1.4 Minnesota. Dysentery 1.7 1.4 .9 1.4 3.6 5.8 1.5 2.3 Nebraska. I Diarrhoea .9 .7 1.0 .8 1.6 3.6 .2 1.3 in. N Cholera .9 .8 1.1 2.0 3.5 2.4 .9 1.7 Alcoholism .3 .3 .3 .4 .5 .3 .6 .4 New Jersey. Other Zymotic 5.6 3.2 3.7 2.8 6.3 18.2 3.4 5.9 Pennsylvania. IV Dropsy 1.8 1.6 2.6 1.9 2.2 2.2 1.9 2.0 ^ Cancer 2.1 1.5 2.1 1.6 2.2 1.1 1.9 1.8 § Consumption 20.8 16.9 22.2 18.5 21.5 21.0 16.9 19.7 Ohio. u Other Const'l 1.7 1.4 1.0 1.0 1.4 2.0 1.4 1.4 5.8 Indiana- Illinois. Iowa. Kansas. V. Delaware. Maryland. Dist. Columbia. Apoplexy 5.2 3.8 5.0 4.2 7.2 8.2 6.8 i CongestionBrain Paralysis, Soft- / ening Brain ) 1.5 7.2 1.5 5.2 2.1 6.5 2.5 6.2 2.9 7.8 4.9 7.5 1.7 7.3 2.4 6.8 ^ Epilepsy and ) Convulsions ) Other Nervous .3 1.2 .3 .7 .6 1.2 .4 1.1 .5 1.7 .7 1.0 1.5 1.1 .6 1.1 ►. Diseasesof Heart 6.1 4.1 7.1 4.4 6.3 6.6 7.4 6.0 Other Circulat'y .5 .3 .2 .2 .4 .2 1.8 .5 Virginia. Kentucky. Pneumonia 7.3 8.5 6.6 9.8 10.8 12.6 9.0 9.2 >> Congest'n Lungs 1.8 1.7 1.2 1.7 2.2 2.2 .9 1.7 Missouri. 1 .a Bronchitis and ) Pleurisy ) 1.8 2.0 1.5 2.0 1.8 3.4 2.2 2.1 VI. 09 Abscess, Hem- ) or' age Lungs ) Other Respirat'y 1.9 .4 2.0 •4 2.0 .8 2.4 .7 2.4 .9 1.9 1.4 1.8 .7 2.1 .8 South op 36° 30'. North Carolina. South Carolina. Tennessee. . Dis. of Stomach 1.6 2.3 2.0 1.9 2.1 5.1 1.3 2.3 > Dis. of Bowels 1.9 2.2 2.5 2.5 2.5 5.8 1.6 2.7 Georgia. s Peritonitis .7 .8 1.0 .8 .7 .5 .3 .7 Florida. bo P Dis. of Liver 2.7 2.4 3.7 3.7 3.9 4.8 4.2 3.6 Alabama. Other Digestive 1.7 2.3 2.0 2.3 2.7 5.8 1.7 2.6 Mississippi. Arkansas. Diabetes .5 .5 .5 .4 .8 .3 .6 .5 Dis. of Kidneys 3.5 1.4 3.1 1.9 3.1 2.4 2.6 2.4 Louisiana. Other Urinary .9 .8 1.1 .5 .9 1.2 .7 .9 Texas. S Childbirth,Pu- ) erperal Dis. J •• •• •• •• VII. s Dis. Breast and Uterus ' '. , Pacific, etc. r"! Abscess, Hem- or'age.OldAge 1.2 .8 1.1 1.1 1.3 1.0 .3 1.0 Washington. Oregon. M Debility, Ex- | haustion, etc. | 1.0 1.0 2.4 .7 1.3 1.1 1.2 1.2 California. Utah. Accid'ts, Injuries 7.2 9.7 6.0 9.0 9.3 13.3 12.8 9.6 Dakota. Suicides 1.3 2.1 1.3 1.3 1.8 1.4 3.3 1.8 New Mexico. UnknownCauses 1.8 1.0 1.1 1.1 1.4 2.5 1.3 863 1.5 Actual Deaths 15,273 2,716 3,976 6,239 3,306 2,153 303 TABLE XYIII. CALENDAR AND DECIMAL PARTS OF A YEAR. Days, Cal. Decimal. Days. Cal. Decimal. Da3r8. Cal. Decimal. Days. Cal. Decimal. Jan. .00000000 50 Fb*19 .13698630 100 Apr.lO. 27397260 150 May 30 .41095890 1 *1 .00273973 51 20 .13972603 101 11 .27671233 151 31 .41369863 2 2 .00547945 52 21 .14246575 102 12 .27945205 162 June 1 .41643836 3 3 .00821918 53 22 .14520548 103 13 .28219178 153 2 .41917808 4 4 .01095890 54 23 .14794521 104 14 .28493151 154 3 .42191781 5 5 .01369863 55 24 .15068493 105 15 .28767123 155 *4 .42465753 6 6 .01643836 56 25 .15342466 106 *16 .29041096 156 6 .42739726 7 7 .01917808 57 *26 .15616438 107 17 .29315068 157 6 .43013699 8 *8 .02191781 58 27 .15890411 il08 18 .29589041 158 7 .43287671 9 9 .02465753 59 28 .16164384 109 19 .29863014 159 8 .43561644 10 10 .02739726 60 Mar. 1 .16438356 110 20 .30136986 160 9 .43835616 11 11 .03013699 61 2 .16712329 111 21 .30410959 161 10 .44109589 12 12 .03287671 62 3 .16986301 112 22 .30684932 162 *11 .44383562 13 13 .0.3561644 1 63 4 .17260274 113 *23 .30958904 163 12 .44(:57534i 14 14 .03835616 64 *5 .17534247 114 24 .31232877 164 13 .44931507| 15 *15 .04109589 ! 65 6 .17808219 115 25 .31506849 165 14 .45205479 16 16 .04383562 66 7 .18082192 116 26 .31780822 166 15 .45479452 17 li .04657534 67 8 .18356164 117 27 .32054795 167 16 .45753425 18 18 .04931507 68 9 .18630137 118 28 .32328767 168 17 .46027397 19 19 .05205479 69 10 .18904110 119 29 .32602740 169 *18 .46301370 20 20 .05479452 70 11 .19178082 120 *30 .32876712 170 19 .46575342 21 21 .05753425 71 *12 .19452055 121 May 1 .33150685 171 20 .46849315 22 *22 .06027397 72 13|. 19726027] 122 2 .33424658 172 21 .47123288 23 23 .06301370 73 14 .20000000 123 3 .33698630 173 22 .47397260 24 24 .06575342 74 15 .20273973 124 4 .33972603 174 23 .47671233 25 25 .06849315 75 16 .20547945 125 5 .34246575 176 24 .47945205 26 26 .07123288 76 17 .20821918 126 6 .34520548 176 *25 .48219178 27 27 .07397260 77 18 .21095890 127 *7 .34794521 177 26 .48493151 28 28 .07671233 78 *19 .21369863 128 8 '.35068493 178 27 .48767123. 29 *29 .07945205 79 20 .21643836 129 9 .35342466 179 28 .49041096 30 30 .08219178 80 21 .21917808 130 10 .35616438 180 29 .49315068 31 31 .08493151 81 22 .22191781 131 11 .35890411 181 30 .49589041 32 Feb. 1 .08767123 82 23 .22465753 132 12 .36164384 182 July 1 .49863014 33 2 .09041096 83 24 .22739726 133 13 .36438356 183 *2 .50136986 34 3 .09315068 84 25 .23013699 134 *14 .36712329 184 3i.50410959| 35 4 .09589041 85 * *26 .23287671 135 15 .36986301 186 4 .50684932 36 *5 .09863014 86 27 .23561644 136 16 .37260274 186 5 .50958904 37 6 .10136986 87 28 .23835616 137 17 .37534247 187 6 .51232877 38 7 .104109591 88 29 .24109589 138 18 .37808219 188 7 .51506849 39 8 .10684932 89 30 .24383562 139 19 .38082192 189 8 .51780822 40 9 .10958904 90 31 .24657534 140 20 .38356164 190 *9;.52054795| 41 10 .11232877 91 April 1 .24931507 141 *21 .38630137 191 10 .52328767 42 11 .11506849 92 *2 .25205479 142 22 .38904110 192 11 .52602740 43 *12 .11780822 93 3 .254794521 143 23 .39178082 193 12 .52876712 44 13 .12054795 1 94 4j. 25753425 144 24 .39452055 194 13 .53150685 45 14 .12328767 i 95 5{. 26027397 145 25 .39726027 196 14 .53424658 46 15 .12602740 96 61.26301370 146 26 .40000000 196 15 ..53698630 47 16 .12876712 1 97 71.26575342 147 27 .40273973 197 *16 ..53972603 48 17 .13150685 98 8 .26849315 148 *28 .40547945 198 17 .54246575 49 18 .13424658 99 *9 .27123288 149 29 .40821918 199 18 .54520548 TABLE XYill. CALENDAR AND DECIMAL PARTS OF A TEAR. 203 204 205 206 207 Days. 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 1242 243 244 245 246 247 248 249 Cal. Decimal. Julyi9 .54794521 20 .55068493 21 .55342466 22 .55616438 *23 .55890411 24 .56164384 25 ,56438356 26 .56712329 27 .56986301 28 .57260274 29 .57534247 *30 .57808219 31 .58082192 Au?. 1 .58356164 2 .58630137 3 .58904110 4 .59178082 5 .59452055 *6 .59726027 7 .60000000 8 .60273973 9 .60547945 10 .60821918 11 .61095890 12 .61369863 *13 .61643836 14 .61917808 15 .62191781 16 .62465753 17 .62739726 18 .63013699 19 .63287671 *20 .63561644 21 .63835616 22.64109589 23 24 25 26 *27 28 29 30 31 Sept. 1 2 *3 4 5 6 64383562 64657534 64931507 65205479 65479452| 65753425 66027397 66301370 66575342 66849315 67123288 67397260 67671233 ,67945205 ,68219178 Days. Cal. 250 Sept. 7 251 8 252 9 253 *10 254 11 255 12 256 13 257 14 258 15 259 16 260 *17 261 18 262 19 263 20 264 21 265 22 266 23 267 *24 268 25 269 26 270 27 271 28 272 29 273 30 274 Oct. *1 275 2 276 3 277 4 278 5 279 6 280 7 281 *8 282 9 283 10 284 11 285 12 286 13 287 14 288 *15 289 16 290 17 291 18 292 19 293 20 294 21 295 *22 296 23 297 24 298 25 299 26 Decimal. .68493151 ,68767123 ,69041096 ,69315068 ,69589041 ,69863014 ,70136986 ,70410959 ,70684932 ,70958904 ,71232877 ,71506849 .71780822 ,72054795 ,72328767 72602740 72876712 73150685 73424658 73698630 73972603 74246575 74520548 74794521 75068493 ,75342466 75616438 75890411 76164384 76438356 76712329 76986301 77260274 77534247 77808219 78082192 78356164 78630137 78904110 79178082 79452055 79726027 80000000 80273973 80547945 80821918 81095890 81369863 81643836 81917808 Days, 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 Cal. Oct. 27 28 *29 30 31 m, 1 2 3 4 *5 6 7 8 9 10 11 *12 13 14 15 16 17 18 *19 20 21 22 23 24 25 *26 27 28 29 30 Dec. 1 2 *3 4 5 6 7 8 9 *10 11 12 13 14 15 Decimal. .82191781 .82465753 .82739726 .83013699 .83287671 .83561644 .83835616 .84109589 .84383562 .84657534 .84931507 .85205479 .85479452 .85753425 .86027397 .86301370 .86575342 .86849315 .87123288 .87397260 .87671233 .87945205 .88219178 .88493151 .88767123 .89041096 .89315068 .89589041 .89863014 .90136986 .90410959 .90684932 .90958904 .91232877 .91506849 .91780822 .92054795 .92328767 .92602740 .92876712 .93150685 .93424658 .93698630 .93972603 .94246575 •94520548 •94794521 •95068493 .95342466 .95616438 Days. Cal. 350 Dec. 16 351 *17 352 18 353 19 354 20 355 21 356 22 357 23 358 *24 359 25 360 26 361 27 362 28 363 29 364 30 365 *31 Decimal. .95890411 .96164384 .96438356 .96712329 .96986301 .97260274 .97534247 .97808219 .98082192 .98356164 .98630137 .98904110 .99178082 .99452055 .99726027 1.00000000 YEAR AND DAY OF THE WEEK DENOTED BY *. 1881 Saturday. 1882 Sunday. 1883 Monday. 1884 *Tues(iay. Wcdn. 1885 Thursday. 1886 Friday. 1887 Saturday. 1888 *Sunday. Mond. 1889 Tuesday. 1890 Wednesday. 1891 Thursday. 1892 *Friday. Satur. 1893 Sunday. 1894 Monday. 1895 Tuesday. 1896 *Wedn. Thurs. 1897 Friday. 1898 Saturday. 1899 Sunday. 1900 Monday. 1901 Tuesday. 1902 Wednesday. 1903 Thursday. 1904 I *Friday. Satur. 1905 Sunday. 1906 ' Monday. 1907 Tuesday. 1908 *Wedn. Thurs. 1909 Friday. 1910 Saturday. * In 1884 and other leap years, the first star day applies only in January and February, and the laat applies only in the remaining ten months. 208 TABLE XX. THE AMOUNT OF $1 AT THE END OF ANY NUMBER OF TEARS. si 2 2\ 3 U 4 4^ i tM Per Cent. Per Cent. Per Cent. Per Cent, Per Cent. Ver Cent. ^ 1 1.02000000 1.02500000 1.03000000 1.03500000 1.04000000 1.04500000' 1 2 1.04040000 1.05062500 1.06090000 1.07122500 1.08160000 1.09202500 i 2 3 1.06120800 1.07689063 1.09272700 1.10871787 1.12486400 1.14116613 3 4 1.08243316 1.10381289 1.12550881 1.14752300 1.16985856 1.19251860 4 5 1.10408080 1.13140821 1.15927407 1.18768631 1.21665290 1.24618194 6 6 1.12616242 1.15969342 1.19405230 1.22925533 1.26531902 1.30226012 6 7 1.14868567 1.18868575 1.22987387 1.27227926 1.31593178 1.36086183 7 8 1.17165938 1.21840290 1.26677008 1.31680904 1.36856905 1.42210061 8 9 1.19509257 1.24886297 1.30477318 1.36289735 1.42331181 1.48609514 9 10 1.21899442 1.28008454 1.34391638 1.41059876 1.48024428 1.55296942 10 11 1.24337431 1.31208666 1.38423387 1.45996972 1.53945406 1.62285305 11 12 1.26824179 1.34488882 1.42576089 1.51106866 1.60103222 1.69588143 12 13 1.29360663 1.37851104 1.46853371 1.56395606 1.66507351 1.77219610 13 14 1.31947876 1.41297382 1.51258972 1.61869452 1.73167645 1.85194492 14 15 1.34586834 1.44829817 1.55796742 1.67534883 1.80094351 1.93528244 15 16 1.37278571 1.48450562 1.60470644 1.73398604 1.87298125 2.02237015 16 17 1.40024142 1.52161826 1.65284763 1.79467555 1.94790050 2.11337681 17 18 1.42824625 1.55965872 1.70243306 1.85748920 2.02581652 2.20847877 18 19 1.45681117 1.59865019 1.75350605 1.92250132 2.10684918 2.30786031 19 20 1.48594740 1.63861644 1.80611123 1.98978886 il9113314 2.41171402 20 21 1.51566634 1.67958185 1.86029457 2.05943147 2.27876807 2.52024116 21 22 1.54597967 1.72157140 1.91610341 2.13151158 2.36991879 2.63365201 22 23 1.57689926 1.76461068 1.97358651 2.20611448 2.46471554 2.75216635 23 24 1.60843725 1.80872595 2.03279411 2.28332849 2.56330416 2.87601383 24 25 1.64060599 1.85394410 2.09377793 2.36324498 2.66583633 3.00543446 25 j 26 1.67341811 1.90029270 2.15659127 2.44595856 2.77246978 3.14067901 26 27 1.70688648 1.94780002 2.22128901 2.53156711 2.88336858 3.28200956 27 28 1.74102421 1.99649502 2.28792768 2.62017196 2.99870332 3.42969999 28 29 1.77584469 2.04640739 2.35656551 2.7118779€ 3.11865145 3.58403649 29 30 1.81136158 2.09756758 2.42726247 2.80679370 3.24339751 3.74531813 30 31 1.84758882 2.15000677 2.50008035 2.90503148 3.37313341 3.91385745 31 32 1.88454059 2.20375694 2.57508276 3.00670759 3.50805875 4.08998104 32 33 1.92223140 2.25885086 2.65233524 3.11194235 3.64838110 4.27403018 33 34 1.96067603 2.31532213 2.73190530 3.22086033 3.79431634 4.46636154 34 35 1.99988955 2.37320519 2.81386245 3.33359045 3.94608899 4.66734781 35 36 2.03988734 2.43253532 2.89827833 3.45026611 4.10393255 4.87737846 36 37 2.08068509 2.49334870 2.98522668 3.57102542 4.26808986 5.09686049 37 38 2.12229879 2.55568242 3.07478348 3.69601131 4.43881345 5.32621921 38 39 2.16474477 2.61957448 3.16702698 3.82537171 4.61636599 5.56589908 39 40 2.20803966 2.68506384 3.26203779 3.95925972 4.80102063 5.81636454 40 41 2.25220046 2.75219043 3.35989893 4.09783381 4.99306145 6.07810094 41 42 2.29724447 2.82099520 3.46069589 4.24125799 5.19278391 6.35161548 42 43 2.34318936 2.89152008 3.56451677 4.38970202 5.40049527 6.63743818 43 44 2.39005314 2.96380808 3.67145227 4.54334160 5.61651508 6.93612290 44 45 2.43785421 1 3.03790328 3.78159584 4.70235855 5.84117568 7.24824843 45 46 2.48661129 3.11385086 3.89504372 4.86694110 6.07482271 7.57441961 46 47 2.53634352 3.19169713 4.01189503 5.03728404 6.31781562 7.91526849 47 48 2.58707039 3.27148956 4.13225188 ' 5.21358898 6.57052824 8.27145557 48 49 2.63881179 3.35327680 4.25621944 1 5.39606459 6.83334937 8.64367107 49 50 2.69158803 3.43710872 4.38390602 5-58492686 7.10668335 9.03263627 50 1 Formula, Amount = (1 + *)"• TABLE XX. ^09 THE AMOUNT OF $1 AT THE END OF ANY NUMBER OF YEARS, i 5 6 7 8 9 10 e tS Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. ^ 1 1.05000000 1.06000000 1.07000000 1.08000000 1.09000000 1.10000000 1 2 1.10250000 1.12360000 1.14490000 1.16640000 1.18810000 1.21000000 2 3 1.15762500 1.19101600 1.22504300 1.25971200 1.29502900 1.33100000 3 4 1.21550625 1.26247696 1.31079601 1.36048896 1.41158161 1.46410000 4 5 1.27628156 1.33822558 1.40255173 1.46932808 1.53862395 1.61051000 5 6 1.34009564 1.41851911 1.50073035 1.58687432 1.67710011 1.77156100 6 7' 1.40710042 1.50363026 1.60578148 1.71382427 1.82803912 1.94871710 7 8 1.47745544 1.59384807 1.71818618 I 1.85093021 1.99256264 2.14358881 8 9 1.55132822 1.68947896 1.83845921- : 1.99900463 2.17189328 2.35794769 9 10 1.62889463 1.79084770 1.96715136 i 2.15892500 2.36736367 2.59374246 10 11 1.71033936 1.89829856 2.10485195 1 2.33163900 2.58042641 2.85311671 11 12 1.79585633 2.01219647 2.25219159 2.51817012 2.81266478 3.13842838 12 13 1.88564914 2.13292826 2.40984500 2.71962373 3.06580461 3.45227121 13 14 1.97993160 2.26090396 2.57853415 2.93719362 3.34172703 3.79749834 14 15 2.07892818 2.39655819 2.75903154 3.17216911 3.64248246 4.17724817 15 16 2.18287459 2.54035168 2.95216375 3.42594264 3.97030588 4.59497299 16 17 2.29201832 2.69277279 3.15881521 3.70001805 4.32763341 5.05447028 17 18 2.40661923 2.85433915 3.37993228 \ 3.99601950 4.71712042 5.55991731 18 19 2.52695020 3.02559950 3.61652753 4.31570106 5.14166125 6.11590904 19 20 2.65329771 3.20713547 3.86968446 4.66095714 5.60441077 6.72749995 20 21 2.78596259 3.39956360 4.14056237 5.03383372 6.10880774 7.40024994 21 22 2.92526072 3.60353742 4.43040174 5.43654041 6.65860043 8.14027494 22 23 3.07152376 3.81974966 4.74052986 5.87146365 7.25787447 8.95430243 23 24 3.22509994 4.04893464 5.07236695 6.34118074 7.91108317 9.84973268 24. 25 3.38635494 4.29187072 5.42743264 6.84847520 8.62308066 10.83470594 25 26 3.55567269 4.54938296 5.80735292 7.39635321 9.39915792 11.91817654 26 27 3.73345632 4.82234594 6.21386763 7.98806147 10.24508213 13.10999419 27 28 3.92012914 5.11168670 6.64883836 8.62710639 11.16713952 14.42099361 28 29 4.11613560 5.41838790 7.11425705 9.31727490 12.17218208 15.86309297 29 ! 30 4.32194238 5.74349117 7.61225504 10.06265689 13.26767847 17.44940227 30 31 4.53803949 6.08810064 8.14511290 10.86766944 14.46176953 19.19434250 31 32 4.76494147 6.45338668 8.71527080 11. 73708300 15.76332879 21.11377675 32 33 5.00318854 6.84058988 9.32533975 12.67604963 17.18202838 23.22515442 33 34 5.25334797 7.25102528 9.97811354 13.69013361 18. 72841093 25.54766986 34 35 5.51601537 7.68608679 10.67658148 14. 78534429 20.41396792 28.10243685 35 36 5.79181614 8.14725200 11.42394219 15.96817184 22.25122503 30.91268053 36 37 6.08140694 8.63608712 12.22361814 17.24562558 24.25383528 34.00394859 37 38 6.38547729 9. 15425235 13.07927141 18.62527563 26.43668046 37.40434344 38 1 39 6.70475115 9.70350749 13. 99482041 20.11529768 28.81598170 41.14477779 39 40 7.03998871 10.28571794 14.97445784 21.72452150 31.40942005 45.25925557 40 1 41 7.39198815 10.90286101 16.02266989 23.46248322 34.23626786 49.78518112 41 42 7.76158756 11.55703267 17.14425678 25.33948187 37.31753197 54.763699241 42 i |43 8.14966693 12.25045463 18.34435475 27.36664042 40. 67610984 60.24006916 43 1 44 8.55715028 12.98548191 19.62845959 29.55597166 44.33695973 66.26407608 44 l45 8.98500779 13.76461083 21.00245176 31.92044939 48.32728610 72.89048369 46 46 9.4,3425818 14.59048748 22.47262338 34.47408534 52.67674185 80.17953205 46 47 9.90597109 15.46591673 24.04570702 37.23201217 57.41764862 88.19748526 47 48 10.40126965 16.39387173 25.72890651 40.21057314 62.58523700 97.01723378 48 49 10.92133313 17.37750403 27.52992997 43.42741899 68.21790833 106.71895716 49 50 11.46739979 18.42015427 29.45702506 46.90161251 74.35752008 117.39085288 60 1 210 TABLE XX. THE AMOUNT OF $1 AT THE END OF ANY NUMBER OF TEARS. 51 62 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 2 Per Cent. 2.74541979 2.80032819 2.85633475 2.91346144 2.97173067 3.03116529 3.09178859 3.15362436 3.21669685 3.28103079 3.34665140 3.41358443 3.48185612 3.55149324 3.62252311 3.69497357 3.76887304 3.84425050 3.92113551 3.99955822 4.07954939 4.16114038 4.24436318 4.32925045 4.41583546 4.50415216 4.59423521 4.68611991 4.77984231 4.87543916 4.97294794 5.07240690 5.17385504 5.27733214 5.38287878 5.49053636 5.60034708 5.71235402 5.82660110 5.94313313 6.06199579 6.18323570 6.30690042 6.43303843 6.56169920 6.69293318 6.82679184 6.96332768 7.10259423 7.24464612 2| Per Cent. 3 Per Cent. 3.52303644 4.51542320 3.61111235 4.65088590 3.70139016 4.79041247 3.79392491 4.93412485 3.88877303 5.08214859 3.98599236 5.23461305 4.08564217 5.39165144 4.18778322 5.55340098 4.29247780 5.72000301 4.39978975 5.89160310 4.50978449; 6.06835120 4.62252910 6.25040173 4.73809233 6.43791378 4.85654464 6.63105120 4.97795826 6.82998273 5.10240721 7.03488222 5.22996739 7.24592868 5.36071658 7.46330654 5.494734491 7.68720574 5.632102861 7.91782191 5.77290543,' 8.15535657 5.91722806 8.40001727 6.06515876 8.65201778 6.21678773 8.91157832 6.372207431 9.17892567 6.5315126l| 9.45429344 6.69480043; 9.73792224 6.8621704410.03005991 7.0337247010.33096170 7.2095678210.64089056 7.3898070110.96011727 3| Per Cent. 7.57455219 7.76391599 7.95801389 8.15696424 11.28892079 11.62758841 11.97641607 12.33570855 8. 36088834!l2. 70577981 8.56991055113.08695320 8.7841583213.47956180 9.0037622813.88394865 9.2288563314.30046711 9.4595777414.72948112 9.6960671815.17136556 9. 93846886 10.18693058 10.44160385 10.70264395 10.97021004 11.24446530 15.62650652 16.09530172 16.57816077 17.07550559 17.58777076 18.11540388 11.52557693118.65886600 11.8137163519.21863198 5.78039930 5.98271327 6.19210823 6.40883202 6.63314114 6.86530108 7.10558662 7.35428215 7.61168203 7.87809090 8.15382408 8.43920792 8.73458020 9.04029051 9.35670068 9.68418520 10.02313168 10.37394129 10.73702924 11.11282526 11.50177414 11.90433624 12.32098801 12.75222259 13.19855038 13.66049964 14.13861713 14.63346873 15.14564013 15.67573754 16.22438835 16.79224194 17.37997041 17.98826938 26. 18.61785880 28. 4 Per Cent. ^ Per Cent. 7.39095068 9.43910490 7.686588711 9.86386463 7.9940522610.30773853 8.31381435110.77158677 8.6463669211.25630817 8.9922216011.76284204 9.35191046{12.29216993 9.7259868812.84531758 10.11502635'l3.42335687 10.51962741114.02740793 10.9404125014. 11.3780290015. 11.83315016jl6. 12.3064761716. 12.79873522|l7. 65864129 31828014 00760275 72794487 48070239 13.31068463 13.84311201 14.39683649 14.97270995 18.26733400 19.08936403 19.94838541 20.84606276 15.57161835 21.78413558 1944830822. 84226241123. 51595290i24. 2I659IO2I25. 94525466127. 76442168 78882066 85931759 97798688 14699629 19.70306485 28.36861112 20.49118744 29.64519862 21.31083494 22.16326833 23.04979907 23. 24. 25. 30.97923256 32.37329802 33.83009643 97179103 35. 93066267i36. 92788918{38. 96500475140. 04360494 42. 19.26948386 29 19.94391580 30 20.64195285 21.36442120 22.11217594 22.88610210 23.68711567 24.51616472 25.37423049 26.26232856 27.18151005 28.13286291 29.11751311 30.13662607 16534913 33196310 54524162 80705129 11933334 35245077 94331106 60576006 34301926 15845513 44.05558561 46.03808696 48.10980087 50.27474191 52.53710530 48410667 54 90347094|57, 37960978:59, 91479417:62. 51138594|65. 43.17I84137I68, 44.89871503 71 46.69466363 48.56245017 31.19140798|50.50494818,81 90127503 37183241 95356487 65147529 47079168 41697730 49574128 71304964 07513687 58851803 TABLE XX. THE AMOUNT OF $1 AT THE END OF ANY NUMBER OF YEARS. 2X1 § 5 6 7 8 10 1 ^ Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. ^ 51 12.04076978 19.53536353 31.51901683 50.65374151 139.13993817 51 52 13.64380836 30.69688534 33.73534799 54.70604083 143.04393198 52 53 13.37494868 31.93869846 36.08613335 59.08252410 156.24722518 53 54 13.93869611 23.25502037 38.61315092 63.80912603 171.87194770 54 55 14.63563093 24.65032159 41.31500148 68.91385611 189.05914247 55 56 15.36741346 26.12934089 44.20705159 74.42696460 207.96505672 56 57 16.13578309 37.69710134 47.30154520 80.38112177 228.76156239 57 58 16.94357334 39.35893742 50.61265336 86.81161151 251.63771863 58 59 17.78970085 31.12046307 54.15553910 93.75654043 276.80149049 59 60 18.67918589 32.98769085 57.94643683 101.25706367 304.48163954 60 61 19.61314519 34.96695230 63.00367671 109.35762876 334.92980350 61 62 30.59380345 37.06496944 66.34386408 118.10623906 368.42378385 62 63 31.63349357 39.28886761 70.98686457 127.55473819 405.36506333 63 64 33.70466730 41.64619966 75.95594509 137.75911734 445.79156845 64 65 33.83990056 44.14497164 81.37386134 148.77984662 - 490.37073530 65 66 35.03189559 46.79366994 86.96196153 160.68223435 539.40779783 66 67 36.38349037 49.60129014 93.04939884 173.53681310 593.34857761 67 68 37.59766488 52.57736755 99.56374976 187.41975814 653.68343537 68 69 38.97754813 55.73300960 106.53314334 303.41333880 717.95177891 69 70 30.43643554 59.07593018 113.98939330 318.60640590 789.74695680 70 71 31.94774681 63.63048599 131.96864965 336.09491837 868.73165348 71 72 33.54513415 66.37771515 130.50645513 254.98251184 955.59381773 72 73 35.33339086 70.36037806 139.64190699 375.38111379 1,051.15319950 73 74 36.98351040 74.58300074 149.41684047 397.41160181 1,156.36851945 74 75 38.83368593 79.05693078 159.87601931 331.30453996 1,271.89537140 75 76 40.77433033 83.80033603 171.06734066 346.90089335 1,399.08490853 76 77 43.81303633 88.83835619 183.04305450 374.65396374 1,538.99339939 77 78 44.95368804 94.15805756 195.85499833 404.63520084 1,692.89273933 78 79 47.30137344 99.80754103 309.56484820 436.99531691 1,862.18201336 79 80 49.56144107 105.79599348 224.23438758 471.95483436 2,048.40021459 80 81 53.03951313 113.14375309 239.93079471 509.71133100 2,253.34033604 81 82 54.64148878 118.87337837 256.72595034 550.48811868 3,478.56435965 82 83 57.37356333 136.00473097 274.69676686 594.53716818 3,726.42068561 83 84 60.34334138 133.56500433 393.93554054 643.08934163 3,999.06375418 84 85 63.25435344 141.57890448 314.50032838 693.45648896 3,398.96903959 85 86! 66.41707113 150.07363875 336.51535136 748.93300808 3,638.86593355 86 87 1 69.73793467 159.07805708 360.07142596 808.84764873 3,991.75353581 87 88 73.33483091 168.63374050 385.27642578 873.55546063 4,390.92777839 88 89 • 76.88606195 178.74010493 412.24577558 943.43989747: 4,830.02055633 89 90 80.73036505 189.46451133 441.10297987 1,018.91508937: 5,313.03361185 90 91 84.76688330 300.83338190 471.98018846 1,100.43829641 5,844.33487303 91 92 89.00533747 313.88333481 505.01880166 1,188.46356013 6,438.75736034 92 93 93.45548884 335.65536430 540.37011777 1,383.53956494 7,071.63309637 93 94 98.13826328 339.19458016 578.19602602 1,386.33373013 7,778.79640601 94 95 103.03467645 353.54635497 618.66974784 1,497.13054854 8,556.67604661 95 96 108.18641027 368.75903037 661.97663019 1,616.89019343 9,412.34365127 96 97 113.59573078 384.88457308 708.31499430 1,746.34140783 10,353.57801640 97 98 119.27551733 301.97764641 757.89704390 1,885.94073044 11,388.93581804 98 99 135.339393191330.09630519 810.94983697 3,036.81597808 12,527.82939984 99 100 131.50135785 339.30308351 867.71632556 3,199.76135633 13,780.61233982 100 213 TABLE XXI. THE PRESENT VALUE OF |1 DUE AT THE END OF ANY NUMBER OF YEARS. g 2 3i 3 n 4 H i ^ Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. ^ 1 .98039216 .97560976 .97087379 .96618357 .96153846 .95693780 1 2 .96116878 .95181440 .94259591 .93351070 .924556'21 .91572995 2 3 .94232233 .92859941 .91514166 .90194270 .88899636 .87629660 3 ! 4 .92384543 .90595064 .88848705 .87144223 .85480419 .83856134 4 5 .90573081 .88385429 .86260878 .84197317 .82192711 .80245105 5 6 .88797138 .86229687 .83748426 .81350064 .79031453 .76789574 6 7 .87056018 .84126524 .81309151 .78599096 .75991781 .73482846 7 8 .85349037 .82074657 .78940923 .75941155 .73069020 .70318513 8 9 .83675527 .80072836 .76641673 .73373097 .70258674 .67290443 9 10 .82034830 .78119840 .74409391 .70891881 .67556417 .64392768 10 11 .80426304 .76214478 .72242127 .68494571 .64958093 .61619874 11 12 .78849318 .74355589 .70137988 .66178330 .62459705 .58966386 12 13 .77303253 •72542038 .68095134 .63940415 .60057409 .56427164 13 14 .75787502 .70772720 .66111781 .61778179 .57747508 .53997286 14 15 .74301473 .69046556 .64186195 .59689062 .55526450 .51672044 15 16 .72844581 .67362493 .62316694 .57670591 .53390818 .49446932 16 17 .71416256 .65719506 .60501645 .55720378 .51337325 .47317639 17 18 .70015937 .64116591 .58739461 .53836114 .49362812 .45280037 18 19 .68643076 .62552772 .57028603 .52015569 .47464242 .43330179 19 20 .67297133 .61027094 .55367576 .50256588 .45638695 .41464286 20 21 .65977582 .59538629 .53754928 .48557090 .43883360 .39678743 21 22 .64683904 .58086467 .52189250 .46915063 .42195539 .37970089 22 23 .63415592 .56669724 .50669175 .45328563 .40572633 .36335013 23 24 .62172149 .55287535 .49193374 .43795713 .39012147 .34770347 24 25 .60953087 .53939059 .47760557 .42314699 .37511680 .33273060 25 26 .59757928 .52623472 .46369473 .40883767 .36068923 .31840248 26 27 .58586204 .51339973 .45018906 .39501224 .34681657 .30469137 27 28 .57437455 .50087778 .43707675 .38165434 .33347747 .29157069 28 29 .56311231 .48866125 .42434636 .36874816 .32065141 .27901502 29 30 .55207089 .47674269 .41198676 .35627841 .30831867 .26700002 30 31 .54124597 .46511481 .39998715 .34423035 .29646026 .25550241 31 32 .53063330 .45377055 .38833703 .33258971 .28505794 .24449991 32 33 .52022873 .44270298 .37702625 .32134271 .27409417 .23397121 33 34 .51002817 .43190534 .36604490 .31047605 .26355209 .22389589 34 35 .50002761 .42137107 .35538340 .29997686 .25341547 .21425444 35 36 .49022315 .41109372 .34503243 .28983272 .24366872 .20502817 36 37 .48061093 .40106705 .33498294 .28003161 .23429685 .19619921 37 38 .47118719 .39128492 .32522615 .27056194 .22528543 .18775044 38 39 .46194822 .38174139 .31575355 .26141251 .21662061 .17966549- 39 40 .45289042 .37243062 .30655684 .25257247 .20828904 .17192870 40 41 .44401021 .36334695 .29762800 .24403137 .20027793 .16452507 41 42 .43530413 .35448483 .28895922 .23577910 .19257493 .15744026 42 43 .42676875 .34583886 . .28054294 ,22780590 .18516820 .15066054 43 44 .41840074 .33740376 .27237178 .22010232 .17804635 .14417276 44 45 .41019680 .32917440 .26443862 •21265924 .17119841 .13796437 45 46 .40215373 .32114576 .25673653 .20546787 .16461386 .13202332 46 47 .39426836 .31331294 .24925876 .19851968 .15828256 .12633810 47 48 .38653761 .30567116 .24199880 .19180645 .15219476 .12089771 48 49 .37895844 .29821576 .23495029 .18532024 .14634112 .11569158 49 50 .37152788 .29094221 .22810708 .17905338 .14071261 .11070965 50 FoBMULA, Present Value (1 + 0" = v"*. TABLE XXI. 213 iTHE PRESENT VALUE OF $1 I^UE AT THE END OF ANY NUMBER OP YEARS. i 5 6 7 8 9 10 1 Sh Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. 1 1 .95238095 .94339623 .93457944 .92592593 .91743119 .90909091 2 -.90702948 .88999644 .87343873 .85733882 .84167999 .82644628 2 3 .86383760 .83961928 .81629788 .79383224 .77218348 .75131480 3 4 .82270247 .79209366 .76289521 ': .73502985 .70842521 .68301346 4 5 .78352617 .7472581ir .71298618 j .68058320 .64993139 .62092132 5 6 .74621540 .70496054 .66634222 .63016963 .59626733 .56447393 6 7 .71068133 .66505711 .62274974 i .58349040 .54703424 .51315812 7 8 .67683936 .62741237 .58200910 i .54026888 .50186628 .46650738 8 9 .64460892 .59189846 .54393374 j .50024897 .46042778 .42409762 9 10 .61391325 .55839478 .50834929 .46319349 .42241081 .38554329 10 11 .58467929 .52678752 .47509280 1 .42888286 .38753285 .35049390 11 12 .55683742 .49696936 .44401196 1 .39711376 .35553473 .31863082 12 13 .53032135 .46883902 .41496445 .36769792 .32617865 .28966438 13 14 .50506795 .44230096 .38781724 .34046104 .29924647 .26333125 14 15 .48101710 .41726506 .36244602 .31524171 .27453804 .23939205 15 16 .45811152 .39364628 .33873460 .29189047 .25186976 .21762914 16 17 .43629669 .37136442 .31657439 .27026895 .23107318 .19784467 17 18 .41552065 .35034379 .29586392 .25024903 .21199374 .17985879 18 19 .39573396 .33051301 .27650833 .23171206 .19448967 .16350799 19 20 .37688948 .31180473 .25841900 .21454821 .17843089 .14864363 20 21 .35894236 .29415540 .24151309 .19865575 .16369806 .13513057 21 22 .34184987 .27750510 .22571317 .18394051 .15018171 .12284597 22 23 .32557130 .26179726 .21094688 .17031528 .13778139 .11167816 23 24 .31006791 .24697855 . 1971466a .15769934 .12640494 .10152560 24 25 .29530277 .23299863 .18424918 .14601790 .11596784 .09229600 25 26 .28124073 .21981003 .17219549 .13520176 .10639251 .08390545 26 27 .26784832 .20736795 .16093037 .12518682 .09760781 .07627768 27 28 .25509364 .19563014 .15040221 .11591372 .08954845 .06934335 28 29 .24294632 .18455674 .14056282 .10732752 .08215454 .06303941 29 30 .23137745 .17411013 .13136712 .09937733 .07537114 .05730855 30 31 .22035947 .16425484 .12277301 .09201605 :06914783 .05209868 31 32 .20986617 .15495740 .11474113 .08520005 .06343838 .04736244 32 33 .19987254 .14618622 .10723470 .07888893 .05820035 .04305676 33 34 .19035480 .13791153 .10021934 .07304531 .05339481 .03^14251 34 35 .18129028 .13010522 .09366294 .06763454 .04898607 .03558410 36 36 .17265741 .12274077 .08753546 .06262458 .04494135 .03234918 36 37 .16443563 .11579318 .08180884 .05798572 .04123059 .02940835 37 38 .15660536 .10923885 .07645686 .05369048 .03782623 .02673486 38 39 .14914797 .10305552 .07145501 .04971341 .03470296 .02430442 39 40 .14204568 .09722219 .06678038 .04603093 .03183758 .02209493 40 41 .13528160 .09171904 .06241157 .04262123 .02920879 .02008630 41 42 .12883962 .08652740 .05832857 .03946411 .02679706 .01826027 42 43 .12270440 .08162962 .05451268 .03654084 .02458446 .01660025 43 44 .11686133 .07700908 .05094643 .03383411 .02255455 .01509113 44 45 .11129651 .07265007 .04761349 .03132788 .02069224 .01371921 45 46 .10599667 .06853781 .04449859 .02900730 .01898371 .01247201 46 47 .10094921. .06465831 .04158747 .02685861 .01741625 .01133819 47 48 .09614211 ,06099840 .03886679 , .02486908 .01597821 .01030745 48 49 .09156391 .05754566 .03632410 , .02302693 .01465891 .00937041 49 50 .08720373 .05428836 .03394776 .02132123 .01344854 .00851855 50 214 TABLE XXI. THE PRESENT VALUE OF $1 DUE AT THE END OF ANY NUMBER OF YEARS. g 2 2i 3 3i 4 4.1 13 ^ Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. 51 .36424302 .28384606 .22146318 .17299843 .13530059 .10594225 51 52 .35710100 .27692298 .21501280 .16714824 .13009672 .10138014 52 53 .35009902 .27016876 .20875029 .16149589 .12509300 .09701449 53 54 .34323433 .26357928 .20267018 .15603467 .12028173 .09283683 54 55 .33650425 .25715052 .19676717 .15075814 .11565551 .08883907 55 56 .32990613 .25087855 .19103609 .14566004 .11120722 .08501347 56 57 .32343738 .24475956 .18547193 ! .14073433 .10693002 .08135260 57 58 .31709547 .23878982 .18006983 .13597520 .10281733 .07784938 58 59 .31087791 .23296568 .17482508 .13137701 .09886282 .07449701 59 60 .30478227 .22728359 .16973309 .12693431 .095016040 .07128901 60 61 .29880614 .22174009 .16478941 .12264184 .09140423 .06821915 61 62 .29294720- .21633179 .15998972 .11849453 .08788868 .06528148 62 63 .28720314 .21105541 .15532982 .11448747 .08450835 .06347032 63 64 .28157170 .20590771 .15080565 .11061591 .08125803 .05978021 64 65 .27605069 .20088557 .14641325 .10687528 .07813272 .05720594 65 66 .27063793 .19598593 .14214879 .10326114 .07512761 .05474253 66 67 .26533130 .19120578 .13800853 .09976922 .07223809 .05238519 67 68 .26012873 .18654223 .13398887 .09639538 .06945970 .05012937 68 69 .25502817 .18199241 .13008628 .09313563 .06678818 .04797069 69 70 .25002761 .17755358 .12629736 .08998612 .06421940 .04590497 70 1 71 .24512511 .17322300 .12261879 .08694311 .06174942 .04392820 71 72 .24031874 .16899805 .11904737 .08400300 .05937444 .04203655 72' 73 .23560661 .16487615 .11557997 .08116232 .05709081 .04022637 73 1 74 .23098687 .16085478 .11221357 .07841770 .05489501 .03849413 74 75 .22645771 .15693149 .10894521 .07576590 .05278366 .03683649 75 76 .22201737 .15310389 .10577205 .07320376 .05075352 .03525023 76 i . 77 .21766408 .14936965 .10269131 .07072827 .04880146 .03373228 77' 78 .21339616 .14572649 .09970030 .06833650 .04692449 .03227969 78, 79 .20921192 .14217218 .09679641 .06602560 .04511970 .03088965 79 80 .20510973 .13870457 .09397710 .06379285 .04338433 .02955948 80 81 .20108797 .13532153 .09123990 .06163561 .04171570 .02828658 81 82 .19714507 .13202101 .08858243 .05955131 .04011125 .02706850 82 83 .19327948 .12880098 .08600236 .05753750 .03856851 .02590287 83 84 .18948968 .12565949 .08349743 .05559179 .03708510 .02478744 84 85 .18577420 .12259463 .08106547 .05371187 .03565875 .02372003 85 86 .18213157 .11960452 .07870434 .05189553 .03428726 .02269860 86 87 .17856036 .11668733 .07641198 .05014061 .03296852 .02172115 87 88 .17505918 .11384130 .07418639 .04844503 .03170050 .02078579 88 89 .17162665 .11106468 .07202562 .04680679 .03048125 .01989070 89 90 .16826142 .10835579 .06992778 .04522395 .02930889 .01903417 90 91 .16496217 .10571296 .06789105 .04369464 .02818163 .01821451 91 92 .16172762 .10313460 .06591364 .04221705 .02709772 .01743016 92 93 .15855649 .10061912 .06399383 .04078942 .02605550 .01667958 93 94 .15544754 .09816500 .06212993 .03941006 .02505337 .01596132 94 95 .15239955 .09577073 .06032032 .03807736 .02408977 .01527399 95 96 .14941132 .09343486 .05856342 .03678972 .02316324 .01461626 96 97 .14648169 .09115596 .05685769 .03554562 .02227235 .01398685 97 98 .14360950 .08893264 .05520164 .03434359 .02141572 .01338454 98 99 .14079363 .08676355 .05359382 .03318222 .02059204 .01280817 99 100 .13803297 .08464737 .05203284 .03206011 .01980004 .01225663 100 TABLE XXI. 215 THE PRESENT VALUE OF $1 DUE AT THE END OP ANT NUMBER OF YEARS. 2 5 6 7 8 9 10 1 1 Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. ^ 1 1 51 .08305117 .05121544 .03172688 .01974188 .01233811 .00774414 51 52 .07909635 .04831645 .02965129 .C1827952 .01131937 .00704013 52 53 .07532986 .04558156 .02771148 .01692548 .01038474 .00640011 53 54 .07174272 .04300147 .02589858 .01567174 .00952728 .00581829 54 55 .06832640 .04056742 .02420428 .01451087 .00874063 .00528935 55 56 .06507276 .03827115 .02262083 .01343599 .00801892 .00480850 56 57 .06197406 .03610486 .02114096 .01244073 .00735681 .00437136 57 58 .05902291 .03406119 .01975791 .01151920 .00674937 .00397397 58 59 .05621230 .03213320 .01846533 .01066592 .00619208 .00361270 59 60 .05353552 .03031434 .01725732 .00987585 .00568081 .00328427 60 61 .05098621 .02859843 .01612834 .00914431 .00521175 .00298570 61 62 .04855830 .02697965 .01507321 .00846696 .00478142 .00271427 62 63 .04624600 .02545250 .01408711 .00783977 .00438663 .00246752 63 64 .04404381 .02401180 .01316553 .00725905 .00402443 .00224320 64 65 .04194648 .02265264 .01230423 .00672134 .00369214 .00203927 65 66 .03994903 .02137041 .01149928 .00622346 .00338728 .00185388 66 67 .03804670 .02016077 .01074699 .00576247 .00310760 .00168535 67 68 .03623495 .01901959 .01004392 .00533562 .00285101 .00153214 68 69 .03450948 .01794301 .00938684 .00494039 .00261560 .00139285 69 70 .03286617 .01692737 .00877275 .00457443 .00239963 .00126623 70 71 .03130111 .01596922 .00819883 .00423558 .00220150 .00115112 71 72 .02981058 .01506530 .00766246 .00392184 .00201972 .00104647 72 73 .02839103 .01421254 .00716117 .00363133 .00185296 .00095134 73 74 .02703908 .01340806 .00669269 .00336234 .00169996 .00086485 74 75 .02575150 .01264911 .00625485 .00311328 .00155960 .00078623 75 76 .02452524 .01193313 .00584565 .00288267 .00143082 .00071^75 76 77 .02335737 .01125767 .00546323 .00266914 .00131268 .00064978 77 78 .02224511 .01062044 .00510582 .00247142 .00120430 .00059070 78 79 .02118582 .01001928 .00477179 .00228835 .00110486 .00053700 79 80 .02017697 .00945215 .00445962 .00211885 .00101363 .00048819 80 81 .01921617 .00891713 .00416787 .00196190 .00092994 .00044381 81 82 .01830111 .00841238 .00389520 ..00181657 .00085315 .00040346 82 83 .01742963 .00793621. .00364038 .00168201 .00078271 .00036678 83 84 .01659965 .00748699 .00340222 .00155742 .00071808 .00033344 84 85 .01580919 .00706320 .00317965 '.00144205 .00065879 .00030313 85 86 .01505637 .00666339 .00297163 .00133523 .00060440 .00027557 86 . 87 .01433940 .00628622 .00277723 .00123633 .00055449 .00025052 87 88 .01365657 .00593040 .00259554 .00114475 .00050871 .00022774 88 89 .01300626 .00559471 .00242574 .00105995 .00046670 .00020704 89 90 .01238691 .00527803 .00226704 .00098144 .00042817 .00018822 90 91 .01179706 .00497928 .00211873 .00090874 .00039282 .00017111 91 92 .01123529 .00469743 .00198012 .00084142 .00036038 .00015555 92 93 .01070028 .00443154 .00185058 .00077910 .00033063 .00014141 93 94 .01019074 .00418070 .00172952 .00072138 .00030333 .00012855 94 95 .00970547 .00394405 .00161637 .00066795 .00027828 .00011687 95 96 .00924330 .00372080 .00151063 .00061847 .00025530 .00010624 96 97 .00880315 .00351019 .00141180 .00057266 .00023422 .00009658 97 98 .00838395 .00331150 .00131944 .00053024 .00021488 .00008780 98 99 .00798471 .00312406 .00123312 .00049096 .00019714 .00007982 99 100 .00760449 .00294722 .00115245 .00045459 .00018086 .00007257 100 216 TABLE XXII. THE AMOUNT OF $1 ANNUITY AT THE END OF ANY NUMBER OF YEARS. § 2 2h 3 H 4 ^ § ^ Per Cent, Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. tS 1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1 2 2.020000 2.025000 2.030000 2.035000 2.040000 2.045000 2 3 3.060400 3.075625 3.090900 3.106225 3.121600 3.137025 i 3 4 4.121608 4.152516 4.183627 4.214943 4.246464 4.278191 4 5 5.204040 5.256329 5.309136 5.362466 5.416323 5.470710 5 6 6.308121 6.387737 6.468410 6.550152 6.632975 6.716892 6 7 7.434283 7.547430 7.662462 7.779408 7.898294 8.019152 i 7 8 8.582969 8.736116 8.892336 9.051687 9.214226 9.3800141 8 9 9.754628 9.954519 10.159106 10.368496 10.582795 10.802114' 9 10 10.949721 11.203382 11.463879 11.731393 12.006107 12.288209! 10 11 12.168715 12.483466 12.807796 13.141992 13.486351 13.841179 11 12 13.412090 13.795553 14.192030 14.601962 15.025805 15.464032 i 12 13 14.680332 15.140442 15.617790 16.113030 16.626838 17.159913 13 14 15.973938 16.518953 17.086324 17.676986 18.291911 18.932109 1^ 1 15 17.293417 17.931927 18.598914 19.295681 20.023588 20.784054 15 16 18.639285 19.380225 20.156881 20.971030 21.824531 22.719337 16 17 20.012071 20.864730 21.761588 22.705016 23.697512 24.741707 17 18 21.412312 22.386349 23.414435 24.499691 25.645413 26.855084 18 19 22.840559 23.946007 25.116868 26.357181 27.671229 29.063562 19 20 24.297370 25.544658 26.870374 28.279682 29.778079 31.371423 20 21 25.783317 27.183274 28.676486 30.269471 31.969202 33.783137 21 22 27.298984 28.862856 30.536780 32.328902 34.247970 36.303378 22 23 28.844963 30.584427 32.452884 34.460414 36.617889 38.937030 23 24 30.421862 32.349038 34.426470 36,666528 39.082604 41.689196 24 25 32.030300 34.157764 36.459264 38.949857 41.645908 44.565210 25 26 33.670906 36.011708 38.553042 41.313102 44.311745 47.570645 26 27 35.344324 37.912001 40.709634 43.759060 47.084214 50.711324 27 28 37.051210 39.859801 42.930923 46.290627 49.967583 53.993333 28 29 38.792235 41.856296 45.218850 48.910799 52.966280 57.423033 29 30 40.568079 43.902703 47.575416 51.622677 56.084938 01.007070 30 31 42.379441 46.000271 50.002678 54.429471 59.328335 64.752388 31 32 44.227030 48.150278 52.502759 57.334502 62.701469 68.666245 32 33 46.111570 50.354034 55.077841 60.341210 66.209527 72.756226 33 34 48.033802 52.612885 57.730177 ' 63.453152 69.857909 77.030256 34 35 49.994478 54.928207 60.462082 66.674013 73.652225 81.496618 35 36 51.994367 57.301413 63.275944 70.007603 77.598314 86.163906 36 37 54.034255 59.733948 66.174223 73.457869 81.702246 91.041344 37 38 56.114940 62.227297 69.159449 77.028895 85.970336 96.138205 38 39 58.237238 64.782979 72.234233 80.724906 90.409150 101.464424 39 40, 60.401983 67.402554 75.401260 84.550278 95.025516 107.030323 40 41 62.610023 70.087617 78.663298 88.509537 99.826536 112.846688 41 42 64.862223 72.839808 82.023196 ! 92.607371 104.819598 118.924789 42 43 67.159468 75.660803 85.483892 ■ 96.848629 110.012382 125.276404 43 44 69.502657 78.552323 89.048409 101.238331 115.412877 131.913842 44 45 71.892710 81.516131 92.719861 \ 105.781673 121.029392 138.849965 45 46 74.330564 84.554034 96.501457; 110.484031 126.870568 146.098214 46 47 76.817176 87.667885 100.396501 115.350973 132.945300 153.672633 47 48 79.353519 90.859582 104.408396 120.388257 139.263206 161.587902 48 1 49 81.940590 94.131072 108.540648 \ 125.601846 145.833734 169.859357 49 50 84.579401 97.484349 112.796867 i 130.997910 152.667084 178.503028 50 Formula, Amount — (1 + ^')" — 1 l + (l + 0+... + (l-fi)«-i. TABLE XXII. ai"? THE AMOUNT OF $1 ANNUITY AT THE END OF ANY NUMBER OF YEARS. OB 5 6 7 8 9 10 d « Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. Per Cent, 1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1 2 2.050000 2.060000 2.070000 2.080000 2.090000 2.100000 2 3 3.152500 3.183600 3.214900 3.246400 3.278100 3.310000 3 4 4.310125 4.374616 4.439943 4.506112 4.573129 4.641000 4 5 5.525631 5.637093 5.750739 5.866601 5.984711 6.105100 5 6 G.801913 6.975319 7.153291 7.335929 7.523335 7.715610 6 7 8.142008 8.393838 8.654021 8.922803 9.200435 9.487171 7 8 9.549109 9.897468 10.259803 10.636628 11.028474 11.436888 8 9 11.026564 11.491316 11.977989 1 12.487558 13.021036 13.579477 9 10 12.577893 13.180795 13.816448 1 14.486562 15.192930 16.937425 10 1 11 14.206787 14.971643 15,783599 16.645487 17.560293 18.631167 11 , 12 15.917127 16.869941 17.888451 18.977126 20.140720 21.384284 12 13 17.712983 18.882138 20.140643 21.495297. 22.953385 24.622712 13 i 14 19.598632 21.015066 22.550488 24.214920 26.019189 27.974983 14 15 21.578564 23.275970 25.129022 27.152114 29.360916 31.772482 15 16 23.657492 25.672528 27.888054 30.324283 33.003399 35.949730 16 17 25.840366 28.212880 30.840217 33.750226 36.973705 40.544703 17 i 18 28.132385 30.905653 33.999033 37.450244 41.301338 45.599173 18 19 30.539004 33.759992 37.378965 41.446263 46.018458 61.169090 19 20 33.065954 36.785591 40.995492 45.761964 61.160120 57.274999 20 21 35.719252 39.992727 44.865177 50.422921 56.764530 64.002499 21 22 38.505214 43.392290 49.005739 ! 55.456755 62.873338 71.402749 22 23 41.430475 46.995828 53.436141 60.893296 69.531939 79.543024 23 24 44.501999 50.815577 68.176671 66.764759 76.789813 88.497327 24 25 47.727099 54.864512 63.249038 1 73.105940 84.700896 98.347059 25 .26 51.113454 59.156383 68.676470 79.954415 93.323977 109.181766 26 27 54.669126 63.705766 74.483823 87.350768 102.723135 121.099942 27 ' 28 58.402583 68.528112 80.697691 1 95.338830 112.968217 134.209936 28 29 62.322712 73.639798 87.346529 103.965936 124.135356 148.630930 29 30 66.438848 79.058186 94.460786 113.283211 136.307539 164.494023 30 31 70.760790 84.801677 102.073041 123.345868 149.575217 181.943426 31 32 75.298829 90.889778 110.218154 134.213537 164.036987 201.137767 32 33 80.063771 97.343165 118.933425 i 145.950620 179.800315 222.251644 33 34 85.066959 104.183755 128.258765 158.626670 196.982344 245.476699 34 35 90.320307 111.434780 138.236878 172.316804 215.710755 271.024368 35 36 95.836323 119.120867 148.913460 187.102148 236.124723 299.126805 36 37 101.628139 127.268119 160.337402 203.070320 258.375948 330.039486 37 38 107.709546 135.904206 172.561020 220.315945 282.629783 364.043434 38 39 114.095023 145.058458 185.640292 238.941221 309.066463 401.447778 39 i 40, 120.799774 154.761966 199.635112 259.056519 337.882445 442.592656 40 41 127.839763 165.047684 214.609570 280.781040 369.291865 487.851811 41 42 135.231751 175.950545 230.632240 304.243523 403.528133 637.636992 42 43 142.993339 187.507577 247.776496 329.583005 440.845665 692.400692 43 . 44 151.14.3006 199.758032 266.120851 356.949646 481.621775 652.640761 44 45 159.700156 212.743514 285.749311 386.505617 525.858734 718.904837 45 : 46 168.685164 226.508125 306.751763 418.426067 574.186021 791.796321 46 47 178.119422 241.098612 329.224386 452.900152 626.862762 871.974863 47 48 188.025393 256.564529 353.270093 490.132164 684.280411 960.172338 48 49 198.426663 272.958401 378.999000 530.342737 746.865648 1057.189672 49 50 209.347996 290.335905 406.528929 573.770156 815.083556 1163.908529 50 218 TABLE XXII. THE AMOUNT OF SI ANNUITY AT THE END OF ANY NUMBER OF YEARS. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 2 Per Cent. 87.270989 90.016409 92.816737 95.673072 98.586534 101.558264 104.589430 107.681218 110.834843 114.051539 117.332570 120.679222 124.092806 127.574662 131.126155 Per Cent. 661134.748679 67 ' 138.443652 68 ' 142.212525 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 146.056776 149.977911 153.977469 158.057019 162.218159 166.462522 170.791773 175.207608 179.711760 184.305996 188.992115 193.771958 198.647397 203.620345 208.692752 213.866607 219.143939 224.526818 230.017354 235.617701 241.330055 247.156656 253.099789 259.161785 265.345021 271.651921 278.084960 284.646659 291.339592 298.166384 305.129712 312.232306 100.921458 104.444494 108.055606 111.756996 115.550921 119.439694 123.425687 127.511329 131.699112 135.991590 140.391380 144.901164 149.523693 154.261780 159.118330 164.096289 169.198696 174.428663 179.789380 185.284114 190.916217 196.689123 202.600351 208.671509 214.888297 221.260504 227.792017 234.486818 241.348988 248.382713 255.592280 262.982087 270.556640 278.320556 286.278570 294.435534 302.796422 311.366333 320.150491 329. 154253 338.383110 347.842687 357.538755 367.477223 377.664154 388.105758 398.808402 409.778612 421.023077 432.5486541 3 Per Cent. 117.180773 121.696197 126.347082 131.137495 136.071620 141.153768 146.388381 151.780033 157.333434 163.053437 168.945040 175.013391 181.263793 187.701707 194.332758 201.162741 208.197623 215.443551 222.900858 230.594004 238.511886 246.067242 255.067259 263.719277 272.630856 281.809781 291.264075 301.001997 311.032057 321.363019 332.003909 342.964026 354.252947 365.880536 377.850952 390.192000 402.898440 415.985393 429.404955 443.348904 457.049371 472.378852 487.550217 503.170724 519.272020 535.850180 552.925092 570.513403 588.628867 607.287733 Per Cent. 136.582837 142.363236 148.345950 154.538058 160.946890 107.580031 174.445332 181.550919 188.905201 190.516883 204.394974 212.548798 220.988006 229.722586 238.702876 248.119577 257.803702 267.826894 278.200835 288.937865 300.050690 311.552464 323.456800 335.777788 348.530011 361.728561 375.389061 389.527678 404.161147 419.306787 434,982524 451.206913 467.999155 485.379125 503.367394 521.985253 541.254737 561.198653 581.840606 603.205027 625.317203 648.203305 671.890421 696.406585 721.780816 748.043144 775.224654 808.357517 832.475030 862.611657 4 Per Cent. 159.773767 167.164718 174.851306 182.845359 191.159173 199.805540 208.797762 218.149072 227.875059 237.990085 248.510313 259.450725 270.828754 282.001904 294.908380 307.707110 321.077800 334.920912 349.317749 364.290459 379.862077 396.056500 412.898823 430.414770 448.631367 467.576621 487.279086 507.770873 529.081708 551.244977 574.294770 598.266567 623.197230 649.125119 676.090123 Per Cent. 704.133728 733.299078 763.63104] 795.176282 827.983334 862.102667 897.586774 934.490244 972.869854 1012.784648 187.535665 196.974769 206.838634 217.146373 227.917959 239.174268 250.937110 263.229279 276.074597 289.497954 303.525362 318.184003 333.502283 349.509886 366.237831 383.718533 401.985867 421.075231 441.023617 461.869680 483.653815 506.4182371 530.207057! 555.066375 581.044362 608.191358 636.559909 066.205168 697.184400 729.557699 763.387795 798.740246 835.683557 874.289317 914.632336 956.790791 1000.846377 1046.884464 1094.994265 1145.269007 1197.806112 1252.707387 1310.079219 1370.032784 1432.684259 1054.296034J1498. 155051 1097.467876:1566.572028 1142.3665911638.067770 1189.06125411712.780819 1237. 623705J1790. 855956 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88, 89 90 I 91 I 92 93 94 95 96 97 98 99 100 TABLE XXII. 219 THE AMOUNT OF |1 'ANNUITY AT THE END OF ANY NUMBER OP YEARS, .- 5 6 7 8 10 tS Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. 51 220.815396 308.756059 435.985955 620.671769 1281.299382 51 52 232.856165 328.281422 467.504971 671.325510 1410.429320 52 53 245.498974 348.978308 501.230319" 726.031551 \ 1552.472252 63 54 258.773922 370.917006 537.316442 785.114075 1708.719477 54 55 272.712618 394.172027 575.928593 848.923201 1880.591425 55 56 287.348249 418.822348 617.243594 917.837058 2069.650567 56 57 302.715662 444.951689 661.450646 1 992.264022 2277.615624 57 58 318.851445 472.648790 708.752191 1072.645144 2506.377186 58 59 335.794017 502.007718 759.364844 1159.456755 2758.014905 59 60 353.583718 533.128181 813.520383 1253.213296 3034.816395 60 61 372.262904 566.115872 871.466810 1354.470360 3339.298035 61 62 391.876049 601.082824 933.469487 1463.827988 3674.227838 62 63 412.469851 638.147793 999.812351 1581.934227 4042.650622 63 64 434.093344 677.436661 1070.799216 1709.488966 4447.915685 64 65 456.798011 719.082861 1146.755161 1847.248083 4893.707253 65 66 480.637912 763.227832 1228.028022 1996.027929 5384.077978 66 67 505.669807 810.021502 1,314.989983 2156.710164 5923.485776 67 68 531.953298 859.622792 1408.039282 2330.246977 6516.834354 68 69 559.550963 912.200160 1507.602032 2517.666735 7169.517789 69 70 588.528511 967.932170 1614.134174 2720.080074 7887.469568 70 71 618.954936 1027.008100 1728.123566 2938.686480 8677.216525 71 72 650.902683 1089.628586 1850.092216 3174.781398 9545.938177 72 73 684.447817 1156.006301 1980.598671 3429.763910 10501.531995 73 74 719.670208 1226.366679 2120.240578 3705.145023 11552.685195 74 75 756.653719 1300.948680 2269.657419 4002.556624 12708.953714 75 76 795.486404 1380.005601 2429.533438 4323.761154 13980.849085 76 77 836.260725 1463.805937 2600.600779 4670.662047 15379.933994 77 78 879.073761 1552.634293 2783.642833 5045.315011 16918.927393 78 79 924.027449 1646.792350 2979.497831 5449.940211 18611.820133 79 80 971.228821 1746.599891 3189.062680 5886.935428 20474.002146 80 81 1020.790262 1852.395885 3413.297067 6358.890263 22522.402360 81 82 1072.829776 1964.539638 3653.227862 6868.601484 24775.642596 82 83 1127.471264 2083.412016 3909.953812 7419.089602 27254.206856 83 84 1184.844828 2209.416737 4184.650579 8013.616770 29980.627542 84 85 1245.087069 2342.981741 4478.576120 8655.706112 32979.690296 85 86 1308.341422 2484.560646 4793.076448 9349.162601 36278.659326 86 87 1374.758493 2634.634285 5129.591799 10098.095609 39907.525258 87 88 1444.496418 2793.712342 5489.663225 10906.943258 43899.277784 881 "89 1517.721239 2962.335082 5874.939651 11780.498718 48290.205562 89 90 1594.607301 3141.075187 6287.185427 12723.938616 53120.226119 90 91 1675.337666 3330.539698 6728.288407 13742.853705 58433.248730 91 92 1760.104549 3531.372080 7200.268595 14843.282002 64277.573603 92 93 1849.109777 3744 254405 7705.287397 16031.744562 70706.330964 93 94 1942.565266 3969.909669 8245.657515 17315.284127 77777.964060 94 95 2040.693529 4209.104250 8823.853541 18701.506857 85556.760466 95 96 2143.728205 4462.650505 9442.523288 20198.627405 94113.436513 96 97 2251.914616 4731.409535 10104.499919 21815.517598 103525.780164 97 98 2365.510346 5016.294107 10812.814913 23561.759006 113879.358180 98 99 2484.785864 5318.271753 11570.711957 25447.699726 125268.293998 99 100 2610.025157 5638.368059 12381.661794 27484.515704 137796.123398 100 220 TABLE XXIII. THE PRESENT VALUE OF $1 PER ANNUM FOR ANY NUMBER OF YEARS. e 2 2i 3 H 4 H g tH Per Cent, Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. 1 1 .980392 .975610 .970874 .966184 .961538 .956938 2 1 1.941561 1.927424 1.913470 1.899694 1.886095 1.872668 2 3 2.883883 2.856024 2.828611 2.801637 2.775091 2.748964 3 4 3.807729 3.761974 3.717098 3.673079 3.629895 3.587526 4 5 4.713460 4.645828 4.579707 4.515052 4.451822 4.389977 5 6 5.601431 5.508125 5.417191 5.328553 5.242137 5.157872 6 7 6.471991 6.349391 6.230283 6.114544 6.002055 5.892701 7 8 7.325481 7.170137 7.019692 6.873956 6.732745 6.595886 8 9 8.162237 7.970866 7.786109 7.607687 r. 435332 7.268790 9 10 8.982585 8.752064 8.530203 8.316605 8.110896 7.912718 10 11 9.786848 9.514209 9.252624 9.001551 8.760477 8.528917 11 12 10.575341 10.257765 9.054004 9.663334 9.385074 9.118581 12 13 11.348374 10.983185 10.634955 10.302738 9.985648 9.682852 13 14 12.106249 11.690912 11.296073 10.920520 10.563123 10.222825 14 15 12.849264 12.381378 11.937935 11.517411 11.118387 10.739546 15 16 13.577709 13.055003 12.561102 12.094117 11.652296 11.234015 16 17 14.291872 13.712198 13.166118 12.651321 12.165669 11.707191 17 18 14.992031 14.353364 13.753513 13.189682 12.659297 12.159992 18 19 15.678462 14.978891 14.323799 13.709837 13.133939 12.593294 19 20 16.351433 15.589162 14.877475 14.212403 13.590326 13.007936 20 21 17.011209 16.184549 15.415024 14.697974 14.029160 13.404724 21 22 17.658048 16.765413 15.936917 15.167125 14.451115 13.784425 22 23 18.292204 17.332111 16.443608 15.620410 14.856842 14.147775 23 24 18.913926 17.884986 16.935542 16.058368 15.246963 14.495478 24 25 19.523456 18.424376 17.413148 16.481515 15.622080 14.828209 25 26 20.121036 18.950611 17.876842 16.890352 15.982769 15.146611 26 27 20.706898 19.464011 18.327031 17.285365 16.329686 15.451303 27 28 21.281272 19.964889 18.764108 17.667019 16.663063 15.742874 28 29 21.844385 20.453550 19.188455 18.035767 16.983715 16.021889 29 30 22.396456 20.930293 19.600441 18.392045 17.292033 16.288889 30 31 22.937702 21.395407 20.000428 18.736276 17.588494 16.544391 31 32 23.468335 21.849178 20.388766 19.068865 17.873552 16.788891 32 33 23.988564 22.291881 20.765792 19.390208 18.147646 17.022862 33 34 24.498592 22.723786 21.131837 19.700684 18.411198 17.246758 34 36 24.998619 23.145157 21.487220 20.000661 18.664613 17.461012 35 36 25.488842 23.556251 21.832252 20.290494 18.908282 17.666041 36 37 25.969453 23.957318 22.167235 20.570525 19.142579 17.862240 37 38 26.440641 24.348603 22.492462 20.841087 19.367864 18.049990 38 39 26.902589 24.730344 22.808215 21.102500 19.584485 18.229656 39 40 27.355479 25.102775 23.114772 21.355072 19.792774 18.401584 40 41 27.799489 25.466122 23.412400 21.599104 19.993052 18.566109 41 42 28.234794 25.820607 23.701359 21.834883 20.185627 18.723550 42 43 28.661562 26.166446 23.981902 22.062689 20.370795 18.874210 43 44 29.079963 26.503849 24.254274 22.282791 20.548841 19.018383 44 45 29.490160 26.833024 24.518713 22.495450 20.720040 19.156347 45 46 29.892314 27.154170 24.775449 22.700918 20.884654 19.288371 46 47 30.286582 27.467483 25.024708 22.899438 21.042936 19.414709 47 48 30.673120 27.773154 25.266707 23.091244 21.195131 19.535607 48 49 31.052078 28.071369 25.501657 23.276564 21.341472 19.651298 49 50 31.423606 28.362312 25.729764 23.455618 21.482185 19.762008 50 FOEMULA. I 'resent Valuf > = i-\ — Xg) = 4.475241. 5 = 0.9936957, 5^ = 0.9993122, ^ = 1.099713375. As = L99725340, a(7 = —0.000298773, A^ = 0.041279507. Nap. log. s = —0.00632429, Nap. log. g = —0.0006879505 Nap. log. ^ = 0.095050036. 338 LIFE ANNUITIES AND INSURANCES. The integer results computed from this formula, terminated with only 2 persons living at the age of 101 years ; and their close agreement with the uugraduated values secured their entire adoption, with a slight change above the age 90, so that the Table ends at the age of 99 years. Here the inquiry arises, why should this Table of Male Life from the experience of Thirty Offices, spontaneously conform to Makeham's law of mortality, while previous Tables showed very considerable dcA-iations? In explanation, the following circum- stances may be noted ; first, that the observations were far more numerous than any previous collection of life insurance statistics, and comprised nearly a million of entrants. Second, that the observations were nearly homogeneous, in respect to the agencies, medical examinations, climate, nationality, education, and habits of living. Third, that the observations were reduced with the greatest care by select clerks, and the results always verified by two independent operations. Fourth, that they contain the first application of the improvement in construction by final series, already men- tioned. Fifth, that the observations are essentially free from the mis-statement of ages adjacent to 30, 35, 40, 45, 50, years, a common source of error in population returns. The present collection also determines the fluctuating nature of climacterics, or periods of health and sickness, at the different ages of life. Physicians assert the existence of periodic alternations, more or less regular, of health and disease, observed in their own persons, and in their professional practice. It is like the surface of a body of water, disturbed by the wind or other causes, and taking the form of waves, having like periods of oscillation. And as the force of gravitation by insensible degrees, causes the waves gradually to subside, so does the innate vital force tend, from conditions of disease, to the original conditions of health. Detached portions of the present statistics, will show these fluctuations ; an increase of mortality at one age, being followed by a decrease at a future age, and conversely. The manifestations prove to be so different in particular individuals, that in the average of several hundred thousands, the opposite fluctuations compensate each other, leaving in adult life, only the mathematical law before stated. Down to the present time, the statistics of life tables have been regarded in practice merely as observations, with irregularities to be smoothed down and graduated like any other set of observations. For a particular account of the earlier life tables, reference can be made to the original publications, or to Walforcls Cyclopcedia, or to the Journal of the Institute of Actuaries. In respect to the mathematical law of mortality, the celebrated Demoivre first observed that the portion of the life table between the ages of 22 and 86 years, was approximately represented by the simple formula, I = 86— a:; where I denotes the number living at the age x. This appeared in the year 1727. Different formulas were subsequently prepared or published by Lambert in 1776; by Babbage in 1823; by Thomas Young in 1826; by Littrow in 1832; by Moser in 1839; and by Orchard before 1856. But the chief interest centers, upon the formula of Gompertz, published in the Philosophical Transactions of 1825 ; and upon Makeham's modification of it, which appeared in 1859 in the Assurance Magazine or Journal, Vol. 8, p. 301. Gompertz's formula is Ix = h-gi^'', Makeham's, Ix = k.g^.S^. Here I denotes the number living at the age x, and k, g, q, s are constants to be derived from the observations. Now with reference to the usual expression of force LIFE ANNUITIES AND INSURANCES. 239 in the science of Mechanics, taking the Napierian logarithms of either formula, and diflferentiating twice, we find the relative Force of Mortality That is, in the average of a great number of adult males, the relative force of mortality, increases regularly with advancing age, in geometrical progression. But the chief value of this law, like that of the law of gravitation, is found in its relations to analysis. Integrating the last formula, we obtain by adding the usual constant, the Instantaneous Rate of Mortality, computed in Table XLIII : ^"^ ~ ^^ ~ -log.S-log.^r.log.gr.^a'. ^a.= — ^ = 1.187753 X A— +1.114833x^1. IxdX J)3^1 px f^x = Hk=l-A-Az^^^:=J^^ nearly. When the quantities are determined from four fixed ages, as 25, 40, 55, 70 years, the common Log. q is found to have almost the same value 0.04 in all collections. But the constant s changes from one table or latitude to another, and representing the uniform influence of exterior causes, may be named the climatic constant. Also in certain cases, s may be used to represent an auxiliary or relative rate of interest. Having found the law, par excelletice, of adult mortality, above the age of 22 years, we have extended the same law, in Table XXVI, down to the age of ten, to represent the very few and exceptional insurances contracted at the youthful ages. With more complete statistics, like the results of the census, we might extend the same law down to the birthday, and compare with the ungraduated values, to indicate the residual series or law in infancy and youth. A numerical example of this kind was given by the writer in the thirteenth Report of the Insurance Commissioner of Massachusetts, for the year 1868, page 106. Aided by this and some further data, Mr. Makeham happily completed his formula down to the birthday, as described in the Journal of the Institute for 1871, Vol. 16, page 344. Thus, for the entire period of Male Life, the number living I in the life table, at any age X is lx = k. g.{ax-\-a'). But in adult life, considered separ/itelv, these waves are so discordant, dissimilar and evanescent in different Tables, that they must disappear in the average of a very great number, as already shown numerically by the present statistics. SECTION III. Present Value of Life Annuities. The early history of Annuities on Lives is given in the various memoirs of the Assurance Magazine, and in Walford's Insurance CyclopcBdia, art. Annuities on Lives. Like some occasional applications in modern times, the earliest dealings rested on mere conjectural estimates. The growth of a correct system is associated with the early Table of the Prefect Ulpian, the unpublished State papers of Johan de Wit in 1671, the papers of Dr. Halley in the Philosophical Transactions of 1693, and with the noted formula of derivation, ax = vpx{l + ax+TL), published by Demoivre in 1725, and independently by Euler in 1760. The system of Commutation Columns was next perfected; and in 1849 Mr. Peter Gray in his admirable Tables and Formulce, gave new derivative formulas, and improved the form of Gauss's Tables for finding Log.(l + a,) and Log.(l— a;) from the argument Log. x. The com- putation of Tables of temporary annuities in the present work, for example, was especially facilitated by Gray's Tables. For the calculation of a large class of applied results, tl'ie process by temporary annuities or their logarithms, is found to be consid- erably shorter than by commutation columns. Further improvements relative to Annuities on Joint Lives and Survivorships have been successively derived from the formulas of Gompertz and Makeham. Also " the theory of continuous annuities " has been successfully introduced by Mr. Woolhouse in connection with Euler's formula of summation or quadrature, as described hereafter. A parallel method by Mr. Makeham, for determining all values of Life Annuities by integration, has been given under an improved form by Mr. Emory McClintock. The equivalent formula or series will be found on page 293. In order to explain how Life Annuity Tables are calculated, let us compare the Present Values year by year, as follows : Years, 1 2 3 n Annuity Certain, Value = v -\- v^ + v^ -\- + V^ + . . . Life Annuity, Ux = -^— • v + -j- • v^ + -j- • v^ + ... + -^— -tf^ + ... Ix ix ix 'x Here r, r*, v^, etc. represent the present value of $1 due at the end of the Ist year, of II due at the end of the 2d year, of $1 due at the end of the 3d year ; and so on to the last. If i denotes the rate of interest, v = - — -:. 1 + 1 In the lower formula, -^ is the proportion by the life table, or probability of Ix surviving the 1st year to receive II whose present value is v ; hence by Principle 7, Sect I, the product -^ • v is the mathematical Expectation relative to the Ist year. LIFE ANNUITIES AND INSURANCES. 241 Again, -— is the probability of surviving the first two years to receive the next Ix payment of $1, whose present value is vK Hence the product -^ • v^ is the mathemat- Ix ical Expectation relative to the 2d year; and so on. The sum of these yearly Expectations is the present Value of the Life Annuity, Table XXVIII. When the series extends over but 5, 7, 10 or n years, the sum is called a Temimrary Life Annuity^ or Temporary Annuity, denoted by a^^S^, to continue n years if the annuitant shall so long live. When the series extends to the end of the life table, near 100 years, the sum is termed a Life Annuity, denoted by ax. " With present pay- ment," the expression is \-\-ax, given by logarithms in Tables XLIV-XLVII. From another point of view, let us suppose l^ persons of the common age 30 to buy each a Life Annuity of $1. By the life table Z31 survive to the age 31 to receive each the first payment of $1, of which the present value is v, and total present value, /31 • V. In like manner l^ will, on the average, survive to the end of the second year, when the present value of the second disbursements will be l^-v^ ', and so on. Dividing the sum of all the present values by the original number of persons /30, to find the average, ax, we obtain the same formula as before, «30 = ^30 Recurring to the general formula, let us observe that on multiplying and dividing by /a;+i, which does not alter the value, fa;-*-2 *a; + l ^a;+2 ta;+3 fx+1 ^x+2 ^+3 ^— = -7— -7 =i?a;-i>a;+i; 'T~ — ~i~ ' 1 T — = Px-Px+l-Px+i'y ••• ix tx I'X+l IX ix <'x+l tx+2 Hence when the annuity is expressed in annual probabilities, ax = V.px + v'^.px.px+l-\-1^'Px'Px+l'Px^2+ Deferred Annuity. Let the present value of $1 annuity deferred m years on a life now aged x, be denoted by "^Ux. At the advanced age x + m, the value of a life annuity will be ax+m. Multiplying this by v"^ to reduce it to present value, and by -^^, the probability of attaining it, we have the value of the deferred annuity, „ 'x+m ^ ""(ix = -J— . V"' . ax^m. Ix By examination, it will be seen that the sum of the temporary and the deferred annuity is equal to the whole life annuity; that is, a^ + "*ffx = ttx ; ax"* = ax— '^ax- Joint Lives'. The probability that two lives aged x and y, shall both survive one year, is ^^^~—, being the product of their separate probabilities. Hence by the same reasoning as for single lives, the present value of $1 annuity payable at the end of tach year, during the joint continuance of both lives, is as follows: „ _ Ix+l'^y + i .V-\-lx+2.ly-i.2.v'^ \-lx^^' ^y+S-V^ f axy — j — J tx' iy For three or more joint lives, the formula depends on the same simple principle. 242 LIFE ANNUITIES AND INSURANCES. Annuity on Three Lives hy Simpson's Rule. From Tables for one and two lives, the annuity for three lives was approximately determined by the author T. Simpson in 1701, as follows : " Take the value of an annuity on the joint lives of the two oldest, and find the age of a single life of the same value. Then find the annuity on the joint lives of the one just found, and the remaining life of the three, and the result, (which Francis Baily diminished by .05), will be very nearly the true value." Table, page 260. Professor De Morgan proved that Simpson's method would be strictly true, if the Table followed the law of mortality of Gompertz. {Journal, Vol. 10, p. 27.) Half-yearly and Quarterly Aiinuities. When the annuity is changed from annual to half-yearly installments, add .25 ; and when to quarterly installments, add .375 to fifi \ the tabular value of the annuity. Or, generally add ~ , when the installment is payable m times a year, as demonstrated on page 70. Thus the value $14. 662 of II annual payment is changed to 14.912 for half-yearly installments of $0.50, and to 51.037 for quarterly installments of $0.25. The more exact formula by Woolhouse for the value of an annuity payable m times a year is {Journal, Vol. 11, p. 327), m — 1 m^ — \, .. «-+-2^-l2^('^ + ^)- And Mr. Spra^Tie's formula for the value of a complete annuity, payable by m equal installments in each year, with a proportionate part to the day of death is {Journal, Vol. 13, pp. 377-380), . i.'iiiUi-— U-- "^^ 12 / ^ 2m/"^12m2* Here ii-= — -r- log. Ix ■= {lx—\—lx+\)-^'^lx nearly ; 6 = Nap. log. (1-f 0' ^"^ * = rate of interest. The values of /*, d are given in Table XLIII. From the present value for 1, the value for any other sum is found by multiplying the value for 1, by the actual annuity or yearly income. The further consideration of Life Annuities is deferred till after the system of Commutation Columns described in Section V. SECTION IV. Expectation of Life and Probable Life. The phrase Expectation of Life has long been used to denote the after-life- time, or average number of years and the decimal part, lived after a given age. Let Ix denote the number living shown in the life table at the age x. Assuming the deaths of each separate year to occur uniformly, the number of years lived in the first year, second year, third year, will be Ix- ^lx+l . lx+l-\-lx+i . lx+2-\- Ix+S , '2 "^ 2 "^ 2' +•••■ Taking the. sum of the contiguous terms continued to the oldest age or end of the life table, and then dividing by Ix to give the average, we obtain the common formula for the expectation of life, denoted by e : 1 , ix+i-\-lx+si-\-lx+3-\- ' • • • l. LIFE ANNUITIES AND INSURANCES. 243 Hence the Rule : Divide the sum of the tabular numbers living above the given age, by the number living at the age, and to the quotient add ^ year. For example, the expectation of life at the age 95 by Table XXVI is found to be eyn = IH j^ = 1.338 or 1^ years, nearly. To demonstrate the derivative formula, let a: + l be substituted in place of re in the formula above; then clearing of fractions, and subtracting one equation from the other, we find ex+i = o + 1 , 'a; + 8+^a!+3 + ^a;+4 + 2 lx+1 In the formulas of Section III for Life Annuities, if we make v = 1, that is, 1 = 0; we obtain what is termed "the curtate expectation of life." In the case of single lives, it is the quantity here denoted by ex—^, and expresses thei average number of complete years to be lived, or of annuities received, or of annual premiums to be finally paid. Again making v = 1 in the formula for Joint Life Annuities, we obtain the curtate expectation of the jomt life, denoted by exy—\ ; where the complete expectation is denoted by exy. According to " the continuous theory," the tabular expectations usually given, slightly exceed the true values, by one-twelfth of p^x- For comparing the longevity of different Life Tables, let the particular expectation of life which is just equal to the age past, be termed the even expectation. By the Thirty Offices Table XXVI, at the age 33.37 years, for example, the even expectation or mean duration of future life is also 33.37 years. In like manner, the even expecta- tion of the Carlisle Table is 32.62 years ; of the table of Des Parcieux 32.49 ; of the Equitable Society by Davies 32.38 ; of the British government Male Annuitants 32.00; of the Female Annuitants 34.58 ; and of the Northampton Table by Dr. Price 28.86 years. The shorter the even expectation or half-interval, the shorter will be the whuie and other connected intervals of life in the same Table, as shown in Tables XXVI and LXXX. A prevalent error on the part of persons imperfectly versed in these matters, con- sists in calculating life annuities and other similar results, from the expectation of life. At the age of 29 years, for example, the true present value of a life annuity by the Carlisle Table at 4 per cent, is 16.997 ; as found empirically from the expectation of hfe, it is 18.665, which is too great by one-tenth part. The nature of this erroneous practice may be further explained by comparison of the preceding formulas for life annuities and for expectations, which are evidently designed for separate objects. Probable Life. At any given age, " the probable life " or " vie probable " denotes the future period, which there is the chance ^ of surviving, and the chance ^ of not surviving ; these chances being equal. Let x and x-{- 7" denote any two ages of the life table, connected by the relation, \lx = Ix-^t', then will the difference of ages T denote the probable life at the age x. The column of Probable Life in Table XXVI was computed by the preceding formula. In this Table at the age of thirty, for example, the probable life is 38.48 years, which is 2.63 years more than the expectation of life at that age. But at the age of 57|, the expectation and the probable life become equal to each other in the common period of 16.61 years; and at older ages, the expectation exceeds the probable life. 244 LIFE ANNUITIES AND INSURANCES. A remarkable agreement of the probable life of males and females combined, was observed by the distinguished Professor Quetelet, in comparing the tables for England by Farr, Sweden by Berg, Belgium by Quetelet, Netherlands by Baumhauer, and Bavaria by Hermann. All Europe, he observes, as will be seen by the following columns, follows a law of mortality nearly the same ; and the slight differences between the results above five years of age, appear due to particular advantages arising from opulence, and a more regular mode of life. {Physique Sociale, 1869, Vol. 1, page 307.) YEARS OF PROBABLE LIFE IN EUROPE. Ages. Sweden. England. Belgium. Netherlands. Bavarla. Ate RAGE. 51 45 42 34 27 40 5 56 55 53 53 53 54 10 53 51 50 50 50 51 15 48 47 46 46 45 46 20 43 43 43 42 41 42 30 35 35 35 34 34 35 40 27 27 27 26 26 27 60 19 20 20 19 18 19 60 18 13 13 12 13 13 70 7 8 7 7 7 7 80 4 4 4 3 4 4 SECTION V. Comimitation Tables. Statistics are ancient, but their scientific forms are modem, as Professor Quetelet has observed. The origin of commutation columns, or preparatory researches from which the system was at length perfected, were published by William Dale in 1772, William Morgan in 1779, .John Nicholas Tetens in 1785, George Barrett in 1812, and Griffith Davies (who added the M and R columns), in 1825. The work of David Jones in 1844 was the first to give an extended series of commutation tables ; and his notation is still followed in the Journal of the Institute of Actuaries. Indeed all subsequent writers retain the D, 0, M, R columns on the same plan. But it is very needful to bear in mind, that in the Tables of Dr. Farr, Chisholm, Henry and most American actu- aries, the N and S columns only are shifted down one space ; so that Na; and Sx denote the same quantities, which David Jones had designated by N»i_i and S^-i- However, by simply adding 1 to the suffix of N^^-i and S^-i, the formulas of Jones and of the Journal will evidently coincide with the change of Dr. Farr, which under present circumstances, is adopted in this collection. The name of Commutation Tables was proposed by Professor De Morgan in 1840, He had noticed the singular property, that on transposing or commuting any two ages, say 30 and 34, whatever may be the present age, if less than 30, the person might now give up IDm due at the age of 30, to receive IDao if he lives to be 34. Similar prop- erties belong to the N and other columns. Journal, Vol. 12, p. 332. COMMUTATION TABLES. 245 Proceeding now to explain the D and N commutation columns, let us refer to Sect. Ill, and multiply both numerator and denominator of the expression for the Life Annuity by the same factor v^, which does not alter the value, Ox = Ox Ix.V^ Dx ~ D«, • Thus the formula is entirely resolved into similar elements, of the type 'Dx=^lx-if^. And Na5+i denotes the sum of the terms of the numerator continued to age 99, or to the end of the life table. The letters N, D, appear to be chosen from the words Numerator and Denominator. A Temporary Life Annuity for 3, or more generally n years, will evidently have the expression, where the exponents 3, n are used, (not in the Algebraic sense of powers), but simply to denote so many years after entry, < = Da; + l + Dx+2 + Da;+3 _ Na;+1 — Na!+4 ^ _ Na;+i — ]S"a;+n+l D;. - Da, ' ''«'- D^ • Again, referring to Section III, and multiplying both numerator and denominator by If', the Deferred Life Annuity takes the surprisingly simple form, n/» — ^«+» • ■^''"" « _ ^x+n Nar+n+l _ '^x+n+1 lx-1^ yJx Dar+n Dx The results already described, will exemplify the superior advantage of commuta- tion columns, for determining the value of Life Annuities, whether immediate or deferred, for the whole life, or a portion of it. So manifest have these advantages appeared, that all the standard formulas of Life Annuities and Assurances have been resolved into similar elements. In all, six columns are required ; whereof D, N, S are termed annuity columns, and 0, ,M, E are termed insurance columns. These columns are connected with each other, by the following simple relations : Let V = r, where i denotes the rate of interest ; 1+* dx = Ix — ^^+1, where x denotes the age, d the deaths, / the living in the -life table ; then, 'Dx = Ix.v^; Cx = dx. v^+i ={lx- Ix+i) v»+^. .Na, = D«, + D«,+i + Da,+2+ Ma,= Ca,+ Ca,+1+ 0«,j.2+ • • Sir = N'«, + N"a,+i + N«,+2+ .... Ra, = Ma, + Ma,+i + Ma,+2 + .... Each series is to be continued to the end of the life table. By writing x+1 ia place of X, and then eliminating, we find, Nfl, = Da, + N«,+i ; Ma, = Co, + Ma,+1. Sa, = Na5+ Sa,+1 ; Ea, = M«, + Ea,+1. In the former equations for S and R, substituting in place of N and M the series of D or C which they represent, and adding, we have Sa, = Da,-l-2Da,+i + 3Da,+2+ . . . . ; £«, = Ca, + 20«,+i + 3Ca,+8+ .... 246 LIFE ANNUITIES AND INSURANCES. Note. — By differentiating the life annuity in Section III, etc. with respect to v, we also have ^ ^Sx+i, d/Nx\ Sx. dm^\ Rr dv «Dx ' dv\v^^} vx' dv\vx/~'cf+^' Again, for temporary periods, Na, — Na,+„ = Da, + Da,+l + + Dx+t^h Mflj — Mx+n = Cas + Ca;+i + . . . . + Caj+n—l. = Da, + 2Da,+i+ +n. Da,+n-l+W. Na^Hi. Ee— Ea>+n= Ma,+ Ma.+1 + • • • +M«,+„_i = Cfl, + 2Ca!+l+ +Jl.Cx+n-l + n.'M.g>+n. When C, M, R are not tabulated, their values can be derived from v, D, N, or S, by the following relations : Making x to represent in separate equations, x, x-{-l,x-{-2, .... and taking the sum of results, M«, = f. No,-Na,+i = Da,-(l-i;)Na> Again making x to represent x,x-\-l,x + 2, and taking the sum as before, Ea) = 'y . Sxy—T>x+l.y+l' Na^y = Da:y + Da;+l.y+l+ . . . . M^y = Cxy+Ca;+l.y+l+ . Sajy = Na:y + Na;+i.y+i4- . . . . Eary = Ma:y + Ma;+i.y+i+ . . . . • For two Equal Ages, Dxx = IxlxV^ = Ix^x, ''^xx = ^xx 4- Da;+i.a;+i + Two joint lives of unequal ages x, y have the same life annuity as two equal lives, w, u, defined in one way by the equation axy = auu, and in other ways under Make- ham's law of mortality, 'by the relations, PxPy=PuPu, Pu=- VpxPy', (f-^qy — %q^, w = 2; + Cor. Table LXV. Since any two joint lives are thus reducible to the case of two equal ages, the corresponding D and N columns, annuities and premiums may be found in Tables LXVI-LXXIV. The radix of these Tables was taken as 100,000 at the age of ten; or the usual D and N values were divided by the constant factor Dio- All the preceding Commutation formulas, for single life, become applicable to joint lives, by merely extending the suffixes to every life. For illustration, multiplying numera.tor and denominator of the formula in Section III for joint lives, by v*, or by x+y V 8 , we obtain in either way, ttxy Da;+i.y-|.l-|-Dx+2.y+2-f- • • ' • N^+l^y+1 iJxy Djcy 248 LIFE ANNUITIES AND INSURANCES. The law is perfectly regular for two, three, or any number of joint lives ; thus, l ^g+l.y+l.z+l 4' Da;+2.y+2.z+2+ • • • • ^x+\.y+l.z+l O^CyZ — =R ■ — TT . y ^xyz ^xyz Ma^ = Darys— (1— r)Na;y2 = ■yNxys— Na;+l.y+1.3+l. Note 1. — It will be proper to observe, that formulas on the plan of De Morgan, when applied to the order of survivorship will involve the square root of c. Thus the single premium for an insurance of $1 " on x against y," that is, payable on the death of x provided y be then living, is The annual premium will be found by simply changing Dxy to N^y. In each term of the numerator, the sum of the suffixes added to the exponent of i o counted as 1, gives the homogeneous sum x + y + 2. Note 2. — Repeated additions of At). There are two methods of correcting the final decimal. For example, at 4 per cent, /.n is 1.982,966,6607, which falls between the six-decimal values 1.982,967 and 1.982,966. Firstly then, if no more than ten or fifteen results are to be computed so as to be exact to six decimal places, the computation can be made with four extra decimals, that is, with ten places, retaining only six at the end of the process. But if /» is to be implicitly added a hundred times in computing so many results exact to six places, the computation can be made with five extra decimals, in order that the first six places may be accurate in all the results. Secondly, the computations can be correctly made with six decimals by suitable changes of the sixth decimal. In using 1.982,967 for 7.v, we commit an error of .338 in the sixth place, in excess ; and in using 1.982,966, we commit an error of ,661 in deficiency. A compensation will ensue by using the endings 7 and 6 in frequency, inversely as those errors ; that is, as 661 : 338, or 2 : 1 nearly. Hence the sixth decimal of ~Av at 4 per cent, should be taken as 7, 7, 6, 7, 7, 6 in order. At 5 per cent, the ratio is 7 : 3, so that the sixth decimal of Ic should be 9, 9, 8, 9, 9, 8, 9, 9, 8,9, when the series recommences. An adjustment can be made at every tenth value. These corrections have already been applied to the sixth decimal place in Table XXXIII, and in all the other regular Tables. SECTION VI. Single and Annual Premiums of Insurance. By general custom, the sum insured is assumed in theory, to be payable at the end of the year of decease. The age at which the insurance is eflfected is often termed the age of issue, or the age of entry. The contract between the insuring party, and the insured is named the policy. The first premium thereon is paid at the time when the contract is made ; and the subsequent payments of premium are due at the commence- ment of each policy year. The premiums, however, may be single, annual, half-yearly or quarterly. The single premium pays for the insurance in full at the beginning, in one payment. "By '^ deferred premium" is meant, the remaining semi-annual or quarterly pre- miums to be paid during the current policy year. In the estimate of liabilities, these deferred premiums, with some reduction for the expense of collection, and for discon- tinuances, are accounted as^ Assets ; and in case of death, the Company usually deducts them from the policy. To provide for expenses, fluctuations of mortality and other contingencies, the net premium of theory is augmented in practice, by a margin or loading, such as one-fifth, one-fourth or one-third, and is then termed the gross or Office premium. SINGLE AND ANNUAL PREMIUMS, 249 In order to explain the computation of net premiums, let Ix denote the number living at the age x in the life table, and let i and v represent the given rate of interest. 1. SINGLE PREMIUMS. Let Aa, denote the single premium at the age of entry x, to insure $1 payable at the end of the year of death. Then by the principles of the system of Probabilities, . _ {l x—1x+l)f>^-{lx^rl — lx+^V^ + lx+i—h+^^-\- Ax — ; J -. ix Here the tabular probability of the life failing in the 1st year is — -, or y- ; Ix l-x and this multiplied by the sum at risk $1, and again by v, to reduce to present value, gives the first term. In like manner the remaining terms express the present value of the mathematical expectation for the 2d year, the 3d year, and so on to the end of the life table. This sum is therefore the present value of the life insurance. Writing c4 for its equal Ix—lx+i, and similarly for other ages, then multiplying both numerator and denominator by if, which does not alter the value, the fraction is changed to the commutation terms : A-x — Ixlf Ma, A, = l_(l-^& In the first formula, the positive and the negative terms are easily identified with the series for the life annuity, so that we have also A^ — v—ax+vax = l — {l—v){l + ax) = —-^^. 1 -ft To facilitate practical applications, the single premium for life is already computed in the four different forms of Tables XXX and XXXI, L and LI. 2. ANNUAL PREMIUMS. Instead of a single premium, the payment for insurance is more usually made by equal annual premiums at the beginning of each policy year. Let Px denote such annual payments on $1 insured. The present value of all these premiums being equal to the single premium, we have for common life policies, on $1, Px(l +«a:) = Ax, whence Px = :f—^-. 250 LIFE ANNUITIES AND INSURANCES. For common life policies, the annual premium P or rr is already computed in Table XXIX, for each 11000 insured. 3. INSURANCE BY A LIMITED NUMBER (») OP ANNUAL PREMIUMS. The rule proceeds on the same general principle as before. That is, " divide the single premium by unity added to the present value of a temporary annuity for one year less than {n) the number of premiums which are to be paid": P.= Aa; M« l+Oa,"-! Na, — Na,+„' Example. A person at the age of 39 years insures his life for $7000 by ten equal premiums, to be paid annually at the beginning of the first ten years ; required this premium when interest is 4 per cent. : Table LI, - - - - AA39 = 1.552218 Table XXXVI, - - aM39 = 3.796864 « XLIV, - A(l+a^i) = 0.908205 ^39= 293,818.6) ^49= 151,634.5) X = 5.152851 ...2.644013 ...2.644013 « LXXXII, - - Pa, - 0.0440568 142,184.1 . Pa, - '0.0440568 X7000 x7000 Annual Premium - - - - = 1308.3976 Annual Premium - - - - =$308.3976 For five, ten, fifteen or twenty payments, the premiums are already computed in Table LXI, for each $1000 insured. Note. — If a special payment is made at the beginning, in order to reduce the future annual premiums, it can be regarded as a single premium, for one part of the sum insured ; and the annual premium will be computed on the remaining part. 4. TEMPORARY INSURANCES. Eequired the single premium to insure $1 for 7 or w years. By reference to the explanation of the first three formulas of this Section, it will be seen that the single premium will be the sum of the first m terms of the series there described ; thus, JXnf. Cx4-C!ar+1+ . . • +Ca;+wi— 1 ^x — ^x+t Dx Da The equivalent annual premium is obtained by dividing the single premium by 1 plus the temporary annuity for one year less : ie = l+«: m— 1 N-E+i— Nar+m+1 ^x — ^x+m n ttx l+«! n— 1' pw» Ma; — Ma;+ffi Da; — JJx+m /-, \ ^x — '-i^ ^ij- — -vr _-»T - — \^ '^r ISx — ^x^m ^x — -lNx+»» SINGLE AND ANNUAL PREMIUMS. 251 The Premium for Temporary Insurance on each $1 for One Year only, is Car /^ N Da;+1 PI ^X /^ N ^x+1 X = p- = v{l—px) = vqx = V ^. For example, by Tables LII and LIII, at the age of 35 years, on $1000 insured for 7 years, the single premium is $51.70, or the annual premium is $8,486. 5. PURE ENDOWMENTS. The single premium or present value of $1 to be received at the end of m years, it the party now aged x be then living, is Ix i^x The annual premium for the same, payable at the beginning of each year, unless the risk is voided by death, is found on the principle before described : Ptn Aflj ^x+m *-, l+a»i-i Na,— N«,+«' Table LXIII, APS' = 0.-ABSV on $1 Pure Endowment. For example, at the age 16, the annual premium is $173.88 on $1000, payable at the age of 21 if the person be then living. The same result is also found by sub- tracting the premium of term insurance. Tables LII and LIII, from the premium of endowment insurance, next to be described. 6. ENDOWMENT INSURANCES. The sum insured is payable on attaining a specified age, or sooner in the event of death. By simply adding the single premiums of temporary insurance and of pure endowment for the term of m years, after the age of entry x, we have at once the single premium of Endowment Insurance on $1 : * (w») Mju — Mas+^-j-l^as+m -. /-. \/i , m— 1\ Aaj = =;r = \ — {l—v){l-\-ax ). To find the equivalent annual premium, we divide by 1 plus the annuity: For a limited number {n) of equal annual premiums to be paid at the beginning of the first {n) years of the whole period m, the divisor is changed as below : Pn Mas — Ma!+w» + Da5+f» „^ *-, a, = — n~:zn ' °° ^^' ^» — -i-ia!+n The various species of Endowment Premiums are already computed in the seven Tables LIV to LX. m LIFE ANNUITIES AND INSURANCES. Computation of Tables. — By taking the difference of single premiums, and observing that Mxfm+i— Mj+TO = — Cj+m = — i'I>j+OT + I*x+in-ri, we obtain the simplified annual difference : ■ (m) . On+1) _ {l — v)'Dx+m •»« — -O^c — =r . This difference can be easily computed by the aid of two moveable slips of paper, the one con- taining /.D, and the other 'a{\—v) + AD. When the period of insurance is only 1 year, the premium is v, as shown in Table LIV. From v, we subtract the difference (1 — v)T>x+i-^^x to give the single pre- mium for the term of 3 years. From this, we subtract (1— cjDa;-r2-i-Ds to give the single premium for the term of 3 years ; and so on. But in single Examples, the shortest method will be to find the logarithm of 1 plus the tem- porary annuity for one year less, in Table XLIV-XLVII. This will guide to the required premium, by mere inspection, in the next or Conversion Tables. 7. DEFERRED INSURANCES. On the principle of deferred Annuities in Sections III and V, and by Section VI, ,the present values of the temporary and of the complementary deferred insurance, added together, equal the present Value for the whole life. Hence by subtraction, the present value or single premium to insure a person now aged x, for the period of life which may remain after the age x-\-m, will be on each $1 insured, P,» -M^g+wi ^x+m — (1 — v)^x+m The equivalent annual premium payable m years, will be m^^ = ^^^-^ , on each $1. The annual premium payable during the whole life, on $1 insured, is ^X x+m -^x Na; 1+ffa; For a Deferred Temporary Insurance, or to be deferred m years, then to continue in force during r years ; if the annual premiums are payable during m -^ r years, we have on each II insured. Single Premium = ^x^m-^x^m^r ^ Annual Premium = -Na; — ^x+m+r 8. RETURN PREMIUMS. Required the net annual premium P to secure a Pure Endowment of $1 at the end of m years, if the life now aged x should live so long, or the Return of all Premiums paid, at the end of the year of death, if it occur before that time. This plan may be regarded as an endowment of II, added to a series of m deferred temporary insurances of P. By making the collective annual premiums equal to P, then substituting R, and reducing the equation, we find SINGLE AND ANNUAL PREMIUMS. 253 =P., T>:c^m + I'xO^x+'M.jc + l + + Mjc + m-J—rnMjc + m) _ '-^x + m . Also by changing T):c^m to ^x+m+\ we determine the annual premium, pay- able from age x to age x + m — 1, to provide a Deferred Life Annuity of $1 per annum, first payment at age x + m+l. The premiums paid are to be returned without interest, six months after death, in the event of death before the age x+m. Or if the premiums are not to be returned, cancel R and M in the above formula. By the same mode of solution, the Single Premium of a Pure Endoiument of $1, with return of the premium six months after death, if it occur before the age X i-m, will be D^— (M;,— M^+;„) Required the gross premium to insure $1 for the Whole Life, and the return of gross premiums equal to g times the net, without interest, six months after death. Let fifP denote the gross annual premium to provide for the ordinary insurance of $1, together with the series of deferred insurances of grP, all for the whole life. Then, as before described: p _ gM:^+ gT':c {M:c+'M.^+i+ ■ • • ) __ gM.:^ + g'Px^x . ^j. g'^x . Annual Premium gPx = ^^ p> Single Premium gA^ = '^ 1, > by similar solution. i-/x -'•"-jr By making (/ to be 1, the solutions will apply to the case of net return premiums, without interest. Again, if the insurance of $1 for the whole life is to be secured by m Limited Annual Payments of P^, with return of the gross premiums, six months after death, or at the end of the year of death : g'^x 9^^ (N^— R^)— (N;^+„— R^+^) 9 INCREASING AND DECREASING SCALES OF PREMIUMS. Suppose the annual premium to increase b}'^ a certain fraction of the first premium every n years, and after r such additions, the premium to continue constant during the remainder of life. Required the annual premium during the first n years. On each $1 insured let P denote the first annual premium, and q its pro- portional increase, added at the end of every n years. The present value of all the premiums P, continued for the whole life, is F{l+ax). And the present value of all the increments P^isthat of so many deferred 254 LIFE ANNUITIES AND INSURANCES. life annuities. By equating the expression of the single premium to the sum of these present values, and then reducing the equation : A _ M? — P ^^a' + g(Na,-t-raH-Na. + 2n+ +Na, + r») p _ M« ■■ \ Instead of a proportional increase q of the premium, let there be an absolute increase h in every n years, the premium to remain constant after rn years, to find the first premium Pa;. We have only to substitute h in place of its equal Yq in the primary formula, then reducing, we find again on $1 insured : ^ _ Ma, — ^(Na,.,.n + Na,+2w4- +N^+m) When the scale of premiums is to be decreasing, let the sign of q and of h be reversed in the given formulas. The above method is evidently applicable to premiums reduced or increased arbitrarily. But as David Jones has observed, the value of q or h should be chosen so as to make the premium for the first interval more than the premium for a risk to be determined at the expiration of that term, as the party assured, has the option of continuing or discontinuing the risk. In the case of annual premiums diminished at the end of every successive year, by the mih. part of the first premium, so that after m payments, the policy is paid up, the first solution will apply by making w = 1, g = , and substituting S for its equal. Thus the first premium on $1 insured, will be M. P„ = 10. INCREASING AND DECREASING INSURANCES. The single premium for a temporary Insurance for m years, commencing at la and increasing %h each year, will be a(Ma, — M^-,.m)+^(Ma, + i — Ma, + ;ra) + 6(Ma,+2 — Ma,+m)+ • » • +hQS.x+m-l — '^xArm) Da, This aggregate is evidently made up of the present insurance of $« together with the series of deferred temporary insurances of %b. By substituting R for its well known equal, we have the required formula : Single Premium ^«(M.-M..^) + 3[R..i-R..^-(m-l)M..^] By merely reversing the sign of 5, we obtain the single premium for a similar insurance, commencing with %a and decreasing %h annually. SINGLE AND ANNUAL PREMIUMS, 255 lu either case, the divisor to give the annual premium will be ' 1. 7/*— 1 + ax , or D. By extending x-\-m to the end of the life table, we evidently have for an insur- ance commencing at %a and increasing $b each year. Single Premium ^ ^M^^lkll. Dx Annual Premium = ^M£+*_^i. Nx When the similar insurance is to decrease $b each year, we have only to make b negative. Finally, making a = b = 1, we obtain the more simple formulas for a life insurance commencing with $1, and increasing $1 annually : Single Premium = ^r • Annual Premium = ^' Ux -Na; 11. PRICE OF ANNUITIES SECURED BY LIFE INSURANCE. An ordinary life annuity of $a per annum is to be sold. Required the price $s which will allow the purchaser a given rate i of interest, besides p&,ying the premium of life insurance to secure the return of his outlay after the death of the annuitant. Solution. — The purchaser must insure the price s, and a final annuity a, both payable at the end of the year of death of the annuitant. An annuity a will then be receivable at the end of each year, for which, the premium will be payable at the beginning of the year. But each premium may be transferred from the beginning, to its equivalent at the end of the year, by adding the year's interest. When this equivalent is subtracted from the annuity a, the remainder must be sufficient to pay «i the current year's interest on the price. Accordingly, if p denote the office annual premium to insure $1 for life, then the premium payable will be p{8 + a). Adding the year's interest, and subtracting from a, we have a—p{8 + a) (1 + i) = si. l-p{l + i) Hence, the Price s = a. i+p{l+i) The Rate of Interest j = ^^1±^J . s+p{8 + a) In case the Problem is changed in any of its conditions, a corresponding change will be made in the solution. Thus, if the annuity a is to be continued to the end of the year of death, the purchaser need only insure the price s, and the solution will be %+p{l + i) 8(1+P) Or if the annuity a is to be paid at the beginning of the first, and each subsequent year < of life, the price « alone need be insured, and the equations will evidently be / \/-< . % • cUl + t) . a—p8 (a--^«)(l+j) = M; 8= .^ ,. .\ ; = :jr-r^ 256 LIFE ANNUITIES AND INSURANCES, 12. INSURANCE ON FEMALE LIFE. From the Observations on Female Life in Table II, a Life Table has been con- structed and graduated on the plan mentioned at page 237. The usual columns are given in Table LXXVII, in which the column of annual mortality q exhibits the very singular feature of the almost unifonn value 0.011 from the age of 20 to 50 years. The principal cause is indicated on page 34. On account of this and other peculiarities, a simple correction of the age has appeared sufficient for entering and employing the general Tables for male life, to give the values for female life. Preparatory to this object, the 4 per cent, commutation and annuity columns were computed in Table LXXVIII. The simple Formula for determining the correction or column c, and an illustrative Example are given on the first page of this Table. The sum of the actual Age and Correction may be termed the Equivalent or Relative Age. And the Tables of Male Life entered with this Relative Age, give the true Annuity and Premium for Female Life. On account of the smallness and fluctuating nature of the corrections above the age of 39, and the irregularities of the data at the oldest ages, it appears that above the ar/e of 39, the "rating up," or correction of the age can safely he omitted. That is, the actual age when above 39, may be taken as the Relative Age of Female Life. In relation to Months, and under the age 39, the actual age, from the birth-day to the (^ate of entry, is to be expressed in years and months, as well as the correction, and their sum, taken to the nearest integral year, will be the Relative Age. Thus if a female life at entry has the age 25 Y. 8 M., and from Table LXXVIII, or page *226, the correction is 4 Y. 3 M., the nearest Relative Age will be 30 years. 13. INSimkNCE ON JOINT LIVES. Required the single premium to insure $1 during the joint existence of two lives, X and y, the insurance to be paid at the end of the year of the first death. The mathematical expectation (pages 235, 234) for each year, as the third for example, will be the product of the present value of the insurance i^, by the compound probability ~- -^j— , that both persons will be alive at the beginning of the third year, Ix ty multiplied by the probability (l — ^^ ^^ that both will not live through that year, or that one or both will die in the third year: The sum of the series of similar expectations for each and every year, to the end of life, gives the total present value, or Single Premium k^y ; thus, A _ ^a-^y/i lx + lly + l\ l^c+^y+\l. /ar+2 ?y+2\ o Axy — ry|i 7 7~/^H T/ \^~~7 — 7 — r + • • •• Performing the multiplications indicated, we have 1 (?x?y — ^x-t-l^y + l)^'-f-(^x+l^y+l — ^x+ily+2)v^ -h {ix+jly+i — ^x43?y+3)^^+ » Axy — j-j x+y Again, multiplying and dividing by w 2 , in order to substitute 0, according to the notation of page 247 : A C'xy + Ca. + l.y+l-|-Ca;+2.y+2-|- • • • . Mxy Aary — ^^ = jy SINGLE AND ANNUAL PREMIUM3, 257 A,^ = 1-^^^ = l-(l-.)(l + «a^). \ Annual Premium, Pj^ = =~ = =^— (1— v). A 1 Annual Premium, Pxy = . ^ = :j— (1— ^)« For a joint insurance of $1 on three lives, x, y, and z, we have only to substitute xyz in place of the suffix xy in the last formulas ; and so for four or more lives. The insurance on joint lives continues only while they are all living. Also, the Premiums may be found from Conversion Tables XLVIII and XLIX. For a Temporary hisurance of $1 to be paid at the end of the year of the first death of the two lives a; or y, if it occurs within n years : Single Premium, in -M-xy — -M-x+n.y + ra V\^xy — ^x + w.y + «) — (JNar+i.y+i — -INx+w + l.y + n + l) ^xy — - Y) ~ ~ T) ^ -L'ary iJxy Annual Premium, P^y = ^r-r = v =~ -^ • '^+Cixy ^ — ^x+n.y+n For a Deferred Insurance of $1 payable on the first death of x or y, if it occur , after the expiration of n years : Single Premium, ^A^y = ^"J^ = Dx^..y.n-(l-^^)Nx^n.y^n ^ Single Premium, ^Kxy — -^^^— .Ax+n.^+n- iJxy ■M-x+n.y+n. Annual Premium for n payments, "P^y = -Nary — ^x+n.y+n Annual Premium during joint life, "P^y = — '^■^"- — • For an Insurance of $1 on the Longest of Two Lives, x and y : Single Premium, A^y = Ax + Ay— Axy = 1 — (1 — v) (1 + «« + ay—a^y). Annual Premium, Pxy = ^^+^y- ^^j ^ — _J /|_^n l+«a; + «y — «a!y l + «a; + flfy — «xy For a Temporary Insurance of $1 for n years on the Longest or Survivor of Two Lives, X and y : Single Premium, A^y = Ax+Aj— A^. Annual Premium, Pj, = —A^+Av^I^S' , ±-\-ux "vay — uxy 258 LIFE ANNUITIES AND INSURANCES. Tables foe Two Joint Lives, Equal Ages. — The agreement of the Thirty Offices Experience with Makeham's law of mortality, offers a great advantage, in super- seding the old and cumbrous system of joint life tables, by the new method of equal lives or ages, shown in Tables LXV-LXXVII. For, under Makeham's law, if the joint probability pxPy is equal to puPu, that of two equal lives, at the ages of entry, the relation of equality will always continue during their joint existence. Example. When interest is 4 per cent., required the annual and the single premium to insure $1000, payable at the end of the year of the first death of two lives, X aged 30 and y aged 40 years. Entering Table LXV, with the difference of ages y — X, or 10 years, we find the correction 6 Y. 1.8 M., which being added to the younger age 30, gives the nearest Equal Age 36. That is, 36, 36 may be substituted for the given ages 30 and 40. Entering Table LXXITI with 36 years, we find the annual premium $29.74; and in the next Table, the single premium $436.06. Method foe Three Joint Lives. — By Table LXV, change the given Ages x, y, z, to the two equal Ages u, u, which increase equally in equal times, like the actual ages, from the date of entry. The current value of u, in any year, increased by the Annual Correction below, under the proper rate of interest, will be the argument for entering Tables LXV-LXXVII, for premiums, reserves, etc. The correction can be adjusted to other rates of interest by the Difference column. THREE LIVES.- -ANNUAL CORRECTION (c) TO THE EQUAL AGE u. Equal 3 4 5 &4 1 Equal 3 4 5 Ut Equal 3 4 5 Ages. Pbr Pbk Per 1 Ages. Feb Feb Feb eI! Ages. Feb Feb Feb U. Cent. Cent. Cent. ft U. Cent. Cent. Cent. Q u. Cent. Cent. Cent. Yr. Y. M. Y. M. Y. M. M. Yr. Y. M. Y. M.Y. M. ,'M. Yr. Y. M. Y. M Y. M. 10 13 8.7 14 11.7 16 1.6 13.9 40 3 3.2 3 4.93 6.61.7 70 5.5 5.5 5.5 11 13 2.2,14 4.5 15 5.813.3 41 3 0.9 3 2.53 4.1,1.6 71 5.1 5.1 5.1 J2 12 8.1 13 9.7 14 10.412.7 42 2 10.8 3 0.33 1.7il.4 72 4.7 4.7 4.7 13 12 2.2 13 3.0 14 3.1112.1 43 2 8.9 2 10.2;2 11.51.3 73 4.3 4.3 4.3 14 11 8.4 12 8.5 13 8.011.5 44 2 7.0 2 8.2|2 9.31.1 74 4.0 4.0 4.0 15 11 2.7 12 2.1 13 l.llll.O 45 2 5.2 2 6.32 7.31.0 75 3.7 3.7 3.7 16 10 9.1 11 7.9 12 6.410.5 46 2 '3.4 2 4.42 5.3i .9 76 3.4 3.4 3.4 17 10 3.7 11 1.9 11 11. 8| 9.9 47 2 1.8 2 2.712 3.51 .8 77 3.2 3.2 3.2 18 9 10.5 10 8.1 11 5.5 9.4 48 2 0.3 2 1.1 2 1.9 .8 78 2.9 2.9 2.9 19 9 5.4 10 2.4 10 11.2 8.8j 49 1 10.9 1 11.6 2 0.3 .7 79 2.7 2.7 2.7 20 9 0.5 9 8.9 10 5.2 8.3 50 1 9.5 1 lO.lll 10.7 .6 80 2.5 2.5 2.5 21 8 7.7 9 3.6 9 11.4 7.8 51 1 8.1 1 8.7 1 9.2 .5 81 2.3 2.3 2.3 22 8 3.0 8 10.5 9 5.9 7.4! 52 1 6.9 1 7.4 1 7.9 .5 82 2.1 2.1 2.1 23 7 10.5 8 5,5 9 0.4 6.9 53 1 5.6 1 6.1 1 6.5 .4 83 1.9 1.9 1.9 24 7 6.2 8 0.8 8 7.3 6.5! 54 1 4.6 1 5.0 1 5.4 .4 84 1.8 1.8 1.8 25 7 2.0 7 8.1 8 2.1 6.0 55 1 3.5 1 3.9 1 4.2 .3 85 1.6 1.6 1.6 26 6 9.9 7 3.6 7 9.2 5.6 56 1 2.4 1 2.8 1 3.1 .3 86 1.4 1.4 1.4 27 6 5.9 6 11.3 7 4.4 5.1 57 1 1.6 1 1.9 1 2.1 .2 87 1.2 1.2 1.2 28 6 2.1 6 7.1 6 11.8 4.7 58 1 0.6 1 0.9 1 1.1 .2 88 .9 .9 .9 29 5 10.5 6 3.2 6 7.4 4.2j 59 11.9 1 0.1 I 0.2 .1 89 .8 .8 .8 30 o 7.1 5 11.4 6 3.2 3.8| 60 11.1 11.3 11.4 .1 90 .8 .8 .8 31 5 3.7 5 7.7 5 11.3 3.6 61 10.3 10.5i0 10.6 .1 91 .8 .8 .8 32 5 0.4 5 4.1 5 7.5 3.4 62 9.7 9.80 9.9 .1 92 .7 .7 .7 33 4 9.5 5 0.8 5 3.9 3.1 63 9.0 9.1I0 9.2 .1 93 .6 .6 .6 34 4 6.5 4 9.5 5 0.4 2.9 64 8.5 8.5 8.7 .2 94 .5 .5 .5 35 4 3.7 4 6.4 •4 9.1 2.7 *65 7.8 7.8 8.0 .2 95 .4 .4 .4 36 4 1.0 4 3.5 4 6.0 2.5 66 7.3 7.30 7.5 .2 96 .4 .4 .4 37 3 10.3 4 0.6 4 2.9 2.3 67 6.8 6.80 7.0 .2 97 .2 .2 .2 38 3 7.8 3 9.9 4 0.0 2.1 68 6.3 6.30 6.4 .1 98 .1 .1 .1 39 3 5.4 3 7.3 3 9.2 1.9 69 5.9 5.90 6.0 1 .1 Formula, for 3.654, 4.660, 5.666, or the rate of vs, Uuu = au+c.u+c for 3, 4, 5 per cent, interest respectively, or the rate of v. SINGLE AND ANNUAL PREMIUMS. 259 Method fob Foqb Join^t Lives, — By Table LXV, change the three youngest Ages to two equal Ages, u, u. With w, u, and the fourth or oldest age, re-enter Table LXV, and find the equal ages, u', ic, for record, at the date of entry. In any year, the current value of u' augmented by the Annual Correction below, will be the argument for entering Tables LXV-LXXVII, for the life annuity or other monetary value. FOUR LIVES —ANNUAL CORRECTION (c) TO THE EQUAL AGE ti • E<)UAL 3 4 5 fn Equai 3 4 5 ^ Equal 3 4 5 Aess. Per Peb Peb Pu Ageb. Peb Peb Per ^ Ages. Pee Peb Per ^l'. Cent. Cent. Cent. fi u'. Cent. Cbkt. Cent. 5 ! u\ Ce»t. Cent. Cent. Yr. T. M. Y. M. Y. M. M. Yr. Y. M.|Y. M. Y. M. M. Yr. Y. M. Y. M. Y. M. 10 21 10.4 23 1.6 24 4.2 14.6 40 6 .56 3.2 6 5.9 2.7 70 10.6 10.7 10.7 11 21 1.8 22 4.5 23 6.7 14.2 41 5 9.15 11.6 6 2.12.5 71 9.8 9.9 9.9 12 20 5.4 21 7.6 22 9.3 13.7 42 5 4.95 7.2 5 9.5|2.3 72 9.0 9.1 9.1 13 19 9.1 20 10.7 23 0.0 13.3 43 5 1.5 5 3.6 5 5.712.1 73 8.6 8.6 8.6 14 19 0.9 20 2.0 21 2.8 12.8 44 4 10.15 0.0 5 1.91.9 •74 7.8 7.8 7.8 15 18 4.8 19 5.4 20 5.8 12.4 45 4 7.04 8.7 4 10.4|l.7! 75 7.3 7.3 7.3 16 17 9.0 18 9.1 19 9.0 11.9 46 4 3.94 5.5 4 7.11.61 76 6.8 6.8 6.8 17 17 1.2 18 0.8 19 0.3 11.5 47 4 1.0 4 2.4 4 3.8jl.4i 77 6.2 6.2 6.2 18 16 5.6 17 4.7 18 3.7 11.0 48 3 10.2 3 11.5 4 0.8 1.3 78 5.8 5.8 5.8 19 15 10.1 16 8.7 17 7.3 10.6 49 3 7.5i3 8.6 3 9.7 1.1 79 5.3 5.8 5.3 20 15 2.9 16 1.0 16 11.1 10.1 50 3 5.0 3 6.0 3 7.0 1.0 80 4.9 4.90 5.0 1 21 14 7.8 15 5.4 16 3.0 9.6 51 3 2.5 3 3.4 3 4.3 .9 81 4.4 4.4 4.4 22 14 1.0 14 10.1 15 7.2 9.1 52 3 0.33 1.1 3 1.9 .8 82 4.0 4.0 4.0 23 13 6.0 14 2.7 14 11.3 8.6 53 2 10.0 2 10.8 2 11.6 .8 83 3.7 3.7 3.7 24 13 11.5 13 7.7 14 3.8 8.1 54 3 8.02 8.7 2 9.4 .7 84 3.3 3.3 3.3 1 25 12 5.1 13 0.8 13 8.4 7.6 55 2 6.0:2 6.6 2 7.2 .6 85 3.1 3.1 3.0 26 ill 10.8 12 6.1 13 1.4 7.3 56 2 4.12 4.6 2 5.1 .5 86 2.7 2.7 2.7 27 11 4.6 11 11.5 12 6.4 6.9 57 2 2.3 2 2.8 2 3.2 .4 87 2.4 2.4 2.4 28 10 10.8 11 5.2 11 11.8 6.6 58 3 0.72 1.1 2 1.5 .4 88 2.1 2.1 2.1 29 10 5.0 10 11.0 11 5.2 6.2 59 1 11.1:1 11.5 1 11.8 .3 89 1.8 1.8 1.8 30 9 11.4 10 5.0 10 10.9 5.9 60 1 9.6'l 9.9 1 lO.li .2 90 1.6 1.6 1.8 31 9 6.0 9 11.3 10 48 5.5 61 1 8.2il 8.5)1 8.7 .2 91 1.4 1.4 1.4 32 9 0.6 9 5.5 9 10.7 5.2 62 1 6.81 7.1 1 7.3 .2 92 1.3 1.3 1.3 33 8 7.6 9 0.2 9 5.0 4.8 63 1 5.7il 5.9 i 6.0 93 1.2 1.2 1.2 34 8 2.6 8 6.8 8 11.3 4.5 1 64 1 4.41 4.6!l 4.7 94 1.1 1.1 1.1 35 7 9.9 8 1.8 8 5.9 4.1 65 1 3.31 3.51 3.6 95 1.0 1.0 1.2 36 7 5.2 7 8.9 8 0.7 3.8 i 66 1 2.21 2.4 1 2.5 96 .8 .8 .8 37 7 0.8 7 4.2 7 7.7 3.5 1 67 1 1.21 1.4 1 1.5 97 .7 .7 .7 38 6 8.4 6 11.6 7 2.9 3.3 68 1 0.31 0.4 1 0.4 !o 98 .3 .3 .3 39 6 4.4 6 7.3 6 10.3 3.0 69 11.4|0 11.5 11.5 .0 Formula, for 4.311, 5.324, 6.337, or the rate of vs\ au'u' cent, interest respectively, or the rate of v. ciu'+c.u'+c, for 3, 4, 5 per Equalization of Annuity Elements. — In explanation of the preceding Tables, the single life annuity a^, on page 240, is or may be resolved into the sum of products of elementary factors of the form vpat . In like manner, the annuity aan/ on two lives, may be resolved into elementary factors of the form vp^Py ; on three lives, into vpxpypz ; and so on. In the logarithmic form of Makeham's law of mortality, from page 237, we find these Annuity Elements: X (vp.,./) = A {vs)+q'''(q—'i) Xg. A {vp^py) = A (vs^) + {q^+ qy) {q-\) Xg. l{vpuPu) = X{vs^) + {q'' + q^){q-l)Xg. A {v'p^pypz) = A (v's^) + (r + qv + q'){q-l) ^g, etc. o2 _ y'^. ajj(j qu-[.qu z= 2q'" = q^-^-q^ + q^, the annuity For a Single Life, For Two Joint Lives, For Two Equal Lives, For Three Joint Lives, Now if V = v's, that is US'' element of three lives, z, y, z, with the rate v', evidently becomes equal to that of two equal lives, u, u, with the rate vs. Also the same element with the rate v' equals that 260 LIFE ANNUITIES AND INSURANCES. of the single age x (defined by g*' = q'^ + qy ^-q'^) with the rate vs^. By the last two Tables giving the correction c, the true annuities first computed for the ages u, u, and rate vs, or vs^, are more conveniently found through the ages u + c, w + c, and the rate of v. Method for Simpson's Eule on Page 242. — For greater convenience, two Tables are here brought together; the first, from Table LXV, gives the Correction to change two joint lives to two equal ages, which does not involve the rate of interest. The second, computed from auu = ctu+c", gives the correction c" to change from two equal ages, u, u, to the equivalent single age, u + c", which last depends on the rate of interest CORRECTION TO BE ADDED TO THE YOUNGER AGE FOR REDUCING TWO JOINT LIVES TO THE CASE OF TWO EQUAL AGES. DUT. AaiES. 1 2 UNITS. 3 4 5 6 7 8 1 9 Yrs. Y. M. Y. M. J. M. Y. H. Y. M. Y. M. Y. M. Y. M. Y. M. Y. M. 0.0 6.1 1 0.6 1 7.3 2 2.3 2 9.5 3 5.0 4 0.8 4 8.9 5 5.2 1 6 1.8 6 10.6 7 7.6 8 4.7 9 2.0 9 11.6 10 9.5 11 7.3 12 5.5 13 3.7 2 14 2.0 15 0.6 i 15 11.2 16 10.0 17 8.8 18 7.7 19 6.7 20 5.9 21 5.0 22 4.3 ?;3 23 3.6 24 3.0 ! 25 2.4 26 1.9 27 1.4 28 1.0 29 0.6 30 0.2 30 11.9 31 11.5 ^4 32 11.3 33 11.0 ; 34 10.8 35 10.6 36 10.4 37 10.2 38 10.1 39 9.9 40 9.8 41 9.7 5 42 9.6 43 9.5 ! 44 9.4 45 9.3 46 9.2 47 9.2 48 9.1 49 9.0 50 9.0 51 9.0 6 52 8.9 53 8.9 54 8.8 55 8.8 56 8.8 57 8.7 58 8.7 59 8.7 60 8.7 61 8.7 7 62 8.7 63 8.6 64 8.6 65 8.6 66 8.6 67 8.6 68 8.6 69 8.6 70 8.6 71 8.6 CORRECTION (c") TO REDUCE AGE . TWO EQUAL .—INTEREST 4 AGES TO THE PER CENT. EQUIVALENT SINGLE Equal Agbs. 1 2 UNITS. 3 4 5 6 7 8 • 9 Yrs. Y. M. Y. M. Y. M. Y. M. Y. M. Y. M. Y. M. Y. M. Y. M. Y. M. 1 2 3 c«4 2 21 16 13 10 5.2 6.8 .1 7.2 20 10.7 16 1.8 12 8.6 10 4.9 20 4.3 15 9.0 12 5.2 10 2.8 19 10.0 15 4.3 12 2.1 10 ■ .8 19 3.9 14 11.9 11 11.0 9 10.9 18 10.0 14 7.5 11 8.1 9 9.0 18 4.2 14 3.3 11 5.2 9 7.2 17 10.6 13 11.3 11 2.5 9 5.6 17 13 11 9 5.2 7.4 .0 40 16 1L9 13 3.6 10 9.5 9 2.5 H 6 6 7 8 9 9 8 7 7 5 1.1 2.5 8.4 .5 7.6 8 11.8 8 1.8 7 8.0 6 10.4 5 3.8 8 10.5 8 1.0 7 7.5 6 7.6 4 11.6 8 9.3 8 .3 7 7.0 6 4.8 4 6.1 8 8.1 7 11.7 7 6.4 6 2.5 3 9.5 8 7.1 7 11.1 7 5.8 6 0.9 3 1.0 8 6.0 7 10.5 7 5.2 5 11.4 2 3.8 8 5.1 7 10.0 7 4.4 5 10.1 8 7 7 5 4.2 8 3.3 9.4 7 8.9 3.4 7 2.2 9.2 5 8.2 i 1 Thus for two Equal Ages of 38 years each, the 4 per cent, annual correction addi- tive, is 11 Y., giving the equivalent single Age 49 Years. Ten years after this, the equivalent single age for finding the life annuity will be 48 Y. + 9 Y. 4 M., or 57 Y. 4 M. For. a rate (i) of Interest other than Jf per cent. — First find the 4 per cent, correc- tion c", and re-enter the Table with the corrected argument m-|-10(.04 — i)c", which will guide to the correction sought. Here 10 (.04 — t)c"isfor 3 per cent, ^-^ig^" ; 3^ per cent. -1-^V^" ; 4| per cent — ^V ' j ^ P^^ cent— ^c'' ; 6 per cent— ^". 14. SURVIVORSHIP INSURANCES. Required the single premium to insure $1, to be received at the end of the year in which X dies, provided another life y be then living. This is sometimes termed the insurance of " x against y." The probability of receiving the $1 at the end of any specified year, as the second, depends on the probability -~ • -y- that both are alive at the beginning of the year, multiplied into the sum of the two probabilities, (1) that x may die in said second year SINGLE AND ANNUAL PREMIUMS. 261 and y survive through it, or {l—px+i)py+\- (3) that both may die in the second year, X dying first, or i(l— J^ar+i) (1— JOy+i). This last expression is multiplied by ^, since it is considered an even chance for so short a period as one year, whether x or y will die first, whatever be the difference of their ages. The sum of (1) and (2) is ^{\-p^^-^) {l+py^i), or \ (l-^) (l + ^^--). And the product of this sum by the previous factor and by v^ gives the second term of the series or formula for the single premium : A?_ ix — 'x+1 Iy-\-iy+l . ix+l — 'x+2 'y + l-\-iy+2 .gi Ix '^'y ^x '^'y In this notation for A, the 1 written over the x, expresses the condition that x is to die before y: otherwise the insurance is void. Performing the multiplications indicated, we have Aa^ = „, , l\tx'y — lx + l'y + l-{-lx'>/ + l — 'yix+l)V 4)1 xly + (Jx+liy+X — ix+fliy+2 + Ix+liy+fi — iy+l^x+yf^ + ]. This expression can be resolved into three simpler series, as shown on page 184 of Milne's treatise. Thus, the sum of the- terms characterized by the first factor ixiy — ^x+i^y+i is identical with the half joint life premium ^Aa^. The factor Ix^y+i is a type of the next series, which being multiplied and divided by lx-\ly, takes the form of \ax-i.y-^Px-i • And the remaining series is like it ; so that L _ 1 / A 1 ^x-l.y ax.y—\ \ ' 2 \ Px-\ py-l / Px-1 Py- Aa;y -f" -^xy =^ Aa^ ', A.xx ^^^ i-^xx' When x and y are both of the same age, the expression is thus simplified by the evident cancelling of the last two terms. Also, by changing each term into the Com- rmUation notation, and observing that Vvpx-iT)x—\.y = ^tey, we verify the formula already given on page 248, Note 1, for the single and the annual premium. Another formula can be obtained by substituting the middle value /y+| in place of i(^y + ^y+i)j 3,nd so forth, in the primary formula, which will be sufiiciently exact. The coefiicient of v divided by Ixly, thus becomes {Ix—iee+ijly+i , or ljy+i—lm+iiy+^ , the result for the first year. And the sum of the series for the first and following years, will be, if ~^ is replaced by \ / : ly ^ \ Py-i A^ = . /X. («^-^i_«,.^_A V Py-l ^ Px-i / Example. — Let ar = «/ == 35 years, with interest 4 per cent. The equal ages to give the two annuities, are found by Table LXV to be 34^ and 34|- years, for entering Table LXXII. And the logarithm of the radical factor is \ of 0.003500 found in Table XXVII. Completing the calculation, we find the single premium 0.21416 on each $1 insured. Verification is obtained by the previous formula, which gives iA35.35 = .21408 ; and this appears a trifle more exact than the result above, which depends substantially on the difference of annuities tabulated to three or four decimals. Dividing the single premium by 1+aa-y, that is, by l + ass.ss or by 14.8678, gives the annual premium 0.014399 on each $1 insured. Changing now the last formula, or rather its previous series, into the notation of De Morgan and of Chisholm, we may find the values of D, N, in the Tables for Two 262 LIFE ANNUITIES AND INSURANCES. Joint Lives, or the values of D, N, S, in the Tables of Single Life, entered with the equivalent single age from page 259, and thence the value of M^ or E^ below : 1 M — 1 ^.tiLti x+y+A A^ = Yr^; '^s^-={lx—lx+\){ly^\)v 2 +{lx+l — lx+^){ly+\\)v 2" + SJxy E^ = M^ + Ma; + l.y+l+ Ma;+2.j,+2+ • ' •• Mi^ = vlNa-.y+i — ■y^Na;+i.y+4 ; Ei^ = vi^x.y+\—V^^x+\.y+\. Insurance of $1 on the life of x, if y be the survivor : Single Premium, N^ = ~ ; Annual Premium = =--^- '^xy -Nary Insurance of 11 on the life x, if y be the survivor, with return of net premium or premiums paid, if y die first : Single Premium, ^— ; Annual Premium = ^ -L'a^ — -iVlxy ^xy — -tncy Insurance of $1 on the life of x, provided he die last of the two lives x, y : 1 ^ j^ Single Premium = Aas— A^ ; Ani;iual Premium = -- — ^^- l+a. xy Insurance of $1 upon the life x, deferred n years, and contingent upon v surviving that term : Mx+n ly+n^ A„„„„i T>„„^:: ^x + iJy + n Single Premium = -^:c — . ^ — ; Annual Premium = -p^ , '^x ' ly ' ^xly{l + a^) Deferred and Temporary Survivorship Insurances. — For the single pre- mium of the contingent insurance deferred n years, and of the temporary insurance during n years, we have respectively, „ A L _ Dx + n.y +n j^A iJxy An- — A- «A- — A- ^x+n.y+ n .— !^ ■"•ay — -ti-ary — -^scy — ^^xy — j^ '-i^x+n.y+n' ■L'a ^x+n.y+n 'xy The divisor to give the annual premium will be 1 plus the temporary annuity on the joint lives for one year less than the whole period of payment. Attention has been called to this advantageous method of determining the single premium of deferred or of temporary insurances, by Mr. Peter Gray, who states that it applies to all insurances, depending on the first death from among any assigned number of lives. The commutation formula for temporary annuities will furnish another illustration. An equation between two or more benefits is first presupposed. Then by transposition, or solution, each benefit may be expressed in terms of the others. SINGLE AND ANNUAL PREMIUMS. 263 General Solution. — For the more general case of an insurance of $1, payable at the end of the year of first death among the {m) lives x, x", x'", .... provided all the other (n) lives y', y", y'" , .... are then surviving, Mr. Makeham has given a strikingly original formula in the Journal, vol. 9, page 363, reproduced by Mr. Woolhouse in vol. 15, page 403. First, two auxiliary single ages x, y, are to be found such that, gx — q If the premium has just been paid, by adding it, and accenting V, y x+n '= S .Ax+n — i^x-(^x+n- When the valuation is made by the same table of mortality and rate of interest with the premium, the former expression will be simplified by substituting, for A, P, their equivalents in terms of v, a; the result is, on each $1 insured, V'l ^-\-ttx + n ttx — flx^n \ri . (h;+n x+n = J- -. , ^ — -^TT 7. — ' V ar+n = V—z. - , l + «a: l+«« l + Oflj 266 LIFE ANNUITIES AND INSURANCES, ^j _ -, 1— Aa.+n_ Vx+l—V Va:+re — -L ^ i — J-- In the case of two joint lives, x+n.y And on the survivor of two lives, -( 1 -{-(Ix+n.y+n x+n.y+n — J- -. , " * 1-t-ttary 1 1 -T-Q'X+n~\~((y-¥n — flx+n.y+n x+n.y+n — J- ^ r~~ j * -\-ttx-\-tty — ttxy By another mode of reasoning, if Fx+n denote the annual premium at the advanced age x-{-n, then the difference Fx+n — Paj measures the annual saving in premium of the former policy over an assumed new policy. The present value of these sa"sdngs will of course be the present value of the former policy ; that is, if the premium is just due, but not paid, Va;+« = {"x+n — ^x) (l+Cfx+n)- EuLE. — Take the difference between the net annual premium, which would be required at the advanced age, and the net annual premium charged in the pohcy, multiply it by unity added to the value of the annuity at the advanced age of the life insured ; the product will be the Reserve. The last formula is equal to the first one, since Ax+n = Pa!+n (l + «ar+»). Again, since Ax+n = 1 — (1 — v) (1 +«x+w)> we have from the first formula on page 265 : Var+« = 1 — (1— v + Par) (l+ctx+n), ou each $1 insured. By proper change of the annuity, as noted above, this formula will likewise apply to two joint lives, or to the survivor of them. On all Temporary and on Endowment Insurances of $1 insured for the term of m years, after the age of entry x, the Reserve at the intermediate age x-\-n, when the next premium is just due but unpaid, will be V — A'""'* P n 4-^"*""'"^^ ^x + n — ^x + n — J:x\^n^"x + n )• Here A denotes the single premium of the given species of policy, and a the temporary annuity at the age x-{-n for the remaining m—n years. The same Reserve with the annuity expressed in commutation terms, will be m—n T» ^x + n — ^x+m y x+n — -i^x+n — ^x- -r\ iJx+n ) The value of an Endowment Insurance for m years, with premiums payable annually, may be simplified like that of the common life policy, as follows : -, 1 -{-dx +n 1 ^x+n — ^x + m ■L'a; 1+af'-^ ^x-^x+m "Ox + n Valuation by Gross Premiums. — Let Pa; denote the net annual premium on any policy, and P'a; the loaded, gross, or office premium. Then will the difference between the gross and the net valuation be expressed by {P'x-Fx){li-a^+r'), or (P',_P,)^^=-_%.-- VALUATION OF POLICIES. 267 This is the present value of the loading receivable in the future. Since many policies will be discontinued, it belongs to the uncertain or "unrealized assets ;" which, as they accrue, may be the material of future dividends. In the early years of life insurance in the United States, the Carlisle 4 per cent, mutual premiums were loaded 35 per cent. RETROSPECTIVE FORMS OP NET VALUATION. In apparent contrast with " Prospective Valuation " which has just been shown to be based on the present single premium, and on future annual premiums, " Ketrospec- tive Valuation " is based only on past premiums already received. Yet by a singular coincidence, in net valuation, the Keserve found by the first method, from premiums future and unpaid, is precisely the same as the Reserve of the second method from premiums past and paid. This will be the more evident from the direct transformation of Prospective formulas into Retrospective. Let us take, for example, the Prospective formula, giving the Reserve at the end of 6,. or more generally, n years, on a ten-pre- mium life policy for $1 insured : Substituting for Ma;+w its equal, M^— (Ma;— Ma;+n), Ma;— Ma,+« „ _ Ma; / Na;+ra — Na; + 10 \ ^^■^"-Da;.„r N^-Na:.10 / Da:^ ^^ Pa;(Na; — Na; + n) — (Ma; — Maj+n) M3. Here the premium P^, = ^r^: ^ ; and the formula now involves no future age, -Ma; — JNa; + 10 but only the past or retrospective ages x and x-\-n ', hence the name of Retrospective Formula, proposed in the original memoir by James Meikle, Journal, Vol. XI, p. 245. If Tx denote the annual premium of Temporary Insurance, and E^ that of Pure Endowment, for w years; then dividing both numerator and denominator of the formula by 'Nx—l^x+n, and substituting equivalents, we have the second form, on each $1 insured : Va+n — X — J-g; En 3! By substituting for Ma; its equal Da;— (1 — y) Na; , and so for Ma;-t.«, we obtain a third form ; and then by substituting 773,+ (1—v) Na, for its equal D^,, the fourth form following,: V _(Pa, + l-v)(Na,-N«, + n)-(Da,-D.+«) VtB+re — =P^ • iJx + n XT _ (P«— 7ra,)N"a,— (P„ + l — v)N let its former equal V+(Pa; — Tr-c) Bj be here substituted. Then observing that the (1— /i) (v— t;j9a;+n-i) + y/" denotes the 270 LIFE ANNUITIES AND INSURANCES. interpolated value of the ordinary Life Policy, hereafter denoted simply by V, we find, for each $1000 insured, as on page *154. or Table LXIII : The last equation but one^ may be transformed, by substituting for Yx+n its equal (Vx+n — 1) + 1 ; thus we obtain another formula when Yx+n is known : Yi^n=h + {l-h)v-{l-Yxin)f=f'-{l-Yx,n)f. f = h + {l-h)v; f=h + {l-h)vpx+n-i.^ The value of/' and the common logarithm of/, for 4 per cent., are given in Table LXIV, for the beginning and middle of each year, and the middle of each month. Intermediate values can be found from these by proportioning the differences. Like the last valuation formula, /' is adapted to 1 or unity, as the amount insured on a single life. By changing the decimal point three places to the right in the final result for $1, it will be adapted to $1000 insured or the value of/" below. It may be proper to observe in the preceding expressions, that the exponent n or h is merely an extension of the suflSx under it; Yx+n meaning the same as Yx+n+h' VALUATION TABLES FOR JOINT LIFE POLICIES. It will be advantageous to apply the correction to give two equal ages, at the time of entry, by first taking account of months, and applying the correction from page 260, or from Table LXV, so far as to determine the nearest integral age for the Office Equal Age. All future calculations on the Policy, can then be made by single entry in the Tables LXV-LXXVI, in the same manner as for a single life. In the case of three or four joint lives, the method of current " equal ages" with an Annual Correc- tion (c) for calculations only, has been given on pages 258, 259. Paid-up Policies on Two Joint Lives admit of Valuation by inspection or inter- polation in Table LXXIV. On Annual Premium Policies, the Eeserve can be calculated from the single entry factors in Table LXXVI, when the net Premium has been prepared, as illustrated in Table LXXV. The formula of valuation having precisely the same form as for a single life, is demonstrated as follows: Substituting the expression for Va?+w, from the fourth form of Retrospective Valuation, into the preceding formula containing/' and / we obtain on $1 insured: Vj n =/'+-^ [(P^_Tr^)]^^^_(P^ + l_^;)N^+„]. Assuming P^^, ttx, to be premiums on $1000 insured, and adjusting the other terms accordingly, let /" hereafter denote the former 1000/'. Let the former /-r-Da;+« = Jx+n ; and let h'x^n^=hx+n^x+n- Also let the prepared premiums V'x = (Px— ^x) Nx ; P"a; = Px + 1000 il—v). Then as computed in Tables LXXV and LXXVI: Yl+n =f" + ^'x.hx^n-^"x-Vx^n, ou $1000 insured. Example 1. — An ordinary Policy issued on the Joint Lives aged 37 and 37 has been in force 5 years 4^ months. Required the 4 per cent. Reserve. L 84012 = AP"37, Table LXXV. - /" = 975.962 1.09963 = AJ'^, Table LXXVI. - 870.46 ....2.93975 Sum, Table LXXXIL $105.50 = Reserve on $1000 Insured. VALUATION OF POLICIES, 271 Example 2. — A Ten Payment Policy on two Joint Lives was issued at the ages of entry 43 and 43. Kequired the 4 per cent. Reserve at the end of 8 years 7^ months. 6.76553 = AP'43, 2.01450 = AF'43, Table LXXV. /" = 985.577 4.01727 = uli, 1.0134 5 = Xb'ii, Table LXXVI. + 606. 46 .... 2. 78280 .... 3. 02795 Sums, Table LXXXII. — 1066.5 $525.54 = Reserve on $1000 Insured. Value of a Sukvivorship Insurance. — At the end of n years, the annual premium P being just due but unpaid, the Reserve on a Policy of $1 payable on the death of x, provided y be then living, will be Reserve = AiT^:^-P(l+«x+n.,+«) = {a' + h') (i_i+^±^J^\, = {a'-\-h') X Reserve on Ordinary Joint x, y, at the end of n years. This expression is based on the Single Premium a' -\-h'Kxy ; and a', V , being con- stant during the continuance of the same Policy, are tabulated on page 263. The age X is presupposed to be younger than y. When x is the older age, a! , h', must be replaced by — a' and 1 — b' respectively, in all the expressions. These formulas, first investigated in the present collection, depend on the comparison of different expres- sions for equal reserves. To correct for Months, whether the insurance be for the whole Life, or Temporary, let V denote the Reserve on 1000, computed from Table LXXVI, as if for ordinary Joint Lives, or for the same term, if Temporary ; then, on the corresponding Survivorship Policy: Reserve = («' + *') V + a'(1000— /"). Insurance of II on the life x, if y be the survivor, with return of wP the net annual premiums paid, if y die first. At the end of n years, when the next premium is just due, but unpaid : Reserve = ^^^^n.y^n _ P(l+a^^„.^,„). l — {l + 7l?)kx + ny^n For other rare varieties, reference may be made to the work of Chisholm, who has tabulated Da^, N^-y', M^, and R^^, for every yearly difference of the Carlisle Table. Short Method of Determining the Equivalent Amount Insured by Paid-up Policy. — A remarkably simple formula was devised by Sprague, and extended by Macfadyen, showing at once the result of the double process of finding the reserve, and thence the equivalent amount insured by a paid-up policy. In the common formula of valuation, the term kx+n is the single premium on $1 of paid-up insurance, page 266. Dividing the Reserve Yx+n or the whole equation by this term, will consequently give the equivalent sum insured, which is ■1 -p l+^x+n -, ^x k-^f+n 272 LIFE ANNUITIES AND INSURANCES. By dividing the first numerator and denominator by the numerator, the denomi- nator takes the well-known expression for ^xau, given in the last fraction. Let s denote the amount insured by the first policy. Multiplying the above result for $1 by 5, we have the equivalent Amount Insured by the Paid-up Policy = s (l — :p — I = Vx + n S -T • Here P denotes the net premium, or the oflBce premium when loaded with a uniform percentage. Example 1. — A temporary insurance of $5000, for ten years, is eflPected at the age of 40. After four annual premiums have been paid, at the end of four years, the equivalent amount of paid-up policy for the remaining six years will be, from Table LIII, s (l-ps) = s (l-n^) = 5000 (1-.91799) = $410.05. Example 2. — A person at the age of thirty-five years has an ordinary endowment insurance of $10,000 on his life, payable at the age sixty-five or at previous death. At the end of ten years, the sum insured by paid-up policy, equivalent to the ten annual premiums paid, will be, from Table LV, This sum is payable at " Death or 65," as under the original policy. NoN-FORFEitiNG Plan. — In comparison with this correct result $4042.28, " the non-forfeiting plan," so called, would give a paid-up policy for ten-thirtieths or one- third of the original $10,000, which is only $3,333.33. This is in the ratio of the total number of premiums payable, to the number already paid. For shorter periods of insurance, not much exceeding ten years, the results of this empiric rule will approach nearer to accuracy. Further discussion of this subject may be found in the fifteenth and sixteenth volumes of the Journal, and an able article by Mr. McClintock in the seventeenth volume. Surrender Value. — The Massachusetts non-forfeiture law provides that on lapsed policies, four-fifths of the net value at the time of lapsing, less the outstanding note, if any, shall be applied as a single premium of temporary insurance to continue the policy. And in case of death during such term of extension, the unpaid premiums may be deducted from the sum insured. We may observe that the premiums of term insurance in this collection, are given in Table LII ; from which, corresponding result? may be found by proportion. In ordinary practice, a deduction is made from the Net Value to giv6 the Surren- der Value, depending on the health of the insured, on the cost of replacing the risk, or other circumstances. The office system of deductions generally refers to surrenders made, as they usually are, at the end of the policy year. Todd's Tables, Edinburgh, 1852, may illustrate this topic. If a surrender is made earlier in the year, as at the middle of it, a portion of the Premium last paid, will be returnable to the insured, for risk not incurred in the unexpired part of the year. For illustration, let the tabular net value during the second year, be separated. into two parts thus described : (1) The I VALUATION OF POLICIES. 273 net value 117.64 at the end of the first year, increasing uniformly to $35.68 at the end of the second year, as noted at the beginning of this Section. (2) The net premium of $28 paid at the beginning of the second year, gradually decreases to during the ensuing second year. The full reserve at any time in the second year, is made up of these two continuous parts, (1) and (2) ; and the office percentage is deducted by Todd from the former part only, while the latter or premium part is directly returnable. Thus at the middle of a policy year, half the net premium which was paid six months before, is subtracted from the present tabular reserve ; then the office reduction is made for surrender value, to which the half premium is again restored. Annual Valuation of an Insurance Office. — The general plan is thus stated in De Morgan's Essay on Probabilities, page 275 : — " The most simple theoretical way of conducting the process, is to ascertain the value of every policy; that is, to ascertain how much should be given to the holder of each policy to renounce his claim, the office also abandoning the future premiums. When this is done, it is obvious that the office is not solvent, unless the assets arising from the accumulation of former years be sufficient to pay the values of all the policies, and thus to buy them all up. Otherwise, calculate the present value of all premiums due to the office, and also the present value of all claims to which it is liable. To the former add the sum total of the assets of the office, and to the latter add the present value of a perpetuity equal to the expenses of management. Thus, let P = present value of all premiums. = present value of all claims. i A = total assets of the office. M = present value of all expenses of management. If then P and A together 6xceed C and M together, the office is solvent. On each of these items a few remarks may be made. (P.) All the parties who are of the same office age, may have their several policies considered as one collective policy, in respect of which the sum of the premiums is paid as one premium, and the sum of the possible claims is one claim. But as these pre- miums are payable at all periods of the year, they may be considered^ as, one with another, due at six months after the valuation, at which time the present office age of the parties may be considered to be their real age. (C.) All bonuses which have actually been added to the policies must be included in the claims ; and the value of each claim must be carefully found, with reference to the time after death at which it is paid. (A.) The principal of the assets must be deduced entirely by means of the income it yields, and must be ascertained from the income by njieans of the rate of interest assumed. (M.) Against the expenses of management may be set, as far as they go, the incidental profits, when- they can be tolerably well ascertained." Distribution of Surplus. — The methods employed in upwards of seventy British Life Offices, are very plainly exhibited by Hewat in the Assurance Magazine, Vol. XXII, page 286. It appears that nearly all the Companies allow the option of applying the dividend to reduce the current premium ; and in the general average, the " loading " on the premiums is practically refunded. The Thirteenth Eeport, 1868, of the Insurance Commissioner of Massachusetts, contains a series of communications on the Contribution plan of dividends, originated 274 LIFE ANNUITIES AND INSURANCES. by Messrs Homans and Fackler in 1863. A single illustration and the formula below are recalled and revised from a former article in the Insurance Times (New York, 1868), Vol. I, page 545 : A party aged thirty-five years, takes out an ordinary life policy of $10,000, at an annual premium of $273 (Carlisle four per cent, net, with 35 per cent, loading). Eequired his annual dividends on the contribution plan, according as the Reserve is reckoned at three, four, five, or six per cent, interest, by the Carlisle Table ; the Com- pany receiving six per cent on investments, above expenses. RESERVE AT THREE PER CENT. RESERVE AT FOUR PER CENT. End of Resebve Contribution Premium less End oi" Reserve Contribution Premium less Ykab. V. Dividend. Dividend. Year. V. Dividend. Dividend. 1 $128.70 $63.25 $209.75 i 1 $108.50 $81.26 $191.74 2 259.90 67.19 205.81 ! 2 230.00 83.69 189.31 3 393.20 71.19 201.81 3 334.60 85.78 187.22 4 539.10 76.56 196.44 4 451.20 88.11 184.89 5 663.80 79.30 190.70 5 567.40 90.44 182.56 10 1,322.60 99 07 173.93 10 1,136.06 101.89 117.19 20 3,100.50 152.40 120.60 20 2.782.10 134.73 138.27 30 4,898.00 207.83 65.17 30 4,538.40 169.86 103.14 RESERVE AT FIVE PER CENT. RESERVE AT SIX PER CENT. End of Reserve Contribution Premixtm less End of Reserve Contribution Premium less Ykab. V. Dividend Dividend. Year. V. Dividend. Dividend. 1 $92.20 $96.16 $176.84 1 $79.20 $108.33 $164.67 2 188.60 97.12 175.88 2 161.10 108.33 164.67 3 285.90 98.12 174.88 3 245.80 108.38 164.67 4 386.90 99.11 173.89 4 333.60 108.33 164.67 5 486.90 100.11 172.88 5 420.80 108.33 164.67 10 977.90 105.02 167.98 ! 10 843.60 108.33 164.67 20 2,498.90 120.23 152.77 20 2,246.60 108.33 164.67 30 4,205.90 137.30 136.70 30 3,898.90 108.33 164.67 In general terms, let x denote the age of entry. Let V denote the Eeserve or net value on each SI insured at the end of n policy years, when the next premium is just due but not paid. Also let Vi denote the corresponding Reserve twelve months after, at the date of investigation. Let i denote the rate of interest assumed in computing the Reserve; and let i' be the rate of interest realized on the Office assets. And let (Pa;— e) denote the Office premium less its current expense, and px, ihe, net premium. The following three-fold expression will then denote the Contrihition to Surplus : (V+i?x) {i'—i) + {Px-e-p:c) (1 + *') + iqx+n-g'x+n) (1-Vi). Here the first term makes restitution for the difference of interest i' — i on the renewal value of the policy. The next term restores the loading on the premium, with a year's interest. And the last term c6rrects for the excess of the tabular rate of mor- tality qx+n above the realized rate q'x-im- In the preceding illustration, these two rates of mortality were assumed to be equal. The original formula differs from the one above by the subtraction of a zero equation, so that the numeric results will be precisely the same. LIFE ANNUITim AND imURANCES, ^75 SECTION VIII. Life Annuities. Survivorship and Reversionary Annuities. Successive Lives. In the received system of life annuities, the annual payment is made at the end of each year during the life of the annuitant, with no allowance for the proportional part of the year of death. In the case of a survivorship annuity on the life x to commence on the death of y, the first yearly payment is made at the end of the year, in which y dies, and the last payment at the end of the last year before x dies. But special formulas have been investigated, which give the price or present value of an annuity payable half-yearly or quarterly, etc., and with a proportional part to the day of death. Life Annuity. To recapitulate from Sections III and V, the Present Value of an annuity of $1 payable at the end of each year during life, is fla; = %^- On two Joint Lives, «^ = ^^^jH, When payable at the beginning of each year, the Present Value is xy Temporary Annuity. The Present Value of an annuity of 11 for n years, on a life aged x, is n Nai+l— Na;+»+l _ . , ^ra— 1 N"a; — Na;+n «« = fv = ax—^ax. !+«« = fi • -L'a! i-'a; Here and in such connections, n denotes the term of years, and is not used as an Algebraic exponent. On Two Joint Lives aged x and y, the Present Value for n years, is n Na;+l.y+l — Nj;+fi+i.y+n+i „«— 1 ^■n/ — ^x+n.y+n O'Ty — j^; — i--\-axy — ^^r Deferred Annuity. The Present Value of a life annuity of $1, deferred for n years, the first payment being due at the age- x-{-n-\-lfis Nx+n+l T)x+n Bx ~ Bx (^x — T=i — ~T=; . fl^x+n- ^x+n+l.y+n+1 ^x+n.y+n 'xy, iJxy "Oxy — pT — — -p. -^x+re.y+ft. ^xv . ^xv The divisor to give the annual premium for n payments, will be 1 + aT'^. • And for a change to the beginning of the year, or first payment of annuity at the age x-\-n : Present Value = ^^. I -L'a; On Joint Lives, Present Value = ^y-^^" . 276 LIFE ANNUITIES AND INSURANCES. Deferred Temporary Annuity. The Present Value of a temporary annuity of $1, to be entered upon at the expiration of t years, and then to continue 7i years, if the annuitant now aged x shall so long live, is D^ - D^ * Increasing or Decreasing Annuity. The Present Value of an annuity on the life x, for the whole term of life, commencing at $a and increasing ^h annually, is readily proved by summations founded on the previous formula, to be Single Premium = =:: -L'a; And for a similar life annuity decreasing %b annually, we have Single Premium = yt~" — " Ux The Present Value of a temporary annuity on the life x for n years, commencing at $a and increasing $b each year, is a ■ (Na;+i — Nx+n+l) + b {Sx+i—Sx+n+i—n. Nay+n+i) When the similar annuity decreases $b annually, simply reversing the sign of b in the last- formula, will give the present value. But in this case b must not exceed , that the annuity may not finally become negative. For the Present Value of an annuity of $1, increasing annually by II for n years, and then continuing constant at $n through life, we have Single Premium = §£±l^£±n±i. -L'a; The Present Value of an Annuity certai7i for n years, commencing at $a and increasing $b each year, was not given in Part I. But by taking the derivative D of the last equation on page 59, with respect to v, we find the required value of this increasing annuity: {a—b)Y + bvDY = \a + nb + -.j. — -. r- The value of the similar annuity decreasing $b each year, is found by reversing the sign of b. Valuation of Annuity Policies. The Present Value of Single and Joint Life Annuities at entry, being ax, a^y, etc., the Eeserve after n years will evidently be Ox+n, ax+n.y^n, ctc. In like manner, the Present Value of a Sur\'ivorship Annuity of $1 on the life of x after the death of y, being ax—axy, the Eeserve n years after entry, will be aa;+n—«a;+n.y+n • But if the Survivorship Annuity of |1 on the life of.?: after the death of y, is con- ditioned with the return of the single premium, without interest, should x die before y, this premium and the Reserve n years after entry, are proved to be LIFE ANNUITIES AND INSURANCES. 277 ttx — dxy Single Premium = — 1 — A-xy Reserve = dx+n — x, n + Vy, n —Px, nPy, n , expresses the probability that both will not be extinct, or that one at least will be living to receive the nth annual payment Therefore multiplying each term by v^, and 378 LIFE ANNUITIES AND INSURANCES. taking the sum of the products for w = 1, 2, 3, , we find the required value on the longest of two lives to be the sum of the two single annuities, minus the joint annuity ; or, The same mode of solution, illustrated by the theory of the higher equations of Algebra, gives the value of an annuity on the longest of three lives, denoted hy a line over the suffixes : ^^ = (^x+ay + az—{axy^-axz+ayz)+aryz* And generally, the value of an annuity on the longest of any number of lives, is equal to the sum of the annuities on the single lives, minus the sum of annuities on the joint lives combined by twos, phis the sum of annuities on the joint lives combined by threes, and so on ; the odd orders being positive, and the even negative. Possibly this species of annuity might be resolved into binomial or single life factors, like the elementary probabilities above, aided by assumed constants. In another expression, if C^ denote the sum of joint life annuities on m lives combined n in a set, then evidently, extending to n such terms, ^m — ^wi + Cffj — Cffi-|- .... xyz. VALUE OP AN ANNUITY ON THE LAST n SURVIVORS OF m LIVES. For an annuity to commence immediately and terminate with the last death but one, of m lives, the present value is demonstrated by Milne, Vol. I, pages 45, 126, to be ^m — 2.C»i-(-3. Cm, — 4. C,»+ .... Thus the present value of an annuity to end with the last death but one of two lives, is a^y-, and similarly of three lives, x, y, z^i\\Q present value is ttpsy -f- Clxz + tt'yz — lidayz • In every case, the algebraic sum of the coefficients of a is 1, to indicate one annuity in present possession. And generally, to commence immediately and terminate with the last death but n — \ of m given lives, the present value of the annuity is the series of m—n-\-l terms of C, nn n^ + l , ^ ^^ + 1 pn + 2 W + 1 n-\-2 pw+3 , This general formula is proved by Milne, pages 42, 126, by multiplying ont the probabilities of five lives, and applying the rules of combination. By introducing an auxiliary t, we obtain the following shorter deynonstration : Let Px> Py, • • . ■, denote the tabular probabilities that the m lives x, y, z, , will survive h years. And let the continued product of the m probabilities of decease {\—px) (1— i?j/) (1— />«) .... be temporarily denoted by P. The required probability that " n or more " out of m lives will survive h years, will be the sum of elementary products of the form 2w factors x {m—n) factors = 'S.p^py {'^—Pz) (1— jo«) = P2 ^ . _^^ + 2 (w + 1) factors X {m—n—1) factors = etc., + LIFE ANNUITIES AND INSURANCES. 279 The right-hand expressions after P represent the combinations of the fractions taken w in a set, w + 1 in a set, and so on to m in a set. By comparison with the Algebraic theory of the roots of the higher equations, these combinations will be the coefficients of t^, ^'*+i, ^"» in the development of = + {t—iy^ CW+ (if— 1)'^+! c(»+i) + (jf— 1)«+2. c('*+2) ^ + (if— 1)»». cc"). Here C(") denotes the sum of the combinations of the m factors px^ Py, , taken n in a set; etc., etc. Developing and omitting all powers of t lower than the wth ; then making ^ = 1, as above indicated, Ave find the precise result sought, here denoted by P' : < ^ 1-2 1.2-3 ^ Lastly multiplying by v'^, and taking the sum for ^ = 1, 2, 3, , we change from probabilities C^"^^ to annuities Cm , and the formula on the preceding page, from Milne, is verified. The symbolic form is evidently C^ (1 + C)-'*, developed as far as C»». REVERSIONARY ANNUITIES. General Problem. — To determine the present value of an annuity dependent upon the joint existence of the last n survivors out of m lives x, y, z, . . . ., after the failure of the joint existence of the last n' survivors out of m' other lives x, y', z', (Milne, p. 130). The leading principle of solution is very simple. As just described, let P' denote the probability that n or more of the m lives will survive h years. And let P" denote the like probability that n' or more of the m' lives will survive h years; then 1 — P" is the opposite probability, that they will fail. The joint probability is therefore P' (1 — P"). Multiplying this by v^, and taking the sum for all values of h, we have the required value of the annuity in terms of single and joint annuities. Also the required annuity and its components may all be temporary or all deferred for equal periods. Example 1. — Eequired the value of an annuity on the life of x to commence after the death of y. Single Premium r:: ax—axy Annual Premium = ^' 1 + flfary By aid of the above type P' (1— P"), this solution and more complex ones yet to follow, are easily written by inspection. To verify the present result, cancel the annuity on x, during the joint lives. This latter method in De Morgan's Essay on Probabilities, is termed '' the balancing of annuities." ^ For future reference, let another form also be given, which is easily proved by substi- tuting for N below, the equivalent series of D and canceling : dx — Ctxy — YY\ ~~~ * lyLfx r. _ (?y— ?y+l) Na;-hl + (?y+l — ^y+g ) J!^x+2 4- • »♦« 280 LIFE ANNUITIES AND INSURANCES. 2. Annuity on the life of x to commence after the first death oi y or z' the present value or Single Premium = ttx—Oxyz'y Annual Premium = ^^^~^xy z 1 -vClxyz v-T Ux — uxiiz — n r -i-v .X-U^Z- ^^^jy X S. Annuity on the joint lives x, y, to commence after the death of z : Single Premium = (ixy—dxyz'. Annual Premium = -— ^. 1 '\-^xyz 4. Annuity on the life of x after the death of both y and 2 : Single Premium = «a— «ary— «ar3 + «aT^« Annual Premium Ctx — (Ixy — dxz -\- ttxyz 1 + (^xy "I" ^X3 — (^xyz 5. Annuity on the survivor of x and y after the death of 2 : Single Premium z= ax-\-ay— ctxy —axz— Oyz + cixyz- Annual Premium = ^^+^y-«^ -^^^ gy£±fgJg. J- ~r f^xz ~r i^yz — Oxyz Verification of Formulas. — In their present state, the algebraic sura of the coefficients of a in the single premium is zero, indicating no immediate possession of the annuity. And the only change from the present state must occur by the first death of either x, y, or z; by making this supposition in the last five cases, and noting the results, the solutions given above are again verified. The same methods can be further applied to the solutions following. 6. Annuity on the life of the survivor of af and y, that is on either one, after the death of the other. Give an annuity to both, to be restored as long as both are alive: Single Premium = aa; + «y—2«a;y, Annual Premium = ^-r-^ '-^' 1 + a.. xy 7. Annuity to commence with the first death of x, y, or z, and to end with ue joint life of the other two, that is with the second death: Grant each pair a joint life annuity to be restored as long as all three are alive : Single Premium =• axy-{-(ixz-\-Ch/z — ^^xyzf Annual Premium = ^M±l^y^-^^^. -»• -v^xyz LIFE ANNUITIES AND INSURANCES. 281 PARTITION OF ANNUITIES. 8. An annuity on the last survivor of two lives x, y, is to be equally divided between them during joint life, and afterwards to go to the survivor. The present value of x'b interest = a^—^axy, y's interest = %— ^«a^. Their sum evidently makes up the whole value of the annuity. 9. An annuity for the longest life is to be divided at each payment equally between the survivors of three lives, x, y, z: Value of afs share = ax—^ {axy + dxz) + i^xyz • Here x may denote either one of the three lives. And generally for equal division of such annuity between the survivors of m lives successively, if [ ] denote that we exclude from Cm as before defined, every combination into which x does not enter* x's share = ax-i[Cl]+^[Cl]-i[(fm] + ....±-[Cl]. ft 10. Two lives x, y, possess an annuity on the survivor of them. If either dies before a third person z, the remaining annuity is to be equally divided between z and the survivor during their joint lives. Required the value of z's interest • z's interest = ^axz + ^yz—axyz. (Milne, p. 139.) 11. An annuity on the last two survivors of three lives x, y, z, is to be divided equally among them while all are living; and after the decease of any one of them is to be divided equally between the two survivors during the remainder of their joint lives. The value of x's interest = ^xy+h^xz—^dxyz- (Milne, p. 140.) 12. An annuity, after the decease of x, is to be divided equally between y and 2 during their joint lives ; and is then continued to the last survivor for his life. (Milne, p. 141.) y'& interest = ap—axy—^yz-i-jaxyz. 13. An annuity on the last survivor of x, y, z, is to be divided equally between x and y during their joint existence. After the decease of either of them, it is to be divided equally between z if then living, and the survivor during their joint existence. And when there is but one survivor left he shall receive the whole of the remaining annuity. (Milne, p. 142.) a:'s interest = ax—\axy—\axz-\-\cixyz. CONDITIONAL PERIODS. H. To find the value of an annuity payable during the joint lives of a;, y, and also during t years after the death of y, provided x shall so long live. During the first t years, whether y be dead or living, x will be entitled to the temporary life annuity, Nar-t-i— Na;4.< -n _ Da;_+« 282 LIFE ANNUITIES AND INSURANCES. After this, or at the end of any given year denoted by ^ + n, the annual payment will depend on the combined chance that x is living, and that y was alive t years previous. This expectation, on both multiplying and dividing by ?a;+<, also summing with respect to n, to the end of the life table, gives 'x 'y 'x ix + tiy i^x Adding this to the former result, we have the total value, ^X + t , V (*x f=x — \(tx+t — (tx+t.y)' 15. An annuity certain for the term of 15 years is to be enjoyed by P and his heirs, during the joint existence of two lives x, y. If that joint existence fail before the expiration of 8 years, the annuity is to go to Q and his heirs for the remainder of the terra of 15 years. To find the present value of P's interest, and that of Q. The value of P's interest for the first 8 years is evidently the temporary annuity on the joint lives x, y\ for the next 7 years, P's interest is the deferred annuity certain for 7 years multiplied by the chance that x and y will both live through the first period of 8 years to secure it. That is, Fs interest = «^_?^t8 [a,,^^^^-_:=f). The value of Q's interest will be obtained by subtracting P's interest from the value of the entire annuity — ; 16. A reversionary annuity to commence on the death of x and continue for the remainder of the life of y, is also made payable for t years, whether y is alive or dead : y's interest = Aa,.— ^^ ^tT {(^y+t—ax.y+t)- I Uy 11. To find the single and annual premiums for an annuity to commence at the death of y, and continue payable during the remainder of a^'s life, but to be payable only if y dies within t years. In this problem, which is quoted from the Journal, Vol. 9, page 302, the single premium will be iy + fiJx + t / \ Ux — ttxy pfj ydx^t — (ix+t.y+t)- • fy^x ^ And the divisor to give the annual premium, payable through t years, if the joint life so long continues, will be l + a^^ . We may observe that the single premium is evidently the first t terms of the series before stated on page 279. 18. Problem. — To determine the present value of an annuity on the life of x, to commence with the failure of the joint existence of y and z provided it be z who dies first. In case the ages of y and z are equal, the probability that z will die first of the two is ^. Therefore | the result of the preceding Problem 2, on p. 280, gives the value; Annuity on a: = ^ {ax—dxyy)- LIFE ANNUITIES AND INSURANCES. 283 Secondly, if the ages of y and z are unequal, in case x is the oldest life, and approximately in other cases, the chance that z will die first of the two is usually taken as ^ in practice. So that \ of the result of Problem 4 gives the value of the annuity on X, after the death of y. Problem 3 gives the value until the death of y ; and the sum of the two will express the total value : Annuity on x =z ^{ax+axy— axz — axyz) ; Annuity on a; = 2 .-^ -TT"? w = 1, 2, 3, ; Iz idly Ux That IS, Annuity on a; = 2 j ^ " • tz iy Ux Another solution of this Problem will be given in the next Section. For other combinations of Survivorship, reference may be made to the works of Morgan, Bailey, Milne, Sang, Farr ; and to the admirable papers of Makeham in the Journal, Vols. 9, 10, 12. ON SUCCESSIVE LIVES. Problem 1. — To determine the present value of a deferred income of $1 per annum to continue forever, the first payment to commence at the end of the death year of x. The principal - when multiplied by the rate i gives the interest $1 in each year perpetually. Withdrawing from this principal, the annuity of |1 during the life of x, we have the required result : Present Value z= -—ax = (1 + -I Aa;. Here the last member is found by merely substituting for a its equal, from the common formula 1—iax Ax = 1 — (1— v)(l + ax) = -J + * Problem 2. — " An annuity is to be enjoyed during the existence of a life aged x, and at the end of the year of his decease, a successor aged y is to be nominated, who is to enjoy the annuity during his life. Required the present value of the annuity on the second life y." lu the system of compound interest, if v" is the factor to discount a sum from the end, to its present value at the beginning of the third year, v' the factor to discount through the second year, and v the factor to discount through the first year; the factor to discount through three years is proved to be the product vv'v". So, the single premium A is evidently the factor to discount the insurance of $1, from the end, to its present value at the beginning of the whole term of insurance; and AA'A" would represent the factor to discount through three successive periods of life. In the Problem above, the value of II annuity during the life y, with first payment on the day of nomination, 1 + %. The factor to discount this sum from the future date of nomination, to the present time, being A^., we have the Present Value of the Annuity 284 LIFE ANNUITIES AND INSURANCES. During the second life = A^ (l + fly) = A^; (1+ •) (1— Ay). During the two successive lives = rt^ + Ax (1 + ay). For Oy, we have above substituted its equal, as in Problem 1. If a;, 1, 2, n — 1, n designate the successive lives in order, the present value of $1 annuity to continue during the nth. successive life-time beginning with an immediate payment, will be Aa;AiA2 A„_i(l + «n); or (1+-.J Aa;(l— A„) Ai Aa A„_i. Problem 3. — Required the total present value of 11 per annum during n suc- oessive lives, the first payment to commence at the end of the death year of the present life x. The result sought will be found by making m = 1, 2, 3, n, in the last expres- sion, and taking the sum of all the terms. Thus, omitting the common factor /1 + -.) A^, and making w = 1, 2, 3, n, the first result is 1 — Ai ; the second is Ai— Ai A2; the third is Ai A3— Ai A2A3, etc. From these, we see that all the terms will cancel each other, except the first and the last. Hence on restoring the omitted factor. Present Value = (l + •) ^x (1— A1A2A3 A»). Wlien n is very great or infinite, the right-hand product vanishes, since each A is less than unity, leaving a result which is independent of the future ages to be nominated : Present Value = ( 1 +~.j Aj When the successive lives, after the life x, are all to be of the same age y at nomination, the formula for the value of the wth successive life, given at the end of Problem 2, becomes Aa;(l-{-rty) Ay"\ - In like manner, the formula of Problem 3, for the sum of n lives, will be (l + i)A,(l-Aj). Problem 4. — " Instead of an annuity, a fine of II is to be paid at the end of the year in which each life in possession fails, being the time when the next sncceeding life is nominated. Required the present value of each fine. By Problem 2, when w = 1, Value of the first fine = Kx, w = 2, " " second fine = Aa; Ai , n = 3, « « third fine =Aa;AiA2. w = 4, " " fourth fine = Ac Ai As A3, etc. LIFE ANNUITIES AND INSURANCES. 285 The sum of the terms written, will evidently be the present value of the first four fines. Each term is readily derived from that which precedes, to any extent. The present value of the first n fines, if the lives are all of the same age at nomination, will be the sum of n terms above, as a geometrical series, of which the ratio is Ay ; and the sum is -~ — 7—^ ; or if w be infinite, the sum is :; -. — 1— Ay 1— Ay By subtracting the former expression from the latter, we obtain the present value of all the fines after the nth, that is, of the (w4-l)th and following fines to perpetuity : -A-a; Ay I^a;' The prece4ing summary is based chiefly on two original papers by Mr. Peter Gray in the Assurance Magazine, Vol. 2. In Vol. 21, a memoir and Table for the Enfran- chisement of Copyholds, is given by Edward Smyth. Further problems relating to English church livings and copyholds are resolved in De Morgan's Essay on Proba- bilities, pages 228-236. SECTION IX. Derivative Formulas. The Continuo-as Method. Besides the regular Commutation Formulas, several principles of mathematical analysis have conducted to other formulas, which occasionally offer new advantages. Especially the common method of Algebraic elimination has been the guide to valuable improvements. For example, writing x and then a: + l in the formula for the expectation of life, we obtained two equations, from which the series common to both was easily eliminated, leaving the simple equation between ex and ex-^-x shown on page 243. By the same general method, and in other ways, the following Formulas of Annual Derivation have been obtained : For the Expectation of Life, <3a;— ^ = i'x (^a;+i + i)- Annuity on a Single Life, ax = vpx (l + «a;+i). Annuity on Two Joint Lives, axy = vpxPy (l + «a;+i.j/+i)- Annuity on Three Joint Lives, axyz = vpxPyPz (l-f«x+i.y+i.a+i)- Temporary Annuity, a^ = vpx (1 +«"~i). Temporary Annuity on Joint Lives, a^ = vpxPy{l + ax+l.y+t). Insurance by Single Premium for Life, A^ = vpx \ — — l + A^+i i • By Single Premium for Term, A^ = vpx \ 1 + A^Ti {• • 286 LIFE ANNUITIES AND INSURANCES. By Single Premium on Joint Lives, Aay = vpxPy (— — 1 + A^+i y+i)- Single Premium of Survivorship Insurance of II to be paid at the end of the year in which x dies, provided y be the survivor, * Ab = vv.Py I \ {^-\) (^^ + 1) + A^^i I . The values of a;?^. and A {ipo^) are already given in Tables XXVI and XLIIL And at the oldest age of the life table, where these computations usually commence, ^99 is ; consequently ^99 = 1^; the annuity 099 = ; the single premium A99 = v ; and A^ = \v (l-f-/)y). From this limit, we can determine the other values in a connected series, by making x ■=^ 98, 97, 96, 95, ...., in successive formulas. The operations will be greatly aided by the Addition and Subtraction Logarithms given in the Tables and Formulm of Peter Gray, whose able researches first directed general attention to this method. For practical illustration. Tables XLIV-XLVII in the present series, were com- puted by the formula of Temporary Annuities, a^ = vfxi^-k-oi^x). First the ages 99, 98, 97, 96, . . . . , were written at the head of so many columns ; under these, the values of A {v'p) from Table XLIII for the ages one year younger, were copied, once for all, to be repeatedly added each in the same column. TEMPORARY ANNUITIES. COMPUTATION OF JL(l + aD. PATKEirrs n 96 97 98 99 Agb X. L681936 1627579 1.585026 L505846 "^{vpx-x) = Xa. 2 3 4 0.153574 0.141329 0.120743 A(l4-ai). L835510 T.768908 L705769 Sum = Aa. 0.200677 0.178369 A(l + a2). 1.882613 L805948 Sum = Aa. 0.214754 A(l + a3). Here at 4 per cent, interest, A (vj^gg) = L505846, A (r;?^) = L 585026, , A (H-«98) = 0.120743, A (1 + «^) = 0.141329, .... At the first, regarding A (vp) as Xa, we look out A (1 + a) in Gray's Table and copy it below, one column to the left. Every logarithm A (1 + a) so found is next added to A (vp) at the head of the column, and the sum written underneath. Again regarding each sum as Aa, the corresponding A (1 + a) is then found by Gray's Table, and copied as before, on the next lower line, one column to the left ; and so on. The alternate lines present a complete Table of A(l + «x~^) for Temporary Annuities. When n is 1, the tabular result is uniformly 0; and at the foot of the columns, the last A (1 + a) must coincide with the result of com- mon life annuities. To prove the correctness of the work, while in- progress, every tenth or fifteenth column can be computed by the independent Commutation formula or otherwise, for comparison and correction. It appears scarcely necessary to observe in the use of Gray's Tables, when the negative index of the argument is 4, 5, 6, etc., we find Log (1 + a;) as if the index LIFE ANNUITIES AND INSURANCES. 387 were 3, and prefix to the result of Table I, as many decimal cyphers ; or in the use of Table II, as many extra decimal 9's, as the excess of the given index above 3, denotes. Thus when Log a: is 5.500, Log {l-\-x) is 0.0000137, and Log {l—x) is 1.9999862. There are Tables to seven places by the German author Zech, and by Wittstein; Gray's to six places are more conveniently arranged for practical use. Formula of Valtjation. — Three different forms have been published, the first by Hon. Elizur Wright, under the name of " the accumulation formula," a second by Mr. D. P. Fackler, and a third by Mr. Emory McClintock. To demonstrate the last, let i denote the rate of interest; and v tl^e ratio \-^{l-\-i). Let Yx+n denote the Keserve at the end of n years on each $1 insured at the age of entry x, with the net annual Premium P^, ; so that Vx+w + Px will be the Reserve just after the Premium is paid. Assuming Ix+n persons to be thus insured for $1 each, the death losses in the ensuing twelve months, according to the life table, will be Ix+n—lx+n+i- Hence the Reserve with a year's interest, after payment of the $1 death losses, evidently gives '«+« (Vaj+w + Po;) (1 + *) — {Jx+n — '«+n+l) = '«+«+! *a;+n+l' Dividing by lx+n+\ and reducing, we have the required formula: Va;+n+l = 1— — [(v— Px)— Va;+n]. f^Px + n At the age x, just before the first premium is paid, the Reserve V is evidently 0, which is the initial value on the right of the last formula, when n is 0. Also at the end of the last year, V on the left becomes when a temporary insurance expires, or 1 when an endowment matures, or 1 when a common life policy attains the oldest age of the life table. Thus the computation of the formula may commence at the begin- ning, or at the end of the policy. Table XLIII will give the values of A {vp). By making the annual Premium P to be 0, the formula becomes applicable to paid-up Policies. Consequently one year before maturity, or just after the last annual premium is paid, the Reserve on every $1 of endowment, will be v. In the higher analysis, all the preceding formulas would be classed among "equations of finite differences of the first order;" which are always integrable, — Boole's Finite DiJEEerences, Oh. IX, Art. 8. But the integrals are already known in the Commutation and other forms. THE CONTINUOUS METHOD. Maclaurin, Euler, and many others have demonstrated the following standard formula, which will serve the double purpose of summation, and conversely of quadra- ture. First, let the values of the ordinate u be taken at the distance h from each other; and let TJa denote the special sum indicated; then (A) hJJh = h (iWo + Uh-ht(ih+ + U(n- l)A + iUnh) = C+fudx+-^i^.^£-^y¥.^,+^hn^¥.'^^-..., For the present applications, let nh fall beyond the oldest age of the life table, where Unh and its differential coefficients all vanish. For, by Taylor's Theorem, when a continuous function becomes permanently zero beyond a certain limit, its derivatives 288 LIFE ANNUITIES AND INSURANCES. % also vanish. Taking the integral of the right-hand member between the limits nh and 0, and making A to be 1, ^nh rill. d^uQ ^1 = y ^^^-^^'^-^^ dx ' ^ " ' d3^ Subtracting this from the former equation integrated between the same limits, and transposing, (B) Ui = hJ5,+i, {h^-1) ^^-^i^ ih^-l) ^+ . . . . Application- to Life Annuities.— Referring back to the common formula Da;«x = Na;+i = Dar+i + Daj+2+ , let US here make uo = T)x, wi = D^+i, etc. Then will Ui = p^+Da;+i+ . . . . = iD,, + Nx+i, and Ua = iT>x+'D^+h + 'Dx+u+ Substituting in the formula (B), dividing by Da;, and reducing, (C) aa; = ^{T)x+h + 'Dx+2h + 'Dx+sh+ .■.•)-^^{h-l)—^\{k^-l){fi^ + d). The values of D are given in the Commutation Tables, and the values of /* and 5, in Table XLIIL The latter represent the diflferential coefficient of D^ or IxV^, divided by Da;, that is, of log Da;; since n = —dlx-—l:^x, and d — —log v = log (l + O- Ac- cording to W. S. B. Woolhouse, the pioneer of this mode of calculation, in the Journal, Vol. XI, page 323, the best intervals to adopt for h, will be 5, 7, or 11 years. In the case of two Joint LivQS, x, y, the derivative ft^. will become [loey ov (tx-\-fiy; and so on. Example. — Required the value of the life annuity ax, when x is 60 years, and interest is 5 per cent. A = 11 ; flreo = i-(D7i + D82+D93) + 5.-9.94(j«60 + <5) = 9.1322. A = 7; «6o = ^-(D67 + D74 + D8i + D88+D95) + 3.-4(|«60 + tJ) = 9.1336. The standard value from Table XXVIII is 9.1322. In the above formula for 11 year intervals, the last coefficient was arbitrarily changed from 10 to 9.94 to compensate for omitted terms. The annuity so computed for age 40 is 13.7601, which differs very slightly from the true value 13.7602. With this mode of amendment, devised by the writer, the intervals h may probably be taken so large that only two or three values of D will be needed. Thus for thirty-five or other ages adjacent : 20 A = 20; ax = TT-P*+20+^a'+4o+Dx+6o) + 9.5-32.7023 {iix+H). Thus with slight correction of the last coefficient, through a single central value of a, this method (B) may also be extended to joint lives and survivorships; and further trials are invited. Instalments. — Let h now designate a fraction — of 1 year or of II annuity; and let the equal instalments be payable at the end of each interval A or — of the year. In formula (C), the present value of the 1st instalment is Da;+A-^Da; when the LIFE ANNUITIES AND INSURANCES. 289 instalment is 1 ; or it \s> hy.'^x+h^^x when the instalment is %h px -; and so for the following terms of D, which evidently make up the required present value «(»*). Therefore, substituting 1-4-m in place of h, and transposing the terms, we find the present value of the annuity payable in instalments of $- at the end of equal intervals — of each year, ♦ (D) «' =''«+^2^-Tayf'*-+')- By adding - , we evidently obtain the present value of the annuity payable m times a year in advance. This wfll also be the divisor of the single premium to give the premium of insurance by m annual instalments. The present addition of — gives Continuous Annuities dx- When the equal instalments are so small as to be payable momently, the corresponding value of the annuity, denoted by Ax , is readily found from formula (D), by making the number of intervals m very great or infinite : (F) «ar = «a; + i — 1^ {l^x + (^). ^ axy = «ccy + i — tV (Ma; + /^y + (J). On the Survivor of Two Lives, d^ = dx + dy—dxy, , 6 = ax + tty—axy-'r^—zrx' Continuous Expectation of Life, ex = ex—^f^x' ^ = ^xy — ^ {f^x-hH'y)' Complete or Apportionable Annuities cox. The characteristic c is prefixed, when the annuity is to be paid with a proportionate part to the instant of death. A close and very simple approximation for this case has already been given in the Note of page 277; that is, cax = ax + ^Ax = {l—yv)ax-\-^v. Here the proportionate part, defined by the actual day of death, is to be paid at what would be the time of the next regular instalment of annuity. The more exact solution first investigated by Mr. Sprague is verified by Mr. Wool- house in the Journal, Vol. XY, page 107, substantially as follows : Let x' denote the age at the beginning, and x'-}-- at the end of any regular interval. And let x'-i-f denote an intermediate age, increasing uniformly to x' -{-- , so that if death should m occur, U will be the proportionate payment. By page 252, ante, and since M is a decreasing function, the single premium to insure this ^t during the instant dt, is —t.dMx'+t^'Dx- Or integrating by parts from ^ = 0, to t = ~, we find the single premium to insure through this interval to be 290 LIFE ANNUITIES AND INSURANCES. Ma^+l + fmUaZ+tdt, 1" m both terms to be divided by Da; . From this expression we next find the sum or single premium for all the intervals by formula (A), which gives the total augmentation of annuity : .2m '^^ 12w2 dx ••• j • -^ 2?w -^ l%m? The mark over M denotes its continuous value ; and so of the single premium A. It will presently be shown that Ma;-^D^ = Aa; = \—6.dx, where <5 := log (l + «) ; also • that d^x = — v^.dlx. Therefore, adding this augmentation to formula (D), we obtain the complete annuity payable with a proportionate part to the day of death : (G) cai"'> = (l-2^)«-.+ ' 12m2 =(-.^.)(''-*-''^V 12m^ Change from Ordinary to Contintjous Annuities and Premiums. — By simply changing the common annuity «x to «a; + i> which will rarely require the slight further correction shown in formula (F), we obtain the continuous annuity ax- And conversely, Ox = (fx — i? very nearly. In respect to Premiums, the operations already explained on page 264, namely, A := A-^-Vv, and A = A Vv, prove to be sufficiently exact in common practice. For illustration, a large variety of single premiums are represented by the wxll known for- mula, A := 1 — (1 — v){l-\-a). Eliminating the annuity a by comparison with the third tjrpe of continuous equation stated below, we have (H) A = jA_A + ;;^ = Anearly. At the age of 40 years, with 4 percent, interest, /x(J-i-12 is 0.000030; and «y-f-(l— v) differs from ^/v by *2-^24 nearly, Avhich is 0.00006. These small terms being neglecti- ble except in rare instances, the formula A = A-^Vv, is so far verified, with its con- verse A = A Vv' Forms of Continuous Single Premiums. — Besides these simple modes of trans- formation, the continuous method presents the single premium under four new forms. As will be demonstrated presently, according to Mr. Woolhouse's analysis in the Jour- nal, Vol. XV, pages 123, 409, the Jirsi form, separating the symbols of operatioiT, is (I) A=(^,-^)«- + (^,-|)a+.... This type will be of general application, if A, a, depend on all the lives ; while X, y, , designate only each life, by whose death the insurance would become payable. Thus, A -ad-^^- A -f«4-uU/ _^_^. etc LIFE ANNUITIES AND INSURANCES. 291 A^ = l^xttxy ^ ; A- _ HyCl^cy ^- -j-l . "'ttxyz T" 1 - ddxyz , A— = ^« axyz ^— • ^aya + ^xyz + ^xyz — -^'^«- The second and i(AiV<^ forms following apply exclusively to absolute assurances, having the common relation A = 1 — (1— r) (1+a) : (J) Xx = l—6dx' 'Kxy=l—Ux,j', Aseyz = 1—^dxyz; etc. (K) ^. = cj(J-e/.+g); Axy = 6(]-axy+^-''^); etc. Here « denotes the ordinary tabular annuity ; and in both, S = Nap. log. (l + «). The fourth type is but another form of the first, where x, y, , designate only each particular life, by whose death the insurance would become payable. Thus, (L) A, = -|-^(«; A^ = -t^(^-«^)-r/|(^^^'^>' "^• The derivatives with respect to age, as x, which occur in the preceding equations, (unlike derivatives with respect to v), are here determined with sufficient accuracy by first differences. Thus, „. da A^rt , A% Since -J- — A« K--{- dx~ 2 ' 3 ddx ^ , \ dttxy dOmi ^ I , , — -^z=.\yax-\ — flfoj+lj; -^ t — =: ^ (Oas— l.y— l — «a! + l.y + l) ; CtC. "" ~^ ~ ^ (^'^^-i y~^+iy)' — ~^ — ^ i^^v-i — (^x.y+i); etc. dlx 'x—1 — Ix + t a\^ 'x—iax—l.y — 'x+lttx+Ly fix — J- — ^j A(ry — ^ As heretofore, the single premium, when divided by 1 + ««, for a single life, or by 1+axy for two joint lives, will give the annual premium. Demonstrations.— It is shown in the Differential and Integral Calculus, in relation to the area of a curve whose ordinate is denoted by //, that fydx represents the sum of yAx, when the difference of abscissas a a; is diminished to its limit. Accordingly, let the commutation Da; = ,y. In the common sum of annual values, Na, = D^+Dai+i + Da!+2+ , the value of Arr is 1 year or 1. Diminishing Aa; to its limit dx or dt, we have the continuous value N = fDx+tdt. In the similar sum, Ma, = (4,— /x+i) ?^^^ + (/a5+i — ^aj+a) v*+^+ , the value of AX is represented by Ix—^x+i. Diminishing this a to its limit —dlx, since l^ is a decreasing function, we obtain the continuous value ; then integrating by parts, 292 LIFE ANNUITIES AND INSURANCES. M = —fdl.if = —lif^log V flvf^dx. Dividing by Da,, we verify the second type of equations, before noted, A=l— (J.«. For another standard example or application : loi.ax = Ix+t-vKdf; Ix'-^x = —fvKdlx+t' Integrating by parts, and observing that v = e^og^; 6 = — log v = log (l + ^); ^ Aa, = — v" ■lx+t + log vf Ix+t- v*dt. Applying the limits ^ = 00 and t = 0, also substituting l» dx for its equal, ?a,Aa, = 4 + log V.Zaj-«a,; A^=zl— 6. da,. In another form, noting the principle of the Calculus that —^ = j±^ ^ a,nd differentiating the first equation with respect to x^ then substituting in the second, , -T- _ d {Ix dx) -T- _ . ddx 1 1 1 A By substitution of the common annuity, since ^ = -.+ - — —^ nearly, •^ *" <5 t 2 12 *' ^x = l-6.d^-^6(^j-ax + l^' In this manner the four previous forms of equation are demonstrated, and the methods of integration are general. For the insurance of $1 payable on the death of x if y be living : l9ilydxy = flx + tly^^tV*dt', IxlyA-sy ^ — J iy + tV^ -d Ix + t • For three lives, Ix ly h k^z = — / ^y+< hrt v^ • d Ix+t • ApPLiCATioi^s OF THE Law OF MORTALITY. — The preceding results are evidently general, and adapted to any life table. But according to the law of Makeham, on page 237, with hyperbolic logarithms, the value of ^ is found as follows: Na, = /Bx.dx = fh{vsY.e-^.dx', if w = log - . g* ; du ■= log q . udx. Let r = l^i-(^ + 1. (vsY = e" ^""^ ^""^ = n^-^ . (log -) "^ Changing entirely from the variable a; to w, we obtain the known form of the Gamma function, or second Eulerian integral, tabulated by Legendre, with an outline Table in De Morgan's Calculus, page 587. By the latter treatise, pages 577, 590: LIFE ANNUITIES AND INSURANCES. 293 /« E-^.u^-^.du = r(r-i) = r(r)-j-(r— 1) = r(i+r)-i-r(r— 1). /=0 /TOO />|^ — J — j , we have log? log a J u 1 "^, 1 r'^r'^r(r + l)"^r(r+l)(r + 2)'^'*" f r.loff — log — ^ vs ° vs Dividing by Do.- , of course, gives the annuity dx . For numeric calculations, reference may be made to more full developments in the Journal, by Mr. Makeham, in volume XVII, page 305, and by Mr. McClintock, in volume XVIII, page 242. An interesting feature is the rapid convergence of the series in the earlier ages of manhood. At age 40, u is 0.030813; at 60, u is 0.20622; at 80, u is 1.3801. Integration of (N) by parts will easily show the following Relation between two life annuities, at equal ages, but at rates of interest differing nearly 10 per cent., or as the rate of v differs from that of v'; the logarithms are hyperbolic: :^ 1 , log q , s" log q For ; I joint lives ; log --;,' a^ + - — y' log a^ = 1 ; - — t- = 0. 9532326, to q J- pxy-'- q ■■- For one life ; log — . cby + u. log q. dy' = l\v' = vq; u = log—. g-*'= — - — Z' Demonstration of the Formula of Survivorship Insurance on P. 263. Having tabulated the coefficients a' and b', let us here adopt the notation, and the two auxiliary ages, X, y, stated on page 263. The probability of the m lives all surviving the time t is J J - — — — or s'^Kg^^^^ -i); and the probability of the n other lives surviving the toj/ . ta// .... time t is entirely similar. Hence by the same reasoning as for two single lives, x, y : -T-l r ^ {^<^+t • ^'«f'^t • • . ) ly'+t ■ ly"+t • • • • , ^ la/ • Ix" • ■ • • h/.ly't . . . . = —/{m log s.dt-\-\ogg.q^. dq^) . 6('"+'») Kg^*^^) (s'-i). Here the differential has been more conveniently found by the latter form, of the identity rf? = l.d log ? = '..y'... Again, eliminating a, by the relation A.r'..y'.. = 1 — 6da/,.y>,,, we obtain the general formula on page 263, which was to be demonstrated. Multiplying by ■y/v as there described, , 1^ ,, _ m mgv-^—n/ 1 1 1\ ^^s' ~ m-{-n qy-'' + \\m + n'^'6' ^^ ~s)' 5^^ = a' + i'Aa^ xy The Table of a' and AJ' on page 263 is based on the values m z= n = 1. But if a" and h" denote the values in any case where the numbers of lives m and w are equal, we have a" = na' ; b" = b'—{n — l) a'-^Vv. Probability of Survivorship.- For two lives, the probability that y will sur- vive X is strictly the value of A— when v is 1, and «'«;r, we should be able to calculate the numeric value of double survivor- ships in any case with the greatest facility. " "A satisfactory solution has never 296 LIFE INSURANCE AND ANNUITIES. yet been published^ determining the value of a reversion on x subject to y surviving during the life-time of j;." — 3Ii1ne, Problem V. Among numerous applications to wills and probate business the following illustration is here adjusted from Griffith Davies: " Suppose z, aged 45, to be entitled to an estate in the event of his surviving his father, y, aged 60, and desirous of providing an annuity of $500 to his wife, x, aged 30, to commence at his death in the event of his dying before his father; it is required to determine the present value of his wife's contingent annuity." This value $1,176 shows at how little cost a provision can be secured against the contingency to which the wife is exposed. It is surprising that such insurances are not more frequent. SECTION X. Insurance on Lives according to the Order of Survivorship. Problem I. — Insurance on the Longest or Survivor of Two Lives. The premiums are already defined on pages 257 and 278, including the reserves on page 271, to be reviewed, and especially the general mode of solution shown on page 261. Problems II.-VII.— A General Insurance of $1 on One of Several Lives. — Such are the insurances payable at the end of the year of fiy^st deaths or of second death, or of third death, or of the last death hut one. Their annuities, given on page 278, may be used to enter Orchard's Tables (or the common formulas of single life), to obtain the proper single or annual premium. Note especially Problem V., already resolved. These general insurances are comparatively few in number. The standard treatise of Milne (London, 1815) resolves the more numerous class of problems which depend on a particular or assigned order of survivorship. Many of his solutions have since been simpli- fied by Makeham and others; and his system of numbering the particular problems is conveniently retained here, with select formulas. Problem VIII. — Required the Single Premium to insure $1, to be received at the end of the year in which x dies, provided another life y be then living. Especially note that the life y can be replaced hy y z, or any number of joint lives, as in Problem X. Extensions to four lives are given by Makeham in the Journal (London), Vol. Xll. The accurate formula of Milne, already copied on page 261, is expressed in three terms, followed by our new method: Then another was discovered by the writer for correction, in combining page 291, lines 1 3 and 19, with the well-known 2/jr= ?;r— i + Ix+i an + (ix + 1^. Thus: Single premium=Ai;y== '^~ ^~ — a^-y-i + C. On trial, 6=0.01 ; c =0.4. Px—\ Computed A 0.5 0.6 0.7 0.8 A. Correction C, add 0.0005 .0012 .0021 .0032 6 A (A— c). LIFE INSURANCE AND ANNUITIES. 297 Problem IX. — Insurance of $1 on the particular life a-, provided he die last of the two lives a;, y. Verify below, or on page 262, by supposing x to die first or last: Single premium=A;,r — Ai^. Annual premium divisor, 1-^axy. Survivorships of Three Lives. — For more convenient reference, the following Problems of Milne, with improved solutions hy Makeham and others, are here denoted by index notation. The condition of first, second or third death is defined by the suffixes 1, 2, 3, under the proper letter; also, the life z is not insured like that of x or y. In {3'^) either x or y may fail first or second. Problem (10) xyz ' 1 (11) xyz 2 (12) XIJZ 3 (13) ' xyz 1.2 (14) xyz 1.3 (15) xyz 2.3 (16) X'yz 1 1 (17) (18) X'yz X'yz 2 2 3 3 (19) (20> (21) (22) (23) (24) (25) (26) (2_7) xyz 2 1 X yz 1.21 xyz 32 xyz 1.3 1 xyz 2.3 1 xyz 1 xyz 3 xy z xyz 2 1.2 (28) (^9) (30) (31) (32). (33) (34) (35) xyz 1-3 xyz 2-3 xyz 2 xyz 1 x-y z 1-2 1-2 X'y z 1-3 1-3 ; X'y z X yz. 2-3 2-3 1.2 1 Problem X. — Insurance of $1, payable if x dies first of the three lives x, y^ z. For a new method of solution, discovered by the writer in 1898, simply annex z to y and z — I to y — 1, and then proceed as in Problem VIII. By the Method of pages 263, 294; while the divisor to give the annual Premium is l + a;ry^; and at 4 percent, interest, ^ \ocr -=0.15812: Nap.log -=0.0027466 S Ai ri'-\-h'A I ^ (m+n) q^ -/v , _1 -,, 1 ivz=«+^^^>^. a-^m— ^^, /^ .log-. h'=——-_ q^'+qy+q^' <^ ' s' q'^+qy-hq^ -/v Problem XI. — Insurance of $1, payable if x dies second of the three lives x^ y, z. The solution of Milne by symbols is readily verified by deaths. The divisor to give the annual premium is 1 + €1;^^ + ci^z — a^^yz- Single premium = A^^+ A^^— 2 A^^^. Problem XII. — To determine the present value of $1, payable upon x failing the last of three proposed lives, a-, y, z. Solution: Substituting from Problem XI. and transposing, evidently, A^^^+ A^^^-f-A3^^==A.^; page 280, line 9: Problem XIII. — To determine the present value of $1, payable upon x failing either first or second of the lives x, i/, z. Grant a single insurance on x, less another by Problem XII. if y and z die before x. Single premium = A^ — A|. ^ 298 LIFE INSURANCE AND ANNUITIES. Problem XIV, — Insurance of $1, payable upon x failing ihe first or last of the lives x, y, z. Grant a single insurance on a*, less another by Problem XI, Single premium ^ A^ — ^ly^« Problem XV. — Insurance of $1, payable upon x failing either the second or third of the lives x, y, z. Single premium = A^ — ^L^. (Problem X, ) Problem XVI, — Insurance of $1, payable on either x or y failing the first of three lives x, y, z. Evidently, Single premium = A^^^ + A^i^ == A^^^ — A^^]. Problem XVII, — Insurance of $1, payable on either a; or ^ failing the second of three lives x, t/, z. Single premium = A^^^ + A^^^. (Problem XI.) Problem XVIII, — Insurance of $1, payable on either x or ^failing the last of. three lives x, y, z. Single premium = A^^^ + A^j^. (Problem XII, ) Problem XIX, — Insurance of $1, payable on x failing second of the lives X, y, z, provided z fail first. If either x or y dies first, the two terms of the premium below cancel or vanish. If z dies first, the last term vanishes; leaving the first subject to the death of x, as the second in order. Single premium = A^^ — A^^^. Problem XX. — Insurance of $1, payable upon x failing either first or second of the lives x,y,3j provided, in the latter case, that y fail first. By symbols, ^(A]^^ + A^] = A^). . Single premium =A^^. Problem XXI. — We come now to the more difficult problems of Milne, simplified by Makeham in the Journal, Vols. X and XII, for computation, as follows : To determine the present value of $1 payable upon x failing last of the three lives x, y, z, provided that y fail second. Single premium = A^^^ — A^j^ — A^^ — (l—v) ^^a^. Problem XXII. — To determine the present value of $1, payable upon x failing either first or last of the three lives x, y, z, provided that in the latter case z fail first. Single premium = 2A^^^ + A^^i — K\y — (1— ^•)^y«;r« t LIFE INSURANCE AND ANNUITIES. 299 Problem XXIII. — To determine the present value of $1, payable upon x failing either second or third, provided that z fail first of the three lives x, y, z. Single premium == A^^i — ( 1 — ^' ) ^^ a^. Problem XXIV. — To determine the present value of $1, payable upon the failure of the last survivor of the two lives x^ ij^ provided that they both survive a third life, z. Single premium = A^^^ - A^^ + A^ + ^^a^ — (1 -^)^^a^ Problem XXV. — To determine the present value of $1, payable at the failure of the last survivor of the two lives x^ y, provided that they both fail before a third life, z. , Single premium = A^^ + A^^ — A^^^ — A^^^. Problem XXVI. — To find the present value of $1, payable upon the failure of the last survivor of the two lives x, i/, provided that one of them fail before and the other after a third life, z. Single premium = A^ + A^ — A^^ — AJ,^ — 2 A^^^ + (1— v) {^^a^ — ^^,a^) Problem XXVII. — Insurance of $1 on the survivor of two lives, x, y, provided that a third life z fail, either first or second of the three. Single premium =A ^ + A^ — A^^ + A^^^ — A^^ — A^^ — A^^^. Problem XXVIII. — Insurance of $1 on the survivor ©f *W0 lives, x, y^ provided that a third life z fail, either first or last of the three. Single premium = A^^+ Aj,^— A^^ + 2A^^^ — {l-v) {y^a^ + ^^a^). Problem XXIX. — Insurance of $1 on the survivor of two lives, x, y, provided a third life z fail, either second or last of the three. Single premium == A^ — A^^l — (1— ^0(^y«;r + :,^y) • Problem XXX. — Insurance of $i on the survivor of two lives, x, y, provided that y fail after a third life, z. See page 257. Single premium == A- — AJ^ + (1— v) ^^a^. Problem XXXI. — Insurance of $1 on the survivor of Xy t/, provided that y fail before a third life, z. Single premium = A j,, — (1 — v) ^^a^ 300 LIFE INSURANCE AND ANNUITIES. Problem XXXII. — Insurance of $1 upon either x or y failing either first or second of three lives, x, y, z. Evidently, ^ Single premium = A^^^ + A^j^ + Ay + A^] = A^^. Problem XXXIII. — Insurance of $1, payable upon either x or y failing either first or last of the three lives, x, y, z. See page 257. Single premium = A- + (1—?;) i^ya^ + ^^A^). Problem XXXIV. — Insurance of $1, payable upon either x or y failing either second or third qf the three lives, x, y, z. Single premium = A^^ — {l—v) {^^a^ + ^^,a^). NOTES AND ADDITIONS. Among the ADDITIONS to the present Revised Edition of 1899 will be found the critical correction of the usual few Errata discovered during the past twelve years; the instructive Draw^ing on page 2; Return Premiums developed on page 253; Paid-up Insurance on pages 271, 272; new modes of Integration on pages 293, 295; new explanation of Least Squares on page '*2o4; and especially Section X. On Survivorships, wherein the new formulas open a wide field for this species of Life Insurance. On the Law of Mortality. — Besides Final Series and other new researches given in the text, a comparison of pages *212 and 301 suggests that the convenience of Makeham's Formula may be united to entire generality of Formula, by vanishing exponentials. Let us begin with the primitive equation on page 239, line 4, which has the form of Newton's Law of the Radiation of Heat. In its second integral on page 237, line 38, after the manner of Lagrange's variable parameters, substitute thus, x\s = xXsi 4- x^Xs^ + X^XSg + Then the equations of condition for the whole life are reducible by Least Squares. And as in Meteorology the climatic cycle of 35 years is needed to de- termine the constants, so in Mortality the experience of one, two, (three score years and ten) or more cycles may be required for the highest accuracy. Historical. — The Life Insurance Year Book (Spectator Co., New York) has the following notes by the author: " For 1887, pages 321-324, American Life Tables from combinations of the Census. For 1888, pages 117-123, On the Measure of Mortality and on Recent Vital Statistics. For 1897, pages 140- 143, A Perpetual Calendar, and Notes on Ancient Chronology." Continued from The Spectator (Weekly): For Sept. 17, 1891, Reductions of the Thirty American Offices. Conversion from Calendar to Policy years. For April 1 1, 1889, Ages of Entry in Life Insurance. For May 17, 1894, Waves in Meteorology and Mortality— Compare page 240, lines 2-5. — Cycle of 35 years. — Deaths by Brain and Nervous Diseases are one-seventh part of the whole. For Dec. 11, 1890, Man and Longevity in America. '212a FEW TABLE LXXII. 301 PRESENT VALUE (ajrx) OF JOINT LIFE ANNUITIES ON TWO EQUAL LIVES. CONTAINING MONTHLY DIFFERENCES, M. D., AND THE CORRECTION FROM PAGE 258. Is 3 M.D. 3/2 M.D. 4 M.D. •3 « ® 3 M.D. ^y^ M.D. 4 M.D. a< be Per Ct. Subt. Per Ct. Subt. Per Ct. Subt. Per Ct. Subt. Per Ct. Subt. Per Ct. Subt. 10 18.1553 75,5 16.7903 61,1 15.5954 49,8 55 8.9127 262,0 8.5937 245,0 8.2936 229,5 11 1S.0647 78,3 16.7170 63,7 15.5357 52,1 56 8.5983 262,6 8.2997 246,3 8.0182 231,0 12 17.9707 82,0 16.6406 66,9 15.4732 54,9 57 8.2832 262,4 8.0041 246,4 7.7410 232,0 13 17.8723 85,3 16.5603 69,9 15.4073 57,6 58 7.9683 261,9 7.7084 246,8 7.4626 232,7 14 17.7699 88,7 16.4764 72,8 15.3382 60,3 59 7.6540 261,1 7.4123 246,4 7.1834 232,9 15 17.6635 92,4 16.3890 76,2 15.2658 63,2 60 7.3407 259,1 7.1166 245,7 6.9039 232,6 16 17.5526 96,3 16.2976 79,8 15.1900 66,5 61 7.0298 258,0 6.8218 244,1 6.6248 231,8 17 17.4370 100,2 16.2018 83,3 15.1102 69,6 62 6.7202 255,2 6.5289 242,6 6.3467 230,6 18 17.3168 104,2 16.1019 86,9 15.0267 72,8 63 6.4140 252,2 6.2378 240,1 6.0700 229,0 19 17.1918 108,4 15.9976 90,8 14.9393 76,4 64 6.1114 248,8 5.9497 237,3 5.7952 226,7 20 17.0617 112,5 15.8887 94,6 14.8476 79,8 65 5.8130 244,7 5.6649 233,9 5.5232 223,8 21 16.9267 116,8 15.7752 98,5 14.7518 83,5 66 5.5194 240,3 5.3842 230,3 5.2546 220,8 22 Id. 7865 121,4 15.6570 102,8 14.6516 87,3 67 5.2310 235,4 5.1078 225,9 4.9896 217,0 23 16.6408 125,9 15.5337 106,9 14.5468 91,3 68 4.9486 230,4 4.8367 221,6 4.7292 213.2 24 16.4897 130^5 15.4054 111,2 14.4373 95,2 69 4.6721 224,4 4.5708 216,3 4.4733 208,5 25 16.3331 135,6 15.2719 115,9 14.3230 99,4 70 4.4028 218,6 4.3113 211,0 4.2231 203,8 26 16.1704 140,1 15.1328 120,3 14.2037 103,7 71 4.1405 212,1 4.0581 205,1 3.9786 198,4 27 16.0023 145,3 14.9885 125,0 14.0793 108,2 72 3.8860 208,8 3.8120 199,1 3.7405 192,9 28 15.8280 150,2 14.8385 129,8 13.9495 112,7 73 3.6393 198,8 3.5731 192,9 3.5090 187,2 29 15.6477 155,1 14.6827 134,5 13.8142 116,9 74 3.4007 191,3 3.3416 186,0 3.2843 180,8 30 15.4616 160,6 14.5213 139,5 13.6739 121,8 75 3.1711 184,3 3.1184 179,3 3.0674 174,5 31 15.2690 165,6 14.3539 144,5 13.5277 126,6 76 2.9500 177,0 2.9033 172,5 2.8580 168,2 32 15.0704 171,1 14.1805 149,8 13.3758 131,5 77 2.7376 169,2 2.6963 165,2 2.6562 161,2 33 14.8651 176,2 14.0008 154,8 13.2180 136,4 78 2.5345 161,7 2.4981 158,0 2.4627 154,5 34 14.6537 181,5 13.8151 159,9 13.0543 141,4 79 2.3405 154,1 2.3085 150,8 2.2773 147,6 35 14.4359 186,5 13.6232 164,8 12.8846 146,2 80 2.1556 146,5 2.1276 143,6 2.1002 140,7 SQ 14.2121 192,0 13.4254 170,2 12.7092 151,4 81 1.9798 139,3 1.9553 136,7 1.9314 134,1 37 13.9817 197,5 13.2211 175,7 12.5275 156,8 82 1.8126 131,7 1.7913 129,3 1.7705 127,0 38 13.7447 202,4 13.0103 180,7 12.3394 161,7 83 1.6546 125,2 1.6361 123,1 1.6181 121,1 39 13.5018 207,7 12.7935 185,8 12.1454 166,7 84 1.5044 118,5 1.4884 116,6 1.4728 114,7 40 13.2526 212,5 12.5705 190,8 11.9453 171,8 85 1.3622 113,2 1.3485 111,5 1.3351 109,8 41 12.9976 217,8 12.3416 196,1 11.7392 176,9 86 1.2264 110,5 1.2147 108,8 1.2033 107,3 42 12.7362 222,3 12.1063 200,7 11.5269 181,7 87 1.0938 109,6 1.0841 108,1 1.0745 106,8 43 12.4695 227,2 11 8655 205,8 11.3089 186,7 88 0.9623 119,1 0.9544 117,8 0.9464 116,5 44 12.1968 231,4 11.6185 210,2 11.0848 191,4 89 .8194 101,7 .8131 100,2 0.8066 99,7 45 11.9191 235,9 11.3662 214,8 10.8551 196,1 90 .6974 79,9 .6929 79,8 0.6870 78,3 43 11.6360 240,8 11.1085 219,2 10.6198 200,6 91 .6015 71,8 .5972 71,2 0.5931 70,6 47 11.3481 243,7 10.8455 223,3 10.3791 204,8 92 .5153 67,7 .5118 67,1 0.5084 66,5 48 11.0557 247,2 10.5776 227,0 10.1334 208,9 93 .4341 61,1 .4313 60,6 0.4286 60,1 49 10.7591 250,5 10.3052 230,6 9.8827 212,8 94 .3608 55,5 .3586 55,0 0.3565 54,9 60 10.4585 253,2 10.0285 233,9 9.6273 216,3 95 .2942 62,0 .2920 61,5 0.2906 61,3 51 10.1546 255,9 9.7478 236,8 9.3678 219,6 96 .2198 40,8 .2188 39,5 0.2170 39,1 52 9.8476 257,9 9.4636 239,4 9.1043 222,6 97 .1708 63,9 .1714 53,3 0.1701 45,6 53 9.5381 259,8 9.1763 241,8 8.8372 225,3 98 .1061 00,8 .1074 00,3 0.1154 23,8 54 9.2263 261,3 8.8862 243,7 . 8.5668 227,7 99 0.0868 NOTATION. i = interest on 11 or any other monetary unit for one year. V = :. : = the present value of 1 due a year hence. 6 — log (l + i) = 'Nap\og {l-\-i), the nominal yearly rate of interest when compounded momently. Log A = AA = notation for the common logarithm of A. log A =: Nap log A = notation for the hyperbolic or Napier logarithm. e = 2.718 281 828 = the number whose hyperbolic logarithm is 1. M = A£ = 0.434 294 482. AM = 1.637 784 311. 4 = number living according to the Life Table, at the precise age x. dx = lx—lx+\ = the number of persons dying between the ages x and x + 1. Ig-y =: Ix.ly = uumbeT living in the Joint Life Table at the combination of ages x, y. dxy = lxy—lx+\.y+Ti = annual decrement in the Joint Life Table. Px = -4^ = -^— ^ = the probability of a person whose age is x living one year. 'x Ix g'ar = y = ^ 7 ^^^ = 1 —Px = the probability of x dying within one year. 'a; ix Pxy = '^t ' j^ - = the probability of the joint existence of two lives x and t/, con- Ix- iy tinuing one year. qxy = 1—pxy = the probability that one of the lives x, y, may die within one year. ^x = — 1—^ = ~ j~ log ^x = limiting ratio of mortality at the age x< Ix ax dx l^x = ^^ = - — ^j—— » approximately. Px, n = -y^ = the probability that a life aged x, may survive n years. 1 —Px, n = the probability of x dying within n years. Pxy, n = '""^;"''/ — = the probability that both x and y will survive n years. NOTATION. 303 Co, = 1+ '^"^ '»+^-r ■ • •; _ |.j^g « expectation " or average duration of life. e^—^ = the "curtate expectation " of life after the age x. Tx = the "probable life" after the age x, defined by ?as+T = i?<». ttx = the present value of $1 Annuity due at the end of each year during the life x. l-\-ax = the present value of 11' annuity due at the beginning of each year during the life x. a^ = the present value of a temporary annuity for the next n years, if x shall so long live. 1 -t-a-^-i = the value of the same, when the annuity is due at the beginning of each year. %a. = the present value of a deferred annuity ; that is, of a life annuity on a; to commence at the age x-\-n. ttxy = the value of an annuity during the joint existence of x and y. a~ = ax + tty — ttxy = the value of an annuity on the longest" life. «(»») = the value of $1 annuity, when payable by m instalments in each year. — + «("*> = the value of the same, when the instalments are payable in advance. ca^''*' = complete annuity providing for a proportionate payment to the day of death. dx = the value of a continuous annuity of |1 per annum, for instalments supposed payable momently. Ax = the single premium to insure $1 payable at the end of the policy year in which X shall die. TT-B or Pa; = annual premium to insure II on the life of x. In some cases it denotes the premium to insure 11000, j^(^) _ siijgie premium of Endowment Insurance of $1 for m years. A— = single premium of insurance " on x against y," that is, of $1 payable on the death of x, if y be the survivor. Note 1. — Generally x denotes the age of entry, when the risk commences j x + n the age of valuation of the policy ; and m or n + m the total period of insurance. The suffixes x, xy, . . . ., are often omitted where the meaning is obvious. Note 2. — The notation relative to other conditions is usually explained on the several pages where it is introduced and applied. ao4 INTERPOLATION BY FINITE DIFFERENCES. From the series r^ h, c, d, e, etc., whose terms are a unit's distauce from each other, to find any intermediate term denoted by u : Let X be the integral or the fractional distance of u from the first term a, and A, A^ A^ etc., the first terms of the differences. That is, A =zh—a, the first difference; a'^ = c— 2J + ff, the first of the second differences; A^ = cl—Zc-\-Zh—a, the first of the third differences; etc. Then, u = a-\-x^ 4 x{x-l ) x{x-l){x-2) x{x-l){x-^){x-d) ^^ 2 "^ 2x3 "^ 2x3x4 aH Note. — The derivative du-i-dx at any point as at c, may be found by differentiating with respect to u and x as the only variables, and at c making x = 2. The well known values of /^ were thus found. The first terra a can generally be so located that the required term ^^ shall be near the middle of the employed portion of the series. DOUBLE INTERPOLATION. Let U denote the term to be interpolated at the junction of a;, y; where x is the horizontal, and y the vertical distance from the term a. We might apply the preceding formula to insert a term u at the distance x from a, then a term u' at the distance x from a', etc. The same formula applied to u, u, u", would evidently give U at the vertical distance y from u. But the result will be the same, if A = h—a, a2 = {c—h) — (b—a), A' = b'—a'. etc. 6 = a— a, (J2 =(«"—«') — (a' — a), x(x — 1) h V h" d d' d" U = a-{-xA + yd + xy{A'—A)-\-' 2 AH 3^(^-1) =Na, + Na,+i-f TABLE XXXIII. COMMUTATION COLUMNS. 3 PER CEFT. ♦15 AGE.. 55 D^. No,. ■ Sx. 0^. M^. Rx. AGE. 13,044.8 177,175.0 1,809,176.1 239.750 7,884.441 124,480.552 55 56 12,425.2 164,130.2 1,632,001.1 243.339 7,644.691 116,596.111 56 57 11,820.0 151,705.0 1,467,870.9 247.416 7,401.352 108,951.420 57 58 11,228.3 139,885.0 1,316,165.9 251.398 7,153.936 101,550.068 58 59 10,649.8 128,656.7 1,176,280.9 255.448 6,902.538 94,396.132 i 59 1 60 10,084.2 118,006.89 1,047,624.22 259.708 6,647.090 87,493.594 60 61 9,530.75 107,922.69 929,617.33 263.983 6,387.382 80,846.504 61 62 8,989.18 98,391.94 821,694.64 267.944 6,123.399 74,459.122 62 63 8,459.43 89,402.76 723,302.70 271.903 5,855.455 68,335.723 63 64 7,941.12 80,943.33 633,899.94 275.550 5,583.552 62,480.268 64 65 7,434.26 73,002.21 552,956.61 278.896 5,308.002 56,896.716 65 66 6,938.84 65,567.95 479,954.40 281.537 5,029.106 51,588.714 66 67 6,455.22 58,629.11 414,386.45 283.789 4,747.569 46,559.608 67 68 5,983.40 52,173.89 355,757.34 284.889 4,463.780 41,812.039 68 69 5,524.25 46,190.49 303,583.45 285.558 4,178.891 37,348.259 69 70 5,077.78 40,666.24 257,392.96 284.844 3,893.833 33,169.368 70 71 4,645.05 35,588.46 216,726.72 283.095 3,608.489 29,276.035 | 71 72 4,226.65 30,943.41 181,138.26 279.935 3,325.394 25,667.546 | 72 73 3,823.61 26,716.76 150,19^.85 275.260 3,045.459 22,342.152 73 74 3,436.99 22,893.15 123,478.09 269.313 2,770.199 19,296.693 74 75 3,067.56 19,456.1.6 100^584.94 261.468 2,500.886 16,526.494 75 76 2,716.76 16,388.60 81,128.78 251.902 2,239.418 14,025.608 76 77 2,385.73 13,671.84 64,740.18 240.876 1,987.516 11,786.190 77 78 2,075.36 11,286.11 51,068.34 228.052 1,746.640 9,798.674 78 79 1,786.86 9,210.75 39,782.23 213.704 1,518.588 8,052.034 79 80 1,521.12 7,423.887 30,571.484 197.991 1,304.884 6,533.446 80 81 1,278.82 5,902.767 23,147.597 181.063 1,106.893 5,228.562 81 82 1,060.51 4,623.947 17,244.830 163.404 925.830 4,121.669 82 83 866.216 3,563.437 12,620.883 145.035 762.426 3,195.839 83 84 695.951 2,697.221 9,057.446 126.705 617.391 2,433.413 84 85 548.976 2,001.270 6,360.225 108.533 490.686 1,816.022 85 86 424.452 1,452.294 4,358.955 91.0068 382.153 1,325.336 86 87 321.083 1,027.842 2,906.661 74.7799 291.1461 943.1827 87 88 236.951 706.759 1,878.819 59.7092 216.3662 652.0366 88 89 170.341 469.808 1,172.060 48.3900 156.6570 435.6704 89 90 116.9892 299.4671 702.2519 37.2722 108.2670 279.0134 90 91 76.3096 182.4779 402.7850 26.5633 70.9948 170.7464 91 92 47.5237 106,1683 220.3071 17.9823 44.4315 99.7516 92 93 28.1573 58.6446 114.1388 11.6183 26.4492 55.3201 93 94 15.7189 30.4873 55.4942 7.05748 14.83089 28.87092 94 95 8.20357 14.76840 25.00690 3.98231 7.77341 14.04003 95 96 3.98231 6.56483 10.23852 2.16059 3.79110 6.26662 96 97 1.70573 2.58252 3.67369 .993631 1.630512 2.475523 97 98 .662419 .876794 1.091169 .428751 .636881 .8450111 98 99 .214375 .214375 .214375 .208130 .208130 .208130 99 Cx^ilx—lx+ljv^"-^; Ma,=Ca.+Ca,+i + Ra.=Ma, + Ma;+i + ^16 TABLE XXXiri. LOGARITHMS OF COMMUTATION COLUMNS. 3 PER CENT. 1 AGE. - AiNa;. 1 1 A{jx. AMa;. Axis. A&E 10 10 4.871627 6.265518 7.579931 2.670366 4.316615 5.866757 11 4.855968 6.247619 7.558349 2.656186 4.306695 5.854345 11 12 4.840297 6.229627 7.536576 2.641327 4.296874 5.841861 12 13 4.824622 6.211535 7.514609 2.627137 4.287167 5.829298 13 14 4.808935 6.193341 7.492440 2.613622 4.277561 5.816652 14 15 4.793237 6.175039 7.470062 2.599425 4.268042 5.803916 15 16 4.777526 6.156625 7.447467 2 585224 4.258626 5.791084 16 17 4.761808 6.138093 7.424648 2.571704 4.249314 5.778148 17 18 4.746073 6.119438 7.401596 2.558182 4.240093 5.765103 18 19 4.730323 6.100654 7.378303 2.544659 4.230964 5.751940 19 20 4.714559 6.081736 7.354760 2.531822 4.221928 5.738653 20 21 4.698776 6.062677 7.330959 2.518985 4.212971 5. 725233 21 22 4.682971 6.043471 7.306889 2.506148 4.204094 5.711672 22 23 4.667146 6.024110 7.282539 2.493996 4.195300 5.697962 23 24 4.651296 6.004587 7.257901 2.481843 4.186573 5.684093 24 25 4.635421 5.984894 7.232962 2.469689 4.177916 5.670056 25 26 4.619518 5.965024 7.207711 2.458896 4.169328 5.655842 26 27 4.603580 5.944966 7.182135 2.447416 4.160787 5.641439 27 28 4.587608 5.924714 7.156223 2.436606 4.152301 5.626837 28 29 4.571601 5.904255 7.129961 2.427128 4.143862 5.612025 29 30 4.555544 5.883581 7.103334 2.416959 4.135443 5.596989 30 31 4.539445 5.862681 7.076327 2.407435 4.127059 5.581718 31 32 4.523292 5.841542 7.048926 2.398540 4.118696 5.566200 32 33 4.507087 5.820154 7.021114 2.390257 4.110344 5.550418 33 34 4.490817 5.798501 6.992874 2.382567 4.101990 5.534359 34 35 4.474481 5.776573 6.964187 2.376079 4.093625 5.518008 35 36 4.458061 5.754352 6.935035 2.368878 4.085221 5.501346 36 37 4.441565 5.731825 6.905398 2.362832 4.076795 5.484359 37 38 4.424981 5.708974 6.875355 2.357885 4.068321 5.467027 38 39 4.408294 5.685783 6.844582 2.353389 4.059779 5.449330 39 40 4.391497 5.662231 6.813357 2.349892 4.051154 5.431248 40 41 4.374575 5.638300 6.781554 2.346198 4.042427 5.412760 41 42 4.357528 5.613969 6.749148 2.344526 4.033597 5.393841 42 48 4.340328 5.589213 6.716110 2.342574 4.024618 5.374468 43 44 4.322971 5.564011 6.682410 2.341926 4.015492 5.354616 44 45 4.305434 5.538336 6.648017 2.341962 4.006183 5.334255 45 46 4.287706 5.512159 6.612898 2.342611 3.996672 5.313356 46 47 4.269761 5.485453 6.577017 2.344284 3.986931 5.291888 47 48 4.251582 5.458185 6.540337 2.346408 3.976928 5.269817 48 49 4.233143 5.430321 6.502816 2.349365 3.966638 5.247109 49 50 4.214422 5.401826 6.464416 2.353049 3.956026 5.223723 50 51 4.195383 5.372659 6.425086 2.357360 3.945054 5.199619 51 52 4.176002 5.342781 6.384782 2.362203 3.933684 5.174753 52 1 53 4.156242 5.312144 6.343450 2.367488 3.921875 5.149080 53 54 4.136069 5.280702 ! 6.301037 2.373134 3.909586 ! 5.122548 54 TABLE XXXIII. LOGARITHMS OF COMMUTATION COLUMNS, 3 PER CENT. *X7 AGE. 55 Wx. ^^x. ASic. ACa,. AM^. AEa;. AGE. 55 4.115439 5.248403 6.257481 2.379759 3.896771 5.095102 56 4.094302 5.215188 6.212720 2.386212 3.883360 5.066684 56 57 4.072616 5.181000 6.166688 2.393428 3.869311 5.037233 57 58 4.050312 5.145771 6.119311 2.400363 3.854545 5.006680 58 59 4.027342 5.109432 6.070511 2.407303 3.839009 4.974955 59 60 4.003641 5.071906 6.020205 2.414486 3.822632 4.941976 60 61 3.979127 5.033113 5.968304 2.421576 3.805323 4.907662 61 62 3.953720 4.992959 5.914710 2.428044 3.786993 4.871918 62 63 3.927341 4.951351 5.859320 2.434413 3.767561 4.834648 63 64 3.899882 4.908181 5.802021 2.440200 3.746910 4.795743 64 65 3.871238 4.863336 5.742691 2.445442 3.724931 4.755087 65 66 3.841287 4.816692 5.681200 2.449536 3.701491 4.712555 66 67 3.809911 4.768113 5.617405 2.452995 3.676471 4.668009 67 68 3.776948 4.717453 5.551154 2.454676 3.649703 4.621301 68 69 3.742273 4.664552 5.482278 2.455695 3,621061 4.572270 69 70 3.705674 4.609234 5.410597 2.454C06 3.590321 4.520737 70 71 3.666990 4.551309 5.335912 2.451932 3.557326 4.466513 71 72 3.625996 4.490568 5.258010 2.447057 3.521843 4.409384 72 73 3.582474 4.426784 5.176655 2.439743 3.483653 4.349125 73 74 3.536178 4.359706 5.091590 2.430257 3.442511 4.285483 74 75 3.486794 4.289057 5.002533 2.417419 3.398094 4.218181 75 76 3.434051 4.214542 4.909175 2.401231 3.350135 4.146922 76 77 3.377621 4.135827 4.811174 2.381793 3.298311 4.071373 77 78 3.317094 4.052544 4.708152 2.358035 3.242203 3.991167 78 79 3.252091 3.964295 4.599690 2.329813 3.181440 3.905906 79 80 3.182162 3.870631 4.485317 2.296645 3.115572 3.815142 80 81 3.106809 3.771056 4.364506 2.257828 3.044105 3.718382 81 82 3.025515 3.665013 4.236659 2.213264 2.966531 3.615073 82 83 2.937626 3.551869 4.101090 2.161473 2.882198 3.504585 83 84 2.842579 3.430916 3.957006 2.102795 2.790560 3.386216 84 85 2.739553 3.301306 3.803473 2.035563 2.690804 3.259121 85 86 2.627829 3.162055 3.639382 1.959073 2.582237 3.122326 86 87 2.506618 3.011926 3.463394 1.873785 2.464111 2.974596 87 88 2.374659 2.849271 3.273885 1.776042 2.335189 2.814272 88 89 2.231318 2.671920 3.068950 1.684756 2.194950 2.639158 89 90 2.068146 2.476349 2.846493 1.571385 2.034496 2.445625 90 91 1.882579 2.261210 2.605074 1.424280 1.851228 2.232352 91 92 1.676910 2.025995 2.343028 1.254844 1.647691 1.998920 92 93 1.449591 1.768228 2.057433 1.065143 1.422413 1.742883 93 94 1.196422 1.484119 1.744248 0.848650 1.171168 1.460461 94 95 0.914003 1.169334 1.398060 0.600135 0.890611 1.147368 95 96 0.600135 0.817223 1.010237 0.334573 0.578765 0.797033 96 97 0.231910 0.412043 0.565103 1.997225 0.212324 0.393667 97 98 1.821133 1.942898 0.037892 1.632205 1.804059 1.926863 98 99 1.331175 1.331175 1.331175 1.318338 1.318338 1.318338' 99 ♦18 TABLE XXXIY. C50MMUTATI0N COLUMNS. 3^ PER CENT. — AGE. Da;. Na.. S.. Cx. M^. Ra>. AOE. 10 1 70,892.0 1,601,484.9 31,058,393.9 443.845 16,735.405 551,200.709 10 11 68,050.8 1,530,592.9 29,456,909.0 427.512 16,291.560 534,465.304 11 12 65,332.1 1,462,542.1 27,926,316.1 411.138 15,864.048 518,173.744 12 13 1 62,701.9 1,397,220.0 26,463,774.0 395.998 15,452.910 502,309.696 13 14 60,185.5 1,334,518.1 25,066,554.0 382.010 15,056.912 486,856.786 14 15 1 57,768.4 1,274,332.6 23,732,035.9 367.938 14,674.902 471,799.874 15 16 55,446.7 1,216,564.2 22,457,703.3 354.382 14,306.964 457,124.972 16 17 53,217.4 1,161,117.5 21,241,139.1 341.859 13,952.582 442,818.008 17 18 51,076.0 1,107,900.1 20,080,021.6 ' 329.779 13,610.723 428,865.426 18 19 49,019.0 1,056,824.1 18,972,121.5 ; 318.124 13,280.944 i 415,254.703 19 20 47,043.2 1,007,805.1 17,915,297.4 307.366 12,962.820 401,973.759 20 21 45,145.0 960,761.9 16,907,492.3 i 296.972 12,655.454 | 389,010.939 21 22 43,321.3 915,616.9 15,946,730.4 ! 286.930 12,358.482 : 376,355.485 22 23 41,569.4 872,295.6 15,031,113.5 277.665 12,071.552 i 363,997.003 23 24 39,886.1 830,726.2 14,158,817.9 268.698 11,793.887 1 351,925.451 24 25 38,268.6 790,840.1 13,328,091.7 260.021 11,525.189 i 340,131.564 25 26 36,714.5 752,571.5 12,537,251.6 252.413 11,265.168 328,606.375 26 27 35,220.5 715.857.0 11,784,680.1 1 244.640 11,012.755 ! 317,341.207 | 27 28 33,784.8 680,636.5 11,068,823.1 : 237.474 10,768.115 306,328.452 1 28 29 32,404.8 646,851.7 10,388,186.6 231.225 10,530.641 295,560.337 j 29 30 31,077.8 614,446.9 9,741,334.9 224.782 10,299.416 285,029.696 30 31 29,802.1 583.369.1 9,126,888.0 218.844 10,074.634' 274,730.280 31 32 ; 28,575.4 553,567.0 8,543,518.9 213.372 9,855.790 ; 264,655.646 i 32 1 33 ! 27,395.8 524,991.6 7,989,951.9 208.330 9,642.418 i 254,799.856 i 33 | 34 26,261.0 497,595.8 7,464,960.3 , 203.684 9,434.088 245,157.438 34 35 25,169.3 471,334.8 6,967,364.5 199.695 9,230.404 235,723.350 j 35 36 24,118.4 446,165.5 6,496,029.7 195.462 9,030.709 1 226,492.946 i 36 37 ! 23,107.3 422,047.1 6,049,864.2 191.828 8,835.247 i 217,462.237 i 37 38 22,134.2 398,939.8 5,627,817.1 188.740 8,643.419 : 208,626.990 38 39 21,196.9 376,805.6 5,228,877.3 185.893 8,454.679 199,983.^71 39 40 20,294.2 355,608.7 4,852,071.7 183.512 8,268.786 191,528.892 40 41 19,424.4 335,314.5 4,496,463.0 181.078 8,085.274 183,260.106 i 41 42 18,586.5 315,890.1 4,161,148.5 179.511 7,904.19C i 175,174.832 42 | 43 ! 17,778.4 297,303.6 3,845,258.4 177.843 7,724.685 167,270.636 : 43 44 16,999.4 279,525.2 3,547,954.8 176.720 7,546.842 159,545.951 44 45 16,247.8 262,525.8 3,268,429.6 175.881 7,370.122 151,999.109 1 45 46 15,522.5 246,278.0 3,005,903.8 175.293 7,194.241 144,628.987 ^ 46 47 14,822.3 230,755.5 2,759,625.8 175.119 7,018.948 137,434.746 : 47 48 14,145.9 215,933.2 2,528,870.3 175.128 6,843.829 130,415.798 i 48 49 13,492.4 201,787.3 2,312,937.1 175.473 6,668.701 ' 123,571.969 1 49 50 12,860.7 188,294.9 2,111,149.8 176.112 6,493.228 116,903.268 ! 50 51 12,249.7 175,434.2 1,922,854.9 177.010 6,317.116 110,410.040 ' 51 52 11,658.4 163,184.5 1,747,420.7 178.130 6.140.106 104,092.924 52 53 11,086.1 151,526.1 1,584,236.2 179.440 5,961.976 97,952.818 53 54 10,531.7 140,440.0 1,432.710.1 180.910 5,782.536 91,990.842 54 1 '19 TABLE XXXIY. C50MMUTATI0N COLUMNS. 3^ PER CENT. AQB. D,, N:.. Sx. Cx Mx. ^ 65 55 9,994.66 129,908.28 1,292,270.07 182.803 5,601.626 86,208.306 ' 56 9,473.86 119,913.62 1,162,361.79 184.644 5,418,823 80,606.680 56 57 8,968.85 110,439.70 1,042,448.17 186.830 5,234.179 75,187.857 57 58 8,478.73 101,470.91 932,008.41 188.920 5,047.349 69,953.678 58 59 8,803.10 92,992.18 830,537.50 191.036 4,858.429 64,906.329 59 60 7,541.42 84.989.08 737,545.32 193.284 4,667.393 60,047.900 60 61 7,093.12 77,447.66 652,556.24 195.516 4,474.109 55,380.507 61 62 6,657.73 70,354.54 575,108.58 197.491 4,278.593 50,906.398 62 63 6,235.11 63,696.81 504,754.04 199.440 4,081.102 46,627.805 63 64 5,824.80 57,461.70 441,057.23 201.139 3,881.662 42,546.703 64 65 5,426.69 51,636.90 383,595.53 202.598 3,680.523 38,665.041 65 66 5,040.59 46,210.21 331,958.63 203.529 3,477.925 34,984.518 66 67 4,666.62 41,169.62 285,748.42 204.165 3,274.396 31,506.593 67 68 4,304.63 36,503.00 244,578.80 203.967 3,070.231 28,232.197 68 69 3,955.10 32,198.37 208,075.80 1 203.459 1 2,866.264 25,161.966 69 70 3.617.90 28,243.27 175,877.43 ! 201.969 2,662.805 22,295.702 70 71 13,293.58 24,625.37 147,634.16 i 199.759 2,460.836 19,632.897 71 72 2,982.44 21,331.79 123,008.79 196.575 2,261.077 17,172.061 72 73 2,685.02 18,349.35 101,677.00 i 192.359 2,064.502 14,910.984 73 74 2,401.86 15, 664. .33 83,327.65 i 187.293 1,872.143 12,846.482 74 75 2,133.34 13,262.47 67,663.32 ! 180.960 1,684.850 10,974.339 75 76 1,880.24 11,129.13 54,400.85 1 173.496 1,503.890 9,289.489 76 77 1,643.16 9,248.89 43,271.72 165.101 1,330.394 7,785.599 77 78 1,422.49 7,605.73 34,022.83 155.556 1,165.293 6,455.205 78 79 1,218.832 6,183.245 26,417.108 ! 145.065 ! 1,009.737 5,289.912 79 80 1,032.552 4,964.413 20,233.863 ' 133.749 864.672 4,280.175 80 81 863.885 3.931.861 15,269.450 121.723 730.923 3,415.503 81 82 712.948 3,067.976 11,337.589 109.321 609.200 2,684.580 82 83 579.518 2,355.028 8,269.613 96.5629 499.8792 2,075.3797 83 84 463.358 1,775.510 5,914.585 83.9517 403.3163 1,575.5005 84 85 363.737 1,312.152 4,139.075 1 71.5639 319.3646 1,172.1842 85 86 279.872 948.415 2,826.923 59.7175 247.8007 852.8196 86 87 210.691 668.543 1,878.508 48.8326 188.0832 605.0189 87 88 154.733 457.852 1,209.965 38.8028 139.2506 416.9357 88 89 110.698 303.119 752.113 31.2950 100.4478 277.6851 89 90 75.6598 192.4206 448.9945 23.9884 .69.1528 177.2373 90 91 49.1127 116.7608 256.5739 17.0135 45.1644 108.0845 91 92 30.4385 67.6481 139.8131 1 11.4618 28.1509 62.9201 92 93 17.9474 37.2096 72.1650 ! 7.36969 16.68907 34.76921 93 94 9.97075 19.26219 34.95535 1 4.45506 9.31938 18.08014 94 95 5.17852 9.29144 15.69316 ' 2.50170 4.86432 8.76076 95 96 2.50170 4.11292 6.40172 i 1.35074 2.36262 3.89644 96 97 1.06637 1.61122 2.28880 \ .618184 1.011883 1.533823 97 1 98 \ .413123 .544852 .677581 ! .265458 .393699 .521940 98 1 aft- ' .132729 .132729 .132729 .128241 .128241 .128241 99 *ao TABLE XXXIY. LOGARITHMS OF COMMUTATION COLUMNS. 3|- PER CENT. AQE. 10 ADa;. ANa,. ASa,. AC^. AMa:. AR-c. AGE. 10 4.850597 6.204523 7.492179 2.647231 4.223636 5.741310 11 4.832833 6.184860 7.469187 2.630948 4.211963 5.727919 11 12 4.815060 6.165108 7.446014 2.6L3987 4.200414 5.714475 12 13 4.797281 6.145265 7.422652 2.597693 4.189010 5.700972 13 14 4.779492 6.125324 7.399095 2.582075 4.177736 5.687401 14 15 4.761690 6.105283 7.375335 2.565775 4.166575 5.673758 15 16 4.743876 6.085135 7.351365 2.549471 4.155547 5.660035 16 17 4.726054 6.064876 7.327178 2.533847 4.144655 5.646225 17 18 4.708217 6.044501 7.302764 2.518223 4.133881 5.632321 18 19 4.690364 6.024002 7.278116 2.502597 4.123229 5.618315 19 20 4.672497 6.003376 7.253224 2.487656 4.112700 5.604198 20 21 4.654610 5.982616 7.228079 2.472716 4.102278 5.589962 21 22 4.636702 5.961714 7.202672 2.457776 4.091966 5.575598 22 23 4.618774 5.940664 7.176991 2.443521 4.081763 5.561097 23 24 4.600822 5.919458 7.151027 2.429265 4.071658 5.546451 24 25 4.582842 5.898089 7.124768 2.415008 4.061648 5.531647 25 26 4.564837 5.876548 7.098202 2.402111 4.051738 5.516676 26 27 4.546796 5.854826 7.071318 2.388528 4.041896 5.501527 27 28 4.528721 5.832915 7.044101 2.375616 4.032140 5.486188 28 29 4.510609 5.810805 7.016540 2.364034 4.022454 5.470646 29 30 4.492450 5.788484 6.988618 2.351762 4.012812 5.454891 30 31 4.474247 5.765943 6.960323 2.340135 4.003229 5.438906 31 32 4.455993 5.743170 6.931637 2.329137 3.993692 5.422681 32 33 4.437684 5.720152 6.902544 2.318750 3.984186 5.406199 33 34 4.419311 5.696877 6.873028 2.308958 3.974700 5.389445 34 35 4.400871 5.673330 6.843068 2.300367 3.965221 5.372402 35 36 4.382349 5.649496 6.812648 2.291063 3.955722 5.355055 36 37 4.363750 5.625361 6.781746 2.282913 3.946219 5.337383 37 38 4.345063 5.600907 6.750340 2.275864 3.936686 5.319371 38 39 4.326272 5.576117 6.718409 2.269264 3.927097 5.300994 39 40 4.307372 5.550972 6.685927 2.263664 3.917442 5.282234 40 41 4.288348 5.525452 6.652871 2.257866 3.907695 5.263068 41 42 4.269197 5.499536 6.619213 2.254091 3.897858 5.243472 42 43 4.249893 5.473200 6.584925 • 2.250036 3.887881 5.223420 43 44 4.230434 5.446421 6.549978 2.247285 3.877765 5.202886 44 45 4.210794 5.419172 6.514339 2.245218 3.867475 5.181841 45 46 4.190961 5.391426 6.477975 2.243764 3.856985 5.160255 46 47 4.170915 5.363152 6.440850 2.243334 3.846272 5.138097 47 48 4.150631 5.334319 6.402927 2.243355 3.835299 5.115330 48 49 4.130090 5.304894 6.364164 2.244209 3.824041 5.091920 49 50 4.109265 5.274839 6.324519 2.245790 3.812461 5.067827 50 51 4.088124 6.244114 6.283946 2.247998 3.800519 5.043008 51 52 4.066640 5.212679 6.242398 2.250737 3.788176 5.017421 52 53 4.044777 5.180487 6.199820 2.253919 3.775391 4.991017 53 54 4.022499 6.147491 6.156158 2.257462 3-. 762119 4.963744 54 *21 TABLE XXXIY. LOGARITHMS OP COMMUTATION COLUMNS. 3|- PER CENT. A&E. ADa,. ANa,. XSx AOa;. AM«,. AR^. 1 AGE. 55 55 3.999768 5.113637 6.111353 2.261984 3.748314 4.935550 56 3.976527 5.078868 6.065341 2.266334 3.733905 4.906371 56 57 3.952737 5.043126 6.018055 2.271446 3.718849 4.876148 57 58 3.928331 5.006342 5.969420 2.276278 3.703063 4.844811 58 59 3.903258 4.968446 5.919359 2.281116 3.686496 4.812287 59 60 3.877453 4.929363 5.867789 2.286195 3.669074 4.778498 60 61 3.850837 4.889008 5.814618 2.291182 3.650707 4.743357 61 62 3.823326 4.847292 5.759750 2.295547 ?. 631301 4.706773 62 63 3.794844 4.804118 5.703080 2.299813 3.610777 4.668645 63 64 3.765281 4.759379 5.644495 2.303497 3.589018 4.628866 64 65 3.734535 4.712960 5.583873 2.306636 3.565909 4.587318 65 66 3.702481 4.664738 5.521084 2.308627 3.541321 4.543876 66 67 3.669002 4.614577 5.455984 2.309982 3.515132 4.498401 67 68 3.633936 4.562329 5.388419 2.309560 3.487171 4.450745 68 69 3.597157 4.507834 5.318222 2.308476 3.457316 4.400744 69 70 3.558456 4.450915 5.245210 2.305284 3.425340 4.348221 70 71 3.517668 4.391383 5.169187 2.300507 3.391083 4.292985 71 72 3.474572 4.329027 5.089936 2.293529 3.354316 4.234823 72 73 3.428947 4.263620 5.007223 2.284112 3.314815 4.173506 73 74 3.380547 4.194912 4.920789 2.272522 3.272338 4.108784 74 75 3.329060 4.122625 4.830353 2.257582 3.226561 4.040379 75 76 3.274213 4.046461 4.735606 2.239291 3.177216 3.967992 76 77 3.215680 3.966090 4.636204 2.217750 3.123979 3.891292 77 78 3.153050 3.881141 4.531770 2.191888 3.066434 3.809910 78 79 3.085944 3.791316 4.421885 2.161563 3.004208 3.733448 79 80 3.013912 3.695868 4.306079 2.126291 2.936850 3.631462 80 81 2.936456 3.594598 4.183823 2.085372 2.863872 3.533454 81 82 2.853058 3.486852 4.054521 2.038705 2.784760 3.428877 82 83 2.763067 3.371996 3.917485 1.984810 2.698865 3.317098 83 84 2.665917 3.249323 3.771924 1.924029 2.605646 3.197418 84 85 2.560787 3.117984 3.616903 1.854694 2.504287 3.068996 85 86 2.446960 2.976999 3.451314 1.776101 2.394103 2.930857 86 87 2.323646 2.825129 3.273813 1.688710 2.274350 2.781769 87 88 2.189584 2.660725 3.082773 1.588863 2.143798 2.630069 88 89 2.044140 2.481613 2.876283 1.495475 2.001940 2.443553 89 90 1.878865 2.284251 2.652241 1.380000 1.839811 2.248555 90 91 1.691194 2.067297 2.409213 1.230793 1.654796 2.033763 91 92 1.483423 1.830256 2.145548 1.059254 1.449492 1.798790 92 93 1.254001 1.570655 1.858327 0.867449 1.222433 1.541195 93 94 0.998728 1.284706 1.543513 0.648853 0.969388 1.257201 94 95 0.714206 0.968083 1.195710 0.398235 0.687022 0.942542 95 96 0.398235 0.614150 0.806297 0.130570 0.373394 0.590668 96 97 0.027907 0.207154 0.359608 1.791118 0.005129 0.185775 97 98 1.615027 1.736279 1.830961 1.423995 1.595164 1.717621 98 99 1.122965 1.122965 1.122965 1.108025 1.108025 1.108025 99 *22 TABLE XXXY. COMMUTATION COLUMNS. 4 PER CENT. A&E. D.. N^. S.. Co.. M^. K.. AQE. 10 10 67,556.5 1,400,554.9 25,570,207.5 420.929 13,688.944 417,085.094 11 64,537.2 1,332,998.4 24,169,652.6 403.490 13,268.015 403,396.150 11 12 61,651.5 1,268,461.2 22,836,654.2 386.169 12,864.525 390,128.135 12 13 58,894.1 1,206,809.7 21,568,193.0 370.161 12,478.356 377,263.610 13 14 56,258.7 1,147,915.6 20,361,383.3 355.369 12,108.195 364,785.254 14 15 53,739.7 1,091,656.9 19,213,467.7 340.634 11,752.826 352,677.059 15 16 51,332.1 1,037,917.2 18,121,810.8 ; 326.505 11,412.192 340.924.233 16 17 49,031.3 986,585.1 17,083,893.6 313.454 11,085.687 329,512.041 17 18 46,832.0 937,553.8 16,097,308.5 300.923 10,772.233 318,426.354 18 19 44,729.9 890,721.8 15,159,754.6 288.893 10,471.310 307,654.121 19 20 42,720.6 845,991.9 14,269,032.8 277.782 10,182.417 297,182.811 20 21 40,799.7 803,271.3 13,423,040.9 267.098 9,904.635 287,000.394 21 22 38,963.4 762,471.6 12,619,769.6 256.825 9,637.537 277,095.759 22 23 37,207.9 723,508.2 11,857,298.0 247.337 9,380.712 267,458.222 23 24 35,529.6 686,300.3 11,133,789.8 238.199 9,133.375 258,077.510 24 25 33,924.9 650,770.7 10,447,489.5 229.398 8,895.176 248,944,135 25 26 32,390.6 616,845.8 9,796,718.8 221.616 8,665.778 240,048.959 26 27 30,923.2 584,455.2 9,179,873.0 213.759 8,444.162 231,383.181 27 28 29,520.1 553,533.0 8,595,417.8 206.499 8,230.403 222,939.019 28 29 28,178.2 524,011.9 8,041,885.8 200.099 8,023.904 214,708.616 29 30 26,894.3 495,833.7 7,517,873.9 193.589 7,823.805 206,684.712 30 31 25,666.4 468,939.4 7,022,040.2 187.568 7,630.216 198,860.907 31 32 24,491.6 443,273.0 6,553,100.8 181.999 7,442.648 191,2.30.691 32 33 23,367.6 418,781.4 6,109,827.8 176.843 7,260.649 183,788.043 33 34 22,292.1 395,413.8 5,691,046.4 172.069 7,083.806 176,527.394 34 35 21,262.6 373,121.7 5,295,632.6 167.888 6,911.737 169,443.588 35 36 20,276.9 351,859.1 4,922,510.9 163.539 6,743.849 162,531.851 36 37 19,333.4 331,582.2 4,570,651.8 159.727 6,580.310 155,788.002 37 38 18,430.2 312,248.8 4,239,069.6 156.400 6,420.583 149,207.692 38 39 17,564.9 293,818.6 3,926,820.8 153.301 6,264.183 142,787.109 39 40 16,736.0 276,253.7 3,633,002.2 150.609 6,110.882 136,522.926 40 41 15,941.7 259,517.7 3,356,748.5 147.898 5,960.273 130,412.044 41 42 15,180.7 243,576.0 3,097,230.8 145.913 5,812.375 124,451.771 42 43 14,450.9 228,395.3 2,853,654.8 143.862 5,666.462 118,639.396 43 44 13,751.2 213,944.4 2,625,259.5 142.266 5,522.600 112,972.934 44 45 13,080.1 200,193.2 2,411,315.1 140.909 5,380.334 107,450.334 45 46 12,4.36.1 187,113.1 2,211,121.9 139.764 5,239.425 102,070.000 46 47 11,818.0 174,677.0 2,024,008.8 138.954 5,099.661 96,830.575 47 48 11,224.5 162,859.0 1,849,331.8 138.292 4,960.707 91,730.914 48 49 10,654.5 151,634.5 1,686,472.8 137.898 4,822.415 86,770.207 49 50 10,106.8 140,979.99 1,534,838.27 137.736 4,684.517 81,947.792 50 51 9,580.36 130,873.19 1,393,858.28 137.773 4,546.781 77,263.275 51 52 9,074.11 121.292.83 1,262,985.09 137.978 4,409.008 72,716.494 52 53 8,587.13 112,218.72 1,141,-692.26 138.324 4,271.030 68,307.486 53 54 8,118.54 103,631.59 1,029,473.541 138.787 4,132.706 64,036.456 54 • 1 TABLE XXXY. COMMUTATION COLUMNS. 4 PER CENT. ♦23 AGE. 55 Do, N.. Sa.. 0^. Ma,. K«. AGE. 7,667.49 95,513.05 925,841.95 139.565 3,993.919 59,903.750 55 56 7,233.03 87,845.56 830,328.90 140.292 3,854.354 55,909.831 56 57 6,814.55 80,612.53 ' 742,483.34 141.271 3,714.062 52,055.477 57 58 6,411.17 73,797.98 661,870.81 142.165 3,572.791 48,341.415 58 59 6,022.43 67,386.81 588,072.83 143.066 3,430.626 44,768.624 59 60 5,647.73 61,364.38 520,686.02 144.053 3,287.560 41,337.998 60 61 5,286.45 55,716.65 459,321.64 145.016 3,143.507 38,050.438 61 62 4,938.11 50,430.20 403,604.99 145.777 2,998.491 34,906.931 62 63 4,602.42 45,492.09 353,174.79 146.508 2,852.714 31,908.440 63 64 4,278.88 40,889.67 307,682.70 147.046 2,706.206 29,055.726 64 65 3,967.26 36,610.79 266,793.03 147.400 2,559.160 26,349.520 65 66 3,667.28 32,643.53 230,182.24 147.366 2,411.760 23,790.360 66 67 3,378.87 28,976.25 197,538.71 147.116 2,264.394 21,378.600 67 68 3,101.79 25,597.38 168,562.46 146.266 2,117.278 19,114.206 68 69 2,836.23 22,495.59. 142,965.08 145.200 1,971.012 16,996.928 69 70 2,581.94 19,659.36 120,469.49 143.444 1,825.812 15,025.916 70 71 2,339.19 17,077.42 100,810.13 141.193 1,682.368 13,200.104 71 72 2,108.03 14,738.23 83,732.71 138.274 1,541.175 11,517.736 72 73 1,888.68 12,630.20 68,994.48 134.657 1,402.901 9,976.561 73 74 1,681.38 10,741.52 56,364.28 130.481 1,268.244 8,573.660 74 75 1,486.23 9,060.14 45,622,76 125.463 1,137.763 7,305.416 75 76 1,303.60 7,573.91 36,562.62 119.710 1,012.300 6,167.653 76 77 1,133.76 6,270.31 28,988.71 113.369 892.590 5,155.353 77 78 976.78 5,136.55 22,718.40 106.302 779.221 4,262.763 78 79 832.910 4,159.77 17,581.85 98.6560 672.9185 3,483.5424 79 80 702.220 3,326.861 13,422.083 90.5231 574.2625 2,810.6239 80 81 584.688 2,624.641 10,095.222 81.9874 483.7394 2,236.3614 81 82 480.212 2,039.953 7,470.581 73.2802 401.7520 1,752.6220 82 83 388.462 1,559.741 5,430.628 64.4168 328.4718 1,350.8700 83 84 309.105 1,171.279 3,870.887 55.7346 264.0550 1,022.3982 84 85 241.481 862.174 2,690.608 47.2821 208.3204 758.3432 85 86 184.911 620.693 1,837.434 39.2655 161.0383 550.0228 86 87 138.534 435.782 1,216.741 31.9541 121.7728 388.9845 87 88 101.251 297.248 780.959 25.2689 89.8187 267.2117 88 89 72.088 195.997 483.711 20.2818 64.5498 177.3930 89 90 49.0339 123.9092 287.7138 15.4717 44.2680 112.8432 90 91 31.6761 74.8754 163.8046 10.9204 28.7963 68.5752 91 1 92 19.5375 43.1993 88.9292 7.32159 17.87594 39.77885 92 93 11.4644 23.6618 45.7299 4.68498 10.55435 21.90291 93 94 6.3385 12.1974 22.0681 2.81850 5.86937 11.34856 94 95 3.27621 5.85884 9.87072 1.57510 3.05087 5.47919 95 96 1.57510 2.58263 4.01188 .846350 1.475769 2.428324 96 97 .668170 1.00753 1.42925 .385483 .629419 .952555 97 98 .256988 .339356 .421724 .164736 .243936 .323136 98 99 .082368 .082368 .082368 .079200 .079200 .079200 99 *24 TABLE XXXY LOGARITHMS OF COMMUTATION COLUMNS. 4 PER CENT. AGE. AD„. ANo,. ASa,- aLix- AM». AEa, AGE. 10 4.829667 6.146300 7.407734 i 2.624208 4.136370 5.620225 10 11 4.809810 6.124829 7.383270 2.605832 4.122806 5.605732 11 12 4.789944 6.103277 7.358632 2.586777 4.109394 5.591207 12 13 4.770072 6.081639 7.333814 2.568391 4.096158 5.576645 13 14 4.750190 6.059910 7.308807 2.550679 4.083080 5.562037 14 15 4.730295 6.038086 7.283606 2.532287 4.070142 5.547377 15 16 4.710389 6.016162 7.258202 2.513890 4.057369 5.532658 16 17 4.690473 5.994135 7.232587 2.496174 4.044763 5.517871 17 18 4.670543 5.971996 7.206756 2.478456 4.032306 5.503009 18 19 4.650598 5.949742 7.180692 2.460737 4.020001 5.488063 19 20 4.630637 5.927366 7.154394 2.443704 4.007850 5.473024 20 21 4.610657 5.904862 7.127851 2.426670 3.995839 5.457883 21 22 4.590657 5.882224 7.101051 2.409637 3.983966 5.442630 22 23 4.570635 5.859443 7.073985 2.393289 3.972236 5.427256 23 24 4.550590 5.836514 7.046643 2.376940 3.960632 5.411750 24 25 4.530518 5.813428 7.019012 2.360590 3.949155 5.396102 25 26 4.510419 5.790177 6.991081 2.345601 3.937808 5.380300 26 27 4.490285 5.766751 6.962837 2.329925 3.926557 5.364332 27 28 4.470118 5.743143 6.934267 2.314919 3.915421 5.348186 28 29 4.449913 5.719341 6.905358 2.301245 3.904386 5.331849 29 30 4.429661 5.695336 6.876095 2.286880 3.893418 5.315308 30 31 4.409365 5.671117 6.846463 2.273159 3.882537 5.298549 31 32 4.389017 5.646671 6.816447 2.260068 3.871728 5.281558 32 33 4.368615 5.621987 6.786029 2.247589 3.860976 5.264317 33 34 4.348150 5.597052 6.755192 2.235703 3.850267 5.246812 34 35 4.327616 5.571850 6.723918 2.225019 3.839587' 5.229025 35 36 4.307002 5.546369 6.692187 2.213622 3.828908 5.210938 36 37 4.286309 5.520591 6.659978 2.203379 3.818246 5.192534 37 38 4.265529 5.494501 6.627271 2.194237 3.807574 5.173791 38 39 4.244646 5.468079 6.594041 2.185544 3.796864 5.154689 39 40 4.223652 5.441308 6.560266 2.177851 3.786104 5.135206 40 41 4.202535 5..414167 6.525919 2.169961 3.775266 5.115321 41 42 4.181292 5.386634 6.490974 2.164093 3.764354 5.095001 42 43 4.159894 5:358687 6.455401 2.157945 3.753312 5.074229 43 44 4.138342 5.330301 6.419172 2.153101 3.742144 5.052974 44 45 4.116610 5.301449 6.382254 2.148940 3.730809 5.031208 45 46 4.094683 5.272104 6.344613 2.145394 3.719284 5.008898 46 47 4.072544 5.242236 6.306212 2.142871 3.707541 4.986013 47 48 4.050168 5.211812 6.267015 • 2.140798 3.695544 4.962516 48 49 4.027533 5.180798 6.226979 2.139559 3.683265 4.938371 49 50 4.004615 5.149157 6.186062 2.139048 3.670665 4.913537 50 51 3.981382 5.116851 6.144219 2.139162 3.657704 4.887973 51 52 3.957804 5.083835 6.101398 2.139809 3.644341 4.861633 52 53 3.933848 5.050065 6.057549 2.140898 3.630533 4.834468 53 54 3.909478 5.015492 6.012615 2.142348 3.616235 4.806427 54 TABLE XXXY. LOGARITHMS OF COMMUTATION COLUMNS. 4 PER CENT. *25 AGE. 55 ADa,. AN«,. Ahx' AUa)' AM«,- A Raj- AGE. 3.884653 4.980063 5.966537 2.144777 3.601399 4.777454 55 56 3.859320 4.943720 5.919250 2.147034 3.585951 4.747488 56 57 3.833437 4.906403 5.870687 2.150053 3.569849 4.716466 57 58 3.806937 4.868044 5.820773 2.152792 3.553008 4.684319 58 59 3.779772 4.828575 5.769431 2.155536 3.535374 4.650974 59 60 3.751874 4.787916 5.716576 2.158523 3.516874 4.616349 60 61 3.723164 4.745985 5.662117 2.161417 3.497415 4.580360 61 62 3.693561 4.702691 5.605956 2.163689 3.476903 4.542912 62 63 3.662986 4.657936 5.547990 2.165862 3.455258. 4.503906 63 64 3.631330 4.611614 5.488103 2.167453 3.432362 4.463232 64 65 3.598491 4.563609 5.426174 2.168499 3.408097 4.420773 65 66 3. 534344 4.513797 5.362072 2.168397 3.382334 4.376401 66 67 3.528771 4.462042 5.295651 2.167659 3.354951 4.329979 67 68 3.491613 4.408196 5.226761 2.165144 3.325778 4.281356 68 69 3.452741 4.352097 5.155230 2.161967 3.294689 4.230371 69 70 3.411946 4.293569 5.080877 2.156682 • 3.261456 4.176841 70 71 3.369066 4.232422 5.003504 2.149812 3.225922 4.120577 71 72 3.323877 4.168445 4.922895 2.140740 3.187853 4.061367 72 73 3.276158 4.101410 4.838814 2.129230 3.147027 3.998981 73 74 3.225666 4.031066 4.751004 2.115548 3.103203 3.933166 74 75 3.172086 3.957135 4.659182 2.098515 3.056051 3.863645 75 76 3.115146 3.879320 4.563037 2.078130 3.005309 3.790120 76 77 3.054520 3.797289 4.462229 2.054496 2.950652 3.712258 77 78 2.989797 3.710672 4.356378 2.026542 2.891660 3.629691 78 79 2.920598 3.61906& 4.245065 1.994123 2.827963 3.542021 79 80 2.846473 3.522035 4.127820 1.956759 2.759111 3.448803 80 81 2.766924 3.419070 4.004116 1.913747 2. 6846] 1 3.349542 81 82 2.681433 3.309620 3.873354 1.864986 2.603958 3.243688 82 83 2.589349 3.193052 3.734850 1.808999 2.516498 3.130614 83 84 2.490106 3.068660 3.587811 1.746125 2.421694 3.009620 84 85 2.382883 2.935595 3.429850 1.674697 2.318731 2.879866 85 86 2.266963 2.792877 3.264212 1.594011 2.206928 2.740381 86 87 2.141556 2.639269 3.085198 1.504527 2.085551 2.589932 87 88 2.005401 2.473119 2.892628 1.402587 1.953368 2.426855 88 89 1.857863 2.292249 2.684587 1.307105 1.809896 2.248936 89 90 1.690496 2.093104 2.458961 1.189538 1.646090 2.052475 90 91 1.500732 1.874339 2.214326 1.038238 1.459332 1.836167 91 92 1.290868 1.635477 1.949044 0.864606 1.252269 1.599652 92 93 1.059352 1.374048 1.660200 0.670708 1.023431 1.340502 93 94 0.801987 1.086267 1.343765 0.450019 0.768591 1.054941 94 95 0.515372 0.767812 0.994349 0.197308 0.484424 0.738716 95 96 0.197308 0.412062 0.603348 1.927550 0.169018 0.385307 96 97 1.824887 0.003258 0.155108 1.586005 1.798940 1.978890 97 98 1.409913 1.530656 1.625028 1.216789 1.387276 1.509385 98 99 2.915759 2.915759 2.915759 2.898726 2.898726 2.898726 99 ♦36 TABLE XXXYI. COMMUTATION COLUMNS. 4^ PER CENT. A&E. D.. N«,. s.. a. M„. Ra;. AGE. 10 64,392.8 1,232,047.6 21,206,324.6 399.297 11,338.107 318,856.514 10 11 61,320.7 1,167,654.8 19,974,277.0 380.923 10,938.810 307,518.407 11 12 58,203.5 1,106,434.1 18,806,622.2 362.827 10,557.887 296,579.597 12 13 55,334.1 1,048,230.6 17,700,188.1 346.123 10,195.060 286,021.710 13 14 52,605.2 992,896.5 16,651,957.5 330.701 9,848.937 275,826.650 14 15 50,009.3 940,291.3 15,659,061.0 315.471 9,518.236 265,977.713 15 16 47,540.2 890,282.0 14,718,769.7 300.940 9,202.765 256,459.477 16 17 45,192.2 842,741.8 13,828,487.7 287.528 8,901.825 247,256.712 17 18 42,958.6 797,549.6 12,985,745.9 274.714 8,614.297 238,354.887 18 19 40,833.9 754,591.0 12,188,196.3 262.469 8,339.583 229,740.590 19 20 38,813.1 713,757.1 11,433,605.3 251.166 8,077.114 221,401.007 20 21 36,890.5 674,944.0 10,719,848.2 240.351 7,825.948 213,323.893 21 22 35,061.5 638,053.5 10,044,904.2 230.001 7,585.597 205,497.945 22 23 33,321.7 602,992.0 9,406,850.7 220.444 7,355.596 197,912.348 23 24 31,666.4 569,670.3 8,803,858.7 211.284. 7,135.152 190,556.752 24 25 30,091.5 538,003.9 8,234,188.4 202.504 6,923.868 183,421.600 25 26 28,593.2 507,912.4 7,696,184.5 194.698 6,721.364 176,497.732 26 27 27,167.2 479,319.2 7,188,272.1 186.897 6,526,666 169,776.368 27 28 25,810.4 452,152.0 6,708,952.9 179.686 6,339.769 163,249.702 28 29 24,519.3 426,341.6 6,256,800.9 173.283 6,160.083 156,909.933 29 30 23,290.1 401,82^.3 5,830,459.3 166.843 5,986.800 150,749.850 30 31 22,120.4 378,532.2 5,428,637.0 160.881 5,819.957 144,763.050 31 32 21,007.0 356,411.8 5,050,104.8 155.357 5,659.076 138,943.093 32 33 19,947.0 335,404.8 4,693,693.0 150.234 5,503.719 133,284.017 33 34 18,937.8 315,457.8 4,358,288.2 145.479 5,353.485 127,780.298 34 35 17,976.8 296,520.0 4,042,830.4 141.264 5,208.006 122,426.813 35 36 17,061.5 278,543.2 3,746,310.4 136.947 5,066.742 117,218.807 36 37 16,189.7 261,481.7 3,467,767.2 133.115 4,929.795 112,152.065 37 38 15,359.5 245,292.0 3,206,285.5 129.719 4,796.680 107,222.270 38 39 14,568.4 229,932.5 2,960,993.5 126.539 4,666.961 102,425.590 39 40 13,814.5 215,364.1 2,731,061.0 123.723 4,540.422 97,758.629 40 41 13,095.9 201., 549. 6 2,515,696.9 120.914 4,416.699 93.218.207 41 42 12,411.0 188,453.7 2,314,147.3 118.721 4,295.785 88,801.508 42 43 11,757.9 176,042.7 2,125,693.6 116.492 4,177.064 84,505.723 43 44 11,135.0 164,284.8 1,949,650.9 114.648 4,060.572 80,328.659 44 45 10,540.9 153,149.8 1,785,366.1 113.012 3,945.924 76,268.087 45 46 9,973.97 142,608.93 1,632,216.26 111.555 3,832.912 72,322.163 46 47 9,432.90 132,634.96 1,489,607.33 110.380 3,721.357 68,489.251 47 48 8,916.33 123,202.06 1,356,972.37 109.329 3,610.977 64,767.894 48 49 8,423.02 114,285.73 1,233,770.31 108.496 3,501.648 61,156.917 49 50 7,951.85 105,862.71 1,119,484.58 107.849 3,393.152 57,655.269 50 51 7,501.55 97,910.86 1,013,621.87 107.362 3,285.303 54,262.117 51 52 7,071.17 90,409.31 915,711.01 107.007 3,177.941 50,976.814 52 53 6,659.66 83,338.14 825,301.70 106.762 3,070.934 47,798.873 53 54 6,266.11 76,678.48 741,963.56 106.607 2,964.172 44,727.939 54 TABLE XXXYI. ■^7 COMMUTATION COLUMNS. 4J PER CENT. AGE. Da,. N^. s^. Cx- Mo,. Ra;- AGE. 55 55 i5,889.67 70,412.37 665,285.08 106.692 2,857.565 41,763.767 56 15,529.36 64,522.70 594,872.71 106.735 2,750.873 38,906.202 56 57 5,184.51 58,993.34 530,350.01 106.965 2,644.138 36,155.329 57 58 4,854.29 53,808.83 471,356.67 107.127 2,537.173 33,511.191 58 59 4,538.14 48,954.54 417,547.84 107.290 2,430.046 30,974.018 59 60 4,235.43 44,416.40 368,593.30 107.513 2,322.756 28,543.972 60 61 3,945.52 40,180.97 324,176.90. 107.715 2,215.243 26,221.216 61 62 3,667.90 36,235.45 283,995.93 107.761 2,107.528 24,005.973 62 63 3,402.20 32,567.55 247,760.48 107.784 1,999.767 21,898.445 63 64 3,147.90 29,165.35 215,192.93 107.662 1,891.983 19,898.678 64 65 2,904.68 26,017.45 186,027.58 107.405 1,784.321 18,006.695 65 66 2,672.'20 23,112.77 160,010.13 106.866 1,676.916 16,222.374 66 67 2,450.27 20,440.57 136,897.36 106.174 1,570.050 14.545.458 67 68 2,238.58 17,990.30 116,456.79 105.056 1,463.876 12,975.408 68 69 2,037.12 15,751.72 98,466.49 103.791 1,358.820 11,511.532 69 70 1,845.61 13,714.60 82,714.77 102.045 1,255.029 10,152.712 70 71 1.664.09 11,868.99 69,000.17 99.9629 1,152.9835 8,897.6832 71 72 1,492.47 10,204.90 57,131.18 97.4282 1,053.0206 7,744.6997 72 73 1,330.77 8,712.43 46,926.28 94.4261 955.5924 6,691.6791 73 74 1,179.04 7,381.66 38,213.85 91.0598 861.1663 5,736.0867 74 75 1,037.20 6,202.62 30,832.19 87.1385 770.1065 •4,874.9204 75 76 905.403 5,165.420 24,629.570 82.7453 682.9680 4,104.8139 76 77 783.670 4,260.017 19,464.150 77.9877 600.2227 3,421.8459 77 78 671.933 3,476.347 15,204.133 72.7760 522.2350 2,821.6232 78 79 570.223 2,804.414 11,727.786 67.2183 449.4590 2,299.3882 79 80 478.451 2,234.191 8,923.372 61.3819 382.2407 1,849.9292 80 81 396.465 1,755.740 6,689.181 55.3280 320.8588 1,467.6885 81 82 324.064 1,359.275 4,933.441 49.2155 265.5308 1,146.8297 82 83 260.894 1,035.211 3,574.166 43.0558 216.3153 881.2989 83 84 206.604 . 774.317 2,538.955 37.0744 173.2595 664.9836 84 85 160.632 567.713 1,764.638 31.3014 136.1851 491.7241 85 86 122.413 407.081 1,196.925 25.8699 104.88371 355.5390 86 87 91.2723 284.6678 789.8437 20.9521 79.0138 250.6553 87 88 66.3897 193.3955 505.1759 16.4894 58.0617 171.6415 88 89 47.0415 127.0058 311.7804 13.1716 41.5723 113.5798 89 90 31.8442 79.9643 184.7746 9.99977 28.40072 72.00746 90 i 91 20.4731 48.1201 104.8103 7.02435 18.40095 43.60674 91 92 12.5671 27.6470 56.6902 4.68696 11.37660 25.20579 92 93 7.33902 15.07986 29.04321 2.98476 6.68964 13.82919 93 94 4.03822 7.74084 13.96335 1.78706 3.70488 7.13955 94 95 2.07726 3.70262 6.22251 i .993905 1.917819 3.434666 95 96 .993905 1.62536 2.51989 .531500 .923914 1.516847 96 97 .419605 .631453 .894534 .240922 .392414 .592933 97 98 .160615 .211848 .263081 .102465 .151492 .200519 98 99 .051233j .051233 .051233 .049027 .049027 .049027 99 -I *28 TABLE XXXYI. LOGARITHMS OF COMMUTATION COLUMNS. 4|^ PER CENT. AGE. 10 ADo,. ANa;. ASo,. AC^. AMa;. AB«. AGE. 4.808837 6.090628 7.336465 3.601396 4.054541 5.503595 10 11 4.786898 6.067315 - 7.300471 3.580837 4.038970 5.487871 11 12 4.764949 6.043936 7.374311 2.559699 4.033576 5.472141 12 13 4.742993 6.030457 7.347978 2.539230 4.008390 5.456399 13 14 4.731029 5.996904 7.331465 2.519436 3.993390 5.440636 14 15 4.699051 5.973362 7.194766 2.498960 3.978557 5.424845 15 16 4.677061 5.949528 7.167871 2.478480 3.963919 5.409019 16 17 4.655063 5.925695 7.140775 2.458681 3.949479 5.393148 17 18 4.633050 5.901758 7.113467 2.438880 3.935220 5.377224 18 19 4.611021 5.877712 7.085939 2.419078 3.921144 5.361238 19 20 4.588978 5.853551 7.058183 2.399962 3.907256 5.345180 20 21 4.566915 5.829268 7.030189 3.380845 3.893537 5.329040 21 22 4.544831 5.804857 7.001946 3.361739 3.879990 5.312808 22 23 4.522727 5.780312 6.973444 3.343398 3.866618 5.296473 23 24 4.500599 5.755624 6.944673 2.324866 3.853403 5.280024 24 25 4.478444 5.730785 6.915631 2.306433 3.840349 5.263451 25 26 4.456262 5.705789 6.886375 2.289361 3.827457 5.246739 26 27 4.434045 5.680625 6.856634 2.271602 3.814693 5.329878 27 28 4.411795 5.655284 6.836655 2.254514 3.802074 5.212852 28 29 4.389508 5.629758 6.796353 2.338756 3.789586 5.195651 29 30 4.367172 5.604034 6.765703 3.333308 3.777195 5.178257 30 31 4.344793 5.578103 6.734691 2.206505 3.764920 5.160658 31 32 4.322363 5.551952 6.703300 2.191330 3.752746 5.142837 32 33 4.299877 5.525569 6.671515 3.176769 3.740656 5.134778 33 34 4.277329 5.498941 6.639316 3.163800 3.728637 5.106464 34 35 4.254713 5.473054 6.606686 2.150033 3.716672 5.087876 35 36 4.232016 5.444893 6.573604 2.136553 3.704729 5.068998 36 37 4.209240 5.417441 6.540050 2.124227 3.693829 5.049807 37 38 4.186377 5.389686 6.506003 2.113002 3.680941 5.030385 38 39 4.163411 5.361600 6.471437 2.102226 3.669034 5.010408 39 40 4.140334 5.333173 6.436331 2.092450 3.657096 4.990155 40 41 4.117134 5.304383 6.400658 2.082477 3.645098 4.969501 41 42 4.093808 5.375305 6.364391 3.074536 3.633043 4.948431 42 43 4.070328 5.345618 6.327501 2.066295 3.620871 4.936886 43 44 4.046692 5.315597 6.289957 2.059368 3.608587 4.904870 44 45 4.022877 5.185117 6.351727 2.053124 3.596148 4.883343 45 46 3.998868 5.154147 6.313778 2.047495 3.583529 4.859271 46 47, 3.974645 5.122658 6.173073 2.042889 3.570702 , 4.835623 47 48 3.950186 5.090618 6.133571 2.038734 3.557625 4.811359 48 49 3.925468 5.057992 6.091334 2.035413 3.544273 4.786445 49 50 3.900468 5.024743 6.049018 3.033817 3.530603 4.760839 50 51 3.875151 4.990834 6.005876 3.030849 3.516575 4.734497 51 52 3.849491 4.956314 5.961759 3.039413 3.502146 4.707372 52 53 3.823453 4.920844 5.916613 3.028418 3.487270 4.679418 53 54 3.796998 4.884674 5.870383 3.037785 3.471903 4.650578 54 *29 TABLE XXXTl. LOGARITHMS OF COMMUTATION COLUMNS. 4|^ PER CENT. A^E. ADa:. ANo, ASa,. AOor. AMa, ARa;. AGE. 55 3.770091 4.847649 5.823008 2.028131 3.455997 4.620800 55 56 3.742675 4.809712 5.774424 2.028305 3.439470 4.590019 56 57 3.714708 4.770803 5.724563 2.029242 3.422285 4.558172 57 58 3.686126 4.730854 5.673350 2.029898 3.404350 4.525189 58 59 3.656878 4.689793 5.620706 2.030559 3.385615 4.490997 59 60 3.626897 4.647543 5.566547 2.031463 3.366004 4.455515 60 61 3.596104 4.604020 5.510782 2.032274 3.345421 4.418653 61 62 3.564418 4.559134 5.453312 2.032463 3.323774 4.380319 62 63 3.531760 4.512785 5.394032 2.032553 3.300980 4.340413 63 64 3.498021 4.464867 5.332828 2.032061 3.276917 4.298824 64 65 3.463099 4.415265 5.269577 2.031024 3.251473 4.255434 65 66 3.426869 4.363852 5.204148 2.028839 3.224512 4.210114 66 67 3.389214 4.310493 5.136395 2.026018 3.195914 4.162728 67 68 3.349972 4.255038 5.066165 2.021420 3.165504 4.113121 68 69 3.309017 4.197328 4.993288 2.016160 3.133162 4.061133 69 70 3.266139 4.137183 4.917583 2.008793 3.098654 4.006582 70 71 3.221176 4.074414 4.838850 1.999839 3.061823 3.949277 71 72 3.173904 4.008809 4.756873 1.988685 3.022437 3.889005 72 73 3.124103 3.940139 4.671416 1.975092 2.980273 3.825535 73 74 3.071528 3.868154 4.582221 1.959327 2.935087 ^886551 3.758616 74 75 3.015864 3.792575 4.489004 1.940210 3.687968 75 76 2.956842 3.713106 4.391457 1.917743 2.834400 3.613293 76 77 2.894133 3.629412 4.289235 1.892026 2.778313 3.534260 77 78 2.827326 3.541123 4.181962 1.861988 2.717866 3.450499 78 79 2.756045 3.447842 4.069216 1.827487 2.652690 3.361612 79 80 2.679837 3.349120 3.950529 1.788040 2.582337 3.267155 80 81 2.598205 3.244460 3.825373 1.742945 2.506314 3.166634 81 82 2.510631 3.133307 3.693150 1.692102 2.424115 3.059499 82 83 2.416464 3.015029 3.553175 1.634031 2.335087 2.945123 83 84 2.315138 2.888918 3.404655 1.569074 2.238697 2.822811 84 85 2.205832 ' 2.754129 3.246656 1.495563 2.134129 2.691722 85 86 2.087829 2.609681 3.078067 1.412794 2.020708 2.550887 86 87 1.960339 2.454338 2.897541 1.321227 1.897703 2.399077 87 88 1.822101 2.286446 2.703443 1.217205 1.763890 2.234623 88 89 1.672481 2.103823 2.493849 1.119640 L 618804 2.055301 89 90 1.503030 1.902896 2.266642 0.999990 1.453329 1.857378 90 91 1.311184 1.682327 2.020404 0.846606 1.264840 1.639553 91 1 92 1.099236 1.441648 1.753508 0.670891 1.056013 1.401500 92 93 0.865638 1.178398 1.463045 0.474910 0.825403 1.140797 93 94 0.606190 0.888788 1.144990 0.252138 0.568774 0.853671 94 95 0.317491 0.568509 0.793966 T. 997345 0.282808 0.535885 95 96 1.997345 0.210950 0.401382 1.725503 1.965632 0.180942 96 97 1.622841 1.800341 1.951597 1.381876 1.593744 1.773005 97 98 1.205785 1.326024 1.420089 1.010577 1.180390 1.302156 98 99 "2.709547 2. 709547 2.709547 2.690431 2.690431 2.690431 99 ♦30 TABLE XXXVII. COMMUTATION COLUMNS. 5 PER CENT. Aa£. J>x- N^. s^. C.:. M^. E«. 1 1 AQE. 10 61,391.3 1,089,674.9 17,710,636.6 378.872 9,502.066 246,311.154 1 10 11 58,089.1 1,028,283.6 16,620,961.7 359.717 9,123.194 236,809.088 11 12 54,963.2 970,194.5 15,592,678.1 340.997 8,763.477 227,685.894 12 13 52,004.9 915,231.3 14,622,483.6 323.749 8,422.480 218,922.417 13 14 49,204.8 863,226.4 13,707,252.3 307.851 8,098.731 210,499.937 14 15 46,553.9 814,021.6 12,844,025.9 292.275 7,790.880 202,401.206 15 16 44,044.6 767,467.7 12,030,004.3 277.485 7,498.605 194,610.326 16 17 41,669.8 723,423.1 [11,262,536.6 263.856 7,221.120 187,111.721 17 18 39,421.7 681,753.3 10,539,113.5 250.895 6,957.264 179,890.601 18 19 37,293.6 642,331.6 9,857,360.2 238.571 6,706.369 172,933.337 19 20 35,279.2 605,038.0 9,215,028.6 227.210 6,467.798 166,226.968 20 21 33,372.0 569,758.8 8,609,990.6 216.391 6,240.588 159,759.170 21 22 31,566.4 536,386.8 8,040,231.8 206.087 6,024.197 153,518.582 22 23 29,857.1 504,820.4 7,503,845.0 196.583 5,818.110 147,494.385 23 24 28,238.9 474,963.3 6,999,024.6 187.517 5,621.527 141,676.275 24 25 26,706.6 446,724.4 6,524,061.3 178.869 5,434.010 136,054.748 25 26 25,256.0 420,017.8 6,077,336.9 171.155 5,255.141 130,620.738 26 27 23,882.2 394,761.8 5,657,319.1 163.515 5,083.986 125,365.597 27 28 22,581.4 370,879.6 5,262,557.3 156.457 4,920.471 120,281.611 28 29 21,349.6 348,298.2 4,891,677.7 150.164 4,764.014 115,361.140 29 30 20,182.8 326,948.6 4,543,379.5 143.895 4,613.850 110,597.126 30 31 19,077.9 306,765.8 4,216,430.9 138.092 4,469.955 105,983.276 31 32 18,031.3 287,687.9 3,909,665.1 i 132.715 4,331.863 101,513.321 32 33 17,039.9 269,656.6 3,621,977.2 i 127.728 4,199.148 97,181.458 33 34 16,100.8 252,616.7 3,352,320.6 123.096 4,071.420 92,982.310 34 35 15,211.0 236,515.9 3,099,703.9 118.961 3,948.324 88,910.890 35 36 14,367.7 221,304.9 2,863,188.0 114.776 3,829.363 84,962.566 36 37 13,568.7 206,937.2 2,641,883.1 i 111.033 3,714.587 81,133.203 37 38 12,811.6 193,368.5 2,434,945.9 i 107.685 3,603.554 77,418.616 38 39 12,093.8 180,556.9 2,241,577.4 104.546 3,495.869 73,815.062 39 40 11,413.4 168,463.1 2,061,020.5 101.732 3,391.323 70,319.193 40 41 10,768.2 157,049.7 1,892,557.4 ' 98.9489 3,289.591 66,927.870 41 42 10,156.4 146,281.46 1,735,507.74 96.6911 3,190.6419 63,638.2786 42 43 9,576.08 136,125.06 1,589,226.28 94.4239 3,093.9508 60,447.6367 43 44 9,025.68 126,548.98 1,453,101.22 92.4874 2,999.5269 57,353.6859 44 45 8,503.39 117,523.30 1,326,552.24 90.7332 2,907.0395 54,354.1590 45 46 8,007.72 109,019.91 1,209,028.94 89.1382 2,816.3063 51,447.1195 46 47 7,537.27 101,012.19 1,100,009.03 87.7777 2,727.1681 48,630.8132 47 48 7,090.58 93,474.92 998,996.84 86.5279 2,639.3904 45,903.6451 48 49 6,666.40 86,384.34 905,521.92 85.4596 2,552.8625 43,264.2547 49 50 6,263.50 79,717.94 819,137.58 84.5461 2,467.4029 40,711.3922 50 51 5,880.69 73,454.44 739,419.64 83.7630 2,382.8568 38,243.9893 51 52 5,516.89 67,573.75 665,965.20 83.0888 2,299.0938 35,861.1325 52 53 5,171.09 62,056.86 598,391.45 82.5041 2,216.0050 33,562.0387 53 54 4,842.34 56,885.77 536,334.59 81.9917 2,133.5009 31,346.0337 1 54 TABLE XXXVII. COMMUTATION COLUMNS. 5 PER CENT. *tt-< AQE. Dec. Na.. fear* Ca,. Ma,. Rs. AQE. 1 55 65 4,529.77 52,043.43 479,448.82 81.6663 2,051.5092 29,212.5328 56 4,232.39 47,513.66 427,405.39 81.3100 1,969.8429 27,161.0236 56 57 3,949.55 43,281.27 379,891.73 ! 81.0975 1,888.5329 25,191.1807 57 58 3,680.37 39,331.72 336,610.46 i 80.8333 1,807.4354 23,302.6478 58 59 3,424.28 35,651.35 297,278.74 80.5710 1,726.6021 21,495.2124 59 60 13,180.65 32,227.07 261,627.39 80.3543 1,646.0311 19,768.6103 60 61 3,948.84 29,046.42 229,400.32 80.1212 1,565.6768 18,122.5792 61 6? 2,728.30 26,097.58 200,353.90 79.7744 1,485.5556 16,556.9024 62 63 2,518.61 23,369.28 174,256.32 79.4110 1,405.7812 15,071.3468 63 64 2,319.26 20,850.67 150,887.04 78.9433 1,326.3702 13,665.5656 64 65 2,129.88 18,531.41 130,036.37 78.3800 1,247.4269 12,339.1954 65 66 1,950.07 16,401.53 111,504.96 77.6153 1,169.0469 11,091.7685 66 67 1,779.60 14,451.46 95,103.43 76.7456 1,091.4316 9,922.7216 67 68 1,618.11 12,671.86 80,651.97 75.5758 1,014.6860 8,831.2900 68 69 1,465.48 11,053.75 67,980.11 74.3104 939.1102 7,816.6041 69 70 1,321.38 9,588.27 56,926.36 72.7125 864.7998 6,877.4939 70 71 1,185.75 8,266.89 47,338.09 70.8896 792.0873 6,012.6941 71 72 1,058.39 7,081.14 39,071.20 68.7631 721.1977 5,220.6068 72 73 939.232 6,022.748 31,990.065 66.3269 652.4346 4,499.4091 73 74 §28.180 5,083.516 25,967.317 63.6577 586.1077 3,846.9745 74 75 725.085 4,255.336 20,883.801 60.6264 522.4500 3.260.8668 75 76 629.931 3,530.251 16,628.465 57.2956 461.8236 2,738.4168 76 77 542.639 2,900.320 13,098.214 53.7442 404.5280 2,276.5932 77 78 463.054 2,357.681 10,197.894 49.9138^ 350.7838 1,872.0652 78 79 391.090 1,894.627 7,840.213 45.8824 300.8700 1,521.2814 79 80 326.585 1,503.537 5,945.586 41.6991 254.9876 1,220.4114 80 81 269.334 1,176.952 4,442.049 37.4075 213.2885 965.4238 81 82 219.102 907.618 3,265.097 33.1163 175.8810 752.1353 82 ,83 175.551 688.516 2,357.479 28.8336 142.7647 576.2543 83 84 138.358 512.965 1,668.963 24.7098 113.9311 433.4896 84 85 107.0600 374.6070 1,155.9980 20.7627 89.2213 319.5585 85 86 81.1989 267.5473 781.3912 17.0782 68.4586 230.3372 86 87 60.2541 186.3484 513.8439 13.7658 51.3804 161.8786 87 88 43.6191 126.0943 327.4955 10.7822 37.6146 110.4982 88 89 30.7598 82.4752 201.4012 8.57174 26.8324 72.88355 89 90 20.7233 51.7154 118.9260 6.47659 18.26065 46.05116 90 91 13.2599 30.9921 67.2106 4.52782 11.78406 *27. 79151 91 92 8.10065 17.73217 36.21852 3.00676 7.25624 16.00745 92 93 4.70813 9.63152 18.48635 1.90567 4.24948 8.75121 93 94 2.57826 4.92339 8.85483 1.13554 2.34381 4.50173 94 95 1.31995 2.34513 3.93144 .628545 1.208272 2.157920 95 96 .628544 1.02518 1.58631 .334520 .579727 .949648 96 97 .264094 .396640 .561125 .150911 .245207 .369921 97 98 .100607 .132546 .164485 .063878 .094296 .124714 98 99 .031939 .031939 .031939 .030418 .030418 .030418 99 *32 TABLE XXXVII. LOGARITHMS OF COMMUTATION COLUMNS. 5 PER CENT. A0£. ADa,. ANa,. ASas. AUa;, AM,. ARa,. AGE. 10 4.788107 6.037297 7.248234 2.578493 3.977818 5.391484 10 11 4.764095 6.012113 7.220656 2.555961 3.960147 5.374398 11 12 4.740073 5.986859 7.192921 2.532750 3.942677 5.357336 12 13 4.716044 5.961531 7.165021 2.510208 3.925440 5.340290 13 14 4.692007 5.936125 7.136950 2.488341 3.908417 5.323252 14 15 4.667956 5.910636 7.108701 2.465792 3.891587 5.306214 15 18 4.643893 5.885060 7.080266 2.443230 3.874981 5.289166 16 17 4.619822 5.859393 7.051636 2.421366 3.858605 5.272101 17 18 4.595736 5.833627 7.022804 2.399493 3.842438 5.255008 18 19 4.571634 5.807759 6.993760 2.377618 3.826488 5.237878 19 20 4.547518 5.781783 6.964496 2.356428 3.810757 5.220701 20 21 4.523382 5.755691 6.935003 2.335239 3.795226 5.203466 21 22 4.499225 5.729478 6.905269 2.314050 3.779899 5.186161 22 23 4.475048 5.703137 6.875284 2.293546 3.764782 5.168775 23 24 4.450847 5.676660 6.845037 2.273041 3.749855 5.151297 24 25 4.426619 5.650039 6.814518 2.252535 3.735120 5.133713 25 26 4.402364 5.623268 6.783714 2.233390 3.720584 5.116012 26 27 4.378074 5.596335 6.752611 2.213558 3.706205 5.098178 27 28 4.353751 5.569232 6.721197 2.194396 3.692007 5.080199 28 29 4.329390 5.541951 6.689458 2.176566 3.677973 5.062059 29 30 4.304982 5.514479 6.657379 2.158045 3.664064 5.043744 30 31 4.280530' 5.486807 6.624945 2.140168 3.650304 5.025237 31 32 4.256026 5.458922 6.592139 2.122921 3.636674 5.006523 32 33 4.231468 5.430811 6.558945 2.106286 3.623161 4.987583 33 34 4.206847 5.402462 6.525346 2.090244 3.609746 4.968401 34 35 4.182158 5.373861 6.491321 2.075404 .3.596412 4.948955 35 36 4.157387 5.344991 6.456850 2.059851 3.583126 4.929228 36 37 4.132539 5.315838 6.421913 2.045453 3.569911 4.909199 37 38 4.107603 5.286386 6.386489 2.032155 3.556731 4.888846 38 39 4.082563 5.256614 6.350554 2.019306 3.543555 4.868145 39 40 4.057414 5.226505 6.314082 2.007457 3.530369 4.847074 40 41 4.032141 5.196037 6.277049 1.995411 3.517142 4.825607 41 42 4.006741 5.165189 6.239427 1.985386 3.503878 4.803718 42 43 3.981188 5.133938 6.201186 1.975082 3.490513 4.781379 43 44 3.955480 5.102258 6.162296 1.966083 3.477053 4.758561 ■ 44 45 3.929592 5.070124 6.122724 1.957766 3.463451 4.735233 45 46 3.9035(39 5.037506 6.082436 1.950064 3.449680 4.711361 46 47 3.877214 5.004373 6.041397 1.943384 3.435712 1 4.686911 47 48 3.850682 4.970694 5.999564 1.937156 3.421504 4.661847 48 49 3.823891 4.936435 5.956899 1.931761 3.407027 4.636130 49 50 3.796817 4.901556 5.913357 1.927093 3.392240 4.609716 50 51 3.769428 4.866018 5.868891 1.923052 3.377098 4.582563 51 52 3.741694 4.829778 5.823452 1.919543 3.361557 4.554624 52 53 3.713582 4.792790 5.776985 1.916476 3.345571 4.525848 53 54 3.685056 4.755003 5.729436 1.913770 3.329093 4.496183 54 TABLE XXXYII. LOGARITHMS OF COMMUTATION COLUMNS. 5 PER CENT. "33 AGE. AD^. ANa,. ASx- .AC«,. AMa;. AKa;. AGE. 55 55 3.656076 4.716366 5.680742 1.912043 3.312073 4.465569 56 3.626586 4.676818 5.630840 1.910144 3.294432 4.433946 56 57 3.596547 4.636300 5.579660 1.909007 3.276125 4.401248 57 58 3.565892 4.594743 5.527128 1.907590 3.257063 4.367405 58 59 3.534570 4.552076 5.473164 1.906179 3.237192 4.332341 59 60 3.502516 4.508221 5.417684 1.905009 3.216438 4.295976 60 61 3.469651 4.463092 5.360594 1.903747 3.194702 4.258220 61 62 3.435892 4.416600 5.301798 1.901863 3.171889 4.218979 62 63 3.401160 4.368645 5.241189 1.899881 3.147918 4.178152 63 64 3.365349 4.319120 5.178652 1.897315 3.122665 4.135627 64 65 3.328354 4.267908 5.114065 1.894205 3.096015 4.091287 65 66 3.290050 4.214884 5.047294 1.889947 3.067832 4.045001 66 67 3.250322 4.159912 4.978196 1.885054 3.037997 3.996631 67 68 3.209008 4.102840 4.906615 1.878383 3.006332 3.946025 68 69 3.165979 4.043509 4.832382 1.871050 2.972717 3.893018 69 70 3.121029 3.981740 4.755314 1.861609 2.936916 3.837430 70 71 3.073992 3.917342 4.675211 1.850582 2.898773 3.779069 71 72 3.024648 3.850103 4.591857 1.837355 2.858055 3.717721 72 73 2.972773 3.779794 4.505015 1.821689 2.814537 3.653156 73 74 2.918125 3.706164 4.414427 1.803851 2.767978 3.585119 74 75 2.860389 3.628933 4.319810 1.782662 2.718045 3.513332 75 76 2.799293 3.547806 4.220852 1.758122 2.664476 3.437499 76 77 2.734511 3.462446 4.117212 1.730332 2.606948 3.357286 77 78 2.665632 3.372485 4.049136 1.698221 2.545040 3.272321 78 79 2.592277 3.277524 3.894328 1.661647 2.478379 3.182209 79 80 2.513996 3.177114 3.774195 1.620127 2.406519 3.086506 80 81 2.430291 3.070759 3.647583 1.572958 2.328967 2.984718 81 82 2.340645 2.957903 3.513896 1.520042 2.245219 2.876296 82 83 2.244404 2.837914 3.372448 1.459899 2.154621 2.760614 83 84 2.141005 2.710088 3.222446 1.392869 2.056642 2.636979 84 85 2.029627 2.573576 3.062957 1.317285 1.950468 2.504550 85 86 1.909550 2.427401 2.892869 1.232443 1.835428 2.362364 86 87 1.779987 2.270325 2.710831 1.138802 1.710797 2.209189 87 88 1.639677 2.100695 2.515205 1.032707 1.575356 2.043355 88 89 1.487983 1.916323 2.304062 0.933069 1.428659 1.862629 89 90 1.316459 1.713619 2.075277 0.811346 1.261516 1.663240 90 91 1.122540 1.491251 1.827438 0.655890 1.071295 1.443912 91 92 0.908520 1.248762 1.558930 0.478101 0.860712 1.204322 92 93 0.672848 0.983695 1.266851 0.280048 0.628336 0.942069 93 94 0.411327 0.692264 0.947181 0.055202 0.369922 0.653380 94 95 0.120556 0.370167 0.594551 1.798336 0.082165 0.334035 95 96 1.798336 0.010800 0.200388 1.524422 1.763223 1.977563 96 97 1.421759 1.598397 1.749060 1.178721 1.389533 1.568109 97 98 1.002630 1.122367 1.216127 2.805349 2.974493 1.095915 98 99 2.504319 2.504319 2.504319 2.483130 2.483130 2.483130 99 *34 TABLE XXXYIII. COMMUTATION COLUMNS. 6 PER CENT. AGE. D^. Na,. Sa;. C.,. Ma,. R^. AOE. 10 10 55,839.4 864,777.4 12,598.316.8 341.358 6,889.834 151,664.904 11 52,337.5 808,938.0 11,733,539.4 321.042 6,548.476 144,775.070 11 12 49,054.0 756,600.5 10,924,601.4 i 301.464 6,227.434 138,226.594 12 13 45,975.8 .707,546.5 10,168,000.9 i 283.515 5,925.970 131,999.160 13 14 43,089.8 661,570.7 9,460,454.4 267.050 6,642.455 126,073.190 14 15 40,383.8 618,480.9 8,798,883.7 251.146 5,375.405 120,430.735 15 16 37,846.7 578,097.1 8,180,402.8 : 236.188 5,124.259 115,055.330 16 17 35,468.3 540,250.4 7,602,305.7 222.468 4,888.071 109,931.071 17 18 33,238.2 504,782.1 7,062,055.3 209.545 4,665.603 105,043.000 18 19 31,147.3 471,543.9 6,557,273.2 197.372 4,456.058 100,377.397 19 20 29,186.8 440,396.6 6,085,729.3 186.200 4,258.686 95,921.339 20 21 27,348.5 411,209.8 5,645,332.7 175.661 4,072.486 91,662.653 21 22 25,624.8 383,861.. 3 5,234,122.9 165.718 3,896.825 87,590.167 22 23 24,008.6 358,236.5 4,850,261.6 156.584 3,731.107 83,693.342 23 24 22,493.1 334,227.9 4,492,025.1 147.954 3,574.523 79,962.235 24 25 21,071.9 311,734.8 4,157,797.2 139.799 3,426.569 76,387.712 25 26 19,739.4 290,662.9 3,846,062.4 132.508 3,286.770 72,961.143 26 27 18,489.6 270,923.5 3,555,399.5 125.399 3,154.262 69,674.373 27 28 17,317.6 252,433.9 3,284,476.0 118.855 3,028.863 66,520.111 28 29 16,218.4 235,116.3 3,032,042.1 112.998 2,910.008 63,491.248 29 30 15,187.5 218,897.9 2,796,925.8 107.258 2,797.010 60,581.240 30 31 14,220.5 203,710.4 2,578,027.9 101.962 2,689.752 57,784.230 31 32 13,313.6 189,489.9 2,374,317.5 97.0677 2,587.7904 55,094.4779 32 33 12,463.0 176,176.3 2,184,827.6 92.5386 2,490.7227 52,506.6875 33 34 11,665.0 163,713.3- 2,008,651.3 88.3414 2,398.1841 50,015.9648 34 35 10,916.4 152,048.3 1,844,938.0 84.5684 2,309.8427 47,617.7807 35 36 10,213.9 141,131.9 1,692,889.7 80.8236 2,225.2743 45,307.9380 36 37 9,554.90 130,918.01 1,551,757.77 77.4503 2,144.4507 43,082.6637 37 38 8,936.61 121,363.11 1,420,839.76 74.4061 2,067.0004 40,938.2130 38 39 8,356.36 112,426.50 1,299,476.65 71.5555 1,992.5943 38,871.2126 39 40 7,811.80 104,070.14 1,187,050.15 68.9727 1,921.0388 36,878.6183 40 41 7,300.66 96,258.34 1,082,980.01 66.4531 1,852.0661 34,957.5795 41 42 6,820.97 88,957.68 986,721.67 64.3241 1,785.6130 33,105.6134 42 43 6,370.54 82,136.71 897,763.99 62.2233 1,721.2889 31,319.9004 43 44 5,947.73 75,766.17 815,627.28 60.3722 1,659.0656 29,598.6115 44 45 5,550.68 69,818.44 739,861.11 58.6684 1,598.6934 27,939.5459 45 46 5,177.82 64,267.76 670,042.67 57.0933 1,540.0250 26,340.8525 46 47 4,827.64 59,089.94 605,774.91 55.6915 1,482.9317 24,800.8275 47 48 4,498.69 54,262.30 546,684.97 54.3806 1,427.2402 23,317.8968 48 49 4,189.66 49,763.61 492,422.67 53.2026 1,372.8596 21,890.6656 49 50 3,899.32 45,573.95 442,659.06 52.1373 1,319.6570 20,517.7960 50 51 3,626.46 41,674.63 397,085.11 51.1671 1,267.5197 19,198.1390 51 52 3,370.02 38,048.17 355,410.48 50.2765 1,216.3526 17,930.6193 52 53 3,128.99 34,678.15 317,362.31 49.4517 1,166.0761 16,714.2667 53 54 2,902.42 31,549.16 282,684.16 48.6809 1,116.6244 15,548.1906 54 TABLE XXXYIII. COMMUTATION COLUMNS. 6 PER CENT. *35 AGE. Do,. Na> Sx' Cx. Ma,. Hr. AQiE. 55 55 2,689.45 28,646.74 251,135.00 48.0303 1,067.9435 14,431.5662 56 2,489.19 25,957.29 222,488.26 47.3696 1,019.9132 13,363.6227 56 57 2,300.93 23,468.10 196,530.97 46.8001 972.5436 12,343.7095 57 58 2,123.88 21,167.17 173,062.87 46.2075 925.7435 11,371.1659 58 59 1,957.46 19,043.29 151,895.70 45.6231 879.5360 10,445.4224 59 60 1,801.03 17,085.83 132,852.41 45.0711 833.9129 9,565.8864 60 61 1,654.02 15,284.80 115,766.58 44.5164 788.8418 8,731.9735 61 62 1,515.88 13,630.78 100,481.78 43.9056 744.3254 7,943.1317 62 63 1,386.17 12,114.90 86,851.00 43.2933 700.4198 7,198.8063 63 64 1,264.41 10,728.73 74,736.10 42.6323 657.1265 6,498.3865 64 65 1,150.21 9,464.32 64,007.37 41.9288 614.4942 5,841.2600 65 66 1,043.18 8,314.11 54,543.05 41.1279 572.5654 5,226.7658 66 67 943.001 7,270.934 46,228.937 40.2835 531.4375 4,654.2004 67 68 849.339 6,327.933 38,958.003 39.2952 491.1540 4,122.7629 68 69 761.967 5,478.594 32,630.070 38.2728 451.8588 3,631.6089 69 70 680.564 4,716.627 27,151.476 37.0965 413.5860 3,179.7501 70 71 604.947 4,036.063 22,434.849 35.8253 376.4895 2,766.1641 71 72 534.878 3,431.116 18,398.786 34.4228 340.6642 2,389.6746 72 73 470.180 2,896.238 14,967.670 32.8900 306.2414 2,049.0104 73 74 410.676 2,426.058 12,071.432 31.2686 273.3514 1,742.7690 74 75 356.161 2,015.382 9,645.374 29.4987 242.0828 1,469.4176 75 76 306.503 1,659.221 7,629.992 27.6150 212.5841 1,227.3348 76 77 261.538 1,352.718 5,970.771 25.6589 184.9691 1,014.7507 77 78 221.075 1,091.180 4,618.053 23.6054 159.3102 829.7816 78 79 184.956 870.105 3,526.873 21.4942 135.7048 670.4714 79 80 152.993 685.149 2,656.768 19.3502 114.2106 534.7666 80 81 124.982 532.156 1,971.619 17.1949 94.8604 420.5560 81 82 100.713 407.174 1,439.463 15.0788 77.6655 325.6956 82 83 79.9335 306.4614 1,032.2888 ; 13.0049 62.5867 248.0301 83 84 62.4041 226.5279 725.8274 11.0398 49.5818 185.4434 84 85 47.8320 164.1238 499.2995 9.18883 38.54196 135.86156 85 86 35.9357 116.2918 335.1757 7.48689 29.35313 97.31960 86 87 26.4147 80.3561 218.8839 5.97784 21.86624 67.96647 87 88 18.9417 53.9414 138.5278 4.63802 15.88840 46.10023 88 89 13.2315 34.9997 84.5864 3.65240 11.25038 30.21183 89 90 8.83015 21.76824 49.58671 2.73362 7.59798 18.96145 90 91 5.59670 12.93809 27.81847 1.89306 4.86436 11.36347 91 92 3.38684 7.34139 14.88038 1.24526 2.97130 6.49911 92 93 1.94988 3.95455 7.53899 .781790 1.726036 ' 3.527809 93 94 1.05772 2.00467 3.58444 .461455 .944246 1.801773 94 95 .536392 .946947 1.579772 .253015 .482791 .857527 95 96 .253015 .410555 .632825 .133387 .229776 .374736 96 97 .105306 .157540 .222270 .059607 .096389 .144960 97 98 .039738 .052234 .064730 .024993 .036782 .048571 98 99 .012496 .012496 .012496 .011789 .011789 .011789 99 *36 TABLE XXXYIII. LOGABITHMS OP COMMUTATION COLUMNS. 6 PER CENT. AGE. 10 AD,. ANa,. Aba?. ACa,. AMa,. ARai. AGE. 4.746941 5.936905 7.100313 2.533211 3.838209 5.180885 10 11 4.718813 5.907915 7.069429 : 2.506562 3.816140 5.160694 11 12 4.690674 5.878866 7.038405 ■ 2.479235 3.794309 5.140591 12 13 4.662529 5.849755 7.007236 2.452576 3.772759 5.120571 13 14 4.634375 5.820576 6.975912 2.426592. 3.751468 5.100623 14 15 4.606207 5.791327 6.944427 ^ 2.399927 3.730411 5.080738 15 16 4.578028 5.762000 6.912775 2.373257 3.709631 5.060907 16 17 4.549840 5.732595 6.880945 2.347268 3.689138 5.041120 17 18 4.521637 5.703104 6.848931 2.321278 3.668908 5.021367 18 19 4.493420 5.673522 6.816723 2.295286 3.648951 5.001636 19 20 4.465187 5.643844 6.784313 2.2699,81 3.629276 4.981915 20 21 4.436934 5.614064 6.751690 2.244675 3.609860 4.962192 21 22 4.408661 5.584174 6.718844 2.219369 3.590711 4.942456 22 23 4.380367 5.554170 6.685766 2.194749 3.571838 4.922691 23 24 4.352049 5.524042 6.652442 2.170127 3.553218 4.902885 24 25 4.323704 5.493786 6.618864 2.145505 3.534859 4.883023 25 26 4.295334 5.463389 6.585016 2.122242 3.516769 4.863092 26 27 4.266927 5.432847 6.550888 2.098294 3.498898 4.843073 27 28 4.238487 5.402148 6.516466 2.075016 3.481280 4.822953 28 29 4.210009 5.371283 6.481735 2.053069 3.463894 4.802714 29 30 4.181485 5.340242 6.446681 2.030431 3.446694 4.782338 30 31 4.152916 5.309013 6.411288 2.008438 3.429712 4.761810 31 32 4.124296 5.277586 6.375539 1.987075 3.412929 4.741108 32 33 4.095621 5.245948 6.339418 1.966323 3.396325 4.720214 33 34 4.066884 5.214084 6.302904 1.946164 3.379883 4.699109 34 35 4.038078 5.181981 6.265982 1.927208 3.363583 4.677769 35 36 4.009191 5.149625 6.228628 | 1.907538 3.347384 4.656174 36 37 3.980226 5.116999 6.190824 1.889023 3.331316 4.634303 37 38 3.951173 5.084087 6.152545 1.871609 3.315341 4.612129 38 39 3.922017 5.050869 6.113768 1.854643 3.299419 4.589628 39 40 3.892751 5.017327 6.074469 1.838677 3.283536 4.566774 40 41 3.863362 4.983438 6.034620 1.822515 3.267656 4.543541 41 42 3.833846 4.949183 5.994195 1.808374 3.251787 4.519901 42 43 3.804176 4.914537 5.953162 1.793953 3.235854 4.495821 43 44 3.774351 4.879475 5.911492 1.780837 3.219864 4.471271 44 45 3.744346 4.843970 5.869151 1.768404 3.203765 4.446219 45 46 3.714147 4.807993 5.826103 1.756585 3.187528 4.420630 46 47 3.683735 4.771513 5.782311 1.745789 3.171121 4.394466 47 48 3.653086 4.734498 5.737737 1.735444 3.154497 4.367689 48 49 3.622179 4.696912 5.692338 1.725933 3.137626 4.340259 49 50 3.590989 4.658717 5.646069 1.717149 3.120461 4.312131 50 51 3.559483 4.619872 5.598884 1.708991 3.102955 4.283259 51 52 3.527633 4.580334 5.550731 1.701365 3.085060 4.253595 52 53 3.495404 4.540056 5.501555 1.694181 3.066727 4.223087 53 54 3.462761 4.498988 5.451301 1.687359 3.047907 4.191680 54 i TABLE XXXYIII. LOGARITHMS OF COMMUTATION COLUMNS. 6 PER CENT. *37 AQrB. W„. AN«. Aba). AOa,. AMa,. AE«,. AGE, 55 55 3.429664 4.457075 5.399907 1.681515 3.028548 4.159313 56 3.396059 4.414259 5.347307 1.675499 3.008563 4.125924 56 57 3.361903 4.370478 5.293431 1.670247 2.987909 4.091446 57 58 3.337131 4.325663 5.238204 1.664713 2.966491 4.055805 58 59 3.291693 4.279742 5.181546 1.659185 2.944254 4.018926 59 60 3.255522 4.232636 5.123370 1.653898 2.921121 3.980725 60 61 3.218540 4.184259 5.063583 1.648520 2.896990 3.941113 61 . 62 3.180664 4.134521 5.002088 1.642520 2.871763 3.899992 62 i 63 3.141817 4.083319 4.938775 1.636420 2.845358 3.857260 63 64 3.101889 4.030548 4.873531 1.629738 2.817649 3.812805 64 65 3.060777 3.976089 4.806230 1.622512 2.788518 3.766507 65 66 3.018357 3.919816 4.736739 1.614137 2.757825 3.718233 66 67 2.974512 3.861590 4.664913 1.605127 2.725452 3.667845 67 68 2.929081 3.801262 4.590597 1.594339 2.691218 3.615188 68 69 2.881936 3.738669 4.513618 1.582890 2.655003 3.560099 69 70 2.832869 3.673631 4.433794 1.569333 2.616566 3.502393 70 71 2.781717 3.605958 4.350923 1.554190 2.575753 3.441878 71 72 2.728255 3.535435 4.264789 • 1.536846 2.532327 3.378338 72 73 2.672264 3.461835 4.175154 1.517064 2.486064 3.311544 73 74 2.613499 3.384901 4.081758 1.495109 2.436721 3.241240 74 75 2.551646 3.304357 3.984319 1.469803 2.383964 3.167145 i 75 76 2.486434 3.219904 3.882525 1.441146 2.327531 3.088963 76 77 2.417535 3.131207 3.776031 1.409239 2.267099 3.006359 77 78 2.344540 3.037896 3.664459 1.373012 2.202244 2.918964 78 79 2.267069 2.939571 3.547390 1.332321 2.132595 2.826380 79 80 2.184671 2.835785 3.424354 1.286685 2.057706 2.728164 80 81 2.096849 2.726039 3.294823 1.235400 1.977085 2.623824 81 82 2.003086 2.609780 3.158200 1.178367 1.890228 2.512812 82 83 1.902729 2.486376 3.013801 1.114107 1.796482 2.394504 83 84 1.795213 2.355122 2.860833 1.042961 1.695322 2.268211 84 85 1.679719 2.215172 2.698362 0.963260 1.585934 2.133097 85 86 1.555526 2.065549 2.525272 0.874302 1.467654 1.988200 86 87 1.421846 1.905019 2.340214 0.776544 1.339774 1.832295 87 88 1.277419 1.731922 2.141537 0.666332 1.201070 1.663703 88 89 1.121609 1.544064 1.927300 0.562578 1.051167 1.480177 89 90 0.945968 1.337823 1.695365 0.436739 0.880698 1.277871 90 91 0.747932 1.111870 1.444333 0.277165 0.687025 1.055511 91 92 0.529795 0.865778 1.172614 0.095261 0.472947 0.812854 92 93 0.290008 0.597098 0.877313 1.893090 0.237050 0.547505 93 94 0.024370 0.302043 0.554421 1.664129 1.975085 0.255700 94 95 1.729482 T. 976326 0.198594 T. 403146 T. 683759 T. 933248 95 96 1.403146 1.613372 1.801283 1.125115 1.361304 1.573725 96 97 1.022452 1.197391 1.346881 2.775298 2.984027 1.161248 97 98 2.599206 2.717953 2.811106 2.397809 2.565635 2.686377 98 99 2.096779 2.096779 2.096779 2.071474 2.071474 2.071474 99 ^38 TABLE XXXIX. COMMUTATION COLUMNS. 7 PER CENT Aa£. • Bo-,. N«,. S«,. c«,. M„. R«,. AGE. 10 50,834.9 697,841.8 9,179,521.1 307.8601 5,181.6666 '97,313.2662 10 11 47,201.4 647,006.9 8,481,679.3 286.8318 4,873.8065 92,131.5996 11 12 43,826.7 599,805.5 7,834,672.4 266.8222 4,586.9747 87,257.7931 12 13 40,692.6 555,978.8 7,234,866.9 248.5908 j 4,320.1525 82,670.8184 13 14 37,781.9 515,286.2 6,678,888.1 231.9654 4,071.6617 78,350.6659 14 15 35,078.3 477,504.3 6,163,601.9 216.1127 3,839.5963 74,279.1042 15 16 32,567.3 442,426.0 5,686,097.6 201.3413 3,623.4836 70,439.5079 16 17 30,235.4 409,858.7 5,243,671.6 187.8736 3,422.1423 66,816.0243 17 18 28,069.5 379,623.3 4,833,812.9 175.3063 3,234.2687 63,393.8820 18 19 26,057.9 351,553.8 4,454,189.6 163.5792 3,058.9624 60,159.6133 19 20 24,189.6 325,495.9 4,102,635.8 152.8778 2,895.3832 '57,100.6509 20 21 22,454.2 301,306.3 3,777,139.9 142.8764 2,742.5054 [54,205.2677 21 22 20,837.6 278,852.1 3,475,833.6 133.5294 2,599.6290 51,462.7623 22 23 19,345.3 258,014.5 3,196,981.5 124.9910 2,466.0996 1 48,863. 1333 23 24 17,954.7 238,669.2 2,938,967.0 116.9982 2,341.1086 146,397.0337 24 25 16,666.9 220,714.5 2,700,297.8 109.5163 2,224.1104 44,055.9251 25 26 15,463.5 204,047.6 2,479,583.3 102.8345 2,114.5941 141,831.8147 26 27 14,349.1 188,584.1 2,275,535.7 96.4078 2,011.7596 ; 39,717.2206 27 28 13,313.9 174,235.0 2,086,951.6 90.5224 1,915.3518 37,705.4610 28 29 12,352.4 160,921.1 1,912,716.6 85.2573 1,824.8294 35,790.1092 29 30 11,459.0 148,568.72 1,751,795.48 80.1708 1,739.5721 33,965.2798 30 31 10,629.2 137,109.72 1,603,226.76 75.4997 1,659.4013 32,225.7077 31 32 9,858.32 126,480.52 1,466,117.04 71.2038 1,583.9016 30,566.3064 32 33 9.142.19 116,622.20 1,339,636.52 67.2472 1,512.6978 28,982.4048 33 34 8,476.86 107,480.01 1,223,014.32 63.5971 1,445.4506 27,469.7070 34 35 7,858.71 99,003.15 1,115,534.31 60.3119 1,381.8535 26,024.2564 35 36 7,284.27 91,144.44 1,016,531.16 57.1026 1,321.5416 24,642.4029 36 37 6,750.61 83,860.17 925,386.72 54.2079 1,264.4390 23,320.8613 37 38 6,254.78 77,109.56 841,526.55 51.5905 1,210.2311 22,056.4223 38 39 5,794.01 70,854.78 764,416.99 49.1504 1,158.6406 20,846.1912 39 40 5,365.81 65,060.77 693,562.21 46.9335 1,109.4902 19,687.5506 40 41 4,967.83 59,694.96 628,501.44 44.7964 1,062.5567 18,578.0604 41 42 '4,598.04 54,727.13 568,806.48 42.9560 1,017.7603 17,515.5037 42 43 4,254.28 50,129.09 514,079.35 41.1647 974.8043 16,497.7434 43 44 3,934.80 45,874.81 463,950.26 39.5668 933.6396 15,522.9391 44 45 3,637.81 41,940.01 418,075.45 38.0908 894.0728 14,589.2995 45 46 3,361.73 38,302.20 376,135.44 36.7217 855.9820 13,695.2267 46 47 3,105.08 34,940.47 337,833.24 35.4854 819.2603 12,839.2447 47 48 2,866.47 31,835.39 302,892.77 34.3263 783.7749 12,019.9844 48 49 2,644.61 28,968.92 271,057.38 33.2688 749.4486 11,236.2095 49 50 2,438.33 26,324.31 242,088.46 32.2979 716.1798 10,486.7609 50 51 2,246.51 23,885.98 215,764.15 31.4007j 683.8819 9,770.5811 51 52 2,068.15 21,639.47 191,878.17 30.56581 652.4812 •9,086.6992 52 53 1,902.28 19,571.32 170,238.70 29.7834{ 621.9154 8,434.2180 53 54 1,748.05 17,669.04 150,667.38 29.0451 592.1320 7,812.3026 54 TABLE XXXIX. COMMUTATION COLUMNS. 7 PER CENT. ♦39 AGE. D«,. N«,. Si. c„. M„. Ka,. AGR 55 1,604.65 15,920.99 132,998.34 28.3891 563.0869 7,220.1706 55 56 1,471.38 14,316.34 117,077.35 27.7369 534.6978 6,657.0837 56 57 1,347.29 12,845.06 102,761.01 27.1474 506.9609 6,122.3859 57 58 1,232.00 11,497.77 89,915.95 26.5532 479.8135 5,615.4250 58 59 1,124.85 10,265.77 78,418.18 25.9723 453.2603 5,135.6115 59 60 1,025.29 9,140.916 68,152.409 25.4183 427.2880 4,682.3512 60 61 932.797 8,115.626 59,011.493 24.8708 401.8697 4,255.0632 61 62 846.904 7,182.829 50,895.867 24.3003 376.9989 3,853.1935 62 63 767.199 6,335.925 43,713.038 23.7374 352.6986 3,476.1946 63 64 693.269 5,568.726 37,377.113 23.1566 328.9612 3,123.4960 64 65 624.758 4,875.457 31,808.387 22.5616 305.8046 2,794.5348 65 66 561.326 4,250.699 26,932.930 21.9239 283.2430 2,488.7302 66 67 502.680 3,689.373 22,682.231 21.2730 261.3191 2,205.4872 67 68 448.521 3,186.693 18,992.858 20.5572 240.0461 1,944.1681 68 69 398.621 2,738.172 15,806.165 19.8352 219.4889 1,704.1220 69 70 352.709 2,339.551 13,067.993 19.0459 199.6537 1,484.6331 70 71 310.588 1,986.842 10,728.442 18.2213 180.6078 1,284.9794 71 72 272.048 1,676.254 8,741.600 17.3443 162.3865 1,104.3716 72 73 236.906 1,404.206 7,065.346 16.4172 145.0422 941.9851 73 74 204.991 1,167.300 5,661.140 15.4620 128.6250 796.9429 74 75 176.118 962.309 4,493.840 14.4504 113.1630 668.3179 75 76 150.146 786.191 3,531.531 13.4013 98.71266 555.15498 76 77 126.922 636.045 2,745.340 12.3357 85.31136 456.44232 77 78 106.283 509.123 2,109.295 11.2423 72.97566 371.13096 78 79 88.0874 402.8396 1,600.1718 10.1412 61.73336 298.16530 79 80 72.1835 314.7522 1,197.3322 9.04428 51.59216 236.42194 80 81 58.4168 242.5687 882.5800 7.96179 42.54788 184.82978 81 82 46.6334 184.1519 640.0113 6.91672 34.58609 142.28190 82 83 36.6660 137.5185 455.8594 5.90966 27.66937 107.69581 83 84 28.3576 100.8525 318.3409 4.96979 21.75971 80.02644 84 85 21.5326 72.4949 217.4884 4.09788 16.78992 58.26673 85 86 16.0260 50.9623 144.9935 3.30768 12.69204 41.47681 86 87 11.6699 34.9363 94.0312 2.61630 9.38436 28.78477 87 88 8.29016 23.26642 59.09493 2.01094 6.76806 19.40041 88 89 5.73687 14.97626 35:82851 1.56879 4.75712 12.63235 89 90 3.79277 9.23939 20.85225 .16319 3.18833 7.87523 90 91 2.38145 5.44662 11.61286 .797990 2.025136 4.686905 91 92 1.42767 3.06517 6.16624 .520014 1.227146 2.661769 92 93 .814258 1.637496 3.101066 .323420 .707132 1.434623 93 94 .437568 .823238 1.463570 .189116 .383712 .727491 94 95 .219827 .385670 .640332 .102723 .194596 .343779 95 96 .102723 .165843 .254662 .053648 .091873 .149183 96 97 .042354 .063120 .088819 .023750 .038225 .057310 97 98 .015833 .020766 .025699 .009865 .014475 .019085 98 99 .004933 .004933 .004933 .004610 .004610 .004610 99 *40 TABLE XXXIX. LOGARITHMS OF COMMUTATION COLUMNS. 7 PER CENT. AOE. 10 ADa,. AN«. Aba;. ACa,. AMa,. AEa,. AGE. 4.706162 5.843757 6.962820 2.488353 3.714470 4.988172 10 11 4.673955 5.810909 6.928482 2.457627 3.687868 4.964409 11 12 4.641739 5.778010 6.894021 2.426222 3.661526 4.940804 12 13 4.609516 5.745058 6.859431 2.395485 3.635499 4.917352 13 14 4.577284 5.712049 6.824704 2.365423 3.609761 4.894043 14 15 4.545038 5.678977 6.789834 2.334680 3.584286 4.870867 15 16 4.512782 5.645841 6.754814 2.303933 3.559126 4.847816 16 17 4.480516 5.612635 6.719636 2.273866 3.534298 4.824881 17 18 4.448235 5.579353 6.684290 2.243798 3.509776 4.802047 18 119 4.415939 5.545992 6.648769 2.213728 3.485574 4.779305 19 20 4.383628 5.512545 6.613063 2.184344 3.461706 4.756641 20 21 4.351298 5.479008 6.577163 2.154961 3.438147 4.734041 21 22 4.318847 5.445374 6.541059 2.125577 3.414911 4.711493 22 23 4.286575 5.411644 6.504740 2.096879 3.392011 4.688981 23 24 4.254179 5.377796 6.468195 2.068179 3.369422 4.666490 24 25 4.221856 5.343831 6.431412 2.039479 3.347156 4.644004 25 26 4.189308 5.309732 6.394379 2.012139 3.325227 4.621507 26 27 4.156823 5.275505 6.357084 1.984112 3.303576 4.598979 27 28 4.124305 5.241135 6.319512 1.956756 3.282249 4.576404 28 29 4.091750 5.206613 6.281651 1.930731 3.261222 4.553763 29 30 4.059148 5.171927 6.243484 1.904016 3.240442 4.531035 30 31 4.026501 5.137068 6.204995 1.877945 3.219951 4.508202 31 32 3.993803 5.102024 6.166169 1.852503 3.199728 4.485243 32 33 3.961050 5.066781 6.126987 1.827674 3.179752 4.462134 33 34 3.928235 5.031328 6.087431 1.803438 3.160004 4.438854 34 35 3.895351 4.995649 6.047483 1.780403 3.140462 4.415378 35 36 3.862386 4.959730 6.007120 1.756656 3.121081 4.391683 36 37 3.829343 4.923556 5.96G323 1.734062 3.101898 4.367745 37 38 3.796212 4.887108 5.925068 1.712570 3.082868 4.343535 38 39 3.762979 4.850369 5.883330 1.691527 3.063949 4.319027 39 40 3.729635 4.813319 5.841085 1.671483 3.045123 4.294192 40 41 3.696167 4.775938 5.798306 1.651243 3.026352 4.269000 41 42 3.662573 4.738203 5.754964 1.633024 3.007645 4.243423 42 43 '3.628826 4.700090 5.711030 1.614525 2.988917 4.217425 43 44 3.594923 4.661574 5.666471 1.597331 2.970179 4.190974 44 45 3.560840 4.622628 5.621254 1.580820 2.951373 4.164034 45 46 3.526563 4.588224 5.575344 : 1.564923 2.932465 4.136569 46 47 3.492073 4.543329 5.528702 1.550050 2.913422 4.108539 47 48 3.457347 4.502910 5.481289 1.535627 2.894191 4.079904 48 49 3.422362 4.461933 5.433061 1.522037 2.874742 4.050620 49 50 3.387093 4.420357 5.383974 1.509175 2.855022 4.020641 50 51 3.351509 4.378143 5.333979 1.496940 2.834981 3.989920 51 52 3.315582 4.335247 5.283026 1.485235 2.814568 3.958406 52 53 3.279275 4.291620 5.231059 1.473974 2.793731 3.926045 53 54 3.242554 4.247213 5.178019 1 1.463073 2.772419 3.892779 54 TABLE XXXIX. LOGARITHMS OP COMMUTATION COLUMNS. 7 PER CENT. HX kQB. 55 XD^. AN:,. ASa;. aC^. AMa,. ARa,. AGE. 55 3.205379 4.201970 5.123846 1.453152 2.750575 3.858547 56 3.167695 4.155832 5.068473 1.443058 2.728108 3.823284 56 57 3.129462 4.108736 5.011828 1.433728 2.704975 3.786920 57 58 3.090612 4.060614 4.953837 1.424116 2.681073 3.749383 58 59 3.051096 4.011391 4.894417 1.414510 2.656348 3.710592 59 60 3.010847 3.960990 4.833481 1.405146 2.630721 3.670464 60 61 2.969787 3.909322 4.770936 1.395690 2.604085 3.628906 61 62 2.927834 3.856295 - 4.706683 1.385611 2.576340 3.585821 62 63 2.884908 3.801810 4.640611 1.375434 2.547404 3.541104 63 64 2.840902 3.745756 4.572605 1.364674 2.517145 3.494641 64 65 2.795712 3.688015 4.502541 1.353370 2.485444 3.446310 65 66 2.749215 3.628460 4.430284 1.340917 2.452159 3.395976 66 67 2.701292 3.566953 4.355685 1.327829 2.417171 3.343505 67 68 2.651783 3.503340 4.278591 1.312963 2.380295 3.288734 68 69 2.600560 3.437461 4.198826 1.297436 2.341413 3.231501 69 70 2.547416 3.369133 4.116209 1.279801 2.300277 3.171019 70 71 2.492185 3.298163 4.030537 1.260580 2.256737 3.108896 71 72 2.434645 3.224340 3.941591 1.239158 2.210550 3.043115 72 73 2.374576 3.147431 3.849134 1.215298 2.161494 2.974044 73 74 2.311734 3.067183 3.752904 1.189265 2.109326 2.901427 74 75 2.245803 2.983314 3.052618 1.159881 2.053705 2.824983 75 76 2.176513 2.895528 3.547963 1.127147 1.994373 2.744414 76 77 2.103536 2.803488 3.438596 1.091162 1.931007 2.659386 77 78 2.026462 2.706823 3.324137 1.050857 1.S63178 2.569527 78 79 1.944914 2.605132 3.204166 1.006088 1.790520 2.474442 79 80 1.858438 2.497969 3.078215 0.956374 1.712584 2.373688 80 ' 81 1.766538 2.384835 2.945754 0.901011 1.628878 2.266772 81 82 1.668697 2.265176 2.806188 0.839900 1.538901 2.153150 82 83 1.564263 2.138361 2.658830 0.771562 1.441999 2.032199 83 84 • 1.452669 2.003687 2.502892 0.696338 1.337653 1.903233 84 85 1.333096 1.860307 2.337436 0.612559 1.225049 1.765421 85 86 1.204825 1.707249 2.161349 0.519523 1.103531 1.617805 86 87 1.067067 1.543277 1.973272 0.417688 0.972405 1.459163 87 88 0.918563 1.366730 1.771550 0.303398 0.830464 1.287811 88 89 0.758675 1.175403 1.554229 0.195566 0.677344 1.101484 89 90 0.578956 0.965644 1.319153 0.065649 0.503563 0.896263 90 91 0.376842 0.736127 1.064939 1.901997 0.306454 0.670886 91 92 0.154628 0.486455 0.790021 1.716015 0.088896 0.425170 92 93 1.910762 0.214181 0.491511 1.509767 1.849510 0.156738 93 94 1.641046 1.915525 0.165413 1.276727 1.584005 1.861828 94 95 1.342080 T. 586216 1.806405 1.011666 1.289134 1.536279 95 96 1.011666 1.219697 1.405964 2.729557 2". 963188 1.173719 96 97 2.626895 2.800167 2.948506 2.375662- 2.582347 2.758230 97 98 2.199571 2.317353 2.409916 3.994096 2.160619 2.280692 98 99 3.693066 3.693066 3.693066 3.663682 3.663682 3.663682 99 *42 TABLE XL. COMMUTATION COLUMNS, 8 PER CENT. AOE. 10 J)x- N^. i^x- Co,. Mo,. Rx. AGE. 46,319.3 571,113.6 6,833,398.9 277.916 4,014.670 64,935.891 10 11 42,610.4 524,794.3 6,262,285.3 256.536 3,736.754 60,921.221 11 12 39,197.6 482,183.9 5,737,491.0 236.430 3,480.218 57,184.467 12 13 36,057.5 442,986.3 5,255,307.1 218.235 3,243.788 53,704.249 13 14 33,168.4 406,928.8 4,812,320.8 201.755 3,025.553 50,460.461 14 15 30,509.8 373,760.4 4,405,392.0 186.226 2,823.798 47,434.908 15 16 28,063.5 343,250.6 4,031,631.6 171.891 2,637.572 44,611.110 16 17 25,812.9 315,187.1 3,688,381.0 158.908 2,465.681 41,973.538 17 18 23,741.9 289,374.3 3,373,193.9 146.906 2,^06.773 39,507.857 18 19 21,836.4 265,632.3 3,083,819.7 135.809 2,159.867 37,201.084 19 20 20,083.0 243,795.9 2,818,187.4 125.749 2,024.058 35,041.217 20 21 18,469.6 223,712.9 2,574,391.5 116.434 1,898.309 33,017.159 21 22 16,985.1 205,243.3 2,350,678.6 107.810 1,781.875 31,118.850 22 23 15,619.1 188,258.2 2,145,435.3 99.9814 1,674.0644 29,336.9753 23 24 14,362.2 172,639.1 1,^57,177.1 92.7214 1,574.0830 27,662.9109 24 25 13,205.6 158,276.9 1,784,538.0 85.9883 1,481.3616 26,088.8279 25 26 12,141.4 145,071.3 1,626,261.1 79.9944 1,395.3733 ' 24,607.4663 26 27 11,162.1 132,929.9 1,481,189.8 74.3007 1,315.3789 I 23,212.0930 27 28 10,260.9 121,767.8 1,348,259.9 69.1189 1,241.0782 : 21,896.7141 28 29 9,431.72 111,506.91 1,226,492.13 64.4959 1,171.9593 20,655.6359 29 30 8,668.58 102,075.19 1,114,985.22 60.0865 1,107.4634 19,483.6766' 30 31 7,966.40 93,406.61 1,012,910.03 56.0616 1,047.3769 18,376.2132 31 32 7,320.22 85,440.21 919,503.42 52.3823 991.3153 17,328.8363 32 33 6,725.60 78,119.99 834,063.21 49.0134 938.9330 16,337.5210 33 34 6,178.39 71,394.39 755,943.22 45.9239 889.9196 16,398.5880 34 35 5,674.82 65,216.00 684,548.83 43.1483 843.9957 14,508.6684 35 36 5,211.31 59,541.18 619,332.83 40.4740 800.8474 13,664.6727 36 37 4,784.80 54,329.87 559,791.65 38.0666 760.3734 12,863.8253 37 38 4,392.31 49,545.07 505,461.78 35.8931 722.3068 12,103.4519. 38 39 4,031.07 45,152.76 455,916.71 33.8788 686. 4137* 11,381.1451 39 40 3,698.59 41,121.69 410,763.95 32.0512 652.5349 10,694.7314 40 41 3,392.56 37,423.10 369,642.26 30.3084 620.4837 10,042.1965 41 42 3,110.96 34,030.54 332,219.16 28.7942 590.1753 9,421.7128 42 43 2,851.72 30,919.58 298,188.62 27.3380 561.3811 8,831.5375 43 44 2,613.15 28,067.86 267,269.04 26.0335 534.0431 8,270.1564 44 45 2,393.54 25,454.71 239,201.18 24.8303 508.0096 7,736,1133 45 46 2,191.41 23,061.17 213,746.47 23.7162 483.1793 7,228.1037 46 47 2,005.37 20,869.76 190,685.30 22.7055 459.4631 6,744.9244 47 48 1,834.12 18,864.39 169,815.54 21.7604 436.7576 6,285.4613 48 49 1,676.50 17,030.27 150,951.15 20.8948 414.9972 5,848.7037 49 50 1,531.42 15,353.77 133,920.88 20.0972 394.1024 5,433.7065 50 51 1,397.88 13,822.35 118,567.11 19.3580 374.0052 5,039.6041 51 52 1,274.98 12,424.47 104,744.76 18.6688 354.6472 4,665.5989 52 53 1,161.87 11,149.49 92,320.29 18.0225 335.9784 4,310.9517 53 54 1,057.78 9,987.62 81,170.80 17.4130 317.9559 3,974.9733 54 TABLE XL. COMMUTATION COLUMNS. 8 PER CENT. H3 AGE. Do,. N.. bfl)' 0^ M«,. RfB- AGE. 55 962.011 8,929.843 71,183.180 16.8622 300.5429 3,657.0174 55 56 873.891 7,967.832 62,253.337 16.3222 283.6807 3,356.4745 56 57 792.835 7,093.941 54,285.505 15.8274 267.3-585 3,072.7938 57 58 718.279 6,301.106 47,191.564 15.3376 251.5311 2,805.4353 58 59 649.736 5,582.827 40,890.458 14.8632 236.1935 2,553.9042 59 60 586.745 4,933.091 35,307.631 14.4114 221.3303 2,317.7107 60 61 528.870 4,346.346 30,374.540 13.9705 206.9189 2,096.3804 61 62 475.724 3,817.476 26,028.194 13.5236 192.9484 1,889.4615 62 63 426.962 3,341.752 22,210.718 13.0881 179.4248 3,696.5131 63 64 382.247 2,914.790 18,868.966 1 12.6496 166.3367 1,517.0883 64 65 341.283 2,532.543 15,954.176 12.2104 153.6871 1,350.7516 65 66 303.792 2,191.260 13,421.633 11.7554 141.4767 1,197.0645 66 67 269.534 1,887.468 11,230.373 11.3008 129.7213 1,055.5878 67 68 238.268 1,617.934 9,342.905 10.8195 118.4205 925.8665 68 69 209.798 1,379.666 7,724.971 10.3428 107.6010 807.4460 69 70 183.916 1,169.868 6,345.305 9.83926 97.25822 699.84498 70 71 160.452 985.952 5,175.437 9.32613 87.41896 602.58676 71 72 139.241 825.500 4,189.485 8.79508 78.09283 515.16780 72 73 120.132 686.259 3,363.985 8.24783 69.29775 437.07497 73 74 102.985 566.127 2,677.726 7.69603 61.04992 367.77722 74 75 87.6605 463.1417 2,111.5992 7.12596 53.35389 306.72730 75 76 74.0413 375.4812 1,648.4575 6.54739 46.22793 253.37341 76 77 62.0095 301.4399 1,272.9763 5.97096 39.68054 207.14548 77 78 51.4451 239.4304 971.5364 5.39136 33.70958 167.46494 78 79 42.2430 187.9853 732.1060 4.81826 28.31822 133.75536 79 80 34.2957 145.7423 544.1207 4.25731 23.49996 105.43714 80 81 27.4979 111.4466 398.3784 3.71307 19.24265 81.93718 81 82 21.7480 83.9487 286.9318 3.19582 15.52958 62.69453 82 83 16; 9412 62.2007 202.9831 2.70523 12.33376 47.16495 83 84 12.9811 45.2595 140.7824 2.25393 9.62853 34.83119 Si 85 9.76558 32.27839 95.52286 1.84129 7.37460 25.20266 85 86 7.20090 22.51281 63.24447 1.47247 5.53331 17.82806 86 87 5.19504 15.31191 40.73166 1.15391 4.06084 12.29475 87 88 3.65633 10.11687 25.41975 .878699 2.906926 8.233909 88 89 2.50679 6.46054 15.30288 .679154 2.028227 5.326983 89 j 90 1.64194 3.95375 8.84234 .498897 1.349073 3.298756 90 ! 91 1.02142 2.31181 4.88859 .339094 .850176 1.949683 91 92 .606667 1.290385 2.57678 .218926 .511082 1.099507 92 93 .342803 .683718 1.286398 .134899 .292156 .588425 93 94 .182511 .340915 .602680 .078150 .157257 .296269 94 95 .090841 .158404 .261765 .04^056 .079107 .139012 95 96 .042056 .067563 .103361 i .021761 .037051 .059905 96 97 .017180 .025507 .035798 .009544 .015290 .022854 97 98 .006363 .008327 .010291 .003928 .005746 .007564 98 99 .001964 .001964 .001964 .001818 .001818 .001818 99 ♦44 TABLE XLI. COMMUTATION COLUMNS. 9 PER CENT. AQtE. ^x ^x- Saf. G^. Ma;. Ba,. AGE. 10 42,241.1 473,006.5 5,184,561.6 251.121 i ; 3,185.523 44,923.415 10 11 38,502.2 430,765.4 4,711,555.1 1 229.675 2,934.402 41,737.892 11 12 35,093.4 392,263.2 4,280,789.7 1 209.733 : 2,704.727 38,803.490 12 13 31,986.0 357,169.8 3,888,526.5 i 191.817 2,494.994 36,098.763 13 14 29,153.2 325,183.8 3,531,356.7 j 175.704 2,303.177 j 33,603.769 14 15 26,570.4 296,030.6 3,206,172.9 160.693 2,127.473 31,300.592 15 16 24,215.8 269,460.2 2,910,142.3 146.963 i 1,966.780 29,173.119 16 17 22,069.4 245,244.4 2,640,682.1 134.616 1,819.817 27,206.339 17 18 20,112.5 223,175.0 2,395,437.7 123.306 1,685.201 25,386.522 18 19 18,328.6 203,062.5 2,172,262.7 ! 112.947 1,561.895 j 23,701.321 19 20 16,702.2 184,733.9 1,969,200.2 ' 103.621 1.448.948 22.139.426 20 21 15,219.5 168,031.7 1,784,466.3 95.0650 1,345.3274 1 20,690.4780 1 21 22 13,867.8 152,812.2 1,616,434.6 ^ 87.2156 1,250.2624 ■ 19,345.1506 22 23 12,635.5 138,944.4 1,463,622.4 I 80.1407 1,163.0468 18,094.8882 23 24 11,512.1 126,308.9 1,324,678.0 73.6396 1,082.9061 16,931.8414 24 25 10,487.9 114,796.8 1,198,369.1 1 67.6656 1,009.2665 15,848.9353 ! 25 26 9,554.26 104,308.94 1,083,572.28 i 62.3714 941.6009 14,839.6688 ^ 26 27 8,703.02 94,754.68 979,263.34 1 57.4006 879.2295 13,898.0679 '. 27 28 7,927.00 86,051.66 884,508.66 i 52.9075 821.8289 13,018.8384 28 29 7,219.58 78,124.66 798,457.00 48.9159 768.9214 12,197.0095 29 30 6,574.55 70,905.08 720,332.34 45.1535 720.0055 11,428.0881 30 31 5,986.56 64,330.53 649,427.26 41.7425 674.8520 10,708.0826 31 32 5,450.50 58,343.97 585,096.73 38.6450 633.1095 10,033.2306 32 | 33 4,961.82 52,893.47 526,752.76 j 35.8279 594.4645 9,400.1211 33 34 4,516.29 47,931.65 473,859.29 33.2615 558.6366 8,805.6566 34 35 4,110.13 43,415.36 425,927.64 30.96,46 525.3751 8,247.0200 35 36 3,739.80 39,305.23 382,512.28 28.7789 494.4105 7,721.6449 36 37 3,402.23 35,565.43 343,207.05 26.8188 465.6316 7.227.2344 37 38 3,094.49 32,163.20 307,641.62 25.0555 438.8128 6,761.6028 38 39 2,813.93 •29,068.71 275,478.42 23.4325 413.7573 6,322.7900 39 40 2,558.15 26,254.78 246,409.71 21.9650 390.3248 5,909.0327 40 41 2,324.96 23,696.63 220,154.93 20.5801 368.3598 ! 5,518.7079 41 42 2,112.41 21,371.67 196,458.30 19.3726 347.7797 : 5,150.3481 42 43 1,918.62 19,259.26 175,086.63 18.2241 328.4071 i 4,802.5684 43 44 1,741.98 17,340.64 155,827.37 17.1952 310.1830 4,474.1613 44 45 1,580.95 15,598.66 138,486.73 16.2501 292.9878 4,163.9783 45 46 1,434.16 14,017.71 122,888.07 15.3786 276.7377 3,870.9905 46 47 1,300.37 12,583.55 108,870.36 14.5881 ! 261.3591 3,594.2528 47 48 1,178.41 11,283.18 96,286.81 13.8527 i 246.7710 3,332.8937 48 49 1,067.26 10,104.77 85,003.63 13.1796 232.9183 3,086.1227 49 50 965.955 9,037.511 74,898.859 12.5602 219.7387 2,853.2044 50 51 873.636 8,071.556 65,861.348 11.9872 207.1785 j 2,633.4657 51 52 789.514 7,197.920 57,789.792 11.4544 195.1913 : 2,4^6.2872 52 53 712.871 6,408.406 50,591.872 10.9564 183.7369 2,231.0959 53 54 643.053 5,695.535 44,183.466 10.4888 172.7805 2,047.3590 54 TABLE XLI. COMMUTATION COLUMNS. 9 PER CENT. ♦45 A&E. D.. N.. s.. c„. M.. R«. AaE. 55 55 579.469 5,052.482 38,487.931 10.06385 162.2917 1,874.5785 56 521.559 4,473.013 33,435.449 9.65214 152.22788 1,712.28676 56 57 468.843 3,951.454 28,962.436 9.27363 142.57574 1,560.05888 57 58 420.857 3,482.611 25,010.982 8.90421 133.30211 1,417.48314 58 59 377.204 3,061.754 21,528.371 8.54962 124.39790 1,284.18103 59 60 337.508 2,684.550 18,466.617 8.21372 115.84828 1,159.78313 60 61 301.427 2,347.042 15,782.067 7.88935 107.63456 1,043.93485 61 62 268.649 2,045.615 13,435.025 7.56693 99.74521 936.30029 62 63 238.901 1,776.966 11,389.410 7.25604 92.17828 836.55508 63 64 211.918 1,538.065 9,612.444 6.94860 84.92224 744.37680 64 65 187.472 1,326.147 8,074.379 6.64584 77.97364 659.45456 65 66 165.347 1,138.675 6,748.232 6.33950 71.32780 581.48092 66 67 145.355 973.328 5,609.557 6.03843 64.98830 510.15312 67 68 127.315 827.973 4,636.229 5.72817 58.94987 445.16482 68 69 111.074 700.658 3,808.256 5.42557 53.22170 386.21495 69 70 96.4773 589.5838 3,107.5975 5.11408 47.79613 332.99325 70 71 83.3973 493.1065 2,518.0137 4.80291 42.68205 285.19712 71 72 71.7082 409.7092 2,024.9072 4.48787 37.87914 242.51507 72 73 61.2997 338.0*010 1,615.1980 4.17001 33.39127 204.63593 73 74 52.0681 276.7013 1,277.1970 3.85533 29.22126 171.24466 74 75 43.9136 224.6332 1,000.4957 3.53700 25.36593 142.02340 75 76 36.7507 180.7196 775.8625 3.22001 21.82893 116.65747 76 77 30.4963 143.9689 595.1429 2.90958 18.60892 94.82854 77 78 25.0686 113.4726 451.1740 2.60305 15.69934 76.21962 78 79 20.3957 88.4040 337.7014 2.30500 13.09629 60.52028 79 80 16.4067 68.0083 249.2974 2.01796 10.79129 47.42399 80 81 13.0340 51.6016 181.2891 1.74385 8.77333 36.63270 81 82 10.2140 38.5676 129.6875 1.48715 7.02948 27.85937 82 83 7.88346 28.35357 91.11993 1.24731 5.54233 20.82989 83 84 5.98522 20.47011 62.76636 1.02969 4.29502 15.28756 84 85 4.46134 14.48489 42.29625 .833461 3.265334 10.992535 85 86 3.25950 10.02355 27.81136 .660399 2.431873 7.727201 86 87 2.32997 6.76405 17.78781 .512777 1.771474 5.295328 87 88 1.62481 4.43408 11.02376 i .386898 1.258697 3.523854 88 89 1.10376 2.80927 6.58968 .296293 .871799 2.265157 89 90 .716326 1.705505 3.780411 .215656 .575506 1.393358 90 91 .441525 .989179 2.074906 .145234 .359850 .817852 91 92 .259835 .547654 1.085727 .092906 .214616 .458002 92 93 . 145475 .287819 .538073 .056722 .121710 .243386 93 94 .076742 .142344 .250254 .032559 .064988 .121676 94 95 .037846 .065602 .107910 .017361 .032429 .056688 95 96 .017361 .027756 .042308 .008900 .015068 .024259 96 97 .007027 .010395 .014552 .003868 .006168 .009191 97 98 .002579 .003368 .004157 .001577 .002300 .003023 98 99 .000789 .000789 .000789 .000723 .000723 .000723 99 *46 TABLE XLII. COMMUTATION COLUMNS. 10 PER CENT. AGE. J)x. ^x- Sa.. C;.. Mo,. Kx- AGE. 10 38,554.3 395,760.5 4,000,286.2 227.120 2,576.107 32,098.169 10 11 34,822.3 357,206.2 3,604,525.7 205.836 2,348.987 29,522.062 11 12 31,450.8 322,383.9 3,247,319.5 186.254 2,143.151 27,173.075 12 13 28,405.3 1 290,933.1 2,924,935.6 168.795 1,956.897 25,029.924 13 14 25,654.2 262,527.8 2,634,002.5 153.211 1,788.102 23,073.027 14 15 23,168.9 236,873.6 2,371,474.7 138.847 1,634.891 21,284.925 15 16 20,923.7 213,704.7 2,134,601.1 125.829 1,496.044 19,650.034 16 17 18,895.8 192,781.0 1,920,896.4 114.210 1,370.215 18,153.990 17 18 17,063.8 173,885.2 1,728,115.4 103.664 1,256.005 16,783.775 18 19 15,408.8 156,821.4 1,554,230.2 94.0914 1,152.3412 15,527.7698 19 20 13,913.9 141,412.6 1,397,408.8 85.5376 1,058.2498 14,375.4286 20 21 12,563.5 127,498.7 1,255,996.2 77.7615 972.7122 13,317.1788 21 22 11,343.6 114,935.2 1,128,497.5 70.6923 894.9507 12,344.4666 22 23 10,241.7 103,591.6 1,013,562.3 64.3673 824.2584 11,449.5159 23 24 9,246.26 93,349.9 909,970.7 58.6080 759.8911 10,625.2575 24 25 8,347.07 84,103.61 816,620.82 53.3639 701.2831 9,865.3664 25 26 7,534.87 75,756.54 732,517.21 48.7414 647.9192 9,164.0833 26 27 6,801.16 68,221.67 656,760.67 44.4491 599.1778 8,516.1641 1 27 1 28 •6,138.41 61,420.51 588,539.00 40.5974 554.7287 7,916.9863 28 29 5,539.77 55,282.10 527,118.49 37.1933 514.1313 7,362.2576 29 30 4,998.96 49,742.33 471,836.39 34.0205 476.9380 6,848.1263 30 31 4,510.50 44,743.37 422,094.06 31.1645 442.9175 6,371.1883 31 32 4,069.28 40,232.87 377,350.69 28.5897 411.7530 5,928.2708 32 33 3,670.76 36,163.59 337,117.82 26.2646 383.1633 5,516.5178 33 34 3,310.79 32,492.83 300,954.23 24.1616 356.8987 5,133.3545 34 35 2,985.64 29,182.04 268,461.40 22.2886 332.7371 4,776.4558 35 36 2,691.94 26,196.40 239,279.36 20.5270 310.4485 4,443.7187 36 37 2,426.69 23,504.46 213,082.96 18.9550 289.9215 4,133.2702 37 38 2,187.13 21,077.77 189,578.50 17.5478 270.9665 3,843.3487 38 39 1,970.75 18,890.64 168,500.73 16.2619 253.4187 3,572.3822 39 40 1,775.33 16,919.89 149,610.09 15.1049 237.1568 3,318.9635 40 41 1,598.83 15,144.56 132,690.20 14.0239 222.0519 3,081.8067 41 42 1,439.46 13,545.73 117,545.64 13.0810 208.0280 2,859.7548 42 43 1,295.52 12,106.27 103,999.91 12.1936 194.9470 2,651.7268 43 44 1,165.55 10,810.75 91,893.64 11.4007 182.7534 2,456.7798 44 45 1,048.19 9,645.20 81,082.89 10.6761 171.3527 2,274.0264 45 46 942.224 8,597.01 71,437.69 10.0116 160.6766 2,102.6737 46 47 846.555 7,654.785 62,840.682 9.41070 150.66503 1,941.99711 47 48 760.185 6,808.230 55,185.897 8.85503 141.25433 1,791.33208 48 49 682.221 6,048.045 48,377.667 8.34818 132.39930 1,650.07775 49 50 611.854 5,365.824 42,329.622 7.88353 124.05112 1,517.67845 50 51 548.346 4,753.970 36,963.798 7.45549 116.16759 1,393.62733! 51 52 491.041 4,205.624 32,209.828 7.05933 108.71210 1,277.45974! 52 53 439.342 3,714.583 , 28,004.204 6.69103 101.65277 1,168.74764 53 54 392.711 3,275.241 1 24,289.621 6.34722 94.96174 1,067.09487 54 TABLE XLII. ••47 COMMUTATION COLUMNS. 1 10 PER CENT. AGE. 55 D,,. No,. • s^. Ca,. Ma,. E^. AGE. 350.662 2,882.531 21,014.380 6.03467 88.61452 972.13313 1 55 56 312.750 2,531.869 18,131.849 5.73523 82.57985 883.51861 56 57 278.583 2.219.119 15,599.980 5.46023 76.84462 :800. 93876 57 58 247.796 1,940.536 13,380.861 5.19506 71.38439 724.09414 58 59 220.075 1,692.740 11,440.325 4.94283 66.18933 652.70975 59 60 195.125 1,472.664 9,747.585 4.70546 61.24650 586.52042 60 61 172.681 1,277.539 8,274.921 4.47855 56.54104 525.27392 61 62 152.504 1,104.858 6,997.382 4.25647 52.06249 468.73288 62 63 134.384 952.354 5,892.524 4.04449 47.80602 416.67039 63 64 118.122 817.970 4,940.170 3.83791 43.76153 368.86437 64 65 103.546 699.848 4,122.200 3.63732 39.92362 325.10284 65 66 90.4955 596.3028 3,422.3523 3.43811 36.28630 285.17922 66 67 78.8306 505.8073 2,826.0495 3.24507 32.84819 248.89292 67 68 68.4191 426.9757 2,320.2422 3.05034 29.60312 216.04473 68 69 59.1488 358.5566 1,893.2665 2.86294 26.55278 186.44161 69 70 50.9087 299.4078 1,534.7099 2.67404 23.68984 159.88883 70 71 43.6066 248.4991 1,235.3021 2.48851 21.01580 136.19899 71 72 37.1539 204.8925 986.8030 2.30414 18.52729 115.18319 72 73 31.4721 167.7386 781.9105 2.12148 16.22315 96.65590 73 74 26.4895 136.2665 614.1719 1.94356 14.10167 80.43275 74 75 22.1378 109.7770 477.9054 1.76687 12.15811 66.33108 75 76 18.3585 87.6392 368.1284 1.59390 10.39124 54.17297 76 77 15.0956 69.2807 280.4892 1.42714 8.79734 43.78173 77 78 12.2961 54.1851 211.2085 1.26518 7.37020 34.98439 78 79 9.91311 41.88903 157.02340 1.11014 6.10502 27.61419 79 80 7.90178 31.97592 115.13437 .963058 4.994877 21.509173 80 81 6.22038 24.07414 83.15845 .824671 4.031819 16.514296 81 82 4.83022 17.85376 59.08431 .696884 3.207148 12.482477 82 83 3.69422 13.02354 41.23055 .579181 2.510264 9.275329 83 84 2.77920 9.32932 28.20701 .473785 1.931083 6.765065 84 85 2.05276 6.55012 18.87769 .380009 1.457298 4.833982 85 86 1.48614 4.49736 12.32757 .298365 1.077289 3.376684 86 87 1.05267 3.01122 7.83021 .229564 .778924 2.299395 87 88 .727410 1.958552 4.818985 .171635 .549360 1.520471 88 89 .489646 1.231142 2.8f)0433 .130246 ■ .377725 .971111 89 90 .314886 .741496 1.629291 .093937 .247479 .593386 90 91 .192323 .426610 .887795 .062687 .153542 .345907 91 92 .112152 .234287 .461185 .039736 .090855 .192365 92 93 .062221 .122135 .226898 .024040 .051119 .101510 93 94 .032524 .059914 .104763 .013674 .027079 .050391 94 95 .015894 .027390 .044849 .007226 .013405 .023312 95 96 .007225 .011496 .017459 .003670 .006179 .009907 96 97 .002898 .004271 .005963 .001580 .002509 .003728 97 98 .001054 .001373 .001692 .000639 .000929 .001219 98 99 .000319 .000319 .000319 .000290 .000290 .000290 99 ♦48 TA.BLE XLIII. COMMON LOGARITHMS FOR SPECIAL FORMULAS. Yalne of A {i)p^), of fXj:, A ju^,, v, (5, and X6. AOE, X. ^ (Vi?x). l^x- A^a;- AGE. X. 3 Per Cent. Per Cent. 4 Per Cent. Per Cent. 5 Per Cent. 6 Per Cent. 10 1.984341 1.982236 1.980143 1.978061 1.975988 1.971872 0.006493 3.81245 10 11 .984329 .982227 .980134 .978051 .975977 .971861 .006513 .81378 11 12 .984325 .982221 .980128 .978044 .975972 .971855 .006530 .81491 12 13 .984313 .982211 .980118 .978036 .975963 .971846 .006547 .81604 13 14 .984302 .982198 .980105 .978022 .975949 .971832 .006572 .81770 14 15 1.984289 1.982186 1.980094 1.978010 1.975937 1.971821 0.006600 3.81954 15 16 .984282 .982178 .980084 .978002 .975929 .971812 .006624 .82112 16 17 .984265 .982163 .980070 .977987' .975914 .971797 .006653 .82302. 17 18 .984250 .982147 .980055 .977971 .975898 .971783 .006687 .82523 18 19 .984236 .982133 .980039 .977957 .975884 .971767 .006721 .82743 19 20 1.984217 1.982113 1.980020 1.977937 1.975864 1.971747 0.006762 3.83008 20 21 .984195 .982092 .980000 .977916 .975.843 .971727 .006807 .83296 21 22 .984175 .982072 .979978 .977896 .975823 .971706 .006854 .83594 22 23 .984150 .982048 .979955 .977872 .975799 .971682 .006906 .83923 23 24 .984125 .982020 .979928 .977845 .975772 .971655 .006965 .84292 24 25 f. 984097 1.981995 1.979901 1.977818 1.975745 T.971630 0.007027 3.84677 25 26 .984062 .981959 .979866 .977783 .975710 .971593 .007098 .85114 26 27 .984028 .981925 .979833 .977750 .975677 .971560 .007176 .85588 27 28 .983993 .981888 .979795 .977713 .975639 .971522 .007258 .86082 28 29 .983943 .981841 .979748 .977664 .975592 .971476 .007354 .86652 29 30 1.983901 1.981797 f. 979704 1.977621 1.975548 1.971431 0.007460 3.87274 30 31 .983847 .981746 .979652 .977570 .975496 .971380 .007569 .87904 31 32 .983795 .981691 .979598 .977514 .975442 .971325 .007692 .88604 32 33 .983730 .981627 .979535 .977452 .975379 .971263 .007829 .89371 33 34 .983664 .981560 .979466 .977384 .975311 .971194 .007978 .90189 34 35 f. 983580 f. 981 478 1.979386 1.977303 1.975229 1.971113 0.008148 3.91105 35 36 .983504 .981401 .979307 .977224 .975152 .971035 .008330 .92065 36 37 .983416 .981313 .979220 .977137 .975064 .970947 .008524 .93064 37 38 .983313 .981209 .979117 .977034 .974960 .970844 .008744 .94171 38 39 .983203 .981100 .979006 .976923 .974851 .970734 .008987 .95361 39 40 1.983078 1.980976 r. 978883 1.976800 1.974727 1.970611 0.009254 3.96633 40 41 .982953 .980849 .978757 .976674 .974600 .970484 .009544 .97973 41 42 .982800 .980696 .978602 .976520 .974447 .970330 .009866 .99414 42 43 .982643 .980541 .978448 .976364 .974292 .970175 .010221 2.00949 43 44 .982463 .980360 .978268 .976185 .974112 .969995 .010606 .02555 44 45 1.982272 1.980167 1.978073 1.975991 1.973917 1.969801 0.011035 2.04277 45 46 .982055 .979954 .977861 .975777 .973705 .969588 .011504 .06085 46 47 .981821 .979716 .977624 .975541 .973468 .969351 .012021 .07994 47 48 .981561 .979459 .977365 .975282 .973209 .969093 .012589 .09999 48 49 .981279 .979175 .977082 .975000 .972926 .968810 .013213 .12100 49 50 1.980961 1.978859 1.976767 1.974683 1.972611 1.968494 0.013901 2.14305 50 51 .980619 .978516 .976422 .974340 .972366 .968150 .014658 .16607 51 52 .980240 .978137 .976044 .973961 .971888 .967771 .015488 .19000 52 > 53 .979827 .977722 .975630 .973546 .971474 .967357 .016402 .21490 53 i 54 .979370 .977269 .975175 .973093 .971020 .966903 , .017402 .24060 54 J i Vlx + Dx+i Formula, vp^ = —^ — = -^^ — i^x =^ — dlx L dx ♦49 TABLE XLIII. COMMON LOGARITHMS FOR SPECIAL FORMULAS. AGE. X. A (?)i?x). V-^ A^^- AGE. X. 3 Per Cent. Per Cent. 4 Per Cent. Per Cent. 5 Per Cent. 1.970510 .969961 .969345 .968678 .967946 6 Per Cent. 55 56 57 58 59 1.978863 .978314 .977696 .977030 .976299 1.976759 .976210 .975594 .974927 .974195 1.974667 .974117 .973500 .972835 .972102 1.972584 .972033 .971418 .970752 .970019 1.966395 ..965844 .965228 .964562 .963829 0.018508 .019725 .021064 .022541 .024151 2.26736 .29502 .32354 .35297 .38294 55 56 57 58 59 60 61 62 63 64 r. 975486 .974593 .973621 .973541 .971356 r. 973384 .972489 .971518 .970437 .969254 1.971290 .970397 .969425 .968344 .967161 1.969207 .968314 .967342 .966261 .965078 1.967135 .966241 .965268 .964189 .963005 1.963018 .962124 .961153 .960072 .958888 0.025921 .027883 .030027 .032385 .034988 2.41365 .44534 .47751 .51034 .54392 60 61 62 63 64 65 66 67 68 69 1.970049 .968624 .967037 .965325 .963401 1.967946 .966521 .964934 .963221 .961299 f. 965853 .964427 .962842 .961128 .959205 1.963770 .962345 .960758 .959045 .957122 1.961696 .960272 .958686 .956971 .955050 1.957580 .956155 .954569 .952855 .950933 0.037853 .040996 .044457 .048249 .052427 2.57810 .61274 .64794 .68349 .71956 65 66 67 68 69 70 71 72 73 74 1.961316 .959006 .956478 .953704 .950616 1.959212 .956004 .954375 .951600 .948513 r. 957120 .954811 .9.52281 .949508 .946420 1.955037 .952728 .950199 .947425 .944336 1.952963 .950656 .948125 .945352 .942264 1.948848 .946538 .944009 .941235 .938147 0.057039 .062089 .067652 .073746 .080484 2.75617 .79301 .83028 .86774 .90571 70 71 72 73 74 75 76 77 78 79 1.947257 .943570 .939473 .934997 .930071 1.945153 .941467 .937370 .932894 .927968 r. 943060 .939374 .935277 .930801 .925875 r. 940978 .937291 .933193 .928719 .923792 1.938904 .935218 .931121 .926645 .921719 1.934788 .931101 .927005 .922529 .917602 0.087902 .095996 .104940 .114801 .125606 2.94400 .98225 1.02094 .05995 .09901 75 76 77 78 79 80 81 82 83 84 1.924647 .918706 .912111 .904953 .896974 r. 922544 .916602 .910009 .902850 .894870 1.920451 .914509 .907916 .900757 .892777 1.918368 .912426 .905833 .898674 .890694 1.916295 .910354 .903759 .896601 .888622 1.912178 .906237 .899643 .892484 .884500 0.137506 .150570 .164978 .180795 .198181 1.13832 .17774 .21743 .25719 .29706 80 81 82 83 84 85 86 87 88 89 1.888276 .878789 .868041 .856659 .836828 1.886173 .876686 .865938 .854556 .834725 1.884080 .874593 .863845 .852462 .832633 1.881997 .872510 .861762 .850380 .830549 r. 879923 .870437 .859690 .848306 .828476 1.875807 .866320 .855573 .844190 .824359 0.217377 .238266 .261501 .286671 .321451 1.33721 .37706 .41747 .45738 .50711 85 86 87 88 89 90 91 92 93 94 1.814433 .794331 .772681 .746831 .717581 1.812329 .792229 .770578 .744727 .715478 r. 810236 .790136 .768484 .742635 .713385 1.808154 .788052 .766402 .740552 .711301 1.806081 .785980 .764328 .738479 .709229 1.801964 .781863 .760213 .734362 .705112 0.371233 .421033 .468238 .522329 .585986 1.56965 .62432 .67047 .71794 .76789 90 91 92 93 94 95 96 97 98 1.686132 .631775 .589223 .510042 r. 684029 .629672 .587120 .507938 1.681936 .627579 .585026 .505846 1.679854 .625496 .582944 .503762 1.677780 .623423 .580871 .501689 1.673664 .619306 1 .576754 .497573 0.655714 .748812 .871770 .988400 1.81671 .87437 .94040 .99493 95 96 97 98 V & 6 X6 0.970874 .029559 2.470687 0.966184 0.961538 .034401 .039221 2.536577 2.59.3516 0.956938 .044017 2.643619 0.952381 .048790 2.688332 0.943396 .058269 1 2.765437 Formula, 8 = Na'p. log (I + i) = — Naj}. log v. *50 TABLE XLIY. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF fi ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. The first Value under the Age X is a(1 + ag.) for the whole Life. 4 PER CENT. Pay- a:=:10 11 12 13 14 15 16 17 ! Fay ments. n. ments. 1.316633 1.315019'l.313333 j 1.311567 1.309720 1.307791 1.305773 1.303662 W. 1 0.000000 O.OOOOOolo. 000000 0.000000 !o. 000000 0.000000 0.000000 0.000000 1 1 2 .291215 .291210 .291207 .291203'! .291196 .291191 .291187 .2^1179 1 2 3 .457564 .457556 .457549 .457538 .457527 .457516 .457505 .457491 ! 3 4 .572834 .572822 .572810 .572794 .572777 .572761 .572743 .572722 4 5 0.660148 0.660132 0.660114 0.660093 0.660071 0.660048'o. 660024 0.659995 5 6 .729805 .729785 .729763 .729736 .729707 .729679! .729647 .729610 6 7 .787302 .787276 .787248 .787216 .787181 .787145 .787106 .787060 7 8 .835915 .835884 .835851 .835813 .835771 .835728 .835681 .835626 8 9 .877761 .877725 .877686 .877642 .877593 .877542 .877486 .877423 9 10 0.914284 0.914242 0.914198 0.914146 0.914090 0.914032 0.913967 0.913893 10 11 .946514 .946466 .946415 .946357 .946293 .946226 .946152 .946068 11 12 .975210 .975157 .975099 .975033 .974962 .974886 .974802 .974708 12 13 1.000952 1.000892 1.000827 1.000754 1.000673 1.000588 1.000494 1.000388 13 14 .024186 ,024119 .024047 .023966 1 .023877 ] .023782 .023677 .023560 14 15 1.045269 1.045196 1.045116 1.045026 1 1.044928 1.044822 1.044707 1.044576 15 16 .064488 .064407 064319 .064220 .064112 .063996 .063868 .063725 16 17 .082076 .081987 .081891 .081783 .081665 .081537 .081397 .081240 17 18 .098228 .098131 .098026 .097908 .097779 .097640 .097486 .097316 18 19 .113106 .113001 .112886 .112758 .112618 .112466 .112299 .112113 19 20 1.126848 1.126733 1.126610 1.126470 1.126318 1.126153'l.l25972 1.125770 20 21 .139570 .139446 .139311 .1391611 .138996 .138818! .138622 .138404 21 22 .151373 .151239 .151094 .150932 .150754 .150561 .150349 .150114 22 23 .162344 .162201 .162044 .161869 .161678 .161470 .161241 .160988 23 24 .172559 .172405 .172237 a 72049, .171843 .171620 .171374 .171102 24 25 1.182086 1.181921 1.181740 1.181538 1.181317 1.181077 1.180814 1.180521 25 26 .190983 .190805 .190610 .190395 .190158 .189901 .189618 .189305 26 27 .199299 .199108 .198901 .198671 .198418 .198142 .197840 .1975051 27 28 .207084 .206879 .206659 .206412 .206142 .205847! .205524 .205166: 28 29 .214377 .214160 .213922 .213660 .213371 .213057! .212712 .212330! 29 30 1.221216 1.220984 1.220731 1.220451 1.220144 1.219809*1. 219441 1.219034 30 31 .227633 .227386 .227116 .226818 .226491^ .2261351 .225743 .225310 31 32 .233659 .233396 .233109 .232792 .232444! .232065 .231648 .231188 32 33 .239320 .239041 .238736 .238399 .238029; .237626 .237183 .236695 33 34 .244642 .244345 .244022 .243663 . 243271 r .242843 .242374 .241854 34 35 1.249645 1.249332 1.248988 1.248607 1.248192 1.247738!l.247239 1.246690 35 36 .254352 .254019 .253655 .253252 .2528111 .252331 .251802 .251220 36 37 .258780 .258427 .258042 .257616 .257148 .256640 .256081 .255465 37 38 .262948 .262574 .262165 .261715 .261221 .260682 .260091 .259440 38 39 .266868 .266473 .266042 .265565 .2650431 .264473 .263848 .263161 39 40 1.270557 1.270140 1.269684 1.269180 1.2686271.268027 1.267367 1.266642 40 41 .274028 .273587 .273105 .272574 .271992 .271357 .270661 .269897 41 ' 42 .277293 .276827 .276318 .275758 .275144 .274474 .273742 .272936 42 43 .280363 .279872 .279335 .278744 .278098' .277391 .276620 .275772 43 ^ 44 .283247 .282730 .282160 .281543 .280862, .280119 1 .279307 .278415 44 1 10 11 12 13 14 15 16 17 Continued on Page *60. Formula, ^l + a^^^) = a ^x-^x^n ^x^^3!=L D« Vpa>-t TABLE XLIY. '5i LIFE AND TEMPORARY ANNUITIES. LOGARITHM OP THE PRESENT VALUE OF W ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4 PER CENT. Pay- 18 19 20 21 22 23 24 25 Pay- ments. n. 1.301453 ments. n. 1.299144 1.296729 1.294205 1.291567 1.288808 1.285924 1.282910 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .291172 .291165 .291155 .291145 .291134 .291123 .291110 .291096 2 3 .457477 .457460 .457441 .457421 .457400 .457376 .457349 .457320 3 4 .572699 .572674 .572645 .572615 .572581 .572546 .572504 .572459 4 5 0.659963 0.659929 0.659890 0.659848 0.659803 0.659753 0.659698 0.659636 5 6 .729569 .729525 .729476 .729422 .729364 .729300 .729229 .729150 6 7 .787011 .786956 .786896 .786830 .786758 .786679 .786591 .786494 7 8 .835567 .835502 .835429 .835351 .835265 .835170 .835065 .834949 8 9 .877353 .877276 .877193 .877100 .876999 .876888 .876765 .876629 9 10 0.913813 0.913725 0.913628 0.913521 0.913404 0.913276 0.913134 0.912977 10 11 .945977 .945876 .945765 .945644 .945510 .945364 .945202 .945024 11 12 .974604 .974491 .974365 .974228 .974078 .973912 .973730 .973528 12 13 1.000273 1.000145 1.000005 .999851 .999683 .999498 .999293 .999068 13 14 .023431 .023289 .023133 1.022963 1.022775 1.022569 1.022342 1.022091 14 15 1.044434 1.044277 1.044105 1.043916 1.043709 1.043481 1.043229 1.042953 15 16 .063569 .063396 .063207 .062999 .062771 .062520 .0622431 .061939 16 17 .081069 .080880 .080672 .080444 .080194 .079920 .0796171 .079283 17 18 .097128 .096922 .096695 .096447 .096174 .095874 .095543 .095180 18 19 .111909 .111685 .111438 .111168 .110871 .110545 .110185 .109790 19 20 1.125549 1.125306 1.125038 1.124745 1.124423 1.124069 1.1236791.123251 20 21 .138164 .137901 .137612 .137295 .136946 .136564 .136142 .135678 21 22 .149856 .149572 .149159 .148917 .148541 .148128 .147674 .147173 22 23 .160710 .160404 .160168 .159699 .159294 .158850 .1583601 .157822 23 24 .170803 .170474 .170113 .169717 .169282 .168804 .168278 .167700 24 25 1.180200 1.179848 1.179460 1.179035 1.178569 1.178056 1.177492 1.176873 25 26 .188961 .188584 .188169' .187714 .187214 .186665 .186062 .185399 26 27 .197137 .196733 .196289 .195803 .195269 ..194682 .194037 .193329 27 28 .204773 .204342 .203868 .203349 .202778 .202152 .201464 .200709 28 29 .211911 .211452 .210946' .210392 .209784 .209117 .208384 .207579 29 30 1.218588 1.218098 1.2175591.216970 1.216322 1.215612 1.214832 1.213976 30 31 .224835 .224314 .223741; .223114 .222425 .221670 .220841 .219932 31 32 .230683 .230129 .2295211 .228854 .228123 .227321 .226441 .225476 32 33 .236159 .235571 .234925; .234218 .233442 .232591 .231659 .230636 33 34 .241286 .240662 .239978 .239228 .238406 .237506 .236518 .235436 34 35 1.246087 1.245427 1.244701 1.243908 1.243038 1.242085 1.241040 1.239896 35 36 .250583 .249883 .249116 .248276 .247357 .246349 .245246 .244037 36 37 .254790 .254051 .253240! .252352 .251381 .250317 .249153 .247879 37 38 .258727 .257946 .2570891 .256153 .255127 .254006 .252778 .251436 38 39 .262408 .261584 .260680 .259692 .258612 .257430 .256138 .254726 39 40 1.265848 1.264979 1.264027 1.262986 1.261848 1.260605 1.2592461.257762 40 41 .269060 .268144 .267141 .266046 .264850 .263543 .262116! .260559 41 42 .272055 .271091 .270037 .268886 .267629 .266257 .264760 .263128 42 43 .274845 .273832 .272724 .271515 .270196 .268758 .267190 .265481 43 1 44 .277441 .276377 .275214 .273946 .272564 .271057 .269416 .267631 44 18 19 20 21 22 23 24 25 Continued on Page *59. *53 TABLE XLIV. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF W ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4 PER CENT. Pay- ments. n. x=2Q 27 28 29 30 31 32 33 Pay- ments n. 1 1.279758 1.276466 1.273025 1.269428 1.265675 1.261752 1.257654 1.253372 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 .291079 .291064 .291045 .291022 .291001 .290975 .290948 .290919 2 3 .457288 .457255 .457214 .457170 .457126 .457073 .457019 .456957 3 4 .572410 .572357 .572296 .572229 .572159 .572081 .571995 .571899 4 5 0.659568 0.659496 0.659413 0.659322 0.659227 0.659119 0.659000 0.658868 5 6 .729063 .728970 .728864 .728748 .728626 .728486 .728335 .728167 6 7 .786388 .786273 .786143 .786001 .785849 .785678 .785492 .785285 7 8 .834821 .834684 .834530 .834359 .834177 .833973 .833750 .833503 8 9 .876480 .876318 .876137 .875937 .875724 .875485 .875223 .874934 9 10 0.912805 0^.912619 0.912409 0.912180 0.911933 0.911657 0.911355 0.911021 10 11 .944827 .944614 .944376 .944115 .943833 .943518 .943174 .942794 11 12 .973307 .973066 .972798 .972503 .972184 .971830 .971441 .971013 12 13 .998820 .998551 .998250 .997921 .997564 .997167 .996732 .996253 13 14 1.021816 1.021516 1.021182 1.020816 1.020419 1.019977 1.019494 1.018962 14 15 1.042648 1.042317 1.041948 1.041543 1.041104 1.040617 1.040083 1.039494 15 16 .061604 .061239 .060834 .060389 .059906 .059370 .058783 .058136 16 17 .078917 .078517 .078073 .077586 .077057 .076471 .075828 .075120 17 18 .094780 .094344 .093861 .093329 .092751 .092112 .091411 .090640 18 L9 .109355 .108881 .108355 .107778 .107150 .106455 .105694 .104856 19 20 1.122779 1.122265 1.121695 1.121070 1.120389 1.119637 1.1188121.117905 20 21 .135169 .134613 .133997 .1333^22 .132586 .131773 .130882 .129903 21 22 .146624 .146024 .145361 .144633 .143839 .142963 .142004 .140949 22 23 .157231 .156585 .155872 .155089 .154236 .153294 .152263 .151130 23 24 .167066 .166372 .165606 .164766 .163850 .162840 .161734 .160520 24 25 1.176193 1.175449 1.174629 1.173729 1.172748 1.171667 1.170483 1.169184 25 26 .184671 .183876 .182998 .182037 .180988 .179833 .178568 .177181 26 27 .192552 .191702 .190765 .189739 .188620 .187388 .186039 .184561 27 28 .199880 .198974 .197976 .196882 .195689 .194377 .192942 .191370 28 29 .206696 .205731 .204669 .203506 .202237 .200843 .199318 .197648 29 30 1.213037 1.212011 1.210883 1.209647 1.208299 1.206819 1.205202 1.203432 30 31 .218935 .217846 .216648 .215337 .213908 .212340 .210627 .208753 31 32 .224419 .223264 .221995 .220607 .219094 .217434 .215623 .213643 32 33 .229516 .228293 .226950 .225482 .223882 .222129 .220217 .218128 33 34 ..234251 .232957 .231538 .229987 .228298 .226449 .224433 .222234 34 35 1.238644 1.237278 1.235780 1.234144 1.232364 1.230416 1.228294 1.225982 35 36 .242716 .241275 .239697 .237974 .236099 .234051 .231822 .229395 36 37 .246486 .244968 .243306 .241494 .239524 .237374 .235035 .232492 37 38 .249970 .248373 .246627 .244723 .242656 .240402 .237953 .235292 38 39 .253185 .251507 .249674 .247678 .245512 .243152 .240592 .237814 39 40 1.256144 1.254384 1.252462 1.250372 1.248106 1.245641 1.242969 1.240074 40 41 .258862 .257017 .255006 .252821 .250455 .247883 .245100 .242089 41 42 .261351 .259421 .257320 .255039 .252571 .249894 .247001 .243875 42 43 .263623 .261608 .259415 .257038 .254470 .251688 .248686 .245447 43 44 .265691 .263588 .261305 .258832 .256164 .253279 .250170 .246822 44 / 1 1 26 27 28 29 30 31 32 33 Contiuued on Page *58. TABLE XLIY. ♦53 LIFE AND' TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4 PER CENT. Pay- niQiits. 34 35 36 37 38 39 40 41 Pay- ments. n. 1.248903 1.244234 1.239367 1.234282 1.228972 1.223433 1.217656 1.211632 71. 1 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 .290884 .290845 .290806 .290765 .290713 .290660 .290600 .290538 2 3 .456885 .456808 .456729 .456641 .456537 .456426 .456306 .456176 3 4 .571790 .571674 .571551 .571413 .571255 .571087 .570901 .570699 4 5 0.658721 0.658564 0.658395 0.658205 0.657991 0.657761 0.657507 0.657231 5 6 .727980 .727778 .727559 .727317 .727043 .726748 .726424 .726069 6 7 .785055 .784808 .784538 .784239 .783904 .783540 .783142 .782706 7 8 .833229 .832932 .832609 .832251 .831851 .831418 .830941 .830419 8 9 .874614 .874265 .873886 .873467 .872999 .872491 .871934 .871322 9 1 10 0.910652 0.910251 0.909813 0.909329 0.908792 0. 908205 0.907561 0.906857 10 11 .942375 .941917 .941418 .940867 .940254 .939587 .938854 .938051 11 12 .970540 .970024 .969460 .968838 .968148 .967396 .966570 .965665 12 13 .995724 .995146 .994515 .993820 .993049 .992208 .991285 .990272 13 14 1.018374 1.017733 1.017032 1.016259 1.015403 1.014470 1.013444 1.012321 14 15 1.038846 1.038137 1.037363 1.036510 1.035565 1.034535 1.033404 1.032165 15 16 .057424 .056645 .055794 .054856 .053820 .052689 .051448 .050088 16 17 .074341 .073490 .072558 .071533 .070401 .069164 .067808 .066323 17 18 .089791 .088863 .087848 .086731 .085499 .084154 .082678 .081062 18 19 .103935 .102927 .101825 .100613 .099277 .097817 .096218 .094466 19 20 1.116908 1.115817 1.114624 1.113313 1.111869 1.110291 1.108563 1.106671 20 21 .128827 .127649 .126362 .124948 .123391 .121691 .119830 .117793 21 22 .139791 .138524 .137138 .135617 .133944 .132117 .130117 .127931 22 23 .149886 .148525 .147038 .145406 .143612 .141654 .139512 .1371711 23 1 24 .159186 .157729 .156136 .154389 .152471 .150377 .148088 .145589 24 25 1.167759 1.166201 1.164499 1.162633 1.160585 1.158352 1.155913 1.153250 25 26 .175660 '.173998 .172183 .170195 .168014 .165638 .163044 .160214 26 27 .182941 .181172 .179240 .177126 .174809 .172286 .169532 .166533 27 28 .189648 .187767 .185716 .183472 .181016 .178341 .175427 .172254 28 29 .195820 .193825 .191651 .189274 .186675 .183847 .180768 .177420 29 30 1.201495 1.199383 1.197082 1.194569 1.191824 1.188841 1.185595 1.182070 30 31 .206705 .204473 .202043 .199392 .196348 .193357 .189943 .186240 31 32 .211481 .209126 .206564 .203772 .200729 .197428 .193845 .189964 32 33 .215849 .213369 .210674 .207739 .204544 .201083 .197332 .193274 33 34 .219835 .217228 .214397 .211319 .207973 .204351 .200433 .196199 34 35 1.223463 1.220727 1.217760 1.214538 1.211039 1.207259 1.203174 1.198769 35 36 .226753 .223888 .220784 .217417 .213768 .209831 .205583 .201010 36 37 .229727 .226732 .223491 .219981 .216183 .212092 .207685 .202950 37 38 .232404 .229279 .225902 .222252 .218307 .214065 .209505 .204614 38 39 .234802 .231548 .228038 .224248 .220161 .215774 .211067 .206029 39 40 1.236940 1.233559 1.229916 1.225992 1.221767 1.217241 1.212394 1.207218 40 41 .238834 .235328 .231558 .227503 .223146 .218488 .213510 .208205 41 42 .240502 .236874 .232980 .228800 .224319 .219537 .214437 .209014 42 43 .241959 .238214 .234202 .229904 .225305 .220407 .215197 .209668 43 44 .243222 .239366 .235241 .230832 .226124 .221122 .215811 .210187 44 34 35 36 37 38 39 40 41 Continued on Page *57. N *54 TABLE XLIY. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OP n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4 PER CENT. Pay. a;=42 43 44 45 46 47 48 49 Pay- ments. ments. n. 11. 1 1.205342 1.198793 1.191959|1.184839 1.177421 1.169692 1.161644 1.153265 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0. 000000 0.000000 1 2 .290463 .290386 .290301 .290205 .290101 .289986 .289861 .289722 2 3 .456026 .455868 .455691 .455496 .455283 .455047 .454790 .454507 3 4 .570471 .570227 .569956 .569656 .569329 .568968 .568573 .568137 4 5 0.656921 0.656587 0.656216 0.655808 0.6553610.654869 0.654329 0.653735 5 6 .725673 .725245 .724770 .724248 .723676 .723046 1 .722356 .721596 6 7 .782220 .781693 .781110 .780468 .779765 .778992 .778144 .777212 7 8 .829838 .829208 .828512 .827746 .826906 .825983 .824972' .823860 8 9 .870643 .869904 .869089 .868194 .867212 .866133 .864951 .863650 9 10 0.906075 0.905224 0.904285 0.903253 0.902123 0. 900882 0.8995210.898025 10 11 .937162 .936194 .935126 .933953 .932668 .931257 .929710 .928011 11 12 .964664 .963572 .962370 .961050 .959604 .958016 .956275 .954366 12 13 .989154 .987935 .986593 .985120 .983506 .981735 .979794 .977665 13 14 1.011081 1.009729 1.008241 1.006608 1.004820 1.002858 1.000711 .998356 14 15 1.030798 1.029307 1.027667 1.025869 1.023900 1.021742 1.019379 1.016790 15 16 .048589 .046954 .045157 .043187 .041030 .038668 .036084 .033255 16 17 .064687 .062902 .060942 .058794 .056444 .053871 .051058, .047981 17 18 .079284 .077344 .075214 .072882 .070333 .067543 .064496 .061164 18 19 .092541 .090440 .088136 .085614 .082859 .079847 .076558 .072965 19 20 1.104593 1.102326 1.099842 1.097126 1.094159 1.090918 1.0873821.083525 20 21 .115561 .113119 .110450 .107533 .104349 .100874 .097087 .092959 21 22 .125532 .122919 .120059 .116936 .113531 .109818 .105775 .101374 22 23 .134605 .131811 .128756 .125423 .121792 .117837 .113536 .108859 23 24 .142852 .139872 .136618 .133071 .129211 .125011 .120449 .115496 24 25 1.150337 1.147169 1.143712 1.139948 1.135856 1.131410 1.126586 1.121356 25 26 .157122 .153761 .150098 .146115 .141790 .137096 .132012 .126509 26 27 .163258 .159702 .155832 .151628 .147069 .142129 .136786 .131012 27 28 .168794 .165041 .160962 .156536 .151745 .146561 .140963 .134925 28 29 .173774 .169822 .165533 .160888 .1558651 .150441 .144593 1 1 .138298 29 30 1.178236 1.1740861.169589 1.164725 1.1594751.153815:1.147726 1.141185 30 31 .182218 .177871 .173168 .168089 .162617 .156729 .150408 .143630 31 32 .185755 .181213 .176307 .171018 .165331 .159224 .152681 .145682 32 33 .188880 .184146 .179041 .173550 .167656 .161340 .154589 .147383 33 34 .191624 .186703 .181406 .175719 .169629 .163117 .156170 .148775 34 35 1.194016 1.188913 1.183433 1.177560 1.171285 1.164590 1.1574661.149897 35 36 .196086 .190809 .185153 .179107 .172659 .165797 .158510; .150789 36 37 .197862 .192420 .186599 .180390 .173785 .166770 .159339 .151484 37 38 .199370 .193773 .187799 .181442 .174693! .167543 .159986 .152015 38 39 .200637 .194896 .188782 .182290 .175414 .168146 .160431 .152413 39 40 1.201690 1.195816 1.189576 1.182964 1.175977 1.168607 1.160852 1.152703 40 41 .202553 .196559 .190206 .183490 .176408 .168952 .161122 .152910 41 42 .203249 .197150 .190698 .183892 .176730 .169204 .161314 .153050 42 43 .203803 .197610 .191074 .184193 .176965 .169384 .161445 .153141 43 44 .204234 .197963 .191356 .184413 .177133 .169505 .161530 .153197 44 42 43 44 45 46 47 48 49 Continued on Page *56. TABLE XLIY. ♦55 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OP THE PRESENT VALUE OF n ANNUAL PAYMENTS OP 1 AT THE BEGINNING OP EACH YEAR. 4 PER CENT. Pay- 50 51 52 53 54 55 1 56 57 Pay- ments 71. ! ments, 1.14454S 5 1.135469 1.12603l|l.ll6217 1.106014 1.09541C U.08440C 1.072966 11. 1 1 O.OOOOOolo. 000000 !o.OOOOOO|o. 000000 O.OOOOOC 0.000000 O.OOOOOC O.OOOOOC 2 .389571 .289400 .289216 .289015 .288795 .2885481 .288281 .28798!^ 2 3 .454196 .453849 .453474 .453064 .452609 .4521081 .451558 .450952 3 4 .567659 .567130 .566556 .565925 .56522!^ .564458 .563617 .562691 4 6 0.653082'0. 652363 0.651580 0.650718 0.649767 0.6487210.647576 0.646315 5 6 .720763 .719845 .718842 .717740 .716526 .715191 1 .713729 .712123 6 7 .776190 .775063 .773833 .772481 .770993 .769358 .767569 .765604 7 8 .822640 .821297 .819829 .818217 .816445 .814499 .812371 .810033 8 9 .862225 .860656 .858942 .857061 .854994 .852727 .850245 .847525 9 10 0.896385l0. 894583 0.892613 0.890452 0.888081 0.885479 0.882637 0.879518 10 11 -.926149 .924104 .921869 .919419 .916732 .913789 .910569 .907045 11 12 .952275 .949978 .947469 .944720 .941709 .938409 .934807 .930865 12 13 .975335 .972779 .969987 .966931 .963583 .959922 .955925 .951558 13 14 .995779 .992955 .■989871 .986498 .982807 .978774 .974375 .969578 14 15 1.013960 1.010859 1.007475 1.003778 0.999736 0.995322 0.990519 0.985279 15 16 .030162 .026777 .023087 .019057 1.014656 1.009861 1.004640 .998959 16 17 .044619 .040945 .036941 .032573 .027811 .022622 .016986 1.010859 17 18 .057527 .053556 .049232 .044522 .039390 .033811 .027756 .021188 18 19 .069049 .064775 .060127 .055071 .049569 .043596 .037127 .030117 19 20 1.079322il.074743'l.069768 1.064362 1.058491 1.052128 1.045244 1.037800 20 21 .088468 .083579 .078276 .072521 .066284 .059531 .052239 .044373 21 22 .096590 .091392 .085760 .079657 .073056 .065920 .058234 .049957 22 23 .103782 .098274 .092315 .085870 .078908 .071404 .063332 .054663 23 24 .110126 .104310 .098030 .091247 .083938 .076071 .067634 .058591 24 25 1.115697 1.109578 1.102979 1.095870 1.088222 1.080014 1.071227 1.061835 25 26 .120563 .114144 .107239 .099814 .091845 .083309 .074197 .064480 26 27 .124787 .118078 .110875 .103148 .094873 .086035 .076619 .066608 27 28 .128426 .121439 .113953 . 105940 .097381 .088259 .078569 .068295 28 29 .131537 .124284 .116531 .108251 .099428 .090050 .080116 .069610 29 30 1.134173 1.126669 1.118665 1.110140 1.101077 1.091471 1.081322 1.070611 30 31 .136383 .128644 .120411 .111662 .102385 .092580 .082240 .071362 31 32 .138214 .180260 .121817 .112869 .103407 .093423 .082929 .071909 32 33 .139713 .131563 .122933 .113809 .104183 .094057 .083430 .072299 33 34 .140922 .132596 .123803 .114528 .104767 .094518 .083788 .072563 34 35 1.141880 1.133402 1.124468 1.115067 1.105193 1.094847 1.084031 1.072733 35 36 .142628 .134018 .124966 .115459 .105495 .095070 .084187 .072838 36 37 .143200 .134479 .125329 .115739 .105701 .095214 .084284 .072899 37 ,38 .143628 .134816 .125587 .115928 .105834 .095303 .084339 .072935 38 39 .143940 .135055 .125763 .116051 .105916 .095354 .084373 .072953 39 40 1.144163 1.1352181.125876 1.116127 1.105963 1.095385 1.084388 1.072961 40 41 .144314 .135323 .125946 .116171 .105991 .095399 .084396 .072964 41 42 .144412 .135388 .125987 .116195 .106004 .095406 .084399 .072965 42 43 .144472 .135426 .126010 .116208 .106011 .095409 .084399 .072966 43 44 .144507 .135447 .126022 .116214 .106013 .095409 .084400 44 45 1.144526 1.135458 1.126028 1.116216 1.106013 1.095410 45 46 .144537 .135463 .126029 .116217 .106014 46 47 .144541 .135466 .126030 .116217 47 48 .144541 .135467 .126031 • 48 49 .144542 .135469 49 50 51 52 53 54 55 56 57 *56 TABLE XLIV. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. VALUE OF 4 PER CENT. Pay- ments. n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 37 38 39 40 41 42 51 50 49 48 47 46 45 x=bS 1.06110 59 60 0.000000 .287660 .450294 .561680 0.644940 .710369 .763457 .807481 .844552 0.876115 .903200 .926569 .946804 .964355 0.979585 992791 1.004222 .014080 .022546 1.029777 .035912 .041075 .045381 .048935 1.051830 .054158 .056003 .057441 .058535 1.059355 .059953 .060379 .060667 .060853 35 11.060968 36 .061035 .061074 .061094 .061102 1.061106 .061107 .061107 1.0488031.036042 61 1.022821 0.000000 0.000000 .286912 .448766 .559343 .287305 .449567 .560568 0.643427 708441 761100 804679 841294 0.872389 .898995 .921878 .941615 .958664 0.973391 .986096 .997024 1.006388 014370 1.021132 .026815 .031550 .035454 .038632 1.041187 043210 044786 045985 046884 1.047539 .048005 .048320 .048524 .048650 1.048723 1.036001 1.022802 62 63 64 1. 009130 0. 994950 0. 980284 65 0.965118 0.000000 10.000000 0.000000 0.000000 0.000000 .2864811 .286011 .285490 .2849211 .284292 .447888! .446929 .4458691 .444707| .443431 .558003,1 .556534 .5549151 .553142: .551197 0.641765 0.639943 .706324; .704004 .758512 .755680 .801608i .798249 .8377271 .833828; 0.8683130.863865 .894402! .889392; .916756' .9111761 .9359591 .929808! .952472 .9457481 0.966663 0.9593671 .978832 .9709621 .989233 .980817! .998078 .989123J 1.005558! .996082! 1.0118351.001865' .017058 .006623 .0213611 .010490 .024861! .013593! 1.001641 .027672 .016048 .003753 0.637947 0.63575l!0.633349|0. 630712 .701465 .698676! .695623! .692286 .752582 .749182 .7454711 .741409 .794577 .790557 .786166 .781369 .829572 .824914 .819834 .814292 Pay- ments. n. 0.859010 0. .883930 .905105 .923125 .938453 0.951469 0, .9624771 .9717431 .979490' .985917| 0.991199 .995485 .998923 853706 0.847930 0.841641 877975^ .871501 .864458 898497 .891319 .883533 915860: .907991 .899465 ,930543^ .921985 .912738 1.0298971.017958 .031629 .019410' .032946 .020498 .033934 .021291 .034654 .021855 1.035166'l. 022237 .035512 .022484 .035736 .022636 .035875 .0227251 .035955 .022775! .048766 .048788 .048797 .048801 .036025 .022813 .036035 .022820 .036041 .022820 .036041; .022821 1.048802' 1.036042! .048803 49 1.153265 .153265 1.153264 .153262 .153258 .153248 .153230 49 48 47 46 1.177421 1.1696921 .177420 1.161644 .169691! .177420 161644 .1696911 .177418 .161644 1.161642 .161638 .161629 .161612 .161582 48 .169689 .1774141 1.169685 .169677 .169661 .169633 .169584 1.005360 0. .006562 . .007439 . .008062 . .008484 . 1.008758 .008925 .009024 .009079 .009108 ,942918 0. ,9533041 . ,9619671 . .969141; , .975027! . ,9797980. .983621: . ,9866411 . ,988985: . ,9907691 . 992103 0. 993075' . 993766! . 994234; . 9945381 . 994723'o. 994833 994894 994926 994941 ,933690 0.923742 ,943427 .932811 ,951472 .940221 ,958060 .946210 ,963393, .950996 9676610.954769 ,971029 .957693 ,973642 .959917 .975629 .961578 .9771151 .962787 ,978197 0.963646 .978966 .964228 ,979487 .964606 ,979825' .964837 ,9800321 .964974 9801540.965049 ,9802221 .965088 980257 .965107 980274 .965116 9802821 .965117 45 1.184839 .184838 .184838 .184836 .184833 .184826 1.177407 1.184812 .177392 .177365 .177320 .177246 47 46 .184787 .184745 .184676 .184570 45 44 43 42 1.191959 .191960 1.205342 1.198793 .205342 .198792 .198792 .1919591.198790 .1919581 .198787 .191955 .198781 .191948 .198769 .191935 .198747 1.1919121.198710 .191872 .198650 .1918081 .198557 .191708' .198419 .191562 .198227 44 43 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 1.009121 0. 994948 0.980282lo. 965118 35 .009128! .9949481 .9802841 .009128' .994950 .009130 58 57 .205342i 56 .2053411 55 1.205338! 54 .2053321 53 .205321 52 .205300, 51 .205266 50 1.205209 .205122 .204993 .204812 .204565 49 48 47 46 45 Continued from Page *54. TABLE XLIY. *57 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ITT YP.AR 4 PER CENT. Pay- ments. 7 8 9 10 11 12 13 14 15 16 17 18 19 27 28 29 30 31 32 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 66 0.949453 67 68 0.9332710.916583 0.000000 .283608 .442034 ,549005 0.627836 .688638 .736976 .776139 .808262 0.834803 .856820 .875100 .890254 .902769J 0.91305010 .921430' .928190J .933583 ^ .937829| 20 J0.941117!0 21 i .9436161 22 .945481! 23 I .946838! 24 i .947802J- 25 0.948455 26 \ .948878 .949138 .949291 .949375 0.949419 .949440 .949450 33 .949451 34 .949453 0000000. 282848J . 440493} . 546733j . 6246790. 684643 732127 770433 801690 827369 0. 848531 865969 880304 892036 901568 0. 909241 . 915351 . 920154 . 923868 . 926690 0. 928793 . 930324 . 931411 . 932147 . 932623 0. 932916 . 933089 . 933183 . 933232 . 9332560. 933267 . 933269 . 933271 - 69 0.899356 70 71 0.88162310.863356 000000 280119 434958 538313 000000 282028 438825 544183 621234 680284 726850 764225 794553 819311 839564 856116 869599 880516 889279 896241 901704 905923 909125 911510 0. 89555^1 10. 878878 913245 .896956! .879966 914476 .8979081 .880668 915310 .898522; .881100 915849 .898899 .881354 .000000 .281111 .436973 .541374 .617447 .675511 .721076 .757451 .786785, .810560' .829851 .845477 .858073 .868150 .876134'|o. .8823861 . .8872061 . .890861' . .893580!! . 0.000000 .279023 .432742 .534961 0.613328 0.608817 .670316 .714804 .750105 .778375 .801106 .819388 .834046 .845724 .854946 862149 867691 871888 875008 877272 916181!0.899121 916376' .899243 916483 .899307 916539 916566 .899338 .899351 916578 0.899355 916581 .899356 916583 38 41 40 1.217656 1.2116321.217656 211632 .217656 217655 .211632 .211630 .211628 1.211622 .211612 .211592 .211560 .217652 .217647 1.217637 .217619 .217588 .217538 .2115071 .217461 1.2114251.217348 .2113041 .217189 .211134 .216971 .210902! .216679 .2105921 .216299 41 40 39 — — — 1.228972 .223433 .228971 .223433 .228970 1.223433 .223432 .223430 .223425 .223415 1.223398 .223369 .223323 .223250 .223144 1.222994 .222789 .222515 .222157 .221699 1.228969 .228967 .228963 .228954 .228938 1.228911 .228867 .228799 .228698 .228557 1.228364 .228111 .227770 .227339 .226796 39 38 0.881494 i .881567 I .881602 ! .881617 j .881622 0.881623 .664644 .707970 .742118 .769254 0.790885 .808109 .821761 .832499 .840860 0.847278 .852130 .855732 .858343 .860194 0.861448 .862256 .862754 .863046 .863208 863291 863332 863349 863355 863356 37 1.234282 .234283 .234283 .234282 1.234280 .234276 .234267 .234252 .234227 1.234185 .234121 .234027 .233894 .233712 1.233470 .233153 .232746 .232236 .231603 36 239367 ,239367 ,239366 ,239366 ,239364 37 72 73 0.844568 0.825252 0.000000 0.000000 .277826 .430325 .531297 .276518 .427673 .527292 0.603900 0.598536 .658470 .700546 .733460 .759395 0.779868 795989 808606 818391 825882 0.831533 .835721 .838754 .840902 .842356 0.843294 .843870 .844209 .844396 .844493 0.844540 844560 844567 844568 .651748 .692480 .724082 .748748 0.768008 .782987 .794547 .803367 .810003 0.814911 .818460 .820971 .822670 .823765 0.824438 .824833 .825051 .825164 .825219 0.825242 .825250 .825252 35 1.244234 1.244234 .244234 .244233 .244231 .244228 1.2393601.244220 34 1.248902 .248901 1.248901 .248900 .248898 62 Pay- ments. n. .239352 .239338 .239314 .239274 .244207 .244184 .244147 .244090 1.2392141.244006 .239125' .243889 .239000 .243727 .238829 .243512 .238600? .243230 1.2383011.242869 .237919 .242415 .237437 .241854 .236842 .241169 .236115 .240345 36 35 .248894 .248888 1.248875 .248853 .248819 .248765 .248686 1.248574 .248422 .248219 .247953 .247613 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 65 64 63 61 60 59 58 57 56 55 54 53 52 51 50 1.247184 49 .246654 48 .246009! 47 .2453211 46 .244308; 45 Continued from Page *53. *58 TABLE XLIY. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OP THE PRESENT VALUE OF fi ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. . 4 PER CENT. Pay- ments. n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 :74 i 76 76 0. 805400 0. 785049 0. 764174 0.000000 .275065 .424759 .522908 0.592676 .644425 .683722 .713931 .737262 0.755259 .769063 .779550 .787413 .793214 0.797403 .800361 .802362 .803651 .804443 0.804907 .805163 .805295 .805360 .805387 0.805397 .805400 33 0.000000 .273492 .421598 .518143 0.586317 .636495 .674263 .702999 .724930 0.741621 .754228 .763641 .770563 .775551 0.779068 0.759860 781446, .761692 0.000000 .271774 .418136 .512941 0.579393 627890 664030 691214 711690 0.727040 .738439 .746788 .752788 .757010 77 0.742769 .782975 ,783914 .784464 0.784768 .784925 .785001 .785033 .785046 0.785049 0.000000 .269873 .414336 .507254 0.571857 .618553 .652969 .678528 .697499 0.711486 .721678 .728977 .734101 .737552 0.739769 .741127 .741923 .742363 .742590 764026'0. 742699 7641171 .742746 764156 .742764 7641711 .742769 764174 .762816 .763475 .763838 31 32 1.257654 1.2533721 .257654 .253373 .253373 .257654 .257653 1.2533721.257652 .253371! .257649 .253367 .253361 .253349 .257642 .257631 .257612 1.2533291.257581 .253296' .257533 .253245 .257462 .253170i .257363 .2530651 .257227 1.25292l'l.257046 .252729 .256809 .252478 .252157 .251753 .256505 .256124 .255651 1.2512531.255075 .250643 .254383 249910 .253560 .249038 .248014 33 .252592 .251467 32 1.261752 .261752 .261752 .261751 .261749 1.261746 .261740 .261730 .261712 .261682 30 1.265675 1.265676 .265676 .265675 .265674 .265671 1.265665 .265655 .265638 .265610 .265568 1.2616371.265504 .261570 .261476 .261348 .265416 .265294 .265132 .261177 .264920 1.2609531.264649 .264308 .263886 .263371 .262752] 1.262017 .261154 .260149 .258991 .257667 .260666 .260305 .259858 .259314 1.258660 .257882 .256968 .255906 .254680 31 78 0.720875 0.000000 .267807 .410203 .501088 0.563698 .608478 .641076 .664943 .682372 0.694987 .703979 .710270 .714497 79 80 81 0.698471 0.675562 0.652146 0.000000 0.000000 .2655471 .263074 .4057001 .400793 .494380| .487095 0.554856 0.545289 .597600 .628292 .650410 .666274 .585882 .614587 .634915 .649192 0.677512 0.65907' .685338 .690583 .693941 .71720R .695994 0.718869 .719842 .720379 .720656 .720789 72084610. 720868 . 720875 = 29 .665675 .669890 .672462 .673966 0.674796 .675225 .675430 .675517 .675551 698463'o. 675562 698471 0.697196 .697859 .698202 .698366 .698436 27 1.269428 .269428 1.2694281 .269427 .269426 .269423 .269418 1.269409 1 .269392! .269366 .269325 _?? 1.276466 273025' .276467 ,273025 .276467 30 .269266 1,269182 .269067 .268913 .268713 .268456 1.2681331, .267734 .267247 .266662 .265967 1.265150 .264200 .263105 .261854 .260433 .273025| .276467 ,273024!l.276465 ,273023! .276463 ,273020, .276458 273015' .276450 ,273006i .276435 ,27299lll. 276412 .272966 .276375 .272928' .276322 .272871' .276246 .272792 .276143 1.272683'l.276006 .2725371 .275826 .272348; .275596 .2721051 .275307 Pay- ments, n. 26 1.279753 .279749 .279741 .279727 .279705 1.279670 .279620 .279548 .279451 .279320 27142111 270961! 270407 269749 268976J 268077!l 267041! 265858 264514 263000 29 28 ,2745131 ,273988 ,273365 ,272634 ,271783 2708031 ,269682; 2684111 266979 265375i 276819 276126 275321 274392 273332 272128 270773 269255 267565 27 26 0.000000 .260385 .395450 .479197 0.534963 .573294 .599946 .618441 .631143 0.639575! 10 .644944 11 .648213 12 .650123 1 13 .651175 14 0.651718' 15 .651979 16 .652090 17 .652133 18 .652146 19 1.279758 74 .279758 73 .279758 .279757 .279756 72 71 70 69 68 67 66 65 64 63 62 61 60 1.279150 59 .278932 58 .278658 57 .278319 56 .271799 .274949 .277906 1 55 ,277409' 54 53 52 51 50 49 48 47 46 45 Continued from Page *52. TABLE XLIT. *59 LIFE AND TEMPORARY ANNUAL PAYMENTS ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF fl OF 1 AT THE BEOINNING OF EACH YEAR. - 4 PER CENT. Pay- ments. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 82 0.628187 83 0.603703 0.000000 .257424 .389631 .470629 0.523818 .559791 .584318 .600972 .611946 0.618899 .623121 .625583 .626938 .627637 0.627973 .628115 .028170 .628187 25 0.000000 .254238 .383339 .461402 0.511880 .545384 .567754 .582338 .591516 0.597065 .600294 .602070 .602984 .603424 0.603609 .603682 .603703 24 1.285924 1.282910 .285924 1.282909 1.285924 1.288808 .282909 .2859241 .288807 84 0.578554 0.000000 .250719 .376475 .451421 0.499024 .529994 .549865 .562249 .569691 0.574008 .576377 .577596 .578182 .578428 0.578525 .578554 23 1.288808 .288808 .288808 .282908, .285923 .28290?! .285921 .282905! .285917 1.282901 .282893 .282880 .282859 .282826 1.282778 .282711 .282618 .282495 .282333 1.282127 .281867 .281546 .281154 .280683 1.280124 .279468 .278704 .277825 .276820 1.275681 .274397 .272960 .271360 .269587 25 1.285909 .285897 .285877 .285846 .285800 1.285736 .285648 .285531 .285378 .285182 1.284936 .284631 .284260 .283814 .283284 1.282661 .281938 .281105 .280153 .279073 1.277857 .276496 .274981 .273302 .271450 24 .288805 .288801 .288794 1.288783 .288764 .288734 .288691 .288630 1.288547 .288436 .288291 .288105 .287871 1.287582 .287230 .286807 .286304 .285714 1.285028 .284238 .283336 .282312 .281160 1.279870 .278435 .276844 .275091 .273165 23 85 86 0.55271210.525914 0.000000 0.000000 .246926 .369109! .440707 1 0.485329i .513258 .530408 .540626 .546524 0.549752 551409 552205 552541 552671 0.552712 22 .291567 .291567 .291567 .291566 1.291565 .291563 .291560 .291553 .291542 1.291524 .291496 .291455 .291397 .291319 1.291213 .291075 .290899 .290677 .290403 1.290069 .289668 .289191 .288631 .287981 1.287231 .286375 .285405 .284312 .283090 1.281729 .280222 .278561 .276736 .274740 .242837 .361116 .429204 0.470086 .494585 .509034 .517295 .521796 0.524104 .525209 .525677 .525857 .525914 21 22 87 0.497713 0.000000 238267 352422 415840 0.452376 473468 485385 491835 495133 88 89 0.46771810.434386 496709 497375 497632 497713 20 296729 296729 296729 296729 296728 296726 0. 000000 0. .2334961 , .341682! , .399517' , 0.4316950. .449522! . .459073 .463930 .466244 0.467223 .467599 .467718 19 1.299144 .299144 1.299144 .299144 .299143 .299141 .299138 1.2967231.299133 .296717 .299123 .296707 .296691 .296666 .299108 .299084 .299049 1.294205 .294205 .294205 ^ .294205 .294204 1.294202 .294199 .294193 .294182 .294165 1.294138 .294099 .294045 , .293970 .293870 1.293739 .293572 .293361 .293101 .292784 1.292403 .291950 .291419 .290802 .290091 1.289279 1.29118011.292942 .288359 .290197 .291898 1.2966291.298999 .2965771 .298932 .296506| .298841 .296410 .298724 .2962871 .298573 1.29612811.298383 .295928 .298148 .295681 .297863 .295380 .297519 .295018 .297111 1.2945891.296632 .294084; .296076 .293498 .295435 .2928241 .294704 .292053! .293875 .287322 .286163 .284873 1.283444 .281869 .280141 .278250 .276188 21 .289097 .287873 .286518 1.285025 .283386 .281593 .279640 .277516 20 .290737 .289451 .288035 1.286481 .284781 .282929 .280916 .278735 19 000000 225360 326255 378512 406483 .421206 .428627 .432148 .433635 0.434205 .434386 18 1.301453 .301453 .301453 1.301453 .301452 .301450 .301448 .301442 1.301433 .301419 .301396 .301362 .301316 1.301251 .301166 .301054 .300910 .300730 1.300507 .300236 .299909 .299522 .299067 1.298539 .297931 .297236 .296449 .295563 1.294572 .293470 .292250 .290906 .289431 1.287819 .286063 .284154 .282087 .279852 18 Pay- ments, n. 1 2 3 4 5 6 7 8 9 10 11 82 81 80 79 78 77 76 75 73 74 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 Continued from Page *51. ♦60 TABLE XLIV. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OP n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4 PER CENT. Pay- I a;=90 ments. 78 77 76 75 73 72 71 70 69 68 67 66 65 64 63 62 61 57 56 55 54 53 52 51 50 49 48 47 46 45 0.402608 1 2 3 4 5 6 7 8 9 10 0.000000 .216431 .310577 .357604 0.381571 .393460 .399061 .40141? .402319 0.402608 17 83 82 81 80 79 1.303660 1.303662 .303662 .303662 .303661 91 0.373607 0.000000 .208653 .296383 .338221 358362 367723 371634 373129 373607 16 92 0.344609 93 0.314696 0.000000 .200520 .281311 .317835 0.334359 .341183 .343779 .344609 15 0.000000 .191140 .264501 .295797 0.308421 .313181 .314696 14 .303659 .303656 .303651 .303643 1.305773 .305774 .305774 .305773 .305772 1.305771 .305769 .305764 .305755 .305742 1.309720 1.307791| .309720 1.30779l!l.309720 30779l! .309720 .307791 .309719 .3077901 .309718 .3077891 .309716 1.3077861.309712 .3077821 .309705 .307775 .309692 .307761 .309674 .307742 .309646 74 1.303629 1.305722 1.307713 1.309608 .303607 .303575 .303531 .303470 .305692! .307673! .309556 .'3056491 .307618; .309486 .305591 .307544; .309395 .305514 .307448! .309277 1.303388 1.305413 1.3073251.309130 .303^82 .303146 .302975 .302763 1.302505 .302195 .301827 .301395 60 i .300893 59 1 1.300315 58 i .299655 .298907 .298066 .297125 1.296078 .294920 .293644 .292245 .290715 1.289049 .287239 .285278 .283160 .280875 17 .305283! .307170 .308948 .305120 .306979] .308727 .304919 .3067461 .308460 .3046741 .306466 .308144 1.304379 1.306133'l.307773 3040291 .305743 .307342 .303619 .303142 .302593 .305290 .304768 .304172 1.3019661.303497 .301255j .302737 .300456 .301888 .299563 .300944 .298568 .299898 1.297469! 1.298748 .296257| .297486 .294928 .296107 .293476 .294606 291895 .292976 1.290178 .288318 .286309 .284142 .281811 16 1.291211 .289304 .287250 .285039 .282666 15 .306846 .306279 .305638 1.304916 .304109 .303211 .302219 .301125 1.299926 .298617 .297191 .295643 .293968 1.292159 .290209 .288112 .285861 .283446 14 94 0.284280 0.000000 .180950 .246836 .272024 0.281336 .284280 13 1.311567 .311567 .311567 1.311567 .311566 .311565 .311563 .311558 1.311551 .311540 .311522 .311496 .311460 1.311410 .311343 .311257 .311145 .311005 95 0.252440 0.000000 .170487 .226526 .246291 0.252440 12 1.313333 .313333 .313333 .313333 1.313333 .313331 .313329 .313325 .313318 1.313308 .313291 .313266 .313231 .313184 96 0.214754 0.000000 .153574 .200677 .214754 11 1.315019 .315020 .315020 .315019 .315019 1.315018 .315016 .315011 .315006 .314995 1.314979 .314956 .314923 .314878 .314817 97 0.178369 000000 ,141329 ,178369 Pay- ments, 1.316633' 89 .3166331 88 .316G33I 87 .316632 86 .316631 85 1.316629 .316626 .316620 .316610 .316595 1.316572 .316541 .316498 .316441 .316367 1.313120 1.314739,1.316271 .313038 .314638 .316150 .312932 .312799 .312634 .3145111 .316000 .314354 .315819 .314163 .315601 1.3108321.3124341.3139341.315342 .310621: .312193 .310368' .311906 .310067 .309714 1.309305 .308833 .308295 .307685 .306998 1.306231 .305378 .304435 .303395 .302256 1.301012 .299658 .298188 .296597 .294879 1.293029 .291038 .288903 .286613 .264163 13 .311571 3136621 .315038 .313343 .312972 311181 .312545 1.310732 .310220 .309640 .308987 .308258 1.307447 .306550 .305562 .304480 .303298 1.312058 .311506 .310885 .310191 .309420 1.308567 .307628 .306599 .305475 .304252 1.3020101.302926 .300614 .299103 .297472 .295714 1.293825 .291798 .289626 .287302 .284817 12 .301489 .299939 .298270 .296476 1.294552 .292490 .290285 .287928 .285413 11 .314685 .314279 .313814 1.313289^ 64 .312699 63 .312039 62 .311305 61 .310494 60 1.309601 .308622 .307554 .306391 .305130 1.303765 .302293 .300707 .299003 .297176 1.295218 49 .293124! 48 .290888 47 .288502 .285958 10 Continued from Page *50. TABLE XLY. ♦61 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. The first Value under the Age a; is A(l + Oas) for the whole Life. 4i PER CENT. Pay- a;=10 11 12 13 14 15 16 1 17 Pay- ments. n. 1 ments. n. 1 1.281791 1.280417 1.278977 1.277464 1.275875!l.274211^1.272467 i 1 1.270632 0.000000 0.000000 0.000000 0.000000 0.000000 o.oooooo'o.ooodoo 0.000000 2 .290199 .290194 .290191 .290187 .290180 .290174 .290170 .290163 2 3 .455548 .455540 .455533 .455523 .455510 .455500 .455490 .455475 3 4 .569836 .569823 .569811 .569796 .569778 .569762 .569745 .569723 4 5 0.656184 0.656166 0.656149 0.656129 0.656106 0.656083 0.656060 0.656031 5 6 .724891 .724869 .7248471 .724822 .724792 .724764 .724734 .724696 6 7 .781454 .781427 .781400 .781369 .781333 .781299 .781260 .781213 7 8 .829151 .829119 .829087 .829049 .829008 .828964 .828918 .828863 8 9 .870097 .870061 .870022 .869979 .869929 .869879 .869825 .869761 9 10 0.905737 0.905694 0.905650 0.905600 0.905544 0.905486 0.905423 0.905350 10 11 .937099 .937051 .937001 .936944 .936880 .936814 .936742 .936657 11 12 .964945 .964891 .964834 .964770 .964699 .964624 .964542 .964448 12 13 .989852 .989792 .989728 .989657 .989577 .989493 .989401 .989296 13 14 1.012269 1.012202 1.012131 1.012052 1.011963 1.011870 1.011767 1.011651 14 15 1.032550 1.032477 1.032398 1.032310 1.032214 1.032109 1.031996 1.031867 15 16 .050984 .050902 .050815 .050721 .050612 .050498 .050374 .050235 16 17 .067801 .067712 .067620 .067513 .067396 .067271 .067137 .066984 17 18 .083199 .083105 .083000 .082885 .082758 .082624! .082476 .082306 18 19 .097341 .097235 .097123 .096998 .096862 .096715 .096551 .096368 19 20 1.110358 1.110246 1.110124 1.109991 1.109842 1.109680 1.109504 1.109307 20 21 .122374 .122252 .122123 .121978 .121815 .12164] .121451 .121239 21 22 .133486 .133357 .133216 .133057 .132884 .132696 .132493 .132263 22 23 .143783 .143643 .143489 .143322 .143134 .142935 .142713 .142466 23 24 .153339 .153186 .153024 .152842 .152644 .152427 .152190 .151925 24 26 1.162219 1.162058 1.161882 1.161690 1.161476 1.161243 1.160989 1.160705 25 26 .170486 .170312 .170127 .169919 .169689 .169441 .169169 .168867 26 27 .178187 .178005 .177804 .177582 .177338 .177072 .176783 .176459 27 28 .185374 .185177 .184962 .184727 .184465 .184183 .183874 .183528 28 29 .192081 .191871 .191644 .191392 .191115 .190814 .190483 .190119 29 30 1.198348 1.198126 1.197884 1.197617 1.197322 1.197000 1.196653 1.196259 30 31 .204210 .203974 .203716 .203433 .203118 .202781 .202404 .201993 31 32 .209695 .209443 .209171 .208868 .208539 .208174 .207780 .207340 32 33 .214828 .214564 .214272 .213955 .213600 .213219 .212798 .212332 33 34 .219639 .219355 .219050 .218709 .218339 .217932 .217486 .216993 34 1 35 1.224142 1.223846 1.223519 1.223163 1.2227681.222337 1.221866 1.221347 35 36 .228367 .228049 .227708 .227329 .226910 .226455 .225960 .225410 36 37 .232323 .231991 .231629 .231226 .230785 .230308 .229782 .229201 37 38 .236034 .235684 .235298 .234875 .234412 .233905 .233351 .232739 38 39 .239513 .239141 .238735 .238291 .237800 .237267 .236684 .236038 39 40 1.242772 1.242380 1.2419551.241485 1.240969 1.240408 1.239792 1.239114 40 41 .245827 ,245417 .244966 .244473 .243930 .2433381 .242691' .241977 41 42 .248692 .248257 .247785 .247265 .246693 .246072 .245391 .244640 42 43 .251372 .250917 .250420 .249872 .249273 .248619 .247903 .247117 43 44 .253883 .253405 .252881 .252308 .251677 .250990 .250240 .249414 44 10 11 12 13 14 15 16 17 Continued on Page *71. FOEMULA, A(l-f ^^-1) = A ^=^~^^^^ = Agg=l. Dx VPx—l ♦62 TABLE XLV. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF W ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4|^ PER CENT. 1 Pay- niPTits a;=18 19 20 21 22 23 24 25 Pay- ments. n. i n. 1* 1.268708|l.366691 1.264573 1.262353 1.260026 1.257585,1.255025 1.252341 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .290155 .290148 .290139 .290129 .290119 .290107 .290094 .290081 2 3 .4554601 .455445 .455426 .455406 .455385 .455361 .455335 .455306 3 4 .569701 .569677 .569647 .569617 .569584 .569547 ,569506 .569462 4 5 0.655999 0.655965 0.655926S0. 655884 0.655840 0.655789 0.655733 0.655673 5 6 .724655 .724612 .724562 .724509 .724452 .724387 .724316 .724239 6 7 .781164 .781111 .781050 .780984 .780914 .780835 .780748: .780653 7 8 .828805 .828741 .828668 .828590 .828506 .828411 .828307 .828194 8 9 .869692 .869617 .869532 .869441 .869342 .869231 .869110 .868976 9 10 0.905269 0.905182 0. 905086 0.904980 0.904866 0.904739 0.904598 0.904444 10 11 .936567 .936469 .936358 .936238 .936109 .935963 .935802 .935627 11 n .964346 .964234 .964110 .963976 .963828 .963663 .963483 .963285 12 13 .989182 .989057 .988920 .988768 .988602 .988418 .988217 .987997 13 14 1.0115231.011387 ;. 011233 1.011063 1.010880 1.010676 1.010453 1.010207 14 15 1.031729 .031577 1.031404 1.031219 1.031017 1.030792 1.030544 1.030273 15 16 .050082 .049911 .049724 .049521 .049298 .049051 .048779 .048481 16 17 .066813 .066629 .066425 .066202 .065958 .065687 .065391 .065065 17 18 .082122 .081922 .081699 .081456 .081189 .080896 .080572 .080217 18 19 .096170 .095952 .095710 .095445 .095157 .094836 .094486 .094102 19 20 1.1090921.108856 1.108593 1.108308 1.107993 1.107649 1.107270 1.106852 20 21 .121006 .120750 .120468 .120157 .119821 .119449 .119037 .118586 21 22 .132011 .131737 .131430 .131099 .130736 .130332 .129891 .129406 22 23 .142196 .141899 .141573 .141217 .140823 .140392 .139917 .139397 23 24 .151634 .151318 .150968 .150583 .150164 .149700 .149192 .148633 24 25 1.1603961.160056 1.159680 1.159270 1.158819 1.158325 1.157780 1.157181 25 26 .168535 .168171 .167771 .167331 .166852 .166321 .165738 .165099 26 27 .176104 .175718 .175288 .174822 .174308 .173741 .173120 .172441 27 28 .183153 .182737 .182283 .181783 .181235 .180633 .179973 .179247 28 29 .189714 .189277 .188791 .188258 .187676 .187037 .186333 .185563 29 30 1.195835 1.195367 1.194849 1.194284 1.193668 1.192987 1.192240 1.19.1422 30 31 .201539 .201041 .200493 .199896 .199239 .198517 .197725 .196859 31 32 .206858 .206332 .205753 .205118 .204421 .203656 .202820 .201900 32 33 .211823 .211266 .210652 .209978 .209241 .208434^ .207546 .206577 33 34 .216456 .215865 .215214 .214503 .213726 .212870 .211935 .210909 34 35 1.220777 1.220152 1.219465 1.218714 1.217892 1.216991 1.216002 1.214921 35 36 .224807 .224148 .223423 .222631 .221765 .220810 .219771 .218631 36 37 .228566 .227870 .227107 .226272 .225356 .224355 .223258 .222059 37 38 .232069 .231337 .230532 .229651 .228690 .227635 .226482 .225222 38 39 .235334 .234563 .233714 .232789 .231777 .230669 .229458 .228136 39 40 1.2383721.237559 1.2366691.235696 1.234634 1.233471 1.232201 1.230815 40 41 .2411961 .240344 .239408] .238388 .237273 .236054 .234723 .233272 41 42 .243822 .242926 .241946 .240875 .239707 .238430 .2370371 .235520 42 43 .246257 .245319 .244290 .243169. .241946 .240610 .239156 .237571 43 44 .248515 .247530 .246454 .245280, .244002 .242608 .241088 .239436 44 18 19 20 21 22 23 24 25 Continued on Page *70. TABLE XLY. ♦63 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4^ PER CENT. Pay. 26 27 28 29 30 31 32 . 33 Pay- ments, n. ments. 1.249527 1.246580 1.243489 1.240250 1.236862 1.233310 1.229589 1.225692 n. 1 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 .290064 .290047 .290029 .290005 .289984 .289960 .289933 .289903 2 3 .455372 .455239 .455200 .455154 .455111 .455060 .455003 .454943 3 4 .569412 .569360 .569299 .569232 .569163 .569084 .568999 .568903 4 5 0.655606 0. 655533 0.655452 0.655360 0.655265 0.655159 0.655039 0.654910 5 6 .724152 .724060 .723955 .723838 .723718 .723578 .723428 .723261 6 7 .780547 .780431 .780303 .780162, .780009 .779842 .779656 .779452 7 8 .828066 .827929 .827777 .827605 .827427 .827224 .827003 .826758 8 9 .868827 .868667 .868486 .868290 .868077 .867843 .867582 .867296 9 10 0.904273 0.904086 0.903882 0.903653 0.903410 0.903137 0.902838 0.902507 10 11 .935431 .935221 .934987 .934729 .934449 .934140 .933798 .933423 11 12 .963067 .962828 .962565 .962273 .961958 .961609 .961225 .960801 12 13 .987751 .987485 .987190 .986864 .986512 .986122 .985691 .985221 13 14 1.009935 1.009639 1.009312 1.008950 1.008560 1.008125 1.007650 1.007127 14 15 1.029973 1.029647 1.029285 1.0288881.028454 1.027977 1.027452 1.026874 15 16 .048152 ♦.047793 .047398 .046958; .046485 .045961 .045384 .044750 16 17 .064705 .064314 .063878 .063401 .062883 .062309 .061679 .060985 17 18 .079827 .079397 .078926 .078406 .077841 .077217 .076539 .075775 18 19 .093675 .093211 .092699 .092135! .091522 .090843 .090099 .089381 19 20 1.106391 1.105889 1.105335 1.104725 1.104060 1. 103327 1.103533 1.101637 20 21 .118090 .117548 .116950 .116291! .115574 .114785 .113915 .113964 21 22 .128872 .128288 .127644 .126935 .126165 .125313 .134383 .133356 22 23 .138823 .138195 .137503 .136744 .135915 .135004 .134001 .133903 23 24 .148018 .147344 .146605 .145789 .144904 .143925 .142855 .141679 24 25 1.156522 1.155804 1.155011 1.154142 1.153192 1.152150 1.151004 1.149749 25 26 .164398 .163628 .162784 .1618541 .160844 .159730 .158510 .157173 26 27 .171690 .170873 .169970 .1689831! .167905 .166720 .165422 .163999 27 28 .178451 .177578 .176621 .1755691 .174424 .173165 .171785 .170374 28 29 .184715 .183790 .182772 .181657 .180440 .179104 .177641 .176040 29 30 1.190524 1.189540 1.188463 1.187279 1.185990 1. 184574 1.183026 1.181333 30 31 .195905 .194866 .193723 .192471! .191105 .189609 .187973 .186184 31 32 .200894 .199792 .198584 .197259; .195818 .194238 .192512 .190635 32 33 .205511 .204348 .203071 .201674 .200153 .198489 .196669 .194684 33 34 .209785 .208556 .207211 .2057381 1 .204137 .202384 .200471 .198385 34 35 1.213735 1.212440 1.211023 1.209475: 1.207790 1.205948 1.203940 1.301749 35 36 .217382 .216020 .214532 .212904; .211134 .209202 .207094 .304803 36 37 .220746 .219316 .217752 .216043! .214189 .212162 .209959 .307560 37 38 .223844 .222343 .220702 .218913 .216968 .214851 .212540 .310046 38 39 .226691 .225117 .223399 .221525 .219495 .217275 .214880 .212273 39 40 1.229301 1.227653 1.225856 1.223900 1.221773 1.219474 1.216972 1.214262 40 41 .231688 .229966 .228090 .226043 .223840 .221440 .218840 .216027 41 42 .233865 .232068 .230107 .227987 .225689 .233196 .220500 .217586 42 43 .235845 .233967 .231936 .229727 .227341 .224757 .221965 .318953 43 44 .237634 .235690 .233575 .231282 .228809 .226135 .223249 .330140 44 26 27 28 29 30 31 32 33 Continued on Page *69. ♦64 TABLE XLY. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4^ PER CENT. Pay- mcnts. a;=34 35 36 37 38 1.203309 39 40 41 Pay- ments. 11. n. 1.2216121.217341 1.212877 1.208201 1.198189 1.1928391.187248 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .289869 .289829 .289792 .289749 .289699 .289644 .289585 .289524 2 3 .454871 .454795 .454715 .454627 .454525 .454414 .454295! .454164 3 4 .568797 .568679 .568557 .568419 .568263 .568095 .567910! .567709 4 5 0.654765 0.654607 0.654438 0.654249 0.654038 0.653808 0.653556 0.653281 5 6 .723078 .722875 .722658 .722416 .722146 .721852 .7215301 .721178 6 7 .779226 .778977 .778710 .778412: .778081 .777721 .777325 .776892 7 8 .826489 .826192 .825872 .825516, .825123 .824691 .824218 .823701 8 9 .866980 .866634 .866258 .865844 .865382 .864877 .864325 .863717 9 10 0.902145 0.901745 0.901314 0.900835 0.900303 0.899723 0.899084 0.898388 10 1 11 .933010 .932558 .932065 .931518 .930915 .930253 .929530 .928738 11 : 12 .960338 .959827 .959270 .958656 .957975 .957233 .956419 .955525 12 13 .984702 .984130 .983509 .982821 .982064 .981235 .980325 .979329 13 14 1.006550 1.005917 1.005226 1.004465 1.0036251.002706 1.001698 1.000592 14 15 1.0262391.025539 1.024779 1.023939 1.02301411.022001 1.020890 1.019673 15 16 .044051 .043286 .042450 .0415301 .040516 .039405 .038187 .036854 16 17 .060224 .059387 .058475^ .057470 .056362 .055150 .053823 .052366 17 18 .074947 .074037 .073045; .071951 .070747 .069432; .067987 .066407 18 19 .088382 .087397 .086320^ .085135^ .083834 .0824061 .080844 .079135 19 20 1.100668 1.099602 1.0984381.097161 1.095753 1.094215 1.092530 1.090686 20 21 .111917 .110769 .109518 .108140 .1066261 .104971 .103159 »101175 21 22 .122232 .121000 .119654: .118175 .116551 .114775 .112830 .110708 22 23 .131699 .130377 .128935 .127351 .125612 .123711 -121635 .119364 23 24 .140390 .138977 .137436 .135743 .133886 .131860 .129643 .127224 24 25 1.148374 1.146867 1.145221 1.143417 1.141441 1.139282 1.136925 1.134351 25 26 .155707 .154101 .152351 .150434 .148332 .146040 .143535 .140805 26 27 .162440 .160735 .158877 .156840 .154613 .152181 .149530 .146642 27 28 .168621 .166813 .164842 .162687 .160327 .157756 .154955 .151907 28 29 .174289 .172375 .170291 .168011 ' .165519 .162806 .159854 .156641 29 30 1.179481 1.177459 1.175258 1.172852 1.170225!l. 167370 1.164262 1.160891 30 31 .184230 .182098 .179777 .177243 .174483' .171480 .168222; .164682 31 32 .188567 .186321 .183879 .181220 .178318 .175174 .171756, .168056 32 33 .192519 .190158 .187597 .184803 .181768 .178472 .174904 .171042 33 34 .196110 .193636 .190948 .188028 .184851 .181412 .177691 .173673 34 35 1.1993681.196774 1.193966 1.190912 1.187599 1.1840171.180148 1.175973 35 36 .202308: .199601 .196666 .193483 .190035 .186313 .182297 .177971 36 37 .204958 .202131 .199075 .195764 .192183 .188323 .184163 .179690 37 38 .207330 .204389 .201212 .197775 .194065 .190068 .185770 .181162 38 39 .209449 .206393 .203096 .199538 .195698 .191571 .187146 .182407 39 40 1.211329 1.208160 1.204749 1.201068 1.1971061.192859 1.188311 1.183447 40 41 .212988 .209710 .206183 .202387 .198311 .193949 .189283 .184307 41 42 .214443 .211056 .207420 .203517 .199333 .194860 .190088 .185011 42 43 .215707 .212218 .208481 .204475 .200186 .195613 .190746 .185576 43 44 .216799 .213213 .209379 .205275 .200892 .196229 .191275 .186022 44 34 35 36 37 38 39 40 41 Continued on Page *68. TABLE XLY. *65 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OP THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4^ PER CENT Pay- 42 43 44 45 46 47 48 49 Pay- ments. n. ments. n. 1 1.181397 1.175290J1.168905 1.162240 1.155279 1.148013 1.140432 1.132524 1 0.000000 0.000000 0.000000 0.000000 0.000000,0.000000 0.000000 0.000000 2 .289449 .289373 .289286 .289192 ' .2890881 .288973 .288847 .288710 2 3 .454015 .453858 .453681 .453487 .453275 .453040 .4527841 .452501 3 4 .567483 .567240 .566968 .566672 .566344 .565986 .565592 .565158 4 5 0.652973 0.652640^0. 652271 0.651865 0.6514210.650931 0.650393 0.649804 6 6 .720784 .720358 .719885 .719369 .7187991 .718173 .717488; .716733 6 7 .776409 .775885 .775307 .774671 .773972J .773206 .772363 .771439 7 8 .823124 .822501 .821809 .821051 .820219 .819303 .818300 .817198 8 9 .863047 .862316 .861507 .860623 .859649! .858581 .857409 .856121 9 10 0.897616 0.896773 0.895845 0.894825 0.8937070.892480 0.8911310.889653 10 11 .927859 .926903 .925846 .924690 .923421 .922025 .920497i .918819 11 12 .954538 .953460 .952274 .950975 .949545 .947980 .9462621 .944379 12 13 .978226 .977025 .975703 .974251 .972661 .970916 .969003 .966906 13 14 .999373 .998042 .996575 .994972 .993212 .991281 .989166 .986851 14 15 1.018330 1.016862 1.015252 1.013488 1.011551 1.009429 1.007108 1.004566 15 16 .035381 .033778 .032014 .030082 .027964 .025648 .023112 .020338 16 17 .050764 .049016 .047094 .044990 .042689 .040168 .037413 .034397 17 18 .064668 .062770 .060685 .058408 .055912 .053185 .050201 .046944 18 19 .077254 .075201 .072953 .070491 .067800 .064857 .061645 .058136 19 20 1.088658 1.086451 1.084027 1.081380 1.078486 1.075327 1.071877 1.068119 20 21 .099001 .096628 .094029 ..0911901 .088092 .084708! .0810231 .077005 21 22 .108377 .105838 .103057 .100025 .096714; .093108 .0891781 .084904 22 23 .116876 .114165 .111201 .107969 .104448' .100610 .096439 .091903 23 24 .124573 .121688 .118536 .115104 .111365 .107301 .102882 .098085 24 25 1.131535 1.128473 1.125132 1.121495 1.117542 1.113243 1.108579'l.l03526 25 26 .137823 .134581 .131047 .127208; .123033| .118504' .1135981 .108290 26 27 .143489 .140065 .136340 .132291 .127900; .123143 .117998 .112438 27 28 .148581 .144978 .141054 . 136802 ii .132195 .127213 .121831 .116027 28 29 .153147 .149356 .145240 .140784 1 .135967 .130761 .125150 .119106 29 30 1.157219 1.153246 1.148938 1.144285 '1.139256 1.133836 1.127999 1.121733 30 31 .160839 .156685 .152192 .147339! .142108 .136476 .130431 .123949 31 32 .164042 .159713 .155031 .149988 .144558 .138731 .132484 .125799 32 33 .166863 .162356 .157496 .152266 .146652 .140635 .134197 .127326 33 34 .169327 .164652 .159615 .154212 .148421 .142224 .135612 .128572 34 35 1.171468 1.166626 1.161427 1.155857 1.149897 1.143537 1.1367671.129572 35 36 .173309 .168315 .162959 .157230 .151117 .144609 .1376941 .130360 36 37 .174885 .169743 .164237 .158365 .152114 .145471 .138425; .130975 37 38 .176217 .170935 .165295 .159294 .152915 .146149 .1389951 .131442 38 39 .177330 .171922 .166160 .160039 .153545 .146678 .139428 .131789 39 40 1.178251 1.172728 1.166854 1.160625 1.154037 1.147080 1.139750 1.132039 40 41 .179004 .173376 .167401 .161083 .154411 .147380 .139982 .132219 41 42 .179609 .173886 .167827 .161432 .154690 .147595 .140149 .132342 42 43 .180086 .174284 .168153 .161691 .154890 .147750 .140263 .132419 43 44 i .180457 .174588 .168394 .161877 .155034 .147856 .140335 .132467 49 44 42 43 44 46 46 47 48 Cont nued oa Vs ige *67. *66 TABLE XLV. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4|^ PER CENT. Pay- x=50 51 52 53 54 55 56 57 Pay- ments. n. 1 ments. n. 1.1242741.115680 1.1067231.097392 1.0876761.077558 1.0670371.056095 0. 000000 0. OOOOOOJO. 000000 0. 000000 1 0.000000 0.000000 0.000000 0.000000 1 2 .288556 .288390 .288206 .288005 .287785 .287538 .287272 .286974 2 3 .452188 .451848 .451473 .451062 .450610 .450110 .449562 .448958 3 4 .564681 .564158 .563584 .562954 .562262 .561495 .560657 .559736 4 5 0.649153 0.648441 0.647660 0. 646801 i 0.645857 0.644815 0.643677 0.642425 5 6 .715902 .714995 .713995 .7128991 .711694 .710368 .708916 .707320 6 7 .770422 .769307 .768083 .766740 .765266 .763642 .761866 .759916 7 8 .815985 .814658 .813200 .811601 .809846 .807916 .805805 .803489 8 9 .854707 .853158 .851458 .849594 .847549 .845302 .842846 .840152 9 10 0.888028 0.886250 0.884299 0.882161 0.879816 0.877243 0.874430 0.871348 10 11 .916977 .914961 .912751 .910330 .907677 .904767 .901588 .898107 11 12 .942311 .940051 .937573 .934860 .931889 .928634 .925079 .921195 12 13 .964608 .962094 .959341 .956327 .953031 .949422 .945488 .941187 13 14 .984313 .981539 .978501 .975182 .971551 .967585 .963259 .958538 14 15 1.001782 0.998739 0.995413 0.991777 0.987811 0.983476 0.978755 0.973610 15 16 .017299 1.013984 1.010359 1.006408 1.002094 .997388 .992268 .986694 16 17 .031102 .027505 .023584 .019304 .014642 1.009560 1.004038! .998040 17 18 .043382 .039506 .035274 .030668 .0256521 .020193 .0142731.007849 18 19 .054311 .050142 .045603 .040663 .035293 .029461 .023143 .016300 19 20 1.0640191.059560 1.054708 1.049435 1.0437151.037510 1.030799J1.023545 20 21 .072632 .067879 .062713 .057113 .051043 .044470 .037375 .029718 21 22 .080253 .075205 .069732 .063803 .057388 .050456 .042984 .034942 22 23 .086975 .081638 .075856 .069604 .062853 .055568 .047738 .039328 23 24 .092885 .087257 .081173 .074606 .067525 .059906 .051733 .042973 24 25 1.098053 1.092142 1.085762 1.078886 1.071493 1.063554 1.055055 1.045971 25 26 .102550 .096362 .089692 .082525 .074833 .066590 .057791 .048407 26 27 .106438 .099978 .093036 .085590 .077614 .069092 .060014 .050357 27 28 .109772 .103058 .095855 .088143 .079907 .071126 .061794 .051896 28 29 .112614 .105655 .098204 .090249 .081772 .072756 .063200 .053089 29 30 1.115011 1.107820 1.100142 1.091963 1.083268ll.074042 1.064290 1.053994 30 31 .117011 .109607 .101720 .093338 .084448! .075041 .065117 .054670 31 32 .118661 .111063 .102987 .094423 .085865 .075798 .065735 .055158 32 33 .120007 .112232 .103986 .095266 .086060 .076364 .066181 .055506 33 34 .121087 .113154 .104763 .095906 .086579 .0767731 .066499 .055742 34 35 1.121939 1.113871 1.105353 1.096383 1.086955 1.0770641.066714 1.055891 35 36 .122602 .114416 .105792 .096729 .087222 .0772611 .066851 .055985 36 37 .123106 .114821 .106111 .096975 .087403 .077387 .066937 .056038 37 38 .123481 .115115 .106338 .097141 .087519 .077466 .066986 .056069 38 39 .123752 .115325 .106491 .097248 .087591 .077511 .067014 .056082 39 40 1.123946 1.115467 1.106589 1.097314 1.087633 1.077536 1.067026 1.056091 40 41 .124078 .115557 .106650 .097353 .087656 .077547 .0670341 .056094 41 42 .124161 .1156131 .106686 .097374 .087666 .077555 .067037^ .056094 42 43 .124213 .115646 .106706 .097383 .087673 .077557 .067037' .056095 43 44 .124244 .115665 .106714 .097389 .087675 .077557 .067037 44 45 1.124261 1.115672 1.106720 1.097390 1.087675 1.077558 45 46 .124268 .115678 .106721 .097391 46 47 .124273 .115679 .106722 47 48 .124274 .115679 48 49 .124274 49 50 51 52 53 54 55 56 57 TABLE XLY. *61 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4i PER CENT. Pay- ments. n. 58 59 60 61 62 63 64 65 Pay- ments. n. 1 1.044728 1.03291o'l.020646 1.007916 0.994716 0.981025 0.966846 0.952166 0.000000 O.OOOOOO'0. 000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 .286653 .286298 .285906 .285476 .285008 .284488 .283919 .283293 2 3 .448302 .447578 .446779 .445905 .444948 .443891 .442734 .441463 3 4 .558730 .557622 .556403 .555067 .553605 .551993 .550229 .548291 4 5 0.641055 0.639550 0.637896 0.636084 0.634099 0.631916 0.629525 0.626906 5 6 .705576 .703660 .701557 .699252 .696730 .693958 .690928 .687611 6 7 .757784 .755444 .752876 .750067 .746991 .743619 .739934 .735903 7 8 .800957 .798179 .795136 .791806 .788167 .784181 .779828 .775073 8 9 .837207 .833982 .830450 .826591 .822376 .817764 .812734 .807248 9 10 0.867981 0.864297 0.860267 0.855867 0.851067 0.845822 0.840110 0.833891 10 11 .894310 .890158 .885620 .880672[ .875279 .869397 .863002 .856049 11 12 .916957 .912327 .907277 .901773 .895785 .889266 .882188 .874509 12 13 .936501 .931391 .925819 .9197611 .913178 .906024 .898272 .889876 13 14 .953401 .947802 .941713 .935098 .927923 .920143 .911726 .902632 14 15 0.968015 0.961932 0.955322 0.948156^ 0.940400 0.932002 0.922939 0.913173 15 16 .980646 .9740761 .966953 .959245 .950914 .941918 .932235 .921823 16 17 .991536 .984488 .976859 .968617 .959731 .950161 .939883 .928862 17 18 1.000898 .993377 .985252 .976495 .967078 .956958 .946119 .934533 18 19 .008908 1.000926 .992321 .983072 .973146 .962510 .951152 .939045 19 20 1.015723 1.007294 0.998233 0.988513 0.978111 0.966998 0.955160 0.942585 20 21 .021481 .012629 1.003131 .992971 .982129 .970575 .958310 .945317 21 22 .026311 .017054; .007148 .996582 .985335 .973388 .960742 .947387 22 23 .030321 .020686 .010405 .999466 .987858 .975563 .962586 .948926 23 24 .033616 .023634 .013009 1.001737 .989810 .977212 .963958 .950040 24 25 1.036292 1.0259911.015060 1.003495 0.991290 0.978440 0.964950 0.950829 25 26 .038433 .027850! .016649 .004829 .992393 .979328 .965654 .951364 26 27 .040122 .029290! .017854 .005822 .a93191 .979957 .966131 .951703 27 28 .041431 .0303821 .018752 .006542 .993756 .980385 .966433 .951914 28 29 .042424 .031197 .019403 .007051 .994140 .980655 .966622 .952037 29 30 1.043165 1.0317871.019863 1.007398 0.994383 0.980825 0.966731 0.952105 30 31 .043701 .032205 .020178 .007616 .994536 .980922 .966792 .952138 31 32 .044082 .032489 .020375 .007754 .994623 .980977 .966821 .952155 32 33 .044340 .032669 .020500 .007833 .994672 .981003 .9668361 .952163 33 34 .044504 .032782 .020571 .007877 .994696 .981016 .9668431 .952165 34 35 36 37 38 39 40 41 42 1.044607 .044666 .044699 .044714 .044723 1.044727 .044727 .044728 1.032847 .032883 .032900 .032909 .032914 1.032914 1.020611 .020630 .020640 .020645 .020645 1.020646 1.007899 .007909 .007915 .007915 .007916 0.994707 .994714 .994714 .994716 0.981023 .981024 .981025 0.966845 .966846 0.952166 35 58 57 56 55 54 53 42 43 44 1.181397 .181395 .181395 .181395 1.181393 .181389 45 1.175290 .175289 .175289 1.175288 .175286 46 1.168905 .168905 1.168905 .168904 .032915 1.162240 1.162239 .162239 47 1.155279 .155278 48 49 1.148013 1. 140432 .148012 .155278 .162238 .168902 .175282 .181380 52 51 1.132524 .140431 .148012 .155277 .162235 .168897 .175272 .181362 51 50 .132523 .140431 .148011 .155273 .162230 .168886 .175253 .181334 50 49 1.132523 1.140430 1.148007 1.155268 1.162218 1.168866 1.175223 1.181287 49 48 .132522 .140426 .148002 .155255 .162197 .168834 .175173 .181211 48 47 .132518 .140420 .147988 .155233 .162162 .168780 .175091 .181102 47 46 .132511 .140405 .147964 .155195 .162104 .168692 .174975 .180951 46 45 .132495 .140379 .147923 .155133 46 ( .162011 .168568 .174813 .180741 45 49 48 47 45 44 43 42 Continued from Page *65, ♦68 TABLE XLT. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 41 „^^ ^^^.^ Pay- ments. 71. 1 2 3 i 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 3^=66 67 0.936983,0.921279 0.000000 .282611 .440070 .546174 0.624044 .683985 .731504 .769890 '.801278 0.827131 848506 866193 880805 892832 0.902672 910659 917083 922185 926183 0.929266 .931600 .933334 .934590 .935478 0.936080 .936462 .936700 .936838 .936914 0.936951 .936971 .936979 .936982 .936983 41 1.187248 .187246 .187246 .187246 .187244 1.187241 .187232 .187215 .187189 .187145 1.187074 186972 186831 186635 186369 41 0.000000 .281852 .438541 .543847 0.620902 .680015 .726692 .764233 .794771 0.819779 .840319 .857190 .871009 .882275 0.891393 .898708 .904508 .909047 .912543 0.915188 .917150 .918572 .919577 .920257 0.920690 .920959 .921115 .921201 .921243 0.921265 921274 921278 921279 68 0.905066 69 70 0.888311 0.871044 71 72 73 0.853238 0.834905 0.816036 0. 000000 0. OOOOOOi |0. 000000 0. 000000 .2810351 .2801201 .279130 .278037 .436872 .435027L .433019 .430811 .541309 .538512! .535465 .532128 0. 61747610. 613710 0. 609614 ,675686 ,721455 .758081 ,787707 ,6709441: .665786 ,715727: .709502 . 7513671 ' .744086 .7800171 .771693 .811813 .831469 .847477 .860468 .870947 .879328 .885960 .891141 .895127 .898140 ,8031631 ,821880 ,836986 ,849118 ,858787 866419 872370 876940: 880391! 882946 0.605131 .660148 .702722 .736172 .76266 0.900372^0.884796 40 1.192839 1.192837 .192837 .192837 .192835 .192833 1.192824 .192808 .192784 .192743 .192677 1.192581 .192449 .192266 .192017 .191692 40 .901990 .903132 .903904 .904397 0.904702 .904879 .904977 .905025 .905050 0.905060 .905065 .905066 39 1.198189 .198187 1.198187 .198187 .198186 .198184 .198175 1.198160 .198138 .198099 .198037 .197948 1.197824 .197653 .197419 .197115 .196724 39 .886101 .886983 .887547 .887895 0.888097 .888209 .888264 .888293 .888304 0.888309 .888311 38 1.203309 .203304 .203304 1.203304 .203303 .203301 .203292 .203278 1.203258 .203222 .203164 .203080 .202964 1.202804 .202584 .202299 .201933 .201469 38 793818 0.783714 811550| .800415 825715' .813603 836958! .823938 845804; .831952 8526850.838081 8579581 .842695 861935! .846104 8648771 .848567 867004 .850301 0.868504'o. 851473 869518: .852224 .870167 .870566 .870799 0.870927 0, .87099li .871023! .871036! .871042 0.871044 .852685 .852955 O.OOOOOOlO. 000000 .276843 .275538 .428403 .425760 .528482 .524492 0.600238 0.594904 .654017 .647340 .695357 .687355 .727594 .718300 .752909 .742371 0.772824 0.761101 .788450 .775614 .800034 .786774 .810047 .795255 .817225! .801609 0.822618 0.806288 .826596! .809660 .8294671 .812031 .8314871 .813631 .832852! .814656 0.833726 0.815284 .834262 .834575 .815650 .815852 853102 .834834 .816005 37 1.208201 .208199 .208199 .208199 1.208198 .208196 .208188 .208175 .208156 1.208122 .208068 .207989 .207880 .207730 1.207524 .207256 .206913 .206478 .205937 37 853177 853213 853229 853235 853238 .834747 .815954 0.83487710.816025 .834895! .816034 .834902' .816036 .834905 34 36 1.212877 .212875 .212875 .212875 .212874 1.212872 .212864 .212852 .212834 .212802 1.212751 .212677 .212575 .212434 ."212241 1.211989 .211667 .211259 .210751 .210130 36 35 1.221612 1.217341! .221612 1.2173401.221612 .217340! .221611 .217339; .221610 .217338! .221608 61 .217336' .221601 60 Pay- ments, n. 1.217329!l. 221591 .217318! .221575 .217301 .217271 .217223 .221547 .221502 .221437 1.2171541.221346 .217058! .221221 .216925 .216744 .216508 1.2162051.220184 .215821 .215345 .214761 .214056 36 .219736 .219187 .218526 .217733 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 66 65 64 63 62 .221051 .220829 .220545 50 59 58 57 56 55 54 53 52 51 49 48 47 46 45 Continued from Page *64. TABLE XLY. ^69 LIFE AND TEMPORAET ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4^ PER CENT. Pay- ments. n. 1 74 75 76 77 78 79 80 81 -... - - ] ' Pay- ments. 0.796626 0.776711 0.756264 0.735279 0.713797 0.691797 ( 3.669283 ( 3. 646255' 0.000000 o.oooooolo.ooooool 0.000000 0.000000 0. 000000 3.000000 3. OOOOOo' 1 2 .274089 .272521 .270807 .268910 .266851 .264595 .262129 .2594471 2 3 .422855 .419707 .416256 .412470 .408356 .403865 .398975 .393652 3 4 : 520129 .515387 .510208 .504550 .498410 .491733 .484479 .476622 4 5 0.589077 0.582756 0.575872 0.568376 0.560266 0.551472 0.541963 0.531696 5 6 .640067 .632192 .623643 .614371 .604364 .593563 .581926 .569428 6 7 .678666 .669282 .659130 .648157 .636363 .623682 .610091 .595572 7 8 .708241 .697409 .6857321 .673164 .659708 .645312 .629962 .613646 8 9 .731003 .718801 .705699 .691659 .676693 .660765 .643866 .626012 9 10 0.748499 0.735023 0.720614 0.705242 0.688938 0.671670 0. 653455 0.634186 10 11 .761871 .747230 .731645 .715100 .697634 .679234 .659827 .639362 11 12 .771990 .756308 .739691 .722134 .703690 .684280 .663875 .642509! 12 13 .779547 .7629541 .745453 .727048 .707743 .687494 .666342 .644336 13 14 .785096 .767728 .749487 .730344 .710329 .689456 .667776 .645339: 14 1 15 0.789090 0.771075 0.752198 0.732450 0.711910 0.690597 0.668563 0.645853 15 16 .791895 .773328 .753933 .733738 .712829 .691223 .668967 .646101 16 17 .793785 .774771 .754994 .734488 .713334 .691545 .669162 .646203 17 h 18 .794996 .775654 .755612 .734900 .713594 .691701 .669242 .646243 18 19 20 21 22 23 24 25 26 .795738 0.796170 .796408 .796529 .796589 .796613 0.796623 .776169 0.776452 .776596 .776668 .776696 .776708 0.776711 .755952 0.756126 .756211 .756245 .735112 0.735215 .735256 .735273 .713719 0.713770 .713791 .713797 .691764 0.691790 .691797 .669274 0.669283 .646265 19 74 73 72 71 70 69 68 26 27 1.249527 .249527 .249527 .249527 .249526 1.249524 .249519 28 .756259 .735279 29 1.246580 .246580 .246580 .246580 1.246579 .246577 .756264 30 1.243489 .243489 .243489 1.243489 .243488 31 1.240250 .240249 1.240249 .240249 .796626 1.236862 1.236861 .236861 oo 33 Oti 1.233310 1.229589 .233309 67 1.225692! .229589 .233309 .236861 .240248 .243486 .246572 .249514 67 66 .225692; .229589 .233309 .236860 .240246 .243481 .246566 .249502 66 65 .225692 .229588 .233308 .236858 .240241 .243474 .246554 .249485 65 64 1.225691 1.229587 1.233306 1.236853 1.240233 1.243462 1.246536 1.249459 64 63 .225690 .229585 .233300 .236845 .240221 .243443 .246508 .249418 63 62 .225688 .229579 .233292 .236832 .240201 .243413 .246465 .249363 62 61 .225682 .229570 .233278 .236810 .240169 .243368 .246406 .249285 61 60 .225672 .229556 .233255 .236776 .240121 .243305 .246323 .249178 60 59 1.225657 1.229531 1.233218 1.236725 1.240054 1.243217 1.246211 1.249039 59 58 .225631 .229492 .233164 .236654 .239961 .243098 .246063 .248863 58 57 .225589 .229434 .233089 .236556 .239835 .242942 .245877 .248641 57 56 .225528 .229354 .232985 .236422 .239669 .242745 .245641 .248361 56 55 .225442 .229244 .232843 .236246 .239460 .242494 .245345 .248022 55 54 1.225325 1.229093 1.232656 1.236024 1.239194 1.242181 1.244985 1.247613 54 53 .225165 .228895 .232420 .235741 .238862 .241799 .244552 .247121 53 52 .224955 .228644 .232119 .235389 .238456 .241339 .244030 .246546 52 51 .224688 .228325 .231746 .234958 .237969 .240786 .243420 .245868 51 50 .224349 .227928 .231289 .234441 .237382 .240139 .242702 .245088 50 49 1.223928 1.227443 1.23074C 1.233819 1.336695 1.239378 1.241875 1.244192 49 48 .223412 .226859 .23007S .233089 .235888 .238500 .240924 .243168 48 47 .222790 .226157 .22930S .232233 .234956 .237491 .239839 .242011 47 46 .222045 .225332 .22839^ t .231243 .233886 .236340 .238612 .240712 46 45 .22116^ 33 n .224366 .22734^ ; .230107 .^32664 : .235038 .237234 .239258 45 1 32 31 30 1 29 28 27 26 Continued from Page *63. ^70 TABLE XLV. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OP « ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 4|^ PER CENT. .Pay- ments. n. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 2^=83 83 0.622676 0.598565 84 85 0.573780 0.548297 0.000000 0.000000 0.000000 0.000000 2 .256494 3 i .387854 4 .468093 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 .2533161 .249806 .3815841 .374746 .458911 .448980 .246024 .367409 .438319 0.520618 0.508753i0.495976 0.482365 .556023 .580075 .596346 .607023 .541721 .563652 .577893 .586809 0.613752 0.592192 .617831! .595305 .6201961 .597009 .621492; .597883 .6221571 .598301 0.622476 .622609 .622661 .622676 25 75 74 73 72 71 70 69 1.252334 68 .252329 1.252341 1.252342 .252342 .252342 .252341 .252339 .526442 .545916 .557991 .565240 .509831 .526616 .536606 .542334 0.569416 0.545455 .571699 .572869 .573427 .573662 0.598476 0.573752 .598544 .573780 .598565! 24 28 .252318 .252302 .252277 1.252239 252187 252113 252012 1.255025 .255025 1.255025 .255025 .255024 .255022 .255017 1.255013 .255002 .254987 .254964 .254928 1.254879 .254809 .254714 .254590 .251881 .254433 1.251715 .251505 .251241 .250921 .250535 1.250071 .249528 .248888 .248152 .247307 1.246341 .245249 .244024 .242653 .241122 1.257585 .257585 .257585 1.257585 .257584 .257582 .257578 .257574 .547053 .547814 .548135 .548258 0.548297 22 1.260026 .260027 .260027 .260027 1.260026 .260024 .260020 .260016 .260006 1.257563 1.259993 .257549 .259972 .257527' .259940 .257493 .259896 .257447 .259833 1.257381 1.259749 86 0.521852 I- 87 0.493999 88 0.464345 89 0.431342 0. 000000 0. 000000:0. 000000 .241946: .359446 .426874 .237388 .350786 .413580 .232632 .340091 .397349 Pay- ments, n. 0.000000 .22451 .324725 .376450 0.467218 0.449629 0.429089 0.404031 .491308' .470362! .446604 .418493 .505459 .513513 .517883 .482025 .488313 .491513 .455950 .460680 .462925 0.520115 0.493035 0.463870 .521178 .521627, .521798! .521852! .4936771 .464230 .493921 .464345 .493999 21 20 1.264573 1.262353 .262354 .262354 .262354 .262353 19 1.266691 .266692 1.2645741.266692 .264574| .266692 .264574! .266691 .264573! .266689 .264571: .266686 .257291 .257174 .257026 .259638 .259498 .259322 1.254235 .253986 .253683 .253319 .252881 1.252368 .251764 .251068 .250271 .249360 1.248329 .247174 .245880 .244437 242847 1.262351 1.264567 1.266682 262347! .264563: .266674 262343 .264555 .266663 .262334 .262322 .264543; .266646 .264525! .266618 .425749 .429178 .430618 0.431167 431342 18 1.262302 1.2644961.266580 .2622711 .264456' .266528 .262229| .264400 .266457 .262170| .264325' .266363 .262091! .264226 .266244 1.2619861.264100 .2618531 .263943 .261687! .263744 25 24 .256839 .259100 1.256604 1.258828 .256317 .258504! .255973 .258113: .255560 .257655 .255075 .257116| 1.254505 1.256496 .253848 .255785; .253095 .254972 .252235 .254054 .251262 .253023 1.250171 1.251871 .248950 .250586j .247589 .249170; .246089 .247.605! .244430 .245886j 23 22 .261477 .261220 .263501 .263212 1.260914 1.262863 1.264687 .260545 .260112 .259602 1.266095 .265907 .265677 .265404 .265074 259016 .260784 1.258345 .257577 .256709 .255736 .254648 .262454 .264231 .261972 .261418 .263707 .263108 .262422 1.2600581.261646 .2592381 .260776 .258317 .259805 .257290 .256143 .258722 .257529 1.2534341.254881 1.256211 .252097 .2534861 .254765 .2506201 .251955 .253180 .248998^ .250276: .251452 .247220 .248445! .249572 21 1.268708 .268709 .268709' 1.268709 .268708 .268706 .268703 .268699 1.268692 .2^8681 .268665 .268639 .268603 1.268554 .268487 .268398 .268285 .268144 1.267966 .267749 .267490 .267178 .266812 1.266381 .265885 .265319 .264670 .263936 1.263114 .262195 .261172 .260045 .258799 1.257433 .255935 .254304 .252529 .250601 20 19 18 Continued from Page *62. TABLE XLY. m LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEOmNING OF EACH YEAR, ^j ^^^ ^^^.j. Pay- ments. n. 90 91 92 93 94 95 96 97 0.177500 Pay. ' ments.j n. 1 1 0.399866 0.371143 0.342412 0.312760 0.282598 0.251018 0.213605 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .215615 .207859 .199751 .190399 .180240 .169812 .152954 .140751 2 3 4 5 6 7 8 9 10 .309111 .355648 0.379276 .390948 .396423 .398714 .399587 0.399866 .294974 .336370 0.356220 .365406 .369226 .370680 .371143 .279967 .316098 0.332378 .339071 .341606 .342412 .263229 .294182 0.306615 .311281 .312760 .245642 .270547 0.279714 .282598 .225430 .244967 0.251018 .199694 .213605 .177500 3 90 89 88 87 86 85 10 11 12 1.281791 1.281791 .281791 .281791 .281790 .281789 13 1.280417 .280417 .280417 .280417 .280416 14 1.278977 .278977 .278977 .278977 15 1.277464 .277464 .277464 16 1.275875 .275876 1.274211 17 1.272467 1.274213 1.275876 1.277464 1.278976 1.280415 1.281787 84 83 1.270632 .272468 .274213 .275876 .277463 .278975 .280413 ■ .281785 83 82 .270633 .272468 .274213 .275875 .277462 .278973 .280411 .281780 82 81 .270633 .272468 .274212 .275873 .277460 .278971 .280405 .281774 81 80 .270633 .272467 .274210 .275871 .277458 .278965 .280399 .281761 80 79 1.270632 1.272465 1.274208 1.275869 1.277452 1.278958 1.280385 1.281745 79 78 .270630 .272463 .274205 .275863 .277444 .278944 .280369 .281723 78 77 .270628 .272460 .274199 .275854 .277430 .278927 .280345 .281691 77 76 .270624 .272453 .274190 .275840 .277412 .278901 .280312 .281648 76 75 .270617 .272444 .274175 .275820 -.277384 .278866 .280267 .281591 75 74 1.270607 1.272428 1.274154 1.275791 1.277348 1.278819'l. 280206 1.281518 74 73 .270591 .272406 .274123 .275752 .277297 .2787541 .280128 .281426 73 72 .270567 .272373 .274081 .275699 .277229 .278672! .280032 .281310 72 71 .270533 .272329 .274025 .275627 .277142 .278571 .279908 .281170 71 70 .270486 .272270 .273950 .275535 .277036 .278440 .279762 .281003 70 69 1.270423 1.272190 1.273852 1.275423 1.276898 1.278286:1.279585 1.280801 69 68 .270339 .272087 .273734 .275278 .276735 .278099! .279373 .280563 68 67 .270231 .271962 .273581 .275106 .276537 .277875 .279122 .280287 67 66 .270098 .271801 .273399 .274897 .276301 .277610 .278829 .279966 66 65 .269929 .271608 .273179 .274647 .276022 .277301 .278491 .279599 65 84 1.269724 1.271376 1.272915 1.274352 1.275695 1.276945 1.278105 1.279181 64 63. .269479 .271097 .272604 .274007 .275319 .276536 .277662 .278707 63 62 .269184 .270769 .272239 .273610 .274888 .276069 .277162 .278178 62 61 .268838 .270383 .271819 .273156 .274394 .275542 .276604 .277587 61 60 .268430 .269939 .271340 .272634 .273838 .274952 .275980 .276928 60 59 1.267961 1.269433 1.270789 1.272046 1.273215 1.274293 1.275284 1.276203 59 58 .267426 .268851 .270167 .271388 .272519 .273558 .274519 .275403 58 57 .266811 .268194 .269472 .270652 .271742 .272750 .273674 .274526 57 56 .266117 .267459 .268694 .269832 .270889 .271857 .272748 .273566 56 55 .265340 .266637 .267828 .268931 .269945 .270880 .271734 .272521 55 54 1.264471 1.265722 1.266876 1.267934 1.268913 1.269809 1.270629 1.271387 54 53 .263503 .264715 .265822 .266843 .267781 .268641 .269432 .270157 53 52 .262438 .263601 .264668 .265646 .266547 .267376 .268133 .268827 52 51 .261260 .262380 .263403 .26434.2 .265210 .266003 .266726 ;267391 51 50 .259969 .261043 .262024 .262928 .263758 .264516 .265208 .265845 50 49 1.258554 1.259585 1.260528 1.261392 1.262186 1.262911 1.263574 1.264181 49 48 .257012 .258002 .258903 .259730 .260489 .261183 .261815 .262395 48 47 .255336 .256282 .257145 .257934 .258660 .259322 .259924 .260481 47 46 .253516 .254422 .255244 .255999 .256691 .257322 .257901 .258429 46 45 .251,546 .252409 .253196 .253915 .254575 .255181 .255729 .256233 45 17 16 16 14 13 12 11 10 Continued from Page *61. *72 TABLE XLTI LIFE AND TEMPORARY ANI^UITIES. LOGARITHJtf OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. The first Value under the Age X is A(l + Ox) for the whole Life. 5 PER CENT. Pay- niATltlS a;=10 11 12 13 14 15 16 17 Pay- ments. UIVUUo* n. 1 1.249190 1.248018 1.2467871.2454871 1.244118 1.242680 1.241167 1.239571 7?. 1 0.000000 0.000000 0. 000000 0. 000000 0. 000000 0.000000 0.000000 0.000000 2 .289190 .289185 .2891821 .289178l| .289171 .289165 .289161 .289154 2 3 .453548 .453539 .4535341 .453523 .453510 .453500 .453490 .453475 3 4 .566862 .566850 .566838 .566823 .566805 .566789 .566773 .566751 4 5 0.652256 0.652239 0.652223,0.652202 0.652179 0.652157 0.652133 0.652104 5 6 .720028 .720006 .719985 .719959! .719930 .719901 .719871 .719834 6 7 .775673 .775646 .775620^ .775588 .775553 .775518 .775480 .775435 7 8 .822470 .822438 .822407' .822370 .822328 .822285 .822240 .822186 8 9 .862534 .862497 .862461 .862417! .862368 .862319 .862265 .862201 9 10 0.897309 0.897268 0.897225;0.897175 0.897120 0.897062 0.896999 0.896928 10 11 .927826 .927779 .9277301 .927674i .927611 .927544 .927474 .927390 11 12 .954845 .954791 .954737 .954673 .954601 .954528 .954446 .954353 12 13 .978941 .978883 .978821 .978749 .978671 .978587 .978496 .978394 13 14 1.000565 1.000500 1.000431 1.000353 1.000264 1.000172 1.000073 .999957 14 15 1.020071 1.019999 1. 019924 1.019835 1.019740 1.019639 1.019527 1.019401 15 16 .037746 .037668 .037582 .037488 .037384 .037271 .037149 .037010 16 17 .053824 .053737 .0536461 .053543 .053427 .053305 .053171 .053022 17 18 .068498 .068405 .0683061 .068192 .068067 .067933 .067790 .067625 18 19 .081933 .081831 .081722 .081600 .081463 .081322 .081163 .080983 19 20 1.094262 1.094152 1.094035 1.093901 1.093758 1.093601 1.093428 1.093237 20 21 .105604 .105486 .105358 .105218 .105061 .104890 .104707 .104497 21 22 .116060 .115932 .115798 .115644 .115474 .115293 .115092 .114870 22 23 .125716 .125581 .125434 .125268 .125088 .124891 .124678 .124439 23- 24 .134649 .134503 .134344 .134169 .133974 .133765 .133537 .133281 24 25 1.142924 1.142766 1.142598 1.142409 1.142202 1.141980 1.141734 1.141460 25 26 .150597 .150431 .150250 .150049 .149830 .149590 .149329 .149037 26 27 .157725 .157546 .157355 .157141 .156905 .156651 .156367 .156061 27 28 .164349 .164160 .163956 .163728 .163478 .163203 .162911 .162579 28 29 .170513 .170312 .170095 .169853 .169584 .169300 .168983 .168631 29 30 1.176252 1.176039 1.1758081.175549 1.175270 1.174964 1.174628 1.174257 30 31 .181599 .181373 .181127 .180859. .180559 .180234 .179880 .179483 31 32 .186585 .186344 .186089 .185800; .185483 .185141 .184762 .184343 32 33 .191235 .190985 .190710 .190405 .190072 .189706 .189307 .188867 33 34 .195578 .195310 .195019 .194699 .194344 .193958 .193539 .193069 34 35 1.1996281.199345 1.199041 1.198700 1.198325 1.197921 1.197473 1.196983 35 36 .203409 .203114 .202790 .202430; .202038 .201606 .201140 .200619 36 37 .206942 .206629 .206287 .205911 .205493 .205044 .204548 .204001 37 38 .210240 .209909 .209552 .209152 .208717 .208240 .207720 .207147 38 39 .213319 .212974 .212594 .212177 .211716 .211217 .210671 .210067 39 40 1.216197 1.2158301.215434 1.214992 1.214511 1.213987 1.213412 1.212777 40 41 .218880' .218498 .218078 .217618 .217112 .216561 .215957 .215294 41 42 .221388 .220983 .220546 .220061; .219530 .218951 .218322 .217623 42 43 .223724 .223303 .222842 .222334 .221777 .221174 .220510 .219779 43 44 .225906 .225462 .224979 .224447 .223867 .223231 .222537 .221773 44 10 11 12 13 .14 15 16 17 Continued on Page 82. Formula, A,(l4-a!J-i).= k ?£:z??£±«=A^£=i. * Da; vpx-t TABLE XLVl. ^73 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 5 PER CENT. Pay- 18 19 20 21 22 23 24 25 Pay- i ments. n. 1 1.223420 ments. 1.23789l!l.236125'l.234365 1 1 1.232309 1.230253 1.228089 1.225813 n. 1 0. OOOOOO'0. 000000 0. 000000 0. 000000 ''o. 000000 0.000000 0. 000000 0. 000000 2 .289146 .289139 .289130 .289120 .289110 .289099 .289085! .289072 2 3 .453460 .453445' .453426 .453406 .453386 .453362 .453336 .453307 3 4 .566728 .566704 .566675 .566645 .566613 .566577 .566536 .566492 4 5 0.652073 0.652039 0.652000 0.651959: 0.651916!o. 651865 0,651810 0.651750 5 6 .719794 .719751 .719702 .719650 .719592 .719528 .719458 .719380 6 7 .775386 .775333 .775274 .775207 .775137 .775059 .774972 .774877 7 8 .822127 .822065 .821991 .821914 .821831 .821736 .821633 .821520 8 9 .862134 .862058 .881975 .861885 .861786 .861677 .861555 .861424 9 10 0.896847'o. 896762 0.896667 0.896562 0.896449 0.896322 0.896184 0.896030 10 11 .927301^ .927205 .927095 .926976 .926847i .926704 .926545 .926371 11 12 .954254^ .954144 .954021 .953887 .953742 .953579 .953401 .953207 12 13 .978281^ .978159! .978021 .977873 .977709: .977528! .977330 .977111 13 14 .999833 .999697 .999545! .999379 j .999198: .998998 1 .998777 .998536 14 15 1.0192631.019114 1.0189461.018763 1.0185641.018342 1.018100 1.017833 15 16 .036861! .036695 .036511! .036311 i .036091! .035850 .035583 .035291 16 17 .052856' .052675 .052475 .052255] .052017j .051752 .051461 .051143 17 18 .087444' .067249 .067029 .066793 .0665321 .066244 .065929 .065581 18 19 .080790, .080575 .080340 .080082' .079800! .079489 .079145 .078770 19 20 1.093024 1.092796 1.0925401.092262 1.091957 1.091619 1.091250 1.090842 20 21 .104271 .104023 .103749! .1034481 .103119 .102756 .102356 .101916 21 22 .114626 .114360 .114064 .1137411 .113388 .112996 .112565 .112097 22 23 .124178 .123891 .123574 .123229 .122847 .122427 .121969 .121463 23 24 .133000 .132693 .132355 .131982 .131575 .131128 .130635 .130094 24 25 1.141161 1.140833 1.140469 1.140072 1.139639 1.139160 1.138632 1.138056 25 26 .148717 .148366 .147978 .147557 .147093 .148581 .146021 .145404 26 27 .155720 .155346 .154936 .1544841 .153990 .153447 .152848 .152194 27 28 .162216 .161821 .161382 .160902; .160378 .159798 .159164 .158466 28 29 .168248 .167825 .167359 .166852 .166293 .165680 .165004 .164266 29 30 1.173847 1.173399 1.172908 1.172366 1.171776 1.171123 1.170410 1.169626 30 31 .179049 .178578; .178053 .177483 .176855 .176166 .175412 .174584 31 32 .183888 .183384' .182833 .182226 .181564 .180837 .180039 .179163 32 33 .188380 .187852 .187266 .186628 .185929 .185161 .184317 .183395 33 34 .192561 .191999 .191383 .190710 .189972 .189161 .188273 .187300 34 35 1.196442 1.195852 1.195204 1.194493 1.193715 1.192862 1.191926 1.190902 35 36 .200051 .199431 .198747 .197998 .197180 .196282 .195298 .194222 36 37 .203405 .202751 .202030 .201244 .200384 .199439 .198406 .197278 37 38 .206518 .205829 .205074 .204247 .203342 .202352 .201270 .200085 38 39 .2094051 .208685 1 .207889 .207021 .206073 .205037 .203901 .202663 39 40 1.212086 1.211326 1.210492 1.209583 1.208592 1.207504 1.206318 1.205022 40 41 .214566 .213769 .212897 .211947 .210907 .209772 .208530 .207174 41 42 .216860 .216027 .215116 .214120 .213035 .211847 .210549 .209139 42 43 .218981 .218112 .217157 .216118 .214984 .213743 .212393 .210921 43 44 .220940 .220030 .219034 .217949 .216765 .215474 .214066 .212531 44 18 19 20 21 22 23 24 26 Continued on Page *81. 74 TABLE XLYl. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF fl ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 5 PER CENT. Pay- ments. n. a:=26 j 27 28 29 30 31 32 ! 33 Pay- ments. 1.2209041.218261 1.215481 1.212561 1.209497 1.206277|l.202896 1.199343 n. 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .289055 .289039 .289020 .288997 .288976! .288951 .288925 .288894 2 3 .453274 .453240 .453200 .453156; .453111! .453061 .453006 .452944 3 4 .566442 .566388 .566328 .566262 .566194: .566115 .566031 .565934 4 5 0.651681 0.651609 0.651528 0.651439 0.651343i0. 651237 0.651120 0.650988 5 6 .719293 .719201 .719098 .718982 .718861 .718725 .718574 .718408 6 7 .774771 .774658 .774530 .774390 .774241 .774072! .773889 .773684 7 8 .821394 .821258 .821107i .820940 .820759 .820559! .820340 .820097 8 9 .861276 .861117 .860941 .860744 .860534 .860300 .860044 .859760 9 10 0.8958610.895679 0.895474 0.895250 0.895007 0.894738 0.894444 0.894116 10 11 .926180 .925971 .925739 .925484 .925209 .924903, .924567 .924195 11 12 .952990i .952755 .952494: .952209 .951897 .951553: .951175 .950756 12 13 .976871 .976607 .976318 .975997 .975650 .9752661 .974843 .974377 13 14 .998267 .997977 .997654 .997301 .996916 .996488 .996021 .995504 14 15 1.017539 1.017217 1.0168631.016473 1.016046 1.0155771.015062 1.014493 15 16 .034968 .034616! .0342281 .033798' .033332 .032818 .032252 .031628 16 1 17 .050790; .050407 .049980 .049513. .049005 .048442 .047825 .047147 17 18 .065199 .064779 .064317 .063810 .063256 .062644 .061976 .061235 18 19 .078353 .077899 .077400 .076849 .076249 .0755891 .074861 .074063 19 1 20 1.090392 1.089904 1.089363 1.088769 1.088122 1.0874054.086624 1.085758 20 21 .101435 .100907 .100324 .099686' .098985 .098218: .097372 .096444! 21 22 .111577 .111009 .110385 .109695 .108947 .1081191 .107214 .106217 22 23 .120906 .120299 .119626 .118891 .118086 .11720l| .116231 .115166 23 24 .129499 .128846 .128131 .127342 .126483 .125537! .124501 .123361 24 25 1.137418 1.136724 1.135958 1.135118 1.13420lk.l33193'l.l32087 1.130873 25 26 .144727 .143985 .143170 .142275 .141299 .140225' .139049 .137758 26 27 .151471 .150684 .149816 .148866 .147827 .146687! .145438 .144066 27 28 .157701 .156862 .155944 .154934 .153833 .152623 .151297 .149846 28 29 .163452 .162567 .161591 .160522 .159355 .158072 .156672 .155136 29 30 1.168768 1.167827 1.166796 1.165665 1.164429 1.163076 1.161596 1.159973 30 31 .173673 .172681 .171591 .170394 .169091 .167664 .166101 .164394 31 32 .178204 .177155 .176003 .174743 .173370 .171864 .170221 .168423 32 33 .182382 .181275 .180063 .178737 .177290 .175708 .173979 .172090 33 34 .186233 .185069 .183794 .182398 .180879 .179216 .177402 .175422 34 35 1.189780 1.188557 1.187216 1.185752 l.i84157 1.182413 1.180513 1.178438 35 36 .193044 .191758 .190354 .188818 .187145 .185320 .183330 .181164 36 37 .196041 .194695 .193222 .191612 .189864 .187954 .185879 .183620 37 38 .198792 .197381 .195837 .194157 .192328 .190338 .188175 .185820 38 39 .201308 .199830 .198221 .196464 .194550 .192487 .190233 .187785 39 40 1.203603 1.2020641.200382 1.198545 1.196572 1.194413 1.192071 1.189534 40 41 .205698 .204089 .202332 .200439 .198375 .196133 .193708 .191077 41 42 .207597 .205918 .204107 .202129 .199986 .197667 .195152 .192435 42 43 .209312 .207583 .205692 .203639 .201424 .199019 .196424 .193621 43 44 .210875 .209070, .207108 .204987 .202691 .200211 .197534 .194646; 44 26 27 1 28 29 1 30 31 32 33 Continued on Page *80. TABLE XLYL ♦75 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 5 PER CENT. Pay- ments. n. 34 35 36 37 38 39 40 41 Pay- j ments. n. 1.195615 1.191703 1.187604 1.183299 1.178783 1.174051 1.169091 1.163896 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .288861 .288821 .288783 .288741 .288690 .288637 .288577 .288516 2 ! 3 .452874 .452797 .452719 .452629 .452528 .452418 .452299 .452168 3 ! 4 .565828 .565712 .565590 .565452 .565297 .565131 .564945 .564745 4 ! 5 0.650846 0.650688 0.650521 0.650332 0.650122 0.649894 0. 649642 0.649369 5 6 .718225 .718025 .717809 .717569 .717300 .717008 .716688 .716337 6 7 .773461 .773214 .772950 .772653 .772324 .771967 .771573 .771143! 7 8 .819830 .819536 .819219 .818865 .818474 .818047 .817578 .817064 8 9 .859447 .859104 .858733 .858320 .857862 .857365 .856815 .856216 9 [ 10 0.893757 0.893362 0.892935 0.892459 0.891934 0.891359 0.890729 0.890039! 10 11 .923787 .923339 .922852 .922312 .921715 .921064 .920348 .919565 11 12 .950298 .949793 .949245 .948637 .947967 .947235 .946431 .945549: 12 13 .973865 .973302 .972690 .972013 .971266 .970450 .969552 .9685711 13 14 .994937 .994313 .993636 .992885 .992059 .991155 .990162 .989074| 14 15 1.013868 1.013183 1.012434 1.011610 1.010699 1.009706 1.0086121.007416 15 16 .030946 .030192 .029375 .028469 .027475 .026385 .025190 .023879 16 17 .046399 .045581 .044686 .043701 .042616 .041430 .040125 .038701 17 18 .060427 .059535 .058565 .057495 .056318 .055028 .0536171 .0520721 18 19 .073184 .072221 .071171 .070014 .068738 .067349 .065823! .0C4153 19 20 1.084812 1.083773- 1.082642 1.081390 1.080021 1.078522 1.0768791.075080 20 21 .095425 .094310 .093088 .091749 .090276 .088667 .086900: .0849721 21 22 .105126 .103926 .102620 .101183 .099606 .097879 .095991, .093921 22 23 .113995 .112714 .111317 .109780 .108092 .106252 .104229 .102035 23 24 .122113 .120746 .119254 .117614 .115818 .113850 .111711 .109366 24 1 25 1.129544 1.128087 1.126498 1.124755 1.122839 1.120761 1.118479!l.ll5994 25 26 .136344 .134796 .133110 .131254 .129233 .127020 .124606 .121973 26 27 .142567 .140926 .139134 .137178 .135030 .132692 .130139 .127358 27 28 .148258 .146519 .144630 .142554 .140289 .137821 .135127 .132199! 28 | 29 .153456 .151623 .149623 .147437, .145049 .142448 .139616 .136537 29 30 1.158204 1.156267 1.154162 1.151860 1.149346 1.146614 1.143641 1.140412 30 31 .162527 .160491 .158276 .155855 1 .153218 .150354 .147238 .143859 31 32 .166461 .164322 .161995 .159458 1 .156696 .153697 .150441 .146913! 32 1 33 .170032 .167786 .165350 .162696 .159807 .156676 .153279 .149605 33 34 .173263 .170915 .168368 .165594 .162580 .159317 .155783 .151967 34 35 1.176182 1.173729 1.171070 1.168178 1.165041 1.161648 1.157980 1.154022 35 36 .178810 .176251 .173480 .170473 i .167212 .163694' .159893 .155800 36 37 .181165 .178500 .175622 .172497 1 .169119 .1654761 .1615481 .157326 37 38 .183266 .180501 .177511 .174276 1 .170782 .1670181 .162969! .158626 38 39 .185136 .182266 .179172 .175828 i .172219 .168342 .164179 .159718 39 40 1.186786 1.183818 1.180621 1.177169 1.173454 1.169470 1.165197 1.160629 40 41 . .188237 .185172 .181874 .178322 .174507 .170419 .166046 .161380 41 42 .189504 .186343 .182952 .179306 .175392 .171211 .166746 .161989 42 43 .190599 .187351 .183871 .180132 .176131 .171864 .167313 .162474 •43 44 .191542 .188210 .184643 .180822 .176741 .172393 .167765 .162861 44 34 35 36 37 38 39 40 41 Continued on Page *79. *76 TABLE XLTI. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 5 PER CENT Pay- x=^2 43 44 45 46 47 48 49 Pay. ments. n. 1 ments. n. 1.158448 1.152750 1.146778 1.140532 1.133997 1.127159 1.120012 1.112544 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0. 000000 1 2 .288442 .288366 .288279 .288184 .2880811 .287967 .287841 .287704 2 3 .452020 .451864 .451686 .451494 .451283 .451048 .450792 .450511 3 4 .564520 .564278 .564007 .563710 .563388 .563029 .562637 .562206 4 5 0.649063 0.648732 0.648362 0.647962 0.647519 0.647032 0.646498 0.645909 5 6 .715947 .715523 .715054 .714539: .713975 .713353 .712670 .711921 6 7 .770664 .770146 .769569 .7689391 .768247 .767485 .766649 .765730 7 8 .816495 .815875 .815189 .814438 .813614 .812705 .811709 .810616 8 9 .855551 .854828 .854026 .853150 .852187 .851127 .849966 .848693 9 10 0.889276!o. 888444 0.887524 0.886516 0.885410 0.884194 0.882862 0.881400 10 11 .9186981 .917753 .916709 .915566 .914311 .912933 .911424 .909766 11 12 .944576 .943514 .942341 .941057 .939649 .938105 .9364091 .934551 12 13 .9674861 .966301 .934996 .963567 .962002 .960281 .958396 .956332 13 14 .987874 .986564 .985121 .983544' 1 .981812 .979912 .977833 .975556 14 15 1.006096(1.004656 1.003073 1.001338; 0.999437 0. 997354|0. 995073 0. 992575 15 16 .022438; .020865 .019133 .01723811.015164 1.0128901.010401 1.007676 16 17 • .037133 .035420 .033537 .031478, .029223 .026753 .024051 .^21100 17 18 .050372 .048515 .046478 .044248 .041809 .039138 .036222 .033036 18 19 .062316 .060315 .058107 .055713' 1 .053084 .050211 .047074 .043650 19 20 1.0731071.070944 1.068590 1.0660071 1.063188 1.06010711.056746 1.053080 20 21 .082848 .080546 .078013 .075253! .072237 .0689451 .0653571 .061448 21 22 .091665 .089195 .086495 .083548' .080334 .076828; .073011 .068857 22 23 .099320 .096992 .094116 .090982' .087567 .083846! .079799^ .075400 23 24 .106802| .104008 .100956 .097633 .094016 .090079 .085800 .081156 24 25 1.1132731.110314 1.107083 1.103570 1.099750 1.09559511.091087 1.086204 25 26 .119096' .115969 .112559 .108856 .104830 .100460 .095728 .090606 26 27 .124324; .121028 .117440 .113542 .109315 .104734 .099779 .094424 27 28 .129005 .125541 .121770 .117684 .113258 .108468 .103294 .097711 28 29 .133185 .129549 .125600 .121328 .116706 .111711 .106323 .100525 29 30 1.136899:1.133093 1.128973 1.124515 1.119702 1.114507 1.108917 1.102914 30 31 .140188 .136221 .131924 .127287 .122286 .116901 .111121 .104919 31 32 .143087! : 138957 .134492 .129679 .124500 .118938 .112971 .106586 32 33 .145628; .141339 .136709 .131730 .126384 .120647 .114510 .107957 33 34 .147839 .143397 .138610 .133475 .127966 .122070 .115776 .109070 34 35 1.149751 1.145161 1.140229 1.134940 1.129283 1.123240 1.116804 1.109957 35 36 .151390 .146665 .141588 .136161 .130366 .124191 .117623 .110657 36 37 .152788 .147926 .142721 .137165 .131247 .124949 .118270 .111196 37 38 .153981 .148979 .143653 .137982 .131949 .125547 .118768 .111605 38 39 .154940 .149845 .144411 .138632 .132503 126007 .119145 .111910 39 40 1.155746 1.150550 1.145014 1.139147 1.132929 1.126356 1.119427 1.112128 40 41 .156401 .151110 .145493 .139542;' .133252 .126617 .119628 .112283 41 42 .1569221 .151555 .145859 .139841} .133494 .126803 .119772 .112388 42 43 .157337 .151895 .146137 .140065] .133666 .126937 .119869 .112455 43 44 1 .157653 1 .152154 .146345 .140225 1 .133790 .127026 .119931 .112496 44 i 42 43 44 45 1 46 47 48 49 Continued on Page *78. TABLE XLYI. *n LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. , „„^ __^ d PER CENT. Pay. 50 51 52 53 64 66 56 57 Pay- 1 ments. n. 1 ments. n. 1.104739 1.096590 1.088084 1.079208 1.069947 1.060290 1.050232 0.000000 1.039753 O.OOOOOOi 0.000000 0.000000 0.000000 0.000000 0. 000000 0.000000 1 2 .287551 .287384 .287201 .287001 .286782 .286535 .286270 .285973 2 3 .450200 .449859 .449486 .449078 .448627 .448128 .447584 .446980 3 4 .561729 .561209 .560639 .560012 .559322 .558559 .557725 .556805 4 5 0.645263 0.644555 0.643778 0.642925 0. 641986 0. 640948 0.639816 0.638570 5 6 .711096 .710193 .709202 .708114 .706914 .705595 .704155 .702568 6 7 .764721 .763615 .762403 .761068 .759603 .757993 .756230 .754294 7 8 .809416 .808101 .806655 .805069 .803329 .801415 .799323 .797025 8 9 .847294 .845758 .844074 .842230 .840S03 .837978 .835545 .832878 9 10 0.879793 0.878033 0.876106 0.873990 0.871670 0.869124 0.866344 0.863295 10 11 .907947 .905956 .903773 .901381 .898758 .895885 .892746 .889308 11 12 .932514 .930282 .927838 .925160 .922230 .919019 .915515 .911680 12 13 .954068 .951590 .948876 .945910 .942661 .939107 .935229 .930992 13 14 .973058 .970325 .967340 .964074 .960503 .956598 .952343 .947700 14 15 0.'989837 0.986849 0.983583 0.980013' 0.976113 0.971854 0.967218 0.962164 15 16 1.004699 1.001446 .997893 .994014 .989782 .985165 .980146 .974678 16 17 .017873 .014351 1.010508 1.006317 1.001749 .996773 .991366 .985490 17 18 .029556 .025761 .021625 .017120 .012215 1.006876 1.001088 .994807 18 19 .039913 .035843 .031411 .026590 .021345 .015651 .009484 1.002802 19 20 1.049085 1.044737 1.040009 1.034869; 1.029293 1.023244 1.016704 1.009629 20 21 .057192 .052566 .047540 .042090 .036181 .029785 .022878 .015427 21 22 .064340 .059435 .054118 .048357 .042125 .035386 .028129 .020313 22 23 .070621 .065443 .059836 .053773 .047220 .040155 .032560 .024399 23 24 .076122 .070673 .064783 .058420 .051563 .044185 .036269 .027781 24 25 1.080915 1.075201 1.069031 1.062385 1. 055238!l.047560ll. 039340 1.030548 25 26 .085069 .079093 .072659 .065743 .058316 .0503571 .041855 .032789 26 27 .088643 .082420 .075734 .068558 .060869 .052648, .043894 .034575 27 28 .091700 .085242 .078313 .070893 .062962' .054507, .045519 .035978 28 29 .094294 .087610 .080454 .072809 .064660; .055988 .046795 1 .03^059 29 30 1.096471 1.089575 1.082211 1.074363 1. 06601311. 057153|1.047779 1.037879 30 31 .098279 .091190 .083637 .075602 .067078 .058051 .048525 .038487 31 32 .099766 .092501 .084774 .076578 .067900 .058731 .049080 .038924 32 33 .100973 .093546 .085670 .077331 .068522 .059238 .049477 .039232 33 34 .101935 .094370 .086360 .077901 .068985, .059600 .049758 .039441 34 35 1.102694 1.095004 1.0868841.078325 1.0693161.059857 1.049948 1.039576 35 36 .103278 .095486 .087273 .078629 .069552 .060030 .050072 .039657 36 37 .103722 .095844 .087553 .078845 .069710 .060144 .050145 .039703 37 38 .104052 .096101 .087751 .078990 .069814 .060210 .050187 .039730 38 39 .104288 .096284 .087884 .079085 .069875 .060248 .050212 j .039744 39 40 1.104457 1.096406 1.0870711.079141 1.069910 1.0602711.050225 1.039750 40 41 .104570 .096486 .088023, .079173 .069931 .060283' .050230 .039753 41 42 .104643 .090534 .088052 .079193 .069942 .060287 .050233 .039753 42 43 .104687 .096561 .088071 .079203 .069945 .060290 .050233 .039753 43 44 .104712 .096578 .088080 .079205 .069948 .060290 .050230 45 1.104728 1.096586 1.088082 1.079208 1.069948 1.060290 46 .104735 .096588 .088083 .079208 47 .104737 .096590 .088084 48 .104739 .096590 49 .104739 60 51 52 53 64 55 66 67 '78 TABLi: XLYI. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF ANNUAL PAYMENTS OP 1 AT THE BEGINNING OF EACH YEAR. ~ O PER CENT. ^»y- a;=58 r 59 ments. . n. '1.0288511.017506 36 37 38 39 40 41 42 51 50 49 48 47 46 45 1 iO. 000000 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.956668 16 .968745 17 .979124 18 ! .988009 19 .995584 .285651 .446325 .555804 0.637208 .700835 .752177 .794515 .829963 0.859966 .885556 .907499 .926377 .942646 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 1.028745 1.002005 .007410 .011925 .015658 .018711 1.021181 .023150 .024695 .025885 .026788 1.027457 .027939 .028277 .028508 .028656 .028796 .028825 .028841 .028847 1.028851 .028851 .028851 0.000000 .285299 .445605 .554702 0.635712 .698932 .749855 .791763 .826771 0.856322 .881455 .902934 .921344 .937140 0.950694 .962305 .972221 .980655 .987792 0.993790 .998794 1.002928 .006306 .009036 1.011212 .012919 .014233 .015230 .015968 1.016500 .016873 .017128 .017291 60 1.005705 0.000000 .284908 .444810 .553489 0.63400: .696841 61 0.993441 0.000000 .284479 .443937 .552159 0.632263 .694552 .747308 .744519 .788747 .823271 0.852337 876974 897952 915855 931150 0.944203 .955318 .964748 .972711 .979391 0.98495'6 ,989548 .993297 .996325 .998736 1.000627 .002083 .003186 .004003 .004592 1.005005 .005286 .005467 .005575 .017389 .005637 1.017445 .017477 .017494 .017501 .017505 1.017506 .017506 49 1.112544 .112544 1.112544 .112542 .112540 .112534 .112519 49 48 1.120012 .120013 .120013 1.120011 .120010 .120004 .119990 .119969 48 1.005673 .005691 .005699 .005703 .005704 1.005705 47 1.127159 .127160 .127160 .127158 1.127157 .127151 .127138 .127119 .127084 47 .785443 .819450 0.847985 .872086 .892522 .909888 .924644 0.937165 .947756 .956677 .964147 .970359 0.975478 .979653 .983022 .985703 .987804 0. 989422 .990646 .991553 .992207 .992665 0.992977 .993177 .993297 .993366 .993406 0. 993426 .993435 .993439 .993441 .993441 46 1.133997 .133997 .133997 .133995 .133994 1.133989 .133977 .133959 .133927 .133873 46 62 0.980708 63 0.967485 0.000000 .284011 .442984 .550702 0.630289 .692045 .741465 .781837 .815276 0.843237 .866758 .886618 .903405 .917588 0.929544 .939587 .947976 .954939 .960667 0.965334 .969096 .972087 .974429 .976232 0.977596 .978606 .979334 .979843 .980191 0.980413 .980547 .980624 .980669 .980691 0.9807010 .980706 .980708 .980708 0.000000 .283494 .441932 .549098 0.628118 .689292 .738121 .777885 .810711 0.838051 .860953 .880190 .896360 .909935 0.921299 .930766 .938605 .945043 .950280 0.954497 .957846 .960467 .962484 .964008 0.965136 .965950 .906519 .966908 .967156 45 1.140532 1.140532 .140532 .140530 .140529 .140525 1.140513 .140497 .140467 .140417 .140340 45 967305 967392 ,967442 967467 967477 967483 ,967485 ,967485 44 1.146778 .146779 1.146779 .146777 .146776 .146772 .146761 1.146746 .146718 .146672 .146600 .146493 44 64 65 0.95377l!0.939554 0.000000 .282926 .440776 .547340 0.625740 .686282 .734461 .773570 .805730 0.000000 .282300 .439509 .545411 0.623135 .682984 .730461 .768856 .800300 0.832404 0.826254 Pay- ments. n. .854637 .873209 .888724 .901658 0.912399 .921268 .928538 .934441 .939187 0.942953 .945897 .948161 .949872 .951137 0.952050 952688 953124 953402 953569 ,953667 .953722 .953750 .953762 .953768 .953770 .953771 43 1.152750 .152750 .152750 1.152748 .152747 .152744 .152733 .152719 1.152693 .152651 .152584 .152484 .152347 43 .847771 .865635 .880456 .892715 0.902800 .911058 .917744 .923111 .927303 0.930685 .933238 .935165 .936589 .937618 0.938335 .938827 .939139 .939328 .939438 939499 939531 939544 939551 939553 939554 35 42 1.158448 .158448 .158448 .158440 1.158445 .158443 .158432 ,158419 .158395 50 1.158356 .158294 .158201 .158073 .157894 42 Continued from Page *76, TABLE XLYL *79 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. VALUE OFn 5 PER CENT. 25 26 27 28 29 30 31 32 33 34 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 Pay. 66 ments. n. 1 0.924834 0.000000 2 .381620 3 .438123 4 .543304 5 0.620287 6 .679383 7 .726095 8 .763720 9 .794390 10 0.819572 11 .840323 12 .857436 13 .871525 14 .883078 15 0.892496 16 .900110 17 .906210 18 .911035 19 .914800 20 0.917691 21 .919872 22 .921482 23 .922646 24 .923456 67 0.909590 0.000000 .280864 .436599 .540986 0.617164 .675438 .721322 .758113 .787949 0.812303 .832240 .848558 .861879 .872697 0.881419 888390 893894 898184 901474 0.924013 924365 924579 924703 924772 0.924808 924823 924831 924833 924834 41 0.903953 .905783 .907105 .908025 .908657 0.909057 .909300 .909441 .909519 .909560 0.909577 .909586 .909588 .909590 68 0.893832 69 0.877530 0.000000 0.000000 .280048 .279136 .434934 .433097 .538460 .535675 0.613755 0.610013 .671139 .666429 .716125 .710441 .752015 .745362 .780955 .773345 0.804427 0.795875 .823499 .814035 .838981 .828638 .851498 .840321 .861555 .849600 1.86957010.856894 .875886; .862558 .880800 .884565 .887398^ .889489 0. .890998 , .892047, . .892768; . .893224 . .8935010. .893663| . .893752; , .893798 , .893818 , .866890 .870146 .872547 874279 875483 876311 876833 .877151 877337 877438 ,877491 ,877514 ,877525 40 1.163896 .163896 .163896 .163894 .163893 1.163891 .163881 .163869 .163846 .163810 1.163752 .163666 .163546 .163379 .163155 41 1.169091 1.169090 .169090 .169089 .169088 .169086 1.169076 .169065 .169044 .169010 .168956 1.168876 .168765 .168609 .168400 .168126 40 0.893828:0.877528 .8938301 .877530 .893832 39 1.174051 .174050 1.174050 .174049 .174048 .174046 .174037 1.174027 174007 173976 173926 173851 1.173747 .173602 .173406 .173151 .172814 38 39 1.178783 .178781 .178781 1.178780 .178780 .178778 .178769 .178760 1.178741 .178712 .178666 .178596 .178499 1.178363 .178180 .177942 .177628 .177235 70 0.860711 0.000000 .278148 .431094 .532643 0. 605938 .661302 .704264 .738146 .765103 0. 786638 .803836 .817523 .828349 .836833 0. 843404 .848421 .852187 .854962 .856960 0.858350 .859305 .859907 .860274 .860488 0.860605 .860666 .860693 .860705 .860709 0.860711 38 37 1.183299 .183298 .183298 .183297 1.183297 .183295 .183287 .183278 .183261 1.183234 .183191 .183125 .183035 .182907 1.182736 .182514 .182221 .181853 .181392 37 71 73 0.843350 0.825455 0.000000 0.000000 .277058 .428896 .529321 0.601478 .655705 .697538 .730302 .756171 0.776651 .792842 .805584 .815530 .823211 0.829062 .833448 .836674 .838996 .840610 0.841718 .842417 .842843 .843091 .843227 843297 843329 843343 843348 843350 36 1. 187604 .187603 .187603 .187612 .187602 1.187602 .187592 .187584 .187568 .187548 1. 187502 .187441 .187357 .187237 .187077 1.186869 .186596 .186252 .185820 .185288 36 73 0.807021 .275867 .426495 .525690 0.596614 .649613 .690228 .721801 .746513 0.765885 .781031 .792797 .801851 .808728 0.813873 .817651 .820367 .822254 .823550 0.824366 .824863 .825153 .825311 .825393 0.825431 .825447 .825453 .825455 0.000000 .274565 .423862 .521720 0.591309 .642980 .682291 .712593 .736083 0.754299 .768362 .779133 .787287 .793372 0.797832 .801033 .803255 .804781 .805741 0.806326 .806666 .806852 .806948 .806993 0.807012 .807019 .807021 35 1.191703 1.191701 .191701 .191701 .191701 .191700 1.191692 .191684 .191668 .191845 .191607 1.191550 .191471 .191359 .191210 .191015 1.190759 .190437 .190033 .189536 .188933 36 34 1.195615 .195614 1.195614 .195614 .195614 .195613 .195606 1.195598 .195583 .195562 .195526 .195472 1.195398 .195294 .195154 .194971 .194732 1.194437 .194052 .193587 .193023 .192346 34 Pay. ments. n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 5a 49 48 47 46 45 Continued from Page *75. ^80 TABLE XLYI. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 5 PER CENT, Pay- ments. n. 1 X = 74: 75 76 77 0.727935 78 79 80 81 Pay-| ments. 1 n. 1 0.788039 0.768544 0.748513 0. 706853 0.685247 0.663118 0.640468 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 O.OOOOOOiO. 000000 2 .273121 .271556 .269845 .267955 .265900 .263651 .261191 .258516 2 3 .420968 .417828 .414391 '.410619 .406519 .402044 .397173 .391869 3 4 .517374 .512654 .507501 .501868 .495755 .489111 .481890 .474070 4 5 0.585515 0.579229 0.572385 0.564931 0.556868 0.548124 0.53867l!o. 528466 5 6 .635757 .627935 .619444 .610236 .600297 .589573 .578019 .565609 6 7 .673671 .664359 .654291 .643406 .631708 .619132 .605652: .591253 7 8 .702624 .691893 .680323 .667874 .654542 .640282 .625077, .608915 8 9 .724834 .712758 .699796 .685902 .671094 .655336 .638617 .620953 9 10 0.741844 0.728525 0.714284 0.699094 0.682982 0.665917 0.647917 0.628878 10 11 .754796 .740342 .724961 .708631 .691389 .673227 .654072 .633875 11 12 .764558 .749097 .732717 .715406 \697167 .678084 .657966 .636899 12 13 .771820 .755481 .738246 .720077 .701106 .681163 .660327 .638648 13 14 .777130 .780045 .742067 .723269 .703574 .683032 .661695 .639602 14 15 0.780935 0.763204 0.744683 0.725271 0.705074 0.684118 0.662441 0.640089 15 16 .783573 .765370 .746325 .726489 .705946 .684709 .662822 .640322 16 i 17 .785383 .766731 .7473251 .727198 .706421 .685012 .663004 .640418 17 18 .786522 .767560 .747907 .727584 .706665 .685156 .663079 .640456 18 19 20 21 22 23 24 25 26 .787315 0.787618 .787839 .787952 .788005 .788028 0.788036 .788039 .768042 0.768306 .768441 .768504 .768531 .768541 0.768544 .748225 0.748387 .748464 .748496 .748508 .748513 .727782 0.727876 .727915 .727930 .727935 .706780 0.706828 .706847 .706853 .685215 0.685239 .685247 .663109 0.663118 .640468 19 74 73 72 71 70 89 26 27 1.220904 .220905 .220905 .220905 .220905 1.220904 28 29 1.218261 .218262 30 1.215481 31 1.212561 .212562 1.212562 .215483, .218262 .215483, .218262 1.215483ll. 218262 1.209497 33 32 1.206277 1.209498 1.202896 .206277 .209498 212562 2154831 -2l85»fii .220899 68 67 1.199343 .202895 .206277 .209498' .212562 .215482 .218256 .220893 67 66 .199342 .202895 .206277 .209498 .212561 .215476 .218250 .220885 66 65 .199342 .202895 .206277 .209497 .212555 .215470 .218241 .220873 65 64 1.199342 1.202895 1.206276 1. 209491 j 1.212548 1.215460 1.218228 1.220851 64 63 .199342 .202894 .206270 .2094841 .212538 .215446 .218205 .220817 63 62 .199341 .202888 .206263 .209473: .212523 .215422 .218169 .220772 62 61 .199335 .202881 .206251 .209457 .212497 .215384 .218121 .220711 61 60 .199327 .202868 .206234 .2094291 .212457 .215333 .218056 .220627 60 1 59 1.199313 1.202850 1.206204 1.209387 1.212403 1.215263 1.217967 1.220516 59 ! 58 .199294 .202818 .206159 .209329 .212328 .215169 .217849 .220374 58 ! 57 .199260 .202770 .206098 .209249 .212228 .215043 .217697 .220191 57 56 .199209 .202705 .206012 .209142; .212094 .214881 .217503 .219965 56 55 .199140 .202614 .205898 .208999 .211922 .214675 .217262 .219685 55 54 1.199043 1.202491 1.205746 1.208816 1.211702 1.214419 1.216964 1.219347 54 53 .198911 .202330 .205551 .208583 .211429 .214101 .216605 .218940 53 52 .198740 .202121 .205302 .208291' .211091 .213719 .216172 .218461 52 51 .198516 .201856 .204991 .207931 .210683 .213258 .215661 .217895 51 50 .198234 .201524 .204607 .207496 .210193 .212713 .215059 .217238 50 49 1.197879 1.201115 1.204143 1.206974 1.209612 1.212073 1.214360 1.216481 49 48 .197443 .200620 .203587 .206355 .208930 .211328 .213554 .215615 48 47 .196915 .200027 .202927 .205628 .208137 .210470 .212631 .214632 47 46 .196282 .199322 .202152 .204783; .207223 .209487 .211585 .213518 46 45 .195529 .198495 .201251 .203808 .206175 .208372 .210398 .212271 45 33 32 31 30 29 28 27 26 Continued from Page *74, TABLE XLVI. *81 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 5 PER CENT. Pay- ments. n. i 2 3 4 82 83 84 85 86 87 88 89 Pay- ] ments. n. 1 2 3 4 0.617258 0.593510 0.569083 0.543949 0.517851 0.490338 0.461018 0.428340 0.000000 .255571 .386091 .465583 0.000000 .252400 .379847 .456446 0.000000 .248902 .373034 .446561 0.000000 .245128 .365722 .435952 0.000000 .241062 .357791 .424566 0.000000 .236517 .349165 .411342 0. 000000 .231773 .338513 .395201 0.000000 .223681 .323210 .374409 5 6 7 8 9 0.517453 .552297 .575884 .591780 .602169 0.505659 .538098 .559600 .573508 .582179 0.492958 .522932 .542018 .553805 .560851 0.479431 .506445 .522889 .532637 .538204 0.464382 .488069 .501928 .509785 .514027 0.446913 .467291 .478711 .484838 .487942 0.426510 .443722 .452865 .457472 .459663 0.401609 .415812 .422907 .426267 .427642 5 6 7 8 9 10 11 12 13 14 15 16 17 18 75 0.608689 .612625 .614896 .616135 .616767 0.617070 .617194 .617243 .617258 0.587392 .590394 .592029 .592863 .593262 0.593427 .593491 .593510 0.564895 .567093 .568214 .568750 .568971 0.569057 .569083 0.541222 .542759 .543493 .543796 .543913 0.543949 0.516183 .517212 .517636 .517799 .517851 0.489422 .490030 .490264 .490338 0.460562 .460908 .461018 0.428172 .428340 10 11 82 81 80 79 78 77 76 75 18 19 20 1.237891 .237893 .237893 1.237893 .237893 .237892 .237889 .237886 21 1.236125 .236127 1.236127 .236127 .236127 .236126 .236123 1.234265 1.234266 .234266 .234266 .234266 .234265 22 1.232309 .232310 .232310 .232310 .232310 23 24 1.230253 .230254 .230254 .230254 26 1.223420 1.228089 .228090 .228090 1.225813 .225814 74 73 72 71 70 1.223422 .223422 .223422 .223422 .223421 1.225814 .225814 .225814 .225813 .225809 1.228090 .228090 .228089 .228086 .228081 1.230254 .230253 .230250 .230246 .230238 1.232309 .232306 .232303 .232295 .232288 1.234262 .234259 .234251 .234245 .234229 1.236120 .236112 .236107 .236091 .236068 1.237879 .237874 .237859 .237837 .237810 74 73 72 71 70 69 68 67 66 65 1.223416 .223411 .223403 .223392 .223371 1.225804 .225796 .225786 .225766 .225736 1.228073 .228064 .228045 .228016 .227979 1.230230 .230212 .230185 .230150 .230102 1.232271 .232245 .232212 .232167 .232105 1.234204 .234173 .234131 .234073 .233995 1.236039 .235999 .235945 .235871 .235778 1.237772 .237721 .237652 .237564 .237453 69 68 67 66 65 64 63 62 61 60 1.223339 .223297 .223240 .223161 .223056 1.225697 .225643 .225569 .225470 .225345 1.227928 .227859 .227766 .227648 .227497 1.230037 .229950 .229838 .229696 .229517 1.232023 .231918 .231785 .231616 .231410 1.233896 .233771 .233612 .233418 .233184 1.235660 .235510 .235328 .235108 .234842 1.237312 .237140 .236933 .236683 .236388 64 63 62 61 60 59 58 57 56 55 1.222923 .222751 .222538 .222275 .221957 1.225184 .224982 .224735 .224436 .224077 1.227307 .227074 .226794 .226455 .226058 1.229298 .229034 .228715 .228342 .227899 1.231161 .230862 .230510 .230093 .229609 1.232902 .232571 .232178 .231723 .231199 1.234530 .234160 .233732 .233238 .232674 1.236040 .235636 .235172 .234641 .234037 59 58 57 56 55 54 53 52 51 50 1.221575 .221125 .220593 .219975 .219263 1.223654 .223154 .222573 .221904 .221140 1.225588 .225041 .224412 .223694 .222877 1.227385 .226794 .226118 .225350 .224480 1.229053 .228417 .227694 .226877 .225962 1.230600 .229920 .229152 .228290 .227327 1.232034 .231311 .230500 .229593 .228581 1.233357 .232593 .231740 .230787 .229741 54 53 52 51 50 49 48 47 46 45 1.218450 .217526 .216480 .215309 .213998 1.220271 .219288 .218188 .216956 .215579 1.221953 .220919 .219761 .218467 .217045 1.223508 .222420 .221204 .219867 .218391 1.224938 .223795 .222538 .221151 .219623 1.226251 .225070 .223765 .222329 .220755 1.227470 .226242 .224891 .223412 .221794 1.228586 .227315 .225923 .224401 .222743 49 48 47 46 45 25 24 23 22 21 20 19 18 Continued from Page *73. *82 TABLE XLYL LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF » ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 5 PER CENT. Pay- ments. n. 1 ;r=90 91 92 93 1 94 95 96 97 Pay- ments. 0.397160 0.368711|0.340242 0.310847 0.280937 0.249611 0.212464 0.176638 n. 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 .214806 .207072 .198987 .189664 .179538 .169142 .152340 .140179 2 3 4 5 6 7 8 9 10 .307656 .353712 0.377006 .388464 .393850 .396046 .396892 0.397160 .293579 .334538 0.354101 .363173 .366849 .368262 .368711 .578635 .314378 0.330417 .336984 .339458 .340242 .261970 .292583 0.304829 .309403 .310847 .244460 .2^9086 0.278110 i .280937 .224344 .243655 0.249611 .198721 .212464 .176638 3 90 89 88 87 86 85 84 10 11 12 1.249190 1.249191 .249191 .249191 .249190 1 13 1.245487 .245488 .245488 1.24,'S488 1.248018 .248019 .248019 .248019 14 1.246787 .246788 .246788 .246788 1.246788 15 16 1.244118 .244119 1.244119 1.242680 1.242682 17 1.241169 .248019 1.248018 .249190 1.2491S8 83 1.239571 .241169 .242682 .244119' .245488 .246787 .248016 .249186 83 82 .239572 .241169 .242682 .244119 ; .245487 .246785 .248014 .249182 82 81 .239572 .241169 .242682 .244118 1 .245485 .246783 .248010 .249179! 81 1 80 .239572 .241169 .242681 .244116! .245482 .246778 .248007 .249170 80 79 1.239572 1.241168 1.242678 1.244113 1.245477 1.246775'l.247997 1.249156 79 78 .239571 .241165 .242675 .244107 .245474 .246764 .247982 .249139 78 77 .239568 .241162 ,242669 .244104 .245463 .246749 .247964 .249115 77 76 .239565 .241156 .242666 .244092 .245447 .246729 .247939 .249083 76 75 .239558 .241152 .242653 .244075 .245426 .246703 .247905 .249041 75 74 1.239554 1.241139 1.242635 1.244053 1.2453981.246667 1.247860 1.248986 74 73 .239540 .241119 .242612 .244023, .245360! .246620 .247802 .248917 73 72 .239519 .241095 .242580 .243983; .245310 .246558 .247730 .248827 72 71 .239494 .241061 .242537 .243930' .245244 .246481 .247633 .248721 71 70 .239458 .241016 .242481 .243860' .245162 .246379 .247521 .248591 70 69 1.239410 1.240955 1.242406 1.243773' 1.245055'l.246260'l.247383 1.248433 69 68 .239345 .240877 .242314 .243660 .244928 .246113 .247216 .248250 68 67 .239262 .240779 .242195 .243525, .244773! .245937 .247023 .248032 67 66 .239158 .240652 .242051 .243360i .2445861 .245732 .246791 .247781 66 65 .239024 .240500 .241877 .243163' .244368 .245486 .246525 .247491 65 64 1.238863 1.240315 1.241668 1.242932 1.244108 1.245204 1.246217 1.247160 64 63 .238667 .240094 .241422 .242656! .243809 .244878 .245866 .246782 63 62 .238432 .239832 .241130 .242338] .243463 .244506 .245466 .246359 62 61 .238154 .239523 .240792 .2419721 .243068 .244082 .245017 .245882 61 60 .237826 .239164 .240404 .241553 .242619 .243606 .244512 .245352 60 59 1.237445 1.2387521.239959 1.241076 1.242114 1.243070 1.243950 1.244761 59 58 .237008 .238281! .239454 . 240541 i .241546 .242474 .243322 .244109 58 57 .236508 .237745' .238886 .239938 .240914 .241808 .242632 .243391 57 56 .235939 .237142! .238247 .239268 .240207 .241076 .241871 .242604 56 55 .235299 .236464 .237535 .238518i .239431 .240269 .241035 .241741 55 54 1.234580 1.2357081.236739 1.237694' 1.238575 1.239383 1.240120 1.240799 54 53 .233776 .234863; .235865 .236786 .237604 .238411 .239120 .239775 53 52 .232879 .2339351 .234901 .235787 .236603 .237350 .238034 .238662 52 51 .231894 .232912 .233840 .234693 .235477 .236198 .236853 .237455 51 50 .230807 .231785| .232679 .233498 .234254 .234944 .235572 .236152 50 49 1.229610 1.2305521.231410 1.232199 1.232923 1.233585 1.234189 1.234744 49 48 .228301 .229205 .230030 .230785 .231479 .232116 .232694 .233225 48 47 .226869 .227738, .228528 .229252 .229920 .230528 .231081 .231588 47 46 .225310 .226142 .226899 .227595 .228232 .228815 .229342 .229827 46 45 .223614 .224410 .225138 .225801 .226412 .226967 .227472 .227936 45 17 16 16 14 13 12 11 10 Continued from Page *73. TABLE XLYII. ♦83 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OP THE PRESENT VALUE OP « ANNUAL PAYMENTS OF 1 AT THE BEGINNING OP EACH YEAR. The first Value under the Age X is A(l + aa,) for the whole Life. 6 PER CENT. Pay. ments. n. 1 a;=10 11 13 13 14 15 16 17 Pay- ments. 1.189964 1.189103|l. 188192 1.187336 1.186301 1.185130 1.183972 1.183755 m. 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .387194 .387188 .387185 .387181 .387174 .387169 .287165 .387158 2 3 .449596 .449587 .449581 .449571 .449559 .449549 .449538 .449534 3 4 .560996 .560984 .560972 .560957 .560939 .560933 .560907 .560886 4 5 0.644517 0.644499 0.644483 0. 644462 0. 644439 0.644419 0.644395 0.644366 5 6 .710457 .710435! .710413 .710387 .710360 .710332 .710301 .710364 6 7 .764310 .764383 .764356 .764337 .764193 .764157 .764119 .764074 7 8 .809356 .809333 .809394 .809357 .809316 .809174 .809138 .809075 8 9 .847709 .847674 .847637 .847594 .847546 .847497 .847444 .847383 9 10 0.880815 0.880773 0.880731 0.880683 0.880637 0.880573 0.880511 0.880439 10 11 .909701 .909654! .909606 .909550 .909490 .909436 .909355 .909277 11 12 .935138 .935076 .935031 .934960 .934891 .934819 .934743 .934650 12 13 .957674 .957615 .957555 .957486 .957409 .957331 .957341 .957143 13 14 .977784 .977731 .977654 .977577 .977495 .977404 .977308 .977197 14 15 0.995818 0.995747 0.995673 0.995591 0. 995496 0.995400 0.995393 0.995171 15 16 1.013057 1.0119811.0119011.011807 1.011708 1.011603 1.011483 1.011349 16 17 .036739 .036656 .036565 .036468 .036359 .026343 .036113 .035973 17 18 .040054 .039963 .039869 .039763 .039643 .039514 .039378 .039318 18 19 .053165 .053069 .051966 .051849 .051719 .051585 .051431 .051263 19 20 1.063311 1.063107 1.0639951.063867! 1.063733 1.062581 1.063430 1.063335 20 21 .073305 .073193 .073070 .073939! .072788 .073630 .073453 .072356 21 22 .083549 .083436 .083300 .083153 .081995 .081832 .081635 .081431 22 23 .091036 .090900 .090759 .090606 .090434 .090352 .090047 .089837 23 24 .098818 .098678 .098533 .098366 .098185 .097987 .097776 .097535 24 25 1.105983 1.105838 1.105679 1.105504 1.105308 1.105104 1.104873 1.104631 25 26 .113585 .113439 .113361 .113073 .111871 .111649 .111407 .111134 26 27 .118671 .118506 .118337 .118133 .117914 .117680 .117420 .117139 27 28 .134389 .134113 .133937 .133716 .133488 .123338 .133960 .133656 28 29 .139478 .139395 .139095 .138875 .138630 .128363 .128073 .137751 29 30 1.134377 1.134081 1.133873 1.133637 1.133376 1.133097 1.132791 1.133449 30 31 .138713 .138509 .138386 .138035 .137764 .137470 .137145 .136787 31 32 .143831 .143604 .143366 .143105 .141820 .141508 .141168 .140788 32 33 .146633 .146391 .146144 .145870- .145568 .145343 .144881 .144484 33 34 .150141 .149901 .149640 .149353 .149036 .148690 .148313 .147891 34 35 1.153403 1.153150 1.153877 1.153574' 1.153240 1.151880 1.151479 1.151038 35 36 .156434 .156160 .155873 .155554 .155306 .154833 .154405 .153943 36 37 .159336 .158948 .158645 .158313 .157944 .157545 .157106 .156631 37 38 .161831 .161539 .161313 .1608631 .160478 .160058 .159599 .159093 38 39 .164334 .163930 .163587 .163331 .163817 .163380 .161900 .161371 39 40 1.166453 1.166133 1.165785 1.165400 1.164980 1.1645331.164021 1.163468 40 41 .168514 .168180 .167815 .167415 .166977 .166498 .165975 .165396 41 42 .170433 .170073 .169694 .169377 .168818 .168319 .167771 .167170 42 43 .173188 .171835 .171430 .170994 .170515 .169994 .169435 .168798 43 44 .173833 .173444 .173033 .173576 .172077 14 .171537 .170943 .170390 44 10 11 13 13 15 16 17 Cont maed on Page 93. -v. ,^ n F( )EMULA, m+ar' )=A^ ~^x^n^ :A^^-». Da; Vpj^x *84 TABLE XLTII. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 6 PER CENT. Pay- a:=18 19 20 21 22 23 24 25 Pay- ments. n. 1 1.181467 1.180102 1.178657 1.177130 1.175513 1.173803 1.171993 1.170082 n. 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .287151 .287143 .287133 .287123 .287113 .287101 .287088 .287076 2 3 .449510 .449493 .449474 .449455 .449433 .449410 .449385 .449358 3 4 .560863 .560838 .560809 .560779 .560748 .560712 .560672 .560629 4 5 0.644335 0.644301 0.644262 0.644223 0.644179 0.644129 0.644075 0.644017 5 6 .710235 .710181 .710134 .710082 .710025 .709962 .709894 .709818 6 7 .764026 .763974 .763914 .763851 .763781 .763705 .763620 .763526 7 8 .809019 .808955 .808885 .808809 .808727 .808635 .808533 .808423 8 9 .847316 .847243 .847160 .847072 .846975 .846868 .846750 .846621 9 10 0.880364 0.880277 0.880184 0.880083 0.879971 0.879848 0.879713 0.879565 10 11 .909188 .909092 .908986 .908870 .908745 .908605 .908452 .908282 11 12 .934553 .934445 .934325 .934196 .934055 .933898 .933724 .933534 12 13 .957033 .956912 .956779 .956636 .956478 .956301 .956109 .955898 13 14 .977076 .976943 .976797 .976638 .976462 .976267 .976055 .975816 14 15 0.995038 0.994897 0.994732 0.994556 0.994362 0.994149 0.993910 0.993659 15 16 1.011210 1.011046 1.010870 1.010676 1.010465 1.010228 1.009977 1.009696 16 17 .025811 .025637 .025443 .025234 .025001 .024752 .024472 .024171 17 18 .039048 .038857 .038649 .038419 .038174 .037898 .037600 .037264 18 19 .051076 .050872 .050644 .050403 .050133 .049839 .049508 .049149 19 20 1.062036 1.061813 1.061575 1.061310 1.061023 1.060699 1.060346 1.059960 20 21 .072039 .071807 .071546 .071266 .070950 .070605 .070227 .069813 21 22 .081196 .080942 .080667 .080359 .080023 .079654 .079251 .078806 22 23 .089580 .089313 .089012 .088685 .088327 .087934 .087501 .087030 23 24 .097276 .096985 .096665 .096318 .095937 .095516 .095058 .094549 24 25 1.104340 1.104030 1.103692 1.103323 1.102916 1.102472 1.101979 1.101446 25 26 .110835 .110508 .110149 .109756 .109327 .108850 .108334 .107758 26 27 .116815 .116469 .116087 .115672 .115213 .114714 .114157 .113549 27 28 .122323 .121955 .121553 .121110 .120629 .120093 .119506 .118864 28 29 .127397 .127011 .126582 .126119 .125603 .125038 .124419 .123741 29 30 1.132078 1.131667 1.131219 1.130723 1.130180 1.129585 1.128932 1.128216 30 31 .136393 .135963 .135484 .134963 .134392 .133764 .133076 .132325 31 32 .140376 .139917 .139415 .138867 .138265 .137605 .136884 .136091 32 33 .144045 .143564 .143037 .142461 .141828 .141137 .140376 .139542 33 34 .147432 .146928 .146373 .145767 .145106 .144378 .143578 .142706 34 35 1.150557 1.150026 1.149445 1.148812 1.148116 1.147351 1.146516 1.145601 35 36 .153436 .152881 .152275 .151609 1 .150878 .150080 .149205 .148247 36 37 .156092 .155513 .154875 .154177 .153415 .152579 .151664 .150663 37 38 .158540 .157932 .157264 .156536 .155739 .154866 .153910 .152866 38 39 .160792 .160155 .159459 .158698 157867 .156955 .155959 .154871 39 40 1.162862 1.162199 1.161472 1.160679 1.159811 1.158861 1.157824 1.156690 40 41 .164765 .164073 .163316 .162489 .161585 .160598 .159517 .158343 41 42 .166512 .165791 .165002 .164141 .163203 .162174 .161055 .159825 42 43 .168113 .167362 .166542 .165649 .164670 .163606 .162435 .161174 43 44 1 .169577 .168798 .167947 .167016 .166005 .164892 .163692 .162381 44 18 19 20 21 22 23 24 25 Continued on Page *93. TABLE XLTIL ♦85 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OP THE PRESENT VALUE OF ft ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 6 PER CENT. Pay- 26 27 28 29 30 31 32 33 Pay- ments. Jl. ments.! ft. 1.168055 1.165920 1.163661 1.161274 1.158757 1.156097 1.153290 1.150327 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .287059 .287043 .287025 .287002 .286980 .286955 .286929 .286899 2 3 .449323 .449290 .449251 .449207 .449162 .449113 .449058 .448997 3 4 .560579 .560526 .560467 .560401 .560333 .560256 .560172 .560076 4 5 0.643948 0.643876 0.643796 0.643708 0.643614 0.643509 0.643393 0.643263 5 6 .709730 .709639 .709538 .709425 .709304 .709170 .709021 .708858 6 7 .763421 .763309 .763185 .763046 .762898 .762733 .762552 .762350 7 8 .808298 .808165 .808015 .807851 .807674 .807478 .807261 .807023 8 9 .846476 .846318 .846146 .845954 .845748 .845518 .845267 .844989 9 10 0.879396 0.879218 0.879018 0.878800 0.878561 0.878299 0.878010 0.877689 10 11 .908093 .907889 .907664 .907415 .907145 .906855 .906518 .906156 11 12 .933322 .933094 .932841 .932561 .932266 .931923 .931554 .931148 12 13 .955656 .955408 .955125 .954822 .954477 .954104 .953694 .953242 13 14 .975561 .975279 .974973 .974624 .974250 .973838 .973385 .972884 14 15 0.993372 0.993068 0.992720 0.992342 0.991932 0.991480 0.990980 0.990433 15 16 1.009388 1.009044 1.008669 1.008257 1.007809 1.007313 1.006770 1.006170 16 17 .023826 .023456 .023049 .022602 .032113 .021575 .020982 .020330 17 18 .036894 .036494 .036054 .035568 .035039 .034454 .033813 .033107 18 19 .048751 .048320 .047843 .047324 .046746 .046117 .045424 .044661 19 20 1.059533 1.059067 1.058559 1.057989 1.057375 1.056696 1.055949 1.055129 20 21 .069353 .068858 .068302 .067698 .067037 .066307 .065507 .064628 21 1 22 .078318 .077778 .077190 .076542 .075832 .075053 .074198 .073257 22 23 .086500 .085936 •.085301 .084606 .083850 .083020 .082105 .081104 23 24 .093996 .093381 .092708 .091970 .091166 .090278 .089309 .088243 24 25 1.100846 1.100196 1.099482 1.098699 1.097840 1.096902 1.095871 1.094736 25 26 .107124 .106436 .105679 .104845 .103939 .102943 .101846 .100645 26 27 .112880 .112151 .111346 .110468 .109508 .108449 .107291 .106024 27 28 .118156 .117383 .116537 .115607 .114589 .113472 .112252 .110912 28 29 .122993 .122180 ,121285 .120301 .119228 .118053 .116765 .115353 29 30 1.127430 1.126571 1.125625 1.124591 1.123462 1.122224 1.120868 1.119385 30 31 .131495 .130588 .129596 .128508 .127321 .126019 .124596 .123040 31 32 .135216 .134267 .133224 .132081 .130834 .129469 .127978 .126346 32 33 .138627 .137629 .136535 .135336 .134029 .132601 .131038 .129332 33 34 .141745 .140700 .139553 .138298 .136932 .135436 .133804 .132022 34 35 1.144595 1.143500 1.142301 1.140990 1.139561 1.138000 1.136297 1.134440 35 36 .147195 .146052 .144799 .143430 .141940 .140313 .138539 .136607 36 37 .149566 .148372 .147064 .145639 .144086 .142393 .140549 .138540 37 38 .151722 .150476 .149117 .147622 .146017 .144258 .142342 .140261 38 39 .153678 .152385 .150959 . .149425 .147749 .145922 .143939 .141784 39 40 1.155454 1.154097 1.152635 1.151034 1.149295 1.147406 1.145353 1.143127 40 41 .157046 .155655 .154131 .152471 .150674 .148719 .146600 .144305 41 42 .158496 .157047 .155467 .153754 .151895 .149877 .147695 .145328 42 43 .159791 .158290 .156661 .154889 .152970 .150895 .148644 .146216 43 44 .160948 .159402 .157717 .155889 .153917 .151777 .149470 .146977 44 26 27 28 29 30 31 32 33 Continued on Page *91. *86 TABLE XLYIL LIFE AND TEMPORARY ANNUITIES. LOGARITHM OP THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. XXXI X 6 PER CENT. Pay- ments. n. a:=34 35 36 37 38 39 40 41 Pay- ments. 1.1472001.143903 1.140434 1.136773 1 1.132914 1.128852 1.124576 1.120076 n. 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .286866 .286826 .286789 .286747 .286697 .286643 .286584 .286522 2 3 .448926 .448850 .448773 .448684 .448583 .448475 .448355 .448225 3 4 .559971 .559856 .559735 .559598 .559446 .559279 .559096 .558898 4 5 0.643122 0.642965 0,642799 0.642614 0.642406 0.642179 0.641931 0.641661 5 6 .708676 .708478 .708266 .708028 .707762 .707475 .707159 .706811 6 7 .762130 .761888 .761626 .761333 .761011 .760659 .760269 .759846 7 8 .806761 .806471 .806158 .805812 .805429 .805007, .804546 .804042 8 9 .844681 .844344 .843980 .843577; .843127 .842638' .842102 .841512 9 10 0.877337 0.876951 0.876533 0.876067^ 0.875554 0.874994 0.874376 0.873702 10 11 .905757 .905320 .904844 .904318 .903737 .903101 .902403 .901637 11 12 .930701 .930209 .929677 .929086; .928434 .927722! .926937 .926080 12 13 .952744 .952199 .951605 .950947; .950224 .949429 .948559 .947603 13 14 .972336 .971733 .971076 .970351 .969550 .968675 .967712 .966658 14 15 0.989829 0.989167' 0.988446 0.987648^ 0.986771 0.985809!o. 984754 0.983601 15 16 1.005511 1.0047881.003999 1.003129 1.002170 1.001121i .999973 .998712 16 17 .019614 .018827j .017970 .017023 .015983 .0148461.013596 1.012227 17 18 .032330 .031480! .030550 .029527 .028403 .027171; .025819 .024338 18 19 .043824 .0429051 .041903 .040802 .039589 .038260 .036804 .035217: 19 1 20 1.054229 1.0532421.052167 1.050982 1.049677 1.0482511.046694 1.044982 20 21 .063663 .062607J .061453 .060182 .058786 .0572651 .055591 .053763 21 22 .072227 .071097! .069863 .068505 .067020 .065390 .063606 .061657 22 23 .080005 .078799 .077483 .076042 .074455 .072721 .070825 .068755 23 24 .087072 .085789 .084395 .082858 .081173 .079334 .077324 .075128 24 25 1.093493 1.0921361.090654 1.089024 1.087241 1.085295^1.083167 1.080844' 25 26 .099333 .097893 .096322 .094602 .092718 .090661 .088414 .085965 26 27 .104634 .103112 .101456 .099641 .097653 .095484 .093119 .090545 27 28 .109446 .107843 .106098 .104185 .102093 .099814 .097332 .094629 28 29 .113810 .112125 .110288 .108278 .106081 .103693 .101091 .098256 29 30 1.117764 1.115993 1.114065 1.111956 1.109658:1.107157 1.104432 1.101474 30 31 .121339 .119482 .117462 .115258 .112854] .110238 .107397 .104311 31 32 .124565! .122622 .120512 .118210 .115698 .112974 .110013 .106807 32 33 .127470 .125442 .123241 .120837 .118226 .115389 .112316 .108990 33 34 .130080 .127967 .125671 .123175 .120457 .117516 .114331 .110885 34 35 1.132419 1.130217 1.127834 1.125238 1.122423 1.119377 1.116080 1.112524 35 36 .134503 .132220 .129744 .127057 .124144 .120994 .117594 .113929 36 37 .136359 .133990 .131429 .128650 .125639' .122393 .118892 .115126 37 38 .138000 .135551 .132904 .130034 .126934 .123593 .119998 .116131 38 39 .139448 .136919 .134186 .131233 .128044 .124616 .120927 .116975 39 40 1.140717 1.138107 1.135297 1.132261 1.128991 1.125476 1.121707 1.117669' 40 41 .141819 .139137 .136250 .133138J .129788 .126197 .122349 .118236' 41 42 .142775 .140021 .137063 .133877 .130455 .126791 .122873 .118690 42 43 .143595 .140775 .137748 1 .134495 .131005 .127276 .123293 .119054; 43 44 .144295 .141411 .13832] .135004 .131454 .127665 .123629 .119335 44 34 35 36 37 38 39 40 41 Continued on Page *90. ^87 TABLE XLYIL LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF »? ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 6 PER CENT. Pay- 42 43 44 45 46 47 48 49 Pay- ments. ments. 1.115337 1.110361 1.1051241.099624 1.093846 1.087778 1.081412 1.074733 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1 2 .286448 .286374 .286287' .286192 .286090 .285975 .285852 .285715 2 3 .448079 .447923 .447747 .447556 .447345 .447113 .446859 .446580 3 4 .558674 .558434 .558166 .557873 .557551 .557196 .556808 .556380 4 5 0.641356 0.641030 0.640665:0.6402671 0.639830 0.639347 0.638819 0.638237 5 6 .706426 .706008 .705544 .705037 .704479 .703865 .703191 .702450 6 7 .759374 .758863 .758296 .757675 .756992 .756241 .755417 .754513 7 8 .803481 .802873 .802198! .801459 .800647 .799755 .798777 .797703 8 9 .840861 .840150 .839364 .838504 .837558 .836521 .835384 .834132 9 10 0.872953 0.872139 0.871238 0.870252 0.869171 0.867982 0.866678 0.865245 10 11 .900791 .899867 .898848 .897735 .896510 .895165 .893689 .892073 11 12 .925129 .934095 .922956 .921708 .920337 .918829 .917182 .915374 12 13 .946549 .945402 .944136 .942749 .941224 .939566 .937729 .935723 13 14 .965498 .964231 .962834 .961303 .959637 .957791 .955777 .953571 14 15 0.982327 0.980937 0.979405 0.977741 0.975899 0.973886 0.971685 0.969273 15 16 .997323 .995806 .994148 .992319J .990321 .988132 .985737 .983119 16 17 L010718 1.009083 1.007271 1.0052951 1.003131 1.000761 .998173 .995338 17 18 .022718 .020938 .018988 .016856 .014523 .011972 1.0091801.006134 18 19 .033462 .031552 .029455 .027166 .024662 .021920 .018931 .015668 19 20 1.043104 1.041056 1.038813 1.036362 1.033679 1.030753 1.027562 1.024079 20 21 .051756 .049571 .047176 .044558 .041702 .038587 .035190 .031487 21 22 .059522 .057194 .054643 .051863 .048831 .045523 .041920 .037999 22 23 .066485 .064012 .061308 .058363 .055151 .051651 .047845 .043706 23 24 .072723 .070106 .067248 .064135 .060743 .057053 .053044 .048690 24 25 1.078304 1.075544 1.072529 1.069247 1.065678 1.061798 1.057589 1.053028 25 26 ,083291 .080384 .077210 .0737641 .070017 .065951 .061548 .056778 26 27 .087734 .084677 .081351 .077738 .073818 .069572 .064975 .060005 27 28 .091677 .088479 .084996 .081222 .077135 .072708 .067924 .062764 28 29 .095172 .091828 .088196 .084265 .080009 .075408 .070448 .065105 29 30 1.098253 1.094769 1.090991 1.086903 1.082485 1.077721 1.072591 1.067073 30 31 .100961 .097341 .093415i .0891771 .084607 .079685 .074393 .068713 31 32 .103329 .099572 .095506 .091126 .086410 .081337 .075895 .070066 32 33 .105383 .101497 .097299 .092784 .087927 .082715 .077134 .071165 33 34 .107159 .103148 .098825 .094179 .089193 .083851 .078141 .072049 34 35 1.108682 1.104554 1.100109 1.095342 1.090236 1.084776 1.078952 1.072752 35 36 .109978 .105733 .101180 .096302 .091086 .085520 .079596 .073298 36 37 .111066 .106724 .102064 .097084 .091770 .086111 .^80097 .073718 37 38 .111980 .107539 .102784 .097713 .092313 .086571 .080481 .074032 38 39 .112732 .108203 .103363 .098213 .092736 .086923 .080770 .074263 39 40 1.113345 1.108737 1.103824 1.098602 1.093059 1.087189 1.080982 1.074429 40 41 .113838 .109162 .104182 .098899 .093304 .087383 .081133 .074544 41 42 .114231 .109492 .104456 .099125 .093482 .087522 .081239 .074622 42 43 .114535 .109745 .104664 .099289 .093610 .087620 .081311 .074669 43 44 .114769 .109937 ,104815 .099407 .093700 .087686 .081353 .074698 44 42 43 44 45 46 47 48 49 Cont inued on F •age *89. *88 TABLE XLYII. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 6 PER CENT. Pay- x=50 51 52 53. 54 55 56 57 Pay. ments. n. ments. 1.067728 1.060389 1.052701 1.0446521 1.036227 1.027411 1.018200 1.008575 n. 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 o.ocooco' 1 2 .285563 .285397 .285214 .285015 .284796 .284553 .284288 .283991 2 3 .446270 .445932 .445561 .445155 .444709 .444213 .443670 .443073 3 4 .555907 .555391 .554825 .554205 .553521 .552763 .551937 .551026| 4 5 0.637596 0.636896 0.636128 0.635284 0.634353 0.633329 0.632207 0.630974' 5 6 .701636 .700745 .699766 .6986881 .697509 .696206 .694781 .693215 6 7 .753519 .752429 .751230 .749920 .748476 .746889 .745152 .743244 7 8 .796521 .795225 .793807 .7922491 .790537 .788657 .786599 .784343 8 9 .832757 .831254 .829604 .827793: .825807 .823625 .821241 .818624| 9 10 0.863674 0.861954 0.860066 0.857998 0.855728 0.853240 0.850518 0.847537 10 11 .890299 .888355 .886225 .883890 .881334 .878529 .875466 .872112 11 12 .913389 .911218 .908838 .906235 .903382 .900259 .896847 .893117 12 13 .933525 .931119 .928488 .925607 .922457 .919007 .915244 .9111321 13 14 .951153 .948512 .945620 .942461 .939005 .935228 .931109 .9266151 14 15 0.966634 0.963748 0.960596 0.957152 0.953390 0.949281 0. 944806 0.939930 15 16 .980250 .977121 .973703 .969973; .965901 .961460 .956629 .951371! 16 17 .992243 .988866 .985180 .981160; .976779 .972006 .966820 .9611861 17 18 1.002807 .999179 .995223 .990914 .986224 .981121 .975587 .969579' 18 19 .012106 1.008226 1.004001 .9994051 .994408 .988981 .983100 .976731J 19 20 1.020281 1.016150 1.011657 1.006776' 1.001478 0.995729 0.989513 0. 9827921 20 21 .027455 .023074 .018316 .013155 .007558 1.001499 .994957 .987898 21 22 .033734 .029106 .024088 .018649 .012764 .006404 .999550 .992168 22 23 .039211 .034342 .029065 .023359 .017195 .010546 1.003394 .9957071 23 ' 24 .043972 .038862 .033336 .027372 .020941 .014017 .006583 .998615 24 25 1.048085 1.042744 1.036979 1.030767 1.024082 1.016898 1.009206 1.000974 25 26 .051621 .046058 .040064 .033617 .026692 .019269 .011334 .002867 26 27 .054642 .048867 .042655 .035986 .028841 .021194 .013043 .004363 27 28 .057204 .051227 .044810 .037938 .030585 .022741 .014394 .005528; 28 | 29 .059357 .053191 .046586 .0395221 .031988 .023964 .015447 .006418 29 30 1.061150 1.054810 1.048028 1.040797 1.033098 1.024917 1.016251 1.007085 30 31 .062629 .056125 .049189 .041806 .033963 .025645 .016853 .007576 31 32 .063830 .057183 .050108 .042593 .034624 .026191 .017297 .007928 32 33 .064797 .058022 .050824 .043194 .035119 .026593 .017616 .008173 33 34 .065564 .058675 .051372 .043644 .035484 .026882 .017837 .008336 34 35 1.066161 1.059176 1.051783 1.043976 1.035746 1.027082 1.017985 1.008440l 35 36 .066619 .059551 .052085 .044215 .035928 .027216 .018079 .008503 36 37 .066962 .059827 .052302 .044380 .036049 .027301 .018135 .008539 37 38 .0Q7214 .060025 .052453 .044490 .036127 .027352 .018169 .008558; 38 39 .067395 .060163 .052554 .044561 .036173 .027383 .018185 .008567; 39 40 1.067521 1.060255 1.052618 1.044603 1.036201 1.027397 1.018193 1.008572' 40 41 .067606 .060313 .052656 .044629' .036214 .027404 .018198 .008575 41 42 .067658 .060348 .052680 .044641 .036220 .027409 .018200 .0085751 42 43 .067690 .060370 .052691 .0446461 .036225 .027410 .018200 .008575 43 44 .067710 .060380 .052695 .0446511 .036226 .027410 .018200 44 1 45 1.067719 1.060383 1.052700 1.044652 1.036226 1.027411 45 46 .067722 .060388 .052701 .044652 .036227 46 47 .067727 .060389 .052701 47 48 .067728 .060389 48 49 .067728 49 50 51 52 53 54 55 56 57 TABLE XLYII. *89 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF W ANNUAL PAYMENTS OF 1 AT THE BEGINNINe OF EACH YEAR. ^ 6 PER CENT. Pay- ments. n. 58 59 60 61 62 63 64 65 Pay- ments. 0.998532 0.988049 0.977114 0.965719 0.953857 0.941502 0,928659 0.915312 7i. 1 2 3 4 0.000000 .283673 .442422 .550033 0.000000 .283321 .441706 .548939 0.000000 .282933 .440916 .547736 0.000000 .282505 .440049 .546419 0. 000000 .282040 .439104 .544975 0.000000 .281525 .438057 .543385 0.000000 .280960 .436911 .541642 0.000000 .280338 .435652 .539730 1 2 3 4 5 6 7 8 9 0.629627 .691504 .741161 .781875 .815767 0.628145 .689626 .738873 .779171 .812635 0.626519 .687562 .736365 .776205 .809208 0. 624735 .685303 .733616 .772962 .805460 0. 622783 .682828 .730611 .769416 .801368 0.620634 .680110 .727313 .765533 .796893 0.618282 .677137 .723711 .761293 .792012 0.615705 .673883 .719772 .756662 .786688 5 6 7 8 9 10 11 12 13 14 0.844280 .868454 .889049 .906654 .921727 0.840718 .864452 .884607 .901771 .916400 0.836820 .860080 .879762 .896447 .910605 0.832563 .855314! .874480' .890658 .904312 0.827923 .850119 .868737 .884370 .897487 0.822851 .844455 .862484 .877535 .890083 0.817329 .838297 .855695 .870129 .882077 0.811318 .831601 .848329 .862111 .873430 10 11 12 13 14 15 16 17 18 19 0.934630 .945665 .955078 .963081 .969852 0.928866 .939468 .948458 .956050 .962422 0.922604 .932748 .941293 .948452 .954415 0.915815 .925476 .933552 .940266 .945802 0.908466 .917619 .925210 .931459 .936563 0.900511 .909135 .916220 .921996 .926654 0. 891928 .900000 .906567 .911854 .916073 0.882677 .890181 .896212 .901017 .904792 15 16 17 18 19 20 21 22 23 24 0.975546 .980303 .984242 .987476 .990096 0.967739 .972136 .975742 .978662 .981004 0.959340 .963374 .966638 .969254 .971319 0.950331 .953992 .956924 .959236 .961032 0.940684 .943982 .946579 .948596 .950137 0. 930376 .933306 .935580 .937315 .938615 0.919392 .921966 .923927 .925396 .926471 0.907716 .909944 .911611 .912831 .913705 20 21 22 23 24 25 26 27 28 29 0.992199 .993860 .995153 .996140 .996880 0.982853 .984291 .985389 .986213 .986818 0.972923 .974149 .975068 .975742 .976225 0.962404 .963432 .964185 .964726 .965103 0.951291 .952135 .952742 .953165 .953445 0.939565 .940248 .940724 .941040 .941241 0.927242 .927780 .928137 .928364 .928501 0.914315 .914720 .914978 .915133 .915222 25 26 27 28 29 30 31 32 33 34 0.997425 .997815 .998087 .998266 .998383 0.987252 .987555 .987754 .987883 .988028 0.976563 .976785 .976929 .977091 .977065 0.965352 .965512 .965694 .965665 .965681 0. 953624 .953828 .953795 .953813 .953843 0.9414710.928580 .941433 .928623 .941454 .928641 .941487 .928655 .941499 .928660 0.915271 .915292 .915307 .915313 .915313 30 31 32 33 34 35 36 37 38 39 40 41 42 51 50 0.998513 .998492 .998503 .998523 .998529 0.998532 .998532 .998532 0.988005 .988017 .988039 .988046 .988049 0.988049 988049 0.977079 .977103 .977111 .977114 .977114 0.977114 0:965708 .965717 .965720 .965720 .965719 0.953853 .953857 .953857 .953857 0.9415030.928660 0.915312 35 58 67 56 55 54 53 52 51 50 .941503' .941502 .928659 42 43 44 1.115337 .115337 .115337 .115337 1.115333 .115333 .115328 .115317 .115302 45 1 OQQfi'Pl 1.110361 .110361 .110361 1.110361 .110,357 46 1.105124 ,105124 1.105124 .105124 48 47 1.093846 .093846 .093846 .093845 .093841 1.099624 .099624 .099623 .099619 .099619 49 1.087778 .087778 .087778 .087777 1.081412 .081412 .081412 .105120' 110.S57 1.074733 .074733 .105120 .105113 .110351 .110340 49 48 47 46 45 1.074733 .074732 .074727 .074725 .074716 1.081411 .081406 .081405 .081396 .081380 1.087773 .087772 .087764 .087749 .087725 1.093840 .093833 .093819 .093797 .093761 46 1.099612 .099599 .099579 .099546 .099490 1.105101 .105083 .105053 .105001 .104924 1.110323 .110296 .110248 .110176 .110076 1.115277 .115233 .115167 .115074 .114946 42 49 48 47 46 45 49 48 47 46 44 43 Continued from Page *87. *90 TABLE XLYII. LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 6 PER CENT. n Pay- ments. n. a: = 66 67 68 69 70 0.840762 71 72 73 Pay- ments. 0.901459 0.887078 0,872181 0.856733 0.824241 0.807180 0.789571 n. 1 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 .279660 .278908 .278097 .277190 .276207 .275121 .273936 .272640 2 3 .434275' .432761 .431109 .429283 .427295 .425110 .422726 .420111 3 4 .537641 .535345 .532840 .530080 .527074 .523780 .520183 .516248 4 5 0.612888 0.609799 0.606427 0.602724 0.598694 0.594284 0.589474 0.584227 5 6 .670330 .666436 .662192 .657542 .652485 .646959 .640949 .634404 6 7 .715472 .710770 .705653 .700056 .693975 .687351 .680156 .672342 7 8 .751614 .746106 .740116 .733580 .726492 .718788 .710440 .701403 8 9 .780896 .774584 .767731 .760272 .752199 .743446 .733991 .723764 9 10 0.804784 0.797679 0.789981 0.781623 0.772595 0. 762839 0.752312 0.741000 10 11 .824339 .816459 .807939 .798712 .788774 .778052 .766548 .754203 11 12 .840356 .831725 .822413 .812359 .801549 .789946 .777520 .764238 12 13 .853450 .844096 .834033 .823190 .811586 .799157 .785895 .771776 13 14 .864109 .854071 .843293 .831734 .819388 .806213 .792208 .777355 14 15 0.872733 0.862047 0.850622 0.838395 0.825383 0.811546 0.796892 0.781410 15 16 .879649 .868376 .856350 .843526 .829923 .815510 .800302 .784295 16 17 .885148 .873334 .860770 .847417 .833302 .818401 .802731 .786299 17 18 .889463 .877164 .864127 .850318 .835770 .820462 .804421 .787631 18 19 .892801 .880078 .866633 .852439 .837531 .821896 .805544 .788473 19 20 0.895343 0.882254 0.868466 0.853952 0.838757 0.822850 0.806255 0.788982 20 21 .897243 .883846 .869774 .855007 .839572 .823454 .806684 .789275 21 22 .898633 .884984 .870687 .855708 .840090 .823819 .806931 .789433 22 23 .899628 .885779 .871294 .856154 .840402 .824029 .807065 .789500 23 24 .900323 .886307 .871680 .856423 .840581 .824143 .807122 .789552 24 25 0,900785 0.886643 0.871912 0.856577 0.840678 0.824191 0.807165 0.789568 25 26 .901078 .886844 .872046 .856660 .840720 .824228 .807179 .789573 26 27 28 29 30 31 32 33 .901254 .901356 .901412 .886961 .887024 .887052 .872118 .872150 .872172 0.872181 .872182 .872181 .856697 .856723 .840751 .840762 .824240 .824243 .807183 .807180 .789571 27 34 .856733 0.856734 .856733 .840764 0.840762 .824241 35 66 65 64 63 0.901436 0.887Q71 .901453 .887079 .901460 .887079 .901460 .887078 36 1.147200 .147198 1.147198 .147198 37 1.140434 .140433 1.143903 1.143902 .143902 38 1.136773 34 .901459 40 39 1.132914 .136773 .136773 .140433 .140433 .143902 143901 .147197 .147197 62 61 41 1.128852 .132913 1.124576 .128851 .132913 .136773 .140432 143901 147195 fin 59 1.120076 1.124575 1.128851 1.132913 1.136771 1.140432 1.143898 1.147188 59 58 120076 .124575 .128851 .132911 .136771 .140428 .143891 .147181 58 57 .120076 .124575 .128849 .132911 .136767 .140421 .143883 .147167 57 56 .120076 .124572 .128849 .132907 .136760 .140412 .143868 .147142 56 65 .120073 .124572 .128844 .132899 .136750 .140397 .143842 .147107 55 54 1. 120073 1.124567 1.128836 1.132888 1.136733 1.140368 1.143804 1.147056 54 53 .120068 .124558 .128824 .132870 .136702 .140327 .143749 .146987 53 52 .120058 .124545 .128804 .132837 .136658 .140268 .143675 .146891 52 51 .120044 .124524 .128769 .132790 .136594 .140189 .143570 .146765 51 50 .120021 .124486 .128718 .132720 .136509 .140075 .143435 .146599 50 49 1.1199801.124430 1.128643 1.132628 1.136386 1.139929 1.143257 1.146385 49 48 .119919 .124350 .128543 .132496 .136228 .139737 .143026 .146119 48 47 .119833 .124241 .128401 .132325 .136022 .139489 .142739 .145786 47 46 .119715 .124089 .128216 .132103 .135754 .139179 .142381 .145379 46 45 .119551 .123889 .127976 .131814 .135420 .138793 .141943 .144886 45 41 40 39 38 37 36 36 34 Continued from Page *86. TABLE XLYII. *91 LIFE AND TEMPORAKY ANNUITIES. LOGARITHM OP THE PRESENT VALUE OF fl ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 6 PER CENT. Pay- I meuts. 74 75 76 n. 0.771402 0.7527110.733470 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 1 :o. 000000 .271204 .417238 .511942 0.578498 .627277 .663864 .691612 .712756 0.728834 .740985 .750074 .756780 .761642 0.765097 .767494 .769086 .770092 .770700 0.771049 .771239 .771319 .771380 .771400 0.000000 .269646 ,414120 .507265 0.572282 .619569 .654696 .681078 .700933 0.715826 .726906 .735048 .740935 .745110 0.000000 .267945 .410707 .502157 0.565525 .611186 .644789 .669724 .688242 0.701919 .711919 .719124 .724219 .727744 0.748002 0.730080 .749922 .731553 .751134 .732443 .751866 .752287 0.752515 .752611 .752684 .752708 .752716 .732955 .733232 0.733348 .733437 .733466 .733476 .733470 0.771406 0.752711 .771402 33 32 1.153290 1.1503271 .153289 ,150325 .150325 .153289 .153289 1.1503251.153288 .150324 .153288 .150324 .153286 .150322 .153279 .150315 .153274 1.1503091.153262 .150296| .153240 .150273 .153209 .150240 .153166 .150193 .r53106 1.150129,1.153023 .150039 .152914 .149922 .152770 .149769 .152585 ,149569 .152357 1.1493221.152069 .149013 .151717 .148634 .151292 .148177 .150782 .147628 .150177 33 32 0.000000 266064 406961 496577 0.558147 .602107 .634083 .657506 .674642 0. 687086 .696009 .702298 .706638 .709510 0.711320 . 712413 .713040 .713380 .713523 31 1.156097 .156096 .156096 .156096 .156095 1.156095 .156093 .156087 .156082 .156071 1.156051 .156022 .155982 .155926 .155849 .155747 .155614 .155441 .155229 .154962 154634 154239 153765 153202 152545 31 77 0.713672 713632 713667 713679 713672 30 1.158757 1.158755 .158755 .158755 .158754 .158754 1.158752 .158747 .158742 .158732 .158714 1.158686 .158649 .158597 .158525 .158431 1. 158307 158146: 157949 157700 157395; 157027 156586 156063 155452 154737 30 78 0.693356 0.000000 .264019 .402891 .490520 0.550179 .592309 .622576 .644426 .660148 0.671351 .679211 .684619 .688190 .690438 0.691795 .692572 .692994 .693171 .693306 0.693350 .693365 .693356 29 1.161274 .161272 1.161272 .161272 .161271 .161271 .161269 1.161265 .161260 .161251 .161234 .161208 1.161173 .161125 .161058 .160971 .160855 ,160705 ,160522 ,160290 ,160006 ,159664 .159254 .158766 .158198 .157533 .156769 29 79 0.672502 80 0.651114 0.000000 0.000000 .261782 .259335 .398447 .393621 .483943 .476798 0.541542 0.532205 .581737 .570351 .610209 .596956 .630436 .615521 .644724 .628364 0.654689 0.637113 .661516 .642856 .666013 .646457 .668840 .648623 .670543 .649863 0.671518 0.650536 .672048 .650818 .672270 .651035 .672440 .651104 .672495 .651128 0.672514 0.651114 .672502 81 0.629190 0.000000 .256690 .388367 .469061 0.522129 .558124 .582800 .599672 .611079 0.618529 .623185 .625980 .627578 .628445 0.628809 .629089 .629177 .629208 .629190 28 1.163661 .163659 .163659 1.163659 .163658 .163658 .163657 27 26 1.168055 .168054 .168054 1.165920 .165918 .165918 ,168054 .165918 .168053 1.165917 .165917 .165916 .165912 .163653: .165908 1.163648ll. 165900 .163640 .165885 .163624 .163599 .163566 1.163522 .163459 .163378 .163271 .163131 1.162961 .162744 .162480 .162162 .161780 1.161326 .160797 .160177 .159467 .158647 28 .165862 .165831 .165790 1.165732 .165656 .165557 .165431 .165268 1.165066 .164820 .164523 .164167 .163745 163252 162674 162013 161250 160385 27 1.168053 .168052 .168049 .168045 .168037 1.168024 .168002 .167973 .167935 .167881 1.167810 .167718 .167600 .167448 .167260 1.167031 .166754 .166422 .166029 .165570 1.165031 .164416 .163705 .162899 .161983 26 Pay- ments, n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 Continued from Page *85. *9$ TABLE XLYIl LIFE AND TEMPOKARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OP n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 6 PER CENT. Pay a:=82 ments. n. 0.606694 83 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 75 74 73 72 71 70 52 51 50 49 48 47 46 45 0.583647 84 0.559909 85 0.535453 o.oooooolo.oooooo .253744 .350591 .382611 .376412 .460632 .451581 0.511220,0.499566 .544974 .530980 .567662 .551649 .582833 .564914 .592671 .573117 0.598793 0.578007 602457 .580794 604549 .582302 .605683 .606159 0.606526 .606640 .606681 .606694 25 1.170082 .582934 .583423 0.583574 .583629 .583647 24 1.171993 .171993 1.1700801.171993 I7OO8O; .171993 170080 .171992 .170079 .170079 .171992 .171991 69 1.1700781.171988 .000000 .247111 .369650 .441793 .487020 .516032 .534368 .545601 .552261 .556044 .558089 .558944 .559606 .559810 .559885 .559909 0.000000 .243358 .362393 .431288 0.473656 .499792 .515579 .524861 .530109 0.532939: 534121 535035 535316 535420! 0.535453 23 .173803 ,173802 .173802 1.173802 .173802 .173802 .173801 .173798 22 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 1.168868 53 .168558 .1700751 .171984 .170071 .171978 .170064 .171967 .170052 .171947 1.170031 .170004 .169968 .169918 .169852 1.175513 .175512 .175512 .175512 1.175512 .175512 .175512 .175509 .175505 1.1737941.175499' 86 0.510023 0.000000 .239313 .354522 .420014 0.458801 .481703 .494998 .502462 .506472 87 0.483173 88 0.454503 89 0.422455 0. 000000 0. 000000 0. 000000 .234793! .230075 .222029 .345962 .406931 0.441565 .461257 .472187 .478024 .480948 508142 0.482321 509434 .482891 .509829 .509977 .510023 21 .171888 .171842 .171780 .171700 1.171598 .171466 .171302 .171102 1.169767 .169657 .169515 .169340 .169126' .17086;^ 1.170573 .170232 .169833 .169365 .168831 1.168213 .167513 .166718 .165819 .164816 .168192 .167764 .167262 1.166689 .166027 .165275 .164422 .163458 25 24 .173788 .175490 .173778 .175472 .173759 .175451 .173736 .175421 1.173704 1.175381 .173661 .175327 .173604 .175258 .173529 .175168 .173433 .175055 1.173311 1.174911 .173157 .174737 .172971 .174528 .172747 .174276 .172477 .173979 1.172159 1.173632 .171787 .173226 .171351 .172761 .170853 .172223 .170276 .171615 1.169624 1.170924 .168883 .170142 .168045 .169271 .167110 .168294 .166063 .167203 23 22 1.177130 .177128 .177128 .177128 .177128 1.177128 .177128 .177125 .177122 .177116 1.177108 .177091 .177071 .177043 .177006 1.176955 .176891 .176807 .176701 .176567 1.176405 .176210 .175975 .175698 .175373 1.174995 .174561 .174059 .173491 .172847 1.172117 .171305 .170394 .169377 .168260 .483106 .483173 .335400 .390968 .320218 .370386 0.421437 0.396806 .438034 .410543 .446811 .451185 .453233 .417331 .420496 .421807 Pay- ments, 20 19 1.180102 1.178657! .180101 1.178656 .178656 .178656 .178656 .178656 1.180101 .180101 .180101 .180101 .180101 18 1.181467 .181466 .181406 1.181466 .181466 .181466 .181466 .181464 1.181461 .181457 .181450 .181436 .181420 1.181396 .181366 .181325 .181273 1798941 .181205 1.1784341.1798211.181118 1.1786561.180099 .180096 .180091 .180084 .180069 .178653 .178650 .178645 .178637 1.178621 .178603 .178577 .178542 .178494 1.180052 .180027 .179994 .179950 .178356 .178257 .178132 .177980 1.177799 .177579 .177321 .177017 .176664 1.176259 .175791 .175260 .174659 .173979 1.173221 .172371 .171423 .170381 .169221 21 20 .179729 .179612 .179470 .179301 1.179095 .178855 .178571 .178241 .177863 1.177426 .176930 .176369 .175735 .175028 1.174234 .173350 .172378 .171296 .170108 19 .181009 .180876 .180719 .180526 1.180302 59 .180037 .179729 .179376 .178967 1.178504 .177980 .177389 .176728 .175988 1.175163 .174256 .173246 .172138 .170916 18 0.454083 0.422301 10 .454403' .422455 11 .454503 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 58 57 56 55 54 53 52 51 50 49 48 47 46 45 Continued from Page *84, TABLE XLYII. *93 LIFE AND TEMPORARY ANNUITIES. LOGARITHM OF THE PRESENT VALUE OF n ANNUAL PAYMENTS OF 1 AT THE BEGINNING OF EACH YEAR. 6 PER CENT Pay- ments. n. 90 91 93 93 94 95 96 97 Pay- ments. 0.391855 0.000000 0.363938 0.335983 0.307090 0.277673 0.246844 0.000000 0.210226 0.174939 n. 1 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2 .213204 .205516 .197480 .188212 .178148 .167818 .151126 .139047 2 3 4 5 6 7 8 9 10 .304786 .349841 0.372540 .383586 .388701 .390812 .391606 0.391855 .290824 .330929 0.349933 .358617 .362181 .363519 .363938 .276008 .310990 0.326562 .332882 .335244 .335983 .259484 .289433 0.301314 .305714 .307090 .242126 .266205 0.274956 .277673 .222201 .241074 0.246844 .196801 .174939 3 90 89 88 87 86 85 .210226 10 11 12 1.189964 1.189962 .189962 .189962 .189962 .189962 13 1.189102 .189101 .189101 .189101 .189101 14 1.188192 .188191 .188191 .188191 15 1.187226 .187225 .187225 16 1.186201 .186200 1.185120 83 17 1.183972 .183972 1.185118 .185118 1.186200 .186200 1.187225 .187225 1.188191 .188191 1.189101 .189101 1.189962 .189962 84 83 1.182755 82 .182754 .183972 .185118 .186200 .187225 .188191 .189101 .189960 82 81 .182754 .183972 .185118 .186200 .187225 .188191 .189099 .189957 81 80 .182754 .183972 .185118 .186200 .187225 .188189 .189096 .189952 80 79 1.182754 1.183972 1.185118 1.186200 1.187222 1.188186 1.189091 1.189945 79 78 .182754 .183972 .185118 .186197 .187219 .188181 .189082 .189936 78 77 .182754 .183971 .185115 .186194 .187214 .188171 .189073 .189922 77 76 .182753 .183968 .185112 .186188 .187203 .188161 .189057 .189905 76 75 .182750 .183965 .185105 .186177 .187192 .188144 .189040 .189881 75 74 1.182746 1.183958 1.185094 1.186165 1.187174 1.188126 1.189014 1.189850 74 73 .182739 .183946 .185081 .186146 .187155 .188098 .188981 .189809 73 72 .182726 .183932 .185061 .186125 .187125 .188063 .188938 .189760 72 71 .182711 .183910 .185038 .186093 .187088 .188017 .188885 .189697 71 70 .182688 .183885 .185004 .186054 .187039 .187960 .188817 .189618 70 69 1.182661 1.183849 1.184962 1.186.001 1.186977 1.187887 1.188733 1.189527 69 68 .182622 .183804 .184905 .185935 .186899 .187798 .188635 .189414 68 67 .182574 .183744 .184835 .185852 .186804 .187693 .188515 .189284 67 66 .182510 .183668 .184746 .185750 .186692 .187565 .188376 .189129 66 65 .182429 .183574 .184637 .185630 .186555 .187416 .188210 .188950 65 64 1.182328 1.183457 1.184508 1.185484 1.186395 1.187239 1.188018 1.188744 64 63 .182203 .183319 . 184352 .185313 .1862061 .187034 .187799 .188506 63 62 .182056 .183152 .184169 .185110 .1859871 .186799 .187544 .188236 62 61 .181876 .182956 .183952 .184876 .185735 .186526 .187255 .187931 61 60 .181667 .182724 .183701 .184607 .185444 .186218 .186929 .187588 60 59 1.181419 1.182455 1.183414 1.184295 1.185114 1.185869 1.186562 1.187204 59 58 .181131 .182148 .183080 .183942 .184741 .185476 .186152 .186773 58 57 .180802 .181791 .182702 .183543 .184320 .185038 .185691 .186295 57 56 .180420 .181386 .182275 .183093 .183851 .184544 .185179 .185769 56 55 .179987 .180929 .181794 .182591 .183324 .183996 .184617 .185184 55 54 1.179498 1.180414 1.181216 1.182027 1.182736 1.183395 1.183991 1.184542 54 53 .178946 .179838 .180653 .181398 .182094 .182725 .183304 .183837 53 52 .178329 .179193 .179980 .180710 .181377 .181991 .182549 .183063 52 51 .177638 .178473 .179243 .179943 .180591 .181183 .181722 .182218 51 50 .176868 .177683 .178422 .179101 .179726' .180297 .180817 .181297 50 49 1.176021 1.176804 1.177520 1.178175 1.1787771.179328 1.179831 1.180293 49 48 .175079 .175838 .176528 .177159 .177740! .178272 .178756 .179200 48 47 .174045 .174775 .175440 .176049 .176610| .177122 .177585 .178014 47 46 .172905 .173609 .174251 .174838 .1753771 .175867 .176316 .176727 46 45 .171656 .172335 .172953 .173517 .174033 13 .174508 .174938 .175333 45 17 16 15 14 12 11 10 Continued from Page *83. *94 TABLE XLVIII. CONVERSION TABLE FROM ANNUITY OR A(l+a) TO ANNUAL PREMIUM ON 1,000. INTEREST 4 PER CENT, Applicable to ordinary Life and Endowment Insurances, to Joint Lives and Survivorships, and all cases of the annexed Formula, where a denotes the Present Value of the Annuity. At 4 per cent. 1—v is .03846154 P is theAnnual Premium. For $1,000 Insured, P = 1,000 j :r^ (1— p) | • For a Bate of Interest other than 4 per cent., apply thefollowing Correction, 1,000(?)— f ') to the Tabular result * Fortherateof. . 3 8i 4 4^ 5 6 7 8 percent. Subtract 4601 P-158 18.142 26.959 35.613 Add 9.335 4645 p.OOO • Example. Interest 5 per cent. A(l + a) = 0.136820 ; the Tabular result is found to be 691.306 ; this, less the above Correction 9.158, leaves the Annual Premium required $682.15. PROPOBTICNAL PARTS, StTBT. A(l+a), 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0,5 5 5 5 0894.79 5884.11 0873.55 5 863.11 961.54 950.09 938.78 927.59 916.53 905.60 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 755.87 1 746.77 6 852.79 842.59 832.50 822.53 0812.68 5 802.93 793.30 783.78 774.37 765.06 1,6 737.79 728.90 720.12 711.43 702.85 694.36 685.97 677.68 0669.48 5 661.38 959.24 947.82 936.53 925.37 914.33 903.43 892.65 881.99 871.45 861.04 850.74 840.56 830. 50 820.55 810.72 801.00 791.39 781.89 772.50 763.22 754.04 744.97 736.00 727.14 718.37 709.71 11701.14 6692.68 653.37 645.45 637.62 629.88 0,5 684.31 676.03 667.86 659.77 651.78 643.88 636.07 628.35 2,7 956.94 945.55 934.29 923.15 912.14 901.26 890.50 879.87 869.36 858.97 848.69 838.54 828.50 818.58 808. 77 799.07 789.48 780.00 770. 63 761.37 752.22 743.17 734.22 725.37 716.63 707.99 699.44 691.00 682.65 674.39 666.23 658.17 650.19 642.31 634.52 626.81 2,7 3,8 4 9 4 9 4 9 4 9 4 9 846.65 4 836.52 9 954.65 943.29 932.05 920.94 909.96 899.10 888.37 877.76 867.27 856.90 826.51 816.61 806.82 797.14 787.58 778.12 4,9 768.77 4 759.53 9 750.40 741.37 732.44 723.62 714.89 706.27 697.75 689.32 680.99 672.75 664.61 656.56 648.61 640.74 632.97 625.28 3,8 952.37 941.03 929.82 918.73 907.78 896.94 886.24 875.65 865.19 854.84 844.62 834.51 824.52 814.64 804.87 795.22 785.68 776.24 766.92 757.70 748.58 739.58 730.67 721.86 713.16 704.56 696.05 687.64 679.33 671.12 662.99 654.96 647.03 639.18 631.42 623.76 4,9 46j69 92 45 68 90 45,67 44 66 23 23 22 22 22 22 21 21 21 21 20 20 20 40 20 39 2o!39 19 39 90 66 65 64 63 63 62 61 61 60 59 59 58 57 56 66 55 55 54 53 53 52 51 51 50 50 49 49 48 48 47 46 46 86 86 85 84 83 82 81 80 79 78 77 76 75 74 74 73 72 71 70 70 69 68 67 66 66 65 64 63 63 62 61 115 113 112 111 110 108 137160 136158 134 133 131 130 107 128 106 127 105 103 102 101 100 99 98 96 95 94 93 92 91 90 89 125 124 122 121 9 183 181 179 177 175 173 157 155 153 151 150 171 148 169 206 204 202 146 144 143 141 167 165 163 161 120 140 118138 117 116 114 113 112 110 109 108 107 105 87 104 86 103 85 84 83 82 81 80 79 78 77 77 102 101 99 98 97 96 95 94 93 92 160179 158177 137 135 133 132 130 129 127 126 124 123 199 197 195 192 190 188 186 184 182 156 154 152 151 149 147 146 144 142 140 122139 120 137 119 117 116 115 113 112 111 110 136 134 176 173 171 169 168 165 164 162 160 158 156 154 153 151 133 149 131 148 130 128 127 125 108124 107122 i_^ 7 8 146 144 143 141^ 139i 138j 9 TABLE XLTTII. *95 CONVERSION TABLE FROM ANNUITY OR X{l + (l) TO ANNUAL PREMIUM ON 1,000. INTEREST 4 PER CENT. PROPORTIONAL PARTS, SUBT. ^l+a). 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0,5 622.23 614.67 607.19 599.80 592.50 585.27 0578.13 5 571.08 564.10 557.20 550.38 543.64 536.98 530.39 523.89 517.44 511.09 504.80 498.57 492.42 486.35 480.34 0474.40 468.53 1,6 0429 0418 462.73 456.99 451.32 445.71 440.17 434.69 27 423.92 63 413.39 22 403.12 398.05 393.06 12 383.24 378.41 373.64 368.93 364.26 359.65 355.10 350.58 346.13 0,5 0388 5 5 620.71 613.17 605.71 598.33 591.04 583.84 576.72 569.67 562.71 555.83 549.03 542.30 535.66 529.08 522.59 516.16 509.82 503.54 497.34 491.20 485.14 479.15 473.22 467.36 461.57 455.85 450.19 444.60 439.07 433.60 428.20 422.86 417.58 412.36 407.19 402.09 397.05 392.07 387.14 382.27 377.46 372.69 367.98 363.34 358.73 354.18 349.69 345.25 2,7 3,8 2i619.20 7611.67 604.23 596.87 589.60 582.41 575.30 568.27 561.33 554.46 547.68 540.97 534.33 527.78 521.30 514.90 508.55 502.29 496.10 489.98 483.93 477.95 472.04 466.20 460.42 454.71 449.07 443.49 437.97 432.52 416.53 411.32 406.18 401.08 2396.06 7391.07 386.16 381.30 376.49 371.74 367.05 362.41 357.82 353.28 348.80 344.36 2,7 617.68 610.17 602.75 595.41 588.15 580.98 573.90 566.89 559.95 553.10 546.33 539.63 533.02 526.48 520.02 513.62 507.30 501.05 494.87 488.77 482.73 4 476.77! 9 4,9 470.88 465.04 459.28 453.58 447.95 442.38 436.87 431.43 426.05 420.74 415.48 410.28 405.15 400.08 395.05 390.09 385.18 380.33 375.54 370.81 366.11 361.48 356.91 352.39 347.91 343.48 3,8 616.17 608.68 601.27 593.95 586.71 579.55 572.48 565.49 558.57 551.74 544.98 538.30 531.70 525.18 518.72 512.35 506.04 499.81 493.65 487.56 481.53 475.58 469.70 463.88 458.13 452.45 446.83 441.27 435.78 430.35 424.99 419.68 414.44 409.25 404.13 399.06 394.05 389.10 384.21 379.38 374.59 369.86 365.18 360.56 356.00 351.48 347.02 342.60 4,9 15 15 15 15 14 14 14 14 14 14 13 13 13 13 13 13 13 12 12 12 12 29'44 1224 10 21 10 21 10 10 10 10 10 10 10 9 9 9 9 9 9 9 1928 60 60 59 58 58 57 56 56 55 55 54 53 53 52 52 51 50 50 49 49 48 48 47 46 46 45 45 44 44 43 43 42 42 41 41 41 40 40 39 39 38 38 37 37 36 36 36 35 5 76 75 74 73 72 71 71 70 69 68 67 67 66 65 65 64 63 62 62 61 60 59 59 58 57 57 56 55 55 54 54 53 52 52 51 51 50 49 49 48 48 47 47 46 46 45 45 44 6 35 4:4 4|5 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 75 74 73 72 71 70 70 69 68 67 66 66 65 64 63 63 62 61 61 60 59 59 58 57 57 56 55 55 54 53 53 106 105 103 102 101 100 99 98 97 95 94 93 92! 9l| 90 89 88 87 86 85 84 83 82 81 80 79 79 78 77 76 75 74 73 72 71 71 70 69 68 68 67 66 65 65 64 63 62 62 8 121 120 118 117 116 114 113 112 110 109 108 107 105 104 103 102 101 100 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 76 75 74 73 72 71 70 9 136 135 133 131 130 129 127 126 124 123 121 120 119 117 116 114 113 112 111 109 108 107 106 104 103 102 101 100 99 98 96 95 94 93 92 91 90 89 88 87 86 '85 84 83 82 81 80 79 8 9 96* . TABLE XLYIII. CONVERSION TABLE FROM ANNUITY OR /(l+r^ TO ANNUAL PREMIUM ON 1,000. INTEREST 4 PER CENT. PKOPORTIONAL PAKTS, 8UBT. U(l+«) 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.5 Oj341.73 5 337.38 1 6 1 6 324.62 1 320.46 6 333.07 328.82 5 5 5 0308.28 5 304.31 1.6 316.35 312.29 300.38 296.50 292.67 288.88 285.13 281.43 277.77 274.15 270.57 267.03 263.53 260.08 256.66 1 253.28 6 249.941 1 246.64 6 236.96 233.81 230.69 227.61 224.57 221.55 218.58 215.64 212.73 209.85 207.01 204.20 201.42 198.68 195.96 193.28 190.63 188.00 185.41 182.85 340.85 336.51 332.22 327.98 323.78 319.63 315.54 311.48 307.48 303.52 299.60 295.73 291.91 288.13 284.39 280.69 277.04 273.43 269.86 266.33 262.84 259.39 255.98 252.61 249.28 245.98 242.73 239.51 236.33 233.18 230.07 227.00 223.96 220.96 217.99 215.05 212.15 209.28 206.44 203.64 200.87 198.13 195.42 192.75 190.10 187.48 184.90 182.34 2,7 3,8 339.98 335.66 331.37 327.13 322.95 318.81 314.72 310.68 306.68 302.73 298.83 294.96 291.15 3290.39 287.38 8286.63 283.6513 279.96 8 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2229.46 7226.39 339.11 334.79 330.52 326.29 322.12 317.99 313.91 309.88 305.89 301.95 298.05 294.20 276.31 272.71 269.15 265.63 262.15 258.71 255.30 251.94 248.62 245.33 242.08 238.87 235.70 232.56 0,5 1,6 223.36 220.36 217.40 214.47 211.57 208.71 205.88 203.08 200.32 197.59 194.88 192.21 189.57 j 186.96 184.38; 181.83 2,7 282.90 279.23 275.59 271.99 268.44 264.93 261.45 258.02 254.63 251.27 247.96 4 244.68 1 9 241.44 4 238.23 9 4,9 235.07 231.93 228.84 225.78 222.75 219.76 216.81 213.89 211.00 208.14 205.32 202.53 199.77 197.04 194.35 191.68 189.05 186.44 338.24 333.93 329.67 325.45 321.29 317.17 313.10 309.07 305.10 301.16 297.28 293.43 289.63 285.88 282.17 278.50 274.87 271.28 267.73 264.23 4 260.77 9 257.34 3183.87 8ll81.32 3,8 253.95 250.61 247.30 244.03 240.79 237.60 234.44 231.31 228.22 225.17 222.15 219.17 216.22 213.31 210.42 207.58 204.76 201.97 199.22 196.50 193.81 191.15 188.53 185.93 183.36 180.82 4,9 3 4 17 17 17 17 17 16 16 16 16 16 16 15 1? 15 15 15 14 14 14 14 14 14 14 13 13 13 13 13 13 12 12 12 12 12 12 12 12 11 11 11 11 11 16 16 16 16 16 15 15 35 34 34 34 33 33 32 32 32 31 31 31 30 30 30 29 29 29 28 28 28 27 27 27 26 26 26 26 25 25 25 24 24 24 24 23 23 23 22 22 22 22 21 21 21 21 20 20 44 43 43 42 42 41 41 40 40 39 39 38 38 38 37 37 36 36 35 35 35 34 34 33 33 33 32 32 32 31 31 30 30 30 29 29 29 28 28 28 27 27 27 27 26 26 26 25 6 7 52 61 52 60 1 51 60 50 59 50 '58 49 58 49 57 56 56 55 54 54 53 53 52 51 51 50 50 49 48 48 47 47 46 46 45 45 44 44 43 43 42 42 41 41 40 40 39 39 38 38 38 37 37 36 36 36 70 78 78 77 76 75 74 73 72 71 71 70 69 68 68 67 66 65 64 64 63 62 62 61 60 59 59 69 68 o7 67 66 65 64 64 63 62 61 61 60 59 59 58 57 57 56 55 55 54 53 53 52 5258 51,57 5057 50 56 49,55 4955 48:54 48J53 47 53 47152 46 52 45 51 4551 44 50 49 49 48 48 47 47 46 46 89 TABLE XLYIII. *9r CONVERSION TABLE FROM ANNUITY OR A(l + a) TO ANNUAL PREMIUM ON 1,000. INTEREST 4 PER CENT. PROPORTIONAL PARTS, SUBT. A(l+a). 0,5 1,6 2,7 3,8 4,9 1 2 3 4 20 20 6 25 25 6 30 30 7 35 35 8 40 40 9 45 45 0.66 3180.31 5177.81 1179.81 6177.31 2 7 179.31 176.83 3 8 178.81 176.32 4 9 178.31 175.83 5 5 10 ]0 15 15 0.67 0175.33 5172.89 1 6 174.84 172.40 2 7 174.35 171.92 3 8 173.86 171.43 4 9 173.37 170.95 5 5 10 10 15 15 20 19 24 24 29 29 34 34 39 39 44 44 0.68 0170.47 5168.08 1 6 169.99 167.60 2 7 169.51 167.13 3 8 169.03 166.65 4 9 168.55 166.18 5 5 10 9 14 14 19 19 24 24 29 28 33 33 38 38 43 43 0.69 0165.71 5163.38 1 6 165.24 162.91 2 7 164.77 162.45 3 8 164.31 161.99 4 9 163.84 161.52 5 5 9 9 14 14 19 19 23 23 28 28 33 32 37 37 42 42 0.70 0161.06 5158.78 1 6 160.61 158.33 2 7 160.15 157.87 3 8 159.69 157.42 4 9 159.24 156.97 5 5 9 9 14 14 18 18 23 23 27 27 32 32 36 36 41 41 0.71 0156.52 jl 5154.29 6 156.07 153.85 2 7 155.63 153.41 3 8 155.18 152.96 4 9 154.74 152.52 4 4 i 13 13 18 18 22 22 27 27 31 31 36! 40 35|40 0.72 0152.08 5149.90 1 6 151.65 149.47 2 7 151.21 149.04 3 8 150.77 148.01 4 9 150.34 148.18 4 4 9 9 13 13 17 17 22 22 26 26 31 30 35 34 39 39 0.73 i 0147.75|1 5145.62 6 147.32 2'l46.89 145.19 7144.77 3 8 146.47 144.35 4 9 146.04 143.93 4 4 9 8 13 13 17 17 21 21 26 25 30 30 34 34 38 38 0.74 ' ) 143.51 1 5141.43 6 143.09 141.01 2 7 142.67 140.60 3 8 142.26 140.19 4 9 141.84 139.78 4 4 8 8 12 12 17 16 21 21 25 25 29 29 33 33 37 37 0.75 ' 3139.37 5137.33 1 6 138.96 136.93 2 7 138.55 136.52 3 8 138.14 136.12 4 9 137.74 135.72 4 4 8 8 12 •12 16 16 20 20 24 24 29 28 33 32 37 36 0.76 < ) 135.32 ) 133.33 1 6 134.92 132.93 2 7 134.52 132.54 3 8 134.12 132.15 4 9 133.73 131.75 4 4 8 8 12 12 16 16 20 20 24 24 28 28 32 32 36 35 0.77 1 ) 131.36 ) 129. 42 1 6 130.97 129.03 2 7 130.58 128.65 3 8 130.19 128.26 4 9 129.81 127.88 4 4 8 8 12 12 16 15 19 19 23 23 27 27 31 31 35 35 0.78 ( ) 127.50 1 ) 125. 60 6 127.12 125.22 2 7 126.73 3 124.84 8 126.35 124.47 4 9 125.98 124.09 4 4 8 8 11 11 15 15 19 19 23 23 27 26 30 30 34 34 0.79 ( ) 123. 72 1 ) 121.86 6 123.35 121.49 2 7 122.97 3 121.13 8 122.60 120.76 4 9 122.23 120.39 4 4 7 7 11 11 15 15 19 18 22 22 26 26 30 29 33 33 0.80 1 ) 120. 03 1 5118.21 6 119.66 117.85 2 7 119.30 117.49 3 8 118.94 117.14 4 9 118.57 116.78 4 4 7 7 11 11 15 14 18 18 22 21 25 25 29 29 33 32 0.81 1 ) 116.42 1 5114.65 6 116.06 114.30 2 7 115.71 113.94 3 8 115.35 113.59 4 9 115.00 113.24 4 4 7 7 11 11 14 14 18 18 21 21 25 25 28 28 32 32 0.82 )112.89 1 5111.16 6 112.55 110.82 2 7 112.20 110.47 3 8 111.85 110.13 4 9 111.51 109.79 3 3 7 7 10 10 14 14 17 17 21 21 24 24 28 27 31 31 0.83 } 109. 45 1 5107.76 6 109.11 107.42 2 7 108.77 107.08 3 8 108.43 106.75 4 9 108.09 106.42 3 3 7 7 10 10 14 13 17 17 20 20 24 24 27 27 80 30, 0.84 ( 106.08 'l 5104.43 6 105.75 104.10 2 7 105.42 103.77 3 8 105.08 103.44 4 9 104.76 103.12 3 3 7 7 10 10 13 13 17 16 20 20 23 23 26 26 30 ' 30 0.85 0102.79 1 5101.18 6 102.47 100.85 2 7 102.14 100.53 3 8 101.82 100.21 4 9 101.50 99.80 3 3 6 6 10 10 13 13 16 16 19 19 28 22 26 26 29 29 0.86 99.58 'l 5 98.00 6 99.26 97.68 2 7 98.94 97.37 3 8 98.63 97.06 4 9 98.31 96.75 3 3 6 6 9 9 13 13 16 16 19 19 22 22 25 25 28 28 0.87 96.43 1 5 94.89 6 96.12 94.58 2 7 95.82 94.28 3 8 95.51 93.97 4 9 95.20 93.67 3 3 6 6 9 9 12 12 15 15 18 18 22 21 25 24 28 28 0.88 93.36 1 5 91.86 6 93.06 91.56 2 7 92.76 91.26 3 8 92.46 90.96 4 9 92.16 90.66 3 3 6 6 9 9 12 12 15 15 18 18 21 21 24 24 27 27 0.89 90.36 1 5 88.89 6 90.07 88.60 2 7 89.77 88.30 3 8 89.48 88.01 4 9 89.18 87.72 3 3 G G 9 9 12 1^ 15 15 18 18 21 20 24 23 26 26 0,6 ; 1,6 2,7 3,8 4,9 T ~2 ¥ 4 5 6 T ~S 9 '98 TABLE XLYIU CONVERSION TABLE FROM ANNUITY OR A(l+a) TO ANNUAL PREMIUM ON 1,000. INTEREST 4 PER CENT. PROPOBTIONAL PAKTS , SUBT. A(l+a) 1 2 3 4 5 6 7 8 9 ! 1 3 213 4 11 5 14 6 17 7 20 8 23 9 26 0.90 8743 87.14 86.85 86.56 86.28 85.99 85.70 85.42 85.13 84.85 6 9 0.91 84-57 84.38 84.00 83.72 83.44 83.16 83.88 82.60 82.32 82.04 13 6 8 11 14 17 20 33 25 0.92 81.77 81.49 81.21 80.94 80.66 80.39 80.13 79.84 79.57 79.30 3 5 8 11 14 16 19 22 25 0.93 79.03 78.76 78.49 78.22 77.95 77.68 77.43 77.15 76.88 76.62 3 5 8 11 13 16 19 21 24 0.94 76.35 76.09 75.83 75.56 75.30 75.04 74.78 74.52 74.26 74.00 3 5 8 10 13 16 18 21 33 0.95 73.74 73.48 73.23 72.97 72.71 72.46 73.30 71.95 71.69 71.44 3 5 8 10 13 15 18 20 33 0.96 71.18 70.93 70.68 70.43 70.18 69.93 69.68 69.43 69.19 68.84 2 5 7 10 12 15 17 20 32 0.97 68.69 68.44 68.30 67.95 67-71 67.46 67.32 66.98 66.73 66.49 2 5 7 10 12 15 17 20 22 0.98 66.25 66.01 65.77 65.53 65.29 65.05 64.81 64.58 64.34 64.10. 2 5 7 10 12 14 17 19 21 0.99 63.87 63.63 63.40 63.16 62.93 62.69 63.46 62.23 62.00 61.77 2 5 7 9 12 14 16 19 21 l.OD 61.54 61.31 61.08 60.85 60-62 60.39 60.17 59.94 59.71 59.49 2 5 7 9 11 14 16 18 21 1.01 59.36 59.04 58.81 58.59 58-37 58.14 57.93 57.70 57.48 57.25 2 4 7 9 11 13 16 18 20 1.02 57.04 56.83 56.60 56.38 56-16 55.95 55.73 55.51 55.30l55.08 3 4 7 9 11 13 15 17 20 1.03 54.86 54.65 54.44 54.22 54.01 53.80 53.58 53.37 53.16 52.95 2 4 6 8 11 13 15 17 19 1.04 53.74 52.53 52.32 53.11 51.90 51.70 51.49 51.28 51.08 50.87 2 4 6 8 10 12 15 17 19 1.05 50.66 50.46 50.25 50.05 49.85 49.64 49.44 49.34 49.04 48.84 3 4 6 8 10 12 14 16 18 1.08 48.64i48.44 48.24 48.04 47.84 47.64 47.44 47.34 47.05 46.85 3 4 6 8 10 12 14 16 18 1.07 46.65|46.46 46.26 46.07 45.87 45.68 45.49 45.39 45.10 44.91 2 4 6 8 10 12 14 15 17 1.08 44.73 44.53 44.33 44.14 43.95 43.76 43.57 43.39 43.20 43.01 2 4 6 8 10 11 13 15 17 1.09 43.83 43.64 42.45 42.26 42.08 41.89 41.71 41.53 41.34 41.15 3 4 6 7 9 11 13 15 17 1.10 40.97 40.79 40.61 40.43 40.24 40.06 39.88 39.70 39.52 39.34 3 4 5 7 9 11 13 14 16 1.11 39.16 38.99 38.81 38.63 38.45 38.28 38.10 37.93 37.75 37.57 3 4 5 7 9 11 13 14 16 1.12 37.40 37.33 37.05 36.87 36.70 36.53 36.36 36.18 36.01 35.84 3 3 5 7 9 10 13 14 16 1.13 35.67 35.50 35.33 35.16 34.99 34.82 34.65 34.48 34.32 34.15 2 3 5 7 8 10 12 14 15 1.14 33.98 33.83 33.65 33.48 33.32 33.15 32.99 33.83 32.66 32.50 2 3 5 7 8 10 12 13 15 1.15 33.33 32.17 33.01 31.85 31.68 31.52 31.36 31.30 31.04 30.88 2 3 5 6 8 10 11 13 14 1.16 30.73 30.56 30.40 30.25 30.09 29.93 29.77 39.63 29.46 29.30 2 3 5 6 8 9 11 13 14 1.17 39.15 38.99 28.84 28.68 28.53 38.37 28.22 28.07 27.91 27.76 2 3 5 6 8 9 11 12 14 1.18 37.61 37.46 27.30 27.15 27.00 36.85 26.70 26.55 36.40 26.25 2 3 5 6 8 9 11 12 14 1.19 36.10 25.96 25.81 25.66 25.51 35.37 25.22 25.07 24.93 24.78 3 4 6 7 9 10 12 13 1.20 34.63 24.49 24.34 24.20 24.06 33.91 23.77 23.63 23.48 23.34 3 4 6 7 9 10 11 13 1.21 33.30 23.06 22.91 33.77 22.63 33.49 32.35 22.21 22.07 21.93 3 4 6 7 8 10 11 13 1.22 31.79 31.66 21.52 31.38 21.24 31.10 20.97 20.83 20.69 20.56 3 4 5 7 8 10 11 12 1.23 30.4330.39 20.15 30.03 19.88 19.75 19.61 19.48 19.35 19.33 3 4 5 7 8 9 11 12 1.24 19.08 18.95 18.82 18.89 18.55 18.43 18.29 18.16 18.03 17.90 3 4 5 7 8 9 10 12 1.25 17.77 17.64 17.51 17.39 17.26 17.13 17.00 16.87 16.75 16.63 3 4 5 6 8 9 10 12 1.26 16.49 16.37 1-6.24 16.11 15.99 15.86 15.74 15.61 15.49 15.37 3 4 5 6 8 9 10 11 1.27 15.34 15.12 14.99 14.87 14.751 14.63 14.50 14.38 14.26 14.14 2 4 5 6 7 9 10 11 1.28 14.03 13.90 13.78 13.66 13.54 13.43 13.30 13.18 13.06 13.94 2 4 5 6 7 8 10 11 1.29 13.83 12.71 12.59 12.47 12.35 13.34 12.12 12.00 11.89 11.77 3 3 5 6 7 8 9 10 1.30 11.66 11.54 11.43 11.31 11.20 11.08 10.97 10.86 10.74 10.63 3 3 5 6 7 8 9 10 1.31 10.53 10.40 10.29 10.1810.07 9.96 9.84 9.73 9.62 9.51 2 3 4 6 7 8 9 10 1.32 9.40 9.29 9.18 9.071 8.96 8.85 8.74 8.64 8.53 8.42 2 3 4 5 •7 8 9 10 1.33 8.31 8.20 8.10 7.99 7.88 7.78 7.67 7.56 7.46 7.35 2 3 4 5 6 7 8 10 1.34 7.35 7.14 7.04 6.93 6.83 6.73 6.62 6.52 6.41 6.31 2 3 4 5 6 7 8 9 1.35 6.31 6.10 6.00 5.90 5.80 5.70 5.59 5.49 5.39 5.29 2 3 4 5 6 7 8 9 1.36 5.19 5.09 4.99 4.89. 4.79 4.69 4.59 4.49 4.39 4.39 2 3 4 5 6 7 8 9 1.37 4.30 4.10 4.00 3.90 3.81 3.71 3.61 3.51 3.42 3.33 2 3 4 5 6 7 8 9 1.38 3.33 3.13 3.03 2.94 2.84 3.75 2.65 2.56 2.46 2.37 2 3 4 5 6 7 8 9 1.39 2.38 2.18 2.09 2.00 1.90 1.81 1.72 1.63 1.53 1.44 T 2 2 3 3 4 4 5 6 ~6 7 7 7 8 1 2 3 4 5 6 7 "s^ 9 TABLE XLIX. *99 CONVERSION TABLE FROM ANNUITY OR A(l-f«) TO SINGLE PREMIUM ON 1,000. INTEREST 4 PER CENT. Applicable to ordinary Life and Endowment Insurance, to Joint Lives and Survivorships, and all cases under the annexed Formula : For $1,000 Insured, A = 1,000 ^ - (1— ») (1 +ffl)j . For other Bates of Interest : Before entering the Table, correct /l(l + a) by }-{\~v') — A(l— ■»), thus: For the rate of. 3 3^ 4 4.V 5 6 7 8 percent. Subtract 0.120743 .055899 .000000 ...'. Add 049070.093754 .167819.230688.284640 Example. Interest 3 per cent. A(l + a) = 0.368943. Subtracting 0.120743. leaves 0.248200. With this, the Table gives the required Single Premium 93189. X{\+a). 1 2 8 4 6 6 7 8 9 *Ref. 0.00 961.54 961.45 961.36 961.27 961.18 961.09 961.00 960.91 960.83 960.73 1.39 0.01 960.64 960.55 960.46 960.37 960.28 960.19 960.09 960.00 959.91 959.82 1.39 0.02 959.73 959.63 959.54 959.45 959.35 959.36 959.17 959.07 958.98 958.88 1.38 0.03 958.79 958.69 958.60 958.50 958.41 958.31 958.21 958.12 958.02 957.93 1.37 0.04 957.83 957.73 957.63 957.54 957.44 957.34 957.24 957.14 957.04 556.94 1.36 0.05 956.85 956.75 956.65 956.55 956.45 956.35 956.25 956.14 956.04 955.94 1.35 0.06 955.84 955.74 955.64 955.53 955.43 955.33 955.23 955.13 955.02 954.93 1.34 0.07 954.81 954.71 954.60 954.50 954.39 954.29 954.18 954.08 953.97 953.87 1.33 0.08 953.76 953.65 953.55 953.44 953.33 953.22 953.12 953.01 952.90 953.79 1.32 0.09 952.68 952.57 952.46 952.35 952.24 952.13 952.02 951.91 951.80 951.69 1.31 0.10 951.58 951.47 951.36 951.24 951.13 951.02 950.91 950.79 950.68 950.57 1.30 0.11 950.45 950.34 950.22 950.11 949.99 949.88 949.76 949.65 949.53 949.41 1.29 0.12 949.30 949.18 949.06 948.95 948.83 948.71 948.59 948.47 948.36 948.24 1.28 0.13 948.12 948.00 947.88 947.76 947.64 947.52 947.39 947.27 947.15 947.03 1.27 0.14 946.91 946.79 946.66 946.54 946.42 946.29 946.17 946.05 945.93 945.80 1.26 0.15 945.67 945.55 945.42 945.29 945.17 945.04 944.92 944.79 944.66 944.53 1.35 0,16 944.41 944.28 944.15 944.02 943.89 943.76 943.63 943.50 943.37 943.24 1.24 0.17 943.11 942.98 942.85 942.72 942.58 942.45 942.32 943.19 943.05 941.92 1.23 0.18 941.79 941.65 941.52 941.38 941.25 941.11 940.98 940.84 940.70 840.57 1.33 0.19 940.43 940.29 940.16 940.02 939.88 939.74 939.60 939.46 939.33 939.18 1.31 0.20 939.04 938.90 938.76 938.62 938.48 938.34 938.19 938.05 937.91 937.77 1.20 0.21 937.63 937.48 •937.33 937.19 937.05 936.90 936.75 936.61 936.46 936.32 1.19 0.22 936.17 936.02 935.88 935.73 935.58 935.43 935.28 935.13 934.98 934.83 1.18 0.23 934.68 934.53 934.38 934.23 934.08 933.93 933.77 933.62 933.47 933.33 1.17 0.24 933.16 933.01 932.85 932.70 933.54 932.39 932.33 932.08 931.93 931.76 1.16 0.25 931.60 931.45 931.29 931.13 930.97 930.81 930.65 930.49 930.33 930.17 1.15 0.26 930.01 929.85 929.69 929.53 939.36 929.201 929.04 938.87 938.71 938.55 1.14 0.27 928.38 928.22 928.05 927.88 937.73 927.55| 927.38 927.22 927.05 936.88 1.13 0.28 926.71 926.54 926.37 926.21 926.03 925.86 935.69 925.53 925.35 925.18 1.12 0.29 925.01 924.83 924.66 924.49 924.31 924.14 933.96 933.79 923.61 933.44 1.11 0.30 923.26 923.08 922.90 922.73 932.55 922.37 933.19 932.01 921.83 931.65 1.10 0.31 921.47 921.29 921.11 920.93 920.74 930.56 930.38 920.20 920.01 919.83 1.09 0.32 919.64 919.46 919.27 919.09 918.90 918.71 918.53 918.34 918.15 917.96 1.08 0.33 917.77 917.581 917.39 917.30 917.01 916.83 916.63 916.43 916.24 916.05 1.07 0.34 915.86 915.66 915.47 915.27 915.08 914.88 914.68 914.49 914.39 914.09 1.06 0.35 913.90 913.70 913.50 913.30 913.10 912.90 913.70 913.50 913.29 912.09 1.05 0.36 911.89 911.69 911.48 911.28 911.07 910.87 910.66 910.46 910.25 910.04 1.04 0.37 909.84 909.63 909.42i 909.21 909.00 908.79 908.58 908.37 908.16 907.95 1 1.03- 0.38 907.74 907.52 907.31 907.10 906.88 906.67 906.45 906.24 906.02 905.81 1.02 0.39 905.59 905.37 905.15 904.93 904.71 904.49 904.27 904.05 903.83 903.61 1.01 0.40 903.39 903.17 902.94 902.72 902.50 902.27 902.04 901.82 901.59 901.37 1.00 0.41 901.14 900.91 900.68 900.45 900.22 899.99 899.76 899.53 899.30 899.07 1.00 0.42 898.84 898.60 898.37 898.13 897.90 897.66 897.43 897.19 896.96 896.72 0.99 1 2 3 4 6 6 7 8 9 * Ref, 1 ■"his refe] PS to Prc )PORTIOI TAL PaB Ta, comi non to b oth, on t he prece ding Pag e. *100 TABLE XLIX. CONVERSION TABLE FROM ANNUITY OR ?i{l-\-a) TO SINGLE PREMIUM ON l.OOa INTEREST 4 PER CENT. PROPOBTIONAL, PARTS, 8TJBT. X{l + a). 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0,5 0896.48 5895.38 0894.07 5 892.84 0891.60 1 5890.35 6 1 6 1,6 889.08 887.79 886.49 885.18 883.85 5J882.50 0:881.14 5|879.77 0878.37 1 5;876.97 6 o!875.54 5J874.10 0872.6411 5871.17 6 869.6811 868.171 6 866.64 1 865.101 863.531 1 861.9516 896.24 895.04 893.82 892.59 891.35 890.09 888.82 887.53 886.23 884.91 883.58 882.23 880.87 879.49 878.09 876.68 1875.25 6873.81 872.35 870.87 869.37 867.86, 2,7 3,8 860.35 858. 74 857.10 855.45 853.77 852.08 850.37 848.63 846.88 845.11 843.32 841.50 839.67 837.81 835.93 834.03 832.11 830.17 828.20 826.21 824.20 822.16 oj" 860.03 858.41 856.77 855.11 853.44 851.74 850.02 848.29 846.53 844. 75 842.95 841.14 839.30 837.44 835.55 833. 65 831.72 829.77 827.80 825.81 823.79 821.75 2896.00 7894.80 2 7 2 7 2 7 866.3312 864. 78 7 863. 22 1 2 861.63 7 893.58 892.35 891.10 ' 889.841 888.56 887.27, 885.97 884.^65 J 883.31 ! 881.96 880.59 879.21 877.81 876.40 874.97 873.52 872.05 870.57 869.07 867.56 866.02 3 864.47 8 862.9013 861.32 8 859.71 858.09 856.44 854. 78 853.10 851.40 849. 68 847.94 846.18 844.39 842.59 840. 77 838.93 837.06 835.17 833.27 831.33 829.38 827.41 825.41 823.39 821.34 2,7 I 895.76 894.55 4,9 893.33 4 892.10 9 890.85 889.59 888.31 887.01 885.71 884.38 883.04 881.69 880.32 878.93 877.53 876.11 874.68 873.23 3871.76 8870.27 868.77 867.25 865.72 864.16 859.39 857.76 856.11 854.45 852.76 851.05 849.33 847.59 845.82 844.04 842.23 840.40 838.55 836.68 834.79 832.88 830.95 828.99 3|827.01 8825.00 3|822.98|4 8|820.93 9 3.8 895.52 894.31 893.09 891.85 890.60 889.33 888.05 886.75 885.44 884.12 882.77 881.42 880.04 878.65 877.25 875.83 874.39 872.93 871.46 869.98 4868.47 9 866.95 862.59 4 861.00 9 865.41 863.85 862.27 860.68 859.06 857.43 855.78 854.11 852.42 850. 71 848.98 847.23 845.47 843.68 841.87 840.03 838.18 836.31 834.41 832.50 830.56 828.59 826.61 824.60 822.57 820.52 4,9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 10 10 10 10 10 10 10 10 10 11 11 11 8|ll 11 11 11 12 12 12 12 12 12 12 13 13 13 13 13 13 13 14 14 14 14 14 14 15 15 15 15 15 15 16 16 16 16 16 16 3 4 12 12 12 12 13 13 13 13 13 13 14 14 14 14 14 14 14 15 15 15 15 15 15 16 16 16 16 16 17 17 17 17 17 18 18 18 18 18 19 19 19 19 19 20 20 20 20 21 6 7 8 9 14 15 15 15 15 15 15 16 16 16 16 16 16 17 17 17 17 18 18 18 18 18 18 19 19 19 19 20 20 20 20 21 21 21 21 21 22 22 22 23 23 23 23 24, 24 24 24 25 6^ 17 17 17 17 18 18 18 18 18 19 19 19 19 20 20 20 20 20 21 21 21 21 22 22 22 22 23 23 23 24 24 24 24 25 25 25 25 26 26 26 27 27 27 28 28 28 29 19 19 20 20 20 20 21 21 21 21 22 22 22 22 22 23 23 23 24 24 24 24 25 25 25 26 22 22 22 22 23 23 23 23 24 24 24 24 25 25 25 26 26 26 26 27 27 28 28 28 28 29 26 29 26 30 2731 28 31 29 33 37 32|36 3337 8 9 TABLE XLIX. *101 CONVEKSION TABLE FROM ANNUITY OR X{l-\-a) TO SINGLE PREMIUM ON 1,000. INTEREST 4 PER CENT. PROPORTIONAL PARTS, SUBT. A(l + «). 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0,5 0820.10 5 818.03 815.91 813.78 811.62 1,6 5 809.44 Oj807.24 5j805.00 0802.75 5 800.46 798.15 795.81 793.45 791.06 788.64 786.19 783.71 781.21 778.68 776.11 773.52 770.90 768.25 765.56 762.85 760.10 757.32 754.51 751.67 748.80 745.89 742.94 739.97 736.96 733.91 730.83 727.71 724.56 721.37 718.14 714.88 711.58 708.24 704.86 701.44 697.99 694.49 690.95 0,5 819.69 817.60 815.49 813.35 811.19 809.00 806.79 804.55 802.29 800.00 797.69 795.34 792.97 790.58 788.15 785.70 783.22 780.71 778.17 775.60 773.00 770.37 767.71 765.02 762.30 759.55 756.76 753.95 751.10 748.22 745.30 742.35 739.37 736.35 733.30 730.21 727.09 723.93 720.73 717.49 714.22 710.91 707.57 704.18 700.76 697.29 693.78 690.24 2,7 819.27 817.18 815.06 813.92 810.75 808.56 2|806.35 7804.10 801.84 799.54 797.33 794.87 793.50 790.09 787.66 785.30 783.73 780.30 777.66 775.08 772.48 769.84 767.18 764.48 761.75 758.99 756.30 753.38 750.53 747.64 744.71 741.76 738.77 735.74 733.68 729.59 726.46 723.29 730.08 716.84 713.56 710.35 706.89 703.50 700.07 696.59 693.08 689.53 2,7 3,8 818.85 816.76 814.64 813.49 810.33 808.13 805.90 803.65 801.38 799.08 796.75 794.40 793.03 789.61 787.17 784.71 783.33 779.69 777.14 774.56 771.95 769.31 766.64 763.94 761.20 758.44 755.64 753.81 749.95 747.05 744.13 741.16 738.17 735.13 733.07 728.96 725.83 722.65 719.44 716.19 712.90 709.58 706.33 703.83 699.37 695.89 3 692.37 8 688.81 3,8 4,9 818.44 816.33 814.21 812.06 809.88 807.68 805.45 803.20 800.93 798.63 796.28 793.92 791.54 789.13 786.68 784.31 781.71 779.19 776.63 774.04 771.43 768.78 766.10 763.39 760.65 757.88 755.08 753.34 749.37 746.47 743.54 740.57 737.56 734.53 731.45 738.34 735.19 733.01 718.79 715.54 713.24 708.91 705.54 702.13 698.68 695.19 691.66, 688.09 4,9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 13 13 13 13 13 13 12 13 13 13 13 13 13 14 14 14 14 14 14 2 16 31 16 31 33 37 33 27 6 35 35 26 26 26 36 37 37 37 38 38 38 39 39 39 30 30 30 31 31 31 33 32 33 33 33 34 34 34 35 35 36 36 37 37 37 38 38 39 39 40 40 41 41 41 42 43 43 8 33 34 34 35 35 35 36 36 41 37 41 37143 37:43 38|43 38 43 39 44 39 44 40 45 39 30 30 30 31 31 31 33 33 32 33 33 33 34 34 35 35 35 36 36 37 37 38 43 48 38 43 49 ♦102 TABLE XL IX. CONVERSION TABLE FROM ANNUITY OR A(l + a) TO SINGLE PREMIUM ON 1,000 PROPORTIONAL PARTS, 8UBT. INTEREST 4 PER CENT. A(l+a). 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 l.OB 1.09 1.10 1.11 1.12 1.13 1.14 0.5 687.37 683.75 680.09 376.39 673.64 668.85 665.01 661.13 657.21 653.24 649.23 645.16 641.06 636.90 632.70 5 628.44 624.14 5 619.79 615.38 610.93 606.43 601.87 597.26 592.59 587.88 583.10 578.28 573.39 568.45 563.46 558.40 553.29 548.12 542.88 537.59 532.24 526.82 521.34 0515.80 5510.19 1,6 504.52 498. 78 492.98 487.11 481.17 475.16 469.08 462.94 686.65 683.02 679.35 675.64 671.88 668.09 664.24 660.35 656.42 652.44 648.42 644.35 l|640.23 61636.06 631.85 6 627.59 623.27 618.91 614.50 610.03 605.52 600.95 596.33 591.66 586.93 582.14 577.31 572.41 567.46 562.45 557.38 552.26 547.07 541.83 536.52 531.16 525.73 520.24 514.68 509.06 503.38 497.63 491.81 485.92 479.97 473.95 2,7 467.86 461.70 0,6 1,6 685.93 682.29 678.61 674.89 671.13 667.32 663.47 659.57 655.63 651.64 647.61 643.53 2639.40 7635.22 2631.00 71626.73 2622.40 618.03 613.61 609.14 604.61 600.03 595.40 590.71 585.97 581.18 576.33 571.43 566.46 561.44 556.36 551.23 546.03 540. 77 535.46 530.08 524.64 519.13 513.56 507.93 502.23 496.47 490.64 7'484.74 478.77 472.74 466.63 460.46 2?r" 3,8 685.21 681.56 677.87 674.14 670.37 666.55 662.69 658.79 654.83 650.84 646.80 642.71 638.57 634.38 630.15 625.87 621.53 617.15 612.72 608.23 603.70 599.11 594.47 589.77 585.02 580.22 575.35 8 570.44 3565.46 8560.43 555.34 550.19 544.98 539.72 534.39 528.99 523.54 518.02 512.44 506.80 501.08 495.31 4,9 4:684.48 9680.83 489.46 483.55 477.57 471.52 465.40 459.21 3^8" 677.13 673.39 669.61 665,78 661.91 658.00 654.04 650.03 645.98 641.88 637.73 633.54 629.30 625.00 620.66 616.27 4611.83 9'607.33 602.78 598.18 593.53 588.82 4584.06 579.25 574.38 569.45 564.46 559.42 554.32 549.16 543.94 538.65 533.31 527.91 522.44 516.91 511.32 505.66 499.93 494.14 488.29 482.36 476.37 470.30 464.17 457.97 4^ 1|2 14 15 15 15 15 15 16 16 16 16 16 16 17 17 17 17 17 18 18 18 18 18 19 19 19 19 20 20 20 20 20 21 21 21 21 22 8 8 8 9 9 9 9 9 9 9 9 b 9 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 22 22 22 23 23 23 23 24 24 24j32 2433 25J33 25^33 25'34 26 34 26134 2635 26|35 27 36 28 38 30,40 30140 31 ki 41 22 22 22|34 23'34 23134 23 23 42 42 43 43 44 44 45 45 46 46 47 48 48 49 49 50 36 37 37 38 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 46 46 47 47 48 48 49 49 50 51 51 52 52 53 54 54 65 55 56 57 57 58 59 59 60 61 61 62 8 56 65 67|66 5767 58 67 59 68 69 63 73 74 6 16 58 69 59 60 61 61 62 63 64 64 65 66 67 67 68 69 70 71 71 72 73 74 76 76 76 77 78 79 80 81 82 83 84 86 86 87 88 89 90 91 92 93 94 96 96 97 98 100 ~8~ 9 65 66 67 68 68 69 70 71 71 72 73 74 75 76 77 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 96 96 98 99 100 101 102- 103 104 106 107 108 109 111 112 TABLE XLIX. *103 CONVERSION TABLE PROM ANNUITY OR ^l+a) TO SINGLE PREMIUM ON 1,000. INTEREST 4 PER CENT. PROPORTIONAL PARTS, SUBT. Hl + a). 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 ).28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 0,5 0456.72 5 450.43 444.06 437.62 431.11 424.52 1,6 0417.86 1 5411.12 6 0404.30 5 397.40 0390.43 5 383.37 376.23 369.00 0361.70 5354.31 0346.83 5 339.27 0331.61 5 323.87 455.46 449.16 442.78 436.33 429.80 423.20 416.52 409.76 402.93 396.01 389.02 381.95 374.79 367.55 360.23 352.82 345.32 337.74 2,7 1330.07 2 6322.32 7 2 7 2 7 2 7 2 7 2401.55 3 7394.62 8 454.21 447.89 441.50 435.03 428.49 421.87 415.17 408.40 316.05 308.13 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 77.37 1 66.6916 300.11 292.01 283.81 275.52 267.13 258.64 250.06 241.38 232.59 223.71 214.72 205.62 196.42 187.12 177.71 168.18 158.55 148.81 138.95 128.98 118.90 108.69 98.37 87.93 0,6 314.47 306.53 298.50 290.38 282.16 273.85 265.44 256.93 248.33 239.63 230.82 221.92 212.91 203.79 194.57 185.25 175.81 166.27 156.61 146.85 136.97 126.97 116.87 106.64 96.29 85.83 75.24 64.54 1,6 3,8 4,9 452.95 4 446.62 9 440.21 433.73 387.61 380.52 373.35 366.09 358.75 351.33 343.81 336.22 328.53 320. 75 312.89 304.93 296.88 288.74 280.51 272.17 263.75 255.22 246.60 237.87 229.05 220.12 211.09 201.96 192.71 183.37 173.91 164.35 154.67 144.88 134.98 124. 96i 114.83 104.58 94.21 83.72 73.11 62.38 2/7~ 427.17 420.54 413.83 407.04 400.17 393.23 386.^ 379.09 371.90 364.63 357.27 349.83 342.30 4 334.69 9 326.98 319.19 311.30 4 303.33 9 3295.26 8287.10 278.85 270.50 262.05 253.50 244.86 236.12 227.27 218.32 209.27 200. 12 190.85 181.48 172.01 162.42 152.72 142.91 132.98 122.94 112.79 102.52 92.12 81.61 451.69 445.34 438.92 432.42 425.85 419.20 412.47 405.67 398.79 391.83 384.79 377.66 370.45 363.16 355.79 348.33 340.79 333.15 325.43 317.62 309.72 301.72 293.64 285.46 ^77.19 268.82 260.35 251.78 243.12 234.36 225.49 216.52 207.45 198.27 188.99 179.60 170.10 160.49 150.77 140.93 4|l30.99 9120.92 70.98 4 60.22 9 110.74 100.45 90.03 79.49 68.83 58.05 3,8 I 4,9 13 13 13 13 13 13 13 14 14 14 14 14 14 15 15 15 15 15 15 16 16 16 16 16 17 17 17 17 17 18 18 18 18 18 19 32148 19 19 19 19 20 20 20 20 21 21 21 21 22 32 33 33 34 34 34 35 35 36 36 36 37 37 38 38 39 39 39 40 40 41 41 42 42 43 43 2 49 49 50 50 51 51 52 53 53 54 55 55 56 56 57 58 58 59 60 60 61 62 63 63 64 65 50 61 52 52 53 53 54 55 55 56 56 57 58 58 59 60 60 61 62 63 63 64 65 66 66 67 68 69 69 70 71 72 73 74 74 75 76 77 78 79 80 81 82 83 84 85 85 86 63 64 64 65 66 67 67 68 69 70 71 71 72 73 74 75 76 77 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 99 100 101 102 103 104 106 107 108 6 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 101 102 103 104 105 107 108 109 110 112 113 114 116 117 89 90 91 92 93 94 95 97 98 99 100 101 102 103 105 106 107 108 109 111 112 113 115 116 117 119 120 122 123 124 126 127 129 130 132 133 135 136 8 101 102 103 104 105 107 108 109 110 112 113 114 116 117 118 120 121 123 124 125 127 128 130 131 133 134 136 137 139 141 142 144 9 113 115 116 117 119 120 121 123 124 125 127 129 130 131 133 135 136 138 139 141 143 144 146 148 149 151 153 154 156 158 160 162 146164 147166 118138 120 140 121 123 124 125 127 128 130 6 141 143 144 146 148 150 J51 149 151 152 154 156 158 160 161 163 165 167 169 171 173 8~ 167 169 172 173 175 177 179 181 184 186 188 190 192 195 9 *104 TABLE L. SINGLE PREMIUM OR RESERVE BY MONTHS, ON PAID-UP INSURANCE OF 1,000 FOR THE WHOLE LIFE. 4 PER CENT. AGE. X+n. Om. |m. l|m. 3im. 3im. Hm. 5^m. AGE. x-\-n. *Dec. Nov. Oct. Sept. Aug. July. 10 202.630 303.753 203.000 303.346 303.493 303.739 303.986 10 11 205.587 305.716 205.972 306.338 306.485 306.741 306.997 11 12 208.665 308.799 209.067 309.334 209.603 209.869 210.137 12 13 211.878 313.017 213.396 313.575 213.854 213.133 313.413 13 14 215.224 315.369 315.658 315.948 316.338 216.538 316.817 14 15 218.699 218.850 319.153 319.454 319.755 330.057 330.359 15 16 223.321 333.478 333.793 333.107 333.433 333. 736 334.051 16 17 226.095 336.358 336.586 336.913 337.339 337.566 337.894 17 18 230.019 330.189 330.539 330. §69 331.310 331.550 331.890 18 19 234.101 334.378 234.633 334.986 335.340 335.694 336.048 19 20 238.349 338.533 338.901 339.369 339.636 340.004 340.373 20 21 242.763 343.954 343.336 343.718 344.100 344.482 344.864 21 22 247.348 347.547 347.944 348.341 348.739 349.136 349.533 22 23 252.116 353.333 353.735 353.147 353.560 353.973 354.385 23 24 257.065 257.379 357.707 358.135 358.564 358.993 259.430 24 25 262.203 363.435 363.870 263.315 363.759 364.304 364.649 25 26 267.540 367.771 368.331 368.693 369.153 369.614 370.074 26 27 273.069 373.308 373.786 374.364 374.743 375.330 375.698 27 28 278.806 379.054 379.550 380.046 380.541 381.037 381.533 28 29 284.756 285.012 385.535 386.038 386.551 387.064 387.577 29 30 290.909 291.175 391.706 393.337 292.768 393.399 393.830 30 31 297.284 397.559 398.109 398.659 399.310 399.760 300.310 31 32 303.886 304.171 304.739 305.308 305.878 306.447 307.015 32 33 310.714 311.008 311.596 313.185 313.773 313.361 313.949 33 34 317.773 318.077 318.685 319.392 319.901 330.508 331.116 3^ 35 335.066 335.379 336.006 336.633 337.360 ^37.887 338.514 35 36 333.588 333.913 333.559 334.307 334.854 335.503 336.149 36 37 340.359 340.693 341.361 343.029 343.696 343.364 344.033 37 38 348.373 348.717 349.405 350.093 350.783 351.470 353.158 38 39 356.630 356.984 .357.693 358.403 359.110 359.819 360.538 39 40 365.134 365.498 366.337 366.956 367.684 368.413 369.143 40 41 373.879 374.354 375.004 375.754 376.504 377.354 378.004 41 42 383.879 383.364 384.034 384.804 385.574 386.344 387.114 42 43 393.119 393.514 393.305 394.096 394.887 395.678 396.469 43 44 401.608 403.013 403.824 403.635 404.446 405.257 406.068 44 45 411.338 411.753 413.584 413.415 414.346 415.077 415.908 45 46 431.309 431.734 433.585 433.435 434.387 435.137 435.988 46 47 431.516 431.951 433.831 433.690 434.560 435.439 436.399 47 48 441.953 443.397 443.386 444.175 445.064 445.953 446.843 48 49 453.618 453.071 453.978 454.885 455.792 456.699 457.606 49 50 463.500 463.963 464.887 465.811 466.736 467.660 468.585 50 51 474.594 475.065 476.006 476.947 477.888 478.839 479.770 51 52 485.889 486.368 487.335 488.383 489.240 490.197 491.154 52 53 497.376 497.863 498.835 499.807 500.780 501.753 503.735 53 54 509.046 509.539 510.536 511.514 513.500 513.488 514.475 54 * Months of Entry for Annual Valuation Dec. 31. ♦105 TABLE L. SINGLE PREMIUM OR RESERVE BY MONTHS, ON PAID-UP INSURAJ^CE OF 1,000 FOR THE WHOLE LIFE. 4 PER CENT. AGE. 6m. 6|m. 7im. 8im. 9^m. lO^m. IHm. AGE, x-\-n. 10 June. May. April. March. Feb. Jan. 10 204.109 204.232 204.479 204.725 204.971 205.217 205.464 11 207.126 207.255 207.511 207.767 208.024 208.280 208.536 11 12 210.271 210.405 210.673 210.940 211.209 211.476 211.744 12 13 213.551 213.690 213.969 214.248 214.527 214.806 215.085 13 14 216.962 217.107 217.396 217.686 217.975 218.265 218.554 14 15 220.510 220.661 220.963 221.265 221.566 221.868 222.170 15 16 224.208 224.365 224.680 224.994 225.309 225.623 225.938 16 17 228.057 228.220 228.548 228.875 229.201 229.528 229.856 17 18 232.060 232.2.30 232.570 232.910 233.251 233.591 233.931 18 19 236.225 236.402 236.756 237.110 237.464 237.818 238.172 19 20 240.556 240.740 241.108 241.476 241.843 242.211 242.579 20 21 245.055 245.246 245.628 246.010 246.393 246.775 247.157 21 22 249.732 249.931 250.328 250.725 251.123 251.520 251.917 22 23 254.591 254.797 255.210 255. 622 256.034 256.446 256.859 23 24 259.634 259.848 260.276 260. 704 261.133 261.561 261.989 24 25 264.871 265.093 265.538 265.983 266.428 266.873 267.318 25 26 270.305 270.536 270.996 271.457 271.917 272.378 272.838 26 27 275.937 276.176 276.654 277.132 277.611 278.089 278.567 27 28 281.781 282.029 282.525 283.021 283.516 284.012 284.508 28 29 287.833 288.089 288.602 289.115 289.627 290.140 290.653 29 30 294.096 294.362 294.893 295.424 295.956 296.487 297.018 30 31 300.585 300.860 301.410 301.960 302.511 303.061 303.611 31 32 307.300 307.585 308.153 308.722 309.292 309.861 310.429 32 33 314.243 314.537 315.125 315.714 316.302 316.891 317.479 33 34 321.420 32L724 322.332 322.939 323.547 324.154 324.762 34 35 828.827 329.140 329.767 330.394 331.021 331.648 332.275 35 36 336.473 336.797 337.444 338.092 338.740 339.388 340.035 36 37 344.366 344.700 345.368 346.036 346.703 347.371 348.039 37 38 352.502 352.846 353.534 354.222 354.910 355.598 356.286 38 39 360.882 361.236 361.945 362.654 363.362 364.071 364.780 39 40 369.506 369.870 370.599 371.328 372.057 372.786 373.515 40 41 378.379 378.754 379.504 380.254 381.004 381.754 382.504 41 42 387.499 387.884 388.654 389.424 390.194 390.964 391.734 42 43 396.864 397.259 398.050 398.841 399.631 400.422 401.213 43 44 406.473 406.878 407.689 408.500 409.311 410.122 410.933 44 45 416.323 416.738 417.569 418.400 419.232 420.063 420.894 45 46 426.413 426.838 427.689 428.539 429.390 430.240 431.091 46 47 436.734 437.169 438.039 438.908 439.779 440.648 441.518 47 48 447.286 447.730 448.619 449.508 450.396 451.285 452.174 48 49 458.059 458.512 459.419 460.326 461.233 462.140 .463.047 49 50 469.047 469.509 470.434 471.358 472.283 473.207 474.132 50 51 480.241 480.712 481.653 482.594 483.536 484.477 485.418 51 52 491.633 492.112 493.069 494.026 494.983 495.940 496.897 52 53 503.211 503.697 504.670 505.642 506.615 507.587 508.560 53 54 514.968 515.461 516.448 517.436 518.422 519.410 520.397 54 *106^ TABLE L. SINGLE PREMIUM OR RESERVE BY MONTHS, ON PAID-UP INSURANCE OP 1.000 FOR THE WHOLE LIFE. 4 PER CENT. ACtE. x+n. Cm. W- l^m. 2^m. S^m. ^m. 5|m. AGE. x + n. *Dec. Nov. Oct. Sept. Aug. July. 55 520.890 521.390 522.389 523.388 524.388 525.387 526.386 55 56 532.882 533.388 534.399 535.411 536.421 537.433 538.444 56 57 545.019 545.530 546.551 547.573 548.594 549.616 550.637 57 58 557.277 557.792 558.823 559.853 560.884 561.914 562.945 58 59 569.642 570.161 571.200 572.238 573.276 574.314 575.353 59 60 582.103 582.625 583.670 584.714 585.759 586.803 587.848 60 61 594.636 595.160 596.208 597.256 598.305 599.353 600.401 61 62 607.214 607.740 608.791 609.842 610.893 611.944 612.995 62 63 619.829 620.355 621.408 622.460 623.513 624.565 625.618 63 64 632.458 632.983 634.034 635.085 636.136 637.187 638.238 64 65 645.069 645.593 646.641 647.689 648.736 649.784 650.832 65 66 657.643 658.165 659.208 660.251 661.295 662.338 663.381 66 67 670.162 670.680 671.716 672.753 673.789 674.826 675.862 67 68 682.598 683.112 684.141 685.169 686.198 687.226 688.255 68 69 694.941 695.450 696.467 697.484 698.502 699.519 700.536 69 70 707.148 707.651 708.656 709.661 710.666 711.671 712.676 70 71 719.211 719.706 720.697 721.688 722.678 723.669 724.660 71 72 731.099 731.586 732.561 733.536 734.510 735.485 736.460 72 73 742.795 743.274 744.232 745.189 746.148 747.105 748.063 73 74 754.288 754.757 755.694 756.631 757.568 758.505 759.442 74 75 765.535 765.994 766.910 767.828 768.744 769.662 770.578 75 76 776.539 776.987 777.882 778.778 779.673 780.569 781.464 76 77 787.285 787.721 788.592 789.464 790.335 791.207 792.078 77 78 797.743 798.167 799.014 799.862 800.710 801.558 802.405 78 79 807.914 808.325 809.148 809.970 810.792 811.614 812.437 79 80 817.783 818.181 818.978 819.775 820.573 821.370 822.167 80 81 827.346 827.732 828.,505 829.277 830.049 830.821 831.594 81 82 836.614 836.987 837.733 838.480 839.225 839.972 840.718 82 83 845.569 845.931 846.655 847.379 848.103 848.827 849.551 83 84 854.256 854.607 855.309 856.010 856.712 857.413 858.115 84 85 862.677 863.019 863.704 864.389 865.073 865.758 866.443 85 86 870.893 871.231 871.908 872.584 873.262 873.938 874.615 86 87 879.012 879.349 880.022 880.695 881.367 882.040 882.713 87 88 887.089 887.437 888.132 888.827 889.523 890.218 890.913 88 89 895.433 895.740 896.354 896.969 897.583 898.198 898.812 89 90 902.805 903.066 903.589 904.111 904.635 905.157 905.680 90 91 909.076 909.321 909.811 910.301 910.792 911.282 911.772 91 92 914.958 915.194 915.665 916.137 916.608 917.080 917.551 92 93 920.617 920.841 921.288 921.735 922.183 922.630 923.077 93 94 925.985 926.203 926.639 927.075 927.512 927.948 928.384 94 95 931.219 931.457 931.934 932.410 932.887 933.363 933.840 95 96 936.936 537.147 937.570 937.992 938.414 938.836 939.259 96 97 942.005 942.305 942.906 943.506 944.108 944.708 945.309 97 98 949.212 949.726 950.758 951.780 952.806 953.833 954.860 98 99 1 961.537 1 963.140 966.345 969.550 972.756 975.961 979.166 99 Months of Entry for Annual Valuation Dec. 31. ♦lO-? TABLE L. SINGLE PREMIUM OR RESERVE BY MONTHS, ON PAID-UP INSURANCE OF 1,000 FOR THE WHOLE LIFE. 4 PER CENT. AGE. x-\-n. 6m. 6im. 7im. 8im. 9|^m. 10|m. 11 |m. AGE. X-\-7l. June. May. April. March. Feb. Jan. 55 526.886 627.386 528.385 529.384 530.384 531.383 532.382 55 56 538.950 539.456 540.467 541.479 542.490 543.502 544.513 56 57 551.148 551.659 552.680 553.702 554.723 555.745 556.766 57 58 563.460 563.975 565.006 566.036 567.066 568.096 569.127 58 59 575.872 576.391 577.430 578.468 579.507 580.545 581.584 59 60 588.370 588.892 589.937 590.981 592.025 593.069 594.114 60 61 600.925 601.449 602.497 603.545 604.594 605.642 606.690 61 62 613.521 614.047 615.098 616.149 617.201 618.252 619.303 62 63 626.144 626.670 627.723 628.775 629.827 630.879 631.932 63 64 638.763 639.288 640.339 641.390 642.442 643.493 644.544 64 65 651.356 651.880 652.928 653.976 655.023 656.071 657.119 65 66 663.903 664.425 665.468 666.511 667.554 668.597 669.640 66 67 676.380 676.898 677.934 678.971 680.007 681.044 682.080 67 68 688.769 689.283 690.312 691.340 692.370 693.398 694.427 68 69 701.045 701.554 702.571 703.588 704.605 705.622 706.639 69 70 713.179 713.682 714.687 715.692 716.698 717.703 718.708 70 71 725.155 725.650 726.641 727.632 728.622 729.613 730.604 71 72 736.947 737.434 738.409 739.384 740.358 741.333 742.308 72 73 748.542 749.021 749.979 750.936 751.894 752.851 753.809 73 74 759.911 '760.380 761.317 762.254 763.192 764.129 765.066 74 75 771.037 771.496 772.412 773.330 774.246 775.164 776.080 75 76 781.912 782.360 783.255 784.151 785.046 785.942 786.837 76 77 792.514 792.950 793.821 794.693 795.564 796.436 797.307 77 78 802.829 803.253 804.100 804.948 805.795 806.643 807.490 78 79 812.848 813.259 814.082 814.904 815.727 816.549 817.372 79 80 822.565 822.963 823.760 824.557 825.354 826.151 826.948 80 81 831.980 832.366 833.139 833.911 834.683 835.455 836.228 81 82 841.091 841.464 842.210 842.957 843.703 844.450 845.196 82 83 849.913 850.275 850.999 851.723 852.446 853.170 853.894 83 84 858.466 858.817 859.519 860.220 860.923 861.624 862.326 84 85 866.785 867.127 867.812 868.497 869.181 869.866 870.551 85 86 874.953 875.291 875.968 876.644 877.321 877.997 878.674 86 87 883.050 883.387 884.060 884.733 885.406 886.079 886.752 87 88 891.261 891.609 892.304 892.999 893.695 894.390 895.085 88 89 899.119 899.426 900.040 900.655 901.269 901.884 902.498 89 90 905.941 906.202 906.725 907.247 907.770 908.292 908.815 90 91 912.017 912.262 912.752 913.242 913.733 914.223 914.713 91 92 917.787 918.023 918.494 918.966 919.438 919.910 920.381 92 93 923.301 923.525 923.972 924.419 924.867 925.314 925.761 93 94 928.602 928.820 929.256 929.692 930.129 930.565 931.001 94 95 934.078 934.316 934.793 935.269 935.745 936.221 936.698 95 96 939.470 939.681 940.104 940.526 940.949 941.371 941.794 96 97 945.609 945.909 946.510 947.110 947.711 948.311 948.912 97 98 955.374 955.888 956.915 957.942 958.969 959.996 961.023 98 99 980.769 982.372 985.577 988.782 991.987 995.192 998.397 99 ♦108 TABLE LI. LOGARITHM OF THE SINGLE PREMIUM BY MONTHS, ON 1 INSURED FOR THE WHOLE LIFE. 4 PER CENT. AGE. Om. ^m. l^m. 2^m. 3^m. 4im. 5Jm. AGE x-\-n. ♦Dec. Nov. Oct. Sept. Aug. July. 10 1.306703 1.306967 1.307496 1.308022 1.308549 1.309074 1.309601 10 11 .312996 .313267 .313808 .314348 .314889 .315427 .315964 11 12 .319450 .319728 .320286 .320839 .321395 .321948 .322503 12 13 .326086 .326371 .326942 .327512 .328082 .328651 .329219 13 14 .332890 .333183 .333766 .334349 .334932 .335514 .336093 14 15 1.339847 1.340147 1.340745 1.341344 "1.341939 T. 342535 1.343131 15 16 .346980 .347287 .347902 .348513 .349126 .349736 .350347 16 17 .354290 .354604 .355233 .355860 .356483 .357107 .357733 17 18 .361763 .362084 .362725 .363366 .364007 .364645 .365282 18 19 .369403 .369731 .370388 .371042 .371696 .372349 .373000 19 20 1.377213 1.377548 1.378218 1.378886 1.379552 1.380218 1.380884 20 21 .385182 .385524 .386207 .386887 .387568 .388247 .388925 21 22 .393309 .393658 .394354 .395049 .395744 .396436 .397128 22 23 .401601 .401955 .402666 .403373 .404081 .404786 .405492 23 24 .410042 .410404 .411126 .411847 .412568 .413286 .414003 24 25 1.418637 1.419005 T. 419741 i. 420476 T.421207 1.421939 1.422670 25 26 .427389 .427764 .428509 .429255 .429999 .430742 .431482 26 27 .436272 .436653 .437412 .438168 .438925 .439680 .440434 27 28 .445303 .445688 .446459 .447230 .447997 .448764 .449530 28 29 .454473 .454863 .455645 .456424 .457202 .457979 .458754 29 30 T. 463757 i. 464155 1.464945 1.465736 1.466524 1.467311 i.468096 30 31 .473172 .473573 .474376 .475176 .475976 .476774 .477570 31 32 .482711 .483118 .483928 .484738 .485548 .486356 .487160 32 33 .492361 .492771 .493592 .494412 .495228 .496045 .496859 33 34 .502117 .502532 .503362 .504188 .505015 .505839 .506662 34 35 1.511971 1.512390 1.513226 1.514060 1.514893 T. 515724 1.516554 35 36 .521906 .522330 .523173 .524015 .524855 .525695 .526532 36 37 .531937 .532363 .533214 .534063 .534909 .535755 .536599 37 38 .542045 .542473 .543329 .544184 .545037 .545888 .546738 38 39 .552218 .552649 .553511 .554370 .555227 .556084 .556939 39 40 T. 562452 1.562885 1.563750 T.564614 1.565475 1.566335 1.567193 40 41 .572731 .573166 .574036 .574904 .575770 .576634 .577496 41 42 .583062 .583498 .584369 .585239 .586107 .586974 .587839 42 43 .593418 .593855 .594730 .595602 .596473 .597342 .598209 43 44 .603802 .604240 .605115 .605989 .606861 .607731 .608599 44 45 1.614199 T. 614637 1.615512 1.616386 1.617259 1.618129 1.618997 45 46 .624601 .625038 .625914 .626787 .627660 .628529 .629397 46 47 .634997 .635434 .636308 \ .637179 .638050 .638917 .639784 47 48 .645376 .645812 .646684 1 .647554 .648423 .649289 .650154 48 49 .655732 .656166 .657035 1 .657901 .658767 .659630 .660492 49 50 1.666050 1.666483 1.667347 1.668210 T. 669071 1.669930 1.670789 50 51 .676322 .676753 .677612 .678470 .679326 .680180 .681033 51 52 .686537 .686965 .687819 .688671 .689522 .690370 .691218 52 53 .696685 .697109 .697957 .698802 .699647 .700489 .701331 53 54 .706757 1 .707177 .708018 .708858 .709694 .710530 .711364 54 * Moutha of Entry for Annual Valuation Dec. 31. ♦109 TABLE LI. LOGARITHM OF THE SINGLE PREMIUM BY MONTHS, ON 1 INSURED FOR THE WHOLE LIFE. 4 PER CENT. AGE. x-\-n. 10 6m. 6^m. 7ini. . 8^in. 9im. lO^m. ll^m. AGE. x-\-n. 10 June. May. April. March. Feb. Jan. 1.309862 1.310124 1.310649 1.311171 i.311692 1.312214 1.312735 11 .316235 .316505 .317041 .317577 .318113 .318648 .319181 11 12 .322779 .323056 .323609 .324159 .324712 .325261 .325811 12 13 .329502 .329784 .330351 o330917 .331482 .332046 .332610 13 14 .336384 .336674 .337252 .337830 .338407 .338984 .339559 14 15 1.343428 1.343726 1.344320 T.344913 1.345503 1.346095 1.346685 16 16 .350651 .350955 .351564 .352171 .352778 .353384 .353989 16 17 .358043 .358354 .358977 .359599 .360217 .360835 .361456 17 18 .365600 .365918 .366544 .367188 .367824 .368456 .369088 18 19 .373326 .373651 .374301 .374950 .375598 .376244 .376891 19 20 1.381216 T.381548 1.382211 1.382874 1.383533 1.384194 1.384853 20 21 .389264 .389602 .390278 .390953 .391628 .392301 .392973 21 22 .397474 .397820 .398510 .399198 .399886 .400573 .401258 22 23 .405843 .406194 .406898 .407598 .408298 .408996 .409694 23 24 .414362 .414720 .415434 .416148 .416862 .417573 .418283 24 25 T. 423035 1.423398 1.424127 1.424854 1.425580 1.426305 1.427028 25 26 .431854 .432225 .432963 .433701 .434436 .435172 .435905 26 27 .440810 .441186 .441937 .442687 .443437 .444184 .444930 27 28 .449912 .450294 .451057 .451819 .452578 .453337 .454095 28 29 .459141 .459527 .460299 .461071 .461840 .462608 .463375 29 30 T. 468489 1.468882 1.469665 1.470446 1.471227 T. 472006 1.472783 30 31 .477967 .478364 .479158 .479949 .480741 .481530 .482317 31 32 .487563 .487965 .488766 .489568 .490369 .491167 .491963 32 33 .497265 .497672 .498483 .499294 .500102 .500910 .501715 33 34 .507073 .507484 .508304 .509121 .509937 .510751 .511566 34 35 L516967 1.517381 1.518207 1.519032 1.519855 T. 520677 T. 521498 35 36 .526950 .527368 .528202 .529035 .529866 .530696 .531524 36 37 .537020 .537441 .538282 .539121 .539958 .540793 .541628 37 38 .547161 .547585 .548431 .549275 .550118 .550960 .551799 38 39 .557365 .557791 .558643 .559493 .560339 .561186 .562031 39 40 1.567621 1.568049 1.568904 1.569758 r. 570609 T.571460 T. 572308 40 41 .577927 .578357 .579216 .580074 .580929 .581783 .582636 41 42 .588271 .588702 .589563 .590422 .591280 .592136 .592991 42 43 .598641 .599074 .599938 .600800 .601659 .602518 .603375 43 44 .609031 .609464 .610329 .611192 .612053 .612913 .613771 44 45 1.619430 1.619863 1.620728 1.621592 1.622454 1.623314 1.624173 45 46 .629830 .630263 .631128 .631990 .632852 .633711 .634569 46 47 .640217 .640649 .641513 .642373 .643234 .644092 .644948 47 48 .650585 .651016 .651878 .652737 .653594 .654451 .655306 48 49 .660921 .661351 .662209 .663065 .663921 .664774 .665625 49 50 1.671216 1.671644 1.672499 1.673351 1.674203 1.675051 1.675899 50 51 .681459 .681885 .682735 .683582 .684428 .685273 .686116 51 52 .691641 .692064 .692908 .693750 .694590 .695429 .696266 52 53 .701750 .702169 .703007 .703843 .704678 .705511 .706342 53 54 .711780 .712196 .713026 .713857 .714684 .715510 .716335 54 *110 TABLE LI. LOGARITHM OF THE SINGLE PREMIUM BY MONTHS, ON 1 INSURED FOR THE WHOLE LIFE. 4 PER CENT. AGE x-\-n. 55 Om. |m. lim. 2^in. 3im. 4im. 5|m. AGE. x-\-n. *Dec. Nov. Oct. Sept. Aug. July. T. 716746 1.717163 1.717994 1.718823 T. 719652 T. 720480 1.721304 55 56 .726631 .727043 .727865 .728687 .729506 .730324 .731140 56 57 .736412 .736819 .737631 .738442 .739251 .740060 .740866 57 58 .746071 .746473 .747274 .748074 .748873 .749670 .750466 58 59 .755602 • .755998 .756788 .757577 .758364 .759149 .759934 59 60 T. 765000 T. 765389 1.766167 1.766944 1.767719 1.768492 T. 769265 60 61 .774251 .774634 .775398 .776160 .776923 .777683 .778442 61 62 .783342 .783718 .784469 .785217 .785965 .786712 .787457 62 63 .792272 .792641 .793377 .794111 .794845 .795578 .796310 63 64 .801032 .801392 .802113 .802832 .803550 .804267 .804983 64 65 1.809606 1.809959 T. 810664 1.811366 1.812068 1.812769 1.813469 65 66 .817990 .818335 .819023 .819709 .820396 .821080 .821763 66 67 .836180 .826515 .827186 .827856 .828524 .829192 .829858 67 68 .834165 .834492 .835146 .835797 .836450 .837100 .837750 68 69 .841948 .842266 .842900 .843534 .844167 .844799 .845431 69 70 T. 849510 T. 849820 1.850436 1.851051 T.851666 1.852280 1.852892 70 71 .856856 .857156 .857753 .858350 .858945 .859540 .860134 71 72 .863976 .864266 .864844 .865422 .865998 .866574 .867149 72 73 .870869 .871149 .871708 .872266 .872825 .873382 .873938 73 74 .877537 .877807 .878346 .878885 .879422 .879959 .880494 74 75 1.883965 T. 884225 T. 884744 1.885264 1.885781 T. 886300 1.886817 75 76 .890163 .890414 .890914 .891414 .891913 .892411 .892909 76 77 .896132 .896373 .896852 .897332 .897811 .898290 .898768 77 78 .901863, .902094 .902554 .903015 .903475 .903935 .904394 78 79 .907365 .907586 .908028 .908469 .908909 .909349 .909790 79 80 1.912638 T. 912850 1.913272 T. 913695 1.914118 T. 914539 1.914960 80 81 .917687 .917890 .918296 .918700 .919104 .919508 .919911 81 82 .922525 .922719 .923106 .923493 .923879 .924265 .924650 82 83 .927149 .927335 .927707 .928078 .928449 .928820 .929190 83 84 .931588 .931767 .932123 .932479 .932835 .933191 .933546 84 85 1.935848 1.936021 1.936365 1.936710 1.937053 1.937397 1.937740 85 86 .939965 .940134 .940471 .940807 .941145 .941481 .941817 86 87 .943995 .944162 .944494 .944826 .945157 .945488 .945820 87 88 .947967 .948138 .948478 .948818 .949158 .949496 .949836 88 89 .952033 .952182 .952480 .952778 .953075 .953372 .953669 89 90 L 955594 1.955719 1.955971 1.956222 T. 956474 T. 956724 1.956975 90 91 .958600 .958718 .958951 .959185 .959419 .959653 .959886 91 92 .961401 .961513 .961737 .961961 .962184 .962407 .962630 92 93 .964079 .964185 .964395 .964606 .964817 .965028 .965238 93 94 .966604 .966706 .966911 .967115 .967320 .967524 .967728 94 95 1.969052 T.969163 T. 969385 T. 969607 1.969829 T. 970051 T. 970272 95 96 .971710 .971808 .972004 .972119 .972395 .972590 .972786 96 97 .974053 .974191 .974468 .974745 .975022 .975298 .975574 97 98 .977363 .977598 .978068 .978537 .979004 .979473 .979940 98 99 .982986 .983689 .985132 .986570 .988004 .989432 .990856 99 * Months of Entry for Annual Valuation Dec. 31. TABLE LI. ♦Ill LOGARITHM OF THE SINGLE PREMIUM BY MONTHS, ON 1 INSURED FOR THE WHOLE LIFE. 4 PER CENT. A&E, x-\-n. 6m. 6|m. 7iin. 8^in. 9^in. lOim. llim. AGE. x-\-n. June. May. April. March. Feb. Jan. 55 1.721717 T. 722129 T. 722951 T. 723771 1.724590 1.725407 1.726224 55 56 .731548 .731956 .732770 '.733581 .734392 .735202 .736008 56 57 .741368 .741670 .742474 .743276 .744076 .744876 .745673 57 58 .750863 .751260 .752053 .752844 .753634 .754422 .755209 58 59 .760326 .760717 .761499 .762279 .763059 .763836 .764612 59 60 T. 769651 T. 770036 T. 770806 T. 771574 1.772340 1.773105 1.773870 60 61 .778821 .779199 .779955 .780710 .781464 .782216 .782967 61 62 .787830 , .788202 .788945 .789685 .790427 .791165 .791903 62 63 .796674 .797039 .797768 .798496 .799221 .799946 .800670 63 64 .805340 .805697 .806410 .807122 .807834 .808544 .809253 64 65 1.813818 T. 814168 T. 814866 T. 815562 T.816257 T. 816951 T. 81 7644 65 66 .822105 .822446 .823127 .823808 .824486 .825164 .825841 66 67 .830191 .830524 .831187 .831852 .832513 .833175 .833835 67 68 .838073 .838398 ..839045 .839692 .840338 .840983 .841626 68 69 .845746 .846061 .846691 .847319 .847946 .848572 .849197 69 70 T. 853198 1.853505 1.854116 1.854726 1.855336 T. 855945 T. 856553 70 71 .860431 .860727 .861320 .861912 .862502 .863093 .863682 71 72 .867436 .867723 .868297 .868870 .869442 .870014 .870584 72 73 .874216 .874494 .875049 .875603 .876156 .876709 .877261 73 74 .880763 .881031 .881666 .882099 .882634 .883167 .883699 74 75 1.887075 1.887334 T. 887849 T. 888365 T. 888879 T. 889393 T. 889906 75 76 .893158 .893407 .893903 .894400 .894895 .895390 .895885 76 77 .899007 .899246 .899723 .900200 .900675 .901151 .901626 77 78 .904623 .904853 .905310 .905768 .906225 .906682 .907137 78 79 .910009 .910229 .910668 .911106 .911545 .911982 .912420 79 80 T. 915171 T. 915381 T.915801 T. 916221 T. 916640 T. 917060 1.917478 80 81 .920113 .920314 .920718 .921120 .921522 .921923 .922325 81 82 .924843 .925035 .925420 .925806 .926190 .926574 .926957 82 83 .929375 .929560 .929929 .930299 .930667 .931036 .931404 83 84 .933723 .933901 .934256 .934610 .934965 .935318 .935671 84 85 T. 937912 1.938083 1.938426 1.938769 1.939111 1.939452 1.939795 85 86 .941985 .942153 .942488 .942823 .943159 .943494 .943828 86 87 .945985 .946151 .946482 .946813 .947142 .947473 • .947802 87 88 .950005 .950175 .950513 .950851 .951190 .951527 .951865 88 89 .953818 .953965 .954262 .954559 .954855 .955151 .955446 89 90 1.957100 T. 957225 T. 957476 T. 957726 T. 957976 T. 968226 T. 958476 90 91 .960003 .960120 .960353 .960586 .960820 .961052 .961285 91 92 .962742 .962854 .963076 .963299 .963522 .963745 .963968 92 93 .965344 .965449 .965659 .965869 .966080 .966289 .966499 93 94 .967830 .967932 .968135 .968339 .968544 .968747 .968950 94 95 1.970383 T. 970494 T.970716 T. 970937 T. 971158 T. 971379 T. 971600 95 96 .972883 .972981 .973176 .973371 .973567 .973761 .973956 96 97 .975712 .975850 .976125 .976400 .976676 .976951 .977226 97 98 .980174 .980407 .980874 .981339 .981805 .982269 .982734 98 99 .991567 .992276 .993691 .995101 .996506 .997907 .999304 99 *112 TABLE LII. SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON- TINUING m YEARS FROM THE PRESENT AGE. m the order oi time, a; denotes the Age of Entry; x + n the Present Age ; m the unexpired Policy Years; and ?i+wi the whole Period of Insurance. a PFR CFNT PRESEUT AGE. L 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65* m =1. 6.339 6.361 6.393 6.426 6.459 6.502 6.547 6.591 6.647 6.704 6.762 6.842 6.913 6.995 7.101 7.198 7.308 7.431 7.568 7.719 7,896 8.065 8.262 8.486 8.728 8.999 9.277 9.612 9.955 10.346 10.773 11.239 11.758 12.321 12.943 13.628 14.381 15.206 16.108 17.095 18.202 19.396 20.731 22.175 23.756 25.506 27.432 29.521 31.833 34.366 37.154 1^. jim.d. 9.376 9.414 9.462 9.510 9.564 9.628 9.694 9.765 9.848 9.933 10.028 10.142 10.251 10.385 10.536 10.685 10.853 11.042 11.249 11.485 11.741 12.004 12.307 12.645 13.015 13.417 13.854 14.350 14.877 15.470 16.115 16.826 17.609 18.463 19.407 20.444 21.582 *22.828 24.189 25.691 27.350 29.162 31.162 33.333 35.715 38.345 41.219 44.356 47.808 51.590 55.727 .253 .254 .256 .257 .259 .261 .262 .265 .267 .269 .272 .275 .278 .283 .286 .291 .295 .301 .307 .314 .320 .328 .337 .347 .357 .368 .381 .395 .410 .427 .445 .466 .488 .512 .539 .568 .600 .635 .673 .716 .762 .814 .869 .930 .997 1.070 1.149 1.236 1.331 1.435 1.548 2. 2i. -J^m.d. 12.414 12.467 12.530 12.594 12.669 12.755 12.841 12.939 13.049 13.161 13.295 13.441 13.590 13.774 13.971 14.172 14.399 14.652 14.931 15.250 15.587 15.943 16.351 16.804 17.302 17.836 18.430 19.088 19.800 20.593 21.458 22.412 23.460 24.606 25.870 27.260 28.783 30.450 32.271 34.286 36.499 38.928 41.593 44.490 47.675 51.183 '64.089 55.007 168.864 59.190 I 74.079 63.783 79.796 68.814 1 86.034 74. 300 192.841 15.330 15.398 15.476 15.560 15.655 15.760 15.873 15.996 16.132 16.279 16.445 16.629 16.826 17.052 17.300 17.556 17.844 18.164 18.524 18.918 19.343 19.799 20.316 20.890 21.512 22.196 22.942 23.774 24.676 25.674 26.770 27.972 29.294 30.741 32.336 34.086 36.002 38.097 40.397 42.926 45.711 48.756 52.090 55.724 59.715 .243 .244 .246 .247 .249 .250 .253 .255 .257 .260 .263 .266 .270 .273 .277 .282 .287 .293 .299 .306 .313 .321 .330 .341 .351 .363 .376 .391 .406 .423 .443 .463 '.486 .511 .539 .569 .602 .637 .677 .720 .768 .819 .875 936 1.003 1.076 1.155 1.241 1.334 1.435 1.545 18.247 18.329 18.422 18.526 18.640 18.766 18.904 19.053 19.215 19.398 19.595 19.817 20.061 20.331 20.628 20.940 21.289 21.677 22.116 22.586 23.099 23.656 24.281 24.976 25.722 26.555 27.455 28.460 29.551 30.756 32.081 33.532 35.128 36.877 38.801 40.912 43.221 45.744 48.523 51.566 54.924 58.583 62.587 66.959 71.754 76.995 82.721 88.968 95.809 103.254 111.382 3i. 21.047 21.143 21.255 21.377 21.511 21.661 21.823 21.997 22.193 22.406 22.639 22.905 23.192 23.508 23.857 24.228 24.641 25.104 25.616 26.169 26.777 27.436 28.176 28.988 29.876 30.853 31.917 33.101 34.387 35.808 37.368 39.076 40.956 43.014 45.276 47.755 50.464 53.435 56.692 60.266 64.195 68.472 73.157 78.268 83.857 89.966 96.629 103.892 111.819 120.445 129.816 ^m. d. PEESENT AGE. .233 15 .235 16 .236 17 .238 18 .239 19 .241 20 .243 21 .245 22 .248 23 .251 24 .254 25 .257 26 .261 27 .265 28 .269 29 .274 30 .279 31 .286 32 .292 33 .299 34 .307 35 .315 36 .325 37 .334 38 .346 39 .358 40 .372 41 .387 42 .403 43 .421 44 .441 45 .462 46 .486 47 .511 48 .540 49 .570 50 .604 51 .641 52 .681 53 .725 54 .773 55 .824 56 .881 57 .942 58 1.009 69 1.081 60 1.159 61 1.244 62 1.334 63 1.433 64 1.536 65 Ages, 66, 67, continued on Page *117. Formula, Ax+» = 1,000 Ma;+n — Ma TABLE LII. *n3 SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CONTINU- ING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PRESENT AOE. 4. 4i. Ym.d.* 5. bh ^m.d. 6. 6i. |m.d. PEE8ENT AQE. 15 23.847 26.535 .224 29.223 31.808 .215 34.392 36.877 .207 15 16 23.957 26.663 .226 29.369 31.970 .217 34.572 37.073 .208 16 17 24.088 26.811 .227 29.535 32.154 .218 34.773 37.296 .210 17 18 24.229 26.971 .229: 29.713 32.354 .220 34.994 37.537 .212 18 19 24.382 27.147 .230 29.911 32.574 .222 35.237 37.801 .214 19 20 24.556 27.344 .232 30.132 32.816 .224 35.501 38.095 .216 20 21 24.742 27.553 .234 30.364 33.080 .226 35.796 38.416 .218 21 22 24.940 27.784 .237 30.628 33.371 .229 36.114 38.764 .221 22 23 25.171 28.043 .239 30.916 33.691 .231 36.466 39.155 .224 23 24 25.415 28.321 .242 31.227 34.043 .235 36.859 39.583 .227 24 25 25.682 28.632 .246 31.581 34.434 .238 37.287 40,051 .230 25 26 25.994 28.982 .249 31.971 34.867 .241 37.762 40.572 .234 26 27 26.322 29.355 .253 32.387 35.330 .245 38.273 41.132 .238 27 28 26.685 29.768 .257 32.851 35.846 .250 38.841 41.755 .243 28 29 27.087 30.225 .262 33.363 36.416 .254 39.469 42.448 .248 29 30 27.515 30.714 .267 1 33.913 37.034 .260 40.156 43.196 .253 30 31 27.993 31.263 .273 34.534 37.720 .266 40.906 44.018 .259 31 32 28.532 31.871 .278 35.210 38.471 .272 41.731 44.924 .266 32 33 29.115 32.533 .285 35.950 39.296 .279 42.643 45.923 .273 33 34 29.752 33.260 .292 36.768 40.206 .287 43.644 47.022 .282 . 34 35 30.455 34.060 .300 37.665 41.207 .295 44.748 48.226 .290 35 36 31.216 34.930 .310 1 38.644 42.291 .304 45.938 49.536 .300 36 37 32.071 35.896 .319 1 39.721 43.494 .314 47.268 50.989 .310 37 38 33.001 36.959 .330 40.918 44.821 .325 48.724 52.583 .322 38 39 34.029 38.124 .341 42.219 46.269 .338 50.319 54.330 .334 39 40 35.151 39.401 .354 43.651 47.861 .351 52.071 56.247 .348 40 41 36.379 40.799 .368 1 45.218 49.602 .365 53.985 58.343 .363 41 42 37.742 42.345 .384 46.949 51.525 .381 56.102 60.657 .380 42 43 39.223 44.031 .401 48.838 53.623 .399 58.408 63.179 .398 43 44 40.861 45.889 .419 50.918 55.932 .418 60.946 65.954 .417 44 45 42.654 47.926 .439 53.197 58.462 .439 63.727 68.994 .439 45 46 44.621 50.159 .462 55.697 61.236 .462 66.775 72.322 .462 46 47 46.783 52.612 .486 58.441 64.279 .487 70.116 75.968 .488 47 48 49. 151 55.297 .512 61.444 67.605 .513 73.767 79.950 .515 48 49 51.751 58.243 .541 64.734 71.247 .543 77.760 84.309 546 49 50 54.598 61.464 .572 68.330 75.235 .575 82.139 89.080 .578 50 51 57.708 64.992 .607 72.276 79.597 .610 86.919 94.292 .614 51 52 61.125 68.855 .644 76.586 84.370 .649 92.154 99.987 .653 52 53 64.861 73.087 .686 81.312 89.590 .690 97.868 106.198 .694 53 54 68.967 77.723 .730 86.479 95.290 .734 104.101 112.973 .739 54 55 73.465 82.794 .777 92.124 101.518 .783: 110.912 120.368 .788 55 56 78.362 88.320 .830 98.278 108.303 .835 118.327 128.405 .840 56 57 83.726 94.367 .887 105.007 115.703 .891 126.399 137.148 .896 57 58 89.578 100.947 .947 112.316 123.742 .952 135.168 146.636 .956 58 59 95.960 108.123 1.014! 120.287 132.495 1.017 144.704 156.942 1.020 59 60 102.936 115.954 1.085 128.972 142.022 1.088 155.071 168.117 1.087 60 61 110.537 124.479 1.162 138.420 152.358 1.162 166.296 180.211 1.160 61 62 118.817 133.738 1.243 148.660 163.556 1.241 178.452 193.262 1.234 62 63 127.829 143.812 1.332 159.794 175.684 1.324 191.574 207.348 1.315 63 64 137.636 154.727 1.424 171.819 188.7861.4141 205.754 222.516 1.397 64 65 148.250 166.550 1.525 184.850 202.929 1.507 i 221.007 238.801 1.483 65 * Note that J m, d. is ^^ of the diflference of tlie integer or m columns, 5 — 4, etc. *1U TABLE LII. SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON- TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PEE8ENI AaE. m = 7. 7i. ^m.d. 8. 8i. ym.d. 9. 9i. |^m.d. PEESENT AGE. 15 39.362 41.751 .199 44.141 46.442 .192 48.743 50.959 .185 15 16 39.575, 41.985 .201 44.394 46.714 .193 49.034 51.269 .186 16 17 39.818 42.247 .202 44.676 47.015 .195 49.354 51.614 .188 17 18 40.081 42.530 .204 44.979 47.345 .197 49.711 51.993 .190 18 19 40.365 42.842 .206 45.320 47.709 .199 50.099 52,407 .192 19 20 40.689 43.191 .209 1 45.693 48.110 .201 50.526 52.868 .195 20 21 41.035j 43.566 .211 46.097 48.549 .204 51.001 53.373 .198 21 22 41.414 43.982 .214 46.550 49.034 .207 51.518 53.925 .201 22 23 41.844' 44.445, .217 47.046 49.566 .210 52.087 54.533 .204 23 24 42.307, 44.947 .220 47.587 50.148 .213 52.709 55.198 .207 24 25 42.816 45.4991 .224 48.181 50.788 .217 53.394 55.930 .211 25 26 43.381 46.110 .227 48.840 51.496 .221 54.153 56.744 .216 26 27 43.991 46.774 .232 49.556 52.271 .226 54.985 57.630 .220 27 28 44.670 47.513 .237 50.357 53.127 .231 55.897 58.602 .225 28 29 45.427 48.329 .242 51.231 54.065 .236 56.899 59.675 .231 29 30 46.236 49.206 .248 52.175 55.083 .242 57.991 60.841 .238 30 31 47.129 50.176 .254 53.223 56.209 .249 59.195 62.129 .245 31 32 48.117 51.246 .261 54.376 57.451 .256 60.526 63.545 .252 32 33 49.203 52.426 .269 55.649 58.813 .264 61.978 65.100 .260 33 34 50.401 53.718 .276 57.035 60.308 .273 63.581 66.808 .269 34 35 51.704 55.135 .286 58.566 61.949 .282 65.332 68.677 .279 35 36 53.134 56.681 .296 60.228 63.737 .292 67.245 70.720 .290 36 37 54.709 58.388 .307 62.067 65.711 .304 69.356 72.970 .301 37 38 56.443 60.266 .319 64.088 67.880 .316 71.672 75.442 .314 38 39 58.341 62.319 .332 66.298 70.254 .330 74.209 78.145 .328 39 40 60.422 64.573 .346 68.724 72.856 .344 76.988 81.108 .343 40 41 62.701 67.039 .362 71.376 75.701 .360 80.026 84.346 .360 41 42 65.212 69.753 .378 74.295 78.832 .378 83.369 87.906 .378 42 43 67.951 72.717 .397 77.482 82.249 .397 87.016 91.790 .398 43 44 70.962 75.971 .417 80.981 85.998 .418 91.015 96.045 .419 44 45 74.260 79.535 .440 84.809 90.096 .441 95.384 100.689 .442 45 46 77.870 83.431 .463 88.993 94.573 .465 100.153 105.764 .468 46 47 81.820 87.692 .489 93.564 99.469 ,.492 105.374 111.309 .495 ' 47 48 86.132 92.349 .518 98.566 L 04. 816 .521 111.065 117.358 .524 48 49 90.859 97.443 .549 104.027 L10.656 .552 117.286 123.957 .556 49 50 96.020 103.009 .582 109.998 117.031 .586 124.064 131.142 .590 50 51 101.665 109.085 .618 116.504 L23.971 .622 131.438 138.956 .627 51 52 107.821 115.704 .657 123.588 131.526 .662 139.463 147.453 .666 52 53 114.528 122.916 .699 131.304 139. 748 .704 148.192 156.680 .707 53 54 121.844 130.776 .744 139.707 148.685 .748 157.663 166.686 .752 54 1 55 129.825 139.331 .792 148.837 158.391 .796 167.945 177.534 .799 56 56 138.482 148.609 .844 158.737 168.902 .847 179.067 189.257 .849 56 57 147.898 158.687 .899 169.476 180.291 .901 191.106 201.919 .901 57 58 158.104 169.600 .958 181.095 192.588 .958 204.081 215.554 .956 58 59 169.179 181.413 1.020 193.648 205.862 1.018 218.076 230.220 1.012 6& 60 181.164 194.189 1.085 207.213 220.162 1.079 233.111 245.966 1.071 60 61 194.125 207.959 1.153 221.793 235.526 1.144 249.259 262.8261.131 61 62 208.072 222.774 1.225 237.476 252.000 1.210 266.524 280.8201.191 62 63 223.123 238.706 1.299 254.290 269.629 1.278 284.968 299.9891 1.252 ! 63 64 239.277 255.776 1.375 272.275 288.432 1.346 304.590 320.325 1.311 64 65 65* 256.596 274.023 1.452 291.450 308.421 1.414 325.392 341.837 1.370 j * Ages 66, 67, . . . . continu< jd on Pa ge *117. TABLE LII. *115 SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON- TINUINe^ m YEARS FROM THE PRESENT AGE, 4 PER CENT. PRESENT A&E, 10. loi iin.d. 11. Hi. Jm.d. 12. 12i. Jm.d. PRESENT AGE. 15 53.176 55.310 .178 57.444 59.506 .172 61.568 63.557 .166 15 16 53.503 55.661 .180 57.820 59.902 .174 61.984 63.995 .168 16 17 53.874 56.054 .182 58.234 60.340 .176 62.446 64.487 .170 17 18 54.275 56.480 .184 58.685 60.821 .178 62.958 65.024 .172 18 19 54.715 56.952 .186 59.189 61.353 .180 63.517 65.614 .175 19 20 55.210 57.476 .189 59.742 61.937 .183 64.132 66.262 .178 20 21 55.746 58.045 .192 60.343 62.574 .186 64.804 66.971 .181 21 22 56.332 58.667 .195 61.003 63.272 .189 65.542 67.750 .184 22 23 56.979 59.356 .198 61.732 64.044 .193 66.356 68.612, .188 23 24 57.686 60.107 .202 62.529 64.892 .197 67.255 69.557 .192 24 25 58.466 60.941 .206 63.415 65.825 .201 68.235 70.589 .196 25 26 59.336 61.860 .210 64.385 66.851 .206 69.316 71.730 .201 26 27 60.274 62.857 .215 65.439 67.968 .211 70.497 72.976 .207 27 28 61.308 63.957 .221 66.606 69.202 .216 71.799 74.350 .213 28 29 62.450 65.170 .227 67.890 70.563 .223 73.235 75.859 .219 29 30 63.691 66.491 .233 69.291 72.040 .229 74.790 77.503 .226 30 31 65.063 67.944 .240 70.826 73.669 .237 76.511 79.313 .234 31 32 66.565 69.544 .248 72.522 75.459 .245 78.396 81.301 .242 32 33 68.222 71.300 .257 74.378 77.422 .254 80.467 83.482 .251 33 34 70.034 73.225 .266 76.416 79.577 .263 82.737 85.872 .261 34 .35 72.023 75.336 .276 78.650 81.937 .274 85.224 88.491 .272 35 36 74.194 77.641 .287 81.087 84.513 .286 87.939 91.350 .284 36 37 76.585 80.178 .299 83.772 87.348 .298 90.925 94.491 .297 37 38 79.211 82.963 .313 86.715 90.456 .312 94.197 97.934 .311 38 39 82.082 86.008 .327 89.933 93.854 .327 97.775 101.696 .327 39 40 85.227 89.342 .343 93.457 97.573 .343 101.689 105.812 .344 40 41 88.666 92.987 .360 97.308 101.636 .361 105.964 110.302 .362 41 42 92.444 96.988 .379 101.533 106.089 .380 110.645 115.216' .381 42 43 96.564 101.350 .399 106.136 110.938 .400 115.740 120.569, .402 43 44 101.074 106.121 .421 111.167 116.242 .423 121.316 126.417 .425 44 45 105.994 111.329 .445 116.665 122.027 .447 127.390 132.791 .450 45 46 111.375 117.016 .470 122.657 128.337 .473 134.016 139.732 .476 46 47 117.245 123.222 .498 129.199 135.213 .501 141.228 147.281 .504 47 48 123.650 129.983 .528 136.316 142.689 .531 149.062 155.479 .535 48 49 130.629 137.343 .560 144.057 150.817 .563 157.577 164.382 .567 49 50 138.219145.346 .594 152.472 159.647 .598 166.821 174.033 .601 50 51 146.474154.042 .631 161.611 169.219 .634 176.827 184.473 .637 51 52 155.4441163.477 .669 171.509 179.582 .673 187.655 195.757 .675 52 53 165.168:173.698 .711 182.229 190.791 .714 199.353 207.936 .715 53 54 175.709184.765 .755 193.821 202.899 .757 211.977 221.053 .756 54 55 187.123196.735 .801 206.347 215.956 .801 225.566 235.159 .799 55 56 199.446209.633 .849 219.820 229.990 .848 240.159 250.270 .843 56 57 212.732 223.526 .900 234.320 245.052 .894 255.784 266.438 .888 57 58 227.028 238.435 .951 249.842 261.166 .944 272.490 283.677 .932 58 59 242.363j254.418 1.005 266.473 278.382 .992 290.291 302.013 .977 59 60 1258.821 271. 52C 1.058 284.219 296.719 1.042 309.219 321.461 1.020 60 61 !276. 393:289. 74? 1.113 303.102 316.180 1.090 329.258 341.994 1.061 61 62 1295.116:309.117 1.167 323.118 336.752 1.136 350.387 363.598 1.101 62 63 ,315.011 329. 640 1.219 344.269 358.445 1.181 372.620 386.250 1.136 63 64 336.060 351.3071.271 366.555 381.215 1.222 395.876 409.865 1.166 64 65 :358.282 374.094 1.318 389.906 404.994 1.257 420.081 434.369 1.191 65 *116 TABLE LII. SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON. TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PRESENT AGE. m—lZ, 13i. im.d. 14. 14i. |m.d. 15. 15i. ^m.d. PB,ERENT A&E. 15 65.546 67.467 .160 69.389 71.250 .155 73.112 74.913 .150 15 16 6.6.007 67.956 .162 69.905 71.791 .157 73.677 75.504 .152 16 17 66.527 68.501 .165 70.475 72.388 .159 74.300 76.156 .155 17 18 67.091 69.094 .167 71.096 73.039 .162 74.983 76.871 .157 18 19 67.710 69.744 .170 71.779 73.756 .165 75.732 77.656 .160 19 20 68.393 70.463 .173 72.532 74.546 .168 76.560 78.525 .164 20 21 69.138 71.247 .176 73.356 75.413 .171 77.471 79.475 .167 21 22 69.958 72.112 .180 74.267 76.366 .175 78.464 80.514 .171 22 23 70.868 73.066 .183 75.264 77.410 .179 79.557 81.658 .175 23 24 71.858 74.106 .187 76.353 78.554 .183 80.755 82.912 .180 24 25 72.943 75.248 .192 77.554 79.813 .188 82.072 84.292 .185 25 26 74.145 76.512 .197 78.878 81.203 .194 83.527 85.811 .190 26 27 75.454 77.889 .203 80.324 82.715 .199 85.107 87.466 .197 27 28 76.901 79.406 .209 81.911 84.383 .206 86.854 89.291 .203 28 29 78.484 81.073 .216 83.662 86.214 .213 88.767 91.292 .210 29 30 80.215 82.890 .223 85.565 88.210 .220 90.854 93.474 '.218 30 31 82.116 84.887 .231 87.659 90.404 .229 93.149 95.871 .227 31 32 84.205 87.082 .240 89.958 92.811 .238 95.665 98.502 .236 32 33 86.497 89.487 .249 92.478 95.451 .248 98.424 101.383 .247 33 34 89.007 92.124 .260 95.240 98.342 .259 101.444 104.537 .258 34 35 91.759 95.011 .271 98.263 101.506 .270 104.748 107.987 .270 35 36 94.760 98.160 .283 101.560 104.956 .283 108.353 111.750 .283 36. 37 98.058 101.620 .297 105.182 108.745 .297 112.308 115.876 .297 37 38 101.670 105.408 .312 109.146 112.889 .312 116.632 120.385 .313 38 39 105.618 109.545 .327 113.473 117.411 .328 121.348 125.299 .329 39 40 109.934 114.067 .344 118.199 122.345 .346 126.491 130.661 .348 40 41 114.641 118.993 .363 123.346 127.723 .365 132.101 136.501 .367 41 42 119.787 124.384 .383 128.981 133.602 .385 138.222 142.875 .388 42 43 125.398 130.252 .405 135.106 139.994 .407 144.882 149.801 .410 43 44 131.518 136.655 .428 141.792 146.961 .431 152.130 157.332 .433 44 45 138.191 143.626 .453 149.060 154.528 .456 159.997 165.504 .459 45 1?/ 145.448 151.200 .479 156.952 162.744 .483 168.536 174.366 .486 46 153.334 159.428 .508 165.523 171.659 .511 177.7941 183.962 .514 47 48 161.896 168.356 .538 174.815 181.309 .541 187.803 194.329 .544 48 49 171.188 178.029 .570 184.870 191.745 .573 198.621 205.522 .575 49 50 181.244 188.492 .604 195.740 203.015 .606 210.290 217.582 .608 50 51 192.119 199.793 .640 207.468 215.161 .641 222.854 230.545 .641 51 52 203.860 211.982 .677 220.104 228.225 .677 236.345 244.451 .676 52 53 216.518 225.099 .715 233.680 242.246 .714 250.812' 259.328 .710 53 54 230.129 239.190 .755 248.250 257.258 .751 266.266 275.209 .745 54 55 244.753 254.291 .795 263.830 273.298 .789 282.767 292.121 .779 55 56 260.381 270.419 .837 280.456 290.372 .826 300.288 310.048 .813 56 57 277.091 287.616 .877 298.141 308.501 .863 318.860 329.006 .846 57 58 294.864 305.875 .918 316.887 327.671 .899 338.455 348.957 .875 58 59 313.736 325.216 .957 336.696 347.875 .932 359.055 369.888 .903 59 60 333.702 345.624 .994 357.545 369.097 .963 380.648 391.755 .926 60 61 354.730 367.071 1.028 379.413 391.279 .989 403.145 414.468 .944 61 62 376.810 389.514 1.059 402.217 414.338 1.010 426.459 437.938 .957 62 63 399.880 412.885 1.084 425.890 438.206 1.026 450.523 462.071 .962 63 64 423.853 437.100 1.104 450.348 462.770 1.035 475.192 486.720 .961 64 65* 448.657 462.055 1.117 475.452 487.886 1.036 500.319 511.728 .951 65 Ages 66, 67, continued on Page *117. TABLE LII. *117 SINGLE PREMIUM OR RESERVE (A), FOR TEMPORARY INSURANCE OF 1,000 CON. TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PBESENT AGE. W=:l. 2. 3. 4. 5. 6. 7 PEESENT ' • AGE. 66 40.184 80.300 120.184 159.777 198.892 237.392 275.097 66 67 43.540 86.829 129.802 172.255 214.042 254.965 294.818 67 68 47.155 93.967 140.213 185.732 230.311 273.724 315.790 68 69 51.195 101.770 151.552 200.305 247.783 293.788 338.024 69 70 55.557 110.241 163.796 215.949 266.485 315.078 361.442 70 71 60.360 119.471 177.037 232.818 286.453 337.628 386.094 71 72 65.594 129.472 191.369 250.886 307.673 361.453 411.880 72 73 71.297 140.383 206.812 270.195 330.221 386.505 438.740 73 74 77.604 152.223 223.420 290.846 354.069 412.745 466.583 74 75 84.417 164.963 241.243 312.767 379.147 440.055 495.220 75 76 91.830 178.796 260.341 336.020 405.461 468.353 524.567 76 77 99.995 193.756 280.772 360.616 432.931 497.565 554.382 77 78 108.829 209.830 302.505 386.441 461.463 527.411 584.471 78 79 118.447 227.130 325.565 413.546 490.885 557.801 614.568 79 80 128.910 245.665 350.020 441.753 521.122 588.455 644.371 80 81 140.224 265.556 375.719 471.043 551.910 619.066 673.718 81 82 152.600 286.742 402.805 501.266 583.033 649.575 702.195 82 83 165.825 309.300 431.016 532.095 614.353 679.402 731.612 83 84 180.310 333.275 460.305 563.681 645.430 711.044 761.097 84 85 195.801 358.404 490.730 595.372 679.361 743.431 788.654 85 PEESENT AGE. m=8. 9. 10. 11. 12. 13. 14. PEESENT AGE. 66 311.816 347.395 381.607 414.250 445.163 474.150 501.052 66 67 333.435 370.566 405.995 439.548 471.009 500.207 526.998 67 68 356.239 394.832 431.382 465.653 497.459 526.643 553.076 68 69 380.231 420.203 457.683 492.467 524.384 553.291 579.128 69 70 405.351 446.522 484.732 519.792 551.547 579.928 604.877 70 71 431.538 473.713 512.411 547.460 578.788 606.326 630.152 71 72 458.680 501.622 540.515 575.278 605.835 632.275 654.704 72 73 486.669 530.079 568.879 602.986 632.496 657.530 678.320 73 74 515.345 558.929 597.240 630.389 658.509 681.863 700.867 74 75 544.526 587.868 625.369 657.183 683.602 705.102 722.104 75 76 573.981 616.735 653.006 683.126 707.638 727.022 742.581 76 77 603.542 645.245 679.879 708.063 730.351 748.240 761.886 77 78 632.877 673.076 705.800 731.669 752.433 768.263 779.443 78 79 661.711 700.075 730.413 754.764 773.339 786.450 795.241 79 80 689.875 725.860 754.742 776.775 792.326 802.752 809.424 80 81 716.936 751.624 778.085 796.763 809.285 817.298 822.118 81 82 744.430 776.649 799.389 814.636 824.392 830.261 833.541 82 83 771.440 799.552 818.399 830.460 837.715 841.770 843.949 83 84 796.426 820.113 835.270 844.388 849.484 852.222 853.469 84 85 818.973 838.374 850.046 856.569 860.073 861.670 862.352 85 ni8 TABLE LIII. ANNUAL PREMIUM (P) OF TEMPORARY INSURANCE OF 1000 FOR m TEARS. 4 PER CENT. ALSO A(7r — P).* AQE. X. m=l. 2. 3. 4. AGE. X. P. Hjr-n P. A(7r-P). P. A(7r-P). P. A(7r-P). 15 6.339 0.646159 6.349 0.645088 6.363 0.643740 6.378 0.642295 15 16 6.361 .666022 6.377 .664539 6.392 .663059 6.408 .661562 16 17 6.393 .685150 6.409 .683844 6.425 .682346 6.443 i .680708 17 18 6.426 .704502 6.442 .703069 6.461 .701461 6.481 .699713 18 19 6.459 .724063 6.480 .722280 6.501 .720520 6.622 .718763 19 20 6.502 0.743023 6.524 0.741335 6.545 0.739587 6.569 0.737749 20 21 6.547 .762213 6.568 .760500 6.594 .758675 6.619 .756706 21 22 6.591 .781648 6.619 .779692 6.646 .777720 6.673 .775721 22 23 6.647 .800593 6.675 .798740 6.703 .796760 6.735 .794541 23 24 6.704 .819800 6.733 .817846 6.767 .815632! 6.801 .813358 24 25 6,762 0.839277 6.801 0.836814 6.836 0.834539' 6.873 0.832U88 25 26 6.842 .857724 6.876 .855689 6.914 .853366: 6.958 .850718 26 27 6.913 .877101 6.953 .874716 7.000 .872014; 7.046 .869338 27 28 6.995 .896179 7.047 .893342 7.095 .890663 7.145 .887858 28 29 7.101 .914412 7.148 .911939 7.199 .909193 7.253 .906320 29 30 7.198 0.933538 7.252 0.930807 7.309 0.927915 7.369 0.924798 30 31 7.308 .952468 7.368 .949558 7.432 .946425 7.498 .943141 31 32 7.431 .971234 7.498 .968128 7.568 .964853 7.644 .961235 32 33 7.568 .989881 7.641 .986585 7.722 .982942 7.802 .979305 33 34 7.719 1.008430 7.805 1.004747 7.888 1.001140 7.975 .997379 34 35 7.896 1.026460 7.979 1.023012 8.068 1.019327 8.166 1.015296 35 36 8.065 .045362 8.161 .041571 8.264 .037507 8.372 .033181 36 37 8.262 .063840 8.371 .059760 8.485 .055417 8.604 .050823 37 38 8.486 .081934 8.604 .077663 8.730 .073089 8.857 .068422 38 39 8.728 .100102 8.860 .095477 8.993 .090880 9.136 .085754 39 40 8.999 1.117984 9.135 1.113515 9.286 1.108366 9.442 1.103102 40 41 9.277 .136381 9.440 .131161 9.604 .125917 9.776 .120288 41 42 9.612 .153845 9.779 .148674 9.959 .143115 10.147 .137178 42 43 9.955 .171864 10.146 .166291 10.344 .160342 10.551 .154077 43 44 10.346 .189423 10.554 .183514 10.771 .177300 10.999 .170673 44 45 10.773 1.206904 11.000 1.200718 11.240 1.194123 11.489 1.187120 45 46 11.239 .224347 11.492 .217759 11.754 .210787 12.028 .203401 46 47 11.758 .241475 12.032 .234588 12.320 .227262 1 12.622 .219402 47 48 12.321 .258627 12.624 .251332 12.941 .243507 13.273 .235204 48 49 12.943 .275547 13.276 .267802 13.625 .259532 13.989 .250767 49 50 13.628 1.292263 13.995 1.284037 14.377 1.275337 14.775 1.266078 50 51 14.381 .308803 14.782 .300163 15.200 .290977 ; 15.635 .281179 51 52 15.206 .325195 15.645 .316105 16.101 .306388 16.583 .295944 52 53 16.108 .341466 16.588 .331851 17.095 .321472 1 17.622 .310432 53 54 17.095 .357628 17.633 .347236 18.187 .336294 18.768 .324502 54 55 18.202 1.373155 18.781 1.362382 19.393 1.350687 20.027 1.338234 55 56 19.396 .388816 20.044 .377181 20.711 .364847 21.404 .351666 56 57 20.731 .403844 21.431 .391695 22.158 .378680 22.918 .364650 57 58 22.175 .418941 22.940 .406085 23.742 .392188 1 24.577 .377242 58 59 23.756 .433834 24.603 .420046 25.485 .405246 26.395 .389423 59 60 25.506 1.448213 26.437. 1.433548 27.397 1.417933 28.394 1.401053 60 61 27.432 .462217 28.441 .446845 29.494 .430164 30.584 .412207 61 62 29.521 .476214 30.637 .459719 31.791 .441965 32.988 .422773 62 63 31.833 .489607 33.054 .472085 34.319 .453137 35.622 .432525 63 64 34.366 .502669 35.708 .483974 37.085 .463577 38.511 .442041 64 65 37.154 .515179 38.610 .495002 40.122 .473918 41.668 .450779 65 Formula, P^= 1000 Ma; — Ma;+wi . -Nx -Nx + TO TTa- = 1000 Ma TABLE LIII. ANNUAL PREMIUM (P) OF TEMPORARY INSURANCE OF 1.000 FOR m YEARS. ALSO A (tt—P).* *119 4 PER CENT. AGE. X. 5. 6. 7. 8. AGE. X. P. 6.393 A(:r-P). P. A (tt-P). P. A(7T-P). 0.637504 P. 6.443 A (TT-P). 15 0.640832 6.409 0.639220 6.426 0.635731 15 16 6.425 .669952 6.443 .658221 6.461 .656452 6.481 .654585 16 17 6.462 .678969 6.481 .677201 6.502 .675334 6.523 .673364 17 18 6.501 .697950 6.523 .696094 6.545 .694120 6.568 .692079 18 19 6.545 .716909 6.569 .714943 6.593 .712909 6.619 .710715 19 20 6.594 0.735785 6.619 0.733765 6.646 0.731583 6.675 0.729294 20 21 6.645 .754701 6.674 .752516 6.704 .750229 6.735 .747838 21 22 6.704 .773521 6.735 .771232 6.767 .768846 6.802 .766238 22 23 6.768 .792264 6.801 .789893 6.838 .787280 6.876 .784583 23 24 6.836 .810996 6.876 .808378 6.915 .805678 6.957 .802847 24 25 6.915 0.829545 6.957 0.826847 7.000 0.824014 7.046 0.821056 25 26 7.002 .848043 7.047 .845218 7.094 .842270 7.144 .839126 26 27 7.094 .866511 7.144 .863582 7.196 .860440 7.251 .857136 27 28 7.197 .884938 7.252 .881811 7.309 .878511 7.371 .874936 28 29 7.310 .903209 7.371 .899934 7.436 .896365 7.502 .892676 29 30 7.433 0.921515 7.501 0.917931 7.571 0.914245 7.644 0.910386 30 31 7.571 .939550 7.644 .935869 7.720 .932026 7.801 .927918 31 32 7.721 .957574 7.801 .953739 7.885 .949646 7.974 .945294 32 33 7.886 .975510 7.974 .971437 8.067 .967094 8.165 .962484 33 34 8.068 .993329 8.165 .988999 8.268 .984402 8.374 .979615 34 35 8.267 1.010988 8.375 1.006399 8.486 1.001641 8.604 0.996488 35 36 8.486 .028590 8.602 .023843 8.727 .018690 8.855 1.013311 36 37 8.726 .046103 8.856 .040956 8.991 .035596 9.133 .029884 37 38 8.993 .063290 9.135 .057964 9.284 .052274 9.439 .046233 38 - 39 9.284 .080462 9.440 .074798 9.604 .068780 9.774 .062407 39 40 9.605 1.097463 9.776 1.091462 9.955 1.085120 10.143 1.078367 40 41 9.956 .114305 10.144 .107983 10.341 .101252 10.547 .094105 41 42 10.345 .130901 10.551 .124215 10.767 .117103 10.993 .109558 42 43 10.769 .147405 10.996 .140321 11.233 .132806 11.481 .124785 43 44 11.237 .163630 11.486 .156159 11.747 .148180 12.019 .139689 44 45 11.751 1.179693 12.025 1.171750 12.311 1.163306 12.610 1.154311 45 46 12.316 .195498 12.617 .187096 12.930 .178153 13.257 .168630 46 47 12.937 .211053 13.267 .202162 13.610 .192696 13.968 .182614 47 48 13.619 .226367 13.980 .216961 14.356 .206942 14.749 .196212 48 49 14.368 .241427 14.763 .231474 15.176 .220813 15.606 .209439 49 50 15.189 1.256204 15.624 1.245613 16.076 1.234321 16.548 1.222199 50 51 16.093 .270661 16.568 .259449 17.065 .247406 17.581 .234542 51 52 17.083 .284815 17.606 .272856 18.150 .260083 18.713 .246429 52 53 18.174 .298553 18.746 .285880 19.339 .272329 19.955 .257783 53 54 19.370 .311934 19.995 .298493 20.645 .284063 21.319 .268563 54 55 20.685 1.324922 21.369 1.310618 22.080 1.295250 22.814 1.278781 55 56 22.125 .337489 22.875 .322259 23.649 .305935 24.452 .288353 56 57 23.708 .349562 24.526 .333402 25.372 .315984 26.247 .297240 57 58 25.439 .361242 26.334 .343990 27.258 .325424 28.212 .305386 58 1 59 27.339 .372347 28.317 .353971 29.326 .334127 ; 30.362 .312640 59 60 29.426 1.382882 30.494 1.363248 31.591 1.341971 32.720 1.319205 60 61 31.715 .392792. 32.876 .371730 34.072 .349229 35.294 .324817 61 62 34.218 .401931 35.486 .379712 36.782 .355582 38.110 .329362 62 63 36.967 .410631 38.341 .386804 39.752 .360895 41.188 .332839 63 64 39.969 .418541 41.470 .392957 42.996 .365247 44.549 .335136 64 65 43.262 .425541 44.887 .398203 46.541 .368478 48.217 .336167 65 * In valuing Term Polidea by Table LXIII, {n — P) Bf will be suUractive. ^120 TABLE LIIl. ANNUAL PREMIUM (P) OF TExMPORARY INSURANCE OP 1000 FOR m YEARS. ALSO A(7T— P).* 4 PER CENT. AQE. X. w=9. 10. 11. 12. AGE. X, P. A(Tr_P). P. 6.482 A(7r-P)- P. A(77 — P). P. A(Tr-P). 15 6.462 0.633870 0.631890 6.502 0.629841 6.523 0.627645 15 16 6.501 .652608 6.522 .650560 6.545 .648358 6.569 .646055 16 17 6.545 .671315 6.569 .669115 6.593 .666813 6.619 .664401 17 18 6.593 .689886 6.619 .687584 6.646 .685183 6.675 .682581 18 19 6.646 .708419 6.674 .706021 6.705 .703421 6.736 .700709 19 20 6.704 0.726900 6.736 0.724295 6.769 0.721592 6.803 0. 718741 20 21 6.768 .745231 6.803 .742532 6.839 .739681 6.877 .736702 21 22 6.838 .763537 6.876 .760694 6.916 .757714 6.957 .754542 22 23 6.916 .781738 6.957 .778765 7.001 .775606 7.046 .772270 23 24 7.000 .799881 7.046 .796722 7.094 .793392 7.145 .789811 24 25 7.094 0.817907 7.144 0.814585 7.197 0.811003 7.252 0.807277 25 26 7.197 .835813 7.253 .832240 7.311 .828519 7.371 .824619 26 27 7.310 .853562 7.371 .849848 7.434 .845953 7.501 .841813 27 28 7.435 .871237 7.501 .867354 7.571 .863218 7.644 .858826 28 29 7.571 .888805 7.645 .884685 7.721 .880305 7.802 .875655 29 30 7.720 0.906271 7.801 0.901898 7.886 0.897254 7.974 0.892401 30 31 7.885 .923554 7.974 .918920 8.066 .914081 8.164 .908872 31 32 8.067 .940668 8.164 .935843 8.266 .930648 8.372 .925198 32 33 8.266 .957678 8.373 .952498 8.485 .947064 8.602 .941282 33 34 8.486 .974448 8.603 .969034 8.726 .963270 8.854 .957146 34 35 8.727 0.991097 8.856 0.985349 8.990 0.979248 9.132 0.972783 35 36 8.990 1.007578 9.132 1.001491 9.280 .995046 9.435 .988189 36 37 9.282 .023816 9.437 .017393 9.599 1.010559 9.769 1.003301 37 38 9.602 .039835 9.772 .033028 9.950 .025796 10.137 .018117 38 39 9.953 .055630 10.140 .048425 10.335 .040775 10.540 .032621 39 40 10.339 1.071190 10.544 1.063574 10.7e59 1.055448 10.982 1.046804 40 41 10.762 .086519 10.988 .078425 11.223 .069814 11.468 .060645 41 42 11.229 .101504 11.476 .092935 11.734 .083808 12.002 .074092 42 43 11.741 .116252 12.011 .107164 12.293 .097490 12.587 .087184 43 44 12.303 .130650 12.599 .121025 12.908 .110770 13.230 .099802 44 45 12.921 1.144737 13.244 1.134535 13.583 1.123621 13.934 1.111980 45 46 13.597 158485 13.953 .147629 14.323 .136048 14.708 .123637 46 47 14.342 .171819 14.730 .160304 15.136 .147960 15.556 .134772 47 48 15.158 .184764 15.584 .172490 16.027 .159378 16.485 .145355 48 49 16.055 .197237 16.520 .184203 17.003 .170264 17.504 .155314 49 50 17.038 1.209248 17.546 1.195400 18.073 1.180543 18.620 1.164597 50 51 18.116 .220788 18.672 .206028 19.247 .190187 19.841 .173207 51 52 19.298 .231771 19.905 .216038 20.531 .199176 21.178 .181035 52 53 20.595 .242162 21.255 .225422 21.938 .207409 22.641 .188040 53 54 22.016 .251953 22.736 .234076 23.478 .214850 24.243 .194125 54 55 23.574 1.261047 24.358 1.241974 25.166 1.221407 25.993 1.199187 55 56 25.280 .269440 26.133 .249038 27.008 .226989 27.906 .203328 56 57 27.149 .277017 28.075 .255146 29.024 .231688 29.992 .206313 57 58 29.191 .283700 30.197 .260457 31.223 .235299 32.269 .208025 58 59 31.428 .289625 32.515 .264700 33.625 .237667 34.750 .208422 59 60 33.872 1.294528 35.050 1.267751 36.245 1.238777 37.455 1.207347 60 61 36.544 .298311 37.815 .269622 39.102 .238491 40.398 .204706 61 62 39.460 .300979 40.830 .270165 42.211 .236713 43.595 .200380 62 63 42.646 .302366 44.118 .269269 45.595 .233311 47.073 .194101 63 64 46.119 .302418 47.697 .266860 49.277 .228062 50.844 .185804 64 65 49.902 .301039 51.5j92 .262680 53.272 .220883 54.929 .175306 65 In valuing Term Policies by Table LXIII, (tt — P) B/" will be stibtractive. TABLE LIII. *121 ANNUAL PREMIUM (P) OP TEMPORARY INSURANCE OF 1000 FOR m YEARS. ALSO A(7r— P).* 4 PER CENT. AQE. X. m- = 13. 14. 15. AGE, z. 15 P. A(7r— P). P. A(7r^P). P. A(7r-P). 15 6.546 0.625336 6.569 0.622924 6.594 0.620319 16 6.593 .643643 6.620 .641039 6.647 .638319 16 17 6.647 .661795 6.675 .659077 6.705 .656216 17 18 6.705 .679867 6.736 .677006 6.769 .674007 18 19 6.769 .697854 6.803 .694858 6.839 .691677 19 20 6.839 0.715751 6.877 0.712573 6.917 0.709226 20 21 6.916 .733532 6.958 .730182 7.002 .726594 21 22 7.001 .751204 7.047 .747616 7.095 .743870 22 23 7.095 .768685 7.145 .764946 7.198 .761015 23 24 7.198 .786078 7.252 .782156 7.310 .777990 24 25 7.310 0.803363 7.371 0.799202 7.434 0.794785 25 26 7.435 .820466 7.501 .816058 7.572 .811378 26 27 7.571 .837412 1 7.644 .832740 7.721 .827846 27 28 7.721 .854168 : 7.801 .849285 7.886 .844048 28 29 7.886 .870786 7.975 .865563 8.067 .860072 29 30 8.067 0.887182 8.164 0.881705 8.265 0.875876 30 31 8.265 .903407 8.372 .897595 8.483 .891421 31 32 8.484 .919399 8.601 .913241 8.723 .906715 32 33 8.725 .935143 8.853 .928636 8.987 .921720 33 34 8.989 .950662 9.130 .943764 9.276 .936437 34 35 9.279 0.965909 9.433 0.958602 9.594 0.950845 35 36 9.596 .980904 9.765 .973169 9.942 .964925 36 37 9.946 .995592 10.132 .987375 10.325 .978631 37 38 10.331 1.009931 10.534 1.001221 10.746 .991945 38 39 10.753 .023943 10.975 .014702 11.207 1.004864 39 40 11.216 1.037598 11.460 1.027799 11.713 1.017357 40 41 11.724 .050885 11.989 .040486 12.267 .029368 41 42 12.282 .063739 12.573 .052671 12.876 .040863 42 43 12.893 .076163 13.211 .064406 13.543 .051815 43 44 13.564 .088101 13.913 .075567 14.274 .062175 44 45 14.301 1.099505 14.681 1.086178 15.074 1.071927 45 46 15.108 .110376 15.522 .096198 15.951 .080996 46 47 15.992 .120667 ' 16.444 .105546 16.911 .089324 47 48 16.961 .130316 1 17.453 .114180 17.961 .096891 48 49 18.022 .139272 18.557 .122080 19.109 .103598 49 50 19.184 1.147512 : 19.765 1.129137 20.364 1.109387 50 51 20.455 .154948 21.086 .135317 21.735 .114174 51 52 21.845 .161532 22.530 . .140519 23.231 .117843 52 53 23.365 .167167 24.106 .144633 24.864 .120438 53 54 25.026 .171744 25.827 .147716 26.643 .121755 54 55 26.842 1.175338 27.704 1.149560 28.583 1.121651 55 56 28.820 .177745 29.750 .150034 30.692 .120077 56 57 30.979 .178815 31.977 .149081 32.985 .116863 57 1 58 33.329 .178529 34.399 .146556 35.475 .111893 58 59 35.888 .176713 37.031 .142323 38.174 .105012 59 ] 60 38.672 1.173251 39.890 1.136245 41.102 1.095957 60 61 41.696 .168027 42.989 .128075 44.269 .084614 61 62 44.978 .160787 46.346 .117699 47.688 .070781 62 63 48.537 .151417 49.975 .104913 51.381 .054124 63 64 52.387 .139747 53.896 .089416 55.358 .034441 64 65 56.552 ■ .125466 58.126 .070998 59.636 .011425 65 *122 TABLE LIY. SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON- TINUING m YEARS FROM THE PRESENT AGE. 4 PER ( :;ent. PEESENT A&E. m=l. li-. l^m. d. 2. 2i. Jm.d. 3. 3i. ^m.d. PRESENT AGE. 15 961.538 943.169 1.531 924.800 907.254 1.462 889.708 872.949 1.397 15 16 961.538 943.169 1.531 924.801 907.256 1.462 889.711 872.953 1.397 16 17 961.538 943.170 1.531 924.802 907.258 1.462 889.715 872.959 1.396 17 18 961.538 943.171 1.531 924.803 907.261 1.462 889.718 872.964 1.396 18 19 961.538 943.171 1.531 924.805 907.264 1.462 889.723 872.971 1.396 19 20 961.538 943.172 1.531 924.806 907.267 1.462 889.727 872.978 1.396 20 21 961.538 943.173 1.530 924.808 907.270 1.462 889.733 872.986 1.396 21 22 961.538 943.174 1.530 924.810 907.274 1.461 889.738 872.994 1.395 22 23 961.538 943.175 1.530 924.812 907.278 1.461 889.7441873.003 1.395 23 24 961.538 945.176 1.530 924.814 907.282 1.461 889.751873.013 1.395 24 25 961.538 943.177 1.530 924.816 907.287 1.461 889.758 873.024 1.395 25 26 961.538 943.178 1.530 924.819 907.292 1.461 889.766i873.037 1.394 26 27 961.538 943.180 1.530 924.822 907.299 1.460 889.775|873.049 1.394 27 28 961.538 943.181 1.530 924.825 907.305 1.460 889.785i873.064 1.393 28 29 961.538|943.184 1.530 924.829 907.312 1.460 889.796 873.082 1.393 29 30 961.538 943.185 1.530 924.833 907.320 1.459 889.808 873.099 1.392 30 31 961.538 943.188 1.529 924.838 907.329 1.459 889.821 873.118 1.392 31 32 961.538 943.190 1.529 924.842 907.339 1.459 889.835 873.139 1.391 32 33 961.538 943.192 1.529 924.847 907.349 1.458 889.851 873.163 1.391 33 34 961.538 943.196 1.529 924.853 907.361 1.458 889.868 873.190 1.390 34 35 961.538 943.199 1.528 924.860 907.374 1.457 889.888 873.219 1.389 35 36 961.538 943.202 1.528 924.867 907.388 1.457 889.908 873.249 1.388 36 37 961.538 943.206 1.528 924.874 907.402 1.456 889.931 873.283 1.387 37 38 961.538 943.210 1.527 924.883 907.420 1.455 889.957 873.322 1.386 38 39 961.538 943.215 1.527 924.892 907.438 1.455 889.985 873.364 1.385 39 40 961.538 943.220 1.527 924.902 907.458 1.454 890.015 873.410 1.384 40 41 961.538 943.225 1.526 924.913|907.480 1.453 890.048 873.460 1.382 41 42 961.538 943.232 1.526 924.926 907.506 1.452 890.086 873.516 1.381 42 43 961.538 943.239 1.525 924.939 907.532 1.451 890.126 873.576 1.379 43 44 961.538 943.246 1.524 924.954 907.562 1.449 890.171 873.644 1.377 44 45 961.538 943.254 1.524 924.971 907.595 1.448 890.220 873.717 1.375 45 46 961.538 943.263 1.523 924.989 907.631[1.447 890.274 873.798 1.373 46 47 961.538 943.273 1.522 925.008 907.670il.445 890.333 873.887 1.371 47 48 961.538 943.284 1.521 925.030 907.7141.443 890.399|873.985 1.368 48 49 961.538 943.296 1.520 925.054 907.7621.441 890.470 874.091 1.365 49 50 961.538 943.309 1.519 925.080 907.8141.439 890.549 874.210 1.362 50 51 961.538 943.323 1.518 925.109 907.872:1.436 890.635 874.338 1.358 51 52 961.538 943.339 1.517 925.141 907.935 1.434 890.730 874.480 1.354 52 53 961.538 943.357 1.515 925.176 908.004 1.431 890.833;874.635 1.350 53 54 961.538 943.376 1.514 925.214 908.080 1.428 890. 9471874. 805 1.345 54 55 961.538 943.397 1.512 925.256 908.164 1.424 891.073'874.994 1.340 55 56 961.538 943.420 1.511 925.302 908.256 1.421 891.211875.199 1.334 56 57 961.538 943.446 1.508 925.354 908.359 1.416 891.363|875.425 1.328 57 58 961.538 943.473 1.505 925.409 908.469 1.412 891.528'875.670 1.322 58 59 961.538 943.504 1.503 925.470 908.589 1.407 891.709 875.940 1.314 59 60 961.538J943.537 1.500 925.537 908.722 1.401 891.908'876.237 1.306 60 61 961.5381943.575 1.497 925.611 908.868 1.395 892.126 876.560 1.297 61 62 961.538 943.615 1.494 925.692i909.028 1.389 892.365 876.915 1.288 62 63 961.538 943.659 1.490 925.781i909.204 1.381 892.627 877.303 1.277 63 64 961.538 943.708 1.486 925.878i909.396 1.374 892.914 877.728 1.266 64 65 961.538 943.761 1.481 925.985,909.606 1.365 893.228,878.192 1.253 65 Formula, Ag^> = 1000 \ | = 1000|l-(l-i;)(l+ar0} TABLE LIY ♦123 SINGLE PREMIUM OR RESERVE (A). FOR ENDOWMENT INSURANCE OF 1,000 CON. TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PRESENT AGE. 4. 4i. ^m.d. 5. 5i |m.d. 6. 6i. im.d. PEESENT AQE. 15 856.191 840.184 1.334 824.177 808.889 1.274 ' 793.602 779.002 1.217 15 16 856.196 840.191 1.334 824.187 808.902 1.274 793.617 779.020 1.216 16 17 856.203 840.201 1.334 824.199 808.917 1.274 793.635 779.041 1.216 17 18 856.211 840.211 1.333 824.212 808.933 1.273 793.654 779.064 1.216 18 19 856.220 840.223 1.333 824.226 808.950 1.273 793.675 779.090 1.215 19 20 856.229 840.235 1.333 824.241 808.970 1.273 793.699 779.118 1.215 20 21 856.239 840.248 1.333 824.258 808.991 1.272 793.724 779.149 1.215 21 22 856.250 840.263 1.332 824.277 809.014 1.272 793.752 779.182 1.214 22 23 856.262 840.279 1.332 824.297 809.039 1.272 793.782 779.218 1.214 23 24 856.276 ■840.297 1.332 824.319 809.068 1.271 793.816 779.259 1.213 24 25 856.290 840.317 1.331 824.344 809.099 1.270 793.853 779.303 1.213 25 26 856.307 840.339 1.331 824.372 809.133 1.270 793.894 779.353 1.212 26 27 856.324 840.363)1.330 824.401 809.170 1.269 793.939 779.407 1.211 27 28 856.344 840.3891.330 824.434 809.211 1.269 793.989 779.467 1.210 28 29 856.367 840.419,1.329 824.471 809.258 1.268 794.044 779.533 1.209 29 30 856.390 840.45oll.328 824.510 809.306 1.267 794.102 779.603 1.208 30 31 856.416 840.485 1.328 824.553 809.360 1.266 794.168 779.682 1.207 31 32 856.444 840.522 1.327 824.601 809.420 1.265 794.240 779.769 1.206 32 33 856.476 840.566 1.326 824.655 809.488 1.264 794.320 779.864 1.205 33 34 856.512 840.612 1.325 824.713 809.560 1.263 794.408 779.970 1.203 34 35 856.550 840.663 1.324 824.777 809.640 1.261' 794.504 780.085 1.202 35 36 856.591 840.718 1.323 824.846 809.726 1.260 794.607 780.209 1.200 36 37 856.636 840. 779 1.321 824.922 809.822 1.258 794.722 780.348 1.198 37 38 856.688 840.848 1.320 825.008 809.929 1.257 794.850 780.501 1.196 38 39 856.744 840.922 1.319 825.101 810.045 1.255 794.990 780.669 1.193 39 40 856.805 841.004 1.317 825.203 810.173 1.253 795.143 780.853 1.191 40 41 856.872 841.093 1.315 825.314 810.312 1.250 795.310 781.054 1.188 41 42 856.947 841.193 1.313 825.439 810.468 1.248 795.497 781.278 1.185 42 43 857.027 841.300 1.311 825.573 810.635 1.245 795.698 781.519 1.182 43 44 857.117 841.419 1.308 825.722 810.822 1.242 795.922 781.788 1.178 44 45 857.215 841.550 1.305 825.886 811.026 1.238 796.167 782.081 1.174 45 46 857.322 841.693 1.302 826.065 811.250 1.235 796.435 782.403 1.169 46 47 857.441 841.851 1.299 826.262 811.496 1.231 796.730 782.757 1.164 47 48 857.571 842.025 1.296 826.478 811.766 1.226 797.054 783.144 1.159 48 49 857.713 842.2141.292 826.715 812.061 1.221 797.408 783.568 1.153 49 50 857.871 842.4231.287 826.976 812.386 1.216 797.797 784.034 1.147 50 51 858.042 842.651 1.283 827.260 812.741 1.210 798.223 784.544 1.140 51 52 858.230 842.901 1.277 827.572 813.130 1.204 798.688 785.101 1.132 52 53 858.437 843.176 1.272 827.915 813.557 1.197 799.199 785.712 1.124 53 54 858.664 843.477 1.266 828.291 814.025 1.189 799.759 786.381 1.115 54 55 858.914 843.809 1.259 828.704 814.539 1.180 800.374 787.115 1.105 55 56 859.187 844.171 1.251 829.155 815.099 1.171 801.044 787.915 1.094 56 67 859.487 844.568 1.243 829.650 815.714 1.161 801.779 788.791 1.082 57 58 859.813 845.001 1.234 830.189 816.383 1.151 802.578 789.743 1.070 58 59 860.172 845.475 1.225 830.779 817.116 1.139 803.453 790.784 1.056 59 60 860.566 845.996 1.214 831.426 817.917 1.126 804.409 791.921 1.041 60 61 860.995 846.563 1.203 832.132 818.791 1.112 805.450 793.158 1.024 61 62 861.465 847.183 1.190 832.901 819.742 1.097 806.584 794.504 1.007 62 63 861.980 847.862 1.177 833.744 820.783 1.080 807.823 795.972 .988 63 64 862.542 848.601 1.162 834.661 821.914 1.062 809.167 797.563 .967 64 65 863.157 849.409 1.146 835.661 823.145 1.043 810.629 799.290] .945 1 65 Note that i^ m. d. is ^ of the difference of the integer or m columns, 4 — 5, etc. *124 TABLE LIT SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON. TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PRESENT AGE. m = 7. 7i. imA. 8. 8i. ^m.d. 9. 9i. |m.d. PEESENT AGE, 15 764.402 750.459 1.162 736.516 723.201 1.110 709.886 697.171 1.060 15 16 764.^23 750.483 1.162 736.544 723.233 1.109 709.923 697.213 1.059 16 17 764.448 750.512 1.161 736.577 723.272 1.109 709.966 697.262 1.059 17 18 764.475 750.544 1.161 736.614 723.313 1.108 710.013 697.314 1.058 18 19 764.505 750.579 1.161 736.653 723.358 1.108 710.063 697.371 1.058 19 20 764.537 750.617 1.160 736.697 723.408 1.107 710.120 697.435 1.057 20 21 764.573 750.659 1.160 736., 745 723.463 1.107 i 710.181 697.504 1.056 21 22 764.612 750.704 1.159 736.797 723.523 1.106 710.249697.581 1.056 22 23 764.654 750.754 1.158 736.854 723.588 1.106 710.323697.664 1.055 23 24 764.702 750.810 1.158 736.918 723.661 1.105 710.405 697.757 1.054 24 25 764.754 750.870 1.157 736.987 723. 741 1.104 710.495 697.858 1.053 25 26 764.812 750.938 1.156 737.065 723.830 1.103 710.595'697.971 1.052 26 27 764.875 751.012 1.155 737.149 723.926 1.102 710. 7031698.093 1.051 27 28 764.945 751.093 1.154 737. 2421724. 032 1.101 710.823l698.228 1.050 28 29 765.022,751.183 1.153 737.345 724.150 1.100 710.956 698.378 1.048 29 30 765.104 751.280 1.152 737.456 724.277 1.098 711,0991698.539 1.047 30 31 765.197 751.388 1.151 737.579 724.418 1.097 711.257 698.718 1.045 31 32 765.297 751.505 1.149 737.713 724.572 1.095 711.431 698.913 1.043 32 33 765.409 751.636 1.148 737.863 724.743 1.093 711.624 699.131 1.041 33 34 765.532 751.779 1.146 738.027 724.931 1.091 711.835 699.368 1.039 34 35 765.667 751.937 1.144 738.207 725.137 1.089 712.067 699.629 1.037 35 36 765.812 752.107 1.142 738.402 725.360 1.087 712.318 699.913 1.034 36 37 765.974 752.295 1.140 738.617 725.606 1.084 712.596 700.226 1.031 37 38 766.153 752.505 1.137 738.857 725.880 1.081 712.904 700.572 1.028 38 39 766.349 752.733 1.135 739.118 726.179 1.078 713.241 700.951 1.024 39 40 766.564 752.984 1.132 739.404 726.506 1.075 713.609 701.366 1.020 40 41 766.798 753.257 1.128 739.717|726.864 1.071 714.012 701.820 1.016 41 42 767.059 753.562 1.125 740.065j727.261 1.067 714.458 702.322 1.011 42 43 767.341753.891 1.121 740.4411727. 692 1.062 714.943 702.868 1.006 43 44 767.654 754.256 1.117 740.858 728.168 1.058 715.478 703.469 1.001 44 45 767.996 754.655 1.112 741.3141728.689 1.052 716.064!704.127 .995 45 46 768.371 755.092 1.107 741.813 729.2591.046 716.705!704.848 .988 46 47 768.783,755.572 1.101 742.362 729.885 1.040 717.408 705.638 .981 47 48 769.235 756.098 1.095 742.962 730.5691.033 718.177:706.502 .973 48 49 769.729 756.674 1.088 743.619 731.319|1.025 719.019 707.447 1 .964 49 50 770.272 757.305 1.081 744.339 732.1401.017 719.941708.482 .955 50 51 770.865 757.995 1.073 745.126 733.0371.007 720.948 709.611 .945 51 52 771.514 758.751 1.064 745.987 734.018 .997 722.049 710.845 .934 52 53 772.225 759.577 1.054 746.929 735.090 .987 723.251:712.192 .922 53 54 773.003 760.481 1.044 747.959 736.2621 .975 724.565.713.663 .908 54 55 773.856 761.471 1.032 749.086 737.542 .962 725.999 715.267 .894 55 56 774.786,762.549 1.020 750.313 738.936 .948 727.560 717.012 .879 56 57 775.803 763.728 1.006 751.653 740.457! .933 729.261:718.912 .862 57 58 776.909 765.008 .992 1 753.108 742.108' .917 731.108 720.973 .845 58 59 778.116 766.406 .976 ; 754.696 743.906 .899 733.117 723.212 .825 59 60 779.434 767.929 .959 i 756.424 745.8621 .880 735.300 725.642 .805 60 61 780.867 769.583 .940 758.300 747.982! .860 737.665 728.272 .783 61 62 782.425 771.380 .920 760.335 750.280 .838 740.225 731.115 .759 62 63 784.121 773.332 .899 762.544 752.770 .815 742.996 734.188 .734 63 64 785.959 775.446 .876 764.933 755.458 .790 745.984^37.495 .707 64 65 787.952 777.733 .852 767.515 758.360 .763- 749.205 741.054 .679 65 TABLE LIT. *J35 SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON- TINUING ni YEARS FROM THE PRESENT AGE. ^ 4 PER CENT. PRESENI AGE. 10. lOj. ^m.d. 1.012 11. Hi iin.d. 12. 12i. im.d. PRESENT AGB. 15 684.45? 672.317 660.177 648.586 .966 636.995 625.929 .922 15 16 684.504! 672.369 1.011 660.235 648.650 .965 637.f^65 626.006 .922 16 17 684.558,' 672.429 1.011 660.301 648.722 .965 637.144 626.092 .921 17 18 684.616 672.494 1.010 660.372 648.801 .964 637.231 626.187 .920 18 19 ; 684.680 672.566 1.009 660.451 648.888 .964 637.326 626.291 .920 19 20 684.751 672.645 1.009 : 660.538 648.984 .963 637.430 626.405 .919 20 21 684.828 672.730 1.008 660.633 649.089 .962 637.545 626.530 .918 21 22 684.913 672.825 1.007 660.737 649.203 .961 637.670 626.667 .917 22 23 685.006 672.928 1.007 660.851 649.329 .960 637.808 626.818 .916 23 24 685.109 673.043 1.006 660.977 649.469 .959 637.960 626.985 .915 24 25 685.222 673.169 1.004 661.116 649.621 .958 638.127 627.167 .913 25 26 685.347 673.308 1.003 661.270 649.791 .957 638.312 627.370 .912 26 27 685.483j 673.460 1.002 661.437 649.975 .955 638.514 627.590 .910 27 28 685.634 673 627 1.001 661.621 650.178 .954 638.736 627.834 .909 28 29 685.800 673.812 .999 661.825 650.403 .952 638.981 628.101 .907 29 30 685.979 674.012 .997 662.045 650.646 .950 639.247 628.392 .905 30 31 686.178 674.233 .995 1 662.289 650.915 .948 639.541 628.713 .902 31 32 686.396 674.476 .993 662.557 651.210 .946 639.863 629.065 .900 32 33 686.638i 674.745 .991 662.852 651.535 .943 640.219 629.454 .897 33 34 686.902, 675.039 .989! 663.177 651.893 .940 640.609 629.880 .894 34 35 687.192 675.362 .986 663.532 652.284 .937 641.037 630.348 .891 35 36 687.508 675.713 .983 663.919 652.710 .934 641.502 630.857 .887 36 37 687.856 676.101 .980 664.346 653.181 .930 642.016 631.418 .883 37 38 688.241 676.529 .976 664.817 653.699 .927 642.582 632.036 .879 38 39 688.662 676.997 .972 665.332 654.267 .922 643.202 632.713 .874 39 40 689.123 677.510 .968 665.897 654.888 .917 643.880 633.453 .869 40 41 689.628 678.071 .963 666.514 655.568 .912 644.621 634.262 .863 41 42 • 690.186 678.691 .958 667.196 656.318 .907 645.440 635.155 .857 42 43 690.792 679.364 .952 667.937 657.133 .900 646.329 636.125 .850 43 44 691.461 680.107 .946 668.753 658.030 .894 647.308 637.192 .843 44 45 692.191 680.918 .939 669.645 659.011 .886 648.377 638.358 .835 45 46 692.991 681.806 .932 670.621 660.083 .878 649.546 639.631 .826 46 47 693.868 682.779 .924 671.690 661.258 .869 650.825 641.025 .817 47 48 694.827 683.843 .915 673.859' 662.540 .860 652.222 642.546 .806 48 49 695.875 685.005 .906 674.135 663.941 .850 653.747 644.205 .795 49 50 697.023 686.276 .896 675.530 665.471 .838 655.413 646.017 .783 50 51 698.275 687.663 .884 677.052 667.139 .826 1 657.227 647.988 .770 51 52 699.641 689.176 .872 678.711 668.957 .813 659.203 650.134 .756 52 53 701.133 690.826 .859 680.519 670.936 .799 661.354 652.469 .740 53 54 702.761 692.625 .845 682.489 673.092 .783 663.695 655.008 .724 54 55 704.535 694.585 .829 684.635 675.437 .767 666.239 657.764 .706 55 56 706.464 696.713 .813 686.963 677.979 .749 668.996 660.749 .687 56 57 708.563 699.028 .795 689.493 680.739 .730 671.986 663.982 .667 57 58 710.838 701.533 .775 692.229 683.721 .709 675.214 667.469 .645 58 59 713.308 704.251 .755 695.194 686.949 .687 678.705 671.235 .623 59 60 715.985 707.193 .733 698.402 690.437 .664 682.472 675.294 .598 60 61 718.880 710.371 .709 701.862 694.193 .639 1 686.525 679.654 .573 61 62 722.006 713.797 .684 705.587 698.231 .613 690.876 684.328 .546 62 63 725.379 717.488 .658 709.596 702.570 .586 695.545 689.335 .518 63 64 729.007 721.450 .630 713.894 707.214 .557 700.535 694.676 .488 64 65 732.904 725.699 .600 718.495 712.176 .527 705.857 700.361 .458 65 ♦126 TABLE LIY. SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON- TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. 1 PRESENT AGE, m=13. 13i. ^mA. 14. 14i. ^m.d. 15. 15i. \mA. PEESENT AGE. 15 614.863 604.299 .880 593.736 583.652 .840 573.569 563.944 .802 15 16 614.947 604.390 .880 593.834 583.758 .840 573.683 564.067 .801 16 17 615.040 604.492 .879 593.944 583.877 ■.839 573.810 564.204 .801 17 18 615.143 604.603 .878 594.064 584.007 .838 573.950 564.354 .800 18 19 615.256; 604.726 .878 594.197 584.150 .837 574.104 564.520 .799 19 20 615.380 604.861 .877 594.342 584.307 .836 574.273 564.701 .798 20 21 615.516 605.009 .876 594.502 584.480 .835 574.458 564.900 .797 21 22 615.665 605.170 .875 594.676 584.668 .834 574.661 565.119 .795 22 23 615.829 605.349 .873 594.869 584.876 .833 574.884 565.358 .794 23 24 616.010 605.546 .872 595.081 585.105 .831 575.130 565.623 .792 24 25 616.208 605.761 .871 595.314 585.357 .830 575.400 565.913 .791 25 26 616.428 605.999 .869 595.571 585.634 .828 575.698 566.233 .789 26 27 616.667 606.259 .867 595.851 585.937 .826 576.023 566.582 .787 27 28 616.931 606.545 .866 596.160 586.271 .824 576.382 566.968 .785 28 29 617.222 606.861 .863 596.501 586.638 .822 576.776 567.392 .782 29 30 617.537 607.204 .861 596.871 587.038 .819 577.205 567.852 .779 30 31 617.886 607.582 .859 597.279 587.479 .817 577.679 568.361 .777 31 32 618.268 607.997 .856 597.727 587.962 .814 578.198 568.918 .773 32 33 618.690 608.455 .853 598.221 588.495 .811 578.769 569.532 .770 33 34 619.152 608.957 .850 598.762 589.079 .807 579.396 570.204 .766 34 35 619.659 609.507 .846 599.355 589.719 .803 580.083 570.941 .762 35 36 620.211 610.106 .842 600.002 590.416 .799 580.831 571.745 .757 36 37 620. 82€ 610.767 .838 600.714 591.184 .794 581.655 572.629 .752 37 38 621.491 611.494 .833- 601.498 592.029 .789 582.561 573.601 .747 38 39 622.224 612.289 .828 602.354 592.952 .784 583.551 574.662 .741 39 40 623.026 613.159 .822 603.292 593.963 .777 584.634 575.823 .734 40 41 623.904 614.110 .816 604.317 595.067 .771 585.818 577.092 .727 41 42 624.871 615.157 .810 605.444 596.281 .764 587.119 578.486 .719 42 43 625.922 616.296 .802 606.671 597.602 .756 588.534 580.002 .711 43 44 627.077 617.547 .794 608.017 599.051 .747 590.086 581.663 .702 44 45 628.339 618.913 .786 609.487 600.632 .738 591.778 583.474 .692 45 46 629.717 620.404 .776 611.092 602.358 .728 593.625 585.450 .681 46 47 631.225 622.035 .766 612.845 604.242 .717 595.640 587.604 .670 47 48 632.870 623.813 .755 614.756 606.295 .705 597.835 589.949 .657 48 49 634.664 625.751 .743 616.838 608.531 .692 600.224 592.501 .644 49 50 636.621 627.863 .730 619.106 610.964 .679 602.823 595.274 .629 50 51 638.750 630.161 .716 621.572 613.608 .664 605.645 598.283 .613 51 52 641.066 632.658 .701 624.251 616.479 .648 608.707 601.546 .597 52 53 643.585 635.372 .684 627.159 619.592 .631 612.025 605.078 .579 53 54 646.321 638.317 .667 630.313 622.966 .612 615.619 608.900 .560 54 55 649.290 641.510 .648 633.731 626.617 .593 619.504 613.028 .540 55 56 652.502 644.961 .628 637.421 630.556 .572 623.691 617.472 .518 56 57 655.978 648.692 .607 641.406 634.804 .550 628.203 622.254 .496 67 58 659.725 652.709 .585 645.692 639.368 .527 633.045 627.375 .473 58 59 663.766 657.034 .561 650.303 644.272 .503 638.242 632.873 .447 59 60 668.116 661.685 .536 655.254 649.529 .477 643.804 638.743 .422 60 61 672.783 666.667 .510 660.551 655.144 .451 649.737 644.995 .395 61 62 677.780 671.992 .482 666.205 661.128 .423 656.051 651.636 .368 32 63 683.125 677.678 .454 672.231 667.493 .395 662.756 658.675 .340 63 64 688.817 683.721 .425 678.626 674.236 .366 669.846 666.102 .312 64 65 694.866 690.131 .395 685.396 681.358 .337 677.321 673.917 .284 65 TABLE LIT. *i2r SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON- TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PBESENT AQE. 16. lei |m.d. 17. 17i. ^m.d. 18. 18i. ■Jm.d. PRESENT AOE. 15 554.320 545.135 .765 535.951 527.186 .730 518.422 510.060 .697 15 16 554.452 545.276 .765 536.101 527.347 .730 518.593 510.241 .696 16 17 554.598 545.433 .764 536.268 527.525 .729 518.782 510.442 .695 17 18 554.759 545.605 .763 536.451 527.720 .728 518.989 510.662 .694 18 19 554.936 545.794 .762 536.653 527.934 .727 519.216 510.904 .693 19 20 555.130 546.002 .761 536.874 528.171 .725 519.469 511.172 .691 20 21 555.343 546.230 .759 537.117 528.430 .724 519.744 511.464 .690 21 22 555.576 546.480 .758 537.384 528.713 .723 520.043 511.782 .688 22 23 555.833 546.754 .757 537.676 529.027 .721 520.378 512.138 .687 23 24 556.116 547.057 .755 537.999 529.369 .719 520.740 512.523 .685 24 25 556.426 547.389 .753 5.38.352 529.746 .717 521.140 512.948 .683 25 26 556.768 547.755 .751 538.742 530.162 .715 521.583 513.419 .680 26 27 557.142 548.155 .749 539.168 530.616 .713 522.065 513.930 .678 27 28 557.554 548.595 .747 539.637 531.115 .710 522.594 514.492 .675 28 29 558.007 549.080 .744 540.153 531.666 .707 523.180 515.114 .672 29 30 558.499 549.606 .741 540.714 532.263 .704 523.813 515.787 .669 30 31 559.043 550.188 .738 541.333 532.922 .701 524.511 516.528 .665 31 32 559.639 550.826 .734 542.012 533.646 .697 525.280 517.345 .661 32 33 560.295 551.526 .731 542.758 534.440 .693 526.122 518.238 .657 33 34 561.013 552.294 .727 543.576 535.310 .689 527.045 519.217 .652 34 35 561.800 553.135 .722 544.471 536.264 .684 528.057 520.290 .647 35 36 562.659 554.053 .717 545.447 537.302 .679 529.157 521.457 .642 36 37 563.603 555.061 .712 546.520 538.445 .673 530.370 522.743 .636 37 38 564.641 556.169 .706 547.698 539.697 .667 531.697 524.149 .629 38 39 565.774 557.379 .700 548.985 541.066 .660 533.147 525.686 .622 39 40 567.013 558.702 .693 550.391 542.560 .653 534.730 527.363 .614 40 41 568.367 560.146 .685 551.926 544.192 .645 536.459 529.194 .605 41 42 569.854 561.732 .677 553.610 545.980 .636 538.351 531.196 .596 42 43 571.470 563.455 .668 555.441 547.925 .626 540.410 533.2r75 .586 43 44 573.241 565.343 .658 557.445 550.052 .616 1542.658 535.752 .576 44 45 575.171 567.398 .648 559.626 552.366 .605 545.106 538.339 .564 45 46 577.275 569.639 .636 562.003 554.886 .593 547.770 541.153 .551 46 47 579.569 572.080 .624 564.591 557.628 .580 550.666 544.210 .538 47 48 582.064 574.733 .611 567.403 560.606 .566 553.808 547.525 .524 48 49 584.777 577.616 .597 570.456 563.837 .552 557.218 551.119 .508 49 50 587.726 580.748 .582 573.770 567.340 .536 560.911 555.009 .492 50 51 590.922 584.139 .565 577.357 571.131 .519 564.905 559.212 .474 51 52 594.385 587.811 .548 581.238 575.227 .501 569.217 563.745 .456 52 53 598.132 591.780 .529 585.429 579.647 .482 573.866 568.627 .437 53 54 602.182 596.066 .510 589.950 584.409 .462 578.868 573.874 .416 54 55 606.552 600.685j .489! 594.818 589.532 .441 584.246 579.509 .395 55 56 611.2531 605.648 .467 600.043 595.022 .418 1 590.001 585.530 .373 56 57 616.305 610.975 .444 605.646 600.900 .396 596.155 591.961 .350 57 58 621.705:616.661 .420 611.618 607.161 .371 602. 704 598.793 .326 58 59 627.504 622.758 .396 618.012 613.849 .347 609.686 606.065 .302 59 60 633.682 629.243 .370 624.804 620.943 .322 617.083 613.757 .277 60 61 640.253:636.128 .344 632.004 628.452 .296 624.900 621.870 .252 61 62 647.221 643.417 .317 639.613 636.369 .270 633.126 630.391 .228 62 63 654.594 651.113 .290 647.633 644.699 .245 641.764 639.320 .204 63 64 662.35S 659.203 .263 656.047 653.420 .219 650.794 648.635 .180 64 65 670.5141 667.67ii .236 664.845 662.518 .194 660.191 658.308 .157 65 *128 TABLE LIV SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF l.OUU COJS- TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PRESENT AGE. w=19. 19i. ^m.d. 20. 20i. |m.d. 21. 21i. ^m.d. PEESEHtI AGE. 15 501.698 493.720 .665 485.743^478.134 .634 470.525 463.269 .605 15 16 501.890 493.924 .664 485.959 478.362 .633 470.766 463.523 .604 16 17 502.103 494.150 .663 486.197:478.614 .632 471.031 463.802 .602 17 18 502.336 494.397 .662 486.458478.890 .631 471.3221464.109 .601 18 19 502.592 494.668 .660 486.745|479.193 .629 471.641 464.446 .600 19 20 502.876 494.969 .659 487.062i 479.528 .628 471.995 464.818 .598 20 21 503.185 495.296 .657 487.4081479.894 .626 472.380 465.225 .596 21 22 503.522 495. 654 .656 487.786 480.293 .624 472.801 465.668 .594 22 23 503.899 496.053 .654 488.207 480.738 .622 473.269 466.162 .592 23 24 504.307 496.485 .652 488.664 481.221 .620 473.778 466.698 .590 24 25 504.757 496.962 .650 489.167 481.752 .618 474.337 467.288 .587 25 26 505.255 497.489 .647 489.723 482.339 .615 474.956 467.939 .585 26 27 505.796 498.062 .645 490.328 482.978 .613 475.629 468.648 .582 27 28 506.391 498.692 .642 490.994 483.681 .609 476.369 469.428 .578 28 29 507.049 499.389 .638 491.729 484.457 .606 477.186 470.288 .575 29 30 507.761 500.142 .635 492.524 485.297 .602 478.070 471.219 .571 30 31 508.545 500.972 .631 493.399 486.221 .598 479.043 472.244 .567 31 32 509.409 501.887 .627 494.3641487.239 .594 480.114 473.371 .562 32 33 510.354 502.886 .622 495.418 488.351 .589 481.285 474.603 .557 33 34 511.389 503.981 .617 496.573 489.569 .584 482.566 475.952 .551 34 35 512.524 505.181 .612 497.838 490.903 .578 483.968 477.426 .545 35 36 513.758 506.486 .606 499.214 492.354 .572 485.494 479.031 .539 36 37 515.J17 507.922 .600 500.728 493.949 ,565 487.171 480.794 .531 37 38 516.602 509.491 .593 502.381 495.692 .557 489.002 482.718 .524 38 39 518.225 511.205 .585 504.186 497.592 .550 490.999 484.816 .515 39 40 519.997 513.076 .577 506.156 499.666 .541 498.177 487.102 .506 40 41 521.929 515.116 .568 508.303 501.926 .531 495.549 489.592 .496 41 42 524.042 517.345 .558 510.648 504.392 .521 498.137 492.307 .486 42 43 526.340 519.769 .548 513.197 507.072 .510 500.948 495.253 .475 43 44 528.847 522.410 .536 515.974 509.990 .499 504.006 498.458 .462 44 45 531.573 525.282 .524 518.991 513.158 .486 507.326 501.934 .449 45 46 534.536 528.401 .511 522.266J 516.595 .473 510.924 505.699 .435 46 47 537.754 531.786 .49? 525.819! 520.321 .458 514.823 509.775 .421 47 48 541.242 535.453 .482 529.6641524.349 .443- 519.035 514.176 .405 48 49 545.021 539.422 .467 533.824 528.704 .427 523.585 518.925 .388 49 50 549.107 543.710 .450 538.3131 533.401 .409 '528.488 524.037 .371 50 51 553.519 548.336 .432 543.153 538.458 .391 533.762 529.530 .353 51 52 558.273 553.315 .413 548.358 543.890 .372 539.423 535.420 .334 52 53 563.388 558.667 .393 553.947 549.717 .353 545.487 541.721 .314 53 54 568.881 564.407 .373 559.933 555.950 .332 551.968 548.447 .293 54 55 574.772 570.554 .352 566.337 562.609 .311 558.882 555.612 .273 55 56 581.060 577.108 .329 573.157 569.691 .289 566.225 563.210 .251 56 57 587.767 584.088 .307 580.410 577.210 .267 574.011 571.254 .230 57 58 594.883 591.482 .283 588.082 685.152 .244 582.222 579.723 .208 58 59 602.445 599.326 .260 596.207 593.547 .222 590.888 588.645 .187 59 60 610.431 607.595 .236 604.759 602.368 .199 599.977 597.986 .166 60 61 618.840 616.285 .213 613.731 611.604 .177 609.477 607.730 .146 61 62 627.657 62^.380 .190 623.103 621.233 .156 619.363 617.850 .126 62 63 636.877 634.870 .167 632.864 631.241 .135 629.618 628.326 .108 63 64 646.477 644.731 .146 642.985 641.596 .116 640.207 639.121 .091 64 65 656.425 654.926 .125 653.428 652.257 .098 651.087 650.190 .075 65 TABLE LIT. 129 SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON- TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PRESENT AGE. 15 22. 22i. |md. 23. 23i Jm.d. 24. 2^. l^m.d. PRESENT AGE, 456.013 449.094 .577 442.176 435.581 .550 428.986 422.700 .524 15 16 456.280 449.375 .575 442.471 435.890 .548 : 429.310 423.040 .523 16 17 456.574 449.685 .574 442.796 436.232 .547 429.667 423.415 .521 17 18 456.897 450.024 .573 443.152 436.605 .546 430.059 423.825 .520 18 19 457.251 450.397 .571 443.543 437.016 .544 430.490 424.277 .518 19 20 457.642 450.808 .570 443.975 437.470 .542 430.965 424.775 .516 20 21 458.070 451.259 .568 444.447 437.965 .540 431.484 425.319 .514 21 22 458.536 451.749 .566 444.962 438.506 .538 432.050 425.912 .512 22 23 459.055 452.294 .563 445.534 439.107 .536 432.679 426.571 .509 23 24 459.618 452.887 .562 446.156 439.759 .533 433.363 427.288 .506 24 25 460.238 453.539 .558 446.840 440.477 .530 434.114 428.074 .503 25 26 460.923 454.259 .555 447.595 441.269 .527 434.943^28.943 .500 26 27 461.668 455.042 .552 448.417 442.131 .524 435.846 429.888 .497 27 28 462.487 455. 903 .549 449.319 443.078 .520 436.837 430.926 .493 28 29 463.391 456.852 .545 450.314 444.121 .516 437.928 432.068 .488 . 29 30 464.369 457.880 .541 451.392 445.252 .512 439.112 433.307 .484 30 31 465.445 459.011 .536 452.577 446.494 .507 440.412 434.667 .479 31 32 466.628 460.253 .531 453.879 447.859 .502 441.838 436.158 .473 32 33 467.922 461.612 .526 455.302 449.349 .496 443.397 437.789 .467 33 34 469.337 463.097 .520 456.857 450.979 .490 445.100 439.569 .461 34 35 470.885 464.721 .514 458.558 452.759 .483 446.961 441.514 .454 35 S6 472.569 466.488 .507 460.408 454.696 .476 448.984 443.628 .446 36 37 474.417 468.426 .499 462.436 456.818 .468 451.200 445.941 .438 37 38 476.434 470.540 .490 464.647 459.131 .460 453.615 448.462 .429 38 39 478.633 472.845 .482 467.057 461.651 .451 456.244 451.205 .420 39 40 481.028 475.354 .473 469.680 464.391 .441 459.103 454.186 .410 40 41 483.635 478.083 .463 472.531 467.369 .430 462.208 457.422 .399 41 42 486.477 481.056 .452 475.636 470.610 .419 465.584 460.9381 .387 42 43 489.559 484.279 .440 479.000 474.120 .407 469.240 464.744 .375 43 44 492.910 487.781 .427 482.653 477.928 .394 473.202 468.864 .362 44 45 496.542 491.574 .414 486.607 482.046 .380 477.486 473.316 .348 45 46 500.474 495.677 .400 490.881 486.496 .365 482.110 478.117 .333 46 47 504.728 500.113 .385 495.498 491.296 .350 487.095 483.289 .317 47 48 509.317 504.893 .369 500.470 496.462 .334 492.454 488.842 .301' 48 49 514.265 510.043 .352 505.821 502.016 .317 498.211 494.802 .284 49 50 519.586 515.575 .334 511.564 507.971 .299 504.377 501.178 .267 50 51 525.299 521.508 .316 517.717 514.342 .281 510.967 507.983 .249 51 52 531.418 527.854 .297 524.291 521.141 .263 517.992 515.229 .230 52 53 537.956 534.627 .277 531.299 528.380 .243 525.461 522.922 .212 53 54 544.921' 541.839 .257 538.751 536.066 .224 533.380 531.066 .193 54 55 552.343 549.499 .237 546.656 544.206 .204 541.756 539.667 .174 55 56 1560.196 557.599 .216 ,555.002 552.787 .185 550.573 548.706 .156 56 57 568.498 566.147 .196 1 563.797 561.815 .165 559.833 558.183 .138 57 58 1 577.225 575.118 .176 1 573.012 571.259 .146 569.505 568.065 .120 58 59 586.403 584.536 .156 1582.669 581.135 .128 579.602 578.361 .103 59 60 595.995 594.360 .136 592.725 591.402 .110 590.079 589.027 .088 60 61 605.983 604.570 .118 603.157 602.033 .094 600.908 600.029 .073 61 62 616.337 615.133 .100 613.930 612.989 .078 612.049 611.329 .060 62 63 627.035 626.026 .084 625.017 624.245 .064 623.472; 622.893 .048 63 64 638.036 637.205 .069 636.374 635.752 .052 635.129 634.674 .038 64 65 j 649.294 648.622 .056 647.951 647.461 .041 646.970 646.621 .029 65 1 *130 TABLE LIY. SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON- TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PBESENT AGE. m=25. 25}. 1 Jm.d. 26. 26i ^m.d. 27 27i ^m.d. PEESENT AOE. 15 416.415 410.426 .499 404.437 398.732 .475 393.027 387.595 .453 15 16 416.770 410.798 .498 404.826 399.139 .474 393.451 388.037 .451 16 17 417.162 411.208 .496 405.254 399.586 .472 393.918 388.525 .449 17 18 417.592 411.658 .495 405.724 400.077 .471 394.431 389.059 .448 18 19 418.064 412.152 .493 406.240 400.617 .469 394.993 389.646 .446 19 20 418.585 412.697 .491 406.808 401.210 .467 395.612 390.292 .443 20 21 419.153 413.291 .489 407.430 401.859 .464 396.289 390.999 .441 21 22 419.775 413.942 .486 408.109 402.569 .462 397.029 391.770 .438 22 23 420.463 414.662 .483 408.860 403.353 .459 397.846 392.622 .435 23 24 421.212 415.445 .481 409.678 404.208 .456 398.738 393.553 .432 24 25 422.035 416.305 .478 410.576 405.146 .453 399.715 394.571 .429 25 26 422.942 417.254 .474 411.566 406.178 .449 400.791 395.693 .425 26 27 423.930 418.287 .470 412.644 407.304 .445 401.963 396.914 .421 27 28 425.015 419.421 .466 413.826 408.537 .441 403.249 398.254 .416 28 29 426.208 420.667 .462 415.126 409.894 .436 404.661 399.725 .411 29 30 427.502 422.019 .457 416.536 411.364 .431 406.192 40;.319 .406 30 31 428.922 423.502 .452 418.083 412.977 .426 407.871 403.068 .400 31 32 430.479 425.129 .446 419.778 414.744 .420 409.710 404.981 .394 32 33 432.180 426.904 .440 421.628 416.672 .413 411.716 407.068 .387 33 34 434.038 428.8421 .433 423.647 418.775 .406 413.903 409.342 .380 34 35 436.067 430.959! .426 425.851 421.069 .398 416.288 411.822 .372 35 36 438.272 433.258 .418 1428.244 423.561 .390 418.878 414.513 .364 36 37 440.683 435.772 .409 430.860 426.282 .382 421.704 417.448 .355 37 38 443.310 438.507 .400 433.705 429.241 .372 424.776 420.636 .345 38 39 446.166 441.482 .390 436.797 432.453 .362 428.110 424.095 .335 39 40 449.269 444.710 .380 440.152 435.938 .351 431.724 427.842 .324 40 41 452.636 448.213 .369 443.788 439.712 .340 435.636 431.894 .312 41 42 456.293 452.013 .357 447.732 443.803 .327 439.874 436.281 .299 42 43 460.247 456.119 .344 451.991 448.217 .315 444.443 441.007 .286 43 44 464.527 460.560 .331 456.594 452.983 .301 449.372 446.101 .273 44 45 469.146 465.350 .316 461.554 458.115 .287 454.676 451.577 .258 45 46 474.124 470.507 .301 466.890 463.630 .272 460.370 457.449 .243 46 47 479.482 476.052 .286 472.621 469.548 .256 1 466.474 463.738 .228 47 48 485.231 481.995 .270 478.759 475.878 .240 472.998 470.452 .212 48 49 491.393 488.358 .253 485.323 482.641 .224 479.958 477.605 .196 49 50 497.978 495.150 .236 492.322 489.841 .207 487.361 485.204 .180 60 51 505.000 502.383 .218 499.766 497.491 .190 495.215 493.254 .163 51 52 512.466 510.063 .200 507.661 505.590 .173 503.520 501.755 .147 52 53 520.383 518.196 .182 516.008 514.143 .155 512.277 510.704 .131 53 54 528.752 526.779 .164 524.806 523.143 .139 521.480 520.095 .115 54 55 537.578 535.817 .147 534.056 532.589 .122 531.123 529.919 .100 55 56 546.839 545.284 .130 543.730 542.454 .106 541.177 540.144 .086 56 57 556.533 555.178 .113 553.823 552.726 .091 551.630 550.758 .073 57 58 566.624 565.459 .097 564.293 563.366 .077 562.439 561.714 .060 58 59 577.121 576.134 .082 575.147 574.376 .064 673.605 573.015 .049 59 60 587.974 587.152 .069 586.330 585.701 .052 585.071 584.599 .039 60 61 599.151 598.478 .056 597.806 597.302 .042 596.798 596.429 .031 61 62 610.609 610.069 .045 609.530 609.135 .033 608.741 608.460 .023 62 63 622.314 621.891 .035 621.468 621.167 .025 620.865 620.661 .017 63 64 634.219 633.895 .027 633.57l|633.350 .018 633.130 632.987 .012 64 65 646.271 646.033 .020 645.795; 645. 642 .013 645.488 645.394 .008 65 TABLE LIT. 131 SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1.000 CON- TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PEE8ENT AGE. 28 28i -J^m.d. 29. 29i ^m.d. 30 30i |m.d PEESENT A&E. 15 15 383.162 376.991 .431 371.820 366.899 .410 361.978 357.397 .390 16 382.624 377.473 .439 372.320 367.430 .408 362.520 357.861 .388 16 17 383.131 378.001 .438 372.871 367.993 .407 363.116 358.480 .386 17 18 383.688 378.583 .436 373.475 368.633 .404 363.769 359.160 .384 18 19 384.399 379.318 .433 374.137 369.313 .402 364.486 359.905 .383 19 20 384.972 379.919 .431 374.867 370.071' .400 365.275 360.735 .379 20 21 385.708 380.686 .419 375.664 370.900! .397 366.136 361.631 .376 21 22 386.511 381.533 .416 376.535 371.806 .394 367.078 363.599 ,373 22 23 387.399 383.448 .413 377.496 373.806 .391 368.116 363.678 .370 23 24 388.367 383.455 .409 378.544373.896 .387 369.248 364.854 .366 24 25 389.427 384.559 .406 379.692 375.090 .384 370.488 366.141 .363 25 26 390.595 385.775 .403 380.955 376.403i .379 371.850 367.556 .358 26 27 391.866 387.097 .397 383.3391377.831 .375 373.333 369.095 .353 27 28 393.259 388.547 .393 383.835 379.396 .370 374.957 370.780 .348 28 29 394.788,390.137 .388 385.486:381.111 .365 376.736 373.635 .343 29 30 396.447'391.863 .383 387.3781383.973 .359 378.666 374.638 .337 30 31 398.264^393.751 .376 389.339385.007' .353 380.776 376.815 .330 31 32 400.252|395.818 .370 391.383!387.332j .346 383.081 379.303 .323 32 33 402.430 398.069 .363 393.719i389.655: .339 385.591 381.804 .316 33 34 404.782 400.533 .355 396.263 393.292 .331 388.321 384.630 .308 34 35 407.356 403.194 .347 399.031395.161 .323 391.291 387.702 .299 35 36 410.148 406.089 .338 402.031;398.268i -^^^ 394.506 391.028 .390 36 37 413.191 409.345 .339 405.2991401. 651 .304 398.003 394.642 .280 37 38 416.497 413.670 .319 408.843'405.318 .394 401.792 398.556 .270 38 39 420.080 416.381 .308 413.681409.385 .383 405.889 403. 784 .259 39 40 423.959 430.395 .397 416.831 413.573 .273 410.313 407.346 .247 40 41 438.153 434.731 .385 431.310 418.195 .260 415.081413.259 .235 41 42 433.688 439.417 .373 436.146 433.183 .347 430.330 417.649 .333 42 43 437.571 434.458 .359 431.345 438.539 .334 435.734 433.331 .309 43 44 443.830 439.883 .346 436.934,434.393 .330 431.651:439. 300 .196 44 45 448.477 445.700 .331 443. 933 '440. 451 .206 437.979'435.794 .183 45 46 454.539 451.939 .317 449.339 447.031 .193 444.733 443.716 .168 46 47 461.003 458.584 .303 ! 456.166^454.045 .177 451. 9331450. 078 .154 47 48 467.905 465.673 .186 463.438'461.496 .162 459.5531457. 879 .140 48 49 475.353 473.206 .171 471.160 469.397 .147 467.634 466.130 .135 49 50 483.047 481.188 .155 479.330 477.745 .132 476.160 474.834 .111 50 51 491.293 489.622 .139 487.950 486:540 .118 485.131483.957 .098 51 52 499.990 498.503 .134 497.014j495.775 .103 494.535 493.518 .085 52 53 509.132 507.833 .109 506.513i505.437 .090 504.363 503.492 .073 53 54 518.710 517.573 .095 516.435 515.515 .077 514.594 513.862 .061 54 55 538.714 537.740 .081 536.766 535.990 .065 525.315 534.609 .051 55 56 539.111 538.389 .069 537.468l536.836 .054 536.184 535.693 .041 56 57 549.886 549.3051 .057 548.533 548.001 .044 547.479 547.088 .033 57 58 560.990 560.435 .046 559.881 559.466 .035 559.050 558.747 .035 58 59 573.424 571.983 .037 571.540 571.316 .027 570.893 570.663 .019 59 60 584.137 583.783 .029 583.438 583.193 .020 583.947 583.780 .014 60 61 596.061 595.799 .022 595.537 595.358! .015 595.180 595.065 .010 61 62 608.179 607.988 .016 607.798 607.675 .010 607.551 307.475 .006 62 63 630.456 630.334 .011 630.191 630.109 .007 630.038 619.980 .004 63 64 633.845 633.757 .007 633.670 633.619 .004 633.567 633.538 .003 64 65 645.399 645.344 .005 645.188 645.157 .003 645.1361645.110 .001 65 '132 TABLE LIY. SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON- TINUING m YEARS FROM THE PRESENT AGE. 4 PER CENT. PEE8ENT AaE. m=31. 3 If |^m.d. 32. 32i. ^m.d. 33. 33i. ^m.d. PKESENT AQE. 15 353.616 348.166 .371 343.716 339.487 .352 335.258 331.241 .335 15 16 353.202 348.774 .369 344.347 340.142 .350 335.937i331.945 .333 16 17 353.845 349.443 .367 345.041 340.862 .348 336.683332.719 .330 17 18 354.551 350.176 .365 345.801 341.651 .346 337.501333.567 .328 18 19 355.324 350.979 .362 346.634 342.515 .343 338.396 334.495 1 .325 19 20 356.175 351.863 .359 347.550 343.465 .340 339.381335.516 .322 20 21 357.105 352.828 .356 348.551 344.504 .337 340.456 336.629 .319 21 212 23 358.121|353.882 .353 349.644 345.637 .334 341.630 337.846 .315 22 359.2401355.044 .350 350.848 346.885 .330 342.922 339.183 .312 23 24 360.459 356.309 .346 352.159 348.244 .326 344.329 340.641 .307 24 25 361.795 357.695 .342 353.594 349.730 .322 345.866342.233 .303 25 26 363.261359.215 .337 355.169 351.363 .317 347.557 343.881 .298 26 27 364.857;360.870 .332 356.883 353.138 .312 349.393 345.880 .293 27 28 366.603 362.680 .327 358.757 355.077 .307 351.398 347.955 .287 28 29 368.515 364.660 .321 360.806 357.199 .301 353.591350.221 .281 29 30 370.589 366.809 .315 363.029 359.498 .294 355.967 352.676 .274 30 31 372.854 369.154 .308 365.454 362.006 .287 358.558 [355.352 .267 31 32 375.328 371.713 .301 368.099 364.739 .280 361.379 358.264 .200 32 33 378.016 374.494 .294 370.973 367.708 .272 364.443 361.425 .252 33 34 380.939 377.517 .285 374.094 370.930 .264 367.766 364.851 .243 34 35 384.114 380.797 .276 377.481 374. 425 .255 371.369 368.564 .234 35 36 387.550 384.345 .267 381.141 378.199 .245 375.257 372,567 .224 36 37 391.281 388.193 .257 385.111 382.290 .235 379.469 376.801 214 37 38 395.319 392.359 .247 389.400 386. 706 .225 384.012i381.571 .203 38 39 399.679 396.852 .236 394.025 391.464 .213 388.903 386.595 .192 39 40 404.379 401.691 .224 399.003 396.581 .202 394.159 391.988 .181 40 41 409.437 406.894 .212 404.351 402.072 .190 399.794 397.766 .169 41 42 414.879 412.487 .199 410.094 407.964 .178 405.834 403.951 .157 42 43 420.707 418.469 .187 416.232 414.255 .165 412.277 410.542 .145 43 44 426.949 424.871 .173 422.792 420. 969 .152 419.146 417.560 .132 44 45 433.609'431.692 .160 429.776 428.109 .139 426.442 425.006 .120 45 46 440.70l'438.948 .146 437.194 435.683 .126 434.173 432.885 .107 46 47 448.233 446.643 .133 445.054 443. 699 .113 442.344 441.201 .095 47 48 456.206 454.779 .119 453.352 452. 149 .100 450.946 449. 945 .083 48 49 464.627 463.359 .106 462.092 461.037 .088 459.981 459.114 .072 49 50 473.488 472.375 .093 471.263 470.349 .076 469.435 468.696 .062 60 51 482.783 481.819 .080 480.855 480.075 .065 479.296 478. 675 .052 51 52 492.500 491.677 .069 490.853 490.198 .055 489.543|489.031 .043 52 53 502.622,501.930 .058 501.238 500. 697 .045 500.156 499.742 .035 53 54 513.130,512.558 .048 511.986 511.548 .037 511.110 510.782 .027 54 55 524.004'523.540 .039 523.076 522.729 .029 522.381522.127 .021 55 56 535.200 534.832 .031 534.464 534.194 .023 533.925 533. 734 .016 56 57 546. 697:546.411 .024 546.126 545.922 .017 545.719 545.580 .012 57 58 558.443 558.227 .018 558.010 557.863 .012 557.716 557.621 .C08 58 59 570.433 570.276 .013 ,570.119 570.018 .008 569.917 569.855 .C05 59 60 582.613 582.505 .009 i 582.397 582.330 .006 582.264 582.225 .003 60 61 594.950 594.879 .006 594.808 594.766 .004 594.724 594. 701 .002 61 62 607.399 607.354 .004 607.309 607.285 .002 607.260 607.247 .001 62 63 619.932 619.905 .002 619.879 619.865 .001 619.851 619.844 .001 63 64 632.510 632.495 .001 632.480 632.473 .001 632.466 632.463 .000 64 65 j 645.094 645.0871 -001 1 645.079 645.076 .000 645.073 645.0721 .000 65 TABLE LIT. ♦133 SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 CON- TINUING ra YEARS FROM THE PRESENT AGE. 4 PER CENT. PEESENl AGE. 34. 34i. im.d. 35. 35i l-m.d. 36. 36i. im.d. PBESENT AGE. 15 337.234 333.411 .318 319.599 j 315.983 .301 312.366 308.937 .286 15 16 337.953 334.167 .316 320.381 316.791 .399 313.302 309.803 .283 16 17 338.755 334.997 .313 331.340 317.681 .397 314.133 310.754 .381 17 18 339.633 335.906 .311 333.180 318.654 .394 315.138 311.794 .278 18 19 330.594 336.903 .308 333.210 319.719 .391 316.339 313.932 .275 19 20 331.650 337.995 .305 334.340 330.888 .288 317.437 314.181 .271 20 21 333.803 339.189 .301 335.574 ■333.165 .384 318.756 315.544 .268 21 22 334.061 330.491 .398 336.923 323.559 .380 330.195 317.031 .264 22 23 335.445 331.933 .394 338.401 335.087 .376 331.774 318.^61 .259 23 24 336.953 333.483 .389 330.013 336.753 .272 333.493 320.436 .355 24 25 338.600 335.186 .385 331.773 338.570 .267 335.369 333.372 .250 25 28 340.405 337.053 .379 333.699 330.561 .262 337.433 334.490 .244 26 27 343.368 339.081 .374 335.793 333.733 .256 339.651 336.789 .239 27 28 344.511 341.394 .368 338.077 335.078 .250 333.080 339.393 .232 28 29 346.851 343.710 .363 340.569 337.649 .343 334.738 333.031 .336 29 30 349.385 346.335 .355 343.366 340.439 .236 337.593 334.969 .319 30 31 353.146 349.174 .348 346.301 343.453 .339 340.705 338.174 .311 31 32 355.149 353.369 .340 349.390 346.737 .221 344.084 341. 648 .303 32 33 358.407 355.636 .333 353.845 350.393 .213 347.740 345.406 .195 33 34 361.937 359.361 .333 356.585 354.138 .304 351.691 349.464 .186 34 35 365.758 363.193 .314 360.627 358.392 .195 355.957 353.843 .176 35 36 369.878 367.439 .304 364.980 363.763 .185 360.543 358.544 .167 36 37 374.333 373.005 .194 369.678 367.581 .175 365.485 363.606 .157 37 38 379.130 376.931 .183 374.731 373.761 .164 370.790 369.035 .146 38 39 384.387 383.319 .173 380.153 378.311 .153 376.470 374.843 .136 39 40 389.818 387.886 .161 385.954 384.346 .142 383.539 381.041 .125 40 41 395.738 393.945 .149 393.153 390.579 .131 389.007 387.639 .114 41 42 403.068 400.417 .138 398.765 397.339 .120 395.893 394.656 .103 42 43 408.807 407.398 .136 405.790 404.490 .108 403.190 403.081 .092 43 44 415.974 414.608 .114 413.343 413.077 .097 410.913 409.931 .083 44 45 433.570 433.346 .103 431.131 430.089 .086 419.056 418.197 .073 45 46 431.597 4.30.511 .091 439.436 438.531 .075 437.617 436.874 .063 46 47 440.058 439.107 .079 438.155 437.374 .065 436.593 435.960 .053 47 48 448.943 448.130 .009 447.397 446.631 .056 445.966 445.437 .044 48 49 458.348 457.547 .058 456.845 456.388 .046 455.730 455.294 .036 49 50 467.957 467.369 .049 466.781 466.331 .038 465.863 465.510 .039 50 51 478.055 477.570 .040 477.085 476.714 .031 476.343 476.065 .033 51 52 488.530 488.138 .033 487.736 487.443 .024 487.149, 486.934 .018 52 53 499.338 499.017 .036 498.707 498.480 .019 498.354 498.093 .014 53 54 510.454 510.314 .030 509.974 509.803 .014 509.633 509.516 .010 54 55 531.873 531.693 .015 531.513 531.389 .010 531.366 531.187 .007 55 56 533.543 533.411 .011 533.381 533.197 .007 533.113 533.061 .004 56 57 545.443 545.353 .007 545.363 545.308 .005 545.153 545.130 .003 57 58 557.536 557.479 .004 557.433 557.399 .003 557.364 557.345 .003 58 59 569.793 569.756 .003 569.719 569.699 .002 569.679 569.669 .001 59 60 583.186 583.164 .003 583.143 583.133 .001 583.131 583.115 .001 60 61 594.678 594.666 .001 594.654 594.648 .001 594.643 594.641 .000 61 62 607.334 607.338 .001 607.333 607.319 .000 607.317: 607.316 .000 62 63 619.838 619.836 .000 619.833 619.831 .000 619.830 619.830 .000 63 64 633.460 633.459 .000 632.458 633.458 .000 64 65 645.070 645.069 .000 65 *]34 TABLE LIT. SINGLE PREMIUM OR RESERVE (A), FOR ENDOWMENT INSURANCE OF 1,000 (X)N. TINUING m YEARS FROM THE. PRESENT AGE. 4 PER CENT. PKTiaENT A&E. m=37. 37i. |rm.(l. 38. 38i. Jm.d. 39. 39i. ^m.d. FBESENT AGE, 15 305.509 302.261 .271 299.014 295.941 .256 292.869 289.963 .242 15 16 306.403 303.186 .268 299.969 296.928 .253 293.886 291.014 .239 16 17 307.386 304.202! .265 301.017 298.010 .251 295.003| 292.166 .236 17 18 308.460 305.311! .262 302.163 299.198 .248 296.223 293.425 .233 18 19 309.636 306.527 .259 303.417 300.487 .244 297.557,294.800 .230 19 20 310.925 307.857 .256 304. 790 801.904 .241 299.018; 296.307 .226 20 21 312.332 309.310 .252 306.288:303.449 .237 300.611 297.949 .222 21 22 313.866 310.894 .248 307.9211305.134 .232 802.346 299.737 .217 22 23 315.548 3ia.629 .243 309.710 806.978 .228 304.246 301.694 .213 23 24 317.379 314.517 .239 311.656 808.983 .223 306.3111303.820 1 .208 24 25 319.376 316.577 .233 313.777 311.168 .217 808.559 306.133 .202 25 26 321.558 318.825 .228 316.093 818.558 .212 311.012 308.657 .196 26 27 323.927 321.266 .222 318.605 816.187 .206 313.670: 311.389 .190 27 28 326.506 323.922 .215 321.337 818.948 .199 816.559; 314.358 .183 28 29 329.313 326.810 .209 324.307j322.001 .192 319.695 317.578 .176 29 30 332.347 329.931 .201 327.515 825.297 .185 323.079 321.051 .169 30 31 335.642 333.318 .194 330.994 328.869 .177 326.7441324.809 .161 31 32 339.213 336.986 .186 334.759 332.782 .169 380.7041328.867 .153 32 33 343.072 340.947 .177 338.822 386.897 .160 384.972 333.237 .145 33 34 347.237 345.219 .168 343.201341.882 .152 339.564 337.985 .136 34 35 351.726 349.819 .159 347.918 346.205 .142 344.496 342.975 .127 35 36 356.545 354.754 .149 352.962 351.867 .138 349. 7731 348.364 .117 36 37 361.727 360.055 .139 358.383 356.905 .123 355.426 354.129 .108 37 38 367.281 365.730 .129 364.179 362.819 .113 361.459 360.276 .099 38 39 373.215 371.788 .119 370.861 869.119 .104 367.878 366.809 .089 39 40 379.543 378.240 .109 376.937 875.815 .094 374.693 373.736 .080 40 41 386.272 385.093 .098 383.915 382.911 .084 381.906 381.058 .071 41 42 393.418 392.363 .088 391.308 390.418 .074 389.529 388.788 .062 42 43 400.973 400.039 .078 399.104 898.826 .065 397.548 396.909 .063 43 44 408.949 408.131 .068 407.3181406.642 .056 405.970 405.427 .045 44 45 417.337,416.631 .059 415.925 415.858 .048 414.782 414.328 .038 45 46 426.132 425.531 .050 424.981 424.453 .040 423.975 423.601 .031 46 47 435.328 434.825 .042 434.822 488.929 .033 433.536 433.235 .025 47 48 444.907 444.493 .035 444.079 448.763 .026 443.446 448.209 .020 48 49 454.858 454.524 .028 454.190 453.940 .021 453. 690| 453.507 .015 49 50 465.158 464.895 .022 464.631 464.488 .016 464.245! 464.108 .011 50 51 475.787 475.583 .017 475.380 475.285 .012 475.091 474.993 .008 51 52 486.719 486.567 .013 486.414 486.310 .009 486.206 486.139 .006 52 53 497.931 497.821 .009 497.711 497.640 .006 1 497.569 497.525 .004 53 54 509.400 509.325 .006 509.250 509.204 .004 509.157 509.130 .002 54 55 521.107 521.058 .004 521.009 520.980 .002 520.951 520.936 .001 55 56 533.009 532.978 .003 532.948 532.931 .001 532.914 532.905 .001 56 57 545.088 545.071 .001 545.053 545.043 .001 545.034 545.029 .000 57 58 557.326 557.316 .001 557.306 557.302 .000 557.297 557.295 .000 58 59 569.658 569.653 .000 569.648 569.645 .000 569.643 569.643 .000 59 60 582 110 582.107 .000 582.105 582.104 .000 582.103 582.103 .000 60 61 594.638 594.637 .000 594.686 594.636 .000 61 62 607.215 607.215 .000 62 TABLE LT. *135 CONTINUED ANNUAL PREMIUM (P), ALSO A (P — rr), FOR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH OR AGE 30, 86, 75. 4 PER CENT. Formula, PS 1000 Ma;— Ma;m + D; x+m Na,-N. = iooo|- _|_am-l (1- ,) 10( -V)h 773.= — lOOOMa D or 30. D or 40. D or 45. AQE. X. P. A (P - n). AGE. X. P. A(P-7r). AGE. X. P. A(P-7r). 10 36.209 1.422188 10 21.626 1.073792 27 42.013 1.440353 11 38.628 1.457505 11 22.610 1.102321 28 45.085 1.480236 12 41.333 1.494033 12 23.674 1.131372 29 48.557 1.521718 13 44.374 1.531914 13 24.828 1.160996 30 52.508 1.565008 14 47.814 1.571314 14 26.080 1.191238 31 57.043 1.610358 15 51.732 1.612430 15 27.444 1.222149 32 62.294 1.658046 16 56.232 1.655497 16 28.933 1.253772 33 68.441 1.708448 17 61.449 1.700814 17 30.564 1.286187 34 75.728 1.762024 18 67.560 1.748737 18 32.357 1.319456 35 84.494 1.819347 19 74.811 1.799719 19 34.333 1.353668 36 95.233 1.881195 20 83.542 1.854341 20 36.521 1.388905 37 108.685 1.948607 21 94.247 1.913374 21 38.955 1.425278 38 126.012 2.023045 22 107.667 1.977848 22 41.674 1.462912 39 149.147 2.106622 23 124. 964 2.049214 23 44.730 1.501943 40 181.574 2.202634 24 148.078 2.129593 24 48.186 1.542552 41 230.259 2.316583 25 180.498 2.222272 25 52.122 1.584929 42 311.463 2.458789 26 27 229.202 310.474 2.332749 2.471330 26 27 56.639 61.873 1.629316 1.676005 43 473.944 2.652376 28 473.167 2.661148 28 68.002 1.725368 D or 50. 29 75.271 1.777851 D or a 5. 30 84.019 1.834038 31 94.742 1.894707 10 15.173 0.732305 32 108.178 1.960887 11 15.680 0.757859 10 27.291 1.243468 33 125.490 2.034036 12 16.219 0.783718 11 28.773 1.274618 34 148.615 2.116277 13 16.795 0.809903 12 30.397 1.306530 35 181.039 2.210894 14 17.410 0.836438 13 32.181 1.339279 36 229.732 2.323388 15 18.066 0.863341 14 34.150 1.372945 37 310.967 2.464075 16 18.769 0.890617 15 16 36.329 38.753 1.407614 1.443390 38 473.559 2.656095 17 18 19.522 20.330 0.918329 0.946482 17 41.464 1.480401 D or 4 •5. 19 21.199 0.975105 18 19 44.510 47.956 1.518786 1.558715 20 21 22.134 23.142 1.004235 1.033906 20 51.882 1.600380 10 17.819 0.905510 22 24.232 1.064158 21 56.388 1.644025 11 18.510 0.932276 23 25.412 1.095030 22 61.611 1.689944 12 19.250 0.959437 24 26.692 1.126579 23 67.730 1. 738495 13 20.045 0.987018 25 28.085 LI 58863 24 74.987 1.790134 14 20.901 1.015049 26 29.605 1.191915 25 83.725 1.845446 15 21.821 1.043559 27 31.268 1.225823 26 94.437 1.905194 16 22.814 1.072577 28 33.093 1.260654 27 107.863 1.970416 17 23.888 1.102152 29 35.105 1.296492 28 125.166 2.042566 18 25.052 1.132321 30 37.328 1.333431 29 148.285 2.123761 19 26.315 1.163126 31 39.799 1.371589 30 180.704 2.217287 20 27.690 1.194622 32 42.558 1.411076 31 229.405 2.328653 21 29.191 1.226870 33 45.655 1.452050 32 310.663 2.468160 22 30.834 1.259935 34 49.153 1.494684 33 473.316 2.658944 23 24 32.639 34.628 1.293878 1.328793 35 36 53.132 57.693 1.539172 1.585758 25 36.830 1.364772 37 62.972 1.634744 26 39.279 1.401919 38 69.148 1.686510 ♦136 TABLE LY. CONTINUED ANNUAL PREMIUM (P), ALSO A (P — rr), FOR ENDOWMENT INSURANCE OP 1,000, PAYABLE AT DEATH OR AGE 30, 35, 75. 4 PEli CENT. D or 50. D or 55. D or 60. Aa£. X. P. A(P-Tr). AQE. X. P. A(P-rr). A&E. X. P. A (P - tt). ' 39 76.463 1.741492 46 97.304 1.840747 ' 48 72.131 1.619831 40 85.258 1.800288 47 110.824 1.911844 49 79.567 1.679106 41 96.025 1.863669 48 128.208 1.990107 50 88.483 1.742373 42 109.504 1.932686 49 151.386 2.077669 51 99.369 1.810412 43 126.851 2.008777 50 183.827 2.177823 52 112.954 1.884252 44 150.003 2.094087 51 232.476 2.296083 53 130.395 1.965369 45 182.437 2,191901 52 313.525 2.442754 54 153.615 2.055898 46 47 48 231.108 312.252 474.564 2.307724 2.451875 2.647485 53 475.564 2.640983 55 56 57 186.071 234.677 315.575 2.159133 2.280580 2.430562 T> rkf df^ u or o \j. 58 477.171 2.632212 D or 6 5. 10 11.984 0.344432 D or 65. 11 12 12.290 12.613 0.368547 0.392838 10 13.304 0.547799 11 13.690 0.572500 13 12.954 0.417355 10 11.069 0.112203 12 14.100 0.597443 14 13.316 0.442119 11 11.321 0.135832 13 14.534 0.622659 15 13.698 0.467164 12 11.586 0.159718 14 14.996 0.648155 16 14.103 0.492439 13 11.867 0.183754 15 15.486 0.673960 17 14.533 0.518014 14 12.163 0.208038 16 16.008 0.700054 18 14.988 0.543869 15 12.474 0.232590 17 16.563 0.726491 19 15.472 0.570029 16 12.804 0.257270 18 17.156 0.753284 20 15.985 0.596509 17 13.152 0.282237 19 17.788 0.780440 21 16.532 0.623363 18 13.520 0.307475 20 18.463 0.807995 22 17.113 0.650570 19 13.909 0.332963 21 19.185 0.835969 23 17.732 0.678154 20 14.320 0.358715 22 19.958 0.864398 24 18.392 0.706146 21 14.756 0.384801 23 20.787 0.893296 25 19.096 0.734600 22 15.217 0.411199 24 21.678 0.922710 26 19.849 0.763488 23 15.706 0.437893 25 22.636 0.952662 27 20.655 0.792868 24 16.225 0.464966 26 23.669 0.983189 28 21.518 0.822782 25 16.776 0.492453 27 24.783 1.014332 29 22.445 0.853236 26 17.362 0.520248 28 25.990 1.046160 30 23.440 0.884274 27 17.984 0.548488 29 27.299 1.078678 31 24.512 0.915959 28 18.646 0.577170 30 28.720 1.111975 32 25.668 0.948305 29 19.352 0.606274 31 30.271 1.146109 33 26:918 0.981366 30 20.103 0.635896 32 31.965 1.181126 34 28.271 1.015213 31 20.906 0.666059 33 33.824 1.217123 35 29.741 1.049865 32 21.764 0.696715 34 35.870 1.254188 36 31.340 1.085416 33 22.683 0.727972 35 38.130 1.292396 37 33.086 1.121911 34 23.668 0.759872 36 40.638 1.331870 38 34.999 1.159474 35 24.724 0.792378 37 43.436 1.372737 39 37.101 1.198149 36 25.859 0.825595 38 46.574 1.415162 ' 40 39.420 1.238041 37 27.082 0.859529 39 50.113 1.459295 i 41 4L990 1.279284 38 28.402 0.894311 40 54.135 1.505350 42 44.852 1.322000 39 29.829 0.929894 41 58.741 1.553575 43 48.055 1.366328 40 31.376 0.966381 42 64.067 1.604276 44 51.663 1.412456 41 33.056 1.003848 43 70.288 1.657804 45 55.756 1.460599 42 34.888 1.042375 44 77.650 1.714635 46 60.435 1.510991 43 36.888 1.082002 \ 45 86.492 1.775362 47 65.834 1.568945 44 39.083 1.122855 TABLE LY. ♦137 CONTINUED ANNUAL PREMIUM (P), ALSO A (P — tt), FOR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH OR AGE 30, 35, 75. 4 PER CENT. D or 65. D or 70. D or 75. AaE, X. P. A(P-7r). AGE. X. P. A(P-Tr). A&E, X. 24 P. A(P-Tr). 45 41.499 1.165036 37 23.520 0.565210 14.001 1.840420 46 44.169 1.208648 38 24.528 0.598353 25 14.405 1.866996 47 47.135 1.253832 39 25.607 0.6.32133 26 14.831 1.893595 48 50.447 1.300737 40 26.762 0.666658 27 15.281 1.920697 49 54.167 1.349551 41 28.001 0.701982 28 15.756 1.948119 50 58.375 1.400484 42 29.335 0.738154 29 16.258 1.975891 51 63.174 1.453807 43 30.769 0.775181 30 16.789 0.004106 52 68.694 1.509788 44 32.317 0.813167 31 17.350 0.032780 53 75.113 1.568822 1 45 33.991 0.852175 32 17.943 0.061754 54 82.673 1.631384 46 35.804 0.892256 33 18.571 0.091210 55 91.711 1.698067 47 37.775 0.933497 34 19.237 0.121330 56 102.713 1.769645 48 39.922 0.975974 35 19.942 0.151737 57 116.408 1.847174 49 42.270 1.019810 36 20.689 0.182472 58 133.941 1.932106 50 44.845 1.065076 37 21.483 0.214208 59 157.224 2.026593 51 47.682 1.111954 38 22.326 0.246474 60 189.696 2.133927 52 50.821 1.160505 39 23.222 0.279233 61 238.231 2.259621 53 54.313 1.210926 40 24.175 0.312622 62 318.870 2.413989 54 58.220 1.263428 41 25.189 0.346764 63 479.753 2.620184 55 56 62.621 67.616 1.318189 1.375481 42 43 26.271 27.424 0.381638 0.417239 D or 7 '0. 57 73.338 1.435614 44 28.655 0.453639 58 59 79.960 87.723 1.498957 1.566005 45 46 29.973 31.383 0.490983 0.529174 10 10.461 1.836830 60 96.960 1.637343 47 32.896 0.568331 11 10.678 1.860158 61 108.151 1.713757 48 34.520 0.608526 12 10.907 1.883718 62 122.019 1.796305 49 36.268 0.649812 13 11.148 1.907411 63 139.702 1.886456 60 38.152 0.692256 14 11.402 1.931356 64 163.086 1.986335 51 40.187 0.735998 ' 15 11.669 1.955688 65 195.577 2.099251 52 42.389 0.780987 16 11.950 1.979912 66 243.984 2.230711 53 44.781 0.827415 17 12.247 0.004536 67 324.205 2.391049 54 47.384 0.875339 18 12.560 0.029343 68 483.901 2.603346 55 50.228 0.924925 19 12.889 0.054422 56 53.345 0.976281 20 13.237 0.079651 D or 7 5. 57 56.777 1.029538 21 13.605 0.105272 58 60.571 1.084855 22 13.992 0.131137 59 64.792 1.142461 23 14.402 0.157275 10 10.088 T. 497206 60 69.517 1.202557 24 14.835 0.183754 11 10.285 1.520221 61 74.844 1.265398 25 15.293 0.210586 12 10.492 1.543571 62 80.903 1.331322 26 15.777 0.237695 13 10.709 1.567026 63 87.867 1.400704 27 16.289 0.265195 14 10.938 1.590842 64 95.969 1.474015 28 16.833 0.293097 15 11.178 1.614897 65 105.537 1.551885 29 17.409 0.321391 16 11.431 1.638988 66 117.041 1.635077 30 18.018 0.350112 17 11.697 1.663324 67 131.194 1.724712 31 18.666 0.379324 18 11.977 1.687975 68 149.103 1.822091 32 19.354 0.408935 19 12.272 1.712734 69 172.639 1.929527 33 20.086 0.439064 20 12.583 1.737670 70 205.135 2.050237 34 20.865 0.469793 21 12.910 1.762978 71 253.307 2.189751 35 21.694 0.500991 22 13.254 1.788593 72 332.796 2.358365 36 22.577 0.532818 23 13.618 1.814248 ■ i *i3d TABLE LYI. FIVE ANNUAL PREMIUM (P), ALSO A(P — tt), FOR ENDOWMENT INSURANCE OP 1,000, PAYABLE AT DEATH OR AGE 30, 35, 75. 4 PER CENT D or 30. D or 40. D or 45. AaE. X. P. A (P - tt). AGE. X. P. A(P-7r). AGE. X. P. A(P-rr). 10 106.053 1.983531 10 78.712 1.838461 31 130.936 2.059429 11 109.592 1.998426 11 80.971 1.851367 32 135.574 2.074759 12 113.295 2.013483 12 83.332 1.864452 33 140.430 2.090231 13 117.171 2.028696 13 85.805 1.877745 34 145.515 2.105851 14 121.226 2.044062 14 88.386 1.891195 35 150.839 2.121609 15 125.469 2.059574 15 91.092 1.904855 36 156.415 2.137509 16 129.909 2.075234 16 93.919 1.918679 37 162.261 2.153558 17 134.558 2.091037 17 96.874 1.932666 38 168.393 2.169764 18 139.423 2.106984 18 99.967 1.946830 39 174.824 2.186120 19 20 144.514 149.845 2.123062 2.139278 19 20 103.201 106.584 1.961159 1.975654 40 181.574 2.202634 21 155.425 2.155624 21 110.123 1.990308 D or 50. 22 23 161.267 167.383 2.172098 2.188697 22 23 113.825 117.697 2.005115 2.020079 24 173.788 2.205423 24 121.750 2.035195 10 61.869 1.716798 25 180.498 2.222272 25 26 125.989 130.427 2.050457 2.065872 11 12 63.338 64.876 1.727415 1.738256 D or a 5. 27 135.068 2.081421 13 66.482 1.749286 28 29 139.927 145.018 2.097112 2.112958 14 15 68.161 69.913 1.760517 1.771933 10 90.774 1.908484 30 150.343 2.128929 16 71.744 1.783539 11 93.597 1.922434 31 155.922 2.145044 , 17 73.659 1.795342 12 96.551 1.936559 32 161.763 2.161289 , 18 75.659 1.807330 13 99.642 1.950861 33 167.890 2.177688 ' 19 77.749 1.819498 14 102.876 1.965331 34 3 74.310 2.194224 20 79.934 1.831855 15 106.257 1.979964 35 181.039 2.210894 21 82.215 1.844383 16 109.796 1.994759 22 84.598 1.857078 17 113.499 2.009715 D or 4 5. 23 87.089 1.869954 18 117.373 2.024826 24 89.690 1.882991 19 121.427 2.040090 25 92.409 1.896195 20 125.669 2.055504 10 69.242 1.774283 26 95.250 1.909563 21 130.108 2.071064 11 71.058 1.786070 27 98.217 1.923083 22 134.754 2.086765 12 72.955 1.798052 28 101.318 1.936761 23 139.616 2.102608 13 74.940 1.810234 29 104.560 1.950596 24 144.707 2.118590 14 77.015 1.822608 30 107.945 1.964569 25 150.036 2.134709 15 79.184 1.835168 31 111.483 1.978692 26 155.616 2.150963 16 81.451 1.847914 32 115.184 1.992966 27 161.458 2.167347 17 83.821 1.860843 33 119.051 2.007377 28 167.578 2.183865 18 86.298 1.873951 34 123.098 2.021947 29 173.989 2.200512 19 88.890 1.887247 35 127.329 2.036649 30 180.704 2.217287 20 91.600 1.900715 36 131.752 2.051484 21 94.431 1.914349 37 136.383 2.066468 22 97.391 1.928147 i 38 141.234 2.081604 23 100.488 1.942117 1 39 146.311 2.096879 24 ■ 103.723 1.956242 i 40 151.632 2.112306 25 107.108 1.970529 41 157.208 2.127887 26 110.648 1.984974 42 163.061 2.143633 27 114.348 1.999566 ' 43 169.200 2.159538 28 118.219 2.014313 i 44 175.653 2.175626 29 122.269 2.029209 i 45 182.437 2.191901 30 126.504 2.044246 i FOEMULA, Pa; = 1000 Ma; — Marfm + Da Nar — Na;+6 TABLE LYI. ♦139 FIVE ANNUAL PREMIUM (P), ALSO a(P — tt), FOR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH OR AGE 30, 35, 75. 4 PER CENT. D or 55. D or 60. D or 65. AGE, X. p. XiV-rr). Aa£. X. P. A (P - tt). AGE. X. P. A (P - tt). 10 56.208 1.666834 16 58.694 1.678507 17 55.747 1.648466 11 57.413 1.676324 17 59.996 1.688058 18 56.906 1.657208 12 58.672 1.686008 18 61.353 1.697785 19 58.113 1.666112 13 59.987 1.695894 19 62.770 1.707688 20 59.371 1.675182 14 61.361 1.705974 20 64.248 1.717770 21 60.682 1.684414 15 62.795 1.716242 21 65.790 1.728022 22 62.049 1.693804 16 64.292 1.726698 22 67.396 1.738435 23 63.472 1.703346 17 65.856 1.737349 : 23 69.074 1.749028 24 64.955 1.713045 18 67.487 1.748169 '\ 24 70.822 1.759771 25 66.500 1.722890 19 69.194 1.759203 25 72.645 1.770675 26 68.110 1.732886 20 70.976 1.770409 26 74.546 1.781737 27 69.781 1.742988 21 72.835 1.781787 27 76.528 1.792950 28 71.528 1.753272 22 74.774 1.793333 28 78.594 1.804308 29 73.345 1.763670 23 76.801 1.805060 29 80.748 1.815814 30 75.232 1.774175 24 78.915 1.816947 30 82.991 1.827444 31 77.198 1.784805 25 81.122 1.829002 31 85.328 1.839207 32 79.243 1.795555 26 83.427 1.841224 32 87.768 1.851120 33 81.372 1.806412 27 85.831 1.853593 33 90.309 1.863151 34 83.586 1.817376 28 88.341 1.866119 34 92.957 1.875306 35 85.889 1.828432 29 90.962 1.878804 35 95.717 1.887577 36 88.281 1.839568 30 93.694 1.891620 36 98.591 1.899954 37 90.771 1.850807 31 96.547 1.904586 37 101.589 1.912455 38 93.362 1.862128 32 99.527 1.917698 38 104.715 1.925070 39 96.055 1.873523 33 102.637 1.930947 39 107.974 1.937790 40 98.855 1.884991 34 105.884 1.944330 40 111.372 1.950617 41 101.767 1.896527 35 109.276 1.957855 41 114.916 1.963548 42 104.798 1.908140 36 112.814 1.971497 42 118.616 1.976595 43 107.949 1.919805 37 116.513 1.985281 43 122.476 1.989744 44 111.225 1.931540 38 120.379 1.999202 44 126.509 2.003011 45 114.644 1.943338 39 124.418 2.013249 45 130.723 2.016394 46 118.200 1.955198 40 128,640 2.027427 46 135.128 2.029898 47 121.904 1.967124 41 133.056 2.041744 47 139.738 2.043533 48 125.766 1.979119 42 137.679 2.056206 48 144.566 2.057300 49 129.794 1.991187 43 142.517 2.070803 49 149.628 2.071237 50 134.001 2.003344 44 147.588 2.085556 50 154.940 2.085333 51 138.398 2.015592 45 152.904 2.100467 51 160.524 2.099620 52 142.996 2.027943 46 158.482 2.115545 52 166.399 2.114107 63 147.814 2.040421 47 164.341 2.130805 53 172.594 2.128833 54 152.872 2.053054 48 170.502 2.146258 54 179.140 2.143829 55 158.191 2.065863 49 50 176.988 183.827 2.161921 2.177823 55 186.071 2.159133 56 163.792 1 /*rv Y*^/\Ci 2.078876 57 58 169.708 2.092141 D or e .5. 175.966 2.105692 D or ( 30. 59 182.614 2.119599 60 189.696 2.133927 10 48.874 1.592176 10 51.955 1.625114 11 49.735 1.599683 11 52.962 1.633556 12 50.635 1.607377 12 54.012 1.642170 13 51.573 1.615248 13 55.109 1.650978 14 52.553' 1.623298 14 56.254 1.659974 15 53.573 1.631514 ^' 57.448 1.669149 1 16 54.637 1.639898 *140 TABLE LVI. FIVE ANNUAL PREMIUM (P), ALSO A(P — tt), FOR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH OR AGE 30, 35. 75, 4 PER CENT. D or 70. D or 70. D or 75. AGE. X. P. MP-tt). AQE. X. P. A(P_Tr). AGE. X. P. A (P - tt). 10 46.763 1.568076 51 123.245 1.946960 32 69.756 1.723993 11 47.526 1.574872 52 126.967 1.957208 33 71.425 1.733094 12 48.322 1.581829 53 130.845 1.967477 34 ■ 73.156 1.742266 13 49.152 1.588960 54 134.884 1.977748 35 74.950 1.751476 14 50.017 1.596258 55 139.099 1.988038 36 76.805 1.760717 15 50.919 1.603719 56 143.499 1.998360 37 78.730 1.770005 16 51.858 1.611327 57 148.107 2.008743 38 80.725 1.779325 17 52.838 1.619113 58 152.933 2.019199 39 82.788 1.788652 18 53.859 1.627054 59 158.008 ■ 2.029783 40 84.923 1.797979 19 54.923 1.635154 60 163.357 2.040535 41 87.132 1.807300 20 56.031 1.643405 61 169.011 2.051507 42 89.418 1.816611 21 57.185 1.651808 62 175,008 2.062769 43 91.780 1.825882 22 58.386 1.660356 63 181.398 2.074414 44 94.224 1.835122 23 59.638 1.669058 64 188.2.30 2.086527 45 96.749 1.844309 24 60.938 1.677877 65 195.577 2.099251 46 99.358 1.853434 25 62.292 1.686848 47 102.055 1.862490 26 63.702 1.695946 D or 7 5. 48 104.841 1.871459 27 65.165 1.705159 49 107.720 1.880339 28 66.690 1.714506 50 110.695 1.889115 29 68.275 1.723966 10 45.444 1.552301 i 51 113.771 1.897785 30 69.919 1.733517 11 46.145 1.558602 ; 52 116.948 1.906322 31 71.629 1.743179 12 46.875 1.565061 53 120.235 1.914738 32 73.404 1.752924 13 47.638 1.571681 54 123.637 1.923025 33 75.252 1.762785 14 48.432 1.578460 55 127.161 1.931181 34 77.169" 1.772715 15 49.259 1.585385 56 130.811 1.939194 35 79.158 1.782714 16 50.121 1.592460 57 134.600 1.947074 36 81.220 1.792769 17 51.019 1.599695 58 138.529 1.954800 37 83.362 1.802892 18 51.955 1.607081 59 142.623 1.962432 38 85.586 1.813074 19 52.929 1.614612 60 146.889 1.969949 39 87.892 1.823293 20 53.943 1.622288 61 151.343 1.977375 40 90.283 1.833548 21 54.998 1.630102 62 156.006 1.984743 41 92.762 1.843827 22 56.096 1.638054 63 160.907 1.992106 42 95.335 1.854139 23 57.238 1.646131 64 166.068 1.999498 43 98.000 1.864456 24 58.426 1.654345 65 171.528 2.007005 44 100.766 1.874787 25 59.661 1.662689 66 177.323 2.014694 45 10.3.634 1.885123 26 60.945 1.671141 67 183.513 2.022728 46 106.607 1.895453 27 62.278 1.679703 68 190.152 2.031155 47 109.692 1.905778 28 63.665 1.688381 69 197.334 2.040271 48 112.891 1.916089 29 65.104 1.697159 70 205.135 2.050237 49 116.212 1.926390 30 66.597 1.706013 50 119.661 1.936679 31 68.147 1.714968 141 TABLE LYII TEN ANNUAL PREMIUM (P), ALSO A (P — rr), FOR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH OR AGE 30, 35, 75. 4 PER CENT. D or 30. D or 40. D or 50. AGE. X. P. A(P.-7r). AGE. X. P. A(P-77). AGE. X. P. A(P-Tr). 10 59.073 1.692836 24 67.925 1.737328 18 42.171 1.486872 11 61.047 1.708370 25 70.306 1.753103 19 43.341 1.499481 12 63.114 1.724049 26 72.800 1.769018 20 44.565 1.512267 ; 13 65.278 1.739872 27 75.410 1.785063 21 45.844 1.525216 14 67.543 1.755833 28 78.146 1.801246 22 47.180 1.538324 15 69.912 1.771927 29 81.015 1.817579 23 48.578 1.551600 16 17 72.393 74.991 1.788154 1.804513 30 84.019 1.834038 24 25 50.039 51.567 1.565032 1.578620 18 77.711 1.821000 D or 45. 26 53.165 1.592361 19 20 80.559 83.542 1.837609 1.854341 27 28 54.835 56.584 1.606247 1.620290 10 38.568 1.459311 29 58.413 1.634480 D or a 5. 11 39.582 1.471713 30 60.325 1.648805 12 40.642 1.484297 31 62.326 1.663279 13 41.750 1.497075 32 64.422 1.677900 10 50.562 1.610534 14 42.910 1.510035 33 66.616 1.692658 11 52.138 1.625153 15 44.122 1.523169 34 68.916 1.707581 12 63.786 1.639928 16 45.389 1.536479 35 71.325 1.722638 13 55.512 1.654872 i 17 46.715 1.549963 36 73.848 1.737844 14 57.318 1.669970 i 18 48.101 1.563619 37 76.496 1.753204 15 59.207 1.685216 ' 19 49.552 1.577441 38 79.275 1.768732 , 16 61.185 1.700611 i 20 51.069 1.591429 39 82.193 1.784423 17 63.255 1.716155 i 21 52.655 1.605572 40 85.258 1.800288 18 65.421 1.731841 22 54.315 1.619876 19 67.689 1.747668 23 56.052 1.634337 D or 5 5. 20 70.063 1.763631 24 57.868 1.648946 21 72.549 1.779731 25 59.770 1.663709 22 75.152 1.795963 26 61.760 1.678621 10 31.308 1.333133 23 77.878 1.812326 27 63.842 1.693674 11 31.982 1.342979 24 80.733 1.828821 28 66.023 1.708878 12 32.685 1.353008 25 83.725 1.845446 29 68.306 1.724224 13 33.420 1.363236 30 70.696 1.739709 14 34.188 1.373646 D or 4 0. 31 73.202 1.755344 15 34.990 1.384237 32 33 75.827 78.579 1.771120 1.787046 16 17 35.827 36.703 1.395008 1.405966 10 43.844 1.532368 34 81.466 1.803123 18 37.616 1.417076 11 45.105 1.545939 35 84.494 1.819347 19 38.572 1.428400 12 46.423 1.559682 20 39.571 1.439877 13 47.803 1.573607 ' D or 5 0. 21 40.613 1.451521 14 49.246 1.587684 22 41.702 1.463319 15 50.757 1.601961 23 42.839 1.475290 16 52.337 1.616392 10 34.462 1.392484 24 44.027 1.487409 17 53.990 1.630969 11 35.283 1.403625 25 45.269 1.499686 18 55.719 1.645712 12 36.141 1.414955 26 46.566 1.512117 19 57.529 1.660609 13 37.038 1.426482 27 47.920 1.524686 20 59.423 1.675660 14 37.976 1.438201 28 49.336 1.537405 21 61.405 1.690861 15 38.956 1.450095 29 50.816 1.550270 22 63.480 1.706204 16 39.980 1.462172 30 52.361 1.563263 23 65.651 1.721694 17 41.051 1.474435 31 53.976 1.576399 Formula, P,. = 1000 Ma, — Mx+m + Da Na- — Na,+io AT >4' *142 TABLE LYII. TEN ANNUAL PREMIUM (P), ALSO A (P — rr), FOR ENDOWMENT INSURANCE OF 1.000, PAYABLE AT DEATH OR AGE 30, 35 75. 4 PER CENT. D or 55. D or 60. D or 65. AGE. X. P. ;,(P-7r). 1.589673 A&E. X. P. A (P - tt). AGE. X. 49 P. 1 A(P-7r). 32 55.665 43 69.090 1.646207 73.955 1.624815 33 57.432 1.603081 44 71.458 1.659392 50 76.525 1.636458 34 59.279 1.616620 45 73.944 1.672733 51 79.233 1.648277 35 61.212 1.630306 46 76.557 1.686238 52 82.090 1.660296 36 63.233 1.644112 47 79.305 1.699929 53 85.110 1.672563 37 65.351 1.658064 48 82.200 1.713828 54 88.311 1.685133 38 39 67.569 69.894 1.672161 1.686402 49 50 85.255 88.483 1.727968 1.742373 55 91.711 1.698067 40 72.331 1.700792 D or 70. 41 74.887 1.715340 D or 65. 42 77.572 1.730049 43 80.395 1.744960 10 26.048 1.211486 44 83.364 1.760054 10 27.223 1.241780 11 26.474 1.218028 45 86.492 1.775362 11 12 27.705 28.207 1.249228 1.256850 12 13 26.919 27.383 1.224712 1.231551 D or e 0. 13 28.732 1.264638 14 27.868 1.238538 14 29.280 1.272591 15 28.372 1.245673 15 29.851 1.280697 16 28.898 1.252928 10 28.939 1.282518 16 30.447 1.288951 17 29.448 1.260341 11 29.502 1.291120 17 31.069 1.297377 18 30.020 1.267882 12 30.089 1.299878 18 31.718 1.305959 19 30.617 1.275560 13 30.702 1.308825 19 32.395 1.314682 20 31.239 1.283360 14 31.343 1.317950 20 33.101 1.323553 21 31.887 1.291287 15 32.010 1.327245 21 33.837 1.332570 22 32.562 1.299331 16 32.708 1.336710 22 34.604 1.341721 23 33.266 1.307502 17 33.437 1.346357 23 35.405 1.351001 24 33.998 1.315752 18 34.197 1.356167 24 36.239 1.360423 25 34.761 1.324126 19 34.991 1.366141 25 37.109 1.369963 26 35.556 1.332590 20 35.820 1.376278 26 38.016 1.379632 27 36.383 1.341134 21 36.685 1.386574 27 , 38.962 1.389409 28 37.245 1.349779 22 37.587 1.397017 28 39.947 1.399291 29 38.142 1.358495 23 38.529 1.407620 29 40.974 1.409287 30 39.074 1.367263 24 39.512 1.418368 30 42.043 1.419366 31 40.045 1.376104 25 40.538 1.429258 31 43.159 1.429547 32 41.055 1.384977 26 41.609 1.440287 32 44.321 1.439816 33 42.108 1.393931 27 42.726 1.451455 33 45.532 1.450171 34 43.202 1.402908 28 43.893 1.462753 34 46.796 1.460606 35 44.341 1.411908 29 45.110 1.474183 35 48.112 1.471107 36 45.525 1.420917 30 46.379 1.485723 36 49.482 1.481668 37 46.757 1.429943 31 47.704 1.497390 37 50.912 1.492299 38 48.040 1.438980 32 49.088 1.509178 38 52.404 1.503000 39 49.375 1.448010 33 50.533 1.521081 39 53.960 1.513757 40 50.764 1.457022 34 52.042 1.533097 40 55.583 1.524562 41 52.209 1.466010 35 53.617 1.545218 41 57.277 1.535424 42 53.716 1.474987 36 55.261 1.557442 42 59.047 1.546353 43 55.283 1.483916 37 56.980 1.569782 43 60.895 1.557327 44 56.917 1.492818 38 58.777 1.582234 44 62.828 1.568371 45 58.621 1.501682 39 60.656 1.594795 45 64.850 1.579485 46 60.398 1.510501 40 62.621 1.607464 46 ^6.966 1.590670 47 62.253 1.519279 41 64.678 1.620253 47 69.184 1.601943 48 64.190 1.528014 42 66.833 1.633167 48 71.511 1.613318 49 66.216 1.536723 TABLE LYll. ^143 TEN ANNUAL PREMIUM (P), ALSO A (P — tt), FOR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH OR AGE 30, 35, 75. 4 PER CENT. D or 70. D or 75. D or 75. AGE. X. P. A(P-7r). AQE. X. P. A(P_:r). AGE. X. P. A (P - tt). 50 68.336 1.545404 19 29.505 1.249176 43 51.774 1.430786 51 70.559 1.554086 20 30.074 1.256198 44 53.222 1.437888 52 72.887 1.562734 21 30.667 1.263328 45 54.727 1.444843 53 75.339 1.571470 22 31.285 1.270560 46 56.291 1.451631 54 77.920 1.580248 23 31.927 1.277873 47 57.919 1.458248 55 80.643 1.589137 24 32.596 1.285289 48 59.613 1.464674 56 83.519 1.598159 25 33.293 1.292792 49 61.377 1.470913 57 86.571 1.607430 26 34.017 1.300356 50 63.216 1.476939 58 89.810 .1.616970 27 34.771 1.307984 51 65.133 1.482745 59 93.265 1.626906 28 35.555 1.315681 52 67.138 1.488374 60 96.960 1.637343 29 36.371 1.323421 53 69.230 1.493740 30 37.217 1.331190 54 71.422 1.498907 D or •; '5. 31 38.099 1.339006 55 73.722 1.503878 32 39.014 1.346826 56 76.135 1.508644 33 39.966 1.354664 57 78.676 1.513255 10 25.313 1.191417 34 40.956 1.362511 58 81.351 1.517700 11 25.705 1.197311 35 41.984 1.370324 59 84.183 1.522105 12 26.113 1.203340 36 43.050 1.378103 60 87.185 1.526473 13 26.540 1.209510 37 44.159 1.385851 61 90.374 1.530890 14 26.985 1.215815 38 45.311 1.393554 62 93.773 1.535485 15 27.448 1.222238 39 46.508 1.401190 63 97.414 1.540400 16 27.930 1.228785 40 47.750 1.408741 64 101.322 1.545788 17 28.434 1.235465 41 49.040 1.416198 65 105.537 1.551885 18 28.958 1.242263 42 50.382 1.423557 ^ ^144 TABLE LYIII. CONTINUED ANNUAL PREMIUM (P), ALSO A(P— tt), FOR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH OR AT THE END OF m YEARS. 4 PER CENT. Formula, P^= 1000 ^x — ^x+m :1000 li+^'-^i-"'! AQiE. X. , m=5. 10. 15. 20. AGE. X. 15 P. A(P-7r). P. X(P-n). P. 51.732 A(P-7r). P. A(P_7r). 15 180.391 2.229234 83.428 1.861309 1.612429 36.3291.407614 16 180.303 .228678 83.447 .860046 51.757 .610247 36.3611 .404235 16 17 180.317 .228095 83.468 .858725 51.783 .607961 36.395 .400686 17 \ 18 180.333 .227483 83.489 .857333 51.813 .605559 36.433; .396956 18 i 19 180.350 .226842 83.514 .855874 51.845 .603035 36.475 .393033 19 1 20 180.370 2.226172 83.543 1.854341 51.881 1.600380 36.522 1.388904 20 21 180.391 .225467 83.572 .852734 51.921 .597591 36.572 .384560 21 22 180.414 .224724 83.605 .851045 51.964 .594660 36.628 .379985 22 23 180.439 .223945 83.641 .849268 52.012 .591575 36.689 .375175 23 24 180.467 .223129 83.682 .847402 52.004 .588336 36.755 .370093 24 25 180.498 2.222271 83.724 1.845446 52.122 1.584928 36.831 1.364770 25 26 180.532 .221370 83.774 .843387 52.186 .581343 36.912 .359150 26 27 180.569 .220425 83.826 .841222 52.254 .577567 37.002 .353220 27 28* 180.611 .219432 83.884 .838949 52.331 .573597 37.101 .346973 28 29 180.657 .218388 83.950 .836557 52.416 .569415 37.210 .340389 29 30 180.705 2.217289 84.019 1.834038 52.509 1.565009 37.328 1.333433 30 31 180.759 .216135 84.097 .831394 52.610 .560372 37.459 .326094 31 ! 32 180.819 .214920 84.182 .828607 52.722 .555482 37.605 .318354 32 33 180.886 .213646 84.277 .825679 52.846 .550330 37.763 .310173 33 34 180.960 .212308 84.380 .822599 52.982 .544901 37.938 .301516 34 35 181.039 2.210894 84.495 1.819347 53.131 1.539171 38.130 1.292396 35 36 181.125 .209405 84.619 .815924 53.295 .533121 38.341 .282731 36 37 181.221 .207839 84.756 .812315 53.476 .526732 38.574 .272498 37 38 181.329 .206197 84.908 .808520 53.676 .520002 38.829 .261674 38 39 181.446 .204462 85.075 .804513 53.894 .512879 39.111 .250205 39 40 181.574 2.202633 85.259 1.800288 54.135 1.505348 39.420 1.238041 40 41 181.714 .200706 85.459 .795826 54.400 .497389 39.760 .225149 41 42 181.871 .198679 85.682 .791126 54.693 .488973 40.136 .211460 42 43 182.041 .196538 85.925 .786154 55.014 .480051 40.547 .196922 43 44 182.229 .194281 86.195 .780906 55.367 .470606 41.001 .181469 44 45 182.437 2.191901 86.491 1.775361 55.756 1.460599 41.498 1.165036 45 46 182.665 .189389 86.818 .769493 56.184 .449981 42.047 .147534 46 47 182.915 .186730 87.177 .763284 56.655 .438712 42.650 .128887 47 48 183.190 .183924 87.571 .756711 57.174 .426745 43.313 .108998 48 49 183.494 .180960 88.006 .749750 57.746 .414025 44.042 .087771 49 50 183.828 2.177825 88.483 1.742373 58.375 1.400488 44.845 1.065079 50 51 184.196 .174508 89.010 .734554 59.069 .386092 45.728 .040832 51 52 184.598 .170989 89.590 .726243 59.832 .370731 46.698 .014843 52 53 185.041 .167261 90.230 .717419 60.672 .354350 47.765 0.986987 53 54 185.531 .163316 90.935 .708040 61.600 .336868 48.938 .957080 54 55 186.071 2.159134 91.711 1.698067 62.621 1.318187 50.228'o. 924935 55 56 186.664 .154692 92.566 .687439 63.746 .298183 51.645 .890354 56 57 187.318 .149973 93.510 .676122 64.986 .276765 53.203; .853083 57 58 188.034 .144951 94.550 .664036 66.351 .253779 54.911 .812780 58 59 188.825 .139615 95.695 .651135 67.856 .229103 56.790 .769348 59 60 189.696 2.133925 96.959 1.637341 69.517 1.202557 58.850 0.722272 60 61 190.655 .127866 98.354 .622568 71.346 .173961 61.110 .671219 61 62 191.711 .121406 99.892 .606746 73.362 .143121 63.586 .615708 62 63 192.878 .114511 101.592 .589771 75.586 .109825 66.299 .555336 63 64 194.161 .107135 103.467 .571521 78.035 .073755 I 69.269 .489396 64 65 195.578 .0992521 105.537 .551884 80.733 .034697 1 72.515! .417223 65 TABLE LYIII. ♦145 CONTINUED ANNUAL PREMIUM (P), ALSO A, (P — tt), FOR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH OR AT THE END OF m YEARS. 4 PER CENT. AGE. 25. 30. 35. 40. AGE. X. X. P. ;i(P-Tr). P. 31.831 X (P-tt). P. A (P-tt). P. A (P-tt). 15 27.444 1.333149 1.043559 18.066 0.863341 15.486 0. 673951 15 16 27.484 .317189 31.873 .036501 18.131 .853449 15.568 .660193 16 17 37.528 .311979 \ 21.939 .0390671 18.303 .843015 1 15.658 .645629 17 18 37.577 .306491 i 31.991 .031331 18.381 .831985 15.758 .630304 18 19 37.631 .300711 i 33.059 .013960 18.368 .830330 15.866 .613843 19 20 37.690 1.194635 33.134 1.004335 18.463 0.807995 15.985 0.596530 20 21 37.755 .188305 33.316 0.995016 18.567 .794948 16.116 .578146 21 22 37.836 .181440 33.307 .985387 18.681 .781138 16.359 .558613 22 23 37.904 .174316 33.406 .975013 18.807 .766517 16.416 .537883 23 24 37.990 .166791 33.51.6 .964146 18.945 .751017 16.588 .515847 24 25 38.085 1.158857 23.636 0.952657 19.096 0.734593 16.776 0.493411 25 26 38.190 .150480 33.768 .940507 19.363 .717163 16.983 .467475 26 27 38.304 .141635 33.913 .927642 19.444 .698666 17.308 .440909 27 28 38.430 .133393 23.073 .914015 19.644 .679001 17.455 .413578 28 29 38.569 .133425 23.248 .899585 19.864 .658136 17.735 .383395 29 30 38.730 1.111978 23.440 0.884380 30.103 0.635906 18.019 0.350113 30 31 38.888 .100936 23,651 .868050 20.366 .613365 18.340 .315656 31 32 39.073 .089348 23.883 .850818 30.655 .587083 18.690 .378799 32 33 39.374 .076870 24.138 .833515 30.970 .560318 19.073 .239335 33 34 39.496 .063765 24.417 .813067 31.316 .531581 19.489 .197087 34 35 39.741 1.049865 24.734 0.793378 31.694 0.500991 19.943 0.151737 35 36 30.009 035131 35.059 .770351 33.106 .468388 30.434 .103051 36 37 30.303 .019461 35.438 .746875 33.557 .433390 30.969 .050650 37 38 30.638 .003853 35.833 .731868 33.050 .395868 31.550 1.994405 38 39 30.984 0.985184 36.377 ] .695175 33.588 .355739 33.178 .933740 39 40 31.375 0.966381 36.763 0.666658 34.175 0.313633 33.859 1. 868393 40 41 31.805 .946383 37.394 .636187 34.814 .366373 33.594 .797683 41 42 33.378 .935070 37.877 .603577 35.509 .316641 34.389 .731333 42 43 33.796 .903340 38.513 .568636 36.366 .163073 35.345 .638589 43 44 33.366 .878079 39.311 .531159 37.088 .105373 36.167 .548635 44 45 33.991 0.853175 39.973 0.490969 27.980 0.043087 37.158 T. 451173 45 46 34.677 .834464 30.805 .447733 38.947 1.975707 38.333 .344785 46 47 35.439 .794803 31.714 .401345 39.994 .903818 39.364 .338400 47 48 36.354 .76.3008 33.705 .351130 31.136 .823670 30.586 .099681 48 49 37.160 728914 33.785 .397082 33.350 .737908 31.894 3.959041 49 50 38.151 0.693356 34.961 0.338648 33.669 T. 644340 33.391 3.797960 50 51 39.338 .653894 36.340 .175599 35.091 .543835 34.784 .634383 51 52 40.439 .610479 37.630 .107108 36.630 .430881 36.377 .436511 52 53 41.730 .564737 39.138 .033830 38.363 .307496 38.076 .306836 53 54 43.155 .515344 40.775 L 952066 40.038 .171141 39.888 3.963788 54 55 44.713 0.461934 43.547 T. 864096 41.930 T. 018384 41.831 3.681341 55 56 46.413 .404115 44.463 .767823 43.947 3.846337 I 43.879 .397940 56 57 48.368 .341355 46.533 .662386 46.118 .656098 46.074 4.954343 57 58 50.389 .373978 48.764 .544564 48.443 .463398 48.414 .477131 58 59 53.490 .198904 51.171 .415808 ' 50.936 .309515 50.910 00 59 60 54.886 0.117735 53.760 T.369513 ' 53.583 3.934498 60 61 57.488 .038815 56.547 .105851 56.434 .633349 61 62 60.311 1.931305 59.543 3.934379 59.461 .378754 62 63 63.374 .833148 63.761 .734376 63.709 4.845098 63 64 66.687 .703344 66.314 .495544 66.184 .477131 1 64 65 70.370 .566084 69.930 .340549 65 ♦146 TABLE LIX. FIVE ANNUAL PBEMIUM (P), ALSO X(P— ?r), poR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH OR AT THE END OF m YEARS. 4 PER CENT. AGE. X. m—\0. 15. 20. 25. AGE. X, P. A(P-7r). p. A (P-tt). P. 106.257 A (P-TT). P. A(P— 7r). 15 149.726 2.142890 125.469 2.059574 1.979964 91.092 1.904855 15 16 149.745 .142233 125.501 .058828 106.310 .979162 91.174 .904060 16 17 149.767 .141547 125.537 .058050 106.370 .978331 91.266 .903252 17 18 149.790 .140822 125.577 .057237 106.434 .977471 91.367 .902423 18 19 149.816 .140068 125.621 .056390 106.506 .976578 91.477 .901575 19 20 149.845 2.139278 125.669'2. 055504 106.584 1.975654 91.600 1.900715 20 21 149.877 .138451 125.7221 .054583 106.670 .974696 91.733 .899833 21 22 149.911 .137579 : 125.779 .053612 106.764 .973702 91.878 .898937 22 23 149.949 .136667 \ 125.842 .052602 106.869 .972680 92.040 .898035 23 24 149.990 .135711 125.912 .051554 106.980 ,971610 92.215 .897117 24 25 150.036 2.134709 125.989 2.050457 107.108 1.970529 92.409 1.896195 25 26 150.087 .133661 126.074 .049315 107.247 .969407 92.622 .895275 26 27 150.142 .132561 126.167 .048127 107.397 .968243 92.853 .894347 27 28 150.203 .131407 126.269 .046885 107.563 .967053 93.109 .893429 28 29 150.271 .130199 126.382 .045593 107.747 .965833 93.390 .892525 29 30 150.343 2.128929 126.504 2.044246 107.945 1.964569 93.694 1.891620 30 31 150.424 .127601 126.639 .042843 108.163 .963278 94.028 .890740 31 32 150.513 .126206 126.788 .041385 108.405 .961964 94.396 .889892 32 33 150.612 .124745 126.951 .039862 108.669 .960618 94.798 .889077 33 34 151^721 .123218 127.132 .038290 108.957 .959240 95.237 .888305 34 35 150.839 2.121609 127.329 2.036649 109.276 1.957855 95.717 1.887577 35 36 150.967 .119918 127.543 .034937 109.621 .956430 96.239 .886898 36 37 151.110 .118149 127.780 .033162 110.001 .954995 96.810 .886293 37 38 151.269 .116299 128.042 .031328 110.419 .953548 97.435 .885774 38 39 151.442 .114351 128.327 .029412 110.874 .952086 98.115 .885333 39 40 151.632 2.112306 128.640 2.027427 111.372 1.950617 98.855 1.884991 40 41 151.839 .110159 128.983 .025372 111.916 .949142 99.659 .884754 41 42 152.671 .107915 129.361 .023244 112.512 .947677 100.536 .884646 42 43 152.321 .105547 129.773 .021036 113.161 .946212 101,485 .884657 43 44 152.599 .103071 130.226 .018755 113.871 .944766 102.517 .884815 44 45 152.904 2.100467 130.723 2.016394 114.644 1.943338 103.634 1.885123 45 46 153.238 .097733 131.266 .013953 115.486 .941934 104.841 .885585 46 47 153.606 .094859 131.861 .011427 116.404 .940561 106.146 .886215 47 48 154.010 .091843 132.511 .008817 117.401 .939223 107.552 .887010 48 49 154.453 .088668 133.223 .006124 118.485 .937929 109.067 ,887979 49 50 154.940 2.085333 134.001 2.003344 119.661 1.936679 110.695 1.889115 50 51 155.476 .081vS29 134.852 .000477 120.937 .935482 112.442 .890421 61 52 156.062 .078137 135.778 1.997507 122.316 .934328 114.310 .891873 52 53 156.705 .074250 1136.789 .994446 123.809 .933228 116.307 .893467 53 54 157.413 .070164 137.894 .991293 125.421 .932178 118.436 .895188 54 55 158.191 2.065863 1 139.099 1.988038 127.161 1.931181 120.704 1.897011 55 56 159.043 .061328 140.409 .984673 129.032 .930214 123.107 .898894 56 57 159.979 .056547 141.836 .981196 131.045 .929276 125.654 .900809 57 58 161.002 .051496 143.382 .977582 133.198 .928319 128.338 .902681 58 59 162.125 .046164 145.063 .973839 135.510 .927370 131.172 .904511 59 60 163.357 2.040535 146.889 1.969949 137.980 1.926372 134.151 1.906207 60 61 164.707 .034576 148.866 .965889 140.616 .925294 137,276 .907713 61 62 166.186 .028279 151.005 .961643 143.421 .924088 140.545 .908951 62 63 167.809 .021607 153.322 .957195 146.406 .922718 143.966 .909865 63 64 169.583 .014521 155.821 .952491 149.573 .921111 147.533 .910359 64 65 171.528J .007005 158.520 .947522 152.928 .919214 151.253 .910363 65 Formula, Pj = 1000 I Mx — Ma,+m + D x+m ( N*— Na;+5 1 + al TABLE LIX. *ui FIVE ANNUAL PREMIUM (P) ALSO A (P— tt), FOR ENDOWMENT INSURANCE OF 1,000, PAYABLE AT DEATH, OR AT THE END OF m YEARS. 4 PER CENT. A&E. X. 30. 35. 40. AOE. X. P. A(P-Tr). P. A(P-Tr). P. ;.(P-Tr). 15 79.184 1.835168 69.913 1.771933 62.795 1.716242 15 16 79.306 834492 70.088 .771532 63.035 .716335 16 17 79.443 .833820 70.281 .771177 63.299 .716526 , 17 18 79.591 .833154 70.491 .770865 63.588 .716821 18 19 79.754 .832495 70.722 .770603 63.903 .717227 19 20 79.934 1.831855 70.976 1.770409 64.248 1.717770 20 21 80.130 .831226 71.253 .770280 64.625 .718453 21 22 80.344 .830618 71.555 .770228 65.034 .719284 22 23 80.581 .830045 71.887 .770273 65.482 .720297 23 24 80.839 .829502 72.249 .770416 65.969 .721491 24 25 81.122 1.829002 72.645 1.770675 66.500 1.722890 26 26 81.433 .828559 73.078 .771070 67.078 .724518 26 27 81.771 .828166 73.549 .771595 67.704 .726372 27 28 82.143 .827845 74.063 .772279 68.385 .728482 28 29 82.550 .827609 74.625 .773145 . 69.123 .730870 29 30 82.991 1.827444 75.232 1.774175 ^ 69.919 1.733517 30 31 83.474 .827388 ' 75.894 .775415 I 70.781 .736471 31 32 84.002 .827446 ! 76.614 .776876 1 71.711 .739740 32 33 84.578 .827632 77.396 .778571 i 72.714 .743326 33 34 85.206 .827958 78.242 .780516 1 73.793 .747239 34 35 85.889 1.828432 ; 79.158 1.782714 74.950 1.751476 35 36 86.628 .829058 80.145 .785175 76.186 .756027 36 37 87.434 .829875 : 81.212 .787932 77.511 .760918 37 38 88.310 .830894 ■ 82.362 .790987 78.925 .766137 38 39 89.258 .832112 1 83.598 .794336 80.429 .771652 39 40 90.283 1.833548 i 84.923 1.797979 82.025 1.777457 40 41 91.391 .835208 i 86.342 .801922 83.713 .783522 41 42 92.588 .837119 ! 87.861 .806168 85.499 .789839 42 43 93.875 .839260 89.478 .810687 87.378 .796356 43 44 95,261 .841661 91.199 .815480 89.354 .803052 44 45 96.749 1.844309 93.025 1.820524 91.424 1.809883 45 46 98.341 .847203 94.957 .825787 93.587 .816806 46 47 100.045 .850342 96.997 .831245 95.841 .823778 47 48 101.861 .853702 99.144 .836855 98.185 .830749 48 49 103.794 .857277 101.399 .842585 100.618 .837683 49 50 105.845 1.861037 103.760 1.848385 103.136 1.844523 50 5t 108.018 .864963 106.226 .854213 105.739 .851239 51 52 110.311 .869000 108.794 .860000 108.423 .857772 52 53 112.726 .873125 111.462 .865709 111.188 .864088 53 54 115.265 .877293 114.230 .871289 114.035 .870148 54 55 117.928 1.881455 117.097 1.876686 116.963 1.875917 55 56 120.709 .885543 120.055 .881833 119.969 .881341 56 57 123.610 .889507 123.109 .886696 123.056 .886392 57 58 126.622 .893259 126.256 .891219 126.224 .891042 58 59 129.756" .896782 129.489 .895309 129.472 .895215 59 60 133.004 1.899981 132.820 1.898975 60 61 136.366 .902797 136.245 .902142 61 62 139.842 .905166 139.765 .904755 62 63 143.437 .907030 143.392 .906786 63 64 147.149 .908303 147.124 .908167 64 65 150.985 .908931 65 *148 TABLE LX TEN ANNUAL PREMIUM (P) ALSO a(P — tt), FOR ENDOWMENT INSURANCE OF 1,000. PAYABLE AT DEATH OR AT THE END OF m YEARS. 4 PER CENT. AQE. mz =15. 20. 25. AQE. 15 P. A(P-7r). P. A (P - 77). P. A (P - 77). 15 69.912 1.771927 59.207 1.685216 50.757 1.601961 16 69.937 .770420 59.243 .683472 50.808 .600017 16 17 69.964 .768843 59.281 .681649 50.864 .598000 17 18 69.994 .767188 59.324 .679742 50.926 .595895 18 19 70.027 .765453 59.371 .677748 50.994 .593706 19 20 70.063 1.763631 59.423 1.675660 51.069 1.591429 20 21 70.103 .761723 59.480 .673478 51.151 .589058 21 22 70.147 .759720 59.542 .671194 51.241 .586594 22 23 70.195 .757617 59.611 .668810 51.340 .584037 23 24 70.248 .755414 59.685 .666304 51.448 .581376 24 25 70.306 1.753103 59.770 1.663709 51.567 1.578620 25 26 70.370 .750677 59.861 .660987 51.698 .575763 26 27 70.440 .748129 59.961 .658133 51.841 .572794 27 28 70.518 .745459 60.071 .655162 51.999 .569729 28 29 70.604- .742655 60.193 .652058 52.173 .566555 29 30 70.696 1.739709 60.325 1.648805 52.361 1.563263 30 31 70.799 .736621 60.470 .645411 52.568 .559867 31 32 70.912 .733376 60.631 .641875 52.796 .556367 32 33 71.037 .729968 60.807 .638180 53.045 .552757 33 34 71.174 .726396 60.999 .634316 53.318 .549045 34 35 71.325 1.722638 61.212 1.630306 53.617 1.545218 35 36 71.489 .718687 61.443 .626103 53.943 .541282 36 37 71.670 .714537 61.698 .621728 54.300 .537246 37 38 71.870 .710185 61.979 .617169 54.691 .533119 38 39 72.090 .705606 62.285 .612419 55.118 .528888 39 40 72.331 1.700792 62.621 1.607464 55.583 1.524562 40 41 72.595 .695729 62.989 .602305 56.091 .520146 41 42 72.887 .690414 63.394 .596938 56.646 .515653 42 43 73.206 .684812 63.835 .591346 57.249 .511067 43 44 73.558 .678925 64.320 .585530 57.906 .506410 44 45 73.944 1.672733 64.850 1.579485 58.621 1.501682 45 46 74.369 .666212 65.429 .573190 59.398 .496877 46 47 74.835 .659343 66.062 .566645 60.241 .492005 47 48 75.346 .652110 66.754 .559836 61.154 .487056 48 49 75.908 .644490 67.511 .552765 62.145 .482041 49 50 76.525 1.636458 68.336 1.545404 63.216 1.476939 50 1 51 77.203 .627994 69.237 .537759 64.374 .471757 51 ' 52 77.946 .619053 70.218 .529794 65.622 .466457 52 53 78.762 .609619 71.288 .521507 66.969 .461029 53 54 79.658 .599658 72.453 .512870 68.418 .455442 54 i 55 80.643 1.589137 73.722 1.503878 69.978 1.449674 55 56 81.721 .578003 75.099 .494476 71.651 .443648 56 57 82.905 .566227 76.598 .484655 73.447 .437335 57 58 84.201 .553741 78.221 .474329 75.367 .430617 58 59 85.624 .540512 79.985 .463526 77.424 .423490 59 60 87.185 1.526473 81.897 1.452135 79.624 1.415803 60 61 88.894 .511543 83.968 .440093 81.973 .407452 61 62 90.767 .495666 86.209 .427329 84.480 .398315 62 63 92.822 .478764 88.635 .413754 87.157 .388268 63 64 95.070 .460703 91.258 .399238 90.014 .377135 64 65 97.534 .441409 94.093 .383658 93.062 .364752 65 1 ^'OBMULA, Pa,= =1000 1^- ) ~ 1 + a 9 * 0! TABLE LX. *149 TEN ANNUAL PREMIUM (P), ALSO A (P— tt), FOR ENDOWMENT INSURANCE OP 1,000 PAYABLE AT DEATH OR AT THE END OF m YEARS. 4 PER CENT. AGE. 15 30. 35. 40. AQE. P. A (P - tt). P. A (P - tt). 1.450095 P. A(P-Tr). 44.122 1.523169 38.956 34.990 1.384237 15 16 44.194 .521124 39.057 .448114 35.127 .382584 16 17 44.274 .519013 39.169 .446104 35.278 .380957 17 18 44.362 .516833 39.290 .444056 35.442 .379355 18 19 44.459 .514582 39.424 .441976 35.622 .377787 19 20 44.565 1.512267 39.571 1.439877 35.820 1.376278 20 21 44.681 .509880 39.731 .437760 36.035 .374833 21 22 44.808 .507425 39.906 .435626 36.269 .373455 22 23 44.948 .504910 40.099 .433496 36.526 .372179 23 24 45.101 .502328 40.308 .431367 36.805 » .371008 24 25 45.269 1.499686 40.538 1.429258 37.109 1.369963 25 26 45.453 .496993 40.790 .427181 37.441 .369069 26 27 45.654 .494238 41.063 .425131 37.800 .368328 27 28 45.875 .491442 41.362 .423140 38.191 .367770 28 29 46.117 .488608 41.689 .421222 38.616 .367421 29 30 46.379 1.485723 42.043 1.419366 39.074 1.367263 30 31 46.667 .482818 42.430 .417615 39.571 .367350 31 32 46.982 .479895 42.850 .415977 40.108 .367691 32 33 47.327 .476963 43.307 .414470 40.688 .368294 33 34 47.702 .474033 43.804 .413112 41.313 .369173 34 35 48.112 1.471107 44.341 1.411908 41.984 1.370324 35 36 48,556 .468193 44.922 .410868 42.703 .371747 36 37 49.041 .465316 45.541 .409858 43.475 .373458 37 38 49.569 .462496 46.230 .409390 44.301 .375458 38 39 50.142 .459726 46.963 .408964 45.182 .377712 39 40 50.764 1.457022 47.750 1.408741 46.120 1.380204 40 41 51.437 .454392 48.596 .408730 47.116 .382904 41 42 52.168 .451863 49.504 .408944 48.174 .385801 42 43 52.956 .449417 50.475 .409344 49.291 .388829 43 44 53.808 .447077 51.513 .409933 50.471 .391960 44 45 54.727' 1.444843 52.620 1.410688 51.715 1.395134 45 46 55.715 .442699 53.798 .411560 53.021 .398287 46 47 56.778 .440651 55.049 .412525 54.393 .401363 47 48 57.918 .438669 56.374 .413523 55.828 .404288 48 49 59.140 .436753 57.776 .414515 57.330 .407009 49 50 60.446 1.434853 59.255 1.415426 58.899 1.409433 50 51 61.841 .432953 60.815 .416198 60.536 .411521 51 52 63.326 .430981 62.456 .416730 62.243 .413174 52 53 64.907 .428896 64.179 .416959 64.021 . .414331 53 54 66.586 .426630 65.988 .416797 65.876 .414920 54 55 68.369 1.424120 67.887 1.416166 67.810 1.414878 55 56 70.255 .421252 69.875 .414943 69.824 .414104 56 57 72.252 .417951 71.959 .413074 71.928 .412545 57 58 74.359 .414073 74.144 .410457 74.126 .410143 58 59 76.589 .409583 76.431 .406912 76.421 .406736 59 60 78.943 1.404301 78.834 1.402431 60 61 81.430 .398116 81.358 .396865 61 62 84.057 .390910 84.011 .390105 62 63 86.837 .382544 86.810 .382053 63 64 89.779 .372844 89.764 .372557 64 65 92.898 .361648 65 ^150 TABLE LXI. LIMITED OR m ANNUAL PREMIUMS (P), ALSO A (P — tt), FOR LIFE POLICIES OF 1,000. 4 PER CENT. AQE. X. m : = 5. 10. 15. 20. AGE. X. p. A (P - tt). P. P. A(P_:r). P. A(P_7r). 15 47.841 1.569080 26.657 19.725 0.952279 16.357 0.747458 15 16 48.636 .575656 27.103 20.057 .957224 16.634 .751210 16 17 49.465 .582385 27.568 20.404 .962251 16.925 .754982 17 18 50.327 .589248 28.051 20.765 .967323 17.227 .758723 18 19 51.224 .596246 28.555 21.141 .972434 17.543 .762431 19 20 52.158 1.603385 29.079 21.533 0.977591 ' 17.872 0.766115 20 21 53.129 .610649 29.625 i 21.942 .982773 18.215 .769739 21 22 54.139 .618034 30.193 1 22.367 .987966 ; 18.573 .773296 22 23 55.188 .625547 30.784 1 22.810 .993176 ; 18.947 .776774 23 24 56.279 .633170 31.398 23.271 .998377 19.336 .780144 24 25 57.412 1.640910 32.038 23.751 1.003564 19.742 0.783410 25 26 58.590 .648762 32.703 24.252 .008732 20.166 .786546 26 27 59.810 .656696 33.393 24.772 ■ .013840 ' 20.607 .789489 27 28 61.079 .664734 34.111 25.314 .018900 21.067 .792273 28 29 62.395 .672861 34.857 25.878 .023891 21.548 .794857 29 30 63.757 1.681044 35.631 26.464 1.028767 22.048 0.797184 30 31 65.171 .689305 36.435 27.074 .033544 22.570 .799272 31 32 66.636 .697631 37.270 27.710 .038195 23.115 .801067 32 33 68.154 .706006 38.136 ■ 28.371 .042690 23.684 .802534 33 34 69.726 .714424 39.036 29.058 .047022 24.278 .803655 34 35 71.352 1.722864 39.969 29.774 1.051141 24.897 0.804364 35 36 73.032 .731309 40.935 30.517 .055026 25.544 .804637 36 37 74.771 .739775 41.938 31.292 .058665 26.220 .804432 37 38 76.569 .748241 42.979 32.098 .062044 26.926 .803723 38 39 78.426 .756679 44.057 32.937 .065094 27.665 .802425 39 40 80.342 1.765086 45.174 33.810 1.067800 28.437 0.800490 40 41 82.319 .773437 46.331 34.719 .070119 29.246 .797884 41 42 84.361 .781742 47.532 35.667 .072033 30.093 .794523 42 43 86.463 .789955 48.775 36.653 .073469 30.981 .790341 43 44 88.631 .798083 50.063 37.682 .074407 31.912 .785266 44 45 90.864 1.806100 51.398 38.755 1.074798 32.891 0.779236 45 46 93.162 .813987 52.781 39.875 .074582 33.919 .772138 46 47 95.527 .821727 54.214 41.045 .073707 35.001 .763885 47 48 97.960 .829300 55.700 42.267 .072118 36.141 .754371 48 49 100.461 .836691 57.241 43.545 .069761 37.343 .743502 49 50 103.031 1.843872 58.839 44.884 1.066531 38.613 0.731121 50 51 105.672 .850829 60.498 46.287 .062413 39.956 .717154 51 52 108.382 .857524 62.219 47.760 .057275 41.378 .701387 52 53 111.165 .863946 64.008 49.307 .051033 42.887 .683659 53 54 114.022 .870073 65.868 50.936 .043625 44.490 .663842 54 55 116.957 1.875879 1 67.806 52.653 1.034937 46.198 0.641682 55 56 119.965 .881320 69.822 54.464 .024810 48.016 .616969 56 57 123.054 .886386 71.927 56.381 .013174 49.959 .589480 57 58 126.221 .891022 74.124 58.410 0.999852 52.035 .558900 58 59 129.472 .895215 76.421 60.564 .984707 54.259 .524954 59 60 132.811 1.898927 78.829 62.855 0.967553 56.646 0.487280 60 61 136.241 .902120 81.355 65.296 .948227 59.209 .445480 61 62 139.764 .904747 84.010 67.901 .926461 61.965 .399033 62 63 143.391 .906781 86.809 70.689 .902085 64.934 .347603 63 64 147.124 .908167 89.764 73.679 .874801 68.135 .290458 64 65 150.971 .908858 92.889 76.889 .844309 71.588 .226806 65 Formula, Pg. = 1000 Na; — Na;+OT 1 -t- «: ■m—1 TABLE LXII. *151 NET VALUE OR RESERVE ON TEN PAYMENT LIFE POLICIES OP 1,000. 4 PER CENT. r Age of Entry. 1st Year. 2d Year. 3d Year. Age of Entry. 6m. 12m. l^m.d. 6m. 12m. |m.d. 6m. 12m. ^m.d. 15 23.965 21.272 .224 45.724 43.520 .184 68.478 66.779 .142 15 16 24.409 21.716 .224 46.619 44.418 .183 69.838 68.154 .140 16 17 24.869 22.169 .225 47.541 45.346 .183 71.247 '69.580 .139 17 18 25.346 22.642 .225 48.504 46.315 .182 72.712 71.059 .138 18 19 25.845 23.135 .226 49.503 47.316 .182 74.231 72.590 .137 19 20 26.360 23.641 .227 50.535 48.349 .182 75.804 74.180 .135 20 21 26.896 24.166 .228 51.609 49.427 .182 77.439 75.826 .134 21 22 1 27.454 24.715 .228 52.724 50.541 .182 79.133 77.532 .133 22 23 ! 28.030 25.277 .229 53.877 51.692 .182 80.887 79.299 .132 23 24 28.631 25.863 .231 55.076 52.891 .182 82.706 81.122 .132 24 25 29.255 26.472 .232 56.314 54.119 .183 84.583 83.010 .131 25 26 29.895 27.088 .234 57.591 55.390 .183 86.527 84.961 .131 26 27 30.566 27.739 .236 58.923 56.715 .184 88.539 86.969 .131 27 28 31.259 28.407 .238 60.290 58.062 .186 90.606 89.040 .130 28 29 31.969 29.080 -.241 61.696 59.454 .187 92.744 91.177 .131 29 30 32.712 29.793 .243 63.164 60.904 .188 94.962 93.389 .131 30 31 33.479 30.524 .246 64.674 62.390 .190 97.244 95.662 .132 31 32 34.272 31.273 .250 66.231 63.918 .193 99.591 97.994 .133 32 33 35.090 32.044 .254 67.832 65.485 .196 102.000 100.380 .135 33 34 35.934 32.832 .258 69.475 67.082 .199 104.478 102.837 .137 34 35 36.801 33.633 .264 71.168 68.733 .203 107.032 105.363 .139 35 36 37.704 34.473 .269 72.922 70.437 .207 109.663 107.954 .142 36 37 38.633 35.328 .275 74.717 72.167 .213 112.351 110.597 .146 37 38 39.585 36.192 .283 76.551 73.932 .218 115.100 113.288 .151 38 39 40.568 37.079 .291 78.434 75.731 .225 117.919 116.050 .156 39 40 41.576 37.977 .300 80.364 77.577 .232 120.802 118.854 .162 40 41 42.621 38.912 .309 82.347 79.452 .241 123.752 121.720 .169 41 42 43.685 39.837 .321 84.360 81.351 .251 126.751 124.620 .178 42 43 44.784 40.794 .333 86.429 83.288 .262 129.818 127.572 .187 43 44 45.910 41.756 .346 88.531 85.244 .274 132.931 130.556 .198 44 45 47.063 42.729 .361 90.675 87.222 .288 136.095 133.570 .210 45 46 48.247 43.713 .378 92.855 89.217 .303 139.307 136.616 .224 46 47 49.458 44.701 .396 95.072 91.229 .320 142.561 139.679 .240 47 48 50.701 45.702 .417 97.326 93.250 .340 145.856 142.762 .258 48 49 51.968 46.696 .439 99.606 95.275 .361 149.179 145.841 .278 49 50 53.269 47.699 .464 101.918 97.297 .385 152.529 148.922 .301 50 51 54.593 48.688 .492 104.247 99.309 .411 155.900 151.994 .326 51 52 55.949 49.678 .523 106.608 101.319 .441 159.298 155.058 .353 52 53 57.336 50.665 .556! 108.995 103.317 .473 162.703 158.080 .385 53 54 58.755 51.641 .593 111.391 105.273 .510 166.102 161.064 .420 54 55 60.194 52.583 .634 113.792 107.195 .550 169.489 163.977 .459 55 • 56 61.673 53.524 .679 116.210 109.074 .595 172.867 166.838 .502 56 57 63.173 54.418 .730 118.620 110.895 .644 176.221 169.620 .550 57 58 64.713 55.302 .784 121.053 112.680 .698 179.563 172.321 .603 58 59 66.289 56.158 .844 123.486 114.392 .758 183.858 174.904 .663 59 60 67.898 56.967 .911 125.902 116.008 .825 186.106 177.374 .728 60 61 69.541 57.726 .985 128.316 117.552 .897 189.308 179.709 .800 61 62 71.236 58.463 1.064 130.741 119.008 .978^ 192.458 181.898 .880 62 63 72.972 59.134 1.153 133.141 120.340 1.067 195.525 183.901 .969 63 64 74.756 59.748 1.251 135.531 121.549 1.165 198.523 185.734 1.066 64 65 76.592 60.295 1.358 137.914 122.644 1.272 201.446 187.359 1.174 65 *152 TABLE LXII. NET VALUE OR RESERVE ON TEN PAYMENT LIFE POLICIES OF 1,000. 4 PER CENT. 1 A^eof Entry. 4th Year. 5th Year. 6th Year. % Age of Entry. 6m. 12m. ^m.d. 6m. 12m. ^m.d. 6m. 12m. |m.d. 15 92.268 91.100 .097 117.145 116.533 .051 143.157 143.124 .003 15 16 94.116 92.974 .095 119.500 118.923 .048 146.039 146.052 .001 16 17 96.031 94.914 .093 121.939 121.397 .045 149.028 149.091 .005 17 I 18 98.017 96.925 .091 124.474 123.971 .042 152.133 152.243 .009 18 1 19 100.080 99.015 .089 127.102 126.635 .039 1 155.350 155.510 .013 19 20 102.218 101.177 .087 129.826 129.396 .036 158.687 158.899 .018 20 21 104.434 103.417 .085 132.652 132.261 .033 162.144 162.402 .022 21 22 106.735 105.744 .083 135.579 135.221 .030 165.724 166.034 .026 22 23 109.112 108.141 .081 138.605 138.286 .027 169.432 169.794 .030 23 24 111.573 110.627 .079 141.743 141.461 .024 173.265 173.671 .034 24 25 114.124 113.199 .077 144.983 144.729 .021' 177.224 177.681 .038 25 26 116.752 115.840 .076 148.325 148.107 .018 181.317 181.824 .042 26 27 119.470 118.578 .074 151.787 151.602 .015 185.549 186.104 .046 27 28 122.275 121.400 .073 155.357 155.203 .013 1 189.915 190.515 .050 28 29 125.171 124.308 .072 159.039 158.913 .010 194.414 195.059 .054 29 30 128.167 127.313 .071 162.842 162.740 .009 199.053 199.734 .057 30 31 131.248 130.399 .071 166.749 166.664 .007 1203.824 204.550 .060 31 32 134.412 133.560 .071 170.768 170.706 .005'! 208.739 209.502 .064 32 33 137.666 136.816 .071 174.906 174.860 .00411 213.791 214.585 .066 33 34 141.016 140.160 .071 179.157 179.119 .003 218.977 219.800 .069 34 35 144.460 143.587 .073 183.522 183.487 .003 224.299 225.142 .070 35 36 147.996 147.104 .074 187.999 187.960 .003 229.761 230.627 .072 36 37 151.612 150.688 .077 192.583 192.539 .004 235.350 236.223 . .073 37 38 155.312 154.358 .079 197.271 197.206 .005 241.064 241.943 .073 38 39 159.102 158.096 .084 202.065 201.977 .007 246.908 247.781 .073 39 40 162.970 161.912 .088 206.961 206.835 .010 252.869 253.730 .072 40 41 166.924 165.798 .094 211.956 211.784 .014 258.952 259.788 .070 41 42 170.945 169.737 .101 217.037 216.804 .019 265.135 265.934 .067 42 43 175.044 173.741 .109 222.209 221.902 .026 271.428 272.179 .063 43 44 179.204 177.789 .118 227.457 227.063 .033 277.812 278.498 .057 44 45 183.426 181.884 .129 232.781 232.279 .042 284.283 284.890 .051 45 46 187.705 186.014 .141 238.169 237.543 .052 290.831 291.338 .042 46 47 192.034 190.174 .155 243.613 242.839 .065 297.442 297.831 .032 47 48 196.406 194.350 .171 249.106 248.162 .079 304.112 304.362 .021 48 49 200.806 198.530 .190 254.634 253.497 .095 310.827 310.916 .007 49 50 205.238 202.715 .210 260.197 258.839 .113 317.571 317.464 .009 50 51 209.691 206.890 .233 265.770 264.153 .135 324.326 324.000 .027 51 52 214.154 211.032 .260 271.348 269.444 .159 331.080 330.497 .049 52 53 218.614 215.139 .290 276.911 274.676 .186 337.812 336.940 .073 53 54 223.054 219.177 .323 282.442 279.839 .217 344.509 343.311 .100 54 55 227.461 22^.138 .360 287.929 284.914 .251 351.148 349.577 .131 55 56 231.845 227.030 .401 293.372 289.891 .290 357.717 355.720 .166 56 57 236.176 230.806 .447 298.726 294.720 .334 364.181 361.716 .205 57 58 240.453 234.461 .499 304.002 299.418 .382 370.545 367.548 .250 58 59 244.654 237.982 .556 309.170 303.938 .436 376.764 373.168 .300 59 60 248.770 241.338 .619 314.208 308.248 .497 382.812 378.548 .355 60 61 252.785 244.505 .690 319.091 312.323 .564 388.672 383.666 .417 61 62 256.687 247.467 .768 323.817 316.156 .638 394.325 388.484 .487 62 63 260.456 250.201 .855 328.348 319.687 .722 399. 742 392.987 .563 63 64 264.089 252.680 .951 332.690 322.936 .813 404.905 397.110 .650 64 65 267.582 254.917 1.055 336.816 325.825 .916 409.779 400.845 .745 65 TABLE LXII. *lo3 NET VALUE OR RESERVE ON TEN PAYMENT LIFE POLICIES OF 1,000. 4 PER CENT. Age of Entry. 7th Year. 8th Year. 9th Year.* Age of Entry. 6m. 12in. ^m.d. 6m. 12m. Jm.d. 6m. 12m. l^m.d. 15 170.354 170.927 .048 198.794 200.004 .101 228.535 230.408 .156 15 16 173.789 174.422 .053 202.805 204.084 .107 233.143 235.100 .163 16 17 177.351 178.043 .058 206.963 208.315 .113 237.928 239.972 .170 17 18 181.047 181.800 .063 211.280 212.709 .119 242.889 245.018 .177 18 19 184.883 185. 700 .068 215.755 217.255 .125 248.030 • 250.251 .185 19 20 188.855 189.732 .073 220.390 221.969 .132 253.363 255.677 .193 20 21 192.970 193.912 .079 225.196 1226.855 .138 258.882 261.284 .200 21 22 197.235 198.244 .084 230.169 231.902 .144 264.593 267.091 .208 22 23 201.646 202.713 .089 235.311 237.125 .151 270.505 273.102 .216 23 24 206.203 207.338 .095 240.633 242.530 .158 276.622 279.316 .225 24 25 210.920 212.121 .100 246.136 248.113 .165 282.943 285.735 .233 25 26 215.792 217.057 .105 251.819 253.878 .172 289.472 292.363 .241 26 27 220.826 222.155 .111 257.688 259.828 .178 296.208 299.195 .249 27 28 226.017 227.409 .116 263.737 265.953 .185 303.156 306.248 .258 28 29 231.365 232.813 .121 269.971 272.273 .192 310.323 313.516 .266 29 30 236.878 238.391 .126 276.404 278.785 .198 317.708 320.999 .274 30 31 242.557 244.129 .131 283.022 285.480 .205 325.307 328.699 .283 31 32 248.398 250.024 .135 289.828 292.363 .211 333.121 336.609 .291 32 33 254.400 1 256.079 .140 296.821 299.427 .217 341.153 344.743 .299 33 34 260.560 1262.285 .144 304.003 306.684 .223 349.401 353.083 .307 34 35 266.885 268.658 .148 311.372 314.117 .229 357.863 361.639 .315 35 36 273.370 275.179 .151 318.925 321.736 .234 366.537 370.403 .322 36 37 280.007 281.852 .154 326.658 329.527 .239 375.418 379.371 .329 37 38 286.793 288.665 .156 334.566 337.488 .244 384.502 388.537 .336 38 39 293.729 295.620 .158 342.646 345.615 .247 393.784 397.896 .343 39 40 300.807 302.709 .159 350.891 353.898 .251 403.258 407.444 .349 40 41 308.022 309.926 .159 359.297 362.337 .253 412.918 417.169 .354 41 42 315.362 1317.257 .158 367.850 370.911- .255 422.753 427.062 .359 42 43 322.827 324.700 .156 376.548 379.622 .256 432.755 437.114 .363 43 44 330.401 332.241 .153 385.377 388.450 .256 442.913 447.313 .367 44 45 338.077 339.866 .149 394.325 397.385 .255 453.216 457.648 .369 45 46 345.842 347.566 .144 403.381 406.416 .253 463.653 468.109 .371 46 47 353.687 355.328 .137 412.538 415.533 .250 474.207 478.668 .372 47 48 361.602 363.142 .128 421.773 424.705 .244 484.862 489.319 .371 48 49 369.566 370.975 .117 431.071 433.925 .238 495.601 500.036 .370 49 50 377.563 378.824 .105 440.415 443.167 .229 506.405. 510.803 .366 50 51 385.577 386.656 .090 449.782 452.411 .219 517.257 521.605 .362 51 52 393.588 394.459 .073 459.162 461.645 .207 528.140 532.417 .356 52 53 401.582 402.216 .053 468.531 470.838 .192 539.026 543.206 .348 53 54 409.537 1409.895 .030 477.859 479.955 .175 549.892 553.961 .339 54 55 417.420 417.458 .003 487.124 488.984 .155 560.721 564.652 .328 55 56 425.224 424.906 .026 496.313 497.898 .132 571.484 575.247 .314 56 57 432.919 432.195 .060 505.387 506.653 .105 582.148 585.716 .297 57 58 440.485 439.297 .099 514.323 515.225 .075 592.693 596.038 .279 58 59 447.881 446.174 .142 523.091 523.587 .041 603.093 606.177 .257 59 60 455.090 452.802 .191 531.660 531.689 .002 613.315 616.112 .233 60 61 462.076 459.132 .245 540.006 539.524 .040 623.336 625.793 .205 61 62 468.830 465.165 .305 548.100 547.026 .090 633.118 635.201 .174 62 63 475.305 '470.815 .374 555.898,554.171 .144 642.635 644.290 .138 63 64 481.476 476.078 .450 563.376 560.911 .205 651.853 653.031 .098 64 65 487.316 480.897 .535 570.499 567.212 .274 660.750 661.399 .054 65 Note. The last Premium at the beginning of the 10th Year changes the Policy to the Paid-up class. *154 TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. INTEREST, 4 PER CENT. I. Common Life Policies. — The Net Value or Reserve is already computed in the column (V, 6n3.) for the middle of each Policy Year, and in the column (V, 12m.) for the end of each Policy Year, when the next Annual Premium is just due and unpaid. For any other time of year, a correction derived from the column of half monthly or 15 days difference (|^m. d.) is to be added or subtracted, according as the Reserve is increasing or decreasing in the direction of the given interval. For example, if the Age of Entry, x, of a common Life Policy of 11000 be 35 years, the 4 per cent. Reserve at the middle of the 24th Policy Year (V||, 6m.) is given $362.47, and at the end of the same Policy Year, $362.37. Hence, at the end of 3 months in the same year, the Reserve i? $362.52, and at the end of the 9th month, $362.42. Formulas, (V, 12m.) = 1000 ] 1 - 1 + ^" i . ., = ^^^. - (V, 6m.) = i \ (Vr\ 12m.) +77^4- (VS, Urn.)}. B^ = ^^-^1^. IL General Valuation. — For Endowment Insurances and all other Annual Premium Policies, the Reserve V is first found from the Table, and corrected for the elapsed fraction of a year, when required, as if for a common Life Policy. The remain- ing part and process is thus indicated : General Formula, Reserve = V 4- (Pa- — ttx) • B^ •/*+„. Middle of Policy Year, Reserve = (V, 6m.) + (Pa, — tt^) • (B/)". End " " " Reserve = (V, 12m.) + (Pa; — tt^,) • B^. V Pure Endowment, Reserve =: — rp^ • B^ • fx+n- Here n -\- m- denotes the expired (n) added to the unexpired (m) years of the whole period of simple endowment. In all cases, X = the Age of Entry or of original Insurance. h = fraction elapsed of the current Policy Year. f = h-{-{l—h) vjJx+n-h Table LXIV. At 12 montiis, or the end of the year, /= 1, since A = 1. Example 1. End of the Year. — A Ten Premium Endowment Insurance of $1000, payable at Death or Age 50, is contracted at the Age of 30. Required the Net Value at the end of the 9th Year. x = dO; n + h = 8 yrs. 13m. 1.648805 = a(P - n), Table LVIL V = 92.685, 1.060738 = /.B30, Table LXIII. + 512.322 . . . .3.709543 Sum. Table LXXXIL Reserve = feoXOOT on $1000 Insured. Example 3. Intermediate Months and Term Policies. — An Insurance of $1000 is entered at the age of 38, for the Term of 7 years. Required the Net Value at the end of 4 years and 8J months (5th year). a; = 38 ; a + n + A = 43 yrs. 8^ m. 1 .05327 = A (tt - P), Table UII. 1.99387 = lf\z, Table LXIV. Adjusted V = 68.993, 0.76363 = 1B\», Table LXIII. -64.531 ■...1.80977 Sum, Table LXXXIL Net Value = $4,462 on $1000 Insured. In this case, (P — n) being negative, since P is less than tt, the computed term is subtractive, which is a general characteristic of Simple Term Policies. Example 3. Middle of the Year.—K Fifteen Premium Insurance of $1000 for the whole Life is entered at the Age of 42. Required the Net Value at the end of 7 years and 6 months, (8th Policy Year). V, 6m. 1.07203 = ?.(P-7r), Table LXI. 133.75 0.99521 = ?^(B/), Table LXIIL + 116.75 . . . .2.06724 Sum, Table LXXXIL Reserve = $250.50 on $1000 Insured. Note 1. Five, Ten, or Limited Premium Policies can be valued by this method only till one year after the last Annual Premium is paid. After the beginning of this year, the Policy being Paid-up, the Reserve is more easily found from the proper Single Premium Table. Note 2. Although six decimals are given, yet in practice five, and in many cases, four decimals will be suflBcient for logarithms or anti-logarithms. Thus the last of the six decimals can generally be omitted, as in the last Examples. TABLE LXIII. *155 GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. n. 10. 11. o o n. V. 6m. V. 12m. ^m.d 6m. AB. 13m. V. 6m. V. 13m. im.d 6m. AB. 12m. 6.742 3.70E .253 0.010043 0.019857 6.914 3.875 .353 0.010046 0.019866 1 10.527 7. 570 .246 .331118 .330938 10.873 7.917 .346 .331136 .330948 1 2 14.470 11.597 .239 .507337 .517159 14.999 13.137 .339 .507349 .517176 2 3 18.582 15.792 .233 .643484 .653311 19.393 16.505 .333 .643504 .653337 3 4 22.860 20.154 .226 .749687 .759530 33.763 31.065 .325 .749714 .759553 4 5 27.312 24.696 .218 0.839344 0.849083 38.415 35.811 .317 0.839378 0.849133 5 6 31.94? 29.425 .310 .916653 .936496 33.359 30.754 .309 .916693 .926543 6 7 36.774 34.349 .203 .985188 .995039 38.301 35.893 .301 .985338 .995096 7 8 41.797 39.470 .194 1.046973 1.056830 43.543 41.340 .193 1.047033 1.056898 8 " 9 47.020 44.796 .185 .103449 .113314 48.995 46.796 .183 .103530 .113395 9 10 52.450 50.331 .177 1.155649 1.165534 54.659 53.568 .174 1.155734 1.165619 10 11 58.094 56.083 .168 .304335 .314330 60.545 58.568 .165 .304436 .314333 11 12 63.959 62.060 .158 .350088 .359984 66.661 64.799 .155 .350305 .360113 12 13 70.051 68.268 .149 .393356 .303363 73.010 71.367 .145 .393491 .303411 13 14 76.377 74.712 .139 .334498 .344418 79.603 77.983 .135 .334654 .344587 14 15 83.944 81.403 .138 1.373803 1.383736 86.441 84.945 .125 1.373983 1.383933 15 16 89.759 88.340 .118 .411508 .431458 93.533 93.167 .114 .411713 .431679 16 17 96.824 95.534 .108 .447810 .45777711100.888 99.655 .103 .448043 .458038 17 18 104.151 102.995 .096 .483875 .493860 108.505 107.403 .093 .483143 .493150 18 19 111.741 110.713 .086 .516846 .536854 116.393 115.438 .080 .517149 .537178 19 20 119.598 118.710 .074 1.549843 1.559873 134.559 133.736 .069 1.550184 1.560339 20 21 127. 735 126.986 .063 .581968 .593033; 133.013 133.333 .057 .583353 .593434 21 22 136.156 135.552 .050 .613315 .633396 141.753 141.318 .045 .613749 .633861 22 23 144.865 144.403 .039 .643964 .654076 150.786 150.399 .033 .644454 .654599 23 !24 153.864 153.552 .036 .673993 .684137 160.110 159.867 .030 .674545 .G84739 24 25 163.155 162.983 .014 1. 703464 1.713648 169.734 169.647 .007 1.704083 1.714306 25 126 172.742 172.727 .001 .733434 .743657 179.669 179.737 .006 .733134 .743389 26 27 182.640 182.779 .012 .760957 .771333 189.911 190.133 .018 .761737 .773043 27 28 192.845 193.137 .024 .789083 .799398 300.460 300.834 .031 .789948 .800318 28 29 203.354 203.798 .037 .816861 .837331 311.315 211.843 ,.044 .817839 .838359 29 30 214.169' 214.766 .050 1.844335 1.854765 333.486 333.175 .057 1.845413 1.855904 30 31 225.299 226.057 .063 .871543 .883034 333.965 334.803 .070 .873745 .883313 31 32 236.735 237.640 .075 .898536 .909093 345.753 346.748 .083 .899867 .910509 32 33 248.479: 249.543 .089 .935335 .935967 357.850 358.997 .096 .936815 •937545 33 34 260.531 261.746 .101 .951973 .963703 370.348 371.546 .108 .953634 .964459 34 35 272.884 274.248 .114 1.978511 1.989336 383.948 384.396 .131 1.980356 1.991385 35 36; 385,536 287.050 .126 2.004974 2.015903 395.943 397.534 .133 2.007035 3.018069 36 37' 298.483 300.141 .138 .031403 .043447 309.333 310.957 .145 .033681 .044851 37 38 311.714 313.513 .150 .057833 .069003 333.785 334.659 .156 .060360 .071668 38 39 325.225 327.164 .162 .084301 .095609 336.618 338.633 .167 .087107 .098568 39 40 339.007 341.075 .172 3.110853 3.133313 350.708 353.840 .178 3.113964 2.135593 40 41 353.044 355.240 .183 .137537 .149156 365.047 367.300 .188 .140976 .153789 41 42 367.331 369.647 .193 .164369 .176183 379.633 381.991 .197; .168190 .180304 42 43 381.852 384.283 .203 .191433 .303436 394.433 396.898 .306' .195653 .307887 43 10. IT = 9.7739. 11. Tr = 9.9535. Continued on Page *199. *156 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. a; = 12. 13. H n. V. 6ni. V. 12m. im.d. 6m. AB. 12m. V. 6m. V. 12m. im.d. A(B/). 6m. AB. 12m. 7.100 4.058 .254 0.010050 0.019872 7.292 4.244 .254 0.010055 0.019882 1 11.243 8.285 .247 .321134 .330901 11.621 8.657 .247 .321147 .330980 1 2 15.553 12.680 .239 .507365 .517198 16.125 13.253 .239 .507382 .517221 2 3 20.039 17.257 .232 .042526 .652365 20.815 18.037 .232 .642549 .652393 3 4 24.711 22.022 .224 .749741 .759585 25.698 23.019 .223 .74977r .759622 4 5 29.574 26.984 .216 0.839313 0.849164 30.779 28.200 .215 0.839352 0.849210 5 6 34.634 32.143 .208 .916736 .926594 36.065 33.589 .206 .916786 .926651 6 7 39.898 37.510 .199 .985293 .995158 41.559 39.189 .197 .985353 .995228 7 8 45.370 43.088 .190 1.047098 1.056973 47.268 45.007 .188 1.047172 1.057057 8 ■ 9 51.056 48.883 .181 .103600 .113485 53.201 51.055 .179 .103687 .113583 9 10 56.966 54.906 .172 1.155828 1.165724 59.365 57.336 .169 1.155932 1.165839 10 11 63.104 61.161 .102 .204546 .214453 65.766 63.855 .159 .204667 .214587 ill 12 69.479 67.654 .152 .250333 .260253 72.410 70.625 .149 .250474 .260407 12 13 76.096 74.396 .142 .293639 .303572 79. .304 77.643 .138 .293803 .303753 13 14 82.962 81.386 .131 .334825 .344775 86.452 84.922 .128 .335013 .344980 14 15 90.082 88.636 .121 1.374178 1.384145 93.866 92.470 .116 1.374394:1.384379 15 16 97.465 96.153 .109 .406004 .416049 101.545 100.279 .105 .412186^ .422194 16 17 105.113 103.930 .099 .448301 .458309 109.494 108.309 .094 .448586 .458615 17 18 113.030 111.988 .087 .483430 .493465 117.726 110.743 .082 .483758 .493813 18 19|131.229 120.328 .075 .517482 .527537 126.246 125.409 .070 .517846 .527927 19, 20 129.714 128.958 .063 1.550559 1.560640 135.057 134.365 .058 1.550971 1.561083 20^ 21138. 488 137.877 .051 .582776 .5928881 144.162 143.619 .045 .583243 .593388 21 22 147.557 147.094 .039 .614227 .624372 153.561 153.163 .033 .614755 .624939 22 231156.917 156.599 .027 .644995 .655179 163.261 163.020 .020 .645589 .655812 23 24 [166.579 166.417 .014 .675152 .685375 173.276 173.191 .007 .675816 .686081 24 25:176.553 176.547 .001 1.704760 1.715025 183.600 183.669 .006 1.705505 1.715821 25 26 186.836 186.982 .012 .733883 .744199 194.232 194.450 .019 .734721 .745091 26 27 197.424 197.725 .025 .762581 .772951 205.175 205.553 .032 .763519 .773949 27 281208.322 208.777 .038 .790901 .801331 216.434 216.976 .045 .791949 .802440 28 29 219.537 220.154 .051 .818892 .829383 228.006 228.695 .057 .820062 .830629 29 30 231.060 231.825 .064 1.846599 1.857166 239.886 240.737 .071 1.847906 1.858548 30 31 243.893 243.819 .077 .874069 .884711 252.080 253.084 .084 .875524 .886254 31 321255.0.38 256.115 .090 .901.340 .912070 264.579 265.733 .090 .902903 .913788 32 33 267.485 268.713 .102 .928458 .939283 277.379 278.685 .109 .930202 .941191 33 34 280.234 281.612 .115 .955459 .966388 290.477 291.929 .121 .957407 .968511 34 35 293.278 294.802 .127 1.982387 1.993431 303.864 305.459 .133 1.984021 1.995791 35 36 306.610 308.277 .139 2.009283 2.020453! 317.534 319.270 .145 2.011765 2.023073 36 37 320.226 322.032 .151 .036186 .047494i .331.477 333.344 .150 .038944 .050405 37 38 334.111 336.049 .162 .063143 .074604 345.680 347.675 .100 .066204 .077833 38 39 348.256 350.322 .172 .090195 .101824 360.133 362.251 .170 .093591, .105404 39 40 362.652 364.839 .182 2.117388 2.129201 374.825 377.059 .186 2.121154' 2.133168 40 41 377.284 379.587 .192 .144770 .156784- 389.742 392.085 .195 .148942 .101177 41 42 392.140 394.552 .201 .172391 .1846261 404.864 407.303 .203 .177014 .189496 42 43 407.201 409.708 .209 .200307 .212789 420.173 422.703 .211 .205430 .318178 43 ■ ■■■..— — y . ' — • 12. 77 = 10.1419. 13. -n- = 10.3400. Continued on Page *198. *157 TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. n. 14. 15. V. 6m. V. 12m. ^m.d. A(B/). 6m. aB. 12m. V. 6m. V. 12m. ^m.d. A(B/). 6m. aB. 12m. n. 7.490 4.432 .255 0.010062 0.019895 7.701 4.636 .255 0.010067 0.019906 1 ol 1 12.013 9.047 .247 .321158 .330997 12.432 9.462 .248 .321169 .331013 1 2 16.724 13.852 .239 .507400 .517244 17.358 14.488 .239 .507417 .517268 2 3 21.627 18.855 .231 .642573 .652424 22.483 19.713 .231 .642600 .652458 3 4 26.731 24.058 .223 .749805 .759663 27.815 25.150 .222 .749841 .759706 4 5 32.038 29.470 .214 0.839395 0.849260 33.357 30.799 .213 0.839442 0.849317 5 6 37.556 35.094 .205 .916839 .926714 39.117 36.668 .204 .916898 .926783 6 7 43.289 40.937 .196 .985419 .995304 45.101 42.768 .194 .985492 .995388 7 8 49.248 47.011 .186 1.047252 1.057148 51.319 49.104 .185 1.047340 1.057247 8 9 55.439 53.318 .177 .103783 .113690 57.775 55.680 .175 .103889 .113809 9 10 61.865 59.865 .167 1.156045 1.165965 64.477 62.509 .164 1.156169 1.166102 10 11 68.539 66.664 .156 .204800 .214733 71.432 69.589 .154 .204946 .214896 11 12 75.462 73.712 .146 .250628 .260578 78.643 76.931 .143 .250798 .260765 12 13 82.641 81.022 .135 .293982 .303949 86.121 84.545 .131 .294179 .304164 13 14 90.086 88.602 .124 .335220 .345205 93.867 92.422 .120 .335448 .345456 14 15 97.797 96.444 .113 1.374633 1.384641 101.885 100.583 .108 1.374897 1.384926 15 16 105.780 104.569 .101 .412461 .422490: 110.190 109.030 .097 .412760 .422815 16 17 114.048 112.979 .089 .448897 .458952 118.784 117.772 .084 .449239 .459320 17 18 122.605 121.682 .077 .484112 .494193 127.671 126.805 .072 .484499 .494611 18 19 131.452 130.675 .065 .518246 .528358 136.856 136.141 .060 .518687 .528832 19 20 140.596 139.969 .052 1.551425 1.561570 146.338 145.768 .048 1.551927 1.562111 20 21 150.035 149.553 .040 .583758 .593942 156.122 155.711 .034 .584324 .594547 21 22 159.777 159.453 .027 .615336 .625559 166.224 165.971 .021 .615971 .626236 22 23 169.835 169.668 .014 .646239 .656504 176.639 176.541 .008 .646953 .657269 23 24 180.203 180.190 .001 .676545 .686861 j 187.365 187.422 .005 .677350 .687720 24 25 190.881 191.024 .012 1.706326 1.716696 198.401 198.615 .018 1.707231 1.717661 25 26 201.869 202.167 .025 .735643 .746073 209.760 210.138 .031 .736654 .747145 26 27 213.177 213.639 .039 .764549 .775040 221.432 221.960 .044 .765681 .776248 27 28 224.798 225.408 .051 .793100 .803667 233.416 234.107 .058 .794368 .805010 28 29 236.728 237.501 .064 .821350 .831992 245.717 246.561 .070 .822764 .833494 29 30 248.975 249.900 .077 1.849341 1.860071 258.324 259.321 .083 1.8509221.861747 30 31 261.526 262.604 .090 .877126 .887951 271.237 272.386 .096 .878887 .889816 31 32 274.381 275.611 .102 .904746 .915675 284.448 285.745 .108 .906709 .917753 32 33 287.535 288.911 .115 .932249 .943293 297.953 299.394 .120 .934435 .945605 33 34|30d.979 302.499 .127 .959679 .970849 311.742 313.325 .132 .962110 .973418 34 35 '.314. 708 316.369 .138 1.987078 1.998386 325.807 327.522 .143 1.989783 ».001244 35 36 328.710 330.502 .149 2.014495 2.025956 340.133 341.979 .154 2.017502 .029131 36 37 342.972 344.895 .160 .041978 .053607 .354.713 356.681 .164 .045316 .057129 37 38 357.488 359.533 .170 .069572 .081385 369.533 371.619 .174 .073276 .085290 38 39 372.243 374.404 .180 .097325 .109339 384.581 386.776 .183 .101434 .113669 39 40 387.223 389.494 .189 2.125294 2.137529 399.834 402.127 .191 2.129850 2.142332 40 41 402.409 404.777 .197 .153532 .166014 415.278 417.662 .199 .158584 .171332 41 42 417.784 420.243 .205 .182103 .194851 430.888 433.348 .205 .187701 .200749 42 43 433.326 435.860 .211 .211067 .224115 446.645 449.177 Ml .217272 .230642 43 14. TT = 10.5480. 15. 7r = 10.7660. Continued on Page *197. ♦158 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. n. X = 16. 17. .s " n. V. 6m. V. 12in. ^in.d. 6m. 12m. V. 6m. V. 12m. ^mA. 6m. AB. 12m. 7.922 4.849 .256 0.010072 0.019916 8.155 5.073 .257 0.010079 0.019930 1 12.871 9.898 .248 .321182 .331033 13.329 10.349 .248 .321196 .331054 1 2 18.021 15.148 .239 .507438 .517296 18.711 15.837 .239 .507462 .517327 2 3 23.376 20.609 .231 .642630 .652495 24.307 21.540 .231 .642663 .652538 3 4 28.944 26.285 .222 .749881 .759756 30.120 27.465 .221 .749926 .759811 4 5 34.731 32.181 .213 0.839494 0.849379 36.163 33.624 .212 0.839552 0.849448 5 6 40.743 38.310 .203 .916964 .926860 42.440 40.020 .202 .917036 .926943 6 7 46.990 44.675 .193 .985573 .995480 . 48.958 46.660 .192 .985661 .995581 7 8 53.476 51.282 .183 1.047437 1.057357 55.725 53.554 .181 1.047544 1.057477 8 9 60.210 58.143 .172 .104004 .113937 1 62.745 60.701 .170 .104131 .114081 9 10 67.196 65.255 .162 1.156306 1.166256 70.026 68.114 .159 1.156456 1.166423 10 11 74.441 72.632 .151 .205106 .215073 77.575 75.800 .148 .205283 .215268 11 12 81.954 80.281 .139 .250985 .260970 85.394 83.752 .137 .251192 .261200 12 13 89.736 88.195 .128 .294397 .304405 93.489 91.991 .125 .294639 .304668 13 14 97.792 96.394 .117 .335702 .345731 101.873 100.519 .113 .335977 .346032 14 15 106.134 104.880 .105 1.375185 1.385240 110.550 109.344 .101 1.375502 1.385583 15 16 114.769 113.663 .092 .413090 .423171 119.522 118.464 .088 .413451 .423563 16 17 123.699 122.739 .080 .449613 .459725 128.794 127.888 .075 .450028! .460173 17 18 132.925 132.117 .067 .484927 .495072 138.365 137.607 .063 .4854001 .495584 18 19 142.451 141.789 .055 .519175 .529359 148.245 147.646 .050 .519711 .529934 19 20 152.281 151.779 .042 1.552479 1.562702 158.443 158.004 .037 1.553083'l.563348 20 21 162.431 162.087 .029 .584944 .595209 168.957 168.675 .024 .585625 .595941 21 22 172.894 172.706 .016 .616668 .626984 179.786 179.660 .010 .617439 .627809: 22 23 183.669183.638 .003 .647741 .658111 190.928 190.960 .003 .648610 .659040 23 24 194.758 194.883 .010 .678238 .688668 202.394 202.593 .017 .679211 .689702 24 25 206.169 206.460 .024 1.708224 1.718715 214.178 214.527 .029 1.709317 1.719884 25 26 217.896 218.336 .037 .737768 .748335 226.277 226.791 .043 .738994 .749636 26 27 229.935 230.540 .050 .766929 .777571 238. 696 239.364 .056 .768299 .779029 27 28 242.294 243.052 .063 .795761 .806491 251.422 252.245 .069 .797295 .808120 28 29 254.959 255.872 .076 .824322 .835147 264.458 265.435 .081 .826034 .836963 29 30 267.932 268.997 .089 1.852659 1.863588 277.797 278.922 .094 1.854571 1.865615 30, 31 281.206 282.419 .101 .880825 .891869 291.429 292.701 .106 .882958 .894128 31' 32 294.772 296.131 .113 .908869 .920039 305.352 306.766 .118 .911245 .922553 32 33 308.627 310.127 .125 .936840 .948148 319.550 321.098 .129 .939484 .950945 33 34 322.756 324.390 .136 .964785 .976246 334.013 335.693 .140 .967730 .979359 34 35' 337.149 338.914 .147 1.992758 2.004387 348. 733 350.536 .150 1.996032 2.007845 35 36 351.797 353.685. .157; 2.020809 .032622 363.694 365.616 .160 2.024446 .036460 36 37: 366.686 368.692 .167 .048988 .061002 378.885 380.918 .169 .053025 .065260 37 38 381.804 383.920 .176 .077349 .089584 394.285 396.416 .178 .081832 .094314 38 39 397.128 399.342 .185 .105954 .118436 409.875 412.099 .185 .110930 .123678 39 40 412.644 414.950 .192 2.134865 2.147613 425.636 427.936 .192 2.140385 2.153433 40 41 428.327 430.709 .198 .164146 .177194 441.543 443.915 .198 .170267 .183637 41 42 444. 158^46. 612 .205 .193867 .207237 457.584 460.017 .203 .200646 .214371 42 43 460.121462.635 .209 .224097 .237822 473.731476.208 .206 .2316061 .245724 43 16. TT = 10.9953. 17. 77 = 11.2364. Continued on Page *196. ♦159 / TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OP 1,000. 4 PER CENT. '•si o o n. 18. 19. s§ V. 6m. V. 13m. -Jm.d. A(B/). 6m. AB. 12m. V. 6m. V. 13m. \mA. A(B/). 6m. AB. 12m. II n. 8.396 5.303 .358 0.010087 0.019945 8.651 5.545 .259 0.010096 0.019961 1 13.806 10.818 .349 .331213 .331078 14.304 11.308 .250 .321231 .331106' 1 2 19.439 16.551 .340 .507488 .517363 30.180 17.395 .240 .507516 .517401 2 3 35.374 33.506 .331 .643700 .652585 36.384 33.518 .331 .642741 .652637 3 4 31.346 38.696 .331 .749975 .759871 33.638 39.981 .221 .750030 .759937 4 5 37.655 35.135 .311 0.839615 0.849522 39.313 36.690 .210 0.839685 0.849605 5 6 44.307 41.798 .301 .917116 .927036 46.051 43.656 .300 .917202 .927135 6 7 51.007 48.737 .190 .985758 .995691 53.145 50.878 .189 .985865 .995815 7 8 58.064 55.911 .179 1.047661 1.057611 60.501 58.368 .178 1.047789 1.057756 8 9 65.381 63.361 .168 .104371 .114238 68.130 66.135 .166 .104425 .114410 9 10 73.969 71.087 .157 1.156633 1.166607 76.031 74.171 .155 1.156805 1.166813 10 11 80.839 79.080 .146 .205478 .215486 84.211 82.496 .143 .205695 .215734 11 12 88.965 87.361 .134 .351422 .3614511 92.683 91.113 .131 .251671 .261726 12 13 97.393 95.933 .133 .294902 .3049571 101.449 100.030 .118 .295191 .305272 13 14 106.113104.803 .109 .336281 .346363 110.516 109.245 .106 .336613 .346725 14 15 115.13o'll3.969 .097 1.375850 1.385962 119.884 118.768 .093 1.376333 1.386378 15 16 134.450133.441 .084 .413851 .4239961 139.557 128.589 .081 .414393 .434476 16 17 134.071sl33.310 .073 .450485 .460669 139.539 138.733 .067 .450988 .461211 17 18 144.000:143.300 .058 .485920 .496143 149.844 149.199 .054 .486489 .496754 18 19 154.350153.710 .045 .520298 .530663 160.468 159.981 .041 .520942 .531258 19 20 164.817164.435 .033 1.553745 1.564061 171.409 171.081 .037 1.554477 1.564847 20 21 175.701175.477 .019 .586377 .596747 183.668 182.500 .014 .587205 .597635 21 22 186.901 186.834 .006 .618288 .628718 194.355 194.254 .000 .619219 .629710 22 23 198.435 198.537 .009 .649563 .660054 306.163 206.313 .013 .650611 .661178 23 24 310.370 310.533 .031 .680282 .690849 318.387 218.705 .026 .681462 .692104 24 25 333.430 333.848 .035 1.710520 1.721162 330.935 231.409 .040 1.711842 1.722572 25 26 334.91l!335.485 .048 .740340 .751070 343. 795 244.426 .053 .741823 .752648, 26 27 347.704 348.433 .061 .769808 .780633 356.968 257.754 .065 .771467 .783396|27 28 360.806 361.690 .074 .798981 .809910 370.446 371.383 .078 .800838 .811883 28 29 374.313 375.345 .086 .827919 .838963 384.222 385^05 .090 .829993 .841163 29 30 387.915 389.094 .098 1.856675 1.867845 298.288 399.516 .102 1.858989 1.870397 30 31 301.907 303.331 .110 .885303 .896611 312.636 313.999 .114 .887884 .899345 31 32 316.179 317.636 .131 .913859 .925320 337.250 338.746 .125 .916736 .938365 32 33 330.715i333.305 .133 .942396 .9540251 342. 123 343.744 .135 .945599 .957413 33 34 345.510 347.334 .143 ■ .9709691 .982782^ 357.241 358.983 .145 .974533 .986547 34 35 360.547 363.381 .153 1.999634'3.011648' 372.591 374.444 .154 2.003593 3.015838 35 36 375.816 377.761 .163 3.038445 .040680! 388.152 390. 104 .163 .032847 .045329 36 37 391.395 393.338 .170: .057468 .069950 403.906 405.951 .170 .062359 .075107 37 38 406.964|409.101 .178 .086766 .099514 419.830 431.953 .177 .092197 .105245 38 39 433.805 435.018 .184 .116406 .129454 435.904 438.100 .1«3 .122435 .135805 39 40 438.794 441.080 .191 2.146461 3.159831 452.113 454.370 .188 2.153143 2.166868 40 41 454.916 457.363 .196 .176999 .190734 468.428 470.730 .192 .184407 .198525 41 42 471.145 473.537 .199 .208106 .333334 484.831 487.155 .195 .316320 .230869 42 43 487.451 489.874 .302 .239874 .354433 501.270 1 1 503.639 .197 .248971 .263989 43 18. 77 = 11.4897. 19. TT = 11.7560. Continued on Page *195. '160 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. &e a:=20. 21. Is Ms V. V. |^m.d. A(B/). AB. V. V. 1 ^ ^(B/). AB. n. 6m. 12m. 6m. 12m. 6m. 12m. ^m.d. 6m. 12m. n. 8.915 5.795 .260 0.010105 0.019980 9.193 6.056 .261 0.010115 0.020000 1 14.833 11.816 .251 .321250 .331135 15.368 12.350 .252 .321271 .331167 1 2 20.963 18.073 .241 .507547 .517443 21.783 18.887 .241 .507581 .517488 2 3 27.340 24.572 .231 .642785 .652692 28.445 25.672 .231 .642834 .652754 3 4 33.963 31.318 .220 .750089 .760009 35.360 32.718 .220 .750153 .760086 4 5 40.839 38.323 .210 0.839761 0.849694 42.535 40.023 .209 0.839844 0.849794 5 6 47.972 45.586 .199 .917298 .927248 49.976 47.599 .198 .917402 .927369 6 7 55.370 53.118 .188 .985981 .995948 57.692 55.454 .186 .986110 .996095 7 8 63.041 60.928 .176 1.047932 1.057917 65.682 63.581 .175 1.048088 1.058096 8 9 70.986 69.008 .165 .104596 .114604 73.957 72.002 .163 .104784 .114813 9i 1 10 79.212 77.380 .153 1.157008 1.167037 82.524 80.717 .151 1.157229 1.167284 10 11 87.731 86.045 .141 .205930 .215985 91.392 89.737 .138 .206189 .216270 11 12 96.546 95.012 .128 .251946 .262027 100.563 99.058 .125 .252246 .262358 12 13 105.663 104.278 .115 .295508 .305620 110.038 108.689 .112 .295859 .306004 13 14 115.085 113.855 .102 .336981 .347126 119.821 118.622 .100 .337387 .347571 14 15 124.810 123.730 .090 1.376658 1.386842 129.917 128.882 .086 1.377124 1.387347 15 16 134.848 133.930 .077 .414777 .425000 140.340 139.468 .073 .415307 .425572 16 17 145.211 144.455 .063 .451538 .461803 151.086 150.374 .059 .452142 .462458 17 18 155.894 155.297 .050 .487113 .497429 162.152 161.601 .046 .487803 .498173 18 19 166.896 166.459 .036 .521653 .532023 173.540 173.149 .033 .522437 .532867 19 20 178.218 177.941 .023 1.555284 1.565714 185.259 185.039 .018 1.556169 1.566660 20 21 189.869 189.761 .009 .588113 .598604 197.303 197.236 .006 .589113 .599680 21 22 201.842 201.888 .004 .620244 .630811 209.667 209.769 .009 .621372 .632014 22 23 214.136 214.348 .018 .651766 .662408 222.359 222.619 .022 .653034 .663764 23 24 226.754 227.124 .031 .682757 .693487 235.372 235.794 .035 .684184 .695009 24 25 239.687 240.213 .044 1.713298 1.724123 248. 694 249.264 .047 1.714898 1.725827 25 26 252.932 253.615 .057 .743453 .754382 262.321 263.048 .061 .745248 .756292 26 27 266.485 267.319 .070 .773293 .784337 276.254 277.130 .073 .775303 .786473 27 28 280.337 281.319 .082 .802880 .814050 290.482 291.505 .085 .805126 .816434 28 29 294.482 295.610 .094 .832273 .843581 304. 994 306.152 .097 .834784 .846245 29 30 308.910 310.173 .105 1.861535 1.872996 319.775 321.068 .108 1.864338 1.875967 30 31 323.606 325.003 .116 .890725 .902354 334. 818 336.239 .118 .893850 .905663 31 32 338.562 340.085 .127 .919901 .931714 350.110 351.651 .128 .923383 .935397 32 33 353.764 355.408 .137 .949123 .961137 365.635 367.289 .138 .952997 .965232 33 34 369.200 370.956 .146 .978450 .990685 381.374 383.128 .146 .982763 .995245 34 35 384.848 386.703 .155 2.007951 2.020433 397.307 399.157 .154 2.012748 2.025496 35 36 400.689 402.639 .162 .037692 .050440 413.415 415.342 .161 .043024 .056072 36 37 416.702 418.730 .169 .067741 .080789 429.672 431.673 .167 .073668 .087038 37 38 432.867 434.967 .175 .098175 .111545 446.067 448.130 .172 .104750 .118475 38 39 449.165 451.328 .180 .129065 .142790 462.568 464.677 .176 .136361 .150479 39 40 465.572 467.779 .184 2.160496 2.174614 479.148 481.289 .178 2.168593 2.183142 40 41 482.055 484.295 .187 .192564 .207113 495.786 497.952 .180 .201539 .216557 41 42 498.596 500.861 .189 .225357 .240375 512.452 514.623 .181 .235303 .250842 42: 43 515.166 517.'436 .189 .258982 .274521 529.117 531.281 .180 .270003 .286112 43 20. n = 12.0361. 21. 77 = 12.3304. Continued on Page *194. TABLE LXIII. *161 GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. &l 22. 23. si Pk(M V. V. -i.m.d. ^(B/). AB. V. V. im.d ^(B/). AB. |{S n. 6m. 12m. ^ 6m. 12m. 6m. 12m. g-m.u.. 6m. 12m. 9.487 6.333 .263 0.010126 0.020022 9.792 6.619 .264 0.010138 0.020045 1 15.941 12.909 .253 .321294 .331201 16.537 13.489 .254 .321320 .331240 1 2 22.642 19.736 .242 .507619 .517539 23.539 20.623 .243 .507659 .517592 2 3 29.601 26.825 .231 .642886 .652819 30.804 28.018 .232 .642946 .652896 3 4 36.819 34.174 .220 .750225 760175 38.336 35.689 .221 .750303 .760270 4 5 44.305 41.796 .209 0.839936 0.849903 46.149 43.643 .209 0.840037 0.850022 5 6 52.068 49.699 .197 .917517 .927502 54.241 51.872 .197 .917645 .927653 6 7 60.107 57.876 .186 .986253 .996261 62.618 60.398 .185 .986411 .996440 7 8 68.432 66.348 .174 1.048262 1.058291 71.293 69.222 .17311.048451 1.058506 8 9 77.053 75.117 .161 .104989 .115044 80.271 78.354 .160 .105215 .115296 9 10 85.974 84.191 .149 1.157471 1.167552 89.556 87.792 .147 1.157737^1.167849 10 11 95.199 93.568 .136 .206473 .216585 99.151 97.544 .134 .206786 .216931 11 12 104.734 103.259 .123 .252579 .262724 109.055 107.601 .121 .252947 .263131 12 13 114.575 113.252 .110 .296246 .306430 119.278 117.989 .107 .296672 .306895 13 14 124.733 123.574 .097 .337834 .348057 129.831 128.707 .094 .338322 .348587 14 15 135.220 134.225 .083 1.377634 1.387899 140.711 139.749 .080 1.378193 1.388509 15 16 146.031 145.197 .070 .415889 .426205 151.916 151.117 .067 .416533 .426903' 16 17 157.164 156.492 .056 .452809 .463179 163.447 162.810 .053 .453544 .4639741 17 18 168.622 168.112 .042 .488563 .498993 175.312 174.848 .039 .489397 .499888 18 19 180.413 180.073 .028 .523297 .533788 187.505 187.197 .026 .524243 .534810 19 20 192.529 192.345 .015 1.557142 1.567709 200.026 199.888 .012 1.558215 1.568857 20 21 204.969 204.954 .001 .590214 .600856 212.876 212.898 .002 .591423 .602153 21 22 217.738 217.882 .012 .622611 .633341 226.046 226.228 .015 .6239771 .634802 22 23 230.825 231.128 .025 .654431 .665256 239.535 239.877 .028 .6559661 .666895 23 24 244.229 244.690 .038 .685753 .696682 253.338 253.833 .041 .687479; .698523 24 25 257.944 258.558 .051 1.716659 1.727703 267.445 268.091 .054 1.718697 1.729767 25 26 271.962 272.726 .064 .747223 .758393 281.851 282.645 .066 .749394 .760702 26 27 286.277 287.188 .076 .777512 .788820 296.544 297.476 .078 .779944 .791405 27 28 .300.877 301.925 .087 .807598 .819059 311.510 312.579 .089 .810319 .821948 28 29 315.748 316.932 .099 .837546 .849175 326.742 327.938 .100 .840586 .852399 29 30 330.883 332.194 .109 1.867421 1.879234 342.223 343.543 .110 1.870813 1.882827 30 31 346.268 347.701 .119 .897288 .909302 357.944 359.378 .120 .901068 .913303 31 32 361.888 363.435 .129 .927211 .939446 373.879 375.414 .128 .931424 .943906 32 33 377.722 379.370 .137 .957261 .969743 390.012 391.644 .136 .961956 .974704 33 34 393.753 395.496 .145 .987510 2.000258 406.320 408.031 .143 .99273512.005783 34 35 409.958 411.780 .152 2.018029 2.031077 422. 782 424.567 .149 2.023842 2.037212 35 36 426.316 428.211 .158 .048896 .062266 439.381 441.229 ..154 .055353 .069078 36 37 442.809 444. 767 .163 .080185 .093910 456.089 457.983 .158 .087359 .101477 37 38 459.411 461.415 .167 .111987 .126105 472.876 474.803 .161 .119955 .134504 38 39 476.092 478.129 .170 .144395 .158944 489.722 491.674 .163 .153236 .168254 39 40 492.831 494.893 .172 2.177503 2.192521 506.597 508. .554 .163 2.187309 2.202848 40 41 509.599 511.666 .172 .211417 .226956 523.469 525.419 .162 .222292 .238401 41 42 526.366 528.425 .172 .246253 .262362 540.309 542.232 .160 .258311 .275049 42 43 543.098 545.131 .169 .282139 .298877 557.069 558.941 .156 .295499 .312921 43 22. TT = 12.6399 23. 7r = 12.9656 Continued on Page *193. *163 TABLE LXIII, GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. s n. x = 24. 25. n. V. 6m. V. 12m. |m.d A(B/). 6m. AB. 12m. V. 6m. V. 12m. |m.d. 6m. AB. 12m. 1 10.112 6,916 .366 0.010153 0.020073 10.450 7.231 .368 0.010167 0.020100 1 17.161 14.097 .255 .331348 .331381 17.814 14.738 .357 .321380 .331330 1 2 24.473 21.542 .244 .507704 .517654 25.450 22.504! .246 .507754 .517721 2 3 32.057 29.264 .233 .643009 .652976 33.370 30.567 .334 .643080 .653065 3 4 39.922 37.271 .221 .750390 .760375 41.572 38.908 .222 .750486 .760494 4 5 48.067 45.555 .209 0.840150 0.850158 50.064 47.550 .209 0.840275 0.850304 5 6 56.500 54.137 .197 .917787 .927816 58.857 56.495 .197 .917941 .927996 6 7 65.233 63.021 .184 .986583 .996638 67.958 65.752 .184 .986772 .996853 7 8 74.271 72.214 .171 1.04865911.058740 77.370 75.319 .171 1.048886 1.058998 8 9 83.618 81.714 .159 .105462 .115574 87.096 85.205 .158 .105735 .115880 9 10 93.277 91.531 .146 1.158031 1.168176 97.137 95.399 .145 1.158357 1.168541 10 11 103.247 101.655 .133 .207134 .217318 107.498 105.929 .131 .207515 .217738 11 12 113.537 112.112 .119 .253351 .263574 118.1961116. 794 .117 .353793 .264058 12 13 124.161 133.903 .105 .297138 .307403 129.225 127.987 .103 .397648 .307964 13 14 135.114 134.018 .091 .338857 .349173 140.583 139.510 .089 .339450 .349820 14 15 146.394 145.461 .078 1.378811 1.389181 152.271 151.363 .076 1.379493 1.389923 15 16 158.000 157.333 .064 .417242 .427672 164.299 163.565! .061 .418019 .438510 16 17 169.945169.350 .050 .454350 .464841 176.659 176.084! .048 .455238 .465805 17 18 182.220181.783 .036 .490314 .500881 189.350 188.947 .034 .491325 .501967 18 19 194.824 194.557 .022 .535385 .535927 302.376 202.136 .020 .536430 .537160 19 20 207.759 307.654 .009 1.559392 1.570133 215.726 215.648 .007 1.560689 1.571514 20 21 221.018 321.073 .005 .593756 .603581 339.400 229.483 .007 .594319 .605148 21 22 334.596 234.812 .018 .635477 .636406 343.391 243.630 .030 .627139 .638173 22 23 348.4911248.861 .031 .657656 .668700 357.691 258.083 .033 .659516 .670686 23 24 363.693 363.315 .044 .689379 .700549 372.295 272.837 .045 .691469 .703777 24 25 377.194 377.866 .056 1.730729 1.733037 287.188 287.870 .057 1.733075 1.734536 25 26 291.985 392.795 .068 .751784 .763245 302.359 303.179 .068 .754415 .766044 26 27 307.051307.999 .079 .783631 .794350 317.798 318.749 .079 .785567 .797380 27 28 333.384 323.461 .090 .813313 .835126 333.493 334.567 .089 .816606 .838630 28 29 337.969 339.170 .100 .843930 .855944 349.427 350.618 .099 .847607 .859842 29 30 353.794 355.109 .110 1.874543 1.886778 365.580 366.874 .108 1.878649 1.891131 30 31 369.835:371.253 .118 .905339 .917711 381.934 383.325 .116 .909810 .922558 31 32 386.076,387.591 .136 .936064 .948812 398.466 399.937 .133 .941170 .954318 32 33 403.4931404.087 .133 .967123 .980171 415.152 416.699 .129 .973813 .986183 33 34 419.064420.733 .139 .998488 3.011858 431.979 433.589 .134 3.004816 3.018541 34 35 435.773 437.506 .144 3.030237 3.043962 448.915 450.573 .138 3.037274 2.051392 35! 36 452.593 454.372 .148 -.062461 .076579 465.931 467.633 .141 .070288 .084837 36 37 469.492 471.304 .151 .095258 .1098071 483.008 484.734 .143 .103951 .118969 37 38 486.450 488.287 .153 .128724 .1437421 500.113 501.834 .143 .1383761 .153915 38 39 503.437 505.279 .153 .162967 .178506 517.317 518.930 .143 .173681 .189790 39 40 520.422 532.357 .153 2.198106 3.214315 534.286 535.973 .141 2.209996 3.336734 40 41 537.373539.183 .151 .234268 .251006 551.376 553.910 .136 .247454 .364876 41 42 554.3461556.003 .146 .271587 .289009! 568.188 569.797 .134 .286206 .304387 42 43 571.042|572.773 .144 .310212 .328393' 584.997 586.538 .138 .326408 .345409 43 24. TT = 13.3082. 26. -n = 13.6687. Continued on Page *192. ^ ♦lea TABLE LXIII. GENERAL VALUATION OP ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. ■u 26. 27. Is V. V. |-m.d. A(B/). AB. V. V. ^m.d. A(B/-). AB. n. 6in. 12m. 6m. 12m. 6m. 12m. 6m. 12m. n. 10.800 7.551 .271 0.010184 0.020134 11.170 7.892 .273 0.010199 0.020166 1 18.492 15.384 .259 .321413 .331380 19.208 16.075 .261 .321450 .331435' 1 2 26.469 23.505 .247 .507809 .517794 i 27.532 24.541 .249 .507870 .517878' 2 3 34.731 31.907 .235 .643160 .653168 36.151 33.313 .237 .643247 .653276 3 4 43.284 40.613 .223 .750593 .760622 1 45.076 42.391 .224 .750708 .760763 4 5 53.143 49.623 .210 0.840410 0.850465 : 54.313 51.787 .210 0.840558 0.850639 5 6 61.309 58.947 .197 .918111 .928192 63.865 61.496 .197 .918295 .928407 6| 7 70.790 68.583 .184 .986978 .997090 73.737 71.530 .184 .987207 .997352 7 8 80.586 78.541 .170 1.049138 1.059283 1 83.928 81.877 .171 1.049416 1.059600 8 9 90.700 88.810 .158 .106038 .116222 94.444 92.564 .157 .106371 .116594 9 10 101.138 99.416 .143 1.158714 1.168937 105.302 103.591 .143 1.159104 1.169369Jl0 11 111.913 110.361 .129 .207932 .218197 116.495 114.952 .129 .208388 .218704J 11 12 123.022 121.635 .116 .254277 .264593 128.024 126.647 .115 .254813 .265183 12 13 134.463 133.242 .102 .298213 .308583 139.886 138.677 .101 .298835 .309265 13 14 146.236 145.181 .088* .340102 .350532 |152.093 151.062 .086 .340818 .351309 14 15 158.352 157.473 .073 1.380240 1.390731 'l64.639 163.768 .073 1.381062 1.391629 15 16 170.802 170.082 .060 .418875 .429442 177.520 176.823 .058 .419817 .430459 16 17 183.585 183.040 .045 .456215 .466857 190.740 190.209 .044 .457288 .468018 17 18 196.707 196.324 .032 .492434 .503164 204.290 203.923 .031 .493657 .504482 18 19 210.154 209.935 .018 .527690 .538515 |218.168 217.965 .017 .529076 .540005 19 20 223.927 223.871 .005 1.562115 1.573044 1232.368 232.324 .004 1.563685 1.574729 20 21 238.021 238.121 .008 .595831 .606875 i246.883 246.993 .009 .597605 .608775 21 22 252.424 252.679 .021 .628947 .640117 i261.704 261.967 .022 .630946 .642254 22 23 267.134 267.540 .034 .661562 .672870 276.820 277.225 .034 .663813 .675274 23 24 282.136 282.683 .046 .693769 .705230 292.218 292.763 .045 .696300 .707929 24 25 297.418 298.104 .057 1.725657 1.737286 307.888 308.566 .057 1.728499 1.740312 25 26 312.970 313.787 .068 .757310 .769123 323.817 324.620 .067 .760494 .772508: 26 27 328.778 329.720 .079 .788807 .800821 339.990 340.911 .077 .792370 .804605 27 28 344.829 345.888 .088 .820227 .832462 356.384 357.410 .085 .824213 .836695 28 29 361.099 362.262 .097 .851654 .864136 372.983 374.107 .094 .856110 .868858 29 30 377.573 378.834 .105 1.883169 1.895917 389.761 390.967 .100 1.888145 1.901193 30 31 394.224 395.566 .112 .914853 .927901 406.697 407.979 .107 .920406 .933776 31 32 411.033 412.450 ^.118 .946793 .960163 423.774 425.122 .112 .952978 .966703 32 33 427.981 429.463 .123 .979071 .992796 440.965 442.359 .116 .985959 2.000077 33 34 445.041 446.570 .127 2.011781 2.025899 458.235 459.664 .119 2.019452 .034001 34 35 462.181 463.744 .130 2.045025 2.059574 475.567 477.021 .121 2.053555 2.068573 35 36 479.382 480.970 .132 .078901 .093919 492.928 494.387 .122 .088383 .103922 36 i 37 496.612 498.205 .133 .113520 .129059 510.287 511.739 .121 .124057 .140166 37 38 513.840 515.426 .132 .149004 .165113 527.612 529.037 .119 .160709 .177447 38 39 531.034 532.593 .130 .185481 .202219 544.856 546.227 .114 .198475 .215897 39 40 548.147 549.653 .126 2.223088 2.240510 562.020 563.366 .112 2.237507 2.255688 401 41 565.183 566.664 .123 .261976 .280157 579.081 580.347 .106 .277963 .296964 41 42 582.115 583.516 .117 .302300 .321301 595.966 597.137 .098 .320028 .339946 42 43 598.872 600.180 .109 .344246 .364164 612.658 613.731 .089 .363895 .384806 43 26. 77 = 14.0485. 27. TT = 14.4479. Continued on Page *1W nu TABLE LXIII, GENEKAL VALUATION OF ANNUAL PREMIUM POLiaES OF 1,000. 4 PER CENT. li n. a; = 28. 29. ■si n. V. 6m. V. 12m. |m.d. MB/). 6m. AB. 12m. V. 6m. V. 12m. im.d. MB/). 6m. AB. 12m. 11.559 8.248 .276 0.010220 0.020205 11.958 8.604 .279 0.010244 0.020252 1 19.949 16.782 .264 .3214941 .331502 20.718 17.519 .267 .321541 .331570 1 2 28.637 25.623 .251 . 507938; .517967 29.788 26.746 .254 .508011 .518066 2 3 37.633 34.774 .238 .643342 .653397 39.177 36.295 .240 .643446 .653527 3 4 46.944 44.244 .225. .750835 .760916 48.885 46.163 .227 .750973 .761085 4 5 56.572 54.031 .212 0.840720 0.850832 58.918 56.361 .213 0.840900 0.851045 5 6 66.522 64.144 .198 .918500 .928645 69.275 66.877 .200 .918728 .928912 6 7 76.793 74.573 .185 .987462 .997646 79.964 77.739 .185 .987740 .997963 7 8 87.394 85.346 .171 1.049723 1.059946 90.998 88.946 .171 1.050056 1.060321 8 9 98.338 96.461 .156 .106733 .116998 102.375 100.492 .157 .107131 .117447 9 10 109.620 107.911 .142 1.159532 1.169848 114.091 112.378 .143 1.160006 1.170376 10 11 121.240 119.699 .128 .208894 ..219264 126.148 124.605 .129 .209451; .219881 11 12 133.197 131.826 .114 .255403 .265833 138.554 137.192 .113 .256051 .266542 12 13 145.502 144.309 .099 .299517 .310008 151.305 150.106 .100 .300268 .310835 13 14 158.147 157.116 .086 .341605 .352172 164.397 163.375 .085 .342472 .353114 14 15 171.130 170.275 .071 1.381968 1.392610 177.833 176.979 .071 1.382962 1.393692 15 16 184.455 183.767 .057 .420851 .431581 191.604 190.917 .057 .421991 .432816 16 17 198.114 197.591 .044 .458471; .469296 205.709 205.189 .043 .459769 .470698 17 18 212.N102 211.744 .030 .495000 .505929 220.141 219.782 .030 .496479 .507523; 18 19 226; 415 226.217 .017 .530601 .541645 :234.893 234.691 .017 .532280 .543450119 20 241.044 241.003 .003 1.565412 1.576582 249.956 249.909 .004 1.567312 1.578620 20 21 255.984 256.096 .009 .599556 .610864 265.319 265.417 .008 .601703 .613164 21 22 271.221 271.476 .021 .633147 .644608 280.969 281.209 .020 .635569 .647198 22 23 286.741 287.137 .033 .666291 .677920 :296.895i297.269 .031 .669018 .680831 23 24 1 302.535 303.065 .044 .699086 .710899 313.084 313.586 .042 .702150 .714164 24 25 318.591 319.248 .055 1.731624 1.743638 329.520 330.143 .052 1.735062 1.747297 25 26 334.893 335.668 .065 .763995 .776230 1346.184 346.912 .061 .767850 .780332 26 27 351.417 352.298 .073 .796292 .808774 363.053;363.882 .069 .800610 .813358 27 28 368.148 369.128 .082 .828602 .841350 380.105 381.017 .076 .833434 .846482 28 29 385.059 386.122 .089 .861016 .874064 397.318 398.307 .082 .866418 .879788 29 30 402.131 403.270 .095 1.893624 1.906994 414.674 415.729 .088 1.899651 1.913376 30 31 419.344 420.549 .100 .926514 .940239 432.145 433.248 .092 .933238 .947356 31 32 436.670 437.923 .104 .959786 .973904 449.698 450.836 .095 .967285 .981834 32 33 454.079 455.366 .107 .993546 2.008095 467.312 468.477 .097 2.001896 2.016914 33 34 471.548 472.861 .109 2.027894 .042912| 484.958 486.127 .097 .037188 .052727 34 35 489.048 490.365 .110 2.062946 2.078485; 502.601 503.762 .097 2.073287 2.089396 35 36 506.544 507.855 .109 .098824 .114933 520.208 521.342 .095 .110325 .127063 36 37 524.007 525.290 .107 .135663 .152401; 537.733 538.813 .090 .148443 .165865 37 38 541.388 542.617 .102 .173599 .191021 555.179 556.233 .088 .187797 .205978 38 39 558.690 559.893 .100 .212786 .230967 572.518 573.491 .081 .228543 .247544 39 40 575.885 577.009 .094 2.253382 2.272383 589.679 590.555 .073 2.270870 2.290788 40 41 592.906 593.933 .086 .295074 .315492 606.644 607.420 .065 .314975 .335886 41 42 609.730 610.658 .077 .339556 .360467 623.386 624.041 .055 .361067 .383074 42 43 626.334 627.142 .067 .385539 .407546 639.875 640.396 .043 .409383 .432587 43 28. n = 14.8689. 29. TT = 15.3125. Continued on Page *190, *165 TABLE LXIII. GENERAL VALUATION OP ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. n. 30. V. 6in. Y. 12m. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 170.865 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 12.386 21.535 31.005 40.798 5J3.917 61.364 72.146 83.276 94.751 106.569 118.730 131.245 144.106 157.311 8.992 18.300 27.931 37.885 48.171 58.778 69.734 81.039 92.685 104.674 117.007 129.704 142.729 156.114 169.836 183.895 198.291 213.010 228.048 243.399 259.041 274.970 291.170 307.629 324.329 340.676 341.243 357.691358.361 374.892 375.644 392.254 393.085 409.761410.658 ^m.d 184. 755 198.982 213.540 228.419 243.613 259.109 274.895 290.960 307.289 323.868 427. 445. 462. 480. 498. 516. 533. 551. 568. 586. 383 428.329 089 446.070 857463.864 655 481.667 450!499.455 211517.188 889 534.810 485J552.382 975,569.789 285587.002 603.397|604.013 620.285'620.778 636.916,637.275 653.269 653.483 .283 .270 .256 .243 .229 6ni. AB. 12m. 0.010266 0.020295 .321589 .331644 .508090 .5181711 .643557 .6536691 .751126 .761271 .216 0.841100 ,201 .918977 ,186 .988043 ,172 1 1.050422 ,158 .107571 .144 .128 .115 .100 .086 .072 .058 .044 .031 .018 .006 .006 .017 .028 .038 .047 .056 .063 .069 .075 .079 .082 .084 .084 .084 .081 .077 .075 .068 .060 .051 .041 .030 .018 1.160528 .210061 .256763 .301095 .343424 0.851284 .929200 .998308 1.060738 .117941 1.170958 .220552 .267330 .311737 .354154 1.384058 1, .423244 , .461199 , .498107 , .534126 . 1.5694031. .604066 . .638235 . .672018 . .705520 . 1.738846 .772095 .805364 .838755 .872360 394883 434173 472243 509277 545434 580864 615695 650048 684032 717755 1.751328 .784843 .818412 .852125 .886085 1.9062861.920404 .940644 .955193 .975538 .990556 2.011089 .047424 2.084677 .122991 .162522 .203430 .245902 2.290138 .336347 .384768 .435645 2.026628 .063533 2.101415 .140413 .180703 .222431 .265820 2.3110491 .358354 .407972! .460158' 31. V. 6m. 12.832 22.387 32.268 42.480 53.022 63.901 75.132 86.712 98.637 110.908 123.536 136.514 149.839 163.515 177.532 191.888 206.578 221.592 236.924 252.561 268.490 284. 700 301.178 317.907 334.868 352.037 369.394 386.914 404. 580 422.362 440. 228 458.157 476.116 494.074 511.995 529.834 547.590 565.238 582.706 599.972 V. 12m. 9.392 19.111 29.155 39.534 50.238 61.293 72.701 84.452 96.550 108.995 121.807 134.950 148.456 162.303 176.490 191.016 205.869 221.044 236.533 252.318 268.391 284.738 301.346 318.198 335.266 352.538 369.979 387.578 405.311 423.142 441.043 458.999 476.963 494.913 512.807 530.589 548.320 565.886 583.254 600.420 4m.d. 30. 71 = 15.7791. 617.014 617.337 633.796|633.984 650.297 650.338 666.477 666.345 6m. .287 0.010293 .273 .321644 .259 .508176 .246 .643685 .2321 .751298 .21710.841319 .2031 .919249 .188' .988376 ,174 .159 1.050828 .108057 .144 1.161100 ,130 .210734 .115 .101 .087 .073 .059 .046 .033 .020 .008 .003 .014 .024 .033 .042 .049 .055 .061 .065 .068 .070 .071 .070 .068 .063 .061 .054 .046 .037 .027 .016 .003 .011 .257548 .302002 .344474 1.385262 .424624 .462774 .499897 .536159 1.571705 .606667 .641167 .675317 .709230 1.743013 .776768 .810600 .844609 , .878902 1.913595 .948795 .984625 2.021214 .058699 2.097223 .136945 .178025 .220653 .265029 AB. 12m. Wei ' O « , 0.020348 .331725 1', .518288; 2 1 .6538301 31 .761482 4i 0.851542 .929514 .998692 1.061198 .118487 1.171591 lOi .221301 11 .268190 .312732 .355299 12i 13 14 15 16 1.396191 .435668 .473944' 17 j .511205! 18 .547620119 1.583334 20 .618480^21 .653181 22 .687552123 .721712 24 1.755761 .789816 .823970 .858334 .893020 1.928144 .963813 2.000164 037323 075437 2.114645 .155126 .197026 .240571 .285940 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 ■>.311364 2.338371 40 .3598971 .3831011 41 .4108741 .435387 42 .4645941 .490558,43 31. 7T = 16.2712. Continued on Page *189. •=166 TABLE LXIIl. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. li x = Z2. 33. m o o 0^>* n. V. 6in. V. 12in. 9.811 ^m.d. A (BO. 6m. AB. 12m. V. 6m. V. 12m. ^m.d. A(B/). 6m. AB. 12m. n. 13.301 .291 0.010321 0.020402 13.789 10.240 .296 0.010352 0.020464 1 23.275 19.950' .277 .321703 .331815 24.200 20.821 .282 .321772 .331917 1 2 33.584 30.428| .263 .508275 .518420 34.946 31.733 .268 .508385 .518569 2 3 44.226 41.2331 .249 .643826 .654010 46.037 43.004 .253 .643981 .654204 3 4 55.208 52.394 .235 .751485 .761708 57.488 54.634 .238 .7516881 .761953 4 5 66.547 63.909 .220 0.841558 0.851823 69.294 66.615 .223 0.841819(0.852135 5 6 78.235 75.772 .205 .919547 .929863 81.450 78.948 .209 .919877 .930247 6 7 90.274 87.985 .191 .988745 .999115 93.961 91.636 .194 .989152 .999582 7 8 102.661 100.548 .176 1.051275 1.061705 106.836 104.697 .178 1.051765 1.062256 8 9 115.410 113.481 .161 .108589 .119080 120.066 118.097 .164 .109174 .119741 9 10 128.510 126.749 .147 1.161730 1.172297 133.650 131.865 .149 1.162424 1.173066 10 11 141.961 140.383 .131 .211474 .222116 147.592 145.982 .134 .212287 .223017 11 12 155.767 154.361 .117 .258409 .269139 161.883 160.445 .120 .259359 .270184 12 13 169.916 168.682 .103 .303003 .313828 176.518 175.254 .105 .304103 .315032 13 14 184.409 183.346 .089 .345627 .356556 191.495 190.397 .091 .346896 .357940 14 15 199.238 198.340 .075 1.386588 1.397632 206.801 205.867 .078 1.388047:1.399217 15 16 214.395 213.659 .061 .426142 .437312 222.432 221.659 .064 .427811 .439119 16 17 229.872 229.295 .048 .464505 .4758131 238.374 237.751 .052 .466411 .477872 17 18 245.657 245.229 .036 .501868 .513329 254.613 254.137 .040 .504037 .515666 18 19 261.737 261.455 .024 .538396 .5500251 271.139 270.803 .028 .540858 .552671 19 20 278.101 277.957 .012 1.574238 1.586051 287.937 287.734 .017 1.5770251.589039 20 21 294.734 294.722 .001 .609529 .621543 304.994 304.915 .007 .612675 .624910 21 22 311.623 311.734 .009 .644392 .6566271 322.284 322.315 .003 .647942 .660424 22 23 328.744 328.964 .018 .678949 .691431 339.788 339.924 .011 .682949 .695697 23 24 346.077 346.400 .027 .713315 .726063; 357.483 357.704 .018 .717813 .730861 24 25 363.598 364.006 .034 1.747600 1.760648 375.344 375.646 .025 1.752653 1.766023 25 26 381.284 381.771 .041 .781914 .795284 393.354 393.724 .031 .787576 .801301 26 27 399.117 399.673 .046 .816360 .830085 411.483 411.903 .035 .822702 .836820 27 28 417.068 417.673 .050 .851053 .865171 429.697 430.153 .038 .858152 .872701 28 29 435.103 435.744 .053 .886109 .900658 447.975 448.459 .040 .894042 .909060 29 30 453.202 453.870 .056 1.921640 1.936658 466.285 466.773 .041 1.930498 1.946037 30 31 471.333 472.0051 .056 .957771 .973310 484.592 485.073 .040 .967657 .983766 31 32 489.460 490.125! .055 .994634 2.010743 502.863 503.315 .038 2.005660 2.022398 32 33 507.551 508.188 .053 2.032368 .049106 521.048 521.444 .033 .044655 .062077 33 34 525.559 526.139 .048 .071118 .088540 539.151 539.520 .031 .084802 .102983 34 35 543.483 544.038 .046 2.111045 2.129226 557.143 557.428 .024 2.126267 2.145268 35 36 561.299 561.770 .039 .152310 .171311 574.951 575.135 .015 .169242 .189160 36 37 578.932 579.303 .031 .195106 .215024 592.554 592.635; .007 .213929 .234840 37 38 596.362 596,631 .022 .239632 .260543 609.927 609.882 .004 .260544 .282551 38 39 613.565 613.710 .012 .286102 .308109 627.037 626.853 .015 .309326 .332530 39 40 630.507 630.514 .001 2.334755 2.357959 643.858 643.526 .028 2.360524 2.385037 40 1 41 647.164 647.023 .012 .385839 .410352 660.355 659.845 .043 .414439 .440403 41 42 663.497 663.182 .026 .439653 .465617 676.495 a75.808| .057 .471378 .498915 42 43 679.481 678.989 .041 .496504 .524041 692.273 1 691.399 .073 .531660 .560916 43 32. 7r = 16.7902. 33. 7r = 17.3376. Continued on Page *188. TABLE LXIII. *167 GENERAL VALUATION OP ANNUAL PREMIUM POLICIES OF 1,000, 4 PER CENT. o » n. 34. 35. n. V. 6m. V. 12m. |-m.d. 6m. AB. 12m. V. 6m. V. 12ra. \mA. 6m. AB. 12m. 14.303 10.691 .301 0.010390 0.020535 14.834 11.144 .308 0.010430 0.020614 1 25.161 21.716 .287 .321849 .332033 ! 26.162 22.655 .292 .321929 .332152 1 2 36.367 33.103 .272 .508504 .518727 I 37.855 34.532 .277 .508630 .518895 2 3 47.936 44.853 .257 .644147 .654412 i 49.912 46.767 .262 .644328 .654644 3 4 59.863 56.958 .242 .751910 .762226 62.327 59.363 .247 .752158 .762528 4 5 72.146 69.419 .227 0.842109 0.852479 75.103 72.320 .232 0.842429 0.852859 5 6 84.786 82.238 .212 .920241 .930671 88.252 85.659 .216 .920641 .931132 6 7 97.793 95.434 .197 .989597 1.000088 101.763 99.3441 ,202 .990087 1.000654 7 8 111.161 108.973 .182 1.052304 .062871 115.637 113.406 .186 1.052897 .063539 8 9 124.886 122.884 .167 .109819 .120461 129.876 127.822 .171 .110527 .121257 9 10 138.973 137.147 .152 1.163186 1.173916 144.470 142.593 .156 1.164025 1.174850 10 11 153.411 151.759 .138 .213183 .224008 159.417 157.717 .142 .214167 .225096 11 12 168.198 166.722 .123 .260402 .271331 174.711 173.182 .127 .261550 .272594 12 13 183.329 182,021 .109 .305313 .316357 190.344 188.981 .114 .306646 .317816 13 14 198.794 197.652 .095 .348294 .359464 206.306 205.108 .100 .349830 .361138 14 15 214.586 213.606 .082 1.389652 1.400960 222.587 221.542 .087 1.391418 1.402879 15 16 230.693 229.865 .069 .429649 .441110 239.172 238.277 .075 .431673 .443302 16 17 247.101 246.421 .057 .468509 .480138 256.049 255.297 .063 .470818 .482631 17 18 263.797 263.259 .045 .506425 .518238 273.205 272.589 .051 .509051 .521065 18 19 280.770 280.365 .034 .543567 .555581 290. 624 290.135 .041 .546545 .558780 19 20 298.002 297.724 .023 1.580090 1.592325 308.282 307.905 .031 1.583463 1.595945 20 21 315.471 315.304 .014 .616140 .628622 326.158 325.888 .022 .619955 .632703 21 22 333.157 333.095 .005 .651852 .664600 344.230 344.047 .015 .656156 .669204 22 23 351.035 351.059 .002 .687352 .700400 362.470 362.370 .008 .692203 .705573 23 24 369.080 369.186 .009 .722768 .736138 380. 864 380.833 .003 .728218 .741943 24 25 387.276 387.452 .015 1.758212 1.771937 399.377 399.398 .002 1.764332 1.778450 25 26 405.593 405.819 .019 .793810 .807928 417.979 418.036 .005 .800678 .815227 26 27 423.996 424.258 .022 .829689 .844238 436. 646 436.731 .007 .837379 .852397 27 28 442.463 442.753 .024 .865967 .880985 455. 345 455.435 .008 .874572 .890111 28 29 460.962 461.256 .025 .902777 .918316 474.041 474.123 .007 .912399 .928508 29 30 479.458 479.745 .024 1.940256 1.956365 492. 700 492. 754 ,004 1.951007 1.967745 30 31 497.918 498.176 .021 .978550 .995288 511.273 511.268 .000 .990549 2.007971 31 32 516.292 516.493 .017 2.017807 2.035229 529. 760 529.728 .003 2.031191 .049372 32 33 534.582 534.756 .015 .058192 .076373 548.135 548.017 .010 .073102 .092103 33 34 552.760 552.849 .007 .099872 .118873 566.321 566.101 ,018 .116479 .136397 34 35 570.752 570.739 .001 2.143040 2.162958 584.299 583.973 .027 2.161527 2.182438 35 36 588.537 588.420 .010 .187901 .208812 602.041 601.586 .038 .208464 .230471 36 37 606.090 605.846 .020 .234671 .256678 619.514 618.918 .050 .257533 .280737 37 38 623.377 622.997 .032 .283591 .306795 636.694 635.946 .062 .308985 .333498 38 39 640.372 639.838 .044 .334912 .359425 653.541 652.612 .077 .363125 .389089 39 40 657.040 656.326 .060 2.388935 2.414899 670.026 668.915 .093 2.420261 2.447798 40 41 673.347 672.454 .074 .445970 .473507 686.138 684.837 .108 .480714 .509970 41 42 689.288 688.207 .090 .506334 .535590 701.846 700.332 .126 .544876 .576033 42 43 704.829 703.535 .108 .570419 .601576 717.128 715.399 .144 .613161 .646384 43 34. -rr = 17.9149. 35. 77 = 18.5241. Continued on Page *187. *168 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. li n. x = 36. 37. 3 3 O 4> n. V. 6m: V. 12m. ^m.d. 6m. AB. 12m. V. 6m. V. 12m. ^m.d. A(B/). AB. 6m. 12m. 15.403 11.640 .314 0.010470 0.020693 15.999 12.152 .321 0.0105140.020779 1 ol 1 37.229 23.651 .298 .322014 .332279 28.334 24.671 .305 .322111 .3324271 1 2 39.420 36.024 .283 .508769 .519085 41.037 37.559 .290 .5089271 .519297 2 3 51.976 48.762 .268 .644531 .654901 54.111 50.817 .275 .644756! .655186 3 4 64.897 61.866 .253 .752432 .762862 67.564 64.465 .258 .752730 .763221 4 5 78.194 75.355 .237 0.842778 0.853269 81.388 78.467 .243 0.843164 0.853731 5i 6 91.857 89.194 .222 .921079 .931646 95.583 92.854 .227 .921563 .932205 6 7 105.887 103.414 .206 .990627 1.001269 110.152 107.605 .212 .991219 1.001949 7! 8 120.287 117.993 .191 1.053548 .064278 125.084 122.718 .197 1.054267 .065092 8 9 135.044 132.930 .176 .111307 .122132 140.378 138.193 .182 .112164 .123093 di 10 150.161 148.225 .161 1.164947 1.175876 156.027 154.016 .168 1.165963 1.177007 10 11 165.627 163.864 .147 .215250 .226294 172.022 170.182 .153 .216443 .227613 11 12 181.436 179.841 .133 .262814 .273984 188.355 186.683 .139 .264205 .275513 12 13 197.578 196.150 .119 .308111 .319419 205.013 203.498 .126 .309724 .321185 13 14 214.043 212.769 .106 .351522 .362983 221.981 220.620 .113 .353385 .365014 14 15 230.814 229.693 .093 1.393363 1.404992 239.250 238.035 .101 1.395503 1.407316 15 16 247.882 246.905 .081 .433899 .445712 256.804 255.727 .090 .436349 .448363 16 17 265.231 264.391 .070 .473358 .485372 274.626 273.680 .079 .476151 .488386! 17 ! 18 282.846 282.135 .059 .511939 .524174 292.693 291.862 .069 .515119 .527601 18 19 300.703 300.105 .050 .549824 .562306 310.985 310.262 .060 .553436 .566184 19 20 318.781 318.291 .041 1.587179 1.599927 329.474 328.842 .053 1.591271 1.604319 20 21 337.055 336.654 .033 .624155 .637203 348.138 347.589 .046 .628783 .642153 21 22 355.502 355.184 .026 .660898 .674268 366.957 366.480 .040 .666115 .6798401 22 23 374.103 373.855 .021 .697539 .711264 385.901 385.476 .035 .703415 ,717533 23 24 392.825 392.629 .016 .734219 .748337 404.933 404.546 .032 .740831 .755380 24 25 411.636 411.478 .013 1.771075 1.785624 424.032 423.674 .030 1.778499!l.793517 25 26 430.514 430.383 .011 .808238 .823256 443.166 442.812 .029 .816565! .83210426 27 449.423 449.298 .010 .845849 .8613881 462.295 461.933 .030 .855180! .871289 27 28 468.331 468.197 .011 .884053 .900162 481.387 480.996 .033 .894500 .911238 28 29 487.200 487.037 .014 .923002 .9397401500.390 1 499.939 .038 .934684j .952106|29 30 505.981 505.760 .018 1.962852,1.980274 519.305 518.827 .040 1.97590611.994087 30 31 524.677 524.428 .021 2.003772 2.021953 538.106 537.540 .047 2.0183382.037339 31 32 543.259 542.923 .028 .045934 .064935 556.714 556.043 .056 .062183 .082101 32 33 561.650 561.211 .037 .089535 .109453 575.109 574.329 .065 .107650 .128561 33 i 34 579.830 579.284 .046 .134785 .155696 593.263 592.351 .076 .154962 .176969 34 35 597.774 597.097 .056 2.181902 2.203909 611.140 610.085 .088 2.204363 2.227567 35 36 615.443 614.624 .068 .231131 .254335 628.718 627.507 .101 .256110 .280623 36 37 632.817 631.843 .081 .282725 .307238 645.956 644.559 .116 .310510 .336474 37 38 649*853 648.697 .096 .336990 .362954 602.822 661.240 .132 .367873 .395410 38 39 666.523 665.183 .112 .394235 .421772 679.308 677.532 .148 .428524 .457780 39 40 682.817 681.285 .128 2.454784 2.484040 695.381 693.385 .166 2.492857 2.524014 40 41 698.702 696.954 .146 .519028 .550185 711.016(708.802 .185 .561287 .594510! 411 42 714.156 712.191 .164 .587383 .620606 726.2061723.764 .203 .634256 .669739 42 43 _ 729.168 726.980 .182 .660287 .695770 740.936 738.264 .22^ .712260 .750216 43 36. 71=19.1663. 37. 77 = 19.8452. Continued on Page *186. *169 TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. 38. 39. !|.g n. V. 6m. V. 12m. ^m.d. A(B/). 6m. aB. 12m. V. 6m. V. 12m. ^m.d. 6m. AB. 12m. n. 16.618 12.673 .329 0.010568 0.020884 17.267 13.214 .338 0.010624 0.020994 0| 1 29.477 25.719 .313 .322221 .332591 30.671 26.807 .322 .32234] .332771 l! 2 42.710 39.140 .298 .509102 .519532 44.463 40.800 .305 .509289 .519780 2 3 56.329 52.956 .281 .645002 .655493 58.638 55.156 .290 .645272 .655839 3 4 70.324 67.130 .266 .753060 .763627 73.192 69.908 .274 .753423 .764065 4 5 84.694 81.695 .250 0.843589 0.854231 88.130 85.032 .258 0.844054 0.854784 5 6 99.442 96.627 .235 .922094 .932824 103.440 100.527 .243 .922678 .933503 6 7 114.557 111.926 .219 .991873 1.002698 119.120 116.393 .227 .992591 1.003520 7| 8 130.040 127.591 .204 1.055056 .065985 135.165 132.617 .212 1.055925 .066969 8 9 145.881 143.609 .189 .113110 .124154 151.564149.191 .198 .114148 .125318 9 10 162.072 159.973 .175 1.167081 1.178251 168.310166.109 .183 1.168310 1.179618 10 11 178.606176.677 .161 .217755 .229063 185.390183.350 .170 .219199 .230660 11 12 195.469 193.699 .147 .265736 .277197 202.788 200.906 .157 .267421 .279050 12 13 212.647 211.033 .135 .311500 .323129 220.493 218.761 .144 .313455 .325268 13 14 230.128 228.661 .122 .355434 .367247 238.491236.901 .133 .357689 .369703 14 15 247.897 246.571 .110 1.397858 1.409872 256.764 255.307 .121 1.400447 1.412682 15 16 265.939 264.745 .100 .439043 .451278 275.288 273.949 .112 .442008 .454490 16 17 284.2281283.150 .090 ! .479227 .491709 294.042 292.815 .102 .482615 .495363 17 18 302.7451301.777 .081 .518622 .531370 312.999 311.864 .095 .522478 .535526 18 19 321.462 320.585 .073 .557414 .570462 332.135 331.086 .087 .561795 .575165119 1 20 340.355 339.564 .066 1.595779 1.609149 351.431350.455 .081 1.600738 1.614463 20 21 359.407 358.687 .060 .633875 .647600 370.853 369.931 .077 .639481 .653599 21 22 378.582 377.916 .055 .671860 .685978 390.367 389.484 .074 .678190 .692739 22 23 397.850 397.221 .052 .709891 .724440 409.950 409.096 .071 .717019 .732037 23 241417. 183 416.584 .050 .748111 .763129 429.567 428.717 .071 .756129 .771668 24 25 436.553 435.957 .050 1.786675 1.802214 449.180 448.323 .071 1.795684 1.811793 25 26 455.916 455.314 .050 .825739 .841848 468.755 467.867 .074 .835850 .852588 26 27 475.244 474.611 .053 .865464 .882202 488.238 487.290 .079 .876794 .894216 27 28 494.480 493.787 .058 .906012 .923434 507.633 506.656 .081 .918699 .936880 28 29 513.628 512.908 .060 .947560 .965741 526.909 525.842 .089 .961745 .980746 29 30 532.661 531.851 .068 1.990136 2.009137 545.988 544.813 .098 2.006139 2.026057 30 31 551.497 550.581 .076 2.034395 .054313 564.847 563.562 .107 .052097 .073008 31 32 570.117 569.092 .085 .080097 .101008 583.461 582.040 .118 .099845 .121852 32 33 588.496 587.337 .097 .127619 .149626 601.791 600.222 .131 .149635 .172839 33 34 606.593 605.288 .109 .177207 .200411 619.814 618.085 .144 .201726 .226239 34 35 624.388 622.925 .122 2.229119 2.253632 637.486 635.568 .160 2.256427 2.282391 35 36 641.837 640.187 .137 .283663 .309627 654.780 652.671 .176 .314055 .341592 36 37 658.911 657.073 .153 .341154 .368691 671.683 669.375 .192 .374935 .404191 37 38 675.600 673.565 .170 .401915 .431171 688.162 685.629 .211 .439466 .470623 38 39 691.870 689.613 .188 .466343 .497500 704.192 701.436 .230 .508066 .541289 39 40 707.697 705.219 .207 2.534855 2.568078 719.767 716.777 .249 2.581178 2.616661 40 41 723.073 720.366 .226 .607893 .643376 734.870 731.643 .269 .659303 .697259 41 42 737.986 735.044 .245 .685955 .723911 749.505 746.047 .288 .742974 .783620 421 43 752.436 749.265 .264 .769575 .810221 763.667 759.967 .308 .832813 .876419 43 i 38. TT = 20.5624. 39. 7r = 21.3199. i Continued on Page *185. ♦170 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OP 1,000. 4 PER CENT. 11 n. a; = 40. 41. n. V. 6m. V. 13m. im.d. MB/). 6m. AB. 13m. V. 6m. V. 13m. l^m.d. 6m. AB. 13m. 17.948 13.775 .348 0.010687 0.031117 18.673 14.379 .358 0.010752 0.031343 1 31.936 37.956 .331 322469 .333960 33.338 39.130 .343 .322612 .333179 1 2 46.291 43.504 .316 509497 .530064 48.193 44.388 .335 .509727 .520369 2 3 61.039 57.453 .399 645569 .656311 63.542 59.839 .309 .645894 .656624 3 4 76.177 73.780 .383 753819 .764549 79.273 75.751 .394 .754258 .765083 4 5 91.691 88.483 .267 844568 0.855393 95.386 93.054 .378 0.845131 0.856060 5 6 107.582 104.561 .252 923321 .934350 111.873 108.734 .363 .924029 .9350731 6 7 123.842 131.003 .237 993381 1.004435 138.723 125.755 .347 .994251 1.0054211 7 8 140.461 137.798 .222 1 056883 .068053 145.931 143.140 .333 1.057934 .069242; 8 9 157.431 154.943 .207 115290 .136598 163.481 160.855 .319 .116549 .128010 9 10 174.739 173.414 .194 1 169663 1.181134 181.358 178.894 .305 1.171153 1.182782 10 11 192.370 190.305 .180 230789 .333418 199.551 197.241 .193 .222539 .234352 11 12 210.312 308.299 .168 369376 .381089 218.044 215.881 .180 .271315 .283329 12 13 238.551 236.682 .156 315604 .337618 336.822 334.795 .169 .317968 .330203 13 14 247.069 345.335 .145 360168 .373403 355.856 353.950 .159 .362898 .375380 14 15 265.842 364.337 .135 1 403398 1.415780 375.136 373.335 .149 1.406438 1.419186 15 16 384.846 383.345 .125 445375 .458033 394.605 292.909 .141 .448873 .461921 16 17 304.058 303.650 .117 486345 .499393 314.369 312.661 .134 .490455 .503825 17 18 323.450 333.139 .110 536738 .540098 334.095 332.563 .128 .531402 .545127 18 19 343.004 341.757 .104 566616 .580341 354.053 352.576 .133 .571924 .586042 19 20 362.686 361.494 .099 1 606300 1.630318 374.105 372.667 .130 1.612218 1.626767 20 21 382.462 381.309 .096 645659 .660308 394.337 393.819 .117 .652462 .667480 21 22 402.306 401.183 .094 685160 .700178 414.383 413.981 .117 .693837 .708376 22 23 422.186 431.067 .093 724871 .740410 434.538 433.137 .118 .733534 .749633 23 24 442.061 440.935 .094 764965 .781074 454.651 453.309 .130 .774703 .791441 24 25 461.899 460.741 .096 1 805614 1.833353 474.673 473.167 .135 1.816556 1.833978 25 26 481.643 480,434 .102 846991 .864413 494.600 493.066 .138 .859374 .877455 26 27 501.297 500.049 .104 889385 .907466 514.407 513.781 .136 .903047 .922048 27 28 520.831 519.493 .112 933678 .951679 534.011 533.374 .145 .948091 .968009 28 29 540.166 538.717 .131 977383 .997301 553.390 551.540 .154 .994628 2.015539j29 30 559.277 557.717 .130 3 033618 3.044539 573.517 570.537 .166 3.042891 2.064898' 30 31 578.141 576.443 .141 071613 .093620 591.353 589.310 .178 .093137 .116341*31 32 596.716 594.868 .154 131633 .144826 609.871 607.565 .193 .145630 .170143 32 33 614.979 612.970 .167 173906 .198419 638.031 635.530 .308 .300684 .336648! 33| 34 632.890 630.688 .184 338776 .254740 645.801 643.104 .335 .358631 .386158 34 35 650.414 648.020 .199 3 386553 2.314089 663.169 660.368 .243 3.319769 3.349035 35 36 667.544 664.947 .316 347561 .376817 680.103 676.970 .361 .384531 .415688 36 37 684.244 681.420 .335 413303 .443360 696.574 693.311 .280 .453338 .486551 37 38 700.490 697.438 .354 480898 .514121 713.576 708.975 .300 .536608 .562091 38 39 716.371 712.984 .374 554090 .589573 738.097 734.351 .321 .604873 .643839 39 40 731.577 728.049 .394 2 633383 2.670338 743.134 739.051 .340 3.688661 3.739307 40 {41 746.408 743.646 .314 716010 .756656 757.687 753.356 .361 .778594 .833300 41 '42 760.760 756.753 .334 805894 .849500 771.778 767.333 .379 .875305 .933097 42 43 774.657 770.439 .351 903565 .949357 785.440 780.679 .397 .979538 3.039839 43 40. 7r = 32.1306. 41. 77 = 33.9667. Continued on Page *184. TABLE LXIII. *in GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. 42. 43. V. V. Jm.d. ^(B/). AB. V. V. Jm.d. ^(B/). AB. n. 6m. 12m. 6m. 12m. 6m. 12m. 6m. 12m. n. !o 19.415 14.966 .371 0.010831 0.021398 20.212 15.613 .383 0.010911 0.021553 1 1 34.587 30.345 .354 .322771 .333413 36.021 31.620 .367 .322941 .333671 1 2 50.160 46.113 .337 .509978 .520708 52.225 48.020 .350 .510255 .521080 2 3 66.122 62.267 .321 .646255 .657080 68.821 64.812 .334 .646649 .657578 3 4 82.469 78.808 .305 .754740 .765669 85.802 81.982 .318 .755270 .766314 4 5 99.196 95.722 .289 0.845753 0.856797 103.158 99.524 .303 0.846437 0.857607 5 61116.293 113.001 .274 .924809 .935979 il20.882 117.430 .288 .925665 .936973 6 7il33.752 130.639 •259 .995207 1.006515 1138.959 135.677 .274 .996260 1.007721 7 8|151.557 148.613 .245 1.059092 .070553 157.372 154.258 .259 i.060366 .071995 8 9 169.696 166.916 .232 .117934 .129563 ;i76.112 173.155 .246 .119458 .131271 9 10 188.155 185.530 .219 1.172793 1.184606 195.159 192.354 .234 1.174597 1.186611 10 11 206.917 204.442 .206 .224464 .236478 214.500 211.835 .222 .226579 .238814 11 12 225.968 223.631 .195 .273558 .285793 234.105 231.565 .212 .276028 .288510 12 13 245.280 243.066 .185 .320571 .333053 '253.953 251.532 .202 .323439 .336187 13 14 264.832 262.734 .175 .365905 .378653 :274.018 271.694 .194 .369217 .382265 14 15 284.595 282.594 .167 1.409896 1.422944 294.271 292.038 .186 1.413707 1.427077 15 16 304.546 302.634 .159 .452837 .466207 314.693 312.537 .180 .457198 .470923 16 17 324.661 322.826 .153 .494977 .508702 335.248 333.150 .175 .499957 .514075 17 18 344.910 343.131 .148 .536551 .550669 '355.903 353.845 .172 .542225 .556774 18 19 365.255 363.515 .145 .577775 .592324 376,628 374.601 .169 .584217 .599235 19 20 385.669 383.961 .142 1.618845 1.633863 397.389 395.368 .168 1.626145 1.641684 20 21 406.121 404.417 .142 .659955 .675494 418.148 416.118 .169 .668214 .684323 21 22 426.568 424.857 .143 .701297 .717406 438.866 436.803 .172 .710624 .727362 22 23 446.976 445.232 .145 .743062 .759800 459.486 457.360 .177 .753574 .770996 23 24 467.288 465.481 .151 .785436 .802858 480.013 477.856 .180 .797270 .815451 24 25 487.508 485.671 .153 1.828620 1.846801 500.415 498.163 .188 1.841914 1.860915 25 26 507.603 505.673 .161 .872808 .891809 520.607 518.241 .197 .887733 .907651 26 27 527.494 525.451 .170 .918222 .938140 540.567 538.084 .207 .934959 .955870 27 28 547.155 544.997 ,180 .965089 .986000 560.268 557.641 .219 .983833 2.005840 28 29 566.561 564.261 .192 2.013644 2.035651 579.667 576.884 .232 2.034619 .057823 29 30 585.670 583.217 .204 2.064148 2.087352 598.742 595.790 .246 2.087587 2.112100 30 31 604.460 601.840 .218 .116868 .141381 !617.447 614.294 .263 .143058 .169022 31 132 622.885 620.067 .235 .172122 .198086 635.750 632.395 .280 .201358 .228895 32 33 640.914 637.897 .251 .230233 .257770 653.639 650.074 .297 .262822 .292078 33 34 '658. 535 655.311 .269 .291532 .320788 671.081667.278 .317 .327854 .359011 34 35 '675.716 672.257 .288 2.356424 2.387581 688.048 684.007 .337 2.396883 2.430106 35 36 692.438 688.736 .308 .425333 .458556 704.530 700.243 .357 .470359 .505842 36 37 708.664 704.729 .328 .498706 .534189 720.515 715.977 .378 .548788 .586744137 38 724.410 720.228 .348 .577049 .615005 736.004 731.222 .398 .632712 .673358! 38 39 739.667 735.244 .369 .660903 .701549 750.994 745.955 .420 .722756 .7663621 39 40 754.432 749.757 .390 2.750890 2.794496 765.507 760.249 .438 2.819556 2.866348*40 41 768.729 763.837 .408 .847643 .894435 779.578 774.098 .457 .923851 .974162 41 42 782.589 777.479 .426 .951901 3.002212 793.262 787.616 .471 3.036438 3.090542; 42 43 796.069 790.795 .440 3.064459 .118563 806.697 800.969 .477 .158109 .216302 43' 42. TT = 23.8627. 43. 7r = 24.8099. Continued on Page *183. *172 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. Is x = 44. 45. m o « n. V. 6m. V. 12m. Jm.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. im.d. ^ (B/). 6m. AB. 12m., « n. 21.037 16.261 .398 0.011001 0.021731 21.906 16.936 .414 0.011103 0.021928 1 37.498 32.921 .381 .323134 .333959 39.044 34.276' .397 .323343 .334272' 1 2 54.356 49.979 .365 .510559 .521488 56.580 52.007 .381 .510895 .521939 2 3 71.607 67.422 .349 .647085 .658129 74.502 70.122 .365 .647564 .658734 3 4 89.239 85.242 .333 .755854 .767024 92.806 88.613 .349 .756496 .7678041 4 1 5 107.243 103.432 .318 0.847188 0.858496 111.472 107.455 .335 0.848016 0.859477 5 6 125.607 121.969 .303 .926608 .938069 130.4871 126. 643' .320 .927646 .939275 6 7 144.313 140.844 .289 .997420 1.009049 149. 838146.1571 .307 .998696 1.010509 7 8 163.349 160.041 .276 1.061769 .073582 169.508:165.983 .294 1.063313 .075327 8 9 182.700 179.545 .263 .121134 .133148 189.479 186.100; .282 ' 1 .122976 .135211 9 10 202.346 199.335 .251 1.176579 1.188814 209. 725J206. 4741 .271 1.178762 1.191244 10 11 222.263 219.378 .240 .228909 .241391 230.222 227.093 .261 .231476 .2442241 11 12 242.426 239.661 .230 .278749 .291497 250.941 247.913 .252 .281746 .294794 12 13 262i»809 260.143 .222 .326597 .339645 271.855 268.921 .245 .330077 .343447 13 14 283.382 280.809 .214 .372866 .386236 292.943 290.090 .238 .376881 .390606 14 15 304.128 301.634 .208 1.417899 1.431624 314.171311.376 .233 1.422516 1.436634 15 16 325.011 322.574 .203 .462001 .476119 335.499 332.746 .229 .467295 .481844 16 17 345.991 343.596 .200 .505445 .519994 356.902 354.181 .227 .511489 .526507 17 18 367.046 364. 682 .197 .548473 .563491 378.341 375.6261 .226 .555356 .570895 18 19 388.136 385.778 .197 .591314 .606853 399.778 397.053' .227 i j .599137 .615246 19 20 409.225 406.858 .197 1.634191 1.65030o' 421.171418.414 .230 1.643062 1.659800 20 21 430.271 427.871 .200 .677318^ .694056 442.466 439.6411 .235 .687354 .704776 21 22 451.218 448. 753 .205 .7209041 .738326 463.661,460.806' .238 .732240 .750421 22 23 472.071 469.575 .208 .765165! .783346 484.730i481.777 .246 .777940 .796941 23 24 492.795 490.203 .810 .810311- .829312, 505.581j502.510 .256 .824695 .844613 24 25 513.308 510.600 .226 1.8565751.876493' 526.194 523.0011 .266 1.872749 1.893660 25 26 533.586 530.758 .236 .904196 .925107 546.537 543.197 .278 .922355 .944362 26 27 553.598 550.625 .248 .953419 .975426 566.571 563.069 .292 .973785 .996989 27 28 573.306 570.174 .261 2.004512 2.027716 586.268 582.592 .306 2.027320 2.051833 28 29 592.683 589.379 .275 .057751 .082264 605.584 601.700 .324 .083286 .109250 29 30 611.684 608.177 .292 2.113459 2.139423'!624. 484 620.392 .341 2.142017 2.169554 30 31 630.278 626.565 .309 .171965 .199502 642.958 638.648 .359 .203853 .233109 31 32 348.451 644.524? .327 .233606 .262862; 660.969:656.414 .380 .269207 .300364 32 33 666.169 662.0001 .347 .298793 .329950! 678. 4901673. 689 .400 .338509 .371732 33 34 683.404 678.995 .367 .367953 .401176 695.510'690.456 .421 .412215 .447698 34 35 700.148 695.489 .388 2.441538 2.477021' 712.018 706.703 .443 2.490838 2.528794 35 36 716.388l711.473l .410 .520060 .558016 728.012 722.446 .464 .574922 .615568 36 37 732. 122 726.959 .430 .604061 .644707 743.491 737.660 .486 .665098 .708704 37 38 747.349 741.926 .452 .694168 .737774 758.478 752.420 .505 .762003 .808795' 38 39 762.093 756.446 .471 .791019 .837811 773.009 766.722 .524 .866381 .916692 39 40 776.387 770.516 .489 2.895353 2.945664 787.140 780.682 .538 2.979034 3.033138 40 41 790.289 784.248 .503 3.007972 3.062076 801.015 794.471 .545 3.100754 .158947 41 42 803.937 797.813 .510 .129666 .187859 814.766 808.186 .548 .232640 .295403 42 43 817.466 811.306 .513 .261533 .324296 828.710 822.357 .529 .375628 .443161 43 44. Tr = 25.8133. 45. 77 = 26.8757. Continued on Page *183. TABLE LXIII '173 GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. g n. 46. 47. 1 ••^1 n. V. 6ra. V. 12m. ^m.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. ^m.d. A(B/). 6m. AB. 12m. 22.821 17.640 .432 0.011211 0.022140 23.778 18.361 .451 0.011332 0.022376 1 40.658 35.676 .415 .323573 .334617 42.337 37.118 .435 .323827 .334997 1 2 58.890 54.103 .399 .511264 .522434 61.289 56.265 .419 .511668 .522976 2 3 77.508 72.912 .383 .648090 .659398 80.618 75.776 .404 .648669 .660130 3 4 96.496 92.079 .368 .757202 .768663 100.308 95.645 .389 .757980 .769609 4 5 115.839 111.597 .354 0.848927 0.860556 'l20.346 115.852 .374 0.849929 0.861742 5 6 135.523 131.448 .340 .928788 .940601 1140.714 136.381 .361 .930044 .942058 6 7 155.532 151.615 .326 1.000098 1.012112 161.395 157.213 .349 1.001639 1.013874 7 8 175.847 172.079 .314 .065008 .077243 182.359 178.310 .337 .066875 .079357 8 9 196.442 192.804 .303 .125005 .137487 203.583 199.661 .327 .127241 .139989 9 10 217.292 213.778 .293 1.181167 1.193915 1225.038 221.220 .318 1.183816 1.196864 lO' 11 238.368 234.957 .284 .234303 .247351 '246.694 242.974 .310 .237418 .250788 11 12 259.642 256.327 .276 .285048 .298418 ;268.532 264.894 .303 .288680 .302405 12 13 281.095 277.861 .2^9 .333905 .347630 '290.512 286.935 .298 .338120 .352238 13 14 302.687 299.513 .265 .381302 .395420 312.597 309.064 .294 .386176 .400725 14 15 324.383 321.251 .261 1.427604 1.442153 334.759 331.259 .292 1.433208 1.448226 15 16 346.153 343.055 .258 .473124 .488142 356.959 353.465 .291 .479546 .495085 16 17 367.963 364.869 .258 .518148 .533687i 379.157 375.653 .292 •525487 .541596 17 18 389.768 386.666 .258 .562943 .579052! i401.310 397.772 .295 .571309 .588047 18 19 411.531 408.395 .261 .607761 .624499' 423.360 419.753 .301 .617269 .634691 19 20 433.192 429.988 .267 1.652840 1.670262' 445.309 441.670 .303 1.663624 1.681805 20 21 454.753 451.518 .270 .698421 .716602 467.124 463.384 .312 .710620 .729621 21 22 476.184 472.849 .278 .744734 .763735 488.716 484.853 .322 .758517 .778435 22 23 497.395 493.940 .288 .792031 .811949 510.060 506.072 .332 .807578 .828489 23 24 518.363 514.784 .298 .840563 .861474 531.126 526.984 .345 .858070 .880077 24 25 539.056 535.327 .311 1.890590 1.912597 551.870 547.561 .359 1.910278 1.933482 25 26 559.435 555.542 .324 .942391 .965595 572.266 567.777 .374 .964494 .989007 26 27 579.472 575.401 .339 .996250 2.020763 592.268 587.564 .392 2.021055 2.047019 27 28 599.120 594.839 .357 2.052499 .078*63 611.839 606.919 .410 .080302 .107839 28 29 618.347 613.853 .375 .111476 .139013 630.969 625.823 .429 .142583 .171839 29 30 637.138 632.423 .393 2.173525 2.202781 649.618 644.219 .450 2.208319 2.239476 30 31 655.4o9'650.494 .414 .239061 .270218 667.761 662.107 .471 .277947 .311170 31 32 673.281j668.067 .434 .308519 .341742 685.385 679.469 .493 .351928 .387411 32 33 690.596|685.123 .456 .382357 .417840 702.479 696.294 .515 .430781 .468737 33 34 707.387 701.651 .478 .461089 .499045 719.042 712.594 .537 .515055 .555701 34 35 723.'658 717.664 .500 2.545264 2.585910 735.069 728.349 .560 2.605386 2.648992 35 36 739.403 733.141 .522 .635514 .679120 750.588,743.633 .580 .702416 .749208 36 37 754.649 748.155 .541 .732479 .779271 765.636 758.443 .599 .806893 .857204 37; 38 769.430 762.704 .560 .836904 .887215 780.267 772.897 .614 .919623 .973727 38 39 783.804 776.903 .575 .949594 3.003698 794.634 787.176 .621 3.041402 3.099595 39 40 797.917 790.930 .582 3.071343 3.129536 808.875 801.379 .625 3.173332 3.236095 40 41 811.906 804.882 .585 .203250 .266013 823.313 816.052 .605 .316352 .383885 41 42 826.090 819.297 .566 .346253 .413786 1837.140 829.032 .676 .475762 .551432 42 43 839.673|832.047 .636 .505651 .581321 849.151 840.075 .756 .656718 .741317 43 46. 7r = 28.0014. 47. 7r = 29.1948. Continued on Page *181. •*iu TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. Is X = 48. 49. > n. n. V. 6m. V. 12m. |m.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. l^m.d. 6m. aB. 12m. 24.784 19.108 .473 0.011464 0.022634 25.844 19.885 .497 0.011610 0.022918 1 44.091 38.614 .456 .324105 .335413 45.919 40.149 .481 .324412 .335873 1 2 63.782 58.490 .441 .512114 .523575 66.366 60.781 .465 .512607 .524236 2 3 83.840 78.730 .426 .649307 .660936 87.177 81.769 .451 .650009 .661822 3 4 104.253 99.315 .412 .758835 .770648 108.331103.090 .437 .759776 .771790 1 4 5 125.001 120.228 .398 0.851031 0.863045 129.808124.724 .424 0.852241 0.864476 5 6 146.069 141.449 .385 .931423 .943658 151.581146.635 .412 .932943 .945425 6 7 167.425 162.941 .374 1.003337 1.015819 173.624168.809 .401 1.005208 1.017956 7 8 189.046 184.691 .363 .068933 .081681 195.905191.198 .392 .071198 .084246 8 9 210.902 206.653 .354 .129703 .142751 218.396 213.791 .384 .132416 .145786 9 10 232.963 228.814 .346 1.186735 1.200105 241.0751236.556 .377 1.189945 1.203670 10 11 255.209 251.144 .339 .240843 .254568 263.903:259.447 .371 .244617 .258735 11 12 277.601 273.598 .334 .292679 .306797 286.839 282.429 .368 .297088 .31163? 12 13 300.099 296.140 .330 .342768 .357317 309.856 305.480 .365 .347885 .362903 13 14 322.676 318.751 .327 .391542 .406560 332.913 328.542 .364 .397454 .412993 14 15 345.291 341.372 .327 1.439382 1.454921 355.965 351.585 .365 1.446188 1.462297 15 16 367.904 363.975 .327 .486625 .502734 378.972 374.557 .368 .494432 .511170 16 17 390.471 386.508 .330 .533581 .550319 401.873 397.385 .374 .542504 .559926 17 18 412.934 408.900 .336 .580532 .597954 424.667 420.147 .377 .590704 .608885 18 19 435.294 431.227 .339 .627755 .645936 447.324 442.697 .386 .639316 .658317 19 20 457.517 453.347 .347 1.675512 1.694513 469.747 464.995 .396 1.688628 1.708546 20 21 479.512 475.218 .358 .724078 .743996 491.915 487.031 .407 .738930 .759841 21 22 501.256 496.833 .369 .773726 .794637 513.791 508.749 .420 .790508 .812515 22 23 523.715 518.137 .382 .824732 .846739 535.336 530.120 .435 .843667 .866871 23 24 543.848 539.099 .396 .877391 .900595 556.519 551.115 .450 .898710 .923223 24 25 564.626 559.693 .411 1.932000 1.956513 577.292 571.665 .469 1.955992 1.981956 25 26 585.001 579.849 .429 .988903 2.014867 597.617 591.766 .488 2.015862 2.043399 26 27 604.938 599.567 .448 2.048447 .075984 617.484 611.399 .507 .078682 .107938 27 28 624.425 618.824 .467 .110984 .140240 636.853 630.504 .529 .144877 .176034 28 29 643.424 637.564 .488 .176939 .208096 655.694 649.082 .551 .214897 .248120 29 30 661.906 655.787 .510 2.246754 2.279977 673.999 667.113 .574 2.289207 2.324690 30 31 679.860 673.474 .532 .320892 .356375 691.751 684.586 .597 .368335 .406291 31 32 697.273 690.613 .555 .399876 .437832 708.952 701.515 .620 .452837 .493483 32 33 714.146 707.219 .577 .484258 .524904 725.598 717.877 .643 .543353 .586959 33 34 730.473 723.268 .600 .574678 .618284 741.715 733.750 .664 .640532 .687324 34 35 746.283 738.838 .620 2.671779 2.718571' 757.342 749.131 .684 2.745128 2.795439 35 36 761.612 753.925 .641 .776312 .826623 772.538 764.143 .700 .857950' .912054 36 37 776.517 768.650 .656 .889086 .943190 787.459 778.972 .707 .979799 3.037992 37 38 791.153 783.195 .663 3.010899 3.069092 802.249 793.722 .711 3.111782, .174545 38! 39 805.660 797.664 .666. .142855 .205618 817.243 808.961 .690 .2548401 .322373 39 40 820.368 812.612 .646 3.285893 3.353426 831.602 822.441 .763 3.414277,3.489947 40 41 834.453 825.834 .718 .445315 .520985 844.077 833.910 .847 .595252 .679851 41 42 846.689 837.084 .800 .626281 .710880J 855.176 844.638 .878 .797429 .889806 42 43 857.576 847.607 .831 .828452 .920829 865.710 854.979 .894 4.020868|4.121378 43 48. TT = 30.4602. 49. 7r = 31.8029. Continued on Page *180. ♦17^ TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. n. 50. 51. n. V. 6m. V. 12m. -^m.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. 4m.d. 6m. -AB. 12m. 26.952 20.675 .523 0.011770 0.023231 28.120 21.497 .552 0.011951 0.023580 1 47.815 41.727 .507 .324751 .336380 49.800 43.361 .537 .325123 .336936 1 2 69.047 63.139 .492 .513148 .524961 71.838 65.574 .522 .513741 .525755 2 3 90.630 84.893 .478 .650780 .662794 94.215 88.113 .509 .651626 .663861 3 4 112.543 106.966 .465 .760807 .773042 116.897 110.940 .496 .761945 .774427 4 5 134.758 129.321 .453 0.853574 0.866056 139.862 134.042 .485 0.855044 0.867792 5 6 157.247 151.945 .442 .934618 .947366 163.076 157.368 .476 .936462 .949510 6 7 179.981 174.789 .433 1.007268 1.020316 186.508 180.905 .467 1.009539 1.022909 7 8 202.928 197.840 .424 .073696 .087066 210.135 204.623 .459 .076441 .090166 8 9 226.068 221.067 .417 .135399 .149124 233.918 228.471 .454 .138685 .152803 9 10 249.358 244.422 .411 1.193480 1.207598 257.813 252.414 .450 1.197378 1.211927 10 11 272.760 267.870 .408 .248778 .263327 281.793 276.429 .447 .253358 .268376 11 12 296.244 291.389 .405 .301944 .316962 305.813 300.456 .446 .307294 .322833 12 13 319.768 314.919 .404 .353523 .369062 329.831 324.463 .447 .359739 .375848 13 14 :J43.288 338.430 .405 .403973 .420082 353.801 348.396 .450 .411161 .427899 14 15 366.763 361.867 .408 1.453693 1.470431 377.659 372.179 .457 1.461968 1.479390 15 16 390.127 385.159 .414 .503039 .520461 401.407 395.893 .459 .512535 .530716 16 17 413.384 408.382 .417 .552346 .570527 425.011 419.387 .469 .563198 .582199 17 18 436.501 431.391 .426 .601920 .620921 448.373 442.617 .480 .614295 .634213 18 19 459.379 454.140 .437 .652071 .671989 471.467 465.575 .491 .666150 .687061 19 20 481.996 476.624 .448 1.703104 1.724015 494.259 488.201 .505 1.719079 1.741086 20 21 504.318 498.783 .461 .755319 .777326 516.704 510.466 .520 .773414 .796618 21 22 526.299 520.587 .476 .809033 .832237 538.774 532.339 .536 .829479 .853992 22 23 547.911 542.008 .492 .864560 .889073 560.414 553.748 .556 .887644 .913608 23 24 569.105 562.974 .511 ' .922260 .948224 581.591 574.691 .575 .948279 .975816 24 25 589.843 583.484 .530 1.982493 2.010030 602.288 595.144 .595 2.011752 2.041008 25 26 610.114 603.515 .550 2.045624 .0748801 622.467 615.048 .618 .078508 .109665 26 27 629.875 623.007 .572 .112085 .143242 642.097 634.403 .641 .149002 .182225 27 28 649.098 641.962 .595 .182329 .215552 661.167 653.189 .665 .223712 .259195 28 29 667.775 660.359 .618 .256830 .292313 679.662 671.393 .689 .303173 .341129 29 30 685.887 678.187 .642 2.336116 2.374072 697.582 689.030 .713 2.387949 2.428595 30 31 703.437 695.459 .665 .420748 .461394 714.925 706.077 .737 .478689 .523295 31 32 720.420 712.153 .689 .511371 .554977 731.716 722.614 .758 .576049 .622841 32 33 736.865 728.349 .710 .608637 .655429 747.997 738.638 .780 .680786 .731097 33 34 752.809 744.041 .731 .713299 .763610 763.829 754.277 .796 .793719 .847823 34 35 768.313 759.358 .746 2.826174 2.880278 779.372 769.726 .804 2.915653 2.973846 35 36 783.537 774.487 .754 .948064 3.006257 794.781 785.094 .807 3.047699 3.110462 36 37 798.626 789.537 .757 3.080077 .142840 810.404 800.971 .786 .190804 .258337 37 38 813.925 805.086 .737 .223157 .290690 825.363 815.014 .862 .350273 .425943 38 39 828.576 818.838 .811 .382610 .458280 838.360 826.963 .950 .531271 .615870 39 40 841.304 830.541 .897 3.563596 3.648195 849.922 838.140 .982 3.733462 3.825839 40 41 852.627 841.486 .928 .765780 .858157 860.898 848.913 .999 .956910 4.057420 41 42 863.376 852.037 .945 .989223 4.089733 871.393 859.132 1.022 4.204932 .314823 42 43 873.655 862.045 .968 4.237242 .347133 1 881.483 869.091 1.033 .481379 .601459 43 50. TT = 33.2283. 51. 7r = 34.7419. Continued on Page *179. ♦176 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. '-a 1 o «> x=52. 53. V. V. 2ni.d. A(B/). AB. V. V. -J^m.d. ^(B/). AB. n. 6m. 12ra. 6m. 12in. 6m. 12m. 6m. 12m. n. 29.347 22.344 .584 0.012143 0.023956 30.640 23.219 .618 0.012355 0.024369 1 51.870 45.045 .569 .325528 .337542 1 54.029i 46.780 .604 .325974 .338209 1 2 74.737 68.079 .555 .514390 .526625 77.741! 70.642 .592 .515109 .527591 2 3 97.918 91.408 .543 .652558 .6650401 101. 747i 94.791 .580 .653587 .666335 3 4 131.388 115.017 .531 .763199 .775947 126.012:119.174 .570 .764580 .777628 4 5 145.111 138.856 .521 0.856661 0.869709 ;i50.507 143.779 .561 0.858445 0.871815 5 6 169.058 162.910 .512 .938495 .951865 Il75.205168.571 .553 .940729 1 .954454 6 7 193.205 187.149 .505 1.012034 1.025759 |200.066:i93.501 .547:1.014782 !l.028900 7 8 217.510 211.521 .499 .079464 .093582 Ij225.045;218.529 .543 .082798 .097347 8 9 241.930 235.990 .495 .142307 .156856 !250. 110:243.632 .540 .146295 .161313 9 10 266.437 260.533 .492 1.201669 1.216687 275.220'268.748 .539 1.206397 1.221936 10 11 290.985 285.087 .491 .258404 .273943 1300.326 293.843 .540 .263967 .280076 11 12 315.^29 309.622 .492 .313191 .329300 325.382 318.861 .543 .319696 .336434 12 13 340.026 334.080 .496 .366593 .383331 350.321 343.722 .550 .374152 .391574 13 14 364.408 358.386 .502 .419086 .436508 375.147 368.511 .553 .427831 .446012 14 15 388.679 382.621 .505 1.471097 1.489278 399.820 393.069 .563 1.481162 1.500163' 15 16 412.801 406.631 .514 .523003 . 542004 i 1424. 240 417.352 .574 .534556 .554474' 16 17 436.676 430.371 .525 .575172 .595090 |448.382|441. 351 .586 .588392 .609303:17 18 460.277 453.834 .537 .627954 .648865' 472.207 465.003 .600 .643034 .665041' 18 19 483.571 476.957 .551 .681688 .703695 495.670 488.277 .616 .698847 .722051 19 20 506.509 499.711 .567 1.736718 1.759922 518.739 511.142 .633 1.756189 1.780702 20 21 529.063 522.065 .583 .793385 .817898! 541.362 533.521 .653 .815454 .841418 21 22 551.179 543.944 .603 .852069 .878033 563.497 555.413 .674 .877034 .904571 22 23 572.821 565.347 .623 .913151 .940688 585.133 576.794 .695 .941318 .970574 23 24 593.973 586.250 .644 .977007 2.0062631 606.227|597.600 .719 2.008763 2.039920 24 25 614.596 606.591 .667 2.044089 2.075246 626.746'617.832' .743 2.079840 2.113063 25 26 634.656 626.371 .690 .114858 .148081! 646.681637.469 .768 .155039 .190522 26 27 654.145 645.570 .715 .189801 .225284 666.013|656.498^ ,793 .234907 .272863 27 28 673.047 664.173 .739 .269455 .307411 684.747674.935: .818 .320019 .360665 28 29 691.360 682.198 .764 .354390 .395036, 702.874 692. 754i .843 .411032 .454638 29 30 709.084 699.619 .789 2.445260 2.488866' 720.428 710. 04ll .866 2.508611 2.555403 30 31 726.244 716.520 .810 .542723 .589515' 737.446 726.791: .888 .613522 .663833 31 32 742.883 732.896 .832 .647543 .697854 753.995 743.140 .905 .726589 .780693 32 33 759.063 748.879 .849 .760540 .814644 770.245 759.289 .913 .848626 .906819 33 34 774.948 764.667 .857 .882523 .940716 786.351 775.353 .917 .980750 3.043513 34 35 790.694 780.372 .860 3.014606 3.077369 802.681 791.949 .894 3.123910 3.191443 35 36 806.660 796.598 .838 .157737 .225270 818.319 806.629: .9741 .283420 .359090 36 37 821.949 810.950 .917 .317225 .392895 '831.904 819.120'1.065! .464444 .549043 37 38 835.231 823.162 1.006 .498236 .582835 843.992 830.803 1.099! .666653 .759030 38 39 847.048 834.584 1.039 .700435 .792812; 855.464 842.065 1.117 .890112 .990622 39 40 858.264 845.594 1.056 3.923888 4.0243981 866.436 852.747 1.141 4.138140 4.248031 40 41 868.991 856.038 1.079 4.171913 .281804' 876.982 863.157 1.152 .414590 .534670 41 42 879.301 866.214 1.091 .448362 .568442 1 1887. 874 874.531! 1.112 .722205 .852747 42 43 889.949 877.334 1.051 .755976 .8865181 898.602 884.614 1.166 5.077717 5.225174 43 44 900.438 887.192 1.104 5.111488 5.2589451 44 52. 7r — 36.3501. 53. 7r = 38.0599. TABLE LXIII. *177 GENEIIAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. 54. 66. ^S V. V. 4m.d. A(B/). XB. V. V. ^m.d. X{Bf). AB. n. 6m. 12m. 6m. 12m. 6m. 12m. 6m. 12ra. n. 32.000 24.121 .657 0.012590 0.024825' 33.424 25.033 .699 0.012851 0.025333 1 56.275 48.550 .644 .326471 .338953 58.608 50.367 .687 .327016 .339764 1 2 80.851 73.273 .632 .515902 .528650 84.064 75.947 .676 .516776 .529824 2 3 10.5.694 98.236 .621 .654721 .667769 109.761 101.759 .667 .655969 .669339 3 4 130.771 123.426 .612 .766103 .779473 135.671 127.768 .659 .767775 .781500 4 5 156.056 148.807 .604 0.860405 0.874130 161.752 153.921 .653 0.862562 0.876680 5 6 181.507174.329 .598 j .943189 .957307 187.956 180.177 .646 .945901 .960450 6 7 207.0801199. 952 .594 1.017813 1.032362 .101486 214.253 206.513 .645 1.021148 1.036166 7 8 232.742|225.652 .591 .086468 240.594|232.861 .644 .090511 .106050 8 9 258.4481251. 365 .590 .150690 .166229 266.932 259.188 .645 .155532 .171641 9 10 284.150 277.057 .591 1.211610 1.227719 293.218 285.433 .649 1.217360 1.234098 10 11 309.803 302.669 .595 .270105 .286843 319.381311.514 .656 .276869 .294291 11 12 335.334 328.121 .601 .326868 .344290 345.425 337.520 .659 .334781 .352962 12 13 360.750 353.499 .604 .382491 .400672 371.309 363.284 .669 .391685 .410686 13 14 386.010 378.642 .614 .437472 .456473 396.929 388.758 .681 .448111 .468029 14 15 411.011 403.502 .626 1.492270 1.512188 422.254 413.935 .693 1.504537 1.525448 15 16 435.726 428.071 .638 .547312 .568223 447.249 438.748 .708 .561391 .583398 16 17 460.118 452.286 .653 .602984 .624991 471.863 463.163 .725 . .619102 .642306 17 18 484.139 476.112 .669 .659685 .682889 496.064 487.150 .743 .678070 .702583 18 19 507.756 499.521 .686 .717790 .742303 519.796 510.628 .764 .738731 .764695 19 20 530.916 522.432 .707 1.777712 1.803676 543.019 533.594 .785 1.801501 1.829038 20 21 553.577 544.844 .728 .839851 .867388 565.716i556.024 .808 .866797 .896053 21 22 575.729 566.734 .750 .904610 .933866 587.846 577.852 .833 .935103 .966260 22 23 597.323 588.034 .774 .972462 2.003619 609.372 599.077 .858 2.006903 2.040126 23 24 618.331 608.748 .799 2.043879 .077102 630.284 619.677 .884 .082711 .118194 24 25 638.739 628.851 .824 2.119367 2.154850 650.566 639.640 .910 2.163082 2.201038 25 26 658.531 648.333 .850 .199471 .237427 670.219 658.982 .936 .248609 .289255 26 27 677.710 667.208 .875 .284780 .325426 689.236 677.676 .963 .339957 .383563 27 281696.269 685.450 .902 .375951 .419557 707.651 695.811 .987 .437805 .484597 28 29 714.238 703.148 .924 .473657 .520449 725.505 713.383 1.010 .542930 .593241 29 30 731.662 720.296 .947 2.578669 2. 628980 742.866 730.534 1.028 2.656166 2.710270 30 31 748.604 737.034 .964! .691818 .745922 759.912 747.476 1.036 .778327 .836520 31 32 1765.240 753.567 .973 .813912 .872105 776.810 764.328 1.040 .910546 .973309 32 33,781.730 770.013 .976 .946081 3.008844 793.941 781.739 1.017 3.053775 3.121308 33 34 798.448 787.004 .954 3.089275 .156808 810.346 797.139 1.101 .213334 .289004 34 35 814.457 802.032 1.035 3.248807 3.324477 824.599 810.243 1.196 3.394392 3.478991 35 36 828.366 814.820 1.129 .429848 .514447 837.279 822.500 1.232 .596622 .688999 36 !37 840.740 826.781 1.163 .632067 .724444 849.314 834.314 1.250 .820094 .920604 37 38 852.485 838.310 1.181 .855532 .956042 860.825 845.521 1.275 4.068129 4.178020 38 39 1863.718 849.247 1.206 4.103563 4.213454 871.889 856.441 1.287 .344586 .464666 39 40 874.515 859.904 1.218 4.380017 4.500097 883.314 868.373 1.245 4.652202 4.782744 40 41 885.665 871.548 1.176 .687632 1 .818174 894.570 878.951 1.302 5.007715 5.155172 41 42 896.649 881.871 1.231 5.043145 :5. 190602 907.380j893.993 1.116 .410448 .570149 42 54. Tr = 39.8789. 55. 7T = 41.8154. ^178 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. ;a3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 X = 56. V. 6m. 30 31 32 34.930 61.041 87.396 113.973 140.723 167.600 194.572 221.589 248.604 275.564 302.400 329.111 355.661 381.938 407.914 433.550 458.796 483.620 507.961 531.779 555.060 577.757 599.836 621.285 642.088 662.245 681.751 700.638 718.950 736.758 V. 12m. 754.241 771.573 789.144 33 805.971 34i820.589 35 '833.594 36 845.939 37 857.745 38 869.093 39 880.812 I 40 892.357 41 42 905.496 926.405 25.984 52.221 78.696 105.373 132.197 159.128 186.139 213.164 240.167 267.086 293.837 320.510 346.936 373.064 398.887 424.337 449.380 473. 983 498.063 521.619 544.625 567.013 588.783 609.912 630.388 650.226 869.400 688.000 706.024 723.615 ^mA. 740.992 758.277 776.135 791.931 805.371 817.942 830.060 841.554 852.755 864. 994 875.843 891.272 917.662 .745 .735 .725 .717 .711 .706 .703 .702 .703 .706 .714 .717 .727 .740 .752 .768 .785 .803 .825 .847 .870 .895 .921 .948 .975 1.002 1.029 1.053 1.077 1.095 A(B/). 6m. 0.013135 .327616 .517736 .657338 .769614 0.864939 .948885 1.024822 .094965 .160875 1.223696 .284333 .343503 .401831 .459862 AB. 12m. 0.025883 .340664 .531106 .671063 .783732 0.879488 .963903 1.040361 .111074 .177613 1.241118 ,302514 362504 421749 480773 57. V. 6m. 1.518076 1.540083 .576944 .600148 36.505 63.564 90.849 118.314 145.907 173.598 i 201. 338 '229.072 256.752 284.302 311.727 338.985 365.963 392.632 418.952 .636897 .698397 .761881 .661410 .724361 .789418 1.827783 1.857039 .896600; .927757 .968831 2.002054 2.0449981 .080481 .125667 .163623 2.211438'2.252084 V. 12m. 26.937 54.118 81.507 109.047 136.695 164.428 192.174 219.897 247.534 274.998 302.383 329.514 356.339 382.851 408.980 im.d, 444. 8721434. 691 470.357 495.348 519.801 459.950 484.673 508.857 543.704 532.477 567.006 589.674 611.696 633.053 653.748 1.104 1.108 1.084 1.170 1.268 1.304 1.323 1.349 1.361 1.318 .302984 .400991 .506242 .619575 2.741811 .874085 3.017354 .176942 .358020 3.560262 .783742 4.031781 .308241 .615858 1.376 4.971372 1.185 15.374105 0.729 I .847673 .346590 .447783 .556553 .673679 2.800004 .936848 3.08488 .252612 .442619 3.652639 884252 4.141672 .428321 .746400 5.118829 533806 6.027960 56. 77 = 43.8764. 555.462 577.813 599.506 620.528 640.895 673.774 660.580 693.165 679.677 711.966 730.248 748.198 698.181 716.242 734.082 765.992:751.828 784.032 801.308 816.316 829.668 842.342 854.464 866.114 878.146 889.998 770.163 786.380 800.179 813.085 825.526 837.328 848.827 861.392 872.531 903.488 888.371 924.955 915.465 .797 .787 .779 .772 .751 .768 .764 .765 .768 .775 .779 .789 .802 .815 .831 .848 .867 .889 .912 .936 .962 .988 1.016 1.044 1.071 1.099 1.124 1.149 1.167 1.176 1.180 1.156 1.244 1:345 1.382 1.401 1.428 1.441 1.396 1.456 1.260 0.791 6m. 0.013452 .328277 .518790 .658846 AB. 12m. 0.026500 .341647 .532515 .672964 .771642 .786191 0.867556 .952172 1.028870 .099880 .166762 1.230688 .292560 .353131 .413038 .472832 0.882574 .967711 1.044979 .116618 .184184 1.248869 .311561 .373049 .433949 .494839 1.533034 1.55G238 15 .594117 .656575 .720871 .787461 1.856856 1 .929573 2.006144 .087148 .173193 o a> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 2.264962 .363147 .468538 .581980 .704299 2.836635 .979950 3.139570 .320669 .522925 3.746413 994458 4.270920 .578540 .934054 5.336787 .810356 .618630116 .6825391 17 .7484081 18 .816717 19 .888013 20 .962796 21 .041627122 .125104 23 .213839 24 .308568' 25 .409939 26 ,518849 27 ,636084 28 ,762492129 2.899398 3.047483 .215240 .405268 .615302 3.846923 4.104349 391000 709082 5.081511 5.496488 .990643 57. 77 = 46.0730. 30 31 32 33 34 35 36 37 38 39 40 41 ♦179 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. o o n. 58. 77 = 48.4132. 59. 7T = 50.9095. n. V. 6m. V. 12m. |m.d. 6m. AB. 12m, V. 6m. V. 12m. l^m.d. 6m. AB. 12m. 38.173 27.933 .853 0.013795 0.027165 39.933 28.956 .915 0.014173 0.027898 1 66.213 56.080 .844 .328998 .342723 68.969 58.071 .908 .329795 .343913 1 2 94.438 84.383 .838 .519949 .534067 98.141 87.302 .903 .521229 .535778 2 3 122.797 112.797 .833 .660507 .675056 '127.417 116.621 .900 .662336 .677354 3 4 151.253 141.297 .830 .773873 .788891 156.742 145.954 .899 .776330 .791869 4 5 179.761 169.811 .829 0.870437 0.885976 186.064 175.264 .900 0.873613 0.889722 5 6 208.263 198.302 .830 .955794 .971903 215.328 204.482 .904 .959790 .976528 6 7 236.709 226.704 .834 1.033336 1.050074 244.455 233.518 .911 1.038258 1.055680 7 8 265.023 254.928 .841 .105296 .122718 273.449 262.469 .915 .111272 .129453 8 9 293.206 283.071 .845 .173258 .191439 302.265 291.152 .926 .180419 .199420 9 10 321.218 310.953 .855 1.238395 1.257396 330.788 319.513 .940 1.246903 1.266821^10 11 348.944 338.521 .869 .301642 .321560 358.982 347.541 .953 .311673 .332584 11 12 376.350 365.767 .882 .363764 .384675 i386.808 375.165 .970 .375503 .397510 12 13 403.400 392.619 .898 .425408 .447415 414.211 402.347 .989 .439074 .462278 13 14 430.037 419.042 .916 .487160 .510364 441.154 429.051 1.009 .502984 .527497 14 15 456.227 445.000 .936 1.549549 1.574062 1467.575 455.189 1.032 1.567818 1.593782 15 16 481.910 470.407 .959 .613109 .639073 1493.428 480.757 1.056 .634113 .661650 16 17 507.041 495.261 .982 .678334 .705871 '518.697 505.728 1.081 .702384 .731640 17 18 531.604 519.5.35 1.006 .745707 .774963 543.333 530.028 1.109 .773188 .804345 18 19 555.552 543.156 1.033 .815760 .846917 567.298 553.658 1.137 .847083 .880306 19 20 578.848 566.126 1.060 1.889028 1.922251 590.580 576.592 1.166 1.924631 1.960114 20 21 601.479 588.419 1.088 .966056 2.001539 613.159 598.816 1.195 2.006442 2.044398 21 22 623.427 610.023 1.117 2.047438 .085394 635.038 620.349 1.224 .093147] .133793 22 23 644.695 630.954 1.145 .133793 .174439 656.210 641.161 1.254 .1854491 .229055 23 24 665.276 651.184 1.174 .225812 .269418 676.710 661.350 1.280 .284061 .330853 24 25 685.203 670.810 1.199 2.324197 2.370989 696.587 680.913 1.306 2.389788 2.440099 25 26 704.525 689.826 1.225 .429746 .480057 715.915 700.007 1.326 .503491 '.557595 26 27 723.313 708.387 1.244 .543311 .597415 734.892 718.868 1.335 .626008 .684201 27 28 741.760 726.721 1.253 .665723 .723916 753.704 737.629 1.340 .758492 .821255 28 29 760.046 744.958 1.257 2.798128 .860891 772.776 757.013 1.314 .901915 .969448 29 30 778.586 763.800 1.232 2.941494 3.009027 791.040 774.158 1.407 3.061611 3.137281 30 31 796.339 780.466 1.323 3.101150 .176820 806.907 788.746 1.513 .242761 .327360 31 32 811.763 794.647 1.426 .282273 .366872 821.024 802.391 1.553 .445051 .537428 32 33 825.486 807.911 1.465 .484545 .576922 834.422 815.544 1.573 .668560 .769070 33 34 838.510 820.696 1.484 .708043 .808553 847.237 828.020 1.601 .916617 4.026508 34 35 850.966 832.824 1.512 3.956094 4.065985 859.554 840.178 1.615 4.193086 4.313166 35 36 862.940 844.642 1.525 4.232559 .352639 872.275 853.462 1.568 .500710 .631252 36 37 38 875.305 887.485 857.555 869.002 1.480 1.540 .540181 .895695 .670723 5.043152 884.805 865.238 1.631 .856225 5.003682 37 51. 903.790 891.899 50. .991 5.176810 5.324267 46 45 902.165 889.617 1.046 5.144503 5.291960 893.739 882.453 .941 4.821300 4.951842 45 44 891.902 879.971 .994 4.788992 4.919534 883.535 871.797 .978 .513687| .633767 44 51. 77 = 34.7419. 50. TT = 33.2283. Continued from Page *175. *180 TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. n. a; = 60. TT =53.5745. 61. 77 = 56.4196. n. V. 6m. V. 12in. Jm.d. 6m. AB. 12m. V. 6m. V. 12m. l^m.d. A(B/-). 6m. AB. 12m. 41.779 29.984 .983' 0.014592 0.028710 43.727 31.033 1.058 0.015054 0.029603 1 71.822 60.086 .978 .330676 .345225 74.806 62.159 1.054 .331641 .346659 1 2 101.970 90.279 .974 .522636 .537654 105.940 93.301 1.053 .524183 .539722 2 3 132.170 120.487 .974 .664348 .679887 137.069 124.418 1.054 .666567 .682676 3 4 162.366 150.671 .975 .779039 .795148 168.138 155.437 1.058 .782025 .798763 4 5 192.502 180.760 .979 0.87711610.893854 199.060 186.263 1.066 0.880975 0.898397 5 6 222.498 210.662 .986 .964193! .981615 229.841 217.000 1.070 .969050 .987231 6 7 252.357 240.477 .990 1.0436881.061869 260.436 247.451 1.082 1.049671 1.068672 7 8 282.033 270.015 1.001 .117859 .136860 290.715 277.560 1.096 .125128 .145046 8 9 311.405 299.221 1.015 .188323 .208241 320.648 307.316 1.111 .197052 .217963 9 10 340.440 328.085 1.030 1.256299 1.277210 350.190 336.643 1.129 1.266672 1.288679 10 11 369.096 356.533 1.047 .322746 .344753 379.282 365.501 1.148 .334978 .358182 11 12 397.316 384.526 1.066 .388471 .411675 407.886 593.852 1.170 .402793 .427306 12 13 425.063 412.026 1.086 .454167 .478680 435.936 421.600 1.195 .470862 .496826 13 14 452.272 438.943 1.111 .520487 .546451 463.382 448.744 1.220 .539848 .567385 14 15 478.895 465.273 1.135 1.588023 1.615560 490.210 475.255 1.246 1.610360 1.639616 15 16 504.918 490.989 1.161 .657331 .686587 516.364 501.053 1.276 .683027 .714184 16 17 530.288 516.014 1.190 .728998 .760155 541.806 526.140 1.305 .758466 .791689 17 18 554.968 540.348 1.218 .803611 .836834 566.524 550.48811.336 .837290 .872773 18 19 578.944 563.966 1.248 .881753 .917236 590.496 574.0831.368 .920149 .958105 19 20 602.197 586.854 1.279 1.964052 2. 002008 613.723 596.9431.398 2.007708 2.048354 20 21 624.729 609.029 1.308 2.051156 .091802 636.200 619.0381.430 .100699 .144305 21 22 646.531 630.460 1.339 .143780 .187386 657.9651640.4721.458 .199859 .246651 22 23 667.643 651.252 1.366 .242648 .289440 679.067|661.24l!l.486 .306018 .356329 23 24 688.112 671.398 1.393 .348577 .398888 699.586 681.5121.506 .420055 .474159 24 25 708.017 691.061 1.413 2.462436 2.516540 719.734 701.5361.516 2.542825 2.601018 25 26 1727.560 710.485 1.423 .585071 .643264 739. 705|721.454 1.521 .675498 .738261 26 27 [746. 9321729. 806 1.427 .717644 .780407 759.9531742.032:1.493 .819059 .886592 27 28 766.574 749.767 1.401 .861132 .928665 779.343 760.2341.592 .978853 3.054523 28 29:785.382 767.423 1.497 3.020874 3. 096544 796.188 775.7221.706 3.160070 .244669 29 30 1801. 722 782. 447 1.606 3.202055 3.286654 811.175 790.2081.747 3.362403 3.454780 30 31i816.260j796.499 1.647 .404365 .496742 825.400 804.172ll.769 .585938 .686448 31 32i830.058!810.043 1.668 .627887 .728397 839.005i817.417|1.799 .834011 .943902 32 33 1843. 2541822. 892 1.697 .875951 .985842 852.081 830.325 1.813 4.110487 4.230567 33 34 855.939 835.412 1.711 4.152423 4.272503 865.586 844.427 1.763 .418116 .548658 34 35 36 869.039 849.092 881.943 861.219 1.662 1.727 4.460049 .815565 4.590591 .963022 878.888|856.929!l.830 4.773633 4.921090 35 49. 906.763 897.099 896.073 886.992 48. .891 .842 5.239465 4.883955 5.386922 .014497 48 47 47s905.321 894.049 .939 5.208451 5.355908 46,895.469 884.790 1 .890 4.852941 4.983483 887.290 876.747 .879 .576344 4.696424 46 45i885.469 874.346 ! .927 .545329 .665409 877.791 867.372 .868 .299901 .409792 45 44 875.785 864.788 J .916 .268885 .378776 867.908 857.750 .847 .051887 .152397 44 49. 77 = 31.8029. 48. 77=30.4602. Continued from Page *174. TABLE LXIII. ♦181 GENERAL VALUATION OP ANNUAL PREMIUM POLICIES OP 1,000. 4 PER CENT. 11 62. 7r= 59.4582. 63., 77 = 62.7079. V. V. ^m.d. ^(B/). AB. V. V. |m.d. MB/). AB. n. 6m. 12m. 6m. 12m. 6m. 12m. 6m. 12m. n. 45.791 32.123 1.139 0.015557 0.030575 47.957 33.206 1.229 0.016117 0.031656 1 77.922 64.263 1.138 .332703 .348242 81.150 66.385 1.230 .333876 .349985 1 2 110.048 96.375 1.139 ,525890 .541999 114.276 99.460 1.235 .527773 .544511 2 3 142.110 128.388 1.144 .669013 .685751 147.249 132. .329 1.243 .671708 .689130 3 4 174.024 160.202 1.152 .785315 .802737 180.070 165.103 1.247 .788943 .807124 4 5 205.791 191.922 1.156 0.885232 0.903413 212.691 197.572 1.260 0.889920 0.908921 5 6 237.365 223.349 1.168 .974401 .993402 244.979 229.677 1.275 .980304 1.000222 6 7 268.614 254.422 1.183 1.056274 1.076192 276.895 261.406 1.291 1.063566 .084477 7 8 299.506 285.132 1.198 .133156 .154067 308.396 292.677 1.310 .142016 .164023 8 9 329.994 315.398 1.216 .206687 .228694 339.416 323.447 1.331 .217330 .240534 9 10 360.018 345.180 1.236 1.278129 1.301333 369.915 353.676 1.353 1.290782 1.315295 10 11 389.538 374.438 1.258 .348487 .373000 399.824 383.264 1.380 .363433 .389^97 11 12 418.486 403.076 1.284 .418636 .444600 429.090 412.208 1.407 .436163 .463700 12 13 446.812 431.089 1.310 .489331 .516868 457.696 440.476 1.435 .509753 .539009 13 14 474.498 458.449 1.337 .561254 .590510 485.584 467.984 1.467 .584950 .616107 14 15 501.490 485.074 1.368 1.635084 1.666241 512.713 494.733 1.498 1.662469 1.695692 15 16 527.748 510.964 1.399 .711483 .744706 539.068 520.695 1.531 .742997 .778480 16 ;i7 553.257 536.092 1.430 .791095 .826578 564.628 545.853 1.565 .827247 .865203 17 18 577.996 560.442 1.463 .874598 .912554 589.395 570.229 1.597 .915934 .956580 18 19 601.968 584.03511.494 .962681 2.003327 613.362 593.788 1.631 2.009829 2.053435 19 20 625.165 606.837 1.527 2.056091 2.099697 636.569 616.642 1.661 2.109709 2.156501 20 21 647.626 628.957 1.556 .155586 .202378 659.069 638.788 1.690 .216433 .266744 21 22 669.403 650.391 1.584 .262008 .312319 680.950 660,403 1;712 .330904 .385008 22 23 690.580 671.312 1.606 .376247 .430351 702.432 681.754 1.723 .454006 .512199 23 24 711.374 691.977 1.616 .499172 .557365 723.727 702.992 1.728 .586925 .649688 24 25 731.984 712.533 1.621 2.631959 2.694722 745.318 724.935 1.699 2.730665 2.798198 25 26 752.881 733.771 1.593 .775604 .843137 765.993 744.343 1.804 .890586 .966256 26 27 772.892 752.555 1.695 .935457 3.011127 783.954 760.857 1.925 .3.071889 3.156488 27 28 790.276 768.539 1.811 3.116714 .201313 799.934 776.304 1.969 .274279 .366656 28 29 805.743 783.489 1.854 .319074 .411451 815.103 791.193 1.993 .497847 .598357 29 30 820.423 797.900 1.877 3.542624 3.643134 829.608 805.316 2.024 3.745941 3.855832 30 31 834.464 811.570 1.908 .790707 .900598 843.552 819.079 2.039 4.022428 4.142508 31 32 847.959 824.890 1.922 4.067188 4. 187268 857.951 834.116 1.986 .330062 .460604 32 33 34 861.897 875.625 839.445 852.347 1.871 1.940 .374819 .730338 .505361 .877795 872.136 847.447 2.057 .685583 .833040 33 49 47. 909.402 900.085 899.781 891.023 46. .802 .755 5.29975815.447215 4.944248 5.074790 50 49 908.121 897.981 .845 5.269889 5.417346 48 898.636 889.067 .797 4.914379 .044921 890.624 881.145 .790 .636639 4.756719 48 47 889.006 879.010 .833 .606769 4.726849 881.465 872.103 .780 .360198 .470089 47 1 46 879.682 869.807 .823 .330327 .440218 871.935 862.825 .759 .112187i .212697 46 1 45 869.981 860.362 .802 .082315 .182825 861.970 853.044 .744 3.888759 3.981136 45 1 44 859.838 850.405 .786 3.858883 3.951260 851.472 842.896 .715 3.686599 3.771198 44 47. 7r= 29.1948. 46. 77 = 28.0014. Continued from Page *173. *183 TABLE LXIII, GEKERAL VALUATION OP ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. •^3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 51 50 49 48 47 46 45 44 a; = 64. 7T = 66.1832. V. 6m. 50.251 84.515 118.621 152.569 186.311 219.707 252.720 285.302 317.387 348.935 379.872 410.142 439.730 468.576 496.636 523.898 550.335 575.953 600.744 624.747 648.020 670.652 692.872 714.899 737.230 758.616 777.195 793.723 809.412 824.417 838.838 853.734 868.404 V. 12m. 34.318 68.530 102.528 136.427 170.012 203.219 236.038 268.383 300:209 331.478 362.082 392.019 421.258 449.711 477.379 504.233 530.255 555.468 579.836 603.475 626.382 648.739 670.823 692.791 715.487 735.562 752.644 768.620 784.021 798.629 812.865 828.419 842.207 ^m.d 1.328 1.332 1.341 1.345 1.358 1.374 1.390 1.410 1.432 1.455 1.482 1.510 1.539 1.572 1.605 1.639 1.673 1.707 1.742 1.773 1.803 1.826 L837 1.842 1.812 1.921 2.046 2.092 2.116 2.149 2.164 2.110 2.183 6m. 0.016730 .335169 .529844 .674678 .792937 0.895089 .986824 1.071612 .151802 .229081 1.304781 .379966 .455545 .532361 .611199 1.692801 .777922 .867311 '.961768 2.062093 2.169165 283905 407210 540281 684131 2.844130 3.025486 227910 451500 699606 3.976100 4.283737 639260 AB. 12m. 0.032839 .351907 .547266 .692859 .811938 0.915007 1.007735 .093619 .175006 .253594 1.330745 .407503 .484801 .563518 .644422 1.728284 .815878 .907957 2.005374 .108885 2.219476 338009 465403 603044 751664 2.919800 3.110085 320287 552010 809497 4.096180 414279 786717 45. 910.611 901.451 892.151 883.147 901.478 892.869 883.157 874.269 873.778l865.148 863.983 853.663 842.062 855.533 845.557 834.892 .761 .715 .750 .740 .719 .704 .675 .598 5.329103 4.973594 4.665985 .389545 .141536 3.918111 .715956 3.535015 5.476560 .104136 4.786065 .499436 .242046 .010488 3.800555 3.610685 45. TT = 26.8757. 65. V. 6m. V. 12m. 35. 70. 52.665 87.981 123.137105. 158.077140. 192.661174. 226.8461208. 260.586 242. 293.8121275. 326.481307. 358.517 339. 427 634 737 516 |m.d. 1.437 1.446 1.450 1.463 77 = 69.9017. |>.g laa o« A(B/). 6m. 903 1.480 382 340 720 411 389.863!370.413 420.503 400.691 450. 3741430. 155 479.431|458.806 507.661486.614 535.0391513.562 561.567(539.670 587.238'564.904 612.095:589.384 636. 1951613. 104 659.631 682. 642 705.450 728.575 750.721 769.960 787.076 803.323 818.860 833. 794 849.219 864.411 636.256 659.125 681.873 705.376 726.164 743.853 760.398 776.345 791.473 806.214 822.321 836.600 1.496 1.517 1.539 1.563 1.592 1.621 1.651 1.685 1.719 1.754 1.790 1.825 1.861 1.893 1.924 0.017409 .336590 .532128 .677946 .797339 0.900800 .994016 1.080498 .162604 .242082 1.320266 .398248 .477002 .557408 .640277 1.726422 .816633 .911746 2.012589 .120066 1.948 12.235117 1.960 .358659 1.965 1.933 2.046 2.176 2.223 2.248 2.282 2.298 2.242 2.318 .491905 .635882 .795971 2.977388 3.179852 .403466 .651587 .928088 4.235729 .591254 AB. 12m. 0.034147 .354012 .550309 .696947 .817257 0.921711 1.016023 .103702 .187117 .268046 n. 1.347803 10 .427504J 11 .5081591 12 .590631 .675760 1.764378 .857279 .955352 2.059381 .170377 2.289221 .416852 .554668 .703415 .871641 3.061987 .272229 .503976 .761478 4.048168 4.366271 .738711 44. 911. 902. 893. 884. 875. 865. 855. 844. 752 740 592 734 518 882 729 318 831.182 903.080 894.611 885.057 876.314 867.341 857.882 848.068 837.577 825.246 .723 .677 .711 .702 .681 .667 .638 .562 .495 5.357955 .002446 4.694837 5.505412 .132988 4.814917 4.418398:4.528289 .170391 .270901 3.946968! .039345 .744818 3.829417 563883 3.404507 .639553 3.472040 44. TT = 25.8133. 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Continued from Page *172. *183 TABLE LXIII, GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OP 1,000. 4 PER CENT. O V n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 66. TT = 73.8817. V. 6ni. V. 12m. 36.499 72.893 108.948 144.598ll.593 179.8321.610 55.191 91.637 127.861 163.714 199.156 234.135 268.582 302.450 335.662 368.160 399.925 430.893 461.018 490.284 518.668 546.170 572.784 598.555 623.539 647.836 671.692 695.338 719.313 742.272 762.217 779.962 751.597 796.805 768.131 812.913l783.814 im.d, 1.558 1.562 1.576 214.556 248.725 282.293 315.149 347.289 378.679 409.225 438.929 467.758 495.695 522.763 548.924 574.303 598.894 622.896 646.605 670.189 694.555 716.107 1.632 1.655 1.680 1.709 1.739 1.771 1.806 1.841 1.877 1.914 1.951 1.988 2.021 2.054 2.078 2.091 2.096 2.063 2.180 828.397 844.387 860.137i830.598 734.445 2.314 2.364 2.389 2.425 2.442 2.383 2.462 799.097 815.795 A(B/). 6m. 0.018151 .338158 .534636 .681545 .802203 0.907098 1.001958 .090304 .174556 .256464 1.337388 .418490 .500777 .585157 .672514 1.763695 .859579 .961029 2.068979 .184394 2.308211 .441661 .585786 .745980 .927468 3.129977 .353620 .601757 .878267 4.185913 4.541440 AB. 12m. 0.035573 .356339 .553637 .701463 .823114 0.929105 1.025162 .114817 .200520 .284001 1.366644 .449647 .534000 .620640 .710470 1.804341 903185 2.007821 .119290 .238498 2.366404 .504424 .653319 .821650 3.012067 3.222354 .454130 .711648 .998347 4.316455 4.688897 43. 53 912.830 52 903.959 51 894.953 50 886.234 49 48 47 46 45 44 877.161 867.676 857.681 846.448 833.518 820.015 904.593 896.256 886.852 878.245 869.412 860.101 850.440 840.112 827.974 814.252 .686 5.386341 .642 I .030832 .675 14.723224 .666 .446786 .646 .631 .603 .528 .462 4.198780 3.975360 .773214 .592286 .432918 480 3.289958 3.352721 5.533798 .161374 4.843304 .556677 4.299290 .067737 3.857813 .667956 .500451 43. -n = 24.8099. 67. 7T = 78.1464. V. 6m. 57.956 95.553 132.763 169.548 205.852 241.604 276.755 311.225 344.955 377.922 410.064 441.329 471.705 501.164 529.707 557.330 584.076 610.008 635.225 659.984 V. 12m. 37.772 75.193 112.194 148.762 184.802 220.265 255.105 289.206 322.563 355.142 386.845 417.674 447.596 476.591 504.684 531.836 558.176 583.699 608.611 633.218 684.527 657.695 709.409 682.984 733.238 705.352 753.939 772.356 789.837 806.556 822.625 839.222 855.569 724.385 742.187 759.347 775.625 791.486 808.817 824.181 im.d. 1.682 1.697 1.714 1.732 1.754 1.778 1.804 1.835 1.866 1.898 1.935 1.971 2.009 2.048 2.085 2.125 2.158 2.192 2.218 2.230 2.236 2.202 2.324 2.463 2.514 2.541 2.578 2.595 2.534 2.616 6m. 0.018977 .339877 .537400 .685527 .807566 0.914052 1.010719 .101154 .187778 .272364 1.356348 .440919 .527119 .615927 .708260 1.805057 .907224 2.015731 .131572 .255712 2.389400 .533699 .694016 .875587 3.078149 3.301825 .549982 .826502 4.134153 .489683 AB. 12m. 0.037168 .358878 .557318 .706438 .829573 0.937256 1.035232 .127118 .215315 .301620 1.387505 .474142 .562602 .653883 .748906 1.848663 .954016 2.066042 .185676 .313905 o « n. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2.452163 .601232 .769686 .960186 23 3.170526 24 20 21 22 3.402335 .659873 .946582 4.264695 .637140 42. 913.846 905.109 896.237 887.648 878.712 869.368 859.523 848.459 835.722 822.421 906.021 897.809 888.545 880.067 871.366 862.194 852.679 842.505 830.549 817.032 809.303 803.948 .652 .608 .641 .632 .612 .598 .570 .496 .431 ,449 5.414289 .058780 4.751172 .474735 .226730 4.003313 3.801170 .620248 .460889 .317940 446 3.186108 5.561746 .189322 4.871252 .584626 .327240 4.095690 3.885769 .695918 .528422 .380703 3.244301 42. n = 23.8627. 25 26 27 28 29 54 53 52 51 50 49 48 47 46 45 44 Continued from Page *171, *184 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. 5*8 e o n. 1 2 3 4 5 6 7 8 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 55 54 53 52 51 50 49 48 47 46 45 44 a; = 68. TT = 82.7146. V. 6in. 60.803 99.474 137.703 175.432 212.587 249.118 284.942 819.994 354.257 387.659 420.153 451.721 482.335 512.001 540.707 568.503 595.453 621.660 647.391 672.896 698.757 723.520 745.034 764.174 782.341 799.716 816.416 833.664 850.653 V. 12m. Jm.d. 38.890 77.344 115.347 152.802 189.657 225.864 261.304 295.970 329.828 362.776 394.815 425.911 456.045 485.241 513.458 540.883 567.358 593.247 618.820 644.258 670.540 693.786 713.566 732.067 749.900 766.817 783.301 801.312 817.279 A(B/). 6m. 1.826 0.019871 1.844 1.863 1.886 1.911 1.938 1.970 2.002 2.036 2.074 2.112 2.151 2.191 2.230 2.271 2.306 2.341 2.868 2.381 2.386 .341777 .540461 .689912 .813485 0.921718 1.020413 .113155 .202390 .289970 1.377356 .465773 .556332 .650050 .747937 AB. 12m. 69. 7r= 87.6177. V. 6m. 0.038872 63.814 361695 103.588 .561372 .711919 .836689 0.946231 1.046877 .140692 .281646 .321127 1.410579 .501256 .594288 .690696 .791543 1.850956 1.897748 .960123 2.010434 2.076469 .200989 .334959 3512. 478 622 676 703 742 760 696 781 479462 .639923 .821592 .024217 .247932 .496111 .7726441 , .0803024 .435835 . .130573 .259182 .397722 J. 546995 .715593 .906191 }. 116594 .348442 J. 606002 .892724 210844 583292 41. 914.809 907.373 906.197 897.453 888.988 880.179 870.970 861.267 850.362 837.807 824.698 811.769 899.278 890.148 881.792 873.216 864.176 854.797 844.770 832.986 819.662 806.767 798.724 793.803 620 5.4418215.589278 .577 .609 .600 .580 .566 .539 .466 .402 .420 .417 .410 5.086313 4.778705 .502269 .254265 .030850 3.828711 .647794 .488443 .345505 .213688 3.092060 5.216855 4.898785 .612160 .354775 .123227 3.913310 .723464 .555976 .408268 .271881 3.146164 142.845 181.503 219.512 256.786 293.256 328.906 363.660 397.469 430.313 462.166 493.082 522.901 551.821 V. 12m. 40.009 79.550 118.521 156.867 194.540 281.413 267.482 302.711 336.992 370.327 402.681 434.034 464.412 493.771 522.253 579.861549.851 607.1281576.788 633.901603.396 660.439|629.863 687.3441657.208 713. 735. 755. 774. 792. 809. 827. 845. 111681. 494 701. 410721. 312739. 390 757. 765 712 388 774. 793. 809. 395 976 226 780 381 532 273 886 ^m.d. 1.984 2.003 2.027 2.053 2.081 2.114 2.148 2.183 2.222 2.262 2.303 2.344 2.385 2.427 2.464 2.501 2.528 2.542 2.548 2.511 2.643 2.793 2.849 2.878 2.917 2.936 2.870 2.960 6m. AB. 12m. 0.020877 .343875 .543830 .694753 .820009 0.930202 1.031134 .126416 .218572 .309480 1.400636 .493338 .588735 .687936 .791977 1.901933 2.018880 .143853 .278157 .422901 2.583533 .765318 .968018 3.191778 .489984 3.716532 4.024198 379735 0.040795 .364786 .565887 .717957 .844522 0.956166 1.058671 .155672 .249729 .342703 O 4> 1.43611P .531294 .629381 .731542 13 .838769 14 1.952244 2.072984 .202046 .340920 .4904341 19 2.659203 20 .849917 21 8.060395:22 .292288 23 .549875 24 3.836612 4.154740 527192 40. 41. 7r = 22.9667. 915.718 907.224 898.601 890.252 881.565 872.483 862.913 852.157 839.777 826.849 814.096 801.232 788.130 908.649 900.666 891.661 883.420 874.963 866.047 856.797 846.908 885.286 822.147 809.429 796.648 783.700 .589 .546 .578 .569 .550 .536 .510 .437 .374 .392 .389 .382 .369 5.468963 .118454 4.805847 .529411 .281409 .057995 3.855859 3.674947 .515604 .372677 .240874 .119264 3.006757 56 55 54 53 52 51 50 .750617 49 .583137 48 .435440 47 .299067 46 .173368 45 .057068 44 5.616420 .243996 4.925927 .689302 .381919 .150372 3.940458 40. 7r = 22.1206. Continued from Page *170. TABLE LXIIT. *186 GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. ^1 n. 70. TT = 92.8725. V. 6m. 67.031 107.923 148.193 187.786 226.612 264.604 301.739 337.941 373.158 407.372 V. 12m. 10 440.553 11 472.705 121503.818 13|533.944 14j563.153 15 1591.557 16 619.445 17i647.089 18|675.116 19 '701.956 20 21 22 23 24 25 26 725.272 746.018 765.707 784.539 802.640 821.333 839.746 41.189 81.784 121.728 160.971 199.381 236.954 273.650 309.359 344.084 377.787 410.447 442.090 472.673 502.342 531.091 559.150 586.867 614.437 642.922 668.117 689.555 709.607 728.935 747.270 765.136 784.657 801.962 im.d. MB/). 6m. AB. 12m. 2.154 0.021969 2.178 .346181 .547542 .700080 .827224 2.205 2.235 2.269 2.304 2.341 2.382 2.423 2.465 2.509 2.551 2.595 2.633 2.672 2.701 2.715 2.721 2.683 2.820 2.976 3.034 3.064 3.106 3.125 0.939579 1.042974 .141097 .236500 .331096 1.426454 .523913 .624715 .729994 .840901 1.958570 2.084085 .218790 .363822 .524658 0.042880 .368188 .570746 .724593 .853188 0.967116 1.072230 .172254 .269723 .366579 1.464410 .564559 .668321 .776786 .891212 2.012674 . 142278 .281553 .431355 .600328 2.706581:2.791180 .909369 3.133184 .381422 .657988 3.056 3.965663 3.149 4.321204 3.001746 .233694 .491313 .778068 4.096205 .468661 39. 57 916.577 56 908.196 55 899.687 1541891.447 53 882.876 52 873.914 |51j864.470 '50 '853.857 49 48 47 46 45 44 841.640 828.882 816.299 803.605 790.675 777.379 909.856 901.979 893.093 884.960 876.615 867.817 858.690 848.931 837.463 824.497 811.947 799.331 786.558 773.472 .560 .518 .550 .541 .522 .508 .482 .410 .348 .374 .363 .356 .343 .325 5.495734 .140226 4.832619 4.556183 .308182 .084770 3.882638 .701730 3.542394 .399476 .267686 .146094 .033609 2.929447 5.643191 .270768 4.952699 4.666074 .408692 .177147 3.967237 .777400 3.609927 .462239 .325879 .200198 .083920 2.976239 71. 7r = 98.5145. V. 6m. V. 12m. ^m.d. A(B/). 6m. 70.427 42.339 2.3410.023182 0.045189 112.426 83.999 2.369 .348727 .371931 1 153.720124.928 2.399 .551629 .576142 2 194.215164.987 2.436 .705977 .731941 3 233.837 204.174 2.472 .835200 .862737 4 272.568 242.447 2.510 0.949934 0.979190 5 310.326 279.691 2.553 1.056082 1.087239 6 347.056 315.907 2.596 .157363 .190586 7. 382.739 351.058 2.640 .256364 .291847 8 417.346 385.120 2.686 .355071 .393027 9 450.879 418.123 2.730 1.455096 1.495742 10 483.328 450.020 2.776 .557872 .601478 11 514.749 480.964 2.815 .664667 .711459 12 545.213 510.947 2.855 .776732 .827043 13 574.836 540.212 2.885 .895277 .94938114 603.923 569.119 2.900 2.021447 2.079640 15 632.753 597.874 2.907 .156634 .219397 16 661.986 627.583 2.867 .302013 .369546 17 689.978 653.860 3.010 .463094 .538764 18 714.297 676.219 3.173 .645183 .729782 19 735.932 697.132 3.233 2.848077 2.940454 20 756.469 717.2913.265 3.071958 3.172468:21 776.100 736.413 3.308 .320234 .430125|22 794.986 755.046 3.328 .596822 .716902 814.483 775.406 3.256 .904507 4.035049 833. 688 793. 455 3.353 4. 260054 4. 407511 AB. 12m. n. 38. 39. 7r = 21.3199. 917.390 909.116 900.714 892.579 884.116 875.267 865.944 855.465 843.402 830.807 818.383 805.849 793.084 789.263 779.957 776.343 766.418|763.009 910.998 903.221 894.448 886.418 878.178 869.492 860.480 850.846 839.522 826.721 814.330 801.874 .533 .491 .522 .513 .495 .481 .455 .385 .323 .341 .338 .331 .318 .301 .284 5.522155 .166647 4.859041 .582606 4.334606 .111196 3.909066 .728163 .568832 3.425923 .294145 .172570 .060106 2.955971 2.859371 5.669612 .297189 4.979121 .692497 4.435116 .203573 3.993665 .803833 .636365 3.488686 .352338 .226674 .110417 .002763 2.902977 23 24 25 38. 77 = 20.5624. 58 57 56 55, 54 53 52 51 50 49 48 47 46 45 44 Continued from Page *169. *186 TABLE LXIII GENERAL VALUATION OP ANNUAL PREMIUM POLICIES OP 1,000. 4 PER CENT. &2 n. a; = 72. TT = 104.570. 73. TT = 111.075. •■si n. V. 6m. V. 12in. ^m.d. A(B/). 6m. aB. 12m. V. 6m. V. 12m. ^m.d. 6m. AB. 12m. 74.036 43.502 2.545 0.024515 0.047719 77.879 44.682 2.766 0.025979 0.050492 1 117.156 86.240 2.576 .351524 .376037 122.086 88.415 2.806 .354626 .380590 1 2 159.441 128.071 2.614 .556152 .582116 165.343 131.196 2.846 .561148 .588685 2 3 200.815 168.990 2.652 .712491 .740028 207.624 172.978 2.887 .719674 .748930 3 4 241.258 208. 955 2.692 .844001 .873257 248.845 213.637 2.934 .853740 .884897 4 5 280.685 247.846 2.737 0.961393 0.992550 288.944 253.175 2.981 0.974085 1.007308 5 6 319.040 285.663 2.781 1.070602 1.103825 327.900 291.549 3.029 1.086682 .122165 6 7 356.300 322.368 2.828 .175381 .210864 365.679 328.735 3.079 .195360 .233316 7 8 392.437 357.936 2.875 .278392 .316348 402.287 364.764 3.127 .302827 .343473 8 9 427.452 392.398 2.921 .381666 .422312 437. 71^ 399.586 3.177 .411211 .454817 9 10 461.337 425.705 2.969 1.486911 1.530517 472.014 433.367 3.221 1.522247 1.569039 10 11 494.146 458.017 3.011 .595585 .642377 505.271 466.100 3.264 .637511 .687822 11 12 525.956 489.326 3.053 .709074 .759385 537.611 498.048 3.297 .758458 .812562 12 13 556.890 519.884 3.084 .828692 .882796 569.365 529.607 3.313 .886412 .944605 13 14 587.262 550.070 3.099 .955661 2.013854 600.840 560.998 3.320 2.022905 2.085668 14 15 617.368 580.096 3.106 2.091434 2.154197 632.752 593.431 3.277 2.169222 2.236755 15 16 647.892 611.118 3.064 .237235 .304768 663.312 622.118 3.433 .330963 .406633 16 17 677.123 638.557 3.214 .398613 .474283 689.861 646.528 3.611 .513497 .598096 17 18 702.516 661.905 3.384 .580902 .665501 713.481 669.359 3.677 .716678 .809055 18 19 725.108 683.742 3.447 .783926 .876303 735.900 691.366 3.711 .940734 3.041244 19 20 746.552 704.792 3.480 3.007885 3.108395 757.341 712.242 3.758 3.189113 3.299004 20 21 767.061 724.760 3.525 .256208 .366099 777.951 732.584 3.781 .465757 .585837 21 22 786.774 744.217 3.546 .532821 .652901 799.235 754.811 3.702 .773472 .904014 22 23 24 59 807.132 765.477 827.185|784.323 3.471 3.572 .840520 1 1 Qfin7^ .971062 820.200 774.514 3.807 4.129033 4.276490 23 Ht,l.V\)\J t rr**j-i:tj*j^\j\ 36. 60 59 37. 918.890 910.810 913.103 905.510 .482 .442 5.574024 5.218516 5.721481 5.349058 918.161 912.080 .507 5.548246 5.695703 58 909.986 904.397 .466 .192738 .323280 902.607 896.944 .472 4.910910 .030990 58 57 901.687 895.730 .496 4.885132 .005212 894.666 889.105 .463 .634476 4.744367 57 56 893.651 887.799 .488 .608697 4.718588 886.402 88L060 .445 .386478 .486988 56 55 885.291 879.659 .469 .360698 .461208 877.763 872.579 .432 .163071 .255448 55 54 876.550 871.078 .456 4.137290 4.229667 868.659 863.780 .407 3.960945 4.045544 54 53 867.339 862.176 .430- 3.935162 .019761 858.429 854.373 .338 .780050 3.855720 53 52 856.988 852.658 .361 .754263 3.829933 846.651 843.318 .278 .620731 .688264 52 51 845.073 841.473 .300 .594939 .662472 834.354 830.819 .295 .477838 .540601 51 50 832.629 828.827 .317 .452038 .514801 822.224 818.722 .292 .346082 .404275 50 49 820.357 816.587 .314 3.320271 3.378464 809.986 806.560 .286 3.224535 3.278639 49 48 807.975 804.281 .308 .198711 .252815 797.523 794.247 .273 .112109 .162420 48 47 795.365 791.824 .295 .086267 .136578 784.707 781.633 .256 .008023 .054815 47 46 782.397 779.061 .280 2.982156 .028948 771.488 768.615 .239 2.911484 2.955090 46 45 769.023 765.889 .261 .885589 2.929195 757.836 755.195 .220 .821765 .862411 45 44 755.211752.312 .242 2.805832 2.836478 743.728 741.310 .201 2.738237 2.776193 44 37. TT = 19.8452. 36. TT = 19.1663. Continued from Page *168. ♦187 TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. 11 n. 74. TT = 118.069. 75. 7r= 135.579. n. V. 6m. V. 13in. -^m.d. A(B/). 6m. AB. 13m.- V. 6m. V. 13m. ^m.d. A(B/). 6m. AB. 13m. 81.924 45. 779 3.013 0.037616 0.053580 86.355 46.930 3.277 0.029403 0.056940 1 137.304 90.560 3.054 .358048 .385585 133.637 93.765 3.323 .361802 .391058 1 2 171.463 134.397 3.097 .566649 .595905 177.855 137.367 3.374 .573730 .603887 2 3 314.613 176.858 3.146 .737630 .758777 331.843 180.740 3.425 .736408 .769631 3 4 356.586 318.344 3.195 .864531 .897744 264.577 333.835 3.479 .876447 .911930 4 5 397.363 358.413 3.346 0.988135 1.033618 306.031 363.638 3.533 1.003701 1.041657 5 6 336.910 397.339 3.398 1.104508 .143464 346.180 303.153 3.586 .134370 .164916 6 7 375.331 335.053 3.348 .317518 .358164 385.041 341.351 3.641 .342130 .285736 7 8 413.313 371.503 3.401 .339973 .373579 433.669 378.409 3.688 .360118 .406910 8 9 448.318 406.864 3.446 .444037 .490819 459.153 414.316 3.736 .480513 .530824 9 10 483.031 441.138 3.493 1.561535 1.611846 494.639 449.363 3.772 1.605247 1.659351 10 11 516.884 474.571 3.536 .684149 .738253 539.463 483.983 3.790 .735978 .794171 11 12 550.133 507.606 3.543 .813330 .871533 563.990 518.419 3.798 .874485 .937248 12 13 583.070 540.466 3.550 .950716 2.013479 598.998 553.998 3.750 2.022241 2.089774 13 14 616.475 574.415 3.505 13.097673 .165306 633.533 585.467 3.921 .184988 .260658 14 15 648.464 604.444 3.668 3.359861 3.335531 661.645i612.244 4.117 2.368201 2.453800 15 16 676.354 639.995 3.855 .443697 .537396 687.557 637.290 4.189 .571816 .664193 16 17 700.9791653.894 3.934 .646073 .738449 712.150 661.431 4.227 .796138 .896648 17 18 734.447 676.931 3.960 .870347 .970757 735.671 684.333 4.278 3.044672 3.154563 18 19 746.893 698.783 4.009 3.118695 3.338586 758.279 706.647 4.303 .321403 .441483 19 20 768.464 730.076 4.033 3.395377 3.515457 781.627 731.039 4.216 3.639161 3.759703 20 21 22 790.743 813.690 743.343 763.968 3.950 4.060 .703111 4.058683 .833653 4.306139 804.626 753.644 4.333 .984743 4.133300 21 34. 62 35. 920.234 914.990 .437 5.634707 5.773164 61 919.579 914.071 .459 5.599505 5.746963 912.330 907.563 .397 .369199 .399741 61 60 911.590 906.563 .419 .343997 .374539 904.305 899.183 .427 4.961593 .081673 60 59 903.478 898.093 .449 4.936393 5.056473 896.535 891.513 .419 4.685161 4.795053 59 58 895 635 890.341 .440 .659958 4.769849 888.453 883.643 .401 .437164 .537674 58 57 887.454 883.385 .433 .411961 .513471 880.000 875.346 .388 .313759 .306136 57 56 878.911 873.999 .409 .188555 .280932 871.094 866.738 .363 .011639 .096338 56 55 869.909 865.398 .384 3.986433 .071031 861.085 857.536 .296 3.830750 3.906430 55 54 859.793 855.996 .316 3.805540 3.881210 849.564 846.730 .237 3.671441 3.738974 54 53 848.146 845.064 .357 .646336 .713759 837.533 834.493 .253 .538561 .591334 53 52 835.986 833.705 .373 .503341 .566104 835.666 833.658 .251 .396834 .455017 52 51 833.990 830.743 .371 .371594 .429787 813.695 810.760 .245 .375303 .339407 51 50 811.890 808.715 .365 .350061 .304165 801.503 798.716 .233 .163910 .313331 50 49 799.565 796.540 .353 3.137653 3.187963 788.965 786.375 .316 3.058866 3.105658 49 48 786.891 784.066 .335 .033587 .080379 776.033 773.639 .199 3.962380 .005986 48 47 773.830 771.193 .319 3.937076 2.980682^ 763.677 760.511 .180 .873736 3.913373 47 46 760.330 757.933 .300 .847391 .8880371 748.876 746.938 .162 .789380 .837336 46 45 746.370 744.193 .181 .763905 .8018611 734.633 733.908 .144 .711436 .746909 45 44 731.973 730.033 .163 3.686005 3.731488 719.945 718.441 .125 3.638638 3.671861 44 35. TT = 18.5341. 34. TT = 17.9149. Continued fio-u Page *167. ♦188 TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OP 1,000, 4 PER CENT. n. .r = 76. 77 = 133.656. 77. 77=142.352. "3 3 n. V. 6m. V. 12m. im.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. im.d. A(B/). 6m. AB. 12m. 90.874 48.092 3.565 0.031370 0.060626 95.758 49.163 3.883 0.033566 0.064723 1 138.320 94.891 3.619 .365966 .397123 144.242 96.970 3.939 .370572 .403795 1 2 184.473 140.399 3.673 .579461 .612684 191.347 143.371 3.998 .586900 .622383 2 3 239.311 184.568 3.729 .746131 .781614 237.029 188.335 4.058 .756894 .794850 3 4 272.797 227.369 3.786 .889659 .927615 281.293 231.900 4.116 .904298 .944944 4 5 314.932 268.839 3.841 1.020957 1.061603 324.128 274.004 4.177 1.040118 1.083724 5 6 355.707 308.919 3.899 .146221 .18982V 365.604 314.851 4.229 .170591 .217383 6 7 395.188 347.801 3.949 .269462 .316254 405.817 354.431 4.282 .299854 .350165 7 8 433.467 385.477 3.999 .393642 .443953 444.922 393.061 4.322 .430952 .485056 8 9 470.691 422.250 4.037 .521119 .575223 483.316 431.220 4.341 .566257 .624450 9 10 507.240 458.574 4.055 1.653836 1.712029 521.375 469.178 4.350 1.708034 1.770797 10 11 543.468 494. 706 4.064 .793770 .856533 559.962 508.394 4.297 .85810* .925634 11 12 580.199 532.036 4.014 .942538 2.010071 596.914 543.081 4.486 2.022455 2.098125 12 13 615.374 565.055 4.193 2.105990 .181660 629.015 572.597 4.702 .206741 .291340 13 14 645.931 593.151 4.398 .289675 .374274 657.576 600.203 4.781 .411044 .503421 14 15 673.118 619.430 4.474 2.493593 2.585970 684.684 626.813 4.823 2.635785 2.736295 15 16 698.923 644. 760 4.514 .718100 .818610 710.610 652.055 4.880 .884565 .994456116 17 723.602 668.788 4.568 .966743 3.076634 735.529 676.651 4.906 3.161431 3.281511' 17 18 747.323 692.202 4.593 3.243532 .363612 761.265 703.527 4.812 .469260 .599802 18 jl9 20 771.822 795.952 717.785 740.464 4.503 4.624 .551322 3.906919 .681864 4.054376 786.615 727.352 4.939 .824875 .972332 19 32. 33. 921.441 916.686 .396 5.674326 K Q01 lyQQ OA O.o-ci too v± 63 920.854 915.861 .416 5.649642 5.797099 913.694 909.406 .357 .318819 .449361 63 62 913.030 908.508 .377 .294135 .424677 905.830 901.193 .386 .011214 .131294 62 61 905.088 900.214 .406 4.986529 .106609 898.216 893.677 .378 4.734781 4.844672 61 60 897.398 892.624 .398 .710097 4.819988 890.293 885.964 .361 .486786 .587296 60 59 889.397 884.834 .380 4.462101 4.562611 882.010 877.833 .348 4.263384 4.355761! 59 58 881.031 876.622 .367 .238698 .331075 873.283 869.397 .324 .061267 .145866 58 57 872.218 868.103 .343 .036579 .121178 863.473 860.378 .258 3.880384 3.956054 57 56 862.311 858.994 .276 3.855693 3.931363 852.181 849.778 .200 .721083 .788616 56 55 850.908 848.290 .218 .696388 .763921 840.391 837.795 .216 .578216 .640979|56 54 839.000 836.188 .234 3.553515 3.616278 828.761 826.196 .214 3.446495 3.504688 54 53 827.255 824.474 .232 .421786 .479979 817.028 814.536 .208 .324996 .379100 53 62 815.406 812.698 .226 .300276 .354380 805.079 802.731 .196 .212632 .262943 52 51 803.339 800.777 .214 .187898 .238209 792.791 790.637 .180 .108624 .155416 61 50 790.929 788.563 .197 .083873 .130665 780.117 778.155 .163 .012186 .055792 50 49 778.130 775.957 .181 2.987412 3.031018 767.028 765.289 .145 2.922589 2.963235 49 48 764.910 762.964 .162 .897788 2.938434 753.502 751.977 .127 .839212 .877168 48 47 751.250 749.519 .144 .814377 .852333 739.543 738.237 .109 .761444 .796927 47 46 737.153 735.643 .126 .736568 .772051 725.149 724.058 .091 .688756 .721979 46 45 722.616 721.324 .108 .663831 .697054 710.333 709.450 .074 .620655 .651812 45 44 707.654 706.571 .090 2.595674 2.626831 695.103 694.427 .056 2.556708 2.585964 44 33. 77 = 17.3376. 32. 77 = 16.7902. Coutinaed from Page *166. TABLE LXIII 189 GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. n. 78. :r = 151.701. 79. u == 161.769. li n. V. 6m. V. 13m. l^m.d. MB/). 6m. AB. 13m. V. 6m. V. 13m. ^m.d. 6m. AB. 13m. 100.990 50.379 4.336 0.035976 0.069199 106.576 51.383 4.599 0.038642 0.074135 1 150.539 99.078 4.388 .375648 .411131 157.164 101.175 4.666! .381365 .419331 1 2 198.573 146.367 4.351 .595130 .633076 306.181 149.418 4.730 .604319 .644865 2 3 345.137 193.185 4.413 .768806 .809453 353.615 196.044 4.798 .783033 .835639 3 4 390.176 336.467 4.476 .930540 .964146 399.545 341.377 4.856 .938656 .985348 4 5 333.797 379.435 4.531 1.061377 1.108169 344.077 385.107 4.914 1.085004 1.135315 5 6 376.088 331.051 4.586 .197679 .347990 387.381 337.886 4.958 .337833 .381937 6 7 417.316 361.679 4.638 .333673 .387777 439.899 370.143 4.980 .371359 .439453 7 8 457.596 401.813 4.649 .473430 .530613 473.044 413.176 4.989 .518708 .581471 8 9 497.633 441.733 4.657 .616630 .679383 514.774 455.604 4.931 .673714 .740347 9 10 538.305 483.978 4.603 1.768380 1.835913 555.695 494.016 5.140 1.839770 1.915440 10 11 577.066 519.456 4.801 .933901 3.009571! 591.343 536.701 5.379 3.035850 3.110449 11 12 610.838 550.498 5.038 3.118963 .3035631 633.870 557.371 5.467; .331394 .333671 12 13 640.865 579.531 5.111 .333761 .416138 653.889 586.738 5.5131 .456730 .557340 13 14 669.374 607.517 5.155 .548804 .649314; 681.599,614.691 5.576! .705916 1 .815807 14 15 696.641 634.064 5.315 3.797761 3.907653 709.195 641.939 5.605 3.983005 3.103085 15 16 733.849 659.933 5.343 3.074734 3.194804; 737.694 671.691 5.500 3.390950 .431493 16 17 18 749.916 776.577 688.198 713.355 5.143 5.377 .383603 fV O O ."i A G\ .513145 .885699 765.767 698.074 5.641 .646630 .794077 17 .738343 66 30. 31. 933.537 914.933 918.311 .360 5 733991 5 870448 65 931.998 917.469 .377 5.698773 5.846339 91l!064 !333 .367484 .498036 65 64 914.335 910.357 .339 5.343364 5.473806 907.303 903.003 .350 5.059879 5.179959 64 63 906.534 903.131 .368 .035659 .155739! 899.737 895.633 .343 4.783447 4.893338 63 62 898.991 894.676 .360 4.759337 4.869118 891. 949 888.051 .335 .535453 .635963 62 61 891.143 887.035 .343 .511333 .6117431 883.817 880.068 .313 .313053 .404430 61 60 883.937 878.980 .330 .387833 .380309 875. 349 871.787 .388 .109939 .194538 60 59 874.393 870.634 .306 4.085716 4.170315 865. 630 863.933 .334 3.939063 4.004733 59 58 864.575 861.689 .341 3.904836 3.980506 854,535 853.538 .167 .769768 3.837301 58 57 853.389 851.189 .183 .745539 .813073 843.960 840.763 .183 .636913 .689675 57 56 841.709 839.318 .199 .603677 .665440 831.543 839.377 .181 .495305 .553398 56 55 830.189 837.839 .197 .470964 .539157 830.035 817.930 .175 .373735 .437839 55 54 818.566 816.378 .191 3.349475 3.403579 808.394 806.341 .163 3.361386 3.311697 54 53 806.739 804.584 .179 .337134 .387435 796.331 794.468 .147 .157411 .304303 53 52 794.557 793.603 .163 .133133 .179935 783.789 783.315 .131 .061013 .104619 52 51 783.001 780.339 .147 .036715 .080331 770.9391769.584 .113 3.971467 .013113 51 50 769.035 767.493 .138 3.947144 3.987790 757.661j756.516 .095 .888151 3.936107 50 49 755.636 754.306 .111 3.863799 3.901755 743.957 743.037 .077 3.810456 3.845939 49 48 741.808 740.695 .093 .786069 .831553 739.837 739.108 .060 .737856 .771079 48 47 737.550 736.650 .075 .713436 .746649 715.383 714.767 .043 .669860 .701017 47 46 713.874 713.179 .058 .645379 .676536 700.330|700.0],9 .036 .606033 .635389 46 45 697.786 697.397 .041 .581495 .610751 684.994 684.863 .011 .545968 .573505 45 44 683.310 683.004 .036 3.531363 3.548899 669.304 669.346 .004 3.489377 3.515341 44 31. 7r = 16.3713. 30. 77 = 15.7791. Continued from Page *165. '190 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. i n. a; = 80. TT = 172.614. 81. 7r = 184.307. 1 n. V. 6in. V. 12m. ^m.d. A(B/). 6m. 12m. V. 6m. V. 12m. |m.d. A(B/). 6m. AB. 12m. 112.553 52.490 5.005 0.041593 0.079549 118.990 53.673 5.443 0.044845 0.085491 1 164.225103.346 5.073 .387468 .428114 171.764 105.548 5.518 .394354 .4379601 1 2 214.229:152.497 5.144 .614311 .657917 222.864 155.873 5.583 .625476 .672268 2 3 262.645 200.180 5.205 .796670 .843462 272.408 204.636 5.648 .812927 .863238 3 4 309.589 246.384 5.267 .958568 1.008879 320.587 252.230 5.696 .980820 1.034924 4 5 355.239 291.480 5.313 1.111288 1.165392 367.890'299.244 5.721 1.140469 1.198662 5 6 400.060 336.026 5.336 .261311 .319504 414.780 346.009 5.731 .298706 .361469 6 7 444.488 380.336 5.346 .413224 .475987 462.320 394.324 5.666 .459969 .527502 7 8 489.533 426.116 5.285 .570269 .637802 507.846 437.060 5.899 .631901 .7075711 8 9 532.670 466.609 5.505 .739384 .815054 547.395 473.424 6.164 .821168 .905767 9 10 570.143 501.064 5.757 1.926817 2.011416 582.584 507.436 6.262 2.028623 2.121000 10 11 603.484 533.290 5.850 2.133118 .225495 615.981 540.220 6.313 .255275 .355785 11 12 635.129 564.354 5.898 .359073 .459583 647.923 571.319 6.384 .505169 .615060 12 13 665.394 593.820 5.964 .608561 .718452 678.625 601.623 6.417 .782647 .902727, 13 14 694.484 622.534 5.996 .885817 3.005897 710.332 634.735 6.300 3.090792 3.221334 14 15 16 724.528 754.121 653.908 681.720 5.885 6.033 3.193848 .549559 3.324390 .697016 741.565 664.088 6.456 3.446559 3.594016 15 29. 923.504 919.583 28. .327 5 770798'-'^ ?»i ^'>^^ 68 67 67 923.028 918.915 .343 5.746996 5.894453 916.027 912.556 !289 .415291 .545833 66 915.488 911.829 .305 .391489 .522031 908.436 904.629 .317 .107686 .227766 66 65 907.835 903.836 .333 .083884 .203964 901.087 897.375 .309 4.831255 4.941146 65 64 900.423 896.521 .325 4.807453 4.917344 893.439 889.929 .293 4.583262 4.683772 64 63 892.713 889.014 .308 .559460 .659970 1885.445 882.081 .280 .359864 .452241 63 62 884.651 881.100 .296 .336060 .428437 1877.020 873.939 .257 .157753 .242352 62 61 876.157 872.890 .272 .133948 .218547 867.552 865.233 .193 3.976880 .052550 61 60 866.610 864.112 .208 3.953072 .028742 856.654 855.002 . .138 .817593 3.885126 60 59 855.621 853.796 .152 3.793783 3.861316 845.272 843.436 .153 3.674746 3.737509 59 58 844.145 842.133 .168 .650931 .713694 '834.047 832.240 .151 .543052 .601245 58 57 832.826 830.845 .165 .519231 .577424 |822.723 820.985 .145 .421588 .475692 57 56 821.408 819.496 .159 .397759 .451863 811.188 809.591 .133 .309272 .359583 56 55 809.779 808.008 .148 .285432 .335743 1799.328 797.917 .118 .205325 .252117 55 54 797.819 796.237 .132 3.181471 3.228263 1787.095 785.870 .102 3.108962 3.152568 54 53 785.484 784.089 .116 .085091 .128697 1774.461 773.451 .084 .019460 .060106 53 52 772.745 771.567 .098 2.995568 .036214 761.406 760.602 .067 2.936199 2.974155 52 51 759.580 758.611 .081 .912280 2.950236 747.981 747.340 .049 .858569 .894052 61 50 745.995 745.238 .063 .834619 .870102 734.038 733.654 .032 .786046 .819269 50 49 731.986 731.439 .046 2.762059 2.795282 719.738 719.553 .015 2.718140 2.749297 49 48 717.566 717.221 .029 .694109 .725266 705.037 705.053 .001 .654419 .683675 48 47 702.744 702.600 .012 .630337 .659593 689.958 690.152 .016 .594476 .622013 47 46 687.539 687.575 .003 .570335 .597872 674.531 674.895 .030 .537926 .563890 46 45 671.983 672.191 .017 .513717 .539681 658.764 659.298 .044 .484453 .508966 45 44 i 656.086 656.464 .032 2.460167 2.484680 642.687 643.362 .056 2.433756 2.456960 44 — 29. TT = 15.3125. 28. TT = 14.8689. Continued from Page *164. U91 TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. li n. 82. 77 = 196.942. 83. :t = 210.594. li 11. V. 6in. v. 12in. Jm.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. \mA. 6m. AB. 12m. 125.880 54.817 5.922 0.048478 0.092084 133.429 56.263 6.431 0.052451 0.099243 1 179.877 107.996 5.990 .401959 .448751 188.818 110.780 6.503 .410393 .460704 1 2 232.232 159.525 6.059 .637870 .688181 242.683 163.991 6.558 .651621 .705725 2 3 283.142 209.818 6.110 .830995 .885099 295.568 216.552 6.585 .851002 .909195 3 4 333.129 259.498 6.136 1.005502 1.063695 347.991 268.836 6.596 1.033065 1.095828 4 5 382.678 308.916 6.147 1.173060 1.235823 401.142 322.853 6.524 1.209337 1.276870 5 6 432.915 359.972 6.079 .340355 .407888 452.039 370.631 6.784 .390937 .466607 6 7 481.023 405.131 6.324 .516239 .591909 496.256|411.287 7.081 .586356 .670955 7 8 522.815 443.558 6.605 .708048 .792647 535.596 449.312 7.190 .797620 .889^97 8 9 560.000 479.499 6.708 .917087 2.009464 572.936 485.965 7.248 2.026552j2.127062 9 10 595.291 514.142 6.762 2.144692 2.245202 608.646 520.734 7.326 2.277765 2.387656 10 11 629.045 547.005 6.837 .395138 .505029 642.971 554.614 7.363 .555967 .676047 11 12 661.487 579.028 6.872 .672919 .792999 678.421 591.633 7.232 .864483 .995025 12 13 14 694.994 727.998 614.018 645.036 6.748 6.914 .981220 3.337062 3.111762 .484519 713.338 624.450 7.407 3.220429J3.367886 13 26. 70 27. 924.385 920.820 .297 5.817832 5.965289 69 923.956 920.218 .312 5.794407 5.941864 917.023 913.901 .260 5.462325 5.592867 69 68 916.538 913.246 .274 .438899 .569441 909.549 906.096 .288 .154720 .274800 68 67 909.007 905.382 .302 .131295 .251375 902.313 898.953 .280 4.878290 4.988181 67 66 901.715 898.184 .294 4.854864 4.^64755 894.783 891.623 .263 .630298 .730808 66 65 894.129 890.798 .278 .606872 .707382 886.911 883.895 .251 .406901 .499278 65 64 886.196 883.012 .265 4.383474 4.475851 '878.617 875.878 .228 4.204793 4.289392 64 63 877.839 874.933 .242 .181365 .265964 869.294 867.306 .166 .023923 .099593 63 62 868.446 866.297 .179 .000493 .076163 858.563 857.233 .111 3.864643 3.932176 62 61 857.633 856.147 .124 3.841210 3.908743 847.357 845.844 .126 .721803 .784566 61 60 846.342 844.671 .139 . .698366 .761129 836.304 834.821 .124 .590121 .648314 60 59 835.205 833.564 .137 3.566678 3.624871 825.154 823.739 .118 3.468672 3.522776 59 58 823.970 822.398 .131 .445223 .499327 813.798 812.520 .106 .356375 .406686 58 57 812.527 811.094 .119 .332916 .383227 802.119 801.026 .091 .252453 .299245 57 56 800.760 799.512 .104 .22JJ982 .275774 790.075 789.164 .076 .156122 .199728 56 55 788.623 787.560 .089 .132636 .176242 777.635 776.936 .058 .066659 .107305 55 54 776.089 775.239 .071 3.043154 3.083800 764.780 764.285 .041 2.983445 3.021401 54 53 763.137 762.491 .054 2.959917 2.997873 751.514 751.227 .024 .905872 2.941355 53 52 749.769 749.334 .036 .882316 .917799 737.8341737.751 .007 .833417 .866640 52 51 735.985 735.756 .019 .809828 .843051 723.753:723.868 .010 .765591 .796748 51 50 721.798 721.767 .003 .741964 .773121 709.279 709.590 .026 .701964 .731220 50 49 707.213 707.381 .014 2.678291 2.707547 694.431694.918 .041 2.642128 2.669665 49 48 692.253 692.597 .029 .618404 .645941 679.242 679.896 .055 .585701 .611665 48 47 676.947 677.460 .043 .561916 .58^880 663.719 664.539 .068 .532368 .556881 47 46 661.305 661.986 .057 .508516 .533029 647.888 648.849 .080 .481830 .505034 46 45 645.355 646.177 .068 .457901 .481105 631.787 632.878 .091 .433790 .455797 45 44 629.132 630.085 .079 2.409775 2.431782 615.439 616.648 .101 2.388007 2.408918 44 27. TT = 14.4479. 26. rr = 14.0485. Continued from Page *163. *193 TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. a; =84. 77 = 225.442. 85. Tr = 241.622. li n. V. 6m. V. 12in. ^m.d. 6m. AB. 12ra. V. 6m. V. 12m. im.d. 6m. AB. 12m„ 141.605 57.768 6.986 0.056912 0.107223 150.731 59.840 7.574 0.061816 0.115920 1 198.680!ll4.150 7.044 .419758 .473862 210.206 118.949 7.605 .430060 .488253 1 2 254.719 169.845 7.073 .666832 .725025 269.158 177.746 7.618 .683828 .746591 2 3 310.266 225.246 7.085 .873363 .936126 328.931 238.493 7.537 .898194 .965727 3 4 366.586 282.483 7.009 1.063734 1.131267 386.169 292.224 7.829 1.102046 1.177716 4 5 420.517 333.110 7.284 1.253934 1.329604 435.895 337.944 8.163 1.310810 1.395409 5 6 467.371 376.189 7.599 .454640 .539239 480.136 380.706 8.286 .530046 .622423 6 7 509.056 416.481 7.715! .669110 .761487 522.127 421.925 8.350 .763647 .864157 7 8 548:621 455.319 7.775 .899935 2.000445 562.286 461.026 8.438 2.017529 2.127420 8 9 586.461 492.161 7.858 2.152236 .262127 600.888 499.127 8.480 .297183 .417263 9 10 622.832 528.061 7.898 2.431031 2.551111 640.753 540.758 8.333 2.606442 2.736984 10 11 12 660.395 567.287 697 395 602 061 7.759 7 944 .739852 3 095944 .870394 3 243401 680.022 577.663 8.530 .962744 3.110201 11 \J tJ 1 •t/Ct/'Jw-^* VJJ. 1 • «/^^ t/« \J tJfJ %J JL J. t7» /v^t/^v/X 24. 72 71 25. 925.180 921.937 .270 5.864170 6.011627 71 924.793 921.393 .283 5.841083 5.988540 917.922 915.115 .234 .508663 5.639205 70 917.484 914.524 .247 .485576 .616118 910.553 907.420 .261 .201059 .321139 70 69 910.064 906.775 .274 5.177972 5.298052 903.418 900.378 .253 4.924629 5.034520 69 68 902.879 899.684 .266 4.901542 .011433' 895.996 893.151 .237 .676637 4.777147 68 67 895.405 892.406 .250 .653550 4.754060; 888.234 885.532 .225 .453242 .545619 67 66 887.590 884.735 .238 .430154 .522531! 1880.056 877.628 .202 .251136 .335735 66 65 879.355 876.775 .215 .228047 .312646 870.866 869.177 .141 .070270 .145940 65 64 870.100 868.266 .153 4.047179 4.122849 860.285 859.246 .087 3.910994 3.978527 64 63 859.446 858.265 .098 3.887901 3.955434 849.238 848.017 .102 .768162 .830925 63 62 848.322 846.959 .114 .745066 .807829 838.341 837.150 .099 .636489 .694682 62 61 837.349 836.015 .111 .613388 .671581 827.347 826.224 .094 .5150541 .569158 61 60 826.279 825.014 .105 .491947 .546051 816.151 815.163 .082 .402774 .453085 60 59 815.005 813.876 .094 3.379658 3.429969 804.638 803.831 .067 3.298874 3.345666 59 58 803.412 802.465 .079 .275748 .322540 792.762 792.136 .052 .202571 .246177 58 57 791.453 790.689 .064 .179431 .223037 780.498 780.081 .035 .113142 .153788 57 56 779.104 778.549 .046 .089986 .130632 767.825 767.607 .018 .029970 .067926 56 55 766.342 765.989 .029 .006793 .044749 754.745 754.734 .001 2.952448 2.987931 55 54 753.171 753.026 .012 2.929246 2.964729 741.259 741.449 .016 2.880053 2.913276 54 53 739.591 739.648 .005 .856822; .890045 727.377 727.761 .032 .812297 .843454 53 52 725.612 725.865 .021 .7890331 .820190 713.105 713.684 .048 .748752 .778008 52 51 711.242 711.690 .037 .725447 .754703 698.468 699.219 .063 .689012 .716549 51 50 696.503 697.125 .052 .665661 .693198 683.493 684.409 .076 .632693 .658657 50 49 681.422 682.211 .066 2.609289 2.635253 668.187 669.268 .090 2.579484 2.603997 49 48 666.011 666.965 .079 .556021 .580534 652.581 653.799 .102 .529085 .552289 48 47 650.295 651.388 .091 .505554 .528758 636.707 638.054 .112 .481202 .503209 47 46 634.311 635.533 .102 .457595 .479002 620.588 622.052 .122 .435594 .456505 46 45 618.080 619.420 .112 .411902 .432813 604.256 605.816 .130 .392028 .411946 45 44 601.634 603.071 .120 2.368242 2.388160 587.734 589.388 .138 2.350298 2.369299 44 25. Tr = 13.6687. 24. TT = 13.3082. Continued from Page *163. ♦193 TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. li n. 86. 77 = 259.448. 87. 77 = 279.436. p4tH n. V. 6m. V. 12m. Jm.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. ^m.d. A(B/). 6m. AB. 12m. 161.160 62.872 8.191 0.067214 0.125407 173.086 66.735 8.863 0.073392 0.136155 1 223.866 125.411 8.205 .441636 .504399 240.927 135.683 8.770 .454427 .521960 1 2 287.441 190.024 8.118 .702683 .770216 305.893 196.668 9.102 .727812 .803482 2 3 348.324 247.175 8.429 .930001 1.005671 362.333 248.561 9.481 .972065 1.056664 3 4 401.214 295.805 8.784 1.151718 .236317 412.546 297.096 9.621 1.210687 .303064 4 5 448.271 341.289 8.915 1.378303 1.470680 460.206 343.880 9.694 1.455162 1.555672 5 6 492.934 385.132 8.984 .616135 .716645 505.788 388.^60 9.794 .715063 .824954 6 7 535.651 426.721 9.077 .872380 .982271 549.600 431.504 9.841 .997939 2.118019 7 8 576.708 467.247 9.122 2.153307 2.273387 594.848 478.756 9.674 2.308839 .439381 8 9 10 619.111 660.879 511.528 550.782 8.965 9.175 .463217 2.819828 .593759 2.967285 639.418 520.644 9.898 .665921 .813378 9 74 22. 23. 925.897'922.944 .246 5.909879 6.057336 73 925.548 922.453 .258 5.887099 6.034556 918.733 916.211 .210 .554372 5.684914 73 72 918.338 915.677 .222 .531592 5.662134 911.460 908.615 .237 .24676* .366848 72 71 911.018 908.033 .249 .223988 .344068 904.417 901.664 .229 4.970339 .080230 71 70 903.930 901.037 .241 4.947558 .057449 897.089 894.530 .213 .722348 4.822858 70 69 896.557 893.858 .225 4.699567 4.800077 889.428 887.009 .202 4.498954 4.591331 69 68 888.846 886.289 .213 .476173 .568550 881.357879.207 .179 .296850 .381449 68 67 880.724 878.438 .191 .274068 .358667 872.284 870.866 .118 .115987 .191657 67 66 871.593 870.043 .129 . .093203 .168873 861.841861.063 .065 3.956716 .024249 66 65 861.083 860.177 .075 3.933930 .001463 850. 9361849. 979 1 .080 .813890 3.876653 65 64 850.108 849.023 .090 3.791101 3.853864 840.179 839.252 .077 3.682227 3.740420 64 63 839.283 838.227 .088 .659433 .717626 829.328 828.467 .072 .560803 .614907 63 62 828.364 827.374 .083 .538004 .592108 818.277 817.549 .061 .448538 .498849 62 61 817.241 816.387 .071 .425732 .476043 806.911 806.364 .046 .344658 .391450 61 60 805.804 805.130 .056 .321842 .368634 795.190 794.819 .031 .248379 .291985 60 59 794.007i793.512 .041 3.225551 3.269157 783.084 782.920 .014 3.158981 3.199627 59 58 781.824 781.536 .024 .136138 .176784 770.574 770.608 .003 .075846 .113802158 57 769.235 769.146 .007 .052985 .090941 757. 663 757.900 .020 2.998369 .033852 57 56 756.241 756.357 .010 2.975486 .010969 744.350 744.786 .036 .926027 2.959250 56 55 742.844 743.160 .026 .903118 2.936341 730.648 731.275 .052 .858334 .889491155 54 729.054 729.562 .042 2.835395 2.866552 716.562 717.381 .068 2.794862 2.824118' 54 53 714.877 715.579 .058 .771887 .801143 702.112 703.102 .083 .735205 .762742 53 52 700.337 701.210 .073 .712190 .739727 687.331 688.483 .096 .678982 .704946 52 51 685.460 686.497 .086 .655921 .681885 672.223 673.538 .110 .625883 .650396' 51 50 670.257 671.457 .100 .602768 .627281 656.817 658.268 .121 .575607 .598811 50 49 654.753 656.091 .112 2.552433 2.575637 641.148 642.726 .131 2.527863 2.549870 49 48 638.984 640.449 .122 .504621 .526628 625.238 626.931 .141 .482409 .503320 48 47 622.972 624.553 .132 .459093 .480004 609.117 610.905 .149 .439015 .458933 47 46 606.748 608.425 .140 .415615 .435533 592.808 594.688 .157 .397476 .416477 46 45 590.335 592.105 .147 .373984 .392985 576.331 578.288 .163 .357599 .375780 45 44 573.754 575.600 .154 2.334006 2.352187 559.752 561.734 .165 2.319204 2.336626 44 23. 77=12.9656. 22. 77 = 12.6399. Continued from Page *161 nu TABLE LXIII GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. li 88. Tr = 302.168. 89. n = 329.342. &»2 V. V. ^m.d. ^(B/). AB. V. V. ^m.d. ^(B/). AB. n. 6m. 12m. 6m. 12m. 6m. 12m. 6m. 12m. n. 188.023 73.878 9.51 0.080005 0.147538 199.951 70.559 10.78 0.091697 0.167367 1 257.635 139.224 9.87 .472731 .548401 265.249 130.597 11.22 .497892 .582491 1 2 318.110 194.827 10.27 .761752 .846351 323.345 186.752 11.38 .800873 .893250 2 3 371.914 246.833 10.42 1.021673 1.114050 378.488 240.881 11.47 1.076513 1.177023 3 4 422.982 296.963 10.50 .277234 .377744 431.225 292.227 11.58 .352468 .462359 4 5 471.823 344.516 10.61 1.543045 1.652936 481.914 342.260 11.64 1.643617 1.763697 5 6 518.769 390.853 10.66 .829022 .949102 534.266 396.930 11.45 .958640 2.089182 6 7 567.252 441.483 10.48 2.141481 2.272023 585.833 445.393 11.70 2.317667 .465124 7 8 615.009 486.367 10.72 .499301 .646758 75 21. 926.546 919.466 923.855 917.201 20. .224 .189 5.955022 .599515 6.102479 5.730057 76 75 926.229 923.411 .235 5.932518 6.079975 74 919.109 916.718 .199 5.577011*5.707553 912.279 909.695 .215 5.291911 5.411991 74 73 911.879 909.169 .226 .2694071 .389487 905.319 902.826 .208 .015482 .125373 73 72 904.879 902.259 .218 4.992978 .102869 898.078 895.776 .192 4.767492 4.868002 72 71 897.596 890.169 .202 .744988 4.845498 890.507 888.344 .180 .544099 .636476 71 70 889.982 887.694 .191 .521594 .613971 882.532 880.635 .158 .341997 .426596 70 69 881.958 879.939 .168 4.319491|4.404090 873.566 872.392 .098 4.161137 4.236807 69 68 872.941 871.648 .108 .138629 .214299 863.246 862.704 .045 .001870 .069403 68 67 862.561 861.904 .055 3.979360 .046893 852.469 851.752 .060 3.859050 3.921813 67 66 851.722 850.888 .069 .836538 3.899301 841.841 841.151 .058 .727394 .785587 66 65 841.030 840.225 .067 .704878 .763071 831.117 830.494 .052 .605980 .660084 65 64 830.245 829.506 .062 3.583460 3.637564 820.196 819.705 .041 3.493730 3.544041 64 63 819.260 818.654 .051 .471202 .521513 808.966 808.651 .026 .389867 .436659 63 62 807.964 807.536 .036 .367331 .414123 797.382 797.244 .011 .293610 .337216 62 61 796.313 796.062 .021 .271063 .314669 785.420 785.485 .005 .204239 .244885 61 60 784.281 784.234 .004 .181679 .222325 773.057 773.318 .022 .121137 .159093 60 59 771.846 771.997 .013 3.098561 3.136517 760.299 760.760 .038 3.043699 3.079182 59 58 759.014 759.366 .029 .021104 .056587 747. 144 747.802 .055 2.971405 .004628 58 57 745.782 746.332 .046 2.948786 2.982009 733. 603 734.450 .071 .903767 2.934924 57 56 733.162 732.902 .062 .881122 .912279 719.683 720.720 .086 .840359 .869615 56 55 718.161 719.092 .078 .817683 .846939 705.405 706.610 .100 .780778 .808315 55 54 703.800 704.900 .092 2.758065 2.785602 690.797 692.164 .114 2.724640 2.750604 54 53 689.107 690.369 .105 .701886 .727850 675.869 677.395 .127 .671638 .696151 53 52 674.091 675.515 .119 .648837 .673350 660.645 662.306 .138 .621472 .644676 52 51 658.779 660.338 .130 .598617 .621821 645.161 646.948 .149 .573850 .595857 51 50 643.205 644.890 .140 .550936 .572943 629.439 631.339 .158 .528533 .549444 50 49 627.391 629.190 .150 2.505553 2.526464 613.507 615.502 .166 2.485291 2.505209 49 48 611.367 613.261 .158 .462237 .482155 597.392 599.477 .174 .443920 .462921 48 47 595.157 597.143 .165 .420784 .439785 581.110 583.271 .180 .404229 .422410 47 46 578.780 580.842 .172 .381004 .399185 564.727 566.912 .182 .366037 .383459 46 45 562.302 564.388 .174 .342714 .360136 548.269 550.506 .186 .329195 .345933 45 44 545.749 547.886 .178 2.305763 2.322501 531.735 533.997 .188 2.293553 2.309662 44 21. TT = 12.3304. 20. TT = 12.0361. Continued from Page *160. TABLE LXIII. ♦195 GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. o® 90. n = 357.261. V. 6m. 310.938 373.435 333.764 389.504 444.043 5 500.3^8 6 555.849 V. 13m. 64.595 135.014 183.353 338.496 393.338 |m.d. 13.19 13.37 13.46 13.58 13.64 351.14713.44 403.39013.71 MB/). 6m. 0.105165 .533683 .841311 1.136333 .436615 1.756106 3.117313 AB. 12m. 0.189764 .616059 .941731 1.346113 .556695 1.886648 3.264670 19. 77 936.847 919.806 913.658 934.377 917.660 910.196 74 905.7381903.365 73 898.537i896.354 72 1891.008 888.964 71 1883.076 881.397 70l874.16l|873.099 )863. 8981863.466 i853.18l!853.574 i 843. 611:843.033 i831. 947 831.434 1831.087 830.705 64 1809.919 809. 713 63 798.399i798.368 786.503 774.309 761.533 59 748.439 581734.974 571731.131 56^706.932 55:693.406 54^677. 53 1663. 52 647. 51 631. 50,615. 49 599. 48 583. 47 46 45 44 567.034:569.314 550. 534 559 431 032 387 544 518 326 786.674 774.575 762.087 749.200 735.923 722.269 708.237 693.871 679.184 664.179 648.906 633.383 617.634 601.698 585.582 .667 .225 517.748 552.998 536.581 520.112 .214 .179 .305 .198 .183 .170 .148 .089 .036 .051 .048 .043 .033 .017 .003 .014 .030 .047 .063 .079 .095 .109 .132 .135 .147 .157 .166 .174 .182 .188 .190 .194 .196 .197 5. 977398'6. 124855 .62189115.752433 .314287 .434367 5.037858 4.789869 .566476 .364375 .183516 5.147749 4.890379 .658853 .448974 .259186 4.0243514.091784 3.881433 3.944196 .749781 .638373 .516138 3.413373 316036 336667 143581 066161 3.993888 .936376 .863898 .803350 .747353 3. 694394 .644178 .596613 .551358 .508185 3.466891 .437385 .389186 .353445 .316914 2.283464 .807974 .682476 .566439 3.459065 .359632 .267313 .181537 .101644 3.037111 3.957433 .893154 .830887 .773316 3.718807 .667382 .618619 .572269 .528103 2.485892 .445466 .406608 .369183 .333023 2.298003 19. 7r = 11.7560. 91. T = 384.586. V. 6m. 224.588 288.013 348.673 V. 12m. 64.590 126.850 185.909 406.976l243.458 ^m.d. 13.33 13.43 13.56 13.63 467.193 306.339 13.40 536.504 363.083 13.70 1.896111 3.043568 A(B/). 6m. 0.117487 .549533 .885337 1.303501 .531344 AB. 13m. •^3 o a> 0.309864 .650033 1 .995138 1.333581 .661786 18. 937.133 930.130 913.019 906.136 898.973 891.484 883.594 874.736 864.518 853.858 843.344 833.737 831.934 810.824 799.366 787.533 775.304 763.685 749.671 736.277 733.508 708.384 693.934 679.168 664.108 648.793 633.340 924.679: .205 918.097| .169 910.672 .196 903.8771 .188 896.904 889.552 881.936 873.772i 864.190 853.3561 842.870| 832.328' 821.656| 810.722 799.437 787.805 775.770 763.349 750.530 737.323 723.741 709.784 695.494 680.885 665.960 650.767 635.328 .173 .161 .139 .079 .037 .042 .040 .034 .023 .008 5.999652.6.147109 .64414515.774687 .336541 .060113 4.812123 .588731 .386631 .205773 .046509 .456621 .170004 4.912633 .681108 .471330 .281443 .114042 3.903695 3.966458 .773045 .830338 .650643 .538403 .434555 .006 3.338318 .033 .248971 617.481:619.662 601.539 6D3. 810 585.433 587.779 569.338 571.597 553.949 555.369 536.593 539.038 530.303j533.656 503.813 506.361 .039 .055 .073 .087 .103' .117 .130 .143 .154 .165 .174 .183 .189 .195 .197 .303 .304 .304 .165899 .088496 .016344 3.948656 .885306 .835791 .769739 .716813 3.666744 .619331 .574036 .530939 .489707 3.450180 .413169 .375534 .340097 .305761 .304 3.2733913.387409 .704746 .588714' 66 .481347 65 3.381934 .389617 .303855 .133979 .049467 3.979813 .914563 .853328 .795693 .741326 2.689948 64 .641238 53 .594947 .550847 .508708 2.468361 .429591 .392262 .356206 .321300 18. rr = 11.4897. 52 51 50 49 48 47 46 45 44 Continued from Page *159. U96 TABLE LXIII, GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OP 1,000. 4 PER CENT. n. x = 92. 7r = 413.801. 93. n = 446.049. n. V. 6m. V. 12m. |m.d. 6m. AB. 12m. V. 6m. V. 12m. ^m.d. MB/). 6m. AB. 12m. 240.180 66.559 14.47 0.131006 0.231516 256.844 67.639 15.77 0.147474 0.257365 1 305.028 129.696 14.61 .5795-10 .689401 323.619 133.549 15.84 .61£040 .735120 1 2 367.358 191.219 14.68 .936727 1.056807 392.582 205.566 15.59 .996003 1.126545 2 3 4 431.731 AQK 1 on 258.442 o-t o noK 14.44 14.76 1.280853 .652883 .411395 .800340 460.511 269.408 15.93 1.382805 .530262 3 ^vO. Loo'^oi-o.yjoo 16. 80 17. 927.663 925.424 .187 6.043818 6.191275 79 927.405 925.061 .195 6.021790 6.169247 920.729 918.907 .152 5.688311 5.818853 79 78 920.436 918.512 .160 5.666283 5.796825 913.690 911.556 .178 .380708 .500788 78 77 913.363 911.125 .187 .358680 .478760 906.874i904.829 .170 .104280 .214171 77 76 906.514 904.365 .179 .082251 .192142 899.783 897.924 .155 4.856291 4.956801 76 75 899.388 897.427 .163 4.834262 4.934772 892.368 890.646 .144 .632899 .725276 75 74 891.937 890.113 .152 4.610871 4.703248 884.555 883.095 .122 4.430800 4.515399 74 73 884.087 882.525 .130 .408771 .493370 875.775 875.021 .063 .249945 .325615 73 72 875.264 874.413 .071 .227914 .303584 865.668 865.534 .011 .090685 .158218 72 71 865.108 864.879 .019 .068652 .136185 855.114 854.807 .026 3.947874 .010637 71 70 854.501 854.100 .033 3.925840 3.988603 844.703 844.425 .023 .816231 3.874424 1 70 69 844.041 843.667 .031 3.794194 3.852387 834.202 833.987 .018 3.694836 3.748940 69 68 833.487 833.178 .026 .672794 .726898 823.505 823.421 .007 .582608 .632919 68 67 822.740 822.561 .015 .560561 .610872 812.506 812.595 .007 .478774 .525566 67 66 811.686 811.682 .000 .456721 .503513 801.162 801.422 .022 .382554 .426160 66 65 800.287 800.455 .014 .360493 .404099 789.4451 789. 906 .038 .293229 .333875 65 64 788.513 788.882 .031 3.271157 3.311803 777.338 777.990 .054 3.210183 3.248139 64 63 776.346 776.908 .047 .188098 .226054 764.843 765.691 .071 .132812 .168295 63 62 763:791 764.549 .063 .110712 .146195 751.958 752.999 .087 .060597 .093820 62 61 750.844 751.796 .079 .038479 .071702 738.697 739.923 .102 2.993054 .024211 61 60 737.517 738.656 .095 2.970914 .002071 725.064 726.476 .118 .929755 2.959011 60 59 723.818 725.143 .110 2.907590 2.936846 711.080 712.657 .131 2.870299 2.897836 59 58 709.766 711.257 .124 .848105 .875642 696.774 698.508 .145 .814305 .840269 58 57 695.390 697.039 .137 .792078 .818042 682.152 684.044 .158 .761465 .785978 57 56 680.697 682.504 .151 .739201 .763714 667.243 669.266 .169 .711483 .734687 56 55 665.714 667.654 .162 .689177 .712381 652.078 654.224 .179 .664068 .686075 55 54 650.477 652.539 .172 2.641714 2.663721 636.679 638.937 .188 2.618980 2.639891 54 53 635.002 637.178 .181 .596574 .617485 621.076 623.426 .196 .575994 .595912 53 62 619.324|621.591 .189 .553529 .573447 605.293 607.732 .203 .534904 .553905 52 51 603.464 605.821 .196 .512375 .531376 589.346 591.859 .209 .495523 .513704 51 50 587.438|589.870 .203 .472924 .491105 573.302 575.838 .211 .457672 .475094 50 49 571.316 573.771 .205 2.434995 2.452417 557.183 559.770 .216 2.421202 2.437940 49 48 555.119 557.625 .209 .398440 .415178 540.988 543.601 .218 .385967 .402076 48 47 538.846 541.377 .211 .363112 .379221 524.762 527.381 .218 .351838 .367377 47 46 522.540 525.078 .211 .328882 .344421 508.534 511.148 .218 .318694 .333712 46 45 506.232 508.766 .211 .295629 .310647 492.333 494.924 .216 .286421 .300970 45 44 489.954 492.463 .209 2.263238 2.277787 476.189 478.748 .213 2.254918 2.269036 44 17. TT = 11.2364. 16, n = 10.9953. Continued from Page *158 *197 TABLE LXIII. GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. II n. 94. n = 481.198. 95. 77 = 520.729. •■si n. V. 6m. Y. 12m. Jm.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. ^m.d. 6m. AB. 12m. 275.945 70.691 17.11 0.166535 0.286615 301.923 83.117 18.23 0.187522 0.318064 1 2 349.910 422.768 147.932 216.406 16.83 17.20 .655087 1 076479 .785629 1 223936 380.323 156.799 18.63 .713515 .860972 1 ^•\j i \j^ 1 t/ J-^/V/Vv tJtJXJ 14. 82 15. 928.144 926.099 .170 6.087566 6.235023 81 927.909 925.770 .178 6.065742 6.213199 921.272 919.641 .136 5.732059 5.862601 81 80 921.008 919.283 .144 5.710235 5.840777 914.297 912.356 .162 .424456 .544536 80 79 914.001 911.966 .170 5.402632 5.522712 907.542 905.689 .154 5.148028 5.257919 79 78 907.216 905.270 .162 .126203 .236094 900.516 898.848 .139 4.900040 .000550 78 77 900.158 898.397 .147 4.878215 4.978725 893.167 891.635 .128 .676650 4.769027 77 76 892.778 891.153 .135 .654825 .747202 885.426 884.152 .106 .474551 .559150 76 75 885.002 883.637 .114 .452725 .537324 876.725 876.152 .048 .293698 .369368 75 74 876.262 875.601 .055 4.271871 4.347541 866.710 866.750 .003 4.134440 4.201973 74 73 866.202 866.157 .004 .112612 .180145 856.250 856.121 .011 3.991634 .054397 73 i 72 855.696 855.480 .018 3.969804 .032567 845.935 845.832 .009 .859997 3.918190 72 71 845.334 845.146 .016 .838164 3.896357 835.528 835.489 .003 .738609 .792713 71 70 834.882 834.757 .010 .716772 .770876 824.929 825.019 .008 .626391 .676702 70 69 824.235 824.240 .000 3.604549 3.654860 814.029 814.291 .022 3.522569 3.569361 69 68 813.286 813.464 .015 .500722 .547514 802.786 803.219 .036 .426365 .469971 68 67 801.995 802.343 .029 .404510 .448116 791.176 791.806 .052' .337059 .377705 67 66 790.332 790.880 .040 .315195 .355841 779.179 779.998 .068 .254037 .291993 66 65 778.281 779.019 .062 .232161 .270117 766.796 767.811 .085 .176694 .212177 65 64 765.844 766.777 .078 3.154805 3.190288 754.029 755.234 .100 3.104513 3.137736 64 63 753.019 754.144 .094 .082607 .115830 740.887 742.276 .116 .037009 .068166 63 62 739.820 741.129 .109 .015084 .046241 727.377 728.950 .131 2.973756 .003012 62 61 726.249 727.744 .125 2.951809 2.981065 713.520 715.256 .145 .914353 2.941890 61 60 712.330 713.989 .138 .892380 .919917 699.342 701.235 .158 .858419 .884383 60 59 698.090 699.905 .151 2.836417 2.862381 684.854 686.902 .171 2.805649 2.830162 59 58 683.536 685.508 .164 .783613 .808126 670.079 672.258 .182 .755744 .778948 58 57 668.696 670.799 .175 .733670 .756874 655.052 657.352 .192 .708415 .730422 57 56 653.602 655.827 .185 .686299 .708306 639.793 642.204 .201 .663424 .684335 56 55 638.275 640.611 .195 .641262 .662173 624.331 626.833 .209 .620545 .640463 55 54 622.744 625.172 .202 2.598329 2.618247 608.690 611.281 .216 2.579573 2.598574 54 53 607.033 609.550 .210 .557301 .576302 592.888 595.552 .222 .540322 .558503 53 52 591.161 593.751 .216 .517987 .536168 576.987 579.675 .224 .502614 .520036 52 51 575.190 577.804 .218 .480209 .497631 561.015 563.752 .228 .466299 .483037 51 50 559.146 561.810 , .222 .443819 .460557 544.968 547.730 .230 .431233 .447342 50 49 543.027 545.716 .224 2.408671 2.424780 528.888 531.657 .231 2.397289 2.412828 49 48 526.876 529.572 .225 .374637 .390176 512.806 515.571 .230 .364345 .379363 48 47 510.723 513.415 .224 .341595 .356613 496.752 499.493 .228 .332289 .346838 47 46 494.598 497.265 .222 .309435 .323984 480. 754 483.463 .226 .301020 .315138 46 45|478.528 481.164 .220 .278052 .292170 464.831 467.497 .222 .270452 .284177 45 44 462.535 465.127 .216 2.247362 2.261087 449.013 451.618 .217 2.240494 2.253864 44 15. 7r= 10.7660. 14. 7: = 10.5480. Continued from Page *157. ♦198 TABLE LXIII, GENERAL VALUATION OF ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. n. 83 a: = 96. tt = 571.434. li n. V. 6m. V. 12m. ^m.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. im.d. 6m. AB. 12m. 325.898 80.362 20.46 924964 n Q 79/1 91 M V.yC'/C'^i/Urr v/.t^ 1 /v^/vjl 12. 13. 928.580 921.766 926.711 920.307 .156 .122 6.130933 5.775427 6.278390 928.367 926.412 .163 6.109295|6.256752 5.905969 83 82 921.525 919.982 .129 5.75378815.884330 914.848 913.083 .147 .467823 .587903 82 81 914.579 912.728 .154 .4461851 .566265 908.150 906.471 .140 .191395 .301286 81 80 907.854 906.090 .147 169757 .279648 901.180 899.686 .125 4.943407 .043917 80 791900.856 899.277 .132 4.921768i5.022278 893.894 892.533 .113 4.720017 4.812394 79 78 893.539 892.095 .120 .698378 4.790755 886.216 885.112 .092 .517921 .602520i 78 77 885.831 884.644 .099 .496281 .580880 877.587 877.178 .034 .337069 .412739! 77 76 877.167 876.678 .041 .315428 .391098 867.654 867.854 .017 .177814 .245347 76 75 867.194 867.316 .010 .156172 .223705 857.283 857.313 .002 .035011 .097774 75 74 856.779 856.731 .004 4.013368 4.076131 847.052 847.110 .005 3.903379 3.961572 74 73 846.507 846.487 .002 3.881733 3.939926 836.731 836.852 .010 .781997 .836101 73 72 836.144 836.187 .004 .760348 .814452 826.220 826.468 .021 .669788 .720099 72 71 825.590 825.761 .014 .648135 .698446 815.410 815.829 .035 .565978 .612770 71 70 814.736 815.079 .029 .544319 .591111 804.261 804.849 .049 .469788 .513394 70 69 803.542 804.054 .043 3.448122 3.491728 792.747 793.531 .065 3.380499 3.421145 69 68 791,981 792.690 .059 .358825 .399471 780.849 781.821 .081 .297498 .335454 68 67 780.034 780.932 .075 .275813 .313769 768.569 769.734 .097 .220181 .255664167 66 767.705 768.796 .091 .198484 .233967 755. 907 757.262 .113 .148029 .181252 66 65 754.991 756.273 .107 .126318 .159541 742.875 744.411 .128 .080561 .111718 65 64 741.906 743.370 .122 3.058832 3.089989 729.476 731.196 .143 3.017349 3.046605 64 63 728.453 730.101 .137 2.995601 .024857 715.734 717.615 .157 2.957993 2.985530 63 62 714.654 716.465 .151 .936222 2.963759 701. 674 703.711 .170 .902114 .928078^ 62 j 61 700.537 702.503 .164 .880317 .906281 687.306 689.496 .183 .849405 .873918! 61 60 686.110 688.231 .177 .827578 .852091 672. 653 674.973 .193 .799569 .822773 60 59 671.397 673.649 .188 2.777708 2.800912 657. 750 660.191 .203 2.752318 2.774325 59 58 656.434 658.806 .198 .730419 .752426 642. 618 645.168 .212 .707414 .728325 58 57 641.240 643.722 .207 .685473 .706384 627. 284 629.925 .220 .664630 .684548 57 56 625.843 628.417 .214 .642643 .662561 611.772 614.501 .227 .623764 .642765 56 55 610.269 612.930 .222 .601725 .620726 596. 101 598.902 .233 .584630 .602811 55 54 594.534 597.268 .228 2.562534 2.580715 580. 333 583.157 .235 2.547049 2.564471 54 53 578.701 581.459 .230 .524891 .542313 564. 493 567.367 .240 .510872 .527610 53 52 562.797 565.604 .234 .488648 .505386 548. 578 551.477 .242 .475958 .492067 52 51 546.817 549.649 .236 .453660 .469769 532. 631 535.537 .242 .442178 .457717 51 50 530.806 533.645 .237 .419800 .435339 516. 683* 519.584 .242 .409412 .424430 50 49 514.792 517.627 .336 2.386947 2.401965 500. 763 503.640 .240 2.377548 2.392097 49 48 498.806 501.617 .234 .354991 .369540 484.897 487.743 .237 .346487 .360605 48 47 482.877 485.656 .232 .323828 .337946 469.106 471.909 .234 .316143 .329868 47 46 467.021 469.757 .228 .293376 .307101 453.418 456.161 .229 .286428 .299798 46 45 451.269 453.945 .223 .263543 .276913 437.861 440.534 .223 .257261 .270309^45 44 435.648 438.254 .217 2.234250 2.247298 422.448 425.046 .216 2.228576 2.241324 44 13. 7r= 10.3400. 12. TT = 10.1419. Continued from Page *156. ♦199 TABLE LXIII. GENERAL VALUATION OP ANNUAL PREMIUM POLICIES OF 1,000. 4 PER CENT. n. 11. 10. ti n. 86 V. 6m. V. 12m. ^m.d. A(B/). 6m. AB. 12m. V. 6m. V. 12m. |m.d. MB/). 6m. AB. 12m. 928.975 927.266 .142 6.173956 6.321413 85 928.783 926.995 .149 6.152485 6.299942 922.212 920.910 .109 5.818449 5.948991 85 84 921.994 920.616 .115 5.796979 5.927521 915.346 913.740 .134 5.510846 5.630926 84 83 915.103 913.419 .140 .489376 .609456 908.699 907.179 .127 .234418 .344309 83 82 908.431 906.833 .133 .212948 .322839 901.783 900.445 .111 4.986431 .086941 82 81 901.488 900.074 .118 4.9649591 .065469 894.551 893.346 .100 .763042 4.855419 81 80 894.230 892.949 .107 .741571 4.833948 886.932 885.982 .079 .560946 .645545 80 79 886.583 885.557 .086 4.539474 4.624073 878.368 878.108 .022 4.380096 4.455766 79 78 877.987 877.654 .028 .358623 .434293 868.51] 868.855 .029 .220843 .288376 78 77 868.093 868.366 .023 .199370 .266903 858.217 858.393 .015 .078044 .140807 77 76 857.761 857.866 .009 .056569 .119332 848.064 848.267 .017 3.946416 .004609 76 75 847.570 847.702 .011 3.924939 3.983132 837.822 838.087 .022 .825041 3.879145 75 74 837.289 837.484 .016 3.803560 3.857664 827.389 827.782 .033 3.712840 3.763151 74 73 826.819 827.141 .027 .691355 .741666 816.662 817.223 .047 .609040 .655832 73 72 816.051 816.543 .041 .587550 .634342 805.598 806.327 .061 .512862 .556468 72 71 804.945 805.605 .055 .491366 .534972 794.170 795.094 .077 .423588 .464234 71 70 793.476 794.331 .071 .402085 .442731 782.362 783.472 .093 .340606 .378562 70 69 781.623 782.666 .087 3.319093 3.357049 770.175 771.478 .109 3.263312 3.298795 69 68 769.392 770.627 .103 .241788 .277271 757.610 759.099 .124 .191188 .224411 68 67 756.778 758.202 .119 .169651 .202874 744.675 746.346 .139 .123751 .1549081 67 1 66 743.796 745.401 .134 .102199 .133356 731.379 733,231 .154 .060576 .089832 66 65 730.450 732.237 .149 .039006 .068262 717.740 719.753 .168 .001263 .028800 65 64 716.761 718.709 .162 3.979672 3.007209 703.787 705.953 .180 2.945431 2.971395 64 63 702.756 704.859 .175 .923818 2.949782 689.527 691.846 .193 .892777 .917290 63 62 688.443 690.699 .188 .871137 .895650 674.985 677.434 .204 .843004 .866208 62 61 673.848 676.233 .199 .621333 .844537 660.196 662.763 .214 .795822 .817829 61 60 659.002 661.508 .209 .774117 .796124 645.177 647.854 .223 .750995 .771906 60 59 643.928 646.543 .218 2.729253 2.750164 629.959 632.726 .231 2.708297 2.728215 59 58 628.654 631.359 .225 .686513 .706431 614.566 617.419 .238 .667526 .686527 58 57 613.203 615.995 .233 .645696 .664697 599.013 601.939 .244 .628495 .646676 57 56 597.591 600.456 .239 .606615 .624790 583.365 586.313 .246 .591028 .608450 56 55 581.885 584.773 .241 .569092 .586514 567.644 570.642 .250 .554976 .571714 55 54 566.106 569.043 .245 3.532980 2.549718 551.849 554.872 .252 2.520197 2.536306 54 53 550.252 553.214 .247 .498136 .514245 536.024 539.053 .252 .486563 .502102 53 52 5 54.369 537.337 .247 .464430 .479969 520.196 523.221 .252 .453956 .468974 52 51 518.481 521.446 .247 .431745 .446763 504.396 507.397 .250 .422264 .436813 51 50 502.623 505.563 .245 .399970 .414519 488. 650 491.620 .248 .391388 .405506 50 49 486.818 489.728 .243 2.369004 2.383122 472.979 475.906 .244 2.361244 2.374969 49 48 471.089 473.955 .239 .338763 .352488 457.410 460.278 .239 .331743 .345113 48 47 455.462 458.269 .234 .309158 .322528 441.971 444.769 .233 .302806 .315854 47 46 439.965 442.701 .228 .280110 .293158 426.674 429.398 .227 .274370 .287118 46 45 424.611 427.274 .222 .251553 .264301 411.542 414.176 .219 .246367 .258849 45 44 409.424 411.995 .214 2.223421 2.235903 396.596 399.135 .212 2.218737 2.230972 44 11. 77 = 9.9535. 10. TT = 9.7739. Continued from Page *155. *300 TABLE LXIY. VALUE OP /' AND LOG / FOR EVERY YEAR AND MONTH OF AGE, 4 PER CENT. Tear of Age. x-\-n. h = Om. im. l4m. 2im. 3jm. 4im. 5|m. Tear of Age. X + n. *Dec. Nov. Oct. Sept. Aug. July. /' =.961538 .963141 .966346 .969551 .972756 .975962 .979167 Log/. * 10 1.980143 T. 980989 T. 982675 1.984355 T. 986029 T.987696 1.989357 10 11 .980134 .980980 .982668 .984348 .986022 .987691 .989352 11 12 .980128 .980974 .982662 .984344 .986019 .987687 .989349 12 13 .980118 .980965 .982654 .984336 .986011 .987681 .989344 13 14 .980105 .980953 .982642 .984326 .986002 .987673 .989337 14 15 1.980094 T. 980942 1.982633 1.984317 T. 985994 T.987666 T.989331 15 16 .980084 .980932 .982624 .984309 .985988 .987660 .989325 16 17 .980070 .980919 .982612 .984298 .985978 .987651 .989318 17 18 .980055 .980904 .982599 .984286 .985968 .987642 .989310 18 19 .980639 .980889 .982585 .984274 .985956 .987633 .989302 19 20 T. 980020 T. 980871 T. 982568 1.984259 1.985943 T.987621 T.989292 20 21 .980000 .980852 .982551 .984244 .985929 .987608 .989281 21 22 .979978 .980831 .982532 .984226 .985914 .987595 .989269 22 23 .979955 .980809 .982512 .984208 .985898 .987580 .989257 23 24 .979928 .980783 .982488 .984187 .985879 .987564 .989243 24 25 T. 979901 1.980758 T. 982465 T. 984166 1.985860 T.987548 1.989229 25 26 .979866 .980724 .982434 .984138 .985835 .987526 .989210 26 27 .979833 .980692 .982406 .984112 .985813 .987505 .989193 27 28 .979795 .980656 .982373 .984083 .985786 .987483 989173 28 29 .979748 .980611 .982332 .984045 .985753 .987454 .989148 29 30 T. 979704 1.980569 T. 982294 1.984011 T. 985722 T.987426 T.989124 30 31 .979652 .980519 .982249 .983970 .985686 .987395 .989097 31 32 .979598 .980468 .982201 .983928 .985648 .987361 .989068 32 33 .979535 .980407 .982147 .983879 .985604 .987323 .989035 33 34 .979466 .980341 .982087 .983825 .985556 .987280 .988998 34 35 T. 979386 T. 980265 T. 98201 7 T. 983 762 1.985500 1.987232 T.988956 35 36 .979307 .980189 .981948 .983700 .985445 .987182 .988914 36 37 .979220 .980106 .981873 .983632 .985384 .987130 .988868 37 38 .979117 .980008 .981783 .983551 .985312 .987066 .988813 38 39 .979006 .979901 .981686 .983464 .985235 .986998 .988754 39 40 1.978883 1.979784 T. 981580 T. 983368 T. 985149 T.986923 1.988689 40 41 .978757 .979663 .981470 .983269 .985061 .986845 .988622 41 42 .978602 .979515 .981335 .983148 .984953 .986750 .988540 42 43 .978448 .979368 .981201 .983027 .984845 .986656 .988459 43 44 .978268 .979196 .981045 .982886 .984720 .986545 .988364 44 45 1.978073 T. 979009 T. 980876 1.982733 1.984583 T.986426 T.988260 45 46 .977861 .978807 .980691 .982567 .984435 .986296 .988149 46 47 .977624 .978580 .980485 .982382 .984271 .986151 .988023 47 48 .977365 .978332 .980260 .982179 .984090 .985992 .987886 48 49 .977082 .978062 .980013 .981957 .983892 .985819 .987736 49 50 T. 976767 T. 977760 1.979740 T.981711 T. 983673 T.985626 1.987570 50 51 .976422 .977430 .979440 .981441 ; .983432 .985414 ' .987388 51 52 .976044 .977069 .979112 .981145 ' .983169 .985183 .987188 52 53 1 .975630 .976673 .978752' 1 .980821 .982881 .984930 . .986970 53 * Months of Entry for Annual Valuation Dec. 31. X-\-tl + h= Present Age in Years and Fraction ; f'=h + {l—h)vj f= h-\-(l—h) vpx + n-L TABLE LXIY. ♦201 VALUE OF / AND LOG / FOR EVERY YEAR AND MONTH OF AGE. 4 PER CENT. Year of Age. x-\-n. 6m. 6|-m. 7im. 8jm. 9iin. 10|m. llim. 1 Year June. May. April. March. Feb. Jan. of Age. X + 71. /'=. 980769 .982372 .985577 .988782 .991987 .995192 .998397 Log/. 1 10 1.990185 T. 991011 T. 992660 1.994302 T. 995937 1.997667 1.999190 10 11 .990180 .991007 .992656 .994300 .995936 .997566 .999190 11 12 .990178 .991004 .992654 .994298 .995934 .997565 .999190 12 13 .990173 .991000 .992651 .994294 .995932 .997564 .999189 13 14 .990167 .990995 .992646 .994291 .995930 .997562 .999189 14 15 T.990161 1.990990 T. 992642 T. 994288 T. 995928 T.997561 1.999188 15 16 .990156 .990985 .992638 .994285 .995926 .997560 .999188 16 17 .990149 .990979 .992633 .994281 .995923 .997558 .999187 17 18 .990142 .990972 .992628 .994277 .995919 .997556 .999187 18 19 .990135 .990965 .992622 .994272 .995917 .997555 .999186 19 20 T.990125 T. 990957 1.992614 T. 994267 T. 995912 T. 997552 T. 999185 20 21 .990115 .990948 .992608 .994261 .995908 .997550 .999185 21 22 .990104 .990938 .992600 .994255 .995904 .997547 .999184 22 23 .990093 .990928 .992591 .994249 .995900 .997545 .999183 23 24 .990080 • .990916 .992581 .994241 .995894 .997541 .999182 24 25 T.990067 T. 990903 T. 992571 T. 994234 L995889 T. 997538 1.999181 25 26 .9900,50 .990888 .992559 .994224 .995882 .997534 .999179 26 27 .990033 .990873 .992547 .994214 .995875 .997530 .999178 27 28 .990015 .990856 .992533 .994204 .995868 .997525 .999176 28 29 .989992 .990835 .992516 .994191 .995858 .997519 .999175 29 30 T.989971 1.990815 T. 992499 1.994178 T. 995849 T. 997514 T. 999173 30 31 .989945 .990792 .992481 .994163 .995839 .997508 .999171 31 32 .989919 .990768 .992461 .994148 .995828 .997501 .999169 32 33 .989888 .990740 .992439 .994130 .995815 .997494 .999166 33 34 .989855 .990709 .992413 .994110 .995802 .997486 .999163 34 35 1.989816 T. 990673 1.992384 T. 994088 T. 995785 T. 997476 T. 999160 35 36 .989777 .990638 .992355 .994065 .995769 .997466 .999157 36 37 .989735 .990599 .992324 .994041 .995752 .997456 .999154 37 38 .989684 .990553 .992286 .994012 .995732 .997444 .999150 38 39 .989630 .990504 .992246 .993981 .995709 .997431 .999146 39 40 1.989570 1.990449 T. 992201 T. 993946 1.995684 1.997416* T. 999140 40 41 .989509 .990392 .992155 .993911 .995660 .997401 .999135 41 42 .989433 .990323 .992099 .993867 .995628 .997382 .999129 42 43 •989358 .990254 .992043 .993824 .995597 .997364 .999123 43 44 .989270 .990175 .991977 .993773 .995562 .997343 .999116 44 1 45 T.989175 1.990088 T. 991907 T. 993718 T. 995522 T.997319 1.999108 45 46 .989071 .989993 .991829 .993659 .995480 .997293 .999099 46 47 .988956 .989887 .991743 .993592 .995432 .997265 .999090 47 48 .988830 .989772 .991649 .993519 .995381 .997234 .999080 48 49 .988692 .989646 .991547 .993440 .995324 .997200 .999069 49 50 T.988539 T. 989506 T. 991433 1.993351 1.995261 1.997164 1.999057 50 51 .988371 .989352 .991308 .993254 .995192 .997122 .999042 51 52 .988187 .989184 .991170 .993149 .995117 .997077 .999028 52 53 .987986 .989000 .991021 .993032 .995035 . 9970^8. 999012 53 *20Z TABLE LXIY. VALUE OF /' AND LOG / FOR EVERY YEAR AND MONTH OP AGE. 4 PER CENT. Year of Age. h = Om. im. l4m. 24m. 34m. 44m. 54ni. Year of Age. x-\-n. * Dec. Nov. Oct. Sept. Aug. July. /' =.961538 .963141 .966346 .969551 .972756 .975962 .979167 , Log/. 54 T. 975175 1.976238 T. 978357 T. 980465 T. 982563 T.984651 1.986729 54 55 1.974667 1.975752 T. 977916 1.980068 1.982210 1.984341 1.986462 55 56 .974117 .975227 .977438 .979638 .981827 .984005 .986172 56 57 .973500 .974637 .976903 .979156 .981397 .983628 .985847 57 58 .972835 .974001 .976325 .978636 .980935 .983222 .985497 58 59 .972102 .973301 .975689 .978064 .980426 .982775 .985111 59 60 1.971290 T. 972525 T. 974984 T. 977430 1.979861 T.982280 1.984685 60 61 .970397 .971672 .974210 .976733 .979242 .981736 .984216 61 62 .969425 .970743 .973367 .975975 .978568 .981144 .983706 62 63 .968344 .969710 .972430 .975132 .977818 .980487 .983140 63 64 .967161 .968580 .971404 .974210 .976998 .979768 .982521 64 65 T. 965853 1.967331 T. 970271 1.973192 1.976093 T.978974 T.981838 65 66 .964427 .965969 .969036 .972082 .975107 .978110 .981093 66 67 .962842 .964456 .967664 .970850 .974012 .977152 .980268 67 68 .961128 .962819 .966181 .969518 .972829 .976116 .979377 68 69 .959205 .960983 .964519 .968026 .971504 .974956 .978379 69 70 1.957120 T. 958994 1.962718 1.966409 T. 970070 1.973700 T.977300 70 71 .954811 .956791 .960724 .964621 .968484 .972312 .976108 71 72 .952281 .954378 .958541 .962664 .966749 .970796 .974805 72 73 .949508 .951734 .956150 .960522 .964851 .969137 973381 73 74 .946420 .948789 .953490 .958140 .962742 .967295 .971800 74 75 T. 943060 T. 945588 T. 950599 1.955554 1.960453 T.965297 T.970086 75 76 .939374 .942077 .947432 .952722 .957947 .963111 .968215 76 77 .935277 .938175 .943915 .949579 .955171 .960692 .966143 77 78 .930801 .933915 .940078 .946154 .952148 .958059 .963891 78 79 .925875 .929230 .935863 .942396 .948832 .955174 .961425 79 80 T. 920451 T. 924073 L 931229 T. 938268 1.945195 T.952013 T.958726 80 81 .914509 .918428 .926162 .933761 .941228 .948569 .955790 81 82 .907916 .912169 .920552 .928777 .936848 .944773 .952554 82 83 .900757 .905378 .914475 .923386 .932117 .940677 .949070 83 84 .892777 .897815 .907720 .917403 .926875 .936145 .945221 84 85 T. 884080 T. 889581 T. 900378 T. 910913 1.921200 T.931247 T.941068 85 86 .874593 .880609 .892397 .903873 .915054 .925954 .936587 86 87 .863845 .870458 .883388 .895945 .908148 .920018 .931573 87 88 .852462 .859722 .873887 .887604 .900902 .913804 .926334 88 89 .832633 .841059 .857435 .873216 .888444 .903156 .917386 89 90 1.810236 1.820041 1.839009 1.857185 1.874629 1.891400 T. 907548 90 91 .790136 .801236 .822619 .842998 .862464 .881095 .898959 91 92 .768484 .781043 .805122 .827936 .849610 .870255 .889961 92 93 .742635 .757028 .784456 .810256 .834607 .857666 .879562 93 94 .713385 .729978 .761375 .790655 .818085 .843885 .868237 94 95 T. 681936 T. 701052 T. 736932 1.770072 1.800862 1.829612 T.856578 95 96 .627579 .651472 .695641 .735731 .772430 .806268 .837660 96 97 .585026 .613055 .664207 .709965 .751358 .789146 .823908 97 98 .505846 .542608 .607936 .664711 .714914 .759910 .800679 98 * Months of Entry for Annual Valuation Dec. 31. TABLE LXIY. *203 VALUE OF /' AND LOG / FOR EVERY YEAR AND MONTH OF AGE. 4 PER CENT. Year of Age. x+n. 6m. 64m. 7im. 84m. 94m. lO^m. Il4m. Jan. Year of Age. x+n. June. May. April. March. Feb. /'=. 980769 .982372 .985577 .988782 .991987 .995192 .998397 Log/. 54 T.987765 T. 988798 L 990856 T. 992905 T. 994944 1.996973 T. 998994 54 55 T.987518 T. 988572 1.990673 1.992763 1.994843 T. 996913 1.998974 55 56 .987252 .988328 .990474 .992609 .994734 .996848 .998952 56 57 .986952 .988055 .990251 .992437 .994611 .996774 .998928 57 58 .986630 .987760 .990011 .992251 .994479 .996696 .998901 58 59 .986275 - .987435 .989747 .992047 .994334 .996609 .998873 59 60 1.985882 T. 987076 T. 989455 T. 991821 1.994173 1.996514 1.998841 60 61 .985451 .986682 .989134 .991573 .993997 .996408 .998806 61 62 .984982 .986253 .988785 .991302 .993805 .996294 .998768 62 63 .984461 .985777 .988397 .991003 .993593 .996167 .998726 63 64 .983891 .985256 .987974 .990676 .993360 .996028 .998680 64 65 T.983262 T. 984682 1.987507 T. 990315 T. 993104 T. 995875 T. 998630 65 66 .982578 .984057 .986999 .989922 .992825 .995710 .998574 66 67 .981819 .983363 .986436 .989486 .992517 .995526 .998514 67 68 .980999 .982614 .985827 .989017 .992184 .995327 .998448 68 69 .980082 .981777 .985147 .988492 .991812 .995105 .998375 69 70 T.979089 T. 980871 T. 984412 T. 987924 T. 991409 T. 994866 T. 998295 70 71 .977993 .979870 .983600 .987299 .990966 .994602 .998208 71 72 .976796 .978777 .982714 .986615 .990482 .994314 .998113 72 73 .975487 .977584 .981746 .985870 .989954 .994000 .998009 73 74 .974036 .976260 .980674 .985044 .989370 .993653 .997894 74 75 T.972463 T. 974825 1.979512 1.984149 1.988737 1.993277 1.997771 75 76 .970744 .973259 .978245 .983174 .988048 .992869 .997636 76 77 .968843 .971526 .976844 .982098 .987288 .992418 .997487 77 78 .966777 .969645 .975324 .980930 .986465 .991929 .997326 78 79 .964517 .967587 .973663 .979656 .985566 .991397 .997151 79 80 T. 962044 T. 965337 T. 971849 T. 978264 T. 984586 T. 990817 1.996961 80 81 .959354 .962891 .969878 .976755 .983524 .990190 .996754 81 82 .956394 .960200 .967713 .975098 .982360 .989502 .996529 82 83 .953208 .957305 .965386 .973320 .981112 .988766 .996287 83 84 .949689 .954111 .962824 .971365 .979741 .987958 .996023 84 85 T.945896 T. 950671 T. 960067 T. 969264 1.978270 1.987093 T. 995740 85 86 .941807 .946966 .957103 .967008 .976693 .986167 .995438 86 87 .937237 .942827 .953798 .964498 .974941 .985139 .995103 87 88 •932467 .938513 .950360 .961892 .973125 .984076 .994757 88 89 .024330 .931164 .944518 .957475 .970056 .982282 .994174 89 90 1.915401 T.923115 T. 938145 T. 952672 T. 966729 T. 980344 T. 993546 90 91 .907623 .916118 .932624 .948525 .963865 .978682 .993009 91 92 .899490 .908813 .926880 .944226 .960905 .976968 .9924571 92 93 .890109 .900407 .920297 .939316 .957536 .975023 .991833 93 94 .879920 .891297 .913193 .934038 .953929 .972948 .991168 94 95 T.869458 T. 881965 T. 905951 T. 928681 T. 950280 1.970856 1.990501 95 96 .852543 .866934 ^ .894359 .920155 .944504 .967560 .989454 96 97 .840299 .856093 ! .886056 .914085 .940414 .965238 .988719 97 98 .819713 .837947 ! 1 .872269 .904075 1 .933710 .961452 1 .987527' 98 *204 TABLE LXY. JOINT LIVES MALE. CORRECTION TO BE ADDED TO THE YOUNGEST AGE FOR REDUCING TWO LIVES .C, y, AND ALSO THREE LIVES x, y, z TO THE CASE OF TWO EQUAL AGES. Y. M. denote Years and Months. For Two Lives, the Correction is given in the Second Column only. For Three Lives, if a" be the Youngest Age, enter the Table jointly with y—x, z—x to find the Correc- tion of the Age x, to give u, that is, one of the two Equal Ages. With u, which increases uniformly with the age, proceed as before explained on Pages 358, 259. 'si Two Difference of Years or Ages z — X. !bI i«^ Lives. P Da.,. Nara;. XBaxc- ANa^o:. 10 100,000 1,695,834.4 5.000000 6.339381 55 6,063.75 56,930.00 3.783741 4.755265 11 94,457.8 1,595,834.4 4.975338 .202985 56 5,585.03 50,856.35 .747035 .706344 12 89,218.7 1,501,366.6 .950456 .176487 i 57 5,131.07 45,371.33 .710208 .655833 13 84,267.7 1,412,147.9 .935661 .149880 158,4,700.67 40,140.16 .672160 .603579 14 79,588.3 1,327,880.3 .900849 .133159 59 4,393.19 35,439.49 .632780 .549487 15 75,163.8 1,248,291.9 4.876009 6.096316 60 '3,907.83 31,146.30 3.591934 4.493407 16; 70,981.6 1,173,128.1 .851146 .069345! 6li3,543.76 37,338.48 .549465 .435183 17 i 67,039.5 1,102,146.5 .836366 .043339 62 3,300.44 33,694.73 .505209 .374653 18 i 63,293.0 1,035,117.0 .801356 .014989 63j3,877.46 30,494.28 .459009 .311633 19 59,760.6 971,834.0 .776415 5.987588 64 3,574.33 17,616.82 .410648 .345937 20 56,421.6 913,063.4 4.751445 5.960035 65 2,290.45 15,042.59 3.359930 4.177333 21 53,364.1 855,641.8 .726435 .933393 !66|3,035.70 12,752.14 .306576 .105583 22 50,278.7 802,377.7 .701384 .904379 67 1,779.84 10,726.44 .350383 .030456 23 47,456.1 752,099.0 .676393 .876375 68 1,552.44 8,946.60 .191015 3.951658 24 44,787.0 704,643.9 .651153 .847969 69 1,343.45 7,394.16 .138331 .868889 25' 43,363.8 659,855.9 4.625958 5.819449 70 1,152.34 6,050.71 3.061583 3.781806 26 39,875.9 617,593.1 .600711 .790703 71 978.976 4,898.367 2.990772 .690051 27 37,617.8 577,717.3 .575393 .761715 72 833.894 3,919.391 .915344 .593319 28 35,482.1 540,099.4 .550009 .733474 73 1 683.688 3,096.497 .834858 .490871 29 33,461.9 504,617.3 .524551 .703963 74 560.831 2,412.809 .748824 .383533 30 31,549.8 471,155.4 4.498996 5.673164 75 453.537 1,851.988 2.656613 3.367638 31 29,740.9 439,605.6 .473354 .643063 76 361.148 1,398.451 .557685 .145647 32 28,029.2 409,864.7 .447610 .613640 77 383.737 1,037.303 .451383 .015906 33 26,409.2 381,835.5 .421755 .581876 78 317.313 754.566 .336886 3.877697 34 24,875.7 355,436.3 .395775 .550750 79 163.471 537.353 .213440 .730360 35 23,423.9 330,550.6 4.369659 5.519338 80 130.365 373.883 2.080140 3.573735 36 22,048.6 307,136.7 .343381 .487317 81 86.2963 353.6169 1.935992 .404178 37 20,746.6 385,078.1 .316946 .454964 82 60.3506 167.3306 .779961 .223550 38 19,513.5 364,331.5 .290336 .422149 83 40.8078 107.0700 .610743 .029668 39 18,345.1 344,818.0 .263520 .388843 84 36.7438 66.3623 .427207 1.821366 40 17,237.9 326,472.9 4.236483 5.355016 85 16.8933 39.5194 1.337713 1.596810 41 16,188.2 209,335.0 .209199 .320634 86 10.2533 23.6362 .010832 .354612 42 15,193.7 193,046.8 .181663 .285663 87 5.95604 12.37393 0.774958 .092508 43 14,250.1 177,853.1 .153819 .350061 88 3.29306 6.41789 .517599 0.807392 44 13,355.6 163,603.0 .125664 .213791 89 1.737736 3.124835 .337475 .494826 45 12,506.9 150,347.4 4.097150 5.176807 90 .8373495 1.3971089 T.917689 0.145230 46 11,701.7 137,740.5 .068248 .139062 91 .3573668 .5697594 .553113 1.755693 47 10,937.5 136,038.8 .038919 .100504 92 .1407139 .3133926 .148334 .337139 48 10,212.1 115,101.3 .009117 .061080 93 .0501480 .0716797 2.700354 3.855396 49 9,533.53 104,889.19 3.978798 .030731 94 .0158663 .0315317 .300474 .333078 50 8,869.78 95,365.66 3.947913 4.979393 95 .0043873 .0056654 3.643193 3.753334 51 8,348.90 86,495.88 .916396 .936995 96 .0010496 .0012783 .031017 .106595 52 7,659.35 78,346.98 .884192 .893468 97 .0001954 .0003386 4.391135 4.359084 53 7,099.54 70,587.63 .851230 .848739 98 .0000299 .0000331 5.476139 5.530012 54 6,568.09 63,488.09 .817439 .803693 99 .0000031 .0000033 6.503770 6.503770 J *210 TABLE LXX COMMUTATION COLUMNS FOR TWO EQUAL AGES. MALE LEFK 5 PER CENT. ^ jy^ N«B. XT)^. ANaa:. M ^- N«^ AjUaa;. ANasr. 10 100,000 1,587,002.3 5.000000 6.200578 55 4,891.66 44,516.96 3.689456 4.648526 11 94,008.0 1,487,002.3 4.973165 .172312 56 4,484.00 39,625.30 .65M66 .5979731 12 88,369.2 1,392,994.3 .946309 .143949 57 4,099.94 35,141.30 .612777 .545818 13 83,069.6 1,304,625.1 .919442 .115486 58 3,738.14 31,041.36 .572656 .491941 14 78,083.1 1,221,555.5 .892557 .086913 59 3,397.83 27,303.22 .531202 .436214 15 73,391.2 1,143,472.4 4.865644 6.058226 60 3,078.10 23,905.39 3.488283 4.378496 16 68,977.6 1,070,081.2 .838708 .029417 61 '2,778.06 20,827.29 .443742i .318633 17 64,826.9 1,001,103.6 .811755 .000479 62 i2, 496.97 18,049.23 .397413 .256459 18 60,921.7 936,276.7 .784772 5.971404 63 2,234.29 15,552.26 .349139 .191792 19 57,247.7 875,355.0 .757758 .942184 64 1,989.33 13,317.97 .298706 .124438 20 53,791.7 818,107.3 4,730715 5.912810 65 1,761.59 • 11,328.64 3.245905'4.054178 21 50,539.6 764,315.6 .703632 .883273 66 1,550.55 9,567.05 .190487 3.980778 22 47,479.7 713,776.0 .676508 .853562 67 1,355.88 8,016.50 .132220 .903985 23 44,600.8 666,296.3 .649343 .823667 68 1,177.01 6,660.62 .070781! .823515 24 41,891.9 621,695.5 .622130 .793578 69 1,013.71 5,483.61 .005913 .739067 25 39,342.6 579,803.6 4.594863 5.763281 70 865.370 4,469.901 2.937202 3.650298 26 36,943.9 540,461.0 .567543 .732764 71 731.675 3,604.531 .864318 .556849 27 34,685.8 503,517.1 .540152 .702014 72 612.094 2,872.856 .786818 .458314 28 32,560.8 468,831.3 .512695 .671017 73 506.125 2,260.762 .704258 .354255 29 30,560.7 436,270.5 .485163 .639756 74 413.191 1,754.637 .616151 .2441871 30 28,677.1 405,709.8 4.457536 5.608215 75 332.558 1,341.446 2.521868 3.1275731 31 26,904.3 377,032.7 .429821 .576379 76 263.552 1,008.888 .420866 .003843 32 25,235.0 350,128.4 .402003 .544227! 77 205.348 745.336 .312491 2.872352 33 23,663.3 324,893.4 .374076 .511741' 78 157.008 539.988 .195922 .732384 34 22,183.1 301,230.1 .346023 .478898; 79 117.599 382.980 .070402 .583176 35 20,789.0 279,047.0 4.317834 5.445677 80 86.1051 265.3809 1.935029 2.423870 36 19,475.2 258,258.0 .289482 .412054 81 61.4905 179.2758 .788808 .253522 37 18,237.9 238,782.8 .260975 .378003 82 42.7273 117.7853 .630705 .071091 38 17,072.3 220,544.9 .232292 .343497 83 28.8014 75.0580 .459413 1.875397 39 15,973.9 203,472.6 .203402 .308506 84 18.7847 46.2566 .273804 .665174 40 14,938.0 187,498.7 4.174293 5.272998 85 11.8096 27.4719 1.072237 1.438889 41 13,961.6 172,560.7 .144936 .236942 86 7.13301 15.66231 0.853273 .194856 42 13,041.5 158,599.1 .115326 .200301' 87 4.12417 8.52930 .615336 0.930913 43 12,173.3 145,557.6 .085409 .163035 88 2.26937 4.40513 .355905 .643959 44 11,354.9 133,384.3 .055182 .125104 89 1.18497 2.13576 .073707 .329552 45 10,582.7 122,029.4 4.024595 5.086464 90 .564739 .950788 T.751848 T.978084 46 9,854.15 111,446.70 3.993619 .0470671 91 .242772 .386049 .385199 .586642 47 9,166.80 101,592.55 .962218 .006858 92 .0951367 .1432769 2.978348 .156176 48 8,518.11 92,425.75 .930343 4.965793 93 .0337438 .0481402 .528194 2.682508 49 7,905.89 83,907.64 .897951 .923801 94 .0106253 .0143964 .026341 .158254 50 7,328.11 76,001.75 3.864992 4.880824 95 .0029241 .0037711 3.465988 3.576466 51 6,782.71 68,673.64 .831403 .836790 96 .0006962 .0008470 4.842738 4.927886 52 6,267.95 61,890.93 .797125 .791627 97 .0001291 .0001508 .110773 .178401 53 5,782.16 55,622.98 .762090 .745254 98 .0000197 .0000217 5.293704 5.337379 54 5,323.86 49,840.82 .726227 .697585 99 . .0000021 .0000021 6.318272 6.318272 ♦^11 TABLE LXXI. COMMUTATION COLUMNS FOR TWO EQUAL AGES. MALE LIFE. 6 PER CENT. DaxB. N«^ ^iJxX' A^a-g. T>a^. ^xx- ADa:aj. ANa^. 10 100,000.0 1,405,298.1 5.000000 6.147768 55 3,193.08 27,382.81 3.504210 4.437478 11 93,121.3 1,305,298.1 4.969049 .115710 56 2,899.38 24,189.73 .462305! .3836311 12 86,711.6 1,212,176.8 .938077 .083566 57 2,626.03 21,290.35 .419299 .328183 13 80,740.8 1,125,465.2 .907093 .051332 58 2,371.71 18,664.32 .375061 .271012 14 75,178.0 1,044,724.4 .876091 .019002 59 2,135.46 16,292.61 .329491 .211991 15 69,994.0 969,546.4 4.845061 5.986569 60 1,916.26 14,157.15 3.282455 4.150976 16 65,164.2 899,552.4 .814009 .954027 6111,713.16 12,240.89 .233797! .087813] 17 60,665.1 834,388.2 .782939 .921368 62 1,525.29 10,527.73 .183351 .022335 18 56,472.8 773,723.1 .751839 .888585 63 1,351.95 9,002.44 .130962 3.954360 19 52,566.6 717,250.3 .720710 .855671 64 1,192.37 7,650.49 .076412 .883689 20 48,927.2 664,683.7 4.689550 5.822615 65 1,045.91 6,458.12 3.019494 3.810106 21 45,535.5 615,756.5 .658350 .789409 66 911.927 5,412.212 2.959960 .733375 22 42,375.0 570,221.0 .627110 .756043 67 789.907 4,500.285 .897576 .653240 23 39,430.1 527,846.0 .595828 .722507 68 679.235 3,710.378 .832020 .569418! 24 36,685.8 488,415.9 .564498 .688790 69 579.477 3,031.143 .763036 .481606 25 34,128.3 451,730.1 4.533114 5.654879 70 490.013 2,451.666 2.690208 3.389461 26 31,745.3 417,601.8 .501679 .620762 71 410.402 1,961.653: .613209 .292622 27 29,523.7 385,856.5i .470171 .586426 72 340.088 1,551.251 .531591 .190682 28 27,453.5 356,332.8 .438597 .551856 73 278.558 1,211.163 .444915 .083203 29 25,524.0 328,879.3 .406948 .517036 74 225.264 932.605 .35269112.969698 30 23,724.9 303,355.3 4.375205 5.481952 75 179.594 707.341 2.254291 2.849629 31 22,048.2 279,630.4 .343373 .446584 76 140.985 527.747 .149173 .722426 32 20,485.1 257,582.2 .311439 .410916 77 108.813 386.762 .036681 .587444 33 19,028.1 237,097.1 .279395 .374926 78 82.4131 277.9488il.915996 .443965 34 17,669.6 218,069.0 .247226 .338594 79 61.1449 195.5357 .786360 .291226 35 16,402.9 200,399.4 4.214920 5.301896 80 44.3476 134.3908 1.646870 2.128370 36 15,221.3 183,996.5 .182452 .264810 81 31.3713 90.0432 .496532 1.954451 37 14,119.8 168,775.2 .149828 .227309 82 21.5930 58.6719 .334312 .768430 38 i 13,092.7 154,655.4 .117028 .189365 83 14.4180 37.0789 .158904 .569127 39 12,134.5 141,562.7 .084022 .150949 84 9.314896 22.660866 0.969178 .355277 40 11,240.8 129,428.2 4.050796 5.112029 85 5.800895 13.345970 0.763495 1.125350 41 10,406.9 118,187.4 .017323 .072571 86 3.470683 7.545075 .540415 0.877664 42 9,629.35 107,780.51 3.983597 .032540 87 1.987747 4.074392 .298361 .610063 43 8,903.55 98,151.16 .949563 4.991895 88 1.083460 2.086645 .034813 .319448 44 8,226.57 89,247.61 .915219 .950597 89 .560401 1.003185 1.748499 .001381 45 7,594.78 81,021.04 3.880515 4.908598 90 .264559 .442783 1.422523 1.646191 46 7,005.24 73,426.26 .845423 .865851 91 .112657 .178224 .051757 .250966 47 6,455.13 66,421.02 .809905 .822306 92 .043731 .065567 2.640789 2.8166851 48 5,941.73 59,965.89 .773913 .777904 93 .015364 .021836 .186520 .339173 49 5,462.67 54,024.16 .737405 .732588' 94 .004792 .006472 3.680550 3.811038 50 5,015.68 48,561.49 3.700330 4.686292 95 .001306 .001680 3.116080 3.225309 51 4,598.58 43,545.81 .662624 .638946 96 .000308 .000374 4.488714 4.572872 52 4,209.50 38,947.23 .624230 .590477^ 97 .000057 .000066 5.752632 5.819544 53 3,846.61 34,737.73 .585078 .540801' 98 .0000085 .0000094 6.931446 6.974733 54 3,508.31 30,891.12 .545098 .4898341 1 99 .0000009 .0000009 7.951898 7.951898 ♦212 TABLE LXXII. JOINT LIFE ANNUITIES. PRESENT VALUE {ttxx) OF 1, PAYABLE AT THE END OF EACH YEAR DURING THE JOINT CONTINUANCE OF TWO MALE LIVES. 1 ^•4 ; Per Ct. Per Ct. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 20.237 20.117 19.992 19.863 19.729 19.591 19.447 19.298 19.144 18.985 18.820 18.649 18.473 18.291 18.102 17.908 17.708 17.501 17.288 17.069 16.844 16.612 16.373 16.128 15.876 15.618 15.354 15.084 14.807 14.524 14.235 13.940 13.639 13.334 13.023 12.707 12.388 12.061 11.733 11.400 4 PerCt. PerCt. 5 PerCt. 6 PerCt. 18.610 17.1956 15.9583 14.870 13.053' 18.514ll7.1174'l5.8946,14.818|13.017 18.413:17.0359115.827914.76312.979; 18.309'l6.9504 15. 7579il4.705 12.939' 18.200'l6.8612ll5.6844'l4.644|12.897| 18.08616.7683 15.6076 14.58l!l2.852! 17.969!l6.6713jl5.5272|l4.514ll2.804| 17.846!l6.5698il5.4427|l4. 443112.754 17.719!l6.4640l5.3543 14.369^12. 70l| 17.587J16.3537il5.262014.291il2.645i I7.449!l6.2387 15.1652 17.307|16.1190il5.0641 17.159 15.9944!l4 9586 17.005 16.846 15.864714.8483 15.7297il4.7332 16.68215.589414.6132 16.511 16.335 16.152 15.963 15.769 15.568 15.360 15.443514.4878 15.292214.3579 15.135l!l4.2218 14.209ll2.585 14.12312.523 14.03312.457 13.939ll2.387 13.841 12.314 13.73712.236 13.629 13.517 13.399 14.9721 14.0803 13.276 14.8035 14.6287 14.4479 15.147|14.2607 14.927114.0674 14.700ll3.8678 14.467 13.6623 14.228 13.983 13.731 13.473 13.209 12.939 12.664 12.383 13.4503 13.2318 13.0072 12.7765 12.5401 12.2967 12.0482 11.7936 13.9337 13.7812 13.6228 13.4584 12155 12.069 11.980 11.885 13.148 11.786 13.014 11.683i 12.875 12.730 11.574 11.460; 13.2881 12.57911.342 12.097J11.5.336 11.805 11.2684 11.509 11.209 10.904 ll.€65 10.596 10.72710.284 10.3871 9.970 10.045! 9.653 9.702' 9.334 10.9979 10.7227 10.4430 101591 9.8715 9.5805 9.2864 8.9898 131117 12.9295 12.7410 12.5461 12.3451 121381 11.9251 11.7057 11.4808 11.2497 11.0131 10.7710 10.5235 10.2710 10.0137 9.7518 9.4857 9.2159 8.9426 8.6661 12.42311.217 12.26l!ll.088 12.093il0.953| 11.91810.812 11.73810.666 11.552 11.360 11.161 10.957 10.747 10.531 10.310 10.083 9.851 9.613 9.371 9.125 8.874 8.620 8.362 10.514! 10.3571 10.1931 10.024| 9.849 9.668 9.482 9.290 9.092 8.890| 8.682! 8.469' 8.252: 8.031| 7.805 ! 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 3 ^ PerCt. PerCt. 9.3589.014 9.014 8.693 8.67l!8.372 8.3298.051 7.988 7.730 4 PerCt. 8.6909 8.3907 8.0893 7.7880 7.4862 7.650 7. 41117.1851 7.3157.095 6.8853 6.983:6.780 6.65616.469 6.333:6.162 6.016:5.859 O.704 5.399 5.101 4.810 4.527 4.252 3.986 3.729 3.480 3.242 3.013 2.793 2.583 2.383 2.193 2.012 1.841 1.679 1.526 380 242 1.107 .973 .828 .704 .607 .520 .438 .364 .296 .221 .172 .107 5.562 5.270 4.984 4.704 4.431 4.166 3.909 3.660 3.419 3.187 2.965 2.750 2.546 2.350 2.164 1.987 1.819 1.660 1.509 1.366 1.230 1.097 .965 .821 .699 .603 .516 .435 .361 .295 .220 .171 .108 .001 6.5874 6.2918 5.9992 5.7103 5.4257 5.1456 4.8711 4.6019 4.3394 4.0835 3.8348 3.5935 3.3598 3.1346 2.9177 2.7091 2.5094 2.3184 2.1362 1.9629 1.7979 1.6419 PerCt. 8.3869 8.1058 7.8230 7.5392 5 PerCt. 8.101 7.837 7.571 7.304 7.254817.035 6 PerCt. 7.576 7.343 .107 6.870 6.630 6.9703 6.766 6.68636.497 6.4036:6.228 6.1224 5.9615.659 5.8435 5.695 5.416 6.388 6.145 5.902 5.5675:5.431 5.29525.170 5.0266;4.912 4.7629 4.659 4.5039 4.409 4.2508 4,165 4.0036L3.926 3.762913.693 3.5291:3.467 3.3023!3.247 5.175 4.935 4.697 4.463 4.231 4.003 3.780 3.561 3.348 3.140 3.0834;3.034 2.939 2.8722!2;828;2.743 2.6688!2.630 2.554 2.4739 2. 439j2. 373 2.2872-2.257 2.198 2.108812.082 1.93891.916 1.777l|l.757 1.62381.606 1.49341.47781.462 1.35281.3394 1.2183il.2069 1.08711.0775 .9568 .8150 .6938 .5985 .5128 .4322 .3593 .2930 .2190 .1703 .1069 .0000 .9489 .8086 .6887 .5943 .5094 .4294 .3571 .2913 .2178 .1694 .1063 .0000 1.326 1.196 1.068 .941 2.0301 1.870 1.717 1.572 1.433 1.301 1.174 1.050 926 .802 .790 .684 .590 .506 .427 .355 .290 .217 .169 .106 .000 .674 .582 .499 .421 .350 .286 .214 .167 .105 .000 K x+l .x+1 FOBMIJLA, Oxa; = fT^- Vqcx TABLE LXXIII. 213 ANNUAL PREMIUM (rr) FOR THE INSURANCE OF 1,000 ON TWO JOINT LIVES MALE, PAYABLE AT THE END OF THE YEAR OF THE FIRST DEATH. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 3 Per Ct, 17.97 18.23 18.52 18.80 19.11 19.44 19.78 20.14 20.52 20.91 21.33 21.77 22.23 22.71 23.22 23.76 24.33 24.93 25.56 26.21 26.92 27.66 28.44 29.26 30.13 31.05 32.02 33.05 34.14 35.30 36.51 37.81 39.18 40.64 42.19 43.83 45.58 47.44 49.41 51.52 34 Per Ct. 17.18 17.43 17.69 17.98 18.27 18.58 18.90 19.25 19.61 19.99 20.39 30.81 21.26 21.72 22.22 22.74 23.29 23.87 24.49 25.14 25.82 26.54 27.31 28.12 28.98 29.88 30.84 31.85 32.93 34.07 35.28 36.56 37.92 39.37 40.91 4 PerCt. 16.49 16.74 16.98 17.25 17.53 17.82 18.13 18.45 18.80 19.17 19.55 19.95 20.38 20.84 21.31 21.82 22.36 22.92 23.51 24.15 24.82 25.53 26,27 27.07 27.91 28.80 29.74 30.74 31.80 32.93 34.13 35.39 36.75 38.18 39.70 42.54 41.32 44. 28:43.05 46.12 48.09 50.19 53.76 52.43 56.1554.80 58.70 57.34 61.4160.05 64.3262.95 44.88 46.84 48.93 PerCt, 15.91 16.12 16.37 10.61 16.87 17.15 17.44 17.76 18.08 18.43 18.80 19.19 19.60 20.04 20.50 20.99 21.51 22.05 22.63 23.25 23.90 24.59 25.33 26.10 26.93 27.80 28.73 29.71 30.76 31.87 33.05 .34.31 35.65 37.06 38.57 5. Per Ct. 15.39 15.60 15.82 16.06 16.30 16.57 16.84 17.14 17.45 17.78 18.13 18.51 18.90 19.32 19.77 20.23 20.73 21.27 21.84 22.43 23.06 23.74 24.46 25.21 26.02 26.88 27.79 28.76 6 PerCt. 14.55 14.73 14.93 15.13 15.36 15.59 15.84 16.10 16.39 16.69 17.00 17.35 17.71 18.10 18.51 18.95 19.41 19.91 20.44 21.01 21.60 22.24 22.93 23.65 24.42 25.25 26.12 27.06 29.79128.05 30.88 29.12 32.05 33.29 34.61 36.01 37.51 40.18 39.10 41.89 40.80 43.7242.61 45.66 44.55 ^7.74 46.61 51.15 49.95 48.79 53.52 52.3151.14 56.06 54.83 53.65 58.75|57.51|56.33 61.64 60. 40159. 20 30.25 31.45 32.74 34.11 35.58 37.14 38.80 40,58 42.48 44.51 46.69 49.00 51.48 54.13 56.97 5 .257479 21 22 .274604 .204952 .276564 .206912 .278515 .208863 , .280459 22 23 .297615 .201416 .299580 .203381 .301536 .205337 .303482 2G 24 .320675 .197708 .322644 .199677 .324605 .201638 .326556 24 25 5.343789 1.193821 5.345762 1.195794 5.347727 1.197759 5.349683 25 26 .366956 .189739 .368935 .191718 .370907 .193690 .372868 26 27 .390193 .185467 .392179 .187453 .394156 .189430 .396124 27 28 .413498 .180986 .415490 .182978 .417473 .184961 .419448 28 29 .436878 .176286 .438878 .178286 .440870 .180278 .442852 29 30 5.460353 1.171372 0.462361 1.173380 5.464360 1.175379 5.466350 30 31 .483916 .166219 .485933 .168236 .487941 .170244 .489939 31 32 .507583 .160820 .509610 .162847 .511627 .164864 .513635 32 33 .531360 .155164 .533397 .157201 .535425 .159229 .537444 33 34 .555263 .149242 .557313 .151292 .559352 .153331 .561383 34 35 5.579303 1.143043 0.581367 1.145107 5.583422 1.147162 0.585466 35 36 .603506 .136568 .605583 .138645 .607650 .140712 .609708 36 37 .627865 .129783 .629958 .131876 .632040 .133958 .634113 37 38 .652401 .122677 .654512 .124788 i .656613 .126889 .658703 38 39 .677144 .115253 .679274 .117383 .681394 .119503 .683502 39 40 5.702109 1.107501 5.704260 1.109652 0. 706401 1.111793 5.708531 40 41 .727321 .099381 : .729495 .101555 .731657 .103717 .733809 41 42 .752788 .090893 ^ .754988 .093093 .757178 .095283 .759356 42 43 .778564 .082040 .780791 .084267 .783008 .086484 .785212 43 44 .804650 .072780 .806909 .075039 .809157 .077287 .811393 44 45 5.831099 1.063118 0.833392 1.065411 5.835673 1.067692 5.837942 45 46 .857937 .053030 .860268 .055361 .862586 .057679 1 .864892 46 i 47 .885203 .042501 .887575 .044873 ! .889934 .047232 .892281 47 48 .912945 .031519 .915363 .033937 .917767 .036341 .920158 48 49 .941206 .020066 .943674 .022534 .946127 .024987 .948567 49 1 50 5.970037 1.008123 5.972560 1.010646 5.975069 1.013155 5.977563 50 51 .999501 0.995680 4.002085 0.998264 : 4.004653 .000832 4.007207 51 52 4 029657 .982718 .032307 .985368 i .034942 0.988003 .037561 52 53 .060573 .969218 .063297 .971942 ' .066003 .974648 .068693 53 Formula, V = /" f P; • b^^^ - P^ • 5;,„ ; F^ = (P. - -x) N.. The suffix is not here repeated for equal Ages, Page 270. I^ = Pa: + 1000 (1 — v). nil TABLE LXXYI. GENERAL VALUATION OP JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000. 4 PER CENT. Equal *Sept. 4|-rn. Aug. 5^in. July. 6m. Equal Ages. x-^n. 975.962 979.167 980.769 Ages. x+n. 10 Xb'. Xb. Xb'. Xb. Xb'. Xb. Xb'. 10 1.242155 0. 008645 1.244061 5.010542 1.245958 5.011488 1.246904 11 .240182 .031332 .242090 .033232 .243990 .034178 .244936 11 12 .238112 .054035 .240020 .055936 .241921 .056883 • .242868 12 13 .235934 .076754 .237845 .078656 .239747 .079604 .240695 13 14 .233651 .099494 .235563 .101398 .237467 .102348 .238417 14 15 1.231258 0. 122260 1.233172 0.124166 1.235078 5.125116 1.236028 15 16 .228743 .145046 .230659 .146954 .232567 .147905 .233518 16 17 .226101 .167855 .228019 .169766 .229930 .170718 .230882 17 18 .223328 .190694 .225249 .192607 .227162 .193560 .228115 18 19 .22.0418 .213563 .222342 .215479 .224258 .216433 .225212 19 20 1.217366 5.236467 1.219293 0.238384 1.221210 5.239341 1.222167 20 21 .214165 .259409 .216095 .261330 .218016 .262289 .218975 21 ^ 22 .210807 .282393 .212741 .284318 .214666 .285278 .215626 22 23 .207283 .305421 .209222 .307350 .211151 .308312 .212113 23 24 .203589 ! .328499 .205532 .330433 .207466 .331397 .208430 24 25 1.199715 5.351630 1.201662 5.353569 1.203601 6.354535 1.204567 25 26 .195651 .374822 .197605 . .376766 .199549 .377735 .200518 26 27 .191398 .398082 .193356 .400032 .195306 .401004 .196278 27 28 .186936 .421412 . .188900' .423369 .190857 .424344 .191832 28 29 .182260 .444825 .184233 .446790 .186198 .447769 .187177 29 30 1.177369 5.468331 1.179350 0. 470302 1.181321 5.471285 1.182304 30 31 .172242 .491930 .174233 .493910 .176213 .494897 .177200 31 32 .166872 .515633 .168870 .517623 .170860 .518614 .171851 32 33 .161248 .539453 .163257 .541453 .165257 .542450 .166254 33 34 .155362 .563404 .157383 .565415 .159394 .566418 .160397. 34 35 1.149206 5.587500 1.151240 5.589525 1.153265 5.590534 1.154274 35 36 .142770 .611756 .144818 .613795 .146857 .614810 .147872 36 37 .136031 .636176 .138094 .638229 .140147 .639252 .141170 37 38 .128979 .660783 .131059 .662853 .133129 .663885 .134161 38 39 .121611 .685601 .123710 .687690 .125799 .688731 .126840 39 40 1.113923 5.710651 1.116043 0.712762 1.118154 5.713813 1.119205 40 41 .105869 .735951 .108011 .738082 .110142 .739143 .111203 41 42 .097461 .761524 .099629 .763681 .101786 .764755 .102860 42 43 .088688 .787406 .090882 .789589 .093065 .790676 .094152 43 44 .079523 .813617 .081747 .815830 .083960 .816933 .085063 44 45 1.069961 5.840200 1.072219 5.842446 1.074465 5.843564 1.076583 45 46 .059985 .867186 .062279 .869467 .064560 .870604 .065697 46 47 .049579 .894615 .051913 .896937 .054235 .898093 .066391 47 48 .038732 .922536 .041110 .924901 .043475 .926079 .044653 48 49 .027427 .950993 .029853 .953406 .032266 .954607 .033467 49 50 1.015649 0. 980043 1.018129 5.982508 1.020594 0.983736 1.021822 50 51 .003386 4.009745 .005924 4.012269 .008448 4.013525 .009704 51 52 0.990622 .040164 0.993225 .042751 0.995812 .044039 0.997100 52 53 .977338 .071367 .980012 .074024 .982669 .075347 .983992 53 ■* Months of Entry corresponding to Annual Valuation, Dec. 31. Here x + n + h= Present Age in years and fraction ; f" =: 1000 J7i + (1 — ^) V^ ; _ h + (l — h) vpx+n~l . b' = b 'N *x+n = *2id TABLE LXXTI GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000. 4 PER CENT. Equal AgM. X+n. 7^=6Jra . June. 7^in. May. 8^in. April. 9|in. Equal 1 Ages, j x-\-n. /" = 982.372. 985.577. 988.782. 991.987. Xb. Xb'. Xb. Xb'. Xb. Xb'. Xb. 10 5.012432 1.247848 O.014313 1.249729 0.016187 1.251603 5.018052 10 11 .035122 .245880 .037006 .247764 ■ .038880 .249638 .040747 U 12 .057828 .243813 .059712 .245697 .061588 .247573 .063456 12 13 .080549 .241640 .082435 .243526 .084313 .245404 .086182 13 14 .103294 .239363 .105182 .241251 .107062 .243131 .108933 14 15 5.126063 1.236975 ' 0.127954 1.238866 0.129835 1.24074t 5.131708 15 16 .148853 .234466 .150744 .236357 .152628 .238241 .154502 16 17 .171667 .23183] .173561 .233725 ' .175446 .235610 .177323 17 18 .194511 .229066 .196407 .230962 .198295 .232850 .200175 18 19 .217386 .226165 .219284 .228063 .221174 .229953 .223056 19 20 5.240295 1.223121 0.242196 1.225022 5.244090 1.226916 5.245975 20 21 .263244 .219930 .265149 .221835 .267046 .223732 .268935 21 22 .286235 .216583 .288143 .218491 .290044 .220392 .291936 22 23 .309271 .213072 .311183 .214984 .313087 .216888 .314984 23 24 .332358 .209391 ; .334275 .211308 .336183 .213216 1 .338084 24 25 5.355499 1.205531 0. 357420 1.207452 0.359333 1.209365 5.361238 25 26 .378702 .201485 ; .380629 .203412 .382548 .205331 .384458 26 27 .401974 .197248 i .403906 .199180 .405830 .201104 ; .407746 27 28 .425316 .192804 .427255 .194743 .429185 .196673 ! .431107 28 29 .448745 .188153 .450691 .190099 .452630 .192038 ; .454559 29 30 5.472265 1.183284 5.474218 1.185237 5.476164 1.187183 5.478101 30 31 .495881 .178184 .497844 .180147 .499797 .182100 .501742 31 32 .519604 .172841 1 .521575 .174812 .523537 .176774 .525491 32 33 .543444 .167248 i .545426 .169230 .547399 .171203 ,549363 33 34 .567418 .161397 .569411 .163390 .571394 .165373 .573370 34 35 5.591541 1.155281 ^ 0.593548 1.157288 5.595544 1.159284 5.597533 35 36 .615823 .148885 .617843 .150905 .619853 .152915 .621853 36 37 .640273 .142191 .642306 .144224 .644330 .146248 .646346 37 38 .664913 .135189 .666965 .137241 .669006 .139282 .671037 38 39 .689769 .127878 i .691839 .129948 .693898 .132007 .695947 39 40 5.714861 1.120253 0. 716950 1.122342 5.719030 1.124422 5.721099 40 41 .740202 .112262 .742313 .114373 .744412 .116472 .746502 41 42 .765827 .103932 .767962 .106067 .770088 .108193 .772202 42 43 .791761 .095237 .793922 .097398 .796073 .099549 .798212 43 44 .818032 .086162 .820223 .088353 .822403 .090533 .824572 44 45 5.844680 1.076699 0. 846903 1.078922 5.849114 1.081133 0.851314 45 46 .871736 .066829 .873994 .069087 .876241 .071334 .878476 46 47 .899246 .056544 .901543 .058841 .903828 .061126 .906100 47 48 .927253 .045827 .929593 .048167 .931919 .050493 .934234 48 49 .955805 .034665 .958192 .037052 .960565 .039425 .962925 49 50 5.984960 1.023046 5.987398 1.025484 5.989822 1.027908 5.992234 50 51 4.014778 .010957 4.017273 .013452 4.019753 .015932 4.022220 51 52 .045323 0.998384 .047881 .000942 .050422 .003483 .052950 52 53 .076665 .985310 .079291 0.987936 .081900 0.990545 .084494 53 ♦219 TABLE LXXYI. GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000, 4 PER CENT. Equal Ages. March. lOjm. Feb. llj^m. Jan. 12m. Equal Ages. x-\-n. 995.192. 998.397. 1000.000. x-\-n. kb'. kb. W. A5. Xb'. U. AJ'. 10 1.253468 5.019909 1.255325 0. 021759 1.257175 5.022680 1.258096 10 11 .351505 ' .042606 .253364 .044456 .255214 .045379 .256137 11 12 .249441 1 .065315 .251300 .067167 .253152 ; .068090 .254075 12 13 .247273 .088043 .249134 .089896 .250987 ! .090820 .251911 13 14 .245002 .110797 .246866 .112653 .248722 .113577 .249646 14 15 1. 242620 5.133573 1.244485 5.135431 1.246343 5.136356 1.247268 15 16 .240115 .156368 .241981 .158228 .243841 .159154 .244767 16 17 .237487 .179192 .239356 .181054 .241218 .181981 .242145 17 18 .234730 .202046 .236601 .203910 .238465 .204838 .239393 18 19 .231835 .224930 .233709 .226796 .235575 .227726 .236505 19 20 1.^^.28801 0.247853 1.230679 5.249722 1.232548 15.250653 1.233479 20 21 .225621 .270815 .227501 .272687 .229373 .273620 .230306 21 22 .222284 ..293819 .224167 .295695 .226043 1 .296630 .226978 22 23 .218785 .316872 .220673 .318751 .222552 ! .319687 .223488 23 24 .215117 .339976 .217009 .341860 .218893 .342798 .219831 24 25 1.211270 5.363134 1.213166 0. 365022 1.215054 5.365962 1.215994 25 26 .207241 .386360 .209143 .388253 .211036 .389197 .211980 26 27 .203020 .409653 .204927 .411552 .206826 .412498 .207772 27 28 .138595 .433020 .200508 .434925 .202413 .435874 .203362 28 29 .193967 .456479 .195887 .458392 .197800 i .459345 .198753 29 30 1.189120 5.480028 1.191047 0.481948 il. 192967 5.482904 1.193923 30 31 .184045 .503678 .185981 .505606 .187909 .506566 .188869 31 32 .178728 .527436 .180673 .529372 .182609 .530337 .183574 32 33 .173167 .551318 .175122 1 .553264 .177068 .554234 .178038 33 34 .167349 .575336 .169315 .577293 .171272 .578268 .172247 34 35 1.161273 0.599512 1.163252 0. 601481 1.165221 5.602463 1.166203 35 36 .154915 .623845 .156907 .625828 .158890 .626815 .159877 36 37 .148264 .648351 .150269 .650347 .152^65 .651342 .153260 37 38 .141313 . .673060 .143336 .675072 .145348 .676075 .146351 38 39 .134056 .697987 .136096 .700018 .138127 .701029 .139138 39 40 1.126491 0.723159 1.128551 5.725209 1.130601 5.726230 1.131622 40 41 .118562 .748582 .120642 .750652 .122712 .751683 .123743 41 42 .110307 .774307 .112412 .776401 .114506 .777445 .115550 42 43 .101688 .800342 .103818 .802461 .105937 .803516 .106992 43 44 .092702 .826730 .094860 .828877 .097007 .829947 .098077 44 45 1.083333 0. 853504 1.085523 0.855683 1.087702 5.856767 1.088786 45 46 .073569 .880698 .075791 .882910 .078003 1 .884012 .079105 46 47 .063398 .908362 .065660 .910611 .067909 .911731 .069029 47 48 .052808 .936537 .055111 .938827 .057401 .939967 .058541 48 49 .041785 .965272 .044132 .967608 .046468 .968770 .047630 49 50 1.030320 5.994631 1.032717 5.997016 1.035102 i 5.998203 1.036289 50 51 .018399 4.024672 .020851 4.027111 .023290 4.028325 i .024504 51 52 .006011 .055463 .008524 .057960 .011021 .059204 1 .012265 52 53 0.993139 .087072 0.995717 .089635 0.998280 .090911 0.999556 53 *320 TABLE LXXYI. GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000. 4 PER CENT. Equal h = ^m. *Dec. l^ni. Nov. 2|-m. Oct. d^m. Eaual Ages. x-^n. /" = 963.141 i 966.346 969.551 972.756 Ages. x-\-n. Xb. Ab'. U. W. Xb. Xb'. Xb. 54 4.092330 0.955158 4.095124 0.957962 4.097911 0.960749 4.100680 54 55 4.124982 0.940521 4.127877 0.943416 4.130752 0.946291 4.133609 55 56 .158664 .925303 .161656 .928295 .164628 1 .931267 .167580 56 57 .193453 .909482 ' .196555 .912584 .199635 . .915664 .202694 57 58 .229479 .893034 .232700 .896255 .235898 ' .899453 .239072 58 59 .266842 .875945 ' .270196 .879299 .273523 .882626 .276824 59 60 4.305679 0.858190 4.309178 0.861689 : 4.312648 0.865159 ! 3.316091 60 61 .346147 .839762 .349806 .843421 .353435 .847050 .357035 61 62 .388409 .820646 .392245 .824482 .396046 .828283 .399815 62 63 .432626 .800802 .436657 .804833 .440651 .808827 .444610 63 64 .479013 .780234 .483260 .784481 .487467 .788688 .491633 64 65 4.527780 0.758918 4.532266 0.763404 4.536708 0.767846 4.541105 65 66 .579173 .736846 .583924 .741597 .588622 .746295 .593270 66 67 .633435 .713979 .638478 .719022 .643465 .724009 .648393 67 68 .690881 .690332 .696244 .695695 .701541 .700992 .706775 68 69 .751776 .665830) .757499 .671553 .763149 .677203 .768725 69 70 4.816533 0.640525 4.822651 0.646643 4.828684 0.652676 4.834634 70 71 .885485 .614346 1 .892043 .620904 .898503 .627364 .904869 71 72 .959079 .587302 1 .966124 .594347 .973056 .601279 .979879 72 73 3.037759 .559356 i 3.045342 .566939 3.052795 .574392 3.060123 73 74 .122022 .530465 i .130211 .538654 .138250 .546693 .146141 74 75 3.212493 0.500686 3.221349 0.509542 3.230029 0.518222 3.238540 75 76 .309721 .469924 .319320 .479523 .328712 .488915 .337904 76 77 .414374 .438131 .424809 .448566 .435000 .458757 .444957 77 78 .527269 .405362 .538632 .416725 1 .549706 .427799 .560504 78 79 .649174 .371527 .661577 .383930 ; .673635 .395988 .685367 79 80 3.781001 0.336594 3.794569 0.350162 3.807726 0.363319 3.820496 80 81 .923750 .300520 .938622 .315392 .953001 .329771 .966919 81 82 2.078479 .263185 2.094829 .279535 2.110586 .295292 2.125791 82 83 .246487 .224618 .264482 .242613 .281761 .259892 .298379 83 84 .428946 .184457 .448823 .204334 .467831 .223342 .486042 84 85 2.627494 0.142647 2.649484 0.164637 2.670414 0.185567 2.690381 85 86 .843590 .098482 .867956 .122848 .891028 .145920 .912936 86 87 1.078896 .050518 1.106049 .077671 1.131603 .103225 1.155737 87 88 .335865 1.996769 .366080 .026984 .394328 .055232 .420851 88 89 .617059 .930245 .652886 1.966072 .685981 1.999167 .716732 89 90 T. 938654 T. 864208 T. 981258 T. 906812 0.020053 T. 945607 0.055665 90 91 0.304967 .803897 0.354059 .852989 .398162 .897092 .438196 91 92 .712147 .741287 .768670 .797810 .818675 .847815 .863513 92 93 1.164049 .672794 1.230048 .738793 1.287328 .796073 1.337926 93 94 .669216 .600063 .746739 .677586 .812499 .743346 .869597 94 95 2.234475 T. 520627 2.325331 T. 611483 2.400428 T. 686580 2.464435 95 96 .872089 .412726 .988191 .528828 3.079712 .620349 3.155262 96 97 3.617270 .320785 3.754903 .458418 .859255 .562770 .943328 97 98 4.470370 .158523 4.651266 .339419 4.778587 .466740 4.876920 98 * Mouths of Entry corresponding to Annual Valuation Dec. 31. n2\ TABLE LXXYI. GENERAL VALUATION OP JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000. 4 PER CENT. Equal Sept. 4^m. Aug. 5-^m. July. 6m. Equal Ages. x-\-n. 975.962 979.167 980.769 Ages. x-\-n. W. Xh. W. U. XV. Xh. XV. 54 0.963518 4.103431 0.966269 4.106165 0.969008 4.107525 0.970363 54 55 0.949148 4.136447 0.951986 4.139266 0.954805 4.140669 0.956208 55 56 .934219 .170513 .937152 .173425 .940064 .174874 .941513 56 57 .918723 .205732 .921761 .208748 .924777 .210248 .926277 57 58 .902627 .242223 .905778 .245352 .908907 .246908 .910463 58 59 .885927 .280101 .889204 .283353 .892456 .284970 .894073 59 60 0.868602 4.319507 0.872018 4.322897 0.875408 4.324582 0.877093 60 61 .850650 .360603 .854218 .364144 .857759 .365904 .859519 61 62 .832052 .403551 .835788 .407256 .839493 .409096 .841333 62 63 .812786 .448532 .816708 .436819 .804995 .454351 .822527 63 64 .792854 .495759 .796980 .499847 .801068 .501876 .803097 64 65 0.772243 4.545457 0.776595 4.549766 0.780904 4.551904 0.783042 65 66 .750943 .597869 .755542 .602420 .760093 .604678 .762351 66 67 .728937 .653267 .733811 .658087 .738631 .660477 .741021 67 68 .706226 .711947 .711398 .717057 .716508 .719590 .719041 68 69 .682779 .774231 .688285 .779669 .693723 .782362 .696416 69 70 0.658626 4.840504 0.664496 4.846296 0.670288 4.849162 0.673154 70 71 .633730 .911143 .640004 .917328 .646189 .920387 .649248 71 72 .608102 .986597 .614820 .993212 .621435 .996482 .624705 72 73 .581720 3.067329 .588926 3.074417 .596014 3.077918 .599515 73 74 .554504 .153892 .562335 .161508 .569951 .165266 .573709 74 75 0.526733 3.246886 0.535079 \ 3.255074 0.543267 3.259112 0.547305 75 76 .498107 .346906 .507109 .355726 .515929 .360069 .520272 76 77 .468714 .454690 .478447 .464210 .487967 .468894 .492651 77 78 .438597 .571041 .449134 .581328 .459421 .586381 .464474 78 79 .407720 .696791 .419144 .707922 .430275 .713382 .435735 79 80 0.376089 3.832902 0.388495 3.844963 0.400556 3.850870 0.406463 80 81 .343689 .980404 .357174 .993484 .370254 .999879 .376649 81 82 .310497 2.140482 .325188 2.154691 .339397 2.161626 .346332 82 1 83 .276510 .314384 .292515 .329820 .307951 .337337 .315468 83 1 84 .241553 .503518 .259029 .520320 .275831 .528483 .283994 84 85 0.205534 2.709470 0.224623 2.727757 0.242910 2.736619 0.251772 85 86 .167828 .933791 .188683 .953691 .208583 .963309 .218201 86 87 .127359 1.178600 .150222 1.200320 .171942 1.210785 .182407 87 88 .081755 .445846 .106750 .469481 .130385 .480833 .141737 88 89 .029918 .745449 .058635 .772384 .085570 .785250 .098436 89 90 T. 981219 0.088577 0.014131 0.119169 0.044723 0.133694 0.059248 90 91 .937126 .474848 1.973778 .508647 .007577 .524609 .023539 91 92 .892653 .904153 .933293 .941312 1.970452 .958764 1.987904 92 93 .846671 1.383239 .891984 1.424268 .933013 1.443415 .952160 93 94 .800444 .920054 .850901 .965254 .896101 .986205 .917052 94 95 T. 750587 2.520208 T. 806360 2.569627 1.855779 2.592377 1.878529 95 96 .695899 3.219597 .760234 3.275620 .816257 3.301135 .841772 96 97 .646843 4.013736 .717251 4.074305 .777820 4.101691 .805206 97 98 .565073 .957048 .645201 5.024669 .712822 5.054902 .743055 98 n%% TABLE LXXYI, GENERAL VALUATION OF JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000, 4 PER CENT. Eaual h = 6^m. June. 7^m. May. 8^m. April. 9^m. 1 Equal 1 Ages. /" = 982.372 985.577 988.782 991.987 Ages. 1 x-\-n. Xb. kb'. 1 - kb. Xb'. Xb. Xb'. Xb. 54 4.108881 0.971719 4.111582 0.974420 4.114264 0.977102 4.116932 54 55 4.142067 0.957606 4.144851 0.960390 4.147616 0.963155 4.150365 55 56 .176318 .942957 .179191 .945830 .182047 .948686 .184883 56 57 .211743 .927772 .214718 .930747 .217673 .933702 .220608 57 58 .248458 .912013 .251543 .915098 .254605 .918160 .257646 58 59 .286581 .895684 .289786 .898889 .292967 .902070 .296123 1 59 60 4.326260 0.878771 4.329597 0.882108 4.332909 0.885420 i 4.336195 60 61 .367656 .861271 .371139 .864754 .374595 .868210 .378023 61 62 .410929 .843166 .414572 .846809 .418184 .850421 .421766 62 63 .456273 .824449 .460092 .828268 .463877 .832053 .467630 63 64 .503897 .805118 .507909 .809130 .511884 .813105 .515824 64 65 4.554033 0.785171 4.558258 0.789396 4.562443 0.793581 4.566588 65 66 .606924 .764597 .611381 .769054 .615793 .773466 .620160 66 67 .662853 .743397 .667568 .748112 .672232 .752776 .676847 67 68 .722108 .721559 .727101 .726552 .732038 .731489 .736918 68 69 .785038 .699092 .790342 .704396 .795583 .709637 .800760 69 70 4.852011 0.676003 4.857652 0.681644 4.863221 0.687213 4.868720 70 i 71 .923425 .652286 .929438 .658299 .935370 .664231 .941221 71 72 .999728 .627951 3.006148 .634371 3.012474 .640697 3.018709 72 73 3.081391 .602988 .088255 .609852 .095013 .616610 .101667 73 74 .168991 .577434 .176349 .584792 .183583 .592026 .190699 74 75 3.263112 0.551305 3.271004 t).559197 3.278755 0.566948 3.286369 75 76 .364369 .524572 .372845 .533048 .381157 .541360 .389315 76 77 .473527 .497284 .482647 .506404 .491580 .515337 .500334 77 78 .591376 .469469 .601198 .479291 .610802 .488895 .620199 78 79 .718776 .441129 .729363 .451716 .739700 .462053 .749795 79 80 3.856698 0.412291 3.868124 0.423717 3.879257 0.434850 3.890112 80 81 2.006181 .382951 2.018518 .395288 2.030514 .407284 2.042187 81 82 -.168451 .353157 .181788 .366494 .194728 .379434 .207293 82 83 .344726 .322857 .359138 .337269 .373087 .351218 .386601 83 84 .536495 .292006 .552090 .307601 .567143 .322654 .581693 84 85 2.745303 0.260456 2.762168 0.277321 2.778403 0.293556 2.794053 85 86 .972718 .227610 .990948 .245840 ' 1.008442 .263334 1.025259 86 87 1.221005 .192627 1.240749 .212371 1 .259635 .231257 .277733 87 88 .491896 .152800 .513211 .174115 .533528 .194432 .552937 88 89 .797745 .110931 .821707 .134893 .844416 .157602 .865997 89 90 0.147748 0.073.302 0.174561 0.100115 0.199815 0.125369 0.223681 90 91 .540005 .038935 .569250 .068180 .596649 .095579 i .622422 91 92 .975541 .004681 1.007268 .036408 1.036834 .065974 1.064516 92 93 1.461753 1.970498 .496258 .005003 .528222 .036967 .557995 93 94 2.006190 .937037 2.043598 1.974445 2.078038 .008885 2.109946 94 95 2.613993 T. 900145 2.654244 1.940396 2.691078 T. 977230 2.725031 95 96 3.325234 .865871 3.369758 .910395 3.410138 .950775 3.447080 96 1 97 4.327452 .830967 4.174798 .878313 4.217486 .921001 4.256352 97 98 5.083166 .771319 5.134712 .822865 5.180783 .868936 5.222432 98 ♦223 TABLE LXXYI. GENERAL VALUATION OP JOINT LIFE POLICIES BY ANNUAL PREMIUM ON 1,000. 4 PER CENT. Equal Ages. x-\-n. March. lO^m. Feb. ll^m. Jan. 12m. Equal Ages. x + n. 995.192 998.397 1000.000 Xb'. AJ. W. U. Xb'. Xb. Xb'. 54 0.979770 4.119582 0.982420 4.122216 0.985054 4.123527 0.986365 54 55 0.965904 4.153096 0.968635 4.155809 0.971348 4.157160 0.972699 55 56 .951522 .187700 .954339 .190500 .957139 .191893 .958532 56 57 .936637 .223523 .939552 \ .226418 .942447 .227859 .943888 57 58 .921201 .260666 .924221 .263665 .927220 .265156 .928711 58 59 .905226 .299259 .908362 .302371 .911474 .303919 .913022 59 60 0.888706 4.339457 0.891968 4.342695 0.895206 "4.344305 0.896816 60 61 .871638 .381425 .875040 .384800 .878415 .386478 .880093 61 62 .854003 .425319 .857556 .428844 .861081 .430595 .862832 62 63 .835806 .471351 .839527 .475040 .843216 .476873 .845049 63 64 .817045 .519727 .820948 .523596 .824817 .525518 .826739 64 65 0.797726 4.570693 0.801831 4.574760 0.805898 4.576779 0.807917 65 66 .777833 .624484 .782157 .628766 .786439 .630891 .788564 66 67 .757391 .681414 .761958 .685933 .766477 .688174 .768718 67 68 .736369 .741744 .741195 .746518 .745969 .748885 .748336 68 69 .714814 .805878 .719932 .810935 .724989 .813441 .727495 69 70 0.692712 4.874149 0.698141 4.879511 0.703503 i. 882168 0.706160 70 71 .670082 .946994 .675855 .952692 .681553 .955513 .684374 71 72 .646932 3.024857 .653080 3.030918 .659141 3.033917 .662140 72 73 .623264 .108221 .629818 .114676 .636273 .117868 .639465 73 74 .599142 .197701 .606144 .204590 .613033 .207995 .616438 74 75 0.574562 3.293853 0.582046 3.301209 0.589402 3.304841 0.593034 75 76 .549518 .397321 .557524 .405183 .565386 .409060 .569263 76 77 .524091 .508913 .532670 .517327 .541084 .521473 .545230 77 78 .498292 .629396 .507489 1 .638403 .516496 .642837 .520930 78 79 .472148 .759662 .482015 .769310 .491663 .774054 .496407 79 80 0.445705 3.900703 0.456296 ! 3.911041 0.466634 3.916119 0.471712 80 81 .418957 2.053555 .430325 ! 2.064633 .441403 2.070067 .446837 81 82 .391999 .219505 .404211 .231383 .416089 .237202 .421908 82 83 .364732 .399708 .377839 .412431 .390562 .418655 .396786 83 84 .337204 .595771 .351282 .609406 .364917 .616067 .371578 84 85 0.309206 2.809158 0.324311 2.823756 0.338909 2.830874 0.346027 85 86 .280151 1.041449 .296341 1.057056 .311948 1.064655 .319547 86 87 .249355 .295107 .266729 .311813 .283435 .319931 .291553 87 88 .213841 .571515 .232419 .589332 .250236 ! .597973 .258877 88 89 .179183 .886555 .199741 .906184 .219370 .915675 .228861 89 90 0.149235 0.246303 0.171857 0.267805 0.193359 0.278169 0.203723 90 91 .121352 .646751 .145681 .669788 .168718 .680864 .179794 91 92 .093656 1.090538 .119678 1.115089 .144229 1.126862 .156002 92 93 .066740 .585856 .094601 .612038 .120783 .624559 .133304 93 94 .040793 2.139669 .070516 2.167488 .098335 2.180756 .111603 94 95 0.011183 2.756522 0.042674 2.785883 0.072035 2.799850 0.086002 95 96 1.987717 3.481126 .021763 3.512695 .053332 3.527659 .068296 96 97 .959867 4.292022 1.995537 4.324985 .028500 4.340573 .044088 97 98 .910585 5.260434 .948587 5.295377 1.983530 5.311848 -.000001 98 *224 TABLE LXXVIT. FEMALE UFE TABLE. FROM THIRTY OFFICES' EXPERIENCE OF LOSSES AND AMOUNTS INSURED. AGE. X. Living. Decre- ment. dx- Expectation of Life. Prob. of Dec. d. Ix Prob. Living. Px — , • Ix ^Px- AZa;. AGE. X. 1 10 10 100,000 314 48.05 0.003140 0.996860 T.998634 5.000000 11 99,686 420 47.21 .004212 .995788 .998167 4.998634 11 12 99,266 510 46.40 .005140 .994860 .997762 4.996801 12 13 98,756 581 45.64 .005882 .994118 .997438 4.994563 13 14 98,175 631 44.91 .006426 .993574 .997200 4.992001 14 15 97,544 644 44.19 0.006603 0.993397 T.997123 4.989201 15 16 96,900 678 43.48 .006998 .993002 .996950 4.986324 16 17 96,222 745 42.79 .007741 .992259 .996625 4.983274 17 18 95,477 792 42.12 .008296 .991704 .996382 4.979899 18 19 94,685 820 41.46 .008659 .991341 .996223 4.976281 19 I 20 93,865 856 40.82 0.009120 0.990880 T.996021 4.972504 20 21 93,009 847 40.19 .009106 .990894 .996027 4.968525 21 22 92,162 943 39.56 .010233 .989767 .995533 4.964552 22 23 91,219 992 38.96 .010873 .989127 .995252 4.960085 23 24 90,227 990 38.38 .010973 .989027 .995208 4.955337 24 25 89,237 1,006 37.80 0.011274 0.988726 T.995076 4.950545 25 26 88,231 1,024 87.23 .011606 .988394 .994930 4.945621 26 27 87,207 998 36.66 .011442 .988558 .995002 4.940551 27 28 86,209 977 36.08 .011333 .988667 .995050 4.935553 28 28 85,232 975 35.49 .011440 .988560 .995003 4.930603 29 30 84,257 971 34.89 0.011524 0.988476 T.994966 4.925606 30 31 83,286 966 34.29 .011599 .988401 .994933 4.920572 31 32 82,320 916 33.69 .011126 .988874 .995141 4.915505 32 i 33 81,404 869 33.06 .010675 .989325 .995339 4.910646 33 34 80,535 903 32.42 .011212 .988788 .995103 4.905985 34 35 79,632 882 31.78 0.011076 0.988924 T.995163 4.901088 35 36 78,750 877 3L13 .011137 .988863 .995136 4.896251 36 1 37 77,873 878 30.47 .011274 .988726 .995076 4.891387 37 i 38 76,995 887 29.81 .011522 .988478 .994967 4.886463 38 i 39 76,108 863 29.16 .011338 .988662 .995048 4.881430 39 i ' 40 75,245 885 28.48 0.011763 0.988237 T.994861 4.876478 40 41 74,360 893 27.82 .012009 .987991 .994753 4.871339 41 1 42 73,467 821 27.15 .011174 .988826 .995120 4.866092 42 1 43 72,646 791 26.45 .010889 .989111 .995245 4.861212 43 1 44 71,855 805 25.74 .011203 .988797 1 .995107 4.856457 44 45 71,050 799 25.02 0.011244 0.988756 T.995089 4.851564 45 46 70,251 792 24.30 .011274 .988726 .995076 4.846653 46 47 69,459 757 23.57 .010900 .989100 .995240 4.841729 47 1 48 68,702 766 22.83 .011149 .988851 .995131 4.836969 48 49 67,936 803 22.08 .011820 .988180 .994836 4.832100 49 50 67,133 821 21.33 0.012230 0.987770 T.994656 4.826936 50 51 66,312 889 20.59 .013407 .986593 .994138 4.821592 51 52 65,423 955 19.87 .014597 ; 985403 .993614 4.815730 52 53 64,468 . 976 19.15 .015139 .984861 .993375 4.809344 53 54 63,492 1,009 18.44 .015891 .984109 .993043 4.802719 54 TABLE LXXVIl. *Z26 FEMALE LIFE TABLE. FROM THIRTY OFFICES' EXPERIENCE OF LOSSES AND AMOUNTS INSURED. Decre- Prob. of Dec. Prob. Living. AGE. 'X. Living. ment. dx. Expectation of Life. dx n — ^*^^ Xpx. Xlx. A&E. X. 55 62,483 1,082 17.73 0.017318 0.982682 T. 992413 4.795762 55 56 61,401 1,138 17.03 .018533 .981468 .991876 4.788175 56 57 60,363 1,221 16.35 .030363 .979738 .991110 4.780051 57 58 59,042 1,302 15.67 .033051 .977949 .990316 4.771161 58 59 57,740 1,360 15.03 .023554 .976446 .989648 4.761477 59 60 56,380 1,429 14.37 0.025345 0.974655 T. 988851 4.751135 60 61 54,951 1,501 13.73 .027315 .972685 .987972 4.739976 61 62 53,450 1,617 13.10 .030254 .969746 .986658 4.737948 62 63 51,833 1,690 13.49 .032605 .967395 .985604 4.714606 63 64 50,143 1,771 11.90 .035318 .964682 .984384 4.700310 64 65 48,372 1,856 11.31 0.038370 0.961630 1.983008 4.684594 65 66 46,516 1,940 10.74 .041705 .958295 .981499 4.667603 66 67 44,576 2,024 10.19 .045405 .954595 .979819 4.649101 67 68 42,552 2,107 9.65 .049515 .950485 .977945 4.638930 68 69 40,445 2,185 9.13 .054034 .945976 .975880 4.606865 69 70 38,260 2,257 8.63 0.058991 0.941009 T. 973594 4.583745 70 71 36,003 2,323 8.13 .064495 .935505 .971046 4.556339 71 72 33,681 2,376 7.65 .070544 .929456 .968229 4.537385 72 73 31,305 3,416 7.30 .077176 .922824 .965119 4.495614 73 74 28,889 2,442 6.76 .084533 .915468 .961643 4.460733 74 75 26,447 2,449 6.34 0.093599 0.907401 T. 957799 4.422376 75 76 23,998 2,435 5.93 .101467 .898533 .953534 4.380175 76 77 21,563 2.397 5.55 .111163 .888837 .948823 4.333709 77 78 19,166 2,335 5.18 .131839 .878171 .943579 4.282531 78 79 16,831 2,246 4.83 .133445 .866555 .937796 4.226110 79 80 14.585 2,133 4.49 0.146345 0.853755 T. 931333 4.163906 80 81 12,453 1,994 4.17 .160134 .839866 .924210 4.095239 81 82 10.458 1,833 3.88 .175373 .824727 .916310 4.019449 82 83 8,635 1,654 3.59 .191768 .808232 .907536 3.935759 83 84 6,971 1,462 3.33 .209725 .790275 .897778 3.843295 84 35 5,509 1,262 3.08 0.229081 0.770919 T. 887009 3.741073 85. 86 4,247 1,062 3.84 .250059 .749941 .875027 3.628082 86 87 3,185 868 2.62 .272527 .727473 .861817 3.503109 87 88 2,317 687 2.42 .296503 .703497 .847262 3.364936 88 89 1,630 526 2.23 .322700 .677300 .830781 3.313188 89 90 1,104 387 2.05 0.350544 0.649456 T. 812550 3.042969 90 91 717 272 1.89 .379358 .620643 .792841 2.855519 91 92 445 183 1.73 .411236 .588764 .769941 2.648360 92 93 262 116 1.59 .442747 .557353 .746052 2.418301 93 94 146 70 1.46 .479452 .530548 .716461 2.164353 94 95 76 39 1.34 0.513158 0.486842 1.687388 1.880814 95 96 37 20 1.33 .540541 .459459 .662247 1.568202 96 97 17 10 1.09 .588235 .411765 .614649 1.330449 97 98 7 4 .93 .571429 .428571 .632023 0.845098 98 99 3 3 .50 1.000000 .000000 00 ! 0.477131 99 ^336 TABLE LXXYIII. FEMALE LIFE. D, N, Ox, AND CORRECTION OF AGE. Example. An actual Age, 32 years 3 months, added to its Correction, gives 34 years, the equivalent. Age at which the equal Life Annuity and Premium are found in the preceding tables for Male Life. 4 PER CENT. AGE. X. Life An'y.. Correction AGE. X. ^x. XDx- ■Na,. XEx- AJN /p — AxJ/p, ttx- (.). 10 67,556.5 4.829667 1,372,039.3 6.137366 1.307699 19.3095 Y. 5 M. 0.6 10 11 64,754.1 .811267 1,304,482.8 .115438 .304171 19.1452 5 9.1 11 12 62,001.3 .792401 1,239,728.7 .093327 .300926 18.9952 6 2.7 12 i 13 59,310.3 .773130 1,177,727.4 .071045 .297915 18.8571 6 6.1 13 14 56,693.6 .753534 1,118,417.1 .048604 .295070 18.7274 6 7.9 14 15 54,162.8 4.733701 1,061,723.5 6.026012 1.292311 18.6025 6 8.6 15 16 51,735.8 .713791 1,007,560.7 .003271 .289480 18.4751 6 9.1 16 17 49,397.7 .693707 955,824.9 5.980378 .286671 18.3496 6 8.9 17 18 47,130.2 .673299 906,427.2 .957333 .284034 18.2324 6 7.5 18 19 44,941.5 .652648 859,297.0 .934144 .281496 18.1204 6 5.4 19 20 42,838.8 4.631837 814,355.5 5.910814 1.278977 18.0098 6 2.9 20 21 40,815.5 .610825 771,516.7 .887345 .276520 17.9025 5 11.8 21 22 38,888.3 .589819 730,701.2 -863740 .273921 17.7898 5 8.9 22 23 37,009.9 .568318 691,812.9 .839989 .271671 17.6927 5 4.5 23 24 35,199.5 .546537 654,803.0 .816111 .269574 17.6026 4 11.5 24 25 33,474.3 4.524712 619,603.5 5.792113 1.267401 17.5098 4 6.5 25 26 31,823.9 .502754 586,129.2 .767993 .265239 17.4179 4 1.3 26 27 30,244.8 .480651 554,305.3 .743749 .263098 17.3273 3 7.9 27 28 28,748.8 .458620 524,060.5 .719381 .260761 17.2289 3 2.9 28 29 27,329.6 .436633 495,311.7 .694879 .258246 17.1237 2 10.3 29 30 25,978.0 4.414606 467,982.1 5.670229 1.255623 17.0145 2 5.7 30 31 24,691.0 .392539 442,004.1 .645426 .252887 16.9014 2 1.3 31 32 23,465.9 .370438 417,313.1 .620462 .250024 16.7838 1 9.0 32 33 22,312.4 .348546 393,847.2 .595328 .246782 16.6515 1 5.5 33 34 21,225.2 .326852 371,534.8 .570000 .243148 16.5044 1 2.7 34 35 20,180.0 4.304921 350,309.6 5.544452 1.239531 16.3593 11.6 35 36 19,188.9 .283051 330,129.6 .518684 .235633 16.2041 8.8 36 37 18,245.4 .261153 310,940.7 .492678 .231525 16.0422 6.3 37 38 17,345.9 .239196 292,695.3 .466416 .227220 15.8741 3.8 38 39 16,486.6 .217130 275,349.4 .439884 .222754 15.7014 1.4 39 40 15,672.7 4.195144 258,862.8 5.413070 1.217926 15.5168 .0 40 41 14,892.7 .172972 243,190.1 .385946 .212974 15.3295 .0 41 42 14,147.9 .150692 228,297.4 .358501 .207809 15.1365 .0 42 43 13,451.7 .128778 214,149.5 .330717 .201939 14.9199 .0 43 44 12,793.5 .106990 200,697.8 .302542 .195552 14.6874 .0 44 45 12,163.7 4.085064 187,904.3 5.273937 1.188873 14.4480 .0 45 46 11,564.3 .063119 175,740.6 .244872 .181753 14.1968 .0 46 47 10,994.2 .041162 164,176.3 .215310 .174148 13.9330 .0 47 48 10,456.1 .019369 153,182.1 .185208 .165839 13.6500 .0 48 49 9,941.82 3.997466 142,725.97 .154503 .157037 13.3561 .0 49 50 9,446.46 3.975269 132,784.15 5.123146 1.147877 13.0565 .0 50 51 8,972.06 .952892 123,337.69 .091096 .138204 12.7469 .0 51 52 8,511.30 .929996 114,365.63 .058295 .128299 12.4369 .0 52 53 8,064.49 .906577 105,854.33 .024709 .118132 12.1260 .0 53 54 7,636.93 .882919 97,789.84 4.990293 .107374 11.8048 .0 54 Formula, Dj^ = «;« Za, j l + «. = P^; Female Ox = %+< Male Life. lABLE LXXYIII. *227 FEMALE LIFE, D, N, a,., AND CORRECTION OF AGE. 4 PER CENT. A OTI Life An'y. Correction AGE. X. AftE. X. r>x. ADa,. ^x. ANa,. ANa; — ADa;. ax- {c). Y. M. 55 7,226.50 3.858928 90,152.91 4.954980 1.096052 11.4753 .0 55 56 6,828.23 .834308 82,926.41 .918693 .084385 11.1447 .0 56 57 6,443.93 .809151 76,098.18 .881374 .072223 10.8093 .8 67 58 6,070.54 .783227 69,654.25 .842948 .059721 10.4742 1.4 58 59 5,708.34 .756510 63,583.71 .803346 .046836 10.1387 1.9 59 60 5,359.51 3.729125 57,875.37 4.762494 1.033369 9.7986 2.5 60 61 5,022.76 .700942 52,515.86 .720291 .019349 9.4556 3.1 61 62 4,697.65 .671881 47,493.10 .676631 .004750 9.1100 3.7 62 63 4,380.32 .641506 42,795.45 .631398 0.989892 8.76994 4.2 63 64 4,074.52 .610076 38,415.13 .^84502 .974426 8.42814 4.7 64 65 3,779.44 3.577427 34,340.61 4.535808 0.958381 8.08617 5.2 65 66 3,494.64 .543402 30,561.17 .485170 .941768 7.74517 5.8 66 67 3,220.08 .507867 27,066.53 .432432 .924565 7.40553 6.3 67 68 2,955. 6o .470653 23,846.45 .377424 .906771 7.06810 6.9 68 69 2,701.25 .431565 20,890.80 .319955 .888390 6.73375 7.5 69 70 2,457.03 3.390411 18,189.55 4.259822 0.869411 6.40306 8.1 70 71 2,223.17 .346972 15,732.52 .196799 .849827 6.07664 8.7 71 72 1,999.79 .300985 13,509.35 .130634 .829649 5.75537 9.3 72 73 1,787.23 .252180 11,509.56 .061059 { .808879 5.43990 i ^ 9.9 73 74 1,585.86 .200266 9,722.33 3.987770 .787504 5.13061 10.6 74 75 1,395.97 3.144876 8,136.47 3.910436 0.765560 4.82854 ! 11.2 . 75 76 1,217.98 .085641 6,740.50 .828692 .743051 4.53415 1 11.8 76 77 1,052.31 .022142 5,522.52 .742138 .719996 4.24803 .5 77 78 899.355 2.953931 ; 4,470.212 .650328 .696397 3.97047 1.1 78 79 759.410 .880476 3,570.857 .552773 .672297 3.70216 1.7 79 80 632.760 2.801239 2,811.447 3.448930 0.647691 3.44315 2.3 80 81 519.444 .715539 2,178.6^7 .338194 .622655 3.19426 2.8 81 82 419.484 .622715 1,659.243 .219910 .597195 2.95544 3.2 82 83 332.653 .521992 1,239.759 .093338 .571346 2.72689 3.4 83 84 258.520 .412495 907.106 2.957658 .545163 2.50884 3.5 84 85 196.444 2.293239 648.586 2.811968 0.518729 2.30163 3.1 85 86 145.618 .163215 452.142 .655275 .492060 2.10499 2.3 86 87 105.005 .021208 306.524 .486465 .465257 1.91915 .9 87 88 73.4501 1.865992 201.5189 .304316 .438324 1.74362 10.6 88 89 49.6845 .696221 128.0688 .107443 .411222 1.57764 8.8 89 90 32.3571 1.509969 78.3843 1.894229 0.384260 1.42248 7.7 90 91 20.2062 .305485 46.0272 .663015 .357530 1.27788 6.8 91 92 12.0585 .081293 25.8210 .411973 .330680 1.14131 5.7 92 93 6.82653 0.834200 13.76251 .138698 .304498 1.01604 4.1 93 94 3.65779 .563219 6.93598 0.841108 .277889 .89622 2.5 94 95 1.83083 0.262647 3.27819 0.515634 0.252987 .79055 .0 95 96 .857040 1.933001 1.447356 .160576 .227575 .68879 I .0 96 97 .378630 .578215 .590316 1.771084 .192869 .55908 1 .0 97 98 .149910 .175831 .211686 .325692 .149861 .41209 .0 98 99 .061776 2.790820 .061776 2.790820 .000000 .00000 .0 99 \ ' ♦238 TABLE LXXIX. COMPARISON OF DIFFERENT LIFE TABLES. 1 X. Caxlisle. 17 Offices. Am. 1858. English Life Table, No. 3. 20 Offices . HM. AGE. X. Ix- Ix' Males, Ix' dx- Fern. Ix. dx. Ix. dx. 10,000 1,539 511,745 83,719 488,255 65,774 1 8,461 682 428,026 27,521 422,481 26,159 1 2 7,779 505 400,505 14,215 396,322 14,023 2 3 7,274 276 386,290 9,213 382,299 9,243 3 4 6,998 201 377,077 6,719 373,056| 6,596 4 5 6,797 121 370,358 5,033 366,460 4,866 5 6 6,676 82 365,325 3,953 361,594 3,815 e 7 6,594 58 361,372 3,310 357,779 3,249 7 8 6,536 43 358,062 2,734 354,530 2,724 8 9 6,493 33 355,328 2,297 351,806 2,328 9 10 6,460 29 100,000 100,000 353, 03r 1,983 349,478 2,045 100,000 490 10 11 6,431 31 99,324 99,251 351,048 1,776 347,433 1,861 99,510 397 11 12 6,400 32 98,650 98,505 [ 349,272 1,666 345,572 1,765 99,113 329 12 13 6,368 33 97,978 97,762 i 347,606 1,637 343,807 1,745 98,784 288 13 14 6,335 35 97,307 97,022 345,969 1,679 342,062 1,789 98,496 272 14 15 6,300 39 96,636 96,285 1 344,290 1,781 340,273 1,888 98,224 282 15 16 6,261 42 95,965 95,550 11342,509 1,928 338,385 2,029 97,942 318 16 17 6,219 43 95,293 94,818 : 340,581 2,112 336,356 2,205 97,624 379 17 18 6,176 43 94,620 94,089 338,469 2,320 334,151 2,400 97,245 466 18 19 6,133 43 93,945 93,362 1 336,149 2,541 331,751 2,609 96,779 556 19 20 6,090 43 93,268 92,637 333,608 2,764 329,142 2,819 96,223 609 20 21 6,047 42 92,588 91,914 330,844 2,801 326,323 2,867 95,614 643 21 22 6,005 42 91,905 91,192 1 328,043 2,836 323,456: 2,912 94,971 650 22 23 5,963 42 91,219 90,471 i 325,207 2,868 320,544 2,952 94,321 638 23 24 5,921 42 90,529 89,751 1 322,339 2,897 317,592 2,989 93,683 622 24 25 5,879 43 89,835 89,032 319,442 2,926 314,603 3,024 93,061 617 25 26 5,836 43 89,137 88,314 1 316,516 2,954 311,579 3,055 92,444 618 26 27 5,793 45 88,434 87,596 1 313,562 2,981 308,524 3,084 91,826 634 27 28 5,748 5o; 87,726 86,878 ; 310,581 3,009 305,440 3,112 91,192 654 28 29 5,698 56 87,012 86,160 ; 307,572 3,038 302,328 3,138 90,538 673 29 30 5,642 57 86,292 85,441 304,534 3,068 299,190 3,163 89,865 694 30 31 5,585 57 85,565 84,721 301,466 3,100 296,027 3,187 1 89,171 706 31 32 5,528 56 84,831 84,000 298,366 3,134 292,840 3,209 1 88,465 717 32 33 5,472 55 84,089 83,277 295,232 3,171 289,631 3,233! 87,748 727 33 34 5,417 55 83,339 82,551 292,061 3,211 286,398 3,255 1 87,021 740 34 35 5,362 55 82,581 81,822 288,850 3,254 283,143 3,279 86,281 757 35 36 5,307 56 81,814 81,090 285,596 3,300 279,864 3,301 85,524 779 36 37 5,251 57 81,038 80,353 282,296 3,352 276,563 3,326j 84,745 802 37 38 5,194 58 80,253 79,611 278,944 3,406 273,237 3,350!i 83,943 821 38 39 5,136 61 79,458 78,862 275,538 3,465 269,887 3,376 83,122 838 39 40 5,075 66 78,653 78,106 272,073 3,529 266,511 3,402 82,284 848 40 41 5,009 69 77,838 77,341 268,544 3,596 263,109 3,431 81,436 854 41 42 4,940 71 77,012 76,567 264,948 3,668 259,678 3,459 80,582 865 42 43 4,869 71 76,173 75,782 261,280 3,746 256,219 3,490 79,717 887 43 44 4,798 71 75,316 74,985 257,534 3,826 252,729 3,522 78,830 911 44 45 4,727 70 74,435 74,173 253,708 3,912 249,207 3,555 77,919 950 45 46 4,657 69 73,526 73,345 249,796 4,001 245,652 3,591 76,969 996 46 47 4,588 67 72,582 72,497 245,795 4,095 242,061 3,627 75,973 1,041 47 48 4,521 63 71,601 71,627 241,700 4,192 238,434 3,665 74,932 1,082 48 49 4,458 61 70,580 70,731 237,508 4,292 234,769 3,705 73,850 1,124 49 TABLE LXXIX. COMPARISON OF DIFFERENT LIFE TABLES. *229 AGE. X. Caxlisle. 17 Offices. Am. 1858. Ix. English Life Table, No. 3. 20 Offices. HM. AaE. X. Ix' (*X' Males, ly.. dx. Fem. Ix' dx. Ix. dx. 50 4,397 59 69,517 69,804 233,216 4,395 231,064 3,746 72,726 1,160 50 51 4,338 62 68,409 68,842 228,821 4,626 227,318 3,788 71,566 1,193 51 52 4,276 65 67,253 67,841 224,195 4,758 223,530 3,832 70,373 1,235 52 53 4,211 68 66,046 66,797 219,437 4,885 219,698 3,876 69,138 1,286 53 54 4,143 70 64,785 65,706 214,552 5,013 215,822 4,246 67,852 1,339 54 55 4,073 73 63,469 64,563 209,539 5,144 211,576 4,439 66,513 1,399 55 56 4,000 76 62,094 63,364 204,395 5,281 207,137 4,628 65,114 1,462 56 57 3,924 82 60,658 62,104 199,114 5,428 202,509 4,817 63,652 1,527 57 58 3,842 93 59,161 60,779 193,686 5,584 197,692 5,009 62,125 1.592 58 59 3,749 106 57,600 59,385 188,102 5,752 192,683 5,206 60,533 1,667 59 60 3,643 122 55,973 57,917 182,350 5,929 187,477 5,409 58,866 1,747 60 61 3,521 126 54,275 56,371 176,421 6,118 182,068 5,619 57,119 1,830 61 62 3,395 127 52,505 54,743 170,303 6,314 176,449 5,835 55,289 1,915 62 63 3,268 125 50,661 53,030 163,989 6,515 170,614 6,057 53,374 2,001 63 64 3,143 125 48,744 51,230 157,474 6,720 164,557 6,282 51,373 2,076 64 65 3,018 124 46,754 49,341 150,754 6,921 158,275 6,509 49,297 2,141 65 66 2,894 123 44,693 47,361 143,833 7,115 151,766 6,731 47,156 2,196 66 67 2,771 123 42,565 45,291 136,718 7,297 145,035 6,947 44,960 2,243 67 68 2,648 123 40,374 43,133 129,421 7,458 138,088 7,149 42,717 2,274 68 69 2,525 124 38,128 40,890 121,963 7,593 130,939 7,332 40,443 2,319 69 70 2,401 124 35,837 38,569 114,370 7,695 123,607 7,489 38,124 2,371 70 71 2,277 134 33,510 36,178 106,675 7,756 116,118 7,613 35,753 2,433 71 72 2,143 146 31,159 33,730 98,919 7,770 108,505 7,698 33,320 2,497 72 73 1,997 156 28,797 31,243 91,149 7,733 100,807 7,736 30,823 2,554 73 74 1,841 166 26,439 28,738 83,416 7,639 93,071 7,724 28,269 2,578 74 75 1,675 160 24,100 26,237 75,777 7,483 85,347 7,653 25,691 2,527 75 76 1,515 156 21,797 23,761 68,294 7,268 77,694 7,521 23,164 2,464 76 77 1,359 146 19,548 21,330 61,026 6,990 70,173 7,329 20,700 2,374 77 78 1,213 132 17,369 18,961 54,036 6,655 62,844 7,071 18,326 2,258 78 79 1,081 128 15,277 16,670 47,381 6,266 55,773 6,755 16,068 2,138 79 80 953 116 13,290 14,474 41,115 5,832 49,018 6,382 13,930 2,015 80 81 837 112 11,424 12,383 35,283 5,361 42,636 5,959 11,915 1,883 81 82 725 102 9,694 10,419 29,922 4,862 36,677 5,496 10,032 1,719 82 83 623 94 8,112 8,603 25,060 4,349 31,181 5,003 8,313 1,545 83 84 529 84 6,685 6,955 20,711 3,834 26,178 4,490 6,768 1,346 84 85 445 78 5,417 5,485 16,877 3,328 21,688 3,972 5,422 1,138 85 86 367 71 4,306 4,193 13,549 2,840 17,716 3,458 4,284 941 86 87 296 64 3,348 3,079 10,709 2,384 14,258 2,962 3,343 773 87 88 232 51 2,537 2,146 8,325 1,965 11,296 2,494 2,570 615 88 89 181 39 1,864 1,402 6,360 1,590 8,802 2,063 1,955 495 89 90 142 37 1,319 847 4,770 1,260 6,739 1,673 1,460 408 90 91 105 30 892 462 3,510 979 5,066 1,331 1,052 329 91 92 75 21 570 216 2,531 744 3,735 1,037 723 254 92 93 54 14 339 79 1,787 553 2,698 790 469 195 93 94 40 10 184 21 1,234 401 1,908 588 274 139 94 95 30 7 89 3 833 285 1,320 428 135 86 95 96 23 5 37 548 196. 892 304 49 40 96 97 18 4 13 352 132 588 210 9 9 97 98 14 3 4 220 86 378 142 96 99 11 2 1 134 55 236 92 99 100 9 2 79 33 144 69 100 1 Beyond 100 Tears : Carlisle, 1-, 7, 5, 3, 1. English, No. 3, Males l^, 46, 35, 14, 7, 4, 3, 1. " _ Females l^, 86, 49. 27, 15, 8, 4, 2, 1, ♦230 TABLE LXXX. EXPECTATION OF LIFE BY VARIOUS TABLES— YEARS AND DECIMALS. AGE. North- Carlisle. 17 Am. i English Life, No. 3. j 20 Offices. HM. 30 Am. Offices. AGE. X. ampton. Offices. 1858. Males. Females. Hales. Females, Males. Females. X. 25.18 38.72 39.91 41.85 1 32.74 44.68 46.65 47.31 1 2 37.79 47.55 48.83 49.40 2 3 39.55 49.82 49.61 50.20 3 4 40.58 50.76 49.81 50.43 4 5 40.84 51.25 49.71 50.33 5 6 41.07 51.17 49.39 50.00 6 7 41.03 50.80 48.92 49.53 7 8 40.79 50.24 48.37 48.98 8 9 40.36 49.57 47.74 48.35 9 10 39.78 48.82 48.36 48.72 47.05 47.67 I 50.29 48.20 49.99 48.05 10 11 39.14 48.04 47.68 48.08 46.31 46.95 49.54 47.35 49.32 47.21 11 12 38.49 47.27 47.01 47.45 45.54 46.20 48.73 46.54 48.64 46.40 12 13 37.83 46.51 46.33 46.80 44.76 45.44 47:89 45.78 47.95 45.64 13 14 37.17 45.75 45.64 46.16 43.97 44.66 47.03 45.05 47.26 44.91 14 15 36.51 45.00 44.96 45.50 43.18 43.90 46.16 44.34 46*. 57 44.19 15 16 35.85 44.27 44.27 44.85 42.40 43.14 45.29 43.64 45.88 43.48 16 17 35.20 43.57 43.58 44.19 41.64 42.40 44.44 42.95 45.18 42.79 17 18 34.58 42.87 42.88 43.53 40.90 41.67 43.61 42.25 44.48 42.12 18 19 33.99 42.17 42.19 42.87 40.17 40.97 42.82 41.54 43.78 41. 4e 19 20 33.43 41.46 41.49 42.20 39.48 40.29 42.06 40.82 43.07 40.82 20 21 32.90 40.75 40.79 41.53 38.80 39.63 41.33 40.09 42.36 40.19 21 22 32.39 40.04 40.09 40.85 38.13 38.98 40.60 [ 39.39 41.65 39.56 22 23 31.88 39.31 39.39 40.17 37.46 38.33 39.88 38.70 40.93 38.96 23 24 31.86 38.59 38.68 39.49 36.79 37.68 39.15 38.04 40.21 38.38 24 25 30.85 37.86 37.98 38.81 36.12 37.04 38.41 37.41 39.49 37.80 25 26 30.33 37.14 37.27 38.12 35.44 36.39 37.66 36.81 38.77 37.23 26 27 29.82 36.41 36.56 37.43 i 34.77 35.75 36.91 36.23 38.04 36.66 27 28 29.30 35.69 35.86 36.73 : 34.10 35.10 36.16 35.66 37.31 36.08 28 29 28.79 35.00 35.15 36.03 33.43 34.46 35.42 35.09 36.58 35.49 29 30 28.27 34.34 34.43 35.33 32.76 33.81 34.68 34.50 35.85 34.89 30 31 27.76 33.68 33.72 34.63 32.09 33.17 33.95 33.91 35.12 34.29 31 32 27.24 33.03 33.01 33.92 31.42 32.53 33.21 33.31 1 34.38 33.69 32 33 26.72 32.36 32.30 33.21 30.74 31.88 32.48 32.69 33.65 33.06 33 34 26.20 31.68 31.58 32.50 30.07 31.23 31.75 32.07 32.91 32.42 34 35 25.68 31.00 30.87 31.78 29.40 30.59 31.02 31.45 32.17 31.78 35 36 25.16 30.32 30.15 31.07 28.73 29.94 30.29 30.81 31.43 ai.i3 36 37 24.64 29.64 29.44 30.35 28.06 29.29 1 29.56 30.18 30.70 30.47 37 38 24.12 28.96 28.72 29.62 27.39 28.64 • 28.84 29.54 29.96 29.81 38 39 23.60 28.28 28.00 28.90 26.72 27.99 \ 28.12 28.90 29.22 29.16 39 40 23.08 27.61 27.28 28.18 26.06 27.34 1 27.40 28.25 28.48 28.48 40 41 22.56 26.97 26.56 127.45 25.^^ 26.69 ' 26.68 27.61 27.75 27.82 41 42 22.04 26.34 25.84 26.72 24.73 26.03 i 25.96 26.96 27.01 27.15 42 43 21.54 25.71 25.12 26.00 24.07 25.38 i 25.23 26.30 26.28 26.45 43 44 21.03 25.09 24.40 1 25.27 23.41 24.72 \ 24.51 25.65 25.55 25.74 44 45 20.52 24.46 23.69 24.54 22.76 24.06 123.79 24.99 24.82 25.02 45 46 20.02 23.82 22.97 23.81 22.11 23.40 23.08 24.33 24.09 24.30 46 47 19.51 23.17 22.27 23.08 21.46 22.74 22.38 23.66 23.38 23.57 47 48 19.00 22.50 21.56 i 22.36 i 20.82 22.08 21.68 22.98 22.66 22.83 48 49 18.49 21.81 20.87 |21.63 20.17 21.42 20.99 22.30 21.95 22.08 49 TABLE LXXX. ♦231 EXPECTATION OF LIFE BY VARIOUS TABLES— YEARS AND DECIMALS. 1 AGE. North- 17 Am. English Life, No. 3. 20 Offices. HM. 30 Am. Offices. AGE. Carlisle. X. ampton. Offices. 1858. Males. Females. Males. Females. Males. Females X. 50 17.99 21.11 20.18 20.91 19.54 20.75 20.31 21.62 21.24 21.33 50 51 17.50 20.39 19.50 20.20 18.90 20.09 19.63 20.93 20.54 20.59 51 52 17.02 19.68 18.82 19.49 18.28 19.42 18.95 20.24 19.84 19.87 52 53 16.54 18.97 18.16 18.79 17.67 18.75 18.28 19.55 19.15 19.15 53 54 16.06 18.28 17.50 18.09 17.06 18.08 17.62 18.87 18.47 18.44 54 55 15.58 17.58 16.86 17.40 16.45 17.43 16.96 18.19 17.80 17.73 55 56 15.10 16.89 16.22 16.72 15.86 16.79 16.32 17.52 17.13 17.03 56 57 1463 16.21 15.59 16.05 15.26 16.17 15.68 16.85 16.47 16.35 57 58 1415 15.55 1497 15.39 1468 15.55 15.05 16.18 15.83 15.67 58 59 13.68 1492 1437 1474 1410 1494 1444 15.52 15.19 15.02 59 60 13.21 1434 13.77 1410 13.53 1434 13.83 1485 1456 1437 60 61 12.75 13.82 13.18 13.47 12.96 13.75 13.24 1420 13.94 13.73 61 62 12.28 13.31 12.61 12.86 12.41 13.17 12.66 13.56 13.34 13.10 62 63 11.81 12.81 12.05 12.26 11.87 12.60 12.10 12.95 12.74 12.49 63 64 11.35 12.30 11.51 11.67 11.34 12.05 11.55 12.35 12.16 11.90 64 65 10.88 11.79 10.97 11.10 10.82 11.51 11.01 11.77 11.60 11.31 65 66 10.42 11.27 10.46 10.54 10.32 10.98 10.49 11.21 11.04 10.74 66 67 9.96 10.75 9.96 10.00 9.83 10.47 9.98 10.66 10.50 10.19 67 68 9.50 10.23 9.47 9.47 9.36 9.97 9.48 10.12 9.97 9.65 68 69 9.05 9.70 9.00 8.97 8.90 9.48 8.98 9.59 9.46 9.13 69 70 8.60 9.18 8.54 8.48 8.45 9.02 8.50 9.08 8.97 8.62 70 71 8.17 8.65 8.10 8.00 8.03 8.57 8.03 8.59 8.49 8.13 71 72 7.74 8.16 7.67 7.55 7.62 8.13 7.58 8.12 8.02 7.65 72 73 7.33 7.72 7.26 7.11 7.22 7.71 7.15 7.69 7.57 7.20 73 74 6.92 7.33 6.86 6.68 6.85 7.31 6.75 7.29 7.14 6.76 74 75 6.54 7.01 6.48 6.27 6.49 6.93 6.38 6.93 6.72 6.34 75 76 6.18 6.69 6.11 5.88 6.15 6.56 6.02 6.60 6.32 5.93 76 77 5.83 6.40 5.76 '5.49 5.82 6.21 5.67 6.31 5.93 5.55 77 78 5.48 6.12 5.42 5.11 5.51 5.88 5.34 6.02 5.57 5.18 78 79 5.11 5.80 5 09 474 5.21 5.56 5.03 5.74 5.21 482 79 80 475 5.51 478 439 493 5.26 472 5.45 487 449 80 81 441 5.21 448 405 466 498 443 5.14 455 417 81 82 409 493 418 3.71 441 471 417 479 424 3.88 82 83 3.80 465 3.90 3.39 4,17 445 3.93 444 3.95 3.59 83 84 3.58 439 3.63 3.08 3.95 421 3.71 411 3.67 3.33 84 85 3.37 4.12 3.36 2.77 3.73 3.98 3.51 3.81 3.40 3.08 85 86 3.19 3.90 3.10 2.47 3.53 3.76 3.31 3.57 3.14 2.84 86 87 3.01 3.71 2.84 2.18 3.34 3.56 3.10 3.39 2.89 2.62 87 88 2.86 3.59 2.59 1.91 3.16 3.36 2.88 3.29 2.64 2.42 88 89 2.66 3.47 2.35 1.66 3.00 3,18 2.63 3.27 2.39 2.23 89 90 2.41 3.28 2-11 1.42 2.84 3.01 2.36 3.30 2.17 2.05 90 91 2.09 3.26 1.89 1.19 • 2.69 2.85 2.08 3.37 1.98 1.89 91 92 1.75 3.37 1.67 .98 2.55 2.70 1.80 3.42 1.81 1.73 92 93 1.37 3.48 1.47 .80 2.41 2.55 1.50 3.30 1.64 1.59 93 94 1.05 3.53 1.28 .64 2.29 2.42 1.20 2.91 1.49 1.46 94 95 .75 3.53 1.12 .50 2.17 2.29 .93 2.47 1.34 1.34 95 96 .50 3.46 .99 2.06 2.17 .68 1.98 1.18 1.23 96 97 3.28 .89 1.95 2.06 .50 1.49 1.03 1.09 97 98 3.07 .75 1.85 1.96 1.00 .83 .93 98 99 2.77 .50 1.76 1.86 50 .50 V .50 99 100 2.28 1.68 1.76 100 *23a TABLE LXXXI; PERCENTAGE OP MORTALITY BY SEVERAL LIFE TABLES North- English Life, No. 3. i 17 20 Offices. American, 30 Offices, AGhE. Carlisle. Aa£. ampton. Males. Females. Offices. HM 1858. Am.Hales. 25.7511 15.3900 18i326 14.749 1 15.8035 8.0605 6.680 6.436 1 2 6.8928 ^6.4918 3.624 3.603 2 3 4.9403 3.7943 2.416 2.450 3 4 3.0562 2.8723 1.799 1.785 4 5 2.9445 1.7810 1.369 1.337 5 6 2.3084 1.2283 1.088 1.061 6 7 1.8565 0.8796 0.920 0.912 7 8 1.3757 .6579 .767 .771 8 9 1.0462 .5082 .649 .664 9 10 0.9163 .4489 .563 .587 0.6760 0.4900 0.7490 0.6479 10 11 .8892 . .4820 .507 .537 .6786 .3990 .7516 .6502 11 12 .8973 .5000 .478 .512 .6812 .3319 .7542 .6516 12 13 .9053 .5182 .472 .509 1 .6848 .2915 .7569 .6536 13 14 .9136 .5525 .486 .524 .6896 .2762 .7596 .6568 14 15 .9220 .6191 .519 .556 .6944 .2871 .7633 .6593 15 16 .9864 .6708 .564 .601 .7003 .3247 ..7660 .6614 16 17 1.0902 .6914 .622 .658 .7062 .3882 .7688 .6648 17 18 1.1972 .6962 .688 .721 .7134 .4792 .7726 .6683 18 19 1.2887 .7011 .759 .789 .7206 .5745 .7765 .6717 19 20 1.4030 .7061 .832 .860 .7291 .6329 .7804 .6763 20 21 1.4822 .6946 .850 .882 .7377 .6725 .7855 .6808 21 22 1.5045 .6994 .868 .904 .7464 .6844 .7906 .6856 22 23 1.5275 .7043 .886 .925 .7564 .6764 .7958 .6911 23 24 1.5512 .7093 .903 .946 .7666 .6639 .8011 .6973 24 25 1.5756 .7314 .920 .966 .7770 .6630 .8064 .7032 25 26 1.6009 .7368 .938 .985 .7887 .6685 .8130 .7115 26 27 1.6269 .7768 : .955 1.005 .8006 .6904 .8196 .7190 27 28 1.6538 .8699 .974 1.024 .8139 .7172 .8264 .7275 .28 29 1.6816 .9828 .993 1.043 .8275 .7473 .8344 .7385 29 30 1.7104 1.0103 1.013 1.063 .8425 .7723 .8426 .7485 30 31 1.7401 1.0206 1.034 1.082 .8578 .7917 .8510 .7602 31 32 1.7710 1.0130 1.056 1.102 .8747 .8105 .8607 .7727 32 33 1.8029 1.0051 1.080 1.123 .8919 .8285 .8717 .7871 33 34 1.8360 1.0151 1.105 1.143 .9095 .8504 .8830 .8027 34 35 1.8704 1.0257 , 1.133 1.165 .9288 .8774 .8946 .8212 35 36 1.9060 1.0552 1.162 1.186 .9485 .9109 .9088 .8390 36 37 1.9430 1.0855 1.194 1.210 .9687 .9464 .9234 .8591 37 38 1.9815 1.1167 1.229 1.234 .9906 .9780 .9408 .8826 38 39 2.0216 1.1877 1 1.265 1.259 1.0131 1.0082 .9586 .9077 39 40 2.0908 1.3005 ' 1.306 1.285 1.0362 1.0306 .9794 .9360 40 41 2.1635 1.3775 1.348 1.313 1.0612 1.0487 1.0008 .9647 41 42 2.2401 1.4373 1.394 1.341 1.0894 1.0734 1.0252 .9998 42 43 2.2914 1.4582 1.444 1.371 1.1251 1.1127 1.0517 1.0351 43 44 2.3452 1.4798 i 1.497 1.403 j 1.1697 1.1556 1.0829 1.0761 44 45 2.4015 1.4809 ' 1.554 1.437 1.2212 1.2192 1.1163 1.1203 45 46 2.4606 1.4816 1.615 1.473 1.2839 1.2940 1.1562 1.1688 46 47 2.5227 1.4603 1.680 1.510 1.3516 1.3702 1.2000 1.2227 47 48 2.5879 1.3935 1.749 1.549 1.4260 1.4440 1.2509 1.2814 48 49 2.6908 1.3683 1.823 1.591 j 1.5061 1.5220 1.3106 1.3459 49 Formula, 1002 53284 53297 53309 53321 6 7 .727 53333 53346 53358 53370 53383 53395 53407 53420 53432 53444 7 9 .728 53456 53469 53481 53493 53506 53518 53530 53543 53555 53567 8 10 .729 53580 53592 53604 53617 53629 53641 53654 53666 53678 53691 9 11 .730 53703 53716 53728 53740 53753 53765 53777 53790 53802 53815 .731 53827 53839 53852 53864 53877 53889 5390] 53914 53926 53939 1 1 .732 53951 53963 53976 53988 54001 54013 54026 54038 54051 54063 2 3 .733 54075 54088 54100 54113 54125 54138 54150 54163 54175 54188 3 4 .734 54200 54213 54225 54238 54250 54263 54275 54288 54300 54313 4 6 .735 54325 54338 54350 54363 54375 54388 54400 54413 54425 54438 5 6 .736 54450 54463 54475 54488 54500 54513 54526 54538 54551 54563 6 8 .737 54576 54588 54601 54613 54626 54639 54651 54664 54676 64689 7 9 .738 54702 54714 54727 54739 54752 54765 54777 54790 54802 54815 8 10 .739 54828 54840 54853 54866 54878 54891 54903 54916 54929 64941 9 11 .740 54954 54967 54979 54992 55005 55017 55030 55043 55055 55068 .741 55081 55093 55106 55119 55132 55144 55157 55170 55182 55195 1 1 .742 55208 55220 55233 55246 55259 55271 55284 55297 55310 55322 2 3 .743 55335 55348 55360 55373 55386 55399 55412 55424 55437 55450 3 4 .744 55463 55475 55488 55501 55514 55526 55539 55552 55565 55678 4 5 .745 55590 55603 55616 55629 55642 55654 55667 55680 55693 55706 5 6 .746 55719 55731 55744 55757 55770 55783 1 55796 55808 55821 55834 6 8 .747 55847 55860 55873 55886 55898 55911 1 55924 55937 55950 i 55963 7 9 .748 55976 55989 56002 56014 56027 56040 56053 56066 56079 56092 8 10 .749 56105 56118 56131 56144 56156 56169 56182 56195 56208 56221 9 12 Log. 1 2 3 4 5 6 7 8 9 TABLE LXXXII. ANTI-LOGARITHMS, OR NUMBERS TO LOGARITHMS. *249 Log. 1 2 3 4 5 6 7 8 9 P.P. .750 56234 56247 56260 56273 56286 56299 56312 56325 56338 56351 .751 56364 56377 56390 56403 56416 56429 56442 56455 56468 56481 1 1 .752 56494 56507 56520 56533 56546 56559 56572 56585 56598 66611 2 3 .753 56624 56637 56650 56663 56676 56689 56702 56715 56728 66741 3 4 .754 56754 56768 56781 56794 56807 56820 56833 56846 56859 66872 4 5 .755 56885 56898 56911 56925 56938 56951 56964 56977 56990 67003 5 7 .756 57016 57030 57043 57056 57069 57082 57095 57108 57122 57135 6 8 .757 57148 57161 57174 57187 57201 57214 57227 57240 67253 67266 7 9 .758 57280 57293 57306 57319 57332 57346 57359 57372 57385 67398 8 10 .759 57412 57425 57438 57451 57465 57478 57491 57504 57517 67531 9 12 .760 57544 57557 57570 57584 57597 57610 57624 57637 57650 57663 .761 57677 57690 57703 57717 57730 57743 57756 57770 67783 57796 1 1 .762 57810 57823 57836 57850 57863 57876 57890 57903 57916 57930 2 3 .763 57943 57956 57970 57983 57996 58010 5S023 58036 58050 58063 3 4 .764 58076 58090 58103 58117 58130 58143 58157 58170 58184 68197 4 5 .765 58210 58224 58237 58251 58264 58277 58291 58304 58318 58331 5 7 .766 58345 58358 58371 58385 58398 58412 58425 58439 68452 58466 6 8 .767 58479 58492 58506 58519 58533 58546 58560 58573 58587 58600 7 9 .768 58614 58627 58641 58654 58668 58681 58695 58708 58722 68735 8 11 .769 58749 58762 58776 58790 58803 58817 58830 58844 58857 58871 9 12 .770 58884 58898 58911 58925 58939 58952 58966 58979 58993 69007 .771 59020 59034 59047 59061 59074 59088 59102 59115 59129 69143 1 1 .772 59156 59170 59183 59197 59211 59224 59238 59252 59265 69279 2 3 .773 59293 59306 59320 59334 59347 59361 59375 59388 59402 59416 3 4 .774 59429 59443 59457 59470 59484 59498 59511 59525 59539 69553 4 6 .775 59566 59580 59594 59607 59621 59635 59649 59662 59676 69690 5 7 .776 59704 59717 59731 59745 59759 59772 59786 59800 69814 59827 6 8 .777 59841 59855 59869 59883 59896 59910 59924 59938 69951 69965 7 10 .778 59979 59993 60007 60021 60034 60048 60062 60076 60090 60104 B 11 .779 60U7 60131 60145 60159 60173 60187 60200 60214 60228 60242 < 9 12 .780 60256 60270 60284 60298 60311 60325 60339 60353 60367 60381 .781 60395 60409 60423 60437 60451 60464 60478 60492 60506 60520 ] I 1 .782 60534 60548 60562 60576 60590 60604 60618 60632 60646 60660 i I 3 .783 60674 60688 60702 60716 p0730 60744 60758 60772 60786 60799 { J 4 .784 60814 60828 60842 60856 60870 60884 60898 60912 60926 60940 ^ L 6 .785 60954 60968 60982 60996 61010 61024 61038 61052 61066 61080 t ) 7 .786 61094 61108 61122 61136 61150 61165 61179 61193 61207 61221 i J 8 .787 61235 61249 61263 61277 61291 61306 61320 61334 61348 61362 : f 10 .788 61376 61390 61404 61419 61433 61447 61461 61475 61489 61504 i 1 11 .789 61518 61532 61546 61560 61574 61589 61603 61617 61631 61645 i 1 13 .790 61660 61674 61688 61702 61716 61731 61745 61759 61773 61787 .791 61802 61816 61830 61844 61859 61873 61887 61901 61916 61930 1 1 .792 61944 61958 61973 61987 62001 62015 62030 62044 62058 62073 2 3 .793 62087 62101 62116 62130 62144 62158 62173 62187 62201 62216 a 4 .794 62230 62244 62259 62273 62287 62302 62316 62330 62345 62359 4 6 .795 62373 62388 62402 62417 62431 62445 62460 62474 62488 62503 6 7 .796 62517 62532 62546 62560 62575 62589 62604 62618 62633 62647 6 9 .797 62661 62676 62690 62705 62719 62734 62748 62762 62777 62791 7 10 .798 62806 62820 62835 62849 62864 62878 62893 62907 62922 62936 8 11 .799 62951 62965 62980 62994 63009 63023 63038 63052 63067 63081 9 13 Loo. 1 2 3 4 5 6 7 8 9 - I *250 TABLE LXXXII. ANTI-LOGARITHMS, OR NUMBERS TO LOGARITHMS. Log. 1 2 3 4 5 6 7 8 9 P.P. .800 63096 63110 63125 63139 63154 63168 63183 63198 63212 63227 .801 63241 63256 63270 63285 63299 63314 63329 63343 63358 63372 1 1 .802 63387 63402 63416 63431 ' 63445 63460 63475 63489 63504 63518 2 3 .803 63533 63548 63562 1 63577 63592 63606 63621 i 63636 63650 63665 3 4 .804 63680 63694 63709 63724 63738 63753 63768 63782 63797 63812 4 6 .805 63826 63841 63856 63870 63885 63900 63915 63929 63944 63959 5 7 .806 63973 63988 64003 64018 64032 64047 64062 64077 64091 64106 6 9 .807 64121 64136 64150 64165 64180 64195 64210 64224 64239 64254 7 10 .808 64269 64284 64298 64313 64328 64343 64358 64372 64387 64402 8 12 .809 64417 64432 64447 64461 64476 64491 64506 64521 64536 64551 9 13 .810 64565 64580 64595 64610 64625 64640 64655 64670 64684 64699 .811 64714 64729 64744 64759 64774 64789 64804 64819' 64834 64849 1 2 .812 64863 64878 64893 i 64908 64923 64938 64953 64968 64983 64998 2 3 .813 65013 65028 65043 65058 65073 65088 65103 65118 65133 65148 3 5 .814 65163 65178 65193 65208 65223 65238 65253 65268 65283 65298 4 6 .815 65313 65328 65343 65358 65373 65388 65403 65418 65433 65449 5 8 .816 65464 65479 65494 65509 65524 65539 65554 ! 65569 65584 65599 6 9 .817 65615 65630 65645 65660 65675 65690 65705 65720 65736 65751 7 11 .818 65766 65781 65796 65811 65826 65842 65857 -65872 65887 65902 B 12 .819 65917 65933 65948 65963 65978 65993 66009 66024 66039 66054 9 14 .820 66069 66085 66100 66115 66130 66145 66161 66176 66191 66206 .821 '66222 66237 66252 66267 66283 66298 66313 66328 66344 66359 1 2 .822 66374 66390 66405 66420 66435 66451 66466 66481 66497 66512 2 3 .823 66527 66543 66558 66573 66589 66604 66619 66635 66650 66665 3 5 .824 66681 66696 66711 66727 66742 66757 66773 66788 66804 66819 ' 1 6 .825 66834 66850 66865 66881 66896 66911 66927 66942 66958 66973 5 8 .826 66988 67004 67019 67035 67050 67066 67081 67097 67112 67127 < 5 9 .827 67143 67158 67174 67189 67205 67220 67236 67251 67267 67282 • 7 11 .828 67298 67313 67329 67344 67360 67375 67391 67406 67422 67437 J 3 12 .829 67453 67468 67484 67499 67515 67531 67546 67562 67577 67593 1 J 14 .830 67608 67624 67639 67655 67671 67686 67702 67717 67733 67749 .831 67764 67780 67795 67811 67827 67842 67858 67873 67889 67905 ] I 2 .832 67920 67936 67952 67967 67983 67999 68014 68030 68046 68061 i I 3 .833 68077 68093 68108 68124 68140 68155 68171 68187 68202 68218 i ) .834 68234 68250 | 68265 68281 68297 68312 68328 68344 68360 68375 4 L 6 .835 68391 68407 68423 68438 68454 68470 68486 68501 68517 68533 I t ) 8 .836 68549 68565 68580 68596 68612 68628 68644 68659 68675 68691 | ( } 9 .837 68707 68723 68738 68754 68770 68786 68802 68818 68834 j 68849 !1 r 11 .838 68865 68881 68897 68913 68929 68945 68960 68976 68992 ! 69008 i 1 13 .839 69024 69040 69056 69072 69088 69103 69119 69135 69151 69167 £ ) 14 .840 69183 69199 69215 69231 69247 69263 69279 69295 69311 69327 .841 69343 69359 69375 ! 69390 69406 69422 69438 69454 69470 69486 1 2 .842 69502 ! 69518 69534 69550 69566 69582 69599 69615 69631 69647 2 1 3 .843 69663 | 69679 69695 69711 69727 69743 69759 69775 69791 69807 a 5 .844 69823 69839 69855 69871 69888 69904 69920 69936 69952 69968 4 6 .845 69984 70000 70016 70033 70049 70065 70081 70097 70113 70129 5 8 .846 70146 70162 70178 70194 70210 70226 70243 70259 70275 70291 6 10 .847 70307 70323 70340 1 70356 70372 70388 70404 70421 70437 70453 7 11 .848 70469 70486 70502 1 70518 70534 70550 70567 70583 70599 70615 8 113 .849 70632 70648 70664 70681 70697 70713 70729 70746 70762 70778 9 14 Loo. 1 2 3 4 6 6 7 8 9 TABLE LXXXII. ♦251 ANTI-LOGARITHMS, OR NUMBERS TO LOGARITHMS. Log. 1 2 3 4 5 6 7 8 9 P. P. .850 70795 70811 70827 70843 70860 70876 70892 70909 70925 70941 .851 70958 : 70974 70990 71007 71023 71040 71056 71072 71089 71105 1 2 .852 71121 71138 71154 71170 71187 71203 71220 71236 71252 71269 2 3 .853 71285 71302 71318 71335 71351 71367 71384 71400 71417 71433 3 5 .854 71450 71466 71483 71499 71515 71532 71548 71565 71581 71598 4 7 .855 71614 71631 71647 71664 71680 71697 71713 71730 71746 71763 5 8 .856 71779 71796 71812 ' 71829 71846 71862 71879 71895 71912 71928 6 10 .857 71945 71961 71978 71995 72011 72028 72044 72061 72078 72094 7 12 .858 72111 72127 72144 ; 72161 72177 72194 72210 72227 72244 72260 8 13 .859 72277 72294 72310 72327 72344 72360 72377 72394 72410 72427 9 15 .860 72444 72460 72477 72494 72510 72527 72544 72560 72577 72594 .861 72611 72627 72644 72661 72678 72694 72711 72728 72744 72761 1 2 .862 72778 72795 72812 1 72828 72845 72862 72879 72895 72912 72929 2 3 .863 72946 72963 72979 : 72996 73013 73030 73047 73063 73080 73097 3 6 .864 73114 73131 73148 73164 73181 73198 73215 73232 73249 73266 4 7 .865 73282 73299 73316 73333 73350 73367 73384 73401 73418 73434 5 8 .866 73451 73468 73485 73502 73519 73536 73553 73570 73587 73604 6 10 .867 73621 73638 73655 73672 73689 73706 73722 73739 73756 73773 7 12 .868 73790 73807 73824 73841 73858 73875 73892 73909 73926 73943 8 13 .869 73961 73978 73995 74012 74029 74046 74063 74080 74097 74114 9 15 .870 74131 74148 74165 74182 74199 74216 74234 74251 74268 74285 .871 74302 74319 74336 74353 74370 74388 74405 74422 74439 74456 1 2 .872 74473 74490 74508 74525 74542 74559 74576 74593 74611 74628 2 3 .873 74645 74662 74679 74696 74714 74731 74748 74765 74783 74800 3 5 .874 74817 74834 74851 74869 74886 74903 74920 74938 74955 74972 - 4 7 .875 74989 75007 75024 75041 75059 75076 75093 75110 75128 75145 j . 5 9 .876 75162 75180 75197 75214 75232 75249 75266 75284 75301 75318 i ( 5 10 .877 75336 75353 75370 75388 75405 75422 75440 75457 75474 75492 ' 7 12 .878 75509 75527 75544 75561 75579 75596 75614 75631 75648 75666 i J 14 .879 75688 75701 75718 75736 75753 75770 75788 75805 75823 75840 i ) 16 .880 75858 75875 75893 75910 75928 75945 75963 75980 75998 76015 .881 76033 76050 76068 76085 76103 76120 76138 76155 76173 76190 ] L 2 .882 76208 76225 76243 76261 76278 76296 76313 76331 76348 76366 S I 4 .883 76384 76401 76419 76436 76454 76472 76489 76507 76524 76542 I 2 t 5 .884 76560 76577 76595 76613 76630 76648 76666 76683 76701 76718 4 I 7 .885 1 76736 76754 76771 76789 76807 76825 76842 76860 76878 76895 S » 9 .886 76913 76931 76948 76966 76984 77002 77019 77037 77055 77073 € » 11 .887 77090 77108 77126 77144 77161 77179 77197 77215 77232 77250 7 12 .888 77268 77286 77304 77321 77339 77357 77375 77393 77411 77428 8 14 .889 1 77446 77464 77482 77500 77518 77535 77553 77571 77589 77607 9 16 .890 ! 77625 77643 77660 77678 77696 77714 77732 77750 77768 77786 .891 1 77804 77822 77839 1 77857 77875 77893 77911 77929 77947 77965 1 2 .892 77983 78001 78019 78037 78055 78073 78091 78109 78127 78145 2 4 .893 78163 78181 78199 78217 78235 78253 78271 78289 78307 78325 3 5 .894 78343 78361 78379 78397 78415 78433 78451 78469 78487 78505 4 7 .895 78524 78542 78560 78578 78596 78614 78632 78650 78668 78686 5 9 .896 78705 78723 78741 1 78759 78777 78795 78813 78832 78850 78868 6 11 .897 78886 78904 78922 : 78941 78959 78977 78995 79013 79031 79050 7 13 .898 79068 79086 79104 1 79122 79141 79159 79177 79195 79214 79232 8 14 .899 79250 79268 79287 79305 79323 79341 79360 79378 79396 79415 9 16 Log. 1 2 3 4 5 6 7 8 9 i ♦262 TABLE LXXXII. ANTI-LOGARITHMS, OR NUMBERS TO LOGARITHMS. Log. 1 2 3 4 5 6 7 8 9 P.P. .900 79433 79451 79469 79488 79506 79524 79543 79561 79579 79598 1.901 79616 79634 79653 79671 79689 79708 79726 79744 79763 79781 1 2 .902 79799 79818 79836 79855 79873 79891 79910 79928 79947 79965 2 4 .903 79983 80002 80020 80039 80057 80076 80094 80112 80131 80149 3 6 .904 80168 80186 80205 80223 80242 80260 80279 80297 80316 80334 4 7 .905 80353 80371 80390 80408 80427 80445 80464 80482 80501 80519 5 9 .906 80538 80556 80575 80593 80612 80631 80649 \ 80668 80686 80705 6 11 .907 80724 80742 80761 80779 80798 80816 80835 80854 80872 80891 7 13 .908 80910 80928 80947 80965 80984 81003 81021 81040 81059 81077 8 15 .909 81096 81115 81133 81152 81171 81190 81208 81227 81246 81264 9 17 .910 81283 81302 81320 81339 81358 81377 81395 81414. 81433 81452 .911 81470 81489 81508 81527 81546 81564 81583 81602 81621 81639 1 2 .912 81658 81677 81696 81715 81733 81752 81771 81790 81809 81828 2 4 .913 81846 81865 81884 81903 81922 81941 81960 81979 81997 1 82016 3 6 .914 82035 82054 82073 82092 82111 82130 82149 82167 82186 82205 4 8 .915 82224 82243 82262 82281 82300 82319 82338 82357 82376 82395 5 9 .916 82414 82433 82452 82471 82490 82509 82528 82547 82566 82585 6 11 .917 82604 82623 82642 82661 82680 82699 82718 82737 82756 82775 7 13 .918 82794 82813 82832 82851 82871 82890 82909 82928 82947 82966 8 15 .919 82985 83004 83023 83042 83062 83081 83100 83119 83138 83157 9 17 .920 83176 83196 83215 83234 83253 83272 83291 83311 83330 83349 .921 83368 83387 83407 83426 83445 83464 83483 83503 83522 83541 1 2 .922 83560 83580 83599 83618 83637 83657 83676 83695 83714 83734 2 4 .923 83753 83772 83792 83811 83830 83849 83869 83888 83907 i 83927 3 6 .924 83946 83965 83985 84004 84023 84043 84062 84081 84101 84120 4 8 .925 84140 84159 84178 84198 84217 84236 84256 84275 84295 84314 5 10 .926 84333 84353 84372 84392 84411 84431 84450 84470 84489 84508 6 12 .927 84528 84547 84567 84586 84606 84625 84645 84664 84684 84703 7 14 .928 84723 84742 84762 84781 84801 84820 •84840 84859 84879 84898 8 16 .929 84918 84938 84957 84977 84996 85016 85035 85055 85075 85094 9 17 .930 85114 85133 85153 85173 85192 85212 85231 85251 85271 85290 .931 85310 85330 85349 85369 85389 85408 85428 85448 85467 85487 1 2 .932 85507 85526 85546 85566 85585 85605 85625 85645 85664 85684 2 4 .933 85704 85724 85743 85763 85783 85803 85822 85842 85862 86882 3 6 .934 85901 85921 85941 85961 85981 86000 86020 86040 86060 86080 4 8 .935 86099 86119 86139 86159 86179 86199 86218 86238 86258 86278 5 10 .936 86298 86318 86338 86357 86377 86397 86417 86437 86457 86477 6 12 .937 86497 86517 86537 86557 86576 86596 86616 86636 86656 86676 7 14 .938 86696 86716 86736 86756 86776 86796 86816 86836 86856 86876 8 16 .939 86896 86916 86936 86956 86976 86996 87016 87036 87056 87076 9 18 .940 87096 87116 87136 87157 87177 87197 87217 87237 87257 87277 .941 87297 87317 87337 87357 87378 87398 i 87418 | 87438 87458 87478 1 2 .942 s 87498 87519 87539 1 87559 87579 87599 87619 87640 87660 87680 2 4 .943! 87700 87720 87740 87761 87781 87801 87821 87842 87862 87882 3 6 .944 J 87902 87922 87943 87963 87983 88004 88024 88044 88064 88085 4 8 .945^ 88105 88125 88145 88166 88186 88206 88227 88247 88267 88288 5 10 .946 88308 88328 88349 88369 88389 88410 88430 88450 88471 88491 6 12 .947 88512 88532 88552 88573 88593 88614 88634 88654 88675 88695 7 14 .948 88716 88736 88756 88777 88797 88818 88838 88859 88879 88900 8 16 .949 Log. 88920 88941 88961 88982 89002 89023 89043 89064 89084 89105 9 18 1 2 3 4 5 6 7 8 9 TABLE LXXXII. ANTI-LOGARITHMS, OR NUMBERS TO LOGARITHMS. *^53 Log. 1 2 3 4 5 6 7 8 9 P.P. .950 89125 89146 89166 89187 89207 89228 89248 89269 89289 89310 .951 89331 89351 89372 89392 89413 89433 89454 89475 89495 89516 1 2 .952 89536 89557 89578 89598 89619 89640 89660 89681 89702 89722 2 4 .953 89743 89764 89784 89805 89826 89846 89867 89888 89908 89929 3 6 .954 89950 89970 .89991 90012 90033 90053 90074 90095 90116 90136 4 8 .955 90157 90178 90199 90219 90240 90261 90282 90303 90323 90344 5 10 .956 90365 90386 904:07 90427 90448 90469 90490 90511 90532 90552 6 12 .957 90573 90594 90615 90636 90657 90678 90698 90719 90740 90761 7 15 .958 90782 90803 90824 90845 90866 90887 90908 90928 90949 90970 8 17 .959 90991 91012 91033 91054 91075 91096 91117 91138 91159 91180 9 19 .960 91201 91222 91243 91264 91285 91306 91327 91348 91369 91390 .961 91411 91432 91453 91474 91496 91517 91538 91559 91580 91601 1 2 .962 91622 91643 91664 91685 91706 91728 91749 91770 91791 91812 2 4 .963 91833 91854 91876 91897 91918 91939 91960 91981 92003 92024 3 6 .964 92045 92066 92087 92109 92130 92151 92172 92193 92215 92236 4 8 .965 92257 92278 92300 92321 92342 92363 92385 92406 92427 92449 5 11 .966 92470 92491 92512 92534 92555 92576 92598 92619 92640 92662 6 13 .967 92683 92704 92726 92747 92768 92790 92811 92832 92854 92875 7 15 .968 92897 92918 92939 92961 92982 93004 93025 93046 93068 93089 8 17 .969 93111 93132 93154 93175 93197 93218 93240 93261 93282 93304 9 19 .970 93325 93347 93368 93390 93411 93433 93454 93476 93498 93519 .971 93541 93562 93584 93605 93627 93648 93670 93691 93713 93735 1 2 .972 93756 93778 93799 93821 93843 93864 93886 93907 93929 93951 2 4 .973 93972 93994 94016 94037 94059 94081 94102 94124 94146 94167 3 7 .974 94189 94211 94232 94254 94276 94297 94319 94341 94363 94384 4 9 .975 94406 94428 94450 94471 94493 94515 94537 94558 94580 94602 5 11 .976 94624 94646 94667 94689 94711 94733 94755 94776 94798 94820 6 13 .977 94842 94864 94886 94907 94929 94951 94973 94995 95017 95039 7 15 .978 95060 95082 95104 95126 95148 95170 95192 95214 95236 95258 8 17 .979 95280 95302 95324 95345 95367 95389 95411 95433 95455 95477 9 20 .980 95499 95521 95543 95565 95587 95609 95631 95653 95675 95697 .981 95719 95741 95763 95786 95808 95830 95852 95874 95896 95918 1 2 .982 95940 95962 95984 96006 96028 96051 96073 96095 96117 96139 2 4 .983 96161 96183 96206 96228 96250 96272 96294 96316 96339 96361 3 7 .984 96383 96405 96427 96450 96472 96494 96516 96538 96561 96583 4 9 .985 96605 96627 96650 96672 96694 96716 96739 96761 96783 96805 5 11 .986 96828 96850 96872 96895 96917 96939 96962 96984 97006 97029 6 13 .987 97051 97073 97096 97118 97140 97163 97185 97208 97230 97252 7 16 .988 97275 97297 97320 97342 97364 97387 97409 97432 97454 97477 8 18 .989 97499 97521 97544 97566 97589 97611 97634 97656 97679 97701 9 20 .990 97724 97746 97769 97791 97814 97836 97859 97881 97904 97926 .991 97949 97972 97994 98017 98039 980G2 98084 98107 98130 98152 1 2 .992 98175 98197 98220 98243 98265 98288 98311 98333 98356 98378 2 5 .993 98401 98424 98446 98469 98492 98514 98537 98560 98583 98605 3 7 .994 98628 : j 98651 98673 1 98696 98719 98742 98764 98787 98810 98833 4 9 .995 98855 98878 98901 98924 98946 98969 98992 99015 99038 99060 5 11 .996 99083 99106 99129 99152 99174 99197 99220 99243 99266 99289 6 14 .997 99312 99334 99357 99380 99403 99426 99449 99472 99495 99518 7 16 .998 99541 99563 99586 99609 99632 99655 99678 99701 99724 99747 8 18 .999 99770 99793 99816 99839 99862 99885 99908 99931 99954 99977 9 20 Log. 1 2 3 4 5 6 7 8 9 ADDITION TO SECTION VII. Tontines. Dividends of Annual Interest may be made to survivors of all ages, proportional to their deposits. Dividends from Mortality can be appor- tioned to the products of each Deposit X g:t+n-^px + n from Table XXYI; the shares being as if the Tontines were separate for each age. ADDITION TO THE FIRST FORM, Page *256. The Method of Least Squares and of Least Product. To explain the fundamental principle of Least- Squares, let j) denote the probability of the error e, regarded as the radius of a circle. Then as the probability changes by — dp, the mathematical expection of error increases by the ring p.2nede. Among the possible hypotheses, the theory of least squares assumes these increments to be in a constant ratio, as in the equation integra- ted below. The maximum probability po is evidently attained when e = 0, as at the center of the circle. For the rest, including weights and probable error, reference may be made to ivoi^lcs on Least Squares. -dp=kp.2^ede, p=poe-*^'^, pjj j9 3=^0' e— *"■('>' +'^2" +'3'.) Method of Least Product. In either of the forms of equations from the Census, although positive and negative errors are equally likely, as re- quired by least squares, yet the square roots of the weights are unequal, and difficult of estimation. Therefore to diminish the large errors in these and other Examples, let the square of each error be replaced by its logarithm, before differentiating for the minimum ; thus, ^ aiX-{-h^y+ ...+k^v+l^ =6^, .... a„x-\-h^y+...+l\v+In^ e^. (1) IT = i (log 61' +log e," + ... -f- log e„*) = Log {e^e^...e^) (2) (3) d\J a, a- ttn ^ dU = — _| j- .. + — =0, ., 'Zdx e. e. Cn Mv 1 /A A \ For two equations of condition, x = — -I — •""^l' i^^* i^ is only m these limiting cases, where the n equations have n — 1 unknown quantities, that the true value of x, y, ... is the mean of the n particular values, found by omitting one equation at a time and solving the others. In all other cases, the true roots are different, and must be approximately known in advance. For example, in the eight equations from the Census, making x= 1.431 + x' and y = 0.703 + y', we first take x' and i/'=0, 0.001, .002, ... and so compute U= common log {e^e^...e^) to represent a, a', b, h', c, c', on page 304 text. Then differentiating U on page 304, for the minimum, we find x' and y' from the resulting equations, by common elimination. g= ^+y' (A'_A) + (y _i) A« =0. 6+x' (A'_A) + (y'_i) J*=0=f . AMERIOAIsr LIFE TABLES. From Combinations of the Census. The American Life Tables on the following pages, being derived from the general census of the United States, will be the proper standard for civil purposes, especially for the youthful ages, and the valuation of Legacies, Reversions and widows* right of Dower. They diflfer to some extent from Life Insurance Tables, which are modified by selection, and from which a considerable number of unhealthy or diseased lives have been rejected by "the medical examination." For greater accuracy, separate Tables have been constructed for Males and for Females of the White Population. In regard to other classes, the census of 1880 gives the relative Deaths per cent, of Whites 1.474 to 1.728 of Colored. Further comparisons indicate the average mortality of the Colored to be nearly that of the Northampton Table. Hence, as shown by the text page 49, the Annuity for the Colored at 5 per cent, interest may be taken from the following Table at G per cent, by accounting in this and in other cases, the difference or excess of 1 per cent, interest as an offset against the extra mortality. A similar difference of 1 or 2 per cent, interest evidently appertains to the Irish popidation ; for by the census of 1870, the relative Deaths per cent, were for total foreign as 1.474, Irish 1.813, German 1.371. A similar correction has been applied in the preceding Climatic Tables. Since 1850, a series of systematic trials have shown that in the present state of the national statistics, the average rates of Mortality are most correctly determined from repeated enumerations of the people, by the survivorships from one census to another. For this object, the following mode of analysis was found superior to all others in precision, and is entirely similar to that which astronomers have proved to be the most advantageous in similar cases. During a tour in Europe in 1869, the whole process was submitted to the highest authority. Prof. De Morgan of London, and fully endorsed by him, repeating " it is a very fine method indeed." In order to obtain permanent results between fluctuations of high or low mortality, the present Tables are based on four enumerations from 1830 to 1860, including thirty years' experience and more than seventy millions of observations. Compared with the experience twenty years later by the census of 1880, the one series verifies the other, without either increase or decrease of mortality. Thus, omitting certain States in which the census of 1870 was deficient, the number of annual survivors calculated by the Life Table, for males between the ages of 20 and 75, was to the survivors found from the census of 1870 and 1880, as 1 to 1.00014. And the corresponding ratio for surviving females was as 1 to 1.00005. Had the census of 1870 been com- plete, the ratios in verification would have approached stUl nearer to equality. Equations of Condition. In any large population, let Dx denote the Deaths and ly the Immi- gration registered during twelve months at any specified age or ages, as from ten to fifteen. If the registry of Deaths be complete, the value of a; is 1. But if incomplete, and still D and I are proved to be proportional to the true numbers by ages, let D' denote the true result found from correct cen- suses of the living ; and let P denote the population thus exposed, in the middle of the same twelve months. Also admitting averages, let L denote the value of P exposed in the death registry. By the common method, the correlative values in the equation will be : Mrst Form, = J^ = tT- Second Form, — ly = D'. \t r r L D' In order to determine ^ , let a; denote any age specified above, at the calendar time t, between two censuses ; then by Taylor's Theorem, if La;,< — L, _dL_dL_ La;-i, <-}— La; + i,/ + i _ dx dt (p-) La;,< Here the second derivatives have disappeared by opposite signs, and the third by their smallness. For if we assume L = a6*c<, observing that c is nearly 1.03, we find by the next data that the third Copyright, 1886, by Lbvi W. Mbbch. ♦255 ♦256 AMERICAN LIFE TABLES, derivatives could only augment the numerator in the fifth decimal place, which is here insensible. Therefore summing the numerator and the denominator by double integration between the proper limits of X and t, we obtain the mean ratio : D[ _ f(U -U')dt-f(Jj^-lj^)dx _ F-F^ -(I860) + (1830) P ~ SS\.dxdt ~ P P = ^ U (1830) + (1840) + (1850) + i (1860) + 2 V(1830 '40 '50) (1840 '50 '60)}. Those versed in the Integral Calculus will observe that the second member corresponds to a gen- eral Theorem, the denominator representing the contents of a solid P, of which the terms of the numerator are the four lateral surfaces, comparable to the walls of a four square city. For example, if the specified ages are from 10 to 15, as before, the difference of two opposite walls may be denoted by —(I860) + (1830), being the difference of population at those ages by the census of 1830 and of 1860. While P—P" denotes the difference of the other two walls. Thus if we suppose intermediate cen- suses to fill the spaces between (1830), (1840), (1850), and (1860), this continuous solid will represent P the denominator, here found by Legendre's quadratures; the last term denoting twice the sum of population in 1835, '45, '55. And if A, B, C denote three continuous five-year values of P, then P' will denote the boundary between A and B, and P" that between B and C. Hence by parabolic inter- polation, F -P" r= =^ : P" = Z° — . D' = P' -P'' -(I860) + (1830). 10 90 In this manner the fallowing equations were derived from the U. S. Census of White Males in 1830, 1840, 1850, and 1860 ; and another parallel solution for Females, not here inserted. The Deaths of Whites are those given by the census of 1860, adjusted by weights to one aggregate of P above the age of 40 ; and the Immigration returns are those of the State Department, of which a full summary down to 1855 was given in Bromwell's History of Immigration. The proportionality by ages was proved by comparing the statistics of different decades or periods, excluding those imder five years of age : lOOD'-^P. S€C c^ 5.36 5.92 6.54 7.22 7.98 8.80 9.72 10.74 11.88 13.10 14.48 16.00 17.68 19.58 21.70 23.96 65 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 26.52 29.34 32.44 35.90 39.74 44.12 48.74 54.04 60.04 66.52 73.86 81.54 90.14 99.58 109.04 119.20 129.44 141.12 154.42 168.88 183.84 199.50 216.92 235.82 255.88 277.88 301.14 327.30 353.90 383.42 416.02 448.28 490.76 3 Per Ct. 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 4 Per Ct. 5 PerCt. 11.648 11.333 11.008 10.681 10.351 10.019 9.683 9.343 9.003 8.665 8.323 7.981 7.638 7.282 6.937 6.598 6.268 5.938 5.615 5.299 4.989 4.686 4.382 4.082 3.789 3.513 3.253 3.003 2.765 2.536 2.346 2.167 2.018 1.863 1.708 1.562 1.431 1.312 1.201 1.097 0.998 0.906 0.820 0.739 0.666 0.600 0.530 0.476 0.415 0.385 6 PerCt. 10.649 10.385 10.110 9.832 9.551 9.266 8.976 8.681 8.385 8.088 7.787 7.484 7.179 6.860 6.550 6.243 5.944 5.643 5.348 5.057 4.771 4.491 4.208 3.927 3.641 3.392 3.146 2.908 2.682 2.464 2.282 2.111 1.968 1.819 1.670 9.786 9.563 9.329 9.091 8.850 8.603 8.351 8.094 7.834 7.573 7.307 7.038 6.765 6.478 7 Per Ct 9.035 8.845 8.645 8.440 8.231 8.017 7.797 7.572 7.343 7.112 6.875 6.635 6.390 6.131 8.378 8.215 8.042 7.865 7.684 6.198 5.877 1.528 1.401 1.286 1.178 1.077 0.980 0.891 0.807 0.727 0.656 0.591 0.522 0.470 0.410 0.381 5.920 5.647 5.372 5.101 4.834 4.569 4.309 4.045 3.782 3.523 3.278 3.045 2.819 2.604 2.395 2.221 2.057 1.920 1.776 1.632 1 1 495 373 1.261 1.156 1.058 0.963 0.876 0.794 0.716 0.646 0.583 0.515 0.464 0.404 0.375 5.625 5.375 123 4.874 4.627 4.382 4.140 3.893 3.646 3.402 3.170 2.950 2.735 2.529 2.329 2.163 2.005 1.874 1.736 1.597 1.464 1.345 1.237 1.135 1.039 0.947 0.862 0.782 0.705 0.637 0.574 0.508 0.457 0.399 7.497 7.304 7.105 6.902 6.697 6.486 6.270 6.050 5.815 5.584 5.354 5.126 4.894 4.664 4.436 4.208 3.982 3.751 3.519 3.289 3.069 2.860 2.656 2.459 2.267 2.108 1.956 1.830 1.697 1.562 1.434 1.319 1.214 1.114 1.021 0.931 0.848 0.770 0.695 0.628 0.566 0.502 0.452 0.396 0.3720.370 Formula, ajxc — Da ♦270 AMERICAN FEMALE LIFE TABLE. PRESENT VALUE OF WIDOW'S TX)WER IN AN ESTATE WORTH $100, ACCORDING TO A GIVEN RATE OF INTEREST. Example. A widow whose age is 39 is entitled to Dower in an Estate worth ^7,200, at 5 per cent, interest. Opposite age 39 is 23.475, to be multiplied by 72 (the number of hundreds in 7,200) ; the product $1,690.20 is the present value of the Dower. By the laws of most of tlie States, on the death of the husband, the widow is entitled to " the life use of one-third of the real estate," of which^he present value is given by this Table. In some States, where the Dower, instead of one-third, is one-half of the life use, multiply the present result by 3 and divide the product by 2. For greater accuracy, the present Table is based on " complete annuities " of the Female Life Table, continuing to the day of death. For page *255, line 12, take | of the 6, to be the 5 per cent. Dower. AGE. 3 4 5 6 7 A&E. 3 4 5 6 7 X. Per Cent. Per Cent. Per Cent. Per Cent Per Cent. X. 50 Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. 5 23.840 26.266 27.800 28.818 29.525 15.257 18.201 20.510 22.338 23.800 6 23.773 26.225 27.780 28.813 29.534 51 14.926 17.850 20.156 21.993 23.471 7 23.699 26.177 27.754 28.806 29.538 52 14.586 17.486 19.787 21.630 23.121 8 23.622 26.127 27.727 28.797 29.543 53 14.241 17.114 19.408 21.257 22.760 9 23.539 26.070 27.694 28.782 29.544 54 13.892 16.736 19.021 20.873 22.386 10 23.449 26.006 27.653 28.762 29.539 55 13.538 16.350 18.623 20.476 21.998 11 23.339 25.921 27.592 28.720 29.514 56 13.179 15.956 18.214 20.065 21.594 12 23.188 25.791 27.483 28.630 29.439 57 12.814 15.553 17.794 19.642 21.176 13 22.977 25.594 27.303 28.465 29.288 58 12.447 15.145 17.366 19.208 1 20.745 14 22.733 25.359 27.082 28.258 29.092 59 12.078 14.733 16.931 18.764 20.303 15 22.482 25.115 26.849 28.037 28.882 60 11.703 14.312 16.485 18.307 19.845 16 22.259 24.902 26.649 27.850 28.705 61 11.327 13.886 16.031 17.839 19.372 17 22.055 24.708 26.470 27.683 28.550 62 10.946 13.454 15.567 17.358 18.886 18 21.896 24.565 26.344 27.573 28.452 63 10.556 13.006 15.083 16.853 18.371 19 21.736 24.421 26.217 27.461 28.353 64 10.170 12.561 14.600 16.348 17.854 20 21.578 24.280 26.094 27.354 28.259 65 9.786 12.118 14.116 15.839 17.331 21 21.418 24.136 25.968 27.244 28.161 66 9.406 11.676 13.632 15.327 16.803 22 21.261 23.994 25.844 27.136 28.068 67 9.025 11.230 13.141 14.806 16.262 23 21.104 23.854 25.723 27.032 27.977 68 8.648 10.787 12.651 14.282 15.717 24 20.963 23.732 25.620 26.948 27.908 69 8.274 10.345 12.326 13.756 15.166 25 20.820 23.606 25.516 26.862 27.837 70 7.904 9.905 11.-668 12.226 14.610 26 20.672 23/477 25.408 26.773 27.764 71 7.536 9.468 11.177 12.696 14.051 27 20.521 23.344 25.296 26.680 27.688 72 7.170 9.027 10.680 12.155 13.477 28 20.361 23.202 25.174 26.578 27.601 73 6.806 8.588 10.181 11.610 12.897 29 20.196 23.053 25.046 26.469 27.512 74 6.448 8.154 9.686 11.067 12.315 30 20.024 22.897 24.910 26.354 27.414 75 6.104 7.735 9.207 10.539 11.749 31 19.847 22.736 24.769 26.233 27.311 76 5.774 7.333 8.745 10.028 11.198 32 19.666 22.569 24.622 26.107 27.203 77 5.454 6.939 8.291 9.525 10.652 33 19.480 22.396 24.470 25.975 27.091 78 5.146 6.560 7.852 9.036 10.123 34 19.289 22.219 24.312 25.838 26.973 79 4.849 6.193 7.426 8.559 9.603 35 19.096 22.039 24.152 25.700 26.854 80 4.587 5.869 7.048 8.137 9.143 36 18.902 21.858 23.993 25.562 26.737 81 4.3^6 5.558 6.685 7.729 8.698 37 18.704 21.672 23.827 25.418 26.615 82 4.112 5.279 6.360 7.364 8.298 38 18.500 21.480 23.654 25.268 26.486 83 3.884 4.994 6.026 6.987 7.884 39 18.291 21.281 23.475 25.112 26.352 84 3.658 4.711 5.693 6.611 7.470 40 18.075 21.075 23.288 24.948 26.211 85 3.443 4.441 5.374 6.248 7.069 41 17.841 20.847 23.078 24.759 26.045 86 3.245 4.191 5.079 5.913 6.698 42 17.594 20.603 22.850 24.552 25.859 87 3.061 3.958 4.802 5.598 6.349 43 17.337 20.348 22.609 24.331 25.660 88 2.884 3.735 4.537 5.295 6.012 44 17.068 20.077 22.351 24.092 25.442 89 '2.717 3.523 4.285 5.007 5.691 45 16.790 19.795 22.080 23.839 25.210 90 2.556 3.319 4.041 4.727 5.378 46 16.501 19.499 21.793 23.569 24.960 91 2.405 3.126 3.811 4.462 5.083 47 16.204 19.194 21.494 23.286 24.696 92 2.261 2.942 3.591 4.209 4.799 48 15.898 18.876 21.181 22.987 24.416 93 2.124 2.759 3.380 3.965 4.525 49 15.582 18.545 20.853 22.670 24.117 94 1.998 2.605 3.185 3.740 4.272 FoJRMULA, Dower on 100 = f Or-T,^-: V I UNlVc, . \ UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. SENT ON ILL SEP 9 ]m U. C. BERKELEY LD 21-95»n-ll,'50(2877sl6)476 YE n?6l4 / !V\ e-e c f I