ox S«>ME METHODS OF INTEUPOLATIOX APPLICABLE TO THE GliADUATIOX 

 OF lUREUl'lAH SERIES, Sll'H AS TABLES OF MORTALITY, *fcc., &C. 



By Erasti's L. De Fokkst, 51. A., 

 Of Watertowu, Connecticut. 



[The portions of the follo\viug methods of interpolation comprising the formulas 2, 

 8, A, B, C, D, E, F, 11, 1-3, 13, 17, 19, 20, 21, 24, 25, 2<5, 27, 28, 30, 43, 44, 45, 46, 48, 49, and 

 50, were presentev^ to the Smithsonian Institution for i^uhlication in the year 18GS. The 

 method of constructing tables of mortality from two successive census enumerations 

 ■was first given in January, 1869, and the formulas 40, 41, 42, 53, 54, 55, 56, and 59, 

 in January, 1870. — J. II.] 



We have no analytical formula wliicli expresses tlie law of mortality 

 Avith precision, and at the same time with such simplicity as to be prac- 

 tically useful. For all the purposes of life insurance and life annuities, 

 it is expressed by numerical series. The law is known to vary in dif- 

 ferent localities, and even iu the same locality at different epochs. That 

 which ])revails in any community, at a given period, can be ascertained 

 by enumerating the persons living at the various ages, and the deaths 

 which annually occur among them. Eeduced to one of its usual forms, 

 it is expressed in a statistical table, showing-, out of a certain number of 

 persons born, how many survive to complete each successive year of 

 their age. These numbers of the living form a diminishing series of 

 about one hundred terms, whose first dilferences are the numbers dying 

 during each year of age. We have reason to believe that a true law of 

 mortality is a continuous function of the age, free from sudden irregu- 

 larities, so that in a perfect table the second, third, &c., orders of dilfer- 

 ences of the series ought to go on continually diminishing, and each 

 order by itself ought to show a (pertain degree of regularity ; in other 

 words, the table should be well graduated. But, in point of fact, 

 all purely statistical tables are irregular, especially when the popula- 

 tion observed has been small, and every table of mortality now iu use 

 has been graduated artificially. It was not strange that the Carlisle 

 t^ble, derived from records of population and deaths in a single town, 

 should show many irregularities. They have been adjusted to some 

 extent, but very imperfectly. The Combined Experience table, also, 

 which was compiled from the records of seventeen British life insurance 

 oflices, owes its better graduation to art rather than to nature. Farrs 

 English life-table, iSTo. .'>, lor males, derived from the census returns of 

 1811 and 1851, and from the registry of deaths in England and Wales 



