276 METHODS OF INTERPOLATION. 



for the seventeen years from 1838 to 1854, thongh perhaps the best ex- 

 pression we have for the hiw of general mortality, is by no means well 

 graduated. In this case the population observed was so large that if 

 the tables had been formed directly from the enumeration of persons 

 living and persons dying in each single year of age, and if these obser- 

 vations conld have been relied upon as accurate, any irregularities then 

 existing in the series might possibly have been thought to result from 

 something peculiar in the law of life at certain ages. But it was neces- 

 sary to combine the single years of age into groups, owing to the impos- 

 sibility of ascertaining ages with i)recision. All persons were rerjuired 

 to give their exact ages at last birthday, but the reports state that 

 round numbers, such as 50, 60, &c., were disproportionately numerous, 

 showing that the ages were not always correctly given. In forming the 

 life-table No. 3 the years of age were grouped together into decennial 

 periods chiefly, and the whole term of life was then divided into five 

 unequal parts, so as to form a chain of sub-series, each of the fourth 

 order, and not continuous at their points of junction. We must con- 

 clude, then, that the great irregularities now found at certain points in 

 the series result from imperfect distribution, and not from any irregu- 

 larity in the true law of mortality. 



A good system of distribution or adjustment, though not positively 

 essential in practice, is nevertheless desirable, first, because a judiciously 

 adjusted table probably comes nearer to the truth than an unadjusted 

 or ill-adjusted one; that is, nearer to what the statistics would show if 

 the population observed could be made in<lefinitely large, and if the 

 numbers for each year of age could be independently determined. 

 Secondly, if the primary table is well graduated, all the various series 

 of numbers derived from it, forming the usual " commutation tables" 

 and tables of premiums and valuations of assurances and annuities, 

 will be well graduated also, and this will sometimes facilitate the 

 computation of such tables, because a part of the tabular numbers 

 can be accurately found by ordin^,ry interpolation, and errors of com- 

 putation can be discovered by the method of differences. Many writers 

 on the law of mortality have treated of the subject of adjustnunit, as 

 may be seen in the pages of the London Journal of the Institute of 

 Actuaries and Assurance Magazine, and elsewhere. The rule ot least 

 squares was used to adjust the American table given in the report of 

 tbe United States census of 1800. (See the Appendix on Average Ilate 

 of Mortality, pages 518 and 521.) The series there given, however, is 

 not very thoroughly graduated, as can easily be shown by taking its 

 successive orders of difierences. In England, the "law of Gompertz" 

 has been chiefly taken as a basis. But it is not necessary to adopt any 

 exclusive theory respecting the precise nature of that function which 

 expresses the law of mortality. The following system of distribution 

 and graduation is based upon j)rinciples which apply to any continuous 

 series of numbers, and is analogous to the ort^inary methods of inter- 



