342 METHODS OF INTERPOLATION. 



shown by comparing the respective ratios of the probable errors of the 

 adjusted and unadjusted terms, computed according to formula (92). 



The adjusted series of probabilities of dying within a year at each 

 age, given at page xciv of the Institute of Actuaries' Life-Tables, was 

 not obtained directly from the corresponding observed series; but the 

 sufficiency of its adjustment can be tested all the same ; and on compar- 

 ing it with the observed series, we find that the arithmetical mean of 



{ - ) between the ages 20 and 89, inclusive, is only 0.82. This value 



falls short of unity by more than its probable error, and indicates that 

 the series in question has not been smoothed out quite enough to give 

 it the greatest accuracy attainable. 



The earliest publication of adjustment-formulas of the kind we have 

 been considering, so far as I am aware, is that of Schiaparelli, already 

 referred to, (Smithsonian Report of 1871, page 335.) As his work may 

 not be generally accessible, it is perhaps well to state what that geceral 

 relation is, which he discovered to exist between the numerical co-effi- 

 cients which we have called local weights. The notation of formula (88) 

 being used, it amounts to this: that if the formula is such as to give 

 exact results when applied to a series of the third or any lower order, 

 we shall have the two conditions — 



lH, + 2^k + 3H,-{. +mH^ = 



But if the formula is exact in the case of a series of the fifth or any 

 lower order, like our formulas (22) and (59), then to the above condi- 

 tions this third one is added — 



Vl, + 2U, + 3'h+ +mU,,^0 



If the formula holds good for a series of the seventh or any lower order, 

 as in the case of (23), we have a fourth condition — 



lHi + 2U,-^3U3-\- , . . . + m« Z^ = 



and so on for formulas of higher orders. 



INTENSITY OF MORTALITY. 



Writers on the law of mortality have often made use of what is called 

 the intensity or force of mortality, meaning thereby the ratio of deaths 

 to population at any given instant of age. If we consider a stationary 

 population, and denote by y the number of persons who annually attain 

 a given year of age or birthday, and denote by x the age in years, then 

 y dx will represent the number living at the exact age x, and —dy will 

 represent the number dying at that exact age. The intensity of mor- 

 tality then is — , 



dji 



^' ~ y dx 

 If it were possible to discover the true analytical form of the function 

 which expresses the relation between mortality and age, this quantity /x 

 would naturally be the essential element in it. But if the hope of dis- 



