METHODS OF INTERPOLATION. 295 



Hero ?/i, ?/2, ?/:!, and 7/4 are four equidistant ordinates to a curve of the 

 third or any h)wer order, and n' is the area between the twoniiddh' ordi- 

 nates. Hence, when the mean numbers of i)ersons annually attaining 

 ea(;h of four consecutive ages are known, the mean iJOpulation living 

 between the two middle ages can l)e comi)uted by this formula. For 

 instance, if ^i, i/-2, y?., and ^4 denote the numbers annually attaining the 

 ages JS), 40, 41, and lU, then w' is the population living between the ages 

 40 and 41. 



Let us now make a practical apjdication of the second m«4hod of 

 a<ljustment, in graduating the iriegular rate of mortality given in column 

 {(■i) of Table XL This is a new experience table (juite recently ])ublished 

 in England in an unadjusted form. It is jn'obably correct in its essential 

 features, and suited ibr practical use, having been prepared by the Insti- 

 tute of Actuaries, from the experience of twenty British life insurance 

 companies, all of which had been in existence more than twenty years, so 

 that the average duration of the policies observed was about nine years. 

 The original publication not being at hand, the data have been taken 

 as they are given in the Massachusetts and New York State Insurance 

 lte[)orts of 18G9. The probabilities of dying within a year at ea'di 

 age, according to these data, and multiplied by 100, are as they stand 

 in column (//), for the ages 18 to 01 inclusive. The original series ex- 

 tends froiu the age 10 to 90, but a few of the earliest and latest terms 

 show such irregularities as to be evidently worthless for the ])nrpose of 

 graduation. This is owing to the paucity of observations at those ages. 

 There were no deaths at all at the ages 11, 10, and 94, and no survivors 

 at the age 97. The eight terms from 10 to 17 are therefore rejected 

 here, and their jdaces supplied b}' others taken from the English life- 

 table, No. 3, for males, reduced a little to correspond with the new rate. 

 The sum of the terms for the eight ages 18 to 25 is 5.1802 by the new 

 table, and is 0.G775 by the table No. 3. Accordingly, each of the first 

 eight terms in series [d) is taken from the table No. 3, but diminished 

 in the ratio of GG775 to 518G2. The eight last terms, from 92 to 99, 

 have been obtained in a similar way, using the sums of the terms for 

 the eight ages next i^receding, so as to increase the values given by 

 the table No. 3 in the ratio of 1843G to 1845G. Series {d) thus com- 

 pleted, has been approximately adjusted by means of Ibrmuhi (20), 

 which reaches all the terms excei)t the tirst lour and last four. The 

 result is given in column (c). For instance, at the age 30 the adjusted 

 term is — 



«5=-iJ8.2341 + 7(.74000+.72027) + 4(.77808+.83G35) 

 -f (.G5324-f .83200) — 2(.G9197 + .87340)] 

 =.77770 



At the ages 13 and 9G the adjustment has been made by formula (18), 

 at the ages 12 and 97 by (10), at 1 1 and 98 by (2G), aiul at 10 and 99 by 

 (25). Todiunnish some irregularitiesstill existing in series (c), the adjust- 

 meut has been repeated, only this time formula (IG) was used throughout. 



