298 METHODS OF INTERPOLATION. 



The foregoiniif metliod affords a ready means of diiuinishing the irregu- 

 larities of a series witliout removing them altogether. It can be proved 

 that in a series of the mth order, if any >n +1 or more eonseeutive terms 

 are adjusted by any single formula, such as (10) or (liO), the adjnsted 

 values will themselves form a series of the mth order. But, although the 

 order of the series remains unchanged, the absolute values of the ditter- 

 euces are in general diminished, and thus an approximate graduation is 

 secured. 



THIRD METHOD OF ADJUSTMENT. 



The second method can be combined with ordinary interpolation in 

 such a way as to fnrnish an adjusted series of any given order, extending 

 to any desired number of places of decimals. For example, let the terms 

 of series (/) in Table II be grouped together by decades of age, as was 

 done in forming (c) in Table I. The ninety terms form nine groups of 

 teu terms each. Their sums are — 



Si=4.r)0r)21 84 = 12.20.340 St= 05.22130 



52 = 0.80r)8 1 S5 = 20.80420 S^ = 199.05500 



53 = 8.79041 Se = 42.51440 Sg = 333.99100 

 These nine values tx)rm a series which has eight orders of differences, 

 as follows: 



Ji= 2.30000 J3 = 1.78039 ^ = 2.38714 J, = -10.70885 

 J2 = —.31000 J4 = 1.87103 Jg = 3.44040 Js=- 7.75719 

 Using the ordinary formula for interpolation by finite differences, we 

 can obtain nine equidistant values between every two terms of this 

 series, so as to make 81 terms in all, forming a perfectly graduated 

 series of the eighth order. The terms of this series are approximately 

 the sums of the terms in (/) for every possible decade of age, commencing- 

 with 10 to 19, 11 to 20, 12 to 21, &c., and ending with 90 to 99. To (con- 

 struct the series, nine consecutive terms were carefully computed, their 

 eight orders of differences were taken, and the rest of the series was 

 constructed therefrom by simple^additions and subtractions. One great 

 advantage of this mode of procedure is, that the agreement of the 

 values thus found for the decades 10-19, 20-29, &c., with the given 

 values S], S2, Sec, furnishes a convenient test of the accuracy of the 

 whole work. It is necessary, however, to carry out the values of the 

 function and the differences to a large number of places of decimals, 

 otherwise the error represented by th(5 neglected figures will accumulate 

 so as finally to vitiate some of the results. In the present case, the 

 decimals were carried out as far as they would go ; that is, to twenty 

 jjlaces. 



The series is readily extended by the same law, so as to comprise all 

 the possible decades of age from 4-13 to 90-105. Thus completed, it is 

 given in column (g). Now let Si, 83, 83, 84, be any four consecutive 

 terms in it, and in formula ( 8 ) take — 



Wi = «2 = 10, «i = |, 02=^, ir=0, 'n^l, 8 = m' 



