METHODS OF INTERPOLATION. 309 



is +.r' there sliall be a group of au equal uuuiber of terms whoso 

 abscissa is — x'y and vice versa. 



Cases will often occur where the. whole number of terms in a .series is 

 not an exact multiple of the number of groups we wish to assume, and 

 therefore will not form the desired number of consecutive groui>s con- 

 taining each an equal and entire number of terms. But it is not neces- 

 sarj^ that the number of terms in a group should be a whole number. 

 If we suppose it to have a fractional part, then certain terms in the 

 given series must be divided eacli into two portions, and each portion 

 must be joined to its proper group. Every such term being geometric- 

 ally represented by an area whose base is unity, and the two parts into 

 which this unit is divided being known, the problem is, to divide the 

 area into its two corresponding parts. We can often do this accurately 

 enough for practical purposes by assuming that the two portions of the 

 area are proportional to the two portions of the base ; but a much closer 

 approximation will be made by taking the term in question and the two 

 others nearest to it as data for an interpolation by formula (A). Let S,, 

 S2, S3, be the three terms, and let n denote the first one of the two parts 

 into which the base of S2 is divi<led; then if we take — 



^^1 = 1, x=-),{l-n) 



formula (A) reduces to — 



S=^[2Si+5S2-S3+o(S2-Si)« + (Si+S3-2S2);r] . . . (39)* 



where S is that portion of the area S2 which corresponds to the first 

 fractional part of the base. The other portion is of course S2 — S. For 

 example, if we wish to divide the ninety terms of series {/) into seven 

 consecutive groups of an equal nund)er of terms each, the number of 

 terms in a group will be -Y-=12|. The sum of the terms in the first 

 group will be composed of the twelve terms for the ages 10 to 21 inclu- 

 sive, together with so much of tlie term for the age 22 as corresponds 

 to the fractional interval n=^j. The three terms for the ages 21, 22, and 

 23 are— 



Si=. 07530, 8.= . 08445, S,= . 07104 



and formula (39) gives for that part of Ss which belongs to the first 

 group the value S=.58095, and the sum of the terms in the first group 

 is tlicrefore 0.42804. The jjortion S^ — S = .0975U belongs to the second 

 group. After the sums of the terms in all tlie other groups have been 

 formed in the same way, the etpiation of a graduated series of the sixth 

 order can be obtained by means of formuhii (E), just as when «i is a 

 whole number. The accuracy of this iast part of the work can l)e tested 

 by the condition that the sum of all the terms in the graduated series 

 must be precisely equal to the sum of all the terms in the original 

 series (/). 



* This foriimlu can p.Iko 1)o -writtuu — 



s^^-^(s. + 8.+«A,-l=:i!:A,) 



where Aj and Az are the Guitc difl'crence.s of the series Si, 8;, S3. 



