METHODS OF INTERPOLATION. 279 



are constautly used as data, tlie single terms interpolated from tliein 

 will themselves form a series of tlie second order. Assuming auytliree 

 groups of terms in any given series, regular or irregular, we can con- 

 struct a new series of the second order, such that the arithmetical means 

 ^f the terms in the three corresponding groups in it shall be severally 

 equal to those in the given series. 



In the special case in which the three groups are consecutive, and con- 

 tain Hi terms each, taking formula (1), which expresses the sum S of any 

 n terms in a group, the abscissa of the middle point of the group being 

 a', we may assign to x' its three values — Hi, 0, and -j-Wi in succession, 

 obtaining the three equations — 



S2-=»i(A-f-LC'«i-) 



S,=»i(A+C«i+ifC»,2) 



These suffice to determine the three constants A, B, C ; and dropping 

 the accent from x' in (1), we have — 



A=^[2CS,-(S,+S,)] 



S = n (A + ^i^C )f-+B x-i- C a-) 



This can be used in place of the more general formula (2), in all cases 

 where the three groups are consecutive and of equal extent. 



We have here a means of ajiproximating to the population living 

 within each single year of age when the statistics are given by decades 

 or other intervals of age, as is Often the case in census re])orts. If we 

 take ??i=10, and let u represent what S becomes when n=l, then form- 

 ula (A) will reduce to — 

 « = sc?oo[86GS3-33(S, + S3) + 40(S3-Si).r+4(Si-fS3-2S.,),r2] . . (3) 



If, for example. Si, Sg, S3 are the numbers aged 30 and under 40, 40 

 and under 50, 50 and under GO, respectively, then giving *• the values 

 — ■!, + },+!, &c., in succession, the resulting values of u will be the num- 

 bers aged 44 and under 45, 45 and under 4G, 4G and under 47, &c. If 

 instead of taking n=l we take n=i^ or «=^, then by assigning the 

 proper values to x we may find the poi)ulation living within any desired 

 half year or quarter of a year of age. (See Milne on Annuities, Vol. 1, 

 Ch. 3.) The same formula (3) enables us to distribute among the single 

 years of age the deaths which occur within any three consecutive de- 

 cades of age during a given period of time. If the population or deaths 

 were ihus distributed within every decade by means of the totals for 



(A) 



