284 



METHODS 01* INTERPOLATION. 



A=j^^J2134 S3+9(Si+S5)-llG(S2+S4)] ^ 

 E=j^i34(S,-S,)-5(S5-S,)] 



D=^J(S5-Si)-2(S4-S2)] 



M^) 



"12 Ml'' 



This, in connection witli formula (12), enables ns to express tlie 

 sum S of any group of w terms in a series of the fourth order by means 

 of the sums Si, Ss, S3, S4, S5, of the terms in any five consecutive groups 

 of ?fi terms each. In case the given series is of a higher order than the 

 fourth, or irregular, we can find adjusted values for each term, and for 

 any given set of groups assumed these values will form a series of the 

 fourth order. If we take Wi=10, formulas similar to (3) and (5) may be 

 obtained, by which to interpolate numbers for each single year when 

 statistics of population and mortality are given by decades of age. 



Particular relations exist between the numerical coeflScients of Si, S2, &c., 

 in the values of the constants A, B, &c., in this and similar formulas. In 

 the expression for A, the factor +2131 belongs to a single quantity S3, while 

 the factors +9 and -^IIG belong each to two quantities. So we have— 



2131 + 2x9 - 2x110 = 1920 

 and 1920 is the numerical part of the denominator of the fraction out- 

 side the bracket. In the expression for B a different relation appears. 

 From the middle of the group S2 to that of S4 is a distance of two inter- 

 vals, while from Si to S5 there are four intervals. We have accordingly— 



2x31 - 4x5 = 48 

 and 48 is the numerical part of the denominator of the fraction without 

 the bracket. Similar relations are found in the expressions for 0, D, 

 and E, except that the totals are equal to zero instead of to the denomi- 

 nator of the fraction. 



Again, assuming six consecutive groups of equal extent, with a curve 

 of the fifth degree, whose origin of coordinates is at the point of division 

 between the third and fourth groups, and pursuing the same method as 

 before, we find that the six constants are — 



A=^j^J37(S3+S4)+(Si+Sg)-8(S2+S5)] ^ 



B=.^^2[245(S4-S3)+2(S6-Si)-25(S5-S2)] 



"180 7tr 



C=j^i7(S2+S5)-6(S3+S4)-(Si+Sc)] 



D=3g^,[ll(S,-S2)-28(S4-S3)-(SG-Si)] 

 E=^,[2(S3+S4)+(Si+So)-3(S2+S5)] 

 _L_^.[10(S4-S3)+(S6--Si)-5(85-S2)] 



F= 



(D) 



J 



