294 METHODS OF INTERPOLATION. 



These formulas give exact results wbeu applied to auy series of tbe 

 second order. 



Let us now make iiz an even number in formula (2), and assume as 

 before — 



«3 = »1, «3 = ^'l 



s 



and let y' represent wbat - becomes wben we take — 



then y' is tbe middle ordinate of tbe middle area S2, and we bave tbis 

 formula : 



Wben Si, S2, S3 denote stationary population living witbin tbree inter- 

 Aals of age, tbe two lateral intervals being of ??i years eacb, and tbei'r 

 middle points being eacb distant «i years from tbe middle point of tbe 

 middle interval, wbicb consists of 923 years, tben y' is an adjusted value 

 for tbe number of persons wbo annually attain tbe exact middle age of 

 tbe middle interval. Tbe simplest case is wbere we bave tbe populations 

 Wi, U2, 1(3, «4, living witbin four consecutive years of age, and take tbe 

 two middle ones as tbe middle group, and eacb of tbe otbers as a lat- 

 eral group ; tben — 



"1 = 1, «2=2, fli=f 



and (27) reduces to — 



2/'=-rV[7S.,-(S, + S:0J ) 



[(28) 

 =tL[7(m2+^v,)-(«, + «,)J ) 



For example, i( iii, ?/•>, «3, «4, denote stationary population living witbin 

 tbe ages 38 to 39, 39 to 40, 40 to 41, and 41 to 42, tben ?/' is tbe nund)er 

 annually attaining tbe age 40. And even if tbe population is not 

 stationary, but increases or diminisbes from natural causes or by migra- 

 tion, still, if Wi, U2, &c., denote tbe mean population living witbin tbe 

 ages named during a given number of years, tben y' will be tbe mean 

 nmnber annually attaining tbe age 40, as before. 



Adjustment formulas analogous to (13) and (27) can also be derived 

 from (8) by taking x=0 and n = l or 11=0. It can be sbown tbat (13) 

 and (27) are particular cases under tbese, so tbat all tbe special adjust- 

 ment formulas derived from tbem will give exact results wben rpplied 

 to series of tbe tbird order as well as tbe second. 



If in formula (8) we take ?ii = «2=0, tben — , "^ " , -- , and -^ will rep- 



iii 712 ^'2 "1 



resent ordinates to tbe curve, and may be denoted bj^ 2/1? Jhj i/.h '^^^^^ !/*• 



If we also take — 



a-=0, fl,=f, «2=i, 'J=l, S = 2<' 



tben (8) reduces to — 



u'=U^o{if2+y.)-{yi+i!.o] .*. . (29) 



