METHODS OF INTERPOLATION. 28o 



T\"lien the siiius Si, S2, S3, S4, denote popnlatioii living- or deaths occur- 

 riiiii: within four consecutive decades of age, and n denotes the numbers 

 for IX single year of age, then we have — 



and consequently — 



+^rn^,[-^^'(«^-S.)-100(S,-S0], 



X- 



+ innn[(^i + »^)-(»^ + ^:>)] 



■(0) 



40(K> 



.[(S,-S0-3(S3-S,)] 



' OOOOO' 



"When the values of A, B, C, D, are substituted in the equation of the 

 curve, the number of persons who annually attain the age indicated by 

 w is expressed thus : 



2/=i^[7(SH-S.)-(S: + S,)]+j|^J15(S3-S,)-(S,-S0J^ 



(10) 



These last two formulas may be used instead of (3) and (5) when 

 greater accuracy is desired. It will be easy to obtain similar ones for 

 cases in which the ages of a population are taken by intervals of five, 

 twenty, or any other number of years. 



Let us now assume five or more groups, with a curve of the gci^eral 

 form — 



7/=A4-B ,r+C ,x-+J} ,r+E .v'+F .c^+G *«+H .r^+&c. 



and, to make the case as simple as possible, let the groups be consecu- 

 tive and composed of iii terms each. The sum of any n terms in a 

 group will be — 



ydx 



which, arranged according to the powers of ir, is — 



S = 4 A+ J,C li'-Vi^ «^+7-{^G ^^H (B + iD n^+^\-F n^+ JjH if')x \ 



+ (C+AE7t2+3i.G7i^).t^^+(B + F^^'+Tyi^^V+(E+f<^^«V > (12) 

 + (F+|II «2).c5+G .«G+n *'^+&c.] ) 



If we assume only five groups, the series will be of the fourth order, the 

 constants F, G, &c., will be zero, and by substituting for x in fornuda 

 (11) the five values — 2«i,— «i, 0, -\-ni^ and +2ui, in succession, and put- 

 ting 111 for «, we shall ol)tain five equations by which to determine the 

 five constants as follows: 



