306 METHODS OF INTERPOLATION. 



number of persons Tvbo are born. It is quite possible, too, tbat tbe 

 form of the series may be imj)roved by this mode of procedure. 



The foregoing method of reduction will evidently apjilj' to cases where 

 the interval between the two census enumerations is any whole number of 

 years other than ten, or even a fractional number. Suppose it to be ten 

 and one-half years for instance, and take — 



7Jl = 7?2=«3=%S «i = «3 = l, 11 = 1, S = u 



then formula (2) reduces to — 



S=ToW970S2-437(Si+S3)+48(S3-Si)j^+48(Si+S3-2S2)^] . . (35) 



Let Si, S2, and S3 be the logarithms of the probabilities of living ten and 

 one-half years at the ages m+^, wi+li, and m+2i respectively; then 

 if we assign to x the values — f, — ^, + f, «S:c., in succession, the result- 

 ing values of u will be the logarithms of the probabilities of living one 

 year at the ages m+S, 7n-j-6, w+T, &c. When x= — I, the formula 

 becomes — 



logi>u. + 6=5k(482S2-211Sx-223S3) . . (3G) 



from which values of log^ can easily be found for all but the extreme 

 ages of life. 



If the interval is either exactly or approximately an odd number of 

 years, there will be a slight advantage in deriving the formula of reduc- 

 tion from (S) rather than from (2). Suppose, for instance, that the second 

 census is taken five years after the first one. In the series of logarithms 

 of the probabilities of living one year at each age, any eight consecu- 

 tive terms will form four groups of five terms each, and formula (8) will 

 enable us to find any single term by means of the sums of the terms in 

 these groups. If we take — 



w.=W2=5j <^i=fj (f2=h n=l, S=M 



then (8) reduces to — 



:1 [17 (S2+S3)-9(Si+S4)J + J^[405(S3-S2)-103(S4-Si)] 



+ ^[(Sl-fS,)-(S2+S3)]+j^[(S4-Si)-3(S3-S2)] 



(37) 



Let Si, S2, S3, S4, denote the logarithms of the probabilities of living 

 five years at the ages w-f i, w-f 1^, wi+2.^, ?»+3.^, respectively ; then if 

 X takes the values —1, 0, +1, &c., in succession, the resulting values of 

 u will be the logarithms of the probabilities of living one year at the 

 ages ?«+3, w-f-l, w+o, «S:c. For x=0 we have the simple formula — 



logi>,n + 4=«V[17(S2+S3)-9(Si + S4)] . . . (38) 



which affords a ready means of determining log p for all the birthdays 

 except the extreme ones of childhood and old age. 



