METHODS OF INTEKPOLATION. 299 



then we have — 



w'=JJ21(S,+S,)-17(S, + S,)] . . . (30) 

 Tliis forumla gives an adjusted value tor any term iu series (./') by 

 means of the sums of the terms in tlie four nearest decades as given in 

 series {<j). For instance, at the age '.'>') tlie vahic obtained is — 

 u' = J^[2l (8.79G4+<J.UL'i:S)— 17 (S.5«4S+l>.'JG7li)] 

 = .88399 

 Cohiinn (//) shows the gradnated series, carried to as many places of 

 decimals as are needed in order to give live sigiiilicaut tigures. It is of 

 the eighth order, and the arithmt^tical means of the terms in the nine 

 decades 10-19, 20-29, &c., are ap])roxinrately equal to those in sei-ies (^■), 

 though not precisely so. This method of adjustment, however, has one 

 advantage, namely, that it enables us to divide a given series into a 

 large number of gronps, and make the graduated series of as high an 

 order as we please, without previously o1)taining formnlas like (E) and 

 (F), which require some labor when the number of grou[)s is increased. 

 If the number of terms in a group is other than ten, it will be easy to 

 find a corresponding formula similar to (30). When it is an odd num- 

 ber the formula will l)e derived from (13) instead of from (8). For ex- 

 ample, with eleven terms in a group we have — 



7Ji = }J2 = ll, fli^l 



and (13) becomes — 



«' = S.-A(S. + S,) .... (31) 



giving the adjusted value of a term by means of the sunjs of the terms 

 in the three nearest groups of eleven terms each. 



Series {h) shows a mujiunim at the age 12, and increases continu- 

 onsly thereafter. It terminates at the age 99, and must not be ex- 

 tended farther by the same law, for since {{/) is a series of an even order 

 with the hnal difference, Ja, negative, it will, if produced far enough, 

 diminish at both ends instead of increasing as the rate of mortality 

 does. The limit of old age is evidently not reached until one year after 

 the point where the i)robability of <lying within a year becomes nnity, 

 that is, certainty. The position of the limit is very doubtful. The old 

 Combined E\i»erience table places it at 100, the Carlisle table at 105, 

 the ICnglish Life Table No. 3 at 108, the French table of Deparcienx at 

 95, the tables of Duvillard and De Montferrand at 110, and the United 

 States census table of 18(J0 at lOG. Owing to the paucity of reliable 

 observations at the greatest ages, the termination of series (h), or that 

 of any other graduated table, mnst necessarily be somewhat artilicial. 

 This is not of much consequence in practice, for the chance of attaining 

 any age beyond 100 is so small as to make but little difference in the 

 valne of an assurance or annuity for a person in middle life. If we 

 assume 110 as the limit in the i)resent case, then from the three known 

 values of the probability for the ages 98, 99, and 109, the values for the 



