304 METHODS OF INTERPOLATION, 



inately adjusted series. The problem which remains to be solved is, to 

 find the probabilitj^ of living one year at each age when the above-men- 

 tioned probabilities of living ten years are given. 



It is an interesting point in rehition to the whole subject of graduation 

 of numerical series, that, instead of gradu;Uing a given series directly, 

 we can take a constant function of each term in it, thus forming a new 

 series, and, having graduated this, we can inversely derive from each of 

 its terms a graduated value for the corresponding term in the original 

 series. One consequence of this principle is, that if we take the loga- 

 rithm of each term in the given series, and divide the series of logarithms 

 thus formed into groups and graduate it by the tirst method, and then ' 

 take the numbers corresponding to the graduated logarithms, we shall 

 have a graduated series representing the given one, and possessing this 

 l^roperty, that the products of the terms in the assumed groups in it will 

 be severally equal to the products of the terms in the corresponding 

 groups in the given series. This is evidently the case, because the suras 

 of the lognrithms of the terms in the assumed groups are equal in the two 

 series. Furthermore, since the equation of the graduated series of loga- 

 rithms enables us to interpolate the sum of the logarithms of the terms 

 in any group when the sums of the logarithms of the terms in the assumed 

 groups are given, it follows that when the products of the terms, in any 

 assumed groups in a numerical series, are known, we can find, by interpo- 

 lation, the product of the terms in any other group, or any single term. 



Now let jhn + i.i l>m H-it,, Pm + 2!,i &'"•? dcuotc the probabilities of livingone 

 year at the exact ages w+i, w-fl.^, w + 2^, &c. The chance of living 

 through any one year of age is contingent upon having lived through the 

 years which precede it, so that the probability that a person aged ?«-f^ 

 will live two years is equal to the product j)m + i^Xi^m + u,7 <iiitl the proba- 

 bility that he will live ten years is equal to the continued product — 



I'm + >. X J^m + 11. X /^m + 2!,; X 'XPm + Sii 



It appears, then, that the probabilities of living one year at each age 

 form a series such that the product of any n terms taken in a group 

 is equal to the probability of living 7i years at the age corresponding to 

 the first term in the group ; and hence, according to the principles which 

 have been stated, we can find, by interpolation, the probabilities of liv- 

 ing one year when the probabilities of living ten years are known. 



Any twelve consecutive terms in a series will form three groups often 

 terms each, and fornuda (2) will enable us to find any single term by 

 means of the sums of the terms in the three groups. If we take — 



»J^ = 5?2 = ^'3 = 10, ai = rt3 = l, ?l = l, S = M 



then (2) reduces to — 



« = J [74So— 33(Si.fS3)4-4(S3— Si)x+4(Si-fS3— 2S,),r=] . . (33\ 



Let Si, S2, and S3 represent the logarithms of the probabilities of living 

 ten years at the ages7« + i, w-f lA, and w + 2^, respectively ; then if we 



