280 METHODS OF INTERPOLATION. 



that decade aud tlie two others nearest to it, the result would be a chain 

 of sub-series of the second order extending throughout the term of life, 

 but not forming a well-graduated series, because in general it would 

 not be continuous at the points of junction between the decades. It 

 might, however, be made approximately continuous afterward by means 

 of the second method of adjustment, which will soon be explained. We 

 must observe, too, that at the ages before 20 or after 80 the population 

 and deaths vary so rapidly, that, in order to secure a good distribution 

 by these methods, the data for those ages ought to bo given by intervals 

 of five years, or some other number less than a decade. In the ages of 

 infancy they should be given for each single year. 

 Eeverting now to the general formula (2), we observe that the quan- 



Q a g Q 



titles - — \ — , — , are the mean values of the ordinate within the 



W' Oii^ M2' 713 



several areas, so that the formula enables us not only to interpolate the 

 arithmetical mean of a group of n terms in a series when the means of 

 the terms in three other groups are known, but also to interpolate the 

 mean value of a function within any interval n when its mean values 

 within three other intervals ih, »2, ^3, are known ; so that if we know 

 the mean annual rate of mortality for three consecutive decades of age, 

 we can find the mean rate for each single year of age by formula (3), 

 since Si, S2, S3, are simply ten times the given mean rate for their 

 respective decades. 



When any one of the intervals Wj, «2, ih or n is diminished, the mean 

 value of the ordinate within such interval will evidently approximate to 

 the value of the middle ordinate of the interval, and will become equal 

 to it at the limit, when the interval becomes zero. Hence, making w= 0, 



we have — for the ordinate corresponding to the abscissa x, and (2) 



K-^Xi:XrTX^>(iX^) • • • <^> 



When Si, S2, S3, denote the population living within given intervals of 

 age, the area 7/ f7,r may be regarded as denoting the number living at 

 the'fexactage indicated by a;, and if the population is a stationary one— 

 that is, neither increasing nor diminishing, the product n'y will repre- 

 sent the number of persons w^ho attain that exact age during the interval 

 of time 01' ; so that when the ages are grouped by decades, and we 

 have «= 0, formula (A) will give for the number of persons who annually 

 attain the age indicated by x, since n' is nuity, 

 2/=i3(7(n7[650S2-25(Si+S3)4-30(S3-Si)j? + 3(Si+S3-2S2)J?2] . (5) 



For example, when Si, S3, S3, denote the population aged 30 and under 

 40, 40 and under 50, 50 and under GO, respectively, if we assign to x the 

 values — 1, 0,+l, &c., in succession, the resulting values of y will be the 

 numbers annually attaining the ages 44, 45, 46, &c. It has usually- 

 been the i)ractice to consider these numbers as being represented by 



