METHODS OF INTERPOLATION. 303 



w+11 years, nud the difference P,„ — P,„ + lo between these two ennnier- 



ations will be the number of deaths vviiich have occurred out of the i)Oi»- 



ulation P,„ within the ten year period, if there has been no immigration 



or emigration, or if the immigration and emigration have been equal, so 



as to balance each other. ♦If we regard P^ and P,„ + lo nn representing 



the numbers annually attaining the exact ages m-\-A and ?» + 10.j, then 



p 

 the fraction — "^^^ will denote the probability that a person aged 



7» + A will live ten years. 



In the United States, however, the number of immigrants contin- 

 ually entering the country is so large as to becon^e very important in 

 this connection. Emigration from the country is comparatively snnill ; 

 but assuujing, for the sake of generality, that there has been a registry 

 kept of the ages of both immigrants and emigrants, let us denote by I 

 the number of persons who have entered the country during the ten- 

 year period, and who are of such age as to have been m and under t)i-\-i 

 years old at the time of the first census, and let I'j denote the number 

 of persons of similar age who have left the country during the same 

 period. Also let D be the nund)er of deaths which have occurred in the 

 country out of the excess I — E of immigrants over emigrants, ami let 

 Pm + ]„ denote the population returned as aged )u-\-l() and under )«-flI 

 at the second census. Then the portion of I — I] surviving at the second 

 census is I — E — I), and the difference P,„ + lo — (I — E — D) is equal to 

 that portion of the initial po})ulation P,„ which survives at the tinu' of 

 the second census. The probability that a i»erson aged w/ + i will live 

 ten years is therefore expressed by — 



P,„^io-(I-E-D) 



All the quantities involved in this fraction are known excepting the 

 deaths D ; and as this is a small number compared with the others, the 

 result will not be seriously affected if we compute the value of D, or, 

 what amounts to the same thing, comi>ute the survivors (1 — E — D), 

 by means of any good table of general mortality, considering separately 

 the excess of immigrants of the sui)posed age who have entered the 

 country in each one of the ten years. (See the Assurance IMagazinc for 

 April, 1SG7, page 2S1).) 



We can thus obtain the probability of living ten years for the middle 

 of every year of age throughout the whole term of life. If the statistics 

 of i)oi)ulatiou and migration are given in the first place by dec;!des or 

 other intervals of age, the numbers can be distributed among the single 

 years by means of (3) or sonu^ similar formula derivable from (2), (8), or 

 (C). On the other hand, if th(^ statistics are given for singh^ years, the 

 irregulaiities of the series can be diminished by using some formula 

 under the second method of adjustment. We may assnme, then, that 

 the probability of living ten years has been ascertained for the middle 

 of each single year of age, and that these probabilities form an a[)proxi. 



