October 25, 1912] 



SCIENCE 



567 



sent the population the next year. Likewise, 

 if B represents the number of births in any 

 year, then I.OIB will represent the number 

 the next year. In general, if B represents the 

 number of births in any year and r the annual 

 rate of increase in population, then (1 + f)B 

 will represent the number of births the first 

 year thereafter, (l + r)^J5 the number of 

 births the second year thereafter, and 

 (l-\-r)'^B the number of births the nth year 

 thereafter. 



Eeturning now to formula (A), where N^ 

 represents the number of individuals in the 

 first year of life, iV, the number in their sec- 

 ond year, and so on, we have already seen that 

 in a constant population these numbers bear 

 such relation to each other that N, represents 

 the number of the present N^'s that will live 

 to enter their second year. But in an increas- 

 ing population this is not the case, for the 

 number of individuals born in the year in 

 which the present NJs were born was smaller 

 than the number born in the year in whish the 

 present N^'a were born — that is, the number 

 born last year is smaller than the number 

 born this year. Hence, in an increasing pop- 

 ulation N, is smaller than the number of N^'s 

 that will live to enter their second year. But 

 if we increase N^ in proportion as the number 

 bom this year is greater than the number born 

 last year, this increased value of N, will repre- 

 sent the number of the present N^'s that will 

 live to enter their second year. 



If we let B stand for the number bom in 

 the year in which the present N.'s were born, 

 then (1 -\- r)B will represent the number born 

 the year the present iV/s were born, which of 

 course is just one year later. The increased 

 value of N„ for which we are seeking, may 

 now be found from the proportion 

 B : {l+r)B -.-.N.-.X, 



from which 



X= (1 + r)N„. 



In similar manner it can be shown that if 

 we substitute for N^ the expression (1 + rYN^ 

 this new value will represent the number of 

 present N^'s that will live to enter their third 

 year, and so on for all of the various N's in 



the numerator of formula (A). This gives us 

 L = 



+ ...-t-(14,,.)n-lJV„ (C) 



In this new formula the terms of the nu- 

 merator represent, respectively, the number of 

 the present iV^'s that will live to enter the vari- 

 ous years of life indicated by the subscripts 

 after the iV's. Hence the sum of the terms of 

 the numerator is equal to the sum of the ages 

 the present iV/s will reach at death, and the 

 value of the whole fraction becomes the aver- 

 age length of life of the population. 



To use formula (C), which applies to pop- 

 ulations that are increasing or decreasing at 

 a constant rate, r, we must know the number 

 of individuals in each of the various years of 

 life at the present time and the annual rate of 

 increase or decrease in population. Such data 

 are usually not available except in the cases 

 of human beings in restricted areas where 

 births and deaths are accurately recorded. 

 In some cases, however, it may be possible to 

 obtain data of this kind concerning a class of 

 articles of farm equipment. When this is pos- 

 sible, the average length of life may be calcu- 

 lated where the number of objects is increas- 

 ing or decreasing at a constant rate per year. 

 It will be noticed that when r is equal to 

 zero, which it is in a constant population, 

 formula (C) becomes identical with for- 

 mula (A). 



Formula (C) applies only to populations in 

 which the rate of increase or decrease is the 

 same from year to year. It is possible to de- 

 velop another formula for the average length 

 of life which is independent of the rate of in- 

 crease and which therefore applies to any kind 

 of population, no matter what the rate of in- 

 crease or decrease, and whether this rate is 

 the same from year to year or not. 



Let ^3 represent the number of individuals 

 born the year the present N, individuals were 

 born, and B^ the number born the present year. 

 Then the proportion B,:B^::N,:X, in which 

 X is equal to (B,/BJN„ gives a value which 

 if used instead of N^ makes the third term of 

 the numerator of formula (A) represent the 



