566 



SCIENCE 



[N. S. Vol. XXXVI. No. 930 



number of dwellings of the oldest age repre- 

 sented in the group. 



In any group of objects which last for 

 varying lengths of time but in which the num- 

 ber of objects is kept constant by replacing 

 discarded ones by new ones the following 

 principles apply: 



1. The number of old objects discarded each 

 year is, on the average, equal to the number 

 of new ones introduced. 



2. The average number of objects in the 

 second year of their life at a given time is 

 equal to the average number of those in their 

 first year that will live to enter their second 

 year. The average number in their third year 

 is on the average equal to the number of those 

 in their first year that will live to enter their 

 third year, and so on. In general, the num- 

 ber of objects in their nth year is equal to the 

 number of those in their first year that will 

 live to enter their nth year. 



3. Hence N,, N^, etc., which represent the 

 number of objects now in their second, third, 

 etc., years of life, may also be taken to repre- 

 sent the number of the objects now in their 

 first year that will ultimately reach their sec- 

 ond, third, etc., years of life. 



4. If now we add together N^, iV,, N,, etc., 

 this is equivalent to counting each object now 

 in its first year as many times as it will live 

 years. Hence the sum of iV,, N^, N,, etc., 

 which represents the total number of objects 

 of all ages, also represents the sum of the ages 

 that will be attained by all the objects now in 

 their first year. 



5. Therefore, if we divide the total number 

 of objects of all ages in the group by the aver- 

 age number in their first year the quotient 

 will be the average length of life that those 

 now in their first year will live. But since 

 the average number of objects in their first 

 year is the same from year to year, this aver- 

 age is a general one and applies to the whole 

 population. We may thus express the average 

 length of life of any constant population by 

 means of the following formula: 



This formula may be expressed more simply 

 by writing for the numerator simply the total 

 population instead of the sum of individuals 

 of different ages. We thus have 



P 



(S) 



L = 





(^) 



In this formula L equals the average length 

 of life, P the total population, and 2Vj the 

 average number in their first year of life at 

 a given time. 



In applying either of the above formulae to 

 eases like those of farm houses and most kinds 

 of farm implements the fact that very few 

 such objects are discarded until they are at 

 least four or five years old makes N^, N„, N^, 

 N^ and iV, approximately equal. That is, the 

 number of objects one year old is about the 

 same as the number two years old, or three 

 years, etc., up to about five years, and some- 

 times even longer. In making a study of such 

 objects with a view to determining the aver- 

 age leng-th of their life it is usually possible 

 to get quite accurately the number of objects 

 in the group in each year of life up to five or 

 six years of age, and where these numbers are 

 about the same for each year their averages 

 will represent quite accurately the average 

 number of new objects introduced in a year, 

 which is the same as the average number of 

 old ones discarded. Hence, in populations 

 where the number of objects in each of the 

 earlier years of life is approximately the same, 

 the average length of life in the population 

 may be obtained by dividing the total number 

 of objects by the average number in each of 

 the early years of life. 



POPULATIONS THAT ABE DECREASIING OR LNCKEAS- 

 ING 



The principles stated above do not apply in 

 a population that does not remain constant 

 from year to year. It is not difficult, however, 

 to work out a formula based on formula (A) 

 above that does apply to such populations. 

 This may be done as follows: 



Suppose the rate of increase in population 

 is 1 per cent, a year. Then if P represents the 

 population in any one year, I.OIP will repre- 



