IN determining the cost of farm operations one of the most difficult items to determine accurately is the rate of depreciatIOn of farm equipment. Recently Mr. H. H. Mowry, of the Office of Farm Management, who has col lected extensive data on the problem of depreciation of farm equipment, suggested to the writer the possibility of determining the average length of life of farm imple ments from data relating to the number of years, each implement has been used. Apparently a solution for this

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... lem has been found. The solution applies to all objects, either animate or inanimate, lasting for vary ing lengths of time. Two cases are to be considered, namely, (1) when the number of the objects under consideration is approxi mately constant from year to year, and (2) when their number is increasing or decreasing. The first case may be conveniently considered ·in its application to farm dwellings. Suppose that on a given gr oup of farms there is a definite number of farm dwellings of various ages, and that as fast as old dwellings become unsuited to their purpose they are replaced by new ones. For convenience of reference let us reduce the numbers with which we haye to deal to symbols. Let Nl represent the number of dweUings in their first year of life, N. the number in their second year, N I the number in their third year, and so on, Nn representing the number of dwellings of the oldest age represented in the group. In any group of objects which lasts for varying lengths of time but in which the number of objects is kept con stant by replacing discarded ones by new ones the follow ing principles apply: 1. The number of old objects discarded each year is, on the average, equal to the number of new ones intro duced. 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 sec ond 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 number 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 Nt, N" etc., which represent the number of objects now in their second, third, etc., 'years of life, may aiso be taken to represent the number of the objects now in their first year that will ultimately reach their second, third, etc., years of life. 4. If now we add together N 1, N 2, N I, etc., this is eq ui valent to counting each object now in its first year as many times as it will live years. Hence the sum of N 10 N I, Na, 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, it we divide the total number of objects of all ages in the group by the average 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 average is a general one and ap plies to the whole population. We may thus express the average le� of life of any constant population by means of the following formula: . Nl (A) 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 In this formula L equals the average length of life, P the total population, and N 1 the average number in their first year of life at a given time. In applying either of the above formulm to cases like those of farm houses and most kinds of farm implements the' fact �h8.t very few such objectS are discarded until they are a.t Jeast four or five years old makes N I, N 2, N 3, N( �d'Nl approXimately equal. That is, the number of obj!lCts one year old is about the same as the number two :veab 014, or thi-ee years, etc., up to about five years, and BOn?-etinies even longer. In making a study of Buch objects Witli a View'to deterIniiil lig the average length of their'liftHill!1 "usu�y possibie to get quite accfuately the nUln� of objects.iii the group in each year of life up t� five or SUt -ye'a.r8 of' age, and' where these numbers are about the same tor�ch year their averages will represent quite accurately the average number of new oojects introduced 'iii a year, wmch'is tlie Saine as' the average • Reproduced from &:Ietau number of old ones discarded. Hence, in populations where the number of objects in e�ch 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 numb e r of objects by the avera,ge nUmber in each of the ea.rly years of life. POPULATIONS THAT ARE DECREASING OR INCREASING. The principles sta.ted above do not apply In a popula tion 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 popula, tions. This may be done as follows: Suppose the rate of increase in population is 1 per cent e. year. Then it P represents the population in any one year, 1.01P will represent the population the next year. Likewise, if B represents the number of births in any year, tnen 1.01B 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 +r)B will represent the number of births the first year thereafter, (1 +r)2B the nUmber of births the second year thereafter, and (1 +r) n B the number of births the nth year thereafter. Returning now to formula (A), where Nl represe»,ts the number of individuals in the first year of life,.Nt the number in their second year, and so on, we have�y seen that in a constant population these num�rs bear such relation to each other that N. represents the num ber of the present Nl'S that will live to enter their second year. But in an increasing populam,on this' is not the case, tor the number of indivlduals born in the year in whic)l the present N.'s were born was. smaller than the number born in th� year in wHich the preSent N I'S were born-that is,' the number bOnit8.Bt year: is smaller than the number' born 'this year. Hence; in'·aD.' increasing population' N. is smaller than the number of N l'S that will live to' ente� their second year. But if we increase N. in proporti6n as the numPer !:lorn this year is greater than the number born last year, this increased value of N2 will represent the number of the present Nl'S that will live to enter their second year. If we let B stand for the number born in the year in which the present Nt's were born, then (1 +r)B will represent the number born the year the present N I'S were born, which of course is just one year later. The increased value of Nt, for which we are seeking. may now be found from the proportion B : (1+r)B :: N, : X, from which X = (1+r)Nt. In similar manner it can be !lhown that it we substitute for N I the expression (1 +r)1 N I this new value will repre sent the number of present Nl'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 = N1+Cl+r)Nt+(1+r)'Na+ ... +(1+r) n -1 N n . (C) Nl In this new formula the terms of the numerator repre sent, respectively, the number of the present Nt's that will live to enter the various years of life indicated by the subscripts after the N's. Hence the sum oC the terms of the numerator is equal to the sum of the ages the present N1's will reach at death, and the value of the whole fraction becomes the average length of life of the population. To use formula (C), which applies to populations that are increasing or decreasing at a constant .rate, r, we must know the number of individuals in each of the vari ous 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 possible, the average length' of life may be cl!lculated where the number of objects is increasing or decreasing at a constant rate per year. �t will be noticed that when r is equal to zero, which it is in a constant population, formula (C) becomes iden tic8J. 'with formula (A). ]"orini.ila: (C) a:pplies only to populat, ions in which the rate of.ilicrease or decrease is the' same from year to'year� It"is�possible to develop anothedormUla. for the average .length of life which is independent of the rate of increase and which therefor applies to any kind of popul�tion; no_matter .w:hat t]l�,J:ate of' increase or decrease, and whether this rate is the same from year to year or not. Let Ba represent the number of individuals born th� year the present N. individuals were born, and Bl the number born the present year. Then the proportion Ba:B1::N.:X, in which X is equal to (BIIBa)Na, gives a 'value which it used instead of N a makes the third term of the numerator of formula (A) represent the number or the present N I'S who will live to enter their third y()ar. The other terms of the numerator of formula (A) may be modified in similar manner, givingthe fomlula Nl which is applicable to all populations for which we have the following data: the number of individuals born each year since and including the year in which the oldest individuals now living were born, and the number of people of various ages now living. While formula ( D) has very wide applicability, its use fulness is greatly limited by the fact that it requires so large an amount of data which is usually difficult to obtain. Before applying any of these formulre It is necessary to eliminate the effect of immigratIOn and emigration. This means that only those indiVIduals should be con sidered whose whole life is to be spent as a part of the population und!>r consideration. In using any of the methods here presented in determirnng the average rate of depreciation of, say, a farm implement of a given kind, only those implements !tre to be counted that were . bought new (not second hand) and which will presumably be replaced when destroyed or worn out by new' ones. Note by the Editor of the SCIENTIFIC AMERICAN SUPPLEMENT. Formulre of the kind developed here by Mr. Spill man are calculated to prove very, valuable to the pro ducers and consumers of a great variety of commodi ties. Their utility is in fact limited only by the diffi culty which is likely to be experienced in obtaining the requisite 'data for substitution in such formulre. This difficulty may be considerable in dealing with cases lying in the past, but will vanish in the future, if·the need of the data is properly appreciated. It seems well worth while to go a step further than Mr. Spillman. has done, and to employ infi nitesimals in place of the finite intervals (of one year each) which he introduces. In place of the numbers Nl N.. etc., which appear in Mr. Spillman's formulre we must then introduce a coefficient of age-distribution c (a) which is defined as follows: Of the total population P, let the number whose ages lie between a and a + da, be given by P c (a) da. With this notation it can readily be seen that Mr. Spillman's formula (D) takes the form Bt P r a �C(a)da J 0 Bt_a L= --------Bt (1) or, since --is the annual birth rate per head, which P we may denote by bt r a �C(a)da J 0. " Bt_a L= (2) bt In the special case in which the population increases at a constant fractional rate r (geometric increase), we have Bt ___ = c ra (3) Bt-ll Hence a f. era c(a)da (4) L= -------b The correctness of this formula may be checked by comparing it wi� the form developed for c ( a) for a population increasing in geometric progression. This has been given by Lotka1 as follows: c(a) = b (Tn p(a) where p(a) is the probability, at birth, of r�aching age a. If :we substitute (5) in (4) we obtain a L = [ p(a)da which' is indeed the definition of' the mean length of life.