DR. FARR ON THE CONSTRUCTION OF LIFE-TABLES. 
841 
this is more evident when they are exhibited on the larger scale of Plate XLIL 
fig. 2. The decrement in the first year is large ; in the first five years the decrements of 
life are considerable; at the age of 10 to 15 they fall to their minimum; slowly increase 
to the age of 56 ; increase more rapidly until the maximum is attained at the age of 75 ; 
then decline gradually to 85, and after that more rapidly until every life is extinct at 
the age 107 by this Table. 
II. PRINCIPLES OF CONSTRUCTION. THE FUNDAMENTAL COLUMN L. 
The conditions of the hypothesis upon which the preceding reasoning rests are never 
precisely realized in nature ; in the first place the number of births fluctuates, increases, 
01 decreases from year to year, and the deaths fluctuate still more ; rarely equalling the 
births in number. Immigration and emigration interfere. Under these circumstances, 
Tables such as those which Halley, Pbice and others made from the observations 
on the deaths alone are never accurate, and require correction to give approximate 
results. If it be assumed that the law of mortality remains invariable, and that migra- 
tion does not interfere, then the nature of the correction to be applied to a Table 
framed from the deaths alone will become immediately apparent by an example. The 
births increase in England. Let the annual births in a portion of the community be 
doubled in sixty years, thus be 50,000 in 1796, and 100,000 in 1856; then the deaths 
of persons of the age of 60 in 1856 must be doubled to obtain the deaths which 
would have happened at that age if the annual births sixty years before these deaths 
had been 100,000. If the births have been accurately registered, formulae for correct- 
ing the ordinary Table drawn up from the deaths at different ages will be suggested by 
the above considerations. 
I now proceed to describe another method which has been adopted in framing the 
Table C, and is applicable wherever (1) the number of annual births, (2) the numbers 
of the population living at definite periods of age, (3) the deaths at the corresponding 
ages during a certain number of years, in any community are ascertained by observation. 
This method is not open to the preUous objections. 
The aim is to obtain equations which wiU describe the curve lines (Plate XLII. 
fig. 1) of the Life-Table, in the most direct way ; and these equations may be deduced 
from the determined rate of mortality at certain intervals of age. 
The relative numbers living at two ages, 20 and 21, can evidently be found from an 
equation which expresses the relation of the average numbers living and dying between 
those ages during a given time. This can be determined very nearly ; for although the 
ages of the living are not ascertained with exact precision at the census, still by taking 
all the numbers living at the ages 15, 16, 17 years up to 24 and under 25, together, 
the aggregate represents very nearly the numbers living in that decenniad of life. The 
deaths at the same ages are obtained with at least equal accuracy from the registers of 
deaths. By this process, and by extending the observations over five or more years, a 
number of facts is obtained sufficiently great to yield average results ; and it may be 
