DE. FAEE ON THE CONSTEUCTION OE LIFE-TABLES. 847 
The inequalities in the second ditFerences vary in every separate class of observations ; 
u ere is generally a tendency in the first and in the second differences to increase' 
over a certam extent of the series. The error of the hypothesis is slight if the rate oi 
increase {r) of which X -00677 is the logarithm in the case in hand, is only assumed to 
remam uniform for the ten years 20 to 30, or for the one year 20 to 21. Now let the 
number living at toe age 20 be represented by l„, and the number living at the age 21 
by 4.: then put Here it is evident that if 4. and ji,. be known, 4, is deter- 
mmed immediately by the equation 4.=4. Xy>=.. But is the value of y in the equa- 
tnm y,_10" \ when 2 is put =1. Taking the numbers from Table A., w'e have 
m=-00765^t the precise age 20=(19i+20i)i ; and km=3-8835130; Ar=-0067728; 
modulus of the common logarithms! 
i^/P0b8b; k{\r) is the complement of the logarithm of (Ar). 
1-2755686 
\m 
3-8835130 
k(\r) 
2-1692317 
2-1963697 
--0033472 
3-5246830 
T-9966528 
As the factor (1-r) is negative it makes the exponent of 10 negative, and upon 
a ' ng the complement of this the logarithm of y is found to be 1-9966528. This is 
a so the logarithm of ^20 =-992 32 ; and it enables us to pass, in the construction of a Life- 
able, from the livmg at the age of 20 to the Ihdng at 21. If we obtain the several 
values^ at every year of age, the whole of the Life-Table can be constructed. 
It will be found that is always a fraction, and it does not differ very much from 
1 m But while m. * shows the deaths in a year out of a unU of life (which may con- 
of individual lives constantly kept up), shows how much out of a 
unit of the same life at the beginning of a year, the dead not being replaced, survives 
a ymr after the age and l~p^ is the amount of loss which occurs in the same vear 
ou JJ“t of life at Its commencement. Thus, as -99232, it follows that 
.no£?r ■ mortality is m,„ = -00771, or 
, onoQo^ If the unit of life is made 100,000 living at the age 20, 
then 99.32 will survive, and 768 will die in the ensuing year of age. But if it is 
assumed that the deaths take place at equal intervals, it may also be assumed that the 
number of lives (100,000) being constantly sustained, the accessions of 768 new lives 
ake place at equal intervals, consequently that they are under observation half a year 
on an average, giving the equivalent of ^ =384 years of lifetime at the age 20 to 21 ; 
* m serves to indicate the mean mortality in the year following the exact age a-. 
5 T 2 
