DE. FAEE ON THE CONSTEUCTION OF LIFE-TABLES. 
855 
IV. CONSTEUCTION OF THE COLUMNS d^, 4, L^, Q^, T^, AND NOTICES OF SOME 
OF THEIE PEACTICAL APPLICATIONS. 
Th.6 S6ri6s has bcGii coustructGd i and from that serios othors aiG deduced to com- 
plete the Life-Table, consisting now of six columns. 
(1.) 4“ 4+1 = number of deaths in the year of age following, out of 4 alive at the 
age w. By taking a; successively at 0, 1, 2, 3, ... . to the last age in the Table, the numbers 
dying in every year of age are obtained. The numbers dying of the age x and under the 
age 4+„ are immediately derived from the column 4; as (2.) 4-4+n=^^.+<^.+i • ■ • 
When x-^n>u=.i\ie oldest age in the Table, 4=f^^+^^^+i 
(3.) L^=4+4+i +4- The series is formed by the successive addition of the 
series 4, from 4 upwards. 
(3 a.) — L^+„=L^i„=4+4+i — +4+n-i- 
} “d (5) 
T*+ 1 = 4+1 ^d ^^ , = 4+2 + i^?^+ 1 < 
The series in column is constructed from the two columns 4 and d,, or from the 
single column 4, as 2P, = 4+4,,; and .*. P.=^-^, .-. 4=2P.-4+i; so, conversely, 
the series 4 can be constructed from the series P^. The P^ is assumed to represent the 
population, as expressed by the Life-Table, living at the age x and under the age x-\-l. 
Thus P 2 o= the population of the age 20 and under 21 years. 
By substituting the successive values of P^ in the equation (5a), P^-fP^+i ... P^+„, we 
have 14+4+1 — + 4 +n +'2 4+«+)- 
(^■) Qjr T^ + B^ + 1 + P»+2 P i'+n-l+P^+n +Pu 
Q^+» Pj^+« + Pi' + ra + l + P«+» + 2 • • • • +Pw" 
(7.) . . Qjr+„ Q+|«=Pa:+P^+i+P^+ 2 .. .. P^+„_,. The column is constructed 
by adding up the column P^, and transferring the successive sums to the column Q,. 
By substituting for the series P^ its values in 4? we have 
(8.) Q^=^4+4+i + 4+2 +4* 
And by again substituting for the series 4 its corresponding values in d^^ we have 
(9.) 
(10.) Thus IS equal to the numbers dying in each year of age after the age x, 
multiplied by the time (expressed in years and fractions of a year) that they have 
respectively lived over that age; and if x=z0, then Qo=id„ + lld, + 2^d,....(7i+^)d,^^, 
when {x-f-n) becomes >^y. 
(11.) This column Q, represents, therefore, two distinct orders of facts: it represents 
the sum of the number of years that will be lived after the age x by the 4 persons then 
Imng, and .*. -~= the mean after-lifetime; of which ^ will be enjoyed before the age 
x-\-n is attained, and — ^ after the age x-f-n is attained. At birth the mean after-life- 
, Qo , 
time IS -j-, the unit here being one year of individual life. 
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