DE. FAEE ON THE CONSTEHCTION OF LIFE-TABLES. 
857 
and keeping them apart in a separate community, subject to a definite law of mortality. 
Any population living in the tabular proportions at each year of age may, for the sake 
of distinction, be called a normally constituted population. 
The ages of the population represented by the Life-Table amount, in the aggregate, 
to Yo years ; it is the aggregate number of years which they have already lived^ and, sin- 
gularly enough, it is also, if the law of mortality remain constant, the number of years 
which they will live. Thus persons in such a population have lived on an average 
Y 
^ years ; that is their siean age, and it is also their mean after-lifetime. Y, is the 
number of years that persons have lived over the age x ; and the mean age of such 
Y 
persons is their after-lifetime is 
The series is formed by successively adding up a series of the form 
commencing at the oldest age in the Table. 
(15.) .*. Yo=^Qo+Qi-|-Q 2 • fi-Qw? 
By substituting for Qo, for Qj, for Q 2 , and so on, their values in P„ it will be found 
that 
(16.) Yo=iPo+liPi+2iP2+3-|-P3 .... --i-(^i-j-i-)P^^ -]-(^_|_A)p^. 
(17.) But the mean age of the persons (Po) of the age of 0 and under 1 is nearly i; 
and so the series i, li, 2^, 3i, 4^, 5^, 01 . . . . (n-\-^} expresses nearly the mean age of 
all the persons in the first (Po), second (P,), third (P^), and (wH-l)th (PJ years of age, 
and so for all other ages; consequently the sum of the series (16) Yo is the sum of the 
ages of all the persons living contemporaneously, as they are represented in the Life- 
Table. 
In like manner it is shown that 
(18.) Y^=-|-P^-J-(l-|--i)P^,_^,-f-(2 + i)P^_^2 
is the sum of the number of years that the Q,, persons in the Table have lived over the 
Y 
age w. They have all lived x years ; and consequently gives their average age 
Y 
precisely as gives the average age of the whole community. 
(19.) It has been shown that expresses the number of years that 4 persons will 
live; in the same manner it may be shown that i expresses the number of years that 
persons will live; .-. (/,+4+,) persons will live (Q^ + Q,,^J years, .•. 
persons will live -|(Qj,+ Q^+i) years. And the same may be demonstrated for each 
successive value of x. 
But the sum of the series P.,, is the number of persons living of all ages. And 
the sum of the series i(Q^+Q^+i) is Y^=the number of years that Q^, persons will live ; 
Qj— mean after -lifetime of all the persons living simultaneously of the age x and 
upwards. Thus by the Table D, 4,899,665 persons are living contemporaneously; their 
. Yq 166209701 „„ 
mean age is T899665 ‘ “ 3d-92 years; and they will live on an average 33-92 years. 
