846 
DE. EAEE ON THE CONSTEECTION OF LIEE-TABLES, 
Upon passing to the numbers, equation (4.) becomes 
=the value of y (taken as 1 at the origin) at the end of z years. 
Let \ denote the common logarithm with the base 10; then where 1c is the 
modulus of the common system of logarithms ; as also 
Equation (2.) becomes, 
km - 
Ac=— , and 
® Kr ’ 
mr^ kmr^ 
KiV Kr 
after the required substitutions. 
and 
Ky km kmr^ 
k~~ Kr Kr 
k^m.^ , 
(5.) 
so the equation becomes finally y. 
This is the form given by Mr. Edmonds, and is convenient for use. 
( 6 .) 
By making 2 successively 1, 2, 3, up to any number less than the number 
of years of age within which r remains constant, the number being known, the 
number living at any other age within that range will be obtained by multiphfing by 
the corresponding value of y. Thus, if y^^ is the value of y when 2=10 in equation (6.); 
then putting Zjo foi’ the numbers living at the age 20, the living at the age 30 '^ill be 
3/10 X ^20“ ^ 30 - 
This hypothesis does not express the facts deduced from the observations exactly. 
If could be expressed exactly over more than 20 years by m^— 111 ^ 1 '^, the first differ- 
ences (^*) of the logarithms in the series following would in a certain number of cases 
be equal. 
Females in Healthy Disteicts of England. 
Precise age. 
Annual rate of 
mortality. 
Logarithms of the 
annual mortality. 
First decennial 
differences of 
Krtix. 
Second decennial 
differences of 
Xmj.. 
m*. 
\m. 
P. 
20 
•00765 
3-8835 
•0677 
-•0197 
30 
•00894 
3-9512 
•0480 
•0290 
40 
•00998 
3-9992 
•0770 
•1817 
50 
•01192 
2-0762 
•2587 
•1047 
60 
•02162 
2-3349 
•3634 
•0126 
70 
•04992 
2-6983 
•3760 
— •0236 
80 
•11866 
1-0743 
•3524 
-•1259 
90 
•26711 
1-4267 
•2265 
100 
•45000 
1-6532 
* Here, at the age 20, m is the mean mortality that rules over the age 19|- to 201 years of exact time. 
