851 
DE. FAEE ON THE CONSTEUCTION OF LIFE-TABLES. 
In constructing the Life-Table, x was made 10 from the age of 20, and on inserting 
the numbers, the equations (2, 3, 4, 5, 6) became 
"J* M 40 4m3o 'I' 6^20 4mio Uf. 
— + — + 
10,000 ' (7.) 
M 30 3^20 3Mjo Ur) 
)i3_- + - + "-TOIV. 
Tddo +^^ 2 ^ (8.) 
~i~ ^20 2mjq "j" Uq 
wo ;9^’I44F ( 9 .) 
^10 ^0 
® 1 12s - 21s * 
The checking equation is 
«ho= + Wo ^ 40^‘ + 780^^ + 9880^^ + 91390^^ (H.j 
If three orders of differences are used, the checking equation is 
W30= X Uo 1 30§> + 435^^ + 4060^^ (12 ) 
•After adding or subtracting any constant to or from a series of numbers, the differences 
remam the same ; and if consecutive terms are multiplied or divided by the same factor, 
the differences are multiplied or divided by that factor. Thus (b-]-a)~(c-\-a)~b — c, 
and ab — ac=a(i — c). Advantage is taken of these properties to reduce any one of the 
terms in the equations to zero. 
Thus let the logarithms to be interpolated be the following— values of j,.., and 
taken from the column headed males. Table B; then they may, among other ways, 
be interpolated as follows : — 
As 1-9969724 is the contracted expression of (-9969724 — 1), we have 
Age 
20 
30 
40 
50 
1 - 9969724 = — -0030276 
1 - 9964260= --0035740 
1-9959051 = --0040949 
r -9943048 = — -0056952 
(1) Multiplying each term by 10,000,000, 
that is, striking out the decimal point 
and the two adjoining ciphers, and (2) 
then subtracting from each 30,276, the 
values of to be operated on 
become 
Wo = -00000 
Wjo^^- 5464 
W2o=- 10673 
«^3o= -26676 
By inserting these values with their negative signs in the equations, and taking the 
upper signs, the three differences are found ; that is, 
— 11'049: ^"=101-991; and S’= — 872-7715. 
Ihe differences are now divided by 10,000,000, that is, ciphers are added to their left- 
hand side, so that the above decimal point may be moved seven places in that direction. 
