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analytical process known as "interpolation by the method of finite differences." The latter method
has been used in the construction of the present life table, and a brief description of the assumptions
involved in this method may here be given.
Suppose we have an arithmetical series (x; x + a; x + 2a; x + 3a, etc.) it will be seen that
the first differences are constant and equal to (a) while the second diSerences equal zero. Now
suppose the squares of this series are taken (x2; x2 + 2 ax + a2; x2 + 4 ax + 4 a2 etc.) it will be
seen that the first differences = 2 ax + a2; 2 ax + 3 a2 ; 2 ax + 5a2 etc., the second differences are
constant and equal to 2a2 while the third differences equal zero; and generally it may be shown that
if the nth power of the series be taken the nth differences are constant and the n,+ 1th differences vanish.
If then we assume that Ux = A + Bx + Cx2 . ... + Nx2, we assume that the nth differences of the
series of quantities represented by Ux (when x is made successively x; x + a; x + 2a, etc.) are all
equal while the n+1th differences vanish. This then is the assumption made in interpolating for
Ux by the method of finite differences given n + 1 values of Ux for equi-distant values of X.
As an instance suppose a set of figures represents the population belonging to five equidistant
ages, say ages 25 and upwards, 35 and upwards, 45 and upwards, 55 and upwards and 65
and upwards and assume that the number representing the population at each of these ages (Ux) is of
the form A + Ba; + Cx2 + Dx3 + Ex4 then we have a series of 5 equations from which the values of the
co-efficients A, B, C, D and E, may be calculated; thus for convenience let x = 0 in the case of the
middle term, age 45 and upwards, then—
U_20 = A — 20B + 400 C — 8000 D -t- 160,000 E
U_10 = A — 10B + 100 C — 1000 D + 10,000 E
U0 = A
U10 = A + 10B + 100 c + 1000 D + 10,000 E
U20 = A + 20B + 400 C + 8000 D + 160,000 E
From these equations the numerical values of the co-efficients may be calculated and if it is
required to interpolate say age 44 and upwards the equation would be U_1=A — B + C — D + E
or say for age 47 and upwards, U2 =A + 2B + 4C + 8D + 16E.
In practice less laborious methods are adopted for interpolating intermediate values when a
succession of such values are required although the principle is exactly the same. Thus the given
terms of the series may be set down and differenced and the problem then becomes one of subdividing
these 10 yearly differences represented by the symbols ∆, ∆2, ∆3, etc.> into smaller differences
represented by the symbols 8, etc., corresponding to one year.
The eeneral formula used for finding the values of δ is as follows : five orders of differences—
From these equations the sub-divided differences can be calculated for any value of m. In life
table construction this value generally = 10, and the following equations show the values of δ
when m is equal to 10—
This formula has been used for interpolation from age 25 onwards; for ages 5 to 25, which
were calculated from the series u4, u10, ul0, u15, u25, and (six orders of differences), it was
necessary to use Lagrange's formula, as the given terms are not equi-distant. In the formula of
Lagrange another but equivalent form is substituted for the expression ux = A + Bx + Cx2, etc.,
viz.