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the expression thns formed is like A + Bx + Cx2, etc., rational and integral and contains n undetermined
co-efficients. From this expression it may be shown that
This formula was used in calculating the values of u6, v7, u8, and u9, and from the seven
equi-distant terms thus obtained, viz., u4, v5, w6, w7, w8, u9, and u10 the successive differences were
calculated and the series carried on as far as u.25.
The justification for the assumption made in interpolating by the method of finite differences
in connection with life table construction, viz., that the general expression of the values under consideration
is a rational and integral function of x, the independent variable, lies in the fact that this
assumption rests upon the supposition that the successive orders of differences rapidly diminish; and
this supposition has been verified in the case of all tabulated functions of a similar nature to those
under consideration. It has been shown that in the case of a rational and integral function of x of
the nth degree the n+lth differences vanish, therefore if in any other function such differences
become extremely small the function may be approximately represented by a function which is
rational and integral, and the error introduced by the assumption is so small that, in practice, it may
be disregarded. It is an essential condition in the problem, however, that the value of x for which
that of the function is required is not far removed from those functions for which the values are
given. In life table construction this condition is fulfilled by, as far as possible, using only those
values which form the central portion of a given series.
It may be further stated in justification for the use of the method of Unite differences in life
table construction that this method gives a curve or series which passes by small gradations, from one
fixed value to another. It is a known fact that life contingencies also change by small gradations,
i.e., that after the perils of birth there is nospecial age at which the chances of death or survival
suddenly alter. If there be any critical periods of life these probably occur earlier in some individuals
than in others, so that their effect on the life curve is modified, producing only a slight and gradual
distortion. It is true that an infinite number of curves may be drawn through a given series of
points, but the curves that pass by small gradations will lie very near each other except perhaps at
or near the extreme sections of the curves. The gradually changing interpolated curve, therefore, will
not differ very greatly from the gradually changing life curve.
In constructing the present life table from the data already mentioned, viz., population and
deaths in the decennium 1891-1900, two columns were first obtained, one showing population X 2
minus deaths at each given age and upwards, and the other, population X 2 plus deaths at each given
age and upwards, the logarithms of these numbers were then taken out and from the values so
obtained the logarithms of the intervening terms were interpolated.. By this process two series were
obtained, viz., 2P—D at age x and upwards and 2P + D at age x and upwards, the values of 2P—D
and 2P + D at ages x, x+1, x+2, etc., being obtained by subtracting each term from the one
immediately preceding. It has been previously shown that the probability of living a year at age x
2P—D
(px) = 2p+D so that the px values are easily obtained from the two columns showing 2P—D and
2P+D at each age. The following table shows the facts relating to males in London upon which the
present life table is based, arranged in the manner above described ; the same method was adopted in
the case of females, but from considerations of space the figures relating to females are omitted—
Males.
Age x and upwards.
2P — D at age x
and upwards.
Log(2P-D)
at age x and
upwards.
2P + D at age x
and upwards.
Log (2P + D)
at age x and
upwards.
4 and upwards
36,875,597
7-5667390
37,389,295
7-5727473
5 „
35,947,627
7-5556702
36,449,769
7-5616948
10 „
31,497,920
7-4982819
31,977,880
7-5048497
15 „
27,369,031
7-4372594
27,838,841
74446511
25 „
19,243,096
7-2842750
19,680,064
7-2940265
35 „
12,254,474
7-0882947
12,639,790
7 1017399
45 „
7,054,615
6-8484733
7,365,477
6-8672008
55 „
3.434,498
6-5358632
3,660,598
6-5635521
65 „
1,320,986
6-1208982
1,459,330
61641535
75 „
312,391
54946985
369,109
5-5671546
85 „
27,717
4 4427462
36,895
4-5669675
The interpolations were carried out in four series which may be set out as follows, using the
symbol Ux to denote the logarithm of 2 P — D at age x and upwards.
Series (1) U4, U5, U10, U15, U25, U35, (six orders of di0fferences).
Series (2) U15, U25, U35, U45, U45, U65 (five orders of differences).
Series (3) U25, U35, U45, U55, U65, U75 (five orders of differences).
Series (4) U35, U45, U55, U65 U75, U85 (five orders of differences).
. There is a special reason in favour of interpolation of logarithms in that the interpolation of numbers must finish up
at the higher ages either with a rapidly increasing positive value or a rapidly increasing negative value of the function—the
one being absurd and the other meaningless.