On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies
Author(s)
Benjamin Gompertz
Year
1825
Volume
115
Pages
72 pages
Language
en
Journal
Philosophical Transactions of the Royal Society of London
Full Text (OCR)
XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of Life Contingencies. In a Letter to Francis Baily, Esq. F.R.S. &c. By Benjamin Gompertz, Esq. F.R.S.
Read June 16, 1825.
Dear Sir,
The frequent opportunities I have had of receiving pleasure from your writings and conversation, have induced me to prefer offering to the Royal Society through your medium, this Paper on Life Contingencies, which forms part of a continuation of my original paper on the same subject, published among the valuable papers of the Society, as by passing through your hands it may receive the advantage of your judgment.
I am, Dear Sir, yours with esteem,
9th June 1825.
Benjamin Gompertz.
CHAPTER I.
Article 1. In continuation of Art. 2. of my paper on the valuation of life contingencies, published in the Philosophical Transactions of this learned Society, in which I observed the near agreement with a geometrical series for a short period of time, which must pervade the series which expresses the number of living at ages in arithmetical progression, pro-
ceeding by small intervals of time, whatever the law of mortality may be, provided the intervals be not greater than certain limits: I now call the reader's attention to a law observable in the tables of mortality, for equal intervals of long periods; and adopting the notation of my former paper, considering $L_x$ to express the number of living at the age $x$, and using $\lambda$ for the characteristic of the common logarithm; that is, denoting by $\lambda(L_x)$ the common logarithm of the number of persons living at the age of $x$, whatever $x$ may be, I observe that if $\lambda(L_n) = \lambda(L_{n+m})$, $\lambda(L_{n+m}) = \lambda(L_{n+2m})$, $\lambda(L_{n+2m}) = \lambda(L_{n+3m})$, &c. be all the same; that is to say, if the differences of the logarithms of the living at the ages $n$, $n+m$, $n+2m$, $n+3m$, &c. be constant, then will the numbers of living corresponding to those ages form a geometrical progression; this being the fundamental principle of logarithms.
Art. 2. This law of geometrical progression pervades, in an approximate degree, large portions of different tables of mortality; during which portions the number of persons living at a series of ages in arithmetical progression, will be nearly in geometrical progression; thus, if we refer to the mortality of Deparcieux, in Mr. Baily's life annuities, we shall have the logarithm of the living at the ages 15, 25, 35, 45, and 55 respectively, 2,9285; 2,88874; 2,84136; 2,79379; 2.72099, for $\lambda(L_{15})$; $\lambda(L_{25})$; $\lambda(L_{35})$; &c. and we find $\lambda(L_{25}) - \lambda(L_{15}) = , 04738$ $\lambda(L_{35}) - \lambda(L_{25}) = , 04757$, and consequently these being nearly equal (and considering that for small portions of time the geometrical progression takes place very nearly) we observe that in those tables the numbers of
living in each yearly increase of age are from 25 to 45 nearly, in geometrical progression. If we refer to Mr. Milne's table of Carlisle, we shall find that according to that table of mortality, the number of living at each successive year, from 92 up to 99, forms very nearly a geometrical progression, whose common ratio is $\frac{3}{4}$; thus setting out with 75 for the number of living at 92, and diminishing continually by $\frac{1}{4}$, we have to the nearest integer 75, 56, 42, 32, 24, 18, 13, 10, for the living at the respective ages 92, 93, 94, 95, 96, 97, 98, 99, which in no part differs from the table by $\frac{1}{37}$th part of the living at 92.
Art. 3. The near approximation in old age, according to some tables of mortality, leads to an observation, that if the law of mortality were accurately such that after a certain age the number of living corresponding to ages increasing in arithmetical progression, decreased in geometrical progression, it would follow that life annuities, for all ages beyond that period, were of equal value; for if the ratio of the number of persons living from one year to the other be constantly the same, the chance of a person at any proposed age living to a given number of years would be the same, whatever that age might be; and therefore the present worth of all the payments would be independent of the age, if the annuity were for the whole life; but according to the mode of calculating tables from a limited number of persons at the commencement of the term, and only retaining integer numbers, a limit is necessarily placed to the tabular, or indicative possibility of life; and the consequence may be, that the value of life annuities for old age, especially where they are
MDCCCXXV.
deferred, should be deemed incorrect, though indeed for immediate annuities, where the probability of death is very great, the limit of the table would not be of so much consequence, for the present value of the first payment would be nearly the value of the annuity.
Such a law of mortality would indeed make it appear that there was no positive limit to a person's age; but it would be easy, even in the case of the hypothesis, to show that a very limited age might be assumed to which it would be extremely improbable that any one should have been known to attain.
For if the mortality were, from the age of 92, such that $\frac{1}{4}$ of the persons living at the commencement of each year were to die during that year, which I have observed is nearly the mortality given in the Carlisle tables between the ages 92 and 99,* it would be above one million to one that out of three millions of persons, whom history might name to have reached the age of 92, not one would have attained to the age of 192, notwithstanding the value of life annuities of all ages above 92 would be of the same value. And though the limit to the possible duration of life is a subject not likely ever to be determined, even should it exist, still it appears interesting to dwell on a consequence which would follow, should the mortality of old age be as above described. For, it would follow that the non-appearance on the page of history of a single circumstance of a person having arrived
* If from the Northampton tables we take the numbers of living at the age of 88 to be 83, and diminish continually by $\frac{1}{4}$ for the living, at each successive age, we should have at the ages 88, 89, 90, 91, 92, the number of living 83; 61.3; 45.9; 34.4; 25.8; almost the same as in the Northampton table.
at a certain limited age, would not be the least proof of a limit of the age of man; and further, that neither profane history nor modern experience could contradict the possibility of the great age of the patriarchs of the scripture. And that if any argument can be adduced to prove the necessary termination of life, it does not appear likely that the materials for such can in strict logic be gathered from the relation of history, not even should we be enabled to prove (which is extremely likely to be the state of nature) that beyond a certain period the life of man is continually becoming worse.
Art. 4. It is possible that death may be the consequence of two generally co-existing causes; the one, chance, without previous disposition to death or deterioration; the other, a deterioration, or an increased inability to withstand destruction. If, for instance, there be a number of diseases to which the young and old were equally liable, and likewise which should be equally destructive whether the patient be young or old, it is evident that the deaths among the young and old by such diseases would be exactly in proportion of the number of young to the old; provided those numbers were sufficiently great for chance to have its play; and the intensity of mortality might then be said to be constant; and were there no other diseases but such as those, life of all ages would be of equal value, and the number of living and dying from a certain number living at a given earlier age, would decrease in geometrical progression, as the age increased by equal intervals of time; but if mankind be continually gaining seeds of indisposition, or in other words, an increased liability to death (which appears not to be an unlikely supposition with respect to a great part of life, though
the contrary appears to take place at certain periods) it would follow that the number of living out of a given number of persons at a given age, at equal successive increments of age, would decrease in a greater ratio than the geometrical progression, and then the chances against the knowledge of any one having arrived to certain defined terms of old age might increase in a much faster progression, notwithstanding there might still be no limit to the age of man.
Art. 5. If the average exhaustions of a man's power to avoid death were such that at the end of equal infinitely small intervals of time, he lost equal portions of his remaining power to oppose destruction which he had at the commencement of those intervals, then at the age $x$ his power to avoid death, or the intensity of his mortality might be denoted by $aq^x$, $a$ and $q$ being constant quantities; and if $L_x$ be the number of living at the age $x$, we shall have $a L_x \times q^x \dot{x}$ for the fluxion of the number of deaths $= -(L_x)'$; $\therefore abq^x = -\frac{\dot{L}_x}{L_x}$,
$\therefore abq^x = -\text{hyp. log. of } b \times \text{hyp. log. of } L_x$, and putting the common logarithm of $\frac{1}{b} \times \text{square of the hyperbolic logarithm of } 10 = c$, we have $c.q^x = \text{common logarithm of } \frac{L_x}{d}$; $d$ being a constant quantity, and therefore $L_x$ or the number of persons living at the age of $x = d.g^q$; $g$ being put for the number whose common logarithm is $c$. The reader should be aware that I mean $g^q$ to represent $g$ raised to the power $q$ and not $g^q$ raised to the $x$ power; which latter I should have expressed by $g^{q^x}$, and which would evidently be equal to $g^{qx}$. I take this opportunity to make this observation, as algebraists are sometimes not sufficiently precise in their notation of exponentials.
This equation between the number of the living, and the age, becomes deserving of attention, not in consequence of its hypothetical deduction, which in fact is congruous with many natural effects, as for instance, the exhaustions of the receiver of an air pump by strokes repeated at equal intervals of time, but it is deserving of attention, because it appears corroborated during a long portion of life by experience; as I derive the same equation from various published tables of mortality during a long period of man's life, which experience therefore proves that the hypothesis approximates to the law of mortality during the same portion of life; and in fact the hypothesis itself was derived from an analysis of the experience here alluded to.
Art. 6. But previously to the interpolating the law of mortality from tables of experience, I will premise that if, according to our notation, the number of living at the age $x$ be denoted by $L_x$, and $\lambda$ be the characteristic of a logarithm, or such that $\lambda(L_x)$ may denote the logarithm of that number, that if $\lambda(L_a) - \lambda(L_{a+r}) = m$, $\lambda(L_{a+r}) - \lambda(L_{a+2r}) = mp$, $\lambda(L_{a+2r}) - \lambda(L_{a+3r}) = m^2p$; and generally $\lambda(L_{a+n-r}) - \lambda(L_{a+n}) = m \cdot p^{n-1}$; that by continual addition we shall have $\lambda(L_a) - \lambda(L_{a+n}) = m(1 + p + p^2 + p^3 + \ldots + p^{n-1}) = m \cdot \frac{1-p^n}{1-p}$; and therefore if $p^r = q$, and $\varepsilon$ be put equal to the number whose common logarithm is $\frac{m}{1-q^n}$, we shall have $\lambda(L_{a+n}) = \lambda(L_a) - \lambda(\varepsilon) \times (1 - q^n) = \lambda\left(\frac{L_a}{\varepsilon}\right) + \lambda(\varepsilon) \cdot q^n$;
$\therefore L_{a+n} = \frac{L_a}{\varepsilon} \times \varepsilon q^n$; and this equation, if for $a + n$ we write $x$, will give $L_x = \frac{L_a}{\varepsilon} \cdot \varepsilon^{-a} \times q^x$; and consequently if $\frac{L_a}{\varepsilon}$ be put
Mr. Gompertz on the nature of the function
\[ d = \varepsilon^q - a \] and \( g = q^{-a} \), the equation will stand \( L_x = d \cdot g^{q^x} \), and
\[ \lambda(g) = \lambda(\varepsilon) \times q^{-a} = \frac{m}{1-q^a} \]; and I observe that when \( q \) is affirmative, and \( \lambda(\varepsilon) \) negative, that \( \lambda(g) \) is negative. The equation \( L_x = d \cdot g^{q^x} \) may be written in general \( \lambda(L_x) = \lambda(d) \pm \) the positive number whose common logarithm is \( \{\lambda^2(g) + x \lambda(g)\} \), the upper or under sign to be taken according as the logarithm of \( g \) is positive or negative, \( \lambda^2 \) standing for the characteristic of a second logarithm; that is, the logarithm of a logarithm, \( \lambda(q) = \frac{1}{r} \times \lambda(p) \), \( \lambda^2(g) = \lambda^2(\varepsilon) - a \lambda(q) = \lambda\left(\frac{m}{1-p}\right) - a \lambda(q) = \lambda(m) - \lambda(1-p) - a \lambda(q) \); also \( \lambda(d) = \lambda(L_a) - \frac{m}{1-p} \).
Art. 7. Applying this to the interpolation of the Northampton table, I observe that taking \( a = 15 \) and \( r = 10 \) from that table, I find \( \lambda(L_a) - \lambda(L_a + r) = ,0566 = m \), \( \lambda(L_a + r) - \lambda(L_a + 2r) = ,0745 \), \( \lambda(L_a + 2r) - \lambda(L_a + 3r) = ,0915 \), and \( \lambda(L_a + 3r) - \lambda(L_a + 4r) = ,1228 \); now if these numbers were in geometrical progression, whose ratio is \( p \), we should have respectively \( m = ,0566 \); \( mp = ,0745 \); \( mp^2 = ,0915 \); \( mp^3 = ,1228 \). No value of \( p \) can be assumed which will make these equations accurately true; but the numbers are such that \( p \) may be assumed, so that the equation shall be nearly true; for resuming the first and last equations we have \( p^3 = \frac{1228}{566} \);
\[ \therefore \text{logarithm of } p = \frac{1}{3} (\text{logarithm of } 1228 - \text{logarithm of } 566) = ,11213, \therefore \lambda(q) = ,011213 \text{ and } p = 1,2944. \]
And to examine how near this is to the thing required, continually to the logarithm of ,0566 namely \( \bar{2},75282 \), adding ,11213 which is the logarithm of \( p \), we have respectively for the
logarithms of \( mp \), of \( mp^2 \), of \( mp^3 \) the values \( 2,8649, 2,9771, 1,0892 \); the numbers corresponding to which are \( ,07327; ,09486; ,1228 \); and consequently \( m, mp, mp^2, \) and \( mp^3 \) respectively equal to \( ,0566; ,07327; ,09486, \) and \( ,1228 \) which do not differ much from the proposed series \( ,0566; ,07327; ,09486, \) and \( ,1228 \); and according to our form for interpolation, taking \( m = ,0566 \) and \( p = 1,2944 \); we have \( \frac{m}{1-p} = \frac{,0566}{,2944} = -,1922 \); and \( \lambda(L_{15}) \) agreeably to the Northampton tables, being \( = 3,7342 \) we have \( \lambda(d) = 3,7342 + ,1922 = 3,9264, d = 8441, \lambda^2(q), \) that is to say, the logarithm of the logarithm of \( q = \lambda\left(\frac{m}{1-p}\right) - a \lambda(q) = 1,28375 - ,16819 = 1,1156, \lambda(g) = - ,130949 = 1,8695, \) the negative sign being taken because \( \lambda(g) = \lambda(\varepsilon) \times q^{-a} = \frac{m}{1-q} \cdot q^{-a}, \) and \( g = ,7404. \)
And therefore \( x \) being taken between the limits, we are to examine the degree of proximity of the equation \( L_x = 8441 \times \sqrt[7404]{1,0261} \) or \( \lambda(L_x) \), that is, the logarithm of the number of living at the age \( x = 3,9264 \)—number whose logarithm is \( (1,11556 + x \times .011213) \), as the logarithm of \( g \) is negative. The table constructed according to this formula, which I shall lay before the reader, will enable him to judge of the proximity it has to the Northampton table; but previously thereto shall show that the same formula, with different constants, will serve for the interpolations of other tables.
Art. 8. To this end let it be required to interpolate Deparcieux’s tables, in Mr. Baily’s life annuities, between the ages 15 and 55.
The logarithms of the living at the age of
15 are $2,92840$ differences $= ,03966 = \lambda(L_{15}) - \lambda(L_{25})$
25 $2,88874$ $,04738 = \lambda(L_{25}) - \lambda(L_{35})$
35 $2,84136$ $,04757 = \lambda(L_{35}) - \lambda(L_{45})$
45 $2,79379$ $,07280 = \lambda(L_{45}) - \lambda(L_{55})$
55 $2,72099$
Here the three first differences, instead of being nearly in geometrical progression are nearly equal to each other, showing from a remark above, that the living, according to these tables, are nearly in geometrical progression; and the reader might probably infer that this table will not admit of being expressed by a formula similar to that by which the Northampton table has been expressed between the same limits, but putting,
\[
\begin{align*}
\lambda(L_{15}) &= \ldots \ldots \ldots = 2,92840 \\
\lambda(L_{25}) &= \lambda(L_{15}) - m \ldots \ldots = 2,88874 \\
\lambda(L_{35}) &= \lambda(L_{15}) - m - mp \ldots \ldots = 2,84136 \\
\lambda(L_{45}) &= \lambda(L_{15}) - m - mp - mp^2 \ldots \ldots = 2,79379 \\
\lambda(L_{55}) &= \lambda(L_{15}) - m - mp - mp^2 - mp^3 = 2,72099
\end{align*}
\]
and we shall have
\[
\lambda(L_{15}) - \lambda(L_{35}) \text{ or its equal } m + mp = ,08704, \text{ and } \lambda(L_{35}) - \lambda(L_{55}) \text{ or its equal } p^3 \times m + pm = ,12037; \quad \therefore p^2 = \frac{12037}{8704} \text{ and }
\]
the log. of $p = \frac{\log. \text{ of } 12037 - \log. \text{ of } 8704}{2} = ,0703997$ and
$p = 1,176, m = \frac{,08704}{1 + p} = \frac{,08704}{2,176} = ,04$. And to see how these values of $m$ and $p$ will answer for the approximate determination of the logarithms above set down of the numbers of living at the ages 15, 25, 35, 45, and 55, we have the following easy calculation by continually adding the logarithm of $p$.
expressive of the law of human mortality, &c.
Logarithm of \( m = 2,6020600 \)
\[
\begin{align*}
\text{Log. of } p &= 0.0703997 \\
\text{therefore } mp &= 0.047039 \\
\text{Log. of } mp &= 2.6724597 \\
mp^2 &= 0.055317 \\
\text{Log. of } mp^2 &= 2.7428594 \\
mp^3 &= 0.065051 \\
\text{Log. of } mp^3 &= 2.8128591 \\
\end{align*}
\]
These logarithms of the approximate number of living at the ages 15, 25, 35, 45 and 55, are extremely near those proposed, and the numbers corresponding to these give the number of living at the ages 15, 25, 35, 45 and 55, respectively, 848; 773,4; 694; 612,3; and 526; differing very little from the table in Mr. Baily's life annuities; namely, 848; 774; 694; 622 and 526. And we have \( a = 15, r = 10, m = 0.04; \lambda(m) = 2,60206; 1 - p = 0.176; \lambda q = \frac{1}{10} \lambda(p) = 0.00703997; \lambda(g) = \frac{mg - a}{1 - p} = \frac{0.04 \times g - a}{0.176}, \) and is negative;
\[
\lambda \lambda(g) = \lambda(0.04) - 15 \times 0.00704 - \lambda(0.176) = 1,25095;
\]
\[
\lambda(d) = \lambda(L_a) - \frac{m}{1-p} = 2,9284 + 0.22727 = 3,1557; \therefore \lambda(L_x) = 3,1557 - \text{number whose log. is } (1,25095 + 0.00704x), \]
for the logarithm of living in Deparcieux' table in Mr. Baily's annuities, between the limits of age 15 and 55. The table which we shall insert will afford an opportunity of appreciating the proximity of this formula to the table.
Art. 9. To interpolate the Swedish mortality among males between the ages of 10 and 50, from the table in Mr. Baily's annuities:
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Here \( \lambda(L_{10}) = 3.779091 \)
\( \lambda(L_{20}) = 3.746868 \) to be assumed \( = \lambda(L_{10}) - m \)
\( \lambda(L_{30}) = 3.703205 \) \( = \lambda(L_{10}) - m - mp \)
\( \lambda(L_{40}) = 3.648165 \) \( = \lambda(L_{10}) - m - mp - mp^2 \)
\( \lambda(L_{50}) = 3.564192 \) \( = \lambda(L_{10}) - m - mp - mp^2 - mp^3 \)
Consequently \( m + mp = \lambda(L_{10}) - \lambda(L_{30}) = 0.075886 \), and
\( \lambda(L_{30}) - \lambda(L_{50}) = p^a \times m + mp = 1.39013 \); therefore
\( p^a = \frac{139013}{75886} \), and \( \lambda(p) = 1.314468 \); \( \therefore p = 1.3535; m = \frac{0.075886}{1 + p} = \frac{0.075886}{2.3535} \); \( \lambda(m) = 2.5084775; m = 0.032244; a = 10; r = 10; \)
\( \lambda(q) = 0.01314468; \lambda g = \frac{m \cdot q^{-10}}{-0.3535} \), negative; \( \lambda \lambda(g) = \lambda(m) \)
\(-10 \lambda(q) - \lambda(0.3535) = 2.82861; \lambda(d) = \lambda L_a - \frac{m}{1-p} = 3.779091 + 0.091218 = 3.8703 \); consequently this will give
between the ages 10 and 50 of Swedish males,
\( \lambda(L_x) \) or the logarithm of the living at the age of \( x = 3.8703 \) — number, whose logarithm is \( (2.82861 + 0.013145 \cdot x) \).
A table will also follow to show the proximity of this with Mr. Baily’s table.
Art. 10. For Mr. Milne’s table of the Carlisle mortality we have, as given by that ingenious gentleman,
\( \lambda(L_{10}) = 3.81023 \)
\( \lambda(L_{20}) = 3.78462 \)
\( \lambda(L_{30}) = 3.75143 \)
\( \lambda(L_{40}) = 3.70544 \)
\( \lambda(L_{50}) = 3.64316 \)
\( \lambda(L_{60}) = 3.56146 \)
And the difference of these will form a series nearly in geometrical progression, whose common ratio is \( \frac{4}{3} \), and in consequence of this, the first method may be adopted for the
interpolations. Thus because \( \lambda (L_{10}) - (L_{20}) = 0.02561 \), the first term of the differences, and \( \lambda (L_{50}) - \lambda (L_{60}) = 0.0817 \), the fifth term of the differences: take the common ratio \( = \frac{817}{256} \), and \( m = 0.0256 \); \( \therefore \lambda(m) = 2.40824 \). These will give \( \lambda(p) = 0.126; p = 1,3365; a = 10, r = 10, \lambda(q) = 0.0126, \lambda(\varepsilon) = \frac{m}{1-p} = \frac{0.0256}{0.3365}; \therefore \lambda(g) \text{ negative}; \lambda \lambda g = 2,40824 - \lambda(0.3365) - 0.126 = 2,75526; \text{ and } \lambda(d) = \lambda(L_{10}) + \frac{0.0256}{0.3365} = 3,88631, \text{ and accordingly, to interpolate the Carlisle table of mortality for the ages between 10 and 60, we have for any age } x, \)
\[
\lambda(L_x) = 3,88631 - \text{number whose logarithm is } (2,88126 + 0.0126x).
\]
Here we have formed a theorem for a larger portion of time than we had previously done. If by the second method the theorem should be required from the data of a larger portion of life, we must take \( r \) accordingly larger; thus if \( a \) be taken 10, \( r = 12 \), then the interpolation would be formed from an extent of life from 10 to 58 years; and referring to Mr. Milne’s tables, our second method would give \( \lambda(L_x) = 3,89063 - \text{the number whose logarithm is } (2,784336 + 0.0120948x) \); this differs a little from the other, which ought to be expected.
If the portion between 60 and 100 years of Mr. Milne’s Carlisle table be required to be interpolated by our second method, we shall find \( p = 1,86466; \lambda(m) = 1,30812; m = 0.20329, \&c. \text{ and we shall have } \lambda(L_x) = 3,79657 - \text{the number whose logarithm is } (3,74767 + 0.02706x).
This last theorem will give the numbers corresponding to the living at 60, 80, and 100, the same as in the table; but for the ages 70 and 90, they will differ by about one year:
the result for the age of 70 agreeing nearly with the living corresponding to the age 71; and the result for the age 90, agreeing nearly with the living at the age 89 of the Carlisle tables.
Art. 11. Lemma. If according to a certain table of mortality, out of \(a\), persons of the age of 10, there will arrive \(b, c, d, \&c.\) to the age 20, 30, 40, \&c.; and if according to the tables of mortality, gathered from the experience of a particular society, the decrements of life between the intervals 10 and 20, 20 and 30, 30 and 40, \&c. is to the decrements in the aforesaid table between the same ages, proportioned to the number of living at the commencement of those intervals respectively, as 1 to \(n\), 1 to \(n'\), 1 to \(n''\), \&c. it is required to construct a table of mortality of that society, or such as will give the above data.
Solution. According to the first table, the decrements of life from 10 to 20, 20 to 30, 30 to 40, \&c. respectively, will be found by multiplying the number of living at the commencement of each period by \(\frac{a-b}{a}, \frac{b-c}{b}, \frac{c-d}{c}, \&c.\), and therefore, in the Society proposed, the corresponding decrements will be found by multiplying the number of living at those ages by \(\frac{a-b}{a} n; \frac{b-c}{b} n'; \frac{c-d}{c} n'' \&c.\); and the number of persons who will arrive at the ages 20, 30, 40, \&c. will be the numbers respectively living at the ages 10, 20, 30, \&c. multiplied respectively by \(\frac{1-n.a+nb}{a}, \frac{1-n'.b+n'c}{b}, \frac{1-n''.c+n''d}{c}, \&c.\); hence out of the number \(a\), living at the age 10, there will arrive at the age 10, 20, 30, 40, 50, \&c. the numbers \(\frac{1-n.a+nb}{a} \times \frac{1-n'.b+n'c}{b} \times \frac{1-n''.c+n''d}{c}\); \&c. and the numbers for the intermediate ages must be found by interpolation.
In the ingenious Mr. Morgan's sixth edition of Price's Annuities, p. 183, vol. i. it is stated, that in the Equitable Assurance Society, the deaths have differed from the Northampton tables; and that from 10 to 20, 20 to 30, 30 to 40, 40 to 50, 50 to 60, and 60 to 80, it appears that the deaths in the Northampton tables were in proportion to the deaths which would be given by the experience of that society respectively, in the ratios of 2 to 1; 2 to 1; 5 to 3; 7 to 5, and 5 to 4. According to this, the decrements in 10 years of those now living at the ages 10, 20, 30, and 40, will be the number living at those ages multiplied respectively by ,0478; ,0730; ,1024; ,1284; and the deaths in twenty years of those now living at the age of 60, would be the number of those living multiplied by ,3163. And also, taking, according to the Northampton table, the living at the age of 10 years equal to 5675, I form a table for the number of persons living at
| Ages | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
|------|------|------|------|------|------|------|------|------|
| Being | 5675 | 5403.5 | 5010 | 4496 | 3919 | 3116 | * | 1197 |
| Log. of Persons Living | 3.75612 | 3.73268 | 3.69984 | 3.65283 | 3.59318 | 3.49360 | * |
Consequently, if \(a = 20\), \(r = 10\), we have \(\lambda(L_{20}) = 3,73268\); \(\lambda(L_{40}) = \lambda(L_{20}) - m - mp = 3,65283\); \(\lambda(L_{60}) = L_{20} - m - mp - mp^2 - mp^3 = 3,49360\); \(m.1 + p = ,07985\); and \(mp^2 \times 1 + p = 3,65283 - 3,49360 = ,15923\); hence \(\lambda(p) = \frac{1}{2} \lambda(,15923) = ,149875\); and \(p = 1,412131\); \(\lambda(m) = \lambda(,07985) - \lambda(2,41243) = 2,519874\); and \(m = ,033013\); \(\therefore \lambda(e) = \frac{-m}{,412131}\) negative; \(\therefore \lambda(g)\) is negative; \(\lambda(g) = \lambda m - \lambda,412131 - ,0149875 \times 20 = 2,6051\); \(\lambda(d) = \lambda(L_{20}) - \lambda(e) = 3,73268 - ,080302 = 3,813\) sufficiently near; and our formula for the
mortality between the ages of 20 and 60, which appears to me to be the experience of the Equitable Society, is $\lambda (L_x) = 3,813$ —the number whose log. is $(\log_{10} 2.6051 + 0.0149875 x)$.
This formula will give
| At the ages | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
|-------------|------|------|------|------|------|------|------|------|
| No. of living | 5703.2 | 5403.5 | 5007 | 4496 | 3862 | 3116 | * | 1500 |
| Differs from the proposed by | 28.2 | 0 | +3 | 0 | -57 | 0 | * | 303 |
In the table of Art. 12, the column marked 1, represents the age; column marked 2, represents the number of persons living at the corresponding age; column marked 3, the error to be added to the number of living deduced from the formula, to give the number of living of the table for which the formula is constructed; column marked 4, gives the error in age, or the quantity to be added to the age in column 1, that would give the number of living in the original table, the same as in column 2. It may be proper to observe, that where the error in column 3 and 4 is stated to be 0, it is not meant to indicate that a perfect coincidence takes place, but that the difference is too small to be worth noticing.
Art. 12. \( \lambda (L) = \lambda (d) \) — number whose logarithm is \( (\lambda^2(g) + x \lambda q) \).
| Northampton | Deparcieux | Sweden | Carlisle |
|-------------|------------|--------|---------|
| 1 | 2 | 3 | 4 |
| 10 | 6013 | 0 | 6460 | 5703 |
| 11 | 5974 | -16 | 6427 | 5677 |
| 12 | 5935 | -22 | 6393 | 5650 |
| 13 | 5894 | -26 | 6358 | 5622 |
| 14 | 5852 | -24 | 6322 | 5594 |
| 15 | 5423 | 848 | 5810 | 5654 |
| 16 | 5360 | 13 | 5767 | 5564 |
| 17 | 5297 | 23 | 5722 | 5534 |
| 18 | 5233 | 29 | 5677 | 5503 |
| 19 | 5168 | 31 | 5630 | 5470 |
| 20 | 5102 | 30 | 5583 | 5437 |
| 21 | 5036 | 34 | 5534 | 5403 |
| 22 | 4969 | 16 | 5484 | 5368 |
| 23 | 4899 | 11 | 5434 | 5333 |
| 24 | 4830 | 15 | 5382 | 5295 |
| 25 | 4762 | 2 | 5339 | 5258 |
| 26 | 4689 | 4 | 5275 | 5218 |
| 27 | 4616 | 6 | 5220 | 5178 |
| 28 | 4545 | 10 | 5164 | 5137 |
| 29 | 4472 | 12 | 5107 | 5095 |
| 30 | 4403 | 22 | 5049 | 5051 |
| 31 | 4325 | 15 | 4989 | 5007 |
| 32 | 4250 | 15 | 4929 | 4961 |
| 33 | 4174 | 14 | 4868 | 4914 |
| 34 | 4098 | 13 | 4805 | 4866 |
| 35 | 4021 | 11 | 4741 | 4817 |
| 36 | 3944 | 11 | 4676 | 4767 |
| 37 | 3866 | 6 | 4611 | 4715 |
| 38 | 3788 | 3 | 4544 | 4662 |
| 39 | 3709 | 11 | 4476 | 4608 |
| 40 | 3630 | 8 | 4407 | 4553 |
| 41 | 3551 | 8 | 4337 | 4496 |
| 42 | 3473 | 11 | 4266 | 4438 |
| 43 | 3392 | 12 | 4194 | 4379 |
| 44 | 3312 | 14 | 4121 | 4319 |
| 45 | 3235 | 13 | 4047 | 4257 |
| 46 | 3152 | 18 | 3973 | 4194 |
| 47 | 3072 | 20 | 3897 | 4130 |
| 48 | 2991 | 23 | 3821 | 4065 |
| 49 | 2911 | 25 | 3744 | 3998 |
| 50 | 2831 | 26 | 3660 | 3931 |
| 51 | 2752 | 24 | 3587 | 3862 |
| 52 | 2672 | 22 | 3508 | 3793 |
| 53 | 2593 | 19 | 3428 | 3721 |
| 54 | 2514 | 16 | 3348 | 3657 |
| 55 | 2436 | 12 | 3266 | 3590 |
| 56 | 2358 | 8 | 3184 | 3521 |
| 57 | 2280 | 4 | 3105 | 3454 |
| 58 | 2206 | 4 | 3026 | 3387 |
| 59 | 2123 | 3 | 2947 | 3320 |
| 60 | 2052 | 14 | 2868 | 3253 |
\[
\begin{align*}
\lambda(d) &= 3.9265 \\
\lambda^2(g) &= 1.11556 \\
\lambda(q) &= 0.011213 \\
\lambda(d) &= 3.1557 \\
\lambda^2(g) &= 1.25095 \\
\lambda(q) &= 0.00704 \\
\lambda(d) &= 3.8703 \\
\lambda^2(g) &= 2.82861 \\
\lambda(q) &= 0.0131457 \\
\lambda(d) &= 3.89063 \\
\lambda^2(g) &= 2.784336 \\
\lambda(q) &= 0.0120948 \\
\lambda(d) &= 3.813 \\
\lambda^2(g) &= 2.6051 \\
\lambda(q) &= 0.0149875 \\
\lambda(d) &= 3.86743 \\
\lambda^2(g) &= 2.6051 \\
\lambda(q) &= 0.0149875 \\
\end{align*}
\]
CHAPTER II.
Article 1. The near proximity to the geometrical progression of the series expressing the number of persons living at equal small successive intervals of time during short periods, out of a given number of persons living at the commencement of those intervals, affords a very convenient mode of calculating values connected with life contingencies, for short limited periods; by offering a manner of forming general tables, applicable (by means of small auxiliary tables of the particular mortalities) to calculations for any particular mortality; and by easy repetition, to calculate the values for any length of period for any table of mortality we please.
If, for instance, it were required to find the value of an annuity of an unit for $p$ years, on three lives of the age $b$, $c$, $d$, the rate of interest being such that the present value of an unit to be received at the expiration of one year, be equal to $r$, then the value of the first payment would be $\frac{L_{b+1}}{L_b} \times \frac{L_{c+1}}{L_c} \times \frac{L_{d+1}}{L_d} \times r$;
and of the $p^{th}$ payment the present value would be $\frac{L_{b+p}}{L_b} \times \frac{L_{c+p}}{L_c} \times \frac{L_{d+p}}{L_d} \times r^p$; but if $L_{b+p} = L_b \times \left(\frac{L_{b+1}}{L_b}\right)^p$ whether $p$ be 1, 2, 3, &c. which will be the case when $L_b$, $L_{b+1}$, $L_{b+2}$, &c. form a geometrical progression, and similarly, if $L_{c+p} = L_c \times \left(\frac{L_{c+1}}{L_c}\right)^p$, and also, $L_{d+p} = L_d \times \left(\frac{L_{d+1}}{L_d}\right)^p$, the pre-
sent value of the \( p \)th payment will be \( \left( \frac{L_{1:b,c,d}}{L_{b,c,d}} r \right)^p \); hence, if \( \frac{L_{1:b,c,d}}{L_{b,c,d}} r \) be put \( = a \), the value of the annuity will be
\[
a + a^2 + a^3 + a^4 \ldots a^p = \frac{a - a^{p+1}}{1-a} = \frac{1-a^p}{a-1}.
\]
Art. 2. Consequently, let a general table be formed of the logarithm of \( \frac{1-a^p}{a-1} \) for every value of the log. of \( a^p \); and also let a particular table be formed for every value of the log. of \( \frac{L_{x+p}}{L_x} \) according to the particular table of mortality to be adopted; from the last table take the log. of \( \frac{L_{b+p}}{L_b}, \frac{L_{c+p}}{L_c}, \frac{L_{d+p}}{L_d} \); and also from a table constructed for the purpose, take the log. of \( r^p \), add these four logs. together, and the sum will be the log. of \( \frac{1-a^p}{a-1} \), which being sought for in the general table, will give the log. of \( \left( \frac{1-a^p}{a-1} \right) \) which will be the log. of the annuity sought for the term \( p \), on supposition of the geometrical progression being sufficiently near. Here I remark, that were it not for more general questions than the above, it would be preferable to have general tables formed for the values of \( \frac{1-a^p}{a-1} \), instead of the log. of such values; but from the consideration that for most purposes a table of the logs. of \( \frac{1-a^p}{a-1} \) will be found most convenient, I have had them calculated in preference.
Art. 3. The shorter the periods are, the nearer does the series of the number of persons living at the equal intervals of successive ages approximate to the geometrical progression; and consequently this mode, by the assumption of sufficiently short periods, and frequent repetitions, will answer
for any degree of accuracy the given table of mortality will admit of, but then the labour will be increased in proportion.
Art. 4. There are different modes of obviating, in a great measure, this inconvenience, by assuming an accommodated ratio for the given age, instead of the real ratio, from amongst which I shall only for the present select a few. The first is as follows: find for every value of \(a\), the log. of \(\frac{L_{x+y}}{L_x}\), that is, the log. of \(\frac{L_{x+1}}{L_x} + \frac{L_{x+2}}{L_x} + \frac{L_{x+3}}{L_x} \ldots \frac{L_{x+p}}{L_x}\); seek this value in the general table, which will give the corresponding value of the log. of \(a^p\); and construct a table of such values for every value of \(x\), and adopt these values for log. of \(a^p\), instead of the abovenamed values of the log. of \(\frac{L_{x+p}}{L_x}\), for the determination of the values of the limited periods: the preference of this to the first proposed method consists in this; that if the series \(\frac{1}{L_b} \times (L_{b+1} + L_{b+2} + L_{b+3} \ldots L_{b+p}) = \varepsilon + \varepsilon^2 + \varepsilon^3 + \&c. \ldots \varepsilon^p\), the series \(\frac{L_{b+1}}{L_b}, \frac{L_{b+2}}{L_b}, \&c.\) being nearly in geometrical progression, and \(\frac{L_{b+1}}{L_b} - \varepsilon = \varepsilon_1, \frac{L_{b+2}}{L_b} - \varepsilon^2 = \varepsilon_2, \&c.\varepsilon_1, \varepsilon_2, \&c.\) will be small, and \(\varepsilon_1 + \varepsilon_2 + \varepsilon_3 \ldots \varepsilon_p = 0\), and therefore, if the series \(\frac{L_{c+1}}{L_c}, \frac{L_{c+2}}{L_c}, \frac{L_{c+3}}{L_c}, \&c.\) and \(\frac{L_{d+1}}{L_d}, \frac{L_{d+2}}{L_d}, \&c.\) formed accurately geometrical progressions, and the value of \(\frac{L_{c+1}}{L_c} \times \frac{L_{d+1}}{L_d} \cdot r = m\), the value of the annuity for the term, would be accurately equal to \(m\varepsilon + m^2\varepsilon^2 + m^3\varepsilon^3 \ldots + m^p\varepsilon^p + m\varepsilon_1 + m^2\varepsilon_2 + m^3\varepsilon_3 \ldots + m^p\varepsilon_p\), but because in
general \( \frac{L_{c+1}}{L_c} \), \( \frac{L_{d+1}}{L_d} \) and \( r \) differ very little from unity, \( m \) will not differ much from unity; and therefore if \( p \) be not great, \( m, m^2, m^3, \&c. \) will not differ much from unity; and consequently, as \( \varepsilon_1, \varepsilon_2, \varepsilon_3, \&c. \) are small, \( m \varepsilon_1 + m^2 \varepsilon_2 + m^3 \varepsilon_3 + \ldots + m^p \varepsilon_p \) will not differ much from \( \varepsilon_1 + \varepsilon_2 + \varepsilon_3 + \ldots + \varepsilon_p \); but this has been shown to be 0; consequently \( m \varepsilon_1 + m^2 \varepsilon_2 + m^3 \varepsilon_3 + \ldots + m^p \varepsilon_p \) differs very little from 0, or in other words is very small; and consequently, the value of the annuity differs very little from \( m \varepsilon + m^2 \varepsilon^2 + m^3 \varepsilon^3 + \ldots + m^p \varepsilon^p \); and the same method of demonstration would apply with any one of the other ages, the remaining ages being supposed to possess the property of the accurate geometrical progression; notwithstanding this, however, as none of them probably will contain that property, but in an approximate degree, a variation in the above approximations may be produced of a small quantity of the second order; that is, if the order of the product of two small quantities; but, as in this approximation, I was only aiming at retaining the quantities of the first order, I do not consider this as affecting the result as far as the approximation is intended to reach: thus far with regard to the first accommodated ratios.
Art. 5. Moreover, on the supposition that \( L_c, L_{c+1}, L_{c+2}, \ldots, L_{c+p} \), and also \( L_d, L_{d+1}, L_{d+2}, \ldots, L_{d+p} \) are series in geometrical progression, and that \( r \cdot \frac{L_{c+1}}{L_c} \times \frac{L_{d+1}}{L_d} = m = n.q \). Since the annuity for \( p \) years on the three lives is equal to \( \frac{L_{b+1}}{L_b} \cdot m + \frac{L_{b+2}}{L_b} \cdot m^2 + \ldots + \frac{L_{b+p}}{L_b} \cdot m^p \) it follows
that if \( \frac{L_{b+1}}{L_b} \cdot n + \frac{L_{b+2}}{L_b} \cdot n^2 + \frac{L_{b+3}}{L_b} \cdot n^3 \ldots \ldots \frac{L_{b+p}}{L_b} \cdot n^p = \) \( \varepsilon \cdot n + \varepsilon^2 \cdot n^2 + \varepsilon^3 \cdot n^3 \ldots \ldots + \varepsilon^p \cdot n^p \) that if \( n \) be very nearly equal to \( m \), \( \frac{L_{b+1}}{L_b} \cdot n \cdot q + \frac{L_{b+2}}{L_b} \cdot n^2 \cdot q^2 + \&c. \ldots \ldots \frac{L_{b+p}}{L_b} \cdot n^p \cdot q^p \) which will be the value of the annuity on the three lives, will be nearly \( \varepsilon \cdot n \cdot q + \varepsilon^2 \cdot n^2 \cdot q^2 + \&c. \ldots \ldots \varepsilon^p \cdot n^p \cdot q^p \). If \( q \) were equal to unity, or, which is the same thing, \( m = n \), the equality would be accurate; but it may not be so when \( m \) differs from 1; but the nearer \( n \) is to \( m \), at least when the difference does not exceed certain limited small quantities, the nearer will be the coincidence. It appears therefore, that if instead of taking the accommodated ratio for \( \varepsilon^p \) so that \( \frac{1}{L_b} \times (L_{b+1} + L_{b+2} + L_{b+3} \ldots L_{b+p}) = \varepsilon + \varepsilon^2 + \varepsilon^3 \ldots \varepsilon^n \) it will be preferable generally to take it so that \( \frac{1}{L_b} \times (n \cdot L_{b+1} + n^2 \cdot L_{b+2} + n \cdot L_{b+3} \&c. \ldots n^p \cdot L_{b+p}) = \varepsilon + \varepsilon^2 + \varepsilon^3 \&c. \ldots \varepsilon^p \) in which \( n \) is between \( m \) and 1, the nearer \( m \) the better generally, though possibly not universally so throughout the whole limit. And the second method I use for increasing the accuracy, is to adopt an accommodated ratio, or \( \varepsilon^p \), so that \( \frac{1}{L_b} \times (1,05^{L_{b+1}} + 1,05^{-2} L_{b+2} + \&c. \ldots 1,05^p L_{b+p}) = 1,05^{-1} \varepsilon + 1,05^{-2} \varepsilon^2 + 1,05^{-3} \varepsilon^3 \ldots 1,05^{-p} \varepsilon^p \). Another method which might have its peculiar advantage, is to assume \( \varepsilon^p = \frac{L_{b+\frac{1}{2}p}}{L_b} \) under the idea of using a mean ratio.
The General Tables.*
Art. 6. I have had three general tables calculated for fixed periods, Numbers 1, 2, and 3. Number 1, for pe-
* The chief of the arithmetical operations in the constructions of most of the tables were performed under my direction, by Mr. David Jones, of No. 10, King-street, Soho; and, as far as my leisure would allow, I have endeavoured to assure myself of their accuracy by different inspections.
riods of ten years; that is, for $\lambda \left( \frac{1-a^{10}}{a^1-1} \right)$, corresponding to a given value of $\lambda (a^{10})$. No. 2, for seven years, or for $\lambda \left( \frac{1-a^7}{a^1-1} \right)$, corresponding to $\lambda (a^7)$, and the 3d for five years, or for $\lambda \left( \frac{1-a^5}{a^1-1} \right)$, corresponding to $\lambda (a^5)$; calculated (whether $p = 10, 7$ or $5$) for every value of $\lambda (a^p)$, answering to $3,00; 3,01; 3,02$, &c. . . . o. The first column containing the aforesaid value of $\lambda (a^p)$, corresponding to which, in an horizontal line, is placed the log. of $\frac{1-a^p}{a^1-p}$, and between each successive value is placed the difference, retaining a decimal figure more; at the head of the other columns for the proportional parts of the differences, are placed a column showing the number of cyphers to be prefixed to the differences entered in the column following, which are headed
$\{1\ 2\ 3\ 4\ 5\}$ nearest under $1, 2, 3, 4$ and $5$, and opposite the number; suppose $2,16$ of table log. of $a^{10}$, stands
$0275, 0550, 0826, 1101, 1376\}$, the upper, with the addition of the two cyphers, give the proportional parts for $,001, 002, 003, ,004, 005$: and the under, with the two cyphers, shows the proportional parts for $,009, 008, 007, 006$; and the reason of choosing this arrangement, is the advantage which it offers of proof of correctness; thus the sum of the higher an lower numbers of each of the above row with the two cyphers $= 002752$, which is double $,001376$, and equal to the whole difference between the successive terms.
Let it be required to find the logarithm of $\left( \frac{1-a^{10}}{a^1-1} \right)$, corresponding to log. of $a^{10} = 1.7954$. In the General Table I,
Opposite to $1.79$ we have . . . ,88868
For $,005$ we have proportional part . 256
For $,0004$ . ditto . . . . 20
The sum . ,89144 is the answer.
If log. of \(a^p\) is less than \(\bar{3},00\), then it will be necessary to calculate \(\lambda \left( \frac{1-a^p}{a^{-1}-1} \right)\) by common methods, as the tables do not go lower. And generally it will be then sufficient, omitting \(a^p\), only to calculate the value of \(-\lambda (a^{-1}-1)\); but from this, if more accuracy be required, subtract the number whose common logarithm is \((1,6378 + \lambda (r)^p)\).
If \(\lambda \left( \frac{1-a^p}{a^{-1}-1} \right)\) be given, and \(\lambda (a)\) be required, proceed thus, \(\lambda \left( \frac{1-a^{10}}{a^{-1}-1} \right)\) being = ,89144 for example. In Table I, the next value of \(\lambda \left( \frac{1-a^{10}}{a^{-1}-1} \right)\) is ,88868 to which \(\lambda (a^{10})\) corresponding is 1,79
Difference ,00256 belonging to 1,79 gives . . . ,005
Difference ,00020 . . . ditto . . . . ,0004
\[\therefore \text{if } \lambda \left( \frac{1-a^{10}}{a^{-1}-1} \right) = ,89144 \quad \text{then we have } \lambda (a^{10}) = 1,7954\]
If \(\lambda (a^p)\) is less than \(\bar{3}\), proceed thus: put the given value of \(\lambda \left( \frac{1-a^p}{a^{-1}-1} \right) = \lambda q\), and we have the common logarithm of \(a = -p \times \lambda (1 + q^{-1}) + a\) small correction if great accuracy be required; which correction is nearly equal to \(p \times\) the number whose common log. is \(\{1,6378-\lambda q-p+1,(1+q^{-1})\}\)
These methods and tables only apply immediately to \(\lambda \left( \frac{1-a^p}{a^{-1}-1} \right)\) when \(a\) is a proper fraction; but if \(a\) be greater than unity, put it equal to \(b^{-1}\), then will \(b\) be a proper fraction; but \(\frac{1-a^p}{a^{-1}-1} = \frac{a^p-1}{1-a^{-1}} = \frac{b^p-1}{1-b} = b \times \frac{1-b^p}{b^{-1}-1} = a \times \left( \frac{1-b^p}{b^{-1}-1} \right)\); consequently \(\lambda \left( \frac{1-a^p}{a^{-1}-1} \right) = p+1.\lambda (p)+\lambda \left( \frac{1-b^p}{b^{-1}-1} \right)\) I have likewise had Table IV. calculated, which is a general table, for the common log. of \(\left( \frac{1}{a^{-1}-1} \right)\), corresponding to a given value of \(\lambda a\),
commencing with $\lambda (a) = \frac{1}{7}; \frac{1}{701}; \frac{1}{702}, \&c.$ with the differences between them. I have not, in this table, had the proportional parts inserted, though it would be attended with advantage, as the table is not meant to be of general use; but only given to be applied for rough purposes, or where accuracy is not particularly required for calculating at once the value of a life annuity for the whole term of life, or the whole remaining terms of life, after a given term, by considering the present value of each successive payment to form the successive terms of a geometrical progression whose first term and common ratio are each equal to $a$. And as $\lambda \left( \frac{1}{a^L} \right)$ will represent the log. of the sum of the said geometrical progression, it will likewise express approximatively the logarithm of the value required. For many purposes, a table of $\frac{1}{a^L}$, answering to given values of $a$, would be preferable, but not for general purposes.
Art. 7. I have already, in Art. 4 and 5, Chap. II, introduced the term accommodated ratios, or chances, and endeavoured to explain the methods to be adopted to reap the advantage of the ideas there expressed. Table V, for Carlisle, Deparcieux, and Northampton, are the logarithms of tenth terms of the accommodated ratios, or the logarithms of the accommodated chances for living ten years, calculated according to a mode laid down in Art. 5, Chap. II; that is, it expresses for every age, or value of $b$, the logarithm of $\epsilon^{10}$, when $\frac{1}{L_b} \times (1.05^{-1} L_{b+1} + 1.05^{-2} L_{b+2} + \&c. \ldots 1.05^{-p} L_{b+p})$ is equal to $1.05^{-1}\epsilon + 1.05^{-2}\epsilon^2 + \&c. \ldots 1.05^{-10}\epsilon^{10}$. and to show, by example, how these are calculated, let it be required to find the logarithm of the accommodated chance for living
ten years, for the age 20, calculated according to the Carlisle table upon the consideration of interest at 5 per cent. According to the Carlisle tables, I find \( \lambda_{10}^{1} \); that is, the logarithm of the annuity of one pound on a life of 20, for ten years, at 5 per cent = ,87176, and putting \( a = 105^{\beta} \), by hypothesis we shall have \( \lambda_{10}^{a} \); that is the logarithm of \( (a + a^2 + a^3 \ldots a^{10}) = ,87176 \); that is, \( \lambda \left( \frac{1-a^{10}}{a-1} \right) = ,87176 \); hence proceeding, as shown above, to find from General Table I. \( \lambda(a^{10}) \)
Having given . . . ,87176 = \( \lambda \left( \frac{1-a^{10}}{a-1} \right) \)
We have next less = ,86842 corresponding to . . . 1.75
,00334 difference
,00302 proportional part . . . ,006
30 . . . ditto . . . : ,0006
2 . . . ditto . . . : ,00004
,87176 corresponds to . \( \lambda(a^{10}) = 1.75664 \)
\( \lambda(1.05^{10}) = ,21189 \)
1.96853 for the log. of the accommodated chance to live 10 years at the Carlisle mortality.
In the same way may the accommodated chance be found for any other term, when general tables for the term are constructed, and from any other base of interest. I may observe, that by using different rates of interest, as a base for determining the accommodated chances, different degrees of accuracy may be obtained. See Art. 5. Chap. II.
Art. 8. Table VI. is the logarithm of the accommodated chances \( \epsilon \) at every age, \( b \) for living one year, where \( \epsilon \) is of such value that the sum of the geometrical progression \( \frac{\epsilon}{1.05} + \frac{\epsilon^2}{1.05^2} + \&c. \) ad infinitum, or, which is the same thing,
shall be equal to the value of the whole life annuity at five per cent. at such age, namely \( \frac{1}{1.05} \times (1 + \frac{1}{1.05})^b \); consequently \( \frac{1}{1.05} \times (1 + \frac{1}{1.05})^b = \frac{1}{1.05} \times (1 + \frac{1}{1.05})^b \).
This table is constructed for Carlisle, Deparcieux, and Northampton, and is to be used in conjunction with Table IV., where only a rough value of the contingency is required; and though this table applies as the other tables of accommodated chances, to different rates of interest, still it would be of advantage more particularly here for the greater approximation to have similar tables constructed from the formula \( \lambda(\varepsilon) = \lambda(1|b) + \lambda(r^{-1}) - \lambda(0|b) \) for different values of \( r \).
Art. 9. In calculating the value of life annuities for long periods, by means of adding together the values of portions of those periods, the portions of the distant periods contain factors of the real chance of living to these periods, and likewise of the discounted value of the money of which the payment is not immediate; thus if \( t \) be greater than 10,
\[
\begin{align*}
\frac{1}{r} \left[ a, b, c \right] &= \frac{1}{10} \left[ a, b, c \right] + \frac{1}{10} \left[ a, b, c \right] + \frac{L_{10}}{L_{a, b, c}} \cdot r^{10} \times \\
&\quad \frac{(t-10)}{10} \left[ a+10, b+10, c+10 \right].
\end{align*}
\]
It will be therefore convenient to have a table of the logarithm of the real chance of living 10, 20, 30 years, &c. and also for other terms; and some of these are given by Tables VII., VIII., IX.
MDCCCXXV.
Time will not allow me, for the present, to offer more than a very few examples of the method to be employed in calculating by these tables, which are as follow:
Example 1. Required, according to the Carlisle table, the value of a life annuity, for ten years, on the joint lives 30 and 40, at 3 per cent interest.
In Table VIII. for Carlisle, log. of accommodated chance for 10 years, at the age 30 . . . = 1.9552
Ditto 40 . . . = 1.9383
Ditto λ 1.03 . . . = 1.8716
Sum . . 1.7651 = λ (a^10)
In Table I, 1.76 corresponds to . . . .8734
In proportional parts ,005 corresponds to .253
Ditto . . 0001 corresponds to .5
Consequently 1.7651 corresponds to .87604
which is the log. of the required value: the number corresponding to this is 7,5169, for the value of the annuity, according to the Carlisle mortality, at 3 per cent. on the joint lives 30 and 40; and by calculation from Mr. Milne's tables, I find the value should be 7,5168; the difference of the two is evidently insignificant. In this way I calculated the log. of the value of the life annuity, at the Carlisle mortality, at 3 per cent. for 10 years, for the joint lives 0 and 10, 10 and 20, 20 and 30, 30 and 40, 40 and 50, 50 and 60, to be ,76580; ,90247; ,89139; ,87604; ,86295; ,81067; and the annuity, or the numbers corresponding to the said logarithms,
5,8318; 7,9874; 7,7874; 7,5169; 7,2937; 6,4665;
and, according to calculation from Mr. Milne's tables, I get
5,8595; 7,999; 7,7906; 7,5168; 7,2916; 6,4679.
The difference between the two sets is insignificant, except
perhaps in the values of \( \frac{1}{10} \left[ \frac{1}{9}, \frac{1}{10} \right] \); that is, the value of the annuity on the joint life of a child just born, with one of the age of 10, at 3 per cent. Had we divided the period in portions, the value might have been obtained as near as we pleased; or we should likewise have obtained greater accuracy, had we assumed an accommodated chance deduced at a more appropriate interest than 5 per cent. See Art. 5, Chap. II.
Example 2. Let it be required to find the value of a life annuity at 3 per cent. for 10 years, at the Carlisle mortality, for the five lives of the age 20, 30, 40, 45 and 50.
In Table VIII. log. of accom. chance for 10 years at age 20 = 1.9685
Ditto . . . . 30 = 1.9552
Ditto . . . . 40 = 1.9383
Ditto . . . . 45 = 1.9367
Ditto . . . . 50 = 1.9292
\( \lambda_{1.05^{-10}} = 1.8716 \)
\( \lambda(a^{10}) = 1.5995 \)
This sought in Table I.; thus, 1.59 giving .79035
,009 427
,0005 23
gives .79485 the No to which log. is 6,2352
for the value of \( \frac{1}{10} \left[ \frac{1}{20}, \frac{1}{30}, \frac{1}{40}, \frac{1}{45}, \frac{1}{50} \right] \).
Example 3. Let it be required to find the value of \( \frac{1}{10} \left[ b, b+10 \right] \) Carlisle mortality, when \( b = 10 \), that is, for the whole joint lives of 10 and 20. By dividing the whole in portions of ten years, the operation will stand thus for \( \frac{1}{10} \left[ b, b+10 \right] \).
And the present worth of each, or the numbers corresponding to the last logarithms are arranged below.
| For first 10 years | 7.9886 |
|-------------------|--------|
| 2nd ditto | 5.0607 |
| 3d d° | 3.0291 |
| 4th d° | 1.7044 |
| 5th d° | .8071 |
| 6th d° | .2492 |
| 7th d° | .0303 |
| 8th d° | .0007 |
| sum | 18.8701|
As the method by which the logarithms of the present worth of the different portions are found, may not be seen by every reader, I will explain the operation in the third portion; that is, when the logarithm of the portion first found is anticipated for 20 years.
Resume
Table VII. log. of real chance for age
10 living 20 years: 1.94120
Ditto 20 years living: 1.92082
\( \lambda (1,03^{-20}) \): 1.74325
which differs but insignificantly from Mr. Milne's table, which gives 18.873. In a similar way, I find the value of the joint lives for ages 20 and 30, at 3 per cent. and Carlisle mortality to be 16.745; which, according to Mr. Milne's table, should be 16.749; which appears to be an insignificant difference.
Example 4. To find, when particular accuracy is not required, according to the formula for the whole of life,
\[
\frac{1}{a} \left[ \frac{a}{a+10} \right] \text{ at the Carlisle mortality, when } a = 10, 20, 30, \&c.
\]
call the logarithm of accommodated ratios for an unlimited time at the age \( a \), \( R_a \) standing for the accommodated ratio in Table VI. at the age \( a \).
To find the value corresponding to \(1.66658\), not in the table, find the number corresponding complement of the log. \(1.6658\), which number is \(2,159\); subtract \(1\), and find the complement of the log. which is \(= 1.9359165\), whose number is ,8628. Mr. Milne's table gives .979. But as it is not always the same rate of interest which gives the best accommodated ratios, in order to try when, for instance, the interest of money is 3 per cent. what rate of interest should be used in determining the ratios, use the following table:
| Interest | \(\lambda (1.08^{-1} \times 1.03) = 1.979\) |
|----------|---------------------------------------------|
| 1.07 | \(\lambda (1.07^{-1} \times 1.03) = 1.983\) |
| 1.06 | \(\lambda (1.06^{-1} \times 1.03) = 1.987\) |
| 1.05 | \(\lambda (1.05^{-1} \times 1.03) = 1.991\) |
| 1.04 | \(\lambda (1.04^{-1} \times 1.03) = 1.996\) |
* This is not given as a perfect and unerring rule, but as a method in many cases useful, and which would be perfect for the accommodated ratio of one of the lives, if the other lives followed an exact geometrical ratio throughout; and that the real geometrical ratios were in that case used for them, provided that instead of comparing the said sum with the small table, we take for the base of interest the number whose logarithm is — \(\lambda (1.03)\), when the interest is 3 per cent.; and it is to be recollected that the methods is only given as a rough approximation.
Add the logarithm of accommodated ratios, as given in the Table VI. of all the lives but one in question, together, and see which of those rates of interest it nearest agrees with, and use that to calculate the life left, and proceed so for every life; thus for \( \frac{1}{30} \), \( \frac{40}{30} \); to find the rate of interest for 30, I observe that \( R = 1.9899 \) agrees nearest with 6 per cent. in the little table, and \( R = 1.99265 \) agrees nearest with 5 per cent., I therefore take 6 per cent. for the age 30, and for the other I take 5 per cent.: proceed thus:
**Example 5.**
| R if calculated at 6 per cent. | 1.99316 |
|-------------------------------|---------|
| R per table | 1.98991 |
| \( \lambda^{1.05^{-1}} \) | 1.98716 |
| \( \lambda^{1.03^{-1}} \) | 1.97023 |
| Proportionate parts | .00327 |
| To which logarithm | 1.14885 |
| The No corresponding is | 14.088 |
| Instead of | 14.449 |
**Example 6.**
| R at 6 per cent. | 1.99060 |
| R at 6 per cent. | 1.98632 |
| \( \lambda^{1.03^{-1}} \) | 1.98716 |
| \( \lambda^{1.05^{-1}} \) | 1.96408 |
| \( \lambda^{1.05^{-1}} \) | 1.05336 |
| \( \lambda^{1.03^{-1}} \) | .00102 |
| \( \lambda^{1.03^{-1}} \) | 1.06438 |
| Instead of | 11.598 |
**Example 7.**
| \( \lambda^{1.03^{-1}} \) | 1.98759 |
| \( \lambda^{1.03^{-1}} \) | 1.97599 |
| \( \lambda^{1.03^{-1}} \) | 1.98716 |
| \( \lambda^{1.03^{-1}} \) | 1.95074 |
| \( \lambda^{1.03^{-1}} \) | .91357 |
| \( \lambda^{1.03^{-1}} \) | .00687 |
| \( \lambda^{1.03^{-1}} \) | .92044 |
| which log. corresponds | .8318 |
| instead of | .8729 |
I observe that I have not given any table of the logarithm of the accommodated ratios for an unlimited term, except that calculated with 5 per cent. as a radix; but by the assistance of a table of life annuities, for single life at different rates per cent., this will enable us, independent of certain exceptions, to derive the quantity for the same rates per cent. for any radix at the per cent. contained in the second table; thus to find $R$ Carlisle mortality, radix 8 per cent. I look to the Carlisle table of single lives at 8 per cent., and I find the value of the annuity on the life of $50 = 8.987$, I search the age to which this will correspond at 5 per cent. and I find sufficiently nearly $59,82$ for the age corresponding, to which from my table (with the radix at 5 per cent.) for the log. of ratios I find $1.97536$; to this I add log. of $\frac{1.08}{1.05}$; that is, $0.01223$, and we get $1.98759$, the same as given on the other side. This method is accurately consistent with the definition of accommodated ratios for unlimited periods; and if this description of accommodated ratios at a certain rate per cent. be given for one table, for which at the same rate per cent. we have the value of single lives, we may find the same description of accommodated ratios for any other table of mortality for which, at the same rate per cent. we have a table of the value of single lives: thus, suppose the logarithm of this description of accommodated ratios be given for the Carlisle table at five per cent., and the same be required for
$$\frac{1.05^{-1}}{1} \text{ Northampton for the age } 60, \text{ at the same rate}; \frac{1}{1} \text{ Northampton } = 8,392, \text{ this being sought in the Carlisle}$$
table for $\frac{1}{1-x}$ gives $x = 62,41$ for the corresponding age; seek the logarithm of accommodated ratios for an unlimited term, corresponding to this for Carlisle, for the age 6,241, and we have 1.9723, agreeing with the table given.
Previously to concluding this chapter, I shall add a small table, which will be found very useful in the application of the methods here proposed.
| $n$ | Log. of $1,03^{-n}$ | Log. of $1,035^{-n}$ | Log. of $1,04^{-n}$ | Log. of $1,045^{-n}$ | Log. of $1,05^{-n}$ |
|-----|-------------------|-------------------|-------------------|-------------------|-------------------|
| 1 | 1.9871628 | 1.9850597 | 1.9829667 | 1.9808837 | 1.9788107 |
| 2 | 1.9743256 | 1.9701193 | 1.9659333 | 1.9617674 | 1.9576214 |
| 3 | 1.9614883 | 1.9551790 | 1.9489000 | 1.9426511 | 1.9364321 |
| 4 | 1.9480511 | 1.9402386 | 1.9318666 | 1.9235348 | 1.9152428 |
| 5 | 1.9358139 | 1.9252983 | 1.9148333 | 1.9044185 | 1.8940535 |
| 6 | 1.9229767 | 1.9103579 | 1.8978000 | 1.8853023 | 1.8728642 |
| 7 | 1.9101394 | 1.8954176 | 1.8807666 | 1.8661860 | 1.8516749 |
| 8 | 1.8973022 | 1.8804772 | 1.8637333 | 1.8470697 | 1.8304856 |
| 9 | 1.8844650 | 1.8655369 | 1.8466999 | 1.8279534 | 1.8092903 |
| 10 | 1.8716278 | 1.8505965 | 1.8296666 | 1.8088371 | 1.7881070 |
expressive of the law of human mortality, &c.
General Table I. $\lambda(a^{10}), \lambda\left(\frac{1-a^{10}}{a-1}\right)$:
| $a$ | $\lambda(a^{10})$ | $\lambda\left(\frac{1-a^{10}}{a-1}\right)$ |
|-----|------------------|---------------------------------------------|
| 3.00 | 0.0163 | 0.00200 |
| 3.01 | 0.01996 | 0.00200 |
| 3.02 | 0.02003 | 0.00200 |
| 3.03 | 0.02006 | 0.00200 |
| 3.04 | 0.02015 | 0.00200 |
| 3.05 | 0.02020 | 0.00200 |
| 3.06 | 0.02024 | 0.00200 |
| 3.07 | 0.02028 | 0.00200 |
| 3.08 | 0.02034 | 0.00200 |
| 3.09 | 0.02038 | 0.00200 |
| 3.10 | 0.02043 | 0.00200 |
| 3.11 | 0.02047 | 0.00200 |
| 3.12 | 0.02052 | 0.00200 |
| 3.13 | 0.02057 | 0.00200 |
| 3.14 | 0.02061 | 0.00200 |
| 3.15 | 0.02065 | 0.00200 |
| 3.16 | 0.02070 | 0.00200 |
| 3.17 | 0.02073 | 0.00200 |
| 3.18 | 0.02076 | 0.00200 |
| 3.19 | 0.02081 | 0.00200 |
| 3.20 | 0.02086 | 0.00200 |
| 3.21 | 0.02091 | 0.00200 |
| 3.22 | 0.02096 | 0.00200 |
| 3.23 | 0.02101 | 0.00200 |
| 3.24 | 0.02106 | 0.00200 |
MDCCCXXV.
### General Table I. \( \lambda (a^{10}), \lambda \left( \frac{1-a^{10}}{a-1} \right) \)
| \( \lambda (a^{10}) \) | \( \lambda \left( \frac{1-a^{10}}{a-1} \right) \) |
|------------------------|----------------------------------|
| \( \frac{3}{5} \) | \( \frac{10749}{226} \) |
| \( \frac{3}{5} \) | \( \frac{10975}{226} \) |
| \( \frac{3}{5} \) | \( \frac{11201}{227} \) |
| \( \frac{3}{5} \) | \( \frac{11428}{228} \) |
| \( \frac{3}{5} \) | \( \frac{11655}{228} \) |
| \( \frac{3}{5} \) | \( \frac{11883}{229} \) |
| \( \frac{3}{5} \) | \( \frac{12112}{229} \) |
| \( \frac{3}{5} \) | \( \frac{12341}{230} \) |
| \( \frac{3}{5} \) | \( \frac{12571}{231} \) |
| \( \frac{3}{5} \) | \( \frac{12802}{231} \) |
| \( \frac{3}{5} \) | \( \frac{13033}{232} \) |
| \( \frac{3}{5} \) | \( \frac{13265}{233} \) |
| \( \frac{3}{5} \) | \( \frac{13497}{233} \) |
| \( \frac{3}{5} \) | \( \frac{13730}{234} \) |
| \( \frac{3}{5} \) | \( \frac{13964}{235} \) |
| \( \frac{3}{5} \) | \( \frac{14199}{235} \) |
| \( \frac{3}{5} \) | \( \frac{14434}{236} \) |
| \( \frac{3}{5} \) | \( \frac{14670}{237} \) |
| \( \frac{3}{5} \) | \( \frac{14906}{237} \) |
| \( \frac{3}{5} \) | \( \frac{15143}{238} \) |
| \( \frac{3}{5} \) | \( \frac{15381}{239} \) |
| \( \frac{3}{5} \) | \( \frac{15620}{239} \) |
| \( \frac{3}{5} \) | \( \frac{15859}{240} \) |
| \( \frac{3}{5} \) | \( \frac{16099}{241} \) |
| \( \frac{3}{5} \) | \( \frac{16339}{241} \) |
---
This table provides values for \( \lambda (a^{10}) \) and \( \lambda \left( \frac{1-a^{10}}{a-1} \right) \) for various fractions \( \frac{3}{5} \).
### General Table I.
\[ \lambda (a^{10}), \lambda \left( \frac{1-a^{10}}{a-1} \right) \]
| \( \lambda (a^{10}) \) | \( \lambda \left( \frac{1-a^{10}}{a-1} \right) \) | 1 | 2 | 3 | 4 | 5 |
|------------------------|----------------------------------|---|---|---|---|---|
| 2.00 | 0.02856 | 0.0261 | 0.0253 | 0.0784 | 1.046 | 1.307 |
| | | 0.00261 | 0.00261 | 0.00261 | 0.00261 | 0.00261 |
| 2.01 | 0.02856 | 0.0262 | 0.0254 | 0.0786 | 1.048 | 1.310 |
| | | 0.00262 | 0.00262 | 0.00262 | 0.00262 | 0.00262 |
| 2.02 | 0.02856 | 0.0263 | 0.0256 | 0.0789 | 1.052 | 1.315 |
| | | 0.00263 | 0.00263 | 0.00263 | 0.00263 | 0.00263 |
| 2.03 | 0.02856 | 0.0264 | 0.0258 | 0.0791 | 1.055 | 1.319 |
| | | 0.00264 | 0.00264 | 0.00264 | 0.00264 | 0.00264 |
| 2.04 | 0.02856 | 0.0265 | 0.0259 | 0.0794 | 1.059 | 1.324 |
| | | 0.00265 | 0.00265 | 0.00265 | 0.00265 | 0.00265 |
| 2.05 | 0.02856 | 0.0266 | 0.0261 | 0.0797 | 1.062 | 1.328 |
| | | 0.00266 | 0.00266 | 0.00266 | 0.00266 | 0.00266 |
| 2.06 | 0.02856 | 0.0267 | 0.0263 | 0.0799 | 1.065 | 1.332 |
| | | 0.00267 | 0.00267 | 0.00267 | 0.00267 | 0.00267 |
| 2.07 | 0.02856 | 0.0268 | 0.0264 | 0.0801 | 1.068 | 1.335 |
| | | 0.00268 | 0.00268 | 0.00268 | 0.00268 | 0.00268 |
| 2.08 | 0.02856 | 0.0269 | 0.0265 | 0.0803 | 1.071 | 1.339 |
| | | 0.00269 | 0.00269 | 0.00269 | 0.00269 | 0.00269 |
| 2.09 | 0.02856 | 0.0270 | 0.0266 | 0.0805 | 1.074 | 1.343 |
| | | 0.00270 | 0.00270 | 0.00270 | 0.00270 | 0.00270 |
| 2.10 | 0.02856 | 0.0271 | 0.0267 | 0.0807 | 1.077 | 1.347 |
| | | 0.00271 | 0.00271 | 0.00271 | 0.00271 | 0.00271 |
| 2.11 | 0.02856 | 0.0272 | 0.0268 | 0.0809 | 1.080 | 1.351 |
| | | 0.00272 | 0.00272 | 0.00272 | 0.00272 | 0.00272 |
| 2.12 | 0.02856 | 0.0273 | 0.0269 | 0.0811 | 1.083 | 1.355 |
| | | 0.00273 | 0.00273 | 0.00273 | 0.00273 | 0.00273 |
| 2.13 | 0.02856 | 0.0274 | 0.0270 | 0.0813 | 1.086 | 1.359 |
| | | 0.00274 | 0.00274 | 0.00274 | 0.00274 | 0.00274 |
| 2.14 | 0.02856 | 0.0275 | 0.0271 | 0.0815 | 1.089 | 1.363 |
| | | 0.00275 | 0.00275 | 0.00275 | 0.00275 | 0.00275 |
| 2.15 | 0.02856 | 0.0276 | 0.0272 | 0.0817 | 1.092 | 1.367 |
| | | 0.00276 | 0.00276 | 0.00276 | 0.00276 | 0.00276 |
| 2.16 | 0.02856 | 0.0277 | 0.0273 | 0.0819 | 1.095 | 1.371 |
| | | 0.00277 | 0.00277 | 0.00277 | 0.00277 | 0.00277 |
| 2.17 | 0.02856 | 0.0278 | 0.0274 | 0.0821 | 1.098 | 1.375 |
| | | 0.00278 | 0.00278 | 0.00278 | 0.00278 | 0.00278 |
| 2.18 | 0.02856 | 0.0279 | 0.0275 | 0.0823 | 1.101 | 1.379 |
| | | 0.00279 | 0.00279 | 0.00279 | 0.00279 | 0.00279 |
| 2.19 | 0.02856 | 0.0280 | 0.0276 | 0.0825 | 1.104 | 1.383 |
| | | 0.00280 | 0.00280 | 0.00280 | 0.00280 | 0.00280 |
| 2.20 | 0.02856 | 0.0281 | 0.0277 | 0.0827 | 1.107 | 1.387 |
| | | 0.00281 | 0.00281 | 0.00281 | 0.00281 | 0.00281 |
| 2.21 | 0.02856 | 0.0282 | 0.0278 | 0.0829 | 1.110 | 1.391 |
| | | 0.00282 | 0.00282 | 0.00282 | 0.00282 | 0.00282 |
| 2.22 | 0.02856 | 0.0283 | 0.0279 | 0.0831 | 1.113 | 1.395 |
| | | 0.00283 | 0.00283 | 0.00283 | 0.00283 | 0.00283 |
| 2.23 | 0.02856 | 0.0284 | 0.0280 | 0.0833 | 1.116 | 1.399 |
| | | 0.00284 | 0.00284 | 0.00284 | 0.00284 | 0.00284 |
| 2.24 | 0.02856 | 0.0285 | 0.0281 | 0.0835 | 1.119 | 1.403 |
| | | 0.00285 | 0.00285 | 0.00285 | 0.00285 | 0.00285 |
Mr. Gompertz on the nature of the function
General Table I. $\lambda(a^{10}), \lambda\left(\frac{1-a^{10}}{a-1}\right)$.
| $a$ | $\lambda(a^{10})$ | $\lambda\left(\frac{1-a^{10}}{a-1}\right)$ |
|-----|------------------|---------------------------------------------|
| 2.50 | 37058 | 0.00 |
| | | 0.003103 |
| 2.51 | 37368 | 0.003115 |
| | | 0.003126 |
| 2.52 | 37680 | 0.003134 |
| | | 0.003138 |
| 2.53 | 37993 | 0.003149 |
| | | 0.003156 |
| 2.54 | 38306 | 0.003161 |
| | | 0.003173 |
| 2.55 | 38621 | 0.003185 |
| | | 0.003196 |
| 2.56 | 38937 | 0.003208 |
| | | 0.003220 |
| 2.57 | 39255 | 0.003233 |
| | | 0.003245 |
| 2.58 | 39573 | 0.003256 |
| | | 0.003268 |
| 2.59 | 39893 | 0.003279 |
| | | 0.003283 |
| 2.60 | 40213 | 0.003294 |
| | | 0.003306 |
| 2.61 | 40536 | 0.003319 |
| | | 0.003321 |
| 2.62 | 40859 | 0.003332 |
| | | 0.003334 |
| 2.63 | 41183 | 0.003337 |
| | | 0.003338 |
| 2.64 | 41509 | 0.003339 |
| | | 0.003344 |
| 2.65 | 41836 | 0.003347 |
| | | 0.003349 |
| 2.66 | 42164 | 0.003351 |
| | | 0.003355 |
| 2.67 | 42493 | 0.003357 |
| | | 0.003359 |
| 2.68 | 42833 | 0.003361 |
| | | 0.003362 |
| 2.69 | 43156 | 0.003364 |
| | | 0.003365 |
| 2.70 | 43490 | 0.003367 |
| | | 0.003368 |
| 2.71 | 43824 | 0.003369 |
| | | 0.003371 |
| 2.72 | 44159 | 0.003373 |
| | | 0.003375 |
| 2.73 | 44496 | 0.003377 |
| | | 0.003378 |
| 2.74 | 44835 | 0.003379 |
| | | 0.003380 |
| $a$ | $\lambda(a^{10})$ | $\lambda\left(\frac{1-a^{10}}{a-1}\right)$ |
|-----|------------------|---------------------------------------------|
| 2.75 | 45174 | 0.003408 |
| | | 0.003423 |
| 2.76 | 45515 | 0.003435 |
| | | 0.003435 |
| 2.77 | 45857 | 0.003450 |
| | | 0.003450 |
| 2.78 | 46201 | 0.003461 |
| | | 0.003461 |
| 2.79 | 46546 | 0.003475 |
| | | 0.003475 |
| 2.80 | 46892 | 0.003489 |
| | | 0.003489 |
| 2.81 | 47239 | 0.003502 |
| | | 0.003502 |
| 2.82 | 47588 | 0.003516 |
| | | 0.003516 |
| 2.83 | 47938 | 0.003530 |
| | | 0.003530 |
| 2.84 | 48290 | 0.003544 |
| | | 0.003544 |
| 2.85 | 48643 | 0.003558 |
| | | 0.003558 |
| 2.86 | 48997 | 0.003571 |
| | | 0.003571 |
| 2.87 | 49353 | 0.003585 |
| | | 0.003585 |
| 2.88 | 49710 | 0.003600 |
| | | 0.003600 |
| 2.89 | 50060 | 0.003614 |
| | | 0.003614 |
| 2.90 | 50420 | 0.003627 |
| | | 0.003627 |
| 2.91 | 50790 | 0.003642 |
| | | 0.003642 |
| 2.92 | 51153 | 0.003656 |
| | | 0.003656 |
| 2.93 | 51517 | 0.003671 |
| | | 0.003671 |
| 2.94 | 51883 | 0.003685 |
| | | 0.003685 |
| 2.95 | 52250 | 0.003698 |
| | | 0.003698 |
| 2.96 | 52618 | 0.003714 |
| | | 0.003714 |
| 2.97 | 52988 | 0.003729 |
| | | 0.003729 |
| 2.98 | 53360 | 0.003744 |
| | | 0.003744 |
| 2.99 | 53732 | 0.003759 |
| | | 0.003759 |
expressive of the law of human mortality, &c.
General Table I. \( \lambda (a^{10}), \lambda \left( \frac{1-a^{10}}{a^{-1}-1} \right) \).
| \( \lambda (a^{10}) \) | \( \lambda \left( \frac{1-a^{10}}{a^{-1}-1} \right) \) | 1 | 2 | 3 | 4 | 5 |
|----------------------|---------------------------------|---|---|---|---|---|
| 1.00 | 0.54107 | 0.00 | 0.0376 | 0.0752 | 0.1127 | 0.1503 | 0.1879 |
| | | 0.0037558 | 0.03382 | 0.0606 | 0.0963 | 0.1325 | 0.1684 |
| 1.01 | 0.54483 | 0.00 | 0.0377 | 0.0755 | 0.1132 | 0.1510 | 0.1887 |
| | | 0.003774 | 0.03397 | 0.0610 | 0.0968 | 0.1330 | 0.1692 |
| 1.02 | 0.54860 | 0.00 | 0.0379 | 0.0757 | 0.1136 | 0.1515 | 0.1894 |
| | | 0.003787 | 0.03408 | 0.0614 | 0.0972 | 0.1334 | 0.1700 |
| 1.03 | 0.55239 | 0.00 | 0.0380 | 0.0761 | 0.1141 | 0.1521 | 0.1902 |
| | | 0.003803 | 0.03423 | 0.0618 | 0.0976 | 0.1340 | 0.1706 |
| 1.04 | 0.55619 | 0.00 | 0.0382 | 0.0764 | 0.1146 | 0.1528 | 0.1910 |
| | | 0.003819 | 0.03437 | 0.0622 | 0.0980 | 0.1348 | 0.1712 |
| 1.05 | 0.56001 | 0.00 | 0.0383 | 0.0767 | 0.1150 | 0.1533 | 0.1917 |
| | | 0.003833 | 0.03450 | 0.0626 | 0.0984 | 0.1356 | 0.1716 |
| 1.06 | 0.56384 | 0.00 | 0.0385 | 0.0770 | 0.1153 | 0.1540 | 0.1925 |
| | | 0.003849 | 0.03464 | 0.0630 | 0.0988 | 0.1360 | 0.1720 |
| 1.07 | 0.56769 | 0.00 | 0.0386 | 0.0773 | 0.1159 | 0.1546 | 0.1932 |
| | | 0.003864 | 0.03478 | 0.0634 | 0.0992 | 0.1374 | 0.1724 |
| 1.08 | 0.57156 | 0.00 | 0.0388 | 0.0776 | 0.1164 | 0.1552 | 0.1940 |
| | | 0.003880 | 0.03492 | 0.0638 | 0.0996 | 0.1388 | 0.1728 |
| 1.09 | 0.57544 | 0.00 | 0.0389 | 0.0779 | 0.1168 | 0.1558 | 0.1947 |
| | | 0.003894 | 0.03505 | 0.0642 | 0.1000 | 0.1402 | 0.1732 |
| 1.10 | 0.57933 | 0.00 | 0.0391 | 0.0782 | 0.1173 | 0.1564 | 0.1955 |
| | | 0.003910 | 0.03519 | 0.0646 | 0.1004 | 0.1416 | 0.1736 |
| 1.11 | 0.58324 | 0.00 | 0.0393 | 0.0785 | 0.1178 | 0.1570 | 0.1963 |
| | | 0.003925 | 0.03533 | 0.0650 | 0.1008 | 0.1430 | 0.1740 |
| 1.12 | 0.58717 | 0.00 | 0.0394 | 0.0788 | 0.1183 | 0.1577 | 0.1971 |
| | | 0.003942 | 0.03548 | 0.0654 | 0.1012 | 0.1444 | 0.1744 |
| 1.13 | 0.59111 | 0.00 | 0.0396 | 0.0792 | 0.1188 | 0.1584 | 0.1980 |
| | | 0.003959 | 0.03563 | 0.0658 | 0.1016 | 0.1458 | 0.1748 |
| 1.14 | 0.59507 | 0.00 | 0.0397 | 0.0794 | 0.1191 | 0.1588 | 0.1986 |
| | | 0.003971 | 0.03574 | 0.0662 | 0.1020 | 0.1472 | 0.1752 |
| 1.15 | 0.59904 | 0.00 | 0.0399 | 0.0798 | 0.1196 | 0.1595 | 0.1994 |
| | | 0.003988 | 0.03589 | 0.0666 | 0.1024 | 0.1486 | 0.1756 |
| 1.16 | 0.60303 | 0.00 | 0.0401 | 0.0801 | 0.1202 | 0.1602 | 0.2003 |
| | | 0.004005 | 0.03605 | 0.0670 | 0.1028 | 0.1500 | 0.1760 |
| 1.17 | 0.60703 | 0.00 | 0.0402 | 0.0804 | 0.1206 | 0.1608 | 0.2011 |
| | | 0.004021 | 0.03619 | 0.0674 | 0.1032 | 0.1514 | 0.1764 |
| 1.18 | 0.61105 | 0.00 | 0.0404 | 0.0827 | 0.1211 | 0.1615 | 0.2019 |
| | | 0.004037 | 0.03633 | 0.0678 | 0.1036 | 0.1528 | 0.1768 |
| 1.19 | 0.61509 | 0.00 | 0.0405 | 0.0831 | 0.1216 | 0.1621 | 0.2027 |
| | | 0.004053 | 0.03648 | 0.0682 | 0.1040 | 0.1542 | 0.1772 |
| 1.20 | 0.61914 | 0.00 | 0.0407 | 0.0841 | 0.1221 | 0.1628 | 0.2035 |
| | | 0.004069 | 0.03662 | 0.0686 | 0.1044 | 0.1556 | 0.1776 |
| 1.21 | 0.62321 | 0.00 | 0.0409 | 0.0847 | 0.1226 | 0.1634 | 0.2043 |
| | | 0.004085 | 0.03677 | 0.0690 | 0.1048 | 0.1570 | 0.1780 |
| 1.22 | 0.62729 | 0.00 | 0.0410 | 0.0850 | 0.1230 | 0.1640 | 0.2051 |
| | | 0.004101 | 0.03691 | 0.0694 | 0.1052 | 0.1584 | 0.1784 |
| 1.23 | 0.63140 | 0.00 | 0.0412 | 0.0854 | 0.1235 | 0.1647 | 0.2059 |
| | | 0.004118 | 0.03706 | 0.0698 | 0.1056 | 0.1600 | 0.1788 |
| 1.24 | 0.63551 | 0.00 | 0.0413 | 0.0860 | 0.1239 | 0.1652 | 0.2066 |
| | | 0.004131 | 0.03718 | 0.0702 | 0.1060 | 0.1614 | 0.1792 |
Mr. Gompertz on the nature of the function
General Table I. $\lambda (a^{10}), \lambda \left(\frac{1-a^{10}}{a-1}\right)$.
| $a^{10}$ | $\frac{(1-a^{10})}{a-1}$ | 1 | 2 | 3 | 4 | 5 |
|----------|------------------------|---|---|---|---|---|
| 1.50 | 74849 | 00 | 0458 | 0916 | 1374 | 1832 | 2290 |
| | 00458 | 4122 | 3664 | 3206 | 2748 | 2299 |
| 1.51 | 75307 | 0460 | 0920 | 1379 | 1839 | 2299 |
| | 00459 | 4138 | 3678 | 3219 | 2759 | 2308 |
| 1.52 | 75766 | 0462 | 0923 | 1385 | 1846 | 2317 |
| | 00461 | 4154 | 3693 | 3231 | 2770 | 2317 |
| 1.53 | 76228 | 0463 | 0927 | 1390 | 1853 | 2317 |
| | 00463 | 4170 | 3706 | 3243 | 2780 | 2326 |
| 1.54 | 76691 | 0465 | 0930 | 1396 | 1861 | 2326 |
| | 00465 | 4187 | 3722 | 3256 | 2791 | 2326 |
| 1.55 | 77156 | 0467 | 0933 | 1400 | 1867 | 2334 |
| | 00466 | 4200 | 3734 | 3267 | 2800 | 2345 |
| 1.56 | 77623 | 0469 | 0938 | 1407 | 1876 | 2345 |
| | 00469 | 4221 | 3752 | 3283 | 2814 | 2352 |
| 1.57 | 78092 | 0470 | 0941 | 1411 | 1882 | 2352 |
| | 00470 | 4234 | 3763 | 3293 | 2822 | 2362 |
| 1.58 | 78562 | 0472 | 0945 | 1417 | 1890 | 2362 |
| | 00472 | 4252 | 3779 | 3307 | 2834 | 2371 |
| 1.59 | 79035 | 0474 | 0948 | 1422 | 1896 | 2371 |
| | 00474 | 4267 | 3793 | 3319 | 2845 | 2380 |
| 1.60 | 79509 | 0476 | 0952 | 1428 | 1904 | 2380 |
| | 00476 | 4284 | 3808 | 3332 | 2856 | 2389 |
| 1.61 | 79985 | 0478 | 0956 | 1433 | 1911 | 2389 |
| | 00477 | 4300 | 3823 | 3345 | 2867 | 2398 |
| 1.62 | 80463 | 0480 | 0959 | 1439 | 1918 | 2398 |
| | 00479 | 4316 | 3837 | 3357 | 2878 | 2408 |
| 1.63 | 80942 | 0482 | 0963 | 1445 | 1926 | 2408 |
| | 00481 | 4334 | 3852 | 3371 | 2889 | 2417 |
| 1.64 | 81424 | 0483 | 0967 | 1450 | 1933 | 2417 |
| | 00483 | 4350 | 3866 | 3383 | 2900 | 2429 |
| 1.65 | 81907 | 0486 | 0971 | 1457 | 1943 | 2429 |
| | 00485 | 4371 | 3886 | 3400 | 2914 | 2432 |
| 1.66 | 82393 | 0486 | 0973 | 1459 | 1946 | 2432 |
| | 00486 | 4378 | 3891 | 3405 | 2918 | 2444 |
| 1.67 | 82879 | 0489 | 0978 | 1466 | 1955 | 2444 |
| | 00488 | 4399 | 3910 | 3422 | 2933 | 2459 |
| 1.68 | 83368 | 0492 | 0983 | 1475 | 1967 | 2459 |
| | 00491 | 4425 | 3934 | 3442 | 2950 | 2463 |
| 1.69 | 83859 | 0493 | 0985 | 1478 | 1970 | 2463 |
| | 00492 | 4433 | 3940 | 3448 | 2955 | 2472 |
| 1.70 | 84351 | 0494 | 0989 | 1483 | 1978 | 2472 |
| | 00494 | 4450 | 3955 | 3461 | 2966 | 2482 |
| 1.71 | 84846 | 0496 | 0993 | 1489 | 1985 | 2482 |
| | 00496 | 4467 | 3970 | 3474 | 2978 | 2491 |
| 1.72 | 85342 | 0498 | 0996 | 1494 | 1992 | 2491 |
| | 00498 | 4483 | 3985 | 3487 | 2989 | 2500 |
| 1.73 | 85840 | 0500 | 1000 | 1500 | 2000 | 2500 |
| | 00499 | 4499 | 3999 | 3499 | 2999 | 2510 |
| 1.74 | 86340 | 0502 | 1004 | 1506 | 2008 | 2510 |
| | 00501 | 4517 | 4005 | 3513 | 3011 | 2519 |
expressive of the law of human mortality, &c.
General Table II. $\lambda (a^7), \lambda \left( \frac{1-a^7}{a-1} \right)$.
| $a$ | $\lambda (a^7)$ | $\lambda \left( \frac{1-a^7}{a-1} \right)$ |
|-----|-----------------|---------------------------------------------|
| 3 | 1,77356 | 1,83164 |
| | 0,0227 | 0,002385 |
| 3.01| 1,77583 | 1,83403 |
| | 0,0227 | 0,00239 |
| 3.02| 1,77810 | 1,83641 |
| | 0,0227 | 0,002395 |
| 3.03| 1,78038 | 1,83881 |
| | 0,0228 | 0,00240 |
| 3.04| 1,78266 | 1,84121 |
| | 0,0228 | 0,002405 |
| 3.05| 1,78495 | 1,84361 |
| | 0,0229 | 0,002409 |
| 3.06| 1,78724 | 1,84602 |
| | 0,0229 | 0,002415 |
| 3.07| 1,78954 | 1,84844 |
| | 0,0230 | 0,002419 |
| 3.08| 1,79184 | 1,85086 |
| | 0,0230 | 0,002425 |
| 3.09| 1,79414 | 1,85328 |
| | 0,0230 | 0,002432 |
| 3.10| 1,79645 | 1,85571 |
| | 0,0231 | 0,002436 |
| 3.11| 1,79877 | 1,85815 |
| | 0,0231 | 0,002441 |
| 3.12| 1,80108 | 1,86059 |
| | 0,0232 | 0,002447 |
| 3.13| 1,80341 | 1,86304 |
| | 0,0232 | 0,002452 |
| 3.14| 1,80573 | 1,86549 |
| | 0,0233 | 0,002458 |
| 3.15| 1,80806 | 1,86795 |
| | 0,0233 | 0,002462 |
| 3.16| 1,81040 | 1,87041 |
| | 0,0234 | 0,002469 |
| 3.17| 1,81274 | 1,87288 |
| | 0,0234 | 0,002474 |
| 3.18| 1,81509 | 1,87535 |
| | 0,0235 | 0,002479 |
| 3.19| 1,81744 | 1,87783 |
| | 0,0235 | 0,002485 |
| 3.20| 1,81979 | 1,88032 |
| | 0,0236 | 0,002491 |
| 3.21| 1,82215 | 1,88281 |
| | 0,0236 | 0,002497 |
| 3.22| 1,82452 | 1,88530 |
| | 0,0237 | 0,002502 |
| 3.23| 1,82689 | 1,88780 |
| | 0,0237 | 0,002508 |
| 3.24| 1,82926 | 1,89031 |
| | 0,0238 | 0,002514 |
Mr. Gompertz on the nature of the function
General Table II. $\lambda (a^7), \lambda \left( \frac{1-a^7}{a-1} \right)$.
| $a(a^7)$ | $\lambda \left( \frac{1-a^7}{a-1} \right)$ |
|----------|---------------------------------------------|
| 3.50 | 1,89283 |
| | ,00252 |
| | 2268 |
| | 2016 |
| | 1764 |
| | 1512 |
| | 1260 |
| | 3.75 |
| | 1,95767 |
| | ,00268 |
| | 2412 |
| | 2144 |
| | 1876 |
| | 1608 |
| | 1340 |
| 3.51 | 1,89535 |
| | ,00252 |
| | 2273 |
| | 2021 |
| | 1768 |
| | 1516 |
| | 1263 |
| | 3.76 |
| | 1,96035 |
| | ,00268 |
| | 2417 |
| | 2148 |
| | 1880 |
| | 1611 |
| | 1343 |
| 3.52 | 1,89787 |
| | ,00252 |
| | 2278 |
| | 2025 |
| | 1772 |
| | 1519 |
| | 1266 |
| | 3.77 |
| | 1,96303 |
| | ,00269 |
| | 2424 |
| | 2154 |
| | 1885 |
| | 1616 |
| | 1347 |
| 3.53 | 1,90040 |
| | ,00253 |
| | 2283 |
| | 2030 |
| | 1776 |
| | 1522 |
| | 1269 |
| | 3.78 |
| | 1,96573 |
| | ,00270 |
| | 2430 |
| | 2160 |
| | 1890 |
| | 1620 |
| | 1350 |
| 3.54 | 1,90294 |
| | ,00254 |
| | 2289 |
| | 2034 |
| | 1780 |
| | 1526 |
| | 1272 |
| | 3.79 |
| | 1,96843 |
| | ,00270 |
| | 2436 |
| | 2166 |
| | 1895 |
| | 1624 |
| | 1354 |
| 3.55 | 1,90548 |
| | ,00255 |
| | 2295 |
| | 2040 |
| | 1785 |
| | 1530 |
| | 1275 |
| | 3.80 |
| | 1,97113 |
| | ,00271 |
| | 2442 |
| | 2170 |
| | 1899 |
| | 1628 |
| | 1357 |
| 3.56 | 1,90803 |
| | ,00255 |
| | 2300 |
| | 2044 |
| | 1789 |
| | 1533 |
| | 1278 |
| | 3.81 |
| | 1,97385 |
| | ,00272 |
| | 2449 |
| | 2177 |
| | 1905 |
| | 1633 |
| | 1361 |
| 3.57 | 1,91059 |
| | ,00256 |
| | 2306 |
| | 2052 |
| | 1793 |
| | 1537 |
| | 1281 |
| | 3.82 |
| | 1,97657 |
| | ,00272 |
| | 2456 |
| | 2183 |
| | 1910 |
| | 1637 |
| | 1365 |
| 3.58 | 1,91315 |
| | ,00256 |
| | 2311 |
| | 2054 |
| | 1798 |
| | 1541 |
| | 1284 |
| | 3.83 |
| | 1,97930 |
| | ,00273 |
| | 2461 |
| | 2187 |
| | 1914 |
| | 1640 |
| | 1367 |
| 3.59 | 1,91572 |
| | ,00257 |
| | 2317 |
| | 2059 |
| | 1802 |
| | 1544 |
| | 1287 |
| | 3.84 |
| | 1,98203 |
| | ,00274 |
| | 2469 |
| | 2194 |
| | 1920 |
| | 1646 |
| | 1372 |
| 3.60 | 1,91829 |
| | ,00258 |
| | 2322 |
| | 2064 |
| | 1806 |
| | 1548 |
| | 1290 |
| | 3.85 |
| | 1,98477 |
| | ,00275 |
| | 2475 |
| | 2200 |
| | 1925 |
| | 1650 |
| | 1375 |
| 3.61 | 1,92087 |
| | ,00258 |
| | 2329 |
| | 2070 |
| | 1811 |
| | 1552 |
| | 1294 |
| | 3.86 |
| | 1,98752 |
| | ,00275 |
| | 2482 |
| | 2206 |
| | 1931 |
| | 1655 |
| | 1379 |
| 3.62 | 1,92346 |
| | ,00259 |
| | 2336 |
| | 2074 |
| | 1814 |
| | 1555 |
| | 1296 |
| | 3.87 |
| | 1,99028 |
| | ,00276 |
| | 2489 |
| | 2212 |
| | 1936 |
| | 1659 |
| | 1383 |
| 3.63 | 1,92605 |
| | ,00259 |
| | 2339 |
| | 2079 |
| | 1819 |
| | 1559 |
| | 1300 |
| | 3.88 |
| | 1,99305 |
| | ,00277 |
| | 2496 |
| | 2218 |
| | 1941 |
| | 1664 |
| | 1387 |
| 3.64 | 1,92865 |
| | ,00260 |
| | 2345 |
| | 2085 |
| | 1824 |
| | 1564 |
| | 1303 |
| | 3.89 |
| | 1,99582 |
| | ,00278 |
| | 2502 |
| | 2224 |
| | 1946 |
| | 1668 |
| | 1390 |
| 3.65 | 1,93125 |
| | ,00261 |
| | 2351 |
| | 2090 |
| | 1828 |
| | 1567 |
| | 1306 |
| | 3.90 |
| | 1,99860 |
| | ,00278 |
| | 2509 |
| | 2230 |
| | 1952 |
| | 1673 |
| | 1394 |
| 3.66 | 1,93387 |
| | ,00261 |
| | 2356 |
| | 2094 |
| | 1833 |
| | 1571 |
| | 1309 |
| | 3.91 |
| | 1,00139 |
| | ,00279 |
| | 2516 |
| | 2230 |
| | 1957 |
| | 1677 |
| | 1398 |
| 3.67 | 1,93648 |
| | ,00262 |
| | 2363 |
| | 2100 |
| | 1838 |
| | 1575 |
| | 1313 |
| | 3.92 |
| | 1,00418 |
| | ,00280 |
| | 2523 |
| | 2242 |
| | 1962 |
| | 1682 |
| | 1402 |
| 3.68 | 1,93911 |
| | ,00263 |
| | 2369 |
| | 2106 |
| | 1842 |
| | 1579 |
| | 1316 |
| | 3.93 |
| | 1,00699 |
| | ,00281 |
| | 2530 |
| | 2249 |
| | 1968 |
| | 1687 |
| | 1406 |
| 3.69 | 1,94174 |
| | ,00263 |
| | 2374 |
| | 2110 |
| | 1847 |
| | 1583 |
| | 1319 |
| | 3.94 |
| | 1,00980 |
| | ,00281 |
| | 2537 |
| | 2255 |
| | 1973 |
| | 1691 |
| | 1410 |
| 3.70 | 1,94438 |
| | ,00264 |
| | 2380 |
| | 2115 |
| | 1851 |
| | 1586 |
| | 1322 |
| | 3.95 |
| | 1,01262 |
| | ,00282 |
| | 2544 |
| | 2262 |
| | 1979 |
| | 1696 |
| | 1414 |
| 3.71 | 1,94702 |
| | ,00265 |
| | 2386 |
| | 2121 |
| | 1856 |
| | 1591 |
| | 1326 |
| | 3.96 |
| | 1,01544 |
| | ,00283 |
| | 2551 |
| | 2267 |
| | 1984 |
| | 1700 |
| | 1417 |
| 3.72 | 1,94967 |
| | ,00266 |
| | 2392 |
| | 2126 |
| | 1861 |
| | 1595 |
| | 1329 |
| | 3.97 |
| | 1,01828 |
| | ,00284 |
| | 2558 |
| | 2274 |
| | 1989 |
| | 1705 |
| | 1421 |
| 3.73 | 1,95233 |
| | ,00266 |
| | 2399 |
| | 2132 |
| | 1866 |
| | 1599 |
| | 1333 |
| | 3.98 |
| | 1,02112 |
| | ,00285 |
| | 2565 |
| | 2280 |
| | 1995 |
| | 1710 |
| | 1425 |
| 3.74 | 1,95500 |
| | ,00267 |
| | 2405 |
| | 2138 |
| | 1870 |
| | 1603 |
| | 1336 |
| | 3.99 |
| | 1,02397 |
| | ,00285 |
| | 2573 |
| | 2287 |
| | 2001 |
| | 1715 |
| | 1430 |
### General Table II. \( \lambda (a') \), \( \lambda \left( \frac{1-a^7}{a^{-1}-1} \right) \).
| \( \lambda (a') \) | \( \lambda \left( \frac{1-a^7}{a^{-1}-1} \right) \) | 1 | 2 | 3 | 4 | 5 |
|-------------------|---------------------------------|---|---|---|---|---|
| 2.00 | ,02683 | 00 | 0287 | 0573 | 0860 | 1147 | 1434 |
| | ,00286,7 | 2580 | 2294 | 2007 | 1720 | 1438 |
| 2.01 | ,02969 | 0288 | 0575 | 0863 | 1150 | 1438 |
| | ,00287,5 | 2588 | 2300 | 2013 | 1725 | 1442 |
| 2.02 | ,03257 | 0288 | 0577 | 0865 | 1153 | 1442 |
| | ,00288,3 | 2595 | 2306 | 2018 | 1730 | 1450 |
| 2.03 | ,03545 | 0289 | 0578 | 0867 | 1156 | 1446 |
| | ,00289,1 | 2602 | 2313 | 2024 | 1735 | 1450 |
| 2.04 | ,03834 | 0290 | 0580 | 0870 | 1160 | 1450 |
| | ,00290,0 | 2610 | 2320 | 2030 | 1740 | 1450 |
| 2.05 | ,04124 | 0291 | 0582 | 0872 | 1163 | 1454 |
| | ,00290,8 | 2617 | 2326 | 2036 | 1745 | 1459 |
| 2.06 | ,04415 | 0292 | 0583 | 0875 | 1167 | 1459 |
| | ,00291,7 | 2625 | 2334 | 2043 | 1750 | 1463 |
| 2.07 | ,04707 | 0293 | 0585 | 0878 | 1170 | 1463 |
| | ,00292,5 | 2633 | 2340 | 2048 | 1755 | 1467 |
| 2.08 | ,04999 | 0293 | 0587 | 0880 | 1174 | 1467 |
| | ,00293,4 | 2641 | 2347 | 2054 | 1760 | 1471 |
| 2.09 | ,05293 | 0294 | 0588 | 0883 | 1177 | 1471 |
| | ,00294,2 | 2648 | 2354 | 2059 | 1765 | 1471 |
| 2.10 | ,05587 | 0295 | 0590 | 0885 | 1180 | 1476 |
| | ,00295,1 | 2656 | 2361 | 2066 | 1771 | 1480 |
| 2.11 | ,05882 | 0296 | 0592 | 0888 | 1184 | 1480 |
| | ,00296,0 | 2664 | 2368 | 2072 | 1776 | 1484 |
| 2.12 | ,06178 | 0297 | 0594 | 0890 | 1187 | 1484 |
| | ,00296,8 | 2671 | 2374 | 2078 | 1781 | 1489 |
| 2.13 | ,06475 | 0298 | 0596 | 0893 | 1191 | 1489 |
| | ,00297,8 | 2680 | 2382 | 2085 | 1787 | 1493 |
| 2.14 | ,06772 | 0299 | 0597 | 0896 | 1194 | 1493 |
| | ,00298,6 | 2687 | 2389 | 2090 | 1792 | 1493 |
| 2.15 | ,07071 | 0300 | 0599 | 0899 | 1198 | 1498 |
| | ,00299,5 | 2696 | 2396 | 2097 | 1797 | 1502 |
| 2.16 | ,07370 | 0300 | 0601 | 0901 | 1202 | 1502 |
| | ,00300,4 | 2704 | 2403 | 2103 | 1802 | 1507 |
| 2.17 | ,07671 | 0301 | 0603 | 0904 | 1205 | 1507 |
| | ,00301,3 | 2712 | 2410 | 2109 | 1808 | 1511 |
| 2.18 | ,07972 | 0302 | 0604 | 0907 | 1209 | 1511 |
| | ,00302,2 | 2720 | 2418 | 2115 | 1813 | 1516 |
| 2.19 | ,08274 | 0303 | 0606 | 0909 | 1212 | 1516 |
| | ,00303,1 | 2728 | 2425 | 2122 | 1819 | 1521 |
| 2.20 | ,08577 | 0304 | 0608 | 0912 | 1216 | 1521 |
| | ,00304,1 | 2737 | 2433 | 2129 | 1825 | 1525 |
| 2.21 | ,08882 | 0305 | 0610 | 0915 | 1220 | 1525 |
| | ,00305,0 | 2745 | 2440 | 2135 | 1830 | 1530 |
| 2.22 | ,09187 | 0306 | 0612 | 0918 | 1224 | 1530 |
| | ,00305,9 | 2753 | 2447 | 2141 | 1835 | 1535 |
| 2.23 | ,09493 | 0307 | 0614 | 0921 | 1228 | 1535 |
| | ,00306,9 | 2762 | 2455 | 2148 | 1841 | 1539 |
| 2.24 | ,09799 | 0308 | 0616 | 0923 | 1231 | 1539 |
| | ,00307,8 | 2770 | 2462 | 2155 | 1847 | 1544 |
MDCCXXXV.
Mr. Gompertz on the nature of the function
General Table II. $\lambda (a^7), \lambda \left( \frac{1-a^7}{a^{-1}-1} \right)$.
| $a$ | $\lambda (a^7)$ | $\lambda \left( \frac{1-a^7}{a^{-1}-1} \right)$ |
|-----|-----------------|-----------------------------------------------|
| 2.50 | 18130 | 00335 |
| | 00334.7 | 0669 |
| | | 1004 |
| | | 1339 |
| | | 1674 |
| 2.51 | 18464 | 0336 |
| | 00335.8 | 0672 |
| | | 1017 |
| | | 1343 |
| | | 1679 |
| 2.52 | 18880 | 0337 |
| | 00336.9 | 0674 |
| | | 1011 |
| | | 1348 |
| | | 1685 |
| 2.53 | 19137 | 0338 |
| | 00338.1 | 0676 |
| | | 1014 |
| | | 1352 |
| | | 1691 |
| 2.54 | 19475 | 0339 |
| | 00339.2 | 0678 |
| | | 1018 |
| | | 1357 |
| | | 1696 |
| 2.55 | 19815 | 0340 |
| | 00340.4 | 0681 |
| | | 1021 |
| | | 1362 |
| | | 1702 |
| 2.56 | 20155 | 0342 |
| | 00341.5 | 0683 |
| | | 1025 |
| | | 1366 |
| | | 1708 |
| 2.57 | 20496 | 0343 |
| | 00342.7 | 0685 |
| | | 1028 |
| | | 1371 |
| | | 1714 |
| 2.58 | 20839 | 0344 |
| | 00343.9 | 0688 |
| | | 1032 |
| | | 1376 |
| | | 1720 |
| 2.59 | 21183 | 0345 |
| | 00345.0 | 0690 |
| | | 1035 |
| | | 1380 |
| | | 1725 |
| 2.60 | 21528 | 0346 |
| | 00346.2 | 0692 |
| | | 1039 |
| | | 1385 |
| | | 1731 |
| 2.61 | 21874 | 0347 |
| | 00347.4 | 0695 |
| | | 1042 |
| | | 1390 |
| | | 1737 |
| 2.62 | 22222 | 0349 |
| | 00348.7 | 0697 |
| | | 1046 |
| | | 1395 |
| | | 1744 |
| 2.63 | 22570 | 0350 |
| | 00349.7 | 0699 |
| | | 1049 |
| | | 1399 |
| | | 1749 |
| 2.64 | 22920 | 0351 |
| | 00351.1 | 0702 |
| | | 1053 |
| | | 1404 |
| | | 1756 |
| 2.65 | 23271 | 0352 |
| | 00352.3 | 0705 |
| | | 1057 |
| | | 1409 |
| | | 1762 |
| 2.66 | 23623 | 0354 |
| | 00353.5 | 0707 |
| | | 1061 |
| | | 1414 |
| | | 1768 |
| 2.67 | 23977 | 0355 |
| | 00354.7 | 0709 |
| | | 1064 |
| | | 1419 |
| | | 1774 |
| 2.68 | 24332 | 0356 |
| | 00356.0 | 0712 |
| | | 1068 |
| | | 1424 |
| | | 1780 |
| 2.69 | 24688 | 0357 |
| | 00357.2 | 0714 |
| | | 1072 |
| | | 1429 |
| | | 1786 |
| 2.70 | 25045 | 0358 |
| | 00358.4 | 0717 |
| | | 1075 |
| | | 1434 |
| | | 1792 |
| 2.71 | 25403 | 0360 |
| | 00359.7 | 0719 |
| | | 1079 |
| | | 1439 |
| | | 1799 |
| 2.72 | 25763 | 0361 |
| | 00361.0 | 0722 |
| | | 1083 |
| | | 1444 |
| | | 1805 |
| 2.73 | 26124 | 0362 |
| | 00362.2 | 0724 |
| | | 1087 |
| | | 1449 |
| | | 1811 |
| 2.74 | 26486 | 0364 |
| | 00363.5 | 0727 |
| | | 1091 |
| | | 1454 |
| | | 1817 |
| 2.75 | 26850 | 0365 |
| | 00364.7 | 0729 |
| | | 1094 |
| | | 1459 |
| | | 1824 |
| 2.76 | 27214 | 0366 |
| | 00366.2 | 0732 |
| | | 1099 |
| | | 1465 |
| | | 1831 |
| 2.77 | 27581 | 0367 |
| | 00367.4 | 0735 |
| | | 1102 |
| | | 1470 |
| | | 1837 |
| 2.78 | 27948 | 0369 |
| | 00368.7 | 0737 |
| | | 1106 |
| | | 1475 |
| | | 1844 |
| 2.79 | 28317 | 0370 |
| | 00370.1 | 0740 |
| | | 1110 |
| | | 1480 |
| | | 1851 |
| 2.80 | 28687 | 0370 |
| | 00371.3 | 0743 |
| | | 1114 |
| | | 1485 |
| | | 1857 |
| 2.81 | 29058 | 0373 |
| | 00372.8 | 0746 |
| | | 1118 |
| | | 1491 |
| | | 1864 |
| 2.82 | 29431 | 0374 |
| | 00373.9 | 0748 |
| | | 1122 |
| | | 1496 |
| | | 1870 |
| 2.83 | 29805 | 0375 |
| | 00375.4 | 0751 |
| | | 1126 |
| | | 1502 |
| | | 1877 |
| 2.84 | 30180 | 0377 |
| | 00376.7 | 0753 |
| | | 1130 |
| | | 1507 |
| | | 1884 |
| 2.85 | 30557 | 0378 |
| | 00378.1 | 0756 |
| | | 1134 |
| | | 1512 |
| | | 1891 |
| 2.86 | 30935 | 0379 |
| | 00379.4 | 0759 |
| | | 1138 |
| | | 1518 |
| | | 1897 |
| 2.87 | 31314 | 0381 |
| | 00380.8 | 0762 |
| | | 1142 |
| | | 1523 |
| | | 1904 |
| 2.88 | 31695 | 0382 |
| | 00382.2 | 0764 |
| | | 1146 |
| | | 1529 |
| | | 1911 |
| 2.89 | 32077 | 0384 |
| | 00383.6 | 0767 |
| | | 1151 |
| | | 1534 |
| | | 1918 |
| 2.90 | 32461 | 0385 |
| | 00385.0 | 0770 |
| | | 1155 |
| | | 1540 |
| | | 1925 |
| 2.91 | 32846 | 0386 |
| | 00386.3 | 0773 |
| | | 1159 |
| | | 1545 |
| | | 1932 |
| 2.92 | 33232 | 0388 |
| | 00387.8 | 0776 |
| | | 1163 |
| | | 1551 |
| | | 1939 |
| 2.93 | 33620 | 0389 |
| | 00389.2 | 0778 |
| | | 1168 |
| | | 1557 |
| | | 1946 |
| 2.94 | 34009 | 0391 |
| | 00390.6 | 0781 |
| | | 1172 |
| | | 1562 |
| | | 1953 |
| 2.95 | 34400 | 0392 |
| | 00392.0 | 0784 |
| | | 1176 |
| | | 1568 |
| | | 1960 |
| 2.96 | 34792 | 0394 |
| | 00393.5 | 0787 |
| | | 1181 |
| | | 1574 |
| | | 1968 |
| 2.97 | 35185 | 0395 |
| | 00394.9 | 0790 |
| | | 1185 |
| | | 1580 |
| | | 1975 |
| 2.98 | 35580 | 0396 |
| | 00396.3 | 0793 |
| | | 1189 |
| | | 1585 |
| | | 1982 |
| 2.99 | 35976 | 0398 |
| | 00397.8 | 0796 |
| | | 1193 |
| | | 1591 |
| | | 1989 |
| λ (a^7) | λ (1 - a^7) / (a^-1 - 1) | 1 | 2 | 3 | 4 | 5 |
|---------|--------------------------|---|---|---|---|---|
| 1.00 | 0.36374 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.01 | 0.36773 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.02 | 0.37174 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.03 | 0.37576 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.04 | 0.37980 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.05 | 0.38385 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.06 | 0.38792 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.07 | 0.39200 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.08 | 0.39610 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.09 | 0.40021 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.10 | 0.40434 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.11 | 0.40842 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.12 | 0.41254 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.13 | 0.41681 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.14 | 0.42100 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.15 | 0.42520 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.16 | 0.42942 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.17 | 0.43366 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.18 | 0.43791 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.19 | 0.44218 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.20 | 0.44646 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.21 | 0.45076 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.22 | 0.45507 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.23 | 0.45940 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
| 1.24 | 0.46375 | 0.00 | 0.399 | 0.799 | 1.198 | 1.597 | 1.997 |
...
## General Table II. $\lambda (a^7), \lambda \left(\frac{1-a^7}{a-1}\right)$
| $a$ | $\lambda (a^7)$ | $\frac{1-a^7}{a-1}$ | $a$ | $\lambda (a^7)$ | $\frac{1-a^7}{a-1}$ |
|-----|----------------|---------------------|-----|----------------|---------------------|
| 1.50 | 0.58262 | 0.00 | 1.75 | 0.70810 | 0.00 |
| | 0.00480 | | | 0.00526 | |
| 1.51 | 0.58742 | | | 0.00573 | |
| | 0.00482 | | | 0.00527 | |
| 1.52 | 0.59225 | | | 0.00527 | |
| | 0.00483 | | | 0.00529 | |
| 1.53 | 0.59709 | | | 0.00531 | |
| | 0.00485 | | | 0.00533 | |
| 1.54 | 0.60194 | | | 0.00533 | |
| | 0.00487 | | | 0.00533 | |
| 1.55 | 0.60682 | | | 0.00534 | |
| | 0.00489 | | | 0.00534 | |
| 1.56 | 0.61171 | | | 0.00536 | |
| | 0.00491 | | | 0.00536 | |
| 1.57 | 0.61662 | | | 0.00536 | |
| | 0.00492 | | | 0.00538 | |
| 1.58 | 0.62155 | | | 0.00540 | |
| | 0.00494 | | | 0.00540 | |
| 1.59 | 0.62649 | | | 0.00542 | |
| | 0.00496 | | | 0.00542 | |
| 1.60 | 0.63146 | | | 0.00544 | |
| | 0.00498 | | | 0.00544 | |
| 1.61 | 0.63644 | | | 0.00546 | |
| | 0.00500 | | | 0.00546 | |
| 1.62 | 0.64144 | | | 0.00548 | |
| | 0.00501 | | | 0.00548 | |
| 1.63 | 0.64646 | | | 0.00550 | |
| | 0.00503 | | | 0.00550 | |
| 1.64 | 0.65150 | | | 0.00551 | |
| | 0.00505 | | | 0.00551 | |
| 1.65 | 0.65655 | | | 0.00553 | |
| | 0.00507 | | | 0.00553 | |
| 1.66 | 0.66162 | | | 0.00555 | |
| | 0.00509 | | | 0.00555 | |
| 1.67 | 0.66671 | | | 0.00557 | |
| | 0.00511 | | | 0.00557 | |
| 1.68 | 0.67182 | | | 0.00559 | |
| | 0.00512 | | | 0.00559 | |
| 1.69 | 0.67695 | | | 0.00561 | |
| | 0.00514 | | | 0.00561 | |
| 1.70 | 0.68209 | | | 0.00563 | |
| | 0.00516 | | | 0.00563 | |
| 1.71 | 0.68726 | | | 0.00565 | |
| | 0.00518 | | | 0.00565 | |
| 1.72 | 0.69244 | | | 0.00568 | |
| | 0.00520 | | | 0.00568 | |
| 1.73 | 0.69764 | | | 0.00570 | |
| | 0.00522 | | | 0.00570 | |
| 1.74 | 0.70286 | | | 0.00572 | |
| | 0.00523 | | | 0.00572 | |
### General Table III. \( \lambda (a^5), \lambda \left( \frac{1-a^5}{a-1} \right) \).
| \( \lambda (a^5) \) | \( \lambda \left( \frac{1-a^5}{a-1} \right) \) | 1 | 2 | 3 | 4 | 5 |
|---------------------|---------------------------------|---|---|---|---|---|
| 3.00 | 1,52519 | 0.0266 | 0.533 | 0.799 | 1.065 | 1.332 |
| | 0.0026663 | 2.397 | 2.130 | 1.864 | 1.598 | |
| 3.01 | 1,52786 | 0.0267 | 0.533 | 0.800 | 1.066 | 1.333 |
| | 0.0026665 | 2.399 | 2.132 | 1.866 | 1.599 | |
| 3.02 | 1,53052 | 0.0267 | 0.534 | 0.802 | 1.069 | 1.336 |
| | 0.002672 | 2.405 | 2.138 | 1.870 | 1.603 | |
| 3.03 | 1,53319 | 0.0267 | 0.535 | 0.803 | 1.070 | 1.338 |
| | 0.002675 | 2.408 | 2.140 | 1.873 | 1.605 | |
| 3.04 | 1,53587 | 0.0267 | 0.536 | 0.804 | 1.072 | 1.340 |
| | 0.002679 | 2.411 | 2.143 | 1.875 | 1.607 | |
| 3.05 | 1,53855 | 0.0268 | 0.537 | 0.805 | 1.073 | 1.342 |
| | 0.002683 | 2.415 | 2.146 | 1.878 | 1.610 | |
| 3.06 | 1,54123 | 0.0269 | 0.537 | 0.806 | 1.075 | 1.344 |
| | 0.002687 | 2.418 | 2.150 | 1.881 | 1.612 | |
| 3.07 | 1,54392 | 0.0269 | 0.538 | 0.807 | 1.076 | 1.346 |
| | 0.002691 | 2.422 | 2.153 | 1.884 | 1.615 | |
| 3.08 | 1,54661 | 0.0270 | 0.539 | 0.809 | 1.078 | 1.348 |
| | 0.002695 | 2.426 | 2.156 | 1.887 | 1.617 | |
| 3.09 | 1,54930 | 0.0270 | 0.540 | 0.810 | 1.080 | 1.350 |
| | 0.002699 | 2.429 | 2.159 | 1.889 | 1.619 | |
| 3.10 | 1,55200 | 0.0270 | 0.541 | 0.811 | 1.081 | 1.352 |
| | 0.002703 | 2.433 | 2.162 | 1.892 | 1.622 | |
| 3.11 | 1,55470 | 0.0271 | 0.541 | 0.812 | 1.083 | 1.354 |
| | 0.002707 | 2.436 | 2.166 | 1.895 | 1.624 | |
| 3.12 | 1,55741 | 0.0271 | 0.542 | 0.814 | 1.085 | 1.356 |
| | 0.002712 | 2.441 | 2.170 | 1.898 | 1.627 | |
| 3.13 | 1,56012 | 0.0272 | 0.543 | 0.815 | 1.086 | 1.358 |
| | 0.002716 | 2.444 | 2.173 | 1.901 | 1.630 | |
| 3.14 | 1,56284 | 0.0272 | 0.544 | 0.816 | 1.088 | 1.360 |
| | 0.002720 | 2.448 | 2.176 | 1.904 | 1.632 | |
| 3.15 | 1,56556 | 0.0273 | 0.545 | 0.818 | 1.090 | 1.363 |
| | 0.002725 | 2.453 | 2.180 | 1.908 | 1.635 | |
| 3.16 | 1,56828 | 0.0273 | 0.546 | 0.819 | 1.092 | 1.365 |
| | 0.002729 | 2.456 | 2.183 | 1.910 | 1.637 | |
| 3.17 | 1,57101 | 0.0273 | 0.547 | 0.820 | 1.093 | 1.367 |
| | 0.002733 | 2.460 | 2.186 | 1.913 | 1.640 | |
| 3.18 | 1,57375 | 0.0274 | 0.548 | 0.821 | 1.095 | 1.369 |
| | 0.002738 | 2.464 | 2.190 | 1.916 | 1.643 | |
| 3.19 | 1,57648 | 0.0274 | 0.548 | 0.823 | 1.097 | 1.371 |
| | 0.002742 | 2.468 | 2.194 | 1.919 | 1.645 | |
| 3.20 | 1,57923 | 0.0275 | 0.549 | 0.824 | 1.099 | 1.374 |
| | 0.002747 | 2.472 | 2.198 | 1.923 | 1.648 | |
| 3.21 | 1,58197 | 0.0275 | 0.550 | 0.825 | 1.100 | 1.376 |
| | 0.002751 | 2.476 | 2.201 | 1.926 | 1.651 | |
| 3.22 | 1,58472 | 0.0276 | 0.551 | 0.827 | 1.102 | 1.378 |
| | 0.002756 | 2.480 | 2.205 | 1.929 | 1.654 | |
| 3.23 | 1,58748 | 0.0276 | 0.552 | 0.828 | 1.104 | 1.380 |
| | 0.002760 | 2.484 | 2.208 | 1.932 | 1.656 | |
| 3.24 | 1,59024 | 0.0277 | 0.553 | 0.830 | 1.106 | 1.383 |
| | 0.002765 | 2.489 | 2.212 | 1.936 | 1.659 | |
---
**Note:** The table provides values for \( \lambda (a^5) \) and \( \lambda \left( \frac{1-a^5}{a-1} \right) \) for various values of \( a \).
Mr. Gompertz on the nature of the function
General Table III. $\lambda (a^5), \lambda \left( \frac{1-a^5}{a-1} \right)$.
| $a$ | $\lambda (a^5)$ | $\lambda \left( \frac{1-a^5}{a-1} \right)$ |
|-----|----------------|---------------------------------------------|
| | | |
| 3.50| 1.66371 | 0.00 |
| | 0.02896 | |
| 3.51| 1.66661 | 0.00 |
| | 0.02901 | |
| 3.52| 1.66951 | 0.00 |
| | 0.02907 | |
| 3.53| 1.67242 | 0.00 |
| | 0.02914 | |
| 3.54| 1.67533 | 0.00 |
| | 0.02918 | |
| 3.55| 1.67825 | 0.00 |
| | 0.02924 | |
| 3.56| 1.68117 | 0.00 |
| | 0.02929 | |
| 3.57| 1.68410 | 0.00 |
| | 0.02937 | |
| 3.58| 1.68704 | 0.00 |
| | 0.02941 | |
| 3.59| 1.68998 | 0.00 |
| | 0.02947 | |
| 3.60| 1.69293 | 0.00 |
| | 0.02953 | |
| 3.61| 1.69588 | 0.00 |
| | 0.02959 | |
| 3.62| 1.69884 | 0.00 |
| | 0.02965 | |
| 3.63| 1.70180 | 0.00 |
| | 0.02971 | |
| 3.64| 1.70477 | 0.00 |
| | 0.02977 | |
| 3.65| 1.70775 | 0.00 |
| | 0.02984 | |
| 3.66| 1.71074 | 0.00 |
| | 0.02988 | |
| 3.67| 1.71372 | 0.00 |
| | 0.02996 | |
| 3.68| 1.71672 | 0.00 |
| | 0.03002 | |
| 3.69| 1.71972 | 0.00 |
| | 0.03008 | |
| 3.70| 1.72273 | 0.00 |
| | 0.03014 | |
| 3.71| 1.72574 | 0.00 |
| | 0.03020 | |
| 3.72| 1.72876 | 0.00 |
| | 0.03027 | |
| 3.73| 1.73179 | 0.00 |
| | 0.03034 | |
| 3.74| 1.73483 | 0.00 |
| | 0.03040 | |
| $a$ | $\lambda (a^5)$ | $\lambda \left( \frac{1-a^5}{a-1} \right)$ |
|-----|----------------|---------------------------------------------|
| | | |
| 3.75| 1.73787 | 0.00 |
| | 0.03047 | |
| 3.76| 1.74091 | 0.00 |
| | 0.03053 | |
| 3.77| 1.74393 | 0.00 |
| | 0.03060 | |
| 3.78| 1.74702 | 0.00 |
| | 0.03066 | |
| 3.79| 1.75009 | 0.00 |
| | 0.03074 | |
| 3.80| 1.75316 | 0.00 |
| | 0.03080 | |
| 3.81| 1.75624 | 0.00 |
| | 0.03088 | |
| 3.82| 1.75933 | 0.00 |
| | 0.03093 | |
| 3.83| 1.76243 | 0.00 |
| | 0.03101 | |
| 3.84| 1.76553 | 0.00 |
| | 0.03108 | |
| 3.85| 1.76863 | 0.00 |
| | 0.03114 | |
| 3.86| 1.77175 | 0.00 |
| | 0.03122 | |
| 3.87| 1.77487 | 0.00 |
| | 0.03130 | |
| 3.88| 1.77800 | 0.00 |
| | 0.03136 | |
| 3.89| 1.78113 | 0.00 |
| | 0.03143 | |
| 3.90| 1.78428 | 0.00 |
| | 0.03150 | |
| 3.91| 1.78743 | 0.00 |
| | 0.03158 | |
| 3.92| 1.79059 | 0.00 |
| | 0.03164 | |
| 3.93| 1.79375 | 0.00 |
| | 0.03173 | |
| 3.94| 1.79692 | 0.00 |
| | 0.03180 | |
| 3.95| 1.80010 | 0.00 |
| | 0.03187 | |
| 3.96| 1.80329 | 0.00 |
| | 0.03195 | |
| 3.97| 1.80648 | 0.00 |
| | 0.03203 | |
| 3.98| 1.80969 | 0.00 |
| | 0.03210 | |
| 3.99| 1.81290 | 0.00 |
| | 0.03218 | |
expressive of the law of human mortality, &c.
General Table III. $\lambda (a^5), \lambda \left( \frac{1-a^5}{a-1} \right)$.
| $a$ | $\lambda (a^5)$ | $\lambda \left( \frac{1-a^5}{a-1} \right)$ |
|-----|----------------|---------------------------------------------|
| 2.00 | 1,81612 | 0323 |
| | ,00322,5 | 0645 |
| | | 0968 |
| | | 1290 |
| | | 1613 |
| 2.01 | 1,81934 | 0323 |
| | ,00323,4 | 0647 |
| | | 0970 |
| | | 1294 |
| | | 1617 |
| 2.02 | 1,82257 | 0324 |
| | ,00324,2 | 0648 |
| | | 0973 |
| | | 1297 |
| | | 1621 |
| 2.03 | 1,82582 | 0325 |
| | ,00324,9 | 0650 |
| | | 0975 |
| | | 1300 |
| | | 1625 |
| 2.04 | 1,82907 | 0326 |
| | ,00325,8 | 0652 |
| | | 0977 |
| | | 1303 |
| | | 1629 |
| 2.05 | 1,83232 | 0327 |
| | ,00326,5 | 0653 |
| | | 0980 |
| | | 1306 |
| | | 1633 |
| 2.06 | 1,83559 | 0327 |
| | ,00327,3 | 0655 |
| | | 0982 |
| | | 1309 |
| | | 1637 |
| 2.07 | 1,83886 | 0328 |
| | ,00328,2 | 0656 |
| | | 0985 |
| | | 1313 |
| | | 1641 |
| 2.08 | 1,84214 | 0329 |
| | ,00329,0 | 0658 |
| | | 0987 |
| | | 1316 |
| | | 1645 |
| 2.09 | 1,84543 | 0330 |
| | ,00329,8 | 0660 |
| | | 0989 |
| | | 1319 |
| | | 1649 |
| 2.10 | 1,84873 | 0331 |
| | ,00330,6 | 0661 |
| | | 0992 |
| | | 1322 |
| | | 1653 |
| 2.11 | 1,85204 | 0332 |
| | ,00331,5 | 0663 |
| | | 0995 |
| | | 1326 |
| | | 1658 |
| 2.12 | 1,85533 | 0332 |
| | ,00332,4 | 0665 |
| | | 0997 |
| | | 1330 |
| | | 1662 |
| 2.13 | 1,85868 | 0333 |
| | ,00333,2 | 0666 |
| | | 1000 |
| | | 1333 |
| | | 1666 |
| 2.14 | 1,86201 | 0334 |
| | ,00334,0 | 0668 |
| | | 1002 |
| | | 1336 |
| | | 1670 |
| 2.15 | 1,86535 | 0335 |
| | ,00334,9 | 0670 |
| | | 1005 |
| | | 1340 |
| | | 1675 |
| 2.16 | 1,86870 | 0336 |
| | ,00335,7 | 0671 |
| | | 1007 |
| | | 1343 |
| | | 1679 |
| 2.17 | 1,87205 | 0337 |
| | ,00336,6 | 0673 |
| | | 1010 |
| | | 1346 |
| | | 1683 |
| 2.18 | 1,87542 | 0338 |
| | ,00337,5 | 0675 |
| | | 1013 |
| | | 1350 |
| | | 1688 |
| 2.19 | 1,87880 | 0338 |
| | ,00338,4 | 0677 |
| | | 1015 |
| | | 1354 |
| | | 1692 |
| 2.20 | 1,88218 | 0339 |
| | ,00339,3 | 0679 |
| | | 1018 |
| | | 1357 |
| | | 1697 |
| 2.21 | 1,88557 | 0340 |
| | ,00340,2 | 0680 |
| | | 1021 |
| | | 1361 |
| | | 1701 |
| 2.22 | 1,88897 | 0341 |
| | ,00341,0 | 0682 |
| | | 1023 |
| | | 1364 |
| | | 1705 |
| 2.23 | 1,89238 | 0342 |
| | ,00342,1 | 0684 |
| | | 1026 |
| | | 1368 |
| | | 1711 |
| 2.24 | 1,89581 | 0343 |
| | ,00342,8 | 0686 |
| | | 1028 |
| | | 1371 |
| | | 1714 |
| 2.25 | 1,89923 | 0344 |
| | ,00343,8 | 0688 |
| | | 1031 |
| | | 1375 |
| | | 1719 |
| 2.26 | 1,90267 | 0345 |
| | ,00344,7 | 0692 |
| | | 1034 |
| | | 1379 |
| | | 1724 |
| 2.27 | 1,90612 | 0346 |
| | ,00345,6 | 0695 |
| | | 1037 |
| | | 1383 |
| | | 1729 |
| 2.28 | 1,90958 | 0347 |
| | ,00346,6 | 0699 |
| | | 1040 |
| | | 1386 |
| | | 1733 |
| 2.29 | 1,91304 | 0348 |
| | ,00347,6 | 0703 |
| | | 1043 |
| | | 1390 |
| | | 1738 |
| 2.30 | 1,91652 | 0349 |
| | ,00348,5 | 0707 |
| | | 1046 |
| | | 1394 |
| | | 1743 |
| 2.31 | 1,92000 | 0349 |
| | ,00349,3 | 0711 |
| | | 1048 |
| | | 1397 |
| | | 1747 |
| 2.32 | 1,92349 | 0351 |
| | ,00350,7 | 0715 |
| | | 1052 |
| | | 1403 |
| | | 1754 |
| 2.33 | 1,92700 | 0351 |
| | ,00351,4 | 0720 |
| | | 1054 |
| | | 1406 |
| | | 1757 |
| 2.34 | 1,93052 | 0352 |
| | ,00352,4 | 0725 |
| | | 1057 |
| | | 1410 |
| | | 1762 |
| 2.35 | 1,93404 | 0353 |
| | ,00353,3 | 0730 |
| | | 1060 |
| | | 1413 |
| | | 1767 |
| 2.36 | 1,93757 | 0354 |
| | ,00354,3 | 0735 |
| | | 1063 |
| | | 1417 |
| | | 1771 |
| 2.37 | 1,94112 | 0355 |
| | ,00355,4 | 0740 |
| | | 1066 |
| | | 1422 |
| | | 1777 |
| 2.38 | 1,94467 | 0356 |
| | ,00356,4 | 0745 |
| | | 1069 |
| | | 1426 |
| | | 1782 |
| 2.39 | 1,94823 | 0357 |
| | ,00357,3 | 0750 |
| | | 1072 |
| | | 1429 |
| | | 1787 |
| 2.40 | 1,95181 | 0358 |
| | ,00358,4 | 0755 |
| | | 1075 |
| | | 1434 |
| | | 1792 |
| 2.41 | 1,95539 | 0359 |
| | ,00359,4 | 0760 |
| | | 1078 |
| | | 1438 |
| | | 1797 |
| 2.42 | 1,95898 | 0360 |
| | ,00360,4 | 0765 |
| | | 1081 |
| | | 1442 |
| | | 1802 |
| 2.43 | 1,96259 | 0361 |
| | ,00361,4 | 0770 |
| | | 1084 |
| | | 1446 |
| | | 1807 |
| 2.44 | 1,96620 | 0362 |
| | ,00362,4 | 0775 |
| | | 1087 |
| | | 1450 |
| | | 1812 |
| 2.45 | 1,96983 | 0363 |
| | ,00363,4 | 0780 |
| | | 1090 |
| | | 1454 |
| | | 1817 |
| 2.46 | 1,97346 | 0365 |
| | ,00364,6 | 0785 |
| | | 1094 |
| | | 1458 |
| | | 1823 |
| 2.47 | 1,97711 | 0366 |
| | ,00365,6 | 0790 |
| | | 1097 |
| | | 1462 |
| | | 1828 |
| 2.48 | 1,98076 | 0367 |
| | ,00366,7 | 0795 |
| | | 1100 |
| | | 1467 |
| | | 1834 |
| 2.49 | 1,98443 | 0368 |
| | ,00367,8 | 0800 |
| | | 1103 |
| | | 1471 |
| | | 1839 |
Mr. Gompertz on the nature of the function
General Table III. \( \lambda (a^5), \lambda \left( \frac{1-a^5}{a-1} \right) \).
| \( \lambda (a^5) \) | \( \lambda \left( \frac{1-a^5}{a-1} \right) \) | 1 | 2 | 3 | 4 | 5 |
|---------------------|----------------------------------|---|---|---|---|---|
| 2.50 | 1.98811 | 0.00 | 0.369 | 0.738 | 1.106 | 1.475 | 1.844 |
| | 0.003688 | | 0.3319 | 0.750 | 1.109 | 1.480 | 1.850 |
| 2.51 | 1.99179 | | 0.370 | 0.740 | 1.109 | 1.480 | 1.850 |
| | 0.003699 | | 0.3329 | 0.759 | 1.109 | 1.480 | 1.850 |
| 2.52 | 1.99549 | | 0.371 | 0.742 | 1.113 | 1.484 | 1.855 |
| | 0.003710 | | 0.3339 | 0.768 | 1.113 | 1.484 | 1.855 |
| 2.53 | 1.99920 | | 0.372 | 0.744 | 1.116 | 1.488 | 1.861 |
| | 0.003721 | | 0.3349 | 0.777 | 1.116 | 1.488 | 1.861 |
| 2.54 | 1.99920 | | 0.373 | 0.747 | 1.120 | 1.493 | 1.867 |
| | 0.003733 | | 0.3360 | 0.786 | 1.120 | 1.493 | 1.867 |
| 2.55 | 0.00666 | | 0.374 | 0.749 | 1.123 | 1.497 | 1.872 |
| | 0.003743 | | 0.3369 | 0.794 | 1.123 | 1.497 | 1.872 |
| 2.56 | 0.01040 | | 0.376 | 0.751 | 1.127 | 1.502 | 1.878 |
| | 0.003755 | | 0.3380 | 0.804 | 1.127 | 1.502 | 1.878 |
| 2.57 | 0.01416 | | 0.376 | 0.753 | 1.129 | 1.506 | 1.882 |
| | 0.003764 | | 0.3388 | 0.811 | 1.129 | 1.506 | 1.882 |
| 2.58 | 0.01792 | | 0.378 | 0.756 | 1.134 | 1.512 | 1.890 |
| | 0.003779 | | 0.3401 | 0.823 | 1.134 | 1.512 | 1.890 |
| 2.59 | 0.02170 | | 0.379 | 0.758 | 1.136 | 1.515 | 1.894 |
| | 0.003783 | | 0.3409 | 0.830 | 1.136 | 1.515 | 1.894 |
| 2.60 | 0.02549 | | 0.380 | 0.760 | 1.140 | 1.520 | 1.900 |
| | 0.003800 | | 0.3420 | 0.840 | 1.140 | 1.520 | 1.900 |
| 2.61 | 0.02929 | | 0.381 | 0.762 | 1.144 | 1.525 | 1.906 |
| | 0.003812 | | 0.3431 | 0.850 | 1.144 | 1.525 | 1.906 |
| 2.62 | 0.03310 | | 0.382 | 0.765 | 1.147 | 1.530 | 1.912 |
| | 0.003824 | | 0.3442 | 0.860 | 1.147 | 1.530 | 1.912 |
| 2.63 | 0.03692 | | 0.383 | 0.767 | 1.150 | 1.534 | 1.917 |
| | 0.003834 | | 0.3451 | 0.870 | 1.150 | 1.534 | 1.917 |
| 2.64 | 0.04076 | | 0.385 | 0.769 | 1.154 | 1.539 | 1.924 |
| | 0.003847 | | 0.3462 | 0.878 | 1.154 | 1.539 | 1.924 |
| 2.65 | 0.04460 | | 0.386 | 0.772 | 1.157 | 1.543 | 1.929 |
| | 0.003858 | | 0.3472 | 0.886 | 1.157 | 1.543 | 1.929 |
| 2.66 | 0.04846 | | 0.387 | 0.774 | 1.161 | 1.548 | 1.936 |
| | 0.003871 | | 0.3484 | 0.897 | 1.161 | 1.548 | 1.936 |
| 2.67 | 0.05233 | | 0.388 | 0.776 | 1.165 | 1.553 | 1.941 |
| | 0.003882 | | 0.3494 | 0.906 | 1.165 | 1.553 | 1.941 |
| 2.68 | 0.05621 | | 0.389 | 0.779 | 1.168 | 1.558 | 1.947 |
| | 0.003894 | | 0.3505 | 0.915 | 1.168 | 1.558 | 1.947 |
| 2.69 | 0.06010 | | 0.391 | 0.781 | 1.172 | 1.562 | 1.953 |
| | 0.003906 | | 0.3515 | 0.924 | 1.172 | 1.562 | 1.953 |
| 2.70 | 0.06401 | | 0.392 | 0.784 | 1.176 | 1.568 | 1.960 |
| | 0.003919 | | 0.3527 | 0.935 | 1.176 | 1.568 | 1.960 |
| 2.71 | 0.06793 | | 0.393 | 0.786 | 1.179 | 1.572 | 1.965 |
| | 0.003930 | | 0.3537 | 0.944 | 1.179 | 1.572 | 1.965 |
| 2.72 | 0.07186 | | 0.394 | 0.789 | 1.183 | 1.577 | 1.972 |
| | 0.003943 | | 0.3549 | 0.954 | 1.183 | 1.577 | 1.972 |
| 2.73 | 0.07581 | | 0.396 | 0.791 | 1.187 | 1.582 | 1.978 |
| | 0.003955 | | 0.3560 | 0.964 | 1.187 | 1.582 | 1.978 |
| 2.74 | 0.07976 | | 0.397 | 0.794 | 1.190 | 1.587 | 1.984 |
| | 0.003968 | | 0.3571 | 0.974 | 1.190 | 1.587 | 1.984 |
expressive of the law of human mortality, &c.
General Table III. $\lambda (a^5)$, $\lambda \left(\frac{1-a^5}{a-1}\right)$.
| $\lambda (a^5)$ | $\lambda \left(\frac{1-a^5}{a-1}\right)$ |
|-----------------|------------------------------------------|
| $I_{0.00}$ | $0.8717$ |
| $I_{0.01}$ | $0.9148$ |
| $I_{0.02}$ | $0.9581$ |
| $I_{0.03}$ | $0.9943$ |
| $I_{0.04}$ | $0.9943$ |
| $I_{0.05}$ | $0.9943$ |
| $I_{0.06}$ | $0.9943$ |
| $I_{0.07}$ | $0.9943$ |
| $I_{0.08}$ | $0.9943$ |
| $I_{0.09}$ | $0.9943$ |
| $I_{0.10}$ | $0.9943$ |
| $I_{0.11}$ | $0.9943$ |
| $I_{0.12}$ | $0.9943$ |
| $I_{0.13}$ | $0.9943$ |
| $I_{0.14}$ | $0.9943$ |
| $I_{0.15}$ | $0.9943$ |
| $I_{0.16}$ | $0.9943$ |
| $I_{0.17}$ | $0.9943$ |
| $I_{0.18}$ | $0.9943$ |
| $I_{0.19}$ | $0.9943$ |
| $I_{0.20}$ | $0.9943$ |
| $I_{0.21}$ | $0.9943$ |
| $I_{0.22}$ | $0.9943$ |
| $I_{0.23}$ | $0.9943$ |
| $I_{0.24}$ | $0.9943$ |
MDCCXXV.
Mr. Gompertz on the nature of the function
General Table III. $\lambda(a^5), \lambda\left(\frac{1-a^5}{a-1}\right)$.
| $a$ | $\lambda(a^5)$ | $\lambda\left(\frac{1-a^5}{a-1}\right)$ |
|-----|----------------|------------------------------------------|
| 1.50 | 0.42174 | 0.00 |
| | 0.00510 | 0.00 |
| 1.51 | 0.42684 | 0.00 |
| | 0.00512 | 0.00 |
| 1.52 | 0.43197 | 0.00 |
| | 0.00514 | 0.00 |
| 1.53 | 0.43711 | 0.00 |
| | 0.00516 | 0.00 |
| 1.54 | 0.44227 | 0.00 |
| | 0.00517 | 0.00 |
| 1.55 | 0.44745 | 0.00 |
| | 0.00519 | 0.00 |
| 1.56 | 0.45264 | 0.00 |
| | 0.00521 | 0.00 |
| 1.57 | 0.45786 | 0.00 |
| | 0.00523 | 0.00 |
| 1.58 | 0.46309 | 0.00 |
| | 0.00524 | 0.00 |
| 1.59 | 0.46833 | 0.00 |
| | 0.00526 | 0.00 |
| 1.60 | 0.47360 | 0.00 |
| | 0.00528 | 0.00 |
| 1.61 | 0.47888 | 0.00 |
| | 0.00530 | 0.00 |
| 1.62 | 0.48418 | 0.00 |
| | 0.00531 | 0.00 |
| 1.63 | 0.48950 | 0.00 |
| | 0.00533 | 0.00 |
| 1.64 | 0.49484 | 0.00 |
| | 0.00535 | 0.00 |
| 1.65 | 0.50019 | 0.00 |
| | 0.00537 | 0.00 |
| 1.66 | 0.50556 | 0.00 |
| | 0.00539 | 0.00 |
| 1.67 | 0.51095 | 0.00 |
| | 0.00540 | 0.00 |
| 1.68 | 0.51636 | 0.00 |
| | 0.00542 | 0.00 |
| 1.69 | 0.52178 | 0.00 |
| | 0.00544 | 0.00 |
| 1.70 | 0.52722 | 0.00 |
| | 0.00546 | 0.00 |
| 1.71 | 0.53269 | 0.00 |
| | 0.00547 | 0.00 |
| 1.72 | 0.53816 | 0.00 |
| | 0.00549 | 0.00 |
| 1.73 | 0.54366 | 0.00 |
| | 0.00551 | 0.00 |
| 1.74 | 0.54918 | 0.00 |
| | 0.00553 | 0.00 |
| $a$ | $\lambda(a^5)$ | $\lambda\left(\frac{1-a^5}{a-1}\right)$ |
|-----|----------------|------------------------------------------|
| 1.75 | 0.55471 | 0.00 |
| | 0.00555 | 0.00 |
| 1.76 | 0.56026 | 0.00 |
| | 0.00556 | 0.00 |
| 1.77 | 0.56583 | 0.00 |
| | 0.00558 | 0.00 |
| 1.78 | 0.57142 | 0.00 |
| | 0.00560 | 0.00 |
| 1.79 | 0.57702 | 0.00 |
| | 0.00562 | 0.00 |
| 1.80 | 0.58265 | 0.00 |
| | 0.00564 | 0.00 |
| 1.81 | 0.58829 | 0.00 |
| | 0.00565 | 0.00 |
| 1.82 | 0.59395 | 0.00 |
| | 0.00567 | 0.00 |
| 1.83 | 0.59963 | 0.00 |
| | 0.00569 | 0.00 |
| 1.84 | 0.60532 | 0.00 |
| | 0.00571 | 0.00 |
| 1.85 | 0.61104 | 0.00 |
| | 0.00573 | 0.00 |
| 1.86 | 0.61677 | 0.00 |
| | 0.00575 | 0.00 |
| 1.87 | 0.62252 | 0.00 |
| | 0.00577 | 0.00 |
| 1.88 | 0.62829 | 0.00 |
| | 0.00578 | 0.00 |
| 1.89 | 0.63408 | 0.00 |
| | 0.00580 | 0.00 |
| 1.90 | 0.63989 | 0.00 |
| | 0.00582 | 0.00 |
| 1.91 | 0.64572 | 0.00 |
| | 0.00584 | 0.00 |
| 1.92 | 0.65156 | 0.00 |
| | 0.00586 | 0.00 |
| 1.93 | 0.65742 | 0.00 |
| | 0.00587 | 0.00 |
| 1.94 | 0.66330 | 0.00 |
| | 0.00590 | 0.00 |
| 1.95 | 0.66920 | 0.00 |
| | 0.00591 | 0.00 |
| 1.96 | 0.67512 | 0.00 |
| | 0.00593 | 0.00 |
| 1.97 | 0.68105 | 0.00 |
| | 0.00595 | 0.00 |
| 1.98 | 0.68701 | 0.00 |
| | 0.00597 | 0.00 |
| 1.99 | 0.69298 | 0.00 |
General Table IV. For the whole of life. — $\lambda(a^{-1}-1)$.
| $\lambda(a)$ | $-\lambda(a^{-1}-1)$ | $\lambda(a)$ | $-\lambda(a^{-1}-1)$ | $\lambda(a)$ | $-\lambda(a^{-1}-1)$ | $\lambda(a)$ | $-\lambda(a^{-1}-1)$ | $\lambda(a)$ | $-\lambda(a^{-1}-1)$ |
|-------------|---------------------|-------------|---------------------|-------------|---------------------|-------------|---------------------|-------------|---------------------|
| 1.700 | 0.0206 | 1.725 | 0.0537 | 1.750 | 0.1088 | 1.775 | 0.1682 | 1.800 | 0.2329 |
| | 0.0201 | 1.726 | 0.0558 | 1.751 | 0.1115 | 1.776 | 0.1707 | 1.801 | 0.2356 |
| | 0.0201 | 1.727 | 0.0579 | 1.752 | 0.1134 | 1.777 | 0.1732 | 1.802 | 0.2383 |
| | 0.0202 | 1.728 | 0.0614 | 1.753 | 0.1157 | 1.778 | 0.1757 | 1.803 | 0.2411 |
| | 0.0202 | 1.729 | 0.0622 | 1.754 | 0.1180 | 1.779 | 0.1782 | 1.804 | 0.2438 |
| | 0.0203 | 1.730 | 0.0645 | 1.755 | 0.1203 | 1.780 | 0.1807 | 1.805 | 0.2466 |
| | 0.0203 | 1.731 | 0.0666 | 1.756 | 0.1227 | 1.781 | 0.1832 | 1.806 | 0.2493 |
| | 0.0203 | 1.732 | 0.0687 | 1.757 | 0.1250 | 1.782 | 0.1857 | 1.807 | 0.2521 |
| | 0.0204 | 1.733 | 0.0709 | 1.758 | 0.1273 | 1.783 | 0.1883 | 1.808 | 0.2549 |
| | 0.0205 | 1.734 | 0.0731 | 1.759 | 0.1297 | 1.784 | 0.1908 | 1.809 | 0.2577 |
| | 0.0205 | 1.735 | 0.0753 | 1.760 | 0.1320 | 1.785 | 0.1934 | 1.810 | 0.2605 |
| | 0.0205 | 1.736 | 0.0775 | 1.761 | 0.1344 | 1.786 | 0.1959 | 1.811 | 0.2634 |
| | 0.0206 | 1.737 | 0.0797 | 1.762 | 0.1367 | 1.787 | 0.1985 | 1.812 | 0.2662 |
| | 0.0207 | 1.738 | 0.0819 | 1.763 | 0.1391 | 1.788 | 0.2011 | 1.813 | 0.2690 |
| | 0.0207 | 1.739 | 0.0841 | 1.764 | 0.1415 | 1.789 | 0.2037 | 1.814 | 0.2719 |
| | 0.0207 | 1.740 | 0.0863 | 1.765 | 0.1439 | 1.790 | 0.2063 | 1.815 | 0.2748 |
| | 0.0208 | 1.741 | 0.0885 | 1.766 | 0.1463 | 1.791 | 0.2089 | 1.816 | 0.2777 |
| | 0.0208 | 1.742 | 0.0908 | 1.767 | 0.1487 | 1.792 | 0.2115 | 1.817 | 0.2806 |
| | 0.0209 | 1.743 | 0.0930 | 1.768 | 0.1511 | 1.793 | 0.2142 | 1.818 | 0.2834 |
| | 0.0210 | 1.744 | 0.0953 | 1.769 | 0.1535 | 1.794 | 0.2168 | 1.819 | 0.2864 |
| | 0.0210 | 1.745 | 0.0975 | 1.770 | 0.1559 | 1.795 | 0.2195 | 1.820 | 0.2894 |
| | 0.0211 | 1.746 | 0.0998 | 1.771 | 0.1584 | 1.796 | 0.2222 | 1.821 | 0.2923 |
| | 0.0211 | 1.747 | 0.1020 | 1.772 | 0.1608 | 1.797 | 0.2248 | 1.822 | 0.2953 |
| | 0.0212 | 1.748 | 0.1043 | 1.773 | 0.1633 | 1.798 | 0.2275 | 1.823 | 0.2983 |
| | 0.0212 | 1.749 | 0.1066 | 1.774 | 0.1658 | 1.799 | 0.2302 | 1.824 | 0.3013 |
| | 0.0213 | 1.750 | 0.1089 | | | | | 1.825 | 0.3043 |
| | 0.0213 | 1.751 | 0.1112 | | | | | 1.826 | 0.3073 |
| | 0.0214 | 1.752 | 0.1135 | | | | | 1.827 | 0.3103 |
| | 0.0215 | 1.753 | 0.1158 | | | | | 1.828 | 0.3134 |
| | 0.0215 | 1.754 | 0.1181 | | | | | 1.829 | 0.3164 |
| | 0.0216 | 1.755 | 0.1204 | | | | | 1.830 | 0.3195 |
| | 0.0216 | 1.756 | 0.1227 | | | | | 1.831 | 0.3226 |
| | 0.0217 | 1.757 | 0.1250 | | | | | 1.832 | 0.3257 |
| | 0.0217 | 1.758 | 0.1273 | | | | | 1.833 | 0.3289 |
| | 0.0218 | 1.759 | 0.1297 | | | | | 1.834 | 0.3320 |
| | 0.0218 | 1.760 | 0.1320 | | | | | 1.835 | 0.3351 |
| | 0.0219 | 1.761 | 0.1344 | | | | | 1.836 | 0.3383 |
| | 0.0220 | 1.762 | 0.1367 | | | | | 1.837 | 0.3415 |
| | 0.0220 | 1.763 | 0.1391 | | | | | 1.838 | 0.3447 |
| | 0.0221 | 1.764 | 0.1415 | | | | | 1.839 | 0.3479 |
| | 0.0221 | 1.765 | 0.1439 | | | | | 1.840 | 0.3512 |
| | 0.0222 | 1.766 | 0.1463 | | | | | 1.841 | 0.3544 |
| | 0.0222 | 1.767 | 0.1487 | | | | | 1.842 | 0.3577 |
| | 0.0223 | 1.768 | 0.1511 | | | | | 1.843 | 0.3610 |
| | 0.0223 | 1.769 | 0.1535 | | | | | 1.844 | 0.3643 |
| | 0.0224 | 1.770 | 0.1559 | | | | | 1.845 | 0.3676 |
| | 0.0224 | 1.771 | 0.1584 | | | | | 1.846 | 0.3709 |
| | 0.0226 | 1.772 | 0.1608 | | | | | 1.847 | 0.3743 |
| | 0.0226 | 1.773 | 0.1633 | | | | | 1.848 | 0.3777 |
| | 0.0227 | 1.774 | 0.1658 | | | | | 1.849 | 0.3811 |
| | 0.0228 | 1.775 | 0.1682 | | | | | 1.850 | 0.3844 |
Mr. Gompertz on the nature of the function
General Table IV. For the whole of life. — $\lambda(a^{-1}-1)$.
| $a$ | $\lambda(a)$ | $-\lambda(a^{-1}-1)$ | $a$ | $\lambda(a)$ | $-\lambda(a^{-1}-1)$ | $a$ | $\lambda(a)$ | $-\lambda(a^{-1}-1)$ | $a$ | $\lambda(a)$ | $-\lambda(a^{-1}-1)$ |
|-----|-------------|----------------------|-----|-------------|----------------------|-----|-------------|----------------------|-----|-------------|----------------------|
| 1.850 | 38454 | 0.0343 | 1.875 | 47688 | 0.0401 | 1.900 | 58683 | 0.0488 | 1.925 | 72468 | 0.0635 |
| 1.851 | 38797 | 0.0345 | 1.876 | 48088 | 0.0405 | 1.901 | 59171 | 0.0493 | 1.926 | 73103 | 0.0642 |
| 1.852 | 39142 | 0.0348 | 1.877 | 48493 | 0.0407 | 1.902 | 59664 | 0.0497 | 1.927 | 73745 | 0.0650 |
| 1.853 | 39490 | 0.0349 | 1.878 | 48900 | 0.0410 | 1.903 | 60161 | 0.0502 | 1.928 | 74395 | 0.0659 |
| 1.854 | 39839 | 0.0351 | 1.879 | 49310 | 0.0412 | 1.904 | 60663 | 0.0507 | 1.929 | 75054 | 0.0668 |
| 1.855 | 40190 | 0.0353 | 1.880 | 49722 | 0.0416 | 1.905 | 61170 | 0.0511 | 1.930 | 75722 | 0.0676 |
| 1.856 | 40543 | 0.0355 | 1.881 | 50138 | 0.0419 | 1.906 | 61681 | 0.0516 | 1.931 | 76398 | 0.0685 |
| 1.857 | 40899 | 0.0357 | 1.882 | 50557 | 0.0422 | 1.907 | 62197 | 0.0521 | 1.932 | 77083 | 0.0695 |
| 1.858 | 41256 | 0.0360 | 1.883 | 50979 | 0.0425 | 1.908 | 62718 | 0.0527 | 1.933 | 77778 | 0.0704 |
| 1.859 | 41616 | 0.0362 | 1.884 | 51404 | 0.0428 | 1.909 | 63245 | 0.0531 | 1.934 | 78482 | 0.0715 |
| 1.860 | 41978 | 0.0364 | 1.885 | 51832 | 0.0431 | 1.910 | 63776 | 0.0537 | 1.935 | 79197 | 0.0724 |
| 1.861 | 42342 | 0.0366 | 1.886 | 52263 | 0.0435 | 1.911 | 64313 | 0.0543 | 1.936 | 79921 | 0.0735 |
| 1.862 | 42708 | 0.0368 | 1.887 | 52698 | 0.0438 | 1.912 | 64856 | 0.0548 | 1.937 | 80565 | 0.0746 |
| 1.863 | 44076 | 0.0371 | 1.888 | 53136 | 0.0442 | 1.913 | 65404 | 0.0554 | 1.938 | 81402 | 0.0758 |
| 1.864 | 44447 | 0.0373 | 1.889 | 53578 | 0.0445 | 1.914 | 65958 | 0.0559 | 1.939 | 82160 | 0.0769 |
| 1.865 | 44820 | 0.0376 | 1.890 | 54023 | 0.0449 | 1.915 | 66517 | 0.0566 | 1.940 | 82929 | 0.0781 |
| 1.866 | 44196 | 0.0378 | 1.891 | 54472 | 0.0452 | 1.916 | 67083 | 0.0572 | 1.941 | 83710 | 0.0793 |
| 1.867 | 44574 | 0.0380 | 1.892 | 54924 | 0.0456 | 1.917 | 67655 | 0.0578 | 1.942 | 84503 | 0.0807 |
| 1.868 | 44954 | 0.0383 | 1.893 | 55380 | 0.0460 | 1.918 | 68233 | 0.0584 | 1.943 | 85310 | 0.0820 |
| 1.869 | 45337 | 0.0385 | 1.894 | 55840 | 0.0464 | 1.919 | 68817 | 0.0591 | 1.944 | 86130 | 0.0833 |
| 1.870 | 45722 | 0.0388 | 1.895 | 56304 | 0.0467 | 1.920 | 69408 | 0.0598 | 1.945 | 86963 | 0.0848 |
| 1.871 | 46110 | 0.0390 | 1.896 | 56771 | 0.0472 | 1.921 | 70006 | 0.0605 | 1.946 | 87811 | 0.0863 |
| 1.872 | 46500 | 0.0393 | 1.897 | 57243 | 0.0476 | 1.922 | 70611 | 0.0611 | 1.947 | 88674 | 0.0878 |
| 1.873 | 46893 | 0.0396 | 1.898 | 57719 | 0.0479 | 1.923 | 71222 | 0.0620 | 1.948 | 89552 | 0.0894 |
| 1.874 | 47289 | 0.0399 | 1.899 | 58198 | 0.0485 | 1.924 | 71842 | 0.0626 | 1.949 | 90446 | 0.0911 |
| | | | | | | | | | | | |
| 1.950 | 91357 | 0.0929 | 1.951 | 92286 | 0.0946 | 1.952 | 93232 | 0.0966 | 1.953 | 94198 | 0.0984 |
| 1.954 | 95182 | 0.1006 | 1.955 | 96188 | 0.1027 | 1.956 | 97215 | 0.1049 | 1.957 | 98264 | 0.1073 |
| 1.959 | 99337 | 0.1097 | 1.960 | 101557 | 0.1150 | 1.961 | 10207 | 0.1179 | 1.962 | 103886 | 0.1209 |
| 1.964 | 106336 | 0.1274 | 1.965 | 107610 | 0.1309 | 1.966 | 108919 | 0.1348 | 1.967 | 110267 | 0.1387 |
| 1.969 | 113083 | 0.1475 | 1.970 | 114558 | 0.1523 | 1.971 | 116081 | 0.1574 | 1.972 | 117055 | 0.1530 |
| 1.973 | 119285 | 0.1690 | 1.974 | 120975 | 0.1753 | 1.979 | 123728 | |
TABLE V.—Logarithms of the accommodated chances of living 10 years, deduced from the value of an annuity for 10 years, at 5 per cent. from the actual tables of mortality, and considered equal to a geometrical series of ten terms, of which the common ratio is the same as the first term, and the tenth term the accommodated chance; and to find the accommodated chance for 5, 7 years, &c. without a table calculated for the purpose, it may be considered sufficient to multiply by \( \frac{5}{10} \), \( \frac{7}{10} \), &c. the accommodated ratio in this table when extreme accuracy be not required.
| Age | Carlisle | Deparcieux | Northampton |
|-----|----------|------------|-------------|
| 0 | 1.6892 | — | — |
| 1 | 1.6703 | — | 1.7044 |
| 2 | 1.8699 | 1.8356 | 1.8790 |
| 3 | 1.9159 | 1.9166 | 1.8790 |
| 4 | 1.9401 | 1.9315 | 1.9081 |
| 5 | 1.9586 | 1.9411 | 1.9220 |
| 6 | 1.9686 | 1.9486 | 1.9369 |
| 7 | 1.9737 | 1.9544 | 1.9476 |
| 8 | 1.9764 | 1.9592 | 1.9559 |
| 9 | 1.9773 | 1.9637 | 1.9586 |
| 10 | 1.9768 | 1.9669 | 1.9592 |
| 11 | 1.9754 | 1.9679 | 1.9582 |
| 12 | 1.9742 | 1.9669 | 1.9566 |
| 13 | 1.9729 | 1.9658 | 1.9546 |
| 14 | 1.9716 | 1.9704 | 1.9521 |
| 15 | 1.9704 | 1.9628 | 1.9490 |
| 16 | 1.9698 | 1.9609 | 1.9455 |
| 17 | 1.9694 | 1.9600 | 1.9419 |
| 18 | 1.9693 | 1.9586 | 1.9388 |
| 19 | 1.9690 | 1.9574 | 1.9358 |
| 20 | 1.9685 | 1.9559 | 1.9337 |
| 21 | 1.9679 | 1.9554 | 1.9321 |
| 22 | 1.9670 | 1.9549 | 1.9311 |
| 23 | 1.9659 | 1.9544 | 1.9298 |
| 24 | 1.9644 | 1.9540 | 1.9289 |
| 25 | 1.9628 | 1.9534 | 1.9277 |
| 26 | 1.9573 | 1.9531 | 1.9264 |
| 27 | 1.9591 | 1.9524 | 1.9257 |
| 28 | 1.9570 | 1.9521 | 1.9238 |
| 29 | 1.9556 | 1.9518 | 1.9226 |
| 30 | 1.9552 | 1.9514 | 1.9211 |
| 31 | 1.9548 | 1.9514 | 1.9196 |
| 32 | 1.9540 | 1.9514 | 1.9180 |
| 33 | 1.9528 | 1.9515 | 1.9164 |
| 34 | 1.9513 | 1.9517 | 1.9146 |
| 35 | 1.9485 | 1.9522 | 1.9126 |
| 36 | 1.9477 | 1.9528 | 1.9104 |
| 37 | 1.9452 | 1.9534 | 1.9083 |
| 38 | 1.9437 | 1.9527 | 1.8057 |
| 39 | 1.9406 | 1.9517 | 1.9031 |
| 40 | 1.9383 | 1.9506 | 1.9001 |
| 41 | 1.9372 | 1.9488 | 1.8973 |
| 42 | 1.9365 | 1.9466 | 1.8943 |
| 43 | 1.9365 | 1.9438 | 1.8915 |
| 44 | 1.9366 | 1.9403 | 1.8882 |
| 45 | 1.9367 | 1.9361 | 1.8848 |
| 46 | 1.9366 | 1.9308 | 1.8810 |
| 47 | 1.9358 | 1.9263 | 1.8767 |
| 48 | 1.9351 | 1.9200 | 1.8740 |
| 49 | 1.9328 | 1.9158 | 1.8631 |
| 50 | 1.9292 | 1.9098 | 1.8621 |
| 51 | 1.9233 | 1.9027 | 1.8571 |
| Age | Carlisle | Deparcieux | Northampton |
|-----|----------|------------|-------------|
| 52 | 1.9172 | 1.9006 | 1.8523 |
| 53 | 1.9098 | 1.8957 | 1.8471 |
| 54 | 1.9013 | 1.8901 | 1.8417 |
| 55 | 1.8915 | 1.8853 | 1.8357 |
| 56 | 1.8803 | 1.8799 | 1.8294 |
| 57 | 1.8680 | 1.8732 | 1.8228 |
| 58 | 1.8513 | 1.8673 | 1.8156 |
| 59 | 1.8435 | 1.8601 | 1.8081 |
| 60 | 1.8318 | 1.8511 | 1.7998 |
| 61 | 1.8243 | 1.8398 | 1.7908 |
| 62 | 1.8171 | 1.8264 | 1.7811 |
| 63 | 1.8090 | 1.8120 | 1.7699 |
| 64 | 1.7974 | 1.7946 | 1.7576 |
| 65 | 1.7860 | 1.7735 | 1.7431 |
| 66 | 1.7703 | 1.7510 | 1.7267 |
| 67 | 1.7506 | 1.7270 | 1.7083 |
| 68 | 1.7107 | 1.7017 | 1.6879 |
| 69 | 1.7005 | 1.6754 | 1.6651 |
| 70 | 1.6689 | 1.6480 | 1.6402 |
| 71 | 1.6319 | 1.6167 | 1.6126 |
| 72 | 1.5936 | 1.5841 | 1.5823 |
| 73 | 1.5563 | 1.5500 | 1.5487 |
| 74 | 1.5269 | 1.5119 | 1.5117 |
| 75 | 1.4940 | 1.4711 | 1.4723 |
| 76 | 1.4642 | 1.4218 | 1.4308 |
| 77 | 1.4344 | 1.3684 | 1.3846 |
| 78 | 1.4007 | 1.3134 | 1.3307 |
| 79 | 1.3538 | 1.2497 | 1.2644 |
| 80 | 1.3134 | 1.1876 | 1.1900 |
| 81 | 1.2582 | 1.1214 | 1.1101 |
| 82 | 1.2043 | 1.0609 | 1.0234 |
| 83 | 1.1765 | 2.9088 | 2.9341 |
| 84 | 1.0727 | 2.8536 | 2.8592 |
| 85 | 2.9939 | 2.7199 | 2.7813 |
| 86 | 2.9166 | 2.5736 | 2.7003 |
| 87 | 2.8490 | 2.4254 | 2.6149 |
| 88 | 2.8055 | 2.1943 | 2.5369 |
| 89 | 2.7537 | 3.9129 | 2.4179 |
| 90 | 2.6695 | 3.5265 | 2.2414 |
| 91 | 2.6658 | 3.0266 | 3.9356 |
| 92 | 2.7323 | 4.3694 | 3.5037 |
| 93 | 2.8031 | 5.2971 | 4.7375 |
| 94 | 2.8355 | — | 5.5769 |
| 95 | 2.8107 | — | 7.0496 |
| 96 | 2.8279 | — | — |
| 97 | 2.7589 | — | — |
| 98 | 2.6695 | — | — |
| 99 | 2.5111 | — | — |
| 100 | 2.1629 | — | — |
| 101 | 3.5689 | — | — |
| 102 | 4.3245 | — | — |
| 103 | 6.0595 | — | — |
TABLE VI.—Accommodated annual ratio for an unlimited period for every age \( \alpha \)
\[
\lambda r = \lambda^1 \left[ \frac{a}{1 + \lambda} \right] + \lambda^{1.05}.
\]
| \( \alpha \) | \( \lambda r \) Carlisle | \( \lambda r \) Deparcieux | \( \lambda r \) Northampton |
|---|---|---|---|
| 0 | 1.98665 | — | — |
| 1 | 1.99121 | — | 1.98517 |
| 2 | 1.99313 | — | 1.98997 |
| 3 | 1.99458 | 1.99399 | 1.99151 |
| 4 | 1.99528 | 1.99446 | 1.99244 |
| 5 | 1.99577 | 1.99473 | 1.99284 |
| 6 | 1.99599 | 1.99493 | 1.99324 |
| 7 | 1.99606 | 1.99505 | 1.99346 |
| 8 | 1.99606 | 1.99513 | 1.99357 |
| 9 | 1.99600 | 1.99519 | 1.99354 |
| 10 | 1.99529 | 1.99519 | 1.99341 |
| 11 | 1.99576 | 1.99514 | 1.99323 |
| 12 | 1.99563 | 1.99503 | 1.99304 |
| 13 | 1.99549 | 1.99491 | 1.99284 |
| 14 | 1.99535 | 1.99478 | 1.99262 |
| 15 | 1.99522 | 1.99464 | 1.99240 |
| 16 | 1.99509 | 1.99450 | 1.99213 |
| 17 | 1.99487 | 1.99438 | 1.99190 |
| 18 | 1.99484 | 1.99425 | 1.99167 |
| 19 | 1.99471 | 1.99413 | 1.99145 |
| 20 | 1.99455 | 1.99398 | 1.99124 |
| 21 | 1.99442 | 1.99388 | 1.99106 |
| 22 | 1.99435 | 1.99375 | 1.99088 |
| 23 | 1.99408 | 1.99364 | 1.99070 |
| 24 | 1.99390 | 1.99350 | 1.99051 |
| 25 | 1.99370 | 1.99338 | 1.99030 |
| 26 | 1.99349 | 1.99323 | 1.99009 |
| 27 | 1.99328 | 1.99308 | 1.98988 |
| 28 | 1.99306 | 1.99293 | 1.98965 |
| 29 | 1.99290 | 1.99277 | 1.98943 |
| 30 | 1.99265 | 1.99259 | 1.98917 |
| 31 | 1.99245 | 1.99241 | 1.98892 |
| 32 | 1.99224 | 1.99223 | 1.98865 |
| 33 | 1.99201 | 1.99202 | 1.98837 |
| 34 | 1.99176 | 1.99181 | 1.98808 |
| 35 | 1.99149 | 1.99158 | 1.98777 |
| 36 | 1.99110 | 1.99134 | 1.98745 |
| 37 | 1.99090 | 1.99109 | 1.98811 |
| 38 | 1.99058 | 1.99077 | 1.98675 |
| 39 | 1.99025 | 1.99043 | 1.98637 |
| 40 | 1.98991 | 1.99006 | 1.98597 |
| 41 | 1.98958 | 1.98967 | 1.98556 |
| 42 | 1.98924 | 1.98924 | 1.98513 |
| 43 | 1.98890 | 1.98878 | 1.98469 |
| 44 | 1.98853 | 1.98828 | 1.98423 |
| 45 | 1.98814 | 1.98771 | 1.98375 |
| 46 | 1.98771 | 1.98714 | 1.98323 |
| 47 | 1.98725 | 1.98655 | 1.98270 |
| 48 | 1.98673 | 1.98590 | 1.98209 |
| 49 | 1.98612 | 1.98526 | 1.98148 |
| 50 | 1.98546 | 1.98456 | 1.98083 |
| 51 | 1.98471 | 1.98386 | 1.98017 |
| \( \alpha \) | \( \lambda r \) Carlisle | \( \lambda r \) Deparcieux | \( \lambda r \) Northampton |
|---|---|---|---|
| 52 | 1.98390 | 1.98216 | 1.97950 |
| 53 | 1.98305 | 1.98240 | 1.97878 |
| 54 | 1.98212 | 1.98156 | 1.97802 |
| 55 | 1.98112 | 1.98073 | 1.97721 |
| 56 | 1.98005 | 1.97982 | 1.97635 |
| 57 | 1.97887 | 1.97882 | 1.97542 |
| 58 | 1.97763 | 1.97780 | 1.97445 |
| 59 | 1.97637 | 1.97668 | 1.97341 |
| 60 | 1.97514 | 1.97545 | 1.97230 |
| 61 | 1.97400 | 1.97408 | 1.97111 |
| 62 | 1.97281 | 1.97254 | 1.96983 |
| 63 | 1.97154 | 1.97093 | 1.96843 |
| 64 | 1.97014 | 1.96912 | 1.96693 |
| 65 | 1.96818 | 1.96707 | 1.96526 |
| 66 | 1.96685 | 1.96489 | 1.96346 |
| 67 | 1.96491 | 1.96250 | 1.96150 |
| 68 | 1.96273 | 1.96009 | 1.95936 |
| 69 | 1.96029 | 1.95750 | 1.95703 |
| 70 | 1.95755 | 1.95480 | 1.95448 |
| 71 | 1.95440 | 1.95181 | 1.95171 |
| 72 | 1.95111 | 1.94870 | 1.94869 |
| 73 | 1.94784 | 1.94542 | 1.94541 |
| 74 | 1.94467 | 1.94180 | 1.94186 |
| 75 | 1.94185 | 1.93790 | 1.93812 |
| 76 | 1.93888 | 1.93330 | 1.93423 |
| 77 | 1.93591 | 1.92833 | 1.92994 |
| 78 | 1.93265 | 1.92314 | 1.92503 |
| 79 | 1.92803 | 1.91715 | 1.91916 |
| 80 | 1.92461 | 1.91124 | 1.91246 |
| 81 | 1.91891 | 1.90493 | 1.90509 |
| 82 | 1.91491 | 1.89856 | 1.89697 |
| 83 | 1.90939 | 1.89069 | 1.88844 |
| 84 | 1.90344 | 1.88005 | 1.88110 |
| 85 | 1.89657 | 1.86782 | 1.87361 |
| 86 | 1.88978 | 1.85416 | 1.86599 |
| 87 | 1.88369 | 1.84011 | 1.85803 |
| 88 | 1.87972 | 1.81788 | 1.85069 |
| 89 | 1.87481 | 1.78941 | 1.85454 |
| 90 | 1.86660 | 1.75223 | 1.82243 |
| 91 | 1.86560 | 1.70265 | 1.79303 |
| 92 | 1.87056 | 1.63694 | 1.74992 |
| 93 | 1.87595 | 1.52971 | 1.67376 |
| 94 | 1.87840 | — | 1.55753 |
| 95 | 1.87967 | — | 1.30505 |
| 96 | 1.77774 | — | — |
| 97 | 1.77140 | — | — |
| 98 | 1.76333 | — | — |
| 99 | 1.84829 | — | — |
| 100 | 1.81282 | — | — |
| 101 | 1.75663 | — | — |
| 102 | 1.65421 | — | — |
| 103 | 1.40266 | — | — |
expressive of the law of human mortality, &c.
TABLE VII.—Logarithm of Carlisle chance of living 5 years at every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|------|-------------------|------|-------------------|------|-------------------|------|-------------------|
| 0 | 1.83232 | 20 | 1.98469 | 40 | 1.96915 | 60 | 1.91826 |
| 1 | 1.89709 | 21 | 1.98457 | 41 | 1.96836 | 61 | 1.91483 |
| 2 | 1.92823 | 22 | 1.98439 | 42 | 1.96790 | 62 | 1.91180 |
| 3 | 1.95354 | 23 | 1.98405 | 43 | 1.96780 | 63 | 1.90864 |
| 4 | 1.99747 | 24 | 1.98333 | 44 | 1.96808 | 64 | 1.90492 |
| 5 | 1.97792 | 25 | 1.98213 | 45 | 1.96857 | 65 | 1.90067 |
| 6 | 1.98376 | 26 | 1.98091 | 46 | 1.96918 | 66 | 1.89586 |
| 7 | 1.98703 | 27 | 1.97967 | 47 | 1.96941 | 67 | 1.88838 |
| 8 | 1.98869 | 28 | 1.97863 | 48 | 1.96915 | 68 | 1.87746 |
| 9 | 1.98930 | 29 | 1.97804 | 49 | 1.96818 | 69 | 1.86279 |
| 10 | 1.98911 | 30 | 1.97789 | 50 | 1.96676 | 70 | 1.84362 |
| 11 | 1.98836 | 31 | 1.97783 | 51 | 1.96477 | 71 | 1.82305 |
| 12 | 1.98754 | 32 | 1.97767 | 52 | 1.96269 | 72 | 1.80220 |
| 13 | 1.98670 | 33 | 1.97736 | 53 | 1.96017 | 73 | 1.78348 |
| 14 | 1.98593 | 34 | 1.97687 | 54 | 1.95660 | 74 | 1.76877 |
| 15 | 1.98528 | 35 | 1.97611 | 55 | 1.95155 | 75 | 1.75508 |
| 16 | 1.98490 | 36 | 1.97490 | 56 | 1.94461 | 76 | 1.74231 |
| 17 | 1.98479 | 37 | 1.97349 | 57 | 1.93711 | 77 | 1.72712 |
| 18 | 1.98476 | 38 | 1.97194 | 58 | 1.92973 | 78 | 1.71062 |
| 19 | 1.98472 | 39 | 1.97044 | 59 | 1.92343 | 79 | 1.68963 |
Logarithm of the Carlisle chance of living 10 years at every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|------|-------------------|------|-------------------|------|-------------------|------|-------------------|
| 0 | 1.81023 | 19 | 1.96805 | 38 | 1.93973 | 57 | 1.84801 |
| 1 | 1.88086 | 20 | 1.96682 | 39 | 1.93851 | 58 | 1.83836 |
| 2 | 1.91526 | 21 | 1.96548 | 40 | 1.93772 | 59 | 1.82835 |
| 3 | 1.94223 | 22 | 1.96406 | 41 | 1.93754 | 60 | 1.81893 |
| 4 | 1.96767 | 23 | 1.96268 | 42 | 1.93731 | 61 | 1.81070 |
| 5 | 1.96702 | 24 | 1.96136 | 43 | 1.93694 | 62 | 1.80018 |
| 6 | 1.97213 | 25 | 1.96002 | 44 | 1.93626 | 63 | 1.78610 |
| 7 | 1.97457 | 26 | 1.95873 | 45 | 1.93533 | 64 | 1.76771 |
| 8 | 1.97540 | 27 | 1.95734 | 46 | 1.93395 | 65 | 1.74430 |
| 9 | 1.97523 | 28 | 1.95598 | 47 | 1.93211 | 66 | 1.71891 |
| 10 | 1.97438 | 29 | 1.95490 | 48 | 1.92932 | 67 | 1.69058 |
| 11 | 1.97326 | 30 | 1.95400 | 49 | 1.92478 | 68 | 1.66094 |
| 12 | 1.97233 | 31 | 1.95273 | 50 | 1.91830 | 69 | 1.63157 |
| 13 | 1.97146 | 32 | 1.95116 | 51 | 1.90938 | 70 | 1.59870 |
| 14 | 1.97065 | 33 | 1.94929 | 52 | 1.89980 | 71 | 1.56536 |
| 15 | 1.96996 | 34 | 1.94730 | 53 | 1.88990 | 72 | 1.52932 |
| 16 | 1.96947 | 35 | 1.94526 | 54 | 1.88003 | 73 | 1.49411 |
| 17 | 1.96918 | 36 | 1.94326 | 55 | 1.86981 | 74 | 1.45840 |
| 18 | 1.96881 | 37 | 1.94138 | 56 | 1.85944 | 75 | 1.42434 |
### TABLE VII.—continued.
Logarithm of the Carlisle chance of living 15 years for every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|-------|---------------------|-------|---------------------|-------|---------------------|-------|---------------------|-------|---------------------|
| 0 | 1.79934 | 18 | 1.94744 | 36 | 1.91244 | 54 | 1.78495 | 72 | 1.14027 |
| 1 | 1.86922 | 19 | 1.94609 | 37 | 1.91080 | 55 | 1.77048 | 73 | 1.06511 |
| 2 | 1.90280 | 20 | 1.94471 | 38 | 1.90888 | 56 | 1.75530 | 74 | 2.99263 |
| 3 | 1.92893 | 21 | 1.94331 | 39 | 1.90669 | 57 | 1.73729 | 75 | 2.92827 |
| 4 | 1.94270 | 22 | 1.94173 | 40 | 1.90448 | 58 | 1.71582 | 76 | 2.84078 |
| 5 | 1.95230 | 23 | 1.94004 | 41 | 1.90231 | 59 | 1.69114 | 77 | 2.74184 |
| 6 | 1.95702 | 24 | 1.93823 | 42 | 1.90000 | 60 | 1.66256 | 78 | 2.64853 |
| 7 | 1.95936 | 25 | 1.93613 | 43 | 1.89712 | 61 | 1.63375 | 79 | 2.56823 |
| 8 | 1.96015 | 26 | 1.93364 | 44 | 1.89284 | 62 | 1.60238 | 80 | 2.49803 |
| 9 | 1.95995 | 27 | 1.93082 | 45 | 1.88687 | 63 | 1.56958 | 81 | 2.43900 |
| 10 | 1.95907 | 28 | 1.92792 | 46 | 1.87856 | 64 | 1.53648 | 82 | 2.39494 |
| 11 | 1.95784 | 29 | 1.92534 | 47 | 1.86922 | 65 | 1.49937 | 83 | 2.35164 |
| 12 | 1.95672 | 30 | 1.92316 | 48 | 1.85905 | 66 | 1.46123 | 84 | 2.31794 |
| 13 | 1.95551 | 31 | 1.92108 | 49 | 1.84820 | 67 | 1.41770 | 85 | 2.30588 |
| 14 | 1.95397 | 32 | 1.91905 | 50 | 1.83656 | 68 | 1.37157 | 86 | 2.28043 |
| 15 | 1.95209 | 33 | 1.91709 | 51 | 1.82421 | 69 | 1.32120 | 87 | 2.22768 |
| 16 | 1.95038 | 34 | 1.91538 | 52 | 1.81160 | 70 | 1.26797 | 88 | 2.11163 |
| 17 | 1.94885 | 35 | 1.91383 | 53 | 1.79853 | 71 | 1.20730 |
Logarithm of the Carlisle chance of living 20 years for every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|-------|---------------------|-------|---------------------|-------|---------------------|-------|---------------------|-------|---------------------|
| 0 | 1.78462 | 17 | 1.92652 | 34 | 1.88356 | 51 | 1.72007 | 68 | 2.94257 |
| 1 | 1.85412 | 18 | 1.92479 | 35 | 1.88059 | 52 | 1.69998 | 69 | 2.85542 |
| 2 | 1.88759 | 19 | 1.92295 | 36 | 1.87721 | 53 | 1.67599 | 70 | 2.77190 |
| 3 | 1.91369 | 20 | 1.92082 | 37 | 1.87349 | 54 | 1.64744 | 71 | 2.66383 |
| 4 | 1.92742 | 21 | 1.91821 | 38 | 1.86906 | 55 | 1.61410 | 72 | 2.54404 |
| 5 | 1.93699 | 22 | 1.91521 | 39 | 1.86329 | 56 | 1.57835 | 73 | 2.43202 |
| 6 | 1.94160 | 23 | 1.91197 | 40 | 1.85602 | 57 | 1.53949 | 74 | 2.33701 |
| 7 | 1.94376 | 24 | 1.90866 | 41 | 1.84692 | 58 | 1.49930 | 75 | 2.25311 |
| 8 | 1.94421 | 25 | 1.90528 | 42 | 1.83711 | 59 | 1.45991 | 76 | 2.18132 |
| 9 | 1.94328 | 26 | 1.90199 | 43 | 1.82684 | 60 | 1.41763 | 77 | 2.12205 |
| 10 | 1.94120 | 27 | 1.89872 | 44 | 1.81628 | 61 | 1.37606 | 78 | 2.06227 |
| 11 | 1.93874 | 28 | 1.89572 | 45 | 1.80513 | 62 | 1.32950 | 79 | 2.00757 |
| 12 | 1.93639 | 29 | 1.89342 | 46 | 1.79339 | 63 | 1.28021 | 80 | 3.97515 |
| 13 | 1.93414 | 30 | 1.89172 | 47 | 1.78101 | 64 | 1.22611 | 81 | 3.92237 |
| 14 | 1.93201 | 31 | 1.89027 | 48 | 1.76768 | 65 | 1.16864 | 82 | 3.83863 |
| 15 | 1.92999 | 32 | 1.88847 | 49 | 1.75312 | 66 | 1.10317 | 83 | 3.68263 |
| 16 | 1.92821 | 33 | 1.88624 | 50 | 1.73723 | 67 | 1.02866 |
### TABLE VII.—continued.
Logarithm of the Carlisle chance of living 25 years at every age \(a\).
| \(a\) | \(a\) | \(a\) | \(a\) |
|-------|-------|-------|-------|
| 0 | 1,76930 | 16 | 1,90311 |
| 1 | 1,83969 | 17 | 1,90001 |
| 2 | 1,87198 | 18 | 1,89673 |
| 3 | 1,89774 | 19 | 1,89339 |
| 4 | 1,91075 | 20 | 1,88997 |
| 5 | 1,91912 | 21 | 1,88657 |
| 6 | 1,92251 | 22 | 1,88311 |
| 7 | 1,92342 | 23 | 1,87977 |
| 8 | 1,92283 | 24 | 1,87674 |
| 9 | 1,92131 | 25 | 1,87335 |
| 10 | 1,91909 | 26 | 1,87117 |
| 11 | 1,91657 | 27 | 1,86813 |
| 12 | 1,91406 | 28 | 1,86487 |
| 13 | 1,91150 | 29 | 1,86159 |
| 14 | 1,90888 | 30 | 1,85848 |
| 15 | 1,90610 | 31 | 1,85504 |
Logarithm of the Carlisle chance of living 30 years at every age \(a\).
| \(a\) | \(a\) | \(a\) | \(a\) |
|-------|-------|-------|-------|
| 0 | 1,75143 | 15 | 1,87525 |
| 1 | 1,81960 | 16 | 1,87146 |
| 2 | 1,85164 | 17 | 1,86790 |
| 3 | 1,87637 | 18 | 1,86453 |
| 4 | 1,88879 | 19 | 1,86147 |
| 5 | 1,89701 | 20 | 1,85854 |
| 6 | 1,90033 | 21 | 1,85575 |
| 7 | 1,90109 | 22 | 1,85253 |
| 8 | 1,90019 | 23 | 1,84892 |
| 9 | 1,89818 | 24 | 1,84492 |
| 10 | 1,89520 | 25 | 1,84061 |
| 11 | 1,89147 | 26 | 1,83594 |
| 12 | 1,88755 | 27 | 1,83083 |
| 13 | 1,88344 | 28 | 1,82504 |
| 14 | 1,87931 | 29 | 1,81819 |
Logarithm of the Carlisle chance of living 35 years for every age \(a\).
| \(a\) | \(a\) | \(a\) | \(a\) |
|-------|-------|-------|-------|
| 0 | 1,72933 | 14 | 1,84739 |
| 1 | 1,79743 | 15 | 1,84382 |
| 2 | 1,82932 | 16 | 1,84005 |
| 3 | 1,85373 | 17 | 1,83732 |
| 4 | 1,86505 | 18 | 1,83308 |
| 5 | 1,87312 | 19 | 1,82964 |
| 6 | 1,87523 | 20 | 1,82530 |
| 7 | 1,87458 | 21 | 1,82052 |
| 8 | 1,87213 | 22 | 1,81522 |
| 9 | 1,86861 | 23 | 1,80909 |
| 10 | 1,86435 | 24 | 1,80152 |
| 11 | 1,85983 | 25 | 1,79216 |
| 12 | 1,85544 | 26 | 1,78055 |
| 13 | 1,85123 | 27 | 1,76794 |
| 14 | 1,84739 | 28 | 1,75476 |
| 15 | 1,84382 | 29 | 1,74162 |
| 16 | 1,84005 | 30 | 1,72829 |
| 17 | 1,83732 | 31 | 1,71448 |
| 18 | 1,83308 | 32 | 1,70007 |
| 19 | 1,82964 | 33 | 1,68477 |
| 20 | 1,82530 | 34 | 1,66850 |
| 21 | 1,82052 | 35 | 1,65106 |
| 22 | 1,81522 | 36 | 1,63251 |
| 23 | 1,80909 | 37 | 1,61078 |
| 24 | 1,80152 | 38 | 1,58488 |
| 25 | 1,79216 | 39 | 1,55443 |
| 26 | 1,78055 | 40 | 1,51858 |
| 27 | 1,76794 | 41 | 1,48066 |
| 28 | 1,75476 | 42 | 1,43949 |
| 29 | 1,74162 | 43 | 1,39042 |
| 30 | 1,72829 | 44 | 1,35277 |
| 31 | 1,71448 | 45 | 1,30451 |
| 32 | 1,70007 | 46 | 1,25462 |
| 33 | 1,68477 | 47 | 1,19872 |
| 34 | 1,66850 | 48 | 1,13925 |
| 35 | 1,65106 | 49 | 1,07432 |
| 36 | 1,63251 | 50 | 1,00520 |
| 37 | 1,61078 | 51 | 2,92738 |
| 38 | 1,58488 | 52 | 2,84025 |
| 39 | 1,55443 | 53 | 2,74110 |
| 40 | 1,51858 | 54 | 2,64036 |
| 41 | 1,48066 | 55 | 2,54237 |
| 42 | 1,43949 | 56 | 2,41913 |
| 43 | 1,39042 | 57 | 2,28133 |
| 44 | 1,35277 | 58 | 2,14784 |
| 45 | 1,30451 | 59 | 2,02815 |
| 46 | 1,25462 | 60 | 3,91566 |
| 47 | 1,19872 | 61 | 3,81506 |
| 48 | 1,13925 | 62 | 3,72443 |
| 49 | 1,07432 | 63 | 3,63185 |
| 50 | 1,00520 | 64 | 3,54405 |
| 51 | 2,92738 | 65 | 3,47452 |
| 52 | 2,84025 | 66 | 3,38360 |
| 53 | 2,74110 | 67 | 3,25633 |
| 54 | 2,64036 | 68 | 3,05420 |
MDCCCXXV.
### Table VII—continued.
Logarithm of the Carlisle chance of living 40 years for every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|-------|---------------------|-------|---------------------|-------|---------------------|-------|---------------------|
| 0 | 1,70544 | 13 | 1,82038 | 26 | 1,69538 | 39 | 1,32320 |
| 1 | 1,77233 | 14 | 1,81557 | 27 | 1,67973 | 40 | 1,27366 |
| 2 | 1,80280 | 15 | 1,81057 | 28 | 1,66340 | 41 | 1,22298 |
| 3 | 1,82567 | 16 | 1,80542 | 29 | 1,64654 | 42 | 1,16661 |
| 4 | 1,83609 | 17 | 1,80001 | 30 | 1,62896 | 43 | 1,10705 |
| 5 | 1,84227 | 18 | 1,79385 | 31 | 1,61034 | 44 | 1,04240 |
| 6 | 1,84359 | 19 | 1,78624 | 32 | 1,58845 | 45 | 2,97377 |
| 7 | 1,84247 | 20 | 1,77684 | 33 | 1,56223 | 46 | 2,89056 |
| 8 | 1,83992 | 21 | 1,76513 | 34 | 1,53130 | 47 | 2,80967 |
| 9 | 1,83669 | 22 | 1,75233 | 35 | 1,49469 | 48 | 2,71025 |
| 10 | 1,83292 | 23 | 1,73882 | 36 | 1,45556 | 49 | 2,60854 |
| 11 | 1,82901 | 24 | 1,72494 | 37 | 1,41298 | 50 | 2,50913 |
| 12 | 1,82486 | 25 | 1,71042 | 38 | 1,36836 | 51 | 2,38390 |
Logarithm of the Carlisle chance of living 45 years for every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|-------|---------------------|-------|---------------------|-------|---------------------|-------|---------------------|
| 0 | 1,67459 | 12 | 1,78755 | 24 | 1,62987 | 36 | 1,19787 |
| 1 | 1,74068 | 13 | 1,78055 | 25 | 1,61109 | 37 | 1,14010 |
| 2 | 1,77070 | 14 | 1,77217 | 26 | 1,59125 | 38 | 1,07899 |
| 3 | 1,79346 | 15 | 1,76212 | 27 | 1,56812 | 39 | 1,01283 |
| 4 | 1,80417 | 16 | 1,75002 | 28 | 1,54086 | 40 | 2,94292 |
| 5 | 1,81084 | 17 | 1,73712 | 29 | 1,50933 | 41 | 2,86492 |
| 6 | 1,81277 | 18 | 1,72357 | 30 | 1,47258 | 42 | 2,77756 |
| 7 | 1,81190 | 19 | 1,70967 | 31 | 1,43339 | 43 | 2,67805 |
| 8 | 1,80907 | 20 | 1,69510 | 32 | 1,39065 | 44 | 2,57662 |
| 9 | 1,80487 | 21 | 1,67996 | 33 | 1,34571 | 45 | 2,47770 |
| 10 | 1,79968 | 22 | 1,66412 | 34 | 1,30007 | 46 | 2,35308 |
| 11 | 1,79378 | 23 | 1,64745 | 35 | 1,24977 | 47 | 2,21344 |
Logarithm of the Carlisle chance of living 50 years for every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|-------|---------------------|-------|---------------------|-------|---------------------|-------|---------------------|
| 0 | 1,64316 | 11 | 1,73839 | 22 | 1,55251 | 33 | 1,05634 |
| 1 | 1,70987 | 12 | 1,72466 | 23 | 1,52491 | 34 | 2,98970 |
| 2 | 1,74011 | 13 | 1,71028 | 24 | 1,49266 | 35 | 2,91903 |
| 3 | 1,76261 | 14 | 1,69560 | 25 | 1,45471 | 36 | 2,83982 |
| 4 | 1,77234 | 15 | 1,68038 | 26 | 1,41430 | 37 | 2,75105 |
| 5 | 1,77760 | 16 | 1,66486 | 27 | 1,37032 | 38 | 2,65000 |
| 6 | 1,77754 | 17 | 1,64892 | 28 | 1,32434 | 39 | 2,54705 |
| 7 | 1,77458 | 18 | 1,63222 | 29 | 1,27810 | 40 | 2,44685 |
| 8 | 1,76925 | 19 | 1,61459 | 30 | 1,22766 | 41 | 2,32144 |
| 9 | 1,76147 | 20 | 1,59577 | 31 | 1,17570 | 42 | 2,18133 |
| 10 | 1,75123 | 21 | 1,57582 | 32 | 1,11777 | 43 | 2,04495 |
expressive of the law of human mortality, &c.
TABLE VII.—continued.
Logarithm of the Carlisle chance of living 55 years for every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|------|---------------------|------|---------------------|------|---------------------|------|---------------------|
| 0 | 1,60991 | 10 | 1,66649 | 20 | 1,43940 | 30 | 2,89693 |
| 1 | 1,67464 | 11 | 1,65322 | 21 | 1,39887 | 31 | 2,81764 |
| 2 | 1,70281 | 12 | 1,63646 | 22 | 1,35471 | 32 | 2,72872 |
| 3 | 1,72278 | 13 | 1,61892 | 23 | 1,30840 | 33 | 2,62734 |
| 4 | 1,72894 | 14 | 1,60051 | 24 | 1,26143 | 34 | 2,52392 |
| 5 | 1,72914 | 15 | 1,58105 | 25 | 1,20979 | 35 | 2,42296 |
| 6 | 1,72215 | 16 | 1,56072 | 26 | 1,15661 | 36 | 2,29634 |
| 7 | 1,71169 | 17 | 1,53730 | 27 | 1,09743 | 37 | 2,15482 |
| 8 | 1,69897 | 18 | 1,50967 | 28 | 1,03497 | 38 | 2,01689 |
| 9 | 1,68490 | 19 | 1,47738 | 29 | 2,96773 | 39 | 3,89144 |
| | | | | | | | |
Logarithm of the Carlisle chance of living 60 years for every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|------|---------------------|------|---------------------|------|---------------------|------|---------------------|
| 0 | 1,56146 | 9 | 1,58982 | 18 | 1,29315 | 27 | 2,70839 |
| 1 | 1,61925 | 10 | 1,57016 | 19 | 1,24615 | 28 | 2,60597 |
| 2 | 1,63992 | 11 | 1,54908 | 20 | 1,19448 | 29 | 2,50196 |
| 3 | 1,65251 | 12 | 1,52484 | 21 | 1,14119 | 30 | 2,40086 |
| 4 | 1,65237 | 13 | 1,49038 | 22 | 1,08183 | 31 | 2,27417 |
| 5 | 1,64740 | 14 | 1,46331 | 23 | 1,01902 | 32 | 2,13249 |
| 6 | 1,63698 | 15 | 1,42467 | 24 | 2,95106 | 33 | 3,99425 |
| 7 | 1,62349 | 16 | 1,38377 | 25 | 2,87906 | 34 | 3,86830 |
| 8 | 1,60761 | 17 | 1,33950 | 26 | 2,79855 | 35 | 3,74779 |
| | | | | | | | |
Logarithm of the Carlisle chance of living 65 years for every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|------|---------------------|------|---------------------|------|---------------------|------|---------------------|
| 0 | 1,47972 | 8 | 1,48507 | 16 | 1,12608 | 24 | 2,48528 |
| 1 | 1,53408 | 9 | 1,45261 | 17 | 1,06662 | 25 | 2,38298 |
| 2 | 1,55171 | 10 | 1,41378 | 18 | 1,00378 | 26 | 2,25507 |
| 3 | 1,56115 | 11 | 1,37213 | 19 | 2,93578 | 27 | 2,11216 |
| 4 | 1,55729 | 12 | 1,32704 | 20 | 2,86374 | 28 | 3,97288 |
| 5 | 1,54807 | 13 | 1,27986 | 21 | 2,78313 | 29 | 3,84634 |
| 6 | 1,53285 | 14 | 1,23208 | 22 | 2,69278 | 30 | 3,72569 |
| 7 | 1,51187 | 15 | 1,17975 | 23 | 2,59002 | 31 | 3,61471 |
| | | | | | | | |
Logarithm of the Carlisle chance of living 70 years for every age \(a\).
| \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. | \(a\) | \(\lambda\) chance. |
|------|---------------------|------|---------------------|------|---------------------|------|---------------------|
| 0 | 1,38039 | 7 | 1,31407 | 14 | 2,92171 | 21 | 2,23965 |
| 1 | 1,42994 | 8 | 1,26855 | 15 | 2,84902 | 22 | 2,09655 |
| 2 | 1,44010 | 9 | 1,22138 | 16 | 2,76802 | 23 | 3,95693 |
| 3 | 1,43861 | 10 | 1,16886 | 17 | 2,67757 | 24 | 3,82907 |
| 4 | 1,42008 | 11 | 1,11445 | 18 | 2,57478 | 25 | 3,70782 |
| 5 | 1,39170 | 12 | 1,05416 | 19 | 2,47001 | 26 | 3,59561 |
| 6 | 1,35590 | 13 | 2,99049 | 20 | 2,36767 | 27 | 3,49237 |
| | | | | | | | |
### TABLE VII.—continued.
Logarithm of the Carlisle chance of living 75 years for every age $a$.
| $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. |
|-----|------------------|-----|------------------|-----|------------------|-----|------------------|
| 0 | 1.22402 | 6 | 1.99821 | 12 | 2.66511 | 18 | 3.94169 |
| 1 | 1.25299 | 7 | 1.94119 | 13 | 2.56149 | 19 | 3.81439 |
| 2 | 1.24230 | 8 | 2.97918 | 14 | 2.45593 | 20 | 3.69250 |
| 3 | 1.22209 | 9 | 2.91109 | 15 | 2.35295 | 21 | 3.58019 |
| 4 | 1.18885 | 10 | 2.83813 | 16 | 2.2455 | 22 | 3.47076 |
| 5 | 1.14678 | 11 | 2.75639 | 17 | 2.08134 | 23 | 3.37066 |
| | | | | | | | |
Logarithm of the Carlisle chance of living 80 years for every age $a$.
| $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. |
|-----|------------------|-----|------------------|-----|------------------|-----|------------------|
| 0 | 2.97909 | 5 | 2.81604 | 10 | 2.34206 | 15 | 3.67778 |
| 1 | 2.99530 | 6 | 2.74015 | 11 | 2.21291 | 16 | 3.56508 |
| 2 | 2.96941 | 7 | 2.65214 | 12 | 2.06888 | 17 | 3.46155 |
| 3 | 2.93272 | 8 | 2.55018 | 13 | 2.92839 | 18 | 3.35542 |
| 4 | 2.87848 | 9 | 2.44523 | 14 | 3.80031 | 19 | 3.25372 |
| | | | | | | | |
Logarithm of the Carlisle chance of living 85 years for every age $a$.
| $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. |
|-----|------------------|-----|------------------|-----|------------------|-----|------------------|
| 0 | 2.64836 | 4 | 2.41271 | 8 | 2.91708 | 12 | 3.44909 |
| 1 | 2.63724 | 5 | 2.31997 | 9 | 2.78962 | 13 | 3.34213 |
| 2 | 2.58037 | 6 | 2.19667 | 10 | 2.66689 | 14 | 3.23965 |
| 3 | 2.50372 | 7 | 2.05591 | 11 | 2.55345 | 15 | 3.15490 |
| | | | | | | | |
Logarithm of the Carlisle chance of living 90 years for every age $a$.
| $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. |
|-----|------------------|-----|------------------|-----|------------------|-----|------------------|
| 0 | 2.15229 | 3 | 3.87062 | 6 | 3.53721 | 9 | 3.22895 |
| 1 | 2.09377 | 4 | 3.75799 | 7 | 3.43612 | 10 | 3.14401 |
| 2 | 3.98414 | 5 | 3.64480 | 8 | 3.33082 | 11 | 3.03682 |
| | | | | | | | |
Logarithm of the Carlisle chance of living 95 years for every age $a$.
| $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. | $a$ | $\lambda$ chance. |
|-----|------------------|-----|------------------|-----|------------------|-----|------------------|
| 0 | 3.47712 | 2 | 3.56435 | 4 | 3.19642 | 6 | 3.02058 |
| 1 | 3.43431 | 3 | 3.28436 | 5 | 3.12193 | 7 | 4.87982 |
| | | | | | | | |
Logarithm of the Carlisle chance of living 100 years for every age $a$.
| $a$ | $\lambda$ chance. |
|-----|------------------|
| 0 | 4.95424 |
| 1 | 4.91768 |
| 2 | 4.80805 |
| 3 | 4.61535 |
TABLE VIII.—Logarithm of Deparcieux chance of living for every age.
| | 10 years | 20 years | 30 years | 40 years | 50 years | 60 years | 70 years | 80 years | 90 years |
|---|----------|----------|----------|----------|----------|----------|----------|----------|----------|
| 0 | | | | | | | | | |
| 1 | | | | | | | | | |
| 2 | | | | | | | | | |
| 3 | 1.93450 | 1.89763 | 1.85126 | 1.80346 | 1.73957 | 1.62634 | 1.39967 | 2.85126 | 3.30103 |
| 4 | 1.94469 | 1.90644 | 1.85957 | 1.81188 | 1.74401 | 1.62495 | 1.37684 | 2.78408 | 3.01323 |
| 5 | 1.95159 | 1.91193 | 1.86455 | 1.81638 | 1.74418 | 1.61979 | 1.34747 | 2.70443 | |
| 6 | 1.95683 | 1.91575 | 1.86784 | 1.82040 | 1.74248 | 1.61130 | 1.31482 | 2.61130 | |
| 7 | 1.96027 | 1.91825 | 1.86981 | 1.82177 | 1.73928 | 1.59608 | 1.27663 | 2.50098 | |
| 8 | 1.96282 | 1.91985 | 1.87151 | 1.82222 | 1.73410 | 1.58512 | 1.23231 | 2.38721 | |
| 9 | 1.96495 | 1.92101 | 1.87278 | 1.82146 | 1.72821 | 1.56781 | 1.18415 | 2.25473 | |
| 10| 1.96614 | 1.92122 | 1.87309 | 1.81970 | 1.72110 | 1.54688 | 1.12740 | 2.09691 | |
| 11| 1.96582 | 1.92042 | 1.87239 | 1.81607 | 1.71269 | 1.52337 | 1.06380 | 3.90458 | |
| 12| 1.96448 | 1.91860 | 1.87060 | 1.81067 | 1.70296 | 1.49545 | 1.09190 | 3.66454 | |
| 13| 1.96313 | 1.91676 | 1.86896 | 1.80507 | 1.69184 | 1.46517 | 1.01676 | 3.36653 | |
| 14| 1.96175 | 1.91488 | 1.86719 | 1.79932 | 1.68026 | 1.43215 | 1.08393 | 3.06854 | |
| 15| 1.96034 | 1.91296 | 1.86539 | 1.79259 | 1.66820 | 1.39588 | 1.05447 | | |
| 16| 1.95892 | 1.91101 | 1.86357 | 1.78595 | 1.65447 | 1.35799 | 1.02742 | | |
| 17| 1.95798 | 1.90954 | 1.86150 | 1.77901 | 1.63941 | 1.31636 | 1.00249 | | |
| 18| 1.95703 | 1.90869 | 1.85940 | 1.77128 | 1.62230 | 1.28949 | 1.00249 | | |
| 19| 1.95606 | 1.90783 | 1.85651 | 1.76326 | 1.60286 | 1.21920 | 1.00249 | | |
| 20| 1.95508 | 1.90695 | 1.85356 | 1.75496 | 1.58074 | 1.16126 | 1.00249 | | |
| 21| 1.95460 | 1.90657 | 1.85030 | 1.74687 | 1.55755 | 1.09798 | 1.00249 | | |
| 22| 1.95412 | 1.90621 | 1.84619 | 1.73848 | 1.53097 | 1.02742 | 1.00249 | | |
| 23| 1.95363 | 1.90583 | 1.84194 | 1.72871 | 1.50204 | 1.02742 | 1.00249 | | |
| 24| 1.95313 | 1.90544 | 1.83757 | 1.71851 | 1.47040 | 1.02742 | 1.00249 | | |
| 25| 1.95262 | 1.90505 | 1.83225 | 1.70786 | 1.43554 | 1.02742 | 1.00249 | | |
| 26| 1.95209 | 1.90465 | 1.82673 | 1.69555 | 1.39907 | 1.02742 | 1.00249 | | |
| 27| 1.95156 | 1.90352 | 1.82103 | 1.68143 | 1.35838 | 1.02742 | 1.00249 | | |
| 28| 1.95166 | 1.90237 | 1.81425 | 1.66527 | 1.31246 | 1.02742 | 1.00249 | | |
| 29| 1.95177 | 1.90045 | 1.80720 | 1.64180 | 1.26314 | 1.02742 | 1.00249 | | |
| 30| 1.95187 | 1.89648 | 1.79988 | 1.62566 | 1.20618 | 1.02742 | 1.00249 | | |
| 31| 1.95197 | 1.89570 | 1.79227 | 1.60295 | 1.14338 | 1.02742 | 1.00249 | | |
| 32| 1.95209 | 1.89207 | 1.78436 | 1.57655 | 1.07330 | 1.02742 | 1.00249 | | |
| 33| 1.95220 | 1.88838 | 1.77508 | 1.54841 | 1.00000 | 1.02742 | 1.00249 | | |
| 34| 1.95231 | 1.88444 | 1.76538 | 1.51727 | 1.92451 | 1.02742 | 1.00249 | | |
| 35| 1.95243 | 1.87953 | 1.75524 | 1.48292 | 1.83988 | 1.02742 | 1.00249 | | |
| 36| 1.95256 | 1.87464 | 1.74346 | 1.44698 | 1.74346 | 1.02742 | 1.00249 | | |
| 37| 1.95196 | 1.86947 | 1.72987 | 1.40682 | 1.60311 | 1.02742 | 1.00249 | | |
| 38| 1.95071 | 1.86259 | 1.71361 | 1.36080 | 1.51570 | 1.02742 | 1.00249 | | |
| 39| 1.94868 | 1.85543 | 1.69503 | 1.31137 | 1.38195 | 1.02742 | 1.00249 | | |
| 40| 1.94661 | 1.84801 | 1.67379 | 1.25431 | 1.22382 | 1.02742 | 1.00249 | | |
| 41| 1.94373 | 1.84030 | 1.65098 | 1.19141 | 1.03219 | 1.02742 | 1.00249 | | |
| 42| 1.93998 | 1.83227 | 1.62476 | 1.12121 | 1.79385 | 1.02742 | 1.00249 | | |
| 43| 1.93611 | 1.82288 | 1.59641 | 1.04780 | 1.49757 | 1.02742 | 1.00249 | | |
| 44| 1.93213 | 1.81307 | 1.56496 | 2.97220 | 1.20135 | 1.02742 | 1.00249 | | |
| 45| 1.92720 | 1.80281 | 1.53049 | 2.88745 | | 1.02742 | 1.00249 | | |
| 46| 1.92208 | 1.79090 | 1.49442 | 2.79090 | | 1.02742 | 1.00249 | | |
| 47| 1.91751 | 1.77791 | 1.45486 | 2.67921 | | 1.02742 | 1.00249 | | |
| 48| 1.91188 | 1.76290 | 1.41000 | 2.50429 | | 1.02742 | 1.00249 | | |
| 49| 1.90675 | 1.74635 | 1.36269 | 2.43327 | | 1.02742 | 1.00249 | | |
| 50| 1.90140 | 1.72718 | 1.30770 | 2.27721 | | 1.02742 | 1.00249 | | |
| 51| 1.89657 | 1.70725 | 1.24768 | 2.08846 | | 1.02742 | 1.00249 | | |
| 52| 1.89229 | 1.68478 | 1.18123 | 2.85387 | | 1.02742 | 1.00249 | | |
[continued.]
TABLE VIII. continued.—Logarithm of Deparcieux chance of living for every age $a$.
| $a$ | 10 years. | 20 years. | 30 years. | 40 years. | 50 years. | 60 years. | 70 years. | 80 years. | 90 years |
|-----|-----------|-----------|-----------|-----------|-----------|-----------|-----------|-----------|-----------|
| 53 | 1.88677 | 1.66010 | 1.11169 | 3.56146 | | | | | |
| 54 | 1.88094 | 1.62283 | 1.04007 | 3.26922 | | | | | |
| 55 | 1.87561 | 1.60329 | 2.96025 | | | | | | |
| 56 | 1.86882 | 1.57234 | 2.86882 | | | | | | |
| 57 | 1.86040 | 1.53735 | 2.76170 | | | | | | |
| 58 | 1.85102 | 1.49821 | 2.65311 | | | | | | |
| 59 | 1.83960 | 1.45594 | 2.52052 | | | | | | |
| 60 | 1.82578 | 1.40630 | 2.37581 | | | | | | |
| 61 | 1.81068 | 1.35111 | 2.19189 | | | | | | |
| 62 | 1.79249 | 1.28894 | 3.96158 | | | | | | |
| 63 | 1.77333 | 1.22492 | 3.67469 | | | | | | |
| 64 | 1.75189 | 1.15913 | 3.38828 | | | | | | |
| 65 | 1.72768 | 1.08464 | | | | | | | |
| 66 | 1.70352 | 1.00000 | | | | | | | |
| 67 | 1.67695 | 2.90130 | | | | | | | |
| 68 | 1.64719 | 2.80209 | | | | | | | |
| 69 | 1.61634 | 2.68692 | | | | | | | |
| 70 | 1.58052 | 2.55003 | | | | | | | |
| 71 | 1.54943 | 2.38121 | | | | | | | |
| 72 | 1.49645 | 2.16909 | | | | | | | |
| 73 | 1.45159 | 3.90136 | | | | | | | |
| 74 | 1.40724 | 3.63639 | | | | | | | |
| 75 | 1.35696 | | | | | | | | |
| 76 | 1.29048 | | | | | | | | |
| 77 | 1.22435 | | | | | | | | |
| 78 | 1.15490 | | | | | | | | |
| 79 | 1.07058 | | | | | | | | |
| 80 | 2.96951 | | | | | | | | |
| 81 | 2.84078 | | | | | | | | |
| 82 | 2.67264 | | | | | | | | |
| 83 | 2.44977 | | | | | | | | |
| 84 | 2.22915 | | | | | | | | |
TABLE IX.—Logarithm of the Northampton chance for living at every age \(a\).
| \(a\) | 10 years. | 20 years. | 30 years. | 40 years. | 50 years. | 60 years. | 70 years. | 80 years. | 90 years. |
|------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
| 0 | 1.68764 | 1.64396 | 1.57564 | 1.49417 | 1.38958 | 1.24287 | 1.02428 | 2.60484 | 3.59643 |
| 1 | 1.81295 | 1.76713 | 1.69746 | 1.61431 | 1.50604 | 1.35435 | 1.12443 | 2.67151 | 3.59446 |
| 2 | 1.88378 | 1.83356 | 1.76454 | 1.67952 | 1.56809 | 1.41064 | 1.16788 | 2.67677 | 3.51790 |
| 3 | 1.91089 | 1.85979 | 1.78780 | 1.70070 | 1.58568 | 1.42229 | 1.16522 | 2.62961 | 3.37283 |
| 4 | 1.92894 | 1.87511 | 1.80190 | 1.71263 | 1.59383 | 1.42421 | 1.15070 | 2.55993 | 3.14495 |
| 5 | 1.93843 | 1.88180 | 1.80733 | 1.71581 | 1.59300 | 1.41601 | 1.12431 | 2.47370 | 4.80625 |
| 6 | 1.94739 | 1.88788 | 1.81211 | 1.71823 | 1.59118 | 1.40806 | 1.09339 | 2.37854 | |
| 7 | 1.95322 | 1.89101 | 1.81390 | 1.71755 | 1.58601 | 1.39522 | 1.05661 | 2.27263 | |
| 8 | 1.95660 | 1.89203 | 1.81382 | 1.71549 | 1.57827 | 1.37909 | 1.01505 | 2.15453 | |
| 9 | 1.95739 | 1.89080 | 1.81084 | 1.70923 | 1.56781 | 1.35940 | 2.96901 | 2.03386 | |
| 10 | 1.95632 | 1.88800 | 1.80653 | 1.70194 | 1.55523 | 1.33664 | 2.91720 | 3.90879 | |
| 11 | 1.95418 | 1.88451 | 1.80136 | 1.69345 | 1.54140 | 1.31148 | 2.85856 | 3.78151 | |
| 12 | 1.95158 | 1.88076 | 1.79574 | 1.68431 | 1.52668 | 1.28410 | 2.79299 | 3.63412 | |
| 13 | 1.94890 | 1.87691 | 1.78681 | 1.67479 | 1.51140 | 1.25433 | 2.71872 | 3.46194 | |
| 14 | 1.94617 | 1.87296 | 1.78369 | 1.66489 | 1.49527 | 1.22176 | 2.63099 | 3.21602 | |
| 15 | 1.94337 | 1.86890 | 1.77738 | 1.65457 | 1.47848 | 1.18588 | 2.53527 | 4.86782 | |
| 16 | 1.94049 | 1.86472 | 1.77084 | 1.64379 | 1.46067 | 1.14600 | 2.43115 | | |
| 17 | 1.93779 | 1.86068 | 1.76433 | 1.63279 | 1.44200 | 1.10339 | 2.31941 | | |
| 18 | 1.93543 | 1.85692 | 1.75799 | 1.62167 | 1.42249 | 1.05845 | 2.19793 | | |
| 19 | 1.93341 | 1.85345 | 1.75184 | 1.61042 | 1.40201 | 1.01162 | 2.07647 | | |
| 20 | 1.93168 | 1.85021 | 1.74562 | 1.59891 | 1.38032 | 2.96088 | 3.95247 | | |
| 21 | 1.93033 | 1.84718 | 1.73927 | 1.58722 | 1.35730 | 2.90438 | 3.82733 | | |
| 22 | 1.92918 | 1.84417 | 1.73274 | 1.57511 | 1.33253 | 2.84141 | 3.68255 | | |
| 23 | 1.92801 | 1.84091 | 1.72589 | 1.56250 | 1.30543 | 2.76982 | 3.51304 | | |
| 24 | 1.92679 | 1.83752 | 1.71872 | 1.54910 | 1.27559 | 2.68482 | 3.26084 | | |
| 25 | 1.92553 | 1.83401 | 1.71120 | 1.53511 | 1.24251 | 2.59191 | 4.92445 | | |
| 26 | 1.92423 | 1.83035 | 1.70330 | 1.52018 | 1.20551 | 2.49066 | | | |
| 27 | 1.92289 | 1.82654 | 1.69500 | 1.50421 | 1.16560 | 2.38162 | | | |
| 28 | 1.92149 | 1.82256 | 1.68624 | 1.48706 | 1.12302 | 2.26250 | | | |
| 29 | 1.92004 | 1.81843 | 1.67701 | 1.46860 | 1.07821 | 2.14306 | | | |
| 30 | 1.91853 | 1.81394 | 1.66723 | 1.44864 | 1.02920 | 2.02079 | | | |
| 31 | 1.91685 | 1.80894 | 1.65689 | 1.42697 | 1.07405 | 2.89700 | | | |
| 32 | 1.91498 | 1.80355 | 1.64592 | 1.40334 | 1.01223 | 3.75336 | | | |
| 33 | 1.91290 | 1.79788 | 1.63449 | 1.37742 | 1.04181 | 3.58503 | | | |
| 34 | 1.91073 | 1.79193 | 1.62231 | 1.34880 | 1.07580 | 3.34305 | | | |
| 35 | 1.90848 | 1.78567 | 1.60958 | 1.31698 | 1.16637 | 4.95892 | | | |
| 36 | 1.90612 | 1.77907 | 1.59595 | 1.28128 | 1.25643 | | | | |
| 37 | 1.90365 | 1.77211 | 1.58132 | 1.24271 | 1.34587 | | | | |
| 38 | 1.90107 | 1.76475 | 1.56557 | 1.20153 | 1.43101 | | | | |
| 39 | 1.89839 | 1.75697 | 1.54856 | 1.15817 | 1.52230 | | | | |
| 40 | 1.89541 | 1.74870 | 1.53011 | 1.11067 | 1.62226 | | | | |
| 41 | 1.89209 | 1.74004 | 1.51012 | 1.05720 | 1.98015 | | | | |
| 42 | 1.88857 | 1.73094 | 1.48836 | 1.99725 | 3.83838 | | | | |
| 43 | 1.88498 | 1.72159 | 1.46452 | 2.92891 | 3.07213 | | | | |
| 44 | 1.88120 | 1.71158 | 1.43807 | 2.84730 | 3.43232 | | | | |
| 45 | 1.87719 | 1.70110 | 1.40850 | 2.75759 | 3.09044 | | | | |
| 46 | 1.87295 | 1.68983 | 1.37516 | 2.66031 | | | | | |
| 47 | 1.86846 | 1.67707 | 1.33906 | 2.55508 | | | | | |
| 48 | 1.86368 | 1.66450 | 1.30046 | 2.43994 | | | | | |
| 49 | 1.85858 | 1.65017 | 1.25978 | 2.32463 | | | | | |
| 50 | 1.85329 | 1.63470 | 1.21526 | 2.20685 | | | | | |
| 51 | 1.84795 | 1.61803 | 1.16511 | 2.08806 | | | | | |
| 52 | 1.84237 | 1.59979 | 1.10868 | 3.94981 | | | | | |
[continued.]
TABLE IX. continued.—Logarithm of the Northampton chance for living at every age \(a\).
| \(a\) | 10 years. | 20 years. | 30 years. | 40 years. | 50 years. | 60 years. | 70 years. | 80 years. | 90 years. |
|------|-----------|-----------|-----------|-----------|-----------|-----------|-----------|-----------|-----------|
| 53 | 1.83661 | 1.57954 | 1.04393 | 3.78715 | | | | | |
| 54 | 1.83038 | 1.55687 | 2.96610 | 3.55112 | | | | | |
| 55 | 1.82391 | 1.53131 | 2.88070 | 3.21325 | | | | | |
| 56 | 1.81688 | 1.50221 | 2.78736 | | | | | | |
| 57 | 1.80921 | 1.47060 | 2.68662 | | | | | | |
| 58 | 1.80082 | 1.43678 | 2.57626 | | | | | | |
| 59 | 1.79159 | 1.40120 | 2.46605 | | | | | | |
| 60 | 1.78141 | 1.36197 | 2.35356 | | | | | | |
| 61 | 1.77008 | 1.31716 | 2.24011 | | | | | | |
| 62 | 1.75742 | 1.26631 | 2.10744 | | | | | | |
| 63 | 1.74293 | 1.20732 | 3.95054 | | | | | | |
| 64 | 1.72649 | 1.13572 | 3.72074 | | | | | | |
| 65 | 1.70740 | 1.05679 | 3.38934 | | | | | | |
| 66 | 1.68533 | 2.97048 | | | | | | | |
| 67 | 1.66139 | 2.87741 | | | | | | | |
| 68 | 1.63596 | 2.77544 | | | | | | | |
| 69 | 1.60961 | 2.67446 | | | | | | | |
| 70 | 1.58056 | 2.57215 | | | | | | | |
| 71 | 1.54708 | 2.47003 | | | | | | | |
| 72 | 1.50889 | 2.35002 | | | | | | | |
| 73 | 1.46439 | 2.20761 | | | | | | | |
| 74 | 1.40923 | 3.99425 | | | | | | | |
| 75 | 1.34939 | 3.68194 | | | | | | | |
| 76 | 1.28515 | | | | | | | | |
| 77 | 1.21602 | | | | | | | | |
| 78 | 1.13948 | | | | | | | | |
| 79 | 1.06485 | | | | | | | | |
| 80 | 2.99159 | | | | | | | | |
| 81 | 2.92295 | | | | | | | | |
| 82 | 2.84113 | | | | | | | | |
| 83 | 2.74322 | | | | | | | | |
| 84 | 2.58502 | | | | | | | | |
| 85 | 2.33255 | | | | | | | | |
How the value of particular assurances may be determined from the value of annuities, is shown in my Paper in the Philosophical Transactions for the year 1820, many of the cases of which are solved by methods essentially the same as those which have been long adopted; but when such assurances are but for terms, which are not of great extension, very near approximations may be had by using a geometrical progression, without confining the arithmetical operations to the same route, since the chance of extinction of the joint lives of the present age \(a, b, c, \&c.\) taking place between the period commencing with the time \(n + t - 1\), and finishing with the time \(n + t\), from the present, is
\[
\left( \frac{L_{n+t-1}}{L_n} : a, b, c, \&c. - \frac{L_{n+t}}{L_n} : a, b, c, \&c. \right) \div L_{a, b, c, \&c.}
\]
it follows that if \(r\) be the present value of unity, to be received certain in the time 1, and \(L_{n+t-1} : a, b, c, \&c. = L_{n-1} : a, b, c, \&c. \times \pi^t\), whatever \(t\) may be, that \(m \left[ a, b, c, \&c. \right]\) or the assurance of unity to be received at the first of the equal periods 1, from the commencement of the time \(n - 1\) to the expiration of the time \(m\), which shall happen after the extinction of the joint lives, is equal to
\[
\frac{L_{n-1}}{L_{a, b, c, \&c.}} \times \left\{ r^n \times (1 - \pi) + r^{n+1} (\pi - \pi^2) + r^{n+2} (\pi^2 - \pi^3) \ldots r^m (\pi^m - \pi^{m-1}) \right\}
\]
\[
= (1 - \pi) \times \frac{L_{n-1}}{L_{a, b, c, \&c.}} \times \left\{ r^n + r^{n+1} + r^{n+2} + \ldots + r^m \right\}
\]
\[
= (1 - \pi) \times r \times \frac{L_{n-1}}{L_{a, b, c, \&c.}} \times \left\{ r^n + r^{n+1} + r^{n+2} + \ldots + r^m \right\}
\]
If the assurance be not deferred, \(n\) will be equal to 1, and we shall have, according to the hypothesis,
\[
m \left[ a, b, c, \&c. \right] = (1 - \pi) \times r \times \frac{L_{n-1}}{L_{a, b, c, \&c.}} ; \text{ and also } \frac{1 - \pi}{\pi} \times m \left[ a, b, c, \&c. \right]
\]
If \(t\) be MDCCCXXV.
taken equal to 1, we shall have from the equation \( L_{n+t-1:a,b,c,\&c} = L_{n-1:a,b,c,\&c} \times \pi^i \), \( \pi = \frac{L_{n:a,b,c,\&c}}{L_{n-1:a,b,c,\&c}} \), and this would be the real value which should be taken for \( \pi \), if the geometrical progression coincided perfectly with the fact; and it would be indifferent whether we made it equal to \( \frac{L_{n+t:a,b,c,\&c}}{L_{n-1+t:a,b,c,\&c}} \), or \( \frac{L_{n:a,b,c,\&c}}{L_{n-1:a,b,c,\&c}} \), as the two would be the same; but this not being the case, there will be a preference; and generally, if not always, \( \pi \) should be taken an intermediate value between the two; and when the term is not very long, it will answer a good purpose to take it about the middle between them, inclining generally, though perhaps not always, rather nearer the last than the first, as the first terms are generally of more consequence than the last. If the said assurance be not deferred, and instead of being paid for immediately, be to be paid for by equal periodic payments, at an unite of time from each other, up to the time \( m-1 \) inclusive, and the first payment be to be made immediately, then will the present value of such periodic payment be \( \frac{r}{m-1} \left[ a, b, c, \&c. \right] \), and consequently each payment, from what is shown above, is equal to \( \frac{r}{m} \left[ a, b, c, \&c. \right] - \frac{r}{m-1} \left[ a, b, c, \&c. \right] = (1-\pi). r \). From whence we may draw an inference worthy of remark, namely; when an assurance of joint lives is meant to commence immediately, and to continue for a term of \( t \) years, which is not large, and to be paid for by \( t \) annual payments, that those payments will not differ much with the increase of the time \( t \), provided, as I have said, that \( t \) be not large, and the ages
be not at the extremes of life, a consequence which follows from the near agreement to a geometrical progression which takes place in the number of living at each small equal increment of time; that is to say, from the near coincidence of \( \frac{L}{n} : a, b, c, &c. \) with \( \frac{L}{n+1} : a, b, c, &c. \), or the small variation of \( \pi \) for the different values of \( t \); and also, that when the number of years for which an assurance continues be not very long, and the ages be not at the extremes of life, the annual premiums will not differ widely from the premiums to be paid for an assurance of one year of a life older than the proposed life by about half the term: thus, according to the Northampton table, at three per cent. to assure 100 l. at the
| Age | 15 | 20 | 30 | 40 | 50 | 60 | 64 |
|-----|----|----|----|----|----|----|----|
| For 7 years, the annual premium by the common modes of calculation | £1..2..11 | 1..9..5 | 1..14..11 | 2..4..1 | 3..0..8 | 4..7..1 | 5..4..10 |
| And the premium for one year assurance for an age 3 years older | 1..3..3 | 1..9..8 | 1..15..0 | 2..4..6 | 3..1..0 | 4..7..8 | 5..5..6 |
the difference of which is very small.—As another example, let
| Age | 10 | 20 | 30 | 40 | 50 | 60 |
|-----|----|----|----|----|----|----|
| For 10 years, the annual premium will be, by common modes of calculation | £0..19..2 | 1..9..1 | 1..15..8 | 2..5..8 | 3..3..4 | 4..12..6 |
| Premium for one year assurance, age 5 years older | 0..17..11 | 1..10..7 | 1..16..4 | 2..6..8 | 3..5..1 | 4..15..2 |
Here, except at the age 10, the excess is rather more in the approximation than in the first set of examples; but it should be recollected, that we took the exact middle, instead of inclining to the early age.
According to the Carlisle table of mortality at 3 per cent. to assure 100l. at the
| Age | 10 | 20 | 30 | 40 | 50 | 60 |
|-----|----|----|----|----|----|----|
| For 7 years, the annual premium, by common modes of calculation | £0 10 | 5 0 13 | 10 0 19 | 10 1 7 | 8 1 11 | 0 3 13 | 8 |
| For one year, the premium | 0 10 | 5 0 13 | 9 0 19 | 2 1 8 | 6 1 12 | 1 3 15 | 9 |
| For 10 years, the annual premium, by common modes of calculation | 0 11 | 3 0 14 | 7 1 0 4 | 1 7 | 7 1 14 | 11 3 17 | 8 |
| For one year, at an age 5 years older | 0 12 | 0 0 14 | 2 0 19 | 11 1 9 | 0 1 14 | 10 3 19 | 9 |
Moreover, because \( \frac{a, b, c, &c.}{m} \), or the single premium for the assurance of unity, on the joint lives \( a, b, c, &c. \) for \( m \) years, is \( \frac{\sum_{r=0}^{m-1} a, b, c, &c. \cdot r}{\sum_{r=0}^{m-1} a, b, c, &c.} = \frac{\sum_{r=0}^{m-1} a, b, c, &c. \cdot r + 1 - L_m : a, b, c, &c. \cdot r^m}{L_a, b, c, &c.} = 1 - \frac{L_m : a, b, c, &c. \cdot r^m}{L_a, b, c, &c.} \cdot (1-r)^{m-1} \cdot a, b, c, &c.; \)
if this be divided by \( \frac{a, b, c, &c.}{m} \), we shall have the annual premium for such assurance; that is, \( \frac{\sum_{r=0}^{m-1} a, b, c, &c.}{\sum_{r=0}^{m-1} a, b, c, &c.} = \frac{1 - L_m : a, b, c, &c. \cdot r^m}{L_a, b, c, &c.} = 1 + r. \)
The said annual premium may be expressed by
\[ \left( 1 - \frac{L_m : a, b, c, &c. \cdot r^m}{L_a, b, c, &c.} \right) \div \left( \frac{\sum_{r=0}^{m-1} a-1, b-1, c-1, &c. \cdot L_{a-1, b-1, c-1, &c.}}{\sum_{r=0}^{m-1} a, b, c, &c. \cdot r \cdot L_a, b, c, &c.} \right) = 1 + r. \]
This last mode is well adapted to logarithms in the use of our general tables; and this method, supposing the annuities were accurately determinable by our general tables, would be accurate. The last formula is derived from that immediately before, in consequence of \( \frac{a, b, c, &c.}{m} \) being identical with \( \frac{\sum_{r=0}^{m-1} a-1, b-1, c-1, &c. \cdot L_{a-1, b-1, c-1, &c.}}{\sum_{r=0}^{m-1} a, b, c, &c. \cdot r \cdot L_a, b, c, &c.} \).
Example. To find the annual premium to assure a life, at the age \(a\) years, for 10 years, according to the Carlisle mortality, and three per cent. interest.
| \(a\) | 20 | 30 | 40 | 50 | 60 | 70 |
|-------|----|----|----|----|----|----|
| Log. of the accommodated chance for living 10 yrs. at the age \(a\), Tab.V. \(x_{1,03}^{-1}\) | 1.9690 | 1.9556 | 1.9406 | 1.9328 | 1.8435 | 1.7005 |
| Sum | 1.8406 | 1.8272 | 1.8122 | 1.8044 | 1.7151 | 1.5721 |
| \(\frac{r}{L}\) corresponding To this we get from Ta.I. \(x_{10}^{a-1}\) | .91443 | .90407 | .89892 | .89379 | .84846 | .78092 |
| Therefore, \(x_{10}^{a+10}(T.VII.)\) \(x_{1,03}^{-10}\) | 1.96682 | 1.95400 | 1.93772 | 1.91830 | 1.81893 | 1.59873 |
| Sum = the log. | 1.83845 | 1.82563 | 1.80935 | 1.78993 | 1.69056 | 1.47036 |
| The No corresponding | .68937 | .66932 | .64469 | .61650 | .49041 | .29536 |
| Its complement to unity | .31063 | .33068 | .35531 | .38350 | .50959 | .70464 |
| The log. of the last \(r\) | 1.49224 | 1.51941 | 1.55061 | 1.58377 | 1.70722 | 1.84797 |
| Complement of \(x_{10}^{a-1}\) | 1.08526 | 1.09221 | 1.09995 | 1.10395 | 1.14901 | 1.21809 |
| \(\frac{r}{L}\) \(x_{10}^{a-1}\) | 1.99694 | 1.99571 | 1.99481 | 1.99402 | 1.98754 | 1.97813 |
| \(\frac{r}{L}\) \(x_{10}^{a-1}\) | 1.98716 | 1.98716 | 1.98716 | 1.98716 | 1.98716 | 1.98716 |
| Sum = logarithm | 2.56160 | 2.59449 | 2.63253 | 2.66890 | 2.83093 | 1.03135 |
| Number corresponding \(x_{1,03}^{-1}\) | .03644 | .03931 | .04291 | .04666 | .06775 | .10749 |
| Ann. premium for an assurance of 1l. | .00732 | .01018 | .01378 | .01753 | .03862 | .07836 |
| Ditto for 100l. | £0..14..8 | 1.. 0.. 4 | 1.. 7.. 7 | 1.. 15.. 1 | 3.. 1 .. 3 | 7.. 16.. 9 |
The reader has here an opportunity of comparing the results from my tables, with those above calculated by Mr. Milne's Carlisle tables.—I may probably be able at a future period to add examples, which I regret time will not at present permit.