I. On the Law of Human Mortality expressed by a New Formula. By THOMAS ROWE EDMONDS, B.A. Cantab.*

THE

HE love of life being the strongest of human passions, the most interesting of all laws ought to be the law which regulates the rate of decrement of human life from the time of birth until extreme old age.

Tables of mortality, in great variety, have been constructed for the purpose of exhibiting particular rates of decrement according to age appropriate to particular populations at different times. In nearly all these Tables there is exhibited a uniformly decreasing rate of decrement from the time of birth until the age of about 10 years, and a uniformly increasing rate of decrement from the age of about 15 years until the last age in the Tables. The appearances presented by all these Tables are such as to warrant the supposition that the rates of decrement in the period of childhood, as well as in the period of manhood, are functions of the age measured from birth or some other fixed time in the age of It will presently be seen that such supposition is well founded and in conformity with the facts given by observation. The rate of decrement of life will be found to be a certain function of the age, which is remarkable for simplicity and novelty.

man.

If P, be taken to represent the number living or surviving at the age t years, out of a given number born alive (Po), the number dying in the (t+1)th year of age will be P-Pt+1=APt. The mean rate of decrement, or mean ratio of dying to living, during this year of age will be represented by AP, divided by the mean number living throughout the year. Such mean number is (P+P+1), or Pt++ very nearly; so that the mean rate * Communicated by the Author. Phil. Mag. S. 4. Vol. 31. No. 206. Jan. 1866.

B

of decrement in the (t+1)th year of age is

y also be represented by Pt+

The mean annual rate

AP the mean infinite

of decrement in the (¿+1)th year being P+

simal rate of decrement during the same year of age will be ΔΡ. 1

X if q be taken to represent the infinitely great
Pitt

q
number of equal parts into which a unit of age is supposed to be
divided so that the infinitely small given time dt multiplied by
q is equal to unity.

The infinitesimal rate of decrement (or ratio of dying to living) will at the precise age (t) be represented by

Pt-Pt+dt
Pt

dP

=d.log Pt

P

dP

(since=d.log P, according to a well-known property of hyperbolic logarithms). That is to say, the infinitesimal rate of decrement at any age is identical with the differential of the hyperbolic logarithm of the number living or surviving at that age. Consequently, if the infinitesimal rate of decrement is a known function of the age, then the logarithm of the number living (log, P), being the sum of such infinitesimals, will also be a function of the age, which may be found by integration. The infinitesimal rate of decrement at the precise age (t+) years is +, which is the differential of log P+ Piti be taken to represent very nearly the mean infinitesimal rate of

the (t+1)th year of age, for which the expression

This may

ΔΡ. 1
Pets

has already been found. On equating these two approximate values of the mean infinitesimal rate of decrement in the (t + 1) th

year of age, it will ensue that dPt+== very nearly; i. e., the

APt
q

differential of the number living at the age (t+) years is equal to the number of deaths in the (t+1)th year of age divided by 4, the infinitely great number of equal parts into which the unit one year is supposed to be divided.

When P is a function of the age (known or unknown), it will be found that all three of the following important quantities are

nial interval of age, in the period of life between the ages of 15 and 80 years. These three quantities are (1) the mean ratio of the dying to the living throughout the interval, (2) the differential coefficient of the hyperbolic logarithm of the living at the middle point of the interval, and (3) the finite difference between the hyperbolic logarithms of the numbers living at the ages tand (+1). That is,

The first and second of the above quantities have already been shown to be equal to one another very nearly. The near coincidence in value of the second and third quantities will be obvious on consideration that, if the rate of decrement throughout the given unit interval of age varies continuously and equably from a at the beginning to at+1 at the end of the interval, the total effect produced in the diminution of the number living (Pt) will be very nearly the same as the total effect in the same interval which would be produced by the rate of decrement a+ at the middle of the interval, assumed to be constant for the whole interval. On reference to Table V. hereunto annexed, it will be seen, for example, how slightly the three quantities above mentioned differ from perfect equality when the unit interval of age is five years. The remarkable property now mentioned, being possessed in common by all good Tables of mortality, is of great practical importance. For observations, hitherto supposed to give ratios of dying to living only, may henceforth be used as giving directly the finite differences of the logarithms of the living (A log, Pt), which finite differences are the essential parts of the Tables of mortality sought to be constructed. In columns 5 and 6 of another Table (VII.) hereunto annexed, the reader may see how closely the values of A log P in the English Life Table No 2 for males approach the approximate values obtained as above stated directly from observation.

When for a particular population the rates of decrement at every age are known, a Table of mortality may be constructed therefrom which will correctly represent the number living or surviving at the end of any entire number of years from birth, out of a given number born alive. Similarly, when the Table of mortality is given, and the number of survivors at every year of age is known, there may be deduced from such Table the rate of decrement for every age.

Observations for the purpose of determining the laws of mortality according to age of particular populations are made in one

or the other of the two ways just mentioned. The direct method, by observing the number of survivors at particular ages out of a given number living at birth-time, or out of a given number living at any other age, is applicable chiefly to selected classes distinct from the general population of a nation or district, such as classes of persons on whose lives annuities or assurances have been granted for money by trading companies or by governments.

The most common method of observation, by observing the rate of decrement at every age, is indirect, and is the only method used for determining the law of mortality, according to age, of the total population of a nation, or of the constituent parts of such population distinguished as village, town, or city population. Observations according to this method are made by periodical enumerations of the numbers living and the numbers who have died in annual, quinquennial, or decennial intervals of age. The ratios of the numbers dying to the numbers living (which are the rates of decrement of life) are thus obtained for every age. These rates being known, the consequent Table of mortality representing survivors at every year of age may be deduced by calculation.

The law of human mortality may be most simply expressed by means of the rates of decrement and the relation between these rates at different ages. If at two known ages the rates of decrement are a。 and at, then, if the rate of decrement at one age is known, the rate of decrement at the other age may be found from the proportion

at
=
do

1

a

wherein t is the difference of age, a is a constant representing distance (in time or age) from a fixed point, which is the ideal zero of life or vital force, and is the hyperbolic logarithm of

1

k

10, and equal to 2:302585. There are two ideal zeros of human life- -one belonging to the period of childhood, and the other to the period of manhood. The curve which indicates the law of decrease of human life consists of two branches-one on each side of the period of puberty. There apparently exists a short intermediate branch at or near the junction of the two others. The existence of such short branch may be due to the differences of age at which puberty is attained by different individuals.

The complete expression for the law of human mortality is contained in two similar formulæ,-one for the period of increase of vital force, extending from birth to the age of about 9 years;

the age of about 12 years to the end of life. Both formulæ are similar to the formula which has been shown to represent the law, according to temperature, of the elastic force of steam of maximum density, and the formula which has been shown to represent the law of density of saturated steam (Philosophical Magazine, March and July 1865). All four formulæ are deduced -from a differential of the form following:

In the two formulæ for human life, the quantity P represents survivors at the absolute age (a+t) out of a unit of population existing at the absolute age (a) measured from one of the two zeros of life. The quantity a is the rate of decrement in a unit of time at the absolute age (a) whence t is measured, on the assumption that the infinitesimal rate of decrement at age a continues constant for the unit of time. The above equation for d.log P, on integration, yields

The most remarkable difference between the two new formulæ for human life, and the two formulæ for elastic force and density of saturated steam, will be found to consist in the relative positions of the ideal zeros of the forces of life and steam. In the case of steam the constant a (which indicates distance from the ideal zero) is very great, and marks a position of such zero far beyond the reach of observation, viz. 276° Centigrade below the temperature of melting ice. In the case of the two formulæ for human life, both of the ideal zeros are close at hand, and one of the two zeros may be passed by a living person. In the formula for the period of immaturity or childhood, the value of the constant a at birth-time is 2 years very nearly; that is to say, one of the zeros of life is an ideal point 21 years before the time of birth. In the formula for the period of maturity or manhood, the place of the ideal zero of vital force is at the age 102 years from birth-time nearly.

When the places of the two ideal zeros of life have been determined (say at -2 years and at +102 years from birth), there remains only to be determined the point of meeting of the two periods, and the rate of mortality or decrement of life common to both periods at the point of junction. According to the