842 DE. PAEE ON THE CONSTETJCTION OE LIFE-TAELES. 
assumed that the ratio of the living at the ages 15—25 * to the dying in a year at the 
same ages 15 — 25 represents the annual rate of mortality at the exact age 20. So also 
the mortality rate at the ages 30, 40, 60 and other ages maybe determined. As observa- 
tions grow more exact, and the facts are multiplied, the inter^’als of age may be dimi- 
nished to 5 years, and ultimately to 1 year. 
In determining the rate of mortality, a given number of persons living a year is 
considered equivalent to twice that number living half a year, or to half the number 
living two years. 
Thus if nd represent the deaths in n years out of a number amounting on an average 
to P during the same years, then ^ the rate of mortality, or the proportions of 
death in a year (always taken as the unit of time) out of one year of lifetime. It is 
found from all the observations hitherto made on a large scale, that the rate of mortality 
varies at every interval of age ; but at the same age it may for the present pui’pose be 
considered invariable under similar circumstances. 
Ill:, therefore varies in every moment of age ; but I have employed it to express the 
mean annual rate of mortality during the year following the year of age x, .'. ^ = 
where d, indicates the deaths, P^ the year of lifetime, after the year of age x. The 
m, is the expression of the force of the causes that induce death, of the death-force, ns 
mortalis; and its reciprocal — '=u, measures the forces that sustain hfe, the us vitalis. 
771 ^ 
The vital force under natural circumstances may by one hypothesis be sufficient to 
sustain a whole generation alive for seventy or eighty years, and then suddenly collapse. 
The Life-Table, if this hypothesis were true, would be represented by the jmrallelogram 
in which the curve of the Life-Table is inscribed (Plate XLII. fig. 1). 
By the hypothesis of DEMOiVEEf the rate of mortality is such, that at the age of 20 one 
in 66 living at the beginning dies before the end of the year, leading 65, 64, 63, 62, 61 
to enter on each year of age until at the age of 86 all are dead. 
Upon this hypothesis the relative numbers living up to the age 86 form an aiith- 
metical progression : and the deaths in the equal times are equal out of the diminishing 
numbers living. The rate of mortality increases on this hypothesis as age advances hi 
the same ratio as w--|: 1 ; where n is the difference between the actual age x and 86. 
It is called the complement of life. The Life-Table, upon this hypothesis, has equal 
decrements, and might be represented on Plate XLII. fig. 1, by di-anfing a diagonal line 
through the parallelogram. Its deviation from the true curve on this scale is eiident ; 
but it is also evident that a series of straight lines, which would neai’ly represent the 
true curve, may be drawn from point to point of all the ordinates. 
If the causes of death act with equal intensity at all ages, they may be represented 
by any simple external cause, destroying an equal jpro^ortion of the numbers Ihing in 
equal intervals of time. Thus, if 1600 men were distributed equally over ground where 
* By this 15 and under 25 years of age is understood, and so in all similar cases, 
t See Treatise of Annuities on Lives, Preface to 2nd Edition. 
