ON PROBABILITY. 
- d, +1 
Lr Ta. LT 
dtd tri. 101-1 
RR a IT 
in this case d+ do + &eoovvue.. = 1000 
7 dy Fd + &e +101 —n 
Yon = T7071" Po, n == or ‘ 
59. Unfortunately, no registers of this kind have been kept, and we are 
obliged to have recourse to those sources which best supply this deficiency. If 
the population of any district were subject to no fluctuations arising from an 
influx of the inhabitants of the neighbouring countries, and if it were con- 
stant, that is, if the births and deaths were always the same and equal to each 
other, a register of the ages at which deaths took place would alone be 
wanted to determine the values of p and ¢ for this place. For it is evident 
that if the population were large the probability of an individual dying at the 
n" age would be equal to the deaths of persons at that age (n) divided by 
the births z years previously, but the births years previously are, upon the 
hypothesis we have made, exactly equal to the present number of yearly 
deaths = d, + dy &c. = = d,, using the letter = as before, 
and Gow == ie ; 
=(d,) 
and in the same way g,, , may be determined. 
The parish books, therefore, if they were accurately kept, and if the popu- 
lation were subject to the conditions we have mentioned, would furnish the 
information required, and they were used as a first approximation. 
60. When the population is not stationary, the preceding results require 
modification. 
Let d, be the deaths observed at the age n in a given place, b, the births 
in that place » years previously; then 
. d,+1 
CG Ty 
Let us suppose that the births m years ago were equal to the total num- 
ber of deaths now, and that the births increase in a geometric progression, 
of which the common ratio is »; then 
Bh=b=017=2({)1", 
Co bpm (Dh 
a 5 d.+1 
an Go, n — = (d) Fr-n + 101 = n 
m must be found, from the consideration that ¥ (g,) = 1, which, in the 
present form of the equation, would be troublesome: the labour may be 
much simplified by observing, that when # does not differ much from unity, 
this value of gq, , does not sensibly differ from 
_ d, +1 Jil (d, +1) 
bE EDF nr 7 Sth 110i = 
In this form g,,, . 2" may be calculated without any previous knowledge of 
m ; let the value of q,,, « 7" be called D,; then - 
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