r 
r 
230 Mr. Gompertz’s analysis applicable to the 
Section II. Art. 1. If there be any function M x , which de- 
creases as x increases, and for x we write x-\-n, the function 
will be transformed into M r+w ; and may be developed into 
M x —n M M "-f-w 3 M"', &c. where 1VT is positive; and 
if m be taken sufficiently small we shall have = 
M -mM' sufficiently near. And «M' will be nearly the 
decrement produced in M x by writing x-\-n for x. And this 
approximative decrement is proportional to n. Hence it ap- 
pears, that the number of persons living in any table of mor- 
tality indicating the number of persons living at every pos- 
sible age, that is to any fraction of a year or unity of time, 
the intervals of age may be taken so small, that whatever the 
law of mortality may be, during any portion of any the same 
interval, the decrements may be considered proportional to 
the time. Observation informs us, that this proportionality 
of decrement may be admitted as affording a tolerable degree 
of accuracy during very long intervals, and in that respect, it 
gives us some idea of the nature of the function of mortality ; 
but independently of observations from known results, we 
see that we may approximate to any degree of accuracy con- 
tained in the tables, by dividing long intervals into shorter 
intervals, and taking, whatever may be the functions of mor- 
tality or of living, the decrements proportional to the por- 
tions of time between the separate intervals, and thus if we 
wished to find the value of 
m 
a, b, c, See . , that is the value of 
an annuity of one pound on the joint lives whose present ages 
are a , b, c, &c. ; the first payment to be made in the time n , 
and the last in the time m, at a rate per cent, indicated by the 
