232 Mr. Gompertz’s analysis applicable to the 
annuity on the whole possible joint existence: and where 
there are many lives concerned, perhaps this may answer 
with as little trouble as by the common methods by the 1 h 
mited tables ; and an)r accuracy may be obtained that the 
tables of mortality will afford, b) using sufficiently small in- 
tervals ; which is not the case when the common method is 
used, of searching in the tables of two joint lives, and then of 
a single life, and then of another life with this single life, &c. 
till we have comprehended the whole number of lives. 
Art. 2. Moreover any functions may be developed 
rniWV -1- ^ M" 4- &.C 
into the form M x .tt x x and, consequently, if 
n be sufficiently small to admit of all powers of n above 
the first being omitted, we shall have M , = M 
0 x m 
7T 
mM'x 
and if this remark be applied to the function L a+n of the 
living at the age a-\-n, we see that we may take an interval 
m, from n so small, whatever be the real constitution of the 
function, that the number of the living, during that interval, 
shall decrease so as to form a geometrical progression very 
nearly, whilst the portions of time increase in an arithmetical 
progression ; and that the decrements of living are also in 
consequence very nearly in geometrical progression : and 
we may moreover in this, as in the former case, accom- 
modate the function so as to be accurate at the two ex- 
tremes. Thus, for instance, if we wish to have an approxi- 
mative expression in a geometrical progression, for the num- 
ber of living for ages from 20 to 30 years, which shall be 
exact for the age 20 and 30, according to the Northampton 
tables of mortality. Assume it any convenient number at 
pleasure, say 10, take a == 20, and a -f n = 30 ; n = 10, 
