120 
LIFE ANNUITIES. 
121. If money produced no interest, the formula in Art. 112 would 
become 
fn+1 4 l m +2 4 4i+8 + lm+i 4 
• * • L 
this expression shows the average number of years that each individual 
completes. 
122. The number of years expectation of life of an individual whose 
prospect of longevity is the same with that of individuals of the same 
age, at any particular place where observations of the rate of mortality 
have been made, is usually taken as the average number of years 
enjoyed by each individual at that place, as shown in the tables. 
Let us suppose those who complete their mth year, but die before 
completing their (m + l)th year, to die at equal intervals therein, so 
that, for every one who dies before the expiration of a half of the year, 
some other will survive so much more than the half-year; each individual 
who dies in the year survives therefore upon an average one-half of that 
year. 
Of l m persons who complete the mth year of their age, l m — l m+l die 
in their (m 4 l)th year, and l m+l survive their (m 4 l)th year; —— 
the number of years enjoyed by all those who die in the (m+ l)th year, 
added to the l m+ i years enjoyed by those who complete their (m-fl)th 
year, gives ( ■ + C+i— the number of years that will 
be enjoyed in the first year by these l m persons or the survivors. 
And in the same manner may be’shown that 
l-m+X+L 
lm+2 T* In 
, &c. 
is the number of years that will be enjoyed in the 2nd, 3rd, 4th, &c., 
years by these l m persons or the survivors. 
If we continue these values to the oldest age in the table, and sum 
them together (making z as before tire difference between the age m and 
the oldest in the table), we obtain the total number of years enjoyed by 
these l m persons until they all cease to exist: viz., 
lm 4“ lm+\ , An+l 4“ lm+2 
X r — 
. 4ii+2 4 l'm+3 , lm+3 ~1~ In 
4 « i ^— 
I lm+z-1 4- l,n+s , 4i+* I'm , , , , 
i q > 2 “ 9 l,n+1 lm+ ' 2 ^ m+3 lm+i ’ 
4- lm+z-i l m +z) this expression divided by l m , the number of indivi 
duals amongst whom this quantity of existence is divided, gives the 
share of each, or in other words, the expectation of life of an individual 
aged m, which will be expressed by the symbol e m ,