540 Mr. Gompertz on the nature of the function 
Time will not allow me, for the present, to offer more 
than a very few examples of the method to be employed in 
calculating by these tables, which are as follow : 
Example i . Required, according to the Carlisle table, the 
value of a life annuity, for ten years, on the joint lives so and 
40, at 3 per cent interest. 
In Table VIII; for Carlisle, log. of accommodated 
chance for lo years, at the age 30 . . = 1.9552 
Ditto 40 . . . . r= 1,9383 
Ditto X 1,03 . . . ' = 1.8716 
Sum . . T.7651 =x(a*®) 
In Table I, T.76 corresponds to . . . .8734 
In proportional parts ,005 corresponds to .253 
Ditto . . 0001 corresponds to . 5 
Consequently 7,765 1 corresponds to . .87604 
which is the log. of the required value : the number corres- 
ponding to this is 7,5169, for the value of the annuity, 
according to the Carlisle mortality, at 3 per cent, on the joint 
lives 30 and 40; and by calculation from Mr. Milne's tables, 
I find the value should be 7,5168 ; the difference of the two 
is evidently insignificant. In this way I calculated the log. 
of the value of the life annuity, at the* Carlisle mortality, at 
3 per cent, for 10 years, for the joint lives o and 10, 10 and 
20, 20 and 30, 30 and 40, 40 and 50, 50 and 60, to be ,76580 ; 
,90247 ; ,89139 ; ,87604 ; ,86295 ; ,81067 ; and the annuity, 
or the numbers corresponding to the said logarithms, 
5,8318; 7,9874; 7,7874; 7,5169; 7,2937; 6,4665; 
and, according to calculation from Mr. Milne's tables, I get 
5,8595; 7,992; 7,7906; 7,5168; 7,2916; 6,4679. 
The difference between the two sets is insignificant, except 
