An algebraical Expression of the Values of Lives. 7 
value, from which it is not possible to deduce immediately a 
practical value, without further reference to the comparative 
numbers of the survivors at different ages, in order to estimate 
the various operation of interest, and particularly of compound 
interest. But the same mode of computation may readily be 
extended, so as to comprehend the effect of interest also, by 
supposing the ordinates, expressing the number of survivors, to 
be reduced in the proportion of the present value of a given 
sum payable at the corresponding time: that is, by multiplying 
the expressions denoting them by v7; v being the present value 
of a unit payable at the end of a year. 
But the, expression containing the circular are becomes in- 
tractable, even when applied to the termination of the present 
valug fof a single life: we may therefore take, instead of the 
> some members of the equivalent 
primary expression 
1+ xx 
ceed 1 1 erg Tt Meena a 
e i Ey cok, an : ae) (ee ee 
series — ee oe and make the fluxion z G suis 
- i 1 1 ghd 
of which the fluent is C— — + ———; and this, if we put 
42 1223 
C = +405, will be sufficiently accurate even in infancy. We 
shall then have, for the number of survivors at the age a, 
595 + — — — — -0002005 «*—-0000014.3 ; which, mul- 
tiplied by v7, will give the fluxion of the area. 
The fluent may be found by means of the following theorems 
of Euler’s Calculus Integralis (§ 228, 225, 224) ya ay 
Bi => 
z°*(HLv)? 3(aLv)3 4 4 
hes apogee et Sk ae 10+ (=X); 
wr i we r fl 
f= = aes b ees ee ey ve; 
[oreua ve (= — eae of aa and fv®six = 
ve ( —~. rs ao aay = ane When » is large, the 
series converges somewhat slowly: but its utility is chiefly con- 
fined to the earlier ages; and the ultimate value, being once 
only computed, for x = 100, will serve for the correction in all 
other cases: but after the age of 10, the formula a~bx? + cx?, 
already mentioned, will be more convenient. The fluent, when 
corrected, by subtracting it from its ultimate value, must be di- 
vided by the number of survivors, and multiplied by , since 
7] 
A4 it 
