estimation of the value of life contingencies. 231 
present worth r of one pound to be received, certain in one 
one year ; because « 
b > &c - = ( r ” • L « : a, b, c, Sec. + 
rnL m:a,b,c,Sec.) d^ided by 
7 * M + 1 _ L„ + 1 :a,b, c, &c.+ &c - * * • * ' 
c, Sec., it follows that if the interval be taken sufficiently 
small, the decrements being considered proportional to the 
time, and (> be taken not greater than m—n, we shall have 
T n ^ r ^ . L„ g : a, b, c, ~ = r ^ n+a £ L n + a X L b + n £ ^ b + n * 
L r+n — ^ . h'c+n x &c., by using, mutatis mutandis, the notation 
above, and this is =r n+ ?. (L„ . a> b,c,sec. — ? (L»+«- 
-j- L'„ + & . L n: a, C,&c.“l“^ cc *)" , h? + ; c,Sec. "j” &C.) 
_ &c. ) = r“+« . I*, bi c _ &c . x<A„ f + B„ f — &c -) ! A «> B », 
&c. being put for the coefficients of the different powers of f, 
are constant during the interval of uniform decrement, and 
consequently, by writing o, 1,2, 3, &c. for £, we see how we 
may obtain the value of annuities approximately for por- 
tions of time sufficiently small, and that we therefore have 
a, b, c. Sec. = r* 
J a, b > c, Sec ' 
r> 
1 r 
l 
-j-B . r— 
n 
n 
r % 
2 r a 
4 r * 
y>3 
yi 
9 r3 
Sec. 
Sec. 
&c. 
n — m 
r n — m n — m r n — n 
> 
and if T n _ m be put = 1 -f r r z r n ~ m , T' Mn ==r-}-2r‘ 
+ 3 r * + 4 / • • • • n — m - r n ~ m , T . . 
n — m\ 
r«— &c., numerical values of which may be arranged in a 
r 
L 
a , b, c. Sec. 
‘ n : a, b, c, Sec. 
tty by Cy 
small table, we shall have 1 
* n 
m 
x ( T n-rn — \ T n-m + — &c - ) and by repetition 
r 
and addition we may obtain T a ,b,c,sec., that is the value of the 
