estimation of the value of life contingencies. 233 
Log. of L —Log. of L 
and we have L' ; = L' 20 == — — — 3 ° io ° — ,0068317 
according to the Northampton tables. And when n is any- 
thing from o to 10, L 20+n = L 2Q . 10 — oo68 3 « 7 « ? and the loga- 
rithm of the numbers of living at the age 20+^=3,7102866 
— 0063 1772. Hence we have respectively 
Living at the ages 
According to the Nor- 7 
thampton Table ( 
Ditto, Geometrical 7 
progression - - ) 
Deficiency of the Geo- \ 
metrical progression j 
20 
21 
22 
23 
24 
2 S 
26 
27 
28 
29 
3 ° 
5132 
5060 
49 8 S 
4910 
483s 
4760 
*-0 
OO 
VO 
4610 
4535 
4460 
vo 
OO 
eo 
5132 
505I 
4972 
ON 
OO 
xh 
4819 
4744 
4670 
9457 
4525 
4454 
4385 
0 
9 
»3 
*5 
16 
16 
J S 
‘3 
10 
6 
0 
•L' 
Art. 3. Hence, if we consider L x+T = L x . To] ‘ r , whilst r 
is not greater than m—n as a sufficient approximation ; and 
we wish to find the value of « 
m 
a, b, c, See. we have by writing ^ 
for the common logarithm of r, and putting ^ + L' fl+n + L ' b+n 
+ L' c+n -f & c - 
P 
p, n 
a, b, c, See. 
— x (l + 
J a, b, c, Sec. 
+ ^ + = T5\ nt . 
J a, b, c, Sec. 
1— 10 A 1 
„^n: a, b, c. Sec. 
r H - r X 
; and by restorations this may be written 
tn—n+p L , . , „ „ 
1 — r m+p : a, b, c. Sec. 
J n : a, b, c, Sec. 
1 —r? . L 
« + /> : a. b, c. Sec. 
1, 
d 1 b > c , Sec. 
f j 
J n : a, b, c. Sec. 
Art. 4. If the number of living, corresponding to times in 
arithmetical progression, form a series in geometrical pro- 
gression, we should have 
