579 
expressive of the law of human mortality, &c. 
How the value of particular assurances may be deter- 
mined from the value of annuities, is shown in my Paper in 
the Philosophical Transactions for the year 1820, many of 
the cases of which are solved by methods essentially the 
same as those which have been long adopted ; but when 
such assurances are but for terms, which are not of great 
extension, very near approximations may be had by using 
a geometrical progression, without confining the arithmeti- 
cal operations to the same route, since the chance of extinc- 
tion of the joint lives of the present age a, b, c, &c. taking 
place between the period commencing with the time n-\-t — 1, 
and finishing with the time n t, from the present, is 
= ^ - 1 : a, 6, c, &c. \ ^ t : b,Cy &c.) b, c, &c. » h folloWS that 
if r be the present value of unity, to be received certain in 
% 
the time i, and L = L_^ j x whatever t 
may be, that m L 
a, b, c, &c. 
or the assurance of unity to be 
received at the first of the equal periods 1, from the com- 
mencement of the time w*— 1 to the expiration of the time m, 
which shall happen after the extinction of the joint lives. 
“* I • Cf ScCw 
• 1 j. — 1 . u., i/, ut r « / \ I 
IS equal to — r x 4 r x (1 — tt) -f 7 x ('tt — tt®) 4- 
rt^rL^TT^) r-x(7T^‘”:ii == ( 1 x 
' a, b, c, &c. 
K-r^rr+TTr +7rr....r. -T ) ^r n-i:a,b,cAcT 
m~l 1 ^ r . 
ft m 
a, b, c,&c. + ^ • h-i:a,b,c, &c. 5 \g,b,c,Scz . 
If the assurance be not deferred, n will be equal to i, and 
r 
1 I 
we shall have, according to the hypothesis, ^ g* — 
r 
( 1 — fr). rx « and also = If t be 
MDCCCXXV. 
4 F 
