on Survivor/hips . 34.9 
depending on the younger of the two lives dying lad; is eafily 
obtained, by fubtrading the value fir ft found from the whole 
value of the reverfion after the extinction of both lives. 
The anfwers computed by this rule differ rather more from 
thofe computed by Mr. Simpson’s approximation than they do 
in the preceding problem. But I am fearful of becoming te~ 
dious, and therefore fhall defift from inferting the comparative 
values in this cale. It may not, however, be improper to 
exemplify the truth of this demonftration, by (hewing in 
what manner the feries may be refolved into the plain 
fimple rule for computing the value of the reverfion, 
when the ages are equal. The feries in this cafe become 
S . b — d S 
+ - 
2 bbr 
2 bbr 
a . S,o—c.c—d t §.b—d.d—e 
&C. 4 - 77 -* + - 
bbr z 
bb, 3 
+ &C. = 
S bb cc—lbc . dd^lbd 9 S 2 bc — tc 2bd'—dcL 
+ &c - + i * 
<SCC. =r 
c 1 , 1 
b X — + - 
2 r r 
213 - HB 2B-BB 
- = (putting L for 2B - BB, the 
value of an annuity on the longeft of the two equal lives) 
s 1 , x 
- x — r - 
2 r 
r . L S P . / - 1 L . r- 1 
= - x — — : — ■ 
r 2 r r 
QJE. D * 
The exaB value of all reverfions depending on furvivorfhips 
between two lives might be found in the lame manner as the 
values in the preceding problems. With regard to the values 
of reverfions depending on furvivorfhips between three lives, 
I am fenfible that the folutions of thofe cafes would be rather 
difficult when deduced from the real probabilities of life. But 
they certainly might be effeded ; and thofe are the more necef- 
fary, inafmuch as the folutions derived from the expeditions 
of life are often fo very defective as not to defer ve the name of 
approximations. 
Chatham Place, Feb. 2 , 1788. 
V t V. 1 1 I. 
* See the latter part of the foluticn of the fecond problem* p. 347. 
