Mr. Morgan on Survivorships. 
141 
f 1 a'+a"+ tip 
zacr 3 1 CXL - 
and — 
mda' 
zabcr 
ne . a 0/. tz'-f a " -f a"' 
8cc. — 
C_ AC BC — ABC 
zabcr 
zabcr 3 
If be the oldest of the three lives 
this rule will be insufficient. Let z, C', B'C' and <p denote the 
the same quantities as in the second part of the preceding pro- 
blem ; then will the value of the annuity in this case, for the 
first z vears hp ^ B'C' — ABC , . . 
3 ’ 2 2 j and its value for the re- 
maining years of C's life = i~ . C^Ty ; for the payment 
of it during this last term depends on the contingency of C's 
living so long, and of A's having survived B, which probabi- 
lity is = 1 <p ; therefore the whole value will be = C — 
C'+B'C 
— <P . C — C' 
AC-ABC 
If B be the oldest of the three lives , let x, tv, C r , A'C' £ g r 
See. denote the same quantities as in the third part of the’fore- 
going problem ; also let x' denote the sum of the decrements of 
the life of A for * years, and &c. the decrements of 
the same life in the x + rst, ~ ad, x^d, &c. years. The 
value of the annuity for the first x years will it is evident be 
= C - AfC BC — ABC T : 
2 2 * ^ the -£ + 1st year the payment of 
it wilLdepend on the contingency of A's having died after B 
m x + 1 years, and C's having lived to the end of this term. 
s the life of B becomes necessarily extinct in x years, it is 
plain that the probability of A's dying after him in T +~ 1 
years must be = and therefore that the value of 
the annuity in this vear will bp — .. * «■ . * 
* — 7 ^+t- In 
the same manner the value of the annuity in the TT~«d 
MDCCXCIV. J j ‘ 5 
