H 8 
Mr. Morgan on Survivorships. 
year, gdly, If B and A shall have both died in the foregoing 
years (B dying last), and C died in that year. 4 ,thly, If C and 
A shall have both died in the foregoing years (C dying last), 
and B died in that year. From the fractions expressing these 
several contingencies the value of the reversion will be found 
„ JL v _1_&C. - 
— yibc X r ' r x ' r 3 ' 
s x nd • < _l fillip 4 - &C. 
2 abcr 
&C. 
S me . a' , nf . a'-]- u" 
X — ^ 
2 abcr 
&c. 
' 3 abc* r ^ r 1 “ r 3 1 
S 
zabc 
S * a'md , <fne , a" of 
s 
abd . a 0 me . a'"nf , 
X "TtTT T “T3 T 
. „ S i a' + a".of , & _j_ 
-j- &c. + — x — 4 h &c - -r 
4- &c. + 
-xbcr 
S 
zabr 
-e . a ‘ \ 
-ABC 
B + C 
4- BC 
+ 
- 0 . a-\- a 
— 4- &c. = S into r —^~ 
AB + AC m 
PC — APG 
—J- X 1 + AP “ 
zbr 1 3 
+ 
d xi + AT 
BT-ABT 
2 Cl 
4 m. i + AP 1' 
7b 
(3 . FC — AF'C 
7 
X 
3 C 
BK^ABiC 4-( 3 -^. If B and C are both of them older 
than A, and also are nearly of the same age, this general rule 
will be sufficiently correct. But if the ages of B and C differ 
much from each other, it is evident that the annuity on the 
single life of the younger of them (suppose C), and on the joint 
lives of AC and AT, ought to be continued only for as many 
years as are equal to the difference between the age of B and 
of the oldest life in the table of observations. In this case also 
there is a further value of S, after the necessary extinction of 
the life of B, arising from the contingency of that life s having 
failed after the life of A, and of C's having failed after both of 
them. Let x, *, C', and A'C' respectively denote the same 
