250 Mr. Morgan on Survivorships . 
and B'T', respectively denote the values of annuities on those 
single and joint lives for , years, then will the first and second 
za.HC— HSC d 
series be = 
3« 
6 cr 
at — A l b, the third series 
ds 
ABC 
_^ x NT_NBT = ^xt + NBT- , 
AC — ABC 
the fourth and fifth series — 
c.HK -HBK. 
3 r 
. HBK i_ _j_ ABC 
UT~ 6 r ir 
— , the sixth series 
—v. . AK— ABIC 
3c 
+ 
5 • nc-nbc ^ t ^ e seven th series = 
r — i . V-B' 
the eighth series = 
3«r 
.H-HB 
A-AB 
.HB 
; -f AB 
2 r 
the ninth series 
- . K'— B'K' 
C-B'C' 
the tenth series = — C' — B'C' + — x 1 ' B'l , 
and the eleventh series = 
C-BC' 
T' — B'T'. In order 
to obtain the value of S after the necessary extinction of the 
life of A, let 7 r and p denote the probability that B and C re- 
spectively die after A ; k, l h £ &<=• the number of P erS011S 
living in the table opposite the age of C at the end of y, y + 1 , 
Zj- - &c years ; p the number of persons living opposite the 
age of B at the end ofy years ; andfr.g", P", &c. the decre- 
ments of life at the age of B after y + i, y + 2 > y + 3. &c - 
years. In the y + ist year the given sum may be received, 
provided either of three events shall have happened, ist, 
B and C shall have both died in that year, adly, If C only 
shall have died, B having died after A in the first y years, 
odly If B only shall have died, C having died after A in the 
first y years. The value of S depending on these contingen- 
+ 
cies will be = — x 
s p , s • * .k-c . Inthey-f^d, 
• br > + 1 
bcr 
cr> + 
