2 A1 
Mr. Morgan on Survivorships. 
y -f 3d, &c. years, the given sum may be received, provided 
either of five events shall have happened. 1st, If both the 
lives of B and C shall have become extinct in the year. adly, 
If C only shall have failed, B having died after A in the first 
y y ears - 3 dl y> If B only shall have failed, C having died after 
A in the first y years. 4thly, If B shall have failed, C having 
died in any of the preceding years after the first y years : and 
5thly, If C shall have failed, B having died in any of the pre- 
ceding years after the first y years. The series therefore ex- 
pressing the value of S after the necessary extinction of A's 
life will be x — - s ' g 
ben r 
+ 
*-« . 0 ' 
+ 
+ &c * + Tir x 
k—» 
+ 
■ + ^+ &C. +^X 
bn 
T +£+£+&»• + -E 7 
S—e . 0 
+ 
^■0' , k. 
r 2 I ~ 
• 0 " 
+ &C. 
f • it k—S . S—i 
bn 
0 1 0 ' , 0 , - , 
■ T + “ + - + &c - + 
Sk 
+ 
0' , 0" 
* + + &c. 
s 
ben 
+ ?r + &c --^x? + f + ? + &c.+ 
The three first of these series are 
S . irk 
-i . V_Q S p 
~ ” T bn + l X f‘ + 7 X ^ — B^, supposing O 
and By respectively to be the values of annuities on the single 
lives of persons y years older than C and B. The other two 
series are a continuation of the tenth and eleventh series in the 
former part of this solution, so that the sum of those four se- 
ries will be = —■ x T 
BT — C — BC + 
C-BC 
“ n X 
BT — 
C'— B'C' , d. T'-B'I 
zer 
=^ xT '- B, T' 
C'—B'C' 
Kk 
