34 s 
Mr. Morgan 
will be exprefifed by the fraction 
S . a' . b — c 
labr 
The payment of 
the fum S at the end of the fecond year will depend on either 
of two events happening. Firft, that A and B both die in 
the fecond year after having furvived the firft, retrained, as 
above, to the contingency of B’s having died laft ; fecondly, 
that B dies in the fecond year and A in the firft year. The 
value therefore of S for this year will be exprefied by the two 
fractions ■ * ‘ * - 4 -1- LilZ ill , Again, the payment of S in 
the third year will depend either on A and B’s both dying in 
that year, and B having died laft ; or on B’s dying in that 
year, and A’s dying in the firft or fecond years. The value 
therefore of S for this year will be=: 
S . a" .a — e S .d—e.a'-\-a' 
+ 
2 abr* ‘ abr 3 
By proceeding in this manner for the other years the whole 
S Q? t) C 
value of the reverfion will be found = - x — — P 
2 abr 
a" . c-d 
abr 
+ 
a"' . d-e 
abr 3 
nn r 
H f~A — ~ +&C. + SX 
abr 4 
a', c-d a' + d'.d-e a' + a" + a"\e-f 
abr z abr 3 abr 4 
-P&c.. The firft of thefe feries by proceeding in the fame 
manner as in the folution of the fecond problem may be found = 
®xK-AK- 
lb 
13 -AB 
-i . B-AB + 4-x C-AC; and the fe- 
ir 2 br 
cond feries may be found = - x C - AC + 
hr 
B- AB 
Hence the 
whole value of the reverfion will be = S x K — — — 1 c £2 
a br 
r— I . B-AB 
» 
2 r 
Q. E. D.. 
Having now the value of the fum S depending on the older 
of the two lives dying laft, the value of the fame fum 
depending 
