Mr. Morgan on Survivorjhips. 43 
manner the payment of the given fum in the third year wiil 
depend on the contingency of the fame number of events 
as in the fecond year; that is, it will, firft, depend on the 
contingency of all the three lives dying in that year, A having 
died firft, B next, and C laft ; 2dly, on the contingency of 
B’s dying in that particular year, C’s living to the end of it, 
and A’s dying in the firft or fecond years ; 3dly, on the con- 
tingency of B’s dying after A in the third year (both of them 
having furvived the two preceding years), and of C’s living 
to the end of that year ; and, 4thly, on the contingency of 
A’s dying in the firft or fecond year, and of B and C’s both 
dying in the third year, C having died laft, Thefe feve- 
ral contingencies are exprefled by the refpedive fractions 
a'" . n — 0 . e—f a! +.a" * n — o . f a" .n—o.f 
, T " ' " • • • , • • • * 
0 . abc 
a' -j- a" . n — o . e—f 
9 9 9- 
and 
2 abc 
abc 2.abc 
. Confequently the value of the fum S for 
s 
the third year will be = -7—= x 
J abcr * 
a' 1 2 * ' . en a'" . eo a'" . fn a'" . fo 
-h 
a ' +a " .d? _ a ' + a " -f° , And by reafoning 
2 2 T 2 2 J ° 
in the fame manner the value of the fum S for the fourth 
, , S a"".fp a" ".go i"".gp 
year may be found = i—. t ; — 1 — — j — “ d “ - r -r 
abcr 4 
3 
W -f a" + a!" .go a' + a " + a "' . gp , a' + a" -f a f " . of a' -f a" + a" - f p 
I. ■ -- — OK- ■ ■ 11 ■ ■ ■ —f** “ O 
2 2 2 2 
If either B or C be the oldeft of the three lives, thefe 
feries continued to the extremity of that life will exprefs 
5 
the whole value of the reverfion, which will be — — - x 
+ ^ + & c . + - x ~ + a b±f-^+ 
abcr abcr 2, abcr a abcr 4 2? ci^ct • 
as. -t a 
fp.- 
G % 
