Mr. Morgan on 
obferve, that this rule is immediately obtained from the dif- 
ferent fractions which exprefs the feveral contingencies in each 
year. For in this cafe, the value of S for the firft year becomes 
— S X 1 — — \ 2dl 
CC 
\ . r S dd ee nd l 
for the fecond year = -r * - - - - — + - 
and fo on for the other years. Thefe feries b ing added 
S i 
together will be found =-x- 
\d 3 2£ : 
+ &c. - 
7 dd . ye 
X — + 
ccr 
. S yid , ye n S 2 d> 2e* 
. &C. -1 X — + ; l + - 7 ,T7 + TT> 
« . r x 3 r ccr cc sr y c . r c .r 
+ &C. 
S r — 1 
3 X "T“ 
X V— 3CC-2CCC Q. E. D. 
P R O B I- E M HI. 
To determine the value of a given fum payable on the 
death of A, if his life fhould be the lajl that fails of the three 
lives A, B, C. 
SOLUTION. 
The given fum can be received in the firft year only upon 
the extinction of the three lives, A having died laft. In the 
fecond and following years it may be received provided either 
of four events fhould happen : ift,- if all the three lives fhould 
fail in the year, A dying laft ; zdly, if A fhould die after C in 
the year, B having died in either of the foregoing years; 
3dly, if A fhould die after B in the year, C having died in 
either of the foregoing years ; 4thly, if only A fhould die m 
the year, B and C having both died before the beginning of it. 
The value therefore of the reverfion (when B or C are older 
