266 Mr. Morgan on 
fhould die in the year, B and C having died in either of the 
preceding years. From the fra&ions denoting thefe feveral 
contingencies the whole value of the reverfion will be found n: 
- x- + *^ + -rr + &c - 
a r r t 6 
2 ac r 
GdO,C q 
+ — + — r + &C. - 
S a'd , a"e a"'f . S a'b 
— X +-T+-T + &C. , X + 
2 ac r r r 5 'lab r 
+ &c. - 
+ +&c. + ^ 
2 ab r r t 5 %abc r 
"t* &CC. -f- 
S a' me a"nd a f oe 2 S a'bd a" me a"'rf Q 
— x +—- + - 3 -+ See. + — -X — +-- + —T-+ &C. + 
%abc r r r 5 3^ r r r 3 
2 S _ . a! md , a"ne a!" of , c ^ 7“ . ** 
— X — + --+-— +&c. = G-D + h+2M. 
3^ r r r 3 
This general theorem will give the exaft value in all cafes ; 
but when the lives are equal, it is rendered more Ample, by 
fubftituting the feveral values of G, D, E, and M, and will 
g r j 
then become — — -x 2V-3C— 3CC+2CCC; which expref- 
fion may alfo be derived in this particular cafe from the dif- 
ferent feries denoting the value of S in each year. 
The folution of this, like thofe of the two preceding pro- 
blems, may likewife be obtained by the affiftance of the firfl 
three problems. For the value of this contingent reverfion is 
either equal to the fum of the two values of S payable on the 
death of A, if his life fhould be the firft, or if it fhould be 
the laft that fails (found by Prob. I. and 3.), or it is equal to 
the difference between the value of the ablolute reverfion after 
A’s death, and the value of the contingent reverfion after A’s 
death, provided he fhould be the fecond that fails of the three 
lives (found by Prob. 2.). In both cafes the general rule be- 
comes =: G-D + K + 2M. Q. E. D. 
PRO 
