Survivorships* 269 
from felf* evident principles. This exprefli '011 may alfo be 
immediately derived from the different fractions which denote 
the value of S in each year. For in the firft year thefe fractions 
will be reduced to m the fecond year to 
c~r 
* X in the third year to 2 ± x and fo on in the 
3 A 
other years. Hence the whole value of the reverfion will, as 
a bo, e ,be = 5 -^*TWCC. Q.E.D*. 
PROBLEM IX. 
To determine the value of a given fum payable on the death 
of A or B, fhould either of them be th & fecond that fails of the 
three lives A, B, and C. 
solution. 
The payment of the given fum in the firft year will depend 
■upon either of four events happening, ift. That the three 
lives fhould fail, A or B having been the fecond that failed. 
2 dly, That A fhould die after C, and B live. 3 dly, That B 
fhould die after C, and A live. 4-thly, That A and B fhould 
both die, and C live. In the fecond and following years the 
given fum will become payable, provided either of eleven 
events fhould happen, ift, If the three lives fhould drop m 
the year, A or B having been the fecond that failed, 2dly, If 
A fhould die after C in the year, and B live, sdly, It 8 
fhould die after C in the year, and A live. 4thly, If A and B 
* I do not know that any folution has been attempted before, either of this or 
•f the two following problems* fhould 
