Mr. Morgan on Survivorships. 43 



the reversion in this case may therefore be found = S . into 



Izil x V — B — 2 C -f 2 BC + AC — ABC + B ' C ^ AB -f 



x HB + HC - HBC x B'K'+ AK — ABK + <p + \ x 



P ' r 7rZ~* X + b ' P - k i:^; V - C ^ B'C, and B'K', denotingthe 

 values of annuities on those joint lives for z years. 



When the three lives are of equal age, the value by the first 



two rules will be = S into — x V— 3C + 3CC — CCC + 



d , „rr, dd 



— xi + CT 2p» X 1 + CTT -xCK- CCK, in 



2 cr ' 2 c cr ' 2c ' 



which the last three fractions destroy each other ; and by the 



S.r—i 



last rule the value will be = -^p- x V—3C + 3CC —CCC; 



that is, in both cases, " half the reversion after the extinction of 

 " the three lives/' which from self-evident principles is known 

 to be the true value. 



The solution of this problem may also be derived from those 

 of the 3d problem in my first Paper,* and of the 1st problem in 

 my last Paper,-f <( by deducting the value of an estate after the 

 " death of C, provided that should happen after the death of A, 

 " from the value of an annuity on the life of B after C, pro- 

 " vided C should die before A." Thus, in the case of equal 



lives, the value by the first of these problems being ^~ 



and the value by the second being = -=-~ : ~ , their 



difference, or ■ ~~ 3 +3 ~ , is the number of years pur- 



eldest of the three lives, has been investigated much in the same manner with the pre- 

 sent case; but the operation was omitted merely for the sake of conciseness* 



• Phil, Trans. Vol. LXXVIII. page- 347. 



t Phil. Trans, for the year 1794, P a 2 e 2 35- 



Ga 



