Mr. Morgan on Survivorships. 25 



r J 



. rJ S b a! , m . a'+ a" ■ r_s i S 



a b r ' r 2 ' ' ' 2«k 



b c . a' . m d a'-f a" . rj S m c a! , n d . a' + a" , rj , 



r_ ■ r 2, • ' zab c r ' r ' 



+ S 6 rf a' , m e . a'+ a" , c js S m d . a! , 



2a be r ' r 7 - ' ' zab c r ' 



r * -f, <-><-• -f 2abr x - -f- r * , vt. 2abr 



n.d . . a'-\- a" . ?■> , S md.d . ne.d-{-a" , r-> 



-7- 1 ? b &?. + — T7T x — h 3 b ® c - 



2 ab c r 



x 



nd . d . e . a' -J- a" 



2 abc r r 



_j l _-Z fl ? &? c# which may at last be re- 



duced to S into ~xV+ ABC - A + B + c _l_ AC 



r • 2 



2 r 



(3 . „ ■ S=?TS , WZ 



AB — — xAK + BK- 2 ABK + -%-xAF - AFC + 



d . PT _ APT 



x 1 + AP -f • B ut > unless A and B are very 



nearly of the same age, and both older than C, this rule will 

 not be sufficiently accurate. If B be the oldest of the three 

 lives, the annuities A, AC, and AK, should be continued only 

 for as many years (x) as are equal to the difference between 

 the age of B and that of the oldest life in the table of observa- 

 tions. Let those annuities be respectively denoted by A', A'C, 

 and A'K'; also let q> denote the probability that C survives B,* 

 q the number of persons living opposite to the age of A at 

 the end of x years, then will the value of S, after x years, be 



(p q ■ r— 1 . V - A* 



r 



x - — - , and the whole value of the reversion will 



be=S into r -=l x V + ABC- ±Jii+±^ x iLL _ A B _ 



— x A'K'+BK - 2 BK -f- -4- x AF — At C + -4-x 1 + AP 



'.* ' '2D- ' 2 b r ' 



* By the Table, page 337, Phil. Trans. Vol. LXXVIII. 

 MDCCC. E 



