Mr. Morgan on Survivorships. 31 



given sum may be received, provided either of three events shall 

 happen; 1st, if A shall have died after C in the first z* years, 

 and B dies in the z 4 1 . % 4 2, &c. year ; 2dly, if B having 

 died after C in the first z years, A dies in the % 4- 1 . z 4 2, 

 &c. year ; 3dly, if both A and B having survived the first z years, 

 the survivor of them dies in any of the following years. Let 7r de- 

 note the probability that A dies after C, and <p the probability 

 that B dies after C, (both found by the Table in Phil. Trans. Vol. 

 LXXXIV. page 229) and let all the other symbols be the same 

 as in the latter part of the preceding problem, then will the 



value of S, after z years, be = S into - '^Jj~ '■ x ^ — A z 4" 



<P-P- r - 1 v , v Si P ■ 9 ., _ a o* 1 1, . D* P.9 



xV-B-+-fc£ r xi-AB" + A- + B" — 



br x +i ' abr xJ fi * * abr 



A z -{- B* — AB 2 , and the whole value of the reversion will be 



=S into^xV_ABC-iA'+B'_i-- A ' F - AFC 



T 2 • 



2 b z a 



. m i+PC s.i+NPC AC + BC . AT3 . s , XT p; 



-f -r — x — h AB A x 1 4- NC 



1 b r z a 2 r • ' zar ' 



V — A 55 4- B* j^r- x AB* 



If the three lives be equal, the first of these rules becomes 



= s into — X V- C + CC — CCC 4- — x CK — CCK + 



r • • 2 c 



dd , ^^o d 



x 1 4 CTT — x 1 + CT ; and the second = S 



zc c r ' 2 c r 



dd 



into— x V — C + CC — CCC - x CK — CCK- 



r • c 



x 1 4. CTT -f- — x 1 4- CT; but the three last fractions in 



c c r 



d_ 

 cr 



* z being, as usual, the number of years between the age of C and the age of the 

 eldest person in the table. 



