26 Mr. Morgan on Survivorships. 



H — '■ — c f- -~r * ~~~~~ » ( A* denoting the value of 



an annuity on a life x years older than A). 



If C be the oldest of the three lives, let a — s, s — t, t — u, 

 &c. be substituted for their equals a\ a", a'", &c. and c — c\ 



c — c'-\- c", c — c' -f c" -\- c'", &c. for their equals e,f, g. &c. 

 then will the value of the reversion, by pursuing the same 

 steps as in the former case, be found = S into r ~' x 



V-XA + B + ABC + ■£#£-- AB + e - A ^ AFC + 



k . HB — HBC s ™v Tr , m ; — iF>7T , ms 



x l -f NC -, — x 1 -f PC -4 r— x 



1 2 b r ■ ' a or 



2. a z a r 



i + NPC. 



But, as these series are to be continued only during C's life, 

 it is evident that the annuities A, B, AB, &c. should also be 

 continued only during this term ; and therefore, if % be the dif- 

 ference between the age of C and the oldest person in the 

 Table, A', B', A'B', &c. the values of annuities respectively on 

 the single and joint lives of A, B, A and B, &c. for z years, 

 these several symbols should be substituted above, in lieu of A, 

 B, AB, &c. to denote the value of the reversion during the 

 first z years. After the extinction of the life of C, the given 

 sum may be received upon either of three events; ist, if A 

 should have died before C, and B died after him in the z -\- l, 



z -\- 2, &c. years; 2dly, if B should have died before C, and 

 A died after him in those years respectively ; 3dly, if both the 

 lives of A and B should die after the first z years. Let <p 

 denote the probability that C dies after A, and it the probability 



