Mr. Morgan on Survivorships. 35 



/3*.AFK m o?^ Tor d.i + APT 



— x BK - ABK -L-£i^£±-+ ™ x PC— APC- 



3c T66c^3&r 2c 



, A'C'—AB . p q.r— i . V_A* 



When C is the oldest of the three lives, let <p denote the pro- 

 bability that C dies after B ; * let % be the difference between 

 the age of C and that of the oldest person in the table ; A', A'F', 

 and A'B' the values of annuities on those single and joint lives 

 for z years ; and k the number of persons living opposite a life 

 z years older than A ; then will the value of the reversion in 



this case be = S into r -=^ x 2 V — 3 A'— ABC -xBK 



— ABK + 4t- x AFK + -4- x PC — APC- ^- 1+APr 



1 6 c ' 3 b r zc 



d tvT^ Hrr . $ 



6c 



- x BT-ABT + - 1 k-xFC — AFC + 3 A'F--£- x 



AK 4- AC - A ' B ' » <pk-r-i . V_A 



If Abe the oldest of the three lives, let b', b", b'", &c. be sub- 



stituted for b — m, m — n, n — 0, &c. and a — s, s — t, &c. 

 for a', a", &c. then will the value of the given sum for the 1st 



1 S a — s.c — d . b' , a — s . d b' c j , 



year be = — E — x ; lor the 2d year = 



•J abc r ? ' 2 J 



S s—t.d—e . b" , s—t . eb" 



__ x .... + ±^__ +t _ t . d _ t , v + s _ t , tV 



-f- - — — — '— , and so on for the other years. Hence the whole 



value may be found = S into r -r^- x 2 V — 3 A — ABC 



6r 3C 



* By the Table in Phil. Trans, for the year 1794, page 229. 



F 2 



