36 Mr. Morgan on Survivorships. 



HBK 



x AK - ABK — "•"** _i_ • x HC — HBC — •£- x AT- 



3 a c ' 6 <j 6 c r 



s.i+NBT s NC — NBC — xTi 3 , AC— AB 



ABT+ V-' 1 ""^ — x -— 1 4-NB + 



/F/fow £/j£ lives are equal, the value, by the two first rules, will 



S. r-i 



be= °- r ~ l xaV- 3C— CCC + Sinto— ^- + CK— CCK 



CKK + -/- x CT - CCT — J*- 



6cc 6 cr 6 c c r 



' — 1 

 ~6~7 



+ _JL1_CKK + -/-xCT-CCT— ^-x 1 + CTT; 



n r. r. • n r r h r. r. r 



and by the third rule it will be = S into ^ x 2 V — 3 C— CCC 



x CKK + ~^— x CT + — — x CCT + 



6 c c ' 6cr ■ 3 c r ^ zc r 



6 c cr 



x 1 + CTT. In both cases, all the fractions after the 



first destroy each other, so that the general rule is = — '-£ — - 



2 V — 3C — CCC, which may be proved, from other principles, 

 to be the true value. 



As the reversion, in this problem, consists of two parts ; 1st, 

 of the contingency of receiving the given sum on the death of 

 A, provided B should be then dead and C living ; and 2dly, of 

 the contingency of receiving it on the death of A, provided B 

 should be the first, C the second, and A the third, that fails, it 

 follows, that the present solution may be obtained from those 

 of the problem in my second Paper,* and of the 6th problem -f 

 in my last Paper on this subject. But the computations derived 

 from the addition of those two problems would be too tedious 

 and complicated, and therefore the preceding rules are pre- 

 ferable. In the particular case of the equality of the lives, 



the general rule becomes the same as above, (or = S x -j~- x 



* Phil. Trans. Vol. LXXIX. page 41. -j- Phil. Trans, for the year 1794, page 253. 



