Mr. Morgan on Survivorships. 27 



that C dies after B,* then will the value of the reversion de- 

 pending on these several contingencies (putting^ for the num- 

 ber of persons living opposite the age of B at the end of % 

 years, and q for the same number opposite the age of A) be 



. p . <p . r—i . V — B* . air . r—i . V — A^ . * a i+AB* 



= S into bfK+t + 2 ^p + -^ x — 



AB% and the whole value of the reversion will be = S into 



A'+B 



r _! . V - ! —- + ABC 



, BC-f AC * o 1 ff-A'F'— AFC ■ * . H'B'— HBC 



' zr ; I z b ' z a 



m j.i + PNC i+PC , p.tp. r— i . V— B* 



zar ' 1 r a 2 ' b r*T' 



+ - ? -;^. V - A " + fe x 7+A&.+ 



If the lives be all equal, the value, according to the first rule, 

 will be= S into — x V — C— '- CC + CCC 4- — — CC 



~xV- C— '-CC + CCC 4- — 



-f-Ji-xKCC — KC-r--f-xi +CT + -^-xTT- CTT; 



* zc ' 2 cr * ' 2 c c r ' 



and, according to the 2d rule, it will be = S into 1 —^- x 



CC „„ , * CK-CCK 



V — C + CCC + — — CC + * ^-^ L_ x 1 1 CT 



1 ' r ' c cr ' 



-| xi + CTT. If these expressions be resolved into their 



respective series, the value in each case will be found = ' r ' 



x V — C — CC -j- CCC, which is known to be the true value, 

 from self-evident principles. 



But the solution of this problem may be obtained by the 



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



f In this and the following problems, A x , B^, AB*, A*, B*, &c. signify the value 

 of an annuity on the single or joint lives of persons zor x years older than A and B, &e. 



E 2 



