34 Mr. Morgan on Survivorships. 



S . a" . m — n . d — e 



a . m — n . e 



abcr* 



into 1 yh— m.d — e.a"-\- 



b—m . c—d . a" „• ^ , S 



b — m . e . a" -j ; in the 3d year = — - — — into 



a" 1 , n—o . e—f . a'",n — o.f . , ... > . # /// r 



— -\ — -{- — n . a"' . e — J -\- b — n . a . j 



b ov w c p /j « 



-J '■ '■ — , and so on in the other years. These several 



fractions being expanded, will form seven series, whose sum 



r— i.V—A r — i.BC— ABC 



will be = S into — T x BK — 



zr or 3 c 



ABK + '■«-"« + = Fc - Al ' c _ -1- „ BT - ABT 



™ . PT — APT . (3 I'C—ArC -^ TF^" i B C+AC — AB , 



r r X i^ — At 1 -{ f- 



*.K-AK 



2 c 



When B is the oldest of the three lives, let x be the number of 

 years between the age of B and of the oldest person in the table ; 

 p the number of persons living opposite the age of C ; and q the 

 same number opposite the age of A, at the end of x years ; A', 

 A"C, and A'K, the values of annuities on those single and joint 

 lives for x years. The given sum may be received after the 

 first x years, on the death of A, provided C shall have died 

 after B in that time, or if C shall have survived the said term 



of x years. Let it denote the first probability,* and -£— will 



denote the second ; then will the value of this part of the re- 



version be = S . into n -\ — £— x — -V — ■ — ,and the whole 



1 c a r*-T* 



value of the reversion will be = S . into r -^ x 2V — 3 A' — ABC 



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



