38 Mr. Morgan on Survivorships. 



years, and A's dying in the x -f 1, x + 2, &c. years, or on 

 the event of C's having lived x years, and A's dying in the 

 x -f- 1, x -f- 2, ©V. years, it is obvious that the preceding rule 

 in this case will not express the whole value of the reversion. 

 Retaining the same symbols as in the first case of Prob. V. the 

 value depending on the former of those two events will be 



= S x - ' q ' r y , , * - — , and on the latter of them it will be = 



a r ■+' 



AC * ( j__ AT* , pq-r-i . V-A * pg-r-i ■ V -AC- . 



"'+ 1 2crM-i T" acr*+i 2acr+" 



therefore, the whole value of the reversion will be = S into 

 1=1 x V - A'- ABC + ^S._AL + JZL. x J + apt - 



(3* 



x 1 + AT + BT - ABT p -j— x AFK + -^- x AF 





+ FC-A FC- ^- r -,X'' AC ' +^ + -^-x f - f -- v - A ' . 



' zacr'+i ' ' c oc'+J 



When C is the oldest of the three lives, let q> denote the proba- 

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

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

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

 for % years ; and q the number of persons living at the age of A 

 after % years ; then will the value in this case be = S into 



I X V-A'-ABC + — - — +-^xrTAPT--^- 



■2 2 r ■ 2 c r ' 2 c r 



zr 



x 1 + AT + BT - ABT - -^- x AFK + -±- x A'F'+FC 



Qq . r—\ . V — A J 



__ AFC + ^-^TT 



When A is the oldest of the three lives, let the symbols be 

 changed in like manner as in the corresponding case in the 



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



