42 Mr. Morgan" on Survivorships. 



After the extinction of A's life, the given sum may be received 

 on either of three events ; ist. on the death of B in the z -f- 1, 

 z -{- 2, &c. years, [z denoting the difference between the ages 

 of A and of the oldest person in the table,) C having died after 

 A in the first z years ; 2dly, on the death of C in the z -f 1, 

 z -j- 2, &c. years, B having died in the first % years ; 3dly, on 

 the extinction of both the lives of B and C after the first z 

 years. Let <p denote the probability that C dies after A in z 

 years ;* p the number of persons living opposite the age of B ; 

 and k the same number opposite the age of C at the end of % 

 years ; and the value on the two first of these contingencies will 



»e = r pq Il 1 bcrX+l . Again, let 



greek letters be substituted for the corresponding italic letters in 

 the first part, and the value on the third of those contingencies 



will be = -7-- x — — — H —> & c - ; — - * r- 



br^ r ' t ' r 3 b c r~ r ' r 



tr . jsk . s A? i <-l±l «, -=* 



4. lf_ 4- ft?/- 4- ° v -12- 4- i,p ^ p £Tc 



JJL 4- ilfilS 4^, &V. The ist of these series being added to 

 the first series in the former part of the solution, their sum will 



be = S into r ~ I \ r ~~ h P ' r ~ b 1 r L +1 ~ ~ ' tne sd and 4 th bein g 



added to the 4th and 5th, their sum will be = — S x C J* BC; 

 and the 3d series being added to the 3d series in that part of the 



solution, their sum will be = -^— — ; so that these six series 

 last mentioned are = — S ' r ~ t ' r ~ -f The whole value of 



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



t The solution of the latter part of the case in the 6th problem, in which B is the 



