VOL. XC.] PHILOSOPHICAL TRANSACTIONS. 5/0 



by the solution of the 2d problem in my first paper on this subject, in the year 

 1788, being no more than " the value of a given sum on the death of a, should c 

 survive him." 



Prob. 5. To determine the value of a given sum, payable on the death of a, 

 should his life be the 2d or 3d that fails, and should b's life, when it fails, become 

 extinct before the life of c. — The analytical investigation gives for the general value 



, i .• flV ii • ■ (r— l).(v — a) (r — 1). (bc — abc) k 



in the solution of this problem, s into a ^ * t-^£ 1 __ x 



/3.(fk — afk) n . m.(pc- apc) d , »j.(pt — apt). 

 (BK - ABK + Urrm > + "3^ Wr X ( BT ~ ABT ~ -*— I 9 



& ,FC AFC , N , B — C + AC — AB , *.(K-Ak) 



+ Tt x (—3 f + ap) + - r + -i-j . 



Prob. 6. To determine the value of a given sum, payable on the death of a, 

 should his life be the first or last that shall fail, of the 3 lives; and should b's life, 

 if it fail, become extinct before the life of c. — In like manner, the solution of this 

 problem gives, for the value required, the general expression, s into 



r— 1 , \ . ac ab , md i ' V. v '"'. x d , 



__ X (v - A - ABC) + --- + -_ X (l + APT) - - X (1 + AT + BT 



— ABT) — ~ X AFK + ~ b X (AF + PC — APC). 



Prob. 7. To determine the value of a given sum, payable on the death of a, b, 

 and c, provided c shall die after one life in particular, a. — The expression for the 

 solution in this case, is s into^- X (bc — b — abc + y — £7 + ^. X + 

 ap — rf-(1 + APT) )— ^ X (bk- abk) -f r; r denoting the value of s, by the 3d 

 problem in the first paper, on the contingency of c's dying after a, anno 1788. 

 This general rule gives the true value of the reversion, when b is the oldest of the 

 3 lives. But, when c is the oldest of the 3 lives, the general rule will be = s into 



r — 1 / / \ 1 BC a'b' . m , , , . d . (I 4- apt) *. 



___ x (BC - B - ABC) + - - —+ - X (1 + AV - jffiffi? - I X 



(bk — abk) + r -j- - p ' r br3C+l V — — ; — oc denoting the difference between the 

 ages of c and of the oldest person in the table ; p the number of persons living at 

 the age of b after oc years; B r , a'b', and aV, the values of annuities on those single 

 and joint lives for x years; and p the probability that c dies after a, in the paper 

 anno 1794. 



The foregoing problems, together with those which have been investigated in my 

 former papers, adds Mr. M., comprehend, as far as I can perceive, all the different 

 cases of survivorship between 3 lives. The great number of contingencies on which 

 these reversions depend, must necessarily render the solutions intricate, and conse- 

 quently the general rules complicated and laborious. It would not however be a 

 difficult task to abridge these rules very considerably, without destroying their accu- 

 racy in any great degree ; but this would be foreign to my purpose in these papers, 

 which has uniformly been confined to the investigation of the correct values of the 

 different reversions. Nor do I think that such an abridgment is necessary, as the 

 operations of even the longest of the present rules, may be completed in very 



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