32 Mr. Morgan on Survivorships. 



each of those rules destroy one another, and therefore, in both 



of them the value of the reversion is = — '-^- x V — C -f CC 



— CCC Q.E.D. 



PROBLEM IV. 



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

 of A, should his life be the first or second that fails, and should 

 B's life, if it fail, become extinct before the life of C. 



Solution. 



In this case, the payment of the given sum can only be pre- 

 vented by the contingency of C's dying before A, and therefore 

 its value is immediately found by the solution of the 2d pro- 

 blem in my first Paper on this subject,* being no more than 

 " the Value of a given Sum on the Death of A, should C sur- 

 " vive him." 



The accuracy of this solution will appear from the following 

 investigation. In the 1st year, the given sum will become pay- 

 able should either of four events take place, the probabilities of 



i • 1 i 1 ,i r r .' a' .b — m . c — d , a' b — tn . d 



which are expressed by the lour fractions ; r 



1 J 2 ab c ' ab c 



+ a'md . d.c — d.m a' c , a' d T ^ -, .. .i, •, 



— , = , i n the 2d year, it will be- 

 a b c ' 2 ab c z ac ' 2 ac J 



come payable provided either of six events should take place, 



1 1 . i r .• a" • m — n . d — e . a" . m — n . e . a" n e 



expressed by the tractions z -j , r— 



r •> z ab c ' a b c ' a b c 



+ a" . d — e . n , b — m . d — e ■ a" . b — m . e . a" a" d , a" e T 

 . = . In 

 2 a be ' 2 ab c ' a b c 2 ac ' 2 a c 



the third year, the payment of the given sum will depend on the 



* Phil. Trans. Vol. LXXVIII. page 34.1. 



