Mr. Morgan on Survivorships. 37 



2 V — 3C — CCC) and consequently affords an additional 

 proof of the truth of the investigation of the three problems. 



PROBLEM VI. 



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 

 three lives ; and should B's life, if it fail, become extinct before 

 the life of C. 



Solution. 

 The value of the given sum, in the 1st year, is abcr into 



a'.b—m.c — d , , 7 , a'.b — m.d . a' . c — d.m • ,i -, 



±-- [- a'md -\ 1 ; in the 2d year, 



, . S • . a" . m — n . d — e , ,. . a" . m — n . e , 



its value is = -j^ into \- a"ne-\ * \- 



a" . d—e . n . b — m . d — e . a" . b — m . c — d ■ a" . , -, S 



+ + ; m the 3d year, 7 ^ 



i nto *'*-~-*~ + a '"of+ a '"~°- f -f ^j3i£ + Eli2if! 



2 



b — n . c — e ■ a 



_j — —n. — - — ^ an( j SQ Qn m t ^ e ot ^ er y earSi The sum of these 

 several fractions, after reducing them into their proper series, 

 may be found = S into ^—5- x V — A — ABC -] — - -f- 



-4^- x 1 + APT — x 1 + AT + BT — ABT B ~ x 



2bcr ' zcr ' l ib c 



AFK + -?j x AF + FC — AFC. 



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

 between his age and that of the oldest person in the table ; and, 

 as the given sum may be received after the necessary extinction 

 of his life, either on the event of C's having died after him in x 



