Mr. Morgan on Survivorships. 33 



same number of events, and the probabilities of the several con- 

 tingencies may be reduced to -4~ -J- -7—-. The value, there- 

 fore, of the reversion will be = 2ac x -~ + -jjr + ~~pr-> & c ° 



a — x — 4- ±4- 4- —r-> &c. which two series are known 



to express the value of S, on the contingency of C's surviving A.* 

 Were a further proof necessary, it might be observed, that 

 the value of the given sum, in this problem, is equal to the sum 

 of the values of the two reversions depending on the con- 

 tingency of A's being the first that shall fail, of the three lives ; 

 and on the contingency of C's surviving A, in case B shall be 

 then dead. Supposing the three lives to be equal, these values 



will be= b 6 r ~ I xV — 3CC— aCCC-f- 8, - r ~ l xV-CCCf 

 x V — CC, or the value of the reversion depending 



2 r 



on one life's surviving the other Q. E. D. 



PROBLEM V. 



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

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

 B's life, when it fails, become extinct before the life of C. 



Solution. 

 The value of the given sum, in the 1st year, will be = -~ — 



-. a' . b — m.c — d . a'.b—m.d .■> j •, mi -l 



mto J ; in the 2d year, it will be = 



• See Phil. Trans. Vol. LXXVIII. page 342. 



t See Phil. Trans. Vol. LXXIX. page 49 ; and Vol. LXXXI. page 253. 

 MDCCC F 



