Mr. Morgan on Survivorships. 23 



of A or B, should either of them be the first or second that fails, 

 of the three lives, A, B, and C. 



Solution. 



In this case, the payment of the given sum must certainly 

 take place on the extinction of the joint lives of A and B, inde- 

 pendent of C, and therefore the value of the reversion will be 



_S . r — 1 . V— AB # 



The fractions expressing the contingencies on which the pay- 



s 



abcr 



s — * — - 

 ment of S depends, in the 1st year, are — r — x d.b — m.c — d 



-j- a' . b — m . d -f- a' . c — d .m -J- b — m . 



c — d . a — a' = 



s * ■■ 



S // 



.... . -ii" n 7YI 71 



..... X Wl/t- "~ — libit .lb CI , 111 Lilt "Zi\X V tell — — 



a 6 c r 



^ — ^ -}- a" . m — n . e -j- a" x d — e . n -f- a" . n e -\- m — n . 



e . a — a' -{■ a" -\- m — n . a — a' -{■ a", e -j- c — d . n .a" 



-f- c — d . m — n a — a! -\- a" -\- c — d . m — n . a" = 7^7^ 



\mc . a — a — nc . a — a' -\- a", and so on in the other years ; 

 hence the whole value is = — x r — - 1 .V — r. AB 4- AB = 



r *■ 



S.7Z", 



V — AB, as before. Q . E . D . 



PROBLEM II. 



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

 decease of A or B, should either of them be the second or third 

 that shall fail, of the three lives, A, B, and C. 



* The same symbols are uniformly retained in this, as in my last two papers on the 

 subject. See Phil. Trans. Vol. LXXXI. page 247. 



