Mr. Morgan on Survivorships. 29 



preceding one, cannot be prevented ultimately from taking 

 place; and there is only the single contingency of C's being 

 the first that fails, of the three lives, which can postpone the 

 possession of it after the extinction of the joint lives. The whole 

 value, therefore, of the reversion, after the joint lives of A and B, 

 must in this case be lessened by the sum of the values of an 

 annuity on A's life after B, provided B should survive C ; and 

 of an annuity on B's life after A, provided A should survive C, 

 (both found by the 2d problem in my last Paper.*) Let these 

 two values be respectively denoted by W and Z, then will the 

 general rule expressing the value of an estate be = V — AB 



s . r-i 



— W -|- Z, and the value of a given sum = — '■ x V — AB 



— W -f- Z. When the lives are all equal, the value of the 

 reversion, by substituting the values of W and Z, becomes = 



S . r. 



V -j- CC - C — CCC, or ';•" x V + CC - C — CCC, ac- 

 cording as it consists of an estate, or a given sum. 



But the solution of this, like that of the preceding problem, 

 may be obtained without having recourse to any other. In the 

 first year, the value of the given sum will be = S into 



a! . b — m c — d . , a . b—m . d , a' . c — d . m ■ b — m c — d . a — a' . 

 a b c r a b c r z ab c r ' z a b c r ' 



a'md . b—m.a — d.d .■> , .- M1 , S 



~ t ; m the 2d year it will be == — t — - into 



a b c r ' ab c r J abcr x 



j ,, , ,, . d—e n . a" , m—n.d—e.a — u'4-a" 



m — n.d — e . a'-\-m —n .e.a -\ 1 -J— 



-}- a" n e -f- m — n . e.a — a' -j- a" -j- c — d . m — n . a" -|- 



c — d . m — n . a' . b — m . c — d . a! • ,i j S • . 



1 ; in the 3d year = b t • into 



* Phil. Trans, for the year 1794, page 240. 



