on Successive Lives. 



noting the value of an annuity of £\ during the life of A^^^, 

 so that we have (r being the interest of £\ tor a year) u 



, . 1 -rA^P^ ,, , 



and 



l-'^^r»^-L-{l^AiP)), (2.) 



1+A^^) 



(l+i)(l_a(.)) (3.) 



We have evidently a^ = a'; also when A" shall be nominated 

 the value of .£1 payable at his death will be a"; but the pre- 

 sent value of £l payable when A" shall be nominated, that is, 

 at the end of the year in which A' shall die, is a^; hence the 

 present value of £l payable at the death of A" is ;^* 



Jf — ^Ut 



a xa" = a'a"; ,1| 



. , ag — a a . 



And generally when the life A^p'> shall be nominated, the 

 then value of £l payable at his death will be aO'); but the 

 present value of j61 payable at the death of A^^*"'^, that is, at 

 the nomination of A(p), is ap_i, 



Hence taking ^ = 1, 2^ 3 .... n in succession, we have " 



a| = a' 

 ag^aja 

 do^^^ aoa 



III 



a„=a„_i.aW. 



Multiply these equations together and cancel the common 

 factors 



a„=a'a"a"' a^ . ..... (4.) 



Hence the present value of an assurance payable on the 

 failure of the last of any number of successive lives is very 

 readily computed, providing we have a table of the present 

 values of assurances on single lives ; and such tables are given 

 in D. Jones's work on Annuities and Reversionary Payments 

 (Tables IX. and XXII.), according to both the Northampton 

 and Carlisle Tables of Mortality. If however only a table of 

 annuities be at hand, it will be better to modify (4.) as fol- 

 lows:— , 



