hy means of Tables of Single and Joint Lives. 43 



can be constructed out of a single set of existing lives_, treating 

 tliem always as a set, and abstaining, as we have done, from 

 dividing the set into individuals ; and the results are obtained by 

 merely handling the formulae without the necessity of entering 

 into any consideration of the various contingencies that may 

 arise. Among the more obvious results are the following, which 

 I presume are well known. The value of a perpetuity expectant 

 upon the death of the survivor of a set of lives is 



1-:S(A) H-2;(AA) -S(AAA) + . . ; 



and the value of an annuity for their lives and the life of the 

 survivor is 



S(A)-2:(AA)+S(AAA)-..; 



the former being, (perpetuity) + (annuities for joint lives in even 

 combinations) — (annuities for single lives and joint lives in odd 

 combinations) ; and the latter being, (annuities for single lives 

 and joint lives in odd combinations) — (annuities for joint lives 

 in even combinations). 



Let us now introduce another set of lives B^, Bg . . B^, and 

 construct a status out of them which is to coexist with the status 

 constructed out of the former set. We have merely to find the 

 probability of the new status being in force at a given time, and 

 multiply it by the probability of the old status ; and when the 

 multiplications are perforjned, change the small letters into large 

 ones as before. The same process may be extended to any num- 

 ber of coexistent status. 



Thus, suppose we require the value of an annuity for the lives 

 of A J and Ag, and the survivor commencing at the death of B^ ; 

 we have merely to multiply a^ + ac^—a^a^ by 1 — ^j ; which gives, 

 changing the letters, 



Ai + Ag-AiAg-AiBi-AgBi + AiA^Bi. 



The complexity of the status only increases the labour of working 

 out the results, without in any way adding to the real difficulty 

 of the problem. Thus, suppose we require the value of an 

 annuity for the lives of Aj and Ag and the survivor of them, 

 commencing at the death of such one of the set Bj, Bg and Bg as 

 shall die second, with the proviso that it shall only be payable 

 so long as there are either exactly two or exactly three of the 

 set Cj, Cg, Cg and C4 living. We have to multiply together 



ai + fi'a^^i^j 

 1 - (^^1^2 + ^1^3 + ^2^3) + 2^] KK 



^1^2 ^] ^3 ^1 ^4 2^3 ^2^4 3^4 \,^1^2^3 1^2^4 1^3^4 2^3^4/ 



