12 On Annuities and Assurances on Successive Lives. 



If the lives, at the time of the nomination, be all of the 

 same age (which must be assumed in practice), then, P de- 

 noting the value at the time of nomination of an annuity of 

 £\ on each renewal life, and p the assurance on the same, we 

 shall haV6^*>**' '* 5, ....iiiit luin t«liJtiiu>i c.jiiisi4 .iLL 'a .i^,i.iu 



and each horizontal row in (11.) now constitutes a geometrical 

 progression of which the common ratio is p ; and the sum of 

 the whole is evidently 



Hence 



•^ ll-p^l-p^l-p^ J 



^^ a + b + c+ .^^^^^^^ i^ (^: ^ 

 ,^.qoiD^CJ iiHO} -^aiy 1 — p {fi|^t3 aid* nrrr%u*r.'!^ n*! 



is'the present vaiiie of all the fines. ' 



The preceding expression (12.) may also be exhibited in 

 terms of the annuities on the lives instead of the assurances 

 on the same. For 



(i-)'''=TTr'''=T?r.-'™'',(2.),i-p=j^^a+B» 



heace (12.) becomes '^' ^^^ 



:'^^ «i-(A+B + C+....) 



f-— 1 + p ' (13.) 



where n denotes the number of lives A, B, C . . . on which 



the estate is held, and it may be observed that - is a perpe^^^ 



tuity of jei. 'li >at>' 



The formulas (6.), (10), (II.) and (12.) agree with those 

 given by Mr. Milne, but they are here expressed in a way 

 better adapted to computation. That these coincide with 

 Mr. Milne's expressions, will readily appear, if instead of 

 A'..,, we introduce annuities certain of equal values; thus 

 l^p^ ^ff^. be equivalent to an annuity certain for tp years, so that 



A .\iii^iW>^'yi+ -){ir'.v'fV), and, (2.), a^^)=t;<^+>, 



hence (10.) becomes , ^^^^^ j^^^^j aoii-qood odi o.ob -le. 

 i'uil si; /, . 1\>^ ,j.,v '^il'-fO'in? Yboil iznq 0,t im ba-iaikic 



