228 ESSAY ON PROBABILITIES. 



of a deferred annuity on A's life, divided by a perpetuity 

 due : the remainder is the present value of*U. to be re- 

 ceived^ if A outlive t years, at the end of the year in 

 which he dies. 



The preceding pages contain all those cases which 

 most usually occur in practice : but it is to be noticed 

 that various modifying circumstances will present them- 

 selves in different problems, for which no general rule 

 can be given. I now proceed to problems in which 

 successions occur ; that is, in which a benefit depends 

 upon the continuance of a status which does not begin 

 until another status is finished. 



In the case of an annuity for a certain term of years 

 t t to begin payment at the end of the year in which A 

 dies, we have obviously to consider a benefit which, at 

 the end of the said year, will amount to an annuity due, 

 of t payments; or II. augmented by the present (as it 

 will be then) value of t 1 future payments. This 

 then value can be found as in page 185., and, being mul- 

 tiplied by the present value of II. receivable at the end of 

 the year in which A dies (found in page 214.), gives the 

 present value of this contingently deferred annuity. 



In the case of A|B, an annuity on the life of B after 

 the death of A, we certainly have a succession, but it is 

 one which may never exist. To make a problem which 

 may come under the present division of our subject, we 

 must imagine that, at the end of the year in which 

 A dies, a new life may be nominated at pleasure, which 

 is then to be of a given age. If P be the value of an 

 annuity upon such a life, then, according as the benefit 

 is an annuity, or an annuity due at the end of A's year 

 of death, we find the present value of P or 1 -f P to be 

 received at the end of that year. The result is the pre- 

 sent value of the succession. This problem includes 

 that of finding the value of the next presentation to a 

 living. The patron of a living of 500/. a year may con- 

 sider that he gives the clergyman whom he presents 

 100/. a year (or whatever may be called liberal remuner- 

 ation for a curate) for work and labour, and the remain- 



