202 ESSAY ON PROBABILITIES. 



Brackets, as distinguished from colons,, will serve the 

 same purpose as in algebra, namely,, to give compound 

 terms the meaning of single ones. Thus, 



(AB) C|D : E denotes the last-mentioned annuity, 

 on the supposition that payment is to begin when either 

 of two events happens, the failure of the joint existence 

 of A and B, or that of C. 



AB|A : B the present value of an annuity to begin 

 when one of the two, A and B, dies, and to continue 

 during the life of die survivor. There is in this par- 

 ticular case the expression of an event which cannot 

 happen ; for if B die first, it is only A who can receive 

 the annuity. Thus B|B is an expression for nothing ; 

 for the present value of an annuity on the life of B, to 

 begin at the death of B, is nothing. In the expression 

 A B | A : B, part is nothing, and the rest has a value. 



A 1 1 is the present value of 1 1. to be received at the 

 end of the year in which A dies; -|l is the present 



value of I/, due t years hence; AB|lC is the pre- 

 sent value of an annuity which, commencing with the 

 first death out of the two, A and B, lasts till either one 

 payment, or the life of C drops: that is to say, the 

 value of I/, to be received at the end of the year in 

 which the joint existence of A and B fails, provided C 

 be then alive. 



Let a colon placed after the final letter denote that 

 a perpetuity is one of the status during which the 

 annuity is to last. Thus, 



A|(C:) denotes the present value of an annuity to 

 last for ever, after the death of A. The symbol denotes 

 that C being alive at the time of the first payment is a 

 necessary condition. This being satisfied, the longest 

 of the two, C, or a perpetuity, is of course a perpetuity. 



The presence of the colon always indicates the 

 longest of the two status, and when the colon is a final 

 symbol, one of the status is an infinite number of years, 

 or a perpetuity. 



A! : B 2 |C denotes the present value of an annuity 



