hy means of Tables of Single and Joint Lives. 45 



payable after the death of n such lives, each life being nominated 

 at the commencement of the year in which his predecessor died, 

 is M(l — P)""; from which we at once perceive that the present 

 value of a series of payments of M pounds, one made now, 

 another expectant on death of P, the next expectant on the death 

 of another nominee P^ and so on ad infinitum, is 



M(l+(1-P) + (1-P)2+..) or^j 



and if such an interest be expectant on the fall of an existing life 

 A„ , its value is M(l — A„ ) -^P. Payments of this nature, which 

 go by the name of fines, are in practice made payable at the end 

 of the year in which the life falls, and the new life is not nomi- 

 nated until that time j this adds one year certain to the value of 

 every life, so that (l—AJ becomes v (1— A„), and P becomes 

 1 -[\—V)v, V being l^{l-i-r), where r is the rate of interest 

 per pound per annum. These give as the present value of such 

 a fine M(l-- A„)-J-(P + r-), which is in substance the same for- 

 mula as that adopted by Mr. Milne. 



The reader will be cautious not to confound (1—Ai)(l—Ac^ 

 with 1— (Ai + A2) + AiA2; the former is the value of a per- 

 petuity expectant on the death of two successive lives, the latter 

 is a perpetuity on the death of the survivor of two concurrent 

 lives of the same respective values. If the former expression be 

 multiplied out, it should be written thus : 1 — (Aj -f Ag) + Aj . Ag, 

 so as to show that the last term is a product, and not an expres- 

 sion of the value of joint lives. It is evident that in dealing with 

 successive lives of difi'erent values, it is immaterial in what order 

 they are taken. 



The formulae at the head of this paper *may be applied not 

 merely to lives, but to any other set of events, the individual 

 probabilities of which are known. Thus if there are n events 

 whose respective probabilities are «i, a^ ... «„, then 



±,^W(l)^(3.3..j») 



is the probability that exactly p will happen ; and 



is the probability that p or less will happen ; and so on: 

 Dublin, Dec. 18, 1852. 



