PHILOSOPHICAL TKANSACTIONS. 



487 



rot. XVII.} 



or dead. And consequently, as the sum of both, or the number of persons 

 living of the age first proposed, is to the number remaining after so many 

 years (both given by the table) so the present value of the yearly sum payable 

 after the term proposed, to the sum which ought to be paid for the chance the 

 person has to enjoy such an annuity after so many years. And this being 

 repeated for every year of the person's life, the sum of all the present values of 

 those chances is the true value of the annuity. The following table, computed 

 on this principle, shows the value of annuities for every 5th year of age to 

 the 70th. 



Use VI. Two lives are likewise valued by the same rule ; for the number of 

 chances of each single life, found in the table, being multiplied together, become 

 the chances of the two lives. And after any certain term of years, the product 

 of the two remaining sums is the chances that both the persons are living. The 

 product of the two differences, being the numbers of the dead of both ages, 

 are the chances that both the persons are dead. And the two products of the 

 remaining sums of the one age, multiplied by those dead of the other, shoiv 

 the chances there are, that each party survives the other ; whence is derived the 

 rule to estimate the value of the remainder of one life after another. Now as 

 the product of the two numbers in the table for the two ages proposed, is to 

 the difference between that product and the product of the two numbers of per- 

 sons deceased in any space of time, so is the value of a sum of money, to be 

 paid after so much time, to its value under the contingency of mortality. And 

 as the aforesaid product of the two numbers, answering to the ages proposed, 

 is to the product of the deceased of one age multiplied by those remaining alive 

 of the other ; so is the value of a sum of money to be paid after any time pro- 

 posed, to the value of the chances that the one party has that he survives the 

 other, whose number of deceased you made use of in the second term of the 

 proportion. This perhaps may be better understood, by putting N for the 

 number of the younger age, and rj for that of the elder; Y,y the deceased of 

 both ages respectively, and R, r for the remainders ; also R + Y = N, and 

 r -\- y =: w. then shall N « be the whole number of chances ; N « — Y^ the 



