ON ANNUITIES. 205 



The tables, of which an abstract has been given, 

 will enable us to find |A and |AB, the values of an 

 annuity on one or two lives, by simple inspection or 

 easy interpolation, but not |ABC, the value of an 

 annuity on three joint lives. A sufficient approximation 

 to | ABC is found by finding the single life Z, whose 

 annuity is equal, or as nearly equal as the tables will 

 give, to the annuity on the joint lives of the two elder 

 of A, B, and C. This new life must be placed instead 

 of those of B and C, which reduces the three lives to 

 two. That is |Z being equal to |BC (A being the 

 youngest of the three) | A B C is | AZ very nearly. Thus, 

 the ages of A, B, and C being 45, 50, and 65, we 

 have |BC (Carlisle table, 5 per cent.) = 6*8, and in 

 the table of single lives 7*8 and 6*3 are the values of a 

 single life at 65 and 70 years, giving a diminution of 

 1-5 in five years or '3 per year of age. Hence 7*8 

 will become 6*8 in about three years, or 68 is the age 

 of Z. Again, writing the ages at the feet of the letters 

 signifying lives, !A 45 Z 65 is 7'0 and |A 45 Z 70 is 5'8, 

 giving a diminution of '24 for every year of Z's age, so 

 that |A 45 Z 68 will be 7*0 '7 or 6'3, which is near 

 the value of |A 45 B 50 C f)5 . 



The values of annuities upon single or joint lives 

 being thus found, it is easy to solve many other pro- 

 blems of annuities. Such annuities may either be re- 

 quired for a temporary purpose, or as a provision for 

 future years, or for some parties after the death of others. 

 I write the result required in symbols at the beginning 

 of each question, and leave the reader to refresh his 

 memory of them by observing the demonstration. 



PROBLEM. An annuity of II. on the life of A, aged n, 

 is to be bought, to begin payment at the end of t years, 

 if A should live so long : required its value. 



The question is proposed in the most usual form, 

 but a little change will facilitate its solution. The 

 right of A is evidently an annuity to begin in t 1 

 years, should he live so long. And the rule is expressed 

 thus, 



