APPENDIX THE FIFTH. 



have a method of finding the value of the required 

 annuity, which may be described in the following 



Take out the value of an annuity of 1 /. at the given 

 rate of interest, and at several successive higher rates : 

 take the successive differences, the difference of the dif- 

 ferences, and so on. To the first difference add half of 

 the second difference, one-third of the third, and so on : 

 the sum of these, multiplied by the amount of WOL in 

 one year at the first named rate, is the value of the 

 annuity required. 



I take examples from the Northampton tables, at 4 

 per cent., because Mr. Morgan has given a table of the 

 annuities required, which will serve to find verifications. 

 First suppose the age to be 5 years. 



060 



2-421 + ^ of -556 + of -164 + \ of -060 = 2'769 

 2-769 x 104 = 288-0 answer: in Morgan 288-4 



Next suppose the age to be 80 years. 



Annuity 4 p. c. 3*643 , 9R 



5 3-515 if: -007 



6 3-394 *. '008 



7 3-281 .jlj* -006 



8 3-174 



128 -- \ of -007 = -132 ; -132 x 104 = 13 -7 answer 



13*8 in Morgan. 



PROBLEM. A life annuity is m at the end of the 

 first year, and diminishes n every year, until nothing 

 is due, after which it ceases entirely. Required its 

 present value. 



RULE. When n is so small, that the annuity cannot 



