322 Mr. Lubbock on the Comparison of 



living atn + \o\ — n 

 P""" ~ living at w + lOl-m' 



the value of a payment of unity after « years is 



(1 +»•)»' 



and the value of an annuity is 



^l(l+r)")' 

 m being constant in this expression and n variable. 



Instead however of interpolating values of p,„, „ between those 

 values which are known, it is better to interpolate at once be- 

 tween the values of j»,„.„ x (i +r)—' which are given, but even this 



labour is unnecessary, because 2 ^"'" or the value of the 



annuity is a function of those terms only of the series which 



are given. 



Let s/o' ilo y".i^—y«i> 3/("*})i> &c. 

 be successive values of any variable j/, 



0> 



2/o=y 

 = n{y„ + J/ni+y^,. + y) 



+ i {1 + 2 + 3 n-l.\ >,A2/„ + A_y„i + &c. + i^3/„„_„„,j 



+ -^{l.i"-l +2.i-2 + &c. + n-l.lS \d.%-\-A'y,„ + &c.\, 



