2 PROCEEDINGS OF THE AMERICAN ACADEMY 



Mr. Ferrel made the following communication on certain 

 FormuliB of Interpolation. 



The necessity of frequent interpolations in almost all kinds of com- 

 putations renders it important that the most convenient formulae pos- 

 sible should be devised for that purpose. The following formula? are 

 especially designed to facilitate interpolations where a number of them 

 are to be made at equal intervals between values of a function given or 

 computed for equal intervals of the variable : — 



Let Fy. be any function of x, given or computed, for the equal inter- 

 vals of a: = — o), a; = 0, a? = £D, a; = 2 a, «fec., and let A'^, A^, A^, «fec. 

 express the different orders of finite differences. By writing A/ for 

 ^ (ALi + AjO = Aji - ^ Ao^ A/ for J (Ai, + a/) = A,« - ^ L,\ 

 &c., we have, 



x^ 



(1.) F^ = F^^ Arx -^A,x'-^ As 



in which 



^, = ^(A/-iA,« + ^VA/-Tl^A/ ) 



-^2 = IT. 2 (^0 tV ^0 ~r -gV A) bFTT ^0 ) 



(2-) ^4 = 2^^^ (^0* - i Ao'' + jW ) 



^« ^ 2 • 3 • 4 • 5 • 6 . 0,° ^^' — i V ) 



^^ = O.^.A .P^.R. 7. „7 (A/ ) 



The preceding formula may be used for interpolating F^ for any 

 value of X positive or negative within a certain range, but the greater 

 the value of x the greater the effect of the neglected orders of differ- 

 ences upon the interpolated numbers, and if i orders of differences are 

 used, it may become quite inaccurate if x is taken greater than ^ i a. 

 If the value of x is confined within the limits of =F s- <", instead of the 

 preceding formula, we may use the following of only four variable terms 

 without sensible error : — 



