Z PROCEEDINGS OF THE AMERICAN ACADEMY 



Mr. Ferrel made the following communication on certain 

 Formula of Interpolation. 



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

 putations renders it important that the most convenient formulas pos- 

 sible should be devised for that purpose. The following formulae 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 F x be any function of x, given or computed, for the equal inter- 

 vals of x = — a, x = 0, x = co, x — 2 co, &c, and let A 1 , A 2 , A 3 , &c. 

 express the different orders of finite differences. By writing A, 1 for 

 i (Ai, + A. 1 ) = A, 1 - i Ao 2 , A/ for £ (Ai 4 + A,*) = A, 3 - £ V, 

 &c, we have, 



(1.) F x = F + A lX + A 2 x> + A :i x 3 



in which 



CO 



-^2 = 7T. — 2 ( A 12 A ~T~ FiJ A) S^U ^o ) 



- CO 



A = 9.0.3 ( A / 3 — ¥ A / 5 ~f~ T2"ff A/ . . . . .) 



& o CO 



< 2 '> ^ = = 2-3- 1 4-eo* (Ao4 -* Ao6 + AA ° 8 > 



' ^ = 2-3-4-5-co^ ^-^ A/ - ••-•> 



^6 = 9 . o . A . R . ft . „6 (A) 6 -iA) 8 ) 



2-3-4-5-6- 



u> 



^ 7 — 2 • 3 • 4 • 5 • 6 • 7 • co 7 (A/ ) 



The preceding formula may be used for interpolating F x 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 co. 

 If the value of x is confined within the limits of ^^ co, instead of the 

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

 without sensible error : — 



