OF ARTS AND SCIENCES : SEPTEMBER 12, 1865. 



9 



^a; -^x -^(as-l)? 



If we put 

 (10.) 

 we obtain from (8) 



(11.) 8^ = B^-{-{1x — \)Bo^-\-{Zx' — Zx-\-\)Bs, 



which may be put into the following form : — 



(12.) <5. = ^ + (a, + 1 — 2 a;) ^ + (^ 0,2 -I- 3 „ 4- 1 4- 3 a;=' 

 — 3 0) a; — 3 x) 0, 



in which 



^ = - A^ 



O) 



(13.)^=-i, |_A2_iA^+(^4--V . |)A^ + JtA^-^VA« \ 



G = 



\^' 



h^' 



By this formula we obtain the first differences, di, 82, ds, &c., of the 

 interpolated numbers, and from them the numbers themselves. The 

 formula can be used for all values of a ; but when it is taken greater 

 than 12, the numerical coefficients of ^ and C obtained from (12) for 

 all values of x are not all small enough to be convenient in computa- 

 tion. For all values of a not greater than 12, by making some slight 

 modifications, we can obtain numerical coefficients convenient in com- 

 putation. The following are some of the forms most frequently used. 



For interpolating to thirds, putting « = 3, (12) gives, 



.1 



(14.) 



z=A 



+ 11 

 0\2B 



-ij 



For interpolating to fourths, putting to = 4, we get 



(15.) 



82 



= A 



+ 3 



+ 1 



— 1 



— 3 



rB 



For interpolating to fifths, putting w = 5, we get 



(16.) 



82 



83 > 



85. 



VOL. VII 



= A 



+ 2 



+ 1 







— 1 



— 2 



2B 



