OF ARTS AND SCIENCES : SEPTEMBER 12, 1865. 



h = F x — F {x _ V)i 



If we put 

 (10.) 

 we obtain from (8) 



(11.) d x = B 1 -\- (2x — 1) B 2 + (3X 2 — 3x-\- 1) B 3 , 



which may be put into the following form : — 



(12.) d x = A + (<o + 1 — 2 x) B -f Q w 2 + f <o -f 1 + 3 x 2 

 — 3 a x — 3 x) G, 



in which 



A = 



A 1 . 



to 



(13.) B=— 2 ^_A 2 _^A 3 +a + T V . DA' + AA* 



THJ' 



0=«b **'-*** 



■I 



By this formula we obtain the first differences, 3 I? S 2 , 5 3 , &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 B 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 a> — 3, (12) gives, 



(14.) 



A 



2B 



(15.) 



+ 1 







• j -1 



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



+ 3 



+ 1 

 — 1 



S 4 J — 3 



For interpolating to fifths, putting a> = 5, we get 



= A 



>B 



(16.) 



8i 



VOL. VII. 



z=A 



+ 2 



+ 1 

 



— 1 



— 2 



\2B 



