Prof. Encke on Ititei'polation. 33 



[0] = 0, [1] = to [« — 2] = 0. But we obtain likewise 



[7i — 1] = 1, and generally [n + r— 1] ='"(0)". 



The first result [« — !]= 1 is sufficient for our present pur- 

 pose. 



Let it be supposed that the complete expression forj'x con- 

 tained besides the terms employed, one of the fourth power 

 + £(x—ay which would be the most considerable. P, Q, R, S 

 being calculated from the complete expression would contain 

 these additional terms e(j)— «)*, s{q — ay,s{r—ay, e(s— a)*, 

 and the expression (I) would, besides the terms arising from 

 the lower powers, have this increment : 



—e . (x—p) (x—q) (x—r) (x—s) \ . , ' ° r, -, 



+ 

 + 

 + 



(g-ay 



{q-x) {q-p) (q-r) (y-s) 



(r-gy 



(r— i) (r~}j) {r — q) {r—s) 



(s-aY 



i-q){s-r) 5 



(s-x) (s-;;) (s- 



The denominators of these fractions will not be changed if 

 in these denominators the quantities x, p, q, r, s are all dimi- 

 nished by a, and consequently by the theorem just demon- 

 strated the part inclosed in { } will be 



J O^ay 



{x—p) (,x—q) (x — r) (x—s) 



and the formula (1) gives this value 



X = a + /3 (x—a) + y {x—af + 8 {x—aY 



+ s{x—aY — s{x—p) {x—q) (x—r) (x—s) 

 In reference to the 4th powers only, the error of interpola- 

 tion, that is to say, that which is to be added in order to ob- 

 tain the true value, is, therefore, 



+ e(^— P) i^-g) {x — r) (x—s) 



The conditions of the problem do not contain anything re- 

 specting the value of e ; but as the arguments for the interpo- 

 lation may be chosen arbitrarily, the solution may be made in 

 such a manner as to make the product 



(x—p) {jc — q) {x—r) {x—s) a minimum. 

 This will clearly be the case if oneof the values p,<7,r, s is as near 

 as possible to x, and the others distributed as equally as possible 

 on both sides of it. If, for instance, it were required to interpolate 

 for X = 4. 1^, the error would be smallest by interpolating from 

 the arguments 40, 41, 42, 43 ; viz. = + b" * ; for 41, 42, 43, 44, 



New Series. Vol. 5. No. 25. Jrt«. 1829. F it 



