Prof. Encke ow Interpolationi 31 



V _ p (J-?) Qf-r) ix- s) Q (x-p) (x-r) (x—s) 



(p-g) O'-O (P-s) (g-p) {g-r) (g-s) 



j^ (x-p) (x-q)(x-s) g (x-p) (x-g ) (x-r) 



(r-jj) [r-q) {r-s) {s-p) (s-q) (s-r) ' 



which not only fulfils the given conditions, but is likewise the 

 only one which does it completely, if there is no higher power of 

 .r than the third. For it is clear that the difference between any 

 other expression not identical with (I), and the expression (I) 

 must be = for the four values iv = p, x = q, x = 7; x = s, 

 and must consequently contain together the four factors x—p, 

 x—q, x — r, x — Si or x raised to the fourth power, which is 

 against our supposition. 



The equation (I) may be expressed in a neater form if di- 

 vided on both sides by {x—p) (x—q) {x—r) {x—s). It will then 

 become 



= ; w w. ^.,. „ ^ + 



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



Q. R 



+ 



(g-x)(q-p) (g-r) (q-s) (r-x) (r-p) (r—q) (r-s) 



S 



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



The form of y^ being altogether arbitrary, we may assume 

 for X, .r"*, so that P = p"^, Q = q"'' &c. and our equation may 

 be thus expressed : n quantities a, b, c, d being given (instead 

 of the preceding x, p, q, r), and the mth power of each of 

 them being divided by the product of all the differences be- 

 tween the quantity raised to the power and each of the others, 

 the sum of all these « quotients is always = 0, as long as ?w is 

 between and 71 — 2, both included. This latter restriction is 

 necessary on account of the condition, that in deducing the 

 series, the powers higher than x^ should be excluded. 



The investigation of the value of the series, 



(a-b)(a-c)(a-d)... + (b^a) (b-c) (b-d)7.. + '(7~a) (c-b)(c-d)... + &C.( A) 



there being ?? quantities and the value of ?« arbitrary, leads to 

 an estimation of the error of an interpolation. With this view 

 we will expand in a twofold manner this fraction : 

 1 1 



y ~ {y-"-)i>f-b)Q/-c)(y-d)... 



At first we will consider it as the product of the simple 

 fractions — — , -, , &c. every one expanded sepa- 

 rately, and then all nniltiplicd together. As 



