Prof. Encke on Interpolation. 93 



{a ..ab..b ] !lA_(^r„ „ab..b ] - [a ,...ab..b ,]) 



M + 1 n 



r« ..ai..i 1— ( [a ..aJ..6 1 — [« , 1 ••"^••^ il) 



where again the factor ^ is positive and less than 1. 



The same exception takes place here as above. 



If we apply these genei^al formulae to the case most fre- 

 quent in astronomical calculations in which p, q, r, s form an 

 arithmetical progression, it will be immediately seen that the 

 functions denoted by [ ] will then become the first, second, 

 third, and higher differences, every one respectively divided 

 by the product of all entire numbers to its index inclusively. 

 We shall have 



Ip, ?] = A P, [p, q, r] = -^-, [p, q, r, s] = -^-^ 



where the equal intervals q—p, r — q, &c. are considered as 

 unities. If we put x—]} = t expressed in these unities, and 

 write throughout for x—q .... {x—p) —{q—p) 

 x—r .... x—p —{r—p) 

 the formula II. will become 



(II)* X=P+^.AP+-^^' A^P+— ,^^'a'P 



the common formula for interpolation. 



But if we assume that A, A% A^ denote the differences 

 which lie alternately below and above the horizontal line which 

 is drawn from the place of x, we shall have from (III) 

 if r — X = t, or if a; is between q and r : 



(III)* X=R-;.aQ + -^^A^R--4^=^ A^Q 



or from (IV) if :r— ;• = t, or ii x is between r and 5. 



(IV)* X=R+^AR+ ^A^R + ^=^^A^R 



If the argument in the last formulae were descending instead 

 of ascendi?ig, It would only be necessary to change the signs of 

 the terms in both, if / is always to be considered as positive. 

 For the successive correction of the differences, we obtain 



X =R -/ 5 AQ-lZi I A'R- i±i { A'Q- !Z! I Mi ... ; } I ? 



X = H + . JaR+^' JA'R+ i±i)A.R4- i-ll JA^n...}} I I 



If 



