34 Pi"of. En eke on Interpolatioti. 



it would be = — f^j, and for the arguments 39, 40, 41, 42, 

 it would be = — If e ; so that, independently of the sign, the 

 ratio of the errors would be 10 : 20 : 14. This rule, that the 

 values of j9, q, r, s should always be so chosen that x is as nearly 

 as possible equal to the mean of them, ought never to be neg- 

 lected. 



In single interpolations the formula (I) may sometimes be 

 advantageously employed. It has the advantage that in case 

 one of the quantities P, Q, R, S should be erroneous, one can 

 see at once what influence this will have on the value of X. 

 But it has the disadvantage, that in general it is not known how 

 many terms P, Q, R, S will be sufficient, and be required for 

 an accurate interpolation ; and one is, therefore, not certain in 

 individual cases whether the greatest accuracy has been at- 

 tained. In order to facilitate this object, let the formula (I) be 

 expanded into a series which successively proceeds from the 

 use of two quantities to three, &c. Let the value of X de- 

 rived from n quantities be X„ , we have 



-X.4 — X3 ^ -b 



(x—p) (x—q) (x—r) 



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

 (x-p) (x-g) (x-s) (x-p) (x-g) 



4. R / (.^-P) jx-g) jx- s) _ (x-p) (x- g) \ 



\ ('•-;>) ('•-?) (»•-«) (»•-;') (»•-?) / 



^ V ig-p) (g-r) (g-s) {q-p) {g-r) ) 



, p / (j-g) i^-r) {x-s) _ (j- g) {x- r) \ 



\(p-?) Q'-O {p-s) ip-g) (p-r) ) 

 or, 



X4-X3 = (x-p) (x-q)(x-r) \ + r 



* ^ ^ -^'"^ '^'^ ')^(p-gXp-r)ip-s)^ig-p){g-r){g-s) 



- s 7 



(>•-;)) {r-g) {r-s) {s-p){s-g){s-r) 



In the same manner, 

 X3 - X2 = (x-p) {x-q) \ (^_^)^^,_,) + 



+ 



(j-iO {g-r) 



1 I 



(r- p) {r-g) ) 



X, - X, = {x-p) { ^-^ + ^) } , X. = P because 



there can be no question of interpolation if one value only is 

 given; Xg — X, is nothing but the simple proportional part. 

 The quantities inclosed in brackets are symmetrical functions 

 of 2, 3, 4, quantities ; and, as will be easily seen, without further 

 demonstration, of five and more quantities. Each value of the 

 function is divided in these expressions by the product of all 

 the differences between the argument to which the functional 

 value belongs, and all the other arguments. Let these expres- 



sions 



