30 Prof. Encke on Interpolation. 



from which the four coefficients, and no more, may be deter- 

 mined by elimination. We obtain, therefore, only the terms of 

 the expansion of/x, as far as that involving {x—af, and the 

 others must be assumed as evanescent. The elimination is 

 clear, and requires no further explanation. Instead of investi- 

 gating, however, the values «, |3, y, S for every single case, and 

 substituting them in fx, another consideration will effect this 

 purpose more concisely. 



In looking at the process of elimination, it becomes at once 

 evident how the values of a, (5, y, S must be with regard to the 

 powers of P, Q, R, S. The equations being linear with regard 

 to these quantities, P, Q, R, S will occur only in the first power 

 in the expressions for «, /3, y, S; but at the same time every term 

 will contain one of these quantities as a factor. Accordingl}'^, 

 the form of «, |3, y, 8 will be generally this : 

 cP 4- c,Q + cM + c^S. 

 If these values are substituted in J'x, J'x will likewise con- 

 tain no term without one of the quantities P, Q, R, S; or we 

 shall have 



X = 7rP +xQ+ ?R + crS 

 where tt, y(^, g, tr are coefficients, which, however they may be 

 formed, can involve no higher power of x than the third, as 

 the original equation contains none higher than (.r — «)\ 



Applying now the conditions of the problem to this last 

 form, we have evidently for 



X =: p, TT = 1 X = 



X = q, TT = X = 



j; = ;•, TT = X = 



X =i Si 7r = % = 



But if w is to be equal Q ior x = q, x z=. r, x = s, it is clear 

 by the principles of algebra that it must contain the factors 

 x — q, X — r, x—s; and if, as we have liere supposed, q, ?•, s are 

 different quantities, it must contain all three. In the other 

 factors of 71, x cannot be contained, because otherwise % would 

 contain a higher power of x than the third. If we call, there- 

 fore, the product of the other constant factors of tt... C, we have 



TT = C.{x—q) [x—r) (x — s) 



But by the fii'st condition is tt = 1 for x = p ; consequently, 



1 = C . {p-q) (p-r) ip-s), or 



Q 1 j^„ J ^ _ (..•-?) (x—r) (I— >) 



U>-i) (/'-'■) 0>-') ' ' (/'-?) (!'-'■: O'-*) 



The same reasoning applied to Xi §> '^ ^^^^^ ^^'^^^ ^° '^'"^ S^" 

 neral expression : 



(1) X = 



