Prof. Encke o/i Interpolatio7i. 29 



an argument,) the value of this same function for any other given 

 value of this argument, without knowing the form of the func- 

 tion, and even without the intention of knowing it. For this 

 purpose Taylor's theorem is an auxiliary proposition, as it 

 contains the general development of every function. By this 

 theorem every function of a binomial quantity may be expanded 

 into a series which has for its first term the function of the first 

 part itself, and whose following terms proceed according to the 

 powers of the second part, while the coefficients are formed 

 from the derived functions or differentials of the first part. In 

 order to render the series quickly convergent, the second part 

 is usually assumed very small. In applying it to interpolation, 

 it is therefore necessary to bring the value from which the 

 result is to be derived as near as possible to the given one. 



The cases in which Taylor's theorem ceases to be true, do 

 not occur in interpolation, if this operation is only applied 

 where it solely ought to be applied, viz. in cases in which the 

 required numerical value of the function lies between the given 

 ones, and in which the function within those limits becomes 

 neither infinite nor impossible. In order to facilitate the ex- 

 planation, let us assume that four values are given. The pro- 

 cess may without difficulty be extended to any higher number. 

 Let J), q, r, s denote the four values of the argument, and 

 P, Q, R, S the corresponding values of the function. It is re- 

 quired to find the numerical value of the function X for the 

 argument a: 



We have by Taylor's theorem 



y (a + w) = c -f- c, o) + Cj CO® + Cj w^ . . . . 

 Assuming « as a value not much differing from jr, and w = x — a, 

 we have, 



fx = u + /3(^-a) + y{x-af+ ^{x-af 



where a, /3, y, 8 are unknown coefficients. For deterniining 

 them we have these four conditions ; viz. 



for X = p must be J'x = P 



X = q fx = Q &c. 



We have therefore these four equations : 



P = « + ^{p-a) + y{p-af + ^{p-af 

 Q = « + ^{q-a) + y{q-af + l\q-aY 

 R = « + /3 {r-a) + y {r-af + I {r-af 

 S = « + /3 (s —a) + y {s—af + 8 (5 —af 



which I had the good fortune to liear in 1812, In the whole course of the 

 development I have followed, as far as my menmry served nie, the steps of 

 my tmich-csteemed preceptor, who combines tlie greatest strictness of rea- 

 soning with the greatest bim|)licity and elegance. 



from 



