32 Prof. Encke on Interpolation. 



— _^ =y~^ + by~'^ + b^ij~^ &c. we have 



(B) 4- =i/-'' + A3/-("+i) + B3/-("+2) + .... 



For our purpose it is unnecessary to know the values of 

 A, B, &c. According to the doctrine of combinations they are 

 for the — (« + /•)th power of j/, the ?'th class of combinations with 

 repetitions formed of n elements; agreeably to Posselt's* no- 

 tation, who has more closely investigated the series (A), it is 



If we dissolve, secondly, ^ into the sum of the partial frac- 

 tions, whose denominatoi's are respectively y — a, y — b,y—c, 

 the well-known process show^s that a, h, c, d, being all different, 

 the numerators are obtained for each partial fraction by sub- 

 stituting in the product of all the other factors for y the value 

 which makes the denominator of the partial fraction = 0. We 

 have consequently 



i- = \ L. + i I_ &c 



Y (a-6) {a-c) (a-U) ' y-a ^ {b-a) {b-c) (b-d) ' y-b 



Expanding again the fractions -^-, —^Zi &c. into series, 



we 



have 



+ (6Z7) (6-c)(6-^ l y~' + ^^'^ + ^ ''y'^"" 

 + (c-a)(c-6) i^dy.-{ y~' + 'y~^ + ^^ y~^"" 



These quantities being summed up, the coefficients of the dif- 

 ferent powers of 3/ are all series of the same form as (A). If 

 we denote the sum of such a series by [0], [1] [?«], ac- 

 cording to the degree of the power to which the numerators 

 of the fractions are raised, we shall have 



(C) 4 - [0]j/-V [lj3/-'+ i2-]y-' 



+ [«-2]i/"^""'^ + [«-l3i/-"....+C« + r-lly-(''+'-^ 

 and the comparison of the coefficients of the same powers of j/ 

 in (B) and (C) gives immediately in accordance with what was 

 proved above, 



* In his excellent dissertation : De Funclionibus qiiibusdam st/mmetncis. 

 Aiict. Posselt. GottingjB, 1818. 



[0] = 



