198 



let k be the extreme quote corresponding to 1, and — A,, — k„ 

 — /fg, &c. the extreme differences of those quotes of the first, 

 second, third, &c. orders, by continuing the differences be- 

 ginning with the penultimate differences, we shall continue 

 the terms of the series, and the term opposite to cypher will 

 be found k + k\ + /:,, + A:^+ &c. ; and because the member which 

 alone is not multiplied by the substituted number in the 

 quotes, is the co-efficient of the penultimate term of the di- 

 visor, that co-efficient should be the term found opposite to 

 cypher. • 



In general, the co-efficient whose distance from the first is 

 m, along with. the co-efficient whose distance from the last is 

 also=rm, will be derived, in like manner as above, from the 

 quotes which result from dividing by the corresponding sub- 

 stituted numbers, the remainders, after the quotes in the pre- 

 ceding step are diminished by the co-efficient whose distance . 

 from the last is m, and by the product of the co-efficient 

 whose distance from the first is the same, multiplied into the 

 powers of the correspondent natural numbers, whose index is 

 n — 2m. 



If the last differences of the quotes in the first, second, 

 &c. steps, be 1.2.3...?/ — 2p, 1.2.3. ..7i — 4q, &c. ; and if, in the 

 series of the first, second, &c. quotes or depressed remainders, 

 the terms opposite to cypher be k + k^ + k^ + Scc. l + l^ + l^ + 8cc. 



&c. the polynomial divisor is Dx'+px'~^ + qx"~^ + Sec. + (1+ 



/, + &c,>+ (k + k^+&c.)x+abcd&c, 



CHAP. 



