the rational divisors, viz. if the roots of the given equatioa 



be ~^, jr, -pr, &c. and A is any substituted term, tlie 



correspoiiding terms in the different series of the increasing 

 numerators of the roots will, with their signs changed, be 

 found among the corresponding terms of the decreasing series 

 of divisors, which terms, by putting D generally for any de- 

 nominator, will be A X D + a, Ax D+b, A x D+c, &c. ; and 



the products or contents of these, will be A"xD"-{-A"~ x 



D'~'a+6+c+&c. + A"-^ X D"-^a6+ac-&c. + A"-' x D"~' 



abc+ abd-\- Sec. + &.C. + abcde &CC.. ; and here all the powers of the 

 substituted number, descending from the number of dimen- 

 sions of the polynome required, are connected with the different 

 combinations of the numerators of n number of roots with their 

 signs changed, that is with the numerators of the co-efficients 

 of a polynomial divisor of the proposed equation, their com- 

 mon denominator being the product of the denominators of 

 n roots, since the numerators of such co-efficients are made 

 up of members each of which is a product of the numerators 

 bf roots, with their signs changed, multiplied into all the 

 rest of the n denominators but their own. 



When the substituted numbers are the terms of an arith- 

 metical series, we shall have their powers multiplied, in the 

 successive results, by the several co-efficients of the divisor 

 required, and if we take the differences of such results, and 



the 



