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reduce to narrower limits the divisors of the other absolute 

 terms which stand opposite to the even substituted numbers ; 

 let the even divisors alone be compared with the even divisors 

 of the absolute term of the given equation, and the odd alone 

 with the odd. For, ahcde Sec. or the divisor of the abso- 

 lute term of the given equation, being subducted from 



A X D + &c. + A X Dabc &:c.+bcd6i.c.+8cc.+abcde &c. or from 

 the divisors of the absolute term corresponding to A, the re- 

 mainder should be divisible by A. Hence, that the even re- 

 mainders may correspond to even substituted terms, the di- 

 visors to be compared should be together even, or together 

 odd ; for, the sum or difference of an odd and an even number 

 cannot be even, and therefore cannot be divisible by an even 

 number. 



Thus, in Newton's example, (which shall be given imme- 

 diately) 14, which is opposite to cypher, cannot be compared 

 with 19, which stands opposite to 2; nor 7» opposed to cypher, 

 Avith — 38, opposed to 2. 



From the above general statement, we are enabled, a pn'on, 

 (as soon as we shall have obtained those numeral divisors 

 "whose differences afford arithmetical series) to try, without 

 the trouble of division, whether polynomes can from thence 

 be deduced, which shall really divide. For, if an equation 

 can be divided by any compound divisor, it will also be di- 

 visible -by the quotient, or by the polynome whose index is the 

 difference of the indices of the equation, and of the compound 



VOL. XI. 2 E divisor. 



