194 



of the polynomial divisor, or the co-eflicient of the highest 

 term of tliat divisor. Subduct the divisors of the above re- 

 sults from the powers of the correspondent terms of the arith- 

 metical series (n being the index of the powers) multiplied 

 into the last found numeral divisor of the highest term. Thus 



jou will have subducted the divisors from their own first 



th 

 members : hence there will be found only the n — 1 powers for 



th 



the highest powers in the remainders, and the^i — 1 dift'erences 

 of the remainders will be constant, and being divided by 

 1.2.3...n — 1, the quote is the numerator of the co-efficient of 

 the second term with the signs changed, (for the signs of the 

 co-efficients are changed by the subduction of the divisors 

 from their own first members). Subduct the last found nu- 

 merator, with the sign changed, multiplied into the correspon- 

 dent powers of the natural numbers whose index is n — 1, 

 from the first remainders. If the order of dift'erences of these 

 or of the second remainders be constant when the number 

 of orders is equal to n — 2, that difference being divided by 

 1.2.3. ..n — 2, the quote is the numerator of the third co-effi- 

 cient with the sign changed. ' 

 In general, if the numerator of a co-efficient with the sign 



changed, Avhose distance from the first term is ?m — 1, be de- 



II, 

 rived from the last dift'erences of the/« — 1 remainders, divided 



by 1.2.3, ..(n — m — 1) as we have shewn where that distance 



is 



