204 



divisor. And, in a similar manner as by the general rule we 

 deduced a divisor from the series of contents of the nvune- 

 rators of some roots, the signs of those roots being changed, 

 so should the quote be discoverable from the series of contents 

 of the remaining roots, with their signs changed. Therefore, 

 when we have found a series of numeral divisors whicli have a 

 constant order of differences, if such coincidence with the 

 rule be not casual, the numeral co-factors of those divisors 

 should also coincide with the rule for finding a polynomial 

 divisor whose index is the complement of that of the index of 

 -the polynome sought, to that of the given equation. The 

 nighest co-efficient of the new polynome should be the highest 

 co-efficient of the given equation divided loy the highest co- 

 efficient of the polynome required, and the co-efficients of 

 the second terms of both polynomes should constitute a sum 

 equal to the co-efficient of the second term of the equation. 



5 4 3 2 



Thus, in Newton's example, 5j/ — %+j/ — 8j/ — 14y+l4, sub- 

 stitute for 1/ the terms of the arithmetical series: 



3 I 170 1 34, 10, 17, 5,'i-. 5 | ■; 17 



2|-38|19, 1, -38, -2,i|| -2 _ 8 ||-38 . 98 



1 CO 10- lu, --, a, -1,-g-j 1 _3 ~ ^^ _j -3^ i _g -it _,„ 



0^ 14 I 7, 1, 14, 2,|5 2 , -4 1.2.3 -Si 14 -22 



_l| 10| 10, 10, 1, l,|o 1 ^■^ 1 



Here the divisors 34, 19, 10, 7, 10, multiplied by the cor- 

 responding divisor 5, — 2, — 1, 2, i, give the respective 

 absolute terms 170, — 38, — 10, 14, 10. Now, since the 



divisors 



