]go 



CHAR I. 



The general Theori/ from whence the Rules for finding Divisors 

 of Equations are deduced. 



F there is given any equation, as /«x + PA: + Qjf + Ra; + Sx + 



1 



Tjf + Vx+W = 0, by substituting for tlic unknown letter .r, 

 any given quantity A, we shall have a result which is the ab- 

 solute term of a transformed equation, whose roots are indi- 

 vidually those of the given equation diminished by A. For, 



if we substitute y + A for x, we shall have t))(7/+ A) + Pf'^+A) 



+ Qr3/ + A) + R(^^+A) + Sfj/ + A) + T(^ + A) + V(?/ + A) + W = 

 and when the powers of the binomials arc expanded, the last 

 terms of those powers will be the similar powers of A, and 



7 6 5 4 



will contain no dimensions of ij. Hence m A + PA -1- Q A -I- R A 



+ SA + TA + VA + W, the sum of those members which con- 

 tain no dimension of ?/, will be the absolute term in an equa- 

 tion in which y+A=x and y = x — A. 



If for the unknown letter there be severally substituted the 

 terms of a decreasing arithmetical series, 3, 2, 1, 0, -1, -2, -3, 

 the results are the several absolute terms of transformed 

 equations, whose roots are those of the given equation, res- 

 pectively diminished by the substituted quantities ; and since 



the 



1 



