Resolution of Algebraic Equations. 275 
operation. To shew this, let us recur to the general equation 
before given (x* — px n ~ 1 + qx n ~ 2 — rx n ~z + sx n ~ * = o) ; 
suppose its (n) roots to be represented by ( a , b, c , d, &c. n *), 
then, by the construction of equations, we have (n) distinct 
equations from the several coefficients in succession ; viz. 
a + b + c + d &c + 72 - - in number ( n) —p, 
ab + ac + ad &c. - - - - - - [ nx ~ir -) — q, 
abc + abd &c. ‘ - n x —7— x ~J~ = r > 
(abode &c. n), or the product of them all, being the coeffi- 
cient of the last term. Now, as we have (n) equations, and (n) 
indeterminate quantities, it is evident, that by employing each 
equation successively to determine one quantity, the whole will 
be determined. But the equations are not all of the same degree : 
the first, is a simple equation : the second, being composed on 
one side wholly of products by two, is in degree a quadratic ; 
the third, for the same reason, a cubic; and so on. If the first 
of these equations be used to determine (a), we shall have 
(a=p — b — c — d &c. — n ) ; inserting that value for (a) 
in the second equation, it becomes the quadratic (pb — b z -\- pc 
— c' + pd — d z — be — bd &c. =*q). If that quadratic be 
solved to determine (b), and the values of (a and b) be inserted 
in the third equation, it becomes the cubic (c 3 &c. ,. = r). 
Moreover, the quadratic having two roots, its solution will have 
* The nature of the roots is not material in this place ; whether affirmative or 
negative, real or imaginary, they have just the same operation in forming the coeffi- 
cients of the equation. I have however, throughout chosen, wherever I could, to 
give examples capable of being tried by real and affirmative roots ; and, for that pur- 
pose, have uniformly made the signs of the coefficients alternately affirmative and 
negative. 
