Of CUBIC EQUATIONS. 105 



the two roots being 1 and £ : Comparing this formula with 

 our equation, we have n = 3N — AM and m = BN — 3CM : 

 And fo we get, 



3N — AM 



y — BN- 3 CM' 



Now, a* = 3 + 2 ^ +B y and £ = a X y : therefore, 



'M 



<3_(BN — 3 CM ) X + V 3 (BN— 3CM)'+2A( 3 N--AM)(BN--3CM)+B( s 3N--AM) 



^— (3 N — AM ) X — V 3 (BN— 3CMy+2A(3N-AM)(BN- 3 CM)+B(3N-AM) 

 The quantities a and 3 are therefore found by a {ingle ex- 

 traction of the fquare root : and they have each two values, one 

 pofitive, and the other negative. It is indifferent which of 

 thefe two values of a and b be taken, provided they are corre- 

 fpondent values, fo that b r= a X y. It is to be remarked too, 

 that a and b have always real values, on account of the double 

 fign prefixed to M ; for that fign is to be taken that will render 

 the radical quantity pofitive. And it is to be carefully noted 

 which of the two figns is neceffary, that a and b may have real 

 values : becaufe on this depends the characleriilic of the reduced 

 equation, and whether it is to be referred to the fir ft or fecond 

 form, and, confequently, whether it has three roots, or only one. 

 If the fign — is requifite that a and b may have real values, then 

 the reduced equation will have the characteriftic of the firft 

 form, and will have three roots. But if the fign •+• is requifite for 

 that end, the reduced equation will have the characleriftic of 

 the fecond form, and will have only one root. All this is ma- 

 nifeft from the ftatement in Art. 5. 



7. The rule, or law, according to which the preceding for- 

 mulae for a and b are conftituted, is fufficiently fimple and per- 

 fpicuous ; and the formulas are therefore, in that refpect, con- 

 venient for practice. But in examining the expreflion in the 



Part I. O denominator 



