Of CUBIC EQUATIONS. 105 



the two roots being i and - : Comparing this formula with 



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



And fo we get, 



_ 3N — AM 

 y — BN — 3CM' 



Now, a^ ■=. — i — ± — [— n-5 and b =. a X y : therefore, 

 rt=:(BN— 3CM) ><+ V ^bn^3CM)*+^:aX3N=3mxbN3:Jc^^ 



"—{3^ AM) X_V^^Biv^,_^Q]yiy_^2AC3N-AM)CBN-3CMKB(3N-AiVl)* 

 The quantities a and b are therefore found by a fingle ex- 

 tradion of the fquare root : and they have each two vahies, one 

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

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

 fpondent values, fo that b =z a Xj. 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 neceflTary, that a and b may have real 

 values : becaufe on this depends the charadterifliic 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 eqviation will have the characSteriftic of the firft 

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

 that end, the reduced equation will have the charafteriflic of 

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

 nifeft from the fiatement in Art. 5. 



7. The rule, or law, accoi'ding to which the preceding for- 

 mula for a and b are conflituted, is fufiiciently fimple and per- 

 fpicuous ; and the formulse are therefore, in that refpecl, con- 

 venient for pra<flice. But in examining the exprefllon in the 

 Part I. O denominator 



