Of CUBIC EQUATIONS. 107 



What was remarked above, with regard to the double fign 

 prefixed to M, is now to be applied to the double fign prefixed 



to Q^ 



8. The preceding inveftigation fupplies us with the follow- 

 ing rule, or criterion, by which to determine, whether any pro- 

 pofed cubic equation has three real roots or not : 



The propofed equation will have three real roots, when the 

 amount o£ the expreffion 



(4B ' + 27C^-) — 2AC (9B — A') + A' (2AC — BO 

 is negative : But if this expreffion is pofitive, the equation will 

 have only one i"eal root : And, (as will afterwards be fhewn), 

 when the amount of the expreflion is z=. o, the equation will 

 have two equal roots. 



g. Havin G now found a and b, if we fubflitute V^^ for x in 



o-f- z 



the propofed equation, we (hall have an equation for z that will 



come under one of our two forms, and from which z (and con- 



fequently the root or roots of the propofed equation) may 



therefore be found. But fuch fubftitution is not neceffary. For 



if we go back to Art. 5. and compare the transformed equation 



with the forms, we lliall find R zz i -f A + B -f C, and 



3R = M + N : alfo — 3r =: (3^ -f 2A^ + B^) 



+ {Ab + 2'Bb-\- T,Cb) ^Ma + m-. Therefore 



- — Mf+N-J 

 R ~ M + JN ' 



Whence the value or values of z are found by what is obfer- 

 ved in Art. 4. 



10. I SHALL now give the refult of the whole analyfis in tlie 

 form of a general rule for the refolution of cubic equations, 

 and add a few examples by way of illuftration. 

 Let the propofed equation be 



x^ -f Av^ + B.v + C=:o. 



I. Com- 



