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 investigation 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 of the exprefllon 



(4B 3 -f 27O) — 2 AC ( 9 B — A 1 ) -f A 2 (2 AC — BO 

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

 have only one real root : And, (as will afterwards be Ihewn), 

 when the amount of the expreffion is zz o, the equation will 

 have two equal roots. 



o. Having now found a and Ik if we fubftitute c ^— for x in 



-' ' b -J- z 



the propofed equation, we fhall 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 fubflitution is not neceffary. For 

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

 with the forms, we fhall find R= i + A + B-f-C, and 

 3R r= M + N : alfo — t> t — (Z a + 2A ^ + Ba) 

 -f (Ab -f 2 Bl> -f 3 Cb) = Ma + Nb: Therefore 



— — Mfl-f -lS3^ 



R ~ 5T+1T' 



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 the 

 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 



* J + A* 2 + B* + C = o. 



t. Com- 



