Of CUBIC EQUATIONS. 



103 



1 



The terms of this expreffion are now to be evolved and ar- 

 ranged, according to the powers of z : which being done, we 

 mall find, 



+ Aa 2 b -\-2AabXz + AbXz- 



+ Aa 2 Xz +2AaXz 2 + AXz 3 



-{-Bab 2 -{- Bb 2 Xz 



-f BaX-z 2 -f BX2 ! 

 -fC£' -f- zCb'Xz +3axz ^ +CXz , 

 In order to reduce this equation to our forms, we muft c- 

 quate three times the coefficient of z ! to the coefficient of z, 

 either with the fame or different figns : and alfo the coefficient 

 of z i to three times the abfolute term, likewife with the fame 

 or different figns. For in the forms the coefficient of z 3 and z 

 are R and Z 3R : and the coefficient of z 2 and the abfolute 

 term are, — $r and + r. Now, in the transformed equation 

 above, three times the coefficient of z 1 is 3 -f 3 A + 3 B + 3^> 

 which I write thus, (3 + 2A -f B) -f- (A + 2B -j- 3C) : And in 

 like manner, for three times the abfolute term, I write 

 (3^ + 2Aa 2 b + Bab 2 ) + (Aa 2 b+2Bab* + $Cb*). This be- 

 ing obferved, we mall have thefe two equations for determining 

 a and b : 



3 + 2A+B 7 _„(*3^ 2 + 2Aab + Bb 2 

 + A+2B+3C3 + LAa 2 -\- 2Bab + 3 Cb 2 y 

 $a -f- 2Aa-\-Ba 1 ___ C $a 5 -J- 2 Aa % b -f Bab * 

 -\-Ab-\-2Bb + 3 Cby~ + l+Aa* -f 2 Bab* + $Cb>. 

 6. It is manifeft, from the manner in which I have written 

 the two equations for determining a and b, that they depend 



upon 



