Of CUBIC EQUATIONS. 



103 



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

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

 fliall find, 



rt= + 3^'Xx + 3^X2'+ iX; 



1 



> 



J 



o. 



+ Aa'b -\-2AahXz + Ab-Kz- 



+ Aa'Xz +2AaX3' -f Axz^ 



-\-Bab- + Bb^y.z 



-f- I'^abXz 4- 2Bi^Xz' 



+ BaXz^ + BX2' 



+ C^' -{- 2,Cb'Xz +3C3XZ' -fCXz' 



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

 quate three times the coefficient of z ' to the coefBcient of z, 

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

 of 2 ' to three times the abfolute term, hkewife with the fame 

 or different figns. For in the forms the coefficient of 2' and z 

 are R and Ijl 3R : and the coefficient of z ^ and the abfolute 

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

 above, three times the coefficient of z' is 3 + 3A -(- 3B + 3C, 

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

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

 (3«»-f 2A^z = ^ + B^3 = )-(-(Art'^+2Brt3^-f-3C3'). This be- 

 ing obferved, we fliall have thefe two equations for determining 

 a and b : 



3-f 2A+B 7__r3rt' -|-2Art^ + B^^ 

 + A + 2B + 3CJ1 ~ "^ lAa^ + 2^ab -f 2,Cb\ 



■\-Ab-\- 2B5 + 2,Cb\ ~ + 1+ A^' + 2^ab- + 30^'. 

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

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



upon 



