Young — Algebraical Equations of the Third Degree, ^c. 37 



If two roots, whether real or imaginary, of a cubic equation be ex- 

 pressed in the form p + q (and any two numbers may be so expressed),* 

 it is impossible that, by diminishing the three roots, each by p, the 

 resulting equation can ever be of the form 



For, putting a for the diminished third root of the equation, the three 

 roots of the transformed equation will be 



a^q-q; 



and, therefore, the coefficient of x, in that transformed equation, will be 



and, consequently, this coefficient cannot be zero, so long as j is a 

 significant quantity — whether real or imaginary. 



[If q be zero, that is, if the cubic equation have two roots, each 

 equal to p, then, of course, not onlj^ is the coefficient of x, in the 

 transformation by p zero, but the absolute term in that transformation 

 is also zero]. 



(13.) It is deserving of remark, finally, that — 



If the same pair of imaginary roots enter into two different equa- 

 tions, that pair may be indicated, by the signs of the coefficients of one 

 of those equations, as belonging to the positive region of the roots, and 

 by the signs of the coefficients of the other equation, as belonging to 

 the negative region. 



For, first, let there be the equation 



{x''' + a'){x+h)=^^-bx'+ax'^cH^O •••[!]; 



and, next, the equation 



(x^ + a^){x - c) = x^ - cx^ + arx - a^c = . . . [2], 



in which equations h and c are both positive numbers. Then the 

 three roots of [1] will all belong to the negative region of the roots, 

 and the three roots of [2] will all belong to the positive region ; and 

 yet the pair of imaginary roots, namely, the two roots of the equation 



X' + a' = 



are the same in both equations. 



* For >•, >■', being any two numbers, if we put p for ^r + ^r', and q for \r - \)-' , 

 we shall have r =p + q, and r' =p - q. 



