Young — Algebraical Equations of the Third Degree, Sfc. 29 



that all the roots of the cubic equation must be real roots (" Proceed- 

 ings," vol. X., page 374) ; yet we have no means of extricating these 

 real values from the imaginary forms under which they lie concealed. 

 (See T^'oTE art the end of this Paper). 



The more restricted formula of Cardan is in the like predicament. 

 Por the application of this formula, the equation must be reduced to 

 the form 



x^+ A^x + ^0=0. 



The expression for sc, as furnished by the formula (4), when the pro- 

 posed equation takes this more simple form, is 



X = r — - — ^ : \ 



^r-V-'dA^r 1/ - 9A^r - 3r. 



(5) 



This expression is " irreducible " under precisely the same circum- 

 stances that the expression of Cardan is ; namely, whenever 11 A^ 

 + ■^A^< ; or, which is the same thing, whenever 



This may be easily proved by deducing the equation Q^ - 3PF '= 0, 

 and thence the expression for r, from the incomplete cubic equation 

 P = : thus : 



P = x^ + AfX + Ao = 11 

 Q = 3x^+Ai 

 P' = 3x, 



from which we get the quadratic equation, 



Q' ~ 3PP' = - QA.x"' - 9AaX + A,^ = ; 



-9A,±V{{9A,y+l2A,'} 

 ''■ ''^'"' QA, ' 



showing that r is imaginary, and, therefore, all the roots of P= real 

 whenever, and only whenever 



81^0^+ 124i3<0; or whenever 27^o' + 44i'<0; 



under which condition the formula (5), like that of Cardan, is irre- 

 ducible to a finite numerical expression for the real roots of the equa- 

 tion P. = Q, which shall be unencumbered with imaginary quantities. [It 



