OF THE ROOTS OF ALGEBRAIC EQUATIONS 355 



AB = — , BC = h - —, which in the figure is supposed positive: X^ CX^ is the imaginary 

 blanch cutting the plane of xy in X, and X^, so that AX^, AX^ are the roots of the equation. 



I may observe, that the mode of viewing the subject which is explained in this paper, though 

 rather complicated when considered generally, is of very easy application in the case of a quadratic ; 

 for the ordinary solution gives us the roots 



a /a- a^ 



.r = - ± V 6, if 6 be less than -, 



2 4 4 



J ^ /' — / ^' c-' 



and .r = - ± ^/ _ 1 \/ h , if 6 be greater than — . 



Now ,r = - corresponds to the minimum value of ,1?'' - aoe + b, and therefore the usual mode of 



interpreting the symbol \/ - 1 would lead us to consider the preceding expressions as the distance 

 of the minimum point from the origin ± a distance measured along the axis of x or perpendicular 



to it, according as h is less or greater than — . 



4 



21. Let us take the case of a cubic, which I shall suppose to be deprived of its second term 

 for reasons heretofore assigned. We have then 



f{x) = x^ - qx + r = (26), 



f {x) = Sx' - q, 



f" (j?) = 6.r, 



/'" (v) = 6. 



Hence the equations of the curve will be 



z = x'* — qx + r - Sxif 1 



., (27). 



= 3x - q - y j 



The curve will assume different forms according to the nature of the parameters q and r. 

 Let us consider the real branch of the curve ; then the condition dz = gives us 



Sx' - q = 0, or .r = ± \/ - ; 



hence in order that there may be a maximum or minimum point q must be positive ; suppose this 

 to be the case, then there will be one maximum and one minimum, and for the value of z we 

 have 



.V^; 



= r =F : 



27 



r" q' 

 I shall sup|)()se r to be positive, and — > — , so that both values of z may be positive. 



The curve in the plane of asy is evidently an hyperbola, the asymptotes to which are inclined 

 at an angle of 6o" to the axis of w, and in the case here supposed of q being positive the real 

 principal axis will be the axis of <r. 



z z 2 



