OF THE ROOTS OF ALGEBRAIC EQUATIONS. 



f'i.v) = \ix' + 2q, 

 f"{x) = 24.r, 



and the two equations to the curve are therefore, 



ss = x' + qx^ + rx + s - y^(6x'^ + ?) + y^ 1 

 = 4cjr' + 2qx + r — iy^x J 



357 



(29). 



Now the sign of s need not be considered, since (as has been observed before) a change in its 

 sign will only correspond to a change in position of the plane of xy, the figure of the curve 

 remaining the same ; the combinations of sign of q and r will be as under, 



and these different cases must be considered. 



The equation for determining the maxima and minima of the real branch of the curve is 



x^ + - X + - = 0, 



2 4 



which has one real root if q is positive, and if q is negative it has one or three, accordino 



)•-' . , q^ 



— IS > or < than — . 

 8 27 



First then, let q be positive and let also r be 

 positive, then it will be found that the curve will 

 be such as is represented in the annexed figure. 

 P is the minimum point of the real branch; the 

 dotted lines represent the imaginary branches, which 

 cut the plane of xy in the points A',, A';, and in two 

 other similarly situated points on the other side of 

 the plane of xz which are not represented for fear 

 of complicating the figure. 



If r be negative, the figure will be essentially 

 the same, but must be supposed to revolve through 

 two ritrht angles about the axis of z. 



Secondly, let q be negative; then if — be > — , 



8 27 



there will be no difference in the figure but this, 

 that the curve of ])rojection on the plane of xy 

 will lie nearer to the axis of x than the asymptotes, instead of lying further away, as in the 

 last case. 



