3.56 



Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION 



These indications ai'e sufficient to siiew the whole course of the curve which is represented 

 in the annexed figure : 



P„ P„, are respectively the mininunii and maximum point ; the real branch of the curve 



•necessarily cuts the axis of .v to the left of the origin ; from the minimum point Pj goes off an 



imaginary branch which meets the plane of xy in X,, X^, thus giving two imaginary roots. It will 



be remembered that the conventions which have been made are that q shall be positive, r positive, 



r' q' . 

 and — > — ; it will be easily seen that the form of the curve will remain essentially the same 



4 27 



so long as the first condition is fulfilled, and the changes introduced by varying the latter conditions 

 may be represented by supposing the plane of xy shifted into different positions. Suppose for 

 instance the plane of aiy to cut the real branch between P, and D (the point of intersection of the 



curve with the axis of z) ; this will correspond to r positive, and — < — , then there are three 



real roots, two positive and one negative ; if the plane of xy cuts the real branch between 



J.- n' 



D and P., we have the case of r negative, and — < — , and there are one positive and two 

 negative roots ; lastly, if the plane of acy cuts the real branch above P„ we have the case of r 

 negative, and — > — , and we have one positive real root and two imaginary. I may just observe 



that all the imaginary roots here spoken of are of the class which I have termed connected. 



If we suppose q negative we have an entirely different form of curve, for in this case the real 

 branch has no maximum or minimum point, and therefore it is clear that one of the roots will be 

 real and the other two isolated and imaginary. Also the real principal axis of the hyperbola in the 

 plane of ley will be the axis of y, and not the axis of x as in the preceding instance. It is not 

 necessary to trace the curve, as its form is easy to imagine and it presents no varieties. 



The preceding discussion includes everj^ case of cubic equations. 



22. We may discuss in like manner the general biquadratic equation. In this case, 



f{x) = a' + qx'' + rx + s = (28), 



/'(*•) = ix^ + iqx + r. 



