354 



Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION 



and proved in Art. (17), /Or) = has at least as many imaginary roots as any one of its derived 

 equations; hence it will have at least ,- imaginary roots if v be even, and at least r + 1 or .- - 1, 

 according as M and JV have the same or opposite signs, if .. be odd. 



20. I will now illustrate what precedes by discussing some actual cases and tracing the 

 corresponding curves. 



Let the equation be a quadratic, that is, let 



/(,r) = x' - a.v + b = (2+), 



••• /' Cr) = 2* - a, 

 f" {V) = 2, 

 and the equations of the curve of double curvature are 

 z = x' - a,v + b - y'' 1 

 = 2 .r - a J 



if we eliminate x by means of the second of these equations, we have 



a' 

 S! = b- --f. 



(25), 



Hence the complete locus of the equation z=f(w) will be in this case two parabolas in planes 

 perpendicular to each other, with their vertices coincident and their curvatures in opposite senses : 



the height of the vertex above the plane of xy will be 6 - - , if this be positive the roots of the 



given equation are imaginary, if negative they are real, because in the former case the plane of ,vy 

 cuts the imaginary branch, in the latter the real. We see in this simple instance what has already 

 been proved generally, namely, that x: does not admit of a maximum or minimum value properly 

 speaking, because at the minimum point an imaginary branch goes off" along which .j^ still 

 decreases. 



The figure represents the curve corresponding to a quadratic equation, 



