144 Trans. Acad. Sci. of St. Louis. 



cusp of the evolute, one of the asymptotes is tangent to the 

 parabola at its vertex ; this is a critical case for which the 

 three real normals coincide. 



The constructions for these cases can easily be supplied by 

 the reader. 



THE NORMAL TO THE ELLIPSE. 



The equation of the ellipse referred to its rectangular axes 

 as axes of co-ordinates is, in the usual notation, 



The equation of the normal to the ellipse is 



a^xy^ — b'^yx, = (a^ — b'^) x^y^ (16) 



in which x^ y^ are the co-ordinates of the point in which the 

 normal crosses the curve. 



If the normal is to be drawn through any point in the 

 plane, P, whose co-ordinates are $, 77, these co-ordinates must 

 satisfy equation (16), so that 



a^c^i — b'^'nx^ = {a^ — 6^) x^y^. (17) 



The co-ordinates Xj, y^ also satisfy equation (16). 



If now we make cc^ y^ variables, equation (17) becomes 



a'^^y — b^-qx = (a"^ — 6^) xy. (18) 



This equation is satisfied by the co-ordinates $, 77 and also by 

 the co-ordinates aSj, y^, and, therefore, represents a curve 

 which passes through the given point P and intersects the 

 given ellipse at the intersection of the normals through P 

 with the ellipse. 



Equation (18) represents an equilateral hyperbola. The 

 equations of its asymptotes are 



