Engler — Tlie Normal to the Conic Section. 



143 



points, and there are three normals, PN^, PN^j, P-IV^; and 

 for any point in the plane on the concave side of the evolute 

 (Fig. 4), only one branch of the hyperbola cuts the parabola 

 and that in a single point, and there is but one normal, PN. 



Figure 4. 



Special Cases. 1. When the point P lies on the given 

 parabola one of the normals through P coincides with the 

 tangent to the hyperbola at that point and is found by joining 

 with P a point on the axis of JY" at a distance p to the left of 

 the foot of the ordinate through P; which is, in fact, the 

 usual construction for this normal to the parabola based on 

 the principle that the sub-normal is constant. 



The other normals can be obtained by the construction 

 already given. 



2. When the point P lies on the axis of the parabola, the 

 auxiliary hyperbola degenerates into its asymptotes. The 

 intersections of these asymptotes with the parabola give the 

 normals. It will be observed that when the point P is at the 



