142 



Trans. Acad. Sci. of St. Louis. 



point on the evolute is always one greater than the number 

 which can be drawn through a point on its concave side and 

 one less than the number which can be drawn through a point 

 on its convex side ; that is to say, that for a point on the 

 evolute, two of the theoretically possible normals coincide, 



Figure 3. 



and for a point on its concave side, two of them are imagi- 

 nary, while for a point on its convex side, all of them are real. 

 From this property and by reference to the figures, it is 

 apparent that for any point on the evolute (Fig. 2) one 

 branch of the hyperbola is tangent to the parabola, while the 

 other branch cuts it in a single point, and there are two 

 normals, PN V PiV 2 ; for any point in the plane on the con- 

 vex side of the evolute (Fig. 3), both branches of the hyper- 

 bola cut the parabola, one in one point and the other in two 



