158 



Dr. J. BOTTOMLEY 



In the present enquiry it will only be necessary to consider 

 the internal portion of the curve. The radius of curvature 

 at the vertex of the parabola is 2a ; provided c be not 

 greater than this quantity, the internal curve will cut all 

 the normals to the parabola in the first quadrant in the 

 same quadrant, but if greater, it will cut some of these 

 normals in the lower quadrant, as in Fig. 2, where EB cuts 



all the normals in the first quadrant, but the remaining 

 portion of this branch of the curve cuts them in the second 

 quadrant ; having descended some distance below the axis 

 of the parabola, this branch will cease to cut the normals, 

 but at the point A there is a cusp, and the portion AC cuts 

 the remaining normals drawn to the parabola in the first 

 quadrant; hence OC the intercept on the axis is equal to c. 

 From symmetry, we may infer that the branch AC will 

 be continued above the axis of x to a point D where there 

 will be another cusp, and that there will be a branch DF 

 corresponding to AB, and passing through E, which will 

 therefore be a double point. The position of the cusps is 

 given by the equations 



