NEWTON'S FIVE TYPES OF PLANE CUBICS 



283 



J,-. 



''■-^- -.1.^^ J>.. 



Fig. 5 



V. The Cuspidal Cubic. 



In this case three of the four points of a real fundamental quad- 

 ruple coincide. Constructively such an arrangement can be realized 

 by assuming as the pencil of conies a pencil through a fixed point A, 

 and with its conies osculating each other at another fixed point which 

 evidently may be considered as the representative of the three coin- 

 cident points A^AgA^. 



To construct a pencil of osculating conies we may start with 

 the fact that the picture of a circle in a perspective coUineation whose 



