NEWTON'S FIVE TYPES OF PLANE CUBICS 



281 



AjAj at Bj. Bj and B,^ coincide with A2A^, Fig. 3. In the Steiner- 

 ian transformation we find the point C corresponding to a point B, 

 by joining B to B,, Bj, B^ and constructing the fourth harmonic rays 



Fig. 3 



to these joining lines with respect to the pairs of sides of the quad- 

 ruple through the points B. The three fourth harmonic rays concur 

 at the required point C. In our case the rays BjC and B3C coincide, 

 as can easily be seen by passing over to the limit. As in the general 

 case of a real quadruple, they cut the fourth harmonic ray through 

 Bj at C, the point through which the asymptote passes. The pencil 

 of conies through the quadruple cuts every ray through B to the left 

 of A3 and the right of Aj in elliptic involutions, and only the rays 

 between Aj and A3 contain hyperbolic involutions. The only branch 

 of the cubic is therefore contained between two lines through Aj and 

 A3 parallel to the direction of B. The ray through A2A^ carries a 

 parabolic involution and AjA^ represents an isolated point of the cubic. 



IV. The Nodal Cubic. 



Assuming in the fundamental quadruple A, and A^ real and co- 



