NEWTON'S FIVE TYPES OP PLANE CUBICS 277 



To the line at infinitely (a?=oo , y=oo ) corresponds the circle 



..-+y'^-i 



4' 



i. e., the circle passing through B, B^ B,. 



In order to get the analytical expression for the most general 

 Steinerian transformation we simply have to apply a general collin- 

 eation to the above expressions, since by a collineation a quadrilateral 

 may be transformed into any other quadrilateral. 



From this it is easily seen that to a curve of the ti^^ order cor- 

 responds a curve of the 2/^"' order. To a straight line corresponds a 

 conic through Bj B2 B3. To the line at infinity corresponds also a 

 conic through Bj Bj B3 which moreover cuts the sides A^ A^ at their 

 middle -points. Thus, nine points of this conic are known at the 

 outset. 



2. In the Steinerian transformation cubics appear by the fol- 

 lowinor considerations: To a straight line g corresponds a conic G. 

 (I shall use j7 = 0, G = as abbreviated equations of these lines) which 

 may cut g in two real, or two imaginary points X, X', which corres- 

 pond to each other in the transformation. The same straight line 

 cuts the pencil P+\Q=0 through the fundamental quadruple in an 

 involution whose double-points coincide with X and X'. Taking a 

 pencil of straight lines, g-{-\h = 0, through a point B, then on every 

 Line of this pencil there are two points X and X' . The locus of these 

 points is a cubic through A, A^ A, A^, B, B2 B3, B and its corres- 

 pondiuo- C. The lines from B to the fundamental points are tangents 

 to the cubic. Transforming this cubic by the same transformation 

 it is found that it is transformed into itself. If the fundamental 

 quadrilateral is such that Bj B^ B3 are the foot-points of the altitudes 

 of A, Aj A3 and if B is infinitely distant, the cubic associated with 

 B is circular. In what follows I shall, for the convenience of their 

 construction, consider only such cubics. The results obtained 

 thereby may be immediately extended to the general case by collin- 

 eations. 



