NEWTON^S FIVE TYPES OF PLANE CUBICS OB- 

 TAINED BY THE STEINERIAN 

 TRANSFORMATION 



By Arnold Emch 



Introduction. 



1. The transformation by which plane cubics may be studied 

 in a very simple and elegant manner was investigated by Steiner in 

 his " Systematische Entwickelungen, etc."^'^ 



Without giving it Steiner's original form, the transformation may 

 be defined as follows: Let P-f-\Q=0 represent a pencil of conies 

 whose fundamental points AjA^AgA^ may be all real, or all imaginary, 

 or two real and two conjugate imaginary. Now it is known that the 

 polars of any point X with respect to the conies of the given pencil 

 are concurrent at a point X'. Thus, in general, to every point X of 

 the plane of the pencil corresponds a point X ' . This correspondence 

 is called the Steinerian transformation and is evidently involutoric. 



In Fig. 1 the fundamental quadruple has been assumed entirely 

 real, in such a manner that AjAjAg is an equilateral triangle and A^ 

 the point of concurrence of its altitudes. This special assumption 

 has no bearing upon the subsequent reasoning. In this case the 

 diagonal points BjBjBg of the quadruple are the foot-points of the 

 altitudes. 



To a point Q of a side A^ A^ corresponds the harmonic point Q' 

 on this side with respect to A; and A., so that (QQ'AiAj)= — 1. To 

 a diagonal point B, corresponds every point of the opposite diagonal 

 B.^ B3. The points A are self-corresponding. Otherwise the corres- 

 pondence in the whole plane is uniform. 



Assuming AjA^ as the x-axis and the perpendicular through 



(') See also his "Werke," Vol.1, pp. W-421 and M. Disteli: Die Metrik der circularen 

 Curven dritter Ordnungr im Zusammenhangr mit geometrischen Lehrsatzen Jakob 

 Steiners. Also Poncklet: Trait6 des propri6t6s projectives dea fiffures, 1 ed., 1822. 

 p. 198, and Transon (projection gauche. Nouv. Ann, II, 4 & 5). 



