90 



UNIVEKSITY OF COLORADO STUDIES 



§3. Problems of Closure on the Plane Cubic. 



1. In 1845 Steiner published an important memoir, "Geo- 

 metrische Lehrsiitze", in Crelle's Journal, Vol. 32, pp. 371-373, con- 

 taining the now famous problem on closing polygons on a cubic. 

 Since that time a great number of geometers have studied these 

 problems'. Of all these Clebsch^ has probably made the most im- 

 portant contribution to the subject. From a purely geometrical 

 stand-point the work of Desteli, "Die Steinerschen Schliessungs- 

 probleme nach darstellend geometrischer Methode'" ranks undoubt- 

 edly among the best contributions to the subject. 



2. We shall first establish Steiner's theorem in its generalized 

 form. 



Assume n distinct arbitrary points on a plane cubic, with the 

 arguments Vj, v^, . . ., v„. For the sake of simplicity designate the 

 points themselves by the same symbols. Through Vy^ pass any straight 

 line cutting the cubic in two other points with the arguments u^ and 

 •Mj (designate these points by their arguments as before and hereafter). 



Through u^ and %\ pass another straio;ht line, cutting the cubic 

 at -w,; through ti^ and v^ pass a line, cutting the cubic at u^\ and so 

 forth; finally connect -?/„ with y„, cutting the cubic at u^j^^. The 

 jpToblem is to -find the condition for which w„^, is identical with ?/„ 

 no matter how u^ may have heen chosen. We evidently have: — 



1. 'Wi+«j,+^, = 0, 



2. u.i-\-v^-]-u., = 0, 



3. U3+V3-\-U^ = 0, 



• (23) 



/I— 1. '?^„_i+^„-i+w„ = 0, 



n. 



'The reader is referred to the interesting: and valuable pamphlet of Prof. Gino Loria: "I 

 Poligoni di Poncelet, discorso pronunziato nell' Universitia di Genova. Turin, 1889, 

 and to the list of references at the end." 



'(a) tjber einen Satz von Steiner und einige Punkte der Theorie der Curven dritter Ordnung, 

 Crelle's Journal. Vol. 63. 1864. pp. 94-121. 



(6) Vorlesnueen iiber Geometrie, Vol. I, pp. 615-627. 



'Teubner, Leipzig, 1888. 



