THE KANSAS UNIVERSITY 

 SCIENCE BULLETIN. 



VOL, IX, No. 6.] DECEMBER, 1914. [^SSSIe 



On Cubic Surfaces and Their Nodes. 



BY S. LEFSCHETZ. 



1. The present paper is chiefly concerned with the development 

 of a construction of non-ruled cubic surfaces given by R. Sturm ( x ) 

 It is first shown rapidly how from this construction the whole theory 

 of the surface can be obtained by applying the principle of corres- 

 pondence, then the modifications necessary to obtain the 21 different 

 species are given. While very few of the results obtained are new, 

 the general treatment is, as well as the use of synthetic methods to 

 obtain constructions of nodal cubic surfaces. Furthermore, these 

 processes are susceptible of extension to surfaces of higher order, 

 but we reserve this for a later occasion. 



2. Before entering upon our main subject we will treat the 

 following problem: To find the number of common bisecants to two 

 arbitrary twisted curves. ( 2 ) Let R m , R n be the two curves, m, n their 

 order, 8 m , 8 n the number of double edges of their projecting cone 

 from an arbitrary point in space. It will be sufficient for our purpose 

 to consider the case where three of the common bisecants are tri- 

 secants for either curve. Let then A be a point on R m , from which 

 we draw one of the bisecants of R72 that go through it, and meet 

 R n in a' and a". Through either of the latter points draw one of the 

 bisecants of R m , which will meet it in 2 points, one of which we will 

 call R. To every point A there correspond %8 n points a, and there- 

 fore 4>8 m 8n points R. Also to every point R there correspond all 

 the points in which R n meets the cone of order (m — 1) projecting 

 R m , from R, and they are n(m— l)in number. Similarly to every 



Received for publication May 17, 1914. 



1. Flachen dritter Ordnung. p. 338. 



2. Another solution given in Basset Geometry of Surfaces, p. 230. 



(69) 



