96 UNIVERSITY OF COLOEADO STUDIES 



Taking a second quadruple 



four points 



x^ = 2v,-\-v' +v" —w, (34) 



Making the same construction for each of the points v^' ,v^' , vj , and 

 the four points of (33), the same points of intersection a?,, x^^ «„ x^ 

 are obtained in every case. Thus the theorems of Clebsch,loc. cit.(a), 

 follow immediately: — 



The straight lines joining any point of a cubic with a Stein- 

 erian quadruple meet the curve in a second quadruple. From a 

 single quadruple all others are obtained ly letting the variable 

 point describe the whole curve. 



The sixteen lines joining the points of any two quadruples 

 intersect each other four by four in four points of a third quad- 

 ruple. 



In §§8, 9 it will be seen what connection these theorems have 

 with the twisted curve of the fourth order and of the first species. 



§5. Case of Degenerating Cubic. 



Suppose the cubic degenerates into a straight line and a conic 

 (in Figs. 1, 2, 3, a circle). Steiner's generalized and special theorems 

 still hold. Thus, a great number of theorems may be derived of 

 which only three shall be mentioned. From the previous consider- 

 ations the proofs for these are evident. On the other hand they may 

 be easily derived from well-known projective properties. 



1. If a quadrilateral u^ii.^u^u^ is inscribed in a circle^ Fig. ly 

 and its sides by any 8t7'aight line I are cut in the four points 



