M.-P.— Vol. I.] HASKELL— QUADRATIC TRANSFORMATIONS. 5 



are then nomographic, and generate a conic. But the bi- 

 rational one-to-one correspondence in which a straight line 

 in general is converted into a conic is a quadratic Cremona 

 transformation. 



Since a pair of homographic pencils determine a conic 

 through their vertices as the locus of the intersections of 

 corresponding rays, and, vice versa, a conic determines a 

 pair of homographic pencils whose vertices are any two 

 points of the conic, we can restate Theorem I in a form 

 which may be more convenient for practical work: 



Theorem II : Let a and /3 be any two conies. Let A 

 and A' be any two -points on a, B and B' any two points on 

 /3. P and P' are corresponding points in a birational quad- 

 ratic transformation if P A and P' A' meet on a, while 

 P B and P' B' meet on /3. 



Tigl 



The construction is illustrated by fig. 1. The four 

 points, Di, D2, D 3 , D4, in which a and /3 meet are evidently 

 self-corresponding points of the transformation. They are 

 in general the only such'points, but if a and /3 coincide, all 



