THE QUADRATIC CREMONA TRANSFORMA- 

 TION. 



BY LEONARD E. DICKSON, P«. D., 



Instructor in Mathematics, University of California. 



I. The importance of the quadratic Cremona (i. e., 

 birational) transformation is enhanced by the fact that any 

 Cremona transformation of arbitrary order can be obtained 

 by applying a succession of quadratic ones. The theorem 

 that the deficiency of an algebraic curve is unchanged 

 under a Cremona transformation thus requires proof only 

 for quadratic transformations. 



A quadratic transformation T 



x :y ' :z'=U (x,y, z) : V (x, y, z) : W (x, y, z) 



will be birational (i. e., to any point x':y':z' will corre- 

 spond but one point x:y:z) if and only if the three conies 

 U=o, V=o, W=o have in common three points of inter- 

 section, say I\, f 2 , Is, called principal points. The most 

 general correspondence thus required between the straight 

 lines of one plane and the conies through three fixed points 

 in a second plane may be obtained geometrically by 

 Steiner's Construction. An arbitrary line of the first plane, 

 together with two fixed lines a and b not lying in either 

 plane, determine an hyperboloid of one sheet which is cut 

 by the second plane in a conic passing through three fixed 

 points, one on a, one on b, and a third on that line of the 

 first plane meeting both a and b. 



In applications to the resolution of higher singularities of 

 plane curves, we desire geometric constructions in a single 

 plane. 



2. Analytic reduction of T to a normal type Q. 



By a proper choice of f\q:r, the line px J r qy J r rz=o is 

 transformed by T" 1 (the inverse of T) into a conic p U-\- 

 qV-\-rW=o which breaks up into a pair of lines through 



[ 13 1 January 31, 1898. 



