ON RATIONAL QUADRATIC TRANSFORMA- 

 TIONS. 



BY M. W. HASKELL, Ph. D., 



Associate Professor of Mathematics, University of Calif ornia. 



In my inaugural dissertation 1 certain quadratic trans- 

 formations played an important part. Their generation by 

 the combination of conjugate imaginary collineations (in 

 point co-ordinates) was a novelty that seemed capable of a 

 generalization which I sought in vain at that time. It was 

 also desirable to invert the transformations. This inversion 

 was accomplished for some of the transformations, but not 

 for all, although it was evident that they were really bira- 

 tional. I had always intended to investigate the inversion 

 of quadratic transformations in general, but a press of other 

 work had caused the subject to be abandoned until very 

 recently, when Dr. Dickson's interesting construction* of a 

 special transformation recalled my interest to this field. 



The main object of this paper was to have been the in- 

 version of the general quadratic transformation, which is 

 accordingly presented at the close of the paper. But the 

 method of treatment led me also to the generalization men- 

 tioned above, furnishing a simple geometrical construction 

 for the general transformation, and this is to be found in § 2. 



§ 1. Transformations in the reduced form. 

 That a quadratic transformation 

 (1) pyi=(f) (x u x 2 , x 3 ), pyz=^r (xi,x 2 , x 3 ), pyz=X (#11 #2, *s) 

 shall admit a rational inversion, it is necessary and suffi- 



1 "Ueber die zu der Curve: X 3 /x + /iv 4- i/ 3 A = o im projectiven Sinne 

 gehorende mehrfache Ueberdeckung der Ebene." — American Journal of 

 Mathematics, Vol. XIII, 1890. 



2 L. E. Dickson, "A quadratic Cremona transformation defined by a 

 conic." — Rendiconti del Circolo Matematico di Palermo, T. IX, 1895. See 

 also his paper in the current volume of the Proceedings of the California 

 Academy of Sciences. 



L 1 ] January 31, 1898. 



