4 CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



Theorem I: Let A mid A' be homographic -pencils of 

 rays, and let B and B' be two other homographic -pencils of 

 rays. If to each intersection of a ray of A with a ray of B 

 correspond the intersection of the corresponding rays of A' 

 and B ', this correspondence can be represented by a quadratic 

 transformation . 



The transformation is evidently birational, and the most 

 general birational quadratic transformation can be repre- 

 sented in this way. It might indeed seem at first that the 

 position of the pencils is slightly too general, but this is not 

 the case. For, denoting the rays of the various pencils by 



(13) {A) p — \ q = ; (A') k—\l=o; 

 (B) r — /jl t = o; (B') m — /x n = o, 



it is easily shown that 



(X x — X 2 ) (p — \ q) 



= (X — X 2 ) (p—X.q) — (X— Xx) (p — \ 2 q), 



with like formulas for the other pencils. If now we choose 

 Xi, X 2 , fii, ft, 2 so that 



p — X 2 q = r — p 2 t=g 

 k — X\ I = m — fii n = h 



and furthermore write 



X — X x ix — /*! 



— = A = fx , 



A — A 2 [A — yu* 2 



then formulas (13) become 



(A) (p — \ 1 q)—\'g=o; 



(14) (A') h — X'{k — \ 2 l)=o; 



(B) ( r - n l t)—/*'g=o;' 



(B') h — fjb (m — /x 2 n) = o , 



and these are of the same form as (11) and (12). 



Theorem I can also be proved directly without difficulty 

 by the method of synthetic geometry. To a ray of A or 'of 

 B corresponds a ray of A' or of B' respectively. But a 

 straight line in general will be generated by pencils A and 

 B in perspective. The corresponding pencils A' and B' 



