1 4 CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



I\, lo Iz. Thus if the latter are distinct points, we can find 

 three lines 



Z=flix + q x y-\-nz=o 



(1) ri=.fax-\-q 2 y-\ r r%Z=o 



%=f&-\- q z y-\-rsZ=o 



which T~ x transforms into the degenerate conies Li'Lz=o, 

 L\'Lz=o, L\-Li-=o respectively, where L\ (,v, y, z)=o is 

 the line joining I 2 with 7 3 , etc. We may write, after sol- 

 ving (1), 



A (*,y', z) =\ (f, v, (i= 1, 2, 3). 



Hence 



(2) p^=p(J> l x'J r q l yJ r r x z')=p l U+q x V+n W=L % ' Z 3 =X 2 - \ 3 , 



and similarly for 77' and £". Thus T written in the coordi- 

 nates £, r], £ becomes 



(3) f : V '• f = X 2 X 3 : Xi X 3 : Xi X 2 , 



which we denote as the transformation T, geometrically 

 identical with T. If S denotes the linear transformation 

 [see (1)]: 



p£=$ix-\-giy + ri0, etc. , 

 we have T= S" 1 T S ; 



for S~ x transforms the point (f :??: £) into (x:y.z), which 

 T transforms into (x : y: z'), which finally S transforms to 

 (I '• V : £")> the intermediate eliminations being expressed by 



To obtain a further reduction, denote by Si the linear 

 transformation 



r=xi (i, v , 0, v=^2 (f, if, 0. ?=n (*,*, 0. 



Then -S'i -1 Z 1 becomes the normal type of transformation Q: 



1 l 1 



f : 7] : £'=7; £: £ £ : f 77 = -^- : — : -^ 



» 



as we may readily verify that T = -^i Q. 



The most general quadratic Cremona transformation with 

 three distinct principal points may thus be written 



T= S SiQS- 1 . 



