M.-P.— Vol. I.] DICKSON— QUADRATIC TRANSFORMATIONS. 1 7 



The point P' (£': rj : £") corresponding by the construction 

 to P (£: ?/ : £) is readily found to have the coordinates 1 



The vertices and sides of triangle A O D are transformed 

 into the sides and vertices of triangle COB. Repeating the 

 transformation, we find for T\\ 



r_f,*=,,r- — r <»* + '"> , 



a c a 



the proportionality factor in 71 dropping out of 71 2 . 

 The point which 71 2 leaves fixed are given by 



viz., the base conic and the line O K joining O to the inter- 

 section K of A D and B C, upon which therefore an invo- 

 lution is marked out by 71. For the special case d = c = o, 

 when C coincides with A, B with D, we have Hirst's 

 Construction 2 , by which P and P' are conjugate points 

 with respect to the base conic; thus 71 2 = 1 in this case. 

 In general, if S' is the linear transformation 



£'•' V ' £'= 17 : f : — (b% -\- cq + d £) 



we find that 71 S' reduces to the normal type Q. 



7. Investigation of the most general quadratic Cremona 

 transformation , 



,_ U (x,y) ,_ V (x,y) 

 J ' X — W{x,y) ' y — W{x,y) 



such that corresponding joints P : (x,y) and P : (x',y) are 



l l exclude the trival case a = o, since then the whole plane is transformed into BC, a 

 part of the degenerate base conic. Also I exclude d—o, when O is on the base conic. If A 

 and D have coincided (viz.: at O), every point of the plane is transformed into the base 

 conic; if C and B have coincided (viz.: with O), every point is transformed with the single 

 point O. Not trivial are the cases when D and C, or when A and B, coincide at O. The 

 two cases are quite analogous. For either, the transformation and its inverse have but 

 two distinct principal points. The formulae are readily obtained, taking for example 

 A O B as triangle of reference. 



2 Cf. Scott, "Modern Analytical Geometry," pp. 219-224. 



