M.-P.— Vol. I.] DICKSON— QUADRATIC TRANSFORMATIONS. 21 



The conic which T transforms into this straight line is 

 L { (y — b)—w{x — a) \ = 1{C — M). 



These two conies meet in four points given by 



(Z — M) { {y — b) — m{x — a) — l} = o, 



two being the intersections of the straight line with the base 

 conic and two lying on OK, viz.: O itself and, say, H. 

 The point H' in which the given line intersects OK and H 

 mutually correspond, so that an involution is marked out 

 on OK. 



The compartments of the plane under the transformation 

 T given by (3) and (31) are given in the accompanying 

 figure. 



The line BC transforms into a conic tangent to the base 

 conic A B CD at B and C and tangent to OK at O. The 

 conic transforming into A D is tangent to the base conic at 

 A and D and to OK at O. 



The line at infinity transforms into the conic C = jL (lying 

 at the right in the figure) whose tangent at O is parallel to 

 AD (L=o). The conic C = M (lying at the left) with 

 its tangent at O parallel to B C (Af^o) transforms into 

 the line at infinity. 



