M.-P.-Vol. L] HASKELL— QUADRATIC TRANSFORMATIONS. 7 



theorem, which is easily deduced from the relations given 

 above : 



Theorem III: Let a, /3, 7 be three conies of a -pencil 

 throiigh four -points, and let A, B, C be three points chosen 

 arbitrarily on a, /3, <y respectively. Let A B meet a again 

 in Qi and /3 again in P\ ; let B C meet j3 again in R\ and 

 7 again in Q 2 ; let C A meet 7 again in P 2 and a again in 7? 2 - 

 Then P\ Pi , Q1Q2 , Pi Ri are the sides of a triangle whose 

 vertices A' B' C lie on a, /3, 7 respectively. 



This construction is illustrated in Figure 2. 



A simple construction for a special case in which, as in 

 Dr. Dickson's transformation, there is a self-corresponding 

 conic, is shown in Fig. 3. A, B and C are any three 

 points on a conic. All pairs of corresponding points P 



B 



and P 1 are collinear with A. Furthermore, P B and P' C 

 meet on the conic. All points on the conic evidently cor- 

 respond to themselves. A counts twice as a vertex of each 

 fundamental triangle, B and C are the other vertices. If 

 \:fi:v are the co-ordinates of P with reference to the tri- 

 angle ABC, V : fi' : v those of P' and \x.v -\- v\ -\-\ p = o 

 the equation of the conic, the formulas of the transformation 



will be 



X' : fi ; v == fj> ( v -\- \ ) : — fi v : — v 2 



