M.-P— Vol. I.] DICKSON— QUADRATIC TRANSFORMATIONS. 19 

 For the X, Y coordinates of A and D we have 



A: (=* !.«); /?: (o, M 



\ a 2 / \ " «i / 



so that the equation to ^4 Z> is 



Z.= -K- 2* — -K- JT — i = c, 



the multiplicative constant in LjW of (3) being taken into 

 W. The base conic W= L is met by X= o and 2"= o in 

 respectively 



Our transformation (3) becomes for the ^, 2~ coordinates : 



u; ^ — w(X,r) ' ~~ w(x,r)  



We may verify that ^4 Z 3 and B P' meet in a point on the 

 base conic (4), From Pascal's Theorem on the inscribed 

 hexagon, it follows that 2? jP and C P' meet on this base 

 conic. We may thus invert the relations (3') by inter- 

 changing A with C, B with D , and P with P' , the base 

 conic remaining fixed, viz. : 



a 2 X 2 + \Xr+ a, T 2 —L — M—i =0 



where the equation oi B C has been written in the form 



M=B 1 r— hX— i = o. 



We have therefore to replace in (3') o\ by -*-, o 2 by ns- 

 (leaving a 2 , X. and a x unchanged) and interchange X with 



x\ r w ith t. 



Hence 

 where 



w,{x, r f )=a 2 x hi + xj' r'+ o, r' + -^ x — -£- r\ 



