M.-P.-Vol. I.] HASKELL— QUADRATIC TRANSFORMATIONS. 9 



In other words, the sides of the triangle \ , fi , v are con- 

 verted into the above three points by the transformation. 

 These points are then the vertices of the triangle Y\ Y 2 Y 3 ; 

 that is to say, they are the three points common to the 

 conies <& = o , M* = , X = <? of the inverse transforma- 

 tion. Indeed, this can readily be seen by substituting the 

 co-ordinates of these points in (6). Our first problem will 

 be to establish the equations which determine these co- 

 ordinates. 



Again, writing x 2 = o, x 3 = o in (15), we find 



J\ : Ji : yz = a x : a 2 : a 3 . 



In other words, ^ = o and X = o meet also in the point 

 ( d\ : a 2 : a 3 ). Similarly, X — o and <& = o meet also in 

 ( b\ : b 2 : b 3 ) , and <I> = o and M r = o meet in ( C\ : c$ : c% ) . 

 We have therefore found five points on each of these conies. 

 In what follows, we shall frequently denote these points 

 by the single letters^, q, r, a, b, c. 



§ 4. Determination of the equations for fi , q, r. 



We have just seen that the points fi, q, r are the funda- 

 mental points of the inverse transformation. Our first 

 problem is therefore to establish the equations for these 

 points. We shall find that it will not be necessary to solve 

 the equations, but that they appear directly in the formula? 

 of the inverse transformation. 



A comparison of the general form (15) with the reduced 

 form (3) gives us the following identities: 



a\ = fii m x «i + q x n x h + n h m x 

 bi = fii m 2 « 2 -f- q\ n 2 l 2 -f- n l 2 m 2 

 Cl = fix m 3 n 3 -f- q x n 3 l 3 -j- n h ni 3 

 2f = fix ( m 2 n 3 + m 3 n 2 ) + q x ( n 2 l 3 -f- n 3 h) -\- 

 (16) r x (l 2 m 3 -f h m 2 ) 



2 g x = fix ( m 3 ii x -f nix n 3 ) -f- q x ( n 3 l x -f n } l 3 ) + 



rx ( h m>\ + h m 3 ) 

 2 hx = fii ( nix n 2 -f m 2 n x ) + q x ( #1 / 2 -f #2 A ) + 

 f\ ( A ^2 + h mi ) 



