M.-P.— Vol. I.] DICKSON— QUADRATIC TRANSFORMATIONS. 23 



Thus SiQ (which is geometrically equivalent to T= 

 S (S\ Q) S _1 ) becomes 



, , , yz xy xz 

 x : v : z = — : — f : — — . 

 a c b 



For b" = c, z'/y =z/y, and the transformation is virtually 

 that of Dr. Hirst (§ 6), the fixed conic here being 



2 b" 



x* — — y z = o. 



a 



The general case b"^.c is reduced to the latter by first 

 applying the transformation x :y: z' = x: y : az,a = b" jc , 

 which transforms any point P into a point P' such that the 

 cross-ratio 



OP . OP' 



RP * RP 



O being the vertex (x = o, y = o) and R the intersection of 

 OP with the side z — o. 



(I b ) Suppose however B'—o. Then by (4), A" = o, 

 a" C" = o. Thus C" ^io, a" = o. By (5) b" = o. Thus 

 B' = ac'^j), so that by (2), a=o, A' = 0. Thus A^.o, 

 whence by (6), b = 0. From B" = 0, ^4" = follows c' = o, 

 c = o. Hence 



1 : ,r : jy : z = a.v : b y : c ^. 



ri-,1 c /^ v. ' ' ' y z x z x y 



1 hus Oi Q becomes x : y : z = <— - : — r : ~. 



a b c 



If a, b', c" have the same sign, this is the transformation 

 of § 5 by means of harmonic conjugates. If, for example, 

 a alone be negative we apply first the transformation 

 x';y''.z' = — x:y:z, which transforms any point P into a 

 point P' separating harmonically the vertex (y = o, z = o) 

 and the side opposite, x = 0, from the point P. 



Case II: a = o. By a similar investigation we find that 

 S\ is either x : y ' : z = by :d x : c" z or x : y : z'= c z : b'y : a x, 

 as the symmetry of the problem for x,y, z would lead us to 

 expect from the result of the case (I a ). 



