NEWSON: COLLINEATIONS IN THE PLANE. 2^ 



the transformation ■/ z is a perspective transformation of order 2; 



(y 



hence T is of order 4. T^ is the inverse of T and is given by 



T3~py,=x+a2y+z, (26) 



pZj=x-\-ay-\-a^z. 



The vertex of the perspective transformation T- is the point of 

 inflection A=(o, I, — i), and its axis is the harmonic polar of A, 

 viz: y — zz=o. Hence the other two invariant points B and C of T 

 are on the line y — z=o. To find B and C we proceed as follows: 

 Assume the coordinates of B to be (i,a,a); these satisfy y — z=o. 

 Substitute these assumed coordinates of B in T and we must get 

 again (i,a,a). Substituting we have 



pX^=:l-\-2SL 



py, = i+aa+aa2; (27) 



pz^=l-f aa^-j-aa 



whence — ; ; s^=a; solving for a we find a= ~^ . Since 



we have found two values of a, it follows that we have the coordi- 

 nates of both B and C. The invariant triangle of T is therefore 



b. I, — I 



ji ^ + 13. — i + l 3 

 given by ^ ' 2 2 " We may check the correctness 



^ — 1 3. "i-l 3 



I. 



2 



of this result by using these values in formulas (10) along with 

 k=i and k'= — i. We deduce thereby the transformation T. 



The invariant points B and C are somehow related to the har- 

 monic cubics of equation (25). We readily find that the point B 

 is on the cubic 



azEEx-^ f 3'M zH-6( ^ - ^); 



and C is on the cubic 

 b^^ 



===x3^y^+z3^ 6^ ilLKJ\xyz=o. 



The SIX points in which the line y — z=:o cuts the tjwo cybics a and 



