newson: collineations in the plane. 23 



( x,=a*x 

 Hence we have px^^=- — 2u2x, pyi= — 2z, /3Z,=— 2y; or Jy^^=z. 



We have seen how the two transformations T and T^ are related 

 to the point of inflection A. In like manner there are two trans- 

 formations of order 6 related in the same way to each of the nine 

 points of inflection. We have thus eighteen transformations of 

 order 6. Their equations may readily be found from their known 

 invariant triangles by means of formulas (10). These eighteen 

 transformations are all given in rows 2, 3, and 4, of the table. 



There still remain to be investigated the twelve transformations 

 found in the last six places in rows 5 and 6. Take, for example, 



(Xi=a2y 



-/yj=z and denote it by S. Taking the powers of S we find 

 (z,:=x 



a*y, a^Z, Z, a^X 



S= z, 52= x, or ax, S^=a2y=i, S and S^ are thus of 

 x, a^y, y, u^z 



order 3. We proceed to find the vertices of the invariant triangle 



x=:a2y 

 of S by solving the equations y=z. We find the coordinates of 



z=x 



the invariant points to be A=(i,\8,X*), B==(I,A^A^), C=^(i,\2,A), 

 where X is an irreducible 9th root of unity. Since i-f-X'^ -f A.3=o, 

 it follows that these points all lie on the cubic x^-fy^+^^^o. It 

 may easily be verified that the tangent to this cubic A cuts the 

 cubic again at B; the tangent at B cuts again at C; and the tangent 

 at C cuts the cubic again at A. Hence the cubic is both inscribed 

 and circumscribed to the triangle ABC. 



In order to determine the number of such triangles both inscrib- 

 ing and circumscribing the cubic, x^-j-y^-f z3=o, we write down 

 the tangent at the point x', y', z'; viz: xx'^-fvy'^-j-zz'^r^o. If this 

 passes through the point x", y", z", we have x"\'" -\-y"y'^ -\-z"z'' =0; 

 and similar results for the other points. Thus we have six equa- 

 tions, viz: 



x"x'2+y"y'^+z"z'"^=^o» x'3+y'3 -f z'3=o, 



x"'x"2+y"'y"2+z"'z"2=o, x"3-^y"3+z"3:=o. (17) 



x'x"'2-fy'y"'2+z'z"'2=o, x"'3-f-y"'3-f z'"3=o, 



from which to determine the coordinates of the invariant points. 

 These equations have eighteen solutions; hence there are six such 

 triangles. The coordinates of these eighteen points are as follows: 



