newson: collineations in the plane. 21 



We observe that the eighteen transformations given in the first 

 three columns of the table form the group Gjg discussed above. 

 We shall find that of the thirty-six remaining transformations 

 eighteen are of order 3 and eighteen of order 6; we shall further 

 find that there are two distinct varieties of these transformations of 

 order 3. 



We first consider the last six transformations of the first row of 



the table. Take first the transformation -I yjr=y; it evidently 

 leaves invariant each side of the triangle of reference. It may also 





be Vv^ritten in the form J ay, from which we see that the cross-ratios 



( az 

 along the sides y and z are each equal to a and that along x is 

 unity. The transformation is, therefore, of type IV, the axis of 

 invariant points being x=o; and the single invariant point or vertex 

 being the point (i, o, o), the opposite vertex of the invariant tri- 

 angle. This transformation is evidently of order 3; its inverse is 



i ax 



also its square viz: -' y. In like manner it is seen thatthe two inverse 



j x, (x 

 transformations -' a'y, -< ay, are also of type IV and of order 3, 



( z, ( z . 



the axis being y:=o and the vertex being the opposite vertex of the 

 triangle of reference. A similar results holds also for the pair 



( X, ( x 



J y, J y. We have thus found six perspective transforma- 



( a^Z, ( aZ 



tions, each of order 3; these are easily identified with the last six 

 transformations of the first row of the table. 



We next consider the fourth transformation of the second row of 

 (a^x 

 the table, viz: -I z. Calling it T we have, 



(y 



a*X, aX, X, a^X, aX, X, 



T :=z, T«=.y, T3=z, T*=y, T5::^z, T6^y, = i. (15) 

 y z y z y z 



The transformation T is therefore of order 6; T^ and T* are of 

 order 3 and T^ is of order 2. T'^, T^, T* have been studied above 

 and their characteristics are already known. 



T and its inverse T^ are now to be investigated. The invariant 



