newson: collineations in the plane. 25 



The invariants of this group are the cubic and triangle (2) of 

 equations (4) and (5) as follows: 



(2) x3-]-y3-(-z3+6xyz=:o 



and (18) 



(2) (x + y + z)(x + ay-fa3z)(x + a2y4-aZ)z=0. 



The triangle and cubic (i) are transformed into triangle and cubic 

 (2) by the transformation. 



Xi=z+y-fz 

 T=Eyi=x+ay-|-a2x (19) 



Zj=X-(-a2y-(-aZ. 



Hence the group Gg^ (i) is transformed into Gg^ (2) by means of 

 the transformation T. Thus we have TS^T'^Sg, where Sg is a 

 transformation of G-^ (2). By this operation the invariant points 

 of the individual transformations in Gj^ (i) are transformed into 

 those of Gj^ (2). We shall now verify this for a few cases. 



The eighteen transformations of Gjg are contained in Gg^ (2) 

 and hence we expect that the operations symbolized by TG^g T'^ 



will give us again G,^. The transformation S= -< z leaves invari- 



(y 



ant the point of inflection (0,1, — i) and its harmonic polar y — z=o. 

 The coordinates (0,1, — i) substituted in T give again (0,1, — i). 

 From the form of T we have y — z=(a — a'^){y — z). The operation 

 TS^T-i gives Sj as may easily be verified. 



Again take the transformation S = -< ay whose invariant triangle 



(a^z 



(1,0,0 (y 



is ■} 0,1.0. The operation TS,T-i gives S' -l z; and the above co- 

 (0,0,1 (x 



( ^'^'^ 

 ordinates substituted in T gives -! i,a,a3 as the invariant triangle 



( I,a^,a 



of the new transformation. This is as it should be (see equation 

 9). In this way it may be verified in detail. that Gjg is an invari' 

 ant sub-group of Gg^ (i;. 



The perspective transformation J y, leaving invariant a vertex 



(z 

 and opposite side of triangle (i), is transformed by T into a new 

 perspective transformation, leaving invariant a vertex and opposite 



