(236. KANSAS UNIVERSITY QUARTERLY. 



side of triangle (2). The vertex (1,0,0) and opposite side x:^o of 

 triangle (i) are transformed by T into (1,1,1) and x-f-5^+zr=o, 

 vertex and opposite side of triangle (2). Transforming this per- 

 spective transformation by T we get 



/jxi=ax4-y+z 

 TST-i^/5yj=x+ay+z (20) 



pZi=x+y-|-ax, 



which is likewise a perspective transformation of order 3. 



We next take the transformation designated by T in §5, viz: 



(a«x 



< z. Call it Sg for convenience. Its invariant triangle was found 



(y 



ro,i,— I jo,i,— I 



to be -< 1,0,0. Substituting these values in T we get ■! 1,1,1 as 



(0,1,1 (2,1,-1 



the vertices of the invariant triangle of the transformation TSgT~^- 

 Performing the operation indicated by TSgT"^ we get 



/3Xj=ax+y+z, 

 TS6T-i=/,yi=x+y+az, (21) 



. . . pz^=x-fay+z, 



which is accordingly the transformation in Gj^ (2) corresponding 

 to Sg in G54 (i). It is easy to verify that TSgT-i is of order 6 



(0,1, — I 

 and that its invariant triangle is ■/ 1,1,1. It should be remarked 



( 2,-1 — 1 



that the point (0,1, — i) is a point of inflection, (1,1,1) the vertex 

 of the triangle (2) which lies on the polar of (0,1, — i), and 

 (a, — I, — i) is the intersection of the harmonic polar and the side 

 of triangle (2) opposite (1,1,1). (See the discussion of the trans- 

 formation Sg in §5.) In a similar manner the equations of the 

 other seventeen transformations of order 6 in Gg^ (2) may be writ- 

 ten down and their invariant triangles determined. 



We come finally to the consideration of the twelve transforma- 

 tions in Gg^ (2), corresponding to those in Gg^ (i) whose invariant 

 triangles are both inscribed and circumscribed to the cubic (i). 



Take, for example, the transformation S3 = ^ z, the coordinates of 



- (x 

 ■ " ( IA^A* 



whose invariant points are ■! i,X*,X''. (See §5.) The transforma- 



tion TS3T-* is found to be 



