newson: collineations in the plane. '27 



pxj=z+ay-fz, 

 TST-i^py,=a2x+a3y + z, (22) 



pZ J =a2x-(-ay-|-aZ. 



The coordinates of the vertices of the invariant triangle of TS3T-' 

 are found by substituting in T the coordinates of those of S3. We 

 thus find 



J i+X5 + \^ I4A8 + AS I+X2-I-A , 

 ( I+X2-I-A , iH-A«+X\ i+X»+AS 



as the vertices of the invariant triangle of TS3T"^. This triangle 

 is both inscribed and circumscribed to the cubic (2). There are 

 six such triangles and their vertices may be found in a manner 

 similar to the above. The equations of the tvv^elve transformations 

 corresponding to these triangles may be obtained by transforming 

 the twelve corresponding transformations in Gg^ (0 ^Y means of T. 

 This completes the discussion of the group Gg^ (2). The groups 

 Gg^ (3) 3.nd G54 (4) may be treated in a similar manner. The in- 

 variant triangle xyz^o of Gg^ (i) is transformed into the invariant 

 triangle of Gg^ (3) by the transformation 



/)XjZ=ax-j-y4-z, 

 T,^pyi==x+ay+z, (23) 



pzi=x+y+az, 



and into the invariant triangle of Gg^ (4) by the transformation 



pXi=a2x + y+z, 

 T2^py,=x-{-a3y-f-z, (24) 



pZi=x + y+a2z. 



Hence, if we change T into Tj in the above discussion, all of the 

 results thus obtained hold true for the group Gg^ (3); replacing T 

 by Tg we get the corresponding results for Gg^ (4). These four 

 groups Gg^(i) (i=i, 2, 3, 4) contain 4.36-f 18=162 different transfor- 

 mations. 



Theorem 2. — Every transformation of the group G3g(i) (1=1,2,3,4) 

 transforms into itself the inflectional triangle i and its correspond- 

 ing equianharmonic cubic. G3g(i) contains . (i) the eighteen 

 transformations of G, g, (2) six of type IV and order 3, (3) twelve 

 of type I and order 3, eighteen of type I and order 6. 



(2). Each vertex and opposite side of the inflectional triangle i 



