newson: collineations in the plane. 31 



B is on the curve d and C on c. By means of formulas (10) we 

 find the corresponding inverse pair of transformations to be 



pXjr^a^x + y-l-z faSx-fy-j-Z 



pyj=ax4-y+az and j ax-fay+z (^28) 



pZ,r=aX+ay-j-Z [ aX + y + ciZ. 



All other transformations of order 4 in this group may be obtained 

 in the same way. 



The group Ggg (2) may also be obtained from Ggg (i) by trans- 

 forming the latter by some transformation that changes the har- 



( ax 

 monic cubics a and b into c and d. ^^-^ Y is such a transforma- 



tion. By substituting the coordinates of the vertices of the invariant 

 triangles of Ggg (i.) in S we obtain those of Gg^ (2). The opera- 

 tion STS"' applied to the transformations of Gjg (i) give those of 

 G38 (2). 



The group G3g (3) may be obtained in the same way from 



^3 6 (O by using the transformation-/ y with the transformations 



( z 



of Gjg (i); a detailed discussion is not necessary. 



Theoretn j. — Every transformation of the group GggCj) (3=1,2,3) 

 leaves invariant a pair of harmonic cubics j; G3g(j) contains the 

 eighteen transformations of G^^ and also eighteen others of type I 

 and order 4. Each of the nine triangles formed by the point of 

 inflection k and the double points of the involution which, the har- 

 monic polar of k cuts from the pair of harmonic cubics j is the 

 invariant triangle of a pair of inverse transformations of type I and 

 order 4. 



§9. The Group G72. 



The three groups G3g(j) (j=i,2,3) contains in all fifty-four trans- 

 formations of order 4. These, together with the eighteen of the 

 group Gjg, constitute the group G,g. This group G,g contains 

 therefore no new transformations; accordingly we shall not con- 

 sider this group at length. These fifty-four transformations of 

 order 4, added to the 162 determined above and found in the four 

 groups Gg 4 (i) (i=i,2,3,4), give the 216 transformations of Gjig- 



