newson: collineations in the plane. 19 



This group contains one transformation of order i and eight of 

 order 3. 



Theorc//i i. — Every transformation of the group G^j, transforms 

 into itself every cubic of the pencil C-f 6mH=o, where H is the 

 Hessian of C; Gj,, contains (i) one transformation of order i, (2) 

 nine transformations of order 2, (3) eight transformations of order 

 3. These are as follows: 



(i). The identical transformation. 



(2), Each of the nine points of inflection and its corresponding 

 harmonic polar are the vertex and axis respectively of a perspective 

 transformation of order 2. 



(3). Each of the four inflectional triangles is the invariant tri- 

 angle of a pair of inverse transformations of type I and order 3. 



§5. The Group G54 (1). 



As remarked above, the group Gg^g contains four sub-groups 

 Gj^Ci), (i=i, 2, 3, 4), one for each equianharmonic cubic. We 

 begin with the most simple one, which has for invariant figure the 

 triangle of reference xyz=o and the cubic x^-(-y^-)-z^=o. This 

 group Gg^ contains, of course, the eighteen transformations of 

 Gjg and hence thirty-six other transformations which we must 

 investigate. 



Since x^-|-y^-|-z3 is an invariant of our group it is evident that 

 the group contains all transformations of the form of those con- 

 tained in G,g, where x, y, and z are interchanged in all possible 

 ways and combined with the coefficients a, a^, a^, in all possible 

 ways which give rise to different transformations. We can readily 

 write down a table of all such transformations, and we find that it 

 contains just fifty-four transformations and no more. These, then, 

 constitute the group Gj^. The table is as follows, in which the 

 number placed above each formula indicates the order of the 

 transformation: 



