l6 KANSAS UNIVERSITY QUARTERLY. 



§3. The Group G216. 



It is a well known fact* in the theory of plane cubics that every 

 non-singular cubic C can be projectively transformed into itself in 

 eighteen different ways and that these eighteen collineations form 

 a group Gjg which also transforms into itself every cubic of the 

 pencil C-|- 6kH=o. We shall investigate this group in detail in 

 the next section, but here we wish to make use of the generally 

 known fact. 



It was shown in §2 that a cubic C is one of a set of twelve cubics 

 Ci (i = i .... 12) which can be projectively transformed into one 

 another. Since each cubic of the set may be transformed into 

 itself in eighteen different ways, we infer that each cubic of the set 

 may also be transformed into any other cubic of the set in eighteen 

 different ways. If this be true, there are 12.18 transformations 

 which leave invariant the set of twelve cubics. These 216 colli- 

 neations form a group Gg^^. 



The configuration of four equianharmonic cubics, four degenerate 

 cubics and six harmonic cubics — shown in §2 — is analogous to a 

 tetrahedron which has four vertices, four faces and six edges. Ac- 

 cordingly the structure of the group Ggjg is analogous to the 

 structure of the tetrahedron group G^g' which we assume as 

 known. To the identical substitution in Gjg corresponds the 

 group Gjg in Gg^g. To the four sub-groups G3 in Gjo, each 

 leaving a vertex and opposite face invariant, correspond four sub- 

 groups G^^ in Ggig, each leaving invariant a degenerate and an 

 equianharmonic cubic. To the three sub-groups G^ in Gj,, each 

 leaving invariant a pair of opposite edges, correspond three sub- 

 groups Ggg in Gg^g, each leaving invariant a pair of conjugate 

 harmonic cubics. To the invariant sub-group Gj in Gjg corres- 

 ponds an invariant sub-group G,, ^" ^sie- 



We shall now take up the study of these sub-groups of G^, g and 

 examine into their structure and determine the properties of the 

 individual transformations found in them. We shall determine in 

 particular the order and the invariant triangle of each transforma- 

 tion occurring in Ggig- 



§4. The Group Gis. 



The harmonic polar 1 of a point of inflection I is characterized 

 by the following property: Every line through I cuts the cubic. C 

 in two other points P and Q and 1 in L. The cross-ratio of 



*Cleb3ch, Vorlesungen uebcr Geonietrie, I, S. 51:2. 



