32 KANSAS UNIVERSITY QUARTERLY. 



§10. Conclusion. 



The transformations contained in the group Gjig may be divided 

 into three classes as follows: (i) the eighteen transformations be- 

 longing to the Group G,g; (2) the 144 transformation in the four 

 groups G5^(i) and not belonging to Gjgj (3) the fifty-four trans- 

 formations in G^g not belonging to G,g. 



The cubics of the pencil C-f 6kH=o are distributed into sets of 

 twelve each, such that each set of twelve cubics is an invariant of 

 the group Gg^^. Each set of twelve is divided into four sub-sets 

 of three cubics each; thus: m^, amj, a^mj, (i=T,2,3,4) where the 

 m's are the roots of equation (3). 



The effect of a transformation of the first class is to transform 

 every cubic of the pencil C-f 6kH=o into itself. The effect of a 

 transformation of the second class is to cyclically interchange the 

 three cubics of one sub-set in each set of twelve and to cyclically 

 interchange the other three sub-sets. As a special case of this one 

 of the equianharmonic cubics is invariant and the other three are 

 cyclically interchanged; also one of the inflectional triangles is 

 invariant and the other three is cyclically interchanged. The three 

 pairs of harmonic cubics are cyclically interchanged by a trans- 

 formation of this class. A transformation of the third class inter- 

 changes by twos the four equianharmonic cubics and also inter- 

 changes by twos the four inflectional triangles. It leaves invariant 

 one pair of harmonic cubics and interchanges the other two pairs. 



Since the Hessian of a cubic is a covariant of the cubic, every 

 transformation that leaves a cubic invariant must leave its Hessian 

 also invariant. Thus every transformation of the first class leaves 

 both cubic and Hessian invariant. The Hessian of an equianhar- 

 monic cubic is its corresponding inflectional triangle. These are 

 invariant together under a transformation of the second class. 

 The Hessian of a harmonic cubic is the other harmonic cubic of 

 the same pair; these are invariant together under a transformation 

 of the third class. 



