28 KANSAS UNIVERSITY QUARTERLY. 



are the vertex and axis respectively of a pair of inverse perspective 

 transformations of order 3. 



(3). Each of the six triangles both inscribed and circumscribed 

 to the equianharmonic cubic i is the invariant triangle of a pair of 

 inverse transformations of type I and order 3. 



(4). Each of the nine triangles, k, (k;=i . . 9), described below, 

 is the invariant triangle of a pair of inverse transformations of type 

 I and order 6. A point of inflection k and its corresponding har- 

 monic polar form one vertex and the opposite side of the invariant 

 triangle; that vertex of the inflectional triangle i which is on this 

 harmonic polar and that side of the triangle i which passes through 

 the point of inflection k complete the invariant triangle. 



§7. The Group G36 (1). 



It was pointed out in §2 that the group Ggig contains three 



equivalent sub-groups G3 g(j) (j=i,2,3), one corresponding to each 



pair of harmonic cubics in the pencil C-|-6kH=o. We first take 



up the group Ggg (i), which leaves invariant the pair of cubics 



— I ±1/3 . 

 given by m== — in the pencil x^-f-y^z^-f-bmxyz^o. 



We learned in §6 that the transformation 



pXir^x+y-j-'z, 

 T^py^^z+ay+a^z, 

 pZj^x-fa^y-faz, 



transforms the cubic x^-\-y^-\-z^=o into x^-fy'+2^+6xyz=o. If 

 we make the substitution T in cubic (2) we find that it is trans- 

 formed into (i); thus T interchanges the two equianharmonic 

 cubics (i) and (2). It may also be verified that T interchanges 

 the equianharmonic cubics (3) and (4). If, however, we make the 

 substitution T in the pair of harmonic cubics 



x^+y^+z3-f6mxyzrz::o, (m— ~^~^ ^ ) (25) 



we find that both of these cubics remain invariant. Thus T is a 

 transformation belonging to the group GjgQ), since it leaves invari- 

 ant a pair of harmonic cubics. 



Since T interchanges the equianharmonic cubics (i) and (2), T' 

 must leave both of them invariant; hence T^ is a transformation of 



the group G^^. We readily find T^=-/ z. We found in §4 that 



(y 



