30 KANSAS UNIVERSITY QUARTERLY. 



b are in involution, and the invariant points B and C, one on each 

 cubic, constitute the double points of the involution. The other 

 two points on each cubic form pairs of points in the involution. 

 The transformation T^ evidently has the same invariant triangle 

 as T. 



We have now seen how the two transformations T and T^ of 

 order 4 are related to the point of inflection A and the pair of har- 

 monic cubics a and b. It may be shown in the same way that 

 there are two transformations of order 4 related in the same manner 

 to each of the nine points of inflection and the cubics a and b. 

 The invariant triangle having one vertex at any given point of in- 

 flection may easily be found and the corresponding transformations 

 written down by means of formulas (to). In this way we see that 

 there are eighteen transformations of order 4, each of which leaves 

 invariant the pair of harmonic cubics a and b. These eighteen 

 transformations, together with G,g, constitute the group G35 

 which we set out to investigate. 



§8. The Groups Gm (2) and Gs, (3) 



The three groups G3g(j) (j^i,2,3) are eqivalent sub-groups of 

 Gg,g and hence are similar in structure. Knowing the structure 

 of Gjg (i) we infer at once the structure of the other two groups. 

 Take the group whose invariants are the pair of harmonic cubics 



c^x'-j-y'-f-z^+dal |xyzr=ro, 



d^x^+y»-fz=^-f6a^ ~^~^^^ jxyz^o. 



and 



The eighteen transformations of order 4 are distributed so that two 

 of them correspond to each point of inflection. The invariant tri- 

 angle of such a pair of transformations consists of a point of inflec- 

 tion A and the pair of double points in the involution which the 

 harmonic polar of A cuts from the two cubics c and d. 



We shall here follow out one example. Take the point of inflec- 

 tion A=(o, I, — i); its harmonic polar is y — z=o. The double 

 points of the involution cut from the cubics c and d by y — z=o are 



fo 



und to be B = i, a^ ^'^"^ ^ ], J ^ ;^ ' ^-^ and 



