newson: collineations in the plane. 



15 



Between these four degenerate cubics and the four equianhar- 

 monic cubics there exists a very simple relation. Each eqiiianhar- 

 monic cubic is gotten by taking the sum of the cubes of the linear 

 factors of the corresponding degenerate cubic. Thus, for example, 

 we have 



3rx3+y3+z:^+6xyz)EEi:(x+y+z)3-|-(x+ay+tt2z)3 



-f(x + a2y-|-ax)3. (6) 



The coordinates of the nine points of inflection of the pencil of 

 cubics is best found by eliminating between x^-j-y^-j-^z^^o and 

 xyz=o. The}' are as follows: 



(l) O, I, — i; (2) o. a, — i; (3) o, I, —a: 



(4) —I, o, i; (5) —I, o, a: (6) —a, o, i; (7) 



(7) i> — 1> o; (8) a, —I, o; (9) i, —a, o. 



The harmonic polars of these nine points of inflection are given 

 by the following equations: 



(i) y— z^o, (2) a2y— z^o, (3) ay— z=o, 



(4) Z — X=0, (5) tt~'z— x=o, (6) az— x = o, (8) 



(7) X— y=Oj (8) a^x— y=o, (9) ax— y=o. 



Each of the inflectional triangles intersect the pencil of cubics in 

 the same nine points; hence the nine points of inflection lie three 

 by three on twelve right lines. Each harmonic polar passes 

 through a vertex of each of the four inflectional triangles; hence 

 the twelve vertices of the inflectional triangles lie four by four on 

 nine right lines. 



The vertices of the four inflectional triangles may be designated 

 by Aj, Bi, Ci, (i=i, 2, 3, 4). The coordinates of these twelve 

 points are as follov^s: 



I I, o, o, 



I 

 1. -\ O, I, o. 



I, I, I, I a", I, I, I a, I, I, 



I I 



I, a-, a, 3. J I, a-, I, 4. ^ I, a, I, (9) 



[ O, O, I. [ I, a, a-. 



I, I, a-. 



I, I, a, 



