14 KANSAS UNIVERSITY QUARTERLY 



:<^(k). (2) 



[(k4-i)(k— 2)(k— ^)]2 (8m«+2om3 — 1)3 



For ail}' given value of k this is an equation of the twelfth degree 

 in m; hence there are twelve cubics in the pencil, all having the 

 same cross-ratio k. Given any cubic of the pencil it can be pro- 

 jected into itself and into only eleven other cubics of the pencil. 

 ""Expanding equation (2) we have 



C3) 

 64(1— <^)m' 2—64(3 + 5</,)m« + 192(1— 2<^)m«— 8(8— 5<^)nv'^—(^^o. 



The solution of this equation depends upon that of a quartic; its 

 roots may be written mi, ami, a^T^i, (i=ij 2, 3, 4) and a^^^i. Hence 

 our set of twelve is composed of four sub-divisions of three cubics 

 each. 



For certain special values of k our system of twelve cubics re- 

 duces to a smaller number. Thus when k= — a or — a^ we have 

 m=o, i,a,a"; here the three cubics of each sub-division ha\'e coin- 

 cided and our twelve cubics have reduced to four. These four 

 cubics are called the cquianJiannouic cubics of the pencil. Their 

 equations are 



(i") x^-|-y^+z2^=o, 



(2) x^-[~y^+z3-|-6xyzr^o, 



(3) x=^+y3+z3 4-6axyz=o, (4) 



(4) x3-f~y^H"Z^+6a"xyz=o. 



When k= — i, 2, \, the twelve values of m reduce to six; viz: 

 I ±13 



m: 



■i^'^y-^-')- ^^^^ 



These six cubics are called the iianiionic cubics of the pencil. 



When k=i, o, (do, we have m=oo , — \, — l^j — \o-^- These four 

 cubics are characterized by the common property that each of 

 them breaks up into three linear factors. Thus 



(i) xyz3=o, 



(2) x-'^-fy^+zS— 3xyzE^(x+y-f z)(x+ay+aSz)(x+a«y+az)=:o, 



(3) x3+y3^z3— 3axyzEEE(ax+y + z)(x-fay+z)(x-fy+az)=o, (5) 



(4) x^+y^-f-z^ — 3a"xyz^(a2x+y-(-z)(x-f a2y+z)(x-(-y-|-a2z)=o. 



Each of these degenerate cubics consists of three straight lines 

 which form a triangle, one of them being the triangle of reference. 

 They are the inflectional triangles of the pencil of cubics. 



