NF.VVSON: LINEAR GEOMETRY. 93 



Again let us give to k the values i, o, CC: 



when k -i, we have u^-j-w-v=:o; 

 when k o, we have u-^-Lwv:=:o; 



2 



when k=;0o, we have U'^-' v--:o. 

 These expressions are factors of u2-]-v^=:o. Substituting the values 

 of u and v, this expression reduces to I-'* — 27]"^— o. 



This expression, P — 27J-, is an invariant of the quartic, and its 

 vanishing is the condition that two of the four points constituting the 

 ((uartic should coincide. IJut it is not u fundamental invariant of the 

 {[uartic, since it is composed of I and J. We conclude, therefore, 

 that the quartic has only two fundamental invariants, I and J. 



