202 THEORY OF COLLINEATIONS. 
invariant triangle (see Fig. 25). Evidently the conic K may 
have any one of three positions with reference to the invari- 
ant triangle. Each side in turn may be the chord of contact 
with the other two sides as tangents. 
236. The k-Relation for G,(K). We found in Art. 182 
that the determinant A of the normal form of T is kk’ A’; 
we also found in Art. 210 that the determinant of T in G,(K) 
is A=1. Hence we have for every collineation in G,(K) the 
condition kk’/A’=1. When A is the invariant point of T not 
on the conic, we have A’=1; substituting this value in 
kk' A’ = 1, we get 
ih =a- (48) 
This shows that the cross-ratios along the two invariant 
tangent lines to k from the invariant point A have reciprocal 
values. In the same way when A’ (or A’’) is not on the 
conic, we get 
Hino puget aay (48’) 
These relations have the same interpretation along invariant 
tangent lines to the conic. Hence in G,(K) the cross-ratios k 
and k’ are not independent of each other. 
Equation X of Art. 1795 reduces to the identity of 1 = 1 for 
the group G,(k) and we can get from it no relation between 
the k’s of T, T,and T,. Hence for the group G,(K) the sec- 
ond k-relation turns out to be not a relation between the k’s 
of two components and their resultant, but between the k’s 
of each collineation in the group. 
237. k-Relations for GAlK) and G,(A'A”’K). The 
group G,(A/lK) is defined by sets of both linear and quadratic 
relations, hence the results of the last article and of theorem 
27 both hold. Since this group has an invariant lineal ele- 
ment, we have k,=kk, and k,’=k’k,'’; but since vs, 
we have only one k-relation, viz.: 
e—Kice = oinnular reasoning applies to the group G,(A A’A’ Kk) 
and we reach the same result, viz.: one k-relation k, = kk,. 
