k-RELATIONS. 203 
238. The Insufficiency of the k-Relations. We have seen 
that the existence of the k-relations in the subgroup of G, is 
a necessary consequence of the vanishing of the function 
A(1). Hence we conclude that the existence of a k-relation 
is a necessary condition for a subgroup of G,; but we can not 
infer that it is also a sufficient condition. For if we assume 
the existence of the k-relation k,k,’=kk’k,k,/ and combine it 
with equation X, we get A,’=A’A,’ and this alone is insuffi- 
cient to restrict the group G,. 
239. Wesum up the results of our investigation on the 
k-relations as follows : 
THEOREM 29. A 4é-relation between the 4’s of two components 
and their resultant holds in reducible groups; if the subgroup is ir- 
reducible the relation is between the 4’s of each collineation in the 
group. 
240. Reduced Normal Form of G,(A). Sinee the group 
G,(A) is reducible in the general form we may expect its 
normal form to possess the same property. The geometrical 
significance of the reduced form of G,(A) is that the inva- 
riant point coincides with one vertex of the triangle of refer- 
ence. If T and 7, have one vertex of their invariant triangles 
in common, it is evident geometrically that 7, will also leave 
the same point invariant. To show this analytically we let 
T and T, have the invariant point A in common and let this 
common point be chosen as one vertex of the triangle of ref- 
erence. For example, let the co-ordinates of A be (0,0, C); 
then in the normal form of T we must put A = B=0 and in 
T,, A,=56,=0. Substituting these values in the equations 
I-[X, Art. 179, we see that the right-hand sides of III and VI 
vanish and the right-hand side of IX reduces to 
Clan “pris Calan Bil: 
We now have the three following equations from which to 
determine the values of A., B, and C.: 
A, B. As A, B, Bz 
A f B,! kp A.! = ’ A)! B,! kp Bo! — OF 
Ap! Be! klAo! Ao! Bol! Keo! Bo! 
