k-RELATIONS. 201 
THEOREM 28. Every subgroup of type I in G; which is defined 
by linear relations “ on the elements of M, and whose invariant 
figure contains a lineal element has a double 4-relation. 
235. The Normal Form of G,(K). If we replace a,, b,, 
etc., in the relations R, by their values from the norma! form 
Of Tt Viz: 
BuiG- Ash [A CA 
€,=|B) C kA’ |, O=|A C ka’ |, ete., 
Br Cc’ kA’ | Al Cc" kl A!” 
we get the normal form of the conditions for the group G,(k). 
If these new relations be multiplied by 2’, y’, ete., and added, 
we get 
ce meen ee nO 4 Cee Ze) 10 
AvseB a Co “A ANE Gass 
Ll Boo Ra | ty B o pp |t ete. 
All Bi C’ kA" All Br Cc ki A!” 
= la? + my? + nz? + 2pxey + 2quz+ 2ryz =f, (47) 
which shows that the function f is invariant in form under 
the normal form of 7, (and also of 7') and has the same value 
at a pair of corresponding points of the plane. It is therefore 
invariant in value at an invariant point of the plane. The 
function f vanishes at every point of a certain conic K. 
We must now examine the relation of the conic K to the 
invariant triangle of T. Evidently the position of the conic 
is not independent of the position of the invariant points 
A, A’, A”. Since the properties of pole and polar with re- 
spect to a conic are projective properties, it follows that if a 
point P and a conic K are invariant under 7, the polar of P is 
also invariant. Suppose the invariant point A is not on the 
conic K; its polar pis an invariant line of T and cuts k in 
two points, B and C, which are also invariant points. But 7 
leaves invariant only these points; hence B and C coincide 
with A’ and A” respectively. The polar of A’ (or A”) which 
is on the conic is the tangent at A’ (or A’’), and this passes 
through A. Hence K passes through two vertices of the in- 
variant triangle and at these points touches two sides of the 
