186 THEORY OF COLLINEATIONS. 
It remains to be pointed out that the groups G,(A/K) and 
G,(A A'A"K) are each reducible and that G,(K) is irredu- 
cible. We have thus far found two irreducible groups, viz: 
G, and G,(K). It will be shown later that these are the only 
irreducible groups of plane collineations. 
221. The Orthogonal Group. The group G,(K) is 
irreducible and hence its matrix can not be brought to a 
canonical form containing two or more zeros; but nevertheless 
its defining equations R, may be reduced to certain canonical 
forms. These we now proceed to find. 
The group G,(K) leaves invariant in form the quadratic 
function 
fealv’t+ my + nz + 2pxry + 2quz+2ryZ. 
From the theory of quadratic forms we know that f may be 
brought by linear transformation to the form 
ety t 2. 
In this case the triangle of reference is one of the self-polar 
triangles of the invariant conic f=0. We may, therefore, 
without loss of generality set 1 =m =n =1 andp=q=r=0 
in the relations R,, equations (24). The relations R, reduce to 
a,?+ a.2+a32=1, a,b, + dsb. + a3b3 = 0, 
b,? + bo? + b3? =1, Q\C; + AxC2 + a3¢3 = 0, (36) 
cy? + ¢22 +3? =1, biey + bees +b3¢3 = 0. 
These are the well-known relations on the elements of M 
which define the orthogonal group in three variables. 
For the same values of the constants /, m, etc., equations 
(27) reduce to 
A teu 13 (nt A Ag Bibs Cis = 0) 
A,?+ Bo? + Co? = 1, A,\A3+ B\B;+ OC; =0, (36’) 
A;?+ B3?+ C3? =1, Av,A;+ BoB; + C.C3=0; 
equations (27’’) reduce to 
A,?+ A2?+ A? =1, A,B, + A2B2+ A3B3=0, 
B+ B24 B;?=1, A,C, + AsC2 + A3C; =0, (36” ) 
C2 + C.2 + Co” =1, B,C, + BC» + B3C3 =0; 
and (27) become 
a,?+6,?+¢,*=1, ad. + bbs + c\¢2 =0, 
as? + bo? + co? = 1, aa; + 6163+ ce; =0, (ea0uee) 
24 6.24 ¢,?= 1, Qo03 + bobs + coc; =0. 
