REDUCIBLE GROUPS. 183 
ducible group of plane collineations is therefore a subgroup 
of G,. We wish to determine all varieties of such subgroups. 
219. Reduced Matrices of Subgroups.of G,. The matrix 
of each of the subgroups of G, enumerated in Art. 209 may be 
written in a reduced form.. For example, the group G,(A) is 
defined by the homogeneous and symmetrical relations 
la; + mb, + ne, = l, 
ie > las+mbs+ne2=™m, (22) 
la; + mb;+nce3 =n. 
If we make 1=1, m=0, n=O, in R,, these equations 
reduce to a,=1, a,=0,a,=0. The matrix M reduces to 
1 6b; cy 
0 bs Ce 
0 bs c3 
and the invariant point (1, m, 7) becomes (1, 0,0), one vertex 
- of the triangle of reference. It is easily verified that if a= 1, 
CAO 1G: 10 an dua, — lt oe—0 og — OP thenniA, —ile As — 0) 
A,=0. There are three equivalent reduced forms for the 
matrix of this group, viz.: 
’ 
Hn OG | || a 0 G4 | || a1 b, 0|| 
0 be ee | ) I a2 1 ¢2\| , |do b. O|| 
0 b3 ¢s]| la3 0 c¢s| lla3 63 1 
It may readily be verified by multiplication of matrices that 
each of these reduced matrices has both group properties, and 
represents therefore a reduced group. 
In like manner if we make /=1, m=0, n=O in the rela- 
tions R, which define G,(/), we get a,=1, b,=0, c,=0. The 
matrix M reduces to 
ie OO 
is bs C2 
b 
a3 b3 C3 
and the invariant line becomes the line, x =0, of the triangle 
of reference. Without going further into details we may 
write down at once one reduced form of the matrix of each 
of the other subgroups of G, defined by linear relations. 
it > (0) 10) Che ln (0) 
G,(Al) = & . Onle G,(A,l) = | bp oe 
Gin (Ny (5 ih \ 
