182 THEORY OF COLLINEATIONS. 
We may also eliminate /, m, , p, etc., from the quadratic 
family and obtain the condition, 
a,2—1 b;2 C)2 2a,b, Zajc, 2bic; 
as? b;2-1 - - 
a3" = = = = = 
BN (18) : ; _1=0. (35) 
aa; : - Z = 
Ao; 
This condition factors into 
a,—1 by C} a,+1 b; Cy 
(A*t—7 | do bs—1 C2 , | ae bo+1 Co ; 
) | ds bs ¢3—1 a3 bs e3;+1 ( 30 ) 
=(A+1) (A—1) A(1) A(—1)=0. 
Here we must discard the first and last factors, since they 
come from the introduction of the transformations 7, Art. 
181, when the quadratic family is formed by the process of 
Art. 193. Since \—1 always vanishes for the relations R,, 
we cannot assert here that A(7Z) will also vanish. This ques- 
tion will be settled in § 6. 
C. REDUCIBLE GROUPS AND CANONICAL FORMS OF GROUPS. 
218. Reducible.Groups of Plane Collineations. We shall 
now introduce the important conception of reducible groups 
of collineations. Groups of collineations may be divided into 
two distinct classes, reducible and irreducible. If a group G 
has the property that for each collineation in the group cer- 
tain elements of its matrix not in the principal diagonal are 
always zero while the determinant of the matrix does not 
vanish, the group is said to be reducible. Furthermore 
every group G’, equivalent to G according to the formula, 
G’=S~'GS, where S is any collineation with non-vanishing 
determinant, is also said to be reducible. 
Each group G’ in the infinite system of equivalent reducible 
groups may by a suitable transformation of coordinates be 
brought into the form of G having certain zero elements in 
its matrix, which therefore may be called the reduced form 
of the reducible group. The general projective group G, is 
according to the definition not a reducible group. Every re- 
