LIST OF GROUPS. 181 
and 4/e7-+u’y+r’z=0; this system of collineations does not 
form a group. 
216. List of Groups. We have now found the entire list 
of subgroups of G, of type I which are defined by linear rela- 
tions on the elements of M, or by quadratic relations, or by 
any combination of linear and quadratic relations. In addi- 
tion to the list given in Art. 209, we have three more groups 
one set of whose defining relations is quadratic, viz.: G,(K), 
G.(AlK) and G,(AA’A”K). Our list is now in exact agree- 
ment with that given in Theorem 12. We may now restate 
Theorem 12 in this form : 
THEOREM 16. The general projective group Gs has only eleven 
varieties of continuous subgroups of type I which are detined by 
linear and quadratic relations on the elements of M; these are the 
groups enumerated in Theorem 12. 
217. The Condition A(1)=0. If we have given the 
family of automorphic forms linear in the rows of M, 
la, + mb, + ne, =l, 
laz+mbs+ne.=m, ( 929 ) 
la;+mb3+ ne; =n, 
we can eliminate /, m, and n, and thus obtain the condition 
|a,—1 by C1 
A(1) = |® bo—-1 Cy = (0). (33) 
| ds b3 ¢3—1 
The family linear in the columns of M gives the same condi- 
tion. This necessary condition for a group defined by a 
linear set of relations is not a new independent condition on 
the elements of M. It is a sufficient condition for a group 
defined by a set of linear relations. For if the condition (83) 
be given in determinant form, there exists a set of numbers 
l, m, n, such that if the columns (or rows) be multiplied re- 
spectively by the numbers and the rows (or columns) added, 
each of the three sums will be zero. These sums give us at 
once the family of automorphic forms linear in the rows (or 
columns ) of M. 
