162 THEORY OF COLLINEATIONS. 
Hence the system of collineations satisfying equations (24) 
has both group properties. 
This method is applicable in general to any set of equations 
9;=1, where ?; is a complete family of automorphic forms of 
degree 7 in the elements of the rows or columns of M. For 
every set of relations R, given by ¢;=1; there is a correspond- 
ing set R,’ given by ¥;=A’l;, where J; is obtained from ¢; by 
substituting for the elements of M their cofactors in the mat- 
rix conjugate to M. Hence the system of collineations sat- 
isfying a set of relations ¢;=/; has both group properties. 
THEOREM 10. Every set of relations ¢; =1;, where¢; is a com- 
plete family of automorphic forms in the elements of the rows (or 
columns) of M, implies another set ¢; = A7l; where ¢; is a complete 
family of automorphic forms in the elements of the columns (or 
rows) of M-?. The first set of relations ¢; = 1; establishes for the 
system of collineations satisfying them the first group property; the 
second set ¢; = Atl; establishes for the system the second group 
property. 
196. Analytic Conditions for a Subgroup of G,. Weare 
now in position to determine the form which the relations R 
must assume in order to define a subgroup of G,. We have 
shown in Theorem 9 that the functions ¢; must constitute a 
complete family of automorphic forms of degree rv. It fol- 
lows that among the relations R which define a subgroup of 
G, there must be a complete family of automorphic forms 
each equated to the corresponding coefficient of its family. . 
THEOREM 11. A necessary and sufficient condition for the ex- 
istence of a subgroup of G, is that the elements of the matrix M sat- 
isfy a set of equations ¢; =1; consisting of a complete family of 
automorphic forms in the elements of the rows or columns of M, 
each equated to the corresponding coefficient of the family. 
