152 THEORY OF COLLINEATIONS. 
Next let us suppose that ~, contains only 7, i<j, of the 
parameters /;. From », we are able to generate a set of 7 of 
the equations ~;=/,. No equation in the set 9,;=l; contain- 
ing a parameter not in }, can be generated from the 7 equa- 
tions generated from 7. This smaller set of 7 equations is a 
self-generating set and also defines a system of collineations 
having the first group property. 
Hence the set of equations ¢;=1; is a self-generating set or 
contains within itself one or more self-generating sets »;=1;. 
Thesystem of collineations defined by ¢;=/; has the first 
group property; and if the set of relations FR is identical 
with this subset ¢;=/;, the two systems of collineations are 
the same. But if the set R contains more relations than 
~;=l,, then the system of collineation defined by the subset 
is the larger and contains as a subsystem the system of col- 
lineations defined by the set R. We shall confine our atten- 
tion for the present to the larger system of collineations 
defined by ~;=1/;. We can now state the following theorem : 
THEOREM 8. The set of relations R which defines a system of 
collineations within Gs, having the first group property. contains a 
subset in the form of a set of functions of the elements of the matrix 
M, each equated to a constant, and each of these constants occurs 
among the parameters of the functions. 
185. The Functions o;. Our next concern is to deduce the 
properties of the functions 7;. These functions must satisfy 
two conditions, if the set of equations ~;=/; define a system 
of collineations having the first group property. 
lst, they must be unchanged in form, or as we shall say 
automorphic, under the linear transformation U (or V); and 
2d, the parameters of the transformed functions must re- 
produce in some order the original functions and no others. 
The theory of linear transformation leads us at once to the 
most general class of functions that satisfy the first condition, 
viz., that are automorphic under linear transformation. They 
must be functions only of homogeneous polynomials in the 
elements of M. Since our transformation U (or V) involves 
