156 THEORY OF COLLINEATIONS. 
Therefore when the set of functions f; is transformed by T, 
the coefficients of each of the transformed functions repro- 
duce precisely the original functions f;. The process therefore 
closes and the set of functions f, is identical with the set f.. 
188. A Complete Family of Automorphic Forms. This re- 
sult is independent of the degree v.of the function f with 
which we started, and it is also independent of the numerical 
values of the coefficients of f/ Such a set of functions f; with 
arbitrary coefficients therefore exists for every positive in- 
teger r. The set of functions f; must have the following 
properties : 
1st. Each function of the set must be of the same degree 
r in the combined three sets of variables. 
2d. Each function of the set must be symmetrical in each 
of the three sets of variables. 
3d. The set of functions must contain all the functions ob- 
tainable by all possible combinations of the three sets of va- 
riables consistent with the degree r, viz. : “ tO +*) 
Ath. The corresponding terms of each function of the set 
must have the same coefficient. 
A set of functions having these properties we shall call a 
complete family of automorphic forms. 
THEOREM 9. The most veneral set of functions f of degree r in 
three sets of three variables each which satisfy these two conditions. 
viz.: (1) that they are automorphic under a linear transformation 7 
in three sets of cogredient variables. and (2) that the coefficients of 
the transformed functions reproduce just the original functions. isa 
complete family of automorphic forms of degree r. 
189. Examples of Complete Families. We give a few ex- 
amples of complete families of automorphicforms. Let r= 1; 
in this case we have three linear functions of three terms 
each in three sets of variables, the coefficients of the corre- 
sponding terms being the same in all three functions, thus 
le +my +nz , 
fk > lal+my! +nz! , (20) 
le’+my"+n2". 
