160 THEORY OF COLLINEATIONS. 
the coefficients of the transformed form are polynomials in 
the elements of the matrix of 7; these constitute a complete 
family of automorphic forms of degree r. 
A complete family of automorphic forms may be expressed 
in another very convenient manner. Let us take two linear 
transformations T and T’ such that the matrix of T’ is the 
conjugate of that of T. Thus 
G=He+biy+eiz, t=a,2+ary+a32z, 
T : y=matbytez, and T’: y=baetbytbsz, (25) 
2=03%+b3y+ C32 , Z= CU +e0y+C32. 
A complete family of linear automorphic forms in the rows 
(or columns) of M may be obtained by replacing the variables 
x,y,z, of T (or T’) by l, m, n, respectively. 
A complete family of quadratic automorphic forms in the 
rows (or columns) of M is obtained by squaring the three 
equations of 7 (or 7’) and forming their products two at a 
time and then replacing «’ byl, y* by m, z* by n, xy by p, ete. 
A complete family of cubic automorphic forms is obtained in 
an analogous manner and the process holds for the family of 
degree 7. 
194. Determinant of an Automorphic Family. Associated 
with every complete family of automorphic forms is a certain 
determinant which requires attention. The determinant of 
a linear automorphic family is equal to \ the determinant of 
T or T’ (25). It is evident from the last mode of expression 
for a complete family that the determinant of the family can- 
not be independent of A. It is not difficult to prove the fol- 
lowing identity : 
a2 db 2 2a,b; Zajc, 2bicy 
As” - - - 
a;? 
ad» 
a\As 
A203 
In like manner the determinant of a cubic family is equal to 
A’; and in general the determinant of an automorphic family 
of degree 7 is equal to A to the power 7°. 
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