ANALYTIC CONDITIONS. 159 
Hence if these relations are imposed upon the elements of M 
and M,, they are also satisfied by the elements of M.. 
In addition to the relations of equations (22) there may be 
other relations at the same time on the elements of M, M,, 
and M,. But the relations (22) are alone sufficient to insure 
that the system of collineations satisfying them has the first 
group property. 
A second family linear in the columns of Mis 
la,+mact+na;=l, 
R, :  lbitmb+nbs=m, (23) 
le; +men.+ne3 =7n. 
The sufficiency of these relations may be shown in a manner 
similar to that for R,. 
192. The Quadratic Families R,. If r=2 in our complete 
family of automorphic forms, we can write down two sets of 
quadratic relations ¢;=1; on the elements of M, one in the 
rows and the other in the columns of M@. The set in the col- 
umns of / is as follows: 
lay? + may? + na;2 + 2p ayao+ 2G aia; + 2raca; =l, 
1b\2? + mb.? + nb;2 + 2p bibs + 2qb1b3 + 2rbob;=m, 
Re: Le; + mes? + nes? + 2p C)€2+ 2G ee; + 27resxe,=N, 24 
2 + Llayb\+marzb.+na3b; +p (aibs +26) +¢q (ab; +a3b1) +7 (2b; +a3b0) =p, ( ) 
laye,+mar2es+ nase, +p (deo + d2e1) +q (Gi¢es+azc1) + 7 (Ave; +a3¢2) =q, 
Lb,e, + mboes + nb3c3 + p (dies + boe:) +. (dies + B31) +7 (boe3+ b3¢2) =r. 
The system of collineations defined by these relations has 
the first group property. The sufficiency of the conditions 
may be shown by actually transforming the relations in 4,, B,, 
etc., by V and equating coefficients; or by substituting as 
above the relations on the elements of , in those of M. 
These examples suffice to illustrate the general theorem ; 
the same consideration may be extended to complete families 
of any degree. 
193. Other Expressions for »;. We have thus far only 
one method of expressing a complete family of automorphic 
forms which may be stated precisely as follows: When a 
complete homogeneous polynomial of degree 7 in one set of 
three variables is transformed by a linear transformation 7, 
