LINEAR RELATIONS. IL: 
many conditions on the elements of M. If a relation Sal=0 
exists between the constants of R, and R,, the number of in- 
dependent conditions on M is decreased by one. The number 
of varieties of subgroups of G, defined by linear relations is 
therefore a definite finite number, since the number of such 
combinations is a definite finite number. The following 
groups are defined each by its proper set of relations: 
G,(A) defined by R,. 
G, (1) ‘é 66 R,. 
G,(Al) eee eh AN Owo A — Oe 
Gia) CEs SS Troster Ure 
G,(AA’) ae each ao lal fence 
Gy) BO CONT Feen nok aire 
G,(AA'l’) ee alte aly and SAL —Os0t SAL) — 10: 
GAA) ee sac lte. Clem and tc 
This list is in exact agreement with the list given in Theo- 
rem 12 except as to the groups G,(K), G,(AlK) and 
G,(AA’A’K) which remain to be investigated. 
We are able to infer from these results that there is no 
seven-parameter group of plane collineations. The smallest 
possible number of relations in a set R defining a group is 
three and the group so defined is a six-parameter group. 
THEOREM 13. There is no seven-parameter group of plane col- 
lineations. 
210. The Group G,(K). We come now to the problem of 
finding all continuous groups of collineations defined by quad- 
ratic relations on the elements of M. There are two possible 
complete families of quadratic automorphic forms in the ele- 
ments of M; one of them is homogeneous in the elements of 
the columns of M and the other in the rows. We shall 
first consider the former system. The set of relations is 
lay? + mas? + na32?+ 2paia2. + 2qai\a;+ 2ra.a; =I, 
lb,2 + mb? +nb32 + 2pbib. + 2qbibz + 2rbo.b; =m, 
. le;2 + mes? + ne32 + 2peies + 2qeie3+2re2c;=N, 2 
Te: layb; + masbs + naz3b3+ p(aibs+azb;) + q(a,b3+a3b1) +r (a2b3+a3b2) =P, ( 4 ) 
laye, + marc, +nase3 +p (a\C2+42€)) + g(Aie3+G3¢1) +7 (A2e3+a3C2) =Q, 
lb\e, + mbse2 + nb3e3 + p(bie2 + boe;) +. (bie3 + b3¢1) + 7 (b2€3+-b3¢2) =P. 
