72, THEORY OF COLLINEATIONS. 
Multiply the second equation of R, by the third of R,,, the 
second of R,, by the third of R,, and subtract; we thus get 
L’A,+M’B,4+N’C,=L", 
where A,, B,, ete., are cofactors of a,, b,, ete., nm Mand L”, 
ete., are cofactors of /’’, etc., in 
Um! a 
Wanatany, 
In like manner we get the entire set 
LA, + MB, +N’ CG,=L", 
R! :) L"A,+M"B,+N'CG=M", (30) 
L" A, +M"B;+N"C;=N". 
From these three equations we readily derive the set 
L!a,+M"4,+N"as;=AL", 
R,. Livbi+ Mb, +N"b3= AM", (31) 
L"e,+ M"ez + N"c3=AN". 
This secures the invariance of the line L’*x+ M"y+N’z=0, 
which is the line joining the two points (/, m, 1) and (1’, m’, n’). 
In the same way two sets R, and R, imply the existence of 
a set of the other kind R,, which means that the point of in- 
tersection of the invariant lines | and l’ is an invariant 
point P. 
The three relations R,, R,, R,-, which define the group 
G,(AA'A”), taken two at a time give us three other rela- 
tions R,, R,, R,, which define the group G,(ll/l’’). Hence 
these two groups G,(AA’A”) and G,(Il/l’’) are equivalent 
groups. 
Two relations of different kinds R, and R, may be wholly 
independent of each other or a relation may exist between 
them. The invariant point (1, m, 7) will lie on the invariant 
line 2¢+ uy+rz=0, if 
Sal=al+um+rvn=0. 
209. Subgroups Defined by Linear Relations. We are 
now in position to enumerate all the subgroups of G, of type 
I that are defined by linear relations among the elements of 
the matrix M. It is clear that we may have groups defined by 
one, two, or three sets of relations of the kind R, or R, and 
any combination of R,’s and R,’s that does not impose too 
. 
