LINEAR RELATIONS. 169 
transformations of the group. This isa six-parameter group, 
since the three relations among the nine elements of M leave 
six independent parameters. It will be designated by G,(A). 
206. Linear Functions of the Elements of the Columns. 
Let »; be linear homogeneous functions of the elements of 
the columns of M. Then by Art. 191 we have three relations 
of the form 
la,+masz+na;=l, 
R, : lbi+mb.+nb;=m, (28) 
le, +me+ne3=n, 
i. €., the » function of the elements of each column is equated 
to the coefficient of the elements of the corresponding row. 
Relations of this form satisfy the necessary conditions for a 
subgroup of G,. 
In order to show that this set of relations is sufficient to 
define a subgroup of G, we assume that they hold for T and 
ests 
laj+maz,+na;=l, la, 4+moa+na3,=1, 
1b,+mb,+nb3=m , lf, +mio+nf33=m , 
ley +me.+ne3 =n , fi tmre+ny3=n. 
Substituting for /, m, n, on the left hand side of the first set 
their values from the second set, we get 
lA, +mA,+nAz=1, 
1B,+m8,+nB;=m , (23) 
1C,+mC.+2C3=Nn. 
These relations hold therefore for T, and they are sufficient 
to define a group. 
If equations (23) are multiplied respectively by w, y, and 
z, and then added we get 
f= lx+my+nz=l (a,r+biy+eiz) +m(acx+boy+e.z) +n(azx+bzsy+¢32), 
=lx+myi+nz. 
This shows that the function f=la+my-+nz is invariant in 
form under all the transformations of the group and that this 
function has the same value at a pair of corresponding points 
of the plane. This function vanishes at all points of a cer- 
tain line of the plane and hence the geometric invariant of the 
group is the line / whose equation is 
let+my+nz=0. 
