LIMITING CASES. 179 
relations when the determinant of the invariant conic Art. 210 
vanishes. 
If we have three sets of linear relations R,, R,, and R,,, so 
related that the determinant 
Cains aa i 
U om wn 
“om” ni 
A(l)= =). 
then these three sets of numbers are linearly related. Hence 
there exist three relations, as follows : 
aL feU =", 
Am+tm! =m", 
An jin! =n". 
R,, may therefore be replaced by Ryp+np. But the determi- 
nant of R,, R,, and Rjp+4p vanishes for all values of % and wu. 
Hence if T leaves invariant three points which lie on a line J, 
all points on / are also invariant. In this case the three 
equations 
xc=a,cx+by+ec2, 
y= a." + boy +22, 
2=a30+bsy +c32, 
have an infinite number of linearly related solutions, and 
hence their determinant, 
a,;—1 b Cy 
EN (Gi) ==) a2) 9 beeen) |=) (38 ) 
ay b, C3—1 
is of rank 1, 7. e., its first minors all vanish. But this is just 
the condition Art. 112, that the collineations be perspectives. 
Therefore the group defined by R,, R,, R,, and A(l)=i0. 
a group in which the collineations are all perspective collinea- 
tions. It is a three-paramenter group, since the condition 
(1) = 0 decreases by one the number of independent condi- 
tions on the elements of M, and its geometric invariant is a 
line of invariant points. 
If we have given three linear relations R,, R, and R,,, with 
the condition that the determinant 
ie VG ae 
a) = Yl | =p). 
| lt omit nit 
