194 THEORY OF COLLINEATIONS. 
invariant the same lineal element. It also follows that 
G,'(A1L) is the only group of type III defined by linear rela- 
tions on the elements of M. 
THEOREM 25. A necessary and sufficient condition for the ex- 
istence of a group of collineations of type III is that the invariant 
figures of all the collineations in the system have in common the 
same lineal element. There is but one group of type III defined by 
linear relations on the elements of M. 
AENQed oy IINYe 
In order that a collineation be of type IV it is necessary 
and sufficient that its characteristic equation A(p)=0 have a 
double root for which all the first minors of A(p) vanish. 
This is equivalent to three non-linear relations D, D’, D’ on 
the elements of M. lt follows that there are ~’ collineations 
of type IV and no four- or five-parameter groups of this type. 
228. Three-parameter Groups of Type IV. Let us exam- 
ine for the first group property the system of ~° collineations 
defined by R,, D, D’, D’. We may take Mand A(¢) as before 
in the reduced forms: 
Ja, bi 0| a,—? dy 0 
M= |e: oland A(p)= le te 0 | =0. 938) 
| a3 bs 1 | a3 bs 1 |\¥ 
One root of A(p)=0 is 7 and this value makes six of the first 
minors of A(p) vanish identically. It must make the re- 
maining three vanish also, viz.: 
a,—/ by Q—P 6} a2 bo—p 
a» bo—P , a3 b3 | ; a; bs 
This gives us three conditions as follows: 
(a, —1) (bs —1) =azb,, 
(a, —1) bs =azh, (40) 
(6b: —1) a3=a2b;. 
These three are not independent, since the product of the sec- 
ond and third gives the first. 
Applying the first condition exactly as in case (1) Art. 224 
we find that a necessary and sufficient condition for a group 
is either b,=0 ora.=0in M. If b,=0, a,—1 can not vanish 
