LINEAR RELATIONS. TYPE IV. 195 
without A(p) vanishing identically. Substituting 6,=0 in 
(40) we get the conditions b,=0, b,=1, 6,=0. In like man- 
ner if a,=0, we have the conditions a,=1, a,=0,a,=0. 
The matrix M may now be written in either of the equivalent 
forms: 
a 0 O 1 bh O 
a 1 Oj|}or||O 5 Oj}, 
az; O 1 (tip al 
These show that all collineations of the group leave invariant 
the line corresponding to the single root of A(¢) and the points 
on it corresponding to the double root ; 7. e., all points on the 
line. The group may therefore be designated by H, (1). 
In exactly the same manner it may be shown that the sys- 
tem of collineations defined by R,, D,D’,D” form a three- 
parameter group whose matrix may be written in the 
equivalent reduced forms : 
gl (0), 00) a, bh «4 
Qebo Coll OF 0: “2 10 
OMROT ed O Oil 
These forms show that the collineations of the group all 
leave invariant the point corresponding to the single root of 
A(p) and the lines through it corresponding to the double 
root; 7. e., all lines through it. This group will be desig- 
nated by H,(A). 
THEOREM 24. A necessary and sufficient condition for the ex- 
istence of a group of collineations of type IV is that the invariant 
figures of all collineations of the system have in common either the 
same vertex or the same axis. 
Other Groups of Type IV. In a similar manner we can 
prove the existence of a two-parameter group H.(AA’) in 
which all the collineations of the group have a common vertex 
A, and axes intersecting in a common point A’; also a two- 
parameter group H,(/l’) in which all the collineations have a 
common axis /, and vertices on a common line l’; also a one- 
parameter group H,(A,/), in which all the collineations have 
a common vertex and a common axis. 
