196 THEORY OF COLLINEATIONS. 
THEOREM 25. There are five varieties of groups of type IV de- 
fined by linear relations on the elements of J, viz.: H;(4), H;(¢), 
H,(AA), He(tl’), H,(A,). 
TYPE V. 
229. The corresponding results for type V may be readily 
deduced from those for type I[V. There is one added condi- 
tion, viz.: That A(p)=0 have a triple root instead of a 
double root and a single root. We infer at once that there 
are «’ collineations of type V and no three- or four-parame- 
ter groups of type V. 
If to the analysis of Art. 228 we add the condition that 7 is 
a triple root of A(p) =0, we reach the conclusion that there 
is a two-parameter group of type V whose matrix may be 
written in either form, 
a Oe @ i (i @ 
iy TE ON OVE HO ah ON 
Gs) (One IN OM Nie ik 
also another two-parameter group whose matrix may be either 
i (a) (0) L* by ‘cy 
a 1 c¢s|) OF |\O0 ft O 
ON Ome? 00" 2 
All collineations belonging to the first group have in com- 
mon the line.of,invariant points; all belonging to the second 
group have in common the pencil of invariant lines. We des- 
ignate the first group by H.’(l) and the second by H,’/(A). 
We also find a one-parameter group H,/(A1/) in which all the 
collineations have in common the same axis and the same 
vertex. We also see that theorem 24 holds word for word 
for type V as well as for type IV. 
THEOREM 26. There are three varieties of groups of type V de- 
fined by linear relations on the elements of J/, viz.: H.’ (A), 4,’ (2), 
H,! (Al). 
